A25363	----	Dary's miscellanies examined and some of his fundamental errors detected by authority of ancient and modern mathematicians ... : to which is added a task for Mr. Dary of his own setting / by Robert Anderson. Anderson, Robert, fl. 1668-1696. 1670 Approx. 21 KB of XML-encoded text transcribed from 8 1-bit group-IV TIFF page images. Text Creation Partnership, Ann Arbor, MI ; Oxford (UK) : 2003-11 (EEBO-TCP Phase 1). A25363 Wing A3102 ESTC R9335 12642925 ocm 12642925 65041 This keyboarded and encoded edition of the work described above is co-owned by the institutions providing financial support to the Early English Books Online Text Creation Partnership. This Phase I text is available for reuse, according to the terms of Creative Commons 0 1.0 Universal . The text can be copied, modified, distributed and performed, even for commercial purposes, all without asking permission. Early English books online. (EEBO-TCP ; phase 1, no. A25363) Transcribed from: (Early English Books Online ; image set 65041) Images scanned from microfilm: (Early English books, 1641-1700 ; 340:16) Dary's miscellanies examined and some of his fundamental errors detected by authority of ancient and modern mathematicians ... : to which is added a task for Mr. Dary of his own setting / by Robert Anderson. Anderson, Robert, fl. 1668-1696. [3], 13 p. Printed for Philip Brooksby ..., London : 1670. Reproduction of original in Bodleian Library. Created by converting TCP files to TEI P5 using tcp2tei.xsl, TEI @ Oxford. Re-processed by University of Nebraska-Lincoln and Northwestern, with changes to facilitate morpho-syntactic tagging. Gap elements of known extent have been transformed into placeholder characters or elements to simplify the filling in of gaps by user contributors. EEBO-TCP is a partnership between the Universities of Michigan and Oxford and the publisher ProQuest to create accurately transcribed and encoded texts based on the image sets published by ProQuest via their Early English Books Online (EEBO) database (http://eebo.chadwyck.com). The general aim of EEBO-TCP is to encode one copy (usually the first edition) of every monographic English-language title published between 1473 and 1700 available in EEBO. EEBO-TCP aimed to produce large quantities of textual data within the usual project restraints of time and funding, and therefore chose to create diplomatic transcriptions (as opposed to critical editions) with light-touch, mainly structural encoding based on the Text Encoding Initiative (http://www.tei-c.org). The EEBO-TCP project was divided into two phases. The 25,363 texts created during Phase 1 of the project have been released into the public domain as of 1 January 2015. Anyone can now take and use these texts for their own purposes, but we respectfully request that due credit and attribution is given to their original source. Users should be aware of the process of creating the TCP texts, and therefore of any assumptions that can be made about the data. Text selection was based on the New Cambridge Bibliography of English Literature (NCBEL). If an author (or for an anonymous work, the title) appears in NCBEL, then their works are eligible for inclusion. Selection was intended to range over a wide variety of subject areas, to reflect the true nature of the print record of the period. In general, first editions of a works in English were prioritized, although there are a number of works in other languages, notably Latin and Welsh, included and sometimes a second or later edition of a work was chosen if there was a compelling reason to do so. Image sets were sent to external keying companies for transcription and basic encoding. Quality assurance was then carried out by editorial teams in Oxford and Michigan. 5% (or 5 pages, whichever is the greater) of each text was proofread for accuracy and those which did not meet QA standards were returned to the keyers to be redone. After proofreading, the encoding was enhanced and/or corrected and characters marked as illegible were corrected where possible up to a limit of 100 instances per text. Any remaining illegibles were encoded as <gap>s. Understanding these processes should make clear that, while the overall quality of TCP data is very good, some errors will remain and some readable characters will be marked as illegible. Users should bear in mind that in all likelihood such instances will never have been looked at by a TCP editor. The texts were encoded and linked to page images in accordance with level 4 of the TEI in Libraries guidelines. Copies of the texts have been issued variously as SGML (TCP schema; ASCII text with mnemonic sdata character entities); displayable XML (TCP schema; characters represented either as UTF-8 Unicode or text strings within braces); or lossless XML (TEI P5, characters represented either as UTF-8 Unicode or TEI g elements). Keying and markup guidelines are available at the Text Creation Partnership web site . eng Dary, Michael. -- Dary's miscellanies. Mathematics -- Early works to 1800. 2003-07 TCP Assigned for keying and markup 2003-07 Aptara Keyed and coded from ProQuest page images 2003-09 Mona Logarbo Sampled and proofread 2003-09 Mona Logarbo Text and markup reviewed and edited 2003-10 pfs Batch review (QC) and XML conversion DARY'S Miscellanies Examined ; And some of his Fundamental Errors DETECTED . BY Authority of Ancient and Modern MATHEMATICIANS . The Ancient : Euclid , Diophantus , Apollonius and Archimedes . The Modern : Xylander , Bachetus , Stevin , Albert Girard , Torricellius and Regeomontanus . To which is added , A Task for Mr. Dary of his own setting . By Robert Anderson . London , Printed for Philip Brooksby , next to the Golden Ball , near the Hospital gate , in West-Smithfield . 1670. DARY'S Miscellanies EXAMINED . And some of his Fundamental Errours DETECTED . WHilst those miscellanies were printing , I met with two of Master Darys friends together , at their office in Holborn and they related to me that Master Dary had a Book in the press , in the Preface whereof he had a quarrel with me , and that he was resolved to vex me ; my answer was , that if he gave me bad language I would lay it under my feet ; but if he gave bad mathematicks I would return it to him again ; therefore , all those calumnies , that bad and scurrilous language , ( for such are their only demonstrations ) either by him or by any of his crew in that preface given , or may be hereafter given in any of their writings , I shall take no further notice of , but shall ever lay them as dirt under foot ; but I shall prosecute a close conviction of their erronious principles in Geometry . That preface those authors divides into two parts , the first against Stereometrical Propositions , the second in defence of the ART of practical gaging ; and as they have little to say against the first , they have as little to say in defence of the latter , but in both I shall easily subvert their crippled arguments . To the first , In the first page of the preface , Master Dary hath it thus ; in the tail of which Book there is a whole broad side . Here he is outragious because he was so perfectly confuted in the tail of the guide to the young gager : truly , as it was the confutation of the ART of practical gaging ; it deserved no better preferment , than to be put in the tail of the young gagers guide : however , if I find Master Dary's understanding improved by my instructions there given , in these his Miscellanies ; I shall to encourage him , commend him in the tail of this . In the second page of the Preface , he hath these words , the word frustum pyramide I cannot understand , But if he had said frustum of a pyramid , &c. this complaint may consist of three parts , first , frustum pyramide ; second , I cannot understand ; third , frustum of a pyramide ; if we compare the first with the last , we shall find them both of one signification , for frustum signifies a broken piece , therefore it is as well sense to say a broken pyramid , as to say a piece of a pyramid ; one familiar example for many , a broken knife , as to say a piece of a knife . Such expressions are brief and well understood , both signifying the same thing , and he himself using the same expressions in the 31 and 32. 89 pag , of the Art of practical gaging : thus , is the content of the frustum pyramid , and in 29 page of that Book of Art , you have it thus , Master Michael Dary , an ingenious Artist and practised gager : when it is the frustum of a Cylindriod : here we find him a giving himself a good character ; and then telling us of a frustum of a Cylindriod , and in page 32 of that Book of Art , thus , to cut this Cylindriod , here we may observe this ingenious artist , how in one page he calls it a frustum of a Cylindriod , and immediately he calls the same solid a Cylindriod , so he makes no difference betwixt the part and the whole . Further , as for those mighty words of art , to wit , Cylindroid , prismoid and peripetasma ; I shall say only this , Oft have I known some men of no great parts , Stuff up their mouths with mighty words of arts . For his 2. complaint , that is , I cannot understand ; it troubles me to hear it , yet I see his understanding mend a little , as we shall observe hereafter . In the same page , he is angry because that irregular frustum is cut into so many parts , if that do not please his worship he may take one of the other ways which cuts that solid into fewer parts ; for there are four ways every one lesse work than other . But the gunner and his crew must be a shooting though but with pot-guns . In the third page , our gunner gives the seventh prop. of the 5 of Diophantus a Broad-side ; thus , the stress of his argument is weak and infirm . Though we should grant Z equal to ⅗ , it is yet to demonstrate that Z is 3 and A 5. A by supposition was an unit , then reduce them to one denomination , and that denomination being rejected , Z will be equal to 3 , and A equal to 5 ▪ this he looks upon as an hard demonstration , which I am not bound to tell him how to do , saith he . Further , he dwindles to his Reader , hoping for glory and would know , Whether this proposition hath any relation at all to gaging . I answer yes , and argue thus , numbers have relation to gaging , this Proposition is of Numbers ; therefore this Proposition hath relation to Gaging . Again , triangles have relation to gaging , this proposition is of triangles ; therefore , this proposition hath relation to gaging . This prop. which he quarrels with , is the 7. of 5. of Diophantus , as is cited in the 106 page of Stereo . Prop. and seeing he hath so much immodesty as to say his arguments are weak and infirm , I shall set down the text as Bachetus hath it . Esto primus 1 N. secundus unitatum quotlibet , puta 1. & est productus eorum multiplicatione 1 N. summa vero quadratorum est 1 Q + 1. adde 1 N. fit 1 Q + 1 N + 1 aequalis quadrato . Esto latus ejus 1 N − 2. fit quadratus 1 Q + 4 − 4 N. aequalis 1 Q + 1 N + 1. & fit 1 N. ⅗ ad positiones , Erit primus ⅗ , secundus 5 / 5 : & abjecto denominatore , erit primus 3. secundus 5. & postulatis respondent . So then I have these witnesses on my side , 1. Diophantus the author of the proposition , 2. Xylander . 3. Bachetus . 4. Stevin . 5. Albert Girard . Those four commentators upon Diophantus every one of them setting it as above . 6. Truth it self , and it will prevail . So then , if Mr. Dary cannot bring better authority then these on his side for the stress of his argument ; I shall conclude , that pride and ignorance is baffled ; and where he saith I fling dirt in the face of Van Schooten , I may very well say he flingeth dirt in the face of these 5 authors , yea and in the face of truth it self . Further , had this 7. and 8. prop. of the 5 of Diop been observed by the proposer and resolver of that question , it is very likely it would not have been proposed by the one , nor resolved by the other ; however , what I said concerning Van-Schooten and des Cartes is true and just , therefore no dirt . In the fourth page , our gunner hath more fire-works , to wit , his note for progressions is invalid and of no force . For saith he , there is no need of unity for the first terme of this progression . My answer is , that note for progression is of force and truth , and unity of use ; thus , the question it self requires whole numbers ; the seventh of the fifth of Dioph. finds whole numbers ; therefore greathan an unit ; therefore well limited . The second part , in defence of the Art of Practicall gaging , and it begins in the fifth page , and there he telleth his Reader how he hath been commended by divers artists in this City . Here he appeals to men as ignorant , as himself is vain glorious . In the 6 pag. he flingeth dirt in the face of the printer , thus , in which I see there are many press-faults ; that is false , they are the segment makers faults , for the segments are the complement of one to the other to 100000 &c. therefore no printers faults . In the sixth and seventh pages , he sheweth how to calculate a table of segments , and here his understanding mends a little , for he works pretty well since the last time I taught him ; so then , as one mends in his Rules , so I hope the other will mend in his calculation , ( with that instruction I formerly gave him ) so we may expect a better table of segments some time or other . In the 8. page he again dwindles and would fain insinuate into the affection of his Reader ; and make him believe that I did not know that there was a third &c. differences in the table of segments , to speak the truth , that table of segments was calculated so falsly that the first differences did manifestly shew it ; further , If Mr. Dary had known that way , or any other way better to examine Tables by , before he published those segments ; more shame to him to publish such false tables , without examination . In the 9 and 10. pa. the Gunner has fire and gun-powder , viz. know ye not that the Table for wine , ale and beer , are capable but only of the first and second differences . If so , more shame to the Calculator that they have more diff . and they so much confusedly put . As for that Book entituled A guide to the young gager , I knew not the man nor heard of the Book untill a great part of it was printed , neither did I see one line of that part of it , till it was publickly exposed to sale . Thus have I passed through this fiery conflict , and have not heard the bounce of one gun , nor received any harme , which makes me conclude our gunner and his crew are as bad marks men , as they are segment makers , for he promi●ed at the beginning of his preface to charge his guns and pepper me . Thus have I considered him as a Gunner with his Crew ; now will I consider him as a Geometer , with his famous Companions . These famous men , whose true descent doth run From aged Neptune , and the glorious Sun. AN EXAMINATION OF Dary's Miscellanies . IN the first page of the Preface , he saith , Most whereof have lain by me many years : If so , I hope very true . 1 In the second page of the Preface , saith he , For although the sides thereof be continued , they would never be included or terminated in one point , as the Pyramide is ; that is , the sides of a Pyramide are included in one point , which I deny , thus ; a point hath no part , by 1 def . 1 Euclid . A Superfices ( for such are the sides of a Pyramide ) have length and breadth 5 def . 1 Euclid . That which hath no part , to include that which hath length and breadth , is absurd ; that 's a lumping point for an able Anylist . 2 In the fourteenth page , saith he , The 3 Angles of any Spherical Triangle being given , there are likewise three sides of another Spherical Triangle given , whose Angles are equal to the sides of the former Triangle . Here the Gentlemen forgot to complement , and I presume in the next they will forget all good manners . Further , the sum of the sides of any spherical triangle , are less then two semi-circles , Reg. 39 ▪ of 3. The sum of the three angles of any spherical triangle , are greater then two right angles , but less then six , Reg. 49 of 3. therefore the Rule is false , except the sum of the three sides be greater then two right angles ; but the Rule is set down general , therefore a general error . 3 In page 21. we have it thus ; If a sphere be by a plain touch'd , and the eye be placed at the center of the sphere , then a right line infinitely extended from the eye to any assigned point in the spherical surface , shall project the assigned point upon the plain . Here the Radius of the Sphere is taken to be infinite , for , saith he , then a right line infinitely extended from the eye to any assigned point in the spherical surface ; but the plain is without the sphere , therefore beyond infiniteness it self , which is absurd : however this proves them to be infinite Projectors . 4 In page 29. at the 18th it is thus ; If a sphere be inclosed in a cylinder , and that cylinder be cut with plains parallel to its base , then the intercepted rings of the cylinder are equal to the intercepted surfaces of the respective segments of the sphere ; that is false : For ▪ Hemisphaerii superficies aequalis est superficiei curvae cylindri eadem ipsi basim , & eadem altitudinem habentis , saith Torricellius at the 18. Prop. de sphaera , & solidis sphaeralibus lib. prim . and as the whole , so the parts , by the 19. Prop. of the same . Here we have a combat betwixt Torricellius and our Geometers : First , they say the intercepted rings of the cylinder are equal to the intercepted surfaces of the respective segments of the sphere . Torricellius proves , that the intercepted superfices of the cylinder , are equal to the intercepted superfices of the respective segments of the sphere . 2. These Geometers say , if a sphere be inclosed in a cylinder , here we may make the Diameters of the base of the cylinder of any magnitude , greater then the diameter of the sphere , and yet the sphere be inclosed ▪ Torricellius proves , that the cylinder and hemisphere must have the same base . 3. These Geometers regard it not , whether the sphere and cylinder are upright or inclining . Torricellius by construction makes them upright . Thus do these Geometers make solid superfices , for a ring is solid . 5 In page 33. they set down a rule for the sphere , and conclude it will hold in the spheroid ; this rule will also hold if it were the Frustum of a Spheroid , putting d●d equal to the fact of the right angled conjugates in the base . That is false , by 21 of 1 of Apollonius , and 31 and 33 of Archimedes of conoid and spheroid ; for the diameter of the base one way , or the right angled conjugates of the base the other , with the height of either , will not limit a spheroid , as the diameter of the base and the height doth a sphere . This very rule Crowns all their endeavours ; for before they had made a point bigger then any superfice , a line longer then infiniteness , a solid superfice , but now they are come to an unlimited solid . 6 In page 39. they write thus ; But if such a solid have not its Zons made by circles or ellipses , but by four flat sides at right angles to the foresaid conjugates , then it is a prismoid ; nevertheless , the rules before prescribed , hold to all intents and purposes : that is false to all intents and purposes at the first appearance ; for if two right lines be at right angles , and they be at right angles with four plains , those plains wil be the 4 sides of a Parallelepipedon , by the 2 , 3 , and 30 def . of 11 Euclid . a Parallelepipedon being calculated gradually , can have but a first difference , and not a second and third : But this is like the rest of these Famous Geometers works . Our Master Geometer telleth his Reader thus , Most whereof have lain by me many years . And in the Title Page he saith , they are brief Collections from divers Authors : If so , why so many Fundamental Errors ? Further , seeing M. Dary , and his Companions will assert any thing , and demonstrate nothing , except they are required in print ; therefore I desire them to demonstrate these six following assertions of their own , and and shall call it A Task for Mr. Dary of his own setting . To wit : 1 The sides of a Pyramide being 32000. I desire Mr. Dary to give one point to include that superfice , as he asserts in page 2. 2 The three sides of a Spherical Triangle , being 6 , 8 , and 10 degrees , their sum 24 degrees , I desire Mr. Dary to give a Spherical Triangle , whose sum of the three angles are 24 degrees , as he asserts in page 14. 3 In the Gnomonick Projection , the Radius of the Sphere being infinite , and the arch from the touch point to the assigned point be 30 degrees , I desire Mr. Dary to extend a line from the center of that sphere , by the assigned point , to the touching plain , that is further then infiniteness , as he asserts in page 21. 4 If a Cylinder and Hemisphere be of one height , but the diameter of the base of the Cylinder be greater then , or equal to the diameter of that Sphere , and they concentrick , this Hemisphere is inclosed in that Cylinder ; let that Hemisphere and Cylinder be cut with Plains parallel to their bases I desire Mr. Dary to prove , that those intercepted rings of the Cylinder ( that is solid rings ) are equal to the intercepted surfaces of the respective segments of the Sphere , as he asserts in page 29 5 In a Spheroid , let 6 be the perpendicular height of the Frustum , 8 the diameter of the base , when cut by a Plain at right angles with the Axis ; let 10 and 12 be the right angled conjugates in the base ( as he calls them ) when the cutting Plain is parallel to the axis , the altitude of the Frustum 4. I desire Mr. Dary to give one example in each , if but one ; if more then one , to give them all ; that is , to prove it a limited Proposition , as he asserts in page 33. 6 If two right lines , to wit , one 6 , the other 8 , be at right angles , and these two lines be at right angles with four Plains , the height of these plains may be 12. those will be the limits of a solid , which Euclid names a Parallelepipedon , at the 30 def . of 11. I desire Mr. Dary to prove such a solid to be a Prismoid , and to have second and third differences , as he asserts in page 39. Now to commend him . THose six Assertions of Mr. Dary's , may well be termed A Task for him for six daies ; which Assertions being performed according to the Rules of Geometry , I shall ever conclude Mr. Dary to be great , yea greater ; nay the greatest Geometer of all mortal men . But if Mr. Dary , with the help of his Companions , cannot or will not fairly demonstrate these their Assertions , but still cavil and quarrel it out . I may well conclude , his or their Geometry is not , nor will not be worth taking notice of for the future ; for that Micellanea Riff-raff having lain by him many years ; and we may be sure , often thumb'd over with much care and prudence , like an ingenious Artist , and a practised Gager ; being his whole stock of Mathematical knowledge , is now made publick , to prove himself what he is , to wit , a Geometer full of errors , and a Mathematician altogether without demonstration ; therefore I shall imploy my idle time better then in confuting such unwise ridiculous Assertions ; for this we may be sure of ▪ that whatever Mr. Dary writes , will be full of Fundamental Errors . Although I am well assured , that whatever Mr. Dary writes will be so full of Fundamental Errors , that it will not be worth taking notice of : yet seeing one deeply swears by his Maker he would have us never agree , because it will be good sport for them ; and another of Mr. Dary's friends is desirous to see Paper Battels , therefore I shall the rather desist : However , if I take pen in hand again , I shall be as ready to bring them into the Lift , by examining their works , as they are desirous that we should make them sport . Further , Mr. Dary hath related to several of my acquaintance , that those his Miscellanies were published as a snare for me ; and one of his Crew hath told me to my face , that he could be revenged on me , and never appear in it himself : I asked him how ; He answered , he could hires a stab to be given for a very small matter : My answer to these two , and the rest of them is , I value the snares of one , the stab of the other , and the envy of the rest , no more then the dirt of my shoes ; my seconds shall be Euclid , Diophantus , Apollonius and Archimedes , and my Weapons Truth and Demonstration . FINIS . 
A29755	----	[A description of a mathematical instrument] made by John Browne. Brown, John, philomath. 1671 Approx. 1 KB of XML-encoded text transcribed from 1 1-bit group-IV TIFF page image. Text Creation Partnership, Ann Arbor, MI ; Oxford (UK) : 2009-03 (EEBO-TCP Phase 1). A29755 Wing B5037 ESTC R30254 11276087 ocm 11276087 47251 This keyboarded and encoded edition of the work described above is co-owned by the institutions providing financial support to the Early English Books Online Text Creation Partnership. This Phase I text is available for reuse, according to the terms of Creative Commons 0 1.0 Universal . The text can be copied, modified, distributed and performed, even for commercial purposes, all without asking permission. Early English books online. (EEBO-TCP ; phase 1, no. A29755) Transcribed from: (Early English Books Online ; image set 47251) Images scanned from microfilm: (Early English books, 1641-1700 ; 1456:20) [A description of a mathematical instrument] made by John Browne. Brown, John, philomath. 1 sheet. s.n.,] [London : 1671. In the drawing: 1671 Made by John Browne at the Sphear & Sun Diall in the Great Minories neer Aldgate London. Reproduction of the original in the Chetham's Library. Created by converting TCP files to TEI P5 using tcp2tei.xsl, TEI @ Oxford. Re-processed by University of Nebraska-Lincoln and Northwestern, with changes to facilitate morpho-syntactic tagging. Gap elements of known extent have been transformed into placeholder characters or elements to simplify the filling in of gaps by user contributors. EEBO-TCP is a partnership between the Universities of Michigan and Oxford and the publisher ProQuest to create accurately transcribed and encoded texts based on the image sets published by ProQuest via their Early English Books Online (EEBO) database (http://eebo.chadwyck.com). The general aim of EEBO-TCP is to encode one copy (usually the first edition) of every monographic English-language title published between 1473 and 1700 available in EEBO. EEBO-TCP aimed to produce large quantities of textual data within the usual project restraints of time and funding, and therefore chose to create diplomatic transcriptions (as opposed to critical editions) with light-touch, mainly structural encoding based on the Text Encoding Initiative (http://www.tei-c.org). The EEBO-TCP project was divided into two phases. The 25,363 texts created during Phase 1 of the project have been released into the public domain as of 1 January 2015. Anyone can now take and use these texts for their own purposes, but we respectfully request that due credit and attribution is given to their original source. Users should be aware of the process of creating the TCP texts, and therefore of any assumptions that can be made about the data. Text selection was based on the New Cambridge Bibliography of English Literature (NCBEL). If an author (or for an anonymous work, the title) appears in NCBEL, then their works are eligible for inclusion. Selection was intended to range over a wide variety of subject areas, to reflect the true nature of the print record of the period. In general, first editions of a works in English were prioritized, although there are a number of works in other languages, notably Latin and Welsh, included and sometimes a second or later edition of a work was chosen if there was a compelling reason to do so. Image sets were sent to external keying companies for transcription and basic encoding. Quality assurance was then carried out by editorial teams in Oxford and Michigan. 5% (or 5 pages, whichever is the greater) of each text was proofread for accuracy and those which did not meet QA standards were returned to the keyers to be redone. After proofreading, the encoding was enhanced and/or corrected and characters marked as illegible were corrected where possible up to a limit of 100 instances per text. Any remaining illegibles were encoded as <gap>s. Understanding these processes should make clear that, while the overall quality of TCP data is very good, some errors will remain and some readable characters will be marked as illegible. Users should bear in mind that in all likelihood such instances will never have been looked at by a TCP editor. The texts were encoded and linked to page images in accordance with level 4 of the TEI in Libraries guidelines. Copies of the texts have been issued variously as SGML (TCP schema; ASCII text with mnemonic sdata character entities); displayable XML (TCP schema; characters represented either as UTF-8 Unicode or text strings within braces); or lossless XML (TEI P5, characters represented either as UTF-8 Unicode or TEI g elements). Keying and markup guidelines are available at the Text Creation Partnership web site . eng Calculators. Mathematical instruments -- England. 2008-07 TCP Assigned for keying and markup 2008-07 SPi Global Keyed and coded from ProQuest page images 2008-08 John Pas Sampled and proofread 2008-08 John Pas Text and markup reviewed and edited 2008-09 pfs Batch review (QC) and XML conversion diagram of a triangular mathematical instrument 1671 Made by John Browne at the Sphear & Sun Diall in the Great Minories neere Aldgate London WD S M T W T F S LYe 68 80 92 76 88 ●2 84 Mon ● 9 6 3 11 8 5 2 7 10 4 ● Day 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31         1 Pleiad 2 Buls eye 3 Ori● gir 4 Lit dog 5 Lyon har 6 Lyon tay 7 Arcturus 8 Vult har 9 Dolfin 10 Peg mou 11 Fomahaut 12 Peg low W 
A29754	----	A collection of centers and useful proportions on the line of numbers by John Brown ... Brown, John, philomath. 1670 Approx. 26 KB of XML-encoded text transcribed from 9 1-bit group-IV TIFF page images. Text Creation Partnership, Ann Arbor, MI ; Oxford (UK) : 2004-08 (EEBO-TCP Phase 1). A29754 Wing B5036 ESTC R33266 13117535 ocm 13117535 97768 This keyboarded and encoded edition of the work described above is co-owned by the institutions providing financial support to the Early English Books Online Text Creation Partnership. This Phase I text is available for reuse, according to the terms of Creative Commons 0 1.0 Universal . The text can be copied, modified, distributed and performed, even for commercial purposes, all without asking permission. Early English books online. (EEBO-TCP ; phase 1, no. A29754) Transcribed from: (Early English Books Online ; image set 97768) Images scanned from microfilm: (Early English books, 1641-1700 ; 1545:9) A collection of centers and useful proportions on the line of numbers by John Brown ... Brown, John, philomath. 16 p. s.n., [London : 1670] Caption title. Place and date of publication suggested by Wing. Reproduction of original in the British Library. Created by converting TCP files to TEI P5 using tcp2tei.xsl, TEI @ Oxford. Re-processed by University of Nebraska-Lincoln and Northwestern, with changes to facilitate morpho-syntactic tagging. Gap elements of known extent have been transformed into placeholder characters or elements to simplify the filling in of gaps by user contributors. EEBO-TCP is a partnership between the Universities of Michigan and Oxford and the publisher ProQuest to create accurately transcribed and encoded texts based on the image sets published by ProQuest via their Early English Books Online (EEBO) database (http://eebo.chadwyck.com). The general aim of EEBO-TCP is to encode one copy (usually the first edition) of every monographic English-language title published between 1473 and 1700 available in EEBO. EEBO-TCP aimed to produce large quantities of textual data within the usual project restraints of time and funding, and therefore chose to create diplomatic transcriptions (as opposed to critical editions) with light-touch, mainly structural encoding based on the Text Encoding Initiative (http://www.tei-c.org). The EEBO-TCP project was divided into two phases. The 25,363 texts created during Phase 1 of the project have been released into the public domain as of 1 January 2015. Anyone can now take and use these texts for their own purposes, but we respectfully request that due credit and attribution is given to their original source. Users should be aware of the process of creating the TCP texts, and therefore of any assumptions that can be made about the data. Text selection was based on the New Cambridge Bibliography of English Literature (NCBEL). If an author (or for an anonymous work, the title) appears in NCBEL, then their works are eligible for inclusion. Selection was intended to range over a wide variety of subject areas, to reflect the true nature of the print record of the period. In general, first editions of a works in English were prioritized, although there are a number of works in other languages, notably Latin and Welsh, included and sometimes a second or later edition of a work was chosen if there was a compelling reason to do so. Image sets were sent to external keying companies for transcription and basic encoding. Quality assurance was then carried out by editorial teams in Oxford and Michigan. 5% (or 5 pages, whichever is the greater) of each text was proofread for accuracy and those which did not meet QA standards were returned to the keyers to be redone. After proofreading, the encoding was enhanced and/or corrected and characters marked as illegible were corrected where possible up to a limit of 100 instances per text. Any remaining illegibles were encoded as <gap>s. Understanding these processes should make clear that, while the overall quality of TCP data is very good, some errors will remain and some readable characters will be marked as illegible. Users should bear in mind that in all likelihood such instances will never have been looked at by a TCP editor. The texts were encoded and linked to page images in accordance with level 4 of the TEI in Libraries guidelines. Copies of the texts have been issued variously as SGML (TCP schema; ASCII text with mnemonic sdata character entities); displayable XML (TCP schema; characters represented either as UTF-8 Unicode or text strings within braces); or lossless XML (TEI P5, characters represented either as UTF-8 Unicode or TEI g elements). Keying and markup guidelines are available at the Text Creation Partnership web site . eng Mathematics -- Formulae. Weights and measures -- Tables. 2004-01 TCP Assigned for keying and markup 2004-03 Apex CoVantage Keyed and coded from ProQuest page images 2004-04 Mona Logarbo Sampled and proofread 2004-04 Mona Logarbo Text and markup reviewed and edited 2004-07 pfs Batch review (QC) and XML conversion A Collection of Centers and useful Proportions on the Line of Numbers , by Iohn Brown at the Sphear and Sun-Dial in the Minories . In Multiplication . AS 1 to the Multiplicator , so is the Multiplicand to the Product , consisting of as many Figures , as the Multiplicator and Multiplicand , and sometimes of one less , when the 2 first Figures of the Product are greatest . And more than 4 figures in the Product , must be adjusted by multiplication of the last figures by the Pen or Head. In Division As the Divisor to 1 , so is the Dividend to the Quotient , containing as many figures as the Divisor may be set times under the Dividend by the Pen , and the fraction remaining is a Decimal Fraction , and reduced to the Vulgar Fraction , by setting the sa●e extent the contrary way from the Decimal Fraction , gives the Vulgar Fraction . Of mean Proportions The Arithmetical mean , is the half sum of the two extreams added . The Geometrical mean , is the Square Root of the Product of the two extreams multiplyed together : Or the middle between the two extreams ▪ counted on a line of Numbers . The Musical is thus found ; As the Arithmetical mean , to one extream , so is the other extream to the musical mean proportion required . Of Square and Cube Roots . The middle between 1 , and any other Number more or less than 1 , is the Square Root of that Number , being measured on a fit Line of Numbers ; the exact third part between 1 and any other Number , more or less then 1 , counting from 1 is the Cube Root of that Number , counted on a true Line of Numbers . The exact fourth part , between 1 and any other Number ▪ counted on a Line of Numbers , from 1 , is the Byquadrat Root of that Number , &c. Of Reduction . As one Denominator to his Numerator , so is the other Denominator to the inquired Numerator . Or , As one Denominator to the other ; so is one Numerator to the other required . Note ▪ That the Line of Numbers and Pence together , reduceth Decimals of a pound , to shillings and pence , and the contrary , by inspection only . Thus , ,001 is neer 9 farthings ⅗ ▪ ,0●●● just 2 s. or 1 , 10th , of al. or 20 s. ,0125 is 2 s. 6 d. and 1,532 is 1 l. 10 s. 7 d. 2 f. ¾ and so for any other ; for 1 is 1 l. ,05 is 10 s. and just against ,032 is 7 d. 2 f. ¾ the exact Reduction . In the Rule of Three Direct . As the first term , to the second , so is the third to the fourth required . Or , As a greater to a less , so is another greater to another less , extending the same way , from the third to the fourth , as from the first to the second . And so the contrary , As a lesser term to a greater , so is another less to another greater extending the same way . In Superficial Measure . 1. As the breadth in inches to 12 , so is 12 to the inches long to make 1 superficial foot . 2. As the breadth in foot measure to 1 , so is 1 to the length in foot measure , to make 1 foot of Superficial measure . 3. As 1 to the length in feet , so is the breadth in feet to the content in feet . 4. As 12 to the breadth in inches , so is the length in feet to the content in feet . 5. As 1 inch to the breadth in inches , so is the length in inches to the content in inches . 6. As 144 to the breadth in inches , so is the length in inches to the content in feet . Of Land-measure . 1. As 9 to the length in feet ; so is the breadth in feet to the content in yards . 2. As 14f . 0625 the feet in a Square Ell to the breadth in feet , so is the length in feet , to the content in Superficial Ells. 3. As 1 Perch to the breadth in Perches , so is the length in Perches , to the content in like Perches ; 16 foot and a half to a Perch . 4. As 160 ( the Perches in 1 Acre ) to the breadth in Perches , so is the length in Perches to the content in Acres . 5. As 10 to the breadth in Chains and Links , so is the length in Chains and Links , to the content in Acres , ( because 10 Square Chain is 1 Acre ) . 6. As the length of a customary Pearch , is to the length of the Statute Perch , so is the content in Statute Acres known , to the content in customary Acres required , at two repetions or turning the Compasses twice . 7. To find by what Scale a Field is plotted , measure by a Scale as near as you can estimate , then the middle between the Area , found by the estimated Scale , and the Area set down in the Plot ; and the Area found by the estimated Scale , shall reach f●om the estimated Scale to the true Scale by which the draught was plotted . 8. As the length of any Oblong in Perches to 160 , so is 1 Acre to the breadth in Perches required to make 160 Perch or 1 Acre . Or , As the length in Chains to 1 , so is 1 to the breadth in Chains for 1 Acre . Or , As the length in feet to 43560 , ( the feet contained in a Square Acre ) so is 1 to the breadth in feet to make up 1 Acre . Of Triangles . 1 As 2 to the Base , or longest side of any Triangle , so is the Perpendicular to the Area of the Triangle required , in like parts . 2 The sides given , to find the Perpendicular , square each side severally , then add the greatest , and least Squares together , and from the half Sum of them , substract the remaining Square , noteing the difference or remainder ; half this Remainder divide by the longest side , and the quote sheweth where the Perpendicular will fall on the Base . Then square the Quotient , and substract it from the least Square first added , then the Square Root of the remainder is the Perpendicular required for that Triangle . 3. To find the Area by the sides only , Add the 3 sides together , to get the sum and then the half sum , and then find the difference between the half sum and each side severally . Then multiply one difference by another difference , and then this Product by the remaining difference ; and lastly , this second Product , by the half Sum , after the manner of continual Multiplication . Then Lastly , the square Root of this last Product , is the Area or Content of the Triangle required ▪ The three sides 6 , 8 , and 10 , have 24 for the Area . 4 All irregular many sided figures , are reduced to Triangles , by drawing of Lines , and then measured by the first Rule of Triangles , or two Triangles at once , by one common Base and two Perpendiculars . 5 Regular Polligon , of 5 , 6 , 7 , 8 , or more sides , are measured at one Multiplication , thus , As 1 to the Radius or Measure from the Center to the middle of one side , so is the half Sum of the measure of the sides to the content . Of the Circle and Elipsis . 1 As 10 , as a fixed Diameter , to any other Diameter ; so is 31,416 the fixed circumference where 10 is Diameter , to the circumference required . 2 As 31,416 the fixed circumference , to any other circumference , so is 10 a fixed Diameter to the Diameter required ▪ 3 As 10 to 8,862 , the fixed side of a square equal to a Circle , whose Diameter is 10 ▪ so is any other Diameter given , to his proportional Square equal required . 4 As 10 to 7 ●71 the side of the inscribed Square , so is any other Diameter to his proportional side of the inscribed square required . 5 As 8,862 , to 10 , so is the side of any other square equal to his Diameter required . 6 As 10 to any other Diameter , so is 78,540 the Area of a Circle whose Diameter is 10 , twice repeated to the Area required . 7 As 31,416 to any other Circumference given , so is 78,540 the fixed Area , to the Area required at twice turning the Compasses . 8 The half distance on a line of Numbers , between 78,540 and any other Area given , shall reach from 10 the fixed Diameter , to the Diameter required . And from 31,416 the fixed circumference to the circumference required . And from 8,862 the fixed side of the square equal , to the side of the square equal required . And Lastly , from 7,071 to the side of the square inscribed in the Circle required . 9 A mean Proportion Geometrical , between the longest and shortest Diameters of an Elipsis , or Oval , is the diameter of a circle , equal in Area to the Oval . Then having the Diameter , you have the rest . 10 As 100 the square of 10 a fixed Diameter , is to the Rectangle Figure made of the longest and shortest Diameters of a Oval ; so is 78,540 the Area for 10 , to the Area of the Oval or Elipsis required . 12 As 452 , to 355 , or as 14 to 11 , so is the square of any Circles Diameter to the Area . 13 As 1 to the Radius of a Circle , so is the semicircle to the content of a Circle . 14 As 1 to the Radius of a Circle , so is half the arch of any portion , to the Area or Superficial Content of that Portion , proceeding to the Center , as a Quadrant , a Sextance , or the like part . 15 For other segments find the Diameter of the Circle or the Radius and length of the Arch , by measuring : Or thus , First , for the Diameter of the whole Circle , Square half the Diameter , or Chord of the segment , divide the Product by the segments Altitude or Side , then the quote and sine added together shall be the Diameter required , and half the Diameter is Radius . 2 And the Square Root of the Sum of the Segments Altitude , and half the Segments Chord , is the Chord of half the Segments Arch. 3. Lastly , the length of the Arch of the whole Segment is equal to twice the Chord of the half Segments Arch , more 1 third part of the difference between them , and the whole Segments Diameter . Then measure the whole Segment as before by Radius and half the Arch , and then the Triangle substracted remains the Area of the Segment required . Of a Globe , or Sphear , and his Segments , the Diameter and Circumference is the same as in a Circle ; the rest thus , 1 As 1 to the Diameter in inches , so is the Circumference in inches , to the superficial Content round about the Sphear in inches likewise . 2 Or as 10 to the Diameter , so is 31,416 twice , to the Superficial Content round about . 3 As 1 to the Diameter , so is the Diameter twice to the Cube of the Diameter ; then as 21 to 11 , so is the Cube of the Diameter to the solidity . 4 Or the extent from 10 to any other Diameter , being thrice repeated from 523,60 , ( the solid content of a Globe of 10 inches Diameter ) gives the solid content required . 5 The extent from 1 to 1,90986 , the Arithmetical Complement of 523,5987 , the Area of that Globe whose Diameter is 10 , shall reach from the Solid Con●ent , to the Cube of the Diameter , whose Cube Root ●s the Diameter required . 6 As any Gauge point , for Wine or Beer Gallons , to the Sphears Diameter , so is two third parts of the Sphears Diameter twice , to the solid content according to the Gauge Point . 7 As 1 to the Segments Diameter in Inches , so is the ●egments half Altitude , less by 1 , 8th thereof twice ●epeated , to the neer solidity in inches required . 8 As 1 to the Sphears Circumference , so is the Seg●ents Altitude , to the Superficial Content of the sphe●cal part of the Segment ▪ 9 As the Gauge Point for Wine or Ale Gallons , to ●●e Segments D●ameters , so is the Segment half Altitude ●ore by one eighth of the half Altitude twice repeated 〈◊〉 the Segments near Content in like Gallons , to the ●auge Point . 10 As for Solid Angles of a Sphere , they are reduced to Cones or Pyramids , and so measured , but for the Superficies of Spherical Triangles , by Mr. Iohn Leak thus , As 1 to the excess of the three Angles above 180 degso is Radius , to the Superficial Content of the Spherical Triangle . Of Cones , and Pyramids , and Prismes . As 1 to the Area of the Base , so is one third of the Perpendicular Altitude , to the Solidity . 1 A Cone is best Gauged thus , As the Gauge Point for round Vessels , to the Cones Diame●er at the Base , so is one third of the Cones Perpendicular Altitude twice , to the Content in Wine or Beer Gallons : as the Gage Point was . 2 To get the Perpendicular height , Square half the Diameter of the Base ; also square the Hypothenuse or slaunt height , then take the lesser square from the greater , and the Square Root of the residue , is the Perpendicular height of the Cone required . 3 For Prismes and Pyramids say , As the Gauge Point for square Vessels to the side of the Square , equal to the Base of any Prisme or Piramid , so is one third part of the Perpendicular Altitude twice , to the content in Gallons , according to the Gauge Point used . Or , As the Gauge Point for Square Vessels is to the breadth of any Prisme , so is the length to a 4th . Again , As the same Gauge Point to the 4th . so is one third of the depth , to the content in Gallons acording to the Gauge Point used . A Table of Cube inches and Gauge Points for these Vessels . Names of the Vessels . Cube Inches GP Round ▪ GP Square Wine Gallon at 231 0 17 1485 15 199 Corn Gallon at 272 25 18 6168 16 500 Ale Gallon at 288 00 19 1485 16 972 Ale Gallon at 282 00 18 9468 16 793 Firkin at 282 Ale 2256 00 53 5960 47 498 Firkin at 282 Beer 2538 00 56 846 48 552 Kilder . at 282 Ale 4512 00 75 798 67 180 Kilder . at 282 Beer 5076 00 80 395 7 246 Barrel at 282 Ale 9024 00 107 191 95 000 Barrel at 282 Beer 10152 00 113 690 100 58 Barrel at 288 Ale 9216 00 108 380 96 000 Barrel at 288 Beer 10368 00 114 88 101 823 Corn Bushel at 272 1 , 4th , for a Gallon ; 2178 00 52 6666 46 668 Coal Bushel at 280 ● , 4th , to a gallon , 2246 00 53 500 47 395 The first column is the Number of Cube inches con●ained in any of these Vessels , and serve as Divisors to bring any great Number of Inches , to Gallons , Barrels , or Bushels , accordingly . The Second is the Gauge Points for round Vessels , be●ng the Diameters of Circles , whose Content at 1 inch deep , contains a Gallon , or Barrel , &c. The Third column are Square Roots of the First Co●umn , and thus used as Gauge Points for Square Vessels . 1 As the Gauge Point to the side of the Square of any ●quare Vessel , 10 is the depth twice to the content in Gallons or Barrels , according to the Gauge Point used . 2 As the Gauge Point to the breadth of any square Vessel , so is the length to a 4th . Again , As the Gauge Point to the 4th so is the depth to the content as the Gauge Point was . 3. As the Gauge Point for a round Vessel is to the equated Diameter , so is the length twice , to the content as the Gauge Point was . 4. The most ready Rule for all ordinary Casks for the mean Diameter is , as 10 to 7 , so is the difference of Diameters to a fourth , to be added to the least Diameter to make a mean. Note , That in Cask neer a Cyllender , you may say as 10 to 8,50 ; in Cask like to Cones , say as 10 to 5,30 . In Cask like a Parabolick Spindle , or Oval , say , as 10 to 6. But for very much swelling Cask , say , as 10 to 7,3 ; or 7,4 , or 7,5 , so is the difference of Diameters to a 4th , to add to the lesser Diameter , to make a mean Diameter . Of Square Solid or Timber Measure . 1 A mean Proportion Geometrical , or the middle between the breadth and thickness , measured on a line of Numbers , is the side of the square equal . 2 As the inches square to 12 , so is 12 twice , to the inches long to make 1 foot . 3 As 12 to the breadth in inches , so is the depth in inches to a 4th . Again , as the 4th . to 12 , so is 12 to the inches long to make 1 foot . 4. For small timber , say , As 1 to the breadth , so is the depth in inches to a fourth . Again , As that 4th . to 12 , so is 12 to the feet and parts long , to make 1 foot . 5. As 12 to the inches square , so is the inches square to the quantity in 1 foot in inches . 6. As 12 to the breadth in inches , so is the depth in inches , to the quantity in a foot in inches . 7. As 12 to the inches square , so is the feet long twice , to the content in feet . 8. As 12 to the breadth in inches , so is the depth in inches to a 4th . Again , As 12 to that 4th . so is the length in feet , to the content in feet . 9. As 1 to the bredth in inches , so is the depth in inches to a 4th . Again , As 1 to tha 4th . so is the length in inches , to the content in Cube inches required . 10. As 1 to the thickness in inches , so is the bredth in inches to a 4th . Again , As 144 to the 4th . in inches , so is the length in feet , to the conteet in feet . Of round Timber measure , by the Diameter . 1 As the Diameter in inches to the 13,54 , ( the Diameter , when one foot long makes 1 foot of Timber ) so is 12 twice , to the length in inches to make 1 foot . 2. As 13,54 , to the Diameter in inches , so is 12 twice to the quantity in a foot long in inches , or so is 12 twice , to the feet in a foot long . ( With foot measure use 1,128 . ) 3. As 13,54 , to the Diameter in inches , so is the length in feet twice , to the content in feet . 4. As 1,128 to the Diameter in inches , so is the length in inches twice , to the content in inches . 5. As 1,128 , to the Diameter in feet and parts , so is the length in feet and parts , to the content in feet and parts . By the Circumference . 1. As the Circumference in inches to 147,36 ( the inches about , when one inch long makes a foot ) so is 1 twice , to the inches long to make a foot . 2. As the Circumference in inches to 42,54 ( the inches about , when 12 inches long makes one foot ) so ●s 12 twice , to the inches long to make 1 foot : Or so ●s 1 twice to the feet long to make one foot of Timber . 3. As 42,54 to the Circumference in inches , so is 12 twice , to the inches in a foot long : Or so is 1 ●wice to the feet in a foot long . 4. As 42,54 , to the inches about , so is the length in feet twice , to the content in feet . 5. As 3,545 the feet about , when 1 foot in length is a foot of Timber ) to the girt in feet , so is the length in feet twice , to the content in feet . 6. As 3,545 , to the girt in inches , so is the length in inches twice , to the content in solid inches . To measure Brickwork and reduce it . 1. Note 272 foot 1 fourth is a rod of Brickwork , at a Brick and half thick , and the center for it is 1. Centers for other thicknesses are thus found : 2. As half a Brick , at 05 , or as ( 1 Brick at 1,2 Bricks at 2,2 and a half at 2,5,3 Bricks at 3 , ) is to 1,5 ; so is 1 the Center for a Brick and half to 3 the Center for half a Brick required . 3. As the Center for any thickness , is to the length in feet of any Brick-wall , so is the bredth in feet to the content in feet , at a Brick and a half thick , 4. As 272 , 1 fourth to 1 , so is any number of feet to the Rods , and Quarters , and Decimals ; and the same extent laid the contrary way , from the Decimal Fraction gives the odd feet . For the extent from 272 , 1 fourth to 1 , laid the same way from 1528 foot , gives 5 Rod and a half and 11 parts over , then the same extent laid the contrary way from 11 , gives 30 foot over , in all 5 Rod and half 30 foot . 5. As 272,1 fourth the foot in one Rod , to 5 l. the price of one Rod , so is 1528 foot to 28 l. 1 s. the price of 1528 foot at 5 l. per Rod. 6. As 0666 , a point for once and half , is to the breadth of any roo , so is the length , to the content in feet on both sides ( being measured in feet . ) For Digging of Earth in Square Yards 1st . As 9 to the length in feet , 10 is the bredth to the superficial Yards . Again , As 3 foot the depth of one Yard , to the content in superficial Yards , so is the depth 11 feet , to the content in solid Yards . 2ly . 1. Or plainly thus , As ● to the length in feet , so the breadth in feet , to the superficial ●eet . 2. As 1 to the superficial feet , so is the depth in feet to the solid content in feet . 3. As 27 the feet in 1 yard to 1 , so is the solid content in feet , to the content in solid yards . Of Simple Interest 1st . As 100 is to 6 l. the Interest due for 100 l. in 1 year , so is any other sum to his proportional interest in 1 year . 2ly . For months count thus , If 6 be 12 months or 1 year , then 5 is 10 months , 4 is 8 months , 3 is 6 months , &c. decreasing less than 1 year ; so likewise 12 is 2 year , 18 is 3 year , &c. So that for 30 months , take 15 the half of 30 ▪ &c. As 100 to 15 the interest of 100 l. in 30 months , so is any other sum to his simple interest in the like time at 6 per Cent. 3ly . For principle and interest either increase or present worth , thus at any time . If 106 is 1 years principle and increase , 112 is 2 years , 118,3 years : thus every tenth on the line of Numbers : is 2 months : thus , What is the increase of 125 l. interest and principle in 40 months , or its present worth being due 40 months to come . As 100 to 120 , so is 125 to 150 l. the principle and increase in 40 months . And if the same extent be laid the contrary way from 125 , it gives 104 l. 4 s. the present worth of 125 l. due 40 months to come at 6 per Cent. simple interest . 4ly . 1. For dayes thus , As 365 , the dayes in a year to the rate ; so is the number of dayes , to the interest of 200 l. in so many dayes , as a 4th . 2. As this 4th , with 100 added to it , is to 100 , so is the sum propounded to the present worth ; if the sum be pounds , shillings and pence , reduce it to a Decimal Fraction , and the years to dayes : Note also the same extent laid the other way , gives the principal and increase in so many dayes . Equation of Payments 1. By the last rule find all the present worths , at the times to comes , then say , as all the present worths to all the payments , so is 1 l. to a fourth , from which taking a unite , the remainder is the interest of 1 l. the time sought , which divided by the inetrest of 1 l. for 1 day , viz. 000164384 , the quote is the number of dayes required . Example , If 220 l. be due at a years end , the present worth is 207 l. 11 s. 6 d. for the extent from 106 to 100 laid from 220 , gives 207 l. 11 s. 6 d. But if it should be paid at 55 l. per quarter , when should it be paid at once to be equal between Debtor and Creditor ? As 101,5 3 months 103,0 6 months 104,5 9 months 106,0 12 months to 1003 ∷ So is 55 to 55,20 53,41 52,63 51,90 The sum of them 212,14 Then as 212 l. 14 s. the sum of the present worths , to 220 the sum of the payments , so is 1 to 103,70 , from which tak●ng 100 , rests 03,70 , the interest of 100 l. the time sought . Then the 03,70 , divided by 00016438 the interest of 1 l. 1 day , hath in the quotient 225 dayes the true time to pay the 220 l. at once . Compound Interest The extent from 100 to 106 , the principal and increase in one year , repeated as many times as there be years , gives the principle and increase of any sum , and being turned as many times the decreasing way gives the present worth . FINIS 
A25364	----	Gaging promoted an appendix to stereometrical propositions / by Robert Anderson. Anderson, Robert, fl. 1668-1696. 1669 Approx. 60 KB of XML-encoded text transcribed from 19 1-bit group-IV TIFF page images. Text Creation Partnership, Ann Arbor, MI ; Oxford (UK) : 2004-03 (EEBO-TCP Phase 1). A25364 Wing A3103 ESTC R23633 07869628 ocm 07869628 40180 This keyboarded and encoded edition of the work described above is co-owned by the institutions providing financial support to the Early English Books Online Text Creation Partnership. This Phase I text is available for reuse, according to the terms of Creative Commons 0 1.0 Universal . The text can be copied, modified, distributed and performed, even for commercial purposes, all without asking permission. Early English books online. (EEBO-TCP ; phase 1, no. A25364) Transcribed from: (Early English Books Online ; image set 40180) Images scanned from microfilm: (Early English books, 1641-1700 ; 1194:27) Gaging promoted an appendix to stereometrical propositions / by Robert Anderson. Anderson, Robert, fl. 1668-1696. 33, [1] p. Printed by J.W. for Joshua Coniers, London : 1669. "Faults escaped in the impression of Stereometrical proposition": p. [34] Reproduction of original in the Cambridge University Library. Created by converting TCP files to TEI P5 using tcp2tei.xsl, TEI @ Oxford. Re-processed by University of Nebraska-Lincoln and Northwestern, with changes to facilitate morpho-syntactic tagging. Gap elements of known extent have been transformed into placeholder characters or elements to simplify the filling in of gaps by user contributors. EEBO-TCP is a partnership between the Universities of Michigan and Oxford and the publisher ProQuest to create accurately transcribed and encoded texts based on the image sets published by ProQuest via their Early English Books Online (EEBO) database (http://eebo.chadwyck.com). The general aim of EEBO-TCP is to encode one copy (usually the first edition) of every monographic English-language title published between 1473 and 1700 available in EEBO. EEBO-TCP aimed to produce large quantities of textual data within the usual project restraints of time and funding, and therefore chose to create diplomatic transcriptions (as opposed to critical editions) with light-touch, mainly structural encoding based on the Text Encoding Initiative (http://www.tei-c.org). The EEBO-TCP project was divided into two phases. The 25,363 texts created during Phase 1 of the project have been released into the public domain as of 1 January 2015. Anyone can now take and use these texts for their own purposes, but we respectfully request that due credit and attribution is given to their original source. Users should be aware of the process of creating the TCP texts, and therefore of any assumptions that can be made about the data. Text selection was based on the New Cambridge Bibliography of English Literature (NCBEL). If an author (or for an anonymous work, the title) appears in NCBEL, then their works are eligible for inclusion. Selection was intended to range over a wide variety of subject areas, to reflect the true nature of the print record of the period. In general, first editions of a works in English were prioritized, although there are a number of works in other languages, notably Latin and Welsh, included and sometimes a second or later edition of a work was chosen if there was a compelling reason to do so. Image sets were sent to external keying companies for transcription and basic encoding. Quality assurance was then carried out by editorial teams in Oxford and Michigan. 5% (or 5 pages, whichever is the greater) of each text was proofread for accuracy and those which did not meet QA standards were returned to the keyers to be redone. After proofreading, the encoding was enhanced and/or corrected and characters marked as illegible were corrected where possible up to a limit of 100 instances per text. Any remaining illegibles were encoded as <gap>s. Understanding these processes should make clear that, while the overall quality of TCP data is very good, some errors will remain and some readable characters will be marked as illegible. Users should bear in mind that in all likelihood such instances will never have been looked at by a TCP editor. The texts were encoded and linked to page images in accordance with level 4 of the TEI in Libraries guidelines. Copies of the texts have been issued variously as SGML (TCP schema; ASCII text with mnemonic sdata character entities); displayable XML (TCP schema; characters represented either as UTF-8 Unicode or text strings within braces); or lossless XML (TEI P5, characters represented either as UTF-8 Unicode or TEI g elements). Keying and markup guidelines are available at the Text Creation Partnership web site . eng Mathematics -- Early works to 1800. 2003-07 TCP Assigned for keying and markup 2003-07 Aptara Keyed and coded from ProQuest page images 2004-01 Jonathan Blaney Sampled and proofread 2004-01 Jonathan Blaney Text and markup reviewed and edited 2004-02 pfs Batch review (QC) and XML conversion GAGING PROMOTED . AN APPENDIX TO Stereometrical Propositions . By ROBERT ANDERSON . LONDON , Printed by I. W. for Ioshua Coniers , at the Raven in Ducklane , 1669. Gaging Promoted AN APPENDIX TO Stereometrical Propositions . I. Note . AS an Abstract from the undoubted Axioms of Geometry , it is generally observed , that in a Rank of numbers , having equal difference , the second differences of the squares of those numbers are equal ; the third differences of the cubes of those numbers are equal : and so in order in the higher powers . Thus , In Squares 1 1 1     3 2 4 2 5 3 9 2 7 4 16 2 9 5 25     1 2 3 4 2 1 1     8 3 9 8 16 5 25 8 24 7 47 8 32 9 81     1 2 3 4 Observe in the first of these Examples , in the first collum are the numbers of a progressition , having equal difference , to wit , a unite . In the second colum , the squares of those numbers . In the third colum , the first differences . In the fourth colum , the second differences , to wit , 2 , 2 , 2. In the second Example , in the first colum are a Rank of numbers , having equal difference , to wit , 2. In the second colum , their squares . In the third colum , the first difference . In the fourth colum , the second difference . II. Note . Hence it follows , that by the help of such differences the table of squares may be calculated : thus , in the first Example , the sum of 1 and 3 , is 4 ; the square of 2. The sum of 2 , 3 and 4 , is 9 ; the square of 3. The sum of 2 , 5 and 9 , is 16 ; the square of 4. The sum of 2 , 7 and 16 , is 25 ; the square of 5. The sum of 2 , 11 and 36 , is 49 ; the square of 7. &c. III. Note . Like plain numbers are in the same proportion one to another , that a square number is in , to a square number : Euclide the 26 Proposition of the Eighth Book . Therefore the second difference in such a Rank of plane numbers are equal . Further , what planes and solids are either equal or proportionable to such Ranks may be gradually calculated ; as in the last . IV. Note . 1 1       7 2 8 12   19 6 3 27 18 37 6 4 64 24 71   5 125       1 2 3 4 5 1 1       26 3 27 72 98 48 5 125 120 218 48 7 343 186 386   9 729     1 2 3 4 5 In the first Example , in the first colum are the numbers in a Rank having equal difference , to wit , a unite . In the second colum , the cubes of those numbers . In the third colum , the first differences of those cubes . In the fourth colum , the second differences . In the fifth colum , the third differences , to wit , 6 , 6. The like in the second Example . V. Note . Hence it follows , that the table of cubes may be made thus : In the first Example , 1 and 7 , is 8 ; the cube of 2. The sum of 8 and 19 , is 27 ; the cube of 3. The sum of 18 , 19 and 27 , is 64 ; the cube of 4. The sum of 6 , 18 , 37 and 64 , is 125 ; the cube of 5. The sum of 6 , 24 , 61 and 125 , is 216 ; the cube of 6. The sum of 6 , 30 , 91 and 216 , is 343 ; the cube of 7. The like in the second Example . VI. Note . Like solid numbers are in the same proportion one to another , that a cube number is in , to a cube number ▪ Euclide the XXVII Prop. of the Eighth Book . Therefore the third differences in such a Rank of solid numbers are equal : further , such planes and solids as are either equal or proportionable to such Ranks , may be gradually calculated , as in the last . VII . Note . If a Rank of Squares , whose Roots have equal differences , be multiplied by any number , the second differences of such a Rank of proucts are equal . Let the number multiplying be 10. 0 0 00     10 1 1 10 20 30 2 4 40 20 50 3 9 90 10 70 4 16 160 20 90 5 25 250     1 2 3 4 5 In the first colum are the numbers bearing equal difference . In the second colum are the squares of those numbers . In the third colum the products . In the fourth colum the first differences . In the fifth colum the second differences and they equal . VIII . Note . If unto such a Rank of Products , as in the last , there be added a Rank of Cubes , whose Roots are equal to the Roots of the Squares , the third differences of such a Rank will be equal . 0 00 0 00       11 1 10 1 11 26   37 6 2 40 8 48 32 69 6 3 90 27 117 38 107 6 4 160 64 224 44 151   5 250 125 375       1 2 3 4 5 6 7 In the first colum are the numbers having equal difference . In the second colum the Products of their squares by a given number . In the third colum the cube of the numbers in the first colum . In the fourth colum the sum of the products and cubes . In the fifth colum their first difference . In the sixt colum the second differences . In the seventh colum the third differences which are equal . IX . Note . Let a constant number be added to a Rank of Products , so that one of the numbers multiplying be a constant number , and the other of the numbers be the squares of numbers having equal difference , and this Rank of sums be added to a Rank of cubes , whose roots are the same with the roots of the squares ; such a compounded Rank will have their third difference equal . Thus , 1 8 10 1 19       37 2 8 40 8 56 32   69 6 3 8 90 27 125 38 107 6 4 8 160 64 232 44 151   5 8 250 125 383       1 2 3 4 5 6 7 8 In the first colum are the numbers having equal difference . In the second colum is constant number to be added . In the third colum are the Rank of products , that is , the squares of the numbers in the first colum multiplied by a given number . In the fourth colum are the cubes of the numbers in the first colum . In the fifth colum are the sum of the numbers in the second , third and fourth colums . In the sixth colum are the first differences of their sums . In the seventh colum are the second differences . In the eighth colum the third differences , and they equal . X. Note . In a Rank of numbers , having equal difference , and equal in number ; if the third part of the cubes of each of these numbers , be substracted from the products of the squares of each of these numbers , in half the greatest number of that Rank , the remainders will be a Rank of numbers , equal to all the squares in the several portions of one fourth of a sphere , whose diameter is equal to the greatest number in that Rank , and the third differences of this Rank of portions are equal ; but the first and second differences will increase and decrease , differently one from another . Thus , 0 00 00 00       99 3 09 108 99 162   261 54 6 72 432 360 108 369 54 9 243 972 729 54 423 54 12 756 1728 1152 00 423 54 15 1125 2700 1575 54 369 54 18 1944 3888 1944 108 261 54 21 3087 5292 2205 162 96   24 4608 6912 2304       1 2 3 4 5 6 7 In the first colum are the numbers having equal difference . In the second colum are the pyramides adscribed within the cubes of the numbers in the first colum . In the third colum are the products of the squares of the numbers in the first colum , by half the greatest number in the first colum . In the fourth colum are the differences of the numbers in the second and third colums , that is , all the squares in several portions of one fourth of a sphere , whose diameter is 24. In the fifth colum are the first differences . In the sixt colum are their second differences . In the seventh colum are the third differences , and they equal . XI . Note . The Application or Vse of the Preceeding Notes . The application or Use may be , to calculate Pyramides and cones , either the whole or their parts ▪ as also to calculate the parabolick and hyperbolick conoides , either the whole , or their frustums ; yet also , to calculate the sphere or spheroide , either the whole or their portions or Zones , and that gradually , that is , to find the solidity upon every inch or foot . To find the solidity of a parabolick Conoide upon every two inches . To do which , consider the Diagram of the 18 Prop. of my Stereometrical Prop. Let PA be 16 ; AR 12 ; therefore AV or PH will be 9 ; for it ought to be as PA , is to AR ; so is AR , to AV. Let the axis AP be divided into eight equal parts ; viz. 2 , 4 , 6 , 8 , 10 , 12 , 14 , 16. Let there be planes drawn parallel to the base , through every one of these divisions , though in the Diagram there is not so many . From P to the first Q suppose to be 2 , its square 4 ; the half thereof 2 , which multiplied by 9 equal to PH , the Product will be 18 ; that is , the Prism QZGIHP ; equal to all the squares in the portion of the conoid QOP . Let from P , to the second Q be 4 , its square 16 , the half is 8 ; which multiplied by 9 , the Product is 72 ; equal to the Prism QZFIHP , equal to all the squares in the portion of the conoid QOP . Let from P to the third Q be 6 , its square 36 , the half of it is 18 , which multiplied by 9 , the product is 162 ; the Prism QZEIHP ; equal to all the squares in the third portion of the conoid QOP . Having obtained two portions , the rest may be obtained thus : having obtained the second difference , which is 36 , we may proceed to find the rest by the Seventh Note ; thus , add 36 to 54 , and it makes 90 : which added to 72 , the sum is 162 , equal to all the squares in that portion , and so in order ; 36 , 90 and 162 ; the sum is 288. 36 , 126 and 288 ; the sum is 450. 36 , 162 and 450 , the sum is 648 , &c. 0 00     8 2 18 36 54 4 72 36 90 6 162 36 126 8 28● 36 62 10 45● 36 198 12 648 36 234 14 882 36 270 16 1152     1 2 3 4 In the first colum are the parts of the altitude of the conoid . In the second colum are all the squares in several portions of one fourth of a conoid . In the third colum the first differences . In the fourth colum the second differences . These portions in the second colum may be reduced to circular portions , thus , as 14 is to 11 ; so are all the squares in these portions to the portions themselves . I. Scholium : The use of this gradual calculation may be thus : Suppose a Brewers Copper be in form of a parabolick conoid ; the quantity of liquor therein contained may be found , thus , having calculated a table upon every inch , or two inches , or as is thought convenient ; then having a straight Ruler divided equally into inches , putting the Ruler into the liquor to the bottom of the Copper , see how many inches of the Ruler is wet ; with the number of wet inches enter the first colum of your table , and in the next colum are the number of cubick inches which that portion contains ; the number of cubick inches thus found , being divided by the number of cubick inches in a Gallon , the quotient shews the number of Gallons in that portion of the Copper . II. Scholium : To compose several works into one . As 14 , is to 11 ; so are all the squares in one fourth of the conoid , to one fourth of the conoid it self . because this one fourth ought to be divided by the number of cubick inches in a Gallon , suppose it 288 , to shew the number of Gallons in each portion , we may multiply 14 by 288 , that is 4032. Then as 4032 is to 11 ; so are all the squares in one fourth of the conoid , to the Gallons in that one fourth . Further , because this one fourth ought to be multiplied by 4 , to reduce it to a whole conoid ; therefore , divide the constant divisor , that is , 4032 , by 4 , and it will be 1008. Then , as 1008 , is to 11 ; so are those several portions in the second colum of the last table , to the number of Gallons in those several portions of a parabolick conoid . By such compositions may the Practitioner compose constant divisors or dividends , which will much breviate the work ; this is onely for an Example . Every parabolick conoid hath its second differences equal . To find the second differences , work thus , Square one of the equal segments of the axis , and multiply that Square by the Parameter , that product will be the second difference . In this Example , the equal segment of the axis is 2 , the square of it 4 ; which multiplied by the Parameter 9 , the product is 36 , the second difference . Half of the second difference , is always the first of the first difference . Half 36 the second difference , is 18 , the first of the first difference , &c. Here note , this 36 is the second difference of one fourth of all the squares in a parabolick conoid ; if 36 be multiplied by 4 , it makes the second difference 144 ; whose half is 72 , the first of the first differences . Or , the first differences are found by taking half the difference of the squares of any two segments , which multiplied by the Parameter , thereby the first differences are obtained . Thus , to find the first difference answerable to the Segments 6 and 8 ; the Square of 8 , is 64 ; the Square of 6 , is 36 ; the difference of those Squares is 28 , whose half is 14 ; which multiplied by the Parameter 9 , the product is 126 ; the first difference answerable betwixt 6 and 8. XII . Note . To find the solidity of an hyperbolick conoid gradually , to wit , upon every three inches . For the performance of which , take notice of the XVI Prop. of Stereom . Prop. in that Diagram , Let AM equal to AB be 9. Let ML equal to AF , be 6. Let AE be 15 : therefore FE will be 9. Let the rest of the construction be as in that Proposition . Let from M to the first K be 3 , whose square is 9 , whose half is 4½ , the area KHM ; which being multiplied by ML , 6 ; the product will be 27 , the prism KHNOLM . Because FE , FC and FL are equal , that is , each of them 9 : Therefore , the first pyramid ONXILM will be 9. Then this prism and pyramid being added , will make 36 , the whole prism KHXILM , equal to all the squares in the portion KZM. Let from M to the second K be 6 , whose square is 36 , its half 18 , the area KHM ; which being multiplied by ML , 6 ; the product will be 108 , equal to the prism KHNOLM . The cube of 6 , is 216 ; a third part is 72 , the pyramid ONXIL , this prism and pyramid being added together , the sum will be 180 ; the prism KHXILM : equal to all the squares in the portion KZM. Let from M to A , be 9 ; its square 81 , the half 40½ , equal to the area ABM , which being multiplied by ML , 6 ; the product will be 243 : the cube of 9 , is 729 ; a third part thereof is 243 ; equal to the pyramid FCDEL : this prism and pyramid being added together , is 486 ; the whole prism ABDELM , equal to all the squares of the portion AZM . These three portions being obtained , they may be continued by the VIII Note , thus : 0 00       36   3 46 108 144 54 6 180 162 306 54 9 486 216 522 54 12 1008 270 792 54 15 1800 324 1116 54 18 2216 378 1494 54 21 4410 432 1926   24 6336       1 2 3 4 5 For if the third differences which are equal , and in this Example is 54 , be added to the first of the second differences , being 108 , it makes 162 , and by such additions , the second differences in the fourth colum are made . Further , by adding these second differences to the first of the first differences which is 36 , it makes 144 , &c. So the numbers in the third colum are made . Yet further , by adding these first differences to the first number in the second colum , the Rank of portions of such a conoid is made . Then , By making use of the directions in the first and second Scholiums , the number of Gallons are obtained . The parabolick and hyperbolick conoides may well be made use of for Brewers Coppers ; the parabolick , when the crown is somewhat blunt ; but the hyperbolick conoid when the crown is more sharp . XIII . Note . To calculate a Sphere gradually , to wit , upon every three Inches . Consider the XV Prop. of Stereom . Prop. Let ED equal to EF , be 24. The rest of the construction as in that Prop. Let from E , to the first R be 3 , whose square is 9 ; whose half is 4½ , the area RXE , which being multiplied by 24 , the product will be 108 ; the prism KHXREF . The cube of 3 , is 27 ; a third part thereof is 9 , the pyramid KHOIF ; this pyramid taken from the former prism , leaves the prism RXOIFE , 99 : equal to all the squares in the portion RQE. From E , to the second R , 6 ; its square 36 , the half 18 , which multiplied by 24 , makes 432 ; the prism RXHKFE . The cube of 6 , is 216 , a third part of it is 72 ; the pyramid KHOIF : this pyramid being taken from that prism , there rest 360 ; the prism RXOIFE , equal to all the squares in the portion RQE. Let from E to the third R be 9 ; its square 81 , the half thereof 40½ , the area RXE , this area being multiplied by 24 , the product will be 972 , the prism KHYREF : the cube of 9 , is 729 , a third part of it is 243 , the pyramid KHOIF : this pyramid being substracted from that prism , the remainder is 729 ; the prism RXOIFE , equal to all the squares in the third portion RQE. Having obtained these three portions , the rest may be found by their third difference , according to the X. Note . 0 00 0 0       99   3 09 108 99 162 261 54 6 72 432 360 108 369 54 9 243 972 729 54 423 54 12 576 1728 1152 00 423 54 15 1125 2700 1575 54 369 54 18 1944 3888 1944 108 261 54 21 3087 5292 2205 162 99   24 4608 6912 2304       1 2 3 4 5 6 7 The numbers in the seventh colum are the third differences , and they equal ; the numbers in the sixt colum are the second differences , and are composed by substracting the numbers in the seventh from the first and last numbers in the sixt colum ; the numbers in the fifth colum are the first differences , and are composed by adding those numbers in the sixt colum to the first and last of those in the fifth colum ; the numbers in the fourth colum are all the squares in several portions of one fourth of a sphere whose diameter is 24 , those portions are made by adding the numbers in the fifth colum to the numbers in the fourth , thus , 261 , and 99 , is 360. 369 , and 360 , is 729. 423 , and 729 , is 1152 , &c. Then making use of the first and second scholium the number of gallons are obta●ned . Or if it be made , as 14 , is to 11 , so is 54 , to a fourth number , with that fourth number proceede to make tables of the second and first differences , and then the table of portions it selfe . Every sphere hath its third differences equall . To find the third difference , doe thus . Cube one of the equal segments of the axis and multiply that cube by 2 , and that product will be the third difference , thus , the cube of three is 27 , which multiplyed by 2 , the product is 54 ; the third difference of all the squares in one fourth of a sphere . Here note , that it is to be understood , that the axis of the sphere is equally divided into an equal number of segments ; so then , if the number of segments in the semiaxis , less by one ; be multiplyed by the third difference , it gives the first of the second differences . Thus , the number of segments in the semiaxis is 4 , then 4 less 1 , is 3 ; which being multiplyed by 54 , the product is 162 : the first of the second differences . To find the third difference in one fourth of all the squares in a spheroid , do thus : The axis being divided as above in the sphere ; cube the difference betwixt two Segments , which being multiplyed by 2 , makes a product ; then , as the square of the semiaxis , is to the square of the other semidiameter ; so is that former product to a fourth number , which will be the third difference . For the second differences , use the Rules given for the sphere . XIV . Note . To calculate a pyramid or cone gradually . To find the third difference in a pyramid work thus , the Altitude of the pyramid being equally divided ▪ cube the difference of the two segments , which being doubled , makes a number ; then , as the square of the Altitude of the pyramid , is to the area of the base of that pyramid ; so is that former number , to the third difference of that pyramid . To find the second differences in a pyramid : As the difference of two of the segments of the Altitude , is to the following segment ; so is the third difference , to the second difference answerable to that segment . To find the first differences in a pyramid . Cube two of the segments , and take a third part of their difference . Then , as the square of the Altitude of the pyramid ; is to the area of the base of that pyramid ; so is that former difference ; to the first difference answerable to those two segments . Let there be a pyramid whose Altitude is 10 , and one side of the base is 40 , and the other side 5 ; therefore the area of the base is 200. Let the Altitude be divided into five equall parts , and to calculate accordingly . To find the third difference , the cube of 2 , is 8 ; whose double is 16. Then , as 100 the square of the Altitude , is to 200 the area of the base ; so is 16 , to the third difference 32. To find any of the second differences at demand , to find the second difference answerable to 8. As 2 , the difference betwixt the segments 6 , and 8 , is to 8 ; so is the first difference 32 , to 128 the second difference answerable to 8. The second differences are in proportion one to another , as their answering segments ; as 2 , is to 3 ▪ 2 ; so is 8 , to 128. To find any of the first differences , cube the two Segments , to wit , 2 and 4 , and the cubes will be 8 and 64 ; then take 8 from 64 and the Remainder is 56 , a third part is 18⅔ . then , as the square of the Altitude 100 , is to the area of the base 200 ; so is 18⅔ , to 37⅓ , the first difference , answering to 2 and 4. Then by a continuall adding of the third difference to the second differences they are made , and by adding the first of the second differences to the first of the first differences and so in order the first differences are made . Lastly by adding the first differences the Segments of the pyramid , are made according to the III. Note ; or thus . 0 0       5⅓   2 5⅓ 32 37⅓ 32 4 42⅔ 64 101⅓ 32 6 144·· 96 197⅓ 32 8 341⅓ 128 325⅓   10 666⅔       1 2 3 4 5 The numbers in the fifth colum are the third differences , the first number in the fourth colum being found by the Rule before given , all the numbers in that fourth colum may be made by adding the third difference , thus , to 32 adde 32 , the summe is 64. adde 32 , to 64 ; the summe is 96. adde 32 , to 96 ; the summe is 128. The first number in the third colum being found by the Rule above , then 5⅓ added to 32 ; the summe is 37⅓ . adde 64 , to 37⅓ ; the summe is 101⅓ . adde 96 , to 101⅓ , the sum is 197⅓ . adde 128 , to 197⅓ ; the sum is 325⅓ . further , adde the first of the third colum , to the first of the second colum ; thus , adde 5⅓ , to 0 ; the sum will be 5⅓ , adde 57⅓ , to 5⅓ , the sum is 42⅔ , adde 101⅓ , to 42⅔ ; the sum is 144. adde 197⅓ , to 144 ; the sum is 341⅓ , adde 325⅓ , to 341⅓ ; the sum is 666⅔ . If it be to calculate a cone whose diameters of the base are 40 and 5. Let it it be made , as 14 , is to 11 ; so is 32 , to the third difference of the same cone . Then proceede with the third difference to make the second and first ; and lastly , the table it self . XV. Note . The calculation of frustum pyramides whose bases are unlike , To the performance of which consider the third case of the second proposition of Stereom . Prop. Every such solide hath its third differences equall , but the second and first differences will be complicated according to the IX . Note . To find the third difference proper to the pyramid BCDHF , Let the construction and numbers be the same as in that diagram , and let it be to calculate it upon every two inches , thus . The cube of 2 , is 8 ; the double thereof is 16 , Then , as the square of the Altitude 40 , that is 1600 , is to the area of the base BCDH , 336 ; so is 16 , to 336 / 100. by the Rule delivered in the 14 Note , the first of the second differences is 336 / 100. and the first of the first differences is 56 / 100. The solide HDEGVF hath its second differences equall by the VII . Note . To find its first and second differences . The square of 2 , is 4. which multiplyed by FV , 26 ; the product will be 104. then , as 40 the Altitude , is to HD , 28 ; so is 104 , to 7280 / 100. the second difference . Therefore the first of the first differences will be 3640 / 100. To find the second differences of the solide ABHOIF the square of 2 is 4 , which multiplyed by IF , 30 ; the product is 120 , then , as 40 , the Altitude ; is to OA , 12 : so is 120 , to 36 , the second difference . Therefore the first of the first differences are 18. For the complication of these differences . 1 2 3   ●6 / 100 336 / 100 3●6 / 100 in the pyramid BCDHF 3640 / 100 7280 / 100   in the prism HGEDFV 18 36   in the prism ABHOIF 5496 / 100 11216 / 100 336 / 100 their summe . Rejecting the denominators they may be written Thus , 5496 11216 336 Because the denominators are Rejected , therefore the two last figures toward the Right hand are decimals . 0         161496   2 161496 11216 172712 336 4 334208 11552 184264 336 6 518472 11888 196152 336 8 714624 12224 208376 336 10 923000 12560 220936 336 12 1143936 12896 233832 336 14 1377768 13232 247064 336 16 1624832 13568 260632 336 18 1885464 13904 274536 336 20 2160000 14240 288776 336 22 2448776 14576 303352 336 24 2752128 14912 318264 336 26 3070392 15248 333512 336 28 3403904 15584 349096 336 30 3753000 15920 365016 336 32 4118016 16256 381272 336 34 4499288 16592 397864 336 36 4897152 16928 414792 336 38 5311944 17264 432056   40 5744000       1 2 3 4 5 The construction of the table may be thus ; the numbers in the first colum are the third differences ▪ The first number in the fourth colum is the complicated second difference , and the other number in that fourth colum are made thus , to the first 11216 , adde 336 ; the sum is 11552. Then to that 11552 , adde 336 ; the sum is 11888 , &c. The first number in the third colum is complicated from the first complicated difference and a parallelipepidon whose base is the plane RIFV , and the Altitude the first Segment of the Altitude of the frustum , thus , the plane RIFV , is 780 ; which being doubled is 1560 ; then , 156000 more 5496 is 161496 ; the first of the first differences , then 161496 more 11216 , is 172712. Further , 172712 more 11552 , is 184264. Yet further 184264 more 11888 , is 196152 , &c. The numbers in the second colum are made thus , the first number in the second colum , is the same as the first number in the third colum , then , 161496 more 172712 , is 334208 , and 334208 more 184264 , is 518472 , &c. Then makeing use of the first and second scholium , the quantity of Liquor that such vessels contain may easily be obtained . XVI . Note . To calculate Elliptick solides whose bases are unlike . The calculation of such solides are the same as in the 15 , note for if the first , second and third complicated differences be found , then makeing use of this propotion as 14 , is to 11 ; so is 336 ; to the third difference . And As 14 , is to 11 ; so is 11216 , to the first of the second differences . Further As 14 , is to 11 ; so is 161496 , to the first of the first differences , then proceede to make the table it self , as in the 15 note . Or make use of the secund scholium of the 11 note and you will have the quantity in Gallons . Or Such Elliptick solides may be calculated by the 12 note : for every such Elliptick solide is equall to a frustum hyperbolick conoide whose circular bases of the conoide , are equall to the Elliptick bases of the Elliptick solide ; and the Altitude of one frustum is equall to the Altitude of the other . XVII . Note . Every hyperbolick conoid hath its third differences equal . To find the third , second and first differences in an hyperbolick conoid , and consequently to calculate that conoid gradually . In the forementioned diagram of the 17. prop. Stereom . Prop. Let GM , the Transverse diameter be 12. ML , the parameter 6. MA , the axis of the conoid 24. To calculate the solidity of this conoid vpon every three Inches . To find the third difference of this conoid . Take the difference of two of the Segments , to wit , 3 ; whose cube is 27 : whose double is 54. Then , as GM , 12 ; is to ML , 6 : so is 54 , to 27. The third difference of all the squares in one fourth of that conoid proper to that pyramid FCDEL . By the Rule in the last note the first of the second differences is 27. For the first of the first differences , worke thus ; take the first Segment which is 3 ; whose cube is 27 ; a third part is 9 , then , as GM , 12 ; is to ML , 6 : so is 9 , to 4½ . the first of the first differences proper to the pyramid FCDEL . The second and first differences of all the squares in the fourth of this conoid , is complicated from the second and first differences of the pyramid FCDEL , and the second and first differences of the prism ABCFLM . Every such prism hath it second difference equall . To find the second and first difference of the prism ABCFLM . Square the difference of two of the Segments of the axis , to wit , 3 ▪ that is 9 , which being multiplyed by the parameter ML , 6 ; the product is 54 , the second difference . The first of the first d●fferences of every such prism is half of the second difference ; therefore the first of the first differences is 27. To complicate these differences . 1 2 3 differences 4½ 27 27 in the prism FCDEL . 27 54   in the prism ABCFLM . 31½ 81 27 in all the squares of one fourth of that hyperbolick conoid . 0 0       31½   3 31½ 81 112½ 27 6 144 108 220½ 27 9 364½ 135 355½ 27 12 720 162 517½ 27 15 1237½ 189 706½ 27 18 1944 216 922½ 27 21 2866½ 243 1165½   24 4032       1 2 3 4 5 In the first colum are the third differences . In the fourth colum the second differences . In the third colum the first differences . In the second colum the portions of all the squares of one fourth of an hyperbolick conoid , upon every three inches , whose Transverse diameter is 12 , and parameter is 6 , and axis is 24. The construction of this table is the same as the former ; thus , 81 more 27 ; is 108. more 27 ; is 135. more 27 ; is 162. &c. 31½ . more 81 ; is 112½ . more 108 ; is 220½ . &c. 0 more 31½ ; is 31½ . more 112½ ; is 144. more 220½ . is 364½ . more 355½ ; is 720. Here remember that the Transverse diameter is found , by the 9 of the 23 Proposition . of Stereom . Prop. Also the parameter found by the converse of the first part of the 11 Prop. of Stereom . Prop. The parameter of the parabolike conoid is found , by the converse of the 9 Prop. Of Stereom . Prop. XVIII . Note . Cautions Concerning Reduction . 1 If it be to calculate pyramids whether Regular or Irregular , whole or frustums ; the third , second and first differences are to be found as above : then Reduce those differences into Gallons and parts of a Gallon , or Barrells , or parts of a barrels ; Thus Suppose 288 cubick inches make one Gallon , and 36 Gallons make one Barrell . Then , If the measure be taken in inches , divide the third , second and first differences by 288 , and so there will be three quotients in Gallons or parts of a Gallon , then with those three quotients proceede to make the table of solid Segments , and that table will be in Gallons or parts of a Gallon . If it be to calculate a table in Barrells multiply 288 by 36 and the product will be 10368 the number of cubick inches in one Barrell . Then divide the third , second and first differences by 10368 , there will be three quotients in Barrells or parts of a Barrell : Then with these three quotients proceede to make the table of solides Segments . That table being so made will be in Barrells or parts of a Barrell . 2 ▪ To calculate Cones and Elliptick solids , whether the whole or their frustums . Haveing found their third second and first differences , as above , and it be to calculate them in cubick inches , Let it be made as 14 , is to 11 ; so is the third difference , to a fourth , And , As 14 , is to 11 ; so is the second difference , to a fourth , Further , As 14 , is to 11 ; so is the first difference , to a fourth with these three number thus found , proceede to make the table of solid Segments , and that table will be in cubick inches . To calculate these solids in Gallons . Multiply 14 by 288 the product will be 4032. Then , As 4032 , to 11 ; so is the third difference , to a fourth . And , As 4032 , to 11 ; so is the second difference , to a fourth . Further , As 4032 , to 11 ; so is the first difference , to a fourth . With these three numbers thus found , proceed to make the table of solid Segments . So that table will be in Gallons . To calculate these solids in Barrells . Suppose 288 cubick inches makes one Gallon , and 36 gallons makes one Barrell , then multiply 288 , 36 and 14 one into another and they make 145152. Then , As 145152 , is to 11 ; so is the third difference , to a fourth . And , As 145152 , is to 11 ; so is the second difference , to a fourth . Further , As 145152 , is to 11 ; so is the first difference , to a foruth . With these three numbers thus found make the table of solid Segments : that table will be in Barrells . III. Having found the third , second and first differences of all the squares of one fourth of a sphere , spheroid and hyperbolick Conoid , as in the 12 and 13 notes and the second and first differences of all the squares of one fourth of a parabolick conoid as in the 11 note : they may be Reduced to Circular differences . Thus ▪ As 14 , is to 11 ; so is the third difference , to a fourth . And , As 14 , is to 11 , so is the second difference , to a fourth . Further , As 14 , is to 11 ; so is the first difference , to a fourth . With these numbers thus found make a table , of solid Segments of cubical inches of one fourth of any of these solids . These solid Segments ought to be multiplyed by four , to reduce them to solid Segments of a whole sphere , spheroid , hyperbolick and parabolick conoid : but to shun that work divide 14 , by four , and then find the new differences ; but because 14 cannot be just divided by four , therefore divide 14 , by two , and multiply 11 , by two , and then work ; Thus , Then , As 7 , to 22 ; so is that third difference , to a fourth . And , As 7 to 22 ; so is that second difference , to a fourth . Further , As 7 , to 22 ; so is that first difference , to a fourth . With these numbers thus found , proceed to make tables as is taught in those Notes : tables so made , will be tables of solid Segments of those solids , in cubick inches . To calculate these solids in Gallons . Multiply 288 by 14 , whose product is 4032 ; one fourth thereof is 1008 ; Then , As 1008 , is to 11 ; so is the third difference , to a fourth . And As 1008 , is to 11 ; so is the second difference , to a fourth . Further , As 1008 , is to 11 ; so is the first difference , to a fourth . Tables being made , with numbers thus found ; according to the former directions in the sphere , spheroid , hyperbolick and parabolick conoids , will be tables of solid Segments of a whole sphere , spheroid , hyperbolick and parabolick conoid , in Gallons or parts thereof . To calculate these solids in Barrells . Multiply 4032 by 36 , the product will be 145152 , one fourth thereof will be 36288 ; Then , As 36288 , is to 11 ; so is the third difference , to a fourth . And , As 36288 , is to 11 ; so is the second difference , to a fourth . Further , As 36288 , is to 11 ; so is the first difference , to a fourth . Tables being made , with numbers thus found , according to the former directions , will be tables of solid Segments in Barrells . &c. Then , Using a Rod or Ruler equally divided into inches as in scholium the first , the number of Gallons or Barrells may speedily be obtained . As for the just magnitude of the Gallon , it i● not my businesse to dispute ; that being determined by custom or Authority : I took 288 onely for Example sake . XIX . Note . In a Rank of numbers having equal differences . Let the first term in the Rank be Z , its square ZZ . the second term 2Z , its square 4ZZ , therefore the first of the first differences is 3ZZ , the third term in that Rank 3Z , its square 9ZZ , then 9ZZ , Less 4ZZ , the second of the first differences 5ZZ , therefore 5ZZ Less 3ZZ the second difference will be 2ZZ . Further , The fourth term in that Rank is 4Z , its square is 16ZZ , then 16ZZ Less 9ZZ the third of the first differences 7ZZ ; again , 7ZZ Left 5ZZ the second difference is 2ZZ . Hence it follows , That the second difference is equall to the square of the first term doubled . Or also , The second difference is equall to the squar● of the difference of two of the terms , ( in order taken ) doubled . By the same method we find that the third difference in a Rank of cubes are equall , and the third difference is equal to the first term multiplyed by 6. Or , The third difference is equal to the cube of the difference of two of the terms , taken in order , multiplyed by 6. The index and equal difference , of every power agrees ; to wit , the index of the square is 2 , and the second differences are equal . The index of the cube is 3 , and the third differences are equal . The index of the square squared is 4 , and the fourth differences are equal . &c. The equal difference of every power , is complicated from the index of that power , and the equal difference of the next Lesser power . Let the Rank be in naturall order , Thus ; 1 , 2 , 3 , 4. &c. The indices of the powers , Thus. 1 2 3 4 5 Z ZZ ZZZ ZZZZ ZZZZZ A unity the equal difference in that naturall Rank , whose square is 1 , which multiplyed by 2 the index of the square the product is 2 , the equal difference in the squares . 3 , the index of the cube multiplyed by 2 the equal difference in the squares , the product is 6 , the equal difference in the cubes ▪ 4 the index of the square squared multiplyed by 6 the product is 24 the equall difference in the square squared , &c. If the Rank be in order thus , 2 , 4 , 6 , 8 , &c. 2 the equal difference of this Rank whose square is 4 ; multiplyed by 2 the index of the square the product is 8 ; the equal difference of the squares in such a Rank . Because the equal difference of the Rank is 2 , therefore the indices are to be doubled , &c. And the equall difference of the powers in such a Rank will be 8 , 48 , 384 , &c. XX. Note . For the more easier calculation of the second sections of the sphere and spheroid ; worke , Thus. From the double of the superficies of the triangle BZN , substract the superficies of the triangles BZGD and NZPA , the Remainder will be the superficies of the triangles BZPA and NZGD , the areas of these two triangles being substracted from the area of the traingle NZB , the Remainder will be the superfice of the triangle ZGDAP . FINIS . By Iohn Baker , living in Barmonsey-street in Southwark , over against the Princes-Armes , is Taught Arithmetick , both in whole numbers and fractions , Decimal , Logarithmetical and Algebraical , Geometry , Trigonometry , Astronomy , the use of Globes , Navigation , Measuring , Gageing , Dyalling , &c. Also the Construction and use of all the usual lines put upon Rules or Scales , He also teacheth how to find the ( Length and ) Spreading of a Hip-rafter , only by a Line of Chords of singular use for Carpenters , a way not as yet vulgarly known amongst Workmen . Faults Escaped in the Impression of Stereometrical Propositions . Page 1 , line 18 , for and Z Read and H. p. 10 , l. 23 , for 56 , r. 58. p. 34 , l. 25 , after RI put . p. 43 , for 297232 , r. 297432. p. 45 , for 18 , r. V432 ▪ p. 46 , l. 1. for XVI , r. XVII . and l. 21 , for AE , 16 ; r. AF , 6 ; p. 48 , l. 6 , for 634. r. 624 , p. 51. l. 22 , for diameter , r. semidiameter . p. 58 ▪ l. 25 , for ZB , r. XB ▪ p. 63 , l. 1 , for XIII . r. XXIII . p. 100. l. 17 , for parameter , r. diameter . p. 102 and 103 for as 4 to 3 , r. as 3 to 2. p , 105. l. vlt. for Z+2 ▪ r. Z-2 . p. 106 , l. 4 , for Z = ● r. Z-⅗ . and l. 25 , for 89 , r. 98. p. 7 against 12 in the first col . in the sec. r. 576. and in the fifth colum f. 96 r. 99. p. 13. in the second col . f. 46. r. 36. and in the same col . f. 2216 , r. 2916. 
A29764	----	The triangular quadrant, or, The quadrant on a sector being a general instrument for land or sea observations : performing all the uses of the ordinary sea instruments, as Davis quadrant, forestaff, crosstaff, bow, with more ease, profitableness, and conveniency, and as much exactness as any or all of them : moreover, it may be made a particular and a general quadrant for all latitudes, and have the sector lines also : to which is added a rectifying table to find the suns true declination to a minute or two, any day or hour of the 4 years : whereby to find the latitude of a place by meridian, or any two other altitudes of the sun or stars / first thus contrived and made by John Brown ... Brown, John, philomath. 1662 Approx. 38 KB of XML-encoded text transcribed from 14 1-bit group-IV TIFF page images. Text Creation Partnership, Ann Arbor, MI ; Oxford (UK) : 2004-08 (EEBO-TCP Phase 1). A29764 Wing B5043 ESTC R33264 13117453 ocm 13117453 97765 This keyboarded and encoded edition of the work described above is co-owned by the institutions providing financial support to the Early English Books Online Text Creation Partnership. This Phase I text is available for reuse, according to the terms of Creative Commons 0 1.0 Universal . The text can be copied, modified, distributed and performed, even for commercial purposes, all without asking permission. Early English books online. (EEBO-TCP ; phase 1, no. A29764) Transcribed from: (Early English Books Online ; image set 97765) Images scanned from microfilm: (Early English books, 1641-1700 ; 1545:11) The triangular quadrant, or, The quadrant on a sector being a general instrument for land or sea observations : performing all the uses of the ordinary sea instruments, as Davis quadrant, forestaff, crosstaff, bow, with more ease, profitableness, and conveniency, and as much exactness as any or all of them : moreover, it may be made a particular and a general quadrant for all latitudes, and have the sector lines also : to which is added a rectifying table to find the suns true declination to a minute or two, any day or hour of the 4 years : whereby to find the latitude of a place by meridian, or any two other altitudes of the sun or stars / first thus contrived and made by John Brown ... Brown, John, philomath. [2], 24, [1] p. : ill. To be sold at [his, i.e. Brown's] house, or at Hen. Sutton's ..., [London] : 1662. Added illustrated t.p. Place of publication suggested by Wing. Reproduction of original in the British Library. Created by converting TCP files to TEI P5 using tcp2tei.xsl, TEI @ Oxford. Re-processed by University of Nebraska-Lincoln and Northwestern, with changes to facilitate morpho-syntactic tagging. Gap elements of known extent have been transformed into placeholder characters or elements to simplify the filling in of gaps by user contributors. EEBO-TCP is a partnership between the Universities of Michigan and Oxford and the publisher ProQuest to create accurately transcribed and encoded texts based on the image sets published by ProQuest via their Early English Books Online (EEBO) database (http://eebo.chadwyck.com). The general aim of EEBO-TCP is to encode one copy (usually the first edition) of every monographic English-language title published between 1473 and 1700 available in EEBO. 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Understanding these processes should make clear that, while the overall quality of TCP data is very good, some errors will remain and some readable characters will be marked as illegible. Users should bear in mind that in all likelihood such instances will never have been looked at by a TCP editor. The texts were encoded and linked to page images in accordance with level 4 of the TEI in Libraries guidelines. Copies of the texts have been issued variously as SGML (TCP schema; ASCII text with mnemonic sdata character entities); displayable XML (TCP schema; characters represented either as UTF-8 Unicode or text strings within braces); or lossless XML (TEI P5, characters represented either as UTF-8 Unicode or TEI g elements). Keying and markup guidelines are available at the Text Creation Partnership web site . eng Quadrant. Dialing. Mathematical instruments. 2004-01 TCP Assigned for keying and markup 2004-03 Apex CoVantage Keyed and coded from ProQuest page images 2004-04 Mona Logarbo Sampled and proofread 2004-04 Mona Logarbo Text and markup reviewed and edited 2004-07 pfs Batch review (QC) and XML conversion THE TRIANGULAR QUADRANT : OR The QUADRANT on a SECTOR . Being a general Instrument For Land or Sea Observations . Performing all the Uses of the ordinary Sea Instruments ; as Davis Quadrant , Forestaff , Crossstaff , Bow , With more ease , profitableness , and conveniency , and as much exactness as any or all of them . Moreover , It may be made a particular , and a general Quadrant for all latitudes , and have the Sector lines also . To which is added a Rectifying Table , to find the Suns true Declination to a minute or two , any day or hour of the 4 years : Whereby to find the latitude of a place by a Meridian , or any two other altitudes of the Sun or Stars . First thus Contrived and made by Iohn Browne at the Sphere and Dial in the Minories , and to be sold at 〈◊〉 house , or at Hen. Sutton's in Thredneedle-street behind the Exchange . 1662. triangular quadrant THE TRIANGULAR QUADRANT : Being a GENERAL INSTRUMENT for Observations at Land or Sea , performing all the Uses of all ordinary Sea Instruments for Observations , with more speed , ease and conveniencie than any of them all will do . Contrived and made by Iohn Browne at the Spheare and Sun-dial in the Minories , and sold there or at Mr H. Suttons behind the Exchange . 1662. THE Description , and some uses of the Triangular Quadrant , or the Sector made a Quadrant , or the use of an excellent Instrument for observations at Land or Sea , performing all the uses of the Forestaff , Davis quadrant , Bow , Gunter's crosstaff , Gunter's quadrant , and sector ; with far more convenience and as much exactness as any of them will do . The Description . First , it is a jointed rule ( or Sector ) made to what Radius you please , but for the present purpose it is best between 24. and 36 inches Radius , and a third peice of the same length , with a tennon at each end , to make it an Equilateral Triangle ; from whence it is properly called a Triangular Quadrant . ●●Secondly , as to the lines graduated thereon , they may be more or lesse as your use and cost will please to command , but to make it compleat for the promised premises , these that follow are necessary thereunto . 1. A Line of degrees , of twice 90. degrees on the moveable leg , and outer edge of the cross piece : for quadrantal and back observations . 2. Such another Line of 60. degrees , for forward observations on the inside of the cross piece . 3. On the moveable leg ▪ a Kallender of Months , and dayes in 2 lines . 4. Next to them , a Line of the Suns place in degrees . 5. Next to that , a Line of the Suns right Assention , in degrees or hours . 6. Next above the Months on the same Leg , an hour and Azimuth Line , fitted to a particular Latitude , as London , or any other place , for all the uses of Gunter's Quadrant as you may find in the former discourse ( called a Joynt-rule . ) 7. On the head leg , and same side , a particular scale of altitudes , for the particular Latitude . 8. Next to that a general scale of Altitudes , for all Latitudes . 9. A Line of 360 degrees , divided so as to serve for 360 degrees of 12 Signs ; and 24 hours , ( and in foot and 2 foot rules for inches also . ) 10. A Line of 29½ laid next to the former , serving to find the Moons coming to South , and her age , and place , and the time of the Night , by the fixed Stars . 11. A perpetual Almanack , and the right Assension , and declination of several fixed Stars . 12. A Line of Lines next the inside , may be put on without Trouble , or incumbering one another , all these on one side . Secondly , on the other side may be put the Lines of natural Signs , Tangents and Secants , to a single , double , and treble Radius , and by this means more then a Gunters Sector , ( the particular Lines being inscribed between the general Lines . ) Thirdly on the other edge there may be Artificial Numbers , Tangents , Signs , and versed Signs , and by this means it is a Gunters Rule or a Crosstaff . Fourthly , on the insides , inches , foot-measure , and a Line of 112. parts , and a large meridian Line , or the like : as you please . Lastly , two sliding nuts with points in them , fitted to the Cross piece , makes it a proper beam Compass to use in working by the Numbers Signs and Tangents on the edge , or flat side , also it must have four or five sights , a Thred , and Plummet , and Compasses , as other instruments have , thus much for the Description , the uses follow . An Advertisement . First , for the better understanding and brevity sake , there are ten things to be named and described , as followeth ; 1. the head leg , in which the brass Revet is fixed , and about which the other turn ; 2. the moveable leg , on which the Months and Days must always be ; 3. the Cross piece , that is fitted to the head , and moveable Leg , by the two Tennons at the end ; 4. the quadrant side where the degrees and Moneths are , for observation . 5. the other or Sector side for operation , 6thly the head center , being Center to the degrees on the inside of the crosse peece , for a forward observation as with the forestaffe . 7 The other Center , near the end of the head leg , being the Center to the moveable leg , for backward observations , ( as the Davis quadrant is used , and the bow ) which you may call the foot center , or leg center for backward observations . 8thly the sights , as first the turning or eye sight , which is alwayes , set on one of the centers , with a screw to make it fast there , which I call the turning sight , 9thly the Horizon sight that cuts the degrees of altitude , and sometimes is next to the eye , and sometimes remote from the eye , yet called the horizon ( slideing ) sight , 10thly . the object or shaddow sight of which there may be 3 for convenience sake , as two fixed and one moveable to slide as the horizon sight doth : the other two do serve also to pin the crosse peece , and the two legs together , through the two tennons , all whose names in short take thus : 1. The head leg : 2. the moveable leg : 3. the crosse peece : 4. the quadrant side : 5. the sector side , 6. the head ( or forward ) center . 7 the leg ( or backward ) center . 8. the turning sight . 9. the ( slideing ) horzon sight . 10. the object ( or shaddow ) sight . of which there be 3. all differing according to your use and occasions : one to slide to any place , the other 2. to be put into certain holes . nigher , or further off : as will afterwards largely appear . THE USES : I. To find the suns declination , true place , right assention , and rising , the day of the moneth being given . First open the rule to an angle of 60. degrees , which is alwayes done when the cross peece is fitted into the Mortesse holes , and the pins of the object sights put in the holes through the tennons , or else by the second Chapter of the Joynt-rule : then extend a thred from the center pin in the head leg , to the day of the month , & on the degrees it cuts the suns declination , in the line of right assention his right assention , in the line of true place his true place , and in the hour line his true rising and setting , in that latitude the line is m●de for : Example , on the first of May I would know the former questions , the rule being set by the crosse peece , and the thred on the leg center pin ; and drawn straight and laid over May 1. it cuts in the degrees 18. 4. north declination , and 20. 58. in ♉ Taurus for his place , and 3 hours 14. minutes right assention in time , or 48. 32. in degrees : and the rising of the sun that day is at 4. 23 , and sets at 5. 37 , in 51. 32 latitude . The finding of hour and azimuth , either particularly , or generally , with other Astronomical propositions , are spoken enough of before in the Joynt-rule , and in all other authors that write of the Sector , or Gunter's rule , so that all I shall speak of now , shall be onely what was forgot in the first part , and what is new as to the using the instrument in sea observations . II. To find the Suns or a Stars Altitude , by a forward Observation . Skrew the turning sight to the head Center , and set that object sight , whose holes answer to the Sliding horizon sight , in the hole at the end of the head leg , and put the horizon sight on the crosse peece next the inside ; Then holding the crosse peece with your right hand , and the turning sight close to your eye , and the moveable leg against your body , with your thumb on the right hand thrust upwards , or pull downwards , the horizon sight : till you see the sun through the object sight , and the horizon through the horizon sight , then the degrees cut by the middle of the horizon sight , on the crosse peece shall be the true altitude required : III. To perform the same another way . If your instrument be parted , that is to say the crosse peece from the other , and an altitude be required to be had quickly , then set the two object sights , in two holes at the end of the line of naturall signs , then set the head of the rule to your eye , so as the sight of the eye may be just over the Center , then open or close the Joynt , till you see the horizon through one sight , and the sun or star through the other , then is the sector set to the angle required , to find which angle do thus , take the parallel sign of 30 and 30 , and measure it from the Center , and it shall reach to the sign of half ●he angle required . Example . Suppose I had observed an altitude , and the distance between 30 and 30 , should reach from the center to 10. degrees on the signs , then is the altitude of the sun 20. degrees for 10 doubled is 20. IIII. To find the suns Altitude by a back observation , Skrew the Turning sight to the leg center , ( or center to the degrees on the moveable leg ) and put one of the object sights , in the hole by 00. on the outer edge of the crosse peece , and set the edge of it just against the stroke of 00 , or you may use the sliding object sight and set the edge or the middle of that , to the stroke of 00 , as you shall Judge most convenient ; and the horizon sight to the moveable leg , then observe in all respects as with a Davis quadrant , till looking through the small hole of the horizon sight , you see the crosse bar and button , in the turning sight , cut the horizon : and at the same instant the shadow of the edge or middle of the object or shadow sight , fall on the middle of the turning sight , by sliding the horizon sight higher or lower , then the middle stroke of the horizon sight , shall cut on the moveable leg , the suns true altitude required . As f 〈…〉 t stay at 50 degrees , then is the sun 50 degrees above the horizon . V. But if the sun be near to the Zenith or 90 degrees high , then it will be convenient to move the object sight , to a hole or two further as suppose at 10 , 20 , 30 degrees more , toward the further end of the crosse peece and then observe as you did before in all respects , as with a Davis quadrant , and then whatsoever degrees the horizon sight cuts , you must ad so much to it , as you set the object sight forwards , as suppose 30 , and the horizon sight stay at 60 , then I say 60 , and 30 , makes 90 : the true altitude required . Note that by this contrivance , let the altitude be what it will , you shall alwayes have a most steady observation : with the instrument leaning against your brest , a considerable thing , in a windy day , when you may have a need of an observation in southern voyages , when the sun is near to the zenith at a Meridian observation . VI. To find the suns distance from the zenith , by observing the other way , the sun being not above 60 degrees high , or 30 from the zenith . Set the turning sight as before on the leg Center , then set an object sight in one of the holes in the line on the head leg , nigher or further of , the turning sight : as the the brightnesse or dimnesse of the sun will allow to see a shadow , then looking through the small hole on the horizon sight , till you see the horizon cut by the crosse bar of the great hole , in the turning sight , turning the foreside of that sight , till it be fit to receive the shadow of the middle of the object sight ; then the degrees cut by the horizon sight , shall be the suns true distance from the zenith , or the complement of the Altitude . VII . Note that by adding of a short peece about 9 inches long on the head leg , whereon to set the slideing shadow sight , you may obtain the former convenience of all angles , this way also , at a most steady and easie manner of observation ; but note whatsoever you set forwards on that peece , must be substracted from that the sight sheweth , and the remainder shall be the suns distance from the zenith required . As suppose you set forward 30 degrees , and the horizon sight should stay at 40 , then 30 from 40 rest 10 , the suns distance from the zenith required ; thus you see , that by one and the same line , at one manner of figuring , is the suns altitude , or coalitude acquired and that at a most certain steady manner of observation . VIII . To find an observation by thred and plummet , without having any respect to the horizon , being of good stead in a misty or cloudy day at land or sea . Set the rule to his angle of 60 degrees by putting in the crosse peece , then skrew the turning sight to the head center , then if the sun or star be under 30 degrees high , set the object sight in the moveable leg , then looking through the small hole in the turning sight , through the object sight , to the middle of the star or sun , as the button in the crosse bar will neatly shew ; then the thread and plummet , hanging on the leg center pin , and playing evenly by the moveable leg , shall shew the true alti●ude of the sun , or star required counting the degrees as they are numbred , for th : north declinations from 60 toward the head with 10 20 , As if the thred shall play upon 70 10 then is the altitude 10 degrees . IX But if the sun or star be above 30 degrees high , then the object sight must be set to the hole in the end of the head leg : then looking as before , and the thred playing evenly by the moveable leg , shall shew the true altitude required , as the degrees are numbred . Note that if the brightnesse of the sun should offend the eye , you may have a peice of green , blew , or red glasse , fixed on the turning sight , or else remove the object sight nearer to the turning sight , and then let the sun beams pierce through both the small holes , according to the usuall manner and the thred shall shew the true altitude required . Note also if the thred be apt to slip away from his observed place , as between 25 and 40 it may : note a dexterious handling thereof will naturally shew you how to prevent it : X. To find a latitude at Sea by forward meridian Observation or Altitude . Set the moving object Sight to the Suns declination , shewed by the day of the Month , and rectifying Table , and skrew the turning sight to the leg center , and the Horizon sight to the moveable leg , or the outside of the Crosse piece , according as the Sun is high or low ( but note all forward observations respecting the Horizon , ought to be under 45 degrees high , for if it be more it is very uncertain , by any Instrument whatsoever , except you have a Plummet and then the Horizon is uselesse ) then observe just as you do in a forward observation , moving the Horizon sight till you see the Sun through the Horizon sight , and the Horizon through the object sight , or the contrary . ( moving not that sight that is set to the day of the Month or Declination , ) then whatsoever the moving sight shall shew , if you add 30 to it , it shall be the latitude of the place required ; observing the difference in North and South Latitudes ; that is , setting the sight to the proper declination , either like , or unlike , to the latitude - Example . Suppose on the 10. of March when the Suns declination is 0 — 10. North , as in the first year after leap year it will be , set the stroke in the middle of the moving object sight to 10 of North Declination , and the Horizon sight on the moveable leg , then move it higher or lower , till you see the Horizon through one , and the Sun through the other , then the degrees between , is the Suns meridian altitude , if it be at Noon , as suppose it stayed at 21 30 ▪ then by counting the degrees between , you shall find them come to 38. 40. then if you add 30. to 21. 30. it makes 51. 30. the Latitude required , for if you do take 0 10′ minutes from 38. 40. there remains 38. 30. the complement of the Latitude . Note , that this way you may take a forward observation , and so save the removing of the ●urning sight . Note also , That when the Horizon sight shall stay about the corner , you may move the object sight 10. or 20. degrees towards the head , and then you must add but 20. or 10 degrees to what the sight stayed at ; or if you shall set the sight the other way 10 or 20. degr . then you must add more then 30 so much . As suppose in this last observation , it had been the latitude of 45 or 50 degrees , then you shall find the sight to play so neer the corner , that it will prove inconvenient , then suppose instead of 0 10. I set it to 20 degrees 10′ North declination , which is 20. degrees added to the declination , then the Suns height being the same as before , the sight will stay at 41. 30. to which if you add 10 degrees , it doth make 51. 30. as before ; here you must add but 10 degrees , because you increased the declination 20. degrees ; but note by the same reason , had you set it to 19. 50. South declination , then it had been diminished 20 degrees , and then instead of 30 you must add 50 ▪ to 1 - 30. the place where the sight would have stayed . Thus you see you may very neatly avoid this inconvenience , and set the sights to proper and steady observations , at all times of observation . XI . To find the latitude by a backward Meridian Observation at Sea. This is but just the converse of the former , for if you set one sight to the declination , either directly , ( or augmented or diminish'd as before , when the moving sight shall stay , about the corner of the Triangular Quadrant ) then the other being slipt to and fro , on the outside of the Crosse peece , till the shadow of the outer edge , shall fall on the middle of the turning sight , then 30 just , or more or lesse added , to that number the moving sight stayed at , ( according as you set the first Horizon sight to the declination ) shall be the true latitude required . Example . Suppose on the same day and year as before , at the same Noon time , I set my Horizon sight to just 10′ of North declination , you shall find the moving sight to stay at 21. 30. neer to the corner , now if the Sun shine bright , and will cast the shadow to the turning sight , then set the Horison sight at the declination , forward 10 or 20. degrees , then the moving sight coming lower you , add but 20. or to that it shall stay at , and the summe shall be the latitude . But it is most likely that it will be better to diminish it 20. degrees , then the moving sight will stay about 2 - 30. on the Crosse piece , and so much the better to cast a shadow ; for if you look through the Horizon and turning sight to the Horizon , you shall find the shadow of the former edge of the moving shadow sight , to stay at 2 - 30. to which if you add 20. the degrees diminished , and 30. it makes 51 30. the latitude required as before . Note also for better convenience of the shadow sight , when you have found the true declination , as before is taught , set the moving object sight to the same , on the Crosse piece , counted from 00. towards the head leg , for like latitudes and declinations ; and the other way for unlike latitudes and declinations , then observing as in a back observation , wheresoever the sight shall stay , shall be the complement of the latitude required . If you add or diminish consider accordingly . Note likewise , when the declination is nere the solstice , and the same way as the latitude is , and by diminishing , or otherwise the moving sight shall fall beyond 00 on the Crosse piece ; Then having added 30. and the degrees diminished together , whatsoever the sight shall stay at beyond 00 must be taken out of the added sum , and the remainder shall be the latitude required . Example . Suppose on the 11. of Iune , in the latitude of 51 30 north , for the better holding sake I diminish the declination 30 degrees , that is in stead of setting it to 23. 32. north declination , I set it to 6 : 28 south then the sun being 62 degrees high will stay at 08. 30. beyond 00. the other way now 30 to be added , and 30 diminished , makes 60 , from which take 8. 30 , rest 51 30 the latitude required . XII . To find a latitude with thred and plummet , or by an observation made without respecting the Horizon . Count the declination on the cross peece ( and let 00 be the equinoctiall and let the declination which is the same with the latitude be counted toward the moveable leg , and the contrary the other way , as with us in north latitude , north declination is toward the moveable leg , and south declination the contrary and contrarily in south latitudes ) and thereunto set the middest of the sliding object or horizon sight , then is the small hole on the turning sight and the small hole on the horizon sight , two holes whereby the sun beams are to pierce to shine one on the other : then shall the thred shew you the true latitude of the place required . Example . Suppose on the 11 of Decem. 1663 , at noon I observe the noon altitudes set the middle of the Horizon sight to 23 ▪ 32 counted from 00. toward the head leg end , then making the sun beams to peirce through the hole of this , and the turning sight , you shall find the plummet to play on 51. 30 , the latitude required , holding the turning sight toward the sun . Note also that here also you may avoid the inconvenience of the corner , or the great distance between the sights , by the remedy before cited , in the back and forward observation . For if you move it toward the head leg , then the thred will fall short of the latitude , if toward the moveable leg then it falls beyond the latitude , as is very easie to conceive of : Thus you see all the uses of the forestaff , and quadrant , and Mr. Gunter's bow are plainly and properly applyed to this Triangular quadrant , that the same will be a Sector is easie to perswade you to believe , and that all the uses of a Gunter's quadrant , are performed by it , is fully shewed in the use of the Joynt-rule , to which this may be annexed , the numbers signs and tangents and versed signs makes it an excellent large Gunter's rule , and the cross peece is a good pair of large compasses to operate therewith ; lastly , being it may lie in so little roome it is much more convenient for them , with whom stowage is very precious , so I shall say no more as to the use of it , all the rest being fully spoke to in other Authors , to whom I refer you : only one usefull proposition to inure you to the use of this most excellent instrument , which I call the Triangular Quadrant . Note that in finding the latitude , it is necessary to have a table of the suns declination for every of the four years , viz. for the leap year and the 1 , 2 and 3d. after , now the table of the suns declination whereby the moneths are laid down , is a table that is calculated as a mean between all the 4 years , and you may very well distinguish a minute on the rule ; now to make it to be exact I have fitted this rectifying table for every Week in the year , and the use is thus : hang the thred on the center pin , and extend the thred to the day of the moneth , and on the degrees is the suns declination , as near as can be for a common year , then if you look in the rectifying table for that moneth , and week you seek for you shall find the number of minutes you must add to or substract from the declination found for that day and year : Example , suppose for April 10. 1662. the second after leap year , the rule sheweth me 11. 45 , from which the rectifying table saith I must substract 3′ then is the true declination 11 42 , the like for any other year . Note further that the space of a day in the suns swiftest motion being so much , you may consider the hour of the day also , in the finding of a latitude , by an observation taken of the Meridian , as anon you shall see that as the instrument is exact , so let your arithmetical calculation be also : by laying a sure foundation to begin to work upon , then will your latitude be very true also . A Rectifying Table for the Suns declination .   D 1 year 2 year 3 year Leap year Ianuary 7 Sub 5 Sub 2 Add 1 Add 4 15 s 6 s 2 a 2 a 5 22 s 7 s 3 a 1 a 6 30 s 7 s 3 a 2 a 7 Februar 7 s 8 s 3 a 2 a 7 15 s 8 s 4 a 2 a 8 22 s 9 s 3 a 2 a 7 March 1 s 4 s 1 a 7 Sub 11 7 s 3 Ad 3 Add 9 Sub 9 15 Add 3 ad 1 Sub 9 Add 9 22 a 2 Sub 4 Sub 9 ad 8 30 a 1 Sub 4 Sub 9 ad 8 April 7 a 2 Sub 3 s 8 a 8 15 a 2 s 3 s 8 a 7 22 a 1 s 3 s 8 a 6 30 a 1 s 3 s 6 a 5 May 7 a 0 s 2 s 6 a 5 15 a 0 s 2 s 5 a 3 22 a 0 s 1 s 3 a 3 30 a 0 s 1 s 2 a 1 Iune 7 a 0 s 1 s 1 ad 0 15 a 0 s 0 s 0 Sub 0 22 s 1 s 0 ad 2 s 2 30 s 1 Ad 1 ad 3 s 3 Iuly 7 Sub 2 Add 1 Add 3 Sub 4 15 s 1 a 1 a 5 s 5 22 s 2 a 1 a 5 s 6 30 s 2 a 2 a 6 s 7 August 7 s 3 a 2 a 7 s 8 15 s 3 a 2 a 7 s 8 22 s 3 a 3 a 8 s 9 30 s 3 a 3 a 9 s 9 September 7 s 3 a 3 Add 9 Sub 9 15 Add 3 Sub 3 Sub 9 Add 9 22 a 2 s 4 s 9 a 8 30 a 2 s 4 s 9 a 8 October 7 a 3 s 3 s 9 a 8 15 a 2 s 3 s 8 a 7 22 a 2 s 2 s 7 a 8 30 a 2 s 2 s 7 a 8 November 7 a 1 s 1 s 6 a 7 15 a 1 s 2 s 5 a 5 22 a 1 s 3 s 4 a 3 30 a 0 s 2 s 3 a 2 December 7 0 s 1 s 1 Add 1 15 Sub 1 s 0 s 0 Sub 1 22 s 1 s 0 Add 1 s 3 30 s 3 s 0 Add 2 s 5 The Declination of the Sun being given , or rather the Suns Distance from the Pole , and the Complement of two Altitudes of the Sun , taken at any time of the day , knowing the time between : to find the Latitude . Suppose on the 11. of Iune , the Sun being 66 degrees , 29′ distant from the North Pole , and the Complement of one altitude be 80. 30. and the Complement of another altitude 44. 13. and the time between the two observations just four hours ; then say , As the Sine of 90 00 To Sine of Suns dist . f. the Pole 66 28 So is the Sine of ½ time betw . 30 00 To the Sine of ½ the 3d. side 27 17½   27 17½ of a Triangle as A B. 54 35 The side A P 66 28 The side P A 66 28 The side A B 54 35 whole sum 187 31 half sum 93 46 The differ . betw ½ sum and AP side op . to inqu . Triangle 27 18 Then say , as S. of 90 90 00 To S. of Suns dist . from Pole 66 28 So is the Sine of A B 54 35 To the Sine of a fourth Sine 48 23 Then as that 4 to Sine of ½ sum 93 46 So is S. of the difference 27 18 To a seventh Sine 37 44½ or the versed Sine of P A B 77 02 Then to find Z A B Z B is 80 30 And Z A is 44 13 The former side A B is 54 35 Sum 179 18 half sum 89 39 difference 09 09 As S. of 90 90 00 To Sine of A B 54 35 So is S. of Z A 44 13 To a 4th . Sine 34 38 As S. 4th . 34 38 To S. of ½ sum 89 39 So is S. diff . 09 39 To a 7th . Sine 16 15¼ or to the vers . sine of Z A B 116 07 Then if you take P A B from Z A B there will remain Z A P 39 05 Then say again by the rule as before , As the sine of 90 00 To Co-sine of Z A P 50 55 So is the Tang of A Z 44 13 To the Tangent of A C 37 04 which taken from A P 66 28 remaineth P C 29 24 Then lastly say ,   As the Co-sine of A C 52 56 To the Co-sine of C P 60 36 So is the Co-sine of Z A 45 47 To the Co-sine of Z P 51 30 The Latitude required to be found . This Question or any other may be wrought by the Sines and Tangents and versed sines on the rule , But if you would know more as concerning this or any other , you may be fully satisfied by Mr. Euclid Spidal at his Chamber at a Virginal Makers house in Thred-needle Street , and at the Kings head neer Broadstreet end . Vale. FINIS . The Triangle explained . S P Z N a Meridian circl . Ae Ae the Equinoctial . ♋ B A ♋ the , tropick of Cancer . P B 5 P the hour circle of Five , ante , mer. P A 9 P the hour circle of Nine , ante , mer. Z B the Suns coaltitude at 5 — 80 — 30 Z A the Suns coaltitude at 9 — 44 — 13 P A & P B the Suns distance from the Pole on the 2 h. li. of 5 & 9 66 — 28 5-9 the equinoctial time between the two Observations — 60 — 00 B A in proper measure is as found by the first measure working — 54 — 35 Z C a perpendicular on A P from Z 29 — 24 P Z the complement of the latitude that was to be found — 38-30 
A44017	----	Three papers presented to the Royal Society against Dr. Wallis together with considerations on Dr. Wallis his Answer to them / by Tho. Hobbes. Hobbes, Thomas, 1588-1679. 1671 Approx. 18 KB of XML-encoded text transcribed from 7 1-bit group-IV TIFF page images. Text Creation Partnership, Ann Arbor, MI ; Oxford (UK) : 2004-08 (EEBO-TCP Phase 1). A44017 Wing H2263 ESTC R25546 09012078 ocm 09012078 42220 This keyboarded and encoded edition of the work described above is co-owned by the institutions providing financial support to the Early English Books Online Text Creation Partnership. This Phase I text is available for reuse, according to the terms of Creative Commons 0 1.0 Universal . The text can be copied, modified, distributed and performed, even for commercial purposes, all without asking permission. Early English books online. (EEBO-TCP ; phase 1, no. A44017) Transcribed from: (Early English Books Online ; image set 42220) Images scanned from microfilm: (Early English books, 1641-1700 ; 1287:5) Three papers presented to the Royal Society against Dr. Wallis together with considerations on Dr. Wallis his Answer to them / by Tho. Hobbes. Hobbes, Thomas, 1588-1679. Wallis, John, 1616-1703. [3], 3, [1], 4 p. Printed for the author, London : 1671. "To the Right Honorable and others, the learned members of the Royal Society for the Advancement of the Sciences" on leaf preceding t.p. Reproduction of original in the Bodleian Library. To the Right Honourable and others, the learned members of the Royal Society for the Advancement of the Sciences--Considerations upon the Answer of Dr. Wallis to the Three papers of Mr. Hobbes. Created by converting TCP files to TEI P5 using tcp2tei.xsl, TEI @ Oxford. Re-processed by University of Nebraska-Lincoln and Northwestern, with changes to facilitate morpho-syntactic tagging. 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Square root -- Early works to 1800. 2004-04 TCP Assigned for keying and markup 2004-05 SPi Global Keyed and coded from ProQuest page images 2004-06 Mona Logarbo Sampled and proofread 2004-06 Mona Logarbo Text and markup reviewed and edited 2004-07 pfs Batch review (QC) and XML conversion To the Right Honourable and others , the Learned Members of the Royal Society , for the Advancement of the Sciences . PResenteth to your Consideration , your most humble servant Thomas Hobbes , a Confutation of a Theoreme which hath a long time passed for Truth ; to the great hinderance of Geometry , and also of Natural Philosophy , which thereon dependeth . The Theoreme . The four sides of a Square being divided into any number of equal parts , for example into 10 ; and straight lines drawn through the opposite points , which will divide the Square into 100 lesser Squares ; The received Opinion , and which Dr. Wallis commonly useth , is , that the root of those 100 , namely 10 , is the side of the whole Square . The Confutation . The Root 10 is a number of those Squares , whereof the whole containeth 100 , whereof one Square is an Vnitie ; therefore the Root 10 , is 10 Squares : Therefore the Root of 100 Squares is 10 Squares , and not the side of any Square ; because the side of a Square is not a Superficies , but a Line . For as the root of 100 Vnities is 10 Vnities , or of 100 Souldiers 10 Souldiers : so the root of 100 Squares is 10 of those Squares . Therefore the Theoreme is false ; and more false , when the root is augmented by multiplying it by other greater numbers . Hence it followeth , that no Proposition can either be demonstrated or confuted from this false Theoreme . Upon which , and upon the Numeration of Infinites , is grounded all the Geometry which Dr. Wallis hath hitherto published . And your said servant humbly prayeth to have your Judgement hereupon : And that if you finde it to be false , you would be pleased to correct the same ; and not to suffer so necessary a Science as Geometry to be stifled , to save the Credit of a Professor . Three PAPERS Presented to the ROYAL SOCIETY Against Dr. WALLIS . Together with CONSIDERATIONS ON Dr. Wallis his ANSWER to them . By THO. HOBBES of Malmsbury . LONDON : Printed for the Author ; and are to be had at the Green Dragon without Temple-bar . 1671. To the Right Honourable and others , the Learned Members of the Royal Society , for the Advancement of the Sciences . YOur most humble servant Thomas Hobbes presenteth , That the quantity of a Line calculated by extraction of Roots , is not to be truely found . And further presenteth to you the Invention of a Straight Line equal to the Arc of a Circle . A Square Root is a number which multiplied into it self produceth a number . And the number so produced is called a Square number . For example : Because 10 multiplied into 10 makes 100 ; the Root is 10 , and the Square number 100. Consequent . In the natural row of Numbers , as 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , &c. every one is the Square of some number in the same row . But Square numbers ( beginning at 1 ) intermit first two numbers , then four , then six , &c. So that none of the intermitted numbers is a Square number , nor has any Square root . PROP. I. A Square root ( speaking of quantity ) is not a Line , such as Euclide defines , without Latitude , but a Rectangle . Suppose A B C D be the Square , and AB , BC , CD , DA be the sides ; and every side divided into 10 equal parts , and Lines drawn through the opposite points of division ; there will then be made 100 lesser Squares , which taken all together are equal to the Square ABCD. Therefore the whole Square is 100 , whereof one Square is an Unit ; therefore 10 Units , which is the Root , is ten of the lesser Squares , and consequently has Latitude ; and therefore it cannot be the side of a Square , which according to Euclide is a Line without Latitude . Consequent . It follows hence , that whosoever taketh for a Principle , That a Side of a Square is a meer Line without Latitude , and That the Root of a Square is such a Line , ( as Dr. Wallis continually does ) demonstrates nothing . But if a Line be divided into what number of equal parts soever , so the Line have bredth allowed it , ( as all Lines must , if they be drawn ) and the length be to the bredth as Number to an Unite , the Side and the Root will be all of one length . PROP. II. Any Number given is produced by the greatest Root multiplied into it self , and into the remaining Fraction . Let the Number given be two hundred Squares , the greatest Root is 14 4 / 14 Squares . I say , that 200 is equal to the product of 14 into it self , together with 14 multiplied into 4 / 14. For 14 multiplied into it self , makes 196. And 14 into 4 / 14 makes 56 / 14 , which is equal to 4. And 4 added to 196 maketh 200 ; as was to be proved . Or take any other Number 8 , the greatest Root is 2 ; which multiplied into it self is 4 , and the Remainder 4 / 2 multiplied into 2 is 4 ; and both together 8. PROP. III. But the same Square calculated Geometrically by the like parts , consisteth ( by Eucl. 2.4 . ) of the same numeral great Square 196 , and of the two Rectangles under the greatest side 14 , and the Remainder of the side , or ( which is all one ) of one Rectangle under the greatest side , and double the Remainder of the side ; and further of the Square of the less Segment ; which all together make 200 , and moreover 1 / 49 of those 200 Squares , as by the operation it self appeareth thus . The side of the greater Segment is 14¼ . 14¼ . Which multiplied into it self , makes 200. The product of 14 the greatest segment , into the two Fractions 4 / 14 , that is , into 4 / 14 ( or into twice 2 / 14 ) is 56 / 14 ( that is 4 ) and that 4 added to 196 makes 200. Lastly , the product of 2 / 14 into 2 / 142 or 1 / 7 into 1 / 7 , is 1 / 49. And so the same Square calculated by Roots , is less by 1 / 49 of one of those two hundred Squares , then by the true and Geometrical Calculation ; as was to be demonstrated . Consequent . It is hence manifest , That whosoever calculates the length of an Arc or other Line by the extraction of Roots , must necessarily make it shorter then the truth , unless the Square have a true Root . The Radius of a Circle is a Mean Proportional between the Arc of a Quadrant and two fifths of the same . DEscribe a Square ABCD , and in it a Quadrant DCA . In the side DC take DT two fifths of DC ; and between DC and DT a Mean Proportional DR ; and describe the Quadrantal Arcs RS , TV. I say , the Arc RS is equal to the streight line DC . For seeing the proportion of DC to DT is duplicate of the proportion of DC to DR , it will be also duplicate of the proportion of the Arc CA to the Arc RS ; and likewise duplicate of the proportion of the Arc RS to the Arc TV. Suppose some other Arc less or greater then the Arc RS to be equal to DC , as for example rs : Then the proportion of the Arc rs to the streight line DT will be duplicate of the proportion of RS to TV , or DR to DT . Which is absurd ; because Dr is by construction greater or less then DR . Therefore the Arc RS is equal to the side DC . Which was to be demonstrated . Corol. Hence it follows that DR is equal to two fifths of the Arc CA. For RS , TV , DT being continually proportional ; and the Arc TV being described by DT , the Arc RS will be described by a streight line equal to TV. But RS is described by the streight line DR . Therefore DR is equal to TV , that is , to two fifths of CA. And your said servant most humbly prayeth you to consider ( if the demonstration be true and evident ) whether the way of objecting against it by Square Roots , used by Dr. Wallis ; and whether all his Geometry , as being , built upon it , and upon his supposition of an Infinite Number , be not false . Considerations upon the Answer of Dr. Wallis to the Three Papers of Mr. Hobbes . DR . Wallis sayes , All that is affirmed , is but , If we SVPPOSE That , This will follow . But it seemeth to me , that if the Supposition be impossible , then that which follows will either be false , or at least undemonstrated . First , this Proposition being founded upon his Arithmetica Infinitorum , If there he affirm an absolute Infiniteness , he must here also be understood to affirm the same . But in his 39th Proposition he saith thus : Seeing that the number of terms increasing , the excess above sub-quadruple is perpetually diminished , so as at last it becomes less than any proportion that can be assigned ; If it proceed in infinitum it must utterly vanish . And therefore if there be propounded an Infinite row of quantities in triplicate proportion of quantities Arithmetically proportional ( that is , according to the row of Cubical numbers ) beginning from a point or 0 ; that row shall be to a row of as many , equal to the greatest , as 1 to 4. It is therefore manifest that he affirms , That in an Infinite row of quantities the last is given ; and he knows well enough that this is but a shift . Secondly , he sayes , That usually in Euclide and all after him , by Infinite is meant but , more than any assignable Finite , or the greatest possible . I am content it be so interpreted . But then from thence he must demonstrate those his conclusions , which he hath not yet done . And when he shall have done it , not only the Conclusions , but also the Demonstration will be the same with mine in Cap. 14. Art. 2 , 3 , &c. of my Book De Corpore . And so he steals what he once condemn'd . A fine quality . Thirdly , he sayes ( by Euclides 10th Proposition , but he tells not of what Book ) That a Line may be bisected , and the halves of it may again be bisected , and so onwards infinitely ; and that upon such supposed Section Infinitely continued , the parts must be supposed Infinitely many . I deny that ; for Euclide , if he sayes a Line may be divisible into parts perpetually divisible , he means , That all the divisions , and all the parts arising from those divisions , are perpetually Finite in number . Fourthly , he sayes , That there may be supposed a row of quantities infinitely many , and continually increasing , whereof the last is given . 'T is true , a man may say ( if that be supposing ) that white is black ; but if Supposing be Thinking , he cannot suppose an Infinite row of quantities whereof the last is given . And if he say it , he can demonstrate nothing from it . Fifthly , He sayes ( for one absurdity begets another ) That a Superficies or Solid may be supposed so constituted , as to be Infinitely long , but Finitely great ( the breadth continually decreasing in greater proportion than the length increaseth ) and so as to have no center of gravity . Such is Toricellio 's Solidum Hyperbolicum acutum , and others innumerable discovered by Dr. Wallis , Monsieur Fermat , and others . But to determine this , requires more of Geometry and Logick ( whatsoever it do of the Latine Tongue ) than Mr. Hobbes is master of . I do not remember this of Toricellio , and I doubt Dr Wallis does him wrong , and Monsieur Fermat too . For to understand this for sense , 't is not required that a man should be a Geometrician or a Logician , but that he should be mad . In the next place he puts to me a Question as absurd as his Answers are to mine . Let him ask himself ( saith he ) if he be still of opinion , That there is no Argument in Natural Philosophy to prove that the World had a beginning : First , whether in case it had no beginning , there must not have passed an Infinite number of years before Mr. Hobbes was born . Secondly , whether at this time there have not passed more , that is , more than that Infinite number . Thirdly , whether , in that Infinite ( or more than Infinite ) number of years , there have not been a greater number of dayes and hours , and of which hitherto the last is given . Fourthly , whether , if this be an Absurdity , we have not then ( contrary to what Mr. Hobbes would perswade us ) an Argument in Nature to prove the World had a beginning . To this I answer , not willingly , but in service to the Truth , that by the same Argument he might as well prove that God had a beginning . Thus : in case he had not , there must have passed an Infinite length of time before Mr. Hobbes was born ; but there hath passed at this day more than that Infinite length ( by eighty four years ) . And this day , which is the last , is given . If this be an Absurdity , have we not then an Argument in Nature to prove that God had a beginning ? Thus 't is when men intangle themselves in a Dispute of that which they cannot comprehend . But perhaps he looks for a Solution of his Argument to prove that there is somewhat greater than Infinite ; which I shall do so far , as to shew it is not concluding . If from this day backwards to Eternity be more than Infinite , and from Mr. Hobbes his birth backwards to the same Eternity be Infinite , then take away from this day backwards to the time of Adam , which is more than from this day to Mr. Hobbes his birth , then that which remains backwards must be less than Infinite . All this arguing of Infinites is but the ambition of School-boyes . To the latter part of the first Paper . There is no doubt , if we give what Proportion we will of the Radius to the Arc , but that the Arc upon that Arc will have the same Proportion . But that is nothing to my Demonstration . He knows it , and wrongs the Royal Society in presuming they cannot find the Impertinence of it . My proof is this ; That if the Arc on TV , and the Arc RS , and the streight Line CD , be not equal , then the Arc on TV , the Arc on RS , and the Arc on CA , cannot be proportional . Which is manifest by supposing in DC a less than the said DC ; but equal to RS , and another streight Line , less than RS , equal to the Arc on TV ; and any body may examine it by himself . I have been asked by some that think themselves Logicians , Why I proceeded upon 2 / 5 rather than any other part of the Radius . The reason I had for it was , That long ago some Arabians had determined , That a streight Line whose square is equal to 10 squares of half the Radius , is equal to a quarter of the Perimiter ; but their demonstrations are lost . From that Equality it follows , that the third proportional to the Quadrant and Radius , must be a mean proportional between the Radius and 2 / 5 of the same . But my answer to the Logicians was , That though I took any part of the Radius to proceed on , and lighted on the Truth by chance , the Truth it self would appear by the Absurdity arising from the denial of it . And this is it that Aristotle meant , where he distinguisheth between a Direct demonstration , and a demonstration leading to an absurdity . Hence it appears , that Dr. Wallis his objections to my Rosetum are invalid , as built upon Roots . To the second Paper . First , he sayes , That it concerns him no more than other men . Which is true . I meant it against the whole Herd of them who apply their Algebra to Geometry . Secondly , He sayes , That a bare Number cannot be the Side of a Square figure . I would know what he means by a Bare Number . Ten Lines may be the side of a Square figure . Is there any Number so bare , as by it we are not to conceive or consider any thing numbred ? Or by ten Nothings understands he Bare 10 ? He struggles in vain , his Conscience puzzles him . Thirdly , He sayes , Ten Squares is the Root of 100 Square-squares . To which I answer , first , That there is no such Figure as a Square-square . Secondly , That it follows hence that a Root is a Superficies , for such is 10 Squares . Lastly , He sayes , That neither the Number 10 , nor 10 Souldiers is the Root of 100 Souldiers ; because 100 Souldiers is not the Product of 10 Souldiers into 10 Souldiers . That last I grant , because nothing but Numbers can be multiplied into one another . A Souldier cannot be multiplied by a Souldier . But no more can a Square-figure by a Square-figure , though a Square-number may . Again , If a Captain will place his hundred Men in a square Form , must not he take the Root of 100 to make a Rank or File ? And are not those 10 Men ? To the third Paper . He objects nothing here , but that , The Side of a Square is not a Superficies but a Line , and that a Square Root ( speaking of quantity ) is not a Line but a Rectangle , is a contradiction . The Reader is to judge of that . To his Scoffings I say no more , but that they may be retorted in the same words , and are therefore childish . And now I submit the whole to the Royal Society , with confidence that they will never ingage themselves in the maintenance of these Unintelligible Doctrines of Dr. Wallis , that tend to the suppression of the Sciences which they endeavour to advance . Notes, typically marginal, from the original text Notes for div A44017-e250 Definition . 
A51544	----	Mechanick dyalling teaching any man, though of an ordinary capacity and unlearned in the mathematicks, to draw a true sun-dyal on any given plane, however scituated : only with the help of a straight ruler and a pair of compasses, and without any arithmetical calculation / by Joseph Moxon ... Moxon, Joseph, 1627-1691. 1668 Approx. 89 KB of XML-encoded text transcribed from 28 1-bit group-IV TIFF page images. Text Creation Partnership, Ann Arbor, MI ; Oxford (UK) : 2005-12 (EEBO-TCP Phase 1). A51544 Wing M3009 ESTC R20066 12354007 ocm 12354007 60072 This keyboarded and encoded edition of the work described above is co-owned by the institutions providing financial support to the Early English Books Online Text Creation Partnership. This Phase I text is available for reuse, according to the terms of Creative Commons 0 1.0 Universal . The text can be copied, modified, distributed and performed, even for commercial purposes, all without asking permission. Early English books online. (EEBO-TCP ; phase 1, no. A51544) Transcribed from: (Early English Books Online ; image set 60072) Images scanned from microfilm: (Early English books, 1641-1700 ; 643:12) Mechanick dyalling teaching any man, though of an ordinary capacity and unlearned in the mathematicks, to draw a true sun-dyal on any given plane, however scituated : only with the help of a straight ruler and a pair of compasses, and without any arithmetical calculation / by Joseph Moxon ... Moxon, Joseph, 1627-1691. 49, [6] p. : ill. Printed for Joseph Moxon ..., London : 1668. Advertisement: p. [1]-[6] at end. Reproduction of original in Huntington Library. Created by converting TCP files to TEI P5 using tcp2tei.xsl, TEI @ Oxford. Re-processed by University of Nebraska-Lincoln and Northwestern, with changes to facilitate morpho-syntactic tagging. 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Understanding these processes should make clear that, while the overall quality of TCP data is very good, some errors will remain and some readable characters will be marked as illegible. Users should bear in mind that in all likelihood such instances will never have been looked at by a TCP editor. The texts were encoded and linked to page images in accordance with level 4 of the TEI in Libraries guidelines. Copies of the texts have been issued variously as SGML (TCP schema; ASCII text with mnemonic sdata character entities); displayable XML (TCP schema; characters represented either as UTF-8 Unicode or text strings within braces); or lossless XML (TEI P5, characters represented either as UTF-8 Unicode or TEI g elements). Keying and markup guidelines are available at the Text Creation Partnership web site . eng Sundials. Mathematical instruments. 2005-02 TCP Assigned for keying and markup 2005-03 SPi Global Keyed and coded from ProQuest page images 2005-04 Judith Siefring Sampled and proofread 2005-04 Judith Siefring Text and markup reviewed and edited 2005-10 pfs Batch review (QC) and XML conversion Mechanick Dyalling : TEACHING Any Man , though of an Ordinary Capacity and unlearned in the Mathematicks , to draw a True SUN-DYAL On any Given Plane , However scituated : Only with the help of a straight RVLER and a pair of COMPASSES ; And without any Arithmetical Calculation . By Joseph Moxon , Hydrographer to the Kings most Excellent Majesty . LONDON . Printed for Joseph Moxon on Ludgate-hill , at the Sign of Atlas . MDCLXVIII MECHANICK DYALLING . Description of Dyalling . DYalling originally is a Mathematical Science , attained by the Philosophical contemplation of the motion of the Sun , the motion of the Shaddow , the Constitution of the Sphere , the Scituation of Planes , and the consideration of Lines . Explanation . THE motion of the Sun is regular , it moving equal Space in equal Time ; But the motion of the Shaddow irregular in all parts of the Earth , unless under the two Poles , and that more or less according to the Constitution of the Sphere and scituation of the Plane . And therefore Scientifick Dyalists by the Geometrick considerations of Lines , have found out Rules to mark out the irregular motion of the Shaddow in all Latitudes , and on all Planes , to comply with the regular motion of the Sun. And these Rules of adjusting the motion of the Shaddow to the motion of the Sun may be called Scientifick Dyalling . But though we may justly account Dyalling originally a Science , yet such hath been the Generosity of many of its studious Contemplators , that they have communicated their acquired Rules ; whereby it is now become to many of the Ingenious no more difficult than an Art , and by many late Authors so intituled : Nay more , by this small Treatise it will scarce be accounted more than a Manual Operation ; for , though ( hitherto ) all the Authors I have met with seem to presuppose their Reader to understand Geometry , and the projecting of the Sphere already , or else endeavour in their Works to make him understand them , as if they were absolutely necessary to be known by every one that would make a Dyal , when as in truth ( the contemplative pains of others aforesaid considered ) they are not ; but indeed are only useful to those that would know the reason of Dyalling . Thus they do not only discourage young beginners , but also disappoint many Gentlemen and others that would willingly either make them themselves , or set their Workmen about them , if they knew how to make them . This little Piece I have therefore composed for the help of those who understand neither the Projection of the Sphere , or Geometrical Operations : Only , if they know how to draw a straight Line between two Points by the side of a Ruler , describe a Circle with a pair of Compasses , erect a Perpendicular , and draw one Line parallel to another , they may know how to draw a Dyal for any given Plane , however scituated in any Latitude . But perhaps these two last little Tricks are not known to all new beginners , therefore I shall shew them . First , How to erect a Perpendicular . For Example , in Fig. 1. Upon the Line AB you would erect a Perpendicular to the Point C : Place one Foot of your Compasses upon the point C , and open the other to what distance you please ; For Example , to the point A , make there a mark ; then keeping the first Foot still in C , turn the other Foot toward B , and make there another mark ; then open your Compasses wider , suppose to the length AB , and placing one Foot in the point A , with the other Foot describe a small Arch over the point C , and removing the Foot of your Compasses to the point B , with the other Foot describe another small Arch , to cut the first Arch , as at D. Then lay your straight Ruler to the point where the two small Arches cut each other , and upon the point C , and by the side of the Ruler draw the Line CD , which shall be a Perpendicular to the Line AB . Another way with once opening the Compasses , as by Fig. 2. Draw the Line AB , and place one Foot of your Compasses upon the point you would have the Perpendicular erected , as at the point C , and with the other foot describe the Semi-Circle A ab B , then placing one Foot in B , extend the other Foot to b , in the Semi-Circle ; and keeping that Foot in b , extend the other Foot to D , and make there a small Arch : Then remove one Foot of your Compasses to A , and extend the other Foot to a in the Semi-Circle , and keeping that Foot in a , extend the other to D , and make there another small Arch , to cut the first small Arch ; and laying a straight Ruler to the point where these two small Arches cut each other , and upon the point C , draw by the side of the Ruler the Line CD , which shall be Perpendicular to the Line AB . To erect a Perpendicular upon the end of a Line , as by Fig. 3. On the point B , at one end of the Line AB , place one Foot of your Compasses in the point B , and extend the other on the Line towards A , as to b , and with it describe the Arch ba C ; then placing one Foot in b , extend the other to a in the Arch , and make there a mark ; Divide with your Compasses the Arch ba into two equal parts , and keeping the Feet of your Compasses at that distance , measure in the Arch from a to C , then draw a straight Line from the point C to the end of the Line B , and that straight Line shall be Perpendicular to the end of the Line AB . To draw a Line Parallel to another Line , as by Fig. 4. Example . If you would draw a Line Parallel to the Line AB , open your Compasses to the distance you intend the Lines shall stand off each other , and placing one Foot successively near each end , describe with the other Foot the small Arches CD ; lay a straight Ruler to the top of these Arches , and draw a Line by the side of it , and that Line shall be Parallel to the Line AB . Definitions . A Dyal Plane is that Flat whereon a Dyal is intended to be projected . Of Dyal Planes some be Direct , other Decliners , others Oblique . Of Direct Planes there are five sorts : 1. The Horizontal whose Plane lies flat , and is parallel to the Horizon , beholding the Zenith . 2. The South Erect , whose Plane stands upright , and directly beholds the South . 3. The North Erect , whose Plane stands upright , and directly beholds the North. 4. The East Erect , whose Plane stands upright , and directly beholds the East . 5. The West Erect , whose Plane stands upright , and directly beholds the West . Of Decliners there are infinite : and yet may be reduced into these two Kinds : 1. The South Erect Plane , declining more or less towards the East or West . 2. The North Erect Plane , declining more or less towards the East or West . Of Oblique Planes some are Direct , others Declining ; and are of four sorts : 1. Direct Inclining Planes , which lean towards you , and lie directly in the East , West , North , or South quarters of Heaven . 2. Direct Reclining Planes , which lean from you , and lie directly in the East , West , North , or South quarters of Heaven . 3. Inclining Declining Planes , which lean towards you , but lie not directly in the East , West , North , or South quarters of Heaven : But decline more or less from the North or South , towards the East or West . 4. Reclining Declining Planes , which lean from you , but lie not directly in the East , West , North , or South quarters of Heaven : But Decline more or less from the North or South , towards the East or West . If the Scituation of the Plane be not given , you must seek it : For , there are several wayes how to know these several kinds of Planes used among Artists ; But the readiest and easiest is by an Instrument called a Declinatory , fitted to the variation of your Place : And if it be truly made , you may as safely rely upon it as any other . OPERATION I. The Description of the Clinatory . THE Clinatory is made of a square Board , as ABCD , of a good thickness ; , and the larger the better ; between two of the side is described on the Center Aa Quadrant as EF devided into 90 equal parts or degrees , which are figured with 10 , 20 , 30 , to 90 ; and then back again with the Complements of the same numbers to 90 : between the Limb and the two Semi-diameters is made a round Box , into which a Magnetical Needle is fitted ; and a Card of the Nautical Compass , devided into four Nineties , beginning their numbers at the East , West , North , and South points of the Compass , from which points the opposite sides of the Clinatory receives their Names of East , West , North and South . But , Note , that the North point of the Card must be placed so many degrees towards the East or West sides of the Clinatory as the Needle varies from the true North point of the world , in the place where you make your Dyal ; which your Workman that makes your Clinatory will know how to fit . Upon the Center A , whereon the Quadrant was described , is fastened a Plumb-line , having a Plumbet of Lead or Brass fastned to the end of it , which Plumb-line is of such length that the Plumbet may fall just into the Grove GH , below the Quadrant , which is for that purpose made of such a depth that the Plumbet may ride freely within it , without stopping at the sides of it . See the Figure annexed . With this Clinatory you may examine the scituation of Planes . As if your Plane be Horizontal , it is direct : and then for the true scituating your Dyal you have only the true North and South Line to find : which is done only by setting the Clinatory flat down upon the Plane , and turning it towards the right or left hand , till you can bring the North point of the Needle to hang just over the Flower-de-luce , for then if you draw a Line by either of the sides parallel to the Needle , that Line shall be a North and South Line . If Your Plane either Recline or Incline , Apply one of the sides of your Clinatory parallel to one of the Semi-diameters of the Quadrant to the Plane , in such sort that the Plumb-line hanging at liberty , may fall upon the Circumference of the Quadrant , for then the number of degrees of the Quadrant comprehended between the side of the Quadrant parallel to the Plane , and the Plumb-line shall be the number of degrees for Reclination , if the Center of the Quadrant points upwards ; or Inclination , if the Center points downwards . If your Reclining or Inclining Plane Decline , Draw upon it a Line parallel to the Horizon , which you may do by applying the back-side of the Clinatory , and raising or depressing the Center of the Quadrant , till the Plumb-line hang just upon one of the Semi-diameters , for then you may by the upper side of the Clinatory draw an Horizontal Line if the Plane Incline , or by the under side if it Recline . If it neither Incline or Recline , you may draw a Horizontal Line both by the upper and under sides of the Clinatory . Having drawn the Horizontal Line , apply the North side of the Clinatory to it , and if the North end of the Needle points directly towards the Plane , it is then a South Plane . If the North point of the Needle points directly from the Plane , it is a North Plane : but if it points towards the East , it is an East Plane : if towards the West , a West Plane . If it do not point directly either East , West , North , or South , then so many degrees as the Needle declines from any of these four points to any of the other of these four points , so many degrees is the Declination of the Plane . You may find a Meridian Line another way ; thus , If the Sun shine just at Noon , hold up a Plumb-line so as the shaddow of it may fall upon your Plane , and that shaddow shall be a Meridian Line . OPERAT. II. To describe a Dyal upon a Horizontal Plane . FIrst draw a North and South Line ( which is called a Meridian Line ) through the middle of the Plane : Thus , Set your Declinatory flat upon the Plane , and turn it to and fro till the Needle hang precisely over the Meridian Line of the Declinatory ; then by the side of the Declinatory parallel to its Meridian Line , draw a straight Line on the Plane , and if that straight Line be in the middle of the Plane , it shall be the Meridian Line , without more ado : But if it be not in the middle of the Plane , you must draw a Line parallel to it through the middle of the Plane for the Meridian Line , or twelve a Clock line : And it shall be the Meridian Line , and also be the Substilar Line ; then draw another straight Line through the middle of this Line , to cut it at right Angles for the VI. a Clock Lines ; and where these two Lines cut one another make your Centre , whereon describe a Circle on your Plane as large as you can , which by the Meridian Line , and the Line drawn at right . Angles with it will be devided into four Quadrants ; one of the Quadrants devide into 90 degrees thus , Keeping your Compasses at the same width they were at when you described the Quadrant , place one Foot in the twelve a Clock Line , and extend the other in the Quadrant , and make in the Quadrant a mark with it ; so shall you have the sixtieth degree marked out : then place one Foot of your Compasses in the six a Clock Line , and extend the other in the Quadrant , and make in the Quadrant another mark with it ; so shall that Quadrant be divided into three equal parts ; each of these three equal parts contains 30 degrees : Then with your Compasses devide one of these three equal parts into three parts , and transfer that distance to the other two third parts of the Quadrant , so shall the whole Quadrant be devided into nine equal parts . Then devide one of these nine equal parts into two equal parts , and transfer that distance to the other eight equal parts , so shall the Quadrant be devided into eighteen equal parts . Then devide one of these eighteen equal parts into five equal parts , and transfer that distance to the other seventeen equal parts , so shall the whole Quadrant be devided into 90 equal parts . Each of these 90 equal parts are called Degrees . Note , That you may in small Quadrants devide truer and with less trouble with Steel Deviders , ( which open or close with a Screw for that purpose , ) than you can with Compasses . In this Quadrant ( thus devided ) count from the Substilar or Meridian Line the Elevation of the Pole , that is , the number of Degrees that the Pole of the World is elevated above the Horizon of your Place , and draw a Line from the Center through that number of Degrees for the Stilar Line . Then on the Substilar Line choose a point ( where you please ) and through that point draw a Line at right Angles to the Substilar Line as long as you can , for the Line of Contingence , and from that point in the Substilar Line measure the nearest distance any part of the Stilar Line hath to that point ; and keeping one Foot of your Compasses still in that point , set off that distance in the Substilar Line , and at that distance describe against the Line of Contingence a Semi-Circle , which devide from either side the Meridian or Substilar Line into six equal parts thus ; Draw a Line through the Center of this Semi-Circle parallel to the Line of Contingence , which shall be the Diametral Line , and shall devide this Semi Circle-into two Quadrants ; one on one side the Substilar Line , and the other Quadrant on the other side the Substilar Line : Then keeping your Compasses at the same distance they were at when you described the Semi-Circle , place one Foot first on one side the Diametral Line at the Intersection of it and the Semi-Circle , and then on the other side , at the Intersection of it and the Semi-Circle , and extend the other in the Semi-Circle , and make marks in the Semi-Circle on either side the Substilar Line : Then place one Foot of your Compasses at the Intersection of the Semi-Circle and the Substilar Line , and turn the other Foot about on either side the Semi-Circle and make marks in the Semi-Circle , so shall the Semi-Circle be devided into six equal parts : Devide one of these equal parts into two equal parts , and transfer that distance to the other five equal parts , so shall the whole Semi-Circle be devided into twelve equal parts . These twelve Devisions are to describe the twelve Hours of the Day , between six a Clock in the Morning , and six a Clock at Night . If you will have half Hours you may devide each of these twelve into two equal parts , as before : If you will have Quarters you may devide each of these twenty four into two equal parts more , as before . For thus proportioning the Devisions in the Semi-Circle , you may proportion the Devisions and Sub-devisions of Hours upon the Dyal Plane ; for a straight Ruler laid upon each of these Devisions , and on the Center of this Semi-Circle , shall shew on the Line of Contingence the several Distances of all the Hours and parts of Hours on the Dyal Plane : And straight Lines drawn from the Center of the Dyal Plane , through the several Devisions on the Line of Contingence shall be the several Hour Lines and parts on the Dyal Plane . But an Horizontal Dyal in our Latitude will admit of four Hours more , viz. V , IV , in the Morning , and VII , VIII , in the Evening . Therefore in the Circle described on the Center of the Dyal Plane transfer the distance between VI and V , and VI and IV , on the other side the six a Clock Line ; And transfer the Distances between VI and VII , and VI and VIII on the other side the opposite six a Clock Hour Line , and from the Center of the Dyal Plane draw Lines through those transferred Distances for the Hour Lines before and after VI. Then mark your Hour Lines with their respective numbers . The Substiler Line in this Dyal ( as aforesaid ) is XII , from thence towards the right hand mark every successive Hour Line with I , II , III , &c. and from XII towards the left hand with XI , X , IX , &c. The Stile must be erected perpendicularly over the Substilar Line , so as to make an Angle with the Dyal Plane equal to the Elevation of the Pole of your Place . Example . You would draw a Dyal upon a Horizontal Plane here at London ; First draw the Meridian ( or North and South Line ) as XII B , and cross it in the middle with another Line at right Angles , as VI , VI , which is an East and West Line ; where these two Lines cut each other as at A , make the Center , whereon describe the Semi-Circle B , VI. VI ; but one of the Quadrants , viz. the Quadrant from XII to VI , towards the right hand you must devide into 90 equal parts ( as you were taught in Fol. 12. ) and at 51 ½ degrees ( which is Londons Latitude ) make a mark , and laying a straight Ruler to the Center of the Plane , and to this mark draw a Line by the side of it for the Stiler Line . Then on the Substilar Line chuse a point as at C , and through that point draw a Line as long as you can perpendicular to the East and West Line VI , VI , as EF , ( which is called the Contingent Line , ) where this Contingent Line cuts the Substilar Line place one Foot of your Compasses , and from thence measure the shortest distance between the point C and the Stilar Line . And keeping one Foot of your Compasses still in the point C , set off the shortest distance between the point C and the Stilar Line on the Substilar Line , as at D ; which point D shall be a Center , whereon with your Compasses at the same width you must describe a Semi-Circle to represent a Semi-Circle of the Equinoctial . This Semi-Circle devide into six equal parts ( as you were taught Fol. 13. ) to each of which equal parts , and to the Center of the Equinoctial Semi-Circle lay a straight Ruler , and where the straight Ruler cuts the Line of Contingence make marks in the Line of Contingence . Then lay the straight Ruler to the Semi-Circle of the Dyal Plane , and to each of the marks in the Line of Contingence , and by the side of it draw twelve straight Lines for the twelve Fore and Afternoon Hour Lines , viz. from VI in the Morning to VI in the Evening . Then in the Quadrant VI B , measure the distance between the VI a Clock Hour Line , and the V a Clock Hour Line , and transfer the same distances from the VI a Clock Line to VII , and V on both sides the VI a Clock Hour Lines , and through those distances draw from the Center of the Plane the VII and V a Clock Hour Lines , and measure the distance between the VI a Clock Hour Line and the IV a Clock Hour Line , and transfer the same distance from the VI a Clock Line to VIII and IV , and through those distances draw from the Center of the Plane the VIII a Clock and IV a Clock Hour Lines . If you will have the half Hours and quarter Hours , or any other devision of hours , you must devide each six devisions of the Equinoctial into so many parts as you intend , and by a straight Ruler laid to the Center of the Equinoctial , and those devisions in the Equinoctial Circle make marks in the Line of Contingence , as you did before for the whole Hour Lines ; and Lines drawn from the Center of the Plane through those marks shall be the sub-devisions of the Hours : But you must remember to make all sub-devisions short Lines , and near the verge of the Dyal Plane , that you may the easier distinguish between the whole Hours and the parts of Hours ; as you may see in the Figure . Having drawn the Hour Lines , set the number of each Hour Line under it , as you see in the Figure . Last of all sit a Triangular Iron , whose angular point being laid to the Center of the Dyal Plane , one side must agree with the Substilar Line , and its other side with the Stilar Line ; so is the Stile made . And this Stile you must erect perpendicularly over the Substilar Line on the Dyal Plane , and there fix it . Then is your Dyal finished . OPERAT. III. To describe an Erect Direct South Dyal . YOU may know an Erect Direct South Plane by applying the North side of the Declinatory to it ; For then if the North end of the Needle hang directly over the North point of the Card in the bottom of the Box , it is a South Plane ; but if it hang not directly over the North point of the Card , it is not a Direct South Plane , but Declines either East or West , and that contrary to the pointing of the Needle Easterly or Westerly from the North point of the Card : for if the North point of the Needle points Easterly , the Plane Declines from the South towards the West : if it point Westerly , the Plane Declines from the South towards the East . You may know if the Plane be truly Erect or upright , by applying one of the sides AB or AD to it ; for then by holding the Center A upwards , so as the Plumb-line play free in the Grove , if the Line falls upon 0 , or 90 , the Plane is upright ; but if it hang upon any of the intermediate Degrees , it is not upright , but Inclines or Reclines . If you find it Incline , apply the side AB to it , and see what number of Degrees the Plumb-line falls on , for that number of Degrees counted from the side AB , is the number of Degrees of Inclination . If you find the Plane Reclines , apply the side AD to it , and see what number of Degrees the Plumb-line falls on , for that number of Degrees counted from the side AD is the number of Degrees of Reclination . These Rules being well understood , may serve you to find the scituation of all other sorts of Planes . But for the making a Dyal on this Plane , you must first draw a Meridian Line through the middle of the Plane , by applying a Plumb-line to the middle of it , till the Plumbet hang quietly before it : for then if the Plumb-line be black't ( for a white Ground , or chalked for a dark Ground ) and strained as Carpenters do their Lines , you may with one stroak of the string on the Plane describe the Meridian Line , as A XII : This Meridian is also the Substilar Line . Then on the top of this Meridian Line , as at A , draw another Line athwart it to cut it at right Angles , as VI , VI , for an East and West Line . At the meeting of these two Lines on the top , make your Center , whereon describe a Semi-Circle on your Plane , as large as you can , which by the Meridian Line and the East and West Line will be devided into two Quadrants . One of these Quadrants devide into 90 Degrees ( as you were taught Fol. 12. ) and from the Substilar Line count the Complement of the Poles Elevation , which ( here at London where the Pole is elevated 51 ½ Degrees , its Complement to 90 ) is 38 ½ Degrees , and make there a mark , as at E. Then on the Substilar Line chuse a point ( where you please ) as at F , for the Line of Contingence to pass through : which Line of Contingence draw as long as you can , so as it may cut the Substilar Line at right Angles , and from the point F in the Substilar Line measure the shortest distance between it and the Stilar Line , and keeping one Foot of your Compasses still in the point F , transfer that distance into the Substilar Line , as at G ; then on the point G describe a Semi-Circle of the Equinoctial against the Line of Contingence , which Semi-Circle devide into twelve equal parts , ( as you were taught by the Example in the Horizontal Dyal , Fol. 13. ) and by a straight Ruler laid to each of these Devisions , and to the Center of the Semi-Circle make marks in the Line of Contingence by the side of the Ruler : For straight Lines drawn from the Center of the Dyal Plane through these marks in the Contingent Line shall be the 12 Hour Lines before and after Noon . Then mark your Hour Lines with their respective Numbers : The Substilar or Meridian Line is XII , from thence towards the right hand with I , II , III , &c. and from thence towards the left hand with XI , X , IX , &c. The Stile must be erected perpendicularly over the Substilar Line , so as to make an Angle with the Dyal Plane equal to the Complement of the Poles Elevation , viz. 38 ½ Degrees . OPERAT. IV. To make an Erect Direct North Dyal . THE Erect Direct North Dyal , Stile and all , is made by the same Rules , changing upwards for downwards , and the left side for the right , the Erect Direct South Dyal is made : for if the Erect Direct South Dyal be drawn on any transparent Plane , as on Glass , Horn , or an oyled Paper , and the Horizontal Line VI , VI , turned downwards , and the Line VII mark't with V , the Line VIII with IIII , the Line V with VII , and the Line IIII with VIII , then have you of it a North Erect Direct Dyal . All the other Hour Lines in this Dyal are useless , because the Sun in our Latitude shines on a North Face the longest Day only before VI in the Morning , and after VI at Night . OPERAT. V. To describe an Erect direct East Dyal . HAng a Plumb-line a little above the place on the Wall where you intend to make your Dyal , and wait till it hang quietly before the Wall : Then if the Line be rubbed with Chalk ( like a Carpenters Line ) you may by holding the Plumbet end close to the Wall , and straining it pretty stiff , strike with it a straight Line , as Carpenters do : This Line shall be a perpendicular , as AB . Then chuse a convenient point in this Perpendicular , as at C , for a Center , whereon describe an occult Arch , as DE ; This Arch must contain the number of Degrees of the Elevation of the Equinoctial , counted between D and E , which in our Latitude is 38 ½ , or ( which is all one ) the Complement of the Poles Elevation . Therefore in a Quadrant of the same Radius with the occult Arch measure 38 ½ Degrees , and set them off in the Plane from E to D : Then from D to the Center C in the Perpendicular draw the prick't Line DC ; this prick't Line shall represent the Axis of the World. Then cross this Line at right Angles with the Line CF , and draw it from C to F , so long as possibly you can : This Line shall be the Contingent Line . Then chuse a point in this Contingent Line , as at VI , draw a Line through that point at right Angles for the Substilar Line , as G VI H for the Substilar Line ; then open your Compasses to a convenient width , ( as to VIG ) and pitching one Foot in the point G , with the other Foot describe a Semi-Circle of the Equinoctial against the Line of Contingence , which Semi-Circle devide from VI both wayes into six equal parts , as you were taught by the Example in the Horizontal Dyal : and laying a straight Ruler on the Center of this Semi-Circle of the Equinoctial , and to each of those equal parts mark on the Contingent Line where the Ruler cuts it , for those marks shall be the several points from whence Lines drawn parallel to the Line CD shall be the respective Hour Lines . The reason why the Contingent Line is drawn from VI. to F , so much longer than from VI to C is ; because the Hour Lines from VI towards XII are more in number towards Noon , than they are from VI backward towards IIII : for this Dyal will only shew the Hours from a little before IV in the Morning to almost Noon : For just at Noon the Shaddow goes off the Plane ; as you may see if you apply a straight Ruler to the Center of the Equinoctial Semi-Circle G , and lay it to the point 12 in the Semi-Circle ; for the straight Ruler will then never cut the Line of Contingence , because the Line of Contingence is parallel to the Line G XII on the Equinoctial Circle , and Lines parallel , though continued to never so great a length never meet . To these Hour Lines , set Figures as may be seen in the Scheme . The Stile IK of this Dyal as well as of all others must stand parallel to the Axis of the World ; and also parallel to the Face of the Plane , and parallel to all the Hour Lines , and stand directly over the Substilar or VI a Clock Hour-Line , and that so high as is the distance of the Center of the Equinoctial Semi-Circle from the Contingent Line . OPERAT. VI. To describe a Dyal on an Erect Direct West Plane . AN Erect Direct West Dyal , is the same in all respects with an Erect Direct East Dyal : Only as the East Dyal shews the Forenoon Hours , so the West shews the Afternoon Hours . Thus if you should draw the East Dyal on any transparent Plane , as on Glass , Horn , or oyled Paper , on the one side will appear an East Dyal , on the other side a West : Only the numbers to the Hour Lines ( as was said before in the North Dyal ) must be changed ; for that which in the East Dyal is XI , in the West must be I ; that which in the East Dyal is X , in the West must be II ; that which in the East Dyal is IX , in the West must be III , &c. The Stile is the same . OPERAT. VII . To describe a Dyal on an Erect North , or Erect South Plane Declining Eastwards or Westwards . THese four Dyals , viz. the Erect North Declining Eastwards , the Erect North Declining Westwards , the Erect South Declining Eastwards , and the Erect South Declining Westwards , are all projected by the same Rules ; and therefore are in effect but one Dyal differently placed , as you shall see hereafter . First draw on your Plane a straight Line to represent the Horizon of your place , and mark one end of it W for West , and the other end E for East . Chuse a point in this Horizontal Line for a Center , as at A , whereon you may describe a Circle to comprehend all these four Dyals : Draw a Line as MAM perpendicular to the Horizontal Line WE , through the Center A for a Meridian Line , and on that Center describe a Circle , which by the two Lines WAE , and MAM will be devided into four Quadrants , which will comprehend the four Dyals aforesaid : for if it be a North declining West you are to draw , the upper Quadrant to the left hand serves your purpose : If a South Declining West , the same Lines continued through the Center A into the lower Quadrant to the right Hand serves your turn ; if a North Declining East , the upper Quadrant to the right Hand serves your turn ; or if a South Declining East , the same Lines continued through the Center A into the lower Quadrant to the left hand serves your turn ; and you must draw the Declination , Complement of the Poles Altitude , Substile , Stile and Hour Lines in it ; but the Hour Lines must be differently marked as you shall see hereafter . I shall onely give you an Example of one of these Dyals ; viz. A South Declining East . We will suppose you are to draw a Dyal that declines from the South 50 Degrees towards the East ; here being but one Dyal , you need describe but one Quadrant of a Circle . Set off in the lower Quadrant WAM 50 degrees from the Meridian Line M towards W , and from the Center A draw a straight Line through that mark in the Quadrant as DA , which may be called the Line of Declination ; then set off from the Meridian Line the Complement of the Poles Elevation , which in our Latitude is 38 ½ degrees , and there draw another Line from the Center as AP , which we will call the Polar Line . Then take in the Horizontal Line a convenient portion of the Quadrant , as AB , and from the point B draw a Line parallel to the Meridian Line AM , and continue that Line till it intersect the Polar Line , as at P , from which Point P draw a Line parallel to WA , as PC : Then measure the distance of AB in the Horizontal Line , and set off that distance in the Line of Declination , as from A to D , and from that point of distance draw a Line parallel to the Meridian AM through the Horizontal Line at R , and through the Point D , and continue it through the Line PC , as at S ; then laying a straight Ruler to the Center A , and the Intersection of the Line PC , at S draw the Line AS for the Substile : Then upon the Point S erect a Line perpendicularly as ST ; Then measure the distance between R and D , and set that distance off from S to T , and from the Center to the point T draw the Line AT for the Stile or Gnomon ; and the Triangle SAT made of Iron or Brass and erected perpendicularly over the Substile SA shall by its upper side TA cast a shaddow upon the Hour of the day . But you will say the Hour Lines must be drawn first : It is true ; Therefore to draw them you must chuse a point in the Substile Line where you think good , and through it draw the Line FF as long as you can for the the Line of Contingence : then with your Compasses take the shortest distance between this point and the Stile , and transfer that distance below the Line of Contingence on the Substile as at Ae , and with your Compasses at that distance describe on the Center Ae a Circle to represent the Equinoctial ; Then ( as you were taught in the Example of the Horizontal Dyal ) devide the Semi-Circle of the Equinoctial into twelve equal parts , beginning at the point in the Equinoctial Circle , where a straight Line drawn from the Center of it to the Intersection of the Line of Contingence with the Meridian Line cuts the Equinoctial Line , as here at the Point G ; Then lay a straight Ruler to the Center of the Equinoctial Circle , and to every one of the Devisions in the Semi-Circle , and mark where the straight Ruler cuts the Contingent Line ; for straight Lines drawn from the Center A of the Dyal to those several marks on the Contingent Line shall be the Hour Lines ; and must be numbred from the Noon Line or Meridian A M backwards , as XII , XI , X , IX . &c. towards the left hand . So is your Dyal finished . This Dyal drawn on any transparent matter as Horn , Glass , or an oyled Paper , shall on the other side the transparent matter become a South Declining West , ( Stile and all ) but then the I a Clock Hour Line must be marked II , the XII XII , the XI a Clock Hour Line I , X , II , IX , III , &c. If you project it anew , you must describe the Quadrant MW on the other side the Meridian Line , on the Center A from M to E , and then count , ( as before ) the Declination , Altitude of the Pole , Substile , and Stile in the Quadrant , beginning at M towards E , and work in all respects as with the South Declining East ; only number this South Declining West as in the foregoing Paragraph . If you project a North Declining East , you must describe the Quadrant above the Horizontal Line from M upwards , towards E on your right hand , and count ( as before ) the Declination , Altitude , Complement of the Pole , Substile , and Stile from the Meridian Line , and work as with the South Declining East : It must be numbred from the Meridian Line M towards the right hand with XI , X , IX , VIII , &c. If this Dyal were drawn on transparent matter , the other ▪ side would shew a North Declining West : But if you will project it anew , you must describe the Quadrant above the Horizontal Line , from M upwards towards W , and count from the Meridian Line AM the Declination , Complement , Altitude of the Pole , Substile and Stile , and work with them ( in all respects ) as with the South Declining East ; but then the XI a Clock Hour Line must be marked I , the X , II ; the IX , III , &c. OPERAT. VIII . To draw a Dyal on an East or West Plane Reclining , or Inclining . DRaw a straight Line parallel to the Horizon , to represent a Meridian , or XII a Clock Line , and mark one end N , the other S ; Chuse a point in this Line , as at A for a Center : then if your Plane be an East or a West Incliner , let fall a Perpendicular upon this Center , ( that is , the Perpendicular must stand above the Meridian Line NS . ) as AE , and upon the Center A describe a Semi-Circle above the Meridian Line NS ; ) But if your Plane be an East Incliner , or a West Recliner , let fall a Perpendicular from the Center A under the Meridian Line , and upon the Center A describe a Semi-Circle under the Meridian Line . If your Plane be a West Incliner , work ( as shall be taught ) in the Quadrant on the left hand above the Meridian Line . If an East Recliner , in the Quadrant on the right hand above the Meridian Line . If it be a West Recliner , work in the Quadrant on the left hand under the Meridian . If an East Incliner , in the Quadrant under the Meridian Line the right hand . For Example , An East Dyal Reclining 45 Degrees . You would draw a Dyal on an East Plane Reclining 45 Degrees : Therefore in the Quadrant on the right hand above the Meridian Line , set off from the Perpendicular AE 45 Degrees on the Quadrant , for the Reclination of the Plane ; and set off also in the Quadrant 38 ½ Degrees from the Perpendicular for the Complement of the Poles Elevation , and at these settings off make marks in the Quadrant : Then lay a straight Ruler to the Center A , and to the marks in the Quadrant , and draw straight Lines through them from the Center . Then chuse in the Meridian Line NS a convenient point , as at B , and through that point draw a Line parallel to the Perpendicular AE , which will intersect the Line drawn for the Complement of the Poles Elevation AP in P ; from which point P , draw a Line parallel to the Meridian Line NS , to cut the Perpendicular AE in C , and also the Line of Obliquity AO in O. Then measure the length AO , and set off that length in the Perpendicular ACE from A to E , and draw the Line EG parallel to the Meridian Line NS , which will cut the Line BP prolonged in G. Measure also the length of CO , and set that length off from A to Q on the Line of Obliquity AO , and draw the Line QR parallel to the Perpendicular ACE . Then measure the distance of AR , and upon the Line GPB set it off from G to S ; and laying a straight Ruler to the point S and the Center A , draw by the side of it the Line AS ; for the Substile Line . Then measure the length of QR , and from S raise a Perpendicular , and in that Perpendicular set that length off from S to T ; and laying a straight Ruler to the Center A and the point T , draw the Line AT for the Stilar Line , which Stilar Line being perpendicularly erected over the Substilar Line AS , will stand parallel to the Axis of the World , and cast its shaddow on the Hour of the Day . To draw the Hour Lines on this Plane , you must ( as you have several times before been directed ) chuse a point in the Substilar Line , and through that point draw at right Angles with the Substilar Line the Line of Contingence so long as you can : Then measure the shortest distance between that Point and the Stilar Line , and transfer that distance below the Line of Contingence in the Substilar Line , as at Ae , and with your Compasses at that distance describe against the Line of Contingence the Equinoctial Circle ; Then divide the Semi-Circle of the Equinoctial next the Line of Contingence into twelve equal parts , ( as you have formerly been taught ) beginning at the Point in the Equinoctial Circle , where a straight Line drawn from the Center of it to the Intersection of the Line of Contingence with the Meridian Line NS cuts the Equinoctial Circle , as here at the point D : Then lay a straight Ruler to the Center of the Equinoctial Circle , and to every one of the Devisions in the Equinoctial Semi-Circle , and mark where the straight Ruler cuts the Contingent Line : for straight Lines drawn from the Center A of the Dyal through these several marks in the Contingent Line shall be the Hour Lines , and must be numbred from the Meridian or Noon-Line NS which is the XII a Clock Line upwards , with XI , X , IX , VIII , &c. The Center of this Dyal must stand downward . If this Dyal were turned with its Center upwards , it would shew a West Inclining 45 degrees , only the numbers to the Hour Lines must be changed ; for to XI you must set I , to X , II ; to IX , III , &c. and the Substile over which the Stile must stand , must be placed in the Semi-Circle ( at first described ) as much to the right hand the Perpendicular AE , as it doth on the left hand . If this Dyal were drawn on Glass , Horn , or an oyled Paper , and you turn the Meridian Line NS upwards , the backside shall be an East Inclining 45 degrees , and the Hour Lines must be numbred as they are on the East Reclining : But the Substile over which the Stile must stand , must be placed , in the Semi-Circle ( at first described ) as much to the left hand the Perpendicular AE , as it is on the oyled Paper to the right hand . If you turn the Meridian Line NS downwards , the backside shall be a West Recliner 45 degrees , and the Hour Lines must be numbred from the XII a Clock Line upwards , with I , II , III , &c. You must note that all the Hour Lines of the Day will not be described in this single Quadrant , nor does the Quadrant at all relate to the Hour Lines ; but is described onely for setting off the Complement of the Poles Elevation and Reclination of the Plane , that by working ( as hath been shewn ) you may find the place of the Substilar Line , and the Angle the Stile makes with it : For having the Substilar Line , you know how to draw the Line of Contingence , and to describe the Equinoctial Circle , by which all the Hours are described on the Plane . To draw a Dyal on a Direct South or North Plane Inclining or Reclining . Direct Reclining or Inclining Dyals are the same with Erect Direct Dyals that are made for the Latitude of some other Places ; the Latitude of which Places are either more than the Latitude of your Place , if the Plane Recline ; or less , if the Plane Incline : and that in such a proportion as the Arch of Reclination or Inclination is . Thus a Direct South Dyal Reclining 10 degrees in London's Latitude , ( viz. 51 ½ degrees ) is an Erect Direct South Dyal made for the Latitude of 61 ½ degrees . And a Direct South Dyal Inclining 10 in the Latitude of 51 ½ is an Erect Direct South Dyal in the Latitude of 41 ½ degrees : and is to be made according to the Direction given in Operat . III. OPERAT. IX . To draw a Dyal on a South or North Inclining Declining , or Reclining Declining Plane . THese four sorts of Dyals viz. the South Inclining Declining , and South Reclining Declining , and North Inclining Declining , and South Reclining Declining , are all projected by the same Rules ; and therefore are in effect but one Dyal differently placed , as you shall see hereafter . First draw on your Plane a straight Line parallel to the Horizon , and mark one end W for West , and the other E for East . On South Incliners and Recliners , E on the right hand , and W on the left : on North Incliners and Recliners E on the left hand and W on the right . Chuse a point in this Horizontal Line for a Center , as at A ; Through this point A draw a Line Perpendicular to the Horizon , and on this point ( as on a Center ) describe a Semi-Circle , one Quadrant above , and another below the Horizontal Line . ( though for this Example I describe but one . ) Then if the Plane respect the South , set off in the lower Quadrant from the Perpendicular the Declination , the Inclination , or the Reclination , and the Complement of the Altitude of the Pole ; and through these several settings off in the Quadrant , draw straight Lines from the Center A ; then take in the Horizontal Line towards the Semi-Circle , a convenient distance from the Center A , as B , and through the point B draw a straight Line parallel to the Perpendicular , and prolong it through the Polar Line , as BP : Through the point P , draw a Line parallel to the Horizontal Line , as PC ; this Line will cut the Line of Obliquity in the point O : Then measure the distance of AO , and set off that distance on the Perpendicular from A to F , and through the point F draw a straight Line parallel to the Horizontal Line , as FG , for the Horizontal Intersection . Then measure the distance of CO , and set off that distance on the Perpendicular from A to I ; from the point I , draw the Line ID parallel to the Horizontal Line , to cut the Line of Declination in the point D. Then measure the distance of AB , and set off that distance in the Line of Declination from A to E ; and from the point E draw a straight Line parallel to the Horizontal Line WE , to cut the Perpendicular in the point K. Measure the distance of EK , and set off that distance on the other side the Perpendicular in the Horizontal Intersection , from F to H , and from the point H draw HN parallel to the Perpendicular to cut the Horizontal Line in the point N. Then to find the Meridian Line , Substile and Stile , do thus . If your Plane be a Southern Incliner , or a Northern Recliner , measure the distance of LD , and set off that distance in the Horizontal Intersection from F to M , and through the point M draw the Line AM for the Meridian Line . Then add the distance of AL to AK , thus : measure the distance of AL , and place one Foot of your Compasses in the point K in the Perpendicular Line , and extend the other to X , and measuring the distance of AX , set it off in the Line of Obliquity from A to Q ; and from the point Q draw the Line QR parallel to the Perpendicular , and cutting the Horizontal Line in the point R. Then measure the distance of AR , and set off that distance from H in the Horizontal Intersection to S on the Line HN , and to the point S draw the Line AS for the Substile . Then measure the distance of QR , and set off that distance perpendicularly from the point S to T ; and lastly , from the point A , draw the straight Line AT for the Stilar Line , which Stilar Line being perpendicularly erected over the Substilar Line AS , will stand parallel to the Axis of the World , and cast its shadow on the Hour of the Day . But if the Plane be a Southern Recliner , or Northern Incliner , measure ( as before ) the distance of LD , and ( as before you were directed ) to set it off from F in the Horizontal Intersection on the right hand the perpendicular Line ; So now , set that distance from F to m in the Horizontal Intersection on the left hand in the Perpendicular Line , and draw the Line A m for the Meridian Line . Then as before you were directed to add AL to AK : So now , substract the distance of AL from AK , and the remainder will be LK : Set therefore the distance of IK from A to q in the same Line of Obliquity , and from the point q , draw the Line qr parallel to the perpendicular . Measure then the distance of A r , and set off that distance in the Line HN , from H to s for the Substilar Line : Then erect on the point s a Perpendicular , and on that Perpendicular set off from s to t the distance of qr : And lastly , from A draw the Line A t for the Stilar Line . If K falls upon L the Plane is parallel to the Axis of the World , and the Dyal drawn upon it will have no Center : But s will fall upon H , and AH ( or A s ) will be the Substile . I shall give you two Examples of these Rules : One of a Dyal with a Center , and the other of a Dyal without a Center . And first , OPERAT. X. How to draw a Dyal with a Center , Declining 20 Degrees , and Inclining 30 Degrees . HAving by the foregoing Precepts of the last Operat . found the Substile , Stile , and Meridian , you must ( as you have often been directed ) chuse a point in the Substilar Line , through which , at right Angles to the Substilar Line draw the Line of Contingence as long as you can : Then measure the shortest distance between the point of Intersection and the Stilar Line , and transfer that distance on one side the Line of Contingence upon the Substilar Line , and so describe the Equinoctial Semi-Circle against the Line of Contingence : Then lay a straight Ruler to the Center of the Equinoctial Circle , as at Ae , and to the point where the Line of Contingence cuts the Meridian Line , as at Z , and mark where the straight Ruler cuts the Equinoctial Circle , and from that mark begin to devide the Semi-Circle into twelve equal parts , and by a straight Ruler laid to those devisions and the Center of the Equinoctial , make marks in the Line of Contingence . Then shall straight Lines drawn from the Center A of the Dyal through every one of those marks in the Contingent Line be the Hour Lines of the Dyal , and must be numbred from the XII a Clock Line towards the right Hand with I , II , III , IV , &c. And the other way with XI , X , IX , &c. OPERAT. XI . How to draw a Dyal without a Center , on a South Plane ; Declining East 30 Degrees , Reclining 34 Degrees 32 Minutes . HAving by the Precepts of Operat . IX found the Substile , you must find the Meridian Line otherwise than you were there taught : For , having drawn the Lines of Latitude , Declination and Reclination , and found the Substile , measure the distance of BP , and set it off on the Line of Declination from A to K , and draw from the Perpendicular AF the Line KQ parallel to AB : Then measure the length of KQ , and set it off on the Polar Line AP , from A to V ; then take the nearest distance between the point V and the Line AB , and set it off on the Line QK from Q to M ; through which point M , draw a Line from the Center A : Then measure with your Compasses in the Semi-Circle WNE ( which in this Dyal may represent the Equinoctial ) the distance of the Arch N m , and set off that distance from the Intersection of the Substile with the Semi-Circle at S to T in the Semi-Circle , which point T shall be the point in the Equinoctial that you must begin to devide the Hours at , for the finding their distances on the Line of Contingence . Then consider ( according to the bigness of your Plane ) what heighth your Stile shall stand above the Substile , and there make a mark in the Substile : For the distance between the Center A and that mark must be the heighth of the Stile perpendicularly erected over the Substile , as at I. Draw through this point I a Line of Contingence , as long as you can to cut the Substile at right Angles , and then laying a Ruler to the Center A , and successively to each Devision of the Equinoctial make marks in the Line of Contingence , and through those marks draw straight Lines parallel to the Substile , which shall be the Hour Lines ; and must be numbred from the left hand towards the right , beginning at the XII a Clock Line with I , II , III , &c. and from the right hand towards the left on the XII a Clock Line with XI , X , IX , &c. The Stile to this Dyal may be either a straight Pin of the length of AI , or else a Square of the same heighth , erected perpendicularly upon the point I , in the Substile Line . OPERAT. XII . To make a Dyal on the Ceeling of a Room , where the Direct Beams of the Sun never come . FInd some convenient place in the Transum of a Window to place a small round piece of Looking-Glass , about the bigness of a Groat , or less , so as it may lie exactly Horizontal . The point in the middle of this Glass we will mark A , and for distinction sake call it Nodus . Through this Nodus you must draw a Meridian Line on the Floor , Thus , Hang a Plumb-line in the Window exactly over Nodus , and the Shadow that the Plumb-line casts on the Floor just at Noon will be a Meridian Line ; or you may find a Meridian Line otherwise by the Clinatory . Having drawn the Meridan Line on the Floor , find a Meridian Line on the Ceeling , thus , Hold a Plumb-line to the Ceeling , over that end of the Meridian Line next the Window ; If the Plumbet hang not exactly on the Meridian Line on the Floor , remove your hand on the Ceeling one way or other , as you see cause , till it do hang quietly just over it , and at the point where the Plumb-line touches the Ceeling make a mark , as at B ; that mark B shall be directly over the Meridian Line on the Floor : then remove your Plumb-line to the other end of the Meridian Line on the Floor , and find a point on the Ceeling directly over it , as you did the former point , as at C , and through these two points B and C on the Ceeling , strain and strike a Line blackt with Smal-Coal or any other Colour ( as Carpenters do ) and that Line BC on the Ceeling shall be the Meridian Line , as well as that on the Floor : Then fasten a string just on the Nodus , and remove that string , forwards or backwards , in the Meridian Line on the Ceeling , till it have the same Elevation in the Quadrant on the Clinatory above the Horizon that the Equinoctial hath in your Habitation , and through the point where the string touches the Meridian Line in the Ceeling shall a line be drawn at right Angles with the Meridian , to represent the Equinoctial Line . Thus in our Latitude the Elevation of the Equator being 38 ½ degrees ; I remove the string fastned to the Nodus forwards or backwards in the Meridian Line of the Ceeling , till the Plumb-line of the Quadrant on the Clinatory , when one of the sides are applied to the string , falls upon 38 ½ degrees : and then I find it touch the Meridian Line at D in the Ceeling : therefore at DI make a mark , and through this mark strike the line DE ( as before I did in the Meridian Line ) to cut the Meridian Line at right Angles : This Line shall be the Equinoctial Line , and serve to denote the Hour Distances , as the Contingent Line does on other Dyals , as you have often seen· Then I place the Center of the Quadrant on the Clinatory upon Nodus , so as the Arch of the Quadrant may be on the East side the Meridian Line , and underprop it so , that the flat side of the Quadrant may lie parallel to the string , when it is strained between the Nodus and the Equinoctial , and also so as the string may lie on the Semi-diameter of the Quadrant , when it is held up to the Meridian Line on the Ceeling . Then removing the string the space of 15 degrees in the Quadrant , and extending it to the Equator on the Ceeling , where the string touches the Equator , there shall be a point through which the I a Clock Hour line shall be drawn : and removing the string yet 15 degrees further to the Eastwards in the Semi-Circle of Position , and extending it also to the Equator , where it touches the Equator , there shall be a point through which the II a Clock Hour Line shall be drawn . Removing the string yet 15 degrees f●rther , to the Eastwards in the Semi-Circle of Position , and extending it to the Equator , there shall be a point through which the III a Clock Hour Line shall be drawn : The like for all the other After-noon Hour Lines . So oft as the string is removed through 15 degrees on the Quadrant , so oft shall it point out the After-Noon distances in the Meridian Line on the Ceeling . Having thus found out the points in the Equator through which the After-noon Hour Lines are to be drawn , I may find the Fore-noon Hour distances also the same way , viz. by removing the Arch of the Quadrant to the West side the Meridian , as before it was placed on the East , and bringing the string to the several 15 degrees on the West side the Quadrant ; or else I need only measure the distances of each Hours distance found in the Equator from the Meridian Line on the Ceeling ; for the same number of Hours from XII , have the same distance in the Equinoctial Line on the other side the Meridian , both before and after-noon : The XI a Clock Hour distance is the same from the Meridian Line , with the I a Clock distance on the other side the Meridian ; the X a Clock distance , the same with the II a Clock distance ; the IX with the III , &c. And thus the distances of all the Hour lines are found out on the Equator . Now if the Center of this Dyal lay within doors , you might draw lines from the Center through these pricks in the Equator , and those Lines should be the Hour lines , as in other Dyals : But the Center of this Dyal lies without doors in the Air , and therefore not convenient for this purpose : So that for drawing the Hour Lines , you must consider what Angle every Hour Line in an Horizontal Dyal makes with the Meridian ; that is , at what distance in Degrees and Minutes the Hour Lines of an Horizontal Dyal cut the Meridian ; which you may examine , as by Operat . II. For an Angle equal to the Complement of the same Angle , must each respective Hour Line with the Equator on the Ceeling have . Thus upon the point markt for each Hour distance in the Equinoctial Line on the Ceeling , I describe the Arches I , II , III , IV , as in the Figure , and finding the distance from the Meridian of the Hour Lines of an Horizontal Dyal to be according to the Operat . II. Thus , The 1 a clock Hour line 11.40 whose Complement to 90 is 78.20 The 2 a clock Hour line 24.15 whose Complement to 90 is 65.45 The 3 a clock Hour line 38.14 whose Complement to 90 is 51.56 The 4 a clock Hour line 53.36 whose Complement to 90 is 36.24 I measure in a Quadrant of the same Radius with those Arches already drawn from the Equinoctial Line for the 1 a Clock Hour 78.20 for the 2 a Clock Hour 65.45 for the 3 a Clock Hour 51.56 for the 4 a Clock Hour 36.24 and transfer these distances to the Arches drawn on the Ceeling : For then straight Lines drawn through the mark in the Arch , and through the mark in the Equator , and prolonged both ways to a convenient length , shall be the several Hour Lines ( aforesaid ; ) And when the Sun shines upon the Glass at Nodus , its Beams shall reflect upon the Hour of the Day . Some helps to a young Dyalist for his more orderly and quick making of Dyals . IT may prove somewhat difficult to those that are unpractised in Mathematical Projections , to devide a Circle into 360 Degrees ( or which is all one ) a Semi-Circle into 180 , or a Quadrant into 90 degrees ; and though I have taught you in the projecting the Horizontal Dyal the original way of doing this , yet you may do it a speedier way by a Line of Chords , which if you will be curious in your Practise , you may make your self ; or if you account it not worth your while , you may buy it already made on Box or Brass of most Mathematical Instrument-Makers . This Instrument is by them called a Plain Scale , which does not only accommodate you with the devisions of a Quadrant , but also serves for a Ruler to draw straight Lines with : the manner of making it is as follows . Describe upon a smooth flat even-grain'd Board a quarter of an whole Circle , as BC , whose Radius AB or AC may be four inches , if you intend to make large Dyals , or two inches if small ; but if you will , you may have several Lines of Chords on your Scale or Rule . Devide this Quadrant into 90 equal parts as you were taught in the making the Horizontal Dyal . Then draw close by the edge of your straight Ruler a Line parallel to the edge , and at about 1 / 20 part of an Inch a second Line parallel to that , and at about ⅛ of an Inch a third Line parallel to both . Then place one Foot of your Compasses at the beginning of the first degree on the Quadrant descibed on the Board , as at B , and open the other Foot to the end of the first degree , and transfer that distance upon your Rule , from B to the first mark or devision , between the two first drawn Lines . Then place one Foot of your Compasses again at the begining of the first degree on the Quadrant described on the Board , as at B , and open the other Foot to the end of the second Degree , and transfer that distance upon your Rule from B to the second mark or devision between the two first drawn Lines ; And thus measure the distance of every Degree from the first Degree described on the Quadrant , and transfer it to the Rule . But for distinction sake , you may draw every tenth devision from the first Line parallel to the edge of the third Line , and mark them in succession from the beginning with 10 , 20 , 30 , to 90 : and the fifth Devisions you may draw half way between the second and the third parallel Lines ; the single Devisions only between the two first parallel Lines . So is your Line of Chords made . The Vse of the Line of Chords . AS its use is very easie , so its convenience is very great ; for placing one Foot of your Compasses at the first Devision on the Scale , and opening the other to the 60 th Degree , you may with the points of your Compasses ( so extended ) describe a Circle , and the several Devisions , on the Scale shall be the Degrees of the four Quadrants of that Circle , as you may try by working backwards , to what you were just now taught in the Making the Scale : For as before you measured the distance of the Degrees of the Quadrant , and transferr'd them to the Scale , so now you only measure the D●visions on the Scale , and transfer them to the Quadrant , Semi-Circle , or whole Circle described on your Paper . For Example : If you would measure 30 Degrees in your described Circle , place one Foot of your Compasses at the begining of Devisions on the Scale , as at A , and extend the other Foot to the Divisions marked 30 , and that distance transferred to the Circle , shall be the distance of 30 degrees in that Circle . Do the like for any other number of Degrees . You may draw your Dyal first on a large sheet of Paper , if your Dyal Plane be so large , if it be not so large , draw it on a smaller piece of Paper ; Then rub the back-side of your Paper-Dyal with Smal-coal , till it be well black't ; and laying your Paper Dyal on your Dyal Plane , so that the East , West , North , or South Lines of your Paper agree exactly with the East , West , North , or South scituation of your Dyal Plane . Then with Wax or Pitch fasten the Corners of the Paper on the Plane , and laying a straight Ruler on the Hour-Lines of your Dyal , draw with the blunted point of a Needle by the side of the Ruler , and the Smal-coal rub'd on the back-side the Paper will leave a mark of the Lines on the Plane . If you will have the Lines drawn Red , you may rub the back-side of your Paper with Vermillion ; if Blew , with Verditer ; if Yellow , with Orpment , &c. Then draw upon these marked Lines with Oyl Colours , as you please . An Explanation of some Words of Art used in this BOOK . ANgle . The meeting or joyning of two Lines . Arch. A part of a Circle . Axis . The straight Line that runs through the Center of a Sphere , and both ways through the Circumference : though in Dyalling it is all one with the Diameter of a Circle . Clinatory . See Fol. 8 , 9 , 10. Chord . See Fol. 44 , 45 , 46. Complement . The number that is wanting to make up another number 90 Degr. or 180 Degr. or 360 Degrees . Contingent . A Line crossing the Substile at right Angles . Degree . See Fol. 12. Diameter . The longest straight Line that can be contained within a Circle , viz. the Line that passes through the Center to the Circumference both ways . Dyal Plane . See Fol. 7. Elevation of the Pole. So many degrees as the Pole is elevated above the Horizon . Equinoctial . The Equinoctial is a great Circle that runs evenly between the two Poles of the World. But when we name the Equinoctial in this Book , we mean a small Circle which represents it , and is the Circle or Arch of a Circle which is divided into equal parts to find thereby the unequal parts on the Line of Contingence . In the Horizontal Dyal it is that Arch of a Circle marked GCH . Horizon . Is a great Circle encompassing the place we stand upon ; but in Dyalling it is represented by a straight Line , as in Operat . III. In the South Dyal the Line VI A VI is the Horizontal Line . Latitude . The Latitude of a Place is the number of Degrees contained between the Equinoctial and the place inquired after . Line of Contingence . See Contingent . Magnetick Needle . The Needle touch'd with the Loadstone , to make it point to the North. Meridian , is a great Circle of Heaven passing through the North and South points of the Horizon ; but in Dyalling it is represented by a straight Line , as in Operat . II. in the Horizontal Dyal the Line XII A is a Meridian Line . Nadir . The point directly under our Feet . Nautial Compass , Is the Compass used by Navigators , whereon is marked out all the 32 Winds or Points of the Compass . Oblique Plane . See Fol. 7. Parallel . See Fol. 6 Perpendicular . See Fol. 5. Pole. The North or South Points on the Globe of the Earth , are called North or South Pole. Quadrant . The fourth part of a Circle . Radius . Half the Diameter of a Circle . Right Angle . A straight Line that falls perpendicularly upon another straight Line , makes at the meeting of those two Lines a Right Angle . Semi-Circle . Half a Circle . Semi-Diameter , The same Radius is . Sphere . The highest Heaven with all its imagined Circles is called the Sphere . Stile . The Gnomon or Cock of a Dyal . Substile . The Line the Stile stands on upon a Dyal Plane . Triangle . A figure consisting of 3 Sides and 3 Angles . Zenith . The Point directly over our Head. FINIS . A Catalogue of GLOBES Coelestial and Terrestrial , Spheres , Mapps , Sea-Platts , Mathematical Instruments , and Books , made and sold by Joseph Moxon , on Ludgate-Hill , at the Sign of Atlas . GLOBES 26 Inches Diameter . The price 20 l. the pair . GLOBES , near 15 Inches Diameter . The price 4 l. GLOBES , 8 Inches Diameter . The price 2 l. GLOBES , 6 Inches Diameter . The price 1 l. 10 s. CONCAVE HEMISPHERES of the Starry Orb ; which serves for a Case to a Terrestrial Globe of 3 Inches Diameter , made portable for the Pocket . Price 15 s. SPHERES , according to the Copernican Hypothesis , both General and Particular , 20 Inches Diameter . Price of the General 5 l. Of the Particular 6 l. Of both together 10. SPHERES , according to the Ptolomaick Systeme , 14 Inches Diameter . Price 3 l. SPHERES , according to the Ptolomaick Systeme , 8 Inches Diameter . Price 1 l. 10 s. Gunter's Quadrant , 13 Inches Radius , printed on Paper , and pasted on a Board , with a Nocturnal on the backside . Price 5 s. Gunter's Quadrant , 4 Inches Radius , printed on Paper , and pasted on Brass , with a Nocturnal on the backside , and a Wooden Case covered with Leather fit for it : A new invention contrived for the Pocket . Price 6 s. A large Mapp of the World , 10 Foot long , and 7 Foot deep , pasted on Cloath and coloured . Price 2 l. A Mapp of all the World , 4 Foot long , and 3 Foot deep , pasted on Cloath and coloured . Price 10 s. In sheets 2 s. 6 d. A Mapp of the English Empire in America , describing all places inhabited there by the English Nation , as well on the Islands as on the Continent . Price 15 s. Six Scriptural Mapps , 1. Of all the Earth : And how after the Flood it was divided among the Sons of Noah . 2. Of Paradise , or the Garden of Eden ; with the Countries circumjacent inhabited by the Patriarchs . 3. The 40 years travel of the Children of Israel through the Wilderness . 4. Of Canaan , or the Holy Land : and how it was divided among the twelve Tribes of Israel , and travelled through by our Saviour and his Apostles . 5. The Travels of St. Paul , and others of the Apostles , in their propagating the Gospel . 6. Jerusalem , as it stood in our Saviour's time ; with a Book of Explanations to these Mapps ▪ entituled Sacred Geography Price 6 s. Useful to be bound up with Bibles . A Sea-Platt , or Mapp of all the World , according to Mercator , in two large Royal Sheets of Paper ; set forth by Mr. Edward Wright , and newly corrected by Joseph Moxon Hydrogr . &c. Price 2 s. Sea Platts for sailing to all parts of the World. Price 6 d. the sheet . The famous City of Batavia in the East-Indies , built and inhabited by the Duth ; curiously engraved , and printed on four large Sheets of Royal Paper . Price 2 s. 6 d. A small Mapp of all the World , with Descriptions , on one Sheet . Price 6 d. BOOKS . A Tutor to Astronomy and Geography , or the Use of both the GLOBES Coelestial and Terrestrial ; by Joseph Moxon Hydrographer to the Kings most Excellent Majesty . Price 5 s. The Vse of the Copernican Spheres , teaching to salve the Phaenomena by them , as easily as by the Ptolomaick Spheres ; by Joseph Moxon Hydrographer &c. Price 4 s. Wright's Correction of Errors in the Art of Navigation . Price 8 s. New and rare Inventions of Water-works . Teaching how to raise Water higher than the Spring . By which Invention the perpetual Motion is proposed , many hard labours performed , and varieties of Motion and Sounds produced . By Isaac de Caus , Engineer to King Charles the First . Price 8 s. Practical Perspective , or Perspective made easie . Teaching by the Opticks how to delineate all Bodies , Buildings and Landskips , &c. By the Catropticks , how to delineate confused Appearances ▪ so , as when seen in a Mirrour or Polisht Body of any intended shape , the Reflection shall shew a design . By the Dioptricks , how to draw part of many Figures into one , when seen through a Glass or Christal cut into many Faces . By Joseph Moxon Hydrographer , &c. Price 7 s. An exact Survey of the Microcosine . Being an Antaomy of the Bodies of Man and Woman ; wherein the Skin , Veins , Nerves , Muscles , Bones , Sinews and Ligaments are accurately delineated . Engraven on large Copper Plates , Printed and curiously pasted together , so as at first sight you may behold all the parts of Man and Woman ; and by turning up the several Dissections of the Papers , take a view of all their Inwards : with Alphabetical referrences to the Names of every Member and part of the Body . Set forth in Latine by Remelinus , and Michael Spaher of Tyrol : and Englished by John Ireton Chyrurgeon : and lastly , perused and corrected by several rare Anatomists . Price 14 s. Vignola , or the Compleat Architect . Shewing in a plain and easie way , the Rules of the five Orders in Architecture , viz. Tuscan , Dorick , Ionick , Corinthian and Composite : whereby any that can but read and understand English , may readily learn the proportions that all Members in Building have to one another : set forth by Mr. James Barrozzio of Vignola , and translated into English by Joseph Moxon Hydrographer , &c. Price 3 s. 6 d. Christiologia , or a brief , but true Account of the certain year , Month , Day and Minute of the Birth of Jesus Christ. By John Butler B. D. and Chaplain to his Grace James Duke of Ormond , &c. and Rector of Lichborough , in the Diocess of Peterburgh . Price 3 s. 6 d. A Tutor to Astrology , or Astrology made easie ; being a plain Introduction to the whole Art of Astrology . Whereby the meanest Apprehension may learn to erect a Figure , and by the same give a determinate Judgment upon any Question of Nativity whatsoever . Also new Tables of Houses , calculated for the Latitude of 51 deg . 32 min. Also Tables of Right and Oblique Ascensions to 6 deg . of Latitude . Whereunto is added an Ephemeris for three years ; with all other necessary Tables that belong to the Art of Astrology . Also how to erect a Figure the Rational way by the Tables of Triangles , more methodically than hath yet been published ; digested into a small Pocket Volume , for the conveniency of those that erect Figures abroad . By W. Eland . Price 2 s. The Use of a Mathematical Instrument called a Quadrant , shewing very plainly and easily to know the exact height and distance of any Steeple , Tree , or House , &c. Also to know the Hour of the Day by it ; the heighth of the Sun , Moon or Stars ; and to know the time of the Sun-rising and setting , and the length of every day in the year , the place of the Sun in the Ecliptick , the Azimuth , right Ascension , and Declination of the Sun : with many other necessary and delightful Conclusions , performed very readily . Also the use of a Nocturnal , whereby you may learn to know the Stars in Heaven , and the hour of the Night by them . With many other delightful Operations . Price 6 d. A brief Discourse of a passage by the North-pole to Japan , China , &c. Pleaded by three Experiments , and Answers to all objections that can be urged against a passage that way . As 1. By a Navigation into the North-pole , and two degrees beyond it . 2. By a Navigation from Japan towards the North-pole . 3. By an Experiment made by the Czar of Muscovy : whereby it appears that to the Northward of Nova Zembla is a free and open Sea as far as Japan , China , &c. With a Mapp of all the discovered Land nearest to the Pole. By Joseph Moxon Hydrographer &c. Price 6 d. Regulae Trium Ordinum Literarum Typographicarum : Or the Rules of the three Orders of Print-Letters , viz. The Roman , Italick , English , Capaitals and Small . Shewing how they are compounded of Geometrick Figures , and mostly made by Rule and Compass . Useful for Writing-Masters , Painters , Carvers , Masons , and others that are lovers of Curiosity . By Joseph Moxon Hydrographer &c. Price 5 s. The Use of the Astronomical Playing Cards . Teaching an ordinary Capacity by them to be acquainted with all the Stars in Heaven : to know their Places , Colours , Natures and Bignesses . Also the Poetical Reasons for every Constellation ; very useful , pleasant and delightful for all lovers of Ingeniety . By Joseph Moxon Hydrogr . &c. Price 6 d. The Astronomical Cards . By Joseph Moxon Hydrographer , &c. Price plain 1 s. Coloured 1 s. 6 d. Best coloured and the Stars gilt 5 s. The Genteel House-keepers Pastime : Or , the Mode of Carving at the Table represented in a Pack of Playing Cards . By w●ich , together with the Instructions in this Book , any ordinary Capacity may easily learn how to Cut up , or Carve in M●de all the most usual Dishes of Flesh , Fish , Fowl , and Baked M●●●● ; and how to make the several Services of the same at the Table ; with the several Sawces and Garnishes proper to each Dish of Meat . Set forth by several of the best Masters in the Faculty of Carving , and published for publick Use. Price 6 d. Carving Cards . By the best Carvers at the Lord Mayors Table . Price 1 s. Compendium Euclidis Curiosi : Or Geometrical Operations . Shewing how with one single opening of the Compasses , and a straight Ruler all the Propositions of Euclids first Five Books are performed . Translated out of Dutch into English. By Joseph Moxon . Hydrogr . &c. Price 1 s. An Introduction to the Art of Species . By Sir Jonas Moore . Price 6 d. Two Tables of Ranges , according to degrees of Mounture . By Henry Bond , Senior . Price 6. d. Mechanick Exercises : Or the Doctrine of Handy-Works , in six Monethly Exercises ; began January 1. 1677. and monethly continued till June 1678. The first three , viz. the Numb . I. Numb . II. Numb . III. teaching the Art of Smithing . The other three , viz. Numb . IIII. Numb . V. Numb . VI. teaching the Art of faynery . Accommodated with suitable engraved Figures . By Joseph Moxon Hydrographer , &c. price 6 d. each Exercise . At the place abovesaid , you may also have all manner of Mapps , Sea-Platts Drafts , Mathematical Books , Instruments , &c. At the lowest prizes . ADVERTISEMENT . THere is invented by the Right Honourable the Farl of Castlemain , a new kind of Globe , call'd ( for distinction sake ) the English Globe ; being a fix'd and immovable one , performing what the Ordinary ones do , and much more , even without their usual Appendancies ; as Wooden Horizons , Brazen Meridians , Vertical Circles , Horary Circles , &c. For it Composes it self to the site and Position of the World without the Marriners Compass , or the like forreign Help ; and besides other useful and surprising Operations ( relating both to the Sun and Moon , and performed by the Shade alone ) we have by it not only the constant proportion of Perpendiculars to their Shades , with several Corollaries thence arising , but also an easie , new , and most compendious way of describing Dyals on all Planes , as well Geometrically , as Mechanically : most of which may be taught any one in few Hours , though never so unacquainted with Mathematicks . To this is added on the Pedestal a Projection of all the appearing Constellations in this Horizon , with their Figures and Shapes . And besides , several new things in it differing from the common Astrolabe , ( tending to a clearer and quicker way of Operating ) the very Principles of all Steriographical Projections are laid down , and Mathematically demonstrated ; as is every thing else of Moment throughout the whole Treatise . These Globes will be made and exposed to Sale about August next , ( God willing : ) against which time the Book for its use will also be Printed , and sold by Joseph Moxon , on Ludgate-Hill , at the Sign of Atlas . 
A67225	----	The description and uses of the general horological-ring: or universal ring-dyal Being the invention of the late reverend Mr. W. Oughtred, as it is usually made of a portable pocket size. With a large and correct table of the latitudes of the principal places in every shire throughout England and Wales, &c. And several ways to find a meridian-line for the setting a horizontal dyal. By Henry Wynne, maker of mathematical instruments near the Sugar-loaf in Chancery-lane. Wynn, Henry, d. 1709. 1682 Approx. 44 KB of XML-encoded text transcribed from 23 1-bit group-IV TIFF page images. Text Creation Partnership, Ann Arbor, MI ; Oxford (UK) : 2005-10 (EEBO-TCP Phase 1). A67225 Wing W3778B ESTC R221060 99832437 99832437 36910 This keyboarded and encoded edition of the work described above is co-owned by the institutions providing financial support to the Early English Books Online Text Creation Partnership. This Phase I text is available for reuse, according to the terms of Creative Commons 0 1.0 Universal . The text can be copied, modified, distributed and performed, even for commercial purposes, all without asking permission. Early English books online. (EEBO-TCP ; phase 1, no. A67225) Transcribed from: (Early English Books Online ; image set 36910) Images scanned from microfilm: (Early English books, 1641-1700 ; 2143:12) The description and uses of the general horological-ring: or universal ring-dyal Being the invention of the late reverend Mr. W. Oughtred, as it is usually made of a portable pocket size. With a large and correct table of the latitudes of the principal places in every shire throughout England and Wales, &c. And several ways to find a meridian-line for the setting a horizontal dyal. By Henry Wynne, maker of mathematical instruments near the Sugar-loaf in Chancery-lane. Wynn, Henry, d. 1709. [4], 30, [2] p., [1] folded leaf :bill. (engraved) printed by A. Godbid and J. Playford, for the author, London : 1682. With a final leaf of advertisements. Reproduction of the original in the British Library. Created by converting TCP files to TEI P5 using tcp2tei.xsl, TEI @ Oxford. Re-processed by University of Nebraska-Lincoln and Northwestern, with changes to facilitate morpho-syntactic tagging. Gap elements of known extent have been transformed into placeholder characters or elements to simplify the filling in of gaps by user contributors. EEBO-TCP is a partnership between the Universities of Michigan and Oxford and the publisher ProQuest to create accurately transcribed and encoded texts based on the image sets published by ProQuest via their Early English Books Online (EEBO) database (http://eebo.chadwyck.com). The general aim of EEBO-TCP is to encode one copy (usually the first edition) of every monographic English-language title published between 1473 and 1700 available in EEBO. 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Understanding these processes should make clear that, while the overall quality of TCP data is very good, some errors will remain and some readable characters will be marked as illegible. Users should bear in mind that in all likelihood such instances will never have been looked at by a TCP editor. The texts were encoded and linked to page images in accordance with level 4 of the TEI in Libraries guidelines. Copies of the texts have been issued variously as SGML (TCP schema; ASCII text with mnemonic sdata character entities); displayable XML (TCP schema; characters represented either as UTF-8 Unicode or text strings within braces); or lossless XML (TEI P5, characters represented either as UTF-8 Unicode or TEI g elements). Keying and markup guidelines are available at the Text Creation Partnership web site . eng Oughtred, William, 1575-1660 -- Early works to 1800. Scientific recreations -- Early works to 1800. Sundials -- Early works to 1800. Mathematical instruments -- Early works to 1800. 2005-02 TCP Assigned for keying and markup 2005-02 Aptara Keyed and coded from ProQuest page images 2005-03 Mona Logarbo Sampled and proofread 2005-03 Mona Logarbo Text and markup reviewed and edited 2005-04 pfs Batch review (QC) and XML conversion THE DESCRIPTION and USES Of the General HOROLOGICAL-RING : OR Universal Ring-Dyal . BEING The invention of the late Reverend Mr. W. Oughtred , as it is usually made of a portable pocket size . With a large and correct Table of the Latitudes of the principal Places in every Shire throughout England and Wales , &c. And several ways to find a Meridian-line for the setting a Horizontal Dyal . By HENRY WYNNE , Maker of Mathematical Instruments near the Sugar-loaf in Chancery-lane . London , Printed by A. Godbid and J. Playford , s for the Author , 1682. TO THE READER . I Formerly published half a Sheet on this Subject , and having disposed of all I printed , I found my self necessitated to Print more , to gratify those who bought the Instruments of me , but considering with my self the scantiness of that paper , I took the pains to write a larger which should be more effectual , and as I hope will give better satisfaction . 'T is confest that there is very little new in this ( as in most other Books written now a days ) but what may be found among former Authors . My chiefest care herein hath been to collect and alter so that it might serve my present purpose . As for the Instrument it self , being carefully made and graduated as is here described , I know of none for Portableness , Universality , and exactness , that doth exceed it , I mean with respect to its finding the hour , whereby it becomes absolutely useful for any Gentleman to carry in his pocket , or to rectify his Watch or Pendulum by it , &c. I have endeavoured to be as plain as possible for the sake of young beginners , that the reading of this might Create in some a farther Inclination to the Mathematicks , which I heartily wish may flourish not only as they are my Trade , and consequently it is my Interest to promote them , but because they are of so great and general use and advantage to the Kingdom . H. W. THE DESCRIPTION and USES Of the General HOROLOGICAL RING : OR Universal Ring-Dyal . 1. Of the Name . THis Instrument serveth as a Dyal to find the hour of the day , not in one place only ( as most sorts of Dyals do ) but generally in all Countries whether Northern or Southern ; and therefore it is called the General Horological Ring , or Vniversal Ring-Dyal . 2. The Parts . It consists of these parts , viz. 1. A little Ring and its slider to hang it by . 2. Two circles which fold one within the other . 3. A Diameter a cross in the middle . 4. To this Diameter there is another slider . 3. The Name of each part . The names given to the parts are : 1. The little Ring and its slider is called the Cursor of the Meridian , and is represented ( figl . ) by the letter Z. 2. Of the two Circles , the outermost M M M M , is called the Meridian , and the innermost Ae Ae Ae Ae , is called the Aequinoctial . 3. That which crosseth the middle noted with A A is called the Bridge , or more properly the Axis . 4 The slider within it noted C is called the Cursor of the Bridge or Axis . 4. The Divisions on each part . One side of this Instrument according to fig. I. is thus divided . 1. The Cursor of the Meridian hath but one division or Notch as at O. 2. One half of the Meridian is divided into twice 90 degrees , which are again subdivided into halfs , and these halfs are distinguished from the Degrees , by a shorter line , these Degrees are numbered at every ten , from their middle O both wayes , by 10 , 20 , 30 , &c. to 90 , and in these Degrees are the Latitudes of places reckoned when you would find the hour of the day . 3. The Aequinoctial is divided into 24 hours , and each hour is subdivided into eight parts , viz. halfs , quarters , and half quarters , and some of them have the hours divided into 12 parts , and then every division stands for five Minutes of time , whereof 60 make one hour , these hours are numbred with I. II. III. &c. to twice XII . from the two opposite points in the Meridian where this Circle is fastned . 4. On this side the Axis is divided into months and dayes , every division expressing 2 days , except in June and December , at which time the alteration of the Suns course is almost insensible for several days together , these Months are known on one side the slit by these Letters , I. F. M. A. M. I. Signifying , January , February , March , April , May , June , on the other side by these , I. A. S. O. N. D. for July , August , September , October , November , December . 5. The Cursor of the Axis hath a little hole through it and a line a cross the hole , which line when it is used is to be set to the day of the Month. The other side according to figure II. hath only the Meridian and the Axis divided 1. The Meridian hath a quadrant or 90 Degrees divided on it , whose center is at H. These Degrees are again subdivided into halfs , and this I call the Quadrant of Altitudes , it serving to give the Altitude of the Sun , by the shadow of a pin , or such like wire , which shall be stuck upwright in the Center or hole H. 2. The Axis on that side the slit D is divided into twice 23 ½ and numbered both ways from the middle O by 10 , 20 , &c and this is called the Line of Declination , its use being to give the Declination of the ⊙ , &c. On the other side the slit R , are divided four hours and a half , which are again subdivided , Numbred by IIII / 8 V / 7 VI / 6 VII / 5 VIII / 4 and this line is to shew the Sun 's rising and setting at London , but because it is particular this Line is left out in most Dyals . The Cursor on this side as on the other hath the little hole and a line a cross it . Besides these divisions on each side , on the inside the Aequinoctial , in the middle , is a Line upon which is graduated the 24 hours , and parts agreeable to those on the side described in fig I. Note that the Instrument thus made is general , and will serve wheresoever you are , and therefore most proper for Seamen and those that Travel far . But for such as shall use them about these his Majesties Dominions , it will be sufficient to have but one Quadrant of Latitudes graduated , and no more than 18 hours or thereabouts , viz. from 3 in the morning to 9 at night , and then the Instrument may be afforded so much the cheaper . Vses of the Instrument . THe Principal Uses of this Instrument ( although larger may be made to perform many more ) are as followeth . 1. Knowing the day of the month to find the Suns Declination . 2. To find the Altitude of the Sun at the Meridian and all Hours . 3. By knowing the Suns Declination and Meridian Altitude , to find the Latitude of any place . 4. To find the hour of the day . 5. To find at what time the Sun rises and sets on any day at London or any other place lying under the same Latitude . 6. To find what days and nights throughout the year are equal . USE I. To get the Suns Declination by knowing first the day of the month . Explanation . THe Sun moves not alwayes in the Aequinoctial , but Declines from it sometimes toward the North , and sometimes towards the South , every day , either moving in it or in Circle parallel to it , this diversity of motion is called the Suns Declination , now about the 10 day of March and 13 of September the Suns course is in the Aequinoctial , and then he is said to have no declination , and from the 10 of March to the 13 of September , the sun moves on the North side the Aequinoctial , and it called his Northern Declination , also from the 13 of September to the 10 of March his motion being on the South side , is called Southern Declination . By this variety of the Suns motion , is caused the diversity of Seasons and inequalities of day and night . Note also , that the greatest declination on either side exceeds not 23 Degrees and ½ . Now to find it , The Rule is : Slide the Cursor of the Axis to the day the month , and then turn it on the other side , and the division crossing the same hole will shew the Suns Declination in the Line D. Note that the Axis may be turned without turning the whole Dyal . Example 1. March the 10 , I slide the Cursor to the day of the month , and turning the other side , the division stands at O , which shews the Sun hath no Declination that day , but moves in the Aequinoctial . Example 2. April the 8 , I slide the Cursor to the day of the month , and turning the otherside , the division shews 11 Degrees to be the Suns declination on that day Northward . Example 3. October the 20 , the Cursor being set to the day , on the other side it will shew 14 Deg. for the Suns declination on that day to the Southward . USE II. To find the Suns Altitude on the Meridian and all Hours . Explanation . THe Altitude or height of the Sun is the the number of deg . contained between the middle or Center of the Sun , and the Horizon or Circle which bounds our sight , and the Meridian Altitude is its height every day just at 12 a Clock , the Sun at that time coming to touch the Meridian . To find it , The Rule is : When the Sun shines slide the division on the Cursor of the Meridian to the beginning of the Degrees in fig. I. marked with O , then turn the Dyal and stick a wire or pin upright in the hole H , fig. II. and holding it by the little Ring turn it gently towards the Sun , so that the shadow of the Pin may fall among the Degrees in the Quadrant of the Altitudes , now the Deg. whereon the shadow falleth is the Suns Altitude at that time , but to know the Meridian Altitude you must observe the Suns height just it 12 , now that you may be sure to have it right make several observations just about 12 , and the greatest is the truest , for as the Sun all the morning from its rising grows higher and higher untill it comes to the Meridian where it is highest , so having past the Meridian , all the Afternoon it grows lower and lower until it sets : Wherefore the Suns greatest Altitude on any day is the Meridian Altitude for that day . Examples .     deg . m. March the 10 th . the Suns MeridianAltitude , at Londonwill be found by the foregoing Rule to be 38 28 April the 8 th . 49 28 October the 20 th . 24 28 June the 11 th . 61 58 Now before I proceed further to shew the uses , it will be necessary to explain some terms in Astronomy , such as I shall here make use of , that the young Practitioner may with more ease understand what follows . 1. Degrees and Minutes . And first what is meant by Degrees and Minutes . All Circles according to Astronomy are conceived to be divided into 360 parts , which are called Degrees , every Degree is subdivided into 60 Minutes , every Minute into 60 Seconds , &c. So that one Degree is the three hundred and sixtieth part of a Circle , and one Minute the 60th part of a Degree , &c. Now the whole Circle containing 360 Degrees , the half must contain 180 deg . the Quadrant , or quarter part of a Circle , contains 90 deg . so likewise one deg . containing 60 Minutes , 45 Min. are 3 quarters , 30 Min. are one half , 20 Min. one third part , 15 Min. are one quarter , 12 Min. are one 5 part , 10 Min. are one 6 part , 5 Min. are one 12 part , &c. On the Meridian of the Dyal Fig. I. there are two Quadrants , or twice 90 Deg. graduated , one of which next N P is called the Northern Quadrant of Latitudes , and serves for those places whose Latitudes are on the North side the Aequinoctial , the other is the Southern Quadrant , and serves in South Latitudes . 2. Meridian . It is a great Circle imagined in the Heavens , lying directly North and South , dividing them into two equal parts , the Eastern and Western , passing through both Poles , and the Zenith and Nadir ; to this Circle when the Sun cometh at all times it is noon or midnight , and note that every place hath a several Meridian , except such as ly directly North and South one from the other . 3. Poles . The Poles are two imagined points in the Heavens opposite to each other , one North the other South . 4. Axis . A Right Line imagined to run from one Pole to the other , is called the Axis . 5. Zenith . The Zenith or Vertex is the Point in the Heavens directly over our heads . 6. Nadir . The Nadir is the opposite Point to the Zenith , it being directly under our feet . 7. Equinoctial . The Equinoctial is a great Circle imagined to run directly East and West , it exactly crosseth the Meridian , and lyeth in the middle between the Poles , and divideth the Heavens into two equal parts , the Northern and Southern , when the Sun moves in this Circle , which is twice a year , the days and nights are of an equal length throughout the world . 8. Tropicks . The Tropicks are two lesser Circles dividing the Heavens into two unequal parts , they are Parallel to the Equinoctial , and distant from it 23 deg . 30 min. one on the North side of it the other on the South , these Circles are the utmost bounds of the Suns Declination . 9. Latitude and Eleva●●●● of the Pole. The Latitude of any 〈◊〉 is the Number of Degrees contained between the Zenith of that place and the Aequinoctial , which Degrees are counted in the Meridian , either on the North or South side of the Aequinoctial , according as the place is situated . This Latitude is always equal to the elevation of the Pole , which is the number of Degrees in the Meridian contained between the Pole and the Horizon ; thus those that live under the Aequinoctial are said to have no Latitude , and those that live under the Pole , if any such there be , are in 90 Deg. of Latitude ; hence also it is manifest , that those places which are situate directly East and West one from the other , have one and the same Latitude . 10. Colatitude . The Compliment of the Latitude is the number of degrees contained between the Zenith and the Pole , which is also the same with the distance between the Aequinoctial and the Horizon , or it is so much as the Latitude wants of 90 Deg. for subtract the Latitude from 90 , the remainder is the Colatitude . USE . III. By knowing the Suns Declination and Meridian Altitude to find the Latitude . The Rule . If the Suns declination be North , subtract it from the Meridian Altitude , and the remainder is the Colatitude , but if the Suns Declination be South add it to the Meridian Altitude , and the Sum shall be the Colatitude , which subtracted again from 90 Deg. the remainder is the Latitude . Example 1. March the 10. the Sun hath no Declination , and I find the Meridian Altitude at London , to be 38 deg . 28 min. therefore 38 deg . 28 min. subtracted from 90 deg . the remainder is 51 d. 32 m. the Latitude of London , and by this we see when the Sun is in the Aequinoctial , its Meridian Altitude is equal to the Compliment of the Latitude . Example 2. April the 8. the Suns declination is 11 deg . North and its Meridian Altitude 49 deg . 28 m. now subtract 11 deg . from 49. 28. there rests 38 deg . 28 min. which subtracted again from 90 there rests 51 deg . 32. min. the Latitude required . Example 3. October the 20. the Suns Declination is 14 d. South , and the Meridian Altitude is 24 d. 28 m. then add 14 d. to 24 d. 28 m. the sum is 38 d. 28 m. which subtracted from 90 d. there rests 51 d. 32 m. as before . Example 4. Thus if the declination were 23 d. 30. m. North and the Meridian Altitude 65 d. 10 m. the Latitude would be found to be 48 d. 20. m. Example 5. Let the Declination be 12 d. 15 m. South , and the Meridian Altitude 39 d. 40 m. the Lat. would be 38d . 5 m. Note that these Rules hold good only for finding the Latitudes of such places as ly to the North of the Aequinoctial , for South Lat. the contrary are true , for there if the declination be North , you must add it as you do now when it is South , and if the Suns Declination be South , you must subtract it as you do here when it is North. And least it be thought troublesome to find the Lat. there is added at the end of this Book a Table of the Latitudes of the principal Places in England , Scotland , and Ireland . So that being near any of those places you may make use of the Lat. of that place , for 10 or 20 miles in this case will make a very insensible or no Alteration . USE IV. To find the Hour of the day . NOte that although the Equinoctial fold up within the Meridian to render the instrument the more portable , yet when you would find the hour , the Aequinoctial must be drawn forth according to fig. III. and 't is a little Ray or speck of light that coming through the hole of the Cursor of the Axis falleth upon the line in the middle of the Aequinoctial and sheweth the hour . The Rule . First the Latitude being got by the foregoing Rules , or by the Table at the end of this book , slide the division on the Cursor of the Meridian to it , either in the North or South Quadrants , according as the place is situated . Secondly slide the Cursor of the Axis to the day of the month . Thirdly open the Equinox as far as 't will go , which is just to cross the Meridian , then guess as near as you can at the hour , and turn the Axis towards the hour you guess , that the Sun may the better shine through the hole , and holding the Instrument by the little ring so that it may hang freely , move it gently this way and that , till the Sun shining through the hole you can discern a little Ray or speck of light to fall upon the Aequinoctial within side among the hours and parts , now the point in the middle line whereon the Ray falleth is the true hour . A little practice will make it very easie . Fi. III. representeth the Dyal as it is when you would find the hour , where the Cursor Z is set to the Lat. of London , 51 32. the Cursor of the Axis is set to the day being April the 8 , and the Aequinox is drawn open to cross the Meridian . Now when the Dyal is thus set , and shews the true hour , the Meridian of it hangeth directly North and South , according to that imagined in the Hea ens , the point N P represents the North Pole , S P Represents the South , the Cursor Z Represents the Zenith , and its oposite point N represents the Nadir ; the Axis lyeth according to that of the World passing from Pole to Pole , the points of VI and VI in the Aequinoctially directly East and West , and the middle line within lyeth according to the true Aequinoctial in the Heavens . USE V. To find the Suns Rising and Setting . NOte this line of Rising and setting is particularly for the Latitude of London , or any other place , situated directly East or West from it , but it may indifferently serve the whole Kingdom . Note also that the great figures stand for the Rising and the other for the setting . The Rule . Slide the Cursor of the Axis to the day of the Month , then turn the other side , and the division crossing the hole , shews the Suns Rising , and Setting in the line R. Example 1. I slide the Cursor to March the 10 , and on the other side it shews VI. and 6 , for then the Sun rises at 6 and sets at 6. Example 2. April the 8 , I set the Cursor to the day , and on the other side it shews V. and 7 , which is 5 for the Suns Rising and 7 for its Setting . Example 3. October the 20 , the Cursor being set to the day , on the other side it will shew the Rising to be at a quarter after VII , and the Setting three quarters after 4. Now having found the Suns Rising and Setting , you may likewife from thence find the length of the day and night , for double the time of the Suns Rising , and you have the length of the Night , and double the time of its setting , gives you the length of the Day , as will appear by the three following Examples . Example 1. March the 10 , the Sun rises at 6 and sets at 6 , now twice 6 is twelve for the length both of day and night . Example 2. April the 8 , the Sun rises at 5 and sets at 7 , now twice 5 is 10 the length of the Night , and twice 7 is 14 the length of the day . Example 3. October the 20 , The Sun rises at a quarter after 7 and sets at 3 quarters after 4 , now twice 7 and a quarter is 14 and a half for the length of the Night , and twice 4 and 3 quarters is 9 and an half for the lenghth of the day ; in all which Examples it appears that both the sums of the length of the day and night being added together will make 24 , the hours contained in a natural day . USE VI. To find what days and Nights throughout the year are Equal . The Rule . THe Days on one side the slit are equal to the days on the other . Example . Slide the Cursor to March the 10 , and the day equal to it will be found on the other side Sept. the 13 , So equal to April the 8 is August the 14. And the day equal to the 20 of October is February the Second . Now these days are said to be equal each to the other , in these respects ; 1. in respect of the Suns Declination , it being on both the same . 2. Of the Suns Altitude , for what Altitude the Sun has on any hour on one , the same will be its Altitude on the same hour on the other . 3. The Time of the Suns Rising and Setting is on both the same . 4. They are equal in length both of Day and Night . A Table shewing the Latitudes of most of the Principal Places in every Shire throughout England and Wales . Shires . Places Names . d. m. Anglesey , Beaumaris , 53 27 Holy-head , 53 33 Berkshire , Abington , 51 42 Newbery , 51 25 Reading . 51 28 Bedfordshire , Bedford , 52 09 Dunstable . 51 53 Brecknockshire , Bealt , 52 12 Brecknock . 52 04 Buckinghamshire , Alesbury , 51 45 Buckingham . 52 00 Cambridg●hire , Cambridge , 52 06 Ely. 52 30 Cardiganshire , Aberistwith . 52 35 Cardigan . 52 20 Carmarthenshire , Carmarthen , 51 58 Kidwelley . 51 50 Carnarvonshire , Arberconway , 53 30 Bangor , 53 21 Carnarvon , 53 18 Cheshire , Chester , 53 15 Nantwich . 53 03 Clamorganshire , Cardiff , 51 30 Landaff . 51 34 Cornwall , Fallmouth , 50 20 The Lizard , 50 10 Truro . 50 25 Cumberland , Carlisle , 55 00 Cockermouth . 54 45 Derbyshire , Chesterfield , 53 20 Derby . 53 00 Denbighshire , Denbigh , 53 18 Ruthyn . 53 12 Devonshire , Dartmouth , 50 20 Exeter , 50 41 Plymouth . 50 30 Dorset shire , Dorchester , 50 40 Shaftsbury , 50 58 Weymouth , 50 32 Durham , Aukland , 54 45 Durham . 54 50 Essex , Colchester , 52 00 Harwich . 52 05 Flintshire , St. Asaph , 53 25 Flint . 53 20 Gloucestershire , Gloucester , 51 56 Tewxbury . 52 15 Hampshire , Portsmouth , 50 45 Southampton , 50 54 Winchester . 51 03 Hertfordshire , Hertford , 51 50 Ware. 51 48 Herefordshire , Hereford , 52 12 Lemster . 52 24 Huntingtonshire , Huntington , 52 15 St. Ives . 52 20 Isles of Gernsey , 49 38 Jersey , 49 28 Man , Douglas , 54 25 Wight , Newport . 50 45 Kent , Canterbury , 51 15 Dover , 51 25 Rochester . 51 30 Lancashire , Lancaster . 54 15 Manchester , 53 39 Preston . 53 55 Leicestershire , Harborough , 52 33 Leicester , 52 40 Lincolnshire , Boston , 53 16 Lincoln , 53 16 Stamford . 52 48 Merionethshire , Bala , 52 57 Harlech . 53 00 Middlesex , LONDON , 51 32 Stanes , 51 30 Uxbridge 51 35 Monmouthshire . Chepstow , 51 42 Monmouth . 51 54 Montgomery , 52 40 Montgomeryshire . Montgomery , 52 40 Welchpool . 52 50 Norfolk , Linn , 52 52 Norwich , 52 44 Yarmouth . 52 40 Northamptonshire . Northampton , 52 15 Peterborough . 52 38 Northumberland , Barwick , 55 50 Newcastle . 55 03 Nottinghamshire , Nottingham , 53 00 Workensope . 53 25 Oxfordshire , Banbury , 51 57 Oxford . 51 45 Pembrookshire , St. Davids , 52 00 Pembrook . 51 48 Radnorshire . Prestein , 52 30 Radnor . 52 25 Rutland , Okeham , 52 43 Uppingham . 52 38 Shropshire , Ludlow , 52 28 Shrewsbury . 52 48 Somersetshire , Bath , 51 20 Bristoll . 51 30 Staffordshire . Lichfield , 52 48 Stafford . 52 52 Suffolk , St. Edm. Bury , 52 22 Ipswich . 52 20 Surrey , Guilford . 51 14 Suffex , Chichester , 50 49 Lewis . 50 46 Warwickshire , Coventry , 52 32 Warwick . 52 28 Westmoreland , Apleby , 54 40 Kendal . 54 24 Wiltshire , Marlborough , 51 25 Malmsbury , 51 35 Salisbury , 51 04 Worcestershire , Kidderminster 52 28 Worcester . 52 15 Yorkshire , Bridlington , 54 50 Doncaster , 53 38 Hull , 53 48 Leeds , 53 50 York . 54 00 The Latitudes of the most Eminent places in Scotland . Places names . d. m. Aberdeen . 57 06 St. Andrews . 56 24 Barwick . 55 50 Dunblain . 56 20 Dunbriton . 56 10 Dunbar . 56 03 Dundee . 56 31 Dunfrees . 55 03 Edenburgh . 56 04 Fair-head . 58 43 Glascow . 56 05 Irwin . 55 50 Isles of Orkney . 58 50 Kaithness . 57 48 Larnack . 55 51 Montross . 56 44 Nairn . 57 30 Perth or St. Johns Town , 56 32 Sterlin . 56 15 Withern . 54 57 The Latitudes of the most Eminent places in Ireland . Places names . d. m. Armagh . 54 23 Athloon . 53 21 Bantry . 51 30 Belfast . 54 41 Cashell . 52 24 Casherlash . 52 46 Clare . 52 44 Corke . 51 43 Craven . 54 01 Droughdagh . 53 44 Dublin . 53 20 Dundalk . 54 02 Dungarvan . 51 57 Dunnagall . 54 40 Galloway . 53 12 James Town . 53 53 Kildare . 53 08 Kilkenny . 52 34 Kingsail . 51 30 Knockfergus . 54 50 Limrick . 52 33 Londonderry . 55 04 Longford . 53 42 Slego . 54 17 Waterford . 52 09 Wexford . 52 17 How to place an Horizontal Dyal upon a levell Plane , and to find the Meridian several wayes . 1. PRepare a smooth board or Stone , and place it truly Horizontal or levell , which may be done with such an Instrument as the Artificers call a Plumb-Rule , or otherwise , then find the hour of the day by such an Instrument as is before described , or by some other as true , or having a good Watch go to some Sun-Dyal that you know to go true , and set the Watch by it , afterwards turn the Dial ( which you are to place ) about , untill it shews the same hour with your Instrument or Watch , and there fasten it . 2. Or having prepared your plain as before , near the middle of it set up a wire which shall stand exactly perpendicular or upright , and the Sun Shining clear , observe a little before Noon when the shadow of the wire is at the shortest , and there make a point , and through that point and the center where the wire stood draw a line , upon which place the 12 a Clock line of your Dyal , and fix it . 3. And which is better , near the middle of your Plain choose a point as a center , and thereon describe a Circle of a convenient bigness , and erect a wire at Right Angles to your plain as before , then observe in the forenoon when the shadow of the top of the Pin just toucheth the Circle , and there make a mark , and again in the Afternoon watch when the shadow of the top of the Pin just toucheth the Circle , and there make another Mark , then with a pair of Compasses divide the space between those two Markes into two equal parts , and there make a third Mark , through this last point and the center of the Circle where the wire stood , draw a line and it shall be a true Meridian-line . This last conclusion may be done with more ease , if there be several Circles described one within another on the same center , also then you may make several observations for the doing it with more certainty . 4. The Meridian may be found by the help of a good magnetical needle , well made and fitted to a square Box , if in the useing of it there be an allowance made for the Variation , the use of which is so plain , even to those that have but seen them , that I think it needless here to treat of . I shall set down only two ways more , which will require more knowledge in the Mathematicks than any of the Former , and so conclude . The first is in Dary's Misscellanies , page 22 , thus . 1. Let a piece of Mettal or Wood be made a true Plain , then in some convenient point thereof ( taken as a Center , ) erect a Gnomon of sufficient length at right angles to the plain , this done , fix the Plain truly Horizontal ; secondly if you take the Suns Co-altitude ( that is his distance from the Zenith ) 3 several times in one day , and according to the Stereographick Projection having a line of Tangents by you set off from the center of your plain or foot of the Gnomon , the Tangent of half each arch upon his respective Azimuth or Shadow ( continued if need be ) made by the Gnomon , at that Instant when the Co-altitude is taken , so shall you insert three points upon the plain . Thirdly if you find out the Center to those 3 inserted points , then a right line infinitely extended by this Center found and the foot of the Gnomon or the Center of the plain , is the true Meridian line . 2. The other way , is by the help of the Suns Azimuth , and it is hinted in most Books of Dyalling , thus , 1. Your plain being prepared as before , hold up a string and Plummet , so that the shadow of the string may fall a cross an assigned point in the plain , and in the same line of shadow make another point at a convenient distance from the first , then through these two points draw a right line , secondly at the same instant get the Suns Azimuth or Horizontal distance from the south part of the Meridian , and having a line of chords by you , set off the angle of the Azimuth from the assigned point , either on the west side of the line drawn , if your observation be made in the Morning , or on the east side if your observation be in the Afternoon , and draw the line . Thirdly this last line so drawn shall be in the true Meridian . FINIS . ERRATA . PAge 4 line 9 read subivided and numbered . Page 8 line 15 dele the. Page 13 and 14 read Complement . Page 17 line 14 read Heavens . Page 19 line 14 read length . All sorts of Mathematical Books are sold and Instruments made relating to Arithmetick . Trigonometry . Surveying . Stereometry . Gauging . Astronomy . Geography . Navigation . Opticks . Dyalling . Geometry . Architecture . Fortification . Gunnery . Mechanicks , &c. At reasonable rates : By Henry Wynne , near the Sugar-loaf in Chancery-Lane . 
A44015	----	Stigmai ageōmetrias, agroichias, antipoliteas, amatheias, or, Markes of the absurd geometry, rural language, Scottish church-politicks, and barbarismes of John Wallis professor of geometry and doctor of divinity by Thomas Hobbes. Hobbes, Thomas, 1588-1679. This text is an enriched version of the TCP digital transcription A44015 of text R28097 in the English Short Title Catalog (Wing H2261). Textual changes and metadata enrichments aim at making the text more computationally tractable, easier to read, and suitable for network-based collaborative curation by amateur and professional end users from many walks of life. The text has been tokenized and linguistically annotated with MorphAdorner. The annotation includes standard spellings that support the display of a text in a standardized format that preserves archaic forms ('loveth', 'seekest'). Textual changes aim at restoring the text the author or stationer meant to publish. This text has not been fully proofread Approx. 117 KB of XML-encoded text transcribed from 18 1-bit group-IV TIFF page images. EarlyPrint Project Evanston,IL, Notre Dame, IN, St. Louis, MO 2017 A44015 Wing H2261 ESTC R28097 10409648 ocm 10409648 44957 This keyboarded and encoded edition of the work described above is co-owned by the institutions providing financial support to the Early English Books Online Text Creation Partnership. This Phase I text is available for reuse, according to the terms of Creative Commons 0 1.0 Universal . The text can be copied, modified, distributed and performed, even for commercial purposes, all without asking permission. Early English books online. (EEBO-TCP ; phase 1, no. A44015) Transcribed from: (Early English Books Online ; image set 44957) Images scanned from microfilm: (Early English books, 1641-1700 ; 1386:2) Stigmai ageōmetrias, agroichias, antipoliteas, amatheias, or, Markes of the absurd geometry, rural language, Scottish church-politicks, and barbarismes of John Wallis professor of geometry and doctor of divinity by Thomas Hobbes. Hobbes, Thomas, 1588-1679. [2], 31 p. Printed for Andrew Crooke, London : 1657. Reproduction of original in the Trinity College Library, Cambridge University. eng Wallis, John, 1616-1703. -- Due correction for Mr. Hobbes. Mathematics -- Early works to 1800. A44015 R28097 (Wing H2261). civilwar no Stigmai ageōmetrias, agroichias, antipoliteas, amatheias or Markes of the absurd geometry rural language Scottish church-politicks and barb Hobbes, Thomas 1657 22722 625 795 19 0 0 0 633 F The rate of 633 defects per 10,000 words puts this text in the F category of texts with 100 or more defects per 10,000 words. 2004-09 TCP Assigned for keying and markup 2004-11 Apex CoVantage Keyed and coded from ProQuest page images 2005-01 Emma (Leeson) Huber Sampled and proofread 2005-01 Emma (Leeson) Huber Text and markup reviewed and edited 2005-04 pfs Batch review (QC) and XML conversion ΣΤΙΓΜΑΙ {non-Roman} {non-Roman} {non-Roman} {non-Roman} {non-Roman} OR MARKES Of the Absurd Geometry Rural Language Scottish Church-Politicks And Barbarismes OF JOHN WALLIS Professor of Geometry and Doctor of Divinity . By THOMAS HOBBES of MALMESBURY . LONDON , Printed for Andrew Crocke , at the Green Dragon in Pauls Church-yard , 1657. To the Right honourable Henry Lord Pierrepont Vicount Newarke , Earle of Kingston , and Marquis of Dorchester . My most Noble Lord , I Did not intend to trouble your Lordship twice with this Contention between me and Doctor Wallis . But your Lordship sees how I am constrained to it ; which ( whatsoever reply the Doctor makes ) I shall be constrained to no more . That which I have now said of his Geometry , Manners , Divinity , and G●ammar all together is not much , though enough . As for that which I here have written concerning his Geometry , which you will look for first , is so clear , that not only your Lordship and such as have proceeded far in that Sicence , but also any man else that doth but know how to adde and substract Proportions ( which is taught at the twentieth third Proposition of ●the sixth of Euclide ) may see the Doctor is in the wrong . That which I say of his ill Language and Politicks is yet shorter . The rest ( which concerneth Grammar ) is almost all another mans , but so full of Learning of that kinde , as no man that taketh delight in knowing the proprieties of the Greek and Latine Tongues , will think his time ill bestowed in the reading it . I give the Doctor no more ill words ; but am returned from his manners to my own . Your Lordship may perhaps say my Complement in my Tittle page is somewhat course ; And 't is true . But , my Lord , it is since the writing of the Title page , that I am returned from the Doctors manners to my own ; which are such as I hope you will not be ashamed to own me , my Lord , for one of Your Lordships most humble and obedient Servants Thomas Hobbes . TO DOCTOR WALLIS In answer to his SCHOOLE DISCIPLINE . SIR , WHen ●nprovok'd you addressed unto me in your Elenchus your ha●sh complement with great security , wantonly to shew your wit , I confesse you made me angry , and willing to put you into a better way of considering your own forces , and to move you a little as you had ●moved me ; which I perceive my Lessons to you have in some measure done ; But here you shall see how easily I can bear your reproaches , now they proceed from anger , and how calmly I can argue with you about your Geometry , and other parts of Learning . I shall in the first part confer with you about your Arit●metica Infinitorum , and afterwards compare our manner of Elocution ; then your Politicks ; and last of all your Grammar and Criticks . Your spirall line is condemned by him whose Authority you use to prove me a Plagiary , ( that is , a man that st●aleth other mens inventions , and arrogates them to himself ) whether it be Roberval or not that w●it that paper , I am not certain . But I think I shall be shortly , but whosoever it be , his authority will serve no lesse to shew that your Doctrine of the sp●rall line from the fi●th to the eighteenth proposition of your Arithmetica Infinitorum , is all false ; and that the principal fault therein ( if all faults be not principal in Geometry , when they proceed from ignorance of the Science ) is the same that I objected to you in my Lessons . And for the Author of that paper , when I am certain who it is , it will be then time enough to vindicate my self concerning that name of Plagiary ; And whereas he challenges the invention of your Method delivered in your Arithmetica Infinitorum to have been his before it was yours , I shall ( I think ) by and by say that which shall make him a shamed to own it ; and those that writ those Encomiastick Epistles to you ashamed of the Honour they meant to you : I passe therefore to the ninteenth proposition , which in L●tine is this . Your Geometry . Si proponatur series Quantitatum in duplicata ratione Arithmetice . proportionalium ( sive juxta seriem numerorum Quadraticorum ) continu● crescentium , à puncto vel o inchoa●arum , ( puta ut 0. 1. 4. 9. 16. &c. ) propositum sit , inquirere quam habeat illa rat●onem ad seriem totid●m maximae aequalium . Fiat Investigatio per modum inductioni● ut ( in prop. 1. ) Eritque & sic dein●eps . Ratio proveniens est ubique major quam subtripla seu ⅓ ▪ ; Excessus autem perpetuo decresci● prout ▪ numerus terminorum augetur ( puta 1 / 6 1 / 12 1 / 18 1 / 30 &c. ) aucto nimirum fractionis denominatore sive consequente rationis in singulis locis numero senario ( ut pa●et ) ut ●it rationis provenientis excessus supra subtriplam , Ea quam habet unitas ad sextuplum numeri terminorum posto ; adeoque . That is , if there be propounded a row of quantities in duplicate proportion of the quantities Arithmetically proportional ( or proceeding in the order of the square numbers ) continually increasing ; and beginning at a point or 0 ; let it be propounded to finde what proportion the row hath ; to as many quantities equal to the greatest ; Let it be sought by induction ( as in the first proposition ) The proportion arising is every where greater then subtriple , or ⅓ , And the excesse perpetually decreaseth as the number of termes is augmented , as here , 1 / 6 1 / 12 1 / 18 1 / 24 1 / 30 &c. the denominator of the fraction being in every place augmented by the number six ( as is manifest ) so that the excesse of the rising proportion above subtriple is the same which unity hath to six times the number of termes after 0 ; and so . Sir , In these your Characters I understand by the crosse + that the quantities on each side of it are to be added together and make one Aggregate ; and I understand by the two parallel lines = that the quantities betwen which they are placed are one to another equall ; This is your meaning , or you should have told us what you meant else : I understand also , that in the first row 0 + 1 is equal to 1 , and 1 + 1 equal to 2 ; And that in the second row 0 + 1 + 4 is equal to 5 ; and 4 + 4 + 4 equal to 12 ; But ( which you are too apt to grant ) I understand your Symboles no further ; but must confer with your self about the rest . And first I ask you ( because fractions are commonly written in that manner ) whether in the uppermost row ( which is ) 0 / 1 be a fraction , 1 / 1 be a fraction , ½ be a fraction , that is to say , a part of an unite , and if you will ( for the cyphers sake ) whether 0 / 1 , be an infinitely little part of 1 ; and whether 1 / 1 , or 1 divided by 1 signifie an unity ; if that be your meaning , then the fractio● 0 / 1 added to the fraction 1 / 1 is equal to the fraction ½ : But the fraction 0 / 1 is equal to 0 ; therefore the fraction 0 / 1 + 1 / 1 is equal to the fraction 1 / 1 ; and 1 / 1 equal to ½ which you will ▪ confesse to be an absurd conclusion , and cannot own that meaning . I ask you therefore again if by 0 / 1 you mean the proportion of 0 to 1 ; and consequently by 1 / 1 the proportion of 1 to 1 , and by ½ the proportion of 1 to 2 : if so , then it will follow , that if the proportions of 0 to 1 and of 1 to 1 ●e compounded by addition , the proportion arising will be the proportion of 1 to 2. But the proportion of 0 to 1 is infinitely little , that is , none . Therefore the proportion arising by composition will be that of 1 , to 1 , and equall ( because of the symbol = ) to the proportion of 1 to 2 and so 1 = 2 : This also is so absurd that I dare say that you will not own it . There may be another meaning yet : perhaps you mean that the uppermost quantitie 0 + 1 is equal to the uppermost q●antity 1 ; and the lowermost quantity 1 + 1 equal to the lowermost quantity 2 : Which is true : but how then in this equation ½ = ⅓ + ⅙ is the uppermost quantity 1 equal to the uppermost quantity 1 + 1 ; or the lower most quantity 〈◊〉 equal to the lowermost quantity 3 + 6 ? Therefore neither can this be your meaning ; unlesse you make your symbols more significant , you must not blame me for want of understanding them . Let us now try what better successe we s●all have where the places are three , as here : If your Symbols be fractions , the compound of them by addition is 5 / 4 ; For 0 , ¼ and 4 / 4 make 5 / 4 ; and consequently ( because of the Symbole = ) 5 / 4 equal to 5 / 12 which is not to be allowed , and therefore that was not your meaning . If you meant that the proportions of 0 to 4 & of 1 to 4 & of 4 to 4 compounded is equal to the proportion of 5. to 12. you will fall again into no less an inconvenience . For the proportion arising out of that Composition will be the proportion of 1. to 4. For the proportion of 0. to 4. is infinitely little . Then to compound the other two , set them in this order 1. 4. 4. and you have a proportion compounded of 1. to 4. and of 4. to 4. namely , the proportion of the first to the last , which is of 1. to 4. which must be equal ( by this your meaning ) to the proportion of 5. to 12. and consequently as 5. to 12. so is 1. to 4. which you must not own . Lastly , if you mean that the uppermost quantities to the uppermost , and the lowermost to the lowermost in the first Equation are equal , t is granted , but then again in the second Equation it is false . It concerns your fame in the Mathematicks to look about how to justifie these Equations which are the premises to your conclusion following , namely , that the proportion arising is every where greater then subtriple , or a third ; and that the excess ( that is , the excess above subtriple ) perpetually decreaseth as the number of terms is augmented , as here 1 / 6 1 / 12 1 / 18 1 / 24 1 / 30 &c. which I will shew you plainly is false . But first I wonder why you were so angry with me for saying you made proportion to consist in the Quotient , as to tell me it was abominably false , and to justifie it , cite your own words Penes Quotientem , do not you say here , the proportion is every where greater then subtriple , or ⅓ ? And is not ⅓ the quotient of 1. divided by 3 ? You cannot say in this place that Penes is understood ; for if it were expressed you would ●ot be able to proceed . But I return to your conclusion , that the ex●ess of the proportion of the increasing quantities above the third part of so many t●mes the greatest , decreaseth as 1 / 6 1 / 12 1 / 18 1 / 24 1 / 30 &c. For by this accompt in this row where the quantitie above exceeds the third part of the quantities below by ⅓ ' you make ⅓ equal to ⅙ ' which you do not mean . It may be said your meaning is that the proportion of 〈◊〉 . to the subtriple of 2. which is ⅔ exceedeth what ? I cannot imagine what , nor proceed further where the ter●s be but two . Let us therefore take the second row , that is , The summe above is 5. the summe below is 12. the third part whereof is 4. if you mean , that the proportion of 5. to 4. exceeds the proportion of 4 ▪ to 12. ( which is subtriple ) by 1 / 12 ' you are out again . For 5. exceeds 4. by unitie , which is 12 / 12 I do not think you will own such an equation as . Therefore I believe you mean ( and your next proposition assures me of it ) that the proportion of 5. to 4. exceeds subtriple proportion by the proportion of 1. to 12. if you do so , you are yet deceived . For if the proportion of 5. to 4. exceeds subtriple proportion by the proportion of 1. to 12. then subtriple proportion , that is , of 4. to 12. added to the proportion of 1. to 12. must make the proportion of 5. to 4. But if you look on these quantities 4. 12. 144. you will ●ee and must not dissemble that the proportion of 4. to 12. is subtriple , and the proportion of 12. to 144. is the same with that of 1 ▪ to 12. Therefore by your assertion it must be as 5. to 4. so 4. to 144. which you must not own . And yet this is manifestly your meaning as apppeareth in th●se words ; u● 〈◊〉 rationis pr●venien●is excessus supra subtriplam ●a quam ●abet unitas ad sextuplum numeri terminorum post 0 , adeoque , which cannot be rendered in English , nor need to be . For you express your self in the 20th . pr●position very clearly ; I noted it only that you may be more merciful hereafter to the stumblings of a hasty Pen . For excessus ea quam does not well , nor is to be well excused by subauditur ratio . Your 20th . proposition is this ▪ Siprop●natur serie● quantitatum in duplicata ratione Arithmeticè proportionalium ( sive jux●a seriem Numerorum Quadraticorum ) continuè crescentium , a puncto vel O in choatarum , ratio quam habet illa ad seriem to●idem maximae aequalium subtriplam superabit ; eritque excessus ea ratio qua● habet unitas ad sex●uplum n●meri terminorum post 0 , sive quam habet radix Quadraticae termini primi post 0 ad sextuplum radicis Quadraticae termini maximi . That is , if there be propounded a row of quantities in duplicate proportion of Arithmetically-proportionals ( or according to the row of square numbers ) continually increasing , and beginnin gwith a point or 0. The proportion of that row to a row of so many equals to the greatest , shall be greater then subtriple proportion , and the excesse s●all be that proportion which unity hath to the sextuple of the number of termes after 0 , or the same which the square roo● of the first number after 0 , hath to the sextuple of the square root of the greatest . For proof whereof you have no more here , then pa●e● ex praeceden●●bus ; and no more before but adeoque . You do not we●l to passe over such curious propositions so slightly ; none of the Antients did so ; nor , that I remember , any man before your self . The proposition is false , as you shall presently see . Take for example any one of your rows ; as . By this proportion of yours 1 + 4 which makes 5 is to 12 in more then subtriple proportion ; by the proportion of 1 to the sextuple of 2 which is 12. Put in order these three quantities 5. 4. 12. And you must see that the proportion of 5 to 12 is greater then the proportion of 4 to 12 , that is , subtriple proportion , by the proprtion of 5 to 4. But by your account the proportion of 5 to 4 is greater then that of 4 to 12 by the proportion of 1 to 12. Therefore as 5 to 4 so is 1 to 1● . which is a very strang Parodox . After this you bring in this Consectary . Cum autem cresente numero terminorum excessus ille supra rationem sub●riplam conninuò minuatur , ut tandem quovis assignabili minor eva●●t ( un p●tet ) si in in●ini●um producatur prorsus evaniturus est . Adeoque . That is , seeing as the number of tearms encreaseth , that excesse above subtriple proportion continually decre●seth , so as at length it be●omes lesse then any assignable ( as is manifes● ) if it be produced infinitely , it shall utterly vanish , and so . And so what ? Sir , This consequence of yours is false . For two quantities being given , and the excesse of the greater above the lesse , that excesse may continually be decreased , and ye● never quite vanish . Suppose any two unequal quantities differing by more then an unite , as 3 and 6 , the excesse 3 , let three be diminished , fi●st by an unite , and the ezcesse will be 2 and the quantities will be 3 and 5. 5 is greater then 4 ; the excesse 1. Again , let 1 be diminished and made ½ . the excesse ½ and the quantities 3 and 4½ . 4½ is yet greater then 4. Again diminish the excesse to ¼ , the quantities will be 3 and 4¼ yet still 4¼ is greater then 4. In the same manner you may proceed to ⅛ , 1 / 16 , 1 / 32 , &c. Infinitely ; and yet you shall never come within an unite ( though your unite stand for 100 mile ) of the lesser quantity propounded 3 , if that 3 stand for 300 Mile . The exce●ses above subtriple proportion do not d●creas● in the manner you say it does , but in the manner which I shall now shew you . In this first row a third of the quantities below is ⅔ . set in order these thre3 quantities 1. 2 / 9. ⅔ . The first is 1 equal to the sum above , the last is ⅔ equal to the subtriple of the sum below . The middlemost is 2 / 9 subtriple to the last quantity ⅔ . The excesse of the proportion of 1 to ⅔ above the subtriple proportion of2 / 9 to ⅔ is the proportion of 1 to 2 / 9 , that is , of 9 to 2 , that is of 18 to 4. Secondly in the second row which is a third of the sum below is 4 the sum above is 5. Set in order these quantities 15. 4. 12. There the proportion of 15 to 12 is the proportion of 5 to 4. The proportion of 4 to 12 is subtriple ; the excesse is the proportion of 15 to 4 , which is lesse then the proportion of 18 to 4. as it ought to be ; b●t not lesse by the proportion of ⅙ to 1 / 12 , as you would have it . Thirdly , in the third row , which is . A third of the sum below is 12 , the su● above is 14. Set in order these quantities 4 〈◊〉 . 4. 12. There the proportion of 42 to 12. is the same , with that of 14 to 4. And the p●oportion of 4 to 12 subtriple , lesse then the former excesse of 15 to 4. And so it goes on deceasing all the way in this manner , 18 to 4. 15 to 4. 14 to 4 &c. which differs very much from your 1 to 6. 1 to 12. 1 to 18 &c. and the cause of your mistake is this ; you call the twelfth part of twelve 1 / 12 , and the eighteenth part of thirty ●ix , you call 1 / 18 , and so of the rest . But what need all those equations in Symbols , to shew that the proportion decreases ; is there any man can doubt , but th●t the propartion of 1 to 2 is greater then that of 5 to 12 , or that of 5 to 12 greater then that of 14 to 36 , and so continually forwards ; or could you have fallen into this errour , unlesse you had taken , as you have done in very many places of your Elenchus , the Fra ▪ ctions ⅙ and 1 / 12 , &c. which are the quotients of 1 divided by 6 and 12. for the very proportions of 1 to 6 , and 1 to 12. But notwithstanding the excesse of the proportions of the encreasing quanti●ies , to subtriple proportion decrease stil , as the number of tearms increaseth , and that what proportions soever I shall assigne , the decrement will in time ( in time I say without proceeding in in●initum ) produce a lesse , yet it does not follow that the row of the increasing qu●ntities shall ever be equall to the third part of the row of so many equalls to the last or greatest . For it is not , I hope , a Paradox to you , that in two rows of quantities the proportion of the excesses may decrease , and yet the excesses them selves encrease , and do perpetually . For in the second and third rows , which are and 5 exceeds the third part of 12 , by a quarter of the square of fo●r , and 14 exceedes the third part of 36 by 2 quarters of the square of 4 , and proceeding on , the sum of the increasing quantities where the termes are 5 ( which sum is 30 ) exceedeth the third part of those below ( those below are 80 and their third part 26⅔ by 3 quarters and ½ a quarter of the square of 4. and when the tearms are 6 the quantities above will exceed the third part of them below by 5 quarte●s of the square of four . Would you have ●en beleeve , that the forther you go , the excesse of the increasing quantities above the third p●r● of those below shall be so much the lesse ? And yet the proportions of those above , to the the thirds of those below , shall decrease eternally ; and therefore your 〈◊〉 proposit●on is ●alse , namely this . Siproponatur series Infinita quantitatum in duplic●t● ratione Arithmeticè proportionalium ( sive juxta seriem numerorum quadraticorum ) continué crescentium 〈◊〉 puncto ●ive O inchoata●um ; eri● illa ad seriem to●idem maximae aequalium , ut 1 ad 3. That i● , if an infinite row of quantities be propounded in duplicate proportion of Arithmetically proportio●alls ( or ●ccording to the row of quadratick numbers ) continually increasing and beginning from a point or 0 ; that row shall be to the row of as many equalls to the greatest , as 1 to 3. This is false , ut patet ex praecedentibus , and consequently all that yo● say in proof ●f the proportion of your Parabola to a parallelogram , or of the spiral ( he true spiral ) to a circle is in vain . But your spiral puts me in mind of what you h●ve underwritten to the diagramme of your prop 5. The spirall in both f●gures was to be continued whole to the middle , but by the carelessnesse of the Graver it is in one figure manca , in the other intercis● . T●uly Sir , you will hardly make your Reader beleeve that a Graver could ●ommit those faults without the help of your own Coppy , nor that it had been in your coppy , if you had known how to describe a spiral line then as now . This I had not said , though truth , but that you are pleased to say , though not truth , that I attributed to the Printer some f●ults of mine ; I come now to the thirty ninth proposition which is this Si proponatur series quantitatum in triplicata ratione Arithmeticè proportionalium ( sive juxta seriem numerorum cubicorum ) continuè cresentium a puncto sive O inc●oata●um , ( puta ut O. 1. 8. 27. &c. ) propositum sit inquirere quam habeat series illa rationem ad seriem totidem maximae ●qualium . Fiat investigatio per modum Inductionis ( ut in prop. 1 & prop 19. ) Eritque Et sic d●inceps Ratio proveniens est ubique major quam subquadrupla , sive ¼ . Excessus autem perp●●uo decrescit , p●o ut numerus terminor●m augetur , puta ¼ . ⅛ . 1 / 12. 1 / 16 &c. Aucto nimirum fractionis denominator●●ive cansequente rationis in ●ingulis locis numero quaternatio ( ut patet ) ut sit rati●nis provenie●…●xcess●… supra subquadruplam e● quam habet unitas ad Quadruplum numeri terminorum post 0. Adeoque . That is , if a row of quantities be propounded in triplicate proportion of Arithmetically proportionalls ( or according the row of cubiqūe numbers ) continually encreasing , and beginning from a point or o ( as 0. 1. 8. 27. 64. &c. ) let it be propounded to enquire what proportion that row hath to a row of as many equalls to the greatest● Be it sought by way of induction , ( as in prop. 1. 19 ) . The proportion arising is every where greater then subquadruple , or ¼ , and the excesse perpetually decreaseth as the number of tearms increaseth as ¼ , ⅛ , 1 / 12 , 1 / 16 , 1 / 20 , &c. The denominator of the fraction , or consequent of the proportion being in every place augmented by the number 4 ( as is manifest ) so that the excesse of the arising proportion above subquadruple is the same with that which an unite hath to the quadruple of the number of the tearms after 0. And so . Here are just the same faults which are in prop. 19. For if 0 / 1 be a fraction , and 1 / 1 be a fraction and ½ be another fraction , then this equation is false . For this fraction 0 / 1 is equal , to 0 ▪ and therefore we have 1 / 1 = ½ , that is the whole equal to half . But perhaps you do not mean them fractions , but proportions ; and consequently that the proportion of 0 to 1 , and of 1 to 1 compounded by addition ( I say by addition ; not that I , but that you think there is a composition of proportions by multiplication , which I shall shew you anon is f●lse ) must be equal to the proportion of 1 to 2. which cannot be , For the proportion of 0 to 1 is infinitely little , that is , none at all ; and consequently the proportion of 1 to 1 is equall to the proportion of 1 to 2 ▪ which is again absurd , There is no doubt , but the whole number of 0 + 1 is equal to 1 , and the whole number 1 + 1 equal to 2. But reckoning them as you do , not for whole numbers , but for fractions , or proportions , the equations are false . Again your second equation 2 / 4 = ¼ + ¼ though meant of fractions , that is of quotients it be true , and serve nothing to your purpose , yet if it be meant of proportions , it is false . For the proportion of 1 to 4 and of 1 to 4 being compounded are equal to the proportion of 1 to 16 , and so you make the proportion of 2 to 4 equal to the proportion of 1 to 16 , where as it is but subquaduplicate , as you call it , or the quarter of it as I call it . And in the same manner you may demonstrate to yourself , the same fault in all the other rows of how many tearms soever they consist , Therefore you may give for lost this 39. prop. as well as all the other 38 that went before . As for the conclusion of it , which is , that the excesse of the arising proportion &c. They are the words of your 40 ▪ proposition , where you ▪ e●presse your self better , and make your errour more easie to be detected . The proposition is this . Si proponatur series quantitatum in triplicata ratione Arithmeticè proportionalium ( sive juxta seriem numerorum cubicorum ) continue crescentium à puncto velo inchoa●arum , ratio quam habet illa ad seriem totidem maximae aqualiùm subquadruplam superabit ; eritque excessus ea ratio quam habet unitas ad quadruplum numeri terminorum post 0 ; sive quam habet radix cubica termini primi post 0 ad quadruplum radicis cubicae te●mini maximi . Pat●● ex praecedente . Quum autem crescente numero terminorum excessus ille supra rationem subquadruplam i●a continuo minuatur , ut tandem qu●libet assignabili minor evadat , ( u● pa●et ) si in Infinitum procedatur , prorsus evaniturus est . Ade●que . Paret ex prop. praecedent . That is ; If a row of quantlties be propounded in triplicate proportion of Arithmetically proportionalls ( or according to the row of cubick numbers ) continually encreasing ; and beginning at a point or 0 ; The proportion which that row ●ath to a row of as many equals to the greatest is greater then subquadruple proportion ; and the excesse is that proportion which one unite hath to the Quadruple of the number of termes after 0. Or which the cubick root of the first term after 0 hath to the quadruple of the root of the greatest tearm . It is manifest by the precedent propositions . And seeing , the number of tearmes increasing , that excesse above quadruple proportion doth so continually decrease , as that at length it becomes lesse then any proportion that can be assigned ( as is manifest ) if the proceeding be infinite , it shall quite vanish ; And so . This conclusion was annexed to the end of your 39 proposition ; as there prooved . What cause you had to make a new proposition of it , without other proof then pate● ex praecedente , I cannot imagine . But howsoever the proposition is false . For example , set forth any of your rowes , as this of fower termes . The row above is 36 ▪ the fourth part of the row below is 27. The quadruple of the number of termes after 0 is 12 ▪ then by your accompt , the proportion of 36 to 108 is greater then subquadruple proportion by the proportion of 1 to 12. For trial whereof set in order these three quantities 36. 27. 108. The proportion of 36 ( the uppermost row ) to 108 ( the lowermost row ) is compounded by addition of the proportion● 36 to 27 , and 27 to 108. And the proportion of 36 to 108 exceedeth the proportion of 27 to 108 by the proportion of 36 to 27. But the proportion of 27 to 108 is subquadruple proportion . Therefore the proportion of 36 to 108 exceedeth subquadruple proportion , by the proportion of 36 to 27. And by your account by the proportion of 1 to 12 ; and consequently as 36 to 27 so is 1 to 12. Did you think such demonstrations as these , should alwayes passe ? Then for your inference from the decrease of the proportions of the excesse to the vanishing of the excesse it self , I have already shewed it to be false ; and by consequence that your next Proposition , namely , the 40 is also false . The proposition is this . Si pr●ponatur series infinita quantitatum in triplicata ratione Arithmeticé proportionalium ( sive juxta seriem numerorum cubicorum ) continuè crescentium à puncto sive 0 inchoatarum , erit illa ad seriem totidem maximae aequalium , ut 1 ad 4. pa●et ex praecedente . That is , If there be propounded an infinite row of quantities in triplicate proportion of Arithmetically proportionalls ( or according to the row of cubick numbers ) continually increasing , and beginning at a point or 0 ; it shall be to the row of as many equalls to the greatest as 1 to 4. Manifest out of the precedent proposition . Even as manifest as that 36. 27. 1. 12. Are proportionalls : seeing therefore your Doctrine of the spiral lines and spaces is given by your self for lost , and a vaine attempt , your first 41 propositions are undemonstrated , and the grounds of your demonstrations all false . The cause whereof is partly your taking quotient for proportion , and a point for 0 , as you do in the 1. 16 and 40. propositions and in other places where you say beginning at a point or 0 ; though now you denie you ever said either . There be very manay places in your Elenchus , where you say both ; and have no excuse for it , but that in one of the places , you say the proportion is p●nes quotientem , which is to the same or no sense . Your 42 proportion is grounded on the 40 ; and therefore though true , and demonstrated by others , is not demonstrated by you . Your 43 is this . Pari methodo invenietur ratio seriei infinitae quantitatum Arithmeticè proportionalium in ratione quadruplicata , quintuplicata , sextuplicata , etc Arithmeticè proportionalium à puncto seu 0 in choatarum , ad seriem totidem maximae aequalium . Nempe in quadruplicata ●●it , ut 1 ad 5 ; in quintuplicata ut 1 ad 6 ; in sextuplicata ut 1 ad 7. Et sic d●inceps . That is , By the same method will be found the proportion of an infinite row of Arithmetically proportionalls in proportion quadruplicate , quintuplicate , sextuplicate &c. of Arithmetically proportionalls beginning at a point or 0 , to the row of as many equalls to the greatest ; Namely , in quadruplicate , it shall be as 1 to 5 , in quintuplicate as 1 to 6 ; in sextuplicat● as 1 to 7 ; and so forth . But by the same method that I have demonstrated that the 19. 20. 21. 39. 40. 41. Propositions are false , any man else that will examine the 43 may finde it false also . And because all the rest of the propositions of your Arithmetica infinitorum depend on these , they may safely conclude that there is nothing demonstrated in all that Book , though it consist of 194 propositions . The proportions of your Parabolo●ides to their Parallellogrammes are true , but the demonstrations false , and infer the contrary . Nor were they ever demonstrated ( at least the demonstrations are not extant ) but by me ; nor can they be demonstrated , but upon the same grounds concerning the nature of proportion , which I have clearly laid , and you not understood . For if you had , you could never have fallen into so grosse an errour as is this your Book of Arithmetica Infinitorum , or that of the Angle of Contact . You may see by this that your symbolick Method is not onely , not at all inventive of new Theormes , but also dangerous in expressing the old . If the best Masters of Symbolicks think for all this you are in the right , let them declare it . I know how far the Analysis by the powers of the lines extendeth , as well as the best of your half-learn't Epistlers , that approve so easily of such An●logismes as those . 5. 4. 1 12 , and 36. 27 ▪ 1. 12 ▪ &c. It is well for you that they who have the disposing of the professors places take not upon them to be Judges of Geometry , For if they did , seeing you confesse you have read these Doctrines in your School● you had been in danger of being put out of your place . When the Author of the paper wherein I am called Plagiary , and wherein the honor is taken from you of being the first inventor of these fine Theoremes , shall read this that I have here written , he will look to get no credit by it ; especially if it be Roberval , which me thinkes it should not be . For he understands what proportion is , better then to make 5 to 4 the same with 1 to 12. Or to make again , the proportion of 36 to 27 the same with that of 1 to 12 ; and innumerable disproportionalites that may be inferred from the grounds you go on . But if it be Robeval indeed , that snatches this invention from you , when he shall see this burning coal hanging at it , he will let it fall again , for fear of spoiling his reputation . But what shall I answer to the Authority of the three great Mathemiticians that sent you those Encomiastick letters . For the first , whom you say I use to praise , I shall take better ●eed hereafter of praising any man for his Learning whilst he is young , further then that he is in a good way . But it seemes he was in too ready a way of thinking very well of himself , as you do of your self . For the muddinesse of my brain , I must confesse it . But ●r , Ought not you to confesse the same of yours ? No , men of your tenets use not to do so . He wonders , ( say you ) you thought it worth the while to ●o●●l your fingers about such a piece . T is well ; Every man abounds in his own sense . If you and I were to be compared by the complements that are given us in p●ivate letters , both you and your Complementors would be out of Countenance ; which ●omplements , besides that which has been printed and published in the Commendations of my writings , if it were put together would make a greater volume then either of your Libels . And truely Sir , I had never answered your Elenchus as proceeding from Dr. Walli● ▪ if I had not considered you also as the Minister to execute the malice of that sort of people that are offended with my L●viathan . As for the judgement of that Publick Professor that makes himself a witnesse of the goodnesse of your Geometry , a man may easily see by the letter it selfe that he is a dun● . And for the English person of quality whom I know not , I can say no more yet , then I can say of all three , that he is so ill a Geometrician , as not to detect those grosse Paralogismes as in●er that 5 to 4 and 1 to 12 are the same proportion . He came into the cry of those whom your title had deceived . And now I shall let you see that the composition of proportion by multiplication as it is in the 5 d●f . of the 6 Element , is but another way of adding proportions one to another . Let the proportions be of 2 to 3 , and of 4 to 5. Multiply 2 into 4 and 3 into 5 the proportion arising is of 8 to 15. Put in order these three quantities 8. 12. 15. The proportion therefore of 8 to 15 compounded of the proportions of 8 to 12 ( that is , of 2 to 3 ) and of 12 to 15. that is , of 4 to 5 by addition . Again , let the proportion be of 2 to 3 and & 4 to 5 multiply 2 into 5 and 3 into 4 the proportion arising is of 10 to 12. Put in order these 3 numbers 10 ▪ 8. 12. The proportion 10 to 12 is compounded of the proportions of 10 to 8 that is of 5 to 4 , and of 8 to 12 , that is of 2 to 3 by addition ▪ I wonder you know not this . I finde not any more clamour against me for saying the proportion of 1 to 2 is doublé to that of 1 to 4. Your Book you speak of concerning proportion against Maybonius is like to be very useful when neither of you both do understand what proportion is . You take e●ceptions at that I say , that Eucilde has but one word for double and duplicate ; which neverthelesse was said very truely , and that word is sometimes {non-Roman} {non-Roman} {non-Roman} {non-Roman} {non-Roman} and sometimes {non-Roman} {non-Roman} {non-Roman} {non-Roman} {non-Roman} ▪ And you think you come of handsomly with asking me whether {non-Roman} {non-Roman} {non-Roman} {non-Roman} {non-Roman} and {non-Roman} {non-Roman} {non-Roman} {non-Roman} {non-Roman} be one word . Nor are you now of the minde you were , that a point is not quantity unconsidered ; but that in an infinite series it may be safely neglected . What is neglected but unconsidered ? Nor do you any more stand to it , that the quotient is the proportion . And yet were these the main grounds of your Elenchus . But you will say perhaps I do not answer to the defence you have now made in this your School Discipline , T is true . But 't is not because you answer never a word to my former objection against these prop. 19 ▪ 39. But because you do so shift and wriggle and throw out ink , that I cannot perceive which way you go ; nor need I , especially in your vindication of your Arithmetica Infinitorum . Onely I must take notice that in the end of it , you have these words , well , Arithmetica Infinitorum is come off clear . You see the contrary . For sprawling is no defence . It is enough to me , that I have clearly demonstrated both , before sufficienly , and now again abundantly , that your Book of Arithmetica In●●ni●orum is all nought from the beginning to the end , and that thereby I have effected that your Authority shall never hereafter be taken for a prejudice . And therefore they that have a desire to know the truth in the questions between us , will henceforth , if they be wise , examine my Geometry by attentive reading me in my own writings , and then examine , whether this writing of yours confute or enervate mine . There is in my 5 lesson a proposition , with a diagramme to it to make good , ( I dare say , ) at least against you , my 20. Chapter concerning the dimension of a Circle . If that demonstration be not shewn to be false , your objections to that Chapter ( though by me rejected ) come to nothing . I wonder why you passe it over in silence . But you are not , you say , bound to answer it . True , nor yet to defend what your have written against me . Before I give over the examination of your Geometry , I must tell you that your words ( pag. 〈◊〉 of this your Schoole Discipline ) again the first Corollary are untrue . Your words are these , you aff●…n that the proportion of the parabola A B I to the parabola A F K is triplicate to the proportion of the time A B to A F , as it is in the English . This is not so . Let the Reader turn to the place and judge . And going on you say , or of the imp●… B I to F K , as it is in the Latine . Nay , as it is in the English , and the other in the Latine . T is but your mistake ; but a mistake is not easily excused in a false accusation . Your exception to my saying , That the differences of two quantities is their proportion ( when they differ , as the no difference , when they be equal ) might have been put in amongst other marks of your not sufficiently understanding the Latine tongue . Differre and Differentia differ no more then vivere and vita , which is nothing at all , but as the other words require that go with them , which other words you do not much use to consider . But differre and the quantity by which they differ , are quite of another kinde . Di●ferre ( {non-Roman} {non-Roman} {non-Roman} {non-Roman} {non-Roman} ) differing , exceeding , is not quantity , but relation . But the quantity by which they differ is alwayes a certain and determined quantity , yet the word differrentia serves for both , and is to be understood by the coherence with that which went before . But I had said before , and expressly to prevent cavil , that relation is nothing but a comparison , and that proportion is nothing but relation of quantities and so defined them , and therefore ▪ I did there use the word differentia for differing , and not for the quantity which was le●t by substraction . For a quantity is not a differing . This I thought the intelligent Reader would of himself understand without putting me , instead of differentia to use ( as ●ome do , and I shall never do ) the mongrell word {non-Roman} {non-Roman} {non-Roman} {non-Roman} {non-Roman} differre . And whereas in one onely place for differre ternario I have writ ternarius , If you had understood what was clearly exprest before , you● might have been sure , it was not my meaning , and therefore the excepting against it , was either want of understanding , or want of Candou● ; chuse which you will . You do not yet clear your Doctrine of Condensation and Rarefa●lion . But I beleev● you will be degrees become satisfied that they who say the same Numerical Body may ●e sometimes greater , sometimes lesse , speak absurdly , and that Condensation and Rarefaction here , and definitive and ci●cumscriptive and some other of your distinctions elsewhere are but ●nares , such as School-Divines have invented — {non-Roman} {non-Roman} {non-Roman} {non-Roman} {non-Roman} , {non-Roman} {non-Roman} {non-Roman} {non-Roman} {non-Roman} to intangle shallow wits . And that that distinction which you bring here , that it is of the same quantity , while it is in the same place , but it may be of a different quantity , when it goes out of its place , ( as if the place added to , or took any quantity from the body placed ) is nothing but mee● words . T is true that the Body which swells changeth place ; but it is not by becoming it self a greater body , but by admixtion of Aire or other body ; as when water riseth up in boyling , it taketh in some parts of Aire . But seeing the first place of the body is to the body equal , and the second place equal to the same body , the places must also be equal to one another , and consequently the dimensions of the body remain equal in both places . Sir , When I said that such Doctrine was taught in the Universities , I did not speak against the Universities , but against such as you . I have done with your Geometry , which is one {non-Roman} {non-Roman} {non-Roman} {non-Roman} {non-Roman} . As for your El●quence let the Reader judge , whether your● or mine be the more muddy , though I in plain s●olding should have out done you , yet I have this excuse which you have not , that I did but answer your challenge at that weapon which you thou●ht fit to choose . The Catalogue of the hard language which you put in at the 3. and 4. pages of your School-discipline , I acknowledge to be mine , and would have been content you had put in all . The titles you say I give you of Fooles , Beasts , and Asses I do not give you , but drive back upon you ; which is no more then not to own them ; ●or the rest of the Catalogue I like it so well as you could not have pleased me better , then by setting those passages together to make them more conspicuous , that 's all the de●ence I will make to your accusations of that kinde . And now I would have you to consider whether you will make the like de●ence against the ●aults that I shall ●inde in the Language of your School-Discipline . I observe first the facetiousnesse of your Title page . Due Correction for Mr. H●●bs , or Schoole Discipline , for not saying his lessons right , what a quibble is this upon the word Lesson ; Besides , you know it has taken winde ; for you vented it amongst your ac quaintance at Oxford then when my Lessons were but upon the Presse . Do you think if you had pretermitted that peece of wit , the opinion of your judgement would have been ●re the lesse . But you were ●ot content with this but must make this Metaphor from the rod , to take up a considerable part of your book ; in which there is scarce any thing that your self can think wittily said besides it . Consider also these words of yours ; It is to be hoped that in time you may come to learn the Language , for you be come to great A already . And presently after , were I great A before I would be willing to be so used , I should wish my self little a , a hundred times . Sir , you are a Doctor of Divinity , and a professor of Geometry , but do not deceive your self ; this does not passe for witt in these parts ; no nor generally at Oxford . I have acquaintance there that will blush at the reading it . Again , in another place you have these words , Then you catechize ut , what 's your name . Are you Geometricians ? who gave you that name &c. Besides in other places such abundance of the like insipid conc●ipts as would make men think , if they were no otherwis● acquainted with the Universitv but by reading your Books , that the dearth there of salt were very great . If you have any passage more like to salt then these are ( excepting now and anon ) you may do wel to shew it your acquaintance , lest they despise you ; For ( since the detection of your Geometry ) you have nothing left you else to defend you from contempt . But I passe over this kinde of eloquence ; and come to somewhat yet more rurall . Page 27. line 1. You say I have given Euclide his Lurry . And again pag. 129. l. 11. A●d And now he is lest to learn his Lurry . I understand not the word Lurry . I never read it before , nor heard it , as I remember , but once , and that was when a Clown threatning another Clown said he would give him such a Lurry come poope &c. Such words as these do not become a learned mouth , much lesse are fit to be Registred in the publick writings of a Doctor of Divinity . In another place you have these words , just the same to a Cows thumb , a pretty Adage . Page 2. But pree-thee tell me . And again page 95. pree-thee tell me , why doest thou ask me such a question , and the like in many other places . You cannot but know how casy it is and was for me ; to have spoken to you in the same language . Why did I not ? Because I thought that amongst men that were civilly bred it would have redounded to my shame , as you have cause to fear that this will redound to yours . But what moved you to speak in that manner ? were you angry ? If I thought that the cause , I could pardon it the sooner , but it must be very great anger that can put a man that pro●esseth to teach good manners , so much out of his wits as to ●all into such a language as this of yours ; It was perhaps an imagination that you were talking to your inferiour , which I will not gran● you , nor will the Heralds I beleeve trouble themselves to decide the question . But howsoever I do not finde that civil men use to speak so to their inferious . If you grant my learning but to be equal to yours ( which you may certainly do without very much disparageing of your self abroad in the world ) you may think it lesse insolence in me to speak so to you in respect of my age , then for you to speak so to me In respect of your young Doctor-ship . You will finde that for all your Doctor-ship , your elders , if otherwise of as go●d repute as yo● , will be respected before you . But I am not sure that this language of yours proceeded from th●t cause ; I am rather inclined to think you have not been enough in good company , and that there is still somewhat le●t in your manners for which the honest youths ●of Hedington and Nincsey may compare with you ●or good Language as great a Doctor as you ar● . For my verses of the Peak , though they be as ill in my opinion as I bel●eve they are in yours , and made long since , yet are they not so obscene , as that they ought to be blamed by Dr. Wallis . I pray you Sir , whereas you have these words in your Schoole-Discipline page 96. unlesse you will say that one and the same motion may be now , and anon too ; what was the reason you put these words now and anon too in a different Character , that makes them to be the more taken notice of ; Do you think that the story of the Minister that uttered his affection ( if it be not a sl●nder ) not unlawfully but unseasonably , is not known to others as well as to you ? what needed you then ( when there was nothing that I had said could give the occasion ) to use those words ; there is nothing in my verses that do olerehircum , so much a● this of yours , I know what good you can receive by ruminating on such Ideas , or cherishing of such thoughts . But I go on to other words of 〈◊〉 by you reproached , you may as wel se●k the Focu● of the Parabola of Dives and Lazarus , which you say is mocking of the Scripture . To which I answer onely , that I intended not to mock the Scripture , but you ; and that which was not meant for mocking was none . And thus you have a second {non-Roman} {non-Roman} {non-Roman} {non-Roman} {non-Roman} . Grammar and Critickes . I come now to the comparison of our Grammar and Criticks . You object first against the signification I give of {non-Roman} {non-Roman} {non-Roman} {non-Roman} {non-Roman} , & say thus , what should come into your cap ( that if you markit in a man that wears a square cap , to one that wears a Hatt is very witty ) To make you think that {non-Roman} {non-Roman} {non-Roman} {non-Roman} {non-Roman} signifies a mark or brand with a hot iron , I perceive where the businesse lies . T was {non-Roman} {non-Roman} {non-Roman} {non-Roman} {non-Roman} run in your minde when you talked of {non-Roman} {non-Roman} {non-Roman} {non-Roman} {non-Roman} ; and because the words are somewhat alike you jumble them both together . Sir , I tould you once before , you presume too much upon your first cogitations . Aristophanes , in R●nis . Act. 5. Scen. 5. {non-Roman} {non-Roman} {non-Roman} {non-Roman} {non-Roman} &c. The old Commentator upon the word {non-Roman} {non-Roman} {non-Roman} {non-Roman} {non-Roman} saith thus {non-Roman} {non-Roman} {non-Roman} {non-Roman} {non-Roman} . That is , {non-Roman} {non-Roman} {non-Roman} {non-Roman} {non-Roman} for {non-Roman} {non-Roman} {non-Roman} {non-Roman} {non-Roman} for he ( Adimantus ) was not a Citizen . I hope the Commentator does not here mock Aristophanes for jumbling {non-Roman} {non-Roman} {non-Roman} {non-Roman} {non-Roman} and {non-Roman} {non-Roman} {non-Roman} {non-Roman} {non-Roman} together for want of understanding Greek . No , {non-Roman} {non-Roman} {non-Roman} {non-Roman} {non-Roman} and {non-Roman} {non-Roman} {non-Roman} {non-Roman} {non-Roman} signifie the same , save that for branding I seldom read {non-Roman} {non-Roman} {non-Roman} {non-Roman} {non-Roman} but {non-Roman} {non-Roman} {non-Roman} {non-Roman} {non-Roman} . For {non-Roman} {non-Roman} {non-Roman} {non-Roman} {non-Roman} does no more signifie a brand with a hot iron then {non-Roman} {non-Roman} {non-Roman} {non-Roman} {non-Roman} a point made also with a hot iron . They have both one common theam {non-Roman} {non-Roman} {non-Roman} {non-Roman} {non-Roman} , which does not signifie pungo , nor interpungo nor inuro ( for all you Lexicon ) but notam inprimere , or pungendo notare , without any restriction to burning or punching . It is therefore no lesse proper to say that {non-Roman} {non-Roman} {non-Roman} {non-Roman} {non-Roman} is a mark with a hot iron , then to say the same of {non-Roman} {non-Roman} {non-Roman} {non-Roman} {non-Roman} . The difference is onely this , that when they marked a slave , or a rascal as you are not ignorant is usually done here at the Assizes in the hand or shoulder with a hot iron they called that {non-Roman} {non-Roman} {non-Roman} {non-Roman} {non-Roman} , not for the burning but for the mark . And as it would have been called {non-Roman} {non-Roman} {non-Roman} {non-Roman} {non-Roman} that was imprinted on a slave , though made by st●yning or incision , so it is {non-Roman} {non-Roman} {non-Roman} {non-Roman} {non-Roman} though done with a hot iron ▪ And therefore there was no jumbling of those two words together as for want of reading Greek Authors , and by trusting too much to your Dictionaries ( which you say are proofes good enough for such a businesse ) you were made to imagine . The use I have made thereof was to shew that a point ( both by the word {non-Roman} {non-Roman} {non-Roman} {non-Roman} {non-Roman} in Euclide , and by the word {non-Roman} {non-Roman} {non-Roman} {non-Roman} {non-Roman} in some others ) was not nothing , but a visible mark ; the ignorance where of hath thrown you into so many Paralogismes in Geometry . But do you think you can defend your Adducis Malleum aswel as I have now defended my {non-Roman} {non-Roman} {non-Roman} {non-Roman} {non-Roman} ? You have brought , I confess , above a hundred places of Authors where there is the word Duco , or some of its compounds ; but none of them will justifie Adducis Malleum and ( excepting two of those places ) you your self seem to condemn them all , comparing yours , with none of the rest but with these two only ; both out of Plautus , by you , not well understood . The first is in Casina Act. 5. Scen. 2. ubi intro hanc Novam Nuptam deduxi , via recta , Clavem abduxi . Which you presently presuming of your first thoughts ( a peculiar fault to men of your principles ) assure your self is right . But if you look on the place as Scaliger reads it cited by the commentator , you will finde it should be obduxi , and that Clavis is there used for the bolt of the lock . Besides he bolted it within . Whither then could he carry away the key ? The place is to be rendered thus , when I had brought in this new bride I presently lockt the door , and is this as bad every whit as Adducis Malleum . The second place is it Amphytryo Act. 1. Scen. 1. Eam ( Cirneam ) ut à matre ●uerat natum plenum vini Eduxi me●i . Which you interpret I brought out a flag●n of Wine . Unlearnedly . They are the words of Mercury transformed into Sofia . And to try whether Mercury were So●ia or not , So●ia asked him where he was and what he did during the Battle ; to which Mercury answered , ( who knew where Sofia then was and what he did ) I was in the Cellar where I filled a Cirnea , and brought it up full of Wine , pure as it came from its mother . By the mother of the Wine meaning the Vine , and alluding to the Education of Children , for Ebibi said Eduxi , and with an Emphasis in meri , because Cirnea ( from {non-Roman} {non-Roman} {non-Roman} {non-Roman} {non-Roman} , Misceo ) was a vessel wherein they put water to temper to their Wine . Intimating that th●ugh the vessel were Cirnea , yet the Wine was meru●… . This is the true sense of the place ; but you will have Eduxi to be I brought out , though he came not out himself . You see Sir , that nei●her this is so ●ad as Addu●●s Malleum . But suppose out of some one place in some one blind Author you ●ad pa●ralled your Addu●is M●ll●um , do you think it must therefore presently be held for good Latin , why more then learn his lurry must be therefore thought good English a thousand years hence , because it will be read in Dr. Wallis long ▪ liv'd works . But how do you construe this passage of the Greek Testament {non-Roman} {non-Roman} {non-Roman} {non-Roman} {non-Roman} . You construeit thus : she shall be saved notwithstanding Child bearing , if ( the w●men ) remain in the faith . Is child-bearing any obstacle to the salvation of women ? You might aswel have translated the first verse of Rom. 5. in this manner , Being then justified by fait● , ●e have pea●e with God notwithstanding our Lord Jesus Christ . I let pass your n●t ●●nding in {non-Roman} {non-Roman} {non-Roman} {non-Roman} {non-Roman} ▪ as good a Gra●arian as you are , a Nominative case to {non-Roman} {non-Roman} {non-Roman} {non-Roman} {non-Roman} . If you had remembred the place . 1. Pet. Chap. 3. verse 20. {non-Roman} {non-Roman} {non-Roman} {non-Roman} {non-Roman} , that is , They were saved in the wa●●rs , you would have thought your construstion justified then very well ; but you h●d been deceived , for {non-Roman} {non-Roman} {non-Roman} {non-Roman} {non-Roman} does not there signifie causam , nor ablationem impedimenti but tran●…m ; not cause or removing an impediment but passage . Being come thu●●●r I found a friend that hath eased me of this dispute ; for he shewed me a letter written to himself from a learned man , that hath out of very good Authors collected enough to decide all the Grammatical questions betwen you and me both Greek and Latin . He would not let me know his na●e , nor any thing of him but only this , that he had better ornaments then to be willing to go clad abroad in the habit of a Grammarian . But he gave me leave to make use of so much of the letter as I thought fit in this dispute . Which I have done and have added it to the end of this writing . But before I come to that , you must not take it ill ( though I have done with your School Discipline ) if I ex●mine a little some other of your printed writings as you have examined mine ; for neither you in Geometry nor such as you in Church politicks can not expect to publish any unholesome doctrine without some Antidotes from me , as long as I can hold a pen . But why did you answer nothing to my sixth Lesson ? because ( you say ) it concerned your Colle●gue onely ; No Sir , It concerned you also , and chiefly . For I have not heard that your Colleague holdeth those dangerous principies which I take notice of in you . In my sixth Lesson page 62. upon the occasion of these words , not his but yours . Perhaps you t●ke the whol● History of the fall of Adam for a fable , which is no wonder , seeing you say the rules of honouring and worshiping of God are to be taken form the laws . In answer to which I said thus . You that take so hainously , tha● I would have the rule of Gods worship in a Christian Common-wealth to be taken from the Laws , tell me from whom you would have th●m taken . From your selfe ? Why so , more then from me ? From the Bishops ? Right , if the Supreme power of the Common-wealth will have it so ; if not , why from them rather then from me ? From a Consistory of Pres●yt●rs ▪ themselves or joyned with Lay-Elders , whom they may sway as they please ? Good , If the supream Governour of the Common-wealth will have it so . If not , why fr●m them rather then from me , or from any man else ? They are wiser and learneder then I ▪ It may be so ; but it ha● n●t yet appeared . Howsoever , let that be granted . Is ●h●re any man so very a fool a● to subje●● himself to the rules of other men in those things which do so neerly concern himself , for the title they assume of being wise and learne● , unlesse they also have the sword which must portect them . But it seems you understand the sword as comprehended . If so , Do not you then r●ceive the rules of Gods worship from the Civil Power ? Yes doubtlesse ; and you would expect , if your Consistory had that sword , that no man should dare to ex●rcise or teach any rules concerning Gods worship which were not by you allowed . This will be thought strong arguing , if you do not answer it . But the truth is , you could say nothing against it without too plainly discove●ing your disaffection to the Goverme●t . And yet you have discovered it pretty well in your second Thesis maintained in the Act at Oxford 1654 , and since by your self published . This Thesis I shall speak briefly to . Scotch-Church Politicks . You define Ministers of the Gospel to be those to whom the preaching of the Gospel by their o●fice is injoyned by Christ . Pray you , first , what do you mean by saying preaching ex Offici● is enjoyned by Christ . Are they Preachers ●x Officio , and afterwards enjoyned to Preach ? Ex Officio adds nothing to the definition ; but a man may ●asily see your purpose to disjoyn your self from the State by inserting i● . Secondly , I desire to know in what manner you will be able out of this d●●inition to prove your self a Minister ? Did Christ hi●self immediately enjoyn you to preach , or give you orders ? No . Who then , some Bishop , or Minister or Ministers ? Yes ; by what Authority ? Are you sure they had Authority immediately from Christ ? no . How then ar● you sure but that they might have none ? At least , some of them through whom your Authority is derived might have none . And therefore if you run b●ck for your Authority towards the Apostles times but a matter of sixscore years , you will ●inde your Authority derived from the Pope ; which words have a ●ound very unlike to the voice of the Laws of England . And yet the Pope will not own you . There 's no man doubts but that you hold that your Office comes to you by successive impo●ition o● hands ●rom the time of the Apostles . Which opinion in those gentle terms passeth well enough ; But to say you derive your Authority from thence , not through the Authority of the Soveraign power civill , is too rude to be endured in a state that would live in peace . In a word you can never prove you are a Minister , but by the Supream Autho●ity of the Common-wealth . Why then do you not put some such clause into your definition ? As thus , Ministers of the Gospel are those to whom the preaching of the Gospel is enjoyned by the Soveraign power in the name of Christ . What harm is there in this definition , saving onely it crosses the ambition of many men that hold your p●inciples ? Then you d●●ine the power of a Minister thus ; The power of a Minister is that which belongeth to a Minister of the Gospel in vertue of the Office he holds ; in as much as he holds a publick Station , and is distinguished from private Christians . Such as is the power of preaching the Gospel ; administring the Sacrament ; the use of Eccles●astical censure● ; and Ord●ining of Ministers &c. Again ; how wil you prove out of this definition that you or any man ●lse hath the power of a Minister , i● it be not given him , by him that is the Soveraign o● the Common-wealth ▪ For seeing ( as I have now proved ) it is from him that you must d●rive your Ministery , you can have no oth●r power then that which is limited in y●ur Orders , ●or ●ha● neithe● longer then he thinks fit . For if he give it you for the instruction of his subjects in their duty , he may take it from you again whensoever he shal see you instruct them wi●h undutiful and seditious principles . And if the Sogeraign power give me command ( though without the ceremony of imposition of hands ) to teach the Doctrine of my Leviathan in the Pulpit , why am not I if my Doctrine and life be as good as yours , a Minister as well as you , and as publick a person as you are ? For publick person primarily is none but the civil Soveraign , and so seconda●ily all that are imployed in the execution of any part of the publick Charge . For all are his Ministers , and therefore also Christs Ministers because he is ●o ; and other Ministers are but his Vicars , and ought not to do or say any thing to his people contrary to the intention of the Soveraign in giving them their Commission , Again , if you have in your Commission a power to Ex●ommunicate , how can you think that your Soveraig● who gave you that Commission intended it for a commission to Excommunicate himself , that is , ( as long as he stands Excommunicate ; to deprive him of his Kingdom ) If all Subjects were of your minde , as I hope ▪ they will never be , they will have a very unquiet life . And yet this has ( as I have often heard ) been practised in Scotland , when Ministers holding your principles had power enough ( though no right ) to do it . And for Administration of the Sacraments , if by the Supream power of the Common-wealth it were commited to such of the Laity as know how it ought to be done as well as you , they would ips● facto be Ministers as good as you . Likewise the right of Ordination of Ministers depends not now on the Imposition of hands of a Minister or Presbytery , but on the authority of the Christian Soveraign Christs immedi●t Vicar and supream Governour of all Persons and Judge of all causes both spiritual and temporal in his own Dominions , which I beleeve you will not denie . This being evident , what Acts are those of yours which you call Authoritative , and receive not from the Authority of the civil power ? A Constable does the acts of a Constable authoritatively in that sense . There●ore you can no otherwayes claime your power then a Constable claimeth his , who does not exercise his Office in the Con●●ablery of another . But you ●orget that the Scribes and the Pharisees ●it no more in Moses Chaire . You would have every Minister to be a Minister of the Universal Church , and that it be lawful for you to preach your Doctrine at Rome ; if you would be pleased to try , you would finde the Contrary . You bring no argument for it that looks like reas●n . Examples prove nothing , where persons , times , and other circumstances differ ; as they differ very much now when Kings are Christians , from what they were then when Kings per●ecuted Christians . It is ea●ie to perceive what you aime at . You woul fain have Market-day Lectures set up by authority ( not by the authority of the Civil power but by the authority of example of the Apostles in the Emission of Preachers to 〈◊〉 I●fidels ) not knowing that any Christian may lawfully preach to the Infidels . that is to say , proclaime unto them that Jesus is the Mesiah without need of being other wayes made a Minister ; as the Deacons did in the Apostles time ; nor that many teachers unlesse they can agree better , do any thing else but prepare men for faction , nay , rather you know it well enough ; but it conduces to your end upon the Market-dayes to dispose at once both Town and Country , under a false pretence of obedience to God , to a Neglecting of the Commandments of the Civil Soveraign , and make the Subject to be wholly ruled by your selves ; wherein you have already found your selves deceived . You know how to trouble and sometimes und●e a ●lack Government , and had need to be warily lookt to , but are not fit to hold the rei●es . And how should you , being men of so little judgement as not to see the Necessity of unity in the Governour , and of Absolute obedience in the Governed , as is manifest out of the place of your Elenchus above recited . The Doctrine of the duty of private men in a Common-wealth is much more difficult , not onely then the knowledge of your symbols , but also then the knowledge of G●ometry it self ▪ How then do you think , when you erre so grosly in a few Equations , and in the use of most common words , you should be fit to Govern so great Nations as England , Ireland , and Scotland , or so much as to teach them . For it is not reading but judgement that enables one man to teach another . I have one thing more to adde , and that is the disaffection I am charged with all to the Universities . Concerning the Universities of Oxford and Cambridge , I ever held them for the greatest and Noblest means of advancing learning of all kinds , where they should be therein imployed , as being ●urnished with large endowments , and other helps of Study , and frequent ▪ with abundance of young Gentlemen of good families ●●d good breeding from their childe hood . On the other side , in case the same means and the same wits should be imployed in the advancing of the Doctrines that tend to the weakning of the publick , and strengthning of the power of any private ambitious party , they would also be very effectual for that ; And consequently that if any Doctri●e tending ●o the diminishing of the civil power were taught there , not that the Universities were to blame , but onely those men that in the Universities either in Lectures , Sermons , printed Books , or Thesis did teach such Doctrine to their hear●rs or readers . Now yo● know very well that in the time of the Roman Religion , the power of the Pope in England was upheld principally by such teachers in the Universities . You k●ow also how much the Divines that held the ●ame princi●les in Church Government with you , have contributed to our late troubles . Can I therefore be justly taxed with disaffection to the Universities for wishing this to be reformed . And it hath pleased God of late to reform it in a great measure , and indeed as I thought totally , when out comes this your Thesis boldly maintained to shew the contrary ; Nor can I yet cal this your Doctrine the Doctrine of the Vniver●ity , b●t surely it wil not be unr●asonable to think so , if by publick act of the Vniversity it be not disavowed , which done , and that as often as there shall be need , there can be no longer doubt but that the Vniversities of England are not onely the Noblest of all Christian Vniversities , ●ut also absolutly , & of the greatst benefit to this Common wealth that can be imagined , except that benefit of the head it self that uniteth and ruleth all . I have not here perticularized at length all the ill consequences that may be deduced fromthis Thesis of yours , because I may , when further provoked , have somewhat to say that is new . So much for the 3. {non-Roman} {non-Roman} {non-Roman} {non-Roman} {non-Roman} . An extract of a Letter concerning the Grammaticall part of the Controversie between Mr. HOBBES and Dr. WALLIS . Mr. Hobbes hath these words LOngi●udinem percu●sam motu uniformi , cum impetu ubique ipsi B D ●quali . Dr. Wallis saith cum were better out , unlesse you would have impe●us to be onely a companion , not a cause . Mr. Hobbes answered it was th●Abla●ive case of the manner . The truth is the Ablatives case of the manner , and cause both , may be used with the conjunction cum , as may be justified . Cicero in the 2. de Nat. Deor. moliri aliquid cum labore operoso ac molesto ; and in his Oration for C●cinn● . De se autem hoc praedicat , Antiocho Ebul●j se●vo imperâsse ut in Caecinā advenien●em cum serro invaderet . Let us see then what Dr. Wallis objects against Tully ; where a Causality is imported ; Though we may use with in English , yet not cum in Latine , to kill with a sword ( importing this to have an instrumental or causal in●luence , and not onely that it h●ngs by the mans side whilest some other weapon is made ●se of ) is not in Latine , occidere cum gladio , but gladio occidere . This shews that the Dr. hath not forgot his Grammer , for the subsequent examples as well as this rule are borrowed thence . But yet he might have known that great Personages have never confined themselves to this Pedantry , but have chosen to walk in a greater latitude . Most of the Elegancies and Idioms of every Language are exceptions to his Grammar . But since Mr. Hobbes saith it is the Ablative case of the manner , there is no doubt it may be expressed with cum . The Doctor in the mean time knew no more then what Lilly had taught him ; Alvarez would have taught him more . And Voss●us in his Book de constructione Cap. 47. expressly teacheth , Ablativ●s causae , instrument● , vel modi , non à verbo regised à praepositione omiss● , à vel ab , de , è , vel ex , prae , aut cum , ac praepositiones eas quandoque exprimi nisi quòd cum ablativis inst●umenti haud ●emerè inve●ias ; and afterwards he sath non timerè imitandum . If this be so , then did Mr. Hobbes speak Grammatically , and with Tully , but not usually . And might not one retort ●pon the Doctor , that Vossius is as great a Critick a●●ie . His next reflection is upon praetendi● scire , this he saith is an Angli●isme . If this be all his Accusation , upon this score , we shall lose ma●y expres●ions that are used by the best Authors , which I take to be good Lati●…smes , though they be also Anglicismes , the latter being but an imitation of the former . The Do●tor therefore was too fierce to condemn upon so general an account , that which was not to have been censured for being an Anglicisme , unlesse also it had been no Latinisme . Mr. Hobbes replies , that the Printer had omitted se . He saith , this mends the matter a little It is very likely , for then it is just such another Anglicisme as that of Quintilian ; Cùm loricatus in foro ambularet , praetendebat se id metu facere . The Doctor certanily was very negligent , or else he could not have missed this in Robert Stephen . Or h●ply he was resolved to condemn Quintilian for this and that other Anglicisme , Ignorantia p●●tendi non potest , as all those that have used praetendo , which are many and as good Authors as Doc●●r Wallis that makes his own Encomiasts ( not an English ma● ) amongst them to w●●te A●glicismes . Then ●e bl●mes Tractatus ●●jus partis t●rtiae , in q●â mo●us & magnitudo per se & abstract● consideravimus , te●minum hic statuo . Here I ●●st con●esse the exception is colonrable , yet I can parallel it with the like objection made by Erasm●s aga●●st Tully , out of whom Erasmus q●otes this passage ; D●ut ùs comm●tans Athents ; quo●●am ven●… negabant solvendi facultatem , e●at ani●us ad ●e scr●bere : and excuses it hu● , that Tully might have had at fir●t in his thoughts v●…bam or statuebam , which he afterwards relinquished for erat animus , and did not remember what he had antecedently w●itt●n , which did not vary fr●m his succeeding , though●● , but words . A●d this excuse may passe with any who knows that Mr. Hobbes values not the study of word● , but as it s●rves to expresse his thoughts , which were the same whe●her he wrot● ; in quâ metus & m●g●i●udo per se a● abstractè considerati sunt , or consideravimus . And if the Dr ▪ will make this so capital ; he must prove it volunta●y , and shew that it is greater then what is legible in the p●ny Letter of his Encomiast , whom he would have to be beyond except●●n . Now ●ollowes his r●diculous apol●gi● for adducis malleum , ut occidas muscam ▪ The cause why he did use that prov●rb ( of his own phrasing ) was this . Mr. Hobbes had taken a great deal of paines to demonstrate what Dr. Wallis thought he could have prov'd in shor● ; upon this occasion he objects , ad ducis malleum ut occidas muscam , which I shall suppose he intended to English thus , you bring a beetle to kill a ●●y . Mr. Hobbes retor●ed that adduco was not used in that sense . The Dr. vindicates himself thu● ; duco , dedu●● , reduco , perduco , produco , &c. signifie s●range things , ergo , adduco may be used in that sense ; whi●h is 〈◊〉 most ridic●lo●● kinde of arguing , where we are but to take up our Language from others , and not to coyue new phr●ses . It is not the Grammar that shall secure the Dr. nor weak Analogies , where Elegance comes in contest . To jus●i●ie his expression he must have shewed it usu tritum , or alledged the Authori●y of some Author of great note for i● . I have not the leasure to exam●…e his impertinent citations about those other compounds , nor yet of that simple verb duco ; nay , to justifie his saying he hath not brought one parallel example . He talkes indeed very high , that duco ( ●ith its compounds ) i● a word of a large signification , & amongst the rest to Bring , fetch , carry &c. i● so exceeding frequent ●n all Authors ( Plautus , Terence , Tully Ca●ar , Tacitus , Pliny , Seneca , Virgil , Horace , Ovid , Claudian , &c that he must needs be either maliciously blind , or a very stranger to the Latine tongue that doth not know it , or c●n have the face to denie it . 〈◊〉 ●ead what will ●e my doom for not allowing his Latine ; yet 〈◊〉 must professe I dare secure the Dr , for having read all Authors , notwithstanding his assertion , and I hope he will do the like for me . And for those which he hath read , had he brought no better proofes then these , he had , I am sure , been whipped soundly in Westminster-School , for hi● impudence as well as ig●or●nce , by the learned M●ster therof a● present . But I da●e f●●ther a●…m , the Dr. hath no● read in this point any , ▪ but onely consulted with Robe●t S●ephen's Thesaurus Li●guae La●inae , whence he hath borrowed his allegations in adduco ; and for the other I had not so much Idle ●ime as to compare them . And lest the fact might be discovered he hath sophi●●icated those Authors whence Stephen cites the expressions , and imposed upon th●m others . If it be not so , o● that the Dr. could not write it right wh●n the c●py was right before him , let him ●ell me where he did ever read in Plautus , adducta res in fastidium . I fin●e the whole se●tence in Plinie's preface to Vespas●an ( out of whom in t●e precedent Paragraph he cites it ) about the ●iddle : alia verò ita multis prodita , ut in factidium sint adducta , which is the very example Stephanus useth , although he doth p●emise his adducta res in fastidium . Let the Dr. tell where he ever did read 〈◊〉 Horace Ova noctuae &c. ●aedium vini adducunt ▪ Did he ( or any else ) with the interposition of an &c ) make Trochaicks ? I say , and Step●anus saies so too , that it is in Pliny lib. 13. cap. 15. neer the end ; the whole sentence runs thus ; Ebriosis Ova noctuae per ●●id●um data in vino , ●aedium ejus adducunt . I doubt not but these are the places he aimed at , although he disguised and min● ▪ d the Quotatio●s ; if they be not , I should b● glad to augment my Stephanus with his Additions . These things pr●mis●d , I come to consider the Doctors proofes : Res eò adducta est : adducta vita in ex●remum : adducta res in fas●idium : rem ad muc●ones & ●…s adducere : contracta res & adducta in a●gus●um : ●es ad concordiam adduci potest : in ordinem adducere : adducere febres , s●●im , taedium vini ( all in Robert Stephen ) betwixt which and adducere malleum , what a vast difference there is , I leave them to umpier ●…iteretes & religiosas nacti sunt aures ▪ who are the competent Judges of Elegancy , and onely cast in the verdict of one or two , who are in any place ( where the purity of the L●tin ●ongue flourisheth ) of great esteeme ▪ L●saeus in his Scop● Linguae Latinae , ad purgandam Linguam à barbarie , &c. ( would any think that the Doctors elegant expression , frequent in all Authors , which none but the malicious or ignorant can deny , should suffer so contumelious an expurgation ? ) Losaeus ( I say ) hath these words , Adferre plerique minus attentiutuntur pro adducere . Quod Plautus , in Pseudolo , insigni exem●lo notat . C. Attu●i hunc . P. Quid attulisti ? T. Adduxi volui dic●re P ▪ Quis istic est ? C. Charinus . Satis igitur admonet discriminis inter ducere , reducere , adducere & abducere , quae de pers●nâ ; et ●erre , adferre &c. quae de re dicuntur . Idem . Demetrium , quem ego novi , adduce ; argentum non mo●or qu●… feras Cavendum igitur est ne vulgi more , ( let the Dr. mark this ▪ and know that this Author is authentick amongst the Ciceronians ) adferre de personâ dicamus , s ●●adducere ; licet et hoc de certis quibus●am rebus non ineptè dicatur . In this last clause he ●aith as ●uch as Mr. Hobbes saith , and what the Dr. proves ; but th●t ever the Dr. brought an example which might resemble adducis malleum , is denied ; for I have mentioned alre●dy his allegations every one , of adduco . Another Author ( a fit Antagonist for the elegant Dr. ) is the ●arrago ●ordidorum Verborum , ●oyned with the Epitome of L. Valla's elegancies . He saith , Acce●se , adduc Petrum , latinè dicitur pro eo quod pueri dicunt ▪ adfer Petrum . And this may suffice to justifie Mr. Hobbes's exception , who proceeded no further th●n this Author to tell the Doctor that adduco was used of Animals . But the Dr. replies , this signification is true , but so may the other be also ▪ I s●y , if it never have been used so , it cannot be so , for we cannot coine new Latine words , no more then French or Spanish who are Forreign●rs . Mr. Hobbes was upon the negative , and not to disprove the contrary ●p●nion . If the Dr. would be believed , he must prove it by some example ( which is all the proofe of elegancy ) and till he do so , not to beleev● him , it is sufficient not to h●ve cause . But Dr Wallis , why not adduco for a ●ammer as well as a tree ? I answer , yes , equally for either , and yet for neither ; Did ever any body go about to mo●k his Reade●s thus solemnly ? I do not finde , ( to my best remembrance ) any example of it in S●ephen , and the Dr. is not wiser then his Book ; if there be , it is strange the Dr. should omit the onely pertinent example , and trouble us with such impertinences for three or four pages . In Stephen there are adducer● habenas and adducere lorum , but in a different sense . It is not impossible I may guesse 〈◊〉 the Doctors aime . In Tully de Nat ▪ D●or ( as I remember ) there is this passage ; Quum autem ille respondisset , in agro ambu●anti ra●…ulum adductum , ut remissu● esset , in oculum suum recidisse , where it signifies nothing else but to be bent , bowed , pulled back , and in that sense , the hammer of a Clock ▪ or , that of a Smith , when he fetcheth his strok● ▪ may be said addu●i . And this I conceive the Doctor would have us in the close think to have been his me●ning ; else wh●t doth he drive at in these words ? Wh●n you have done the best you can , you will not be able to find better words then adducere mall●um and reduce●… ▪ to signifie the two contrary motions of the Hammer , the one when you strike with it , ( excellently trivial ! ) The other when you take it back ( better and better ) What to do ? to fetch another stroak . If any can believe that this was his meaning , I shall justifie his Latine , but must leave it to him to prove it sense . If he intended no more , why did he go about to defend the other meaning , and never meddle with this ? Which yet might have been proved by this one example of mine ? May not therefore his own saying be justly retorted upon him in this case , Adducis malleum , us occidas muscam ? Another exception is ▪ Falsae sunt et mult● istiusmodi ( propositiones ) I wish the Doctor could bring so good Parallels , and so many , out of any Author , for his Adducis malleum , as Tully affords in this case . Take one for all out of the beginning of his Paradoxes ; Animadverti saepe Catonem , ●um in senatu sententiam diceret , Locos graves ex Philosophia tractare , abhorrentes ab hoc usu forensi et publico , sed dicendo consequi tamen , ut illa etiam populo pro●abilia viderentur . This is but a Solaecophanes , and h●th ▪ many p●●sidents mo●e , as in the second ●ook of his Academical questions , &c. I cannot now stay upon each particular p●ssage ; I do not see any necessitie of tracing the Dr. in all hi● figaries . Now he dis●llows tanquam diceremus , a● if we should say . But why is that l●ss tolerable then tanquam feceris , as if you had done ? It should be quasi ( forsooth ! ) Or ac si , or tanquam si , which is Tulli's own word . ( What is tanquam si become but one word ? ) tanquam si tua res agatur &c. Good Dr. leave out Tully and all Ciceronians , or you will for ever suffer for this , and your Adducis malleum . Is not this to put your self on their verdict , when you oppose Mr. Hobs with Tully ? But the Dr. gives his reason . And though he hath had the luck in his Adducis malleum , to follow the first part of that saying , Loquendum cum vulgo , yet now it is , sentiendum cum sapientibus . For tanquam without si signifies but , a● , not , as if . It is pity the Dr. could not argue in Symbols too , that so we might not understand him , but suppose all his Papers to carry evidence with them , because they are Mathematically scratch'd . How does he construe this , Plance , tu●es alto Drusorum sanguine , tanquam Feceris ipse aliquid , propter quod nobilis esses . So Coelius , one much esteemed by Cicero , who hath inserted his Epistle● into his works , saith in his fifth Epistle ( Tul. Epist. sam . lib. 8. Ep ▪ 5 ) Omnia desiderantur ab eo tanquam nihil denegatum sit ei quo mi●us paratissimus esset qui publico negotio pr●positus est . But it was not possible the Dr. should know this , it not being in Stephen , where his examples for tanquam si are . But the Dr. having pitched upon this Criticisme , and penned it , some body , I believe , put him in mind of the absurditie th●reof ▪ and yet the generous Professour ( who writes running hand and never tra●scrib●d his papers , if I am not misinformed ) presumed no body else could be more intelligent then he , who had perused Stephen . He would not retract any thing , but subjoyns , That he will allow it as passable , because other modern writers , and some of the Antients , have so used it , as Mr. Hobs hath done . I know not what Authors the Dr. meant , for , if I am not much mistaken , I do not find any in Stephen . His citation of Colum●lla is not right L. 5. cap. 5. ( nor can I deduce any thing thence till I have read the passage ) but if he take Juvenal and Coelius for modern Authors , I hope he will admit of Accius , N●vius , and Ca●me●ta for the only antients . Let him think upon this ▪ Criticisme , and never hope pardon for his Adducis malleum , which is not half so well justified , and yet none but mad men or fools reject it . But certainly the Dr. should not have made it his businesse to object Anglicisines , in whose Elench●s I doubt not but there may be found such phrases as may serve to convince him that he is an English man . However Scottified in his principles . If the Dr. doubt of it , or but desire a Catalogue , let him but signifie his minde , and he shall be furnished with a Florilegium . But I am now come to the main controversie about Empusa . The Dr. saith nothing in defence of his quibble , nor gives any reason why he jumbled Languages to make a silly clinch , which will not passe for wit either at Oxford , or at Cambridge ; no nor at Westmi●ster . It seems he had derived Empusa from {non-Roman} {non-Roman} {non-Roman} {non-Roman} {non-Roman} and {non-Roman} {non-Roman} {non-Roman} {non-Roman} {non-Roman} , and said it was a kinde of Hobgoblin that hopped upon one legge ; and hence it was that the boyes play ( Fox come out of thy hole ) came to be called Empusa . I suppose he means Ludus Empusae . This derivation he would have to be good , and that we may know his reading ( though he hath soarce consulted any of the Authors ) he saith Mr. Hobbes did laugh at it , until somebody told him that it was in the Scholiast of Aristophanes ( as good a Critick as Mr. Hobbes ) Eust●thius , Erasmus , Coel●… Rhodiginus , Step●anus , Scapula , and Calepine . But sure he doth not think to seape so . To begin with the last , Calepine doth indeed say , uno incedit pede , unde et non●●n . But he is a Moderne , and I do not see why his Authority should outweigh mine , if his Authors Reasons do not . He refers to Erasmus and Rhodiginus . Erasmus in the adage . Proteo mutabilior , hath these words of Empusa . Narrant autem uno videripedi ( this is not to hopp ) unde et nomen inditum putant , {non-Roman} {non-Roman} {non-Roman} {non-Roman} {non-Roman} . He doth not testifie his approbation of the derivation at all , onely lets you know what Etymologies some have given before him . And doth any body think that Dr. Harmar was the first which began to shew his wit ( or folly ) in e●ymologizing words ? Coelius Rhodiginus doth not own the derivation , onely ●aith Nominis ratio est , ut placet Eustathio , quia uno incedit pede ; ( is this to hopp ? ) sed nec desunt qui alterum interpretentur habere ●neum pedem , & inde appella●… Empusath ; quod in Batrachis Aristophanes expressit . And then he recites the interpretation that Aristophanes's Scholiast doth give upon the text , of which by and by . If any credit be to be attributed to this allegation , his last thoughts are opposite to Dr. Wallis ; and Empusa must be so called , not because she hopped upon one legg , but because she had but one , the other , being brasse . But for the former derivation he refers to Eustathius . As to Eustathius , I do easily conjecture that the Reader doth beleeve ; that Rodiginus doth mean Eustathius upon Homer , for that is the book of most repute and fame , his other piece being no way considerable for bulk or repute . But it is not that book nor yet his History of Ismenias , but his notes upon the 23 verse of Dionysius {non-Roman} {non-Roman} {non-Roman} {non-Roman} {non-Roman} . The Poet had said of the stone Jaspis , that it was , {non-Roman} {non-Roman} {non-Roman} {non-Roman} {non-Roman} , Upon which Eustathius thus remarks ; {non-Roman} {non-Roman} {non-Roman} {non-Roman} {non-Roman} ( fortè {non-Roman} {non-Roman} {non-Roman} {non-Roman} {non-Roman} Steph. ) {non-Roman} {non-Roman} {non-Roman} {non-Roman} {non-Roman} . This testimony doth not prove any thing of hopping , and as to the derivation , I cannot but say that Eustathius had too much of the Grammarian in him , and this is not the first time , neither in this book , nor elsewhere , wherein he hath trifled . It is observable out of the place that there were m●re Empusa's then one , as indeed the name is applied by several men to any kinde of ▪ frightful phantasme . And so it is used by several Authors , and for as much as phantasmes are various , according as the persons affrighted have been severally educated , &c every man did impose this name upon his own apprehensions . This gave men occasion to fain Empusa as such ( for who will beleeve that she was not apprehended as having four leggs , when she appeared in the form of a Cow , dog , &c. ) but as apprehended by Bacchus and his man at that time . I do not finde that she appeared in any shape , but such as made use of legges in going , whence I imagine that Empusae might be opposite to the {non-Roman} {non-Roman} {non-Roman} {non-Roman} {non-Roman} , which appellation was anciently fixed upon the gods ( propitious ) upon a twofold account , first for that they were usually essigiated as having no feet ( which is evident from ancient sculpture ) and secondly , for that they are all said not to walk , but rather swim , if I may so expresse that non gradiuntur , sed fluunt , which is the assertion of all the commentators I have ever seen upon that verse of Virgil , Et vera incessu patuit dea — This whole discourse may be much illustrated from a Passage in Heliodorus , Aethiop . L. 3. Sect. 12 , 13. Calastris told Cnemon that the gods Apollo and Diana did appear unto him , Cnemen replied , {non-Roman} {non-Roman} {non-Roman} {non-Roman} {non-Roman} ; upon this the old Priest andswered , that both gods and demons , when they appear to men , may be discovered by the curious observer , both in that they never shut their eyes , {non-Roman} {non-Roman} {non-Roman} {non-Roman} {non-Roman} {non-Roman} {non-Roman} {non-Roman} {non-Roman} {non-Roman} {non-Roman} {non-Roman} {non-Roman} {non-Roman} {non-Roman} {non-Roman} {non-Roman} {non-Roman} {non-Roman} {non-Roman} . Farnaby upon the place in Virgil observes , that , Deorum incessus est continuus & aequalis , non dimotis pedibus , neque transpositis , {non-Roman} {non-Roman} {non-Roman} {non-Roman} {non-Roman} . Cernelius Schrevelius in the new Leyden notes , saith , Antiquissima quaeque Deorum simulachra , quod observarunt viri magni , erant {non-Roman} {non-Roman} {non-Roman} {non-Roman} {non-Roman} , dijque ipsi non gradiuntur sed fluunt . Their statues were said to stand rather upon Columnes then upon legges ; for they seeme to have been nothing but columnes shaped out into this or that figure , the base whereof carrying little of the representation of a foot . These things being premised I suppose it easie for the intelligent Reader to finde out the true Etymology of Empus● , quasi {non-Roman} {non-Roman} {non-Roman} {non-Roman} {non-Roman} , or {non-Roman} {non-Roman} {non-Roman} {non-Roman} {non-Roman} from going on her feet , whereas the other gods and demons had a different gate . If any can dislike this deduction , and think her so named from {non-Roman} {non-Roman} {non-Roman} {non-Roman} {non-Roman} , whereas she alwaies went upon two leggs ( if her shape permitted it ) though she might draw the one after her , as a man doth a wooden legge ; I say if any , notwithstanding what hath been said , can joyn issue with the Doctor , my reply shall be , {non-Roman} {non-Roman} {non-Roman} {non-Roman} {non-Roman} . Now as to the words of Aristophanes upon which the Scholiast descants , they are these . Speaking of an Apparition strangely shaped , sometimes like a Camel , sometimes like an oxe , a beautiful woman , a dog &c. Bacchus replies , {non-Roman} {non-Roman} {non-Roman} {non-Roman} {non-Roman} . The scholiast hereupon tells us that Empusa was {non-Roman} {non-Roman} {non-Roman} {non-Roman} {non-Roman} And this is all that is material in the Scholiast , except that he addes by and by , that {non-Roman} {non-Roman} {non-Roman} {non-Roman} {non-Roman} is all one with the leg of an Asse , And this very text and Scholiast is that to which all the Authors he names , and more , do refer . I come now to Stephen , who in his Index and in the word {non-Roman} {non-Roman} {non-Roman} {non-Roman} {non-Roman} , gives the derivation of Empusa . {non-Roman} {non-Roman} {non-Roman} {non-Roman} {non-Roman} , gradior , inced● , ( not to hop ) sic Suidas {non-Roman} {non-Roman} {non-Roman} {non-Roman} {non-Roman} dictam ait {non-Roman} {non-Roman} {non-Roman} {non-Roman} {non-Roman} . In the Index thus ; sunt qui dictam putent {non-Roman} {non-Roman} {non-Roman} {non-Roman} {non-Roman} , quod uno incedat pede , quasi {non-Roman} {non-Roman} {non-Roman} {non-Roman} {non-Roman} , alterum enim pedem aeneum habet . But neither Stephen , nor any else ( except Suida● , whom the Hypercritical Dr. had not seen ) no not the Scholiast of Aristophanes ( a better Critick then Mr. Hobbes ) doth relate the etymology as their own . Nay , there is not one saith , Empusa hopped on one legg , which is to be proved out of them . The great Etymological Dictionary deriveth it {non-Roman} {non-Roman} {non-Roman} {non-Roman} {non-Roman} , to binder , let &c. its apparition being a token of ill luck . But as to the Doctors deduction , it saith {non-Roman} {non-Roman} {non-Roman} {non-Roman} {non-Roman} . It doth onely seem so . And it is strange that {non-Roman} {non-Roman} {non-Roman} {non-Roman} {non-Roman} should not alter onely ▪ its aspiration , but change its ν into μ , which I can hardly beleeve admittable in Greek , least there should be no difference betwixt its derivatives and those of {non-Roman} {non-Roman} {non-Roman} {non-Roman} {non-Roman} . When I consider the several {non-Roman} {non-Roman} {non-Roman} {non-Roman} {non-Roman} which the Grecians had , some whereof did fly , some had no leggs &c. I can think that the origine of this name may have been thus . Some amazed person saw a spectrum , and giving another notice of it , his companion might answer , it is {non-Roman} {non-Roman} {non-Roman} {non-Roman} {non-Roman} , &c. but he meeting with a new phantasme , cryes , {non-Roman} {non-Roman} {non-Roman} {non-Roman} {non-Roman} or {non-Roman} {non-Roman} {non-Roman} {non-Roman} {non-Roman} , for which apprehension of his some body coyned this expression of {non-Roman} {non-Roman} {non-Roman} {non-Roman} {non-Roman} . It may also be possibly deduc'd from {non-Roman} {non-Roman} {non-Roman} {non-Roman} {non-Roman} might afterwards be reduced to the single terme of Empusa . Nor do I much doubt , but that those who are conversant in Languages , and know how that several expressions are often jumbled together to make up one word upon such like cases , will think this a probable origination . I beleeve then that Mr. Hobbes's friend did never tell him it was in Eustathius , or that Empusa was an hopping phantasme . It had two legges , and went upon both , as a man may upon a wooden legge . {non-Roman} {non-Roman} {non-Roman} {non-Roman} {non-Roman} is also a name for Lamia , and such was that which Menippus might have married , which I suppose did neither hop nor go upon one legge , for he might have discovered it . But Mr. Hobbes did not except against the derivation ( although he might justly , derivations made afterwards carrying more of fancy then of truth , and the Dr. is not excused for asserting what others barely relate , none approve ) but asked him , where that is , in what Authors he read that boyes play to be so called . To which question the Dr. ( to shew his reading and the good Authors he is conversant in ) replies , In Junius's Nemenclator , Rider and Thomas's Dictionary , sufficient Authors in such a businesss , which me thinks no man should say that were neer to so copious a Library . It is to be remembred that the trial now is in Westminster School , & amongst Ciceronians , neither whereof wil allow those to be sufficient Authors of any Latine word . Alas they are but Vocabularies ; and if they bring no Author for their allegation , all that may be allowed them is , that by way of allusion our modern play may be called Ludus Empusae . But that it is so called , we must expect till some Author do give it the name . These are so good Authors that I have not either of them in my Library . But I have taken the paines to consult , first Rider ; I looked in him ( who was onely Author of the English Dictionary ) and I could not finde any such thing . T is true in the Latine Dictionary which is joyned with Rider , but made by Holyoke ; ( O that the Doctor would but mark ! ) in the Index of obsolete words , there is Ascoliasmus , Ludus Empusae , Fox to thy hole , for which word , not signification , he quoteth Junius . The same is in Yhomasius , who refers to Junius in like manner . But could the Dr. think the word obsolete when the play is still in fashion ? Or doth he think that this play is so ancient , as to have had a name so long ago , that it should now be grown obsolete ? As for Junius's interpretation of Empusa , it is this , Empusa , spectrum quod se in●elicibus ingerit , uno pede ingrediens . Had the Dr. ever read him , he would have quoted him for his derivation of Empusa , I suppose . In Ascoliasmus he saith , Ascoliasinus , Ludus Empusae fit ubi , altero pede in aere librato unico sub●●liunt pede . {non-Roman} {non-Roman} {non-Roman} {non-Roman} {non-Roman} , Pollux . Almanicè . Hinctelen . Belgi●è , Op 〈◊〉 been springhen . Hinctepin●●en Flandris . But what is it in English he doth not tell● , although he doth so in other places often . What the Dr. can pick out of the Dutch I know not ; but if that do not justifie him , as I think it doth not , he hath wronged 〈◊〉 , and grossely imposed upon his Readers . But to illustrate this controversie further , I cannot be perswaded the Dr. ever looked into Junius , for if he had , I am confident , according to his wonted accuratnesse , he would have cited Polluxs's onomasticon into the bargin , for Junius refers to him , and I shall set down his words , that so the Reader may see what Ascoliasmus was , and all the Drs. Authors say Ludus Empusae and Ascoliasmus were one and the same thing . Jul. Pollux , Lib. 9. Cap. 7. {non-Roman} {non-Roman} {non-Roman} {non-Roman} {non-Roman} . ( old editions read it {non-Roman} {non-Roman} {non-Roman} {non-Roman} {non-Roman} ) {non-Roman} {non-Roman} {non-Roman} {non-Roman} {non-Roman} . So that Ascoliasmus , and consequently , Ludus Empusae , was a certain sport which consisted in hopping , whether it were by striving who could hoppe surthest , or whether onely one did pursue the rest hopping , and they fled before him on both leggs , which game he was to continue till he had caught one of his fellows , or whether it did consist in the boyes striving who could hop longest . Or lastly , whether it did consist in hopping upon a certain bladder , which being blown up , and well oyled over , was placed upon the ground for them to hop upon , that so the unctuous bladder might slip from under them and give them a fall . And this is all that Pollux holds forth . Now of all these wayes there is none that hath any resemblance with our , Fox to thy hole ; but the second : and yet in its description there is no mention of beating him with gloves , as they do now adayes , and wherein the play consists as well as in hopping . It might notwithstanding be called Ludus Empusae , but not in any sort our Fox to thy hole , So that the Dr. and his Authors are out ; imposing that upon Junius and Pollux , which they never said . And thus much may suffice as to this point . I shall onely adde out of Meu●…'s Ludi Geaeci , that Ascolia were not Ludus Empus● but Bacchisacra , and he quotes Aristophanes's Scholiasts in Plutus , {non-Roman} {non-Roman} {non-Roman} {non-Roman} {non-Roman} . As also Hesychius ; {non-Roman} {non-Roman} {non-Roman} {non-Roman} {non-Roman} . But I could have told the Dr. where he might have read of Empusa as being the name of a certain sport or game , and that is , in Tu●nebus adve sar . Lib. 27. Cap. 33 There he speaks of several games mentioned by I●stinian in his C●de , at the latter end of the third book ; one of which he takes to be named Empusa ; addding with all , that the other are games , it is indisputable , onely Empusa , in li●e & causa erit , quod nemo nobis ●a ite ass●… sit Ludum esse , eùm constet spectrum queddam fuisse formas variè mutans . Sed quid vetat eo nomine Ludum fuisse ? certè ad vestigia vittatae Scri●… quàm proxin è accedit . Yet ●e onely is satisfied in this conjecture , till somebody else shall produce a ●etter . And now what shall I say ? was not Turnebus as good a Critick , and of as great Bead●… ? as Dr. Wall●s ? who had read over Pollux , and yet is afraid that no body will beleeve 〈◊〉 to have been a game , and all he alledgeth for it is , quid vetat ? Truely all I shall say , and so conclude this businesse , is , that he had read over an infinitie of ●ooks , yet had not had the happinesse which the Dr. had , to consult with Junius's Nomenclator , Thom●…us and Riders Dictionary , Authors sufficient in such a case . I now come to the Doctors last and greatest triumph , at which I cannot but stand 〈◊〉 admiration , when I consider he hath not got the victory . Had the Dr. been pleased to have conversed with some , of the fift form in Westminster Schoole ( for he needed not to have troubled the learned Master ) he might have been better informed then to have exposed himself thus . Mr. Hobbes had said that {non-Roman} {non-Roman} {non-Roman} {non-Roman} {non-Roman} signified a mark with a hot iron ; upon which saying the Dr. is pleased to play the droll thus , Prethee tell mee , good Tho ▪ before we leave this point . ( O the wit of a Divinity Dr. ! ) who it was told thee that {non-Roman} {non-Roman} {non-Roman} {non-Roman} {non-Roman} was a mark with an hot iron ? for t is a notion I never heard til now ( and do not beleeve it yet . ) Never beleeve him again that told thee that lye , for as sure as can be , he did it to abuse thee , {non-Roman} {non-Roman} {non-Roman} {non-Roman} {non-Roman} signifies a distinctive point in writing made with a pen or quill , not a mark made with a hot iron , such as they brand rogues withal ; and accordingly {non-Roman} {non-Roman} {non-Roman} {non-Roman} {non-Roman} , distinguo , interstinguo , are often so used . It is also used of a Mathematical point , or somewhat else that is very small , {non-Roman} {non-Roman} {non-Roman} {non-Roman} {non-Roman} , a moment , or the like . What should come in your cap , to make you think that {non-Roman} {non-Roman} {non-Roman} {non-Roman} {non-Roman} signifies a mark or brand with a hot iron ? I perceive where the businesse lies ; 't was {non-Roman} {non-Roman} {non-Roman} {non-Roman} {non-Roman} ran in your minde when you talked of {non-Roman} {non-Roman} {non-Roman} {non-Roman} {non-Roman} , and because the words are somewhat alike , you jumbled them both together , according to your usual care and a●curatenesse , as if they had been the same . When I read this , I cannot but be astonished at the Doctors confidence , and applaud him who said , {non-Roman} {non-Roman} {non-Roman} {non-Roman} {non-Roman} . That the Dr. should never hear that {non-Roman} {non-Roman} {non-Roman} {non-Roman} {non-Roman} signifies a mark with a hot iron , is a manifest argument of his ignorance . But that he should advise Mr. H●bbes not to beleeve his own Readings or any mans else that should tell him it did signifie any such thing , is a piece of notorious impudence . That {non-Roman} {non-Roman} {non-Roman} {non-Roman} {non-Roman} signifies a distinctive point in writing made with a pen or quill , ( is a pen one thing and a quill another to write with ? ) no body denies . But it must be withal acknowledged it signifies many things else . I know the Dr. is a good Historian ( else he should not presume to object the want of History to another ) let him tell us how long ago it is , since men have made use of pens or quills in writing ; for if that invention be of no long standing ; this signification must also be such , and so it could not be that from any ●allusion thereunto the Mathe maticians used it for a point . Another thing I would fain know of this great Historian , how long ago {non-Roman} {non-Roman} {non-Roman} {non-Roman} {non-Roman} and {non-Roman} {non-Roman} {non-Roman} {non-Roman} {non-Roman} began to signifie inter●ungo ? for if the Mathematicks were studied before the mystery of Printing was found out ( as shall be proved when ever it shall please the Dr. out of his no reading to maintain the contrary ) then the Mathematical use thereof should have been named before the Grammatical . And if this word be translatitions , and that Sciences were the effect of long contemplation , the names used wherein are borrowed ▪ from talk , Mr. Hobbes did wel to say that {non-Roman} {non-Roman} {non-Roman} {non-Roman} {non-Roman} precedaneously to that indivisible signification , which it afterwards had , did signifie a visible mark , made by a hot iron , or the like . And in this procedure , he did no more then any man would have done ; who considers that all our knowledge proceeds from our senses ; as also that words do , primarily , signifie things obvious to sense , and only secondarily , such as men call incorporeal . This leads me to a further consideration of this word . Hesychius ( of whom it is said that he is Legendus non ●anquam L●xicographus , sed tanquam justus Author . ) Interprets {non-Roman} {non-Roman} {non-Roman} {non-Roman} {non-Roman} , which is a point of a greater or lesser size , made with any thing . So {non-Roman} {non-Roman} {non-Roman} {non-Roman} {non-Roman} signifies to prick or marke with any thing in a●y manner , and hath no impropriated signification in it self , but according to the writer that useth it . Thus in a Grammarian {non-Roman} {non-Roman} {non-Roman} {non-Roman} {non-Roman} signifies to distinguish by poin●ing often , sometimes , even in them , it is the same with {non-Roman} {non-Roman} {non-Roman} {non-Roman} {non-Roman} , sometimes it signifies to set a ma●k , that something is wanting in that place , which marks were called {non-Roman} {non-Roman} {non-Roman} {non-Roman} {non-Roman} . In maters of policy {non-Roman} {non-Roman} {non-Roman} {non-Roman} {non-Roman} signifies to disallow , because they used to put a {non-Roman} {non-Roman} {non-Roman} {non-Roman} {non-Roman} ( not {non-Roman} {non-Roman} {non-Roman} {non-Roman} {non-Roman} ) before his name who was either disapproved , or to be mulcted . In punishments it signifies to mark or brand , whereof I cannot at present remember any other wayes then that of an hot iron , which is most us●al in Authors , because most practised by the Antients . But that the marke which the Turks●nd others do imprint without burning may be said {non-Roman} {non-Roman} {non-Roman} {non-Roman} {non-Roman} , I do not doubt ; no more then that H●rodian did to give that term to the antient Britaines , of whom he sayes , {non-Roman} {non-Roman} {non-Roman} {non-Roman} {non-Roman} . Thus Horses that were branded with {non-Roman} {non-Roman} {non-Roman} {non-Roman} {non-Roman} and {non-Roman} {non-Roman} {non-Roman} {non-Roman} {non-Roman} ( {non-Roman} {non-Roman} {non-Roman} {non-Roman} {non-Roman} and {non-Roman} {non-Roman} {non-Roman} {non-Roman} {non-Roman} ) were said {non-Roman} {non-Roman} {non-Roman} {non-Roman} {non-Roman} . Thus in its origine {non-Roman} {non-Roman} {non-Roman} {non-Roman} {non-Roman} doth signifi● a b●end or mark with an hot Iron , ●r the like , and that must be the proper signification of {non-Roman} {non-Roman} {non-Roman} {non-Roman} {non-Roman} , which is proper to {non-Roman} {non-Roman} {non-Roman} {non-Roman} {non-Roman} , none but such as Dr. Wallis●an doubt . In its descendants it is no less evident , for from {non-Roman} {non-Roman} {non-Roman} {non-Roman} {non-Roman} comes Stigmosus , which signifies to be branded ; Vitellianâ cicatrice Stig●●sus not Stigmatosus . So Pliny in his Epistles , as Robert Stephen cites it . And {non-Roman} {non-Roman} {non-Roman} {non-Roman} {non-Roman} ( the Derivative of {non-Roman} {non-Roman} {non-Roman} {non-Roman} {non-Roman} , which signifies any mark , as well as a brand , even such as remain after stripes , being black and blew ) was a Nick-name imposed upon the Grammarian Nicanor , {non-Roman} {non-Roman} {non-Roman} {non-Roman} {non-Roman} ▪ And though we had not any examples of {non-Roman} {non-Roman} {non-Roman} {non-Roman} {non-Roman} being used in this sense , yet from thence for any man to argue against it ( but he who knows no more then Stephen tells him ) is madness , unless he will deny that any word hath lost its right signification , and is used only ( by the Authors we have , although neither the Dr. nor I have read all them ) in its analogical signification . I have alwayes been of opinion , that {non-Roman} {non-Roman} {non-Roman} {non-Roman} {non-Roman} signified a single point , big or little , it matters not ; and {non-Roman} {non-Roman} {non-Roman} {non-Roman} {non-Roman} , a composure of many ; as {non-Roman} {non-Roman} {non-Roman} {non-Roman} {non-Roman} signifies a line , and {non-Roman} {non-Roman} {non-Roman} {non-Roman} {non-Roman} a l●tter , made of several lines . For {non-Roman} {non-Roman} {non-Roman} {non-Roman} {non-Roman} signified the Owle , the S●mana , the letter K , yea whole words , lines , Epigrams engraven in mens faces ; and {non-Roman} {non-Roman} {non-Roman} {non-Roman} {non-Roman} , I doubt not , had signified a single point , had such been used , and so it became translatitiously used by Grammarians and Mathematicians . I could give grounds for this conjecture , and not be so impertinent as the Doctor in his Sermon ( where he told men that {non-Roman} {non-Roman} {non-Roman} {non-Roman} {non-Roman} was not in Homer ; that from {non-Roman} {non-Roman} {non-Roman} {non-Roman} {non-Roman} came Ebrius ; that Sobrietas was not bad Latine , and that Sobrius was once ( as I remember ) in Tully . Is this to speak suitably to the oracles of God , or rather to lash out into idle words ? Hath the Dr. any ground to think these are not impertinences ? Or are we , poor mortals , accountable for such idle words as fall from us in private discourses , whilest these Ambassadours of Heaven droll in the Pulpit without any danger of an after reckoning ? But I proceed to a further survey of the Doctors intolerable ignorance . His charge in the end of the School ▪ Masters Rant , is , that he should remember {non-Roman} {non-Roman} {non-Roman} {non-Roman} {non-Roman} & {non-Roman} {non-Roman} {non-Roman} {non-Roman} {non-Roman} are not all one . I complained before that he hath not cited Robert Stephen aright , now I must tell him he hath been negligent in the reading of Henry Stephen ; for in him he might have found that {non-Roman} {non-Roman} {non-Roman} {non-Roman} {non-Roman} was sometimes all one with {non-Roman} {non-Roman} {non-Roman} {non-Roman} {non-Roman} though there be no example in him wherein {non-Roman} {non-Roman} {non-Roman} {non-Roman} {non-Roman} is ●sed for {non-Roman} {non-Roman} {non-Roman} {non-Roman} {non-Roman} . Hath not Hes●od ( as Stephen rightly citeth it ) in his S●u●um . {non-Roman} {non-Roman} {non-Roman} {non-Roman} {non-Roman} ubi Scholiastes ; {non-Roman} {non-Roman} {non-Roman} {non-Roman} {non-Roman} . So Johannes Di●conus upon the place , a man , who ( if I may use the Drs. phrase ) was as good a critick , as the Geometry Professour . Thus much for the Doctor . To the understanding Reader , I say , that {non-Roman} {non-Roman} {non-Roman} {non-Roman} {non-Roman} is ●sed for burning with a hot iron ; 〈◊〉 ▪ Macchab. 9. 11. where speaking of Antiochus's lamentable death , his body putrifying , and breeding worms , he is said {non-Roman} {non-Roman} {non-Roman} {non-Roman} {non-Roman} ; being pained as if he had been pricked , or burned with hot ir●ns . And that this is the meaning of that elegant writer , shall be made good against the Doct●r , when he shall please to defend the vulgar Interpretation ▪ Pausanias , in Baeoticis , speaking of Epaminondas , who had taken a town belonging to the Sicyonians called Phaebia ( {non-Roman} {non-Roman} {non-Roman} {non-Roman} {non-Roman} ) wherein were many Baeotian fugitives , who ought by Law to have been put to death , saith , he dismissed them under other names , giving them onely a brand or mark {non-Roman} {non-Roman} {non-Roman} {non-Roman} {non-Roman} . It is true {non-Roman} {non-Roman} {non-Roman} {non-Roman} {non-Roman} is here put adverbially , but that doth not alter the case . Again Zonaras , in the third ●ome of his History , in the l●fe of the Emperour Th●ophilus , saith , that when Theophanes and another Monk had reproved the said Emperour for demolishing images , he took and s●igmatised each of them with twelve lambicks in their faces . {non-Roman} {non-Roman} {non-Roman} {non-Roman} {non-Roman} . A place so evident , that I know not what the Dr. can reply . This place is just parallel to what the same Author saith in the li●e of Irene , {non-Roman} {non-Roman} {non-Roman} {non-Roman} {non-Roman} . If the Dr. object that he is a moderne Author , he will never be able to render him as inconsiderable as Adrianus Junius's Nomenclator , Th●mafius and Rider . If any will deny that he writes good Greek , Hieronymus W●lfius will tell them , his onely fault is , {non-Roman} {non-Roman} {non-Roman} {non-Roman} {non-Roman} , redundancy in words , and not the use of bad ones . Another example of {non-Roman} {non-Roman} {non-Roman} {non-Roman} {non-Roman} used in this sense , is in the Collections out of Diodorus Siculus lib. 34. as they are to be found at the end of his works , and as Photius hath transcr●bed them into his Bibliotheca . He ●aith that the Romans did buy multitudes of servants and employ them in Sicily ; {non-Roman} {non-Roman} {non-Roman} {non-Roman} {non-Roman} . These are the words but of one Author , but ought to passe for the judgement of two , seeing Photius , by inserting them , hath made them his own . Besides , it is the judgement of a great Master of the Greek tongue , that stigmata non tam puncta ipsa qudm punctis variatam super●●ciem Gr●ci vocaverun● : I need not I suppose , name him , so gre●t a Critick as the Dr. cannot be ignorant of him . Nor were {non-Roman} {non-Roman} {non-Roman} {non-Roman} {non-Roman} commonly , but upon extraordinary occasions , imprinted with an hot iron . The letters were first made by incision , then the blood pressed , and the place filled up with ink , the composition whereof is to be seen in Ae●ius . And thus they did use to matriculate Souldiers also in the hand . Thus did the Grecian Emperour in the precedent example of Zonaras . And if the Dr. would more , let him repair to Vin●tas's Comment upon the fifteenth Epigram of Ausonius . And now I conceive enough hath hath been said to vindicate Mr. Hobbes , and to shew the insufferable ignorance of the puny professor , and unlearned Critick . If any more shall be though● necessary , I shall take the paines to collect more examples and Authorities , though I confess I had rather spend time otherwise then in matter of so little moment . As for some other pssages in his book , I am no competent judge of Symbolick Stenography . The Dr. ( Sir Reverence ) might have used a cleanlier expression then that of a s●itten piece , when he censures Mr. Hobbes's Book . Hitherto the letter . By which you may see what came in to my ( not square ) cap to call {non-Roman} {non-Roman} {non-Roman} {non-Roman} {non-Roman} a mark with a hot iron , and that , they who told me that , did no more tell me a lie then they told you a lie that said the same of {non-Roman} {non-Roman} {non-Roman} {non-Roman} {non-Roman} ; and if {non-Roman} {non-Roman} {non-Roman} {non-Roman} {non-Roman} be not right as I use it now , ●hen call these notes not {non-Roman} {non-Roman} {non-Roman} {non-Roman} {non-Roman} , but {non-Roman} {non-Roman} {non-Roman} {non-Roman} {non-Roman} I will not contend with you for a tris●e . For howsoever you call them you are like to be known by them . Sir , the ▪ calling of a Divine hath justly taken from you some time that might have been imployed in Geometry . The study of Algebra hath taken from you another part , for Algebra and Geometry are not all one ; and you have cast away much time in practising and trusting to Symbolical writ writings , and for the Authors of Geometry you have read , you have not examined their demonstrations to the bottom . Therefore you perhaps may be , but are not yet 〈◊〉 Geometrician , much lesse a good Divine . I would you had but so much Ethicks , as to be civil . But you are a notable Critick So fare you well , and consider what honour you do either to the University where you are received for professor , or to the Vniversity from whence you came thither , by your Geometry ; and what honour you do to Emanuel Colledge by your Divinity ; and what honour you do to the degree of Dr. with the manner of your Language . And take the counsel which you publish out of your Encomiast his letter ; think me no more worthy of your pains , you see how I have ●ouled your fingers . FINIS . 
A67916	----	An introduction of the first grounds or rudiments of arithmetick plainly explaining the five common parts of that most useful and necessary art, in whole numbers & fractions, with their use in reduction, and the rule of three direct. Reverse. Double. By way of question and answer; for the ease of the teacher, and benefit of the learner. Composed not only for general good, but also for fitting youth for trade. / By W. Jackson student in arithmetick. Jackson, William, 1636 or 7-1680. This text is an enriched version of the TCP digital transcription A67916 of text R210093 in the English Short Title Catalog (Wing J94). Textual changes and metadata enrichments aim at making the text more computationally tractable, easier to read, and suitable for network-based collaborative curation by amateur and professional end users from many walks of life. The text has been tokenized and linguistically annotated with MorphAdorner. The annotation includes standard spellings that support the display of a text in a standardized format that preserves archaic forms ('loveth', 'seekest'). Textual changes aim at restoring the text the author or stationer meant to publish. This text has not been fully proofread Approx. 80 KB of XML-encoded text transcribed from 56 1-bit group-IV TIFF page images. EarlyPrint Project Evanston,IL, Notre Dame, IN, St. Louis, MO 2017 A67916 Wing J94 ESTC R210093 99868922 99868922 121279 This keyboarded and encoded edition of the work described above is co-owned by the institutions providing financial support to the Early English Books Online Text Creation Partnership. This Phase I text is available for reuse, according to the terms of Creative Commons 0 1.0 Universal . The text can be copied, modified, distributed and performed, even for commercial purposes, all without asking permission. Early English books online. (EEBO-TCP ; phase 1, no. A67916) Transcribed from: (Early English Books Online ; image set 121279) Images scanned from microfilm: (Thomason Tracts ; 241:E2110[3]) An introduction of the first grounds or rudiments of arithmetick plainly explaining the five common parts of that most useful and necessary art, in whole numbers & fractions, with their use in reduction, and the rule of three direct. Reverse. Double. By way of question and answer; for the ease of the teacher, and benefit of the learner. Composed not only for general good, but also for fitting youth for trade. / By W. Jackson student in arithmetick. Jackson, William, 1636 or 7-1680. [6], 102 p. Printed for R.I. for F Smith, neer Temple-Bar, London : 1661 [i.e. 1660] Annotation on Thomason copy: "Oct:"; the second '1' in imprint date is altered to "0". Reproduction of the original in the British Library. eng Arithmetic -- Early works to 1800. Mathematics -- Study and teaching -- Early works to 1800. A67916 R210093 (Wing J94). civilwar no An introduction of the first grounds or rudiments of arithmetick; plainly explaining the five common parts of that most useful and necessary Jackson, William 1660 14334 23 0 0 0 0 0 16 C The rate of 16 defects per 10,000 words puts this text in the C category of texts with between 10 and 35 defects per 10,000 words. 2000-00 TCP Assigned for keying and markup 2002-02 Aptara Keyed and coded from ProQuest page images 2002-03 TCP Staff (Michigan) Sampled and proofread 2002-03 John Latta Text and markup reviewed and edited 2002-04 pfs Batch review (QC) and XML conversion AN Introduction of the First Grounds or Rudiments OF Arithmetick ; Plainly explaining the five Common parts of that most useful and Necessary Art , In whole Numbers & Fractions , With their use in Reduction , and The Rule of three : Direct . Reverse . Double . By way of Question and Answer , for the ease of the Teacher , and benefit of the Learner . Composed not only for general good , but also for fitting Youth for Trade . By W. Iackson Student in Arithmetick . LONDON , Printed by R. I. for F Smith , neer Temple-Bar . 166● Courteous Reader , I Have on purpose omitted Progression , as also many other Rules following , partly because that these being well learned , not only by rote , but also by reason , the young learner ( for whose sake I wrote this ) will be inabled hereby in a good measure to understand what hee findes in other books concerning such ; And if this prove but as useful , as I wish it may , and hope it will ( by the Teachers care , and Scholars diligence ) I may be incouraged to add somewhat to it hereafter , that may bee of further use , or else these weak indeavours may provoke some others of better parts to bring them to the publick Treasurie of Art . In the mean time accept of this mite from him that is one that would count it an honour to bee but one of the meanest of those that might present any thing on the behalf of this most Noble , and most Necessary Art of Arithmetick ; that might further the growth of such as are entring upon the practice of the same , which I presume , if this small Tract may bee as a small Table wherein to see the first Rudiments in , briefly and plainly , which being by the Masters discretion appointed the young Scholar to get by heart , may prove an ease to both ; to the Master , in that ( if hee please to spend some set time in examining his Scholars , as they use to catechize little ones ) hee by that means may teach the Rules to twenty in teaching one , and not only print the Rules in the memory of such as are past such Rules , who perhaps may bee apt to forget , but also teach the Rudiments to other , even before they come to the practice of them , whereby hee may save the pains of often telling them , and may only fit them with examples suitable to the Rule , sometimes descanting a little upon the Rules as they lye in order , as he findes occasion ; and by this course , being observed , he will with the blessing of God finde by the childrens growth in knowledge , that the pains bestowed will not be in vain ; but not to be tedious , I leave each to use his own discretion , how to use this or any other help , only I have thoughts , that a thing of this nature will bee profitable , and have its use ; So wishing this most Noble Art , and all those that love it , to flourish in our Land , I bid thee farewel . W. J. AN Introduction of the First Grounds and Rudiments of Arithmetick . NUMERATION . Quest . WHat is Arithmetick ? Answ. It is the Art of numbring . Q. What is the subject of this Art ? A. The subject of it is Number . Q. Whereof doth Number consist ? A. It consisteth of unites . Q. What is an unite ? A. It is the original , or beginning of number , and is of it self indivisible , so that it still remaineth one . Q. Is not one a Number then ? A. No , for Number is a collection of unites . Q. How are Numbers said to bee divided into kinds ? A. They are divided into many sorts , but vulgarly into whole numbers , and broken numbers called fractions . Q. Are fractions numbers ? A. Not properly , for number consisteth of a multitude of unites , but every fraction is lesser than its unite . Q. How many several parts are accounted in common Arithmetick ? A. These five , Numeration , Addition , Substraction , Multiplication , and Division . Q. What then is extraction of roots ? A. Although it bee another part of Arithmetick , yet it is not so common . Q. What teacheth Numeration ? A. It teacheth how to set down any number in figures , and also to express , or read any such number so set down . Q. How many figures are there ? A. Nine significant figures , and a cipher , which cipher signifieth nothing of it self , only it serveth to supply a place , and thereby increaseth the value of the other figures . Q. Which be the significant figure ? A. 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9. Q. What do these signifie ? A. They signifie only each their own simple value being alone . Q. What if they be joyned with other figures , or ciphers , is their signification altered ? A. Yea , their value is thereby much increased , according to the place they stand in , removed from the place of unites . Q. Do Ciphers then only supply places that are void , and so increase the value of the other figures ? A. No , they do also in decimal fractions , diminish the value of those figures that stand toward the right hand of them , according to the place they stand in removed from the unite place . Q. Which is the place of unites ? A. In whole numbers it is the first place towards the right hand , and any figure standing in that place , signifieth only its own simple value . Q. Why make you a distinction here of whole numbers ? doth it differ infraction ? A. Yea , for in decimal fractions , the unite place standeth to the left hand of the fraction . Q. Why say you its own simple value ? A. Because a figure by being put in the second , third , or fourth place , &c. may signifie ten times , an hundred times or a thousand times its own value , &c. Q. What is the reason of that ? A. Because every place exceeds the place next before it ten times in value , so that the figure that signifies but four in the first place , signifies ten times 4 in the second , and an hundred times four in the third place , and a thousand times four in the fourth place , &c. and in that proportion increaseth infinitely , according as its place is further removed from the unite place . Q. Is the proportion of diminishing decimal fractions , like this of augmenting whole numbers ? A. Yea ; for as these are augmented in a decupled proportion , so those are diminished , or made less , in a decupled proportion , by being removed from the unite place . Q. What must you do when you have a number to set down , where some have not a significant figure to stand in it ? A. I must supply that place with a cipher ( 0 ) for no place must bee void . Q. How will you set down , one thousand six hundred and sixty ? A. First I consider the fourth place is the place of thousands , and there I set down 1 , then 6 in the third place , which betokeneth so many hundreds , then 6 in the second place , signifying six tens , or sixty , and because there is no figure to set in the place of unites , I supply it with a ( 0 ) cipher , to make the number consist of its due number of places , thus ( 1660 ) Q. How set you down four thousand five hundred and six ? A. First I set 4 in the place of thousands , then five in the place of hundreds , then in regard thee is no tens , I supply that place with a ( 0 ) and lastly I place 6 in the unite place thus , ( 4506 ) Q. How value you your places in order to the tenth place ? A. Thus , Unites , Tens , Hundreds , Thousands , Tens of Thousands , Hundreds of Thousands , Millions , Tens of Millions , Hundreds of Millions , Thousands of Millions . Q. Can you repeat your places backwards ? A. Yea , thus , Thousands of Millions , Hundreds of Millions , Tens of Millions , Millions , Hundreds of Thousands , Ten of Thousands , Thousands , Hundreds , Tens , Unites . Addition . Q. Now shew what Addition teacheth . A. Addition teacheth of several numbers to make one total equal to them all . Q. How is that done ? A. First I set my numbers down one right under another , observing still to set the first place or figure toward the right hand of each number under the first place or figure of the uppermost number , and so the second under the second , and the third under the third , &c. Q. After you have set down your numbers each in his due place , what do you then ? A. I must begin at the first place toward the right hand , and count all the figures in that place together , and if they bee less than ten , set it down under the first place , a line being first drawn under my sum to place my total below . Q. What if it come to ten or above ? A. Then I must consider how many tens it contains , and carry so many unites in my mind to the next place , and set down the over-plus if there bee any , but if it bee even tens , then set down a ( 0 ) in that place . Q. And what do you next ? A. I must remember to reckon the tens that I bear in mind for unites , and add them to the figures in the next place , and then do in all points as I did in the former place . Q. Why do you count tens in one place , but for unites in the next ? A. because the place answers to the value thereof , being ten times the value of the former place . Q. What is further to bee considered ? A. When I have gone thorow all the places , if at the last I have any Tens to carry , seeing there are no figures to add them withall , I set a figure signifying the number of Tens , in a place neerer the left hand . Q. Give an example hereof . A. Then thus , Four men owe my Master mony ; A. oweth 4560 l. B. oweth 5607 l. C. oweth 6078 l. D : oweth 385 l. I would know how much these four debts amount to in all . Q. And how will you do that ? A. First I set the several sums right under each other thus , 4560 5607 6078 0385 16630 Then I begin with those figures next the right hand , and say , 5 and 8 is 13 , and 7 makes 20 , now in regard it is just 2 times Ten , I set a cipher underneath the line in that place , and bear in minde 2 to reckon with the figures in the next place , and say , 2 that I bear in my mind , and 8 is 10 , and 7 is 17 , and 6 makes 23 , then I set down the odd 3 in the second place below the line , and for the 2 Tens , I carry 2 in minde to the next place , then I say , 2 that I carry and 3 is 5 , and 6 is 11 , and 5 is 16 , the odd 6 I set below the line , and carry one in lieu of the ten to the next place , and say , 1 I carry , and 6 is 7 , and 5 , is 12 , and 4 is 16 , so I set the 6 below the line , and in regard there is not another place to reckon the one , I bear in minde withall , I set that 1 a place neerer the left hand , and so the total is 16630 l. the sum of those four debts . Q. What if you have numbers of several kinds or denominations to add together ? A. I must set down each number under the denomination of the same kinde , as pounds under pounds , shillings under shillings , and pence under pence , &c. and the like is to bee observed of weight , measure , or any other kinde . Q. And how must they bee added together ? A. I must begin with the smallest denomination , which is next the right hand , and count all those figures together , and consider how many of the next denomination is contained in them , and carry so many unites to the next place , and set down the over-plus ( if there bee any ) right under beneath the line , and it there bee no over-plus , I set a cipher in the place ; And the like I observe in every several denomination . Q. Shew two or three examples of several denominations . A. First , then for pounds , shillings , and pence , take this , l. s. d. 365 6 8 456 7 6 567 8 4 1389 2 6 Then beginning with the smallest denomination toward the right hand , which is pence , I say , 4 and 6 is 10 , and 8 is 18 , which is 1 shilling and 6 pence , the 6 pence I set down in its place below the line , under its own denomination , and carry the one shilling in my minde to the next place , which is the place of shillings , and say , 1 I carry and 8 is 9 , and 7 is 16 , and 6 is 22 , that is one pound and two shillings , the 2 odd shillings I set in its place under its own denomination below the line , and carry one pound in minde to reckon with the pounds , then I come to the first place of pounds , and say , 1 I carry and 7 is 8 , and 6 is 14 , and 5 is 19 , so I set down 9 , and carry 1 , then I say , one I carry and 6 is 7 , and 5 is 12 , and 6 makes 18 , the 8 I set down , and carry one , and then I say , one I carry and 5 is 6 , and 4 is 10 , and 3 is 13 , the odd 3 I set right under , and the one ten I set one place further towards the left hand , so the total is 1389 l. 2 s. 6. d. Q. What is your next example ? A. Take this , of haberdupoize weight , wherein note that 16 ounces make a pound , 28 pound make a quartern , 4 quarterns make a hundred weight , and 20 hundred make a tun weight : Example . tun . C. q quartern . l. o ℥ 123 9 3 16 10 234 8 2 12 8 345 7 1 08 6 703 5 3 09 8 Where as before , I begin with the least denomination next the right hand , and say , 6 and 8 is 14 , and 10 is 24 , which is one pound and eight ounces , the 8 ounces I set down below , and I carry one in minde to the pounds , then I say , one I carry and 8 is 9 , and 12 is 21 , and 16 is 37 , that is one quartern , and 9 pound , the 9 I set down and carry one , and say , one that I carry , and one is two , and two makes 4 , and 3 is 7 , that is one hundred and three quarterns , the three quarterns I set down , and carry one , and say further , one I carry and 7 is 8 , and 8 is 16 , and 9 makes 25 , that is one tun and five hundred , the five hundred I set down , and carry one to the tuns , and say , one that I carry and 5 is 6 , and 4 is 10 , and 3 makes 13 , then I set down 3 , and carry one , saying , one I carry and 4 is 5 , and 3 is 8 , and 2 makes 10 , now being it is just 10 , I set down a cipher , and carry one , saying , one I carry and 3 is 4 , and 2 is 6 , and one makes 7 , which I set down in its place below the line , and so the total is tun . C. q quartern l. o ℥ . 703 5 3 09 8 as in the example . A third Example shall bee of liquid measure , in which note , that one tun is 4 hogs-heads , one hogs-head is 63 gallons , one gallon is 8 pints . tuns . hhd . gal. pints . 234 3 24 6 345 2 21 4 456 1 18 2 1036 3 01 4 Here I only set down the example , and cast it up , without describing the work , to move the learner to take some pains to do the like for his practice . Substraction . Q. Tell mee now what Substraction teacheth ? A. Substraction teacheth to abate , or withdraw a lesser sum or number out of a greater , and to shew the remainder or over-plus . Q. How is that performed ? A. First I set down the greater sum or number uppermost , and under it I draw a line , then I set the lesser sum under the line , observing to set each figure in its due place , under the greater sum , and then I begin at the first place , and abate the lower figure out of the higher , setting the remainder under it , a line being first drawn to separate them . Q. But what if the lower figure bee greatest , how then shall it bee abated from the higher ? A. Then I must borrow one of the next place , which here signifies the value of ten , and abate it from ten , and the uppermost figure added together , and set down the over-plus , or which is all one , abate it from ten , and adding the over-plus with the uppermost figure , set the same down beneath the line for remainder . Q. What is next to bee done ? A. Then for the unite I borrowed , I add one to my lower figure in the next place , abating the same out of the figure over it , doing in all respects as before . Q. Give an example hereof . A. Then thus , If I have borrowed 4567 l. of which I have repaid 3675 , what is behinde ? borrowed 4567 l. repaid 3675. rest behinde 0892 proof 4567 Where having placed my numbers duly under each other , I begin as before at the right hand , and say , 5 out of 7 , there rests 2 , which 2 I set underneath , as in the example , then I come to the next figure , saying , 7 from 6 , I cannot , wherefore I borrow one of the next place , which signifies ten here , and so abate 7 out of 16 , and set down the rest , which is 9 below the line ; and insomuch as I borrowed one , therefore I carry one in minde , and say , in the next place , one that I borrowed and 6 is 7 , which I should abate from 5 over it , which seeing I cannot do , I borrow one as I did before , and say , 7 out of 15 , there rests 8 , which I set underneath the line , and go on as before , saying , one that I borrowed , and 3 is 4 , which being abated from the 4 above it , rests nothing to set below , so that there remains behinde 892 l. as in the example , the proof hereof is by adding the sum paid , and the remainder together , if they make up the sum borrowed , it is right , or else not . Q. But when you have a sum of several denominations to substract from another , how do you then ? A. As in Addition I began at the smallest denomination to add , so here I begin with the same to substract , abating the lowest from the highest . Q. What if the upper figure bee the least ? A. Then I borrow one of the next denomination , and considering how many of the smaller is contained in one of those , I abate my figure or number to bee abated , out of that which I borrowed , and the uppermost number being added together , and set the over-plus below the line for the remainder . Q. Shew by an example or two what you mean ? A. To substract 8 d. from 1s . 4d . I say 8d . from 4d . I cannot , then I borrow one shilling , being the next place , which is 12d . to which I add the 4d . that is above the line , it makes 16d . and say , 8d . from 16d . rests 8d . Again , 17s . from 2l . 12s . I say , 17s . from 12s . I cannot , but 17s . from 1l . 12s . or 32s . there rests 15s . then considering I borrowed one pound , I say one pound that I borrowed from 2l . rests one pound , so that there rests 1l . 15s . to set below the line , And I must alwaies remember to reckon the one I borrowed to the figure that is to bee substracted in the next place . Q. Shew this by an example or two of several denominations . A. Then here is one , if I abate 345l . — 16s , — 8d . from 476l . — 13s . — 4d . I would see what remains . I set my sums thus ,   l. s. d.   476 13 4   345 16 8 rest 130 16 8 proof 476 13 4 Then beginning with the least denomination , which is pence , I say , 8d . from 4d . I cannot , but I borrow one of the next denomination , which is shillings , and say , 8d . from 1s . 4d . and there rests 8d . which I set under the pence , and then I say , one shilling I borrowed and 16 makes 17s . from 13s . I cannot , but I borrow one pound , and say , 17s . from 1l . 13s . rests 16s . then I say , 1l . that I borrowed , and 5l . is 6l . from 6l . rests nothing , so I set down a cipher in this place under the line , and go forward , saying 4 from 7 , rests 3 , which I set down , and then say , 3 from 4 , rests 1 , which I set beneath ; and so there rests 130l . 16s . 8d . And as in Addition I left one example onely cast up , for the learner to pause upon himself , and to imitate , so here I do the like .   tun C. q quartern l. o ℥   345 11 2 24 4 Substr . 256 13 3 20 8 rests 088 17 3 03 12 proof 345 11 2 24 4 Multiplication . Q. What doth Multiplication teach ? A. Multiplication teacheth after a brief & compendious way , to increase or augment any number , by so many times it self , as is any number propounded , as 4 times , 10 times , &c. Q. What is considerable in this Rule ? A. Three numbers are specially considerable , to wit , the multiplicand , or number that is to bee multiplied , secondly , the multiplier , or number wee multiply by , and thirdly , the product , which is the number produced by the multiplication of those two numbers each by other . Q. How many times doth the product contain the multiplicand ? A. Just so many times as there is unites in the multiplier . Q. How is Multiplication done ? A. First I set down the multiplicand , which customarily is the greater number , and under it I set the multiplier , each figure in its due place , and draw a line underneath , then I begin at the first figure of the multiplier toward the right hand , and multiply it by the first figure of the multiplicand , and set the product right under it beneath the line , if it exceed not nine . Q. If it exceed nine , what then ? A. Then I must keep in minde how many tens is in it , and carry so many unites to the next place , and set down the odd figure that is more than even tens underneath , but if it bee even tens , then set down a cipher underneath . Q. And what is then to be done ? A. Then I multiply the said first figure of the multiplier by the second figure of the multiplicand , and to the product add the unites reserved in my mind , & then do in all respects as I did before , and so I continue my work , till I have multiplied the first figure of the multiplier by all the figures of the multiplicand in order . Q. And what do you next ? A. Then I multiply the second figure of the multiplier by all the figures of the multiplicand , in like sort as I did the first , only I must observe to set my first place in this second work , one place nearer the left hand , that it may fall right under the figure I multiply by . Q. What is the reason of that ? A. Because every unite in the second place signifies 10 , in the third place , 100 , &c. Q. Is this order then to be kept in a sum of many figures or places ? A. Yea , the same order is to bee observed in any sum , be the places never so many , I must still set my first figure right under the figure I multiply by , and then the rest in order toward the left hand . Q. Having so set down all your figures , what remains further to bee done ? A. Only to add the several numbers together in order , beginning still at the first place next the right hand . Q. Give one Example . A. Let this bee it then . 2345 234 9380 7035 4690 548730 Where first I say , 4 times 5 is 20 , where I set a cipher below the line , and carry 2 , then I say , 4 times 4 is 16 , and 2 that I carried is 18 , the 8 I set down below , and carry 1 , then 4 times 3 is 12 , and 1 that I carried is 13 , then I set down 3 , and carry one , 4 times 2 ( or 2 times 4 , for it is all one ) makes 8 , and one that I carried is 9 , which I set down in its place , and cancel the first figure of my multiplier , with a dash through it , to signifie that it hath done its office , then I begin with the next figure , saying , 3 times 5 is 15 , the five I set down right under the 3 I multiply by , and carry one in minde , then I say , 3 times 4 is 12 , and one that I carried is 13 , the 3 I set down in the second place , and carry one , and say , 3 times 3 is 9 , and one I carried is 10 , where I set down a ( 0 ) and carry one , then I say , 3 times 2 is 6 , and one I carry is 7 , which I set down , and cancel my second figure of the multiplier , and begin with the third , saying , 2 times 5 is 10 , then I set down a cipher in that place right under my multiplier 2 , and carry one in mind to the next place , then I say , 2 times 4 is 8 , and one I carried is 9 , which I set down in the next place , in order , then I say , 2 times 3 is 6 , which I set in its due place , and lastly , I say , 2 times 2 is 4 , which I write down also , so have I multiplied all my figures of the multiplier , by all the figures of the multiplicand , there remains to add up all into one sum , which to do I begin at the right hand , and work as in Addition , and so the product is 548730 , as in the example . Here is another Example for Imitation . 963852 3741 963852 3855408 6746964 2891556 3605770332 Q. What proof is for Multiplication ? A. The truest proof is by Division , but it is ordinarily proved thus , they make a cross X And then cast away so many nines as can bee found in the multiplicand , and set the remainder on the upper side of the cross , and do the like with the multiplier , & set the remainder under the cross , then multiply the 2 remaidners 1 by another , and cast out the nines out of the product of them , setting the rest at one side of the cross ; and last of all cast out the nines out of the product of the Multiplication , and mark the rest , if it be like that which is placed on the side of the cross , it appears to bee right , or else it is not well done . A Table for Multiplication to bee got by heart . 2 times 2 is 4 2 3 6 3 4 8 2 5 10 2 6 12 2 7 14 2 8 16 2 9 18 3 times 3 is 9 3 4 12 3 5 15 3 6 18 3 7 21 3 8 24 3 9 27 4 times 4 is 16 4 5 20 4 6 24 4 times 7 is 28 4 8 32 4 9 36 5 times 5 is 25 5 6 30 5 7 35 5 8 40 5 9 45 6 times 6 is 36 6 7 42 6 8 48 6 9 54 7 times 7 is 49 7 8 56 7 9 63 8 times 8 is 64 8 9 72 9 times 9 is 81 Division . Q. Now shew mee what Division teacheth ? A. Division teacheth to finde how many times one number is contained in another number . Q. How many numbers are to bee noted in any Division ? A. Three , namely , the dividend , or number to bee divided , secondly , the divisor , or number dividing , thirdly , the quotient , which sheweth how often the divisor is contained in the dividend . Q. In what manner is Division performed ? A. First I set down my dividend , and under it I place my divisor , in such sort , that the figures next the left hand stand right under one another , and so each following place in order , except the divisor bee a greater number than so many figures of the dividend as stand over it , for then the divisor must bee removed a place nearer the right hand . Q. And what do you then ? A. Then I draw a crooked line to the right hand of my figures , to place my quotient beyond , and I consider how often I can take the divisor , out of the number over it , and set the number of times in the quotient , and multiplying the said quotient figure by the divisor , I substract the product from the figures above the divisor , setting the remainder over head , cancelling the other figures that were over the divisor , and also the divisor . Q. And how proceed your further ? A. Then I remove my divisor one place nearer the right hand , and consider as before how often I may take it out of the figures over head , and work in all points as before . Q. If there bee many removings of the divisor , is that order still to bee observed ? A. Yea , where the divisor can bee substracted once or oftner out of the dividend . Q. But what if you cannot take the divisor out of the figures over it ? A. I must then place a cipher in the quotient , and cancel the divisor , and remove it a place nearer the right hand , without cancelling the figures over head , and continue the work as before . Q. What else is to bee observed in Division ? A. If the divisor have any ciphers in the first places , they may bee placed under the first places of the dividend , and divide only by the other figures , till I come to those ciphers . Q. What must be done with the number that remains after the division is ended ? A. If any remainder be , I set it after the quotient , and the divisor under it , with a line drawn betwixt them , to express it in a fraction . Q. Give an Example or two in Division . A. Take this for one , to divid 30038 by 23 , I set it down thus , 30038 ( 23 Then having drawn a crooked line , to set the quotient in , I consider how often I can have my divisor , 23 in the number over it , which is 30 , which I can have but once , therefore I say , once 2 is 2 from 3 that is over it , and there remains I , which I set over the 3 , and cancel the 3 , and also the 2 under it , and it stands thus , 1 30038 ( 1 23 Then I say , once 3 is 3 , from 10 that is over it , and there rests 7 , and stands thus , 17 30038 ( 1 23 Then I remove the divisor one place nearer the right hand , And it stands thus , 17 30038 ( 1 233 2 Now I consider again as before , how often I can take 23 out of 70 that is over it , which I finde I may do 3 times , therefore I put 3 down in the quotient , and say , 3 times 2 is 6 , 6 out of 7 , rests one , which I set over the 7 , and cancel the 7 , and the 2 under it , and say , 3 times 3 is 9 , from 10 over it , rests one , and that I set over head , and cancel the 10 , and the 3 under it , And then it stands thus , 1 171 30038 ( 13 233 ● Then I remove the divisor again , And it stands thus , 1 171 30038 ( 13 2333 22 Then I consider that I cannot take my divisor 23 out of the number over it being but 13 , so I set a cipher in the quotient , and cancel the divisor , and remove it one place more , and let the figures over it stand as they were , And then it stands thus , 1 171 30038 ( 130 23333 222 Now I consider again how often I may take my divisor out of the number over it , which I finde I may do 6 times , wherefore I set 6 in the quotient , and say , 6 times 2 is 12 , from 13 that is over it , and there rests one , which I set over head , and cancel 13 , and 2 under it , And so it stands thus , 1 1711 30038 ( 1306 23333 222 Then I say 6 times 3 is 18 , from 18 that is over it , rests nothing , And the whole work stands thus , 1 1711 30038 ( 1306 23333 222 Here also I set an Example or two , for Imitation . Example . 1 4231 20673 ( 5 456780 ( 18271 255555 2222 ( 1 135 49253 ( 5 3692580 ( 82057 455555 4444 Q. What proof is for Division ? A. This , multiply the quotient by the divisor , and to the product add what remained after the Division was ended , if any such remainder , if then it amount justly to the dividend , it is well done , or else not . Q. You said Multiplication was best proved by Division , how is that proof done ? A. By dividing the product of the Multiplication by the multiplier , if then , the quotient comes justly to the multiplicand , it is well done , or else you have failed . Q Now having spoken of the five first kinds or rules of Arithmetick , let us come to the application of them , to use , therefore now tell mee what use may bee made thereof ? A The uses are so many , and so necessary , that it would require a large volume to declare them , and I resolve brevity . Q. Yet I desire to hear some of them , where the same may bee made profitable ? A. Then for as much as many of the applications hereof require Reduction , I think it needful to begin first with it . Reduction . Q. What doth Reduction teach ? A. It teacheth to turn or change numbers of one denomination , into another denomination , as pounds into shillings , or shillings into pence , or pence into farthings ; or contrarily , farthings into pence , pence into shillings , or shillings into pounds , &c. The like may bee said of weight , measures , time , &c. Q. How do you turn pounds into shillings ? A. I consider 20s . is one pound , therefore there must bee 20 times so many shillings , as there is pounds , so that multiplying the number of pounds by 20 , the product shews the number of shillings . Q. Is the worth or value of the things so reduced ( changed or ) altered ? A. No , only the number and denomination is changed , but the first value remaineth still , like as 20s . is just equal to one pound , and 12d . equal to one shilling , &c. Q. How change you a number of shillings into a number of pence ? A. For as much as 12d . is in each shilling , there must bee 12 times so many pence as there is shillings , therefore I multiply the number of shillings by 12 , and the product is my desire . Q. How reduce you farthings into pence ? A. Seeing 4 farthings make but one penny , therefore there is but a fourth part so many pence as there is farthings , wherefore I divide the number of farthings by 4 , and the quotient is my desire . Q. And how will you turn pence into shillings , and shillings into pounds ? A. I divide pence by 12 , to turn them into shillings , and shillings by 20 to turn them into pounds . Q. Why so ? A. Because 12d . is but one shilling , and 20s . is but one pound . Q. Is there the like reason for weight , measures , time , & c ? A. Yea , altogether , for I am to consider how many of the one sort or denomination will make one of the other , and so multiply by that number , to turn the greater parts into the smaller , or divide by the same number to turn the smaller parts into the greater . Q. Give an Example in weight . A. To turn 256 tuns , into C. q quartern . l. o ℥ . I do thus , First I multiply 256 by 20 , because there are 20C . in each tun , and then the product is Cds. then to turn those into q quartern I multiply by 4 , and that product is my desire ; to turn that into ls . I multiply by 28 , for that each q quartern is 28l . and the product is pounds , then to turn those pounds into ounces , I multiply by 16 , and the product is the solution , &c. Example . 256 tuns . 20 5120 C. 4 20480 q quartern s. 28 163840 4096 573440 l. 16 3440640 57344 9175040 o ℥ . To turn ounces into pounds , divide by 16 , the quotient is ls . divide pounds by 28 , the quotient is q quartern s , divide the q quartern s by 4 , the quotient is C. divide Cds. by 20 , the quotient is tuns , as may be proved by working the Example above backwards , and this I judge will suffice to explain this part of the Rule . Q. Is there no kinde of Reduction that requireth both Multiplication and Division ? A. Yes , when a certain number of the one sort makes another number of the other sort ; As for example , when 3 marks makes 2 pounds , then to turn marks into pounds , I must multiply by 2 , and divide by 3 , the quotient is my desire , but to turn pounds into marks , I must multiply by 3 , and divide by 2 , and the quotient is my desired number . Q. Shew the like instance in long measures . A. Four Ells is equal to five Yards , therefore any number of Yards given , to know how many Ells it contains , I multiply by 4 , and divide by 5 , but if the number given bee Ells , and I would know how many Yards it is , I multiply by 5 , and divide by 4 , and the quotient is my desire . Q. May the like Reduction bee made in other things ? A. Yea , for 3l . starling is worth 5l . Flemmish , therefore any number of pounds starling being multiplied by 5 , and the product divided by 3 , the quotient shews the number of pounds Flemmish , or-any number of pounds Flemmish , being multiplied by 3 , and the product divided by 5 , declares the number of pounds starling in the quotient , the like proportion is in Flemmish Ells , and English Ells , because 5 Ells Flemmish is but 3 Ells English , &c. Q. Is there the like reason for other things ? A. Yea , whether coins , weights , measures , &c. Q. But what if any remainder be in the Division ? A. It must bee exprest in a fraction as before . The Rule of Three . Q. What other use is it for ? A. The next use I shall apply it to , is the Rule of proportion , commonly called , the Rule of Three , and for the usefulnes of it , the Golden Rule . Q. What doth the Rule of Three teach ? A. It teacheth by 3 known numbers to finde a fourth , either in continual proportion , or discontinual proportion . Q. What call you continual proportion ? A. When the numbers are such as hold such proportion among themselves , that what proportion the first hath to the second , the like hath the second to the third , and the third to the fourth , as are 1 , 3 , 9 , 27 , and also 2 , 4 , 8 , 16 , &c. Q. How finde you a number in continual proportion ? A. I multiply the second number by it self , and divide the product by the first number , and the quotient is my desire . Q. But how agrees this with your former words , where you said , that this Rule teacheth by three numbers to finde a fourth ? A. Very well , for the second number here is taken twice , that is , both for the second and third , and the fourth number that is found out , is the third number in continual proportion . Q. Shew an Example . A. Thus , if 3 give 9 , what gives 9 , here I multiply 9 by 9 , gives 81 , which divided by 3 , the first number , the quotient is 27 , being the third number in continual proportion , so that the 3 numbers are here , 3. 9. 27. Q. And how finde you a fourth number in continual proportion ? A. Here I may either multiply the third by it self , and divide by the second , or else multiply the second and third together , and divide by the first , and the quotient is my desire . Q. Give an Example . A. Take the former numbers , and first the former way , I multiply the 3 by it self ( viz. ) 27 by 27 thus , 27   27   189 0 54 729 ( 81 729 99 comes 81 for the fourth . Or secondly , multiply the second and third together , and divide by the first thus , 27 00 9 243 ( 81 243 33 comes 81 as before . Q. What is discontinual proportion ? A. Where the first and second hold like proportion each to other , as the third and fourth do each to other , but the second and third hold not that proportion together . Q. How is the fourth number in discontinual proportion found ? A. By multiplying the second number by the third , and dividing by the first , as before . Q. What use may bee made of these proportionals thus found ? A. They may bee applied to many uses , according to the several imployments men are exercised in ; as in Merchandizing , Measuring heights or distances , magnitudes or quantities , &c. Q. How may they bee so applied ? A. Thus , if 3 yards of cloth cost 12s . what will 5 yards cost at that rate ? here 3 yards is the first number , and 12s . the second , and 5 yards the third , therefore I multiply the second by the third , comes 60 , which divided by the first , the quotient is 20s . my demand . Q. How shall I know which number ought to bee first , that I may not mistake , and so work false ? A. You must observe that the first number and the third are of one kinde , and are , or ought to bee reduced to one denomination , and the second number is of that kinde which is sought in the question , so that the first and second declare the proportion , and the third number is annexed with the demand . Q. Make this plainer by Example . A. In the former Example 3 yards and 5 yards are both of one kinde , to wit , yards , the second or middle number is 12 , and is of another kinde , namely , shillings , now 3 yards and 12s . declare the proportion , and are the first and second numbers ; Now , 5 being that with which the demand is joyned ( as what cost 5 yards ) is the third number , and the fourth number ( resolving the question ) is of the same kinde the second is of , to wit , here shillings . Q. What if the numbers bee of several denominations , as of l. — s. — d. & c ? A. They must bee reduced to one denomination first , before you apply them to the Rule , and the second ( or middle ) number must bee reduced into the denomination of its smallest parts , in which parts the question is resolved by the fourth number . Q. Give an Example hereof . A. If 3 yards and 3 quarters cost 11s . 3d. what cost 16 yards ; here the first and third numbers being both measure , must bee turned into q quartern s , and the first will bee 15 , and the third 64 , the second or middle number must be turned into pence , and will bee 135d . now work by the Rule , multiply the second and third together , being set in order thus , If 15 q quartern s. cost 135d . what cost 64 q quartern s ? 64 540 810 8640 Divide the product by the first . 143 3190 8640 ( 576 1555 11 Comes 576 for solution , which are pence , like the middle number , which I reduce into shillings and pounds thus , 1 190 ( 0 576 ( 4 8 ( 2 122 ( 20 1 Comes for solution — 2l . — 8s . ●d . Q. And what do you further ? A. I reduce those smallest parts into a denomination of greater parts of its own kinde if I can , for the more easie estimation of their value . The Backer Rule of Three . Q. Is there any other manner of Work in the Rule of Three ? A. Yea , there is another manner of work called the Backer Rule of Three . Q. Why is it called the Backer Rule of Three ? A. Because in the former wee multiply the second by the third , and divide by the first , but in this wee multiply the first by the second , and divide by the third number . Q. What is the use of this Rule ? A. It serveth to finde out a number that holds proportion to the first , as the third doth to the second . Q. Of what use is such proportional numbers ? A. The use is manifold , as by example may appear . Shew 2 or 3 Examples . A. First , if 5 men do a peece of work in 15 daies , how many men will do the like in 3 daies ? here I multiply the first by the second , comes 75 , which divided by the third , yeelds 25 in the quotient , being the number of men demanded . Q. What is a second Example ? A. This , if a quantity of provision serve 40 men 30 daies , how many men will it serve 80 daies ? here I multiply and divide according to rule , and finde 15 men . Q. Give one example more . A. Then thus , If 9 yards of cloath , of yard broad , make a man a suit & cloak , how much broad cloath of yard and half broad will make another so large ; here I multiply 4 q quartern , the breadth of the first cloth , by 9 yards the length thereof , and divide by 6 q quartern the second breadth , and it yeelds 6 yards for the length of the second cloth . Q. Is there any other use hereof ? A. Yea , many , which the studious may finde by practice , but these shall serve at present for an entrance . Q. What if any thing remain after the Division is ended ? A. It must bee annexed to the whole number in the quotient , with the divisor under it , and a small line drawn between them , so expressing it in a fraction . Numeration in fractions . Q. Now tell mee what use fractions are of in Arithmetick ? A. They are of like use with whole numbers . Q. And are there the same kinds or species in fractions , as in whole numbers ? A. Yea , only some put a difference in the order of teaching them , that the easiest may bee first taught . Q. But what mean you by species ? A. I mean several kindes of working , or several Rules , as some call them . Q. Then rehearse the order of Rules as they are taught . A. Numeration , Reduction , Multiplication , Division , Addition , Substraction . Q. What sheweth Numeration in fractions ? A. It sheweth how to set down , or express any fraction , part or parts of an unite . Q. how is that done ? A. It is done by setting down two numbers one over another , with a line drawn betwixt them , whereof the lower number signifieth how many parts the whole unite is divided ( or supposed to bee divided ) into ; and the uppermost number sheweth how many of those parts the fraction contains . Q. How are those two numbers called ? A. The uppermost , ( or number above the line ) is called the numerator , and the other below the line is called the denominator . Q. Shew an Example or two to explain this . A. Three quarters is set down , with a 4 under the line , signifying the number of parts the unite is divided into ; and 3 above the line , shewing how many of those parts the fraction expresseth or signifieth . Q. Give another Example . A. Five seventh parts is exprest by 5 above the line , and 7 below it , thus 5 / 7. Q. Is the greatest number alwaies set lowest ? A. Yea , in such as are proper fractions . Q. Are there then any improper fractions ? A. There are sometimes whole numbers or mixt numbers exprest in form of fractions , which are not properly fractions , because a fraction is alwaies lesser than an unite , but these are either equal to , or greater than an unite . Q. Explain this by an Example or two . A. Two halfs 2 / 2 , three thirds 3 / 3 , five fifths , 5 / 5 , &c. are whole unites , onely exprest like fractions ; also nine quarters is a mixt number exprest thus , 9 / 4 , and signifies two unites and a quarter more . Q. Why are such exprest like fractions ? A. For aptness , or for ease in working . Q. What else is considerable in Numeration ? A. This , that as numbers increase infinitely above an unite , so fractions decrease or grow less infinitely under an unite . Q. I remember you mentioned decimal fractions before , how are such exprest ? A. They are exprest by an unite , and 1 , 2 , 3 , 4. or more ciphers below the line , according to the number of places , or parts the fraction is exprest in , and with figures and ciphers above the line , expressing the number of such parts that the fraction contains . Q. Make this plain by an Example , two or three . A. One half or 5 tenths is exprest by 5 above the line , and an unite with one cipher , signifying ten or tenths under the line thus 5 / 10. Secondly , 1 / 4 , or 25 hundreds , is writ with 25 above the line , and 100 under the line thus 35 / 100. Q. How set you down 75 thousand parts ? A. Thus with a cipher , a 7 , and a 5 above the line , and an unite and three ciphers below the line , 075 / 1000. Q. Are decimals alwaies exprest thus ? A. They are often exprest by their numerator , onely separated from the unite place by a prick , and the denominator is understood to consist of so many ciphers , as there are places in the numerator , and an unite before them to the left hand . Q. Shew mee one Example or two . A. First , Five hundreths is writ with a cipher , and a 5 thus 05 , where 100 is understood for denominator . Secondly , 34 ten thousand parts is exprest thus 0034 , where 10000 is understood for denominator . Q. Is there any thing more herein to bee noted , before wee leave numeration ? A. Yea , that not an unite only may bee divided infinitely into fractions or parts , but also any of those parts or fractions may bee divided also infinitely into other parts , called fractions of fractions , and those also again subdivided infinitely , &c. Reduction . Q. Now tell me what is Reduction ? A. Reduction is a changing of numbers or fractions out of one form or denomination , into another . Q. Why are they so reduced ? A. Either for more ease in working , or for the more easie estimation of the value of two or more fractions , either compared one with another , or adding them together , or substracting one from another . Q. How are fractions of several denominations reduced to one denomination ? A. First multiply the denominators together , and set the product for a common denominator ; then multiply the numerator of the first , by the denominator of the second , and set the product for a new numerator for the first fraction ; and multiply the numerator of the second , by the denominator of the first , and set the product for the new numerator of the second fraction , and so are both those fractions brought into one denomination . Q. Give an Example hereof ? A. Two thirds and three quarters , being so reduced , make 8 / 12 for 2 / 3 , and 9 / 12 for 3 / 4 , which yet still retain their first value , but are now both of one denomination . Q. You have shewed how to reduce two Fractions into one denomination , but what if there bee three or more ? A. Then I must multiply all the denominators together , and set the product down so many times as there bee fractions , for a common denominator to them ; and then multiply the numerator of the first , by the denominator of the second , and the product by the denominator of the third , and that by the denominator of the fourth , if I have so many , and so forward , and the product is a new numerator for the first fraction ; then multiply the numerator of the second , by the denominator of the first , and the product by the denominators of the third and fourth , and so forward , if you have so many , and set the product for a new numerator for the second fraction , and multiply your third numerator by the first and second denominators , and the product by the denominator of the fourth , if you have to many , and that product is your third numerator ; then if you have so many , multiply the numerator of the fourth by the other three denominators , the product is a new numerator for the fourth fraction , &c. Q. Must this order then bee observed still , when you have many fractions ? A. Yea , alwaies multiply all the denominators together for a new denominator , and one numerator by all the other denominators , except its own , the product is a new numerator for that fraction whose numerator was taken to multiply by . Q. Is there any other form of reducing to one denomination ? A. Yea , several varieties . Q. What is one way ? A. This is one , when you have found a new denominator as above , then divide the same by the denominator of any of your fractions , and multiply the quotient by the numerator of the same , and the product shall bee a new numerator for that fraction , &c. Q. What is another way ? A. If the lesser denominator will by any multiplication make the greater , then note the multiplier , and by it multiply the numerator over the lesser denominator , and in place of the lesser put the greater denominator , and so it is done without any of the other fractions . Q. What other sort of Reduction is there ? A. A second sort is when fractions of fractions are to bee reduced to one denomination . Q. How is that done ? A. By multiplying the numerators each into other , and setting the product for a new numerator , and in like sort multiply all the denominators each into other , and take that product for a new denominator , and then they express it in the parts of a simple fraction . Q. What if I have a mixt number of unites , and parts to bee reduced into fraction form ? A. Multiply the unites or whole number by the denominator of the fraction , and thereto add the numerator of the fraction , and set the offcome above the line over the said denominator . Q And how reduce you such an improper fraction into its unites and parts ? A. I must divide the numerator by the denominator , and the quotient shews how many unites it contains , and the remainder , if any bee , is the numerator of a fraction , over and above the said unites in the quotient , to which the divisor is denominator . Q. How is a whole number reduced into the form of a fraction ? A. By multiplying it by that number , which you would have denominator to it . Q. What is next in Reduction of Fractions ? A. To reduce a fraction into its smallest or least t●rms . Q. What bee the terms of a Fraction ? A. The terms bee the numerator and denominator whereby it is exprest . Q. What mean you by greatness and smalness of terms ? A. By great , I mean , when a fraction is exprest in great numbers , as 480 / 960 which in its smallest terms is ( 1 / 2 ) one half . Q. How are such reduced into their smallest terms ? A. If they be both even numbers by halfing them both so often as you can , but if they come to bee odd , or either of them odd , then by dividing them by 3 , 5 , 7 , 9 , &c. which will divide them both , without any remainder , and take the last numbers for the terms of the fraction . Q. But is there no way to discover what number would reduce a fraction into its smallest terms , but by halfing or parting in that sort ? A. Yes , thus , divide the denominator by the numerator , and if any thing remain , divide the numerator by it , and if yet any thing remain , divide the last divisor by it , and so do till nothing remain , and with your last divisor , which leaves no remainder , divide the numerator of the fraction , and the quotient is a new numerator , and divide the denominator in like sort by it , and the quotient is a new denominator . Q. What if no number will divide them evenly , till it come to one ? A. Then the fraction is in its smallest terms already . Q. How reduce you fractions of one denomination , into another denomination ? A. I multiply the numerator by the denominator , into which I would reduce it , and divide the product by the first denominator , and the quotient is the new numerator . Q. Give an Example of this . A. If 3 / 4 bee to bee turned into twelfth parts , I multiply 12 by 3 comes 36 , which I divide by 4 , the quotient is 9 , so it is 9 / 12 , equal to 3 / 4. Multiplication in Fractions . Q. How do you multiply in Fractions ? A. I multiply the numerators together for a new numerator , and multiply the denominators together for a new denominator , which numerator and denominator , so found , express the product of that multiplication . Q. What other thing of note it observable in Multiplication of Fractions ? A. That two Fractions multiplied together , the product is lesser than either of the fractions . Q. How comes that to bee , seeing the very name of Multiplication signifies to augment or increase a thing manifold , or many times ? A. That is true , in whole numbers . where a number is increased by so many times as the multiplier contains unites , but in fractions wee must note , that the multiplier being less than one , it makes the product lesser than the multiplicand ; for so often as the multiplier contains unites , just so often doth the product contain the multiplicand , therefore ( in fractions ) feeing the multiplier doth contain but such a part or parts of an unite , even so the product doth contain but the like part or parts of the multiplicand . Division in Fractions . Q. How is Division in Fractions performed ? A. Thus , I multiply the numerator of the dividend by the denominator of the divisor , the product is a new numerator , then multiply the numerator of the divisor by the denominator of the dividend , the product is a new denominator , and this third fraction is the quotient of that division . Q. Shew an Example . A. If I divide 3 / ● by 1 / ● , the quotient is 3 / 4. Q. How comes it to pass that in Division by a fraction the quotion is greater than the dividend ? A. Because the divisor being lesser than an unite , is consequently oftener contained in the dividend , for alwaies the quotient shews how often the divisor is contained in the dividend . Q. How is a whole number divided by a fraction ? A. I multiply the whole number by the denominator of the fraction , and set the product for numerator , and for a denominator , I set the numerator of the fraction . Q. How is a fraction divided by a whole number ? A. By multiplying the denominator by the whole number , setting the product for a new denominator , without changing the numerator at all . Q. May this bee done otherwise ? A. Yea , if the whole number will evenly divide the numerator of the fraction , then divide it by it , and set the quotient for numerator , and change not the denominator at all . Addition in Fractions . Q. How are two or more fractions added together ? A. If they bee of one denomination , then add the numerators together in one , and under it place the common denominator , and that fraction represents the total of that Addition . Q. But what if they bee of several denominations ? A. Then I first reduce them to one denomination , and then add their numerators together . Substraction in Fractions . Q. How substract you one fraction from another ? A. If they be not of one denomination , I reduce them to one , and then substract the lesser numerator from the greater , and set the rest for a new numerator over the common denominator . Q. But what if you bee to substract a mixt number from another , or from a whole number ? A. I may as before reduce them to one denomination , and then substract one numerator from the other , or I may substract the fraction of it from an unite converted into the same denomination , and carry one in minde to the whole number , and then substract it out of the other whole number . Several other means may be used in these works of fractions , but I forbear to mention them , for brevity sake , and come to the Rule of Three . The Rule of Three in Fractions . Q. How work you the Rule of Three in Fractions ? A. First I place the numbers as was shewed in whole numbers , and then multiply the numerator of the first by the denominator of the second , and the product by the denominator of the third , and the product thereof must bee my divisor ; then I multiply the denominator of the first , by the numerator of the second , and the product by the numerator of the third , and the offcome is my dividend , then I divide the dividend by the divisor , and the quotient is the fourth number , and answereth the question . Q. Is there any other way to work the Rule of Three ? A. Yea divers , whereof this is one , finde the divisor or first number , as before , then for the second number , take the numerator of the second fraction , and for the third number , take the number that cometh by Multiplication of the numerator of the third by the denominator of the first fraction , and then work as in whole numbers . Q. What proof is there for the Rule of Three ? A. Multiply the second and third numbers together , and multiply the first and fourth numbers together , and if the products be equal , it is right , or else it is not right . Q. Give an Example in the Rule of Three . A. If 4 / 5 of an Ell cost 3 / ● of a pound , what is 1 / ● of an Ell worth ? here I multiply 4 by 8 , comes 32 , & that by 3 comes 96 for divisor , then multiply 5 by 3 is 15 , and that by 2 makes 30 , for dividend or numerator , so it is 30 / ●6 , or in the smallest terms , 5 / 16 of a pound . Q. Examine this by the proof . A. Multiply 3 / ● by 2 / ● comes 6 / 24 , or 1 / 4 , again multiply 4 / 5 by 5 / 16 comes 20 / 80 , or 1 / 4 likewise . The Backer Rule of Three . Q. How work you the Backer Rule of Three in Fractions ? A. Thus , I multiply the numerators of the first and second numbers together , and the offcome by the denominator of the third , and the product is my dividend , then I multiply the denominators of the first and second together ; and the offcome by the numerator of the third , and that product is my divisor , wherewith I divide my dividend , and the quotient resolves the question . Q. Show an Example hereof . A. If my friend lend mee 4 / ● of a pound for 2 / 3 of a year , or 8 months , how long ought I to lend him 2 / ● of a pound to requite his courtesie ? A. I say , if 4 / ● give 2 / ● , what shall 2 / ● give , where I multiply 4 by 2 yeelds 8 , and 8 by 3 comes 24 for dividend , then I multiply 5 by 3 comes 15 , and that by 2 gives 30 for divisor , and so placing the dividend over the divisor , I have 24 / ●● , or in the smallest terms 4 / ● of a year , for the resolution of the question , which is the time I ought to lend him 2 / ●l . to requite his courtesie . Q. I observe , that where the Rule of Three Direct would give the fourth number more than the second , this gives it less , and where it would give it less , this gives it more , what is the reason of that ? A. We are to consider in this Rule , that the less mony lent , requires the more time forbearance to ballance the other ; and in like sort , the less breadth a thing is of , the more length it requires to make it equal with a quantity of more breadth ; in like manner , the more men that are imployed to do a peece of work in , the less time they will do it ; so the fewer men that are to live upon a quantity of provision , the longer time it will last , &c. The Double Rule of Three . Q. Is there any other form in the Rule of Three , besides the above said ? A. Yea , there is divers which resolve double questions , and therefore are called , the Double Rule of Three , or the Rule of Three composed of 5 numbers . Q. What manner of questions doth this Rule resolve ? A. Either such as are uncompound , or compound . Q. What mean you by uncompound , or compound ? A. I mean by uncompound , such as are done by the Rule of Three Direct at two workings , by compound , such as are done by once working by the Rule of Three Direct , and another by the Backer Rule of Three . Q. What is one question of the former sort ? A. If 5 men in 6 daies earn 3l . how much will 10 men earn in 12 daies ? Q. And how is this done ? A. Either by two several workings by the Rule of Three , or it may be resolved at once . Q. First shew mee how it is done at two workings ? A. First I say , if 5 men earn 3l . what will 10 men earn , it gives 6l . for 6 daies ; then again , if 6 daies gives 6l . what gives 13 daies ? facit 12l . Or secondly , I may say 6 daies gives 3l . what gives 12 daies ? comes 6l . for 5 men ; and then , if 5 men earn 6l . what will 10 men earn ? and it comes to 12l . as before . Q. How is this performed at one working ? A. Thus , I say according to the question , If 5 men in 6 daies earn 3l . what earn 10 men in 12 daies , then I multiply the first by the second , viz. 5 by 6 comes 30 , which I keep for my divisor , then I multiply the other three numbers each into other , comes 360 for my dividend , which being divided by my divisor , gives 12 in the quotient , which signifies so many pounds , being of the denomination of the middle number . Q. Now shew an Example of a compound question . A. Take this , if 5 men in 6 daies earn 3l . in how long time will 3 men earn 5l . where first I say , if 3l . give 6 daies , what will 5l . give ? comes 10 daies ; then by the Backer Rule of Three , If 5 men bee 10 daies in earning it , how long will 3 men bee ? and it gives 16 2 / ● daies . Q. And how is this done at one working ? A. I say , if to earn 3l . 5 men bee imployed 6 daies , then to earn 5l . by 3 men , what time is required ? where I multiply the first number and the fifth number together ( being the least sum of mony , and the least number of men ) comes 9 for divisor , then I multiply the other three numbers together , comes 150 for my dividend , which being divided , gives 16 in the quotient , and 6 remains , which abreviated or reduced to the least terms , is 2 / 3 so the whole is 16 2 / 3 daies , as before . Q. What is another question of this sort ? A. This , 30 men work 40 yards of Arras in 6 daies , in how long time will 15 men work 80 yards of the like Arras ? Q. How is this done at two workings ? A. First I say , if 40 yards require 6 daies , how long time will 80 yards require ? ( by the Rule of Three Direct ) comes 12 daies ; then I say , if 30 men bee 12 daies about it , how long will 15 men bee about it by the Backer Rule ? and it comes to 24 daies . Q. How is this performed at one working ? A. I say thus , if 40 yards require 30 men for 6 daies , then to do 80 yards by 15 men , how long time will it require ? where I multiply the first number , being 40 yards , by the fifth number , being 15 men , that is the least number of yards , and the least number of men together , that is , 40 by 15 comes 600 for my divisor , then I multiply the other three numbers together , ( viz. ) 30 by 6 comes 180 , and that by 80 comes 14400 for my dividend , which I divide by my divisor , the quotient is 24 as before , being so many daies , according to the denomination of the middle number . Thus having briefly and plainly explained the Rules , I shall set down some questions , with their resolutions , omitting the work , that the young learner may practise himself in them , or such like , to make himself the more ready in this Art . In Reduction . In 264l . how many shillings is there ? facit 5280s . In 10560s . how many pence is there ? facit 126720d . In 63360d . how many shillings is there ? facit 5280s . In 10560s . how many pounds is there ? facit 528l . In 2650 Ells Flemmish , how many Ells English ? facit 1590. In 3180 English Ells , how many Flemmish Ells is there ? facit 5300. In the Rule of Three . If 42 yards cost 28l . what cost 30 yards ? facit 20l . If 20l . buy 30 yards , how much will 28l . buy ? facit 42 yards . If 30 yards cost 20l . what will 42 yards cost ? facit 28l . If 28l . buy 42 yards , how much will 20l . buy ? facit 30 yards . I have varied this question purposely to shew the learner how hee may do the like with any other . If 3 marks be worth 2l . what is 369 marks worth ? facit 246l . If 20 nobles be worth 6l . 13s . 4d . what is 1000 nobles worth ? facit 333l . 6s . 8d . If 6 o ℥ . of cloves cost 2s . 6d . what cost 16 o ℥ . facit 6s . 8d . If 1C weight cost 18s . 8d . what cost 12l . facit 2s . If 6l . cost 1s . 6d . what cost 112l . facit 1l . 8s . For the Backer Rule of Three . If 5 men do a peece of work in 8 daies , how many men will do the like in 2 daies ? facit 20 men . If 20 men do a peece of work in 2 daies , in how long time will 8 men do the like ? facit 5 daies . If a quantity of provision serve 360 men 45 daies , how long will it serve 288 men ? facit 56 ●● / ●●● daies , or 65 1 / 4 daies . If 5 yards of cloth that is yard and half broad , make a man a gown , how much baize of yard broad will line it throughout ? facit 7 1 / 2 yards . If a foot of board be 12 inches long , and 12 inches broad , how much will make a foot of that board that is but 9 inches broad ? facit 16 inches in length . If I have a plot of ground that is 36 foot broad , and 64 foot long , which I would exchange for so much of another field that is 48 foot broad , how much ought I to have in length of the second ? facit 6● foot . FINIS . 
A29756	----	The description and use of a joynt-rule fitted with lines for the finding the hour of the day and azimuth of the sun, to any particular latitude, or, to apply the same generally to any latitude : together with all the uses of Gunters quadrant applyed thereunto ... / contriv'd & written by J. Brown, philomath. Brown, John, philomath. 1661 Approx. 191 KB of XML-encoded text transcribed from 105 1-bit group-IV TIFF page images. Text Creation Partnership, Ann Arbor, MI ; Oxford (UK) : 2004-08 (EEBO-TCP Phase 1). A29756 Wing B5038 ESTC R33265 13117505 ocm 13117505 97766 This keyboarded and encoded edition of the work described above is co-owned by the institutions providing financial support to the Early English Books Online Text Creation Partnership. This Phase I text is available for reuse, according to the terms of Creative Commons 0 1.0 Universal . The text can be copied, modified, distributed and performed, even for commercial purposes, all without asking permission. Early English books online. (EEBO-TCP ; phase 1, no. A29756) Transcribed from: (Early English Books Online ; image set 97766) Images scanned from microfilm: (Early English books, 1641-1700 ; 1545:10) The description and use of a joynt-rule fitted with lines for the finding the hour of the day and azimuth of the sun, to any particular latitude, or, to apply the same generally to any latitude : together with all the uses of Gunters quadrant applyed thereunto ... / contriv'd & written by J. Brown, philomath. Brown, John, philomath. [24], 168 p., [8] leaves of plates : ill., charts. Printed by T.J. for J. Brown and H. Sutton, and sold at their houses, London : 1661. Woodcut illustration of man sighting with sextant: T.p. verso. Errata: p. 168. Imperfect: pages stained and tightly bound with slight loss of print. Reproduction of original in the British Library. Created by converting TCP files to TEI P5 using tcp2tei.xsl, TEI @ Oxford. 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Understanding these processes should make clear that, while the overall quality of TCP data is very good, some errors will remain and some readable characters will be marked as illegible. Users should bear in mind that in all likelihood such instances will never have been looked at by a TCP editor. The texts were encoded and linked to page images in accordance with level 4 of the TEI in Libraries guidelines. Copies of the texts have been issued variously as SGML (TCP schema; ASCII text with mnemonic sdata character entities); displayable XML (TCP schema; characters represented either as UTF-8 Unicode or text strings within braces); or lossless XML (TEI P5, characters represented either as UTF-8 Unicode or TEI g elements). Keying and markup guidelines are available at the Text Creation Partnership web site . eng Quadrant. Dialing. Mathematical instruments. 2004-01 TCP Assigned for keying and markup 2004-02 Aptara Keyed and coded from ProQuest page images 2004-04 Mona Logarbo Sampled and proofread 2004-04 Mona Logarbo Text and markup reviewed and edited 2004-07 pfs Batch review (QC) and XML conversion The Description and Use OF A JOYNT-RULE : Fitted with Lines for the finding the Hour of the Day , and Azimuth of the Sun , to any particular Latitude ; Or to apply the same generally to any latitude Together with all the uses of Gunters quadrant applyed thereunto , as Sun-rising , Declination , Amplitude , true place , right Ascension , and the hour of the Night by the Moon , or fixed Stars ; A speedy and easie way of finding of Altitudes at one or two stations ; Also the way of making any kinde of erect Sun-Dial to any Latitude or Declination , by the same Rule : With the Description and Use of several Lines for the mensuration of Superficies , and Solids , and of other Lines usually put on Carpenters Rules : Also the use of Mr. Whites Rule for measuring of Board and Timber , round and square ; With the manner of using the Serpentine-line of Numbers , Sines , Tangents , and Versed Sines . Contriv'd & written by J. Brown , Philom . London , Printed by T. I. for I. Brown , and H. Sutton , and sold at their houses in the Minories , & Thredneedle-street . 1661. TO THE READER . Courteous Reader , AMong the multitude of Books which are printed and published , in this scribling Age , some serious , some seditious ; some discovering or savouring of Art , others of ignorance , possibly every one endeavoring to bring their Male : Among the rest of the crowd , I , like the widow , throw in my mite . If it be ( or seem to be ) little , it . is like the Giver , and therefore I presume will of some be accepted , as little as it is ; and as little worth as it is , it is like enough to be challenged : but I shall endeavor to prevent prejudice , by the following Discourse . Having for some time been enquiring to find out a way , whereby Work-men might on their Rules ( their constant Companions ) have a way easily and exactly to finde the hour of the Day , and Suns Altitude and Azimuth , and the like ; and have at several times for several men , at their request , used one and the other contrivance , to finde the Hour : as that of the Cillender , Quadrant , or the like , as the Altitude by a Tangent on the inside of a Square , or Joynt-Rule , and the line of Sines on the flat side ; but still one inconvenience or other of trouble in adding of complements , or difficulty of taking of Aititudes , or trouble to the memory , did accrue to the work ; or else the Radius was small , and so much the more short of exactness : at last there came to my sight a Quadrant made by Mr. Thomson , and as I was informed , was first drawn or contrived to that form , by Mr Samuel Foster , that ingenious ●rtist , and laborious Student , and Reader of the Mathematicks in Gresham Colledge : And considering of the ease and speed in the using thereof , I set my self to the contriving thereof to a more portable form , at last took some pains in delineating one , and another in several forms , and enquired after the uses thereof , and in effect have done , as Mr. Gunter with Stofflers Astrolabe , and Nepeirs Logarithms , and as Mr. Oughtred with Gunters Rule , to a sliding and circular form ; and as my father Thomas Brown into a Serpentine form ; or as Mr. Windgate in his Rule of Froportion , and as of late Mr. Collins with Mr. Gunters Sector on a Quadrant , so may this not unfitly be called , The Quadrant on a Sector . And in fine , the Invention will be valued for the learned Authors sake , and never a whit the worse for the new Contrivers sake : For first , hereby it is made large in little room , and as well on wood as on brass , which is an incommunicable property to broad Quadrants , though of never so good matter , as experienced Workmen know right well ; and by a Tangent of 30 degrees laid together , is gotten all kinde of Angles or Altitudes under 90 degrees , and to be afforded for a low price , in comparison of other Instruments , which will not perform the same operations any better . Having made some Joynt-Rules in the manner following , and exposing them to sale , I have been many times solicited to write somewhat of the use , and now at last after near a years suspence , have committed the following Discourse to publique view , partly to save the labor of tedious transcribings , and also to make so useful , cheap , and exact an Instrument , ( if it be truly made ) to be more known or occupied . In which business , I desire to disclaim all vain-glorious os●entation , and therefore have nakedly and plainly asserted the manner how , and why it comes to be published to the world by me . It is a mechanical thing , and mechanically applied , and of mechanical men will be humanically accepted , I doubt not . Having begun to write , I could not break off so short and abruptly , as at first I did intend to do ; therefore have added this short Discourse of ordinary Dyalling , the exact Method of which I finde in no other Author that ever I met with , ( and indeed I have not time to read many ) yet I dare presume , that for speed , ease , convenience , and exactness , inferior to none , especially the way of making far declining Dyals ; As for other declining reclining Dyals , I referre you to other Authors , or to a Discourse thereof by it self : if I finde encouragement , and ability to perform the same , a Copy whereof I have had a long time by me , written by a very ingenious Artist ) the demonstration of which Dyals is most excellently and easily shewed by the Figure inserted , page 77. As for the other part for taking of Altitudes and Angles , it may also be very conveniently done , if the Rule be fitted to a three-leg staff , with a small Ball-socket to set it level , or upright , as other Surveighing-instruments be , as will be amply found , if a tryal be made thereof . That of Ma●er White 's Rule is a thing that hath given very good content to several Gentlemen in the Counties of Essex , Suffolk , and Norfolk , and indeed is a very neat and accurate way of operation , well becoming a Gentleman ; for while a Workman shall take measure , his Rule keeps the count of length , or breadth ; and having the length first given , the girt or squareness is no sooner agreed on , but you have the content without Pen or Compasses . As for the other lines , as Decimal-board , and Timber-measure , Inches , and Foot , in the way of Reduction , Girt-measure , Circles , Diameter , Circumference , Squares inscribed and equal : The Use of them will be very grateful to many a learner . Lastly , this brief touch of the Serpentine-line I made bold to assert , to see if I could draw out a performance of that promise , that hath been so long unperformed by the promisers thereof . These Collections , courteous Reader , I have printed at my friends and my own proper charges , and if they prove to be ( as I do hope they will ) of publique benefit , I shall enjoy my expectation , and be ready at all times to serve you further , as I may , in these or other Mathematical instruments , at my house at the Sun-dyal in the Minories , and remain to you much obliged , February 8 ▪ 1660. Iohn Brown. A TABLE of the things contained in this BOOK . CHAP. I. Page THe description of the rule for the hour onely 1. 2. 3 CHAP. II. To rectifie or set the Rule to his true angle for observation 4 To finde the Suns altitude 5 To finde the hour of the day 6 To finde the Suns rising and setting 7 To finde a level or perpendicular 7 CHAP. III. A further description of the rule for hour and Azimuth generally 10 , 11 , 12 CHAP. IIII. To finde the Suns declination 13 To finde the Suns true place , and right ascention 14 To finde the Suns amplitude 14 CHAP. V. To finde the Suns Azimuth at any altitude and declination , in this particular latitude in Summer 16 To finde the Azimuth in Winter , and Equinoctial 17 , 18 CHAP. VI. To finde the hour of the night by the Moon . 19 To finde the Moons Age 20 To finde the Moons place 21 To finde the Moons hour by the 11 chap. 2 and 3 Proposition 23 To finde the true hour of the night thereby 24 CHAP. VII . To finde the hour of the night by the sixed Stars 25 Three examples thereof 26 , 27 , 28 CHAP. VIII To finde the Amplitude , Azimuth , rising and so●●hing of the fixed Stars , and Examples thereof in page 29 , 30 , 31 , 32 CHAP. IX . To finde the hour and Azimuth , &c. in any latitude 33 To finde the Suns rising , setting , and ascentional difference ibid. To finde his amplitude in any latitude 34 To finde the Suns altitude at six in any latitude . 35 To finde the hour when the Sun is in the Equinoctial 36 To find the hour in any l●titude , altitude , and declination 37 To finde the Suns Azimuth in any latitude , at any declination , and altitude in summer 38 To finde the same in winter 39 CHAP. X. To finde the inclination of Meridians , substile , stile , and angle , between 12 and 6 for erect decliners three ways , one particular , and two general 40 To finde the substile , stiles elevation , inclination of Meridians 41 To finde the angle between 12 and 6 for a particular latitude 42 To perform the same in general for any latitude by the general scale of altitudes 43 , 44 Five canons to finde the same by the artificial sines and ta●gents , and how to work them on the rule 45 , 46 CHAP. XI . To draw a Horizontal Dyal to any latitude 47 To draw a vertical direct North or South Dial to any latitude 48 , 49 To draw a direct erect East or West Dial 50 , 51 CHAP. XIII . To finde the declination of a Plain 52 To do it by the needle 53 , 54 To finde the quantity of an angle the sector or rule stands at 55 An example of the work 56 To finde a declination by the Sun at any time 57 , 58 Some precepts and examples for the same 59 , 60 , 61 To finde a declination at two choice particular times , viz. when the Sun is in the Meridian of place , or Plain 62 , 63 How to supply a deficiency in one line of the rule , by another line on the rule 64 CHAP. XIIII . To draw a vertical declining Dial to any declination and latitude . 65 To perform it another way 66 , 67 To supply a defect on the parallel contingent 68 CHAP. XV. To draw the hour-lines on an upright declining Dial , declining above 60 degrees 96 To make the table for the hours 70 To finde the substile , stiles augmentation , and Radius , to fit and fill the Plain with any certain number of hours 71 , 72 , 73 , 74 , 75 An advertisement relating to declining reclining Plains 76 , 77 , 78 , 79 CHAP. XVI . To finde a perpendicular altitude at one or two stationi 80 To do it at one station 81 To perform it at two stations 82 , 83 , 84 To work it by the sector lines 85 , 86 CHAP. XVII . The use of several lines inserted on rules for the use of several workmen , for the mensuration of superficial , and solid measure , and reduction , &c. 87 The use of inches , and foot measure laid together , in giving the price of one , to know the price of a hundred , or the contrary 88 The use in buying of timber , knowing the price of a load , or 50 foot , to know the price of one foot , or the contrary 89 , 90. The like work for the great hundred or 112 l. to the C. 90 , 91 The use of the line of decimal board measure 92 The use of the line of decimal Timber measure 93 The use of the line of decimal yard measure 94 Or that which agreeth with feet and inches 95 The use of the line of decimal round or Girt measure 96 The use of the line of decimal sollid measure by the diameter 97 The table of decimal superficial under measure 98 , 99 The table for decimal sollid under measure 100 , 101 The table of under-yard measure for foot measure or inches 102 , 103 The table of under girt measure to inches 104 The table of under diameter measure to inches and quarters 105 A table of brick measure 106 The use of the lines of Circumference , diameter , and squares equal and inscribed 108 CHAP. XVIII . The use of Mr. Whites rule or the sliding rule in Arithmetique , and measuring superficial and sollid measure from 110 to 118 CHAP XIX . To finde hour and Azimnth by that sector To finde the Azimuth by having latitude , Suns declination and altitudes , cemplements , and the hour from noon 119 To finde the hour , by having the same complements , and the Azimuth from south 120 Having the complements of the latitude and altitude , and the Suns distance from the pole , to finde the Suns Azimuth 121 Having the same complements , to finde the hour by the sector 122 Having latitude , declination , meridian , and present altitude , to finde the hour of the day . 123 Having latitude , declination , and altitude , to finde the Suns Azimuth at one operation by the sector 124 Having the length of the shadow , to find the altitude , and the contrary 126 Having latitude and declination , to find the Suns rising and setting 127 To finde the Suns altitade at any hour generally 129 To finde when the Sun shall be due East or West 130 To finde the quesita in erect Dials 131 , 132. CHAP. XX. The use of the serpentine line . The description thereof . 134 , 135 , 136 , 137 Some observations in the use thereof 138. 139 , 140 To finde the hour of the day thereby , according to Mr. Gunter 142 To finde the hour thereby . according to Mr. Collins 145 To finde the Azimuth thereby Mr. Collins his way 146 To finde the Azimuth Mr. Gunters way 148 To finde the hour and Azimnth at one operation , by help of the natural sines and versed sines 149 to 154 To finde the Suns altitude at any hour or Azimuth 155 , 156 To finde the hour by having the Azimuth , and the contrary 157 Five other useful propositions 159 To square and cube a number , and to finde the square , and cube root of a number 162 , 163 To work questions of interest and annuities 164 The use of the everlasting Almanack 167 The right Ascension and Declination of 12 principal fixed Stars in the heavens ; most of which are inserted on the Rule : or if room will allow , all of them .   R. Asc. Declina . Stars Names H. M. Deg. M. Pleiades , or 7 Stars 03 24 23 20 Bulls-Eye 04 16 15 48 Orions Girdle 05 18 01 195 Little Dog 07 20 06 08 Lyons Heart 09 50 13 40 Lyons Tayl 11 30 16 30 Arcturus 14 00 21 04 Vultures Heart 19 33 08 00 Dolphins Head 20 30 14 52 Pegasus Mouth 21 27 08 19 Fomahant 22 39 31 17s Pegasus lower wing 23 55 13 19   1   3   5 7 4     6   8       Moneths 9   11   2 10 12   1 2 3 4 5 6 7   8 9 10 11 12 13 14 Days 15 16 17 18 19 20 21   22 23 24 25 26 27 28   29 30 31         Week-days S M T W T F Sat Dom. Letter d c b a g f e Leap years 68 80 64 76 60 72 84 E●acts 26 9 12 25 28 11 23 THE Description and Use OF A JOYNT-RULE . CHAP. I. The Description of the Lines on the Rule , as it is made onely for one Latitude , and for the finding the hour of the day onely . FIrst open the ( Joynt of the ) Rule , then upon the head-leg , being next to your right hand , you have a line beginning at the hole , which is the Center of the quadrantal lines , and divided from thence downward toward the head , into as many degrees as the Suns greatest altitude in that latitude will be , which with us at London is to 62 degrees ; which line I call the Scale of Altitudes , divided to whole , halfs , and sometime quarters of degrees . 2. Secondly , On the other leg , and next to the inside is the line of hours , usually divided into hours , quarters , and every fifth minute , beginning at the head with 4 , and so proceeding to 5 , 6 , 7 , 8 , 9 , 10 , 11 , and 12 at the end , and then back again with 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , for the morning and afternoon hours . 3. Next to this is a Kalendar of Moneths and Days in two lines ; the uppermost contains that half year the days lengthen in , and the lowermost the shortning days , as by the names of the moneths may appear ; the name of every moneth standing in the moneth , and at the beginning of the moneth : and all but the two moneths that have the longest , and the shortest days , viz. Iune and December , are divided into single days , the tenth day having a figure 10 , or a point or prick on the head of the stroke , and the fifth onely a longer stroke without a prick , and the beginning of every moneth a long stroke , and every single day all alike of one shortness , according to the usual manner of distinguishing on lines . 4. And lastly you have a line of degrees , for so they be most properly called , and they are the same with the equal limb on quadrants , and serve for the same use , viz. for taking of Altitudes , or Horizontal Angles , and are divided usually to whole , and half degrees of the quadrant , and figured with 30 , 40 , 50 , 600 , 7010 , 8020 , and 90 , just on the head , cutting the center or point , where the Scale of Altitudes and the Line of Hours meet ; which point , for distinction sake , I call The rectifying point . And the reckoning on this line , as to taking of Altitudes , is thus : At the number 600 is the beginning , then towards the head count 10 , 20 , 30 , where the 90 is ; then begin at the end again , & count as the figures shew you to 90 at the head , as before . CHAP II. The Uses of the Rule follow . 1. To Rectifie or set the Rule to his true Angle . OPen the Rule to 60 degrees , which is done thus , ( indifferently : ) make the lines on the head , and the lines on the other leg , meet in a streight line ; then is the Scale of Altitudes and the line of Hours set to an Angle of 60 degrees , the rectifying point , being the center of that Angle ; Or to do it more exactly , do thus : put one point of a pair of Compasses into the rectifying point , then open the other to 10 , 20 , 30 , or 40 , on the Scale of Altitudes , the Compasses so opened , and the point yet remaining in the rectifying point , turn the other to that margenal line in the line of hours , that cuts the rectifying point , and there stay it ; then remove the point that was fixed in the rectifying point , and open or shut the Rule , till the point of the Compasses will touch 10 , 20 , 30 , or 40 , being the point you set the Compasses too in the Scale of Altitudes , in the innermost line that cuts the center , and the rectifying point , then is it set exactly to 60 degrees , and fitted for observation . 2. To finde the Suns Altitude at any time . Put a pin in the center hole , at the upper end of the Scale of Altitudes , and on the pin hang a thread and plummet ; then if the Sun be low , that is to say , under 25 degrees high , as in the winter it will always be , then lift up the moveable leg , where the moneths and the degrees be , till the shadow of the end fall just on the meeting of that leg with the head , then the thread shall shew the Suns altitude , counting from 600 towards the head , either 10 , 20 , 25 , or any degree between . But if the Sun be above 25 or 30 degrees high , lift up the head leg till the shadow of that play as before , or make the shadow of the pin in the center hole play on the innermost line of the Scale of Altitudes where the pin standeth , then the thread will fall on the degree , and part of a degree that his true altitude shall be . But if the Sun be in a cloud , and can not be seen so as to give a shadow , then look up along by the head-leg , or moveable leg , just against the middle of the round body of the Sun , and the thread playing evenly by the degrees , shall show the true altitude required . The like must you do for a Star , or any other object , whose altitude you would find . 3. Having found the Suns altitude , and the day of the moneth , to finde the hour of the day . Whatsoever you finde the altitude to be , take the same off from the Line of Altitudes , from the center downwards with a pair of Compasses , then lay the thread ( being put over the pin ) on the day of the moneth , then put one foot of the Compasses in the line of hours , in that line that cuts the rectifying point , and carry it further off , or nigher , till the other foot of the Compass being turned about , will just touch the thred , at the nearest distance , then the point of the Compasses on the line of hours , shall shew the true hour and minute of the day required . Example on the 2. of July . 1. I observe the altitude in the morning , and I finde it to be 30 degrees high , then laying the thread on the day of the moneth , and taking 30 degrees from the Scale of Altitudes , and putting one point in the line of hours , till the other point turned about , will but just touch the th●ead , and I finde it to 23 minutes past 7 , but if it had been in the afternoon , it would have been 37 minutes past 4. 2. Again , on the tenth of August in the Afternoon , at 20 degrees high , I take 20 degrees from the Scale of Altudes , and laying the thread on the day of the moneth , viz. the tenth aforesaid , counting from the name at the beginning of August , toward September , and carrying the Compasses in the line of hours , till the other point doth but just touch the thread , and you shall finde it to be 54 minutes past 4 a clock . 3. Again , on the 11. of December at 15 degrees high , work as before , and you shall finde it to be just 12 a clock ; but to work this , you must lay the Rule down on something , and extend the thread beyond the Rule , for the nighest distance will happen on the out-side of the Rule . 4. Again , on the 11 of Iune at noon I finde the altitude to be 62 degrees high , then laying the thread on the 10 th or 11 th of Iune , for then a day is unsensible , and working as before , you shal finde the point of the Compasses to stay at just 12 a clock , the time required for that altitude . 4. To finde the Suns rising any day in the year . Lay the thread on the day of the month , and in the line of hours it sheweth the true hour and minute of the Suns rising or setting ; for the rising , count the morning hours ; and for the setting , count the evening hours . 5. To finde if any place lye level , or nor . Open the rule to his true angle of 60 degrees , then set the moveable leg upon the place you would make level , and if the thread play just on 60 degrees , it is a true level place , or else not . 6. To try if any thing be upright or not . Hang a thread and plummet on the center , then aply the head leg of the rule to the wall or post , and if it be upright , the thread will play just on the innermost line of the scale of altitudes , or else not . CHAP. III. A further description of the Rule , to make it to shew the Suns Azimuth , Declination ; True place , right Ascention , and the hour of day or night , in this , or any other Lattitude . 1. FIrst in stead of the scale of Altitudes to 62 degrees , there is one put to 90 degrees in that place , and that of 62 is put by in some other place where it may serve as well 2. The line of hours hath a double margent , viz , one for hours , and the other for Azimuths , & then every 5 th minute is more properly made 4 , or else every 2 minutes , and in a large rule to every quarter of a degree of Azimuth , or to every single minute of time . 3. The degrees ought to be reckoned after 3 maner of wayes : first as before is exprest ; secondly from 60 toward the end , with 10 , 20 , 30 , 40 , 50 , 60 , &c. to be so accounted in finding the Azimuch for a particular latitude ; and and thirdly from the head or 90 , toward the end , with 10 , 20 , 30 , 40 , 50 , 60 , 70 , 80 , &c. for the general finding of Hour and Azimuth in any latitude , and many other problems of the Sphere besides ; to which may be added , where room will alow , a line of hours , beginning at 6 at the head , and 12 at the end , but reckoning 15 degrees for an hour , and 4 minutes for every degree , it may do as well without it . 4. To the Kalender of moneths and days , is added a line of the Suns true place in the Zodiack , or where room fails , the Characters of the twelve Signs put on that day of the moneth the Sun enters into it , and counting every day for a degree , may indifferently serve for the use it is chiefly intended for . 5. Under that is a line of the Suns right Ascension , to hours and quarters at least , or rather every fifth minute , numbred thus : 12 and 24 right under ♈ and ♎ , or the tenth of March , and so forward to the tenth of Iune , or ♋ , where stands 6 , then backwards to 12 where you began , then backwards still to the eleventh of December , with 13 , 14 , 15 , 16 , 17 , 18 , to ♑ , then from thence forward to 24 where you first began : but when you are streightned for room , as on most ordinary Rules you will be , then it may very well suffice to have a point or stroke , shewing when the Sun shall gradually get an hour of right Ascension , and from that for every day count four minutes of time , till it hath increased to an hour more , and this computation will serve very well ; and in stead of saying 13 , 14 , 15 hours of right Ascension , say 1 , 2 , 3 , &c. which will perform the work as well , and reduce the time to more proper terms . 6. There is fitted two lines , one containing 24 houres , and the other 29 days , and about 13 hours , and they serve to finde the time of the Moons coming to the South , before or after the Sun , and by that , the time of high-water at London-bridge , or any other place , as is ordinary . CHAP. IV. The Uses follow in order . 1. To finde the Suns Declination . LAy the Thread on the day of the moneth , then in the line of degrees you have the declination . From March the tenth toward the head is the Declination Northward , the other way is Southward , as by the time of the year is discovered . Example : On the tenth of April it is 11d 48 ' toward the North ; but on the tenth of October it is 10d 30 ' toward the South . 2. As the thread is so laid on the day of the moneth , in the line of the Suns place , it sheweth that ; and in the line of the Suns right Ascension , his right Ascension also , onely you must give it its due order of reckoning , as thus : it begins at ♈ Aries , and so proceeds to ♋ , then back again to ♑ at the eleventh of December , then forwards again to ♈ Aries , where you began . 3. To finde the Suns right Ascension in hours and minutes . Lay the thread as before , on the day of the moneth , and in the line of right Ascension you have the hour and minute required , computing right according to the time of the year , that is , begin at the tenth of March , or ♈ Aries , and so reckon forwards and backwards as the moneths go . Example . On the tenth of April the Suns place is 1 degree in ♉ Taurus , and the Suns right Ascension 1 hour 55 minutes : on the tenth of October 27d 1 / 4 in ♎ Libra , and his right Ascension is 13 hours and 42 minutes . 4. To finde the Suns Amplitude at rising or setting . Take the Suns Declination out of the particular Scale of Altitudes , and lay it the same way as the Declination is , from 90 in the Azimuth Scale , and it shall shew the Amplitude from the east or west , counting from 90. Example : May the tenth it is 33. 37. CHAP. V. Having the Suns Declination , or day of the moneth , to finde the Azimuth at any Altitude required for that day . FIrst finde the Suns Declination , by the first Proposition of the fourth Chapter , then take that out of the particular Scale of Altitudes , or scale to 62 degrees ; then whatsoever the Altitude shall happen to be , count the same on the degrees from 60 toward the end of the Rule , according to the second maner of counting , in the third Proposition of the third Chapter , and thereunto lay the thred , then the Compasses set to the Declination , carry one point along the line of hours on the same side of the thread the Declination is ; that is to say , if the day of the moneth , or Declination be on the right side the Aequinoctial , then carry the Compasses on the right side ; but if the Declination be on the South side , that is , toward the end , ( counting from the tenth of March , or Aries or Libra , then carry the Compasses along the line of hours and Azimuths on the left side of the th●ead , as all win●er time it will be , and having set the Compasses to the least distance to the thread , it sh●ll stay at the Suns true Azimuth from the South required , counting as the figures are numbred ; or from East or West , counting from 90. Example 1. On the tenth of Iuly I desire to finde the Suns Azimuth at any Altitude , first on that day I finde the Suns Declination to be 20. 45 , which number count from the beginning of the particular Scale of Altitudes toward 62 , and that distance take between your Compasses , then are they set for all that day ; then supposing the Suns height to be ten degrees , lay the thread on 10 , counted from 60 toward the left end , then carrying the Compasses on the right side of the thread , ( because it is summer or north declination ) on the line of Azimuths , it shall shew 110. 40 , the Azimuth from the south required ; but if you count from 90 , it is but 20. 40. from the east , or west point northward , according to the time of the day , either morning or evening . Example 2. Again , on the 14. of November , or the 6. of Ianuary , when the Sun hath the same declination south-ward , and the same Altitude , to work this you must lay the Rule down on something , then lay the thread on the Altitude , counted from 60 toward the end ( as before ) and carrying the Compasses on the south-side of the Aequinoctial , along the Azimuth-line , till the other point do but just touch the th●ead , and it shall stay at 36. 45 , the Azimuth from south required ; if it be morning , it wants of coming to south ; if it be after-noon , it is past the south . Example 3. But if the Sun be in the Aequinoctial , and have no declination , then it is but laying the thread to the Altitude , and in the line of Azimuths the thread shall shew the true Azimuth required . As for instance : at 00 degrees of altitude , the Azimuth is 90 , at 10 degrees it is 77. 15 , at 20 degrees 62. 45 , at 30 degrees high 43. 15 , at 35 degrees high 28. 10 , at 38 degrees 28 ' high , it is just south , as by practice may plainly appear . But if the Suns altitude be above 45 , then the degrees will go beyond the end of the Rule : To supply this defect , do thus : Substract 45 out of the number you would have , and double the remainder , then lay the Rule down with some piece of the same thickness , in a streight line with the moveable leg ; then take the distance from the tangent of the remainder doubled ( counted from 60 to the end of the Rule , in the line next the edge ) to the Center , lay that distance in the same streight line from the tangent doubled , and that shall be the tangent of the Angle above 45 , whereunto you must lay the thread for the finding the Azimuth , when the Sun is above 45 degrees high . CHAP. VI. To finde the hour of the Night by the Moon . FIrst by the help of an Almanack , get the true time of the New Moon , then compute her true place at that time , which is always the place of the Sun ( very nigh ) at the hour and minute of conjunction ; then compute how many days old the Moon is , then by the line of Numbers say : If 29 dayes 13 hours , ( or on the line 29. 540 ) require 860 degrees , or 12 signs , what shall ●ny less number of days and part of a day require ? The answer will be : The Moons true place at that age . Having ●ound her true place , then take her al●itude , and lay the thred on the Moons place found , and work as you did for the Sun , and note what hour you finde ; then consider if it be New Moon , the hour you finde is thētrue hour , likewise in the Full ; but if it be before or after , you must substract by the Line of Numbers thus : If 29 days 540 parts require 24 hours , what shall any number of days and parts require ? The answer is : What you must take away from the Moons hour found , to make the true hour of the Night which was required . But for more plainness sake , I will reduce these Operations to so many Propositions , before I come to an Example . PROP. 1. To finde the Moons Age. First , it is most readily and exactly done by an Ephemerides , such a one as you finde in Mr. Lilly's Alman●ck , or ( as to her Age onely ) in any book or Sheet-Almanack ; but you may do it indifferently by the Epact thus , ( by the Rules of the 152 page in the Appendix to the Carpenters Rule . ) Adde the Epact , the moneth , and the day of the mone●h together , and the sum , if under 30 , is the Moons age ; but if above , consider if the moneth have 30 or 31 days , then substract 29 or 30 out , and the remainder is the Moons age in days . Example . August 2. 1660. Epact 28. Month 6. day 2. added makes 36. Now August or sixt moneth , hath 31 days , therefore 30 being taken away , 6 days remains for the moons age required . PROP. 2. To finde the Moons place . By the Ephemerides aforesaid in Mr. Lilly's Almanack , you have it ser down every day in the year ; but to finde it by the Rule , do thus : Count six days back from August 2. viz. to Iuly 27. there lay the thread , and in the line of the Suns place , you have the Moons place required , being then near alike ; then in regard the Moon goes faster than the Sun , that is to say , in 29 days 13 hours , 12 signs , or 360 degrees ; in 3 days 1 sign , 6 degr . 34 min. 20 sec. in one day o signs , 12 degr . 11 min. 27 sec. in one hour , 30 min. 29 sec. ( or half a minute : ) adde the signs , and degrees , and minutes the Moon hath gone in so many days and hours , if you know them together , and the Sun shall be the Moons true place , being added to what she had on the day of her change ; but far more readily , and as exactly by the line of Numbers ( or Rule of Three ) say , if 29. 540. require 360. what 6 facit 73 1 / 4 , that is , 2 signs , 13 degrees , 15 minutes , to be added to 14 degrees in ♌ , and it makes 27 1 / 4 in ♎ , the Moons place for that day . Or thus , multiply the Moons age by 4 , divide the product by 10 , the quotient sheweth the signs , and the remainder multiplied by 3 , sheweth the degrees which you must adde to the Suns place on the day required , and it shall be the Moons true place required for that day of her Age. Example . Iuly 27. the Sun and Moon is in Leo 14 degr . 0. August 2. being 6 days , adde the Moons motion 2. 13d 9 ' . makes being counted , Virgo ♍ , Libra ♎ . 27 deg . 9 the place required ; which on the Rule you may count without all this work or trouble : but for plainness sake I am constrained so to do . Or thus . PROP. 3. To finde the Moons hour . To do this , you must do the work of the second Chapter , and second and third Proposition ; where note that the Moons place found , is to be used as the day of the moneth , or Suns declination . Example . The Moon being 27 degrees in Libra , and 20 degrees high , I finde the our to be 31 ' past 9 , if on the east-side ; or 29 ' past 2 , if on the west-side ●f the Meridian . PROP. 4. To finde the true hour of the Night . Having found the Moons hour , as before , consider the Moons age , then say by the Line of Numbers , or Rule of Three , if 29. 540 part require 24 , what shall 6 days require facit 4 hours and 52 minutes ? which taken from 9. 31. rest ● . 29. the true time required . Example . Moons hour — 9 31 Time to be subst . ● 52 True time remain 3 39. the time required . This work is done more readily by the two lines fitted for that purpose ; for look for 6 the Moons age in one , and you shall finde ● hours 52 minutes , the time to be substracted in the other . CHAP. VII . To finde the hour of the Night by the fixed Stars . FOr the doing of this , I have made choice of twelve principal fixed Stars , all within the two Tropicks ; many more might be added , but these will very well serve the turn : The names of them , and their right Ascension in hours and minutes , is set on the Rule , and the star is placed in his true Declination on , or among the moneths ; and for to know the stars next to a Tutor , a Celestial Globe , or a Nocturnal , of all the chief stars from the Pole to the Equinoctial , and to be had at the Sun-Dial in the Mi●nories , is the best ; the uses whereof do follow . First know the star you observe , then observe his Altitude , and laying the thread on the star by the second Chapter , second and third Proposition , get the stars hour , then out of the righ● Ascension of the star , take the righ● Ascension of the Sun , ( found by the fifth Proposition of the third Chapter ) for that day , and note the difference for this difference added to the star hours found , shall shew the hour o● the night . Example . On the first of November I observe the Altitude of the Bulls-eye , and find it to be 30. then by second and third of the second Chapter , I finde the hour to be 7. 54 past the Meridian the Suns right Ascension that day finde to be 15 hours 8 minutes , the stars right ascension is 4 hours 16 minutes ; which taken , the greatest from the least , by adding of 24 hours , re● 13. 08. then 7. 54. the stars hou● added , makes 21. 02. from which taking 12 hours , rest 9. 02. the hour o● the night required . For more plainness , note the work of two or thre● Examples . Stars right Ascension being set on the Rule — 4 — 16 Suns right Asc. Nov. 1. — 15 — 08 — Substraction being made , by adding 24 , remain — 13 — 08 To which you must adde the stars hour found — 7 — 54 Then the remainder , taking away 12 hours , is — 9 — 02 the true hour . Again May 15 by Arcturus , at 50 0 ' high westwards . The right Asc. of Arcturus , is 14h● — 0 ' Suns right Asc. May 15. is — 4 — 10 — The right Asc. of the Sun taken from the R. Asc. of the star , rest ● — 50 The stars hour at 50 degr . high , found to be — 02 — 13 — Which being added to the difference before , makes — 1● — 03 or — 11 — 03. Again January 5. by the Great-dog at 15 degrees high east . The right Asc. of Great-dog , is 06-29 The right Asc. of the Sun , is 19 — 50 — Substr . made , by adding 24 h. is 10-39 Stars hour at 15 degr . high , is 9 — 32 Ante M. or P. Septen . Which added to the difference found , and 12 substracted , remains 8. 11 ' . for the true Hour of the Night required to be found , and so of any other star se● down in the Rule , as by the trial and pr●ctice , will prove easie and ready to the ingenuous practitioner . But by the line of 24 , or twice 12 hours , and the help of a pair of Compasses , you may perform it withou● writing it down , thus : Take the righ● Ascension of the Sun , out of that lin● of hours between your Compasses ( being always counted under 12 ) an● set the same from the right Ascensio● of the star , toward a lesser number , o● the beginning of the hours , and th● point shall stay at the remainder that is to be added to the stars hour found , then open the Compasses from thence to the beginning of the hours , and adde that to the stars hour found , and it shall reach to the hour of the night required . Example . Feb. 6. 1660. by Arcturus 20 degrees high : Take 10 hours 1 minute between your Compasses , and set it from 14 hours , or 2 hours beyond 12 , and it shall stay at 3. 59 : then take 3. 59 , and adde it to 6. 24 , the stars hour at 20 degrees high , and it shall be 10. 23 , the hour of the night required . CHAP. VIII . To finde the Amplitude or Azimuth of the fixed Stars ; also their rising , setting , and southing . 1. FIrst for the Amplitude , take the stars declination from the particular Scale of Altitudes , and lay it from 90 in the Azimuth-line , and it shall shew his amplitude from East or West toward South or North , according to the declination , and time of day , morning or evening . The same work is for the sun . Example . The Bulls-eye hath 25. 54 degrees of amplitude , so hath the Sun at 15 degrees 48 minutes of declination . 2. To finde the Azimuth , work as you did for the Sun at the same declination the star hath , by Chapter 5. and you shall have your desire . Example . December 24. at 6 degrees high , by the Bulls-eye I finde the Azimuth to be 107. 53. from the south . 3 To finde the stars rising and setting , lay the thread on the star among the moneths , and in the line of hours it shews the stars rising and setting , as you counted for the Sun ; but yet note this is not the time in common hours , but is thus found : Adde the complement of the Suns Ascension , and the stars right Ascension , and the stars hour last found together , and the Sun , if less than 12 ; or the remain 12 being substracted , shall be the time of his rising in common hours ; but for his setting , adde the stars setting last found to the other numbers , and the sum or difference shall be the setting . Example . For the Bulls-eye on the 23 of December , it riseth at 2 in the afternoon , and sets at 4. 46 in the morning . 4. To finde the time of the southing of any star on the Rule , or any other whose right ascension and declination is known , Substract the Suns right ascension from the stars , increased by 24 , when you cannot do without , and the remainder , if less than 12 , is the time required , in the afternoon or night before 12 ; but if there remain more than 12 , substract 12 , and the residue is the time from mid-night to mid-day following . Example . Lyons-heart on the tenth of March , the Suns Ascension is 0 2 ' . Lyons-heart whole right asc . is 9 50 ' Time of southing is 9 48 ' at night . 5. To finde how long any Star will be above the Horizon . Lay the thread to the star , and in the hour-line it sheweth the ascensional difference , counting from 90 ; then note if the star have North declination , adde that to 6 hours , and the sum is half the time ; if south , substract it from 6 , and the residue is half the time ; and the complement of each to 24 being doubled , is the whole Nocturnal Arch under the Horizon . Example . For the Bulls-eye , his Ascensional difference will be found to be one hour , 23 minutes , which added to 6 hours and doubled , makes 14. 46 , the Diurnal Ark of the Star , and the residue from 24 is 9. 14. for the Nocturnal Ark , or the time of its being under the Horizon . CHAP. IX . To perform the fore-going work in any latitude , as rising , amplitude , ascensional difference , latitude , hour , and azimuth , wherein I shall give onely the rule , and leave out the examples for brevity sake . 1. FOr the rising , and setting , and ascensional difference , being all one , do thus : Take the Suns declination out of the general Scale of Altitudes , then set one foot of the Compasses in the colatitude on the same scale , and with the other lay the thred to the nighest distance ; then the thred so laid , take the nighest distance from the latitude to the thread , with that distance set one foot in the Suns declination , counted from 90 toward the center , and the thread laid to the nearest distance , shall in the degrees shew the ascensional difference required , counting from 90 at the head toward the end of the Rule ; and if you reduce those degrees and minutes to time , you have the rising and setting before and after 6 , according to the declination and time of the year . 2. To finde the Suns amplitude . Take the Suns declination , and setting one foot in the colatitude , with the other lay the thread to the nearest distance , and on the degrees it sheweth the Suns amplitude at rising or setting , counting as be●ore from 90 to the left end of the Rule . 3. Having amplitude and declination , to finde the latitude . Take the declination from the general scale , and set one foot in the amplitude , the thread laid to the nearest distance in the line of degrees , it sheweth the complement of the latitude required , or the converse . 4. Having latitude , Suns declination and altitude , to find the height at 6 , and then at any other time of the day and year . Count the declination in the degrees from 90 toward the end , thereto lay the thread , the least distance from which to the latitude in the general Scale , shall be the Suns height at 6 in the summer , or his depression in the winter . The Compasses standing at this distance , take measure on the general Scale of altitudes , from the beginning at the pin towards 90 , keeping one point there , open the other to the Suns altitude , thus have you substracted the height at 6 , out of the Suns altitude ; but in winter you must adde the depression at 6 , which is all one at the same declination with his height at 6 in summer , and that is done thus : Put one point of the Compasses so set in the general Scale to the Suns Altitude , then turn the other outwards toward 90 , there keep it , then open the Compasses to the beginning of the Scale , then have you added it to the Suns altitude ; having this distance , set one foot in the colatitude on the general Scale , lay the thread to the nearest distance ; the thread so laid , take the nearest distance from 90 to the thred , then set one foot in the declination , counted from 90 , and on the degrees it sheweth the hour from 6 , reckoning from the head , or from 12 , counting from the end of the Rule . I shall make all more plain , by making three Propositions of it , thus : Prop. 1. To finde the hour in the Aequinoctial . Take the Altitude from the beginning of the general Scale of altitudes , and set one foot in the colatitude , the thread laid to the nearest distance ( with the other foot ) in the degrees , shall shew the hour from 6 , counting from 90 , and allowing for every 15d 1 hour , and 4 min : for every degree . Prop. 2. To finde it at just 6. Is before exprest by the converse of the first part of the fourth , which I shall again repeat . Prop. 3. To finde it at any time do thus . Count the Suns declination in the degrees , thereunto lay the thred , the least distance , to which from latitude in the general Scale , shall be the Suns altitude at 6 ; which distance in summer you must substract from , but in winter you must add to the Suns present altitude ; having that distance , set one foot in the coaltitude , with the other lay the thread to the neerest distance , take again the neerest distance from 90 to the thread , then set one foot in the Suns diclination counted from 90 , and lay the thread to the neerest distance , and in the degrees it shall shew the hour required . Example . At 10 declination north , and 30 high , latitude 51. 32 , the hour is found to be 8. 25 , counting 90 for 6 , and so forward . Again at 20 degrees of declination South , and 10 degrees of altitude , I finde the hour in the same latitude to be 17 minutes past 9. Having latitude , delination , and altitude , to finde the Suns Azimuth . Take the sine of the declination , put one foot in the latitude , the thread laid to the neerest distance : in the degrees , it sheweth the Suns height at due East or West , which you must in summer substract from the Suns altitude , as before on the general Scale of Altitudes , with which distance put one foot in the colatitude , and lay the thread to the neerest distance , then take the neerest distance from the sine of the latitude , fit that again in the colatitude , and the thread laid to the heerest distance , in the degrees shall shew the Suns Azimuth required . 6. But in winter you must do thus : By the second Proposition of the ninth Chapter , finde the Suns Amplitude for that day , then take the altitude from the general Scale of altitudes , and putting one point in colatitude , lay the thread to the neerest distance , then the neerest distance from the latitude must be added to the Suns Amplitude ; this distance so added must be set from the coaltitude , and the thread laid to the highest distance , and in the line of degrees , it gives the Azimuth from south , counting from the end of the rule , or from the East or West , counting from the head or 90 degrees . Example . At 15 degrees of declination and 10 altitude , latitude 51. 32. the Azimuth is 49. 20. from the South , or 40 degrees and 40 ' from East or West . CHAP. X. To finde all the necessary quesita for any erect declining Sun-dial both , particularly and general , by the lines on the Dial side , also by numbers , sines , and tangents artificial , being Logarithms on a Rule . 1. First a particular for the Substile . COunt the plains declination on ●he Azimuth scale , from 90 toward the end , and thereunto lay the thread , in the line of degrees it shews the distance of the substile from 12. Example . At 10 degrees declination , I find 7. 51. for the substile . 2. For the height of the stile above the substile . Take the Plaines Declination from 90 in the Azimuth line , but counted from the South end , between your compasses : and measure it in the particular scale of altitudes , and it shall give the height of the stile required . Example . At 30 declination is 32. 35. 3. For the inclination of Meridians : Count the substile on the particuler scale of Altitudes , and take that distance between your compasses , measure this distance on the Azimuth line from 90 toward the end , and counting that way it sheweth the inclination of Meridians required . Example . At 15 the substile , the inclination of Meridians will be found to be 24. 36. 4. To finde the Angle of 6 from 12. Take the plaines declination from the particular scale of altitudes , and lay it from 90 on the Azimuth scale , and to the Compasses point lay the thread : then on the line of degrees you have the complement of 6 from 12 , counting from 60 toward the end . Note this Rule ( as this line is drawn ) doth not give this Angle exactly , neither will it be worth the while to delineate another line for this purpose . But if it be required it may be done , but I rather prefer this help , the greatest error is about the space of 45 minutes of the first degree in the particular scale of altitudes ; so that if you conceive those 45 minutes to be divided as the particular scale of altitudes is , like a natural sine , and if your declination be 30 , then take half the space of the 45 minutes less , and that shall be the true distance to lay on the Azimuth line from 90 whereunto to lay the thread . Example . A plaine declining 30 degrees , the angle will be found to be 32. 21. whose complement 57. 49. is the angle required . 5. To perform the same generally by the general scale of altitudes ; and first for the stile . Lay the thred to the complement of the latitude , counted in the degrees from the head toward the end , then the nightest distance from the complement of the plaines declination to the thread , taken and measured on the general scale , from the center , shall be the stiles height required . 6. To finde the inclination of Meridians Take the plaines declination , from the general scale , and fit it in the complement of the stiles elevation , and lay the thread to the neerest distance , and on the degrees it sheweth the inclination of Meridians required . 7. For the substile , Count in the inclination of Meridians on the degrees from 90 , and thereto lay the thread , then take the least distance from the latitudes complement to the thread , set one foot of that distance in 90 , and lay the thread to the neerest distance , and in the degrees it shall shew the substile from 12 required . 8. For the angle of 6 from 12. Take the side of the square , or the measure of the parallel from 12 , and fit it in the cosine of the latitude , and lay the thread to the nighest distance , then take out the nearest distance from the sine of the latitude to the thread , then fit that over in the sine of 90 , and to the nearest distance lay the thread , then take the nearest distance from the sine of the plains declination to the thread , and it shall reach on the parallel line , or side of the square , from the Horizon to 6 a clock line required . Four Canons to work the same by the artificial sines & tangents . Inclination of Meridians . As the Sine of the latitude , To the Sine of 90 : So the Tangent of the Declination , To the Tangent of inclination of Meridians . Stiles Elevation . As the Sine of 90 , To the Cosine of the Declination : So the Cosine of the latitude , To the Sine of the Stiles elevation , Substile from 12. As the Sine of 90 , To the Sine of the Declination : So the Cotangent of Latitude , To Tangent of the Substile from 12 ▪ For 6 and 12. As the Co●tangent of the Latitude , To the Sine of 90 : So is the sine of Declination , To the Cotangent of 6 from 12. For the hours . As the Sine of 90 , To the Sine of the Stiles height : So the Tangent of the hour from the proper Meridian , To the Tangent of the hour from the Substile . The way to work these Canons on the Sines and Tangents , is generally thus : As first , for the inclination of Meridians , set one point in the Sine of the latitude , open the other to the Sine of 90 , that extent applied the same way , from the Tangent of the Plains declination , will reach to the Tangent of the inclination of Meridians required . CHAP. XI . To draw a Horizontal Dyal to any latitude . FIrst draw a streight line for 12 , as the line A B , then make a point in that line for a Center , as at C , then through the Center C , raise a perpendicular to A B , for the two six a clock hour-lines , as the line D E ; then draw two occult lines parallel to A B , as large as the Plain will give leave , as D E , and E G then fit C D in the Sine of the Latitude , in the general Scale , and lay the thread to the nighest distance , then take the nearest distance from 90 to the thread , and set it from D and E in the two occult lines , to F and G , and draw the line F and G parallel to the two sixes , ( or make use of the Sines on the other side , thus : Fit A D , or C D in the Sine of the latitude , and take out the Sine of 90 , and lay it as before from D and E , ) then fit D F , or E G in the Tangent of 45 degrees , on the other side of the Rule , and lay off 15 , 30 , and 45 , for every whole hour , or every 3 degrees and 45 minutes for every quarter , from D and E , toward F and G , for 7 , 8 9 , and for 3 , 4 , and 5 a clock hour points . Lastly , set C D , or B E in the Tangent of 45 , and lay the same points of 15 , 30 , 45 , both wayes , from B or 12 , for 10 , 11 , and 1 , 2 , and to all those points draw lines for the true hour-lines required , for laying down the Stiles height ; if you take the latitudes complement , out of the Tangent-line as the Sector stood , to prick the noon hours , and set it on the line D F , or E G , from D or E downwards from D to H , it will shew you where to draw C H for the Stile , then to those lines set figures , and plant the Dial Horizontal , and the Stile perpendicular , and right north and south , and it shall shew when the sun shineth , the true hour of the day . Note well the figure following . CHAP. XII . To draw a Vertical , Direct , South , or North Dyal . FIrst draw a perpendicular line for 12 a clock , then in that line at the upper end , in the south plain : and at the lower end in a north plain , appoint a place for the center , through which point cross it at right angles , A Horizontall Diall A South Diall for 6 and 6 , as you did in the Horizontal Plain , as the lines A B , and C D , on each side 12 make two parallels , as in the Horizontal , then take A D the parallel , and fit it in the sine of the latitudes complement , and take out the sine of 90 and 90 , and lay it in the parallels from D and C , to E and F , and draw the line E F , then make D E , and B E tangents of 45 , and lay down the hours as you did in the horizontal , and you shall have points whereby to draw the hour lines . For the north you must turn the hours both ways for 4 , 5 , 8 and 7 in the morning and 4 , 5 , 7 and 8 , at night the height of the stile must be the tangent of the complement of the of the latitude when the sector is set to lay off the hours from D , as here it is laid down from C to G , and draw the line A G for the stile . For illustration sake note the figure . CHAP. XII . To draw an erect East or West Dial. FIrst by the fifth Proposition of the second Chapter draw a horizontal line , as the line A B at the upper part of the plaine . Then at one third part of the line A B , from A the right end if it be an East plaine , or from B the left end , if it be a West Plain , appoint the center C , from which point C draw the Semicircle A E D , and fit that radius in the sine of 30 degrees , ( which in the Chords is 60 degrees ) then take out the sine of half the , latitude , and lay it from A to E , and draw the line C E for six in the morning on the East , ( or the contrary way for the West . ) Then lay the sine of half the complement of the latitude , from D to F , and draw the line C F , for the contingent or equinoctial line , to which line you must draw another line parallel , as far An East Diall . A West diall . assunder as the plaine will give leave , then take the neerest distance from A to the six a clock line , or more or less as you best fancy , and fit it in the tangent of 45 degrees , and prick down all the houres and quarters , on both the equinoctial lines , both ways from six , and they shall be points , whereby to draw the hoor lines by , but for the two houres of 10 and 11 there is a lesser tangent beginning at 45 , and proceeding to 75 , which use thus : fit the space from six to three , in the little tangent of 45 , and then and then lay of 60 in the little tangents from 6. to 10 and the tangent of 75 from 6 to 11 , and the respective quarters also if you please , so have you all the hour●s in the East , or west Diall , the distance from six to nine or from six to three , in the West , is the height of the stile , in the East and West Diall , and must stand in the six a clock line , and parallel to the plaine . CHAP. XIII . To finde the declination of any Plain . FOr the finding of the declination of a Plain , the most usual and easie way , is by a magnetical needle fitted according to Mr. Failes way , in the index of a Declinatory ; or in a square box with the 90 degrees of a quadrant on the two sides , or by a needle fitted on the index of a quadrant , after all which ways , you may have them at the Sign of the Sphere and Sun-Dial in the Minories , made by Iohn Brown. But the work may be very readily and exactly performed by the rule , either by the Sun or needle in this manner following , of which two ways that by the Sun is always the best , and most exact and artificial , and the other not to be used ( if I may advise ) but when the other failes , by the Suns not shining , or as a proof or confirmation of the other . And first by the needle because the easiest . For this purpose you must have a needle well touched with the Loadstone of about three or four inches long , and fitted into a box somewhat broader then one of the legs of the sector , with a lid to open and shut ; and on the inside of the lid may be drawn a South erect Dial , and a wire to set the lid upright , and a thread to be the Gnomon or stile to that Dial : it will not be a miss also to extend the lines on the Horizontal part for the same thread is a stile for that also . Also on the bottom let there be a rabbit , or grove , made to fit the leg of the rule or sector ; so as being pressed into it , it may not fall off from the rule , if your hand should shake , or you cease to hold it there . This being so fitted , the uses follow in their order . Put your box and needle on that leg of the rule , that will be most fit for your purpose , and also the north end of the needle toward the wall , if it be a south wall ; and the contrary , if a north , as the playing of the needle will direct you , better then the way how in a thousand words , then open or close the Rule , till the needle play right over the north and south-line , in the bottom of the Box : then the complement of the Angle that the Sector standeth at ( which may always be under 90 degrees ) is the declination of the Plain . But if it happen to stand at any Angle above 90 , then the quantity thereof above 90 , is the declination of the wall . To finde the quantity of the Angle the Sector stands at may be done two ways : first by protraction , by laying down the Rule so set on a board , and draw two lines by the legs of the Sector , and finde the Angle by a line of Chords . Secondly , more speedily and artificially , thus : By the lines of Sines being drawn to 2. 4 , or 6 degrees asunder . The Sector so set , take the parallel sine of 30 and 30 , and measure it on the lateral sines from the center , and it shall reach to the sine of half the Angle the line of sines stands at , being more by 2 , 4 , or 6 degrees then the sector stands at , because it is drawn one , two , or three degrees from the inside . Or else take the latteral sine of 30 from the center , and keeping one foot fixed in 30 , turn the other till it cross the line of sines on that line next the inside , and counting from 90 , it shall touch at the Angle the line of sines standsat , being two degrees more then the Sector stands at ; the lines being drawn so , will be ( as I conceive ) most convenien . Take an Example . I come to a south-east-wall , and putting my box and needle on my Rule , with the cross or north-end of the needle toward the wall , and the Rule being applyed ●lat to the wall , on the edge thereof , on the evenest place thereof , and held level , so as the needle may play well with the head of the Rule toward your right hand , as you shall finde it to be in an east-wall most convenient ; then I open or close the Rule , till the needle play right over the north and south line in the bottom of the box ; then having got the Angle , ( take off the box , or if you put it on the other side that labor may be saved . ) I take the parallel sine of 30 , and measuring it from the center it reaches , suppose to the sine of 20 , then is the line of sines at an Angle of 40 degrees , but the Sector at two degrees less , viz. 38 degrees , whose complement 52 is the declination ; then to consider which way , minde thus : First it is south , because the sun being in the south , shines on the wall . Secondly consider , the sun being in the east , it shines also on the wal , therefore it is east plain : thus have you got the denomination which way , and also the quantity how much that ways . Or if you take the latteral sine of 30 from the center , and turn the point of the Compasses from 30 towards 90 , on the other leg you shall finde it to reach to the sine of 50 degrees , whose complement , counting from 90 , is 40 , or rather 38 , for the reason before-said , or else adde 2 to 50 , and you have the angle required , without complementing of it , being the true declination sought for . Thus by the needle you may get the declination of any wall , which in cloudy weather may stand you in good stead , or to examine an observation by the Sun , as to the mis-counting or mistaking therein ; but for exactness the Sun is alwayes the best , because the needle , though never so good ▪ may be drawn aside by iron in the wall , and also by some kinde of bricks , therefore not to be too much trusted unto . To finde a Declination by the Sun. First open the Rule to an Angle of 60 degrees , as you do to finde the hour of the day , and put a pin in the hole , and hang the thread and plummet on the pin ; also you must have another thread somewhat longer and grosser then that for the hour in a readiness for your use . Then apply the head leg to the wall , if the sun be coming on the Plain , and hold the Rule horizontal or level , then hold up the long thread till the shadow falls right over the pin , or the center hole , at the same instant the shadow shall shew on the degrees , how much the sun wants of coming to be just against the Plain , which I call the Meridian or Pole of the Plain , which number you must write down thus , as suppose it fell on 40 , write down 40-00 want : then as soon as may be take the Suns true altitude , and write that down also , with which you must finde the Suns Azimuth ▪ then substract the greater out of the less , and the remainder is the declination required . But for a general rule , take this : if the Sun do want of coming to the Meridian of the place , and also want of coming to the Meridian of the Plain , then you must alway substract the greater number out of the less ▪ whether it be forenoon or after-noon , so likewise when the Sun is past the Meridian , and past the Plain also . But if the signes be unlike , that is to say , one past the Plain , or Meridian , and the other want either of the Plain , or Meridian , then you must add them together , and the sum is the declination from the South . Which rule for better tenaciousness sake take in this homely rime . Signs both alike substraction doth require , But unlike signs addition doth desire . The further illustration by two or three Examples . Suppose on the first of May , in the forenoon , I come and apply the Rule , being opened to his Angle of 60 degrees to the wall , ( viz. the head leg , or the leg where the center is ) and holding up my thickest thread and plummet , so as the shadow of it crosseth the center , and at the same instant also on 60 degrees , then I say the sun wants 60 deg . of coming to the Meridian of the Plain ; at the same instant , or as soon as possible I can , I take the suns altitude , as before is shewed , and set that down , which suppose it to be 20 degrees , then by the rules before , get the Suns azimuth for that day , and altitude ; which in our example will be found to be 94 degrees from the ( south or ) Meridian , then in regard the signs are both alike , i. e. want , if you substract one out of the other , there remains 34 the declination required ; but for the right denomination which way , either north or south , toward either east or west , observe this plain rule : First , if the Sun come to the Meridian or Pole of the Plain , before it come to the Meridian or Pole of the place , then it is always an East-plain ; but if the contrary , it is a West-plain , that is to say , if the Sun come to the Meridian or Pole of the place , before it comes to the Meridian or Pole of the Plain , then it is a West-plain . Also if the sum or remainder , after addition or substraction , be under 90 , it is a South-plain ; but if it be above 90 , it is a North-plain . Also note , that when the sum or remainder is above 90 , then the complement to 180 , is always the declination from the north toward either east or west ; So that according to these rules in our example it is 34 degrees South-east . Again in the morning , Iune 13. I apply my rule to the wall , and I finde the Sun is past the Pole or Meridian of the Plain 10 degrees , and the altitude at the same time 15 degrees , the Azimuth at that altitude , and day in this latitude , will be found to be 109 degrees want of south or pole of the place ; therefore unlike signs , and to be added , and they make up 119 degrees , whose complement to 180 is 61 ; for 61 and 119 added , make up 180 , therefore this Plain declineth 61 degrees from the north toward the east . Again the same day in the afternoon , I finde the Azimuth past the south or meridian of the place 30 degrees , and at the same time the Sun wants in coming to the meridian or pole of the Plain 10 degrees , here by addition I finde the declination to be 40 degrees south-west . Note what I have said in these three examples , is general at all times ; but if it be a fair day , and time and opportunity serve , to come either just at 12 a clock , when the Sun is the meridian or pole of the place ; or just when the Sun is in the meridian or pole of the Plain , then your work is onely thus : First if you come to observe at 12 , then applying your rule to the wall , and holding up the thread and plummet , how much so ever the Sun wants or is past the pole of the Plain , that is the declination , if it be past it is east-wards ; if it wants , it is south-west-wards ; if neither , a just South Plain , and then the poles , or Meridians of place and Plain , are the same . But secondly , if you come when the Sun is just in the pole of the Plain , then whatsoever you finde the Suns Azimuth to be , that is the declination ; if it wants of south , it declineth East-wards ; if it be past , it declineth West-wards . Thus I have copiously , ( and yet very briefly ) shewed you the most artificial way of getting the declination of any wall , howsoever situated . Note if the Sun be above 15 degrees wanting of the Meridian of the Plain , your rule will prove defective in taking the Plains Meridian when the center leg is next the wall , then you must turn the other leg to the wall , and then you finde a supply for all angles to 45 degrees past the Plain . But for the supply of the rest which is 45 degrees , do thus : open the rule till the great line of tangents & the outside of the leg make a right angle , for which on the head you may make a mark for the ready setting , then making the inside of the leg at the end of 45 , as a center , the tangents on the other leg supply very largely the defect of the othersides . Or if you set on the box and needle on the rule , and open or shut the rule till the shadow of the thread shew just 12 , then the Angle the Sector stands at , is the complement of what the sum wants , or is past the meridian or pole of the Plain . CHAP. XIV . To draw a vertical declining Plain to any declination . FIrst draw a perpendicular-line for 12 , as A B then design a point in that line for the center , as C , at the upper end , if it be a South Plain ; or contrary , if it be a North Plain , then on that center describe an Arch of a Circle on that side of 12 , which is contrary to the Plains declination , as D E , and in that Arch lay off from 12 the substile , and on that the stiles height , and the hour of 6 , being found by the tenth chapter , and draw those lines from C the center , then draw two parallels to 12 , as in the direct south : then fit the distance of the parallels in the secant of the declination , and take the secant of the latitude , and set it from the center C on the line of 12 , to F , and on the parallel from 6 to G , and draw a line by those two points F and G , to cut the other parallel in H : then have you found 6 , 3 , and 9 , then fit 6 G in the tangent of 45 , and prick off 15 , 30 , and the respective quarters both ways from 6 , for the morning , and afternoon hours , then fit F G in 45 , and lay off the same points from F both ways for 10 , 11 , 1 , and 2 , and the quarters also , if you please , and those shall be points to draw the hour-lines by . The stile must be set perpendicular over the substile , to the Angle found by the rules in the tenth chapter , and then the Dial shall shew the true hour of the day , being drawn fit to his proper declination . Another way to perform this Geometrically for all erect Dyals with centers . When you have drawn a line of 12 , and appointed a center , make a Geometrical square on that side 12 , as the stile must stand on , as A B C D , the perpendicular side of which square may also be the parallel as before . Again , fit the side of the square in the cosine of the latitude , and take out the sine of the latitude , and fit that over in the sine of 90 , then take out the 〈◊〉 sine of the declination , and lay it from D to G for the hour of 6 , and draw the line A G for the 6 a clock hour line . Then again , fit the side of the square ( or the distance of the parallel in the other way , when you want a secant , or your secant too little ) in the sine of 90 , and take out the cosine of the declination , fit that in the cosine of the latitude , then take out the sine of 90 , and lay it from the center on the line of 12 , and 〈◊〉 6 in the side of the square , and by those two points draw the contingent line ▪ and then fit those points or distances in the tangent of 45 , and lay down the hours as in the former part of this chapter ; but if you want the hours before 9 in west-decliners , or the hours after 3 in east-decliners , and the 6 fall too high above the horizontal line on that side , you may supply this defect thus : Take the measure with your Compasses from 3 to 4 and 5 , upwards in the west-decliners , or from 9 to 7 and 8 in east-decliners , and lay it upwards from 9 or 3 on the deficient side , in the parallel as it should have been from 6 downward in south or upward in north Plains , and you shall see all your defective points to be compleatly supplied , whereunto draw the hour-lines accordingly . CHAP. XV To draw the Hour-lines on an upright declining Dial , declining above 60 degrees ▪ IN all erect Decliners , the way of finding the stile , substile 6 and 12 , and inclination of Meridians , is the same ; but when you come to protract , or lay down the hour-lines , you shall finde them come so close together , as they will be useless , unless the stile be augmented . The usual way for doing of which , is to draw the Dial on a large floor , and then cut off so much and at such a distance , as best serveth your turn , but this being not always to be affected , for want of conveniency , and large instruments , it may very artificially be done by a natural tangent of 75 or 80 degrees , fitted on the legs of a sector in this manner . The example I shall make use of , let be a Plain declination 75 degrees , South , West . First on the north edge of the Plain draw a perpendicular-line A B , representing 12 a clock line ; then on the center draw an occult Arch of a circle as large as you can , as B D , therein lay off an angle of 37. 30 for the substile , ( though indeed this line will not prove the very substile , yet it is a parallel to it : ) then cross it with two perpendiculers for the two contingent lines , at the most convenient places , one in the upper part , and the other in the lower part of the Plain , as the two lines C E , and F G do shew , then by help of the inclination of Meridians , make the table for the hours in this manner ; for this declination , it is 78 : 09 , Now every hour containing 15 degrees , and every quarter three degrees 45 ' I find that the substile will fall on neer a quarter of an hour past five at night , therefore if you take 75 , the measure of five hours out of 78. 9 , there remaines 03. 09. for the first quarter from the substile next 12. Again , if you take 78. 09 , the inclination of Meridians out of 78. 45 , the measure of 5 hours one quarter , and there remaines 00. 36. for the first quarter on the other side of the substile , then by continual adding of 03. 45 , to 03. 09 , and to 00. 36 , I make up the table as heer you see or else you may against 12 , set down 78. 09 , and then take out 3. 45 , as often as you can , till you come to the substile , and then what the remainder wants of . 3. 45 , must be set on the other side of the substile , and 3. 45 added to that , till you come to as late as the sun will shew on the Plain , as here you see to 1 / 4 past 8. 45 36 . sub . 48 09 2 41 51 8 03 095 51 54 . 38 06 . 06 54. 55 39 . 34 21 . 10 39. 59 24 . 30 36 . 14 24. 63 09 1 26 51 7 18 094 66 54 . 23 06 . 21 54. 70 39 . 19 21 . 25 39. 74 24 . 15 36 . 29 24. 78 09 12 11 51 6 33 093     08 06 . 36 54.     04 21 . 40 39.     00 36 . 44 24.     The stile or gnomon is to be erected right over the substile , to the angle of 9. 16 , and augmented as much as from H to ● , and from I to ● , at the nighest distance from the point H and I , drawnin the substile line on the Plain , this way is as easy , speedy , artificial and true , as any extant , ( if your scales be true , ) and improveth the Plain at the first , to the most possible best advantage that can be . Note also , if in striving to put as many hours as you can on the Plain , the sum of the two natural numbers added together , comes to above 40000 : then you must reject something of one of them , for they will not be comely nor convenient , nor far enough asunder . I might have enlarged to declining , reclining plaines , out my intention is not to make a business of it , but onely to give a taste of the usefulness , and convenience of a Joynt-rule , as now it is improved . An Advertisement relating to Dialling . These directions are sufficient for any Horizontal , direct , and declining vertical Dials , but for all other , as East or West inclining and reclining , direct , or reclining and inclining , and declining polars ; and all declining and inclining : and reclinimg Plains , the perfect knowledge of their affections and scituations is very hard to conceive , but much more hard to remember by roat , or knowingly . A representation of which instrument take in this place , let the circle E S W N , represent the Horizon S AE P N the Meridian , AE W AE E the Equinoctial , P C P the Pole , being the points of an Axes made of th●ead , and passing through the center C W C E being shaddowed , represents the Plain , set to a declination , and to a reclination 45 degrees , F the foot that doth support it ; moreover the Plain hat● an Arch fitted to it , to get the stiles height , and to set it to any angle ; also the Horizon and Plain is made to turn round , which could not well be expressed in this figure , or representation of the instrument : a more liveley and easie help cannot likely be invented , and for cheapness the like may be in a half circle , or made of Past-board . The use and application of which figure , I shall not now speak of , partly because of the facillity in the using thereof , and partly because of the difficulty in description thereof ; and lastly , wheresoever you shall buy the same , at either of the three places , the use will be taught you gratis . Also note , that half of the Instrument may be made either in b●ass , or pastboard , and be made to fold down in a book , and perform the uses thereof indifferently well , for many purposes , as to the Affections of Dials . Then having discovered the affections , you may by the Canno●s in the seventh Chapter of Mr. Windgates R●le of proportion , finde all the requisites , and then to speed the laying down the hours , you may do it by help of the tangent of 45 , as you did in erect decliners ; for after you have drawn the hours of 6 , and three , or nine , and made a parallelogram by two lines parallel to 12 , and one parallel to six , then making the distance from six to nine , or three in the line parallel to 12 , a tangent of 45 , and the distance from 12 to 9 , or 3 in the parallel to 6 , likewise a tangent of 45 , in the sector , and laying off 15 , 30. 45 , or the respective quarters from 6 , and 12 , in those two lines , and they shall be the true points to draw the hour lines , by laying a ruler to them , and the center of the Dial ; but for those that have no centers , the rule of augmenting the stile in far decliners , serveth for these also . CHAP. XVI To finde a perpendicular altitude at one or two stations , and observations by by the degrees on the rule . OPen the rule to his Angle of 60 degrees , then looking up to the top or point of altitude you would observe the height of , as you do when you take the Suns altitude , and note the degrees and parts the thread cuts , and write it down in chaulk or ink , that you forget it not . Then measure from the place you stand to the foot or base of the object , ( being right under the top of the object , whose height you would measure ) in feet , yards , or any other parts . Thirdly , consider this being a right angled Plain triangle , if you have the angle at the top , the angle at the base is always the complement thereof . These things being premised , the proportion holds , as the fine of the angle opposite to the measured side , ( or base ) being the complement of the angle found , is to the base or measured side : so is the angle found , to the height required . Always remember to adde the height of your eye from the ground , ( at the time of taking the angle ) to the altitude found . For the operation of this , extend the compasses from the sine of the complement of the Angle found to the number of the measured side , on the line of numbers that distance applied the same way , from the sine of the Angle found , shall reach on the line of numbers to altitude required . Example at one station . I open my rule , and hang on the thread and plummet on the center , and observing the Angle at C , I finde it to be 41. 45 , and the Angle at B the complement of it 48. 15 , and the measure from C to A 271 feet : then the work being so prepared , is thus : As the Sine of 48. 15 , Is to 271 the measure of the side opposite to it : So is the Sine of the Angle 41. 45 , To 242 the measure of the side A B , opposite to the Angle at C , the height required . Again , at the station D , 160 foot from A , I observe and finde the Angle D to be 56. 30 , the Angle at A is the complement thereof , viz. 33. 30. This being prepared , I extend my Compasses from the sine of 33. 30 to 160 , on the line of numbers the same extent will reach from the sine of 56. 30 , to 242 on the line of numbers , lacking a small fraction , with which I shall not trouble you . An example at two stations . As the Sine of the difference , which is the Angle C B D 14. 45 , Is to the side measured , viz. D C 111 feet on the numbers : So is the sine of the Angle at C 41. 45 , To the measure of the side B C , the hypothenusa , or measure from your eye to the top of the object , viz. 290 feet . Again for the second Operation . As the sine of 90 the right angle at A , Is to 290 the hypothenusa B D : So is the sine of 56. 30 , the Angle at the first station D , To 242 , the Altitude B A , the thing required : : So also is the sine of the Angle at B 33. 70 , the complement of 56 , 30. To 160 the distance from D to A. To perform the same by the line of sines , drawn from the center on the flat-side , and the line of lines , or equal parts or inches in ten parts . To work these or any other questions by the line of natural Sines and Tangents , on the flat-side , drawn from the center , it is but changing the terms , thus : As the measured distance taken out of the line of lines , or any scale of equal parts , is to the sine of the angle , opposite to that measured side , fitted across from one leg to the other , the Sector so standing , take out the parallel sine of the angle opposite to the enquired side , and that measure shall reach on the line of equal parts , to the measure of the Altitude requi●●● ▪ Example as before . Take out of the lines or inches 2. 71 , and fit it in the sine of 48. 15 , across from one legge to the other , which I call A parral sine , ( but when you measure from the center onwards the end , I call it A latteral sine , ) then take out the parallel sine of 41. 45 , and measure it on the line of inches , or equal parts , and it shall reach to 2 inches 42 parts , or 242 , the Altitude required . After the same manner may questions be wrought on the line of lines , sines , or tangents alone , or any one with the other , by changing the Logarithmetical Canon from the first to the second , or third , and the second or third to the first to second ; as the case shall require , from a greater to a less , and the contrary ; for the fourth is always the same ; of which in the use of the Sector , by Edmund Gunter , you may finde many examples , to which I refer you . Also without the lines of sines either natural or artificial , you may find altitudes , by putting the line of quadrat , or shadows , on the Rule as in a quadrant , then the directions in the use of the quadrat , page ▪ 146 of the Carpenters Rule , will serve your turn , which runs thus : As 100 ( or 50 according as it is divided ) to the parts cut by the thread , so is the distance measured , to the height required ; which work is performed by the line of numbers onely . Or again , As the parts cut , to 100 or 50 : so is the height to the distance required . But when the thread falls on the contrary shadow , that is , maketh an Angle above 45 , then the work is just the contrary to the former . What is spoken here of taking of Altitudes , may be applied to the taking of distances ; for if the Sector be fitted with a staff , and a ball-socket , you may turn it either horizontal , or perpendicular , and so take any Angle with it , very conveniently and readily by the same rules and directions as were given for the finding of Altitudes . CHAP. XVII . The use of certain lines for the mensuration of superficial and solid bodies , usually inserted on Ioynt-Rules for the use of Work-men , of several sorts and kindes . FIrst the most general and received line for mensuration of Magnitudes , is a foot divided into 12 inches , and those inches into 8 , 10 , 12 , or more parts ; but this being not so apt for application to the numbers , I shall not insist of it here , but rather refer you to the Carpenters Rule ; yet nevertheless those inches , laid by a line of foot measure , doth by occular inspection onely , serve to reduce foot measure to inches , and inches likewise to foot measure , and some other conclusions also . 1. As first , The price of any commodity at five score to the hundred , either tale or weight , being given , to finde the price of one in number , or one pound in weight . As suppose at two pence half-peny a pound , ( or one ) I demand to what cometh the hundred weight , ( or five-score , ) counting so many pound to the hundred weight ? If you look for two inches and a half , representing two pence two farthings , right against it on the foot measure , you have 21 very near ; for if you conceive the space between 20 and 21 , to be parted into 12 parts , this will be found to contain ten of them , for the odde ten pence . But for the more certain computation of the odde pence , look how many farthings there is in the price of one pound , twice so many shillings , and once so many pence is the remainder , which if it be above 12 , the 12 or 12s , being substracted , the remainder is the precise number of pence , above the shillings there expressed ; and on the contrary , at any price the C hundred , or 5 score , to finde the price of one , or 1l . As suppose at 40s . the C. or 5 score , look for 40 in the foot measure , and right against it in the inches , you have 4 inches , 3 quarters , and 1 / 4 of a quarter , which in this way of account is 4 pence 3 farthings , and about a quarter of one farthing . Thus by the lines , as they are divided , it proceeds to 12 pence a pound ; but if you conceive the inches to be doubled , and the foot measure also , you shall have it to 24 d. or 48 d. the pound , or one in tale , of any commodity . As at 18 d. a piece , or pound , the price comes to 7l . 10s . the C. for then every ten strokes is 20s in the foot measure , and every inch is 2 pence , and every eighth one farthing . 2. Secondly , for the buying of Timber at 50 foot to a Load , at any price the load , how much a foot . Here in resolving this , the inches are to be doubled , and the foot measure taken as it is : As at 40 shillings the Load , 40 in the foot measure stands right against 4 inches 3 quarters and better , which being doubled , is 9d . 2 far . 1 / 2 far . near , for the price of one foot ; and on the contrary , at 5d . a foot , is 41s . 8 d. a load , &c. 3. For the great Hundred of 112l . to the Hundred , let the space of 12 inches be divided into 112 parts , then the like rule holds for that also . For the inches being divided into quarters , every quarter is a farthing , and every eighth half a farthing , and every division of the 112 is a shilling , and every alteration of a farthing in the price of a pound , makes a groat in the Hundred , as thus : At 3 pence a pound is 28s . the C. At 3d. 1 q. a pound , 30s . and 4d . the C. At 3d. 1 / 2 the pound , 32s . 8 d. the C. At 3 d. 3 farthings , 35s . the C. Thus you see that every fraction at a farthing advance , is 4 pence in the Hundred ; but for any other account , as 3 pence farthing half farthing , then count the fraction , as 1 , 12th part of a shilling , and nearer you cannot come by a bare occular inspection ; but the price of the Hundred being given , the price of the pound you have as near by this occular inspection , as any usual Coin is reducable , viz. to the 32 part of a peny , or nigher if you please . Again note here also , you may double , or quadruple the price : as to 24d . or 48d . the pound , or any price between . As for example : At 13d . a pound , is 6l . 1s . 4d . the C. At 32d . or 2s . 8d . the pound , is 14l . 19s . the C. and the like by dupling and quadrupling the inches , and the 112 parts , that layeth by it . 4. These lines of equal parts serve as Scales , for the protracting of any Draught of house or field , or the like ; also for addition or substraction of any small number . 5. Note that the line of foot measure , may be applied for the reducing of any odde fraction to a decimal fraction , as you may fee it in page 64. of Mr. Windgats Arithmetick made easie . 2. The use of the lines of decimal Timber and Board measure . The lines of decimal Timber and Board measure , are fitted to agree with the tenths , or foot measure , as those lines in the first chapter of the Carpenters Rule , are fitted to the inches , and the use of them is thus : And first for the decimal Board measure . Suppose a Board is 1 foot 50 broad , I look for 150 on that line , and from that place to the end of the Rule forwards , toward 100 , so much in length must you have to make a foot of superficial or board measure . 2. Or else thus : If you apply the end of the Rule next 100 , to one edge of the breadth of a board , or glass , then right against the other edge of the board , on that line of decimal board measure , you shall finde the 10ths and 100s , ( or feet 10ths & 100 ) parts of a foot , that you must have in length to make a foot superficial at that breadth . Example . I come to a board and applying the upper end next 100 , even to one edge of the board , the other reacheth to 0. 8 tenths , then I say that 8 tenths of a foot length at that breadth , makes a foot . 3. The use of decimal Timber measure . The use of this is much like the Board measure , onely here you must have a respect to the squareness of the piece , and not to the breadth onely ; for after you know how much the piece of timber or stone is square , in feet and 100 parts , then look that number on the line of decimal Timber measure , and from thence to the end of the Rule , is the length that goes to make a foot of timber . Example . At 14 , or 1. 40. parts of a foot square , look the same on the rule , and from thence to the end where 40 is , is the length of a foot of Timber , at that squareness , being about 51 parts of a foot divided into a 100 parts . 5. The use of the line of decimal yard measure , also running yard measure , according to the inches or decimal parts of a foot . The decimal yard measure , is nothing else but a yard or 3 foot divided into a 100 parts , and used in the same manner as the foot measure is , for if you take the length , and the breadth in that measure , and multiply it together , you shall have the content , in yards and 100 parts of a yard . Example . Suppose a peece of plastering is 4 yards 78 parts one way ; and 7. 35 parts another way : being multiplied together makes 35 yards , and 9954. of 10000 , which is very neer 36 yards . 5. But the decimal running yard measure is fitted to the foot measure , and the use is thus : Suppose a room is to be measured that is 7 foot 8 tenths high , and I would know how much makes a yard , at that breadth or height ; look foor 7f . 8 10ths on the line of decimal running yard measure , and the space on the rule , from thence to the end next 100 is the true length , that goeth to make up a yard of superficial measure , at that breadth or height . But if the peece be between 4 foot 5 10 broad , and 2 foot , then the table at the end of the line , will supply the defect : or you may change the terms , and call the length the breadth , and the contrary . But if it be under 2 foot broad , then if you do as you did with the board measure , you shall have your desire . Example . At 1 foot 3 10th broad , 6 foot 9 10ths make a yard . 6. But if the running yard measure be made to agree with the inches , then measure the height of the room in feet and inches , and if you take a pair of compasses , and measure from that place , to the end of the rule , then turn the compasses set at that distance as many times as you can about the room , so many yards is there in the room . 7. The use of the line of decimal round measure , commonly called Girt-measure , which is when the circumference of a round Cillender , or piller given in inches or ten parts of a foot . First ( for Girt-measure according to inches , being the most usual measure ) now much the pillar is about , then look for the same number on the line of Girt-measure , and from thence to the end of the rule , is the length that goeth to make a foot of Timber . But if it be under 30 inches about , then you must have above two foot in length , and then a table at the end of the line , or a repetition in another line , will supply the defect . But if the line of Girt-measure be divided according to foot measure , then use it as before , seeking the decimal part on the line , and from thence to the end is a foot . 8. The use of a line of solid measure , by having the Diameter of a round piece given in inches , or foot measure . Take the diameter with a rule , or a pair of Callipers , and learn the measure either in inches , or foot measure according as your line of Diameter is divided . Then look for the same number on the line of Diameter , and from thence to the end of the rule forward , is the length that makes a foot of timber at that diameter ( or measure cross the end of the round piece of Timber or stone . ) The Tables of all the under measure for all these lines follow . Decimal Superficiall under M.   10th . F. 1000   10 F. 1000 p.   1 100. 00     3. 848   2 50. 000     3. 706   3 33. 300     3. 570   4 25. 000     3. 450   5 20. 000 3 3 3. 332   6 16. 600     3. 217   7 14. 300     3. 115   8 12. 500     3. 025   9 11. 120     2. 940 1 1 10. 000   5 2. 850   1 9. 100   6 2. 780   2 8. 340   7 2. 700   3 7. 720   8 2. 628   4 7. 150   9 2. 560   5 6. 670   4 2. 500   6 6. ●60     ● . 440   7 5. 888     2. 382   8 5. 5●5     2. 336   9 5. 260     2. 273 2 2 5. 000     2. 213     4. 760     2. 173     4. 5●6     2. 127     4. 350     ● . 083     4. 170     2. 042   5 4 ▪ 000   5 2. 000 Decimall Superficiall . M.   F. 1000. p. 01 F. 1000. p.   1. 962   1. 320   1. 923   1. 304   1. 816   1. 286   1. 850   1. 268   1. 820 8 1. 250   1. 785   1. 237   1. 756   1. 220   1. 726   1. 207   1. 697   1. 192 6 1. 669   1. 178   1. 640   1. 164   1. 615   1. 151   1. 589   1. 138   1. 563   1. 125   1. 538 9 1. 112   1. 516   1. 100   1. 493   1. 087   1. 472   1. 076   1. 450   1. 063 7 1. 430   1. 052   1. 409   1. 041   1. 391   1. 030   1. 373   1. 020   1. 353   1. 011   1. 337 10 1. 000 Decimal Solid under Measure .   F. 1000. p. 10 F. 1000. p. 1 10000. 000   14. 805 2 2500. 000   13. 735 3 1100. 000   12. 780 4 630. 000   11. 916 5 400. 000 3 11. 125 6 277. 900   10. 415 7 200. 430   9. 760 8 150. 660   9. 125 9 120. 350   8. 625 1 100. 0000   8. 150   82. 800   7. 700   96. 500   7. 310   59. 390   6. 900   51. 100   6. 565   44. 500 4 6. 250   39. 150   5. 945   34. 650   5. 664   30. 850   5. 404   27. 750   5. 465 2 25. 000   4. 938   22. 700   4. 720   20. 675   4. 530   18. 920   4. 342   17. 400   4. 162   16. 000 5 4. 000 Decimall Solid under measure .   F. 1000. p. 01. F. 100. p.   3. 825   1. 738   3. 7●0   1. 694   3. 524   1. 651   3. 430   1. 608   3. 310 8 1. 568   3. 188   1. 528   3. 078   1. 493   2. 968   1. 458   2. 873   1. 420 6 2. 780   1. 390   2. 688   1. 356   2. 602   1. 323   2. 521   1. 297   2. 442   1. 266   2. 366 9 1. 236   2. 294   1. 208   2. 227   1. 185   2. 160   1. 160   2. 100   1. 131   2. 043   1. 109 7         1. 985   1. 084   1. 93●   1 , 061   1. 878   1. 041   1. 830   1. 021   1. 781 10. 1. 000 Vnder Yard-measure for feet and inches from one inch'to four feet six inches F. F. 1000. F. F. 1000. In.   In.   1 108.000   3. 850 2 54. 000   3. 720 3 36. 000   3. 600 4 27. 000 6 3. 482 5 21. 600   3. 373 6 18. 000   3. 271 7 15. 420   3. 175 8 13. 520   3. 085 9 12. 000 3 3. 000 10 10. 300   2. 922 11 9. 820   2. 842 1. 9. 000   2. 769   8. 320   2. 710   7. 740   2. 633   7. 201 6 2. 572   6. 760   2. 512   6. 350   2. 455 6 6. 000   2. 400   5. 680   2. 345   5. 400   2. 298   5. 140 4 2. 250   4. 906 1 2. 203   4. 695 2 2. 160 2 5. 500 3 2. 119   4. 320 4 2. 073   4. 160 5 2. 037   4. 000 9 2. 000 Vnder yard measure according to Decimal or Foot measure F. 10. F. 1000. p.   F. 1000. p. 1 90. 000 4 3. 7●0 2 45. 000 5 3. 600 3 30. 000 6 3. 461 4 22. 500 7 3. 332 5 18. 000 1 3. 211 6 15. 000 9 3. 104 7 12. 880 3 3. 000 8 11. 200 1 2. 903 9 10. 000 2 2. 812 1 9. 000 3 2. 728 1 8. 190 4 2. 648 2 7. 510 5 2. 572 3 6. 930 6 2. 502 4 6. 430 7 2. 435 5 6. 000 8 2. 370 6 5. 625 9 2. 310 7 5. 290 4 2. 250 8 5. 000 1 2. 195 9 4. 735 2 2. 142 1 4. 500 3 2. 093 1 4. 285 4 2. 046 2 4. 092 5 2. 000 3 3912     Vnder Girt-measure . Inc. about F. in . 100.   F. in . 100 1 1809.6.81 24 3.1.87 2 452. 4. 74 25 2. 10. 74 3 201. 0. 77 26 2. 8. 12 4 113. 1. 18 27 2. 5. 87 5 72. 4. 60 28 2. 3. 70 6 50. 3. 19 29 2. 1. 83 7 39. 3. 22 30 2. 0. 13 8 28. 4. 00 31 1. 10. 60 9 22. 4. 09 32 1. 9. 21 10 18. 1. 15 33 1. 7 : 94 11 14. 11. 46 34 1. 6. 78 12 12. 6. 80 35 1. 5. 72 13 10. 8. 09 36 1,4,75 14 9. 2. 79 37 1. 3 , 86 15 8. 0. 51 38 1 , 3 , 04 16 7. 0. 82 39 1 , 2 , 28 17 6. 3. 14 40 1 , 1 , 57 18 5. 7. 30 41 1 , 0 , 92 19 5. 0. 15 42 1 , 0 , 31 20 4. 6. 28 43 0 , 11 , 74 21 4. 1. 24 44 0 , 11 , 22 22 3. 8. 87 45 0 , 10 , 72 23 3. 5. 04     Vnder measure for the Diameter in Inches and quarters . In. over F. in 10   F. in 100 , 29 3 0 , 0 , 00 , 4 , 8 , 31 , 73● , 1 , 00 , 4 , 4 , 06 , 326 , 0 , 10 , 4 , 0 , 27 1 183 , ● , 80 7 3 , 8 , 89 , 118 , 2 , 16 , 3 , 5 , 85 , 81 , ● , 00 , 3 , 3 , 10 , 59 , 10 , 24 , 3 , 0 , 61 2 45 , 10 , 90 8 2 , 10 , 36 , 36 , 2 , 41 , 2 , 8 , 31 , 29 , 3 , 93 , 2 , 6 , 44 , 24 , 3 , 86 , 2 , 4 , 73 3 20 , 4 , 40 9 2 , 3 , 15 , 17 , 4 , 25 , 1 , 1 , 70 , 15 , 0 , 00 , 1 , 0 , 37 , 13 , 1 , 86 , 1 , 11 , 13 4 11 , 5 , 14 10 1 , 9 , 99 , 10 , 1 , 78 , 1 , 8 , 93 , 9 , 0 , 62 , 1 , 7 , 95 , 8 , 1 , 80 , 1 , 7 , 04 5 7 , 3 , 98 11 1 , 6 , 18 , 6 , 7 , 80 , 1 , 5 , 38 , 6 , 0 , 72 , 1 , 4 , 63 , 5 , 6 , 53 , 1 , 3 , 93 6 5 , 1 , 10 12 1 , 3 , 27     13 1 , 1 , 02 A Table of the number of bricks in a rodd of Walling at any Feet high , from 1 to 20 for 1 and 1 1 / 2 Feet high . at 1 brick thick . at 1 brick & 1 / 2 thick 1 176 264 2 352 528 3 528 792 4 704 1056 5 880 1320 6 1136 1704 7 1232 1848 8 1408 2112 9 1584 2376 10 1760 2640 11 1936 2904 12 2112 3168 13 2288 3432 14 2464 3696 15 2640 3960 16 2816 4224 16 1 / 2 2904 4356 17 -2992 4488 18 -3168 4752 19 -3344 5010 20 -3520 5280 If you would have this Table for 1 / 2 a brick , take the half of the table for one brick . If for two bricks then double it . If for two and a 1 / 2 then ad both these together ; if for three , double that for one brick and 1 / 2. If you have any number of feet of brick work , at half a brick , one brick , or two bricks , or more , and you would reduce it to one brick and half , then say by the line of numbers as 1. 2. 4. 5 or 6 is to three , so is the number of feet at 1 / 2 1. 2. 2 1 / 2 or three bricks to the number of feet at one and 1 / 2. The use of four scales , called Circumfence , Diameter , Square equal , Square inscribed . Suppose you have a circle whose diameter is 10 inches , or 10 feet : and to this circle you would finde the Circumference , or the side of a square equal , or inscribed , or having any one of the three , to finde the other three , do thus : Take the measure of the Circumference , Diameter , or either of the squares , which is first given , and open the compasses to the number of the given measure , in its respective scale : the compasses so set , if you apply it in the scale whose number you would know , you shall have your desire . Example . Suppose a circle whose diameter is 10 inches , and to it I would know the Circumference , take 10 out of the diameter scale , and in the Circumference scale it shall reach to 31 42 , and on the line of square equal 8 86 , and on square inscribed 7. 07. For illustration sake , note the figure . The use of the line to divide a circle into any number of parts . Take the Semidiameter , or Radius of the circle between your compasses , and fit it over in six and six of the line of circles , then what number of parts you would have , take off from that point by the figure in the line of circles , and it shall divide the circle into so many parts . As suppose I would have the former circle divided into nine parts , take the measure from the center to the circle as exactly as you can , fit that over in 6 and 6 , then take out 9 and 9 , and that shall divide it into so many parts ; but if you would divide a wheel into any odde parts , as 55. 63. or 49 parts , you shall finde it an almost impossible thing , to take a part so exact that in turning about so many times , shall not miss at last : to help which the parts the rule giveth shall fit you exact enough for all the odde parts , then the even will easie be had by dividing , therefore usually the rule is divided but to 30 or 40 parts . So that for this use as the finding the side of an 8 or 10 square piece , as the mast of a ship , or a newel , or a post , this will very readily , and exactly help you . CHAP XVIII . The use of Mr. Whites rule , for the measuring of Timber and Board , either by inches or foot measure . 1. ANd first for superficial or board measure , by the inches , the breadth and length being given in inches , and feet and parts , slide or set 12 on any one side , to the breadth in inches or parts on the other side , then just against the length found on the first side , where 12 was on the second side you shall have the content in feet , and 10ths , or 100 parts required . Which by the rules of reduction by the foot measure , you may reduce to inches and 8 parts . Example . At three inches broad , and 20 foot long , you shall finde it to be 5 foot just : but at 7 inches broad , and the same length it will come to 11 foot 7 10th fere , or 8 inches . 2. The breadth being given in inches , to finde how many inches in length goes to make a foot of board or flat measure . Set 12 on the first side , to the breadth in inches on the second side , then look for 12 on the second side , and right against it on the first side , is the number of inches , that goes to make a foot Superficial , at that breadth . Example . At three inches broad you shall finde 48 inches to make a foot . 3. To work multiplication on the s●iding , or Whites rule . Set one on the first side , to the multiplicator on the second side , then seek the multiplicand on the first , and right against it on the second , you shall finde the product . Example . If 9 be multiplied by 16 , you shall finde it to be 144. 4. To work division on the same rule . Set the divisor on the first side to one on the second , then the dividend on the first , shall on the second shew the quotient required . Example . If 144 be to be divided by 16 , you shall finde the quotient to be 9. 3. To work the rule of 3 direct . Set the first term of the question , sought out on the first line , to the second term on the other , ( or second line : ) then the third term sought on the first line , right against it on the second , you shall finde the fourth proportional term required . Example . If 15 yard 1 / 2 , cost 37s . 6d . what cost 17 3 / 4 ? facit 42s . 10d . 3 q. for if you set 15 1 / 2 right against 37 1 / 2 , then look for 17 3 / 4 on the first line , ( where 15 1 / 2 was found , and right against it on the second line , is neer 42 the fractions are all decimal , and you must reduce them to proper fractions accordingly To work the rule of 3 reverse . 4. Set the first term sought out on the first line , to the second being of the same denomination or kind to the second line , or side . Then seek the third term on the second side , and on the first you shall have the answer required . Example . 5. If 300 masons build an edifice in 28 days , how many men must I have to perform the same in six days , the answer will be found to be 1400. 6. To work the double rule of 3 direct . This is done by two workings : As thus for Example . If 112 l. or 1 C. weight , cost 12 pence the carraiage for 20 miles , what shall 6 C , cost , 100 miles ? Say first by the third rule last mentioned , as 1 C. weight to 12 , so is 6 C. weight to 72. pence , secondly say if 6 C. cost 72 pence or rather 6s , for 20 miles ? what shall 100 miles require ? the answer is 30 s. for if you set 20 against 6 then right against 100 is 30 , the answer required . The use of Mr. Whites rule in measuring Timber round , or square , the square or girt being given in inches , and the length in feet and inches . 1. The inches that a piece of Timber is square , being given : to finde how much in length makes a foot of Timber , look the number of inches square on that side of the Timber line , which is numbred with single figures from 1 to 12 , and set it just against 100 on the other or second side , then right against 12 at the lower , ( or some times the upper ) end , on the first line , in the second you have the number of feet and inches required . Example . At 4 1 / 2 inches square , you must have 7 foot 1 inch 1 / 3 to make a foot of Timber . But if it be above 12 inches square , then use the sixth Problem of the 5th chapter of the Carpenters Rule , with the double figured side and Compasses . 2. But if it be a round smooth stick , of above 12 inches about , and to it you would know how much in length makes a true foot , then do thus . Set the one at the beginning of the double figured side , next your left hand , to the feet and inches about , counted in the other side , numbred with single figures from 1 to 12 , then right against three foot six 1 / 2 inches , in the single figures side next the right hand , you have in the first side the number of feet , and inches required . Example . A piece of 12 inches about , requires 11 f 7 in : fere to make a foot . Again a piece of 15 inches about , must have 8 foot 1 / 2 an inch in length , to make a foot of timber . 3. But if you would have it to be equal to the square , made by the 4th part of a line girt about the piece , then instead of three foot 6 1 / 2 inches make use of four foot , and you shall have your desire . 4. The side of a square being given in inches , and the length in feet , to find the content of a piece of timber . If it be under 12 inches square then work thus : set 12 at the beginning or end of the right hand side , to the length counted on the other side , then right against the inches square on the right side is the content on the left side Example . At 30 foot long , & 9 inches square , you shall find 16 foot 11 inches for the working this question , 12 at the end must be used . But if it be above 12 inches square , then ser one at the beginning , or 10 at the end of the right hand side , to the length counted on the other side , then the number of inches or rather feet and inches , counted on the first side , shall shew on the second the feet and parts required . Example . At 1 f. 6 inch . square , and 30 foot long , you shall finde 67 feet and about a 1 / 2. 5. To measure a round piece by having the length , and the number of inches about , being a smooth piece , and to measure true , and just measure , then proceed thus : Set 3 f. 6 1 / 2 inches on the right side , to the length on the other side , then the feet and inches about , on the first side , shall shew on the second or left , the content required . As at 20 inches about , and 20 foot long , the content will be found to be about 4 foot 5 inches . But if you give the usual allowance , that is made by dupling the string 4 times , that girts the piece : then you must set 4 foot on the right side , to the length on the other , then at 1 foot 8 inches about , the last example you shall finde but three foot 6 inches . 6. ● astly , if the rule be made fit for foot measure , onely then the point of 12 is altogether neglected , and one onely made use of as a standing number : and the point at three foot 6 1 / 2 will be at three foot 54 parts ; and the four will be the same , and the same directions in every respect , serve the turn . And because I call it Mr. Whites rule , being the contriver thereof , according to feet and inches , I have therefore fitted these directions accordingly , and there are sufficient to the ingenious practitioner . CHAP. XIX . Certain Propositions to finde the hour , and the Azimuth , by the lines on the Sector . PROP. 1. HAving the latitude , and complements of the declination , and Suns altitude , and the hour from noon , to finde the Suns Azimuth , 〈◊〉 that time . Take the right sine of the complement of the Suns altitude , and mak 〈◊〉 it a parallel sine in the sine of th 〈◊〉 hour from noon , ( counting 15 degree 〈◊〉 for an hour , and 1 degree for for minutes ) counted from the center . The Sector so set , take the right sine of the complement of the declination , and carry it parallel till the compasses stay in like sines , and the sine wherein they stay shall be the sine of the Azimuth required . Or else thus : Take the right sine of the declination , make it a parallel in the cosine of the Suns altitude , then take the parallel sine of the hour from noon , and it shall be the latteral or right sine of the Azimuth from the south required . If it be between six in the morning , and 6 at night ; or from the north , if it be before or after six : and so likewise is the Azimuth . PROP. 2. Having the Azimuth from south or north , the complement of the Suns altitude , and declination , to finde the hour . Take the latteral , or right sine of the complement of the Suns altitude , make it a ga●●llel in the cosine of the declination : the sector so sett , ake out the parallel sine of the Azimuth , and measure it from the center , and it shall reach to the right sine of the hour from noon required . Or else as before . As the right sine of the complement of the Suns declination : is to the parallel sine of the Azimuth , so is the right sine of coaltitude , to the parallel sine of the hour from noon , counting as before . PROP. 3. Having the complements of the latitude , Suns altitude , and declination . to finde the Suns Azimuth from the north part of the horizon . 1. First of the complement of the latitude , and Suns present altitude finde the difference . 2. And secondly count it on the line of sines from 90 toward the center . 3. Take the distance from thence , to the sine of the Suns declination ; but note when the latitude and declination differ , as in winter you must counte the declination beyond the center , and you must call it the Suns distance from the pole . 4. Fourthly , make that distance a parallel sine in the complement of the latitude . 5. Fifthly , then take out the parallel sine of 90. 6. And sixthly , make it a parallel sine in the coaltitude . 7. Seventhly , then the sector so set , take out parallel , Radius , or sine of 90. And eighthly , measure it on the line of sines from 90 towards , ( and if need be beyond ) the center : and it shall reach to the versed sine of the Suns Azimuth from the north , or if you count the other way from the south , note that in working of these , if the line of sines be too big , then you have two or three smaller sines on the rule , where on to begin and end the work . Example . Latitude 51. 32 , Declination 18 ▪ 30 , Altitude 48. 12 , you shall finde the Azimuth to be 130 from the North , or 50 from the South . PROP. 4. Having the complements of the latitude and declination , or Suns distance from the Pole , and the Sun altitude given , to finde the hour from East or West , or else from noon . 1. First of the complement of the latitude , and Suns distance from the Pole , finde the difference . 2. Count this from the sine of 9● toward the center . 3. Take the distance from thence to the sine of the Suns altitude . 4. Make that distance a parallel sine of the complement of the latitude . 5. Take out the parallel sine of 90 degrees , and 6. Make that a parallel sine in the codeclination , then 7. Take out the parallel sine o● 90 again , and 8. Measure it from the sine of 90 toward the center , and it shall shew the versed sine of the hour from the North , or the sine of the hour from East or West ; or if you reckon from 90 , the hour from noon required . Example . Latitude 51. 32 , Declination North 20. 14 , Altitude 50. 55 , you shall finde the hour from the North to be 10 houres , or 10 a clock in the forenoon , or 4 hours past 6 , or two short of noon , according to each proper reckoning . PROP. 5. Having the latitude , the complement of the Suns declination , the Suns present altitude , and Meridian altitude for that day , to finde the hour . Make the lateral secant of the latitude , a parallel sine in the codeclination , then take the distance from the Suns Meridian altitude , to his present altitude , and lay it from the cer on both sides of the line of sines , and take the parallel distance between those two points , and measure it from 90 on a line of sines , of the same Radius the secants be , ( as the small adjoyning sine is , ) and it shall shew the versed sine of the hour from noon , or the right sine before or after 6 , towards noon or night . PROP. 6. Having the Latitude , Declination , and Suns Altitude , to finde the Suns Azimuth . Take the latteral secant of the latitude , and make it a parallel in the complement of the Altitude : Then take the distance , between the sun● of the complement● of of Museum the latitude and altitude , ( if under 90 , ) and the sons declination , and lay it from the center on the line of sines , that parallel distance taken , and measured on the sines ( of the same Radius the secant was ) from 90 , shall shew the versed sine of the Azimuth from noon . But if the sum of the colatitude , and coaltitude exceed 90 , then take the excess above 90 , out of the natural sine from the center toward 90 , and add that to the sine of the Suns declination towards 90 , and then the parallel distance between those two points , shall be the Azimuth required , from noon , But when the latitude and declination are unlike , as with us ( in the northern parts ) in winter , then you must take the declination out of the excess , or the lesser out of the greater , and lay the rest from the center , and the parallel distance , shall be versed sine of the Azimuth from noon . Example . At 18 15 altitude , latitude 51 32 , declination 13 15 South . The sum of the colatitude , and the coaltitude is 109. 37 , then count the center for 90 , the right sine of 10 for a 100 and 19 for 109 , and 37 minutes forwarder , there set the point of the Compasses , then take from thence to the right sine of the declination , and lay this distance from the center on the line of sines , and the parallel space between , is the versed sine of the Azimuth required . PROP. 7. Having the length of the shaddow of any object standing perpendicular , and the length of that object , to finde the altitude . Take the Tangent of 45 , and make it a parallel in the length of the shaddow in the line of lines , then the parallel distance between the length of the object that casts the shaddow , taken from the line of lines , and measured on the line of tangents from the center , shall reach to the Suns altitude required . Example . If the object be 40 parts long , and the shaddow 80 parts , the altitude will be found to be 26. 35. But if you have the altitude , and shaddow , and would know the height of the object , then work thus : Take the length of the shaddow out of the line of lines , or any other equal parts , and make it a parallel tangent of 45 , then take out the parallel tangent of the Suns altitude , and measure it on the line of lines , ( or the same equal parts ) and it shall shew the length of the object that caused the shaddow : the same rule doth serve in taking of altitudes by the rule , as in the 18 chapter , accounting the measure from the station to the object , the length of the shaddow , and the Suns altitude , the angle at the base . PROP. 8. To finde the Suns rising and setting in any latitude . Take the latteral cotangent of the latitude , make it a parallel in the sines of 90 and 90 , then take the latteral tangent of the Suns declination , and carry it parallel in the sines till it stay in like sines , that sine shall be the asentional difference between six , and the time of rising before or after 6 , counting 15 degrees to an hour . PROP. 9. To finde the Amplitude in any latitude . Take the latteral sine of 90 , and make it a parallel in the cosine of the latitude , then the parallel sine of the declination , taken and measured in the line of sines from the center , shall give the amplitude required . PROP. 10. To finde the Suns height at six in any latitude . Take the lateral or right sine of the declination , and make it a parallel in the sine of 90 , then take out the parallel sine of the latitude , and measure it in the line of sines from the center , and it shall reach to the altitude required . Note in working of any of these Propositions , if the sines drawn from the center , prove too large for your Compasses , or to make a parallel sine or Tangent to a small number of degrees , then you may use the smaller sine or tangent adjoyning , that is set on the Rule , and it will answer your desire . And note also in these Propositions , the word right , or latteral sine or tangent , is to be taken right on from the center or beginning of the lines of sines , or tangents ; and the word parallel always across from one leg to the other . PROP. 11. To finde the Suns height at any time , in any latitude . As the right Sine of 90 , Is to the parallel cotangent of the latitude : So is the latteral or right Sine of the hour from 6 , To the parallel tangent of a fourth ark ; which you must substract from the suns distance from the Pole , and note the difference . Then , As the right of the latitude , To the parallel cosine of the fourth ark : So is the parallel cosine of the remainder , To the latteral sine of the Altitude required . PROP. 12. To finde when the Sun shall come to due East , or West . Take the tangent of the latitude from the smaller tangents , make it a parallel in the Sine of 90 , then take the latteral tangent of the declination from the smaller tangents , and carry it parallel in the Sines , till it stay in like Sines , and that Sine shall be the Sine of the hour required from 6. PROP. 13. To finde the Suns Altitude at East or West ( or Vertical Circle . ) As the latteral sine of declination , Is to the parallel sine of the latitude : So is the parallel sine of 90 , To the latteral sine of the Altitude required . PROP. 14. To finde the Stiles height in upright declining Dials . As the right Sine of the complement of the latitude , To the parallel sine of 90 : So the parallel sine complement of the Plains declination , To the right sine of the Stiles elevation PROP. 15. To finde the Substiles distance from the Meridian . As the lateral tangent of the colatitude , To the parallel sine of 90 : So the parallel sine of the declination , To the latteral tangent of the Substile from the Meridian . PROP. 16. To finde the Inclination of Meridians . As the latteral tangent of the declination , To the parallel sine of 90 : So is the parallel sine of the latitude , To the latteral cotangent in the inclination of Meridians . PROP. 17. To finde the hours distance from the Substile in all Plains . As the latteral tangent of the hour from the proper Meridian , To the parallel sine of 90 : So is the parallel sine of the Stiles elevation , To the latteral tangent of the hour from the substile . PROP. 18. To finde the Angle of 6 from 12 , in erect Decliners . As the latteral tangent of the complement of the latitude , To the parallel sine of the declination of the Plain : So is the parallel sine of 90 : To the latteral tangent of the Angle between 12 and 6. Thus you see the natural Sines and Tangents on the Sector , may be used to operate any of the Canons that is performed by Logarithms , or the artificial Sines and Tangents , by changing the terms from the first to the third , and the second to the first , and the third to the second , and the fourth must always be the fourth , in both workings being the term required . CHAP. XX. A brief description , and a short-touch of the use of the Serpentine-line , or Numbers , Sines , Tangents , and versed sine contrived in five ( or rather 15 ) turn . 1. FIrst next the center is two circles divided one into 60 , the other into 100 parts , for the reducing of minutes to 100 parts , and the contrary . 2. You have in seven turnes two in pricks , and five in divisions , the first Radius of the sines ( or Tangents being neer the matter , alike to the first three degrees , ) ending at five degrees and 44 minutes . 3. Thirdly , you have in 5 turns the lines of numbers , sines , Tangents , in three margents in divisions , and the line of versed sines in pricks , under the line of Tangents , according to Mr. Gunters cross staff : the sines and Tangents beginning at 5 degrees , and 44 minutes where the other ended , and proceeding to 90 in the sines , and 45 in the Tangents . And the line of numbers beginning at 10 , and proceeding to 100 , being one entire Radius , and graduated into as many divisions as the largeness of the instrument will admit , being from 10 to 50 into 50 parts , and from 50 to 100 into 20 parts in one unit of increase , but the Tangents are divided into single minutes from the beginning to the end , both in the first , second , and third Radiusses , and the sines into minutes ; also from 30 minutes to 40 degrees , and from 40 to 60 , into every two minutes , and from 60 to 80 in every 5th minute , and from 80 to 85 every 1oth , and the rest as many as can be well discovered . The versed sines are set after the manner of Mr. Gunters Cross-staff , and divided into every 10th minutes beginning at 0 , and proceeding to 156 going backwards under the line of Tangents . 4. Fourthly , beyond the Tangent of 45 in one single line , for one turn is the secants to 51 degrees , being nothing else but the sines reitterated beyond 90. 5. Fifthly , you have the line of Tangents beyond 45 , in 5 turnes to 85 degrees , whereby all trouble of backward working is avoided . 6. Sixthly , you have in one circle the 180 degrees of a Semicircle , and also a line of natural sines , for finding of differences in sines , for finding hour and Azimuth . 7. Seventhly , next the verge or outermost edge is a line of equal parts to get the Logarithm of any number , or the Logarithm sine and Tangent of any ark or angle to four figures besides the carracteristick . 8. Eightly and lastly , in the space place between the ending of the middle five turnes , and one half of the circle are three prickt lines fitted for reduction . The uppermost being for shillings , pence , and farthings . The next for pounds , and ounces , and quarters of small Averdupoies weight . The last for pounds , shillings , and pence , and to be used thus : If you would reduce 16 s. 3 d. 2 q. to a decimal fraction , lay the hair or edge of one of the legs of the index on 16. 3 1 / 2 in the line of l. s. d. and the hair shall cut on the equal parts 81 16 ; and the contrary , if you have a decimal fraction , and would reduce it to a proper fraction , the like may you do for shillings , and pence , and pounds , and ounces . The uses of the lines follow . As to the use of these lines , I shall in this place say but little , and that for two reasons . First , because this instrument is so contrived , that the use is sooner learned then any other , I speak as to the manner , and way of using it , because by means of first , second , and third radiusses , in sines and Tangents , the work is always right on , one way or other , according to the Canon whatsoever it be , in any book that treats of the Logarithms , as Gunter , Wells , Oughtred , Norwood , or others , as in Oughtred from page 62 to 107. Secondly , and more especially , because the more accurate , and large handling thèreof is more then promised , if not already performed by more abler pens , and a large manuscript thereof by my Sires meanes , provided many years ago , though to this day not extant in print ; so for his sake I claiming my intrest therein , make bold to present you with these few lines , in order to the use of them : And first no●e . 1. Which soever of the two legs is set to the first term in the question , that I call the first leg always , and the other being set to the second term , I call the second leg . 2. Secondly , if one be the first or second term , then for the better setting the index exactly , you may set it to 100 , for the error is like to be the least neerest the circumference . 3. Thirdly , be sure you keep a true account of the number of turnes between the first and second term . 4. Fourthly , observe which way you move , from the first to the second term . To keep the like from the third to the fourth , exept in the back rule of three , and in such cases as the Canon requires the contrary . 5. Fifthly in multiplication , one is always the first term , and the multiplycator or multiplycand the second , and the product always the fourth . Also note that in multiplycation the product of two numbers multiplyed , shall be in as many places as both the multiplycator , and multiplycand , except the least of them , be less then the two first figures of the product ; moreover , for your more certain assinging of the two last figures of four or six , which is as many as you can see on this instrument , multiply the two last in your minde , and the product shall be the figure , as in page 28 of the Carpenters Rule . 6. In division , the multiplicator is always the first term , and one the second , the dividend the third , and the quotient the fourth ; also the quotient shall have as many figures as the dividend hath more then the divisor , except the first figures of the divisor be greater then the dividends , then it shall have one less . Also note , the fraction after division , is a decimal fraction , and to be reduced as before . 7. Note carefully whether the fourth proportional ought to be a greater or a less , and resolve accordingly , and note if one cometh between the third and fourth term , then must the fourth be raised a Radius or a figure more , and be careful to set the hairs exactly over the part representing the number or minutes of any degree . 8. Always in direct proportion , and Astronomical calculation , set the first leg to the first term , and the second leg to the second term , and note how many circles is between , then set the first leg to the third term , and right under the second leg , the same way , and so many turnes between the third and fourth , is always the fourth term required . Example . As 1 , to 47 , so is 240 , to 11280. As the sine of 90 , to 51 degrees 30 minutes , so is the sine of 80 to 50. 26. And so of all other questions according to their respective Canons by the Logarithms in other books as in Mr. Oughtreds Circles of Porportions , from page 62 to 107 , and others . Here followeth the working of certain Propositions by the Serpentine-line . Those that I shall insert , are onely to shew the manner of working , and knowing of that once , all the Canons for all manner of questions , either in Arithmetick , Geometry , Navigation , or Astronomy , by any other Author , as Mr. Gunter , Mr. Oughtred , Mr. Windgate , Mr. Norwood , or others , may be speedily resolved , and as exactly as by the Tables , if the instrument be well and truly made . And first ●or the hour , according to Gunter . PROP. 1. Having the Latitude , Declination , and the Suns Altitude , to finde the hour . Add the complement of the altitude , the complement of the Suns present altitude , and the distance of the Sun , from the elevated Pole together , and nore their sum , and half sum , and find the difference between their half sum and the complement of the Suns present altitude , then work thus : for 36 42 degrees high , at 23. 32 declination , lat . 51. 32. Lay the first leg , viz. that next your right hand being here most convenient , on the Sine of 90 , keeping that fixed there , lay the other leg to the cosine of the latitude , viz. 38. 28. and note the turns between , which here is none between , but it is found in the next over it , then set the first leg to the Sine of 66 , 28 , the suns distance from the Pole , and in the circle just over it you shall have the Sine of 34. 47. for the fourth Ark or Sine . Then in the second Operation . Set the first or left leg to the sine of 34. 47. of the fourth Sine last found , then keeping that fixed there , set the other leg to the Sine of the half sum , viz. 79. 37. then remove the first leg to the Sine of the difference between the half sum , and the coaltitude , viz. 25. 49. and then in the next circle , the other leg shall shew the sine of 48. 34. whose half distance toward 90 , being found by the Scale of Logarithms on the outermost circle , will discover the Sine of an Ark , whose complement being doubled , and turned into time ( by counting 15 degrees to an hour ) will give the hour required ; but by help of the versed Sines all this trouble is saved ; for when the index or second leg cuts the Sine of 48. 34. at the same instant it cuts the versed Sine of 60 , the hour from noon required , being 8 in the morning , or 4 in the afternoon , at 23. 32. of declination , in the latitude of 51. 32. PROP. 2. To work the same another way , according to Mr. Collins . The Latitude and Declination given , to finde the Suns height at 6 a clock , Dec. 23. 31. Lat. 51. 32. Lay the left leg on the Sine of 90 , and the other to the Sine of 23. 32 , and you shall finde one turn between upwards , then the first leg laid on the Sine of the latitude 51 , 32 , the other leg shall shew the Sine of 18. 3. ( in the second circle above 51 , 32. ) for the Suns height at 6 required , and this is fixed for one day . Then in summer time , or north declinations , by help of the line of natural sines , in the second line , finde the difference between the Suns present altitude , and the latitude at 6 , but in Winter or Southern ( signs or ) declinations , add the two altitudes together , in this manner . Lay one leg of the index to the natural sine of the altitude at 6 , and the other to the altitude proposed , the two legs so set , bring one of them ( viz the right ) to the beginning of the line of natural sines , and the other shall stay at the difference required , but in Winter set one leg to the beginning of the sines , and open the other to the height at 6 , or rather depression under the Horizon at 6 , ( which is all one at like declinations , North and South ) then set the first leg to the present altitude of the Sun , and the other shall shew the Sine of the sum of both added together ; which sum or difference is thus to be used : Lay the left leg to the Cosine of the declination , and the other to the secant of the latitude , counted beyond 90 , as far as the secant of 9 , 40 ; or rather lay the left leg on the Co-sine of the latitude , and the other to the secant of the suns declination , then the first leg laid on the sine of the sum in winter , or difference in summer , shall cause the other leg to fall on the sine of the hour from 6 , toward noon in winter and summer , except the altitude in summer be less then the altitude at 6 , then it is the hour from 6 , toward mid-night . PROP. 3. Having the latitude , Suns altitude and declination , to finde the Suns Azimuth from east or west . Lat. 51. 32. Declin . 23. 30. Alt. 49 56. First you must get the Suns altitude , or depression in the vertical Circle by this Cannon . Lay the first leg to the sine of the latitude , and the second to the sine of 90 , and you shall finde them both to be on the same line , then the first leg laid on the sine of the declination , shall cause the second ( being carried with the first , without moving the Angle first set ) to fall on the sine of 30. 39. the Suns altitude in the Vertical Circle ; with which you must do , as you did before with the altitude at 6 , and the present altitude , to finde the sum and difference by help of the line of natural Tangents , then this proportion holds . Lay the first leg to the cosine of the Altitude , ( by counting the Altitude from 20 ) and the second leg to the tangent of the latitude , and observe which way , and the turns between ; then the first leg removed and laid to the sine of the sum ( before found ) in winter , or the difference in summer , shall cause the second leg to fall on the sine of the Azimuth of the Sun , from east or west toward noon , if winter ; and also in summer , when the Suns altitude is more than his altitude at the Vertical Circle ; but if less from the east or west , toward north or mid-night meridian : Thus in our Example , it will be found to be the sine of 30 degrees , or 60 from the south , the sine of the difference being found to be 14. 49. PROP. 4. To finde the Azimuth , according to Mr. Gunter , by having the Latitude , Suns Altitude , and Declination given . First by the Suns declination get his distance from the Pole , which in summer or North declinations , is always the complement of the declination , ( likewise in south latitudes , and south declinations ) but when the latitude and declination is unlike , then you must adde 90 to the declination , and the sum is the distance from the elevated Pole. Having found the distance from the Pole , adde that and the complement of the latitude , and Suns altitude together , finde the difference between their half sum , and the Suns distance from the Pole , then the proportion will be thus , as in this Example : 13 Declination , 41. 53. Alt. Lat. 51. 32. Lay one leg on the sine of 90 , and the other to the sine of 38 , 28 , being so set , remove the first leg to the sine complement of the altitude 48. 07. and the second legg shall fall on a fourth sine , which will be found to be 27. 36. then set the first or left leg to 27. 36. the fourth sine , and the second to 81. 47. the sine of the half sum , then removing the first leg to the sine of the difference , shall cause the second to shew two circles lower , the versed sine of 130 , the Azimuth required , being counted from the North part of the horizon , whose complement to 180 from the South is 50 degrees . Two other Canons to finde the hour of the day , and Azimuth of the Sun , by one operation , by help of the natural sines : and first for the hour . Having the latitude , the Suns Meridian , and present altitude , and declination , to finde the hour from noon . First lay one leg to the Meridian altitude , in the line of natural sines , and the other to the sine of the altitude in the same line , then bring the right leg to the beginning of the line of sines , and the other 〈…〉 difference , which difference 〈…〉 keep . Then lay one leg on the 〈◊〉 the declination , and the other 〈◊〉 secant of the latitude , and note the turnes between , or rather lay the first leg to the cosine of the latitude , and the other to the secant of the declination , then the legs being so set , bring the first or left leg to the sine of the difference first found , and the other leg shall shew the versed sine of the hour from noon , if the versed sines had been set thus , i. e. the versed sine of 90 , against the sine of 90 , as in some instruments it is : But to remedy this defect , do thus : keep the right leg there , and open the other to the versed sine of 0. or sine of 90 , and note the turnes between , then lay the leg that was on the sine of 90 , ( or versed sine of 0 , ) to the versed sine of 90 , and the other leg shall shew the versed sine of the hour from noon , counting from 90. Example . At 45 13 degrees altitude , declination North 23 32. latitude 51 32 , the Meridian altitude is 62 , ( being found by adding colatitude , and declination together , and in southern declinations by substraction . ) Then the natural sine of 45. 42 , taken from 62 , shall be the sine of 9. 38. then as the cosine of the latitude 38. 28 , is to the secant of declination 23. 32 , so is the sine of 9. 38. to the sine of 17. 05 ) or versed sine of 45 ▪ if they were placed and numbred , as in some instruments they be : but to help i● in this , say : as the versed sine of 0 , is to the verses of 114. 26 , so is the versed sine of 90 , to the versed sine of 135 , whose complement to 180 is the angle or hour from noon required . Secondly , for the Suns Azimuth . Having the latitude , declination , and Suns altitude , to finde the Azimuth from South or North. First add the complements of the coaltitude and cola●itude together , then if the sum be under 90 , take the distance between the cosine of it , and the sine of the declination , in the line of natural sines , and measure it in the line of sines from the beginning , and it shall give the sine of the difference ; but if the sum exceed 90 , then when the latitude and declination is alike , add the excess to the declination ; but if contrary substract one out of other , and measuring the sum or remainder from the beginning of the sines , you have the difference which you must keep . Then lay the first or left leg to the cosine of the altitude , and the second to the secant of the latitude , or else lay the first to the cosine of the latitude , and the other to the secant of the altitude , and more the turnes between , then lay the first leg to the sine of the difference before found , and the other shall shew the versed sine of the Azimuth from noon required , if the versed sines be set as before is expressed , that is to say 90 of right sines , and versed sines together , and numbred forwards as the sines be : but in the use of this instrument , the remedy aforesaid supplyeth the defect . Example . At 10. 19. altitude , 23. 32. declination , 51. 32. latitude , to finde the Azimuth . The sum of the coaltitude and co-latitude is 118.09 , the excess above 90 , with right sine of declination added is 60. 30 , found by natural sines , then say , As the cosine of latitude , to the secant of altitude , so is the sine of the difference 60. 30 , to the versed sine of the Azimuth , but here to the versed sine of an ark beyond Radius unknown , then as the versed sine of 0 , to that ark , so is the versed sine of 90 , to the ver sine of 60 , the azimuth from noon , whose complement to 180 , is 125 the Azimuth from Sonth required . Having been so large in these , I shall in the rest contract my self as to the repetition , and onely give the Canon for the propositions following , the way of working being the same in all other , as in these before rehearsed , and note also what is to be done by the Serpentine-line , is to be done by the three same lines of numbers , sines and Tangents on the edge of the Sector , by altering the term leg , to to the point of the Compasses . The Canons follow . PROP 4. Having latitude , declination , and hour given , to finde the Suns altitude at that hour or quarter . And first for the hour of 6. As the sine of 90 , To the sine of the latitude 51. 30 : So is the sine of the declination 23. 30 , To the sine of the altitude at 6. 18. 13. Secondly , for all hours in the Equinoctial . As the Radius or sine of 90 , to the cosine of the latitude 51. 32 : so is the sine of the Suns distance from 6 ( in hours and minutes , being turned into degrees and minutes 30 , for 8. or 4 , ) To the sine of the altitude of the Sun at the time required 18. 07 , But for all other times say , As the sine of 90 , To the cotangent of the latitude 38. 28. So is the sine of the Suns distance from 6 , 30 , 0 , To the tangent of the 4 arke 21 40. Which fourth arke must be taken out of the Suns distance from the Pole 66 , 21 , ( in Cancer ) leaveth a residue 44 48 , which is called a fift ark . But for the hours before and after 6 , you must add the fourth arke to the Suns distance from the Pole , and the Sum is the fifth ark . Then say , As the cosine of the fourth ark 78 , 20 , Is to the sine of the latitude 51 , 32 : So is the cosine of the residue 45 , 12 , To the sine of the Suns altitude at 8 , 36 , 42 , at that declination . PROP. 6. Having the latitude , declination , and Azimuth , to finde the Suns altitude at that Azimuth . And first to finde the Suns altitude at any Azimuth in the aequator . Then , As the sine of 90 , to the cosine of the Suns Azimuth from the South 50 , 0 , So is the cotangent of the latitude 38 , 28 , to the tangent of 27 , 03 ▪ the Suns altitude , at that Azimuth required . Secondly , to find it at all other times , do thus : As the sine of the latitude 51 , 32 , To the sine of the Suns declination , 23 , 32 : So is the cosine of the Suns altitude in the aequator , at the same Azimuth from the vertical , viz. 30 , to the sine of a 4th . ark 28 , 16. Which fourth ark must be added to the Suns altitude at the aequator in all Azimuths under 90 , from the meridian , where the latitude and declination are alike . But in Azimuths more then 90 from the meridian , take the altitude in the aequator out of the fourth ark , and the sum or remainder shall be the altitude required , viz. 42 , 56. But when the latitude and declination are unlike , as with us in winter time , then take the fourth ark out of the altitude at the aequator , and you shall have the altitude belonging to that Azimuth required . PROP. 7. Having the hour from noon , and the altitude to find the Suns Azimuth at that time . As the cosine of the altitude , To the sine of the hour , So is the cosine of the Suns declination , To the sine of the Azimuth required . PROP. 8. Having the Suns Azimuth , Altitude , and declination , to find tho hour of the day . As the cosine of the declination , To the sine of the Suns Azimuth : So is the cosine of the altitude , To the sine of the hour . PROP. 9 , Having the latitude and declination , to find when the Sun shall be due East or West . As the tangent of the latitude , To the tangent of the Suns declination , So is the sine of 90 , To the cosine of the hour from noon . PROP. 10. Having the latitude and Suns declination , to find the Amplitude . As the cosine of the latitude , To the sine of the declination : So is the sine of 90 , To the sine of the amplitude from the East or West , toward North or South , according to the time of the day and year . PROP. 11. The latitude and declination given , to find the time of the Suns rising before or after 6. As the cotangent of the latitude , To the sine of 90 : So is the tangent of the Suns declination , to the sine of the Suns ascentional difference between the hour of 6 and the Suns rising . PROP. 12. Having the Suns place , to find his declination , and the contrary . As the sine of 90 , To the suns distance from the next equinoctial point , So is the sine of the suns greatest declination , To the sine of his present declination required . PROP. 13. The greatest and present declination given , to find the Suns right ascension . As the tangent of the greatest declination , To the sine of 90 : So the tangent of the present declination , To the right ascension required . Onely you must regard to give it a right account by considering the time of the year , and how many 90s . past . PROR . 14. To find an altitude by the length , and shadow of any perpendicular object . Lay the hair on one legg to the length of the shadow found on the line of numbers , and the hair of the other leg to the length of the object that caused the shadow found on the same line of the numbers ; then observe the lines between , and which way when the legs are so set , bring the first of them to the tangent of 45 , and the other leg shall ●hew on the line of tangents , so many turns between , and the same way the tangent of the altitude required . Thus may you apply all manner of quest . to the Serpentine-line & work them by the same Canons , that you use for the Logarithms in all or most Authors . PROP. 15. To square , and cube a number , and to findethe square root , or cube roat of a number . The squaring of a number , is nothing else but the multiplying of the number by it self , as to square 12 is to multiply 12 by 12 , and then the cubing of 12 , is to multiply the square 144 by 12 , & that makes 1728 , and the way to work it , is thus : Set the first leg to 1 , and the other to 12 , then set the first to 12 , and then the second shall reach to 144 , then set the first to 144 , and the second shall reach to 1728 , the cube of 12 required : but note , the number of figures in a cube , that hath but one figure is certainly found by the line , by the rule aforegoing : but if there be more figures then one , so many times 3 must be added to the cube , and so many times two to the square . To find the square root of a number , do thus : Put a prick under the first , the third , the 5th , the 7th , & the number of pricks doth shew the number of figures in the root ; and note if the figures be even , count the 100 to be the unit , if odde as 3 , 5 , 7 , 9 , &c. the 10 at the beginning must be th● unit , as for 144 , the root consists of two figures , because there is two pricks under the number , and if you lay the index to 144 in the numbers , it cu●s on the line of Logarithms 15870 , the half of which is 7915 whereunto if you lay the index , it shall shew the 12 the root required ; but if you would have the root of 14+44 , then divide the space between that number , and 100 you shall finde it come to 8 , 4140 that is four turnes , and 4140 for which four turnes , you must count 80000 , the half of which 8,4140 , is 4,2070 , whereunto if you lay the index , and count from 1444 ●r 100 , at the end you shall have it cut at 38 lack four of a 100. To extract the cubique root of a number , set the number down , and put a point under the 1 , the 4th , the 7th , and 10th , and look how many pricks , so many figures must be in the root , but to finde the unity you must consider , if the prick falls on the last figure , then the 10 is the unit at the beginning of the line , as it doth in 1728 , for the index laid on 1728 , in the Log●rithms , sheweth 2,3760 , whose third part 0,7920 counted from 10 , falls on 12 the root , but in 17280 , then you must conceive five whole turnes , or 1000 to be added , to give the number that is to be divided by three , which number on the outermost circle in this place , is 12 , +3750. by conceiving 10000 to be added , whose third part counted from 10 , viz. two turnes or 4.125 , shall fall in the numbers to be near 26. But if the prick falls of the last but 2 , as in 172800 then 100 at the end of the line , must be the unit , and you must count thus : count all the turnes from 172830 to the end of the line , and you shall finde them to amount to 7,6250 , whose third part 2 , 5413 counted backward from 100 , will fall on 55,70 the cubique root required . PROP. 16. To work questions of interest or progression , you must use the help of equal parts , as in the extraction of roots , as in this question , if 100 l. yield 106 in one year , what shall 253 yield in 7 year ? Set the first leg to 10 at the beginning , in this case representing a 100 , and the other to 106 , and you shall finde the legs to open to 253 of the small divisions , on the Logarithms , multiply 253 by 7 , it comes to 1771 , now if you lay the hair upon 253 , and from the place where the index cuts the Logarithms count onwards 1771 , it shall stay on 380 l. 8 s. or rather thus : set one leg to the beginning of the Logarithms , and the other to 1771 either forward or backward , and then set the same first leg to the sum 253 , and the second shall fall on 380. 8 s. according to estimation ; the contrary work is to finde what a sum of money due at a time to ●ome , is worth in ready money : this being premised here , is enough for the ingenious to apply it to any question of this nature , by the rules in other Authors . However you may shortly expect a more ample treatise , in the mean time take this for a taste and farewell . The Use of the Almanack . Having the year , to finde the day of the week the first of March is on in that year , and Dominical letter also . First if it be a Leap-year , then look for it in the row of Leap-year , and in the column of week-days , right over it is the day required , and in the row of dominical letters is the Sunday letters also : but note the Dominical letter changeth the first of Ianuary , but the week day the first of March , so also doth the Epact . Example . In the year 1660 , right over 60 which stands for 1660 , there is G for the Dominical or Sunday letter , beginning at Ianuary , and T for thursday the day of the week the first of March is on , and 28 underneath for the epact that year , but in the year 1661. being the next after 1660 the Leap-year , count onwards toward your right hand , and when you come to the last column , begin again at the right hand , and so count forwards till you come to the next Leap-year , according to this account for 61 , T is the dominical letter , and Friday is the first of March. But to finde the Epact , count how many years it is since the last Leap-year , which can be but three , for every 4th is a Leap-year , and adde so many times 11 to the epact in the Leap-year last past , and the sum , if under 30 , is the Epact ; if above 30 , then the remainder 30 or 60 , being substracted is the Epact for that year . Example for 1661.28 the epact for 1660 , and 11 being added makes 39 from which take 30 , and there remaineth 9 , for the Epact for the year 1661 the thing required . Note that in orderly counting the years , when you come to the Leap-year , you must neglect or slip one , the reason is , because every Leap-year hath two dominical letters , and there also doth the week day change in the first of March , so that for the day of the month , in finding that the trouble of remembring the Leap-year is avoided . To find the day of the Month. Having found the day of the week , the first of March is on the respective year ; then look for the month in the column , and row of months : then all the daies right under the month are the same day of the week the first of March was on , then in regard the days go round , that is change orderly every seven days , you may find any other successive day sought for . Example . About the middle of March 1661 on a Friday , what day of the month is it ? First the week day for 1661 is Friday , as the letter F on the next collumn beyond 60 she●et● ; then I look for 1 among the months , and all the days right under , viz. 1 , 8 , 15 , 22 , 29. in March , and November 61 , are Friday , therefore my day being Friday , and about the middle of the month , I conclude it is the 15th day required . Again in May 1661. on a Saturday about the end of May , what day of the month ? May is the third month , by the last rule I find that the 24 and 31 are Fridays , therefore this must needs be the 25 day , for the first of Iune is the next Saturday . FINIS . ERRATA . PAge 23. l. 4. adde 1660 p. 24. l. 6. for 5 hours r. 4. l. 9. for 3. 29. r. 4. 39. 1. 12 for 5. 52. r. 4 , 52. l. 13. for 3. 39. r. 4. 39. l. 17 for 5 hours 52. r. 4. 52. p. 27. l. ult . dele or 11. 03. p 31. l 4. for sun r. sum . p. 50. l. 8. for B r. A. p. 50 d CHAP. XII . p. 51. r. 16. for 6. 10. 1. 6 to 10. p. 71. l. 6. for 7 / 4 r. 1 / 4. l. penult . for 2 afternoon , r. 1. p. 74 l. ult . for 1. r. 1 , 2. p. 83. l. 18. for BC r. BD. p. 69 l. 17. add measure , p. 129 l. 24. for right of , r. right sine of . p. 114 l. 9 for 18 3. r 18 13. p. 147 1. 2 for 20 , r. 90 p. 163. l. 16. for of , r. on . 
A29760	----	The description and use of the carpenters-rule together with The use of the line of numbers commonly called Gunters-line : applyed to the measuring of all superficies and solids, as board, glass, plaistering, wainscoat, tyling, paving, flooring, &c., timber, stone, square on round, gauging of vessels, &c. : also military orders, simple and compound interest, and tables of reduction, with the way of working by arithmatick in most of them : together with the use of the glasiers and Mr. White's sliding-rules, rendred plain and easie for ordinary capacities / by John Brown. Brown, John, philomath. 1688 Approx. 207 KB of XML-encoded text transcribed from 107 1-bit group-IV TIFF page images. Text Creation Partnership, Ann Arbor, MI ; Oxford (UK) : 2004-08 (EEBO-TCP Phase 1). A29760 Wing B5040 ESTC R37165 16263527 ocm 16263527 105159 This keyboarded and encoded edition of the work described above is co-owned by the institutions providing financial support to the Early English Books Online Text Creation Partnership. This Phase I text is available for reuse, according to the terms of Creative Commons 0 1.0 Universal . The text can be copied, modified, distributed and performed, even for commercial purposes, all without asking permission. Early English books online. (EEBO-TCP ; phase 1, no. A29760) Transcribed from: (Early English Books Online ; image set 105159) Images scanned from microfilm: (Early English books, 1641-1700 ; 1089:4) The description and use of the carpenters-rule together with The use of the line of numbers commonly called Gunters-line : applyed to the measuring of all superficies and solids, as board, glass, plaistering, wainscoat, tyling, paving, flooring, &c., timber, stone, square on round, gauging of vessels, &c. : also military orders, simple and compound interest, and tables of reduction, with the way of working by arithmatick in most of them : together with the use of the glasiers and Mr. White's sliding-rules, rendred plain and easie for ordinary capacities / by John Brown. Brown, John, philomath. [205] p., [1] leaf of plates : ill. Printed for W. Fisher and R. Mount ..., London : 1688. Special t.p. (p. [169]): The use of the line of numbers on a sliding (or glasiers) rule in arithmatick & geometry ... / first drawn by Mr. White ; ... made easie and useful by John Brown. London printed : [s.n.], 1688. Imperfect: some pages tightly bound, with slight loss of print. Reproduction of original in the Huntington Library. Created by converting TCP files to TEI P5 using tcp2tei.xsl, TEI @ Oxford. Re-processed by University of Nebraska-Lincoln and Northwestern, with changes to facilitate morpho-syntactic tagging. Gap elements of known extent have been transformed into placeholder characters or elements to simplify the filling in of gaps by user contributors. EEBO-TCP is a partnership between the Universities of Michigan and Oxford and the publisher ProQuest to create accurately transcribed and encoded texts based on the image sets published by ProQuest via their Early English Books Online (EEBO) database (http://eebo.chadwyck.com). 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Keying and markup guidelines are available at the Text Creation Partnership web site . eng Mensuration -- Early works to 1800. Mathematical instruments. 2004-01 TCP Assigned for keying and markup 2004-03 Apex CoVantage Keyed and coded from ProQuest page images 2004-04 Emma (Leeson) Huber Sampled and proofread 2004-04 Emma (Leeson) Huber Text and markup reviewed and edited 2004-07 pfs Batch review (QC) and XML conversion The Description and Use OF THE CARPENTERS-RULE : Together with the vse of the LINE of NUMBERS Commonly called GUNTERS-LINE . Applyed to the Measuring of all Superficies and Solids , as Board , Glass , Plaistering , Wainscoat , Tyling , Paving , Flooring , &c. Timber , Stone , Square or Round , Gauging of Vessels , &c. ALSO Military Orders , Simple and Compound Interest , and Tables of Reduction , with the way of working by Arithmatick , in most of them . Together with the Use Of the Glasiers and Mr. White 's Sliding-Rules . Rendred plain and easie for ordinary Capacities . By Iohn Brown. London , Printed for W. Fisher , and R. Mount , at the Postern on Tower-hill , 1688. Courteous Reader , THis Little Book was first written by me several years since , and hath been accepted of among many , that ●ave had the Perusal-thereof . And several Impressions , in that time , be●●g sold off ; and it being now out ●f Print , and none to be had , I have ●●vised it , and left out what might ●ell be spared ; and added that which ●ight make it more plain , and easier 〈◊〉 be remembred . As for Instance , in the using of ●●e Line of Numbers ( commonly ●●lled Gunters Line ) for the mea●●ring of Board , or Timber , or Stone , ●●e fixed Points or Centers is only 10 ●●d 12 , for square Timber or Stone . And in measuring of round Timber or Stone ( as round Timber ) ●●ere is used only 13. 54 for Inches , ●●d 1. 128 for Foot-measure , being ●e Diameter in Inches , and Foot●easure of any round solid , when one ●oot in length makes one solid Foot of 12 Inches every way , or 1728 so ▪ lid Cube Inches , which is a foot 〈◊〉 Timber , or Stone . And if the Circumference , or Gi●● of the Piece about is given , then t●● fixed Point or Center used , is at 4●● 54 the Inches , and 100 part of ●● Inch about , when one Foot , or 1●● Inches long makes one Foot solid . Or else at 3.545 the Feet , and 100● parts about , when one Foot long mak● one solid Foot equal to 1728 Cube I● Also , after every Problem , is t● brief way of working it by the Pen , 〈◊〉 a proof of the truth of every Oper● ▪ tion by the Rule ; being more th●● was before in the former Impression● Also , the Line of Pence is a● ded to the Line of Numbers , and plain way set forth of the use there by the Line on the Rule . Or the m● plain description thereof in a Pr● of Gunters Line 11 times repeate● which may be had with the Book , Tower-Hill , or the Minories . The Use of the Gunters Line ●he Art of Gauging , is here but brief●y hinted , because there are several Books of Gauging , purposely made for that Imployment , more compleat ●han can be expected in this short Discourse . And Lines of Area's of Circles , in Ale-gallons , at any Diameter given , from 1 Inoh to 200 In●hes ; which may be used for round ●r square Vessels , to give the content ●f every Inch deep in any taper Vessel , as fast as any one can write it down . And Directions for the ready measuring the Drip , or stooping Bottoms of round or square Tuns , and the Liquor about the Crowns of Coppers . Which Books are to be had at the Postern at Tower-Hill , or at the Authors , in the Minories . To this Impression is added the Use of the Lines called Diameter , Circumference , Square-Equal , and Square within a Circle ; and to find the Circles , Area , or Content by them ; or , having the Area , to find the Diameter , or the Circumference or the Square-equal , or Square-within . Also , to this is added the way b● the Pen , to multiply Feet , Inche● and 12 parts of an Inch together whereby any Superficies , as Boar● Floor , Wall , Yard , or Field , way ●● exactly measured by the Pen. Als● by a second Operation , or Multipl● cation , may any Solid , as Timber , 〈◊〉 Stone , or round Vessel , be measured the Arithmetical way whereof , 〈◊〉 worded as plain as in any Bo●● whatsoever . Also the Use of the Glasier's ▪ Sl● ▪ ding-Rule , to measure Glass , or a●● Superficies . And Mr. White 's Sliding-Ru●● to measure Timber ; being as ne●● and ready a way as ever was used . Thus you have a brief Account 〈◊〉 what is contained in this little Boo● and I wish it may be helpful to ma● a Learner , for whom it is prepare● So I remain ready to serve you in th● or the like . John Brown. From my House at the Sphere an● Sun-Dial in the Minories , Londo● ●he Description and Use of the Carpenters Rule . CHAP. I. IT is call'd a Carpenters Rule , ( rather then a Ioyners , Bricklayers , Masons , Glasiers , or the like ) I suppose , because ●●ey find the most absolute necessi●● of it in their way , for they have 〈◊〉 much or more occasion to use 〈◊〉 than most other Trades , though ●●e same Rule may measure all kind ●f Superficies and Solids , which ●wo Measures measure every visi●●e substance which is to be mea●●red . And it is usually made of ●●ox or Holly , 24 Inches in length , ●nd commonly an Inch and half , or 〈◊〉 Inch and quarter in breadth ; ●nd of thickness at pleasure ; and ●n the one side it is divided into ●4 equal Inches , according to the ●tandard at London●nd ●nd every one of those 24 Inches is divided into eight parts , that i● Halfs , Quarters , and Half-quarter● or ten parts , as you please and th● Half-inches are known from th● Quarters , and Quarters from th● Half-quarters , by short , longer and longest strokes , and at ever● whole inch is set figures , proceedin● from 1 to 24 , from the right hand toward the left , and these part● and figures are on both edges 〈◊〉 one side of the Rule both way● numbred , to the intent that howsoever you hold the rule you have th● right end to measure from , provided you have the right side . On the other side you have th● Lines of Timber and Board measure , the Timber-measure is tha● which begins at 8 and a half , tha● is , when the figures of the Timber line stand upright to you , then I sa● it begins at the left end at 8 and 〈◊〉 and proceeds to 36 within an Inc● and ⅜ of an Inch of the end . Also a● the beginning end of the Line o● Timber measure is a Table of figures , which contains the quantity of the under-measure from one Inch square to 8 Inches square , for the figure 9 comes upon the Rule , as you may see near to 8 in the Table . On , or next the other edge , and same side you have the Line of Board-measure , and when those figures stand upright , you have 6 at the left , or beginning end , and 36 at the other ( or right end ) just 4 Inches off the end unless it be divided up to a 100 , then it is nigh an Inch and half off the end . This Line hath also his Table of Under-measure at the beginning end , and begins at 1 and goes to 6 , and then the divisions on the Rule do supply all the rest to 100. Thus much for Description : Now for Use. The Inches are to measure the length or breadth of any Superficies or Solid given , and the manner of doing it were superfluous to speak of , or once to mention , being not only easie , but even natural to every man , for holding the Rule in the left hand , and applying it to the board , or any thing to be measured ▪ you have your desire : But now for the use of the other side , ● shall shew it in two or three examples in each measure , that is , Superficial or Solid . And first in Superficial , or Board-measure . Ex. 1. The breadth of any Superficies ( as Board , or Glass , or the like ) being given , to find how much i● length makes a Square Foot , ( or i● equal to 12 inches broad , and 12 Inches long ; for so much is a ●rne Foo● Superficial . ) To do this , look for the number of Inches your Superficies is broad in the Line of Board-measure , an● keep your finger there , and right against it , on the Inches side , you have the number of Inches that goe● to make up a Foot of Board o● Glass , or any Superficies , Suppose ● have a peice 8 Inches broad , How many Inches make a Foot ? I look for 8 on the Board-measure , and just against my finger ( being set to 8 ) on the Inch side , I find 18 , and so many Inches long , at that breadth , goes to make a Foot Superficial . Again , suppose it had been 18 Inches broad , then I find 8 Inches in length , to make a Foot superficial ; but if 36 Inches broad , then 4 Inches in length makes a Foot. Or you may do it more easie thus : Take your Rule and hold it in your left hand , and apply it to the breadth of your Board or Glass , making the end that is next 36 even with one edge of the board or glass , & the other edge of the board shews how many Inches or Quarters of an Inch go to make a foot of the board or Glass . This is but the converse of the former , and needs no example , for laying the Rule ●o it , and looking on the Board-measure , you have your desire . Or else , you may do thus in all narrow peices under 6 inches broad . As suppose 3¼ , double 3¼ it makes 6½ , then I say , that twice the length from 6½ to the end of the Rule , shall make a Foot Superficial , or so much in length makes a foot . Ex. 2. Having A Superficies of any length and breadth given , to find the Content , that is , how many Foot there is in it . Having found the breadth , and how much makes one Foot , turn that over as many times as you can , for so many Foot is there in that Superficies : But if it be a great breadth , then you may turn it over two or three times , and then take that together , and so say 2 , 4 , 6 , 8 , 10 , &c. or 3 , 6 , 9 , 12 , 15 , 18 , 21 , and till you come to the end of the Superficies . Note that the three short strokes between figure and figure , are the Quarters ; as thus , 8 and a quarter , 8 and a half , 8 and three quarters , then 9 , &c. till you come to 30 , and then 30 and a half , 31 , &c. to 36. And if it be divided any further , it is to whole Inches only to 100. The use of the Table at the beginning end of the Board-measure , First , you have five ranks of figures ; the first , or uppermost is the number of Inches that any Superficies is broad , and the other 4 are Feet , and Inches , and parts of an Inch that goes to make up a Foot of Superficial measure : As for example , at 5 Inches broad you must have 2 Foot , 4 Inches , and 4 Fifths of an Inch more , that is , 4 parts of 5 , the Inch being divided into 5 parts ; but where you have but two figures beside the uppermost , and Ciphers in the rest , you must read it thus , At two Inches broad you must have six Foot in length , no Inches , no parts . Thus much for the use of the line of Superficial or Board measure . The Use of the Line of ( Solid ) or Timber-measure . The use of this Line is much like the former : For first , you must learn how much your piece i● square , and then look for the same number on the Line of Timber-measure , and the space from thence to the end of the Rule , is the tru● length , at that squareness , to make a Foot of Timber . Ex. 1. I have a piece that is 9 Inches square , I look for 9 on the Line of Timber-measure , and ther● I say , the space from 9 to the end of the Rule , is the true length to make a Foot of Timber , and it i● near 21 Inches , 3 eights of an Inch. Again , suppose it were 24 Inches square , then I find 3 Inches i● length makes a Foot , for so I find 3 Inches on the other side , just against 24 : But if it were small Timber ▪ as under 9 Inches square , then you must seek the square in the upper rank in the Table , and right under you have the Feet Inches , and parts that go to make a Foot square , as was in the Table of Board-measure . As suppose 7 Inches square , ●hen you must seek the square Inches in the upper rank in the Table , and right under you have the Feet Inches , and parts that go to make a Foot square , as was in the Table of Board-measure . As suppose 7 Inches square , right under 7 , I find in the Table 2 Foot 11 Inches , and 2 sevenths of an Inch , divided into 7 parts , and at 8 Inches square you find only 2 Foot , 3 Inches , 0 parts , and so for the rest . But if a piece be not just square , but broader at one side than the other , then the usual way is to add them both together , and to take half for the square , but if they differ much , then this way will be very erroneous , and therefore I refer you to the following Rules : But if it be round Timber , then take a string and girt it about , and the fourth part of this is usually allow'd for the side of the square , and then you deal with it as if it were just square . Thus much for the Use of th● Carpenters Plain-rule . I have also added a Table fo● the Under-measure for Timber & Board , to Inches and Quarters ; an● the use is thus : Look on the left side for the number of Inches an● Quarters , your Timber is square● or your Board is broad , and right against it you have the Feet ; Inches tenth part of an Inch , and tenth of ▪ tenth ( or hundredth part of an Inch ) that goeth to make a Foot o● Timber or Board . Ex. 3. A piece of Timber 3 Inches 1 quarter . F. Inch. 10. 100 ▪ square will have 13 7 5 9 parts to make a Foot. And a Board of 3 Inch ▪ and a quarter broad must ▪ F. Inc. 10. 100. have 3 8 3 0 in length to make a Foot ; and so of the rest , as is plain by the Table , and needs no further explication , being common to most Artificers . A Table for the under Timber-measure , to Inches and quarters . in qr . feet . in . 10p . 100   1 2304 0 0 0   2 576 0 0 0   3 256 0 0 0 1 1 144 0 0 0   1 92 1 9 7   2 64 0 0 0   3 47 0 2 4 2 2 36 0 0 0   1 28 4 3 3   2 23 0 4 1   3 19 0 3 1 3 3 16 0 0 0   1 13 7 5 9   2 11 9 0 6   3 10 1 8 8 4 4 9 0 0 0   1 7 11 6 6   2 7 1 3 3   3 6 4 5 9 5 5 5 9 1 2   1 5 2 6 9   2 4 9 1 2   3 4 4 2 6 6 6 4 0 0 0   1 3 8 2 3   2 3 4 9 0   3 3 1 9 3 7 7 2 11 2 8   1 2 8 8 6   2 2 6 7 2   3 2 4 7 7 8 8 2 3 0 0 8¼   2 1 3 9 A Table for the under Board-measure , to Inches and quarters .   fe . in . 10. 100   48 0 0 0   24 0 0 0   16 0 0 0 1 12 0 0 0   9 7 2 0   8 0 0 0   6 10 2 9 2 6 0 0 0   5 4 0 0   4 9 6 0   4 4 3 6 3 4 0 0 0   3 8 3 0   3 5 1 4   3 2 4 0 4 3 0 0 0   2 9 8 8   2 8 0 0   2 6 3 1 5 2 4 8 0   2 3 4 2   2 2 1 8   2 1 0 4 6 2 0 0 0   1 11 0 5   1 10 1 5   1 9 3 3 7 1 8 5 8   1 7 8 6   1 7 2 0   1 6 5 8 8 1 6 0 0 8¼ 1 5 4 5 Note also , that this Table , or any smaller part of under-measure , may be supplyed by the divisions of the board and timber-measure only as thus ; Double the inches and parts of breadth for board-measure , or of squares for timber-measure , and seek it in the Lines of board or timber-measure , and count twice from thence to the rules end , for board , or 4 times for timber , and that shall be the true length that makes a foot of board or timber . Ex. 4. At 4 inches and ½ square or broad 4½ doubled , is 9. then look for 9 on the board-measure , and two times from thence to the end , shall make a foot of board . Or look for 9 on the Line of timber-measure , and 4 times from thence to the end of the Rule , shall be the true length to make a foot of timber , at 4 inches ½ square . But if it be so small a piece , that when it is doubled , the number is not on the divided part of the rule , then double it again , and count 4 times for board-measure , and 16 times for timber . Ex. 5. At 2 inches and a half , a quarter broad , or square , that doubled , is 4¼ , which is not on the rule , therefore I double it again , saying , 4 ¼ and 4¼ is 8½ which , is on the rule ; then for board , count 4 times from 8½ on the board-measure , to the upper end by 36 , to make a foot of board at 2⅛ broad : And for timber , count 16 times from 8½ near the beginning of timber-measure , which will be near 32 foot , to make a foot of timber at 2● square : But if twice doubling will not do , then double again , and count 8 times for board , and 64 times for timber , as in the Table you may see , which will be very slender timber . Also between the two lines of Inches , is set four scales of equal parts , called Circumference Diameter , Square-equal , and Square-within . Whose Use may be thus . The Diameter of any circle being given , to find the circumference or the side of a Square-equal , o● the side of the square within . Example , suppose the Diameter of a circle be 15 inches . Take 15 from the scale called Diameter , and measure it in the scale called , circumference and i● gives 47. 10. Also the same extent measured in the line called square-within and it gives 10. 55. For the side of a square-within in that circle of 51 inches Diameter . Again , the same extent being measur'd in the scale call'd square-equal and it gives 13. 27 for the side of ● square equal to a circle of 15 inches Diameter . Lastly , this 13. 27 the side of the square-equall multiplyed by it sel● gives 176 , the Area of that Circle in Inches , whose Diameter is 15 Inches . The same may be done , if the Circumference be first given , then that taken first from that Line , and measured in the other Lines , you shall have the respective Answers , as before . But if the Area be first given , then to find the Square-equal , find the Square-root of the Area , and that root shall be the side of the Square-equal . Example . Suppose the Area of a Circle be 288 , what is the side of the Square-equal . The middle between 1 and 288 , is at near 17 the side of the Square-equal , for 17 squared is 289. Then 17 taken from the Scale call'd square equal , gives you any of the rest . These Scales serve to draw any Platform of a House or Field very convenlently , being of several bignesses . The Description and Use of the Line of Numbers , ( commonly called Gunter's Line . ) CHAP. II. The definition and description of the Line of Numbers , and Numeration thereon . THE Line of Numbers is only the Logarithmes contrived on a Ruler , and the several ranks of figures in the Logarithms are here express'd by short , and longer , and longest division ; and they are so contrived in proportion one to another , that as the Logarithmes by adding together , and substracting one from another produce the quesita , so here by turning a pair of Compasses forward or backward , according to due order , from one point to another , doth also bring out the quesita in like manner . For the length of this Line of Numbers , know , that the longer it is the better it is , and for that purpose it hath been contrived several ways , as first into a Rule of two Foot long , and three Foot long by Mr. Gunter , and I suppose it was therefore called Gunter's Line . Then that Line doubled or laid so together , that you might work either right on , or cross from one to another , by Mr. Wingate afterwards projected in a Circle , by Mr. Oughtred , and also to slide one by another by the same Author ; and ●ast of all projected ( and that best of all hitherto , for largeness , and conseqenly for exactness ) into a Serpentine , or winding circular Line , of 5 , or 10 , or 20 turns , or more or less , by Mr. Brown , the uses being in all of them in a manner the same , only some with Compasses , as Mr. Gunter's and Mr. Wingat's ; and some with flat Compasses , or an opening Index , as Mr. Oughtred's and Mr. Browne's , and one without either as the sliding Rules ; but the Rules or Precepts that serve for the use of one , will indifferenly serv● for any : But the projection that ● shall chiefly confine my self to , i● that of Mr. Gunter's ; being the most proper for to be inscribed o● a Carpenters Rule , for whose sake● I undertake this collection of the most useful , convenient , and proper applications to the uses in Arithmatick and Geometry . Thus much for definition of what manner of Lines of Numbers there be and of what I intend chiefly to handle in this place . The order of the divisions o● this Line of Numbers , and com● monly on most other , is thus , i● begins with 1 , and proceeds wit● 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 ; and then ● 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 ; whos● proper power or order of numeration is thus : The first 1 dot● signifie one tenth of any whole number or integer ; as one tenth o● a Foot , Yard , Ell , Perch , or the like ; or the tenth of a penny , shil●●g , pound , or the like , either in ●●eight , or number , or measure ; ●●d so consequently , 2 is 2 tenths three tenths ; and all the small ●●termediate divisions , are 100 ●●rts of an integer , or a tenth , of ●●e of the former tenths ; so that 1 the middle , is one whole integer , ●●d 2 onwards two integers , 10 at ●●e end is 10 integers : Thus the ●●e is in its most proper acception natural division . But if you are to deal with a ●●ater number then 10 , then 1 at beginning must signifie● inte●● , and in the middle 10 integers , 10 at the end 100 integers . But if you would have it to a fi●●e more , then the first 1 is ten , the ●●nd a hundred , the last 10 a ●●usand . If you proceed further , ●n the first 1 is a 100. the middle 1000 , and the 10. at the end is ●●00 , which is as great a number you can well discover , on this or most ordinary lines of numbers and so far with convenient car● you may resolve a question very exactly . Now any number bein● given under 100●0 , to find the point representing it on the rul● do thus . Numeration on the Line of number PROB. I. Any whole number being given und●● four figures , to find the point the Line of numbers that doth r●present the same . First , Look for the first figure your number , among the long ●● visions , with figures at them , a●● that leads you to the first figure your number : then for the seco●● figure count so many tenths fro● that long divisions onwards , as th● second figure amounteth to ; th●● for the third figure , count from 〈◊〉 last tenth , so many centesmes the third figure contains ; and for the fourth figure , count fr●● the last centesme , so many millions , as that fourth figure hath unites , or is in value ; and that shall be the point where the number propounded is on the line of numbers : Take two or three Examples . First , I would find the point upon the line of numbers representing 12 , now the first figure of this number is one , therefore I take the middle one for the first figure ; then the next figure being 2 , I count two tenths from that 1 , and that shall be the point representing 12 , where usually there is a brass pin with a point in i● . Secondly , To find the point representing 144. First , as before I take for 1 , the first figure of the number 144 , the middle Figure 1 then for the second Figure ( viz. 4. ) I count 4 , tenths onwards for that : Lastly , for the other 4 , I count 4 , centesmes further , and that is the point for 144. Thirdyl , To find the point representing 1728. First , As before , for 1000. take the middle 1 , on the line Secondly , For 7 , I reckon seventenths onward , and that is 700. Thirdly , For 2 , reckon two centesmes from that 7th . tenth for 20 ▪ And lastly , For 8 , you must reasonably estimate that following centesme , to be divided into 10 parts ( if it be not express'd , which in lines of ordinary length cannot be done ) and 8 , of that supposed 10. is the precise point for 1728 the number propounded to be found , and the like of any number whatsoever . But if you were to find a fraction , or broken number , then you must consider , that properly , or absolutely , the line doth express none but decimal fractions : thus , 1 / 10 or 1 / 100 or 1 / 1000 and more nearer the rule in common acception cannot express ; as one inch , and one tenth or one hundredth or one thousandth part of an inch , foot , yard , perch , or the like , in weight , number , or time , it being capable to be applyed to any thing in a decimal way : ( but if you would use other fractions , as quarters , half quarters , sixteens , twelves , or the like , you may reasonably read them , or else reduce them into decimals , from those fractions , of which more in the following Chapters ) for more plainness sake , take two or three observations : 1. That you may call the 1. at the beginning , either one thousand , one hundred , or one tenth , or one absolutely , that is , one Integer , or whole number , or tenintegers , or a hundred , or a thousand integers , and the like may you call 1 , in the middle , or 10 at the end . 2. That whatsoever value or denominations you put on 1 the same value or denomination all the other figures must have successively , ei●●er increasing forward , or decreasing backwards , and their intermediate divisions accordingly , as for example ; If I call 1 at the beginning of the line , one tenth of any integer , then 2 following must be two tenths , 3 three tenths , &c. and 1 in the middle 1 integer , 2 two integers , & 10 at the end must be ●en integers . But if one at the beginning be one integer , then 1 in the middle must be 10 integers , and 10 at the end 100 integers , and all the intermediate figures 20 , 30 , 40 , 50 , 60 , 70 , 80 , 90 integers , and every longest divsion between the figures 21 , 22 , 23 , 24 , 25 , 26 ; &c. integers , and the shortest divisions tenths of those integers , and so in proportion infinitely : as [ 1 10 1. 10 ] [ 1. 10. 100. ] [ 10. 100. 1000 ] [ 100. 1000. 10000. ] in all which 4 examples , the first order of Figutes , viz. 1 / 10 1. 10. 100. is represented by the first 1. on the line of of numbers , the second order of figures , viz. 1. 10. 100. 1000. is represented by the middle 1. on the ●ine of numbers : the last order or Place of figures , viz. 10. 100. 1000. 10000. is represented by the 10 , at the end of the line of numbers . 3. That I may be plain ( yet further ) if a number be propounded of 4 figures , having two cyph●rs in the middle , as 1005 , it is expressed on the line between that prime to which it doth belong , and the next centesm or small division next to it ; but if you were to take 5005 , where there are not so many divisions , you must imagin them so to be , and reasonably estimate them accordingly . Thus much for numeration on the line , or naming any point found on the Rule , in its proper value and signification . Note , that since this Book was first written , I have provided a Print of the Line eleven times repeated , where the Numerator is certainly expressed from 0001 to 10000000 ; and may be had here , or in the Minories . CHAP. II. PROB. I. Two Numbers being given , to find third Geometrically proportiona● unto them ; and to three , a fourth ▪ and to four , a fifth , &c. GEometrical proportion is when divers numbers being compar'd together , diffe● among themselves , increasing o● decreasing , after the rate or reaso● of these numbers , 2. 4. 8. 16. 32 ▪ for as 2 is half 4 , so is 8 half 16 and as this is continued , so it may be also discontinued , as 3. 6. 14 ▪ 28 ; for though 3 is half 6 , and 14 half 28 , yet 6 is not half 14 ▪ nor in proportion to it , as 3 is to 6. There is also Arithmetical and Musical proportion ; but of that in other more large discourses , being not material to our present purpose ( though I may hint it afterward . ) To find this by the numbers , extend the Compasses upon the ●●ine of numbers , from one number ●o another ; this done , if you ap●ly that extent ( upwards or downwards , as you would either increase ●r diminish ) from either of the numbers propounded , the moveable point will stay on the third proportional number required . Also , the same extent applyed the ●ame way from the third , will give ●ou a fourth , and from the fourth 〈◊〉 fifth . Example . Let these two numbers 2 and 4 be propounded to find a third proportional to them ( that is , to find a number that shall bear the same proportion that 2 doth to 4 ) and then to that third , a fourth , fifth , and sixth , &c. ) Extend the Compasses upon ●he first part of the line of num●ers , from 2 to 4 ; this done , if ●ou apply the same extent upwards ●om 4 , the moveable point will fall upon 8 , the third proportional required ; and then from 8 , it will reach to 16 , the fourth proportional ; and from 16 to 32 , the fifth ; and from 32 to 64 , the sixth proportional . But if you will continue the progression further , then remove the Compasses to 64 in the former part of the line , and the moveable point will stay upon 128 , the seventh proportional ; and from 128 to 256 , the eighth ; and from 256 to 512 , the ninth , &c. Contrarily to this , if you would diminish , as from 4 to 2 , extend the Compasses from 4 to 2 , and the moveable point will fall on 1● ; and from 1 to 1 / 10 , or 5 of ten , which is one half ( by the second Problem of the first Chapter ) and from 5 to 25 , or ¼ , and so forward . But generally in this , and most other work , make use of the small divisions in the middle of the line , that you may the better estimate the fractions of the numbers you make use on ; for observe , look how much you miss in setting of ●he Compasses to the first and se●ond terms , so much , or more will ●ou err in the fourth : therefore ●he middle part will be most use●ul : as for example , as 8 to 11 , ●o is 12 to 16 , 50 , or 5 , if you do ●magine one integer to be divided ●ut into 10 parts , as they are on ●he line on a two foot Rule ; but on the other end you cannot so well express a small fraction , as there you may . PROB. II. One number being given , to be multipl●ed by another number given , to find the Product . Extend the Compasses from 1 to ●he Multiplicator , and the same ex●ent applyed the same way ▪ from ●he Multiplicand , will cause the moveable point to fall on the product . Example ▪ Let 6 be given to be multiplyed by 5 , here if you extend the Compasses from 1 to 5● the same extent will reach from ● to 30 ; which number 30 , though it be numbred but with 3 , yet you● reason may regulate you , to call i● 30 , and not 3 ; for look what proportion the first number bears to 1 , the same must the other number ( or Multiplicand ) bear to the Product ; which in this place , cannot be 3 , but 30. Another Example , for more plainness ▪ Let 125 be given to be multiplied by 144 ; extend the Compasses from 1 to 125 , and the same extent laid the same way from 144 , the moveable point will fall on 18000 : now to read this number 18000 ( so much , and no more ) you must consider , that as in 125 there is two figures more than in 1 , so there must be two figures more in the Product 18000 , than in the Multiplicand 144 : and as for the order of reading the numbers , you may consider well the first Problem of the first Chapter . Some other Examples for more light . 3. As 1 to 25 , so 30 to 750 : as 1 to 8 , so 16 to 48 : as 1 to 9 , so 9 to 81 : as to 12 , so 20 to 240. One help more I shall add , as to the ●ight computation of the last figure in 4 figures ( for more cannot ●e well exprest , on ordinary lines , as that on a two foot Rule is ) but for the true number of figures in the Product , note , that for the most part , there is as many as there ●s in the Multiplicator and Multiplicand put together , when the ●wo first figures of the lesser of ●hem , do exceed so many of the ●●rst figures of the Product ; but ●f the two first figures of the least of them , do not exceed so many of ●he first figures of the Product , then ●t shall have one less than the Multiplicator and Multiplicand put together ; as , 92 and 68 multiplied , ●akes 6256 , four figures , because 68 is more than 62 ; and 12 multiplied by 16 , makes but 192 , three figures , for the reason abovesaid , because 19 is more than 16. Now for right naming the last figure , write them down ; as thus , 75 by 63 ; now you multiply 5 by 3 , that is 15 , for which you , by vulgar Arithme●ick , set down 5 , and carry 1 ; therefore 5 is the last figure in the Product , and it is 4725. PROB. III. Of Division . One number being giuen to be divided by another , to find the Quotient . Extend the Compasses from the Divisor to 1 , and the same extent will reach , the same way , from the Dividend to the Quotient ; or , extend the Compasses from the Divisor to the Dividend , the same extent shall reach , the same way , from 1 to the Quotient ; as for Example . Let 750 be a number given to be divided by 25 ( the Divisor ) I extend the Compasses from 25 to 〈◊〉 , then applying of that extent the same way from 750 , the other ●oint of the Compasses will fall up●n 30 , the Quotient sought : or ●ou may say , as 25 is to 750 , so is to 30. 2. Let 1728 be given , to be di●ided by 12 , say , as 12 is to 1 , so is ●728 to 144. Extend the Compas●es from 12 to 1 , and the same ex●ent shall reach the same way from ●728 to 144 : or as before , as 12 to 1728 , so is 1 to 144. 3. If the number be a decimal fraction , then you work as if it were an absolute whole number ; ●ut if it be a whole number joyned to a decimal fraction , it is wrought here as properly as a whole number : Example , I would divide 111.4 by 1.728 , extend the Compasses from 1.728 to 1 , the same extent applyed from 111.4 , shall reach to 64.5 : so again , 56.4 ▪ being to be divided by 8.75 , the Quotient will be found to be 6.45 . for the extent from 8.75 to 1 being laid the same way from 56.4 , will reach to 6.45 , the Quotient , viz. 6 , and a decimal fraction 45. To reduce this decimal fraction to the vulgar fraction found by the Pen ; the same extent of the Compasses laid the contrary way from 45 the decimal fraction , gives 39 / 75 ▪ the vulgar fraction found by the Pen , as is more apparent in the next Example . In Division by the Pen , the fraction remaining is the Numerator , and the Divisor is the Denominator ; to find which by the line , do thus , with the same extent of the Compasses that wrought the Division , laid the contrary way from the decimal fraction , gives the usual fraction when you work by the Pen. Example . To divide 522 by 34. The extent from 34 to 1 reaches from 522 to 15.35 ; then the same extent laid the contrary way from 35 , gives 12.15 , as by the Pen. Now to know , of how many figures any Quotient ought to consist , it is necessary to write the Dividend down , and the Divisor under it , and see how often it may be written under it , for so many figures must there be in the Quotient ; as in dividing this number ●2231 by 27 , according to the Rules of Division , 27 may be written three times under the Dividend , therefore there must be 3 figures in the Quotient ; for if you extend the Compasses from 27 to 1 ●t will reach from 12231 to 453 , the Quotient sought for . Note , that in this , and also in ●ll other Questions , it is best to ●rder it so , as that the Compasses ●ay be at the closest extent ; for ●ou may take a close extent more easily and exactly than you can a ●rge extent , as by experience you will find . PROB. IV. To three numbers given to find a fourth in a direct proportion ( of the Rule of Three Direct . Extend the Compasses from the first number to the second , that done , the same extent applied the same way from the third , will reach to the fourth proportional number required . Example . If the Circumference of a Circle , whose Diameter is 7 , ●● 22 , what Circumference shall a Circle have , whose Diameter is 14 ? Extend the Compasses from ●● in the first part , to 14 in the second , and that extent applyed the same way from 22 , shall reach t● 44 , the fourth proportional required ; for so much shall the Circumference of a Circle be , whose Diameter is 14. — And the contrary , if the Circumference were given again . Ex. 2. If 8 foot of Timber b● worth 10 shilling , how much is 1● foot worth ? Extend the Compasses from 8 to ●0 ( either in the first part or second ) the same extent applyed ●e same way from 12 , shall reach ●o 15 ; which is the answer to the ●uestion ; for so many shillings is ●2 foot worth at that rate . PROB. V. Three numbers being given , to find a fourth in an inversed proportion , ( or the Back Rule of Three . ) Extend the Compasses from the first of the numbers given , to the second of the same denomination , ●● that distance be applyed from the third number backwards , it shall reach to the fourth number ●ought . If 60 pence be 5 shillings , how much is 30 pence ? facit 2.5 . two ●illings , 5 tenths of a shilling , that ●● , being reduced , 2 s. 6 d. Again , If 60 Men can raise a ●rest-work of a certain length and ●readth , in 48 hours , how long will it ●e ere 40 Men can raise such another ? Extend the Compasses from 60 ▪ to 40 , numbers of like denomination , viz. of Men ; this done , that extent applyed the contrary wa● from 48 , will reach to 72 , the fourth number you look for ▪ Therefore I conclude , that 40 ▪ Men will perform as much in 72 hours , as 60 Men will do in 48 hours . Note , that this Back Rule o● Three , may for the most part , b● wrought by the Direct Rule of Three . If you do but duely consider the order of the Question , for you must needs grant , that fewer Men must have longer time , and the contrary therefore the Answer must be , in proportion to the Question , which might have been wrought thus as well : The extent from 40 to 60 will reach the same way , from 48 to 72 in direct proportion ; or contrarily , as 60 to 40 , so is 72 to 48 : which you see , is but turning the question to ●ts direct operation , according to the true reason of the question . Thus you have the way for the Direct and Reverse Rule of Three : for the Double Rule of Three , and Compound Rule of Three , this is the Rule for it . Always in the Double Rule of Three 5 terms are propounded , and a sixth is required , three of which terms are of supposition , and two of demand ; now the difficulty is in pla●●ng them , which is ●●est done thus , as in this Example . Example . If 5 spend●● ●● l. in 3 months , How many pounds will serve 9 Schollars for 6 months ? Note , here the terms of supposition are the first three , viz. 5 , 20 , and 3 , and the terms of demand ●●e 9 and 6. Then next , for the right placing them , observe which of the terms of supposition is of the same denomination with the term required , as here the 20 l. is of the same denomination with the how many pounds required : set th● always in the second place , and the two terms of supposition o● above another in the first place and the terms of demand one above another in the last place , th●● 5 — 20 — 9 3 — pounds — 6 Then the work is performed ●● two single Rules of Three Dir●● thus : Extend the Compasses from 5●● ▪ 20 , the same extent applyed t●● same way from 9 , shall reach 36 , a fourth ; this is the first o●●ration : Then as 3 to 36 , the 4● so is 6 to 72 , the number of pou●● required . By the Line of Nu●bers , the Double Rule is wroug●● as soon as the Compound : the●●fore I shall wave it now . Four Questions and their Answe●● to shew the various forms of work●● on the Line of Numbers . Quest. 1. If 12 Men raise a Frame in 10 days , in how many days might 8 Men raise the like Frame ? Reason tells me , that fewer Men must have longer time ; therefore the work is thus , as 12 is to 8 , so is 10 to 15 , by the last Rule ; or , as 8 to 10 , so is 12 to 15. Quest. 2. If 60 yards of Stuff , at 3 quarters of a yard broad , would hang a Room about ; how many yards of Stuff , of half a yard broad , will serve to hang about the same Room ? The work is thus ; as 5 10th . to 7 10th . ½ , so is 60 to 90 ; or , as 75 to 5 , so is 60 to 90 , wrought backwards . Quest. 3. If to make a Foot Superficial 12 Inches in breadth , do require 12 Inches in length , the breadth being 16 Inches , how many Inches in length must I have to make a Foot Superficial ? The work is thus ; as 16 is to 12 , so is 12 to 9 , the number of Inches to make a Foot. Quest. 4. If to make a Foot Solid , a Base of 144 Inches require i● Inches in height , a Base being given of 216 Inches , how much in heighth makes a Foot Solid ? The work then is , as 216 is to 144 , so is 12 to 8 : or otherwise thus , as 12 is to 216 , so is 144 to 8 , the heighth sought . PROB. VI. To three numbers given , to find a fourth in a doubled Proportion . This Problem concerns Questions of proportions between Lines and Superficies : now if the denominations of the first and second terms be of Lines , then extend the Compasses from the first term to the second ( of the same kind o● denomination ) this done , that extent applyed twice , the same way from the third term , the moveable point will stay upon the fourth term required . Example . If the Content of a Circle whose Diameter is 14 , be 154 , what will the Content of a Circle be , whose Diameter is 28 ? Here 14 and 28 having the same denomination , viz. of lines , I extend the Compasses from 14 to 28 , then applying that extent the same way , from 154 twice , the moveable point will fall on 616 , the fourth proportional sought , that ●s , first from 154 to 308 , and from ●08 to 616. But if the first denomination be ●f superficial content , then extend the Compasses unto the half of the ●istance between the first and second of the same denomination ; of the same extent will reach from the third to the fourth . Example . being●54 ●54 . have a Diameter that is 14 , ●hat shall the Diameter of a Circle ●e , whose content is 616 ? Divide the distance betwixt 154 and 616 into two equal parts , then ●et one foot in 14 , the other shall each to 28 , the Diameter required . The like is for Squares : For if a Square , whose side is 40 foot , contain 1600 foot ; how much shall ● Square contain , whose side is 60 foot . Take the distance from 40 to 60 , and apply it twice from 160● and the moveable point will sta● on 3600 , the content sought for . PROB. VII . To three numbers given , to find fourth in a triplicated proportion . The use of this Problem consisteth in Questions of proportion between Lines and Solids , wherei● if the first and second terms ha● denomination of Lines , then extend the Compasses from the fir●● to the second , that extent applye● thtee times from the third , w● cause the moveable point to st●● on the fourth proportional required . Example . If an Iron Bullet , whose Diameter is 4 Inches , shall weigh 9 pound what shall another Iron Bullet weigh whose Diameter is 8 Inches ? Extend the Compasses from 4 to 8 , that extent applyed the same way three times from 9 , the moveable point will fall at last on 72 , the fourth proportional & weight required , that is in short , thus , as 4 to 8 , so 9 to 18 , so 18 to 36 , so 36 to 72. But if the two given terms be weight or contents , of solids , and the Diameter or side of a Square or ) a Line is sought for , then divide the space between the two given terms of the same Denomination into three parts , and that ●●sistance shall reach from the third ●o the fourth proportional . Example . Divide the space between 9 and 72 in three parts , that third part shall reach from 8 to 4 , ●or from 4 to 8 , as the Question ●as propounded , either augmenting or diminishing . ) Example . If a Cube whose side is Inches , contains 216 Inches , how many Inches shall a Cube contain whose side is 12 Inches ? Extend the Compasses from 6 to 12 , that extent measured from 2● in the first part of the Line of Numbers three times , shall a last fall upon 1728 , in the second part of the Line of Numbers ; for note , if y●● had begun on the second part , y●● would at three times turning , hat● fallen beyond the end of the line and the contrary , as above , hol● here in Squares also . PROB. VIII . Betwixt two numbers given , to 〈◊〉 a mean Arithmetically Proportional . This may be done without ●● help of the Line of Numbers ; 〈◊〉 vertheless , because i● serves to 〈◊〉 the next following , I sh●ll here 〈◊〉 sert it , though I thought to 〈◊〉 both this and the next over i●●● lence ; yet to set forth the exc●● lency of Number , I have set th● down . and the Rule is this : Add half the difference of the ●iven terms to the lesser of them , and ●hat aggregate ( or summe ) is the Arithmetical mean required : or add ●hem together , and the half summe is the same . Example . Let 20 and 80 be the terms given , now if you substract one out of the other , their difference is 60 , whose half difference 30 , added to ●● the lesser term , makes 50 ; and that is the Arithmetical mean ●ought : also 20 and 80 is 100 , the ●alf is 50 ▪ as before . PROB. IX . ●etween two numbers given , to find a mean Musically proportional . Multiply the difference of the ●erms by the lesser term , and add likewise the same terms together ; ●is done , if you divide that product by the summ of the terms , ●●d to the Quotient add the lesser ●●rm ; that last summ is the Musi●●l mean required . Or shorter , ●●us ; Multiply the terms one by another , and divide the product by their summ , and the Quotient doubled is the Musical mean required ▪ Example . The numbers given being 8 and 12 , multiplied together , make 96 , that divided by 20 ▪ the summ of 8 and 12 , the Quotient is 4 80 , which doubled , i● 9-6 10 s , the Musical mean required . This may be done by the Line of Numbers ; otherwise thus find the Arithmetical mean between 8 and 12 , and then the Analogy or agreement is thus ; As the Arithmetical mean found viz. 10 , is to the greater term ●● so is the lesser term 8 to the Musical mean required , 9 6 / 10. PROB. X. Betwixt two numbers given , to find mean Geometrically proportional ▪ Divide the space on the Line ●● Numbers , between the two e● tream Numbers , into two eq●● parts , and the point will stay at t●● mean proportional required . So the extream numbers being 8 and 32 , the middle point between them will be found to be 16. PROB. XI . Betwixt two numbers given , to find two means Geometrically proportional . Divide the space between the two extream numbers , into three equal parts , and the two middle points dividing the space , shall shew the two mean proportionals . As for Example ; let 8 and 27 be two extreams , the two means will be found to be 12 and 18 ; which are the two means sought for . PROB. XII . To find the Square-Root of any number under 1000000. The Square-Root of every number is always the mean proportional between 1 , and that number for which you would find a Square-root ; but yet , with this general caution , if the figures of the number be even , that is , 2. 4. 6. 8 or 10. &c. then you must look for the unit ( or one ) at the beginning of the Line , and the number in the second part , and the Root in the first part ; or rather reckon 10 at the end to be the unit , and then both Root and Square will fall backwards toward the middle 1 in the second length or part of the line : but if they be odd , then the middle 1 , will be most convenient to be counted th● unity , and both Root and Square will be found from thence forward towards 10. So that according to this Rule , the square of 9 will be found to be 3 , the square of 64 wil● be found to be 8 , the square of 14● to be 12 , the square of 1444 to b● 38 , the square of 57600 to be 240● the square of 972196 will be found to be 986 : and so for any other number . Now to know of how many figures any Root ought to consist , put a prick under the first figure , the third , the fifth , and the seventh , if there be so many ; and look how many pricks , so many figures there must be in the Root . PROB. XIII . To find the Cube-Root of a Number under 1000000000. 1111 The Cube-root is always by the first of two mean proportionals between 1 and the number given , and therefore to be f●und by dividing the space between ●●em into three equal parts : so by this means , the Root of 17●8 will be found to be 12 , the Root of 17280 is near 26 , the Root of 17●800 is almost 56 , although the p●i●● on the Rule repres●ming all t●e square numbers , is in one place , y●● by altering the unit , it p●oduc●th v●rious points and numbers for t●eir respective prop●● Roots . The Rule to find which unit , is i● this manner : You must set ( or suppose to be set ) pricks under the first figure to the left hand , the fourth figure , the seventh , and the tenth ; now , if by this means , the last prick to the left hand shall fall on the last figure , as it doth in 1728 , then the unit will be best placed at 1 in the middle of the line , and the Root , the Square , and the Cube , will all fall forwards toward the end of the line . But if it fall on the last but one , as it doth in 17280 , then the unit may be placed at 10 in the beginning of the line , and the Cube i● the second length . But if the las● prick fall under the last but two● as in 172800 , it doth then pla● the unit always at 10 in the end o● the line ; then the Root , the Square and Cube , will all fall backward and be found in the second par● between the middle 1 and the end of the line . By these Rules it doth appear , that the Cube-root of ●● is 2 , of 27 is 3 , of 64 is 4 , of 125 is 5 , of 216 is 6 , of 345 is 7 , of 512 is 8 , of 729 is 9 , of 1000 is 10. As you may see by this Table of Square and Cube roots . Thus you have the chief Use of the Line of Numbers in general , and they that have skill in the Rule of Three , and a lirtle knowledge in Plain Triangles , may very aptly apply it to their particular purposes ; yet for their sakes for whom it is intended , I shall inlarge , to some more particular applica●ions , in measuring all sorts of Superficies and Solids ; wherein I do judge it will be most serviceable to them that be unskilful in Arithmetick , as before said . A Table of the Square and Cubique Roots Root . Square . Cube . 1 1 1 2 4 8 3 9 27 4 16 64 5 25 125 6 36 216 7 49 343 8 64 512 9 81 729 10 100 1000 12 144 1728 26 676 17576 56 3136 175616 204 41616 8489664 439 192721 84604519 947 896809 849278123 1000 1000000 1000000000 CHAP. III. The Use of the Line of Numbers , in measuring any Superficial measure , as Board , Glass , Plaistering , Paving , Painting , Flooring , &c. THe ordinary measure , and most in use , is a Two-foot Rule divided into 24 Inches , and every Inch in●o 8 parts , that is , Halfs , Quarters , and Half-quarters ; but these parts not agreeing with the parts on the Line of Numbers , which are Decimals or tenth parts , is bred very much trouble ; and there cannot be exactness without taking of small parts , as Ha●f quarters of Inches , or else using o● Reduction ; and it is also as troub●esome by Arithmetick as by the Line of Numbers . To avoid which , I would advise either to measure altogether by Foot-measure ( that is , a Foot divided into 1000 parts ( or rather , as i● sufficient for ordinary use , 100 ) and then the divisions on the numbers will agree fitly to the parts on your Rule , without any trouble for Fractions ; for so doing , Fractions do become whole Numbers as it were , and are wrought accordingly : But if you use i● not in measuring , yet you may have it set for to help you for the ready reducing of such Numbers as shall require it , thoug● I shall apply it to Inches also as it is commonly used , that i● may appear useful both ways ▪ accordingly as any man shall b● affected . The like reason holdeth for inches , Yards , Ells and Perches , o● any other measure ; for thereb● the work is made more easie , a● shall appear anon . First , by Foot-measure only . PROB. I. The breadth of any Oblong Superficies , as a Table , Gravestone , or the like , given in Foot-measure , to find how much in length makes one Foot. Extend the Compasses from the ●readth in Foot-measure to 1 , the ●ame extent applyed the same way ●rom 1 , shall reach to the length ●equired , to make one Foot Su●erficial in Foot-measure . Exam. At 8 tenths broad . Set one point of the Compasses ●n 8 , and extend the other point ●o 1 ; the same extent being laid ●he same way from 1 , shall reach ●o 1.25 , the length required , be●ng 1 Foot , and 25 parts of a hun●red : For 1.25 multiplied by 08 , ●● 100.0 , or just one Foot in De●●mals . PROB. II. The breadth and length of any Superficies given in Foot-measure , to find the superficial Content in Foot-measure . Extend the Compasses from 1 to the breadth , the same extent applyed the same way from the length in Foot-measure , shall reach to the superficial Content in Foo●● measure . Ex. At 8 tenths broad , & 15 f. long Extend the Compasses from 1 t● 8 , the same extent laid the sam● way from 15 foot the length , shal● reach to 12 foot , the superficia● content required . Again , at 1. 75 broad , and 25. 30 lon●● Set one point of the Compass●● in 1 , and open the other to 1. 7 then the same extent laid the sam● way from 25. 30 , will reach to 4● feet 275 parts , the superficial co●tent required . Note , the 275 parts coun●ed ●● foot measure , right against it inches is 3 inches 3 tenths . PROB. III. The breadth of any Oblong Superficies being given in inches , to find how many inches in length makes one foot . The extent from the inches ●road to 12 being laid the same way from 12 , shall reach to the inches long to make 1 foot . Examp. At 9 inches broad . Set one point in 9 , open the other to 12 , then the same extent said the same way from 12 , reaches to 16 inches , the length required to make 1 foot superficial : for 9 multiplied by 12 is 144 , the number of superficial inches in one foot . PROB. IV. The breadth and length of an Oblong Superficies given in inches , to find the superficial content in like inches . Examp. At 30 inches broad , and 83 inches long . Set one point of the Compasses in 1 , and open the other to 183 then the same extent laid the same way from 30 , shall reach to 5490 the true content in inches . Now to call this 5490 , and no more or less , observe , as 183 is two figures more than 1 , so must 549● be two figures more than 30 , as i● observed in the Rule of Multiplication . PROB. V. The length and breadth given i● inches , to find the content in Superficial Feet . The extent from 144 ( th● inches in one foot ) to the breadt● in inches , being laid the same wa● from the length in inches , sha●● reach to the content in feet . Examp. At 30 inches broad , a●● 183 long . Set one point in 144 , and op●● the other to 183 , the same exten● laid the same way from 30 , reache● to 38 foot 12 / 000 , the near conte●● in feet required . BROB. VI. The breadth given in inches , and the length in feet , to find the content in Superficial Feet . The extent from 12 to the breadth in inches , shall reach the same way from the length in feet to the superficial content in feet required . Examp. At 30 inches broad , and 15 foot 3 inches long . Set one point in 12 , and open the other to 30 , the breadth in inches ; then the same extent laid the same way from 15 foot 3 inches the length , reaches to 38 foot 1 inch and ½ , the true content . PROB. VII . The length and breadth being given in feet and inches , to find the superficial content in feet & inches . As 1 to the breadth or length in feet and inches , so is the length or breadth in feet and inches to the superficial content in like feet and inches . Examp. At 18 foot 9 inches broad , and 30 foot 7 inches long . Set one point always in 1 , open the other to the ( length or ) breadth 18 foot 9 inches , then the same extent laid the same way from 30 foot 7 inches the length reaches to 573 foot 5 inches , the superficial content . Note , if you have the line of Numbers divided into twelves for inches as before said , you may work this question more readily , and truely as I have often times made them for my own use and others also . Note , that how broad soever any superficies is , so much is there in a foot long of that Superficies . Examp. If a board be 3 feet 9 inches broad , there is 3 feet 9 inches in every foot long of that board , therefore ( 3 feet 9 inches or ) the breadth in feet and parts , multiplied by the length in feet and parts , give the superficial content . PROB. VIII . Of a Circle and his parts , as Diameter , Circumference , Square-equal and Square within , and Area of a Circle . For the ready finding of any of these , any one being given , there is found out five numbers , in whose places on the li●e may be set five Center-pins for the more ready finding of them and readiness to use ; which are as followeth , Viz. Diameter — 10. 000 Circumference — — 31. 416 Square equal — — 8. 862 Square within — — 7. 071 Area or Content — 78. 538 Thu if the Diameter of a Circle be 10 inches , then the Circumference is near 31 inches and 416 parts of 1000. The side of the Square-equal is 8 inches 862. The side of the Square within is 7. 071. The Area or Content is 78 inches 538 parts . And any one of these being given , all the other four may be readily found by the Line of Numbers , as followeth . Any Diameter of a Circle given to find the Circumference , &c. The extent from 10 a fixed Diameter , to any other Diameter given , shall reach the same way from 31. 416 the fixed Circumference , to the Circumference required for the given Diameter And from the fixed side of th● Square-equal to the required sid● of the Square-equal . And from the fixed side of the Square-wit● in to the required side of th● Square within , &c. 1. Examp. Let the given Diameter of a Circle be 15 , what is t●● Circumference answerable to i● &c. Set one point of the Compass● in 10 the fixed Diameter , and t●● other in 15 the given Diameter ; then the same extent applyed the same way from 31. 416 the fixed Circumference , shall reach to 47. 14 the Circumference required . 2. To find the side of a Square-equal to a Circle of 15 inches Diameter . Also the same extent applyed the same way from 8. 862 , the fixed number for the side of the Square-equal , shall reach to 13. 30 , the ●ide of the Square-equal to the Circle required . To find the side of the Square within . The same extent from 10 to 15 being laid the same way from 7. 071 the fixed side of the Square within , shall reach to 10. 61 , the ●●de of the Square within in a Circle of 15 inches Diameter required . To find the Area or Content of the Circle whose Diameter is 15 inches . The same extent from 10 to 15 being twice repeated , the same way from 78.538 will reach to 176 inches and 7 tenths , the Area of a Circle of 15 inches Diameter ▪ PROB. IX . The Circumference of any Circle being given , to find the Diameter ▪ the Square-equal or Squar● within and the Area . Examp. If the Circumference give● be 47.14 , what is the Diameter the side of the Square-equal , ●● Square within , and Area . 1. Set one point in 31.416 the fixed Circumference , and the other in 47.14 the given Circumference ; then the same extent lai● from 10 the fixed Diameter , sha●● reach to 15 the Diameter required for a Circle of 47 ▪ 14 Circumference . 2. Also the same extent la●● from 8.862 the fixed side of t●● Square-equal , shall reach the sa●● way to 13.30 , the side of t●● Square - equal required . 3. Also the same extent laid the same way from 7.071 , shall reach to 10.61 , the side of the Square-within in a Circle of 15 inches Diameter required . 4. The same extent from 31. 416 the fixed Circumference , to 47.14 the given Circumference , being twice repeated the same way from 78.538 , the fixed Area for 10 inches Diameter , shall reach ●o 176 inches and 7 tenths , the Area of a Circle of 47.14 inches about . PROB. X. The Area of any Circle being given , to find the Diameter , Circumference , Square-equal , or the side of the Square within required . The exact middle , or half di●tance measured on the Line of Numbers , between the fixed Area ●8 . 538 and the given Area , shall reach from the fixed Diameter to the inquired Diameter ; and from the fixed Circumference to the inquired Circumference , and from the sides of the fixed Square-equal or within to the inquired side● of the Squares-equal or within . Examp. Let the given Area ●● 176 inches ●● . The exact middle between 78 ▪ 538 the fixed Area , for a Circle o● 10 inches Diameter , and 176 inches and 7 tenths the given Area measured on a Line of Numbers . 1. Then that extenr laid th● same way from 10 the fixed Diameter , reaches to 15 the inquire● Diameter . 2. Also the same extent lai● the same way from 31.416 the fi●● ed Circumference , gives 47. 1●● the Circumference required . 3. Also the same extent lai● the same way from 8.862 the fix●● side of the Square-equal , reach● to 13.30 , the side of the Squar●● equal required . 4. Lastly , the same extent l●●● the same way from 7.071 the fixed Number for the side of the Square within , shall reach to 10.61 the side of the Square within required for a Circle whose Area is 176 inches and 7 tenths . PROB. XI . Having the Area of any Circle , to find the side of a Square equal so it . The Square root of the given Area ( found by Problem 12 of the 11th . Chapter ) is always the side of the Square-equal to the Circles Area given ; thus the Square-root of a Circle whose Area is 144 is 12 , the Square-root of 144. PROB. XII . To find the Content of a Circle two ways . Multiply the Diameter by it self , and multiply that product by 11 , and divide this last product by 14 , and the Quotient shall be the content required . Or else multiply half the Diameter and half the Circumference together , and the product is the content required . PROB. XIII . How to measure a Circle , a Semicircle , or a quarter of a Circle , or any part that goeth to the Centre of the supposed Circle . First for a Circle . Take half the Diameter and half the Circumference , and measure it then as an Oblong Square ; for the half Circle take half the Diameter and half the Semicircumference and do likewise . Thirdly , for the quarter of a Circle , take half the Arch of that quarter , and the Radius or Semidiameter of the whole Circle , and work as you would d● with an oblong square piece , and you shall have your desire . PROB. XIV . How to measure a Triangle . Take half the Base and the whol● Perpendicular , and work with them two as if it were an oblong square figure , or you may take the whole Base and half the Perpendicular . Or , by the Line of Numbers , the extent from 2 to the Base shall reach from the Perpendicular to the Content . PROB. XV. How to measure a Rhombus , or a Rhomboides . A Rhombus is a Diamond-like figure , as a quarry of Glass is , containing 4 equal sides , and two equal opposite angles : but a Rhomboides is a figure made of two equal opposite sides , and two equal opposite angles : and to measure them you must take any one side , and the nearest distance from that side to its opposite side for the other side , and then reckon it as an oblong Square . PROB. XVI . How to measure a Trapezium . A Trapezium is a figure comprehended of 4 unequal sides , and of 4 unequal angles ; and before you can measure it , you must reduce it into two Triangles , by drawing a line from any two opposite corners , then deal with it as two Triangles ; or you may save some work thus , the line you draw from corner to corner will be the common base to both Triangles : then say , as 1 is to half the Perpendiculars of both the Triangles put together , so the whole Base to the Content . PROB. XVII . How to measure a many sided irregular figure or Polygon . You must reduce it into Triangles or to Trapeziums , by drawing of lines from convenient opposite corners ; and then the work is all one with that of the last Problem . PROB. XVIII . How to measure a many sided regular figure , commonly called a regular Polygon . Measure all the sides , and take the half of the summ of them for one side of a square , and the nearest distance from the center of the Polygon to the middle of one of the sides for the other side of a square ; and with them two numbers work as if it were a square oblong figure , and it will give the content of the Polygon desired . PROB. XIX . How to reduce Feet into Yards , Ell● , or other parts . First for yards , if 9 foot make one yard , how many shall 36 foot make . The extent from 9 to 2 will reach from 36 to 4 , for so many yards is 36 foot . But if you were to measure any quantity by the yard , as the plaistering or painting of a House , then I would advise you to have a yard to be divided into a hundred parts ( which is as near as commonly Workmen go to , or else into 1000 , if you do require more exactness ) and measure all your lengths and breadths with that , and set them down thus , 2. 25 ( or by 1000 , thus 2. 250 ) and the length thus 10. 60 , and multiply them together , and the product is the true content of that long square : the like holds fo● Ells , or Poles , Furlongs , or any other kind of measure . Again , fo● a yard in length , if 3 foot make one yard , then what shall 30 make 〈◊〉 it maketh 10 : or the contrary , i● 10 yards make 30 foot , what shall 12 make ? The extent from 10 to 12 will reach from 30 to 36 foot 〈◊〉 but if it be given in feet or inches ▪ then say , as 9 to the breadth , so i● the length in feet and inches ( o● decimal parts ) to the content i● yards required . CHAP. IV. The Use of the Line of Numbers i● measuring of Land by P●rc●●● and Acres . PROB. I. Having the breadth and length of an Oblong Superficies given in Perches , to find the content in Perches . As 1 perch to the breadth in perches , so the length in perches to the content in perches . Examp. As 1 is to 30 , so is 183 to 5490 perches . PROB. II. Having the length and breadth in perches , to find the content in square acres . As 160 to the breadth in perches so is the length in perches to the content in acres . As 160 unto 30 , so is 183 to 34. ●1 ( in acres and 100 parts . PROB. III. Having the length and breadth of an oblong Superficies given in Chains , to find the content in Acres . It being troublesome to divide ●he content in perches by 160 , we may measure the length and breadth by Chains , each Chain being 4 perches in length , and divided into a hundred links , then the work will be more easie in Arithmatick or by the Rule ; for as 10 to the breadth in Chains , so the length in Chains to the content in Acres . Examp. As 10 to 7. 50 , so is 45. 75 to 34. 81 ( 100 parts of an Acre . ) PROB. IV. Having the Base and Perpendicular of a Triangle given in Perches , to find the content in Acres . If the Perpendicular go for the length , and the whole Base for the breadth , then you must take half of the oblong for the content of the Triangle , by the second Problem , as 160 to 30 , so 〈◊〉 183 to 34. 31 ; or else , without halsing , say , as 320 to the Perpendicular , so is the Base to the content in Acres ; as 320 unto 30 , so is 183 to 17. 15. PROB. V. Having the Perpendicular and Base given in Cains , to find the content in Acres . As 20 to the perpendicular , so is the Base to the content in Acres . As 20 to 7. 50 , so is 45. 75 to 17. 15 parts . PROB. VI. Having the content of a Superficies after one kind of perch , to find the content of the same Superficies according to another kind of perch . As the length of the second perch , is to the first , so is the content in Acres to a fourth number , and that 4th . number to the content in acres required . Suppose a superficies be measured with a chain 66 feet , or with a perch of 16½ , and it contains 34. 31 , and it be demanded how many acres it would contain if it were measured with a perch of 18 foot ? These kind of proportions are to be wrought by the Backward Rule of Three , after a duplicated proportion : wherefore I extend the Compasses from 16.5 unto 18.0 , and the same extent doth reach backward , first from 34.31 to 31.45 , and then from 31.45 to 28.84 , the content in those larger acres of 18 foot to a perch . PROB. VII . Having the plot of a Field with the content in Acres , to find the Scale by which it was plotted . Suppose a plain contained 34 acres 31 centesmes , if I should measure it with a Scale of 10 in an inch , the length should be 38 chains and 12 centesmes , and the breadth 6 chains and 25 centesms , and the content according to that dimension , would by the 3d. Problem of this Chapter be found to be 23 , 82. whereas it should be 34 ▪ 31 ; therefore to gain the truth ▪ I divide the distance between 23 ▪ 82 and 34.31 into two equal parts , then setting one foot of the Compasses upon 10 , the supposed true Scale , I find the other to extend to 12 , which is the length of the Scale required . PROB. VIII . Having the length of the Oblong , to find the breadth of the Acre . As the length in perches to 160 so is one acre to the breadth in perches . As 40 to 160 , so is 1 to 4. Again , as 50 to 160 , so is 1 to 3.20 , so is 2 to 6.40 ; or again , if you measure by chains , as the length in chains to 10 , so is 1 acre to his breadth in chains ; as 12.50 unto 10 , so 1 to 0.80 ; or if the length be measured by foot measure , then as the length in feet unto 43560 , so is 1 acre to his breadth in foot measure . So the length of the oblong being 792 feet , the breadth of one acre will be found to be 55 foot , the breadth of two acres 110 feet . The use of this Table is to shew you how many inches , centesmes of a chain , feet , yards , paces , perches , chains , acres , there is in a mile , either long or square , or consequently any of them all , in any of the other that is less ; as for example , I would know how many inches there is in a long perch , I look on the uppermos● row for perches , and in the nex● row under I find 198 for the quantity of inches in a long perch . B●● if I would know how many inches there is in a square perch , the● look for perch on the left hand ▪ and in the inch column you ha●● 39204 , for if you multiply 196 b● 196 , it will produce 39204. A necessary Table for Mensuration of Superficial Measure .   Inch. Centesme . Feet . Yard . Pace . Perch . Chain Acre . Mile . Inch. ( 1 ) 7 92 12 36 60 198 792 792● 6●360 Centes . 62 7264 ( 1 ) 1●515 4 545 7 575 25 〈◊〉 〈◊〉 8000 Feit . 144 2 295 ( 1 ) 3 5 16 5 66 〈◊〉 5280 Yards . 1296 20 655 9 ( 1 ) 1 66 5 50 22 220 1760 Pace . 3600 57 381 25 2 778 ( 1 ) 3 30 13● 1●2 1056 Perch . 39204 625 272.25 30 25 1● . 89 ( 1 ) 4 40 ●●20 Chain . 627264 10000 4356 484 174. 24 16 ( 1 ) 10 ●0 Acre . 6272640 100000 43560 4840 1742. 4 160 10 ( 1 ) 8 Mile . 101448960 64000000 27878400 3097●00 1115136 1●2400 6●00 640 ( 1 ) Square . Inches . Centesmes . Feet . Yards . Paces . Perch . Chain Acre . Mil● . The like is for any other number in the whole Table , and is of very good use to reduce one number into another , or one sort of measure into another ; as inches into feet , and feet into yards , and yards into perches , and perches to chains , and chains into acres , and acres into miles , or the contrary , either long-wise or square-wise ; as is well known to them that have occasion for these measures . Thus much shall suffice for superficial measure , the practice of which will make it plain to any ordinary capacity . CHAP. V. The Use of the Line of Numbers in measuring of solid Measure , as Timber , or Stone , or such like solids . PROB. I. The breadth and thickness of any solid given , to find the side of a square that shall be equal in area to it . Divide the space on the Line of Numbers , between the breadth and thickness , into two equal parts and the Compass point shall stay at the side of the Square-equal required . Examp. At 6 inches thick , and 18 inches broad . The exact middle between 6 and 18 on a true Line of Numbers , will be at 10 . 39● , which is a Geometrical mean proportion between 6 and 18 , and the side of a Square-equal , to a piece 6 inches thick , and 18 inches broad ; for 10.393 squared , viz. multiplied by 10.393 , produceth 108 very near , being the product of 6 times 18. PROB. II. The side of the square of any solid given , to find how much in length makes a foot of solid or Timber-measure , by foot measure or inches . The extent from the side of the square given in foot-measure to 1 , being twice repeated or laid the same way from 1 , shall reach to the length required to make a foot solid . Examp. At 2 foot and 20 parts square , how much in length makes on● foot solid . The extent from 2.20 to 1 being laid or extended twice the same way from 1 , will stay at o● 2065 , the true length to make ● foot solid . For 2.20 the side of the square , multiplyed by it self is 4.84 , the● 4.84 multiplied by 2065 , the product is 999.46 , which is near one foot solid . 2. If the side of the square b● given in inches , then the exten● from the inches square to 12 , being twice repeated the same wa● from 12 , shall reach to the inche● long to make one ▪ foot . Examp. At 15 inches square . Set one point in 15 and the o●●er in 12 , the same extent laid ●●ice the same way from 12 , shall ●●ach to 7 inches 68 parts , the true ●●●ngth to make one foot solid . For ●●5 inches squared is 225. Then 225 multiplied by 7.68 , ●●oduceth 1728 , the cube-inches 〈◊〉 a foot-solid . 3. For small Timber work thus . The extent from the inches ●●uare to 12 being laid two times ●●e same way from 1 , shall reach ●● the feet and parts in length to ●●ake one foot solid . Examp. At 3 inches and a half ●●uare . The extent from 3 inches and a ●alf to 12 being laid twice the ●●me wa● from 1 shall stay at ●1 foot 76 par●s , the length in ●eet to make a foot solid . For 3½ squared is 12.25 , which ●ultiplied by 11 foot 76 parts , ●roduceth 144 , the number of long inches in a foot solid , fo● 144 pieces a foot long , and on● inch square is a true solid foot o● 1728 cube inches . PROB. III. The breadth and thickness of any solid given in inches or foot measure , to find the length of one fo● solid . 1. The extent from 12 to th● breadth in inches shall reach fro● the depth in inches to a fourth . 2. The extent from that fourt● number to 12 laid the same wa● from 12 , shall reach to the inch●● long to make one foot solid . Examp. At 20 inches broad a●● 9 inches thick , how many inches lo● makes one foot . 1. The extent from 12 to 9 t●● inches thick , being laid the sa● w●y from 20 , the inches broa● gives 14.98 a fourth number . 2. The extent from 1498 t● 12 being laid the same way fro● 12 , gives 9 inches and 60 p●r●● the length to make one foot solid . But for small Timber work ●hus . 1. The extent from 1 to the breadth , shall reach from the depth to a fourth . 2. The extent from that fourth ●umber to 12 , laid the same way from 12 , gives the length in feet ●nd parts to make one foot solid . Examp. At 3 inches thick and 5 inches broad . 1. The extent from 1 to 3 the ●●ickness , laid the same way from 〈◊〉 inches the breadth , reaches to ●4 . 95 a fourth . 2. The extent from 14.95 the ●ourth to 12 , being laid the same ●ay from 12 , gives 9 foot 60 parts ●●e length in feet to make one foot ●●lid . For the product of 20 multipli●d by 9 is 180 , and 180 multiplied ●y 9 inches 6 tenths will produce ●728 the cube-inches in one foot . So also for the small piece . The 3 inches multiplied by is 15 , then 15 multiplied by 9 fo● 60 parts , will produce 144 , as b● fore , the long inches in a foot 〈◊〉 lid . PROB. IV. 1. At any squareness , or brea● and thickness given , to find 〈◊〉 much is in a foot long . The extent from 12 to the 〈◊〉 ches square , laid the same 〈◊〉 from the inches square , gives 〈◊〉 quantity in one foot long . Examp. At 4 inches squan●● Set one point of the Compa● in 12 , and the other point i● the same extent laid the same 〈◊〉 from 4 the inches square , giv● inch and 33 parts , the quantity one foot long . Which being multiplied by 〈◊〉 length of the Tree , gives the 〈◊〉 true content . 2. If the Timber is not sq●● then thus , The extent from 12 to the breadth , shall reach the same way from the depth or thickness to the quantity in one foot long . Examp. At 14 and●8 ●8 inches thick . The extent from 12 to 14 inches the breadth , shall reach the same way from 8 the inches thick , to 9 inches and 33 parts , the quantity or content in 1 foot long . Then if the Tree be 20 foot long , 20 times so much is 15 foot ●7 inches ⅔ , the true content . PROB. V. being most used . The side of the Square given in inches , and the length in feet , to find the content in feet and parts required . The extent of the Compasses from 12 to the inches square , be●ng twice repeated the same way from the length , gives the true content of the Tree required . Examp. At 15 inches square , ●nd 2C foot long , how much Timber 〈◊〉 there . Set one point in 12 , and extend the other to 15 , the inches square . Then the same extent laid twice the same way from 20 the feet long , shall reach to 31 foot 3 inches , the true content required . For 15 inches , or 1 foot 3 inches squared ( or multiplied by it self● is 1. 6. 9 or 1 foot 6 inches ¾ , an● 20 times so much is 31 foot 3 inches . PROB. VI. The breadth and thickness of any piece given in inches , and the length in feet , to find the soll●● content in feet . 1. The extent from 12 to the breadth in inches , laid the same way from the depth in inche● gives a fourth number . 2. The extent from that fourth number to 12 shall reach the same way from the length to the content required . Examp. At 8 inches thick an● 13 inches broad , and 19 foot long what is the solid content in feet . 1. The extent of the Compasses from 12 to 8 inches the thickness , being laid the same way from 13 inches the breadth , sh●ll reach to 8.66 a fourth . 2. The extent from 12 to that fourth , being laid the same way from 19 foot the length , gives 13 foot 73. ( or near 9 inches ) the true content in feet and parts . For 8 inches the thickness , multiplied by 1 foot 1 inch the breadth is of . 8 inc . 8.12 pts . then 19 foot the length , multiplied by of . 8. in . 8.12 makes 13 foot 8 inches and 8.12 , the exact content of a peece 8 inches thick , 13 inches broad , and 19 foot long . PROB. VII . The length , breadth , and depth of any solid given in inches to find the solid content in cube-inches . 1. The extent from 1 to the breadth in inches , being laid the same way from the depth in inches , shall reach to a fourth . 2. The extent from 1 to that fourth , shall reach the same way from the length in inches to the content in cube-inches required . Examp. At 9 inches thick , 15 inches broad , and 40 inches long . 1. Set one point of the Compasses in 1 , and the other in 9 the inches thick , then the same extent laid the same way from 15 the breadth in inches , gives 13.50 a fourth . 2. Then the extent from 1 to 13.50 the fourth , laid the same way from 40 the length in inches , gives 5400 the content in inches . For 9 multiplied by 15 is 135 , and 135 multiplied by 40 , is 5400. PROB. VIII . The length , breadth , and depth of a great solid given in feet and decimals , or in feet and inches , to find the content in feet and parts . 1. the extent from 1 to the breadth in feet and parts , shall reach the same way from the depth in feet and parts to a fourth . 2. The extent from 1 to that fourth being laid the same way from the length in feet and parts , shall reach to the content in feet and decimal parts required . Examp. Suppose a Cistern be 3 foot 75 parts ( or 3 foot 9 inches ) broad , and 4 foot 50 parts ( or 4 foot 6 inches ) deep , and 7 foot 66 parts , or 8 inches long , how many solid feet will it hold . 1. The extent from 1 to 3.75 being laid the same way from 4.50 gives 16.875 for a fourth . 2. The extent from 1 to 16.875 the fourth , being laid the same way from 7.666 the length , gives 129 foot 3637 parts , or 4 inches and ⅓ . For 3.75 multiplied by 4.50 , gives 16.875 the fourth , then 16.875 multiplied by 7.66 the length , gives 129 foot 364 parts , or 4 inches and ⅓ , which you may reduce by a look of your eye on inches and foot measure , laid one by the other on your Rule . Note , every 6 foot is a full Beer Barrel , therefore 129 foot is 11 Barrels and ½ . Note also , if the space on the Line of Numbers between every figure , is divided into 12 proportional parts by small pricks , they will represent inches , and be very ready in use , for any Lea●ner especially . CHAP. VI. To measure round Timber , by having the Diameter given in foot-measure , or inches , and the length in feet . PROB. I. At any Diameter given in inches , to find how many inches in length will make one foot solid . The extent from the Diameter in inches to 13.54 ( the Diameter when one foot long makes 1 foot of Timber ) being laid twice the same way from 12 , shall reach to the inches in length ●o make 1 foot . Examp. At 10 inches Diameter . The extent from 10 to 13.54 , being laid twice the same way from 12 , gives 22 inches , the length to make one foot solid . For 78.54 the area of a Circle of 10 inches diameter , multiplied by 22 produceth 1728 , the cube-inches in 1 foot solid . PROB. II. The Diameter given in foot measure , to find how much in length make one foot solid . The extent from the Diameter in foot-measure to 1.128 ( the Diameter in foot-measure , when one foot long makes one foot ) being laid two times the same way from 1 , gives the length in foot measure to make 1 foot solid , at that given Diameter . Examp. At 0.833 Diameter , how much in length makes 1 foot solid . The extent from 0.833 the given Diameter to 1.128 , laid twice the same way from 1 , gives 1.833 the near length in foot measure to make 1 foot solid . For 54.56 the area of a circle whose diameter is 0.833 , being multiplied by 1.833 , the product is 1000.045 , the decimals in 1 foot solid . PROB. III. The Diameter given in inches , to find how much is in one foot long . The extent from 13.54 , to the Diameter in inches given , being twice repeated from 12 , shall reach to the inches of Timber contained in one foot long . Or the same extent laid twice the same way from 1 , will reach to the feet and inches contained in one foot long . Ex. At 8 inches diameter , how much is in one foot long . Set one point in 13.54 , and open the other to 8 , the given Diameter in inches . Then the same extent laid twice the same way from 12 , gives 41 inches ¾ the long inches of Timber in one foot long , 144 be-being one foot . Or the same extent laid twice the same way from 1 , reaches to 0.348 , 1000 being 1 foot . Then 0.348 multiplied by the length of the Tree in feet gives the true content . Examp. At 15 inches diameter , and 20 foot long . The extent from 13.54 to 15 being laid twice the same way from 12 , gives 14 inches 66 parts , or laid twice from 1 , gives 1 foot 223. Then 20 times so much is 24 foot 56 parts , or 6 inches ¾ . Also 20 times 14.66 inches is 293.20 inches , 12 being one foot , which divided by 12 makes 24 foot near 7 inches . PROB. IV. The Diameter in inches being given of any tree or cillander , & the length in feet , to find the content in feet . The extent from 13.54 to the Diameter in inches , being laid twice the same way from the length , gives the content in feet . Examp. At 15 inches Diameter , and 20 foot long . The extent from 13.54 to 15 , being laid twice the same way from 20 , gives 24 foot and 42 pts . the content in feet . Note , that right against 42 in feet measure is 5 inches in the inch measure . For the area of a Circle 15 inches diameter is 177 , and 20 times this is 3540 , which divided by 144 , hath 24 foot 84 cube inches a Quotient ( for 7 inches the near content . ) PROB. V. The Diameter given in inches , and the length in inches , to find the content in inches . The extent from 1.128 to the Diameter in inches , being laid twice the same way from the length in inches , gives the true content in cube-inches . Examp. A Well , or a Tubb of 40 inches Diameter , and 50 inches deep . The extent from 1.128 to 40 the inches Diameter , being laid twice the same way from 50 the length in inches , gives 62900 inches , the near content in cube-inches . Or the extent from 1.128 to 3 foot 33 parts ( or 4 inches ) the given Diameter laid twice from 4 foot 166 ( or 4 foot 2 inches ) gives 36 foot 40 parts , the content in feet and parts . For 36 times 1728 inches is 62208 inches , then 692 inches in the 40 part over being added makes 62900 the cube inches in that Cilander . PROB. VI. Having the Circumference given in inches , to find how many inchee long makes one foot , or how many feet and parts long . The extent from the Circumference ( or girt ) about any Cillander given in inches , to 42.54 ( the inches about , when one foot long makes one foot solid ) the same extent laid twice the same way from 12 , reaches to the inches long , to make one foot solid measure . And the same extent laid twice the same way from 1 , gives the feet long to make one foot solid . Examp. Suppose a Tree be 30 inches about , how many inches , or feet and inches long , makes one foot solid measure . The extent from 30 inches the girt , to 42.54 , laid twice the same way from 12 , gives 24 inches 1 / 10 , the inches long to make one foot . Also the same extent laid twice the same way from 1 , gives 2 foot 01 or 1 / 100 part , the length in feet and parts to make 1 foot . For 71.7 the area of a Circle of 30 inches about , being multiplied by 24. ● , produceth 1728 , the cube inches in a foot solid . PROB. VII . The Girt of any Cillander given in inches , to find how many inches , or feet and inches is in one foot long . The extent from 42.54 to the girt in inches , laid twice the same way from 12 , gives the number of inches in one foot long . Or the same extent laid twice from 1 the same way , gives the feet and parts in a foot long . Examp. At 40 inches about . The extent from 40 the inches about to 42.54 , being laid twice the same way from 12 , gives 13 inches and 63 parts , the content of one foot long . Or the same extent laid twice the same way from 1 , gives one foot , and 136 parts of 1000 , the quantity in feet and parts in one foot long ; which 136 parts is 1 inch , 6 tenths and a half , as is seen presently by the inches and foot measure laid together on your two foot Rule . Then this 13 inches 63 parts , multiplied by the length of the Tree , gives the true content . PROB. VIII . The Circumference or Girt given in inches , and the length in feet , to find the content of any Cillender in feet and parts . The extent from 42.54 to the inches about , being laid twice the same way from the length in feet , gives the content in solid feet required . Examp. At 36 inches about , and 30 foot long . The extent from 42.54 to 36 the inches girt , being laid twice the same way from 30 the feet long , gives 21 foot and a half , the near content in solid feet . For the Area of a Circle 36 inches about , is 103.1 inches , which multiplied by 30 the length of the Cillender in feet , gives 3093 which summ divided by 144 , produceth 21 foot 69 cube inches , the near content . PROB. IX , Having the Girt and length given in foot measure , to find the content in feet . The extent from 3.545 ( the feet and 100 parts about , when one foot long makes one foot of Timber ) to the Girt in feet and parts . The same extent laid twice from the length in feet , gives the content in feet . Examp. A Brewers Tun of 20 foot about , and 4 foot and ½ deep , how many solid feet is it . The extent from 3.545 to 20 , shall reach the same way at two turnings from 4 foot and ½ to 143 foot and 10 parts , the solid content in feet : and 6 foot being a full Beer-barrel , it contains 24 barrels , Beer-measure . For 31 foot 8 parts , the area of a Circle of 20 foot about , being multiplied by 4 foot 5 parts the depth , gives 143 foot and 10 pts . as before . PROB. X. Having the Girt in inches , and the length of a Cillander given in inches , to find the solid content in cube inches . The extent from 3.545 to the Girt in inches , being twice repeated the same way , from the length in inches , gives the content in inches . Examp. At 48 inches about , and 24 inches in length , how many cube inches is it . The extent from 3.545 to 48 the inches girt , being twice repeated from 24 the length in inches , gives 4398 the content in solid inches . For 183.2 the area of a Circle of 48 inches about , multiplied by 24 the inches deep , produceth 4397 the near content as before . To insure you the number of places , the Print 11 times repeated , doth certainly direct you . PROB. XI . It being an ordinary way in measuring of round Timber , such as Oak , Elme , Beech , Pear-Tree , and the like , ( which is sometimes very rugged , uneven , and knotty ) to take a line and girt about the middle of it , and then to take the fourth part of that for the side of a Square-equal to that Circumference : but this measure is not exact , but more than it should be . But either because of allowance for the faults abovesaid , or for Ignorance , the custome is still used , and Men commonly think themselves wrong'd , if they have not such measure . Therefore I have fitted you with a Proportion for it , both for Diameter and Circumference . And first for Diameter . The Diameter given in inches , and the length in feet , to find the content . As 1.526 to the Diameter , so is the length to a fourth , and that fourth to the content in feet , according to the rate abovesaid . The extent from 1.526 to 9.53 being twice repeated from 8 , shall reach to 3.12 , the content . PROB. XII . Having the Circumference in inches to find the content in the abovesaid measure . As 48 to the inches about , so is the length to a fourth number , and that fourth to the content . The extent from 48 to 30 being twice repeated from 8 , shall fall upon 3. 12 , the content required . PROB. XIII . How to measure Taper Timber , that is , bigger at one end than at the other . The usual way for doing of this is to take the Circumference of the middle or mean bigness ; but a more exact way , is to find the content of the base of both ends , and add them together ; and then to take the half for the mean , which multiplied by the length , shall give you the true content . Examp. A round Pillar is to be measured , whose Diameter at one end is 20 inches , at the other end it is 32 inches Diameter , and in length 16 foot ( or 192 inches ) the content of the little end is 314. 286 , the area or content of the greater end is 773.142 , which put together , make 1087.428 , whose half 543.714 multiplied by 192 the length , gives 104393.143 Cubical inches , which reduced into feet , is 60 feet and 713 cubical inches for the solid content of the Pillar . PROB. XIV . To measure a Cone , such as is a Spire of a Steeple , or the like , by having the height and Diameter of Base . Examp. Let a Cone be to be measured , whose Base is 10 foot , and the height thereof 12 foot , the content of the Base will be found by the 14th . Problem of superficial measure , to be 78. 54 ; then this 78. 54 multiplied by 4 a third part of 12 , the perpendicular or height of the Cone will give 314. 4 , for the content of the Cone required . By the numbers work thus ; the extent from 1 to 4 will reach from 78. 54 to 314. 4. But because there may be some trouble in getting the true perpendicular of a Cone , which is its height , take this Rule : First , take half the Diameter , and multiply it in its self , which here is 25 , then measure the side of the Cone 13 , and multiply that by it self which here is 169 , from which take the Square of half the Base , which is 25 , your first number found , and the remain is 144 , the Square-root of which is the height of the Cone , or length of the perpendicular . PROB. XV. To measure a Globe or Sphere Arithmetically . Cube the Diameter , then multiply that by 11 , and divide by 21 , gives you the true solid content ; let a Sphere be to be measured whose Axis or Diameter is 14 , that multiplied by it self , gives 196 , and 196 again by 14 gives 2744 , this multiplied by 11 gives 30184 , and this last divided by 21 gives 1437. 67 , for the content of the Sphere whose Diameter is 14. But more briefly , by the Numbers thus , the extent from 1 to the Axis , being twice repeated from 3. 142 , will reach to the superficial content , that is , the superficies round about . But if the same extent from 1 to the Axis be thrice repeated from 5238 , it will reach to the solid content ; as 1 to 14 , so 3. 142 to 617 being twice repeated , as 1 to 14 , so 5278 to 1437 being thrice repeated . As for many sided figures , if they have length , you have sufficient for them in the Chapter of superficial measure , to find the base , and then the base multiplied by the length , giveth the content . But as for figures of roundish form , they coming very seldom in use , I shall not in this place trouble you with them , for they may be reduced to Spheres or Cones , or Trirngles , or Cubes , and then measured by those Problems accordingly . And so much for the mensuration of Solids . CHAP. VII . The Use of the Line of Numbers in Questions that eoncern Military Orders . PROB. I. Any number of Souldiers being propounded , to order them into a square Battle of Men. Find by the 12th . Problem of the second Chapter , the square-root of the number given ; for so much as that root shall be , so many Souldiers ought you to place in Rank , and so many likewise in File , to make a square Battle of Men. Examp. Let it be required to order 625 Souldiers into a square Battle of men ; the square-root of that number is 25 ; wherefore you are to place 25 in rank , and as many in file ; for Fractions in this practice are not considerable . For had there been but 3 less , there would have been but 24 in rank and file . PROB. II. Any number of Souldiers being propounded , to order them into a double battle of Men : that is , which may have twice as many in rank as file . Find out the square-root of half the number given , for that root is the number of men to be placed in file , and twice as many to be placed in rank , to make up a double Battle of Men. Examp. Let 1368 Souldiers be propounded to be put in that order : I find by the 12 aforesaid , that 26 , &c. is the square-root of 684 ( half the number propounded ) and therefore conclude , that 52 ought to be in rank , and 26 in file , to order so many Souldiers into a double Battle of men . PROB. III. Any number of Souldiers being propounded , to order them into a quadruple battle of men ; that is , four times so many in Rank as File . Here the Square-root of the fourth part of the number propounded will shew the number to be placed in File , and four times so many are to be placed in Rank . So 2048 being divided by 4 , the quotient is 512 , whose root is 22 ( 6 ) and so many are to be placed in file and 88 in Rank , being four times 22 , &c. PROB. IV. Any number of Souldiers being given , together with their distances in Rank and File , to order them into a Square battle of ground . Extend the Compasses from the distance in File to the distance in Rank ; this done , that extent applyed the same way from the number of Souldiers propounded , will cause the moveable point to fall upon a fourth number , whose square-root is the number of men to be placed in file ; by which , if you divide the whole number of Souldiers , the quotient will shew the number of men to be placed in Rank . Examp. 2500 men are propounded to be ordered in a Square-battle of ground , in such sort that their distance in File being seven foot , and their distance in Rank 3 foot , the ground whereupon they stand may be a just square : To resolve this question , extend the Compasses upon the Line of Numbers downwards from 7 to 3 ( then because the fourth number to be found , will in all likelihood consist of 4 figures ) if you apply that extent the same way from 2500 , in the second part among the smallest divisions , the moveable point will fall upon the fourth number you look for , whose square-root is the number of men to be placed in file . By which square-root , if you divide the whole number of Souldiers , you have the number of men to be placed in Rank . As 7 to 3 , so 2500 to 1072 , whose biggest square-root is 32 , then as 32 is to 1 , so is 2500 to 78. PROB. V. Any number of Souldiers being propounded , to order them in Rank and File , according to the reason of any two numbers given . This Problem is like the former ▪ for as the proportional number given for the file , is to that given for the Rank , so is the number of Souldiers to a fourth number ▪ whose square-root is the number of men to be placed in Rank , by which , if you divide the whole , you may have the number to be placed in File . Examp. So if 2500 Souldiers were to be martialled in sueh order , that the number of men to be placed in File , might bear such proportion to the number of men to be placed in Rank , as 5 bears to 12 , I say then , as 5 is to 12 , so is 2500 to 6000 , whose square-root is 77 the number in Rank ; then as 77 is to ● , so is 2500 to 32 ▪ &c. The number of men to be placed in File . CHAP. IX . The use of the Line in questions of Interest and Annuities . PROB. I. A summ of Money put out to Use , and the Interest forborn for a certain time , to know what it comes to at the end of that time , counting Interest upon Interest at any rate propounded . Take the distance with your Compasses between 100 , and the increase of 100 l. for one year , ( which you must do very exactly ) and repeat it so many times from the principal as it is forborn years and the point of the Compasses will stay on the Principal with the Interest , and increase according to the rate propounded . Examp. I desire to know how much 125 l. being forborn 6 years , will be increased according to the rate of 6 l. per Cent. reckoning Interest upon Interest , or Compound Interest . Extend the Compasses from 100 to 106 , that extent being 6 times repeated from 125 , shall reach to 177 l. the Principal increased with the Interest at the term of 6 years at the rate propounded . But if it were required for any number of months , then first find what 100 is at one month , then say thus , If 100 gives 10 s. at one month , what shall 125 be at 6 months end ? facit 75 s. And the work is thus : First say , If 100 gives 10 s. at one months end , what shall 125 ? and it makes 12 s. 6 d. then say , If one month require 12 s. 6 d. what shall 6 months require ? facit 75 s. that is three pound fifteen shillings , the thing required to be fonnd . PROB. II. A summ of money being due at any time to come , to know what it is worth in ready money . This question is only the inverse of the other ; for if you take the space between 106 and 100 , and turn it back from the summ proposed as many times as there are years in the question , it shall fall on the summ required . Examp. Take the distance between 106 and 100 , and repeat it 6 times from 177 , and it will at last fall on 125 , the summ sought . PROB. III. A yearly Rent , Pension , or Annuity being forborn for a certain term of years , to find what the Arrears come to at any rate propounded . First , you must find the Principal that shall answer to that Annuity , then find to what summ the Principal would be augmented at the rate and term of years propounded ; then if you substract the Principal out of that summ , the remainder is the Arrears required . Examp. A Rent , or Annuity , or Pension of 10 pound the year , forborn for 15 years , what will the Arrears thereof come to at the rate of 6 per Cent. Compound Interest ? The way to find the Principal that doth answer to 10 l. is thus : If 6 l. hath 100 for his Principal , what shall 10 have ? facit 166 l. 16 s. or 166 l. 8 s. for the extent from 6 to 10 will reach from 100 to 166.8 . which is 166 l. 16 s. Then by the first Problem of this Chapter , 166 l. 16 s. forborn 15 years , will come to 398 l. then substract 166 l. 16 s. out of 398 l. and the remainder , viz. 231 l. 4 s. is the summ of the Arrears required . But note , in working this question , your often turning unless your first extent be most precisely exact ▪ you may commit a gross errour , to avoid which , divide your number of turns into 2 , 3 , or 4 parts , and when you have turned over one part , as here 5 , for 3 times 5 is 15 , open the Compasses from thence to the Principal , and then turn the other two turns , viz. 10 — 15 , and this may avoid much errour , or at the least , much mitigate it ; for in these questions , the larger the Line is , the better . PROB. IV. A yearly Rent or Annuity being propounded , to find the worth in ready Money . First , find by the last what the Arrears come to at the term propounded , and then what those Arrears are worth in ready Money , and that shall be the value of it in ready money . Examp. What may a Lease of 10 l. per annum , having 15 years to come , be worth in ready money ? I find by the last Problem , that the Arrears of 10l . per ann . forborn 15 years , is worth 231 l. 4 s. And likewise I find by the second Problem that 231 l. 4 s. is worth in ready money 96 l. 16 s. and so much may a man give for a Lease of 10 l. per ann . for 15 years to come , at the rate of 6 l. per cent . But if it were not to begin presently , but to stay a certain term longer , then you must add that time to the time of forbearance ; as suppose that after 5 years it were to begin , then you must say , 231 l. 4 s. forborn 20 years is worth in ready money , and it is 72 l. 8 s. and that shall be the value of the Lease required . PROB. V. A sum of money being propounded , to find what Annuity to continue any number of years , at any rate propounded , that summ of Money will purchase . Take any known Annuity , and find the value of it in ready money ; this being done , the proportion will be thus ; as the value found out is to the Annuity taken , so is the summ propounded to the Annuity required . Examp. What Annuity to continue 15 years , will 800 l. purchase after the rate of 6 l. per Cent. Here first I take 10l . per ann . for 15 years and find it to be worth in ready money 96 l. 16 s. by the last Problem ; then I say , as 96 l. 8 s. is to 10 , so is 800 to 82-7 , which is 82l . 14 s. and so much near do I conclude will an Annuity of 82 l. 14 s. per ann . be worth for 15 years , after the rate of 6 l. per Cent. viz. 800 l. Also in this Impression is added the use of the Line of Pence , which is added to the Line of Numbers next to it , when it shall be desired by any one , being very convenient for casting up small summs of Money in any concern whatsoever . The Line of Numbers and Pence together , do give the Decimal Fraction of any summ under 20 s. very near , as by the Print to 11 Radixes is most plainly seen . Where 20 s. or 1 l. in Money , is at 1 in the fourth line , and 2 s. one tenth of a pound , at 1 right over it in the third Line ; 9 farthings and 6 tenths of a farthing , being one tenth of two shillings , at 001 in the second line right over 01. Lastly , 96 / 100 parts of a farthing at 0001 in the first Line of Numbers towards the right hand , or the 96 / 1000 parts , at the 00001 at the left end of the same Line . Therefore note , if 20 s. is 1 , 15 s. is 075 , 12 s. is 06 , 10 s. is 05 , 5 s. is at 025 , and 2 s. or 24 d. at 01 , which on Two-foot Rules is set at ▪ 10 at the end of the Rule , though in this case called but one tenth , when 1 is one pound . Then if 1 or 10 at the end , be 24 pence , 18 pence is at 75 , 12 d. at 5 , 6 d. at 25 , 2 d. at 8.33 in the first part , and 1 d. at 4166 , and 1 farthing at 1042 , a little beyond the first 1 on Two-foot Rules . From hence you may see that the Print of the eleven Lines of Numbers sheweth the right Decimal Fraction of any summ under 1 pound Sterling , & by consequence , for any summ above , and was purposely made to explain this on Two-foot Rules . The Use of the Line of Numbers and Pence laid together on Two-foot Rules , may be in a brief manner , thus ; 1. At any price 100 , or 5 score , what cost 1 , counting 10 at the end of the Line , 10 l. Examp. At 5 l. per 100 , counted at 5 in the second part , because 10 is called 10 l. just over or under it in the line of pence , is 12 d. the exact answer , for 100 s. is 5 l. 2. Examp. At 2 l. 10 s. per 100 , right against 2.5 on Numbers representing 2 l. 10 s. right against it on the Line of Pence is 6 d. the Answer , for 100 6 d. is 2 l. 10 s. Now observe , if 1 in the middle is called 1 l. then in the first part , 9 is 18 s. 9.5 is 19 s. 5 towards the beginning-end is 10 s. and 1 at the very beginning end is 2 s. Then to supply all under 2 s. to 1 farthing , begin again at the upper end at 10 , where is set 24 d. the same with 2 s. and back again , 23. 22. 21. 20. 19. 18 pence at 75 , then 17. 16. 15. 14. 13. 12 d. and 5 , then 11 10. 9. 8. 7. 6 d. at 25 ; then 5. 4. 3. 2 d. at 0.834 , then 1 penny at 0.4166 , then 3 farthings , 2 farthings , 1 farthing at 0.1042 , and every 10th . of a farthing in pricks between the farthings . 3. Examp. At 1 l. per 100 , counted at the middle 1 , is 2 d. 1 far . 6 tenths of a far . for 1. 4. Examp. At 10 s. per 100 , counted at 5 in the first part , just against it in the Line of pence is 1 penny ( or 4 farthings ) & 8 tenths of a farthing , the true answer . 5. Examp. At 2 s. per 100 , counted at 1 at the beginning of the Line of Numbers is no farthings , but 96 parts of a farthing in 100 pts . the answer for the half of 96 is 48 , the farthings in 2 s. 6. Examp. But for any price under 2 s. per 100 , count thus , as 12 d. per 100 , just against 12 d. on the Line of pence is 5 on numbers , for 5 tenths of a farthing the near answer ▪ for 5 tenths of a farthing is half a farthing , and 100 half farthings is 50 farthings , being two farthings above 48 , the farthings in one shilling . 7. Examp. At 6 d. per 100 , right against 6 in the Line of pence is 025 , or one quarter of a tenth of a farthing , on the Line of Numbers , the answer near the truth . For a hundred quarters of a farthing is 50 half farthings , or 25 farthings , one more than 6 d. But if you count 1 farthing less for every 6 d. you shall have it right . Example at 18 d. per 100 , count 17 ▪ d. and 1 far . 3 far . less than 18 d. and just against it on Numbers is 12 / 100 pts . of 1 farthing , the true price of 1. Again , at 12 d. counted at 11 d. 2 far . just against it on numbers is 48 / 100 pts . of a farthing , the price of 1. Again , for 6 d. counted at 5 d. 3 far . just against it is 24 / 100 pts . of a farthing , the true answer or price of 1. I have been large in this , that you might see the reason of it the plainer . Again , on the contrary upwards , at any price for 1 ▪ what cost 100 , or 5 score . At 48 / 100 pts . of a farthing for 1 , the price of 100 , is just 12 d. the contrary to the last example but one . Again , at 72 / 100 pts . of a farthing for one , 100 will cost just 18 d. as in the last but two foregoing . Again , at 69 / 100 pts . of a farthing for 1 , 100 is just 2 s. Note , the 96 is just against 23 d. being 4 far . less than 2 s. that is , 4 farthings abated for the four 6 pence's in 2 s. But for all summs between 2 s. at 1 at the beginning-end , and 10 l. at the end of the Line . This in 3 Examples . 1 Ex. At 4 s. per 100 counted at 2 in the first part of the Line of Numbers , just against it on the Line of Pence , is 1 farthing and 92 / 100 more for 1. 2 Ex. At 13 s. per 100 , counted at 6.5 on the Line of Numbers , just against it on the Line of Pence , is 6 farthings , 2 tenths and ● , the answer . 3 Ex. At 3 l. per 100 , counted at 3 in the second part , is 7 d. o far . ● of a farthing , the answer . But if your 100 be 5 score and 12 , then count thus with a pair of Compasses . The extent from 112 to 3 l. counted as before , laid the same way from the middle 1 , gives 6 d. 1 far . 6 / 10 of a farthing more for 1 l. Again , on the contrary , If 1 cost 7 d. farthing , and half a farthing , what cost 112 ? The extent from the middle 1 to 7 d. 1 far . and half a far . ( half way beyond the prick ) being laid the same way from 112 , will reach to 3.45 , or 3 l. 9 s. the answer . For if every figure in the second part be 1 l. then every 10th . cut between is 2 s. then 4 and ● and 45 is 9 s. Also , if every figure in the first part is 2 s. then 5 cuts or tenths beyond any figure is 12 d. 2½ is 6d . 1 is 2 d. farthing , half farthing . Just as you see by counting 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10. in the second part on the Numbers , just against it on the pence is the exact reduction to pence and farthings required . Ex. Against 3 is 7 d. near a farthing , against 6 is 14 d. near 2 farthings ; against 7.5 is just 18 d. And so for all other . Again , when the price of 100 costs above 10 l. then count thus . Ex. At 65 l. 15 s. and 6 d. per 100. First , 65 doubled is 13 s. then the 15 s. 6 d. counted on the first part of the Line of Numbers at 7. 75 , and just against it on the Line of pence is 7 far . and 44 pts . the true answer is 65 l. 15 s. 1d . 3 far . and 44.100 pts . of 1 farthing . Again , at 18 s. 7 d. 2 far . for 1 , what is the price of 100 ? Note , 18 s. is at 9 on the first part of the Numbers , which here stands for 90 l. then seek the odd 7 d. 2 far . on the Line of pence , and just against it on the Line of Numbers is 3.125 , the 3 is 3 l. the 1 more is 2 s. and the 25 more is 6 d. in all 93 l. 2 s. 6 d. the price of 100. For 1800 s. is 90 l. 100 times 7 d. is 2 l. 18 s. 4 d. 100 half pence is 4 s. 2 d. in all 93 l. 2 s. 6 d. Once more for a greater summ at 8l . 5 s. 6 d. for 1 , what costs 100. First , to 8 l. add 2 Ciphers , and it is 800 l. then add 2 ciphers to the 5 s. and it is 500 s. or 25 l. then just against 6d . in the line of pence in the line of Numbers is 2 l. 10 s. In all , 827 l. 10 s. the true price of 100 at 8 l. 5 s. and 6 d. for 1. A further use of the Lines of Pence and Numbers in several Questions . If 2 pound and ½ of any Commodity cost 7 d. 3 f. and ¼ of a far . what cost 32 pound ¾ ? The extent from 2.5 on Numbers to 7 d. 3 f. and ¾ on the Line of Pence , being laid the same way from 32 75 , shall reach to 4.27 . Note , the 4 is 8 s. and the 27 more counted on the line of Numbers beyond the middle 1 , gives on the line of Pence near 6 d. 2 f. in all 8 s. 6 d. 2 f. the true answer . If 3 yards and ½ of Ribbon cost 3 d. 3 f. and ½ , what cost 17 yards and ● ? The extent from 3.5 on the Line of Numbers to 3 d. 3 f. and half farthing on the Line of pence , being laid the same way from 17.5 on the Line of Numbers reaches to 19 d. 1 f. and half on the Line of pence , the true answer . 3. If 1 yard of plaistering cost 8 d. half penny , what cost 142 yards and a half . The extent from 1 on the Line of Numbers to 8 d. half penny on the Line of pence , laid the same way from 142½ on Numbers , reaches to 5.047 ; the 5 is 5l . and the 047 more , counted on Numbers , on the pence , is 11 d. farthing , the true answer . 4. If 112 l. cost 2 l. 10 s. 6 d. what cost 21 C. 3 q. and 12 lb ? The extent from 112 to 2.525 , for 2 l. 10 s. 6 d. laid the same way from 2448 , the pounds in 21 C. 3 q. and 12 lb. shall reach to 55 lb 4 s. the near true answer . For if 2.525 the decimal for 2 l. 10 s. 6 d. be multiplied by 2448 , the pounds in 21 C. 3 q. 12 lb , the product is 6181200 , which divided by 112 , the quotient is 55 lb. 189 , which reduced , is near 4 s. or 3 s. 38 farthings , or 9 d. half-penny . 5. If 5 ounces and a half of Nntmegs cost 20 d. 3 far . what shall 1 C. 1 q. 15 lb come to ? The extent from 5½ on Numbers to 20 d. 3 q. on the Line of pence , shall reach from 16 the ounces in 1 lb , to 02518 , the 25 is 5 s. and the 18 more is 1 far . 7 tenths , the price of 1 lb. Then the extent from 1 on Numbers to 2518 , laid the same way from 155 , the pounds in 1 C. 1 q. and 15 lb , shall reach to 39 l. 0 s. 10 d. the near true answer to the Question . CHAP. X. The application of the Line of Numbers to use in Domestick Affairs , as in Coals , Cheese , Butter , and the like . I have added this Chapter , not that I think it absolutely necessary but only because I would have the absolute applicableness of the Rule to any thing behinted at ; for it may be the answer of some , Do you come with a Rule to measure my Commodities which are sold by weight ? Yea , so far as there is proportion , it concerns that and any thing else , the application of which I leave to the industrious Practitioner ; only I here give a hin● ; as much as ●o say , here is a Treasure , if you dig , you may find ; for some may be apt to think , it being a Carpenters-Rule , it is fit but for Carpenters use only : but know , that in all measures which are either lengths , Superficials or Solids , or as some call , Longametry , Planametry , and Solidametry : and in all Liquids by weight or measure : and in all time , either by Years , Months , Weeks , Days , Hours , Minutes , and Seconds , and almost , ( I think , I may say ) in all things number is used , and in many things , proportional numbers ; why then may not this Line put in for a share of use , seeing it is wholly composed of , and fitted for proportional Numbers , & of so easie an attainment for any ordinary capacity , and chiefly intended for them that be ignorant of Arithmetick , & have not time to learn that noble Science , as some have . And first , for more conveniency of Reduction , take these Rules of Reduction . Rules for English Money . Note , that 4 farthings make a penny ; 16 farthings , 8 half pence , or 4 pence , make a groat ; 48 farthings , 24 half pence , or 12 pence , make a shilling : 40 pence , or 10 groats , is 3 shillings 4 pence ; 80 pence , 20 groats , or 6 shilling 8 pence , is a Noble ; 160 pence , 40 groats , or 13 shillings 4 pence , is a mark ; 20 shillings , 4 crowns , 3 nobles , or 2 angels , is a pound sterling . Rules for Troy-weight Note , that 24 grains is a penny-weight ; 20 penny-weights , or 24 carrots is an ounce Troy ; 12 ounces is a pound ; 25 lib. a quarter of a hundred , 50 lib. half a hundred , 75 lib. three quarters of a hundred , and 100 lib. is a hundred weight Troy. Rules for Aver-du-poize weight . Note , that 20 grains make a scruple ; 3 scruples is a dram ; 8 drams is an ounce ; 16 ounces is a pound ; 8 pound a stone ; 28 lib. a quarter of a hundred ; 56 lib. half a C. 84 lib. 3 quarters of a C. and 112 lib. or 14 stone , or 4 quarters of a C. is an hundred weight : 5 C. is a hogshead weight : 19½ C. is a Fother of lead : and 20 C. is a Tun weight . And note , that l. signifies a pound in money , and lib. signifies a pound in weight , either Troy or Aver-du-poize . Rules for concave Dry-measure . Note , that 2 pints is a quart , 2 quarts a pottle , or quarter of a peck ; 8 pints 4 quarts ; 2 pottles is one gallon , or half a peck ; 2 gallons is a peck ; 2 pecks make half a bushel ; 4 pecks or 56 lib. make a bushel ; 2 bushels is a strike ; 2 strikes a coomb or half-quarter ; 2 coombs 4 strikes ; or 8 bushels , make a Quarter , or a Seame ; 10 Quarters , or 80 bushels make a. Last . Rules for Concave Wet-measure . Note , that 2 pints is a quart ; 2 quarts a pottle ; 2 pottles 4 quarts or 8 pints make a Gallon ; 9 Gallons make a Firkin , or half a Kilderkin ; 18 Gallons make 2 Firkins , a Kilderkin , or a Rundlet ; 36 Gallons is 2 Kilderkins , or a Barrel ; 42 Gallons make a Terce ; 63 Gollons or 3● Rundlets make a Hogshead ; 84 Gallons , or 2 Terces make a Tercion or Punchion ; 126 Gallons is 3 Terces , 2 Hogsheads , 1 Pipe , or a But. A Tun is 252 Gallons , 14 Rundlets , 7 Barrels , 6 Terces , 4 Hogsheads , 3 Punchions , 2 Pipes , or Buts . Note , that in sweet Oyl , 236 Gallons make a Tun ; but of Whale-Oyl 252 goes to the Tun. Water Measure . Note , that 5 pecks is a Bushel ; 3 Bushels a Sack , 4● Bushels a Flat , 12 Sacks , 4 Flats , or 36 Bushels , make a Chaldron of Coals . Rules for Long-Measure . Note , that 3 barley-corns make an inch ; 2¼ inches make a Nail ; 4 Nails , or 9 inches make a quarter of a yard ; 12 inches make a Foot ; 3 foot , 4 quar . 16 nails , or 36 inches make a yard ; 45 inches , or 5 quarters of a yard , make an Ell ; 5 foot is a pace ; 6 feet , or 2 yards is a fathom ; 5 yards and a half , or 16 feet and a half , is a Pole , Rod , or Perch ; 160 perches in length , and 1 in breadth ; or 80 perches in length and 2 in breadth ; or 4 in breadth and 40 in length , make an Acre . 220 Yards , or 40 poles is a furlong ; 1760 yards , 320 poles , or 8 furlongs is an English mile ; 3 miles is a League ; 20 Leagues , or 60 miles is a degree in ordinary account , and every mile a minute . Rules for Motion and Time in Astronomy and Navigation . Note , that a minute contains 60 seconds , and 60 minutes is 1 degree ; and 30 degrees is 1 sign ; 2 signs , or 60 degrees is a sextile ⚹ , 3 signs , or 90 degrees is a Quadrant , or quartile □ ; 4 signs , or 120 degrees a trine △ ; 6 signs , or 180 degrees is one opposition ☍ ; or semicircle ; 12 signs or 360 deg . is a conjunction ☌ , and the Suns Annual or Moon 's monthly motion . Note also , every hour of time hath in motion 15 degrees : and a minute of time hath 15 minutes of motion , and one degree of motion is 4 minutes of time . Note further , that every hour of time hath 60 minutes , therefore 45 is 3 quarters , 30 is half , 15 is a quarter of an hour ; 24 hours a day natural , 7 days a week , 365 days and about 6 hours is a year . Hence it follows , that ¼ of a degree in the Heavens is 5 Leagues on the Earth , or 15 minutes of motion above , is one minute of time below , therefore a degree or 60 minutes of motion is 4 minutes of time , as before is said . All these Rules I shall express more largely and in shorter terms , by these following Tables . Equation for Motion .   Signs . Deg. minutes . Seconds . Note , that the twelve Signs is 12 360 21600 1296000 One Sign is 1 30 1800 108000 One Deg. is   1 60 3600 One Min. is     1 60 Equation for Time.   Mon. VVeek . Day . Hour . Minute . One Year 13 52 305 8760 52560 Month hath 1 4 28 671 40320 Week hath   1 7 168 10080 Day natural     1 24 1440 Hour hath       1 60 Minute is         1 Equation for Long-measure .   Mile . Furl . Perch . Yards . Feet . Inches . Leag . 3 24 960 5971¼ 1●4● 190080 Mil. 1 8 340 1760 5280 63360 Furlong   1 40 220 660 7920 Per. Rod. P●l .     1 5½ 16½ 198 Acre contains of squa . Perc.   160 4840 43560     Acre is in leng     4 20 660 7020 Acre is in breadth     4 2● 66 792 1 Rood , or ¼ of an acre is in len .     40       1 Rood , or ¼ of an acre is in bread .     1 5½ 16½ 198 One Fathom is         6 72 One Ell English is         3¼ 45 One yard is         3 36 One foot is         1 12 One inch is 1 inch 3 grains .           1 Equation of Liquid-measure .   Gall. Pottl . Quarts . Pints . Tun of sweet Oyl 236 472 944 1888 Tun of Wine is 252 504 1008 2016 But or Pipe is 126 252 504 1008 Tertian of Wine 84 168 336 672 Hogshead is 63 126 252 504 A Barrel of Beer or 2 Runlets of Wine — is 36 72 144 288 Kilderkin or one Runlet . is 18 36 72 144 Barrel of Ale is 32 64 128 256 Kilderkin of Ale 16 32 64 128 Firkin of Beer is 9 18 36 72 Firkin of Ale is 8 16 32 64 Equation of small Dry-measure , and then of great measure .   Peck . Gal. Pottl . Qu. Pi. Bushel of water-m . is 5 10 20 40 80 Bushel of Land-m . is 4 8 16 32 64 One peck is 1 2 4 8 16 One Gallon is   1 2 4 8 One Pottle is     1 2 4 One Quart is       1 2   Last . weig . Chal. Qu. Bush. Peck . Pints , Last of Dry-m . 1 2 2½ 10 80 320 5120 One weight is   1 1¼ 5 40 160 2560 Chaldron of coals     1 4 36 144 2098 Quarter of wheat is       1 8 32 512 One Bushel is         1 4 62 Equation for Averdupoize-Weight .   Hogsh . C. Stons . Lib. Ounces . Drams . Scruples . Grains . Tun-w . gross is 4 20 280 2240 35840 286720 860160 17203200 One Hogshead is 1 5 70 560 8900 71680 215040 4300800 One C. or hund . is   1 14 112 1792 14336 43008 860160 One half C. is     7 56 876 7168 21504 430080 One quarter of C. is     3½ 28 448 4584 10752 213040 One Stone is     1 8 128 1024 3072 61440 One Lib. pound is       1 16 128 384 7680 One Ounce is         1 8 24 480 One Dram is           1 3 9 One Scruple is             1 3 Equation for Troy-weight .   Lib. Ounce . Dp. Carrots . Grains C. w. ●00 ●200 ●400● 2●800 ●●6000 ½ . C is 50 600 12000 14400 288000 ¼ C. is 29 300 6000 7●00 1440●0 ⅛ C. is 12½ 150 3000 ●600 72000 Pound 1 12 340 88 5760 Ounce   1 20 24 480 One penny weight     1 1● 2● One Carrot Troy is       1 20 One Grain is         1 Equation of Money .   Mark. An. Nob Cro. Sh. Groat . Penc . Far. Pound ●● 1½ 2 3 4 20 60 240 960 Mark is 1 1⅓ 2 2⅔ 13⅓ 40 160 640 An Angel is   1 1½ 2 10 30 120 410 A Noble is     1 1⅓ 6⅔ 20 80 320 A Crown is     1 5 15   60 240 A Shilling is         1 3 12 48 A Groat is           1 4 16 A Penny is             1 4 The use of which ( to come to our intended purpose ) may be thus . There you see how many farthings , pence , groats , shillings , and the like , is in one , or any usual piece of Coyn , also , how many ounces , scruples , in any kind of weight ; and the like for measure , both liquid and dry ; and also in Time : now if you would know how many there shall be in any greater number than one ; then say by the Rule ( or Line of Numbers ) thus , If 48 farthings be one shilling , how many shillings is 144 farthings ? facit 3 s. for the extent from 48 to 1 , will reach from 144 to 3. and the contrary . Again , If a mark and a half be 1 pound , how many pound is 12 marks ? the extent from 1.50 to 1 , shall reach from 12 to 8. for reason must help you not to call it 80 l. Again , If 3 nobles be 1 l. what is 312 nobles . facit 104 l. the extent from 3 to 1 will reach from 312 to 104. Further , If a Chaldron of Coals cost 36 s. what shall half a Chaldron cost ? facit 18. ( but more to the matter ) If 36 bushel cost 30 s. what shall 5 bushels cost ? facit 4. 16. that is by Reduction 4 s. 2 d. near the matter , or penny , or 3 farthings , half farthing , and better : or on the contrary ; If 1 bushel cost 8 d. then what cost 36 ? facit 288 d. which being brought to shillings , is just 24. which you may do thus ; If 12 d. be 1 s. how many shall 288 be ? facit 24. for the extent from 12 to 1 shall reach the same way from 288 to 24 , as before . The like may be applyed to all the rest of the Rules of weight and measure ; of which take in fine , some Examples in short , and their Answers . If 14 Stone be 1 C. what is 91 Stone ? facit 6 C. and a half . If 1 ounce be 8 drams , how many drams in 9 ounces ? facit 72. The extent from 1 to 8 , reacheth from 9 to 72. If 1 bushel of water-measure be 5 pecks , how many pecks is 16 bushels ? facit 80 pecks . If 1 barrel hold 288 pints , how much will a Firkin hold ? This being the fourth part of a barrel , work thus , if 1 give 288 , what 25 ? facit 72 , the answer sought . If 1 week be 7 days , how many days is 39 weeks ? As 1 is to 7 , so is 39 to 273. So many days in 39 weeks . If 160 perches be 1 acre , how many acres is 395 perches ? facit 2. 492 , that is near 2½ acres . If 8 furlongs make 1 mile , how much is 60 furlongs ? facit 7½ mile . For the extent from 1 to 8 , gives from 60 to 7. 50. CHAP. XI . To measure any superficies or solid by inches only ( or by foot-measure ) without the help of the line , by multiplication of the two sides . PROB. I. Possibly that this little Book may meet with some that are well skilled in Arithmetick , and being much used to that way , are loath to be weaned from that way , being so artificial and exact , yet tho they can multiply and divide very well , yet perhaps they know not this way to save their division , and yet to take in all the fractions together , as if of one denomination : I shall begin first with foot measure , being the more easie , and I suppose my Two-foot Rule to be divided into 200 parts , and figured with 10 , 20 , 30 , 40 , 50 , 60 , 70 , 80 , 90 , 100. and then so again to 200 , as in the 3d. Chap. and then the work is only thus : set down the measure of one side of the square or oblong , thus , as for example , 7. 25 and 9. 88 , and multiply them as if they were whole numbers , and from the product cut off 4 figures , and you have the content in feet , and 1000 parts of a foot , or yard , Ell , Perch , or whatsoever else it be . Note the examples following . For any kind of flat superficies , this is sufficient instruction to him that hath read the first part ; but if it be Timber or Stone , you must thus find the base , and then another work will give you the other side , as in Chap. 5. Prob. 2. or multiply the length by the product of the breadth and thickness , and that product shall be the content required . PROB. II. To multiply feet , inches , and 8 parts of an inch together without Reduction , and so to measure superficial ( and solid ) measure . First , multiply all the whole feet , then all the feet and inches across , and right on , then the parts by the feet , and also the inches and parts across and right on ; then add them together , and you shall have the answer in feet , long inches ( that is in pieces of a foot long , and inch broad ) square inches , and 8 parts of a square-inch : as for Example . Let a piece of board be given to be measured that is 3. 3. 5. i. e. 3 foot , 3 inches , and 5 eights one way , and 2. 3. 4. the other way , I set the numbers down in this manner , and then right on , first as the line in the Scheme from 2 to 3 leads . I say thus , 3 times 2 is 6 , set 6 right under 2 and 3 , as in the example in the left page : for 6 foot , as is clear , if you consider the Scheme over the example , viz. the squares noted with f. then for the next , I say cross-wise , 2 times 3 is 6 , viz. long inches , as you may percieve by the 2 long squares marked with 9 L. and 6 L. which 6 I put in the next place to the right hand , as in the example ; then for the next , viz. 3 times 3 is 9 ( cross wise , as the stroke from 3 to ●● shews ) which 9 is also 9 long inches , as the Scheme sheweth , and must be put under 6 in the second place towards the right hand , in the Scheme it is expressed by the 3 long squares marked with L9 . Then lastly for the inches , 3 times 3 is 9 , going right up , as the stroke from 2 threes lead you : but note , this 9 must be set in the next place to the right hand , because they are but 9 square inches , but had the product been above 12 , you must have substracted the 12 out , and set them in the long inches place , and the remainder , where this 9 now standeth , and this 9 is expressed in the Scheme , by the little square in the corner mark't with ( □ 9. ) Then now for the Fractions , or 8 parts of an inch , first say , cross-wise , as the longest prick line doth lead you to , 3 times 4 is 12 , for which 12 you must set down 1.6 , that is 1 long inch , and 6 square inches ; the reason is , a piece 8 half quarters of an inch broad , and 12 inches long , is a long inch , or the twelfth part of a foot superficial ; and if be 12 square inches , then 4 must needs be 6 square inches : therefore instead of 12 I set down 1.6 , as you may see in the example , and in the least long square of the Diagram or Scheme . Then do likewise for the other long square , which is also multiplied across , as 2 times 5 ▪ is 10 , that is , as I said before , 1.3 as the example and Scheme make manifest , considering what I last said , and it is marked by the 2.00 . But if this or the other had come to a greater number , you must have substracted 8 as oft as you could , and set down the remainder in the place of square inches , and the number of 8 in the place of long inches , as here you see . Then for the two shorter long squares next the corner , say cross-wise again , 3 times 5 is 15 , that is 1.7 , because 8 half quarters an inch long , do make 1 square-inch as well as eight half-quarters a foot long made 1 long inch : therefore I set 1 in the place of square inches and 7 in the next place to the right hand , and it is expressed in the diagram by the small long square , and marked with * 1.7 . Then again for the other little long square , say cross-wise , as the shorter prick line leads you , three times 4 is 12 , that is 1.4 ; and do by this as the last : it is noted in the Scheme by 1.4 . Then lastly , for 5 times 4 , as the short prick line sheweth you , is 20 : out of which 20 take the 8 s. and set them down in the last place , and the 4 remaining you may either neglect ( or set it down a place further ) for you cannot see it on the Rule ; therefore I thus advise , if it be under 4 , neglect it quite , but if above , increase the next a figure more , if 4 , then it is a half , and so may be added ; for note , 64 of these parts make but 1 square inch ; of which parts the little square in the right hand lower corner of the Scheme is 20 , for which I fet down 2.4 , that is two half quarters and 4 of 64 , which is the last work , as you may see by the Scheme and example . Now to add them together , say thus , 4 is 4 , which I put furthest to the right hand , as it were useless , because not to be expres●● ; then 472 are 13 , from which take 8 , and for it carry 1 on to the next place , or as many times 1 as you find 8 , and set down the remainder , which here is 5 , then 1 I carried , and 13619 is 21 , from which I take 12 and set down 9 , because 12 square inches is 1 long inch ; then 1 I carried ( or more , had there been more 12 s. ) and 1169 is ▪ 18 , from which take 12 , as before , there remains 6 , that is 6 long inches ▪ and so had there been more 12. so many you must carry to the next place , because 12 long inches is one foot , lastly , 1 I carried and 6 is 7 foot , so that the work stands thus , F. Lo. In. Sq. In. 8. 64 7 — — 6 — 9 — — 5 — 4. And so for any other measure Superficial or Solid . To multiply feet , inches , and 12 parts of an inch , by feet , inches , and 12 parts of an inch . Although the usual way of dividing the line of inches , on ordinary Two-foot Rules , is into 8 parts , according to which counting the former R●●● is worded , into feet , inches , and 8 parts of an inch . Yet if every inch be divided into 12 parts , o● conceived so to be , which you may easily count by calling every quarter of an inch 3 , every half inch 6 , every three quarters 9. Then the parts between 3 , 6 , and 91 may be easily estimated by help of the halfquarter cut . Then I say , the Arithmetical multiplication will be much easier , being brought to one denomination after the way by decimals , and somewhat more exact , as will appear by this following example . Suppose a Cedar ▪ board be 2 foot 3 inches , and 7 twelve parts of an inch broad , and 9 foot 5 inches and 8 twelve parts of an inch long , how many superficial feet is it ? First , set down the numbers as is usually done in their three denominations of feet , inches , and 12 parts , as in the example is made apparent by the black lines and the pricked lines down right and sloping from figure to figure , done so only for ease and plainness , in wording of it to a Learner . Then proceed in the multiplication thus ; 1. First , 2 times 9 is 18 , as the down right black line from 2 to 9 sheweth , for which set down 18 foot under the 2 and 9 , as in the Example . 2. Secondly , say 9 times 3 is 27 long inches , as the sloping black line from 9 to 3 sheweth , for which 27 you must set down 2 foot in the place of feet , and 3 in the place of long inches , because 27 long inches is 2 foot 3 inches . 3. Thirdly , Say 2 times 5 is 10 long inches , as the sloping black line from 2 to 5 sheweth , for which you set down only 10 long inches under 5 and 3 as in the the example , because it is under 12 long inches , which make 1 foot , as in the second just before . 4. Fourthly , Say 3 times 5 is 15 square inches , as the short black line from 3 to 5 sheweth ; every 12 whereof makes 1 long inch . Therefore I set down 1 long inch in the place of long inches , and the 3 square inches over in the place of square inches right under 7 and 8. 5. Fifthly , Say 9 times 7 is 63 square inches , as the long pricked line from 9 to 7 sheweth , every 12 whereof makes 1 long inch , and every 144 one foot ; therefore I set down 5 long inches in that place and 3 in the place of square inches . 6. Sixthly , Say 2 times 8 is 16 square inches , as the other long sloping pricked line from 2 to 8 sheweth , for which set down one long inch and 4 square inches , each in their proper places , as in the fifth last mentioned . 7. Seventhly , Say 5 times 7 is 35 , 144 or 12 parts of 1 square inch , as the shorter sloping pricked line from 5 to 7 sheweth , every 12 whereof is 1 square inch ; therefore set down 2 in the place of square inches under 7 and 8 and 11 , the parts over in a space beyond , under 144. 8. Eighthly , Say 3 times 8 is 24 , 144 or 12 parts of one square inch , as before , as the other short sloping pricked line sheweth , every 12 whereof is one square inch ; therefore I set down 2 in the place of square inches under 8 and 7 , and no more , because nothing is over 2 halves . Lastly , Say 7 times 8 is 56 , 1728s . as the short down right pricked line from 7 to 8 sheweth , every 12 whereof makes one 144 , or the 12 part of 1 square inch ; therefore I set down 4 in the place of 144 & the 8 parts over in the place of 1728 s. being a place further , as in the example you see done . Feet Lon. inc . Sq. inc . 1●4 . 1728. 18 3 3 11 8 2 10 3 4     1 4       5 2       1 2     21 9 3 3 8 1. Then add them together , saying , 8 under 1728 is 8 and no more . 2. Then 11 and 4 is 15 , set down ● under 144 , and carry 1 to the next place . 3. One I carried and 2. 2. 4. 3 , 3 , under the square inches , or 12 is 15 , for which I set down 3 and carry 1 to the next place . 4. One I carried and 1. 5. 1. 10. 3 is 21 , for which I set down 9 under the long inches , and carry 1 to the next place . 5. One I carried and 2 and 8 is 11 , for which set down 1 and carry 1 to the next place . 6. One I carried and 1 is 2 , in all 21 foot 9 long inches , or 12 parts of a foot ; 3 square inches , or 12 parts of a long inch ; 3 144 or 12 parts of a square inch , and 8. 1728 parts of 1 long inch . If it be a piece of Timber o● Stone ; then having thus gotten the Area of the Base , then multiply that Area by the length in feet , inches , and 12 parts , and the product shall be the solid content required . As in this Example , 1. 2. 4 thick ▪ 2. 1. 6 broad , and 9 foot 7 inches 6 twelves long . The Use of the LINE of NUMBERS ON A SLIDING ( or GLASIERS ) RULE , In Arithmatick & Geometry . AS ALSO A most excellent contrivance of the Line of Numbers , for the measuring of Timber , either Round or Square , being the most easie , speedy , and exact as ever was used . WHEREBY At one setting to the length , all ordinary pieces of Timber , from 1 inch to 100 Foot , is with a glance of the eye resolved , without Pen or Compasses . First drawn by Mr. White , and since much inlarged , and made easie and useful by John Brown. London , Printed in the Year , 1688. The Use of the Line of Numbers on a Sliding-Rule , for the measuring of Superficial or Solid Measures . CHAP. I. A Sliding-Rule is only two Rules , or Rule-pieces fitted together , with a brass socked at each end , that they slip not out of the grove ; and the Line of Numbers thereon , is cut across the moving Joynt on each piece , the same divisions on both sides , only the placing of the Lines differ ; for on one side of the Rule you have 1 set at the beginning , and 10 at the end on each piece ; but on the other side , 1 is set in the middle , and the rest of the figures answerably both ways , on purpose to make it large , and to take in all numbers ; and the reading of this is the very same with the other ; for if you pull out the Rule , and set 10 at the end , right against 1 at the beginning , then on both pieces you have the former Line of Numbers complealy ; therefore I shall say nothing as to description , or reading of it , but come streight to the use . On the edges of the Rule is usually set Foot-measure , being the Foot or 12 inches parted into 100 parts ; and on the flat sides next to the foot-measure , inches in 8 parts , and on the other flat edges on the other side , the Line of Board-measure , and sometimes Timber-measure , whose use is shewed in the first Chapter of the Book ; but note , if the Rule be a just foot when it is shut , as Glasiers commonly is , then the inches are set alike on both sides , and the foot-measure alike on both edges ; and being pulled out as far as the brasses will suffer , it wants about one inch of 2 foot just when pulled out , as it is made for Carpenters use , then the inches on one side , and Foot-measure on the same reciprocal edge , must be figured otherwise , as 13. 14. 15. 16. 17. &c. to 25 inches , and the Foot-measure with 110 , near the end 120 , 130 , 140 , 150 , &c. with 210 at the very end , shewing the measure from end to ▪ end , being drawn out to any distance ; as is very easie to conceive of , and need no example to illustrate it withal . Note also in using the Line of Numbers , that that side or part of it , on which you find the first part or term in the question , shall always call the first side , and then the other must needs be the second , that the Rules and Examples may be shortned and made easie . 1. Multiplication by the Sliding-rule . Set the 1 on any side ( which being found , I call the first side ) to the Multiplicator on the other ( or second ) side ; then seek the Multiplicand on the first side where 1 was ▪ and right against it on the second , is the Product required . Examp. If I would multiply 25 by 28 , set 1 on the first side to 25 on the second , then just against 28 on the first side on the second is 700 : for the right naming the last figure , and the true number of figures , you have a Rule in the 2d . Chapter , and 2d . Problem of the Carpenters-Rule , as in pag. 28. 2. Division by the Rule . Set the divisor found always on the first side , to 1 on the second side then right against the Dividend found out on the first side , on the second is the Quotient required . Examp. If I divide 156 by 12 , the Quotient is 13 ; note , to find how many figures shall be in the Quotient , do thus , if the two first figures of the Divisor be greater than the two first figures of the Dividend , then the quotient hath so many places or figures as there is more in the Dividend than in the Divisor ; but if it be less , that is to say , the Dividends two first figures greater than the Divisors , then the quotient shall have one place or figure more : then the Dividend exceeds the Divisor . Examp. 2964 divided by 39 , makes a quotient 76 , of 2 figures , but if you divide the same number by 18 , you shall have the figures in the quotient , viz. 164 , and 12 remaining , or by the Rule , 2 third parts of one more , for the reason above said , the two first figures of the dividend being greater than the Divisor , it must have one place more than the difference of the number of figures in the Multiplicator and Multiplicand . 3. The Rule of 3 Direct . Set the first term of the question sought out on the first side , to the second term of the question on the second ( or other ) side : then right against the 3d. term found out on the 1st . side , on the 2d . side is the 4th . proportional term required . Examp. If 2 yards of Cl●th cost 8 s. what cost 11 yards● ? The answer is 46 s. for if you s●t 2 on any one side to 8 on the other , th●n look for 11½ on the first side where 2 was , and right against it on the second you shall find 46 , the number required . Note , that all your fractions on the Line of Numbers are decimal fractions , and to work them , you must reduce your proper fractions to them , which for ordinary fractions , you may do it by inches and foot-measure ; but this general Rule by the Numbers will reduce any kind whatsoever , as thus ; suppose I would have the decimal fraction of 9 foot , 7 inch . ¾ , first note that 9 are integers , for the rest , say thus , as 48 the number of quarters in ( 12 inches or ) 1 foot is 1000 , so is 31 the number of quarters in 7 inches 3 quarters to 645 the decimal fraction required , for 9645 is equal to 9 foot 7 inches ¾ ; & so for any other whatsoever . 4. To work the Rule of 3 Reverse . Set the first term sought out on the first side , to the second , being of the same denomination on the 2d . line or side , then seek the third term on the 2d . side , and on the 1st . you shall have the answer requir'd . Ex. If 48 men perform a piece of work in 24 hours , how many men may there be to do the like in 4 hours ? Set 24 on the first side , to 4 on the second , then right against 48 found out on the second , on the first is 288 , the number of men required ▪ 5. To work the double Rule of 3 direct . To perform this , you must have two workings ; as thus , for an Examp. If the increase of 3 bushels of Wheat in one year , be 36 bushels , what shall the increase of 8 bushels be for 7 years ? First , set 3 on the first side , to 36 on the second , then against 8 on the first , on the second you find 96 , then set ●on the first side to 96 , then against 7 on the first side on the second you have 612 , the increase in 7 years , the answer required . CHAP. II. To measure Board or Glass by the Sliding ▪ rule , the length and breadth being given . PROB. I. The breadth given , to find how much makes a foot . If the breadth be given in inches then set 12 on the first side , to the inches on the second ; then right against 12 on the second , on the first is the number of inches required . Examp. At 6 inches broad set 12 to 6 , then against 12 on the second on the first you have 24 But if it be given in foot-measure , then instead of 12 use 1 , and do in like manner as before . Examp. At 0 ▪ 50 broad , set 1 to 0.50 , then right against the other 1 is 2.00 the answer required . But to find how much is in a foot long at any breadth , do thus : First , for foot-measure , just as the Rule stands even look for the breadth on one side , and the quantity in a foot is on the other side ▪ but for inches set 1 to 12 , the● right against the inches broad i● the feet and tenths in a foot long . Ex. At 6 inches broad is 50 o● half a foot in a foot long . Again a● 30 inches broad is 2 foot and a ha●● in a foot long . PROB. II. The length and breadth given , ●● find the content . First , the breadth given in inch●● and the length in feet and inches set 12 on the 1st . side to the breadt● on the second , then right again●● the length on the first , on the s●cond is the content required . Ex. At 16 inches broad and ●● foot long : Set 12 to 16 , then right against 20 you have 26 foot 7 10th . look for the 7 10th . on the foot-measure , and right against it on the inches you have 8 inches and ¼ and ½ quarter , the answer desired . But if the breadth be given in foot-measure , then set 1 to the breadth ; then right against the length on the first side , on the second you shall have the content required . Ex. At 1.20 broad , 20.00 foot long , you shall find 24 foot . For if you set 1 to 1.20 , then right against 20 foot , you have on the 2d . 24 as before . PROB. III. The breadth given in feet and inches and the length also in the same parts , to find the content . Set 1 on the first side to the feet and inches brought to a decimal fraction , or as near as you can guess ( for 6 inches is half , 3 inches is 1 quarter , 9 inches is 3 quarters , 4 inches is one third , 8 inches is two thirds , and 1 inch is somewhat less than 1 tenth on Rule ) on the other or second side ; then right against the length found on the first , on the second is the content required . Ex. At 3 foot 3 inches broad , and 9 foot 9 inches long , you shall have 31 foot 8 inches ½ near , the very same is for foot-measure ( only much easier ) because the divisions on the Line of Numbers , and on the Line of foot-measure on the edge , do agree together . This being premised as to the using of it , you may apply all the former precepts and examples to this Rule as well as the other . CHAP. III. To measure Timber by the Sliding-Rule . PROB. I. To measure Timber by this Rule , is nothing else but to work the Double Rule of Three . Examp. At 8 inches square & 20 foot long , I would know the content . Set 12 , if the side of the square be given in inches ( or 1 , if in foot me●sure ) on the first side , to 8 the inches square on the second : then right against 12 on the second side on the first is 18 , the fourth proportional part : then for the second work , set 18 , the fourth proportional last found to 8 the inches square on the second , then right against 20 the length , is 9 , the content required . Or rather thus ; Set 12 against 8 , then right against 20 on the same side 12 was , is 13. 5 near on , then look for 13. 5 fere on the first side , and right against it on the second is 9 foot the content required . PROB. II. To measure a piece that is not square Set 12 if you use the inches ( or 1 if you use foot-measure ) on the first side , to the inches thick on the second ; then right against the inches broad on the first side , on the second is a fourth proportional : then in the second operation , set 2 on the first side to the fourth proportional on the second , then right against the length on the first side , on the second is the content required . Ex. At 8 inches thick and 16 broad , and 20 foot long , you shall find 18 foot fere . PROB. III. The square given , to find how much makes a foot . Set the inches square on the first side , to 12 on the second ; then right against 12 on the first , on the second is a fourth proportional number : then in the second work , as the inches square to the fourth proportional , so is 12 to the number of inches required , to make a foot of Timber . Ex. At 6 inches square , set 6 to 12 , then against the other 12 is 24 : then set 6 to 24 , then right against 12 you shall have 48 , the length in inches required . After the same manner are other questions wrought , but the Compasses are easier and more ready ; therefore I shall say no more to this but only refer you to the former Rules in the third , fourth and fifth Chapters . Only note , that in those Sliding-rules made for Glasiers use , the one half of the Line of Numbers is on one side of the Rule , and the other on the other side ; and whatsoever leg or piece of the rule is the first on the one side , the same leg or piece is the first , when the Rule is turned , on the other side , which must well be observed ; but note , that for measuring of Timber , those that use it may have one side fitted for that , as I shall more plainly and fully shew in the next chapter , being the easiest , speediest and nearest way that ever yet was used by any man , resolving any Contents by having the length and the diameter , the circumference or square given . A Table of the true size of Glasiers Quarries , both long and square , calculated by J. B. Square-Quarries 77. 19 gr . Quarries . Rang. Sides bread . leng . content in feet content in inc 8 in 100 i. 100 I. p i. pts . F. p ●ts Inc. p. 8 4 20 - 4 30 5 36 6 70 0. 1250 1. 50 10 3 76 3 84 4 80 6 00 0. 1000 1. 20 12 3 43 3 51 4 38 5 47 0. 0833 1. 00 15 3 07 3 13 3 92 - 4 90 0. 0667 0. 80 18 2 80 2 86 3 57 4 47 0. 0555 0. 666 20 2 66 2 72 3 39 4 24 0 , 5000 0. 60 Long Quarries 67. 22 Quarries Rang. Sides bread . leng . content content   in pts . I. pts . ● . 100 ● pts . F. 100 ● . pts . 8 4 09 4 41. 4. 90 7. 34 0. 1250 1. 50 10 3 65 3. 95 4. 38 - 6. 57 0. 1000 1. 20 12 3 34 3 61 4. 00 6. 00 0. 0833 1. 00 15 2 98. 3 23 3. 58 5. 37 0. 0667 0. 80 18 2 58 2 79 ▪ 3. 10 4. 90 0. 0555 0. 666 20 2 72 - 2 94 - 3. 27 4. 65 0. 0500 0. 60 Note , that a prick after the 100 parts of an inch , notes a quarter , and a stroke ( - ) a half of 100 parts of an inch ; to make this Table , work thus by the Line of Numbers . Divide the distance between the content of some known size , as square 10s . or long 12s . and the content of the inquired size , into two equal parts , for that distance laid the right way ( increasing for a bigger , or decreasing for a less ) from the sides of the known size , shall give the reciprocal sides of the inquired size . Example for square 12s . The half distance on the Line of Numbers , between 1000 the content of square 10s . and 0.833 the content of square 12s . shall reach from 6 the length of square 10s . to 5.47 the length of square 12s . and from 4. 80 the breadth of square 10s . to 4 38 - the breadth of square 12s . and from 3 84 - to 3. 51 , and from 3 76 to 3 43 ; and so for all the rest . CHAP. IV. The description of the Line of Numbers on a Sliding-rule , to measure Solid measure only , according to Mr. White 's first contrivance , but much augmented by J. B. First , when the figures on the Timber side ) stand right towards you , fit to read , then that half or piece next to your right hand , I call the right side , the other is of necessity the left . Secondly , the figures on the right side are , first at the lower end , ( where the brass is pin'd fast ) either 3 , or 4 , or 5 , it matters not much which , yet to have 3 there is best ; then upwards , 4. 5. 6. 7. 8. 9. 10. 11. for so many inches then 1. 2 3. 4. 5. 6. 7. 8 9. 10. 11. 12. under the brass at the top , for so many feet , the divisions between to 1 foot , are quarters of inches , the next above 1 foot , are only whole inches , as you may plainly see . Thirdly , at 1 foot you have the word square , at 1 foot , 1 inch ½ is a mark , and right against it is set TD , noting the true diameter of a ●ound Cylinder ; at an inch further ●s 12 set , which I call small 12 , being in small figures . Again at 1 foot 3 inches better , is another mark , and right against it the word diameter , for the diameter of a piece of timber according to the usual english allowance . Then again at 3 foot 6 inches ½ near , is TR , for the true circumference of a round cillander . Lastly , at 4 foot is the word round , noting the circumference according to the usual allowance , whose use followeth . Also at 13 foot 7 inches is TD , and at 3 inches and ½ is TR. Note also , if you put on the Gage-points for Ale or Wine , with the mean diameter , and length , you may gage any Wine or Beer-vessel , the Wine at 17 inches 15. the Ale or Beer at 18. 95. Also the Gage-points for a Beer-barrel at 35 inches and 9525 parts ; and the gagepoint for a barrel of Ale at 33 inches and 89645 parts . Fourthly , the figures on the left side are not much unlike the right , for 1 at the beginning is one inch , and so it proceeds by quarters of inches to 1 foot ; then by figures at the feet , and the divisions all whole inches to 10 foot , then every whole foot , and half , and quarter , or 10th . to 100 , or 140 , or 150 foot ; and this I call the left side , the other the right side ; so that from 1 inch at the lower end to one foot , every inch hath a figure ; from 1 foot to 10 foot , every foot hath a figure , and from 10 foot to 100 , every 10th . foot only is figured . I have been very plain in explaining this , because I would avoid vain repetitions in the following uses , wherein you shall have first the most ordinary and easie questions , and then the more hard and critical , and less useful . The Uses follow . PROB. I. A piece of Timber being not square , to make it square , or to find the Square-equal . Set the breadth on the left side , to the breadth on the right , then right against the inch and quarters thick found on the left side , on the right is the inches square required . Examp. At 18 broad and 6 thick you shall find 10 inches ⅜ , the side of the square required . For if you set 6 inches against 6 inches , on the right and left side : then right against 18 inches , or 1 foot 6 inches on the left , on the right you have 10 inches 1 quarter , and half a quarter ; for the side of the square-equal to 18 one way , and 6 the other way . PROB. II. The side of the square given , to find how much makes a foot . For all pieces between 3 or 4 inches and 42 inches square , which are the most useful : this the best way , set the inches or feet , and inches square , found out on the right side , to one foot on the left , then right against 1 foot on the right , on the left is the inches or feet , and inches required to make a foot of Timber . But when the piece is small , count 1 foot on the right for 1 inch , and call 12 on the right for 1 foot . Example . At 8 inches square set 8 on the right , to 1 foot on the left , then right against 1 foot on the right , on the left is 2 foot 3 inches , the length required . To find how much is in a foot long . Just as the Rule stands even look for the inches the piece is square on the right , and on the left is the inches or feet , and inches required . Example . At 17 inches square , there is 2 foot of timber in 1 foot long , which if you multiply by the length , you shall have the true content . A very good way for large pieces , and very exact . PROB. III. The side of the square , and length given , to find the content . For all pieces between 1 inch or ●● part of a foot , and 100 foot , this is the easiest way . Set the word square or 1 foot to the length on the left , then right against the inches or feet and inches square on the right , on the left you have the content . Examp. At 9 inches square and 20 foot long ; set the ( long stroke by the ) word square , to 20 foot on the left , then right against 9 inches on the right side , on the left side you have 11 foot and a quarter , the content required . But if it be a very great piece , as above 100 foot , then call 1 foot on the left side 10 foot , and 2 foot 20 , &c. then 10 shall be 100 , and 100 a 1000 , that will supply to 1500 foot in a piece . But for all small pieces under 3 inches square , and above 1 quarter of an inch , do thus : Set 12 on the top ( or the small 12 when it is most convenient to use ) to the length on the left side , then right against the inches ( or 12s . of 1 inch ) squares found on the right side , on the left is the true content required . Example . At 2 inches ( 3 twelves or ) 1 quarter square , and 10 foot long , you shall find 4 inches and a quarter ferè . But note , when you use the small 12 , the answer is given in decimals of a foot , therefore the top 12 is best . ROB. IV. The square of a small piecè of Timber given , to find how much makes a Foot. For all pieces from 12 inches to 1 inch square , do thus ; set the inches and ( 12s . or ) quarters square , counting 1 foot on the right side for 1 inch , and 2 foot for 2 inches , &c. found out on the right side to 100 on the left ; then right against the upper , or small 12 on the right , on the left is the length required to make a foot of Timber . Examp. At 2 inches ¼ square , you must have 28 foot 4 inches , to make a foot . PROB. V. Under 1 inch square , to find the length of a foot . Set 1 foot 9 inches , 6 inches o● 3 inches , found on the right side , for 1 inch ¼ , ½ , or ¼ of an inch , against 10 on the left side , counted for 100 : then right against the small 12 you have the feet in length required . Examp. At 1 inch square you find 144 feet , at ½ square 256 feet , at ½ an inch square 576 feet , at ¼ or an inch square , you find 2034 feet in length , to make 1 foot of Timber . Or , if you set the former numbers 12 , 9 , 6 , 1 , against 1 inch on the left , then right against the upper 12 is a number , which multiplied by 12 , is the number of feet required . PROB. VI. A great piece above 3 foot ¼ square , to find the length of a foot . Set the feet and inches on th● right , to 100 on the left ; the● right against small 12 is the inche● and 12s . or 12s . of a 12th . tha● goes to make a foot . Examp. At 4 foot square , yo● have 9. 12ths . or ¾ of an inch t● make a foot of Timber ; at 5 foo● square , 5. 12ths . and 10. 12ths . ●● a 12th . to make a foot . Thus you see the Rule as no● contrived , resolves from 1 quarte●● square , to 12 foot square , the content or quantity of a foot of Timber in length at any squarenes● without Pen or Compasses . CHAP. V. For round Timber . PROB. I. The number of inches that a piece ●● Timber is about , being given , find how much makes a foot . First , for all ordinary pieces , s● 1 foot . on the left , to the inches ●● feet , and inches above on the righ● then right against TR for true measure , or round for the usual measure , is the feet , or feet and inches required to make a foot of Timber at that circumference about . Examp. At 4 inches about , 113 foot 2 inches is for true measure , but for the usual measure , 142 foot goes to make a foot of Timber . At 12 foot 3 inches about , 1 inch is a true foot , but for the usual allowance , as the fourth part of a fine girt about gives : it must be 1 inch ¼ long , to make a foot of Timber at that circumference . But for very large pieces , count 1 foot on the right for 12 foot , 2. 24 , &c. and set 1 foot on the left , as before ; then in the answer , 1 foot on the left is 1. 12th . of an inch , and 1 inch 1. 14th . of an inch . Example . At 144 foot about , 1. 144th . part of an inch , is a foot of Timber . PROB. II. For very small wood , to find a foot in length . But for very small pieces of under 4 inches about , set 1 foot , 2 foot , &c. on the right ( counted for 1 inch , 2 inches , 3 inches , or 4 inches ) to 1 foot on the left , then right against TR or round , you have a number , which multiplied by 12 , is the number of feet required . Example . At 1 inch round true measure is 151 foot ferè , but for the usual allowance 196 , which numbers multiplied by 12 , is the number of feet required , viz. 1809 , and 2352. But note , you must read the 196 , and 151 right , as thus ; 1 foot on the left is 12 , 2 is 24 , &c. so that 12 foot is 144 , and our number by the same account is 151 near . To find how much is in a foot in length , set round or TR to 1 foot on the left , then right against the inches , or feet and inches about , found on the right , on the left is the answer required . PROB. III. The inches , or feet and inches about and length given , to find the content . Set the word round , or TR for the usual or true measure , to the length on the left ; then right against the inches about on the right , on the left is the content required . Example . At 2 foot 3 inches about , and 20 foot long , it is 6 foot 2 inches of the usual allowance , or 8 foot of true measure . But if it be a great Tree , then set TR , or round to 1 called 10 , or to 10 called 100 , then is the content augmented to 1000 foot , as you did in the Rules for square Timber . But if you would have it measure bigger still , then set the 4 inches or a TR set close by the brass on the right side , to the length on the left , either as it is , or augmented , counting at last according ; ( the note 1 foot on the right is 12 foot , and 12 at the top is 144 foot ) then right against the feet about on the right , on the left is the content required . Examp. A Brewers Tun 3 foot long or deep , and 72 foot about , set the TR by the brass to 36 inch . ( which is thus counted ) on the left side ; ( 1 inch is 10 inches , 2 is 20 , 3 is 30 , 3½ is 35 , somewhat more is 36 ; so then 1 foot is 120 inches , o● 10 foot ) then right against 6 times 12 foot on the right ( which is at ● foot ) on the left you have 12 30 foot , as near as the Rule will give it , which counting 6 foot to a barrel , is 205 barrels , the content required . PROB. IV. To find the content of a very smal● piece . Set the word round , or TR to the length on the left , as in the third Problem of this Chapter ; the● right against the inches about o● the right , ( calling 1 foot 1 inch and 6 inches ½ an inch ) on the lef● is the 13s . of 1 inch ) or 12s . of a 12th . required . Example . At half an inch about , and 10 foot long , it is 2 12s . and a half of 1 12th . of an inch , or 2 square inches and ½ true measure . Again , 2 inches ¼ about , and 10 foot long , is half an inch of true measure , 12 inches to a foot solid , or ½ a foot superficial of one inch thick . CHAP. VI. To measure Timber , having the Diameter and the length given . PROB. I. The Diameter given in inches , to find the length of a foot . Set 1 foot on the left to the inches diameter on the right : then right against TD for true diameter or the word diameter for the usual allowance ( of a string girt about and doubled 4 times for the side of the square ) you have the feet and inches required . Example . At 10 inches diameter , 1 foot 10 inches makes a foot . But for very great pieces , set 1 foot as before , but look for TD beyond the upper 12 , & right against it on the left you have the 12 of 1 inch , or the 12s . of a 12 , that makes a foot . But for very small sticks , set 1 , 2 , or 3 foot on the right ( for 1 , 2 , or 3 inches ) to 1 foot on the left , then right against TD or Diameter , you have a number , which multiplied by 12 , is the number of feet required to make a foot of Timber . Examp. At 1 inch Diameter you shall have 15 foot 3 inches and better , which multiplied by 12 is 123 foot 3 inches . Note , that 1 on the left is reckoned 10 foot , and 2. 20 foot , as before in the same Rule , for the circumference , and then note 1 inch is 10 inches . At any diameter , to find how much is in 1 foot long , do thus ; set Diam . or TD to 1 foot , then just against the inches , or feet and inches diameter found on the right , on the left is the answer . Example . At 2 foot diameter is 3 foot 2 inches in 1 foot of length , which multiplied by the length , gives the true content of any round piece , and very exactly . PROB. II. The diameter and length given , to find the true content . For all ordinary pieces , set the word diameter for the usual measure , or TD for true measure , always to the length on the left ; then right against the inches or feet and inches diameter on the right , on the left is the content required . Examp. At 5 inches diameter , and 30 foot long , you shall find 4 foot and ½ an inch true measure for the content required . But for very small pieces set TD or Diam . to the length , as before ; then counting 1 foot on the right for 1 inch , and 6 inches for ⅓ an inch ; on the left you shall have the answer or content required . But note , as the right side is diminished , so is the left , for 1 foot on the left is a 12th . of an inch of Timber , whereof 12 makes a foot , or 1 long inch , a foot long , and 1 inch square , and every ineh on the left is 1 square inch ; thus , at 2 foot long , and half an inch diameter , it is 4 □ inches , ¾ in content . But for a great piece under 1000 foot , set TD or diameter to 1 , 2 , or 3 foot , called 10 , 20 , or 30 foot : then right against the feet and inches diameter , you have a content augmented accordingly , as at 30 foot long , and 7 foot diameter , you have 1140 foot , for the true content using TD . Note , that in large Taper-timber , whether square or round , when i● is measured by the usual way ; that is , by the middle square or girt , or the 2 squares or girts put together , and the half counted for the equal square , or girt ; I say , a square of half the difference of the squares or girts , and one part of the length is to be added to the former measure , as is proved in the circles of proportion , pag. 50. As thus for Example . Suppose a Taper piece be at one end 16 inches square , at the other 30 inches square , and 30 foot long , the square in the middle is like to be 23 inches , the content then is 110 foot ; now half the difference of the two ends square is 7 inches , and 1 third part of the length is 10 foot ; a piece 7 inches square , and 10 foot long , is 3 foot 5 inches which added , is 113 foot 5 inches , the true content of that taper piece abovesaid . The general way of Gauging by this Rule , is thus . Set the W. or the A. for Wine or Ale-measure , always to the length of the Vessel found out on the left . Then right against the mean Diameter found out on the right side , on the left is the answer required . Examp. At 30 inches Diameter and 36 long , you shall find about 90 gallons and a half Ale-measure The Gage-point for a Beer-barrel i● near 3 foot , and the Ale-barre● near 34 inches , which use thus . Set 3 to the depth of the Tun then right against the mean Diam is the content in Barrels . Example . Set 3 to 36 inches , and then right against 5 foot Diam . is 10 barrel of Beer-measure , a very good an● spedy way . FINIS . Notes, typically marginal, from the original text Notes for div A29760-e29120 3. 3. 5. 2. 3. 4. 
A64223	----	The semicircle on a sector in two books. Containing the description of a general and portable instrument; whereby most problems (reducible to instrumental practice) in astronomy, trigonometry, arithmetick, geometry, geography, topography, navigation, dyalling, &c. are speedily and exactly resolved. By J. T. Taylor, John, 1666 or 7-1687. 1667 Approx. 163 KB of XML-encoded text transcribed from 77 1-bit group-IV TIFF page images. Text Creation Partnership, Ann Arbor, MI ; Oxford (UK) : 2005-10 (EEBO-TCP Phase 1). A64223 Wing T533B ESTC R221720 99832990 99832990 37465 This keyboarded and encoded edition of the work described above is co-owned by the institutions providing financial support to the Early English Books Online Text Creation Partnership. This Phase I text is available for reuse, according to the terms of Creative Commons 0 1.0 Universal . The text can be copied, modified, distributed and performed, even for commercial purposes, all without asking permission. Early English books online. (EEBO-TCP ; phase 1, no. A64223) Transcribed from: (Early English Books Online ; image set 37465) Images scanned from microfilm: (Early English books, 1641-1700 ; 2155:11) The semicircle on a sector in two books. Containing the description of a general and portable instrument; whereby most problems (reducible to instrumental practice) in astronomy, trigonometry, arithmetick, geometry, geography, topography, navigation, dyalling, &c. are speedily and exactly resolved. By J. T. Taylor, John, 1666 or 7-1687. [8], 144 p. printed for William Tompson, bookseller at Harborough in Leicestershire, London : 1667. J.T. = John Taylor. In two books; Book II has caption title (the first word of which is in Greek characters); register and pagination are continuous. Pages stained with some loss of print. Reproduction of the original in the British Library. Created by converting TCP files to TEI P5 using tcp2tei.xsl, TEI @ Oxford. Re-processed by University of Nebraska-Lincoln and Northwestern, with changes to facilitate morpho-syntactic tagging. Gap elements of known extent have been transformed into placeholder characters or elements to simplify the filling in of gaps by user contributors. EEBO-TCP is a partnership between the Universities of Michigan and Oxford and the publisher ProQuest to create accurately transcribed and encoded texts based on the image sets published by ProQuest via their Early English Books Online (EEBO) database (http://eebo.chadwyck.com). The general aim of EEBO-TCP is to encode one copy (usually the first edition) of every monographic English-language title published between 1473 and 1700 available in EEBO. 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Understanding these processes should make clear that, while the overall quality of TCP data is very good, some errors will remain and some readable characters will be marked as illegible. Users should bear in mind that in all likelihood such instances will never have been looked at by a TCP editor. The texts were encoded and linked to page images in accordance with level 4 of the TEI in Libraries guidelines. Copies of the texts have been issued variously as SGML (TCP schema; ASCII text with mnemonic sdata character entities); displayable XML (TCP schema; characters represented either as UTF-8 Unicode or text strings within braces); or lossless XML (TEI P5, characters represented either as UTF-8 Unicode or TEI g elements). Keying and markup guidelines are available at the Text Creation Partnership web site . eng Mathematics -- Early works to 1800. Navigation -- Early works to 1800. Dialing -- Early works to 1800. 2000-00 TCP Assigned for keying and markup 2001-00 Aptara Keyed and coded from ProQuest page images 2001-08 Sean Norton Sampled and proofread 2005-03 Olivia Bottum Text and markup reviewed and edited 2005-04 pfs Batch review (QC) and XML conversion THE SEMICIRCLE ON A SECTOR : In two Books . CONTAINING The description of a general and portable Instrument ; whereby most Problems ( reducible to Instrumental Practice ) in Astronomy , Trigonometry , Arithmetick , Geometry , Geography , Topography , Navigation , Dyalling , &c. are speedily and exactly resolved . By J. T. LONDON , Printed for William Tompson , Bookseller at Harborough in Leicestershire . 1667. To the Reader . ALL that is intended in this Treatise , is to acquaint thee with an Instrument , that is both portable and general , of no great price , easie carriage , yet of a speedy and Accurate dispatch in the most difficult Problems in Astronomy , &c. The lines for the most part have been formerly published by Mr. Gunter , the famous Mr. Foster , Mr. White , &c. The reduction of the 28. Cases of Spherical Triangles unto II. Problems , I first learned from the reverend Mr. Palmers Catholick Planisphere . Many of the proportions in the Treatise of Dyalling are taken from ( though first compared with the Globe ) my worthy Friend ( to whom I am indebted in all the Obligations of Civility , and without whose encouragement this had never adventured the publick Test ) Mr. John Collins . The applying Mr. Fosters Line of Versed Sines unto the Sector was first published by Mr. John Brown , Mathematical Instrument-Maker , at the Sphere and Sun-Dial in the Minories , London , Anno 1660. who bath very much assisted me , by making , adding unto , and giving me freely the perusal of many Instruments , according to any directions for Improvement , that was proposed to him . After this account , what hath been my part in this Work , I hazard to thy censure ; and when I see others publish a more convenient , speedy , accurate , and general Instrument , I assure them to have as low thoughts of this , as themselves . But here is so large a Catalogue of Errata's as would stagger my confidence at thy pardoning , had they not been irrevocably committed before I received the least notice of them . The Printer writing me word ( after I had corrected so much as came to my sight ) that he could alter no Mistakes until the whole Book was printed : By which means he enforced me to do pennance in his Sheets , for his own Crimes : Did not one gross mistake of his become my purgation , viz. in lib. 2. ( throughout Chap. 3. ) where instead of the note of equality ( marked thus = ) he hath inserted the Algebraick note of Subtraction , or Minoration ( marked thus - ) Nor hath the Engraver come behinde the Composer , who so miserably mangleth Fig. 13. that ( at first sight ) it would endanger branding of a mans Brains to spell the meaning thereof , either in it self , or in reference to the Book . All that I can help thee herein , is this ; Whereas the Book mentions that Figure for an East Dyal , if you account it ( as now cut ) a West Dyal , and alter the names of the hours , by putting Figures for the afternoon , in the place of those there for the morning ; you will then have a true West Dyal of that Figure . The correction of Punctations would be an endless task ; for I finde some to be resolved , ever since Valentine , to recreate themselves at Spurn-point . What other material mistakes are in the Book ( which ought to be corrected before reading thereof ) you will finde mentioned in the Errata . Farewel . March 29. 1667. J. T. Errata . PAge 4. line 12. signs , r. sines . p. 8. l. 5. seconds , r. secants . p. 12. l. 9. all , r. allone . p. 13. l. 9 , and 10 . sec . r. min. p. 14. l. 17. sec. r. min. p. 22. l. 14. any , r. what . p. 23. l. 3. exact , r. erect . p. 24. l. ult . adde lib. 2. p. 25. l. 2. signs , r. sines . l. 5 . sign . r. sine . p. 35. l. 3 . a mark , r. an ark . p. 37. l. 22. 20. r. 22. and 42. r. 20. p. 44. l. 5. At , r. At. p. 49. l. 1 . divided by , r. dividing . p. 62. l. 15. dele , a. In lib. 1. chap. 9. the pages are false numbred . But in chap. 9. p. 62. l. 11 . next , r. exact . l. 15 . gauger , r. gauge . l.24.the , r. what . p. 73.l . 5 . whereas , r. where I. p. 80. l. 13 . wherein , r. whereof . l.22.pont , r. point . p. 91 . l.1 . the co-tangent , r. half the co-tangent . l. 19 . L. r. P. p. 92. l. 21 . PZ . r. PS . p. 93. l. 12 . NSP , r. NSZ . 6.angle , r. ark . throughout page 96. Fig. 3. r. Fig. 7. and Fig. 4. r. Fig. 8. p. 98. l. 13. serve , r. scrue . p. 99 . l.12 . places , r. plates . l. 13. proportion , r. perforation . l.9.serve , r. scrue . l.4.serve , r. scrue . 21.serve , r : scrue . p. 111. l. 20. lay in , r. laying . l.14.of , ●● and. p. 128. l. 3. Fig. 12. r. Fig. 13. l. 22 . ED. r. EC . In Fig. 12. C. r. A. at the end of the line G. To the Right Honorable , The Lord SHERARD , Baron of Letrim . My Lord , SInce the trifling Treatise of an Almanack hath usurped a custom to pinnion some Honourable Name to the Patronage of the Authors Follies ; had we not certain evidence from the uncertainty of their Predictions , that their Brains ( like their great Oracles the Planets ) are often wandring ; it might be deemed a Crime , beyond the benefit of the Clergy , to prefix before any Book , a dedication to a Noble Person : Or when I read the unreasonableness of others in those Addresses , imploring their Patrons to be their Dii Tutelares , and prostrate their reputes to the unmannerly mangling of every Censurist , under the notion of protecting ( that is adopting ) the Authors Ignorance , or negligence ; it s enough to tempt the whole world to turn Democritians , and hazard their spleens in laughing at such mens madness . My present design is only to give your Lordship my observance of your Commands about the Description and Improvement of the Sector ; and wherein I have erred ( through mistake , or defect ) I despair not , but from your Honor I shall meet with a pardon of Course to be granted unto Your Lordships most humble Servant J. T. THE SEMICIRCLE ON A SECTOR . LIB . I. CHAP. I. A Description of the Instrument , with the several Lines inscribed thereon . THe Instrument consists of three Rulers , or Pieces ; two whereof are joyned together by a River that may open and shut to any Angle , in fashion of the Sector ; or to use a courser comparison , after the manner of Compasses . The third piece is loose , or separable from them , to be put into the Tenons at the end of the inward ledge of the joyned pieces , and thereby constituting an aequilateral triangle . On these Rulers ( after this manner put together ) we take notice ( for distinction sake ) of the sides , ledges , ends , and pieces . The sides are thus differenced , one we call the quadrantal , the other the proportional , or sector side . The ledges are distinguished by naming one the inward , the other the outward ledge . The ends are known in terming one the head ( viz. where the two pieces are riveted together ) the other the end . The pieces are discovered by styling one the fixed piece , viz. that which hath the rivet upon it ; the second the movable piece , which turns upon that rivet , and the last the loose piece , to be put into the tenons , as before expressed . The quadrantal side of the joyned pieces is easily discerned , having the names of the Moneths stamped on the movable piece , and Par. Scale on the fixed piece . The quadrantal side of the loose piece is known by the degrees on the inward and outward limb . These directions are sufficient to instruct you how to put the Instrument together . Imagining the Instrument thus put together , the lines upon the quadrantal side are these . First on the fixed piece , next the outward limb , is a line of 12 equal parts ; and each of those parts divided into 30 degrees , marked from the end towards the head with ♈ ♉ ♊ , &c. representing the 12 Signs of the Zodiack ; the use of this line , with the help of those under it , was intended to find the hour of the night by the Moon . The next line to this is a line of twice 12 , or 24 equal parts , each division whereof cuts every 15th degree of the former line : and therefore if the figures were set to every 15th degree on this side the former line , this second line would be useless , and the former perform its office more distinctly . This line was intended an assistant for finding the hour by the Moon ; but is very ready to find the hour by any of the fixed Stars . The third line is a line of 29● equal parts , serving for the dayes of the Moons age ; in order to find the hour of the night by the Moon . But the operation is so tedious , and far from exactness , that I have no kindness for it ; and should place some other lines in the room of this and the former , did I not resolve to impose upon no mans phantasie . The fourth line is a line of altitudes for a particular latitude , noted at the end , Par. Scale , &c. This helpeth to find the hour and azimuth of the Sun , or any fixed Star very exactly . The fifth line is a line of natural sines ; at the beginning whereof there is a pin , or else an hole to put a pin into , whereon to hang the thread and plummet for taking of altitudes . To this line of sines may be joyned a line of tangents to 45 degr . The use of the sines alone , is to work proportions in signs . The use of the sines with the tangent line may be for any proportion in trigonometry ; but that I leave to liberty . The sixth line , and last on this side the fixed piece is a line of versed sines , numbred from the centre at the head to 180 at the end . On the quadrantal side of the movable piece , the first line next the inward edge , is a line of versed sines answering to that on the fixed piece . The use of these versed sines is various at pleasure . The second line from the inward edge , is a line of hours and Azimuths serving to find the hour by the Sun or Stars , or the Azimuth of the Sun , or any fixed Star from the South . The third and fourth lines are lines of Moneths , marked with the respective names , and each Moneth divided into so many parts as it contains dayes . The fifth line is a line of signs marked ♈ Taurus ; Gemini ; , &c. each sign being divided into 30 degr . and proceeding from ♈ , or Aries ( which answers to the tenth of March ) in the same order as the Moneths . The use of this line , with help of the Moneths , is to find the Suns place in the Zodiack . The sixth line is a line of the Suns right ascension , commonly noted by hours from 00 to 24. but better if divided by degrees or sines from 00 to 360 , and both wayes proceeding backward and forward as the signs of the Zodiack , or dayes of the Moneth . Lastly the outward edge or limb of the movable and loose piece both , is graduated unto 180 degrees , or two quadrants ; whose centre is the pin , or pin-hole before mentioned , at the beginning of the sines on this side the fixed piece . The perpendicular is at 0 / 60 upon the loose piece ; from whence reckoning along the outward edge of the loose piece , till it intersects the produced line of sines at the end of the fixed piece , you have 90 degrees . Or counting from 0 / 60 on the loose piece , and continuing it along the degrees of the outward limb of the movable piece , until they intersect the produced line of sines on the fixed piece at the head , you have again 90 degrees , which compleat the Semicircle . The use of this line is for taking of altitudes , counting upon the former 90. degr . when you hold the head of the fixed piece toward the Sun : and numbring upon this latter , when you hold ( which is best because the degrees are largest ) the end of the fixed piece toward the Sun. There are other wayes of numbring these degrees for finding of the Azimuth , &c. which shall be mentioned in their proper places . On the quadrantal side of the loose piece , the inward edge or limb is graduated unto 60 degrees ( or twice 30 , which you please ) whose centre is a pin at the head . The use of this is to find the altitude of the Sun , or any Star , without thread and plummet ; or to perform some uses of the Cross-staff . This is for large rules or instruments , and therefore not illustrated here . In the empty spaces upon the quadrantal side may be engraven the names of some fixed Stars , with their right ascensions and declinations . On the proportional side the lines issuing from the centre are the same upon the fixed and movable piece , but happily transplaced ( thanks to the first contriver ) after this manner : The line that lies next the inward edge on the fixed piece hath his fellow or correspondent line toward the outward edge on the movable piece ; by which means these lines all meeting at the centre , stand all at the same angle , and give you the freedom from a great deal of trouble , in working proportions by sines and tangents , or laying down any sine or tangent to any Radius given , &c. The lines issuing from the centre toward the outward edge of the movable piece , whose fellow is next the inward edge of the fixed piece , is a line of natural sines on the outward side , marked at the end S , and on the inward side a line of lines or equal parts , noted at the end L ; the middle line serving for both of them . The lines issuing from the centre next the inward edge of the movable piece , whose fellows are toward the outward edge of the fixed piece , are lines of natural tangents , which on the outward , side of the line is divided to 45 the Radius ; and on the inward side of the lines ( the middle line serving for both ) at a quarter of the former Radius from the centre is another Radius noted 45 at the beginning , and continued to the tangent 75. These lines are noted at the end T. The use of these you will find Chap. 3,4,5 . Betwixt the lines of sines and tangents , both upon the fixed and movable pieces , is placed a line of seconds , continued unto 60 , and marked at the end . Se. Next to the outward edge on the fixed and movable piece ( which is best discerned when those pieces are opened to the full length ) is a line of Meridians divided to 85 ; whose use is for Navigation , in describing Maps or Charts , &c. In the vacant spaces you may have a line of chords , sines , and tangents , to any Radius the space will bear ; and what other any one thinks best of , as a line of latitudes and hours , &c. On the proportional side of the loose piece are lines for measuring all manner of solids , as Timber , Stone , &c. likewise for gaging of Vessels either in Wine or Ale measure . On the outward ledge of the movable and fixed piece , both ( which in use must be stretched out to the full length ) is a line of artificial numbers , sines , tangents , and versed sines . The first marked N , the second S , the third T , the fourth VS . On the inward ledge of the movable piece is a line of 12 inches divided into halfs , quarters , half-quarters . Next to that is a prick'd line , whose use is for computing of weight and carriages . Lastly a line of foot measure , or a foot divided into ten parts ; and each of those subdivided into ten or twenty more . On the inward ledge of the loose piece you may have a line of circumference , diameter , square equal , and square inscribed . There will still be requisite sights , a thread and plummet . And if any go to the price of a sliding Index to find the shadow from the plains perpendicular , in order to taking a plains declination , and have a staff and a ballsocket , the Instrument is compleated with its furniture . Proceed we now to the uses . Onely note by the way that Mr. Brown hath ( for conveniency of carrying a pair of Compasses , Pen , Ink , and Pencil ) contrived the fixed piece and movable both to be hollow , and then the pieces that cover those hollows do , one supply the place of the loose piece for taking altitudes ; the other ( being a sliding rule ) for measuring solids , and gaging Vessels without Compasses . CHAP. II. Some uses of the quadrantal side of the Instrument . PROBL. 1. To find the altitude of the Sun or any Star. HAng the thread and plummet upon the pin at the beginning of the line of sines on the fixed piece , and ( having two sights in two holes parallel to that line ) raise the end of the fixed piece , toward the Sun until the rayes pass through the sights ( but when the Sun is in a cloud , or you take the altitude of any Star , look along the outward ledge of the fixed piece , until it be even with the middle of the Sun or Star ) then on the limb the thread cuts the degree of altitude , if you reckon from 0 / 60 on the loose piece toward the head of the movable piece . PROBL. 2. The day of the Moneth given , to find the Suns place , declination , ascensional difference , or time of rising and setting , with his right ascension . The thread laid to the day of the Moneth gives the Suns place in the line of signs , reckoning according to the order of the Moneths ( viz. forward from March the 10th . to June , then backward to December , and forward again to March 10. ) In the limb you have the Suns declination , reckoning from 60 / 0 on the movable piece towards the head for North , toward the end for South declination . Again , on the line of right ascensions , the thread shews the Suns right ascension , in degrees , or hours , ( according to the making of your line ) counting from Aries toward the head , and so back again according to the course of the signs unto 24 hours , or 360 degrees . Lastly on the line of hours you have the time of Sun rising and setting , which turned into degrees ( for the time from six ) gives the ascensional difference . Ex. gr . in lat . 52. deg . 30 min. for which latitude I shall make all the examples . The 22 day of March I lay the thread to the day in the Moneths , and find it cut in the Signs 12 deg . 20 min. for the Suns place , on the limb 4 deg . 43 min. for the Suns declination North. In the line of right ascensions it gives 46 min. of time , or 11 deg . 30 min. of the circle . Lastly , on the line of hours it shews 28 min. before six for the Suns rising ; or which is all , 7 deg . for his ascensional difference . PROBL. 3. The declination of the Sun or any Star given to find their amplitude . Take the declination from the scale of altitudes , with this distance setting one point of your Compasses at 90 on the line of Azimuth , apply the other point to the same line it gives the amplitude , counting from 90 Ex. gr . at 10 deg . declination , the amplitude is 16 deg . 30 min. at 20 deg . declination , the amplitude is 34 deg . PROBL. 4. The right ascension of the Sun , with his ascensional difference , given to find the oblique ascension . In Northern declination , the difference betwixt the right ascension and ascensional difference , is the oblique ascension . In Southern declination take the summ of them for the oblique ascension , Ex. gr . at 11 deg . 30 sec. right ascension , and 6 deg . 30 sec. ascensional difference . In Northern declination the oblique ascension will be 5 deg . in Southern 18 deg . PROBL. 5. The Suns altitude and declination , or the day of the Moneth given to find the hour . Take the Suns altitude from the Scale of altitudes , and laying the thread to the declination in the limb ( or which is all one , to the day in the Moneths ) move one point of the compasses along the line of hours ( on that side the thread next the end ) until the other point just touch the thread ; then the former point shews the hour ; but whether it be before or after noon , is left to your judgment to determine . Ex. gr . The 22 day of March , or 4 deg . 43 min. North declination , and 20 deg . altitude , the hour is either 47 minutes past 7 in the morning , or 13 minutes past 4 afternoon . PROBL. 6. The declination of the Sun , or day of the Moneth , and hour given to find the altitude . Lay the thread to the day or declination , and take the least distance from the hour to the thread , this applyed to the line of altitudes , gives the altitude required . Ex. gr . The 5 day of April or 10 deg . declination North , at 7 in the morning , or 5 afternoon , the altitude will be 17 deg . 10 sec. and better . PROBL. 7. The declination and hour of the night , given to find the Suns depression under the horizon . Lay the thread to the declination on the limb ; but counted the contrary way , viz. from 60 / 0 on the movable piece toward the head for Southern ; and toward the end for Northern declination . This done take the nearest distance from the hour to the thread , and applying it to the line of altitudes , you have the degrees of the Suns depression . Ex. gr . at 5 deg . Northern declination , & 8 hours afternoon , the depression is 13 deg . 30 min. PROBL. 8. The declination given to find the beginning and end of twilight , or day-break . Lay the thread to the declination counted the contrary way , as in the last Problem , and take from your Scale of altitudes 18 deg . for twilight , and 13 deg . for day-break , or clear light ; with this run one point of the Compasses along the line of houres ( on that side next the end ) until the other will just touch the thread , and then the former point gives the respective times required . Ex. gr . At 7 deg . North declination , day breaks 8 minutes before 4 : but twilight is 3 houres 12 minutes in the morning , or 8 hours 52 minutes afternoon . PROBL. 9. The declination and altitude of the Sun or any Star , given to find their Azimuth in Northern declination . Lay the thread to the altitude numbred on the limb of the moveable piece from 60 / 0 toward the end ( and when occasion requires , continue your numbring forward upon the loose piece ) and take the declination from your line of altitude ; with this distance run one point of your Compasses along the line of Azimuths ( on that side the thread next the head ) until the other just touch the thread , then the former point gives the Azimuth from South . Ex. gr . at 10 deg . declination North , and 30 deg . altitude , the Azimuth from South is 64 , deg . 40 min. PROBL. 10. The Suns altitude given to find his Azimuth in the aequator . Lay the thread to the altitude in the limb , counted from 60 / 0 on the loose piece toward the end , and on the line of Azimuths it cuts the Azimuth from South . Ex. gr . at 25 deg . altitude the Azimuth is 53 deg . At 30 deg . altitude the Azimuth is 41 deg . 30 min. fere . PROBL. 11. The declination and altitude of the Sun , or any Star given to find the Azimuth in Southern declination . Lay the thread to the altitude numbred on the limb from 60 / 0 on the moveable piece toward the end , and take the declination from the Scale of altitudes ; then carry one point of your Compasses on the line of Azimuths ( on that side the thread next the end ) until the other just touch the thread , which done , the former point gives the Azimuth from South . Ex. gr . at 15 deg . altitude and 6 deg . South declination the Azimuth is 58 deg . 30 min. PROBL. 12. The declination given to find the Suns altitude at East or West in North declination , and by consequent his depression in South declination . Take the declination given from the Scale of altitudes , and setting one point of your Compasses in 90 on the line of Azimuths , lay the thread to the other point ( on that side 90 next the head ) on the limb it cuts the altitude , counting from 60 / 0 on the moveable piece . Ex. gr . at 10 deg . declination the altitude is 12 deg . 40 min. PROBL. 13. The declination and Azimuth given to find the altitude of the Sun or any Star. Take the declination from the Scale of altitudes ; set one point of your Compasses in the Azimuth given , then in North declinanation turn the other point toward the head , in South toward the end ; and thereto laying the thread , on the limb you have the altitude , numbring from 60 / 0 on the moveable piece toward the end . Ex. gr . At 7 deg . North declination , and 48 deg . Azimuth from South , the altitude is 35 deg . but at 7 deg . declination South , and 50 deg . Azimuth the altitude is onely 18 deg . 30 min. PROBL. 14. The altitude , declination , and right ascension of any Star with the right ascension of the Sun given , to find the hour of the night . Take the Stars altitude from the Scale of altitudes , and laying the thread to his declination in the limb , find his hour from the last Meridian he was upon , as you did for the Sun by Probl. 5. If the Star be past the South , this is an afternoon hour ; if not come to the South , a morning hour ; which keep . Then setting one point of your Compasses in the Suns right ascension ( numbred upon the line twice 12 or 24 next the outward ledge on the fixed piece ) extend the other point to the right ascension of the Star numbred upon the same line , observing which way you turned the point of your Compasses , viz. toward the head or end . With this distance set one point of your Compasses in the Stars hour before found counted on the same line , and turning the other point the same way , as you did for the right ascensions , it gives the true hour of the night . Ex. gr . The 22 of March I find the altitude of the Lions heart 45 deg . his declination 13 d. 40 min. then by Probl. 5. I find his hour from the last Meridian 10 houres , 5 min. The right ascension of the Sun is 46 m. of time , or 11 d. 30 m. of the Circle , the right ascension of the Lions heart , is 9 hour 51 m. fere , of time , or 147 deg . 43 m. of a circle ; then by a line of twice 12 , you may find the true hour of the night , 7 hour 13 min. PROBL. 15. The right ascension and declination of any Star , with the right ascension of the Sun and time of night given , to find the altitude of that Star with his Azimuth from South , and by consequent to find the Star , although before you knew it not . This is no more than unravelling the last Problem . 1 Therefore upon the line of twice 12 or 24 , set one point of your Compasses in the right ascension of the Star , extending the other to the right ascension of the Sun upon the same line , that distance laid the same way upon the same line , from the hour of the night , gives the Stars hour from the last Meridian he was upon . This found by Probl. 5. find his altitude as you did for the Sun. Lastly , having now his declination and altitude by Probl. 8. or 10. according to his declination , you will soon get his Azimuth from South . This needs not an example . By help of this Problem the Instrument might be so contrived , as to be one of the best Tutors for knowing of the Stars . PROBL. 16. The altitude and Azimuth of any Star given to find his declination . Lay the thread to the altitude counted on the limb from 60 / 0 on the moveable piece toward the end , setting one point of your Compasses in the Azimuth , take the nearest distance to the thread ; this applyed to the Scale of altitudes gives the declination . If the Azimuth given be on that side the thread toward the end , the declination is South ; when on that side toward the head , its North. PROBL. 17. The altitude and declination of any Star , with the right ascension of the Sun , and hour of night given to find the Stars right ascension . By Probl. 5. or 14. find the Stars hour from the Meridian . Then on the line twice 12 , or 24 , set one point of your Compasses in the Stars hour ( thus found ) and extend the other to the hour of the night . Upon the same line with this distance set one point of your Compasses in the right ascension of the Sun , and turning the other point the same way , as you did for the hour , it gives the Stars right ascension . PROBL. 18. The Meridian altitude given to find the time of Sunrise and Sunset . Take the Meridian altitude from your particular Scale , and setting one point of your Compasses upon the point 12 on the line of hours ( that is the pin at the end ) lay the thread to the other point , and on the line of hours the thread gives the time required . PROBL. 19. To find any latitude your particular Scale is made for . Take the distance from 90 , on the line of Azimuth unto the pin at the end of that line , or the point 12 : this applyed to the particular Scale , gives the complement of that latitude the Instrument was made for . PROBL. 20. To find the angles of the substile , stile , inclination of Meridians , and six and twelve , for exact declining plains , in that latitude your Scale of altitudes is made for . Sect. 1. To find the distance of the substile from 12 , or the plains perpendicular . Lay the thread to the complement of declination counted on the line of Azimuths , and on the limb it gives the substile counting from 60 / 0 on the moveable piece . Sect. 2. To find the angle of the Stile 's height . On the line of Azimuths take the distance from the Plains declination to 90. This applyed to the Scale of altitudes gives the angle of the stile . Sect. 3. The angle of the Substile given to find the inclination of Meridians . Take the angle of the substile from the Scale of altitudes , and applying it from 90 on the Azimuth line toward the end ; the figures shew the complement of inclination of Meridians . Sect. 4. To find the angle betwixt 6 and 12. Take the declination from the Scale of altitudes , and setting one point of your Compasses in 90 on the line of Azimuths , lay the thread to the other point and on the limb it gives the complement of the angle sought , numbring from 60 / 0 on the moveable piece toward the end . This last rule is not exact , nor is it here worth the labour to rectifie it by another sine added ; sith you have an exact proportion for the Problem in the Treatise of Dialling Chap. 2 . Sect. 5. Paragr . 4. CHAP. III. Some uses of the Line of natural signs on the Quadrantal side of the fixed piece . PROBL. 1. How to adde one sign to another on the Line of Natural Sines . TO adde one sine to another , is to augment the line of one sine by the line of the other sine to be added to it . Ex. gr . To adde the sine 15 to the sine 20 , I take the distance from the beginning of the line of sines unto 15 , and setting one point of the Compasses in 20 , upon the same line , turn the other toward 90 , which I finde touch in 37. So that in this case ( for we regard not the Arithmetical , but proportional aggregate ) 15 added to 20 , upon the line of natural sines , is the sine 37 upon that line , and from the beginning of the line to 37 is the distance I am to take for the summe of 20 and 15 sines . PROBL. 2. How to substract one sine from another upon the line of natural sines . The substracting of one sine from another , is no more than taking the distance from the lesser to the greater on the line of sines , and that distance applyed to the line from the beginning , gives the residue or remainer . Ex. gr . To substract 20 from 37 I take the distance from 20 to 37 that applyed to the line from the beginning gives 15 for the sine remaining . PROBL. 3. To work proportions in sines alone . Here are four Cases that include all proportions in sines alone . CASE 1. When the first term is Radius , or the Sine 90. Lay the thread to the second term counted on the degrees upon the movaeble piece from the head toward the end , then numbring the third on the line of sines , take the nearest distance from thence to the thread , and that applyed to the Scale from the beginning gives the fourth term . Ex. gr . As the Radius 90 is to the sine 20 , so is the sine 30 to the sine 10. CASE 2. When the Radius is the third term . Take the sine of the second term in your Compasses , and enter it in the first term upon the line of sines , and laying the thread to the nearest distance , on the limb the thread gives the fourth term . Ex. gr . As the sine 30 is to the sine 20 , so is the Radius to the sine 43. 30. min. CASE 3. When the Radius is the second term . Provided the third term be not greater than the first , transpose the terms . The method of transposition in this case is , as the first term is to the third , so is the second to the fourth , and then the work will be the same as in the second case . Ex. gr . As the sine 30 is to the radius or sine 90 , so is the sine 20 to what sine ; which transposed is As the sine 30 is to the sine 20 , so is the radius to a fourth sine , which will be found 43 , 30 min. as before . CASE 4. When the Radius is none of the three terms given . In this case when both the middle terms are less than the first , enter the sine of the second term in the first , and laying the thread to the nearest distance , take the nearest extent from the third to the thread : this distance applyed to the scale from the beginning gives the fourth . Ex. gr . As the sine 20 to the sine 10 , so is the sine 30 to the sine 15. When only the second term is greater than the first , transpose the terms and work as before . But when both the middle tearms be greater than the first , this proportion will not be performed by this line without a paralel entrance or double radius ; which inconveniency shall be remedied in its proper place , when we shew how to work proportions by the lines of natural sines on the proportional or sector side . These four cases comprizing the method of working all proportions by natural sines alone , I shall propose some examples for the exercise of young practitioners , and therewith conclude this Chapter . PROBL. 4. To finde the Suns amplitude in any Latitude . As the cosine of the Latitude is to the sine of the Suns declination , so is the radius to the sine of amplitude . PROBL. 5. To finde the hour in any Latitude in Northern Declination . Proport . 1. As the radius to the sine of the Suns declination , so is the sine of the latitude to the sine of the Suns altitude at six . By Probl. 2. substract this altitude at six from the present altitude , and take the difference . Then Proport . 2. As the cosine of the latitude is to that difference , so is the radius to a fourth sine . Again Proport . 3. As the cosine of the declination to that fourth sine , so is the radius to the sine of the hour from six . PROBL. 6. To finde the hour in any Latitude when the Sun is in the Equinoctial . As the cosine of the latitude is to the sine of altitude , so is the radius to the sine of the hour from six . PROBL. 7. To finde the hour in any latitude in Southern Declination . Proport . 1. As the radius to the sine of the Suns declination , so is the sine of the latitude to the sine of the Suns depression at six ; adde the sine of depression to the present altitude by Probl. 1. Then Proport . 2. As the cosine of the latitude is to that summe , so is the radius to a fourth sine . Again , Proport . 3. As the cosine of declination is to the fourth sine , so is the radius to the sine of the hour from six . PROBL. 8. To finde the Suns Azimuth in any latitude in Northern Declination . Proport . 1. As the sine of the latitude to the sine of declination , so is the radius to the sine of altitude at East , or West . By Probl. 2. substract this from the present altitude , then , Proport . 2. As the cosine of the latitude is to that residue , so is the radius to a fourth sine . Again , Proport . 3. As the cosine of the altitude is to that fourth sine , so is the radius to the sine of the Azimuth from East or West . PROBL. 9. To finde the Azimuth for any latitude when the Sun is in the Equator . Proport . 1. As the cosine of the latitude to the sine of altitude , so is the sine of the latitude to a fourth sine . Proport . 2. As the cosine of altitude to that fourth sine , so is the radius to the sine of the Azimuth from East , or West . PROBL. 10. To finde the Azimuth for any latitude in Southern Declination . Proport . 1. As the cosine of the latitude to the sine of altitude , so is the sine of the latitude to a fourth . Having by Probl. 4. found the Suns amplitude , adde it to this fourth sine by Probl. 1. and say As the cosine of the altitude is to the sum , so is the radius to the sine of the Azimuth from East or West . The terms mentioned in the 5th . 7th . 8th . 10th . Problems are appropriated unto us that live on the North side the Equator . In case they be applyed to such latitudes as lie on the South side the Equator . Then what is now called Northern declination , name Southern , and what is here styled Southern declination , term Northern , and all the proportion with the operation is the same . These proportions to finde the hour and Azimuth , may be all readily wrought by the lines of artificial sines , only the addition and substraction must alwayes be wrought upon the line of natural sines . CHAP. IV. Some uses of the Lines on the proportional side of the Instrument , viz. the Lines of natural Sines , Tangents , and Secants . PROBL. 1. To lay down any Sine , Tangent , or Secant to a Radius given . See Fig. 1. IF you be to lay down a Sine , enter the Radius given in 90 , and 90 upon the lines of Sines , keeping the Sector at that gage , set one point of your Compasses in the Sine required upon one line , and extend the other point to the same Sine upon the other Line : This distance is the length of the Sine required to the given Radius . Ex. gr . Suppose A. B. the Radius given , and I require the Sine 40. proportional to that Radius . Enter A. B. in 90 , and 90 keeping the Sector at that gage , I take the distance , twixt 40 on one side , to 40 on the other , that is , C. D. the Sine required . The work is the same , to lay down a Tangent to any Radius given , provided you enter the given Radius in 45 , and 45 , on the line of Tangents . Only observe if the Tangent required be less than 45. you must enter the Radius in 45. and 45 next the end of the Rule . But when the Tangent required exceeds 45. enter the Radius given in 45 , and 45 'twixt the center and end , and keeping the Sector at that Gage , take out the Tangent required . This is so plain , there needs no example . To lay down a Secant to any Radius given , is no more than to enter the Radius in the two pins at the beginning of the line of Secants ; and keeping the Sector at that Gage , take the distance from the number of the Secant required on one side , to the same number on the other side , and that is the Secant sought at the Radius given . The use of this Problem will be sufficiently seen in delineating Dyals , and projecting the Sphere . PROBL. 2. To lay down any Angle required by the Lines of Sines , Tangents , and Secants . See Fig. 2. There are two wayes of protracting an Angle by the Line of Sines , First if you use the Sines in manner of Chords . Then having drawn the line A B at any distance of your Compass , set one point in B , and draw a mark to intersect the Line B A , as E F. Enter this distance B F in 30 , and 30 upon the Lines of Sines , and keeping the Sector at that Gage , take out the Sine of half the Angle required , and setting one point where F intersects B A , turn the other toward E , and make the mark E , with a ruler draw B E and the Angle E B F is the Angle required , which here is 40. d. A second method by the lines of Sines is thus , Enter B A Radius in the Lines of Sines , and keeping the Sector at that Gage , take out the Sine of your Angle required with that distance , setting one point of your Compasses in A , sweep the ark D , a line drawn from B by the connexity of the Ark D , makes the Angle A B C 40 d. as before . To protract an Angle by the Lines of Tangents is easily done , draw B A the Radius upon A , erect a perpendicular , A C , enter B A in 45 , and 45 on the Lines of Tangents , and taking out the Tangent required ( as here 40 ) set it from A to C. Lastly , draw B C , and the Angle C B A is 40 d. as before . In case you would protract an Angle by the Lines of Secants . Draw B A , and upon A erect the perpendicular A C , enter A B in the beginning of the Lines of Secants , and take out the Secant of the Angle , with that distance , setting one point of your Compasses in B , with the other cross the perpendicular A C , as in C. This done , lay a Ruler to B , and the point of intersection , and draw the Line B C. So have you again the Angle C B A. 40. d. by another projection . These varieties are here inserted only to satisfie a friend , and recreate the young practitioner in trying the truth of his projection . PROBL. 3. To work proportions in Sines alone , by the Lines of natural Sines on the proportional side of the Instrument . The general rule is this . Account the first term upon the Lines of Sines from the Center , and enter the second term in the first so accounted , keeping the Sector at that Gage , account the third term on both lines from the Center , and taking the distance from the third term on one line to the third term on the other line , measure it upon the line of Sines from the beginning , and you have the fourth term . Ex. gr . As the Radius is to the Sine 30 , so is the Sine 40 to the Sine 18. 45. There is but one exception in this Rule , and that is when the second term is greater than the first ; yet the third lesser than the first , and in this case transpose the terms , by Chap 3. Probl. 3. Case 3. But when the second term is not twice the length of the first , it may be wrought by the general Rule without any transposition of terms . Ex. gr . As the Sine 30 is to the Sine 50 , so is the Sine 20 to the Sine 31. 30. min. And by consequent , when the third term is greater than the first , provided it be not upon the line , double the length thereof , it may be wrought by transposing the terms , although the second was twice the length of the first . Ex. gr . As the Sine 20 is to the Sine 60 , so is the Sine 42 , to what Sine ? which transposed is , As the Sine 20 is to the Sine 42 , so is the Sine 60 to the Sine 35. 30. This case will remove the inconveniency mentioned , Chap. 3. Probl. 3. Case 4. of a double Radius . I intended there to have adjoyned the method of working proportions by natural Tangents alone , and by natural Sines , and Tangents , conjunctly : But considering the multiplicity of proportions when the Tangents exceed 45. I suppose it too troublesome for beginners , and a needless variety for those that are already Mathematicians . Sith , both may be eased by the artificial Sines and Tangents on the outward ledge , where I intend to treat of those Cases at large , and shall in this place only annex some proportions in Sines alone , for the exercise of young beginners . PROBL. 4. By the Lines of Natural Sines to lay down any Tangent , or Secant required to a Radius given . In some Cases , especially for Dyalling , your Instrument may be defective of a Tangent , or Secant for your purpose , Ex. gr . when the Tangent exceeds 76 , or the Secant is more than 60. In these extremities use the following Remedies . First , for a Tangent . As the cosine of the Ark is to the Radius given , so is the sine of the Ark to the length of the Tangent required . Secondly , for a Secant . As the cosine of the Ark is to the Radius given , so is the Sine 90 to the length of the Secant required . PROBL. 5. The distance from the next Equinoctial Point given to finde the Suns declination . As the Radius to the sine of the Suns greatest declination , so is the sine of his distance from the next Equinoctial Point to the sine of his present declination . PROBL. 6. The declination given to finde the Suns Equinoctial Distance . As the sine of the greatest declination is to the sine of the present declination , so is the Radius to the sine of his Equinoctial Distance . PROBL. 7. The Altitude , Declination , and Distance of the Sun from the Meridian given to finde his Azimuth . As the cosine of the altitude , to the cosine of the hour from the Meridian , so is the cosine of declination to the sine of the Azimuth . CHAP. V. Some uses of the Lines of the Lines , on the proportional side of the Instrument . PROBL. 1. To divide a Line given into any Number of equal parts . See Fig. 3. SUppose A B a Line given to be divided into nine equal parts . Enter A B in 9 , and 9 on the lines of lines , keeping the Sector at that gage , take the distance from 8 , on one side , to 8 on the other , and apply it from A upon the line A B , which reacheth to C ; then is C B a ninth part of the line A B. By this means you may divide any line ( that is not more than the Instrument in length ) into as many parts as you please , viz. 10 , 20 , 30 , 40 , 50 , 100 , 500 , &c. parts according to your reckoning the divisions upon the lines , Ex. gr . The line is actually divided into 200 parts , viz. first into 10 , marked with Figures , and each of those into twenty parts more . Again , if the line represents a 1000 , then every figured division is 100 , the second or shorter division is 10 , and the third or shortest division is 5. In case the whole line was 2000 , then every figured division is 200 , every smaller or second division is 20 , every third or smallest division is 10 , &c. Suppose I have any line given , which is the base of a Triangle , whose content is 2000 poles , and I demand so much of the Base as may answer 1750 poles . Enter the whole line in 10 , and 10 at the end of the lines of lines , and keep the Sector at that gage . Now the whole line representing 2000 poles , every figured division is 200 ; therefore 1700 is eight and an half of the figured divisions , and 50 is five of the smallest divisions more ( for in this case every smallest division is 10 , as was before expressed ) wherefore setting one point of the Compasses in 15 of the smallest divisions beyond 8 on the Rule . I extend the other point to the same division upon the line on the other side , and that distance is 1750 poles in the base of the Triangle proposed . How ready this is to set out a just quantity in any plat of ground , I shall shew in a Scheam , Chap. 12. PROBL. 2. To work proportions in Lines , or Numbers , or the Rule of three direct by the Lines of Lines . Enter the second term in the first , and keeping the Sector at that gage , take the distance 'twixt the third on one line , to the third on the other line , that distance is the fourth in lines , or measured upon the line from the centre , gives the fourth in numbers , Ex. gr . As 7 is to 3 , so is 21 to 9. PROBL. 3. To work the Rule of Three inverse , or the back Rule of Three by the Lines of Lines . In these proportions there are alwayes three terms given to finde a fourth , and of the three given terms two are of one denomination ( which for distinction sake I call the double denomination ; ) and the third term is of a different denomination from those two , which I therefore call the single denomination , of which the fourth term sought must also be . Now to bring these into a direct proportion , the rule is this . When the fourth term sought is to be greater than the single denomination ( which you may know by sight of the terms given ) say , As the lesser double denomination is to the greater double denomination , so is the single denomination to the fourth term sought . The work is by Probl. 2. If 60 men do a work in 5 dayes , how long will 30 men be about it ? As 30 is to 60 , so is 5 to 10. The number of dayes for 30 men in the work . Again , when the fourth term is to be less than the single denomination , say , As the greater double denomination is to the lesser double denomination , so is the single denomination to the fourth term sought . If 30 men do a work in 5 dayes , how long shall 60 be doing of it ? As 60 is to 30 , so is 5 to 2½ . The time for 60 men in the work . PROBL. 4. The length of any perpendicular , with the length of the shadow thereof given , to finde the Suns Altitude . At the length of the shadow upon the lines of lines , is to the Tangent 45 , so is the length of the perpendicular numbred upon the lines of lines , to the tangent of the Suns altitude . PROBL. 5. To finde the Altitude of any Tree , Steeple , &c. at one station . At any distance from the object ( provided the ground be level ) with your Instrument , look to the top of the object along the outward ledge of the fixed piece , and take the angle of its altitude . This done , measure by feet or yards , the distance from your standing to the bottom of the object . Then say , As the cosine of the altitude is to the measured distance numbred upon the lines of lines , so is the sine of the altitude to a fourth number of feet or yards ( according to the measure you meeted the distance ) to this fourth , adde the height of your eye from the ground , and that sum gives the number of feet or yards in the altitude . CHAP. VI. How to work proportions in Numbers , Sines , or Tangents , by the Artificial Lines thereof on the outward ledge . THe general rule for all of these , is to extend the Compasses from the first term to the second ( and observing whether that extent was upward or downward ) with the same distance , set one point in the third term , and turning the other point the same way , as at first , it gives the fourth . But in Tangents when any of the terms exceeds 45 , there may be excursions , which in their due place I shall remove . PROBL. 1. Numeration by the Line of Numbers . The whole line is actually divided into 100 proportional parts , and accordingly distinguished by figures , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , and then , 10 , 20 , 30 , 40 , 50 , 60 , 70 , 80 , 90 , 100. So that for any number under 100 , the Figures readily direct you , Ex. gr . To finde 79 on the line of numbers , count 9 of the small divisions beyond 70 , and there is the point for that number . Now as the whole line is actually divided into 100 parts , so is every one of those parts subdivided ( so far as conveniency will permit ) actually into ten parts more , by which means you have the whole line actually divided into 1000 parts . For reckoning the Figures impressed , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , to be 10 , 20 , 30 , 40 , 50 , 60 , 70 , 80 , 90 , and the other figures which are stamped 10 , 20 , 30 , 40 , 50 , 60 , 70 , 80 , 90 , 100 , to be 100 , 200 , 300 , 400 , 500 , 600 , 700 , 800 , 900 , 1000. You may enter any number under 1000 upon the line , according to the former directions . And any numbers whose product surmount not 1000 , may be wrought upon this line ; but where the product exceeds 1000 , this line will do nothing accurately : Wherefore I shall willingly omit many Problems mentioned by some Writers to be wrought by this line , as squaring , and cubing of numbers , &c. Sith they have only nicety , and nothing of exactness in them . PROBL. 2. To multiply two numbers , given by the Line of Numbers . The proportion is this . As 1 on the line is to the multiplicator , so is the multiplicand to the product . Ex. gr . As 1 is to 4 , so is 7 , to what ? Extend the Compasses from the first term , viz. I unto the second term , viz. 4. with that distance , setting one point in 7 the third term , turn the other point of the Compasses toward the same end of the rule , as at first , and you have the fourth , viz. 28. There is only one difficulty remaining in this Problem , and that is to determin the number of places , or figures in the product , which may be resolved by this general rule . The product alwayes contains as many figures as are in the multiplicand , and multiplicator both , unless the two first figures of the product be greater than the two first figures in the multiplicator , and then the product must have one figure less than are in the multiplicator , and multiplicand both . Ex. gr . 47 multiplied by 25 , is 2175 , consisting of four figures ; but 16 multiplied by 16 , is 240 , consisting of no more than three places , for the reason before mentioned . I here ( for distinction sake ) call the multiplicator the lesser of the two numbers , although it may be either of them at pleasure . PROBL. 3. To work Division by the Line of Numbers . As the divisor is to 1 , so is the dividend to the quotient . Suppose 800 to be divided by 20 , the quotient is 40. For , As 20 is to 1 , so is 800 to 40. To know how many figures you shall have in the quotient , take this rule . Note the difference of the numbers of places or figures in the dividend and divisor . Then in case the quantity of the two first figures to the left hand in your divisor be less than the quantity of the two first figures to the left hand in your dividend , the quotient shall have one figure more than the number of difference : But where the quantity of the two first figures of the divisor is greater than the quantity of the two first figures of the dividend , the quotient will have only that number of figures noted by the difference . Ex. gr . 245 divided by 15. will have two figures in the quotient ; but 16 divided 〈…〉 ●●ve only one figure in the quotient . PROBL. 4. To finde a mean proportional 'twixt two Numbers given by the line of Numbers . Divide the space betwixt them upon the line of numbers into two equal parts , and the middle point is the mean proportional . Ex. gr . betwixt 4 and 16 , the mean proportional is 8. If you were to finde two mean proportionals , divide the space 'twixt the given numbers into three parts . If four mean proportionals divide it into five parts , and the several points 'twixt the two given numbers , will show the respective mean proportionals . PROBL. 5. To work proportions in Sines alone , by the Artificial Line of Sines . Extend the Compasses from the first term to the second , with that distance set one point in the third term , and the other point gives the fourth . Only observe that if the second term be less than the first , the fourth must be less 〈…〉 or if the second term exceed 〈…〉 fourth will be greater than the third . This may direct you in all proportions of sines and tangents singly or conjunctly , to which end of the rule to turn the point of your Compasses , for finding the fourth term . Ex. gr . As the sine 60 is to the sine 40 , so is the sine 20 to the sine 14. 40. Again , As the sine 10 is to the sine 20 , so is the sine 30 to the sine 80. PROBL. 6. To work proportions in Tangents alone by the Artificial Line of Tangents . For this purpose the artificial line of tangents must be imagined twice the length of the rules , and therefore for the greater conveniency , it is doubly numbred , viz. First from 1 to 45 , which is the radius , or equal to the sine 90 : In which account every division hath ( as to its length on the rule ) a proportional decrease . Secondly , it s numbred back again from 45 to 89 , in which account every division hath ( as to its length on the line ) a proportional encrease . So that the tangent 60 you must imagine the whole length of the Rule ; and so much more as the distance from 45 unto 30 or 60 is . This well observed , all proportions in tangents are wrought after the same manner of extending the Compasses from the first term to the second , and that distance set in the third , gives the fourth , as was for sines and numbers . But for the remedying of excursions , sith the line is no more than half the length , we must imagine it . I shall lay down these Cases . CASE 1. When the fourth term is a tangent exceeding 45 , or the Radius . Ex. gr . As the tangent 10 is to the tangent 30 , so is the tangent 20 , to what ? Extending the Compasses from 10 on the line of tangents to 30 , with that distance I set one point in 20 , and finde the other point reach beyond 45 , which tells me the fourth term exceeds 45 , or the radius ; wherefore with the former extent , I set one point in 45 ; and turning the other toward the beginning of the line , I mark where it toucheth , and from thence taking the distance to the third term , I have the excess of the fourth term above 45 in my compass : wherefore with this last distance setting one point in 45 , I turn the other upon the line , and it reacheth to 50 , the tangent sought . CASE 2. When the first term is a tangent exceeding 45 , or the Radius . Ex. gr . As the tangent 50 is to the tangent 20 , so is the tangent 30 , to what ? Because the second term is less than the first , I know the fourth must be less than the third . All the difficulty is to get the true extent from the tangent 50 to 20. To do this , take the distance from 45 to 50 , and setting one point in 20 , the second term , turn the other toward the beginning of the line , marking where it toucheth , extend the Compasses from the point where it toucheth to 45 , and you will have the same distance in your Compasses as from 50 to 20 , if the line had been continued at length unto 89 tangents , with this distance , set one point in 30 the third term , and turn the other toward the beginning ( because you know the fourth must be less ) and it gives 10 the tangent sought . CASE . 3. When the third term is a Tangent exceeding 45 , or the Radius . As the tangent 40 is to the tangent 12 , 40 min. so is the tangent 65 , to what ? Extend the Compasses from 40 to 12 d. 40 min. with distance , setting one point in 65. turn the other toward 45 , and you will finde it reach beyond it , which assures you the fourth term will be less than 45. Therefore lay the extent from 45 toward the beginning , and mark where it toucheth , take the distance from that point to 65 , and laying that distance from 45 toward the beginning it gives 30 , the tangent sought . These Cases are sufficient to remove all difficulties . For when the second term exceeds the Radius , you may transpose them , saying , as the first term is to the third ; so is the second to the fourth , and then it s wrought by the third Case . I suppose it needless to adde any thing about working proportions by sines and tangents conjunctly , sith , enough hath been already said of both of them apart , in these two last Problems ; and the work is the same when they are intermixed . Only some proportions I shall adjoyn , and leave to the practice of the young beginner , with the directions in the former Cases . PROBL. 7. To finde the Suns ascensional difference in any Latitude . As the co-tangent of the latitude is to the tangent of the Suns declination , so is the radius to the sine of the ascensional difference . PROBL. 8. To finde at what hour the Sun will be East , or West in any Latitude . As the tangent of the latitude is to the tangent of the Suns declination , so is the radius to the cosine of the hour from noon . PROBL. 9. The Latitude , Declination of the Sun , and his Azimuth from South , given to finde the Suns Altitude at that Azimuth . As the radius to the cosine of the Azimuth from south , so is the co-tangent of the latitude , to the tangent of the Suns altitude in the equator at the Azimuth given . Again , As the sine of the latitude is to the sine of the Suns declination , so is the cosine of the Suns altitude in the equator ( at the same Azimuth from East or West ) to a fourth ark . When the Azimuth is under 90 , and the latitude and declination is under the same pole , adde this fourth ark to the altitude in the equator . In Azimuths exceeding 90 , when the latitude and declination is under the same pole , take the equator altitude out of the fourth ark . Lastly , when the latitude and declination respect different poles , take the fourth ark out of the equator altitude , and you have the altitude sought . PROBL. 10. The Azimuth , Altitude , and Declination of the Sun , given to finde the hour . As the cosine of declination is to the sine of the Suns Azimuth , so is the cosine of the altitude to the fine of the hour from the Meridian . Proportions may be varied eight several wayes in this manner following . 1. As the first term is to the second , so is the third to the fourth . 2. As the second term is to the first , so is the fourth to the third . 3. As the third term is to the first , so is the fourth to the second . 4. As the fourth term is to the second , so is the third to the first . 5. As the second term is to the fourth , so is the first to the third . 6. As the first term is to the third , so is the second to the fourth . 7. As the third term is to the fourth , so is the first to the second . 8. As the fourth term is to the third , so is the second to the first . By thesse any one may vary the former proportions , and make the Problems three times the number here inserted . Ex. gr . To finde the ascensional difference in Problem 10 , of this Chapter , which runs thus . As the co-tangent of the latitude is to the tangent of the Suns declination , so is the radius to the sine of ascensional difference . Then by the third variety you may make another Problem , viz. As the radius is to the co-tangent of the latitude , so is the sine of the Suns ascensional difference to the sine of his declination . Again , by the fourth variety you may make a third Problem , thus , As the sine of the Suns ascensional difference is to the tangent of the Suns declination , so is the radius to the co-tangent of the latitude . By this Artifice many have stuffed their Books with bundles of Problems . CHAP. VII . Some uses of the Lines of Circumference , Diameter , Square Equal , and Square Inscribed . ALL these are lines of equal parts , bearing such proportion to each other , as the things signified by their names . Their use is this , Any one of them given in inches or feet , &c. to finde how much any of the other three are in the same measure . Suppose I have the circumference of a Circle , Tree , or Cylinder given in inches , I take the same number of parts ( as the Circle is inches ) from the line of circumference , and applying that distance to the respective lines , I have immediately the square equal , square inscribed , and diameter , in inches , and the like , if any of those were given to finde the circumference . This needs no example . The conveniency of this line any one may experiment in standing timber ; for taking the girth , or circumference with a line , finde the diameter ; from that diameter abate twice the thickness of the bark , and you have the true diameter , when it s barked , and by Chap. 9. Probl. 5. you will guess very near at the quantity of timber in any standing Tree . CHAP. VIII . To measure any kinde of Superficies , as Board , Glass , Pavement , Walnscot , Hangings , Walling , Slating , or Tyling , by the line of Numbers on the outward ledge . THe way of accounting any number upon , or working proportions by , the line of numbers , is sufficiently shewn already , Chap. 6. which I shall not here repeat , only propose the proportions for these Problems , and refer you to those directions . PROBL. 1. The breadth of a Board given in inches , to finde how many inches in length make a foot at that breadth , say , As the breadth in inches is to 12 , so is 12 to the length in inches for a foot at that breadth . Ex. gr . At 8 inches breadth you must have 18 inches in length for a foot . PROBL. 2. The breadth and length of a Board given to finde the content . As 12 is to the length in feet and inches , so is the breadth in inches to the content in feet . Ex. gr . at 15 inches breadth , and 20 foot length , you have 25 foot of Board . PROBL. 3. A speedy way to measure any quantity of Board . The two former Problems are sufficient to measure small parcels of Board . When you have occasion to measure greater quantities , as 100 foot , or more , lay all the boards of one length together , and when the length of the boards exceeds 12 foot , use this proportion . As the length in feet and inches is to 12 , so is 100 to the breadth in inches for an 100 foot . Ex. gr . At 30 foot in length 40 inches in breadth , make an 100 foot of board ( reckoning five score to the hundred . ) This found with a rule or line , measure 40 inches at both ends in breadth , and you have 100 foot . When one end is broader than another , you may take the breadth of the over-plus of 100 foot at both ends , and taking half that sum for the true breadth of the over-plus by Probl. 2. finde the content thereof . When your boards are under 12 foot in length , say , As the length in feet and inches is to 12 , so is 50 unto the breadth in inches for 50 foot of board , and then you need only double that breadth to measure 100 foot as before . In like manner you may measure two , three , four , five , a hundred , &c. foot of board speedily , as your occasion requires . PROBL. 4. To measure Wainscot , Hangings , Plaister , &c. These are usually computed by the yard , and then the proportion is . As nine to the length in feet , and inches , so is the breadth or depth in feet and inches to the content in yards , Ex. gr . At 18 foot in length , and two foot in breadth , you have four yards . PROBL. 5. To measure Masons , or Slaters Work , as Walling , Tyling , &c. The common account of these is by the rood , which is eighteen foot square , that is 324 square foot in one rood , and then the proportion is . As 324 to the length in feet , so is the breadth in feet to the content in roods . Ex. gr . At 30 foot in length , and 15 foot in breadth , you have 1 rood 3 / 10 and better , or one rood 126 / 124 parts of a rood . CHAP. IX . The mensuration of Solids , as Timber , Stone , &c. by the lines on the proportional side of the loose piece . THese two lines meeting upon one line in the midst betwixt them ( for distinction sake ) I call one the right , the other the left line , which are known by the hand they stand toward when you hold up the piece in the right way to read the Figures . The right line hath two figured partitions . The first partition is from 3 , at the beginning to the letters Sq. every figured division representing an inch , and each subdivision quarters of an inch . The next partition is from the letters Sq. unto 12 , at the end , every figured division signifying a foot , and each sub-division the inches in a foot . The letters T R , and T D , are for the circumference and Diameter in the next measuring of Cylinders . The letters R and D. for the measuring of Timber , according to the vulgar allowance , when the fourth part of the girt is taken , &c. The letters A and W are the gauger points for Ale and Wine Measures . Lastly , the figures 12 'twixt D and T D. are for an use , expressed Probl. 2. The left line also hath two figured partitions , proceeding first from 1 at the beginning to one foot , or 12 inches , each whereof is sub-divided into quarters . From thence again to 100 , each whereof to 10 foot , is sub-divided into inches , &c. and every foot is figured . But from 10 foot to 100 , only every tenth foot is figured ; the sub-divisions representing feet . The method of working proportions by these lines ( only observing the sides ) is the same as by the line of Numbers , viz. extending from the first to the second , &c. PROBL. 1. To reduce Timber of unequal breadth and depth to a true Square . As the breadth on the left is to the breadth on the right , so is the depth on the left , to the square on the right line . At 7 inches breadth , and 18 inches depth , you have 11 inches ¼ and better for the true square . PROBL. 2. The square of a Piece of Timber given in Inches , or Feet , and Inches to finde how much in length makes a Foot. As the square in feet and inches on the right is to one foot on the left , so is the point Sq. on the right to the number of feet and inches on the left for a foot square of Timber . At 18 inches square , 5 inches ¼ and almost half a quarter in length makes a foot . When your Timber ( if it be proper to call such pieces by that name ) is under 3 inches square , account the figured divisions on the right line from the letters Sq. to the end , for inches , and each sub-division twelve parts of an inch . So that every three of them makes a quarter of an inch . Then the proportion is , as the inches and quarters square on the right is to 100 on the left , so is the point 12 'twixt D , and T D , on the right , to the number of feet in length on the left to make a foot of Timber . As 2 inches ½ square you must have 23 foot 6 inches , and somewhat better for the length of a foot of Timber . PROBL. 3. The square and length of a plece of Timber given to finde the content . As the point Sq. on the right is to the length in feet and inches on the left , so is the square in feet and inches on the right to the content in feet on the left . At 30 foot in length , and 15 inches square , you have 46 foot ½ of Timber . At 20 foot in length , and 11 inches square you have 16 foot , and almost ¼ of Timber . When you have a great piece of Timber exceeding 100 foot ( which you may easily see by the excursion upon the rule ) then take the true square , and half the length , sinde the content thereof by the former proportion , and doubling that content , you have the whole content . PROBL. 4. The Circumference , or girth , of a round piece of Timber , being given , together with the length , to finde the content . As the point R. on the right , is to the length in feet and inches on the left , so is the circumference in feet and inches on the right , to the content in feet on the left . At 20 foot in length , and 7 foot in girth , you have 60 foot of Timber for the content . This is after the common allowance for the waste in squaring ; and although some are pleased to quarrel with the allowancer , as wronging the seller , and giving the quantity less , than in truth it is ; yet I presume when they buy it themselves , they scarcely judge those Chips worth the hewing , and have as low thoughts of the over-plus , as others have of that their admonition . If it be a Cylinder that you would take exact content of , then say , As the point T R , on the right , is to the length on the left , so is the girt on the right to the exact content on the left . At 15 foot in length , and 7 foot in girt , you have 59 foot of solid measure . The Diameter of any Cylinder given , you may by the same proportion finde the content , placing the point D , instead of R , in the proportion for the usual allowance , and the point T D , for the exact compute . PROBL. 5. To measure tapered Timber . Take the square or girt at both ends , and note the sum and difference of them . Then for round Timber , as the point R. on the right , is to the length on the left , so is half the sum of the girt at both ends on the right , to a number of feet on the left . Keep this number , and say again , As the point R. on the right , is to the third part of the former length on the left , so is half the difference of the girts on the right , to a number of feet on the left ; which number added to the former , gives the true content . The same way you may use for square Timber , only setting the feet and inches square instead of the girt , and the point Sq. instead of the point R. At 30 foot in length , 7 foot at one end , and 5 at the other in girt , half the sum of the girts is 6 foot , or 72 inches ; the first number of feet found 67 , half the difference of the girts is 1 foot , or 12 inches , the third part of the length 10 foot ; then the second number found will be 7 foot , one quarter and half a quarter . The sum of both ( or true content ) 74 foot , one quarter , and half a quarter . For standing Timber , take the girt about a yard from the bottom , and at 5 foot from the bottom , by Chap. 7 , set down these two diameters without the bark ; and likewise the difference 'twixt them . Again , by Chap. 6. Probl. 4. finde the altitude of the tree , so far as it bears Timber ( or as we commonly phrase it , to the collar ) this done , you may very near proportion the girt at the collar and content of the tree , before it falls . In case any make choice of the hollow contrivance mentioned , Chap. 1. they need no compasses in the mensuration of any solid ; provided the lines for solid measure , and gauging vessels , be doubly impressed ( only in a reverted order , one pair of lines proceeding from the head toward the end , and the other pair from the end toward the head ) upon the sliding cover , and its adjacent ledges . This done , the method of performing any of the Problems mentioned in this Chapter , is easie . For whereas you are before directed to extend the Compasses from the first term to the second ; and with that distance setting one point in the third term , the other point gave the fourth , or term sought . So here , observing the lines as before , slide the cover until the first term stand directly against the second ; then looking for the third on its proper line , it stands exactly against the fourth term , or term sought on the other line . Only note , that when the second term is greater than the first , it s performed by that pair of lines proceeding from the head toward the end : But when the first term is greater than the second , it is resolved by that pair of lines which is numbred from the end toward the head . CHAP. X. The Gauge Vessels , either for Wine , or Ale , Measure . PROBL. 1 . The Diameter at Head , and Diameter at Boung , given in Inches , and tenth parts of an Inch , to finde the mean Diameter in like measure . TAke the difference in inches , and tenth parts of an inch , between the two diameters . Then say by the line of numbers , As 1 is to 7 , so is the difference to a fourth number of inches , and tenth parts of an inch . This added to the Diameter at head , gives the mean Diameter . Ex. gr . At 27 inches the boung , and 19 inches two tenths at the head , the difference is 7 inches , 8 tenths . The fourth number found by the proportion will be 5 inches , 4 tenths , and one half , which added to the diameter at the head , gives 24 inches , 6 tenths , and one half tenth of an inch for the mean diameter . PROBL. 2. The length of the Vessel , and the mean Diameter given in Inches , and tenth parts of an Inch , to finde the content in Gallons , either in Wine or Ale measure . Note first , that the point A. on the right is the gauge point for Ale measure , and the point W. on the right , is the gauge point for Wine measure . Then say , As the gauge point on the right to the length in inches , and tenth parts of an inch on the left , so is the mean diameter in inches and tenth parts of an inch on the right , to the content in gallons on the left . CHAP. XI . Some uses of the Lines on the inward ledge of the moveable piece . THe Line of inches and foot measure do by inspection ( only ) reduce either of the measures into the other . Ex. gr . Three on the line of inches stands directly against 25 cents of foot measure . Or 75 cents of foot measure is directly against 9 on the inches . Another use of these lines ( welcome perhaps to them that delight in instrumental computations ) is to know the price of carriage for any quantity , &c. by inspection . For this purpose , the line of inches represents the price of a pound , every inch being a penny , and every quarter a farthing . The prickt line is the price of an hundred pound at five score and twelve to the hundred , every division signifying a shilling . The line of foot measure is the price of an hundred pound at five score to the hundred , every division standing for a shilling . Ex. gr . At 3 pence the pound on the inches , is 28 shillings on the prickt line , and 25 shillings on the foot measure , for the price of an hundred , the like for the converse . Otherwise , the price of a pound being given , the rate of an hundred is readily computed without the rule . For considering the number of farthings is in the price of a pound , twice that number of shillings , and once that number of pence , is the price of an hundred , reckoning five score to the hundred ; of twice that number of shillings , and once that number of groats , is the price of an hundred , at five score and twelve to the hundred . Ex. gr . At three half pence the pound , the number of farthings is six . Therefore , twice six shillings , and once six pence ( that is 125. 6d . ) is the price of an hundred , at five score to the hundred . Again , twice six shillings , and once six groats ( that is 14s . ) is the price of an hundred , at five score and twelve to the hundred . But of this enough , if not too much already . CHAP. XII . To divide a plot of Ground into any proposed quantities . See Fig. 4 SUppose A B C D E F G H I K L , a plot of ground , containing 54 acr . 2 roods , 28 poles , from the point A , I am required to shut off 18 acres , next the side B C. Draw A D. and measure the figure A B C D , which is 14 acr : 2 r. 3 p. that is 3 acr . 1. r. 37 p. or 557 poles too little . Draw again A F , and measure A D F , which is 1309. poles . Then by Chap. 5. Probl. 1. entring D F , the base of the triangle in 1309 on the lines of lines , and taking out 557 , set it from D to E , and draw A E. So have you the figure A B C D E 18 acres . Again , from the point G , I would set off 20 acres next the side A E , draw A G , and measure A E F G 15 acr . or . 2 pol. whereof want 4 acr . or . 38 pol. that 678 poles , then draw G K , and measure G A K 1113 poles . Lastly , by Chap. 5. Probl. I. enter A K , the base of the triangle in 1113 upon the lines of lines , and taking out 678 , set it from A , and draw G L. So have you the figure A E F G L , twenty acres . How ready the instrument would be for surveying with the help of a staff , Ballsocket , and Needle , is obvious to any one that considers its graduated into 180 degrees . CHAP. XIII . So much of Geography as concerns finding the distance of any two places upon the Terrestrial Globe . HEre are three Cases , and each of those contains the same number of propositions . CASE 1. When the two places differ in latitude only . PROP. 1. When one place lies under the Equator , having no Latitude . The latitude of the other place turned into miles , ( reckoning 60 miles , the usual compute , for a degree ) is the distance sought . PROP. 2. When both places have the same pole elevated , viz. North , or South . Take the difference of their Latitudes , and reckoning 60 miles for a degree ( as before ) you have their distance . PROP. 3. When the two places have different poles elevated , viz. one North , the other South . Adde two latitudes together , and that sum turned into miles is the distance . CASE 2. When the two places differ in Longitude only . PROP. 1. When neither of them have any Latitude , but lie both under the Equator . Their difference of Longitude turned into miles ( as before ) is their distance . PROP. 2. When the two places have the same Pole elevated . The proportion is thus . As the Radius is to the number 60 , so is the cosine of the common latitude to the number of miles for one degree of longitude . Multiply this number found by the difference of longitude , and that product is the distance in miles . PROP. 3. When the two places have different poles elevated . As the Radius is to the cosine of the common latitude , so is the sine of half the difference of longitude to the sine of half the distance . Wherefore this sine of half the distance doubled and turned into miles , is the true distance . CASE 3. When the two places differ in Longitude and Latitude both . PROP. 1. When one of the places lies under the equator , having no Latitude . As the radius is to the cosine of the difference of longitude , so is the cosine of the latitude to the cosine of the distance . PROP. 2. When both the places have the same pole elevated . As the radius is to the cosine of the difference in longitude , so is the co-tangent of the lesser altitude , to the tangent of a fourth ark . Subtract this fourth ark out of the complement of the lesser latitude , and keep the remain . Then , As the cosine of the fourth ark is to the cosine of the remain , so is the sine of the lesser latitude to the cosine of the distance . PROP. 3. When the two places have different poles elevated . As the radius is to the cosine of the difference in longitude , so is the co-tangent of either latitude to the tangent of a fourth ark . Subtract the fourth out of the latitude not taken into the former proportion , and note the difference . Then , As the cosine of the fourth ark is to the cosine of this difference , so is the sine of the latitude first taken , to the cosine of the distance . CHAP. XIV . Some uses of the Instrument in Navigation , or plain Salling . HEre it will be necessary to premise the explication of some terms , and adjoyn two previous proportions . 1. The Compass being a circle , divided into 32 equal parts , called rumbs ; one point or rumb is 11 d. 15 min. of a circle from the meridian : two points or rumbs is 22 d. 30 min. &c. of the rest . 2. The angle which the needle , or point of the compass under the needle , makes with the meridian , or North and South line is called the course or rumb ; but the angle which it makes with the East , and West line , or any parallel , is named the complement of the course or rumb . 3. The departure is the longitude of that Port from which you set sail . 4. The distance run , is the number of miles , or leagues ( turned into degrees ) that you have sailed . 5. When you are in North latitude , and sail North-ward , adde the difference of latitude to the latitude you sailed from ; and when you are in North latitude , and sail Southward , subtract the difference of latitude from the latitude you sailed , and you have the latitude you are in . The same rule is to be observed in South latitude . 6. To finde how many miles answer to one degree of longitude in any latitude . As the radius is to the number 60 , so is the cosine of the latitude to the number of miles for one degree . 7. To finde how many miles answer to one degree of latitude on any rumb . As the cosine of the rumb from the Meridian , is to the number 60 , so is the radius to the number of miles . The most material questions in Navigation are these four . First , To finde the course . Secondly , The distance run . Thirdly , The difference of latitude . Fourthly , The difference in longitude ; and any two of these being given , the other two are readily found by the Square and Index . These two additional rulers were omitted in the first Chapter of this Treatise ; sith they are only for Navigation , and large Instruments of two , or three foot in length , which made me judge their description most proper for this place : because , such as intend the Instrument for a pocket companion , will have no use of them . The square is a flat rule , having a piece , or plate fastened to the head , that it may slide square , or perpendicular to the outward ledge of the fixed piece . It hath the same line next either edge on the upper side , which is a line of equal parts , an hundred , wherein is equal to the radius of the degrees on the outward limb of the moveable piece . The Index is a thin brass rule on one side , having the same scale as the square . On the other side is a double line of tangents , that next the left edge , being to a smaller ; that next the right edge , to a larger radius . For the use of these rulers , you must have a line of equal parts adjoyning to the line of sines on the fixed piece , divided into 10 parts , stamped with figures , each of those divided into 10 parts more ; so that the whole line is divided into 100 parts , representing degrees . Lastly , let each of those degrees be sub-divided into as many parts , as the largeness of your scale will permit , for computing the minutes of a degree . The Index is to move upon the pin on the fixed piece ( where you hang the thread for taking altitudes ) and that side of the Index ( in any of the four former questions ) must be upward , which hath the scale of equal parts . The square is to be slided along the outward ledge of the fixed piece . Then the general rules are these . The difference of latitude is accounted on the line of equal parts , adjoyning to the sines on the fixed piece . The difference of longitude is numbred on the square . The distance run is reckoned upon the Index . The course is computed upon the degrees on the limb from the head toward the end of the moveable piece . But when any would work these Problems in proportions , let them note , The distance run , difference of longitude , and difference of latitude , are all accounted on the line of numbers ; the rumb or course is either a sine or tangent . This premised . I shall first shew how to resolve any Problem by the square and Index ; and next adjoyn the proportions for the use of such as have only pocket Instruments . PROBL. 1. The course and distance run given , to finde the difference of latitude , and difference of longitude . Apply the Index to the course reckoned on the limb from the head , and slide the square along the outward ledge of the fixed piece , until the fiducial edge intersect the distance run on the fiducial edge of the Index . Then at the point of Intersection you have the difference of longitude upon the square , and on the line of equal parts on the fixed piece , the square shows the difference of latitude . The proportion is thus , As the radius is to the distance run , so is the cosine of the course to the difference of latitude . Again , As the radius is to the distance run , so is the sine of the course to the difference in longitude . PROBL. 2. The course and difference of latitude given to find the distance run , and difference in longitude . Slide the square to the difference of latitude on the line of equal parts upon the fixed piece , and set the Index to the course on the limb . Then at the point of intersection of the square and index , on the square is the difference of longitude , on the index the distance run . The proportion is thus . As the cosine of the course is to the difference of latitude , so is the radius to the distance run . Again , As the radius is to the sine of the course , so is the distance run to the difference of longitude . PROBL. 3. The course and difference in longitude given , to finde the distance run , and difference of latitude . Apply the Index to the course on the limb , and the difference of longitude on the square to the fiducial edge of the Index . Then at the point of intersection you have distance run on the index , and upon the line of equal parts on the fixed piece , the square shows the difference of latitude . The proportion is thus . As the sine of the course is to the difference of longitude , so is the radius to the distance run . Again , As the radius is to the distance run , so is the cosine of the course to the difference of latitude . PROBL. 4. The distance run , and difference of latitude given to finde the course , and difference in longitude . Slide the square to the difference of latitude on the line of equal parts on the fixed piece , and move the index until the distance run numbred thereon , intersect the fiducial edge of the square ; then at the point of intersection you have the difference of longitude on the square , and the fiducial edge of the Index on the limb shows the course . As the distance run is to the difference of latitude , so is the radius to the cosine of the course . Again , As the radius is to the distance run , so is the sine of the course to the difference of longitude . PROBL. 5. The distance run , and difference of longitude given to finde the course , and difference of latitude . Apply the distance run numbred on the fiducial edge of the index , to the difference of longitude , reckoned on the fiducial edge of the square . Then on the line of equal parts upon the fixed piece , the square shows the difference of latitude , and the index shows the course on the limb . The proportion is thus . As the distance run is to the difference of longitude , so is the radius to the sine of the course . Again , As the radius is to the distance run , so is the cosine of the course , to the difference of latitude . PROBL. 6. The difference of latitude , and difference of longitude given , to finde the course , and distance run . Apply the square to the difference of latitude on the scale of equal parts upon the fixed piece , and move the index until its fiducial edge intersect the difference of longitude , reckoned on the square . Then at the point of intersection you have the distance run upon the index , and the fiducial edge of the index upon the limb shows the course . The proportion is thus . As the difference of latitude is to the difference of longitude , so is the radius to the tangent of the course . Again , As the sine of the course is to the radius , so is the difference of longitude to the distance run . PROBL. 7. Sailing by the Ark of a great Circle . For this purpose the tangent lines on the index will be a ready help , using the lesser for small , and the greater tangent line , for great latitudes . The way is thus , Account the pont 60 / 0 , on the outward limb of the moveable piece to be the point , or port of your departure ; thereto lay the fiducial edge of the index , and reckoning the latitude of the Port you departed from upon the index , strike a pin directly touching it , into the table your instrument lies upon . This pin shall represent the Port of your departure . Therefore hanging a thread , or hair , on the center , whereon the index moves ; and winding it about this pin . Count the difference of longitude 'twixt the port of your departure , and the Port you sail toward , from 60 / 0 on the moveable piece toward 0 / 60 , on the loose piece ; and thereto laying the same fiducial edge of the index , reckon the latitude of this last Port upon the index , directly touching of it , strike down another pin upon the table , and draw the thread strait about this pin fastening it thereto . This done , the thread betwixt the two pins represents the ark of your great Circle ; and laying the fiducial edge of the index to any degree of difference of longitude accounted from 60 / 0 on the moveable piece , the thread shows upon the index what latitude you are in , and how much you have raised , or depressed the pole since your departure . On the contrary , laying the latitude you are in ( numbred upon the index ) to the thread , the index shows the difference of longitude upon the limb ; counting from 60 / 0 on the moveable piece . So , that were it possible to sail exactly by the ark of a great Circle , it would be no difficulty to determine the longitude in any latitude you make . But I intend not a treatise of Navigation ; wherefore let it suffice , that I have already shown how the most material Problems therein , may easily , speedily , and ( if the instrument be large ) exactly , be performed by the instrument without the trouble of Calculation , or Projection . CHAP. XV. The Projection and Solution of the sixteen Cases in right angled Spherical Triangles by five Cases . See Fig. 5. THe fundamental Circle N B Z C , is alwayes supposed ready drawn , and crossed into quadrants , and the diameters produced beyond the Circle . CASE 1. Given both the sides Z D , and D R , to project the Triangle . By a line of chords , prick off Z D , upon the limb , and draw the diameter D A E. Again , by a line of tangents , set half co-tangent D R , upon A D , from A to R , then have you three points , viz. N R Z , to draw that ark , and make up the triangle . The center of which ark always lies on AC , ( produced beyond C , if need requires ) and is found by the intersection of the two arks made from R , and Z. CASE 2. Given one side Z D , and the hypothenuse Z R , to project the Triangle . Prick off Z D , and draw D A E , by Case 1. Again , set half the co-tangent Z R , on the line A Z , from A to F , and the tangent Z R , set from F to P , with the extent F P , upon the center P , draw the ark V F I , and where it intersects the diameter D A E , set R ; then have you three points N R Z , to draw that ark , as in the former Case . CASE 3. Given the Hypothenuse Z R , and the Angle D Z R , to project the Triangle . Prick half the co-tangent D Z R , from A to S , and the secant D Z R , from S to T , upon the center T , with the extent T S , draw the ark N S Z. Again , by Case 2 , draw the ark H F I , where these two arks intersect each other , set R. Lastly , lay a ruler to A R , and draw the diameter DRAE , and your triangle is made . CASE 4. Given one side Z D , and its adjacent Angle D Z R , to project the triangle . Prick off Z D , and draw D A E , by Case 1. Again , by Case 3 , draw the ark N S Z , where this ark intersects the diameter DAE , set R , and your triangle is made . Or , Given the side D R , and its opposite angle D Z R , you may project the triangle . Draw the ark N S Z , as before . Again , take half the co-tangent D R , and with that extent upon A , the center , cross the ark N S Z , setting R , at the point of intersection . Lastly , lay a ruler to A R , and draw the diameter D R A E , which makes up your triangle . The triangle projected in any of the four former Cases , to measure any of the sides , or angles , do thus . First , the 〈◊〉 Z D , is found by applying it to a line of chords . Secondly , R A , applyed to a line of tangents , is half the co-tangent D R. Thirdly , A S , applyed to a line of tangents is the co-tangent D Z R. Fourthly , set half the tangent D Z R , from A to L , then is L the pole point , and laying a ruler to L R , it cuts the limb at V , and the ark Z V , upon the line of chords , gives the hypothenuse Z R. Fifthly , prick Z D from C to K , a ruler laid from R to L , cuts the limb at G , then G K , upon a line of chords , is the quantity of Z R D. CASE . 5. The two oblique angles , D Z R , and Z R D given , to present the Triangle . See Fig. 6. This is no more than turning the former triangle . Thus , Draw the ark N S Z , by Case 1 , and set half the tangent of that ark from A to L. Again , set half the co-tangent D R Z , from A to F , and the secant of D R Z , from F to L , upon the center A , with the extent A P. Draw the ark P G , and with the extent F P , from L , cross the ark P G in G. Lastly , upon the Center G , with the extent G L. Draw the ark R D F L , and your triangle is made . The triangle projected you may measure off the sides and hypothenuse . Thus , First , the hypothenuse Z R , is measured by a line of chords . Secondly , a ruler laid to L D , cuts the limb at H , and Z H , upon a line of chords , is the measure of the ark Z D. Thirdly , draw A G , and set half the tangent D R Z , from A to V , apply a ruler to V D , it cuts the limb at E , then R E , upon a line of chords , measure the ark R D. Note . The radius to all the chords , tangents , and secants , used in the projection , and measuring , any ark or angle , is the semidiameter of the fundamental circle . CHAP. XVI . The projection and solution of the 12 Cases in oblique angled spherical triangles in six Cases . See Fig. 7. THe fundamental circle N H Z M , is alwayes supposed ready drawn , and crossed into Quadrants , and the Diameters produced beyond the Circle . CASE 1. The three sides , Z P , P Z , and Z S , given , to project the Triangle . By a line of chords prick off Z P , and draw the diameter P C T , crossing it at right angles in the center with AE C E , set half the co-tangent P S , from C to G , and he secant P S from C to R , upon the center R , with the extent R G , draw the the ark FGL . Again , set half the co-tangent Z S , from C to D , and the tangent Z S , from D to O , with the extent O D , upon the center O , draw the ark B D P , mark where these two arks intersect each other as at S. Then have you three points T S P , to draw that ark , and the three points N S P , to draw that ark , which make up your triangle . CASE 2. Given two sides Z S , and Z P , with the comprehended Angle P Z S , to project the Triangle . Prick off Z P , and draw PCT , and AECE , and the ark B D P , by Case 1. Again , set the tangent of half the excess of the angle P Z S above 90 , from C to W , and co-secant of that excess from W to K , upon the center K , with the extent K W ; draw the ark N W Z , which cuts the ark B D P in S. Then have you the three points T S P , to draw that ark which makes up the triangle . CASE 3. Two Angles S Z P , and Z P S , with the comprehended side Z P , given , to project the Triangle . Prick off Z P , and draw the lines P C T , and AE C E , by Case 1 , and the angle NWZ . by Case 2. Lastly , set half the co-tangent ZPS from C to X , and the secant Z P S , from X to V , upon the center V , with the extent V X , draw the ark T X S P , and the triangle is made . CASE 4. Two sides , ZP , and PS , with the Angle opposite to one of them SZP , given , to project the Triangle . Prick off ZP , and draw PCT , and AECE , by Case 1. and the angle SZP by Case 2. Lastly , by Case 1. draw the ark FGL , and mark where it intersects NWZ , as at S , then have you the three points TSP , to draw that ark , and make up the triangle . CASE 5. Two Angles SZP , and ZPS , with the side opposite to one of them ZS , given , to project the Triangle . Draw the ark BDP , by Case 1. and the ark NWZ , by Case 2. at the intersection of these two arks , set S , with the tangent of the angle ZPS , upon the center C. sweep the ark VΔI . Again , with the secant of the ark ZPS upon the center S , cross the ark VΔI , as at the points V and I. Then in case the hypothenuse is less than a quadrant ( as here ) the point V , is the center , and with the extent VS , draw the ark TSP , which makes up the triangle . But in case the hypothenuse is equal to a quadrant , Δ , is the center ; if more than a quadrant , I , is the center ; in which cases the extent from Δ , or I , to S , is the semidiameter of the ark TSP . CASE 6. Three Angles ZPS , and PZS , and ZSP , given , to project the Triangle . See Fig. 7. and 8. The angles of any spherical triangle may be converted into their opposite sides , by taking the complement of the greatest angle to a Semicircle for the hypothenuse , or greatest side . Wherefore by Case 1. make the side ZP , in Fig. 4. equal to the angle ZSP , in Fig. 3 , and the side ZS , in Fig. 4. equal to the angle ZPS , in Fig. 3. and the side PS , in Fig. 4. equal to the complement of the angle PZS , to a Semicircle in Fig. 3. Then is your triangle projected where the angle ZPS in Fig. 4. is the side ZS , Fig. 3. Again , the angle ZSP , Fig. 4. is the side ZP , in Fig. 3. Lastly , the complement of the angle PZS to a Semicircle in Fig. 4. is the measure of the hypothenuse , or side P S , in Fig. 3. The Triangle being in any of the former Cases projected , the quantity of any side or angle may be measured by the following rules . First , The side Z P , is found by applying it to a line of chords . Secondly , CX , applyed to a line of tangents , is half the co-tangent of the angle ZPS . Thirdly , CW applyed to a line of tangents , is half the co-tangent of the excess of the angle SZP , above 90. Fourthly , set half the tangent of the angle ZPS , from C , to Π , a ruler laid to ΠS , cuts the limb at F ; then PF , applyed to a line of chords , gives the side PS . Fifthly , take the complement of the angle PZS , to a Semicircle , and set half the tangent of that complement from C , to λ , a ruler laid to λS , cuts the limb at B , and ZB , applyed to a line of chords , gives the side ZS . Sixthly , a ruler laid to Sλ , cuts the limb at L. Again , a ruler laid to SΠ , cuts the limb at φ , and L φ , applyed to a line of chords , gives the angle ZSP . The end of the first Book . An Appendix to the first Book . THe sights which are necessary for taking any Altitude . Angle , or distance ( without the help of Thread or Plummet ) are only three , viz. one turning sight , and two other sights , contrived with chops , so that they may slide by the inward or outward graduated limbs . The turning sight hath only two places , either the center at the head , or the center at the beginning of the line of sines on the fixed piece ; to either of which ( as occasion requires ) it s fastened with a sorne . The center at the head serving for the graduations next the inward limb of the loose piece . And the center at the beginning of the line of sines serving for all the graduations next the outward limb of the moveable and loose piece both . Yet because it is requisite to have pins to keep the loose piece close in its place . You may have two sights more to supply their place ( which sometimes you may make use of ) and so the number of sights may be five , viz. two sliding sights , one turning sight , and two pin sights , to put into the holes at the end of the fixed and moveable piece , to hold the tenons of the loose piece close joynted . Every one of these sights hath a fiducial ( or perpendicular ) line , drawn down the middle of them , from the top to the bottom , where this line toucheth the graduations on the limb , is the point of observation . The places of these sights have an oval proportion , about the middle of them , only leaving a small bar of brass , to conduct the fiducial line down the oval cavity , and support a little brass knot ( with a sight hole in it ) in the middle of that bar , which is ever the point to be looked at . There are two wayes of observing an altitude with help of these sights . The one when we turn our face toward the object . This is called a forward observation in which you must alwayes set the turning sight next your eye . This way of observation will not exactly give an altitude above 45 degrees . The other way of observing an altitude is peculiar to the Sun in a bright day , when we turn our back toward the Sun. This is termed a backward observation ; wherein You must have one of the sliding sights next Your eye , and the turning sight toward the Horizon . This serves to take the Suns altitude without thread , or plummet , when it is near the Zenith . PROBL. 1. To finde the Suns altitude by a forward observation . Serve the turning sight to the center of those graduations you please to make use of ( whether on the inward or outward limb ) and place the two sliding sights upon the respective limb to that center ; this done , look by the knot of the turning sight ( moving the instrument upward or downward ) until you see the knot of one of the sliding sights directly against the Sun , then move the other sliding sight , until the knot of the turning sight , and the knot of this other sliding sight be against the horizon ; then the degrees intercepted 'twixt the fiducial lines of the sliding sights on the limb , shew the altitude required . PROBL. 2. To finde the distance of any two Stars , &c. by a forward observation . Serve the turning sight to either center , and apply the two sliding sights to the respective limb ( holding the instrument with the proportional side downward ) and applying the turning sight to your eye , so move the two sliding sights either nearer together , or further asunder , that you may by the knot of the turning sight see both objects even with the knots of their respective sliding sights , then will the degrees intercepted 'twixt the fiducial lines of the object sights on the limb show the true distance . By this means you may take any angle for surveying , &c. PROBL. 3. To finde the Suns Altitude by a backward Observation . Serve the turning sight to the center at the beginning of the line of sines , and apply one of the sliding sights to the outward limb of the loose piece , and the other to the outward limb of the moveable piece ; and turning your back toward the Sun , set the sliding sight upon the moveable piece next your eye ; and slide it upward or downward toward the end , or head , until you see the shadow of the little bar , or edge , of the sight on the loose piece fall directly on the little bar on the turning sight ; and at the same time the bar of the sight next your eye , and the bar of the turning sight to be in a direct line with the Horizon . Then will the degrees on the limb intercepted 'twixt the fiducial lines of the sliding sights ( if you took the shadow of the bar ) or 'twixt the fiducial line of the sliding sight next your eye , and the edge of the other sliding sight ( when you took the shadow of the edge ) be the true altitude required . ΣΚΙΟΓΡΑΦΙΑ , OR , The Art of Dyalling for any plain Superficies . LIB . II. CHAP. I. The distinction of Plains , with Rules for knowing of them . ALL plain Superficies are either horizontal , or such as make Angles with the Horizon . Horizontal plains are those , that lie upon an exact level , or flat . Plains , that make Angles with the Horizon are of three sorts . 1. Such as make right angles with the Horizon , generally known by the name of erect , or upright plains . 2. Such as make acute angles with the horizon , or have their upper edge leaning toward you , usually termed inclining plains . 3. Such as make obtuse angles with the horizon , or have their upper edge falling from you , commonly called reclining plains . All these three sorts are either direct , viz. East , West , North , South . Or else Declining From South , toward East , or West . From North , toward East , or West . All plain Superficies whatsoever are comprized under one of these terms . But before we treat of the affections , or delineation of Dials for them ; it will be requisire to acquaint you with the nature of any plain , which may be found by the following Problems . PROBL. 1. To finde the reclination of any Plain . Apply the outward ledge of the moveable piece to the Plain with the head upward , and reckoning what number of degrees the thread cuts on the limb ( beginning your account at 30. on the loose piece , and continuing it toward 60 / 0 on the moveable piece ) you have the angle of reclination . If the thread falls directly on 60 / 0 upon the moveable piece , it s an horizontal ; if on 30. on the loose piece , it s an erect plain . PROBL. 2. To finde the inclination of any Plain . Apply the outward ledge of the fixed piece to the plain , with the head upward , and what number of degrees the thread cuts upon the limb of the loose piece , is the complement of the plains inclination . PROBL. 3. To draw an Horizontal Line upon any Plain . Apply the proportional side of the Instrument to the plain , and move the ends of the fixed piece upward , or downward , until the thread falls directly on 60 / 0. upon the loose piece ; then drawing a line by the outward ledge of the fixed piece , its horizontal , or paralel to the horizon . PROBL. 4. To draw a perpendicular Line upon any Plain . When the Sun shineth hold up a thread with a plummet against the plain , and make two points at any distance in the shadow of the thread upon the plain , lay a ruler to these points , and the line you draw is a perpendicular . PROBL. 5. To finde the declination of any Plain . Apply the outward ledge of the fixed piece to the horizontal line of your plain , holding your instrument paralel to the horizon . This done , lift up the thread and plummer , until the shadow of the thread fall directly upon the pin hole on the fixed piece ( where you hang the thred to take altitudes ) Then observe how many degrees the shadow of the thread cuts in the limb , either from the right hand , or from the left hand 0 / 60. upon the loose piece ; and immediately taking the altitude of the Sun. By lib. 1. cap. 2. Probl. 9 , 10 , 11. finde the Suns Azimuth from South . And , When you make this observation in the morning , these Cases determine the declination of the plain . CASE 1. When the shadow of the thread upon the limb falls on the right hand 0 / 60 on the loose piece , take the difference of the shadow , and Azimuth ( by subtracting the lesse out of the greater ) and the residue or remain is the plains declination . From South toward East , when the Azimuth is greater than the shadow . From South toward West , when the shadow is greater than the azimuth , when the shadow and azimuth are equal , it s a direct South plain . When the difference is just 90. its a direct East , when above 90. subtract the difference from 180. and the remain is the declination from North toward East . CASE 2. When the shadow falls on the left hand 0 / 60. Adde the azimuth and shadow together , that sum is the plains declination ; from South toward East , when under 90 ; if it be just 90 , it s a direct East . If above 90. subtract it from 180. the remain is the declination from North toward East . When the sum is above 180. subtract 180 from it , and the remain is the declination from North toward West . CASE 3. When the shadow falls upon 0 / 60. the azimuth is the plains declination . When under 90 , its South-East , when equal to 90. direct East , when above 90. subtract it from 180. the remain is the declination from North toward East . If you make the observation afternoon , the following Cases will resolve you . CASE 4. When the shadow falls on the left hand 0 / 60. the difference 'twixt the shadow and azimuth is the declination ; when the shadow is more than the azimuth it declines South-East , when less , South-West . When the shadow and azimuth are equal , it s a direct South plain , when their difference is equal to 90. it s a direct West ; when the difference exceeds 90. subtract it from 180. the remain is the declination North-West . CASE 5. When the shadow falls on the right hand 0 / 60. take the sum of the shadow and azimuth , and that is the declination from South toward West , when under 90. when just 90. its a direct West plain ; when more than 90. subtract it from 180. the remain is the declination North-West ; when the sum is above 180. subtract 180 from it , and the remain is the declination from North toward East . CASE 6. When the shadow falls upon 0 / 60. the azimuth is the quantity of declination . From South toward West , when under 90. when equal to 90. its direct West ; when more than 90. subtract it from 180. The remain is the declination from North toward West . PROBL. 6. To draw a Meridional Line upon a Horizontal Plain . Draw first a circle upon the plain , and holding up a thread and plummet ( when the Sun shines ) so that the shadow of the thread may pass through the center of the circle , make a point in the circumference where the shadow intersects it . At the same time finding the Suns Azimuth from South , by a line of chords , set it upon the limb of the circle from the intersection of the shadow toward the South , and it gives the true South point . Wherefore laying a ruler to this last point and the center , the line you draw is a true Meridian . CHAP. II. The affections of all sorts of Plains . Sect. 1. The affections of an horizontal Plain . 1. THe style , or cock of every horizontal Dial , is an angle equal to that latitude for which the Dial is made . 2. The place of the style is directly upon the meridional , or twelve a clock line , and the angular point must stand in the center of the hour lines . 3. The rule for drawing the hour lines before six in the morning , is to draw the respective hour lines afternoon beyond the center ; or for the hour lines after six at evening , draw the respective hour lines in the morning beyond the center . 4. To place an horizontal Dial upon the plain ; first draw a Meridian line upon the plain by Cap. 1. Probl. 6. and lay in the line of 12. exactly thereon , with the angular point of the style toward the South , fasten the Dial upon the plain . Sect. 2. The affections of erect , direct South and North Plains . 1. The style in both these is an angle equal to the complement of the latitude for which the Dial is made to stand upon the Meridian line , or perpendicular of your plain , with the angular point in the center of the hour lines , and that point in South alwayes upward , in North alwayes downward . 2. To prick off the Dial from your paper draught upon the plain , lay the hour line of 6. and 6. upon the plains horizontal ; and applying a ruler to the center , and each hour line , transmit the hour lines from your paper draught to the plain . Sect. 3. The affections of erect , direct East and West Plains . 1. In both these the style may be a pin or plate , equal in length or heighth to the radius of the tangents , by which you draw the Dial. 2. The style in both of them is to stand directly upon the hour of six , and perpendicular to the plain . Sect. 4. The affections of erect declining Plains . 1. These are of two sorts , either such as admit of centers to the hour lines and style , or such as cannot with conveniency ( because of the lowness of the style , and nearness of the hour lines to each other ) be drawn with a center to those lines . Of this latter sort are all such plains , whose style is an angle less than 15 degrees . For where the angle of the style is more than 15 degrees ; those Dials may be drawn with a center to the hour lines . 2. In all erect declining plains with centers the Meridian is the plains perpendicular . In those that admit not of centers in their delineation , the meridian is parallel to the plains perpendicular . 3. In all declining plains the substile , or line whereon the style is to stand , must be placed on that side the meridian , which is contrary to the coast of declination ; and also in such decliners as admit of centers , the angle 'twixt 12. and 6. is to be set to the contrary coast to that of declination . 4. In all decliners , without centers , the inclination of meridians is to be set from the substyle toward the coast of declination . 5. The proportions in all erect decliners , for finding the height of the style , the distance of the substyle from the meridian , the angle of twelve and six , with the inclination of meridians ( all which may be wrought either by the canon , or exactly enough for this purpose by the instrument ) are as followeth . To finde the Styles height above the Substile . As the radius is to the cosine of the latitude , so is the cosine of declination , to the sine of the styles height . To finde the Substyles distance from the Meridian . As the radius is to the sine of declination , so is the co-tangent of the latitude to the tangent of the substyle from the Meridian . To finde the angle of twelve and six . As the co-tangent of the latitude is to the radius , so is the sine of declination , to the co-tangent of six from twelve . To finde the inclination of Meridians . As the sine of the latitude is to the radius , so is the tangent of declination to the tangent of inclination of Meridians . 6. All North decliners with centers have the angular point of the style downward , and all South have it upward . 7. All North decliners without centers have the narrowest end of the style downward , all South have it upward . 8. In all decliners without centers take so much of the style as you think convenient , but make points at its beginning and end upon the substyle of your paper draught , and transmit those points to the substyle of your plain , for direction in placing your style thereon . 9. In all North decliners the Meridian , or inclination of Meridians is the hour line of twelve at mid-night : in South decliners , at noon , or mid-day . This may tell you the true names of the hour lines . 10. In transmitting these Dials from your paper draught to your plain , lay the horizontal of your paper draught , upon the horizontal line of the plain , and prick off the hours and substyle . Sect. 5. The affections of direct reclining Plains inclining Plains . For South Recliners , and North Incliners . 1. The difference 'twixt your co-latitude , and the reclination inclination is the elevation , or height of the style . 2. When the reclination inclination exceeds your co-latitude , the contrary pole is elevated so much as the excess . Ex. gr . a North recliner , or South incliner 50. d. in lat . 52. 30. min. the excess of the reclination inclination , to your co-latitude is 12. d. 30. min. and so much the North is elevated on the recliner , and the South pole on the incliner . 3. When the reclination inclination is equal to the co-latitude , it s a polar plain . For South incliners , and North recliners . 1. The Sum of your co-latitude , and the reclination inclination is the styles elevation . 2. When the reclination inclination is equal to your latitude , it s an equinoctial plain , and the Dial is no more than a circle divided into 24. equal parts , having a wyer of any convenient length placed in the center perpendicular to the plain for the style . 3. When the reclination inclination is greater than your latitude , take the summe of the reclination inclination of your co-latitude from 180. and the residue , or remain is the styles height . But in this case the style must be set upon the plain , as if the contrary pole was elevated , viz. These North recliners must have the center of the style upward , and the South incliners have it downward . Note . In all South re-in-cliners North re-in-cliners , for their delineation the styles height is to be called the co-latitude , and then you may draw them as erect direct plains , for South , or North ( as the former rules shall give them ) in that latitude , which is the complement of the styles height . For direct East and West recliners incliners . 1. In all East and West recliners incliners , you may refer them to a new latitude , and new declination , and then describe them as erect declining plains . 2. Their new latitude is the complement of that latitude where the plain stands , and their new declination is the complement of their reclination inclination : But to know which way you are to account this new declination , remember all East and West recliners are North-East , and North-West decliners . All East and West incliners are South-East , and South-West decliners . 3. Their new latitude and declination known , you may by Sect 4. par . 5. finde the substyle from the Meridian , height of the style , angle of twelve and six , and inclination of Meridians , using in those proportions the new latitude and new declination instead of the old . 4. In all East and West recliners incliners with centers , the Meridian lies in the horizontal line of the plain ; in such as have not centers , its paralel to the horizontal line . Sect. 6. The affections of declining reclining Plains inclining Plains . The readiest way for these , is to refer them to a new latitude , and new declination , by the subsequent proportions . 1. To finde the new Latitude . As the radius is to the cosine of the plains declination , so is the co-tangent of the reclination inclination to the tangent of a fourth ark . In South recliners North incliners get the difference 'twixt this fourth ark , and the latitude of your place , and the complement of that difference is the new latitude sought . If the fourth ark be less than your old latitude , the contrary pole is elevated ; if equal to your old latitude , it s a polar plain . In South incliners North recliners the difference 'twixt the fourth ark , and the complement of your old latitude is the new latitude . If the fourth ark be equal to your old co-latitude , they are equinoctial plains . 2. To finde the new declination . As the radius is to the cosine of the reclination inclination , so is the sine of the old declination to the sine of the new . 3. To finde the angle of the Meridian with the Horizontal Line of the Plain . As the radius is to the tangent of the old declination , so is the sine of reclination inclination , to the co-tangent of the angle of the meridian with the horizontal line of the plain . This gives the angle for its scituation . Observe , in North incliners less than a polar , the Meridian lyes . above That end of the Horizontall Line contrary to the Coast of Declination .     below   South recliners more than a polar , the Meridian lyes . below That end of the Horizontall next the Coast of Declination .     above   North recliners less than an equinoctial , the Meridian lyes above That end of the Horizontalnext the Coast of Declination .     below In North recliners this Meridian is 12. at midnight .   equal to an equinoctial the Meridian descends below the Horizontal at that end contrary to the coast of Declination , and the substyle lies in the hour line of six .     South incliners more than an equinoctial , the Meridian lyes . below That end of the horizontal contrary to the declination .     above In South incliners this Meridian is only useful for drawing the Dial , and placing the substyle , for the hour lines must be drawn through the center to the lower side . After you have by the former proportions and rules found the new latitude , new declination , the angle and scituation of the meridian , your first business in delineating of the Dial will be ( both for such as have centers , and such as admit not of centers ) to set off the meridian in its proper coast and quantity . This done , by Sect. 4. Paragr . 5. of this Chapter , finde the substyles distance from the Meridian , the height of the style , angle of twelve and six ( and for Dials without centers , the inclination of Meridians ) in all those proportions , using your new latitude and new declination , instead of the old , and setting them off from the Meridian , according to the directions in Paragr . 3. and 4. you may draw the Dials by the following rules , for erect declining plains . In placing of them , lay the horizontal line of your paper draught upon the horizontal line of your plain , and prick off the substyle and hour lines . Only observe . That such South-East , or South-West recliners , as have the contrary pole elevated must be described as North-East , and North-West decliners , and such North-East , and North-West incliners as have the contrary pole elevated , must be described as South-East , and South-West decliners , which will direct you which way to set off the substyle , and hour line of six from the Meridian in those oblique plains , which admit of centers , or the substyle from the Meridian , and the inclination of Meridians from the substyle in such as admit not of Centers . 4. Declining polar plains must have a peculiar calculation for the substyle and inclination of Meridians , which is thus . To finde the Angle of the Substile with the Horizontal Line of the Plain . As the radius is to the sine of the polar plains reclination , so is the tangent of declination to the co-tangent of the substyles distance from the horizontal line of the plain . To finde the inclination of Meridians . As the radius is to the sine of the latitude , so is the tangent of declination to the tangent of inclination of Meridians . 5. The reclining declining polar hath the substyle lying below that end of the horizontal line that is contrary to the coast of declination . The inclining declining polar hath the substyle lying above that end of the horizontal line , contrary to the coast of declination . CHAP. III. The delineation of Dials for any plain Superficies . HEre it will be necessary to premise the explication of some few terms and symbols , which for brevity sake we shall hereafter make use of . Ex. gr . Rad. denotes the radius , or sine 90. or tangent 45. Tang. is the tangent of any arch or number affixed to it . Cos. is the cosine of any arch or number of degrees , or what it wants of 90. Ex. gr . cos . 19. is what 19. wants of 90. that is 71. Co-tang - is the co-tangent of any ark or number affixed to it , or what it wants of 90. Ex. gr . co-tangent 30. is what 30. wants of 90. that is 60. = This is a note of equality in lines , numbers , or degrees . Ex. gr . AB = CD . That is , the line AB . is equal unto , or of the same length as the line CD . Again , ABC = EFG . that is the angle ABC . is of the same quantity , or number of degrees , as the angle EFG . Once more AB = CD = FG = tang . 15. That is AB . and CD . and FG. are all of the same length , and that length is the tangent of 15. d. ♒ This is a note of two lines being paralel unto , or equidistant from each other . Ex. gr . FI. ♒ RS. that is the line FL. is paralel unto , or equi-distant from the line RS. Sect. 1. To delineate an horizontal Dial. See Fig. 9. First draw the square BCDE . of what quantity the plain will permit . Then make AF = AG = HD = HE = sine of the latitude , and AH = Radius . Enter HD . in tang . 45. and keeping the Sector at that gage , set off HI = HK = tang . 15. and HO = HL = tang . 30. Again , enter FD. in tang . 45. and set off FQ = GN = tang . 15. and FP = GM = tang . 30. This done , Draw AQ . AP. AD. AO . AI. for the hour lines of 5. 4. 3. 2. 1. afternoon . Again , Draw AK . AL. AE . AM. AN. for the hour lines of 11. 10. 9. 8. 7. before noon . The line FAG . is for six and six . In the same manner you may prick the quarters of an hour , reckoning three tangents , and 45. minutes , for a quarter . How to draw the hour lines before , and after six , was mentioned , Chap. 2. Sect. 1. Sect. 2. To describe an erect , direct South Dial. See Fig. 10. Draw ABCD. a rect-angle parallellogram . Then make AE = EB = CF = FD = cos . of your latitude . And EF = AC = BD = sine of your latitude . Enter CF. in tang . 45. and lay down FK = FL = tang . 15. and FI = FM = tang . 30. Again , enter AC . in tang . 45. and lay down AG = BO = 15. and AH = BN = tang . 30. with a ruler draw the lines EG . EH . EC . EF. EK . for the hour lines of 7. 8. 9. 10. 11. in the morning . and EO . EN . ED. EM . EL. for 5. 4. 3. 2. 1. afternoon , the line AEB . is for six and six . The line EF. for twelve . The description of a direct North-Dial differs nothing from this , only the hour lines from Sun rise to six in the morning , and from six in the evening , until Sun set , must be placed thereon , by drawing the respective morning and evening hours beyond the center as in the horizontal . See Fig. 11. Sect. 3. To describe an erect , direct East Dial. See Fig. 12. Having drawn ABCD. a rectangle paralellogram , fix upon any point in the lines AB . and CD . for the line of six , provided the distance from that point to A. being entred radius in the line of tangents , the distance from thence to B. may not exceed , nor much come short of the tangent 75. This point being found , enter the distance from thence to A. ( which we shall call 6. A. ) radius in the line of tangents , and keeping the sector at that gage , lay down upon the lines AB . and CD . 6. 11. = tang . 75. 6. 10 = tang . 60. and 6. 9 = 6. A. = tang . 45. and 6. 8. = 6. = 4. = tang . 30. Lastly , 6. 7 = 6. 5 = tang . 15. draw lines from these points on AB . to the respective points on CD . and you have the hours . To place it on the plain , draw the angle DCE . = co-latitude , and laying ED. on the horizontal line of the plain , prick off the hours . The same rules serve for delineation of a West Dial , only as this hath morning , that must be marked with afternoon hours . Sect. 4. To describe an erect declining Dial , having a Center . See Fig. 14. Draw the square BCDE , and make AC = AK of quantity what you please . Again , draw AG. 12. ♒ CE. and KHF. ♒ AG. 12. By a line of chords , set off the angles of the substyle , style , and hour of six from twelve ; having first found these angles by Chap. 2. Sect. 4. Paragr . 5. This done , make a mark in A 6. where it intersects KF - as at H. Then enter AK . in the secant of the plains declination , and keeping the Sector at that gage , take out the secant of the latitude , which place from A to G. upon the the line A. 12. and again , from H. ( which is the intersection of the paralel FK . with the line of six ) unto F. This done , lay a ruler to the points F. and G. and draw a line until it intersects CE. as FG. 3. Lastly , Enter GF . in tang . 45. and set off GL = GN = tang . 15. and GM = GO = tang . 30. Again , enter HF. in tang . 45. and set off HR = tang . 15. and HP = tang . 30. A ruler laid to these points , and the center , you may draw the hours lines from six in the morning unto three afternoon . For the other hour lines , do thus , Produce the line EC . and likewise HA. beyond the center , until they intersect each other as at S. Then setting off ST = HR . and S. 4. = HP . you have the hour points after three in the afternoon , until six , although none are proper beyond the hour line of four ; only by drawing them on the other side the center , they help you to the hour lines before six in the morning . Sect. 5. To describe an erect declining Plain without a Center . See Fig. 12. The delineation of these Dials is the most difficult of any , and therefore I shall be the larger in their description . 1. By Chap. 2. S. 4. Paragr . 5. finde the angles of the style , substyle , and inclination of Meridians . 2. Having found the inclination of Meridians , make the following Table for the distance of every hour line and quarter from the substyle . Where I take for example a North plain declining East 72. d. 45. min. in lat . 52. 30. min. Imcl . mer . hou . quart . 76 = 30. 12 00 20 = 15 3 ... 16 = 30 4 00 12 = 45 4 . 09 = 00 4 .. 05 = 15 4 ... 01 = 30 5 00 02 = 15 5 . 06 = 00 5 .. 09 = 45 5 ... 13 = 30 6 00 17 = 15 6 . 21 = 00 6 .. 24 = 45 6 ... 28 = 30 7 00 32 = 15 7 . 36 = 00 7 .. 39 = 45 7 ... 43 = 30 8 00 47 = 15 8 . 51 = 00 8 .. 54 = 45 8 ... 58 = 30 9 00 62 = 15 9 . 66 = 00 9 .. 69 = 45 9 ... 73 = 30 10 00 The manner of drawing this Table is thus . The inclination of Meridians ( which because its a North decliner , is twelve at midnight . ) I finde 76. d. 30. min. Now considering the Sun never riseth till more than half an hour after three in this latitude , I know that one quarter before four is the first line proper for this plain : Therefore reckoning 15. d. for an hour , or 3. d. 45. m. for a quarter of an hour , I finde three hours , three quarters ( the distance of a quarter before four from midnight ) to answer 56. d. 15. min. which being subtracted from 76. d. 30. min. the inclination of Meridians there remains 20. d. 15. m. for the distance of one quarter before four from the substyle . Again , from 20. d. 15. min. subtract 3. d. 45. min. ( the quantity of degrees for one quarter of an hour ) and there remains 16. d. 30. min. for the distance of the next line from the substyle , which is the hour of four in the morning . Thus for every quarter of an hour continue subtracting 3. d. 45. min. until your residue or remain be less than 3. d. 45. min. and then first subtracting that residue out of 3. d. 45. min. This new residue gives the quantity of degrees for that line on the other side the substyle . Now when you are passed to the other side of the substyle , continue adding 3. d. 45. min. to this last remain for every quarter of an hour , and so make up the table for what hours are proper to the plain . 3. Draw the square ABCD. of what quantity the plain will admit , and make the angle CAG equal to the angle of the substyle with twelve . Again , cross the line AG. in any two convenient points , as E. and F. at right angles by the lines KL . and CM . 4. Take the distance from the center unto 45. the radius to the lesser lines of tangents , which is continued to 76. on the Sector side , enter this distance in 45. on the larger lines of tangents , and keeping the Sector at that gage , take out the tang . 20. d. 15. min. ( which is the distance of the first line from the substyle ) set this from 73. d. 30. min. ( the distance of your last hour line from the substyle , as you see by the Table ) toward the end upon the lesser line of tangents , and where it toucheth as here at 75. 05. call that the gage tangent . 5. Enter the whole line KL . in the gage tangent , which in this example is 75. d. 05. min. and keeping the Sector at that gage , take out the tangent 73. 30. min. which is your last hour , and set from L. on the line KL . unto V. Again , take out the tangent 20. d. 15. min. which is your first line , and set it from K. towards V. and if it meet in V. it proves the truth of your work , and a line drawn through V. paralel unto AG. is the true substyle line . Then keeping the Sector at its former gage , set off the tangents of the hours , and quarters ( as you finde them in the Table ) from V. towards K. and from V. towards L. making points for them in the line KL . Lastly , enter the radius of your tangents to these hour points in the radius of secants , and set off the secant of the styles height from V. to T. Thus have you the hour points and style on one line of contingence . To mark them out upon the other line do thus . Set the radius to the hour points upon the former line of contingence , from h. to p. on the line ChM. and entring hV. as Radius in the line of tangents , take out the tangent of the styles height , and set from p. to r. Again , enter hr. radius in your line of tangents , and keeping the Sector at that gage , take out the tangents for each hour , and quarter , according to the table , and lay them down from h. to the proper side of the substyle toward C. or M. and applying a ruler to the respective points on KL . and CM . draw the lines for the hours and quarters . Lastly , enter hr. radius on the lines of secants , and taking out the secant of the styles height , set it from h. to S. and draw the line ST . for the style . Sect. 6. To describe a direct polar Dial. See Fig. 15. Draw BCDE . a rectangle paralellogram , from the middle of BC. to the middle of DE. draw the line 12. or substyle , appoint what place you please in BC. or CD . for the hour point of 7. in the morning , and 5. afternoon . Then , Enter 12. 7 = 12. 5. In the tangent 75. and set off 12. 1. = 12. 11. = tang . 15. and 12. 2. = 12. 10 = tang . 30. and 12. 3. = 12. 9. = tang . 45. Lastly , 12. 4. = 12. 8. = tang . 60. From these points draw the hour lines of 7. 8. 9. 10. 11. 12. 1. 2. 3. 4. 5. which are all the hours proper for these plains . Sect. 7. To draw a declining Polar . See Fig. 16. 1. By Chap. 2. Sect. 6. Par. 4. finde the inclination of Meridians , and distance of the substyle from the horizontal . 2. By Chap. 3. Sect. 5. Par. 2. make a Table for the distance of the hour points from the substyle . 3. Draw the square BCDE . Set off the angle CAG . for the substyle , and cross that substyle line at right angles in any two convenient places , as at H. and K. with the lines PHS. and RKT. for contingent lines . 4. Take any convenient length for your styles height , and enter it radius in your line of tangents , keeping the Sector at that gage , prick off the hours from the substyle ( by your table ) upon both the contingent lines . Draw lines by the points in both contingents , and you have the hours : For all other declining reclining inclining plains , it would be needless ( I presume ) to insist upon the description of them : Sith so much hath been already mentioned , Chap. 2. S. 6. that there can scarcely be any mistake , unless through meer wilfulness , or grandnegligence . CHAP. IV. To determine what hour lines are proper for any plain Superficles . By projection of the Sphere . See Fig. 17. DRaw the Circle NESW . representing the Horizon , and crossing it into quadrants N. is North. S. South , E. East , W. West , NS . the Meridian ( which let be infinitely produced ) Z. the center represents the Zenith . To finde the pole set half the co-tangent of the latitude from Z. toward N. it gives the point P. for the pole or the point through which all the hour lines must pass . The Suns declination in Cancer subtracted from the latitude , and the tangent of half the remain set from Z. to Π. gives the intersection of Cancer with the Meridian . Again , adde the complement of the Suns delineation in Cancer , unto the complement of your latitude , and the tangent of half that sum set from Z. to ψ , gives the diameter of that tropick , half ψ , is the radius to describe it . Half the tangent of your latitude set from Z. to Q. gives that point for the intersection of the equator with the Meridian ; and the co-secant of the latitude set from oe . toward N. gives the point ζ. the center of the equator . Adde the Suns greatest declination ( or his declination in Capricorn ) to the latitude , and the tangent of half that sum set from Z. toward S. gives the point φ , where Capricorn intersects the Meridian . Subtract the Suns declination in Capricorn from your latitude , and that remain subtract from 180. the tangent of half this last remain , set from Z. toward N. gives the point T. the diameter of Capricorn , and half the distance T φ. is the radius to describe it . Set the secant of the latitude from P. towards S. it gives the point H. the center of the hour line of six cross the line ZSH. at right angles in the point H. Then entring PH. Radius on the lines of tangents , set off the hour centers both wayes from H. reckoning 15. d. for an hour . Lastly , setting one point of your Compasses in these center points , extend the other to P. and with that radius describe the hour lines . Thus have you the sphere projected , the following Sections will determine the hours for all plains . Sect. 1. To determine the hour lines for erect direct plains . Fig. 17. The line NS . represents an erect East , and West plain . That side next W. is West , the other side next E. is East , where you may see that the Sun shines upon the East until twelve , or noon , and at that time comes upon the West . The fine WE . represents a direct North and South plain , the side next N. is North , the other next S. is South , where the North cuts the tropick of Cancer ( which in the hour lines you finde 'twixt 7. and 8. in the morning ; and again 'twixt 4. and 5. afternoon ) is the time of the Suns going off , and coming on that plain . Where the South cuts the equator , which is in the points of six , and six is the time of the Suns going off , and coming on that plain . Sect. 2. To determin the hour lines for direct reclining inclining Plains . Fig. 17. NBS . on the convex side is a West incliner , where it cuts Capricorn , is the time of the Suns coming on that plain , afternoon . On the concave side its an East recliner , where it cuts Cancer , is the time of the Suns going off that plain , afternoon . NCS. On the convex side is an East incliner , where it cus Capricorn , is the time of the Suns going off in the morning . On the concave side it is a West recliner , where it cuts Cancer , is the time of the Suns coming on in the morning . WDE. On the convex side is a South incliner , where until D. reach below oe . it hath all hours from six to six , and until D. reacheth below φ. it may have the twelve a clock line . But when D. reacheth below oe . draw a paralel of declination to pass through the point D. and the intersection of that paralel with the limb of the circle NE SW . doth among the hour lines , shew the time of the Suns coming upon that plain in the morning , and going off again afternoon , when D. reacheth below φ. the intersection of the ark WDE. with the tropick of Capricorn , shews the time of the Suns going off that plain before noon , and coming on again , afternoon . And the intersection of the tropick of Capricorn with the limb shews the first hour in the morning the Sun comes on , and the last hour afternoon , that it staves upon that plain . The convex side of WDE. is a north recliner , where it cuts Cancer , is the time of the Suns going off in the morning , and coming on again afternoon . WFE. On the convex side is a North incliner , where it cuts Cancer , is the time of the Suns going off in the morning , and coming on afternoon . On the concave side is a South recliner , where until F. reach beyond P. it enjoyes the Sun only from six to six . When F. reacheth beyond P. where the ark cuts Cancer , you finde how much before six in the morning the Sun comes on , or after six at evening it goes off . To draw any of these arks , Ex. gr . the ark NBS . do thus . Set the tangent of half the reclination inclination from Z. on the line ZW . and it gives the point B. produce ZE. and set the co-secant of the reclination inclination from B. towards E. which reacheth to G. then G. is the center . & GB . the radius to draw that ark . Note . The Semidiameter of the circle SENW . is radius to all the tangents , and secants , which you make use of for placing any oblique plain upon the Scheme . Sect. 3. To determin the hour lines for erect declining plains . Fig. 18. For South-East , or North-West plains . By a line of chords set the angle of declination on the limb from W. toward S. as H. lay a ruler to HZ . and draw HZK. which on that side next NW . represents the North-West , and where the line cuts Cancer , you have the time of the Suns coming on afternoon , and staying until Sun set . But if it cut Cancer twice , then in the morning hours it shews what time the Sun goes off this plain , having all hours from Sun rise to that time , and in the evening hours you have the time of the Suns coming on again , and staying till sun set . That side HZK. next SE. represents a South-East . Where the line cuts the equator in the evening hours , is the time of the Suns going off , where it cuts Cancer , in the morning hours , is the time of the Suns coming on that plain . For North-East , or South-West , set the declination by a line of chords from E. towards S. as L. lay a ruler to ZL . and draw the line LZR. which on that side next NE. is North-East , where it cuts Cancer , is the time of the Suns going off in the morning . If it cuts Cancer twice , you have in the evening hours the time of the Suns coming on again , and staying until Sun-set . On that side next SW . is the South-west . Where it cuts the equator , or Capricorn in the morning hours , is the time of the Suns coming on , where it cuts Cancer in the evening hours , is the time of the Suns going off . Sect. 4. To determin the hours of declining reclining Plains inclining Plains . Fig. 18. First , set in the plain according to its declination . By Sect. 3. Ex. gr . LZR. a North-East , or South-West declining 50. d. 00. min. This done Cross the line LZR. representing the declination of the plain , at right angles in the point Z. as CZBG . Then for North-East incliners , or South-West recliners , set half the tangent of the reclination inclination from Z. toward C. 〈◊〉 T. and set the co-secant of the reclination inclination from T. toward B. as TG . Then G. is the center , and GT . the radius to describe the ark RTL. Whose convex side represents a North-East incliner , where it cuts Libra or Capricorn , is the time of the Suns going off in the morning ; if it cuts Cancer twice , the intersection of Cancer with the evening hours shews what time the Sun comes again upon such a plain afternoon , and continues till Sun setting . The concave side is a South-West recliner , where it cuts Cancer in the morning hours , is the time of the Suns coming on , in case it intersects Cancer twice in the evening hours , you have the time that the Sun goes off . For a North-East recliner , or South-West incliner , set the point T. from Z. toward B. and the point G. from T. ( so placed ) toward C. and draw the ark on that side RZL. toward B. whose convex side will represent a South-West incliner , and where the ark cuts the equator or Capricorn , you have the time of the Suns coming on that plain . The concave side is a North-East recliner , where the ark cuts Cancer , is the time for the Suns going off that plain . When the ark cuts Cancer twice , the Sun comes on again before it sets . For a North-West recliner , or South-East incliner . Enter the declination by Sect. 3. as HZK. Cross it in the point Z. at right angles , as OZD. set half the tangent of the reclination inclination from Z. toward O. as V. and the co-secant of the reclination inclination from V. toward D. as F. then is F. the center , and FV. the radius to draw the ark HVK. Where it cuts Cancer the hour lines , tell you the time of the Suns going off in the morning , and entring again afternoon , upon the North-West recliner . Where it cuts the equator you have the time of the Suns going off the South-East , where it cuts Cancer in the morning hours is the time of the Suns coming on that plain . For a North-West incliner , or South-East recliner , set the point V. from Z. toward D. and the point F. set from V. ( so placed ) toward O. and draw the ark on that side Z. next D. Then where the convex side cuts Cancer , you have the time of the Suns going off in the morning ; and coming on again afternoon upon the North-West incliner . Where the concave side cuts the equator , you have the time of the Suns going off the South-East recliner ; where it intersects Cancer , is the time of his coming on that plain in the morning . Note . All the precedent rules about plains are appropriated to us that live in Northern Hemisphere , In case any one would apply them to the South Hemisphere : What is here called North , there name South , and what we here term South , there call North , and the rules are the same . — Si quid novisti plenius istis , promptius istis , rectius istis , Candidus imperti : Sinon , His u●ere mecum . FINIS . 
A52120	----	The country-survey-book: or Land-meters vade-mecum Wherein the principles and practical rules for surveying of land, are so plainly (though briefly) delivered, that any one of ordinary parts (understanding how to add, substract, multiply and divide,) may by the help of this small treatise alone and a few cheap instruments easy to be procured, measure a parcel of land, and with judgment and expedition plot it, and give up the content thereof. With an appendix, containing twelve problems touching compound interest and annuities; and a method to contract the work of fellowship and alligation alternate, very considerably in many cases. Illustrated with copper plates. By Adam Martindale, a friend to mathematical learning. Martindale, Adam, 1623-1686. 1692 Approx. 245 KB of XML-encoded text transcribed from 96 1-bit group-IV TIFF page images. Text Creation Partnership, Ann Arbor, MI ; Oxford (UK) : 2006-02 (EEBO-TCP Phase 1). A52120 Wing M854A ESTC R217468 99829133 99829133 33569 This keyboarded and encoded edition of the work described above is co-owned by the institutions providing financial support to the Early English Books Online Text Creation Partnership. This Phase I text is available for reuse, according to the terms of Creative Commons 0 1.0 Universal . The text can be copied, modified, distributed and performed, even for commercial purposes, all without asking permission. Early English books online. (EEBO-TCP ; phase 1, no. A52120) Transcribed from: (Early English Books Online ; image set 33569) Images scanned from microfilm: (Early English books, 1641-1700 ; 1917:11) The country-survey-book: or Land-meters vade-mecum Wherein the principles and practical rules for surveying of land, are so plainly (though briefly) delivered, that any one of ordinary parts (understanding how to add, substract, multiply and divide,) may by the help of this small treatise alone and a few cheap instruments easy to be procured, measure a parcel of land, and with judgment and expedition plot it, and give up the content thereof. With an appendix, containing twelve problems touching compound interest and annuities; and a method to contract the work of fellowship and alligation alternate, very considerably in many cases. Illustrated with copper plates. By Adam Martindale, a friend to mathematical learning. Martindale, Adam, 1623-1686. Collins, John, 1625-1683. [12], 225 [i.e. 226], 229-234 p., plates, folding printed for R. Clavel, at the Peacock in St. Pauls Church-yard, and T. Sawbridge, at the Three Flower-de-luces in Little-Britain, London : 1692. "An appendix containing XII. problems touching compound interest & annuities" has separate title page with imprint: London, printed in the year 1692. Register and pagination are continuous. Errata on p. [12]. "To the reader" signed: John Collins. Page 226 misnumbered 225. Reproduction of the original in the British Library. Created by converting TCP files to TEI P5 using tcp2tei.xsl, TEI @ Oxford. Re-processed by University of Nebraska-Lincoln and Northwestern, with changes to facilitate morpho-syntactic tagging. Gap elements of known extent have been transformed into placeholder characters or elements to simplify the filling in of gaps by user contributors. 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Copies of the texts have been issued variously as SGML (TCP schema; ASCII text with mnemonic sdata character entities); displayable XML (TCP schema; characters represented either as UTF-8 Unicode or text strings within braces); or lossless XML (TEI P5, characters represented either as UTF-8 Unicode or TEI g elements). Keying and markup guidelines are available at the Text Creation Partnership web site . eng Surveying -- Early works to 1800. Interest -- Early works to 1800. Mathematics -- Early works to 1800. 2005-07 TCP Assigned for keying and markup 2005-08 Aptara Keyed and coded from ProQuest page images 2005-12 Judith Siefring Sampled and proofread 2005-12 Judith Siefring Text and markup reviewed and edited 2006-01 pfs Batch review (QC) and XML conversion THE Country-Survey-Book : OR LAND-METERS VADE-MECVM . WHEREIN The PRINCIPLES and practical RULES for Surveying of Land , are so plainly ( though briefly ) delivered , that any one of ordinary parts ( understanding how to add , subtract , multiply and divide , ) may by the help of this small Treatise alone , and a few cheap Instruments easy to be procured , Measure a parcel of Land , and with judgment and expedition Plot it , and give up the Content thereof . WITH An APPENDIX , containing Twelve Problems touching Compound Interest and Annuities ; and a Method to Contract the work of Fellowship and Alligation Alternate , very considerably in many Cases . Illustrated with Copper Plates . By ADAM MARTINDALE , A Friend to Mathematical Learning . Frustra sit per plura quod fieri potest per pauciora . London , Printed for R. Clavel , at the Peacock in St. Pauls Church-yard , and T. Sawbridge , at the Three Flower-de-luces in Little-Britain , 1692. TO THE Right Honourable THE LORD DELAMER , Baron of Dunham-Massie , &c. My Lord , THIS small Tract comes to Your Lordship , not as to a Patron to protect its Errors ( if any such there be ) but as to a critical noble Friend , that will be sure faithfully to tell the Author of them . Which favour , together with a chearful acceptance of this poor present , he humbly hopes for , because of its Relation , being writ at Dunham , by Your humble Servant , who besides his domestick dependance , cannot forbear without ingratitude to tell the World that Your Lordship's kindness hath very much encouraged and assisted him in Mathematical Studies , not only by a free Communication of many a choice notion both vivâ voce , and by the loan of Manuscripts . But also by a Considerable number of excellent Books and costly Instruments bountifully bestowed upon him : Who wanting other ways to express his many singular Obligations , and deep sence thereof , humbly offers this Punie Treatise for Your Lordships diversion at spare hours ; and is ambitious to write himself , Oct : 26. 1681. MY LORD , An humble and faithful ( though unworthy ) Servant to all Your Lordship's Noble Family A. M. Mr. COLLINS TO THE READER . Courteous Reader , THE Learned Mr. Adam Martindale formerly Writ two excellent Almanacks , called Country Almanacks , which were Printed , and esteemed by several Members of the Royal Society very useful , especially for Country Affairs , but meeting with some Discouragements from such as knew not how to judge of the Authors worth , he gave over that undertaking , contrary to the desires of many Ingenious Men. And having since Writ a little Treatise of Survey , in which he hath had experience , as well as Theory , and being willing to have the Approbation or Dislike of others , that it might either be Printed or Stifled , imparted the same to some of the Members of the aforesaid Illustrious Body . Upon the Perusal of whom and divers Experienced Artists who make a Livelihood of it , I find it well approved ( as clear and concise ) only the latter were sorry so much was discovered , as detrimental to their Practice , particularly about setting off the outjettings , where it 's inconvenient intirely to measure them . The Book is small , the price ( though not the worth ) mean , no small Encouragements to young Students and the Vulgar , for whom it was chiefly intended : By which , that they may reap Benefit , is the hearty desire of ( a Well-willer to the Author and them ) John Collins . The PREFACE . READER , THe Title sufficiently informs thee in general of my Designs : But I confess I owe thee a a more particular Account , not only of that , but several other things which I shall briefly give thee . 1. I have observed that the Country aboundeth with such , as by their Inclination and Interest are prevailed with to take pains in measuring Land , that for want of better Instruction use ill divided Chains and tedious Methods of Computation , which makes their work intolerable troublesom , if exactness be required . And some for want of skill in the Fundamentals of Geometry , have imbibed prodigiously false Principles , as this for one , viz. That the Content of any Close , of what Figure soever , may be found by Squaring a quarter of the Perimeter . Mathematical-Schools , where better things might be learned , are very rare , and an able Artist to instruct one in private is hard , and charge able to be procured . Excellent Books indeed there are in our English Tongue , Written by our Famous Rathborn , Wing , Leybourn , and Holwell , to which may be added Industrious Mr. Atwells Treatise , and some part of Capt. Sturmy's : But those I rather esteem fit to be read byan able Artist ( towards his perfecting ) than by a new beginner , for in the best of those Books he will find the most useful and plain Rules so intermixed with others that are less necessary , and more intricate , ( though very excellent for their proper ends ) and so many Curiosities touching Trigonometry , Transmtation of Figures , &c. which his business never calls for , that for want of judgment to pick out that which fits his present purpose , and to study higher Speculations afterwards , he is apt to be confounded and discouraged ; whereby it accidentally comes to pass that plenty , makes him poor . Besides , three of these six Books are in Folio , another in Quarto , and the other two ( though Octavo's ) too large for ordinary carrying in a Pocket , and in that regard not so convenient for one that hath much occasion to be out of his own House ; to say nothing of their Price , which to some poor Youths is not the least discouragement . I have therefore made my Book so little , that the Price can neither much empty the Pocket , nor the Bulk overfill it . And yet so plain , that I doubt not to be understood by very ordinary Capacities . My Method is fitted to my Design : Beginning with the Principles of the Art , and so proceeding gradatim till I have shewed how all ordinary Figures may be Measured , Protracted , and Cast up , without any other Instrument of charge but Chains , Compasses and Scales . Afterwards for such as desire higher Attainments , I have endeavoured to speak so fully ( in a little compass ) of the Plain Table , and given such hints , applicable not only to it , but also to the Peractor , Theodolite , and Semicircle , as that an Ingenious Person may make great use of them . But as touching the Doctrine of Triangles , and Transmutation of Figures into others equipollent , with the large Tables ( referring to the former ) of Logarithms of Numbers , Sines , and Tangents , I thought it improper to cumber this small Manual , or the unlearned Reader with them , having ( as I humbly hope ) sufficiently inform'd him how to find all his Sides and Angles by Instruments , and also the Content of any Figure without such transmutation , reserving such Curiosities and many others touching Drawing and Painting of Maps ; Measuring of Ways , and Rivers , &c. to a Second Part , which I may perhaps hereafter Publish upon due encouragement , but if I do not , the Curious may find themselves good store of work in the Authors even now quoted . 2. It may seem strange to some , that in referring to the Figures , I sometimes use Words seeming to imply that the Figure I speak of is in that very Page , and so it was in my Copy , but the Printer and Gravers have otherwise contrived them for convenience in Copper Cuts by themselves . And to give them their due they are generally done with great accuracy , and none of them having any such error as is like to beget trouble or mistake to the Reader , saving only that fig. 19 hath D instead of O at the Center , and the Line OL in the Margin of Fig. 14. should be of the length from L to the uppermost o in the Scale , and the Figures on the side should be made 1 less than they are , viz. 2 should be 1 , 3 made 2 , &c. And lastly , as to the Errata , though I have not been so anxiously careful , as to correct every literal mistake , I have very diligently perused all from p. 1 to p. 224 inclusive , and hope I have sufficiently restored the Sense to the places wronged , when thou hast done them right by the Pen according to the Directions of the Errata following next after the contents , and that you continue the Line in the Margin of p. 34. to the length of the Line OL in fig. 14. THE CONTENTS OF THE CHAPTERS . Chapt. 1. OF Geometrical Definitions , Divisions , and Remarks . p. 1. Chapt. 2. Of Geometrical Problems . p. 6. Chapt. 3. To find the Superficial Content of any right lined Figure , the lines being given . p. 17. Chapt. 4. Concerning Chains , Compasses , and Scales . p. 26. Chapt. 5. How to cast up the Content of a Figure , the lines being given in Chains and Links . p. 35. Chapt. 6. How to measure a Close , or parcel of Land , and to protract it , and give up the Content . p. 41. Chapt. 7. Concerning the measuring of Circles and their parts . p. 48. Chapt. 8. Concerning Customary measure , and how it may be reduced to Statute measure , & e Contra , either by the Rule of Three , or a more compendious may by Multiplication only . p. 52. Chapt. 9. How a Man may become a ready Measurer by Practice in his private Study , without any ones assistance or observation , till he design to practise abroad . p. 65. Chapt. 10. How to measure a piece of Land with any Chain of what length soever , and howsoever divided ; yea with a Cord or Cart-rope ; being a good Expedient when Instruments are not at hand of a more Artificial ●ake . p. 67. Chapt. 11. Concerning dividing of Land Artificially and ●echanically . p. 70 Chapt. 12. Concerning the Boundaries of Land , where the ●ines to be measured must begin and end . p. 80. Chapt. 13. Containing a Description of the Plain-Table , the ●rotractor , and Lines of Chords . p. 82. Chapt. 14. How to take the true Plot of a Field by the ●lain-Table upon the Paper that covers it , at one or ●ore Stations . p. 85. Chapt. 15. Concerning the plotting of many Closes together , ●hether the ground be even or uneven . p. 99. Chapt. 16. Concerning shifting of Paper . p. 102. Chapt. 17. Concerning the plotting of a Town Field , where 〈◊〉 several Lands , Buts , or Doles , are very crooked : ●●th a Note concerning Hypothenusual or sloping ●●undaries , common to this and the fifteenth Chapter . ● 104. Chapt. 18. Concerning taking the plot of a piece of ground 〈◊〉 the Degrees upon the Frame of the Plain-Table se●●●al ways , and protracting the same . p. 108. Chapt. 19. Concerning taking inaccessible Distances by the ●●ain-Table , and accessible Altitudes by the Protractor . ● 121. Chapt. 20. Of casting up the Content of Land by a Table . ● 193. ERRATA . PAge 3 Line 16 Read Trilaterals . p. 4 〈◊〉 Geodates . p. 5 l. 29 Eneagon . p. 10 l. 6 belong as the other two . p. 16 l. 6 Centers at right A●●gles . p. 28 l. 1 forefinger . p. 36 l. 20 Poles or R●● p. 47 l. 15 fourth Diagonal and the sixth side . p. 〈◊〉 l. 28 as in this figure is ABC . p. 56 l. 23 , 〈◊〉 and l. 24 ● . 22 ½ p. 58 l. 11 28. p. 69 l. 5 side 74 l. 25 FG. p. 77 l. 27 138562 , and l. 28 〈◊〉 33 r. 242030. p. 79 l. 33 triangulate . p. 80 l ▪ former . p. 82 l. 24 fitted . p. 86 l. 2 Stationary ●●●stances . p. 90 l. 15 Chart , or Card. p. 95 l. ● Park , Pond . p. 103 l. 22 Line . p. 104. l. 19 〈◊〉 Fig. 1. Fig. 2. Fig. 3. Fig. 4. Fig. 5. Fig. 6. Fig. 7. Fig. 8. Fig. 9. Fig. 10. Fig. 11. Fig. 12. Fig. 13. Fig. 14 Fig. 15. Fig. 16. Fig. 17. Fig. 18. Fig. 19. Fig. 20. Fig. 21. Fig. 22. Fig. 23. Fig. 24. Fig. 25. Fig. 26. Fig. 27. Fig. 28. Fig. 29. Fig. 30. Fig. 31. Fig. 32. Fig. 33. Fig. 34. Fig. 35. Fig. 36. Fig. 37. Fig. 38. Fig. 39. THE Country Survey-Book : OR , LAND-METER's VADE-MECVM . CHAP. I. Of Geometrical Definitions , Divisions , and Remarks . I. A Point is that which hath no parts , either of longitude or latitude , but is indivisible , ordinarily expressed with a small prick , like a period at the end of a sentence . II. A Line hath length , but no bredth nor depth , whose limits or extremities are Points . This is either right or crooked . III. A right Line lies straight , and equal between its extreme points , being the shortest extension between them ; the crocked or circular not so . IV. A Superficies hath length and bredth , but no depth ; of this Lines are the limits . V. A plain Superficies is that which lieth equally ( or evenly ) between its Lines . VI. An Angle is the Meeting or two Lines in one point , so as not to make one straight Line , and if drawn on past that point , they will intersect or cross one another . This is vulgar English may be called a Corner ; of which there be two sorts , one right , the other oblique . VII . A right Angle is that which is made by two right lines , crossing or touching one another perpendicularly , ( or squarely ) like an ordinary Cross , or Carpenters Square . VIII . An oblique Angle is that which is either greater or less than a right Angle , and this is of two sorts , obtuse and acute . IX . An obtuse Angle is greater than a right Angle , like the left and right Corners of a Roman X. X. An acute Angle is less than a right Angle , like the highest and lowest Corners of the same Letter . XI . A Figure is that which is comprehended under one line or many : Of this there are two kinds , a Circle and a right-lined Figure . XII . A Circle is a perfect round Figure , such as is drawn with a pair of Compasses , the one Foot being turned round in a point , and the other wheeled about it . The point in the precise middle is called the Center ; the round line , the Circumference or Peripheri ; a line going through the Center , and divide the Circle into two equal parts , is called the Diameter ; half of that line is a Somidiameter , or Radius ; half the Circle is stiled a Semicircle ; the quarter , a Quadrant ; any portion of it , cut off by a right Line not touching the Center , is called a Segment . XIII . Right-lined Figures are such as are limited by three right Lines or more , and are either Triangles or Triangulate , that is , such as are compounded of , and resolvable into Triangles . XIV . Triangles are Figures comprehended under three right Lines , and ( as Ramus thinks for a Reason that he gives , lib. 6. pr. 6. ) might be better called Trilaterals ; but the name Triangle from the number of the Angles hath obtained . Also from the nature and Quantity of their Angles these Triangles are distinguished into three sorts : 1. Rectangled , having one right Angle ; 2 Obtuse-angled , having one obtuse Angle ; and 3. Acute-angled ; having all acute Angles ; for no Triangle can have more right or obtuse Angles than one , because by by an old Rule ( easie to be demonstrated ) no Triangle upon a plain Superficies can consist of three greater Angles than such , as being jointly taken are equal to two Right . These three sorts of Triangles may , according to the length and proportion of their sides , be subdistinguished into seven ; for each of them may have either two equal Sides or none , and the Acute-angled may have all three Sides or lines equal : To all which kinds , learned Men give distinct Greek Names , which if mine English Reader have a mind to see , they are to be found in 〈…〉 Practices , Book 1. page . 6. for my present purpose the above-mentioned ●rimembred distinction will abundantly suffice ; for be Triangles of what name or kind soever , they are all capable of being exactly measured by one plain Rule as hereafter shall fully appear . XV. Triangulare Figures are such as have more Angles ( and consequently more Sides or Lines ) than three : and these are either Quadrangular or Multangular . XVI . Quadrangular Figures are such as have fo●● Angles ( and as many Side ) and these are either Parallellograms or Trapezia's . XVII . Parallellograms are Figures that are bounded with parallel Lines , that is , such lines as are every where of the same distance one from another , so as if they were infinitely extended they would never meet , like the upright lines of he Roman H. These Parallellograms are either Rectangular or Obliquangular . XVIII . Rectangular Parallellograms are such as have four right Angles . viz. the Square or Quadrat , and the long Square , otherwise called the Oblong . XIX . The Square is that Figure that hath four right Angles , and four equal Sides , like any of the six Faces of a Die. XX. The long Square hath also four right Angles , and the oposite Sides are equal , but the adjoyning Sides meeting at each Angle differ in length . Of this Figure is a well printed Page in a Book , and the Superficies of a well cut Sheet of Paper , or an ordinary Pane of Glass . XXI . Obliquangled Parallellograms are such as have oblique Angles , viz. two acute , and two obtuse . Of these there are two kinds , the Rhombus , and the Rhomboides . XXII . The Rhombus is a Figure that hath equal Sides , but no right Angles , ( like the form of a Diamond on the Cards , or the most ordinary Cut of Glass in Windows ) whose oposite Angles are equal . XXIII . The Rhomboides is ( as it were ) a defective Rbombus , for if from any side of a Rhombus we cut off a part with a parallel Line , the Remainder will be a Rhomboides , which hath neither equal Sides nor Angles , but yet the opposite Sides and Angles are equal . XXIV . The Trapezium is a Figure that is neither parallellogram , nor ( consequently ) hath equal Sides or Angles , but is irregularly quadrangular , as if drawn at adventure . Of this shape most Fields prove , that seem to the Eye to be Squares or Oblongs . XXV . Multangular Figures are such as contain more Sides and Angles than four , and they are either regular or irregular . XXVI . Regular Multangulars take their names from their Number of Angles , so a Pentagon , Hexagon , Heptagon , Octogon , Encagon , Decagon , signifie a multangular Figure of five , six , seven , eight , nine , ten , Angles , and consequently Sides . XXVII . An irregular Polygon or multangular Figure , is that which hath more Angles ( and Sides ) than four , the Sides ( and Angles ) being unequal to one another . CHAP. II. Of Geometrical Problems . I. To draw a Line parallel to another , at any Distance assigned . Fig. 1 OPen your Compasses to the Distance given and chusing two Points conveniently distant in the Line given , as here at A and B describe the Arches C and D , to whose convixity if you apply a Rule , the parallel Line is easily drawn . II. To raise a Perpendicular upon a Line given , or to cross that Line at right Angles in a Point assigned . Fig. 2 Suppose the point C in the Line AB were assigned for the Perpendicular ; open the Compasses to a convenient distance , and mark out the two points E and F in the line AB , then opening them some what wider , you may ( by setting one Foot in E and F severally ) describe the two Arches cutting one another at the point D , from which if you draw a Line to the point C , the work is done for the raising of a Perpendicular ; but if you be to cross the lines at right Angles , you may continue the line from D through C at pleasure . But if the said line AB had been given to be divided in the precise middle , by another Line crossing it at right Angles , the way were to set one Point of the Compasses in A and B severally , and having described two Arches above the line , intersecting one another as at D , do the like below the line AB from the same points and with the same extent of your Compasses , then through the several intersections ( a Rule being laid upon them ) a line may be drawn , cutting the given line exactly in the middle at right Angles . Note , That when one point of your Compasses stand in A , you may make both the Arches belonging to that Center above and below the line , and then removing the Compasses to B , you may cross them both . III. To raise a Perpendicular at the End of a Line . Fig. 3 Let OR be the line given , then to raise a Perpendicular at R , make five little equal divisions , and taking four of them with your Compasses , set one foot of your Compasses in R , and with the other describe the Arch PP ; then take the distance from R to 5 , and placing one Foot in 3 , with the other describe the Arch BB , intersecting the former in the point S ; then shall the line SR ( being drawn by a straight Rule ) be a Perpendicular to the line OR . IV. To let fall a Perpendicular upon a given Line from any Point assigned . Open your Compasses so as one Foot being set in the assigned point the other may go clear over the line given , and thereby describe an Arch cutting the line at two points ; then shall the half distance between those two points be the point to which the Perpendicular may be drawn from the point assigned . But if you think it too much pains to find the point of half distance by trial , you may help your self by the second Problem : For if you describe two Arches intersecting one another on the farther side of the line from the assigned point , placing ( to that purpose ) the Foot of your Compasses first in one of the Intersections of the given line , and then in the other ; you may by laying a Rule upon the assigned Point , and the Intersection of the two Arches , draw a Perpendicular from the said assigned Point , cutting the given Line at right Angles . Note , that all these Problems touching perpendiculars , aim at no greater matter , than what may be performed in a Mechanical way with exactness enough ( and much more neatly by avoiding unhandsome Pricks and Arches ) by the help of a small Square exactly made , ( or for want thereof a Plate Quadrant , or broad Rule , having a right Angle and true Sides ) for if you apply one Leg of such a Square to any Line , so as the Angle of the Square may touch the end of the said Line , or any other Point where the Perpendicular is to be raised , you may by the other Leg draw the Perpendicular . In like sort to let fall a perpendicular from a point assigned you need only to apply one Leg of the Square to the Line , so as the other may touch ( at the same time ) the assigned point whence you may draw the perpendicular , by that Leg that toucheth the Point : If the Angle of your Square be a little blunt either through ill making or long using , you must allow for it when you apply it to the point in a Line . And when you are drawing a Perpendicular , you must stop before you reach the given line , and then by applying the Leg of your Square to that part of the perpendicular already drawn , so as part of that Leg may pass clearly over the given Line , you may draw the rest of your perpendicular as exactly as if the Angle had been true . The like course is to be taken when a line is to be crossed by another drawn quite through it at right Angles . V. An Angle being given , to make another equal to it . Fig. 4 The Angle XAD being given , and a Line drawn at pleasure as is the lowest from the point E , open your Compasses to any convenient distance , and setting one foot in A describe the Arch BC. Then with the same extent setting one foot in E , with the other describe the Arch GH , long enough to equal or exceed the other . Then taking the distance BC between the points of your Compasses , set one in G , and with the other mark the point H in the Arch GH , through which point H a line being drawn from the point E , will make an Angle with the Line EG equal to the Angle given . Note , when we speak of the quantity of Angles , their equality , or unequality , we never regard the length of the Lines ; for if you extend or contract them at pleasure , the Angle is still the same . But that is the greatest Angle whose lines are farthest distant from one another , at the same distance from the Angular Point , or the place where its lines meet . VI. Any three Lines being given ( equal or unequal ) so as no one of them be longer than the other two joyned together to make a Triangle of them . Fig. 5 The Lines A , B , C , being given , set the line A from D to E ; then with your Compasses take the length of the Line B , and setting one Foot in D describe the Arch PO. This being done , take with your Compasses the length of the line C , and setting one foot in E with the other cross the former Arch at F , from which Intersection drawing Lines by a Rule to D and E , the Triangle is finished . Note , that if all the Sides , or two of them , be equal , the method is the same ; but the labour less , because we need not to take the same length twice over with the Compasses . VII . To find the Perpendicular of the Triangle , in order to the measuring of it . Fig. 6 Let the Line AB be accounted the Base , and from the Angle C let fall a perpendicular as was taught , Probl. 4. Upon that line at D , which is ready for taking off with Compasses and measuring on a Scale , of which hereafter in the Chapters of measuring the Content of Figures . But if we have no occasion to draw the perpendicular , but only to know the length of it , ( as it most frequently falls out in measuring ) no more is needful but to set one foot of the Compasses in the Angular point C , and extend the other to the Base AB , so as it may touch it , but not go beyond it ; then have we the perpendicular between the points of the Compasses . VIII . One Side being given , how to make a Square . Fig. 7 The Line CD being given , raise a perpendicular at C of the length ( at the least ) of the given line ; then taking the line CD between the feet of your Compasses , set it upon the perpendicular from the Angular Point C to A : With the same distance setting one foot in D describe the Arch OP . Lastly , with the same distance ( or extent ) set one foot in A , and with the other describe the Arch crossing the Arch OP in N , from which intersection a line drawn by a Rule to A , and another to D , finish the Geometrical Square or Quadrant ACDN. IX . To make a long Square , the length and bredth being given . This is so like the former , that a particular Figure is not necessary to conceive of it . Suppose each side of the Square in the last Problem to consist of 8 small equal parts , and you were to make a long Square whose length must be equal to a side thereof , viz. 8. and the bredth half so much given in a line thus — 4 ; then when you had drawn the line CD for the length , and raised the Perpendicular at C , you must take the shorter Line given for the bredth , and set it upon the Perpendicular from C upwards to a Point , which for distinction we shall call the Point E , imagining it so marked : With the same extent of the Compasses describe the Arch , placing ( to that purpose ) one foot in D. Lastly , extending your Compasses to the length of the line CD , set one foot in E , and with the other cross the Arch aforesaid . Then a right Line drawn from that Intersection to E , and another from the same to D , complete the long Square . X. To make a Rbombus , the Sides being given . Fig. 8 If the Angles be not limited draw any oblique Angle at pleasure , either Acute or Obtuse , as here the Angle BAC , which is Acute . Then let the Line OP be the length of a Side , which being taken with your Compasses , set it from the angular point A in both Lines to D and E , in which two points place a foot of your Compasses successively without altering them , viz. in D to describe the Arch FG , and in E to describe the Arch HI , crossing one another in the Point K , from which , right lines drawn to D and E , finish the Rhombus DAEK . Note , if any Angle be given , together with the Side ; to limit the shape and content , begin with that , and proceed as before : For you must know , that to make a Rhombus ( or Rhomboides ) like to another for Figure , or equal to it in Content , it is not sufficient to have the same Sides ; for the more oblique the Angles 〈◊〉 the farther will the Rhombus differ from a 〈◊〉 ( and the Rhomboides from a long 〈◊〉 are ) and the less will be the Content . But 〈◊〉 must have an Angle given , ( which will pro●●●e all the rest ) or else a Diagonal Line , which 〈◊〉 right Line passing through the Rhombus ( or ●omboides ) , from one opposite Angle to another , and dividing the Figure into two equal ●●●angles . If the former ( viz. an Angle ) be ●●n , I have shewed what use is to be made 〈◊〉 . If the latter , ( i. e. a Diagonal ) toget●●● with the length of the Sides , you may by ●●●ng the length of the Sides with your Com●●es , and setting a Foot in the ends of the ●●gonal Line , make a Triangle on the one side ●he Diagonal , by Probl. 6. and then another on the other side by the same problem , the ●●gonal being a common Base to them both ; this will give the Figure exactly . To make a Rbomboides , the Sides being given . Fi. 9 〈◊〉 neither Angle nor Diagonal be given , 〈◊〉 if either of them be limited , the case is ●●en to in the last problem ) make any Angle ●dventures as here ABC . Then supposing 〈◊〉 Lines given to be OP and QR , set the 〈◊〉 of the longer upon the Line BC from B 〈◊〉 , and the shorter on the Line BA to E , 〈◊〉 with the Compasses extended from B to E 〈◊〉 one Foot in D and describe the Arch FG. ●●ewise , with the Compasses extended to the 〈◊〉 of the Line OP , setting one Foot in E 〈◊〉 the other describe the Arch HI , intersecting Fig. 9 the former Arch at K , from which Intersection Lines drawn to D and E finish the Rhomboides . XII . To make a Trapezium , the Diagonal and Lines in order being given . Fi. 10 Let the Line HL be the Diagonal of a Trapezium , whose Sides are the Lines A , B , C , D , the Side A being counted the first , as that which takes its beginning from the point H , and the rest in the order as they are marked Alphabetically . Then with your Compasses set to the length of the Line A , place one Foot in H , and with the other describe the Arch EF. Next taking the length of the Line B , with the one Foot , o● your Compasses placed in L , with the other make the Arch GI intersecting the former at K , from which Point of Intersection , Lines drawn to H and L make the Triangle HKL . Then with the extent of the Line C , set one of the Feet of your Compasses at L , and describe the Arch OP . Lastly , setting them to the length of the Line D , and placing one Foot of your Compasses in H , with the other make the Arch SR intersecting the former at Q ; 〈◊〉 shall Lines drawn from Q to L and H make u● the Triangle LQH , and finish the Trapezium HKLQ . I could have been much briefer in this problem by referring to the sixth ; but this being of very great and frequent use , I desired to be very plain . XIII . To make a regular Polygon , otherwise called a regular multangular , or multilateral Figure , consisting of many equal Sides and Angles , viz. above four apiece . Being satisfied what shall be the distance between the Center and every Angle , with that distance describe a Circle , which being equally divided into as many Parts , as the Figure must have Angles ( or Sides , for they are equal in number ) and Lines drawn from the Points of Division within the Circle from Point to Point , ( ordinarily called Chords ) the Polygon is finished as in this Diagram . Fig. 11 Suppose an Heptagon , or multangular Figure of seven Sides , and as many Angles , be to be described , every Angle being designed to be distant from the Center A , seven Eighths , or three quarters and an half of an Inch ; with that distance describe the Circle BCDEFGH , which being divided into seven equal parts , and Lines drawn from Point to Point , the Heptagon BCDEFGH will be therein included . I shall rather leave my unlearned Reader to find out the Points of Division by many tryals , than to puzzle him with the Geometrical way for finding out Chords to that purpose ; nor shall I busie my self to tell him at large how he may divide 360 by the number of his Angles or Sides , and then finding in his Quotient the Degrees and Parts belonging to every Division , set them readily out by a Protractor , or ( for want thereof ) by a Line of Chords ; for I suppose him yet ignorant of such things . I shall therefore only tell him thus much : A Line drawn through the Circle at the Center divides it into two equal parts , which being crossed in the Center by another Line , the Circle will be parted into four equal Parts or Quadrants , and those by halving them into eight Parts . The extent of the Compasses whereby the Circle is drawn ( usually called the Radius or Semidiameter ) will divide it into six equal parts ; two whereof must be a third part , and half of one a twelfth part ; and these still easily capable of farther Division . XIV . Having the Sides of the Triangles whereof it consisteth , orderly given , to make an irregular multangler , or multilateral Figure . This will be more fully handled hereafter , when I come to shew the method of drawing plots of Ground : In the interim I will give you a Specimen of an irregular Pentagon . Fi. 12 Having the Lines of three Triangles given , ( which by a Rule hereafter to be mentioned are necessary to make up a five Figure ) lay down the greatest of the first , viz. 20 , from A to B for a Base , and by Probl. 6. make a Triangle of it and the other Lines 16 and 10. viz. the Triangle ABO . Secondly , you find by the number o over the first Line of the second Triangle that it is the common Base to them both , and therefore by the same Probl. 6. make the Triangle A B P of the Lines 20 , 14 , 18. Fi. 12 Lastly , finding the Base of the third Triangle to be the same with 18 , one of the Sides of the second , make the Triangle PBQ of the Lines 18 , 11 , 12 : So is the quinquangular Figure finished . How every Line is to be found in its due order in this or any other sort of multangular Figures , so as to give a true and exact account , not only of the superficial Content , but also of the Figure ( or shape ) and situation , is to be taught hereafter in the Doctrine and Practice of protraction . CHAP. III. How to find the Superficial Content of any Right-lined Figure , the Lines being given . AS a Foundation to what I shall say upon this Subject , there are some few Geometrical principles or Theorems out of Enclid and Ramus , which I desire may be remembred ; and because understanding is a mighty help to memory , I design for my Country Reader a kind of ocular Demonstration , which though not so strict and artificial as that which is to be found in the Commentators upon Euclid in the quoted places , will be more serviceable to him , because more easily understood . Theor. 1. Every Parallellogram being of the same length with the Base of a Triangle , and of the same height with the Perpendicular of that Triangle , is double to it , Euclid 41. 1. Fi. 13 Here are two equal Oblongs or long Squares , ABCD and BEFC , and within them two Triangles inscribed , whose Bases are of the same length , and their Perpendiculars ( OP and QR ) of the same height with the Oblongs . Now each of these Triangles being parted into two Right-angled Triangles by their Perpendiculars , then it is plain to the Eye ( and from the nature of Diagonals , which ever divide a Parallellogram into equal parts ) that the two new Triangles OPD and OPC , which make up the first of the given Triangles , are equal to the Triangles DAO and OBC , which make up the remainder of the Parallellogram ABCD. Therefore that Parallellogram is double to that Taiangle , which was to be demonstrated : In like manner it is evident , that the Parallellogram BEFC is double to the Triangle CQF , because CRQ is equal to BQC , and QRF is equal to QEF. Theor. 2. All Triangles having the same Base , and lying between the same Parallels , are equal . Euclid 37. 1. So in our last Diagram , the two given Triangles having Bases of the same length , and lying between the same parallels , are evidently equal , because they are demonstrated to contain each of them the exact half of the Parallellograms , wherein they are inscribed ; and the Parallellograms being equal , their halves must be equal also . Theor. 3. The Sides of a Triangulate ( that is , one that hath four or more Sides ) are ever two more than the Triangles of which it is made . Ram. lib. 10. prop. 2. Fi. 10 , 12. This is plain by inspection , if you view again the Figures of the Trapezium and irregular Pentagon , in the 12th and 14th Problems of the second Chapter . These Theorems being allowed to be found ( as nothing more certain ) the Doctrine concerning the Superficial Content of Right-lined Figures might be reduced to a narrow compass ; for he that knoweth how to husband these three Theorems , may easily take up these Corollaries , ordine inverso . 1. Any quadrangular Figure ( regular or irregular ) may by a Diagonal be parted into two Triangles ; any five-sided Figure by two Diagonals into three Triangles ; and six-sided Figures into four by three Diagonals , &c. by Theor. 3. 2. 'T is no matter of what shape the Triangle is , as to the rule for measuring , for whether it be Right-angled , Acute-angled , or Obtuse-angled , and whether it have three , two , or no Lines equal ; 't is only the length of the Base , and height of the Perpendicular , that is considerable ▪ by Theor. 2. 3. The true measure or content of any Triangle , whether alone , or as part of any triangulate Figure of 4 , 5 , 6 , or more Sides ( and consequently of the whole Figure , by summing up the content of all the several Triangles ) is found by multiplying the whele Base of the Triangle by half the Perpendicular , or the whole Perpendicular by half the Base ; which being a Rule of such infinite use in surveying , I desire it may be remembred ; and that it may be understood , I shall give you a plain Example . Fig. 6 Suppose A B the Base of the Triangle , belonging to the seventh Problem of the foregoing Chapter , to be 44 , and the Perpendicular CD to be 20 : Whether you multiply 44 the whole base by 10 the half of the Perpendicular , or 20 the whole Perpendicular by 22 the half Base , the product gives the Content 440 , as is here apparent . 44 10 440 20 22 40 40 440 Having thus given a general method how all Right lined Figures may be reduced to Triangles , and so their Content found out ; I might pass to the next Head concerning Instruments , and their use ; but because there are nearer ways in measuring particular kinds of triangulate Figures proper to those kinds . I shall briefly touch them . I. To find the Contents of a Square , or long Square . Multiply the length by the bredth , the product gives the Area or Content . Example of a Square 17 Inches length . 17 Inches bredth . 119 17 289 Square Inches . Example of an Oblong 25 Feet long . 13 Feet broad . 75 25 325 Square Feet . II. To find the Area or Content in Measure of a Rhombus . Let fall a perpendicular from one of the obtuse Angles upon the opposite side ; that side multiplied by the perpendicular , gives the Area . 20 Yards the Side . 14 Yards the Perpendicular . 80 20 80 Square Yards . III. To find the Area of a Rhomboides . Divide it into two Triangles by a Diagonal drawn between either pair of the opposite Angles , ( as suppose the Acute ) then from either of the other Angles ( for instance , the Obtuse let fall a perpendicular upon that Diagonal then shall that Diagonal , being multiplied by that perpendicular , give the Area . Example . 19 Rods the Diagonal . 5 Rods the Perpendicular . 95 Square Rods the Content . IV. To find the Content of a Trapezium . F. 10 Divide it by a Diagonal into two parts from Angle to Angle ; as the Trapezium , Ch. 2. pr. 12 is divided by the Diagonal HL , then from the other two Angles ( which in that Figure a●● marked with K and Q ) let fall Perpendicular upon the Diagonal , half the Sum of those Perpendiculars being multiplied by the Diagonal ( or common Base ) gives the Superficial Content . Example . Suppose in the Trapezium before mentioned the Diagonal is 33 , the Perpendicular from 〈◊〉 Fi. 10 13 , and that from Q15 , the Area or Superficial Content is thus computed : 13 Chains the first Perpendicular . 15 The second Perpendicular 28 The Sum of both Perpendiculars . 14 Their half Sum. 33 The Diagonal . 42 42 462 Square Chains the Area . V. To find the Content of a regular Polygonial , or multangular Figure , otherwise called multilateral . Draw a Line from the Center to the middle of any Side ; half of the Perimeter ( or of all the Sides ) being multiplied by that Line before-mentioned , gives the Content . VI. To find the Content of an irregular Polygon , or many sided Figure . Divide it into Trapezia's and Triangles by Diagonals , then find their Content severally , and sum up all together ; which that you may better apprehend , Fi. 12 Suppose the Polygon belonging to the last Problem of the second Chapter were given without the Diagonals AB and BP ; then by drawing those Diagonals , Fi. 12 the Figure is divided into three Triangles , whereof two being upon the same Base AB make up a Trapezium , whose Content may be found , just in the same manner as was taught even now in the fourth Rule , having found the perpendiculars from O and P falling upon the Line AB . Then there remains the Triangle BQP , whose Content may be found by the general Rule concerning Triangles , having found the perpendicular falling from Q on the Line BP ; and then having added the Content of that Triangle to the Content of the Trapezium , you have the Area of the whole polygonial Figure . But now methinks I see ( as it were ) my Country Student scratching his Head , and wishing for an opportunity ) to propound two doubts to me . 1. Why I called the Numbers correspondent to my Lines by divers denominations , as Inches , Feet , Yards , Rods ; and sometimes by none at all but propounding the Numbers abstractly . 2. How I came to know how many of those Measures ( whatever they be ) are represented by the Lines given , and perpendiculars found . To the former I answer , I am not yet teaching how to measure Lines ( that work is presently to follow ) but what Lines of Figures are to be measured , and the measures of those Lines being known ( or supposed ) how the Content upon those real ( or supposed ) Grounds may be found , and to this purpose I might call the Numbers represented by the Lines , Inches , Feet , Yards , or any other Measures , at pleasure , provided I called the Squares , to which the Area is equal , by the same names ; for an Inch in length bears the same proportion to a square Inch ( having length and bredth ) that a Mile in length bears to a square Mile . For this reason I profess not in the Title of this Chapter to teach how to measure Figures , ( much less how to measure the Lines of such Figures ) but how to find the Content , the Lines being given . And then to take away the second doubt , know that the Numbers represented by the Lines , were either given by those Learned Artists from whom I borrowed the Figures , or supposed by my self as grounds to go upon ( as in such cases is ordinary ) or , lastly , found out to be agreeable thereto by some Scale of small equal parts , which he is yet supposed ignorant of . But now I am going to shew him the nature and use of two or three plain and cheap Instruments , by the help whereof he may with much exactness , 1. Measure the length of any Lines bounding Right-lined Figures upon the ground . 2. Draw Lines and Figures upon paper proportionable thereunto ( which we call protracting ) . 3. Find upon his paper-figures the true length of all the desired Perpendiculars , which shall also be proportionable to those on the Ground , but much more easie to be obtained . And withal , I intend to give him such farther Instructions and Cautions for the application of the general and more particular Rules of this Chapter to his peculiar use , as will render them ( especially some of them ) singularly advantagious . CHAP. IV. Concerning Chains , Compasses , and Scales . 1. Amongst the many sorts of Chains used for measuring Land , three are most famous ▪ bearing the Names of their Inventors , Mr. Rathborne , Mr. Gunter , and Mr. Wing , all of them Ingeniously divided , and useful in their kind● but my brievity will give me leave only to describe one , and that shall be Mr. Gunter's , being most in use , and easie to be procured . This Chain contains in length four Statute-Poles or Perches , each Perch containing 1● Feet and a half , or 5 Yards and a half ; so that the whole Chain is 66 Feet , or 22 Yards long . This whole Chain is divided into 100 equal parts or Links , whereof 25 are a just Pole or Perch ; and for ready counting , there is usually a remarkable distinction by some Plate or large Ring at the end of every 25 Links , but especially at the precise middle of the Chain , which should differ from the rest in greatness and conspicuousness . Also at the end of every tenth Link 't is usual to hang a small Curtain-Ring and if there be at every five Links end a piece o● Wire made like the bow of a Link , with a little shank an Inch or less long , ( or some such distinction ) 't is still better . When you are to measure any Line by this Chain , you need to regard no other Denomination but only Chains and Links , set down with a prick of your pen betwixt them , e. g. If you found the side of a Close to be 6 Chains and 35 Links long , it is thus to be put down 6. 35. But if the Links be under 10 , a Cypher must be prefixed ; so 7 Chains 9 Links must be thus set , 7. 09. In the using of this ( or indeed of any ) Chain , care must be taken , both to go strait , and to keep a true account ; for which purpose , it is good that he which goeth before carry in his hand a bundle of Rods , to stick down one at the end of the Chain which leads , having first stretched it well , and that he which follows do not only gather up the Rods to keep the Accompt , but also at every remove , mark whether he see the Leader directly between his Eye and the Angle , or other Mark he aims to measure to ; and if need be , call to the Leader to move towards the right or left hand , till he see him in a direct Line to it . II. Compasses are so well known , that I need not describe them ; only they should be of Brass , with Steel Points small and neatly wrought , nine or ten Inches long from the Joint to the points , turning so truly upon the Rivet that they may be easily opened ; and yet stand so firmly , that an Arch or Circle may be without their shrinking described upon a large Radius . For the form , I would commend above all , those that have large Bows , so contrived , that by pressing them with the hinder part of the Hand , they will gently open , and by the Thumb and fourth Finger be put together ( as others will ) so that they are manageable by one hand , which is a great convenience for one that at the same time should hold his Rule with Scales in the other . These might also be contrived with a Screw to take out one of the points , to place in the room ( upon occasion ) a Black-Lead pen , or any pen to draw Circles , with either black or otherwise coloured . And for a Man that would be an Artist indeed , it were convenient he were furnished with dividing Campasses , beam Compasses , and triangular ones , for several uses not here to be mentioned ; but a Country Surveyor may make a good shift with such a plain pair as I first described ; which Mr. Wynn , over against the Rolls in Chancer●-lane , will help him to , with the Chain and Scales , for a small matter . III. Scales are certain Lines divided into equal parts , upon plates or broad Rules of Brass or Box , and they are of two sorts , 1. Plain ; 2. Diagonal . 1. Plain Scales are made up of two small Lines parallel to one another at a little distance , and these are divided into great equal parts , which signifie Tens , and are noted 10 , 20 , 30 , 40 , 50 , &c. according to the length of the Lines . They may be of any convenient length , but these great divisions are seldom more than Inches , or less than third parts a piece . Again , one of the great Divisions ( or parts ) is subdivided into ten equal parts by short Lines , whereof that in the middle standing for 5 is longer than the rest . According to the Numbers of these little parts contained in an Inch , the Scale is named A Scale of 10 , 11 , 12 , 16 , 20 , 24 , 30 , &c. in an Inch. Fi. 13 That short one which I give you the Figure of at A is of 10 in an Inch , so noted at the top , according as is usual upon Rules , and Indices of plain Tables . The Line marked OQ , separates Unites and Tents ; Unites being taken upward from that Line , and Tens downward ; mixt Numbers both ways . As for Example . 7 is the extent of the Compasses upon the Scale A from the Line OQ to K ; 30 is their extent from the Line so marked to OQ ; and 27 is their extent from the Line 20 , to the short Line K aforesaid . Here note , that you must not expect to find the Letters OQ or K upon the Scales which you buy , being only marks used at pleasure , to make my meaning plain ; and likewise that this Scale of 10 in an Inch , and others that are smaller ( all being composed after the same manner ) are usually made for more convenient use , so long as to contain nine or ten ( the more the better ) of the great Divisions , signifying Tens ▪ though the Figure at A being designed for no other use than to help your conceptions , extends but a little beyond 30 , that length being sufficient for my purpose in this place . Fi. 14 These plain Scales , especially the smaller sorts of them ( such as 24 or 30 in an Inch ) are very proper for drawing Figures upon paper , where the Numbers represented by the Lines are not above 100 ( for then every Division may be counted as it is upon the Scale ) or above upon a long Scale . Also in the surveying of Forests , Chases , and great Commons , where the Lines are vastly long , and the mistake of a few Links ( yea , of half a Pole ) is not considerable , they may be conveniently used , accounting the Tens and Unites to signifie so many whole Chains , and so estimating the parts of a Chain with the Compasses upon the small Divisions , which a sagacious Man may do very near upon one of the larger Scales . But it were much better , in my opinion , for ordinary measuring , if the grand Divisions on the Scale were two Inches a piece , as I have one upon the Index of my plain Table ) for then the smaller Divisions being of five in an Inch , would be so large as to be subdivided into five apiece , which represents 20 Links ; and then the half of one of those smaller Divisions signifying 10 Links , and the quarter 5 , a very ordinary judgment may come very near to the truth by estimation . 2. But the Diagonal Scale is so well known to every Mathematical Instrument-maker , so easie to be procured , and every ways so fitted to Gunter's Chain , and our Countryman's use , that I cannot but highly commend it . Of these Diagonal Scales , there are two sorts , the Old and New. Fi. 14 By the Old one , I mean such as is to be found in Mr. Leybourn's Book , whereof I shall present you with a fragment , with such a description as may enable you to understand the whole . 1. It is made ( as appears by the Figure B ) upon eleven parallel Lines equidistant , so as to include ten equal spaces , which are all cut at right Angles by Transwerse Lines dividing them all into four equal parts . 2. One of these Transverse Lines ( viz : PR ) where it toucheth the first and last Lines , separates between the Hundreds ( or whole Chains ) and the Tens , ( representing 10 Links apiece ) the Chains being numbred downwards on the left hand from P only to 3 , but on the Instrument it self they may go on to 9 or 10 , ( the Rule being a Foot long ) but the Tens ( or Decads ) upward from P to 10. 3. From the Points of Division into Tens upon the first Line beginning at P , to the like Points beginning at R in the last Line , are nine Diagonal Lines drawn , the first beginning at P , and ending at the first Division above R. The second beginning at the first Division above P , and ending at the second above R. In a word , they are all drawn from one Division less from P , to one more from R ; by which it comes to pass , that every Diagonal , by that time it hath passed from the first Line to the eleventh is a whole tenth part of an Inch ( which answers to ten Links of the Chain ) farther distant from the Line PR , than at the Point upon the first Line whence it was drawn . Fi. 14 4. Every one of these Diagonals is divided into ten equal parts by the long parallel Lines running through the whole Scale , and numbred on the top from 1 to 9. Whereby it is evident , that the Intersection of any of the nine parallel Lines that are numbred at the head with any Diagonal , must be farther distant from the Line PR , than the Intersection of the Line next before it with the same Diagonal by of that is , by which answereth to a single Link of your Chain . From what hath been said , and inspection of the Figure B , these things plainly follow , which as so many clear instances will help you to understand it fully . 1. The distance from PR to the second Division below it answereth to two Chains . 2. The distance from PR to the eighth Division upward being taken ( with Compasses ) upon the first Line of the eleven from P to 8 , answereth to 80 Links . 3. Consequently the extent of the Compasses from the second grand Division below P to the eighth of the less Divisions upward , is proportionable to 2 Chains 80 Links . 4. The distance from PR to the first Diagonal being taken upon the parallel Line noted with 9 above answereth to 9 Links : Where note , that the first Diagonal is not that which is noted with 1 , but that which is drawn from the point P. 5. The distance upon the same Line from PR to the Diagonal that is marked with 7 , is answerable to 79 Links . Fi. 14 6. The extent of the Compasses from the bottom of the Figure B upon the same Line to the same Diagonal , answereth to 3 Chains 79 Links . Briefly whole Chains may ( by Analogy ) be measured upon any Line from PR to the grand Division noted with the given Number , Decads alone , or Chains and Decads upon the first Line of the eleven where the Diagonals begin . Links alone , Decads with Links , and Chains and De●ads with Links , always upon that Line upon which the number ofodd Links stands at the head of the Scale . And know , that these Directions ( mutatis mutandis ) will as well fit , if half an Inch be only allowed for a Chain , and consequently all the Diagonals drawn within that extent , as it is usual ( and very commodious for longer Lines ) upon the other end of the same Rule , the grand Divisions for Chains going the contrary way , ●nd noted with Numeral Figures in order . It 〈◊〉 good therefore when you furnish your self with Scales , to have Diagonal Scales of both ●imensions on the fore-side of your Rule ; and upon the back-side many plain Scales of equal parts , with a Line of Chords ; all which you may have ( by enquiring only for the Scales described ●n Mr. Leybourn's Book ) of Mr. Wynne aforesaid , ●s likewise all other Mathematical Instruments . Having been so large for my plain Country-man's sake , I shall not proceed to the description of the new Diagonal Scale , of which you may have the Figure and Description in Mr. Wing's Book : For though it be an excellent good one , Fi. 14 as I know by experience ; Mr. Hayes having ( at my desire ) furnished my noble Friend Si● Charles Hoghton with an artificial one of that sort , when I had the honour of assisting him in Mathematical Studies ) yet because 't is pretty cost●● ( if well made ) , and that before described will very well answer its end , I shall at present say no more of it . But my Reader may perhaps object to me , th●● though I have instructed him how he may make a Line of an exact length , to answer to any number of Chains and Links ( given or found by measure ) upon the Diagonal Scales : I have not yet shewed him how to measure a Line ( as suppose a Perpendicular ) whose length is unknown , upon them . To give him therefore all satisfaction ( though what I have writ already , might help him to find this out ) let us suppose , that in some Figure made according to the Diagonal Scale B of 10● in an Inch , we meet in measuring with an unknown Perpendicular equal to the Line in the Margin . Taking it between the Points of my Compasses , I first try whether it be even Chains and finding upon the first view that it is not , 〈◊〉 make a second trial , whether it will prove to be even Decads , or Tens of Links ; to which purpose I set one Foot at 3 Chains in the bottom o● my Scale in the first Line where the Diagonals begin , and the other Foot rests in the same line betwixt 6 and 7 ; whereby I am assured the odd Links above 3 Chains are more than 60 and less than 70. And to find how many above 60 , I remove the Compasses from parallel to Fi. 14 parallel in order , till one Foot in the lowest Line resting in the end of a parallel , the other will touch some Diagonal at the Intersection with that line which falls out to be at L and O in the line marked with 7 , shewing the whole line , being measured by that Scale , to signifie 3 Chains and 67 Links . CHAP. V. How to cast up the Content of a Figure , the Lines being given in Chains and Links . HAving described these plain Instruments , and in some measure shewed the use of them in severals , it were very proper in the next place to teach their joynt use in measuring and protracting ; but because I would have my young Surveyor , before I take him into a Close , able to perform his whole work together , I intend to shew him , 1. How he ought to make his Computations ; 2. The Grounds or Principles that will justifie him in so doing . For the first , take these Rules : 1. Put down your length and bredth of Squares and Oblongs , and your Base and half Perpendicular of Triangles directly under one another , expressed by chains and links with a prick betwixt them , as was taught before , Chap. 4. 2. If the odd links were under ten , put a Cypher before the numeral Figure expressing them , ( as there also was shewed ) and if ther● be no odd links , but all even chains , put tw● Cyphers after the prick . 3. Multiply length by bredth , and Base 〈◊〉 the half Perpendicular , according to the Rul● for finding the Content of Figures , Chap. 3. 4. From their Product cut off 5 Figures ( accoun●ing Cyphers for such ) reckoned from th● right hand backward , with a dash of your 〈◊〉 so shall those to the left hand signifie Acres . 5. If those five cut off were not all Cypher● multiply them by 4 , and cutting off fiv● towar●● the right hand again , the rest will be Roods 〈◊〉 Quarters . 6. If amongst these five Figures towards the right hand that were cut off at the second Multiplication there be any Figures besides Cyphers multiply all the five by 40 , and cutting off fiv● again by a dash of your Pen , those on the left hand signifie square Perches , Poles , or Roods ▪ A few Examples will make all plain . Quest . 1. What is the content of a Square , 〈◊〉 Sides are every one of them 7 Chains , 25 Links ? Length 7.25 Bredth 7.25 3625 1450 5075 525625 525625 4 102500 40 100000 Answ . 5 Acres , 1 Rood , and 1 Perch , as here appears . Quest . 2. In a long Square , whose length is 14 Chains , and the bredth 6 Chains 5 Links , what is contained ? Length 14.00 Bredth 6.05 7000 84000 847000 4 188000 40 3520000 Answ . 8 Acres , 1 Rood , and 35 Perches , as the Work makes it evident . Quest . 3. In a Triangle , whose Base is 3 Chains , and half the Perpendicular 98 Links , what is the Content ? The Base 3.00 Half Perpend . 0.98 2400 2700 29400 4 117600 40 704000 Answ . 0 Acres , 1 Rood , 7 Perches , as here is plain . There be other ways of Computation by Scales , Tables , &c. but that this is sound and demonstrative , I come now to shew by these following Steps . 1. It is evident , that in this way of Multiplication the Product is square Links ; for every Chain being 100 Links , it is all one to multiply 7.25 by 7.25 , or 725 by 725 without pricks , for the pricks signifie something as to Conceptions but nothing at all in Operation . The Product therefore of the first Example was really 525625 Links . 2. Every Chain being 4 Perches long , it follows , that 5 Chains ( or 20 Perches ) in length , and 2 Chains ( or 8 Perches ) in bredth , make an Acre , or 160 square Perches ; for 20 being multiplied by 8 , gives 160. 3. From hence it plainly followeth farther , that there are exactly 100000 square Links in an Acre ; for 5 Chains multiplied by 2 , is the same with 500 Links by 200 , which makes 100000. And he deserveth not the name of an Arithmetician that is ignorant of this old plain Rule , When the Devisor consists of 1 and Cyphers , ( as 10 , 100 , 1000 , 10000 , 100000 , &c. ) cut off from the right hand so many Figures of the Dividend as the Devisor hath Cyphers , accounting them the Remain ; so shall the rest on the left side be the Quotient . It is plain then that 525625 square Links make 5 Acres , and 25625 square Links over . Thus I have made it clear to a very ordinary capacity , that as far as concern Acres , the Rules for Computation are good . Now for Roods and Perches , though I might turn off my Reader with that known Rule in Decimal Arithmetick : Multiplying Decimal Fractions by known Parts , gives those known Parts in Integers , due regard being had to the separation . I shall proceed in my plain way thus : If 25625 square Links , which remain above an Acre , do contain any quarter or quarters of an Acre ; then if they be multiplned by 4 , and divided by 100000 , ( that is , five cut off from the Product ) they will contain so many Acres as now they do quarters ( or Roods ) , for any number of quarters multiplied by 4 , must needs produce the like number of Unites or Integers , and the Division doth only reduce them into the right denomination . Now 25625 being multiplied by 4 , and five Figures being cut off from the product , the result is , 1●02500 , that is an Acre and above ; which shews it was above a quarter before it was multiplied by 4. And to find how much , ( that is , how many square Perches are contained in this last remainder ) you must consider this 2500 , not as square Links remaining above the Rood or Quarter , but as fourth parts or quarters of square Links ; or ( which is all one ) as the true number of square Links multiplied by 4 , and consequently being multiplied by 40 , ( the fourth part of square Perches in an Acre ) it must as often contain 100000 square Links ( or an Acre ) as the quarter of this number 2500 , viz. 625 , signifying square Links , containing square Perches ; and so it doth , for 100000 divided by 160 ( the number of Perches in an Acre ) gives 625 as answerable to 1 Perch ; and 2500 multiplied by 40 , gives 100000 , or 1 Acre ; the five Cyphers being cut off as here is manifest . 2500 40 100000 160 ) 100000 ( 625 960 400 320 800 800 0 Some may perhaps wonder , that in so small a Manual I spend so many words about such ordinary things , as in this and the last Chapter ; but I am most afraid , lest I shall not for all my plainness be sufficiently understood by such as I purposely write for , in things of such necessary and frequent use : And I designed not to make this Treatise small by being obscurely brief in substantial things , but by leaving out such Curiosities as I thought my Country Friend might well spare . CHAP. VI. How to measure a Close , or parcel of Land , and to Protract it , and give up the Content . HItherto we have been like Children learning to spell , now let us set our Syllables together . I mean , let us make use of the Instructions beforegoing to measure a plece of land , to plot it , and to cast up the Content . All Closes , or parcels of land , are either such as need not to be plotted for finding out their true measure , but the Chain alone doth the Work ; or such as cannot be conveniently measured without plotting or protraction . Of the first sort are the Square and long Square , known before-hand to be such , or found so to be by such Instruments as I have not yet described , or by measuring all the Sides and Diagonals . These Squares and long Squares ( I say ) need no protracting , for you need only to multiply the Chains and Links of the length , by the Chains and Links of the bredth , and so proceed as in the first and second Examples of the fifth Chapter : But all others , whether Triangles or Triangulate , are to be protracted . I shall give Examples therefore in the 3 sorts of Figures , triangular , quadrangular , and multangular . But before I proceed to particular Instances , let me advise the young Practitioner thus : Remember , 1. To begin at some notable Angle of the Field , where there is some House , Gate , Stile , Well , or the like ; or if there be none , then to dig up a Clod , drive down a Stake ; or at least , to observe what quarter of the Heavens it pointeth towards , whether East , West , North , or South , and on your Paper mark it with the Letter A , or any other . 2. To go parallel to the side of the Field , 〈◊〉 Pits , Bushes , or the like , hinder not , ( and if they do , to allow for it ) accustoming your self to go either cum Sole , that is , with your left hand towards the Hedges , Walls , or Pales ; or contra Solem , with your right hand towards them ; and when you go contrary to your usual custom , note it on your paper by some mark known to your self . 3. To set down the Chains and Links of every side as you measure them , and not to trust your memory . A Black lead pen will be very proper for this purpose . 4. To take heed ( if you have more Scales than one upon your Rule ) lest you confound your self by taking lines off of several Scales , or measuring perpendiculars upon wrong ones ; for every line of the same Figure must be made by the same Scale , and the perpendiculars measured by it . 5. To make use of a Scale of larger Divisions when you measure small Closes , and of smaller when you measure great ones . 6. To make your lines and points where Angles meet , small , pure , and neat . 7. To set on your Chains and Links at twice , when any line is too long for your Scale . These things being premised , I proceed thus : I. Suppose I measure a triangular Field with my Chain , beginning at the Eastern Angle A , and find the Sides in their order and measures to be severally , thus : ( I going cum Sole ) 2.229 , 3.45 , 4.07 . Fig. 15 Making use of the less Diagonal Scale , because the other would make the Figure too large , otherwise it were more proper for so small a Close ) I first with my Compasses take off the Scale 4 Chains and 7 Links , and setting them from A to C draw that line for the Base , because the longest of the three : Then I take 2 Chains 29 Links off the same Scale , and set them in the Eastern point A where I began , and turning the loose Foot of the Compasses above the line A C , because I went cum Sole , I describe ( at that distance 2. 29. ) the Arch E E. Fi. 15 Next taking with my Compasses upon the same Scale the extent of 3 Chains 45 Links , 〈◊〉 place one Foot in the point C , and with the other make the Arch FF intersecting the former● in the point B ; and drawing the lines A B and B C , ( as was taught in Ch. 2. Probl. 6. ) the Triangle A B C is the Plot of the Triangular Field measured . But before I can give the Content , I must find the length of the Perpendicular , which is done by setting one Foot of the Compasses in B , and extending the other to the Base A C , so as 〈◊〉 touch it and pass not over it , ( according to Ch. ● ▪ Probl. 7. ) for then the length of the Perpendicular is between the Points of the Compasses , and being applied to the same Scale by which the Triangle A B C was made , it appears to be 1 Chain 42 Links . With the half whereof I multiply 4.07 the length of the Base , and proceeding in my Work as was shewed in the last Chapter , the Content appears to be 0 Acres , 1 Rood , 6 Perches , as it is here evident . The Base 4.07 Half Perpend . 0.71 407 2849 28897 4 115588 115588 40 623520 II. Suppose I were to measure a quadrangular or four-corner'd Field , I begin as before at ●ome remarkable Angle ; and going round the ●lose cum Sole , I find the Sides to be 9.04 , 6.72 , ● . 46 , 7.28 , and the Diagonal from that remarkable Angle to the opposite Angle to be 10.02 , I ●●gin therefore to protract it thus . Fi. 16 Having by the help of my Scale and Com●asses drawn my Diagonal 10. 02 from my remarkable Angle A to C the opposite Angle , make a Triangle of it , and the first and second sides 9.04 and 6.72 . according to Ch. 2. Probl. 6. ●nd another after the same method of that Diagonal , and the third and fourth Sides 8.46 and ● . 28 , so have I the Trapezium ABCD. Then by the help of my Scale and Compasses , 〈◊〉 find the Perpendicular of the Triangle A B C 〈◊〉 be 6.02 , and of the other , viz. CDA 6.01 , which added are 12.03 , whereof the half Sum 〈◊〉 6.01 ; by which multiplying the Base 10.02 , and proceeding as formerly hath been shewn , I ●nd the Content of the Field to be 6 Acres , 0 Roods , 3 Perches , as is here apparent . Fi. 16 The Base10 . 02 Half Perpend . 6.01 1002 60120 6●02202 4 ●08808 40 3●52320 Before I pass any further , let me tell you , 1. Any quadrangular Close , or parcel Ground whatsoever , having right Lines , 〈◊〉 be thus measured , protracted , and computed . 2. The odd measure above Perches is 〈◊〉 valuable here , nor in the former Computatio● being always under a square Perch ; but in mu●● angulars where there be many Remainders , th●●● must be summed up , and the Perches contain● in them added to the Content before found . 3. This last , and the following Figures ( wh●● I use any Scale at all ) are made that they mig●● not be too large , by a Scale of 400 in an In●●● i. e. by the less Diagonal Scale , each Chain 〈◊〉 Link being counted two . Fig. 17 III. If this multangular Figure be conceive to represent a Close of seven sides , which is 〈◊〉 be measured , I begin at the remarkable Angle 〈◊〉 and going round the Close ( with the Sun ) find the sides to be in measure 3. 11 , 2. 49 , 2. 4● , 1. 77 , 4. 11 , 2. 29 , 4. 37. Fi. 17 Then I measure the four Diagonals , BD , DF , ●B , and BG , in the order that I named them , ●nd I find them to be 3.45 , 4.97 , 4.13 , and 4.36 , ●hich is as short a way as can be taken , to pre●ent unnecessary Walks . But when I come to protract by the help of ●●y Scale and Compasses , I first make the Trian●●e BCD of the first Diagonal , and the second ●nd third sides . Then the Triangle DEF upon 〈◊〉 second Diagonal , and fourth and fifth sides , 〈◊〉 upon the same Diagonal as a common Base , 〈◊〉 Triangle BDF of the first , second and third Diagonals . Next of the same third Diagonal , together ●ith the fourth and sixth sides I make the Triangle BFG , and upon the fourth Diagonal as ●pon a common Base with the first and last sides 〈◊〉 Triangle ABG , so is the whole Close ●●otted . And now it stands visibly reduced into two ●rapezia's ABFG and BDEF , together with ●he Triangle BCD , which I shall not now cast 〈◊〉 , having so often shewed how such work is 〈◊〉 be done . But I must acknowledge that this sort of plotting of parcels of Land that have many Angles , ●equires not only more care and pains , but better skill and memory than to draw Diago●als upon Paper , when the Plot is already taken by the plain Table , or other standing Instrument . I shall therefore to help my young Practitioner in this case , advertize him of two easie ways to help himself , so as to be out of danger of mistakes . One way is to divide the Multangular Field into two or more parts as the last might have been by the Diagonal BF ; then might each pa●● have been measured severally , as if they had been separated by a Pale , or were sundry Mens Land parted by a Boundary . Another way that much helps both the understanding and memory , is to draw a rude Draugh● of the Figure of the Land you intend to measure , not only as to the sides , but also necessary Diagonals . Then measuring the Lines upo● the Ground correspondent to those on the Paper ( which by the help of the Draught may be easily hit ) set the Lines as you measure them upon the Lines of the Draught , as if it were the true ones , and when you have finished your measuring , protract it truly . Such as you see her● ( but it 's better larger ) will do your business for 't is not a Pin matter how rude or false the Lines or Angles be , resemblance being all that is desired . CHAP. VII . Concerning the measuring of Circles , and their Parts . I Have hitherto abstained of purpose from medling with the Circle and its Parts , that I might lay those things close together without unnecessary mixtures , that are of the greatest use . 'T is wonderful rare , if a Land-meeter ever have occasion to measure any Field or parcel of Land , that will prove either Circle , Semicircle , Quadrant , or Sector . Sometimes indeed there will be a little crook in an old Hedge bowing like an Arch : But I have never seen any offer to measure it as a Segment , but always take it as an Angle or Angles . Yet because it may be expected I should say somewhat of those things , I shall briefly do it . 1. To measere a Circle in the more exact way 〈◊〉 to square the Diameter , and to multiply that Square by . 7854 , so shall the Content be in Integers and Decimals . But the more usual and quick way ( and near enough for any use we shall make of it ) is to multiply the half of the Periphery or Circumference by the Semidiameter . In like manner to find the Content of a Semicircle , Quadrant , or Sector made up of Semidiameters , and arched Lines , multiplying the half Arch by the Semidiameter , gives the Content . Fi. 19 But that which falls out most frequently in Mensuration ( though seldom much regarded ●ave where a curious exactness is required ) is that particular sort of Segment , which we call 〈◊〉 Section , less than a Semicircle , such as this Figure ABC . And to find the Content of it , the center of the Circle whereof this is a Section must be first fou●d out , as here at O , from which Lines drawn to A and B , make up the Sector AOBC ; which being measured according to the last Rule , and from the Content thereof , Fi. 19 the Content of the Triangle AOB sub●racted , the difference or residue is the Content of the Section ABC . But two Questions may be here demanded : 1. How may the Center be found ? 2. How may such a portion of Land be truly protracted and computed ? To the first I answer , that the most exact and artificial way is by making a Mark any where in the Arch. As for Example . At the Point C ; and then ( by a Problem known not only to every Surveyor , but to ordinary Carpenters and Joyners , for finding the center of a Circle , whose Circumference will pass through 3 given points that are not in a right line as ACB ) to find the Center O. But if you know not how to do it so , cross the Line AB in the middle , as here it is done by the Perpendicular OC , so you may by a few trials find both the due extent of your Compasses , and the point in the Perpendicular that will fit your purpose near enough ; for if a little errour be committed in making up the Sector , the most of it goes off again in the substraction of the Triangle . II. For the latter you may take this ready course : Measure the length of both your Lines , ( the Chord and the Arch ) and their distance at the middle of them both . Then when you come to protract , first take the length of your right Line from the Scale , and having laid it down , cross it in the middle at right Angles with a dry Line as in the last Figure , so shall it intersect the Line AB in the point E ; Fig. 19 then from the same Scale take the measured distance between the two lines in the middle , and set it upon that dry line from the Intersection at E to the point C. Then by trials find a due place in the dry line OEC , and such a distance with your Compasses , that the one Foot resting in that line , the other may describe the Arch ACB , and the Section is protracted . These few hints are as much as I thought necessary for my Country Practitioner concerning circular lines ; but if he think otherwise , there are large Treatises enough , and particularly those I mentioned , whose Rules ( though ingenious , sound , and fit to be known by every one that intends to plunge deep into Mathematical Studies ) will not ( I think ) be of that use to him in ordinary measuring , that I should transcend the intended bounds of brevity to transcribe them . CHAP. VIII . Concerning customary Measure , and how it may be reduced to Statute-Measure , & è contra , either by the Rule of Three , or a more compendious way by Multiplication only . WHereas the Statute-Perch or Pole is 16 Feet and a half , and no more , there be Poles of larger measure used in many places , as of 18 , 20 , 21 , 24 , and 28 Feet , yea in some 22 Feet and a half . It were therefore very convenient , that our young Surveyor were furnished with a Chain fitted to the customary measure of the Country where he lives , as I use to make Chains for my self and Scholars of 21 or 24 Feet to the Pole for Lancashire and Cheshire , where those Measures most obtain . But because these are too large and cumbersome for small Closes , it is very convenient , instead of one Chain of 100 Links , to make two of 2 Poles apiece , each Pole divided into 25 Links as that of 100 is , which two half Chains may in measuring large Fields be tied together by the Loops with Pack-thread , or joyned by a buttoning Keyring for more speedy dispatch ; but in smaller we may use the half Chain of 50 Links , only taking care that we count not half Chains for whole ones . And in these cases where the Poles are large and the Closes small , it were still more convenient if you had a Chain of 2 Poles only , divided into 100 Links : Only you must then take notice , that whereas working by whole Chains and Links , the first Multiplication , after five cut off , gives the Content in Acres and Parts . The like work by half Chains and half Links will give the Content in Roods or quarters of Acres , and parts of such Roods . But though it is no hard matter ( for one that can find out the length of a Link by dividing the number of Feet in a Chain by 100 , and provide himself of good Iron-wyre , and Curtain Rings to make it of , and a sharp edged File , and round nosed Plyers to make it with ) to be furnished with such a Chain ; yet because every one cannot do this , I shall shew you how you may easily and yet very truly reduce Statute-measure into customary , that so the Chain before described may do your business all England over . Know therefore ( for a ground to go upon ) that Acres bear proportion to one another , as the squares of their Poles ; and therefore if you multiply 33 , the number of half Feet in the Statute-pole by it self , which gives 1089 , and also multiply the number of half feet contained in a Pole of that measure you would reduce into ; in the same manner you may by the Rule of Three reverse obtain your desire , making to that purpose 1089 the first number , the Statute-measure the second , and the squared half-feet of the Pole given the third . As for example : 9.33 L. 7.21 B. 933 1866 6531 672693 prod . Suppose of a Close measured by the Statute-Pole , the length , breadth , and their product be as here represented in the Margin . And it is desired that the Content may be cast up according to our large Cheshire measure of eight yards or 24 feet to the Pole or Rood ( as we call it : ) The● before I cut off any Figures , I consider that in the Statute-pole are 33 half-feet , and in the Cheshire-pole 48 , then multiplying 33 by 33 , and 48 by 48 , I have these two square numbers , 1089 and 2304 , which together with the said product may be thus placed : 1089. 672693 ∷ 2304 , and so multiplying 672693 by 1089 , and dividing their product being 732562677 by 2304 , the quotient is 317952 , from which if 5 figures toward the right hand be cut off , and dealt withal as was taught in the fifth Chapter , the Content by our customary measure of 24 feet to the Pole , will be 3 acres , 0 roods , 28 perches as here appears 317952 4 71808 40 2872320 But if the lines on the land had been measured according to our custom here , of 24 feet to the pole , and the Content must have been found according to Statute measure ; then I must have multiplied the product by 2304 , and have divided that latter product by 1089 ▪ And in the same method you may proceed in all or any of the rest . But the truth is , that though this way be very exact , plain and comprehensive , suting all the customary measures before-mentioned without fractions , which for my Learners sake I studiously avoid , and for that reason reduced my poles to half-feet : It is something tedious except he knows how to relieve himself by a large Table of Logarithms , or at least a set of Nepair's-bones , which I cannot stand here to treat of : Therefore to contract the work a little , take notice , that all the customary poles before mentioned , ( saving only those of 20 and 28 feet , which I suppose are somewhat rarely used , because I never heard nor read of them ( to my remembrance ) save only in Mr. Holwel ; all the rest I say , are capable of being divided into half-yards : And therefore if instead of squaring the half-feet you square the half-yards of both poles , and work with them , you will attain the same end without any regardable difference , the small diversity that there is being usually in the useless remainders , not at all affecting the desired Quotient that gives the answer near enough for use . As for Example . If I had squared 11 , the number of half-yards in the Statute-pole , which would make 121 , and also 16 the number of half-yards , in our Cheshire-pole , which would make 256 , as appears in the Margin , and then multiplied the first product 672693 by 121 , the second product would have been 81319853 , which being divided by 256 , the Quotient would have been ( as before ) 317952. And this way is in a manner coincident with Mr. Holwells first Method . 11 11 11 11 121 16 16 96 16 256 Take notice also further once for all , that whether you use either of these or the following Methods , you need not reduce the particular Squares , Triangles , or Trapezia's severally ; but sum up all their products together , and then reduce all at once . But if you would reduce Statute measure into Customary by Multiplication only , take notice of this present Table following . The Content by the Statute-pole being multiplied by . 84027 Gives the Content by the Pole of 18 feet . . 68062 20 . 61734 21 ½ . 53777 22 . 47265 24 . 34725 28 The use of this Table . When you have multiplied Lengths by Bredths , or Bases by half Perpendicalars , multiply these Products by the Decimal Fractions answering to the Customary-measure into which you would reduce Statute-measure , and from that latter product , first cut off five places towards the right-hand as not to be regarded , ( being only parts of a square Link ; ) then cutting off 5 more , and proceeding to multiply by 〈◊〉 , and then by 40 , as hath been often shewed , you will have the Content by that Customary-measure , Example . Suppose once more the length of a Close measured by Gunter's Chain , and multiplied by the breadth measured also by the same , produced 672693 square Links ; and it is desired that the Content may be given in Cheshire-measure of 24 to the Pole : You must multiply 〈◊〉 72693 by .47265 , the decimal fraction answering to 24 feet , and from that product being 31794834645 cut off and cast away 5 places , and the rest being ●17948 , are in the usual way easily ●educible into 3 Acres , 0 Roods , 28 Perches , as here appears , agreeable ●o what it amounted to in the former Method . 317948 4 71792 40 2871680 But if you measured by a Chain of Customary Poles , and desire to know what the Content 〈◊〉 in Statute-measure ; this following Table is fo● your purpose . The Content measured by the Pole of 18 Feet . being multiplied by 1. 19008 gives the Content by the Statute-pole . 20 Feet . 1. 46923 21 Feet . 1. 61983 22 ½ Feet . 1. 85950 24 Feet . 2. 11570 26 Feet . 2. 87970 To understand which , take this Example . Suppose the length and breadth of a lo● Square being measured by a Chain of 24 feet 〈◊〉 the Pole , and multiplied together , make the●● product 317952 , le● this be multiplied b● 2.11570 , which answereth to 24 feet , and th● latter product will be 67269104640 , from which if you cut off and cast away 5 places towards the right-hand , the remainder is 672691 , which in the usual way is easily reduced to 6 Acres , 2 Roods , and 36 Perches , as you see here . 67269 29076 4 363056 One thing more and I have done with th● business of Reduction : If the Content to be r●●duced , be given cast up into Acres , Roods , and Perches , reduce all into Perches , and then i● other respects work as before either by th● Rule of Three , or by this last Method of Multi●●lication only . So shall you have the Content in square Perches according to the Measure desired , which you may reduce into Acres by dividing them by 160 , and if any thing remain , that remainder being divided by 40 , will give you the Roods in the Quotient , and the latter remainder the number of square Perches . For tryal of which Rules , mind the Answer ●o these two following Questions wrought all ●hree ways . Quest . 1. How many Acres , Roods and Perches , according to the Pole of 18 Feet , are contained in 5 Acres , 3 Roods , and 11 Perches , Statute-measure ? Answ . 4 Acres , 3 Roods , and 22 Perches , as ●●ere appears : I. Method . 33 36 5 A : 3 R : 11 P. 33 36 4 — — — 99 216 23 99 108 40 — — — 1089 1296 931 1089 . 931 ∷ 1296 1089 8379 7448 9310 1013859 1296 ) 1013859 ( 782 9072 10665 10368 2979 2592 387 160 ) 782 ( 4 640 40 ) 142 ( 3 120 22 II. Method . 11 11 11 11 121 12 12 24 12 144 121 . 931 ∷ 144 121 931 1862 931 144 ) 112651 ( 782 1008 1185 1152 331 288 43 160 ) 782 ( 4 640 40 ) 142 ( 3 120 22 Or thus : 40 ) 782 ( 19 40 382 360 22 4 ) 19 ( 4 16 3 III. Method . 84027 931 84027 252081 756243 782. 291 37 160 ) 782 ( 4 640 40 ) 142 ( 3 120 22 Quest . 2. How many Acres , Roods and Perches , of Statute-measure are contained in 8 Acres , 3 Roods ( or Quarters ) and 21 Perches of 21 Feet to the Pole ? Answ . 14 Acres , 1 Rood , and 21 Perches , as appears by the three following works in the several Methods . I. Method . 8 A : 3 R : 21 P 4 35 40 1421 42 42 84 168 1764 33 33 99 99 1089 1764 . 1421 ∷ 1089 1764 5684 8526 9947 1421 1089 ) 2506644 ( 2301 2178 3286 3267 1944 1089 855 160 ) 2301 ( 14 160 701 640 40 ) 61 ( 1 40 21 II. Method . 14 14 56 14 196 11 11 11 121 196 . 1421 ∷ 121 196 8526 12789 1421 121 ) 278516 ( 2301 242 365 363 216 121 95 160 ) 2301 ( 14 160 701 640 61 40 ) 61 ( 1 40 21 III. Method . 1.61983 1421 161983 323966 647932 161983 2301.77843 160 ) 2301 ( 14 160 701 640 40 ) 61 ( 1 40 21 CHAP. IX . How a man may become a ready Measurer by Practice in his private Study , without any ones assistance or observation , till he design to practice abroad . THis Art above all parts of the Mathematicks , is burdened with two Inconveniences to the young Practitioner : The necessity of having one to assist him in measuring his Lines with the Chain , as oft as he would practice his skill , or get more , and the exposing of his unreadiness to the view of meddlesom people , while he is yet raw and unexperienced , as every one must needs be at first : Both which may in good measure be avoided by this easie knack . Take a small Packthread , and by knots about half Inch asunder divide it into an 100 parts , as Gunter's Chain is divided ; So shall these small divisions between the knots answer to Links ; and if they be not exactly of one length , the matter is not very weighty , but the more equal and short they are , the better . Having this String thus prepared , and marked with longer and shorter pieces of thread tied in the knots , so as you may readily see where is the middle , and where your divisions of 25 Links , and the smaller divisions of 10 links begin and end ; make all sorts of Figures in your Study or Chamber , by marking places , or sticking Knives or Bodkins at pleasure for Angles , accounting the streight lines betwixt them for sides , and so measure the Figures by your knotted string , and cast up the Content by a Scale . This Work you may manage with your own hands in private , and so make your self very quick and ready when you begin to measure for good and all ; as I once made a full experiment , and it was thus : When I first began to instruct Youths in Mathematical Learning in Warrington , some of my Boys Parents desired a sensible demonstration of their Sons proficiency in somewhat that they themselves could in some measure understand ; and particularly pitched upon measuring a Piece of Land : Whereupon I took four or five of my Scholars to the Heath with me , that had only been exercised within the Walls of the School , and never saw ( that I know of ) so much as a Chain laid on the ground : and to the admiration of the Spectators , and especially of a skilful Surveyor then living in the Town , they went about their work as regularly , and dispatched it with as much expedition and exactness , as if they had been old Land-meters . CHAP. X. How to measure a piece of Land with any Chain of what length soever and howsoever divided ; yea with a Cord or Cart-Rope , being a good Expedient when Instruments are not at hand of a more Artificial make . IF you can procure a Chain , and find it is not divided as before hath been shewed , but into Feet or quarters of Yards , or any such vulgar divisions , make no reckoning of the divisions at all , but measure it as exactly as you can to find out the true length of the whole Chain , and if it fit none of those lengths mentioned in the 8th Chapter , nor any of their halfs , make it to fit , by taking off a Link or two , or piecing it out with a string ; then dividing the length of that Chain by 100 , or the half of it by 50 , find the true length of a Link according to our artificial division , and having got a long stick or rod , set as many of those link-lengths upon it as it will hold ; Then may you measure all the whole Chains by your regulated Chain , and the odd links of every line by your divided stick or rod , as is manifest in this Example following . Being far from mine Instruments , and requested by a Friend to measure him a Close , I procure a pair of Compasses , an ordinary Carpenters Rule of two foot , divided into Inches and quarters , and meeting also with a piece of an old Chain seemingly divided into feet , I measure it by the Rule , and finding it to be 45 feet long , and some odd measure , I piece it out with a pretty strong Cord that will not stretch much , to 48 feet exactly ; then it will serve me for half a Chain of 24 feet to the Pole : This 48 I multiply by 12 ( the number of Inches in a Foot ) and that product being 576 , I divide by 50 , the number of links in half a decimal Chain , and the Quotient is 1166-50 Inches , or 11 Inches and an half , and an trifle over : So then dividing a long stick throughout into such parts , each containing 11 Inches and an half , besides the breadth of the nicks , I am provided of Tools to measure Lines to a Link with exactness enough . In like manner would I proceed with a Cord or Rope , having fitted them to some known length or other . And then for protraction it were easie with the Compasses to make a plain Scale of a large sort , either upon Paper , or an even piece of Wood ; this for once may serve a mans turn well enough . Besides there is a way of measuring the Perpendiculars of Triangles and Trapezia`s upon the ground it self , so as to prevent the necessity of a Scale ; for if you have a little Square with an hole in it , to turn upon the head of a little stick , which you may fix where you please , as you are measuring the Base of a Triangle , or the Diagonal of a Trapezium , you may by a very few trials find the place where the one Leg will be just in the Line which you are measuring , and the other point at the Angle from which the Perpendicular falls on it , and then the space between your Stick and that Angle truly measured , is the Perpendicular . If you have not such a Square , a square Trencher , or any end of a Board that hath one right Angle , and two true sided , will supply the want of it . And now that I am mentioning this way of measuring , I shall make bold to add , that this is a good way , ( and as such ordinarily used by that general Scholar and reverend Minister Mr. Samuel Langley of Tamworth in Staffordshire , whose ancient acquaintance I have long esteemed both mine happiness and honour ) to measure a Trapezium thus , though it be protracted afterwards ; for by measuring the Perpendiculars as aforesaid , and observing at how many Chains and Links end the said Perpendiculars meet the common Base , the whole Trapezium may be truly protracted , without going about it ; this little Square competently supplying the place of an Instrument , which is usually called a Cross or Square , made up ( as it were ) of two small Indices , like those for a Plain-Table ( but much less ) with fore-sights and back-sights , and cutting one another at right Angles , put together , and having an hole at the Center , like those things which here in Cheshire we call Yarndles , being used by Country Housewives in winding of their Yarn . CHAP. XI . Concerning dividing of Land Artificially and Mechanically . WEre it sutable to my Design or Humour to be copious or curious , I had here a fait opportunity ; for four or five modern Survey-Books of the best Accompt lying open before me , would tempt me to transcribe abundance of ingenious things ; but for reasons often hinted before , I shall confine my self to a few plain things that will competently do this business . 1. To divide a Triangle into any parts required ; divide the Base as the Demand imports ; then shall Lines drawn from the Points of Division to the opposite Angle finish the Division of the Triangle . Example . Fi. 20 AC , the Base of the Triangle ABC , being divided into 12 equal parts , a Line drawn from the Angular point B to the point 6 divides the Triangle into two equal parts . 2. Lines drawn to 4 and 8 divide it into three equal parts . 3. Lines drawn to the Points noted with 3 , 6 , and 9 , divide it into four equal parts ; and so Lines drawn to 2 , 4 , 6 , 8 , 10 , divide it into six equal parts . Also it is very obvious , that if the same Triangle were so to be divided , that the one part should be double to the other ; a Line drawn from B to 4 or 8 , doth the work . Or if it be required to divide it into two parts , so as the one shall be triple to the other , a Line drawn from B to 3 or 9 , compleats the Work. So also a Line from B to 2 or 10 divides it into two parts , whereof the one is quintuple ( or five-fold ) to the other , and a Line from B to 1 or 11 , divides it into two parts , whereof the one is 11 times as large as the other . Further yet , if it were required this Triangle should be so divided , that the two parts should in quantity bear proportion , as 5 and 7 , a Line from B to 5 or 7 , doth that feat . But to deal plainly with you , I must confess that sometimes the Division will be a little more intricate than thus , yet not such , but that the seeming difficulty may be easily overcome , by observing the method wherein I shall satisfie the following demand . Suppose a large Triangle of common Land be to be divided amongst three Tenants A , B , and C , according to the quantity of their Tenements , A having 19 Acres of Land to his Tenement , B13 , and C7 , the Base of the Triangle being found by measure to be 17 Chains and 27 Links ; and the Dem and is , where the Points of Division must be placed in the Base , so as Lines drawn from thence to the opposite Angle , shall truly limit each mans part ? To answer this , let us add 13 and 07 to 19 , ( as in the Margin ) and they give 39 : So is the work plainly reduced to the Rule of Fellowship ; and therefore to find every mans distinct portion , we need only to multiply the Base by his 19 13 07 39 particular number , and divide that product by 39 , the sum of all their numbers as here is plain ▪ A 39 . 17. 27 ∷ 19 19 15543 1727 39 ) 32813 ( 841 14 / 39 312 161 156 53 39 14 B 39 . 17. 27 ∷ 13 13 5181 1727 39 ) 22451 ( 575 2● / 3● 195 295 273 221 195 26 C 39 . 17. 27 ∷ 7 7 39 ) 12089 ( 309 38 / 39 117 389 351 38 From these Operations it is plain , that if we set off from the Angular point where the Base begins , 8 Chains 41 Links , and a little above the third part of a Link upon the Base for A , and where that ends , 5 Chains and 75 Links and 2 / 3 of a Link for B , and consequently leave between this second division and the other end of the Base 3 Chains and almost 10 Links for C ; Lines drawn from those points of division to the opposite Angle , will give each man his due . What I have said touching the division of Triangles upon their Bases , will with a little variation serve for the dividing of all sorts of Parallellograms , whether Square , Long-squares , Rhombus's , or Rhomboides's : all the difference is , that in stead of drawing Lines from Points in the Base to the opposite Angle , you must draw parallel Lines from Points in one opposite side to another , as will be sufficiently plain by this one Instance . Fig. 7 Suppose the square Figure in the 8th Prop. of the second Chapter to represent a Close of six Acres , and I am to cut off an Acre at the side AC , having set off the 6th part of the Line CD , from C towards D , and also from A towards N , a Parallel drawn between those Points , takes off exactly a 6th part , or an Acre . If it be not thought convenient ( as in some cases it is not ) to cut off a piece so long and narrow , you may by the Rule of Three find what other length of any greater breadth will limit an equal quantity to it . Or you may multiply the breadth by 2 , 3 , or any other , and divide the length by the same number that you multiplied the breadth by . Or lastly , if you set out a double proportion that is 2 / 6 or 1 / 3 , from C towards D , and from the Point where it falleth , draw a Line to the Angle A , you will have a Triangle equal to 1 / 6 of the Square ACDN. But to return to Triangles , ( the most simple and primitive of all Rectilinears , and therefore the most considerable in this case of partition , as giving Laws often to the rest : ) It may fall out , that a Triangle must be divided ( convenience so requiring ) by a Line from some Point in a side , so as that Line may either be parallel to some other side , or not parallel to any . For the former case take this Example following out of Mr. Wing . Lib. 5. Prob. 5. Fig. 21 Let ABC be a Triangle given , and it is required to cut off 3 / 5 by a Line parallel to AB . First , on the Line AC , describe the Semi-circle , AEC , whose Diameter CA divide into five equal parts according to the greater term , and upon three of those parts ( the lesser term ) erect the Perpendicular DE , which cutteth the arch Line in E ; then set the Line CE from C to F , and from thence draw the Line EG parallel to AB ; so will the Triangle CGF contain 3 / 5 of the Triangle ABC , as was required . Fig. 22 Now for the latter case , when the line of partition goes not parallel with any side , take this Example : Let ABC be a Triangle given to be divided into two parts which shall bear proportion to one another , as 3 and 2 , by a Line drawn from the point D in the Base , or , Line AC . From the limited point D , draw a Line to the Angle B ; then divide the Base AC into five equal parts , and from the third point of Division draw the Line to E , parallel to BD. Lastly , from E draw the Line ED. So shall the Trapezium ABED be in content , as 3 to 2 , to the new Triangle DEC . I have now done with the Division of Triangles , when I have added these three Advertisements . 1. You must be sure to take very exactly the distance of every point , where a dividing line cutteth any side , to one of the ends of the same side , as in this last Figure , the distances BE and AD , which distances being applied to the Scale by which the Triangle was protracted , will shew at how many Chains and Links-end you are to make your dividing Line on the Field it self . 2. The proportions by which you are to divide , are not always so formally given as in the former examples , but are sometimes to be found out by Arithmetical working , as in this case . Suppose a Triangular Field of 6 Acres , 2 Roods , and 31 Perches , must be divided , so as the one of the two parts shall be 4 Acres , 3 Roods , and 5 Perches , and the other ( consequently ) 1 Acre , 3 Roods , and 26 Perches , reduce both measures into Perches , and the one will be 765 , and the other 306. Their Sum is 1071 , which by their common measure being reduced into their lowest terms of proportion in whole Numbers , will be 5 , 2 , and 7 , which shews that the Triangle being divided into 7 equal parts , the one must have 5 of those 7 parts , and the other 2. And observe , that it will be sufficient to find the common measure between the Sum of the terms and either of the terms ; the method whereof is shewed in every Arithmetick Book for reducing Fractions into their lowest terms . But if my unlearned Reader cannot skill of that work , he may multiply either of the parts ; ( as suppose 765 ) by the length of the Base , which we will suppose to be 8 Chains and 75 Links , or 875 Links : and that product divided by 1071 , ( the Content of the whole Close in Perches ) gives by the Rule of Three direct , 625 Links , or 6 Chains and 1 Pole , the true distance from either end of the Base , that his mind or occasions may direct him to begin with , to the point of Division ; for the Division must be not only for proportion or quantity , but also as to position or situation of parts upon the Paper , as it is required to be on the Ground . 3. In these and all other divisions of Land , where a strict proportion in quantity is to be observed , you must have respect to the Rules hereafter following , concerning measuring of uneven Ground , Chapter 15. especially if one part prove much more uneven than another : and if there be any useful Pond or Well to draw your Line of Division through it ; but if it be an unuseful Pond , Lake or Puddle ; or if there be an boggy or barren ground , that must be cast out in the divisions ; measure that first , and substract it from the Content of the whole Close , and then lay the just proportion of the remainder on that side that is free from it , that the other may have its just part also , besides that which is useless . What hath been said , with an ordinary measure of discretion , may sufficiently instruct a young Artist to divide Triangles , Parallellograms , and regular Polygonials , in an artificial way : but because many Closes and open Grounds are Trapezia's , and many irregular Polygons , and even those that are regular enough , may fall under an irregular division , in regard of the quality of the Land , Woods upon it , or Quarries in it : Or the conveniences of ways , Currents of Water , situation in respect of adjacent Lands , &c. I shall propose a Method , which though it hath somewhat of the Mechanick in it , will be singularly useful in such cases . Fig. 23 Let ABCD be a Trapezium to be divided betwixt a young Heir and his Mother , so as his part may be double to hers . Having by the Diagonal BC divided it into two Triangles , I find the Content of the Triangle ABC to be 138550 square Links , and the Triangle BCD to contain 103468 , in all 242018 square Links , which if reduced , as hath been formerly taught , would amount to 2 Acres , 1 Rood , and 27 Perches : ( for which see Chap. 5. ) But as to my present Work , they are in a better Order already . Dividing then 242018 into three parts , each of them is 80676 and 2 / 3 , Fig. 23 two therefore of those third parts must contain 161353 and 1 / 3 , which 1 / 3 being inconsiderable , I regard not . Then resolving to lay out the double part towards the Line BD , I strike at adventures the Line EF , and measuring the Trapezium bounded by that Line , and the opposite side BD , together with the Interjacent parts of the Lines AB and CD , I find it to contain 119140 square Links , but because it should have been 161353 , I substract 119140 , out of 161353 , and their difference is 42213 , and perceiving that the Lines AB and CD are very near parallel , and finding their distance where they are cut by the Line EF , to be 326 Links , or 3 Chains and 26 Links , I divide 42213 by 326 , and the quotient is 129 Links and almost half , at which distance I draw the Line GH parallel to EF ; so shall the Trapezium GBDH be the Heirs part . Another way whereby this may be preformed , is thus : Finding the Triangle ABC , to contain 138550 square Links , substract it out of the Heirs part , viz. 161353 , the difference 22803 , shews how many square Links must be taken out of the Triangle BCD , and added to the Triangle ABC : Which to perform with all necessary exactness ; suppose the side or line BD to be the Base , which by measure proves to be 344 Links , or 3 Chains and 44 Links . Say by the Rule of Three direct , If the whole Content of the lesser Triangle , viz. 103468 , give 344 ; what shall 22803 give ? so will the result be 75 links , and somewhat more than 4 / 5 of a Link ; for 22803 multiplied by 344 , gives 7844232 , which being divided by 103468 , the Quotient is 75 84132 / ●03468 ; or ( according to decimal Division ) 75.8131 , which is ( as I said before ) somewhat more than 75 Links and 4 / 5 , wherefore extending your Compasses upon the Scale to almost ' 76 Links , set that distance upon the Line BD , from B to I , and draw the Line CI : so shall the Trapezium ABIC be double to the Triangle ICD , within so small a matter as is not worth regarding , though the Land were a rich Meadow . I hope I need not stand to tell any man of sense , that if he please he may begin with the less part , and take out that : or if there be many Partners , he may divide betwixt any one and all the rest , ( putting their parts together ) and then by the same method subdivide amongst them till each hath his due share ; nor to spend many words in telling him he must substract where he hath by a separating Line at adventures , or by choosing , out a Triangle , taken too much . as I added , when I took too little . Nor lastly , that these methods are not only applicable to Trapezia's , but to any triangular Figure whatsoever , whether regular or irregular . CHAP. XII . Concerning the Boundaries of Land , where the Lines to be measured must begin and end . IF there be no agreement between the Parties concerned , ( for if there be , that must be observed ) Reason and Custom are the Surveyor's Guide . The Farmer speaks loudly , that when a piece of Arable or Meadow-land is let for a year to be sown or mown , no more should be measured nor expected to be paid for , either to the Letter or Workmen , than the Plow or Sythe can go over . So also when a parcel of Land is let for Pasture by measure to a Farmer , it seems very reasonable , that all and only so much should be measured as is useful to that purpose . But Commons to be enclosed are usually measured ( except it be otherwise agreed ) to the uttermost bounds of every mans particular proportion , without any allowance for Ditch or Fence ; every man being to make them upon his own of what breadth he pleases : Nor is this unreasonable , for 't is as good for one as another , and the rate paid to the Lord is usually very little , sometimes nothing . It is also very usual in measuring betwixt Lord and Tenant , in case of Leases for Lives , and long terms of years , to extend the Lines to the utmost bounds of the Tenants claim , taking in the very Walls , Hedges , and Ditches : but this is accounted very hard , and oft proves very unequal among the Tenants of a Lordship ; some being forced to make much more waste of their Ground this way , than others that hold as much or more . But where the Custom obtains , the Survey or must observe it : For it is others work to appoint what must be measured , and his only to measure truly what is so appointed . A good Landlord may ( and will be apt ) to consider it in his Rates , and a bad one 't is like will be tenaC●●us of a Custom to his own advantage . Lastly , In case of Sale by measure at a rate agreed upon per Acre ( no Boundaries being specified in the Bargain ) the Rule ( as I had it from an old famous Surveyor many years ago ) is to extend the Lines to the quick Wood-row , that is , as Reason prompts me to understand him , to the place where the quick-wood actually groweth , or where according to custom it ought to be set . I confess these things are trivial , but yet more necessary to be known , than many artificial things much stood upon , for a young Artist , whom ignorance of these things may expose to considerable mistakes in practice . CHAP. XIII . Containing a description of the Plain-Table , the Protractor , and Lines of Chords . THough what hath been already said , may competently suffice to instruct my young Artist in measuring a Close of Land , yet to advance him a degree higher in useful knowledge , I shall take occasion to describe unto him the Plain-Table , which with Mr. Wing I account the best of all fixed Instruments : This Instrument consists of several parts . 1. The Table it self , which is a Parallellogram of Wood fourteen Inches and an half long , and eleven Inches broad , or thereabouts , and for necessity may be made by an ordinary Country-workman of one Board : but for neatness , convenience of carriage , and freedom from warping , it is usually made of three little Boards joyned together side-ways , with a ledge at each end to hold them fast together , and upon the middle Board a Socket of Brass fixed with three Screws , and with a fourth to be fastned on the head of a three-legged Staff ; of which anon . 2. A Frame of Wood fixed to it , so as a sheet of Paper being laid on the Table , the Frame being forced down upon it , squeezeth in all the edges , and makes it lie firm and even , so as a Plot may be very conveniently drawn upon it : this is usually made with Joynts for more easie carriage , but a plain one may suffice . Upon one side of this Frame should be equal divisions , for drawing parallel Lines both longwise and cross-wise ( as occasion may require ) over your Paper , and on the other side the 360 degrees of a Circle projected from a Center of Brass conveniently placed in the Table . 3. A Box with a Needle and Card , to be fixed with two Screws to the Table , very useful for placing the Instrument in the same Position upon every remove . 4. A three-legged Staff to support it , the Head being made so as to fill the Socket of the Table , yet so as the Table may be easily turned round upon it , when 't is not fixed by the Screw . 5. An Index , which is a large Ruler of wood , ( or rather brass ) at the least sixteen Inches long , and two Inches broad , and so thick as to make it strong and firm , having a sloaped edge , ( by which we draw the Lines ) called usually the fiducial edge , and two sights of one height , ( whereof the one hath a slit above , and a thread below , and the other a thread above and a slit below ) so set in the Ruler , as to be perfectly of the same distance from the fiducial edge . Upon this Index 't is usual to have many Scales of equal parts , and there might be a Diagonal Scale if the Instrument-maker please , and Lines of Chords of sundry lengths : but if you have such a Scale as I before described ; you need not to have them here . The Protractor is an Instrument so well known , and so easie to be made and procured , that I shall be very brief in the description of it . As it is usually made , it consists of two parts , a Scale and a Semicircle , but the Scale is no necessary part of it , but serving ( if you be not otherwise provided ) for other uses before mentioned in the case of Plain-Scales . But the Semicircle is more essential , and it may be made of Brass or other Metal of any convenient size , as four Inches ( more or less ) for the straight sid , this Semicircle being bounded as all others are by two Lines , the one right or strait , the other circular . The right Line is divided in the precise middle by a Point which is in the Center , upon which the Circular boundary is drawn , and two other Arches concentrial with it . The Center , when the Semicircle goes alone without the Scale , should be guarded with two little lips , on each side one , or a little loop , for more convenient turning of the Instrument about upon a Pin fixed in a Paper . The arched or circular edge is divided into 180 Degrees or equal Parts numbred by Tens upon the upper Concentrick Arch , from 0 to 180 , and in the lower from 180 to 360. So that by applying the straight edge of the Protractor twice to any Line , ( keeping the Center right● upon a Pin fixed in the Line ) that is , with the Semicircle first above it , and then below it , or contrarily , you may draw a whole Circle by the guidance of the Arch , or set out any number of Degrees , as will appear more plainly hereafter . A Line of Chords is a Line divided into 90 unequal parts , whereof 60 , and the Radius upon which the Circle was drawn , are equal , and the Divisions upon that Line are equal to the next Extent in a right line , of so many Degrees from the beginning of the Quadrant as answer thereunto . When lines of Chords are cut upon wood , 't is both usual and necessary that there be two Studs of Brass , the one at the beginning , and the other at 60 Deg. with little holes for the feet of the Compasses , when you take the extent of the Radius , to preserve the line from being wounded by the Compasses ; and being thus fenced , it will for need do the work of a Protractor , but not altogether so commodiously . CHAP. XIV . How to take the true Plot of a Field by the Plain-Table , upon the Paper that covers it , at one or more Stations . THere are three ways or methods for doing this work , two more usual and ordinary , the third more unusual and extraordinary , though now pretty well known to most Surveyors , and in late Books published . The first performs the work by measuring every line from the Instrument to every Angle , and is a very sure substantial way where it can be done , as it ordinarily may in most Closes . The second doth it by measuring only the Station or the Distances , and is very quick , but not so sure and exact as the other ; yet if managed by a skilful Artist , that knows how to plant his Instrument , so as to avoid making acute Angles unnecessarily , it will come near enough the matter in many cases ; as in measuring for Workmen , that take the Mowing or Reaping of Fields by the Acre , or when Tythes are let at a small value per Acre , as in poor barren Parishes they usually are . The third is the way of Circuition or Perambulation , the Instrument being oft to be planted , and the Plot to be measured about , by which not only difficult Closes , but even the thickest Woods , yea Bogs , Meres , and Pools of Water , may be plotted , which by neither of the other methods can be performed . In all these Methods , two things are to be performed : 1. At every Angle where there is no perspicuous Mark already , as a Tree , Bush , Stile , &c. one must be placed , as a white Paper , or such like ; or else some one must go from Angle to Angle , and remain there as your Mark to look at , till you bid him remove to another : only when Angles are very near you , this labour may be spared . 2. When ever you have occasion to plant your Instrument more than once , ( as it will often fall out in the first Method , and ever in in the two latter ) you must be sure it stand just as it did the first time for situation , for which your Needle if well touched and hung , will be good direction , but is not thought sufficient without back sight and fore-sight , ( by some practical Surveyors : ) I shall therefore in due season shew you that knack . Now for the first Method . I. When you go about to plot your parcel of Land , find such a place in it if possible , from whence you can see all the Angles , and in that place plant your Instrument covered with a sheet of Paper , and turning it about till the Needle playing at liberty , hang over the Flower-de-luce , ( or any other notable place that you make choice of ) screw it fast . Then choosing any convenient place in your Paper for a Center , and to represent your station ( or place where you fix your Instrument ) make a prick with the small point of your Compasses , to which prick applying the fiducial edge of the Index , ( which is easily done if you keep the Point of the Compasses resting in it ) direct the Index by the sight to all the Angles , and when through the slit or long sight , you see the opposite Thread cut the Mark in the Angle , draw a neat dry line along the fiducial edge to or from the Center : then measuring from the Instrument to every Angle , set the measure by a Scale and Compasses from the Center towards the Angle , upon the line that points at it , making a prick in the line where the Chains and Links ( reckoned from the Center ) do truly end : then shall Lines drawn by a straight Rule from prick to prick , give you the perfect plot upon your paper , which you may divide ( as hath been before shewed ) into Trapezla's and Triangles , and to find the true Content . To make which plain , mark this Example . Fig. 24 Suppose ABCDEFGH to be a Field ; having planted my Plain-Table as before directed at a convenient advantage , so as to see all the Angles , ( as at I ) I make a prick to represent my station in the little Circle ☉ marked with I , upon which laying the fiducial edge of my Index , and directing the Sights to all the Angles , I draw dry lines toward A , and all the rest of the Angles in order from the Center ; and then measuring upon the ground from the Instrument to the Angle A , I find it to be 3. 45 , which I set ( by the help of my Scale ) from the Center to the Point A , and so upon all the rest according to their due measures , and then black lines drawn from Point to Point , as from A to B , from B to C , &c. limit the true Figure of the Field according to the Scale I used , viz. of 400 in an Inch. And now before I pass to further varieties , let my Reader take notice of these following things . 1. From henceforth I shall forbear ( for brevities sake ) to take any notice of the measures of lines measured from the Instrument to the several Angles , having so often shewed how to measure by a Scale . 2. When I speak of measuring from or to the Instrument , I always mean from or to that part of ground that is perpendicularly under the head of the Instrument , where you are to draw your Plot , which will ever be enclosed with the three Legs of your Staff. 3. That it 's usually the quickest way to measure first from the Instrument to the first Angle , and then back from the second Angle to the Instrument , and so the rest in order , still one from the Instrument , and the other to it . 4. It is no matter at all whether your Plain-Table be placed towards the middle of a Field , as was represented in this Figure , or at an Angle , as will appear anon . 5. In all workings by this Instrument , you must have a care that the Instrument be not moved out of its due place , till you have finished the work of the present Station ; for which purpose , cast your eye now and then upon your Needle , observing whether it continue to hang directly over the same point you set it at when you began your work , and to rectifie your Instrument if you see cause . But because all Tables have not Needles , and where Needles are , they are not accounted over-trusty , make use of the following help . When you have planted your Instrument , and made a point or prick in your paper , representing your station , set the fiducial edge of your Index to it , and turning it softly about till you find one remarkable thing or other upon one side of the Close , and another on the opposite side as you look through the sights of your Index ( which we call fore-sight and back-sight ) draw a remarkable Line with Ink , or rather with a Black-lead Pen quite over your Paper , which in this Figure is represented by the black line KL : and then if you suspect that by any accidental jog , or other casuality , the Instrument is any thing removed , you may easily try and rectifie it , by applying the fiducial edge to the same Line , and making use of fore-sights and back-sights again , upon the same Marks which you before observed upon the opposite sides of the Close . But if there be no convenient place for the placing of your Instrument , whence you may see all the Angles of the Field , more stations must be made use of thus : Fig. 25 Let ABCDEFGHIKLMN be a Field whose Angles cannot be all seen from any one Angle , or other place in it : I plant my Instrument at the Angle A , and if it have a Needle , I mark what Degree of the Chord it cuts , or turn about the Table on the head of the Staff , till the Needle hang over over some remarkable place ; ( as suppose the Flower-de-luce ) and screw it fast ; then setting up a Stick with a white paper or cloath on the head of it , where I intend my second Sation ( as here at Q. ) I make a prick or point in my paper , to signifie the point A upon the paper on the Table : to which point I apply the fiducial edge of the Index , and when I see the white at Q , so as looking through the slit , I see the thread , cut it , I draw the Line OP quite through my Paper with a Black-lead Pen , and then keeping the fiducial edge still upon the same point , and turning it round by degrees , I look at the Angles BCDEFLMN , still drawing dry Lines with the points of my Compasses , and setting on the Measures from the Station A to every Angle measured to or from , as I did in the last Example . Then I remove the Instrument to the place of my second station , having set up a mark at A , and laying the fiducial edge to the Line OP , I turn about the Table upon the head of the Staff , till through the slit of the back-sight , I see the thread cutting the mark at A , and then screw it fast , so will my Needle , if a good one , hang directly over the same point that it did at the first station ; but however that be , fore-sight and back-sight will do the business ; for which purpose it is good to take back-marks as well as fore-marks at every station , as was taught in the Example of a single station , only taking notice that the back-mark when the Instrument is planted in an Angle , must needs be out of the Field ; as suppose here at O. But to proceed . Having measured the distance between the first and second station , and finding it to be 7. 10 , I set it upon the Line OP , from A to Q , where I make another point to represent the second station , and turning about my Index with the fiducial edge upon that point , and so looking through the sights at the Angles GHIK , I draw Lines towards them on my Paper , and having measured between every one of those four Angles , and the Instrument , I set those Measures as I did the other , with my Scale and Compasses , from Q towards every Angle upon his proper Line : and then having drawn the black bounding-lines from A to B , from B to C and so round about the Close , the Protraction is finished . But here to make this Figure yet more advantageous , let me ( according to my usual method ) add some Advertisements . 1. Sometimes a Station is so taken , that you may measure towards two Angles at once , ( as here from Q to G and H ) in which case you are to set down the Chains and Links where the first Angle falleth , but still be proceeding to the further Angle , causing the remainder of the Chains at the fore-end to advance beyond the former Angle , so going on with whole Chains so far as you can , to which the odd Links at the end are to be added . 2. If at any of your Stations ( as suppose A ) you can see an Angle ( for example E ) to which you cannot measure in a direct Line without passing the boundaries of your parcel of Land given to be measured ; you may notwithstanding take in that Angle by a strait measured Line , as I have done , provided it may be lawfully done without trespass , and conveniently without troublesom passing of Fences , otherwise it must be taken from another station . 3. I here took one of my Stations at an Angle , and the other within the body of the Field , to shew the variety of working taught by other Authors , and that 't is no great matter where you make your Stations , so you can see the Angles : else it had been full as convenient to have taken my first Station also within the body of the Field , as suppose at R. 4. Though this Figure representeth to your eye only two Stations , A and Q , your fancy may multiply them at pleasure , for suppose the Angle H could not have been seen from A or Q , how easie had it been to have set up a Mark at S , and then to have removed the Instrument thither , observing the same directions ●hat were given at the removal from A to Q. II. In the second Method the Instrument is to be planted twice , or oftener as occasion is , the Rules for removal of the Instrument fore-sight and back-sight , and measuring the distance of Stations , being the same as formerly was ●aught : but instead of measuring to and from ●very Angle , we only view each Angle through ●he Sights from two Stations , having applied ●he fiducial edge to the Points representing ●hose Stations , and having drawn Lines with the point of the Compasses , or a protracting Needle , the Interfections represent the Angles , from which the boundary-lines may be drawn , so is the Field protracted : Which that my Reader may understand , let him note these three Figures . Fig. 26 Here in these three Figures the Angles are marked Alphabetically , ABCDEF , &c. and ●he Stations by a point in a small Circle numbred , 1 , 2 , or 1 , 2 , 3 , according to their number and order . Fig. 26 The first of these Figures represents the Plot●ing of a Field at two Stations within it , from both which all the Angles may be seen . Fig. 27 The second , performs the same work by two Stations taken without the Field , by which 〈◊〉 a Close may be measured , though the present Possessor will not give us leave to come ●nto it . Fig. 28 The third , shews how the Work may be performed at three Stations or more , when two such places cannot be found whence to view al● the Angles ; which last having more of difficulty than the two former , ( though indeed no● very much ) and the Explanation of that will sufficiently help to the understanding of them ▪ I shall a little explicate the meaning of it in these particulars . 1. From the first Station taken acording to former directions , I see the Angles ABCD FGK , and acordingly draw Lines upon my Paper towards them from the point representing that Station , by the fiducial edge of my Inde● with the point of my Compasses . 2. Having removed my Instrument to the second Station , ( and in so doing , observed the Rules before given , touching such removals ) I thence see the Angles ABDEFGHIK ▪ and draw Lines upon my Paper towards them from the point representing the second Station . And now viewing my Work , I find upon my Paper Interfections for the Angles ABDF GK , but only single Lines toward the Angles CEHI : therefore . 3. Removing the Instrument regularly as before , to a third Station , I thence see those four Angles CEHI , and drawing Lines towards them , I have interfections for them also ; so that having drawn the Lines AB , BC , &c. from one Interfection to another , I have the Field perfectly protracted . For these bounding-lines from Angle to Angle , do not only signifie the Boundaries of a piece of Land given to be measured , limiting the Figure or shape thereof , ( and are to that purpose given in this and all other Survey-books ) but also are the true distance by a Scale from Angle to Angle for the Plot upon the Paper : I mean by the same Scale by which the stationary distances were laid down upon their own lines . And this holds true in all kind of true plotting , whether in this Method or any other . III. The third Method is that of Circuition , and this hath several varieties , according to these three following Cases . 1. When the distance from Angle to Angle , ( without any exception ) is measured quite round the Plot , either within or without . 2. When the distance is taken only between some more notable Angles , and the Perpendiculars of the rest measured as you pass along their Bases , within the Plot , proper for plain solid ground . 3. When the like is done without the Plot , as in the case of Plotting thick Woods , Meres , Pools , Bogs , &c. The first of these is very easie , consisting in nothing but planting the Instrument at every Angle ( either within or without , as necessity and convenience determine it ▪ ) observing the former directions for planting and removing the instrument ; and also for measuring the stationary-lines on the ground , and protracting them on the paper , as is manifest in this Example . Fig. 29 Let ABCDEF be a Park-Pond or Close to be protracted , I first plant my Instrument at A , and direct the sights to a Mark in the Angle B , drawing a dry Line from a convenient Poin● on my Paper towards B on the ground , and having measured by my Chain the distance AB I set it by a Scale upon the Correspondent Line from A to B , drawing a black-line between them with Ink or a Lead-pen , the Extremities whereof are the Points A and B on my Paper , and the little pricked Line that goes beyond B , represents the remainder of the dry Line drawn at random ( as to length ) with the point of the Compasses . Then setting up a Mark at A ( if there was none before ) I remove the Instrument to B , and laying the fiducial edge to the Line AB , I turn about the Instrument upon the Saff , till through the sight I perceive the thread cutting the Mark at A , and my Needle ( if I have one ) directly over the same point , that it was when it was planted at A , and so screw it fast . My next work is to lay the fiducial edge to the point B , and direct the sights to C , drawing a dry Line towards it , and setting the distance BC measured by the Chain from B to C. In this manner I proceed , surrounding the Close till I come at last to A , where I began , by planting the Instrument at every Angle , using the help of Back-sight and my Needle , as I did at B , and then from the point representing my present station , directing the sight to the next Angle , as I did from B to C. In the second Case , we do not plant the Instrument at every Angle , but at the more considerable , taking in the smaller by their Perpendiculars from the Base as we pass along , of which this following Figure may be an Instance . Fig. 30 Let ABCDEFGHIKL be a Close to be measured ; by planting the Instrument only at ACF and K , I have the main substance of the Close in the Trapezium ACFK , and for the five small Triangles which must be added to the Trapezium , they may be easily protracted by the help of such a little Square as was mentioned towards the latter end of the Tenth Chapter ; for thereby finding at how many Chains and Links distance from A upon the ground , the Perpendicular B b falleth upon the Line AC ; and having measured the length of that Perpendicular , and taken it between my Compass-points off my Scale , I erect a Perpendicular of that length at b , which is the point upon the Paper , where so many Chains and Links determine , as were measured upon the Ground , from the Angles A , to the place where the Perpendicular fell on Ac , viz. at b. Just in the same manner I raise the other Perpendiculars , Dd , Gg , Ii , and Ll : and then by the help of the Perpendiculars , I draw ( from and to the proper Angles ) the Boundary-lines AB , BC , CD , DE , FG , GH , HI , IK , K L , LA ; which together with the Line EF between the Angles E and F , give the true Plot of the Field in one large Trapezium , and five small Triangles ready for casting up . The third case is so like the second , that there needeth no new Direction concerning it , but to annex one plain Diagram ; all the difference consisting in this , that because we cannot go within it , ( being supposed to be some Pool , Bog , or Thicket ) we must of necessity go on the out-side , and consequently all the Triangles made by inward Angles , and their Lines upon the measured Bases , must be excluded by the boundary-lines from being any parts of the Plot , as here is manifest . Fig. 31 Supposing ABCDEFGHIKLMN OP , to be a great Pool , though here be fifteen Angles , I plant my Table only five times , viz. at AEFH and M , and upon the dry Lines AE , FH , HM , and MA , I raise their Perpendiculars in due places , ( according to measure ) and also of a right height : by which and my five stationary Angles , I draw the bounding-lines of the Plot , excluding all the Triangles as foreign to it , they being no resemblances of any part of the Pool , but of Land adjacent . Where note , 1. That both in this and the former Case , such a little Square as I mentioned in the Second Chapter , will be very useful for speedy raising of Perdendiculars ; but where the Triangles are very small , it needs not be used , not the other mentioned Chap. 10. 2. That if by reason of troublesom Brush-wood , Gorse , or Bogs , &c. I could not have measured close to the sides EF , HI , or LM , it would be the same thing if I went parallel to them . And this is a shift that the practical Surveyor will oft be put to make use of , in other cases as well as this . CHAP. XV. Concerning the Plotting of many Closes together , whether the Ground be even or uneven . THough I design not so high in this Manual as to make my Reader able to Survey Lordships and Forests , much less to draw Maps of Countries , but to measure a parcel of Land with truth and judgment ; yet I would have him so expert , as not to be puzzled , if any should desire him to draw ( as it were ) a true Map of a Tenement or small Demesn , consisting of several Closes ; for which purpose , let him that knows no better observe this Method . Fig. 32 Suppose ABCDEFGHIKLM to be a Tenement or small Demesn divided into fourteen Closes , to be measured and protracted according to their several shapes and situation . I first draw the Plot of the whole by the Method of Circuition , planting mine Instrument either at every Angle , or only the most considerable either within or without , as I find most convenient . This being done , a Line from B to M gives the Triangle ABM from the first Close : In the next place , I go round the second Close beginning at M , then to B , and so about ( cum scle ) to M again : And then for the third Close I plant my Table at C and go round to B , ( the Line BC being protracted already ) and so of all the rest , still observing which are common Lines belonging to several Closes ( representing the Fences ) that I may avoid the trouble of measuring ●hose Lines oftener than once , and lay ever● part of ev●●y Close in its due place ; and that I be su●● to keep the Instrument throughout the whole Work to its true Position by Needle , fore-sight and back-sight . There are I confess divers other ways of doing this Work , but none more sure or plain , especially if the ground be uneven ; for in that case , if you protract according to the length of Lines measured from your Station to the Angles , you will put your Closes into unproportionable shapes , except you reduce Hypothenusal Lines to Horizontal , by instruments or otherwise ( which is somewhat troublesom : ) and the like may be said when you Plot with the Chain only . Indeed the Method of measuring only the stationary distances were very proper for setting out the Figure of each particular Close , provided the distance of the stations be large , and taken ( if possible ) upon pretty even ground , ( which sometimes may be done , though most of the Close be uneven ) and the Work so ordered , as not to make too acute Angles : but because this requireth skill and care , I rather advise my young Artist to use the circling way , as ordinarily most commodious in my poor judgment , ( but not prejudicing other men's that may differ from me in opinion ) and where need requires let him observe the directions in the 17 Chapter . But which way soever you go to work , there is one very necessary Rule to be observed . If the Ground be uneven considerably , you must not give up the Content by measuring the Bases and Perpendiculars of the Triangles on the Paper by your Scale ; but you must measure the Lines correspondent to them on the Ground , and cast up the Content according to that measure . And if it be desired that you should adjoyn to your Plot ( as is usual ) a Scale of Chains to measure distances by ; you must either by making the Forms of Hills erect and reverse , or some other Note in writing , mark out your uneven Ground , lest those that try it by the Scale , judge your work erroneous : for though you make that Scale exactly correspondent to that you protracted by ( as you ought to do ) the Hills and Dales in the ground truly measured , may make a considerable alteration . It is convenient when you plant your Table that the Needle hang just over the North-point of the Compass under it in the Box ; then may you by the Lines overthwart the Frame of the Table , easily draw two Lines quite through the Plot , cutting one another at right Angles , the one pointing at North and South , and the other at East and West . And if your skill serve you to make the Two and thirty Points of the Compass upon the place where they intersect , and to draw the Forms of the Houses , Woods , and other remarkable things upon the Demesn , and the course of Brooks and Rivers running through it , it will add to your commendation . And so it will also , if you take in such parcels of Land bounding it , whether common or peculiar to other men , as will make your Plot to look handsomly , like a perfect Square or Oblong . But however that be , you must be sure to protract truly all Lanes going into it , or through it , and all Closes of other mens mixed with it ; and also all considerable Ponds , Ways and Outlets , with the Names of the Closes and quality of the Ground , whether Meadow , Pasture , Arable , &c. CHAP. XVI . Concerning shifting of Paper . IN such work as that of the last Chapter , it may sometimes fall out ( through the multitude and largeness of Fields ) that one sheet will not hold your whole Plot , in which case you may help your self by shifting Paper ( as we call it ) thus . Fig. 33 Let ABCD represent our sheet of paper that covereth the Table , upon which the Plot of the large piece of Land EFGHIK should be drawn ; having made my first station at E , and the second at F , I find my paper will not receive the Line FG : but however I draw so far as it will go to the edge of the paper , and planting my Table again at E , proceed in my Circuition the contrary way to K and I , where I find my self again at a loss for my Line IH , but draw it also to the edge of the Paper ; Then with the Point of my Compasses striking the Line PO , parallel to the edge of the Paper BC , and the Line QO parallel to DC , and cutting PO in O , I throw aside that paper for a while , covering the Instrument with a new one , which I mark with the figure ( 2 ) for my second sheet . Fig. 34 Upon which second sheet ( the leading part whereof is represented by the three Lines meeting in the Angular points A and B ) I draw PO parallel to AB the leading edge of the paper , and crossing it at right Angles in the point O , by a parallel to BC , viz. the Line OR , being of the same distance from BC , that QO in the former sheet was from DC . Then with a Rule and a sharp Pen-knife I cut off the end of the first sheet at the Line PO , and applying the edge of it to the Line PO of the second sheet , so as it may touch that Line all along , and the Line QO of the former , touch the Line OR in the latter , so as to make one Line with it : I draw the Lines PG , being the Remainder of the Line FG , and the Line OH being the remainder of the Line IH , and from their extremities the Line GH . And if the Plot required it , you might proceed on in the second sheet , and annex a third and a fourth , &c. as there is occasion . These sheets may be pieced together with Mouth-Glew or fine Paste , applying the edge of the former ( as you did upon the Table ) to the Line PO of the latter . And note here once for all , that when I speak of applying the edge of the paper to a Line , I mean the precise edge cut by the Line PO ; but when I speak of drawing Lines to the edge of the Paper upon the Table , I hope none will think me so absurd as to mean the edge that is couched under the Frame , but that my meaning is , that the Lines must be continued on the paper till they touch the Frame . CHAP. XVII . Concerning the Plotting of a Town-Field , where the several Lands , Butts , or Doles , are very crooked : VVith a Note concerning Hypothenusal , or sloaping Boundaries , common to this and the Fift●enth Chapter . Fig. 35 SUppose ABCDE divided in the manner of a common Field , into seven parts or Doles , belonging to seven several men : First Plot the whole as before hath been taught , then measuring from A to B upon the Land : set one Note down ( as you go along ) at how many Chains or Links ( or both ) the Division is between Dole and Dole , and accordingly mark them out by the help of Scale and Compasses in the Line AB on the Paper plot . In the very same manner you must measure and mark out the Lines OC and ED ; which being done , take the Paper off the Instrument , and laying it before you on a Table , with the side AE towards you ; the Compasses must be so opened and placed ( as by a few tryals they may ) that one foot resting upon the Table , the other may pass through the Points of Division upon all the three Lines , viz. AB , OC , and ED , as in this Figure they do . If the Content of any one or more of these Parts , Butts or Doles , be desired without Plotting ; it may easily be done without your Plain-Table thus : Take the breadth by your Chain at the head , middle , and lower end , and adding these Numbers together , the third part of their Sum is the equated breadth : by which multiplying the length measured down the ridge ( or middle ) the Product gives the Content . But both in this case , and that mentioned in the 15th Chapter , the Figure of a Plot may be somewhat disordered , not only by the unevenness of the ground within , for which I have given due caution already ( that being both the more common and more considerable case ) but also by the great diclivity of the Ground where the boundary-lines go , either of the whole Plot or particular parcels . For whereas in Plotting , every Line is presumed to be Horizontal ( or level ) that it may pass from Angle to Angle the shortest way , and that every part may be duly situated , and none thrust another out of its right place : If it be not level , but falling down towards a Valley , or rising up Hill , or compounded of both ; a Line over such Ground ( though true for the measure , and for giving up the Content ) will be false as to the Plot , and therefore must be reduced to a level , and so taken off the Scale and protracted . For the doing of this there are several Instruments very proper , especially Mr. Rathbourne's Quadrant upon the head of his Peractor ( though it were better to have a Semicircle than a Quadrant so placed ) and divers others . But supposing my Country friend to have no other but such as I have already described ; I shall shew him a plain easie way much used by practical Surveyors , especially in Ireland , as some of themselves have told me , being the very same that he may meet with in Mr. Leybourn's Book , Intituled , The Compleat Surveyor ; I mean the second way by him discovered . Fig. 36 Suppose ABC to be part of an Hill falling within my plot , my Boundary-line going crookedly from A to B , following the Surface of the Ground . To find the Horizontal Line ( equal to AC ; ) I cause one to stand at the point A ( the foot of the Hill ) and to hold up the end of the Chain to a convenient height , and gently ascending the Hill , I draw it level and make a mark where it toucheth the Hill , observing the number of Links betwixt mine assistant's hand and that place , where he must take his second standing , and hold it up as before , and so I draw it out level again till it touch the Place , where he must take his third standing , noting the Links as before , and so proceed , till at last from his fifth standing I draw the Chain level to the highest point within my plot , viz. the point B. And now as the pricked Lines of this Figure put together , and evidently equal to the Line AC : So are the Links noted down at every Station , when summed up , equal to the Horizontal Line of that part of the Hill. In the very same manner , only inverting the Order , you may find the Horizontal Line going down-hill , where that is most convenient : And if there be both Ascents and Descents in one Line betwixt two Angles , the Horizontal Lines of both must be found and joyned together in Protraction . All this concerning Declivities of rising or falling Ground , is to be understood when they are considerable , and a very exact plot required : for small ones , especially when much exactness is not expected , are not regardable . CHAP. XVIII . Concerning Plotting a piece of Ground by the Degrees upon the Frame of the Plain-Table several ways , and Protracting the same . HItherto I have shewed the use of the Plain-Table as such , and I think my Directions have been near as plain as the Instrument it self : At which some quarrel for its over-plainness , exposing the Art to proud ignorant people , who judging the rest of the Surveyors work to be as easie as looking through sights at a Mark , and drawing lines by a Rule , are apt to undertake to use it , or slight the skill of such as do . Others say , ( and that truly ) that for vast things , such as Forests , Chases , &c. the Circumferentor is more proper : And every one must grant , that in wet Weather , either that a Peractor , a Theodolite , or Semicircle , must needs be better than the Plain-Table covered with the paper which cannot endure wet . Hence it is that some Artists have to good purpose shewed how the Box screwed to the Index , and that made to turn on the head of the three-legged Staff become a Circumferentor . And if these thus fixed be turned about upon the Center of the Table ; they will ( say some with good reason , Mr. Leybourn for one ) perform the work of the Peractor , much better than the Peractor it self . Others shew , ( as I shall briefly ) that taking the Instrument as it is without the charge of further fitting it , or trouble of removing the Box , the Index turned upon the Center will by help of the Degrees on the Frame , perform the work of the Theodolite , to which the Semicircle is near of kin . And though I might easily answer all these Objections , by saying the first is frivolous ; such foolish Arrogance being easily contemned or checkt ( if worth the while ) by putting such conceited Fools upon the harder part of the work . The second impertinent to our purpose , who design not to plot such vast parcels of Land : And the third concerning only an extraordinary case , and that well provided for otherways , for sure no man that hath not a Body of the same Metal with his Instruments , will ordinarily measure Land in continual Rain , ( a sudden shower may be fenced against by a cover : ) and if any be so eager upon his work , I have shewed ●how it may be done in the former Chapters of this Book , without planting any Instrument at all , by Chain , Scale , and Compasses alone : Yet I shall shew how the plot of a Field may be ta●en by the Degrees on the Frame not every way that I could imagine , nor that I could transcribe , ( for that would be tedious ) but two ways only , whereof the one is proper for an ordinary Close , where all the Angles may be seen from 〈◊〉 Station within it , the other fitting any par●●l of Land though much larger , whatever be the Figure of it . For the former take this Example Fig. 37 Let ABCDE represent the Figure of Field to be plotted by the Plain-Table in rain Weather : I put on the Frame without a Pap●● the graduated side upwards , and plant it i● some convenient place , whence I can see a●● the Angles , as at O ; then placing the Index upon th● Table , so as the fiducial edge do 〈◊〉 the same time go through the Center upon the Table , and the Lines upon the Frame of th● Table cutting it perpendicularly at 360 , ( whe●● the Degrees begin and end ) and 180 ( the 〈◊〉 act half ) I turn about the Table upon the Sta●●●head , till through the Sights ( the Side marke● with 180 being next mine eye ) I see th●● Angle A , and then screw it fast , observin● where my Needle cutteth , and by back-sigh● causing a mark to be set up in the Line CD 〈◊〉 the point F , that the Instrument may be ke●● firm from moving ( or be rectified if it be moved ) during the Work. And now the Li●● AOF passing upon the Land from the Ang●● A , directly under the Sights of the Instrume●● to the Mark at F , is ( as it were ) the pri●● Diameter whence the Degrees of the Angles 〈◊〉 to be numbred , and accordingly I mark th● Angle A in my Table hereafter to be exempl●●fied with 360 Degrees . But to proceed , turning my Index with the fiducial edge upon 〈◊〉 Center , till I see the Thread cutting the Ma●●● at B , the said edge cuts upon the Frame at 〈◊〉 Deg. 15. Min. which I note down for that Angle : The like work I do , turning the Sights to CD and D , ( but not to F , for there is no Angle , but only a Mark in the Boundary ) and I find mine Index to cut for every Angle as I have marked them within the pricked Circle of the last Figure , viz. 157 Deg. 35 Min. for C , 225 Deg. 20 Min. for D , and 278 Deg. and 50 Min. for E. Then I measure ( or cause to be strictly measured by others ) the Distances betwixt the Place where the Instrument stands , and every Angle , and find them to be as I have set them upon the pricked lines in the little Circle , viz. A4 Chains ●0 Links , B4 Chains 3 Links , C3 Ch. 84 Li. D5 Ch. 35 Li. E5 Cha. 6 L. And now my Table both for Lines and Angles is thus perfected , and the Work is ready for Protraction within Doors .   D. M. C. L A 360 00 4 20 B 76 15 4 03 C 157 35 3 84 D 225 20 5 35 E 278 50 6 06 Your judgment will easily inform you , that in such weather we shall hardly stand to make our Table neat and formal , but any thing ( how rude soever ) that we can understand , doth the feat . A Welsh Slate with a sharp Style , ( or for want thereof , a Black-lead Pen and a smooth end of an hard Board like a Trencher ) is more convenient at such a season than Pen , Ink and Paper . But of all I would commend for expedition a Red-lead Pen , whereby you may mark out every Angle neatly with one touch upon the Table it self , just where it toucheth the Frame , by help of the fiducial edge , and close by it the length of the Line from the Center to that Angle : All which may be easily cleared off by a wet Sponge or Cloath so soon as you have protracted . Or if through the sponginess of the wood , the head of the Table ( which we use to cover with paper ) were made a little reddish , what great harm were that ? We are forced to do it more real wrong by the points of the Compasses in the ordinary way . Now to protract our Observations : I draw upon a paper the Line AF at adventures , so it be long enough , and stick a Pin in it at pleasure for the Center O , upon which I place the Center of the Protractor , so as the straight side ( or Diameter ) of the Protractor may just lie upon the Line AF , the Limb or Arched-side being upwards towards B , by help whereof I make a prick or point on the paper at 76 Deg. 15 Min. for B , and at 157 Deg. 35 Min. for C , according to the numbers nearest to the Limb. Then turning the Protractor about on the Pin with the Arch or Limb down towards D and E , till the Diameter lie again just upon the Line AF , I number downwards from the right hand towards the left , by that rank of Figures that are nearer to the Center , beginning 190 , 200 , &c. and over against the places where 225 Deg. 20 Min. and 278 Deg. 50 Min. fall , I prick the Paper at the side of the Limb , and through those four points I draw so many several Lines , ( having laid aside the Protractor ) upon which and also upon the Line AO , I mark out by Points the true measure of every Line ( by a Scale ) from the Center , and from those points drawing the Lines AB , BC , CD , DE , and EA , I have the true Plot of the Field . Where note by the way , that we estimate Minutes as well as we can both upon the Frame of the Plain-Table , and the Protractor , accounting half a Degree , 30 Minutes ; a third part , ●0 ; a fourth part , 15 , &c. And though by this means it is impossible to avoid small errours , 't is easie to avoid sensible ones ; and the like may be said when we protract by a Line of Chords , of which I now come to treat . Having proceeded in the Field as before , and made my Table for Lines and Angles , or done that which is equivolent by a Red-lead Pen , I draw the Line AF , and having extended my Compasses to the Radius ( or 60 Degrees ) on a Line of Chords , I set one Foot towards the middle of the Line AF , and with the other I describe a Circle like that in this Figure of a five-angled Field , but much larger , according to the length of the Radius : Then extending the Compasses from the beginning of the Line to 76 Deg. 15 Min. I set one foot in the Intersection of the Circle by the Line A , and with the other foot make a mark in the Circumference of the Circle upwards towards the right-hand , and through it draw the dry Line BO . In the next place I substract the angle 76. 15 from 157. 35 , where the Index cut for the Angle C , and there resteth 81 Deg. 20 Min. which I take off the Line as before , and set it upon the Circumference from the Intersection by BO , towards the end of the Diameter marked with F , and through the Point were it falleth , draw the dry line CO. In like manner I subtract 157 Deg. 35 Min. from 225 Deg. 20 Min. and the difference is 67 Deg. 45 Min. which I set from the Intersection by the Line CO downwards past the prime Diameter AF , and through the point where it falleth , draw the Line DO . Lastly , Having subtracted 225 Deg. 20 Min. from 278 Deg. 50 Min. there resteth 53 Deg. 30 Min. which must be set downward towards the left-hand from the Intersection by DO , and through the point where that falleth , I draw the Line EO . And now when I have set the particular Measures upon every Line , and drawn the Boundary lines , as I must have done if I had used a Protractor , the Plot is finished . But for better assurance that I have done my Work well , I take the measure of the remaining Angle AOE upon its proper Arch , viz. from the Intersection of the Circumference by AF , to the Intersection by EO , and applying it to the Line of Chords , I find it to be 81 Deg. 10 Min as it ought to be , for it should be the Complement of 276 Deg. 50 Min. to 360 , and so it is , And for further satisfaction , I sum up the Degrees and Minutes of all the five Angles , which for plainness sake I have noted in every one of them on the outside of the Circle in the Figure so often referred to , and their sum is 360 , as it ought to be , and as here is evident . 76. 1● 81. 20 67. 4● 53. 3● 81. 1● 360. 00 My Reader may now perhaps expect that I teach him how to take a Plot at two or more Stations , when all the Angles cannot be seen from one : But because this is so easie from the grounds already laid , to any that is Ingenious , and in part rend●red unnecessary by the Method presently following , I shall content my self to give this general hint . When you have from one Station taken in all the Angles you can see from thence , and then are to remove to your second Station , do just as you would do if the Table were covered with a Paper ; only it is at your choice , whether you would guide your self for back-sight by a Line that may be rubbed off , drawn upon the Table it self from the Center to the Degrees on the Frame along the fiducial edge , or by noting only what Degrees it cuts on either side of the Center , the edge passing through it , that by the help thereof and the Needle , the Instrument may be placed in the same Line and Situation as before , for taking in the rest of the Angles , if it can be , if not , another Station must be taken after the same manner . But now to my second Method ; Fig. 38 Let ABCDE be the Figure of a Field to be plotted , the Weather being bad ▪ I send mine Assistants to find the length of every side , measuring it about , cum sole , beginning at A , who return me such an account of every side , in Chains , and Links , as I have noted them upon the Figure and in the Table following , viz. AB 3 Chains , 73 Links , BC 4 Chains , 91 Links , &c. In the mean season , I make haste to find the Angles , and without curiosity plant the Instrument at B , and laying the Index on the Center , I look at C , and find the Index cutting 10 Deg. 15 Min. and looking at A , it cuts 126 Deg. 45 Min. out of which if I subtract 10 Deg. 15. Min. there resteth 116 Deg 30 Min. for the Angle A : but because I like not my Quarters so well as to subtract there , I set them down thus ; B A 126. 45 C 10. 15. the meaning whereof is , that B notes the Angle , and CA the Lines meeting there , cutting such Degrees on the Frame , and the reason why I set A above , is for more ready subtracting afterwards : then removing to the Angle C , and thence looking at B and D , I find the Index to cut as here expressed , C B 153. 10 C 15. 40. In like manner I find at D thus , D C 96. 05 E 28. 50 At E thus , E D 141. 20 A 11. 45. And lastly , at A I find them thus , A E 98. 30 B. 9. 20 An. D. M Sides . Ch. L. A 89 10 A B 3 73 B 116 30 B C 4 91 C 137 30 C D 1 88 D 67 15 D E 6 64 E 129 35 E A 2 29 This being done , I hast under Covert , and by Subtraction find 116 Deg. 30 M. for the Angle B. 137 Deg. 30 M. for C. 67 Deg. 15 Min. for D. 129 Deg. 35 Min. for E. and 89. Deg. 10 Min. for A , as you find them on the Figure , and in this Table together with the length of the Lines . Note , that there is a way to find the Angles without Subtraction , if at every Station you lay the fiducial edge over the Center , and the Divisions 180 and 360 , turning about the head of the Instrument upon the Staff , till through the Sights you see one of the Neighbouring Angles , for the Index turned upon the Center to the other Angle , will give you the quantity of the Angle you are at , but this exact planting at every Angle is more tedious than the other , and therefore not so fit for wet weather . But now to protract this Plot : First , by my Scale , Rule , and Compass , I draw the Line AB in length 3 Chains , 73 Links , ending at the pont B : then laying the Center of my Protractor upon the Line AB , so as the Center of it be upon the Point A , and that end of the Diameter from which the Numbers are reckoned on the Arch or Limb towards B , I make a point for the Angle A at 89 Deg. 10 Min. by the guidance whereof and the point A , I draw the Line AE , which according to my Scale , must be 2 Chains , 29 Links . In like manner placing the Diameter upon AE , just as it was upon AB , and the Center upon the point E , I mark out by the Limb ( for the Angle E ) 129 Deg. 20 Min. by which I draw the Line ED , 6 Chains , 64 Links . In the next place , I bring the Center of the Protractor to the point D its Diameter , lying on the Line ED , and its Limb towards A , by which I prick out 67 Deg. 15 Min. for the Angle at D and draw the Line 1 Chain , 88 Links . Lastly , the Center being at C , and the Diameter upon the Line DC , in such manner as before at other Angles , I prick out by the Limb or Arch 137 Deg. 30 Min. and draw the Line CB , for at B my Plot should close , and if rightly done , the Angle at B will be 116 Deg. 30 Min. and the side BC 4 Chains 91 Links , which by measure I find so to be . But if I plot by a Line of Chords , I am not bound to this Order , but may go from A to B , and so round that way if I please , which I could not so well do with a Protractor , without reckoning my Numbers backward , yet it must be granted that a Line of Chords neither doth the work so quickly nor conveniently , for this is the way . When I have drawn the Line AB of a right length , I set the Compasses to the Radius , and placing one Foot of the Compasses in the point B , and with the other describe an Arch of a competent length , beginning at that side of the Line AB , that is designed to be the inward-side , and upon this Arch , 116 Deg. 30 Min. must be set , but because my Line of Chords gives me only 90 , I set them first on from the Line AB , and then take off the remainder 26 Deg. 30 Min. I joyn them to the 90 upon the Arch , making a Point , through which the Line B must be drawn of a due length . In the like manner must I do at CE , but the Angles at A and D need no such piecing , being capable of being measured out by a Line of Chords at once . Nor do your Angles only give you trouble in this kind of work , but oft-times your Lines will be found too short to receive the touch of an Arch upon the Radius , especially if the Line of Chords be large and your Scale little , and so it may often fall out when you use the Protractor upon such short lines as AE and CD of this last Figure : In which case a Rule must be applied to them , and they must be extended to a due length that the Arches may meet them without the Figure . And if those Extensions of lines and describing of Arches spoil the beauty of your Plot , the matter is not weighty , 't is so easie to be retrieved , for if you lay it on a clean paper and prick through every Angle , lines drawn between those points will give you the Plot neat and perfect . One thing more before I close this long Chapter ; the Artist sometimes loseth his labour of Protraction through some error in the Field , so as his Plot will not close : 't is therefore good to know before we begin that work , whether it will or no ; for which purpose if we take a Number less by two than the number of Angles in the Plot , and thereby-multiply 180 , that Product being found to be equal to all the Angles , the Plot will close , and so it appears by our Plot in this present work : the Multiplier being 3 , because the Angles are 5 , and the Multiplier must be two less than the number of Angles . 180 3 540 Deg. Min. 89. 10 116. 30 137. 30 67. 15 129. 35 540   This kind of tryal is grounded upon two principles of Euclid and Ramus mentioned in the first and third Chapters of this Book , shewing that in all plain Triangles , all the Angles taken together , are equal to two right Angles , and that the sides ( consequently the Angle also ) of every triangulate Figure , are more by two than the Triangles of which it consisteth : But I think it not proper to be large in such things whereof my young Artist is like to make but little use ; for when all is done , I confess with Mr. Wing in his Art of Surveying , Lib. 6. Chap. 15 , that this way of Plotting by the Degrees , is far more troublesome , tedious , and liable to errour , than the other ways upon a sheet of paper , and therefore not ordinarily to be used , but when necessity compelleth us . CHAP. XIX . Concerning taking inaccessible Distances by the Plain-Table , and accessible Altitudes by the Protractor . Fig. 39 THe Substance of what is to be said for the first of these , is gathered from the Instructions given for Plotting a Field , by measuring only the Stationary-distance ; but to make the case more plain to an ordinary capacity : Suppose the Line AC to be the unknown breadth of a River , over which a Bridge of Boats is to be laid , and the General ( that he may inform himself what store of Boats and Planks is necessary to be brought down ) commands me to tell him the true distance from A where he is at present , to C a little Boat-house on the other side the water . To satisfie his demand , I plant my Table covered with a paper at A , causing one to set me up a mark at B at a good distance from me , along the Bank of the River ( the further the better , if distance do not hinder sight : ) Then having chosen a point to represent A , and laid the fiducial edge upon it , I direct my sights towards C and B , and strike lines towards them . Which done , I set up a mark at A , and from thence measure to B , ( 6 Chains , 32 Links ) and so plant my Instrument at B , laying the fiducial edge to the line AB , and turning about the head of the Instrument upon the Staff , till through the sights I spy the Mark at A , and then screw it fast . In the last place , I take 6 Chains 32 Links off my Scale , and set it on the Line AB , from A to B , and laying the fiducial edge to the point B , from thence direct the sights to C , and draw the Line BC , meeting or cutting the Line AC in C : So shall the space AC measured on the Scale ( viz. 8 Chains , 29 Links be the distance desired : and because the Chain is 22 Yards long , if I multiply 8. 29 by 22 , the Product is 182 Yards and 38 / 100 of a Yard , which by reduction is some little more than 13 Inches and 3 / 5 of an Inch. Now to take the height of a Tree , Tower , or Steeple by a Protractor , without any Arithmetical operation , hang a Plumet with a fine Silk Thread at the Center of it , and hold it stedfastly with that end to your Eye where the Numbers begin , then look streight along the Diameter , as if you were to shoot in a Cross-bow without a Sight ( still removing backward and forward as there is occasion ) till you see the top of the Tree , Tower , or Steeple , and the Thread at the same time fall upon 45 degrees ; 10 shall the distance from your Eye to the Tree , Tower , or Steeple , measured in an horizontal or level Line , together with the height of your Eye above the bottom of it , be equal to the height thereof . If either for convenience of sight , or any other reason , you think good to set the other end of the Diameter to your eye , then the Thread for the tryal aforesaid , must fall upon 135 Deg ▪ instead of 45. Other ways of doing this work by this Instrument ( or a Quadrant ) with the help of Trigonometry , and by other Instruments , I forbear at present ( till I write a second Part ) considering whose benefit is here intended . CHAP. XX. Of casting up the Content of Land by a Table . TO make up the number of my Chapters to an even score , I shall add one at the desire of my worthy Friend Mr. S. L. before mentioned ( to whose Experience and Communicative●ess I acknowledge my self indebted for the notion of measuring crooked Lands or Doles at the middle and both ends , marking every where how the Divisions fall , as is mentioned in the beginning of Chapter 17. ) concerning the use of a Table borrowed out of the 46th Chapter of Mr. Leybourns Compleat Surveyor , second Edition , Page 271 , which with the use take as followeth . Links R. P. 1     00000 4 00 90000 3 24 80000 3 08 70000 2 32 60000 2 16 50000 2 00 40000 1 24 30000 1 08 20000 0 32 10000 0 16 8750 0 14 8125 0 13 7500 0 12 6875 0 11 6250 0 10 5625 0 09 5000 0 08 4375 0 07 3750 0 06 3125 0 05 2500 0 04 1875 0 03 1250 0 02 625 0 01 This Table consists of three Columns , the first containing Links , the second Roods ( or Quarters of Acres ) the third Perches : and the use of it is thus : 7. 25 5. 50 36250 3625 3625 3. 98750 Suppose a Field to be 7 Chains and 25 Links long , and 5 Chains 50 Links broad , these by multiplication make 398750 ( as here is evident ) whereof five Figures being cut off towards the right-hand , the Figure 3 signifies Acres , and the rest , viz. 98750 denote parts , and to reduce them into Roods and A R P 3 3 24     14 3 3 38 Perches , I first subtract from 98750 the greatest number of Links in my Table that can be subtracted from it , viz. 90000 ( and put down for it 3 Roods , 24 Perches which I find over against it in the annexed Columns ) and the remain being 8750 , I look in the Table , and find over against it 14 Perches , which by addition makes 3 Roods 38 Perches : So is the whole Content of the Field 3 Acres , 3 Roods , 38 Perches . But not here , that if the Remainder after the first Subtraction cannot be found in the Table , you may take the nearest to it , so the error will be but part of a Perch . As for Example : 7. 35 , being the half Perpendicular , and 9. 23 , the Base , give for their Product 6.78405 . The 6 signifies Acres and from the rest 70000 being subducted , ( to which 2 Rood , 32 Perches answer ) there resteth 8405 , which because I cannot find in my Table , I take the nearest , which is 8125 , to which 13 Perches answer : So the whole Content of that Triangular Close is 6 Acres , 3 Roods and 5 square Perches , and a little better , A R P 6 2 32     13 6 3 05 But Manum de Tabula — I am at present taking leave of my Country-Man , and supposing he brought with him any competent stock of Natural Capacity , and so much Arithm●tick as enabled him to add , Subtract , Multiply , and Divide ; I dare make him ●udge , after he hath as faithfully laboured to understand me , as I to be understood ; whether I have not performed what I undertook in my Title or elsewhere . FINIS . ADVERTISEMENT . The Author hereof useth in Winter and Spring Seasons to board young Gentlemen and others at his Habitation near Dun●am in Cheshire , and to instruct them in these parts of the Mathematicks , viz. Arithmetick . Vulgar concerning Whole Numbers . Fractions Balancing Accounts . Artificial by Decimals Logarithms Instruments Symbols or Algebra . Geometry and therein The Doctrine of plain Triangles . Measuring Superficials , and particularly Land , by all usual Instruments . Measuring of Solid Bodies . Gauging of Casks . The Doctrine and use of Globes and Spherical Triangle● . The Principles of Astronomy and Navigation . The Art of Dialling , by Logarithms , Scales , and Geometrical Projection , &c. Vivat Rex . Floreat Regnum . AN APPENDIX CONTAINIMG XII . PROBLEMS TOUCHING Compound Interest & Annuities . BEING Part of a Letter sent by the Author to his worthy Friend Mr. John Collins Fellow of the Royal Society . Together with A quick and easie Method to contract the Works of Fellowship and Alligation alternate . LONDON Printed in the Year . 1692. XII . PROBLEMS Touching Compound Interest and Annuities , expressed in Symbols , to be resolved by Logarithms ; first presented in Twelve short Lines , to the Right Honourable the Lord Delamer ; afterwards Explained by the Inventor Adam Martindale , and by his Consent presented to the Royal Society by Mr. Collins , and now applied to pertinent Questions in a Practical way , to make them more plain and useful . THese Problems are distinguished into three Ranks , whose Symbols are thus to be understood . p Principal , r Rate , viz. 1 l. with its Rate , t Time , common to all the three Ranks . a Amount or aggregate , proper to the first Rank . s Sum of Principal and Arrearages , proper to the second Rank . d Difference of Principal and Worth , proper to the third Rank . Their Capitals stand for the Logarithms 〈◊〉 the Numbers signified by the small Symbolic● Letters before mentioned . D. signifies Data , Q. Quaesita , Prob. Problem Res . Resolution . The first Rank , concerning Compound Interest for a single Sum of Money . 1. Prob. D p , r , t. Q. a ? Res . Rt + P = ● 2. Prob. D. a , r , t. Q. p ? Res . A − Rt = ● 3. Prob. D. p , a , t. Q. r ? Res . A − P / t = ● 4. Prob. D. p , a , r. Q. t ? Res . A − P / R = ● Examples relating to the Four Problems in order . Quest . 1. What will 15 l. 10 s. amount to i● 12 yeaps at 6 per Cent. Compound Interest ? Answ . 31 l. 3 s. 9 d. 1 q. as appears by the work . R viz. of 1.06 = 0.25306 t = 12 050612 025306 Rt = 0.303672 P 15.5 = 1.190332 A 31.188 = 1.494004 Quest . 2. What is 31 l. 3 s. 9 d. 1 q. due ●●elve years hence worth in ready money , abating ●●er 6 per Cent. Compound Interest . Answ . 15 l. 10 s. as here appears . A 31.182 = 1.494004 R = 0.025306 t 12 050612 025306 Rt = 0.303672 A − Rt = P = 15.5 = 1.190322 Quest . 3. At what Rate of Compound Interest 〈◊〉 15 l. 10 s. amount to 31 l. 3 s. 9 d. 1 q. in ●●elve years ? Answ . At 6 per Cent. as here . A = 1.494004 P = 1.190382 0.303672 12 ) 0.303672 ( 0.025306 = 1 . 0● 0 Quest . 4. In wha● time will 15 l. 10 s. amount to 31 l. 3 s. 9 d. 1 q. at 6 per Cent. Compound Interest . Answ . In 12 years . A = 1.494004 P = 1.190332 R. 025306 ) 0.303672 ( 12 = t 0 The second Rank , touching Annuities in arrear , grounded upon these two Axioms . 1. The Annuity and Rate of Interest bein● given , the Principal correspondent to that Annuity is in effect given also , being easily found by the Rule of Three , thus ; As the Interest of any Principal ( ex gr . of 1● 10 , 100 , 1000 , &c. ) is to that Principal : S● the Annuity of Pension , to its Principal . 2. The Sum of the Principal and Arrearage● of all the Payments being found , the Arrearage● alone may be obtained by subtracting the Principal from that Sum. 1 Prob. D. p , r , t : Q. s ? Res . Rt + P = S 2 Prob. D. s , r , t. Q. p ? Res . S − Rt = P 3 Prob. D. p , s , t. Q. r ? Res . S − P / t = R 4 Prob. D. p , s , r. Q. t ? Res . S − P / R = t Examples suted to the first and last of these Four Problems . Quest . 1. What will 33 l. per annum amount to in 14 years at 6 per Cent. Compound Interest ? Answ . 693 l. 10 s. as here . 6 . 100 ∷ 33 33 300 300 6 ) 3300 ( .550 30 30 30 00 R = 0.025306 t = 14 0.101224 0.25306 0.354284 Rt = 0.354284 P = 2.740363 S = 3.094647 = 12435 550 6935 Quest . 2 In what time will 33 l. per annu●● raise a stock of 693 l. 10 s. at 6 per cent . Compound Interest ? Answ . In 14 years , as here is evident . 6 . 100 ∷ 33 . 550 550 + 693.5 = 1243.5 = s● S = 3.094647 P = 2.740263 25306 ) 0.354285 ( 14 25306 101224 101224 0 The third Rank of Problems touching Annuities anticipated or bought for a Sum in hand ( o● that which is equivalent ) at compound Interes● discounted , is bottomed upon the former of th●● two Axioms above mentioned , and this that followeth : Axiom . If the difference of the Principal and Worth be once found , the Worth is easily obtained by subtracting that difference from the Principal , which is ever greater , being the Worth of the Annuity at that Rate for ever . 1 Prob. D. p , r , t. Q. d ? Res . P − Rt = D 2 Prob. D. d , r , t. Q. p ? Res . D + Rt = P 3 Prob. D. p , d , t. Q. r ? Res . P − D / t = R 4 Prob. D. p , d , r. Q. t ? Res . P − / R = t Examples fitted to the first and fourth of these Problems . Quest . 1. What is 17 l. 10 s. per annum to continue for 11 years , worth in present Money , at 8 per cent . Compound Interest allowed to the Pur●●aser ? Answ . 124 l. 18 s. 8 d. as here is shewed . 8 . 100 ∷ 17.5 ? 17.5 500 700 100 8 ) 1750.0 ( 218.75 0 〈◊〉 R = 0.033424 11 033424 033424 Rt = 0.367664 P = 2. 339948 Rt = 0.367664 — 21875 D = 1. 972284 = 93817 124933 Note , that if in stead of subducting Rt from P , I had turned Rt into the Arithmetical Complement 9632335 , and added that to the P 2. 339948 , it would have done the same thing in a more convenient manner , ( save that it is not so suited to the letter of the Problems ) as here is evident , 10 being rejected ( as in this case it must ever be ) from the Index . 0.033424 11 033424 033424 0.367664 9.632335 2.339948 1.972283 Quest . 2. In what time will 17l . 10s . pay off a Debt of 124 l. 18 s. 8 d. allowing the Creditor after 8 per cent . Compound Interest ? Answ . In 11 years , thus manifest : 8 . 100 ∷ 17.5 . 218.75 = p 124.933 93 817 = d P = 2.339948 D = 1.972284 R = 33424 ) 0.367664 ( 11 0 Any one that understands the very Elements of Algebra , may contract these Twelve Problems into Three ; for the First of any Rank will by Reduction , Application , and Transposition , produce the rest . Then the work of the second Rank may be performed by the first , if one but understand , that instead of the Pension , he must take the principal correspondent to it , and work with it till he have found the Amount , from which the principal must be subducted when the Arrearages of an Annuity are sought , and so proportionably in the rest of the Problems . My Noble Lord Delamer , only noted down for his own use Rt + P = A , and P − Rt = D : by which and a small Canon of Logarithms he will quickly answer any Question of this kind . But for the help of young Mathematicians , I have set them forth thus explicitely . The only inconvenience of any importance that I can yet discover in this Method , is , that in both Ranks concerning Annuities , the second and third Problems are rather for demonstration of the other , and to compleat the Rounds than for any other great use , proceeding upon such . Data as are seldom given for finding the pensions and rate , ( or if they were , the Work would be rendred useless ) yet we are not left without sufficient help to find out pensions or annuities by plain and proper Data . For in reference to the second Rank , if the Arrearages ( or Stock to be raised ) with the rate of Interest and time , be given for finding out such an Annuity , as at such a rate , and in so much time , will raise a given Stock by its Arrearages ; 't is easily found thus : Suppose an Annuity at pleasure , and by the first Problem of the second Rank , find out its Arrearages : Then say by the Rule of Three . As the Arrearages found , to the supposed Annuity : So the Arrearages given , to the Annuity required . For clearing whereof , let this be the Example . Quest . What Annuity will at 6 per cent . comp . Interest , raise a Stock of 693 l. 10 s. in fourteen years . Answ . 33 l. per annum , as here is manifest . Suppose 30 l. then the Work goeth on thus : 6 . 100 ∷ 30 30 6 ) 3000 ( 500 30 000. R. 0.025306 14 0.101224 0.25306 Rt = 0.354284 2. 698970 S = 3. 053254 = 1130●46 500● 630●46 630.46.30 ∷ 693 . 5● 30 630.46 ) 20805.0 ( 32.999 44046 Here we account 32. 999 for 33 , being so much within an inconsiderable trifle . Likewise in relation to the third Rank , if the Worth ( or Price ) Rate and Time be given , and the Pension to be purchased be required ; find the worth of any supposed Pension by the first Problem of the third Rank , then the proportion stands thus : As the Worth found , to the supposed Annuity : So the Worth ( or price ) given , to the Pension required ; as in the Example follow 〈◊〉 : Quest . What Annuity to continue eleven years will be purchased for 124 l. 18 s. 8 d. at 8 per cent . Comp. Interest . Answ . 17l . 10s . as here . Suppose 12 l. Then 8 . 100 ∷ 12 ? 12 200 100 8 ) 1200 ( 150 8 40 40 00 R = 0.033424 11 0.033424 0.33424 Rt = 0.367664 P = 2. 176091 Rt = 0.367664 — 150 1. 808427 = 64332 85668 85.668.12 ∷ 124.933 ? 12 249866 124933 85.668 ) 1499.196 ( 17. 5 60 If the Interval betwixt every two payments 〈◊〉 less than a year , consider what part it is ; ●hether ½ ¼ 1 / 12 1 / 13 1 / 32 1 / 365 , &c. And by the ●enominator of the Fraction signifying that part , ●ivide the Logarithm of 1 l. and its Rate of In●rest , ( usually called the Logarithm of the Rate ) ●en the absolute Number answering to that ●●otient , being made less by an Unite , will be 〈◊〉 new Divisor , whereby dividing the given Pen●on so payable , half , yearly , quarterly , &c. the ●orrespondent Principal will appear in the Quo●ent , with which you may proceed , as if the payments were so many yearly ones ; as this ●ext Example shews . Quest . 1. What is a quarterly Rent of 3 l. 15 s. 〈◊〉 continue Thirteen Years and one Quarter , worth in ●●dy Money at 6 per cent . Compound Interest ? Answ . 137 l. 10 s : 1 d : as appears here . R = 4 ) 0.025306 0.006326 = 1●01467 - 1● ●01467 .01467 ) 3.75000 ( 235.623 1059 1 / 4 R = 0.006326 Number of Quarters 53 0.018978 0.31630 1 / 4 Rt = 0.335278 Ar. C. 9●664721 P = 2.407600 — 255623 2.072321 = 118-119 137504 Whereas I multiplied the Fourth part of the Log. of the Rate , viz. 0.006326 by 53 ( the Number of Quarters ) I might as well have multiplied the whole Log. of the Rate , viz. 0.025306 ▪ by 13.25 , i. e. 13 years and 1 quarter , and it would have been full as exact , or rather more , ( two places being cut off from the Logarithm so multiplied towards the right hand : ) I say rather more exact , because there remained 2 when I quartered the Log. of the Rate . Quest . 2. What is 5 s. per Week to continue 7 ●ears and three quarters worth in ready Money at 8 per cent . Compound Interest ? Answ . 75 l. 17 s. 8 d. 2 q. 52 ) 0.033424 ( 0.000642 = .00148 40 .00148 ) .25000 ( 168.918 136 R = 0.033424 t = 7.75 0 167120 2 33968 2 33268 Rt = 0.259036.00 Ar. ●9 . 740960 P = 2.227675 — 168918 1.968638 = 913033 75885 Here I multiplied ( not by the Number of Weeks ) but by the Number of Years and Quarters , cutting off two Cyphers from the Product . And I also took the Arithmetical Complement of 0.259036 , viz. 9.740963 , and added it to the Log. of the Principal in stead of substracting the Log. 0.259036 from it . Lastly , As to the Rate of Interest , according to the sense of Dr. Newton , Mr. Dary , and others , I see not why this Method may not b● made use of as well as any other . This universal Rule being first understood : In Questions belonging to the Second Rank , the greater Rate 〈◊〉 Interest , the greater Arrearages ; and the less Rate the less Arrearages . But in Questions belonging to the third Rank , the greater the Rate of Interest the less the present Worth ( or price ) and the less the Rate of Interest , the greater the Worth , & vice versa . I shall shew this in answering the Question following , concerning a yearly payment of 1 l. because that by bare Multiplication may easily be applied to any other . A Field worth 1 l. per annum clearly , is offered to be let 11 years for 15 l. to be paid at the end of those years : What is the Rate of Interest demanded for the several payments ? Answ . First , I suppose 6 per cent . and trying it by the first Problem of the second Rank , it find the Arrearages to be but 19 l. 19 s. 5 d. which is 6 d. 3 q. too little . Then trying what the Arrearages will amount to at 6. 2 ( or 6 l. 4 s. ) per cent . I find 15. 13 , or 15 l. 2 s. 7 d. 1 q. Perceiving then I have over-shot my Mark , I make my third tryal at 6.1 ( or 6 l. 2 s. ) per cent . and the Result is 15.05 , or 15 l. 1 s. which is pretty near , being but 〈◊〉 above the truth . But that I may yet bring it nearer , I first subtract the Result at 6 per cent . which is 14.971 from the Result of the Arrearages at 6.1 . viz. 15.05 ( because these are the Results that come nearest to the truth ) and their difference is .079 . 2. I subtract the Rate 6 from the Rate 6.1 , and there resteth . 1. 3. I substract the nearest Result to 15 , viz. 14. 971 , from 15 the given Arrearages , and their difference is .029 . Then working by the Rule of Three , I find ( as followeth ) .0367 .079 .1 ∷ 029 1 .079 ) .0290 ( .0367 7 Which .0367 being added to 6 , gives 6.0367 , or 6 l. 0 s , 8 d. 3 q. per cent . Compound Interest . And that this Rate exactly fitteth , is plain by the following work . 6.0367.100 ∷ 1 1 6.0367 ) 100.0000 ( 16.565 20245 R. 0.025456 11 0.025456 0.25456 0.280016 1.219191 1.499207 = 31 / 565 16 / 565 15 / 000 But perhaps it will be said , that though in this and all other matters foregoing , I agree well enough with the learned Authors before mentioned , and others that have writ of these Subjects ; yet in my Country-Almanack for the Year 1677 , the Fourth Problem agrees neither with them , nor what I have here written , but apparently clasheth with them . I confess this Charge had been very just , had that Problem been designed for the same end , and produced a different effect : ( for in such a case to differ is to contradict . ) But forasmuch as in my very entrance upon it , I began thus . There is a Problem in some Learned mens Works , seemingly of the same importance with this , but indeed much different both in the Design and Effect : And after in answering Objections show wherein the Design differs ; I cannot but admire the Undertakings of that able Artist , who was so much at leisure as to prove with a great deal of Pomp , that mine will not attain the end for which I declared it was never designed , as clearly as words can utter it . A Sword may be a very good one , yet a very bad Instrument to fell Trees with : But I hear he is dead , and I shall rather lament the loss of him , and divers other famous Mathematicians ( which Death hath of late deprived us of ) than unnecessarily to expose the impertinence of his Paper to the Publick View ; or so much as name him to the prejudice of his Memory . But from henceforth I expect from all ingenious persons , that they neither take for granted what I professedly deny and disprove ; nor urge mine Objections against my self , without taking notice of mine Answers . For understanding of what follows , it were very convenient to have inserted herein the whole Discourse in the Country-Almanack about this business ; but because this little Paper cannot afford it room , I shall only point briefly at the Design and Management ; and answering all the Objections I ever met with of any seeming Importance , make things as clear as I can in so narrow a compass . The Design was to discuss in plain language ( suitable to Country-mens understanding ) this practical Question , wherein they are oft concerned ; viz. Whether it be more advantage for the Lender to receive for 100 l. in hand , Compound Interest at 8 per cent . viz. 71 l. 7 s. 7 d. 3 q. at the end of seven years above the stock of 100 l. supposing it can be legally assured ( as in Ireland it may , and in England by equivalence in Goods or Lands , without mentioning Interest ) Or to enjoy a Farm for seven years in consideration of 100 l. that will clear him just 20 l. per annum ? My Answer was , that the former was the better Bargain ( and consequently the higher Rate of Interest for the whole Stock during the whole term ) for which I gave in these grounds under the feigned persons of A and B. 1. A is to receive ( as is said ) 171 l. 7 s. 7 d. 3 q. and if B , who takes the Farm , receive not equivalent , his rate of Interests is lower . 2. B receives only 140 l. by 20 l. per annum ; only he hath the advantage to improve the several payments from the time they grow due , to the end of the term . 3. These must be computed at some certain rate of Compound Interest : For to Compute them at no rate of Interests , or at Simple Interest , or one taken up arbitrarily , will not sute the case . 4. The Rate tolerated by the Statute , viz. 6 per cent . under which none will take , and above which none dare expresly bind any to give , and at which any responsible man may be fitted , is to be preferred before any other . 5. At this Rate the 7 payments will amount but to 167 l. 17 s. 6 d. q. Of these 5 Pillars , the 132 , and 5 th , were never attempted to be shaken by any that I know of : What assaults have been made against the other two , or design of the whole Fabrick , I shall briefly consider ; and I find them ( besides those answered when I first published that Problem ) to be these three : 1. Ob. The Question is not at all , how the Receiver improves his Payments , but what Improvements was made in the Debtors hands , and the same Rate carried on ? Ans . We are agreed that the Rate is not to be computed according to what the Usurer actually makes of it . If he gives away , loses , or lends freely all the several Payments : Or if he make new Bargains more oppressive than the Original one ; the true value of the loan of so much Mony for such terms , is one and the same . But I utterly deny that it is to be reckoned , as if the same Rate must be carried on , and that for unanswerable Reasons ( as I suppose them to be ) laid down in my former discourse , or easily colligible from it . 1. The Laws of our Nation prohibit upon severe Penalties , the taking of more than 6 per cent therefore that is the utmost of the legal worth , which he that exceeds , runs ( as I take it ) the hazard of all , and a great fine . 2. If he can evade the penalty , no responsible man needs to give more than 6 per cent . and few are willing to give more than they need . 3. If he get over both these blocks , and make ●ew oppressive Bargains , this is nothing to the Question as I propounded it ; for that was , whether A. or B. received the highest Interest for his Money , by vertue of the above mentioned Bargains , not by vertue of occasional after Bargains . When the Debtor hath paid in a years Pension , he hath done with it ; and if he have ●t not ready , he may take it up at ordinary Interest , and the proportion is broken off for so much . 2. Object . If we regard Laws in the case , what need we any Rules concerning Compound Interest , seeing the Laws of the Land allow only simple Interest : Ans . 1. The Laws are not against Compound Interest , as it may be managed : that is , the Usurer may receive his simple Interest at the years end , and put out that as a new Stock , and so undeterminately from time to time ; or if you will call this simple Interest , it 's the same to the purse . 2. I am told they have ways in London for putting out very small Sums to mean Traders upon sufficient security ; and then it is both legal and practical , though I say not how lawful before God ; especially as I hear some use it . 3. Object . It ought to be a Question only o● Art , without dependance upon Laws and Usages ; it being the nature of Art in these kind 〈◊〉 Questions , not to shew so much what ought 〈◊〉 ●e done , as what is really done . Ans . 1. It is one Question , what is the natur● of Art , and another how far Art is concerned that is , whether nothing else save the Rules o● Art be regardable in the case ; which I deny , because Laws and Usages have a great influenc● upon it , hindring the continuance of the proportion . 2. I consider not the Laws as pinching the U●surers Conscience ( as to what is lawful ) but 〈◊〉 tying up his hands , and so obstructing the proportionable increase of his gain . 3. Whatsoever may b● said of Art ( as Art i● the strict notion of it ) the Artist must not be such a Slave to the Rules of it , as not to allow for unavoidable Obstructions and irregularities I hope I may be allowed to tell my Scholars that learn Navigation , that though the direct cours●● from one Port to another be upon such a Point of the Compass ; yet other Courses must be steered sometimes in regard of Rocks , Shallows● crooked Chanels , Currents , Trade-winds 〈◊〉 convenience of Harbour , and fresh Supplies 〈◊〉 Or to avoid Pirats , Enemies , Forts , and Places where great Customs or Payments will be exacted ; and many such things which the experienced Sea-man is better acquainted with , than I with their Names . And to say to such as learn Merchants Accompts , that though it were more Artificial and Rational , that Rebates of Interest for Money paid before ●t be due , should be computed at Compound Interest ( as certainly it is : ) yet forasmuch as it is usual with Merchants to allow no more than simple Interest ( as appears by printed Accompt-Books ) they must submit to the Laws of the great Tyrant Custom . An Artist hath not the same liberty of supposition in answering Questions ( especially such such as are real and practical ) that he hath in proposing Questions , or receiving them from others , when tryal of skill is only or chiefly intended . I may without absurdity in this latter case demand or find the Amount or present worth of any single Sum of Money , or yearly Pension for any terms propounded at any rate of Interest given , though such as is never likely to come in practice : ( ex . gr . at 2 , 3 , 14 , or per cent . ) But if I be put by the nature of my Work , to compute the Amount or present worth of Payments , no rate of Interest being named , I ought prudently to weigh all Circumstances , and pitch upon that which is possible and rational . And this is not a work of Art , but of discreet Judgment , ( wherein great respect is to be had to penal Laws , Usages , and other obstructions ) after which Art takes it own Province , in computing after such a Rate resolved upon . To make all this plainer ( if possible ) than I made it at the first Publication ; Let us for once suppose a rare case , viz. That B a Mathematician turns Usurer , and for 100 l. ready Money , takes a Farm for 7 years , that he lets to another for 30 l. per annum clearly , knowing before hand he could so let it : and A his familiar Friend thus accosts him ; I wonder to bear you are grown such an Extortioner , as to receive 20 l. per annum , seven years together for 100 l. B answers , Before you find fault with the Mote in mine Eye , take the Beam out of your own : You have bought the Reversion of a piece of Land after seven years for 100 l. for which I will bona fide give you 172 l. at the time of your Entrance upon it ; which I find by the Rules of mine Art to be 12 s. 4 d. 1 q. above Compound Interest at 8 per cent . A replies , Learned men , say such a Bargain as yours will clear 9 l. 3 s. 4 d. per cent . and better . B rejoyns , 'T is true they do so , but then they suppose the same Rate to be continued , ( for none will say , that if the Payments lie by unimproved , or be let out at an under Rate , that in such cases that Rate of Interest can be answered ) and this supposition is really impracticable ; or ( to say the least ) that which a rational man cannot depend upon . I did accidentally meet with this Bargain from the hands of a weak man , which yet I durst not have accepted ( the Laws are so strict ) if it had been a Rent-charge in Money . And will you undertake to find me Fools that shall at every years end , take off the several Payments , yea and all the increase of them as they grow due upon the same terms that I put out the 100 l. upon , and to let me Land for it ? If you think you can , I will make you a great Bargain , ( because I know you to be a punctual responsible man ) you shall receive the first yearly Payment , and at the end of the second year , pay me simple Interest for it at 6 per cent . as the Statute allows , and I will instantly return it you , together with the second 20 l. which we will joyn into one Sum , and you shall take it at the same rate : and thus we will do every year to the end of the term , still adding that years growth to what was before , and so keep up all the Accounts to 6 per cent . and if you can make any more profitable use of it , much good may it do you , and I will heartily thank you to boot , for helping me so readily to place out my small parcels without loss . But if you dare not do this , let ingenuity mollifie your charge , and I shall not quarrel with you about your rate of 9 l. 3 s. 4 d. which perhaps you borrowed ( or some one for you : ) from Dr. Newton , his Trigonometria Britannica , P. 37. for I acknowledge it is that and somewhat more , though not much ; but I say ( which I desire you to observe ) that I receive no compound Interest after that rate , but only simple : and that not for the whole Sum during the whole term ( as you do compound Interest for all your Stock ) but only for the whole 100 l. for the first year : for you know that if at the end of that year 20 l. be subtracted from 109 l. 3 s. 4 d. all the Interest is paid , and so much of the Stock as brings it down to 89 l. 3 s. 4 d. This 89 l. 3 s. 4 d. being computed at the same rate of Interest , 20 l. abated from it , clears the Interest again , and brings down the Stock much lower : and so year by year the Stock is dwindled away , till at last the seventh payment ( if the Rate were absolutely exact , as it is near ) clears off 〈◊〉 Stock and Interest . And this is all I receive , save only the benefit of the severa● yearly payments , which I offer you ( or any other solvent man ) at 6 per cent . as aforesaid 〈◊〉 At which rate ( abating not a Farthing for loss of time , but supposing good places ready for eve●ry parcel as it becomes due ) it will but amount● to 167 l. 17 s. 6 d. q. which being the whole Aggregate of Stock and Interest , if we substract from its Logarithm , the Log. of 100 , and divide their difference by 7 , as we are directed by Mr. Wingan's 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 , Chap 5. Prov. 13. the rate of Interest will appear to be 7 l. 13 s. 7 d. 2 q. I profess freely , Sir , I cannot withstand B's Arguments ; but if any other can , I shall neither envy his happiness , nor despise , ( if fairly and ingeniously offered ) his Animadversions ; in that case you shall command a Sheet from me at any time , in consideration of any thing so objected ; but for such injurious and passionate discourses as are apter to provoke than convince , I confess , 't is such a piece of drudgery to answer them , that ( if I could not ease my self by contempt ) I should think it hard measure that my silence should be interpreted as their Victory . Sir , You will p●rdon this Prolix Appendix , when 't is considered that it is not intended for such as your self , but for such to whom nothing can be too plain , by SIR , Your Friend and Servant , Adam Martindale . A Compendious Method for working out many Conclusions in Arithmetick , wherein the Rule of Three is often repeated ; long since invented , practised and taught by the Author , but never till now published . IN all Books of Arithmetick that I have had the happiness to peruse , the works of both Rules ●f Fellowship and Alligation alternate are performed by the Rule of Three , so oft to be repeat●d , as there ere particular proportionable parts ●o be found ; which proving intolerable tedious , when the Partners in Fellowship , or the parcels 〈◊〉 Alligation are many , caused me to think of ●is following course , every way as plain and useful , and much more compendious , which to ●e short is this : The Rule . Divide that which is usually made the second Number in the Rule of Three , by that which is as ●sually made the first and the common Divisor ; That Quotient multiplied severally by the respe●●ive third Numbers , gives the particular results . ●ut here it will be necessary if you use Decimal Division ( which I judge most convenient ) that you continue your Division till either nothing remain , or you have six places of Decimals in the Quotient , accounting Cyphers ( if there be any ) into the number . And if upon the Addition of your Parcels , the Total amount not just to the Sum expected , but an Unite short in the Integers , & three more Figures of 9 immediately following , take it for exact , for so it is within a matter of nothing . All this will be plain by three Examples whereof the first shall be wrought ( for plainness ) both in the old and in this newer way ; the other two ( for brevity ) this latter way only . 1. Example in Fellowship without time . A B C Put into the Common Stock 15 24 33 72 The Sum whereof is 72 l. and their gain 63 l. what is each Mans Part ? The Resolution in the old Method is thus . A 72.63 ∷ 15 15 315 63 72 ) 945 ( 13.125 Or 13l . 2 s. 6 l. B 72.63 ∷ 24 24 252 126 72 ) 1512 ( 21 l. 0 C 72.63 ∷ 33 33 189 189 72 ) 2079 ( 28.875 0 Or 28l . 17 s. 6 d. 13.125 21. 28.875 63.000 Or thus , 13-02-06 21-00-00 28-17-06 60-00-00 In the new way the work is thus done . 72 ) 63.0 ( .875 A .875 — 15 0 — 4375 875 13.125 B. 875 24 3500 1750 21.000 C. 875 33 2625 2625 8.875 I hope by this time mine intelligent Reader is aware , that if the Partners had been many ( as in Voyages and Adventures it oft falls out ) the difference betwixt the two Methods would have been yet more signally conspicuous ; for one single Division sufficeth , be the Partners never so many : Though I confess there is yet a nearer way by the help of Logarithms ; of which I shall present this Specimen . 63 = 1.79934055 72 = 1.85732250 .875 = 1.94201805 .875 = 1.94201805 24 = 1.38021124 21 = 1.32222929 .875 = 1.94201805 15 = 1.17609126 13.125 = 1.11810931 .875 = 1.94201805 33 = 1.51851394 28.875 = 1.46053199 The second Example in Fellowship with time . A laid in 43 for 8 Months They gained 73l . Q. each Mans part ? B 52 5 C 63 7 The Resolution by this new Method is thus . A 43 8 344 B 52 5 260 C 63 7 441 344 260 441 1045 1045 ) 73.00 ( .069856 480 A. .069856 344 279424 279424 209568 24.030464 or 24 l. — 0 s. — 7d . — 2 q. B .069856 260 4191360 139712 18.162560 or 18 l. — 3 s. — 3d. — 0 q. C .069856 441 069856 279424 279424 30.806496 l. s. d. q. or 39 l. — 16 s. — 1 d. — 2 q. Proof 24.030464 18.162560 30.806469 72.999520 or thus 24 l. — 00 s. — 7 d. — 2 q. 18 l. — 03 s. — 30 d. 30 l. — 16 s. — 1 d. — 2 q. 73 l. — 00 s. — 0 d. — 0 q. The third Example in Alligation alternate . Suppose a mixture of Wine of 119 Quarts be required , that must be made up of these several prices 7 d. 8 d. 14 d. and 15 d. so as the whole may be afforded at 12 per Quart , the parts may be found out in thir method ( without decimals ) thus . Having linked 8 to 14 , and 7 to 15 , and Counterchanged their differences from the Common price 12d , I find the Sum of their differences to be 14 , by which dividing 119 the Quotient is 8 / 14 or 8 ½ , which for convenience of Multiplication we shall change into the improper Fraction so ● 2 the Resolution● will be thus . 8 2 14 4 7 3 15 5 14 Quarts . 17 / 2 X 2 = 34 / 2 = 17 17 / 2 X 4 = 68 / 2 = 34 17 / 2 X 3 = 51 / 2 = 25 ½ 17 / 1 X 5 = 85 / 2 = 42 ½ 119 Having our just Measure of Wine , let us try the Prices how they suit our purpose by Alligation medial , for considerable errors may be caused by misapplication of Prices , when the parts were truly taken ; but here-under it is apparent that each parcel multiplied by its price , the Sum of the Products is 1428 pence , which divided by 119 give 12 for the common price . 17 8 136 34 14 136 34 476 136 476 178 ½ 637 ½ 1428 25 ½ 7 178 ½ 42 ½ 15 217 ½ 42 637 ½ 119 ) 1428 ( 12 119 238 238 0 This Proof by Alligation medial I do not account a needless curiosity ; but very useful to be throughly understood , for experience informs me , that young Men being defective in Skill , Care and Memory , are apt to mistake in several points , but especially one : That is so as to esteem the several parcels to be of the several prices from which the differences ( by which they are found ) were originally taken , and not ( as the truth is ) of the prices to whom in the counterchange they were annexed . As to that which seems to look like an unnece●●sary affectation of Novelty , in linking the pric● and differences by separate couples ; I designed 〈◊〉 no higher thing than to free the Printer ( if th●● pass 〈◊〉 hand ) from the trouble of lookin● up his dusty Cuts of Semilunes intersecting or enclosing one another , as in our usual Books of Arithmetic●● Though the truth is , in such cases as this before us , no linking at all is needful ; but when the common Quotient is multiplied severally by all th● differences , any price above the common may b● assigned to any product made by the difference o● riginally belonging to an under price , and contr●●rily , so as true couples be observed . So here 〈◊〉 might have assigned the price 15 to the parcel 3● found out by the difference 4 originally belongi●● to 8 , and the price 14 to the parcel 42 ½ , found b● multiplying the common Quotient by 5 the difference of 7 from the common price . But then 〈◊〉 must be sure to do justice , so as to assign the pri●● 8 to the parcel 25 ½ found by 3 the difference o● 15 from the common price , and the price 7 to the parcel 17 arising from 2 the difference of 14● us here is plain . 17 X 7 = 119 25 ½ X 8 = 204 34 X 15 = 510 42 ½ X 14 = 595 1428 119 ) 1428 ( 12 119 238 238 0 FINIS . Notes, typically marginal, from the original text Notes for div A52120-e2310 ☞ ☜ ☞ ☞ ☞ ☞ ☜ ☜ ☜ ☜ ☞ ☞ ☜ 
A43097	----	The English school-master compleated containing several tables of common English words, from one, to six, seven, and eight syllables, both whole and divided, according to the rules of true spelling; with prayers, and graces both before and after meat, and rules for childrens behaviour at all times and places, with several other necessaries suitable to the capacities of children and youth. Also brief and easie rules for the true and exact spelling, reading, and writing of English according to the present pronunciation thereof in the famous University of Oxford, and City of London. To which is added, an appendix containing the principles of arithmetick, with an account of coins, weights, measure, time, &c. Copies of letters, titles of honour, suitable for men of all degrees, and qualities, bills of parcels, bills of exchange, bills of debt, receipts, and several other rules and observations fit for a youths accomplishment in the way of trade. John Hawkins school-master at St. Georges Church in Southwark. Hawkins, John, 17th cent. 1692 Approx. 439 KB of XML-encoded text transcribed from 73 1-bit group-IV TIFF page images. Text Creation Partnership, Ann Arbor, MI ; Oxford (UK) : 2004-08 (EEBO-TCP Phase 1). A43097 Wing H1175 ESTC R213434 99825818 99825818 30209 This keyboarded and encoded edition of the work described above is co-owned by the institutions providing financial support to the Early English Books Online Text Creation Partnership. This Phase I text is available for reuse, according to the terms of Creative Commons 0 1.0 Universal . The text can be copied, modified, distributed and performed, even for commercial purposes, all without asking permission. Early English books online. (EEBO-TCP ; phase 1, no. A43097) Transcribed from: (Early English Books Online ; image set 30209) Images scanned from microfilm: (Early English books, 1641-1700 ; 1794:20) The English school-master compleated containing several tables of common English words, from one, to six, seven, and eight syllables, both whole and divided, according to the rules of true spelling; with prayers, and graces both before and after meat, and rules for childrens behaviour at all times and places, with several other necessaries suitable to the capacities of children and youth. Also brief and easie rules for the true and exact spelling, reading, and writing of English according to the present pronunciation thereof in the famous University of Oxford, and City of London. To which is added, an appendix containing the principles of arithmetick, with an account of coins, weights, measure, time, &c. Copies of letters, titles of honour, suitable for men of all degrees, and qualities, bills of parcels, bills of exchange, bills of debt, receipts, and several other rules and observations fit for a youths accomplishment in the way of trade. John Hawkins school-master at St. Georges Church in Southwark. Hawkins, John, 17th cent. [6], 124 p., plates printed by A. and I. Dawks for the Company of Stationers, London : 1692. Reproduction of the original in the British Library. Created by converting TCP files to TEI P5 using tcp2tei.xsl, TEI @ Oxford. Re-processed by University of Nebraska-Lincoln and Northwestern, with changes to facilitate morpho-syntactic tagging. Gap elements of known extent have been transformed into placeholder characters or elements to simplify the filling in of gaps by user contributors. 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Copies of the texts have been issued variously as SGML (TCP schema; ASCII text with mnemonic sdata character entities); displayable XML (TCP schema; characters represented either as UTF-8 Unicode or text strings within braces); or lossless XML (TEI P5, characters represented either as UTF-8 Unicode or TEI g elements). Keying and markup guidelines are available at the Text Creation Partnership web site . eng Mathematics -- Early works to 1800. Spellers -- Early works to 1800. Etiquette for children and teenagers -- Early works to 1800. 2004-01 TCP Assigned for keying and markup 2004-02 Apex CoVantage Keyed and coded from ProQuest page images 2004-04 Amanda Watson Sampled and proofread 2004-04 Amanda Watson Text and markup reviewed and edited 2004-07 pfs Batch review (QC) and XML conversion THE ENGLISH School-Master Compleated ; Containing several Tables of Common English Words , from One , to Six , Seven , and Eight Syllables , both whole and divided , according to the Rules of true Spelling ; with Prayers , and Graces both before and after Meat , and Rules for Childrens Behaviour at all times and places , with several other necessaries suitable to the Capacities of Children and Youth . ALSO Brief and Easie Rules for the true and exact Spelling , Reading , and Writing of English according to the present pronunciation thereof in the Famous University of Oxford , and City of London . To which is Added , An Appendix containing the Principles of Arithmetick , with an Account of Coins , Weights , Measure , Time , &c. Copies of Letters , Titles of Honour , suitable for Men of all Degrees , and Qualities , Bills of Parcels , Bills of Exchange , Bills of Debt , Receipts , and several other Rules and Observations fit for a Youths accomplishment in the way of Trade . IOHN HAWKINS School-master at St. Georges Church in Southwark . LONDON , Printed by A. and I. Dawks for the Company of STATIONERS . 1692. THE Education of Children is 〈…〉 to be a thing of as weighty 〈…〉 Commonwealth , as any othe● 〈…〉 ●amed , and by how much the greater 〈…〉 greater ought to be the Care in the 〈…〉 the Principles and Elementary part of 〈…〉 ●ess of a Childs Education , for if the first 〈…〉 t●e Superstructure cannot be so firm and perfect as ( 〈…〉 ●ight be otherwise expected , and as Doctor Newton well 〈…〉 Elementary part of Learning being but weakly performed , the ●rammatical doth too generally fail , and yet ( saith be ) it is the Great DIANA of the EPHESIANS , all other Education of Children being if not totally despised , yet too much neglected , when yt perhaps vpon maturer Consideration it will be found that that is most neglected , which can ( in truth ) in the general be the worst sared ; it is our Mother Tongue that is likely in the practice to be most useful , nor is the preservation thereof in its own purity to be esteemed a small part of our Countreys Honour , since all the parts of Philosophy , and Mathematicks may be easily attained unto without the help of Exotick Languages , which are not gained but with Excessive Pains , and are quickly lost again without Continual Practice , it being a Vulgar Errour ( as Mr. Perkes saith in his Preface to the Art of Spelling ) to think , that to learn over the Grammar and some few Latine Books before a Boy goes to a Trade , are things so very necessary to his reading or writing true English , such a ●attering of Latine being generally useless , and the time spent in it so much 〈…〉 used 〈…〉 ●hall recommend 〈…〉 , viz. It would 〈…〉 , that the Master ●ad to the rest a Leaf or 〈…〉 and so leisurely that they may 〈…〉 ●hey have done , he who read to 〈…〉 in each Paper before they 〈…〉 ●●uld bring Boys to take heed of 〈…〉 Exercises ( after a good founda●●●● 〈…〉 Trades may be brought to a 〈…〉 ●●●ding and writing . 〈…〉 pains herein may redound to the profit of many , the wh●le 〈…〉 which is humbly commended to the Blessing of God , by him who earnestly desires to serve his Generation to the uttermost of his Ability . St. Georges Southwark , June 6. 1692. John Hawkins . Chap 6. Examples of Monosyllables beginning with Three Consonants . scr Scrap scrape scrall scribe scrip scrole scratch scrub . shr Shrank shred shrew shril shrimp shrine shrink shrub shrug . spl Splay splatch splent splice split splint spleen . spr Sprat sprang spread spring sprig spright sprout spruce . str Strand strake strange strap strength stress stretch strew strickt stride strise strike string strip stripe strive stroke strond strong strove struck strung . thr Thrall thread thresh threw thrice thri●t thrive throat throb throng throve through throw throws thrown thrums thrush thrust . thw Thwack thwick thwart thwite . Let the Teacher be very careful that the Scholar ●e perfect in Spelling the foregoing Examples ; it will likewise be necessary that he be well acquainted with the double Consonants for the beginning of words , as in the two last Chapters , viz. bl br ●l ●r &c. for it will be of great advantage to him when he comes to divide the Syllables in the Tenth and Eleventh Chapters following . Let him likewise be made very perfect in the sound of the Diphthongs in the following Chapter ; for whatsoever is to be learnt afterwards will mostly depend upon the knowledge of this and the foregoing Chapters . Chap. 7. Of Diphthongs . ai Pail bait hail nail pain rain gain wait rail sail quail pail trail train strain sprain grain twain plain . ay Flay lay may nay pay ray say stay stray gray spay splay . ei Height streight weight strein blein drein streight veil . ey Hey key grey prey . au Paul saul baulk baud haunt caul cause caught draught fault flaunt fraud fraught haunch hault mault naught paunch pause sauce . aw Bawl brawl 〈◊〉 claw craw crawl draw drawn sawn flaw haw jaw law lawn maw paw pawn raw saw spaw spawl spawn straw . oi Boil bois blois broid broil coil coin coit choice foil foin foist froise hoise noise joint loin moil poise soil spoil . oy Joyn poynt toyl boyl clog foy hoy joy loyn moyl moyst soyl toy . eu ew Feud lewd blew brew chew clew crew few flew glew grew hew jew jews lewd mew new pew sew shew slew spew stew stews screw strew . 〈◊〉 ou Bounce bound bout count chouse doubt fought found foul gout ground hound house hour loud mount mouth noun pounce pound proud round rout sound slouch . ow Clown crown crowd down fowl frown gown how howl now . Improper Diphthongs . ea Bead beat beast breach bread break breast breath clean cream dread dream feast glean knead lead learn least mean stream . ee Bees beef beer bleed breech breed creep deed deep feed feel free green greet keen keep leech leek leet meek meer meet need neer peep peer queen reed steed . ie Brief chief field friend grief grieve liege pierce priest shield chief . eo Feoffe George . oa Boar boat boa●t bloat broach broad coach coast coat cloak croak doat foal ●loat goat hoan hoar hoarse hoast load loaf moat road roam roar roast shoar . oo Book boom boot blood bloom brook broom choose c●ok cool coop crook door droop food fool foot floor good goose groom hood hook hoop hoot look loop loose mood moon moor nook noon poop poor proof proove rood roof rook room root shoot sloop smooth spoon stood stool stoop strook took tool tooth troop whoop wood wool wooe . ui Bruise bruit build built fruit guide guile guilt guire guise juice suit . Chap. 8. Let the Scholar be here taught that e or es at the end of a word doth cause that Syllable to be pronounced long which without it would be short ; as in the following Examples . A A Ac Ace ag age al ale ap ape ar are aps apes at ate B B Bab Babe bad bade bal bale ban bane b●ns banes bar bare bars bares bas base bat bate bats bates bid bide bids bides bil bile bils biles bit bite bits bites blad blade blads blades blam blame blot blote bon bone bons bones bor bore brib bribe brid bride brin brine C C Cag Cage cam came can cane car care cars cares cas case chaf chafe chap chape chid chide clos close cot cote col cole cor core con cone cran crane crans cranes cur cure D D Dal Dale dals dales dam dame dams dames din dine dot dote dots dotes dar dare dars dares F F Fad Fade fac face fam fame far fare fin fine fins fines fil file fils files flam flame flams flames fom fome fum fume fram frame frams frames G G Gap Gape gaps gapes gal gale gals gales gat gate gats gates gor gore grac grace grap grape grat grate grats grates grop grope H H Hal Hale har hare hat hate hast hast her here hid hide hids hides hir hire hol hole hols holes hom home hop hope hops hopes hug huge I I Il Ile ils iles ic ice K K Kin Kine kit kite knif knife L L Lac Lace lad lade lam lame lanc lance lans lanes lat late lic lice lif life lim lime lin line lins lines los lose M M Mad Made mac mace mal male mat mate met mete mit mite mir mire mol mole mor more mot mote mul mule mus muse mals males mits mites mols moles muls mules mak make N N Nam Name nams names net nete nets netes nin nine non none nos nose not note P P Pal Pale pals pales par pare pars pares pat pate pats pates pin pine pins pines pip pipe pips pipes pil pile pils piles plac place plan plane plans planes plat plate plats plates pol pole pols poles prid pride prun prune pur pure purg purge R R Rac Race rag rage rang range rar rare ras rase rat rate rid ride rip ripe ris rise rit rite rits rites rob robe robs robes rop rope rops ropes ros rose rud rude rul rule ruls rules S S Saf Safe sal sale sals sales sam same sat sate scal scale scals scales shad shade shads shades scrap scrape scrib scribe scribs scribes sham shame shap shape shaps shapes shar share shars shares shin shine shins shines shor shore shors shores sid side sids sides slid slide smot smote snar snare snars snares sol sole sols soles sop sope sor sore sors sores spit spite stal stale star stare stol stole ston stone stons stones stor store stors stores strang strange strif strife strip stripe strips stripes sur sure swin swine swar sware T T Tal Tale tals tales tam tame tar tare ther there thin thine thos those thron throne til tile tils tiles tir tire tor tore trad trade trads trades trip tripe trips tripes truc truce twin twine trib tribe tribs tribes trap trape traps trapes V V Val Vale vals vales vil vile vin vine vins vines urg urge us use W W Wad Wade war ware whal whale whil while whit white whor whore whos whose wid wide wif wife wil wile wils wiles win wine wins wines wip wipe wis wise wips wipes writ write wrot wrote Chap. 9. Containing Examples of Words of Two Syllables having one Consonant in the middle thereof , both whole and divided . In this Chapter let the Scholar be taught that when a Consonant is in the middle of a word between two Vowels , such Consonant ought in Spelling to be joined to the latter Syllable , as in the following Examples . A A A-base Abase a-bide abide a-bode abode a-bove above a-bound abound a-bout about a-buse abuse a-far afar a-gain again a-gainst against a-ges ages a-gon agon a-gue ague a-larm alarm a-las alas a-like alike a-live alive a-loft aloft a-lone alone a-long along a-loof aloof a-loud aloud a-men amen a-mend amend a-miss amiss a-mong among a-mongst amongst a-noint anoint a-right aright a-rise arise a-rose arose a-side aside a-venge avenge a-verse averse a-void avoid au-gust august a-wake awake a-ware aware a-way away au-tumn autumn a-maze amaze a-vouch avouch a-board aboard a-part apart B B Ba-bel Babel ba-ked baked ba-ker baker ba-kers bakers ba-keth baketh ba-nish banish ba-ser baser ba-son bason bea-con beacon bea-rer bearer beau-ty beauty be-came became be-come become be-cause because be-fall befall be-fel befel be-fore before be-gan began be-get beget be-gat begat be-got begot be-gin begin be-guile beguile be-gun begun be-half behalf be-have behave be-held beheld be-hold behold be-hind behind be-lieve believe be-lief belief be-long belong be-moan bemoan be-neath beneath be-reave bereave be-seech beseech be-side beside be-sought besought be-times betimes be-wail bewail be-ware beware be-yond beyond bla-med blamed bla-meth blameth ble-mish blemish bra-sen brasen broa-der broader broi-ler broiler broi-led broiled bro-ken broken brui-s●d bruised brui-ses bruises brui-sing bruising bru-tish brutish bu-sy busy C C Ca-mel Camel ca-mest camest ce-dar cedar cho-ler choler cho-sen chosen cau-ses causes cau-sest causest cau-sey causey cau-sing causing cea-seth ceaseth cha-sed chased cha-nel chanel che-rish cherish chi-ding chiding choi-cest choicest ci-ty city cla-mour clamour clo-sed closed clo-ser closer clo-set closet clou-dy cloudy clo-ven cloven clou-ted clouted coa-ches coaches co-lour colour co-meth cometh co-mets comets co-ming coming co-ver cover co-vers covers co-vet covet cou-rage courage cou-sin cousin cu-bit cubit cu-bits cubits cu-red cured cu-reth cureth D D Da-mage Damage dau-bed daubed dau-bing daubing day-ly dayly dea-con deacou de-bate debate de-base debase de-cay decay de-ceit deceit de-ceive deceive de-cent decent de-fame defame de-feat defeat de-fence defence de-fend defend de-fie defie de-file defile de-lay delay de-light delight de-mand demand de-ny deny de-part depart de-pose depose de-pend depend de-pute depute de-ride deride de-sart desart de-serve deserve de-sire desire de-tain detain de-test detest de-vise devise de-vote devote di-ned dined di-rect direct di-vers divers di-vide divide di-vine divine di-vorce divorce dra-gon dragon du-ty duty E E Ea-ger Eager ea-sie easie e-lect elect e-nough enough e-qual equal e-rect erect e-vent event e-ver ever e-vil evil eye-sight eyesight F F Fa-ces Faces fa-deth fadeth fa-mous famous fa-mine famine fa-vour favour fi-gure figure flou-rish flourish fre-quent frequent fro-ward froward fee-ble feeble fe-lons felons fe-male female fea-ver feaver fi-gures figures fi-nish finish fi-ner finer fi-nest finest fra-med framed fra-meth frameth free-dom freedom free-ly freely free-man freeman fre-quent frequent free-will freewill fro-zen frozen fu-ry fury G G Ga-vest Gavest ga-zing gazing gi-ven given gi-ver giver gi-vest givest gi-veth giveth glo-ry glory go-red gored go-vern govern gra-vel gravel gra-ven graven gra-ving graving gray-hound grayhound gree-dy greedy grie-vance grievance grie-veth grieveth grie-vous grievous gro-peth gropeth H H Hai-nous Hainous ha-bit habit hai-ry hairy ha-sel hasel ha-ted hated ha-ters haters ha-test hatest ha-teth hateth ha-ting hating ha-vock havock hea-dy heady hea-ved heaved hea-ven heaven hea-vens heavens hea-vy heavy hei-fer heifer he-rauld herauld hi-dest hidest hi-deth hideth hi-ding hiding hi-red hired hi-rest hirest hoa-ry hoary hoi-sed hoised ho-ly holy ho-nest honest hu-mour humour ho-ney honey ho-ped hoped ho-ping hoping ho-nour honour hou-ses houses I I I-dol Idol i-mage image joy-ful joyful jea-lous jealous ju-ror juror ju-rors jurors L L La-bour Labour la-den laden la-ding lading la-dy lady la-ment lament la-tin latin la-ver laver la-vish lavish le-per leper li-cense license li-ken liken li-king liking li-nage linage li-quor liquor li-ver liver lo-cust locust lo-cal local lo-seth loseth lo-sing losing lo-ver lover lo-vers lovers lo-ving loving lo-vest lovest lo-veth loveth li-zard lizard M M Ma-dest Madest ma-ker maker ma-kest makest ma-king making ma-lice malice ma-son mason mea-sure measure me-lon melon mo-ment moment mo-dest modest mo-ver mover mo-ving moving mu-sed mused mu-sing musing mu-sick musick N N Na-ked Naked na-med named na-tive native na-ture nature na-vel navel na-vy navy nee-dy needy ne-ver never no-ble noble noi-sed noised noi-som noisom no-ses noses nou-rish nourish O O O-bey Obey o-dour odour o-dours odours o-live olive o-lives olives o-mer omer o-nix onix o-pen open o-ven oven o-ver over P P Pa-ces Paces pa-per paper pa-ved paved pe-ril peril pe-rish perish pi-lot pilot pi-ning pining pi-per piper pi-ty pity pla-ces places pla-net planet plea-sant pleasant plea-sed pleased plea-seth pleaseth plea-sing pleasing plea-sure pleasure plea-sures pleasures poi-son poison prai-sed praised prai-seth praiseth prai-sing praising pra-ting prating pre-cept precept pre-cepts precepts pre-fer prefer pre-pare prepare pre-sence presence pre-sent present pre-sume presume pre-tence pretence pre-vail prevail pre-vent prevent pri-son prison pri-vate private pro-ceed proceed pro-cess process pro-cure procure pro-duce produce pro-fane profane pro-fess profess pro-fit profit pro-found profound pro-long prolong pro-mise promise pro-mote promote pro-nounce pronounce pro-per proper pro-test protest pro-verb proverb pro-vide provide pro-vince province pro-voke provoke pru-dence prudence pru-dent prudent pru-ned pruned pru-ning pruning pu-nish punish Q Q Qua-king Quaking qua-keth quaketh qua-ked quaked qui-ver quiver que-ry query qua-ver quaver R R Ra-ging Raging rai-ment raiment ra-vish ravish rea-dy ready rea-son reason re-bel rebel re-buke rebuke re-ceive receive re-cord record re-count recount re-deem redeem re-fine refine re-fuge refuge re-gard regard re-ject reject re-joyce rejoyce re-lease release re-lieve relieve re-ly rely re main remain re-mit remit re-move remove re-new renew re-nounce renounce re-nown renown re-pair repair re-peat repeat ●ent repent re-port report re-quest request re-quire require re-quite requite re-sign resign re-serve reserve re-sist resist re-sort resort re-solve resolve re-tain retain re-tire retire re-turn return re-veal reveal re-venge revenge re-vile revile re-vive revive re-volt revolt re-ward reward ri-der rider ri-gour rigour ri-ver river ro-man roman ro-vers rovers ru-mour rumour ru-ler ruler S S Sa-tan Satan sa-lute salute sa-tyr satyr sa-ving saving sa-vour savour scra-ping scraping sea-son season se-duce seduce se-ver sever sha-dow shadow sha-dy shady sha-king shaking she-riff sheriff si-lent silent si-new sinew so-ber sober so-journ sojourn so-lace solace so-lemn solemn spo-ken spoken sta-tute statute sto-mach stomach sto-ny stony sto-ry story T T Ta-lent Talent te-nour tenour ti-dings tidings to-ken token tray-tor traytor tra-vel travel trea-son treason trea-sure treasure tri-bute tribute trou-ble trouble tru-ly truly tu-mult tumult tu-tor tutor twy-lights twylights ty-rant tyrant V V Va-lour Valour va-lue value va-nish vanish ve-nom venom vi-per viper vi-sage visage vi-sit visit u-nite unite vo-lume volume vo-mit vomit u-surp usurp use-ful useful W W Wa-fer Wafer wa-ges wages wa-king waking wa-ter water wa-ver waver wea-pon weapon wea-ry weary whore-dom whoredom wi-ping wiping whi-ter whiter whi-ting whiting wo-ful woful wo-man woman wa-gon wagon wea-sel weasel who-rish whorish wi-dow widow wi-ser wiser wi-zard wizard wo-ven woven wri-ter writer wri-test writest wri-teth writeth wri-ting writing Y Y Yo-ked Yoked yo-king yoking yo-keth yoketh Z Z Zea-lous Zealous zea-lot zealot Chap. 10. Containing Examples of Words of Two Syllables having Two Consonants at least in the middle thereof , which Consonants are such as may be placed in the beginning of a Word , both whole and divided . Let the Scholar be now taught that when two or three Consonants that can begin a Word , are placed in the middle thereof between two Vowels , they must be joined to the latter Syllable , as in the following Examples . A A A-broad Abroad a-bridg abridg a-fraid afraid a-fresh afresh a-gree agree a-pron apron a-scend ascend a-scribe ascribe a-sleep asleep a-stray astray au-stere austere au-thor author B B Be-stead Bestead be-stir bestir be-stow bestow be-think bethink be-tray betray be-tween between be-troth betroth be-twixt betwixt be-wray bewray bi-shop bishop bre-thren brethren bro-ther brother C C Cha-sten Chasten cha-stise chastise chry-stal chrystal D D Day-spring Dayspring day-star daystar de-clare declare de-cline decline de-cree decree de-creed decreed de-crees decrees de-fraud defraud de-fray defray de-gree degree de-grees degrees de-prave deprave de-scend descend de-coy decoy de-scribe describe de-scry descry de-spight despight de-stroy destroy dou-ble double dry-shod dryshod E E Ea-gle Eagle ea-gles eagles ei-ther either e-phod ephod e-scape escape e-schew eschew e-spy espy e-state estate e-states estates eye-brows eyebrows F F Fa ther Father fa-thers fathers fa-thom fathom fea-ther feather fee-ble feeble fee-bler feebler fire-brand firebrand free stone . freestone G G Ga-ther Gather H H Ha-tred Hatred hea-then heathen hea thens heathens he-brew hebrew hi-ther hither L L Lea-ther Leather lea thern leathern le-prous le prous M M Ma-chine Machine ma ster master ma-sters masters ma-tron matron ma trix matrix mo-ther mother mo-thers mothers N N Nei-ther Neither ne-phew nephew ne-ther nether no ble noble no-stril nostril no-thing nothing O O O ther Other o-thers others ou-ches ouches P P Peo-ple People pine-tree pinetree pla-ster plaster pro-phet prophet pro-phets prophets pro-tract protract R R Ra-ther Rather re-frain refrain re-fresh refresh re-proach reproach re-proof reproof re-scue rescue re-store restore re-strain restrain re-straint restraint re-trive retrive re-dress redress S S Sa-cred Sacred se cret secret se-clude seclude sa-phire saphire sta ble stable sta-blish stablish T T Ta-blet Tablet ta bret tabret ta bring tabring te-trearch tetrearch trou-bled troubled trou-bleth troubleth trou-bler troubler trou-blest troublest trou-bling troubling V V Va-prest Vaprest va-preth vapreth ve-sture vesture vi-prous viprous W W Wa-treth Watreth wa-tring watring wea-ther weather whe-ther whether whi-ther whither wi-ther wither Chap. 11. Containing Words of Two Syllables having Two Consonants at least in the middle thereof , which Consonants are such as cannot be in the beginning of a Word , both whole and divided . Let the Scholar now be Taught , that when such Consonants come in the middle of a Word as cannot be in the beginning , they are to be divided , as also is a double Consonant . Examples . A A Ab-hor Abhor ab-ject abject ab-sence absence ab-stain abstain ac-cept accept ac-cord accord ac-cess access ac-quit acquit ad-ded added ad-jure adjure af-flict afflict af-firm affirm af-fairs affairs af-fect affect al-lure allure al-mond almond al-most almost am-ber amber am-bush ambush an-cle ancle an-chor anchor an-gry angry an-guish anguish an-swer answer an-vil anvil ap-peal appeal ap-ply apply ap-point appoint ar-my army ar-mour armour as-sault assault as-sure assure as-sist assist at-tain attain at-tend attend at-tire attire aug-ment augment B B Back-bone Backbone back-slide backslide back-ward backward bad-ness badness bald-ness baldness ban-ded banded ban-ner banner ban-quet banquet bap-tize baptize bar-ber barber bar-ley barley bar-red barred bar-rel barrel bar-ren barren bas-ket basket bas-tard bastard bat-tel battel bat-ter batter beck-ned beckned bed-stead bedstead beg-gar beggar bel-low bellow bel-ly belly ber-ries berries bet-ter better bib-ber bibber bid-den bidden bil-lows billows bit-ten bitten bit-ter bitter bles-sing blessing bles-sed blessed blos-som blossom blot-ting blotting blot-teth blotteth bond-man bondman bond-maid bondmaid bon-net bonnet bor-row borrow bot-tom bottom bran-dish brandish braw-ler brawler braw-ling brawling bright-ness brightness brim-stone brimstone bub-ble bubble buc-ket bucket buck-ler buckler bud-ded budded buf-fet buffet bul-lock bullock bul-wark bulwark bur-den burden but-ter butter but-ler butler but-tock buttock C C Cab-bin Cabbin cal-dron caldron cal-led called cam-phire camphire can-not cannot cap-tive captive car-kass carkass car-ping car-ping car-nal car-nal cat-tel cattel cen-sure censure cer-tain certain cham-ber chamber chan-ced chanced chan-ged changed chap-man chapman chap-pel chappel char-ger charger chaf-fer chaffer chat-ter chatter cheek-bone cheekbone clap-ping clapping clean-sed cleansed clear-ly clearly clus-ter cluster cof-fer coffer cof-fin coffin col-lar collar col-ledge colledge col-lop collop com-mand command com-mit commit com-pare compare com-pel compel com-plain complain com-pound compound con-fer confer con-demn condemn con-cord concord con-clude conclude con-duct conduct con-firm confirm con-flict conflict con-quer conquer con-sent consent con-stant constant con-sume consume con-tain contain con-temn contemn con-tempt contempt con-test contest con-trite contrite con-vert convert con-verse converse con-vey convey con-voy convoy cop-per copper cor-ner corner cor-net cornet cor-rect correct cor-rupt corrupt cost-ly costly cot-tage cottage coun-sel counsel coun-cil council coun-try country crack-ling crackling crim-son crimson cum-bred cumbred cun-ning cunning cur-tain curtain cus-tom custom cut-teth cutteth cut-ting cutting cym-bal cymbal D D Dag-ger Dagger dain-ty dainty dan-cing dancing dand-led dandled dan-ger danger dark-ness darkness dar-ling darling daugh-ter daughter dear-ly dearly dif-fer differ dig-geth diggeth dim-ness dimness dis-charge discharge dis-close disclose dis-creet discreet dis-grace disgrace dis-guise disguise dis-patch dispatch dis-please displease dis-pute dispute dis-solve dissolve dis-tant distant dis-tress distress doc-tor doctor doc-trine doctrine doubt-ful doubtful doubt-less doubtless down-ward downward dread-ful dreadful dres-ser dresser drop-ped dropped drop-ping dropping drop-sy dropsy drun-kard drunkard drun-ken drunken dung-hil dunghil dung-port dungport dwel-ler dweller dwel-ling dwelling dwel-leth dwelleth dwel-lest dwellest E E Eb-bing Ebbing eb-beth ebbeth ear-ly early ear-nest earnest ef-fect ef-fect el-der elder el-dest eldest em-balm embalm em-pire empire em-ber ember em-ploy employ emp-ty empty en-camp encamp en-cline encline en-close enclose end-less endless en-joy enjoy en-joyn enjoyn en-large enlarge eng-land england en-ter enter en-tire entire en-tring entring en-try entry en-trance entrance er-rand errand er-red erred er-rour errour ex-ceed exceed ex-cel excel ex-cept except ex-cess excess ex-change exchange ex-clude exclude ex-pence expence ex-tream extream ex-pound expound ex-ploit exploit ex-hort exhort ex-pel expel ex-tinct extinct ex-tol extol ex-tream extream F F Fac-tor Factor faint-ness faintness faith-ful faithful faith-less faithless fal-lest fallest fal-leth falleth fals-ly falsly far-ther farther far-thing farthing fas-ter faster far-thest farthest fat-ling fatling fat-ter fatter fat-ted fatted fat-test fattest faul-ty faulty fear-ful fearful fel-ling felling fel-low fellow fen-ced fenced fer-ret ferret fer-vent fervent fet-ters fetters fif-teen fifteen fif-ty fifty fig-tree figtree fil-led filled fil-leth filleth fid-ler fidler fil-thy filthy fir-kin firkin fir-tree firtree fir-wood firwood fish-hook fishhook fish-pool fishpool fit-ly fitly fit-ted fitted flat-ter flatter flesh-ly fleshly flesh-pots fleshpots flin-ty flinty fol-low follow fol-ly folly foot-man footman foot-steps footsteps foot-stool footstool for-bare forbare for-bid forbid for-born forborn for-ced forced for-cing forcing for-get forget for-give forgive for-sake forsake for-sware forsware forth-with forthwith for-ty forty for-ward forward foun-tain fountain four-teen fourteen frag-ment fragment frank-ly frankly fret-ted fretted front-let frontlet fruit-ful fruitful ful-fil fulfil ful-ler fuller ful-ness fulness fur-long furlong fur-nace furnace fur-nish furnish fur-row furrow fur-ther further fun-nel funnel fur-bish furbish G G Gad-ding Gadding gain-say gainsay gal-lop gallop gar-den garden gar-lick garlick gar-ment garment gar-nish garnish gen-tile gentile gent-ly gently get-ting getting glad-ly gladly glad-ness gladness glas-ses glasses glis-ter glister glit-ter glitter glut-ton glutton god-head godhead god-ly godly gold-smith goldsmith good-ly goodly good-ness goodness gos-pel gospel got-ten gotten great-ly greatly great-ness greatness guil-ty guilty guilt-less guiltless gut-ter gutter gun-ner gunner gul-let gullet gul-led gulled gul-ling gulling H H Hal-low Hallow ham-mer hammer hand-ful handful hand-ling handling hand-maid handmaid hap-pen happen hap-ned hapned hap-py happy hard-ned hardned har-lot harlot harm-less harmless har-row harrow har-vest harvest has-ty hasty head-long headlong heark-ned hearkned hem-lock hemlock high-ly highly high-ness highness him-self himself hin-der hinder hid-den hidden his-sing hissing hol-low hollow host-ler hostler hor-net hornet hor-ses horses hot-ly hotly hous-hold houshold hum-bly humbly hun-dred hundred hun-ger hunger hun-ter hunter hun-ting hunting hun-gry hungry hur-ling hurling hurt-ful hurtful hus-band husband hys-sop hyssop I I Jang-ling Jangling jas-per jasper im-brace imbrace im-part impart im-pose impose im-pute impute in-cense incense in-crease increase in-fant infant in-flame inflame in-form inform in-fer infer in-side inside in-stant instant in-struct instruct in-tend intend in-tent intent in-treat intreat in-vade invade in-vent invent in-ward inward jour-ney journey judg-ment judgment jus-tice justice just-ly justly K K Ker-nel Kernel kid-ney kidney kil-led killed kil-lest killest kil-leth killeth kind-led kindled kind-leth kindleth kind-ly kindly kin-dred kindred king-dom kingdom king-ly kingly kins-folk kinsfolk kins-man kinsman kis-ses kisses kis-sed kissed kis-sing kissing know-ledge knowledge L L Lad-der Ladder lan-ces lances lan-guage language lan-guish languish land-man landman land-mark landmark lap-ping lapping lap-wing lapwing large-ness largeness lat-ter latter laugh-ter laughter law-ful lawful law-less lawless les-ser lesser let-ter letter lewd-ness lewdness lewd-ly lewdly light-ly lightly light-ning lightning light-ned lightned lil-ly lilly lil-lies lillies low-ly lowly low-ring lowring lof-ty lofty lus-ty lusty M M Mad-man Madman mad-ness madness main-tain maintain mal-low mallow mam-mon mammon man-ger manger man-ner manner man-hood manhood man-kind mankind mar-ket market mar-row marrow mar-shal marshal mar-tyr martyr mas-ter master mat-ter matter med-led medled med-leth medleth meek-ly meekly mem-ber member mer-chant merchant mer-cy mercy mer-ry merry mes-sage message mid-night midnight mid-wife midwife migh-ty mighty mil-ler miller mil-stone milstone mind-ful mindful min-strel minstrel mis-chief mischief min-gled mingled mis-sed missed mis-tress mistress mix-ture mixture mol-ten molten month-ly monthly mon-ster monster moun-tain mountain mourn-ful mournful muf-ler mufler mum-my mummy mur-der murder mur-mur murmur mur-rain murrain mut-ter mutter myr-tle myrtle N N Nas-ty Nasty need-ful needful neg-lect neglect neigh-bour neighbour net-work network new-ly newly new-ness newness noon-day noonday noon-tide noontide north-ward northward north-west northwest nur-ture nurture num-ber number num-bring numbring nur-sed nursed O O Ob-ject Object ob-scure obscure ob-struct obstruct of-fend offend of-fer offer of-fice office off-spring offspring of-ten often on-ward onward op-pose oppose op-press oppress or-dain ordain or-gan organ or-der order out-ward outward P P Pain-ful Painful pal-sie palsie par-cel parcel parch-ment parchment par-don pardon par-lour parlour part-ly partly part-ner partner part-tridge partridge pas-sage passage pas-sing passing pas-tor pastor pas-ture pasture Pat-tern pattern Pen-ny penny Pen-knife penknife Per-form perform per-fume perfume per-mit permit per-son person per-tain pertain per-vert pervert pic-ture picture pil-grim pilgrim pil-lar pillar pil-low pillow plat-ted platted plat-ter platter plen-ty plenty plot-ting plotting plow-men plowmen plum-met plummet pow-der powder pop-lar poplar por-ter porter pos-sess possess pot-tage pottage prin-ces princes pros-pect prospect pros-per prosper pros-trate prostrate proud-ly proudly pub-lick publick pub-lish publish puf-fed puffed puf-fing puffing pul-pit pulpit pur-chase purchase pur-port purport pur-pose purpose pur-ling purling pur-ses purses pur-sue pursue pur-suit pursuit put-teth putteth put-ting putting Q Q Quar-rel Quarrel quar-ter quarter quick-ly quickly quick-ned quickned quick-neth quickneth quick-ning quickning quick-sand quicksand quick-ness quickness R R Rab-bi Rabbi rain-bow rainbow ram-part rampart ran-som ransom ran-ger ranger ran-dom random rat-leth ratleth rat-ling ratling rec-kon reckon ram-bling rambling rem-nant remnant ren-der render ren-dred rendred res-pit respit red-ness redness rest-less restless rib-bon ribbon right-ly rightly rob-ber robber rob-bing robbing rub-bing rubbing rub-bers rubbers rub-bish rubbish rud-dy ruddy rum-bling rumbling rum-mer rummer rus-ty rusty S S Sab-bath Sabbath sack-but sackbut sack-cloth sackcloth sad-led sadled sad-ly sadly sad-ness sadness saf-fron saffron san-dals sandals scab-bed scabbed scaf-fold scaffold scep-ter scepter scorn-ful scornful scour-ged scourged scour-ging scourging scrip-ture scripture scur-vy scurvy scum-mer scummer sel-ler seller sen-ces sences sen-tence sentence ser-pent serpent ser-vant servant ser-ved served ser-vice service ser-vile servile set-ter setter set-ting setting sel-ler seller sel-ling selling set-led setled sharp-ly sharply sharp-ness sharpness shed-ding shedding sheep-skins sheepskins shel-ter shelter shep-herd shepherd shew-bread shewbread ship-man shipman ship-wrack shipwrack short-ly shortly shoul-der shoulder shut-ting shutting sick-ly sickly sick-ness sickness sig-net signet sig-nal signal sil-ly silly sin-ful sinful sin-ner sinner sin-ning sinning si-ster sister sit-ting sitting six-teen sixteen six-ty sixty skil-ful skilful skip-ping skipping slack-ness slackness slan-der slander sloth-ful slothful slug-gard sluggard slum-ber slumber smel-led smelled smel-ling smelling smit-ten smitten snuf-fers snuffers soc-ket socket sod-den sodden soft-ly softly sor-row sorrow sot-tish sottish sound-ness soundness south-ward southward span-ning spanning spar-row sparrow speech-less speechless spil-led spilled spit-ting spitting spot-ted spotted sprink-led sprinkled stag-ger stagger step-ping stepping stil-led stilled stif-fly stiffly stir-ring stirring stop-per stopper stor-my stormy stout-ness stoutness stran-ger stranger strang-led strangled streng-then strengthen stric-ken stricken strip-ling stripling strong-ly strongly stub-born stubborn stum-bled stumbled sub-due subdue sub-ject subject sub-mit submit sub-scribe subscribe sub-vert subvert suc-ceed succeed suc-cess success suc-cour succour suck-ling suckling sud-den sudden suf-fer suffer sum-mer summer sun-dry sundry sup-ped supped sup-per supper sup-ply supply sup-plant supplant sup-pose suppose sus-tain sustain swad-ling swadling swal-low swallow swel-ling swelling swift-ly swiftly swim-ming swimming T T Tack-ling Tackling tan-ner tanner tar-get target tap-ster tapster tar-dy tardy tar-ry tarry tart-ly tartly tat-ling tatling tem-per temper tem-pest tempest ten-der tender ter-rour terrour thank-ful thankful them-selves themselves thick-ness thickness third-ly thirdly thir-teen thirteen thir-ty thirty thir-sty thirsty threat-ning threatning through-ly throughly thun-der thunder til-lage tillage til-led tilled tim-ber timber tim-brel timbrel tor-ches torches tor-ment torment tor-ture torture tos-sed tossed tos-sing tossing traf-fick traffick trans-gress transgress tran-slate translate trem-bled trembled trem-bling trembling tres-pass trespass trim-mer trimmer trim-meth trimmeth trum-pet trumpet twen-ty twenty V V Vain-ly Vainly val-ley valley ven-ture venture ver-tue vertue ves-sel vessel vil-lage village vil-lain villain vin-tage vintage vir-gin virgin un-just unjust un-known unknown un-lade unlade un-less unless un-done undone un-loose unloose un-ripe unripe un-til until un-true untrue un-wise unwise up ▪ braid upbraid up-hold uphold up-per upper up-right upright up-start upstart up-roar uproar up-side upside up-ward upward ur-ged urged ur-gent urgent ut-most utmost ut-ter utter vul-can vulcan vul-ture vulture vul-gar vulgar W W Wal-low Wallow wan-der wander wan-ton wanton ward-robe wardrobe war-ring warring wash-pot washpot watch-ful watchful watch-man watchman weak-ness weakness weal-thy wealthy wed-ding wedding wed-lock wedlock weigh-ty weighty wel-fare welfare west-ward westward whirl-wind whirlwind whis-per whisper whol-some wholsome wic-ked wicked wil-ling willing wil-low willow wil-ful wilful win-dy windy win-dow window win-ner winner win-ter winter wis-dom wisdom wish-ful wishful with-draw withdraw with-hold withhold with-stand withstand wit-ness witness wit-ty witty won-der wonder work-man workman wor-ship worship wor-thy worthy world-ly worldly wrap-ped wrapped wrap-ping wrapping wrath-ful wrathful writ-ten written wrong-ful wrongful wrest-ling wrestling Y Y Year-ly Yearly yel-low yellow yel-led yelled yon-der yonder youth-ful youthful Chap. 12. Containing Examples of Words which end with le or les after a Consonant . Let the Scholar here be taught to sound le or les in the end of a word coming after a Consonant as if there were no e placed there at all . Examples . A-ble Able nm-ble amble ad-dle addle an-gle angle ap-ple apple an-cle ancle ba-ble bable bea-dle beadle bee-tle beetle bea-gle beagle bun-dle bundle brin-dle brindle bot-tle bottle bri-dle bridle brit-tle brittle bris-tle bristle buc-kle buckle can-dle candle cac-kle cackle ca-ble cable cas-tle castle cau-dle caudle cir-cle circle coc-kle cockle cob-ble cobble cou-ple couple crip-ple cripple dag-gle daggle dan-dle dandle da-zle dazle dou-ble double ea-gle eagle ea-gles eagles fa-ble fable fa-bles fables fee-ble feeble gar-gle gargle gen-tle gentle gir-dle girdle gir-dles girdles gob-ble gobble gog-gle goggle grum-ble grumble han-dle handle han-dles handles hig-gle higgle hob-ble hobble hum-ble humble jan-gle jangle i-dle idle jus-tle justle ket-tle kettle kin-dle kindle lit-tle little man-tle mantle man-tles mantles mar-ble marble med-dles meddles mid-dle middle min-gle mingle muf-fle muffle muz-zle muzzle myr-tle myrtle nee-dle needle no-ble noble net-tles nettles no-bles nobles pad-dle paddle peo-ple people pur-ple purple pim-ple pimple pim-ples pimples prat-tle prattle pud-dle puddle puz-zle puzzle rab-ble rabble rat-tle rattle rid-dle riddle rum-ble rumble sa-ble sable sad-dle saddle sad-dles saddles set-tle settle sham-ble shamble sham-bles shambles sim-ple simple sin-gle single shac-kle shackle shac-kles shackles spin-dle spindle spit-tle spittle sprin-kle sprinkle sta-ble stable sta-bles stables star-tle startle stop-ple stopple stub-ble stubble stum-ble stumble sup-ple supple ta-ble table ta-bles tables tat-tle tattle tem-ple temple this-tle thistle tin-gle tingle ti-tle title tram-ple trample trem-ble tremble tric-kle trickle tic-kle tickle tri-fle trifle trou-ble trouble trou-bles troubles truc-kle truckle tum-ble tumble tur-tle turtle tur-tles turtles un-cle uncle wat-tle wattle wres-tle wrestle wrin-kle wrinkle wrin-kles wrinkles Chap. 13. Containing Examples of Words wherein ti is placed before a Vowel . Let the Scholar be here taught that ti before a Vowel is pronouneed si , but otherwise ty . Examples . Ac-ti-on Action ad-di-ti-on addition a-dop-ti-on adoption af-fec-ti-on affection af-flic-ti-on affliction at-ten-ti-on attention col-lec-ti-on collection com-mo-ti-on commotion con-di-ti-on condition con-sump-ti-on consumption con-ten-ti-on contention dam-na-ti-on damnation de-struc-ti-on destruction de-vo-ti-on devotion di-rec-ti-on direction dis-trac-ti-on distraction e-lec-ti-on election ex-tor-ti-on extortion foun-da-ti-on foundation fac-ti-on faction in-struc-ti-on instruction in-ven-ti-on invention mu-ni-ti-on munition na-ti-on nation ob-la-ti-on oblation o-ra-ti-on oration par-ti-ti-on partition por-ti-on portion re-demp-ti-on redemption sal-va-ti-on salvation sanc-ti-on sanction se-di-ti-on sedition tax-a-ti-on taxation temp-ta-ti-on temptation vex-a-ti-on vexation vo-ca-ti-on vocation Chap. 14. Containing Words of Three Syllables both whole and divided , wherein the former directions are to be observed . A A A-ban-don Abandon a-ba-ted abated ab-strac-ted abstracted a-bun-dance abundance ab-hor-ring abhorring a-but-ting abutting a-bu-ses abuses ac-cep-ted accepted ac-com-plish accomplish ac-cord-ing according ad-mo-nish admonish ac-ti-ons actions ad-van-tage advantage ad-ven-ture adventure af-fir-med affirmed af-fec-ted affected af-fright-ed affrighted af-ter-wards afterwards a-go-ny agony ag-gre-gate aggregate a-gree-ment agreement al-ledg-ed alledged al-li-gate alligate al-ter-nate alternate am-bas-sage ambassage an-ces-tors ancestors an-ci-ent ancient an-ti-pode antipode an-swe-red answered a-no-ther another a-po-stle apostle ap-ply-ed applyed a-po-state apostate ap-pre-hend apprehend ap-pro-ved approved a-ray-ed arayed ar-ri-ved arrived ar-ti-fice artifice ar-ti-choke artichoke ar-ti-cle article a-scen-ded ascended a-sha-med ashamed as-sem-ble assemble as-su-rance assurance as-sun-der assunder at-ten-tive attentive at-trac-tive attractive at-tri-bute attribute at-tai-ned attained a-vai-leth availeth a-vouch-ed avouched au-di-ence audience a-ver-ring averring a-wa-ked awaked a-wa-king awaking B B Back-bi-ting Backbiting back-sli-ding backsliding bal-lan-ced ballanced ba-nish-ment banishment ban-ter-ing bantering bar-ba-rous barbarous bar-ren-ness barrenness bal-der-dash balderdash bat-tle-dore battledore but-te-ry buttery bat-tle-ment battlement beau-ti-ful beautiful be-fore-hand beforehand be-got-ten begotten be-gin-ning beginning be-gui-led beguiled be-ha-ved behaved be-hol-ding beholding be-ho-ved behoved be-lo-ved beloved be-moa-ned bemoaned be-ne-fits benefits be-tray-ed betrayed bet-ter-ed bettered be-way-led bewayled bit-ter-ly bitterly bit-ter-ness bitterness blab-ber-ing blabbering ble-mish-ed blemished bles-sed-ness blessedness blood-thir-sty bloodthirsty bom-ba-sted bombasted bo-di-ly bodily bor-row-ed borrowed boi-ste-rous boisterous bond-wo-man bondwoman bot-tom-less bottomless boun-ti-ful bountiful bra-ve-ry bravery bran-dish-ing brandishing bri-be-ry bribery brick-lay-er bricklayer bri-gan-tine brigantine broi-de-red broidered bro-ther-ly brotherly bru-tish-ly brutishly bur-den-some burdensome bu-ri-al burial bu-si-ly busily bu-si-ness business but-te-ry buttery C C Ca-ni-bal Canibal car-bun-cle carbuncle ca-sti-gate castigate ca-te-chism catechism car-pen-ter carpenter car-ri-age carriage car-nal-ly carnally ce-le-brate celebrate cer-tain-ly certainly cer-ti-fie certifie cham-ber-lain chamberlain cham-pi-on champion charge-a-ble chargeable cheer-ful-ly cheerfully chur-lish-ly churlishly ci-vil-ly civilly clou-di-ness cloudiness cla-mo-rous clamorous cle-men-cy clemency co-lo-ny colony com-li-ness comliness com-men-cing commencing com-mand-ing commanding com-mit-ted committed com-mon-ly commonly com-pel-led compelled con-fes-sing confessing con-fu-ted confuted con-gre-gate congregate con-ju-red conjured con-stan-cy constancy con-stant-ly constantly con-tra-ry contrary con-tro-ler controler con-ver-sant conversant con-vey-ance conveyance con-vin-ced convinced cop-per-smith coppersmith cor-mo-rant cormorant cor-rup-ted corrupted cost-li-ness costliness craf-ti-ly craftily craf-ti-ness craftiness cre-a-ted created cre-a-ting creating cre-di-ble credible cre-di-tor creditor cru-ci-fie crucifie cru-di-ty crudity cru-el-ly cruelly cru-ci-ate cruciate cu-cum-ber cucumber cus-to-med customed cus-to-mer customer cum-ber-land cumberland cu-ri-ous curious cum-be-rance cumberance cu-sto-dy custody cur-sed-ly cursedly cum-min-seed cumminseed D D Da-ma-ges Damages dam-na-ble damnable dan-ge-rous dangerous dar-ken-ed darkened de-cay-ed decayed de-cay-ing decaying de-cla-red declared de-di-cate dedicate de-fen-sive defensive de-for-med deformed de-faul-ter defaulter de-gra-ded degraded de-lay-ing delaying de-lu-ding deluding de-ter-mine determine de-trac-ting detracting di-ges-ted digested di-mi-nish diminish di-rec-ted directed dif-fe-rence difference dig-ni-ty dignity di-li-gence diligence di-sci-ple disciple dis-dain-ed disdained dis-dain-ful disdainful dis-fi-gure disfigure dis-gui-sed disguised dis-sem-ble dissemble dis-tur-bance disturbance dis-man-tle dismantle dis-char-ged discharged di-vi-ding dividing di-vor-ced divorced do-me-stick domestick doc-tri-nal doctrinal do-ci-ble docible doubt-ful-ly doubtfully drun-ken-ness drunkenness drow-si-ly drowsily dread-ful-ly dreadfully dul-ci-mer dulcimer du-ra-ble durable du-ti-ful dutiful dun-ge-on dungeon E E Ea-ger-ly Eagerly e-di-fie edifie e-mi-nent eminent ear-nest-ly earnestly e-lo-quent eloquent em-bol-den embolden em-broi-der embroider em-ploy-ment employment e-ne-my enemy en-mi-ty enmity en-gage-ment engagement en-gra-ver engraver en-ligh-ten enlighten en-tice-ment enticement en-ter-tain entertain en-tan-gle entangle en-ter-prize enterprize en-trap-ping entrapping en-vi-ous envious en-vi-ron environ en-sna-red ensnared e-pis-tle epistle e-qual-ly equally e-qui-ty equity e-qui-nox equinox e-sca-ped escaped e-stran-ged estranged e-sta-blish establish e-spou-sed espoused e-sti-mate estimate e-ter-nal eternal e-ver-more evermore e-ve-ry every e-ven-ly evenly e-ven-ing evening e-vi-dence evidence e-vi-dent evident e-vil-ly evilly ex-ces-sive excessive ex-tream-ly extreamly ex-al-ted exalted ex-pel led expelled ex-am-ple example ex-pec ted expected ex-cel-lent excellent ex-clu-ded excluded ex-cu sed excused ex-pen-ces expences ex-pi red expired ex-po-sed exposed ex-pres-sed expressed ex-ten-ded extended ex-tol-led extolled ex-treamly extreamly F F Fal-li-ble Fallible fa-mous-ly famously fa-cul-ty faculty fa-m●-ly family faith ful-ly faithfully fa-ther-ly fatherly fa vou-rite favourite fear-ful-ly fearfully fee-ble-ness feebleness fel-low-ship fellowship fer-vent-ly fervently fer-men-ted fermented fil-thi-ly filthily fir-ma-ment firmament flat-te-ry flattery fra-ter-nal fraternal fra-grant-ly fragrantly f●a-gi-let f●agilet fruit-ful-ness fruitfulness for sa-ken forsaken for-mer-ly formerly for-ci-ble forcible for-tu-nate fortunate for-get-ful forgetful fur-mi-ty furmitive fu-gi-tive fugitive fur-ni-ture furniture fu-ri-ous furious ful-fil-ling fulfilling fur ther-more furthermore ful-mi-nate fulminate G G Gad-ding-ly Gaddingly gal-le-ry gallery gain-say ing gainsaying gar-di-ner gardiner gal-lant ly gallantly gau-di ly gaudily gar ri son garrison ga sing stock gasingstock ge ne-ral general gen-tle-ness gentleness ger-ma-ny germany glo-ri-ous glorious gloo-mi ness gloominess glo ri-fie glorifie glut-ton-ous gluttonous god-li-ness godliness good-li-ness goodliness gor-ge-ous gorgeous go-vern-ment government go ver-nour governour glut-to-ny gluttony gra-ci-o●s gracious gras-hop-per grashopper gra-vi-ty gravity gree di-ness greediness guil-ti-ness guiltiness guilt-les-ly guiltlesly H H Hab-ber-dine Habberdine hal-low-ed hallowed har mo-ny harmony ha-sti-ly hastily hand-ker-chief handkerchief hate-ful-ly hatefully har-bin-ger harbinger hand mai-den handmaiden hap-pi-ness happiness haugh-ti-ly haughtily hand-som-ly handsomly ha-zar-dous hazardous hear-ti-ly heartily hea-ven-ly heavenly he-re-sie heresie her-mi-tage hermitage he-ri-tage heritage hea-vi-ness heaviness ho-li-ness holiness ho-nou-red honoured home-li-ness homeliness hor-ri-ble horrible hum ble-ness humbleness hus-band-ry husbandry hy-po-crite hypocrite I I I dle-ness Idleness ig-no-ble ignoble ig-no rant ignorant i-ma-ges images i-ma-gine imagine im-po tent impotent im-bra cing imbracing im-pu-ting imputing in-gen-der ingender in-cen sed incensed in-con-stant inconstant in-jus-tice injustice in-ju-ry injury in-fa-mous infamous in-fer-nal infernal in-te-stine intestine in-fi-nite infinite in-fla-ming inflaming in-for-ming informing in-he-rit inherit in no cent innocent in-stant ly instantly in struc ted instructed in-stru-ment instrument in-tan-gle intangle in-trea ted intreated in-ter-pret interpret in-tru-ding intruding in-va-ding invading in-vi-ted invited in-ward-ly inwardly jour-ney-ing journeying ju-bi-lee jubilee ju-sti-fie justifie joy-ful-ly joyfully i-vo-ry ivory K K Kind-nes-ses Kindnesses kins-wo-man kinswoman kna-vish-ly knavishly know-ing-ly knowingly L L La-bou-red Laboured la-men-ted lamented la-bou-rer labourer law-ful-ly lawfully lear-ned-ly learnedly le-pro-sie leprosie li-be-ral liberal li-ber-ty liberty li-bel-ling libelling lu-sti-ness lustiness le-che-ry lechery li-mi-ted limited lof-ti-ness loftiness low-li-ness lowliness loy-te-ring loytering lu-na-cy lunacy lu-na-tick lunatick M M Mag-da-len Magdalen mag-ni-fie magnifie ma-je-sty majesty ma-ni-fest manifest ma-ni-fold manifold mar-ri-age marriage ma-ter-nal maternal ma-ri-ner mariner ma-ster-less masterless ma-scu-line masculine me-di-cine medicine mer-chan-dize merchandize me-di-tate meditate me-mo-ry memory men-ti-on mention mi-ni-ster minister migh-ti ly mightily mys-te-ry mystery mis-for tune misfortune mi-ra-cle miracle mi-ti-gate mitigate mil-li ons millions mo-de-rate moderate mo-de-sty modesty mo-nu-ment monument mor-tal-ly mortally mor-ti fie mortifie mourn-ful-ly mournfully mol-li-fie mollifie mul-ti-tude multitude mul-ti ply multiply mu-si-cal musical mut-te-ring muttering mu-tu-al mutual N N Na-ked-ness Nakedness na-tu-ral natural na-ti-on nation na-vi-gate navigate naugh ti-ness naughtiness na-sti-ness nastiness neg-li-gent negligent neigh-bour-ly neighbourly ne-ther most nethermost nig-gard ly niggardly nim ble ness nimbleness no mi nal nominal nor-ther-ly northerly nu-me-ral numeral nu-me-rous numerous nur-se-ry nursery no-vel-ty novelty num-ber-ing numbering nu-tri-ment nutriment O O O-bey-ed Obeyed o-bey-ing obeying ob-ser-ved observed ob-ser-ver observer of-fen-sive offensive of-fe-ring offering ob-tai-ned obtained of-fen-ded offended ob-sti-nate obstinate o-pen-ly openly o-pe-rate operate of-ten-times oftentimes op-po sed opposed op-po-nent opponent op-pres-sor oppressor o-ra-cle oracle or-na-ment ornament or-dai-ned ordained or-der-ly orderly o-ver-come o●ercome o-ver-much overmuch o-ver-sight oversight o-ver-take overtake o-ver-ture overture o-ver-turn overturn or-di-nance ordinance out go-ing outgoing out-lan dish outlandish out-pas-sed outpassed out-ward-ly outwarly P P Pa-ge-ant Pageant pa-la-ces palaces pa-ter-nal paternal pa-ra-ble parable pa-ra-dice paradice pa-ra-mour paramour par-ta-ker partaker par-ti-al partial pas-sa-ges passages pas-sen-ger passenger pa-sto-ral pastoral pas-si-on passion pa-ti-ence patience pen-ti-on pention per-ti-nent pertinent pes-ter-ing pestering pe-sti-lence pestilence pe-ril-lous perillous per-for mance performance per-ma-nent permanent per-su-med persumed per-ju-ry perjury per-se-cute perse●ute per-mit-ted permitted per-swa-ding perswading pes-ti-lence pestilence phar-ma-cy pharmacy pi-e-ty piety pi-ti-ful pitiful plen-ti-ful plentiful plea-sant-ly pleasantly pro vi-dent provident pro-phe-sie prophesie pro-vi-dence providence pro-mul-gate promulgate pro-mi-sed promised pro-se-lite proselite pro-sti-tute prostitute pro-ven-der provender pub-lick-ly publickly pub-li-can publican pu-nish-ment punishment pur-cha-sed purchased pu-ri fie purifie pu-ri ty purity pur-po-sed purposed pur-su-ant pursuant Q Q Qua li-ty Quality quan-ti-ty quantity quar-ter-ly quarterly qua-king-ly quakingly qui-et-ness quietness qui-e-tude quietude que-sti-on question qui ve-ring quivering quar-rel some quarrelsome qua-ve-ring quavering R R Ram-ping-ly Rampingly ran-so-med ransomed ra-ve-nous ravenous ra-vi-shed ravished ra pi er rapier re-bel-led rebelled re-bu-ked rebuked re-cei-ving receiving re-cei-ver receiver re-com-mend recommend re-cor-der recorder re-co-ver recover re-con-cile reconcile re-for-med reformed re-for-mer reformer re-gi-on region re-gi-ster register re-gar-ded regarded re-hear-sal rehearsal re-gi-ment regiment re joi-ced rejoiced re-lea-sed released re-man-ded remanded re-mo-ved removed re mote-ly remotely re-mem-ber remember re-main-der remainder re-me-dy remedy re-mit-ted remitted re-pu-ted reputed re-por-ted reported re-pro-ved reproved re-pen-tance repentance re-ple-nish replenish re-pro-bate reprobate re-si-due residue re-sem-blance resemblance re-ser-ved reserved re-sol-ved resolved re-sto-red restored re-ti-red retired re-ven-ged revenged re-ve-nue revenue re-ver-sed reversed re-ve-rence reverence re-vi-ling reviling re-vi-ved revived re-vol-ted revolted ri-o-tous riotous ri-val-led rivalled ring-lea-der ringleader rot-ten-ness rottenness roy-al-ly royally roy-al-ty royalty ru-di-ments rudiments S S Sa-cra-ment Sacrament sa-cri-fice sacrifice sanc-ti-fie sanctifie sa-cri-ledge sacriledge sa-lu-ted saluted sa-tis-fie satisfie sa-tur-day saturday sa-vi-our saviour sa-vou-ry savoury se-du-lous sedulous sen-si-ble sensible sen-si-tive sensitive se-ni-or senior scor-pi-on scorpion se-du-ced seduced school-ma-ster schoolmaster se-na-tor senator sen-ten-ces sentences se-pa-rate separate se-pul-chre sepulchre ser-ge-ant sergeant ser-vi-tude servitude se-ven-ty seventy se-vere-ly severely se-ve-ral several shame-les-ly shamelesly sick-nes-ses sicknesses sin-cere-ly sincerely sin-gu-lar singular sin-cere-ness sincereness slan-de-rous slanderous scan-da-lous scandalous scar-ri-fie scarrifie slip-pe-ry slippery sloth-ful-ly slothfully slug-gish-ness sluggishness so-ber-ly soberly so-do-mite sodomite so-lemn-ly solemnly sor-ce-ry sorcery sooth-say-ing soothsaying spa-ring-ly sparingly spe-ci-fie specifie spe-ci-al special spee-di-ly speedily sor-row-ful sorrowful spite-ful-ly spitefully state-li-est stateliest sted-fast-ly stedfastly sto-ma-cher stomacher sto-mach-ful stomachful stub-born-ly stubbornly sub-du-ed subdued suc-cess-ful successful sub-mit-ted submitted sub-mis-sive submissive sub-or-ned suborned sub-scri-bed subscribed sub-sti-tute substitute sub-til-ly subtilly sub-til-ty subtilty suc-ces-sive successive sud-den-ly suddenly suf-fi-ceth sufficeth sul-len-ly sullenly sul-phu-rous sulphurous sump-tu-ous sumptuous sup-per-less supperless sup-pli-cate supplicate s●f-fo-cate snffocate sup-por-ted supported su-ste-nance sustenance sy-ca-more sycamore sy-na-gogue synagogue T T Tar-ri-er Tarrier ta-pi-stry tapistry task-ma-ster taskmaster tem-pe-rance temperance tem-pe-rate temperate tem-po-ral temporal ten-der-ly tenderly ter-ri-ble terrible ter-ri-fie terrifie te-sta-ment testament te-sti-fie testifie te-sta-tor testator thank-ful-ly thankfully to-ge-ther together tor-men-tor tormentor to-tal-ly totally thun-der-bolt thunderbolt tor-tu-red tortured trans-fer-red transferred trans-mi-grate transmigrate trans-gres-sed transgressed tran-spa-rent transparent tran-sla-ted translated tra-vel-ler traveller trea-che-rous treacherous trea-su-ry treasury tres-pas-ses trespasses tu-na-ble tunable ty-ran-ny tyranny tym-pa-ny tympany V V Va-ga-bond Vagabond va-li-ant valiant va-ni-ty vanity va-ri-ance variance ve-he-ment vehement ve-ne-mous venemous ve-ri-ty verity ver-tu-ous vertuous ve-ri-ly verily ve-ni-son venison vic-to-ry victory vice-ge-rent vicegerent vi-li-fie vilifie vic-tu-als victuals vi-gi-lant vigilant vi-go-rous vigorous vil-la-ny villany vo-lu-ble voluble vi-ne-gar vinegar vi-o-lent violent vi-o-late violate vir-ti-go virtigo vi-si-on vision vi-si-ble visible vi-si-ted visited un-be-lief unbelief un-a-ble unable un-der-stand understand un-e-qual unequal un-clean-ly uncleanly un-cer-tain uncertain un-come-ly uncomely un-co-ver uncover un-faith-ful unfaithful un-fruit-ful unfruitful un-god-ly ungodly un-ho-ly unholy u-ni-ted united un-just-ly unjustly un-sta-ble unstable un-law-ful unlawful un-mind-ful unmindful un-ru-ly unruly un-thank-ful unthankful un-time-ly untimely un-skil-ful unskilful un-seem-ly unseemly un-wor-thy unworthy vo-mi-ting vomiting up-right-ly uprightly u-sur-per usurper ut-ter-ly utterly W W Wal-low-ing Wallowing wan-ton-ness wantonness war-ri-our warriour wa-ter-brooks waterbrooks wa-ter-course watercourse wa-ter-flood waterflood wa-ter-house waterhouse wan-ton-ly wantonly wil-ling-ly willingly wil-ful-ly wilfully wea-ri-ness weariness wea-ri-some wearisome whis-per-ing whispering wic-ked-ness wickedness wil-der-ness wilderness wit-nes-ses witnesses wit-nes-sing witnessing won-der-ful wonderful wo-ful-ly wofully wor-ship-ping worshipping wor-ship-per worshipper wor-thi-ly worthily wil-ling-ly willingly wil-ling-ness willingness wrath-ful-ly wrathfully wrong-ful-ly wrongfully wret-ched-ness wretchedness wret-ched-ly wretchedly Y Y Yes-ter-day Yesterday yes-ter-night yesternight yoke-fel-low yokefellow youth-ful-ly youthfully youth-ful-ness youthfulness Z Z Zea-lous-ly Zealously Chap. 15. Containing Words of Four Syllables , both whole and divided , wherein the former Rules are to be observed . A A A-ban-do-ned Abandoned a-bi-li-ty ability a-bo-li-shed abolished a-bo-mi-nate abominate a-bun-dant-ly abundantly ac-cep-ta-ble acceptable ac-com-plish-ed accomplished ac-cor-ding-ly accordingly ab-so-lute-ly absolutely ac-cus-to-med accustomed ac-cep-ta-bly acceptably ac-ti-vi-ty activity a-da-man-tine adamantine ad-di-ti-on addition ad-mi-ni-stred administred a-dop-ti-on adoption ad-ver-sa-ry adversary ad-ven-tu-red adventured ad-ver-si-ty adversity a-dul-te-ry adultery af-fec-ti-on affection af-fi-ni-ty affinity af-flic-ti-on affliction ag-gre-ga-ted aggregated al-le-go-ry allegory a-li-e-nate alienate al-to-ge-ther altogether an-swe-ra-ble answerable a-mi-a-ble amiable an-ti-qui-ty antiquity an-ti-pa-thy antipathy ap-pa-rent-ly apparently ap-per-tain-ed appertained ap-pa-rel-led apparelled ap-pre-hen-ded apprehended ar-ro-gan-cy arrogancy ar-ro-gant-ly arrogantly ar-ti-fi-cer artificer ar-til-le-ry artillery as-su-red-ly assuredly as-sump-ti-on assumption a-sto-ni-shed astonished a-sto-nish-ment astonishment a-stro-lo-gy astrology a-stro-man-cy astromancy a-stro-no-my astronomy at-ten-tive-ly attentively as-si-du-ous assiduous au-tho-ri-ty authority B B Ba-by-lo-nish Babylonish bar-ba-ri-an barbarian bar-ri-ca-do barricado ba-sti-na-do bastinado be-a-ti-tude beatitude be-ne-vo-lence benevolence be-nig-ni-ty benignity boun-ti-ful-ly bountifully boun-ti-ful-ness bountifulness C C Ca-la-mi-ty Calamity cap-ti-vi-ty captivity ca-sti-ga-ted castigated ca-ter-pil-lar caterpillar ce-les-ti-al celestial ca-te-chi-sed catechised ca-tas-tro-phe catastrophe ce-le-bra-ted celebrated cen-tu-ri-on centurion ce-re-mo-nies ceremonies cer-ti-fi-cate certificate cha-ri-ta-ble charitable cir-cum-ci-sed circumcised col-lec-ti-on collection col-le-gi-ate collegiate col-la-te-ral collateral com-for-ta-bly comfortably com-mis-si-on commission com-mu-ni-on communion com-pa-ni-on companion com-mi-se-rate commiserate com-pas-si-on compassion con-ten-ted-ly contentedly con-cep-ti-on conception con-clu-si-on conclusion con-di-ti-on condition con-fes-si-on confession con-fe-de-rate confederate con-fu-si-on confusion con-fi-dent-ly confidently con-gra-tu-late congratulate con-gre-ga-ted congregated con-sump-ti-on consumption con-se-cra-ted consecrated con-si-de-rate considerate con-spi-ra-cy conspiracy con-temp-ti-ble contemptible con-ten-ti-on contention con-ti-nu-al continual con-ta-mi-nate contaminate con-tra-dic-ted contradicted con-tra-ri-ly contrarily con-tro-ver-sie controversie con-tu-ma-cy contumacy con-ve-ni-ent convenient con-ven-ti-on convention con-ver-si-on conversion cor-rec-ti-on correction cor-rup-ti-on corruption co-ve-nan-ted covenanted cru-di-li-ty crudility cur-te-ous-ly curteously cre-a-ti-on creation cre-du-lous-ly credulously cri-ti-cal-ly critically cu-ri-ous-ly curiously cru-ci-fi-ed crucified D D Dai-ry-wo-man Dairywoman dal-li-an-ces dalliances da-mage-a-ble damageable dam-na-ti-on damnation dam-ni-fi-ed damnified dan-de-li-on dandelion de-bi-li-tate debilitate de-ceit-ful-ness deceitfulness de-di-ca-ted dedicated de-bate-ful-ly debatefully de-bo-nair-ly debonairly de-ci-phe-ring deciphering de-clai-ming-ly declaimingly de-col-la-ted decollated de-coc-ti-on decoction de-duc-ti-on deduction de-fa-ti-gate defatigate de-fec-ti-on defection de-fen-so-ry defensory de-fi-ni-tive definitive de-for-med-ly deformedly de-for-mi-ty deformity de-ge-ne-rate degenerate de-gra-ding-ly degradingly de-jec-ted-ly dejectedly de-light-ful-ly delightfully de-li-ne-ate delineate de-li-ve-red delivered de-mi-ca-non demicanon de-mo-cra-cy democracy de-no-mi-nate denominate de-pen-den-cy dependency de-po-pu-late depopulate de-po-si-ted deposited de-pres-si-on depression de-ri-va-tive derivative de-ro-ga-ting derogating de-scrip-ti-on description de-spai-ring-ly despairingly de-spite-ful-ly despitefully de-ter-mi-nate determinate de-ter-mi-ned determined de-tri-men-tal detrimental de-vi-a-ting deviating de-vi-lish-ly devilishly dex-te-ri-ty dexterity di-a-me-ter diameter di-a-go-nal diagonal dif-fi-cul-ty difficulty dif-fi-cult-ly difficultly di-ge-sti-on digestion dis-com-fi-ture discomfiture dis-con-so-late disconsolate dis-cour-te-ous discourteous dis-ho-nou-red dishonoured dis-in-ga-ged disingaged dis-lo-ca-ted dislocated dis-tri-bu-ted distributed di-ver-si-ty diversity dog-ma-ti-cal dogmatical do-mi-ni-on dominion dor-mi-to-ry dormitory E E E-di-fi-ed Edified e-du-ca-ted educated ef-fec-tu-al effectual ef-fe-mi-nate effeminate ef-fi-ca-cy efficacy ef-fi-ci-ent efficient ef-fu-si-on effusion e-gre-gi-ous egregious e-gre-mo-ny egremony e-la-bo-rate elaborate e-le-cam-pane elecampane e-lec-tor-ship electorship e-le-gant-ly elegantly e-le-va-ted elevated e-li-za-beth elizabeth e-lo-quent-ly eloquently e-ma-nu-el emanuel em-broi-de-rer embroiderer e-mi-nent-ly eminently e-mu-la-ting emulating e-na-mou-red enamoured en-cou-ra-ging encouraging en-da-ma-ged endamaged e-ner-va-ted enervated en-ter-tai-ned entertained en-ve-no-med envenomed e-qua-li-zed equalized e-qui-ta-ble equitable e-qui-vo-cate equivocate e-ra-di-cate eradicate es-sen-ti-al essential e-sta-bli-shed established e-sti-ma-ted estimated e-ver-la-sting everlasting e-vi-den-ces evidences eu-ro-pe-an european e-va-cu-ate evacuate ex-a-mi-ned examined ex-a-spe-rate exasperate ex-em-pli-fie exemplifie ex-ces-sive-ly excessively ex-cee-ding-ly exceedingly ex-cel-lent-ly excellently ex-cu-sa-ble excusable ex-e-cu-ted executed ex-er-ci-sed exercised ex-hi-bi-ted exhibited ex-hi-le-rate exhilerate ex-or-bi-tant exorbitant ex-pe-ri-ence experience ex-pe-ri-ment experiment ex-po-stu-late expostulate ex-te-nu-ate extenuate ex-ter-mi-nate exterminate ex-ter-nal-ly externally ex-pul-si-on expulsion ex-tir-pa-ted extirpated F F Fa-bri-ca-ted Fabricated fac-ti-ous-ly factiously fal-la-ci-ous fallacious fa-mi-li-ar familiar fan-ta-sti-cal fantastical fa-tal-li-ty fatallity fa-ther-li-ness fatherliness fa-vou-ra-bly favourably fi-de-li-ty fidelity flat-ter-ing-ly flatteringly for-ma-li-ty formality for-ti-fi-ed fortified for-ni-ca-tor fornicator for-tu-nate-ly fortunately foun-da-ti-on foundation fra-gi-li-ty fragility fra-ter-ni-ty fraternity fu-mi-ga-ted fumigated fun-da-men-tal fundamental fu-ri-ous-ly furiously fu-mi-to-ry fumitory G G Ge-ne-ra-ted Generated gen-ti-li-ty gentility ge-o-gra-phy geography ge-o-me-try geometry gil-li-flow-er gilliflower glis-te-ring-ly glisteringly glo-ri-fi-ed glorified glo-ri-ous-ly gloriously gor-ge-ous-ly gorgeously gra-ci-ous-ly graciously gram-ma-ri-an grammarian gra-ti-fi-ed gratified H H Hal-le-lu-jah Hallelujah har-mo-ni-ous harmonious ha-zar-dous-ly hazardously he-re-ti-cal heretical hy-po-cri-sie hypocrisie his-to-ri-an historian ho-nou-ra-ble honourable hor-ri-ble-ness horribleness ho-spi-ta-ble hospitable ho-sti-li-ty hostility how-so-e-ver howsoever hu-ma-ni-ty humanity hu-mi-di-ty humidity hu-mi-li-ty humility hy-po-cri-sie hypocrisie hy-po-the-sis hypothesis I I Je-o-par-dy Jeopardy ig-no-mi-ny ignominy ig-no-rant-ly ignorantly il-le-gal-ly illegally il-li-te-rate illiterate il-lu-mi-nate illuminate i-mi-ta-ble imitable im-men-si-ty immensity im-mo-de-rate immoderate im-per-ti-nent impertinent im-pe-ri-al imperial im-pi-ous-ly impiously im-per-ti-nence impertinence im-pla-ca-ble implacable im-por-tu-nate importunate im-pos-si-ble impossible im-pri-son-ing imprisoning im-pru-dent-ly imprudently im-pu-ri-ty impurity in-ca-pa-ble incapable in-com-pas-sing incompassing in-cu-ra-ble incurable in-dea-vou-red indeavoured in-dem-ni-fie indemnifie in-dif-fe-rent indifferent in-dig-ni-ty indignity in-du-ra-ble indurable in-dus-tri-ous industrious in-fal-li-ble infallible in-fe-ri-our inferiour in-flic-ti-on infliction in-fi-nite-ly infinitely in-for-tu-nate infortunate in-ge-ni-ous ingenious in-glo-ri-ous inglorious in-gre-di-ent ingredient in-ha-bi-tant inhabitant in-hu-mane-ly inhumanely in-ju-ri-ous injurious in-no-cen-cy innocency in-qui-si-tor inquisitor in-sa-ti-ate insatiate in-scrip-ti-on inscription in-so-len-cy insolency in-spec-ti-on inspection in-struc-ti-on instruction in-ti-ma-ted intimated in-tri-ca-cy intricacy in-tro-du-ced introduced in-va-si-on invasion in-ven-ti-on invention in-ve-te-rate inveterate in-vin-ci-ble invincible in-vi-si-ble invisible jo-vi-al-ly jovially ir-ra-di-cate irradicate ir-re-gu-lar irregular ir-rup-ti-on irruption i-ta-li-an italian ju-di-ca-ture judicature ju-di-ci-al judicial jus-ti-fi-ed justified L L La-bo-ri-ous Laborious la-men-ta-ble lamentable la-men-ta-bly lamentably lan-gui-shed-ly languishedly la-sci-vi-ous lascivious la-va-to-ry lavatory lea-che-rous-ly leacherously le-gi-ti-mate legitimate le-thar-gi-cal lethargical le-vi-ti-cal levitical le-vi-a-than leviathan li-be-ral-ly liberally li-bi-di-nous libidinous li-cen-ti-ate licentiate li-co-rish-ness licorishness lo-qua-ci-ty loquacity lu-gu-bri-ous lugubrious lu-shi-ous-ly lushiously M M Ma-ce-ra-ted Macerated ma-gi-ci-an magician mag-ni-fi-ed magnified mag-ni-fi-cent magnificent mag-ni-fy-ing mag-ni-fy-ing ma-la-pert-ly malapertly ma-le-vo-lence malevolence ma-lig-nant-ly malignantly ma-le-fac-tor malefactor ma-ni-fest-ly manifestly ma-nu-fac-ture manufacture mar-chi-o-ness marchioness mar-ti-a-list martialist ma-tri-cu-late matriculate ma-tu-ri-ty maturity me-cha-ni-cal mechanical me-di-ci-nal medicinal me-di-a-tor mediator me-di-ta-ting meditating me-lan-cho-ly melancholy me-lo-di-ous melodious mer-ci-ful-ly mercifully me-ri-di-an meridian me-tho-di-cal methodical mi-li-ta-ry military mi-li-ti-a militia mi-no-ri-ty minority mi-ra-cu-lous miraculous mis-go-ver-ned misgoverned mis-pri-si-on misprision mi-ti-ga-ted mitigated mo-de-rate-ly moderately mol-li-fi-ed mollified mo-ra-li-zed moralized mor-ta-li-ty mortality mul-ti-pli-ed multiplied mun-di-fy-ing mundifying mu-ni-ti-on munition N N Nar-ra-ti-on Narration na-ti-vi-ty nativity na-tu-ral-ly naturally na-ti-o-nal national na-vi-ga-ble navigable ne-ces-sa-ry necessary ne-ces-si-ty necessity ne-cro-man-cy necromancy neg-li-gent-ly negligently ne-go-ti-ate negotiate ne-gro-man-cer negromancer ne-ver-the-less nevertheless nig-gard-li-ness niggardliness no-mi-na-ted nominated no-bi-li-ty nobility no-to-ri-ous notorious no-ti-o-nal notional nu-me-ra-ry numerary nu-me-ra-ble numerable nu-me-rous-ly numerously O O O-be-di-ence Obedience ob-jec-ti-on objection ob-la-ti-on oblation ob-li-qui-ty obliquity ob-scu-ri-ty obscurity ob-vi-ous-ly obviously oc-ca-si-on occasion o-di-ous-ly odiously of-fer-to-ry offertory of-fi-ci-ous officious ob-sti-nate-ly obstinately om-ni-po-tent omnipotent o-pe-ra-tor operator o-pi-ni-on opinion op-por-tune-ly opportunely op-pres-si-on oppression o-pu-lent-ly opulently o-ra-ti-on oration or-di-na-ry ordinary or-di-nan-ces ordinances or-tho-gra-phy orthography o-ver-char-ged overcharged o-ver-thwart-ly overthwartly o-ver-co-ming overcoming o-ver-flow-ing overflowing P P Pa-ci-fi-ed Pacified p●-la-ta-ble p●latable par-ti-cu-lar particular par-ti-ti-on partition pa ro-chi al parochial pa-ti-ent-ly patiently pas-si-o-nate passionate pa-tri-mo-ny patrimony pas-tu-ra-ble pasturable pa-ter-nal ly paternally pa-ti ent-ly patiently pa-the-ti cal pathetical pe ti-ti-on petition pe-cu li-ar peculiar per-cep ti-ble perceptible per-ti-nent-ly pertinently per-di-ti-on perdition per-so-nal ly personally pe-ni-tent-ly penitently per-fec-ti-on perfection pen-si-o ner pensioner per-mis-si-on permission pe-remp-to-ry peremptory per-pe-tu-al perpetual per-ni-ci-ous pernicious per-se-cu-ted persecuted per-spi-cu-ous perspicuous phy-si ci-an physician phi-lo-so-pher philosopher plen-ti-ful-ly plentifully pi-ti ful-ly pitifully plen-te-ous-ness plenteousness pol-lu-ti-on pollution po-pu-lar-ly popularly pos-ses-si-on possession po-ste-ri-ty posterity po-ste ri our posteriour pre-de-sti-nate predestinate pre-ju-di-cate prejudicate pre me-di-tate premeditate pre-ci-ous-ly preciously pre-sump-ti-on presumption pre-cau-ti-on precaution pre-va-ri-cate prevaricate pre-ven-ti-on prevention pro-fes-si-on profession pro-fi-ta-ble profitable pro-ces-si-on procession pro-mo-ti on promotion pro phe-cy-ing prophecying pro-pri-e-ty propriety pro-por-ti-on proportion pro spe-ri-ty prosperity pro-sti-tu-ting prostituting pro-tec ti-on protection pro-ver-bi-al proverbial pro-vi-si-on provision pu-is-sant-ly puissantly pu-ri-fi-ed purified pu-ni-sha-ble punishable punc-tu-al-ly punctually pur-ga-to-ry purgatory pu-ri-fy-ing purifying py-ra-mi dal pyramidal Q Q Qua-li-fi-ed Qualified quar-rel-som-ly quarrelsomly que-sti-o-ned questioned que-sti-on-less questionless quin-ti-li-an quintilian R R Ra-di-cal-ly Radically ra-pa-ci-ty rapacity ra-ri fi-ed rarified ra-sca-li-ty rascality re-bel-li-on rebellion rea so-na-ble reasonable re-demp-ti-on redemption ra ti-o-nal rational re-co-ver-ing recovering re-cre-a-ted recreated rec-ti-fi-ed rectified re-flec-ti-on reflection re-fri-ge-rate refrigerate re-la-ti-on relation re-len ting-ly relentingly re-li-gi-ous religious re-lin-quish-ed relinquished re-mem-ber-ing remembering re-mu-ne-rate remunerate re-mis-si on remission re-mit-ta-ble remittable re-no va-ted renovated re-pai-ra-ble repairable re-pen-ting-ly repentingly re-pro-ba-ting reprobating re-pro-va-ble reprovable re-pu di ate repudiate re-pug-nant-ly repugnantly re-sol ved-ly resolvedly re-so-lute-ly resolutely re-spec-tive-ly respectively re-spon si-ble responsible re-sto-ra tive restorative re-stric-ti-on restriction re-ti-red ly retiredly re ti-red-ness retiredness re-trac-ti on retraction re-tri-bu-ted retributed re-ver-be-rate reverberate re ve-rent-ly reverently re-ver-si-on reversion re u-ni on reunion re-vi-ling-ly revilingly re-vo-ca-ble revocable re-vul-si-on revulsion ri di-cu-lous ridiculous righ-te-ous-ly righteously ro-tun-di-ty rotundity ruf-fi-an-ly ruffianly ru-mi-na-ting ruminating ru-sti-cal-ly rustically ru-sti-ci-ty rusticity S S Sa-cer-do-tal Sacerdotal sa-cra-men-tal sacramental sa-cri-fi-ced sacrificed sa-ga-ci-ous sagacious sa-git-ta-ry sagittary sa-la-man-der salamander sal-va-ti-on salvation sanc-ti-fi-ed sanctified sanc-tu-a-ry sanctuary san-gui-na-ry sanguinary sa-ti-a-ted satiated sa-tis-fi-ed satisfied sa-ty-ri-cal satyrical se-cu-ri-ty security se-di-ti-on sedition scan-da-lous-ly scandalously sca-ri-fy-ing scarifying schis-ma-ti-cal schismatical scho-las-ti-cal scholastical scru-pu-lous-ly scrupulously se-du-li-ty sedulity se-mi-na-ry seminary sen-si-ble-ness sensibleness sen-ten-ti-ous sententious se-pa-ra-ted separated se-ra-phi-cal seraphical se-ri-ous-ly seriously se-ve-ri-ty severity ser-vice-a-ble serviceable se-ve-ral-ly severally shame-fa-ced-ly shamefacedly sig-ni-fy-ing signifying si-mi-li-tude similitude sim-pli-ci-ty simplicity sin-ce-ri-ty sincerity sin gu-lar-ly singularly slan-de-rous-ly slanderously si-tu-a-ted situated so-bri-e ty sobriety so-lem-ni ty solemnity so-li-ci-ted solicited so-li-ta-ry solitary suf-fi-ci-ent sufficient sump-tu-ous-ly sumptuously T T To-bac-co-pipe Tobaccopipe ta-ber-na-cle tabernacle tar-ta ri an tartarian tau-to-lo-gy tautology tem-pe-rate-ly temperately tem-pe-stu-ous tempestuous te-me-ri-ty temerity tem-po-ri-ser temporiser temp-ta-ti-on temptation ter-mi-na-ted terminated ter-re-stri al terrestrial ter-ri-fi-ed terrified tes-ti-fi-ed testified tes-ti-mo-ny testimony the-o-lo-gy theology ti-me-rous-ly timerously to-le-ra-ble tolerable to-le-ra-bly tolerably to-ta-li-ty totality to-ward-li-ness towardliness trac-ta-ble-ness tractableness tra-di-ti on tradition tran-qui-li-ty tranquility trans-ac-ti-on transaction tran scrip-tion transcription trans-fi-gu-red transfigured trans-gres-si-on transgression trans-la-ti-on translation tri bu-ta-ry tributary tri-um-phant-ly triumphantly trou-ble-som-ly troublesomly tu-mul-tu-ous tumultuous tu-te-la-ry tutelary ty-ran-nous-ly tyrannously V V Va-ca-ti-on Vacation va-cu-i-ty vacuity vain-glo-ri-ous vainglorious va-li-ant-ly valiantly va-lu-a-ble valuable va-ri-a-ble variable va-ri e-ty variety ve-ge-ta-ble vegetable ve-he-ment-ly vehemently ve-ne-ra-ble venerable ve-ne-re-ous venereous ve-ne-mous-ly venemously ven-tu-rous-ly venturously ver-tu-ous-ly vertuously vi-ci-ni-ty vicinity vic-to-ri-ous victorious vi-gi-lan-cy vigilancy vi-o-la-ted violated vi-o-lent-ly violently vi-go-rous-ly vigorously vin-di-ca-ted vindicated vir-gi-ni-ty virginity vi-ti ous-ly vitiously un-ac-cu-stom unaccustom u-na-ni-mous unanimous un cer-tain-ty uncertainty un-ces-sant-ly uncessantly un bu-ri-ed unburied un-ca-pa-ble uncapable un-com-li-ness uncomliness un-con-dem-ned uncondemned un-con-su-med unconsumed un-cor-rec-ted uncorrected un-cor-rup-ted uncorrupted un-de-fi-led undefiled un-co-ve-red uncovered un-der-mi-ned undermined un-der-ta-ken undertaken un-de-ser-ved undeserved un-di-ge-sted undigested un-der-ta-king undertaking un-de-cei-ved undeceived un-de-cei-ving undeceiving un-e-qual-ly unequally un-faith-ful-ly unfaithfully un-fruit-ful-ly unfruitfully un-feign-ed-ly unfeignedly un-god-li-ness ungodliness un-go-ver-ned ungoverned un-lea-ve-ned unleavened un-pu-ni-shed unpunished un-wit-ting-ly unwittingly vo-ca-ti-on vocation vo-lun-ta-ry voluntary vo-lup-tu ous voluptuous W W Wal-low-ing-ly Wallowingly wa-te-rish-ness waterishness wa-ter-cour-ses watercourses wea-ri som-ness wearisomness well-be-lo-ved wellbeloved what-so-e-ver whatsoever where-so-e-ver wheresoever white-li-ve-red whitelivered whom-so-e-ver whomsoever who-so-e-ver whosoever won-der-ful-ly wonderfully won-de-rous-ly wonderously wor-ship-ful ly worshipfully Chap. 16. Containing Words of Five Syllables , both whole and divided , wherein the foregoing Rules are principally to be observed . A A Ab-bre-vi-a-ted Abbreviated abo-mi-na-ble abominable a-bro ga-ti-on abrogation ab so lu-ti-on absolution a-ca-de-mi-an academian ac-ci-den-tal-ly accidentally ac-co-mo-da-ted accomodated ac-com-pa-ni-ed accompanied ac-cu-mu-la-ted accumulated ac-cu-sa-ti-on accusation ac-cu-sto-med accustomed ad-mi-ni stra-tor administrator af fa-bi-li-ty affability af fir-ma-ti-on affirmation ag-gra-va-ti-on aggravation a-li-e-na ted alienated al-le-go-ri cal allegorical al-ter-na ti-on alternation am bi-gu-ous ly ambiguously am-pu-ta-ti-on amputation am-mu-ni-ti-on ammunition a-ni-mo-si-ty animosity an-ni-hi-la-ted annihilated a-po-sto-li-cal apostolical ap-pre-hen-sive-ly apprehensively ar-bi-tra-ti-on arbitration a-rith-me-ti-cal arithmetical as-sas-si-nate assassinate aug-men-ta-ti-on augmentation B B Bac-cha-na-li-an Bacchanalian bar-ri-ca-do-ed barricadoed be-a-ti-fi-cal beatifical be-ne-dic-ti-on benediction be-ne-fi-ci-al beneficial blas-phe-ma-to-ry blasphematory brag-ga-do-chi-o braggadochio bre-vi-a-ti-on breviation C C Ca-ba-li-sti-cal Cabalistical cal-ci-na-ti-on calcination ca-lum-ni-a-ted calumniated ca-no-ni-cal-ly canonically ca-pi-tu-la-ting capitulating ca-pri-ci-ous-ly capriciously ca-sti-ga-ti-on castigation ca-te-go-ri-cal categorical ce-le-bra-ti-on celebration ce-re-mo-ni-al ceremonial cha-rac-te-ri-zed characterized cho-ro-gra-phi-cal chorographical chri-sti-a-ni-ty christianity chro-no-lo-gi-cal-ly chronologically cir-cum-stan-ti-al circumstantial co-a-gu-la-ted coagulated co-es-sen-ti-al coessential co-gi-ta-ti-on cogitation com-bi-na-ti-on combination com-me-mo-ra-ble commemorable com-men-da-ti-on commendation com-mi-sera-ted commiserated com-mo-di-ous-ly commodiously com-pas-si-o-nate compassionate com-pre-hen-si-ble comprehensible con-fe-de-ra-cy confederacy con-fir-ma-ti-on confirmation con-se-cra-ti-on consecration con-sum-ma-ti-on consummation con-tra-dic-ti-on contradiction con-tu-ma-ci-ous contumacious con-ver-sa-ti-on conversation co-ro-na-ti-on coronation cou-ra-gi-ous-ly couragiously . D D De-bi-li-ta-ted debilitated de-cla-ma-ti-on declamation de-cla-ra-ti-on declaration de-fa-ti-ga-ting defatigating de-ge-ne-ra-ted degenerated de-li-be-rate-ly deliberately de-li-ci-ous-ly deliciously de-mon-stra-ti-on demonstration de-no-mi-na-ted denominated de-po-pu-la-ting depopulating de-ro-ga-to-ry derogatory de-so-la-ti-on desolation de-ter-mi-nate-ly determinately di-a-bo-li-cal diabolical dic-ti-o-na-ry dictionary di-la-ce-ra-ting dilacerating di-mi-nu-ti-on diminution dis-ad-van-ta-gi-ous disadvantagious dis-com-mo-di-ty discommodity dis-ho-nou-ra-ble dishonourable dis-lo-ca-ti-on dislocation dis-pen-sa-ti-on dispensation dis-pro-por-ti-on disproportion di-vi-na-ti-on divination do-me-sti-cal-ly domestically E E E-bu-li-ti-on Ebulition e-du-ca-ti-on education ef-fec-tu-al-ly effectually ef-fe-mi-na-cy effeminacy e-gre-gi-ous-ly egregiously e-lec-tu-a-ry electuary e-le-men-ta-ry elementary e-le-va-ti-on elevation en-da-mage-a-ble endamageable e-nig-ma-ti-cal enigmatical en-ter-change-a-ble enterchangeable e-pi-de-mi-cal epidemical e-qui-la-te-ral equilateral e-qui-noc-ti-al equinoctial e-qui-vo-ca-ting equivocating e-ra-di-ca-ted eradicated er-ro-ne-ous-ly erroneously e-sti-ma-ti-on estimation e-thi-o-pi-an ethiopian e-ver-la-sting-ly everlastingly ex-com-mu-ni-cate excommunicate ex-hor-ta-ti-on exhortation ex-po-si-ti-on exposition ex-tra-va-gan-cy extravagancy F F Fa-bri-ca-ti-on Fabrication fa-ce-ti-ous-ly facetiously fa-ci-li-ta-ted facilitated fal-la-ci-ous-ly fallaciously fa-mi-li-ar-ly familiarly fan-ta-sti-cal-ly fantastically fel-lo-ni-ous-ly felloniously fi-gu-ra-tive-ly figuratively fer-men-ta-ti-on fermentation for-ni-ca-ti-on fornication ful-mi-na-to-ry fulminatory fu-mi-ga-ti-on fumigation fun-da-men-tal-ly fundamentally G G Ge-ne-a-lo-gy Genealogy ge-ne-ra-li-ty generality ge-ne-ra-ti-on generation ge-ne-ro-si-ty generosity ge-o-gra-phi-cal geographical ge-o-me-tri-cal geometrical gram-ma-ti-cal-ly grammatically gra-tu-la-ti-on gratulation H H Ha-bi-ta-ti-on Habitation ha-bi-tu-al-ly habitually har-mo-ni-ous-ly harmoniously he-re-di-ta-ry hereditary hi-e-ro-gly-phicks hieroglyphicks his-to-ri-cal-ly historically ho-mo-ge-ne-al homogeneal ho-mo-ge-ne-ous homogeneous ho-spi-ta-li-ty hospitality hu-mec-ta-ti-on humectation hy-dro-gra-phi-cal hydrographical hy-po-chon-dri-ack hypochondriack hy-po-cri-ti-cal hypocritical hy-po-the-ti-cal hypothetical I I Ig-no-mi-ni-ous Ignominious il-le-ga-li-ty illegality il-le-gi-ti-mate illegitimate il-lu-mi-na-ted illuminated il-lu-stra-ti-on illustration im-me-di-ate-ly immediately im-mo-de-rate-ly immoderately im-mor-ta-li-ty immortality im-par-ti-al-ly impartially im-pe-ni-tra-ble impenitrable im-pe-ri-ous-ly imperiously im-per-ti-nent-ly impertinently im-pe-tu-o-si-ty impetuosity im-plan-ta-ti-on implantation im-por-tu-nate-ly importunately im-por-tu-ni-ty importunity im-po-si-ti-on imposition im-po-stu-ma-ted impostumated im-po-ve-rish-ment impoverishment im-pre-ca-ti-on imprecation im-pro-vi-dent-ly improvidently im-pu-ta-ti-on imputation in-ad-ver-ten-cy inadvertency in-ca-pa-ci-tate incapacitate in-car-na-ti-on incarnation in-com-pa-ra-ble incomparable in-con-gru-i-ty incongruity in-con-si-de-rate inconsiderate in-con-ti-nent-ly incontinently in-cor-rup-ti-ble incorruptible in-cre-di-ble-ness incredibleness in-cre-du-li-ty incredulity in-de-cli-na-ble indeclinable in-de-fi-nite-ly indefinitely in-dem-ni-fi-ed indemnified in-dig-na-ti-on indignation in-di-vi-du-al individual in-du-stri-ous-ly industriously in-e-sti-ma-ble inestimable in-ex-pli-ca-ble inexplicable in-flam-ma-ti-on inflammation in-ge-nu-i-ty ingenuity in-ha-bi-ta-ble inhabitable in-hu-ma-ni-ty inhumanity in-ju-ri-ous-ly injuriously in-na-vi-ga-ble innavigable in-nu-me-ra-ble innumerable in-spi-ra-ti-on inspiration in-sti-ga-ti-on instigation in-sti-tu-ti-on iustitution in-suf-fi-ci-ent insufficient in-sur-rec-ti-on insurrection in-tel-lec-tu-al intellectual in-tem-pe-rate-ly intemperately in-ter-ces-si-on intercession in-ter-ro-ga-ted interrogated in-to-le-ra-bly intolerably in-tro-duc-ti-on introduction in-vi-o-la-ble inviolable in-vi-ta-ti-on invitation i-ro-ni-cal-ly ironically ir-re-gu-lar-ly irregularly ir-re-ve-rent-ly irreverently ir-re-vo-ca-ble irrevocable ju-di-ci-al-ly judicially L L Las-ci-vi-ous-ly Lasciviously la-men-ta-ti-on lamentation le-gi-ti-mate-ly legitimately li-be-ra-li-ty liberality li-cen-ti-ous-ly licentiously lu-cu-bra-ti-on lucubration lux-u-ri-ous-ly luxuriously M M Ma-ce-ra-ti-on Maceration ma-chi-na-ti-on machination mag-na-ni-mi-ty magnanimity mag-ni-fi-cent-ly magnificently ma-je-sti-cal-ly majestically ma-le-dic-ti-on malediction ma-li-ci-ous-ly maliciously ma-nu-duc-ti-on manuduction mar-ri-age-a-ble marriageable ma-the-ma-ti-cal mathematical ma-tri-mo-ni-al matrimonial me-cha-ni-cal-ly mechanically me-di-ci-na-ble medicinable me-di-ta-ti-on meditation me-lo-di-ous-ly melodiously me-ri-di-o-nal meridional me-ri-to-ri-ous meritorious me-tho-di-cal-ly methodically mi-ni-stra-ti-on ministration mi-ra-cu-lous-ly miraculously mi-ti-ga-ti-on mitigation mo-de-ra-ti-on moderation mol-li-fi-a-ble mollifiable mul-ti-pli-ci-ty multiplicity mun-di-fi-ca-tive mundificative my-ste-ri-ous-ly mysteriously N N Na-tu-ra-li-zed Naturalized na-vi-ga-ti-on navigation ne-ces-sa-ri-ly necessarily ne-ces-si-ta-ted necessitated ne-fa-ri-ous-ly nefariously no-mi-na-ti-on nomination no-to-ri-ous-ly notoriously nun-cu-pa-to-ry nuncupatory O O Ob-du-ra-ti-on Obduration o-be-di-ent-ly obediently ob-jur-ga-ti-on objurgation ob-li-ga-ti-on obligation ob-li-te-ra-ted obliterated oc-ca-si-o-nal occasional oc-cu-pa-ti-on occupation o-do-ri-fe-rous odoriferous of-fi-ci-ous-ly officiously om-ni-po-ten-cy omnipotency o-pe-ra-ti-on operation op-por-tu-ni-ty opportunity op-po-si-ti-on opposition op-pug-na-ti-on oppugnation or-bi-cu-lar-ly orbicularly or-di-na-ti-on ordination or-di-na-ri-ly ordinarily o-ri-gi-nal-ly originally o-ver-sha-dow-ed overshadowed out-ra-gi-ous-ly outragiously P P Pal-pi-ta-ti-on Palpitation par-ci-mo-ni-ous parcimonious par-ti-a-li-ty partiality par-ti-cu-lar-ly particularly pas-si-o-nate-ly passionately pa-the-ti-cal-ly pathetically pe-cu-li-ar-ly peculiarly pe-cu-ni-a-ry pecuniary pe-remp-to-ri-ly peremptorily per-fi-di-ous-ly perfidiously per-fo-ra-ti-on perforation per-mu-ta-ti-on permutation per-ni-ci-ous-ly perniciously per-pen-di-cu-lar perpendicular per-pe-tu-al-ly perpetually per-se-cu-ti-on persecution per-spi-cu-i-ty perspicuity per-tur-ba-ti-on perturbation phan-ta-sti-cal-ly phantastically phy-si-og-no-my physiognomy pla-ca-bi-li-ty placability po-e-ti-cal-ly poetically pon-ti-fi-ci-al pontificial po-pu-la-ri-ty popularity po-stu-la-ti-on postulation pre-de-sti-na-ted predestinated pre-ju-di-ci-al prejudicial pre-me-di-ta-ting premeditating pre-pa-ra-ti-on preparation pre-sen-ta-ti-on presentation pre-ser-va-ti-on preservation pre-sump-tu-ous-ly presumptuously pro-cla-ma-ti-on proclamation pro-cre-a-ti-on procreation pro-cu-ra-ti-on procuration pro-di-ga-li-ty prodigality pro-di-gi-ous-ly prodigiously pro-mis-cu-ous-ly promiscuously pro-mul-ga-ti-on promulgation pro-pa-ga-ti-on propagation pro-pha-na-ti-on prophanation pro-por-ti-on-ed proportioned pro-po-si-ti-on proposition pub-li-ca-ti-on publication pu-tri-fac-ti-on putrifaction Q Q Qua-dran-gu-lar-ly Quadrangularly qua-dri-par-tite-ly quadripartitely R R Ra-di-a-ti-on Radiation ra-pa-ci-ous-ly rapaciously re-bel-li-ous-ly rebelliously re-can-ta-ti-on recantation re-ci-pro-cal-ly reciprocally re-com-men-da-ble recommendable re-cre-a-ti-on recreation re-e-sta-bli-shed reestablished re-for-ma-ti-on reformation re-fu-ta-ti-on refutation re-ge-ne-ra-ted regenerated re-li-ga-ti-on religation re-li-gi-ous-ly religiously re-no-va-ti-on renovation re-pa-ra-ti-on reparation re-pe-ti-ti-on repetition re-po-si-to-ry repository re-pro-ba-ti-on reprobation re-pu-ta-ti-on reputation re-qui-si-ti-on requisition re-ser-va-ti-on reservation re-so-lu-ti-on resolution re-spi-ra-ti-on respiration re-sti-tu-ti-on restitution re-sur-rec-ti-on resurrection re-tri-bu-ti-on retribution re-vo-lu-ti-on revolution rhe-to-ri-ci-an rhetorician ri-di-cu-lously ridiculously S S Sa-cra-men tal-ly Sacramentally sa-cri-le-gi-ous sacrilegious sa-lu-ta-ti-on salutation sa-tis-fac-ti-on satisfaction sa-ty-ri-cal-ly satyrically scho-la-sti-cal-ly scholastically sea-so-na-ble-ness seasonableness se-con-da-ri-ly secondarily se-di-ti-ous-ly seditiously sen-si-bi-li-ty sensibility se-pa-ra-ti-on separation sig-ni-fi-cant-ly significantly si-tu-a-ti-on situation spe-cu-la-ti-on speculation spi-ri-tu-al ly spiritually se-di-ti-ous-ly seditiously sub-stan-ti-al-ly substantially sub-sti-tu-ti-on substitution suf-fo-ca-ti-on suffocation suf-fi-ci-ent-ly sufficiently su-per-scrip-ti-on superscription su-per-sti-ti-on superstition sup-pli-ca-ti on supplication T T Ta-ci-tur-ni-ty Taciturnity tem-pe-stu-ous-ly tempestuously the-o-lo-gi-cal theological ti-til-la-ti-on titillation to-le-ra-ti-on toleration trans-for-ma-ti-on transformation trans-mi-gra-ti-on transmigration trans-mu-ta-ti-on transmutation trans-pi-ra-ti-on transpiration trans-plan-ta-ti-on transplantation trans-por-ta-ti-on transportation tre-pi-da-tion trepidation tri-bu-la-ti-on tribulation V V Va-cil-la-ti-on Vacillation va le-dic-ti-on valediction va-lu-a-ti-on valuation va-ri-a-ti on variation ve-gi-ta-ti-on vegitation ve-ne-ra-ti-on veneration vic-to-ri-ous ly victoriously vin-di-ca-ti on vindication vi-o-la-ti-on violation u-na-ni-mi-ty unanimity un cha-ri-ta-ble uncharitable un-cir-cum-ci-sed uncircumcised un-cir-cum-spect-ly uncircumspectly un-com-for-ta-ble uncomfortable un-com-mo-di-ous uncommodious un-com-poun-ded-ness uncompoundedness un-con-cei-va-ble unconceivable un con-que-ra-ble unconquerable un-con-se-quent-ly unconsequently un-con-ve-ni-ent unconvenient un-cor-po-re-al uncorporeal u-ni-for-mi-ty uniformity u-ni-ver-sal-ly universally u-ni-ver-si-ty university un-man-ner-li-ness unmannerliness un-mea-su-ra-ble unmeasurable un-mer-ci-ful-ly unmercifully un-na-tu-ral-ly unnaturally un-ne ces-sa-ry unnecessary un-pas-si-o nate unpassionate un-per-cei-va-ble unperceivable un-pow-er-ful ly unpowerfully un-pro-fi-ta-ble unprofitable un-pro-spe-rous-ly unprosperously un-rea-so-na-ble unreasonable un-re-com-pen-sed unrecompensed un-re-mit-ta-ble unremittable un-righ-te-ous-ly unrighteously un-re-tur-na-ble unreturnable un-sa-ti-a-ble unsatiable un suf-fe-ra-ble unsufferable un-sup-por-ta-ble unsupportable un-tem-pe-rate-ly untemperately un-trac-ta-ble-ness untractableness un-va-ri-a-ble unvariable un-wea-ri-a ble unweariable vo-lun-ta-ri-ly voluntarily vo-lup-tu-ous-ly voluptuously u-sur-pa-ti-on usurpation W W What-man-so-e-ver Whatmansoever whi-ther-so-e-ver whithersoever Chap. 17. Containing Words of Six , Seven and Eight Syllable● both whole and divided , wherein the foregoing Rules are principally to be observed . A A A-bo-mi-na-ti-on Abomination ac-ce-le-ra-ti-on acceleration ac-com-mo da-ti-on accomodation ac-cu-mu-la-ti on accumulation ac-cu-sto-ma-rily accustomarily ad-mi-ni-stra-ti-on administration ad-van-ta-gi-ous-ly advantagiously a-li-e-na-ti on alienation al-le-go-ri-cal-ly allegorically am-pli-fi-ca-ti-on amplification a-na-the-ma-ti-zed anathematized a-ni-mad-ver-si-on animadversion an-ni-hi-la-ti-on annihilation an-nun-ci-a-ti-on annunciation an-ni-ver-sa-ri-ly anniversarily a-po-sto-li-cal-ly apostolically a-rith-me-ti-cal-ly arithmetically a rith-me-ti-ci-an arithmetician a-stro-no-mi-cal-ly astronomically as-sas-si-na-ti-on assa ssination as-so-ci-a-ti-on association B B Be-a-ti-fi-cal-ly Beatifically be-ne-fi-ci-a-ry beneficiary be-ne-fi ci-al-ly beneficially C C Ca-no-ni-za-ti-on Canonization ca-pi-tu-la-ti-on capitulation ca-te-go-ri-cal-ly categorically cau-te-ri-za-ti-on cauterization ce-re-mo-ni-ous-ly ceremoniously cer-ti-fi-ca-ti-on certification cir-cum-lo-cu-ti-on circumlocution cir-cum-stan-ti-al-ly circumstantially cir-cum-vo-lu-ti-on circumvolution co-a-gu-la-ti-on coagulation co-es-sen-ti-al-ly coessentially com-me-mo-ra-ti-on commemoration co-mi se-ra-ti-on comiseration com-mu-ni-ca-ti-on communication com-pas-si-o-nate-ly compassionately com-pa-ti-bi-li-ty compatibility con-ca-ti-na-ti-on concatination con-fe-de-ra-ti on confederation con-gra-tu-la-ti-on congratulation con-si-de-ra-ti-on consideration con so-li-da-ti-on consolidation con-ta-mi-na-ti-on contamination con-tu-ma-ci-ous-ly contumaciously cor-ro-be-ra ti-on corroberation D D De-fa-ti-ga-ti-on Defatigation de-bi li ta-ti on debilitation de ge-ne-ra-ti-on degeneration de-li-be-ra-ti-on deliberation de-no-mi-na-ti-on denomination de po-pu la-ti-on depopulation de-ter-mi-na-ti-on determination di-a-bo-li-cal-ly diabolically di la-ce-ra-ti-on dilaceration dis-ad van ta-gi ous-ly disadvantagiously dis-com-men-da-ti-on discommendation dis-con-ti-nu-a ti-on discontinuation dis-in-ge ni-ous-ly disingeniously dis-pro-por-ti-o-ned disproportioned dis si-mu-la-ti-on dissimulation E E Ec-cle-si-a-sti-cal Ecclesiastical ec cle-si-a-sti-cal-ly ecclesiastically e di-fi-ca-ti-on edification e-le-e-mo sy-na-ry eleemosynary e-nig-ma-ti-cal-ly enigmatically e-nu-cle-a-ti-on enucleation e-nu-me-ra-ti-on enumeration e qui-vo-ca-ti-on equivocation e-ra-di-ca-ti-on eradication e-va-cu-a-tion evacuation e-va-po-ra-ti-on evaporation ex-a-mi-na-ti-on examination ex-com-mu-ni-ca-ti-on excommunication ex-tra-or-di-na-ry extraordinary ex-tra-or-di-na-ri-ly extraordinarily ex-o-ne-ra-ti-on exoneration ex-ter-mi-na-ti-on extermination F F Fa-mi-li-a-ri-ty Familiarity for-ti-fi-ca-ti-on fortification fruc-ti-fi-ca-ti-on fructification G G Gra-ti-fi-ca-ti-on Gratification ge-ne-ra-lis-si-mo generalissimo ge-o-gra-phi-cal-ly geographically ge-o-me-tri-cal-ly geometrically glo-ri-fi-ca-ti-on glorification gra-ti-fi-ca-ti-on gratification H H Hy-po-cri-ti-cal-ly Hypocritically hu-mi-li-a-ti-on humiliation hy-po-chon-dri-a-cal hypochondriacal hy-po-chon-dri-a-cal-ly hypochondriacally hy-po-cri-ti-cal-ly hypocritically I I Ig-no-mi-ni-ous-ly Ignominiously il-le-gi-ti-mate-ly illegitimately il-le-gi-ti-ma-ti-on illegitimation il-lu-mi-na-ti-on illumination im-mo-de-ra-ti-on immoderation im-par-ti-a-li-ty impartiality im-pos-si-bi-li-ty impossibility im-pro-ba-bi-li-ty improbability im-pro-pri-a-ti-on impropriation in-au-gu-ra-ti-on inauguration in-com-men-su-ra-ble incommensurable in-com-men-su-ra-bi-li-ty incommensurability in-com-pas-si-o-nate incompassionate in-com-pas-si-o-nate-ly incompassionately in-com-mo-di-ous-ly incommodiously in-com-mu-ni-ca-ble incommunicable in-com-mu-ni-ca-bi-li-ty incommunicability in-com-pre-hen-si-bi-li-ty incomprehensibility in-com-pre-hen-si-ble incomprehensible in-com-pre-hen-si-ble-ness incomprehensibleness in-com-pa-ti-bi-li-ty incompatibility in-con-si-de-rate-ly inconsiderately in-con-si-de-rate-ness inconsiderateness in-con-ve-ni-ent-ly inconveniently in-con-ve-ni-en-cy inconveniency in-cor-rup-ti-bi-li-ty incorruptibility in-de-fa-ti-ga-ble indefatigable in-de-ter-mi-nate-ly indeterminately in-dis-po-si-ti-on indisposition in-di-vi-du-al-ly individually in-fal-li-bi-li-ty infallibility in-sa-ti-a-ble-ness insatiableness in-suf-fi-ci-en-cy insufficiency in-suf-fi-ci-ent-ly insufficiently in-ter-change-a-ble-ness interchangeableness in-ter-lo-cu-to-ry interlocutory in-ter-me-di-ate-ly intermediately in-ter-pel-la-ti-on interpellation in-ter-po-si-ti-on interposition in-ter-pre-ta-ti-on interpretation in-ter-ro-ga-to-ry interrogatory in-vi-o-la-ble-ness inviolableness ir-ra-di-a-ti-on irradiation ir-re-con-cile-a-ble irreconcileable ir-re-con-ci-li-a-ti-on irreconciliation ir-re-gu-la-ri-ty irregularity ir-re-pre-hen-si-ble irreprehensible ju-di-ci-a-ri-ly judiciarily L L Le-gi-ti-ma-ti-on Legitimation M M Ma-the-ma-ti-cal-ly Mathematically ma-the-ma-ti-ci-an mathematician me-di-ter-ra-ne-an mediterranean me-ri-di-o-nal-ly meridionally me-ta-pho-ri-cal-ly metaphorically mis-in-ter-pre-ta-ti-on misinterpretation mo-di-fi-ca-ti-on modification mol-li-fi-ca-ti-on mollification mor-ti-fi-ca-ti-on mortification mul-ti-pli-ca-ti-on multiplication mun-di-fi-ca-ti-on mundification N N Na-tu-ra-li-za-ti-on Naturalization ne-go-ti-a-ti-on negotiation no-ti-fi-ca-ti-on notification O O Ob-li-te-ra-ti-on Obliteration oc-ca-si-o-nal-ly occasionally o-pi-ni-o-na-tive-ly opinionatively P P Pa-ci-fi-ca-ti-on Pacification par-ci-mo-ni-ous-ly parcimoniously per-am-bu-la-ti-on perambulation pe-re-gri-na-ti-on peregrination per-pen-di-cu-lar-ly perpendicularly per-pe-tu-a-ti-on perpetuation phi-lo-so-phi-cal-ly philosophically pre-de-sti-na-ti-on predestination pre-ju-di-ca-ti-on prejudication pre-me-di-ta-ti-on premeditation pre-oc-cu-pa-ti-on preoccupation pro-ble-ma-ti-cal-ly problematically pro-cra-sti-na-ti-on procrastination prog-no-sti-ca-ti-on prognostication pro-nun-ci-a-ti-on pronunciation pro-pi-ti-a-ti-on propitiation pro-por-ti-o-na-bly proportionably pu-ri-fi-ca-ti-on purification pu-tri-fi-ca-ti-on putrification Q Q Qua-dru-pli-ca-ti-on Quadruplication qua-li-fi-ca-ti-on qualification R R Ra-ti-fi-ca-ti-on Ratification ra-ti-o-ci-na-ti-on ratiocination re-ca-pi-tu-la-ti-on recapitulation re-ci-pro-ca-ti-on reciprocation re-com-men-da-ti-on recommendation re-con-ci-li-a-ti-on reconciliation rec-ti-fi-ca-ti-on rectification re-fri-ge-ra-ti-on refrigeration re-ge-ne-ra-ti-on regeneration re-i-te-ra-ti-on reiteration re-mu-ne-ra-ti-on remuneration re-pre-sen-ta-ti-on representation re-ver-be-ra-ti-on reverberation S S Sa-cri-fi-ca-to-ry Sacrificatory sa-cri-le-gi-ous-ly sacrilegiously sanc-ti-fi-ca-ti-on sanctification sca-ri-fi-ca-ti-on scarification sig-ni-fi-ca-ti-on signification so-lem-ni-za-ti-on solemnization stu-pi-fi-ca-ti-on stupification sub-rep-ti-ti-ous-ly subreptitiously sub-si-di-a-ri-ly subsidiarily su-per-nu-me-ra-ry supernumerary su-per-e-ro-ga-ti-on supererogation su-per-fi-ci-al-ly superficially su-per-sti-ti-ous-ly superstitiously sur-rep-ti-ti-ous-ly surreptitiously T T Ter-gi-ver-sa-ti-on Tergiversation the-o-lo-gi-cal-ly theologically te-sti-fi-ca-ti-on testification trans-fi-gu-ra-ti-on transfiguration tran-sub-stan-ti-a-ti-on transubstantiation V V Ve-ri-fi-ca-ti-on Verification un-ac-com-pa-ni-a-ble unaccompaniable un-ac-cu-sto-med-ness unaccustomedness un-a-li-e-na-ble unalienable un-cir-cum-ci-si-on uncircumcision un-com-for-ta-ble-ness uncomfortableness un-com-mo-di-ous-ly uncommodiously un-com-mu-ni-ca-ble uncommunicable un-com-pa-ni-a-ble uncompaniable un-con-sci-o-na-ble unconscionable un-con-cei-va-ble-ness unconceivableness un-con-ta-mi-na-ted uncontaminated un-i-ma-gi-na-ble unimaginable u-ni-ver-sa-li-ty universality un-pas-si-o-nate-ly unpassionately un-pre-me-di-ta-ted unpremeditated un-pro-fi-ta-ble-ness unprofitableness un-que-sti-o-na-ble unquestionable un-rea-son-a-ble-ness unreasonableness un-re-me-di-a-ble unremediable un-sa-ti-a-ble-ness unsatiableness un-sea-son-a-ble-ness unseasonableness vo-ci-fe-ra-ti-on vociferation Chap. 18. The Learner being perfect in Spelling the Examples contained in the several Tables of the foregoing Chapters , let him now learn to read and say by heart the Lords Prayer , the Creed , and the Ten Commandments , as follow . The LORD'S PRAYER . OUR Father which art in Heaven . Hallowed be Thy Name . Thy Kingdom come . Thy Will be done on ●rth as it is in Heaven . Give us this day , our daily Bread. And forgive us our Trespasses , as we forgive them that trespass against us . And lead us not into Temptation , but deliver us from Evil. For Thine is the Kingdom , the Power , and the glory for Ever and Ever . Amen . The CREED . I Believe in God the Father Almighty , Maker of Heaven and Faith : And in Jesus Christ his only Son our Lord , who was Conceived by the Holy Ghost , Born of the Virgin Mary 〈◊〉 under Pontius Pilate , was Crucified , Dead and Buried he descended into Hell ; the third day he rose again from the 〈◊〉 , he ascended into Heaven , and sitteth on the Right Hand of God the Father Almighty ; from thence he shall come 〈◊〉 Judge both the Quick and the Dead I Believe in the Holy Ghost , the Holy Catholick Church ; the Communion of 〈◊〉 the Forgiveness of Sins : the Resurrection of the Body , and the Life Everlasting . Amen . The Ten COMMANDMENTS . GOD spake these words and said , I am the Lord thy 〈◊〉 which brought thee out of the Land of Egypt , out of 〈◊〉 House of Bondage . I. Thou shalt have no other Gods but me . II. Thou shalt 〈◊〉 〈◊〉 to thy self any Graven Image , nor the Likeness of any thing that is in Heaven above , or in the Earth beneath , or in the Waters under the Earth ; Thou shalt not bow down to them , nor worship them , for I the Lord thy God am a Jealous God , visiting the Sins of the Fathers upon the Children , unto the Third and Fourth Generation of them that hate me , and shew Mercy un●o Thousands of them that love me , and keep my Commandments III. Thou shalt not take the Name of the Lord thy God in vain , for the Lord will not hold him guiltless that taketh his Name in vain . IV. Remember that thou keep holy the Sabbath day ; six days shalt thou labour , and do all that thou hast to do ; but the seventh day is the Sabbath of the Lord thy God ; in it thou shalt do no manner of Work , thou , and thy Son , and thy Daughter , thy Man-servant , and thy Maid servant , thy Cattel , and the Stranger that is within thy Gates . For in six days the Lord made Heaven and Earth , the Sea , and all that in them is , and rested the seventh day ; wherefore the Lord blessed the seventh day and hallowed it . V. Honour thy Father and thy Mother , that thy days may be long in the Land which the Lord thy God giveth thee . VI. Thou shalt do no Murther . VII . Thou shalt not commit Adultery . VIII . Thou shalt not Steal . IX . Thou shalt not bear False Witness against thy Neighbour . X. Thou shalt not Covet thy Neighbours House , thou shalt not Covet thy Neighbours Wife , nor his Servant ; nor his Maid , nor his Ox , nor his Ass , nor any thing that is his . Chap. 19. Prayers and Graces fit for Children to get by Heart . A Prayer for the Morning . O Most Glorious Lord God , in whom I live , move , and have my Being , thou wast pleased to take me from the Womb wherein I was Conceived , and hast ever since preserved me to this very day ; ever blessed and praised be thy Name , O God , for all thy Mercies bestowed upon me ; for securing and preserving me from the perils 〈◊〉 dangers of the Night past ; and suffering me to enjoy the Glorious Light of another Day , protect me ( I beseech thee ) this day , and all the days of my Life by thy Holy Spirit , from all Sin and Wickedness , and let me be so armed with Faith in Jesus Christ , that I may powerfully resist the Temptations of the World , the Flesh , and the Devil ; let thy Blessing be upon my Endeavours this day , that I may profit both in Religion and Learning , bless my Parents , Friends , and Relations , and be a comfort to all in distress , and grant that when this Mortal Life shall have an end , I may joyfully hear my Blessed Saviour say , Come ye blessed of my Father , inherit the Kingdom prepared for you . Grant these my Requests for Jesus Christ his sake , in whose Name and Words thou hast taught me to pray , saying ▪ Our Father which art in Heaven , &c. A Prayer for the Evening . EVer Blessed and Glorious Lord God , I a poor Sinner most humbly prostrate my self before the Throne of thy Divine Majesty this Evening , beseeching thee to pardon all my Sins and Iniquities , which are many and very great , preserve me , O God , from Evil this Night , watch over me , and bless me this Night , let me lye down in thy fear , and rise in thy favour , bless my Parents and Friends that they may instruct me in thy Truth , so that I may not be taken in the Snares and Temptations of Satan ; these and whatever else I may stand in need of , I humbly beg for Jesus Christ his sake , in whose Name and Words thou hast taught me to pray , saying . Our Father , &c. A Family Prayer for the Morning . O Most Merciful and Glorious Lord God , we bless and praise thy Holy Name for all the benefits of this Life , and heartily thank thee for the Comfortable Rest which thou had been pleased to give us in the Night past , for saving and defending us from all dangers of our Enemies both ghostly and bodily , and that thou hast been Graciously pleased to let us see the Glorious Light of another Day , grant we beseech thee , that we may dedicate this , and all the rest of the days of our Lives to thy Service , and give us Grace so to walk warily among the Snares of our Mortal Enemies , the World , th● Flesh , and the Devil , that all our Thoughts , Words , an● Deeds may redound to the Honour and Glory of thy Hol● Name , and the Good and Comfort of our Precious and Immortal Souls ; and as thou hast been Graciously pleased to preserve and keep us under thy mighty protection from the beginning of our Lives unto this day , so we beseech thee to receive us this and all the remainder of our Lives into thy tuition , ruling and governing of us by thy Holy Spirit , to the utter destruction of Sin in us . We confess that we have every minute of our Lives committed great and manifold Sins against thy Divine Majesty , therefore we humbly beseech thee through Jesus Christ our Saviour , and for his Sake to forgive us , and let our Consciences be certified of the remission and forgiveness thereof , by thy Holy Spirit . Grant we beseech thee these our Prayers , and whatsoever else we stand in need of either relating to this Life , or the Life to come , for Jesus Christ his Sake , who hath taught us to pray , saying , Our Father , &c. A Family Prayer for the Evening . O Eternal and Glorious Lord God , we beseech thee look down from Heaven thy Dwelling place , upon us poor sinful Creatures , Dust and Ashes , and visit us with thy Mercy , Grace , and Salvation , we confess thy Fatherly Goodness towards us through the whole Course and Progress of our Lives , and therefore we bless and praise thy Holy Name . We beseech thee , O Lord , to continue thy Mercies unto us , bless us in our down lying and in our up rising , let thy Holy Angels pitch their Tents about us to save and deferd us this Night and ever hereafter from all our Enemies both ghostly and bodily , give our Bodies rest and quietness , but let our Souls be continually watching unto , and waiting , and thinking upon thee , and thy Holy Commandments , that whensoever our Lord and Saviour Jesus Christ shall come he may find us like wise Virgins with Oyl in our Lamps , ready prepared to receive him ; bless us with the Light of thy Countenance in the joyful appearance of another Day , that we being whole both in Body and Soul may rise again with thankful Hearts unto thee our God , and diligently walk in our Vocations to our own Comfort , and the Praise , and Glory of thy most Holy Name , through Jesus Christ our Lord and Saviour , who hath taught us when we Pray to ●y , Our Father , &c. Grace before Meat . MOst Glorious Lord God ▪ we beseech thee to look upon us with an Eye 〈◊〉 , and forgive us all our Sins , sanctifie these thy good Creatures to our use , make them healthful for our nourishment , and us truly thankful to thee for these and all other thy Mercies for Jesus Christ his sake , Amen . Grace after Meat . THE God of all Majesty , Power , and Glory , who hath Created , Redeemed , and at this time plentifully fed us , his most Holy Name be blessed and praised both now and for evermore . Amen . Grace before Meat . O Eternal and Glorious Lord God , we beseech thee bless these thy good Creatures which thou hast been pleased to provide for us , and help us by thine especial Grace so to improve every Mercy that we receive from thee , as that all may be to the Praise and Glory of thy Holy Name through Jesus Christ our Lord. Amen . Grace after Meat . MOst Glorious Lord God , we bless and praise thy Holy Name for all the Mercies which thou hast been pleased to bestow upon us , especially for feeding our weak and frail Bodies at this time with thy good Creatures . Lord teach us so to make use of thy Mercies , that they may be to the Eternal Comfort and Salvation of our Souls , through Jesus Christ our Lord and Saviour . Amen . God sa●e his Church , the King and Queen's Majesties , and thi● Realm , and send us Peace through Iesus Christ. Amen . Chap. 20. Directions for a Childs Behaviour at all Times and Places . FIrst in the Morning when thou dost awake To God for his Grace thy Petition make , Some Heavenly Prayer use daily to sa● ▪ And the God of Heaven will 〈◊〉 〈◊〉 〈◊〉 . And when ●ou hast prayed to God for his Grace , Observe these Directions in every place . Down from thy Chamber when as thou shalt go , Thy Parents salute and the Houshold also . Thy Hands see thou wash , thy Head also comb , Keep clean thy Apparel both abroad and at home . This done , thy Satchel , and thy Books take , And unto the School see that thou haste make . In thy going to School . IN going your way and passing the Street , Thy Hat being off , Salute those you meet . When unto the School that thou dost resort , Thy Master Salute I do thee exhort : Thy Fellows also in token of Love , Lest of unkindness they do thee reprove . Learn now in thy Youth for it is too true , It will be too late when Age doth ensue . If somewhat thou doubt , desire to be told , To learn is no shame , be thou never so old . And when from the School thou takest the way , Make haste to thy home , and stay not to play . Then entring the House in Parents presence , Them humbly Salute with due Reverence . At the Table . WHen down to the Table thy Parents shall sit , Be ready in place for purpose most fit . Be meek in thy Carriage , stare none in the Face , First hold up thy Hands , and then say thy Grace . The Grace being said if able thou be , To serve at the Table it will become thee . If thou canst not wait presume in no case , But in sitting down to Betters give place . Then suffer each Man first served to . be , For it is a point of great Courtesie . Thy Tongue suffer not at Table to walk , And do not of any thing jangle or talk ; For Cato doth say that in old and young , The first step to Vertue is bridling the Tongue . In the Church . VVHen unto the Church thou shalt take the way ▪ Kneeling or standing to God humbly pray . A Heart that is Contrite he will not despise , But doth account it a sweet Sacrifice . Unto him thy Sins see that thou confess , For them asking pardon and forgiveness . Then ask thou in Faith not doubting to have , And thou shalt receive what e're thou dost crave . He is fuller of Mercy than Tongue can express , The Author and Giver of Grace and Goodness . Thy self in the Church most comely behave , Sober in Carriage , with Countenance grave . The Lord doth it call the House of Prayer , And must not be used like Market or Fair. Chap. 21. Solomon's Precepts . MY Son hear the Instruction of thy Father , and forsake not the Law of thy Mother , Prov. ● . 8. My Son for get not the Law ; but let thine Heart keep my Commandments , Prov. 3. 1. Hear ye Children the Instruction of a ●er , and attend to know understanding , Prov. 4. ● . Hear O my Son and receive my sayings : And the years of thy Life shall be many , Prov. ● . 10. Hear me now therefore O ye Children , and depart not from the words of my Mouth , Prov. ● 7. A wise Son maketh a glad Father ; but a foolish Son is the heaviness of his Mother , Prov. 10. 1. A wise Son heareth his Fathers Instruction , but a Scorner heareth not Rebuke , Prov. 13. 1. He that spareth his Rod hateth his Son : But he that loveth him ●hasteneth him betimes , Prov. 1. 2. A Fool despiseth his Fathers Instruction , but he that regardeth reproof is prudent , Prov. 15. A wise Son maketh a glad Father : But a foolish Man despiseth his Mother , Prov. 1 ▪ 20. A wise Servant shall have rule over a Son that causeth shame ; and shall have part of the Inheritance among the Brethren , Prov. 17. 2. A foolish Son is a grief to his Father , and bitterness to her that bare him , Prov. ●7 . 25. A foolish Son is the Calamity of his Father , Prov. 19. 13. Cease my Son , to hear the instruction that causeth to err from the words of Knowledge , Prov. 1. 27. Even a Child is known by his doing , whether his work be pure , and whether it be right , Prov. 20. 11. Whoso Curseth his Father or his Mother , his Lamp shall be put out in obscur e darkness , Prov. 20 ▪ 2. Train up a Child in the way he should go ; and when he is old , he will not depart from it , Prov. 2. 6. Withhold not Correction from the Child : for if thou beatest him with the Rod he shall not die . Thou shalt beat him with the Rod , and shalt deliver his Soul from Hell , Prov. 23. 13 , 14. Hearken unto thy Father that begat thee , and despise not thy Mother when she is old , Prov. 2. 22. Thy Father and thy Mother shall be glad , and she that bare thee shall rejoice , Prov. 23. 25. My Son fear thou the Lord and the King ; and meddle not with them that are given to change , Prov ▪ 24. 21. The Rod and Reproof give wisdom , but a Child left to himself , bringeth his Mother to shame , Prov. 29 ▪ 15. Correct thy Son and he shall give thee rest : yea he shall give delight unto thy Soul , Prov. 29. 17. The Eye that mocketh at his Father , and despiseth to obey his Mother , the Ravens of the Valley shall pick it out , and the young Eagles shall eat it , Prov. 30. 17. Better is a poor and wise Child , than an old and foolish King , Eccles. ● . 13. Rejoice O young Man , in thy youth , and let thy Heart chear thee in the days of thy youth , and walk in the ways of thy Heart , and in the sight of thine Eyes : But know thou that for all these things , God will bring thee into Iudgment , Eccl. 11. 9. Chap. 2. English Proverbs Alphabetically placed ▪ A A Cat may look on a King. A Fools bolt is soon shot . A Friend is not known but in time of need . A good Tale is spoil'd by ill telling . A good beginning makes a good ending . A groaning Horse and a grunting Wife never fail their Master . A Fool and his Money is soon parted . After Dinner sit a while , after Supper walk a Mile . A Lark is better than a Kite . After a Storm cometh a Calm . After Meat comes Mustard . A little Pot is soon hot . A living Dog is better than a dead Lion. A long Harvest of a little Corn. A low Hedge is easily leaped over . All is not Gold that glisters . An hasty Man never wants woe . All covet all lose . A proud Horse that will not bear his own provender . A short Horse is soon curried . A Traveller may lye with Authority . A wonder lasteth but nine days . All is well that ends well . An ill Cook that cannot lick his own Fingers . An Inch breaks no square . As good play for nothing as work for nothing . Ask my Companion if I be a Thief . As they Brew so let them Bake . B Batchellors Wives and Maids Children are well taught . Be it better , be it worse , be rul'd by him that bears the Purse . Beggars must not he chusers . Better be envied than pitied . Better Eye out than always ake . Better fed than taught . Better sit still than rise and fall . Better half a Loaf than no Bread. Better late than never . Better leave than lack . Better to bend than break . Better unborn than untaught . Between two Stools the Breech goes to the ground . Birds of a Feather flock together . Blind Men must not judge of Colours . Better at the end of a Feast than the beginning of a fray . Burnt Child dreads the Fire . Buy not a Pig in a poke ▪ C CAT after kind . Change of Pasture makes fat Calves . Children and Fools tell truth . Christmas comes but once a year . Curst Cows have short Horns . Cut your Coat according to your Cloth. D DEar bought and far fetched are good for Ladies . Dinner cannot be long where Dainties want . Do well , and have well . E ENough is as good as a Feast . Ever drunk , ever a dry . Even Reckoning makes long Friends . Every Cock will crow on his own Dunghil . Every one as he likes , quoth the Man when he kist his Cow. Every Man can rule a Shrew but he that has her . Every Man for himself and God for us all . Every little makes a mickle . F FAint Heart never won fair Lady . Fair and softly goes far . Fast bind fast find . Few words to the wise are sufficient . Fine Feathers make fine Birds . First come first serv'd . Fools have ●ortune . Fools are pleased with fair words . Foul Water will quench Fire as soon as fair . Fore-warn'd fore-arm'd . G GOD never sends Mouths but he sends Meat . Good Wine needs no Bush. Good to have two strings to a Bow. Good to be merry and wise . Great boast , small roast . Great cry and little wool . H HAste makes waste . He must needs go if the Devil drives . He goes far that never turns . He that fears every Grass must not piss in a Meadow . He must needs Swim that is held up by the Chin. He that has an ill Name is half hang'd . He that is Born to be hang'd shall never be drown'd . He that killeth a Man when he is drunk must be hang'd when he is sober . He that will not when he may , when he will he shall have nay . Hold fast when vou have it . Home is home though never so homely . Hope well and have well . Hot Love is soon cold . How should the Foal amble , when the Horse and Mare trot . Hunger will break Stone Walls . Hungry Dogs will eat dirty Puddings . I IF every Man would mend one , all would soon be mended . Ill gotten Goods never prosper . Ill Weeds grow apace . In space cometh Grace . It is an ill Bird that bewrayeth his own Nest. It is an ill Wind that bloweth no body profit . It is a good Horse that never stumbles . It is better to kiss a Knave than be troubled with him . It is good Fishing in troubled Waters . It is good to beware of other Mens harms . It is good to be merry and wise . It is good sleeping in a whole Skin . It is hard halting before a Cripple . It is hard striving against the Stream . It is better coming at the end of a Feast than the beginning of a Fray. It is merry in Hall , when Beards wag all . It is merry when Knaves meet . It must needs be true that every Man saith . Ill News comes too soon . Ioan is as good as my Lady in the dark . K KA me and I will ka thee . Kissing goes by Favour . Kill two Birds with one Stone . Kind as a Kite . L LEave is light . Learn to creep before you go . Let him Laugh that wins . Light Gains makes a heavy Purse . Like to like quoth the Devil to the Collier . Little Pot soon hot . Look ere you Leap. Like Master like Man. Look not too high left a chip fall in thine Eye . Love cometh in at the Windows and goes out at the Door . Love is blind . Love me little , and Love me long . Love me and Love my Dog. Love will creep where it cannot go . M MAny Hands make light work . Many Kinsfolk , few Friends . Many Kiss the Child , for the Nurses sake . Many stumble at a Straw and leap over a block . Might overcomes Right . More afraid than hurt . Money makes the Mare to go . Make Hay while the Sun shines . Most haste the worst speed . N NAught is never in danger . Necessity has no Law. Need makes the old Wife trot . Never pleasure without repentance . No Man loves his Fetters though made of Gold. No Penny no Pater Noster . Nothing has no savour . Nothing venture nothing have . No longer Pipe no longer Dance . Nothing so certain as Death . New Lords new Laws . None so proud as an enricht Beggar . No Carrion will kill a Crow . O. ONE scabby Sheep will infect a whole Flock . One Swallow makes no Summer . One Bird in Hand is worth two in the Bush. One ill word begetteth another . One good turn deserves another . Out of sight out of mind . Out of Gods Blessing into the warm Sun. Out of Debt out of danger . One may see day through a little hole . P PEnny wise pound foolish . Poor and proud , fye , fye . Pride will have a fall . Pride goes before and shame comes after . Proffered service stinks . Prove thy Friend before thou have need . Put not a Sword into a mad Mans Hand . Q QUality not quantity bears the Bell. Quick at Meat quick at Work. R RIches have Wings . Reckon not without your Host. Rome was not Built on a day . Rob Peter to pay Paul. S. SAying and doing are two things . Seldom comes a better . Seldom seen soon forgotten . Self do self have . Set a Knave to catch a Knave . Shameful asking must have shameful nay . Set a Beggar on Horseback and he will soon ride him out of Breath . Small Pitchers have wide Ears . Soft Fire makes sweet Malt. Something is better than nothing . Soon gotten soon spent . Soon hot soon cold . Soon ripe soon rotten . Spare to speak spare to speed . Store is no sore . Such a Father such a Son. Strike while the Iron is hot . Sue a Beggar and get a Louse . Such a Carpenter such Chips . Sweet Meat must have sour Sauce . Still Sow eats all the Draff . T TAles of Robin Hood are fit for Fools . Teach your Grandam to suck Eggs. That which one will not another will. That the Eye seeth not , the Heart never grieveth at . The Beggar may sing before the Thief . The best is best cheap . The Blind eat many a Fly. The Blind lead the Blind , and both fall into the Ditch . The Cat would eat Fish but dares not wet his Feet . The Crow thinks her own Birds the fairest : Tell truth and shame the Devil . The more the merrier the fewer the better chear . The Fox fa●es well when he is curst . The greatest Talkers are the least Doers . The highest Tree has the greatest fall . The best may mend . The Keys hang not all at one Mans Girdle . The longest East the shortest West . The longest day will have an end . The more Knave the better luck . The Masters Eye makes the Horse fat . The more haste the worst speed . The more you stir a Turd the worse it will slink . The Eye is bigger than the Belly . The new Broom sweeps clean . The nearer the Church the farther from God. The old Woman would not have looked in the Oven for her Daughter if she had not been there her self . The Priest forgetteth that ever he was Clark. The Pitcher goes not so often to the Well but it comes home broken at last . Take Pepper in the Nose . The rouling Stone gathers no Moss . They that are bound must obey . The Stable robs more than a Thief . Time and Tide stays for no Man. Threatned Folks live long . Too much familiarity breeds contempt . The young Cock croweth after the old one . There are more ways to the Wood than one . There is difference between staring and stark mad . There is no Fool like the old one . There is no Smoke but some Fire . The weakest goes to the Wall. Three may keep Counsel if two be away . Time past cannot be recalled . Touch a gall'd Horse on the Back and he 'll wince ▪ Tread on a Worm and he 'll turn again . Trim tram like Master like Man. Two Heads are better than one . Two hungry Meals make the third a Glutton . V VErtue never waxeth old . Under the Rose be it spoken . W WE can have no more of a Cat than her Skin . What is gotten over the Devils Back is spent under his Belly . When the Fox preacheth beware of the Geese . When the Belly is full the Bones would be at rest . What is bred in the Bone will never out of the Flesh. When the Sky falls we shall catch Larks . When the Steed is stolen shut the Stable Door . Were it not for hope the Heart would break . When thy Neighbours House is on Fire take care of thine own . When Thieves fall out true Men hear of their Goods . Where nothing is to be had the King must lose his Right . While the Grass grows the Steed starves . Who is worse shod than the Shoemakers Wife . Who so deaf as he that will not hear . Who wait for dead Mens Shoes may go barefoot . Wishers and Woulders are no good Housholders ▪ Wit is never good till 't is bought . Y YOU cannot hide an Eel in a Sack ▪ Young Saint old Devil . You cannot fare well but you must cry Roast-meat . You cannot see the Wood for Trees . You cannot eat your Cake and have your Cake . You must not look a given Horse in the Mouth . Chap. 23. Some few Examples of Gods Punishment upon Sinners for breach of the several Commandments . OUR Duty towards God , is to beli● in him , to fear him , and to love him with all our Hearts , 〈◊〉 our Minds , with all our Souls , and with all our strength ; to worship him , and him only , to give him thanks , to put our whole tr●st in hi● , to call upon him , to honour his holy Name and his Word , and to serve him truely all the days of ●ur Lives . Therefore is Atheism , the worshipping of false Gods ▪ or the want of a true Knowledge , Faith , Fear , and Love of the true God a Breach of this Commandment . Many Examples we have in Scripture of Gods vengeance upon those who have worshipped strange Gods. Nadab and Abihu the Sons of Aaron for Offering strange Fire before the Lord in the Wilderness , contrary to his express Commandment , were miserably consumed by Fire from Heaven , Lev. 10. 1 , 2. The Children of Israel being enticed by the Moabites , to offer Sacrifice to their Gods , joined themselves to Baal Peor , and therefore the anger of the Lord being kindled against them , their Princes were hanged , and twenty four Thousand Men were slain , amongst whom were Zimri and Cosbi , Numb 25. and many other Examples of the like Nature there are in Scripture . Pherecydes a Philosopher boasted impudently amongst his Scholars , of his Prosperity , Learning and Wisdom , saying , That though he served not God , yet he led a more quiet and prosperous Life than those that were addicted to Religion , and therefore he passed not for any such Vanity , but soon after his Impiety was justly punished , for the Lord struck ▪ him with such a strong Disease , that out of his Body issued such a slimy and filthy Sweat , and ingendred such a number of Lice and Worms , that his Bowels being consumed by them , he most miserably Died. Idolatry the Breach of the Second Commandment . AMongst the many Examples we have in Scripture of Gods punishment upon Sinners for Idolatrous worship take these two , viz. About three Thousand of the Israelites were slain in the Wilderness for making to themselves , and Idolizing the Golden Calf which Aaro● made , Exod. 32. Ahaziah the Son of Ahab King of Israel for serving and worshipping of Baalzebub the God of Ekron was made an Example of Gods wrath , and died , according to the Word of the Lord by the Prophet Elijah , 2 Kings 1. Many other such Examples there are in the Books of the Kings , and the Chronicles . For as a King will not suffer another to bear the Title in his Realm , so God will not permit any other in the World to be honoured but himself only . Per●ury the Breach of the Third Commandment . THE Eternal God hath commanded that we should so bridle and govern our Tongue , that whatsoever we speak may be to his Honour and Glory , and not that we should rashly bind our selves by his most holy Name with Oaths and Execrations , or abuse him by any other imp●ous means in va● M●tters , for he hath threatned condign punishment to thos● that sh●ll so prophane his most holy Name . The Son of Shelomith the Israelitish Woman , when he had Blasphemed the Name of the Lord with Oaths and Curses , was by all the people , and by the immediate command of God stoned to death . Whosoever Curseth his God shall bear his sin , and he that Blasphemeth the Name of the Lord ▪ he shall surely be put to death , and all the Congregation shall certainly stone him ; as well the stranger , as he that is Born in the Land , when he Blasphemeth the Name of the Lord , shall be put to death , Levit. 24. See Matth. 5. 33 , &c. A certain Nobleman being at a Market Town at play , and having lost a great Sum of Money , in great Passion commands his Servant to get the Horses ready in order to go out of Town , in the mean time bel●hing out most horrible and abominable Oaths , and Execrations , his Servant disswades him from going Home , telling him that it would be dangerous Travelling in the Night , because of the uncouthness of the way , and the dangerous Waters that they must pass by ; but he fell into a greater Passion , Swearing and Cursing more and more , and commanded his Servant to be obedient to what he said , the Servant obeyeth , and having mounted their Horses , they depart the ●own , being in all three of them ▪ they had not gone far , but a great Company of Horsemen being Hellish Apparitions , came to them , and making a most horrible noise seized upon the said Nobleman in the middle of them , and flung him violently from his Horse , being senseless , but there being with him ( besides his Servant ) ▪ a young Man of very great Courage , relying upon God and the Integrity of his own good Conscience , run Couragiously into the midst of this Devilis● Cavalry and rescued his Lord , setting him again upon his Horse , but having lost their way they wandred up and down all Night , and still as they rode along they could hear the troublesome noise of these Infernal Troopers , but God preserved the said young Man that they had no power to hurt him ▪ and in the Morning they brought the Nobleman to a Monastery , where he languished three days and died . Such is the end of those that prophane the Holy Name of God by horrible Oaths and Imprecations . There was a Ten years Truce concluded between Ladislaus King of Hungary , and Amurath the Emperor of the Turks , which was confirmed by an Oath between them ; but Ladislaus having a fair opportunity , by the Instigation of Pope Eugenius , breaks the League , thereby violating his Oath , and raiseth a great Army against Amurath , and with very great speed marches through Walachia , and Bulgaria , to a Town called Varna where Amurath met and engaged him , the Battel was very fierce and doubtful , none perceiving for a great while which way it would incline ; at last Amurath finding the Battel like to go against him ▪ lift up his Hands and Eyes towards Heaven , and said , Behold O Iesus Christ ! These are the Articles which thy Christians have made with me , Swearing by thy Name to observe and keep them , and by this their Perjury they deny thee to be their God , wherefore if thou art a God as they say thou art , revenge this Injury done to me , and to thy Holy Name ; and immediately the Scale turned , Amurath gained the Victory , Ladislaus is slain , and with him Eleven Thousand Christians . Sabbath-breaking a Breach of the Fourth Commandment . GOD has commanded to set apart a time for his more publick Worship , though our whole Life should be a serving of him , yet some time is requisite to be set apart , and observed , for an unanimous , solemn , and publick serving , and this time must be a set time , and a seventh part of our time , which we call the Sabbath ; and God has threatned vengeance to those that by their Impiety shall violate this his Holy Commandment . Verily , saith the Lord , my Sabhath ye shall keep , for it is a sign between me and you throughout your Generations , that you may know that I am the Lord that sanctifieth you . Ye shall keep the Sabbath therefore ; for it is holy unto you , every one that defileth it shall be surely put to death , for whosoever doth any Work therein , that Soul shall be cut off from amongst his people , Exod. 31. 13 , 14. A sad Example of Gods displeasure for the Breach of this Commandment we have in the fifteenth of Numbers , where an 〈◊〉 being found gathering of Sticks upon the Sabbath day ▪ was by the immediate Command of God stoned to Death by the people . The History of the Kings of Iudah and Israel contain many Examples of the Almighties Punishment upon those who have not feared , to contemn his Word , and to prophane his Holy Sabbath . And the Histories of latter Times are not wanting in their Examples of Gods Punishment of Sabbath-breakers . In the year of our Salvation 1553. in a City in Switzerland about three Miles distant from Lucerne , three Gamesters playing at Dice upon a Bench in the Fields near the Walls of the City upon a Sabbath day , whereof one when he had lost a considerable Sum of Money , while he was yet provoking the Almighty with Oaths and Curses he chanced to have a fortunate throw according to his wish , and being thereat incouraged , he Swore that if the Dice run against him again he would fling or strike his Dagger as far as he could into the very Body of God. The Dice fail him , and forthwith he draws his Dagger , and taking it by the point he throws it with all his might towards Heaven ; the Dagger vanished in the Air , and was never more seen , and five drops of Blood fell upon the Bench where they were playing , and immediately the Devil seized upon him , and carried him away with that violence and noise that it affrighted the whole City into a Tumult . The other two were extreamly affrighted , and endeavoured to wipe the Blood from off the Bench , but in vain , for the more they endeavoured to clean it , the more plain did the Purple Colour of the Blood appear ; the whole City being filled with the noise of this Wickedness , and every own crowds towards the place , where they find the two Players that were left , endeavouring in vain to clean the Bench of the said five drops of Blood with Water ; and being Examined , the Magistrates decree them to be immediately bound , and cast under the Walls of the City , and as they were carrying them through the Gate , one of them fell down , being deprived of Strength , and such a company of Worms and Lice came from him , that they devoured him , and he died in that very place , a foul and miserable Death ; which the people seeing , without any more ado immediately destroyed the other . That part of the Bench whereon the Blood sell , was cut off , and remains to this day a Testimony and Monument of this great and abominable 〈◊〉 . Disobedience to Parents a Breach of the Fifth Commandment . THIS is the first Commandment that hath a promise annexed to it , whereby the Eternal God promiseth to those that are obedient thereunto Benediction , Prosperity , and long Life . And the Scripture is very plentiful of Examples of Gods Punishments upon those that have been guilty of the Breach thereof . Ham the youngest Son of Noah , seeing his Fathers Nakedness when he was overcome with Wine , called to his Brethren that were without and told them thereof ▪ scoffing , and making a Laughing-stock of his Father , insulting over his Vice and Imbecility , for which he and his Posterity were accursed , and became a Servant of Servants to his Brethren , Gen. 9. Absalom being in Rebellion against his Father King David , and pursued by his Fathers Servants , was taken from his Mule by the Boughs of an Oak under which he rode , and was hanged to Death , 2 Sam. 18. 9. A poor and Ancient Man being grievously oppressed with Poverty , went to his Son who was very Opulent , and Wealthy , praying him not to despise his Poverty , but to relieve him in his great Necessity , but the Son thinking it would be a great disgrace to him , to have it publickly known that he was dedescended of such Poor Parentage , and therefore ordered his Servants to give him harsh and threatning Language , and set him gone , the poor old Man departs grieving , and weeping extreamly at the unkindness , and undutifulness of his Son. But behold the Justice of God overtook him , for the old Man was no sooner gone but his Son fell mad , and so died . Murder the Breach of the Sixth Commandment . CAIN the first Murderer that ever was , run headlong into extream desperation , having no certain place of Abode , wandring up and down upon the Earth , having his Head and Heart filled with Fear and despair . And the Lord said unto Cain , what hast thou done ? The voice of thy Brothers Blood cryeth unto me from the Ground , now therefore art thou Cursed from the Earth , which hath opened her Mouth to r●ceive thy Brothers Blood , from thy hand . When thou Tillest the Ground it shall not henceforth vield unto thee her strength ; a Fugitive and a Vagabond shalt thou be in the Earth , Gen. 4. A Bakers Servant at V●enna in Austria knowing his Master to be very Rich , having good store of Money by him , left his Service , not without a design to come another time and rob him , and a few days afterwards he breaks privily ( in the Night time ) into the House , and finding that he had disturbed the Man-servant , lays wait for him , and kills him , and after that the Maid-servant likewise , and being now fully resolved to destroy all the Family , enters with this Bloody Resolution into the Bakers Lodging Chamber , and Murdered him and his Wife in their Beds , and not being satisfied with the Blood of these four , must needs Murder a little Girl which was their Daughter , the Child seeing his Intentions , or at least fearing that he would serve her as he had her Father and Mother , cryed out , O Paul , Paul , save me , and I will give you all my Play-things , but he would not hear the little Child , but Murdered it . And when he had done all this , he broke open the Chest wherein the Money was , took it out , and went away with his Booty , and made his escape to Ratisbon . The Neighbours admiring to see the Shop shut up all the next day , at last by Authority broke open the Doors , where they found the Murdered Bodies to their great Horrour and Amazement . The Murderer was in a short time after taken at Ratisbon , and brought from thence to Vienna , where he took his Tryal , and he confessed his Charge , and was Condemned to be Hanged Alive in Chains , which was accordingly performed ; he said nothing troubled him more than the cries of the Child , offering him her Play-things to save her Life , which he said , he continually heard , and could by no means put it out of his Mind . The bloody and ▪ deceitful Men shall not live out half their day● Psal. 55. 23. Adultery the Breach of the Seventh Commandment . THE Punishment of David for his committing of Adultery with Bathsheba the Wife of Uriah , was very great , as also was his Repentance , 2 Sam. 1● . 12. The Wife of a certain Nobleman having more than ordinary Familiarity with another besides her Husband , her Lord being absent , she having written two Letters at one time , the one to her Husband , and the other to her Familiar Friend , she chanced to superscribe them both wrong ; viz. that which was for her Husband , to her sweet Heart , and that which was for her Sweet-heart she superscribed to her Husband , by which he discovered her unfaithfulness to his Bed , and went home and killed her with his own Hand . An Honest Citizen of Ulm in Germany having a very Lewd Wife , had often admonished her to mend her sinful Course of Life , but in vain ; and at last being resolved to make a positive proof of her Chastity ▪ he gave out that he would take a Journey into the Countrey for two days at least , and away he went in the Morning , but returned at Night , and ( undiscovered by his Wife ) got into the House , and hid himself in some convenient place where he might easily perceive how passages were , and found the Servants ( who were privy to their Mistresses Lewd Course of Life , ) making great Preparations for a Splendid Entertainment , by and by in comes the Adulterer , who was kindly received , and made very welcome ; the good Man observing what passed , was highly provoked , but yet bore all very patiently ; soon after Supper was ended they go to Bed , which the good Man perceiving , run in great Passion from the place where he lay undiscovered , and first ki●ed the Adulterer , and then his Wife ; for this he was called to Answer , but the Magistrate thinking his Provocation great , and Revenge just , exempted him from Punishment . Marriage is honourable in all , and the Bed undefiled ; but Whoremongers and Adulterers God will judge , Heb. 13. 4. Theft the Breach of the Eighth Commandment . WE have many Examples both in Sacred Writ and other Histories , of Almighty Gods high Resentment of the Breach of this Commandment . But amongst them none more Famous than that of Achan the Son of Carmi , of the Tribe of Iudah , who was found guilty of the accursed thing , in privily Stealing a Babylonish Garment , two hundred Shekels of Silver , and a Wedge of Gold of fifty Shekels , and hid them in the Earth in the midst of his Tent , but the Lord discovered his Theft , and brought him to Condign Punishment , for he was stoned to Death , and his Sons , and his Daughters , his Oxen , and his Asses , his Sheep , his Tent , and all that he had were Burnt with Fire , Iosh. 7. It is Recorded by Martin Luther , that a very wicked but young Thief , was taken in the Town of Belkig in Germany , where being Tryed for his Life , he was found guilty ; but in consideration of his Youth , and in Hopes , and by his Promise , of a Reformation for the future , he was pardoned and set at Liberty ; but in a very little time he fell to his old Trade of Thieving and Stealing , and professed himself an utter Enemy to the said Town of Belkig , and set it on Fire , and Burnt several Houses in it , at length he was taken again by the Brandenburghers , and being asked how he durst be so wicked as to set Fire to that Town that had been so kind to him as to give him his Life ? he answered , ( and no other Answer could they get from him , but ) that he had there received an unjust Sentence for his Thievery , for they ought not to have let him go , but to have Hang'd him . No less Famous is Gods Vengeance upon Urracha Queen of Spain , for her Sacriledge , for being necessitated for Money in her Wars which she had with her Son Alphonsus , she went into the Church of St. Isidore , and Commanded her Souldiers to seize upon the Riches thereof , but they being fearful to lay Hands on the Holy Treasure , refused to obey her , wherefore she pulled it to pieces with her own Hand , but behold the Justice of God overtook her immediately , for as she was going out of the Church she was struck dead in the very place . Hell and Destruction are never full , so the Eyes of Man are never satisfied . False-witness a Breach of the Ninth Commandment . WE have many Examples of Gods Judgment upon Sinners for the Breach of this Commandment ; one of thae most Famous in Holy Writ is that of Ahab and Iezabel , for procuring False Witness against Naboth , and thereby taking away his Life , on purpose that Ahab might have his Vineyard which he had long coveted , 1 Kings 21. It is Recorded in the Chronicles of Scotland how that one Campbel a Fryar by falsly accusing of one Hamilton , caused him to be Burnt to Death , but Hamilton being in the Fire ready to be Executed , Cited or Summoned the said Fryar to appear ( betwixt that and such a day which he then named ) before the most high God , the Righteous Judge of all Men , to Answer to the Innocency of his Death , and whether his Accusation were just or not ; now behold the just Hand of God , for before the day nominated by the said Hamilton came , the Fryar died miserably without any Remorse of Conscience . These six things doth the Lord hate ; yea seven are an Abomination unto him : A Proud Look , a Lying Tongue , and Hands that shed Innocent Blood , an Heart that deviseth Wicked Imaginations , Feet that be swift in running to Mischief , a False Witness that speaketh Lies , and him that soweth Discord among Brethren , Prov. 6. 16 , 17 , 18 , 19. Covetousness a Breach of the Tenth Commandment . THE Jews when they had forsaken the Law of the Lord , were miserably afflicted , for there were slain of Iudah in one day , by Pe●ah the Son of Ramaliah an Hundred and Twenty Thousand ; and afterwards the Children of Israel took of their Brethren of the House of Iudah Two Hundred Thousand Women , Sons and Daughters , and a vast deal of Treasure , and carried it to Samaria , but they were severely reproved by the Prophet Obed , who denounced the heavy Anger of the Lord against them , and they returned their Captives and Prey into Iudea . As you may see at large , 2 Chron. 28. King Zedekiah is reprehended as a Violator of this Commandment , for grievously oppressing his Subjects , by Building Stately and Magnificent Structures , at the Charge and Labour of the Poor ; as you may see in Ierem. 22. The Sons of Samuel being Covetously minded took Bribes , and perverted Judgment , which made the Children of Israel desirous to change their present Government into a Kingdom , 1 Sam. 8. King Ahab coveting Naboth's Vineyard , and being Naboth would not sell it him because it was the Inheritance of his Fathers ; by the counsel , advice , or instigation of his Wife Iezabel Witnesses are Suborned falsly to accuse him , and Naboth is stoned to Death , so that now Ahab may have the Vineyard at his pleasure ; but behold the Hand of God in revenging his Covetousness and false Accusation , fell upon all his Posterity , 1 Kings 21. W● to them that devise iniquity , and work evil upon their Beds : When the Morning is light they practise it , because it is in the power of their hands . And they covet Fields , and take them by violence ; and Houses , and take them away : So they oppress a Man and his House evon a Man and his Heritage . Therefore thus saith the Lord , Behold against this Family do I devise an evil , from which ye shall not remove your Necks , neither shall ye go haughtily : for this time is Evil. Chap. 24. The Names and Order of the Books of the Old and New Testament , with the Number of Chapters contained in each of them . The Books of the Old Testament . Genesis hath Chapters 50 Exodus 40 Leviticus 27 Numbers 36 Deuteronomy 34 Ioshua 24 Iudges 21 Ruth 4 I Samuel 31 II Samuel 24 I Kings 22 II Kings 25 I Chronicles 29 II Chronicles 36 Ezra 10 Nehemiah 13 Esther 1 Iob 42 Psalms 150 Proverbs 31 Ecclesiasies 12 The Song of Solomon 8 Isaiah 66 Ieremiah 52 Lamentations 5 Ezekiel 48 Daniel 12 Hosea 14 Ioel 3 Amos 9 Obàdiah 1 Ionah 4 Micah 7 Nahum 3 Habakkuk 3 Zephaniah 3 Haggai 2 Zechariah 14 Malachi 4 The Books of the New Testament . Matthew hath Chapters 28 Mark 16 Luke 24 Iohu 21 The Acts of the Apostles 28 The Epistle to the Romans 16 I Corinthians 16 II Corinthians 13 Galatians 〈◊〉 Ephesians 6 Philippians 4 Colossians 4 I Thessalonians 5 II Thessalonians 3 I Timothy 6 II Timothy 4 Titus 3 Philemon 1 To the Hebrews 13 The Epistle of Iames 5 I Peter 5 II Peter 3 I Iohn 5 II Iohn 1 III Iohn 1 Iude 1 Revelations 22 Chap. 25. The Penmen of the Holy Scriptures . Of the Old Testament . MOses the Son of Amram , the Son of Levi , when he was full Forty years old , was called of God to be the Leader of the Children of Israel : He wrote the Book of Genesis , about Eight Hundred years after the Flood : He spake by a large measure of Gods Spirit , of sundry things that were done Two Thousand Four Hundred and Fourteen Years before he was Born : He also wrote the Books of Exodus , Leviticus , Num● and Deuteronomy . Ioshua and Eleazer the High-Priest , are supposed to have penned the Book of Ioshua , who governed Israel Victoriously Seventeen years , and died in the Hundred and Tenth year of his Age. Samuel is supposed to have penned the Books of Iudges and Ruth . The first and second Books of Samuel were written by Samuel the Seer . The first and second Books of Kings , were penned by Nathan the Prophet , Ahiah the Shilonite , Iddo the Seer , Iehu the Prophet , and Semeia . The Book of Ezra , was written by Ezra the Priest. He also wrote the Book of Nehemiah , and therefore in the Hebrew they are put both together . The Book of Esther was written by the Chronicler of King Ahasue●us , and it was taken out of the Records of the Medes and Persians . Iob was of Kin to Abraham , and out-lived Moses , for he lived one Hundred and Forty years , after his Temptation . But the Penman of the Book of Iob is not exactly Recorded . The Psalms were penned by divers , as David , Moses , Asaph , and others , and they were collected by Esdras . The Proverbs , Ecclesiastes , and Solomons Song , were written by Solomon , after his Conversion . Isaiah the Son of Amos , prophesied in the days of Uzziah , Iotham , Ahaz , Hezekiah , and Manasses . Ieremiah the Son of Hilkiah the Priest , prophesied in the days of Iosias , Iehojakim , and Zedekiah . Ezekiel was a Priest in Babilon five years of Iehojakims Captivity . Daniel was a Captive in Babilon , and prophesied under Nebuchadnezzar , Evil Meroduck , and Belshazzer . Hosea prophesied in the days of Uzziah , Iotham , Ahaz , and Hezekiah Kings of Iudah . Joel prophesied in the days of Uzziah , and Ieroboam . Amos was a poor simple Shepherd , and prophesied at the same time with Ioel. Obadiah prophesied against Edom , at that time the Palace of the Temple , and City was set on Fire . Ionah prophesied in the days of Amaziah and Ieroboam . Micah Prophesied in the days of ●otham , Ahaz , and Hezekiah Kings of Iudah . Nahum prophesied in the days of Hezekiah King of Iudah , Uzziah King of Israel , and of Salmanazer King of Assyria . Habakkuk prophesied about the same time . Zephaniah prophesied in the days of Iosiah the Son of Amon. Haggai , Zachariah , and Malachi the Prophets were appointed after the Captivity to comfort the people , and wrote all about the same time . Of the New Testament . St. Matthew wrote his Gospel Eight years after Christs Ascention . St. Mark wrote Ten years after Christs Ascention . St. Luke wrote Fifteen years after Christs Ascention . St. Iohn wrote Thirty Two years after Christs Ascention . The Acts of the Apostles were written by St. Luke . The Epistle to the Romans was written by St. Paul , as also the two to the Corinthians , and those to the Galatians , Ephesians , Philippians , Colossians , Timothy , Titus , Philemon , and the Hebrews , as the Learned do generally allow . ●ames the Son of Alpheus , the Brother of Iude , called also the Brother of our Lord , wrote his Epistle . St. Peter one of the chief of the Apostles wrote two Epistles . St. Iohn , the Son of Zebedee , Beloved of Christ , wrote three Epistles , and the Revelation . St. Iude one of the Apostles wrote his Epistle . Chap. 26. Directions for true Spelling and Writing of English. ALL Speech or Language is composed of Words , and every Word is composed of Syllables , except it be a Monosyllable , and every Syllable is composed of one or more Letters . The Letters are in number twenty four , as followeth , viz. a b c d e f g h i k l m n o p q r s t u w x y z , to which may be added j and v which make up the number twenty six . These Letters are divided into Vowels and Consonants . The Vowels are a e i o u and y after a Consonant . The Consonants are b c d f g h k l m n p q r s t w x z and y when it comes before a Vowel in the same Syllable , as in youth , young , yonder , also j and v are Consonants where-ever found . No Syllable can be spelt without a Vowel , and sometimes the Vowels alone make a Syllable , as a-gainst , e-ve-ry , i-vo-ry , o-ver , usury . And as no Syllable can be without a Vowel , so no Syllable hath more than one Vowel , as di-vi-si-on , except when two have one sound , which we call Dipthongs , as au-tho-ri-ty , soon , pro-ceed , neu-ter . And e or es in the end of a word , which for distinction sake may aptly be called e final , as hence , since , con-fute , names , bones . Or in Compound Words , as safe-guard , not sa-fe-guard , there-fore , not the-refore , &c. Wherefore when any word is given to be divided into Syllables , consider how many Vowels and Dipthongs are therein , so many Syllables must there be , except as before excepted . And to divide your Syllables exactly take the following Rules . I. When two Vowels come together in a word , being no Diphthong , but having each his full sound , then must they in Spelling be divided as mu-tu-al , tri-umph , tri-en-ni-al . II. When a double Consonant is in the middle of a word , then is it likewise in Spelling to be divided , as war-rant , common ; spel-ling , ne-ces-sa-ry . III. When a Consonant is in the middle of a word between two Vowels or Dipthongs , then must that Consonant be joined to the latter Vowel or Dipthong , as di-li-gent , re-ve-la-ti-on , de-li-ve-rance , sau-ci-ly , co-ve-tous . IV. When two or more Consonants , being such as can begin a word follow a Vowel , such Consonants must generally be joined to the latter Syllable , as mi-ni stra-ti-on , mi-ni ster , de-tract , de-spise , de-clare . See more Examples of this Rule in the 10th Chapter . V. When two or more Consonants being such as cannot begin a word , come between two Vowels then must they be divided , one to the former , and the other to the latter Vowel , as ab-sence , a●-ter , al-mond , con-tra-ry , con-strain , where note that when three or four Consonants are in the middle of a word between two Vowels , such of them as can begin a word must be joined to the latter Syllable , as con-tra-ry , where you see ntr cannot begin a word , but tr may , wherefore is n joined to the first , and tr to the latter Syllable . So in con strain , ●str cannot begin a word but str may , &c. From the foregoing general Rules there are these following Exceptions , viz. From the third Rule there are two Exceptions . 1. When x followeth a Vowel it must be always joined with the Vowel before it , as wax eth , fix-ed , ex am-ple , not wa-xeth , fi-xed , e xam p e , for the Letter x hath the sound of two Consonants , viz. c and s , which cannot begin a word , but if instead of x you would use c and s , then it would come under the fifth Rule , as for wax-eth , wac-seth , &c. 2 When e is in the end of a word you ought not to stop at the Vowel before it , as lame , not la-me , con-sume , not consu-me , &c. From the third and fourth Rules are excepted all words that begin with these Prepositions , viz. abs , ob , in , un , dis , mis , per , sub , and such as end with these terminations , ly , less , ness , ler , as ab-la-tive , ob-la-ti-on , in-a-bi-li-ty , in-au gu-rate , un-able , dis-a-ble , dis-trust , mis-place , mis-take , per am-bu-late , suborn , sub-lime , ug-ly , help-less , co ve-tous-ness , bab-ler . For these prepositions and terminations must have their full sound , and pronunciation . A Syllable is either long or short . A Syllable is said to be long when it is pronounced by a longer time than ordinary , and a Syllable is said to be short , when it is pronounced by a short time . There are three things which make a Syllable long , viz. 1. When e is placed at the end of a word , it always makes the last Syllable long , as made , bare , cane , note , tune , whereas if the e were neglected , it would be sounded short , as mad , bar , &c. 2. Secondly a Dipthong maketh a Syllable long , as train , cool , caul , feel , &c. where note that e ought never to be written at the end of a word if the last Syllable have a Diphthong in it , as con-strain , un seen , not con straine , unseene . Except when s follows the Diphthong , soft th , and c and g , as hoise , noise , seeth , voice , choice , siege . 3. Thirdly , gh after a Vowel in the end of a word makes the Syllable long , as high , night , thigh , sight . There are likewise three things which make a Syllable short . 1. First many Consonants joined together , as first , durst , distrust , contemptible . 2. Secondly , the doubling of a Consonant , as pil-low , billow , stag-ger , stam-mer . 3. Thirdly , when e is left out at the end of a word , as if from fare , bare , mate , you take away e the Syllable will be short , as far , bar , mat , also bed , quit , knit , &c. Words of one short Syllable need never have the last Consonant doubled , to shorten its sound , as met , trip , flip , at , top , gut , except in some few words where custom has prevailed to make a distinction from other words of the same sound but of different significations , as Ann , cann , butt , inn , interr . Chap. 27. Some Observations of the several Letters of the Alphabet . A A being placed before l and after a Consonant , is sounded broad and long like the Diphthong au , as call , shall , ball , bald , shalt , s●ald , malt , mall . Therefore when the Diphthong au is sounded before l , it is most commonly written with an a only , except in Paul , brawl , caul , assault , fault , bawl . A is seldom or never sounded after e or o in the same Syllable , that is in the improper Diphthongs ea and oa , as goat , great , re peal , gr●an , except in heart , hear-ken , where the e loseth its sound , also the A is scarce sounded in mar-ri-age , car-ri-age , par-li a-ment , these words being sounded marridg , carridg , parliment . Likewise in some words taken from the Hebrew , where it is either placed before its self or before o , as in Isaac , Canaan , Pharaoh , &c. which words are sounded Isac , Canan , Pharo . B B loseth its sound when it happens in the end of a word after m , or before t , as in womb , climb , thumb , dumb , lamb , limb , doubt , doubtful , debt , subtil , &c. C C is sounded like K when it comes before a , o , and u , as camp , come , count , cool , cure , except in some words taken from the Latin where e follows it , thereby making the Latin Diphthong ae or ae as Caesar or Caesar. And C being placed before e , i , and y , is sounded like s , as in place , ice , mice , city , certain , cypress , exceed . Likewise when C comes before l or r , it is always sounded like K , as clout , cream , clear , croud . Ch in words which are purely English hath a peculiar sound , whether placed before or after a Vowel , first before a Vowel , as child , chance , cheap , chuse , chosen , churl ; secondly after a Vowel , as reach , teach , such , touch , preach , breech , rich , roch . But where you find Ch , in some few words that are of an Hebrew , or Greek derivation , it is for the most part sounded like K , as Christ , Christopher , Chorus , Character , Achan , Lachish , Malchus . And when s is written before it , as Scheme , Scholar , &c. Except when a Consonant follows C● , as in Archbishop , Archdeacon , &c. C when written between a Vowel and K is not at all pronounced , as black , stick , sick , thick , beck , block , suck . Also when sc comes before e or i , then C loseth its sound , as Science , descent , conscience . But before a , o , or u , it keeps its sound , as scarce , score , fourscore , scul , scumin . And here note by the way that C is never placed between n and k , as thank , think , brink , not thanck , thinck , brinck , &c. D In all words where g follows d , there is d very scarcely if at all sounded , as badge , badger , bedge , bridge , dodge , budge . E When an e is found in the end of a pure English word it is very seldom sounded , only it serves either to prolong the Syllable , as bare , care , fare spare , cure , cole , which without e would be short , as bar , car , far , spar , cur , col . Also when s follows it in the end of a word it serves to prolong the last Syllable , as sumes , consumes , names , robes . Except me , ye , he , be , we , the. Or when it follows c , or g , it serves to soften their pronunciation , as rage , race , stage , scarce . When it follows l , or r , it is to be sounded deeply as if it went rather before them , as cable , able , noble , candle , acre , tygre . E loseth its sound in George , Tuesday , Scrivener , Beauty . Also it is generally added for beauties sake in the end of words after o , and u , as roe , due . And here note that e must never be written at the end of a short Syllable , as art , defend , convert , not arte , defende , converte . Except in a few short Syllables which are customarily written with an e after them , as come , some , done , gone , behove , shove , glove , live , love , give , above , move . Also e must never be written at the end of a word after a double Consonant , as bless , goodness , not blesse , goodnesse ; except when another Syllable is thereby added to the word , as 〈◊〉 . Likewise when a Syllable is added to a word that endeth in 〈◊〉 , then shall e be left out , as grace , gracious , shame , shaming , blame , blaming . Except the Syllable added thereto beginneth with a Consonant , for then must e be continued , as grace , graceful , not gracful , shame , shameful , not shamful . Except also when ge , and ce , come before the termination able , as charge , chargeable , not chargable , peace , peaceable , not ●acable . Except likewise words that are compounded of there , here , where , as therein , therefore , hereafter , heretofore , wherein , wherefore . E is commonly sounded in the end of such words as are derived from the Greek or Latin , as Phebe , Epitome , premunire , &c. Also many English words that have the sound of e in the end thereof , are written with ey , as countrey , valley , barley , parsley . F is always sounded or pronounced alike . G G is never sounded when it precedes m or n in the same Syllable , as phl●gm , sign , reign , design , sovereign , gnaw . G before e , i , and y , is commonly sounded soft , as ge-ne-ra-ti-on , gi-ant , gyp-sie , spun-gy . Except give , gift , to-ge-ther , be-gin , gir-dle , gird , girl , girt , tar-get , Gil-bert . G before a , 〈◊〉 , and u , or before its self or any Consonant in the same Syllable is always sounded , or pronounced hard , as gave , go-vern , gum , glass , grass , dig-geth , big-ness . When n goes before g , it is likewise pronounced hard , as ●ang , sing , ring , long , fin-ger , an-ger . Except e follows g in the same Syllable , as range , singe , &c. The sound of gh is various , as , 1. If it be in the beginning of a word is sounded like g hard , as Ghost , ghostly , &c. 2. When gh is found in the end of many words it is pronounced like f , as laugh , enough , cough , tough , rough , hough , &c. 3. In some words it is not sounded at all , but only serves to make the Syllable long , as through , dough , night , might , sight . 4. But when two Syllables are parted according to rule between g , and h , then is g sounded hard , as hog-beard , &c. H H is generally defined to be no Letter but only a note of ●spiration , or breathing . H in the beginning of words after g , or r , is not at all sounded , as ghost , ghostly , rheum , Rhe-to-rick , Rhe-nish , Thomas , Scholar , Scheme . I I is not sounded many times when it follows u in the same Syllable , as juice , fruit , bruit , suit , bruise , re-cruit . Likewise its sound is neglected in the improper Diphthong ei , either , neighbour , neither , &c. Also in adieu , cousin , fashion . When i comes before r , it is commonly sounded like 〈◊〉 , as frst , thirst , irk-some , third , fir , bird , thir-ty sir , firm . Except when it begins a word , as ir-ra-ti-o-nal , ir-re-ve-rent , &c. Or when e follows it at the end of a word , as con-spire , de-sire , fire . K When c hard is pronounced hefore e , i , or n , that word must be written with k , as ken-nel , kill , know , knowledge , &c. But when c hard is sounded before a , o , and u , that word must be written with c , and not with k , as co-ver , can-dle , custome ; not kover , kandle , kustome . L L very often loseth its sound when it comes between a and f , and a and k , as calf , half , stalk , walk , balk . Also it loseth its sound in balm , calm , salve , alms , Sal-mon , fal-con , Lin-coln , Bris-tol , Hol-born , folks . When a Monosyllable ends with l , it is commonly doubled , as call , shall , bill , shell , well , will ; except a Diphthong precede it , as boil , fail ; but if a Consonant be added to the end , then it loseth an l , as shalt , wilt , &c. And if a word of more than one Syllable ends with l , it must never be double , as gospel , not gospell , principal , not principall . M In what word soever m is found , it is never neglected , but always pronounced . N When n follows m at the end of a word it is seldom or never sounded , as condemn , contemn , solemn , Autumn , hymn , limn ▪ O O is variously sounded , viz. sometimes short , as not , got , s●t ; and sometimes long , as know , tow , bestow , go , to● , wo , so , &c sometimes like u , as smother , brother . And in some words it is not sounded at all , as youth , courage , courtesie , double , trouble , dou●let , people . And many times it must be written before n in the end of 〈◊〉 word , when it is not pronounced , as Apron , Iron , &c. P Ph whether it is in the beginning , middle , or end of a word , is sounded like f , as Phi-lo so pher , Phi lip , Or-phan , Tri-umph . Except in some few words where the Syllables are divided between p , and h , as Shep-herd , up-hold , Clap-ha● . When p comes between m and t in the end of a word , then p loseth its sound , as contempt , exempt ; also in Symptom , redemption ; also in psalm , psaltery , &c. Q Q is never written without u , as queen , quill , que-sti-on , quar-rel . Sometimes qu is sounded like k , as ob-lique , pub-lique , relique , ex-che-quer , liquor , &c. R R is always sounded , but never variously wheresoever it is found ; as fa-ther , ra-ther , &c. S The Letter S is either long or short , which are always to be observed in their places , viz. Long s must be always written in the beginning and the middle of words , as such , some , con-sume , con-spire . And short s or little s is always in the end of a word , as sins , sons , hands . Also if there be a double ss , the last ought to be a short s , as assurance , sessions , good-ness . S is sometimes pronounced hard , and sometimes soft , hard , as con-se-quence , se-date , con-sume , and soft like z , as bars , sons , sins . In some words it is not sounded at all , as Isle , Viscount , Island , which are to be read I le , Vicount , Iland . T Ti before a Vowel is generally sounded like si , as in redemption , nation , salvation , satiate , patience . Except s , r , or n , go before it , as question , christian , combustion , courtier , voluntier , frontier , Antioch . Except also when a Syllable beginning with a Vowel is annexed to a word ending in ty , then shall ty be changed into ti keeping its sound , as mighty , mightier , lusty , lustier , 〈◊〉 , pitious , lofty , loftier . U U is sounded sometimes short , as full , dull , but , and sometimes long , as ru-ral , bu ri al , and sometimes it is sounded like w , as an-guish , lan-guish , lan-guage . When u comes between g and another Vowel , it is seldom or never sounded , as plague , tongue , guide , guard , gui●t , catalogue , prologue , &c. Note that u is never in the end of a word except e be after it , as ver-tue , is-sue , con-strue . W W after o , is not sounded , as in grow , sha dow , win-dow ; except in vow , bow , sow , cow , &c. Also when w comes before r in the beginning of words , as wrath , wrought , wre-st●e , wretch , write , wran-gle , &c. X X is a Letter compounded of c and s , and like them it is always sounded where-ever it is found , as wax , like wacs , ax , like acs , &c. Y Y before a Vowel is a Consonant , and is to be sounded , as yet , you , youth , yon-der , york . But when it follows a Consonant then it is accounted a Vowel , as migh-ty , lof-ty , beau-ty . Z Z is generally pronounced like soft s , as zeal , zi-on , a-zimuth , &c. J When j Consonant comes before a Vowel , as it always does , then it is pronounced like soft g when it comes before e , i , or y , as jea-lous , joy-ful , joy , judge , E-li-jah , &c. and wheresoever it is found it is to be so pronounced , and its shape as well as pronounciation differs from i Vowel , being always writted thus j. V V Consonant is always placed before a Vowel , and hath a sound peculiar to its self , as in vertue , vile , vain , verily , vice , Saviour ; Likewise v Consonant differs in shape from ● Vowel , as well as in sound , it being always made thus v. Concerning the Diphthongs read the seventh Chapter . Chap. 28. Of the use of Great Letters , commonly called Capitals . THE Capitals in writing are of very great use , and are to be used according to the followng directions . I. All proper Names , whether of Men or Women , as Adam , Iames , Iohn , Mary , and also the Sirnames of Men. Likewise the proper Names of Countreys , Cities , Towns , Arts , Sciences , Dignities , Titles of Honour , Offices , Days , Months , Winds , Places , Heathenish Gods , and Goddesses , ●ivers and Islands . II. Every Sentence beginning after a period is to be begin with a Capital , and in Poetry every Verse or Line must begin with a Capital . Also every Book , Chapter , Verse , Paragraph , and Section . III. When the words of another is quoted , they ought to begin with a great Letter , as Matth. 2. 8. And he sent them to Bethlehem , and said , Go and search diligently for the young Child , &c. IV. The Titles of Books , as THE HOLY BIBLE , and many times very remarkable Sentences are written with great Letters , as Rev. 17. 5. And upon her Forehead was a Name written , MYSTERY , BABILON THE GREAT , THE MOTHER OF HARLOTS AND ABOMINATIONS OF THE EARTH . Also the Numeral Letters are written with Great Letters , as the date of the present year is MDCXCII . Chap. 29. Of the Points , or Pauses , and Marks . IN reading or writing you are always to observe the Points or Stops , for they give great Life , and Light to the understanding in Reading . And they be these which follow , viz. A Comma marked thus ( , ) a Semicolon thus ( ; ) a Colon thus ( : ) a Period or full stop thus ( . ) an Interrogation noted thus ( ? ) and a note of Exclamation or Admiration noted thus ( ! ) and a Parenthesis which is noted thus ( ) with two Semicircles . A Comma is a stop of the smallest time , and requires but very little breathing ; as for Example , But ye are a chosen Generation , a Royal Priesthood , an holy Nation , a peculiar People , &c. A Semicolon is of somewhat longer time than a Comma ; as in this Example , His very Countenance begat in me a trembling ; his Words were terrible as thunder ; his Rage is sufficient to compleat my ruin . A Colon is a middle pause between a Comma and a Period , and is generally in the middle of a Sentence . Example , If I hold my Tongue , I can expect no relief : And if I speak , I fear I shall be rejected : Unless by some powerful Expressions , I can make her really sensible of what I truly feel . A Period is the longest pause , or breathing time , and is always put at the end of a compleat Sentence . Example , Eternity is an undeterminable Circle , wherein the persons of all Ages , shall be encompassed in endless weal or woe . An Interrogation or Interrogative point is always placed after a question . Example , Whither can I fly for redress ? Or to whom shall I apply myself for relief ? A note of Admiration is always put after a Sentence wherein something is exclaimed against or admired at ; as , Oh the vain pleasures of the World ! Oh that ever Man should be charged with the guilt of his own ruin ! A Parenthesis is when a Sentence is inserted in the middle of another Sentence between two Semicircles , which if wholly neglected or omitted , the sense would not be spoiled . As , I shall in a few words ( because many would be too tedious ) give you an information of the whole matter . Here if these words because many would be too tedious , were left out , yet the sense would be perfect . There are also many other marks which you will meet with in Reading , as sometimes when a Vowel is left out in a word , you will find this mark over its place , viz. ( ' ) which is called Ap●strophus , as th' intent for the intent , consum'd for consumed , I 'll for I will , &c. In compound words you shall find a short Line made between them , which is called an Hyphen , as Self-denial , Time-servers , Will-worship ; and likewise when you have not room to write the whole word in the Line , you ought to make the same mark at the end , to signifie the rest of the word to be in the other Line . And you will find such marks as these in the Bible and other Books , viz. ( * ) ( † ) and ( ‖ ) which are called notes of reference , and do serve to refer or direct the Readers to look for some proof , note , or observation , which you will find at the same mark in the Margin . And sometimes the Letters of the Alphabet are inserted in a smaller Character for the same purpose . When another Author is quoted in his own words , commonly the beginning of each Line of the same is distinguisht from the rest by a double Comma reversed , thus ( " ) When in writing any Word or Sentence is forgotten , then must it be written over the Line , and this mark ( ⁁ ) called a Caret must be made under the Line pointing between the words where the said word or sentence must come in . Likewise at the beginning of a new Head or Section , there is by most Authors set this mark § . An Appendix to the English School-master compleated , containing the Principles of Arithmetick , with an Account of Coins , Weights and Measures , Time , &c. Copies of Letters , and Titles of Honour , suitable for Men of all Degrees and Qualities , Bills of Parcels ▪ Bills of Exchange , Bills of Debt , Receipts , and several other Rules and Observations fi● for a Youths Accomplishment in the way of Trade The Principles of Arithmetick . ALL number is expressed by Nine Figures and a Cypher , which are thus Charactered , viz. 1 one , 2 two , 3 three , 4 four , 5 five , 6 six , ● seven , 8 eight , 9 nine , 0 a Cypher When any number is given whose value you would know , you are to consider that the first figure to the Right Hand signifies but its own single value , the second is ten times its value , as if it be 4 it signifies forty ; the third is an hundred times its own value , as if it be 4 it signifies four hundred , according to the following Table . First place 1 Units . Second place 2 Tens . Third place 3 Hundreds . Fourth place 4 Thousands . Fifth place 5 X of Thousands . Sixth place 6 C of Thousands . Seventh place 7 Millions . Eighth place 8 X of Millions . Ninth place 9 C of Millions . To value this or any other number , begin at the 1 saying , Units , Tens , Hundreds , Thousands , &c. But to read it you must begin at the 9 saying , Nine Hundred Eighty Seven Millions , Six Hundred Fifty Four Thousand , Three Hundred Twenty One. In like manner this number , viz. 507 is five hundred and seven , and 3426 is three thousand four hundred twenty six . ADDITION of Integers . ADdition teacheth to add divers numbers together , and to bring them to one total Sum. When two or more numbers are given to be added together you are to place them units under units , tens under tens , hundreds under hundreds , &c. Then add up the first row in the place of units , and if they be under ten set down what they come to , but if it come to ten , or more than ten , or twenty , or thirty , &c. then set down the excess ; and for every ten carry an unite to the next row , and proceed in the same manner with every row till you come to the last , and whatever that comes to you must set it all down . Example , Let it be required to find the Sum of 234 and 341 and 923 ; first I put them one under another as followeth . 234   341   923   1498 Sum. Then I begin at the units , saying 3 and 1 and 4 make 8 , which I put under the Line , then to the next row , saying 2 and 4 and 3 make 9 , which I also set down under its row , then to the next row , saying 9 and 3 and 2 make 1● , which I set down , and the work is done , and I find the Sum of these three numbers to be 1●9 . Again let it be required to find the Sum of 796 , ●87 , 479 , and 316 ; first I set them down in order , as you see in the Margent . 796   587   479   316   2178 Sum. Then I begin with the row of units , and find it to come to 28 , wherefore I place the 8 under the Line , and carry 2 to the next row for the two tens , and I find that row to come to 27 , wherefore I put the 7 under the Line and carry 2 to the next row , then I add up the third row and find it to come to 21 , all which I set down because it is the last row , and so I find the Sum of these 4 given numbers to be 2178. More Examples for Exercise follow . 6548 57432 4246 5807 3721 438 372 2134 4063 7962 486 369 792 431 592 481 413 876 34 234 576 43 7 75 16113 67182 5737 9100 Addition of Money . NOte that 4 farthings is one penny , 12 pence is on● shilling , and 20 shillings is a pound Sterling , or English Money . The Character of pounds is l. of shillings is ● . of pence d. and of farthings q. When it is required to add pounds , shillings , pence and farthings together , you are to place them in rows one under another , viz pounds under pounds , shillings under shillings , pence under pence , and farthings under farthings , and under all draw a Line . Example , let it be required to add these two Sums of Money together , viz. 195 l. 17 s. 10 d. 3 q. and 86 l. 15 s. 09 d. 2 q. First I place the given Sums one under the other and draw a Line , as you see in the Margent . l. s. d. q. 195 17. 10. 3. 86 15 09 2 2●2 13 08 1 Then I begin with the farthings , saying 2 and 3 is 5 farthings , which is 1 peny and a farthing , wherefore I make a dot at the ● for the peny , and put the 1 farthing under the Line , then I go to the row of pence , saying 1 that I carry and 9 is 10 and 10 is 20 pence , which is 1 shilling 8 pence , wherefore I put a dot at 10 for the shilling , and put the 8 pence under the Line , and carry 1 for the dot to the shillings , and say 1 that I carry and 15 is 16 and 17 is 33 shillings , which is 1 pound 13 shillings , wherefore I make a dot at the 17 for the 1 pound , and put the 13 shillings under the Line , then I proceed to add up the pounds , saying 1 that I carry and 6 is 7 , and 5 is 12 , wherefore I put down 2 and carry 1 to the next row , &c. so that I find the Sum to be 282 l. 13 s. 08 d. 1 q. Other Examples for Exercise follow . l. s. d. q. 374 17. 09. ● . 297 16. 10 3 3●2 14 1 1. 462 1● . 04. 2 695 12. 08 3. 83 11 10. 2 2257 12 07 1 l. s. d. q. 892 12 04 1 437 16. 10. 3. 198 09 11. 1 ●76 12. 07. 0 358 10. 09 3. 84 14 11. 2 2●18 17 06 2 The like is to be understood in all additions , whether of Measure , Weight , Time , &c. observing how many of the lesser denominations go to make one of the greater . Subtraction of Integers . SUbtraction teacheth to take a lesser number from a greater , and gives the remainder or difference . Example . Let it be required to subtract 2234 from 4678 , here must I place the lesser number under the greater in such manner as if they were to be added together , and draw a Line under them , as you see in the Margent . 4678   2234   2444 Rem . Then I begin at the units , saying 4 out of 8 and there remains 4 , which I place under the Line , and go to the next , saying 3 out of 7 and there rests 4 , which I also put under the Line , and go to the next , saying 2 out of 6 and there rests 4 , which I also put under the Line , and proceed to the next , saying 2 out of 4 and there remains 2 , which I put also under the Line , and the work is done , and I find the remainder to be 2●44 . But if in the work of this nature , the undermost figure chance to be greater than that which you are to subtract it from , which is the uppermost , then must you borrow 10 and add to the uppermost , and subtract the undermost from their Sum , but then for what you borrowed , you must remember to add 1 to the next lowermost figure , as in this Example , let it be required to subtract 3578 from 8495 , first I place them as is before directed , and as you see in the Margent with a Line drawn under them , then I begin at the place of units , saying 8 from ● I cannot , wherefore I add 10 to the ● , and it makes 1● , wherefore I say 8 from 1● and there remains ● , which I put under the Line , then I proceed to the next figure , saying ● that I borrowed and 7 is 8 out of 9 and there rests 1 which I put down , and proceed , saying 5 out of 4 I cannot , but 5 out of 14 and there remains 9 , which I put under the Line , and proceed , saying 1 that I borrowed and 3 is 4 out of 8 and there remains 4 , which I put under the Line in its place , and the work is done , and I find the remainder to be 49●7 . The like is to be understood of any other . 8495 3578 4917 Examples for Exercise follow . 4735 304● 9706 576418 918 5●8 3907 82443 3●17 2524 5799 49●975 Subtraction of Money . YOU must place the given numbers one under the other as you were directed in Addition , but with this caution , in Subtraction put always the lesser number undermost , and under all draw a little Line , and begin your Subtraction at the right hand , with the least denomination . But if the lower number of any denomination is greater than the uppermost , then borrow 1 of the next greater denomination , and from that subtract the lowermost , and what remains add to the uppermost , and set their Sum under the Line , and for that you borrowed add 1 to the next denomination , and proceed , &c. Example . Let it be required to subtract 346 l. 08 s. 07 d. 2 q. from 723 l. 4 s. 10 d. 1 q. first I set them down as you see in the Margent , and begin at the farthings , saying 2 farthings from 1 I cannot , wherefore I borrow 1 peny of the next denomination which is 4 farthings , and say 2 from 4 and there remains 2 , which I add to the 1 farthing and they make 3 , wherefore I put 3 under the Line in the place of farthings , then I proceed to the pence , saying 1 that I borrowed and 7 is 8 , from 10 and there remains 2 , which I put under the Line , and proceed to the shillings , saying 8 from ● I cannot , wherefore I borrow 1 of the next denomination which is 20 shillings , and say 8 from 20 and there remains 12 , which being added to the uppermost figure 4 makes 16 , which I put under the Line , and proceed , saying 1 that I carried and 6 is 7 , out of 3 I cannot , but 7 out of 13 and there remains 6 , &c. so when the work is finished I find the remainder to be 35 l. 16 s. 02 d 3 q. and the like is to be understood of any other . l. s. d. q. ●23 04 10 1 ▪ 36● 0● 07 2 35● ●6 ●2 3 Examples for further Exercise follow .   l. s. d. q. l. s. d. q. From 794 13 08 0 462 15 00 1 Subtract 84 17 10 2 75 11 07 3 Remains 709 15 09 2 387 ●3 04 2 If there be a Sum of Money lent , and part thereof received at several payments , and you would know how much remains unpaid ; add the several payments into one Sum , which must be subtracted from the Sum lent , and the remainder will give you what remains due . As in the following Example .   l. s. d. q. Borrowed 7.0 00 00 0   124 17 09. 2   48 16. 11. 0 Paid at several Times 34 0● . 09 ● .   68 14 10. 1   38 12. 05. 2.   97 09 08 2 Paid in all 412 19 06 2 Rests due 287 00 05 2 The like is to be understood of other denominations , as Weight , Measure , Time , &c. I might proceed to the other Rules of Arithmetick , but that being more fit for a large Volume than this small Treatise , I shall therefore wave it , and content my self with giving you the several Tables of Coins , Weights , Measure , Time , &c. A Table of English Coin. 4 farthings   a Peny . 4 pence   a Groat . 12 pence   a Shilling . 2 Shillings six pence   half a Crown . 5 Shillings make a Crown . 6 Shillings eight pence   a Noble . 2 Nobles or 13 s. 4 d.   a Mark. 3 Nobles or 20 Shillings   a Pound Sterling . Troy Weight . THE Original Weight used in England is deduced from : Grain of Wheat gathered out of the middle of the Ear and well dried , and this Weight is called a Grain , from whence is deduced the following Table . 2● Grains   a Peny weight . 20 Peny weight make an Ounce . 12 Ounces   a Pound . With the foregoing Weights are weighed Bread , Gold , Silver , Jewels , and Electuaries . Apothecaries Weights . THE Weights used by Apothecaries are no other than Troy Weight , only the Pound is otherwise subdivided , according to the following Table . 20 Grains   a Scruple   ℈ 3 Scruples make a Dram thus marked ʒ 8 Drams   an Ounce   ℥ 12 Ounces   a Pound   lb Averdupois Weight . 16 Drams   an Ounce . 16 Ounces   a Pound . 28 Pounds make a Quarter of an Hundred . 4 Quarters   an 100 weight , consisting of 112 l. 20 Hundred   a Tun Weight . By this Weight is weighed all Grocery Wares , Butter , Cheese , Flesh , Wax , Pitch , Rosin , Tallow , Hemp , Lead , Iron , Copper , Tin , and all other Commodities from whence there may issue a waste . All Measure whether of Longitude or Capacity are deduced from a Barley Corn , whence comes the following Table . Of Long Measure . 3 Barley-Corns   an Inch. 12 Inches   a Foot. ● Foot   a Yard . 3 Foot 9 Inches make an Ell English. 6 Foot   a Fathom 5 yards and an half   a Pole or Perch . 〈◊〉 Poles or Perches   a Furlong . 8 Furlongs   an English Mile . By the foregoing Table you may understand that 5 yards and an half , which is 16 foot and an half make a Pole or Perch , from whence is deduced the following Table . Of Land Measure . 4● square Perches or Poles make a Rood , or quarter of an Acre . 4 Roods   an Acre of Land. Liquid Measure , ( by which is sold Beer , Ale , and other Liquors , ) i● by the Statute in this manner setled , viz. The Beer Gallon to contain 282 Cubick Inches , each Inch being a solid like a Dye , each side of which is an Inch in length , viz. its length , breadth and thickness an Inch. And the Gallon is customarily subdivided into Pottles , Quarts and Pints , whence is deduced the following Table . Of Liquid Measure . 2 Pints   a Quart. 2 Quarts   a Pottle . 2 Pottles   a Gallon of 282 solid Inches . 8 Gallons make a Firkin of Ale , Soap , Herring ▪ 9 Gallons   a Firkin of Beer . 10 Gallons and an half   a Firkin of Salmon , or Eels , 2 Firkins   a Kilderkin . 2 Kilderkins   a Barrel of Beer or Ale. But Wine Measure hath by the same Statute 231 Cubick Inches to the Gallon , which is likewise subdivided into Pottles , Quarts , and Pints , as Beer Measure is , from whence cometh the following Table . Of Wine Measure . 2 Pints   a Quart. 2 Quarts   a Pottle . 2 Pottles   a Gal. of 23● solid Inches ▪ 42 Gallons make a Tierce of Wine . 63 Gallons   a Hogshead . 2 Hogsheads or 3 Tierces   a Pipe or Butt . 2 Pipes , or Butts , or 6 Tierces   A Tun of Wine . Wheat , Barley , and other Grain , Salt , Coals , Sand , all dry Goods , and such as have substance in them are measure ▪ by dry measure , the least of which is a Pint , according to the following Table . Of Dry Measure . 2 Pints   a Quart. 2 Quarts   a Pottle . 2 Pottles   a Gall. 2 Gallons   a Peck . 4 Pecks make a Bushel Land Measure . 5 Pecks   a Bushel Water Measure . 8 Bushels   a Quarter 4 Quarters   a Chaldron . 5 Quarters   a Weigh . Of Time. The Original measure of Time is a year , which is the time where in the Sun performs his natural Motion or Course , through the Ecliptick , beginning at Aries , and so going through the 12 Signs of the Zodiack , till he return again to the first scruple of Aries , which he performs in 365 days , and almost 6 hours , and the 6 hours are reckoned only every fourth year , and then there is a day extraordinary added to the year making in all 366 days , which day is added to February , and that year is called Leap-year ; and according to the foresaid measure is the year divided and subdivided , as in the following Table of Time. 60 Seconds   a Minute . 60 Minutes   an Hour . 24 Hours   a Day natural . 7 Days make a Week . 4 Weeks   a Month. 13 Months , 1 Day and 6 Hours   a Year . But the Year is most commonly divided into 12 unequal Calendar Months , whose Names and the number of days contained in each take as followeth .   days Ianuary 31 February 28 March 31 April 30 May 31 Iune 30 Iuly 31 August 31 September 30 October 31 November 30 December 31 But they are more briefly sum'd up for the Memory in the four following old Verses . Thirty days hath September , April , Iune , and November , But February ▪ hath twenty eight alone , And all the rest have thirty one . Titles of Honour for Superscription , or Appellations in Letters . TO the King , Sir , or may it please your Majesty , Sacred Sir , Dread Sovereign . To the Queen , Madam , or may it please your Majesty . To the Princess , Madam , or may it please your Royal Highness . To a Duke , My Lord , or may it please your Grace . To a Dutchess , Madam , or may it please your Grace . To a Marquess , My Lord , or may it please your Lordship . To a Marchioness , Madam , or may it please your Ladiship . To an Earl , My Lord , or Right Honourable . To a Countess , Madam , or Right Honourable . The same to a Viscount , or Viscountess . To a Baron . My Lord , or may it please your Lordship . To a Baroness , Madam , or may it please your Ladiship . To all Ladies and Gentlewomen indifferently , Madam . To a Baronet , or Knight , Sir , or Right Worshipful . To an Esquire , May it please your Worship . To a Gentleman , Sir , or much Honoured . To the ●lergy , Reverend Sir , the Archbishop of Canterbury ▪ having the Stile of Grace , and the other Bishops the Stile of Right Reverend . Several Examples of Letters , Bills of Exchange , Bills , of Parcels , Receipts , &c. A Letter from a Youth at the Writing School in London , to his Father in the Country . Honoured Father , London , March 11. 1692. MY Humble Duty presented to you , and to my Mother , and I Return you hearty thanks for all your kindnesses shewed to me ; I make bold to present you with these few Lines , being the first fruits of my Endeavours in this kind , and I hope you will please to pardon the imperfections of my performance , and I doubt not but in a short time to ●e so well accomplished as to give you a Better Account of the Expence of your Money , and my own Time , wherefore at present let me crave your acceptance of this from . To his Honoured Father Mr. Gardner at Southampton . These Your Dutiful Son Thomas Gardner . A Letter from one lately gone from School to his School-fellow there . Loving School-fellow , London , May 12. 1692. I Return you many thanks for all the kindnesses which you have been pleased formerly to shew me , and I now heartily wish when too lat● , that while I had the opportunity which you now enjoy , I had made a better Improvement of those precious moments , which I then too much slighted and neglected , therefore I advise you as a Friend , to beware of that harm which I am now too sensible of , and know that the greatest time you can spend in Learning will be too little to gain Perfection in those most Exquisite Arts which you are now Labouring after ; your parents are in Health , as are all other Friends who desire to be Remembred to you , I pray as you proceed let me have now and then a Line or two from you , and you will much oblige To his Loving Friend Mr. Thomas Swingler . These Your Loving School fellow Iohn Clark. A Letter from a young Man newly out of his Apprenticeship , to his Friend for Correspondence . Respected Sir , London , May 10th . 1692. I Have now finished my seven years Apprenticeship , and am by the assistance of God and my Friends , just entring into the World for an Imployment , and being conscious to my self that my Trade depends upon Acquaintance , makes me thus bold to renew our former intimacy ; Sir , if it lyes in your power to be serviceable to me in the way of my Trade , either by your self or other Friends , I shall not only thankfully acknowledge your kindness , but to the uttermost of my power approve my self as I am , To Mr. Lucas at Lambeth . These Your Friend and Servant Iames Dendy . A Letter from a Shopkeeper to another for Goods which he wanteth . Sir , Whitchurch , March 12. 1692. I Am credibly informed by a Friend , both of yours and mine , that you are very well provided and stockt with sundry parcels of Wares , such as I have at present some occasion for , wherefore I am willing to essay a Trade with you , and would pray you for the present to send me about Ten Pounds worth for a sample , and if I find they are for my turn , I shall immediately give you order for Forty or Fifty Pounds worth more , send them , and the lowest price of them by the first opportunity , and I shall be punctual in making payment according to order , in the mean time I remain To Mr. Sherbrook in London . These . Yours to command Tho. Wickstead . The Answer . Sir , London , March 20. 1692. YOurs of the 12th Inst●nt I received , and according to your order have sent you ( by John Jones the Carrier ) a parcel of Goods , which come to 10 l. 7 s. 6 d. The particulars whereof , together with their prizes are inserted in a Bill of Parcels herein inclosed , for the payment whereof , I shall give d● order in my next , in the mean time I hope they will prove to your satisfaction , and be the foundation of a further Acquaintance and Dealing with you , and assur● your self , that whatsoever you shall intrust to my charge shall be performed and managed with the greatest Candour and Fidelity imaginable , and if there happen any miscarriage in packing or ordering of Wares before they come to your hands , upon notice given thereof it shall be amended or allowed for , to your own content , in the mean time I take leave , and subscribe my self To Mr. Tho. Wickstead at Whitchurch These . Your Friend and Servant Tho. Sherbroo● ▪ A Servants Letter to his Master . Sir , Taunton , April 4th 1692 YOurs of the 28th past I received , and shall be as careful in the management of your Affairs as if they were immediately my ow● Concerns , my diligence shall always supply your Room in your absence I have discoursed Mr. Gilbert concerning your Affair , and he seem very inclinable to have an accommodation therein , and intends speedily to write to you himself concerning it ; as for those Goods which 〈◊〉 ordered me to send you , I find them not for your turn ; but Mr. Burgi● has some excellent Perpetuana's , which if you approve of , I shall send you by the first opportunity after Order ; I have inclosed som● Samples with their prices : Thus with my humble Service to yourself , &c. I remain Your faithful Servant to my power Iohn Patteso● Forms of Bills of Exchange . A Copy of an Inland Bill . London , March 14th . 1692. AT 10 days sight of this my only Bill of Exchange , pay to Mr. John Brewer of Salop , or his Order , the Sum of Fifty Pounds , currant Money , for the value Received here of Joseph Pebworth , make good payment , and place it to the Accompt of To Mr. William Compton of Salop This. Your obliged Friend William Costin ▪ When the Bill is accepted , and day of payment cometh , the Receiver gives a Discharge for the same on the out side of the Bill , as followeth . March 27 ▪ 16. 2. REceived then the full contents of the within written Bill of Exchange ; I say Received by me John Brewer . A Copy of an Out-land Bill , the First . London , March 14 , 1692. for 3● l. Sterling , at 3● Shillings 8 d. Flemish . AT Double usance , pay this my first Bill of Exchange unto Hendrick Coopman , or Order , Three hundred pound Sterling , at Thirty four Shillings Eight pe●ce Flemish per l. Sterling , for the value of John Pennington , and place it to the Account , as per advice from Yours Iames Goodman . To Mr. Tho. Corbet Merchant in Amsterdam . A Copy of the Second Out-land Bill . London , March 14. 1692. for 300 l. Sterling , at 34 Shillings 8 Pence Flemish . AT Double usance pay this my second Bill of Exchange , my first not paid , unto Hendrick Coopman , or Order , Three hundred Pounds Sterling , at Thirty four Shillings Eight Pence Flemish per l. Sterling , for the value of John Pennington , and place it to Account , as per advice from Yours Iames Goodman . For Mr. Tho. Corbet Merchant in Amsterdam . A Copy of the Third Bill of Exchange . London , March 14. 1692. for 300 l. Sterling , at 34 Shillings 8 Pence Flemish . AT Double usance pay this my third Bill of Exchange , my first and second not being paid , unto Hendrick Coopman , or Order , Three hundred Pounds Sterling , at Thirty four Shillings Eight Pence Flemish per l. Sterling , for the value of John Pennington , and place it to the Account , as per advice from Yours Iames Goodman . For Mr. Tho. Corbett Merchant in Amsterdam . It is Customary with Merchants and others , when they have sold Goods to the Shop-keeper upon delivery thereof to give in a Bill of Parcels , the Form whereof take as followeth . A Bill of Parcels . Sold May 14. 1692. to Tho. Gardner per John Burgis , viz.   l. s. d. 164 Ells of Holland Cloth at 4 s. 4 d. 35 09 08 236 Pieces of Fine Lawns at 11 s. 6 d. 135 1● 00 286 Ells of Green Bays at 2 s. 4 d. 33 07 04 41● Ells of Linnen Cloth at 2 s. 9 d. 57 04 0● 518 Ells of Dyed Canvas at 1 s. 4 d. 34 10 08 358 Pieces of Dyed Fus●ians at 18 s. 6 d. 331 03 00 290 Pieces of white Ditto at 17 s. 246 10 00 Total Sum 873 18 08 A Shop-keepers Bill Sold to Robert Carpenter of Horsham .   l. s. d ▪ March 27. 2 Barrels of Raisons 06 06 08 April 10. a Box of Cinamon 07 09 0● May 8. 120 l. of Pepper at 2 s. 12 00 00 20. 144 l. of Tobacco at 20 d. 12 00 00 Iune 12. A Box of Sugar Candy 64 l. 3 07 06 30. A Box of Fine Sugar 240 l. at 6 d. 6 00 00 Total Sum 47 03 04 Workmens Bills . When any Bill is paid it is customary to give a Receipt for the same on the back-side of the Bill , viz. July 20. 1692. Received the full contents of the within written Bill , per me l. s. d. 47 03 04 John Burgis . A Carpenters Bill . Mr. Thompson his Bill for Work and Materials .   l. s. d. For 20 Load of Oak at 44 s. per Load 44 00 00 For 30 Load of Firr at 36 s per Load 54 00 00 For 360 Foot of Oak Plank at 3 d. 04 10 00 For 20 Thousand 10 d. Nails at 6 s. the Thous . 06 00 00 For ●5 Thousand 6 d. Nails at 4 s. 4 d. per Thous . 0● 08 04 For 9 Thousand of double Tens at 10 s. 04 10 00 For 40 l. of Large Spikes at 4 d. per l. 00 13 04 For 10●0 of Deals at 6 l. 10 s. per Hundred 65 00 00 For 94 days work for my self at 3 s. ●4 02 00 For 116 days work for my Man at 2 s. 6 d. 14 10 00 For 64 days for another Servant at 2 s. 6 d. 08 00 00 Total Sum 220 13 08 A Bricklayers Bill . Mr. Johnson his Bill for Work and Materials .   l. s. d. For 8 Thousand Bricks at 13 s. a Thousand 05 04 00 For 10 Thousand of Tyles at 17 s. a Thousand 06 10 00 For 4 Thousand of Pan Tyles at 20 s. a Thous . 04 00 00 For 800 of 10 Inch Tyles at 12 s. a Hundred 04 16 00 For 160 Ridg Tyles at 2 d. per Tyle 01 06 08 For 18 Hundred of Lime at 14 s. 6 d. 13 01 00 For 14 Load of Sand at 4 s. 10 d. 03 07 08 For 28 days for my self at 3 s. 04 04 00 For 34 days for my Man at 2 s. 6 d. 04 0● 00 For a Labourer 30 days at 20 d. 02 10 00 Total Sum 49 04 04 A Receipt in part of a Bill . REceived the 17th of March 1692. in part of the within written Bill the Sum of Twenty nine Pounds four Shillings and four Pence ; I say Received l. s. d. 29 04 0● per me Jeremiah Platton A Receipt in part for Rent . REceived July 26. 1692. of James Thompson the Sum of Four Pounds Ten Shillings in part of a Quarters Rent for his dwelling House due at Midsummer last , I say Received l. s. d ▪ 04 10 0● per me Theophilus Johnson An Acq●ittance from one that Receives Rent by vertue of an Order from the Landlord . REceived the 14th of April 1692. of Richard Powel the Sum of Eight Pounds Five Shillings for a Quarters Rent for his dwelling House , due at Lady-day last , Isa● Received for the use and by the special Order and Appointment of my Master , John Robinson Esq. l. s. d ▪ 03 05 0● per me Stephen Steward . A Copy of a full Discharge . REceived the ●th of December 1692. of Mr. James Farringdon the Sum of twenty six pounds , fourteen shillings , and ten pence , being so much due upon Accompt , and is in full of all Reckonings , Dues , Debts , Accompts , Claims and Demands what soever , to the day of the date hereof , I say Received l. s. 〈◊〉 26 14 1● per me Thomas Trumplin . Or thus . May 18th 1692. REceived then of Henry Halfgood the Sum of thirty four pounds , seven shillings , and six ●ence , being for Goods sold him at sundry times , and is in full of all Accompts , Reckonings , Bonds , Bills , Debts , Dues , Claims , and demands whatsoever to the day of the date hereof , I say Received l. s. d ▪ 34 07 06 per me Thomas Mercer . A Copy of a General Release . KNow all Men by these presents , that I Ionathan Webster of Bridgnorth in the County of Salop , Mercer , have Remised , Released , and for ever quit Claim , and by these presents do , for me , my Heirs , Executors , and Administrators , Remise , Release , and for ever absolutely quit Claim unto Ionathan Hawley of Claverley in the County aforesaid , Shoemaker , his Heirs , Executors and Administrators , all , and all manner of Actions , Suits , Bills , Bonds , Writings Obligatory , Debts , Dues , Duties , Accompts , Sum , and Sums of Money , Judgments , Executions , Extents , Quarrels , Controversies , Trespasses , Damages , and Demands whatsoever both in Law and Equity , or otherwise howsoever which against the said Ionathan Hawley I ever had , now have , or which I , my Heirs , Executors , or Administrators shall , or may have , Claim , Challenge , or Demand , for , or by Reason of any Matter , Cause , or Thing whatsoever , from the beginning of the World unto the day of the date of these presents . In witness whereof I have here unto set my Hand and Seal this twentieth day of Iune , Anno Domini 1692. Ionathan Webster . Sealed and Delivered in the presence of Simon Howland , John Thornton . A short Bill of Debt . REceived and Borrowed the 24th of Iune 1692. of Mr. Edward Eveling the Sum of Twenty Pounds of Lawful Money of England , which I promise to pay upon demand . Witness my Hand and Seal the day and year above written . Iohn Iones Teste George Lovelace . Another of the same . KNow all Men by these presents that I Nehemiah Nonesuch of Kingston in the County of Surry , Taylor , do Owe , and am firmly indebted to Nathaniel Nameless of London , Mercer , in the Sum of Forty Pounds , of Lawful Money of England ; all which I promise to pay to him or his Order on the twenty fourth day of August next ensuing the date hereof ; In witness whereof I have hereunto set my Hand and Seal this seventeenth day of May , Anno Domini 1692. Sealed and delivered in the presence of James Careless , John Hunt. Nehemiah Non such . A Bill of Debt with a Penalty . KNow all Men by these presents that I Nehemiah Nonesuch of Kingston in the County of Surry , Taylor , do Owe , and am firmly Indebted to Nathanael Nameless Citizen and Mercer of London in the Sum of Forty Pounds of Lawful Money of England ; all which I promise to pay to him or his Order on the twenty fourth day of August next ensuing the date hereof ; and that the same may be well and truly paid , I bind me , my Heirs , Executors , and Administrators , in the penal Sum of Eighty Pounds of the like Lawful Money of England . In witness whereof I have hereunto set my Hand and Seal this seventeenth day of May , Anno Domini 1692. Sealed and Delivered in the presence of James Careless , John Hunt. Nehemiah Nonesuch . A Bond for the same Sum in English. KNow all Men by these presents that I Nehemiah Nonesuch of Kingston in the Count● of Surry , Taylor , am holden , and firmly do stand bound unto Nathanael Nameless Citizen and Mercer of London , in the Sum of Eighty Pounds of good and Lawful Money of England , to be paid unto the said Nathanael Nameless , or to his certain Attorney , his Heirs , Executors , Administrators , or Assigns . To the which payment well and truly to be made I bind my self , my Heirs , Executors and Administrator● firmly by these presents . Sealed with my Seal , dated the seventeenth day of May in the fourth year of the Reign of our Sovereign Lord and Lady William and Mary , by the Grace of God of England , Scotland , France , and Ireland King and Queen , Defenders of the Faith , &c. Annoque Domini 1692. THE Condition of this Obligation is such , that if the above-bounden Nehemiah Nonesuch , his Heirs , Executors , or Administrators , or any of them do well and truly pay , or cause to be paid unto the above-named Nathanael Nameless , his Heirs , Executors , Administrators , or Assigns , the just Sum of Forty Pounds of like Lawful Money of England upon or before the four and twentieth day of August next ensuing the date hereof , without fraud or further delay , that then this present Obligation to be void and of none effect , but otherwise to be and remain in full power , force and vertue . Sealed and Delivered in the presenc● of James Careless , John Hunt. Nehemiah Nonesuch . O A Table of some Words as are alike in sound but different in signification . A Abel , Adams Son. A Bell , that rings . Able , sufficient . Accidence , Introduction to Grammar . Accidents , Chances . Accompt , Reckoning . Account , esteem . Aims , he levels at . Alms , given to the poor . Ale , Drink made of Malt. Ail , as what ails you . B Bail , Surety . 〈◊〉 bale , a pack . Bair , Cloth so called . Bayes , of the Bay-tree . Bacon , Swines flesh . Beacon , on a Hill. Beckon , to nod at one . ●all , to play with . ●aal , an Idol . ●awl , to make a noise . C ●all , to name , or call any one . ●aul , that covers the Bowels . ●aul , for the Head. ●arrier , one that carries . ●ariere , full speed . ●ease , to leave off . ●eize , to lay hold on . ●ize , bigness , or to Gild with . ●hair , to sit in . D Dam , to stop water . Damn , to condemn . Dame , a Mistress . Dear , Beloved , of great price . Deer , Venison . Defer , to put off . Differ , to vary . E Ear , that you hear with . Year , twelve months . E●re , before . Farly , betimes . Yearly , year by year . Earn , to deserve . Yearn , to compassionate : Yarn , Woollen thread . F Fain , to have a mind to . Feign , to counterfeit . Fair , Beautiful . Fare , to feed . Fear , trouble . Felon , a Thief . Fellon , a sore on the Finger . G Garden , where Herbs grow . Guardian , a Trustee . Gentle , mild . Gentile , like a Gentleman . Gentiles , Heathens . Gentles , Maggots . Grass , that groweth . Grase , to eat Grass . H Hair , on the Head. Hare , a wild Beast . Are , as we are , ye are . Air , the Element so called . Heir , to an Estate . I Idle , Lazy . Idol , a false God. Imply , to signifie . Imploy , Business . Incite , to stir up . Infight , understanding . K Keel , the bottom of a Ship. Kill , to bereave of Life . Kiln , to put Fire under . Kin , related . Ken , within sight . Keen , sharp . L Latine , speech . Latten , Tin. Lines , in writing . Loins , part of the Body . League , a Covenant . Leg , that you stand on . M Male , the he Creature . Mail , a Coat of Mail. Major , a Field Officer . Mayor , of a City , or Town . Manure , to till ground . Mannour , a Lordship , N Naught , bad , Nought , nothing . Neather , lower . Neither , none of them . O Oar , to row with . Ore , unrefined Metal . Our , belonging to us . ●our , of time . P Pail , a Vessel . Pale , Colour , or wa● . Pale , bounds . Pare , to cut off . Pair , a couple . Q Queen , the Kings Wife . Quean , a base Woman . Quarry , of Glass . Query , a Question . Quench , to put out fire , or thirst Quince , a Fruit so called . R Rase , to demolish , blot out . Race , strife in running . Raise , to lift up . Rays , of the Sun. Rake , to scrape together . Wrack , ruin . S Sale , to be sold. Sail , of a Ship. Salve , for Wounds . Save , to preserve or defend . Same , the same thing . Psalm , a Spiritual Song . T Tail of a Beast . Tale of Robin Hood . Time of the day . Thyme , the Herb so called . Tears of the Eyes . Tares , a sort of Grain . V Vacation , time of leisure . Vocation , Employment . Vain , to no end . Vane on the main Top-mast . W Walls of a Garrison . Wales of a Ship. Wait , to attend . Weight , heavy . Usual Christian Names of Men with their Original Significations . A Aaron , a Teacher . Adam , Man Earthly . Alexander , helper of Man. Ambrose , Divine , Immortal . Andrew , Manly . Anthony , flourishing . Arthur , a Bear. Augustine , Majestical . B Barnabas , Son of Comfort . Bartholomew , Son of the Wanes . Benjamin , Son of the Right Hand . Brian , shril voice . Bernard , Lord of many Children . C Charles , Couragious . Christopher , Christ Carrier . Cornelius , an Horn. Constantine , fast , firm . D David , Beloved of God. Daniel , the Judgment of God. Dennis , Divine Mind . E Edmond , Blessed , Pure . Edward , happy keeper . Enoch , taught or dedicated . Ezekiel , seeking the Lord. F Ferdinando , pure peace . Francis , free . Frederick , peaceable Reign . George , Husbandman . Gervas , all sure and firm . Gerrard , well reported . Gilbert , bright pledge . Gregory , watching . Guy , Guide or Leader . H Henry , Rich Lord , Hier●me , Holy Name . Hugh , Comfort . I Iacob , a Supplanter . Iames , a Maintainer . Iohn , gracious . Ioseph , increase of the Lord. Ioshua , a Saviour . Iosiah , Fire of the Lord. Isaac , Laughter . K Kenhelm , defence of his Kindred . L Leonard , Lion-like Disposition . Lawrence , flourishing . Lewis , Refuge of the People . Lodowick , Famous Warriour . M Mark , High. Matthew , Gods Gift . Michael , who is like God. Maurice , Moor. N Nathanael , the Gift of God. Nehemiah , Comfort of the Lord. Nicholas , Conqueror of the People . O Oliver , the Peace bringing Olive . Owen , Noble , well-born . P Paul , little , humble . Peter , a Stone , or Rock . Philip , a Lover of Horses . R Ralph , help . Randal , fair help . Richard , Rich Lord. Robert , Famous in Counsel . Roger , strong Counsel . S Sampson , there the second time . Simon , obedien● , listening . Solomon , peaceable . Stephen , a Crown . T Theodore , Gods Gift . Theophilus , a Lover of God. Thomàs , bottomless deep o● twin . Timothy , honouring God. V Vincent , Victorious . W Walter , a Pilgrim , or General . William , Defence to many . Z. Zachary , the Memory of the Lord. Usual Christian Names of Women with their Original Significations . A Abigail , the Fathers . Alice , Noble . Agnes , Chaste . B Barbara , strange . Beatrix , Blessed . Blanch , white , fair . C Catherine , Pure , Chaste . Constance , constant , firm . Cicely , gray-ey'd . D Dousabel , sweet , fair . Dido , a Man-like Woman . Dorothy , the Gift of God. E Elianor , Pitiful . Elizabith , Peace of the Lord. Emme , a good Nurse . F Frances , free . G Gertrude , all true , amiable . H Helena , pitiful . I Iane , Gracious . Ioan , Gracious . Iudith , praising , confessing . M Magdalen , Majestical . Margaret , Pearl , Precious . Mary , Exalted . P Phebe , clear , bright . Priscilla , Ancient . R Rachel , a Sheep . Rebecka , fat and full . S Sarah , Lady , Dame. Susannab , Lilly , or Rose . U Ursula , a little Bear. W Winifred , Win , get Peace . FINIS . 
A95751	----	The trissotetras: or, a most exquisite table for resolving all manner of triangles, whether plaine or sphericall, rectangular or obliquangular, with greater facility, then ever hitherto hath been practised: most necessary for all such as would attaine to the exact knowledge of fortification, dyaling, navigation, surveying, architecture, the art of shadowing, taking of heights, and distances, the use of both the globes, perspective, the skill of making the maps, the theory of the planets, the calculating of their motions, and of all other astronomicall computations whatsoever. Now lately invented, and perfected, explained, commented on, and with all possible brevity, and perspicuity, in the hiddest, and most re-searched mysteries, from the very first grounds of the science it selfe, proved, and convincingly demonstrated. / By Sir Thomas Urquhart of Cromartie Knight. Published for the benefit of those that are mathematically affected. Urquhart, Thomas, Sir, 1611-1660. 1645 Approx. 338 KB of XML-encoded text transcribed from 70 1-bit group-IV TIFF page images. Text Creation Partnership, Ann Arbor, MI ; Oxford (UK) : 2008-09 (EEBO-TCP Phase 1). A95751 Wing U140 Thomason E273_9 ESTC R212170 99870816 99870816 123211 This keyboarded and encoded edition of the work described above is co-owned by the institutions providing financial support to the Early English Books Online Text Creation Partnership. This Phase I text is available for reuse, according to the terms of Creative Commons 0 1.0 Universal . The text can be copied, modified, distributed and performed, even for commercial purposes, all without asking permission. Early English books online. (EEBO-TCP ; phase 1, no. A95751) Transcribed from: (Early English Books Online ; image set 123211) Images scanned from microfilm: (Thomason Tracts ; 45:E273[9]) The trissotetras: or, a most exquisite table for resolving all manner of triangles, whether plaine or sphericall, rectangular or obliquangular, with greater facility, then ever hitherto hath been practised: most necessary for all such as would attaine to the exact knowledge of fortification, dyaling, navigation, surveying, architecture, the art of shadowing, taking of heights, and distances, the use of both the globes, perspective, the skill of making the maps, the theory of the planets, the calculating of their motions, and of all other astronomicall computations whatsoever. Now lately invented, and perfected, explained, commented on, and with all possible brevity, and perspicuity, in the hiddest, and most re-searched mysteries, from the very first grounds of the science it selfe, proved, and convincingly demonstrated. / By Sir Thomas Urquhart of Cromartie Knight. Published for the benefit of those that are mathematically affected. Urquhart, Thomas, Sir, 1611-1660. [14], 96, [18] p. : ill. Printed by Iames Young., London, : 1645. Reproduction of the original in the British Library. Created by converting TCP files to TEI P5 using tcp2tei.xsl, TEI @ Oxford. Re-processed by University of Nebraska-Lincoln and Northwestern, with changes to facilitate morpho-syntactic tagging. Gap elements of known extent have been transformed into placeholder characters or elements to simplify the filling in of gaps by user contributors. 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Keying and markup guidelines are available at the Text Creation Partnership web site . eng Mathematics -- Early works to 1800. 2007-04 TCP Assigned for keying and markup 2007-05 Apex CoVantage Keyed and coded from ProQuest page images 2007-06 Jonathan Blaney Sampled and proofread 2007-06 Jonathan Blaney Text and markup reviewed and edited 2008-02 pfs Batch review (QC) and XML conversion THE TRISSOTETRAS : OR A MOST EXQUISITE TABLE FOR Resolving all manner of Triangles , whether Plaine or Sphericall , Rectangular or Obliquangular , with greater facility , then ever hitherto hath been practised : Most necessary for all such as would attaine to the exact knowledge of Fortification , Dyaling , Navigation , Surveying , Architecture , the Art of Shadowing , taking of Heights , and Distances , the use of both the Globes , Perspective , the skill of making of Maps , the Theory of the Planets , the calculating of their motions , and of all other Astronomicall computations whatsoever . Now lately invented , and perfected , explained , commented on , and , with all possible brevity , and perspicuity , in the hiddest , and most re-searched mysteries , from the very first grounds of the Science it selfe , proved , and convincingly demonstrated . By Sir THOMAS URQUHART of Cromartie Knight . Published for the benefit of those that are Mathematically affected . LONDON , Printed by Iames Young. 1645. TO THE RIGHT HONOVRABLE , And most noble LADY , My deare and loving Mother , the Lady DOWAGER of Cromartie . MADAM , FILIALL duty being the more binding in me , that I doe owe it to the best of Mothers ; if in the discharge thereof I observe not the usuall manner of other sonnes , I am the lesse to blame , that their obligation is not so great as mine : Therefore in that doe presume to imprint your Ladiships name in the Frontispiece of this Book , and proffer unto you a Dedication of that , which is beyond the capacity of other Ladies ; my boldnesse therein is the more excusable , that in your person the most vertuous Woman in the world is intreated to Patronize that , which by the learnedest men may happily be perused . I am confident ( Madam ) that your gracious acceptance of this Present is the more easily obtainable , in that it is a grand-child of your own , whom I thus make tender of , to be sheltered under the favor of your protection ; and that unto your Ladiship it will not be the more unwelcome , for proceeding from the braines of him , whose body is not more yours by generation , then by a most equitable purchase are the faculties of his mind ; the dominion which over my better halfe you , by your goodnesse , have acquired , being , in regard of my obedience , no lesse voluntary , then that of the other is for procreation naturall . Thus ( Madam ) unto you doe I totally belong , but so , as that those exteriour parts of mine , which by birth are from your Ladiship derived , cannot be more fortunate in this their subjection ( notwithstanding the egregious advantages of bloud , and consanguinity thereby to them accruing ) then my selfe am happy ( as from my heart I doe acknowledge it ) in the just right , your Ladiship hath to the eternall possession of the never-dying powers of my soule . For , though ( Soveraignty excepted ) there be none in this Island more honourably descended then is your Ladiship , nor whose progenitors , these many ages past , have been ( on either side ) of a more Noble extraction : Yet , laying apart Nobility , beauty , wealth , parentage , and friends , which ( together with many other gifts of fortune ) have hitherto served to adorn your Ladiship beyond others of your sex , who for all these have been deservedly renowned ; and ( in some measure ) not esteeming that properly to be yours , the receiving whereof did not altogether depend upon your owne election : it is the treasure of those excellent graces , wherewith inwardly you are enriched , that , in praising of your Ladiship is most to be pitch'd upon , and for the which you are most highly to be commended ; seeing by the means of them , you , from your tenderest yeeres upwards , untill this time , in the state of both Virginity and Matrimony , have so constantly , and indefatigably proceeded in the course of vertue , with such alacrity fixed your gallant thoughts on the sweetnesse thereof , and thereunto so firmly and cheerefully devoted all your words and actions , as if righteousnesse in your Ladiship had been an inbred quality , and that in your Will there had beene no aptitude of declining from the way of reason . This much is sufficiently well known to those , that have at any time enjoyed the honour of your Ladiships conversation , by whose most unpartiall reports , the Splendour of your reputation is both in this , and foraine Nations accounted precious , in the minds even of those , that have never seen you . But in so much more especially , doe the most judicious of either sex admire the rare and sublime endowments , wherewith your Ladiship is qualified , that ( as a patterne of perfection , worthy to be universally followed ) other Ladies ( of what dignity soever ) are truly by them esteemed of the choiser merit , the nearer they draw to the Paragon proposed , and resemble your Ladiship ; for that , by vertue of your beloved society , your neighbouring Countesses , and other greater Dames of your kindred and acquaintance , become the more illustrious in your imitation ; amidst whom , as Cynthia amongst the obscurer Planets , your Ladiship shines , and darteth the Angelick rayes of your matchlesse example on the spirits of those , who by their good Genius have been brought into your favourable presence to be enlightned by them . Now ( Madam ) lest , by insisting any longer upon this straine , I should seeme to offend that modesty , and humility , which ( without derogating from your heroick vertues ) are seated in a considerable place of your soule , I will here , in all submission , most humbly take my leave of your Ladiship , and beseech Almighty God , that it may please his Divine Majesty so to blesse your Ladiship with continuance of dayes , that the sonnes of those whom I have not as yet begot , may attaine to the happinesse of presenting unto your Ladiship a brain-babe of more sufficiencie and consequence ; and that your Ladiship may live with as much health , and prosperity , to accept thereof , and cherish it then , as ( I hope ) you doe now , at your vouchsafing to receive this , which ( though disproportionable , both to your Ladiships high deserts , and to that fervencie of willingnesse in me , sometime to make offer of what is of better worth , and more sutable to the grandour of your acceptance ) in all sincerity of heart ( confiding in that candour and ingenuity , wherby your Ladiship is accustomed to value gifts , according to the intention of the giver ) and in all duty , and lowlinesse of mind , together with my selfe in whole , and all my best endeavours , I tender unto your Ladiship , as becometh , ( Madam ) Your Ladiships most affectionate Sonne and humble servant , THOMAS URQUHART . To the Reader . TO write of Trigonometry , and not make mention of the illustrious Lord Neper of Marchiston , the inventer of Logarithms , were to be unmindfull of him that is our daily Benefactor ; these artificiall numbers by him first excogitated and perfected , being of such incomparable use , that , by them , we may operate more in one day , and with lesse danger of errour , then can be done without them in the space of a whole week . A secret which would have beene so precious to Antiquity , that Pythagoras , all the seven wise men of Greece , Archimedes , Socrates , Plato , Euclid , and Aristotle , had ( if coaevals ) joyntly adored him , and unanimously concurred to the deifying of the revealer of so great a Mystery : and truly ( besides them ) a great many other learned men , who for the laboriousnesse of long and various Multiplications , Divisions , and Radicall extractions of severall sorts , were deterred from the prosecuting , and divulging of their knowledge in the chiefest , and most noble parts of the Mathematicks , would have left behind them diverse exquisite Volumes , of an incomprehensible value , if the Arithmetical equality of difference , agreeable to every continued Geometricall proportion , had been made known unto them . Wherefore , I am infallibly perswaded , that , in the estimation of scientifically disposed spirits , the Philosophers stone is but trash to this invention , which will alwayes ( in their judicious opinions ) be accounted of more worth to the Mathematicall world , then was the finding out of America , to the King of Spaine ; or the discovery of the nearest way to the East-indies , would be to the Northerly occidentall Merchants . What the merit then of the Author is , let the most envious judge : for my owne part , I doe not praise him so much , for that he is my Compatriot , as I must extoll the happinesse of my Countrey , for having produced so brave a spark , in whom alone ( I may with confidence averre , it is more glorious , then if it had beene the conquering Kingdome of five hundred potent Nations : for , by how much the gifts of the mind , are more excellent then those of either body or fortune ; by so much the divine effects of the faculties thereof , are of greater consequence , then what is performed by meer force of Armes , or chance of Warre . I might say more in commendation of this gallant man , but that my discourse being directed to the Reader , he will possibly expect to be entertained with some other purpose then Encomiasticks ; and therefore , to undeceive him of those hopes ( if any such there be ) I will assure him , that to no other end I did require his observance here , but to be informed by me of the laudable endowments of that honourable Baron , whose eminencie above others ( wher-ever he be spoke of ) deserveth such an ample Elogie by it selfe , that the paper , graced with the receiving of his name and character , should not be blurred with the course impression of any other stuffe . However the Reader ought not to conceive amisse for his being detained so long upon this Eulogistick subject , without the variety of any peculiar instruction bestowed on him ; seeing I am certaine there is nothing more advantagious to him , or that more efficaciously can tend to his improvement , then the imitation of that admirable Gentleman , whose immortall fame , in spite of time , will out-last all ages , and look eternity in the face . The Readers well-wisher . T. U. An Epaenetick and Doxologetick Expresse , in commendation of this Book , and the Author thereof . To all Philomathets . SEeing Trigonometry , which handsomly unlocketh the choycest , and most intime mysteries of the Mathematicks , hath beene hitherto exposed to the world in a method , whose intricacy deterreth many from adventuring on it ; We are all , and every one of us , by duty bound to acknowledge our selves beholding to the Author of this Treatise ; who , by reducing all the secrets of that noble Science into a most exquisite order , hath so facilitated the way to the Learner , that in seven weeks , at most , he may attaine to more knowledge therein , then otherwise he could doe for his heart in the space of a twelve-moneth : And who , for the better encouragement of the studious , hath so gentilely expatiated his spirits upon all its Actioms , Principles , Analogies , precepts , and whole subjected matter ; that this Mathematicall Tractate doth no lesse bespeak him a good Poet , and good Orator , then by his elaboured Poems he hath showne himselfe already a good Philosopher , and Mathematician . Thus doth the various mixture of most excellent qualities in him , give such evidence of the transcendent faculties of his mind , that , as the Muses never yet inspired sublimer conceptions in a more refined stile , then is to be found in the accurate strain of his most ingenious Epigrams : so , on the other part , are the abstrusest difficulties of this Science by him so neatly unfolded , and with such exactnesse hath he resolved the hardest , and most intangled doubts thereof ; that , I may justly say , what praise ( in his Epistle , or rather Preface , to the Reader ) he hath beene pleased ( out of his ingenuity , to confer on the learned , and honourable Neper , doth , without any diminution , in every jot , as duly belong unto himselfe . For , I am certainly perswaded , he that useth Logarithms , shall not gaine so much time on the Worker by the naturall Sines and Tangents , as , by vertue of this succinct manner of calculation , shall be got on him that knoweth it not , how compendiously soever else , with Addition and Subtraction , or Addition alone , he frame his Computations . However , he who , together with that of the Logarithms , maketh use of this invention , is in a way which will bring him so straight and readily to the perfect practice of Trigonometry , that , compared with the old beaten path , trod upon by Regiomontanus , Ptolomy , and other ancient Mathematicians , it is like the Sea voyage , in regard of that by Land , betwixt the two Pillars of ( Hercules commonly called , the Straits of Gibraltar ) whereof the one is but of six houres sailing at most , and the other a journey of seven thousand long miles . If we then consider how a great many , despairing ever to get out ( if once entred ) of the confused obscurity wherein the doctrine of Triangles hath beene from time to time involved , have rather contented themselves barely with Scale and Compasse , and other mechanick tooles and instruments , to prosecute their operations , and in any reasonable measure to glance somewhat neare the truth , then , thorough so many pesterments , and harsh incumbrances , to touch it to a point , in its most indivisible and infallible reality . And how others , for all their being more industrious , in proving their Conclusions by the Mediums from which they are necessarily inferr'd , are neverthelesse ( even when they have bestowed half an age in the Trigonometricall practice ) oftentimes so farre to seek , that , without a great deale of premeditation advisement , and recollecting of themselves , they know not how to discusse some queries , corollaries , problems , consectaries , proportions , wayes of perpendicular falling , and other such like occurring debatable matters , incident to the scientifick measuring of Triangles ; We cannot choose ( these things being maturely perpended ) but be much taken with the pregnancy of this device , whereby we shall sooner hit to a minut upon the verity of an Angle or Side demanded , and trace it to the very source and originall , from whence it flowes , then another mechanically shall be able to come within three degrees thereof , although he cannot , for the same little he doth , afford any reason at all ; And so suddenly resolve any Trigonometricall question ( without paines or labour , how perplexed soever it be ) with all the dependances thereto belonging , as if it were a knowledge meerly infused from above , and revealed by the peculiar inspiration of some favourable Angel. Besides these advantages , administred unto us by the meanes of this exquisite Book , this maine commodity accreweth to the diligent Perusers of it , that , instead of three quarters of a yeere , usually by Professors allowed to their Schollers for the right conceiving of this Science , which ( notwithstanding ) through any little discontinuance , is by them so apt to be forgotten , that the expence of a week or two will hardly suffice to reseat it in their memories ; they shall not need , by this method , to bestow above a moneth , and with such ease and facility for retention , when they have learned and acquired it , that , if multiplicity of businesses , or serious plodding upon other studies happen to blot it out of their minds , they may as firmly recover in one quarter of an hour the whole knowledge and remembrance thereof , as when they had it best , and were most punctually versed in it . A secret ( in my opinion ) so precious , that ( as the Author spoke of Marchiston ) I may with the like pertinencie avouch of him , that his Countrey and kindred would not have been more honoured by him , had he purchased millions of gold , and severall rich territories of a great and vast extent ; then for this subtile and divine invention , which will out-last the continuance of any inheritance , and remaine fresh in the understandings of men of profound Literature , when houses and possessions will change their owners , the wealthy become poor , and the children of the needy enjoy the treasures of those , whose heires are impoverished . Therefore , seeing for the many-fold uses thereof in divers Arts and Sciences , in speculation and practice , peace and war , sport and earnest , with the admirable furtherances we reape by it , in the knowledge of Sea and Land , and Heaven and Earth , it cannot be otherwise then permanent , together with the Authors fame , so long as any of those endure ; I will ( God willing ) in the ruines of all these , and when time it selfe is expired , in testimony of my thankfulnesse in particular for so great a benefit , ( if after the Resurrection , there be any complementall affability ) expresse my selfe then , as I doe now , The Authors most affectionate , and most humbly devoted servant J. A. The Diatyposis of the whole Doctrine of Triangles . The plane Triangles have 13. Moods . Planorectangulars 7. 1 Upalem . 2 Uberman . 3 Uphener . 4 Ekarul . 5 Egalem . 6 Echemun . 7 Etena● . Planobliquangulars . 6. 1 Danarele . 2 There●abmo . 3 Zelemabne . 4 Xemenoro . 5 Shenerolem . 6 Pserelema . The Sphericals have 28 Moods . Orthogonosphericals . 16. 1. Upalam . 2. Ubamen . 3. Uphanep . 4. Ukelamb . 5. Ugemon . 6. Uchener 7. E●alum 8. Edamon . 9. Ethaner . 10. Ezolum . 11. Exoman . 12. Epsoner . 13. Alamun . 14. Amaner . 15. Enerul . 16. E●elam . Of these 16. Mood 〈◊〉 Moods of V●●●gen ●re prounded upon the Axi●●re of Supro●●● . The 8. of Pubkutetkepsaler on Sbaprotea : and the 5. of Uchedezexam on Proso By these 16. Representatives , 1. Le● . 2. Yet . 3. R●c 4. Cle. 5. Lu. 6. Tul. 7. Tere. 8. Tol. 9. Le● 10. At. 11. Tul. 12. Clet . 13. Cret . 14. Tur. 15. Tur. 16. Le ( A. signifying an oblique Angle . E. the Perpendicular U. the subtendent C. Initiall , the complemēt of a side to a quadrant ● . finall , the side continued to the Radius or a Quadrant . I. left . R. right . and T. one of the top Triangles of the Scheme ) it is evidenced in what part of the Diagram the Analogy of my of the 16. Moods begins , which being once knowne , the progressive sequence of the proportionable Sides & Angles is easily discerned out of the orderly in volutions of the Figure it selfe . Here it is to be observed , that as the Book explaineth the Trissotetral Table : so this Trigonodiatyposis unfoldeth ▪ all the intricate difficulties of the Book . Loxogonosphericals . 12. That the Schemes and Types of Triangulary Analogies are not seated in the roomes , where they are treated of , I purposely have done it ; to the end , that being all perceived at one view , their multiplicity ( which would appeare confused in their ●ispersed method ) might ●●ot any way discourage 〈…〉 der : besides that , this their ●eing together in their ●u● order , and rancked ●ecording to the exigence of the Sides or Angles , is such a furtherance to the memory , and illustration to the judgement , that it maketh Trigonometry , which of all Sciences was accounted the abstrusest , to be in effect the most 〈◊〉 and 〈◊〉 . Monurgeticks . 4 1 Lamaneprep . 2 Menerolo . 3 Nerelema . 4 Ralam●●● Of the Disergeticks there be 8. Moods , each whereof is divided into foure Cases . Ahalebmane . 1. Alamebna Dasimforaug . Dadisfo●●ug . Dadisgatin . Simomatin . 2. Alamebne . Dasimforauxy . Dadiscracforeug . Dadiscramgatin . Simomatin . Ahamepnare . 〈…〉 Dadissepamforaur . Dadissexamforeur . Dasimatin . Simomatin . 4. Ammaneprela Dadissepamfor . Dadisse●amfor . Dasimin . Simomatin . Ehenabrole . 〈…〉 Dacramfor . Damracfor . Dasimquzin . Simomatin . 6. E●neral●la . Dacforamb . Damforac . Da●imat●m . Simomatin . Eherolabme 〈…〉 Dacracforeur . Dambracforeur . Dacrambatin . Simomatin . 8. E●relome Dakyxamfor . Dambyxamfor . Dakypambin . Simomatin . In Eruditum D. Thomae Vrquharti equitis Trissotetrados librum . SI cupis aetherios tutò peragrare meatus , Et sulcare audes si vada salsa maris : Vel tibi si cordi est terrae spatia ampla metiri , Huc ades , hunc doctum percipe mente librum . Hoc , sine Daedaleis pennis volitare per auras , Et sine Neptuno nare per alta vales . Hoc duce , jam Lybicos poteris superare calores , Atque pati Scythici frigora saeva poli . Perge Thoma ; tali tandem gaudebit alumno SCOTIA , quam scriptis tollis in astra tuis . Al. Ross . POSITIONS . EVery Circle is divided into three hundred and sixty parts , called Degrees , whereof each one is Sexagesimated , Subsexagesimated , Resubsexagesimated , and Biresubsexagesimated , in Minutes , Seconds , Thirds , Fourths , and so far forth as any Computist is pleased to proceed for the exactnesse of a Research , in the calculation of any Orbiculary Dimension . 2. As Degrees are the measure of Arches , so are they of Angles ; but that those are called Circumferentiall , these Angulary Degrees , each whereof is the three hundred and sixtieth part of four right Angles , which are nothing else but the surface of a Plain to any point circumjacent ; for any space whatsoever about a point , is divided in 360. parts : And the better to conceive the Analogie that is betwixt these two sorts of graduall Measures , we must know , that there is the same proportion of any Angle to 4. right Angles , as of an arch of so many circumferential degrees to the whole circumference . 3. Hence is it , that the same number serves the Angle , and the Arch that vaults it , and that divers quantities are measured ( as it were ) with the same graduall measure . Angles and Arches then are Analogicall , and the same reason is of both . 4. Seeing any given proportion may be found in numbers , and that any two quantities have the same proportion that the two numbers have , according to the which they are measured : if for the measuring of Triangles there must be certain proportions of all the parts of a Triangle , to one another known , and those proportions explained in numbers it is most certain , all Magnitudes , being Figures at least in power , and all Figures either Triangles , or Triangled , that the Arithmeticall Solution of any Geometricall question , dependeth on the Doctrine of Triangles . 5. And though the proportion betwixt the parts of a Triangle cannot be without some errour ; because of crooked lines to right lines , and of crooked lines amongst themselves , the reason is inscrutable , no man being able to finde out the exact proportion of the Diameter to the Circumference : yet both in plain Triangles , where the measure of the Angles is of a different species from the sides , and in Sphericalls , wherein both the Angles and sides are of a circular nature , crooked lines are in some measure reduced to right lines by the definition of quantity which right lines , viz. Sines , Tangents , and Secants , applyed to a Circle have in respect of the Radius , o● half-Diameter . 6. And therefore , though the Circles Quadrature be not found out , it being in our power to make the Diameter , or the semi-Diameter , which is the Radius of as many parts as we please , and being sure so much the more that the Radius be taken , the error will be the lesser ; for albeit the Sines , Tangents and Secants , be irrationall thereto for the most part , and their proportion inexplicable by any number whatsoever , whither whole or broken : yet if they be rightly made , they will be such , as that in them all no number will be different from the truth by an integer , or unity of those parts , whereof the Radius is taken : which is so exactly done by some , especially by Petiscus , who assumed a Radius of twenty six places , that according to his supputation ( the Diameter of the Earth being known , and the Globe thereof supposed to be perfectly round ) one should not fail in the dimension of its whole Circuit , the nine hundreth thousand scantling of the Million part of an Inch , and yet not be able , for all that , to measure it without amisse ; for so indivisible the truth of a thing is , that come you never so neer it , unlesse you hit upon it just to a point , there is an errour still . DEFINITIONS . A Cord , or Subtense , is a right line , drawn from the one extremity to the other of an Arch. 2. A right Sine is the half Cord of the double Arch proposed , and from one extremity of the Arch falleth perpendicularly on the Radius , passing by the other end thereof . 3. A Tangent is a right line , drawn from the Secant by one end of the Arch , perpendicularly on the extremity of the Diameter , passing by the other end of the said Arch. 4. A Secant is the prolonged Radius , which passeth by the upper extremity of the Arch , till it meet with the sine Tangent of the said Arch. 5. Complement is the difference betwixt the lesser Arch , and a Quadrant , or betwixt a right Angle and an Acute . 6. The complement to a semi-Circle , is the difference betwixt the half-Circumference and any Arch lesser , or betwixt two right Angles , and an Oblique Angle , whither blunt or sharp . 7. The versed sine is the remainder of the Radius , the sine Complement being subtracted from it , and though great use may be made of the versed sines , for finding out of the Angles by the sides , and sides by the Angles : yet in Logarithmicall calculations they are altogether uselesse , and therefore in my Trissotetras there is no mention made of them . 8. In Amblygonosphericall● , which admit both of an Extrinsecall , and Intrinsecall demission of the perpendicular , nineteen severall parts are to be considered : viz. The Perpendicular , the Subtendentall , the Subtendentine , two Cosubtendents , the Basall , the Basidion , the chief Segment of the Base , two Cobases , the double Verticall , the Verticall , the Verticaline , two Coverticalls , the next Cathetopposite , the prime Cathetopposite , and the two Cocathetopposites : fourteen whereof , ( to wit ) the Subtendentall , the Subtendentine , the Cosubtendents , the Basall , the Basidion , the Cobases , the Verticall , the Verticaline , the Coverticalls , and Cocathetopposites , are called the first , either Subtendent , Base , Topangle , or Cocathetopposite , whither in the great Triangle or the little , or in the Correctangle , if they be ingredients of that Rectangular , whereof most parts are known , which parts are alwayes a Subtendent and a Cathetopposite : but if they be in the other Triangle , they are called the second Subtendents , Bases , and so forth . 9. The externall double Verticall is included by the Perpendicular , and Subtendentall , and divided by the Subtendentine : the internall is included by cosubtendents , and divided by the Perpendicular . APODICTICKS . THe Angles made by a right Line , falling on another right Line , are equall to two right Angles ; because every Angle being measured by an Arch , or part of a Circumference , and a right Angle by ninety Degrees , if upon the middle of the ground line , as Center , be described a semi-Circle , it will be the measure of the Angles , comprehended betwixt the falling , and sustaining lines . 2. Hence it is , that the four opposite Angles made by one line , crossing another , are always each to its own opposite equall ; for if upon the point of Intersection , as Center , be described a Circle , every two of those Angles will fill up the semi-Circle ; therefore the first and second will be equall to the second and third , and consequently the second , which is the common Angle to both these couples being removed , the first will remain equall to the third , and by the same reason , the second to the fourth , which was to be demonstrated . 3. If a right line falling upon two other right lines , make the alternat Angles equall , these lines must needs be Paralell ; for if they did meet , the alternat Angles would not be equall ; because in all plain Triangles , the outward Angle is greater , then any of the remote inward Angles , which is proved by the first . 4. If one of the sides of a Triangle be produced , the outward Angle is equall to both the inner , and opposite Angles together ; because according to the acclining or declining of the conterminall side , is left an Angulary space , for the receiving of a paralell to the opposite side , in the point of whose occourse at the base , the Exterior Angle is divided into two , which for their like , and alternat situation with the two Interior Angles , are equall each to its own conform to the nature of Angles , made by a right line crossing divers paralells . 5. From hence we gather , that the three Angles of a plain Triangle , are equall to two rights ; for the two inward , being equall to the Externall one , and there remaining of the three , but one , which was proved in the first Apodictick , to be the Externall Angles complement to two rights ; it must needs fall forth ( what are equall to a third , being equall amongst themselves ) that the three Angles of a plain Triangle , are equall to two right Angles , the which we undertook to prove . 6. By the same reason , the two acute of a Rectangled plain Triangle , are equall to one right Angle , and any one of them , the others complement thereto . 7. In every Circle , an Angle from the Center , is two in the Limb , both of them having one part of the Circumference for base ; for being an Externall Angle , and consequently equall to both the Intrinsecall Angles , and therefore equall to one another ; because of their being subtended by equall bases , viz. the semi-Diameters , it must needs be the double of the foresaid Angle in the limb . 8. Triangles standing between two paralells , upon one and the fame base , are equall ; for the Identity of the base , whereon they are seated , together with the Equidistance of the Lines , within the which they are confined , maketh them of such a nature , that how long so ever the line paralell to the base be protracted , the Diagonall cutting of in one off the Triangles , as much of bredth , as it gains of length , ( the ones losse accruing to the profit of the other ) Quantifies them both to an equality , the thing we did intend to prove . 9. Hence do we inferre , that Triangles betwixt two paralells , are in the same proportion with their bases . 10. Therefore if in a Triangle , be drawn a paralell to any of the sides , it divideth the other sides , through which it passeth proportionally ; for besides that it maketh the four segments , to be four bases , it becomes ( if two Diagonall lines be extended from the ends thereof , to the ends of its paralell ) a common base to two equall Triangles , to which two , the Triangle of the first two segments , having reference according to the difference of their bases , and these two being equall , as it is to the one , so must it be to the other , and therefore the first base , must be to the second , ( which are the Segments of one side of the Triangle ) as the third to the fourth , ( which are the Segments of the second ) all which was to be demonstrated . 11. From hence do we collect , that Equiangled Triangles have their sides about the equall Angles proportionall to one another . This sayes Petiscus , is the golden Foundation , and chief ground of Trigonometry . 12. An Angle in a semi-Circle is right ; because it is equall to both the Angles at the base , which ( by cutting the Diameter in two ) is perceivable to any . 13. Of four proportionall lines , the Rectangled figure , made of the two extreames , is equall to the Rectangular , composed of the means ; for as four and one , are equall to two and three , by an Arithmeticall proportion : and the fourth term Geometrically exceeding , or being lesse then the third , as the second is more , or lesse then the first ; what the fourth hath , or wanteth , from and above the third , is supplyed , or impaired by the Surplusage , or deficiency of the first from and above the second : These Analogies being still taken in a Geometricall way , make the oblong of the two middle , equall to that of the extreams , which was to be proved . 14. In all plain Rectangled Triangles , the Ambients are equall in power to the Subtendent ; for by demitting from the right Angle a Perpendicular , there will arise two Correctangles , from whose Equiangularity with the great Rectangle , will proceed such a proportion amongst the Homologall sides , of all the three , that if you set them right in the rule , beginning your Analogy at the main Subtendent , ( seeing the including sides of the totall Rectangle , prove Subtendents in the partiall Correctangles , and the bases of those Rectanglets , the Segments of the great Subtendent ) it will fall out , that as the main Subtendent is to his base , on either side ( for either of the legs of a Rectangled Triangle , in reference to one another , is both base and Perpendicular ) so the same bases , which are Subtendents in the lesser Rectangles , are to their bases , the Segment , of the prime Subtendent : Then by the Golden rule we find , that the multiplying of the middle termes ( which is nothing else , but the squaring of the comprehending sides of the prime Rectangular ) affords two products , equall to the oblongs made of the great Subtendent , and his respective Segments , the aggregat whereof by equation is the same with the square of the chief Subtendent , or Hypotenusa , which was to be demonstrated . 15. In every totall square , the supplements about the partiall , and Interior squares , are equall the one to the other ; for by drawing a Diagonall line , the great square being divided into two equall Triangles , because of their standing on equall bases betwixt two paralells , by the ninth Apodictick , it is evident , that in either of these great Triangles , there being two partiall ones , equall to the two of the other , each to his own , by the same Reason of the ninth : If from equall things ( viz. the totall Triangles ) be taken equall things ( to wit , the two pairs of partiall Triangles ) equall things must needs remain , which are the foresaid supplements , whose equality I undertook to prove , 16. If a right line cut into two equall parts be increased , the square made of the additonall line , and one of the Bisegments , joyned in one , lesse by the Square of the half of the line Bisected , is equall to the oblong contained under the prolonged line , and the line of Continuation ; for if annexedly to the longest side of the proposed oblong , be described the foresaid Square , there will jet out beyond the Quadrat Figure , a space or Rectangle , which for being powered by the Bisegment and Additionall line , will be equall to the neerest supplement , and consequently to the other ( the equality of supplements being proved by the last Apodictick ) by vertue whereof , a Gnomon in the great Square , lacking nothing of its whole Area , but the space of the square of the Bisected line , is apparent to equalize the Parallelogram proposed , which was to be demonstrated . 17. From hence proceedeth this Sequell , that if from any point without a circle , two lines cutting it be protracted to the other extremity thereof , making two cords , the oblongs contained under the totall lines , and the excesse of the Subtenses , are equall one to another ; for whether any of the lines passe through the Center , or not , if the Subtenses be Bisected , seeing all lines from the Center fall Perpendicularly upon the Chordall point of Bisection ( because the two semi-Diameters , and Bisegments substerned under equall Angles , in two Triangles evince the equality of the third Angle , to the third , by the fift Apodictick , which two Angles being made by the falling of one right line upon another , must needs be right by the tenth definition of the first of Euchilde ) the Bucarnon of Pythagoras , demonstrated in my fourteenth Apodictick , will by Quadrosubductions of Ambients , from one another , and their Quadrobiquadrequation● with the Hypotenusa , together with other Analogies of equation with the powers of like Rectangular Triangles , comprehended within the same circle , manifest the equality of long Squares , or oblongs Radically meeting in an Exterior point , and made of the prolonged Subtenses , and the lines of interception , betwixt the limb of the circle , and the point of concourse , quod probandum fuit . 18. Now to look back on the eleaventh Apodictick , where according to Petiscus , I said that upon the mutuall proportion of the sides of Equiangled Triangles , is founded the whole Science of Trigonometry , I do here respeak it , and with confidence maintain the truth thereof ; because , besides many others , it is the ground of these Subsequent Theorems : 1. The right sine of an Arch , is to its co-sine , as the Radius to the co-tangent of the said Arch. 2. The co-sine of an Arch , is to its sine , as the Radius to the Tangent of the said Arch. 3. The Sines , and co-Secants : the Secants , and co-Sines : and the Tangents , and co-Tangents , are reciprocally proportionall . 4. The Radius is a mean proportionall , betwixt the Sine , and co-Secant : the Secant , and co-Sine : and the Tangent , and co-Tangent : The verity of all these ▪ ( If a Quadrant be described , and upon the two Radiuses two Tangents , and two or three Sines be erected ( which in respect of other Arches will be co-Sines and co-Tangents ) and two Secants drawn ( which are likewise co-Secants ) from the Center to the top of the Tangents ) will appear by the foresaid reasons , out of my eleaventh Apodictick . The Trissotetras . Plain . Sphericall . Plain Trissotetras . Axiomes four . 1. Rulerst Vradesso : Directory : Enodandas . Eradetul : Vphechet : 3. Orth. 1. Obl. 2. Eproso Directorie : Enodandas 3. Ax. Grediftal : Dir. ● . Pubkegdaxesh : 4. Orth. 4. Ax. Bagrediffiu : Dir. ● . 3. Obl. The Planorectangular Table : Figures four ▪ 1. Va* le Datas . Quaesitas . Resolvers . Vp* Al§em . Rad — V — Sapy ☞ Yr. 2. Ve* mane Vb* em §an . V — Rad — Eg ☞ So.   Praesubserv . Possubserv . Vph* en §er . Vb* em §an . Vp* al§em , or , Eg* al§em . 3. Ena* ve Ek* ar §ul . Sapeg — Eg — Rad ☞ Vr . Eg* al §em . Rad — Taxeg — Eg ☞ Yr.     Praesubserv . Possubserv . 4. Ere* va Ech* em §un . Et* en §ar . Ek* ar §ul . Et* en §ar . E — Ge — Rad ☞ Toge . The Planobliquangular Table : Figures four . 1. Alahe * me Da*na*re §le . Sapeg — Eg — Sapyr ☞ Yr. 2. Emena*role The*re* lab §mo . Aggres — Zes — Talfagros ☞ Talzo .   Praesubserv . Possubserv . Ze*le*mab §ne . The*re* lab §mo . Da*na*re §le . 3. Enero*lome Xe* me* no §ro . E — So — Ge ☞ So.     Praesubserv . Possubserv . She*ne* ro §lem . Xe* me* no §ro . Da*na*re §le     Praesubserv . Possubserv . 4. Erele* a Pse* re* le §ma . Bagreziu . Vb*em §an . Finall Resolver . Vxi●q — Rad — 〈◊〉 — ☞ Sor. The Sphericall Trissotetras . Axiomes three . 1. Suprosca . Dir. uphugen . 2. Sbaprotca . pubkutethepsaler . 3. Seproso . uchedezexam . The Orthogonospherical Table . Figures 6. Datoquaeres 16.   Dat. Quaes . Resolvers . 1. Valam*menep Vp*al§am . Torb — Tag — Nu ☞ Mir. Vb*am§en . Nag — Mu — Torp ☞ Myr. or ,   Torp — Mu — Lag ☞ Myr. Vph*an §ep . Tol — Sag — Su ☞ Syr. 2. Veman*nore Vk*el§amb . Meg — Torp — Mu ☞ Nir. or ,   Torp — Teg — Mu ☞ Nir. Ug*em §on . Su — Seg — Tom ☞ Sir. or ,   Tom — Seg — Ru ☞ Sir. Uch*en §er . Neg — To — Nu ☞ Nyr . or ,   To — Le — Nu ☞ Nyr . 3. Enar*rulome Et*al§um . Torp — Me — Nag ☞ Mur. Ed*am§on . To — Neg — Sa ☞ Nir. Eth*an§er . Torb — Tag — Se ☞ Tyr. 4. Erol*lumane Ez*●l§um . Sag — Sep — Rad ☞ Sur. or ,   Rad — Seg — Rag ☞ Sur. Ex*●● §an . Ne — To — Nag ☞ Sir. or ,   To — Le — Nag ☞ Sir. Eps* on §er . Tag — Tolb — Te ☞ Syr. or ,   Tolb — Mag — Te ☞ Syr. 5. Acha* ve Al* am §un . Tag — Torb — Ma ☞ Nur. or ,   Torb — Mag — Ma ☞ Nur. Am* an §er . Say — Nag — T● ☞ Nyr . or ,   Tω — Noy — Ray ☞ Nyr . 6. Eshe*va En*er §ul . Ton — Neg — Ne ☞ Nur. Er* el §am . Sei — Teg — Torb ☞ Tir. or ,   Torb — Tepi — Rexi ☞ Tir. The Loxogonospherical Trissotetras . Monurgetick Disergetick . The Monurgetick Loxogonospherical Table . Axiomes two . 1. Seproso . Dir. Lame . Figures two . 2. Parses . Dir. Nera . Moods four . Figures . Datas . Quaes . Resolvers . 1. Datamista Lam*an*ep § rep . Sapeg — Se — Sapy ☞ Syr. Me*ne*ro § lo. Sepag — Sa — Sepi ☞ Sir.       ad 2. Datapura Ne*re*le § ma. Hal Basaldileg Sad Sab Re Regals Bis*ir .     ab     Parses — Powto — Parsadsab ☞ PowsalvertiR Ra*la*ma § ne . Kour Bfasines ( ereled ) Kouf Br*axypopyx . The Loxogonospherical Disergeticks Axiomes foure . 1. Na Bad prosver . Dir. Alama . 2. Naverpr or Tes. Allera . 3. Siubpror Tab. Ammena . 4. Niub prodesver . Errenna . Figures 4. Moods 8.   Fig. M. Sub Res . Dat. Praen . Cathetothesis . Final Resolvers . 1. Ab A         Cafregpiq .     La Vp Tag ut * Op § At Dasimforaug Sat-nop-Seud † nob . Kir . A Meb Al Nu ud * Ob § Aud Dadisforeug Saud-nob-Sat † nop . Ir.   Na. Am Mir uth* Oph § Auth Dadisgatin Sauth-noph-Seuth † nops Ir. Leb 2. Sub. Res . Dat. P●ae● . Cathetothesis . Final Resolvers .   Al         Cafyxegeq .   Ma La Vp Tag ut * op § at dasimforauxy nat-mut-naud † mwd   Meb Al Nu ud * ob § aud dadiscracforeug naud-mud-nat † mwt Ne Ne Am Mir uth * oph § auth dadiscramgatin nauth-muth-neuth † mwth Fig. M.       Cathetothesis . Plus minus . A A Sub. Re. Dat. Pr. Cafriq . Final Resolvers . Sindifora .                 At   Ma up Tag ut*Op § At Dadissepamforaur Nop-Sat-Nob ☞ Seudfr Autir . Ha Nep Al Nu ud*Ob § Aud Dadissexamforeur Nob-Saud-Nop ☞ Satfr Eutir .                 Aud   Ra Am Mir uth* Oph § Auth Dasimatin Noph-Seuth-Nops ☞ Soethj Authir . Mep 4.       Cathetothesis . Plus minus .   Am Sub. Re. Dat. . Pr. Cafregpagiq . Final Resolvers . Sindiforiu .                 Aet Na Ma ub Mu Ut* Op § Aet Dadissepamfor Tob-Top-Saet ☞ Soedfr Dyr .   Nep Am Lag Ud* Ob § Aed Dadissexamfor Top-Tob-Saed ☞ Soetfr Dyr .                 Aed Re Reb En Myr Uth* Oph § aeth Dasimin Tops-Toph-SAEth ☞ Soethj aeth Syr. Fig. M. Cathetothesis . Eb En Sub. Re. Dat. Pr. Cafregpigeq . Final Resolvers .   Er Ub Mu Ut* Op § aet Dacramfor Soed-Top-Saet ☞ Tob. Kir . En Ab Am Lag Ud* Ob § ad Damracfor Soet-Tob-Saed ☞ Top. Ir.   Lo En Myr Uth* Oph § aeth Dasimquaein Soeth-Toph-Saeth ☞ Tops . Ir. Ab 6. Cathetothesis .   En Sub. Re. Dat. Q. Pr. Cafregpiq . Final Resolvers . Ro Ne Ub Mu ut* Op § aet Dacforamb Naet-Nut-Noed ☞ Nwd . Yr.   Rab Am Lag ud* Ob § aed Damforac Naed-Nud-Noet ☞ Nwt . Yr. Le Le En Myr uth* Oph § aeth Dakinatam Naeth-Nuth-Noeth ☞ Nwth . Yr. Fig. M.         Cathteothesis . Plus minus . Eb E Sub. Re. Dat. Pr. Cafriq . Final Resolvers . Sindifora .                 At   Re Up Tag Ut* Op § at Dacracforaur Mut-Nat-Mwd ☞ Neudfr Autir . Er Lo Al Nu Ud* Ob § aud Dambracforeur Mud-Naud-Mwt ☞ Natfr Autir .                 Aud   Mab Am Mir Uth* Oph § auth Dacrambatin Muth-Nauth-Mwth ☞ Neuthj Authir Om 8.         Cathetothesis . Plus minus .   Er Sub. Re. Dat. . Pr. Cacurgyq . Final Resolvers . Sindiforiu .                 Aet Ab Re Ub Mu Ut* Op § aet Dakyxamfor Nut-Nat-Nwd ☞ Noedfr Dyr .   Lo Am Lag Ud* Ob § aed Dambyxamfor Nud-Nad-Nwt ☞ Noetfr Dyr .                 Aed Me Me En Myr Uth* Oph §ath Dakypambin Nuth-Nath-Nwth ☞ Noethj Aeth Syr. THe novelty of these words I know will seeme strange to some , and to the eares of illiterate hearers sound like termes of Conjuration : yet seeing that since the very infancie of learning , such inventions have beene made use of , and new words coyned , that the knowledge of severall things representatively confined within a narrow compasse , might the more easily be retained in a memory susceptible of their impression ( as is apparent by the names of Barbara , Celarent , Darii , Ferio , and fifteen more Syllogistick Moods , and by those likewise of Gammuth , A-re , B-mi , C-fa-uth , and seventeen other steps of Guidos Scale , which are universally received by men of understanding , and that have their spirits tuned to the harmony of reason ) I know not why Logick and Musick should be rather fitted with such helps then Trigonometrie , which , for certitude of demonstration , hath been held inferior to no science , and for sublimity and variety of object , is the primest of the Mathematicks . This is the cause why I framed the Trissotetras , wherein the termes by me invented , without regard of the initiall letters of the words by them expressed , are composed of such as , joyned together , are of most easie pronunciation ; as the Tangent complement of a Subtendent is sooner uttered by Mu then by T C S ; and the Secant complement of the side required , by Ry , then ( in the usuall apocopating way ) by the first syllables or letters of Secant complement , side , and required ; and considering that without opening of the mouth no word can be spoken , which overture is performed by the vowel , to all the sides and Angles I designed vowels , that in the coalescencie of syllables , Sines , Tangents , and Secants might the better consound therewith . The explanation of the Trissotetras . A. signifieth an Angle : Ab. in the Resolvers signifieth abstraction , but in the Figures and Datoquaeres the Angle between : Ac. or Ak. the acute Angle . Ad. Addition . AE . the first base : Amb. or Am. an obtuse Angle : As Angles in the plurall number . At. the double verticall , whether externall or internall . Au. the first verticall Angle : Ay , the Angle adjoyning to the side required . B. or Ba. the true base : Bis the double of a thing . Ca. the perpendicular : Cra. the concurse of a given and required side : Cur. the concurse of two given sides . D. the partiall or little rectangle or rectanglet . Da. the datas . Di. or Dif . the difference : Dir. the directories . D. q. Datoquaeres . Diss . of unlike natures . E. a side : Eb. the side between : Enod , enodandas : Ereled . turned into sides : Es , sides in the plurall number : Ei , the side conterminall with the Angle required : Eu , the second verticall Angle . F. the new base , or angularie base , it being an Angle converted into a side : Fig. figures : Fin. Res . finall resolvers : For , or Fo , outwardly , often made use of in the Cathetothesis : Fr. a subducting of a lesser from a greater , whether it be Side or Angle . G. An Angle or Side given : Gre , or aggre , the summe or aggregat . Hal , or Al , the halfe . I. Vowel , an Angle required : I Consonant , the addition of one thing to another used in the clausuls of some of the finall Resolvers . In , intus or inwardly , and sometimes turned into . Iu , the segments of the base , or the segmented base . K. The complement of an Angle to a Semicircle . L. The Secant : Leg , one of the comprehending sides of an Angle . This representative is once only mentioned . M. A Tangent complement . N. A Sine complement . O. An opposite Angle , or rather Cathetopposite : Ob. the next cathetopposite Angle , by some called the first opposite : Op. the prime cathetopposite Angle , by some called the second opposite Oph , the first of the coopposite Angles : Orth , an acute Angle : Ops , the second of the coopposits : Os , opposite Angles in the plurall number . Oe , the second base ; Ou , the Angle opposite to the base . P. Opposite , whether Angle or side : Par. a parallelogram or oblong . Praes . praesubservient : Possub . possubservient : Pro. proportionall : Prod. directly proportionall : Pror , reciprocally proportionall : Pow. the Square of a Line : Pran . praenoscendas . Q. Continued if need be . Quaes . Quaesitas . Quae. Quaere , or Required . R. The Secant complement , and sometimes in the middle of the Cathetothesis signifies required , as alwayes in the latter end of a finall resolver it doth by way of emphasis , when it followes I. or Y. R. likewise in the Axiom of Rulerst stands for Radius . Ra. the Radius , and in the Scheme the middle angularie Radius . S. The Sine , and in the close of some Resolvers , the Summe . Sim. of like affection or nature : Subs . Subservient . T. The Tangent . To. the Radius or total Sine , but in the Diagram it is taken for the left angularie Radius : Tω . the right angularie Radius in the Scheme proposed : Tol. the first hypotenusal Radius thereof . Tom. the second hyp . Radius . Ton. the third hyp . Rad. Tor. the fourth hyp . Rad. Tolb . the basiradius on the left hand . Torb . the basiradius on the right . Tolp. the Cathetorabdos , or Radius on the left . Torp . the Cathetoradius on the right . Th. the correctangle . U. The Subtendent side . V. consonant , to avoid vastnesse of gaping , expresseth the same in severall figures . Ur. the Subtendent required . W. The second Subtendent . X. Adjacent or Conterminal . Y. The side required . Z. The difference of Segments , and is the same with di , or dif . Neverthelesse the Reader may be pleased to observe , that no Consonants in the Figures or Moods are representative save P. and B. and that only in a few ; both these two and all the other Consonants meerly serving to expresse the order and series of the Moods and Figures respectively amongst themselves , and of their constitutive parts in regard of one another . ANIMADVERSIONS . IN the letter T. I have been something large in the enumeration of severall Radiuses ; for there being eleven made use of in the grand Scheme , whereof eight are Circumferentiall , and three Angularie , that they might be the better distinguished from one another , when falling in proportion we should have occasion to expresse them ; I thought good to allot to every one of them its owne peculiar Character : all which I have done with the more exactnesse , that by the variety of the Radiuses amongst themselves , when any one of them in particular is pitched upon , we may the sooner know what part of the Diagram , by meanes thereof , is fittest for the resolving of any Orthogonosphericall problem : though indeed , I must confesse , when sometimes to a question propounded , I adapt a figure apart , I doe indifferently ( excluding all other characters ) make use of To , or Rad , or R onely for the totall Sine , which , without any obscurity or confusion at all , I have practised for brevities sake . Likewise , it being my maine designe in the framing of this Table , to make alcapable trigonometrically-affected Students with much facility and litle labour attaine to the whole knowledge of the noble Science of the doctrine of Triangles , I deemed it expedient , the more firmely and readily to imprint the severall Datoquaeres or praescinded Problems thereof in their memories , to accommodate them accordingly with letters proper for the purpose ; which , if the ingenious Reader will be pleased to consider , he will find , by the very letters themselves , the place and number of each Datoquaere : This is the reason why my Trissotetras ( conforme to the Etymologie of its name ) is in so many divers Ternaries , and Quaternaries divided ; and that the sharp , meane , blunt , double , and Liquid Consonants of the Greek Alphabet , are so orderly bestowed in their severall roomes , being all and every one of them seated according to the nature of the Moods and Figures , whose characteristicks they are . Thirdly , the Moods of the Planotriangular Table , being in all thirteene , whereof there be seven Rectangular , and six Obliquangular , are fitly comprehended by the three blunt , three meane , three sharp , and soure double Consonants , the Hebrew Shin being accounted for one of them . Fourthly , the sixteen Moods of the Orthogonosphericall Trissotetras are contained under three sharp , three mean , three blunt , three double , and foure Liquids , which foure doe orderly particularise the Binaries of the last two Figures . Fifthly , the foure Monurgetick Loxogonosphericals are deciphred by each its owne Liquid in front , according to their literall order . Sixthly , the eight Loxogonosphericall Disergeticks are also distinguished by the foure Liquids , but with this difference from the Monurgeticks , that the Vowels of A and E precede them in the first syllable , importing thereby the Datas of an Angle or a Side . Now because these Disergeticks are eight in all , there being allotted to every Liquid that characteriseth the Figures , the better to diversifie the first and second Datas of each respective binarie from one another , ( in so farre as they have reference to each its own Quaesitum ) the Figurative Liquid is doubled when a Side is required , and remaineth single when an Angle . Furthermore , in the Oblique Sphericodisergeticks , so farre as the sense of the Resolvers could beare it , I did trinifie them with letters convenient for the purpose , according to the severall cases of their Datoquaeres , whose diversity reacheth not above the extent of π. β. φ. and τ. δ. θ. I had almost omitted to tell you , that for the more variety in the last two Figures of the Orthogonosphericals are set downe the two letters of Ch. and Shin , the first a Spanish , and the second an Hebrew letter . Now if to those helps for the memorie which in this Table I have afforded the Reader , both by the Alphabetical order of some Consonants , and homogeneity of others in their affections of sharpnesse , meannesse , obtusity , and duplicity , he joyne that artificiall aid in having every part of th●● Chem●locally in his mind ( of all wayes both for facility in remembring , and stedfastnesse of retention , without doubt , the most expedite ) or otherwise place the representatives of words , according to the method of the Art of memory , in the severall corners of a house ( which , in regard of their paucity are containable within a Parlour or dining roome at most ) he may with ease get them all by heart in lesse then the space of an houre : which is no great expence of time , though bestowed on matters of meaner consequence . The Commentary . THe Axiomes of plain Triangles are foure , viz. Rulerst , Eproso , Grediftal , and Bagrediffus . Rulerst , that is to say , the Subtendent in plain Triangles may be either Radius or Secant , and the Ambients either Radius , Sines , or Tangents ; for it is a maxime in Planangular Triangles , that any side may be put for Radius , grounded on this , that from any point at any distance a Circle may be described : therefore if any of the sides of a plain Triangle be given together with the Angler , each of the other two sides is given by a threefold proportion , that is , whether you put that , or this , or the third side for the Radius ; which difference occasioneth both in plaine and Sphericall Triangles great variety in their calculations . The Branches of this Axiom are Vradesso and Eradetul . Vradesso , that is when the Hypotenusa is Radius , the sides are Sines of their opposit Angles ; so that there be two Arches described with that Hypotenusal identity of distance , whose Centers are in the two extremities of the Subtendent ; for so the case will be made plaine in both the Legs , which otherwise would not appeare but in one . Eradetul , when any of the sides is Radius , the other of them is a Tangent , and the Subtendent a Secant . The reason of this is found in the very definitions of the Sines , Tangents , and Secants , to the which , if the Reader please , he may have recourse ; for I have set them downe amongst my Definitions . Hence it is ( according to Mr. Speidels observation in his book of Sphericals ) that the Sine of any Arch being Radius , that which was the totall Sine becomes the Secant complement of the said Arch , and that the Tangent of any Arch being Radius , what was Radius becomes Tangent complement of that Arch. The Directory of this Axiome is Vphech●t . which sheweth us , that there be three Planorectangular Enodandas belonging thereto , viz. Vphener , Echemun , and Etenar ; as for Pserelema , which is the Loxogonian one pointed at in my Trissotetras , because it is but a partiall Enodandum , I have purposely omitted to mention it in the Directory of Eradetul . The second Axiome is Epros● , that is , the sides are proportionall to one another as the Sines of their opposite Angles ; for seeing about any Triangle a Circle may be circumscribed , in which case each side is a cord or Subtense , the halfe whereof is the Sine of its opposite Angle , and there being alwayes the same reason of the whole to the whole , as of the halfe to the halfe , the sides must needs be proportionall to one another , as the Sines of their opposite Angles , quod probandum erat . The Directory of this second Axiome is Pubkegdaxesh , which declareth that there are seven Enodandas grounded on it , to wit , foure Rectangular , Upalem , Ubeman , Ekarul , Egalem , and three Obliquangular , Danarele , Xemenoro , and Shenerolem . The third Axiom is Grediftal , that is , in all plain Triangles , As the summe of the two sides is to their difference , so is the Tangent of the halfe sum of the opposite Angles to the Tangent of half their difference ; for if a Line be drawne equall to the summe of the two sides , and if on the point of Extension with the distance of the shorter side a Semicircle be described , and that from the extremity of the protracted Line a Diameter be drawne thorough the Circle where it toucheth the top of the Triangle in question , till it occurre with a parallel to the third side , there will arise two Equicrurall Triangles , one whereof having one Angle common with the Triangle proposed , and the three of the one being equall to the three of the other , any one of the equall Angles in the foresaid Isosceles must needs be the one halfe of the two unknowne Angles . This is the first step to the obtaining of what we demand . Then do we find that the third side cutteth the sides of the greatest Triangle according to the Analogie required , which is perceivable enough , if with the distance of the outmost Parallel from the lower end thereof as Center , be described a new Circle ; for then will the Tangents be perspicuous and so much the more for their Rectangularity , the one with the Radius , and the other with its Parallel , which , being touched at an Angle described in a Semicircle , confirmeth the Rectangularity of both . By the Parallels likewise is inferred the equality of the alternate Angles , whose addition and subduction to and from halfe the sum of the two unknown Angles make up both the greater and lesser Angle . Hereby it is evident how the sum of the two sides , &c. which was to be proved . The Directory of this third Axiom is θ. onely ; for it hath no Enodandum but Therelabmo . The fourth Axiom is Bagrediffiu , that is , As the Base or greatest side is to the summe of the other sides , so the difference of the other sides to the difference of the Segments of the Base ; for if upon the Center of the verticall Angle with the distance of the shortest side be described a Circle , it will so cut the two greater sides of the given Triangle , that , finding thereby two Oblongs of the nature of those whose equality is demonstrated in my Apodicticks , we may inferre ( the Oblong made of the summe of the sides , and difference of the sides being equall to the Oblong made of the Base , and the difference of its Segments ) that their sides are reciprocally proportionall ; that is , As the greatest side is to the sum of the other sides : so the difference of the other sides , to the difference of the Segments of the Base , or greater side . The Directory of this Axiom is θ. and its onely Enodandum , ( though but a partiall one ) Pserelema . The Planorectangular Table hath foure Figures . IT is to be observed , that Figure here is not taken Geometrically , but in the sense that it is used in the Logicks , when a Syllogism is said to be in the first , second , or third Figure ; for , as there by the various application of the Medium or mean terme the Figures are constituted diverse : so doth the difference of the Datas in a Triangle distinguish these Trissotetrall Figures from one another , and ( to continue yet further in the Syllogisticall Analogy ) are according to the severall demands ( when the Datas are the same ) subdivided into Moods . The first two vowels give notice of the Data's , and the third of what is demanded , so that Uale ( and euphonetically pronounced Vale ) which is the first Figure , shewes that the Subtendent , and one Angle are given , and that one of the containing sides is required . Vemane is the second Figure , which pointeth out all those problems wherein the Hypotenusa , and one Leg are given , and an Angle , or the other Leg is required . The third Figure is Enave , which comprehendeth all the Problems , wherein one of the Ambients is given with an Oblique Angle , and the Subtendent , or other Ambient required . The fourth and last of the Rectangular Figures is Ereva , which standeth for those Datoquaeres , wherein the including Sides are given , and the Subtendent or an Angle demanded . Now let us come to the Moods of those Figures . THe first Figure Vale hath but one Mood , and therefore of as great extent as it selfe , which is Upalem ; whose nature is to let us know , when a plane right angled Triangle is given us to resolve , whose Subtendent and one of the Obliques is proposed , and one of the Ambients required , that we must have recourse unto its Resolver , which being Rad — U — Sapy ☞ Yr sheweth , that if we joyne the artificiall Sine of the Angle opposite to the side demanded with the Logarithm of the Subtendent , the summe searched in the Canon of absolute numbers will afford us the Logarithm of the side required . The reason hereof is found in the second Axiom , the first Consonant of whose Directory evidenceth that Upalem is Eprosos Enodandum ; for it is , As the totall Sine , to the Hypotenusa : so the Sine of the Angle opposite to the side required , is to the said required side , according to the nature of the foresaid Axiom , whereupon it is grounded . The second Figure Vemane hath two Moods , Ubeman and Uphener ; the first whereof comprehendeth all those questions , wherein the Subtendent and an Ambient being given , an Oblique is required , and by its Resolver V — Rad — Eg ☞ So. thus satisfieth our demand , that if we subtract the Logarithm of the Subtendent from the summe of the Logarithms of the middle termes , we have the Logarithm of the Sine of the opposite Angle we seek for ; for it is , As the Subtendent to the totall Sine , so the containing side given to the Sine of the opposite Angle required . The reason likewise of this Analogy is found in the second Axiom Eproso , upon the which this Mood is grounded , as the second Consonant of its Directory giveth us to understand . The second Mood or Datoquaere of this Figure is Uphener , which sheweth that those questions in plaine Triangles , wherein the Hypotenusa and a Leg being given , the other Leg is demanded , are to be calculated by its Resolver , which ( because the Canon of Logarithms cannot performe it at one operation , there being a necessity to find one of the oblique Angles before the fourth terme can be brought into an Analogie ) alloweth two Subservients for the atchievement thereof , viz. Vbeman , the first Mood of the second Figure , for the finding out of the Angle , and here ( because anterior in the work ) called Praesubservient : then Vpalem , the first Mood of all , for finding out of the Leg inquired , and here called Possubservient , because of its posteriority in the operation : yet were it not for the facility which addition and subtraction only afford us in this manner of calculation , we might doe it with one work alone by the Bucarnon or Pythagorases Diodot , which plainly sheweth us , that by subducing the square of the Leg given , from the square of the Subtēdent , we have for the remainder another square , whose root is the side required . The reason of this is in my Apodicticks : but that of the former Resolver by two operations , is in the first Axiom , as by the first syllable of its Directory is manifest . The third Figure is Enave , which hath two Moods , Ekarul and Egalem . The first comprehendeth all those Problems , wherein one of the including sides , and an Angle being given , the Subtendent is required , and by its Resolver Sapeg — Eg — Rad ☞ Vr , sheweth , that if we subtract the Sine of the Angle opposite to the given side from the summe of the middle termes ( I meane the Logarithms of the one and the other ) which are the totall Sine , and the Leg proposed , we shall have the Hypotenusa required ; for it is , As the Sine of the Angle opposite to the side given , to the foresaid given side : so the totall Sine , to the Subtendent required . The reason of this proportion is grounded on the second Axiom Eproso ; for K. the third Consonant of its Directory , giveth us to understand , that it is one of the Enodandas thereof . The second Mood of Enave is Egalem , which comprehendeth all those Problems , wherein one of the Ambients , and an oblique Angle being given , the other Ambient is required : and by its Resolver Rad — Taxeg — Eg ☞ Yr sheweth , that if we adde the Logarithm of the side given to the Logarithm of the Tangent of the Angle conterminall with that side , and from the summe if we cut off the first digit on the left hand ( which is equivalent to the subtracting of the Radius whether double or single ) The remainder will afford us a Logarithm ( so neare as the irrationality of the termes will admit ) in the Table of equall parts , expressive of the side required ; for it is As the whole Sine to the Tangent of an Angle insident on the given side : so the side proposed , to the side required : The reason hereof is grounded on the second Axiom , for the fourth Consonant of its Directory sheweth , that Egalem is Eprosos enodandum . The fourth Figure is Ereva , whose Moods are Echemun and Etenar . The first , viz. Echemun , comprehendeth all those Problems , wherein the two Ambients being given , the Subtendent is required , and ( not being Logarithmically resolvable in lesse then two operations ) hath for its Prae and Possubservients the Moods of Etenar and Ekarul ; for an Oblique Angle by this Method is to be searched before the Subtendent can be found out , and by reason of these severall work● , this Mood is grounded on the two first Axioms , and is an Enodandum partially depending on Eradetul , and Eproso . Yet , if you will be pleased to be at the paines of extracting the Square root , you may have the Subtendent at one work by a Quadrobiquadraequation as the Bucarnon doth instruct us , whose demonstration you have plainly set downe in the fourteenth of my Apodicticks . The second Mood of this Figure is Etenar , which includeth all those questions wherein the two containing Sides being given , one of the Obliques is required , and by its Resolver E — Ge — Rad ☞ Toge manifesteth , that , if from the Summe of the Radius and Logarithm of the side given , we subtract the Logarithm of the other proposed side , the remainder will afford the Tangent of the Angle opposite to one of the given sides , the Complement of which Angle to a right one is alwayes the measure of the other Angle , by the fifth of my Apodicticks ; for it is , As the one Ambient is to the other Ambient , so the totall Sine to the Tangent of an Angle ; which found out , is either the Angle required , or the Complement thereof to a right Angle . The reason of this Analogie is grounded on the second Branch of the first Axiom , as by the Characteristick of the Directory is perceivable enough to any industrious Reader . Of the Planobliquangular Triangles there be foure Figures : Alaheme , Emenarole , Enerolome , and Erelea , THe first and last of these foure are Monotropall Figures , and have but each one Mood : but the other two have a couple a piece , so that for the Planobliquangulars , all the foure together afford us six Datoquaeres . The Mood of Alaheme is Danarele , which comprehendeth all those Problems , wherein two Angles being given and a Side , another Side is demanded , and by its Resolver Sapeg — Eg — Sapyr ☞ Yr , sheweth , that , if to the summe of the Logarithm of the side given , and of the Sine of the Angle opposite to the side required , we adde the difference of the Secant complement from the Radius , ( by some called the Arithmeticall complement of the Sine , and in Master Speidels Logarithmicall Canon of Sines , Tangents , and Secants with good reason termed the Secant ; for , though it doe not cut any Arch , thereby more Etymologically to deserve the name of Secant , yet worketh it the same effect that the prolonged Radius doth ) the operation will proceed so neatly , that if from these three Logarithms thus summed up , we onely cut off a Digit at the left hand , we will find as much by addition alone performed in this case , as if from the proposed summe the Sine of the Angle had beene abstracted ; for the totall Sine thus unradiused is the Logarithm of the side required . But such as are not acquainted with this compendious manner of calculating , or peradventure are not accommodated with a convenient Canon for the purpose , may , in Gods name , use their owne way , the Resolver being of such amplitude , that it extends it selfe to all sorts of operations , whereby the truth of the fourth Ternary in this Mood may be attaind unto ; for it is Analogised thus , As the Sine of the Angle opposite to the side given is to the same given side ; so the Sine of the Angle opposite to the side required , to the required side . The reason of this proportion is grounded on the second Axiom , the first determinater of whose Directory sheweth , that Danarele is one of Eprosos Enodandas . The second Figure of the Planobliquangulars is Emenarole , whose Moods are Therelabmo and Zelemabne . The first comprehendeth all those Planobliquangular Problems wherein two sides being given with an interjacent Angle , an opposite Angle is demanded , and by its Resolver Aggres — Zes — Talfagros ☞ Talzos , sheweth , that if from the summe of the Logarithm of the difference of the sides , and Tangent of halfe the summe of the opposite Angles , be subduced the aggregat or summe of the Logarithms of the two proposed sides , the remainder thereof will prove the Logarithm of the Tangent of halfe the difference of the opposite Angles ; the which joyned to the one , and abstracted from the other , affords us the measure of the Angle we require ; for the Theoreme is , As the aggregat of the given sides , to the difference of th●se sides : So the Tangent of halfe the summe of the opposite Angles , to the Tangent of halfe the difference of those Angles ; which , without any more adoe , by simple Addition and Subtraction affordeth the Angle we demand . The third Axiom and the Theorem of the Resolver of this Mood being but one and the same thing , I must make bold to remit you to my Apodicticks for the reason of the Analogie thereof , the onely determinater of whose Directory being θ. pointeth out the Mood of Therelabmo for the sole enodandum appropriated thereunto . The second Mood of this Figure is Zelemabue , which involveth all the Planobliquangulary Problemes , wherein two sides being given with the Angle between , the third side is demanded : and not being calculable by the Logarithmicall Canon in lesse then two operations , because it requireth the finding out of another Angle before it can fix upon the side , Therelabmo is allowed it for a Praesubservient , by vertue whereof an opposite Angle is obtained , and Danarele for its Possubservient and finall Resolver , by whose meanes we get the side required . The reason of the first operation is grounded on the third Axiom , and of the second operation on the second : but because this Mood is meerly a partiall Enodandum , neither of the foresaid Axioms affordeth any Directory concerning it , otherwise then in the two Subservients thereof . The third Figure is Eneroloms , whose two Moods are X●monor● and Shenerolem . The first Mood of this Figure includeth all those Planobliquangularie Problems , wherein two sides being given , with an opposite Angle , another opposite Angle is demanded , and by its Resolver E — Sog — Ge ☞ So , sheweth , that if from the aggregat of the Logarithm of one of the given sides , and that of the Sine of the opposite Angle proposed , we subtract the Logarithm of the other given side , the residue will afford us the Logarithm of the Sine of the opposite Angle required ; for it is Analogised thus , As one of the sides , to the Sine of the opposite Angle given : so the other side proposed , to the Sine of the opposite Angle required . The reason of this proportion is from the second Axiom , the sixth characteristick of whose Directory importeth , that Xemenoro is one of Eprosos enodandas . The second Mood of this Figure is Shenerolem , which containeth all those Planobliquangularie Problems , wherein two sides being given with an opposite Angle , the third side is demanded , which not being findable by the Logarithmicall Table upon the foresaid Datas in lesse then two operations ( because an Angle must be obtained first before the side can be had ) Xemenoro Praesubserves it for an Angle , and Danarele becomes its Possubservient for the side required . The reason of both these operations is founded on the second Axiom , the last Characteristick of whose Directory inrolleth Shenerolem for one of Eprosos enodandas . The fourth figure is Erelea , which , being Monotropall , hath no Mood but Pserelema . This Pserelema encompasseth all those Planobliquangulary Problems wherein the three sides being proposed , an Angle is required . This Datoquaere not being resolvable by the Logarithms in lesse then two operations , because the Segments of the Base , or sustaining side must needs be found out , that by demitting of a Perpendicular from the top Angle , we may hit upon the Angle demanded : the Resolver for the Segments is Ba — Gres — Zes ☞ Zius , whereby we learne , that if from the Logarithm of the summe of the sides , joyned to the Logarithm of the difference of the sides , we subtract the Logarithm of the Base , the remainder is the Logarithm of the difference of the Segments , which difference being taken from the whole Base , halfe the difference proves to be the lesser Segment . This Theorem being thus the Praesubservient of this Mood , its Possubservient is Vbeman , whose generall Resolver V — Rad — Eg ☞ Sor , is particularised for this case Uxiug — Rad — Ing ☞ Sor , which sheweth , that if from the summe of the Logarithms of the totall Sine , and of one of the Segments given , we subduce the Logarithm of the Hypotenusa conterminall with the Segment proposed , the remainer will be the Logarithm of the Sine of the opposite Angle required ; for the demitting of the Perpendicular opens a way to have the Theorem to be first in generall propounded thus , As the Subtendent to the totall Sine , so the containing side given to the Sine of the Angle required : or in particular thus , As the Sine of the Cosubtendent adjoyning the Segment given is to the Radius , so is the said Segment proposed to the Sine of the Angle required . Thus farre for the calculating of plaine Triangles , both right and oblique : now follow the Sphericals . THere be three principall Axioms upon which dependeth the resolving of Sphericall Triangles , to wit , Suprosca , Sbaprotca , and Seproso . The first Maxime or Axiom , Suprosca , sheweth , that of severall rectangled Sphericals , which have one and the same acute Angle at the Base , the Sines of the Hypotenusas are proportionall to the Sines of their Perpendiculars ; for , from the same inclination every where of the one plaine to the other , there ariseth an equiangularity in the two rectangles , out of which we may confidently inferre the homologall sides ( which are the Sines of the Subtendents , and of the Perpendiculars of the one , and the other ) to be amongst themselves proportionall . It s Directory is Uphugen , by the which we learn , that Uphanep , Ugemon , and Enarul , are its three enodandas . The second Axiom is Sbaprotca ; whereby we learne , that in all rectangled Sphericals that have one and the same acute Angle at the Base , the Sines of the Bases are proportionall to the Tangents of their Perpendiculars : which Analogie proceedeth from the equiangularity of such rectangled Sphericals , by the semblable inclining of the plaine towards them both . This proportion neverthelesse will never hold betwixt the Sines of the Bases , and the Sines of their Perpendiculars ; because , if the Sines of the Bases were proportionall to the Sines of the Perpendiculars ( the Sines of the Perpendiculars being already demonstrated proportionall to the Sines of the Subtendents ) either the Sine of the Perpendicular , or the Sine of the Base would be the cord of the same Arch , whereof it is a Sine ; which is impossible , by reason that nothing can be both a whole , and a part , in regard of one and the same thing ; and therefore doe we onely say , that the Sines of the Bases , and Tangents of the Perpendiculars , and contrarily , are proportionall . It s Directory is Pubkutethepsaler , which sheweth , that Upalam , Ubamen , Vkelamb , Etalum , Ethaner , epsoner , Alamun , and Erelam , are the eight Enodandas the reupon depending . The third Axiom is , that the Sines of the sides are proportionall to the Sines of their opposite Angles : the truth whereof holds in all Sphericall Triangles whatsoever ; which is proved partly out of the proportion betwixt the Sines of the Perpendiculars substerned under equall Angles , and the Sines of the Hypotenusas : and partly , by the Analogy , that is betwixt the Sines of the Angles sustained by severall Perpendiculars , demitted from one point , and the Sines of the Perpendiculars themselves . The Directory of this Axiom is Vchedezexam , whereby we know that Uchener , Edamon , Ezolum , Exoman , and Amaner , are the five Enodandas thereof . The Orthogonosphericall Table consisteth of these six Figures : Valamenep , Vemanore , Enarulome , Erolumane , Achave , and Esheva . THe first Figure , Valamenep , comprehendeth all those questions , wherein the Subtendent , and an Angle being given , either another Angle , or one of the Ambients is demanded . Of this Figure there be three Moods , viz. Upalam , Ubamen , and Uphanep . The first , to wit Upalam , containeth all those Orthogonosphericall Problems , wherein the Subtendent and one oblique Angle being given , another oblique Angle is required , and by its Resolver Torb — Tag — Nu ☞ Mir , sheweth , that the summe of the Sine complement of the Subtendent side and Tangent of the Angle given , ( the Logarithms of these are alwayes to be understood ) a digit being prescinded from the left , is equall to the Tangent complement of the Angle required ; for the proposition goeth thus , As the Radius , to the Tangent of the Angle given : so the Sine complement of the Subtendent side , to the Tangent complement of the Angle required : and because Tangents , and Tangent complements are reciprocally proportionall , instead of To — Tag — Nu ☞ Mir , or , To — Lu — Mag ☞ Tir , which ( for that the Radius is a meane proportionall betwixt the L. and N. the T. and M ) is all one for inferring of the same fourth proportionall , or foresaid quaesitum ) we may say , Mag — Nu — To ☞ Mir , that is , As the Tangent complement of the given Angle to the Cosine of the Subtendent , so the totall Sine to the Antitangent of the Angle demanded ; for the totall Sine being , as I have told you , a meane proportionall betwixt the Tangents and Cotangents , the subtracting of the Cotangent , or Tangent complement from the summe of the Radius , and Antisine residuats a Logarithm equall to that of the remainder , by abstracting the Radius from the sum of the Cosine of the subtendent , and Tangent of the Angle given , either of which will fall out to be the Antitangent of the required Angle . Notandum . [ Here alwayes is to be observed , that the subtracting of Logarithms may be avoyded , by substituting the Arithmeticall complement thereof , to be added to the Logarithms of the two middle proportionals ( which Arithmeticall complement ( according to Gellibrand ) is nothing else , but the difference between the Logarithm to be subtracted , and another consisting of an unit , or binarie with the addition of cyphers , that is the single , or double Radius ) for so the sum of the three Logarithms , cutting off an unit , or binarie towards the left hand , will still be the Logarithm of the fourth proportionall required . For the greater ease therefore in Trigonometricall computations , such a Logarithmicall Canon is to be wished for , wherein the Radius is left out of all the Secants , and all the Tangents of Major Arches , according to the method prescribed by Mr. Speidel , who is willing to take the paines to make such a new Canon , better then any that ever hitherto hath beene made use of , so that the publike , whom it most concerneth , or some potent man , well minded towards the Mathematicks , would be so generous , as to releeve him of the charge it must needs cost him ; which , considering his great affection to , and ability in those sciences , will certainly be as small a summe , as possibly he can bring it to . ] This Parenthesis , though somewhat with the longest , will not ( I hope ) be displeasing to the studious Reader . The second Mood of the first Figure is Ubamen , which comprehendeth all those Problems , wherein the Subtendent , and one oblique Angle being given , the Ambient adjoyning the Angle given is required , and by its Resolver , Nag — Mu — Torp ☞ Myr , sheweth , that , if to the summe of the Logarithms of the two middle proportionals , we adde the Arithmeticall complement of the first , the cutting off the Index from the Aggregat of the three , will residuat the Tangent complement of the side required : and therefore with the totall Sine in the first place , it may be thus propounded , Torp — Mu — Lag ☞ Myr ; for the first Theorem being , As the Sine complement of the Angle given , to the Tangent complement of the subtendent side : so the totall Sine , to the Tangent complement of the side required : just so the second Theorem , which is that refined , is , As the totall Sine , to the Tangent complement of the Subtendent : so the Secant of the given Angle , to the Tangent complement of the demanded side . Here you must consider , as I have told you already , that of the whole Secant I take but its excesse above the Radius , as I doe of all Tangents above 45. Degrees ; because the cutting off the first digit on the left , supplieth the subtraction , requisite for the finding out of the fourth proportionall ; so that by addition onely the whole operation may be performed , of all wayes the most succinct and ready . Otherwise , because of the totall Sines meane proportionality betwixt the Sine complement , and the Secant ; and betwixt the Tangent , and Tangent complement , it may be regulated thus , To — Tu — Nag ☞ Tyr , that is , As the Radius , to the Tangent of the Subtendent , so the Sine complement of the Angle given , to the Tangent of the side required . The reason of the resolution both of this , and of the former Datoquaere , is grounded on the second Axiom , and the proportion that , in severall rectangled Sphericals which have the same acute Angle at the Base , is found betwixt the Sines of their Perpendiculars , and Tangents of their Bases , as is shewne you by the two first Consonants of the Directory of Sbaprotca . The third and last Mood of the first Figure is Uphaner , which comprehendeth all those Problems , wherein the Hypotenusa , and one of the obliques being given , the opposite Ambient is required , and by its Resolver Tol — Sag — Su ☞ Syr , sheweth , that , if we adde the Logarithms of the Sine of the Angle , and Sine of the Subtendent , cutting off the left Supernumerarie digit from the summe , it gives us the Logarithm of the Sine of the side demanded ; for it is , As the totall Sine , to the Sine of the Angle given : so the Sine of the subtendent side , to the Sine of the side required : and because by the Axiom of Rulerst , it was proved , that when the Sine of any Arch is made Radius , what was then the totall Sine , becomes a Secant ( and therefore Secant complement of that Arch ) instead of Tol — Sag — Su ☞ Syr , we may say , To — Ru — Rag ☞ Ryr , that is , As the totall Sine , is to the Secant complement of the subtendent : so the Secant complement of the Angle given , to the Secant complement of the side demanded . The resolution of this Datoquaere by Sines , is grounded on the first Axiom of Sphericals , which elucidats the proportion betwixt the Sines of the Hypotenusas , and Perpendiculars , as it is declared to us by the first syllable of Suproscas Directory . The second Figure is Vemanore , which containeth all those Orthogonosphericall questions , wherein the subtendent , and an Ambient being proposed , either of the obliques , or the other Ambient is required , and hath three Moods , viz. Ukelamb , Ugemon , and Uchener . The first Mood Ukelamb comprehendeth all those Orthogonosphericall Problems , wherein the subtendent , and one including side being given , the interjacent Angle is demanded , and by its Resolver Meg — Torp — Mu ☞ Nir ( or because of the totall Sines mean proportion betwixt the Tangent , and Tangent complement ) Torp-Teg — Mu ☞ Nir ( which is the same in effect ) sheweth , that if from the summe of the Logarithms of the middle termes , ( which in the first Analogy is the Radius , and Tangent complement of the subtendent ) we subtract the Tangent complement of the given Ambient : or , in the second order of proportionals , joyne the Tangent of the side given , to the Tangent complement of the subtendent , and from the sum cut off the Index ( if need be ) both will tend to the same end , and produce for the fourth proportionall , the Sine complement of the Angle required ; for to subtract a Tangent complement from the Radius , and another number joyned together , whether that Tangent complement be more or lesse then the Radius , it is all one , as if you should subtract the Radius from the said Tangent complement , and that other number ; because the Tangent ( or rather Logarithm of the Tangent ; for so it must be alwayes understood , and not onely in Tangents , but in Sines , Secants , Sides , and Angles , though for brevity sake the word Logarithm be oftentimes omitted ) because I say , the Logarithms of the Tangent , and Tangent complement together , being the double of the Radius ) if first the Tangent complement surpasse the Radius , and be to be subtracted from it , and another number , it is all one , as if from the said number you would abstract the Radius , and the Tangent complements excesse above it , so that the Radius being in both , there will remaine a Tangent with the other number-Likewise , if a Tangent complement , lesse then the Radius , be to be subtracted from the summe of the Radius , and another Logarithm ; it is yet all one , as if you had subtracted the Radius from the same summe ; because , though that Tangent complement be lesse then the Radius : yet , that parcell of the Radius which was abstracted more then enough , is recompensed in the Logarithm of the Tangent to be joyned with the other number ; for , from which soever of the Tangents the Radius be subduced , its Antitangent is remainder : both which cases may be thus illustrated in numbers ; and first , where the Tangent complement is greater then the Radius , as in these numbers 6. 4. 3. 1. and 4. 2. 3. 1. where , let 6. be the Tangent complement , 4. the Radius , 3. the number to be joyned with the Radius , or either of the Tangents , and 1. the remainer ; for 4. and 3. making 7. if you abstract 6. there will remaine 1. Likewise 2. and 3. making 5. if you subtract 4. there will remaine 1. Next , if the Tangent complement be lesse then the Radius , as in 2. 4. 3. 5. and 4. 6. 3. and 5. where , let 2. be the Tangent complement ; for if from 4. and 3. joyned together , you abstract 2. there will remaine 5. which will also be the remainder , when you subtract 4. from 6. and 3. added together . Now to make the same Resolver ( the variety whereof I have beene so large in explaining ) to runne altogether upon Tangents , instead of Meg — To — Mu ☞ Nir , that is , As the Tangent complement of the side given , is to the totall Sine : so the Tangent complement of the subtendent side , to the Sine complement of the Angle required , we may say , Tu — Teg — To ☞ Nir ; that is , As the Tangent for the subtendent , is to the Tangent of the given side ; so the totall Sine , to the Sine complement of the Angle required . All this is grounded on the second Axiom Sbaprotca , and upon the reciprocall proportion of the Tangents and antitangents , as is evident by the third characteristick of its Directory . The second Mood of Vemanore is Vgemon , which comprehendeth all those orthogonosphericall problems , wherein the subtendent , with an Ambient being given , an opposite oblique is required , and by its Resolver , Su-Seg-Tom ☞ Sir , or ( by putting the Radius in the first place , according to Uradesso , the first branch of the first axiom of the Planorectangulars ) To-Seg Ru ☞ Sir , sheweth , that the summe of the side given , and secant of the subtendent ( the Supernumerarie digit being cut off ) is the sine of the Angle required ; for the Theorem is , As the sine of the subtendent , to the sine of the side given : so the Radius , to the sine of the Angle required : or , As the totall sine , to the sine of the side given : so the secant complement of the subtendent , to the sine of the angle required : or , changing the sines into secant complements , and the secant complements into sines , we may say , To — Su — Reg ☞ Rir ; because , betwixt the sine and secant complement , the Radius is a middle proportion . Other varieties of calculation in this , as well as other problems , may be used ; for , besides that every proportion of the Radius to the sine , Tangent , or secant , and contrarily , may be varied three manner of wayes , by the first Axiom of Plaine triangles , the alteration of the middle termes may breed some diversity , by a permutat , or perturbed proportion , which I thought good to admonish the Reader of here , once for all , because there is no problem , whether in Plaine , or Sphericall triangles , wherein the Analogie admitteth not of so much change . The reasons of this Mood of Ugemon , depend on the Axiom of Suprosca , as the second characteristick of Vphugen seemeth to insinuate . The last Mood of the second figure is Vchener , which comprehendeth all those problems , wherein the subtendent , & one Ambient being given , the other Ambient is Required , and by its Resolver , Neg — To — Nu ☞ Nyr , or , To — le-Nu ☞ Nyr , sheweth , that the summe of the sine complement of the subtendent , and the secant of the given side ( which is the Arithmeticall complement of its Antisine ) giveth us the sine complement of the side desired , the Index being removed ; for the theorem is , As the sine complement of the given side , to the total sine ; so the sine complement of the subtendent , to the sine complement of the side required : or more refinedly , As the Radius , to the sine complement of the subtendent : so the secant of the Leg given , to the sine complement of the side required : and besides other varieties of Analogie , according to the Axiom of Rulerst , by making use of the reciprocall proportion of the sine-complements with the secants we may say , To-Ne Lu - ☞ Lyr , that is , As the totall sine , is to the sine complement of the given side : so the secant of the subtendent , to the secant of the side required . The reason of this Datoqueres Resolution is in Seproso the third Axiom of the Sphericals , as is manifest by the first figurative of its Directorie Uchedezexam . The third figure is Enarrulome , whose three Moods are Etalum , Edamon , and Ethaner . This figure comprehendeth all those orthogonosphericall questions , wherein one of the Ambients with an Adjacent angle is given , and the subtendent , an opposite angle , or the other containing side is required . Its first Mood Etalum , involveth all those Orthogono sphericall problems , wherein a containing side , with an insident angle thereon is proposed , and the hypotenusa demanded : and by its resolver Torp-me-nag ☞ mur or ( by inverting the demand upon the Scheme ) Tolp. — me — nag ☞ mur sheweth , that the cutting of the first left digit , from the summe of the Tangent complement of the Ambient proposed , and the sine complement of the given angle , affords us the Tangent complement of the subtendent required ; for the theorem goes thus , As the totall sine , to the tangent complement of the given side ; so the sine complement of the angle given , to the tangent complement of the hypotenusa required . And because the totall sine , hath the same proportion to the tangent complement , which the sine , hath to the sine complement , we may as well say , To-meg-Sa ☞ nur , that is , As the Radius to the tangent complement of the Ambient side ; so the sine of the angle given , to the sine complement of the subtendent required . The progresse of this Mood , dependeth on the Axiom of Sbaprotca , as you may perceive by the fourth consonant of its directorie Pubkutethepsaler . The second Mood of the third Figure , is Edamon , which comprehendeth all those Orthogonosphericall Problems , wherein an Ambient and an Adjacent angle being given , the opposite oblique ( viz. the angle under which the Ambient is subtended ) is required , and by its Resolver To-Neg-Sa ☞ Nir , sheweth that the Addition of the Cosine of the Ambient , and of the sine of the Angle proposed , affordeth us ( if we omit not the usuall presection ) the Cosine of the Angle we seek for ; for it is , As the Radius to the Cosine , or sine complement of the given side : so the sine of the Angle proposed to the Antisins or sine complement of the Angle demanded : now the Radius being alwayes a meane proportionall betwixt the Sine complement , and the Secant , we may for To — Neg — Sa ☞ Nir , say , To — Leg — Ra ☞ Lir , or To — Rag — Le ☞ Lir : that is , As the totall Sine to the Secant , or cutter of the side given , or to the Cosecant , or Secant complement of the given Angle ; so is the Secant complement of the Angle , or Secant of the side , to the Secant , or cutter of the Angle required . The reason of all this is grounded on Seproso , because it runneth upon the proportion betwixt the Sines of the sides , and the Sines of their opposite Angles , as is perspicuous to any by the second syllable of the Directory of that Axiome . The last Mood of the third Figure is Ethaner , which comprehendeth all those Orthogonosphericall Problems , wherein an Ambient with an Oblique annexed thereto , is given , and the other Arch about the right Angle is required , and by its Resolver , Torb — Tag — Se ☞ Tyr , sheweth , that if we joyne the Logarithms of the two middle proportionals , which are the Tangent of the given Angle , and the Sine of the side , the usuall prefection being observed , we shall thereby have the Tangent , or toucher of the Ambient side desired ; for it is , As the Radius to the Tangent of the Angle given , so the Sine of the containing side proposed to the side required : And because the Tangent complement , and Tangent are reciprocally proportionall , the Sine likewise , and Secant complement , for To — Tag — Se ☞ Tyr , we may say , keeping the same proportion , To — Reg — Ma ☞ Myrs that is , As the Radius , to the Secant complement of the given side : so the Tangent complement of the Angle proposed , to the Tangent complement of the side required . The truth of all these operations dependeth on Sbaprotca , the second Axiome of the Sphericals , as is evidenced by θ. the fifth characteristick of its Directory Pubkutethepsaler . The fourth Figure is Erollumane , which includeth all Orthogonosphericall questions , wherein an Ambient , and an opposite oblique being given , the subtendent , the other oblique , or the other Ambient is demanded : It hath likewise , conforme to the three former Figures , three Moods belonging to it ; the first whereof is Ezolum . This Ezolum comprehendeth all those Orthogonosphericall , Problems , wherein one of the Legs , with an opposite Angle being given , the Subtendent is required , and by its Resolver , Sag — Sep — Rad ☞ Sur , or by putting the Radius in the first place , To — Se-Rag ☞ Sur , sheweth , that the abstracting of the Radius from the sum of the Sine of the side , and Secant complement of the Angle given , residuats the Sine of the hypotenusa required ; for it is , As the Sine of the Angle given , to the Sine of the opposite side : so the Radius to the Sine of the subtendent : or more refinedly , As the totall Sine , to the Sine of the side : so the Secant complement of the Angle given , to the Sine of the subtendent side : And because of the Sines and Antisecants , or Secant complements reciprocall proportionality , To — Sag — Re ☞ Ru , that is , As the Radius to the Sine of the Angle given : so the Secant complement of the proposed side , to the Secant complement of the subtendent required . The reason of all this is grounded on the third Axiom Seproso , as is made manifest by the third Syllable of its Directory . The second Mood of this Figure is Exoman , which comprehendeth all those Problems , wherein a containing side , and an opposite oblique being given , the adjacent oblique is required : and by its Resolver , Ne — To — Nag ☞ Sir , or more refinedly , To — Le — Nag ☞ Sir , sheweth , that the summe of the Sine of the Angle , together with the Arithmeticall complement of the Antisine of the Leg , ( which in the Table I have so much recommended unto the Reader , is set downe for a Secant ) the usuall prefection being observed , affordeth us the Sine of the Angle required , and because of the reciprocall proportion betwixt the Sine complement , and Secant ; and betwixt the Sine , and Secant complement , the Theorem may be composed thus : To — Neg — La ☞ Rir , that is , As the Radius , to the Sine complement of the given side : so the Secant of the Angle proposed , to the Secant complement of the Angle demanded . The reason of this is likewise grounded on Seproso , as you may perceive by the fourth characteristick of its Directory . The last Mood of this Figure is Epsoner , which containeth all those Orthogonosphericall Problems , wherein an Ambient and an opposite Oblique being given , the other Ambient is demanded , and by its Resolver , Tag — Tolb — Te ☞ Syr , or more elabouredly , Tolb — Mag — Te ☞ Syr , sheweth , that the praescinding of the Radius from the summe of the Tangent of the side , and Antitangent of the given Angle , residuats the Sine of the side required ; for it is , As the Tangent of the Angle proposed , to the totall Sine : so the Tangent of the given side , to the Sine of the side demanded : or , As the Radius , to the Tangent complement of the Angle given : so the Tangent of the given side , to the Sine of the side required : and because of the reciprocall Analogy betwixt the Tangents , and Co-tangents : and betwixt the Sines , and Co-secants , we may with the same confidence , as formerly , set it thus in the rule , To — Meg — Ta ☞ Ryr , and it will find out the same quaesitum . The reason of the operations of this Mood because of the ingrediencie of Tangents dependeth on Sbaprotca , as is perceivable by the sixth determinater of its Directory Pubkutethepsaler . The fifth Figure of the Orthogonosphericals is Achave , which containeth all those Problems , wherein the Angles being given , the subtendent or an Ambient is desired , and hath two Moods Alamun , and Amaner . Alamun comprehendeth all those Problems , wherein the Angles being proposed , the Hypotenusa is required , and by its Resolver Tag — Torb — Ma ☞ Nur , or more compendiously , Torb — Mag — Ma ☞ Nur , sheweth , that the summe of the Co-tangents , not exceeding the places of the Radius , is the Sine complement of the subtendent required ; for it is , As the Tangent of one of the Angles , to the Radius : so the Tangent complement of the other Angle , to the Sine complement of the Hypotenusa demanded : or , As the totall Sine , to the Tangent complement of one of the Angles : so the Tangent complement of the other Angle , to the Sine complement of the subtendent we seek for . The running of this Mood upon Tangents , notifieth its dependance on Sbaprotca , as is evident by the seventh determinater of the Directory thereof . The second Mood of this Figure is Amaner , which comprehendeth all those Orthogonosphericall Problems , wherein the Angles being given , an Ambient is demanded , and by its Resolver , Say — Nag — Tω ☞ Nyr , or more perspicuously , Tω — Noy — Ray ☞ Nyr , sheweth , that the summe of the Logarithms of the Antisine of the Angle opposite to the side required , and the Arithmeticall complement of the Sine of the Angle , adjoyning the said side , which we call its Secant complement , with the usuall presection , is equall to the Sine complement of the same side demanded ; for it is , As the Sine of the Angle adjoyning the side required , to the Antisine of the other Angle : so the totall sine , to the Antisine of the side demanded : or , As the Radius , to the Antisine of the Angle opposite to the demanded side : so the Antisecant of the Angle conterminat with that side , to the Antisine of the side required : and because of the Analogy betwixt the Antisines , and Secants : and likewise betwixt the Antisecants , and Sines , we may expresse it , To — Say — La ☞ Lyr ; that is , As the Radius , to the Sine of the Angle insident on the required side : so the Secant of the other given Angle , to the Secant of the side that is demanded . Here the Angulary intermixture of proportions giveth us to understand , that this Mood dependeth on Seproso , as is manifested by the last characteristick of Uchedezexam the Directory of this Axiom . The sixth and last Figure is Escheva , which comprehendeth all those Problems , wherein the two containing sides being given , either the subtendent , or an Angle is demanded : it hath two Moods , Enerul and Erelam . The first Mood thereof Enerul , containeth all such Problems as having the Ambients given , require the subtendent , and by its Resolver , Ton — Neg — Ne ☞ Nur , sheweth , that the summe of the Logarithms of the Cosines of the two Legs unradiated , is the Logarithm of the Co-sine of the subtendent ; for it is , As the totall Sine , to the Co-sine of one of the Ambients : so the Co-sine of the other including Leg given , to the Co-sine of the required subtendent ; and because of the Co-sinal , and Secantine proportion , we may safely say , To — Leg — Le ☞ Lur. That is , As the Radius to the Secant of one shanke or Leg : so the secant of the other shanke or Leg , to the secant of the Hypotenusa demanded . The coursing thus upon Sines , and their proportionals evidenceth that this Mood dependeth on Suprosca , the first of the Sphericall Axioms , which is pointed at by the third and last characteristick of Vphugen the directorie thereof . The second Mood of the last figure , and consequently the last Mood of al the Orthogonosphericals , is Erelam , which comprehendeth all those orthogonosphericall problems , wherin the two containing sides being proposed , an Angle is demanded , and by its Resolver , Sei — Teg — Torb ☞ Tir , or by primifying the Radius , Torb — Tepi-Rexi ☞ Tir , giveth us to understand , that the cutting off the Radius from the summe of the Tangent of the side opposite to the Angle demanded , and the cosecant of the side conterminat with the said Angle , residuats the touch-line of the Angle in question ; for it is , As the sine of the side adjoyning the Angle required , to the tangent of the other given side : so the Radius to the tangent of the Angle demanded : or , As the totall sine to the Tangent of the Ambient opposite to the angle sought : so the Antisecant of the Leg adjacent to the said asked Angle , to the Tangent or toucher thereof : and because Sines have the same proportion to cosecants , which Tangents have to Cotangents , we may say , To — Sei — me ☞ mir , that is , As the Radius to the sine of the side conterminat with the angle required : so the Cotangent of the other Leg , to the Cotangent of the Angle searched after : or yet more profoundly by an Alternat proportion , changing the relation of the fourth proportionall , although the same formerly required Angle , thus , To — Rei — me ☞ mor , that is , As the Radius to the Antisecant of the side adjacent to the Angle sought for , so the Antitangent of the other side , to the Antitangent of that sides opposit Angle , which is the Angle demanded . The reason hereof is grounded on Sbaprotca ; for the Tangentine proportion of the terms of this Mood specifieth its dependance on the second Axiom , which is showen unto us by the eight and last characteristick of its directorie Pubkutethepsaler . Here endeth the doctrine of the right-Angled sphericalls , the whole diatyposis wherof is in the Equisolea or hippocrepidian diagram , whose most intricate amfractuosities , renvoys , various mixture of analogies , and perturbat situation of proportionall termes , cannot choose but be pervious to the understanding of any judicious Reader that hath perused this Comment aright . And therefore let him give me leave ( if he think fit ) for his memorie sake , to remit him to it , before he proceed any further . The Loxogonosphericall Triangles , whether Amblygonosphericall or Oxygonosphericall , are either Monurgetick or Disergetick . THe Monurgetick have two figures , Datamista and Datapura . Datamista is of all those Loxogonospherical Monurgetick problems , wherein the Angles and sides being intermixedly given , ( and therefore one of them being alwaies of another kind from the other two ) either an Angle , or a side is demanded : it hath two Moods , Lamaneprep , and Menerolo . The first Mood Lamaneprep , comprehendeth all those Loxogonosphericall problems , wherein two angles being given , and an opposit side , another opposit side is demanded , and by its Resolver , Sapeg — Se — Sapy — ☞ Syr , sheweth , that if to the Logarithms of the sine of the side given , and sine of the Angle opposit , to the side required , we joyne the Arithmeticall complement of the sine of the Angle opposit , to the proposed side ( which is the refined Antisecant ) we will thereby attain to the knowledge of the sine of the side demanded . The reason of this is grounded on the third Axiom , Seprosa , as you may perceive by the first syllable of the Obliquangularie directory , Lame . The second Mood of this figure is Menerolo , wich comprehendeth all those Amblygonosphericall problems , wherein two sides being given with an opposit angle , another opposit angle is demanded , and by its Resolver Sepag — Sa — Sepi ☞ Sir , sheweth , that if to the summe of the Logarithms of the sine of the given angle , and sine of the side opposit to the angle required , we joyne the Arithmeticall complement of the sine of the side opposit to the given angle ( which is the refined Cosecant of the said angle ) it will afford us the sine of the angle required . The reason of this operation is grounded on the third Axiom of Sphericalls , a progresse in sines shewing clearely , how that both this , and the former , doe totally depend on the Axiom of Seprosa , as is evident by the second syllable of its directorie , Lame . The second figure of the Monurgetick Loxogonosphericalls treateth of all those questions , wherein the Datas being either sides alone , or Angles alone , an Angle or a side is demanded . This Figure of Datapura is divided into two Moods , viz. Nerelema , and Ralamane , which are of such affinity , that upon one and the same Theorem dependeth the Analogy that resolveth both . The first Mood thereof , Nerelema , comprehendeth all those Problems , wherein the three sides being given , an Angle is demanded , and is the third of the Monurgeticks , as by its Characteristick the third Liquid is perceivable . The curteous Reader may be pleased to take notice , that in both the Moods of the Datapurall Figure , I am in some measure necessitated for the better order sake , to couch two precepts , or documents , for the Faciendas thereof , and to premise that one concerning the three Legs given , before I make any mention of the maine Resolver , whereupon both the foresaid Moods are founded , to which Resolver , because of both their dependences on it , I have allowed here in the Glosse , the same middle place , which it possesseth in the Table of my Trissotetras . The precept of Nerelema is Halbasalzes * Ad* Ab* Sadsabreregalsbis Ir : that is to say , for the finding out of an Angle when the three Legs are given , as soone as we have constituted the sustentative Leg of that Angle a Base , the halfe thereof must be taken , and to that halfe we must adde halfe the difference of the other two Legs , and likewise from that halfe subtract the half difference of the foresaid two Legs , then the summe and the residue being two Arches , we must , to the Logarithms of the Sine of the summe , and Sine of the Remainer , joyne the Logarithms of the Arithmeticall complements of the Sines of the sides , which are the refined Antisecants of the said Legs , and halfe that summe will afford , us the Logarithm of the Sine of an Arch , which doubled , is the verticall Angle , we demand ; for out of its Resolver , Parses — Powto — Parsadsab ☞ Powsalvertir , is the Analogy of the former work made cleare , the Theorem being , As the Oblong or Parallelogram contained under the Sines of the Legs , to the square , power or quadrat of the totall Sine : so the Rectangle , or Oblong made of the right Sines of the sum , and difference of the halfe Base , and difference of the Legs , to the square of the right Sine of halfe the verticall Angle . The reason hereof will be manifest enough to the industrious Reader , if when by a peculiar Diagram , of whose equiangular Triangles the foresaid Sines and differences are made the constitutive sides , he hath evinced their Analogy to one another , he be then pleased to perpend , how , in two rowes of proportionall numbers , the products arising of the homologall roots , are in the same proportion amongst themselves , that the said roots towards one another are ; wherewithall if he doe consider , how the halfs must needs keep the same proportion that their wholes ; and then , in the work it selfe of collationing severall orders of proportionall termes , both single and compound , be carefull to dash out a divider against a multiplyer , and afterwards proceed in all the rest , according to the ordinary rules of Aequation , and Analogy , he cannot choose but extricat himselfe with ease forth of all the windings of this elaboured proposition . Upon this Theorem ( as I have told you ) dependeth likewise the Document for the faciendum of Ralamane , which is the second Mood of Datapura , and the last of the Monurgetick Loxogonosphericals , as is pointed at by Nera the Directory therof . This Mood Ralamane comprehendeth all those Loxogonosphericall Problems , wherein the three Angles being given , a side is demanded . And by its Resolver , Parses — Powto — Parsadsab ☞ Powsalvertir , according to the peculiar precept of this Mood Kourbfasines ( Ereled ) Koufbraxypopyx , sheweth , that if we take the complement to a semicircle of the Angle opposite to the side required , which for distinction sake we doe here call the Base ; and frame , of the foresaid complement to a semicircle , a second Base for the fabrick of a new Triangle , whose other two sides have the graduall measure of the former Triangles other two Angles : ( and so the three Angles being converted into sides ) the complement to a Semicircle of the new Verticall , or Angle opposite to the new Base , will be the measure of the true Base or Leg required , and the Angle insident on the right end of the new Base in the second Triangle , falleth to be the side conterminall with the left end of the true Base in the first Triangle , and the Angle adjoyning the left end of the false Base in the second Triangle , becomes the side adjacent to the right end of the old Base in the first Triangle . So that thus by the Angles all andeach of the sides are found out , all which works are to be performed by the preceding Mood , upon the Theorem , whereof the reason of all these operations doth depend . The Disergetick Loxogonosphericals are grounded on foure Axioms , viz. 1. NAbadprosver . 2. Naverprortes , Siubprortab , and Niubprodnesver : the foure Directories whereof , each in order to its owne Axiome , are Alama , Allera , Ammena , and Ennerra . The first Axiome is , Nabadprosver , that is , In Obliquangular Sphericals , if a Perpendicular be demitted from the verticall Angle to the opposite side , continued if need be , The Sines complements of the Angles at the Base , will be directly proportionall to the Sines of the verticall Angles , and contrary : the reason hereof is inferred out of the proportion , which the Sines of Angles , substerned by Perpendiculars , have to the Sines of the said perpendiculars , so that they belong to the Arches of great Circles , concurring in the same point , and that from some point of the one , they be let fall on the other Arches ▪ which proportion of the Sines of the said Perpendiculars , to the Sines of the Angles subtended by them , stoweth immediatly from the proportion , which ( in severall Orthogonosphericals , having the same acute Angle at the Base ) is betwixt the Sines of the Hypotenusas , and the Sines of the perpendiculars ; the demonstration whereof is plainly set downe in my Glosse on Suprosca , the first generall Axiome of the Sphericals , of which this Axiome of Nabadprosver is a consectary . The Directory of this Axiome is Alama , which sheweth , that the Moods of Alamebna and Amanepra are grounded on it . The second Disergetick Axiome is Naverprortes , that is to say , the Sines complements of the verticall Angles , in obliquangular Triangles ( a Perpendicular being let fall from the double verticall on the opposite side ) are reciprocally proportionall to the Tangents of the sides : the reason hereof proceedeth from Sbaprotca , the second generall Axiome of the Sphericals ; according to which , if we doe but regulate , after the customary Analogicall manner , two quaternaries of proportionals of the former Sines complements , and Tangents proposed , we will find by the extremes alone ( excluding all the intermediate termes ) that the Sines complements of the verticall Angles ( both forwardly , and inversedly ) are reciprocally proportioned to the Tangents of the sides , and contrariwise from the Tangents , to the Sines . The Directory of this Axiome is Allera , which evidenceth , that the Moods of Allamebne , and Erelomab depend upon it . The third Disergetick Axiome is Sinbprortab , that is to say , that in Obliquangular Sphericals ( if a perpendicular be drawne from the verticall Angle unto the opposite side , continued if need be ) the Sines of the segments of the Base , are reciprocally proportionall to the Tangents of the Angles conterminate at the Base , and contrary : the proofe of this , as well as that of the former confectary dependeth on Sbaprotca , the second generall Axiome of the Sphericals , according to which , if we so Diagrammatise an Ambly gonosphericall Triangle , by Quadranting the Perpendicular , and all the sides , and describing from the Basangulary points two Quadrantall Arches , till we hit upon two rowes of proportionall Sines of Bases to Tangents of Perpendiculars , then shall we be sure ( if we exclude the intermediate termes ) to fall upon a reciprocall Analogy of Sines , and Tangents , which alternatly changed , will afford the reciprocall proportion of the Sines of the Segments of the Base , to the Tangents of the Angles conterminat thereat , the thing required . The Directory of this Axiome is Ammena , which certifieth that Ammanepreb and Enerablo are founded thereon . The fourth and last Disergetick Axiome is Niubprodnesver , that is to say , that in all Loxogonosphericals ( where the Cathetus is regularly demitted ) the Sines complements of the Segments of the Base , are directly proportionall to the Sines complements of the sides of the verticall Angles , and contrary . The reason hereof is made manifest , by the proportion that is betwixt the Sines of Angles , subtended by Perpendiculars and the Sines of these Perpendiculars ; out of which we collation severall proportions , till , both forwardly and inversedly we pitch at last upon the direct proportion required . The Directory of this Axiome is Ennerra , which declareth that Ennerable , and Errelome are its dependents . Of the Disergetick Loxogonosphericals there be in all foure Figures ; two Angulary , and two Laterall . THe two Angulary are Ahalebmane and Ahamepnare : The two laterall are Ehenabrole and Eheromabne . The first Angulary Disergetick Loxogonosphericall Figure , Ahalebmane , comprehendeth all those Problems , wherein two Angles being given with a side betweene , either the third Angle , or an opposite side is demanded ; and accordingly hath two Moods , the first whereof is Alamebna , and the second Allamebne . Alamebna concerneth all those Loxogonosphericall Disergetick Problems , wherein two Angles being proposed , with an interjacent side , the third Angle is required ; which Angle , according to the severall Cases of this Mood , is alwayes one of the Angles at the Base , that is to say ( in the termes of my Trissotetr as ) a prime , or next opposite , or at least one of the co-opposites , to the Perpendicular to be demitted . And therefore , conforme to the nature of the Case of the Datoquaere in hand , and that it may the more conveniently fall within the compasse of the Axiome of Nabadprosver , an Angle by the first operation of this Disergetick is to be found out , which must either be a double verticall , a verticall in the little rectangle , or a verticall , or co-verticall ( as sometimes I call it ) in one of the correctangles . Thus much I have thought fit to premise of the praenoscendum of this Mood , before I come to its Cathetothesis ; because , in my Trissotetrall Table , to avoid the confusion of homogeneall termes ( though the order of doctrine would seeme to require another Method ) the first and prime Orthogonosphericall work is totally unfolded , before I speak any thing of the variety of the Perpendiculars demission , to which , owing its rectangularity , it thereby obtaineth an infallible progresse to the quaesitum : but , seeing in the Glosse I am not to astrict my self to so little bounds , as in my Table , I will observe the order that is most expedient ; and , before the resolution of any operation in this Mood , deduce the diversity of the Perpendiculars prosiliencie in the severall Cases thereof . Let the Reader then be pleased to consider , that the generall Maxim for the Cathetothesis of this Mood is Cafregpiq , the meaning whereof is , that , whether the side whereon the Perpendicular is demitted be increased or not , that is to say , whether the Perpendicular fall outwardly , or inwardly , it must fall from the extremity of the given side , and opposite to the Angle required : however it is to be remarked , that in this Mood , whatever be the affection of the Angles ( unlesse they be all three alike ) the Perpendicular may fall out wardly . The generall maxim for the Cathetothesis of this Mood , as well as for that of all the rest , is divided into foure Tenets , according to the number of the Cases of every Mood . Here must I admonish the Reader , that he startle not at the mentioning of foure especiall Cathetothetick Tenets , and foure severall Cases belonging to each Disergetick Mood , seeing , to the most observant eye , there be but three of either perceptible in my Trissotetr as ; for , the fourth both Tenet and Case being the same by way of expression in all the Moods , and being fully resolved by the third Case of every Mood , it shall suffice to speak thereof here once for all : The Tenet of this common Case is Simomatin , that is to say , when all the three Angles in any of those Disergeticks are of the same affection , either all acute , or all oblique , the Perpendicular falleth inwardly , whether the double verticall be an Angle given , an Angle demanded , or neither . Yet here it is to be considered , that seeing Triangles may be either calculated by their reall and naturall , or by their circular parts , or by both together , and that for the more facility we oftentimes , instead of the proposed Triangle , resolve its opposite ; it is not the reall and given Triangle , that in this case we so much take notice of , as of its resolvable , and equivalent , the opposite Triangle : as for example , If a Sphericall Triangle , with two obtuse Angles , and one acute , be given you to resolve , it will fall within the compasse of Simomatin ; because its opposite Sphericall is simply acute angled : and also if you be desired to calculate a Sphericall Triangle with two acute Angles , and one obtuse , it will likewise fall within the reach of the same Case ; because its opposite Sphericall is simply obtusangled . The reason of both the premisses is from the equality of the opposite Angles of concurring Quadrants , which that they are equall , no man needs to doubt , that will take the paines to let fall a Perpendicular from the middle of the one Quadrant upon the other ; for so there will be two Triangles made equilaterall : and seeing it is an universally received truth , that equall sides sustaine equall Angles , the identitie of the Perpendicular in both the foresaid Triangles , must needs manifest the equality of the two opposite Angles . I have beene the ampler in the elucidating of this Case , that , it over-running all the Moods of the Disergetick Loxogonosphericals , the Reader , in what Mood or Datoquare soever he please to resolve this foresaid Case , may for that purpose to this place have recourse ; to the which , without any further intended reiteration of this Tenet , I doe heartily remit him . The first especiall Tenet of the generall Maxim of the Cathetothesis of this Mood is Dasimforaug , that is , When the given Angles are of the same nature , but different from the required , the Perpendicular falleth outwardly , and the first verticall is a given Angle : the second Tenet belonging to the second Case of this Mood is Dadisforeug , that is , When the proposed Angles are of different affections , the Perpendicular is externally demitted , and one of the given Angles is a second verticall : Yet this discrepance is to be observed betweene the externall prosiliencie of the Perpendicular Arch in this Case , and that other of the former ; that in the former , it is no matter from which of the ends of the proposed side , the Perpendicular be let fall upon one of the comprehending Legs of the Angle required , which Leg must be increased ; for it is a generall Notandum , that the sustentative Leg of a perpendiculars exterior demission must alwayes be continued : but in this Case , the outward falling of the Perpendicular is onely from one extremity of the given side ; for , if it be demitted likewise from the other end , it falleth then inwardly , and so produceth the third Tenet of this Mood , which is Dadisgatin , that is , If the given Angles be of a different quality , and that the Perpendicular be internally demitted , the double verticall is one of the proposed Angles . The nature of the Perpendiculars demission in all the Cases of this Mood being thus to the full explained , we may without impediment proceed to the performance of all the Orthogonosphericall operations , each in its owne order thereto belonging . To begin therefore at the first , whose quasitum ( as I have told you already ) is a verticall Angle , we must know , seeing the work is Orthogonospherically to be performed , that the forementioned praenoscendum cannot be obtained without the help of one of the sixteen Datoquaeres ; and therefore in my Trissotetras ( considering the nature of what is given , and asked in the Cases of this Mood ) I have appointed Upalam to be the subservient of its praenoscendum ; for , by the Resolver thereof , To — Tag — Nu ☞ Mir , ( the subtendent and an Angle being given ; for one of the given sides of every Loxogonosphericall , if the Perpendicular be rightly demitted , becomes a subtendent , and sometimes two given sides are subtendents both ) we frame these three peculiar Problems , for the three praenoscendas ; to wit , Utopat , for the double verticall , by the meanes of the great Subtendent side , and the prime opposite Angle : secondly , Udobaud , for obtaining of the first verticall in the little rectangle , by vertue of the lesser subtendent in the same rectangle , and the next opposite Angle : lastly , Uthophauth , for the first Co-verticall , by meanes of the first Co-subtendent , and first coopposite Angle : all which is at large set downe in the first partition of Alamebna in my Table . The first and chiefe operation being thus perfected , the verticall Angles so found out must concurre with each its correspondent opposite for the obtaining of the Perpendicular , necessary for the accomplishment of the second operation in every one of the Cases of the foresaid Mood ; to which effect Amaner is made the Subservient , by whose Resolver Say — Nag — Tw ☞ Nyr , these three Datoquaeres , Opatca , Obautca , and Ophauthca come to light , and is manifestly shown how , by any paire of three severall couples of different Angles , the Perpendicular is acquirable . Now , though of this work ( as it is a single one ) no more then of the other succeeding it in the same Mood , nor of the last two in any of the Disergeticks in their full Analogy , I doe not make any mention at all in my Table ; but , after the couching of the first operation for the Praenoscendas , supply the roomes of the other two , with an equivalent row of proportionals out of them specified , for attaining to the knowledge of the maine quaesitum : yet in this Comment upon that Table , for the more perspicuities sake , and that the Reader may as well know , what way the rule is made , as how thereby a demand is to be found out , I have thought fit to expatiat my selfe for his satisfaction on each operation apart , and Analytically to display in the glosse , what is compounded in the Trissotetras . And therefore , according to that prescribed method , to proceed in this Mood , the perpendicular by the second operation being already obtained , it is requisite for the promoving of a third work , that the said Perpendicular be made to joyne with the second verticaline , the double verticall , and second co-verticall conforme to the quality of the three Cases , thereby to obtaine the Angles at the Base , for the which all these operations have beene set on foot ; to wit , the next cathetopposite , ( whose complement to a Semicircle is alwayes the Angle required ) the prime cathetopposite , and the second Cocathetopposite : for the prosecuting of this last work , Edoman is the subservient , by whose Resolver , To — Neg — Sa ☞ Nir , we are instructed how to regulate the Problemets of Catheudob , Cathatop , and Catheuthops . Now these two last operations being thus made patent in their severall structures , it is not amisse that we ponder how appositely they may be conflated into one , to the end , that the verity of all the finall Resolvers of the Disergeticks in my Trissotetras ( which are all and each of them composed of the ingredient termes of two different works ) may be the more evidently knowne , and obvious to the reach of any ordinary capacity , for the performance hereof , the Resolvers of these two operations are to be laid before us , Say — Nag — Ta ☞ Nyr , and To — Neg — Sa ☞ Nir : and , seeing out of both these orders of proportionals , there must result but one , it is to be considered , which be the foure ejectitious termes , and which those foure we should reserve for the Analogy required ; all which , that it may be the better understood by the industrious Reader , I will interpret the Resolvers so farre forth as is requisite : and therefore Say — Nag — To ☞ Nyr , being , As the Sine of one of the Angles at the Base , or Cathetopposite , is to the Sine complement of a verticall : so the Radius to the Sine complement of the Perpendicular : And the other , To — Neg — Sa ☞ Nir , being , As the Radius , to the Sine complement of the Perpendicular : so the Sine of a verticall , to the Sine complement of a Cathetopposite or Angle at the Base ; it is perceivable enough how both the Radius , and the Perpendicular are in both the rows : nor can it well escape the knowledge of one never so little versed in the elements of Arithmetick , that the Perpendiculars being the fourth terme in the first order of proportionals , is nothing else but that it is the quotient of the product of the middle termes , divided by the first , or Logarithmically the remainder of the first termes abstraction from the summe of the middle two ; so that the whole power thereof is inclosed in these three termes , whereby it is most evident , that with what terme soever the foresaid Perpendicular be employed to concurre in operation , the same effect will be produced by the concurrence of its ingredients with the said terme , and therefore in the second row of proportionals , where it is made use of for a fellow multiplyer with the third terme to produce a factus , which divided may quote the maine quaesitum , or Logarithmically to joyne with the third terme , for the summing of an Aggregat from which the first terme being abstracted , may residuat the terme demanded , it is all one , whether the work be performed by it selfe , or by its equivalent , viz. the three first termes of the first order of proportionals , in whose potentia it is : whereupon the fourth terme in the second row being that , for the obtaining whereof , both the Analogies are made , we need not waste any labour about the finding out of the Perpendicular ( though a subservient to the chiefe quaesitum ) but leaving roome for it in both the rows , that the equipollencie of its conflaters may the better appeare , go on in work without it , and , by the meanes of its constructive parts , with as much certainty effectuat the same designe . Thus may you see then how the eight termes of the forementioned Resolvers , are reduced unto six ; but there remaining yet two more to be ejected , that both the orders may be brought unto a compound row of foure proportionals : let us consider the Radius , which , being in both the rows as I have once told you already , may peradventure , without any prejudice to the work , be spared out of both . Thus much thereof to any is perceiveable , that in the first Resolver , it is the third proportionall ; and in the second , the first , and consequently a multiplyer in the one , and in the other a divider : or Logarithmically in the second a subtractor , and in the first an adder : now it being well known that division overthrows the structure of multiplication , and that what is made up by addition , is by subtraction cast down ; we need not undergoe the laboriousnesse of such a Penelopaean task , and by the division and abstraction of what we did adde and multiply , weave and unweave , build up , and throw downe the self same thing : but choose rather ( seeing the Radius undoeth in the one , what it doth in the other ( which ineffect is to doe nothing at all ) to dash the one against the other , and race it out of both ) then idely to expend time , and have the proportion pestred with unnecessarie termes . Thus from those two resolvers , foure termes being with reason ejected , we must , for the finding out of the last in the second Resolver , effectuat as much by three , as formerly was on seven incumbent , which three being the first , and second termes in the first row of proportionalls ; and the third in the second , the two Resolvers Say — Nag — To ☞ Nyr , & To — Neg — Sa ☞ nir are comprehended by this one Say — Nag — Sa — ☞ Nir , that is , As the sine of one verticall to the Antisine of an opposite ; so the sine of another verticall to the Antisine of another opposite : and though the second Resolver doth import , that this other opposite is to be found out by the Antisine of the perpendicular , and sine of a secondarie verticall , yet doth it in nothing evince the coincidence of the two operations in one ; because the first two termes of the Resultative Analogie , doe adaequatly stand for the perpendicular , which I have proved already , and therefore these two in their proper places co-working with the third terme , according to the rule of proportion , have the selfe same influence , that the Perpendicular so seconded , hath upon the operatum . Now , to contract the generality of this finall Resolver , Say — Nag — Sa ☞ Nir , to all the particular Cases of this Mood , we must say , When the given Angles are of the same affection , and the required diverse , as in Dasimforaug , the first case , Sat — Nop — Seud ☞ Nob* Kir , that is , As the Sine of the double verticall to the Antisine of the prime Cathetopposite : so the Sine of the second verticaline ( or verticall in the lesser rectangle ) to the Antisine of the next Cathetopposite , whose complement to a semicircle is the Angle required . But when , the affection of the given Angles being different , the perpendicular is made to fall without , as in Dadisforeug , the second Case of this Mood , the Resolver thereof is particularised thus , Saud — Nob — Sat ☞ Nop * Ir , that is , As the Sine of the first verticaline ( or verticall in the rectanglet , ) to the Sine complement of the nearest Cathetopposite : so the Sine of the double verticall , to the Sine complement of the prime Cathetopposite , which is the Angle required . And lastly , if with the different qualities of the given and demanded Angles , the Perpendicular be let fall within , as in Dadisgatin , the third Case of this Mood , then is the finall Resolver to be determined thus , Sauth — Noph — Seuth ☞ Nops * Ir , that is , As the Sine of the first coverticall , to the Co-sine of the first Co-opposite : so is the Sine of the second coverticall , to the Co-sine of the second Co-opposite which is the Angle required . The originall reason of all these operations is grounded on the Axiome of Nabadprosver , as the first syllable of its Directory Alama giveth us to understand , which we may easily perceive by the Analogy , that is onely amongst the Angles without any intermixture of sides in the termes of the proportion . The second Mood of the first Angulary Figure ( that is to say , the first two termes of whose datas are Angles ) is Allamebne , which comprehendeth all those Disergetick questions , wherein two Angles being given and a side betweene , one of the other sides is demanded , which side ( the perpendicular being let fall ) is alwayes one of the second Subtendents , viz. in the first Case a second Subtendent of the lesser Triangle , in the second a second Subtendent in the great rectangle , and in the last a second Co-subtendent . To the knowledge of all these , that we may the more easily attaine , we must consider the generall maxim of the Cathetothesis of this Mood , which is Cafyxegeq that is to say , that in all the Cases of Allamebne the Perpendicular falleth from the side required , and from that point thereof , where it conterminats with the given side upon the third side , continued if need be ; and according to the variety of the second subtendent , which is the side demanded , there be these three especiall Tenets of this generall Maxim , to wit , Dasimforauxy , Dadiscracforeng , and Dadiscramgatin . Dasimforauxy , the first especiall Tenet of the generall Maxim of the Cathetothesis of this Mood sheweth , that , when the proposed Angles are of the same quality and homogeneall , the Perpendicular falleth externally , and the first verticall is one of the given Angles , and annexed to the required side . The second Tenet , Dadiscracforeug , which pertaineth to the second Case of this Mood , sheweth , that when the given Angles are of a discrepant nature , and heterogeneall , and that the concurse of the proposed and required sides is at an acute Angle , that then the Perpendicular must be demitted outwardly , and one of the proposed Angles becomes a second verticall . The third Tenet is Dadiscramgatin , whereby we learne , that if with the various affection of the Angles given , the concurse ( mentioned in the preceding Tenet ) be at an obtuse Angle , the Perpendicular falleth inwardly , and that one of the foresaid Angles is a double verticall . This is the onely Case of Allamebne , wherein the Perpendicular is demitted inwardly , save when the three Angles are qualified all alike , of which Case , because it falleth in all the Moods of the Loxogonosphericall Disergeticks , and that in Alamebna I have spoke at large thereof , I shall not need ( I hope ) to make any more mention hereafter . Having thus unfolded the mysteries of the Perpendiculars demission in all the Cases of this Mood ( as I must doe in all those of every one of the other Loxogonosphericall Disergeticks ; because such Obliquangulars , till they be reduced to a rectangularity ( which without the Perpendicular is not performable ) can never Logarithmically be resolved ) I may safely go on , without any let to the Reader , to the three severall Orthogonosphericall operations thereof , as they stand in order . The quaesitas of the first operation , which are alwayes the praenoscendas of the Mood , are in this Mood the same that they were in the last , to wit , the double verticall , the first verticaline , and the first coverticall : and are likewise to be found out by the same Datas both of side , and Angle here , that they were in the former Mood ; that is , for the side , by the first and great Subtendent : the first but little Subtendent : and the first Co-subtendent : and for the Angle , by the prime Cathetopposite , the nearest Cathetopposite , and the first Co-cathetopposite : so that the Datoquaere sounding thus , the Subtendent , and an Oblique Angle being given , to find the other Oblique , the Subservient of this Computation must needs be Upalam , and its Resolver , To — Tag — Nu ☞ Mir , which sheweth , that the subducing of the Logarithm of the Radius from the summe of the Logarithms of the Sine complement of one of the first Subtendents , and Tangent of one of the Angles at the Base , residuats the Logarithm of the Tangent complement of one of the verticals required , and consequently involveth within so much generality the particular resolutions of the Sub-problems of Upalam , viz. Utopat , Vdoband , and Vthophauth , diversified thus according to the variety of their praenoscendas , whereon , to speak ingenuously , I intend to insist no longer ; for , besides that the peculiar enodation of all the three apart is clearly set downe in my glosse on the last Mood , they are in both the first partitions of the Moods of Ahalebmane to the full expressed in the Table of my Trissotetras . The verticall Angles , according to the diversity of the three Cases being by the foresaid Datas thus obtained , must concurre with each its correspondent first Subtendent ( notified by the Characteristicks of τ. δ. θ. ) for finding out of the perpendicular , requisite for the performance of the second work in every one of the Cases of this Mood . And to this effect Ubamen is made the Subservient , by whose Resolver Nag — Mu — ☞ Torp ☞ Myr , these three Problems , Vtatatca , Vdaudca , and Uthauthca , are made manifest , and the same quaesitum attained unto by the Datas of three severall Subtendents , and verticals . * The Perpendicular being thus found out , must , for the surtherance of the third operation , joyne with the second verticaline , the double verticall , and second co-verticall , according to the nature of the Case in question , ( the Datas being the same with those of the third work of the last Mood ) thereby to attaine unto the knowledge of the second little Subtendent , the second great Subtendent , and the second Co-subtendent , the which are all the maine quaesitas of this Mood : To the performance of this last operation , Etalum is the subservient , whose Resolver Torp — Me — Nag ☞ Mur , teacheth us how to deale with the under datoquaeres of Catheudwd , Cathatwt , and Catheuthwth . Now , the coalescencie of these last two operations in one , proceeding from the casting out of the Radius in both the orders of proportionals , and leaing roome for the perpendicular , without taking the paines to know its value , as hath beene shewne already in the first Mood of the same Figure ; it cannot be much amisse in this place to give a further illustration thereof , and make the Reader , by an Arithmeticall demonstration , feele ( as it were ) how palpable the truth is of compacting eight proportionals into foure ; let there be then these two orders of numbers , 4 — 6 — 8 ☞ 12. and 8 — 12. — 14 ☞ 21. Where , we may suppose eight to be the Radius , and twelve the Perpendicular ( for such like suppositions can inferre no great absurdity ) and then let us consider how those termes doe beare to one another , especially the 12. and 8. which , by possessing foure places , make up halfe the number of the proportionals . First , we see that twelve in the first row , is nothing else but the result of the product of 6. in 8. divided by 4. And secondly , that 8. in the second row , casteth downe , by its division , whatsoever by its multiplication it builded up in the first ; upon which observations we may ground these Sequels , that 12. may be safely left out , both in the fourth , and sixth place , taking instead of it the number of 4. 6. and 8. in whose potentia it is : and next 8. undoing in one place , what it doth in another , may with greater ease void them both . So that by this abbreviated way of Analogising , 4. and 6. alone in their due order before 14. which is the third terme of the second row , conduce as much to the obtaining of the fourth , or if you will eighth proportionall 21. as if the other foure termes of the two eights , and twelves , were concurrent with it . How plaine all this is , no question needs to be made , and therefore , to returne to our Resolvers ( for the explicating whereof , we thought good to make this digression ) we must understand that the finall Resolver , ( in its generall expression ) made out of them ( they being as they are materially displayed , Nag-Mu-Torp ☞ Myr , & Torp-Me-Nag ☞ Mur ) is no other then Nag-Mu-Na ☞ Mur , that is , As the Sine complement of one verticall is to the Tangent complement of a Subtendent : So the Sine complement of another verticall , to the Tangent complement of another Subtendent : and Analytically to trace the running of this operation , even to the source from whence it flowes , by foysting in the Perpendicular , and Radius , we may bring it to the consistence of the former two subordinate Resolvers , whereof the first is , As the Sine complement of a first , or a double verticall , to the Tangent complement of a first Subtendent : so the Radius to the Tangent complement of the Perpendicular ; and the second , As the Radius , to the Tangent complement of the Perpendicular : so the Sine complement of a second , or a double verticall , to the Tangent complement of a second Subtendent , which is the side required , and the fourth proportionall of Nag — Mu — Na ☞ Mur. Whose generality is to be contracted to every one of the three Cases of this Mood thus : If both the Angles given be of the same nature , they being the first verticals , from which the Cathetus fals on either side , increased according to the demand of the side , as in the first Case , Dasimforauxy , we must particularise the common Resolver , in this manner , Nat — Mut — Neud ☞ Nwd * Yr , that is , As the Antisine of the double verticall , is to the Antitangent of the first , and great Subtendent : so the Antisine of the second verticall in the lesser rectangle , to the Antitangent of the second Subtendent in the same little rectangle , which Subtendent is the side required . For the second Case of this Mood , viz. Dadiscracforeug , we must say , Naud — Mud — Nat ☞ Mwt * Yr , that is , As the Sine complement of the first and little verticall to the Tangent complement of the first , and little Subtendent : so the Sine complement of the double verticall , to the Tangent complement of the second and great subtendent . And lastly , for the third Case Dadiscramgatin , the finall Resolver is determinated thus , Nauth — Muth — Neuth ☞ Mwth * Yr , that is , As the Co-sine of the first Co-verticall , is to the Co-tangent of the first Co-subtendent : so the Co-sine of the second Co-verticall , to the Co-tangent of the second Co-subtendent , which is the side in this third Case required . The truth of all these operations is grounded on the Axiome of Naverprortes , as we are certified by the first syllable of its Directory Allera , which we may perceive by the direct Analogy that is betweene the Sines complements of the verticall Angles , and the Tangents complements , ( and consequently reciprocall 'twixt them and the Tangents ) of the verticall sides , which in this Mood are alwayes second Subtendents . The second Disergetick , and Angulary Figure , is Ahamepnare , which embraceth all those Obliquangularie Sphericals , wherein two Angles being given with an opposite side , another Angle , or the side interjacent , is demanded : this Figure , conforme to the two severall Quaesitas , hath two Moods , viz. Amanepra , and Ammanepreb . The first Mood hereof , which is Amanepra , belongeth to all those Loxogonosphericall questions , wherein , two Angles with an opposite side being proposed , the third Angle is required , which is alwayes a first verticall , a second verticall , or a first co-verticall : to the notice of all which , that we may with ease attaine , the generall Maxim of the Cathetothesis of this Mood is to be considered , which is Cafriq that is to say , that in all the Cases of Amanepra , the Perpendicular falleth from the Angle required upon the side opposite to that Angle , and terminated by the other two Angles , which side is to be increased , if need be . Now in regard , that besides the Cathetothesis of this Mood , and some three moe , to wit , all those Loxogonosphericals wherein the quaesitum is either a partiall verticall , or segment at the Base , there is a peculiar Mensurator , pertaining to every one of the foure , called in my Trissotetras the plus minus , because it sheweth by the specieses thereof to the Moods appropriated , whether the summe , or difference of the verticall Angles , and segments at the Base , be the Angle , or side required , and so clearly leadeth us thorough all the Cases of each of the Moods , that either by abstracting the fourth proportionall from an Angle or a segment , or by abstracting an Angle , or a segment from it , or lastly , by joyning it to an Angle , or a segment , with an incredible facility we attaine to the knowledge of the maine quaesitum , whether Angulary , or laterall . Let the Reader then be pleased to know , that the Mensurator , or Plus minus of this Mood , is Sindifora , which evidently declareth ( as by its representatives in the explanation of the Table is apparent ) that , if the demission of the Perpendicular be internall , the summe ; if exterior , the difference of the verticall Angles , is the Angle required . Seeing thus the notice of the manner of the Perpendiculars falling is so necessary , it is expedient , for our better information therein , that we severally perpend the three especiall Tenets of the generall Maxim of the Cathetothesis of this Mood , which are Dadissepamforaur Dadissexamforeur , and Dasimatin . Dadissepamforaur , which is the Tenet of the first Case , sheweth , that when the Angles given are of a different nature , and that the proposed side is opposite to an obtuse Angle , the Perpendicular falleth outwardly , and the first verticall is the Angle required . The second Tenet belonging to the second Case of this Mood , viz. Dadissexamforeur , sheweth , that if the proposed Angles be of discrepant affections , and that the side given be conterminat with an obtuse Angle , the Perpendicular is demitted externally , and the demanded Angle is a second verticall . The third Tenet pertaining to the last Case of this Mood , to wit , Dasimatin , evidenceth , that if the Angles proposed be of the same quality , the Perpendicular falleth interiourly , and the double verticall is the Angle required . Having thus ( as I suppose ) hereby evinced every difficulty of the Perpendiculars demission in all the Cases of this Mood , I may the more boldly in the interim proceed to the three rectangular works thereto belonging . Now , it being manifest that the Praenoscendas of this Mood , or the Quaesitas of the first operation thereof , are the same with those of the two Moods of the first Disergetick Figure , to wit , the double verticall , the first verticaline , and the first co-verticall ; and that , without any alteration at all , they are to be obtained by the same Datas , both of side , and Angle in this Mood of Amanepra , that , they were in the former Moods of Alamebna , and Allamebne , without any further specifying what these given sides , and Angles are ( which are to the full expressed in the last two forementioned Moods ) I must make bold thither to direct you , where you shall be sure also to learne all that is necessary to know of the Subservient and Resolver of the first operation of this Mood , both which , to wit , Upalam and To — Tag — Nu ☞ Mir , are inseparable dependents on all the Angularie Praenoscendas of the Loxogonosphericall Disergeticks : And though within the generality of this Subservient be compreded the peculiar Problemets of Vtopat , Udobaud , and Uthophauth , which are all three at large couched in the Trissotetras of this Mood ; yet , because what hath beene already said thereof in the foresaid Figure , may very well suffice for this place , the Readers diligence ( I hope ) in the turning of a leaf , will save me the labour of any further recapitulation . The Praenoscendas , or the verticall Angles , according to the nature of the Case , being by the foresaid Datas thus found out , must needs joyne with each its correspondent opposite , specified by the characteristicks of π. β. φ. for the obtaining of the Perpendicular , which in all the rest of the Disergetick Moods , as well as this , is alwayes the quaesitum of the second operation , thorough all the Cases thereof . Of this work Amaner is the subservient , by whose Resolver , Say — Nag — To ☞ Nyr , the three sub-problems , Opatca , Obaudca , and Ophauthca , are made known , and the same quaesitum attained unto by the Datas of three several both cathetopposites , and verticals , it being the only Mood which with Alamebna , hath a cathetopposite and verticall catheteuretick identity . The Perpendicular being thus obtained , is , for the effecting of the third and last operation , to concurre with the next cathetopposite , the prime cathetopposite , and the second cocathetopposite , as the Case requires it , thereby to find out the main quaesitum ; which in the first Case by abstracting the fourth proportionall , in the second by abstracting from the fourth proportionall , and in the third by adding the fourth proportionall to another verticall , is easily obtained by those that have the skill to discerne which be the greater , or lesser of two verticals proposed . To the perfecting of this third work , Exoman is the Subservient , whose Resolver Ne-To-Nag ☞ Sir , instructeth us , how to unfold the peculiar Problems of Cathobeud Cathopat , & Cathopseuth . Now , the nature of proportion requiring that of two rowes of proportionals , when the fourth in the first order is first in the second , that then the multiplyers become dividers , and the dividers multiplyers : as by these numbers following you may perceive , viz. 2 — 4 — 6 ☞ 12. for the first row , and 12 — 4 — 15 ☞ 5 , for the second ; of which proportionals , because of the fourth terme in the first rowes being first in the second , if you turne as many multiplyers into dividers as you can , and ( where the identity of a Figure requires it ) dash out a multiplyer against a divider , you will find , the two foures by this reason being raced out , and the two twelves ( because of their being in the power of the three first proportionals of the first row ) likewise left out , that this Analogy of 6 — 2 — 15 doth the same effect , that the former seven proportionals , for obtaining of the quaesitum , viz. 5. the reason whereof is altogether grounded upon the inversion of a permutat proportion , or the Retrograd Analogy of the alternat termes , whereby the Consequents are compared to Consequents , and Antecedents to Antecedents , in the preposterous method of beginning at the second of both the Consequents and Antecedents , and ending at the first : therefore ( as I was telling you ) the nature of proportion requiring that in such a Case the multiplyers and dividers be bound to interchange their places , the Resolvers of the last two operations , viz. Say — Nag — To ☞ Nyr , and Ne — To — Nag ☞ Sir , the first whereof being , As the Sine of a verticall Angle , to the Sine complement of an Angle at the Base , or one of the Cathetopposites : so the Radius to the Sine complement of the Perpendicular : and the second , As the Sine complement of the Perpendicular , to the Radius : so the Sine complement of one of the Cathetopposite Angles , to one of the verticals , may both of them ( according to the former rule ) be handsomely compacted in this one Analogy , Na — Say — Nag ☞ Sir , that is , As the Sine complement of an opposite is to the Sine of a verticall : so the Sine complement of another opposite , to the Sine of another verticall . This foresaid generall Resolver , according to the three severall cases of this Mood , is to be specialised into so many finall Resolvers ; the first whereof for Dadissepamforaur , Nop — Sat — Nob — ☞ Seudfr* At* Aut* ir , that is , As the sine complement of the prime cathetopposite , to the sine of the double verticall : so the sine complement of the nearest cathetopposite , to the sine of the second verticalin ; the which subtracted from the double verticall , leaveth the first and great verticall , which is the Angle required . Next , for the second Case of this Mood , Dadissexamforeur , we must make use of , Nob — Saud — Nop ☞ Satfr , * Aud* Eut* ir , that is , As the sine complement of the next opposite , to the sine of the first verticallet : so the sine complement of the prime opposite , to the sine of the double verticall , from which , if you deduce the first verticalm , there will remaine the second and great verticall for the Angle demanded . Lastly , for the third Case , Dasimatin , we must , say Noph — Sauth — Nop● ☞ Seuth* jauth* ir , that is , As the sine complement of the first co-opposite , to the sine of the first co-verticall : so the sine complement of the second co-opposite , to the sine of the second co verticall , which added to the first co-verticall , maketh up the Angle we desire . The veritie of all these operations is grounded on the Axiome Nabadprosver , as the second syllable of its directorie Alama , giveth us understand , and as we may discerne more easily by the samenesse in species amongst the proportionall termes ; for they are all Angles , the first , and third being Angles at the Base ( for these are alwaies of the opposits ) and the second , and fourth termes of the verticall Angles , which verticall Angles in the finall resolvers of this Mood , are according to the foresaid Axiome , to the Angles of the Base directly proportionall , and contrarily . The second Mood of the second Angularie figure of the Loxogonosphericall Disergeticks , named Ahamepnare is Ammanepreb , which is said of all those obliquangularie problems , wherein two Angles , and an opposite side being given , the side between is required , and is alwaies one of the basal-segments : to the knowledge whereof , that we may the more easily attaine , we must consider the generall maxime of the Cathetothesis of this Mood , which is Cafregpagyq that is , that the perpendicular falleth still from the given side , opposite to both the Angles given , and upon the side required , continued , if need be , in all and every one of the cases of Ammanepreb . The Plusminus of this Mood , is Sindiforiu , that is to say , the summe of the segments of the Base , if the perpendicular fall inwardly , and the difference of the Bases , if exteriorly , is the side demanded . The perpendiculars demission , being a Sine quo non in all disergetick operations , it will not be amisse , that we ponder what the three severall tenets are of the Cathetothesis of this Mood , and what is meaned by Dadissepamfor , Dadissexamfor , and Dasimin . Dadissepamfor , the tenet of the first Case of this Mood , sheweth , that if the given Angles be of severall natures , and that the proposed side be opposite to an obtuse Angle , the perpendicular falleth externally . The second tenet , Dadissexamfor expresseth , that if the proposed Angles be different , and that the side given be conterminat with the obtuse Angle , it falleth likewise outwardly . But Dasimin , which is the third tenet signifieth , that if the given Angles be of the same affection , the falling of the perpendicular is internall . This much being premised of the perpendicular , we may securely goe on to the orthogonosphericall works of the Mood ; and so beginning with the first operation , consider what the praenoscendas are , which are alwaies the quaesitas by the first operation obtainable , and in this Mood the Bases of the Triangle ; but more particularly to descend to the illustration of the Cases of Ammanepreb , the praenoscendum of the first Case , is the first and great Base , of the second , the first but little Base , and of the third , the first co-base . Now , though these three praenoscendas , be totally different from those of the three former Moods , yet are they to be acquired by the same , and no other Datas ; because none of the Angularie figures must differ from one another in the Datas of their praenoscendas , as out of the definition of an Angularie figure in the entrie , of the second Mood set downe , is easie to be collected : these Datas being tendred to us of intermixed circularie parts , that is to say , of both sides and Angles , the side being the first subtendentall , or great subtendent , the first subtendentine , or little subtendent , and the first co-subtendent : and the Angles the prime cathetopposite , the next cathetopposite , and the first co-catheopposite ; so that considering what is demanded , and that the Datoquaere thereof must be expressed thus , the hypotenusa , and an oblique being given , to finde the Ambient conterminate with the proposed Angle , we are , for the calculation of this work , necessitated to have recourse to Vbamen , which , in the Table of my Trissotetras obtaineth the roome of its subservient , to the end , that by its Resolver Torp — Mu — Lag ☞ Myr , being instructed how by cutting off the Logarithm of the Radius , from the summe of the Logarithms of the M. of one of the first subtendents , and secant complement of one of the cathetopposits , or Angles at the Base , residuats the Logarithm of the Tangent complement of the Base required , we may deliveredly extract , out of the generality of that proposition , the peculiar Subordinate resolutions of these three Problemets of Ubamen , viz. Utopaet , Vdobaed , and Uthophaeth , varied ( as you see ) according to the diversity of the Praenoscendas , which being ( as you were told already ) the first Basal , or great Base , the first Baset or little Base , and the first Co-base ; I will not detain you any longer upon this matter , but the rather hasten my transition to the other work , that in the Praenoscendall partition of Ammanepreb , there is enough thereof set downe in the Table of my Trissotetras . The Praenoscendas of Ammanepreb , or the three severall first Bases , conforme to the various nature of the Cases thereof , being by the foresaid Datas happily obtained , must concurre with each its correspondent Cathetopposite ( discernable , in their severall qualities , by the Characteristicks of π. β. φ. ) for finding out of the perpendicular , which is the perpetuall quaesitum of the second operation . The subservient of this work is Ethaner , by whose Resolver , To — Tag — Se ☞ Tyr , we come to the knowledge of Ethaners three Subdatoquaeres , viz. Aetopca , Aedobca , and Aethophca , whereby we may perceive , that the same quaesitum , to wit , the perpendicular is obtained by the Datas of the three severall both Bases , and Cathet opposite Angles . This so often mentioned perpendicular being thus made known , must , for the performance of the last and third work , joyne with the nixt Cathetopposite , the prime Cathetopposite , and the second Co-cathetopposite , as the Case will beare it , the Datas being the same in every point here , that in the last operation of the foregoing Mood ( as by the subservients , Exoman and Epsoner , is obvious to any judicious Reader ) thereby to obtaine the maine quaesitum , which in the first Case , by abstracting the fourth proportionall from the first great Base , in the second by abstracting from the fourth proportionall , the first little Base , and in the third by adding the fourth proportionall to another segment of the Base , is findable by any , that will undergoe the labour of adding , and substracting . For the acomplishment of this last operation Epsoner is the Subservient , by whose Resolver Tag — Tolb — Te ☞ Syr , we are taught how to deale with its three Subproblems , Cathoboed , Cathopoet , and Cathopsoeth . These last two operations being thus to the full extended , it remaineth now to treat how they ought to be in one compacted , or rather , for brevitie of computation , we should compact them both in one , before we take the paines to extend them : yet , because practice requireth one method , & the order of Doctrine another , we will , that we may be the lesse troublesome to the Readers memory , goe on ( by ejecting some , and reserving other proportional termes ) in our usuall course of conflating two Resolvers together . These Resolvers are in this Mood , To — Tag — Se ☞ Tyr , and Tag — To — To ☞ Syr , the first thereof , sounding , As the Radius , to the Tangent of one of the Cathetopposite Angles , or Angles at the Base : so the Sine of one of the first Bases , to the Tangent of the perpendicular : and the second , As the Tangent of one of the other Cathetopposite Angles to the Radius : so the Tangent of the perpendicular , to the sine of the side required . Here may the Reader be pleased to consider , that in all the glosse upon the posterior operations of my Disergeticks , I have beene contented to set downe ( as he may see in the last two propositions ) the bare Theorems of the Resolvers , conforme to the nature of their Analogy , without troubling my selfe , or any body else , with repeating , or reiterating the way , how the Logarithms of the middle , and initiall termes are to be handled , for the obtaining of a fourth Logarithm ; all that can be desired therein , being to the full expressed already in my ample comments upon the Orthogonosphericall Problems ; to the which the industrious Reader , in case of doubting , may ( if he please ) have recourse , without any great losse of time , or labour : however , for his better encouragement , I give another hint thereof in the closure of this Treatise . But to returne where we left , seeing out of these two Resolvers , To — Tag — Se ☞ Tyr , and Tag — To — Te ☞ Syr , according to the rules of coalescency , mentioned in both the Moods of Ahalebmane , both the Perpendicular and Radius may be ejected without any danger of losing our aime of the maine quaesitum , it is evident , that the proportion of the Remanent termes , is , Ta — Tag — Se ☞ Syr , which comprehendeth both the last two Resolvers , and the three foresaid Problemets thereto belonging , and being interpreted , As the Tangent of one Cathetopposite Angle , to the Tangent of another Cathetopposite : so the sine of one of the first Bases , to the sine of a side , which ushers in the side required . This generall Resolver , according to the three severall Cases of this Mood , is to be particularised into so many finall Resolvers ; the first whereof , for Dadissepanefor , is Tob-Top-Saet ☞ Soedfr * , Aet* Dyr , that is , As the Tangent of the next opposite , to the Tangent of the prime opposite : so the Sine of the first great Base , to the Sine of the second little Base ; which subducted from the foresaid first great Base , will for the remainder afford us that segment of the Base , which is the side in the first Case required . Then for the second Case , Dadissexamfor , the finall Resolver is Top — Tob — Saed ☞ Soetfr * Aed* Dyr , that is , As the Tangent of the prime Cathetopposite to the Tangent of the next opposite : so the Sine of the first Baset , or little Base , to the Sine of the second and great Base ; from which if we abstract the foresaid first little Base , the difference or remainer will be that Segment of the Base , which is the side demanded . Lastly , for the Case Dasimin , the finall Resolver is Tops — Toph — Saeth ☞ Soethj* Aeth* Syr , that is , As the Tangent of the second co-opposite , to the Tangent of the first co-opposite : so the Sine of the first co-base , to the Sine of the second co-base ; the summe of which two co-bases is the totall Base or side in the third Case required . The reason of all this is proved by the third Disergetick Axiome , which is Siubprortab , as is pointed at by the first syllable of its Directory Ammena , and manifested to us in all the Analogies of this Mood , every one whereof runneth upon Tangents of Angles , and Sines of Segments , both to the Base belonging : nor can any doubt , that heares the resolution of the Cases of Ammanepreb , but that the habitude , which all the termes thereof have to one another , proceedeth meerly from the reciprocall proportion , which the Tangents of the opposite Angles have to the Basal-segments , and contrariwise . The third Loxogonosphericall Disergetick Figure , and first of the Laterals ( that is , the first two termes of whose Datas are sides , what ere the quaesitum be ) is Ehenabrole , which comprehendeth all those Problems , wherein two sides being given , and an Angle betweene , either a cathetopposite Angle , or the third side is demanded . This Figure , conforme to the two severall Quaesitas , hath two Moods , to wit , Enerablo , and Ennerable . The first Mood hereof , Enerablo , containeth all those obliquangularie questions , wherein two sides with the Angle comprehended within them , being proposed , another Angle is required , which Angle is alwayes one of the Cathetopposites or Angles at the Base , that is , either the complement to a Semicircle of the next Cathetopposite , the prime Cathetopposite , or the second Cocathetopposite : to the knowledge of all which , that we may with facility attaine , let us consider the generall Maxim of the Cathetothesis of this Mood , which is Cafregpigeq that is to say , that the Perpendicular in all the Cases of Enerablo falleth from that given side , which is opposite to the Angle required , upon the other given side , continued , if need be ; and according to the variety of the Angle at the Base which is the Angle sought for , there be these three especiall Tenets of the generall Maxim of this Mood , viz. Dacramfor , Damracfor , and Dasimquaein . Dacramfor , which is the Tenet of the first Case , sheweth , that if the proposed Angle be sharp , and the required flat , the Perpendicular must fall outwardly . Damracfor , the Tenet of the second Case , signifieth , that if a blunt , or obtuse Angle be given , and an acute or sharp demanded , the demission of the Perpendicular must ( as in the last ) be externall . Lastly , Dasimquaein , the Tenet of the third Case , sheweth , that if the given , and required Angles be of the same nature , the Perpendicular must fall inwardly . Having thus unfolded all the intricacies in my Trissotetras of the Cathetothetick partition of this Mood , I may , without breaking order , step back , to explicate what is contained in the preceding partition , and for the accomplishing of the first Orthogonosphericall work of this Mood , consider what its Praenoscendas are , and by what Datas they are to be obtained : but , seeing both the Praenoscendas , and the Datas , together with the subservient , and its Resolver , with all the three Subdatoquaeres ; and in a word , the whole contents of the first partition of this Mood of Enerablo , is the same in all and every jot with the Praenoscendas , Datas , Subservient , Resolver , and Problemets , contained in the first partition of the last Mood Ammanepreb ; I will not need to tell you any more , then that ( the Trissotetras it selfe ( though otherwise short enough ) shewing that Ubamen is the subservient to the Praenoscendas : Torp — Mu — Lag ☞ Myr , its Resolver : and Vtopaet , Vdobaed , and Vthophaeth , the three Subproblems both of this and the next preceding Mood ) you be pleased to have recourse to the glosse upon the last Mood , where this matter is treated of at large ; to the which , for avoyding of repetition , I doe heartily recommend you . The first work being thus expedited , we are to find out the Perpendicular by the second , but so as that my direction to the Reader for the performance thereof shall detaine me no longer here , then the time I am willing to bestow , in telling him , that the whole progresse of this operation , as well as of the preceding , is amply expressed in my comment on the last Mood , from which , what ere is written of the Subservient , Ethaner , its Resolver , To — Tag — Se ☞ Tyr , or the under-problems , Aetopca , Aedobca , and Aethophca , thereby resolved , may conveniently be transplaced hither , and reseated there againe , without any prejudice to either ; Ammanepreb being the onely Mood , which with this of Enerablo hath a basal and opposite catheteuretick identity . The Perpendicular , by these meanes being found out , must be employed in the last work of this Mood , to concurre with the second Basidion , or little Base , the second great Base , and the second Co-base , for obtaining of such Cathetopposites as are , or usher the maine quaesitas , which in the first Case is the complement of the fourth proportionall ( viz. the next Cathetopposite ) to a Semicircle ; in the second Case the prime Cathetopposite , and in the third , the second Cocathetopposite . For the perfecting of this operation , Erelam is the Subservient , by whose Resolver , Sei — Teg — To ☞ Tir , we are instructed how to unfold its peculiar Problemets , oedcathob , oetcathop , and oethcathops . All the three operations being thus singly accomplished , according to our wonted manner , the last two must be inchaced into one , and therefore their Resolvers , To — Tag — Se ☞ Tyr , and Sei — Teg — To ☞ Tir , must be untermed of some of their proportionals : the which , that we may performe the more judicionsly , let us consider what they signifie apart ; the first importeth ( as in the last Mood I told you ) that , As the Radius is to the Tangent of one of the opposite Angles : so the Sine of one of the first Bases , to the Tangent of the Perpendicular : the second soundeth , As the Sine of one of the second Bases , to the Tangent of the Perpendicular : so the Radius , to the Tangent of an Angle , which either ushers , of is the Angle required . Hereby it is evident , how the Radius is a multiplyer in the one , and a divider in the other , and that the Perpendicular , which with the Radius is a multiplyer in the second row , is in the power of the three first termes of the first row , whereof the Radius is one , by vertue of all which , we must proceed just so with these last two operations here , as we have already done with the two last of the Moods of Alamebna , Allamebne , and Ammanepreb , and ejecting the Radius and Perpendicular out of both , instead of To — Tag — Se ☞ Tyr and Sei — Teg — To ☞ Tir , set downe Sei — Tag — Se ☞ Tir , that is , As the Sine of one of the second Bases to the Tangent of one of the Cathetopposites : so is the Sine of one of the first Bases , to the Tangent of one of the other Cathetopposites : which proposition comprehendeth to the full the last two operations , and according to the three severall Cases of this Mood is to be individuated into so many finall Resolvers . The first thereof , for Dacramfor , is Soed — Top — Saet ☞ Tob * Kir , that is , As the Sine of the second Basidion , or little Base , is to the Tangent of the prime Cathetopposite : so the Sine of the first , and great Base , to the Tangent of the next Cathetopposite , whose complement to a Semicircle is the Angle required . The second finall Resolver , is for Damracfor , the Tenet of the second Case , and is Soet — Tob — Saed ☞ Top * Ir , that is to say , As the Sine of the second , and great Base , to the Tangent of the next Cathetopposite : so the Sine of the first Basidion , to the Tangent of the prime opposite , which is the Angle required . The third and last finall Resolver , is for the third Case Dasimquaein , and is couched thus , Soeth — Toph — Saeth ☞ Tops * Ir , that is , As the Sine of the second Co-base is to the Tangent of the first Cocathetopposite : so is the Sine of the first Co-base to the Tangent of the second Co-cathetopposite , which is the Angle required . The fundamentall reason of all this , is from the third Disergetick Axiome Siubprortab , the second Determinater of whose Directory , Ammena , sheweth that the Mood of Enerablo , in all the finall Resolvers thereof , oweth the truth of its Analogy to the Maxim of Siubprortab ; because of the reciprocall proportion tha● is amongst its termes , to be found betwixt the Sines of the basall segments and the Tangents of the Cathetopposite Angles . The second Mood of Ehenabrole is Ennerable , which comprehendeth all those Obliquangulary Problems , wherein two sides being given , with an Angle intercepted therein , the third side ▪ demanded , which side is alwayes one of the second Subtendent● ▪ that is either the second Subtendentine , the second Subtendentall , 〈◊〉 the second Co-subtendent : to the notice of all which , that we may the more easily attaine , let us perpend the generall Maxim of the Cathetothesis of this Mood , Cafregpaq the meaning whereof is , that in this Mood , whatever the Case be , the Perpendicular may fall from the extremity of either of the given sides , but must fall from one of them , opposite to the Angle proposed , and upon the other given side , continued , if need be . Here may the Reader be pleased to observe , that the clause of the Perpendiculars falling opposite to the proposed Angle , though it be onely mentioned in this place , might have as well beene spoke of in any one of the rest of the Cathetothetick comments ; because it is a generall tie incumbent on the demission of Perpendiculars in all Loxogonosphericall Disergetick Figures , whether Amblygonian or Oxygonian , that it fall alwayes opposite to a knowne Angle , and from the extremity of a knowne side . Of this generall Maxim , Cafregpaq according to the variety of the second Subtendent , which is the side required , there be these three especiall Tenets , Dacforamb , Damforac , and Dakinatam . Dacforamb , the Tenet of the first Case , giveth us to understand , that if the given Angle be acute , and that one onely of the other two be an obtuse Angle , the Perpendicular falleth outwardly . Damforac , the Tenet of the second Case , signifieth , that if the given Angle be obtuse , and the other two acute , that the demission of the Perpendicular is externall , as in the first . Thirdly , Dakinatam , the Tenet of the third Case , and variator of the first , sheweth , that if the proposed Angle be of the same affection with one of the other Angles of the Triangle , as in the first Case , the Perpendicular may fall inwardly . The Cathetology of this Mood being thus expeded , the Pranoscendas thereof come next in hand to be discussed , which are the first Bases , whose subservient is Vbamen , and its Resolver , Torp — Mu — Lag ☞ Myr , upon which depend the three Subdatoquaeras of Vtopaet , Vdobaed , and Vthophaeth . Thus much I beleeve is expressed in the very Table of my Trissotetras ; and though a large explication might be with reason expected in this place , of what is but summarily mentioned there , yet because what concerneth this matter , hath beene already treated of in the last two Moods of Enerablo , and 〈◊〉 , the whole discourse whereof may be as conveniently perused , as if it were couched here , I will not dull the Reader with tedious rehearsals of one and the same thing , but , letting passe the progresse of this first work , with the manner of which ( by my former instructions , I suppose him sufficiently well acquainted ) will proceed to the Cathetouretick operation of this Mood , and perpend by what Datas the perpendicular is to be found out . To this effect , the Praenoscendas of Ennerable , to wit , the first Basal , the first Basidion , and the first co-Base , being by the last work already obtained , must concurre with each its correspondent first subtendent , viz. the first Subtendentall , the first Subtendentine , and the first co-Subtendent , discernable in their severall natures , by the figuratives of τ δ θ. for the perfecting of this second operation . The subservient of this work , is Uch●ner , by whose Resolver Neg — To — Nu ☞ Nyr , the three subproblems Utaeta , Vtadca , and Vthaethca , are made manifest : by vertue whereof it is perceivable , how the same quaesitum is attained unto by the Datas of three severall , both first Subtendents , and first Bases . The perpendicular being thus obtained , must assist some other terme in the third operation , for the finding out of the maine quaesitum ; which quaesitum , though it be different from the finall one of the last Mood , yet is the knowledge of them both attained unto , by meanes of the same Datas ; the perpendicular , and the three second Bases , being ingredients in both . It being certaine then , that the perpendicular must concurre in the last work of this Mood with the second Basidion , the second Basal , and second co-Base , for obtaining the second Subtendentine , the second Subtendentall , and second co-Subtendent ; Enerul , is made use of for their subservient , by whose Resolver , To — Neg — Ne ☞ Nur , we are raught how to deale with its subordinat Problems , Catheudwd , Cathatwt , and Catheuthwth . All the three works being thus specified apart , according to our accustomed Method , we will declare what way the last two are to be joyned into one ; for the better effectuating whereof , their Resolvers , Neg — To — Nu — ☞ Nyr : and To — Neg — Ne ☞ Nur , must be interpreted ; the first being , As the sine complement of a first Base to the Radius : so the sine complement of a first subtendent , to the sine complement of the perpendicular . And the second . As the Radius , to the sine complement of a second Base : so the sine complement of the perpendicular to the sine complement of a second subtendent , which is the side required . Now , seeing a multiplier must be dashed against a divider , being both quantified alike , and that all unnecessary pestring of a work with superfluous ingredients is to be avoided ; we are to deale with the Radius , and perpendicular in this place , as formerly we have done in the Moods of Alamebna , Allamebne , Ammanepreb , and Enerablo , where we did eject them forth of both the orders of proportionalls ; and when we have done the like here , instead of Neg — To — Nu ☞ Nyr , and To — Neg — Ne ☞ Nur , we may with the same efficacie say , Neg — Nu — Ne ☞ Nur , that is , As the sine complement of one side , is to the sine complement of a subtendent : so the sine complement of another side , to the sine complement of another subtendent ; or more determinatly , As the sine complement of a first Base , to the sine complement of a first subtendent : so the sine complement of a second Base , to the sine complement of a second subtendent . This theorem comprehendeth to the full both the last operations , and according to the number of the Cases of this Mood , is particularized into three finall Resolvers , the first whereof for the first Case , Dacforamb , is Naet — Nut — Noed ☞ Nwd*yr , that is , As the sine complement of the first Basal , or great Base to the sine complement of the first Subtendentall , or great subtendent : so the sine complement of the second Basidion , or little Base , to the sine complement of the second subtendentine , or little subtendent , which is the side required . The second finall Resolver , is for Damforac , the second Case , and is set downe thus , Naed — Nud — Noet ☞ Nwt*yr , that is , As the sine complement of the first Basidion , to the sine complement of the first subtendentine : so the sine complement of the second Basal , to the sine complement of the second subtendentall , which is the side in this Case required . The third , and last finall Resolver is for Dakinatamb , and is expressed thus , Naeth — Nuth — Noeth ☞ Nwth*yr , that is to say , As the sine complement of the first co-base , to the sine complement of the first co-subtendent : so the sine complement of the second co-base , to the sine complement of the second co-subtendent , which in the third Case is alwayes the side required . The reason of all this is proved out of the fourth , and last disergetick Axiom , Niubprodnesver , whose directer Ennerra , sheweth by its Determinater , the syllable Enn , that the Datoquaere of Ennerable , is bound for the veritie of its proportion , in all the finall Resolvers thereof , to the maxime of Niubprodnesver , because off the direct analogie that , amongst its termes , is to be seen betwixt the sines complements of the segments of the Base & the sines complements of the sides of the verticall Angles ; which in all this Treatise , both for plainesse , and brevity sake , I have thought fit to call by the names of first and second Subtendents . The fourth and last Loxogonosphericall Disergetick figure , and second of the Lateralls , is Eherolabme , which is of all those obliquangularie problems , wherin two sides being given , and an opposite Angle , the interjacent Angle , or one of the other sides is demanded ; and , conforme to its two severall quaesitas , hath two Moods , viz. Erelomab , and Errelome . The first Mood hereof Erelomab comprehendeth all those Loxogonosphericall Problems , wherein two sides with an opposite Angle being proposed , the Angle between is demanded , which Angle is still one of the verticals , that is , the first verticall , the second verticall , or the double vertical : to the notice of all which , that we may the more easily attain , we must consider the general Maxim of the Cathetothesis of this Mood , which is Cafriq the very same in name with the generall Cathetothetick Maxim of Amanepra , and thus far agreeing with it , that the Perpendicular in both must fal from the Angle required , and upon the side opposite to that Angle , increased if need be : but in this point different , that in Amanepra , the Perpendiculars demission is from the Angle required upon the opposite side , conterminat with the two proposed Angles , and in Erelomab , it falleth from the required Angle , upon the opposite side conterminat with the two proposed sides : and , according to the variety of the fourth proportionall , which , in the Analogies to this Mood belonging , ushers in the verticall required , there be those three especiall Tenets of the generall Maxim of this Mood , viz. Dacracforaur , Damraeforeur , and Dacrambatin . Dacracforaur , which is the Tenet of the first Case , sheweth , that if the given and demanded Angles be acute , and the third an obtuse Angle , the Perpendicular falleth outwardly upon the third side , and the required Angle is a first verticall . Dambracforeur , the Tenet of the second Case , importeth , that if the proposed Angle be obtuse , and an acute Angle required , the third Angle being acute , the Perpendicular must likewise in this Case fall outwardly upon the third side , and the Angle demanded be a second verticall . Dacrambatin , the Tenet of the third Case , signifieth , that if the proposed Angle be acute , and an obtuse Angle required , the Perpendicular falleth inwardly , and the demanded Angle is a double verticall . I had almost forgot to tell you , that Sindifora is the Plus-minus of this Mood , whereby we are given to understand , that the summe of the top Angles , if the Perpendicular fall within , and their difference , if it fall without , is the Angle required : and , seeing it varieth neither in name , nor interpretation from the mensurator of Amanepra ( the diversity betwixt them being onely in this , that the verticals there are invested with Sines , and here with Sine complements ) I must make bold to desire the Reader to look back to that place , if he know not why it is that some Moods are Plus-minused , and not others ; for there he will find that Sindiforation is meerly proper to those Cases , in the Analogies whereof the fourth proportionall is not the maine quaesitum it selfe , but the illaticious terme that brings it in . The Praenoscendas of the Mood , or Quaesitas of the first operation , falling next in order to be treated of , it is fitting we perpend of what nature they be in this Mood of Erelomab , that if they be different from those of other Moods , we may , according to our accustomed diligence , formerly observed in the like occasions , appropriate , in this parcell of the comment to their explication , for the Readers instruction , the greater share of discourse , the lesse that before in any part of this Tractar , they have beene mentioned : But if it be so farre otherwise , that for their coincidence with other proturgetick Quaesitas , there can no materiall document concerning them be delivered here , which hath not beene spoke of already in some one or other of our foregoing Datoquaeres , it were but an unnecessary wasting of both time and paper to make repetition of that , which in other places we have handled to the full ; and therefore , seeing the Praenoscendas of this Mood , to wit , the double top Angle or verticall , the first top Anglet or verticalin , and the first Co-top-Angle , or co-verticall , together with the Datas , whereby these are obtained , viz. for the side , the first subtendentall , the first subtendentine , and the first co-subtendent , and for the Angle , the prime Cathetopposite , the next Cathetopposite , and the first Co-cathetopposite , and consequently the subservient Upalam , its Resolver To — Tag — Nu ☞ Mir , and their three peculiar Problemets , Vtopat , Udobaud , and Uthophauth , are all and every one of them the same in this Mood of Erelomab , that they were in the three preceding Moods of Alamebna , Allamebne , and Amanepra ( for these are the foure Moods , which have an Angulary praenoscendall identity ) we will not need ( I hope ) to talk any more thereof in this place , seeing what hath beene already said concerning that purpose , will undoubtedly satisfie the desire of any industrious civill Reader . The praenoscendas of the Mood , or the verticall Angle , according to the nature of the Case , being by the foresaid Datas thus obtained , must needs concurre with each its correspondent first subtendent , determined by the figuratives of τ. δ. θ for finding out of the Perpendicular , of which work , Ubamen being the subservient , by whose Resolver Nag — Mu — Torp ☞ Myr , the sub-problems of Utatca , Vdaudca , and Vthauthca , are made known , if I utter any more of this purpose , I must intrench upon what I spoke before in the second operation of Allaemebne , it being the onely Mood which , with this of Erelomab , hath a verticall , and subtendentine Catheteuretick identity . The second operation being thus accomplished , the perpendicular , which is alwayes an ingredient in the third work , must joyne with one of the rere subtendents for obtaining of the illatitious terme of the maine quaesitum : or , more particularly , by the concurrence of the Perpendicular with the second subtendentine , the second subtendentall , and second Co-subtendent , according to the variety of the Case , we are to find out three verticals , which , by abstracting the first from another verticall , then by abstracting another verticall from the second , and lastly by adding the third verticall to another , afford the summe , and differences , which are the required verticals . All this being fully set downe in my comment upon the Resolutory partition of Amanepra , in which Mood the maine quaesitum is the same as here ( though otherwise endowed ) I need not any longer insist thereon . For the performance of this work , Ukelamb is the subservient , by whose Resolver Meg — To — Mu ☞ Nir , we are taught how to unfold the peculiar problemets of Wdcathaud , Wicatha● , and Wthcatheuth . All the three works being in this manner perfected , according to our accustomed method , we will shew unto you what way the last two are to be compacted in one : for the better expediting whereof , their Resolvers Nag — Mu — To ☞ Myr , and Meg — To — Mu ☞ Nir , must be explained , the first being , As the Sine complement of an Angle , to the Tangent complement of a subtendent : so the Radius , to the Tangent complement of the side required : Or , more particularly , As the Sine complement of a verticall , to the Tangent complement of a first subtendent : so the Radius , to the Tangent complement of the Perpendicular : And the second Resolver being , As the Tangent complement of a given side , to the Radius : so the Tangent complement of a subtendent , to the Sine complement of a required Angle : Or , more particularly , As the Tangent complement of the Perpendicular , to the Radius : so the Tangent complement of a first subtendent , to the Sine complement of a verticall , which ushers the quaesitum . Now , seeing it falleth forth , that the Perpendicular , which is the fourth terme in the first order of proportionals , becometh first in the second row ; and that in such an exigent ( as I proved already for illustration of the same point in the Mood of Amanepra ) the multiplyers and dividers of the first row must interchange their roomes , and consequently make the Radius ejectable , without any prejudice or hindrance to the progresse of the Analogy ; and a place being left for the Perpendicular in both the rowes , without taking the paines to find our its value , because it is but a subordinate quaesitum for obtaining of the maine , and lieth hid in the power of the three first proportionals , instead of Nag — Mu — To ☞ Myr , and Meg — To — Mu ☞ Nir , we may , with as much truth and energy , say , Mu — Nag — Mu ☞ Nir , that is , As the Tangent complement of a subtendent , to the Sine complement of an Angle : so the Tangent complement of another subtendent , to the Sine complement of another Angle : Or , more particularly , As the Tangent complement of a first subtendent , to the Sine complement of a verticall : so the Tangent complement of a second subtendent , to the Sine complement of a verticall illative to the quesitum . This proposition to the full containeth all that is in both the last operations , and , according to the number of the Cases of this Mood , is specialized into so many finall Resolvers ; the first whereof , for the first Case Dacracforaur , is Mutnat — Mwd ☞ Neud-fr*At*Aut*ir , that is , As the Tangent complement of the first subtendentall , to the sine complement of the double verticall : so the tangent complement of the second Subtendentine , to the sine complement of the second verticalin , which subtracted from the double verticall , leaves the first verticall for the Angle required . The second finall Resolver , is for Damracforeur , the second Case , and is expressed thus , Mud — Naud — Mwt ☞ Natfr*Aud*Eut*ir , that is , As the tangent complement of the first Subtendentine , to the sine complement of the first verticalin : so the tangent complement of the second Subtendentall , to the sine complement of the double verticall ; from which if you deduce the first verticalin , there will remaine the second verticall for the Angle required . The last finall Resolver is for the third Case , Dacrambatin , and is couched thus , Muth — Nauth — Mwth ☞ Neuth*jauth*ir , that is , As the tangent complement of the first co-subtendent , to the sine complement of the first co-verticall : so the Tangent complement of the second co-subtendent , to the sine complement of the second co-verticall , which , joyned to the first co-verticall , affordeth the Angle required . The proofe of the veritie of all these Analogies , is taken out of the second Disergetick Amblygonosphericall Axiome , Naverprortes , the second Determinater of whose Directorie sheweth , that this Mood is one of its dependents ; and with reason , because of the reciprocall Analogie , that amongst its termes is perceivable betwixt the Tangents of the verticall sides , which in this Mood are alwayes first subtendents , and the sine-complements of the verticall Angles ; that is tosay ( according to the literal meaning of my finall Resolvers of this Mood ) the direct proportion that is betwixt the tangent-complements of the verticall sides , or rere subtendents , & the sine-complements of the vertical Angles , for the proportion is the same with that , wherof I have told you somewhat already in the Mood of Allamebme , the fellow dependent of Erelomab . The second Mood of Eherolabme , fourth of the Laterals , eighth of the Sphericobliquangularie Disergeticks , twelfth of the Loxogonosphericalls , eight and twentieth of the Sphericals , and one and fourtieth or last of the Triangulars , is Errelome , which comprehendeth all those obliquangularie Problems , wherein two sides being given with an opposite Angle , the third side is required , which side is alwayes either one of the segments of the Base , or the Base it selfe : to the knowledge of all which , that we may reach with ease , we must perpend the generall Maxim of the Cathetothesis of this Mood , which is Cacurgyq that is to say , the Perpendiculars demission , in all the Cases of Errelome , must be from the concurse of the given sides , upon the side required , continued , if need be . The Plus-minus of this Mood is Sindiforiu , which importeth , that if the Perpendicular fall internally , the summe of the segments of the Base , or the totall Base , is the side demanded : and if it fall without , the difference of the Bases ( the little Base , being alwayes but a segment of the greater ) is the maine quaesitum . The Mood of Ammanepreb is sindiforated in the same manner as this is ; because the maine Quaesitas , and fourth proportionals of both doe in nothing differ , but that those are sinused , and these run upon sine-complements . The prosiliencie of the Perpendicular in all sphericall Disergeticks , being so necessary to be knowne ( as I have often told you ) because of the facility thereby to reduce them to Rectangulary operations , it falleth out most conveniently here , according to the method proposed to my selfe , to speak somewhat of the three severall Tenets of the Cathetothesis of this Mood , and what is understood by Dakyxamfor , Dambyxamfor , and Dakypambin . Dakyxamfor , which is the Tenet of the first Case , declareth , that if the proposed Angle be acute , and the side required conterminate with an obtuse Angle , the demission of the Perpendicular is extrinsecall . Dambyxamfor , the Tenet of the second Case , importeth , that if the given Angle be obtuse , and that the side required be annexed thereto , the Perpendicular must , as in the last , fall outwardly . Thirdly , Dakypambin , the Tenet of the last Case , signifieth , that if the angle proposed be sharp , & that the demanded side be subjacent to an obtuse or blunt Angle , the Perpendicular falleth inwardly . Having thus proceeded in the enumeration of the Cathetothetick Tenets of this Mood , according to the manner by me observed in those of all the former Disergeticks , save the first , I am confident the Reader ( if he hath perused all the Tractat untill this place ) will not think strange why , Dakypambin being but the third , I should call it the Tenet of the last Case of this Mood ; for though in Alamebna I spoke somewhat of every Amblygonosphericall Disergetick Moods generall Cathetothetick maximes division into foure especiall Tenets , appropriable to so many severall Cases : yet the fourth Case , viz. that wherein all the Angles are homogeneall , whether blunt or sharp , not being limited to any one Mood , but adaequatly extended to all the eight , it seemed to me more expedient to let its generality be known by mentioning it once or twice , then ( by doing no more in effect ) to make superfluous repetitions ; and , as in the first Disergetick Case , for the Readers instruction , I did under the name of Simomatin , explicate the nature thereof : so , for his better remembrance , have I choosed rather to shut up my Cathetothetick comment with the same discourse wherewith I did begin it , then unnecessarily to weary him with frequent reiterations , and a tedious rehearsall of one and the same thing in all the six severall intermediat Moods . It is not amisse now , that the perpendicularity of this Mood is discussed , to consider what the praenoscendas thereof are , or the Quaesitas of the first operation : but , as I said in the last Mood , that there is no need to insist so long upon the explication of those praenoscendas , whereof ample relation hath beene already made in some of my Proturgetick comments , as upon those others , which , for being altogether different from such as have beene formerly mentioned , claim ( by the law of parity , in their imparity ) right to a large discourse apart , I will confine my pen upon this subject , within those prescribed bounds , and seeing the first Basal , the first Basidion , and first Co-base , together with the Datas , whereby they are found out , viz. for the side , the first subtendentall , the first subtendentine , and the first co subtendent ; and for the Angle the prime Cathetopposite , the next Cathetopposite , and the first Co-cathetopposite ( the Datas being both for side and Angle the same here , that they were in the former Mood ) then the Subservient Ubamen , and its Resolver Torp — Mu — Lag ☞ Myr , with the three peculiar Problemets thereto belonging , Utopat , Vdobaed , and Vthophaeth , are all and every one of them the same in this Mood of Errelome , that they were in the three foregoing Moods of Ammanepreb , Enerablo , and Ennerable , these being the onely foure Moods which have a laterall praenoscendal identity , the Reader will not ( in my opinion ) be so prodigall of his owne labour , nor covetous of mine , that either he would put himselfe , or me to any further paines , then have beene already bestowed upon this matter by my selfe for his instruction ; and therefore , leaving it for a supposed certainty , that the Praenoscendas , or first Bases ( according to the nature of the Case ) cannot escape the Readers knowledge , by what hath beene by me delivered of them ; I purpose here to give him notice , that these foresaid first Bases must concurre with each its correspondent first Subtendent , to wit , the first subtendentall , the first subtendentine , and first co-subtendent , dignoscible by the Characteristicks of τ. δ. θ for obtaining of the Perpendicular , of which operation , Vchener being the Subservient , by whose Resolver Neg — To — Nu ☞ Nyr , the Problemets of Utaetca , Udaedca , and Uthaethca , are made manifest , as to the same effect it remaines couched in my comment upon Ennerable , which is the onely Mood , that , with this of Errelome , hath a subtendentine and Basal Catheteuretick identity . The second work being thus perfected , the perpendicular , thereby found out , is to assist one of the rere subtendents , in obtaining the illatitious terme of the maine quasitum , correspondent thereto , discernable by the Characteristicks or Figuratives of δ. τ. θ or , more plainly to expresse it , the Perpendicular must concurre ( according as the Case requires it ) with the second subtendentine , the second subtendentall , and second co-subtendent , ( as you may see in the last Mood , the Datas of the Resolutory partition whereof are the same as here ) to find out three Bases , which , by abstracting the first from another Base , then by abstracting another Base from the second ; and lastly , by adding the third Base to another , afford the summe and differences , which are the required Bases . For the performance of this operation , the same Subservient and Resolver suffice , which served for the last : so that Uchener subserveth it , by whose Resolver Neg — To — Nu ☞ Nyr , we are instructed how to explicate the Subdatoquaeres of Wdcathoed , Wtcathoet , and Wthcathoeth , or more orderly Cathwdoed , Cathwtoet , and Cathwthoeth . All the three works being thus accomplished , the manner of conflating the last two in one rests to be treated of ; for the better perfecting of which designe , the two Resolvers , or the same in its greatest generality doubled , viz. Neg — To — Nu ☞ Nyr , and Neg — To — Nu ☞ Nyr , must be interpreted : The truth is , both of them , as they sound in their vastest extent of signification , expresse the same Analogy , without any difference , which is , As the Sine complement of a given side , to the Radius : so the Sine complement of a subtendent , to the Sine complement of another side : but when more contractedly , according to the specification of the side , they doe suppone severally , they should be thus expounded ; the first , As the Sine complement of a first Base , to the totall Sine : so the Sine complement of a first subtendent to the Sine complement of the Perpendicular : and the second , As the Sine complement of the Perpendicular , to the totall Sine : so the Sine complement of a second subtendent , to the Sine complement of a second Base , which ushers the main quaesitum . Now , the Perpendicular , and Radius , being both to be expelled these two foresaid orders of proportionall termes , for the reasons which , in the last preceding Mood , and some others before it , I have already mentioned , and which to repeat ( further then that the sympathy of this place with that may be manifested in the tranf-seating of multiplyers and dividers , occasioned by the fourth terme in the first rowes , being first in the second ) is altogether unnecessary : in lieu of Neg — To — Nu ☞ Nyr , and Neg — To — Nu ☞ Nyr , we may say , with as much truth , power , and efficacie , and farre more compendiously , Nu — Ne — Nu ☞ Nyr , that is , As the Sine complement of a subtendent , to the Sine complement of a side : so the sine complement of another Subtendent , to the Sine complement of another side : Or , more particularly , and appliably to the present Analogy , As the Sine complement of a first subtendent , to the Sine complement of a first Base : so the Sine complement of a second subtendent , to the Sine complement of a second Base , illative to the quaesitum . This theorem , or proposition , comprehendeth in every point all that is in the two last operations , and , not transcending the number of the Cases of this Mood , is divided into so many finall Resolvers ; the first whereof for the first Case , Dakyxamfor is , Nut — Naet — Nwd ☞ Noedfr*Aet* Dyr , that is , As the Sine complement of the first subtendent all , to the Sine complement of the first Basall ▪ so the Sine complement of the second subtendentine , to the Sine complement of the second Base ; which subducted from the first Basal , residuats the segment that is the side required . The second finall Resolver of this Mood , and that which is for the second Case thereof , Dambyxamfor , is Nud — Naed — Nwt ☞ Noetfr*Aed* Dyr , that is , As the Sine complement of the first subtendentine , to the Sine complement of the first Basidion : so the Sine complement of the second subtendentall , to the Sine complement of the second Basall ; which , the first Basidion being subtracted from it , leaves , for Remainder , or difference , that segment of the Base , which is the side demanded . The last finall Resolver of this Mood ( belonging to the third Case , Dakypambin , as also to the fourth , Simomatin , ( if what we have already spoke of that matter will permit us to call it the fourth ) for Simomatin , together with the third Case of every Mood , is still resolved by the last finall Resolver thereof ) is Nuth — Naeth — Nwth ☞ Noethj*Aeth* Syr , that is , As the Sine complement of the first co-subtendent , to the Sine complement of the first Co-base : so the Sine complement , of the second Co-subtendent , or alterne subtendent , to the Sine complement of the second Co-base or alterne Base ; which added to the first Co-base , summes an Aggregat of subjacent sides , which is the totall Base , or side required . The fundamentall ground of the truth of these Analogies , is in the fourth and last Amblygonosphericall Axiome , Niubprodnesver ; ( as we are made to understand by the second determinater of its Directory Ennerra ) for by the direct proportion that , amongst the terms thereof , is visible , ( viz. betwixt the Sines complements of the subtendents , or Sides of the verticall Angles , and the segments of the Bases , and inversedly ) it is apparent , that this Mood doth no lesse firmely depend upon it , then that of Ennerable formerly explained . Now , with reason doe I conjecture , that , without disappointing the Reader of his expectation , I may here securely make an end of this Trigonometricall Treatise ; because of that Trissotetrall Table , which comprehendeth all the Mysteries , Axiomes , Principles , Analogies , and Precepts of the Science of Triangular Calculations , I have omitted no materiall point unexplained : yet seeing , for avoyding of prolixity , I was pleased in my comment upon the eighth Loxogonospherical Disergeticks , barely to expresse in their finall Resolvers , the Analogie of the termes , without putting my selfe to the paines I took in my Sphericorectangulars , how to order the Logarithms , , and Antilogarithms of the proportionalls , , for obtaining of the maine Quaesitas , and that by having to the full explicated the variety of the proportions of the foresaid Moods , and upon what severall Axiomes they doe depend , thereby making the way more pervious , thorough Logarithmicall difficulties , for the Readers understanding , I deliberatly proposed to my selfe this method at first , and chose , rather then dispersedly to treat of those things in the glosse ( where , by reason of the disturbed order , the correspondencie or reference to one another of these Sphericobliquangulary Datoquaeres , could not by any meanes have beene so conceivable ) to summon their appearance to the Catastrophe of this Tractat , that , having them all in a front before us , we may the more easily judge of the semblance , or dissimilitude of their proportionalities , , and what affinity , or relation , whether of parity or imparity is amongst their respective proportionall terms : all which , both for intelligibility and memory , are quicklier apprehended , and longer retained , by being accumulatively reserved to this place , then if they had beene each in its proper cell ( though never so amply ) discoursed upon apart . Here therfore , that the Reader may take a generall view at once of all the Disergetick Amblygonosphericall analogised ingredients , ready for Logarithmication , I have thought fit to set downe a List of all the eight forenamed Moods , together with the Finall Resolvers , in their amplest extent thereto belonging , in the manner as followeth . Alamebna . Say-Nag-Sa ☞ Nir Allamebne . Nag-Mu-Na ☞ Mur Amanepra . Na-Say-Nag ☞ Sir Ammanepreb . Ta-Tag-Se ☞ Syr. Enerablo . Sei-Tag-Seg ☞ Tir Ennerable . Neg-Nu-Ne ☞ Nur Erelomab . Mu-Nag-Mu ☞ Nir Errelome . Nu-Ne-Nu ☞ Nyr . These being the eight Disergeticks , attended by their Adaequat finall Resolvers , it is not amisse , that we examine them all one after another , and shew the Reader how , with the help of a convenient Logarithmicall Canon , he may easily out of the Analogie of the three first termes of each of them , frame a computation apt for the finding out of a fourth proportionall , to every severall ternarie correspondent : and so in order , beginning at the first , we will deale with Say — Nag — Sa ☞ Nir ( which is the Adaequat finall Resolver of Alamebna , and composed ( as it is appropriated to the first Mood of the Disergeticks ) of the Sines of verticals , and the Anti-sines of Cathetopposites ) and so proceed therein , that by adding to the summe of the Sine of a verticall , and Co-sine of a Cathetopposite , the Arithmetical complement of the Sine of another verticall , we will be sure ( cutting off the supernumerary digit or digits towards the left ) to obtaine the Co-sine of the Cathetopposite required , which Cathetopposites and verticals are particularised according to the Cases of the Mood . The second is , Nag — Mu — Na ☞ Mur , which , running upon the Anti-sines of verticals , and the Co-tangents of subtendent sides , sheweth , that if to the Aggregat of a first hypotenusall Co-tangent , and verticall Anti-sine , we joyne the Arithmeticall complement of the Anti-sine of another verticall , ( observing the usuall presection ) we cannot misse of the Co-tangent of the second subtendent side required , which both second , and first subtendents have their peculiar denominations , according to the Cases of the Mood . The third Resolver is , Na — Say — Nag ☞ Sir , which , being nothing else but the first inverted , runneth the same very way upon the Anti-sines of Cathetopposites , and sines of verticals : and therefore doth the unradiused summe of the Anti-sine of a Cathetopposite , the sine of a verticall , and the Arithmeticall complement of the Anti-sine of another Cathetopposite , afford the sine of the verticall , illatitious to the Angle required ; which verticals and Cathetopposites are particularised according to the variety of the Cases of this Sindiforating Mood . The fourth generall Resolver is Ta — Tag — Se ☞ Syr , which , coursing on the Tangents of all the Cathetopposites , and sines of all the Bases , evidenceth , that the summe of the Tangent of a Cathetopposite , and sine of a first Base , added to the Arithmeticall complement of the Tangent of another Cathetopposite ( unradiated ) is the sine of the second Base , illative to the segment required ; which Bases ( both first and second ) and Cathetopposites , are specialised conform to the Cases of this Sindiforiuting Mood . The fifth Resolver is , Sei — Tag — Se ☞ Tir , which , composed of the sines of the second and first Bases , and the Tangents of Cathetopposites , giveth us to know , that if to the summe of the sine of a first Base , and the Tangent of a verticall , we adde the Arithmeticall complement of the sine of a second Base , ( not omitting the usuall presection ) we cannot faile of the Tangent of the Cathetopposite required , which Cathetopposites , and Bases , both first and second , are particularised according to the Cases of the Mood . The sixth generall Resolver is , Neg — Nu — Ne ☞ Nur , which , running along the Co-sines of all the Bases and Subtendents , sheweth , that by the summe of the Co-sines of a second Base , and first subtendent joyned with the Arithmeticall complement of the Co-sine of a first Base ( if we observe the customary presection ) we find the second Subtendent required , which both first and second Subtendents , together with the first and second Bases , are all of them particularised conforme to the Cases of the Mood . The seventh Resolver is , Mu — Nag — Mu ☞ Nir , which , coursing along the Anti-tangents of first , and second Subtendents , and the Anti-sines of verticals , sheweth , that the summe of the Anti-tangent of a second Subtendent , and Anti-sine of a verticall , together with the Arithmeticall complement of the Antitangent of a first Subtendent ( the usuall presection being observed ) is the Anti-tangent of that verticall , which ushers in the verticall required ; all which , both Verticals , and Subtendents , both first , and second , have their peculiar denominations conforme to the Cases of this Sindiforating Mood . The eighth and last generall Resolver is , Nu — Ne — Nu ☞ Nyr , which ( running altogether upon Co-sines of Subtendents , and Bases , both first , and second of either , and is nothing else but the sixth inverted ) sheweth , that the summe of the Cosines of a second subtendent , and first Base , with the Arithmeticall complement of the co-sine of a first Subtendent , ( observing the usuall presection ) affords the Co-sine of the second Base , illatitious to the segment required ; which Bases and Subtendents , both first , and second , are peculiarly denominated according to the severall Cases of this Sindiforiuting Mood . Thus have I finished the Logarithmication of the generall Resolvers of the Loxogonosphericall Disergeticks , so farre as is requisite , wherein I often times mentioned the Arithmeticall complement of Sines , Co-sines , Tangents , and Co-tangents : and though I spoke of that purpose sufficiently in my Sphericorectangular comments , yet , for the Readers better remembrance thereof , I will once more define them here . The Arithmeticall complements of Sines are Co-secants ; of Co-sines , Secants ; of Tangents , Co-tangents ; and of Co-tangents , Tangents ; each being the others complement to the double Radius : but if such a Canon were framed , wherein the single Radius is left out of all Secants , and Tangents of major Arches , then would each be the others complement to the single Radius , and all Logarithmicall operations in questions of Trigonometry so easily performable by addition onely , that seldome would the presectionall digit exceed an unit . Having already said so much of these eight Disergeticks , I will conclude my discourse of them with a summary delineation of the eight severall Concordances which I observed amongst them ; for either they resemble one another in the Datas of their Moods , or in their Proturgetick operations , or in their dependance upon the same Axiome , or in the work of perpendicular finding , or in their Datas for the main demand , or in their materiall Quaesitas ( though diversly endowed ) or in their inversion , or lastly in their sindiforation , which affinity is onely betwixt two paires of them , as the first two amongst two quaternaries apeece , and the next five between foure couples each one , the brief hypotyposis of all which is here exposed to the view of the Reader . CONCORDANCES . Datall . Datangulary . Datolaterall . 1. Alamebna . 3. Amanepra . 1. Enerablo . 3. Erelomab . 2. Allamebne . 4. Ammanepreb . 2. Ennerable . 4. Errelome . Praenoscendall . Verticall . Basall . 1. Alamebna . 3. Amanepra . 1. Ammanepreb . 3. Ennerable . 2. Allamebne . 4. Erelomab . 2. Enerablo . 4. Errelome . Theorematick . Nabadprosver Naverprortes . Siubprortab Niubprodnesver . 1. Alamebna . 1. Allamebne . 1. Ammanepreb . 1. Ennerable . 2. Amanepra . 2. Erelomab . 2. Enerablo . 2. Errelome . Catheteuretick . Oppoverticall Hypoverticall . Oppobasall Hypobasall . 1. Alamebna . 1. Allamebne . 1. Ammanepreb . 1. Ennerable . 2. Amanepra . 2. Erelomab . 2. Enerablo . 2. Errelome . Datysterurgetick . Cathetoverticall Oppocathetall . Cathetobasall Hypocathetal . 1. Alamebna . 1. Amanepra . 1. Enerablo . 1. Erelomab 2. Allamebne . 2. Ammanepreb . 2. Ennerable . 2. Errelome . Zetetick . Cathetopposite Hypotenusall . Verticall Basall . 1. Alamebna* S. 1. Allamebne* M. 1. Amanepra* S. 1. ammanepreb . * S 2. Enerablo* T. 2. Ennerable* N. 2. Erelomab* N. 2. Errelome* N. Inversionall . Sinocosinall Sinocotangentall . Tangentosinall Cosinocosinall . 1. Alamebna . 1. Allamebne . 1. Ammanepreb . 1. Ennerable . 2. Amanepra . 2. Erelomab . 2. Enerablo . 2. Errelome . Sindiforall . Sindiforatall . Sindiforiutall . 1. Amanepra . 2. Erelomab . 1. Ammanepreb . 2. Errelome . THE EPILOGUE . WHat concerneth the resolving of all manner of Triangles , whether plain , or Sphericall , Rectangular , or Obliquangular , being now ( conform to my promise in the Title ) to the ful explained , commented on , perfected , and with all possible brevity , and perspicuity , in all its abstrusest and most difficult Secrets , from the very first principles of the Science it selfe , made manifest , proved , and convincingly demonstrated ; I will here shut up my discourse , and bring this Tractat to a period : which I may do with the more alacrity , in that I am confident , there is no Precept belonging to that faculty which is not herein included , or reducible thereto : and therefore ( I beleeve ) the judicious Reader will not be frustrate of his expectation , though by cutting the threed of my Glosse , I doe not illustrate what I have written with variety of examples ; seeing practically to treat of Triangulary calculations , in applying their doctrine to use , were to digresse from the purpose in hand , and incroach upon the subject of other Sciences ; a priviledge , which I must decline , as repugnant to the scope proposed to my selfe , in keeping this book within the speculative bounds of Trigonometry : for , as Logica utens , is the Science to the which it is applyed , and not Logick : So doth not the matter of Trigonometry , exceed the Theory of a Triangle : And as Arithmeticall , Geometricall , Astronomicall , Physicall , and Metaphysicall definitions , divisions , and argumentations , are no part of the Art that instructeth how to define , divide , and argue , nor matter incumbent to him that teacheth it : even so , by divulging this Treatise , doe I present the Reader with a Key , by meanes whereof he may enter into the chiefest treasures of the Mathematicall Sciences ; for the which , in some measure , I deserve thanks , although I help him not to unshut the Coffers wherein they lie inclosed : for , if the Lord chamberlain of the Kings houshold should give me a Key , made to open all the doores of the Court , I could not but graciously accept of it , though he did not goe along with me to try how it might fit every lock . The application is so palpable , that , not minding to insist therein , I will here stop the current of my Pen , and by a circulary conclusion , ending where I begun , certifie the Reader , that if he intend to approve himselfe an Artist in matters of Pleusiotechnie , Poliechyrologie , Cosmography , Geography , Astronomy , Geodesie , Gnomonicks , Sciography , Catoptricks , Dioptricks , and many other most exquisite Arts and Sciences , Practical and Theoretick , his surest course , for attaining to so much knowledge , is to be well versed in Trigonometry , to understand this Treatife aright , revolve all the passages thereof , ruminate on the Table , and peruse the Trissotetras . A Lexicidion of some of the hardest words , that occurre in the discourse of this institution Trigonometricall . BEing certainly perswaded , that a great many good spirits ply Trigonometry , that are not versed in the learned Tongues , I thought fit , for their encouragement , to subjoyne here the explication of the most important of those Greek , and Latin termes , which , for the more efficacy of expression , I have made use of in this Treatise : in doing whereof , that I might both instruct the Reader , and not weary him , I have endeavoured perspicuity with shortnesse : though ( I speak it ingenuously ) to have been more prolixe therin , could have cost but very little labor to me , who have already bin pretty well versed in the like , as may appear by my Etymologicall dictionary of above twenty seven thousand proper names , mentioned in the Lemmas of my severall Volums of Epigrams , the words whereof are for the most part abstruser , derived from moe Languages , and more liable to large , and ample interpretations . However ( caeteris paribus ) brevity is to be preferred ; therefore let us proceed to the Vocabulary in hand . THE LEXICIDION . A. ACute , comes from Acuo , acuere , to sharpen , and is said of an Angle , whose including sides , the more that its measure is lesse then a Quadrant , have their concursive , and angulary point the more penetrative , sharp , keen and pierceing : Whence an acutangled triangle . Adaequat , is that , which comprehendeth to the full , whatever is in the thing to the which it is compared , and for the most part in my Trissotetras is said of the generall finall Resolvers , in relation to the Moods resolved by them . It is compounded of Ad , and aequo , aequare , parem facere , to make one thing altogether like , or equall to another . Adjacent , signifieth to lie neare , and close , and is applyed both to sides , and Angles , in which sense likewise I make use of the words adjoyning the , conterminat , or conterminall with , annexed to , intercepted in , and other such like , for the more variety , as adherent , bounding , bordering , and so forth : It comes from Adjaceo , Adjacere , to lie neere unto , as the words Ad and jaceo , which are the parts whereof it is compounded , most perspicuously declare . Additionall , is said of the Line , which , in my comment , is indifferently called the Line of Addition , the Line of continuation , the extrinsecall Line , the excesse of the Secant above the Radius , the Refiduum , or the new Secant : it comes from Addo , Addere , which is compounded of Ad , and do , to put to and augment . Affection , is the nature , passion , and quality of an Angle , and consisteth either in the obtusity , acutenesse , or rectitude thereof : It is a verball from Afficio , affeci , affectum , compounded of ad , and facio . Aggregat , is the summe , totall , or result of an Addition , and is compounded of Ad , and grex ; for , as the Shepheard gathers his Sheep into a flock , so doth the Arithmetician compact his numbers to be added into a summe . Alternat , is said of Angles , made by a Line cutting two or more parallels , which Angles may be properly called so ; because they differ in nothing else but their situation ; for if the sectionary Line , to the which I suppose the parallels to be fixed , have the highest and lowest points thereof to interchange their sites , by a motion progressive towards the roome of the under Alternat , and terminating in that of the upper one , we will find , that both the inclination of the Lines towards one another , and the quality of the Angles , will , notwithstanding that alteration , be the same as before ; hence it is that they are called alternat , because there is no other difference betwixt them : or , if alternat be taken ( as arithmetically it is ) for that proportion , wherein the Antecedent is compared to the Antecedent , and the Consequent to the Consequent , the sense will likewise hold in the foresaid Angles ; for if by the parallelisme of two right Lines , cut with a third , two blunt , and two keen Angles be produced ( as must needs , unlesse the Secant line be to the parallels a perpendicular ) the keen or acute Angle will be to its complement , or successively following obtuse Angle , as the other acute unto its following obtuse ; therefore alternly , as the Antecedents are to one another , viz. the Acute to the Acute : so the Consequents , the obtuse to the obtuse . And if the Angles be right , the direct , and alternat proportion is one and the same ; the third , and fourth terms of the Analogy being in nothing different from the first , and second . Ambient , is taken for any of the legs of a rectangle , or the including , containing , or comprehending sides of the right Angle : it comes from Ambio , Ambire , which is compounded of Am and eo , i. e. circumeo : and more properly applied to both , then to any one of them , though usually it be usurped for one alone , vide Leg. Amblygonian , is said of obtuse angled Triangles , and Amblygonosphericall of obtuse sphericals : It is composed of 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 , and 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 angulus . Amfractuosities , are taken here for the cranklings , windings , turnings , and involutions belonging to the equisoleary Scheme ; of am and frango , quod sit quasi via crebris maeandris undequaque interrupta . Analogy , signifieth an equality of proportion , a likenesse of reasons , a conveniencie , or habitude betwixt termes : It is compounded of 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 , aequaliter , and 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 , ratio . Analytick , resolutory , and is said of those things that are resolved into their first principles , of 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 , re , and 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 , solvo . Antilogarithm , is the Logarithm of the complement ; as for example , the Anti-logarithm of a Sine is the Logarithm of the Sine complement , vide Logarithm . Anti-secant , Anti-sine , and Anti-tangent , are the complements of the Secant , Sine , and Tangent , and are called sometime Co-secant , Co-sine , and Co-tangent : they have anti prefixed , because they are not in the same colume , and co , because they are in the next to it . Apodictick , is that , which is demonstrative , and giveth evident proofs of the truth of a conclusion ; of 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 , and 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 , monstro , ostendo , unde 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 , demonstratio . Area , is the capacity of a Figure , and whole content thereof . Arch , or Ark , is the segment of a circumference lesse then a semicircle : major Arch is above 45. degrees , a minor Arch , lesse then 45. vide Circle . Arithmeticall complement , is the difference betweene the Logarithm to be substracted , and that of the double , or single Radius . Artificiall numbers , are the Logarithms , and artificiall Sine the Logarithm of the Sine . Axiome , is a maxim , tenet , or necessary principle , whereupon the Science of a thing is grounded : it cometh from 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 , dignus ; because such things are worthy our knowledge . B. BAsall , adjectively is that , which belongeth to the Base , or the subjacent side , but substantively the great Base . Basangulary , is said of the Angles at the Base . Basidion , or baset , is the little Base , all which come from the Greek word 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 . Basiradius , is the totall Sine of that Arch , a Segment whereof is the Base of the proposed sphericall Triangle . Bisected , and Bisegment , are said of lines cut into two equall parts : it comes from biseco , bisecare , bisectum , bisegmen . Bluntnesse , or flatnesse , is the obtuse affection of Angles . Bucarnon , by this name is entitled the seven and fortieth proposition of the first of the elements of Euclid ; because of the oxe , or , ( as some say ) the hecatomb which Pythagoras , for gladnesse of the invention , sacrificed unto the gods : of 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 , bos , and 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 , vicissim aliquid capio ; they being ( as it is supposed ) well pleased with that acknowledgement of his thankfulnesse for so great a favour , as that was , which he received from them : you may see the proposition in the seventeenth of my Apodicticks . C. CAnon , is taken here for the Table of Sines and Tangents , or of their Logarithms : it properly signifieth the needle or tongue of a balance , and metaphorically a rule , whereby things are examined . Cases , are the parts wherein a Mood is divided from cado . Cathetos , is a Perpendicular line , from 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 , demitto , of 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 , and 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 . Catheteuretick , is concerning the finding out of the Perpendicular of 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 , and 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 , invenio . Cathetobasall , is said of the Concordances of Loxogonosphericall Moods , in the Datas of the Perpendicular , and the Base , for finding out of the maine quaesitum . Cathetopposite , is the Angle opposite to the Perpendicular ; it is a hybrid or mungrell word , composed of the Greek 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 , and Latin oppositus . Cathetorabdos , or Cathetoradius , is the totall Sine of that Arch , a Segment whereof is the Cathetos , or Perpendicular of the proposed Orthogonosphericall . Cathetothesis , and cathetothetick are said of the determinat position of the Perpendicular , which is sometimes expressed by cathetology , instructing us how it should be demitted : of 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 , and 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 , from 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 , pono , colloco . Cathetoverticall , is said of the Concordances of Loxogonosphericall Moods in the Datas of the perpendicular , and the verticall Angle in the last operation . Catoptrick , the Science of perspective , from 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 , perspicio , cerno . Characteristick , is said of the letters , which are the notes and marks of distinction , called sometimes figuratives , or determinaters , from 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 , sculpo , imprimo . Circles , great circles are those which bisect the Sphere , lesser Circles those which not . Circular parts , are in opposition to the reall and naturall parts of a Triangle . Circumjacent , things which lie about , of circum and jaceo . Coalescencie , a growing together , a compacting of two things in one ; it is said of the last two operations of the Loxogonosphericals conflated into one , from coalesco or coalco , of con and alo . Cobase , a fellow Base , or that which with another Base hath a common Perpendicular , of con and basis . Cocathetopposite , is said of two , Angles at the Base , opposite to one and the same Cathetos . Coincidence , a falling together upon the same thing , from coincido , of con , and incido , ex in , & cado . Comment , is an interpretation , or exposition of a thing , and comes from comminiscor , comminisci , mentionem facere . Compacted , joyned , and knit together , put in one ; from compingo , compegi , compactum , vide Coalescencie . Complement , signifieth the perfecting that which a thing wanteth , and usually is that , which an Angle or a Side wanteth of a Quadrant , or 90. degrees : and of a Semicircle , or 180. from compleo , complere , to fill up . Concurse , is the meeting of lines , or of the sides of a Triangle , from concurro , concursum . Conflated , compacted , joyned together , from conflo , conflatum , conflare , to blow together , vide Inchased . Consectary , is taken here for a Corollary , or rather a secondary Axiome , which dependeth on a prime one , & being deduced from it , doth necessarily follow . From consector , consectaris , the frequentative of consequor . Consound , to sound with another thing ; it is said of consonants , which have no vocality without the help of the vowell . Constitutive , is said of those things , which help to frame , make , and build up : From constituo , of con and statuo . Constitutive sides , the ingredient sides of a Triangle . Constructive parts , are those , whereof a thing is built , and framed : From construo , constructum , to heap together , and build up , of con and strues . Conterminall , is that which bordereth with , and joyneth to a thing , of con and terminus , vide Adjacent , or Insident . Cordes , and cordall , are said of subtenses metaphorically ; because the Arches and subtenses are as the bow and string : chorda , comes of the Greek word 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 , intestinum , ilia , quia ex illis chordae conficiuntur . Correctangle , that is one which , with another rectangle , hath a common Perpendicular . Correspondent , that which answereth with , and hath a reference to another thing , of con and respondeo . Cosinocosinall , is said of the Concordances of Loxogonosphericall Moods , agreeing , in that the termes of their finall Resolvers run upon Cosines . Cosmography , is taken here for the Science whereby is described the celestiall Globe , of 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 , and 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 . Cosubtendent , is the subtendent of a correctangle , or that which with another is substerned to two right Angles , made by the demission of one and the same Perpendicular . Coverticall , is the fellow top Angle , from whence the Perpendicular falleth . D. DAta , is said of the parts of a Triangle , which are given us , whether they be sides or Angles , or both , of do , datum , dare . Datimista , are those Datas , which are neither Angles onely , nor sides onely , but Angles , and sides intermixedly : of data , and mista , from misceo . Datangulary , is said of the Concordances of those Moods , for the obtaining of whose Praenoscendas , we have no other Datas , but Angles , unto the foresaid Moods common . Datapurall , comes from datapura , which be those Datas , that are either meerly Angles , or meerly sides . Datolaterall , is said of the Concordances of those Moods , for the obtaining of whose Praenoscendas , the same sides serve for Datas . Datoquaere , is the very Problem it selfe , wherein two or three things are given , and a third or fourth required , as by the composition of the word appears . Datisterurgetick , is said of those Moods which agree in the Datas of the last work : of data , 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 , postremum , and 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 , opus . Demission , is a letting fall of the Perpendicular : from demitto ; demissum . Determinater , is the characteristick or figurative letter of a directory : from determinare , to prescribe and limit . Diagonall , taken substantively , or diagonie , is a line drawn from one Angle to another , of 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 . and 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 what the diagonie is in surfaces , the axle is in solids . Diagrammatise , to make a Scheme or Diagram , from 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 , delineo . Diatyposis , is a briefe summary description , and delineation of a thing : or the couching of a great deale of matter , for the instruction of the Reader , in very little bounds , and in a most neat and convenient order : from 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 , instituo , item melius dispono , vide 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 . Diodot , is Pythagorases Bucarnon , or the gift bestowed on him by the gods : of 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 , the genitive of 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 , and 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 , datus , from 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 , do , vide Bucarnon . Dioptrick , the art of taking heights and distances , from 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 pervidendo , altitudinem dimensionemque turrium & murorum exploro . Directly , is said of two rowes of proportionals , where the first terme of the first row , is to the first of the second , as the last of the first , is to the last of the second . Directory , is that which pointeth out the Moods dependent on an Axiome . Discrepant , different , dissonant , id est , diverso modo crepare . Disergeticks , of two operations , of 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 , and 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 . Document , instruction , from doceo . E. ELucidation , a clearing , explaining , resolving of a doubt , and commenting on some obscure passage , from elucido , elucidare . Energie , efficacie , power , force ; from 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 , qui in opere est , of 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 and 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 , opus . Enodandum , that which is to be resolved and explicated , declared , and made manifest , from enodo , enodare , to unknit , or cut away the knot . Equation , or rather aequation , a making equall , from aequo , aequare . Equiangularity , is that affection of Triangles , whereby their Angles are equall . Equicrurall , is said of Triangles , whose legs or shanks are equall ; of aequale , and crus , cruris ; leg being taken here for the thigh and leg . Equilaterall , is said of Triangles , which have all their sides , shanks , or legs equall , of aequale and latus , lateris . Equipollencie , is a samenesse , or at least an equality of efficacie , power , vertue , and energie ; of aequus and polleo . Equisolea , and Equisolearie , are said of the grand Orthogono sphericall Scheme ; because of the resemblance it hath with a horse-shooe , and may in that sense be to this purpose applied with the same metaphoricall congruencie , whereby it is said , that the royall army at Edge-hill was imbatteld in a half-moon . Equivalent , of as much worth and vertue , of aequus and valeo . Erected , is said of Perpendiculars , which are set or raised upright upon a Base , from erigere , to raise up , or set aloft . Externall , extrinsecall , exteriour , outward , or outer , are said oftest of Angles , which being without the Area of a Triangle , are comprehended by two of its shanks meeting or cutting one another , accordingly as one or both of them are protracted beyond the extent of the figure . F. FAciendas , are the things which are to be done : faciendum is the gerund of facio . Figurative , is the same thing as Characteristick , and is applied to those letters which doe figure and point us out a resemblance and distinction in the Moods . Figures , are taken here for those partitions of Trigonometry , which are divided into Moods . Flat , is said of obtuse , or blunt Angles . Forwardly , is said of Analogies , progressive from the first terme to the last . Fundamentall , is said of reasons , taken from the first grounds and principles of a Science . G. GEodesie , the Art of Surveying , of 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 , or 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 , terra , and 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 , divido , partior . Geography , the Science of the Terrestriall Globe , of 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 terra , and 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 , describo . Glosse , signifieth a Commentary , or explication , it cometh from 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 . Gnomon , is a Figure lesse then the totall square , by the square of a Segment : or , according to Ramus , a Figure composed of the two supplements , and one of the Diagonall squares of a Quadrat . Gnomonick , the Art of Dyalling , from 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 , the cock of a Dyall . Great Circles , vide Circles . H. HOmogeneall , and Homogeneity , are said of Angles of the same kind , nature , quality , or affection : from 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 , communio generis . Homologall , is said of sides congruall , correspondent , and agreeable , viz. such as have the same reason or proportion from 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 , similis ratio . Hypobasall , is said of the Concordances of those Loxogonosphericall Moods , which , when the Perpendicular is demitted , have for the Datas of their second operation the same Subtendent and Base . Hypocathetall , is said of those which for the Datas of their third operation have the same Subtendent and Perpendicular . Hypotenusall , is said of Subtendent sides , from 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 and 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 . Hypotyposis , a laying downe of severall things before our eyes at one time , from 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 , oculis subjicio , delineo , & repraesento , vide 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 . Hypoverticall , is said of Moods , agreeing in the same Catheteuretick Datas of subtendent and verticall , as the Analysis of the word doth shew . I IDentity , a samenesse , from idem , the same . Illatitious , or illative , is said of the terme which bringeth in the quaesitum , from infero , illatum . Inchased , coagulated , fixed in , compacted , or conflated , is said of the last two Loxogonosphericall operations put into one , vide Compacted , Conflated , and Coalescencie . Including sides , are the containing sides of an Angle of what affection soever it be , vide , Ambients , Legs , &c. Individuated , brought to the lowest division , vide , Specialised , and Specification . Indowed , is said of the termes of an Analogie , whether sides , or Angles , as they stand affected with Sines , Tangents , Secants , or their complements , vide Invested . Ingredient , is that which entreth into the composition of a Triangle , or the progresse of an operation , from ingredior , of in and gradior . Initiall , that which belongeth to the beginning , from initium , ab ineo , significante incipio . Insident , is said of Angles , from insideo , vide Adjacent , or Conterminall . Interjacent , lying betwixt , of inter and jaceo ; it is said of the Side or Angle betweene . Intermediat , is said of the middle termes of a proportion . Inversionall , is said of the Concordances of those Moods which agree in the manner of their inversion ; that is in placing the second and fourth termes of the Analogy , together with their indowments , in the roomes of the first and third , and contrariwise . Invested , is the same as indowed , from investio , investire . Irrationall , are those which are commonly called surd numbers , and are inexplicable by any number whatsoever , whether whole , or broken . Isosceles , is the Greek word of equicrurall , of 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 , and 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 , crus . L. LAterall , belonging to the sides of a Triangle from latus , lateris . Leg , is one of the including sides of an Angle , two sides of every Triangle being called the Legs , and the third , the Base ; the Legs therefore or shankes of an Angle are the bounds insisting or standing upon the Base of the Angle . Line of interception , is the difference betwixt the Secant , and the Radius , and is commonly called the residuum . Logarithms , are those artificiall numbers , by which , with addition and subtraction onely , we work the same effects , as by other numbers , with multiplication and division : of 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 , ratio , proportio , and 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 , numerus . Logarithmication , is the working of an Analogy by Logarithms , without having regard to the old laborious way of the naturall Sines , and Tangents ; we say likewise Logarithmicall and Logarithmically , for Logarithmeticall , and Logarithmetically ; for by the syncopising of et , the pronunciation of those words is made to the eare more pleasant : a priviledge warranted by all the dialects of the Greek , and other the most refined Languages in the world . Loxogonosphericall , is said of oblique sphericals , of 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 , obliquus , and 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 , ad sphaeram pertinens , from 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 , globus . M. MAjor and Minor Arches , vide Arch. Maxim , an axiome , or principle , called so ( from maximus ) because it is of greatest account in an Art or Science , and the principall thing we ought to know . Meane , or middle proportion , is that , the square whereof is equall to the plane of the extremes : and called so because of its situation in the Analogy . Mensurator , is that , whereby the illatitious terme is compared , or measured with the maine quaesitum . Monotropall , is said of figures , which have one onely Mood , of 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 , and 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 ▪ from 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 . Monurgeticks , are said of those Moods , the maine Quaesitas whereof are obtained by one operation , of 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 , and 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 . Moods , aetermine unto us the severall manners of Triangles , from modus , a way , or manner . N. NAturall , the naturall parts of a Triangle , are those of which it is compounded , and the circular , those whereby the maine quaesitum is found out . Nearest ▪ or next , is said of that Cathetopposite Angle , which is immediatly opposite to the perpendicular . Notandum , is set downe for an admonition to the Reader , of some remarkable thing to follow , and is the Gerund of Noto , notare . O. OBlique , and obliquangulary , are said of all Angles that are not right . Oblong , is a parallelogram , or square more long them large : from oblongus , very long . Obtuse , and obtuse angled , are said of flat , and blunt Angles . Occurse , is a meeting together , from occurro , occursum , Oppobasall , is said of those Moods , which have a Catheteuretick Concordance in their Datas of the same Cathetopposite Angles , and the same Bases . Oppocathetall , is said of those Loxogonosphericals which have a Datisterurgetick Concordance in their Datas of the same Angles at the Base , and the Perpendicular . Oppoverticall , is said of those Moods which have a Catheteuretick Concordance in their Datas of the same Cathetopposites , and verticall Angles . Orthogonosphericall , is said of right angled Sphericals , of 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 , rectus , 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 , angulus , and 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 , gobus . Oxygonosphericall , is said of acute-angled sphericals , of 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 , and 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 . P. PArallelisme , is a Parallel , equality of right lines , cut with a right line , or of Sphericals with a Sphericall , from 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 , equidistans of 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 and 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 . Parallelogram , is an oblong , long square , rectangle , or figure made of parallel lines : of 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 , and 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 , linea . Partiall , is said of enodandas depending on severall Axioms . Particularise , specialise , by some especiall difference to contract the generality of a thing . Partition , is said of the severall operations of every Loxogonosphericall Mood , and is divided in praenoscendall , catheteuretick and hysterurgetick . Permutat proportion , or proportion by permutation , or alternat proportion , is when the Antecedent is compared to the Antecedent , and the Consequent to the Consequent , vide , Perturbat . Perpendicularity , is the affection of the Perpendicular , or plumb-line ; which comes from perpendendo , id est , explorando altitudinem . Perturbat , is the same as permutat , and called so because the order of the Analogie is perturbed . Planobliquangular , is said of plaine Triangles , wherein there is no right Angle at all . Planorectangular , is said of plaine right-angled Triangles . Planotriangular , is said of plaine Triangles , that is , such as are not Sphericall . Pleuseotechnie , the Art of Navigation , of 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 , navigatio , and 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 , ars . Plusminused , is said of Moods which admit of Mensurators or whose illatitious termes are never the same , but either more or lesse then the maine quaehtas . Poliechyrologie , the Art of fortifying Townes and Cities , of 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 , urbs , civit as 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 , munio firmo , and 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 , ratio . Possubservient , is that which after another serveth for the resolving of a question ; of post , and subserviens : of sub and servio . Potentia , is that wherein the force and whole result of another thing lies . Power , is the square , quadrat , or product of a line extended upon it selfe , or of a number in it selfe multiplied . Powered , squared quadrified . Precept , document , from praecipio , praeceptum . Praeroscenda , are the termes , which must be knowne before we can attaine to the knowledge of the maine quaesitas of prae and nosco . Praenoscendall , is said of the Concordances of those Moods , which agree in the same praenoscendas . Praesection , praesectionall , is concerning the digit towards the left , whose cutting off saveth the labour of subtracting the double or single Radius . Praescinded problems , are those speculative Datoquaeres , which are not applied to any matter by way of practice . Praesubservient , is said of those Moods which in the first place we must make use of for the explanation of others ; of prae , and ●ub●ervio . Prime , is said of the furthest Cathetopposite , or Angle at the Base , contained within the Triangle to be resolved . Primifie the Radius , is to put the Radius in the first place , primumque inter terminos collocare proportionales . Problem , problemet , a question or datoquaere , from 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 , unde 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 propositum , objectaculum . Product , is the result , factus , or operatum of a multiplication , from produco , productum . Proportion , proportionality , are the same as Analogy , and Analogisme ; the first being a likenesse of termes , the other of proportions . Proposition , a proposed sentence , whether theorem or problem . Prosiliencie , is a demission , or falling of the Perpendicular , from prosilio , ex pro & salio . Proturgetick , is said of the first operation of every Disergetick Mood , of 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 and 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 , the 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 being Attically contracted into 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 . Q. QUadrant , the fourth part of a Circle . Quadranting , the protracting of a Sphericall side unto a Quadrant . Quadrat , a Square , a forma quadrae , the power or possibility of a line , vide , Power . Quadrobiquadraequation , concerneth the Square of the subtendent side , which is equall to the Biquadrat , or two Squares of the Ambients . Quadrosubduction , is concerning the subtracting of the Square of one of the Ambients from the Square of the Subtendent . Quaesitas , the things demanded from quaero ▪ quaesitum . Quotient , is the result of a division , from quoties , how many times . R. RAdically meeting , is said of those Oblongs , or Squares , whose sides doe meet together . Radius , ray , or beame is the Semidiameter , called so metaphorically , from the spoake of a wheele which is to the limb thereof , as the Semidiameter , to the circumference of a circle . Reciprocall , is said of proportionalities , or two rowes of proportionals , wherein the first of the first is to the first of the second , as the last of the second is to the last of the first , and contrarily . Rectangular , is said of those figures , which have right Angles . Refinedly , is said when we go the shortest way to work by primifying the Radius . Renvoy , a remitting from one place to another , it comes from the French word Renvoyer . Representative , is said of the letters , which stand for whole words ; as E. for side , L. for secant , U. for subtendent . Residuat , is to leave a remainder , nempe id quod residet & superest . Resolver , is that which looseth and untieth the knot of a difficulty , of re and solvo . Resolutory , is said of the last partition of the Loxogonosphericall operations . Result , is the last effect of a work . Root , is the side of a Square , Cube , or any cossick figure . S. SCheme , signifieth here the delineation of a Geometricall figure , with all parts necessary for the illustrating of a demonstration , from 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 , habeo . Sciography , the Art of shadowing , of 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 , umbra , and 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 , scribo . Segment , the portion of a thing cut off , quasi secamentum , quod a re aliqua secatur . Sexagesimat , subsexagesimat , resubsexagesimat , and biresubsexagesimat , are said of the division , subdivision , resubdivision , and reer-resubdivision of degrees into minuts , seconds , thirds , and fourths , in 60. of each other : the devisor of the fore goer being successively the following dividend , and the quotient alwayes sixty . Sharp , is said of acute Angles . Sindiforall , is said of those Moods , the fourth terme of whose Analogie is onely illatitious to the maine quaesitum . Sindiforation , is the affection of those foresaid Moods , whereby the value of the mensurator is knowne . Sindiforatall , is concerning those Moods , whose illatitious terme is an Angle . Sindiforiutall , is of those Moods , whose illatitious terme is a side ; all these foure words are composed of representatives , and ( if I remember well ) mentioned in my explanation . Sinocosinall , is said of the Concordances of those Moods , which agree in this , that their Analogies run upon sines , and sine-complements . Sinocotangentall , is said of those Moods , which agree in that the termes of their Analogie run upon Sines and Tangent-complements . Sinus , is so called ( I beleeve ) because it is alwayes in the very bosome of the Circle . Sinused , is said of termes endowed or invested with Sines . Specialized , contracted to more particular termes , vide , Individuated . Specifying , determinating , particularising . Specification , a making more especiall , by contracting the generality of a thing , vide , Specialized . Sphericodisergeticks , are the Sphericall Triangles of two operations . Structure of an operation , is the whole frame thereof , from struo , structum . Subdatoquaere , is a particular datoquaere , and is applied to the problems of the cases of every Sphericodisergetick Mood , vide , Sub-problems . Sabajcent , is the substerned side or the Base , of sub & jaceo , vide , Sustentative , Sustaining side and Substerned . Subordinate problems , is the same with subdatoquaere . Subproblems , is the same with subordinate problems , or problemets . Subservient , is said of Moods which serve in the operation of other Moods . Substerned , is the subjacent side or Base : or , more generally , any side opposite to an Angle ; of sub and sterno , sternere , vide , Subjacent . Subtendent , is the side opposite to the right Angle , of sub and tendo ; as if you would say , Under-stretched . Subtendentine , is the subtendent of a little rectangled Triangle , comprehended within the Area of a great one , and is sometimes called the little subtendent , and reere subtendent . Subtendent all , is the subtendent of a great rectangled Triangle , within whose capacity is included a little one : it is likewise called the great subtendent , and maine subtendent . Supernumerary , is said of the digit , by the which the proposed number exceeds in places the number of the places of the Radius . Supplements , are the Oblongs made of the Segments of the root of a Square ; and so called , because they supply all that the Diagonals or Squares of the Segments joyned together , want of the whole lines square . Suppone severally , is to signifie severall things . Sustaining side is the substerned , or subjacent side . Sustentative , is the same with sustaining , substerned , subjacent and Base . Sympathie of Angles , is a similitude in their affection , of 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 , and 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 , passio , vide homogeneall . T. TAble , is an Index sometimes , and sometimes it is taken for a Briefe and summary way of expressing many things . Tangentine , is that which concerneth Tangents or touch-lines . Tangentosinall , is said of the Concordance of those Loxogonosphericals , the termes of whose Analogie runne upon Tangents and Sines . Tenet , is a secondary maxim , and is onely said here of Cathetothetick principles . Theorematick , speculative , from 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 , a speculation , which cometh from 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 , or 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 , speculare , or contemplare . Topanglet , and verticalin are the same . Trigonometry , is the Art of calculating , and measuring of Triangles , of 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 , triangulus , and 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 , metior . Trissotetras , is that which runneth all along upon threes and foures of 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 , and in plurali , 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 , tertius , trinus , triplex , tres , and 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 , numerus quaternarius , from 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 , quatuor . U. VAriator , is from vario , variare , to diver sifie , and is said of cases , which upon the same Datas are onely diverse in the manner of resolving the Quaesitum . Verticaline , verticall , verticalet , are the top-angles , and top-anglets , from vertex , verticis . Underproblem , problemet , subordinate problem , sub-problem , under-datoquaere , and sub ▪ datoquaere are , all the same thing . Unradiated , or unradiused , is said of a summe of Logarithms from which the Radius is abstracted . Z. ZEterick , is said of Loxogonosphericall Moods which agree in the same quaesitas , from 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 , quaero , inquiro . The finall Conclusion . IF the novelty of this my Invention be acceptable ( as I doubt not but it will ) to the most Learned , and judicious Mathematicians , I have already reaped all the benefit I expected by it , and shall hereafter , ( God willing ) without hope of any further recompence , cheerfully under-goe more laborious employments , of the like nature , to doe them service : But as for such , who , either understanding it not , or vain-gloriously being accustomed to Criticise on the Works of others , will presume to carp therein at what they cannot amend , I pray God to illuminate their Judgements , and rectifie their Wils , that they may know more , and censure lesse ; for so by forbearing detraction , the venom whereof must needs reflect upon themselves , they will come to approve better of the endeavours of those , that wish them no harme . Sit Deo Gloria . The Diorthosis . THe mistakes of the Presse , can breed but little obstruction to the progresse of the ingenious Reader , if with his Pen , before he enter upon the perusall of this Treatise , he be pleased thus to correct ( as I hope he will ) these ensuing Erratas . Pag. lin . Errata . Emendata . Pag. lin . Errata . Emendata . 8 25 Talfagro Talzo . Talfagros Talzos . 16 25 This Cheme . the Schemes & dining room totall summe . 10 17 Niubprodesver . Niubprodnesver . 16 31 or dining roome totall Sine .   11 29 Natfr . Autir . Natfr . Eutir . 23 23     11 35 Nat. Nad Nath. Naet . Naed . Naeth . 26 6 as the Sine of the cosubtendent . as the cosubtendent . 11 36 Eheromabme . Eherolabme .         11 37 Being Allotted . Being abinarie allotted .       second basidion . 16 8     80 9 second Base .   What errors else ( if any ) have slipt animadversion ( besides their not being very materiall ) are so intelligible , that being by the easiest judgement with as much facility eschewable , as I can observe them , not to mention the commission of such faults is no great omission ; and therefore will I heartily ( without further ceremony ) conduct the Student ( who making this the beginning of the Book , as it is most fit he doe , seeing a Ruler should be made streight before any thing be ruled by it ) is willing to go along with me from hence circularly through the title , to the end of the Treatise in the proposed way , as followes . And so God blesse us both . 
A97051	----	Due correction for Mr Hobbes· Or Schoole discipline, for not saying his lessons right. In answer to his Six lessons, directed to the professors of mathematicks. / By the professor of geometry. Wallis, John, 1616-1703. 1656 Approx. 363 KB of XML-encoded text transcribed from 73 1-bit group-IV TIFF page images. Text Creation Partnership, Ann Arbor, MI ; Oxford (UK) : 2008-09 (EEBO-TCP Phase 1). A97051 Wing W576 Thomason E1577_1 ESTC R204165 99863848 99863848 116063 This keyboarded and encoded edition of the work described above is co-owned by the institutions providing financial support to the Early English Books Online Text Creation Partnership. This Phase I text is available for reuse, according to the terms of Creative Commons 0 1.0 Universal . The text can be copied, modified, distributed and performed, even for commercial purposes, all without asking permission. Early English books online. (EEBO-TCP ; phase 1, no. A97051) Transcribed from: (Early English Books Online ; image set 116063) Images scanned from microfilm: (Thomason Tracts ; 199:E1577[1]) Due correction for Mr Hobbes· Or Schoole discipline, for not saying his lessons right. In answer to his Six lessons, directed to the professors of mathematicks. / By the professor of geometry. Wallis, John, 1616-1703. [12], 130, [2] p., folded plate Printed by Leonard Lichfield printer to the University for Tho: Robinson., Oxford, : 1656. Dedication signed: John Wallis. A reply to: Hobbes, Thomas. Six lessons to the professors of the mathematiques. With a final errata leaf. Annotation on Thomason copy: "7ber [i.e. September] 26". Reproduction of the original in the British Library. Created by converting TCP files to TEI P5 using tcp2tei.xsl, TEI @ Oxford. Re-processed by University of Nebraska-Lincoln and Northwestern, with changes to facilitate morpho-syntactic tagging. 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Understanding these processes should make clear that, while the overall quality of TCP data is very good, some errors will remain and some readable characters will be marked as illegible. Users should bear in mind that in all likelihood such instances will never have been looked at by a TCP editor. The texts were encoded and linked to page images in accordance with level 4 of the TEI in Libraries guidelines. Copies of the texts have been issued variously as SGML (TCP schema; ASCII text with mnemonic sdata character entities); displayable XML (TCP schema; characters represented either as UTF-8 Unicode or text strings within braces); or lossless XML (TEI P5, characters represented either as UTF-8 Unicode or TEI g elements). Keying and markup guidelines are available at the Text Creation Partnership web site . eng Hobbes, Thomas, 1588-1679. -- Six lessons to the professors of the mathematiques. Geometry -- Early works to 1800. Mathematics -- Early works to 1800. 2007-04 TCP Assigned for keying and markup 2007-04 Aptara Keyed and coded from ProQuest page images 2007-05 Emma (Leeson) Huber Sampled and proofread 2007-05 Emma (Leeson) Huber Text and markup reviewed and edited 2008-02 pfs Batch review (QC) and XML conversion Due Correction FOR M R HOBBES . OR Schoole Discipline , for not saying his Lessons right . In Answer To His Six Lessons , directed to the Professors of Mathematicks . By the Professor of GEOMETRY . Hobs Leviathan part . 1. chap. 5. pag. 21. Who is so stupid , as both to mistake in Geometry , and allso to persist in it , when another detects his error to him ? OXFORD , Printed by Leonard Lichfield Printer to the University for Tho : Robinson . 1656. TO THE Right Honourable HENRY Lord Marquesse of Dorchester , Earle of Kingston , Vicount Newark , Lord Pierrepoint , and Manvers , &c. MY LORD , YOUR Honour may perhaps think it strange that a person so wholly a stranger as I , should tender you such a peece as this : Yet will , I doubt not , acquit me of rudenesse and incivility in so doing ; when you consider , That the adverse party , whom it takes to taske , hath made his appeale hither ; and finding himselfe foiled in Latine , hath here put in his English Bill for some reliefe : And it is but reason that Bill and Answer be filed in the same Court. He had the confidence , to tender his book first to another honorable Person the Earle of Devonshire , with this presumption , That though things were not so fully demonstrated as to satisfie every Reader , yet 't was good enough to satisfie his Lordship , he did not doubt . Which presumption of his was then the more tolerable , because he then thought his demonst●a●io●s good . But when he had been so fully convinced what weake stuffe it was ; that now the utmost of his hopes is ( for so I understand from his friends ) that though he be mistaken in the Mathematicks , yet he hopes to prove himselfe an honest man , ( which yet is more I suppose than , by his principles , he need to be : ) To make the world believe , that your Lordship doth approve of his Principles , Method , and Manners in those writing ; and , that this is the only cause of the favours you have expressed towards him ; is so high an affront , as had he not a great confidence of your Lorships Magnanimity , to despise it , or Clemency , to pardon it , he would not have offered to a person of so much honour and worth . Since therefore he hath brought it before you as a controversy , wherein he desires your Lordship to consider and judge , whether he have said his six Lessons aright : I shall not at all demurre to the jurisdiction of the court ; but as readily admit his Umpar , as allow him the choise of his own Weapon ; and so tender your Lordship an English Answer to his English Appeale from my Latine Confutation of his treatise in Latine : That when in the judgement of this own Umpar , he sees himselfe foiled at his own weapons ; he may hereafter make choise of French or Dutch , or some other Language , which he may hope to be more favourable to him , than Latine or English hath yet been . He tells your Lordship , what great feates he hath done in his book ; and your Lordship knows as well , by this and my former answer , how they have been defeated . And then he reckons up certaine positions ( some of them absurd enough ) and would have you believe them to be our Principles at Oxford : But doth not tell your Lordship where they are to be found in any writings of ours . Now , ( that your Lordship may not seek them there in vaine , where they are not to be found , ) I shall briefely shew where the rise of all these accusations lye ; in his own writings , not in ours . First , He had taught us Cap. 13. § . 16. Si ratio detur minoris ad majus , rationesque aliquot addantur ipso aequales non multiplicari proprie , sed submultiplicari dicitur : itaque quando additur primae rationi altera , ratio primae quantitatis ad tertiam , ●emissis est rationis primae ad secundam . That is , in plaine English , If there be any proportion assigned of a lesse quantity to a greater , and to that proportion be added another proportion equall to it ; that proportion that doth result by this addition , is not the double , but the halfe of that assigned proportion . Now , because this is very absurd , and I had told him so ; He would have your Lordship believe , that it was I had said ( not he , ) that Two equall proportions , are not double to one of the same proportions . Which is his first Charge . Secondly ; He had sayd farther , in the same place , Cap. 13. § . 16. Ratio 2 ad 1 vocatur dupla , & 3 ad 1 tripla , &c. ( and he saith true . ) But then ( forgetting that these were his own words ) he would have it thought ( Less . 5. p. 42. ) absurd to say that the proportion of two to one is double ; and asks , is not every double proportion , the double of some proportion ? And doth here intepret that phrase ( of his own ) the proportion of two to one is called double , to be all one as to say , That a proportion is double , triple &c. of a number , but not of a proportion . Which is his second charge . Thirdly he had Cap. 8. § . 13 , 14. ( without any necessity ) layd ●he whole stresse of Geometry , upon this supposition . That , It is not possible for the same body to possesse at one time a greater , at another time a lesser place . ( For , if this be possible , the same body is , by his definition , at the same time equall to a bigger , and to a lesse body than it selfe : as I there shewed by a consequence so cleare that he cannot himselfe deny it . ) Which he there first , attempts to prove , ( as simply as a man would wish , ) but then presently flyes off againe , and say● that a thing in it selfe so manifest needs no demonstration . But sayd I , ( without declaring my own opinion in the case , which what it is he knowe● not ) An assertion of such huge consequence to his doctrine as this is , and being ( as he well knows ) generally denyed ( whatever he or I think of it ) by all those who maintaine Condensation & Rarification in a proper sense , ( without either vacuum , or the admission and extrusion of a forraigne body ; ) ought to be well proved , by him that builds so much upon it , and not be assumed gratis . Now because of this it is , that he tells you in his third charge , That 't is one of our principles , That the same body without adding to it , or taking from it , is sometimes greater and sometimes lesse . So hainous a matter is it , to require a proofe from him , of what he doth affirme though of never so great consequence . Fourthly , He tells us Cap. 14. § . 19. ( and 't is true enough ) that an Hyperbolick line , and its Asymptote , doe still come nearer and nearer till they approach to a distance lesse then any assignable quantity : And consequently if infinitely produced , must be supposed to meet , or to have no distance at all ; ( and so the distance of that hyperbola so produced , from a line parallel to the Asymptote , to be the same with the distance of that Asymptote from the said parallell ; that i● , equall to a given quantity . ) And that this is a good inference , we are taught Less . 5. § . 43. as standing on the same ground with the demonstrations of all such Geometricians , Ancient and Moderne , as have inferred any thing in the manner following , [ viz. If it be not greater nor lesse , then it is equall . But it is neither greater nor lesse . Therefore &c. If it be greater , say by how much . By so much . 'T is not greater by so much : Therefore it is not greater . If it be lesse , say by now much &c. ] which , being good demonstrations are together with this overthrown , if this inference be not good ; that is , if things which differ lesse than any assignable quantity may not be reputed equall , But now , to say thus , That the distance of an Hyperbole , from a streight line drawn beyond its asymptote and the parallell thereunto , doth continually decrease , so as , if it be supposed infinitely produced , it must be supposed to be at length the same with that of the Asymptote from the sayd parallell , because neither greater nor lesse by any assignable quantity ; ( which is but the result of his own assertion ) is all one as to say , That a quantity may grow lesse and lesse eternally , so as at last to be equall to another quantity ; or which is all one , saith he , that there is a last in Eternity ▪ which is his fourth charge : and , what absurdity is in it , falls upon himselfe . Just as , when having told us Cap. 16. § . 20. Punctum inter quantitates nihil est , ut inter numeros Cyphra : And Cap. 14. § . 16. Punctum ad lineam neque rationem habet , neque quantitatem ullam : He railes upon me , twenty times over , as if I had somewhere said A point is nothing ; only because I say with Euclide , 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 . Fiftly 't is his usuall language , in designing an angle , to say , it is contained or comprehended by or between the two sides : As for example Cap. 14. § . 9. ( three times in two lines ) idem angulus est qui comprehenditur inter AB & AC , cum eo qui comprehenditur inter AE & AF , vel inter AB & AF , And § . 15. cor . 1. angul●s comprehensos a duabus rectis . And § . 16. angulus qui continetur inter AB & eandem AB &c. ( And 't is well enough so to speake . ) But now , forgeting that it was himselfe that sayd so , he delivers it as a principle of ours , That the nature of an angle consists in that which lies between the lines which comprehend it ; that is , saith he , An angle is a superficies . Sixtly ; when he had said ( absurdly enough , ) Cap. 11. § 5. Consistit ratio antecedentis ad consequens , in Differentia , hoc est , in ea Parte majoris qua minus ab eo superatur ; sive in majoris ( dempto minore ) Residuo , &c And again , Ratio binarii ad quinarium est ternarius , &c. And Cap. 12. § 8. Ratio inaequalium ( linearum ) EF , IG , consistit in differentia GF , ( and the like elsewhere . ) which is all one as to say , that the nature of proportion consists in a number , a line , an absolute quantity ( which how absurd it was I had let him know ; ) He hath then the impudence to say ( as though it had been I , not hee , had so spoken ) that , I make Proportion to be a Quotient , a number , an Absolute quantity , &c. or , as he here speaks in his sixth charge , that the Quotient is the proportion of the Division to the Dividend , ( as pure non-sense as a body need to read ; ) Only because I affirm Rationis ( Geometricae ) aestimationem esse , not penes residuum , but penes quotum : that Geometricall proportion is to be estimated , not according to the Remainder , but according to the Quotient ( which himselfe now knows , though he did not then , to be true enough ; for he hath now learned to say so too , Less . 2. p. 16. As the Quotient gives us a measure of the Proportion of the Dividend to the Divisor in Geometricall Proportion ; so also the Remainder after Substraction is the measure of Proportion Arithmeticall . ) And by these means he goes about to prove himselfe an honest man : Just like the honest man , who when he had cut a purse , put it slyly into another mans pocket ( after he had taken out the mony ) that so this other might be hanged for it . And I hope , by that time Your Lordship hath perused the peece which I now tender , you will be able to judge , whether M. Hobs be not as well a good Mathematician , as an Honest Man ; much alike . Your Lordship hath now the case fair before you ; if you shall think it worth the while to take cognizance of it . I shall leave it here , and permit it to your Lordships judgement , whether to peruse and consider it , ( which by reason of your good accomplishment in these , as well as in other parts of Learning , you are well able to doe , ) or to lay it by for those that will : as being unwilling , by any importune solicitation , to trespasse upon your Lordships leasure , or divert your thoughts , from matters of more concernment , to consider of such toys as these . Desiring mean while your Lordships favour so far , as to give mee leave to honour you , and ( though I have not hitherto had the honour to be known to you ) to subscribe my selfe , MY LORD , Your Honours Most Humble Servant , John Wallis . Oxford . Oct. 15. 1656. DVE CORRECTION for Mr HOBS . SECT . I. Concerning his Rhetorick and good Languge . IT seems , M. Hobs , ( by the fag end of your Book of Body in English ) that you have a mind to say your lesson ; and that the Mathematick Professors of Oxford should heare you . Truth is , 't is scarce worth the while either for you or us . Yet we could be contented , for once , to hear you ; ( if we thought you would say any thing that were worth hearing ) But to make a constant practice of it , or to entertain you as one of our Schollars , I have n● mind at all . Because , I fear , you are to old to learne , ( though you have as much need as those that be yonger ; ) and yet will think much to be whipt , when you doe not sa● your Lesson right . But , before we go further , I should ask you ; what moved you to say your Lessons in English , when as the Books , against which you doe chiefely intend them , were written in Latine ? But I foresee a faire answer that you might possibly make ; ( and therefore doe nor much wonder at it . ) There be many grave and weighty reasons that might move you thereunto . As first , because you doe presume , that there may be found divers persons , who may understand rayling in English , that yet doe not understand Mathematicks in Latin : and those being the persons on whom you have greatest hope of doing good , you ought to have a speciall regard to them , and apply your selfe to their capacities . Secondly , because in case you should have attempted an Answer in Latin ; you had lost your labour as to the whole design : For then those who should read your answer , would be able also to read that against which you write : and , comparing both together , would presently see to how little purpose all is that you have said . Whereas now your English Readers must be faine to take upon trust what you please to tell them . ( Whereby you gain clearly , as to them , the opportunity of misrepresenting at pleasure what you see good . ) And for this Reason , if you shall think fit to make any reply to this ; I would advise you to doe it in Latin ; that so Forrainers , who understand not English , may take upon trust what you shall please to tell them . But thirdly , and principally ( which is the reason of greatest weight ) because that when ever you have thought it convenient to repaire to Billingsgate , to leane the art of Well-speaking , for the perfecting of your naturall Rhetorick ; you have not found that any of the Oister-women could teach you to raile in Latin , and therefore it was requisite to apply your selfe to such lauguage as they could teach you . But prithee tell me , in good earnest , ( for I cannot think you so simple as you would seem to be , ) Whether you doe indeed believe ( though you thought good to set a good face upon it , and talk big , ) that all that you have said is worth a straw , either as to the defending of your Reputation , or the impairing of ours ? As to the Rhetorick and good language of it , ( with which I shall first begin ) that you can upon all occasions , or without occasion , give the titles of Foole , Beast , Asse , D●gge , &c. ( which I take to be but barking , ) with the rest of your course complements : You may take them , perhaps , to be admirable in their kind ; yet are they no better then a man might have at Billingsgate for a box o' th ear . And of no better alloy are those other garnishes ; That we understand not what is Quantity , Line , Superficies , Angle , and Proportion : ( and truly that 's a sad case : ) That neither of us understand any thing either in Philosophy or in Geometry ; ( A lack a day ! ) That you do verily believe ( it's pitty you can't perswade some body else to be of your fai●h , ) that since the beginning of the World there hath not been ( and who doubts but you are a good Historian , ) nor ever shall be , ( and you hope your Prognosticks may be believed , for you would have us think you have been taken for a Conjurer , ) so much absurdity written in Geometry , as is to be found in these books of mine , ( you should alwaies except your own Learned Works , which doubtlesse are , in this kind , incomparable pieces . But the truth is , you are not altogether out here ; for in my Elenchus , which is one of the Books you mention , you may see that there hath been mach absurdity written in Geometry , and , they that read it , may know by whom . ) But you have confuted them wholly and clearly ( it seems you make cleare work where you come ) in two or three leaves , ( a quick riddance ! ) That , the negligences of your own you need not be ashamed of , ( because you are ashamed of nothing ; ) That you verily believe there was never seen worse reasoning , then in that Philos●phicall Essay , ( and that 's all the confutation of it : ) nor worse Principles then these in our Books of Geometry ; ( and that 's another Article of your Faith ) That , by the use of Symbols , and the way of Analysis by squares and cubes , &c. you never saw any thing added to the Science of Geometry ; ( by which a man may see what a good Geometer you are like to prove ; ) That the Scab of Symbols , or Gambols , ( your tongue is your own , you may call them what you please , ) or the Symbolick tongue is harder to understand then Welch or Irish , ( no marvaile then , you never saw any thing thereby added to Geometry . ) That , to confute your Learned labours , is but to take wing like Beetles , from your egestions ; ( it seems it was but a shitten piece we had to deale with . ) That , what you like not , is worthy to be gilded , but you doe not meane with gold ; That Symbols are pior unhandsome scaf , folds of Demonstration ; and ought no more to appeare in publike , then the most deformed necessary businesse which you doe in your Chamber ; ( one would think , by such stuffe as this , together with the ribauldry in your obscene Poem De Mirabilibus Pecci , that you had not learned all your Rhetorick at Billingsgate , but had gone to Turn-ball-street for part of it . ) That , your faults are not attended with shame , ( It 's no commendations , to be past shame ; ) That , you shall without our leaves be bold to say , ( who ever doubted but that you be bold enough ? ) that your selfe are the first that hath made the grounds of Geometry firm and coherent , ( as if Geometry were no lesse beholden to you , then Civil Philosophy ; which , you say , is not ancienter then your Book de Cive . ) That you have reason to blash ( not for any of your own faults doubtlesse , but ) considering the opinion men will have beyond Sea , of the Geometry taugh ' in Oxford , ( no doubt but the University of Oxford , if men knew all ; are much beholden to you for your tender care of them ; ) yet withall , that the third definition of the fift of Euclide , is as bad as any thing was ever said in Geometry by D. Wallis , ( And , if so , then doubtlesse D. Wallis need not be much dismaid ; for Euclide hath not been accounted hitherto a despicable Author . ) But such bumbast as this , and a great deale more of the same kind , I suppose , you doe not take to be Mathematicall demonstrations ; nor to prove any thing , but the Forehead and Fury of him that speaks it . But because the stresse of all this lies only upon what you verily believe , and what you never saw , and what you feare men will think of us beyond Sea : &c. To ease you of this fcar , I think it will not be amisse to let you heare the opinion of others both concerning your selfe and us , and the businesse of Symbols ( with which I see no reason why you should be so angry , save that you do not understand them . ) that you may see ▪ whether others haze the same belief with you . I need not tell you what Morinus and Tacquet think of the businesse . For those you have heard already . I shall only give you an extract of two or three Letters , which I have received from Persons whose face● I never saw ; nor were they otherwise engaged to deliver an opinion in the case , then that they met with my books abroad : And yet no Clergy men , He assure you . The first is from a Noble Gentleman of good worth , who hath deserved better of the Mathematicks then ever M. Hobs is like to doe ; and whom , I heare , you use to commend . His words are these . Eodem ibi tempore [ Paristis ] a Viro Nobili pagella vestra de Circuli Quadratura , Londino mittebatur ; simulque Hobbii Philosolosophia Nova . Quam ubi primum examinare concessum est , continuo Paralogismum eum animadverti , quo Parabolicae lineae rectam aequare contendit , calculoque refutavi . Deinde alia quoque notavi , quae nihilo saniora erant , authoremque ingenio minime defaecato praeferebant . Miror te hunç dignum judicasse quem tam prolixe refelleres . Etsi non sine voluptate Elenchum tuum pervolvi , doctum equidem atque acutum . You see he hath no great opinion of you : He finds you full of Paralogismes : He takes you to be a man of a muddy brain ; and wonders only that I thought it worth while to foul my fingers abou●●uch a piece as yours . The other is a publick Professor of Mathematicks , of known abilities , and beyond exception ; and he speaks yet somewhat fuller to the whole businesse . Cum aestate praeterita in manus inciderit Thomae Hobbes Elementorum Philosophiae Sectio prima ; abs●inere non potui quin tractatum istum leviter evolverim . Instigabat me ad hoc , tum Authoris hujus celebritas , tum etiam quod plura in eadem tractatu offen debam Geometrica , quae si Philosophiam non excelerent , saltem ut quam maxime illustratura forent , opinabar . Sed me illum perlustrante , cum talia ibi invenerim ejus de Algebra sive Ana●ys● judicia , equibus mihi facile fuit colligere , quod Author hic in eadem Arte parum deberet esse versatus ; ( quandoquidem haec ill● Ars existit , ut si liber suus in Geometria egregii ac ardui quid contineret , qualia se passim invenisse praetendere mihi videhatur , id ipsum huic Arti , judicio meo , in totum deberet ; ) Cumque adhuc in perlustrando dum p●rgebam , non nulla de rectae ac curvae aequalitate , aliaque complura animadvertebam quorum cognitionem nunquam mihi pollicebar , ac inter seponenda not abam , vel certe si spos aliqua inveniendi illa mihi superesset , quin Algebram in partes vocarem non dubitabam : Aliam exinde de ipso ●pinionem concepi , credens quod illa quae illū ante e●proprio penu deprompsisse autumabam , non nisi aliorum inventa esse , sed in alium sensum ab eo traducta aut correpta : Ideoque siquid boni in eo comprehenderetur , id quam maxime esse ventilandum ac excutiendum ; ac proinde illius examen , si vel utile aut necessum judicarem , in commodius tempus mihi esse differendum . Quemadmodum autem haec ita conceperam , ita quoque evenit ut amicus , cui me eo tempore invisenti dictum tractatum exhibueram , falsitatem plurium illius propositionum haud longe post invenerit , illasque uno folio coram omnibus exponere decreverit . Qui edere ista utiliter rotus , ubi se ad hoc accinxerat , tuum interim , vir ▪ Clarissime , Elenchum in lucent proditum vidit , ac postquam te isto munere optime defunctum deprehendit , a proposito suo destitit . Egregie autem te eum , Vir Clarissime , sed pro merito tamen excepisse ibidem agnovi , ita ut credam eum in posterum a te prudentiorem doctioremque factum , licet ille tibi nullas gratias ( judicio meo ) pro beneficio isto sit habiturus , Inter illa quae in Elencho tuo offendi , nihil expectationem majorem mihi excitavit , quam Arithmetica tua Infinitorum , de qua subinde mentionem facis : Quam novissime in lucem proditam , quamprimum cum caeteris tuis tractatibus vidi , mihi comparavi , ac multa praeclara & ingeniosa inventa , qualia mihi proposueram , continere deprehendi . Perpl●● et autem quod tum in Arithmetica tua Infinitorum , tum in Sectionibus tuis Conicis pertractandis , calculum Geometricum ubique adhibueris , tum propter brevitatem , tum quod is ( ut ipse mones ) demonstrationum omnium fons existat , atque demonstrationes omnes , solenni modo factae , certa arte ex illo confi●i possint . Id quod prae aliis Clarissimus D. des Cartes in Demonstrationibus suis est molitus , qui neglecta Theorematum ac Lemmatum longa serie , quibus alias in demonstrando difficulter carere liceret , calculo omnia constare voluit ; atque in eum finem passim aequationes investigat , quibus rei veritas , ac quomodo illa cognosci possit , absque verborum involueris , breviter atque perspicue ob oculos ponatur . Quae autem de Circuli quadratura tradis , utrum scilicet rem acu tetigeris necne nondum examinare mihi contigit : subtilissime autem cum illam prosecutus mihi videaris , atque etiam calculo ipsam inquisiveris , non dubito quin omnium saltem proxime atque accuratissime ad scopum collimaveris . You see what he thinks of you , and mee , and Symbols . He discerns presently by your judgement of Algebra , what a Geometer you are like to prove ; that it must needs be one who understood it not , that rants at that rate ; and will yet talke of squaring a Circle , and find a streight line equall to a crooked , and other fine things , without the help of Algebra . He sees by a little what the rest is like to prove ; either little worth , or not your own . And therefore , though at first he made hast to get it , yet when he sees what is in it , he thinks your book may well be thrown aside , or at least be examined at leisure . He tells you of another , that , had not my Elenchus prevented him , meant to have been upon the bones of you . He tells you , that my Elen●hus , as sharp as it is , is no more then you had deserved . He supposed withall ( though therein it seems he was deceived ) that you would have learned from thence , more Mathematicks , and more discretion for the future ; and yet did believe ( as well he might ) 〈◊〉 you would scarce thank me for that favour . He is well enough satisfied also with my other Pieces , ( what ever you think of them , ) and likes them never the worse for that Scab of Symbols ( as you call it ) but much the better ; ( because , though you understand them not , he doth . ) And much more to that purpose . And by this time , I hope , you be pretty well eased of your feare , least the University of Oxford should suffer in the opinion of Learned men beyond Sea , by reason of the Mathematicks that we have written . ( Nor have you reason to think , that Malmesbury , will be much the more renowned for your skill in that kind . ) And , that you may not despise their Testimonies , the persons are very well known to the World , by what Works they have extant in Print , to be no contemptible Mathematicians . Beside these , I shall , for the satisfaction of your English Readers ( who perhaps may not so well understand the words of the Authors above mentioned , ) adde an extract of one Letter more ; from a noble Gentleman , whom as yet ( to my knowledge ) I never saw , nor had formerly any the lest intercourse with him by letter or otherwise , though I had before heard of his worth and skill , both in Mathematicks and other learning : And which is more , he is neither of the Clergy ; nor any great Admirer of them , beyond other persons of equall worth and Learning . He was pleased , though wholly a stranger to mee , upon view of my Elenchus , to intimate to me by a Letter directed to a third person , That D. Wallis had unhappily guessed , that those propositions which M. Hobs had concerning the measure of Parab●lasters , were not his own , but borrowed from some body else without acknowledging his Author : and signified withall , that they were to be found demonstrated in an exercitation of Cavallerius , De usu Indivisibilium in Potestatibus Cossicis ; ( a piece which I then had never read : ) And that M. Hobs , endeavouring to demonstrate them anew , had missed in it . For which civility from a person of Quality , to mee a meer stranger , I could doe no lesse then returne him a civill answer of thanks for that favour . In reply to which ( having in the mean time seen and perused my Arithmetica Infinitorum ) he was pleased to honour me farther with this . I had not so long deferred &c. but that &c. And I beseech you receive it now from a Person , who much honours your eminent Learning and Humanity , and would egerly imbrace an occasiō to give you most ample testimony of the esteem he hath for you . I had not , ( before &c. ) seen your Arithmetica Infinitorum , which alone , although your other labours were not taken in to make up the value , may equall you with the best deservers in the Mathematicks . I was before acquainted with many excellent Propositions therein by you demonstrated ( as you partly know , ) but admired them , there , as wholly new , not because you had demonstrated them only another way , but by a generall method , so little touched at by others , so in effect wholly new , and of so rare consequence for entring into the secrets and Soul of Geometry ( if my judgement may passe for any thing ) as truly I believe the Art may reckon it among the most confiderable advances given it . Sir , I wish all prosperity to your deservings , and humbly thank you for the fair admittance you have given me to your acquaintance and friendship , which I shall preserve with a tendernesse due to a thing so estimable ; and believe , Sir , you have Power at your own measure in Yours &c. This is English , and therefore needs no exposition ; your English Reader , whether Mathematician or not , may understand it without help . You see all are not of your opinion concerning my scurvy book of Arithmetica Infinitorum . I will not trouble your patience with reciting more testimonies in this kind ; ( though , the truth is , very many persons of Honour and Worth , and eminent for their skill in these studies , have been forward of their own accord to put more honour upon me in this kind , then were fit in modesty for me to own . ) These you have heard already , are more , I presume , then you take any great content in ; and the lest of them , were abundantly sufficient to outway your verily believe ; upon the strength of which , you have the confidence to utter all those reproaches which in your scurrilous piece you endeavour to cast upon us ; but find them to fall back , and foul your self . And you see withall , both how little reason we have to fear the opinion that men will have beyond Sea , of the Geometry taught at Oxford ; and with how much vanity it is that you tel us according to your Rhetorick , that when you think , how dejected we will be for the future ; and how the grief of so much time irrecoverably lost , and the consideration of how much our friends will be ashamed of us , will accompany us for the rest of our life , you have more compassion for us then we have deserved . No doubt Sir , but you are a very pittifull man ! ( who have so much compassion for us : ) And we are much bound to behold you . But since your cōpassion of us , is not only more then you think we deserve , but , likewise , more then we think we stand in need of ; we are loath your good nature should be injurious to your selfe . And therefore , knowing how much your selfe at present nay need compassion , we desire you to suffer that charity to begin at home , and not to be too lavish of that commodity upon us , of which at present we have so little need and you so much . But , that there may be no love lost between us ; know , that we have the like compassion for you , upon the same account . You have but prevented us ; and taught us , by your extreme civility , what might have better beseemed us to say . You tell us somewhere , the reason , why the Ladies at Billingsgate , amongst all their complements , have none readier then that of Whore , because , forsooth , when they remember themselves , they think that likeliest to be true of others . And truly , we have reason to believe , that the anguish of such considerations as those you mention , being so frequently present to your own thoughts , makes you so apt to think that others may be tormented in the like manner . ( For who are more compassionate to those that feele the toothach , then those that are most tormented with it themselves ? ) For , as your words are elsewhere , A man of a tender forehead , after so much insolence , and so much contumelious language as yours , grounded upon arrogance and ignorance , would hardly endure to outlive it . As for our selves ; I do not find , that our friends do yet disowne us ; or , that we need to feare , in this contest , the fury of our foes . And , whatever diseases you may believe my Conick , Sections , and Arithmetica Infinitorum , to be infected with : I do not see , that wiser Physitians can yet discerne , either the one to be troubled with the Scab , or the other with the Scurvy . But you tell us , ( and that may serve for answer to the Testimonies but now recited ) Though the Beasts , that think our railing to he roaring , have for a time admired us ; yet , now that you have shewed them your eares , they will be lesse affrighted . Sir , those Persons ( as they needed not the sight of your eares , but could tell by the voice what kind of creature brayed in your books : so they ) doe not deserve such language at your hands : And , you would not have said it to their faces . I know your Apology will be , that you say it provoked ; and that by Vespasians law , when a man is provoked , it is not uncivill to give ill language . And that we may know you have been provoked , you tell us , how hainous and hazardous a thing it is , to speake against some sorts of men , whether that which is said in disgrace be true or false ; And by all men of understanding it is taken ( not only for a provocation , but for a defiance , and a challenge to open Warre . And truly , so far as that may passe for Law , I cannot deny but that you have been provoked ; for sure it is , that much hath been said against you , and that , as is supposed , to your disgrace , and , I believe , the provocation hath been the greater , because that which hath been said , is true . But is this such a provocation as may warrant you , by Vespasians Law , to rave at the next man you meet with ? and to revenge your selfe upon him that comes next ? Is it such a provocation of M. Hobs , for any man to admire us , that he may thenceforth , without incivility , be called a Beast , or what you please ? Is it not enough for you to involve the two Professors in the same crime , and consider us every where as one Author , and therefore both responsible , joyntly and severally , for what is said by either , because forsooth , we approve , you say , of one anothers doctrine : but must all that doe but admire us be under the same condemnation ? It 's possible that some of them may admire our folly ; ( you see , one of them wonders at my discretion , that I would foule my fingers with you , or think you worth the Answering : ) must they be called Beasts also ? It seems 't is a dangerous businesse for a man to admire any who do not admire you . But I have done with the Rhetorick and good Language . We have had a tast of it ; and that 's enough unlesse 't were better . They that desire to have more of it , may either read over your book , or goe to Billingsgate , whether they please . But when men shall heare you rant it after this rate , and talk high ; surely they must needs think , that you have very good ground for it , must they not ? A shallow foundation would never bear a confidence of such a towring hight . One would hardly believe mee , if I should say , That notwithstanding these Braggadocian words , there is not any one assertion of mine , that you have either overthrown or shaken ; nor any one of your own ( which I charge to be false , ) that you have defended ; Yet that 's the case . A great cry , and a little wooll ! ( as the man said when he shore his hoggs . ) Parturiunt Montes . — And that 's it we have next to shew . SECT . II. Concerning his Grammar , and Criticks . I Shall therefore next after the Rhetorick , consider the Grammar , you 'l say , that Grammar should have gone first . It may be so . But it 's no great matter for method , when a man deales with you ; for you are not so accurate in your own , that you need find fault with anothers . There be six or seven places ( and , I think no more ) where you would play the Critick . First , you tell me pag. 11. that [ Punctum est Corpus , quod non consideratur esse Corpus ] is not Latin , nor the version of it [ a Point is a body , which is not considered to be a body ] English . If you had said , it had not been good sense , I would have agreed with you . But why not that , Latin ? or this , English ? ( Nay stay there ; you are not to give a reason for what you say . It 's enough that you say so . ) Quod esse videmus , id videtur esse . Quod esse sentimus , id sentitur esse . Quod esse putamus , putatur esse . Quod esse cognoscimus , cognoscitur esse . Quod esse dicimus , dicitur esse . And why not as well , Quod esse consideramus , consideratur esse ? But what should it have been , if not so ? Why thus , Punctum est corpus quod non consideratur ut corpus . Very good ! Bur Sir , It 's one thing , to consider a thing as a body , or as if it were a body , ( either of which the words ut corpus may beare ; ) another thing , to consider that it is a body , which was the notion I had to expresse , and therefore your word would not so well serve my turne , but rather the other . And when we have this to expresse , That though it be a body , and we know it to be a body , yet do not at present actually consider it so to be ; ( which I take to be neither Irish , nor Welsh , nor , which is worse then either , the Symbolick tongue ; but good English ; ) it is better rendred in Latin by esse , then by ut . Secondly , you tell me pag. 44. I might have left out [ Tu vero ] to seek an [ Ego quidem . ] ( As though vero might never be used where there is not a quidem to answer it . ) And is not this a worthy objection ? But however , to satisfy you , look again and you may see a quidem which answers directly to this vero . My words are these Articulo quarto ( cap. 17. ) curvilineorum illorum descriptionem aggrederis per puncta . Quae quidem res est non ita magnae difficultatis , ut tanto apparatu , tantisque ambagibus opus sit . Exempli gratia . &c. Tu vero , quasi per planorum Geometriam id fieri non possit , statim imperas mediorum quotlibet Geometricorum inventionem . Doe you see the quidem now ? Very good ! But before I leave this , ( to save my selfe that labour anon , ) I must let your English Reader see , how notoriously you doe here abuse him , ( him , I say ; for the abusing of me in it , is a matter of nothing ) My words were these ; In the 4th Article ( of your 17 Chap. ) you attempt the describing of those curve lines by points , ( that is , the finding out as many points as a man pleaseth , by which the said curve lines are to passe , through which , with a steady hand , those lines may be drawn , not Mathematically , but by aim , ) which is a matter of no great difficulty , and may be performed without so much adoe as you make , and so much going about the bush . As for example , &c. ( and so I go on to shew how those points may be easily found Mathematically , by the Geometry of Plains , that is , by the Rule and Compasse , or by streight lines and circles , without the use of Conick Sections , or other more compounded lines . And , having shewed that , I proceed thus ) But you , as though this work ( the finding of those points ) could not be done by the Geometry of Plains , ( as I had shewed it might , ) require presently the finding of as many mean proportionals as you please ( viz. more or fewer according as the nature of those lines shall be ; ) between two lines assigned : ( which by the Geometry of Plains cannot be done : ) And so , of a Plain Probleme , you make a Solid and lineary Probleme . Which how unbeseeming it is for a Geometrician to doe , you may learne from those words which your selfe cite out of Pappus , pag. 181. ( in the English , pag. 233. ) Videtur autem non parvum peccatum esse apud Geometras , cum Problema Planum per Conica aut Linearia ab aliquo invenitur . It 's judged by Geometers no small fault , for the finding out of a Plain Problem , ( as this is , ) to have recourse ( as you here ) to Solid or Lineary Problems . Now these words , one would think , were plain enough for a man of a moderate capacity to understand . And is it not well owl'd of you , to perswade your English Reader that I had here taught , that a man may find as many mean-proportionalls , as one will , by the Geometry of Plaines ? ( where I said only that the work before spoken of , might be done by the Geometry of Plaines , and therefore needed not the finding of such Mean-proportionals ? ) And then ( because you doe not know whether or no , as many mean proportionals as one will , may be found by the Geometry of Plains , ) you tell us , that you never said it was impossible ; ( truly if you had said so , I should not have blamed you for it ; ) but that the way to doe it was not yet found , ( you might have added , nor ever will be , ) and therefore it might prove a Solid Problem for any thing I know . Nay truly , Sir , I know very well ( though it seems you doe not , ) that it is at lest a Solid Probleme , or rather Lineary ; and that the way to doe it , Mathematically , by the Geometry of Plains , is neither yet found , nor ever will be . For those Problems which depend upon the resolution of a Cubick or Superior Aequation , not reducible to a Quadratick , ( which is the case in hand ) can never be resolved by the Geometry of Plains . Which , if , instead of scorning , you had endeavoured to understand , the Analyticks , you might have known too . But this by the way ; to save my selfe the labour anon . I returne to your Criticks again . Thirdly , whereas it is said c. 16. art . 18. Longitudinē percursam cum impetu u●ique ipsi BD aequali ; I said the word cum were better out , unlesse you would have Impetus to be only a Companion and not a cause . For where a causality is imported , though we may use with in English , yet not cum in Latin. To kill with a sword ( importing this to have an instrumentall or causall influence , and not only that it hangs by the mans side , while some other weapon is made use of ) is not in Latin Occidere cum gladio , but gladio occidere . So ebrius vino ; pallidus ira ; incurvus senectute , or , if you will , prae ir● , ob●iram , &c. not cum vino , cum ira , &c. You say , it is better in ( though for the most part your selfe leave it out in that construction ; ) let the Reader judge ; for it is not worth contending for . All that you say in defence , is that Impetus is the Ablative of the Manner . What then ? the question remains , as it was before , whether this Modus do not here import a causall influence ? And 't is evident it doth ; for the effect here spoken ( that such space be dispatched ) doth equally depend upon two causes ; the one , that the motion be uniforme ; the other , that the Impetus be so great . And therefore ( since you please to insist upon it , which I did but give a touch at by the way , as in many other places where you take it Patiently , ) cum not proper in either place ; but either an Ablative without a Preposition ; or , if you would needs have a preposition , per , prae , pro , propter , ob , or some other which do import a Causality ; not cum , which imports only a Concomitancy . Fourthly , you say , pag. 61. That you think , I did mistake [ praetendit scire ] for an Anglicisme . Your words were these at first , ( as that Paragraph was first printed , pag. 176. ) tamen quia tu id nescis , nec praetendis scire praeter quam ex auditu , &c. as appears in the torne papers . And then , ( after you had new modeld that whole Paragraph , as it now is pag. 174. ) tamen quid id nescit , nec praetendit scire &c. This I did and doe still take ( not mistake ) for an Anglicisme And you cannot deny but that it is so . Where is the mistake then ? You say t is a fault in the Impression . Yes that it is ; and that twice for failing . But was it not a fault in the Copy first ? you say it should have been , praetendit se scire . That , I confesse , helps the matter a little . But why was it not so ? The Printer left out se ( ●es , at both places . ) And why ? but , because the Author had not put it in ? In like manner pag. 222. Tractatus huius partis tertiae , in qua motus & magnitud● per se & abstracte consideravimus , terminum hic statuo . This was the Printers fault too , was it not ? or , at least , a fault in the Impression ? ( Beside much more of the like language up and down ) And if you think it worth while to make a catalogue of such phrases ; tell me against next time , and I shall be able to furnish you with good store . There be two places more ( to make up the halfe dozen ) wherein you would faine play the Critick : of which , I heard from divers persons , you made much boast , long before your book came out ; that you had D. Wallis upon the hip ; &c. The one was that adducere malleum was no good Latin , because that duco and adduco were words not used but of Animals , and signified only to guide or leade , not to bring or carry . The other was , that I had absurdly derived Empusa from 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 & 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 . It 's true , those charges , notwithstanding your first confidences , are not now layd in these words ; but the former extenuated , and the latter vanisht . Yet some nibling there is at both . The former of these , ( which I make the fifth in order ) is pag. 51. where you tell me , that Adducis malleum , ut occidas muscam , is not good Latin ? But why ? when we speak of bodies Animate , Ducere and Adducere , you say , are good . T is very true . Did any body deny it ? But are they not good also , of Bodies inanimate ? or other things ? ( I exspected , that in order to the confutation of my phrase you should have told us , in what cases they had not been good , and that this was one of those ; not , in what other cases they are good , as well as this ; for that hurts not mee . ) May they not be applied as well to a Hammer , as to a Tree ? Though this be Animate ; not that ? You were , I heare , of opinion , when you first made braggs of this notion , ( or else your friends belye you ) that they were not to be used but of bodies Animate : But , that being notoriously false , some body it seems had rectified that mistake , and informed you better , and therefore you dare not say so now . But why , now , is not adducis malleum good Latin ? Because , forsooth , Ducere and Adducere , when used of bodies Animate , signify to guide or lead ; ( and sometimes they doe so . ) Now though a ninny may lead a ninny , yet not a hammer . Very witty ? But I am of opinion , that he who leads M. Hobs , leads both . Or however , if a man may not lead a hammer ; yet , I hope , he that hits the naile at head ( which M. Hobs seldome doth ) may be said to guide his hammer : may he not ? The phrase therefore is good , even by your own law . But heark you , man ; to lead , you told us , is the signification of the word , when it is used of Animates ; why then do you talke of leading a hammer ? do you take the hammer to be animate ? or would have us take you to be the ninny ? But farther ; they singify you say to guide or lead . What then ? did I say they do not ? Prithee tell me , what they do not signify ; not , what they doe ; if you meane to overthrow my use of the word . T is true , sometimes they signify to guide , or lead ; viz. with the parties consent , ( Fata ●●lentem , du●unt , nolentemtrahunt : ) yet sometimes the quite contrary ; as ducere captivum , Claud. Cic. to take a man Prisoner , or carry him captive , against his will : so ducere in carcerem ; ducere ad supplicium , & deducere , Cic. to bring or carry a man to Prison , to execution , &c. ( which for the most part is against his will. ) Filia vi abducta , Ter. my daughter was carried away by force . And so , frequently . But suppose they doe , sometimes , signify to guide , sometimes to lead ; what then ? doe they signify nothing else ? Is Ducere lineam , Plin. to guide a line , or to lead a line ? and not rather to draw a line ? Ducere ux●rem , Cic. to guide a wife ? or to lead a wife ? ( though perhaps you will cavill at that phrase ) and not rather to take a wife ? But you say , Of bodies inanimate Adducere , is good for Attrahere , which is , to draw to . Very good ! But what is it not good for ? is it good for nothing else ? Ducere somnos , soporem , somnium , Virg. Hor. Insomnem ducere noctem . Virg. Ducere somno diem , noctem ludo ; sic horam , horas , tempus , aestatem , aevum , adolescentiam , senectutem , vitam , aetatem , coenam , convivium , &c. ducere , producere , traducere , Hor. Virg. Claud. Propert. Ovid. Cic. Sen. Plin. Liv. &c. Do they signify , to leade , to guide , to dran ? and not rather , to spend , to continue , to passe over , to passe away , &c. Well! but however ▪ ( whatever they may signify else , ) Duco , adduco , &c ( with the rest of its com●ounds , ) you would have us believe , ( for that 's it you drive at , though you dare not speak it out , or be confident to affirme it , ) do not signify to take , carry , fetch , or bring , ( which you suppose to be the sense ( aime at ) unlesse when used of bodies animate . But that 's as false as can be . Adducere fehrem Hor. Adducere Sitim , Virg. Adducere vini taedium . Plin &c. doe they signify to lead a fever ? or to guide a fever ? or to draw a fever , ( with cart-ropes , or a team of horses ? ) and not rather to bring a fever , &c. In my Dictionary , duco & adduco , signify to bring , as well as to draw . The truth is , duco , with its compounds , is a word of as great variety and latitude of signification , as almost any the Latine tongue affords . And , amongst the rest , to bring , fetch , carry , take , ( to , from , about , away , before , together , asunder , &c. according as the praeposition wherewith it is compounded doth require ) is so exceeding frequent in all Authors ( Plautus , Terence , Tully , Caesar , Tacitus , Livy , Pliny , Seneca , Virgil , Ovid , Horace , Claudian , &c. ) that he must needs be either malitiously blind , or a very great stranger to the Latin tongue , that doth not know it , or can have the face to deny it . Rem huc deduxi . Cic. Res eo adducta est , ( deducta perducta , ) in eum locum , in eum statum , in dabium , in certamen , in controversiam , in periculum , in maximum discrimen , &c. Cic. Liv. Caesar . Plancus ad Cic. &c. Add●cta vita in extremum . Tacit. Adducta res in fastidium , Plaut . in judicium , Cic. rem ad mucrones & manus adducere , Tacit. Contracta res est & adducta in augustiam . Cic. Rem co producere . Cic. Ad exitum , ad culmen , ad summum , ad umbilicam , ad extremum casum , &c. Cic. Caesar . Liv. Hor. &c. That is , The matter is brought to that passe , &c. So , Sive enim res ad concerdiam adduci potest , sive ad bonorum victoriam &c. Cic. So , ex inordinato in ordinem adduxit , ( speaking of Gods bringing the World out of the first Chaos , ) and again , Eas primum confusas , postea in ordinem adductas mente divina . Cic. So , aquae ductus , aquarum deductio , rivorum a fontibus deductio , aquam ad utilitatem agri deducere , Cic. Aquam ex aliquo loco perducere , Plin. In urbem induxit , idem . To bring water from place to place . ( not to draw it , attrahere . ) Thus adducere febres , to bring fevers . Officiosaque sedulitas , & opella forensis Adducit febres & testamenta resignat . Hor. So , Ova noctuae , &c. tadium vini adducunt . Hor. Addua●ere sitim tempora , Virgi●i , [ sc . aestiva ) Hor. In like manner , febres deducere , to take them away . Non domus & fundus , non aeris acervus & auri , Aegroto domini deduxit corpore febres , Non animo curas . — Hor. So , deducere fastidium . Plin And then Febrimque reducit , Hor. to bring back again . So , Frondosa reducitur aestas . Virg. Luctus fortuna reduxit . Claud. Reducere exemplum , libertatem , morem , &c. Plin. Aurora diem reduxit . Virg. Collectasque fugat nubes solemque reducit . Virg. that is , restoreth . So , Reducere somnum , Hor. Spem mentibus anxiis reducere Idem . In memoriam reducere , Plin. Cic. Now it would be hard to say , that in all these places Adduco , Deduco , Reduco , &c. are put for Attrabo , Detraho , Retraho , &c. Attrahere febres , attrahere taedium , &c. So Abduxi ●lavem , Plaut . I took or brought away the key ( as had every whit , as adducere malleum , to bring a hammer . ) So Navis a praedonibus abducta , Cic. Ter. The ship taken at Sea by Pyrats , and carried away . Visaque confugiens somnos abduxit imago . Ovid. So ( speaking of Hercules loosing the chains whereby Prometheus was chained to the rock ) Vincula prensa manu saxis abduxerat imis Val. Flac. Quidsi de vestro quippiam orem abducere ? Plaut . What if I should desire to carry away somewhat of yours ? Coeperat intendens , abductis montibus , unda Ferre ratem . Val. Flac. Abducti montes , id est , semoti . — abductaque flumina ponto . Idem . Quod ibidem recte custodire poterunt , id ibidem custodiant ; quod non poterunt , id auferre atque abducere licebit . CIC. Where abducere , all along , is no more then auserre . In like manner , conducere is oft times the same with conferre , congerere . As Veteres quidem scriptores hujus artis , unum in locum conduxit Aristoteles . Cic. Partes conducere in unum , Lucret. ( i. e. in unum corpus componere . ) So deducere , to carry forth . Ducere , deducere , producere , funus , exequias . Plin. Virg. Stat. Lucan . Deducunt socii naves , Virg. And to take away , ( the same with tollere , demere , auferre , ) as in deducere febrem , deducere fastidium , as before . Thus deductio and subtractio for as you use to call it both in English and Latin , Substractio , as if it came from sub and straho ) is contrary to Additio , and signifies all kind of Ablation or taking away . Addendo , deducendoque , videre quae reliqui summa fiat . Cic. Vt , deducta parte tertia , deos reli●ua reddatur Africanus de pactis dotalibus . Vt centum nummi deducerentur . Cic. Sibi deducant drachmam , reddant caetera . Cic. ut beneficia integra perveniant , sine ulla deductione . Sen. So , deducere cibum . Ter. to abate , diminish , or take away ; as also , Cibum subducere , Cic. Subducere vires , Ovid. ●t succus ●ecori & lac subducitur agnis . Virg. Jam mihi subduci facies humana videtur . Ovid. Ignem subdito ; ubi ebullabit vinum , ignem subducito . Cato de re rust . Aurum subducitur rerrae . Ovid. So , Annulum subduco . Plant. Subducere pallium , Mart. to take or steale away . Deducere vela , deducere carbasa , Ovid. Luc. — primaque ab origine mundi , Ad mea perpetuum deducite tempora carmen . Ovid. That is , To bring down from the beginning of the World to his own times . — a pectore postquam Deduxit vestes . Ovid. Deducere sibi galerum , vel pileolum , Sueton. to putt off , or take off . Et cum frigida mors animâ subduxerat artus . Virg. — Seductae ex aethere terrae . Ovid. Where seducere , is no more but separare . So , diducta Britannia mundo . Claud. Ante se fossam ducere & jacere vallum , Liv. Vallum ducere , Idem . fossam , vallum , praeducere , Tacit. Sen. perducere , Caes . to cast up a wall , a bank , a trench before them . Murum in altitudinem pedum sexdecim perduxit . duas fossas ea altitudine perduxit . munitio de castello in castellum perducta . Caesar . So , ducere muros , Virg. to raise up : and educere turrim . aramque educere coelo certant . sub astra educere . molemque educere coelo . idem . to raise up as high as heaven . Thus , educere foetum , Cic. Claud. Plin. Educere , producere , faetum , partus , liberos , sobolem , fructus , &c. Si●ius . Plaut . Hor. &c. To bring forth , So , Educere cirneam vini . Plaut , to bring out a flagon of wine , ( as bad , I trow , as adducere malleum . ) Educere naves ex portu ; and in terram subducere , Caesar . Vn●que conspecta livorem ducit ab uva . Hor. — arborea frigus ducebat ab umbra . Ovid. Animum ducere ( to take courage ) Liv. Ab ipso Ducit opes animumque ferro , Hor. Argumenta ducere , Quintil. Ducere conjecturam , similitudinem , &c. Cic. Initium , principium , exordium ducere . Cic. Ortum , originem ducere , Cic. Quint. Hor. ( i. e. sumee , ) Producere exemplum , Juvenal . Ducere cicatricem . Colum. Liv. Ovid. Cicatricem , crustam , rubiginem , callum , obducere . Plin. Cic. Obducere velum , torporem , tenebras , Plin. Cic. Quintil. Inducere , introducere , consuetudinem , morem , ambitionem , seditionem , discordiam , novos mores , Cic. Stat Plin. Qua ratione haec inducis , e●dē illa possunt esse quae tollis . Cic. Inducere formam membris , Ovid. Cuti nitorem , Plin. Tenebras , nubes , noctem , Ovid. Senectus inducit rugas , Tibul. Tentorium vetus deletum sit , novum inductum , Cic. Introducere , quod & in medium afferre , dicitur . Bud. Cic. Obliviae poenae ducere . Val. Flac. Sollicitae vitae , Hor. Nec podagri●us , nec articularius est , quem rus ducunt pedes , Plaut . ( whose feet can carry him , not lead , guide , or draw him . ) Transducere arbores , ( to transplant or remove from place to place , ) Colum . Quod ex Italia adduxerat . Caes . And if these Authorities be not enough ; it were easy to produce a hundred more , ( to justify my use of the word , and bring your new notion to nothing ; ) wherein Duco ( both in it selfe and its compounds ) signifies to take , bring , fetch , carry , &c. without any regard had at all to your notion of guiding , leading , or 〈◊〉 , that we may see what a deale of impudence and ignorance you discover , when you undertake to play the Critick . And when you have done the best you can , you will not be able to find better words then Adducere malleum , and Reducere , to signify the two contrary motions of the 〈◊〉 ; the one when you strike with it , the other when you take it back to fetch another stroke . To all these examples I might , if need were , adde your own which though it would be but as anser inter olores ; nor would it at all increase the reputation of the phrase , to say 〈◊〉 you use it : Yet it may serve to shew , that it is not out of i●dgement , ( because you think so ; ) but out of malice and a designe of revenge ( that you might seem to say somewhat , though to little purpose , ) that you thus cavill without a cause . For duco , adduco , circumduco , and the rest of the compounds , are frequently used by your selfe , in the same ●●nse and construction which you blame in mee . Lineam ●●cere , producere , &c. a puncto , ad punctum , per punctum , &c. are phrases used by your selfe fourty and fourty times . If 〈◊〉 do not seem to come home to the businesse ; that of ●um-effectum , rem a●i●uam &c. producere , ( to produce , ●ring forth , bring to passe , ) comes somewhat nearer ; which 〈◊〉 at lest twenty times in one page . p. 74. and within three leaves , ( cap 9 & 10 , ) above fifty times : and elsewhere frequently . So , actus educi poterit , p. 78. partes flui●●● educi ●osse . p. 258. deduci hinc potest . ( i. e. inferri ) p. 23. 〈◊〉 inde deducere non possum . p. 248. fluviorum origines 〈◊〉 possunt . p. 278. ratio quaevis ad rationem linearum reduci 〈◊〉 . p. 96. linea in se reducta p. 190. quibus & reduci cogi●●● nes praeteritae possint . p. 8. copulatio cogitationem inducit . p 20. n●men aliquod idoneum inducat . p. 52. phantasma finis 〈◊〉 thantasmata mediorum . p. 229. in animum inducere non 〈◊〉 p. 24● . Parallelismus ob eam rem introductus est . p. 246. 〈◊〉 instantia adduci potest . p. 82. And particularly of 〈◊〉 dies , in flectione laminae ( lege , flexione ) capita ejus addu●●●ur . p. 2●5 . flexio est , manente eadem lineâ , adductio extre●●●●●unctorum , vel diductio , p. 196. terminis diductis , ibid. 〈…〉 adductio extremarū linearum . p. 197. cujus puncta ext●ema diduci non possunt . p. 106. adductio vel diductio terminorum , ibid. and so again five or six times in that and the next page . So ex cujuspiam corporis circumductione . p. 4. corpus circumductum , ibid. si corpus aliquod circumducatur , ibid. in●elligi potest planum circumduci , p. 109. si planum circumducat●● , ibid. punctum ambientis quodlibet ab ipso circumducitur , p. 18● . and the like elsewhere . In all which places , by your law , it should have been circumlatio , circumlatus , circumfer●●● , circumferri , circumfertur , &c. as it is , p 50. p. 108. and 〈◊〉 some other places . Now if circumdaci and circumferri , 〈◊〉 be used promiscuously , and so circumductio and circum●●● , &c. why not as well in the same cases adducere and 〈◊〉 & c. ? And if corpus quodpiam , may , without absurdity , be 〈◊〉 circumduci , why not as well adduci ? In like manner , 〈◊〉 sum est conduci mobile ( i. e. simul ferri ) ad E ad A , concu●●● duorum motuum &c. p. 193. and moti per certam & design 〈◊〉 viam conductio facilis , p. 200. with many the like phras● which are every whit as bad as adducere malleum . And therefore , you had very little reason to quarrell at that phrase ; save that there was nothing else to find fault with , and somewhat you were resolved to say . And the like is to be said of that other phrase , next before , quod non consideratur esse corpus , which , though it be 〈◊〉 Latine , when I speak it ; yet , with you the same construction comes over and over again , as least a hundred times 〈◊〉 simulachrum hominis negatur esse verus homo , p. 23. qu● 〈◊〉 gantur esse verae . p. 26. singulae partes singulas lineas conficere ●●telligantur , p 68. si corpus intelligatur moveri , — redigi — 〈◊〉 escere , ibid. severall times intelligitur quiescere , — 〈…〉 70. agens intelligitur producere effectum , p. 73. du● 〈◊〉 intelliguntur transire , p. 87 ostenderetur ratio esse 〈◊〉 p. 100. lineae extendi intelligautur , p. 108. intelligatur radius ●●veri , p. 111. si partes fractae intelligantur esse minim● , p ▪ 11● ▪ supponatur longitudo esse , p. 131 , altitudo ponitur esse in 〈◊〉 basium triplicata , p. 153. sphaera intelligatur moveri , p. 〈…〉 haesio illa supponatur tolli , p. 188. intelligatur radius 〈◊〉 materia dura , ibid. vis magnetica invenietur esse motus , p ▪ 〈◊〉 ▪ And so punctum , corpus , res aliqua , ponitur , supponitur , inte●●●gitur , ostenditur , &c. esse , quiescere , movere , circum 〈…〉 &c. p. 62 , 64 , 68 , 75. 85 , 106 , 112 , 115 , 110 , 〈…〉 141 , 142 , 147 , 155 , 171 , 182 , 183 , 184 , 188 , 〈…〉 239. and many other places : which are every whit 〈◊〉 as consideratur esse . Yea and consideratur also is by your 〈◊〉 so used p. 87. Eaedem duae lineae — prout considerantur pro ipsis magnitudinibus — poni . &c. So that 't was not judgement , but revenge , that put you upon blaming this phrase also . And you care not , all along , how much you bespatter your self , ( for , you think , you cannot look much fouler then you doe already , ) if you have but hopes to be a little revenged on us . And truly you have that good hap all the way , that there is scarce any thing ( right or wrong ) that you blame in us , but the same is to be found in your selfe also with much advantage . But this fault ( adducis malleum ) you should not , you say , ( though it had been one , ) have taken notice of in an English man ; but that you find me in some places nibling at your Latine . Yes ; I thought , that was the matter . You had a mind to be revenged . And ha'nt you done it handsomely ? Was there nothing else to fasten upon with more advantage then these poor harmlesse phrases ? 'T is very well . It seems my Latine ( though as carelessely written as need to be ; for 't was never twice written , and scarce once read , before it was printed , ) did not much lye open to exception ; for if it had , I perceive I should have heard of it with both eares . But you are offended , it seemes , that I should offer to nibble at your Latine . And truly , if that were a fault , I know not how to help it now . I must needs confesse , I did some times ( when I stumbled upon them , but never went out of my way to seek them ; for , if so , I might have found enough ) correct some phrases , as I went along , ( sometime to make sense , where the sentence was lame ; sometimes to make it Latine , where the phrase was incongruous or barbarous ; ) because I did not know , that your being an English man , had given you a peculiar priviledge above others to speak barbarously without controll . Such as these , nescit , nec pratendit scire praeterquam ex auditu . p. 174. or as it was first printed . p. 176. nescis , ne●p praetendis , &c. And accipiat lector tanquam Problematice dicta . p. 181. And Placuit quoque ea stare quae merito pertinent ad vindicem , ibid. So p. 143. ( at lest in my book ) progressio stabit hoc modo , 0. 1. 2. 3. 4. &c. And diverse other places , which I do not now remember . But you know there be many more , which , had they come in my way , I might have found fault with , as well as these ; As that p. 37. falsae sunt , — & multa istiusmodi ( propositiones . ) And p. 116. definiemus lineam curvam esse eam cujus termini diduci posse intelligimus . And p. 111. quantitas anguli ex quantitate arcus cum perimetri totius quantitate compaeratione aestimatur . ( for ex quantitatis — comparatione , or ex quantitate — cōparata● p. 115. ducatur a'termino primae , ad terminos caeterarum , rectae lineae . And p. 222. partitertiae , in qua motus & magnitudo consideravimus , terminum hic statuo . And p. 224. Ex quo intelligitur esse ea ( phantasmata ) corporis sentientis mutatio aliqua . So p. 269. Exeuns , for exiens . and p. 3. exemplicatum esse , for exemplo explicatum , aut comprobatum . and p. 51. exemplicativum ; and many more of the same stamp ( as barbarous every whit , as those of the Schoolemen , which you blame as such , p. 22 ▪ non sunt itaque eae voces Essentia , Entitas , omnisque illa Barbaries , ad l'hilosophiam necessarius non est . ) I might adde that of p. 20. tanquam diceremus , ( as if we should say , ) and p. 22. tanquam possent , and elsewhere , instead of quasi , acsi , ( or some such word ) or tanquam si , which is Tullies phrase , ( tanquam si tua res agatur . tanquam si Consul esset . tanquam si clausa esset Asia &c. ) for tanquam without si ▪ signifies but as , not as if : But because I know you are not the first , that have so used it , of modern writers ; and that even of the ancients , some of them doe sometimes leave out si , ( as in other cases they doe ut ; ) I shall allow you the same liberty , and passe this by without blame ( as passable , though not so accurate . ) To these we may adde those elegances p. 32. ( syllogismus ) stabit sic . p. 49. sed haec dicta sint pro exemplo tantum , and So , p. 269. Ventus aliud non est quam pulsi aeris motus rectus ; qui tamen potest esse circularis , vel quomodocunque curvus . And a multitude more of such passages , ( which , were it worth while to collect them , might be added as an appendix to Epistolae obscurorum virorum , ) of which some are incongruous , some barbarous , some bald enough , and some manifest contradictions , or otherwise ridiculous But these are but negligences , as you call them , and therefore not attended with shame : for we doubt not but that , if you had particularly considered them , you could have mended them . Only , me thinks , he that is so frequent in such language , need not have quarrelled with such harmelesse phrases as adducere malleum , or consideratur esse . But I go on . The other place ( which makes up the halfe dozen ) you talked much of it at first , yet before it comes to be printed , 't is dwindled to nothing . It was , that I had derived your Athenian Empusa , from 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 and 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 ; and said it was a kind of Hob goblin that hopped upon one legge , ( which you take to be a clinch , forsooth , because your name is Hobs ; ) and hence it was that the Boys play , now a daies in use , ( fox come out of thy hole , ) comes to be called Empusa . This derivation you did , at first , cry out upon as very absurd ; and you meant to pay me for it : Till you were informed , as I hear , by some of your friends , that the Scholiast of Aristophanes ( as good a Critick as M. Hobs ) had the same ▪ ( and so have Eustathius , Erasinus , Caelius Rhodiginus , S●ephanus , Scapula , Calepine , and others : ) and therefore you were advised not to quarrell with it . Whereupon waving your main charge , you only tell mee ( pag. ult . ) that it doth not become my gravity , to tell you that Empusa , your Daem●nium Atheniense , was a kind of Hob-goblin , that hopped upon one legge ; and that thence a boys play , now in use , comes to be called Ludus Empusae . And withall , pray me to tell you , where it was that I read the word Empusa , for the Boys play I spake of ? To the Question , I answer , that I read it so used in Junius's Nomenclator ; Riders , and Thomas's Dictionary ; sufficient Authors for such a businesse . And then as for the Clinch you talk of , in Hobs and Hob-goblins , and the jest you suspect in Hobbius , and Hobbi , which you say , is lost to them beyond sea ; I hope that losse will never undoe mee : and when you can help me to a better English word for your Daemoniū , thē Hob-goblin ; or a better Latin word for Hobbes then Hobbius ( whose vocative case , in good earnest , is Hobbi , ) I shall be content , without any regret , to part with the jest , and the clinch too , to do you a pleasure ; Who tell us presently after , that you meant to try your Witt , to do something in that kind . And then shew your selfe as great a Witt , as hitherto a Critick . There is yet a Seventh passage , p. 14. which may be referred also to this place . The words Mathematicall definition do not please you Those termes or words , which do most properly belong to Mathematicks , we commonly call Mathematicall termes , and the definitions of such termes , in Mathematicks , Mathematicall definitions . And is it not lawfull so to do ? No , you tell us . But why ? Because it doth bewray another kind of Ignorance . What ignorance ? An inexcusable ignorance . How doth it bewray it ? It is a marke of ignorance ; of ignorance inexcusable . Ignorance of what ? Ignorance of what are the proper works of the severall parts of Philosophy . And , I pray , why so ? Because it seems by this , that all this while , I think it is a piece of the Geometry of Euclide , no lesse to make the Definitions he useth , then to inferre from them the Theorems he demonstrates . A great crime , doubtlesse ! But how doth it appeare , that I think so ? May not a man recommend Hellebor to you , as a good Physicall drug , ( because used in Physick , and proper for some diseases , ) unlesse he think , it is the Physitians work to make it , as well as to make use of it ? But suppose I do ; what then ? do you believe no body thinks so , but I ? or do you believe , that any body thinks otherwise but you ? Is it not proper for words of Art , ( voces artis , ) to be defined and explained in that art to which they belong ? is it not proper for a Grammarian to define Gender , Number , Person , Case , Declension , Coniugation &c. in the sense wherein they are used in Grammer ? And for a Logician to define Genus , Species , Vniversale , Individuum , Argumentum , Syllogisinus , &c. in the sense wherein they are used in Logick ? And may not those be called Grammaticall , and these Logicall definitions ? And for a Mathematitian , to define or tell what is a Triangle , a Cone , a Parabolaster , what is Multiplication , Division , Extraction of rootes , what is Binomium , Apotome , Potens duo media , &c. And may not these definitions be called Mathematicall ? No , by no means , you tell us , to call a Definition Mathematicall , Physicall &c. is a marke of ignorance , of unexcusable ignorance . ( And doe you not think then , that Gorraeus was a wise man , to write a large Volumne in folio , intituled Definitiones Medicae ? ) But why a marke of ignorance ? Because a Mathematitian , in his definitions teach you but his language ( not his art ) but teaching language is not Mathematick , nor Logick , nor Phisick , nor any other Science , ( but some Art perhaps , which men call Grammar . ) some men would have thought that to Define , had belonged to Logick ; but let it passe for Grammar at present . Do you think , nothing , is Mathematicall , wherein a man makes use of Grammar ? Can a man teach Mathematicks , in any language , without Grammer ? ( unlesse , perhaps , in the Symbolick Language , which is worse then Welsh or Irish . ) But you say , He that will understand Geometry must understand the termes before he begin : ( because a man ought not to go into the water , before he can swim . ) Well , But if not his Definitions , what then is it , in Euclide , that is Mathematicall ? it is , you tell us , his inferring from them the Theorems he demonstrats . ( And why not the solution of Problems also ; as well as the inferring of Theorems ? ) But to infer and to demonstradte , are , I suppose as much the work of Logick ; as , to define , is the work of Grāmar . And therefore , by the same reason for which you will not allow the Definitions to be Mathematicall , because to teach a language is the work of Grammar , you must also exclude the Propositions and Demonstrations , because to inferre and demonstrate , is the work of Logick . And so , nothing in Euclide will be Mathematicall . 'T will be Grammar and Logick , all of it . And are not these pure Criticismes ; think you ? Do not these wofull notions of yours , and the language that doth accompany them , shew handsomely together ? But enough of this . SECT . III. Concerning Euclide : and the Principles of Geometry . WE have seen your Elegances already , in the first Section , and then your Critsicismes in the second . It 's time now to look upon your Geometry . And I should here begin with your first Lesson ; but that , by what we heard even now , you will not allow me to call it Geometricall , or any peece of Geometry , consisting , as it doth , of Definitions . And yet , what ever the matter is , me thinks you come pretty neer it : for you call them Principles of Geometry . But you 'l say , perhaps , they be Principles of Geometry , but not Geometricall Principles , ( for to call any Definitions Geometricall , were as bad as to call them Mathematicall , which were a marke of ignorance unexcusable . ) Acutely resolved ! But , whatever else they be , Principles they are without doubt . For , as you define p. 4. A Principle , is , the beginning of something : And no man can deny , but that the first Lesson is a beginning of something : And therefore , a Principle . Now contra principia , we know , non est disputandum . I must take heed therefore , what I say here . In this Lesson , you take Euclide to task , and give him his Iurry : ( And when you have lesson'd him , it is to be hoped , wee will not think much to be lesson'd by you : ) And withall intermingle some Principles of your own , for his and our correction and instruction : such as these , That 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 can have no place in solid bodies . p. 2. ( because you know not how to distinguish between a Mechanicall and a Mathematicall 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 , as knowing no other way of measuring but by the Yard and the Bushell , or at least by the Pound . p. 4. & 13. ) And yet you tell us by and by . p. 3. that there may be in bodies , a Coincidence in all points ( which coincidence , had it been Greek , would have been as hard a word as 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 , ) and that this may properly be called 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 : and yet presently p. 4. you tel us again , that 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 hath no place in solids ; nay more , nor in circular , or other crooked lines ; ( as though you did not know , that two equall arches of the same circumference , would 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 . ) That the length of T●me , is the length of a Body . p. 2. ( As though he had not spoken absurdly , that said , Profecto vide , bam fartum , tam Diu , pointing to the length of his arme . ) That an Angle hath quantity , though it he not the Subject of quantity . p. 3. ( for there be octo modi habendi . ) That the quantity of an Angle , is the quantity of an Arch. p. 3. ( And why not as well of a Sector , since Sectors , as well as Archs , in the same circle , be proportionall to their correspondent Angles . ) That 't is a wonder to you , that Euclide hath not any where defined , what are Equalls , at least , what are equall Bodies . p. 4. ( As though every body did not , without a definition , know what the word meanes . Any Clown can tell you , that those bodies are Equall , which are both of the same bignesse . ) That Homogeneous quantities are those which may be compared by 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 , or application of their measures to one another . p. 4. ( And consequently , two solids cannot be Homogeneous ; because , you say , 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 hath no place in solids p. 2. & 4. And also , that incommensurable quantities , cannot be homogeneous ; because by 1 d 10 ▪ they have no common measure . ) That the quantity of Time , and Line are Homogeneous , p. 4. Because Time is to be measured by the Yard ; ( or , in your own words , because the quantity of Time , is measured by application of a line to a line ; ) But why not , by the Pint ? For you know Time may be measured by the Hour-glasse , as well as by the Clock . And though the Hand of a Clock or Diall , determine a Line , yet the sand of an Hour-glasse fills a vessell . That , Line and Angle have their quantity homogeneous , because their measure is an Arch or Arches of a Circle applicable in every point to one another . p 4. ( As though you had forgot , that you told us but now , that 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 , or application , hath no place in circular or crooked lines . ) And All hitherto , you say p. 5. is so plain and easy to be understood that we cannot without discovering our ignorance to all men of reason , though no Geometricians , deny it . Nay more , 'T is new , 'T is necessary , and 'T is yours . very good ! Now have at Euclide . Euclid's first definition , 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 , &c. A Marke is that of which there is no part ; is , you say , to be candidly construed , for his meaning is , that it hath parts , and that a good many . For a marke , or as some put instead of it , 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 , which is a marke with a hot iron , is Visible ; if visible then it hath quantity ; and consequently may be divided into parts innumerable p. 5. ( A witty argument ! 'T is visible , therefore 't is divisible , But could you not as well have said , That A Marke consists of two Nobles ? For that is as much to the businesse , as a marke with a hot iron . ) Nay more Euclids definition , you say is the same with yours , which is , A point is that Body whose quantity is not considered . Lay them both together and look else . A marke is that of which there is no part . A point is that Body , whose quantity is not considered . Just the same to a cow's thumb . They begin both with the letter . As like , as an Apple and a Oyster . But by the way , how comes a Point on a suddaine to be a Body ? you told us just before , in the same page , p. 5. that a Point is neither Substance , nor Quality , and therefore it must be Quantity or else 't is Nothing . If it be no Substance , how can it be a Body in your language ? But we have not done yet . Prithee tell me , good Tho. ( before we leave this point ) who t was told thee , that 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 was a marke with a hot iron ? for 't is a notion I never heard till now , ( and doe not believe it yet . ) Never believe him againe , that told thee that lye ; for , as sure as can be , he did it to abuse thee . 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 signifies a distinctive point in writing , made with a pen or quill , not a mark made with a hot Iron , such as they used to brand Rogues and Slaves with ; ( And accordingly 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 , distinguo , interstinguo , inter●ung● , &c. are oft so used ; ) It is also used of a Mathematicall Point ; or somewhat else that is very small : As 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 , a moment , or point of time , and the like . What should come in your cap , to make you think , that 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 signifies a mark or brand with a hot iron ? I perceive where the businesse lies . 'T was 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 run in your mind , when you talked of 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 , and , because the words are somewhat alike , you jumbled them b●●h together , according to your usuall care and accuratenes● 〈◊〉 as if they had been the same . ( Just as when , in Euclide 〈◊〉 you would have us believe that 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 & 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 is but one word . ) Do you not think now , that a boy 〈◊〉 Westminster Schoole would have been soundly whipt for such a fault ? Me thinks I heare his Master ranting it at this rate ; How now Sirrah ! Is 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 and 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 , all one with you ? I 'le shew you a difference presently . Take him up Boyes . I 'le shew you how 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 may be made without a hot Iron , I warrant you . And after a lash or two , thus goes on : 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 , is a Point made with a Pen , quoth he ( with a lash ) will you remember that ? 'T is 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 , is a mark with a hot Iron , ( lashing again , ) think upon that too . Henceforth , quoth he , ( setting him down , ) Remember the difference between 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 and 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 . The second definition . A line is length which hath no breadth ; you would have to be candidly interpreted also . If a man , you say , have any ingenuity , he will understand it thus , A line is a body &c. very likely ! The Fourth definition , is this , A streight line is that which lies evenly between its own points . p. 6. Well ; how is this to be understood ? Nay , this definition is inexcusable . Say you so ? let it passe then , and shift for its selfe as well as it can . It hath made a pretty good shift hitherto ; perhaps it may outlive this brunt also . But , because you are willing to lend it a helping hand , you say , He meant , perhaps , to call a streight line , that which is all the way from one extreme to another , equally distant from any two or more such lines , as being like and equall have the same extremes . It may be so . Many strange things are possible . But it would have been a great while before I should have thought this to be the meaning of those words . The seventh definition , you say hath the same faults . Then let that passe too ; and answer for it selfe as well as it can . The eighth , is the Definition of a plain Angle . Against which you object onely this of your own , That by this Definition , two right angles taken together are no Angle . And 't is granted . Euclide did not intend to call an aggregate of two right angles , by the name of an Angle : And therefore gave such a definition of an 〈◊〉 , as would not take that in . Where 's the fault then ? The thirteenth definit●●● , A Terme or Bound , is that which is the extreme of any thin● 〈◊〉 you say , is exact , ( very good ? ) But , that it makes against 〈◊〉 doctrine . What doctrine of mine ? viz. that a point is nothing . Who told you , that this is my doctrine ? I have said , perhaps , that a Point hath no hignesse ; or , that a Point hath no parts , ( and so said Euclide in his first definition , ) but when or where did I say , it is nothing ? But how do you prove hence , that a point hath parts ? Because , you say , The extremes of a line are Points . True. What then ? A point therefore , you say , is a part . It doth not follow . How prove you this consequence , If an extreme , then a part ? But , say you , what in a line is the extreme , but the first or last part ? I answer ; A Point , which is no part . Have you any more to say ? — If you have no more to say , then heare mee . A point is the extreme of a line : Therefore it hath no parts . I prove it thus ; because , if that point have parts ; then , either all its parts are extreme , and bound the line , or some one , or more : Not all : For they cannot be all utmost ; but one must stand beyond another : if onely some , or one ; then not the Point , but some part of it , bounds the line , which is contrary to the supposition . You see , therefore , the Definition doth not make against my doctrine . The fourteenth Definition of Euclide , you would have abbreviated thus . A figure is quantity every way determined , and then tell us , it is in your opinion as exact a definition of a Figure as can possibly be given . But I am not of your opinion ; For by this Definition of yours , a streight line ( of a determinate length ) must as well be a Figure , as a circle . For such a line , having no other dimension but length , if its length be determined , it is every way determined ; that is , according to all the dimensions it hath . ( If you object , that it hath no determinate breadth ; I answer , the breadth of a streight line is as much determined , as the thicknesse of a Circle , or other plain figure . ) And , by the same reason , A Pound , a Pint , a Hundred , an Hour , &c. must be Figures , because they are Quantities every way determined , viz. according to all the dimensions that those words import . This Definition of Euclide , — ( stay a while , the Definition mentioned is not Euclides , nor equivalent to it His 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 , imports more then your determined . 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 . should be rendred A figure , is that which is every way encompassed by some bound , or boundes . Which can be only in such a quantity as hath locall extension ; and that , finite . ) But The Definition , you say , ( whose soever it be ) cannot possibly be imbraced by us who carry double , namely Mathematicks and Theology ; ) but by you it seems , it may , who carry simple , and care not how destructive your principles are to Theology . ) Your Definition , we ( whether Theologers or Mathematicians ) cannot admit ; for the reason by us already assigned . But it seems you have a farther reach in it : Le ts hear what it is . For this determination , say you , is the same thing with circumscription . A locall determination , intended by Euclide , is so . But what then ? And whatsoever is any where ( ubicunque ) Definitivè , is there also Circumscriptivé . How do you prove this ? or how doth this follow from the other ? — You cannot but know this is generally denyed . Have you any thing to offer by way of proof ? — Not a word . Well ; but what is it you drive at ? You offer nothing of proofe , for what you affirme ( by your own confession ) against all Divines , or as you call them Theologers . But le ts see what you would gather from it . By this means , you say , the distinction is lost , by which Theologers , when they deny God to be in any place , save themselves from being accused of saying he is nowhere ; for that which is nowhere is nothing . 'T is true , that Divines do ●ay , ( and I hope you 'l say so too ) that God is not bounded , or circumscribed , within the limits of any place ; because they say , and do believe , there is no place where he is not . And he that saies the latter , must needs say the former . For to say that God , who is every where , & fills all places ; is yet bounded within certain limits ; were a contradiction . For , to be concluded within certain limits , is to be excluded from all places without those limits ; And therefore not to be every where . And if this be not your opinion too , speak out , if you can for shame , that the world may see what you are . Do you believe , that what thing soever is at all any where , ( not excepting God himselfe ) must needs be circumscribed within some certain bounds , so as not to be without or beyond them ? And that whatsoever is not , in any place so circumscribed , is no where , and therefore nothing ? If so ; then whether of the two do you affirme ? That God is so circumscribed or concluded within certain limits , and excluded from all others at the same time ? Or , That he is not so concluded , and therefore no where , and so nothing ? If you say the first , you deny God to be Infinite : If the second you deny him to bee . And , either way , you may without injury be affirmed to maintain horrid opinions concerning God. As for that distinction of Definitivè and Circumscriptivè , with which you say the Theologers think to save themselves : You are wholly out in the businesse : Theologers use not that distinction in this case . It 's true , that , in the case of Angells , and the Soules of men , there are that affirme them to be in loco definitivè , but not circumscriptivè : because though they be not bodies , and so locally extended per positionem partis extra partem ; yet neither are they infinite , or every where , but have a definite , determinate existence , as to be here , and not at the same time elsewhere . But as to God , we neither affirme him to be circumscribed , nor to be confined within any bounds ; but to be Infinite and every where . And if any be so absurd as to affirme that God is determined within some place , so as not to be at the same time without or beyond it , whether by Circumscription or Definition , we shall without scruple , ( notwithstanding that we carry double , ) reject the distinction so applied , and your opinion with it , without fear of being cast out from the society of all Divines . But in the mean while , I wonder how this Definition of Euclide comes to have any thing to doe with this businesse . A Figure , saith Euclide , is that which is incompassed within some bound or bounds . Well , what then ? Will you assume But God is a figure ? and then conclude , That , if God be at all any where , he must be so concluded within bounds ? If you do , you argue profanely enough , and deserve as bad Epithites as any have been yet bestowed upon you . We should rather , admitting Euclides definition , argue thus , A figure is concluded within certain bounds ; But God is not so concluded , ( as being infinite , and so without bounds ; ) Therefore God is not a Figure : And be neither in danger of being cast out of the Mathematick Schooles , nor yet , from the Society of Schoole-Divines . The Fifteenth Definition , which is , of a Circle , you grant to be true . And skip over the rest to the five and twentieth , which is , of Parallell streight lines . This Definition you think to be lesse accurate , and think your own to be better : But of this it will be time enough , if need be , to consider in its proper place . After this , you let all the Definitions passe untouched , till the third of the Fift Book . Saving that you touch by the way , on the Fourth of the Third Book , which you grant to be true : and the first of the Fift Book , which , you say , may passe for a Definition of an Aliquot part , as was by Euclide intended . But , the Third Definition of the Fift Book ( the Definition of 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 , Ratio , ) you say , is intollerable . Yea 't is as bad as any thing was ever said in Geometry by D. Wallis . ( Because forsooth , you can make nothing of it , but this , that Proportion is a what-shall-I call it asnesse or sonesse of two magnitudes &c. ) Yet this definition hath hitherto been permitted to passe , and may do still . And when you understand it a little better , perhaps you may think so too . But of this I have discoursed more at large , in a peculiar Treatise against Meibomius : and shall therefore forbear to examine it here . Against the fourth definition , you object nothing , but that the sixt might be spared . The Fourteenth , you say is good . And tell us farther , that the composition here defined , is not the same composition which he defineth in the fourth def . before the sixth book . And you say true ; for this is a composition by Addition , and that is composition by Multiplication . And therefore do not think much if hereafter I shall say , that there be two compositions of proportion . To the rest of his definitions you give a generall approbation . His Postulata you allow also : and so give over Lessoning of Euclide : But tell us before you part , that A man may easily perceive , that Euclide did not intend , That a point should be ( without parts , which you call ) nothing ; or a line , without latitude ; or a Superficies , without thicknesse : though it be evident that he hath defined them so to be . But why must we not think , he meant as he saith ? ( Because , say you , Lines are not drawn but by Motion , and Motion is of Body only . A pretty argument , and worth Marking ! like that above , of 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 , a Mark or brand with a hot Iron . SECT . IV. Concerning the Angle of Contact . HAving dore Schooling of Euclide ; in your second Lesson you fall upon us . Four peeces of mine , you take to task . p. 10. ( My Elenchus of your Geometry ; my Treatise concerning the Angle of Contact ; and that of Conick Sections ; and my Arithmetica Infinitorum . ) Yet have not been able to find , either one false Proposition , or so much as a false Demonstration ; in any one of them . Yet , that you may seem to say something , you 'l blunder on , though you break your shinnes for it . And you 'd have it thought , that you have wholly and clearly confuted them Ep. Ded. ( for you use to make clear work where you goe , ) and that I have performed nothing in any of my books . p. 10 This is the charge . Let 's see how you can make it good . Wee 'l begin with that of the Angle of Contact ; which you undertake in your third Lesson . p. 26. The subject of that treatise , is , a controversy between Clavius and Peletarius . Clavius is of opinion , that the Angle of a Semicircle EAC ( Fig. 1. ) is lesse then the rectilineal Right Angle PAC ; because that is but a part of this ; the other part EAP , the Angle of contact , ( which with that of the Semicircle makes the right Angle PAC , ) being , as he supposeth , an angle of some bignesse . Peletarius is of opinion , that the Angle EAC , is equall to PAC ; and not a part of it , but the whole ; the supposed Angle PAE being , as he thinks , no Angle , or an angle of no bignesse . This being the state of the controversy : I take Peletarius his part . And my first argument is from the nature of a Plain angle , which Euclide defines to be the mutuall inclination of two lines &c. And therefore the lines EA , PA , in the point of concurse A , not being at all inclined each to other ; but in the same coincident position without inclination ; they do not contain an angle . The tendency of the circumference EAN , before it comes at the point A , is towards the tangent PT ; when it 's past that point , the tendency is from it ; but in the point A , it doth neither tend toward it , nor from it , nor crosse it ; and therefore must be either in parallell position , or coincident . And this argument is managed in the 3 and 4 Chapters . You tell us to this , that Peletarius did not well — Clavius did not well — Euclide did not well — That is , You think so . And it 's like , You think , I have done worst of all . But I doe not much stand upon your thoughts . You say particularly , p. 26. That I am more obscure then Euclide . ( It may be so . ) That I am contrary to him , ( That you are to prove . ) That I make two lines when they ly upon one another , to lye 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 , without inclination : I do so . Shew me if you can , where Euclide saith the contrary . Tell mee , where lines , either in the same or in parallell positions , are by Euclide said to incline or be inclined each to other ? to thwart , or crosse each other ? According to Euclide , you say , an angle equall to two right angles should be the greatest inclination , and so the greatest angle , where as , by this 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 , it should be the least that can be , or rather no angle . Shew me where ever Euclide doth acknowledge any angle to be equall to two right angles ? or , which is all one , that too contiguous parts of the same right line , are by Euclide said to be inclined to each other , or to contain an angle ? Nay he says the quite contrary . For in his definition of a plain angle , he makes it one qualification , that the lines containing it , must be such as are non indirectum positae . And therefore two streight lines in directum positae ( such as those must needs be , which are to contain your supposed angle equall to two right angles ) cannot , by Euclides definition , contain an Angle . We do not therefore in this disagree . You adde farther , ( as giving this for lost , Though it be granted ( as it must needs be ) that there be no inclination of the circumference to the tangent ( and consequently no Angle ; by that definition of Euclide . ) yet it doth not follow that they forme no kind of Angle . And why doth it not follow ? Because say you , Euclide there defines but one of the kinds of a plain angle . That Euclide doth not there define , an angle in general or all kinds of angles , is very true ; for there be many other both superficiall and solid angles , which are not plain angles : But that he doth there define a plain angle in generall , and therefore all kinds of plain angles is evident frō his words . For in the eighth Definition he defines a plain Angle , ( as the genus ) A plain Angle , saith he , is the mutuall inclination of two lines , &c. and then in the next definition , defines a right lined plain angle , ( as one species of it ) viz. when both these lines be right lines . It 's manifest therefore that he intended in the former definition to define a plain angle in generall ; whether the lines containing it be streight or crooked . And therefore since the angle of contact falls not within that definition , it is not to be reputed a plain angle . And so my first Argument stands good . The second , is an Argument of Peletarius , drawn from the first Proposition of the tenth of Euclide ; ( and enforced likewise by me , from the second proposition of the first of Archimedes de sphaera & cylindro : ) To which Clavius rejoyns , that the proposition is to be understood only of Homogeneous quantities ; & ; of such , grants the argument to proceed . And you ; supposing these to be Heterogeneous , say , it is like as to seek for the Focus of the Parabola of Dives and Lazarus . To your scoff at Scripture , I reply only this , that the Focus of that Parabola is a bad place to be in , & wish you to take heed of it . With Clavius , we joyn issue ; granting the propositions cited not to be understood of Heterogeneous quantities ; and prove these not to be such ; by this argument : If any thing make the angle of Contact PAE , to be heterogeneous to a rectilineal angle ; it must be the crookednesse of the side AE . ( for if that side were streight ; the angle were rectilineall ; ) But that hinders not , ( for I prove the angles CAE , and SAE , notwithstanding the same side AE , are homogeneous to right lined angles ; as you grant , and Clavius could not deny : ) Therefore nothing hinders . And this is done in my fift Chapter . What Clavius had brought to prove the contrary , is answered in the sixth Chapter . And if you had not thought his arguments to be all answered , you should have done well to have undertaken the managing of some one of them . That you mention , doth only , upon supposition that it is a quantity , prove it to be heterogeneall ; because not Homogeneall . Which is to beg the question . For we , as well as he , deny it to be a Homogeneall quantity ; and therefore conclude it to be no quantity ; for heterogeneous it is not . His argument amounts but to this , 'T is not a quantity Homogeneous , ( by 5 d 5 ) therefore 't is a quantity Heterogeneous . I grant his Antecedent , but deny the Consequence ( which proceeds only upon supposition that it is a Quantity , which is the thing in question . ) He should first have proved it to be a Quantity ; which Peletarius and I deny . In the seventh Chapter I prove , by other arguments , that if the angle of Contact be an angle , it must be homogeneous to rectilineal angles . 1. That which may be added to , or subtracted from , a right lined angle , is homogeneous to it : Because Heterogeneous quantities are not capable of addition , or subduction . ( And this you grant . ) But so here ; For PAE if an angle , may be added to the angle SAP , making the angle SAE ; ( which therefore , saies Clavius , is bigger then SAP ; ) and taken from the angle PAC , leaving the angle EAC , ( which therefore , saies Clavius , is lesse then PAC ; ) Therefore , if an angle , it is homogeneous You grant the major ; and deny the minor : that is , you deny the only foundation upon which Clavius builds his opinion ; and so yeeld the cause . For he doth upon no other ground maintain the angle of the semicircle EAC , to be lesse then the right angle PAC , but because the angle of Contact PAE , is a part of it , and therefore the other part EAC , must be lesse then the whole . 2. Those which are to each other as Greater and Lesse , have proportion each to other ; and are consequently homogeneous ; by the third def . of the fift of Euclide . ( and this you grant . ) But , the angle of Contact PAE , is lesse then the angle SAP ; by the 16 of the third of Euclide ; ( for his words are , that it is lesse then any right lined angle . ) And this Clavius would not deny , but oft affirmes it . Therefore they be homogeneous . All that you have to say is , that though Euclide say it is lesse , yet ( to your understanding ) he doth not mean so . But doth he not , to your understanding prove , that the least right lined Angle is bigger th●n it ? and if so , supposing it to be angle , must it not be Homogeneous ? even by your own concession . To the third and fourth Arguments in that Chapter , You object nothing ; and therefore those , I suppose , you allow to conclude what is contended for . viz. that the angle of Contact is not Heterogeneous to other plain angles : and therefore , this being the only exception , my first main argument stands good . The Eight Chapter you say , contains nothing but the Authority of Sir Henry Savile . And you say true ; for no more was intended . The third main Argument is proposed in the ninth Chapter ; Because the Angles of semicircles ( because like segments ) are equall . Whence Peletarius infers , that the Angle of Contact is no quantity . Clavius grants the consequence of the Argument ; but denies the Antecedent : affirming DAC ( fig. 2 , ) to be lesse then EAC , though both angles of Semicircles , this of the bigger , that of the lesse . To this you say , that in my 9 and 10 Chapters I prove with much adoe , that the Angles of like segments are equall : ( if I prove it , though with much adoe , then I carry the cause ; for that was the only thing denied by Clavius . But you adde ) as if that might not have been taken gratis by Peletarius , without demonstration : ( Implying thereby , that I need not have proved it . ) And this is like your selfe , who care not how you abuse your English Reader . The case is thus . Peletarius had taken it gratis , as a thing that in reason should not have been denied him . Yet 't is denied by Clavius ; and the whole issue of the cause put upon it . Had I not reason then to prove it ? Yet I prove it thus ; First , that Peletarius had reason to take it gratis , and that it was unreasonable in Clavius to put him upon the proofe ; and this is done in the ninth Chapter . But then , because he had denyed it , how unreasonable soever it were so to doe , and withall put the whole issue of the cause upon it ; therefore in the tenth Chapter I undertake to prove it by argument . And you grant , I prove It. What should I doe more ? The 11th Chapter clears the same argument from a seeming difficulty . And you say nothing to it , but that the objection was of no moment , and needed no answer . To the Arguments of the 12 and 13 Chapters , ( and those are a pretty many , for in one of them are contained six , ) your answer is ( and that 's all ) that they are grounded all on this untruth , that an Angle , is that which is contained between the lines that make it , that is to say , is a plain superficies . Which is ( I will not say a lye , though that also be your language , but ) manifestly false ; and you could not but know it so to bee . For there is not , in those whole Chapters any such thing assumed for proofe ; nor doth any one of those arguments depend upon any such notion ; but let your notion of Angle be what it can , my arguments will hold their weight . This therefore is nothing but a notorious untruth , wherewith ( because you had nothing to say to the Arguments ) you meant to abuse your English Reader . But suppose I had said , ( as it is like I may sometimes ) that an Angle is contained by , or between the two sides ; is this any more then to say that the two sides contain the Angle ? And doth not every body say so as well as I ? Are they not Euclide's own words , 9 d 1. When the lines ( 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 ) containing or comprehending the angle be right lines , the angle is called Rectilineall ? Nay are they not your own words , cap. 14. § 7. Anguli qui rectis continentur lineis , rectilinei ; qui curvis , anguli curvilinei sunt ; qui recta & curva continentur , misti ? What a doe then doe you make for nothing ? Perhaps the word between troubles you . But is not by and between in this case all one ? It is to mee ; and if you doe not like the one word take the other ; 't is all one to mee , ( But , by the way , the phrase , contain between , is not so much as once used in either of those Chapters : and therefore that cavill is to no purpose at all , but to abuse your English Reader , who cannot contradict you . ) And doth not Euclide's word 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 signify to contain between ? and 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 , the lines which do comprehend , ( or contain between them ) the Angle ? Nay doe not you your selfe use it again and again , cap. 14. § 9. Ad quantitatem anguli neque longitudo , neque aequalitas aut inaequalitas linearum quae angulum comprehendunt , quicquam faciunt , idem enim angulus est qui comprehenditur inter AB & AC , cum eo qui comprehenditur inter AE & AF , vel inter AB & AF. And again cap. 14. parag . 16. Angulus qui cont●netur inter AB & eandem AB &c. And soon after , angulus qui sit inter GB & BK , aequalis est angulo qui sit inter GB & arcum BC. ( which is also retained in the English . ) And so elsewhere . But say you , To say that an angle is contained between the lines that make it , is as much as to say , that it is a plain superficies . And was it so when you wrote those passages last cited ? Were you then of opinion that the Angle contained or comprehended between the lines AB and AC , ( as you there speak , ) was a plain superficies ? Or , if those words do not import so much when you speak them , why should you think they doe when I speak them ? But , it seems , having nothing else to cavill at , you thought fit to tell your English Reader , who must take it upon trust from you , That I affirme a plain angle , to be a plain superficies , because , forsooth , I say ( as Euclide and all others doe , and your selfe among the rest , ) that it is contained between two lines . You might , with much better Logick , have concluded the contrary For though Euclide , as I doe , said that two streight lines may comprehend an angle , 9 d 1. yet he affirmes , that two streight lines cannot comprehend a superficies , 10 ax 1. And therefore , when I affirme that an angle may be comprehended between two streight lines , you might ( at least a sober-man might ) have concluded , that I did not take it for a superficies , because that cannot be comprehended by fewer streight lines then three . But enough of this . And , if this be all you have to say against the Arguments of the 12 and 13 Chapters , I hope they may passe for current : and be judged to conclude the cause . To that of the last chapters ( as you speak ) where I prove the same from a proposition of Vitellio : ( which proposition of his I doe also vindicate from an exception of Cabbaeus : ) You object nothing , but that I defend Vitellio without need ( and yet I had there told you , that Cabbaeus denies his argument : ) for say you there is no doubt but whatsoever c●ooked line be touched by a streight line , the angle of contingeuce will neither adde any thing to , nor take any thing from a Rectilineall Right Angle ; That is , there is no doubt but that Clavius was in the wrong , and I in the right , all the way : for this was the very thing that was in controversy betwixt us . And so you have brought your confutation to a good Catastrophe . And thus much for the Angle of Contact . SECT . V. Arithmetica Infinitorum , Vindicated . LEt 's see now what you have to say against my Arithmetica Infinitorum . Five propositions you there take to taske ; the first , the third , the fift , the nineteenth , and the thirty ninth . The first you , you say , is this Lemma ; In a series of quantities arithmetically proportionall , beginning with a point or cyphar , ( as for example 0 , 1 , 2 , 3 , 4 , &c. ) to find the proportion of the Aggregate of them all , to the Aggregate of so many times the greatest as there are termes . Very true , this is the first proposition ; what then ? This you say , is to be done by multiplying the greatest into halfe the number of termes . What is to be done thus ? finding the proportion ? No such matter . That 's the way to find the summe , ( upon supposition that the proportion is already known to be , as 1 to 2 , ) not to find out what is the Proportion , ( supposing it yet unknown , ) which the Lemma proposeth to be inquired , and finds it to be as 1 to 2. But 't is well however that you can at length tell how to gather the summe of such a proportion ( after I had taught you in my Elenchus , ) for you were , it seems , of an other opinion , when you said Cap. 16. parag . 20. In hujusmodi progressione ( 0. 1. 2. 3. 4. &c. ) summa numerorum omnium simul sumptorum , aequalis est semissi ejus numeri qui fit a maximo termino ducto in minimum , id est , hoc loco in ciphram . Which you now confesse pag. 41. to be a great error . You go on , and say , The Demonstration is easie . But how , say you , do I demonstrate it ? You should have asked rather , How I find it , ( then how I demonstrate it : ) for that was it the Lemma proposed . But you are so well acquainted with the Analyticks , that you know not how to distinguish between the 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 , and the 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 . by the first we find out the solution of a Problem ; by the second we prove it . Now if you can find a more naturall 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 , or way of finding out the solution of this and the other Problems ( for I was here shewing a generall method for this and others that follow , ) pray let us know it in your next , and I shall thank you for it . But doe not talk of Demonstrating , when I propose the finding out ; for , if you doe , I shall say , that 's nothing to the purpose . You tell us next , that an Induction , without a Numeration of all the particulars is not sufficient to inferre a Conclusion . Yes , Sir , if after the Enumeration of some particulars , there comes a generall clause , and the like in other cases , ( as here it doth ) this may passe for a proofe , till there be a possiblity of giving some instance to the contrary ; which , here , you will never be able to doe . And if such an induction may not passe for proofe , there is never a proposition in Euclide demonstrated . For all along he takes no other course then such , ( or at least grounds his Demonstrations on propositions no otherwise demonstrated . ) As for instance ; he proposeth it in generall 1 e 1. to mak an Equilater triangle on a line given . And then shews you how to doe it upon the line 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 which he there shews you : and leaves you to supply , and the same by the like meanes , may be done upon any other streight line ; and then inferres his generall conclusion . Yet I have not heard any man object , that the induction was not sufficient , because he did not actually performe it in all lines possible . You then aske , whether it be not also true in these numbers , 0 , 2 , 4 , 6 , &c. or 0 , 7 , 14 , 21 , & c ? Yes , and in these also ( which perhaps you would little think ) 0 , √ 2 , √ 8 , √ 18 , √ 32 , √ 50 , &c. But why , say you , doe I then limit it to the numbers 0 , 1 , 2 , 3 , 4 , & c ? I should rather wonder why you think I doe . No wise man would have thought it ; when he sees that I speak in generall of any series in continued Arithmeticall progression , that begins with a point or ciphar . And there had been no colour , for you to aske such a question , if , in reciting my proposition , you had not in stead of saying as , for example , said only as . For doubtlesse those are continued . Arithmeticall progressions beginning with a Ciphar ; and they are also juxta naturalem numerorum consecutionem , that is , like progressions to those of the naturall numbers 0 , 1 , 2 , 3 , &c. You ask then ( very wisely ) whether it will hold in 0 , 1 , 3 , 5 , 7. I answer , no. ( nor is any such thing affirmed . ) because 0 , 1 , 3 , are not Arithmetically proportionall . And when you have done Catechizing us , you then conclude , well , the Lemma is true : ( in good time ! ) that is as much as to say , you were willing to shew your teeth though you cannot bite . What 's next ? The first Theorem that I draw from it , is , you say , that a Triangle to a Parallelogram of Equall base and Altitude , is as one to two . Well ; and what of this ? The conclusion you say is true . Very good , Then two of the five have scaped already . But you doe not like of the Demonstration , because of the words as it were ( and the like exception you took before , at the word scarce , ) which you say , is no phrase of a Geometrician . Yes Sir ; a very good phrase , if the Geometrician doe determine precisely ( as I have done ) how much by that quasi he intends to limit the accuratè . For I doe not suffer either the scarce , or the as it were , to runne at randome without bounds . I tell you that by quasi linea , or vix aliud quam linea , I doe not meane precisely a Line , but a Parallologramme whose breadth is very small , viz. an aliquot part of the whole figures altitude , denominated by the number of Parallelogramms . ( Which is a determination Geometrically precise . ) And by triangulum constat quasi , &c. I tell you that I mean , that a Figure , consisting of such Parallelogramms , inscribed in a Triangle , whose difference in bignesse from that Triangle is lesse then any assignable quantity , is so constituted . As you may see precisely determined in the place to which this Demonstration referres . The words therefore vix and quasi , being thus determined , are here very good Geometricall words ; and your cavills come to nothing . My fift Proposition is , you say , The Spirall line is equall to half the circle of the first Revolution . But , in saying so , you say not true . For that is not my proposition ; but one of your own , patched together , after your fashion , out of my fift and sixth put together . And , as it stands , I cannot own it . The words of the first revolution , should have been adjoyned to the Spirall line , not to the word circle : to shew how much of the Spirall line is intended . And , instead of halfe the circle , you should have said , halfe the circumference of the first circle ; for I did not compare the Spirall line with the Circle ( that is , a line , with a Figure , ) but that Spirall line with the circumference , ( viz. as 1 to 2 , ) and the Spirall Figure , with the Circle , ( viz. as 1 to 3 ) And the circle you intend , is not by mee , or by Archimedes , called the circle of the first revolution , but the first circle ; which is conterminate with the first revolution of that Spirall line . But if you will needs have my fifth and sixth Propositions put together , pray let it be thus , That so much of the Spirall line , in the sense of the proposition , as belongs to the first revolution , is equall to halfe the circumference of the first circle . Now in what sense I take the words Spirall line , in these propositions , is so clearly defined in the Scholium of pag. 10. that it is not possible for any man , unlesse willfully , to mistake mee . viz. That I doe not intend the true spirall of Archimedes , but the aggregate of the arches of infinite like Sectors , constituting a figure inscribed within that Spirall of Archimedes . And thus , both those and the other Propositions are true . Nor can you deny them . But now because you have nothing to say against the Proposition in the true sense of it : you will needs perswade mee , ( because you know what I meant better then my selfe . ) that I did not so mean , nor would be understood , so , as I said , I meant and desired to be understood ; but that I meant somewhat else . And you have this ground for it ; Because in the sense wherein I said I would have it understood , the proposition is true ; but you have a desire that it should be false ; and therefore it must be understood in some other sense . Let 's see therefore what it is , that I may at length know what it was I meant . What Spirall is meant , you say , we shall understand by the construction . Yes , if you take in the whole construction ; but not by a peece of it . My construction begins thus , Let a streight line MA , turned about the center M , be supposed , by a uniforme motion , to describe , with its point A , the circumference AOA ; whilest a point in the same line , so carried about , is supposed to describe a Spirall line MTA . This is the first part of the construction ; ( and from hence I inferre , by the way , that the streight lines MT will be proportionall to the angles AMT , and the Archs AO . ) This therefore , say you , is the Spirall of Archimedes . Very true : and it was intended so to be . But let 's goe on and heare the rest of the construction , ( for hitherto we have had but a part of it . ) Which if you may be believed , is this ; inscribing in the Circle an infinite multitude of Equall angles , and consequently an infinite number of Sectors , whose Archs will therefore be in Arithmeticall proportion ; ( which , you say , is true ; ) and the aggregate of those archs equall to halfe the circumference AOA . Which , you say , is true also . But , if I had said so , I had lyed ; for I know it to be false : ( in you it was only an Error , or , as you use to call it , a Negligence ; because you thought it had been true . ) For this is neither my construction , nor are those things true which you affirme . For , if in a circle , there be a number of Sectors inscribed ( whether finite or infinite ) both those Sectors , and the Archs of them , are proportionall to their Angles ; and therefore , the Angles being equall , the Archs will be equall also , and not Arithmetically proportionall : And the Aggregate of those archs , will not be equall to half the circumference AOA , but , to that whole circumference . But my construction was this ; Within the Spirall line , described as above , supposing an infinite multitude of Sectors continually inscribed on equall angles , their Radii AT will be Arithmetically proportionall , viz. as 0 , 1 , 2 , 3 , and consequently their archs will be so too . And this , I suppose , is that which you intended to grant as true ; Being the result of the second part of my construction . Then followes the third part of the construction ( which hath the nature of a Definition ; which , till thus much of the construction was past , could not conveniently be expressed , ) The Spirall line ( intended in the proposition , not that of Archimedes ) is supposed therefore to be made up , of the archs of those infinite Sectors , arithmetically proportionall ( for so they are already proved to be ) beginning with a point or o. ( And then goes on the Demonstration . ) But the circumference consists of so many archs equall to the biggest of them ; as is evident . Therefore ( by the second Prop. ) that to this , is as one to two . Which is the thing to be proved . Now this , to some capacities , though not to M. Hobs , would have been easy enough to understand . Yet that it might not lye open to any cavill , or misunderstanding ; I thought fit in a particular Scholium , to expresse my meaning so fully , as that there might be no possibility of mistaking what I intended . ( And , the truth is , I would have had that Scholium Printed next after the Fifth Proposition . But finding , that , through some neglect , the Printer had there left it out , I gave him order to put it in , at the next convenient place ; which was , in the next sheet , at the end of the 12 Proposition : a place proper enough for it . ) And you cannot deny , but that my words there , be plain enough to be understood , and not capable of any distortion to any other sense . And that the Proposition in this sense is true , you cannot deny ; and so much ( I suppose ) you intended to grant , when you said , That the aggregate of those archs is equall to halfe the circumference AOA , is true also . Three therefore of the five are already found to be true . My 19. Prop. you say , is this Lemma . In a series of Quantities , beginning from a point or ciphar , and proceeding according to the order of square numbers , ( as for example 0 , 1 , 4 , 9 , 16 , &c. ) to find what proportion the whole series hath , to so many times the greatest . 'T is true ; this is my 19 Proposition . What then ? I conclude , you say , the proportion is that of 1 to 3. No Sir , I do not conclude it to be so . I conclude it to greater then that of one to three . My words are these , Ratio proveniens est ubique major quam subtripla Excessus autem perpetuo decrescit prout numerus terminorum augetur , &c. ut sit rationis provenientis excessus supra subtriplam , ea quam habet unitas ad sextuplum numeri terminorum p●st o. That is in plain English thus . The series so increasing , is alwaies more then a third part of so many times the greatest . For it containes evermore , a third part thereof , and moreover , an aliquot part denominated by six times the number of termes following the o. And is not this true ? can you have the face to deny it ? Wee 'l try if you please ; take your own instances . Let the series be of three termes 0 , 1 , 4 , the aggregate is 5 : the greatest so many times taken , that is 3 times 4 , is 12. I say 5 contains of 12 , a third part ( viz. 4 = ⅓ × 12. ) and moreover a part denominated by 6 times 2 , ( for there are two termes besides o. ) that is a twelfth part of the number 12. ( viz. 1 = 1 / 12 ; × 12. ) And is not this true ? is not 5 = 4 × 1 ? Again , let the termes be four , viz , 0 , 1 , 4 , 9. = 14. and the greatest so many times taken 9 , 9 , 9 , 9. = 36. I say that 14 containes ⅓ of 36 , ( that is 12 , ) and moreover , ( because 3 × 6 = 18 ) 1 / 18 of 36 , ( that is 2. ) And is it not true , that 14 is equall to 12 + 2 ? I think it is . Again , let the termes be five , viz. 0 , 1 , 4 , 9 , 16 , = 30. and therefore so many times the greatest is 16 , 16 , 16 , 16 , 16 , = 80. I say that 30 contains , ⅓ of 80 , that is 26 2 / 1 & moreover 2 ¼ of 80 ; ( because 4 × 6 = 24 ) that is 3 1 / ● . And is it not so ? is not 26 ⅔ + 3 ½ = 30 ? you may try it farther if you please . My skill for yours , 't will hold . ( And that 's fair odds in a wager . ) The Proposition therefore is true thus farre . Well but I said farther ; That though the Proposition be still more then the subtriple ; yet the excesse doth still decrease . Doe you not think that true too ? if not , let 's try . if the termes be three , you see the proportion is as 5 to 12 , that is as ⅓ + 1 ½ to 1. if four , the proportion is as 14 to 36 , that is ½ + 1 ½ to 1. if five , then as ⅓ + 1 / 24 to 1. &c. As we have seen already . But the proportion of ⅓ + 1 1● to 1 , is more then of ⅓ + 1 ½ to 1 , and yet this more then ⅓ + 1 / 24 to 1. and so forward . But you forsooth would faine perswade us , that as the number of termes increase , so the proportion increaseth . As if the proportion of ⅓ + 1 / 24 to 1 , were greater then that of ⅓ + 1 / 18 to one . and yet would pretend to understand proportions , and tell us what M. Oughtreds meaning is &c. as if we did not understand M. Oughtred , and his meaning too , better then you . But , by the way , I wonder how you durst touch M. Oughtred for fear of catching the Scab . For , doubtlesse , his book is as much covered over with the Scah of Symbolls , as any of mine . Which makes me think , you understand his and mine much alike . I adde farther , ( though not in this proportion , ) that the proportion doth so decrease , as that ( though it be never lesse then a subtriple , yet ) the excesse above the subtriple , will by degrees vanish , as the number of termes increaseth , till it grow lesse then any assignable quantity . and it is proved thus : Because the second fraction , which with ⅓ makes up the antecedent of the proportion , whose consequent is 1 ; doth proportionally decrease , as the number of termes doth increase . And therefore , as the number of termes may increase beyond any assignable number : so may the excesse decrease below any assignable quantity . And , if the number of termes be supposed infinite , the proportion will be infinitely near to the subtriple . But you tell us upon this , ( and wittily doubtlesse , as you suppose , by a sly transition from the phrase infinitely near , to that of eternally nearer , ) you tell us , I say , that if the proportions come eternally nearer and nearer to the subtriple , ( supposing them at first bigger then it , which you should have added , for else the case alters , ) they must also come eternally nearer and nearer to the subquadruple , and so to the subquintuple , &c. I grant it . But what then ? it doth not follow , that if it come eternally nearer to the subquadruple , then it will come infinitely neare , or nearer then any assignable difference ; for it can never , upon that supposition , come nearer to it then the subtriple . Like as the Hyperbole , doth eternally come nearer and nearer to its Asymptote , and consequently , will eternally come nearer also to a parallell that lyes beyond it ; but not infinitely near ; for , since that it never pas●es the Asymptote , though it doe eternally approach , yet it never comes nearer to that Parallell , then the Asymptote doth . And indeed if it should , it could not eternally approach to the Asymptote , but so soon as it is passed it , it would then grow farther and farther from the Asymptote , while it doth approach to the parallell beyond it . And , in the present case , this proportion which doth eternally approach , and may come infinitely neer to the subtriple , doth indeed eternally approach , but not come infinitely near , to the subquadruple . For it never comes nearer to it , then is the subtriple . And I would not have you think us such weak Mathematicians , or such young birds , as to be caught with such chaffe , or not see through so weak a fallacy as that is . And therefore when you inferre , that we may as well conclude thence , that the proportion , is as one to four , or one to five , &c. ( supposing the number of termes infinite ) as to conclude , it is as one to three : We suppose that you would have us think withall , either that you doe not speak in good earnest , or else that you are not well in your wits : For otherwise , doubtlesse you cannot be so simple as to believe it . There is but one Proposition more that you undertake to deal with . Which is the 39 , viz. this Lemma , In a series of quantities beginning with a point or cipher , and proceeding according to the series of Cubick Numbers , ( as for example 0 , 1 , 8 , 27 , 64 , &c. ) to find what proportion the whole series hath to so many times the greatest . And you deal with this , just as you did with the last . First you mis-recite it , and then say 't is false . I conclude , you say , that they have the proportion of 1 to 4. Which is false , I do not so conclude ; but that it is more then so ; viz. it contains a fourth part , and moreover another aliquot part , denominable by four times the number of termes following the cipher . That is , if the termes be three , the proportion is as ¼ + 1 / 8 to 1. if four , it is as ¼ + 1 / 12 to 1. if five , it is as ¼ + 1 / 16 to 1. And so forward . And if you make triall , you shall find it so to be . ( For 0 + 1 + 8 = 9 ; and 8 + 8 + 8 = 24. Now 9 is equall to ¼ + 1 / 8 of 24 , viz. to 6 + 3. So 0 + 1 + 8 + 27 = 36. and 27 + 27 + 27 + 27 = 108. Now 36 is equall to ¼ + 1 / 12 of 108 , viz. to 27 + 9. So 0 + 1 + 8 + 27 + 64 = 100 ; and 64 + 64 + 64 + 64 + 64 = 320. Now 100 is equall to 1 〈◊〉 + 1 / 16 of 320 , viz. to 80 + 20. And so of the rest . ) If you think it to be otherwise ; shew , if you can , one instance to the contrary . The Proposition therefore is true ; but you had not the honesty to report it right . ( or else your witts were at wooll-gathering . ) And so of all those five propositions which you have taken to taske , there is not any one faulty . And I should now have done with this businesse , but that I discern , upon these two last Propositions , your reason why you are so much out of charity with the Symbolick tongue . 'T is very hard , you have told us diverse times ; yet here , it seems , you mean to try what you could doe at it . And 't is to be hoped , you may , in time , learne the language ; for you be come to great A already . ( But truly were it not that you must defend your reputation , you tell us , you should not have done so much . ) But such pittifull work dost thou make with poor great A , and to so little purpose , that if there were no better use to be made of Symbols , then so , it 's pitty they should ever be used at all . And truly , were I great A , before I would be willing to be so abused , I should wish my selfe little a , an hundred times . Yet thus much , I confesse you have done : You have clearly convinced me , that you have reason not to be much in love with Symbols . For to what purpose ? since you can neither use , nor understand them . And truly , upon this very account , I am apt to think , that much of your 13 chapter , is none of your own . Well ; Arithmetica Infinitorum is come off clear . Wee 'l see next what you have to say to Conick Sections . SECT . VI. My Treatise of Conick Sections vindicated . AS for my Treatise of Conick Sections , you say , it is so covered over with the Scab of Symbols , that you had not the patience to examine whether it be well or ill demonstrated . A very fine way of confutation ; and with much case . You have not the patience to examine it , ( that is , in plain English , you do not understand it , ) Ergo I have performed nothing in any of my Books ( for that is the inference in the same page , p. 49. ) 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 . But , Sir , must I be bound to tell you a tale , and find you ears too ? Is it not lawfull for me to write Symbols , till you can understand them ? Sir , they were not written for you to read , but for them that can . However , whether you understand it or not , yet somewhat you observe , you say , ( though you have not the patience to examine whether it be well or ill . ) Pray le ts heare your Observations ; ( for they be like to be wise ones . ) You observe , you say , that I find a Tangent to a point given in a Section , by a Diameter given : ( very good ; There 's no hurt in that , I hope , is there ? ) and in the next Chapter , I teach the finding of a Diameter . You should have done well to have told us , where to find those Chapters . For I do not remember , that that Treatise is at all divided into Chapters . Well! but suppose I had in one Chapter , by the help of Diam●ter given , found a Tangent ; in another Chapter , by the help of a Tangent , found a Diameter : Had there been any hurt in all this ? You observe also , you say , that I call the Parameter an Imaginary line , as if the place thereof were lesse determined then the Diameter it selfe . ( But did you observe , whether I did well or ill , so to call it ? ) And then , you say , I take a mean propoirtionall between the intercepted Diameter , and its contiguous ordinate line , to find it . Pray tell me where you observed that . For , had I observed it , I should have observed it as a great fault ; and not said as you doe , And 't is true , I find it . For , believe mee , that is not the way to find a Parameter . Nor doe I give you any such direction . You may ( in a Parabola ) find the Parameter by taking a third Proportionall , but not by taking a Mean Proportionall , to those two lines . You say , The Parameter hath a determined quantity . Yes doubtlesse . And , in some Writers , it hath a determined Position too ( viz. in the Tangent of the vertex : ) But because I make no use of any such position , I give you leave either to draw it where you will , or not to draw it at all . For by a Parameter , I mean only , a line of such a length , where ever it be ; whether at Rome , or Naples , or in M. Hobs his brain . They that make use of the Parameters position , as inferring any thing from it , must assigne it a certain place . I make use only of its bignesse , and therefore care not where it stand . Lastly you observe , you say , that I doe not shew how to find the Focus . ( nor was ) bound to doe . ) And that 's all . And is not this a worthy confutation ? Yes doubtlesse ; worthy of you ; For how could you else inferre , That I have performed nothing in any of my Books ; if you had not confuted them all . And thus much of those three Treatises . Which , you see are come off safe and sound , without the losse of leg or limb . And with this advantage , ( if M. Hobs his testimony , in point of skill , were worth any thing , ) that they have obtained from him as ample a Testimony as he is able to give . viz. That when as he hath imployed the utmost both of his skill and malice , to find what faults he could , he hath not been able to discover any one : ( which Testimony , from a considerable adversary , would have been worth something ; but , from M. Hobs , J confesse , it signifies little : ) and all the attempts he hath made to that purpose , have not been so strong , but that a Butter-fly might have broken through them . SECT . VII . Concerning the Eighth Chapter in M. Hobs his Book of Body . HItherto we have tryed your skill and valour in point of Assault : And found , that , though you charge as furiously as if you meant to look us dead ; yet you come off as poorly as a man could wish . J am apt to think , that your weapons were not well made , and that your Musket was of a bad bore , ( for it hath done no execution , save only in the recoile ; ) or else you held it by the wrong end , ( like the Jack-an-Ape that peep'd in the gunns mouth to see the bullet come out , ) for though it made a great noyse , yet it hath hurt no body but your selfe . My Colleague and I , are both of us alive , and live-like ; and Euclide sleeps as securely as he did before . Wee 'l try now , how good you are in point of Defense ; and see how you can defend your Corpus against my Elenchus . Perhaps you may have better luck at that . But , mee thinks , it begins unluckyly . Before you fall to work with Elenchus ; you traverse your ground , that you may take it to the best advantage : and distinguish , between faults of Ignorance , and faults of Negligence , ( pag. 9. ) you tell us that from right Principles to draw false Conclusions ( which you are very good at ) are but faults of negligence and humane frailty , and such as are not attended with shame , &c. That 't is only as being lesse awake , &c. ( and yet think much to be told , that you discourse as if you were halfe a sleep : ) And much more your preface to that purpose . As if the first consideration to be had , in the choice of your ground , were , whence you might with best advantage runne away ; ( a businesse of ill Omen in the beginning of a Combate ; ) that when you shall be forced to quit your ground , you may , at least , shew a fair pair of heeles . My Elenchus , as I then told you , begins at first with some lighter skirmishes , shewing how unhandsome some of your Definitions and Distributions are , giving instance in a few ; which though faults had enough , yet are but small ones in comparison of those greater which follow , in false Propositions and Demonstrations . I begin with that of Chap. 8. § 12. Where you define a line , a length , a point , in this manner . If when a Body is moved , its magnitude ( though it alwaies have some ) be not all considered , the way it makes is called a Line , or one single dimension ; the space through which it passeth is called Length ; and the Body it selfe a Point . But what if a Body be not moved ? i● there then neither Point , nor Line , nor Length ? A Point there may be , which is not a Body , much lesse A Body moved : and a Line , or Length , through which no body passeth : And therefore the definitions are not good , because not reciprocall . The Axis of the Earth , is a Line , and that line hath its Length ; yet doe I not believe that any Body doth , or ever did , passe directly from the one to the other Pole , to describe that Line . The notion therefore of Motion or Body moved , I then said , was wholly extrinsecall and accidentall to the notion of Line , or Length , or of a Point ; no waies essentiall or necessary to it , or to the understanding of it : and that therefore it was not convenient , to clog the definitions of these , with the notion of that . To this you answer , ( having waved first , what you attempted , as from the example of Euclide , ) That , how ever it may be to others , it was fit for you to define a Line by Motion . And I doe acquiesse in that Answer . For , though it would not become any man else so to define it ; yet it becomes M. Hobs very well ; as well agreeing with his accuratenesse in other things . I said farther , That the distance of two points though resting , was a Length , as well as the measure of a passage , ( and therefore the notion of a body moved , not necessary to the definition of Length . ) To which you answer , that the distance of the two ends of a thread wound up into a Clew , is not the length of the thread . Much to the purpose . I asked , Whoever defined a Line to be a Body ? And you tell mee , you take it for an honour to be the first that doe so . And you may , for ought I know , have also the honour to be the last . And as to that long rant against Euclide ; That if a Point have no parts , and so no magnitude ; A Line can have no breadth , nor can be drawn ( mechanically you mean ; ) and then there is not in Euclide one Proposition demonstrated , or demonstrable . We doe not think , that your asseveration a sufficient argument , more than we take a word of your mouth to be a slander ; but desire some better proofe of that consequence before we assent to it . You tell us else where , that A Point is to Magnitude , as a ciphar is to Number ( cap. 16 art . 20. ) And yet I suppose you will not say that , unlesse a Ciphar have some multitude , as well as a Point some Magnitude , there is not in Euclide any one Proposition demonstrated . And to the same purpose is that Cap. 14. § 16. An angle of contingence , if compared with an angle simply so called how little so ever , hath such proportion to it , as a point to a Line , that is , ( neque rationem , neque quantitatem ullam , ) no proportion , nor any quantity at all . Which how well it agrees with your other doctrines , it concerns you to see to , ( for if a Point to a Line , have no proportion nor any quantity at all , then is it not a Part thereof ; ) and how little this comes short of what you so often rant at , as making a Point to be nothing . Again , whereas in the place cited ( both in Latine and English ) you thus define ; The Way ( of the Body so moved ) is called a Line , or one single Dimension ; and the Space through which it passeth , is called Length . I argued , that Length , doubtlesse , was one single dimension ; and therefore , if one single dimension , as in your definition , be the same with Line ; then Length will be a Line , and not therefore need a second definition . Now , to help the matter , in your Lessons ; you define thus , The Way is called a Line ; and the space gone over by that motion , Length or one single dimension . Whence my argument is yet farther inforced , If one single dimension signify the same with Line , ( as in your Book ; ) and also the same with Length , ( as in your Lesson ; ) then Line and Length signify with you the same thing ; & therefore with you , should not have had two distinct and different definitions . Which I take to be ad hominem , a good argument . You answer , that to say Line is Length , proceeds from want of understanding English . It may be so . But what 's this to the clearing of your Definitions ? where those two words are made equivalent . Yet farther , chap. 12. parag . 1. there are , say you , three dimensions , Line ( or Length , ) Superficies , and Solid . where again Line and Length are made the same . Now whether or no Line be Length , or whether it be for want of understanding English that you affirme it , it concernes you to cleare ; for 't is you , not I , that affirme it so to be . Your next definition is of Equall Bodies ; which you thus define , Equall Bodies , are those which may possesse the same place . Against which definition J objected , That you should rather define a thing , by what it is , then by what it may be : That the notion of Place , was wholly extrinsecall to the notion of Equality ; for Time , Tone , Numbers , Proportions , and many other quantities are capable of Equality , without any connotation of Place ; and the notion of Equality in them , is the same notion with that of Equality in Bodies ; ( else how can you say , that two Equall Numbers , and two Equall Bodies , are in the same Proportion ; ) And therefore , That one good definition of Equality , or Equalls , in generall ; had been much better , then so many particulars , of Equall Bodies , Equall Magnitudes , Equall Motions , Equall Times , Equall Swiftnesse , &c. as you here bring ; and yet , when you have all done , there be a great many more Equalls , which you leave undefined : ( And your bare assertion , That there is no Subject of Quantity , or of Equality , or of any other Accident , but Body , doth not help the matter at all ; for we are not bound to take your word for it : ) That , if you would needs mention place , you should rather have defined them by the place they have , then what they may have ; & so , defined those bodies to be Equall , which do possesse Equall places , rather then , which may possesse the same place : That a Pyramid , remaining a Pyramid , may be Equall to a Cube ; yet cannot , remaining a Pyramid , possesse the place of that Cube : Or , if you will , That a Pyramidall Atome , though so Adamantine as to be incapable of any transmutation , ( as those who teach the doctrine of Atomes doe maintain , ) may yet be equall to a Cubicall Atome , though not possesse the place thereof : That you might as well have defined a Man , to be one who may be Prince of Transilvania , as to define Equall Bodies , to be those which May possesse the same place . ( with much more , of which you take no notice . ) To that last particular , you answer , that 't is wittily objected , as I count witt , but impertinently . And why impertinently ? Is not that definition of a Man ; as good as yours of Bodies Equall ? You think not , Because if so , J must be of opinion , That the possibility of being Prince of Transilvania , is no lesse essentiall to a Man ; then the possibility of being in the same Place , is essentiall to Equall Bodies . And truly J am of that opinion . J think it every way as Possible for for any man living , to be Prince of Transilvania ; as for the Arctick and Antartick Circles , ( or the Segments of the Sphere which they cut off , ) be they never so Equall , to possesse the same place . Nor is that possibility lesse essentiall , than this . You adde , That there is no man ( beside such Egregious Geometricians as we are ) that inquires the Equality of two bodies , but by measure : And , as for liquid bodies , &c. by putting them one after another into the same vessell , that is to say , into the same place ; And , as for hard bodies , they inquire their Equality ●y weight . To which I shall reply nothing at all ; because you speak therein so like a Geometrician . I objected farther , That it is not yet agreed amongst Philosophers ( and your authority will not decide the controversy , ) whether or no , the same body may not , by Rarefaction and Condensation , ( words understood by other men , though you understand them not , ) sometimes possesse a bigger , some times a lesser place . We see , that the same Air in the head of a weather-glasse , doth sometimes possesse a bigger , sometimes a lesser part of the glasse , according as the Weather is cold or hot , and you cannot deny , ( what ever others may ) but that both are filled ; for you doe not allow any Vacuum at all . We know , that into a Wind-gun , though it were full ( you say ) before , yet much more Air may be forced in . And into the Artificiall Fountain , ( which you mention Cap. 26. fig 2. ) though full of Air , may be forced also a great quantity of Water . Now how to salve these Phaenomena , ( with many others of the like kind ) without either allowing Vacuum , which you deny ; or Condensation , which you laugh at ; ( one of which others use to assigne ) because you find it too hard a task for you to undertake , ( as well you may , ) you leave to a m●lius inquirendum p. 144. l. 27. ( or in the English , p. 316. l. 34. ) Now if it be true , that the same body doth , or possible , that it may , possesse , some time a bigger , sometime a lesse space , ( as those who deny Vacuum doe generally affirme , ) then , by your definition , the same body ( I doe not say may possibly become , but ) at present is both bigger , and lesse , and equall to it selfe : Because it hath at present a possibility of possessing hereafter both a larger place , by Rarefaction , and a lesser place , by condensation , than now it doth . And so you , by determining the equality or inequality of Bodies , not by the place they have , but by such place as possibly they may have ( upon any supposed metamorphosis or transmutation , ) doe confound Bigger , and Lesse , and Equall , and so take away the whole foundation of Mathematicks : For if there be no difference between Bigger , Lesse , and Equall , there is no roome either for Mathematicks or Measure . But , whether that opinion of Rarefaction and Condensation be true or not : yet since you cannot deny , but that it is at least a considerable controversy , and , by men as wise , and as good Philosophers as M. Hobs , maintained against you ; yea and a Controversy not belonging to Mathematicks but Physicks , or Naturall Philosophy , and there to be determined ; it was not wisdome to hang the whole weight of Mathematicks , upon so slender a thread , as the decision of that controversy in Naturall Philosophy , which whether way it be determined , is wholly impertinent to a Mathematicall Definition . To which you reply onely this , ( which is easy to say ) that Rarefying and Condensing , are but empty words ; and that ( of which we have spoken already ) Mathematicall Definition , is not a good phrase . To that definition you had annexed this also ; Eadem ratione , magnitudo magnitudini , &c. Vpon the same account one Magnitude is equall , or greater , or lesser , then another , when the bodies whose they are , are greater , equall , or lesse . These words , I said , must bear one of these two ●enses , either , that Equall Bodies , or Bodies equally great , are of equall greatnesse , ( which is no very profound notion : ) or else , that the magnitudes , towlt the lines , superficies , &c. or at least , the length , bredth , &c. of Equall bodies , is Equall , ( taking the words for a definition of Equall Lines , Equall Superficies , &c ) and this , I said , was manifestly false : for no bodies may be equall , whose length , breadth , superficies , &c. are unequall . You say now , that you meant the former , ( and I cannot contradict it , for you know your own meaning best , yet you must give me leave to think : ) and so leave us without any definition of Equall Lines , Plaines , or Superficies Which yet , considering how oft you are afterwards to . make use of , might have been as worthy of a definition , as some of those equalls that you have defined . In the next Paragraph , Cap. 8. parag . 14. you undertake to prove , that one and the same Body , is alwaies of one and the same magnitude , and not bigger at one time then another , or at one time fill a bigger place , than it doth at another time . Let 's heare how you prove it ( for , by what we heard but now , you are much concerned to make good proofe of it , because if there be a possibility of possessing at any time a bigger or lesse place than now it doth , than it is , by your definition , at present bigger or lesse than it selfe . ) Your proofe is in these words , For seeing a Body , and the Magnitude , and the Place thereof , cannot be comprehended in the mind otherwise than as they are coincident , ( observe therefore , that this argument doth no more prove , that a Body cannot change its Magnitude , than that it cannot change its Place , for you make Place as much coincident with Body , as you doe Magnitude , and the argument proceeds equally of both : ) if any Body be understood to be at rest , that is , to remain in the same place during some time , and the Magnitude thereof be in one part of the time greater , and in another part lesse , that Bodies place , which is one and the same , will be coincident sometime with greater , sometime with lesse magnitude , that is , the same Place will be greater and lesse than it selfe , which is impossible . This is your whole proof to a word . Now this , I told you , is no sufficient proof , because it proves only that a Body doth not change its quantity so long as it is at rest , and doth precisely keep the same place ; ( which no body doth affirme . ) And , pray look upon the Argument once again : doth it prove any more than so ? But that which you undertook to prove was , that it doth never change its magnitude , but hath alwaies the same , as well when its place is altered , as when it remains in the same place : ( for , J suppose , you will not deny , but that a Body may change its place . ) Those that hold the contrary opinion , doe not say that a body doth change its greatnesse while it doth precisely keep the same place ; but that , with change of place , it may change its dimensions too : And to this , if you would have said any thing , you should have applied your argument . And is not this then a just exception to your argument ? Will this argument hold , think you , Because a Body doth not change its magnitude so long as it keeps precisely the same place : Therefore , it never changeth its magnitude , but hath alwaies the same ? This argument hath no appearance of consequence , but only upon this supposition , that a Body doth alwaies keep precisely the same place . And , then , I confesse , the Argument looks like an Argument , in this forme , So long as a body keeps precisely one and the same place , it hath precisely one & the same Magnitude : But a Body doth alwaies keep precisely one and the same Place : Therefore it hath alwaies one and the same Magnitude . And if this be your argument , we allow the form , but deny the matter of it , and say , the Minor ought to be proved . For we are of opinion , that it is possible , for the same Body , not to be alwaies in the same place . If you think otherwise , pray prove it . For 'till that be proved , your present argument is to no purpose . Sed rem ita per se manifestam , demonstrare opus non esset , &c. But , say you , a thing of it selfe so manifest , would need no Demonstration at all , ( a fine facile way of Demonstration , that which you know not how to prove needs no demonstration . ) but that you see there are some , whose opinion concerning Bodies and their magnitude , is , that Body may exist separated from its magnitude , ( no not so , but that it may change its magnitude , For they doe no more believe that it can exsist without Magnitude , than that it can exsist without a Figure : It cannot be but that a finite Body must have alwaies some figure , though not alwaies the same : and so alwaies some Magnitude , but whether alwaies the same or no , you should have proved if you could : ) and have greater or lesse magnitude bestowed upon it ; ( as well as different figures : ) Making use of this principle for the explication of the nature of Rarum , and Densum . Since therefore you know there are that do so ; why did not you , ( at least in your English Editition , after you had notice of the weaknesse of your Latine Argument ) bring some good Argument to overthrow that opinion ; and not content your selfe to say that it is so manifest of it selfe , as that it needed no demonstration . Especially , ( as I then told you ) since you doe not allow that Euclide may assume to himselfe gratis without demonstration , That the whole is greater than its part ; ( those were my words , though you recite them a little otherwise . ) But you say , I know this to be untrue , that is , I lye : My words were these ; Non interim Euclidi permittis , ut citra demonstrationem hoc sibi gratis assumat , Totum esse majus sua parte : that is , You do not allow it Euclide , that he may without Demonstration assume to himselfe , or challenge , That the whole is greater then its part . Now let your own words be judge , who is the lyar , you or I. Cap. 6. artic . 12 , 13. The whole method of Demonstration , you say , is Syntheticall , — beginning with Principles , or primary Propositions . Now such Principles are nothing but Definitions , — And , Besides Definitions , there is no other Proposition that ought to be called Primary or ( si paulo severius agere volumus ) be received into the number of Principles . For those Axioms of Euclide , seeing they may be demonstrated , are no Principles of Demonstration . And accordingly art . 16. you define Demonstration , to be a syllogisme , or series of syllegismes , derived and continued from the Definitions of names , to the last conclusion . And parag . 17. You require to a Demonstration , That , the premises of all Syllogismes be demonstrated from the first Definitions . ( And the like cap. 20. parag . 6. diverse times . ) So that these Axioms , being no Definitions , nor any Principles of Demonstration , no Demonstration can take rise from them , nor can they be otherwise assumed in demonstration , than as they are themselves deduced or demonstrated from Definitions . And doth not this come home to what I said ? And cap. 8. parag . 25. Of which Axioms ( omitting the rest ) I will only ( say you ) demonstrate this one , The whole is greater then any part thereof . To the end that the Reader may know , that those Axioms are not indemonstrable , and therefore not Principles of Demonstration . And yet again Less . 1. p. 4. As for the commonly received third sort of Principles , called Common Notions , they are Principles only by permission of him that is a Disciple ; who being ingenuous , and coming not to cavill but to learn , is content to receive them ( though demonstrable ) without their demonstration . And again pag. 9. you exclude those common notions called Axioms , from the number of Principles , as being demonstrable from the definitions of their termes , acknowledging no other Principles , but Definitions , and Postulata , ( those the only principles of Demonstration ; these of Construction . ) If therefore they be no Principles of Demonstration ; if only principles by permission of the Disciple , and only in curtesy ; then , though your selfe possibly may he so gracious or liberall , as to admit of them without their demonstration ; Yet the Teacher cannot , without this favour , assume to himselfe , or require them to be granted , as he may doe Principles , without Demonstration . 'T was not I therefore was the lyar , when I said , You doe not allow that Euclide may assume to himselfe gratis , or require to be granted , without demonstration , That the whole is greater than its part . For 't is but in courtesy , if you grant it him , as you may any other true Proposition , and only upon supposition that it may be demonstrated : upon which supposition , you may also allow all the Propositions in Euclide , for they may be all demonstrated . And thus much concerning your eight Chapter . SECT . VIII . Concerning his 11 , and 13 Chapters . WEE shall next consider what you have to say in defense of your 11 and 13 Chapters , concerning Proportion . And here after a freak ; and then a rant against Euclide ; you have a large discourse about Proportion ; p. 15 , 16. The summe of which , so farre as is to the purpose , is this , That there betwo kinds of Proportion , ( as the word is now adaies taken ; ) the one of which is called Arithmeticall Proportion ; the other , Geometricall Proportion : And as the Quotient gives us a measure of the Proportion of the Dividend to the Divisor , in Geometricall Proportion ; so the Remainder , after subtraction , is the measure of Proportion Arithmeticall . Pag. 16. And thus much is both true and clear , and to the purpose . And had you but thus delivered your doctrine of Proportions , in your Book de Corpore , I should never have found fault with it . But you , not knowing ( till you learned it out of my Elenchus , ) that the Quotient did as well determine Geometricall Proportion , ( and give name to it ) as the Remainder doth Proportion Arithmeticall , were fain to blunder on as well as you could , without it : and put your selfe upon a great many unhandsome shifts , and which will not hold water , to give account , even of Geometricall Proportion , from the Remainder or difference , which was not to be done otherwise then by the Quotient , as you here clearly confesse ; For the Measure , you say , of Geometricall progression , is ( not the Remainder , whether absolutely or comparatively considered , but ) the Quotient . But before you come thus farre ; you tell us by the way , That I say , that you make proportion to consist in the Remainder , and that I make it consist in the Quotient . As to the former of these , I did not then say , that you make proportion to consist in the Remainder ; though if I had said so , I had said true enough , for you doe so , more than once . Cap. 11. parag . 7. In ratione inaequalium , say you , ratio minoris ad majus , Defectus ; ratio majoris ad minus Excessus dicitur . And again par . 5. Consistit ratio antecedentis ad consequens in differentia , &c. sive in majoris ( dempto minore ) Refiduo . And. soon after , Ratio binarii ad quinarium est ternarius , &c. You cannot deny but that these are your words , and that I blamed you for them , as a piece of non sense ; all that you have to say is , that it was too hastily put : & therefore you labour in the English a little to disguise it . So cap. 12. art . 8. Cum Ratio inaequalium , per cap. praeced . art . 5. consistit in differe●tia ipsarum , &c. and again , Ratio inaequalium , EG , EF , consistit in differentia EF , quae est quantitas , ( yes , quantitas absoluta , for 't is a line . ) And these , because I did not particularly tell you of them , are yet uncorrected in your English ; seeing ( by the fifth Article of the precedent Chapter , ) the proportion of two unequall magnitudes consists in their difference , &c. And again , the Proportion of unequalls EG , EF , is quantity ; for the difference GF , in which it consists is quantity . Now when , you say in expresse words , as in the places cited , The proportion of the antecedent to the consequent consists in the Difference , or the Remainder ; it had been no wrong if I had said , as you say I doe , that you make Proportion to consist in the Remainder ; and that absurdly enough . And then , J pray , to whom belong those reproaches , that are so oft in your mouth , as if somebody did affirme , that Proportion is a Number , an Absolute quantity , & c ? is it not your selfe that affirme it so to be ? And doth any body so beside your selfe ? And is not then , that ( by your own law p. 10 , ) in your selfe intolerable , which you cannot tolerate in another ? But you adde farther , that I say , that I make it to consist in the Quotient . And is not this abominably false ? J neither say so , nor doe so , nor did J give any ground at all for any man ( that is in his witts ) to believe J did . My words were these , Videmus igitur Rationis aestimationem esse ( secundum Te ) penes Residuum , non penes Quotum , & Subductione , non Divisione quaerendam esse . ( And what reason J had to say so , they that consult the place will see . ) Now could any man ( who had not a great confidence that his English Reader understands no Latine ) be so impudent as to say , that in those words , I say , you make Proportion to consist in the Remainder ; and I , in the Quotient ? Can any man , that understands , though but a little Latine , ( if he be not either out of his witts , or halfe a sleep , ) think that these words Rationis aestimatio est penes Quotum , ( that is , the Proportion is to be estimated according to the Quotient , or , to use your own words , the quotient gives us the measure of the proportion , ) could be thus Englished , proportion consists in the quotient ? And that then you should raile at us , quite through your Book , for saying that Proportion is a certain quotient , that it is a number , that it is an absolute quantity , &c. as if we had been so ridiculous as to speak like you . For , that you have so spoken you cannot deny , ( and therefore the absurdity what ever it be , lights upon your selfe : ) But , to say , that I said so , or any thing to that purpose , till you can shew where I said it , J take to be , ( so farre as a word of your mouth can be ) a manifest slander . J neither say so , nor think so . Now some men perhaps may wonder , there should be so great a cry and so little wooll ; they would think perhaps , by what you say , that J had somewhere said in expresse termes , that Proportion is a Quotient , or that it consists in the Quotient , or that it is a number , or an absolute quantity , or that the quotient is the proportion , or that a Proportion is the double of a Number , but not of a proportion , or somewhat that sounds like somewhat of these , when they hear me thus charged , again and again , many a time , and oft ; and not that the whole ground of the accusation had been but this , that I said , The proportion is to be estimated by the quotient . And truly 't is somewhat hard to give a good account of it : yet wee 'l try what may be done . J was told , some years a goe , of a man that had told a lye so often , and with so much confidence , that at length he began to believe it himselfe . And J am almost of opinion , that M. ●obs having now said it so often over , doth , by this time , begin to think , that J had indeed said , somewhere , that the quotient was the proportion . And truly there is some reason why he should : For if he had heard any other man so oft and so confidently affirme it , he would no doubt have believed him : and why should he not as well believe himselfe . But moreover ; It did perhaps runne in his mind , that he had somewhere read some such words as these , Consistit autem Ratio antecedentis ad consequens , in Differentia , hoc est in ea parte majoris qua minus ab eo superatur ; sive in majoris ( dempto minore ) Residuo . Or such as these , Ratio binarii ad quinarium est ternarius . Or else this , Ratio minoris ad majus , Defectus ; ratio majoris ad minus , Excessus dicitur . ( And well it might : for they are all his own words , Cap 11. parag . 3. & 5. and Cap. 12. parag . 8. ) And he might think , that to say thus , was all one , as to affirme Proportion to be a Number , or an Absolute quantity : ( And truly I think so too . ) And that therefore the expression was very absurd ; ( For so I had intimated to him in my Elenchus , upon this occasion . ) And therefore ( forgetting , perhaps , that they were his own words , and not mine . ) he doth ( like the Woman that called her daughter Bastard , not minding that in so doing shee called her selfe Whore , ) exclaim against his own words , as most ridiculous non-sense . And who might doe it better ? Or else , to use his own comparison , like Women of poor and evill education , when they scold ; amongst whom the readiest disgracefull word is Whore ; because , when they remember themselves , they think that reproach the likeliest to be true ; at least , if they be called Whore themselves , though never so truly , they will be sure to call Whore again at all adventures , hit or misse . So M. Hobbs , finding himselfe to have been so absurd , as to make Proportion a Number , or Absolute quantity , and that I had blamed him for it ; thought , perhaps , it was possible I might , sometime or other , have been as carelesse in my language : and therefore , however , hee 'l say so , ( 't is easy to say it ) and let me disprove it . If any man , notwithstanding all this , be not satisfied that M. Hobs had reason to say as he doth ; truly I cannot help it ; he must speak for himselfe : These were the best reasons I could think of ▪ And so wee 'l goe on . In your 11 Chap. parag . 3. you gave us in the Latine , ( for in the English there be some things altered , ) this definition of Proportion ; Proportion is nothing else but the aequality or in equality of the Antecedent , compared with the consequent , according to magnitude . With this Explication , As for example , the proportion of Three to Two , is nothing else , but , that Three , is greater then Two , by One : and the proportion of Two to Five , is nothing else , but that Two , is lesse than Five by Three : And therefore in the proportion of Vnequalls , the proportion of the Lesse to the Greater is called the Defect ; and that of the Greater to the Lesse , the Excesse . And this is your generall definition of Proportion , with the Explication of it ; and nor a particular definition of Arithmeticall Proportions , ( nor is it at all by you pretended so to be . ) And therefore should have been so ordered , as at least to take in Geometricall Proportion ; For Geometricall proportion , and simply proportion , are by your selfe made equivalent termes ( Less . 2. p. 16. l. 25. ) and this , you say , is onely taken notice of by the name of Proportion : And , so the word is constantly used in Euclide , and elsewhere : ( And therefore you need not wonder as you doe p. 18. l. 7 , that J should say , If Arithmeticall Proportion , ought to be called Proportion ; implying that though now that phrase be common , yet that it is a departing from the former use of the word ; and that , according to Euclides use of the word Proportion , Arithmeticall Proportion cannot be so called . ) Now your Definition and Explication of Proportion , doth wholly leave out Geometricall Proportion altogether , ( which yet is , if not the only , yet the more principall kind of Proportion . ) For it takes no cognizance of the Quotient at all , but only of the Difference , the excesse or defect . And according to your doctrine the Proportion of 3 to 2 , is + 1 , the excesse of 1 ; and of 2 to 5 , is -3 , the defect of three . From this I inferred , that if the proportion of one quantity to another , be nothing else , but the excesse or defect of this to that , ( as you teach , ) then where ever the excesse or defect is the same , there the proportion is the same ; and so 3 to 2 , must have the same proportion that 5 hath to 4 ; ( You say , p. 17. True , the same Arithmeticall Proportion Very good : But J added farther , of which you did not think fit to take notice , ) and on the contrary , where there is not the same defect or the same excesse there is not the same proportion , and consequently , there is not the same proportion of 3 to 2 and of 6 to 4. To this you have nothing to say , and therefore say nothing , ( but recite halfe my sentence , and leave out the other halfe : ) For though , there be not the same Arithmeticall Proportion ( as you speak ) of 3 to 2 , and of 6 to 4 ; ( that is , not the same excesse , ) yet there is the same Geometricall Proportion ; and that you cannot deny to be Proportion , though it doe not come , within your definition . Now it 's true , ( but that 's another fault , not an excuse ) that you do not hold to this sense alwaies , for in the same page art . 5. ( in the Latine , I mean ) you do clearly contradict what you had but now said in art . 3. The proportion , say you , of the Antecedent to the consequent consists in the Difference , or Remainder , not simply ( yes simply , if that be true which you said before ; for if it be nothing else but the difference , that is it the difference simply : But if not simply ; how then ? ) but as compared with one of the termes related , &c. For though there be the same difference between 2 and 5 , that there is between 9 and 12 , yet not the same Proportion . And why not ? as well as the same proportion between 3 and 2 , and between 4 and 5 ? as we heard you reply but now . May not we as well say here , as you there , ( Les . 2. p. 17. ) Is there not the same Arithmeticall Proportion ? And is not Arithmeticall proportion , proportion ? But it seems , by this time , you had forgotten your former exposition , whereby in the same page , your definition of Proportion must be so understood , as will agree to none but Arithmeticall proportion ; now it must bear such a sense as can agree to none but Geometricall . In the English , I confesse , your Translator hath a little mended the matter , and but a little , ( 't is but Coblers work at the best ; ) But however , 't is good to hear folks mend , though it be but a little : it may come to something in time . But now of those two senses , which you have given , of the Definition of Proportion , ( opposite enough in conscience one to another , though , I suppose , you did not intend therein to contradict your selfe , ) neither of them will serve your turn . For the Proportion here defined , and so explicated as we have heard , is a Genus , which is , in the beginning of your 13 Chapter , to be distributed into its two Species ; Proportion Arithmeticall , and Proportion Geometricall . Now take your definition of Proportion in generall , according to which of your two expositions you please , it cannot be thus distributed . For if Propor●●on ( as you say chap. 11 , ●art . 3. ) be nothing else but the excesse or defect , &c. as 3 is lesse then 2 by 1 ; then it cannot agree to Geometricall proportion , for that is somewhat else . If it be such a comparative difference , as you mention cap. 11. art . 3. it will not agree to Arithmeticall proportion ; for according to that sense , you say , 2 to 5 , and 9 to 12 , are not in the same proportion . I say therefore , that neither of those two expositions , do agree to that generall notion of Proportion , which shall be common to both Arithmeticall and Geometricall . And when I aske , which of the two expositions you are willing to stand to . Whether that of Cap. 11. art . 3. or that of Cap , 11. art . 5. ( shewing withall that neither of them will serve your turne , for neither of them will take in both Arithmeticall and Geometricall Progression , ) you fall a raving in the beginning of your third Lesson , something at Euclide , and something at us , but nothing to the purpose . And then tell us , that when you say the Difference is the Proportion , by Difference , we might if we would , have understood , the act of Differing . That is , wee might understand , as madly as you speak . Your words were these , Cap. 11. art . 5. Consistit autem Ratio in Differentia , sive Residuo , &c. ita ratio binarii ad quinarium est ternarius , &c. Would you have us understand Residuum , and Ternarius , to be the Act of Differing ? And C. 12. art . 8. Ratio inaequaliū ( EG , EF ) consistit in differentia GF . Would you have us understand that line GF , to be the act of differing ? You say , we might if we would . But you 'ld think us very simple if we should . To as good purpose is it , that you tell your English Reader ( for you think you may tell him any thing , ) that ● say , that ( thus much of ) your Definition , Ch. 11. Art. 1. [ Proportion is the Comparison of two Magnitudes one to another , ] agrees neither with Arithmeticall nor Geometricall proportion . For I said nothing of any such words , good or bad . And 't were much if I should : for I can find no such words there . At the second Article ( chap. 13. ) I note , you say , for a fault in method , that after you had used the words , Antecedent , and Consequent of a Proportion , in the precedent Chapters , you now define them . 'T is true , I did take notice of it , but I said withall , that this was but a small fault in comparison of many others . But what if I did ? You do not believe , you say that I spake this against my knowledge . No ; why should you for you know 't is true . Have you not used the words many times before in the precedent chapters ? And doe you not define them here ? And is not this a fault in Method ? Do Mathematicians use , when they have taken a Terme for two or three chapters together , to be of a known signification , and sufficiently understood , come at length to define it ? you say , you had before defined it chap. 11. art . 3. 'T is true you had there defined the Antecedent and Consequent of Correlatives ; ( which definitions might have served well enough for the Antecedent and consequent in Proportions too , for those are Correlatives , and you need not have brought any new ones . ) But where was my oversight ? Did I deny this ? I did not blame you for using the words before you had defined them , ( nor would I have blamed you , if they had not been defined at all ; ) But for defining them after you had thus long used them . For , if they had now , ever since the beginning of the 11 Chapter , been taken for words of a known signification , and as such frequently used , ( which you do not deny , and your definitions at that place do but aggravate , not extenuate , this charge , ) then , I say , it was immethodicall and superfluous to define them in the 13 chapter . Nor was it my oversight to say so . And the like impertinent answer you give p. 51. where I blamed you ( not for omitting in the 19 chapter , but ) for defining in the 24 chapter , those termes which were of frequent use in the 19 chapter . But wee go on . You tell us , Chap. 13. art . 3. That the proportion of Inequality is Quantity , but that of Equality is not . Which I said was very absurd ; and that the one did no more belong to the Praedicament of Quantity than the other ; and that it is to bee , of both equally , either denied or affirmed : And that your argument for it , ( That One equality is not greater or lesse then another ; but of proportions of inequality , one may be more or lesse unequall : ) might as well conclude that Oblique angles , be quantities , but not Right angles , for these be all equall , and equally Right ; but not those . For answer to this , you fall a ranting at Aristotle , at Praedicaments , and the L●gick Schooles , &c. And then you tell us the Greater and Lesser cannot be attributed to Right Angles , because a Right Angle is a Quantity determined , ( as though the quantity of the Proportion of Equality were not so too . ) What you alledge out of Mersennus , was but his mistake . Composition of Proportion is a work of Multiplication , not of Addition , as appears by the definition of it 5 d 6. and to argue , that Proportion of equality is as Nothing , because in composition of Proportions it doth not increase or diminish another proportion ; is but as to conclude that , 1 , a Vnity , is Nothing , because in Multiplication it doth neither increase nor diminish the quantity multiplyed thereby . But of this mistake of Mersennus , I have spoken already in the end of another Treatise , already Printed , against Meibomius ; and vindicated Clavius sufficiently from what both Mersennus and Meibomius allege against him . To the fourth Article , where you define Greater and Lesser Proportion ; I said nothing ( because it were endlesse to note all the faults I see ) though those definitions are liable enough to censure . Greater Proportion , you say , is the proportion of a greater Antecedent to the same Consequent , or of the same Antecedent to a lesse Consequent . And Lesse Proportion , is the proportion of a lesse Antecedent to the same Consequent , or of the same Antecedent to a greater consequent . Yet we know , that the proportion of an Ell to a Yard , is lesse then that of a Pottle to a Pint , ( and this therefore greater then that , ) though neither the Antecedents nor the Consequents , be either the same , or Equall , or Homogeneous . To the 5 and 6 Articles , where you define the same Proportion . I said First , that , had Proportion been well defined before , you might have spared these definitions of the same proportion . For having before defined ( as well as you could ) what is Proportion ( both Arithmeticall , and Geometricall ; ) and withall told us , art . 4. that by the same proportion was meant Equall proportions ; and having also defined before ( after your fashion ) what are Equalls chap. 8. and what is the Same chap. 11. Why should you think ( if those definitions were such as they should have been ) that wee needed another definition of the Same , or Equall Proportions ? But , since you were resolved to doe works of Supererogation ; I ask why , having defined the same Arithmeticall proportion , art : 5. by the Equality of the Differences ; you did not also define the same Geometricall Proportion , art . 6 , by the Equality of the Quotients ? For by the Same , you say , you mean Equall , art . 4. Now universally all quantities are Equall , that are measured by the same number of the same Measures ( Less : 1 p : 4. ) and therefore those are the same or equall Proportions , which have the same or equall Measures : And you know now ( though perhaps you did not then ) that as the Quotient gives us a measure of the Proportion in Geometricall Proportion , so the Remainder is the Measure of Proportion Arithmeticall . ( Les : 2. p. 16. ) And therefore , as , in the one , you define the same or equall proportion , by the Equality of the Remainder ; so you should in the other , by the equality of the Quotient , ( that is , in both places by the equality of its measure : ) And not have brought us such an imbrangled definition as this . viz : One Geometricall progression is the same with another , when a cause in equall times troducing equall effects , determining the proportion , may be assigned the same in both , or as your English hath it , when the same cause producing equall effects in equall times , determines both the proportions . So that , to prove , that 4 to 2 , and 6 to 3 , are in the same Geometricall proportion , we must call in the help of Time , and Motion , and Velocity , and Vniformity , &c. which are wholly extrinsecall to it ; and why , but because , forsooth , there is no effect in Nature which is not produced in Time by Motion , ( as though some Motion , in some Time or other , had made this to be a true Proposition , that 4 is the double of 2 : and therefore if we cannot find what motion did make it so , we must imagine some that might have made it . ) I need not tell you , that , if this be a good reason , you should upon the same account , have found out as bad a definition for the same Arithmeticall proportion : ( for that 8 to 6 , and 12 to 10 , are in the same Arithmeticall proportion , is , doubtlesse , as much as that other of Geometricall proportion , an effect which nature hath at some Time or other produced by Motion . ) But , since you have waved this consideration of nature in the definition of the same Arithmeticall proportion , which you define by the equality of the Remainders ; I said , it might have been expected , that you might have done so in the definition of the same Geometricall proportion● , and accordingly defined it , by the Equality of the Quotients . But you are very angry with me , for saying , It might have been expected . And truly I could almost find in my heart to confesse that this was a fault . For though it might have been expected from another man ; yet it was not to be expected from M. Hobs ; for his witt is not like the witt of other men , He is the First ( he tells us ) that hath made the grounds of Geometry firm and coherent . But why was it not to be exspected ? Because , you say , It is impossible to define ( Geometricall ) proportion universally by comparing Quotients . ( Impossible , I confesse , is a hard word ; but yet , I hope , it may be . ) But why is it impossible ? more than it is impossible to define Arithmeticall proportion universally by comparing of Remainders ? Because , forsooth , In quantities incommensurable there may be the same proportion , where neverthelesse there is no Quotient : ( Very good ! But why no quotient ? ) for quotient there is none but in Aliquot parts . ( Gooder , and gooder ! ) But , I pray , is not A / B as good a Quotient , as A-B is a Remainder ? whether the quantities be commensurable , or Incommensurable ? No , you say ; For setting their Symbols one above another with a line between , doth not make a Quotient . But why not ? as well , as setting their Symbols one after another , with a line between , makes a Remainder ? For , if the quantities be incōmensurable , the Remainder is no more explicable in Rationall numbers , then is the quotient . If from 3 you subduct √ 2 , the Remainder is but 3 − ●2 . If you divide 3 by √ 2 , the quotient is 3 / √ 2 ; . And is not his as much a Quotient , as that a Remainder ? and as well designed ? Yet this is all you have to say to the businesse : The rest is but Ranting , or vapouring . But , however , we are much deceived , you tell us , if we think , with pricking of Bladders to let out their vapour ; for we see , you say , we make them swell more then ever . What ? till they bu●st ? I hope not so . ( Crepent licet , modo non Rumpantur . ) I have heard , I confesse , that a Toad would swell the more for being pricked ; but I never knew that a Bladder would , till now . The next thing that troubles you , is , that I said , that the Corollaries of these two Articles taught us nothing new . ( There be as I recon five and nine ; fourteen in all . ) Yes , you say , the ninth Corollary of the sixth Article is new : ( No ; it is not . We are taught the same by the second of the fifth of Euclid ; and by the converse of the eleventh prop. of the sixth chapter of M. Oughtred's Clavis ; ) and the rest were never before exactly demonstrated . What ? none of them ? That 's much . You mean , I suppose not all . And that I am content to believe : For they are not all true . As for example ; The second Corollary of the fifth Article , is thus delivered Universally , If there be never so many magnitudes Arithmetically proportional , ( whether in continuall or interrupted proportion ; for you doe not limit it to either , more then you had done that next before it , which you cannot deny to be understood of both ) the summe of them all will be equall to the product of halfe number of Terms , multiplied by the summe of the extremes . And then that we may be sure it is not intended only of cōtinual proportion , you give instance in proportion discontinued , For ( say you ) if A. B ∷ C. D ∷ E. F. be Arithmetically proportionall ( though but discontinued , for so your Symbols import , both in the Latine and the English , least we might think it had been the Printers fault , and not the Authors ; ) the couples A + F , B + E , C + D , ( you say ) will be equall to one an other . This , though it be true of continued Arithmeticall proportion , yet of discont●nued proportion , as you here affirme it , it is notoriously false . For how doth it appeare , that C+D , is equall to A + F. For instance , let the termes be these 2. 1 ∷ 20. 19 ∷ 3. 2. in arithmeticall proportion . is 20+19 , equall to 2 + 2 ? or to 1 + 3 ? It 's no marvell then that this was never before exactly demonstrated . But we are taught nothing new by this . For though this be new and be years , yet we cannot learn it . Wee 'l go on therefore : and see what you say next of the thirteenth Article . Wee began , as I said , with slighter skirmishings ; about Definitions &c. The skirmish now growes hotter ; when I charge you with false propositions and demonstrations ; and that you be touched to the quick , we may guesse by the loud out-cry ; In objecting against the thirteenth , and sixteenth Articles , we doe at once bewray both the greatest Ignorance , & the greatest Malice , &c ( and so on , for a whole leafe or more ; ) Now this Ignorance h●wrayd , was your own , viz. that you had given us false demonstrations &c. and then is it not spightfully done of us to discover them ? Well ; let 's see what 't is that makes you cry out so fiercely . The proposition is this , Of three quantities that have proportion to one another , ( suppose AB , AC , AD ; or 6 , 3 , 1 ; ) the proportion of the first to the second , and of the second to the third taken together , are equall to the proportion of the first to the third . That is , said I , The propertion compounded of that of the first to the second , ( suppose 6 to 3. which is double , ) and that of the second to the third ( viz. 3 to 1 , which is treble , ) is equall to that of the first to the third , ( viz. 6 to 1 , which is sextuple . ) And was not this your meaning ? ( I am su●e 't is either thus or worse ) This composition , I said , was such as Euclide defines 5 d 6 ; which is done by multiplying the quantities of the proportions : viz. 6 / 3 × 3 / 1 = 6 / 1 , ( not by adding them ; for so 6 / 3 + 3 / 1 = 2 / 1 + 3 / 1 = 5 / 1. ) Did I not explaine your meaning right ? I●meant no hurt in saying this was your meaning ; for the meaning was a good meaning ; and the proporsion so meant , is a good proporsion ; ( but , if you mean otherwise , the proposition is false : ) and , doubtlesse , 't was a good meaning too , when you meant to demonstrate it ; ( all the mischiefe was , you could not do what , you meant to doe . ) If this be your meaning ( as J am sure it is or should be , ) what is it that troubles you ? You doe not like the word Composition : that 's one thing . Well then let it be called Addition for once , J told you then , J would not content for the name ; ( but you know 't is such an Additon of Proportions , as is made by multiplying of the quantities ; as appeares by the very words of the definition 5 d 6 ) Then you doe not like that J should say the proportion of 6 to 3. is double ; and that of 3 to 1 , treble . Tell me ( say you ) egregious Professors , How is 6 to 3 double proportion ? The answer is easy , ( though perhaps you will not like it ; ) The proportion of 6 to 3 , or 2 to 1 , is that which is commonly called Double ; and that of 3 to 1 , is is commonly called Treble ; And if you will not believe me , pray believe your own words , Corp. pag. 110. l. 5 , 6. Ratio 2 ad 1. vocatur Dupla ; et 3 ad 1 Tripla . You tell us then , We may observe that Euclide never distinguisheth between Double and Duplicate ( no more then other Greek writers do between 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 and 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 . ) one word ( you say , serves him every where for either . You might as well bid us put out our eyes ; or else believe that 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 and 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 ▪ are the same words . Perhaps you thought so when you wrote your booke in Latine ; but , since that time you have been better instructed , and have learned at length to distinguish between Double and Duplicate , as we shall heare anon . But let 's goe on . All this hitherto hath been but scuffling , and little to the purpose , though there you make the greatest out cry , ( like a lapwhing , when shee 's furthest off her nest . ) we are now comming to a close grapple . ( and 't is like to prove as had as a Cornish hugge . ) Your demonstration , I said , was false ( and that greeves you . ) The strength of it , as I told you , lyes in this , The difference of AB , AC , ( be they Lines or Times , chuse you whether , for by construction the times and lines are made proportionall , ) together with the difference of AC , AD , taken together , are equall to the difference of AB , AD ; therefore the proportion of AB , to AC , and of AC , to AD , taken together is equall to that of AB to AD. That this is the strength of your demonstration you doe not deny . Now that consequence I denyed ; affirming that from that equality of the difference , you could not inferre the equality of Geometricall proportion ; ( and , of Arithmeticall , the question is not ; nor is pretended to be . ) And J gave this instance to the contrary , to shew the weaknesse of your Argument ; Taking between A and B , any point at pleasure suppose a ; you may as well conclude the proportion of aB to aD , aS of AB to AD. to be compounded of that of AB to AC , and of AC to AD. For , ( in your own words . ) the difference of AB , AC , with that of AC , AD , are equall to the difference ( not only of AB , AD , but even of ) aB , aD ; and therefore the proportions of those , to that of these . Now all that you have to say against it , ( for I doe suppose , as you would have me , the motion to be equally swift all the way , ) is this , The difference of AB , AC , ●annot be the same with the difference of aB , aC , except AB and aB are equall . And here we joyne issue . The difference of AB , AC , say I , is BC ; and the difference of aB , aC , is the same BC ; though AB , aB , are not equall . The case is ripe for a verdict . Let the Jury judge . And now you may , if you will , go on to rant at Ignorance and Malice , at Symbols and Gambols , at double and duplicate , at asses and eares , at Cla●ius , Orontius , and too learned men , or whom you will ; haeret lateri lethalis arundo . But thus 't is , when men will needs have Geometricall proportion , to be estimated by Differences , and not by Quotients . ( I told you moreover that your demonstration was but Petitio principii , and shewed wherein , with some other faults which you take no notice of , because you had nothing to say to them . And shewed you how your 13 , 14 , and 15 , articles with all their Corollaries , ( which fill up a matter of 4 pages . ) might have been to better purpose delivered in so many lines . But this is no great fault with you , who think the farthest way about , the nearest way home . ) At the 16 Article the case is as bad or worse . The cry goes on still . This is all Ignorance and Malice too . And a huge out cry against Quotients , and Symbols , and a loud On●ethmus as you call it . But not a word to the purpose of what was objected ; ( except only one clause wherein you tell us how absurd you mean to be by and by . ) The businesse is this , Euclide ( 10 d 5 ) defines Duplicate , and Triplicate proportion , &c. in this manner , If three magnitudes be in continuall proportion , the first to the last hath duplicate proportion of what it hath to the second ; if four , triplicate ; &c. ( and that indifferently whether the first or last be the bigger . ) Now you ( that you might shew your selfe wiser then Euclide , and be the first that ever made the grounds of Geometry firm and coherent , ) thought it was to be limited to this case only , when the first quantity is the greatest . And therefore thus define , The proportion of a greater quantity to a lesse ( very warily ) is said to be multiplied by a number , when other proportions equall to it , be added . And therefore if the quantities ( continued in the same proportion ) be three ; the proportion of the first to the last is Double , of what it hath to the second ; if four , Treble , &c. ( which most men , you say , call duplicate , triplicate , &c. ) But if the proportion be of the lesse to the greater ( of which Euclide , it seems , was not aware ) and there be an addition of more proportions equall to it , it is not properly said to be multiplied , but submultiplied ( that is , divided ; which yet you tell us , by and by , is to be done by taking mean proportionalls . ) So that of three quantities ( so continued ) the proportion of the first to the last , is halfe of what it hath to the second ; if four , a third part , &c. which are commonly called subduplicate , subtriplicate , &c. Now this , I told you , was foul great mistake , and such a one as should not have proceeded from a Reformer of the Mathematicks . And , to use your own distinction ( Less . 2. p. 9. ) 't is a fault not of Negligence , but of Ignorance , or want of understanding principles : and therefore an ill favoured fault , and , by your own rule , to be attended with shame . I shewd you there ( and you believe me now ) that in the numbers 1 , 3 , 9 , 27 , &c. the proportion of 1 to 9 , though lesse , was not subduplicate to that of 1 to 3 , but duplicate , as truly as the proportion of 9 to 1 is duplicate to that of 3 to 1 ; and that of 1 to 27 was triplicate , not subtriplicate , of that of 1 to 3 ; Of which I gave you this demonstration , ( though it seems , you did not understand it , and therefore say , I bring no Argument . ) Because 1 / 9 = ⅓ × ⅓ , and 1 / 27 = ⅓ × ⅓ × ⅓ , as well as 9● = 2 / 1 × 3 / 1 , add 27 / 1 = 3 / 1 × 3 / 1 × 3 / 1. And the subduplicate of 1 to 3 , is not , as you suppose , that of 1 to 9 , but of 1 to √ 3. Now this was so unlucky a mistake , or Ignorance , in a thing so fundamentall , that ( as I then told you , and you have since found to be true ) an hundred to one , but it would doe you a deal of mischief all along . And it was the touching upon this fore place , that gawled you so much but now , and put you beside your patience . But let 's see now how you behave your selfe . A loud rant we have , as if it were grievous doctrine I had taught , and your own had been much better . But not a word to the purpose save only this 'T is absurd to say , that taking the same quantity twice , should make it lesse . But though you say so , you doe not think so . For when you have done your rant , you goe slyly , ( without saying a word of it , or acknowledging any error , ) and put out that whole sixteenth Article , which we had in the Latine , giving us in the English another instead of it , quite of another tenour , and quite contrary to what you had before . And now a proportion of the lesse to the greater , ( as well as of the greater to the lesse , ) being twice taken , shall be duplicate , ( not subduplicate as before ; ) and thrice taken , ( not subtriplicate , but ) triplicate . Now ( because you say it , ) it is not absurd to say , that taking the same quantity twice , should make it lesse ; ( though when I said it , it was absurd . ) Now A proportion is said to be multiplyed by number , not submultiplyed , when it is so often taken as there be unities in that number . ( Whether it be of the greater to the lesse , or of the lesse to the greater ; ) And if the proportion be the greater to the lesse , then shall also the quantity of the proportion be increased by the multiplication ; but when the proportion is of the lesse to the greater , then as the number increaseth , the quantity of the proportion diminisheth ; For it is no absurdity now , to say that taking the same quantity twice makes it lesse . And truly now , methinks , thou sayst thy lesson pretty well ; I could find in my heart to spit in thy mouth and make much of thee , hadst thou not railed at him that taught thee ; which is but a trick of an ungratefull schollar : But let 's goe on , and see whether this good fit will hold ? As in these numbers , 4 , 2 , 1. the proportion of 4 to 1 , is not only the duplicate of 4 to 2 , but also twice as great . ( Nay that is good againe ; he hath learned that there is a difference between Duplicate and twice as great . Surely this is not he , ( or else the world 's well amended with him , ) that laughed at the distinction of Duplicate and Double . Well , let 's heare some more of it . ) But , inverting the order of those numbers thus , 1 , 2 , 4 , the proportion of 1 to 2 , is greater than that of 1 to 4 ; and therefore though the proportion of 1 to 4 , be the duplicate of 1 to 2 , yet it is not twice so great as that of 1 to 2 , but contrarily the halfe of it . In good truth ; a prety apt Schollar : for one of his inches ; He says just as I bid him . Well , well ! the world 's well amended with T. H. The●'s hopes he may come to good . Yee see he learnes apace . He may be a Mathematician in time ; though I say 't that should not say 't . I confesse he hath his faults still , as well as other men , ( you must not think he can mend all at once , ) The whole article is not so good throughout , at this bit at the beginning . He hath got a naughty trick of saying The proportion of equality is no quantity , ( but he hath been whipt for already ; ) He makes it stand for a Cyphar , ( but that 's a thing of nothing : It should have been but 1 , and that 's not much more . ) And he tells us that the proportion of 9 to 4 is not onely duplicate , of 9 to 6 , but also the Double , or twice as greate . And again , that the proportion of — 4 to — 6 , is double to the proportion of — 4 to — 9 , &c. which would have deserved whipping at another time ; but because he said the rest so well , I 'le spare him for this once . He doth , it seems , believe there is a difference between double and duplicate , though he doe not yet know what it is ; he will learn against next time . And to the like purpose is that which follows ; If there be more quantities then three ( it 's no matter how many ) as A , B , C , D , in continued proportion , what ever the proportion be , so that A be the least ; it may be made appeare that the proportion of A to B , is triple magnitude , though subtriple in multitude , to the proportion of A to D. But however he shall be spared for this bout ; because I said so ; and I will be as good as my word . SECT . IX . Concerning his 14. and 15 Chapters . IN Your 14 Chapter , Art : 2. I found fault with your definition of a Plain , to be that which is described by a streight line so moved as that every point of it describe a streight line . I told you , it is not necessary , much lesse essentiall , to be so described , ( and you confesse it ; ) and many plains there are which are not so described . The definition therefore is not good . Again . You had said in the first Article : Two streight lines cannot include a superficies . ( Right , ) And then Art : 2. Two plain superficies cannot include a solid . No , said I , nor yet Three . 'T was simply done then to name but two . And you confesse it to be a fault ; but not a fault to be ashamed of . Again , you had said Art : 1. That a streight line and a crooked , cannot be coincident , no not in the least part . And then Art : 3. You tell us of some crooked lines which have parts that are not crooked . This I noted for a contradiction ; because with those parts not crooked , a streight line may be coincident . And you cannot deny it . Therefore in the English , instead of crooked , in the former place , you put perpetually crooked ; which though it be but a botch , helps the matter a little . In the fourth Art. In the description of a circle , by carrying round a Radius ; you define the Center to be that point which is not moved . Now a Point you had before defined cap. 8. art . 12. to be a Body moved &c. So that to say , the Point which is not moved , is as much as to say , the Body moved &c. which is not moved . Which seems to me a contradictiction . To this objection , you say only that which I must say to your answer , viz : It is foolish . You said farther , Crooked incongruous lines cannot touch each other , save only in one point . Yes , said I , a Circle may touch a Parabola in two points . And you confesse it . But say , you meant that each contact is not in a line , but only in one point . Perhaps you meant so , ( though yet I question whether you did then think of more contacts then one : ) but why then did you not say so ? ( I mean , in the Latine ? for in the English , upon this notice it is a little mended ) But I reply , Yes , if those incongruous curve lines , have but some parts which are not crooked , ( as even now you told us , ) they may touch in a line . Yea & incongruous lines continually crooked , may in some pasts of them agree , though not congruous all the way , and therefore touch in a line . And therefore even yet , it is not accurate . But you 'l say ( as pag. 10. ) Such faults as these , are not attended with shame , unlesse they be very frequent . What you mean by very frequent , I cannot tell ; but , mee thinks , 't is very ugly to have them come thus thick . Art 7. you divide a superficiall Angle , into an Angle simply so called , and an Angle of contingence . Which you define in this manner ; Two streight lines applied to each other , and contiguous in their whole length , being separated or pulled open in such manner , that their concurrence in one point remains ; If it be by way of circular motion , whose center is the point of concurrence , and the lines retain their streightnesse ; the quantity of this divergence is an angle simply so called : If by continuall flexion in every imaginable point ; an angle of contingence ▪ I asked ; to which of these two you referre the angle made by a right line cutting a circle ? or whether you doe 〈◊〉 take that to be a superficiall angle . You say , to an angle 〈◊〉 so called , that is , as we heard but now , to an angle made by two lines which retain their streightnesse , ( though one 〈◊〉 them be crooked . ) And then , you tell us that Rectilin●●● and Curvilincall hath nothing to doe with the nature of an angle simply so called : When yet your definition requires , that the lines retain their streightnesse . I will ask , you say , ( yes I do ask ; and do you give a wise answer if you can ; ) How can that angle which is generated by the divergence of two streigh lines , [ whose streightnesse remains , ] be other then Rectilineall ? You say , A house may remain a house , though the carriage of the timber cease . Much to the purpose ! How do you apply the similitude ? Even so , the lines retain their streightnesse , though they be crooked , is that it ? Or is it thus , Even so , the Angle remains an angle made by lines retaining their streightnesse , when they be crooked ? Perhaps you mean thus , The Angle being once made by the divergence of streight lines , remains an Angle though one or both of those lines be afterwards made crooked . Very good ! but doth it remain the same Angle ? the same quantity of divergence ? ( for so you define an angle , ) doth not ( in your account , ) the bowing of one of the lines ( the other remaining as it was ) alter the quantity of divergence , ( measurable by the Arch of a circle , as you determine ) from what it was before such bowing ? though yet that very bowing alone , by your doctrine , be enough to make an Angle of it selfe ? Well , let it be so for once , ( though it should not be so , by your principles . ) But however , though this should be allowed , yet at least , so long as the Angle is in making , the lines must be streight . Tell me then , J prithee , how a Sphericall Angle comes to be an Angle simply so called . Is a sphericall Angle made by the divergence of streight lines or of cooked ? Can it be made a sphericall Angle so long as the lines retain their streightnesse ? It seemes so : for an Angle properly so called , that is , an Angle made by the divergence of streight lines , whose streightnesse remains , is distributed into Plain and others , ( as though all Right lined angles , were not Plain Angles ; ) and then again into Rectilineall , Curvilineall , and mixt ; as though these were , species of Rightlined Angles . Do you think it possible to make an Angle Sphericall , Curvilineall , or mixed , so long as the lines retain their streightnesse ? do you think these things will ever hold together ? or is this to make the principles of Geometry firm and coherent ? You were better say , as the truth is , that when you formed that definition of an Angle simply so called , you had your eye only upon a Right-lined Angle , and fitted your definition thereunto ; but when afterward , under the same name , you took in curvilineall and mixt angles , you should have altered the definition , but neglected it : And then apply your ordinary apology ▪ That it was indeed a fault , but not such an one as you need be ashamed of . But , to goe about to defend it , is more ridiculous then the thing it selfe . At the ninth Article , I had shewed how simply you defined the quantity of an Angle , your definition as you call it , is this : The quantity of an Angle , is an Arch of a circle determined by its proportion to the whole perimeter . An Angle was before defined to be the Quantity of Divergence ; That which you define now is the quantity of an angle , that is , the quantity of the quantity of divergence . Very handsomely ! Then in stead of , the quantity of an Angle is measured by an Arch ; you say , the quantity of an Angle is an Arch. Again , it is , you say the Arch of a circle : But what Arch ? and of what circle ? for you determine neither . You mean , I suppose , that Circle whose center is the Angular point ; but you doe not say so : and , you mean also , the Arch of that circle intercepted between the two streight lines containing the angle ; But then you should have said so , as well as meant so . For , as the definition now runs , neither Arch , nor circle , is determined . Next you say , that this quantity is to be determined ( for so the words must be construed to make sense of them ) by the proportion of that Arch to the whole Perimeter : That is , what proportion that intercepted Arch hath to the whole perimeter ; such proportion hath that Angle to — what ? you do not tell us , to what . As for instance , suppose the Arch be a quadrant or quarter of the whole perimeter ; the Angle is then a quarter of — somewhat no doubt ; but you doe not tell us of what , Is it a quarter of an Angle ? or a quarter of an Arch ? or a quarter of a Circle ? No ; 't is a quarter of four right Angles . 'T is that , you should have said . Now are not these faults enough for one poor definition ? They are but Negligences , you 'l say : but they be scurvy ones ; and there be enough of them , for lesse then two lines . But whether to commit so many negligences , in lesse then two lines , be so very frequent , as that they be attended with shame , I leave for others to judge . You should have said thus , as I then told you , ( but I see you are not alwaies willing to learne ; ) The quantity of a Rectilineall angle , in proportion to four Right angles , is determined by the proportion of an Arch of a Circle ( whose center is the Angular point ) intercepted between the two streight lines containing that angle , to the whole circumference . But , it seems , you had rather keep your own definition , with all its faults , then seem to be taught by mee : Though yet you have nothing to say in defence of any one of them ; and therefore ( as you use to doe in such cases ) take no notice of them in your answer at all ; as if no such exceptions had been made . The like exceptions , I said , ly against the 18 Article . And you take the like care neither to mend them , nor to take notice of them . At the 12 Art. I shewed , what a pittifull definition you had brought of Parallells ; and that the Consectary from it was false , and the Demonstration thereof a sad one . You confesse all : But are not pleased that I should triumph . Your emendation which you intimate , by inserting the same way ; will do some good in the consectary , but will not make good the definition . Your new definition in the English , is little better then that of the Latine . The consectary , as it is now mended in the English , is true ; but the demōstration of it hath many of the same faults , though not all , that I noted in the Latine : and doth not at all conclude the truth of the consectary , from that definition . As appears by what I objected formerly . What you attempt to prove of two lines , you should have proved universally of any two ; for so much your definition requires . At the 13 Art. you bring a sorry argument to prove The Perimeters of Circles to be proportionable to their Semidiameters . The strength of the argument lies in this , The bignesse of the Perimeter is determined by its distance from the center ; and the length of the Semidiameter is determined likewise by the same distance ; therefore , since the same cause determines both effects , the Perimeters are proportionall to their Semidiameters . This consequence I deny ; because , not only the bignesse of the Perimeter , but of the circle also is determined by the same cause ; as also the superficies and the solid content of a spheare . For that distance of the circumference and Center , determines the greatnesse of all these . And therefore , by your argument , circles , and spheares , &c. must be proportionall to their semidiameters : which is absurd . To which retort , because you can answer nothing ; you d●e , according to your usuall Rhetorick , fall to ranting . At the 14 Article , I said , that your argument was but petitio principii . You say , There was a fault in the figure , ( that it was not exactly drawn ) which is now amended . True ; but there is a worse fault in the demonstration , which is not amended yet . For though you have altered your Figure , and your demonstration too ; yet the fault remaines . And 't was this , not the figure , which I found fault with . For you do not prove that BH , BI , BC , ( fig. 6. ) are proportionall to AF , AD , AB , but upon supposition that FG , DE , BC , were so : which was the thing at first to be proved . You say , that AF , FD , DE , are equall by construction . ( True. ) And , that FG , DK , BH , KE , HI , IC , are equall by Parallelism . But this is not true . The Parallelism proves that FG , DK , BH , are equall ; and that KE , HI , are also equall ; but not that either of these two , are equall to either of those three , ( or to IC : ) unlesse you first suppose that DE , is the double of DK , or FG , as AD is the double of AF , which is the very thing to be proved . You tell me ; There was another fault ( yes , three or four for failing ) which I might have excepted against . But the weight of the demonstration did not ly there ; and I did not intend to trouble the Reader with every petty fault ; ( for then I should never have done : ) especially in this and the next Article ; where I did not then repeat your Figure at all ; and therefore did briefly intimate where the fault lay : which had been direction enough for an intelligent man to have ●ound it out : But because J did not point with a festcue to every letter , you had not the wit to understand it . In like manner Art. 15. when I told you the third Corollary was false , and shewed you briefly the ground of your mistake ; because J did not , with a festcue point from letter to letter , you were not able to spell out the meaning ; but , as being lesse awake , thought it had been a dream . You had told us , that ( in your 7 figure ) the angles KBC , GCD , HDE , &c. were as 1 , 2 , 3 , &c. And 't is true . Thence you undertake in your third Corollary to give account of the bending of a streight line into the circumference of a circle ; namely , by its fraction continually increasing according to the sayd numbers 1 , 2 , 3 , &c. But how so ? For , say you , the streight line KB being broken at B according to any angle , as that of KBC , and again at C according to the double of that Angle , and at D according to the treble &c. 't will containe a rectilineall figure ; But if the parts so broken be considered as the least that can be , that is , as so many points , 't will be a circumference . This , I said was false , and that the ground of your mistake was , that for the Angle BDE and its Remainder HDE , you took CDE and its remainder . And J need not say more ; verbum sapienti a word for a wise man , had been enough ; but , for you it seemes , it was not . You , like a man halfe a sleep , took it to be a dreame . Therefore , if you please to rub up your eyes a little , and take a festcue I will , for your better noddification , point to the letters as we goe along , and teach you to spell it out . The tangent line BK , continued indefinitely both ways , being broken at B , according to the Angle KBC , will lye in BCG : Now this line BCG being broken at C , according to the Angle GCD which is the double of KBC , its part CG , will lye in CD continued , CDδ And hitherto you be right . But this continuation of CD , is not DH , as you seem to suppose , but Dδ which will fall between DH , DE. When therefore this line CDδ comes to be broken againe at D , that its continuation may lye in DE , the faction will not be according to the Angle HDE ( which indeed is the triple of KBC ) but according to the angle δDE : which will be lesse then HDE , because it is evident that CD cuts BH , And indeed the very same Angle of fraction with that at C ; For seeing the angle CDE , is equal to BCD , by construction , the subtenses being taken equall ; the adjacent angles ( anguli 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 ) must be equall also , that is δDE = GCD And therefore the angle of fraction at D , precisely equall with that at C ; not as 3 to 2 , as you suppose . And by the same reason the angle of fraction at E must be equall to that at D ; not as 4 to 3 , as you suppose . And so the Angles of fraction at C , D , E &c are not as 2 , 3 , 4 , &c. but are all equall . You see therefore , if you be yet awake , that it was not a dreame of mine , but a reall mistake of yours , to take HDE for the angle of fraction of CD . And consequently that your proposition was false . And this fas●ho●d was the occasion of another falsehood in the 20. Article of the 16. Chapter . ( which since you have blotted out . ) for there you cite this proposition as the foundation of that : And whereas you say , You cannot guesse what that proposition was , ( and yet are very sure that it was true , ) for that you have no coppy of that article either printed or written . If you have not , J am sure you may have , for there be enough that have . For your book sold in sheets unbound , had commonly that article amongst the rest , and by that meanes it came to me . And , rather then you should be farre to seek for it , I have recited that whole article verbatim , yea to a letter , in its due place in my Elenchus ; and proved it to be false . Against your opinion concerning the Angle of Contact , ( in the 16. Article , ) J said little ; because J think it needs no refutation . Your opinion is this , That the angle PAD , ( Fig. 2. Sect. 3 ) is bigger then the angle PAE , as being divided by the line AE . But the angle EAC , is not bigger then the angle DAC , nor is divided by the line DA , but both of them equall as well to each other , as to the angle PAC , and also to the angle GAC . That this is your opinion , is evident . They that like it may imbrace it , for all me : And I hope , they that like it not may leave it . The rest of what concernes this businesse , is considered before in its proper place . At the 18. Art beside what is common to this and the seventh , J noted for a fault , and you doe not deny it so to be , that you deliver it as Euclide's opinion , that a Solid angle is but an Aggregate of plain Angles . Jt may be your opinion ; but surely 't was none of Euclide's . If you had thought it had ; you should have here if you could , produced somewhat out of Euclide where he declares such an opinion . At the 19. Article All the ways by which two lines respect one an other , or all the variety of their position , seem , you sayd , to be comprised under four kinds ; For they are etiher Parallells ; or ( if produced at least ) make an angle ; or ( if bigge enough ) be Contingents ; or lastly are asymptotes . By Asymptots you mean ( not all such as never meet , for then Prallells would fall under this kind ; but ) such as will come always nearer and nearer together , but never touch one another ( you might have added this other character ; that they doe so approach each other , as that at length their distance will be lesse than any assignable quantity . But it seems you allow your Asymptotes a greater latitude : And doe in your English , determine your meaning so to be : And that , I suppose , because you had neglected to put in , that limitation , in the latine ; and therefore were not willing upon my intimation to mend it in the English . For none else that I knew , speak of any other lines under the name of Asymptotes , but such as doe not only eternally approach , but do approach also infinitely neare , And , I have reason to believe , from your simple objection Less . 5. p. 48. l. 23. that you thought those two must needs go together , viz. that whatsoever quantities doe eternally approach , must needs at last come infinitely neare . But however wee 'l be content , if you would have it so , to take Asymptotes at what latitude you will give it them . ) You say now , that I am offended at the word it seems . No , Sir , no offence at all . I am not at all angry , that , to you , it should seem so . I said but , that to mee , it seemed otherwise ; ( And , I hope you are not offended that all things did not seem to me , as they did to you : For I perceive , that by this time , it seems otherwise to you also . Which hath made you in the English , to give us this Article new moulded . ) I shewed you then , many other positions of lines , which doe not agree to any of your four kinds . And you confesse it . And some of them such , as will not be salved with your new botch . As they that please to compare them will soon find . J touched at some other faults ; as , That the definition of points alike situate , ( art . 31. ) seemed very uncouth . That the word Figure , which is defined art . 22. had been oft used long before it was defined ; ( which though it be , with you a small fault , yet a fault it is . ) And you confesse it . That by your definition a solid spheare , and a spheare made hollow within , is the same figure . ( For your definition takes notice of no superficies , but that within which they are included : your words are , intra quam solidum includitur . You say , It is my shall●wnesse , to think , those points which are in the concave superficies of a hallowed sphear not to be contiguous to any thing without it , because that whole concave superficies is within the whole spheare . It may be my shallownesse perhaps ; but it is I confesse , my opinion , that this concave superficies being , as you say all within a spheare , ( and therefore may be contiguous to somewhat within the spheare , ) is not contiguous to any thing without it , ( if it be , tell me to what ? and how it can be contiguous when the whole thicknesse of the spheare is between ? unlesse you think it can touch at a distance : ) Nor , is that superficies intra quam sphaera includitur : For if , as you say , that whole superficies be within the whole sphear , how can the spheare be within that superficies ? You should rather have confest , as the truth is , that you did not think of a solides being contained by two or more superficies , not contiguous to one an another : and 〈◊〉 , had not provided for that case . I excepted likewise against your definition of Like t●●ngs , cited here out of Cap. 9. art . 2. Those things , you de●ine to be Like , which differ only in magitude . They do not , I say , alwaies differ in this ; for it is possible like things may be equall ( And therefore if they differ in nothing else , they differ not ut all . ) And sometimes again they may differ in somewhat else ; at least in position . Else what needs your next definion , of similia similiter posita ? if it were not possible for similia to be dissimiliter posita ? To which exception ( because you had nothing to say ) you say nothing So your definition of like figures alike placed , I said was false : you confesse it is so , ( and therefore amend it in the English . ) You confesse you say , there wants something which should have been added ; but call we Foole for taking notice of it : Or else , you call your selfe Foole , for not supplying it ; For you say , that it might easily be supplyed by any student in Geometry , that is not otherwise a Foole. But , rather then fall out for it , wee 'l divide the Foole between us ; and cry Ambo. 'T was I , like a foole , took notice of that to be wanting , which you like a Foole , omitted , when you should have supplyed it . The 15. Chapter , because it contained but little Mathematicall , I did but touch at ; leaving that for my worthy Collegue to take to taske , with the rest of your Philosophy . Which he hath done to purpose . Yet some few things J noted as a rast of the rest . J noted that ( contrary to others who define Time to be the measure of motion ) you determine Motion to be the measure of time ; And yet ( contrary to your own determination ) you do frequently make Time the measure of motion ; measuring both motion , and its affections ( swiftnesse , slownesse , uniformity , &c. ) by Time. You confesse it to be so : But raile at us for minding Books , more than Clocks and hour-glasses . And then ( contrary to both ) you tell us , that time and motion have but one dimension which is a line . And at last would perswade your English Reader , that I would have you measure swiftnesse and slownesse , by longer and shorter motion : But they that understand Latine , can find nothing to that purpose : I only told you what you did , ( and how absurd that was , ) not , what I would have you do ▪ Then , because it still runnes into you mind , that I had some where said , That a point is nothing ( though no body can tell where ; ) you fall againe upon that . For my part , though I oft affirm that a Mathematicall point , hath no parts ▪ yet J never denyed it to stand for as much at least , as a cyphar doth in numbers ; and you allow it noe more , ( c. 16 ; art . 20. ) your words are these Punctum inter quantitates nihil est , ut inter numeros cyphra . Is it then J , or you ? that say a point is nothing ? You told us soon after , that All endeavour ( for even that is motion ) whether strong or weak , is propogated to infinite distance . As if ( said J ) the sk●pping of a Flea did propagate a motion as farre as the Indies . You ask , how we know it ? If you meane , How we know that it is so ; Truely , J doe not know that at all . If you meane , how we know that it follows from what you affirme ; It is so evident a consequence from the words alleadged , that you need not aske ; Or , if those words be not enough those that follow be yet fuller , Procedit ergo omnis conatus , sive in Va●uo , sive in plano , non modo ad distantiam quantamvis , sed etiam in tempore quantulocunque , id●est , in instanti . That ●s , All endeavour of motion whether the space be Full or Emty , is continued , not only to as great a distance as is imaginable , but in as little a time , that is , in an instant . But if your meaning be , what do I say to the contrary ? Truely I say nothing to the contrary . They that have a minde to believe it , may . Then you goe on to catechise us ; What is your name ? Are you Philosophers ? or Geometricians ? or Logicians ? &c. ( Nay , never aske that question , we know you are good at giving names , without asking ) I hope , the next question will be , Who gave you that name ? And truely as to many of the names you give us , a man might easily believe , yourself were the Godfather , you call us so often by your own names . Lastly , Of two things moving with equall swiftnesse , that , say you , strikes hardest which is bignesse . No , say I , but that which is heaviest . A bullet of Lead , though but with equall speed , strikes harder then a blown Bladder . If any man think otherwise let him try . SECT . X. Concerning his 16 Chapter . IN the 16 Chapter , I said , there were 20 Articles ; you say , but 19. 'T is easily reconciled . There be twenty in my book ; and there were 20 in yours too , before the last was cut or torne out : now , it seems , in yours there are but nineteen . Well ; but , be they twenty , or be they nineteen ; twenty to one but the greatest number of them be naught . I do confidently affirme , you say , that all but three are false . Nay , that 's false , to begin with . I said , that , all but three were unsound . Some of them be non-sense , or absurd ; some be false ; some undemonstrated ; all unsound ; at least , within three : And I have already proved them so to be . But you ( you say ) do affirme , that they are all true , and truly demonstrated . And that 's answer enough to all my arguments . What need you say any more ? If that be true , doubtlesse you have the better on 't . But let 's trie a little , if we cannot find one unsound amongst them . Your first Proposition as it stands yet in the Latine , you say , is this , The velocity of any Body moved , during any Time , is so much , as is the product of the Impetus in one point of Time , multiplied into the whole Time. Well , I hope at least the first is sound , is it not ? In one Point , you say ; but which one ? Is it any one ? or some one ? Nay 't is but some one , not any one ; but , which one , you tell us not . What say you to this ? Is it sound ? This , you confesse , without supplying what is wanting , is not intelligible . Very good ! Habemus confitentem rerum . To the first● Article as it is uncorrected in the Latine , ● object , you say , That meaning by Impetus , some middle impetus , and assigning none , you determine nothing ▪ Well what say you to that ? you say , 't is true . And then you rant at us for not mending it , ( as though we were bound to mend your faults ) yet look again , and you 'l find J did . J told you what you should have said ; as well as what you said amisse . But enough of this . Here 's one fault confessed . In the same Article ; you would have the Impetus applied ordinately to any streight line , making an angle with it . J asked , How an impetus can be ordinately applied to a Line ? or make an Angle with it ? Absurdly , you say ; and that 's the answer . And J told you how this should have been mended too . You tell me that Archimedes and others say , Let such a line be the Time , and again p. 36. l. 16. Let the line AB be the Time. Very likely ! just as when we say , Let the Time be A. That is , Let it be so designed ; or , Let the Line AB , or the letter A , be the Symbole of the Time. What then ? Doth it therefore follow , that either Lines or Letters be homogeneous to Time ? No such matter . Their Symbols may be Homogeneous though the Things be not . You say farther , in the same Article : If the Impetus increase uniformely , the whole velocity of the motion shall be represented by a Triangle , one side whereof is the whole Time , and the other the greatest Impetus , ( Well! & what shall be the third side ? or what angle shall these contain ? Do you think that the assigning of two sides , without an Angle , will sufficiently determine the bignesse of a triangle ? But le ts go on . ) Or else &c. Or lastly by a Parallelogramme having for one side a mean proportionall between the greatest Impetus and the halfe thereof . Well , but what for the other side ? And , what Angle ? Is a Parallelogramme , said J , sufficiently determined , be the assignement of but one side , and never an angle ? what think you ? is this sound ? It was indeed a very great oversight , you confesse , to designe a Parallelogram by one only side . And is not all this sufficient to prove the first Article unsound ? if it be not , wee 'l go on , for there be more faults yet . For , say you , these two parallelograms are equall both each to other , and to the ( fore mentioned ) Triangle ( without having any consideration of Angles at all ) as is demonstrated in the Elements of Geometry . This , I say , is notoriously false : For a Triangle of which nothing is determined but two sides : and a Parallelogramme , of which the sides only are determined , but nothing concerning the Angles : can never by any Geometry , be demonstrated to be equall . This therfore is not only unsound , but false . And all this J told you before . What an impudence then is it , when you knew all this , to affirm , that they be , all true and all truly demonstrated , when the very first of them is thus notoriously faulty ! But we have not done yet . It might be hoped , that this confessed oversight is , at lest mended in the English : ( especially since you tell us that one from beyond sea hath taught you how to mend it ) No such matter . For the Amendment is as bad or worse then what we had before . For now it runs thus . The whole velocity shall be represented by a Triangle &c. ( as before ) or else by a Parallelogram , one of whose sides is the whole Time of motion ; and the other , half the greatest Impetus : Or lastly , by a Parallelogram , having for one si●e a mean proportionall between the whole Time and the halfe of that Time ; and for the other side the halfe of the greatest impetus . For both these Parallelograms are equall to one another , and severally equall to the Triangle which is made of the whole line of Time , and the greatest acquired impetus . As is demonstrated in the Elements of Geometry . Now this , you shall see , is pittifully faise . Let the time be T ; and the greatest impetus , I : and let the Angles be supposed all Right Angles ( for such your Figures represent , though your text says nothing of them . ) The Altitude therefore of the triangle , is T , ( the whole time : ) the Basis I , ( the greatest impet●s : ) and consequently the Area thereof is one halfe of T × I : that is ½ IT . Again the Altitude of the former Parallelogram , T , ( the whole time , ) its Basis , ½ ● , ( half the greatest impetus , ) and therefore the area T × ½ I , or ½ IT ; equall to that of the Triangle Le ts see now whether the last Parallelogram be equall to either of these , as you affirm . The Altitude you will have to be a mean proportionall between the whole Time and its halfe : that is , between T&½ T ; It is therefore the root of T × ½ T , that is the root of ½ Tq , that is √ ½ Tq , or T √ ½ : The Basis you will have to be one half of the greatest Impetus , that is ½ I : And consequently , the Area must be ½ I × √ ½ Tq , or ½ I × T √ ½ , or ½ IT √ ½ . But ½ IT √ ½ is not equall to ½ IT : Therefore this Parallelogram is not equall either to the former , or to the Triangle . 'T is false therefore which you affirmed . Quod erat demonstrandum . Now what do you think of the businesse ? is not the matter well amended ? 'T was bad before , now 't is worse . When you told us but of one side , and left us to guesse the other , 't was at our perill if we did not guesse right , and 't was to be hoped , you meant well , though you forgot to set it down . But , now you tell us , what you meant , we find that you neither said well , nor meant well : For what you now say is clearly false . The two Parallelograms which you affirm to be equall , are no more equall then the Side and the Diagonall of a Square ; but just in the same proportion ; viz. as √ ½ to 1. Nay was it not a pure piece of wisdome in you , that , when you had been taught from beyond Sea , as you tell us , how it should have been mended , you had not yet the wisdome to take good counsell ; but , trusting to your own little wit , have made it worse than it was ? it falls out very unluckily , you see , that when you affirmed so confidently , that they are all true , and all truly demonstrated , the very first of them should be so wretchedly faulty . But enough of this . Wee 'l try whether the next will prove better . In the second Article you give us this Proposition . In every uniform motion , the lengths passed over are to one another , as the product of the ones Impetus multiplied into its time , to the product of the others Impetus multiplied into its time . And why not , said J , ( without any more adoe ) as the time to the time ? Which needed no other demonstration than to cite the definition of Vniform motion , ( viz. which doth in equall or proportionall times , dispatch equall or proportionall lengths . ) What need had you to cumber the Proposition with Impetus and Multiplication , and Products , when they might as well be spared ? and then put your selfe to the trouble of a long and needlesse demonstration , when the bare citing of a definition would have served the turne ? You answer , That the product of the Time and ●mpetus , to the product of the Time and Impetus , is also as the Time to the Time. and therefore the Proposition is true . Yes doubtlesse ; and therefore I did not find fault with it , as false ; but as foolish , to make such a busle to no purpose . For , by your own confession , the proportion of the lengths dispatched , is as well designed by the termes alone , as by those multiplications and products . But there is another fault which J f●●● with your proposition ; ● told you that , instead of , in every uniforme motion , you should have said , ( and , that you might have said it safely , as the rest of the wordsly , ) in all uniform motions ; for you make use of this proposition afterwards , not only in comparing divers parts of any the same uniforme motion , but in comparing divers motions one with another . But at this you are highly offended , that J should understand to what purpose this Proposition is brought , better than your selfe ; and that J should presume to tell you , what you ought to have said . ( And , on the other hand , when J do not do so , you blame mee , that J do not to my reprehension adde a correction : So that , it seems , you are neither well , full nor fasting : J must neither do it , nor let it alone . ) And then you go on to rant , after your fashion , at Wit and Mystery , and times and wayes , and steddy brains , at reading thoughts , and noise of words , at step and stumble , &c. And yet , for all the anger , ( when the heats over ) you think best to take my counsell ; and therefore say in the English , just as J said it should have been in the Latine . The proposition then being thus to be understood , ( though at first , ill worded , ) the demonstration , I said , would not hold . For though it will doe well enough ( yea more then enough ; for you might have spared halfe of it ; ) in comparing severall parts of the same motion , and in comparing severall motions of the same swiftnesse ; yet for the comparing of uniforme motions in generall , it will not serve by no meanes ; for you do assume at the first dash , that the motions compared have the same Impetus . Now this must not be allowed . For it 's very possible ( as you now know , since , J told you , though before you seemed to be ignorant of it , as J then convinced you ; ) that two motions may be both uniforme ; and yet not have both the same Impetus . Your proposition therefore ( as it was to be understood ) was not truly demonstrated . Now , because this was very evident , and not to be denied ; therefore you thought it best to make no words of it , but mend it as well as you could . And so , in the English , you have mended the proposition , as J bid you ; and given us a new demonstration , which is pretty good ; But not yet without fault . For in stead of the length AF ( fig. 1. ) you should have said , the length DG : for the length should have been taken in the line DE , which , according to your construction , is the line of Lengths ; not in the line AB , which is , by construction , the line of Times . So impossible a thing is it , for you to mend one fault and not to make another . But if all these faults be not enough to make this Article unsound , there is yet another , before we leave . Since therefore you say , in uniforme motion , the Lengths dispatched are to one another , as the Times in which they are dispatched ; it will also be , by permutation , as time to length , so time to length . This consequence I denied ; because Permutation of proportion hath place only in Homogenealls , no● in Heterogenealls ; ( and referred you for farther instruction concerning it , to what Clavious hath on the 16. Prop. of the 15. of Euclide . ) You tell me , that I think , line and time are Heterogeneous . Yes , and you think so to if you be not a foole . If not , pray tell me how many yards long is an hour ? Or , How much line will make a day ? Well , le ts try a third Article . ( For the two first you see be nought , that 's a bad begining . ) Art. 3. In motion uniformely accelerated from rest , ( that is , when the Impetus increaseth in proportion to the times ) the length run over in one time , is to the length run over in an other time . ( In the English for Impetus , you have put mean Impetus , and so in some other propositions ; but that neither mends nor mars the businesse . ) To this , first you dream of an objection , and then think of an answer to it . I object you say , that the Lengths run over , are in that proportion which the Impetus hath to the Impetus . Prithee tell me , where I made that objection to this article ; and I 'le confesse 't was simply done . But 'till then , I 'le say 't is done like your selfe , to say so however . ( For 't is lawfull with you to say any thing , true or false ) Your English Reader , perhaps , may think 't is true . Next , You aske , you say , where it is that you say or dreame , that the lengths run over are in proportion of the Impetus to the Times ? But prithee , why dost thou aske me such a question ? Am I bound to give an account of all thy dreames ? Perhaps you dreamed that I had charged you with such a saying ; But , look again , and you 'l find that 's but a dreame as well as the rest . That which I said was this , The parallell line FH , BI , ( fig. 1. ) do shew what proportion the Impetus at F hath to the Impetus at B ; to wit , the same with the time AF , to the time AB : ( And is not this your meaning , when you say the Impetus increaseth in proportion to the times ? ) But , though those ( and other parallell lines ) do define what proportion the severall Impetus have to each other ; yet they do not designe ( by permutation of proportion , as you fancied in the Corollary of the precedent article ) what proportion the severall Impetus have to the Times ; because they be Heterogeneous , and do not admit of that permutation . And these are the words , which gave occasion to those your two dreames . And then ( as if between sleeping and waking ) you ask , if it be you or ● that dream ? Had you been well awake , you needed not have asked the question . The objections that I made to it , were these . First , that in stead of motu accelerato , ( accelerate motion , ) you should have said , motibus acceleratis ( accelerate motions , ) because you speake of more than one . You say , there is no such matter : and bid mee give an instance . J will so , and that without going farther then your present 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 . Let AB ( say you ) represent a Time &c. Againe , let AF represent another time &c. And in each of these times you suppose a Motion , which motions this proposition compares . Therefore , say I , there must be at least two Motions , because two Times ; unlesse you will say , that one and the same motion may be now , and anon too . I objected farther , that the demonstration doth nor prove the proportion ; except only in one case , to which you do not restraine it . For the whose stresse of your demonstration , ( in the Latine ) lyes upon this , that the triangles ABI , AFK , be like triangles ( where you inferre , that the space dispatched in the latter time AK , is to that of the former time AB , as the triangle ABI , to the triangle AFK , that is in the duplicate proportion of the times AB , AF. ) Which supposeth that the second motion in the time AF , doth acquire the same Impetus which the first motion had acquired in equall time . Whereas it is possible , that , of two motions , each of them uniformly accelerated , the one of them may in half the time acquire as great a swiftnesse , as the other doth in the whole time ; If therefore the latter motion in the same AF , do acquire a swiftnesse equall to that of the former in the time AB , ( which may very well be , for the words uniformly accelerated , doe imply only the manner of acceleration , not the degree of celerity ; as your selfe now discern , though then you did not , ) the triangles will be , not ABI , AFK , but ABI , AFH ; which are not like triangles , but unlike ; and so the demonstration falls . You should have provided in your proposition , not only that the two motions , ( the one in the time AB , the other in the time AF , ) be each of them uniformly accelerated , but that they be both equally swift . Which when you have neglected to take care of , you affirm that universally , which will hold only in one case . But the truth is , 't is evident enough , by this and divers other Articles , that you took the manner of acceleration , ( viz. if in the same , in the duplicate , or triplicate , &c. proportion to the times , ) had sufficiently determined the speed also . And therefore took it for granted , that the motion in the time AF , if uniformly accelerated , must needs attain precisely the same degree of the celerity , that the other motion in the time AB , uniformly also accelerated , had attained in equall time . ( Which to be a very great mistake , you now doe apprehend . ) Otherwise you would not have let these Articles ly so naked without such provision ; nor would you , ( as in the 13 Article , and those that follow , ) undertake , by the manner of acceleration , and the last acquired Impetus , to determine the time of motion . Whereas , in the same manner of acceleration ( whether uniformly , or in the duplicate , or triplicate , or quadruplicate proportion ; ) any assignable impetus or degree of celerity , may be attained in any assignable time whatever . I objected farther , that because , as hath been shewed , the Triangle AFK , or AFH , is not necessarily like to the triangle ABI , therefore it doth not follow that the length passed over , will be in duplicate proportion to the time . For unlesse the triangles be alike , the proportion of them will not be duplicate to that of their homologous sides . Now these two Objections were clear and full , ( and did destroy your whole demonstration ; ) and this you discerned well enough , though you did not think fit to make any reply or confession ; ( but invent some other objections , which I never made , that you might seem to answer to somewhat . ) And therefore in the English , without making any words of it , you mend it . And instead of those words in the Latine , As the triangle ABI , to the triangle AFK , that is , in duplicate proportion of the time AB to AF : you say in the English . As the triangle ABI , to the triangle AFK , that is , if the triangles be like in the duplicate proportion of the time AB , to the time AF ; but , if unlike in the proportion compounded &c. ( which is a clear confession of all those objections . But let 's go on . Compounded of what ? ) of AB , to Bi , and of AK , to AF. No such matter ; of AF to FK , ( that 's it you would have said : ) not , of AK to AF. There 's one fault therefore ; but that 's not all . Of AB to AF , and of BI to FK ; that 's it you should have said : for AB to BI , the Time to the Impetus , hath no proportion at all ; but are Heterogeneous , as I have often told you . There 's a second fault therefore in your emendation . And is not this Tinker-like , to mend one hole and make two ? Nay there is a third yet , which is the worst of all . In the mending of this fault , ( though you had not missed in it , ) you have discovered another , which you did your endeavour , but now , to hide . I said in the proposition for motion , you should have said motions ; because it was intended of more than one compared . You tell me , there 's no such matter ; meaning , I suppose , the latter motion in the time AF , was but part of that former motion in the time AB : But if , as you now confesse , the triangle AFK , be not necessarily alike triangle to ABI , ( but that the point K may fall either within or without the line AI , ) then must this be not only another , but an unlike motion to the former : viz. either faster or slower , though uniformly accelerated as that was . Do not you know that old rule ; Oportet esse memorem . But this 't is , when men will commit faults , and then deny them . And yet presently after , by going about to mend them , betray themselves . Much such luck you have in mending the Corollary . You had said in the Latine , In motion uniformly accelerated , the lengths transmitted are in the duplicate proportion of their times . This , I said , was true in one case , ( viz. in equall celerities , ) but not universally . Therefore you , to mend the matter , in the English make it worse ; In motion uniformly accelerated , say you , the proportion of the lengths transmitted , to that of their Times , ( No , but the proportion of the length transmitted , one to the other , ) is compounded of the proportions of the Times to the Times , and Impetus to Impetus . There be more faults in this Article ; but I am weary of the businesse ; let 's go to the next . The fourth Article hath all the faults that the third hath , ( which are enough as wee have seen already , ) and some more . First , for motu accelerato , you should have said motibus acceleratis ; because you compare more motions then one . Secondly , the Motion performed in the time AF , ( Fig. 2. ) though accelerate according to the duplicate proportion of the times , as well as that in the time AB ; yet may that be either swifter or slower than this ; ( because as we have often said , the manner of acceleration doth not determine the degree of celerity ; ) And therefore the point K which determines its greatest Impetus , doth not necessarily fall in the Parabolicall line , but may fall either within or without it : according as the celerity is lesse or more . Thirdly , And therefore it doth not follow , that the Lengths dispatched by such motion , are in triplicate proportion to their Times . For this only depends upon supposition that the point K in the second motion , must needs fall in the Parabola AI , designed by the first motion . Now these two latter faults , in the former Article , you did endeavour to amend in the English : But because , it seems , here it was harder to doe , you have left them as they were before . That these were faults , you were clearly convinced of ; and do as good as confesse , by your attempt to mend them in the third Article . But because you saw it was impossible for one of your capacity to think of mending all ; you resolve to give over mending , and ( which is the easier of the two ) resolve to try the strength of your brow . But , as if there were a necessity of growing worse and worse ; beside those , common to this and the third article , here is an addition of more faults , as foul as any of them . In your demonstration ; your stresse lyes upon this argument , Seeing the proportion of FK to BI , is supposed duplicate to that of AE to AB , ( which yet is a false supposition ; for the ordinate lines in a Parabola are not in duplicate , but in subduplicate proportion to the diameters : But , suppose it true , what then ? ) that of AB to AF , will be duplicate to that of BI to FK . That is , Because the Ordinate lines in a Parabola , are in duplicate proportion to the Diameters ; therefore those Diameters are in duplicate proportion to those Ordinate lines . Which if it be not absurd enough , I would it were . First , the proportion of the Ordinates , must be duplicate to that of the Diameters , ( because M. Hobs will have it so ; ) and then ( by the virtue of Hocus Pocus ) this must be duplicate to that . To this you make no reply : but inslead thereof , disguise the matter in your Lesson , by putting double for duplicate , as if they were all one ; ( though yet Chap. 13. art . 16. wee have , in the English , a long harangue of your own to shew the difference between them ; ) and then raile at those that first brought up the distinction ; and tell us , ( which is notoriously false ) that Euclide never used but one word for Double and Duplicate ; ( that is 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 , and 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 , are with M. Hobs but one and the same word . ) But what is all this to the raking off that absurdity with which you are here charged ? Next I shewed you , that your whole argument was grounded upon a false supposition ; viz. that the velocity of the motion in hand , was to be designed by the Semiparabola AKB ; and that the ordinate lines in that Semiparabola , ( by which you would have the increasing Impetus to be designed ) did increase in duplicate proportion to their Diameters ( by which you designe the Times ▪ ) both which are false . For , these ordinate lines , are well known ( to all but M. Hobs ) to increase in the Subduplicate ( not the Duplicate ) proportion of the Diameters : And consequently that Semiparabola can never expresse the Aggregate of the Impetus thus increasing . I did farther demonstrate , that the point K , ought to fall within the Triangle ABI , not without it ; and therefore not in the Parabolicall line by you designed . The demonstration was easy . For if the time AF be one halfe of AB , that is , as 1 to 2 : the Impetus increasing in duplicate proportion to the times , must be as 1 to 4 ; and therefore FK will be but a quarter of BI . But because AF is halfe of AB , therefore FN will be halfe of BI . And consequently FK ( a quarter ) will be lesse then FN , which is the halfe of BI . Which because you saw too evident to be contradicted ▪ you thought it best ( as your usuall custome is in such cases , ) to raise at it in stead of answering it . I shewed you farther , that the Aggregate of all the Impetus in a motion thus accelerated , or the whole Velocity , was not ⅔ of the Parallelogram AI , but only ⅓ of it . For this aggregate is not to be designed by a Semiparabola , but by the complement of a Semiparabola . And many other mistakes ▪ consequent thereunto . And indeed so many , as that dispairing of mending them all , you resolved to let them stand as they were . Yet I shewed you withall the chiefe ground of all these mistakes , and how they might have been mended . But it appeares you had not the wit to understand it , and therefore durst not venture upon it . But have left this whole article such an Hodge podge of errors , as would turne a quea●ie stomach , but to examine it . And your Corollarys are false also . In the first Corollary , 'T is false which you affirme , that the proportion of the parabola ABI to the parabola AFK , is triplicate to the proportion of the times , AB to AF , ( as it is in the English . ) or of the Impetus BI to FK , ( as it is in the Latine . ) This exception you confesse to be just , yet leave it uncorrected in the English ; because you know not how to mend it ; without giving your selfe the ly in the rest . For as badde as it is , it follows , with the rest of your doctrine . It must all stand or fall together . The second Corollary , ( at least , if understood of the Parabola , ) is also false ; for the segments of a parabola ( of equall height ) successively from the Vertex , are not as the numbers 7 , 19 , 37 , &c. the difference of the Cubes 1. 8 , 27 , 64 , &c. but us the differences of these surd nūbers 1 , √ 8 , √ 27 , √ 24. &c. That which you alledge to justify your selfe ; that the parts of the Parabola cut off are as the cubes of their bases ; is but a repetition of the same error . They are not as the Cubes of their Bases , but as the square roots of such Cubes . The third Corrollary is wholly false , A motion so accelerated doth not dispach two thirds ; but one third , of what a uniform motion would have done , with an Impetus equall to the greatest of those so increasing . You say , I give no demonstration of it . ( It may be so ; and it 's all one to me , whether you believe it to be true or no. You may think , if you please , that the Corollary is true still ; it will not hurt me . ) Yet if you considered what had been said before , you should have seen the reason : viz. because the aggregate of the Impetus did not constitute a semiparabole , but the complement of a semiparabola , which is not ⅔ but ⅓ of the Parallelogram . The fift article hath the same faults with the fourth ; and runnes all upon the same mistakes . The main foundation of all these continued errors , was , I told you , the ignorance of what is proportion duplicate , triplicate , subduplicate , subtriplicate , &c. Of three numbers in continuall proportion , if the first be the lest , the proportion of the first to the second is duplicate , of what it hath to the third , not subduplicate : That was your opinion Cap. 13. § 16. of the Latine . In the English , you have retracted that error in part ; yet retaine all the ill consequences that followed from it . Next , you suppose the Aggregates of the Impetus increasing in the duplicate , triplicate &c. proportion of the times , to be designed by the Parabola , and Parabolasters , ( as if their ordinates did increase in the duplicate , triplicate , &c. proportion of their Diameters ; cujus contrarium verum est ; ) whereas you should have designed them by the complements of those figures , But you aske me what line that ( complement ) is ? No Line , good Sir , but a Figure , which with the figure of the Semiparabola &c. doth compleare the Parallelogram . You ought therefore ( as I then told you , but you understood it not , ) to have described your Parabola the other way ; that the convex ( not the concave ) of the parabolicall line should haue been towards the line of times AB . so should the point K have fallen between N and F ; and the convex of the Parabola with AT ( the tangent ) and BI ( a parallel of the Diameter , ) have contained the complement of that parabola , whose diameter therefore must have been AC , and its Ordinate CI. Next , in pursuance of this error , you make the whole velocity , in these accelerations ( in duplicate , triplicate &c. proportion of the times ) to be ⅔ , ¾ , &c. of the velocity of an uniforme motion with the greatest acquired Impetus , ( because the Parabola and Parabolasters , have such proportion to their Para ●lelograms ) whereas they are indeed but ⅓ , ¼ , &c. thereof ; for such is the proportion of the complements of those figures , to their Parallelograms . Now upon these false principles , with many more consonant hereunto , you ground not only the doctrine of the fourth and fifth Articles , but also most of those that follow ; especially the thirteenth and thenceforth to the end of the Chapter : which are all therefore of as little worth as these . But enough of this . The first five Articles therefore are found to be unsound ; and many ways faulty . The sixth , seventh and eighth Articles , I did let passe for sound : And you quarrell with me for so doing . But I said withall , you might have delivered as much to better purpose in three lines , as there you did in five pages . ( Beside such petty errors all along as it were endlesse every where to take notice of ) which gives you a new occasion to raile at Symbols . After these three , there is not one sound Article to the end of the Chapter , and what those were before , we have heard already . The ninth article is this , If a thing be moved by two Movents at once , concurring in what angle soever , of which the one is moved uniformely , the other with motion uniformely acceleeated from rest , till it acqu●e an Impetus equall to that of the Vniforme motion ; the line in which the thing moved is carried , will be the crooked line of a semiparabola . Very good ! but of what semiparabola ? ( for hitherto , we have nothing but a proporsion of Galilaeo's transcribed . ) You tell us , ●t shall be that Semiparabola , whose Busis is the Impetus last acquired ; And this is the whole designation of your Parabola . To this designation I objected many things . First , that the Basis of a Semiparabala is not an Impetus but a Line : and therefore 't is absurd to talke of a Semiparabola whose Basis is an Impetus . Secondly , if it be said that an Impetus may be designed by a line ; I grant it ; ( a line may be the Symbol of an Impetus , as well as a Letter ▪ ) but this line , is what line you please ; ( for any Impetus may be designed by any line at pleasure : ) & so , to say that It is a Semiparabola , whose basis is that line which designes the Impetus : is all one as to say , it is a Semiparabola , whose basis is what line you please . So that we have not so much as the Basis of this Semiparabola determined . Thirdly , suppose that the Base had been determined , ( as it is not ) yet it is a simple thing to think that determining the basis , doth determine the Parabola . For there may be infinite Parabola's described upon the same Base . You doe not tell us what Altitude , what Diameter , nor what Inclination this Parabola is to have . Now to this you keep a bawling ; but say nothing to the businesse . You tell us , that you had said , what angle soever . That is , you supposed your Mevents to concurre in what angle ●soever ; but you sayd nothing of what was to be the angle of inclination in the Parabola . You might have said indeed , it was to be the same with that of the Movents : But you did not ; and therefore I blam'd you for omitting it . Then , as to the Diameter , you might have said ( but you did not ) that the line of the acccelerate motion , would be the diameter . 'T was another fault therefore not to say so ; for that had been requisite , to the determining of the Parabola . But when you had so said ; this had but determined the Position of the diameter , not its magnitude : it may be long or short , at pleasure notwithstanding this . Then as to the altitude of it ; this remaines as much undetermined as the rest . You tell us neither where the Vertex is , nor how farre it is supposed to be distant from the Base . you might have said , ( but you did not , ) that the point of Rest , where the two motions begunne , was the vertex . ( And t was your fault you did not say so in the latine , as you have now done in the English . ) But had you so said , you had not thereby determined either the Altitude , or the Diameters length . You say , The vertex and Base being given , I had not the wit to see that the altitude of the Parabola is determined . No truely ; nor have I yet . But it seems you had so little wit , as to think it was . Had the vertex and Base been , positione data : I confesse , it had been determined : ( For then I had been told how farre off from the Base , the Vertex had been . ) But when the Base is only magnitudine data , there , is no such thing determined . For a base of such a bignesse , may be within an Inch , and it may be above an E●l from the Vertex , according as the Parameter is greater or lesse . Now you doe not pretend any other designation of the Base , then that it be equall to such an Impetus ; which determines only the bignesse of it , not the distance from the vertex . So that the altit●de , notwithstanding this flamme , remaines undetermined . ( And must do so , whatever you think , till you do determine the degree of celerity , which answers to the Parameter of the Parabola ; as well as the manner of acceleration , which only determines that it is a Parabola , but not what Parabola . The proposition therefore is extreamly imperfect ; nor doth determine that which it did undertake to determine . The figure is yet worse . You suppose the line AB , ( fig. 6. ) by uniforme motion , to have dispatched the length AC , or BD , and so ly in CD ; in the same time that the line AC , by motion uniformely accelerated , dispatcheth , the length AB , or CD , to come and lye on CD . That is , ( because AB , according to your figure , it about twise the length of AC , ) the motion accelerated doth , in the same time , dispatch about twise the length of what is dispatched by the uniforme motion . But it is evident , the accelerate motion is all the way , to the very last point , slower than the uniforme , ( for by supposition , it doth not till the last point , attain to that Impetus or swiftnesse , with which the uniform motion was carryed all the way . ) Therefore according to you , a slower motion doth , in the same time , dispatch a a greater length then the swifter , Which is absurd enough : And to which you make no reply . The demonstration also ( saving what you have from Galileo ) I then shewed you to be faulty ; and you reply nothing in its vindication and therefore I need not repeat it . You have in the English a little disguised the proposition , but to little purpose . The Parabola which you undertake to determine , remaines as undetermined as it was before . And the figure the same with all its faults : And the demonstration no whit mended . So much of this Article as yo● tooke out of Galileo was good , before you spoild it ; but the next is all naught . Your tenth Article doth but repeat all the faults of the ninth , and you have nothing more to say in the vindication of this then of that . The Parabolaster here , remaines as undetermined as the Porabola there ; your Figure ( fig. 6. ) makes the flower motion in the same time to dispatch the greater length ; your demonstration is faulty as that was . Nay you have not here , so much as disguised it in your English , as you did the former ; but left it as it was in the Latine . So that this falls under the same condemnation with the former . I hinted also , that we have here a great talke of Parabolasters which are not to be defined till the next Chapter . But that 's a small fault . Your English helps it , by sending us thither for the definiton . Your eleventh Article undertakes to give us a generall rule , to find what kind of line shall be made by the motion of a body carryed by the concurse of any two Movents , the one of them Vniformely , the other with acccleration , but in such proportion of Spaces and Times as are explicable by Numbers , as Duplicate , Triplicate &c. or such as may be designed by any broken number whatsoever . Your rule for this , sends us to the Table of Chap , 17 ▪ art . 3. to seek there a Fraction whose Denominator is to be the summe of the Exponents of Length and Time ; and its Numerator , the exponent of the Length . Upon this I proposed you a case which falls within your proposition , but not within your Rule : ( to shew that your Rule did not performe what you undertook to performe by it . ) Let the motions , sayd I , be , the one , uniforme ; the other accelerate , so as that the spaces be in subduplicate proportion to the Times ; or , in your language , as 1 to 2. We are therefore , by your Rule , to seeek in the Table the fraction ⅓ . But there 's no such fraction to be found ( nor any lesse then ½ . ) Your rule therefore doth not serve the turne . Well ▪ let 's heare what you have to say for your selfe . Did I not see ( you aske ) that the Table is only of those figures which are described by the concourse of a motion Vniforme , with a motion accelerated . Yes I did , see that the table is only of such : Nay more , I saw ( which is more to your purpose ) that the proposition is only of such ; ( though yet if need be , I could shew you how the same figures might be described by motion retarded as well as motion accelerated , ) & therefore I proposed such a case ; viz. an acceleration in the subduplicate proportion of the times , that is after the rate √ 1 ▪ √ 2. √ 3. √ 4. &c. which is the subduplicate rate of 1 , 2 , 3 , 4 , &c. I had no reason therefore , say you , to look for ⅓ in thae Takle . That is , I had no reason to expect , that your Rule should performe what you undertake . But why no reason to expect it ? For my case is of motion uniforme concurring with motion retarded . No ▪ such matter , ( nor be you so simple to think so , whatever you here pretend ; ) for √ 1. √ 2. √ 3. √ 4. &c. is no decreasing progression , but increasing : for √ 2 , is more then √ 1 & √ 3 , & more then √ 2. & so on . But why should you think it is not so ? Because forsooth . I do not make the proportion of the spaces to that of the times duplicate , but subduplicate . Very good 〈◊〉 But if times be proposed in a series increasing as 1 , 2 , 3 , 4 , &c. will not the subduplicate rate be increasing also , as well as the duplicate ? that is , doe not the Rootes of these numbers continually increase , as well as their Squares ? Think againe and you 'l see they doe . Well , but however , though this table will not serve the turne , yet the ●ase may be solved , you tell us , another way . No doubt of it . I could have told you so before . ( For though you knew not how to resolve it ; I did ; and therefore directed ▪ you to the 64. Prop of my Arithmetica Infinitorum ; where you have the case resolved more universally then it is by you proposed ; viz. where the exponent of the rate of acceleration is not explicable by numbers ; but even by surd ro●tes , or other irrationall quantities . ) But what becomes of your rule in the mean while , which sent us to that Table for solution ? where , you now tell us , ( for I had told you so before ) it is not to be hard ? This eleventh Article therefore , is like the rest . Nor is it at all amended in the English . Your twelfth proposition , I said , was wretchedly false ; And I say so , still . But , you say , you have left it standing unaltered ; ( & yet that 's false too ; for your English hath a considerable alteration from what was in the Latine , though not much for the better ) Your words were these ▪ If motion be made by the concourse of two movents , whereof one is moved uniformely , the other with any acceleration whatever ( for which you say in the English , the other beginning with Rest in the Angle of concurse , with any acceleration whatsoever ) the movent which is moved uniformely shall put forward the thing so moved , in the severall parallel spaces , lesse , than if both motions had been Vniforme . I gave instance to the contrary , ( in fig. 5. ) The streight line AND , may be described by a compound of two uniforme motions ; and the parabolick line AGD , by a motion compounded of two , the one uniforme and the other accelerated , ( neither of which you can deny , for you affirme both , at art . 8 , and 9. ) But within the Paralells AC , EF , the thing moved ( contrary to your assertion ) is more put forward by this , than by that motion ( for EG , is greater then EN , ) The p●oposition therefore , in this case is false . Yonr answer is , that other Geometricians find no fault with it . It may be so . But is there any Geometrician ( who hath well examined it ) will say 't is true ? and that , in all cases ? In some cases I told you , it may happen to be true ; and in in other cases it will be certainely false : ( And I told you also , when , and where . ) And I did in the case proposed prove it so to be ; and you can say nothing to the demonstration . You would indeed tell me of another case wherein , you think it is true . But what 's that to the purpose ? When I give instance to the contrary of a universall proposition , you must allow me to lay the case as I think good ( so as it be within the limites of that universall ) and not as you would have me . The proposition therefore is demonstrated to be false . And you have nothing to say in vindication of it . The thirteenth Article doth propose a Problem as ridiculous as a man would desire to read . 'T is this Let AB ( fig. 8. ) be a Length transmitted with uniforme motion in the Time AC : And let it be required to find another length which shall be transmitted in the same time with motion uniformly accelerated , so as the Impetus ( or , as in the English , the line of the impetus ) last acquired be equall to the streight line AC . The Answer say J to this Probleme , is what length you please . ( And you might as well have propounded , A quantity being assigned which is equall to its foure quarters ; let it be required to find another quantity which is equall to its two halves . Or thus A parallelogram being proposed of a known Base and Altitude ; let it be required to find what may be the altitude of a triangle on the same base . Where , what quantity you will , doth serve for answer to the former : And , what altitude you will for the latter . And , what length you will , is the answer to your Problem . ) For there is no length assignable , which may not , in any assignable Time , be dispatched by a motion uniformly accelerated , whose last Impetus shall be what you please . And 't is but as if you should have asked ; What may be the height of that Parabola , or Triangle , whose Basis is equall to AC ? The Problem being thus ridiculous , it cannot be expected that the construction or demonstration should be better . And truly 't is pittifull stuffe all of it : as J then shewed . And you do not so much as attempt any thing by way of answer , to justify either your construction or demonstration . You ask here , ( for you have no more witt then to propose such a question , ) granting that a Parabola may be described upon a Base given ; and yet have any Altitude , or any Diameter one will : ( which you say who doubts ? ) How it will hence follow , that when a Parabolicall line is described ( is to be described , you should have said ; for the Problem is of somewhat to be done , not , of somewhat done already , ) by two motions , the one uniform , the other uniformly accelerated from rest ; That the determining the Base , doth not also determine the whole Parabola ? J answer . Because every Parabola may be so described ; ( which if you did not know before , you may now learn of me : ) And therefore , since that , upon a Base given , a Parabola may be described of any altitude ( as you grant ; ) and that every Parabola may be so described : the determining of the Base , doth not determine the Altitude of a Parabola so to be described ; more then the Altitude of a Parabola simpliciter . But if you would have done any thing to acquit your selfe of the charge in this Article , ( of proposing a Ridiculous , Nugatorious Problem : ) You should have assigned some Length , which by a motion so accelerated , and acquiring such an Impetus , could not have been dispatched in a Time assigned . Till then ; I say , it may dispatch what length you please : And therefore your Problem is as ridiculous as a man could wish . There be divers other petty faults , that J took notice of by the way ; as that those words , so as the Impetus acquired be equall to a Time ( as if heterogeneous things could be equall . ) And , those words , as duplicate proportion is to single proportion , so let the line AH be to the line AI. ( which is as pure nonsense as need to be : ) As if there were one certain Proportion of the Duplicate proportion , to the single Proportion . You tell us , upon second thoughts , in your English , cap. 13. art . 16. that Duplicate proportion is sometime greater then the single ; and that it is sometimes lesse : And yet you would here have us think that it is alwaies as 2 to 1. The proportion of 9 to 1 , is duplicate of that of 3 to 1 : And the proportion of 4 to 1 , is duplicate of that of 2 to 1. But there is not the same proportion of the proportion 2 / 1 to the propor●ion ● / 1 , that there is of the proportion 4 / 1 to the proportion 2 / 1 ▪ but that is triple this double : ( for nine times as many , is the triple of three times as many ; and four times as many , is but the double of twice as many . ) But this you cannot understand , and therefore call for help from somebody that is more ready in Symbols . It seems a man must speak to you in words at length , and not in figures . And truly , all 's little enough to make you understand it . The 14 , 15 , and 16 Articles are just like the 13 : and as ridiculous as it . What was there objected , you confesse , may as well be objected to these . But that hath been proved to be ridiculous : and therefore so are these . Any length being given , which , in a Time given , is dispatched with uniform motion ; To find out what length will be dispatched in the same time with motion so accelerated , as that the Lengths dispatched be continually in triplicate proportion to that of their times . ( so Art. 14. ) or quadruplicate , quintuplicate , &c. ( ibidem . ) or as any number to any number . ( so Art. 15. 16. ) and the Impetus last acquired equall to the Time given . That 's the Problem . The Solution should have been ; What length you please . Take where you will you cannot take amisse . If you say , 't is an Inch , you say true : If you say , 't is an Ell , you say true : And if you say 't is a thousand miles , no body can contradict you . For it may be what you please . And is it not a wise thing of you then , for the designing of an Arbitrary Quantity , a What-you-will , to bring a parcell of Constructions , and Demonstrations , with finding of Mean Proportionalls , as many as one please ; for a matter of two leaves together ? And , when you have done all , 't is but , ( as you were , ) What you will. J noted farther that in all these Articles 13 , 14 , 15 , 16 , as in those before Art. 9 , 10 , 11. & those following 17 , 18 , 19. You doe every where make the slower motion , in the same time , dispatch the greater length . Which I did clearly demonstrate . To this you reply nothing to the purpose : But cavill , that you might seem to say something . You say , I corrupt your Article by putting Movens for Mobile . But there 's no such matter ; for in the place alleadged ( Art. 1. ) Movens is your own word , not mine . You say , 't is no matter whether AB or AC ( in the fifth figure ) be the greater . Yes it is ; it 's impossible that AB , according to your supposition , should be so bigge as AC ; and yet , you have made it almost twice as big . You say , you speak of the concurse of two movents ; very true . But each of those movents have their severall pace assigned them ; & therefore you should not have made the slower movent to rid more ground . And then you would tell mee , what I think ; and then talk of hard speculations , of edge and wit and malice &c. But nothing to the purpose . For when you have all done , its evident , and you cannot deny , that in your 5 and 6 and 11 Figures , AB is made welnigh twice as long as AC ; and so again in your 8 , 9 , and 10 , Figures AH much longer then AB ; and yet these longer lines designe the length , dispatched by the slower motions in the same time . For the motion accelerate , which doth not till its last moment attain the swiftnesse , with which the uniform motion proceeds all the way , must needs be slower then that uniform motion . But this was a fault which I might safely have let passe ; for these Articles were ridiculous enough before . In the 17 Article , I shewed first , that the Proposition , as it was proposed , was not perfect sense . Then , that , the sense being supplied , the Proposition was false . And lastly , that your Demonstration had at lest fourteen faults , and most of them such , as that any one was sufficient to overthrow the Demonstration . The Proposition was this , If in a time given , a Body run over two lengths , one with Vniform , the other with accelerated motion , in any proportion of the length to the time , And again in a part of that time , it run over parts of those lengths with the same motions ; the excesse of the whole longitude above the whole ( to what ? ) is the same proportion with the excesse of the part above the part , to what ? Is this good sense ? No ; you confesse there was somewhat left out in that Proposition , but say , it was absurdly done to reprehend it . Very good ! It seems you must have the liberty to speak non-sense without controll . Well ; but how is the sense to be supplied ? we made two or three essays the last time , and found never a one would hold water , but which way soever we turned it , the Proposition was false . We have two proportions designed only by their Antecedents , and we are to guesse at the consequents . The best conjecture I could make was this ; As the excesse of the whole above the whole , is to one of those wholes ; so is the ex●esse● of the part above the part , to one of those parts , ( respectively . ) That is ( calling the greatest whole G , and its part g : and the lesser whole L , and its part l. ) as G − L , to G ; so g − l , to g. Or secondly thus ; as G − L , to L ; so g − l , to l. But both these are found false . My next conjecture was from the 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 , ( but there I was fain to leave my proposition quite , and take up new Antecedents , as well as seek new consequents , ) and that directs me to such an Analogisme ( p. 140. l. 39. ) I say that as AH to AB , so AB , to AI ; but this is ambiguous , because , AB comming twice , once as a whole , and another time as a part , sit doth not appear which is which ; therefore here be two conjectures more ; viz. a third thus , as the whole to the whole , so the part to the part ; ( that is G. L ∷ g. l. ) Or fourthly thus , as the whole to its part , so the whole to its part . ( that is G. g ∷ L. l. ) But these two are both false also . My next attempt was from the 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 , ( but here also I must desert the proposition too , and seek new antecedents as well as consequents , ) where I find it thus ( p. 141. l. 7 , 9. ) as AH to AB , so is the excesse of AH above AB , to the excesse of AB above AI : which was to be demonstrated . that sends me to a fifth , sixth , seventh and eighth analogisme ( because it doth not appear which AB is the whole , and which the part ; ) the fifth thus , as the whole to the whole , so the excesse of the whole above the whole , to the excesse of the part above the part , ( taking AB in the two first places for the whole ) that is G. L ∷ G − L. g − l. The sixth thus , as the whole to the whole , so the excesse of the whole above its part , to the excesse of the whole above its part ( taking AB in the first and last place , for the whole , ) that is G. L ∷ G − g. L − l. The seventh thus , as the whole to its part , so the excesse of the whole above the whole , to the excesse of the part above the part , ( taking AB in the first and last place for a part , ) that is G. g ∷ G − L. g − l. The eighth thus , as the whole is to its part , so the excesse of the whole above its part , to the excesse of the other whole above its part , ( taking AB in the two first places , for the part , ) that is G. g ∷ G − g. L − l. But these four be all false likewise , as well as those before . Now all these eight conjectures are of equall probability ( though all false ) it cannot be said which of them is more like to be the sense intended than the other . And yet , forsooth , when , by talking non-sense , you leave us at this uncertainty of conjecture , it is ( you say ) absurdly done to reprehend it . I confesse , if any one of these Analogismes had been true , we might have guessed that to be your meaning : but when they be all equally probable , and equally false , which should we take ? Well , but 't is to be hoped , that now you will tell us . You tell us therefore ( Less . p. 38. ) it should be thus , as the excesse of the whole above its part , to the excesse of the other wh●le above its part , so that whole , to this whole : which affords us a ninth analogisme , G − g. L − l ∷ G. L. which is coincident with my sixth conjecture . And yet again ( Less . p. 39. ) you tell us , that the proposition is now made ( in the English ) according to the demonstration ( that is ; both false , ) and there we find it thus , the whole to the whole , as the part to the part ; that is G. L ∷ g. l. which allso is coincident with our third conjecture . But which soever of all these analogismes you take , the Proposition is false , and therefore the demonstration must needs be so too . Now to prove that this Proposition is false , which way soever you turne it , ( either as it was before , or as it is now , ) I made use of the figure of your first article , and proceeded to this purpose . Let the whole time ( fig. 1. ) be AB , an hour , ( that is , because I would not have you mistake mee , as you doe Archimedes , let the line AB represent an hour , or , be the symboll of an hour ; for I would not have you think that I take a line to be an hour ; but to represent an houre ; and the letters AB to represent that line , not to be that line ; like as at another time we take a letter , without a line , to represent an houre : ) and part of that time AF , halfe an houre . Let also the continued Impetus of the Uniform motion ( I mean the Symboll of it ) be AC , or BI : which BI also is to be ( the Symbol of ) the last acquired Impetus of the motion accelerated . And this acceleration we will suppose at present ( as your selfe do in your 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 ) to be uniform acceleration . The velocity therefore of the whole uniform motion , will be represented by the Parallelogram ACIB ; ( by the first article ; ) and of it's part , ACHF ; ( by the same article ; ) The velocity of the whole uniformly accelerated motion , will be the Triangle AIB ; and of its part , AKF ; ( by the same article . ) Since therefore the lengths dispatched be proportionall to those velocities ; the whole length of uniform motion , to the whole of the accelerate , will be as the Parallelogram ACIB , to the Triangle AIB , that is , as 2 to 1. ( viz. the length of the uniform motion , bigger than that of the accelerate ; whereas your figure and demonstration , do all the way suppose the contrary ; ) so that if the uniform motion do in an houre dispatch 16 yards , the accelerate will in the same time dispatch 8 yards , ( that is G = 16 , & L = 8. ) Again , the length dispatched by the uniform motion in the whole time ; to that in half the time , is as the Parallelogram ACIB , to the Parallelogram ACHF ; that is , as 2 to 1 ; so that if G ( as before ) be 16 , then is g = 8. Lastly ; the length dispatched by the accelerate motion in the whole time , to that in halfe the time , is as the Triangle AIB , to the Triangle AKF ; that is as 4 to 1 , ( for the sides AB , to AF , being as 2 to 1 , and the triangles in duplicate proportion to their sides , the triangles will be as 4 to 1 : ) So that if L ( as before ) be 8 , then is l = 2. Now having thus found the measures of these four lengths ; ( viz. G = 16. L = 8. g = 8. l = 2. ) You shall see that those Analogismes are all false ; not one true amongst them . The first is this , G − L. G ∷ g − l. g. that is 16 − 8 = 8. 16 ∷ 8 − 2 = 6. 8. or 8. 16 ∷ 6. 8. But this is false . The second this , G − L. L ∷ g − l. l. that is 16 − 8. 8 ∷ 8 − 2. 2. or 8. 8 ∷ 6. 2. But this is false also The third this , G. L ∷ g. l. that is 16. 8 ∷ 8. 2. and this also is false . The fourth this , G. g ∷ L. l. that is 16. 8 ∷ 8. 2. and this is also as false as the other . The fifth is this , G. L ∷ G − L. g − l. that is 16. 8 ∷ 16 − 8. 8 − 2. or 16. 8 ∷ 8. 6. which is also false . The sixth this , G. L ∷ G − g. L − l. that is 16. 8 ∷ 16 − 8. 8 − 2 , or 16. 8 ∷ 8. 6. which is like the rest . The seventh is this G. g ∷ G − L. g − l. that is , 16. 8 ∷ 16 − 8. 8 − 2 or 16. 8 ∷ 8. 6. false also . The eighth is this . G. g ∷ G − g. L − l. that is 16. 8 ∷ 16 − 8. 8 − 2. or 16. 8 ∷ 8. 6. which is also false . The ninth is this , G − g. L − l ∷ G. L. that is 16 − 8. 8 − 2 ∷ 16. 8. or 8. 6 ∷ 16. 8. The tenth is like the third , G. L ∷ g. l , that is 16. 8 ∷ 8. 2. all false . The proposition therefore , turne it which way you will , is a false Proposition . And yet you have the Impudence to tell us ( though you knew this before , for I told it you last time , and brought the same demonstration , to which you have not replied one word ) that 't is all true , and truly demonstrated . Do you think 't is worth while after all this , to examine your demonstration ? 'T is a sad one , I confesse ; but t is yours , and therefore it may perhaps be beautifull in your eye . The last time we looked upon it , we found it had at least fourteen grosse faults : ( and most of them such , as were singly enough to destroy it : ) enough in conscience for one poor demonstration . ( And had you not been good at it , a man would have wondred how you could have made so many ex tempore . ) Since that time , 't is quite defunct . And there is a young one start up in stead of it . But 't is of the same breed , and t is not two pence to choose , whether this or that . Your new demonstration runs it self out of breath at the first dash . You had told us ( Art. 3. Coroll . 3 ) In motion Vniformly accelerated from rest , ( such as is one of these ) the length transmitted ( as here AH , fig 8. ) is to another length ( viz. AB , ) transmitted uniformly in the same time , but with such Impetus as was acquired by the accclerated motion in the last point of that time ( just the case in hand ) as a Triangle to a Parallelogram which have their altitude and base common , that is , as 1 to 2 , for the Parallelogram is double of the Triangle . So that AH , in your figure , should be but just half as bigge as AB ; and you have made it allmost twise as big . And upon this foundation depends the whole demonstration . For if that fault were mended , your whole construction comes to nothing . And is not this demonstration then well amended ? especially when you had faire warning of it the last time . And then you send us to the demonstration of the 13 Article for confirmation of this , whereas that Article hath been cashiered long agoe , and the demonstration with it . But thus 't is when men will not take warning . At length you fall to raating , ( as you use to do when you be vexed ; ) about skill , and diligence , and too much trusting ; about discretion , Hyperbole's , and Sir H. Savile , &c. And tell us that when a beast ( Joseph Scaliger ) is slain by a Lion ( Clavius ) 't is easy for any of the fowles of the aire ( Sir H. S. ) to settle upon , and peck him . And Vespasian's law , no doubt , will bear you out in all this . Only this I must tell you , that Sir H. Savile , had confuted Joseph Scaliger's Cyclometry , as well as Clavius ; and , I suppose , before him . Which if you have not seen , I have . In the 18 Article , we have this Proposition . If , in any Parallelogram , ( suppose ACDB , fig. 11. ) two sides containing an angle be moved to the sides opposite to them , ( as AB to CD , and AC to BD , ) one of them ( AB ) with uniform motion , the other ( AC ) with motion uniformly accelerated : that side which is moved uniformly ( AB ) will effect as much , with its concurse through the whole length , as it would do if the other motion were also uniform , ( or were not at all . For what ever the other motion be , the motion of AB to CD , carrieth the thing moved with it from side to side , and that 's all . What point of the opposite side it shall come to , depends upon the other transverse motion , not upon this at all . And this is so easy that no body would deny it . If you mean any thing more then this , that it shall carry it just to the opposite side & no farther , your demonstration doth not at all reach it ▪ But you go on ) and the length transmitted by it in the same time , a mean proportionall between the whole and the halfe , of what ? Till you tell us of what ? I say , as I said before , that these words have no sense . The construction and demonstration of this proposition , I remember , we made sad work with , the last time we had to doe with them , as well as with those of the former Article ; which will be now too long to repeat . The whole weight of the Demonstration lies , severally , upon at lest these three Pillars , of which if any one do but fail , the whole demonstration falls . First , upon the strength of the 13 Article , which we have destroyed long agoe . Secondly upon the 12 Article , which we have also long since proved to be false . Thirdly , upon this learned assertion , the streight line FB will be the excesse by which the ( lesser ) length transmitted by AC with motion uniformly accelerated , till it acquire the impetus BD , will exceed the ( greater ) length transmitted by the same AC in the same time with uniform motion , and with the Impetus every where equall to BD. Which destroys it selfe . For if the accelerated motion , as is supposed , do not till its last moment acquire that speed with which the uniforme motion is moved all the way ; then that must needs be slower than this ; and consequently dispatch a lesser length in the same time : whereas you according to your discretion , make the length dispatched by that slower motion , to be more then that of the swifter in the same time , and tell us the excesse is FB . And then to helpe the matter , when I presse you with this absurdity , you tell us you speak of motions in concurse : as though in concurse , the slower motion did in the same time , caeteris paribus , dispatch a greater length than the swifter , though out of concurse the swifter motion did dispatch a greater length than the slower ▪ Now either of these three , much more all of them , doth wholly destroy the strength of your demonstration . Yet they that desire to see more may consult what I sayd before . The ninteenth Article doth not pretend to any other strength than that of the eighteenth . And therefore falls with it . The twentieth Article I did before prove to be false and frivolous . ( it depended upon Chap. 14. Art. 15. Corol. 3. which Corrollary I have there consuted . ) You say nothing by way of vindication of what I excepted against ; only passe your word for it , that it is true . Yet withall confesse , there is a great error ; and that error say I , though there were nothing else , would make that article unsound . But this article you say , was never published ( yet 't is as good as most of those that were in this Chapter ; for I 'le undertake for it , there he above a dozen worse ; ) and therefore it was inhumanly done , you say , to take notice of it . Truly , if the proposition were a good proposition , as you say it was . J think J did you a courtesy to publish it for you , that you may have the credit of it ; yet J should not have done it , had it not been publike before . If you would not have it taken notice of , you should have taken care not to send it abroad . For it hath been commonly sold with the rest of your book ( to many more persons beside my selfe ; ) they that would , might teare it out ( as some did ) and they that would , might keep it in , as J did . Well , ( be the number of articles 20 , or be they 19 , ) before the sixth there was none sound , ( but either in whole or in part unsound , ) and from the eighth there hath been none sound ; therefore there have not been above three sound at the most . Quod erat demonstrandum . SECT . XI . Concerning his 17. Chapter . THE Reader by this time may perhaps be weary , as well as J ; and think it but dull work to busy himselfe upon such an inquiry , where the result is but this , That M. Hobbs his Geometry is nothing worth ; which ( if he had any himselfe ) he knew before . To save him therefore , and myselfe the labour , wee 'l make quicker work in what 's behind . In the 17. Chapter , some of the Propositions are true and good ; ( and truely I wondred at first where you had them , but since I know : ) But the demonstrations are foolish and ridiculous . The Propositions therefore are your own ( you know where you stole them ; ) and the Demonstrations are of your own making ; ( for there be scarce such to be found any where else . ) What you say to the first Article comes to this result ; that I should say , It is well known , that , in Proportion , Double is one thing , and Duplicate another . And you aske , To whom it is known ? ( it seems it was not known to you : ) And tell us , that they are words that signify the same thing ; and , that they differ ( in what subject soever ) you never heard till now . It 's very possible that this may be true ; that you did never know the difference between those two words till I taught you . ( But this was your ignorance not my fault . ) But now , you know there is a difference . And therefore ( contrary to what you had affirmed in the Latin ) you tell us in your English , Chap. 13. art . 16. p. 121. l. 7. &c. and p. 122. l. 26. &c. that the proportion of 4 to 1 , to that 4 to 2 &c. is not only Duplicate , but also double or twise as great . But on the contrary , the proportion of 1 to 4 , to that of 1 to 2 , &c. though it be duplicate , it is not the double , or twise as great , but contrarily the halfe of it ; and that of 1 to 2 , to that of 1 to 4 , &c. is Double you say , and yet not duplicate but subduplicate . Now if you never heard of such a difference till you heard it from me , then you are indebted to me for that peece of knowledge : and have no reason to quarrell with me , as you use to doe , for saying you did not understand what was duplicate and subduplicate proportion ; for you confesse you did not , but tooke it to be the same with double and subduple , and never heard that they did differ till now . In the second Article , because it is fundamentall to those that follow , I took the paines first to shew how unhandsomely the proposition and 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 were contrived ; and then to shatter your demonstration all to pieces ; and shewed it to be as simple a thing as ever was put together , ( unlesse by you , or some such like your selfe . ) As to the first , you tell us , that , to proceed which way you pleased was in your own choice . And I take that for a sufficient answer . You did it , as well as you could ; and they that can do better may . As to the Demonstration , you keep a vapouring ( nothing to the purpose , ) as if it were a good demonstration . and not confuted . Yet , when you have done , ( because you knew it to be naught ) you leave it quite out in the English , and give us another ( as bad ) in stead of it . That is , you confesse the charge . Your fundamentall Proposition was not demonstrated ; and so this whole chapter comes to nothing . But however , 't is to be hoped , that your new demonstration is a good one ; is it not ? No , 'T is as bad as the other . Only 't is not so long : And of a bad thing , ( you know , ) the lesse the better . It begins thus , The proportion of the complement BEFCD , ( fig. 1. ) to the deficient figure ABEFC , is all the proportions of DB to OE , and DB to QF , and of all the lines parallell to DB , terminated in the line BEFC , to all the parallells to AB terminated in the same points of the line BEFC . Now for this ( besides that it is a piece of non-sense ) you send us for proof to the second Article of the 15. Chapter , where there is nothing at all to that purpose . Then you go on . And seeing the proportion of DB to OE , and of DB to QF , &c , are every where triplicate to the proportion of AB to GE , and of AB to HF &c. the proportions of HF to AB , and of GE to AB , &c. are triplicate ( no , but subtriplicate ) of the proportions of QF to DB , and of OE to DB &c. Now this is but the same Bull that hath been baited fo often . viz. because the diameters ( DB , OE , QF , &c. that is CA , CG , CH , ) are in the triplicate proportion of the Ordinates ( AB , GE , HF , ) therefore these Ordinates are in the triplicate proportion of those diameters . You might as well have sayd , seeing that 6 is the triple of two therefore 2 is the triple of 6. But let 's hear the rest , for there is not much behind . ) And therefore the deficient fig. ABEFC , which is the aggregate of all the lines HF , GE , AB , &c. is triple to the complement BEFCD , made of all the lines QF , OE , DB , &c. A very good consequence ! Because the Ordinates are in triplicate proportion to the diameters ( yet that is false too , for they are in subtriplicate ) therefore the figure is triple to its complement ? But how doe you prove this consequence ? Nay , not a word of proof . We must take your word for it . Well then , of this last Enthymem , ( which was directly to have concluded the question , ) the Antecedent is false ▪ and the consequence at lest not proved ( I might have said false also , for so it is . ) And this is your new demonstration . The third article , I sayd , falls with the second ; for having no other foundation but that , ( nor do you pretend to other ) that being undemonstrated ( for your former demonstration your selfe have thrown away , and your new one we have now shewen to be nothing worth , ) this must be undemonstrated too . In the fourth Article , you attempt the drawing of these Curve lines , by point ; and to that purpose require the finding of as many mean proportionalls as one will , ( like as you had before done Cap. 16. 6. 16. for the finding out an arbitrary line to be taken at pleasure : Which I told you was simply done , because that without such mean proportionalls , ( that is , without the effection of solid & Lineary problems , ) it might have been done by the Geometry of Plains , that is with Rule and Compasse . And I shewed you how . To which you have nothing to reply , but , that I made use of one of your figures ( to save my selfe the labour of cutting a new one , ) that is , I made better use of your figure then you could doe . The fifth proposition ( beside that it is built upon the second , and therefore falls with it , ) is inferred only from the Corrollary of the 28. article of the 13. Chapter , ( nor doth your English produce any other proof , ) where , sayd I , there is not a word to that purpose . And you confesse it . The 6 , 7 , 8 , & 9. Art. do not pretend to other foundation than the second ; & therefore till that be proved , fall with it . The 10. Article is a sad one , as may be seen by what I did object against it , as you say , for almost three leaves together . One fault amongst the rest you take notice of , and you would have your Reader think that 's all ; though there be above twenty more . 'T is this , Because ( in fig. 6. ) B C is to BF for so your words are , though your Lesson mis-recite them , in triplicate proportion of CD , to FE ; therefore , inverting , FE , to CD is in triplicate proportion of BF to CB. And doe you not take this to be a fault ? No , you say , this I did object then ( Yes and doe so still , as absurd enough : ) But now , you say , you have taught me ; ( what a hard hap have I , that I cannot learn ; ) That of three quantities , ( you should rather have taken foure ; but however three shall serve for this turne , ) beginning at the lest , ( suppose 1 , 2 , 8 , ) if the third to the first ( 8 to 1 ) be in triplicate proportion of the second to the first ( of 2 to 1 ) also , by conversion , the first to the second ( 1 to 2 ) shall be in triplicate proportion of the first to the third , of 1 to 8. This is that you would have had me learne . But , good Sir , you have forgotten that , since that time , you have unlearned it your selfe . For your 16. artic . of Chap. 13. as it now stands corrected in the English , teacheth us another doctrine ; viz. that if 1 , 2 , 4 , 8 , bee continually proportionall , 1 to 8 shall be as well triplicate ( though not bigger ) of 1 to 2 , ( not this triplicate of that , ) as 8 to 1 is of 2 to 1. The case is now altered from what it was in the Latine . And therefore you are quite in a wrong box , when , in your English , you cite Chapt. 13. Art. 16 , to patronize this absurdity . For in so doing you doe but cut your own throat . You must now learne to sing another song ; called Palinodia . Well , this is one of the faults of this article . They that have a minde to see the rest of them , may consult what I said before ; where I have noted a parcell of two dozen . In the 11. Article , you doe but undertake to demonstrate a proposition of Archimedes . Your demonstration ( besides that it depends upon the second Article which is yet undemonstrated ) is otherwise also faulty , as I then told you . And therefore to say , that I allow this to be demonstrated , if your second bad been demonstrated ; is an untruth . For I told you then , that your manner of inferring this from that , is very absurd . The 12 Article ( like all the rest , since the second , beside their other faults , ) depends upon the second ; and therefore , till that be demonstrated , this must fall with it . In the 13. Art. you undertake to demonstrate this Proposition of Archimedes ; that the Superficies of any portion of a Sphere , is equall to that circle , whose Radius is a streight line drawn from the pole of the portion to the circumference of its base . Your demonstration , I said , was of no force ; but might as well be applyed to a portion of any Conoeid , Parabolicall , Hyperbolicall , Ellipticall , or any other , as to the portion of a sphere . By the truth of this , say you , let any man judg of your and my Geometry . Content , 'T is but transcribing your demonstration ; & inserting the words Conoeid , Vertex , section by the Axis , &c. where you have Sphere , Pole , great Circle &c. which termes : in the Conoeid , answer to those in the Sphere , and the worke is done . Let BAC , ( in the seventh figure , ) be a portion of a spheare , or Conoeid , Parabolicall , Hyperbolicall , Ellipticall , &c. whose Axis is AE , and whose basis is BC ; and let AB be the streight line drawn from the Pole , or vertex , A , to the base in B : and let AD , equall to AB , touch the Great circle , ( or Section made by a plain passing through the Axis of the Conoeid , ) BAC , in the Pole , or vertex , A. It is to be proved that a Circle made by the Radius AD , is equall to the superficies of the portion BAC . Let the plain AEBD be understood to make a revolution about the Axis AE . And it is manifest , that , by the streight line AD , a circle will be described ; and , by the Arch , or Section , AB , the superficies of a Sphere , or Conoeid mentioned ; and lastly , by the subtense AB , the superficies of a right Cone . Now , seeing both the streight line AB , and the Arch or Section AB , make one and the same revolution ; and both of them have the same extreme points A & B : The cause why the Sphericall or Conoeidicall Superficies which is made by the Arch or Section , is greater then the Conicall superficies which is made by the subtense , is , that AB the Arch or Section , is greater then AB the subtense : And the cause why it is greater , consists in this , that although they be both drawn from A to B , yet the subtense is drawen streight , but the arch or Section angularly ; namely , according to that angle which the arch or Section makes with the Subtense ; which angle is equall to the angle DAB . For the Angle of Contact , whether of Circles or other crooked lines , addes nothing to the angle at the segment : as hath been shewn , as to Circles , in the 14 Chapter of the 16 article : and as to all other crooked lines , Lesson 3. pag. 28. lin . ult . Wherefore the magnitude of the angle DAB , is the cause why the superficies of the portion described by the Arch or Section AB , is greater than the superficies of the right Cone described by the Subtense AB . Again , the cause why the Circle described by the tangent AD , is greater then the superficies of the right Cone described by the subtense AB , ( notwithstanding that the Tangent and Subtense are equall , and both moved round in the same time , ) is this , that AD stands at right angles to the axis , but AB obliquely ; which obliquity consists in the same angle : DAB . Seeing therefore that the quantity of the angle DAB , is that which makes the excesse both of the Superficies of the Portion , and of the Circle made by the Radius AD , above the superficies of the Right Cone described by the Subtense AB : It followes , that both the Superficies of the Portion , and that of the Circle , do equally exceed the Superficies of the Cone . Wherefore the Circle made by AD or AB , and the Sphericall or Conoeidicall Superficies made by the arch or Section AB , are equall to one another . Which was to be proved . Shew me now if you can , ( for you have pawned all your Geometry , upon this one issue , ) where the Demonstration halts more on my part then it doth on yours ? Or , where is it , that it doth not as strongly proceed in the case of any Conoeid , as of a Sphere ? All that you can think of by way of exception ( and you have had time to think on 't ever since I wrote last , ) amounts to no more but this ( which yet is nothing to the purpose ) you ask , In case the crooked line AB , were not the arch of a Circle , whether do I think , that the angles which it makes with the Subtense AB , at the points A & B , must needs be equall ? I say , that ( its possible , that in some cases , it may be so ; and J could for a need , shew you where ; and therefore , at least as to those cases , you are clearely gone ; for you had nothing else to say for your selfe ; but ) this is nothing at all to the purpose whether they be or no ; For the angle at B , what ever it be , comes not into consideration at all ; nor is so much as once named in all the demonstration ; So that its equality or inequallity , with that at A , makes nothing at all to the businesse . And therefore your exception is not worth a straw . Think of a better against the next time ; or else all your Geometry is forfeited . And they are like to have a great purchase that get it , are they not ? At the 14. Article ; ( having before , Art. 4. undertooke to teach the way of drawing and continuing those curve lines , by points : and directed us ( for the word require doth not please you ) for that end to take mean proportionalls ; ) you now tell us how that may be done ; viz. by these curve lines first drawn . I asked , whether this were not to commit a circle ? You tell me , No. But mean while take no notice of that which was the main objection ; viz. That this constructiō of yours was but going about the bush ; for , upon supposition that we had those lines already drawn , the finding of mean proportionalls by them might be performed with much more ease than the way you take . And I shewed you , How. But that which sticks most in your stomach , is a clause in the close of this Chap. I told you that some considerable Propositions of this Chapter ( and I could have told you which ) were true , ( though you had missed in your demonstration , ) however you came by them . But that I was confident they were none of your own . ( and you know , I guessed right . ) And least you should think I dealt unworthily to intimate that you had them elsewhere : unlesse I could shew you where : I told you , that I did no worse than those that a while before , had hanged a man for stealing a horse from an unknown person . There was evidence enough that the horse was stolen ; though they did not know from whom . So , though I knew not whence you had taken them , yet I have ground enough to judge they were not your own . And since that time , ( and before that book was fully printed , ) I found whence you had them ; namely out of Mersennus , ( as I told you then pag. 132 , 133 , 134. ) And to take them out of Mersennus , was all one as to rob a Carrier ; for there were at lest three men had right to the goods , ( and some of them if they had been asked , would scarce have given way that you should publish their inventions in your own name , ) Des Chartes , Fermat , and Robervall : And perhaps a fourth had as much right as any one of these ; and that is Cavallerio , who ( though , I then did not know it ) hath ( contrary to what you affirme , that they were never demonstrated by any but you selfe ; and that as wisely as one could wish : ) demonstrated those propositions in a Tractate of his De usu , Indivisibilium in potestatibus Cossicis . But though the thing be true enough and you cannot deny it , yet you doe not like the Comparison . And would have me consider , who it was , was hanged upon Hamans Gallows ? And truly J could tell you that too , for a need . The first letter of his name was H. But enough of this . SECT . XII . Concerning his 18 , 19 , 20. Chapters . WELL ! We have made pretty quick work with the 17 Chapter . With the 18 we shall be yet quicker . The charge against this Chapter , was , that it was all false . And , you confesse it . Not one true Article in the whole . But , you tell us , in the English 't is all well . It is now so corrected in the English as that I shall not be able ( if I can sufficiently imagine motion , that is , if I can be giddy enough , ) to reprehend . Very well ! ( 'T is a good hearing when men grow better . ) They that have a mind to believe it , may : I am not bound to undeceive them . We have had experience all along , that you have a speciall knack at mending . ( as sowr Ale doth in summer . ) You grant that I have truly demonstrated , what was before , to be all false . You would have me do so again , would you ? Very good ! When I have nothing else to doe I 'le consider of it . They that think it worth the while , may take the pains , to examine it a second time . For my part , I think I have bestowed as much pains upon it already , as it deserves , ( and somewhat more : ) And all the amendment that I find , is this ▪ that whereas before wee had three false articles , now we have but two ; and the number of true ones , just as many as we had before , viz. never a one . In the 19 Chapter there were faults enough in conscience ( for a matter of no greater difficulty than that was ; ) I noted some of them ( and left the Reader to pick up the rest : ) Two or three of the lighter touches , ( about method , ) you take notice of , and make a businesse to justify or excuse them ; and the main exceptions ( as you use to do ) you passe over with a light touch , and a way . I told you , in the beginning of it , that your Chapters hang together like a rope of sand . And 't is true enough , for they have no connexion at all . There are so few hooked atomes , that a man cannot tell how to tacke them together . Next , that having in your 24 Chapter undertaken to shew us , what is the Angle of Incidence ; and , what , the Angle of Reflexion ; and , that the Angles of Incidence and of Reflexion are equall : you do , in pursuance of that assertion , in this 19 chapter , shew us the consequences thereof . Upon this I asked ; why not , either this after that ; or that before this ? You tell me , that ( think I what I will , ) you think that method still the best ; ( to set the Cart before the Horse . ) Then you tell us , that I say , you define not here . ( Nay that 's false , I did not say so ; and 't is not the first time that I have taken you tripping in this kind ; ) but many Chapters after , ( that I said , I do confesse ; and you know 't is true ; ) what an Angle of Incidence , and what an Angle of Reflexion is . And then , talk against hast , and oversight . But if your selfe had not been over hasty , ( or rather willfully perverted my words , ) you might have seen ( and you know it well enough ) that I blamed you here , and two or three times before , not so much for using words , before you had defined them ( for this fault , as J remember , J mentioned but once ; and there you took it patiently : ) but for defining words so long after you had used them . For when words , for two or three chapters together , have been supposed , and frequently so used , as of known signification , ( whether they had been before defined , or not , ) it is ridiculous for a Mathen atician to come dropping in with definitions of them at latter end , ( as your fashion is , ) like mustard after meat . For these definitions should either have come in due time , or else not at all . The two first Articles are very triviall . And yet ( as if it were impossible for you , be the way never so plaine , not to stumble ) there wants , at least in the English , a determination in the second Corollary ; and yet ( as if that were to make amends for t'other ) there 's one too much . If upon any point ( say you ) between B and D , fig. 2. ( yes , or any where else upon the same streight line , produced either way , though not between those points , ) there fall ( from the point A , you should have said , ) a streight line , as AC , whose reflected line is CH , this also produced beyond C , will fall upon F. Here , I say , that limitation between B and D , is redundant ; and that from the point A , is wanting . For though C. be taken at pleasure , yet A is not , And if it come not from A , its reflex will not come at F. The third , fourth , and fifth Articles , I told you were false . ( viz. The Propositions affirme that universally , which holds true but in some particular case . And the demonstrations , proceed ex falsis suppositis , supposing that to be , which is not ; or is , in many cases , impossible . ) And this you confesse to be true ; but take it unkindly to be told of it . You have endeavoured a little to patch up the businesse in the English , but not so as to hold water . For they are yet lyable to divers exceptions if it were worth the while to unravell them . The eight Article was ridiculous enough . It makes a huge businesse to no purpose . ( You spend the best part of two pages to resolve a Problem which might as well have been dispatched in two lines . ) And you doe as good as confesse it All you say against it , is but this , that Adduco is not Latine for to Bring . The Twent●eth Chapter will be soon dispatched . This Chap. all but the two last Art. is wholly new , as it is now in the English : that which you had before in the Latine , being wholly routed & beat out the field ▪ ( & your Problematice dictum into the bargain . ) We had in your Latine three attempts for the squaring of a circle ; but they all came to naught , and are now vanished . In your Lesson , you give us a fourth ; endeavouring to new mould and rally one of the former , which I had before routed ; And pretende to vindicate it from the exceptions I had made to it : But not an answer to any one of them ; nor is this new attempt better than the former , but retaines most of the fundamentall errors therein ; And when you have all done , you cashier it your selfe and dare not insist upon it . Beside this , you have in your English , yet three attempts more ; and much a doe there is with long and perplexed figures to no purpose . They are by your own confession but Aggressions ; and you doe not your selfe believe them to be exact . You doe not , I suppose , think it worth the while for me to confute them , ( or if you doe I doe not ; ) for to what purpose ? That you have attempted it , ( seven times over , ) no man can deny ; That your attempts come to nothing , your selfe confesse ; Only , you think it convenient to let the Reader know what paines you have taken to no purpose . For my part , J doe not intend to follow you in all your new freakes : nor think my selfe ingaged to confute false quadratures as oft as you shall make them . I have done enough already , to let the world see , how little 't is that you understand in Geometry , and how much they deceive them selves who expect any great matter from you . Your two last Articles stand as they were , and so doth my answer to them . Your attempt of finding a streight line equall to a Spirall ; is but an attempt , as well as that of squaring a Circle . Your rant at Analyticks , with which you conclude it , ( like doggs barking at the Moon , ) hurts no body but your selfe . That Art will live when you be dead ; and those that know it , will not think it ever a whit the worse for your not understanding it , or rayling at it . SECT . IV. fig. 1. fig. 2. SECT . V. CAP : XIV fig. 6. fig. 7 Cap. XVI . fig. ● . fig. 2. fig● 6. fig. 8. fig. 11. CAP. XVII . fig. ● . fig. 6. fig. 7. Place this at the end SECT . XIII . Concerning his last Lesson . YOur last Lesson , little concernes mee ; but is directed mainly against my Reverend and Learned Collegue ; Who hath allready answered to it as much as he thinks it doth deserve , yet a touch or two there is wherein I am concerned . You had , in your Latine , a railing rant against Vindex , ( and though you thought fit to 〈◊〉 of that , 20 Chapter , yet placuit ea stare quae pertinent ad Vindicem . But in the English that is expunged also ; And now he is left to learn 〈◊〉 , out of your Lessons . ) And in order to this , J 〈◊〉 in my Elenchus , ( p. 〈…〉 117. 122. ) recited verbatim out of his Vindiciae , those 〈◊〉 , which , it seems , stuck ▪ so much in your stomack ; concerning M. Warners papers ; that the Reader might see how small a matter would put you into a rage . ( Which you knew well enough , and can upon no pretence plead ignorance of it . For it is the very same , which both the●● 〈◊〉 in your Lessons , you referre to , and rant at . ) But 〈◊〉 forsooth , upon this , ( according to your usuall honesty ) you would have your Reader believe , that J had there related some personall discourse , which Vindex , creeping into your company unknown , had sometime had with you : and then rant at the incivility of such a carriage , and ( with a fling at Moranus into the bargain ) raile a● it for allmost two whole pages together , p. 57 , 58 , 59. Wherein , whether your Civility or Honesty , be more com●●cuous , let the Reader judge . In like manner , because J cited a passage concerning Rohervall , out of Mersen●● , you suspect , p. 59. that somebody , you know not who , hath most magnanimously interpreted to me in 〈◊〉 d●sgrace , what passed between you and him in the Cloister of the Convent . Which is a suspicion like to that of p. 57. that some of our Philosophers that were at Paris at the same time with you , may perhaps have accused you to us of bragging or ostentation . As though there were not ground enough in your writings , to evidence that , to any man , without any such relation . But , mean while , J wonder how you behaved your selfe at Paris , that you should be so Jealous least somebody there should tell tales . And all this is but a little to disguise the businesse , as if I had not by what is extant in Print , in those places cited out of Mersennus ( Hydraulic . prop. 25 Cor. 2. Ballistic . prop. 32 , Mechanic . praef . punct . 3. & 4. Reflex . Physico-Math . cap. 1 : art . 5. ) made it evident , that all or most of what was worth any thing in your Mathematicks , was manifestly stollen from Gasilaeo , Robervall , Cartesius , Fermat , &c And 〈…〉 them as I perceive by somewhat but now come to 〈…〉 him , doth not stick to call you 〈…〉 , for so doing : and , if some of 〈…〉 were 〈…〉 doubt ▪ not but they would be ready enough to do the like 〈…〉 ▪ Now this is all , ( 〈…〉 what was sufficiently 〈…〉 at before ) that in this 〈…〉 concerns mee . And , for what concernes my 〈…〉 , you have already from himselfe received sufficient 〈…〉 . I know now no exception remaining , unlesse like his , who putting a Bond in suit when the Defendant made proof of Payment ▪ replyed , 〈…〉 the Condition of the Obligation was that he should 〈…〉 , Satisfy , and Pay ; and therefore , though the 〈…〉 all pay'd , yet forasmuch the Plaintife was not 〈…〉 the Bond was forfeit . Now J hope the Reader can bear witnesse , that you have been , by this time , sufficiently Pay'd ; and , J hope , Satisfyed ; But , if we must never have done till you be Contented , I am afraid we shall dye in your debt . FINIS . ERRATA . PAge 1. line 5. language , p. 2. l. 24. learn. p. 5. l. 32. dele quod . p. 6. l. 32. finding . p. 9. l. 13. suffer your . p. 17. l. 2● . Plin. p. 18. l 24. dos . p. 19. l. 24. sumere . p. 38. l 12. second . p. 45. l. ult . 13. p. 46. l. 31. 4 † 1. p. 52. l , 35. not at all , p. 57. l. 22. for two . p. 61. l. 35. art . 3. p. 64. l. 39. proportion . p. 66. l. 34. art . 5. p. 67. l. 1. proportion . ibid. l. 17. art . 3. p. 68. l. 25. that Greater . p. 71 , l. 33 , half the. p. 72. l. 39. proposition . p. 75 , l. pen. and. p. 78 , l. 23 , the points . p. 80 , l. 3. one another . p. 92 , l. 36. √ ½ , or . p. 95 , l. 13. of the 5● ibid. l. 22. adde , as the product of one Impetus into its Time , to the product of the other Impetus into its Time. p. 97 , l. 13. thought . ibid. l. 18. of celerity . p. 99 , l. 13 , it be . p. 103. l. 32 , proposition . p. 106 , l. 2 , the rest . p. 107 , l. 6 , that Table . ibid. l. 13 , and √ 3 is more . ibid. l. 32 , not to be had ▪ 
A48262	----	Mathematicall recreations. Or, A collection of many problemes, extracted out of the ancient and modern philosophers as secrets and experiments in arithmetick, geometry, cosmographie, horologiographie, astronomie, navigation, musick, opticks, architecture, statick, mechanicks, chemistry, water-works, fire-works, &c. Not vulgarly manifest till now. Written first in Greeke and Latin, lately compi'ld in French, by Henry Van Etten, and now in English, with the examinations and augmentations of divers modern mathematicians whereunto is added the description and use of the generall horologicall ring: and the double horizontall diall. Invented and written by William Oughtred. Récréation mathématique. English. 1653 Approx. 447 KB of XML-encoded text transcribed from 177 1-bit group-IV TIFF page images. Text Creation Partnership, Ann Arbor, MI ; Oxford (UK) : 2005-12 (EEBO-TCP Phase 1). A48262 Wing L1790 ESTC R217635 99829293 99829293 33730 This keyboarded and encoded edition of the work described above is co-owned by the institutions providing financial support to the Early English Books Online Text Creation Partnership. This Phase I text is available for reuse, according to the terms of Creative Commons 0 1.0 Universal . The text can be copied, modified, distributed and performed, even for commercial purposes, all without asking permission. Early English books online. (EEBO-TCP ; phase 1, no. A48262) Transcribed from: (Early English Books Online ; image set 33730) Images scanned from microfilm: (Early English books, 1641-1700 ; 1992:7) Mathematicall recreations. Or, A collection of many problemes, extracted out of the ancient and modern philosophers as secrets and experiments in arithmetick, geometry, cosmographie, horologiographie, astronomie, navigation, musick, opticks, architecture, statick, mechanicks, chemistry, water-works, fire-works, &c. Not vulgarly manifest till now. Written first in Greeke and Latin, lately compi'ld in French, by Henry Van Etten, and now in English, with the examinations and augmentations of divers modern mathematicians whereunto is added the description and use of the generall horologicall ring: and the double horizontall diall. Invented and written by William Oughtred. Récréation mathématique. English. Oughtred, William, 1575-1660. aut [40], 286, [17] p. : ill. printed for William Leake, at the signe of the Crown in Fleetstreet, between the two Temple Gates, London : M D C LIII. [1653] Translation of: Jean Leurechon. Recreation mathematique. Henry Van Etten is a pseudonym of Jean Leurechon. With an added engraved title page reading: Mathematicall recreations or a collection of sundrie excellent problemes out of ancient and moderne phylosophers. "The description and use of the double horizontall dyall" has separate title page dated 1652; register is continuous. 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Understanding these processes should make clear that, while the overall quality of TCP data is very good, some errors will remain and some readable characters will be marked as illegible. Users should bear in mind that in all likelihood such instances will never have been looked at by a TCP editor. The texts were encoded and linked to page images in accordance with level 4 of the TEI in Libraries guidelines. Copies of the texts have been issued variously as SGML (TCP schema; ASCII text with mnemonic sdata character entities); displayable XML (TCP schema; characters represented either as UTF-8 Unicode or text strings within braces); or lossless XML (TEI P5, characters represented either as UTF-8 Unicode or TEI g elements). Keying and markup guidelines are available at the Text Creation Partnership web site . eng Science -- Problems, exercises, etx. -- Early works to 1800. Mathematics -- Problems, exercises, etc. -- Early works to 1800. Fireworks -- Early works to 1800. Scientific recreations -- Early works to 1800. Sundials -- Early works to 1800. 2005-02 TCP Assigned for keying and markup 2005-05 SPi Global Keyed and coded from ProQuest page images 2005-06 Jonathan Blaney Sampled and proofread 2005-06 Jonathan Blaney Text and markup reviewed and edited 2005-10 pfs Batch review (QC) and XML conversion Mathematicall RECREATIONS . OR , A Collection of many Problemes , extracted out of the Ancient and Modern Philosophers , as Secrets and Experiments in Arithmetick , Geometry , Cosmographie , Horologiographie , Astronomie , Navigation , Musick , Opticks , Architecture , Stati●k , Mechanicks , Chemistry , Water-works , Fire-works , &c. Not vulgarly manifest till now . Written first in Greeke and Latin , lately compi'ld in French , by Henry Van Etten , and now in English , with the Examinations and Augmentations of divers Modern Mathematicians Whereunto is added the Description and Use of the Generall Horologicall Ring : And The Double Horizontall Diall . Invented and written by WILLIAM OUGHTRED . LONDON : Printed for William Leake , at the Signe of the Crown in Fleetstreet , between the two Temple Gates , MDCLIII . On the Frontispice and Booke . ALL Recreations do delight the minde , But these are best being of a learned kinde : Here Art and Nature strive to give content , In shewing many a rare experiment , Which you may read , & on their Schemes here look Both in the Frontispice , and in the Book . Upon whose table new conceits are set , Like dainty dishes , thereby for to whet And winne your judgement , with your appetite To taste them , and therein to taka delight . The Senses objects are but dull at best , But Art doth give the Intellect a feast . Come hither then , and here I will describe , What this same Table doth for you provide . Here Questions of Arithmetick are wrought , And hidden secrets unto light are brought , The like it in Geometrie doth unfold , And some too in Cosmographie are told : It divers pretty Dyals doth descrie , With strange experiments in Astronomie , And Navigation , with each severall Picture , In Musick , Opticks , and in Architecture : In Statick , Machanicks , and Chymistrie , In Water-works , and to ascend more hie , In Fire-works , like to Joves Artillerie . All this I know thou in this Book shalt finde , And here 's enough for to content thy minde . For from good Authors , this our Author drew These Recreations , which are strange , and true So that this Book 's a Centre , and t is fit , That in this Centre ; lines of praise should meet W. MATHEMATICALL Recreations Or a Collection of sundrie excellent Problemes out of ancient & moderne Phylosophers Both vsefull and Recreatiue Printed for William Leake and are to be solde at the Crowne in fleet streete betweene the two Temple gates . TO The thrice Noble and most generous Lo. the Lo. Lambert Verreyken , Lo. of Hinden , Wolverthem , &c. My honourable Lo. AMongst the rare and curious Propositions which I have learned out of the studies of the Mathematicks in the famous University of Pont a Mousson , I have taken singular pleasure in certaine Problemes no lesse ingenious than recreative , which drew me unto the search of demonstrations more difficult and serious ; some of which I have amassed and caused to passe the Presse , and here dedicate them now unto your Honour ; not that I account them worthy of your view , but in part to testifie my affectionate desires to serve you , and to satisfie the curious , who delight themselves in these pleasant studies , knowing well that the Nobilitie , and Gentrie rather studie the Mathematicall Arts , to content and satisfie their affections , in the speculation of such admirable experiments as are extracted from them , than in hope of gaine to fill their Purses . All which studies , and others , with my whole indevours , I shall alwayes dedicate unto your Honour , with an ardent desire to be accounted ever , Your most humble and obedient Nephew and Servant , H. VAN ETTEN . By vvay of advertisement . Five or six things I have thought worthy to declare before I passe further . FIrst , that I place not the speculative demonstrations with all these Problems , but content my self to shew them as at the fingers end : which was my plot and intention , because those which understand the Mathematicks can conceive them easily ; others for the most part will content themselves onely with the knowledge of them , without seeking the reason . Secondly , to give a greater grace to the practice of these things , they ought to be concealed as much as they may , in the subtiltie of the way ; for that which doth ravish the spirits is , an admirable effect , whose cause is unknowne : which if it were discovered , halfe the pleasure is l●st ; therefore all the finenesse consists in the dexterity of the Act , concealing the meanes , and changing often the streame . Thirdly , great care ought to be had that one deceive not himselfe , that would declare by way of Art to deceive another : this will make the matter contemptible to ignorant Persons , which will rather cast the fault upon the Science , than upon him that shewes it : when the cause is not in the Mathematicall principles , but in him that failes in the acting of it . Fourthly , in certaine Arithmeticall propositions they have onely their answers as I found them in sundry Authors , which any one being studious of Mathematicall learning , may finde their originall , and also the way of their operation . Fifthly , because the number of these Problemes , and their dependances are many , and intermixed , I thought it convenient to gather them into a Table : that so each one according to his fancie , might make best choise of that which might best please his palate , the matter being not of one nature , nor of like subtiltie : But whosoever will have patience to read on , shall finde the end better than the beginning . To the Reader . IT hath been observed by many , that sundry fine wits as well amongst the Ancient as Moderne , have sported and delighted themselves upon severall things of small consequence , as upon the foot of a fly , upon a straw , upon a point , nay upon nothing ; striving as it were to shew the greatnesse of their glory in the smalnesse of the subject : And have amongst most solid and artificiall conclusions , composed and produced sundry Inventions both Philosophicall and Mathematicall , to solace the minde , and recreate the spirits , which the succeeding ages have imbraced , and from them gleaned and extracted many admirable , and rare conclusions ; judging that borrowed matter often-times yeelds praise to the industry of its author . Hence for thy use ( Courteous Reader ) I have with great search and labour collected also , and heaped up together in a body of these pleasant and fine experiments to stirre up and delight the affectionate , ( out of the writings of Socrates , Plato , Aristotle , Demosthenes , Pythagoras , Democrates , Plinie , Hyparchus , Euclides , Vitruvius , Diaphantus , Pergaeus , Archimedes , Papus Alexandrinus , Vitellius , Ptolomaeus , Copernicus , Proclus , Mauralicus , Cardanus , Valalpandus , Kepleirus , Gilbertus , Tychonius , Dureirus , Josephus , Clavius , Gallileus Maginus , Euphanus Tyberill , and others ) knowing Art imitating Nature that glories alwayes in the variety of things , which she produceth to satisfie the minde of curious inquisitors . And though perhaps these labours to some humourous persons may seeme vaine , and ridiculous , for such it was not undertaken : But for those which intentively have desired and ●ought after the knowledge of those things , it being an invitation and motive to the search of greater matters , and to imploy the minde in usefull knowledge , rather than to be busied in vaine Pamphlets , Play-books , fruitlesse Legends , and prodigious Histories that are invented out of fancie , which abuse many Noble spirits , dull their wits , & alienate their thoughts from laudable and honourable Studies . In this Tractate thou maist therefore make choise of such Mathematicall Problemes and Conclusions as may delight thee , which kinde of learning doth excellently adorne a man ; seeing the usefulnesse thereof , and the manly accomplishments it doth produce , is profitable and delightfull for all sorts of people , who may furnish and adorne themselves with abundance of matter in that kinde , to help them by way of use , and discourse . And to this we have also added our Pyrotechnie , knowing that Beasts have for their object only the surface of the earth ; but hoping that thy spirit which followeth the motion of fire , will abandon the lower Elements , and cause thee to lift up thine eyes to soare in an higher Contemplation , having so glittering a Canopie to behold , and these pleasant and recreative fires ascending may cause thy affections also to ascend . The Whole whereof we send forth to thee , that desirest the scrutability of things ; Nature having furnished us with matter , thy spirit may easily digest them , and put them finely in order , though now in disorder . A Table of the particular heads of this Book , contracted according to the severall Arts specified in the Title-page . Experiments of Arithmetick . PAge 1 , 2 , 3 , 16 , 19 , 22 , 28 , 33 , 39 , 40 , 44 , 45 , 51 , 52 , 53 , 59 , 60 , 69 , 71 , 77 , 83 , 85 , 86 , 89 , 90 , 91 , 124 , 134 , 135 , 136 ▪ 137 , 138 , 139 , 140 , 178 179 , 181 , 182 , 183 , 184 , 185 , 188 , 208 , 210 , 213. Experiments ●n Geometrie . Pag. 12 , 15 , 24 , 26 , 27 , 30 , 35 , 37 , 41 , 42 , 47 , 48 , 49 , 62 , 65 , 72 , 79 , 82 , 113 , 117 , 118 , 119 , 214 , 215 , 217 , 218 , 234 , 235 , 236 , 239 , 240. Experiments in Cosmographie . Pag. 14 , 43 , 75 , 106 , 107 , 219 , 220 , 225 , 227 , 228 , 229 , 230 , 232. Experiments in Horologiographie . Pag. 137 , 166 , 167 , 168 , 169 , 171 , 234. Experiments in Astronomie . Pag. 220 , 221 , 222 , 223 , 224. Experiments in Navigation . Pag. 105 , 233 , 234 , 237 , 238. Experiments in Musick . Pag. 78 , 87 , 126. Experiments in Opticks . Pag. 6 , 66 , 98 , 99 , 100 , 102 , 129 , 131 , 141 , 142 , 143 , 144 , 146 , 149 , 151 , 152 , 153 , 155 , 156 , 157 , 158 , 160 , 161 , 162 , 163 , 164 , 165. Experiments in Architecture . Pag. 16 , 242 , 243. Experiments in Staticke . Pag. 27 , 30 , 32 , 71 , 199 , 200 , 201 , 283 , 204 , 205 , 207. Experiments in Machanicks . Pag. 56 , 58 , 68 , 88 , 95 , 108 , 110 , 128 , 173 , 174 , 176 , 246 , 248 , 258 , 259. Experiments in Chymistrie . Pag. 198 , 255 , 256 , 257 , 260 , 262 , 263 , 264. Experiments in Water-workes . Pag. 190 , 191 , 192 , 193 , 194 , 196 , 247 , 249 , 250 , 252 , 253. Experiments in Fireworkes . From page , 265. to the end . FINIS . A Table of the Contents , and chiefe points conteined in this Book . PROBLEM . II. HOw visible objects that are without , and things that passe by , are most lively represented to those that are within . Page 6 Prob. 1 Of finding of numbers conceived in the minde . 1 , 2 , 3 Prob. 5 Of a Geographicall Garden-plot fit for a Prince or some great personage . 14 Prob. 37 Any liquid substance , as water or wine , placed in a Glasse , may be made to boile by the motion of the finger , and yet not touching it . 54 Prob. 3 How to weigh the blow of ones fist , of a Mallet , a Hatchet or such like . 9. Prob. 30 Two severall numbers being taken by two sundry persons , how subtilly to discover which of those numbers each of them took . 46 Prob. 4 That a staffe may be broken ▪ placed upon two Glasses , without hurting of the Glasses . 12 Prob. 7 How to dispose Lots that the 5 , 6 , 9 , &c. of any number of persons may escape . 16 Prob. 13 How the weight of smoke of a combustible body , which is exhaled , may be weighed . 27 Prob. 12 Of three knives which may be so disposed to hang in the aire , and move upon the Point of a needle . 27 Prob. 17 Of a deceitfull bowle , to bowle withall . 32 Prob. 16 A ponderous or heavy body may be supported in the aire without any one touching it . 30 Prob. 18 How a Peare , or Apple , may be parted into any parts , without breaking the rinde thereof . 33 Prob. 15 Of a fine kinde of dore which opens and shuts on both sides . 30 Prob. 9 How the halfe of a Vessell which containes 8 measures may be taken , being but onely two other measures , the one being 3 , and the other 8 measures . 22 Prob. 8 Three persons having taken each of them severall things , to finde which each of them hath taken . 19 Prob. 6 How to dispose three staves which may support each other in the aire . 15 Prob. 14 Many things being disposed Circular ( or otherwise ) to finde which of them any one thinks upon . 28 Prob. 19 To finde a number thought upon without asking questions . 33 Prob. 11 How a Milstone or other ponderosity may hang upon the point of a Needle without bowing , or any wise breaking of it . 26 Prob. 20 and 21 How a body that is uniforme and inflexible may passe through a hole which is round , square and Triangular ; or round , square and ovall-wise , and exactly fill those severall holes . 35 , 37 Prob. 10 How a stick may stand upon ones finger , or a Pike in the middle of a Court without falling . 24 Prob. 22 To finde a number thought upon after another manner than those which are formerly delivered . 39 Prob. 23 To finde out many numbers that sundry persons or any one hath thought upon . 40 Prob. 24 How is it that a man in one & the same time may have his head upward , and his feet upward , being in one and the same place ? 4● Prob. 25 Of a Ladder by which two men ascending at one time , the more they ascend , the more they shal be asunder , notwith standing the one be as high as the other . 42 Prob. 26 How is it that a man having but a Rod or Pole of land , doth brag that he may in a right line passe from place to place 3000 miles . 42 Prob. 27 How is it that a man standing upright , and looking which way he will , he looketh true North or South . 43 Prob. 28 To tell any one what number remaines after certaine operations being ended , without asking any question . 44 Prob. 29 Of the play with two severall things . 45 Prob. 31 How to describe a circle that shall touch 3 points placed howsoever upon a plaine , if they be not in a right line . 47 Prob. 32 How to change a circle into a square forme . 48 Prob. 33 With one and the same compasses , and at one and the same extent or opening , how to describe many circles concentricall , that is , greater or lesser one than another . 49 Prob. 34 Any number under 10. being thought upon , to finde what numbers they were . 51 Prob , 35 Of the play with the Ring . 52 Prob. 36 The play of 3 , 4 , or more Dice . 53 Prob. 38 Of a fine Vessell which holds Wine or Water being cast into it at a certain height , but being filled higher it will runne all out of its owne accord . 56 Prob. 39 Of a Glasse very pleasant . 58 Prob. 40. If any one should hold in each hand as many pieces of money as in the other , how to finde how much there is . 59 Prob. 41 Many Dice being cast , how artificially to discover the number of the points that may arise . 60 Prob. 42 Two metals as Gold and Silver or of other kinde , weighing alike , being privately placed into two like boxes , to finde in which of them the Gold or Silver is . 62 Prob. 43 Two Globes of divers metals ( as one Gold the other Copper ) yet of equall weight , being put in a Box as B.G. to finde in which end the Gold or Copper is . 65 Prob. 44 How to represent divers sorts of Rainbowes here below . 66 Prob. 45 How that if all the powder in the World were inclosed in a bowle of paper or glasse , and being fired on all parts , it could not break that bowle . 68 Prob. 46 To finde a number which being divided by 2. there will remaine 1. being divided by 3. there will remaine 1. and so likewise being divided by 4 , 5 , or 6. there will still remaine one , but being divided by 7 will remaine nothing . 69 Prob. 47 One had a certaine number of Crownes , and counting them by 2 and 2 , there rested 1. counting them by 3 , and 3 , there rested 2. counting them by 4 , and 4 , there rested 3. counting them by 5 , and 5 , there rested 4. counting them by 6 , and 6 , there rested 5. but counting them by 7 and 7 , there rested nothing , how many Crownes might he have ? 71 Prob. 48 How many sorts of weights in the least manner must there be to weigh all sorts of things betweene one pound and 121 pound , and so unto 364 pound ? 71 Prob. 49 Of a deceitfull balance which being empty seems to be just , because it hangs in Aequilibrio , notwithstanding putting 12 pound in one ballance , and 11 in the other , it will remaine in Aequilibrio . 72 Prob. 50 To heave or lift up a bottle with a straw . 74 Prob. 51 How in the middle of a wood or desert , without the sight of the Sun , starres , shadow , or compasse , to finde out the North , or South , or the 4 Cardinal points of the World , East , West , &c. 75 Prob. 52 Three persons having taken Counters , Cards , or other things , to finde how much each one hath taken . 7● Prob. 53 How to make a consort of Musick of many parts with one voice or one instrument onely . 78 Prob. 54 To make or describe an oval form , or that which is neare resembled unto it at one turning , with a paire of common Compasses . 79 Prob. 55 Of a purse difficult to be opened . 80 Prob. 56 Whether is it more hard and admirable without Compasses to make a perfect circle , or being made to finde out the Centre of it ? 82 Prob. 56 Any one having taken 3 Cards , to finde how many points they containe . 83 Prob. 57 Many Cards placed in divers ranks , to finde which of those Cards any one hath thought . 85 Prob. 58 Many Cards being offered to sundry persons to finde which of those Cards any one thinketh upon . 86 Prob. 59 How to make an instrument that helps to heare , as Gallileus made to help to see . 87 Prob. 60 Of a fine Lamp which goeth not out , though one carries it in ones pocket , or being rolled on the ground will still burne . 88 Prob. 61 Any one having thought a Card amongst many Cards , how artificially to discover it out . 89 Prob. 62 Three Women A , B , C. carried Apples to a Market to sell : A had 20. B had 30. C 40. they sold as many for a penny one as the other , and brought home one as much money as another , how could this be ? 90 Prob. 63 Of the properties of some numbers . 91 Prob. 64 Of an excellent Lamp which serves or furnisheth it selfe with Oile , and burnes a long time . 95 Prob. 65 Of the play at Keyles or Nine-pins . 97 Prob. 66 Of Spectacles of pleasure Of Spectacles which give severall colours to the visage . 98 Of Spectacles which make a Towne seeme to be a City , one armed man as a Company , and a piece of Gold as many pieces . 99 How out of a Chamber to see the objects which passe by according to the lively perspective . 100 Of Gallileus admirable Optick-Glasse , which helps one to see the beginning and ending of Eclipses , the spots in the Sunne , the Starres which move about the Planets , and perspicuously things far remote . Of the parts of Gallileus his Glasse . 102 Prob. 67 Of the Magnes and Needles touched therewith . How Rings of Iron may hang one by another in the aire . 103 Of Mahomets Tombe which hangs in the aire by the touch of the Magnes . 104 How by the Magnes only to finde out North and South 105 Of a secrecie in the Magnes , for discovering things farre remote . 106 Of finding the Poles by the Magnes 107 Prob. 68 Of the properties of Aeolipiles or Bowles to blow the fire . 108 Prob. 69 Of the Thermometer , or that which measures the degrees of heat and cold by the aire . 110 Of the proportion of humane bodies , of statues , of Colosses , or huge Jmages and monstrous Giants . 113 Of the commensuration of the parts of the bodie the one to the other in particular , by which the Lion was measured by his claw , the Giant by his thumbe , and Hercules by his foot . 115 , 116 Of Statues or Colosses , or huge Images ; that mount Athos metamorphosed by Dynocrites into a statue , in whose hand was a Towne able to receive ten thousand men . 117 Of the famous Colossus at Rhodes which bad 70 cubits in height , and loaded 900. Camels , which weighed 1080000 l. 118 Of Nero his great Colossus which had a face of 12 foot large . 119 Of monstrous Giants Of the Giant Og and Goliah . 119 , 120 Of the Carkasse of a man found which was in length 49 foot ; and of that monster found in Creet , which had 46. Cubits of height . 120 Of Campesius his relation of a monster of 300 foot found in Sicile , whose face according to the former proportion should be 30 foot in length . 121 Prob. 71 Of the game at the Palme , at Trap , at Bowles , Paile-maile , and others . 122 Prob. 72 Of the game of square formes . 124 Prob. 73 How to make the string of a Viol sensibly shake without any one touching it . 126 Prob. 74 Of a Vessell which containes 3 severall kindes of liquor , all put in at one bung-hole , and drawne out at one Tap severally without mixture . 128 Prob. 75 Of burning-Glasses . Archimedes his way of burning the ships of Syracuse . 129 Of Proclus his way , and of concave and sphericall Glasses which burne , the cause and demonstration of burning with Glasses . 131 Of Maginus his way of setting fire to Powder in a Mine by Glasses . 131 Of the examination of burning by Glasses . 133 Prob. 76 Of pleasant questions by way of Arithmetick . Of the Asse and the Mule. 134 Of the number of Souldiers that fought before old Troy. 135 Of the number of Crownes that two men had . 136 About the houre of the day . 137 Of Pythagoras Schollers . 137 Of the number of Apples given amongst the Graces and the Muses . 138 Of the testament or last will of a dying Father . 138 Of the cups of Croesus . 139 Of Cupids Apples . 139 Of a Mans Age. 140 Of the Lion of Bronze placed upon a fountaine with his Epigram . ibid. Prob. 77 In Opticks , excellent experiments . Principles touching reflections . 141 Experiments upon flat and plaine Glasses . 142 How the Images seeme to sink into a plaine Glasse , and alwayes are seene perpendicular to the Glasse , an● also inversed . 143 The things which passe by in a street may by help of a plaine glasse be seen in a Chamber , and the height of a tower or tree observed . 143 How severall Candles from one Candle are represented in a plaine Glasse , and Glasses alternately may be seene one within another , as also the back-parts of the body as well as the fore-parts are evidently represented . 144 How an Image may be seene to hang in the aire by help of a Glasse : and writing read or easily understood . 146 Experiments upon Gibbous , or convex Sphericall Glasses . How lively to represent a whole City , fortification , or Army , by a Gibbous Glasse . 147 How the Images are seen in Concave Glasses . 149 How the Images are transformed by approaching to the centre of the Glasse , or point of concourse ; and of an exceeding light that a Concave Glasse gives by help of a Candle . 151 How the Images , as a man , a sword , or hand , doth come forth out of the Glasse . 152 , 153 Of strange apparitions of Images in the aire , by help of sundry Glasses . 152 , 154 Of the wonderfull augmentation of the parts of mans body comming neare the point of inflammation , or centre of the Glasse . 155 How writing may be reverberated from a Glasse upon a VVall , and Read. 156 How by help of a Concave Glasse to cast light into a Campe , or to give a perspective light to Pyoneers in a Mine , by one Candle only . 156 How excellently by help of a Concave Glasse and a Candle placed in the centre , to give light to read by . 157 Of other Glasses of pleasure . 158 Of strange deformed representations by Glasses ; causing a man to have foure eyes , two Mouthes , two Noses , two heads . Of Glasses which give a colour to the visage , and make the face seeme faire and foule . 160 Prob. 78 How to shew one that is suspicious , what is in another Chamber or Roome , notwithstanding the interposition of that wall . 160 Corolary , 1. To see the Besiegers of a place , upon the Rampa●●t of a fortification 161 Corolary 2. and 3. Notwithstanding the interposition of VValls and Chambers , by help of a Glasse things may be seen , which passe by . 162 Prob. 79 How with a Musket to strike a marke not looking towards it , as exactly as one aimed at it . 162 How exactly to shoot out of a Mu●ket to a place which is not seene , being hindred by some obstacle or other interposition . 163 Prob. 80 How to make an Image to be seen hanging in the aire , having his head downward . 164 Prob. 81. How to make a company of representative souldiers seeme to be as a regiment , or how few in number may be multiplyed to seem to be many in number . 165 COROLARIE . Of an excellent delightfull Cabinet made of plaine Glasses . 165 Prob. 82 Of fine and pleasant Dyalls in Horologiographie . Of a Dyall of herbs for a Garden . 166 Of the Dyall upon the finger and hand , to finde what of the Clock it is . 167 Of a Dyall which was about an Obelisk at Rome . 168 Of Dyals with Glasses . 168 Of a Dyall which hath a Glasse in the place of the stile . 169 Of Dyals with water , which the Ancients use● 171 Prob. 83 Of shooting out of Cannons or great Artillery . How to charge a Cannon without powder . 173 To finde how much time the Bullet of a Cannon spends in the Aire before it falls to the ground . 174 How it is that a Cannon shooting upward , the Bullet flies with more violence , than being shot point blanke , or shooting downeward . 174 VVhether is the discharge of a Cannon so much the more violent , by how much it hath the more length ? 176 Prob. 84 Of prodigious progressions , and multiplications of creatures , plants , fruits , numbers , gold , silver , &c. Of graines of Mustardseed , and that one graine being sowne , with the increase thereof for 20 yeares will produce a heap greater than all the earth a hundred thousand times . 178 Of Pigges , and that the great Turke with all his Revenne , is not able to maintaine for one yeare , a Sow with all her increase for 12 yeares . 179 Of graines of Corne , and that 1 graine with all its increase for 12 yeares , will amount to 244140625000000000000 graines , which exceeds in value all the treasures in the World. 183 Of the wonderfull increase af Sheepe . 182 Of the increase of Cod-fish . 182 Of the Progressive Multiplication of soules ; that from one of Noahs Sonnes , from the flood unto Nimrods Monarchie , should be produced 111350 soules . 183 Of the increase of Numbers in double proportion , and that a pin being doubled as often as there are weekes in the yeare , the number of pinnes that should arise is able to load 45930 ships of a thousand Tunne apiece , which are worth more than tenne hundred thousand pounds a day . 183 , 184 Of a man that gathered Apples , stones , or such like upon a condition . 185 Of the changes in Bells , in musicall instruments , transmutation of Places , in Numbers , Letters , Men and such like ▪ 185 Of the wonderfull interchange of the Letters in the Alphabet : the exceeding number of men , and time to expresse the words that may be made with these letters , and the number of Books to comprehend them . 187 , 188 Of a servant hired upon certaine condition , that he might have land lent him to sowe one graine of Corne with its increase for 8 yeares time , which amounted to more than four hundred thousand Acres of Land. 188 Prob. 85 Of Fountaines , Hydriatiques ; Stepticks , Machinecks , and other experiments upon water , or other liquor . First , how water at the foot of a Mountaine may be made to ascend to the top of it , and so to descend on the other side of it 190 Secondly , to finde how much Liquor is in a Vessell , onely by using the tap-hole . 191 Thirdly , how is it , that a Vessell is said to hold more water at the foot of a Mountaine , then at the top of it 191 4 How to conduct water from the top of one Mountaine to the top of another 192 5 Of a fine Fountaine which spouts water very high and with great violence , by turning of a Cock 193 6 Of Archimedes screw which makes water ascend by descending . 194 7 Of a fine Fountaine of pleasure . 196 8 Of a fine watering pot for Gardens . 197 9 How easily to take Wine out of a Vessell at the bung hole without piercing a hole in the Vessell . 198 10 How to measure irregular bodies by help of water . 198 11 To finde the weight of water . 199 12 To finde the charge that a vessell may carry , as Ships , Boats or such like . 200 13 How comes it that a ship having safely sailed in the vast Ocean , and being come into the port or harbour , will sinke down right . 200 14 How a grosse body of metall may swim upon the water . 201 15 How to weigh the lightnesse of the aire . 203 16 Being given a body , to mark it about , and shew how much of it will sink in the water , or swim above the water . 204 17 To finde how much severall metalls or other bodies do weigh lesse in the water than in the aire . 204 18 How is it that a ballance having like weight in each scale , and hanging in Aequilibrio in the aire , being removed from that place ( without diminishing the weights in each balance , or adding to it ) it shall cease to hang in Aequilibrio sensibly , yea by a great difference of weight . 205 19 To shew what waters are heavier one than another , and how much . 206 20 How to make a pound of water weigh as much as 10 , 20 , 30 , or a hundred pound of Lead , nay as much as a thousand or ten thousand pound weight . 207 Prob. 86. Of sundry questions of Arithmetick , and first of the number of sands calculated by Archimedes and Clavius . 208 2 Divers metalls being melted together in one body , to finde the mixture of them . 210 3 A subtile question of three partners about equality of Wine and Vessels . 213 4 Of a Ladder which standing upright against a wall of 10 foot high , the foot of it is pulled out 6 foot from the wall upon the pavement , how much hath the top of the Ladder descended . 214 Prob. 87 Witty suits or debates between Caius and Sempronius , upon the forme of figures , which Geometricians call Isoperimeter , or equall in circuit , or Compasse . 214 1 Incident : of changing a field of 6 measures square , for a long rectrangled fiel of 9 measures in length and 3 in breadth : both equall in circuit but not in quantity . 215 2 Incident : about two sacks each of them ho●ding but a bushell , and yet were able to hold 4 bushels . 217 3 Incident : sheweth the deceit of pipes which conveygh water , that a pipe of two inches diameter , doth cast out foure times as much water as a pipe of one such diameter . 218 7 Heapes of Corne of 10 foot every way , is not as much as one heap of Corne of 20 foot every way . 218 Prob. 88 Of sundry questions in matter of Cosmographie , and Astronomy . In what place the middle of the earth is supposed to be . 219 Of the depth of the earth , and height of the Heavens , and the compasse of the World , how much . 219 How much the starry Firmament , the Sun , and the Moone are distant from the centre of the earth . 220 How long a Mill-stone would be in falling to the centre of the earth from the superficies , if it might have passage thither . 220 How long time a man or a bird may be in compassing the whole earth . 220 If a man should ascend by supposition 20 miles every day : how long it would be before he approach to the Moone . 221 The Sunne moves more in one day than the Moone in 20 dayes . 221 If a milstone from the orbe of the Sun should descend a thousand miles in an houre how long it would be before it come to the earth . 221 Of the Sunnes quick motion , of more than 7500 miles in one minute . 221 Of the rapt and violent motion of the starry Firmament , which if a Horseman should ride every day 40 miles , he could not in a thousand yeares make such a distance as it moves every houre . 221 To finde the Circle of the Sunne by the fingers . 223 Prob. 93 Of finding the new and full Moone in each moneth . 224 Prob. 94 To finde the latitude of Countreys . 225 Prob. 95 Of the Climates of Countreys , and how to finde them . 225 Prob. 96 Of longitude and latitude of the places of the earth , and of the Starres of the Heavens . 227 To finde the Longitude of a Countrey . 228 Of the Latitude of a Countrey . 229 To finde the Latitude of a Countrey . 230 To finde the distance of places . 230 Of the Longitude , Latitude , Declination , and distance of the starres . 231 How is it that two Horses or other creatures comming into the World at one time , and dying at one and the same instant , yet the one of them to be a day older than the other ? 232 Certaine fine Observations . In what places of the World is it that the needle hangs in Aequilibrio , and verticall ? 233 In what place of the world is it the sun is East or West but twice in the yeare ? 233 In what place of the World is it that the Sunnes Longitude from the Equinoctiall paints and Altitude , being equall , the Sunne is due East or West ? That the sunne comes twice to one point of the Compasse in the forenoone or afternoone . 233 That in some place of the World there are but two kindes of winde all the yeare . 233 Two ships may be two leagues asunder under the equinoctiall , and sayling North at a certaine parallell they will be but just halfe so much . 233 To what inhabitants , and at what time the sunne will touch the north-part of the Horizon at midnight . 234 How a man may know in his Navigation when he is under the Equinoctiall . 234 At what day in the yeare the extremitie of the styles shadow in a Dyall makes a right line . 234 What height the Sunne is of , and how far from the Zenith , or Horizon , when a mans shadow is as long as his height . 234 Prob. 97 To make a Triangle that shall have three right Angles . 234 Prob. 98 To divide a line in as many parts as one will , without compasses or without seeing of it . 235 Prob. 99 To draw a line which shall incline to another line , yet never meet against the Axiome of Parallells . 236 Prob. 100 To finde the variation of the Compasse by the Sunne shining . 237 Prob. 101 To know which way the winde is in ones Chamber without going abroad . 238 Prob. 102 How to draw a parallel sphaericall line with great ease . 239 Prob. 103 To measure an height onely by help of ones Hat. 240 Prob. 104 To take an height with two strawes . 240 In Architecture how statues or other things in high buildings shall beare a proportion to the eye below either equall , double , &c. 242 Prob. 106 Of deformed figures which have no exact proportion , where to place the eye to see them direct . 243 Prob. 107 How a Cannon that hath shot may be covered from the battery of the Enemy . 244 Prob. 108 Of a fine Lever , by which one man alone may place a Cannon upon his Carriage . 245 Prob. 109 How to make a Clock with one wheele 246 Of Water-workes . Prob. 110 How a childe may draw up a Hogshead of water with ease . 247 Prob. 111 Of a Ladder of Cords to cary in ones pocket , by which he may mount a wall or Tower alone . 248 Prob. 112 Of a marvelous Pump which drawes up great quantity of water . 249 Prob. 113 How naturally to cause water to ascend out of a Pit. 250 Prob. 114 How to cast water out of a fountaine very high . 252 Prob. 115 How to empty the water of a Pit by help of a Cisterne . 253 Prob. 116 How to spout out water very high . 253 Prob. 117 How to re-animate simples though brought a thousand miles . 255 Prob. 118 How to make a perpetuall motion . 255 Prob. 119 Of the admirable invention of making the Philosophers Tree , which one may see to grow by little and little . 256 Prob. 120 How to make the representation of the great world 257 Prob. 121 Of a Cone , or Pyramidall figure that moves upon a Table 258 Prob. 122 How an Anvill may be cleaved by the blow of a Pistoll . 258 Prob. 123 How a Capon may be rosted in a mans travells at his sa●●le-bowe . 259 Prob. 124 How a Candle may be made to burne three times longer than usually it doth 259 Prob. 125 How to draw Wine out of water 260 Prob. 126 Of two Marmouzets , the one of which lights a Candle , and the other blowes it out . 261 Prob. 127 How to make Wine fresh without Ice or Snow in the height of Summer . 262 Prob. 128 To make a Cement which lastes as marble , resisting aire and water . 262 Prob. 129 How to melt metall upon a shell with little fire . 263 Prob. 130 Of the hardning of Iron and steele . 263 Prob. 131 To preserve fire as long as you will , imitating the inextinguible fire of the Vestales . 264 FINIS . Ad Authorem D.D. Henricum Van Etenium , Alumnum Academiae Ponta Mousson . ARdua Walkeri sileant secreta profundi , Desinat occultam carpere Porta viam . Itala Cardani mirata est Lampada docti Terra , Syracusium Graecia tota senem : Orbi terrarum , Ptolemaei Clepsydra toti , Rara dioptra Procli , mira fuêre duo , Anglia te foveat doctus Pont-Mousson alumnum : Quidquid naturae , qui legis , hortus habet . Docta , coronet opus doctum , te sit tua docto Digna , Syracusii , arca , corona , viri . Arca Syracusiis utinam sit plumbea servis , Aurea sed dominis , aurea tota suis. MATHEMATICAL RECREATION . PROBLEM I. To finde a number thought upon . BId him that he Quadruple the Number thought upon , that is , multiply it by 4 , and unto it bid him to adde 6 , 8 , 10 , or any Number at pleasure : and let him take the halfe of the sum , then ask how much it coms to , for then if you take away half the number from it which you willed him at first to add to it , there shall remain the double of the number thought upon . Example The Number thought upon 5 The Quadruple of it 20 Put 8 unto it , makes 28 The halfe of it is 14 Take away halfe the number added from it , viz 4 , the rest is 10 The double of the number thought upon , viz. 10 Another way to finde what Number was thought upon . BId him which thinketh double his Number , and unto that double adde 4 , and bid him multiply that same product by 5 , and unto that product bid him adde 12 , and multiply that last number by 10 ( which is done easily by setting a Cypher at the end of the number ) then ask him the last number or product , and from it secretly subtract 320 , the remainder in the hundreth place , is the number thought upon . Example . The number thought upon 7 For which 700 account onely but the number of the hundreds viz. 7. so have you the number thought upon . His double 14 For which 700 account onely but the number of the hundreds viz. 7. so have you the number thought upon . To it add 4 , makes 18 For which 700 account onely but the number of the hundreds viz. 7. so have you the number thought upon . Which multiplyed by 5 makes 90 For which 700 account onely but the number of the hundreds viz. 7. so have you the number thought upon . To which add 12 makes 102 For which 700 account onely but the number of the hundreds viz. 7. so have you the number thought upon . This multiplyed by 10 which is only by adding a Cypher to it , makes 1020 For which 700 account onely but the number of the hundreds viz. 7. so have you the number thought upon . From this subtract 320 For which 700 account onely but the number of the hundreds viz. 7. so have you the number thought upon . Rest 700 For which 700 account onely but the number of the hundreds viz. 7. so have you the number thought upon . To finde numbers conceived upon , otherwise than the former . BId the party which thinks the number , that he triple his thought , and cause him to take the half of it : ( if it be odde take the least half , and put one unto it : ) then will him to triple the half , and take half of it as before : lastly , ask him how many nines there is in the last half , and for every nine , account four in your memory , for that shall shew the number thought upon , if both the triples were even : but if it be odde at the first triple , and ev●n at the second , for the one added unto the least halfe keep one in memory : if the first triple be even , and the second odde , for the one added unto the least halfe keepe two in memory ; lastly , if at both times in tripling , the numbers be odde , for the two added unto the least halfes , keep three in memory , these cautions observed , and added unto as many fours as the party sayes there is nines contained in the last halfe , shall never fail you to declare or discern truly what number was thought upon . Example . The number thought upon 4 or 7 The triple 12 or 21 The half thereof 6 or 10 , one put to it makes 11 The triple of the halfe 18 or 33 The halfe 9 or 1● , one put to it makes 17 The number of nines in the last halfe 1 or 1 The first 1. representeth the 4. number thought upon , and the last 1. with the caution makes 7. the other number thought upon . Note . Order your method so that you be not discovered , which to help , you may with dexterity and industry make Additions ▪ Substractions , Multiplications , Divisions , &c. and instead of asking how many nines there is , you may ask how many eights tens , &c. there is , or subtract 8.10 . &c. from the Number which remains , for to finde out the number thought upon . Now touching the Demonstrations of the former directions , and others which follow , they depend upon the 2 , 7 , 8 , and 9 , Books of the Elements of Euclide : upon which 2. Book & 4. proposition this may bee extracted , for these which are more learned for the finding of any number that any one thinketh on . Bid the party that thinks , that he break the number thought upon into any two parts , and unto the Squares of the parts , let him adde the double product of the parts , then ask what it amounteth unto , so the root Quadrat shall be the number thought upon . The number thought upon 5 , the parts suppose 3 and 2. The square of 3 makes 9 the sum of these three nūbers 25 , the squa●e Root of which is 5 , the number thought upon The square of 2 makes 4 the sum of these three nūbers 25 , the squa●e Root of which is 5 , the number thought upon The product of the parts . viz. 3 by 2 makes 6 , which 6 doubled makes 12 the sum of these three nūbers 25 , the squa●e Root of which is 5 , the number thought upon Or more compendiously it may be delivered thus . Break the number into two parts , and to the product of the parts , adde the square of half the difference of the parts , then the Root Quadrat of the aggregate is halfe the number conceived . EXAMINATION . THe Problems which concern Arithmetick , we examine not , for these are easie to any one which hath read the grounds and principles of Arithmetick , but we especially touch upon that , which tends to the speculations of Physick , Geometry , and Optickes , and such others which are of more difficulty , and more principally to be examined and considered . PROBLEM II. How to represent to those which are in a Cham●er that which is without , or all that which passeth by , It is pleasant to see the beautifull and goodly representation of the heavens intermixed with clouds in the Horizon , upon a woody scituation , the motion of Birds in the Aire , of men and other creatures upon the ground , with the trembling of plants , tops of trees , and such like : for every thing will be seen within even to the life , but inversed : notwithstanding , this beautifull paint will so naturally represent it self in such a lively Perspective , that hardly the most accurate Painter can represent the like . But here note , that they may be represented right two manner of wayes ; first , with a concave glasse : secondly , by help of another convex glasse , disposed or placed between the paper and the other Glasse : as may be seen here by the figure . Now I will add here only by passing by , for such which affect Painting and portraiture , that this experiment may excellently help them in the lively painting of things perspectivewise , as Topographicall cards , &c. and for Philosophers , it is a fine secret to explain the Organ of the sight , for the hollow of the eye is taken as the close Chamber , the Ball of the Apple of the eye , for the hole of the Chamber , the Crystaline humor at the small of the Glasse , and the bottome of the eye , for the Wall or leafe of paper . EXAMINATION . THe species being pressed together or contracted doth not perform it upon a wall , for the species of any thing doth represent it selfe not only in one hole of a window , but in infinite holes , even unto the whole Sphere , or at least unto a Hemisphere ( intellectuall in a free medium ) if the beams or reflections be not interposed , and by how much the hole is made less to give passage to the species , by so much the more lively are the Images formed . In convexe , or concave Glasses the Images will be disproportionable to the eye , by how much they are more concave , or convexe , & by how much the parts of the image comes neer to the Axis , for these that are neer are better proportioned then these which are farther off . But to have them more lively and true , according to the imaginary conicall section , let the hole be no greater than a pins head made upon a piece of thin brasse , or such like , which hole represents the top of the Cone , and the Base thereof the term of the species : this practice is best when the sun shines upon the hole , for then the objects which are opposite to that plaine will make two like Cones , and will lively represent the things without in a perfect inversed perspective , which drawn by the Pensill of some artificiall Painter , turn the paper upside down , and it will be direct and to the life . But the apparences may be direct , if you place another hole opposite unto the former , so that the spectator be under it ; or let the species reflect upon a concave Glass , and let that glas reflect upon a paper or some white thing . PROBLEM III. To tell how much waighs the blow of ones fist , of a Mallet , Hatchet , or such like , or resting without giving the blow SCaliger in his 331 exercise against Cardan , relates that the Mathematicians of Maximillian the Emperour did propose upon a day this Question , and promised to give the resolution ; notwithstanding ●caliger delivered it not , and I conceive it to be thus . Take a Balance , and let the Fist , the Mallet , or Hatchet rest upon the scale , or upon the beam of the Balance , and put into the other Scale as much weight as may counterpoyse it ; then charging or laying more waight into the Scale , and striking upon the other end , you may see how much one blow is heavier than another , and so consequently how much it may waigh for as Aristotle saith , The motion that is made in striking adds great waight unto it , and so much the more , by how much it is quicker : therefore in effect , if there were placed a thousand mallets , or a Thousand pounde waight upon a stone , nay , though it were exceedingly pressed down by way of a Vice , by Levers , or other Mechanick Engine , it would be nothing to the rigor and violence of a blow . Is it not evident that the edge of a knife laid upon butter , and a hatchet upon a leafe of paper , without striking makes no impression , or at least enters not ; but striking upon the wood a little , you may presently see what effect it hath , which is from the quicknesse of the motion , which breaks and enters without resistance , if it be extream quick , as experience shews us in the blows of Arrows , of Cannons , Thunder-boults , and such like . EXAMINATION . THis Problem was extracted from Scaliger , who had it from Aristotle , but somwhat refractory compiled , & the strength of the effect he says depends only in the violence of the motion ; then would it follow that a little light hammer upon a piece of wood being quickly caused to smite , would give a greater blow , and do more hurt than a great sledge striking soft ; this is absurd , and contrary to experience : therefore it consists not totally in the motion , for if two severall hammers , the one being 20 times heavier than the other , should move with like quickness , the effect would be much different , there is then some thing else to be considered besides the Motion which Scaliger understood not , for if one should have asked him , what is the reason that a stone falling from a window to a place neer at hand , is not so forceable as if it fell farther 〈◊〉 when a bullet flying out of a peece and striking the mark neer at hand 〈◊〉 not make such an effect as striking 〈…〉 that Scaliger and 〈…〉 this subiect ▪ would not be less troubled to resolve this , than they have been in that . PROBLEM IV. How to break a staffe which is laid upon two Glasses full of water , without breaking the Glasses , spilling the water , or upon two reeds or straws without breaking of them . In like manner may you doe upon two Reeds , held with your hands in the aire without breaking them ▪ thence Kitchin boyes often break bones of mutton upon their hand , or with a napkin without any hurt , in only striking upon the middle of the bone with a knife . Now in this act , the two ends of the staffe in breaking slides away from the Glasses , upon which they were placed ; hence it commeth that the Glasses are no wise indangered , no more than the knee upon which a staffe is broken , forasmuch as in breaking it presseth not : as Aristotle in his Mechanick Questions observeth . EXAMINATION . IT were necessary here to note , that this thing may be experimented , first , without Glasses , in placing a small slender staffe upon two props , and then making tryall upon it , by which you may see how the Staffe will either break , bow , or depart from his props , and that either directly or obliquely : But why by this violence , that one Staffe striking another , ( which is supported by two Glasses ) will be broken without offending the Glasses , is as great a difficulty to be resolved as the former . PROBLEM V. How to make a faire Ge●graphic●ll Card in a Garden Plot , fit for a Prince , or great personage . IT is usuall amongst great men to have faire Geographicall Maps ▪ large Cards , and great Globes , that by them they may as at once have a view of any place of the World , and so furnish themselves with a generall knowledge , not only of their own Kingdoms form , scituation , longitude , latitude , &c. but of all other places in the whole Universe , with their magnitudes , positions , Climats , and distances . Now I esteem that it is not unworthy for the meditations of a Prince , seeing it carries with it many profitable and pleasant contentmen●s : if such a Card or Map by the advice and direction of an able Mathematician were Geographically described in a Garden plot form , or in some other convenient place , and instead of which generall description might particularly and artificially be prefigured his whole Kingdoms and Dominions , the Mountains and hils being raised like small hillocks with turfs of earth , the valleys somwhat concave , which will be more agreeable and pleasing to the eye , than the description in plain Maps and Cards , within which may be presented the Towns , Villages , Castles , or other remarkable edifices in small green mo●●e banks , or spring-work proportionall to the pl●tform , the Forrests and Woods represented according to their form and capacity , with herbs and stoubs , the great Rivers , Lakes , and Ponds to dilate themselves according to their course from some artificiall Fountain made in the Garden to passe through chanels ; then may there be composed walks of pleasure , ascents , places of repose , adorned with all variety of delightfull herbs and flowers , both to please the eye or other senses . A Garden thus accommodated shall farre exceed that of my Lord of Verulams specified in his ●ssayes ; that being only for delight and pleasure , this may have all the properties of that , and also for singular use , by which a Prince may in little time personally visit his whole Kingdom , and in short time know them distinctly : and so in like manner may any particular man Geographically prefigure his own possession or heritage . PROBLEM VI. How three staves , knives , or like bodies , may be conceived to hang in the aire , without being supported by any thing but by themselves . TAke the first staffe AB , raise up in the aire the end B , and upon him cros-wise place the staffe CB , then lastly , in Triangle wise place the third staffe EF ▪ in such manner that it may be under AB , and yet upon CD . I say that these staves so disposed cannot fall , and the space CBE is made the stronger , by how much the more it is pressed downe , if the staves break not , or sever themselves from the triangular forme : so that alwayes the Center of gravitie be in the Center of the Triangle : for AB is supported by EF , and EF is held up by CD , and CD is kept up from falling by AB , therefore one of these staves cannot fall , and so by consequence none . PROBLEM VII . How to dispose as many men , or other things in such sort , that rejecting , or casting away the 6 , 9 , 10 part , unto a certain number , there shall remaine these which you would have . ORdinarily the proposition is delivered in this wise : 15 Christians and 15 Turkes being at Sea in one Shippe , an extreame tempest being risen , the Pilot of the Shippe saith , it is necessary to cast over board halfe of the number of Persons to disburthen the Shippe , and to save the rest : now it was agreed to be done by lot , and therefore they consent to put themselves in rank , counting by nine and nine , the ninth Person should alwayes be cast into the Sea , untill there were halfe throwne over board ; Now the Pilote being a Christian indeavoured to save the Christians , how ought he therefore to dispose the Christians , that the lot might fall alwayes upon the Turkes , and that none of the Christians be in the ninth place ? The resolution is ordinarily comprehended in this verse . Populeam virgam mater regina ferebat . For having respect unto the vowels , making a one , e two , i three , o foure , and u five : o the first vowell in the first word sheweth that there must be placed 4. Christians ; the next vowel u , signifieth that next unto the 4. Christians must be placed 5 Turkes , and so to place both Christians and Turkes according to the quantity and value of the vowels in the words of the verse , untill they be all placed : for then counting from the first Christian that was placed , unto the ninth , the lot will fall upon a Turk , and so proceed . And here may be further noted that this Probleme is not to be limited , seeing it extends to any number and order whatsoever , and may many wayes be usefull for Captaines , Magistrates , or others which have divers persons to punish , and would chastise chiefely the unruliest of them , in taking the 10 , 20 , or 100. person , &c. as we reade was commonly practised amongst the ancient Romans : herefore to apply a generall rule in counting the third , 4 , 9 , 10 , &c. amongst 30 , 40 , 50 , persons , and more or lesse ; this is to be observed , take as many units as there are persons , and dispose them in order privately : as for example , let 24 men be proposed to have committed some outrage , 6 of them especially are found accessary : and let it be agreed that counting by 8 and 8 the eight man should be alwayes punished . Take therefore first 24 units , or upon a piece of paper write down 24 cyphers , and account from the beginning to the eighth , which eighth mark , and so continue counting alwayes marking the eighth , untill you have markt 6 , by which you may easily perceive how to place those 6 men that are to be punished , and so of others . It is supposed that Josephus the Author of the Jewish History escaped the danger of death by help of this Problem ; for a worthy Author of beliefe reports in his eighth chapter of the third Book of the destruction of Jerusalem , that the Town of Jotapata being taken by main force by Vespatian , Josephus being Governour of that Town , accompanyed with a Troop of forty Souldiers , hid themselves in a Cave , in which they resolved rather to famish than to fall into the hands of Vespatian : and with a bloudy resolution in that great distresse would have butchered one another for sustenance , had not Josephus perswaded them to die by lot and order , upon which it should fall : Now seeing that Josephus did save himselfe by this Art , it is thought that his industry was exercised by the helpe of this Problem , so that of the 40 persons which he had , the third was alwayes killed . Now by putting himselfe in the 16 or 31 place he was saved , and one with him which he might kill , or easily perswade to yeild unto the Romans . PROBLEM . VIII . Three things , and three persons proposed , to finde which of them hath either of these three things . LEt the three things be a Ring , a piece of Gold , and a piece of Silver , or any other such like , and let them be known privately to your self by these three Vowels a , e , i , or let there be three persons that have different names , as Ambrose , Edmond , and John , which privately you may note or account to your selfe once known by the aforesaid Vowels , which signifie for the first vowel 1 , for the second vowell 2 , for the third vowell 3. Now if the said three persons should by the mutuall consent of each other privately change their names , it is most facill by the course and excellencie of numbers , distinctly to declare each ones name so interchanged , or if three persons in private , the one should take a Ring , the other a piece of Gold , and the third should take a piece of Silver ; it is easie to finde which hath the Gold , the Silver , or the Ring , and it is thus done . Take 30 or 40 Counters ( of which there is but 24 necessary ) that so you may conceale the way the better , and lay them down before the parties , and as they sit or stand , give to the first 1. Counter , which signifieth a , the first vowell ; to the second 2. Counters , which represent e , the second vowel ; and to the third 3. Counters , which stand for i , the third vowell : then leaving the other Counters upon the Table , retire apart , and bid him which hath the Ring , take as many Counters as you gave him , and he that hath the Gold , for every one that you gave him , let him take 2 , and he that hath the Silver for every one that you gave him , let him take 4. this being done , consider to whom you gave one Counter , to whom two , and to whom three ; and mark what number of Counters you had at the first , for there are necessarily but 24. as was said before , the surpluse you may privately reject . And then there will be left either 1.2.3.5.6 or 7. and no other number can remaine , which if there be , then they have failed in taking according to the directions delivered : but if either of these numbers do remaine , the resolution will be discovered by one of these 6 words following , which ought to be had in memory , viz. Salve , certa , anima , semita , vita , quies· 1. 2. 3. 5. 6. 7. As suppose 5. did remaine , the word belonging unto it is semita , the vowels in the first two syllables are e and i , vvhich shevveth according to the former directions , that to vvhom you gave 2 Counters , he hath the Ring ( seeing it is the second vovvell represented by tvvo as before ) and to vvhom you gave the 3. Counters , he hath the Gold , for that i represents the third vovvel , or 3. in the former direction , and to vvhom you gave one Counter , he hath the Silver , and so of the rest : the variety of changes , in vvhich exercise , is laid open in the Table follovving . rest men hid rest men hid 1 1 a 5 1   2 e 2   3 i 3   2 1 e 6 1   2 a 2   3 i 3   3 1 a 7 1   2 i 2   3 e 3   This feat may be done also without the former words by help of the Circle A. for having divided the Circle into 6 parts , write 1. within and 1. vvithout , 2. vvithin and 5. vvithout , &c. the first 1.2.3 . vvhich are vvithin vvith the numbers over them , belongs to the upper semicircle ; the other numbers both vvithin and vvithout , to the under semicircle ; now if in the action there remaineth such a number which may be found in the upper semicircle without , then that which is opposite within shews the first , the next is the second , &c. as if 5 remains , it shews to whom he gave 2 , he hath the Ring ; to whom you gave ● , he hath the Gold , &c. But if the remainder be in the under semicircle , that which is opposite to it is the first ; the next backwards towards the right hand is the second ; as if 3 remains , to whom you gave 1 he hath the Ring , he that had 3 he had the Gold , &c. PROBLEM IX . How to part a Vessel which is full of wine conteining eight pints into two equall parts , by two other vessels which conteine as much as the greater vessell ; as the one being 5 pints , and the other 3 pints . LEt the three vessels be represented by ABC , A being full , the other two being empty ; first , poure out A into B until it be full , so there will be in B 5 pints , and in A but 3 pints : then poure out of B into C untill it be full : so in C shall be 3 pints , in B 2 pints , and in A 3 pints , then poure the wine which is in C into A , so in A will be 6 pints , in B 2 pints , and in C nothing : then poure out the wine which is in B into the pot C , so in C there is now 2 pints , in B nothing , and in A 6 pints , . Lastly , poure out of A into B untill it be full , so there will be now in A only 1 pint in B 5 pints , and in C 2 pints . But it is now evident , that if from B you poure in unto the pot C untill it be full , there wil remain in B 4 pints , and if that which is in C , viz. 3 pints be poured into the vessell A , which before had 1 pint , there shall be in the vessel A , but halfe of its liquor that was in it at the first , viz. 4 pints as was required . Otherwise poure out of A into C untill it be full , which pour into B , then poure out of A into C again untill it be full , so there is now in A onely 2 pints , in B 3 , and in C 3 , then pour from C into B untill it be full , so in C there is now but 1 pint , 5 in B , and 2 in A : poure all that is in B into A , then poure the wine which is in C into B , so there is in C nothing , in B onely 1 pint , and in 7 A 7 pints : Lastly , out of A fill the pot C , so there will remain in A 4 pints , or be but halfe full : then if the liquor in C be poured into B , it will be the other half . In like manner might be taken the half of a vessell which conteins 12 pints , by having but the measures 5 and 7 , or 5 and 8. Now such others might be proposed , but we omit many , in one and the same nature . PROBLEM . X. To make a stick stand upon the tip of ones finger , without falling . FAsten the edges of tvvo knives or such like of equall poise , at the end of the stick , leaning out somevvhat from the stick , so that they may counterpoise one another ; the stick being sharp at the end , and held upon the top of the finger , vvill there rest vvithout supporting : if it fall , it must fall together , and that perpendicular or plumb-wise , or it must fall side-wise or before one another ; in the first manner it cannot : for the Centre of gravitie is supported by the top of the finger : and seeing that each part by the knives is counterpoised , it cannot fall sidevvise , therefore it can fall no vvise . In like manner may great pieces of Timber , as Joists , &c be supported , if unto one of the ends be applied convenient proportionall counterpoises , yea a Lance or Pike , may stand perpendicular in the Aire upon the top of ones finger : or placed in the midst of a Court by help of his Centre of gravitie . EXAMINATION . THis Proposition seems doubtfull ; for to imagine absolutely , that a Pike , or such like , armed with two Knives , or other things , shall stand upright in the Aire , and so remain without any other support , seeing that all the parts have an infinite difference of propensity to fall ; and it is without question that a staff so accommodated upon his Centre of gravity , but that it may incline to some one part without some remedy be applied , and such as is here specified in the Probleme will not warrant the thing , nor keep it from falling ; and if more Knives should be placed about it , it should cause it to fall more swiftly , forasmuch as the superiour parts ( by reason of the Centricall motion ) is made more ponderous , and therefore lesse in rest . To place therefore this prop really , let the two Knives , or that which is for counterpoise , be longer always then the staffe , and so it will hang together as one body : and it will appear admirable if you place the Centre of gravity , neer the side of the top of the finger or point ; for it will then hang Horizontall , and seem to hang onely by a touch , yet more strange , if you turn the point or top of the finger upside down . PROBLEM XI . How a milstone or other Ponderosity , may be supported by a small needle , without breaking o● any wise bowing the same . LEt a needle be set perpendicular to the Horizon , and the center of gravitie of the stone be placed on the top of the needle : it is evident that the stone cannot fall , forasmuch as it hangs in aequilibra , or is counterpoysed in all parts alike ; and moreover it cannot bow the needle more on the one side then on the other , the needle will not therefore be either broken or bowed ; if otherwise then the parts of the needle must penetrate and sinke one with another : that which is absurd and impossible to nature ; therefore it shall be supported . The experiments which are made upon trencher plates , or such like lesser thing doth make it most credible in greater bodies . But here especially is to be noted , that the needle ought to be uniforme in matter and figure , and that it be erected perpendicular to the Horizon , and lastly , that the Center of gravity be exactly found . PROBLEM XII . To make three Knives hang and move upon the point of a Needle . FIt the three Knives in form of a Ballance , and holding a Needle in your hand , and place the back of that , Knife which lyes cross-wise to the other two , upon the point of the Needle : as the figure here sheweth you ; for then in blowing softly upon them , they will easily turne and move upon the point of the Needle with ●ou falling . PROBLEM XIII . To finde the weight of Smoak , which is exhaled of any combustible body whatsoever . LEt it be supposed that a great heape of Fagots , or a load of straw weighing 500 pound should be fired , it is evident that this grosse substance will be all inverted into smoak and ashes : now it seems that the smoak weighs nothing ; seeing it is of a thin substance now dilated in the Aire , notwithstanding if it were gathered and reduced into the thickest that it was at first , it would be sensibly weighty : weigh therefore the ashes which admit 50 pound , now seeing that the rest of the matter is not lost , but is exhaled into smoake , it must necessarily be , that the rest of the weight ( to wit ) 450 pound , must be the weight of the smoak required . EXAMINATION . NOw although it be thus delivered , yet here may be noted , that a ponderosity in his own medium is not weighty : for things are said to be weighty , when they are out of their place , or medium , and the difference of such gravity , is according to the motion : the smoak therefore certainly is light being in its true medium ( the aire , ) if it should change his medium , then would we change our discourse . PROBLEM XVI . Many things being disposed circular , ( or otherwise ) to finde which of them , any one thinks upon ▪ SUppose that having ranked 10 things , as ABCDEFGHIK , Circular ( as the figure sheweth ) and that one had touched or thought upon G , which is the 7 : ask the party at what letter he would begin to account ( for account he must , otherwise it cannot be done ) which suppose , at E which is the 5 place , then add secretly to this 5 , 10 ( which is the number of the Circle ) and it makes 15 , bid him account 15 backward from E , beginning his account with that number hee thought upon , so at E he shal account to himself 7 , at D account 8 , at C account 9 , &c. So the account of 15 wil exactly fall upon G , the thing or number thought upon : and so of others : but to conceal it the more , you may will the party from E to account 25 , 35 , &c. and it will be the same . There are some that use this play at Cards , turned upside downe , as the ten simple Cards , with the King and Queen , the King standing for 12 , and the Queene for 11 , and so knowing the situation of the Cards : and thinking a certain houre of the day : cause the party to account from what Card he pleaseth : with this Proviso , that when you see where he intends to account , set 12 to that number , so in counting as before , the end of the account shall fall upon the Card : which shall denote or shew the houre thought upon , which being turned up will give grace to the action , and wonder to those that are ignorant in the cause . PROBLEM XV. How to make a Door or Gate , which shall open on both sides . ALL the skill and subtilty of this , rests in the artificiall disposer of foure plates of Iron , two at the higher end , and two at the lower end of the Gate : so that one side may move upon the hooks or hinges of the Posts , and by the other end may be made fast to the Gate , and so moving upon these hinges , the Gate will open upon one side with the aforesaid plates , or hooks of Iron : and by help of the other two plates , will open upon the other side . PROBLEM XVI . To shew how a Ponderosity , or heavy thing , may be supported upon the end of a staffe ( or such like ) upon a Table , and nothing holding or touching it . TAke a pale which hath a handle , and fill it full of water ( or at pleasure : ) then take a staffe or stick which may not rowle upon the Table as EC , and place the handle of the Pale upon the staffe ; then place another staffe , or stick , under the staffe CE , which may reach from the bottom of the Pale unto the former staffe CE , perpendicular wise : which suppose FG , then shall the Pale of water hang without falling , for if it fall it must fall perpendicularly , or plumbe wise : and that cannot be seeing the staffe CE supports it , it being parallel to the Horizon and susteined by the Table , and it is a thing admirable that if the staffe CE were alone from the table , and that end of the staffe which is upon the Table were greater and heavier than the other : it would be constrained to hang in that nature . EXAMINATION . NOw without some experience of this Probleme , a man would acknowledge either a possibility or impossibity ; therefore it is that very touchstone of knowledge in any thing , to discourse first if a thing be possible in nature , and then if it can be brought to experience and under sence without seeing it done . At the first , this proposition seems to be absurd , and impossible . Notwithstanding , being supported with two sticks , as the figure declareth , it is made facile : for the Horizontall line to the edge of the Table , is the Centre of motion ; and passeth by the Centre of gravity , which necessarily supporteth it . PROBLEM XVII . Of a deceitfull Bowle to play withall . MAke a hole in one side of the Bowle , and cast molten Lead therein , and then make up the hole close , that the knavery or deceit be not perceived : you will have pleasure to see , that notwithstanding the Bowle is cast directly to the play , how it wil turn away side-wise : for that on that part of the Bowle which is heavier upon the one side then on the other , it never will go truly right , if artificially it be not corrected ; which will hazard the game to those which know it not : but if it be known that the leady side in rolling be always under or above , it may go indifferently right ; if otherwise , the weight will carry it always side-wise . PROBLEM . XVIII . To part an Apple into 2.4 . or 8. like parts , without breaking the Rinde . PAsse a needle and threed under the kinde of the Apple , and then round it with divers turnings , untill you come to the place where you began : then draw out the threed gently , and part the Apple into as many parts as you think convenient : and so the parts may be taken out between the parting of the Rind , and the rind remaining alwayes whole . PROBLEM XIX . To finde a number thought upon without asking of any question , certaine operations being done . BId him adde to the number thought ( as admit 15 ) halfe of it , if it may be , if not the greatest halfe that exceeds the other but by an unite , which is 8 ; and it makes 23. Secondly , unto this 23. adde the halfe of it if it may be , if not , the greatest halfe , viz. 12. makes 35. in the meane time , note that if the number thought upon cannot be halfed at the first time , as here it cannot , then for it keep 3 in the memory , if at the second time it will not be equally halfed , reserve 2 in memory , but if at both times it could not be equally halved , then may you together reserve five in memory : this done , cause him from the last summe , viz. 35. to subtract the double of the number thought , viz. 30. rest 5. will him to take the halfe of that if he can , if not , reject 1. and then take the halfe of the rest , which keep in your memory : then will him to take the halfe againe if he can , if not , take one from it , which reserve in your memory , and so perpetually halveing untill 1. remaine : for then mark how many halfes there were taken , for the first halfe account 2 , for the second 4 , for the third 8 , &c. and adde unto those numbers the one 's which you reserved in memory , so there being 5 remaining in this proposition , there were 2 halfings : for which last ! account 4 , but because it could not exactly be halved without rejecting of 1. I adde the 1 therefore to this 4 , makes 5 , which halfe or summe alwayes multiplied by 4 , makes 20. from which subtract the first 3 and 2 , because the halfe could not be formerly added , leaves 15 , the number thought upon . Other Examples . The number thought upon . The number thought 12 The halfe of it 6 The summe 18 The halfe of it 9 The summe of it 27 The double of the number , 24 Which taken away , rests 3 The halfe of it 1 For which account 2 and 1 put to it because the 3 could not be halfed , makes 3 this multiplied by 4 makes 12 The number thought 79 The greatest halfe 40 3 The summe 119 The greatest halfe of which is 60 2 The summe of it is 179 The double of 79 is 158 Which taken from it , rests 21 The lesser half 10. which halve :   The halfe of this is 5 which makes   The half of this is 2 which is 10   The half of this is 1 , with 10 and 11 is 21.   this 21 which is the double of the last halfe with the remainder being multiplied by 4. makes 84 , from which take the aforesaid 3 and 2 , ●●st 79 , the number thought upon .   PROBLEM . XX. How to make an uniforme , & an inflexible body , to passe through two small holes of divers formes , as one being circular , and the other square , Quadrangular , and Triangular-wise , yet so that the holes shall be exactly filled . THis Probleme is extracted from Geometricall observations , and seemes at the first somewhat obscure , yet that which may be extracted in this nature , will appeare more difficult and admirable . Now in all Geometricall practises , the lesser or easier Problemes do alwayes make way to facilitate the greater : and the aforesaid Probleme is thus resolved . Take a Cone or round Pyramide , and make a Circular hole in some board , or other hard material , which may be equall to the bases of the Cone , and also a Triangular hole , one of whose sides may be equall to the Diameter of the circle , and the other two sides equall to the length of the Cone : Now it is most evident , that this Conicall or Pyramidall body , will fill up the Circular hole , and being placed side-wise will fill up the Triangular hole . Moreover , if you cause a body to be turned , which may be like to two Pyramides conjoyned , then if a Circular hole be made , whose Diameter is equal to the Diameter of the Cones conjoyned , and a Quadrangular hole , whose sloping sides be equall to the length of each side of the Pyramide , and the breadth of the hol equal to the Diameter of the Circle , this conjoyned Pyramide shall exactly fill both the Circular hole , and also the Quadrangle hole . PROBLEM . XXI . How with one uniforme body or such like to fill three severall holes : of which the one is round , the other a just square , and the third an ovall forme ? THis Proposition seemes more subtill then the former , yet it may be practised two wayes : for the first , take a Cylindricall body as great or little as you please : Now it is evident that it will fill a Circular hole , which is made equall to the basis of it , if it be placed downe right , and will also fill a long square ; whose sides are equall unto the Diameter and length of the Cylinder , and acording to Pergeus , Archimedes , &c. in their Cylindricall demonstrations , a true Ovall is made when a Cylinder is cut slopewise , therefore if the oval have breadth equall unto the Diameter of the Basis of the Cylinder , & any length whatsoever : the Cylinder being put into his owne Ovall hole shall also exactly fill it . The second way is thus , make a Circular hole in some board , & also a square hole , the side of which Square may be equall to the Diameter of the Circle : and lastly , make a hole Oval-wise , whose breadth may be equal unto the diagonall of the Square ; then let a Cylindricall body be made , whose Basis may be equall unto the Circle , and the length equall also to the same : Now being placed downe right shall fall in the Circle , and flat-wise will fit the Square hole , and being placed sloping-wise will fill the Ovall . EXAMINATION . YOu may note upon the last two Problemes farther , that if a Cone be cut Ecliptick-wise , it may passe through an Issoc●●● Triangle through many Scalen Triangles , and through an Ellipsis ; and if there be a Cone cut scalen-wise , it will passe through all the former , only for the Ellipsis place a Circle : and further , if a solid colume be cut Ecliptick-wise it may fill a Circle , a Square , divers Parallelogrammes , and divers Ellipses , which have different Diameters . PROBLEM XXII . To finde a number thought upon ●fter another manner , then what is formerly delivered BId him that he multiply the number thought upon , by what number he pleaseth , then bid him divide that product by any other number , and then multiply that Quotient by some other number ; and that product againe divide by some other , and so as often as he will : and here note , that he declare or tell you by what number he did multiply & divide Now in the same time take a number at pleasure , and secretly multiply and divide as often as he did : then bid him divide the last number by that which he thought upon . In like manner do yours privately , then will the Quotient of your divisor be the same with his , a thing which seemes admirable to those which are ignorant of the cause . Now to have the number thought upon without seeming to know the last Quotient , bid him adde the number thought upon to it , and aske him how much it makes : then subtract your Quotient from it , there will remaine the number thought upon For example , suppose the number thought upon were 5 , multiply it by 4 makes 20. this divided by 2 , the Quotient makes 10 , which multiplyed by 6 , makes 60 , and divided by 4 , makes 1● . in the same time admit you think upon 4 , which multiplied by 4 , makes 16 , this divided by 2 , makes 8 , which multiplied by 6 makes 48 , and divided by 4 makes 1● ; then divide 1● by the number thought , which was 5 , the Quotient is ● ; divide also 12 by the number you took , viz. 4 , the Quotient is also 3. as was declared ; therefore if the Quo●ient ● be added unto the number thought , viz. ● , it makes 8 , which being known , the number thought upon is also knowne . PROBLEM XXIII . To finde out many numbers that sundry persons , or one man hath thought upon . IF the multitude of numbers thought upon be odde , as three numbers , five numbers , seven , &c. as for example , let 5 numbers thought upon be these ● 2 , 3 , 4 , 5 , 6. bid him declare the sum of the first and second , which will be 5 , the second and third , which makes 7 , the third and fourth , which makes 9 , the fourth and fifth , vvhich makes 11 , and so alvvayes adding the tvvo next together , aske him hovv much the first and last makes together , vvhich is 8. then take these summes , and place them in order , and adde all these together , vvhich vvere in the odde places : that is the first , third , and fifth , viz. 5 , 9 , ● , makes 22. In like manner adde all these numbets together , vvhich are in the even places , that is in the second and fourth places , viz. 7 and 1● makes 18 , substract this from the former 22 , then there vvill remaine the double of the first number thought upon , viz. 4. which known , the rest is easily known : seeing you know the summe of the first and second ; but if the multitude of numbers be even as these six numbers , viz. 2 , ● , 4 , 5 , 6 , 7 , cause the partie to declare the summe of each two , by antecedent and consequent , and also the summe of the second and last , which will be 5 , 7 , 9 , 11 , 13 , 10 , then adde the odde places together , except the first , that is 9 , and 13 , makes 22 , adde also the even places together , that is 7 , 11 , 10 , which makes 28 , substract the one from the other , there shall remaine the double of the second number thought upon , which known all the rest are knowne . PROBLEM XXIV . How is it that a man in one and the same time , may have his head upward , and his feet upward , being in one and the same place ? THe answer is very facill , for to be so he must be supposed to be in the centre of the earth : for as the heaven is above on every side , Coelum undique sursum , all that which looks to the heavens being distant from the centre is upward ; and it is in this sense that Ma●●olyeus in his Cosmographie , & first dialogue , reported of one that thought he was led by one of the Muses to hell , where he saw Lucifer sitting in the middle of the World , and in the Centre of the earth , as in a Throne : having his head and feet upward . PROBLEM . XXV . Of a Ladder by which two men ascending at one time ; the more they ascend , the more they shall be asunder , notwithstanding one being as high as another THis is most evident , that if there were a Ladder halfe on this side of the Centre of the earth , and the other halfe on the other side : and that two at the Centre of the World at one instant being to ascend , the one towards us , and the other towards our Antipodes , they should in ascending go farther and farther , one from another ; notwithstanding both of them being of like height . PROBLEM . XXVI . How it is that a man having but a Rod or Pole of Land , doth bragge that he may in a right line passe from place to place above 3000 miles . THe opening of this is easie , forasmuch as he that possesseth a Rod of ground possesseth not only the exterior surface of the earth , but is master also of that which extends even to the Centre of the earth , and in this wise all heritages & possessions are as so many Pyramides , whose summets or points meet in the centre of the earth , and the basis of them are nothing else but each mans possession , field , or visible quantity ; and therefore if there were made or imagined so to be made , a descent to go to the bottome of the heritage , which would reach to the centre of the earth ; it would be above 3000 miles in a right line as before . PROBLEM . XXVII . How it is , that a man standing upright , and looking which way he will , he looketh either true North or true South . THis happeneth that if the partie be under either of the Poles , for if he be under the North-pole , then looking any way he looketh South , because all the Meridians concurre in the Poles of the world , and if he be under the South-pole , he looks directly North by the same reason . PROBLEM XXVIII . To tell any one what number remaines after certaine operations being ended , without asking any question . BId him to think upon a number , and will him to multiply it by what number you think convenient : and to the pro●●ct bid him adde what number you please , or 〈◊〉 that secretly you consider , that it ma● be divided by that which multiplied , and 〈…〉 divide the sum by the number which he 〈…〉 by , and substract from this Quotient the number thought upon : In the same time divide apart the number which was add●d by that which multiplied , so then your Quotient shall be equall to his remainder , wherefore without asking him any thing , you shall tell him what did remaine , which will seem strange to him that knoweth not the cause : for example , suppose he thought 7 , which multiplied by 5 makes 35 , to which adde 10 , makes 45 , which divided by 5 , yields 9 , from which if you take away one the number thought , ( because the Multiplier divided by the Divisor gives the Quotient 1 , ) the rest will be two , which will be also proved , if 10 the number which was added , were divided by 5 , viz. 2. PROBLEM XXIX . Of the play with two severall things . IT is a pleasure to see and consider how the science of numbers doth furnish us , not only 〈…〉 recreate the spirits , but also 〈…〉 knowledge of admirable things , 〈…〉 measure be shewen in this 〈…〉 the meane time to produce alwayes some of them : suppose that a man hold divers things in his hand , as Gold and ●ilver ▪ and in one hand he held the Gold , and in the other hand he held the Silver : to know subtilly , and by way of divination , or artificially in which hand the Gold or Silver is ; attribu●e t● the Gold , or suppose it have a certaine price , and so likewise attribute to the Silver another price , conditionally that the one be odd , and the other even : as for example , bid h●m that the Gold be valued at 4 Crownes , or Shillings , and the Silver at ● Crownes , or 3 Shillings , or any other number , so that one be odde ▪ and the other even , as before ; then bid him triple that which is in the right hand , & double that which is in the left hand , and bid him adde these two products together , and aske him if it be even or odde ; if it be even , then the Gold is in the right hand ; if odde , the Gold is in the left hand . PROBLEM . XXX . Two numbers being proposed unto two severall parties , to tell which of these numbers is taken by each of them . AS for example : admit you had proposed unto two men whose names were Peter and John , two numbers , or pieces of money , the one even , and the other odde , as 10. and 9. and let the one of them take one of the numbers , and the other partie take the other number , which they place privately to themselves : how artificially , according to the congruity , and excellency of numbers , to finde which of them did take 10. and which 9. without asking any qustion : and this seems most subtill , yet delivered howsoever differing little from the former , and is thus performed : Take privately to your selfe also two numbers , the one even , and the other odde , as 4. and 3. then bid Peter that he double the number which he took , and do you privately double also your greatest number ; then bid John to triple the number which he hath , and do you the like upon your last number : adde your two products together , & mark if it be even or odde , then bid the two parties put their numbers together , and bid them take the halfe of it , which if they cannot do , then immediately tell Peter he took 10. and John 9. because the aggregate of the double of 4. and the triple of 3. makes odde , and such would be the aggregate or summe of the double of Peters number and Johns number , if Peter had taken 10. if otherwise , then they might have taken halfe , and so John should have taken 10. and Peter 9. as suppose Peter had taken 10. the double is 20. and the triple of 9. the other ●umber is 27. which put together makes 47. odde : in like manner the double of your number conceived in minde , viz. 4. makes 8. and the triple of the 3. the other number , makes 9. which set together makes 17. odde . Now you cannot take the halfe of 17 , nor 47. which argueth that Peter had the greater number , for otherwise the double of 9. is 18. & the triple of 10. is 30. which set together makes 48. the halfe of it may be taken : therefore in such case Peter the took lesse number : and John the greater , and this being don cleanly carries much grace with it . PROBLEM . XXXI . How to describe a Circle that shall touch 3 : Points placed howsoever upon a plaine , if they be not in a right line . LEt the three points be A.B.C. put one foot of the Compasse upon A. and describe an Arch of a Circle at pleasure : and placed at B. crosse that Arch in the two points E. and F. and placed in C. crosse the Arch in G. and H. then lay a ruler upon G.H. and draw a line , and place a Ruler upon E. and F. cut the other line in K ▪ so K is the Centre of the Circumference of a Circle , which will passe by the said three points A.B.C. or it may be inverted , having a Circle drawne ; to finde the Centre of that Circle , make 3. points in the circumference , and then use the same way : so shall you have the Centre , a thing most facill to every practitioner in the principles of Geometrie . PROBLEM . XXXII . How to change a Circle into a square forme ? M●ke a Circle upon past-board or other materiall , as the Circle A.C.D.E. of which A. is the Centre ; then cut it into 4 quarters , and dispose them so , that A. at the centre of the Circle may alwayes be at the Angle of the square , and so the foure quarters of the Circle being placed so , it will make a perfect square , whose side A.A. is equall to the Diameter B.D. Now here is to be noted that the square is greater then the Circle by the vacuity in the middle , viz. M. PROBLEM . XXXIII . With one and the same comp●sses , and at one and the same extent , or opening , how to describe many Circles concentricall , that is , greater or lesser one then another ? IT is not without cause that many admire how this Proposition is to be resolved ; yea in the judgement of some it is thought impossible : who consider not the industrie of an ingenious Geometrician , who makes it possible , and that most facill , sundry wayes ; for in the first place if you make a Circle upon a fine plaine , and upon the Centre of that Circle , a small pegge of wood be placed , to be raised up and put downe at pleasure by help of a small ho●e made in the Centre , then with the same opening of the Compasses , you may describe Circles Concentricall , that is , one greater or lesser than another ; for the higher the Center is lifted up , the lesser the Circle will be . Secondly , the compasse being at that extent upon a Gibus body , a Circle may be described , which will be lesse than the former , upon a plaine , and more artificially upon a Globe , or round bowle : and this againe is most obvious upon a round Pyramide , placing the Compasses upon the top of it , which will be farre lesse than any of the former ; and this is demonstrated by the 20. Prop. of the first of Euclids , for the Diameter ● . D. is lesse than the line AD.A.E. taken together , and the lines AD.AE. being equall to the Diameter BC. because of the same distance or extent of opening the compasses , it followes that the Diameter E.D. and all his Circles together is much lesse than the Diameter , and the Circle BC. which was to be performed . PROBLEM XXXIV . Any numbers under 10. being thought upon , to finde what numbers they were . LEt the first number be doubled , and unto it adde 5. and multiply that summe by 5. and unto it adde 10. and unto this product add the next number thought upon ; multiply this same againe by 10. and adde unto it the next number , and so proceed : now if he declare the last summe ; marke if he thought but upon one figure , for then subtract only 35. from it , and the first figure in the place of tennes is the number thought upon : if he thought upon two figures , then subtract also the said ●5 . from his last summe , and the two figures which remaine are the number thought upon : if he thought upo● three figures , then subtract 350. and then the first three figures are the numbers thought upon , &c. so if one thought upon these numbers 5.7.9.6 . double the first , makes 1● . to which adde 5. makes 15. this multiplied by 5. makes 75. to which adde 1● . makes 85. to this adde the next number , viz. 7. makes 92. this multiplied by 10. makes 920. to which adde the next number , viz. 9. makes 929. which multiplied by 10. makes 9290. to which adde 6. makes 9296. from which subtract 3500. resteth 5796. the foure numbers thought upon . Now because the two last figures are like the two numbers thought upon : to conceale this , bid him take the halfe of it , or put first 12. or any other number to it , and then it will not be so open . PROBLEM . XXXV . Of the Play with the Ring . AMongst a company of 9. or 10. persons , one of them having a Ring , or such like : to finde out in which hand : upon which finger , & joynt it is ; this will cause great astonishment to ignorant spirits , which will make them beleeve that he that doth it works by Magick , or Witchcraft : But in effect it is nothing else but a nimble act of Arithmetick , founded upon the precedent Probleme : for first it is supposed that the persons stand or sit in order , that one is first , the next second , &c. likewise there must be imagined that of these two hands the one is first , and the other second : and also of the five fingers , the one is first , the next is second , and lastly of the joynts , the one is as 1. the other is as 2. the other as 3. &c. from whence it appeares that in performing this Play there is nothing else to be done than to think 4. numbers : for example , if the fourth person had the Ring in his left hand , and upon the fifth finger , and third joynt , and I would divine and finde it out : thus I would proceed , as in the 34 Problem : in causing him to double the first number : that is , the number of persons , which was 4. and it makes 8. to which add 5. makes 13. this multiplied by 5. makes 65. put 10. to it , makes 75. unto this put ● . for the number belonging to the left hand , and so it makes 77. which multiplied by 10. makes 770. to this adde the number of the fingers upon which the Ring is , viz. 5. makes 775. this multiplied by 10. makes 7750. to which adde the number for the joynt upon which the Ring is , viz the third joynt , makes 7●53 . to which cause him to adde 14. or some other number , to conceale it the better : and it makes 7767. which being declared unto you , substract 3514 ▪ and there will remaine 4.2.5.3 . which figures in order declares the whol mystery of that which is to be known : 4. signifieth the fourth person , 2. the left hand , 5. the fifth finger , and 3. the third joynt of that finger . PROBLEM . XXXVI . The Play of 34. or more Dice . THat which is said of the two precedent Problemes may be applied to this of Dice ( and many other particular things ) to finde what number appeareth upon each Dice being cast by some one , for the points that are upon any side of a Dice are alwayes lesse than 10 and the points of each side of a Dice may be taken for a number thought upon : therefore the Rule will be as the former : As for example , one having thrown three Dice , and you would declare the numbers of each one , or how much they make together , bid him double the points of one of the Dice , to which bid him adde 5 , then multiply that by 5. and to it adde 10 , and to the summe bid him adde the number of the second Dice : and multiply that by 10 : lastly , to this bid him adde the number of the last Dice , and then let him declare the whole number : then if from it you subtract ●50 . there will remaine the number of the three Dice throwne . PROBLEM . XXXVII . How to make water in a Glasse seeme to boyle and sparkle ? TAke a Glasse neere full of water or other liquor ; and setting one hand upon the foot of it , to hold it fast : turne slightly one of the fingers of your other hand upon the brimme , or edge of the Glasse ; having before privately wet your finger : and so passing softly on with your finger in pressing a little : for then first , the Glasse will begin to make a noyse : secondly , the parts of the Glasse will sensibly appeare to tremble , with notable rarefaction and condensation : thirdly , the water will shake , seeme to boyle : fourthly , it will cast it selfe out of the Glasse , and leap out by small drops , with great astonishment to the standers by ; if they be ignorant of the cause of it , which is onely in the Rarefaction of the parts of the Glasse , occasioned by the motion and pressure of the finger . EXAMINATION . THe cause of this , is not in the rarefaction of the parts of the Glasse , but it is rather in the quick locall motion of the finger , for reason sheweth us that by how much a Body draweth nearer to a quality , the lesse is it subject or capable of another which is contrary unto it ? now condensation , and rarefaction are contrary qualities , and in this Probleme there are three bodies considered , the Glasse , the Water , and the Aire , now it is evident that the Glasse being the most solid , and impenitrable Body , is lesse subject and capable of rarefaction than the water , the water is lesse subject than the Aire , and if there be any rarefaction , it is rather considerable in the Aire then in the Water , which is inscribed by the Glasse , and above the Water , and rather in the Water then in the Glasse : the agitation , or the trembling of the parts of the Glasse to the sense appeares not : for it is a continued body ; if in part , why then not in the whole ? and that the Water turnes in the Glasse , this appeares not , but only the upper contiguous parts of the Water : that at the bottome being lesse subiect to this agitation , and it is most certaine that by how much quicker the Circular motion of the finger upon the edge of the Glasse is , by so much the more shall the Aire be agitated , and so the water shall receive some apparant affection more or lesse from it , according to that motion : as we see from the quicknesse of winde upon the Sea , or c●lme thereof , that there is a greater or lesser agitation in the water ; and for further examination , we leave it to the search of those which are curious . PROBLEM . XXXVIII . Of a fine vessell which holds wine or water , being cast in●o it at a certaine height , but being filled higher , it will runne out of its owne accord . LEt there be a vessell A.B.C.D. in the middle of which place a Pipe ; whose ends both above at E , and below at the bottom of the vessell as at ● ▪ are open ; let the end ● be somewhat lower than the brimme of the Glasse : about this Pipe , place another Pipe as H. L , which mounts a little above E and let it most diligently be closed at H , that no Aire enter in thereby , and this Pipe at the bottome may have a small hole to give passage unto the water ; then poure in water or wine , and as long as it mounts not above E , it is safe ; but if you poure in the water so that it mount above it , farewell all : for it will not cease untill it be all gone out ; the same may be done in disposing any crooked Pipe in a vessell in the manner of a Faucet or funnell , as in the figure H , for fill it under H , at pleasure , and all will go well ; but if you fill it unto H. you will see fine sport , for then all the vessell will be empty incontinent , and the subtiltie of this will seeme more admirable , if you conceale the Pipe by a Bird , Serpent , or such like , in the middle of the Glasse . Now the reason of this is not difficult to those which know the nature of a Cock or Faucet ; for it is a bowed Pipe , one end of which is put into the water or liquor , and sucking at the other end untill the Pipe be full , then will it run of it selfe , and it is a fine secret in nature to see , that if the end of the Pipe which is out of the water , be lower then the water , it will run out without ceasing : but if the mouth of the Pipe be higher then the water or levell with it , it will not runne , although the Pipe which is without be many times bigger than that which is in the water : for it is the property of water to keep alwayes exactly levell ▪ EXAMINATION . HEre is to be noted , that if the face of the water without be in one and the same plaine , with that which is within , though the outtermost Pipe be ten times greater than that which is within ; the water naturally will not runne , but if the plaine of the water without be any part lower then that which is within , it will freely runne : and here may be noted further , that if the mouth of the Pipe which is full of water , doth but only touch the superficies of the water within , although the other end of the pipe without be much lower than that within , the water it will not run at all : which contradicts the first ground ; hence we gather that the pressure or ponderosity of the water within , is the cause of running in some respect . PROBLEM . XXXIX . Of a Glasse very pleasant . SOmetimes there are Glasses which are made of a double fashion , as if one Glasse were within another , so that they seem but one , but there is a little space between them . No● poure Wine or other liquor between the two edges by help of a Tunnell , into a little hole left to this end , so vvill there appeare tvvo fine delusions or fallacies ; for though there be not a drop of Wine vvithin the hollovv of the Glasse , it vvill seem to those vvhich behold it that it is an ordinary Glasse full of Wine , and that especially to those vvhich are side-vvise of it , and if any one move it , it vvill much confirme it , because of the motion of the Wine ; but that vvhich vvill give most delight , is that , if any one shall take the Glasse , and putting it to his mouth shall think to drink the Wine , instead of vvhich he shall sup the Aire , and so vvill cause laughter to those that stand by , vvho being deceived , vvill hold the Glass to the light , & thereby considering that the raies or beames of the light are not reflected to the eye , as they vvould be if there vvere a liquid substance in the Glasse , hence they have an assured proofe to conclude , that the hollovv of the Glasse is totally empty . PROBLEM . XL. If any one should hold in each hand , as many pieces of money as in the other , how to finde how much there is ? BId him that holds the money that he put out of one hand into the other vvhat number you think convenient : ( provided that it may be done , ) this done , bid him that out of the hand that he put the other number into , that he take out of it as many as remaine in the other hand , and put it into that hand : for then be assured that in the hand which was put the first taking away : there will be found just the double of the number taken away at the first . Example , admit there were in each hand 12 Shi●lings or Counters , and that out of the right hand you bid him take 7. and put it into the left : and then put into the right hand from the left as many as doth remaine in the right , which is 5. so there will be in the left hand ●4 , which is the double of the number taken out of the right hand , to wit 7. then by some of the rules before delivered , it is easie to finde how much is in the right hand , viz. 10. PROBLEM . XLI . Many Dice being cast , how artificially to discover the number of the points that may arise . SVppose any one had cast three Dice secretly , bid him that he adde the points that were upmost together : then putting one of the Dice apart , unto the former summe adde the points which are under the other two , then bid him throw these two Dice , and mark how many points a paire are upwards , which adde unto the former summe : then put one of these Dice away not changing the side , mark the points which are under the other Dice , and adde it to the former summe : lastly , throw that one Dice , and whatsoever appeares upward adde it unto the former summe ; and let the Dice remaine thus : this done , comming to the Table , note what points do appeare upward upon the three Dice , which adde privately together , and unto it adde ●1 or 3 times 7 : so this Addition or summe shall be equall to the summe which the party privately made of all the operations which he formerly made . As if he should throw three Dice , and there should appeare upward 5 , 3 , 2. the sum of them is 10. and setting one of them apart , ( as 5. ) unto 10 , adde the points which are under 3 and 2 , which is 4 and 5 , and it makes 19. then casting these two Dice suppose there should appeare 4 and 1 , this added unto 19 makes 24. and setting one of these two Dice apart as the 4. unto the former 24 , I adde the number of points which is under the other Dice , viz. under 1 , that is 6 , which makes 30. Last of all I throw that one Dice , and suppose there did appeare 2 , which I adde to the former 30 , and it makes 32 , then leaving the 3 dice thus , the points which are upward will be these , 5 , 4 , 2 unto which adde secretly 21 , ( as before was said ) so have you 32 , the same number whi●h he had ; and in the same manner you may practise with 4 , 5 , 6 , or many Dice or other bodies , observing only that you must adde the points opposite of the Dice ; for upon which depends the whole demonstration or secret of the play ; for alwayes that which is above and underneath makes 7. but if it make another number , then must you adde as often that number . PROBLEM . XLII . Two mettals , as Gold and Silver , or of other kin●● weighing alike , being privately placed into two like Boxes , to finde which of them the Gold or Silver is in . But because that this experiment in water hath divers accidents , and therefore subject to a caution ; and namely , because the matter of the chest , mettall or other things may hinder . Behold here a more subtill and certaine invention to finde and discover it out without weighing it in the water ▪ Now experience and reason sheweth us that two like bodies or magnitudes of equall weight , and of divers mettalls , are not of equal quantity : and seeing that Gold is the heaviest of all mettalls , it will occupie less roome or place ; from which will follow that the like weight of Lead in the same forme , will occupie or take up more roome or place . Now let there be therefore presented two Globes or Chests of wood or other matter alike , & equall one to the other , in one of which in the middle there is another Globe or body of lead weighing 12. l. ( as C , ) and in the other a Globe or like body of Gold weighing 12 pound ( as B. ) Now it is supposed that the wooden Globes or Chests are of equall weight , forme , and magnitude : and to discover in which the Gold or Lead is in , take a broad paire of Compasses , and clip one of the Coffers or Globes somewhat from the middle , as at D. then fix in the Chest or Globe a small piece of Iron between the feet of the Compasses , as EK , at the end of which hang a vveight G , so that the other end may be counterpoysed , and hang in aequilibrio : and do the like to the other Chest or Globe . Novv if that the other Chest or Globe being clipped in like distance from the end , and hanging at the other end the same weight G. there be found no difference ; then clip them nearer tovvards the middle , that so the points of the Compasse may be against some of the mettall vvhich is inclosed ; or just against the extremitie of the Gold as in D , and suppose it hang thus in aequilibrio ; it is certaine that in the other Coffer is the Lead ; for the points of the Compasses being advanced as much as before , as at F , vvhich takes up a part of the Lead , ( because it occupies a greater place than the Gold ) therefore that shall help the vveight G. to vveigh , and so vvill not hang in aequilibrio , except G be placed neare to F. hence vve may conclude , that there is the Lead ; and in the other Chest or Globe , there is the Gold. EXAMINATION . IF the two Boxes being of equall magnitude weighed in the aire be found to be of equall weight , they shall necessarily take up like place in the water , and therefore weigh also one as much as another : hence there is no possibilitie to finde the inequalitie of the mettalls which are inclosed in these Boxes in the water : the intention of Archimedes was not upon contrary mettalls inclosed in 〈…〉 Boxes , but consisted of comparing metta●●● , simple in the water one with another : therefore the inference is false and absurd . PROBLEM . XLIII . Two Globes of diverse mettalls , ( as one Gold , and the other Copper ) yet of equall weight being put into a box , as BG , to finde in which end the Gold or Copper is . THis is discovered by the changing of the places of the tvvo Bovvles or Globes , having the same counterpoyse H to be hung at the other side , as in N. and if the Gold vvhich is the lesser Globe , vvere before the nearest to the handle D● , having novv changed his place vvill be farthest from the handle DE , as in K. therefore the Centre of gravity of the two Globes taken together , shall be farther separate from the middle of the handle ( under which is the Centre of gravity of the Box ) than it was before , and seeing that the handle is alwayes in the middle of the Box , the vveight N. must be augmented ▪ to keep it in equil●●●● and by this way one may knovv , that if at the second time , the counterpoise be too light , it is a signe that the Gold is farthest off the handle , as at the first triall it vvas nearest . PROBLEM . XLIIII . How to represent diverse sorts of Rainebowes here below ? THe Rainbovve is a thing admirable in the vvorld , vvhich ravisheth often the eyes and spirits of men in consideration of his rich intermingled colours vvhich are seen under the clouds , seeming as the glistering of the Starres , precious stones , and ornaments of the most beauteous flovvers : some part of it as the resplendent stars , or as a Rose , or burning Cole of fire ▪ in it one may see Dyes of sundry sorts , the Violet , the Blew , the Orange , the Saphir , the Jacinct , and the Emerald colours , as a lively plant placed in a green soile : and as a most rich treasure of nature , it is a high work of the Sun who casteth his raies or beames as a curious Painter drawes strokes with his pensill , and placeth his colours in an exquisite situation ; and Solomon saith , Eccles. 43. it is a chiefe and principall work of God. Notwithstanding there is left to industrie how to represent it from above , here below , though not in perfection , yet in part , with the same intermixture of colours that is above . Have you not seen how by Oares of a Boate it doth exceeding quickly glide upon the water with a pleasant grace ? Aristotle sayes , that it coloureth the water , and makes a thousand atomes , upon which the beames of the Sunne reflecting , make a kinde of coloured Rainbowe : or may we not see in houses or Gardens of pleasure artificiall fountaines , which poure forth their droppie streames of water , that being between the Sunne and the fountaine , there will be presented as a continuall Rainbowe ? But not to go farther , I will shew you how you may do it at your doore , by a fine and facill experiment . Take water in your mouth , and turne your back to the Sunne , and your face against some obscure place , then blow out the water which is in your mouth , that it may be sprinkled in small drops and vapours : you shall see those atomes vapours in the beames of the Sunne to turne into a faire Rainebowe , but all the griefe is , that it lasteth not , but soone is vanished . But to have one more stable and permanent in his colours : Take a Glasse full of water , and expose it to the Sunne , so that the raies that passe through strike upon a shadowed place , you will have pleasure to see the fine forme of a Rainebovve by this reflection . Or take a Trigonall Glasse or Crystall Glasse of diverse Angles , and look through it , or let the beames of the Sunne passe through it ; or vvith a candle let the appearances be received upon a shadovved place : you vvill have the same contentment . PROBLEM XLV . How that if all the Powder in the world were in closed within a bowle of paper or glasse , and being fired on all parts , it could not break that bowle ? IF the bowle and the powder be uniforme in all his parts , then by that means the powder would presse and move equally on each side , in which there is no possibility whereby it ought to begin by one side more than another . Now it is impossible that the bowle should be broken in all his parts : for they are infinite . Of like fineness or subtiltie may it be that a bowle of Iron falling from a high place upon a plaine pavement of thin Glasse , it were impossible any wise to break it ; if the bowle were perfectly round , and the Glasse flat and uniforme in all his parts ▪ for the bowle would touch the Glasse but in one point , which is in the middle of infinite parts which are about it : neither is there any cause why it ought more on one side than on another , seeing that it may not be done with all his sides together ; it may be concluded as speaking naturally , that such a bovvle falling upon such a glasse vvill not break it . But this matter is meere Metaphysicall , and all the vvorkmen in the vvorld cannot ever vvith all their industrie make a bovvle perfectly round , or a Glasse uniforme . PROBLEM . XLVI . To finde a number which being divided by 2 , there will remaine 1 , being divided by 3 , there will remaine 1 ; and so likewise being divided by 4 , 5 , or 6 , there would still remaine 1 ; but being didivided by 7 , there will remaine nothing . IN many Authors of Arithmetick this Probleme is thus proposed : A vvoman carrying Egges to Market in a basket , met an unruly fellovv who broke them : who vvas by order made to pay for them : and she being demanded what number she had , she could not tell : but she remembred that counting them by 2 & 2 , there remained 1 ▪ likewise by 3 and 3 by 4 and 4 , by 5 and 5 , by 6 and 6 ; there still remained 1. but when she counted them by 7 and 7 , there remained nothing : Now how may the number of Egges be discovered ? Finde a number which may exactly be measured by 7 , and being measured by 2 , 3 , 4 , 5 , and 6 ; there vvill still remaine a unite ▪ multiply these numbers together , makes 720 , to which adde 1 ; so have you the number , viz. 721. in like manner 301 vvill be measured by 2 , 3 , 4 , 5 , 6 ; so that 1 remaines : but being measured by 7 , nothing vvill remaine ; to vvhich continually adde 220 , and you have other numbers vvhich vvill do the same : hence it is doubtfull vvhat number she had , therefore not to faile , it must be knovvn vvhether they did exceed 400 , 800 , &c. in vvhich it may be conjectured that it could not exceed 4 or 5 hundred , seeing a man or vvoman could not carry 7 or 8 hundred Egges , therefore the number vvas the former ●01 . vvhich she had in her Basket : vvhich being counted by 2 and 2 , there vvill remaine 1 , by 3 and 3 , &c. but counted by 7 and 7 , there vvill remaine nothing . PROBLEM . XLVII . One had a certaine number of crownes , and counting them by 2 and 2 , there rested 1. counting them by 3 and 3 , there rested 2. counting them by 4 and 4 , there rested 3. counting them by 5 and 5 , there rested 4. counting them by 6 & 6 , there rested 5. but counting them by 7 and 7 , there remained nothing : how many crownes might he have ? THis Question hath some affinitie to the precedent , and the resolution is almost in the same manner : for here there must be found a number , vvhich multiplied by 7 , and then divided by 2 , 3 , 4 , 5 , 6 ; there may alvvayes remaine a number lesse by 1 than the Divisor : Novv the first number vvhich arrives in this nature is 119 , unto vvhich if 420 be added , makes 539 , vvhich also vvill do the same : and so by adding 420 , you may have other numbers to resolve this proposition . PROBLEM . XLVIII . How many sorts of weights in the least manner must there be to weigh all sorts of things between 1 pound and 40 pound , and so unto 121 , & 364 pound . TO vveigh things betvveen 1 and 40 , take numbers in triple proportion , so that their summe be equall , or somewhat greater than 40 , as are the numbers 1 3.9.27 . I say that with ● such weights , the first being of 1 pound , the second being 3 pound , the third being 9 pound , and the fourth being 27 : any weight between 1 and 40 pound may be weighed . As admit to weigh 21 pound , put unto the thing that is to be weighed the 9 pound weight , then in the other ballance put 27 pound and 3 pound , which doth counterpoise 21 pound and 9 pound , and if 20 pound were to be weighed , put to it in the ballance 9 and 1 , and in the other ballance put 27 and 3 , and so of others In the same manner take those 5 weights , 1 , 3 , 9 , 27 ▪ 81 , you may weigh with them between 1 pound , and 121 pound : and taking those 6 weights ▪ as 1 , 3 , 9 , 2● , 81 , 243 , you may weigh even from 1 pound unto 364 pound : this depends upon the property of continued proportionals , the latter of which containing twice all the former . PROBLEM . XLIX . Of a deceitfull ballance which being c●●●ty seemes i● be just , because it hangs in aequilibrio : not●ithstanding putting 12 pound in one ballance , and 11 in the other , it will remaine in aequilibrio . ARistotle maketh mention of this ballance in his mechanick Questions , and saith , that the Merchants of purpose in his time used them to deceive the world : the subtiltie or craft of which is thus , that one arme of the ballance is longer then another , by the same proportion , that one weight is heavier then another : As if the beame were 23 inches long , and the handle placed so that 12 inches should be on one side of it , and 11 inches on the other side : conditionally that the shorter end should be as heavy as the longer , a thing easie to be done : then afterwards put into the ballance two unequal weights in such proportion as the parts of the beame have one unto another , which is 12 to 11 , but so that the greater be placed in the ballance which hangs upon the shorter part of the beame , and the lesser weight in the other ballance : it is most certaine that the ballances will hang in aequilibrio , which will seem most sincere and just ; though it be most deceitfull , abominable , and false . The reason of this is drawne from the experiments of Archimedes , who shewes that two unequall weights will counterpoyse one another , when there is like proportion betweene the parts of the beame ( that the handle separates ) and the vveights themselves : for in one and the same counterpoise , by hovv much it is farther from the Centre of the handle , by so much it seems heavier , therefore if there be a diversitie of distance that the ballances hang from the handle , there must necessarily be an ineqality of weight in these ballances to make them hang in aequilibrio , and to discover if there be deceit , change the weight into the other ballance , for as soone as the greater vveight is placed in the ballance that hangs on the longer parts of the beame : it vvill vveigh dovvne the other instantly . PROBLEM . L. To heave or lift up a bottle with a straw . TAke a stravv that is not bruised , bovv it that it make an Angle , and put it into the bottle so that the greatest end be in the neck , then the Reed being put in the bovved part vvil cast side-vvise , and make an Angle as in the figure may be seen : then may you take the end vvhich is out of the bottle in your hand , and heave up the bottle , and it is so much surer , by how much the Angle is acuter or sharper ; and the end which is bowed approacheth to the other perpendicular parts which come out of the bottle . PROBLEM . LI. How in the middle of a wood or desert , without the sight of the Sunne , Starres , Shadow or Compasse , to finde out the North or South , or the foure Cardinall points of the world , East , West , & c ? IT is the opinion of some , that the windes are to be observed in this : if it be hot , the South is found by the windes that blow that way , but this observation is uncertaine and subject to much error : nature will help you in some measure to make it more manifest than any of the former , from a tree thus : Cut a small tree off , even to the ground , and mark the many circles that are about the sap or pith of the tree , which seem nearer together in some part than in other , which is by reason of the Suns motion about the tree : for that the humiditie of the parts of the tree towards the South by the heat of the Sun is rarified , and caused to extend : and the S●n not giving such heat towards the North-part of the tree , the sap is lesser rarefied , but condensed ; by which the circles are nearer together on the North-part , than on the South-part : therefore if a line be drawne from the widest to the narrowest part of the circles , it shall shew the North & South of the world . Another Experiment may be thus : Take a small needle , such as women work with : place it gently downe flatwise upon still water , and it will not sink , ( which is against the generall tenet that Iron will not swimme ) which needle will by little and little turne to the North and South-points . But if the needle be great and will not swim , thrust it through a small piece of Cork , or some such like thing , and then it will do the same : for such is the property of Iron when it is placed in aequilibrio , it strives to finde out the Poles of the world or points of North and South in a manner as the magnes doth . EXAMINATION . HEre is observable , that the moisture which aideth to the growth of the tree , is dilated and rarefied by the Meridionall heat , and contracted by the Septentrionall cold : this rarefaction works upon the part of the humour or moisture that is more thinne , which doth easily dissipate and evaporate : which evaporation carries a part of the salt with it ; and because that solidation or condensation , so that there is left but a part of the nourishment which the heat bakes up and consumes : so contrarily on the other side the condensation and restrictive quality of the moisture causeth lesse evaporation and perdition : and so consequently there remaines more nourishment , which makes a greater increase on that side than on the other side : for as trees have their growth in winter , because of their pores and these of the earth are shut up : so in the spring when their pores are open , and when the sappe and moisture is drawne by it , there is not such cold on the North-side that it may be condensed at once : But contrarily to the side which is South , the heat may be such , that in little time by continuance , this moisture is dissipated greatly : and cold is nothing but that which hardneth and contracteth the moisture of the tree , and so converteth it into wood . PROBLEM . LII . Three persons having taken Counters , Cards , or other things , to finde how much each one hath taken . CAuse the third party to take a number which ma● be divided by 4 , and as often as he takes 4 , let the second party take 7 , and the first take 13 , then cause them to put them all together , and declare the summe of it ; which secretly divide by 3 , and the Quotient is the double of the number which the third person did take . Or cause the third to give unto the second and first , as many as each of them hath ; then let the second give unto the first and third , as many as each of them hath ; lastly , let the third give unto the second and first , as many as each of them hath ; and then aske how much one of them hath ; ( for they will have then all alike , ) so halfe of that number is the number that the third person had at the first : which knowne all is knowne . PROBLEM . LIII . How to make a consort of musick of many parts with one voyce , or one instrument only ? THis Probleme is resolved , so that a finger or player upon an instrument , be neare an Echo which answereth his voice or instrument ; and if the Echo answereth but once at a time , he may make a double ; if twice , then a triple , if three times , then an harmonie of foure parts , for it must be such a one that is able to exercise both tune and note as occasion requires . As when he begins ut , before the Echo answer , he may begin sol , and pronounce it in the same tune that ●he Echo answereth , by which meanes you ●ave a fifth , agreeable consort of musick : then in the same time that the Echo followeth , to sound the second note sol , he may sound forth another sol higher or lower to make an eight , the most perfect consort of musick , and so of others , if he will continue his voice with the Echo , and sing alone with two parts . Now experience sheweth this to be true , which often comes to passe in many Churches , making one to beleeve that there are many more parts in the musick of a Quire , then in effect truly there are because of the resounding and multiplying of the voic● , and redoubling of the Quire. PROBLEM . LIIII . T● make or describe an Ovall form , or that which neare resembles unto it , at one turning with a paire of common Compasses . THere are many fine wayes in Geometricall practices , to make an Ovall figure or one neare unto it , by severall centres : any of which I will not touch upon , but shew how it may be done promptly upon one centre only . In which I will say nothing of the Ovall forme , which appeares , when one describeth circles with the points of a common Compasses , somewhat deep upon a skinne stretched forth hard : which contracting it selfe in some parts of the skinne maketh an Ovall forme . But it will more evidently appeare upon a Columne or Cylinder : if paper be placed upon it , then with a paire of Compasses describe as it were a circle upon it , which paper afterwards being extended , will not be circular but ovall-wise : and a paire of Compasses may be so accommodated , that it may be done also upon a plaine thus . As let the length of the Ovall be H. K , fasten 2 pinnes or nailes neare the end of that line as F. G , and take a threed which is double to the length of G. H , or F. K , then if you take a Compasse which may have one foot lower than another , with a spring between his legges : and placing one foot of this Compasse in the Centre of the Ovall , and guiding the threed by the other foot of the Compasses , and so carrying it about : the spring will help to describe and draw the Ovall forme . But in stead of the Compasses it may be done with ones hand only , as in the figure may appeare . PROBLEM . LV. Of a pu●se difficult to be opened . IT is made to shut and open with Rings : first at each side there is a strap or string , as AB . and CD , at the end of which are 2 rings , B & D , and the string CD passeth through the ring B , so that it may not come out againe ; or be parted one from another : and so that the ring B , may slide up and downe upon the string CD , then over the purse , there is a piece of Leather EFGH , which covers the opening of the purse , and there is another piece of Leather AE , which passeth through many rings : which hath a slit towards the end I , so great that the string BC may slide into it : Now all the cunning or craft is how to make fast or to open the purse , which consists in making the string BC slide through the side at I , therefore bring down B to I , then make the end I passe through the ring B , and also D with his string to passe through the slit I , so shall the purse be fast , and then may the strings be put as before , and it will seem difficult to discover how it was done . Now to open the purse , put through the end I through the ring B , and then through the slit I , by which you put through the string DC , by this way the purse will be opened . PROBLEM . LVI . Whether it is more hard and admirable without Compasses to make a perfect circle , or being made to finde out the Centre of it ? IT is said that upon a time past , two Mathematicians met , and they would make tryall of their industry : the one made instantly a Perfect circle without Compasses , and the other immediately pointed out the Centre thereof with the point of a needle ; now which is the chiefest action ? it seems the first , for to draw the most noblest figure upon a plaine Table without other help than the hand , and the minde , is full of admiration ; to finde the Centre is but to finde out only one point , but to draw a round , there must be almost infinite points , equidistant from the Centre or middle ; that in conclusion it is both the Circle and the Centre together . But contrarily it may seem that to finde the Centre is more difficult , for what attention , vivacitie , and subtiltie must there be in the spirit , in the eye , in the hand , which will chuse the true point amongst a thousand other points ? He that makes a circle keeps alwayes the same distance , and is guided by a halfe distance to finish the rest ; but he that must finde the Centre , must in the same time take heed to the parts about it , and choose one only point which is equall distant from an infinite of other points which are in the circumference ; which is very difficult . Aristotle confirmes this amongst his morals , and seems to explaine the difficultie which is to be found in the middle of vertue ; for it may want a thousand wayes , and be farre separated from the true Centre of the end of a right mediocritie of a vertuous action ; for to do well it must touch the middle point which is but one , and there must be a true point which respects the end , and that 's but one only . Now to judge which is the most difficult , as before is said , either to draw the round or to finde the Centre , the round seems to be harder than to finde the Centre , because that in finding of it , it is done at once , and hath an equall distance from the whole ; But , as before , to draw a round there is a visible point imagined , about which the circle is to be drawne . I esteeme that it is as difficult therefore , if not more , to make the circle without a Centre , as to finde the middle or Centre of that circle . PROBLEM . LVII . Any one having taken 3 Cards , to finde how many points they containe THis is to be exercised upon a full Pack of Cards of 52 , then let one choose any three at pleasure secretly from your sight , and bid him secretly account the points in each Card , and will him to take as many Cards as will make up 15 to each of the points of his Cards , then will him to give you the rest of the Cards , for 4 of them being rejected , the rest shew the number of points that his three Cards which he took at the first did conteine . As if the 3 Cards were 7 , 10 , and 4 ; now 7 wants of 15 , 8. take 8 Cards therefore for your first Card : the 10 wants of 15 ▪ 5 , take 5 cards for your second card : lastly 4 wants of 15 , 11 , take 11 Cards for your third Card , & giving him the rest of the Cards , there will be 25 ; from which take 4 , there remaines 21 , the number of the three Cards taken , viz. 7 , 10 , and 4. Whosoever would practise this play with 4 , 5 , 6 , or more Cards , and that the whole number of Cards be more or lesse than 52 ; and that the terme be 15 , 14 , 12 , &c , this generall rule ensuing may serve : multiply the terme by the number of Cards taken at first : to the product adde the number of Cards taken , then subtract this summe from the whole number of Cards ; the remainder is the number which must be subtracted from the Cards , which remaines to make up the game : if there remaine nothing after the Subtraction , then the number of Cards remaining doth justly shew the number of points which were in the Cards chosen . If the Subtraction cannot be made , then subtract the number of Cards from that number , and the remainder added unto the Cards that did remaine , the summe will be the number of points in the Cards taken , as if the Cards were 7 , 10 , 5 , 8 , and the terme given were 12 ; so the first wants 5 , the second wants 2 , the third wants 7 , and the fourth wants 4 Cards , which taken , the party gives you the rest of the Cards : then secretly multiply 12 by 4 , makes 48 ; to which adde 4 , the number of Cards taken makes 52 , from which 52 should be taken , rest nothing : therefore according to the direction of the remainder of the Cards which are 30 , is equall to the points of the foure Cards taken , viz. 7 , 10 , 5 , 8. Againe , let these five Cards be supposed to be taken , 8 , 6 , 10 , 3 , 7 ; their differences to 15 , the termes are 7 , 9 , 5 , 12 , 8 , which number of Cards taken , there will remaine but 6 Cards : then privately multiply 15 by 5 , makes 75 , to which adde 5 makes 80 , from this take 52 the number of Cards , rest 28 , to vvhich add the remainder of Cards , make 34. the summe with 8 , 6 , 10 , 3 , 7. PROBLEM . LVII . Many Cards placed in diverse ranks , to finde which of these Cards any one hath thought . TAke 15 Cards , and place them in 3 heaps in rank-wise , 5 in a heap : now suppose any one had thought one of these Cards in any one of the heaps , it is easie to finde vvhich of the Cards it is , and it is done thus ; ask him in vvhich of the heaps it is , vvhich place in the middle of the other tvvo ; then throvv dovvne the Cards by 1 and 1 into three severall heaps in rank-vvise , untill all be cast dovvne , then aske him in which of the rankes his Card is , which heap place in the middle of the other two heaps alwayes , and this do foure times at least , so in putting the Cards altogether , look upon the Cards , or let their back be towards you , and throw out the eight Card , for that was the Card thought upon without faile . PROBLEM . LVIII . Many Cards being offered to sundry persons , to finde which of these Cards any one thinketh upon . ADmit there were 4 persons , then take 4 Cards , and shew them to the first , bid him think one of them , and put these 4 away , then take 4 other Cards , and shew them in like manner to the second person , and bid him think any one of these Cards , and so do to the third person , and so the fourth , &c. Then take the 4 Cards of the first person , and dispose them in 4 rankes , and upon them the 4 Cards of the second person , upon them also these of the third person , and lastly , upon them these of the fourth person , then shew unto eaeh of these parties each of these ranks , and aske him if his Card be in it which he thought , for infallibly that vvhich the first partie thought upon vvill be in the first rank , and at the bottome , the Card of the second person vvill be in the second ranke , the Card of the third thought upon will be in the third rank , and the fourth mans Card will be in the fourth rank , and so of others , if there be more persons use the same method . This may be practised by other things , ranking them by certaine numbers : allotted to pieces of money , or such like things . PROBLEM . LIX . How to make an instrument to help hearing , as Galileus made to help the sight ? THink not that the Mathematickes ( which hath furnished us with such admirable helps for seeing ) is wanting for that of hearing , it s well knowne that long trunks or pipes make one heare well farre off , and experience shewes us that in certaine places of the Orcades in a hollow vault , that a man speaking but softly at one corner thereof , may be audibly understood at the other end : notwithstanding those which are between the parties cannot heare him speak at all : And it is a generall principle , that pipes do greatly help to strengthen the activitie of naturall causes : we see that 〈◊〉 contracted in a pipe , burnes 4 or 5 foot high , which would scarce heat , being in the open aire : the rupture or violence of water issuing out of a fountaine , shewes us that vvater being contracted into a pipe , causeth a violence in its passage . The Glasses of Galeileus makes us see how usefull pipes or trunkes are to make the light and species more visible , and proportionable to our eye . It is said that a Prince of Italy hath a faire hall , in which he can with facility heare distinctly the discourses of those which walk in the adjacent Gardens , which is by certaine vessels and pipes that answer from the Garden to the Hall. Vitruvius makes mention also of such vessels and pipes , to strengthen the voice and action of Comedians : and in these times amongst many noble personages ▪ the new kinde of trunkes are used to help the hearing , being made of silver , copper , or other resounding materiall ; in funnell-wise putting the widest end to him which speaketh , to the end to contract the voice , that so by the pipe applied to the eare it may be more uniform and lesse in danger to dissipate the voice , and so consequently more fortified . PROBLEM . LX. Of a fine lamp which goes not out , though one carry it in ones pocket : or being rolled upon the ground will still burne . IT must be observed that the vessell in which the oile is put into , have two pinnes on the sides of it , one against another , being included within a circle : this circle ought to have two other pinnes , to enter into another circle of brasse , or other solid matter : lastly , this second circle hath two pinnes , which may hang within some box to containe the whole lamp , in such manner , that there be 6 pinnes in different position : Now by the aid of these pegges or pinnes , the lamp that is in the middle will be alwayes well situated according to his Centre of gravity , though it be turned any way : though if you endeavour to turne it upside downe , it will lie levell ▪ which is pleasant and admirable to behold to those which know not the cause : And it is facil from his to make a place to rest quiet in , though there be great agitation in the outvvard parts . PROBLEM . LXI . Any one having thought a Card amongst many Cards , how artificially to discover it out ? TAke any number of Cards as 10 , 12 , &c. and open some 4 or 5 to the parties sight , and bid him think one of them , but let him note vvhether it be the first , second , third , &c. then vvith promptness learn vvhat number of Cards you had in your hands , and take the other part of the Cards , and place them on the top of these you hold in your hand ; and having done so , aske him whether his Card were the first , second , &c. then before knowing the number of Cards that were at the bottome , account backwards untill you come to it : so shall you easily take out the card that he thought upon . PROBLEM . LXII . Three Women AB.C. carried apples to a marke to sell , A had 20 , B 30 ▪ and C 40 , they sold as many for a penny , the one as the other : and brought home one as much money as another , how could this be ? THe answer to the Probleme is easie ▪ as suppose at the beginning of the Market : A ▪ sold her apples at a penny an apple : and sold but 2. which was 2 pence , and so she had 18 left : but B. sold 17. which was 17 pence , and so had 13 left : C. sold 32. which was 32 pence , and so had 8 apples left ▪ then A said she would not sell her apples so cheap , but would sell them for 3 pence the peece , which she did : and so her apples came to 54 pence , and B having left but 13 apples sold them at the same rate , which came to 39 pence : and lastly ▪ C. had but 8 apples , which at the same rate came to 24 pence : these summes of money which each others before received come to 56 pence , and so much each one received ; and so consequently brought home one as much as another . PROBLEM . LXIII . Of the properties of some numbers . FIrst , any two numbers is just the summe of a number , that have equall distance from the halfe of that number ▪ the one augmenting , and the other diminishing , as 7 and 7 , of 8 and ● , of 9 and 5 , of 10 and 4 , of 11 and 3 , of 12 and 2 , of 13 and ● . as the one is more than the halfe , the other is lesse . Secondly , it is difficult to finde two numbers whose summe and product is alike , ( that is ) if the numbers be multiplied one by another , and added together , will be equall , which two numbers are 2 and 2 , for to multiply 2 by 2 makes 4 , and adding 2 unto 2 makes the same : this property is in no other two whole numbers , but in broken numbers there are infinite , whose summe and product will be equall one to another . As Clavius shewes upon the 36 Pro. of the 9t h book of Euclide . Thirdly , the numbers 5 and 6 are called circular numbers , because the circle turnes to the point from whence it begins : so these numbers multiplied by themselves , do end alwayes in 5 and 6 , as 5 times 5 makes 25 , that againe by 5 makes 125 , so 6 times 6 makes 36 , and that by 6 makes 216 , &c. Fourthly , the number 6 , is the first which Arithmeticians call a perfect number , that is , whose parts are equall unto it , so the 6 part of it is 1 , the third part is 2 , the halfe is 3 , which are all his parts : now 1 , 2 , and 3 , is equall to 6. It is wonderfull to conceive that there is so few of them , and how rare these numbers are ▪ 50 of perfect men : for betwixt 1 & 1000000000000 numbers there is but ten , that is ; 6 , 28 , 486. 8128. 120816. 2096128. 33550336. 536854528. 8589869056 , & 137438691328 ▪ with this admirable property , that alternately they end all in 6 and 8 , & the twentieth perfect number is 151115727451553768931328. Fiftly , the number 9 amongst other priviledges carries with it an excellent property : for take what number you will , either in grosse or in part , the nines of the whole or in its parts rejected , and taken simply will be the same , as ●7 it makes 3 times 9 , so vvhether the nines be rejected of 27 , or of the summe of 2 and 7 , it is all one , so if the nines vvere taken avvay of 240. it is all one , if the nines vvere taken avvay of 2 , 4 , and 0 ; for there vvould remaine 6 in either ; and so of others . Sixtly , 11 being multiplied by 2 , 4 , 5 , 6 , 7 , 8 , or 9 , will end and begin with like numbers ; so 11 multiplied by 5 makes 55 , if multiplied by 8 , it makes 88 , &c. Seventhly , the numbers 220 and 284 being unequall , notwithstanding the parts of the one number do alwayes equalize the other number : so the aliquot parts of 220 are 110 , 54 , 44 , 22 , 20 , 11 , 10 , 5 , 4 , 2 , 1 , which together makes 284. the aliquot parts of 284 , are 142 , 71 , 4 , 2 , 1. which together makes 220 , a thing rare and admirable , and difficult to finde in other numbers . I● one be taken from any square number which is odde , the square o● halfe of it being added to the first square , will make a square number . The square of halfe any even number + . 1 being added to that even number makes a square number , and the even number taken from it leaves a square number . If odde numbers be continually added from the unitie successively , there will be made all square numbers , and if cubick numbers be added successively from the unitie , there will be likewise made square numbers . PROBLEM . LXIV . Of an excellent lamp , which serves or furnisheth it selfe with oile , and burnes a long time . I Speak not here of a common lamp which Ca●danus writes upon in his book de subtilita●● , for that 's a little vessell in columne-wise , which is full of Oile , and because there is but one little hole at the bottome neare the weeke or match ; the oile runnes not , for feare that there be emptinesse above : when the match is kindled it begins to heat the lamp , and rarefying the oile it issueth by this occasion : and so sends his more airie parts above to avoid vacuitie . It is certaine that such a lampe the Atheniaus used , which lasted a whole yeare without being touched : which was placed before the statue of Minerva , for they might put a certaine quantitie of oile in the lamp CD , and a match to burne without being consumed : such as the naturalists write of , by which the lamp will furnish it selfe , and so continue in burning : and here may be noted that the oile may be poured in , at the top of th● vessell at a little hole , and then made fast againe that the aire get not in . PROBLEM . LXV . Of the play at Keyles or nine Pinnes . YOu will scarce beleeve that with one bowle and at one blow playing freely , one may strike downe all the Keyles at once : yet from Mathematicall principles it is easie to be demonstrated , that if the hand of him that playes were so well assured by experience , as reason induceth one thereto ; one might at one blow strike downe all the Keyles , of at least 7 or 8 , or such a number as one pleaseth . For they are but 9 in all disposed or placed in a perfect square , having three every way . Let us suppose then that a good player beginning to play at 1 somewhat low , should so strike it , that it should strike down the Keyles 2 and 5 , and these might in their violence strike downe the Keyles 3 , 6 , and 9 , and the bowle being in motion may strike down the Keyle 4 , and 7 ; which 4 Keyle may strike the Keyle 8 , & so all the 9 Keyles may be striken down at once . PROBLEM . LXIV . Of Spectacles of pleasure . SImple Spectacles of blew , yellow , red or green colour , are proper to recreate the sight , and will present the objects died in like colour that the Glasses are , only those of the greene do somewhat degenerate ; instead of shewing a lively colour it will represent a pale dead colour , and it is because they are not dyed greene enough , or receive not light enough for greene : and colour these images that passe through these Glasses unto the bottome of the eye . EXAMINATION . IT is certaine , that not onely Glasses dyed green , but all other Glasses coloured , yield the app●arances of objects strong or weak in colour according to the quantity of the dye , more or lesse , as one being very yellow , another a pale yellow ; now all colours are not proper to Glasses to give colour , hence the defect is not that they want facultie to receive light , or resist the penetration of the beams ; for in the same Glasses those which are most dyed , give alwayes the objects more high coloured and obscure , and those which are lesse dyed give them more pale and cleare : and this is daily made manifest by the painting of Glasse , which hinders more the penetration of the light than dying doth , where all the matter by fire is forced into the Glasse , leaving it in all parts transparent . Spectacles of Crystall cut with divers Angles diamond-wise do make a marvellous multiplication of the appearances , for looking towards a house it becomes as a Towne , a Towne becomes like a Citie , an armed man seems as a whole company caused solely by the diversity of refractions , for as many plaines as there are on the outside of the spectacle , so many times will the object be multiplied in the appearance , because of diverse Images cast into the eye . These are pleasurable spectacles for avaricious persons that love Gold and silver , for one piece will seeme many , or one heap of money will seeme as a treasury : but all the mischiefe is , he will not have his end in the enjoying of it , for indeavouring to take it , it will appeare but a deceitfull Image , or delusion of nothing . Here may you note that if the finger be directed by one and the same ray or beam , which pointeth to one and the same object , then at the first you may touch that visible object without being deceived : otherwise you may faile often in touching that which you see . Againe , there are Spectacles made which do diminish the thing seen very much , and bring it to a faire perspective forme , especially if one look upon a faire Garden plat , a greater walk , a stately building , or great Court , the industry of an exquisite Painter cannot come neare to expresse the lively forme of it as this Glasse will represent it ; you will have pleasure to see it really experimented , and the cause of this is , that the Glasses of th●se Spectacles are hollow and thinner in the middle , than at the edges by which the visuall Angle is made lesser : you may observe a further secret in these Spectacles , for in placing them upon a window one may see those that passe to and fro in the streets , without being seen of any , for their property is to raise up the objects that it lookes upon . Now I would not passe this Probleme without saying something of Galileus admirable Glasse , for the common simple perspective Glasses give to aged men but the eyes or sight of young men , but this of Galileus gives a man an Eagles eye , or an eye that pierceth the heavens : first it discovereth the spottie and shadowed opacous bodies that are found about the Sunne , which darknet and diminisheth the splendor of that beautifull and shining Luminary : secondly , it shewes the new Planets that accompany Saturne and Jupiter : thirdly , in Venus is seen the new , full , and quartill increase ; as in the Moon by her separation from the Sunne : fourthly , the artificiall structure of this instrument helpeth us to see an innumerable number of stars , which otherwise are obscured , by reason of the naturall weaknesse of our sight , yea the starres in via lactea are seen most apparantly ; where there seem no starres to be , this Instrument makes apparantly to be seen , and further delivers them to the eye in their true and lively colour , as they are in the heavens : in which the splendor of some is as the Sunne in his most glorious beauty . This Glasse hath also a most excellent use in observing the body of the Moone in time of Eclipses , for it augments it manifold , and most manifestly shewes the true forme of the cloudy substance in the Sunne ; and by it is seene when the shadow of the earth begins to eclipse the Moon , & when totally she is over shadowed : besides the celestiall uses which are made of this Glasse , it hath another noble property ; it farre exceedeth the ordinary perspective Glasses , which are used to see things remote upon the earth , for as this Glasse reacheth up to the heavens and excelleth them there in his performance , so on the earth it claimeth preheminency , for the objects which are farthest remote , and most obscure , are seen plainer than those which are neere at hand , scorning as it were all small and triviall services , as leaving them to an inferiour help : great use may be made of this Glass in discovering of Ships , Armies , &c. Now the apparell or parts of this instrument or Glasse , is very meane or simple , which makes it the more admirable ( seeing it performes such great service ) having but a convex Glasse thickest in the middle , to unite and amasse the rayes , and mak the object the greater : to the augmenting the visuall Angle , as also a pipe or trunk to amasse the Species , and hinder the greatness of the light which is about it : ( to see well , the object must be well inlightened , and the eye in obscurity ; ) then there is adjoyned unto it a Glasse of a short sight to distinguish the rayes , which the other would make more confused if alone . As for the proportion of those Glasses to the Trunk , though there be certaine rules to make them , yet it is often by hazard that there is made an excellent one there being so many difficulties in the action , therefore many ought to be tryed , seeing that exact proportion , in Geometricall calculation cannot serve for diversity of sights in the observation . PROBLEM . LXVII . Of the Adamant or Magnes , and the needles touched therewith . WHo would beleeve if he saw not with his eyes , that a needle of steel being once touched with the magnes , turnes not once , not a yeare ▪ but as long as the World lasteth ; his end towards the North and South , yea though one remove it , and turne it from his position , it will come againe to his points of North and South . Who would have ever thought that a brute stone black and ill formed , touching a ring of Iron , should hang it in the aire , and that ring support a second , that to support a third , and so unto 10 , 12 , or more , according to the strength of the magnes ; making as it vvere a chaine without a line , without souldering together , or without any other thing to support them onely ; but a most occult and hidden vertue , yet most evident in this effect , which penetrateth insensibly from the first to the second , from the second to the third , &c. What is there in the world that is more capable to cast a deeper astonishment in our minds than a great massie substance of Iron to hang in the aire in the middest of a building without any thing in the world touching it , only but the aire ? As some histories assure us , that by the aid of a Magnes or Adamant , placed at the roof of one of the Turkish Synagogues in Meca : the sepulchre of that infamous Mah●met rests suspended in the aire ; and Plinie in his naturall Historie writes that the Architect or Democrates did begin to vault the Temple of A●sin●e in Alexandria , with store of magnes to produce the like deceit , to hang the sepulchre of that Goddesse likewise in the aire . I should passe the bounds of my counterpoise , if I should divulge all the secrets of this stone , and should expose my selfe to the laughter of the world : if I should brag to shew others the cause how this appeareth , than in its owne naturall sympathy , for why is it that a magnes with one end will cast the Iron away , & attract it with the other ? from whence commeth it that all the magnes is not proper to give a true touch to the needle , but only in the two Poles of the stone : which is known by hanging the stone by a threed in the aire untill it be quiet , or placed upon a peece of Cork in a dish of water , or upon some thinne board , for the Pole of the stone will then turne towards the Poles of the world , and point out the North and South , and so shew by which of these ends the needle is to be touched ? From whence comes it that there is a variation in the needle , and pointeth not out truly the North and South of the world , but only in some place of the earth ? How is it that the needle made with pegges and inclosed within two Glasses , sheweth the height of the Pole , being elevated as many degrees as the Pole is above the Horizon ? What 's the cause that fire and Garlick takes away the propertie of the magnes ? There are many great hidden mysteries in this stone , which have troubled the heads of the most learned in all ages ; and to this time the world remaines ignorant of declaring the rrue cause thereof . Some say , that by help of the Magnes persons which are absent may know each others minde , as if one being here at London , and another at Prague in Germany : if each of them had a needle touched with one magnes , then the vertue is such that in the same time that the needle which is at Prague shall move , this that is at London shall also ; provided that the parties have like secret notes or alphabets , and the observation be at a set houre of the day or night ; and when the one party will declare unto the other , then let that party move the needle to these letters which will declare the matter to the other , and the moving of the other parties needle shall open his intention . The invention is subtile , but I doubt whether in the world there can be found so great a stone ▪ or such a Magnes which carries with it such vertue : neither is it expedient , for treasons would be then too frequent and open . EXAMINATION . THe experimentall difference of rejection , and attraction proceeds not from the different nature of Stones , but from the quality of the Iron ; and the vertue of the stone consisteth only , and especially in his poles , which being hanged in the Aire , turnes one of his ends alwayes naturally towards the South , and the other towards the North : but if a rod of Iron be touched with one of the ends thereof , it hath the like property in turning North and South , as the magnes hath : notwithstanding the end of the Iron Rod touched , hath a contrary position , to that end of the stone that touched it ; yet the same end will attract it , and the other end reject it : and so contrarily this may easily be experimented upon two needles touched with one or different stones , though they have one and the same position ; for as you come unto them apply one end of the magnes neare unto them , the North of the one will abhorre the North of the other , but the North of the one will alwayes approach to the South of the other : and the same affection is in the stones themselves . For the finding of the Poles of the magnes , it may be done by holding a small needle between your fingers softly , and so moving it from part to part over the stone untill it be held perpendicular , for that shall be one of the Poles of the stone which you may marke out ; in like manner finde out the other Pole : Now to finde out which of those Poles is North or South , place a needle being touched with one of the Poles upon a smooth convex body , ( as the naile of ones finger or such like , ) and marke which way the end of the needle that was touched turneth : if to the South , then the point that touched it was the South-Pole , &c. and it is most certain and according to reason and experience : that if it be suspended in aequilibrio in the aire , or supported upon the water , it will turne contrary to the needle that toucheth it ; for then the pole that was marked for the South shall turne to the North , &c. PROBLEM . LXVIII . Of the properties of Aeolipiles or bowels to blow the fire . THese are concave vessels of Brass or Copper or other material , which may indure the , fire : having a small hole very narrow , by which it is filled with water , then placing it to the fire , before it be hot there is no effect seen ; but assoone as the heat doth penetrate it , the water begins to rarefie , & issueth forth with a hidious and marvelous force ; it is pleasure to see how it blowes the fire with great noise . Novv touching the forme of these vessels , they are not made of one like fashion : some makes them like a bovvle , some like a head painted representing the vvinde , some make them like a Peare : as though one vvould put it to rost at the fire , vvhen one vvould have it to blovv , for the taile of it is hollovv , in forme of a funnell , having at the top a very little hole no greater than the head of a pinne . Some do accustome to put vvithin the Aeolipile a crooked funnell of many foldings , to the end that the vvinde that impetuously rolles ▪ to and fro vvithin , may imitate the noise of thunder . Others content themselves vvith a simple funnell placed right upvvard , somevvhat vvider at the top than elsevvhere like a Cone , vvhose basis is the mouth of the funnell : and there may be placed a bovvle of Iron or Brasse , vvhich by the vapours that are cast out vvill cause it to leap up , and dance over the mouth of the Aeolipile . Lastly , some apply near to the hole smal Windmils , or such like , vvhich easily turne by reason of the vapours ; or by help of tvvo or more bovved funnels , a bowle may be made to turne● these Aeolipiles are of excellent use for the melting of mettalls and such like . Now it is cunning and subtiltie to fill one of these Aeolipiles with water at so little a hole , and therefore requires the knowledge of a Philosopher to finde it out : and the way is thus . Heat the Aeolipiles being empty , and the aire which is within it will become extreamely rarefied ; then being thus hot throw it into water , and the aire will begin to be condensed : by which meanes it will occupie lesse roome , therefore the water will immediately enter in at the hole to avoide vacuitie : thus you have some practicall speculation upon the Aeolipile . PROBLEM . LXIX . Of the Thermometer : or an instrument to measure the degrees of heat and cold in the aire . THis Instrument is like a Cylindricall pipe of Glasse , which hath a little ball or bowle at the toppe ▪ the small end of which is placed into a vessell of water below , as by the figure may be seene . Then put some coloured liquor into the Cylindricall glasse , as blew , red , yellow , green , or such like : such as is not thick . This being done the use may be thus . Those that will determine this change by numbers and degrees , may draw a line upon the Cylinder of the Thermometer ; and divide it into 4 degrees , according to the ancient Philosophers , or into 4 degrees according to the Physicians , dividing each of these 8 into 8 others : to have in all 64 divisions , & by this vvay they may not only distinguish upon vvhat degree the vvater ascendeth in the morning , at midday , & at any other houre : but also one may knovv hovv much one day is hotter or colder than another : by marking hovv many degrees the vvater ascendeth or descendeth , one may compare the hottest and coldest dayes in a vvhole year together vvith these of another year : againe one may knovv hovv much hotter one roome is than another , by vvhich also one might keep a chamber , a furnace , a stove , &c. alvvayes in an equalitie of heat , by making the vvater of the Thermometer rest alvvayes upon one & the same degree : in brief , one may judge in some measure the burning of Fevers , and neare unto what extension the aire can be rarefied by the greatest heat . Many make use of these glasses to judge of the vveather : for it is observed that if the vvater fall in 3 or 4 hours a degree or thereabout , that raine insueth ; and the vvater vvill stand at that stay , untill the vveather change : marke the water at your going to bed , for if in the morning it hath descended raine followeth , but if it be mounted higher , it argueth faire weather : so in very cold weather , if it fall suddenly , it is snow or some sleekey weather that wiil insue , PROBLEM . LXX . Of the proportion of humane bodies of statues , of Colossus or huge images , and of monstrous Giants . PYthagoras had reason to say that man is the measure of all things . First , because he is the most perfect amongst all bodily creatures , & according to the Maxime of Philosophers , that which is most perfect and the first in rank , measureth all the rest . Secondly , because in effect the ordinary measure of a foot , the inch , the cubit , the pace , have taken their names and greatnesse from humane bodies . Thirdly , because the symmetrie and concordancie of the parts is so admirable , that all workes which are well proportionable , as namely the building of Temples , of Shippes , of Pillars , and such like pieces of Architecture , are in some measure fashioned and composed after his proportion . And we know that the Arke of Noah built by the commandement of God , was in length 300 Cubits , in breadth 50 Cubits , in height or depth 30 cubits , so that the length containes the breadth 6 times , and 10 times the depth : now a man being measured you will finde him to have the same proportion in length , breadth , and depth . Vilalpandus treating of the Temple of Solomon ( that chieftaine of works ) was modulated all of good Architecture , and curiously to be observed in many pieces to keep the same proportion as the body to his parts : so that by the greatnesse of the work and proportionable symmetrie , some dare assure themselves that by knowledge of one onely part of that building , one might know all the measures of that goodly structure . Some Architects say that the foundation of houses , and basis of columnes , are as the foot ; the top , and roofe as the head ; the rest as the body : those which have beene somewhat more curious , have noted that as in humane bodies , the parts are uniforme , as the nose , the mouth , &c. these which are double are put on one side or other , with a perfect equality in the same Architecture . In like manner , some have been yet more curious than solid ; comparing all the ornaments of a Corinth to the parts of the face , as the brow , the eyes , the nose , the mouth ; the rounding of Pillars , to the vvrithing of haire , the channells of columnes , to the fouldings of vvomens Robes , &c. Novv building being a vvork of the best Artist , there is much reason vvhy man ought to make his imitation from the chiefe vvork of nature ; vvhich is man. Hence it is that Vitru●ius in his third book , and all the best Architectes , treate of the proportion of man ; amongst others Albert Durens hath made a whole book of the measures of mans body , from the foot to the head , let them read it who wil , they may have a prefect knowledge thereof : But I will content my selfe and it may satisfie some with that which followeth . First , the length of a man well made , which commonly is called height , is equall to the distance from one end of his finger to the other : when the armes are extended as wide as they may be . Secondly , if a man have his feet and hands extended or stretched in forme of S. Andrews Crosse , placing one foot of a paire of Compasses upon his navill , one may describe a circle which will passe by the ends of his hands and feet , and drawing lines by the termes of the hands and feet , you have a square within a circle . Thirdly , the breadth of man , or the space which is from one side to another ; the breast , the head , and the neck , make the 6 part of all the body taken in length or height . Fourthly , the length of the face is equall to the length of the hand , taken from the small of the arme , unto the extremity of the longest finger . Fiftly , the thicknesse of the body taken from the belly to the back ; the one or the other is the tenth part of the whole body , or as some will have it , the ninth part , little lesse . Sixtly , the height of the brow , the length of the nose , the space between the nose and the chinne , the length of the eares , the greatnesse of the thumbe , are perfectly equall one to the other . What would you say to make an admirable report of the other parts , if I should reckon them in their least ? but in that I desire to be excused , and will rather extract some conclusion upon ▪ that which is delivered . In the first place , knowing the proportion of a man , it is easie to Painters , Image-makers , &c. perfectly to proportionate their work ; and by the same is made most evident , that which is related of the images and statues of Greece , that upon a day diverse workmen having enterprised to make the face of a man , being severed one from another in sundry places , all the parts being made and put together , the face was found in a most lively and true proportion . Secondly , it is a thing most cleare , that by the help of proportion , the body of Hercules was measured by the knowledge of his foot onely , a Lion by his claw , the Giant by his thumb , and a man by any part of his body . For so it was that Pythagoras having measured the length of Hercules foot , by the steps which were left upon the ground , found out all his height : and so it was that Phidias having onely the claw of a Lion , did figure and draw out all the beast according to his true type or forme , so the exquisite Painter Timantes , having painted a Pygmey or Dwarfe , which he measured with a fadome made with the inch of a Giant , it was sufficient to know the greatnesse of that Giant - To be short , we may by like methode come easily to the knowledge of many fine antiquities touching Statues , Colossus , and monstrous Giants , onely supposing one had found but one only part of them , as the head , the hand , the foot or some bone mentioned in ancient Histories . Of Statues , of Colossus , or huge images . VItruvius relates in his second book , that the Architect Dinocrates was desirous to put out to the world some notable thing , went to Alexander the great , and proposed unto him a high and speciall piece of work which he had projected : as to figure out the mount Athos in forme of a great Statue , which should hold in his right hand a Towne capable to receive ten thousand men : and in his left hand a vessell to receive all the water that floweth from the Mountaine , which with an ingine should cast into the Sea. This is a pretty project , said Alexander , but because there was not field-roome thereabout to nourish and reteine the Citizens of that place , Alexander was wise not to entertaine the designe . Now let it be required of what greatnesse this Statue might have been , the Towne in his right hand , and the receiver of water in his left hand if it had been made . For the Statue , it could not be higher than the Mountaine it selfe , and the Mountaine was about a mile in height plumb or perpendicular ; therefore the hand of this Statue ought to be the 10th part of his height , which would be 500 foot , and so the breadth of his hand would be 250 foot , the length now multiplyed by the breadth , makes an hundred twenty five thousand square feet , for the quantitie of his hand to make the towne in , to lodge the said 10 thousand men , allowing to each man neere about 12 foot of square ground : now judge the capacitie of the other parts of this Collossus by that which is already delivered . Secondly , Plinie in his 34 book of his natural History , speakes of the famous Colossus that was at Rhodes , between whose legges a Shippe might passe with his sailes open or displayed , the Statue being of 70 cubits high : and other Histories report that the Sarasens having broken it , did load 900 Camels with the mettal of it , now what might be the greatnesse and weight of this Statue ? For answer , it is usually allowed for a Camels burthen 1200 pound weight , therefore all the Collosus did weigh 1080000 pound weight , which is ten hundred and fourescore thousand pound vveight . Novv according to the former rules , the head being the tenth part of the body , this Statues head should be of 7 cubits , that is to say , 10 foot and a halfe , and seeing that the Nose , the brovv , and the thumbe , are the third part of the face , his Nose vvas 3 foot and a halfe long , and so much also vvas his thumbe in length : novv the thicknesse being alvvayes the third part of the length , it should seem that his thumb was a foot thick at the least . Thirdly , the said Plinie in the same place reports that Nero did cause to come out of France into Italy , a brave and bold Statue-maker called Zenodocus , to erect him a Colossus of brasse , which was made of 120 foot in height , which Nero caused to be painted in the same height . Now would you know the greatnesse of the members of this Colossus , the breadth would be 20 foot , his face 12 foote , his thumb and his nose 4 foot , according to the proportion before delivered . Thus I have a faire field or subject to extend my selfe upon , but it is upon another occasion that it was undertaken , let us speak therefore a word touching the Giants , and then passe away to the matter . Of monstrous Giants . YOu will hardly beleeve all that which I say touching this , neither will I beleeve all that which Authors say upon this subject : notwithstanding you nor I cannot deny but that long ago there have been men of a most prodigious greatnesse ; for the holy vvritings vvitnesse this themselves in Deut , Chap. 3. that there vvas a certaine Giant called Og , of the Town of Rabath , vvho had a bed of Iron , the length thereof vvas 9 cubits , and in breadth 4 cubits . So in the first of Kings Chap. 17. there is mention made of Goliah , vvhose height vvas a palme and 6 cubits , that is more then 9 foot , he was armed from the head to the foot , and his Curiat onely with the Iron of his lance , weighed five thousand and six hundred shekels , which in our common weight , is more than 233 pound , of 12 ounces to the pound : Now it is certaine , that the rest of his armes taking his Target , Helmet , Bracelets , and other Armour together , did weigh at the least 5 hundred pound , a thing prodigious ; seeing that the strongest man that now is , can hardly beare 200 pound , yet this Giant carries this as a vesture without paine . Solinus reporteth in his 5 Chap. of his Historie , that during the Grecians warre after a great overflowing of the Rivers , there was found upon the sands the carcase of a man , whose length was 33 Cubits , ( that is 49 foot and a halfe ) therefore according to the proportion delivered , his face should be 5 foot in length , a thing prodigious and monstrous . Plinie in his 7. book and 16 Chap. saith , that in the Isle of Crete or Candie , a mountaine being cloven by an Earth-quake , there was a body standing upright , which had 46 Cubits of height : some beleeve that it was the body of Orion or Othus , ( but I think rather it was some Ghost or some delusion ) whose hand should have beene 7 foot , and his nose two foot and a half long . But that which Plutarch in the l●fe of Sertorius reports of , is more strange , who saith , that in Timgy a Morative Towne , where it is thought that the Giant Antheus was buried , Sertorius could not beleeve that which was reported of his prodigious greatnesse , caused his sepulchre to be opened , and found that his body did containe 60 Cubits in length , then by proportion he should be 10 Cubits or 15 foot in breadth ; 9 foot for the length of his face , 3 foot for his thumb , which is neare the capacitie of the Colossus at Rhodes . But behold here a fine fable of Symphoris Campesius , in his book intituled Hortus Gallicus , who sayes that in the Kingdome of Sicilie , at the foot of a mountaine neare Trepane , in opening the foundation of a house , they found a Cave in which was ●aid a Giant , which held in stead of a staffe a great post like the mast of a Ship : and going to handle it , it mouldered all into ashes , except the bones which remained of an exceeding great measure , that in his head there might be easily placed 5 quarters of corn , and by proportion it should seeme that his length was 200 cubits , or 300 foot : if he had said that he had been 300 cubits in length , then he might have made us beleeve that Noahs Ark was but great enough for his sepulchre . Who can believe that any man ever had 20 cubits , or 30 foot in length for his face , and a nose of 10 foot long ? but it is very certaine that there have been men of very great stature , as the holy Scriptures before witnesse , and many Authours worthy of beliefe relate : Josephus Acosta in his first book of the Indian History , Chap. 19. a late writer , reporteth , that at Peru was found the bones of a Giant , which was 3 times greater than these of ours are , that is 18 foot , for it is usually attributed to the tallest ordinary man in these our times but 6 foot of length ; and Histories are full of the description of other Giants of 9 , 10 , and 12 foot of height , and it hath been seen in our times some which have had such heights as these . PROBLEM . LXXI . Of the game at the Palme , at Trap , at Bowles , Paile-maile , and others . THe Mathematickes often findeth place in sundry Games to aid and assist the Gamesters , though not unknowne unto them , hence by Mathematicall principles , the games at Tennis may be assisted , for all the moving in it is by right lines and reflections . From whence comes it , that from the appearances of flat or convex Glasses , the production and reflection of the species are explained ; is it not by right lines ? in the same proportion one might sufficiently deliver the motion of a Ball or Bowle by Geometrical lines and angles . And the first maxime is thus : When a Bowle toucheth another Bowle ▪ or when a trapstick striketh the Ball , the moving of the Ball is made in a right line , which is drawne from the Centre of the Bowle by the point of contingencie . Secondly , in all kinde of such motion ; when a Ball or Bowle rebounds , be it either against wood , a wall , upon a Drumme , a pavement , or upon a Racket ; the incident Angle is alwayes equall to the Angle of reflection . Now following these maximes , it is easie to canclude , first , in what part of the wood or wall , one may make the Bowle or Ball go to reflect or rebound , to such a place as one would . Secondly , how one may cast a Bowle upon another , in such sort that the first or the second shall go and meet with the third , keeping the reflection or Angle of incidence equal . Thirly , how one may touch a Bowle to send it to what part one pleaseth : such and many other practices may be done . At the exercises at Keyls there must be taken heed that the motion slack or diminish by little and little , and may be noted that the Maximes of reflections cannot be exactly observed by locall motion , as in the beames of light and of other quallities , whereof it is necessary to supply it by industry or by strength , otherwise one may be frustrated in that respect . PROBLEM . LXXII . Of the Game of square formes . NVmbers have an admirable secrecie , diversly applied , as before in part is shewed , and here I will say something by way of transmutation of numbers . It 's answered thus , in the first forme the men were as the figure A , then each of these 4 Souldiers placed themselves at each Gate , and removing one man from each Angle to each Gate , then would they be also 9 in each side according to the figure B. Lastly , these 4 Souldiers at the Gates take away each one his Cumrade , and placing two of these men which are at each Gate to each Angle , there will be still 9 for each side of the square , according to the figure C. In like manner if there were 12 men , how might they be placed about a square that the first side shall have 3 every way , then disordered , so that they might be 4 every way ; and lastly , being transported might make 5 every way ? & this is according to the figures , F. G.H PROBLEM . LXXIII . How to make the string of a Viole sensibly shake , without any one touching it ? THis is a miracle in musick , yet easie to be experimented . Take a Viole or other Instrument , and choose two strings , so that there be one between them ; make these two strings , agree in one and the same tune : then move the Viole-bowe upon the greater string , and you shall see a wonder : for in the same time that that shakes which you play upon , the other will likewise sensibly shake without any one touching it ; and it is more admirable that the string which is between them will not shake at all : and if you put the first string to another tune or note , and loosing the pin of the string , or stopping it with your finger in any fret , the other string will not shake : and the same will happen if you take two Violes , and strike upon a string of the one , the string of the other will sensibly shake . Now it may be demanded , how comes this shaking , is it in the occult sympathie , or is it in the strings being wound up to like notes or tunes , that so easily the other may receive the impression of the aire , which is agitated or moved by the shaking or the trembling of the other ? & whence is it that the Viole-bowe moved upon the first string , doth instantly in the same time move the third string , and not the second ? if the cause be not either in the first or second ? I leave to others to descant on . EXAMINATION . IN this Examination we have something else to imagine , than the bare sympathie of the Cords one to another : for first there ought to be considered the different effect that it produceth by extention upon one and the same Cord in capacitie : then what might be produced upon different Cords of length and bigness to make them accord in a unisone or octavo , or some consort intermediate : this being naturally examined , it will be facill to lay open a way to the knowledge of the true and immediate cause of this noble and admirable Phaenomeny . Now this will sensibly appeare when the Cords are of equall length and greatnesse , and set to an unisone ; but when the Cords differ from their equalitie , it will be lesse sensible : hence in one and the same Instrument , Cords at a unisone shall excite or shake more than that which is at an octavo , and more than those which are of an intermediate proportionall consort : as for the other consorts they are not exempted , though the effect be not so sensible , yet more in one than in another : and the experiment will seem more admirable in taking 2 Lutes , Viols , &c. & in setting them to one tune : for then in touching the Cord of the one , it will give a sensible motion to the Cord of the other : and not onely so but also a harmony . PROBLEM . LXXIIII . Of a vessell which containes three severall kindes of liquor , all put in at one bung-hole , and drawn out at one tap severally without mixture . THe vessell is thus made , it must be divided into three Cells for to conteine the three liquors , which admit to be Sack , Claret , and White-wine : Now in the bung-hole there is an Engine with three pipes , each extending to his proper Cell , into which there is put a broach or funnell pierced in three places , in such sort , that placing one of the holes right against the pipe which answereth unto him , the other tvvo pipes are stopped ; then vvhen it is full , turne the funnel , and then the former hole vvill be stopped , and another open , to cast in other vvine vvithout mixing it vvith the other . Novv to dravv out also vvithout mixture , at the bottome of the vessell there must be placed a pipe or broach , vvhich may have three pipes ; and a cock piersed vvith three holes so artificially done , that turning the cock , the whole vvhich ansvvereth to such of the pipes that is placed at the bottom , may issue forth such vvine as belongeth to that pipe , & turning the Cock to another pipe , the former hole vvil be stopped ; and so there will issue forth another kinde of wine without any mixtures ; but the Cocke may be so ordered that there may come out by it two wines together , or all three kindes at once : but it seems best when that in one vessell and at one Cocke , a man may draw severall kindes of wine , and which he pleaseth to drink . PROBLEM . LXXV . Of burning-Glasses . IN this insuing discourse I will shew the invention of Prom●theus , how to steale fire from Heaven , and bring it down to the Earth ; this is done by a little round Glasse , or made of steele , by which one may light a Candle , and make it flame , kindle Fire-brands to wake them burne , melt Lead , ●inne , Gold , and Silver , in a little time ▪ with as great ease as though it had been put into a Cruzet over a great fire . But this is nothing to the burning of those Glasses which are hollow , namely those which are of steele well polished , according to a par●bolicall or ovall section . A sphericall Glasse , or that which is according to the segment of a Sphere , burnes very effectually about the fourth part of the Diameter ; notwithstanding the Parabolie and Ecliptick sections have a great effect : by which Glasses there are also diverse figures represented forth to the eye . The cause of this burning is the uniting of the beames of the Sunne , which heat mightily in the point of concourse or inflammation , which is either by transmissi●n or reflection ▪ Now it is pleasant to behold when one breatheth in the point of concourse , or throweth small dust there , or sprinkles vapours of hot water in that place ; by which the Pyramidall point , or point of inflammation is knowne . Now some Authors promise to make Glasses which shall burne a great distance off , but yet not seen vulgarly produced , of which if they were made , the Parabolie makes the greatest eff●ct , and is g●nerally held to be the invention of Archimedes or Pro●●us . Maginus in the 5 Chap. of his Treatise of sphericall Glasses , shewes how one may serve himselfe with a concave Glasse , to light fire in the shadow , or neare such a place where the Sunne shines not , which is by help of a flat Glasse , by which may be made a percussion of the beames of the Sun into the concave Glasse , adding unto it that it serves to good use to put fi●e to a Mine , provided that the combustible matter be well applyed before the concave Glasse ; in which he saies true : but because all the effect of the practice depends upon the placing of the Glasse and the Powder which he speaks not of : I will deliver here a rule more generall . How one may place a Burning-glasse with his combust●ble matter in such sort , that at a convenient houre of the day , the Sun shining , it shall take fire and burne : Now it is certaine that the point of inflammation or burning , is changed as the Sun changeth place , and no more nor lesse , than the shadow turnes about the style of a Dyall ; therefore have regard to the Suns motion , and ●is height and place : a Bowle of Crystall in the same place that the top of the style is , and the Powder or other combustible matter under the Meridian , or houre of 12 , 1 , 2 , 3 , &c. or any other houre , and under the Suns arch for that day : now the Sunne comming to the houre of 12 , to ● , 2 , ● , &c. the Sunne casting his beames through the Crystall Bowle , will fire the materiall or combustible thing , which meets in the point of burning : the like may be observed of other Burning-glasses . EXAMINATION . IT is certaine in the first part of this Probleme that Conicall , ●oncave and sphericall Glasses , of what matter soever , being placed to receive the beames of the Sun will excite heat , and that heat is so much the greater , by how much it is neere the point of conc●rse or inflamatio● . But that Archimedes or Proclus d●d fire or burne Shipps with such Glasses , the ancient Histories are silent , yea the selves say nothing : besides the great difficultie that doth oppose it in remotenesse , and the matter that the effect is to work upon : Now by a common Glasse we fire things neare at hand , from which it seems very facil to such which are lesse read , to do it at a farre greater distance , and so by re●ation some deliver to the World by supposition that which never was done in action : this we say the rather , not to take away the most excellent and admirable effects which are in Burning-glasses , but to shew the variety of Antiquity , and truth of History : and as touching to burne at a great distance , as is said of some , it is absolutely impossible ; and that the Parabolicall and Ovall Glasses were of Archimedes and ●roclus invention is much uncertaine : for besides the construction of such Glasses , they are more difficult than the obtuse concave ones are ; and further , they cast not a great heat but neere at hand ; for if it be cast farre off , the effect is little , and the heat weake , or otherwise such Glasses must be greatly extended to contract many beames to amasse a sufficient quantity of beames in Parabolicall and Conicall Glasses , the point of inflammation ought to concur in a point , which is very difficult to be done in a due proportion . Moreover if the place be farre remote , as is supposed before , such a Glasse cannot be used but at a great inclination of the Sunne ▪ by which the eff●ct of ●urning is d●min●shed , by reason of the weaknesse of the Sunne-beames . And here may be noted in the last part of this Probleme , that by r●ason of obstacles if one plaine Glasse be not sufficient , a second Glasse may be applyed to help it : that so if by one simple reflection it cannot be done , yet by a double reflection the Sun-beames may be ●ast into the said Caverne or Mine , and though the reflected beams in this case be weak ▪ yet upon a 〈◊〉 c●mbustible matter it will not faile to do the effect . PROBLEM . LXXVI . Containing m●ny ple●sant Questions by way of Arithmetick● . J Will not in●ert i● this Probleme that which is drawne from the ●reek Epigrams , but proposing the Question immediately will give the an●wer also , without ●●aying to shew the manner how they are answered ; in this J will 〈◊〉 be tied to the ●reek tearms , w●●ch J account no● proper to this place , nei●●er to my purpose : ●et t●o●e ●ead that will Di●phanta S●●●●biliu● upon Eu●li●● and others , and they may be satisf●ed Of the 〈…〉 the Mule. JT 〈◊〉 ●hat ●he Mule and the Asse upon a day 〈◊〉 a voyage each of them carried a Barrell full of Wine : now the las●e Asse f●lt her selfe over-loaden , complained and bowed under her burthen ; which th● Mule seeing said unto her being angry , ( for it was in the time when beasts spake ) Thou great Asse , wherefore complainest thou ? if I had but onely one measure of that which thou carriest , I should be loaden twice as much as thou art , and if J should give a measure of my loading to thee , yet my burthen would be as much as thine . Now how many measures did each of them carry ? Answer , the Mule did carry 7 measures , and the Asse 5 measures : for if the Mule had one of the measures of the Asses loading , then the Mule would have 8 measures , which is double to 4 , and giving one to the Asse , each of them would have equall burthens : to wit , 6 measures apiece . Of the number of Souldiers that fought before old Troy. HOmer being asked by He●iodus how many Grecian Souldiers came against Troy ? who answered him thus ; The Grecians , said Homer , made 7 fires , or had 7 Kitchins , and before every fire , or in every Kitchin there were 50 broaches turning to rost a great quantitie of flesh , and each broach had meat enough to satisfie 900 men : now judge how many men there might be . Answer , 315000. that is , three hundred and fifteen thousand men , which is cleare by multiplying 7 by 50 , and the product by 900 makes the said 315000. Of the number of Crownes that two men had . JOhn and Peter had certaine number of crowns : John said to Peter , If you give me 10 of your crownes , I shall have three times as much as you have : but Peter said to J●hn , If you give me 10 of your crownes I shall have 5 times as much as you have : how much had each of them ? Answere , John had 15 crownes and 5 sevenths of a crowne , and Peter had 18 crownes , and 4 sevenths of a crowne . For if you adde 10 of Peters crownes to those of Johns , then should John have 25 crownes and 5 sevenths of a crowne , which is triple to that of Peters , viz. 8 ▪ and 4 sevenths : and John giving 10 to Peter , Peter should have then 28 crownes , and 4 sevenths of a crowne , which is Quintupla , or 5 times as much as John had left , viz. 5 crownes and 5 sevenths . In like manner two Gamesters playing together , A and B ▪ after play A said to B , Give me 2 crownes of thy money , and I shall have twice as much as thou hast : and B said to A , Give me 2 crownes of thy money , and I shall have 4 times as much as thou hast : now how much had each ? Answer , A had 3 and 5 seventhes , and B had 4 and 6 seventhes . About the houre of the day . SOme one asked a Mathemacian what a clocke it was ; who answered that the rest of the day is foure thirds of that which is past : now judge what a clock it is . Answer , if the day were according to the Jewes and ancient Romanes , which ma●e it alwayes to be 12 houres , it was then the ● houre , and one seventh of an hou●e , so there remained of the whole day 6 , that is , 6 houres , and 6 sevenths of an hour . Now if you take the 1 / ● of 5 ● / 7 it is ●2 / 7 or ● and ● 7 , which multipled by 4 makes 6 and 6 / 7 , which is the remainder of the day , as before : but if the day had been 24 houres , then the houre had been 10 of the clock ▪ and two seventhes of an houre , which is found ▪ out by dividing 12 , or 24 by ● . There might have been added many curious propositions in this kinde , but they vvould be too difficult for the most part of people ▪ therefore I have omi●ted them ▪ Of Pythagoras his Schollers . PYthagoras being asked what number of Schollers he had , ansvvered , that halfe of them studied Mathematickes , the fourth part Physick , the seventh part Rethorick , and besides he had 3 vvomen : novv judge you saith he , hovv many Schollers I have . Ansvver , he had in all 28 , the halfe of vvhich is 14 , the quarter of which is 7 , and the seventh part of which is which 14 , 7 , and 4 , makes 25 , and the other 3 to make up the 28 , were the 3 women . Of the number of Apples given amongst the Graces and the Muses . THe three Graces carrying Apples upon a day , the one as many as the other , met with the 9 Muses , who asked of them some of their Apples ; so each of the Graces gave to each of the Muses alike , and the distribution being made , they found that the Graces & the Muses had one as many as the other : The question is how many Apples each Grace had , and how many they gave to each Muse ? ●o ansvver the qeustion , joyne the number of Graces and Muses together vvhich makes 12 , and so many Apples had each Grace : Novv may you take the double , triple , &c. of 12 that is 24 , 36 , &c. conditionally , that if each Grace had but 12 , then may there be allotted to each Muse but one onely ; if 24 , then to each 2 Apples , if ●6 , then to each Muse 3 Apples , and so the distribution being made , they have a like number , that is one as many as the other . Of the Testament or last Will of a dying Father . A Dying Father left a thousand Crovvnes amongst his tvvo children ; the one being legitimate , and the other a Bastard , conditionally that the fifth part which his legittimate Sonne should have , should exceed by 10 , the fourth part of that which the Bastard should have : what was each 〈◊〉 part ? Answer , the legitimate Sonne had 577 crownes and 7 / ● , and the Bastard 42● crownes and 2 / 9 now the fifth part of 577 and 7 ninthes is 1●5 , and 5 / 9 , and the fourth part of 422 and ● is 105 and ● which is lesse then ●15 ● by 10 , according to the Will of the Testator . Of the Cups of Croesus . CRoesus gave to the Temple of the ●ods six Cups of Gold ▪ which weighed together ●00 Drammes , but each cup was heavier one than another by one Dram : how much did each of them therefore weigh ? Answer , the first weighed 102 Drammes and a halfe ; the second 101 Drammes and a halfe , the third 100 Drammes and ● , the fourth 99 a & halfe , the fifth 98 & a halfe ; and the sixt Cup weighed 97 Drammes and a halfe ▪ which together makes 600 Drams as before . Of Cupids Apples . CVpid complained to his mother that the Muses had taken away his Apples , Clio , said he , took from me the fifth part , Euterp the twelfth part , Thalia the eighth part , M●lp●meno the twentieth part , Erates the seventh part ▪ Terpomene the fourth part , Polyhymnia took away 30 , Vrania 220 , and Calliope 300. so there vvere left me but 5 Appls , hovv many had he in all at the first ? I ansvver 3●60 . There are an infinite of such like questions amongst the Greek Epigrams : but it would be unpleasant to expresse them all : I will onely adde one more , and shew a generall rule for all the rest . Of a Mans Age. A Man vvas said to passe the sixth part of his life in childe-hood , the fourth part in his youth , the ●hird part in Manhood , and 18 yeares besides in old age : what might his Age be ? the ansvver is , 72 yeares : vvhich and all others is thus resolved : multiply 1 / ● ▪ ¼ and ⅓ ▪ together , that is , 6 by 4 makes 24 , and that againe by 3 makes 72 , then take the third part of 72 , vvhich is 24 , the fourth part of it , vvhich is 18 , and the sixth part of it vvhich is 12 , these added together make 54 , vvhich taken from 72 , rests 18 this divided by 18 ( spoken in the Question ) gives 1 , which multiplied by the summe of the parts , viz. 72 , makes 72 , the Ansvver as before . Of the Lion of Bronze placed upon a Fountaine with this Epigramme . OVt of my right eye if I let vvater passe , I can fill the Cisterne in 2 dayes : if I let it passe out of the left eye , it vvill be filled in 3 dayes : if it passe out of my feet , the Cistern vvill be 4 dayes a●filling ; but if I let the vvater passe out of my mouth , I can fill the Cistern then in 6 houres : in vvhat time should I fill it , if I poure forth the vvater at all the passages at once ? The Greeks ( the greatest talkers in the vvorld ) variously apply this question to divers statues , and pipes of Fountaines : and the solution is by the Rule of ● , by a generall Rule , or by ●lgebra . They have also in their Anthologie many other questions , but because they are more proper to exercise , than to recreate the spirit , I passe them over ( as before ) with silence . PROBLEM . LXXVII . Divers excellent and admirable experiments upon Glasses . THere is nothing in the world so beautifull as light : and nothing more recreative to the sight , than Glasses vvhich reflect : therefore I vvill novv produce some experiments upon them , not that vvill dive into their depth ( that vvere to lay open a mysterious thing ) but that vvhich may delight and recreate the spirits : Let us suppose therefore these principles , upon which is built the demonstration of the appar●nces which are made ●n all sort of Glasses . First , that the rayes or beames , vvhich reflect upon a Glasse , make the Angle of incident equall to the Angle of Reflection , by the first Theo. of the Catoptick of Euc. Secondly , that in all plain Glasses , the Images are seen in the perpendicular line to the Glasse , as far within the glass as the object is without it . Thirdly , in Concave , or Convex Glasses , the Images are seen in the right line which passeth from the object and through the Centre in the Glasse . Theo. 17. and 18. And here you are to understand , that there is not meant only those which are simple Glasses or Glasses of steele , but all other bodies , which may represent the visible Image of things by reason of their reflection , as Water , Marble , Mettal , or such like . Now take a Glasse in your hand and make experiment upon that which followeth . Experiment upon flat and plaine Glasses . FIrst , a man cannot see any thing in these Glasses , if he be not directly and in a perpendicular line before it , neither can he see an object in these Glasses , if it be not in such a place , that makes the Angle of incidence equall to the Angle of reflexion : therefore when a Glasse stands upright , that is , perpendicular to the Horizon , you cannot see that which is above , except the Glasse be placed down flat : and to see that on the right hand , you must be on the left hand , &c. Secondly , an image cannot be seen in a Glass if it be not raised above the surface of it ; or place a Glasse upon a wall , you shall see nothing which is upon the plaine of the wall , and place it upon a Table or Horizontal Plaine , you shall see nothing of that which is upon the Table . Thirdly , in a plaine Glasse all that is seene appeares or seemes to sink behinde the Glasse , as much as the image is before the Glasse , as before is said . Fourthly , ( as in water ) a Glasse lying downe flat , or Horizontall , Towers , Trees , Men , or any height doth appeare , inversed or upside downe ; and a Glasse placed upright , the right hand of the Jmage seems to be the left , and the left seems to be the right . Fifthly , will you see in a Chamber that which is done in the street , without being seen ▪ then a Glasse must be disposed , that the line upon which the Jmages come on the Glasse , make the Angle of incidence equall to that Angle of reflexion . Seventhly , present a Candle upon a plaine Glasse , and look flaunting upon it , so that the Candle and the Glasse be neere in a right line , you shall see 3 , 4 , 5 , &c. images , from one and the same Candle . Eightly , take tvvo plaine Glasses , and hold them one against the other , you shall alternately see them oftentimes one vvithin the other , yea vvithin themselves , againe and againe . Ninthly , if you hold a plaine Glasse behinde your head , and another before your face , you may see the h●nder part of your head , in that Glasse vvhich you hold before your face . Tenthly , you may have a fine experiment if you place tvvo Glasses together , that they make an acute angle , and so the lesser the angle is , the more apparances you shall see , the one direct , the other inversed , the one approaching , and the other retiring . Eleventhly , it is a vvonder & astonishment to some , to see within a Glasse an Image vvithout knovving from vvhence it came , and it may be done many vvayes : as place a Glass higher than the eye of the beholder , and right against it is some Image ; so it resteth not upon the beholder , but doth cast the Image upvvards . Then place another object , so that it reflect , or cast the Image downeward to the eye of the spectator ▪ without perceiving it being hid behinde something , for then the Glasse will represent a quite contrary thing , either that which is before the Glasse , or that which is about it , to wit , the other hidden object . Twelfthly , if there be ingraved behinde the backside of a Glasse , or drawne any Image upon it , it will appeare before as an Image , without any appearance : o● portraicture to be perceived . EXAMINATION . THis 12 Article of ingraving an Image behinde the Glasse , will be of no great consequence ▪ because the lineaments will seem so obscure , but if there were painted some Image , and then that covered according to the usuall covering of Glasses behinde , and so made up like an ordinary looking-Glasse having an Image in the middle , in this respect it would be sufficiently pleasant : and that which would admire the ignorant , and able to exercise the most subtillest , and that principally if the Glasse be in an obscure place , and the light which is given to it be somewhat farre off . PLace a Glasse neare the floor of a Chamber , & make a hole through the place under the Glasse , so that those which are below may not perceive it , and dispose a bright Image under the hole so that it may cast his species upon the Glasse , and it will cause admiration to those which are below that know not the cause ; The same may be done by placing the Image in a Chamber adjoyning , and so make it to be seen upon the side of the Wall. 14 In these Channel-Images which shew one side a deaths head , & another side a faire face : and right before some other thing : it is a thing evident , that setting a plaine Glasse sidewise to this Image you shall see it in a contrary thing , then that which was presented before sidewise . 15 Lastly , it is a fine secret to present unto a plaine Glasse writing with such industry , that one may read it in the Glasse , and yet out of the Glasse there is nothing to be known , which will thus happen , if the writing be writ backward : but that which is more strange , to shew a kinde of writing to a plaine Glasse , it shall appear another kinde of writing both against sense and forme , as if there were presented to the Glasse WEL it would shew it MET ; if it were written thus MIV , and presented to the Glasse , it would appeare thus VIM ; for in the first , if the Glasse ly flat , then the things are inversed that are perpendicular to the Glass , if the Glass and the object be upright , then that on the right hand , is turned to the left , as in the latter . And here I cease to speak further of these plaine Glasses , either of the Admirable multiplications , or appearances , which is made in a great number of them ; for to content the sight in this particular , one must have recourse to the Cabinets of great Personages who inrich themselves with most beautifull ones . Experiments upon Gibbous , or convex Sphericall Glasses . IF they be in the forme of a Bowle , or part of a great Globe of Glasse , there is singular contentment to contemplate on them . First , because they present the objects lesse and more gracious , and by hovv much more the Images are separated from the Glasse , by so much the more they diminish in Magnitude . Secondly , they that shew the Images plaiting , or foulding , which is very pleasant , especially when the Glasse is placed downe , and behold in it some Blanching , feeling , &c. The upper part of a Gallerie , the porch of a Hall , &c. for they will be represented as a great vessel having more belly in the middle then at the two ends , and Posts , and Joists of Timber will seeme as Circles . Thirdly , that which ravisheth the spirits , by the eye , and which shames the best perspective Painting that a Painter can make , is the beautifull contraction of the Images , that appeare within the sphericity of these small Glasses : for present the Glasse to the lower end of a Gallarie , or at the Corner of a great Court full of People , or towards a great street , Church , fortification , an Army of men , to a whole Cittie ; all the faire Architecture , and appearances will be seene contracted within the circuit of the Glasse with such varietie of Colours , and distinctions in the lesser parts , that I know not in the world what is more agreeable to the sight , and pleasant to behold , in which you will not have an exact proportion , but it will be variable , according to the distance of the Object from the Glasse . Exptriments upon hollow , or Concave sphericall Glasses . I Have heretofore spoken how they may burne , being made of Glasse , or Metall , it remaines now that I deliver some pleasant uses of them , which they represent unto our sight , and so much the more notable it will be , by how much the greater the Glasse is , and the Globe from whence it is extracted for it must in proportion as a segment of some be made circle or orbe . EXAMINATION . IN this we may observe that a section of 2.3 . or 4. Inches in diameter , may be segments of spheres of 2.3 . or 4. foot ● nay of so many fadome , for it is certaine that amongst those which comprehend a great portion of a lesser sphere , and those which comprehend a little segment of a great spheere , whether they be equall or not in section , there will happen an evident difference in one and the same experiment , in the number , situation , quantitie , and figure of the Images of one or many different objects , and in burning there is a great difference . MAginus , in a little Tractate that he had upon these Glasses , witnesseth of himselfe that he hath caused many to be polished for sundry great Lords of Italy , and Germanie , which were segments of Globes of 2.3 . and 4. foot diameter ; and I wish you had some such like to see the experiments of that which followeth ; it is not difficult to have such made , or bought here in Town , the contentment herein would beare with the cost . EXAMINATION . TOuching Maginus he hath nothing ayded us to the knowledge of the truth by his extract out of Vitellius , but left it : expecting it from others , rather than to be plunged in the search of it himselfe , affecting rather the forging of the matter , and composition of the Glasses , than Geometrically to establish their effects . FIrst therefore in concave Glasses , the Images are seene sometimes upon the surface of the Glasses , sometimes as though they were within it and behinde it , deeply sunk into it , sometimes they are seene before , and without the Glasse , sometimes between the object and the Glasse ; sometimes in the place of the Eye , sometimes farther from the Glasse then the object is : which comes to passe by reason of the divers concourse of the beames , and change of the place of the Images in the line of reflection . EXAMINATION . THe relation of these appearances passe current amongst most men , but because the curious may not receive prejudice in their experiments , something ought to be said thereof to give it a more lively touch : in the true causes of these appearances , in the first place it is impossible that the Image can be upon the surface of the Glasse , and it is a principall point to declare truly in which place the Image is seen in the Glasse those that are more learned in Opticall knowledge affirme the contrary , and nature it selfe gives it a certaine place according to its position being alwayes seen in the line of reflection which Alhazen , Vitellius , and others full of grea● knowledge , have confirmed by their writings : but in their particular they were too much occupied by the authority of the Ancients who were not s●fficiently ci●cumspect in experience upon which the principles of this sub●ect ought to be built , an● searched not fully into the true cause of these appearances , seeing they leave unto posterities many 〈◊〉 in their writings , ●nd those that followed them for the most part fell into the like errors . As for the Jmages to bid● in the eye ▪ it cannot be but is imp●rtinent and absurd ; but it followeth that , by how much neerer the ob●ect appro●cheth to the Glasse , by so much the more the appearances seem to come to the eye : and if the eye be without the point of concourse , and the object also ; as long as the object approacheth thereto , the representation of the Image cometh neere the eye , but passing the point of concourse it goes back againe : these appearances thus approaching do not a little astonish those which are ignorant of the cause : they are inversed , if the eye be without the point of concourse untill the object be within , but contrarily if the eye be between the point of concourse and the Glasse , then the Jmages are direct : and if the eye or the object be in the point of concourse , the Glasse will be enlightened and the Jmages confused , and if there were but a spark of fire in the said point of concourse , all the Glasse would seeme a burning fire-brand , and we dare say it would occurre without chance , and in the night be the most certaine and subtilest light that can be , if a candle were placed there . And whosoever shall enter into the search of the truth of new experiments in this subject without doubt he will confirme what we here speak of : & will finde new lights with a conveniable position to the Glasse , he will have reflection of quantities , of truth , and fine secrets in nature , yet not known , which he may easily comprehend if he have but an indifferent sight , and may assure himselfe that the Images cannot exceed the fight , nor trouble it , a thing too much absurd to nature . And it is an absolute verity in this science , that the eye being once placed in the line of reflection of any object , and moved in the same line : the obect is seene in one and the same place immutable ; or if the Image and the eye move in their owne lines , the representation in the Glasse seemes to invest it selfe continually with a different figure . NOw the Image comming thus to the eye , those which know not the secret , draw their sword when they see an Image thus to issue out of the Glasse , or a Pistoll which some one holds behinde : and some Glasses will shew a sword wholly drawne out , sepa●ated from the Glasse , as though it were in the aire : and it is daily exercised , that a man may touch the Image of his hand or his face out of the Glasse , which comes out the farther , by how much the Glasse is great and the Centre remote . EXAMINATION . NOw that a Pistoll being presented to a Glasse behinde a man , should come out of the Glasse , and make him afraid that stands before , seeming to shoot at him , this cannot be : for no object whatsoever presented to a concave Glasse , if it be not neerer to the G●asse then the eye is it comes not out to the sight of the party ; therefore he needs not feare that which is said to be behinde his back , and comes out of the Glasse ; for if it doth come out , it must then necessarily be before his face , so in a concave Glasse whose Centre is farre remote of a sword , stick , or such like be presented to the Glasse , it shall totally be seen to come forth of the Glasse and all the hand that holds it . And here generally note that if an Image be seen to issue out of the Glasse to come towards the face of any one that stands by , the object shall be likewise seen to thrust towards that face in the Glass and may easily be knowne to all the standers by : so many persons standing before a Glasse , if one of the company take a sword , and would make it issue forth towards any o●her that stands there : let him chuse his Image in the Gl●sse and carry the sword right towards it and the effect will follow . In like manner ones hand being presented to the Glosse as it is thrust towards the Centre , s● the representation of it comes towards it , and so the hands will seeme to be united , or to touch one another . FRom which may be concluded , if such a Glasse be placed at the seeling or planching of a Hall , so that the face be Horizontall and look downward ; one may see under it as it were a man hanging by the feet , and if there were many placed so , one could not enter into that place without great feare or scaring : for one should see many men in the aire as if they were hanging by the feet . EXAMINATION . TOuching a Glasse tyed at a seeling or planching , that one may see a man hang by the feet in the aire , and so many Glasses , many men may be seen : without caution this is very absurd for if the Glasse or Glasses be not so great that the Centre of the sphere upon which it was made , extend not neere to the head of him that is under it , it will not pleasantly appeare , and though the Glasse should be of that capacity that the Centre did extend so farre , yet will not the Images be seene to them which are from the Glasse but on●y to those which are under it , or neere unto it : and to them it will not ably appeare , and it would be most admirable to have a Gallerie vaulted over with such Glasses which would wonderfully astonish any one that enters into it : for a●l the things in the Gallery would be seen to hang in the aire , and you could not walk without incountering airie apparitions . SEcondly , in flat or plaine Glasses the Image is seen equall to his object , and to represent a whole man , there ought to be a Glasse as great as the Image is : In convex Glasses the Images are seen alwayes lesse , in concave Glasses they may be seen greater or lesser , but not truly proportionable , by reason the diverse reflexions which contracts or inlargeth the Species : when the eye is between the Centre and the surface of the Glasse ; the Image appeares sometimes very great and deformed , and those which have but the appearance of the beginning of a beard on their chinne , may cheare up themselves to see they have a great beard ; those that seeme to be faire will thrust away the Glasse with despight , because it will transforme their beauty : those that put their hand to the Glasse vvill seeme to have the hand of a Giant , and if one puts his finger to the Glasse it vvill be seen as a great Pyramide of flesh , inversed against his finger . Thirdly , it is a thing admirable that the eye being approached to the point of concourse of the Glasse , there vvill be seen nothing but an intermixture or confusion : but retiring back a little from that point , ( because the rayes do there meet ▪ ) he shall see his Image inversed , having his head belovv and his feet above . Fourthly , the divers appearances caused by the motion of objects , either retiring or approaching : whether they turne to the right hand or to the left hand , whether the Glasse be hung against a wall , or whether it be placed upon a Pavement , as also what may be represented by the mutuall aspect of concave Glasses with plaine and convex Glasses but I will with silence passe them over , only say something of two rare experiments more as followeth . The first is to represent by help of the Sun , such letters as one would upon the front of a house : so that one may read them : Maginus doth deliver the way thus . Write the Letters , saith he , sufficiently bigge , but inversed upon the surface of the Glasse , with some kinde of colour , or these letters may be written with wax , ( the easier to be taken out againe : ) for then placing the Glasse to the Sunne , the letters which are written there will be reverberated or reflected upon the Wall : hence it was perhaps that Pythagoras did promise with this invention to write upon the Moone . In the second place , how a man may sundry wayes help himselfe with such a Glasse , with a lighted Torch or Candle , placed in the point of concourse or inflammation , which is neare the fourth part of the Diameter : for by this meanes the light of the Candle will be reverberated into the Glasse , and vvill be cast back againe very farre by parrallel lines , making so great a light that one may clearly see that vvhich is done farre off , yea in the camp of an Enemie : and those which shall see the Glasse a farre off , will think they see a Silver Basin inlightened , or a fire more resplendent then the Torch . It is this way that there are made certaine Lanthorns which dazell the eyes of those which come against them ; yet it serves singular well to enlighten those which carry them , accommodating a Candle with a little hollow Glasse , so that it may successively be applyed to the point of inflammation . In like manner by this reflected light , one may reade farre off , provided that the letters be indifferent great , as an Epitaph placed high , or in a place obscure ; or the letter of a friend which dares not approach without perill or suspicion . EXAMINATION . THis will be scarce sensible upon a wall remote from the Glasse , and but indifferently seen upon a wall which is neare the Glasse , and withall it must be in obscuritie or shadowed , or else it will not be seen . To cast light in the night to a place remote , with a Candle placed in the point of concourse or inflammation , is one of the most notablest properties which can be shewne in a concave Glasse : for if in the point of inflammation of a parabolicall section , a Candle be placed , the light will be reflected by parallel lines , as a columne or Cylinder ; but in the sphericall section it is defective in part , the beames being not united in one point , but somewhat scattering : notwithstanding it casteth a very great beautifull light . Lastly , those which feare to hurt their sight by the approach of Lampes or Candles , may by this artifice place at some corher of a Chamber , a Lamp with a hollow Glasse behinde it , which will commodiously reflect the light upon a Table , or to a place assigned : so that the Glasse be somewhat raised to make the light to streeke upon the Table with sharp Angles , as the Sunne doth when it is but a little elevated above the Horizon , for this light shall exceed the light of many Candles placed in the Roome , and be more pleasant to the sight of him that useth it . Of other Glasses of pleasure . FIrst , the Columnary and Pyramidall Glasses that are contained under right lines , do represent the Images as plaine Glasses do ; and if they be bowing , then they represent the Image , as the concave and convex Glasses do . Secondly , those Glasses which are plaine , but have ascents of Angels in the middle , will shew one to have foure Eyes , two Mouthes , two Noses , &c. EXAMINATION . TH●se experiments will be found different according to the diverse meeting of the Glasses , which commonly are made scuing-wise at the end , 〈◊〉 which there will be two divers superficies in the Glasse , making the exteriour Angle somewhat raised , at the interiour onely one superficies , which may be covered according to ordinary Glasses to c●use a reflexion , and so it will be but one Glasse , which by refraction according to the different thicknesse of the Glasse , and different Angles of the scuing forme , do differently present the Images to the eye , as foure eyes , two mouthes , two noses ; sometimes three eyes one mouth , and one nose , the one large and the other long , sometimes two eyes onely : with the mouth and the nose deformed , which the Glasse ( impenetrable ) will not shew . And if there be an interiour solid Angle , according to the difference of it ( as if it be more sharp ) there will be represented two distinct double Images , that is , two entire visages and as the Angle is open , by so much the more the double Images will reunite and enter one within another , which will present sometimes a whole visage extended at large , to have foure eyes , two noses , and two mouthes : and by moving the Glasse the Angle will vanish , and so the two superficies will be turned into one , and the duplicity of Images will also vanish and appeare but one onely : and this is easily experimented with two little Glasses of steel , or such like so united , that they make divers Angles and inclinations . THirdly , there are Glasses which make men seeme pale , red , and coloured in diverse manners , which is caused by the dye of the Glasse , or the diverse refraction of the Species : and those which are made of Silver , Latine , Steele , &c. do give the Images a diverse colour also . In which one may see that the appearances by some are made fairer , younger or older than they are ; and contrarily others will make them foule and deformed : and give them a contrary visage : for if a Glasse be cut as it may be , or if many pieces of Glasse be placed together to make a conveniable reflexion : there might be made of a Mole ( as it were ) a mountaine , of one Haire a Tree , a Fly to be as an Elephant , but I should be too long if I should say all that which might be said upon the property of Glasses . I will therefore conclude this discourse of the properties of these Glasses with these foure recreative Problemes following . PROBLEM . LXXVIII . 1 How to shew to one that is suspitious , what is done in another Chamber or Roome : notwithstanding the interposition of the wall . FOr the performance of this , there must be placed three Glasses in the two Chambers , of which one of them shall be tyed to the planching or seeling , that it may be common to communicate the Species to each Glasse by reflexion , there being left some hole at the top of the Wall against the Glasse to this end : the two other Glasses must be placed against the two Walls at right Angles , as the figure here sheweth at B. and C. Then the sight at E by the line of incidence FE , shall fall upon the Glasse BA , and reflect upon the superficies of the Glasse BC , in the point G ; so that if the eye be at G , it should see E , and E would reflect upon the third Glass in the point H , and the eye that is at L , will see the Image that is at E. in the point of the Cath●r● : which Image shall come to the eye of the suspicious , viz. at L. by help of the third Glasse , upon which is made the second reflexion , and so brings unto the eye the object , though a wall be between it . Corolarie . 1. BY this invention of reflections the besiegers of a Towne may be seene upon the Rampart : notwithstanding the Parapet , which the besieged may do by placing a Glasse in the hollow of the Ditch , and placing another upon the toppe of the wall , so that the line of incidence comming to the bottom of the Ditch , make an Angle equall to the Angle of reflexion , then by this situation and reflexion , the Image of the besiege● 〈◊〉 will be seen to him is upon the Rampart Corolarie 2. BY which also may be inferred , that the same reflexions may be seen in a Regular Polygon , and placing as many Glasses as there are sides , counting two for one ; for then the object being set to one of the Glasses , and the eye in the other , the Jmage will be seen easily . Corolarie 3. FArther , notwithstanding the interposition of many Walls , Chambers , or Cabinets , one may see that which passeth through the most remotest of them , by placing of many Glasses as there are openings in the walls , making them to receive the incident angles equall : that is , placing them in such sort by some Geometricall assistant , that the incident points may meet in the middle of the Glasses : but here all the defect will be , that the Jmages passing by so many reflexions , will be very weak and scarce observable . PROBLEM . LXXIX . How with a Musket to strike a mark , not looking towards it , as exact as one aiming at it . AS let the eye be at O ▪ and the mark C , place a plaine Glasse perpendicular as AB . so the marke C shall be seen in Catheti CA , viz. in D , and the line of reflexion is D , now let the Musket FE , upon a rest ▪ be moved to and fro untill it be seen in the line OD , which admit to be HG , so giving fire to the Musket , it shall undoubtedly strike the mark . Corolaries . From which may be gathered , that one may exactly shoot out of a Musket to a place which is not seen , being hindered by some obstacle , or other interposition . AS let the eye be at M , the mark C , and the wall which keeps it from being seene , admit to be QR , then set up a plaine Glass as AB , and let the Musket by GH , placed upon his rest PO. Now because the marke C is seen at D , move the Musket to and fro , untill it doth agree with the line of reflection MB , which suppose at LI , so shall it be truly placed , and giving fire to the Musket , it shall not faile to strike the said mark at C. PROBLEM . LXXX . How to make an Image to be seen hanging in the aire , having his head downeward . TAke two Glasses , and place them at right Angles one unto the other , as admit AB , and CB , of which admit CB , Ho●izontall , and let the eye be at H , and the object or image to be DE ; so D will be reflected at F , so to N , so to HE : then at G , so to ● and then to H , and by a double reflection ED will seeme in QR , the highest point D in R , and the point L in Q inversed as was said , taking D for the head , and E for the feet ; so it will be a man inversed , which will seem to be flying in the aire , if the Jmage had wings unto it , and had secretly 〈◊〉 motion : and if the Glasse were bigge enough to receive many reflexions , it would deceive the sight the more by admiring the changing of colours that would be seen by that motion . PROBLEM . LXXXI . How to make a company of representative Souldiers seeme to be a Regiment , or how few in number may be multiplyed to seem to be many in number . TO make the experiment upon men , there must be prepared two great Glasses ; but in stead of it we will suppose two lesser , as GH . and FI , one placed right against another perpendicular to the Horizon , upon a plaine levell Table : betvveene vvhich Glasses let there be ranged in Battalia-vvise upon the same Table a number of small men according to the square G , H , I , F , or in any other forme or posture : hen may you evidently see hovv the said battel vvill be multiplyed and seem farre bigger in the appearance than it is in effect . Corolarie . BY this invention you may make a little Cabinet of foure foot long , and tvvo foot large , ( more or lesse ) vvhich being filled vvith Rockes or such like things , or there being put into it Silver , Gold , Stones of luster , Jewels , &c. and the walls of the said Cabinet being all covered , or hung with plaine glasse ; these visibles will appeare manifoldly increased , by reason of the multiplicitie of reflexions , and at the opening of the said Cabinet , having set something which might hide them from being seen , those that look into it will be astonished to see so few in number which before seemed to be so many . PROBLEM . LXXXII . Of fine and pleasant Dyal● . COuld you choose a more ridiculous one than the natural Dyall written amongst the Greek Epigrams , upon which some sound Poet made verses ; shewing that a man carrieth about him alwayes a Dyall in his face by meanes of the Nose and Teeth ? and is not this a jolly Dyall ? for he need not but open the mouth , the lines shall be all the teeth , and the nose shall serve for the style . Of a Dyall of hearbes . CAn you have a finer thing in a Garden , or in the middle of a Compartemeet , than to see the lines and the number of houres represented with little bushie hearbes , as of Hysope or such which is proper to be cut in the borders ; and at the top of the style to have a Fanne to shew which way the winde b●oweth ? this is very pleasant and useful . Of the Dyall upon the fingers and the hand . IS it nor a commoditie very agreeable , when one is in the fie●d or in some vil●age vvithout any other Dyall , to see onely by the hand what of the clock it is ? vvhich gives it very neare ; and may be practised by the left hand , in this manner . Take a stravv or like thing of the length of the Index or the second finger , hold this straw very right betvveen the thumb and the fore-finger , then stretch forth the hand ▪ and turne your back , and the palm of your hand tovvards the Sunne ; so that the shadovv of the muscle vvhich is under the Thumb , touch the line of life , vvhich is betvveen the middle of the tvvo other great lines , vvhich is seen in the palme of the hand , this done , the end of the shadovv vvill shevv vvhat of the clock it is : for at the end of the first finger it is 7 in the morning , or 5 in the evening , at the end of the Ring-finger it is 8 in the morning , or 4 in the evening , at the end of the little finger or first joynt , it is 9 in the morning , or 3 in the after-noone , 10 & 2 at the second joynt , 11 and 1 at the third joynt , and midday in the line follovving , vvhich comes from the end of the Index . Of a Dyall which was about an Obeliske at Rome . WAs not this a pretty fetch upon a pavement , to choose an Obeliske for a Dyall , having 106 foot in height , without removing the Basis of it ? Plinie assures us in his 26 book and 8 Chap. that the Emperour Augustus having accom●odated in the field of Mars an Obeliske of this height , he made about it a pavement , and by the industry of Man●lius the Mathematician , there were enchaced markes of Copper upon the Pavement , and placed also an Apple of Gold upon the toppe of the said Obeliske , to know the houre and the course of the Sunne , with the increase and decrease of dayes by the same shadow : and in the same manner do some by the shadow of their head or other style , make the like experiments in Astronomie . Of Dyals with Glasses . PT●lomie w●ites , as Cardanus reports , that long ago there were Glasses which served for Dyals , and presented the face of the beholder as many times as the houre ought to be , twice if it were 2 of the clock , 9 if it were 9 , &c. But this was thought to be done by the help of water , and not by Glasses , which did leake by little and little out of the vessell , discovering anon one Glasse , then anon two Glasses , then 3 , 4 , 5 Glasses , &c. to shew so many faces as there were houres , which was onely by leaking of water . Of a Dyall which hath a Glasse in the place of the Style . WHat will you say of the invention of Mathematicians , which finde out daily so many fine and curious novelties ? they have now a way to make Dyals upon the wainscot or seeling of a Chamber , and there where the Sunne can never shine , or the beames of the Sunne cannot directly strike : and this is done in placing of a little Glasse in the place of the style which reflecteth the light , with the same condition that the shadow of the style sheweth the houre : and it is easie to make experiment upon a common Dyall , changing only the disposition of the Dyall , and tying to the end of the style a piece of plaine Glasse . The Almaines use it much , who by this way have no greater trouble , but to put their Noses out of their beds and see what a clock it is , which is reflected by a little hole in the Window upon the wall or seeling of the Chamber . EXAMINATION . IN this there are two experiments considerable , the first is with a very little Glasse placed so that it may be open to the beames of the Sunne , the other hath respect to a spacious or great Glasse placed to a very little hole so that the Sun may shine on it , for then the shadow which is cast upon the Dyall is converted into beames of the Sunne , and will reflect and becast upon a plain opposite : and in the other it is a hole in the window or such like , by which may passe the beames of the Sun , which represent the extreamity of the style , & the Glasse representeth the plaine of the Dyall , upon which the beames being in manner of shadowes reflect cast upon a plaine opposite : and it is needfull that in this second way the Glasse may be spacious , as before , to receive the delineaments of the Dyall . Otherwise you may draw the lineaments of a Dyall upon any plaine looking-glasse which reflecteth the Sunne-beames , for the applying a style or a pearle at the extreamitie of it : and placed to the Sunne , the reflexion will be answerable to the delineaments on the Glasse : but here note , that the Glasse ought to be great , and so the delineaments thereon . But that which is most noble , is to draw houre-lines upon the outside of the Glasse of a window , and placing a style thereto upon the outside , the shadow of the style will be seen within , and so you have the hour , more certaine without any difficulty . Of Dyals with water . SVch kinde of Dyals were made in ancient times , and also these of sand : before they had skill to make Sun-dyals or Dyals with wheeles ; for they used to fill a vessell with water , and having experience by tryall thar it would runne out all in a day , they did marke within the vessell the houres noted by the running of the water ; and some did set a piece of light board in the vessell to swimme upon the top of the water , carrying a little statue , which with a small stick did point out the houre upon a columne or wall , figured with houre-notes , as the vessell was figured within . Novv it seemes a safer vvay that the vvater passe out by drop and drop , and drop into a Cylindricall Glasse by help of a Pipe : for having marked the exterior part of the Cylinder in the houre notes , the vvater it selfe vvhich falls vvithin it , vvill shevv vvhat of the clock it is , farre better than the running of sand , for by this may you have the parts of the houres most accurate , vvhich commonly by sand is not had : and to vvhich may be added the houres of other Countreys vvith greater ease . And here note , that as soone as the vvater is out ▪ of one of the Glasses , you may turne it over into the same againe out of the other , and so let it runne anevv . PROBLEM . LXXXIII . Of Cannons or great Artillery . Souldiers , and others would willingly see 〈◊〉 Problems , which containe : three or foure subtile questions : The first is , how to charge a Cannon without powder ? THis may be done vvith aire and vvater , only having throvvn cold vvater into the Cannon , vvhich might be squirted forceably in by the closure of the mouth of the Piece , that so by this pressure the aire might more condense ; then having a round piece of vvood very just , and oiled vvell for the better to slide , and thrust the Bullet vvhen it shall be time : This piece of vvood may be held fast vvith some Pole , for feare it be not thrust out before his time : then let fire be made about the Trunion or hinder part of the Piece to heat the aire and vvater , and then vvhen one vvould shoot it , let the pole be quickly loosened , for then the aire searching a greater place , and having vvay novv offered , vvill thrust out the vvood and the bullet very quick : the experiment vvhich vve have in long trunkes shooting out pellats vvith aire only , shevveth the verity of this Probleme . 2 In the second question it may be demanded , how much time doth the Bull●● of a Cannon spend in the aire before i● falls to the ground ? THe resolution of this Question depends upon the goodnesse of the Piece & charge thereof , seeing in each there is great difference . It is reported , that Tich● Bra●e , and the Landsgrave did make an experiment upon a Cannon in Germany , which being charged and shot off ; the Bullet spent two minutes of time in the aire before it fell : and the distance was a Germane mile , which distance proportionated to an hours time , makes 120 Italian miles . 3. In the third question it may be asked , how it comes to passe , that a Cannon shooting upwards , the Bullet flies with more violence than being shot point-blanke , or shooting downeward ? IF we regard the effect of a Cannon when it is to batter a wall , the Question is false , seeing it is most evident that the blowes which fall Perpendicular upon a wall , are more violent than those which strike byas-wise or glaunsingly . But considering the strength of the blow only , the Question is most true , and often experimented to be found true : a Piece mounted at the best of the Randon , which is neare halfe of the right , conveyes her Bullet with a farre greater violence then that which is shot at point blanke , or mounted parallel to the Horizon . The common reason is , that shooting high , the fire carries the bowle a longer time in the aire , and the aire moves more ●acill upwards , than dovvnevvards , because that the airy circles that the motion of the bullet makes , are soonest broken . Hovvsoever this be the generall tenet , it is curious to finde out the inequality of moving of the aire ; vvhether the Bullet fly upvvard , dovvnevvard , or right forvvard , to produce a sensible dfference of motion ; & some think that the Cannon being mounted , the Bullet pressing the povvder maketh a greater resistance , and so causeth all the Povvder to be inflamed before the Bullet is throvvne out , vvhich makes it to be more violent than othervvise it vvould be . When the Cannon is othervvise disposed , the contrary arives , the fire leaves the Bullet , and the Bullet rolling from the Povvder resists lesse : and it is usually seene , that shooting out of a Musket charged onely vvith Povvder , to shoot to a marke of Paper placed Point blanke , that there are seene many small holes in the paper , vvhich cannot be other than the graines of Powder which did not take fire : but this latter accident may happen from the over-charging of the Piece , or the length of it , or windy , or dampenesse of the Powder . From which some may think , that a Cannon pointed right to the Zenith , should shoot with greater violence , then in any other mount or forme whatsoever : and by some it hath beene imagined , that a Bullet shot in this fashion hath been consumed , melted , and lost in the Aire , by reason of the violence of the blow , and the activity of the sire , and that sundry experiments have been made in this nature , and the Bullet never found . But it is hard to believe this assertion : it may rather be supposed that the Bullet falling farre from the Piece cannot be discerned where it falls : and so comes to be lost . 4. In the fourth place it may be asked , whether the discharge of a Cannon b● so much the greater , by how much it is longer ? IT seemeth at the first to be most true , that the longer the Piece is , the more violent it shoots : and to speak generally , that which is direction by a Trunke , Pipe , or other concavitie , is conveyed so much the more violent , or better , by how much it is longer , either in respect of the Sight , Hearing , Water , Fire , &c. & the reason seems to hold in Cannons , because in those that are long , the fire is retained a longer time in the concavitie of the Piece , and so throwes out the Bullet with more violence ; and experience lets us see that taking Cannons of the same boare , but of diversitie of length from 8 foot to 12 , that the Cannon of 9 foot long hath more force than that of 8 foot long , and 10 more than that of 9 , and so unto 12 foote of length . Now the usuall Cannon carries 600 Paces , some more , some lesse , yea some but 200 Paces from the Piece , and may shoot into soft earth 15 or 17 foot , into sand or earth which is loose , 22 or 24 foot , and in firme ground , about 10 or 12 foot , &c. It hath been seen lately in Germany , where there were made Pieces from 8 foot long to 17 foot of like boare , that shooting out of any piece which was longer than 12 foot ; the force was diminished , and the more in length the Piece increaseth , the lesse his force was : therefore the length ought to be in a meane measure , and it is often seene , the greater the Cannon is , by so much the service is greater : but to have it too long or too short , is not convenient , but a meane proportion of length to be taken , otherwise the flame of the fire will be over-pressed with Aire : whic hinders the motion in respect of substance , and distance of getting out . PROBLEM . LXXXIIII . Of predigious progression and multiplication , of Creatures , Plants , Fruits , Numbers , Gold , Silver , &c. when they are alwayes augmented by certaine proportion . HEre we shall shew things no lesse admirable , as recreative , and yet so certaine and easie to be demonstrated , that there needs not but Multiplication only , to try each particular : and first , Of graines of Mustard-seed . FIrst , therefore it is certaine that the increase of one graine of Mustard-seed for 20 yeares space , cannot be contained within the visible world , nay if it were a hundred times greater than it is : and holding nothing besides from the Centre of the earth even unto the firmament , but only small grains of Mustard-seed : Now because this seems but words , it must be proved by Art , as may be done in this wise , as suppose one Mustard-seed sowne to bring forth a tree or branch , in each extendure of which might be a thousand graines : but we will suppose onely a thousand in the whole tree , and let us proceed to ●0 yeares , every seed to bring forth yearely a thousand graines , now multiplying alwayes by a thousand , in lesse then 17 years you shall have to many graines which will surpasse the sands , which are able to fill the whole firmament : for following the supposition of Archimedes , and the most probable opinion of the greatness of the firmament which ●i●ho Brahe hath left us ; the number of graines of sand will be sufficiently expressed with 49 Ciphers , but the number of graines of Mustard-seed at the end of 17 yeares will have 52 Ciphers : and moreover , graines of Mustard-seed , are farre greater than these of the sands : it is therefore evident that at the seventeenth yeare , all the graines of Mustard-seed which shall successively spring from one graine onely , cannot be contained within the limits of the whole firmament ; what should it be then , if it should be multiplied againe by a thousand for the ●8 yeare : and that againe by a thousand for every yeares increase untill you come to the 20 yeare , it 's a thing as cleare as the day , that such a heap of Mustard-seed would be a hundred thousand times greater than the Earth : and bring onely but the increase of one graine in 20 yeares . Of Pigges . SEcondly , is it not a strange proposition , to say that the great Turke with all his Revenues , is not able to maintaine for one yeares time , all the Pigges that a Sow may pigge with all her race , that is , the increase with the increase unto 12 years : this seemes impossible , yet it is most true , for let us suppose and put , the case , that a Sow bring forth but 6 , two Males , and 4 Females , and that each Female shall bring forth as many every yeare , during the space of 12 yeares , at the end of the time there will be found above 3● millions of Pigges : now allowing a crowne for the maintenance of each Pigge for a yeare , ( which is as little as may be , being but neare a halfe of a farthing allowance for each day ; ) there must be at the least so many crownes to maintaine them , one a year , viz. 33 millions , which exceeds the Turkes revenue by much . Of graines of Corne. THirdly , it will make one astonished to think that a graine of Corne , with his increase successively for the space of 12 yeares will produce in grains 24414062●000000000000 , which is able to load almost al the creatures in the World. To open which , let it be supposed that the first yeare one graine being sowed brings forth 50 , ( but sometimes there is seen 70 , sometimes 100 fold ) which graines sowen the next yeare , every one to produce 50 , and so consequently the whole and increase to be sowen every yeare , until 12 yeares be expired , there will be of increase the aforesaid prodigious summe of graines , viz. 244140625000000000000 , which will make a cubical heap of 6258522 graines every way , which is more than a cubicall body of 31 miles every way : for allowing 40 graines in length to each foot , the Cube would be 156463 foot every way : from which it is evident that if there were two hundred thousand Cities as great as London , allowing to each 3 miles square every way , and 100 foot in height , there would not be sufficient roome to containe the aforesaid quantitie of Corne : and suppose a bushel of Corne were equal unto two Cubicke feet , which might containe twenty hundred thousand graines then would there be 122070462500000. bushells , and allowing 30 bushels to a Tunne , it would be able to load 81380●0833 vessels , which is more than eight thousand one hundred and thirty eight millions , ship loadings of ●00 Tunne to each ship a : quantity so great that the Sea is scarce able to beare , or the universal world able to finde vessels to carry it . And if this Corne should be valued at halfe a crown the bushel , it would amount unto 15258807812500 pounds sterling , which I think exceeds all the Treasures of all the Princes , and of other particular men in the whole world : and is not this good husbandry to sowe one grain of Corne ; and to continue it in sowing , the increase only for 12 yeares to have so great a profit ? Of the increase of Sheep . FOurthly , those that have great flocks of Sheep may be quickly rich , if they would preserve their Sheep without killing or selling of them : so that every Sheep produce one each yeare , for at the end of 16 yeares , 100 Sheepe will multiply and increase unto 6553600 , which is above 6 millions , 5 hundred 53 thousand Sheep : now supposing them worth but a crown a piece , it would amount unto 1638400 pounds sterling , vvhich is above 1 million 6 hundred 38 thousand pounds , a faire increase of one Sheep : and a large portion for a Childe if it should be allotted . Of the increase of Cod-fish , Carpes , &c. FIfthly , if there be any creatures in the vvorld that do abound vvith increase or fertilitie , it may be rightly attributed to fish ; for they in their kindes produce such a great multitude of Eggs , and brings forth so many little ones , that if a great part vvere not destroyed continually , vvithin a ●ittle vvhile they vvould fill all the Sea , Ponds , and Rivers in the vvorld ; and it is easie to shevv hovv it vvould come so to passe , onely by supposing them to increase without taking or destroying them for the space of 10 or 12 yeares : having regard to the soliditie of the waters which are allotted for to lodge and containe these creatures , as their bounds and place of rest to live in . Of the increase and multiplication of men . SIxthly , there are some that cannot conceive how it can be that from eight persons ( which were saved after the deluge or Noahs flood ) should spring such a world of people to begin a Monarchie under Nimrod , being but 200 yeares after the flood , and that amongst them should be raised an army of two hundred thousand fighting men : But it is easi●y proved if vve take but one of the Children of Noah , and suppose that a nevv generation of people begun at every 30 yeares , and that it be continued to the seventh generation vvhich is 200 yeares ; for then of one only family there vvould be produced one hundred and eleven thousand soules , three hundred and five to begin the vvorld : though in that time men lived longer , and vvere more capable of multiplication and increase : vvhich number springing onely from a simp●e production of one yearly , vvould be farre greater , if one man should have many vvives , vvhich in ancient times they had : from vvhich it is also that the Children of Israel , vvho came into Egypt but onely 70 soules , yet after 210 yeares captivity , they came forth vvith their hostes , that there vvere told six hundred thousand fighting men , besides old people , women and children ; and he that shall separate but one of the families of Joseph , it would be sufficient to make up that number : how much more should it be then if we should adjoyne many families together ? Of the increase of numbers . SEventhly , what summe of money shall the City of London be worth , if it should be sold , and the money be paid in a yeare after this manner : the first week to pay a pinne , the second week 2 pinnes , the third week 4 pinnes , the fourth week 8 pinnes , the fifth week 16 pinnes . and so doubling untill the 52 weeks , or the yeare be expired . Here one would think that the value of the pinnes would amount but to a small matter , in comparison of the Treasures , or riches of the whole City : yet it is most probable that the number of pinnes would amount unto the sum of 4519599628681215 , and if we should allow unto a quarter a hundred thousand pinnes , the whole would contain ninetie eight millions , foure hundred thousand Tunne : which is able to load 45930 Shippes of a thousand Tunne apiece : and if we should allow a thousand pins for a penny , the summe of money would amount unto above eighteen thousand , eight hundred and thirty millions of pounds sterling , an high price to sell a Citie at , yet certain , according to that first proposed . So if 40 Townes were sold upon condition to give for the first a penny , for the second 2 pence , for the third 4 pence , &c. by doubling all the rest unto the last , it would amount unto this number of pence , 109951●62●●76 , which in pounds is 4581298444 , that is foure thousand five hundred and fourescore millions of pounds and more . Of a man that gathered up Apples , Stones , or such like upon a condition . EIghtly , admit there were an hundred Apples , Stones , or such like things that were plac'd in a straight line or right forme , a Pace one from another , and a basket being placed a Pace from the first : how many paces would there be made to put all these Stones into the basket , by fetching one by one ? this would require near halfe a day to do it , for there would be made ten thousand and ninety two paces before he should gather them all up . Of Changes in Bells , in musicall Instruments , transmutation of places , in numbers , letters , men or such like . NInethly , is it not an admirable thing to consider how the skill of numbers doth easily furnish us with the knowledge of mysterious and hidden things ? which simply looked into by others that are not versed in Arithmetick , do present unto them a world of confusion and difficultie . As in the first place , it is often debated amongst our common Ringers , what number of Changes there might be made in 5 , 6 , 7 , 8 , or more Bells : who spend much time to answer their owne doubts , entring often into a Labyrinth in the search thereof : or if there were 10 voyces , how many severall notes might there be ? These are propositions of such facility , that a childe which can but multiply one number by another , may easily resolve it , which is but only to multiply every number from the unite successively in each others product , unto the terme assigned : so the 6 number that is against 6 in the Table , is 720 , and so many ( hanges may be made upon 6 Bells , upon 5 there are 120 , &c. In like manner against 10 in the Table is 3618800 , that is , three millions , six hundred twenty eight thousand , eight hundred : which shews that 10 voices may have so many consorts , each man keeping his owne note , but only altering his place ; and so of stringed Instruments , and the Gamat may be varied according to which , answerable to the number against X , viz. 1124001075070399680000 notes , from which may be drawne this , or the like proposition . Suppose that 7 Schollers were taken out of a free Schoole to be sent to an Vniversitie , there to be entertained in some Colledge at commons for a certaine summe of money , so that each of them have two meales daily , and no longer to continue there , then that sitting all together upon one bench or forme at every meale , there might be a divers transmutation of place , of account in some one of them , in comparison of another , and never the whole company to be twice alike in situation : how long may the Steward entertaine them ? ( who being not skilled in this fetch may answere unadvisedly . ) It is most certaine that there will be five thousand and forty several 1 a 1 2 b 2 6 c 3 24 d 4 120 e 5 7●0 f 6 5040 g 7 403●0 h 8 362880 i 9 3628800 k 10 39916800 l 11 479001600 m 12 6227020800 n 13 87178291200 o 14 1307674368000 p 15 20922789888000 q 16 355687537996000 r 17 6402375683928000 s 18 121645137994632000 t 19 2432902759892640000 u 20 51090957957745440000 w 21 1124001075070399680000 x 22 25852024726619192640000 y 23 6●0448593438860623360000 z 24 positions or changings in the seatings , which maks 14 years time wanting 10 weeks and 3 dayes . Hence from this mutability of transmutation , it is no marvell tha● by 24 letters there ariseth and is made such variety of languages in the world , & such infinite number of words in each language ; seeing the diversity of syllables produceth that effect , and also by the interchanging & placing of letters amongst the vowels , & amongst themselves maketh these syllables : vvhich Alphabet of 24 letters may be varied so many times , viz. 620448593438860623360000 vvhich is six hundred tvventy thousand , foure hundred forty eight millions of millions of millions five hundred ninety three thousand , foure hundred thirty eight milions of milions , & more . Novv allovving that a man may reade or speak one hundred thousand vvords in an houre vvhich is tvvice more vvords than there are conteined in the Psalmes of David , ( a taske too great for any man to do in so short a time ) and if there were foure thousand six hundred and fifty thousand millions of men , they could not speak these words ( according to the hourely proportion aforesaid ) in threescore and ten thousand yeares ; which variation & transmutation of letters , if they should be written in bookes , allowing to each leaf 28000 words , ( which is as many as possibly could be inserted , ) and to each book a reame or 20 quire of the largest and thinnest printing paper , so that each book being about 15 inches long , 12 broad , and 6 thick : the books that would be made of the transmutation of the 24 letters aforesaid , would be at least 38778037089928788 : and if a Library of a mile square every way , of 50 foot high , were made to containe 250 Galleries of 20 foot broad apiece , it would containe foure hundred mill●ons of the said books : so there must be to containe the rest no lesse than 9●945092 such Libraries ; and if the books were extended over the surface of the Globe of the Earth , it would be a decuple covering unto it : a thing seeming most incredible that 24 letters in their transmutation should produce such a prodigious number , yet most certaine and infallible in computation . Of a Servant hired upon certaine conditions . A Servant said unto his Master , that he would dvvell vvith him all his life-time , if he would but onely lend him land to sowe one graine of Corne with all his increase for 8 years time ; how think you of this bargaine ? for if he had but a quarter of an inch of ground for each graine , and each graine to bring forth yearely of increase 40 graines , the whole sum would amount unto , at the terme aforesaid , 6553600000000 graines : and seeing that three thousand and six hundred millions of inches do but make one mile square in the superficies , it shall be able to receive foureteene thousand and foure hundred millions of graines , which is 14400000000. thus dividing the aforesaid 6553600000000 , the Quotient will be 455 , and so many square miles of land must there be to sowe the increase of one graine of Corne for 8 yeares , which makes at the least foure hundred and twenty thousand Acres of land , which rated but at five shillings the Acre per Annum , amounts unto one hundred thousand pound ; which is twelve thousand and five hundred pound a yeare , to be continued for 8 yeares ; a pretty pay for a Masters Servant 8 yeares service . PROBLEM . LXXXV . Of Fountaines , Hydriatiques , Machinecke , and other experiments upon water , or other liquor . 1. First how to make water at the foot of a mountaine to ascend to the top of it , and so to descend on the other side ? TO do this there must be a Pipe of lead , which may come from the fountaine A , to the top of the Mountaine B ; and so to descend on the other side a little lower then the Fountaine , as at C. then make a hole in the Pipe at the top of the Mountaine , as at B , and stop the end of the Pipe at A and C ; and fill this Pipe at B with water : & close it very carefully againe at B , that no aire get in : then unstop the end at A , & at C ; then will the water perpetually runne up the hill , and descend on the other side , which is an invention of great consequence to furnish Villages that want water . 2. Secondly , how to know what wine or other liquor there is in a vessell without opening the bung-hole , and without making any other hole , than that by which it runnes out at the top ? IN this problem there is nothing but to take a bowed pipe of Glasse , and put it into the faucets hole , and stopping it close about : for then you shall see the wine or liquor to ascend in this Pipe , untill it be just even with the liquor in the vessel ; by which a man may fill the vessel , or put more into it : and so if need were , one may empty one vessel into another without opening the bung-hole . 3. Thirdly , how is it that it is said that a vessell holds more water being placed at the foot of a Mountaine , than standing upon the top of it ? THis is a thing most certaine , because that water and all other liquor disposeth it selfe sphericaliy about the Centre of the earth ; and by how much the vessel is nearer the Centre , by so much the more the surface of the water makes a lesser sphere , and therefore every part more gibbous or swelling , than the like part in a greater sphere : and therefore when the same vessell is farther from the Centre of the earth , the surface of the water makes a greater sphere , and therefore lesse gibbous , or swelling over the vessell : from whence it is evident that a vessell near the Centre of the Earth holds more water than that which is farther remote from it ; and so consequently a vessel placed at the bottome of the Mountaine holds more water , than being placed on the top of the Mountaine . First , therefore one may conclude , that one and the same vessel will alwayes hold more : by how much it is nearer the Centre of the earth . Secondly , if a vessell be very neare the Centre of the earth , there will be more water above the brims of it , than there is within the vessel . Thirdly , a vessel full of water comming to the Centre wil spherically increase , and by little and little leave the vessel ; and passing the Centre , the vessel will be all emptied . Fourthly , one cannot carry a Paile of water from a low place to a higher , but it will more and more run out and over , because that in ascending it lies more levell , but descending it swels and becomes more gibbous . 4. Fourthly , to conduct water from the top of one Mountaine , to the top of another . AS admit on the top of a Mountaine there is a spring , and at the toppe of the other Mountaine there are Inhabitants which want water : now to make a bridge from one Mountaine to another , were difficult and too great a charge ; by way of Pipes it is easie and of no great price : for if at the spring on the top of the Mountaine be placed a Pipe , to descend into the valley , and ascend to the other Mountlaine , the water will runne naturally , and continually , provided that the spring be somewhat higher than the passage of the water at the Inhabitants . 5. Fifthly , of a fine Fountaine which spouts water very high , and with great violence by turning of a Cock. LEt there be a vessell as AB , made close in all his parts , in the middle of which let CD be a Pipe open at D neare the bottome , and then with a Squirt squirt in the water at C , stopped above by the cock or faucet C , vvith as great violence as possible you can ; and turne the cock immediatly . Novv there being an indifferent quantity of vvater and aire in the vessel , the vvater keeps it selfe in the bottome , and the aire vvhich vvas greatly pressed , seeks for more place , that turning the Cock the water issueth forth at the Pipe , and flyes very high , and that especially if the vessell be a little heated : some make use of this for an Ewer to wash hands withall , and therefore putting a moveable Pipe above C , such as the figure sheweth : which the water will cause to turne very quick , pleasurable to behold . 6. Sixtly , of Archimedes screw , which makes water ascend by descending . THis is nothing else but a Cylinder , about the which is a Pipe in form of a screw , and when one turnes it , the water descends alwayes in respect of the Pipe : for it passeth from one part which is higher to that which is lower , and at the end of the Engine the water is found higher than it was at the spring . This great Enginer admirable in all Mathematicall Arts invented this Instrument to wash King Hieroies great vessells , as some Authors saye , also to water the fields of Egypt , as Diodorus witnesseth : and Cardanus reporteth that a Citizen of Milan having made the like Engine , thinking himselfe to be the first inventer , conceived such exceeding joy , that he be came mad , foll . 2. Againe a thing may ascend by descending , if a spiral line be made having many circulations or revolutions ; the last being alwayes lesser than the first , yet higher than the Plaine supposed it is most certaine that then putting a ball into it , and turning the spirall line so , that the first circulation may be perpendicular , or touch alwayes the supposed Plain : the ball shall in descending continually ascend , untill at last it come to the highest part of the spirall line , & so fall out . And here especially may be noted , that a moving body as water , or a Bullet , or such like , will never ascend if the Helicall revolution of the screw be not inclining to the Horizon : so that according to this inclination the ball or liquor , may descend alwayes by a continuall motion and revolution . And this experiment may be more usefull , naturally made with a thred of ●ron , or Latine turned or bowed Helically about a Cylinder , with some distinction of distances between the Heli●es , for then having drawn out the Cylinder , or having hung or tyed some weight at it in such sort , that the water may easily drop if one lift up the said thred : these Helices or revolutions , notwithstanding will remaine inclining to the Horizon , and then turning it about forward , the said weight will ascend , but backward it will descend . Now if the revolutions be alike , and of equallity amongst themselves , and the whirling or turning motion be quicke , the sight vvill be so deceived , that producing the action it vvill seeme to the ignorant no lesse than a Miracle . 7. Seventhly , of another fine Fountaine of pleasure . THis is an Engine that hath two wheeles with Cogges , or teeth as AB , which are placed within an Ovall CD , in such sort , that the teeth of the one , may enter into the notches of the other ; but so just that neither aire nor water may enter into the Ovall coffer , either by the middle or by the sides , for the wheele must joyne so neare to the sides of the coffer , that there be no vacuitie : to this there is an axeltree with a handle to each wheele , so that they may be turned , and A being turned , that turneth the other wheele that is opposite : by which motion the aire that is in E , & the water that is carried by the hollow of the wheeles of each side , by continuall motion , is constrained to mount and flie out by the funnell F : now to make the water runne what way one would have it , there may be applied upon the top of the Pipe F , two other moveable Pipes inserted one within another ; as the figure sheweth . But here note , that there may acrue some inconveniency in this Machine seeing that by quick turning the Cogges or teeth of the wheeles running one against another , may neare break them , and so give way to the aire to enter in , which being violently inclosed vvill escape to occupie the place of the vvater , vvhose vveight makes it so quick : hovvsoever , if this Machine be curiously made as an able vvorkeman may easily do , it is a most sovereigne Engine , to cast vvater high and farre off for to quench fires . And to have it to raine to a place assigned , accommodate a socket having a Pipe at the middle , vvhich may point tovvards the place being set at the top thereof , and so having great discretion in turning the Axis of the vvheele , it may vvork exceeding vvell , and continue long . 8. Eightly , of a fine watering pot for gardens . THis may be made in forme of a Bottle according to the last figure or such like , having at the bottome many small holes , and at the neck of it another hole somevvhat greater than those at the bottome , vvhich hole at the top you must unstop vvhen you vvould fill this vvatering pot , for then it is nothing but putting the lovver end into a paile of vvater , for so it vvill fill it selfe by degrees : and being full , put your thumb on the hole at the neck to stop it , for then may you carry it from place to place , and it vvill not sensibly runne out , som●thing it vvill , and all in time ( if it vvere never so close stopped ) contrary to the ancient tenet in Philosophy , that aire will not penetrate . 9. Ninthly , how easily to take wine out of a vessell at the bu●g-hole , without piercing of a hole in the vessell ? IN this there is no need but to have a Cane or Pipe of Glasse or such like , one of the ends of which may be closed up almost , leaving some small hole at the end ; for then if that end be set into the vessell at the bung-hole , the whole Cane or Pipe will be filled by little and little ; and once being full , stop the other end which is without and then pull out the Cane or Pipe , so will it be ful of wine , then opening a little the top above , you may fill a Glasse or other Pot with it , for as the Wine issueth out , the aire commeth into the Cane or Pipe to supply vacuity . 10. Tenthly , how to measure irregular bodies by help of water ? SOme throw in the body or magnitude into a vessell , and keep that which floweth out over , saying it is alwayes equal to the thing cast into the water : let i● is more nea●er this way to poure into a vessell such a quantity of water , which may be thought sufficient to cover the body or magnitude , and make a marke how high the water is in the vessell , then poure out all this water into another vessell , and let the body or magnitude be placed into the first vessel ; then poure in water from the second vessell , until it ascend unto the former marke made in the first vessell , so the vvater vvhich remaines in the second vessel is equall to the body or magnitude put into the water : but here note that this is not exact or free from error , yet nearer the truth than any Geometrician can otherwise possibly measure , and these bodies that are not so full of pores are more truly measured this way , than others are . 11. To finde the weight of water . SEeing that 574 / 1000 part of an ounce weight , makes a cubicall inch of water : and every pound weight Haverdepoize makes 27 cubicall inches , and 1 9 / ● ; fere , and that ● Gallons and a halfe wine measure makes a foot cubicall , it is easie by inversion , that knowing the quantity of a vessel in Gallons , to finde his content in cubicall feet or weight : and that late famous Geometrician Master Brigs found a cubical foot of vvater to vveigh neare 62 pound vveight Haverdepoize But the late learned Simon Stevin found a cubicall foot of vvater to vveigh 65 pound , vvhich difference may arise from the inequalitie of vvater ; for some vvaters are more ponderous than others , and some difference may be from the weight of a pound , and the measure of a foot : thus the weight and quantitie of a solid foot settled , it is easie for Arithmeticians to give the contents of vessells or bodies which containe liquids . 12. To finde the charge that a vessell may carry as Shippes , Boates , or such like . THis is generally conceived , that a vessell may carry as much weight as that water weigheth , which is equall unto the vessell in bignesse , in abating onely the weight of the vessell : we see that a barrel of wine or water cast into the water , will not sink to the bottome , but swim easily , and if a ship had not Iron and other ponderosities in it , it might swim full of water without sinking : in the same manner if the vessell were loaden with lead , so much should the watter weigh : hence it is that Marriners call Shippes of 50 thousand Tunnes , because they may containe one or two thousand Tunne , and so consequently carry as much . 13. How comes it that a Shippe having safely sayled in the vast Ocean , and being come into the Port or harbour , without any tempest will sink down right ? THe cause of this is that a vessel may carry more upon some kinde of water than upon other ; now the water of the Sea is thicker and heavier than that of Rivers , Wels , or Fountains ; therefore the loading of a vessell which is accounted sufficient in the Sea , becomes too great in the hurbour or sweet water . Now some think that it is the depth of the water that makes vessells more easie to swimme , but it is an abuse ; for if the loading of a Ship be no heavier than the water that would occupie that place , the Ship should as easily swim upon that water , as if it did swim upon a thousand fathom deep of water , and if the vvater be no thicker than a leafe of paper , and weigheth but an ounce under a heavy body , it vvill support it , as vvell as if the vvater under it vveighed ten thousand pound vveight : hence it is if there be a vessell capable of a little more than a thousand pound vveight of vvater , you may put into this vessell a piece of vvood , vvhich shall vveigh a thousand pound vveight ; ( but lighter in his kinde than the like of magnitude of vvater : ) for then pouring in but a quart of vvater or a very little quantitie of vvater , the vvood vvill svvim on the top of it , ( provided that the vvood touch not the sides of the vessell : ) vvhich is a fine experiment , and seems admirable in the performance . 14. How a grosse body of mettle may swimme upon the water ? THis is done by extending the mettle into a thin Plate , to make it hollovv in forme of a vessel ; so that the greatnesse of the vessell which the aire vvith it containeth , be equal to the magnitude of the vvater , vvhich vveighes as much as it , for all bodies may svvim vvithout sinking , if they occupie the place of vvater equal in vveight unto them , as if it vveighed 12 pound it must have the place of 12 pound of vvater : hence it is that vve see floating upon the vvater great vessells of Copper or Brasse , vvhen they are hollovv in forme of a Caldron . And how can it be otherwise conceived of Islands in the Sea that swim and float ? is it not that they are hollow and some part like unto a Boat , or that their earth is very light and spongeous , or having many concavities in the body of it , or much wood within it ? And it would be a pretty proposition to shew how much every kinde of metall should be inlarged , to make it swim upon the water : which doth depend upon the proportion that is between the vveight of the vvater and each metall . Novv the proportion that is betvveene metalls and water of equall magnitude , according to some Authors , is as followeth . A magnitude of 10 pound weight of water will require for the like magnitude of Gold. 187 ½ Lead . 116 ½ Silver . 104 Copper . 91 Iron . 81 Tinne . 75 From which is inferred , that to make a piece of Copper of ●0 pound weight to swimme , it must be so made hollow , that it may hold 9 times that weight of water and somewhat more , that is to say , 91 pound : seeing that Copper and water of like magnitudes in their ponderosities , are as before , as ●0 to 91. 15. How to weigh the lightnesse of the aire ? PLace a Ballance of wood turned upside downe into the water , that so it may swim , then let water be inclosed within some body , as within a Bladder or such like , and suppose that such a quantitie of aire should weigh one pound , place it under one of the Ballances , and place under the other as much weight of lightnesse as may counter-ballance and keep the other Ballance that it rise not out of the water : by which you shall see how much the lightnesse is . But without any Ballance do this ; take a Cubicall hollow vessell , or that which is Cylindricall , which may swimme on the water , and as it sinketh by placing of weights upon it , marke hovv much , for then if you vvould examine the vveight of any body , you have nothing to do but to put it into this vessell , and marke hovv deep it sinkes , for so many pound it vveighes as the vveights put in do make it so to sinke . 16. Being given a body , to marke it about , and shew how much of it will sink in the water , or swim above the water . THis is done by knovving the vveight of the body vvhich is given , and the quantity of vvater , vvhich vveighes as much as that body ; for then certainly it vvill sink so deep , untill it occupieth the place of that quantitie of vvater . 17. To finde how much severall mettle or other bodies doe weigh lesse in the water than in the aire : TAke a Ballance , & vveigh ( as for example ) 9 pound of Gold , Silver , Lead , or Stone in the aire , so it hang in aequilibrio ; then comming to the vvater , take the same quantity of Gold Silver , Lead , or Stone , and let it softly dovvne into it , and you shall see that you shall need a lesse counterpoise in the other Ballance to counter-ballance it : vvherefore all solids or bodies vveigh lesse in the vvater than in the aire , and so much the lesse it vvill be , by hovv much the vvater is grosse and thick , because the vveight findes a greater resistance , and therefore the vvater supports more than aire ; and further , because the vvater by the ponderositie is displeased , and so strives to be there againe , pressing to it , by reason of the other vvaters that are about it , according to the proportion of his weight . Archimedes demonstrateth , that all bodies weigh lesse in the water ( or in like liquor ) by how much they occupie place : and if the water weigh a pound weight , the magnitude in the water shall weigh a pound lesse than in the aire . Now by knowing the proportion of water and mettles , it is found that Gold loseth in the water the 19 part of his weight , Copper the 9 part , Quicksilver the 15 part , Lead the 12 part , Silver the 10 part , Iron the 8 part , Tinne the 7 part and a little more : wherefore in materiall and absolute weight , Gold in respect of the water that it occupieth weigheth 18 , and ¾ times heavier than the like quantitie of water , that is , as 18 ¾ to the Quicksilver 15 times , Lead 11 and ⅗ , Silver 10 and ⅔ , Copper 9 and 1 / 10 , Iron 8 and ½ , and Tinne 8 and 1 / ● . Contrarily in respect of greatnesse , if the water be as heavy as the Gold , then is the water almost 19 times greater than the magnitude of the Gold , and so may you judge of the rest . 18. How is it that a ballance having like weight in each scale , and hanging in aequilibrio in the aire , being placed in another place , ( without removing any weight ) it shall cease to hang in aequilibrio sensibly : yea by a great difference of weight ? THis is easie to be resolved by considering different mettles , which though they vveigh equall in the aire , yet in the vvater there vvill be an apparant difference ; as suppose so that in the scale of each Ballance be placed 18 pound vveight of severall metalls , the one Gold , and the other Copper , vvhich being in aequilibrio in the aire , placed in the vvater , vvill not hang so , because that the Gold los●eth neare the 18 part of his vveight , vvhich is about 1 pound , and the Copper loseth but his 9 part , vvhich is 2 pound : vvherefore the Gold in the vvater vveigheth but 17 pound , and the Copper 16 pound , vvhich is a difference most sensible to confirme that point . 19. To shew what waters are heavier one than another , and how much . PHysicians have an especiall respect unto this , judging that vvater vvhich is lightest is most healthfull and medicinall for the body , & Sea-men knovv that the heaviest vvaters do beare most , and it is knovvne vvhich water is heaviest thus . Take a piece of wax , and fasten Lead unto it , or some such like thing that it may but precisely swimme , for then it is equal to the like magnitude of water , then put it into another vessell which hath contrary water , and if it sinke , then is that water lighter than the other : but if it sinke not so deep , then it argueth the water to be heavier or more grosser than the first water , or one may take a piece of vvood , and marke the quantitie of sinking of it into severall waters , by vvhich you may judge which is lightest or heaviest , for in that which it sinkes most , that is infallibly the lightest , and so contrarily . 20. How to make a Pound of water weigh as much as 10 , 2● , ●0 , or a hundred pound of Lead ; nay as much as a thousand , or ten thousand and pound weight ? THis proposition seems very impossible , yet water inclosed in a vessell , being constrained to dilate it selfe , doth weigh so much as though there were in the concavitie of it a solid body of water . There are many wayes to experiment this proposition , but to verifie it , it may be sufficient to produce two excellent ones onely : which had they not been really acted , little credit might have been given unto it . The first way is thus . Take a Magnitude which takes up as much place as a hundred or a thousand pound of water , and suppose that it were tied to some thing that it may hang in the aire ; then make a Ballance that one of the scales may inviron it , yet so that it touch not the sides of it : but leave space enough for one pound of water : then having placed 100 pound weight in the other scale , throw in the water about the Magnitude , so that one pound of water shall weigh downe the hundred pound in the other Ballance . PROBLEM . LXXXVI . Of sundry Questions of Arithmetick , and first of the number of sands . IT may be said incontinent , that to undertake this were impossible , either to number the Sands of Lybia , or the Sands of the Sea ; and it vvas this that the Poets sung , and that vvhich the vulgar beleeves ; nay , that vvhich long ago certaine Philosophers to Gelon King of Sicily reported , that the graines of sand vvere innumerable : But I ansvvere vvith Archimedes , that not only one may number those vvhich are at the border and about the Sea ; but those vvhich are able to fill the vvhole vvorld , if there vvere nothing else but sand ; and the graines of sands admitted to be so small , that 10 may make but one graine of Poppy : for at the end of the account there need not to expresse them , but this number 30840979456 , and 35 Ciphers at the end of it . Clavius and Archimedes make it somevvhat more ; because they make a greater firmament than Ticho Brahe doth ; and if they augment the Vniverse , it is easie for us to augment the number , and declare assuredly how many graines of sand there are requisite to fill another vvorld , in comparison that our visible vvorld vvere but as one graine of sand , an atome or a point ; for there is nothing to do but to multiply the number by it selfe , vvhich vvill amount to ninety places , vvhereof tvventie are these , 95143798134910955936 , and 70 Ciphers at the end of it : vvhich amounts to a most prodigious number , and is easily supputated : for supposing that a graine of Poppy doth containe 10 graines of sand , there is nothing but to compare that little bovvle of a graine of Poppy , vvith a bovvle of an inch or of a foot , & that to be compared vvith that of the earth , and then that of the earth vvith that o the firmament ; and so of the rest . 2. Divers metalls being melted together in one body , to finde the mixture of them . THis wat a notable invention of Archimedes , related by Vitrivius in his Architecture , where he reporteth that the Gold-smith which King Hiero imployed for the making of the Golden Crowne , which was to be dedicated to the gods , had stolen part of it , and mixed Silver in the place of it : the King suspicious of the work proposed it to Archimedes , if by Art he could discover without breaking of the Crowne , if there had been made mixture of any other metall with the Gold. The way which he found out was by bathing himselfe ; for as he entred into the vessell of water , ( in which he bathed himselfe ) so the water ascended or flew out over it , and as he pulled out his body the water descended : from which he gathered that if a Bowle of pure Gold , Silver , or other metall were cast into a vessell of water , the water proportionally according to the thing cast in would ascend ; and so by way of Arithmetick the question lay open to be resolved : who being so intensively taken with the invention , leapes out of the Bath all naked , crying as a man transported , I have found , I have found , and so discovered it . Now some say that he took two Masses , the one of pure Gold , and the other of pure Silver ; each equall to the weight of the Crowne , and therefore unequall in magnitude or greatnesse ; and then knowing the severall quantities of water which was answerable to the Crown , and the severall Masses , he subtilly collected , that if the Crowne occupied more place within the water than the Masse of Gold did : it appeared that there was Silver or other metall melted with it . Now by the rule of position , suppose that each of the three Masses weighed 18 pound a piece , and that the Masse of Gold did occupie the place of one pound of water , that of Silver a pound and a halfe ▪ and the Crown one pound and a quarter only : then thus he might operate the Masse of Silver which weighed 18 pounds , cast into the water , did cast out halfe a pound of water more then the Masse of Gold , which weighed 18 pound , and the Crowne which weighed also 18 pound , being put into a vessell full of water , threw out more water than the Masse of Gold by a quarter of a pound , ( because of mixt metall which was in it : ) therefore by the rule of proportion , if halfe a pound of water ( the excesse ) be answerable to 18 pound of Silver , one quarter of a pound of excesse shall be answerable to 9 pound of Silver , and so much was mixed in the Crowne . Some judge the way to be more facill by weighing the Crowne first in the aire , then in the water ; in the aire it weighed 18 pound , and if it were pure Gold , in the water it would weigh but 17 pound ; if it were Copper it would weigh but 16 pound ; but because vve vvill suppose that Gold and Copper is mixed together , it vvill vveigh lesse then 17 pound , yet more than 16 pound , and that according to the proportion mixed : let it then be supposed that it vveighed in the vvater 16 pound and 3 quarters , then might one say by proportion , if the difference of one pound of losse , vvhich is betvveen 16 and 17 ) be ansvverable to 18 pound , to vvhat shall one quarter of difference be ansvverable to , vvhich is betvveen 17 and 16 ¾ , and it vvill be 4 pound and a halfe ; and so much Copper vvas mixed vvith the Gold. Many men have delivered sundry vvayes to resolve this proposition since Archimedes invention , and it vvere tedious to relate the diversities . Baptista Benedictus amongst his Arithmeticall Theoremes , delivers his vvay thus : if a Masse of Gold of equall bignesse to the Crovvne did vveigh 20 pound , and another of Silver at a capacity or bignesse at pleasure , as suppose did vveigh 12 pound , the Crovvne or the mixt body would vveigh more than the Silver , and lesser than the Gold , suppose it vveighed 16 pound vvhich is 4 pound lesse than the Gold by 8 pound , then may one say , if 8 pound of difference come from 12 pound of Silver , from vvhence comes 4 pound vvhich vvill be 6 pound and so much Silver vvas mixed in it , &c. 3. Three men bought a quantitie of wine , each paid alike , and each was to have alike ; it happened at the last partition that there were 21 Barrells , of which 7 were full , 7 halfe full , and 7 empty , how must they share the wine and vessells , that each have as many vessells one as another , & as much wine one as another ? THis may be answered two wayes as followeth , and these numbers 2 , 2 , 3 , or 3 , 3 , 1 , may serve for direction , and signifies that the first person ought to have 3 Barrells full , and as many empty ones , and one which is halfe full ; so he shall have 7 vessells and 3 Barrels , and a halfe of liquor ; and one of the other shall in like manner have as much , so there will remaine for the third man 1 Barrell full , 5 which are halfe full , and 1 empty , and so every one shall have alike both in vessells and wine . And generally to answer such questions , divide the number of vessells by the number of persons , and if the Quotient be not an intire number , the question is impossible ; but when it is an intire number , there must be made as many parts as there are 3 persons , seeing that each part is lesse than the halfe of the said Quotient : as dividing 21 by 3 there comes 7 for the Quotient , which may be parted in these three parts , 2 , 2 , 3 , or 3 , 3 , 1 , each of which being lesse than ha●fe of 7. 4. There is a Ladder which stands upright against a wall of 10 foot high , the foot of it is pulled out 6 foot from the wall upon the pavement : how much hath the top of the Ladder descended ? THe ansvver is , 2 foot : for by Pythagoras rule the square of DB , the Hypothenusal is equall to the square of DA 6 , & AB 10. Novv if DA be 6 foot , and AB 10 foot , the squares are 36 and 100 , vvhich 36 taken from 100 rests 64 , vvhose Roote-quadrate is 8 so the foot of the Ladder being novv at D , the toppe vvill be at C , 2 foot lovver than it vvas vvhen it vvas at B. PROBLEM . LXXXVII . Witty suits or debates between Caius and Sempronius , upon the forme of f●gures , which Geometricians call Isoperimeter , or equall in circuit or compasse . MArvell ●ot at it if I make the Mathematicks take place at the Ba●●e , and if I set forth here B●rtoleus , who witnesseth of himselfe , that being then an ancient Doctor in the Law , he himselfe took upon him to learne the elements and principles of Geometry , by which he might set forth certaine Lawes touching the divisions of Fields , Waters , Islands , and other incident places : now this shall be to shew in passing by , that these sciences are profitable and behovefull for Judges , Counsellors , or such , to explaine many things which fall out in Lawes , to avoid ambiguities , contentions , and suits often . 1. Incident . CAius had a field which was directly square , having 24 measures in Circuit , that was 6 on each side : Sempronius desiring to fit himselfe , prayed Caius to change with him for a field which should be equivalent unto his , and the bargaine being concluded , he gave him for counterchange a piece of ground which had just as much in circuit as his had , but it was not square , yet Quadrangular and Rectangled , having 9 measures in length for each of the two longest sides , and 3 in breadth for each shorter side : Now Caius which was not the most subtillest nor wisest in the world accepted his bargaine at the first , but afterward● having conferred with a Land-measures and Mathematician , found that he was over-reached in his bargaine , and that his field contained 36 square measures , and the other field had but 27 measures , ( a thing easie to be knowne by multiplying the length by the breadth : ) Sempronius contested with him in suite of Law , and argued that figures which have equall Perimeter or circuit , are equall amongst themselves : my field , saith he , hath equall circuit with yours , therefore it is equall unto it in quantitie . Now this was sufficient to delude a Judge which was ignorant in Geometricall proportions , but a Mathematician will easily declare the deceit , being assured that figures which are Isoperemiter , or equall in circuit , have not alwayes equall capacitie or quantitie : seeing that with the same circuit , there may be infinite figures made which shall be more and more capable , by how much they have more Angles , equall sides , and approach nearer unto a circle , ( which is the most capablest figure of all , ) because that all his parts are extended one from anothes , and from the middle or Centre as much as may be : so we see by an infa●lible rule of experience , that a square is more capable of quantitie than a Triangle of the same circuit , and a Pentagone more than a square , and so of others , so that they be regular figures that have their sides equall , otherwise there might be that a regular Triangle , having 24 measures in circuit might have more capacitie than a rectangled Parallelogram , which had also 24 measures of circuit , as if it were 11 in length , and 1 inbreadth , the circuit is still 24 , yet the quantitie is but 11. and if it had 6 every way , it gives the same Perimeter , viz. 24. but a quantitie of 36 as before . 2. Incident . SEmpronius having borrowed of Caius a sack of Corne , which was 6 foot high and 2 foot broad , and when there was question made to repay it , Sempronius gave Caius back two sacks full of Corne , which had each of them 6 foot high & 1 foot broad : who beleeved that if the sackes were full he was repaid , and it seems to have an appearance of truth barely looked on . But it is most evident in demonstration , that the 2 sacks of Corn paid by Sempronius to Caius , is but halfe of that one sack which he lent him : for a Cylinder or sack having one foot of diameter , and 6 foot of length , is but the 4 part of another Cylinder , whose length is 6 foot , and his diameter is 2 foot : therefore two of the lesser Cylinders or sackes , is but halfe of the greater ; and so Caius was deceived in halfe his Corne. 3. Incident . SOme one from a common Fountaine of a City hath a Pipe of water of an inch diameter ; to have it more commodious , he hath leave to take as much more water , whereupon he gives order that a Pipe be made of two inches diameter . Now you will say presently that it is reason to be so bigge , to have just twice as 〈…〉 before : but if the Magistrate of the Citie understood Geometricall proportions , he would soon cause it to be amended , & shew that he hath not only taken twice as much water as he had before , but foure times as much : for a Circular hole which is two inches diameter is foure times greater than that of one inch , and therefore vvill cast out four times as much vvater as that of one inch , and so the deceit is double also in this . Moreover , if there vvere a heap of Corne of 20 foot every vvay , vvhich vvas borrovved to be paid next yeare ▪ the party having his Corne in heapes of 12 foot every vvay , and of 10 foot every vvay , proffers him 4 heapes of the greater or 7 heaps of the lesser , for his ovvne heap of 20 every vvay , vvhich vvas lent : here it seems that the proffer is faire , nay vvith advantage , yet the losse vvould be neare 1000 foot . Infinite of such causes do arise from Geometricall figures , vvhich are able to deceive a Judge or Magistrate , vvhich is not somevvhat seene in Mathematicall Documents . PROBLEM . LXXXVIII . Containing sundry Questions in matter of Cosmography . FIrst , it may be demanded , vvhere is the middle of the vvorld ? I speak not here Mathematically , but as the vulgar people , vvho ask , vvhere is the middle of the vvorld ? in this sence to speak absolutely there is no point vvhich may be said to be the middle of the surface , for the middle of a Globe is every vvhere : notvvithstanding the Holy Scriptures speake respectively , and make mention of the middle of the earth , and the interpreters apply it to the Citie of Jerusalem placed in the middle of Palestina , and the habitable vvorld , that in effect taking a mappe of the vvorld , and placing one foot of the Compasses upon Jerusalem , and extending the other foot to the extremity of Europe , Asia , and Afric● , you shall see that the Citie of Jerusalem is as a Centre to that Circle . 2. Secondly , how much is the depth of the earth , the height of the heavens , and the compasse of the world ? FRom the surface of the earth unto the Centre according to ancient traditions , is 3436. miles , so the vvhole thicknesse is 6872 miles , of which the whole compasse or circuit of the earth is 21600 miles . From the Centre of the earth to the Moone there is neare 56 Semidiameters of the earth , which is about 192416 miles . unto the Sunne there is 1142 Semidiameters of the earth , that is in miles 3924912 ; from the starry firmament to the Centre of the earth there is 14000 Semidiameters , that is , 48184000 miles , according to the opinion and observation of that learned Ticho Brahe . From these measures one may collect by Arithmeticall supputations , many pleasant propositions in this manner . First , if you imagine there were a hole through the earth , and that a Milstone should be let fall down into this hole , and to move a mile in each minute of time , it would be more than two dayes and a halfe before it would come to the Centre , and being there it would hang in the aire . Secondly , if a man should go every day 20 miles , it would be three yeares wanting but a fortnight , before he could go once about the earth ; and if a Bird should fly round about it in two dayes , then must the motion be 450 miles in an houre . Thirdly , the Moone runnes a greater compasse each houre , than if in the same time she should runne twice rhe Circumference of the whole earth . Fourthly , admit it be supposed that one should go 20 miles in ascending towards the heavens every day , he should be above 15 years before he could attaine to the Orbe of the Moone . Fifthly , the Sunne makes a greater way in one day than the Moone doth in 20 dayes , because that the Orbe of the Sunnes circumference is at the least 20 times greater than the Orbe of the Moone . Sixthly , if a Milstone should descend from the p●ace of the Sunne a thousand miles every houre , ( which is above 15 miles in a minute , farre beyond the proportion of motion ) it would be above 163 dayes before it would fall dovvne to the earth . Seventhly , the Sunne in his proper sphere moves more than seven thousand five hundred and seventy miles in one minute of time : novv there is no Bullet of a Cannon , Arrovv , Thunderbolt , or tempest of vvinde that moves vvith such quicknesse . Eightly , it is of a farre higher nature to consider the exceeding and unmoveable quicknesse of the starry firmament , for a starre being in the Aequator , ( which is just between the Poles of the world ) makes 12598666 miles in one houre which is two hundred nine thousand nine hundred and seventy foure miles in one minute of time : & if a Horseman should ride every day 40 miles , he could not ride such a compasse in a thousand yeares as the starry firmament moves in one houre , which is more than if one should move about the earth a thousand times in one houre , and quicker than possible thought can be imagined : and if a starre should flye in the aire about the earth with such a prodigious quicknesse , it would burne and consume all the world here below . Behold therefore how time passeth , and death hasteth on : this made Copernicus , not unadvisedly to attribute this motion of Primum mobile to the earth , and not to the starry firmament ; for it is beyond humane sense to apprehend or conceive the rapture and violence of that motion being quicker than thought ; and the word of God testifieth that the Lord made all things in number , measure , weight , and time . PROBLEM . XCII . To finde the Bissextile yeare , the Dominicall letter , and the letters of the moneth . LEt 123 , or 124 , or 125 , or 26 , or 27 , ( which is the remainder of 1500 , or 1600 ) be divided by 4 , which is the number of the Leape-yeare , and that which remaines of the division shewes the leap-yeare , as if one remaine , it shewes that it is the first yeare since the Bissextile or Leap-year , if two , it is the second year &c. and if nothing remaine , then it is the Bissextile or Leap-yeare , and the Quotient shews you how many Bissextiles or Leap-yeares there are conteined in so many yeares . To finde the Circle of the Sun by the fingers . LEt 123 , 24 , 25 , 26 , or 27 , be divided by 28 , ( which is the Circle of the Sunne or whole revolution of the Dominicall letters ) and that which remaines is the number of joynts , which is to be accounted upon the fingers by Filius esto Dei , coelum bonus accipe gratis : and where the number ends , that finger it sheweth the yeare which is present , and the words of the verse shew the Dominicall letter . Example . DIvide 123 by 28 for the yeare ( and so of other yeares ) and the Quotient is 4 , and there remaineth 11 , for which you must account 11 words : Filius esto Dei , &c. upon the joynts beginning from the first joynt of the Index , and you shall have the answer . For the present to know the Dominicall letter for each moneth , account from January unto the moneth required , including January , and if there be 8 , 9 , 7 , or 5 , you must begin upon the end of the finger from the thumbe and account , Adam degebat , &c. as many words as there are moneths , for then one shall have the letter which begins the moneth ; then to know what day of the moneth it is , see how many times 7 is comprehended in the number of dayes , and take the rest , suppose 4 , account upon the first finger within & without by the joynts , unto the number of 4 , which ends at the end of the finger : from whence it may be inferred that the day required was Wednesday , Sunday being attributed to the first joynt of the first finger or Index : and so you have the present yeare , the Dominicall letter , the letter which begins the Moneth , and all the dayes of the Moneth . PROBLEM . XCIII . To finde the New and Full Moone in each Moneth . ADde to t●e Epact for the yeare , the Moneth from March , then subtract that surplus from 30 , and the rest is the day of the Moneth that it vvill be New Moone , and adding unto it 14 , you shall have that Full Moone . Note THat the Epact is made alwayes by adding 11 unto 30 , and if it passe 30 , subtract 30 , and adde 11 to the remainder , and so ad infinitum : as if the Epact were 12 , adde 11 to it makes 23 for the Epact next year , to vvhich adde 11 makes 34 , subtract 30 , rests 4 the Epact for the yeare after , and 15 for the yeare follovving that , and 26 for the next , and 7 for the next , &c. PROBLEM . XCIV . To finde the Latitude of ● Countrey . THose that dwell between the North-Pole and the Tropicke of Cancer , have their Spring and Summer between the 10 of March , and the 13 of September : and therefore in any day between that time , get the sunnes distance by instrumentall observation from the zenith at noone , and adde the declination of the sun for that day to it : so the Aggragate sheweth such is the Latitude , or Poles height of that Countrey . Now the declination of the sunne for any day is found out by Tables calculated to that end : or Mechanically by the Globe , or by Instrument it may be indifferently had : and here note that if the day be between the 13 of September and the 10 of March , then the sunnes declination for that day must be taken out of the distance of the sunne from the zenith at noone : so shall you have the Latitude , as before . PRBOLEM XCV . Of the Climates of countreys , and to finde in what C●imate any countrey is under . CLimates as they are taken Geographically signifie nothing else but when the l●ngt● of the longest day of any place , is half an houre longer , or shorter than it is in another place ( and so of the sh●rtest day ) and this account to begin from the Equinoctia●l Circle , seeing all Countreys under it have the shortest and longest day that can be but 12 houres ; But all other Countreys that are from the Equinoctiall Circle either towards the North or South of it unto the Poles themselves , are said to be in some one Climate or other , from the Equinoctiall to either of the Poles Circles , ( which are in the Latitude of 66 degr . 30 m. ) between each of which Polar Circles and the Equinoctial Circle there is accounted 24 Climates , which differ one from another by halfe an hours time : then from each Polar Circle , to each Pole there are reckoned 6. other Climates which differ one from another by a moneths time : so the whole earth is divided into 60 Climates , 30 being allotted to the Northerne Hemisphere , and 30● to the Southerne Hemispheare . And here note , that though these Climats which are betweene the Equinoctiall and the Polar Circles are equall one unto the other in respect of time , to wit , by halfe an houre ; yet the Latitude , breadth , or internall , conteined between Climate and Climate , is not equall : and by how much any Climate is farther from the Equinoctiall than another Climate , by so much the lesser is the intervall between that Climate and the next : so those that are nearest the Equinoctial are largest , and those which are farthest off most contracted : and to finde what Climate any Countrey is under : subtract the length of an Equinoctiall day , to wit , 12 houres from the length of the longest day of that Countrey ; the remainder being doubled shews the Climate : So at London the longest day is neare 16 houres and a halfe ; 12 taken from it there remaines 4 houres and a halfe , which doubled makes 9 halfe houres , that is , 9 Climates ; so London is in the 9 climate . PROBLEM . XCVI . Of Longitude and Latitude of the Earth and of the Starres . LOngitude of a Countrey , or place , is an arcke of the Aequator conteined between the Meridian of the Azores , and the Meridian of the place , and the greatest Longitude that can be is 360 degrees . Note . That the first Meridian may be taken at pleasure upon the Terrestriall Globe or Mappe , for that some of the ancient Astronomers would have it at Hercules Pillars , which is at the straights at Gibraltar : Ptolomy placed it at the Canary Isl●nds , but now in these latter times it is held to be neare the Azores . But why it was first placed by Ptolomy at the Canary Islands , were because that in his time these Islands were the farthest westerne parts of the world that vvas then discovered . And vvhy it reteines his place novv at Saint Michaels neare the Azores , is that because of many accurate observations made of late by many expert Navigators and Mathematicians , they have found the Needle there to have no variation , but to point North and South : that , is to each Pole of the world : and why the Longitude from thence is accounted Eastwards , is from the motion of the Sunne Eastward , or that Ptolomy and others did hold it more convenient to begin from the Westerne part of the world and so account the Longitude Eastward from Countrey to Countrey that was then knowne ; till they came to the Easterne part of Asia , rather than to make a beginning upon that which was unknowne : and having made up their account of reckoning the Longitude from the Westerne part to the Eastern part of the world knowne , they supposed the rest to be all sea , which since their deaths hath been found almost to be another habitable world . To finde the Longitude of a Countrey . IF it be upon the Globe , bring the Countrey to the Brasen Meridian , and whatsoever degree that Meridian cuts in the Equinoctiall , that degree is the Longitude of that Place : if it be in a Mappe , then mark what Meridian passeth over it , so have you the Longitude thereof , if no Meridian passe over it , then take a paire of Compasses , and measure the distance betweene the Place and the next Meridian , and apply it to the divided parallel or Aequator , so have you the Longitude required . Of the Latitude of Countreys . LAtitude of a Countrey is the distance of a Countrey from the Equinoctiall , or it is an Arke of the Meridian conteined between the Zenith of the place and the Aequator ; which is two-fold , viz. either North-Latitude or South-Latitude , either of which extendeth from the Equinoctiall to either Pole , so the greatest Latitude that can be is but 90 degrees : If any Northern Countrey have the Artick Circle verticall , which is in the Latitude of 66. gr . 30. m. the Sun will touch the Horizon in the North part thereof , and the longest day will be there then 24 houres , if the Countrey have lesse Latitude than 66. degrees 30. m. the Sun will rise and set , but if it have more Latitude than 66. gr . 30 m. it will be visible for many dayes , and if the Countrey be under the Pole , the Sun will make a Circular motion above the Earth , and be visible for a half yeare : so under the Pole there will be but one day , and one night in the whole yeare . To finde the latitude of Countreys . IF it be upon a Globe , bring the place to the Brasen Meridian , and the number of degrees which it meeteth therewith , is the Latitude of the place . Or with a paire of Compasses take the distance between the Countrey and the Equinoctiall , which applied unto the Equinoctiall will shew the Latitude of that Countrey ; which is equall to the Poles height ; if it be upon a Mappe . Then mark what parallel passeth over the Countrey and where it crosseth the Meridian , that shall be the Latitude : but if ●o parallel passeth over it , then take the distance betweene the place and the next parallel , which applied to the divided Meridian from that parallel will shew the Latitude of that place . To finde the distance of places . IF it be upon a Globe : then with a paire of Compasses take the distance betweene the two Places , and apply it to the divided Meridian or Aequator , and the number of degrees shall shew ●e distance ; each degree being 60. miles . ●f it be in a Mappe ( according to Wrights pro●ection ) take the distance with a paire of Com●asses between the two places , and apply this distance to the divided Meridian on the Mappe right against the two places ; so as many degrees as is conteined between the feet of the Compasses so much is the distance between the two places . If the distance of two places be required in a particular Map then with the Compasses take the distance between the two places , and apply it to the scale of Miles , so have you the distance , if the scale be too short , take the scale between the Compasses , and apply that to the two places as often as you can , so have you the distance required . Of the Longitude , Latitude , Declination , and distance of the Starres . THe Declination of a starre is the nearest distance of a Star from the Aequator ; the Latitude of a Starre is the nearest distance of a Sarre from the Ecliptick : the Longitude of a Starre is an Ark of the Ecliptick conteined between the beginning of Aries , and the Circle of the Starres Latitude , which is a circle drawne from the Pole of the Ecliptick unto the starre , and so to the Ecliptick . The distance between two Sarres in Heaven is taken by a Crosse-staffe or other Instrument , and upon a Globe it is done by taking between the feet of the Compasses the two Starres , and applying it to the Aequator , so have you the distance betweene those two starre● . How is it that two Horses or other creatures being foaled or brought forth into the world at one and the same time , that after certaine dayes travell the one lived more dayes than the other , notwithstanding they dyed together in one and the sam● moment also ? THis is easie to be answered : let one of them travell toward the West and the other towards the East : then that which goes towards the West followeth the Sunne : and therefore shall have the day somewhat longer than if there had been no travell made , and that which goes East by going against the Sunne , shall have the day shorter , and so in respect of travell though they dye at one and the selfe same houre and moment of time , the one shall be older than the other . From which consideration may be inferred that a Christian , a Jew , and a Saracen , may have their Sabbaths all upon one and the same day though notwithstanding the Saracen holds his Sabath upon the Friday , the Jew upon the Saturday , and the Christian upon the Sunday : For being all three resident in one place , if the Saracen and the Christian begin their travell upon the Saturday , the Christian going West , and the Saracen Eastwards , shall compasse the Globe of the earth , thus the Christian at the conclusion shall gaine a day , and the Saracen shall lose a day , and so meet with the Jew every one upon his owne Sabbath . Certaine fine observations . 1 UNder the Equinoctiall the Needle hangs in equilibrio , but in these parts it inclines under the Horizon , and being under the Pole it is thought it will hang verticall . 2 In these Countreys which are without the Tropicall Circles , the Sunne comes East and West every day for a halfe yeare , but being under the Equinoctiall the Sun is never East , nor West ▪ but twice in the yeare , to wit , the 10. of March and the 13 of September . 3 If a ship be in the Latitude of 23 gr . 30 m. that is , if it have either of the Tropicks verticall : then at what time the Sunnes Altitude is equall to his distan●e from any of the Equinoctiall points , then t●e Sunne is due East or West . 4 If a ship be betweene the Equinoctiall and either of the Tropicks , the Sunne will come twice to one point of the compasse in the forenoone , that is , in one and the same position . 5 Vnder the Equinoctiall neare Guinea there is but two sorts of windes all the year , 6 moneths a Northerly winde , and 6 moneths a Southerly winde , and the flux of the Sea is accordingly . 6 If two ships under the Equinoctiall be 100. leagues asunder , and should sayle Northerly untill they were come under the Articke circle , they should then be but 50 leagues asunder . 7 Those which have the Artick circle , verticall : when the Sunne is in the Tropick of Cancer , the Sun setteth not , but toucheth the western part of the Horizon . 8 If the complement of the Sunnes height at noon be found equall to the Sunnes Declination for that day , then the ●quinoctiall is verticall : or a shippe making such an observation , the Equinoctiall is in the Zenith , or direct over them , by which Navigators know when they crosse the line , in their travels to the Indies , or other parts . 9 The Sunne being in the Equinoctiall , the extremity of the stile in any Sunne-dyall upon a plaine , maketh a right line , otherwise it is Eclipticall , Hyperbolicall , &c. 10 When the shadow of a man , or other thing upon a Horizontall 〈◊〉 is equall unto it in length , then is the Sunne in the middle point between the Horizon and the Zenith , that is , 45 degrees high . PROBLEM . XCVII . To make a Triangle that shall have three right Angles . OPen the C●passes at p●easure : and upon A , describe an Arke BC. then at the same opening , place one of the feet in B , and describe the Ark AC . Lastly , place one of the feet of the Compasses in C. and describe the Arke AB· so shall you have the sphericall Aequilaterall Triangle ABC , right angled at A , at B , and at C. that is , each angle comprehended 9● . degrees : which can never be in any plaine Triangle , whether it be Equilaterall , Isocelse , scaleve , Orthogonall , or Opigonall . PROBLEM . XCVIII . To divide a line in as many equall parts as one will , without compasses , or without seeing of it . THis Proposition hath a fallacie in it , & cannot be practised but upon a Maincordion : for the Mathematicall line which proceeds from the flux of a point , cannot be divided in that wise : One may have therefore an Instrument which is called Maincordion , because there is but one cord : and if you desire to divide your line into 3 parts , run your finger upon the frets untill you sound a third in musick : if you would have the fourth part of the line , then finde the fourth sound , a fifth , &c. so shall you have the answer . PROBLEM . XCIX . To draw a line which shall incline to another line , yet never meet : against the Axiome of Parallels . THis is done by help of a Conoeide line , produced by a right line upon one & the same plaine , held in great account amongst the Ancients , and it is drawne after this manner . Draw a right line infinitely , and upon some end of it , as at I , draw a perpendicular line I A. augment it to H. then from A. draw lines at pleasure to intersect the line I. M. in each of which lines from the right line , IM . transferre IH . viz. KB . LC.OD.PE.QF.MG . then from those points draw the line H.B.C.D.E.F.G. which will not meet with the line IM . and yet incline nearer and nearer unto it . PROBLEM . C. To observe the variation of the compasses , or needle in any places . FIrst describe a Circle upon a plaine , so that the Sun may shine on it both before noone and afternoone : in the centre of which Circle place a Gn●●on or wire perpendicular as AB , and an houre before noone marke the extremitie of the shadow of AB , which suppose it be at C. describe a Circle at that semidiamiter CDF . then after noone mark when the top of the shadow of AB . toucheth the Circle , which admit in D ; divide the distance CD into two equall parts , which suppose at E. draw the line EAF . which is the Meridian line , or line of North & South : now if the Arke of the Circle CD . be divided into degrees . place a Needle GH , upon a plaine set up in the Centre , and marke how many degrees the point of the Needle G , is from E. so much doth the Needle vary from the North in that place . PROBLEM . CI. How to finde at any time which way the wind is in ones Chamber , without going abroad ? VPon the Plancking or floore of a Chamber , Parlor , or Hall , that you intend to have this device , let there come downe from the top of the house a hollow post , in which place an Iron rod that it ascend above the house 10 , or 6 foot with a vane or a scouchen at it to shew the winds without : and at the lower end of this rod of Iron , place a Dart which may by the moving of the vane with the wind without , turne this Dart which is within : about which upon the plaister must be described a Circle divided into the 32 points of the Mariners Compasse pointed and distinguished to that end , then may it be marked by placi● to Compasse by it ; for having noted the North point , the East , &c ▪ it is easie to note all the rest of the points : and so at any time comming into this Roome , you have nothing to do but to look up to the Dart , which will point you out what way the winde bloweth at that instant . PROBLEM . CII . How to draw a parallel sphericall line with great ease ? FIrst draw an obscure line GF . in the middle of it make two points AB , ( which serves for Centres then place one foot of the Compasses in B , and extend the other foot to A , and describe the semicircle AC . then place one foot of the Compasses in A , and extend the other foot to C , and describe the semicircle CD . Now place the Compasses in B , and extend the other foot unto D , and describe the semicircle DF , and so ad infinitum ; which being done neatly , that there be no right line seene nor where the Compasses were placed , will seeme very strange how possibly it could be drawne with such exactnes , to such which are ignorant of that way . PROBLEM . CIII . To measure an in accessible distance , as the breadth of a River with the help of ones hat onely . THe way of this is easie : for having ones hat upon his head , come neare to the bank of the River , and holding your head upright ( which may be by putting a small stick to some one of your buttons to prop up the chin ) pluck downe the brim or edge of your hat untill you may but see the other side of the water , then turne about the body in the same posture that it was before towards some plaine , and marke where the sight by the brimme of the hat glaunceth on the ground ▪ for the distance from that place to your standing , is the breadth of the River required . PROBLEM . CIIII. How to measure a height with two strawes or two small stickes . TAke two strawes or two stickes which are one as long as another , and place them at right Angles one to the other , as AB . and AC . then holding AB . parallel to the ground , place the end A to the eye at A. and looking to the other top BC. at C. by going backward or forward untill you may see the top of the Tower or tree , which suppose at E. So the distance from your standing to the Tower or Tree , is equall to the height thereof above the levell of the eye : to which if you adde your ovvne height you have the whole height . Otherwise . TAke an ordinary square which Carpenters or other workemen use , as HKL . and placing H. to the eye so that HK . be levell , go back or come nearer untill that by it you may see the top M. for then the distance from you to the height is equall to the height . PROBLEM . CV . How to make statues , letters , bowles , or other things which are placed in the side of a high building , to be seen below of an equall bignesse . LEt BC. be a Pillar 7 yards high , and let it be required that three yards above the levell of the eye A , viz. at B. be placed a Globe , and 9 yards above B. be placed another , & 22. yards above that be placed another Globe : how much shall the Diameter of these Globes be , that at the eye , at A , they may all appeare to be of one and the same Magnitude : It is thus done , first draw a line as AK . and upon K. erect a perpendicular KX . divide this line into 27 parts ▪ and according to AK . describe an Arke KY . then from K ▪ in the perpendicular KX , account● ▪ par●s , viz at L. which shall represent the former three yardes , and draw the line LA. from L , in the said perpendicular reckon the diameter of the lesser Globe of what Magnitude it is intended to be : suppose SL , and draw the line SA . cutting the Arke VK . in N. then from K. in the perpendicular account 9 yards , which admit at T. draw TA , cutting YK. in O transferre the Arke MN , from A to P. and draw AP. which will cut the perpendicular in V. so a line drawne from the middle of VF . unto the visuall lines AI , and AV , shall be the diameter of the next Globe : Lastly , account from K. in the perpendicular XK . 22 parts , and draw the line WA . cutting YK in Q. then take the Arke MN , and transferre it from Q to R and draw AR ▪ which will cut the perpendicular in X so the line which passeth by the meddle of XW . perpendicular to the visuall line AW , and AX. be the Diameter of the third Globe , to wit 5 , 6. which measures transferred in the Pillar BC. which sheweth the true Magnitude of the Globes 1 , 2 , 3. from this an Architect or doth proportion his Images , & the foulding of the Robes which are most deformed at the eye below in the making , yet most perfect when it is set in his true height above the eye . PROBLEM . CVI. How to disg●is● or disfigure an Image , as a head , an arme , a whole body , &c. so that it hath no proportion the eares to become long : the nose as that of a swan , the mouth as a coaches entrance , &c yet the eye placed at a certaine point will be seen in a direct & exact proportion . I Will not strive to set a Geometricall figure here , for feare it may seeme too difficult to understand , but I will indeavour by discourse how Mechanically with a Candle you may perceive it sensible : first there must be made a figure upon Paper , such as you please , according to his just proportion , and paint it as a Picture ( which painters know well enough to do ) afterwards put a Candle upon the Table , and interpose this figure obliquely , between the said Candle and the Bookes of Paper , where you desire to have the figure disguised in such sort that the height passe athwart the hole of the Picture : then will it carry all the forme of the Picture upon the Paper , but with deformity ; follow these tracts and marke out the light with a Coles black head or Ink : and you have your desire . To finde now the point where the eye must see it in his naturall forme : it is accustomed according to the order of Perspective , to place this point in the line drawne in height , equall to the largenesse of the narrowest side of the deformed square , and it is by this way that it is performed . PROBLEM . CVII . How a Cannon after that it hath shot , may be covered from the battery of the enemy . LEt the mouth of a Cannon be I , the Cannon M. his charge NO , the wheele L , the axletree PB . upon which the Cannon is placed , at which end towards B , is placed a pillar AE· supported with props D , C , E , F , G ▪ about which the Axeltree turneth : now the Cannon being to shoot , it retires to H , which cannot be directly because of the Axletree , but it make a segment of a circle ▪ and hides himselfe behind the wal QR , and so preserves it selfe from the Enemies battery , by which meanes one may avoid many inconveniences which might arise : and moreover , one man may more easily replace it againe for another shot by help of poles tyed to the wall , or other help which may multiply the strength . PROBLEM . CVIII . How to make a Lever , by which one man may alone place a Cannon upon his carriage , or raise what other weight he would . FIrst place two thick boards upright , as the figure sheweth , pierced with holes , alike opposite one unto another as CD , and EF : & let L , and M , be the two barres of Iron which passeth through the holes GH , and F , K , the two supports , or props , AB . the Cannon , OP , the Lever , RS , the two notches in the Lever , and Q , the hooke where the burthen or Cannon is tyed to . The rest of the operation is ●cill , that the youngest schollers or learners cannot faile to performe it : to teach Minerva were in vaine , and it were to Mathematicians injury in the succeeding Ages . PROBLEM . CIX . How to make a Clock with one onely wheele . MAke the body of an ordinary Dyall , and divide the houre in the Circle into 12. parts : make a great wheele in height above the Axletree , to the which you shall place the cord of your counterpoize ▪ so that it may descend , that in 1● houres of time your Index or Needle may make one revolution , which may be knowne by a watch which you may have by you : then put a balance which may stop the course of the Wheele , and give it a regular motion , and you shall see an effect as just from this as from a Clock with many wheeles . PROBLEM . CX . How by help of two wheeles to make a Childe to draw up alone a hogshead of water at a time : and being drawne up shall cast out it selfe into another vessell as one would have it . LEt R be the Pit from whence water is to be drawne , P the hook to throw out the water when it is brought up ( this hook must be moveable ) let AB be the Axis of the wheele SF , which wheele hath divers forkes of Iron made at G , equally fastened at the wheele ; let I , be a Card , which is drawne by K , to make the wheele S , to turne , vvhich vvheele S , beares proportion to the vvheele T , as 8 to ● . let N be a Chaine of Iron to vvhich is tyed the vessel O , and the other vvhich is in the Pit : E● is a piece of vvood vvhich hath a mortes in 1 , and ● , by vvhich the Cord I , passeth , tyed at the vvall , as KH , and the other piece of timber of the little vvheele as M , mortified in likevvise for the chaine to passe through : draw the Cord I , by K , and the wheele will turne , & so consequently the wheele T , which will cause the vessell O to raise ; which being empty , draw the cord againe by Y , and the other vessell which is in the pit ●ill come out by the same reason . This is an invention which will save labour if practised ; but here is to be noted that the pit must be large enough , to the end that it conteine two great vessels to passe up and downe one by another ▪ PROBLEM . CXI . To make a Ladder of Cords , which may be carryed in ones pocket : by which one may easily mount up a Wall , or Tree alone . TAke two Pullies A , & D , unto that of A , let there be fastened a Cramp of Iron as B ; and at D , let there be fastened a staffe of a foot and a halfe long as F , then the Pully A : place a hand of Iron , as E , to vvhich tie a cord of an halfe inch thick ( vvhich may be of silk because it is for the pocket : ) then strive to make fast the Pully A , by the help of the Crampe of Iron B , to the place that you intend to scale ; and the staffe F , being tyed at the Pully D , put it betvveen your legges as though you vvould sit upon it : then holding the Cord C in your hand , you may guide your selfe to the place required ▪ vvhich may be made more facill by the multiplying of Pullies . This secret is most excellent in Warre , and for lovers , its supportablenesse avoids suspition . PROBLEM . CXII . How to make a Pumpe whose strength is marvelous by reason of the great weight of water that it is able to bring up at once , and so by continuance . LEt 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 , be the height of the case about two or three foot high , and broader according to discretion : the rest of the Case or concavity let be O : let the sucker of the Pumpe vvhich is made , be just for the Case or Pumpes head 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 , & may be made of vvood or brasse of 4 inches thick , having a hole at E , vvhich descending raiseth up the cover P , by which issueth forth the water , & ascending or raising up it shuts it or makes it close : RS , is the handle of the sucker tyed to the handle TX , which works in the post VZ . Let A , B , C , D , be a piece of Brasse , G the piece which enters into the hole to F , to keep out the Aire . H , I , K , L , the piece tyed at the funnell or pipe : in which playes the Iron rod or axis G , so that it passe through the other piece MN , which is tyed with the end of the pipe of Brasse . Note , that the lower end of the Cisterne ought to be rested upon a Gridiron or Iron Grate ▪ which may be tyed in the pit , by which means lifting up and putting downe the handle , you may draw ten times more water than otherwise you could . PROBLEM . CXIII . How by meanes of a Cisterne , to make water of a Pit continually to ascend without strength , or the assistance of any other Pumpe . LEt IL , be the Pit where one would cause water to ascend continually to ●●ach office of a house or the places which are separated from it : let there be made a receive● as A , well closed up with lead or other matter that aire enter not in , to which fasten a pipe of lead as at E , which may have vent at pleasure , then let there be made a Cisterne as B , which may be communicative to A , by helpe of the pipe G , from vvhich Cistern B , may issue the vvater of pipe D , vvhich may descend to H , vvhich is a little belovv the levell of the vvater of the pit as much as is GH . to the end of vvhich shall be soldered close a Cock vvhich shall cast out the vvater by KH . Novv to make use of it , let B be filled full of vvater , and vvhen you vvould have it run turne the Cock , for then the vvater in B , vvill descend by K. and for feare that there should be vacuity , nature vvhich abhors it , vvill labour to furnish and supply that emptinesse out of the spring F , and that the Pit dry not , the Pipe ought to be small of an indifferent capacity according to the greatnesse or smalnesse of the spring . PROBLEM . CXIIII . How out of a fountaine to cast the water very high : different from a Probleme formerly delivered . LEt the fountaine be BD , of a round forme ( seeing it is the most capable and most perfect figure ) place into it two pipes conjoyned as EA , and HC , so that no Aire may enter in at the place of joyning : let each of the Pipes have a cock G , & L : the cocke at G , being closed , open that at I ▪ & so with a squirt force the water through the hole at H , then close the Cocke at A , & draw out the squirt , and open the cock at G. the Aire being before rarified will extend his dimensions , and force the water with such violence , that it will amount above the height of one or two Pipes : and so much the more by how much the Machine is great : this violence will last but a little while if the Pipe have too great an opening , for as the Aire approacheth to his naturall place , so the force will diminish . PROBLEM . CXV . How to empty the water of a Cisterne by a Pipe which shall have a motion of it selfe . LEt AB , be the vessell ; CDE , the Pipe : HG , a little vessell under the greater , in which one end of the Pipe is , viz. C , and let the other end of the Pipe E. passing through the bottome of the vessell at F , then as the vessell filleth so will the Pipe , and when the vessell , shall be full as farre as PO , the Pipe will begin to runne at E , of his owne accord , and never cease untill the vessell be wholly empty . PROBLEM CXVI . How to squirt or spout out a great height , so that one pot of water shall last a long time . LEt there be prepared two vessels of Brasse , Lead , or of other matter of equal substance as are the two vessels AB , and BD , & let them be joyned together by the two Pillars MN , & EF : then let there be a pipe HG . which may passe through the cover of the vessell CD , and passe through AB , into G , making a little bunch or rising in the cover of the vessell AB , so that the pipe touch it not at the bottome : then let there be soldered fast another Pipe IL , which may be separated from the bottome of the vessell , and may have his bunchie swelling as the former without touching the bottome : as is represented in L , and passing through the bottome of AB , may be continued unto I , that is to say , to make an opening to the cover of the vessell AB , & let it have a little mouth as a Trumpet : to that end to receive the water . Then there must further be added a very smal Pipe which may passe through the bottome of the vessell AB , as let it be OP , and let there be a bunch ; or swelling over it as at P , so that it touch not also the bottome : let there be further made to this lesser vessell an edge in forme of a Basin to receive the water , which being done poure water into the Pipe IL , untill the vessell CD , be full , then turne the whole Machine upside downe that the vessell CD , may be uppermost , and AB , undermost ; so by helpe of the pipe GH , the water of the vessell CD , will runne into the vessel AB , to have passage by the pipe PO. This motion is pleasant at a feast in filling the said vessel with wine , which will spout it out as though it were from a boyling fountaine , in the forme of a threed very pleasant to behold . PROBLEM . CXVIII . How to practise excellently the reanimation of simples , in case the plants may not be transported to be replanted by reason of distance of places . TAke what simple you please , burne it and take the ashes of it , and let it be calcinated two houres between two Creusets wel luted , and extract the salt : that is , to put water into it in moving of it ; then let it settle : and do it two or three times , afterwards evaporate it , that is , let the water be boyled in some vessel , untill it be all consumed : then there will remaine a salt at the bottome , which you shall afterwards sowe in good Ground wel prepared : such as the Theatre of husbandry sheweth , and you shall have your desire . PROBLEM . CVIII . How to make an infalliable perpetuall motion . M●xe 5. or 6. ounces of ☿ with is equall weight of ♃ , grinde it together with 10. or 12 ounces of sublimate dissolved in a celler upon a Marble the space of foure dayes , and it will become like Oile , Olive , which distill with fire of chaffe or driving fire , and it will sublime dry substance , then put water upon the earth ( in forme of Lye ) which will be at the bottom of the Limbeck , and dissolve that which you can ; filter it , then distill it , and there will be produced very subtill Antomes , which put into a bottle close stopped , and keep it dry , and you shall have your desire , with astonishment to all the world , and especially to those which have travelled herein without fruit . PROBLEM . CXIX . Of the admirable invention of making the Philosophers Tree , which one may see with his eye to grow by little and little . TAke two ounces of Aqua fortis , and dissolve in it halfe an ounce of fine silver refined in a Cappell : then take an ounce of Aqua fortis , and two drams of Quick-silver : which put in it , and mixe these two dissolved things together , then cast it into a Viall of halfe a pound of water , which may be well stopped ; for then every day you may see it grow both in the Tree and in the branch . This liquid serves to black haire which is red , or white , without fading untill they fall , but here is to be noted that great care ought to be had in anointing the haire , for feare of touching the flesh : for this composition is very Corrosive or searching , that as soone as it toucheth the flesh it raiseth blisters , and bladders very painfull . PROBLEM . CXX . How to make the representation of the great world ? DRaw salt Niter out of salt Earth ▪ which is found along the Rivers side , and at the foot of Mountaines , where especially are Minerals of Gold and Silver : mix that Niter well cleansed with ♃ , then calcinate it hermetically ▪ then put it in a Limbeck and let the receiver be of Glasse , well luted , and alwayes in which let there be placed leaves of Gold at the bottome , then put fire under the Limbeck untill vapours arise which will cleave unto the Gold ; augment your fire untill there ascend no more , then take away your receiver , and close it hermetically , and make a Lampe fire under it untill you may see presented in it that which nature affords us : as Flowers , Trees , Fruits , Fountaines , Sunne , Moone , Starres , &c. Behold here the forme of the Limbeck , and the receiver : A represents the Limbeck , B stands for the receiver . PROBLEM . CXXI . How to make a Cone , or a Pyramidall body move upon a Table without springs or other Artificiall meanes : so that it shall move by the edge of the Table without falling ? THis proposition is not so thornie and subtile as it seemes to be , for putting under a Cone of paper a Beetle or such like creature , you shall have pleasure with astonishment & admiration to those which are ignorant in the cause : for this animall will strive alwayes to free herself from the captivity in which she is in by the imprisonment of the Cone : for comming neere the edge of the Table she will returne to the other side for feare of falling . PROBLEM CXXII . To cleave an Anvill with the blow of a Pistoll . THis is proper to a Warrier , and to performe it , let the Anvill be heated red hot as one can possible , in such sort that all the solidity of the body be softned by the fire : then charge the Pistoll with a bullet of silver , and so have you infallibly the experiment . PROBLEM . CXXIII . How to r●st a Capon carried in a Budget at a Saddle-bowe , in the space of riding 5 or 6 miles ? HAving made it ready and larded it , stuffe ●t with Butter ; then heat a piece of steele which may be formed round according to the length of the Capon , and big enough to fill the Belly of it , and then stop it with Butter ; then wrap it up well and inclose it in a Box in the Budget , and you shall have your desire : it is said that Count Mansfield served himse●fe with no others , but such as were made ready in this kinde , for that it loseth none of its substance , and it is dressed very equally . PROBLEM . CXXIV . How to make a Candle burne and continue three times as long as otherwise it would ? VNto the end of a Candle half●burned stick a farthing lesse or more , to make it hang perpendicular in a vessel of water , so that it swimme above the water ; then light it , and it will susteine it self & float in this manner ; and being placed into a fountaine , pond , or lake that runnes slowly , where many people assemble , it will cause an extreme feare to those which come therein in the night , knowing not what it is . PROBLEM . CXXV . How out of a quantitie of wine to extract that which is most windy , and evill , that it hurt not a sick Person ? TAke two vials in such sort that they be of like greatnesse both in th● belly and the neck ; fill one of them of wine , and the other of water : let the mouth of that which hath the water be placed into the mouth of that which hath the wine , so the water shall be uppermost , now because the water is heavier than the wine , it will descend into the other Viall , and the wine which is lowest , because it is highest will ascend above to supply the place of the water , and so there will be a mutuall interchange of liquids , and by this penetration the wine wil lose her vapors in passing through the water . PROBLEM CXXVI . How to make two Marmouzets , one of which shall light a Candle , and the other put it out ? Upon the side of a wall make the figure of a Marmouzet or other animall or forme , and right against it on the other wall make another ; in the mouth of each put a pipe or quill so artificially that it be not perceived ; in one of which place salt peter very fine , and dry and pulverised ; and at the end set a little match of paper , in the other place sulphur beaten smal , then holding a Candle lighted in your hand , say to one of these Images by way of command , Blow out the Candle ; then lighting the paper with the candle , the salt-peter wil blow out the Candle immediatly , and going to the other Image ( before the match of the Candle be out ) touch the sulphur with it and say , Light the Candle , & it will immediatly be lighted , which will cause an admiration to those which see the action , if it be wel done vvith a secret dexterity . PROBLEM . XXVII . How to keepe wine fresh as if it were in a celler though it were in the heat of Summer , and without Ice or snow , yea though it were carried at a saddles bow , and exposed to the Sun all the day ? SEt your wine in a viall of Glasse ; and place it in a Box made of wood , Leather , or such like : about which vial place Salt-peeter , and it will preserve it and keep it very fresh : this experiment is not a little commodious for those which are not neare fresh waters , and whose dwellings are much exposed to the Sunne . PUOBLEM . CXXVIII . To make a Cement which indureth or lasteth as marble , which resisteth aire and water without ever disjoyning or uncementing ? TAke a quantity of strong and gluing Morter vvell beaten , mixe vvith this as much nevv slaked Lime , and upon it cast Oile of Olive or Linseed-Oile , and it vvill become hard as Marble being applyed in time . PROBLEM . CXXIX . How to melt metall very quickly , yea in a shell upon a little fire . MAke a bed upon a bed of metall with pouder of Sulphur , of Salt-peeter , and saw-dust alike ; then put fire to the said pouder with a burning Charcole , and you shall see that the metall will dissolve incontinent and be in a Masse . This secret is most excellent , and hath been practised by the reverend father Mercen●● of the order of the Minims . PROBLEM . CXXX . How to make Iron or steele exceeding hard ? QVench your Blade or other Instrument seven times in the blood of a male Hog mixt with Goose-grease , and at each time dry it at the fire before you wet it : and it will become exceeding hard , and not brittle , which is not ordinary according to other temperings and quenchings of Iron : an experiment of small cost , often proved , and of great consequence for Armorie in warlike negotiations . PRBOLEM CXXXI . To preserve fire as long as you will , imitating the inextinguible fire of Vestales . AFter that you have extracted the burning spirit of the salt of ♃ , by the degrees of fire , as is required according to the Art of Chymistrie , the fire being kindled of it selfe , break the Limbeck , and the Irons which are found at the bottome will flame and appeare as burning Coles as soone as they feele the aire ; which if you promptly inclose in a viall of Glasse , and that you stop it exactly with some good Lute : or to be more assured it may be closed up with Hermes wax for feare that the Aire get not in . Then will it keep more than a thousand yeares ( as a man may say ) yea at the bottome of the Sea ; and opening it at the end of the time , as soone as it feeles the Aire 〈◊〉 takes fi●e ▪ with which you may light a Match . This secret merits to be travelled after and put in practice , for that it is not common , and full of astonishment , seeing that all kinde of fire lasteth but as long as his matter lasteth , and that there is no matter to be found that will so long in●●●e . Artificiall fire-Workes : Or the manner of making of Rockets and Balls of fire , as well for the Water , as for the Aire ; with the composition of Starres , Golden-rain , Serpen●s , Lances , Whee●s of fire and such like , pleasant and Recreative . Of the composition for Rockets . IN the making of Rockets , the chiefest thing to be regarded is the composition that they ought to be filled with ; forasmuch as that which is proper to Rockets which are of a lesse sort is very improper to those which are of a more greater forme ; for the fire being lighted in a great concave , which is filled with a quick composition , burnes with great violence ; contrarily , a weak composition being in a small concave , makes no effect : therefore we shall here deliver in the first place rules and directions , which may serve for the true composition , or matter with which you may charge any Rocket , from Rockets which are charged but with one ounce of Powder unto great Rockets which requireth for their charge 10 pound of Powder , as followeth . For Rockets of one ounce . Vnto each pound of good musket Powder smal beaten , put two ounces of smal Cole dust , and with this composition charge the Rocket . For Rockets of 2 or 3 ounces . Vnto every foure ounces and a halfe of powder dust , adde an ounce of Salt-peter , or to every 4 ounces of powder dust , adde an ounce of Cole dust . For Rockets of 4 ounces . Vnto every pound of Powder dust adde 4 ounces of Salt peter & one ounce of Cole dust : but to have it more slow , unto every 10. ounces of good dust powder adde 3 ounces of Salt-peter , and 3 ounces of Cole dust . For Rockets of 5 or 6 ounces . Vnto every pound of Powder dust , adde 3 ounces and a halfe of Salt peter , and 2 ounces and a halfe of Coledust , as also an ounce of Sulphur , and an ounce of fyle dust . For Rockets of 7 or 8 ounces . Vnto every pound of Powder dust adde 4 ounces of Salt peter , and 3 ounces of Sulphur . Of Rockets of 10 or 12 ounces . Vnto the precedent composition adde halfe an ounce of Sulphur , and it will be sufficient . For Rockets of 14 or 15 ounces . Vnto every pound of Powder dust adde 4 ounces of Salt peter , or Cole dust 2 ¼ ounces of Sulphur and file dust of 1 ¼ ounce . For Rockets of 1 , pound . Vnto every pound of Powder dust adde 3 ounces of Cole dust , and one ounce of Sulphur . Of Rockets of 2 , pound . Vnto every pound of Powder dust adde 9 ½ ounces of Salt peter , of Cole dust 2 1 / ● ounces , filedust 1 ● / 2 ounce , and of Sulphur ¾ of ounce . For Rockets of 3 , pound . Vnto every pound of Salt peter adde 6 ounces of Cole dust , and of Sulpher 4 , ounces . For Rockets of 4 , 5 , 6 , or 7 , pound . Vnto every pound of Salt peter adde 5 ounces of Cole dust , and 2 ½ ounces of Sulphur . For Rockets of 8 , 9 , or 10 pound . Vnto every pound of Salt peter , adde 5 ½ ounces of Cole dust , and of Sulphur 2 ½ ounces . Here note that in all great Rockets , there is no Powder put , because of the greatnesse of the fire which is lighted at once , which causeth too great a violence , therefore ought to be filled with a more weaker composition . Of the making of Rockets and other Fireworkes . FOr the making of Rockets of sundry kindes , divers moulds are to be made , with their Rolling pins , Breaths , Chargers , &c. as may be seen here in the figure . And having rolled a Case of paper upon the Rolling pin for your mould , fill it with the composition belonging to that mould as before is delivered : now may you load it on the top , with Serpents , Reports , Stars , or Golden Raine : the Serpents are made about the bignesse of ones little finger , by rolling a little paper upon a small stick , and then tying one end of it , and filling it with the mixt composition somewhat close , and then tying the other end . The reports are made in their paper-Cases as the Serpents , but the Paper somewhat thicker to give the greater report . These are filled with graine-Powder or halfe Powder and halfe composition , and tying both ends close , they are finished . The best kinde of starres are made with this mixture following ; unto every 4 ounces of Salt-peter , adde 2 ounces of Sulphur , and to it put 1. ounce of Powder-dust , and of this composition make your starres , by putting a little of it within a small quantity of towe ; and then tying it up in the form of a ball as great as an Hasel-Nut or a little Wal-nut , through which there must be drawne a little Primer to make it take fire . Touching the making of the Golden Raine , that is nothing but filling of Quilles with the composition of your Rockets somewhat hard . Now if the head of a Rocket be loaded with a thousand of those Quilles , it s a goodly sight to see how pleasantly they ●pread themselves in the Aire , and come downe like streames of Gold much like the falling downe of Snow being agitated by some turbulent winde . Of recreative fires . 1 PHil●strates saith , that if wine in a platter be placed upon a receiver of burning Coles , to exhale the spirit of it , and be inclosed within a Cupboard or such like place , so that the Aire may not go in , nor out , and so being shut up for 30 yeares , he that shall open it , having a wax Candle lighted , and shall put it into the Cubboard there will appeare unto him the figure of many cleare starres . 2 If Aquavitae have Camphire dissolved in it ; and be evaporated in a close Chamber , where there is but a Charcole fire , the first that enters into the Chamber with a Candle lighted , will be extremely astonished , for all the Chamber will seeme to be full of fire very subtile , but it will be of little continuance . 3 Candles which are deceitful are made of halfe powder , covered over with Tallow , and the other halfe is made of cleane Tallow , or Waxe , with an ordinary week ; this Candle being lighted , and the upper halfe consumed , the powder will take fire , not without great noise and astonishment to those which are ignorant of the cause . 4 A dozen or twenty smal Serpents placed secretly under a Candlestick that is indifferent big , which may have a hole passe through the socket of it to the Candle , through which a piece of primer may be placed , and setting a smal C●ndle in the socket to burne according to a time limited : which Candlestick may be set on a side Table without suspition to any ; then when the Candle is burned , that it fires the primer , that immediately will fire all the Serpents , which overthrowing the Candlestick will flye here and there , intermixing themselves , sometimes in the Aire , sometimes in the Planching , one amongst another , like the crawling of Serpents , continuing for a pretty while in this posture , and in extinguishing every one will give his report like a Pistoll ; This will not a little astonish some , thinking the house will be fired , though the whole powder together makes not an ounce , and hath no strength to do such an effect . How to make fire run up and downe , forward and backward TAke small Rockets , and place the taile of one to the head of the other upon a Cord according to your fancie , as admit the Cord to be ABCDEFG . give fire to the Rocket at A , which will flye to B , which will come back againe to A , and fire another at C , that will flie at D , which will fire another there , and fl●e to E , and that to F , and so from F , to G , and at G , may be placed a pot of fire , viz. GH . which fired will make good sport ▪ bec●u●e the Serpents which are in it will variously ●ntermix themselves in the Aire , and upon the ground , and every one will extinguish with a report : and here may you note that upon the Rockets may be placed fierie Dragons , Combatants , or such like to meet one another , having lights placed in the Concavity of their bodies which will give great grace to the action . How to make Wheels of fire . TAke a Hoop , and place two Lath● acrosse one the other ; upon the crossing of which make a hole , so that it may be placed upon a pin to turne easily , as the figure Q. sheweth upon the sides of which hoope or round Circle place your Rockets , to which you may place Lances of fire between each Rocket : let this wheele be placed upon a standard as here is represented , and place a piece of Primer from one Lance to another , then give fire at G , which will fire F , that B , that will fire D , that C , and that will fire the Rocket at A ▪ then immediatly the wheel will begin to move , and represent unto the spectators a Circle of changeable fire , and if pots of fire be tied to it , you will have fine sport in the turning of the wheele and casting out of the Serpents . Of night-Combatants . CLubbes , Targets , Faulchons , and Maces charged with severall fires , do make your night-Combatants , or are used to make place amongst a throng of people . The Clubbes at the ends are made like a round Panier with small sticks , filled with little Rockets in a spirall forme glu●d and so placed that they fire but one after another ; the Ma●es are of divers fashions , some made oblong at the end , some made of a sp●rall forme , but all made hollow to put in several composition , and are boared in divers places , which are for sundry Rockets , and Lances of weak composition to be fired at pleasure : The Faulchons are made of wood in a bowing forme like the figure A , having their backes large to receive many Rockets , the head of one neare the neck of another , glued and fastned well together , so that one being spent another may be fired . 〈◊〉 Targets are made of wooden thinne boards , which are channeled in spiral lines to containe primer to fire the Rockets one after another , which is all covered with thinne covering of wood , or Pastboard , boared with holes spirally also ; which Rockets must be glued and made fast to the place of the Channels : Now if two men , the one having a Target in his hand , and the other a Falchon , or Mace of fire , shall begin to fight , it will appeare very pleasant to the Spectators : for by the motion of fighting , the place will seem to be ful of streames of fire : and there may be adjoyned to each Target a Sunne or a burning Comet with Lances of fire , which will make them more beautifull and resplendent in that acti●n . Of standing Fires . SVch as are used for recreation , are Collossus , Statues , Arches , Pyramides , Chariots , Chaires of triumph and such like , which may be accommodated with Rockets of fire , and beautified with sundry other artificiall fires , as pots of fire for the Aire which may cast forth several figures , Scutchions , Rockets of divers sorts , Starres , Crownes , Leaters , and such like , the borders of which may be armed with sundry Lances of fire , of small flying Rockets with reports , flames , of small birds of Cypres , Lan●hornes of fire , Candles of divers uses , and colours in burning : and whatsoever the fancie of an ingenious head may allude unto . Of Pots of fire for the Aire , which are throwne out of one Case one after another of a long continuance . MAke a long Trunk as AG , and by the side AH , let there be a Channel which may be fiered with slow primer or composition ; then having charged the Trunk AG , with the Pots of fire for the Aire at IGEC , and make the Trunk AG , very fast unto a Post as IK , give fire at the top as at A , which burning downewards will give fire to C , and so throw out that Pot in the Aire , vvhich being spent , in the meane time the fire vvil-burne from B to D , and so fire E , and throvv it out also into the Ayre , and so all the rest one after another vvill be throvvne out : and if the Pots of fire for the Aire vvhich are cast out , be filled vvith diverse Fire-vvorkes , they vvill be so much the more pleasant to the beholders . These Trunks of fire doe greatly adorne a Firevvorke , and may conveniently be placed at each angle of the vvhole vvorke . Of Pots of fire for the ground . MAny Pots of fire being fired together do give a fine representation , and recreation to the spectators , and cause a vvonderfull shout amongst the common people vv ch are standers by ; for those Pots being filled vvith Balles of fire and flying Serpents for the Aire , they vvill so intermix one vvithin another , in flying here and there a little above the ground , and giving such a volley of reports that the Aire vvill rebound vvith their noise , and the vvhole place be filled vvith sundry streames of pleasant fire ; which serpents will much occupie these about the place to defend themselves in their upper parts , when they will no lesse be busied by the balls of fire , which seeme to annoy their feet . Of Balles of f●re . THese are very various according to a mans fancy ; some of which are made with very small Rockets , the head of one tyed to the neck of another : the ball being made may be covered over with pitch except the hole to give fire to it ; this Ball will make fine sport amongst the standers by , which will take all a fire , and rolle sometimes this way , sometimes that way , between the legs of those that are standers by ▪ if they take not heed , for the motion will be very irregular , and in the motion will cast forth severall fires with reports . In the second kind there may be a channell of Iron placed in divers places in spirall manner , against which may be placed as many small petards of paper as possible may be , the Channell must be full of slow comp●sition , and may be covered a● the former , and made fit with his Rockets in the middle : this Ball may be shot out of a morter Peece , or charged on the top of a Roc●et : for in its motion it will flye here and there , and give many reports in the Aire : because of the discharge of the petards . Of fire upon the Water . Places which are 〈◊〉 upon Rivers or great Ponds , are proper to make Recreative fres on : and if it be required to make some of consequence , such may conveniently be made upon two Bo●ts , upon which may be built two Beasts , Turrets , Pagins , Castles , or such like , to receive or hold the diversity of Fire workes that may be made within it , in which may play 〈◊〉 fires , Petards , &c. and cast out many simple Granadoes , Balls of fire to burne in the water-Serpents and other things , and often times these boates in their incounters may hang one in another , that so the Combatants with the Targets , and Maces may fight ; which will give great ▪ content to the eyes of those which are lookers on , and in the conclusion fire one another , ( for which end they were made : ) by which the dexterity of the one may be knowne in respect of the other , and the triumph and victory of the fight gotten . Of Balles of fire which move upon the water . THese may be made in forme of a Ball stuffed with other little Balls , glued round about and filled with composition for the water , which fiered , will produce marvellous and admirable effects , for which there must be had little Cannons of white Iron , as the ends of small funnels ; these Iron Cannons may be pierced in sundry places , to which holes , may be set small Balles ful of composition for the water which small Balls must be peirced deep and large , and covered with Pitch , except the hole : in which hole must be first placed a little quantitie of grain-Powder ; and the rest of the hole filled up with composition ; and note further that these Iron Cannons , must be filled with a slow composition ; but such which is proper to burne in the water : then must these Cannons with their small Balls be put so together that it may make a Globe , and the holes in the Cannons be answerable to the hollow Balls , and all covered over with Pitch and Tallow ; afterwards pierce this Ball against the greatest Cannon ( to which all the lesser should answer ) unto the composition , then fire it , and when it begins to blow , throw it into the water , so the fire comming to the holes will fire the graine Powder , the which will cause the Balls to separate and fly here and there , sometimes two at a time , sometimes three , sometimes more , which will burne within the water with great astonishment and content to those which see it . Of Lances of fire . STanding Lances of fire , are made commonly with hollow wood , to containe sundry Petards , or Rockets , as the figure here sheweth , by which is easie to invent others occording to ones fancy . These Lances have wooden handles , that so they may be fastned at some Post , so that they be not overthrowne in the flying out of the Rockets or Petards : there are lesser sorts of Lances whose cases are of three or foure fouldings of Paper of a foote long , and about the bignesse of ones finger , which are filled with a composition for Lances . But if these Lances be filled with a composition , then ( unto every 4 ouncs of powder add● 2 ounces of Salt-Peter , and unto that adde 1 ounce of Sulphur ) it will make a brick fire red before it be halfe spent , if the Lance be fiered and held to it : and if 20 such Lances were placed about a great Rocket and shot to a house or ship , it would produce a mischievous effect . How to shoot a Rocket Horizontall , or otherwise . VNto the end of the Rocket place an Arrow which may not be too heavy , but in stead of the feathers let that be of thinne white tinne plate , and place it upon a rest , as here you may see by the Figure , then give fire unto it , and you may see how serviceable it may be . To the head of such Rockets , may be placed Petards , Balls of fire , Granadoes , &c. and so may be applyed to warlike affaires . How a Rocket burning in the water for a certaine time , at last shall fly up in the Aire with an exceeding quickness . TO do this , take two Rockets , the one equall to the other , and joyne them one unto another in the middle at C. in such sort that the fire may easily passe from one to another : it being thus done , tye the two Rockets at a stick in D , and let it be so long and great that it may make the Rockets in the water hang , or lye upright : then take a pack-thread and tye it at G. and let it come double about the stick DM . at 〈◊〉 and at that point hang a Bullet of some weight as K. for then giving fire at A. it will burne to B. by a small serpent filled there and tyed at the end , and covered so that the water injure it hot , which will fire the Rocket BD , and so mounting quick out of the water by the loose tying at C. and the Bullet at the pack-thread , will leave the other Rocket in the vvater : and so ascend like a Rocket in the Aire , to the admiration of such as knovv not the secrecie . Of the framing of the parts of a Fire-Worke , together , that the severall workes may fire one after another . CAuse a frame to be made as ABCD. of tvvo foot square every vvay , or thereabouts ( according to the quantity of your severall vvorkes ) then may you at each angle have a great Lance of fire to stand , vvhich may cast out Pots of fire as they consume : upon the ledges AB.BC. and CD . may be placed small Lances of fire about the number of 30 or 60 , some sidevvise , and others upright , betvveen these Lances may be placed Pots of fire sloping outvvards , but made very fast , and covered very close , that they chance not to fire before they should ; then upon the ledges RE. FG.HI . and AD may be placed your soucisons , and behinde all the vvork may be set your Boxes of Rockets , in each of vvhich you may place 6 , 9 , ●2 . or 20 small Rockets : Novv give fire at A. ( by help of a piece of primer going from one Lance to another ) all the Lances vvill instantly at once be lighted , and as soone as the Lance at A is consumed , it vvill fire the Channell vvhich is made in the ledge of the frame vvhich runnes under the Pots of fire , and as the fire goes along burning , the Pots vvill be cast forth , and so the rank of Pots upon the sides of the frame AB.BC. and CD . being spent , the soucisons vvill begin to play being fiered also by a Channel vvhich runnes under them , upon the ledges AD , HI●G , and RE. then when the Soucisons are spent upon the last ledge RE. there may be a secret Channel in the ledge CD which may fire the Box of Rockets at K. and may fire all the rest one after another , which Boxes may be all charged with severall Fire-Workes : for the Rockets of the first Box may be loaden with Serpents , the second with Stars , the third with Reports , the fourth with Golden raine , and the fifth with small flying Serpents ; these mounting one after another and flying to and fro will much inlighten the Aire in their ascending , but when these Rockets discharge themselves above , then will there be a most pleasant representation , for these fires will dilate themselves in divers beautifull formes , some like the branching of Trees , others like fountaines of water gliding in the Aire , others like flashes of lightning , others like the glittering of starres , giving great contentment , and delight to those which behold them ; But if the worke be furnished also with Balons ( which is the chiefest in recreative Fire-works ) then shall you see ascending in the Aire but as it were onely a quill of fire , but once the Balon taking fire , the Aire will seeme more than 100. foot square full of crawling , and flying Serpents , which will extinguish with a volley of more than 500 reports : and so fill the Aire and Firmament with their rebounding clamour . The making of which with many other rare and excellent Fire-workes , and other practises , not onely for recreation , but also for service : you may finde in a book intituled Artificiall Fire-workes , made by Mr. Malthas ( a master of his knowledge ) and are to be sold by VVilliam Leake , at the Crowne in Fleet-street , between the two Temple-Gates . Conclusion . In this Booke we have nothing omitted what was materiall in the originall , but have abundantly augmented it in sundry experiments : And though the examinations are not so full , and manifold ; yet ( by way of brevitie ) we have expressed fully their substance , to avoid prolixitie , and so past by things reiterated . FINIS . Printed or sold by William Leak , at the Crovvne in Fleetstreet neere the Temple , these Books following . YOrk's Heraldry , Folio A Bible of a very fair large Roman letter , 4● Orlando F●rios● Folio . Callu learned Readings on the Scat. 21. Hen. 80. Cap 5 of Sewer● Perkins on the Laws of England . Wi●kinsons Office of She●●●fs . Vade Mecum , of a Justice of Peace . The book of Fees. Peasons Law. Mirrour of Just●ce . Topicks in the Laws of England . Sken de significatione Verborum . Delaman's use of the Horizontal Quadrant . Wilby's 2d set of Musique , 345 and 6 Parts . Corderius in English. D●ctor Fulk's Meteors . Malthus Fire-workes . Nyes Gunnery & Fire-workes C●to Ma●or with Annotations , by Wil. Austin Esquire . Mel Helliconium , by Alex. Ross● Nosce teipsum , by Sr John Davis Animadversions on Lil●i●s Grammer . The History of Vienna , & Paris Lazarillo de Tormes . Hero and L●ander , by G. Chapman and Christoph. Marlow . Al●ilia or Philotas loving folly . Bishop Andrews Sermons . Adams on ●eter . Posing of the Accidence . Am●dis de Gaule . Guillieliam's Heraldry . Herberts Travels . Bacc●s Tales . Man become guilty , by John Francis Sen●●t , and Englished by Henry Earl of Monmouth . The Ideot in 4 books ; the first and second of Wisdom ; the third of the Mind , the fourth of S●●tick Experiments of the Ballance . The life and Reign of Hen. the Eighth , written by the L. Herbet Cornwallis Essays , & Paradoxes . Clenards greek G●ammar 80 A●laluci● , or the house of light : A discourse written in the year 1651 , by SN . a modern Speculator . A Tragedy written by the most learned Hugo Grotius called , Christus Patience , and translated into Engl. by George Sand ▪ The Mount of Olives : or Sollitary Devotions , by Henry Vaughan Silurist VVith an excellent discourse of Man in glory , written by the Reverend Anselm Arch Bishop of Canterbury . The Fort Royall of Holy Scriptures by I. H. PLAYES . Hen. the Fourth . Philaster . The wedding . The Hollander . Maids Tragedie . King & no K. The gratefull Servant . The strange Discovery . Othello ; the Moor of Venice . The Merchant of Venice . THE DESCRIPTION AND USE OF THE DOVBLE Horizontall Dyall . WHEREBY NOT ONELY THE Houre of the Day is shewn ; but also the Meridian Line is found : And most ASTRONOMICALL Questions , which may be done by the GLOBE : are resolved . INVENTED AND WRITTEN BY W. O. Whereunto is added , The Description of the generall HOROLOGICALL RING . LONDON , Printed for WILLIAM LEAKE , and are to be sold at his Shop at the signe of the Crown in Fleetstreet , between the two Temple Gates . 1652. The description , and use of the double Horizontall Diall . THere are upon the Plate two severall Dyals . That which is outermost , is an ordinary diall , divided into houres and quarters , and every quarter into three parts which are five minutes a piece : so that the whole houre is understood to contein 60 minutes . And for this dyall the shadow of the upper oblique , or slanting edge of the style , or cocke , doth serve . The other diall , which is within , is the projection of the upper Hemisphaere , upon the plain of the Horizon : the Horizon it self is understood to be the innermost circle of the limbe : and is divided on both sides from the points of East and West into degrees , noted with 10.20.30 , &c. As far as need requireth : And the center of the Instrument is the Zenith , or Verticall point . Within the Horizon the middle straight line pointing North and South upon which the style standeth , is the Meridian or twelve a clock line : and the other short arching lines on both sides of it , are the houre lines , distinguished accordingly by their figures : and are divided into quarters by the smaller lines drawn between them : every quarter conteining 15 minutes . The two arches which crosse the houre lines , meeting on both sides in the points of intersection of the sixe a clocke lines with the Horizon , are the two semicircles of the Ecliptick or annuall circle of the sun : the upper of which arches serveth for the Summer halfe yeere ; and the lower for the Winter half yeer : and therefore divided into 365 dayes : which are also distinguished into twelve moneths with longer lines , having their names set down : and into tenths and fifts with shorter lines : and the rest of the dayes with pricks as may plainly be seene in the diall . And this is for the ready finding out of the place of the Sun every day : and also for the shewing of the Suns yeerely motion , because by this motion the Sun goeth round about the heavens in the compasse of a yeer , making the four parts , or seasons thereof ▪ namely , the Spring in that quarter of the Ecliptick which begins at the intersection on the East side of the diall ▪ and is therefore called the Vernall intersection . Then the Summer in that quarter of the Ecliptick which begin at the intersection with the Meridian in the highest point next the Zenith . After that , Autumne in that quarter of the Ecliptick which beginneth at the intersection on the West side of the diall , and is therefore called the Au●umna●l intersection and lastly , the Winter in that quarter of the Ecliptic● , which beginneth at the intersection , with the Meridian i● the lowest point next the Horizon . But desides this yeerely motte● , the Sun hath a diurnall , or daily motion , whereby it maketh day and night , with all the diversities and inaequalities thereof : which is expressed by those other circles drawn crosse the houre lines ; the middlemost whereof , being grosser then the rest , meeting with the Ec●iptick in the points of the Vernall , and Autumnall intersections ▪ is the Equinoctiall : and the rest on both sides of it are called the parallels , or diurnall arch of the Sun , the two outermost whereof are the Tropicks , because in them the sun hath his furthe●t digression or Declination from the Aequinoctiall , which is degrees 23 1 / ● ▪ and thence beginneth againe to return towards the Equinoctiall . The upper of the two Tropicks in this nor Northerne Hemisphere is the Trop●ck of Cancer , and the sun being in it , is highest into the North , making the longest day of Summer : And the lower next the Horizon is the Tropick of Capricorne ; and the sun being in it , is lowest into the South , making the shortest day of winter . Between the two Tropicks and the Aequinoctiall , infinite such parallel circles are understood to be conteined : for the sun , in what point soever of the Ecliptick it is carried ▪ describeth by his Lation a circle parallel to the Aequinoctiall : yet those parallels which are in the instrument , though drawn but to every second degree of Declination , may be sufficient to direct the eye in imagining and tracing out through every day of the whole yeere in the Ecliptick , a proper circle which may be the diurnall arch of the sun for that day . For upon the right estimation of that imaginary parallel doth the manifold use of this instument especially rely : because the true place of the sun all that day is in some part or point of that circle . Wherefore for the bet●er conceiving and bearing in minde thereof , every fift parallel is herein made a little g●osser then the rest . For this inner diall serveth the shadow of the upright edge of the style ; which I therefore call the upright shadow . And thus by the eye and view onely to behold and comprehend the course of the sun ▪ throughout the whole yeere both for his annuall and diurnall motion , may be the first use of this instrument . II Use. To finde the declination of the sun every day . Looke the day of the moneth proposed in the Ecliptick , and mark how many degrees the prick shewing that day , is distant from the Equinoctiall , either on the Summer or Winter side , viz. North or South . Example 1. What will the Declination of the sun be upon the eleven●h day of August ? look the eleventh day of August and you shall finde it in the sixth circle above the ●quinoctiall : Now because each parallel standeth ( as hath been said before ) for two degrees , the sun shall that day decline Northwards 12. degrees . Example 2. What declination hath the sun upon the 24 day of March ? look the 24 day of March , and you shall finde it betweene the second and third northern parallels , as it were an half and one fift part of that di●tance from the second : Reckon therefore four degrees for the two circles , and one de●ree for the halfe space : So shall the Suns declination be five degrees , and about one fift part of a degree Northward that same day . Example 3. What declination hath the sun upon the 13 day of November ? look the 13 day of November , and you shall finde it below the Equinoctiall ten parallels , and about one quarter which is 20 degrees and an halfe southward . So much is the declination . And according to these examples judge of all the rest . III. Use. To finde the diurnall arch , or circle of the suns course every day . The sun every day by his motion ( as hath been said ) describeth a circle parallel to the ●quinoctiall , which is either one of the circles in the diall , or some-where ●etween two of them . First , theref●re se●k the day of the moneth ; and if it fall upon one of those parallels ; that is the circle of the suns course that same day : But if it fall betweene any two of the parallels , imagine in your mind● , and estimate with your eye , another parallel th●ough that point betweene those two parallels keeping still the same distance from each of them . As in the first of the three former examples , The circle of the Suns course upon 11 of August ▪ shal be the very sixt circle above the Equinoctiall toward the cente● . In ●xample 2. The circle of the suns cou●se upon the 24 of March shall be an imaginary circle between the second and third parallels still keeping an half of that space , and one fifth part more of the rest , from the second . In example 3. The circle of the suns course upon the 13 of November : shall be an imaginary circle between the tenth and eleventh parallels below the Equinoctiall , still keeping one quarter of that space from the tenth . IIII Use. To finde the r●sing and setting of the sun eve●yday . 〈…〉 ( as was last shewed ) the imaginary circle or parallel of the suns course for that day , and marke the point where it meeteth with the horizon , both on the East and W●st sides , for that is the very point of the suns r●sing , and setting that same day , and the houre lines which are on both sides of it , by proportioning the distance reasonably , according to 15 minutes for the quarter of the houre , will shew the houre of the suns rising on the East side , and the suns setting on the West side . V Use. To know the reason and manner of the Increasing and decreasing of the nights●hroughout ●hroughout the whole yeere . When the Sun is in the Equinoctiall , it riseth and setteth at 6 a clock , for in the instrument the intersection of the Equinoctiall , and the Ecliptick with the Horizon is in the six a clocke circle on both sides . But if the sun be out of the Equinoctial , declining toward the North , the intersections of the parallel of the sun with the Horizon is before 6 in the morning , and after 6 in the evening : and the Diurnall arch greater then 12 houres ; and so much more great , the greater the Northerne Declination is . Againe , if the sun be declining toward the South , the intersections of the parallel of the sun , with the Horizon is after 6 in the morning , and before 6 in the evening : and the Diurnall arch lesser then 12 houres ; and by so much lesser , the greater the Southerne Declination is . And in those places of the Ecliptick in which the sun most speedily changeth his declination , the length also of the day is most a●tered : and where the Ecliptick goeth most parallel to the Equinoctiall changing the declination , but little altered . As for example , when the sun is neer unto the Equinoctiall on both sides , the dayes increase and also decrease suddenly and apace ; because in those places the Ecliptick inclineth to the Equinoctiall in a manner like a streight line , making sensible declination . Again , when the sun is neere his greatest declination , as in the height of Summer , and the depth of Winter , the dayes keep for a good time , as it were , at one stay , because in these places the Ecliptick is in a manner parallel to the Equinoctiall , the length o● the day also is but little , scarce altering the declination : And because in those two times of the yeer , the sun standeth as it were still at one declination , they are called the summer solstice , and winter solstice . And in the mean space the neerer every place is to the Equinoctiall , the greater is the diversity of dayes . Wherefore , we may hereby plainly see that the common received opinion , that in every moneth the dayes doe equally increase , is erroneous . Also we may see that in parallels equally distant from the Equinoctiall , the day on the one side is equall to the night on the other side . VI. Vse . To finde how far the sun riseth , and setteth from the true east and west points , which is called the suns Amp●itude ortive , and occasive . Seek out ( as was shewed in III Vse ) the imaginary circle , or parallel of the suns course , and the points of that circle in the horizon , on the East and West sides cutteth the degree of the Amplitude ortive , and occasive . VII Use. To finde the length of every day and night . Double the houre of the sunnes setting , and you shal have the length of the day ; & double the hour of the sunnes rising , and you shal have the length of the right . VIII Vse . To finde the true place of the sun upon the dyall , that is , the point of the instrument which answereth to the place of the sun in the heavens at any time , which is the very ground of all the questions following . If the dyall be fixed upon a post : Look what a clock it is by the outward dyall , that is , look what houre and part of houre the shadow of the slanting edge of the style sheweth in the outward limbe . Then behold the shadow of the upright edge , and marke what point thereof is upon that very houre and part in the inner dyall among the parallels , that point is the true place of the Sunne at the same instant . If the dyal be not fixed , and you have a Meridian line no●ed in any window where the Sunne shineth : place the Meridian of your dyal upon the Meridian line given , so that the top of the stile may point into the north : and so the dyal is as it were fixed , wherefore by the former rule you may finde the place of the Sunne upon it . If the dyal be not fixed , neither you have a Meridian line , but you know the true houre of the day exactly : hold the dyal even and parallel to the Horizon , moving it till the slanting edge of the stile cast his shadow justly upon the time or houre given ; for then the dyal is truly placed , as upon a post . Seek therefore what point of the upright shadow falleth upon that very houre , and there is the place of the Sun. But if your dyal be loose , and you know neither the Meridian nor the time of the day . First , by the day of the moneth in the Ecliptique , finde the su●s parallel , or d●urnall arch for that day ▪ then holding the dyal level to the horizon , move it every way untill the slanting shadow of the style in the outward limbe , and the upright shadow in the Sunnes diurnal arch , both shew the very same houre and minute , for that very point of the Sunnes parallel , which the upright shadow cutteth , is the true place of the Sun on the dyal at that present . But note that by reason of the thicknes of the style , and the bluntnesse of the angle of the upright edge , the Sun cannot come unto that edge for some space before and after noone . And so during the time that the Sunne shineth not on that upright edge , the place of the Sunne in the dyal cannot be found . Wherefore they that make this kinde of double dyal , are to be careful to file the upright edge of the style as thinne and sharpe as possible may be . That which hath here bin taught concerning the finding out the Suns true place in the dyal , ought perfectly to be understood , that it may be readily , and dexteriously practised , for upon the true performance thereof dependeth all that followeth . IX Vse . To finde the houre of the day . If the dyal be fastned upon a post , the houre by the outward dyal , or limbe , is known of every one , and the upri●ht shadow in the Suns parallel , or diurnal arch will also shew the very same houre . But if the dyall be loose , either hold it or set it parallel to the Horizon , with the style pointing into the north and move it gently every way untill the houre shewed in both dialls exactly agreeth , or which is all one , finde out the true place of the Sun upon the dyall , as was taught in the former question , for that point among the houre lines sheweth the houre of the day . X Vse . To finde out the Meridian , and other points of the Compasse . First , you must seek the tru● houre of the day ( by the last question ) for in that situation the Meridian of the dyall standeth direct●y north and south : and the east pointeth into the east , and the west into the west , and the rest of the points may be given by allowing degrees 11. 1 / ● unto every point of the compasse . XI Vse . To finde out the Azumith of the sun , that is , the distance of the Verticall circle , in which the sun is at that present , from the Meridian . Set your diall upon any plain or flat which is parallel to the horizon , with the Meridian pointing directly north or south , as was last shewed : then follow with your eye the upright shadow in a streight line , till it cutteth the horizon : for the degree in which the point of intersection is , shal shew how far the suns Azumith is distant from the east and west points , and the complement thereof unto 90 ; shal give the distance thereof from the meridian . XII Vse . To finde out the Declination of any Wall upon which the sun shineth , that is , how far that wall swerveth from the north or south , either eastward or westward . Take aboard having one streight edg ▪ & a line stricken perpendicular upon it ; apply the streight edg unto the wall at what time the sun shineth upon it , holding the board parallel to the horizon : Set the dyal thereon , and move it gently every way , untill the same hour and minute be shewed in both dyals : and so let it stand : then if the dyal have one of the sides parallel to the Meridian strike a line along that side upon the board , crossing the perpendicular , or else with a bodkin make a point upon the board , at each end of the meridian , and taking away the instrument from the board , and the board from the wall , lay a ruler to those two points , and draw a line crossing the perpendicular : for the angle which that line maketh with the perpendicular , is the angle of the decli●nation of the wall . And if it be a right angle , the wall is exactly east or west : but if that line be parallel to the perpendicular , the wall is direct north or south without any declination at all . You may also finde out the declination of a wall , if the dial be fixed on a post not very far from that wall ; in this manner . Your board being applyed to the wall , as was shewed , hang up a thred with a plummet , so that the shadow of the thred may upon the board crosse the perpendicular line : make two pricks in the shadow and run instantly to the dyal and look the horizontal distance of the suns Azumith , or upright shadow from the meridian . Then through the two pricks draw a line crossing the perpendicular : and upon the point of the intersection , make a circle equal to the horizon of your Instrument , in which Circle you shal from the line through the two pricks measure the Horizontal distance of the upright shadow , or Azumith from the meridian , that way toward which the Meridian is : draw a line out of the center , to the end of that arch measured : and the angle which this last line maketh with the perpendicular , shall be equall to the declination of the wall . XIII Vse . How to place the dyall upon a post without any other direction but it selfe . Set the diall upon the post , with the stile into the North , as neere as you can guesse : then move it this way and that way , till the same houre and minute be shewed , both in the outward and inward dials by the severall shadowes , as hath been already taught , for then the diall standeth in its truest situation ; wherefore let it be nailed down in that very place . XIIII Vse . To finde the height of the sun at high noon everyday . Seeke out the diurnall Arch or parallel of the suns course for that day , ( by Vse III. ) and with a paire of Compasses , setting one foot in the center , and the other in the point of intersection of that parallel with the Meridian , apply that same distance unto the Semidiameter divided : for that measure shal therein shew the degree of of the Suns altitude above the the Horizon that day at high noon . XV Vse . To finde the height of the sun at any houre or time of the day . Seeke out the diurnal Arch , or parallel of the suns course for that day : and marke what point of it is in the very houre and minute proposed . And with a paire of Compasses , setting one foot in the Center , and the other in that point of the parallel , apply the same distance upon the Semidiameter divided : for that measure shall shew the degree of the suns altitude above the Horizon at that time . And by this meanes you may finde the height of the Sun above the Horizon at every houre throughout the whole yeere , for the making of rings and cylinders and other instruments which are used to shew the houre of the day . XVI Vse . The height of the sun being given , to finde out the houre , or what it is a clocke . This is the converse of the former : Seeke therefore in the Semidiameter divided , the height of the sun given . And with a paire of Compasses , setting one foot in the center , and the other at that height , apply the same distance unto the diurnall arch , or parallel of the Sun for that day : for that point of the diurnall arch , upon which that same distance lights , is the true place of the sun upon the dial ; and sheweth among the houre lines , the true time of the day . XVII Use. Considerations for the use of the instrument in the night . In such questions as concerne the night ▪ or the time before sun rising , and after sun setting , the instrument representeth the lower Hemisphaere wherein the Southerne pole is elevated . And therefore the parallels which are above the Aequinoctiall toward the center shall be for the Southerne , or winter parallels : and those beneath the Aequinoctiall , for the Northerne or Summer paral●els ; and the East shall be accounted for West , and the West for East ; altogether contrary to that which was before , when the Instrument represented the upper Hemisphaere . XVIII Use. To finde how many degrees the sun is under the Horizon at any time of the night . Seeke the Declination of the sun for the day proposed ( by Vse II. ) And at the same declination the contrary side imagine a parallel for the sun that night ▪ and mark what point of it is in the very houre and minute proposed : And with a pair of compasses , setting one foot in the center , and the other in that point of the parallel , apply that same distance unto the semidiameter divided : for that measure shall shew the degree of the suns depression below the Horizon at that time . XIX Use. To finde out the length of the C●epusculum , or twylight , every day . Seek the declination of the sun for the day proposed ( by Vse II. ) And at the same declination on the contrary side imagine a parallel for the sun that night . And with a paire of compasses setting one foot in the center , and the other at 72 degrees upon the semidiameter divided , apply that same distance , unto the suns nocturnall parallel : for that point of the parallel , upon which that same distance shall light , sheweth among the houre lines , the beginning of the twilight in the morning , or the end of the twilight in the evening . XX Use. If the day of the moneth be not known , to finde it out by the dyall . For the working of this question , either the diall must be fixed rightly on a post , or else you must have a true Meridian line drawn in some window where the sun shineth , wherefore supposing the diall to be justly set either upon the post , or upon the Meridian . Look what a clock it is by the outward diall , and observe what point of the upright shadow falleth upon the very same minute in the inner diall , and through that same point imagine a parallel circle for the suns course ; that imaginary circle in the Ecliptick shall cut the day of the moneth . I The description of it . THis Instrument serveth as a Diall to finde the houre of the day , not in one place onely ( as the most part of Dials do ) but generally in all Countreys lying North of the Aequinoctiall : and therefore I call it the generall H●rologicall ●ing . It consisteth of two br●zen circles : a Diameter , and a little Ring to hang it by . The two circles are so made , that though they are to be set at right angles , when you use the Instrument : yet for more convenient carrying , they may be one folded into the other . The lesser of the two circles is for the Aequinoctiall , having in the midst of the inner side or thicknesse , a line round it , which is the true Aequinoctiall circle , divided into twice twelue hours , from the two opposite points in which it is fastened within the greater . The greater and outer of the two circles is the Meridian : One quarter whereof , beginning at one of the points in which the Aequin●cti●ll is hung , is divided into ninety degrees . The Diameter is fastened to the Meridian in two opposite points or poles , o●e of them being the very end of the Quadrant , and is the North Pole. Wherefore it is perpendicular to the ●quinoctiall , having his due position . The diameter is broad , and slit in the middle : and about the slit on both sides are the moneths and dayes of the yeer . And within this slit is a litt●e sliding plate pierced through with a small hole : which hole in the motion of it , while it is applied to the dayes of the yeer , representeth the Axis of the world . The little Ring whereby the Instrument hangeth , is made to slip up and down along the Quadrant : that so by help of a little tooth annexed , the Instrument may be rectified to any elevation of the Pole. II. The use of it . IN using this Instrument , First , the tooth of the little Ring must carefully be set to the height of the Pole in the Quadrant , for the place wherein you are . Secondly , the hole of the sliding plate within the slit , must be brought exactly unto the day of the moneth . Thirdly , the Aeqinoctiall is to be drawn out , and by means of the two studs in the Meridian staying it , it is to be set perpendicular thereto . Fourthly , Guesse as neer as you can at the houre , and turn the hole of the little plate toward it . Lastly , Hold the Instrument up by the little Ring , that it may hang freely with the North Pole thereof toward the North : and move it gently this way and that way , till the beams of the Sun-shining thorow that hole , fall upon that middle line within the Aequinoctiall : for there shall be the houre of the day : And the Meridan of the Instrument shall hang directly North and South . These Instrument all Dials are made in brasse by Elias Allen dwelling over against St. Clements Church without Temple Barre , at the signe of the Horse-shooe neere Essex Gate . FINIS 
A52264	----	Institutio mathematica, or, A mathematical institution shewing the construction and use of the naturall and artificiall sines, tangents, and secants in decimal numbers, and also of the table of logarithms in the general solution of any triangle, whether plain or spherical, with their more particular application in astronomie, dialling, and navigation / by John Newton. Newton, John, 1622-1678. 1654 Approx. 558 KB of XML-encoded text transcribed from 240 1-bit group-IV TIFF page images. Text Creation Partnership, Ann Arbor, MI ; Oxford (UK) : 2005-12 (EEBO-TCP Phase 1). A52264 Wing N1061 ESTC R20441 12355076 ocm 12355076 60119 This keyboarded and encoded edition of the work described above is co-owned by the institutions providing financial support to the Early English Books Online Text Creation Partnership. This Phase I text is available for reuse, according to the terms of Creative Commons 0 1.0 Universal . The text can be copied, modified, distributed and performed, even for commercial purposes, all without asking permission. Early English books online. (EEBO-TCP ; phase 1, no. A52264) Transcribed from: (Early English Books Online ; image set 60119) Images scanned from microfilm: (Early English books, 1641-1700 ; 217:14) Institutio mathematica, or, A mathematical institution shewing the construction and use of the naturall and artificiall sines, tangents, and secants in decimal numbers, and also of the table of logarithms in the general solution of any triangle, whether plain or spherical, with their more particular application in astronomie, dialling, and navigation / by John Newton. Newton, John, 1622-1678. [9], 420 p., [4] leaves of plates (2 folded) : ill. Printed by R. & W. Leybourn, for George Hurlock ... and Robert Boydel ..., London : 1654. First ed. Cf. NUC pre-1956. "The second part" has special t.p. Errata on p. [9]. Reproduction of original in British Library. 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Understanding these processes should make clear that, while the overall quality of TCP data is very good, some errors will remain and some readable characters will be marked as illegible. Users should bear in mind that in all likelihood such instances will never have been looked at by a TCP editor. The texts were encoded and linked to page images in accordance with level 4 of the TEI in Libraries guidelines. Copies of the texts have been issued variously as SGML (TCP schema; ASCII text with mnemonic sdata character entities); displayable XML (TCP schema; characters represented either as UTF-8 Unicode or text strings within braces); or lossless XML (TEI P5, characters represented either as UTF-8 Unicode or TEI g elements). Keying and markup guidelines are available at the Text Creation Partnership web site . eng Geometry -- Early works to 1800. Trigonometry -- Early works to 1800. Logarithms. Mathematics -- Problems, exercises, etc. 2005-05 TCP Assigned for keying and markup 2005-05 Aptara Keyed and coded from ProQuest page images 2005-06 Simon Charles Sampled and proofread 2005-06 Simon Charles Text and markup reviewed and edited 2005-10 pfs Batch review (QC) and XML conversion Institutio Mathematica . OR , A MATHEMATICAL Institution . Shewing the Construction and Use of the Naturall and Artificiall Sines , Tangents , and Secants , in Decimal Numbers , and also of the Table of Logarithms . In the general solution of any Triangle whether Plain or Spherical . WITH Their more particular application in ASTRONOMIE , DIALLING , and NAVIGATION . By JOHN NEWTON . LONDON , Printed by R. & W. Leybourn , for George Hurlock , at Magnus Church corner , and Robert Boydel in the Bulwark neer the Tower : MDCLIV . TO THE COVRTEOVS Reader . ALthough Mathematicall studies have for these many years been much neglected , if not contemned , yet have there been so many rare inventions found , even by men of our own Nation , that nothing now seems almost possible to be added more : as in other studies so may we say in these , nil dictum quod non dictum prius . We at the least must needs acknowledge that in this we have presented thee with nothing new , nothing that is our own . Ex integrâ Graecâ integram Comoediam , hodie sum acturus , Heautonti morum enon , saith Terence , that famous Comoedian : translation was his apologie , transcription and collection ours : this only we have endeavoured , that the first principles and foundations of these studies ( which until now were not to be known , but by being acquainted with many Books ) might in a due method and a perspicuous manner , be as it were at once , presented to thy view , and serve as a perfect INSTITUTION MATHEMATICAL , unto such as have as yet learned nothing but Arithmetick . To that purpose we have first laid down such propositions Geometricall , out of Euclide , Pitiscus , and others , as must be known to such as would understand the nature and mensuration of all Triangles . Next we have proceeded to the affections of Triangles in the generall , and thence to the composition of the Sines , Tangents , and Secants Naturall , in which we have for the most part followed the Rules prescribed by Pitiscus , in some things we have taken the direction of Snellius , and in the trisection and quinquisection of an angle we have proceeded Algebraically , with those two famous Mathematicians of our age and Nation , Brigges , and Oughtred ; and because the Algebraicall work is of it selfe abstruse and intricate , to those that are not acquainted with it , we have insisted the more upon it , and by our explanation we have endeavoured to make it plain and easie ; and that nothing may be wanting , which either former ages or our own ( by Gods blessing and their industry ) have afforded to us , we have to the composition of the Natural Canon , added out of Briggs and Wintage the construction of the Logarithms of any numbers , and consequently how to make the Logarithms of the Naturall Sines , Tangents , and Secants . This done , the proportions in the usuall Cases of all Triangles both Plain and Sphericall , we have first cleared by Demonstration out of Pitiscus , Gelibrand , Norwood , and others ; and then explained the manner of the work in Natural and Artificial Numbers both , and so conclude the first Part of our Institution . And in the second Part we have made our application of all the former unto Astronomie first , and then to Dialling and Navigation . In our application to Astronomie , we have furnished you with a Table of the Suns Motion , whereby to calculate his place in the Zodiac in Decimal numbers , and without which most of the other Problemes would be found ( if not useless ) yet very intricate and obscure , that being , for the most part , one of the three terms supposed to be given in Astronomical computations . In the Chapter of Dialling , you have the Sphears projection , according to the directions of Wels , in his Art of Shadows , and how to draw the houre-lines of all the severall Dials which he hath contrived from thence , we have briefly shewed ; and in the finding of all the arches in these Cases necessary , we have kept our selves to our own CANON , which doth exhibit the degrees of the Quadrant in Centecimal parts or minutes . In the Chapter of Navigation , you have first the division of the Sea-mans Compasse , next the description and making of the Sea-Chart , as Edward Wright our worthy Countrey-man hath given us the Demonstration thereof in his Book entituled , The correction of Errours in Navigation : to these we have added such other Problemes as are now amongst our Sea-men of most frequent use ; annexing thereunto a Table of meridionall parts , and other Tables usefull as well in Dialling as in Navigation , and all these in Decimall numbers , it being indeed our aim ( as much as in us lieth ) not only to promote these studies by this our Compendium of the first rudiments of Mathematical learning , as in relation to the matter therein to be considered , but by such expeditious and advantagious wayes of working also , as have been lately found , or former ages have commended to us ; amongst which there is none more excellent then that which is performed by Decimal numbers ; fully to explicate the manner and worth whereof were matter enough for a whole Treatise , and therefore not to be expected in a short Epistle : It would indeed be very impertinent to intermeddle any further with it here , then in our Institution it selfe is already explained , in which thou maist perceive Addition and Subtraction of Degrees , Minutes , and Seconds , to be performed as in Vulgar numbers , without any Reduction to their severall Denominations , Multiplication is performed by the addition of Cyphers , and Division by the cutting off of Figures . Others that have either spent more time , or made a farther progresse in these ravishing Studies , might ( if they would have taken the Pains ) have haply presented thee with more , and in a lesser room : The most of this was at the first collected for our private use , and now published for the good of others . John Newton . ERRATA . Page Line   6 25 Read , or termes . 17   For B. 9 A , r. B. 9 C. 16 20 read , 3 times 6 is 18. 22 12 r. one third 22 25 r. one third 30 9 r. triangles in the following figure . 32 21 r. by the 18 th . 34 21 r. are equiangled . 38 24 r. the arch CGK . 44 13 r. 21 of the first . 47 5 r. of 120 degrees . 49 4 r. by the 18. 55 10 r. 18 of the first . 60 9 r. by 18. 69 20 r. 77 16 r. 147. 77 21 r. 3913. 77 22 r. 206087. 91 3 r. 24986. 108 1 r. KP . 112 26 r. BD 156 6 r. BD.   7 r. DF.   8 r. BF A MATHEMATICAL Institution . CHAP. I. Geometricall Definitions . OF things Mathematicall there are two principall kindes , Number and Magnitude ; and each of these hath his proper Science . The Science of Number is Arithmetick , and the Science of Magnitude is commonly called Geometry , but may more properly be termed Megethelogia , as comprehending all Magnitudes whatsoever , whereas Geometry , by the very Etymologie of the word , doth seeme to confine this Science to Land-measuring only . Of this Megethelogia Geometry , or Science of Magnitudes , we will set down such grounds and principles as are necessary to be known , for the better understanding of that which follows , presuming that the reader hereof hath already gotten some competent knowledge in Arithmetick . Concerning then this Science of Magnitude , two things are to be considered : First , the severall heads to which all Magnitudes may be referred : And then secondly , the terms and limits of those Magnitudes . All Magnitudes are either Lines , Plains , or Solids , and do participate of Length , Breadth , or Thicknesse . 1. A Line is a supposed length , or a thing extending it self in length , without breadth or thickness , whether it be a right line or a crooked ; and may be divided into parts in respect of his length , but admitteth no other division : as the line AB . 2. The ends or limits of a line are points , as having his beginning from a point , and ending in a point , and therefore a Point hath neither part nor quantity , it is only the term or end of quantity , as the points A and B are the ends of the aforesaid line AB , and no parts thereof . 3. A Plain or Superficies is the second kind of magnitude , to which belongeth two dimensions , length , and breadth , but not thickness . 4. As the ends , limits or bounds of a line are points confining the line , so are lines the limits , bounds and ends inclosing a Superficies ; as in the figure you may see the plain or Superficies here inclosed with four lines , which are the extreams or limits thereof . 5. A Body or Solid is the third kinde of magnitude , and hath three dimensions belonging to it , length , breadth , and thickness . And as a point is the limit or term of a line , and a line the limit or term of a Superficies , so likewise a Superficies is the end or limit of a Body or Solid , and representeth to the eye the shape or figure thereof . 6. A Figure is that which is contained under one or many limits , Under one bound or limit is comprehended a Circle , and all other figures under many . 7. A Circle is a plain figure contained under one round line , which is called a circumference , as in the Figure following , the Ring CBDE is called the circumference of that Circle . 8. The Center of a Circle is that point which is in the midst thereof , from which point , all right lines drawn to the circumference are equal the one to the other ; as in the following figure , the lines AB , AC , AD , and AE , are equal . 9. The Diameter of a Circle , is a right line drawn through the center thereof , and ending at the circumference on either side , dividing the Circle into two equal parts , as the lines CAD and BAE , are either of them the diameter of the Circle BCDE , because that either of them doth passe through the center A , and divideth the whole Circle into two equal parts . 10. The Semidiameter of a Circle is half the Diameter , and is contained betwixt the center and one side of the Circle ; as the lines AB , AC , AD , and AE , are either of them the Semidiameters of the Circle CBDE . 11. A Semicircle is the one halfe of a Circle drawn upon his Diameter , and is contained by the half circumference and the Diameter ; as the Semicircle CBD is halfe the Circle CBDE , and contained above the Diameter CAD . 12. A Quadrant is the fourth part of a Circle , and is contained betwixt the Semidiameter of the Circle , and a line drawn perpendicular unto the Diameter of the same Circle , from the center thereof , dividing the Semicircle into two equal parts , of the which parts the one is the Quadrant or fourth part of the same Circle . Thus , the Diameter of the Circle BDEC is the line CAD , dividing the Circle into two equal parts , then from the center A raise the perpendicular AB , dividing the Semicircle likewise into two equal parts , so is ABD , or ABC , the Quadrant or fourth part of the Circle . 13. A Segment or portion of a Circle is a figure contained under a right line and a part of the circumference of a Circle , either greater or lesser than the Semicircle ; as in the former figure , FBGH is a segment or part of the Circle CBDE , contained under the right line FHG lesse than the Diameter CAD . 14. By the application of several lines ●terms of a Superficies one to another , are made Parallels , Angles , and many sided Figures . 15. A Parallel line is a line drawn by the side of another line , in such sort that they may be equidistant in all places , and of such there are two sorts , the right lined parallel , and the circular parallel . Right lined Parallels are two right lines equidistant in all places one from the other , which being drawn to an infinite length would never meet or concur ; as may be seen by these two lines , AB and CD . A Circular Parallel is a Circle drawn within or without another Circle , upon the same center , as you may plainly see by the two Circles BCDE , and FGHI , these Circles are both of them drawn upon the same center A , and therefore are parallel one to the other . 16. An Angle is the meeting of two lines in any sort , so as they both make not one line ; as the two lines AB and AC incline the one to the other , and touch one another in the point A , in which point is made the angle BAC . And if the lines which contain the angle be right lines , then it is called a right lined angle ; as the angle BAC . A crooked lined angle is that which is contained of crooked lines ; as the angle DEF : and a mixt angle is that which is contained both of a right and crooked line ; as the angle GHI : where note that an angle is ( for the most part ) described by three letters , of which the second or middle letter representeth the angular point ; as in the angle BAC , A representeth the angular point . 17. All Angles are either Right , Acute , or Obtuse . 18. When a right line standeth upon a right line , making the angles on either side equal , either of those angles is a right angle , and the right line which standeth erected , is a perpendicular line to that upon which it standeth . As the line AB ( in the following figure ) falling upon the line CBD perpendicularly , doth make the angles on both sides equal , that is , the angle ABC is equal to the angle ABD , and either of those angles is therefore a right angle . 19. An acute angle is that which is lesse than a right angle ; as the angle ABE is an acute angle , because it is less than the right angle ABD , in the former figure . 20. An Obtuse Angle is that which is greater than a right angle ; CBE in the former figure is greater than the angle ABC by the angle ABE , and therefore it is an obtuse angle . 21. The measure of every angle is the arch of a Circle described on the angular point , as in the following figure , the arch CD is the measure of the right angle CED . The arch BC is the measure of the acute angle BEC . And the arch BCD is the measure of the obtuse angle BED . But of their measure there can be no certain knowledge , unlesse the quantity of those arches be exprest in numbers . 22. Every Circle therefore is supposed to be divided into 360 equall parts , called Degrees , and every Degree into 60 Minutes , every Minute into 60 Seconds , and so forward . This division of the Circle into 360 parts we shall retain , but every Degree we will suppose to be divided into 100 parts or Minutes , & every Minute into 100 Seconds : and thus all Calculations will be much easier , and no lesse certain . 23. A Semicircle is the halfe of a whole circle containing 180 degrees . A Quadrant or fourth part of a circle is 90 degrees . And thus the measure of the right angle CED is the arch CD 90 degrees . The measure of the acute angle BEC is the arch BC 30 degrees . And the measure of the obtuse angle BED is the arch BD 120 degrees . 24. The complement of an angle to a Quadrant is so much as the angle wanteth of 90 degrees , as the complement of the angle AEB 60 degrees is the angle BEC 30 degrees ; for 30 and 60 do make a Quadrant or 90 degrees . 25. The complement of an angle to a Semicircle is so much as the said angle wanteth of 180 degrees , as the complement of the angle BED 120 degrees , is the angle AEB , 60 degrees ; for 60 and 120 do make 180 degrees . 26. Many sided figures are such as are made of three , four , or more lines , though for distinction sake , those only are so called which are contained under five lines or terms at the least . 27. Four sided figures are such as are contained under four lines or terms , and are of divers sorts . 1. There is the Quadrat or Square whose sides are equall and his angles right . 2. The Long Square whose angles are right , but the sides unequal . 3. The Rhombus or Diamond , having equall sides but not equal angles . 4. The Rhomboides , having neither equal sides nor equal angles , and yet the opposite sides and angles equal . All other figures of four sides are called Trapezia or Tables . The dimension whereof as also of all figures whatsoever , dependeth upon the knowledge of three sided figures , or Triangles , of which in the Chapter following . CHAP. II. Of the nature and quality of Triangles . 1. A Triangle is a figure consisting of three sides and three angles . 2. Every of the two sides of any Triangle are the sides of the angle comprehended by them , the third side is the Base , as in the figure following , the sides AC and BC and sides of the angle BCA , and AB is the Base of the said angle . 3. Every side is said to subtend the angle that is opposite to that side ; as the side AB subtendeth the angle ACB , the side AC subtendeth the angle ABC , and the side BC subtendeth the angle BAC : the greater sides subtend the greater angles , the lesser sides lesser angles , and equal sides equal angles . 4. Of Triangles there are diverse sorts ; as , 1. There are Equilateral Triangles , having three equal sides . 2. There is an Isoscheles , which is a Triangle that hath two equal sides . 3. Scalenum , which is a Triangle whose sides are all unequal . 4. An Orthigonium , or a right angled Triangle , having one right angle . 5. An Ambligonium , or an obtuse angled Triangle , having in it one obtuse angle . 6. An Oxigonium , or an acute angled Triangle , having all his angles acute . 7. All these Triangles are either Plain or Spherical . 8. The sides of Plain Triangles in Trigonometria are right lines only , concerning which we have added these Theorems following . 9. Theorem . If one right line cut through two parallel right lines , then are the angles opposite one against another equal . In the following Scheme the two lines WX and YZ are parallel , and therefore the angles XIC , and ICY are equal . Demonstration . The two angles XIC and WIC are equal to two right angles , as also ICY and ICZ , because on the parallel lines at the points I and C there may be drawn two Semicircles , each of which are the measures of two right angles . If then the angle XIC be lesse than ICY , the angle WIC must as much exceed the angle ICZ , and the angles XIC and ZCI would be lesse than two right angles , and consequently the lines WX and YZ may be extended on the side X and Z til at length they shall concur together , and then the lines WX and YZ are not parallels , as is supposed here , and therefore the angles XIC and ICY are equal . 10. Theor , If four right lines be proportionall , the right angled figure made of the two means , is equal to the right angled figure made of the two extreams . Let the four proportional lines be AB two foot , EF three foot , FG six foot , and BC nine foot : I say then that the right angled figure made of the two means EF and FG , that is , the right angled figure EF GH , is equal to the right angled figure made of the extreams AB and BC , that is , to the right angled figure ABCD ; for as twice 9 is 18 , so likewise three times is 18. 11. Theor. If three right lines be proportionall , the Square made of the mean is equal to the rightangled figure made of the extreams . Demonstration . The Demonstration of this proposition is all one in effect with the former , the difference is , that here is spoken of three lines , there of four , and therefore if we take the mean twice , of which the square is made , the work will be the same with that in the former proposition . As if the length of the first line were two foot , the second four , and the third eight ; it is evident , that as four times four is 16 , so two times eight is 16 , and therefore what hath been said of four proportionals , is to be understood of three proportionals also . 12. Theor. If a right line being divided into two equal parts , shall be continued at pleasure , then is the right angled figure made of the line continued , and the line of continuation , with the square of one of the bi-segments , equal to a square made of one of the bi-segments and the line of continuation . The line PQ is divided into two equal parts , the midst is C , to the same is added a right line , as QN ; and of the whole line PQ , and the added line QN is made PN as one line , and of this line PN , and the added line QN is inclosed the right angled figure ●M , and upon the halfe line CQ and the line of continuation QN is made the square CF. Now if you draw the line QG parallel to NF , and equal to the same , then is the right angled figure ●M with the square of CQ that is , the square IG , equal to the square of CN , that is , the square CF. Demonstration . Forasmuch as CQ is equal unto MF , the which is also equal unto IO or I L , it followeth that IG is a Square , which with the right angled figure PM is equal to the square CF , because the right angled figure GM is equal to CO , which is also equal to PI. 13. Theor. To divide a right line in two parts , so that the right , angled figure made of the whole line and one part shall be equal to the square of the other part . The right line given is AB , upon the same line AB , make a square , as ABCD ; and divide the side AD in two equal parts , the midst is M , from M draw a line to B , and produce AD to H , so that MH be equal to MB ; and upon AH make a square , as AHGF. Then extend GF to E , and then is the right angled figure FC , being made of the whole line FE ( which is equal to AB ) and the part BF , equall to the square of the other part AF , that is , to the square AHGF. Forasmuch as by the last aforegoing , the right angled figure comprehended of HD and HA , or the right angled figure of HD and HG , as the figure GHED , with the square of AM , are together equal to the square of HM , being equall to BM : it followeth , that if we take away the square ●f AM , common to both , that the square ●f AB , that is , the square ABCD is equal ●o the right angled figure HGED , and the ●ommon right angled figure AE being taken from them both , there shall remain the right angled figure FC , equal to the square ●HFG , which was to be proved . 14 Theor. To divide a right line given by extream and mean proportion . A right line is said to be divided by an extream and mean proportion , when the whole is to the greater part , as the greater is to the lesse . And thus a right line being divided , as the right line AB is divided in the preceding Diagram in the point F , it is divided in extream and mean proportion ; that is , As AB , is to AF : so is AF , to BF . Demonstration . Forasmuch as the right lined figure included with AB and FB , as the figure FBCE is equal to the square of AF , that is , to the square AFGH ; it followeth , by the eleventh Theorem of this Chapter , that the line AB is divided in extream and mean proportion ; that is , As AB , is to AF : So is AF , to FB . 15 Theor. In all plain Triangles , a line drawn parallel to any of the sides , cutteth the other two sides proportionally . As in the plain Triangle ABC , KL being parallel to the base BC , it cutteth off from the side AC one fourth , and also it cutteth off from the side AB one third part : the reason is , because the right line EH cutteth off one third part from the whole space DGFB , & therefore it cutteth off one third part from all the lines that are drawn quite through that space . And hereupon parallel lines bounded with parallels are equal ; as the parallels ED and GH being bounded with the parallels DG and HE are equal , for since the whole lines DB and GF are equall , DE and GH being one fourth part thereof , must needs be equal also . 16 Theor. Equiangled Triangles have their sides about the equall angles proportionall , and contrarily . Let ABC and ADE be two plain equiangled Triangles , so as the angles at B and D , at A and A , and also at C and E be equal one to the other ; I say , their sides about the equal angles are proportionall ; that is , 1 As AB , is to BC : So is AD , to ED. 2 As AB , is to AC : So is AD , to AE . 3 As AC , is to CB : So is AE , to ED. Demonstration . Because the angles BAC and DAE are equal by the Proposition ; therefore if A + B be applied to AD , AC shall fall in AE ; and by such application is this figure made . In which , because that AB and AD do meet together , and also that the angles at B and D are equall , by the Proposition ; therefore the other sides BC and DE are parallel ; and , by the last aforegoing , BC cutteth the sides AD and AE proportionally : and therefore , As AB , to AD : So is AC , to AE . Moreover , by the point B , let there be drawn the right line BF parallel to the base AE , and it shall cut the other two sides proportionally in the points B and F , and therefore , 1. As AB to AD : so is EF to ED , Or thus . As AB to AD : so is CB to ED : because that FE and BC are equal , by the last aforegoing . 1. Theor. In all right angled plain Triangles , the sides including the right angle are equal to the the third side . In the right angled plain triangle ABC , right angled at B , the sides AB and BC are equal in power to the third side AC ; that is the squares of the sides AB and BC , to wit , the squares ALMB and BEDC added together , are equal to the square of the side AC , that is to the square ACKI . Demonstration . 18. Theor. The three angles of a right lined Triangle are equal to two right angles . As in the following plain Triangle ABC the three angles ABC , ACB , and CAB are equal to two right angles . Let the side AB be extended to D , and let there be a semicircle drawn upon the point B , and let there be also dawn a line parallel unto AC , from B unto G. Demonstration . I say that the angle GBD is equal to the angle BAC , by the 9 th hereof , and the angle CBG is equal to the angle ACB by the same reason , and the angles CBG and GBD , are together equal to the angle CBD , which is also equal to the angle ABC , by the 18 th . of the first : and therefore ; the three angles of a right lined Triangle are equal to two right angles , which was to be proved . 19. Theor. If a plain Triangle be inscribed in a Circle , the angles opposite to the circumference are halfe as much as that part of the Circumference which is opposite to the angles . As if in the circle ABC the circumference BC be 120 degrees , then the angle BAC which is opposite to that circumference shall be 60 degrees . The reason is , because the whole circle ABC is 360 degrees , and the three angles of a plain triangle cannot exceed 180 degrees , or two right angles , by the last aforegoing , therefore , as every arch is the one third of 360 , so every angle opposite to that arch is the one third of 180 , that is 60 degrees . Or thus , From the angle ABC , let there be drawn the diameter BED , and from the center E to the circumference , let there be drawn the two Radii or semidiameters AE and AC , I say then that the divided angles ABD and DBC are the one halfe of the angles AED and DEC : for the angles ABE and BAE are equall , because their Radii AE and EB are equall , and also the angle AED is equal to the angles ABE and BAE added together , for if you draw the line EF parallel to AB , the angle FED shall be equal to the angle ABE by the 9 th . hereof ; and by the like reason the angle AEF is also equal to the angle BAE . and therefore the angle AED is equal to the angles ABE , and BAE : or , which is all one , the angle AED is double to the angle ABD . In like manner , the angles EBC and ECB are equal , and the angle DEC is equal to them both : therefore the angle DEC is double to the angle DBC . Then because the parts of the angle AEC are double to the parts of the angle ABC ; therefore also the whole angle AEC is double to the whose angle ABC ; and thereupon the angle ABC is half the angle AEC ; and consequently , half the arch ADC ; is the measure of the angle ABC , as was to be proved . Hence it followeth , 1. If the side of a plain Triangele inscribed in a circle be the diameter , the angle opposite to that side is a right angle , that is , 90 degrees ; for that it is opposite to a semicircle , which is 180 degrees . 2. If divers right lined triangles be inscribed in the same segment of a circle upon one base ; the angles , in the circumference are equal . As the two triangles ABD and ACD being inscribed in the same segmeut of the circle ABCD , upon the same base AD are equiangled in the points B and C , falling in the circumference . For the same arch AD is opposite to both those angles ; that is , to the angle ACD , and also to the angle ABD . 20 Theor. If two plaine Triangles inscribed in the same segment of a circle , upon the same base , be so joyned together in the top , ( or in the angles falling in the circumference ) that thereof is made a four-sided figure , intersected with Diagonals , the right angled figure made of the Diagonals , is equall to the right angled figures made of the opposite sides added together . Let ABD and ACD be two triangles , inscribed in the same segment of the circle ABCD upon the same base AD so joyned in the top by the right line BC , that thereupon is made the four sided figure ABCD. I say , that the right angled figures made of the opposit sides AB and DC , and also of the sides BC and AD added together are equall to the right angled figure made of the Diagonals AC and BD. Demonstration . If at the point B you make the angle ABE equall to the angle DBC , and so cut the Diagonall AC into two parts by the right line EB at the point E , then shall the angles ABD and EBC be equall , because the angles ABE and DBC are equal by the proposition , and the angle EBD common to both , and the angles ADB & ECB are equal , because the arch AB is the double measure of them both by the last aforegoing , and therefore the triangles ABD & EBC are equiangled and there sides proportional by the 18 th and 16 th Theoremes of this chapter , that is , as BD to DA , so is BC to CE , and therefore also the rectangles of BD in CE is equal to the rectangle of DA in BC by the 10 th hereof . And because the angles DBC and ABE are equal by the proposition , and the angles BDC and EAB equall because the arch BC is the double measure of them both by the last aforegoing , the triangles BDC , & EAB are equiangled and there sides proportional by the 18 th and 16 Theoremes of this chapter , that is as BD to DC so is AB to AE ; and therefore also the rectangle of BD in AE is equal to the rectangle of DC in AB . And because the rectangled figure 〈…〉 of AD and DF is equall to the two rectangled figures of AD in DC and BC in CF , therefore also the rectangled figure of BD in AC is equal to the rectangled figures of BD in AE and BD in EC . From hen●● and the two former proportions the proposition is thus Demonstrated . 1. BD in CE is equal to DA in BC 2. BD in AE DC in AB And by Composition . BD in CE more by BD in AE is equal to DA in BC more by DC in AB , now then because BD in AC is equal to BD in 〈◊〉 EC more by BD in EA , therefore also BD in AC is equall to DA in BC more by DC in AB which was to be demonstrated . 21. Theor. If two right lines inscribed in a circle cut each other within the circle , the rectangle under the segments of the one , is equall to the rectangle under the segment of the other . Let the two lines be FD and BC , intersecting each other in the point A ; I say , the triangles ABF and ADC are like , because of their equal angles AFB and ACD , which are equal , because the arch BD is the double measure of them both , and because of their equal angles BAF and DAC , which are equal by the ninth hereof , and where two are equal , the third is eqaul by the 18 aforegoing ; therefore AD in AF is equal to AC in AB , which was to be proved . Theor. 22. In a plain right angled Triangle , a perpendicular let fall from the right angle upon the Hypothenuse , divides the triangle into two triangles , both like to the whole , and to one another . The triangle ABC is right angled at B , the hypotenuse or side subtending the right angle is AC , upon which from the point B is drawn the perpendicular BD which divideth the triangle ABC into two triangles , ADB and BDC , each of them like to the whole triangle ABC , and each like to one another also , that is equiangled one to another . Demonstration . In the triangle ABD , the angles ABD and ADB are equal to the angles ACB and ABC , because of their common angle at A , and their right angles at B and D , and in the triangle CDB , the angle . CBD and BDC are equal to the angles ABC and CAB , because of their common angle at C , and their right angles at B and D ; these triangles are therefore each of them like to the whole triangle ABC , and by consequence like to one another . 23. Theor. If two sides of one triangle be equal to two sides of another , & the angle comprehended by the equal sides equal , the third side or base of the one , shall be equal to the base of the other , and the remaining angles of the one equal to the remaining angles of the other . Of these two triangles CBH and DEF , the sides CB & BH in the one are equal , to DE & EF in the other and the angle CBH equal to the angle DEF , therefore CH in one is equal to DF in the other , for if the base CH be greater then the base CF from CH let be taken . CG equal to DF and let there be drawne the right line BG , now if BC and BG be equal to DE and EF , yet the angle CBG cannot be equal to the angle DEF by the angle GBH which is contrary to the Proposition , and therefore CH must be equal to DF , and consequently the angle BCH equal to EDF , and CHB equal to DFE which was to be proved . 24. Theor. An Isocles or triangle of two equal sides , hath his angles at the base equal the one to the other , and contrarily . 25. Theor. If the Radius of a circle be divided , in extreame and mean proportion , the greater segment shall be the side of a Decangle , in the same circle . These foundations being laid we will proceed to the making of the tables , whereby any triangle may be measured . Chap. III of Trigonometria , or the measuring of all Triangles . THe dimension of triangles , is performed by the Golden Rule of Arithmetick , which teacheth of four numbers proportional one to another , any three of them being given , to finde out a fourth . Therefore for the measuring of all triangles there must be certain proportions of all the parts of a triangle one to another , and these proportions must be explained in numbers . 2. And the proportions of all the parts of a triangle one to another cannot be certain unlesse the arches of circles ( by which the angles of all triangles , and of Spherical triangles , also the sides are measured ) be first reduced into right lines , because the proportions of arches one to another , or of an arch to a right line , is not as yet found out . 3. The arches of every circle are after a sort reduced to right lines , by defining the quantity , which the right lines to them applied have , in respect of Radius , or the Semidiameter of the circle . 4. The arches of a circle thus reduced to right lines are either Chords , Sines , Tangents , or Secants , 5. A Chord or Subtense is a right line inscribed in a circle , dividing the whole circle into two segments , and in like manner subtending both the segments . 6. A chord or subtense is either the greatest or not the greatest . 7. The greatest Subtense is that which divideth the whole circle into two equal segments , as the right line GD , and is also commonly called a diameter . 8. A subtense not the greatest , is that which divideth the whole circle into two unequal segments : and so on the one side subtendeth an arch less then a semicircle ; and on the other side subtendeth an arch more then a semicircle , as the right line CK on the one side subtendeth the arch CDK , lesse then a semicircle ; and on the other side subtendeth the arch CGK more then a semicircle . 9. A sine is either right or versed . 10. A right sine is the one half of the subtense of the double arch , as the right sine of the arches CD and CG is the right line AC , be●●● half the chord or subtense of the double arches of CD and CG , that is , half of the right line CAK , which subtendeth the arches CDK and CGK ; whence it is manifest , that the right sine of an arch lesse then a Quadrant , is also the right sine of an arch greater th●n a Quadrant . For as the arch CD is lesse then a Quadrant by the arch CE ; so the arch CG doth as much exceed a Quadrant , the right line AC being the right sine unto them both . And hence instead of the obtuse angle GBC , which exceeds 90 degrees , we take the acute angle CBA , the complement thereof to 180 : and so our Canon of sines doth never exceed a quadrant or 90 d. 11. Againe , a right 〈◊〉 is either Sinus totus , that is the Radius or whole sine , as in the triangle ABC , AC is the Radius , semidiameter , or whole sine , Or else the right sine is the Sinus simpliciter , that is , the first sine , as CA or BA , the one whereof is alwayes the complement of the other to 90 degrees ; we usualy call them sine and co-sine . 12. The versed sine of an arch is that part of the diameter , which lieth between the right sine of that arch and the circumference . Thus AD is the versed sine of the arch CD , and AG the versed Sine of the arch CEG ; therefore of versed Sines some are greater , and some are lesse . 13. A greater versed Sine is the versed Sine of an arch greater then a Quadrant , as AG is the versed Sine of the arch CEG greater then a Quadrant . 14. A lesser versed sine is the versed Sine of an arch lesse then a Quadrant , as AD is the versed sine of the arch CD less then a Quadrant . 15. A tangent of an arch or angle is a right line drawn perpendicular to the Radius or semidiameter of the circle of the triangle , so as that it toucheth the outside of the circumference , And thus the right line FD is the tangent of the arch DC . 16. A secant is a right line proceeding from the center of the circle , and extended through the circumference to the end of the tangent ; and thus BF is the Secant of the arch DC . 17. The definition of the quantity which right lines applyed to a circle have , is the making of the Tables of Sines , tangents and secants ; that is to say , of right Sines and not of versed ; for the versed Sines are found by the right without any labour . 18. The lesser versed sine with the sine of the complement is equal to the Rad●●● as the lesser versed sine AD with the right sine of the complement AB is equal to the Radius BD ; therefore if you substract the right sine of the complement AB from the Radius BD , the remainder is the versed sine AD. 19. The greater versed sine is equal to the Radius added to the right sine of the excesse of an arch more then a Quadrant , as the greater versed sine AG is equal to the Radius BG with the sine of the excess AC : therefore if you adde the right sine of the excess AB to the Radins BG , you shal have the versed sine of the arch CEG , & so there is no need of the table of versed sines , the right sines may thus be made . 20. The Tables of Sines , Tangents , and Secants may be made to minutes , but may , by the like reason , be made to seconds , thirds , fourths , or more , if any please to take that paines : for the making whereof the Radius must first be taken of a certain number of parts , and of what-parts soever the Radius be taken , the Sines , Tangents , and Secants are for the most part irrational●● i● , that is , they are inexplicable in any true whole numbers or fractions precisely , because there are but few proportional parts to any Radius , 〈…〉 , whose square root multiplied in it self will produce the number from whence it was taken , without some fraction still remaining to it , and therefore the Tables of Sines , Tangents , and Secants cannot be exactly made by any meanes ; and yet such may and ought to be made , wherein no number is different from the truth by an integer of those parts , whereof the Radius is taken , as if the Radius be taken of ten Millions , no number of these Tables ought to be different from the truth by one of ten Millions . That you may attain to this exactnesse , either you must use the fractions , or else take the Radius for the making of the Tables much greater then the true Radius , but to work with whole numbers and fractions is in the calculation very tedious ; besides here no fractions almost are exquisitely true : therefore the Radius for the making of rhese Tables is to be taken so much the more , that there may be no errour , in so many of the figures towards the left hand as you would have placed in the Tables ; and as for the numbers superfluous , they are to be cut off from the right hand towards the left after the ending of the supputation , Thus , to finde the numbers answering to each degree and minute of the Quadrant to the Radius of 10000000 or ten millions , I adde eight ciphers more , and then my Radius doth consist of sixteen places . This done , you must next finde out the right Sines of all the arches lesse then a Quadant , in the same parts as the Radius is taken of , whatsoever bignesse it be , and from those right Sines the Tangents and secants must be found out . 21. The right Sines in making of the Tables are either primary or secondary . The primarie Sines are those , by which the rest are found , And thus the Radius or whole Sine is the first primary Sine , the which how great or little soever is equall to the side of a six-angled figure inscribed in a circle , that is , to the subtense of 60 degrees , the which is thus demonstrated . Out of the Radius or subtense of 60 degrees the sine of 30 degrees is easily found , the halfe of the subtense being the measure of an angle at the circumference opposite thereunto by the 19 of the second ; if therefore your Radius consists of 16 places being 1000.0000.0000.0000 . The sine of 30 degrees will be the one half thereof , to wit , 500.0000.0000.0000 . 22. The other primary sines are the sines of 60 , 45 , 36 , and of 18 degrees , being the halfe of the subtenses of 120 , 90 , 72 , and of 36 degrees . 23. The subtense of 120 degrees is the side of an equilateral triangle inscribed in a circle , and may thus be found . The Rule . Substract the Square of the subtense of 60 degrees , from the Square of the diameter , the Square root of what remaineth is the side of an equilateral triangle inscribed in a circle● or the subtense of 120 degrees . The reason of the Rule . The subtense of an arch with the subtense of the complement thereof to 180 with the diameter , make in the meeting of the two subtenses a right angled triangle . As the subtense AB 60 degrees , with the subtense AC 120 degrees , and the diameter CB , make the right angled triangle ABC , right angled at A , by the 19 of the second . And therfore the sides including the right angle are equal in power to the third side , by the 〈◊〉 of the second . Therefore the square of AB being taken from the square of CB , there remaineth the square of AC , whose squar root is the subtense of 〈◊〉 degrees or the side of an equilateral triangle inscribed in a circle , Example . Let the diameter CB be 2000.0000 . 0000.0000 . the square thereof is 400000. 00000.00000.00000.00000.00000 . The subtense of AB is 100000.00000.00000 . The square thereof is 100000.00000.00000 . 00000.00000.00000 , which being substracted from the square of CB , the remainder is 300000.00000.00000.00000.00000.00000 , whose square root 173205.08075.68877 . the subtense of 120 degrees . CONSECTARY . Hence it followeth , that the subtense of an arch lesse then a Semicircle being given , the subtense of the complement of that arch to a Semicirc●e is also given . 24. The Subtense of 90 degrees is the side of a square inscribed in a circle , and may thus be found . The Rule . Multiply the diameter in it self , and the square root of half the product is the subtense of 90 degrees , or the side of a square inscribed in a circle . The reason of this Rule . The diagonal lines of a square inscribed in a circle are two diameters , and the right angled figure made of the diagonals is equal to the right angled figures made of the opposite sides , by the 20 th . of the second , now because the diagonal lines AB and CD are equal , it is all one , whether I multiply AC by it self , or by the other diagonal CD , the p●oduct will be still the same , then because the sides AB , AC , and BC do make a right angled triangle , right angled at C , by the 〈◊〉 of the second , & that the 〈◊〉 AC and ●B are equal by the work , the half of the square of AB must needs be the square of AC or CB , by the 17 th . of the second , whose square rootes the subtense of CB , the side of a square or 90 degree . Example . Let the diameter AB be 200000.00000 . 00000 , the square thereof is 400000.00000 . 00000.00000.00000.00000 , the half whereof is 200000.00000.00000.00000.00000 . 00000. whose square root 14142● . 356●3 . 73095. is the subtense of 90 degrees , or the side of a square inscribed in a Circle . 25. The subtense of 36 degrees is the side of a decangle , and may thus be found . The Rule . Divide the Radius by two , then multiply the Radius by it self , and the half thereof by it self , and from the square root of the summe of these two products substract the half of Radius , what remaineth is the side of a decangle , or the subtense of 36 degrees . The reason of the rule . For example . Let the Radius EB be 100000.00000.00000 . then is BH , or the half thereof 500000. 00000.00000 . the square of EB is 100000 00000.00000.00000.00000.00000 . and the square of BH 250000.00000.00000.00000 . 00000.00000.00000 . The summe of these two squares , viz 125000.00000.00000 . 00000 , 00000. 00000 , is the square of HE or HK , whose square root is 1118033● 887●9895 , from which deduct the halfe Radius BH 500000000000000 , and there remaineth 618033988749895 , the right line KB , which is the side of a decangle , or the subtense of 36 degrees . 26 The subtense of 72 degrees is the side of a Pentagon inscribed in a circle , and may thus be sound . The Rule . Substract the side of a decangle from the diameter , the remainer multiplied by the Radius , shall be the square of one side of a Pentagon , whose square root shall be the side it self , or subtense of 72 degrees . The Reason of the Rule . In the following Diagram let AC be the side of a decangle , equal to CX in the diameter , and let the rest of the semicircle be bisected in the point E , then shall either of the right lines AE or EB represent the side of an equilateral pentagon , for AC the side of a decangle subtends an arch of 36 degrees the tenth part of a circle , and therefore AEB the remaining arch of a semicircle is 144 degrees , the half whereof AE or EB is 72 degrees , the fift part of a circle , or side of an equilateral pentagon , the square whereof is equal to the oblong made of DB and BX . Demonstration . Draw the right lines EX , ED , and EC , then will the sides of the angles ACE and ECX be equal , because CX is made equal to AC , and EC common to both ; and the angles themselves are equal , because they are in equal segments of the same circle by the 19 of the second ; and their bases AE and EX are equal by the 23 of the second ; and because EX is equal to AE , it is also equal to EB , and so the triangle EXB is equicrural , and so is the triangle EDB , because the sides ED and DB are Radii , and the angles at their bases X and B , E and B , by the 24th . of the second , and because the angles at B is common to both , therefore the two triangles , EXB and EDB are equiangled , and their sides proportional , by the 18 th . and 16 th . Theoremes of the second Chapter , that is as DB to EB ; so is EB to BX , and the rectangle of DB in BX is equal to the square of EB , whose square root is the side EB , or subtense of 72 degrees , Example . Let AC , the side of a decangle or the subtense of 36 degrees , be as before : 618033988749895 , which being substracted from the diameter BC 200000.00000 , 00000. the remainer is XB , 1381966011151105 , which being multiplied by the Radius DB , the product 1381966011251105 00000.00000.0000 , shall be the square of EB whose square root 1175570504584946 is the right line EB , the side of a Pentagon or subtense of 72 degrees . CONSECTARY . Hence it followes , that the subtense of an arch lesse then a semicircle being given , the subtense of half the complement to a semicircle is given also , Thus much of the primarie Sines , the secondary Sines or all the Sines remaining may be found by these and the Propositions following 27. The subtenses of any two arches together lesse then a semicircle being given , to finde the subtense of both those arches . The Rule . Finde the subtense of their complements to a semicircle , by the 23 hereof ; then multiply each subtense given by the subtense of the complement of the other subtense given , the sum of both the products being divided by the diameter , shall be the subtense of both the arches given . The reason of the Rule . Example . Let AI , the side of a square or subtense of 90 degrees be 141421.35623.73059 . And EO , the side of a triangle , or subtense of 120 degrees , 173205.08075.68877 , the product of these two will be 2449489742783 ▪ 77659465844164315. Let AE , the side of a sixangled figure , or the subtense of 60 degrees be 100000 00000.00000 . And IO , the side of a square , or subtense of 90 degrees 141421.35623.73059 the product of these two will be 141421.35623.73059 . 00000.00000.00000 . the summe of these two products 3863703305156272659465844164315 . And this summe divided by the diameter AO , 200000.00000.00000 . leaveth in the quotient for the side EI , or subtense of 150 degrees , 1931851652578136. the half whereof 965925826289068 , is the Sine of 75 degrees . 28 The subtenses of any two arches lesse then a Semicircle being given , to finde the subtense of the difference of those arches , The Rule . Finde the subtenses of their complements to a semicircle , by the 23 hereof , as before , then multiply each subtense given , by the subtense of the complement of the other subtense given ; the lesser product being substracted from the greater , and their difference divided by the diameter , shall be the subtense of the difference of the arches given . The Reason of the Rule . Let the subtenses of the given arches be AE and EI , and let the subtense sought be the right line EI ; then because the right angled figure made of the diagonals . AI and EO is equal to the right angled figures made of their opposite sides , by the 20 of the second ; therefore if I subtract the right angled figure made of AE and IO , from the right angled figure made of AI and EO the remainer will be the right angled figure of AO and EI , which being divided by the diameter AO , leaveth in the quotient EI. Example , 29. The sine of an arch lesse then a Quadrant being given , together with the sine of half his complement , to finde the sine of an arch equal to the commplement of the arch given , and the half complement added together . The Rule . Multiply the double of the sine given , by the sine of half his complement , the product divided by the Radius , will leave in the quotient , a number , which being added to the sine of the half complement shall be the sine of the arch sought . The reason of the Rule . That is , as AI , is to AM : so is CP , to PM ; and so is PS , to PN , and then by composition , as AI , AM : so is CS , to MN . Now then let ES be the arch given , and SI the complement thereof to a Quadrant , then is CG or IB , being equal to EY , the half of the said complement SI , and AM is the Sine thereof , and the Sine of ES is the right line HS , and the double CS , MN is the difference between AM , the Sine of CG or IB , and AN the Sine of SB , and AI is the Radius , and it is already proved , that AI is in proportion to AM , as CS , is to MN , therefore if you multiply AM by SC , and divide the product by AI , the quotient will be NM , which being added to AM , doth make AN , the Sine or the arch sought . Example . Let ES , the arch given , be 84 degrees , and the Sine thereof 9945219 , which doubled is 19890438 , the Sine of 3 degrees , the halfe complement is 523360 , by which the double Sine of 84 degrees being multiplied , the product will be 104098●9 . 631680 , which divided by the Radius , the quotient will be 10409859 , from which also cutting off the last figure , because the Sine of 3 degrees was at first taken too little , and adding the remainer to the Sine of 3 degrees , the aggregate 1564345 is the Sine of 6 degrees , the complement of 84 , and of 3 degrees , the halfe complement added together , that is , it is the sine of 9 degrees . 30. The subtense of an arch being given , to find the subtense of the triple arch . The Rule . Multiply the subtense given by thrice Radius square , and from the product substract the cube of the subtense given , what remaineth shall be the subtense of the triple arch . The reason of the Rule . Now then we have already proved , that the square of AO divided by Radius , is equal to OX , and also that OX is equal to SA , and therefore SN is less then twice Radius by the right line AS ; or thus , NS is twice Radius less by AO square divided by Radius : and NS multiplied by SA is the same with twice Radius lesse by AO square divided by Radius , multiplied into AO square divided by Radius , and NS multiplied by SA is equal to SC multiplied by OS ; and therefore twice Radius less AO square divided by Rad. multiplied by AO square divided by Radius , is equal to SC , multiplied by SO : or thus , 2 Radius less AO square divided by Radius , multiplied into AO square divided by Radius , and divided by AO or SO is equal to SC. All the parts of the first side of this Equation are fractions , except AO and the two Radii , as will plainly appear , by setting it down according to the form of Symbolical or specious Arithmetick ; thus . . Which being reduced into an improper fraction , by multiplying 2 Radius by Radius , the Equation will run thus : And then these two fractions having one common denominator , they may be reduced into one after the manner of vulgar fractions , that is , by multiplying the numerators , the product will be a new numerator , and by multiplying the denominators the product will be a new denominator ; thus multiplying the numerators , 2 Rad. aa − AO aa by the numerator AO aa , the product is 2 Rad. square into AO square , less AO square square , as doth appear by the operation ; And then the denominators being multiplied by the other , that is , Radius being multiplied by Radius , the product will be Radius aa for a new denominator ; and then the Equation will run thus ; : but before this fraction can be divided by AO , AO being a whole number , must be reduced into an improper fraction , by subscribing an Unite , and then the Equation will be ; . Now as in vulgar fractions , if you multiply the numerator of the dividend by the denominators of the divisor , the product shall be a new numerator ; again , if you multiply the denominator of the dividend by the numerator of the divisor , their product shall be a new denominator , and this new fraction is the Quotient sought in this example , the numerator will be still the same , and the denominator will be Radius square multiplied in AO , and the fraction will be . And in its least termes it is . In words thu● : Twice Radius square multiplied in AO , lesse by the cube of AO divided by Radius square is equal to SC. And by adding AO to both sides of the Equation , it will be , twice Radius square in AO , lesse AO cube divided by Radius square , more AO , is equal to SC more AO , that is , to OC . Here again AO , the last part of the first side of this Equation is a whole number , and must be reduced into an improper fraction , by being multiplied by Radius square , the denominator of the fraction ; and then it will be Radius square in AO divided by Radius square , which being added to twice Radius square in AO , divided by Radius , the summe will be 3 Radius square in AO divided by Radius square , and the whole Equation , the subtense of the triple arch . For Example . Let AO or AB , 17431. 14854. 95316. the subtense of 10 degrees be the subtense given , and let the subtense of 30 degrees be required ; the Radius of this subtense given consists of 16 places , that is , of a unite and 15 ciphers , and therefore thrice Rad. square is 3 , and 30 ciphers thereunto annexed , by which if you multiply the subtense given , the product will be 52293. 44564. 85948. 00000. 00000. 00000. 00000. 00000. 00000. the square of this subtense given is 3038449397 . 55837.60253.85793.9856 , and the cube 529.63662.80907.48519.77452.00270 . 23994. 54977. 14496 , which being substracted from the former product , there will remain 51763.80902.05040.51480.22547.99729 . 76005.45022.85504 . this remainer divided by the square of Radius , will leave in the quotient , 51763.80902.05040 . for the subtense of 30 degrees . 31. The subtense of an arch being given , to finde the subtense of the third part of the arch given . The Rule . Multiply the subtense given by Radius square , and divide the product by thrice Radius square , substracting in every operation the cube of the figure placed in the quotient from the triple thereof ; so shall the quotient in this division be the subtense of the third part of the arch given . The reason of the Rule . The reason of the rule is the same with the triple arch , but the manner of working is more troublesome , the which I shall endeovour to explain by example . Let there be given the subtense of 30 degrees , 517638090205040 , and let the subtense of 10 degrees be required : First , I multiply the subtense given by the square of Radius , that is , I adde 30 ciphers thereunto , and for the better proceeding in the work , I distinguish the subtense given thus inlarged by multiplication into little cubes , setting a point between every third figure or cipher , beginning with the last first , and then the subtense given will stand thus : 517.638.090.205.040.000.000.000.000.000 000.000.000.000.000 . And so many points as in this manner are interposed , of so many places the quotient wil consist , the which in this example is 15 , and because here are too many figures to be placed in so narrow a page , we will take so many of them onely as will be necessary for our present purpose ; as namely , the 15 first figures , which being ordered , according to the rules of decimal Arithmetick , may be divided into little cubes , beginning with the first figure , but then you must consider whether the number given to be thus divided be a whole number or a fraction , if it be a whole number , you must set your point after or over the head of the first figure , if it be a fraction , place as many ciphers before the fraction given , as will make it consist of equal places with the denominator of the Fraction given ; thus the subtense given being a fraction , part of the supposed Radius of a circle , the which , as hath been said doth consist of 16 , and the subtense given but of fifteen , I set a cipher before it , and distinguish that cipher from the subtense given by a point or line , and every third figure after , so will the subtense given be distinguished into little cubes , as before . This done , I place my divisor thrice Radius square , that is , 3 with ciphers ( or at least supposing ciphers to be thereunto annexed ) as in common division under the first figure of the subtense given , that is , as we have now ordered it under the cipher , and ask how often 3 in nought , which being not once , I put a cipher in the margine , and move my divisor a place forwarder , setting it under 5 , and ask how often 3 in 5 , which being but once , I place one in the quotient , and the triple thereof being 3 , I place under 3 my divisor , and the cube of the figure placed in the quotient , which in this case is the same with the quotient it self , I set under the last figure of the first cube , and supposing ciphers to be annexed to the triple root , I substract this cube from it , and there doth remain 299 , which is my divisor corrected ; with this therefore I see whether I have rightly wrought or not , by asking , how often 299 is contained in the first cube of the subtense given , 517 , which being but once , as before , the former work must stand , & this divisor corrected must be subtracted from the first cube in the subtense given , and there will rest 218 , and so have I wrote once . To this remainer of the first cube 218 , I draw down 638 , the figures of the next cube & moving my divisor a place forwarder , I ask , how often 3 in 21 , which being 7 times , I put 7 in the quotient , and under the first figure of this second cube , that is , under 6 I set the triple square of the first figure in the quotient , that is , 3 , for the quotient being but one , the square is no more , and the triple thereof is 3 ; under the second figure of this second cube I set the triple quotient , the which in this example is likewise 3 , and both these added together , do make 33 , which being substracted from my divisor 3000 , there will remain 2967 , for the divisor corrected , and by this also I finde the quotient to be 7 , and yet I know not whether my work be right or not , I must therefore proceed , and set the triple of the figure last placed in the Quotient under the first figure of the remainer of the first cube , that is , I must set 21 , the triple of 7 under 2 , the first figure of 218 , and now having two figures in the quotient , for distinction sake I call the first a , and the second e , that so the method of the work may the better be seen in the margine , and I set 3 aae , that is , 3 , the square of the first figure noted with the letter a , viz. 1. multiplied by the second figure , noted with the letter ( e ) to wit , 7 , under the first figure of the next cube , now the square of ( a ) that is , of one is one , and the triple of this square is 3 , and 3 times 7 is 21 , which is ( 3 aae ) or thrice ( a ) square in e , the last figure whereof , to wit , one , I place under 6 , the first figure of the next cube 638 : next I set ( 3 aee ) that is , three times one multiplied by the square of 7 , that is , 3 multiplied by 49 , which is 1●● under the 2 figure of the cube 638 : and lastly , I set ( eee , that is ) the cube of e , that is , the cube of 7 , viz. 343 , under the last figure of the cube 638 , and these 3 sums added together do make 3●●3 , which being substracted from the triple root , that is , from 21 , supposing ciphers to be thereunto annexed , as before , there will remain 〈…〉 , and because this may be subtracted from the 2d . cube , & the remainer of the first , I finde that 7 is the true figure to be placed in the quotient , and such a subtraction being made , the remainer will be 12511 , and so have I wrote twice . The work following must be done in all things , as this second , save onely in this particular , that both the figures in the quotient are reckoned but as one , which for distinction sake I called a , and the figure to be found by division I called e , and therefore in this third work 3 aa , or thrice a square is the square of 17 , that is 289 , 3 a or thrice a is 3 times 17 , that is , 51 , and so of the rest , in the fourth work the three first figures must be called a , in the fifth work the four first figures found , and so forward , till you have finished your division , and therefore this second manner of working being well observed , there can be no difficulty in that which followes . 32. The subtense of an arch being given to finde the subtense of the arch quintuple , or of an arch five times as much . The Rule . From the product of the subtense given , multiplied by 5 times Radius square square , subtract the cube of the subtense given multiplied by 5 times Radius square , the squared cube of the subtense given being first added thereunto , the remainer divided by Radius square square , shall leave in the quotient the subtense of the arch quintuple , or the arch 5 times as much . The reason of the Rule . Multiply the numerator of the fractions in the second place by the numerator of the fraction in the third , and their product will be a new numerator , the numerator of the fraction in the second term is 2 Rad. − AO aa And in the third , 3 Rad. aa × AO − AO aaa That one of these termes may be the better multiplied by the other , the first of the second term , 2 Rad. must be reduced into an improper fraction , by the multiplication thereof by Radius , the denominator of that fraction , and then the 2d . term will be 2 Rad. aa − AO aa , and because this second term is the lesse , we will multiply the third thereby , the work stands thus : Thrice Radius square in AO multiplied by twice Radius square , doth make 6 Rad. square squares , and AO cube multiplied by 2 Radius square is 2 Rad. square in AO cube , and because it hath the signe lesse , therefore the first product is 6 Rad. aaaa × AO − 2 Rad. aa × AO aaa . Again , 3 R. aa in AO , multiplied by AO aa , doth make 3 R. aa in AO aaa , & AO aa multiplied by AO aaa , doth make AO aaaaa , & because it hath the sign less , therefore the 2. product is 3 R. square × AO aaa + AO aaaaa , and so both the products will be 6 Rad. square of squares multiplied by AO lesse by 5 Rad. square in AO cube more by AO square cube . And if you multiply Rad. square , the denominator of the third term by Rad. the denominator of the second , the product will be Rad. cube , and the whole product will stand thus , To divide this product by twice Radius , twice Radius being a whole number must be first reduced into an improper fraction , by subscribing an unite thus , . then if you multiply the numerator of the product by one , the denominator of this fraction , the product will be still the same , and if you multiply the denominator of the product Rad. aaa by 2 Radius , the numerator of this improper fraction , the product will be 2 Rad. square square for a new denominator , and the Quotient will be the quantity of the right line OT , the double whereof is which is the quantity of the right line OE more by CB , and therefore CB or AO being deducted , the remainer will be the right line OE , which is the quintuplation of an angle , and to this end AO must be reduced into an improper fraction of the same denomination , that is , by multiplying thereof by 2 Rad. aaaa , and then the fraction will be and this being deducted from the remainer will be . And this reduced into its least terms , will be , which was to be proved . For example . Let AO or AB 349048 , the subtense of 2 degrees be given , and let the subtense of 10 degrees be demanded , 5 times Radius square square is 50000000.000000.00000 00.0000000 . by which if you multiply the subtense given , the product will be 1745240 0000000.0000000.0000000.0000000 . The Cube of the subtense given multiplied by 5 times Radius square is 212630453781992960 . 0000000.0000000 . the squared cube of the subtense given is 5184639242824921385360723968 , the which being added to the product of 5 Rad. aa in AO , that is , to 212630453781992960.0000000.0000000 the summe will be 21268230017442120921385360723968 . And this being subtracted from the product of the subtense given multiplied by 5 times Radius square square , the remainer will be 17431141769982557879078614639276032 , and this remainer divided by Radius square square , that is , cutting off 28 figures , their quotient will be 1743114 , the subtense of 10 degrees . 33. The subtense of an arch being given to finde the subtense of the fift part of the arch given . The Rule . Divide the subtense given by five roots , lesse 5 cubes , more one Quadrato cube , the quotient shall be the fift part of the arch given . The reason of the rule depends upon the foregoing Probleme , in which we have proved , that the subtense of five equall arches is equall to 5 roots , lesse 5 cubes , more by one quadrato cube , of which 5 roots one of them is the subtense of the fift part of the arch given . And consequently , if I shall divide the subtense of five equall arches by 5 roots , lesse 5 cubes , more one quadrato cube , the quotient shall be the subtense of the fift part of the arch . The manner of the work is thus : First , consider whether the subtense given to be divided doth consist of equal , or of fewer places then the Radius thereof , if it consist of equal places , set a point over the head of the first figure of the subtense given , if of fewer places , make it equal , by prefixing as many ciphers before the subtense given as it wanteth of the number of places of the Radius thereof . For example . Let the subtense of 10 degrees be given , viz. 0.17431.14854.95316.34711 . This is lesse then the Rad. by one place , and therefore I have set one cipher before , and have distinguished it from the subtense given by a point set between , the which is all one , as if it had been put over the head thereof : next you must distinguish the subtense given into little cubes , & into quadratocubes , which may be conveniently done thus ; having found the place of the first point , which is alwayes the place of the Radius , the subtense given must be distinguished into little cubes , by putting a point under every third figure , as in the trisection of an angle : thus in this example the first cubick point will fall under the figure 4 , and the subtense given must be distinguished into quadrato cubes , by setting a point over the head , or else between every fift figure from the place of the Radius : thus in this example the first quadrato cubick point must be set over the head , or after the figure of 1 , the second after 4 , as here you see . After this preparation made , you must place your two divisors , 5 roots and 5 cubes in this manner , the first as in ordinary division under the first figure of the subtense given , the other 5 under the first cubick point , and they will stand as in the work you see ; then ask how often 5 in one , which being not once , I put a cipher in the quotient , and remove my first divisor a place forwarder , as in ordinary division , but the other 5 I remove to the next cubick point , then , as before , I ask how often 5 in 17 , which being 3 times , I set 3 in the quotient , and of this quotient I seek the quadrato cube , and finde it to be 243 , the last figure whereof , namely , 3 , I set under the last figure of the second quadrato cubick point ( because there are but 3 figures between my divisor 5 and the first cubick point , whereas there must be alwayes four at the least ) then I multiply the figure 3 placed in the quotient by my divisor 5 , and the product thereof is 15 , the first figure whereof I place under my said divisor 5 , to which having annexed ciphers , or at least supposing them to be annexed , ( as to the triple root in the trisection ) I draw the quadrato cube of the figure in the quotient , and these 5 roots or 5 quotients into one summe , the which is 1500000243 , under this summe I draw a line , so have we five roots more one quadrato cube , from which I must subtract 5 cubes , I therefore seek the cube of 3 , the figure placed in the quotient , and finde it to be 27 , which multiplied by 5 , the product will be 135 , the last figure of these five cubes , viz. 5 , I set under my second 5 or cubick divisor , and substracting these 5 cubes from the 5 roots more one quadrato cube , the remainer will be 〈…〉 , which remainer being also substracted from the figures of the subtense given standing over the head thereof , the remainer of the subtense given will be 244464611 , and so have I wrought once . To this remainer of the two first quadrato cubes , I draw down 95316 , the figures of the next quadrato cube , and setting my first divisor a place forwarder , I ask how often 5 in 24 , which being four times , I set 4 in the quotient , not knowing yet whether this be the true quotient or not , but with this I proceed to correct my divisor , and first I seek the quadrato quadrat of 3 , the first quotient , and finde it to be 81 , this multiplied by 5 , will make 405 , this product I set under my divisor , and 5 the last figure thereof I set under 9 , the first figure of the 3 quadrato quadrate ; next I seek the cube of 3 , & finde it to be 27 , which being multiplied by 10 , the product will be 270 , and this I set a place forwarder under the former product : thirdly , I seek the square of 3 which is 9 , and this multiplied by 10 is 90 , which I set a place forwarder under the second product 270. Lastly , I multiply 3 , the figure in the quotient by 5 my divisor , this product which is 15 , I set a place forwarder under 90 , the third product , and now these 4 products together with my divisor and ciphers thereunto annexed , being gathered into one summe , will be 500000432915 , under which I draw a line . And thrice the square of 3 , multiplied by 5 , which is 135 , I set under this summe , the last figure thereof 5 , under the first figure of the third cubick point , that is , under 4 , and the triple of 3 multiplied by 5 , which is 45 , I set under the former summe 135 , a place forwarder , and my cubick divisor 5 under the last summe a place forwarder , that is , under the third cubick point , these drawn into one summe will be 13955 , and being substracted from the former summe 500000432915 , the remainer 498.60493 . + 2915 is my divisor corrected , and yet I know not whether I have a true quotient or not ; under this remainer therefore I draw a line , and work with 4 , which I suppose to be the true quotient in manner following ; and that the manner of the work may be the more perspicuous , ( as in the trisection of an angle , so here ) 3 the first figure found I call ( a ) and 4 the second figure I call ( e ) the square of three I note with aa , the cube with aaa , the quadrate quadrat with aaaa , the quadrato cube with aaaaa , so likewise the square of 4 the second figure I note with ee , the cube with eee , the quadrato quadrate with eeee , the quadrato cube with eeeee ; my first divisor I note with ffff , because this Equation is quadrato quadratick , and 5 my second divisor , I note with cc , because the divisor it self is cubick : these things premised , I proceed thus : First , I multiply 405 , which is 5 aaaa or 5 times the quadrato quadrate of 3 by e , that is , by 4 , and the product thereof 1620 , I set under my divisor corrected , so as the last figure thereof may stand under the first figure of the third quadrato cubick number , and against this number I put in the margine 5 aaaae , that is , five times the quadrato quadrate of 3 multiplied by 4 : next 270 , ten times the cube of 3 , by 16 the square of 4 , and this product 4320 , I set under the former a place forwarder , and 90 , which is 10 times the square of 3 , I multiply by 64 , the cube of 4 , & this product 5760 I set under the last a place forwarder then that , and 15 , which is 5 times 3 , I multiply by 256 , the quadrato quadrate of 4 , & the product thereof 3840 , I set under the third product a place forwarder , and 1024 , the quadrato cube of four under that : lastly , I multiply four , the last figure placed in the quotient by 5 my divisor , and the last figure of this product I set under 5 my divisor , and supposing ciphers to be thereunto annexed , I collect these several products into one summe , and their aggreagate 20000021135424 , is five roots more one quadrato quadrate , under which I draw a line , and seek the five cubes to be substracted , thus . First , I multiply 135 ( which is thrice the square of three multiplied by five my cubick divisor ) by four , the figure last placed in the quotient , and the product thereof 540 I set under the last summe , so as the last figure thereof may be under the first figure of the third cube ; next I multiply 45 that is , five times the triple of three , by 16 the square of four , and this product 720 I set under the former a place forwarder , and under that 320 , which is five times the cube of 4 , a place forwarder too , these products drawn into one summe do make 61520 , the five cubes to the substracted from the five roots more one quadrato quadrate before found , which being done , the remainer will be 19938501135424 , and this remainer being substracted from the figures of the subtense given over the head thereof , the remainer will be 450.79600 . 59892 , and because such a substraction may be conveniently made , I conclude , that I have found the true quotient , and so have I wrought twice . 34 The Sines of two arches equally distant on both sides from 60 degrees , being given , to finde the Sine of the distance . The Rule . Take the difference of the Sines given , and that difference shall be the Sine of the arch sought . The reason of the Rule . Let CN and PN be the two arches given , and equally distant from 60 deg . MN , that is equally distant on both sides from the point M. And let the right lines CK and PL be the Sines of those arches , being drawn perpendicular to the right line AN , and thereupon parallel to one another . Moreover , let the right line PT be drawn perpendicular upon the right line CK , and so parallel to the right line KL , then this right line TP cutteth from the right line CK another line TK , equal unto PL , by the 15 of the second , and leaveth the right line TC for the difference of the Sines CK and PL. Lastly , the Sines of the distance of either of them from 60 degrees let be the right line CD or DP , I say , that the right line TC is equal to the right line CD or DP . Demonstration . Example . Let the arches CN be 70 degrees , PN 50 , CM or PM 10 degrees ; for so many degrees are the arches of 70 degrees ; and 50 degrees distant from the arch of 60 degrees on both sides . And let first the Sines of 70 degrees and 10 degrees be given , and let the Sine of 50 degrees be demanded . From the Sine of 70d. CK 9396926 Subtract the Sine of 10d. CD or CT , 1736482     The Remainer will be the Sine of 50d. TK or PL , 7660444 Then let the Sine of 70 degrees and 50 degrees be given , and let the Sine of ten degrees be demanded . From the Sine of 70 degrees CK , 9396926 Substract the Sine of 50d. TK or PL , 7660444     Remainer is the Sine of 10d . CD , 1736482 Lastly , let the Sines of 50 degrees and 10 degrees be given , and let the Sine of 70 degrees be demanded . To the Sine of 50d . PL or TK , 7660444 Adde the Sine of 10d . DP or TC , 1736482     Their sum will be the Sine of 70d . 9396926 And thus far of the making of the Tables of right Sines , the Tables of versed Sines are not necessary , as hath been said CHAP. IV. By the Tables of Sines to make the Tables of Tangents and Secants . 1. AS the Sine of the complement , Is to the Sine of an arch : so is the Radius , to the tangent of that arch . 2. As the Sine of the complement , is to the Radius ; so is the Radius , to the secant of that arch . For , by the 16 th . of the second : 1. As the Sine of the complement AB , is to the Sine CA : so is the Radius BD or BC , to DF the tangent . 2. As the sine of the complement AB , is to the Radius BD or BC : so is the Radius BC , to the secant BF . Example . Let the tangent and secant of the arch CD 30 degrees be sought for . The sine AC 30 degrees is 5000000 , the sine of the complement AB 60 degrees is 8660254. Now then if you multiply the sine AC 5000000 , by the Radius CB 10000000 , the product wil be 50000000000000 , which divided by the sine of the complement AB 8660254 : the quotient will be 5773503 , the right line FD or the tangent of the arch of 30 degrees . 2. As the sine of the complement AB 8660254 , Is to the Radius DB 10000000 : so is the Radius BC 10000000 , to FB , the secant of the arch of 30 degrees : and so for any other : but with more ease by the help of these Theorems following . Theorem 1. The difference of the Tangents of any two arches making a Quadrant , is double to the tangent of the difference of those arches The Declaration . Let the two arches making a Quadrant be CD and BD , whose tangents are CG and BP , and let BS be an arch made equall to CD ; and then SD will be the arch of the difference of the two given arches CD or BS , and BD. And also let the tangent BT be equal to the tangent CG , and then the right line TP will be the difference of the tangents given CG or BT , and BP . Lastly , let the arches BL and BO ( whose tangents are BK and BM ) be made equal to the arch SD ; I say , the right line TP being the difference of the two given tangents , CG and BP is double to the right line BK , being the tangent of the difference of the two given arches ; or which is all one , I say , that the right line TP is equal to the right line MK . Demonstration . Then that the right line MT is equall to the right line KA is thus proved ; the right line MA is equall to the right line KA , by the work , but the right line MT is equal to the right line MA , and therefore it is also equall to the right line KA . That the right line MT is equal to the right line MA doth thus appear : for that the angles MAT and MTA are equall ; and therefore the sides opposite unto them are equal , for equall sides subtend equall angles : and the angles MTA and MAT are equal , because the angle MTA is equal to the angle TAC , by the like reason , that the angle KPA is equal to the angle DAC ; and the angle MAT is equall to the angle TAC , by the proposition : for the arches CS and SO are put to be equal : therefore it followes , that they are also equal one to another . Generally therefore , the difference of the tangents of two arches , making a Quadrant , is double to the tangent of the difference of those arches , which was to be demonstrated . And by consequence , the tangents of two arches being given , making a Quadrant , the tangent of the difference of those arches is also given . And contrarily , the tangent of the difference of those two arches being given , together with the tangent of one of the arches ; the tangent of the other arch is also given . Example . Let there be given the Tang. of 72 de . 94 m. And the Tang. of its complement , that is , of 17 6 Halfe the difference of these two arches is 27 94 Tangent of 72 de . 94 m. is 32586438 Tangent of 17 6 306●761 Their difference is 29517677 The halfe whereof is 14758838 The Tangent of 55 de . 88 min. Or let the tangent of the greater arch 72 d. 94 m. be given , with the Tangent of the difference 55 de . 88 m. and let the lesser arch 17 de . 6 m. be demanded . Tangent of 72 de . 94 m. is 32586438 Tang. of 55 de . 88 m. doubled is 29517676     Their difference is 03068762 The Tangent of 17 de . 6 m. Or lastly , let the lesser arch be given , with the Tangent of the difference , and let the greater arch be demanded . Tang. of 55 de . 88 m. the diff . is 14758838     Which doubled is 29517676 To which the tang . of 17 d. 6 m. ad . 3068761 Their aggregate is 32586437 the tangent of 72 degrees , 94 minutes . Theor. 2. The tangent of the difference of two arches making a Quadrant , with the tangent of the lesser arch maketh the secant of the difference . The Reason is Because the tangent of the difference BL or BO , that is , the right line BK or BM with the tangent of the lesser arch BS , that is , with the right line BT , maketh the right line MT , which is equall to the Secant AK , by the demonstration of the first Theorem . Therefore , the tangent of the difference of two arches making a Quadrant , and the tangent of the lesser arch being given , the secant of the difference is also given . And contrarily . For example . Let the tangent of the former difference 55 degrees , 88 minutes , and the tangent of the lesser arch 17 degrees , ●● minutes , be given ; I say , the secant of this difference is also given . Tang. of the diff . 55 de . 88 m. is 14758838 The tangent of 17 06 is 3068762     Their sum is the secant of 55 88 , 17827600 Theor. 3. The tangent of the difference of two arches making a Quadrant , with the secant of their difference , is equal to the tangent of the greater arch . Because the tangent of the arch BL , being the difference of the two arches BC and DC , making a Quadrant with the secant of the same arch BL , that is , the right line BK with the right line AK , is equal to the right line BP , by the demonstration of the first Theorem : therefore the tangent of the difference of two arches making a Quadrant being given , with the secant of their difference , the tangent of the greater arch is also given . For example . Let the tangent of the difference be the tang . of the arch of 55 de . 88 m. viz.   14758838 The secant of this difference is 17827600 Their sum is the tang . of 72 94 , 32586438 the greater of the two former given arches . And now by the like reason these Rules may be added by way of Appendix . Rule I. The double tangent of an arch , with the tangent of half the complement , is equall to the tangent of the arch , composed of the arch given and half the complement thereof . For if the arch BL be put for the arch given , the double tangent thereof shall be TP , by the demonstration of the first Theorem . And the complement of the arch BL , shall be the arch LC , whose half is the arch LD or DC , whose tangent is the right line GC or BT , but TP added to BT maketh BP , being the tangent of the arch BD , composed of the given arch BL , and half the complement LD , therefore the double tangent , &c. Rule II. The tangent of an arch with the tangent of half the complement is equal to the secant of that arch . For if you have the arch BL or BO for the arch given , the tangent of the arch given shall be BM , the tangent of half the complement shall be BT , which two tangents added together , make the right line MT , but the right line MT is equal to the right line AK , by the demonstration of the first Theorem ; which right line AK is the secant of the arch given BL , by the proposition : Therefore the tangent of an arch , &c. Rule III. The tangent of an arch with the secant thereof is equal to the tangent of an arch composed of the arch given , and half the complement . For if you have the arch BL for the arch given , BK shall be the tangent , and AK the secant of that arch . But the right line AK and KP are equal , by the demonstration of the first Theorem : therefore the tangent of the arch given BL , that is , the right line BK , with the secant of the same arch , that is , AK is equall to the right line BP , which is the tangent of the arch BD , being composed of the given arch , BL and LD being half the complement . These rules are sufficient for the making of the Tables of natural Sines , Tangents , & Secants . The use whereof in the resolution of plain & spherical triangles should now folow ; but because the Right Honourable John Lord Nepoir , Baron of Marchiston , hath taught us how by borrowed numbers , called Logarithmes : to perform the same after a more easie and compendious way : we will first speak something of the nature and construction of those numbers , called Logarithmes ; by which is made the Table of the artificial Sines and Tangents , and then shew the use of both . CHAP. V. Of the nature and construction of Logarithmes . LOgarithmes are borrowed numbers , which differ amongst themselves by Arithmetical proportion , as the numbers that borrow them differ by Geometrical proportion : So in the first column of the ensuing Table the numbers Geometrically proportional being 1 , 2 , 4 , 8 , 16 , 32 , 64 , 128 , 256 , 512 , 1024 , &c. you may assigne unto them for bo●rowed numbers or Logarithmes , the numbers subscribed under the letters A , B , C , D , or any other at pleasure ; provided , that the Logarithmes so assigned still differ amongst themselves by Arithmetical proportion , as the numbers of the first column differ by Geometrical proportion : For example . In the column C , if you will appoint 5 to be the Logarithme of one , 8 the Logarithme of 2 , and 11 the Logarithme of 4 , 14 must needs be the Logarithme of 8 , the next proportional , because the numbers 5 , 8 , 11 , and 14 differ amongst themselves by Arithmetical proportion , as 1 , 2 , 4 , and 8 ( the proportional numbers unto which they are respectively assigned ) differ by Geometrical proportion , that is , as the numbers 5 , 8 , 11 , and 14 have equal differences : so the numbers 1 , 2 , 4 , and 8 have their differences of the same kinde : for as the difference between 5 and 8 , 8 and 11 , 11 and 14 , is 3 : so in the other numbers , as 1 is half 2 , so 2 is half 4 , 4 half 8 , &c. The same observation may be made of the Logarithmes placed in the columns , A , B , and D , or of any other numbers which you shall assigne as Logarithmes unto any rank of numbers , which are Geometrically proportional , and these Logarithmes or borrowed numbers you may propound to increase , and to be continued upwards , as those of the columnes A , B , C , or otherwise to decrease , and to be continued downwards , as those of the column D.   A B C D 1 1 5 5 35 2 2 6 8 32 4 3 7 11 29 8 4 8 14 26 16 5 9 17 23 32 6 10 20 20 64 7 11 23 17 128 8 12 26 14 256 9 13 29 11 512 10 14 32 8 1024 11 15 35 5   Log Log Log Log The numbers continually proportional , which Mr. Briggs ( after a conference had with the Lord Nepeir ) hath proposed to himself in the Calculation of his C●ili●des , are 1 , 10 , 100 , 1000 , &c. to which numbers he hath assigned for Logarithmes 000 , &c. 1000 , and 2000 , and 3000 , that is to say , to 1 , the Logarithme 0.000 , and to 10 , the Logarithme 1,000 , and to 100 the Logarithme 2.000 , as in the table following you may perceive . In the column marked by the letter A , there is a rank of numbers continually proportional from 1 , and over against each number his respective Logarithme in the other column , signed by the letter B. A B 1 0.00000 10 1.00000 100 2.00000 1000 3.00000 10000 4.00000 Having thus assigned the Logarithme to the proportional numbers of 1 , 10 , 100 , 1000 , &c. in the next place , it is requisite to finde the Logarithmes of the mean numbers situate amongst those proportionals of the same table , viz. of 2 , 3 , 4 , &c. which are numbers scituate betwixt 1 and 10 , of 11 , 12 , 13 , &c. which are placed betwixt 10 and 100 ; and so consequently of the rest : wherefore how this also may be done we intend to explain by that which followeth . 1. § . Make choice of one of the propotional numbers in the Table AB , and by a continued extraction of the square root create a rank of continuall meanes betwixt that number and 1 , in such sort , that the continuall mean which cometh nearest 1 may be a mixt number , lesse then 2 , and so near 1 , that it may have as many ciphers before the significant figures of the numerator , as you intend that the Logarithmes of your Table shall consist of places . Example . In the premised Table AB , I take 10 , the second proportional of that Table , then annexing unto it a compent company of ciphers , as twenty and four , thirty and six , fourty and eight , or any other number at pleasure ; onely observe , that the more ciphers you annex unto the number given , the more just and exact the operation will prove ; to make the Logarithmes of a Table to seven places 28 ciphers will be sufficient , they being therefore added to 10 , I extract the square root thereof , and finde it to be 3.16227766016837 ; again , annexing unto this root thus found 14 ciphers more , and working by that entire number so ordered , as if it were a whole number , I extract the root thereof , which I finde to be 1.77827941003892 : and so proceeding successively by a continued extraction , I produce 27 square roots , or continual means betwixt 10 and 1 , and write them down in the first column of the Table hereunto annexed , in which you may observe , that the three last numbers marked by the letters G , H , and L , viz. 1.00000006862238 1.00000003431119 1.00000001715559 are each of them mixt numbers lesse then 2 , and greater then 1 , and likewise to have seven ciphers placed bef●re the significant figures of their numerators , according to the true meaning and intention of this present rule . 2. § . Having thus produced a great company of continual meanes , annex unto them their proper Logarithmes , by halfing first the Logarithme of the number taken , and then successively the Logarithme of the rest . For example . 1.000000000000000 being assigned the Logarithme of 10 , the number taken 0.500000 , &c. marked by the letter D , in the second column of the following Table , which is the half of 1.0000 , &c. is the Logarithme of the number A , the square root of 10 : in like manner 0.25000 , &c. being half 0.5000 , and is the Logarithme of the number B , and 0.125000 , &c. is the Logarithme of the number C , and so of the rest in their order . So that at last , as you have in the first column of the following Table 27 continuall meanes , betwixt 10 and 1 , as aforesaid : So in the other column you have to each of those continuall meanes , his respective Logarihme . 3. § . When a number which being lesse then 2 , and greater then 1 , comes so neer to 1 , that it hath seven ciphers placed before the significant figures of the numerator , the first seven significant figures of the numerator of such a number , and the first seven significant figures of the numerator of his square root lessen themselves like their Logarithmes , that is , by halfes . This is proved by the Table following ; for there in the second column thereof , the number N being the Logarithme of the number G , I say , as the Logarithme K is half the Logarithme N , so 3431119 , the first seven figures of the numerator of the number H , are half 6862238 , the first seven significant figures of the numerator of the number G. Any two numbers of this kinde therefore being given , their Logarithmes and the significant figures of their numerators are proportional . Example . The numerators G and H being given , I say , as 6862238 , the significant figures of the numerator of the number G ; are to 3431119 , the significant figures of the numerator of the number H ; so is 29802322 , the Logarithme of the number G , to 140901161 , the Logarithme of the number H. In like manner , G and L being given , as 6862238 , is to 1715559 , so is 29802322 , the Logarithme of the number G , to 7450580 , the Logarithme of the number L. This holdeth also true in any other number of this kinde , though it be not one of the continual means betwixt 10 and 1 , for the significant figures of the numerator of any such number bear the same proportion to his proper Logarithme , that the significant figures of any of the numbers marked by the letters G , H , or L bear to his .   10.0000 , &c. 1.000000000000000   A 3.16227766016837 0.500000000000000 D B 1.77827941003892 0.250000000000000   C 1.33352143216332 0.125000000000000     1.15478198468945 0.062500000000000     1.07460782832131 0.031250000000000     1.03663292843769 0.015625000000000     1.01815172171818 0.007812500000000     1.00903504484144 0.003906250000000     1.00450736425446 0.001953125000000     1.00225114829291 0.000976562500000     1.00112494139987 0.000488281250000     1.00056231260220 0.000244140625000     1.00028111678778 0.000122070312500     1.00014054851694 0.000061035156250     1.00007027178941 0.000030517578125     1.00003513527746 0.000015258789062     1.00001756748442 0.000007629394531     1.00000878270363 0.000003814697265     1.00000439184217 0.000001907348632     1.00000219591867 0.000000953674316     1.00000109795873 0.000000476837158     1.00000054897921 0.000000238418579     1.00000027448957 0.000000119209289     1.00000013724477 0.000000059604644   G 1.00000006862238 0.000000029802322 N H 1.00000003431119 0.000000014901161 K L 1.00000001715559 0.000000007450580 M 4. § . These things being thus cleared , it is manifest , that a number of this kinde being given , the Logarithme thereof may be found by the Rule of three direct . For as the significant figures of the numerator of any one of the numbers ( signed in the first column of the last Table by the letters G , H , or L ) are to his respective Logarithme : so are the significant figure of the numerator of the number given , to the Logarithme of the same number . Example . The number 1.00000001021301 being given , I demand the Logarithme thereof : I say then , As 6862238 , the significant figures of the numerator of the number G , are to 29802322 , the logarithme of the same number G : so are 1021301 , the significant figures of the numerator of the number given , to 4357281 , the Logarithme sought ; before which if you prefix 9 ciphers , to the intent it may have as many places as the Logarithme in the last premised Table , ( viz. 16 ) the true and entire Logarithme of 1 . 00000001021301 , the number given is 0 . 000000004357281 , as before . And to every Logarithme thus found , you must prefix as many ciphers as will make the said Logarithme to have as many places as the other Logarithmes in the same table : for though you make your Table of Logarithmes to consist of as many places as you please , yet when you are once resolved of how many places the Logarithmes of your Table shall consist , you must not alter your first resolution , as to make the Logarithme of 2 to consist of six places , and the Logarithme of 16 to have seven , but if the significant figures of the numerator of the Logarithme of 2 have not so many places as the significant figures of the Logarithme of 16 , you must prefix a cipher or ciphers to make them equal ; because ( as hath been said , the Logarithmes of this kinde ought all to consist of equal places in the same Table . 5. § . Now then to finde the Logarithme of any number whatsoever , you are first to search out so many continual means betwixt the same number and 1 , till the continual mean that cometh neerest 1 hath as many ciphers placed before the significant figures of his numerator , as you intend the Logarithmes of your Table shall consist of places ; Again , this being done , you are to finde the Logarithme of that continual mean : And lastly , by often doubling and redoubling of that Logarithme so found ( according to the number of the continual meanes produced ) in conclusion you shall fall upon the Logarithme of the number given . Example . the number 2 being given , I demand the Logarithme thereof to seven places : Here first in imitation of that which is before taught in the first rule of this Chapter , I produce so many continual meanes between 2 and 1 , till that which cometh nearest 1 hath seven ciphers before the significant figures of the numerator , which after three and twenty continued extractions , I finde to be 1.00000008262958 This continual mean being thus found ( by the direction of the last rule aforegoing ) I finde the Logarithme thereof to be 0.000000035885571 . for , As 6862238 , is to 29802322 : So 8262958 , is to 35885571. This Logarithme being doubled will produce the Logarithme of the continual mean next above 1.00000008262958 , and so by doubling successively the Logarithme of each continual mean one after another , according to the number of the extractions ( viz. three and twenty times in all ) at last you shall happen upon the Logarithme 0.301029987975168 , which is the Logarithme of 2 the number propounded : The whole frame of the work is plainly set down in the table following ; for in the first column thereof you have 23 continual meanes betwixt 2 and 1 , and in the other column their respective Logarithmes , found by a continual doubling and redoubling of 0.000000035885571 , the Logarithme of the last continual mean in the table . 2.0000 , &c. 0 301029987975168 1.41421356237309 0.150514993987584 1.18920711500272 0.075257496993792 1.19050713266525 0.037628748496896 1.04427378243220 0.018814374248448 1.02189714865645 0 009407187124224 1.01088928605285 0.004703593562112 1.00542990111387 0.002351796781056 1.00271127505073 0.001175898390528 1.00135471989237 0.000587949195264 1.00067713069319 0.000293974597632 1.00033850805274 0.000146987298816 1.00016923970533 0.000073493649408 1.00008461627271 0.000036746824704 1.00004230724140 0.000018373412352 1.00002115339696 0.000009186706176 1.00001057664255 0.000004593353088 1.00000528830729 0.000002296676544 1.00000264415015 0.000001148338272 1.00000132207420 0.000000574169136 1.00000066103688 0.000000287084568 1.00000033051838 0.000000143542284 1.00000016525917 0.000000071771142 1.00000008262958 0.000000035885571 But now because the Logarithme of the number propounded was to con●ist onely of seven places ; therefore of the Logarithme so found I take onely the first seven figures rejecting the rest as superfluous , and then at the last the proper Logarithme of 2 , the number given will be found to be 0.301029 , and because the eighth figure being 9 , doth almost carry the value of an unit to the same seventh figure , I adde one thereto , and then the precise Logarithme of 2 will be 0.301030 . And thus as the Logarithme of 2 is made , so may you likewise make the Logarithme of any other number whatsoever : Howbeit , the Logarithmes of some few of the prime numbers being thus discovered , the Logarithmes of many other derivative numbers may be found out afterwards without the trouble of so many continued extractions of the square root , as shall appear by that which followes . 6. § . When of four numbers given , the second exceeds the first as much as the fourth exceeds the third ; the summe of the first and fourth is equal to the summe of the second and third ; and contrarily . As 8 , 5 : 6 , 3. here 8 exceeds 5 , as much as 6 exceeds 3 : therefore the summe of the first and fourth , namely , of 8 and 3 is equall to the summe of the second and third ; namely of 5 and 6 : from whence necessarily followes this Corollary ; When four numbers are proportionall , the summe of the Logarithmes of the mean numbers is equal to the summe of the Logarithmes of the extreams . Example . Let the four proportional numbers be those exprest in the first column of the first Table in this Chapter , viz. 4 , 16 , 32 , 128 , in which Table the Logarithme of 4 under the letter A is 3 , the Logarithme of 16 , 5 , the Logarithme of 32 , 6 ; and the Logarithme of 128 is 8. Now as the summe of 5 and 6 , the Logarithmes of the mean numbers do make 11 , so the summe of 3 and 8 , the Logarithmes of the extreames , do make 11 also . 7. § . When four numbers be proportional , the Logarithme of the first substracted from the summe of the Logarithmes of the second and third , leaveth the Logarithme of the fourth . Example . Let the proportion be , as 128 , to 32 ; so is 16 , to a fourth number : here adding 5 and 6 , the Logarithmes of the second and third , the sum is 11 , from which substracting 8 , the Logarithme of 128 , the first proportional , the remainer is 3 , the Logarithm of 4 , the fourth proportional . 8. § . If instead of substracting the aforesaid Logarithme of the first , we adde his complement arithmetical to any number , the totall abating that number , is as much as the remainer would have been . The complement arithmetical of one number to another , ( as here we take it ) is that , which makes that first number equall to the other ; thus the complement arithmetical of 8 to 10 is 2 , because 8 and 2 are 10. Now then whereas in the example of the last Proposition , substracting 8 from 11 , there remained 3 , if instead of substracting 8 , we adde his complement arithmeticall to 10 , which is 2 , the totall is 13 , from which abating 10 , there remains 3 , as before : both the operations stand thus : As 128 , is to 32 : So is 16 , Logar 8 compl . arithmetical 2 6   6 5   5 The aggreg . of 1.2 .   11 Their aggregate is 13 To 4 ,   3     from which abate 10 , there remaines 3 , and the like is to be understood of any other . The reason is manifest , for whereas we should have abated 8 out of 11 , we did not onely not abate it , but added moreover his complement to 10 , which is 2 , wherefore the total is more then if should be by 8 & 2 , that is by 10 ; wherefore abating 10 from it , we have the Logarithme desired ; which rule , although it be generall , yet we shall seldome have occasion to use any other complements , then such as are the complements of the Logarithmes given either to 10,000000 , or to 20,000000 , the ● complement arithmetical of any Logarithme to either of these numbers , is that which makes the Logarithme given equal to either of them . Thus the complement arithmetical of the Logarithme of 2 viz. 0301030 , is 9698970 , because these two numbers added together , do make 10.000000 , and thus the complement thereof to 20 . 000000is 19698970 : if therefore 0301030 be substracted from 10.000000 , the remainer is his complement arithmetical . But to finde it readily , you may instead of substracting the Logarithme given from 10.000000 , write the complement of every figure thereof unto 9 , beginning with the first figure toward the left hand , and so on , till you come to the last figure towards the right hand , and thereof set down the residue unto 10. Thus for the complement arithmetical of the aforesaid Logarithme , 0301030 ; I write for 0 , 9 : for 3 , 6 : for 0 , 9 : for 1 , 8 : for 0 , 9 : for 3 again I should write 6 : but because the last place of the Logarithme is a cipher , and that I must write the complement thereof to 10 , instead of 6 I write 7 , and for 0 , 0 : and so have I this number , 9698970 , which is the complement arithmetical of 0301030 , as before . 9. § Every Logarithme hath his proper Characteristick , and the Character or Characteristicall root of every Logarithme is the first figure or figures towards the left hand , distinguished from the rest by a point or comma . Thus the Character of the Logarithmes of every number lesse then 10 is 0 , but the Character of the Logarithme of 10 is 1 ; and so of all other numbers to 100 , but the Character of the Logarithme of 100 is 2 ; and so of the rest to 1000 ; and the Character of the Logarithme of 1000 is 3 ; and so of the rest to 10000 : in brief , the Characteristick of any Logarithme must consist of a unite lesse then the given number consisteth of digits or places , And therefore by the Character of a Logarithme you may know of how many places the absolute number answering to that Logarithme doth consist . 10. § . If one number multiply another , the summe of their Logarithme is equal to the Logarithme of the product . As let the two numbers multiplied together be 2 , and 2 the products is 4 , I say then that the summe of the Logarithmes of 2 and 2 , or the Logarithme of 2 doubled is equal to the Logarithme of 4 , as here you may see . 2. 0.301030 2. 0.301030     4. 0.602060 Again , let the two numbers multiplied together be 2 , and 4 , the product is 8 , I say then that the summe of the Logarithmes of 2 and 4 is equall to the Logarithme of 8 , as here you may also see , 2. 0.301030 4. 0.602060     8. 0.903090 And so for any other . The reason is , for that ( by the ground of multiplication ) as unit is in proportion to the multiplier : so is the multiplicand , to the product : therefore ( by the sixth of this Chapter ) the sum of the Logarithmes of a unit , and of the product is equall to the summe of the Logarithmes of the multiplier and multiplicand , but the Logarithme of a unit is 0 , therefore the Logarithme of the product alone is equal to the summe of the Logarithmes of the multiplier and multiplicand . And by the like reason , it three or more numbers be multiplied together , the summe of all their Logarithmes is equall to the Logarithme of the product of them all . 11. § . If one number divide another , the Logarithme of the Divisor being substracted from the Logari●hme of the Dividend , leaveth the Logarithme of the Quotient . As let 10 be divided by 2 , the quotient is 5. I say then , if the Logarithme of 2 be substracted from the Logarithme of 10 , there will remain the Logarithme of 5 , as here is to be seen . 10. 1.000000 2. 0.301030     5. 0.698970 For seeing that the quotient multiplied by the divisor produceth the dividend , therefore , by the last proposition , the sum of the Logarithmes of the quotient and of the divisor is equal to the Logarithme of the divi●●● if therefore the Logarithme of the divid●●ol , be substracted from the Logarithme of the divi●●● there remaines the Logarithme of the quotient . 12. § . In any continued rank of numbers Geometrically proportionall from 1 , the Logarithme of any one of them being divided by the denomination of the power which it challengeth in the same rank , the quotient will give you the Logarithme of the root . In the rank of the proportional numbers of the Table ABCD , 2 being the root , or first power ; 4 the square or second power , 8 the cube , or third power , 16 the bi-quadrate or fourth , 32 the fifth power , 64 ▪ the sixth power , &c. I say , the Logarithme of 4 , 8 , 16 , 32 , 64 , or of any of the other subsequent proportionals in that rank , being divided by the demonination of the power that the same proportional claimeth in the same rank , you shall finde in the quotient the Logarithme of 2 the root . For example . In the same Table the Logarithme of 4. the square or second power , viz. 3. being given , I demand the Logarithme of 2 , the root : here the denomination of the power that the proportional 4 challengeth in that rank ( being the square or second power ) is 2 , wherefore if 3 , the Logarithme of 4 be divided by 2 , the quotient will be 1 , and there will remain 1 for a fraction ; so that you see it cometh very near in the Logarithmes of but one figure , but if you take it to seven places , as in this table is intended , you shall finde it exactly : for then the Logarithme of 4 will be 0.602060 , and this being divided by 2 , the quotient will be 0.301030 , the Logarithme of 2 the root . So likewise 0.903090 , the Logarithme of 8 the third power , being divided by 3 , leaves 0.301030 in the quotient , as before , and so of any other . 13. § . In any rank of numbers Geometrically proportionall from 1 , the Logarithme of the root being multiplied by the denomination of any of the powers , the product is the Logarithme of the same power . This Rule is the inverse of the last . For example . In the rank produced in the last rule 0.301030 , ( the Logarithme of 2 the root ) being doubled , or multiplied by 2 , produceth 0.602060 , the Logarithme of 4 , the square or second power , and the same Logarithme of 0.301030 , being trebled or multiplied by 3 , produceth 0.903090 , the Logarithme of 8 , the cube or third power , and so of the rest . The truth of these two last rules may thus be proved . In arithmeticall proportion , when the first term is the common difference of the terms , the last term being divided by the number of the terms , the quotient will give you the first term of the rank : again , in this case , the first term multiplied by the number of the termes produceth the last term . So this rank 3 , 6 , 9 , 12 , 15 , 18 , 21 being propounded , wherein three is both the first term and also the common difference of the terms : I say , 21 , the last term being divided by 7 , the number of the termes , the quotient is 3 , the first term . Contrariwise , 3 the first term multiplied by 7 , produceth 21 , the last term ; and by the like reason , 0.301030 being the first term , and also the common difference of the termes , that is , of the Logarithmes of 4 , 8 , 16 , 32 and 64 , the Logarithme of 2 the first term , being multiplied by 6 , the number of the terms , produceth the Logarithme of 64 , the last term , and the garithme of 64 , the last term , being divided by 6 , leaveth in the quotient the Logarithme of 2 the root . Hence it also followes , that if you adde the Logarithme of 2 , the common difference of the termes , to the Logarithme of any term , their aggregate shall be the Logarithme of the next term . Thus if I adde 0.301030 , the Logarithme of 2 the root or first term , to 0.903090 , the Logarithme of 8 , the third term , their aggregate is 1 . 204120 , the Logarithme of 16 , the fourth term ; and so of the rest . 14. § . Thus having shewed the construction of the Logarithmetical Tables , the converting of the Table of natural Sines , Tangents , and Secants into artificiall cannot be difficult , the artificiall Sines and Tangents being nothing but the Logarithmes of the naturall . 15. § . In the conversion whereof Mr. Briggs in his Trigonometria Britannica , thought fit to make the Radius of his natural Canon to consist of 16 places , and to confine his artificiall to the Radius of eleven , whose Characteristick is 10 , but the Characteristick of the rest of the Sines till you come to the sine of 5 degrees and 73 centesmes is 9 , and from thence to 57 centesmes , the Characteristick is 8 , and from thence 7 , till you come to 5 centensmes , and from thence but 6 , to the beginning of the Canon . The Characteristick still decreasing in the same proportion with the naturall numbers , and the number of the places in the naturall Canon , do therefore exceed the Characteristick in the artificiall , that so the artificiall numbers might be the more exact . 16. § . In the Canon herewith printed , the Characteristick in the artificiall numbers doth exceed the number of places in the naturall , which is not done so much out of necessity as conveniency , for the artificiall numbers in this Canon might in all respects have been made answerable to the natural , and so the Characteristick of the Radius , or whole Sine would have been seven , the Characterick of the first minute 3 , but thus the subduction of the Radius would not have been so ready as now it is , nor yet the Canon it self altogether so exact , and therefore as Master Briggs confined the Radius of his artificiall Canon to eleven places for conveniency sake , though he made the Logarithmes to the Radius of sixteen : so here for conveniency and exactnesse both , the same Characterick is here continued , though the naturall numbers do not require it , if any think this a defect , I answer , that it could not well be avoided here , but may be supplied by Master Briggs his Canon , of which this is an abbreviation : and yet even here there is so small a difference between the Logarithmes of these naturall numbers , and the Logarithmes in the Canon , that any one may well perceive the one to be nothing else but the Logarithme of the other , if they do but change the Characteristick . And hence we may gather , that the making of this Canon is not so difficult as laborious , and the labour thereof may be much abridged by this Proposition following . 17. § . The Sine of an arch and half the Radius are mean proportionals between the Sine of halfe that arch , and the Sine complement of the same half . In the annexed Diagram , let DE be the sine of 56 degrees , BC the sine of 28 , AC the sine complement thereof , that is , of 62. DB the subtense of 56. CF perpendicular to the Radius , then are ABC and ACF like triangles , by the 22 of the second , and their sides proportional that is , AB   AB BC   AO       AC   DE CF   CF And therefore the oblongs of BC×AC , AO × DE , and AB × CF are equal , and the sides of equal rectangled figures reciprocally proportional , that is , as BC , AO ∷ DE , AC . or as AO , BC ∷ AC , DE. If therefore you multiply AO , the half Radius , by DE , the sine of the arch given , and divide the product by BC , the sine of half the arch given , the quotient shall be AC , the sine complement of half the given arch . Or if you multiply BC , the sine of an arch by AC , the sine complement of the same arch , and divide the product by AO , the half Radius , the quotient shall be DE , the sine of the double arch . And therefore the sines of 45 degrees being given , or the Logarithmes of those sines , the rest may be found by the rule of proportion . For illustration sake we will adde an example in naturall and artificiall numbers . Naturall , As BC 28 , 46947 Is to AO 30 ; 50000 So is DE 56 , 82903 To AC 62 : 88294 Logarith . As BC 28 , 9.671609     Is to AO 30 ; 9.698970 So is DE 56 , 9.918574     To AC 62. 9.945935 18. § . The composition of the naturall Tangents and Secants , by the first and second of the fourth are thus to be made . 1. As the sine of the complement , is to the sine of an arch : So is the Radius , to the tangent of that arch . 2. As the sine of the complement , is to the Radius : so is the Radius , to the Secant of that arch ; and by the same rules may be also made the artificiall ; but with more ease , as by example it will appear . Let the tangent of 30 degrees be sought .   Logarith . As the co-sine of 60 degrees , 9.937531     Is to the sine of 30 ; 9.698970 So is the Radius , 10.000000     To the tangent of 30 : 9.761439 And thus having made the artificiall Tangents of 45 degrees , the other 45 are but the arithmeticall complements of the former , taken as hath been shewed in the eighth rule of the fifth Chapter . Again , let the secant of 30 degrees be sought . As the co-sine of 60 degrees , 9.937531     Is to the Radius , 10.000000 So is the Radius , 10.000000       20.000000     To the secant of 30 : 10.062469 And thus the Radius being added to the arithmetical complement of the sine of an arch , their aggregate is the secant of the complement of that arch . And this is sufficient for the construction of the naturall and artificiall Canon . How to finde the Sine , Tangent or Secant of any arch given in the Canon herewith printed , shall be shewen in the Preface thereunto : here followeth the use of the naturall and artificiall numbers both ; first , in the resolving any Triangle , and then in Astronomy , Dialling , and Navigation . CHAP. VI. The use of the Tables of natural and artificial Sines , and Tangents , and the Table of Logarithmes . In the Dimension I. Of plain right angled Triangles . THe measuring or resolving of Triangles is the finding out of the unknown sides or angles thereof by three things known , whether angles , or sides , or both ; and this by the help of that precious gemme in Arithmetick , for the excellency thereof called the Golden Rule , ( which teacheth of four numbers proportional one to another , any three of them being given , to finde out a fourth ) and also of these Tables aforesaid . Of Triangles , as hath been said , there are two sorts ; plain and sphericall . A triangle upon a plain is right lined , upon the Sphere circular . Right lined Triangles are right angled or oblique . A right angled , right-lined Triangle we speak of first , whose sides then related to a circle are inscribed totally or partially . Totally , if the side subtending the right angle be made the Radius of a Circle , and then all the sides are called Sines , as in the Triangle ABC . Partially , if either of the sides adjacent to the right angle be made the Radius of a circle , and then one side of the Triangle is the Radius or whole Sine , the shorter of the other two sides is a Tangent , and the longest a secant . Now according as the right angled Triangle is supposed , whether to be totally or but partially inscribed in a circle ; so is the trouble of finding the parts unknown more or lesse , whether sides or angles ; for if the triangle be supposed to be totally inscribed in a circle , we are in the solution thereof confined to the Table of Sines onely , because all the sides of such a triangle are sines : but if the triangle be supposed to be but partially inscribed in a circle , we are left at liberty to use the Table of Sines , Tangents , or Secants , as we shall finde to be most convenient for the work . In a right angled plain Triangle , either all the angles with one side are given , and the other two sides are demanded , I say , all the angles , because one of the acute angles being given , the other is given also by con●●quence . Or else two sides with one angle , that is , the right angle are given and the other two angles with the third side are demanded . In both which cases this Axiome following is well nigh sufficient . The first AXIOME . In all plain Triangles , the sides are in portion one to another , as are the sines o● the angles opposite to those sides . As in the triangle ABC , the side AB is in proportion to the side AC , as the sine of the angle at B is in proportion to the sine of the angle at c and so of the rest . Demonstration . The circle ADF being circumscribed about the Triangle ABC , the side AB is made the chord or subtense of the angle ACB , that is , of the arch AB , which is opposite to the angle ACB . The side AC is made the subtense of the angle ABC ; and the side BC is made the subtense of the angle BAC , and are the double measures thereof , by the 19 Theorem of the second Chapter : therefore the side AB is in proportion to the side AC , as the subtense of the angle ACB is in proportion to the subtense of the angle ABC , but half the subtense of the angle ACB is the sine of the angle ACB , and half the subtense of the angle ABC is the sine of the angle ABC ; now as the whole is to the whole ; so is the half , to the half . Therefore in all plain Triangles , &c. The first Consectary . The angles of a plain triangle , and one side being given , the reason of the other sides is also given . The second Consectary . Two sides of a plain Triangle , with an angle opposite to one of them being given , the reason of the other angles is also given , by this proportion . If the side of a Triangle be required , put the angle opposite to the given side in the first place . If an angle be sought , put the side opposite to the given angle in the first place . For the better understanding whereof we will adde an example , and to distinguish the sides of the Triangle , we call the side subtending the right angle , the Hypothenusall , and of the other two the one is called the perpendicular , and the other the base , at pleasure , but most commonly the shortest is called the perpendicular , and the longer the base . As in the former figure , the side BC is the Hypothenusal , AC the base , and AB the perpendicular . Now then in the Triangle ABC , let there be given the base AC 768 paces , and the angle CBA 67 degrees , 40 minutes , ( then the angle ACB is also known , it being the complement of the other ) and let there be required the perpendicular : because it is a side that is required , I put the angle opposite to the given side in the first place , and then the proportion is : As the sine of the angle at the perpendicular , is in proportion to the base : So is the sine of the angle at the base , to the perpendicular . Now if you work by the natural Sines , you must multiply the second term given , by the third , and divide the product by the first , and then the quotient is the fourth term required , and the whole work will stand thus : As sine the ang . at the perpend . ABC 67 degrees 40 minutes 9232102     Is in proportion to the base AC ; 768 So is sine the angle at the base , ACB 22 degrees 60 minutes 3842953       30743624   23057718   26900671     The product of the 2d . & 3d. 2951387904 Which divided by 9232102 , the first term given , leaveth in the quotient 320 ferè . But if you work by the artificiall sines , that is , by the Logarithmes of the natural , then you must adde together the Logarithmes of the second and third terms ; given , and from their aggregate substract the Logarithme of the first , and what remaineth will be the Logarithme of the fourth proportional , whether side or angle : the work standeth thus . As sine the angle at the perpendicular B 67 deg . 40 min. 9.9653006     Is in proportiō to the base AB 768 ; 2.8853612 So is sine the angle at the base C 22 degree , 60 minutes . 9.5846651     The aggregate of the 2d . & 3d. 12.4700263 From which I substract the first , 9.9653006 , and the remainer which is 2.5047257 , is the Logarithme of the fourth : wherefore looking in the Table for the absolute number answering thereunto , I finde the nearest to be 320 , which is the length of the perpendicular , as before . The operation it self may yet be performed with more ease , if instead of the Logarithme of the first proportional , we take his complement arithmetical , as hath been shewed in the eighth raile of the fifth Chapter : for then the totall of the arithmaticall complement , and the Logarithme of the second and third proportionals , abating Radius , is the Logarithme of the fourth proportionall , as doth appear in this example . As sine of ABC 67 de . 40m . co . ar . 0.0346994 To the base AC 768 ; 2.8853612 So the sine of ACB 22 de . 60m . 9.5846651     To the perpendic . AB 320 ferè 2.5047257 Thus having sufficiently explained the operation in this first example , we shall be briefer in the rest that follow , understanding the like in them also . In this manner may all the cases of a plain right angled Triangle be resolved by this proportion , except it be when the base and perpendicular with their contained angle ( that is the right angle ) is given , to finde either an angle or the third side ; in this case therefore we must have recourse to the 17 th . Theorem of the second Chapter , by help whereof the Hypothenusall may be found in this manner : square the sides , and from the aggregate of their squares extract the square root , that square root shall be the length of the Hypothenusal . For example . Let the base be four paces , and the perpendicular 3 , the square of the base is 16 , the square of the perpendicular is 9 , the summe of these two squares is 25 , the square root of this summe is 5 paces , and that is the length of the hypo●enusal ; and this hypothenusal being thus found , the angles also may be ●ound , as before . Nor are we tied to this way of finding the hypothenusal , unlesse we confine our selves to the Tables of Sines onely ; if we would make use of the Tables of Tangents or Secants , the hypothenusal may not onely be found with more ease , but all the cases of a right angled plain triangle may be also found several wayes , by the help of this Axiome following . The second AXIOME . In a plain right angled triangle , any of the three sides may be made the Radius of a circle , and the other sides will be as Sines , Tangents , or Secants . And what proportion the side put as Radius hath unto Radius ; the same proportion hath the other sides unto the Sines , Tangents , or Secants of the opposite angles by them represented . If you make the hypothenusal Radius , the triangle will be totally inscribed in the circle , and consequently the other two sides shall represent the sines of their opposite angles , that is , the base shall represent the sine of the angle at the perpendicular , and the perpendicular shall represent the sine of the angle at the base , as in the preceding Diagram . If you make the base Radius , the triangle will be but partially inscribed in the circle , and the other two sides shall be one of them a tangent , and the other a secant . Thus in the first Diagram of this Chapter , the base BD is made the Radius of the circle , the perpendicular D● is the tangent of the angle at the base , and B● is the secant of the same angle . If you make the perpendicular Radius , the triangle will be but partially inscribed in the circle , as before , and the other two sides will be also the one a tangent and the other a secant . As in this example , the perpendicular AB is made the Radius of the circle , the base AC is the tangent of the angle at the perpendicular , and the hypothenusal BC is the secant of the same angle . Hence it followes , that if you make AB the Radius , the base and perpendicular being given , the angle at the perpendicular may be found by this proportion . As the perpendicular , is in proportion to Radius : So is the base , to tang●nt of the angle at the perpendicular ; for the perpendicular being made the Radius of the circle , it must of necessity bear the same proportion unto Radius , as the hypothenusal doth , when that is made the Radius of the circle : and if the perpendicular be the Radius , the base must needs represent the tangent of the angle at the perpendicular . And the angle at the perpendicular be-being thus found , the hypothenusal may be found by the first Axiome . For , As the sine of the angle at the perpendicular , is in proportion to his opposite side the base ; So is Radius , to his opposite side the hypothenusal : and thus you see that the hypothenusal may be found without the trouble of squaring the sides , and thence extracting the square root . And hence also all the cases of a right angled plain triangle may be resolved several wayes : that is to say , 1. In a plain right angled triangle : the angles and one side being given , every of the other sides is given , by a threefold proportion , that is , as you shall put for the Radius , either the side subtending the right angle , or the greater or lesser side including the right angle . 2. Any of the two sides being given , either of the acute angles is given by a double proportion , that is , as you shall put either this or that side for the Radius : to make this clear , we will first set down the grounds or reasons for varying of the termes of proportion : and then the proportions themselves in every case , according to all the variations . The reasons for varying of the termes of proportion are chiefly three . The first reason is , because the Radius of a circle doth bear a threefold proportion to a sine , tangent , or secant ; and contrariwise , a sine , tangent , or secant hath a threefold proportion to Radius , by the second Axiome of this Chapter . For As sine BC , to Rad. AC in the 1. triangle So Rad. BC , to secant AC in the 3d. tri . So tang . BC , to secant AC in the 2d . tri . & contra Again , As tang . BC , to Rad. AB in the 2d . triang . So Rad. BC , to tang . AB in the 3d. trian . So sine BC , to sine BA in the first triang . & contra Lastly , As secant AC , to Rad. BC in the 3d. tri . So Rad. AC , to sine BC in the first trian So secant AC , to tang . BC in the 2d . tri . & contra Hence then As the sine of an arch or ang . is to Rad. So Rad. to the secant comp . of that arch & so is the tang . of that arch , to his sec . & contr . Also As the tang . of an arch or ang . is to Rad. So is Rad. to the tangent compl . thereof . And so is the sine thereof , to the sine of its complement . & contra . Lastly , As the secant of an arch or ang . to Rad. So is Radius , to the sine compl . thereof And so is secant complement to tangent complement thereof . & contra . Example . Let there be given the angle at the perpendicular 41 degrees 60 minutes , and the base 768 paces , to finde the perpendicular . First , by the natural numbers , As the secant of BAC 41 d. 60m . 13372593     Is to Radius , 10000000 So is the base AB 768     To the perpendicular BC ●74 574 By the Artificiall . As the secant of BAC 41.60 . 10.1262157     Is to Radius ; 10.0000000 So is the base 768 , 2.8853612       12.8853612 To the perpendicular 574 : 2.7591455 Secondly , by the naturall numbers . As the Radius , 10000000 To the co-sine of BAC 41.60 . 7477981 So is the base AB 768 To the perpendicular BC 574 By the Artificiall . As the Radius 10.0000000 To the co-sine of BAC 41.60 . 9.8737843 So is the base AB 768 , 2.8853612 To the perpendicular BC 574 : 2.7591455 Thirdly , by the natural numbers . As the co-secant of BAC 41.60 . 15061915 Is to the co-tang . of BAC 41.60 . 11263271 So is the base AB 768 To the perpendicular BC 574 By the artificiall . As the co-secant of BAC 41.60 . 10. 1778802 Is to the co-tang . of BAC 41.60 10.0516645 So is the base AB 768 2.8853612 To the perpendicular BC 574 2.7591455 COROLLARY . Hence it is evident , that Radius is a mean proportional between the sine of an arch , and the secant complement of the same arch ; also between the tangent of an arch , and the tangent of the complement of the same arch . The second Reason . The sines of several arches , and the secants of their complements are reciprocally proportional , that is , As the sine of an arch or angle , is to the sine of another arch or angle : So is the secant of the complement of that other , to the co-secant of the former . For by the foregoing Corollary , Radius is the mean proportional between the sine of any arch and the co-secant of the same arch . Therefore , whatsoever sine is multiplied by the secant of the complement , is equall to the square of Radius ; so that all rectangles made of the sines of arches and of the secants of their complements are equal one to another ; but equall rectangles have their sides reciprocally pro portional , by the tenth Theorem of the second Chapter . Therefore the sines of several arches , &c. The third Reason . The tangents of severall arches , and the tangents of their complements are reciprocally proportional , that is , As the tangent of an arch or angle , is to the tangent of another arch or angle , so is the co-tangent of that other , to the co-tangent of the former . For by the foregoing Corollary , Radius is the mean proportionall between the tangent of every arch and the tangent of his complement . Therefore the Rectangle made of any tangent , and of the tangent of his complement , is equall to the square of Radius : so that all rectangles made of the tangents of arches , and of the tangents of their complements are equall one to another , but equal rectangles , &c. as before . To these three reasons a fourth may be added . For in the rule of proportion ; wherein there are alwayes four termes , three given , the fourth demanded : It is all one , whether of the two middle terms is put in the second or third place . For it is all one , whether I shall say ; As 2 , to 4 ; so 5 , to 10 : or say , as 2 , to 5 ; so 4 , to 10 : and from hence every example in any triangle may be varied , and thus you see the reasons of varying the termes of proportion , we come now to shew you the various proportions themselves of the severall Cases in right angled plain triangles . Right angled plain triangles may be distinguished into seven Cases ; whereof those in which a side is required , viz. three , may be found by a triple proportion ; and those in which an angle is required , viz. three , may be found by a double proportion . CASE 1. The angles and base given , to finde the perpendicular . First , As sine the angle at the perpendicular , is to the base : so is sine the angle at the base , to the perpendicular . Or secondly , thus : As Radius , to the base ; so tangent the angle at the base , to the perpendicular . Or thirdly , thus : As the tangent of the angle at the perpendicular , is to the base : so is Radius , to the perpendicular . CASE 2. The angles and base given , to finde the hypothenusal . First , As the sine of the angle at the perpendicular , is to the base ; so is Radius , to the hypothenusal . Or secondly thus : As Radius , is to the base ; so the secant of the angle at the base , to the hypothenusal . Or thirdly , thus : As the tangent of the angle at the perpendicular , is to the base : so is the secant of the same angle in proportion to the hypothenusal . CASE 3. The angles and hypothenusal given , to finde the base . First , As Radius , to the hypothenusal : so the sine of the angle at the perpendicular , to the base . Or secondly , thus : As the secant of the angle at the base , to the hypothenusal : so is Radius , to the base . Or thirdly , thus : As the secant of the angle at the perpendicular , to the hypothenusal : so the tangent of the same angle , to the base . CASE 4. The base and perpendicular given , to finde an angle . First , As the base , to Radius : so the perpendicular , to the tangent of the angle at the base . Or secondly , thus : As the perpendicular , is to Radius : so the base , to the tangent of the angle at the perpendicular . CASE 5. The base and hypothenusal given , to finde an angle . 1. As the hypothenusal , is to Radius : so is the base , to the sine of the angle at the perpendicular . Or secondly thus , As the base is to Radius ; so is the hypothenusal , to the secant of the angle at the base . CASE 6. The base and perpendicular given , to finde the hypothenusal . First , finde the angle at the perpendicular , by the fourth Case : Then , As the sine of the angle at the perpendicular , is to the base : so is Radius , to the hypothenusal . Otherwise by the Logarithmes of absolute numbers . From the doubled Logarithme of the greater side , whether base or perpendicular , substract the Logarithme of the lesse , and to the absolute number answering to the difference of the Logarithmes adde the lesse , the half summe of the Logarithmes of the summe , and the lesse side , is the Logarithme of the hypothenusal inquired . The Illustration Arithmetical . Let the base be 768 , and the perpendicular 320. The Logarithme of 768 is 2.8853612     This Logarithme doubled is 5.7707224 From which substr . the Log. of 320 , 2.5051500 The remain . is the Log. of 1843 : 3.2655724 To which the lesser side being added 320 , their aggregate is 2163. The Logarithme of 2163 is 3.3350565 The Logarithme of 320 is 2.5051500 The summe is 5.8402065 The half sum is the Log. of 832. 2.9201032 which is the length of the hypothenusal inquired . CASE 7. The base and hypothenusal given , to finde the perpendicular . The resolve this Probleme by the Canon , there is required a double operation : First , by the 5 Case , finde an angle . Secondly , by the first Case , finde the perpendicular . But Mr. Briggs resolves this Case more readily , by the Logarithmes of the absolute numbers , Briggs Arithmetica Logarith . cap. 17. Take the Logarithmes of the summe and difference of the hypothenusal and side given , half the summe of those two Logarithmes , is the Logarithme of the perpendicular , or side inquired . As let the hypothen . be   832   The side given   768         Logarith . The summe is   1600 3.2041200 The difference is   64 1.8061800           The summe is ,   5.0103000 The half sum is the Logarith . of 320 the side inquired .     2.5051500 The two Axiomes following are true in all plain triangles , but are chiefly intended for the oblique angled ; which now we come to handle . II. Of plain oblique angled Triangles . In a plain oblique angled triangle , there are four varieties . 1. All the angles may be given , ( for when two are given , the third is given by consequence ) and one side , and the other two sides demanded . 2. Two sides with an angle opposite to one of them may be given , and the angle opposite to the other , with the third side are demanded . In both which cases the first Axiome is fully sufficient . 3. Two sides with an angle comprehended by them may be given , and the other two angles with the third side demanded . For the solution whereof we will lay down this Axiome following . The third AXIOME . As the summe of the two sides , is to their difference : so is the tangent of half the summe of the opposite angles , to the tangent of half the difference . Let ABC be the oblique angled triangle , in which let the side AB be continued to H , and let the line of continuation BH be made equall to BC , and BK equal to AB ; then is AH the summe of the sides , AB , BC , and KH is their difference , now if you draw the lines BD and KG parallel unto AC , then shall the angle CBH be equal to the two angles of the triangle given ACB and CAB , because the angle CBA common to both is their complement to a Semicircle , and DB being parallel to CA , the angle DBH shall be equall to the angle CAB , and the angle DBC equall to the angle ACB , if therefore you let fall the perpendicular BE , and draw the periphery MEL , the right line CE shall be the tangent of half the summe of the angles ACB and CAB , it being the tangent of half the angle CBH . Again , if you make E● equal to DE , and draw the right line FB , then shall the angle DBF be the difference between the angles CBD and DBH , or between the angles ACB and CAB , and DE the tangent of half the difference . And because the right sines AC , DB , and KG are parallel , and CD , DG , and FH are equall , and DF equal to GH , and the triangles ACH and KGH are like , and therefore ; As AH is in proportion to HK : so is CH , to HG : or as AH , the summe of the sides , is in proportion to HK , their difference : so is CE the tangent of the half summe of the angles ACB and CAB , to DE , the tangent of half their difference . Consectary . Hence it followes , that in a plain oblique angled Triangle ; if two sides and the angle comprehended by them be given , the other two angles and the third side are also given . As in the triangle ABC , having the sides AC 189 , and AB 156 , whose summe is 345 , and difference 33 , with the angle BAC 22 degrees , 60 minutes , to finde the angle ABC or ACB . The proportion is As the sum of the sides given 345 , 2.5378190     Is to their difference 33 , 1.5185139 So the tangent of half the angles at B & C 78de . 70m . 10.6993616     To the tangent of half their difference 25 degr . 58 minutes 9.6800565 Which being added to the half sum 78 degrees , 70 minutes , the obtuse angle at B , is 104 degrees , 28 minutes ; and substracted from the half summe , it leaveth 53 degrees , 12 minutes for the quantity of the acute angle ACB . Then to finde the third side BC , the proportion , by the first Axiome , is , As the sine of the angle at ● , is in proportion to his opposite side AB ; so is the sine of the angle at A , to his opposite side BC. 4. And lastly , all the three sides may be given , and the angles may be demanded ; for the solution whereof we will lay down this Axiome . The fourth AXIOME . As the base , is to the summe of the sides : So is the difference of the sides , to the difference of the segments of the base . Let BCD be the triangle , CD the base , BD the shortest side ; upon the point B describe the circle ADFH , making BD the Radius thereof , let the side BC be produced to A , then is CA the summe of the sides , because BA and BD are equal , by the work , CH is the difference of the sides , CF the difference of the segments of the base . Now if you draw the right lines AF and HD , the triangles CHD and CAF shall be equiangled , because of their common angle ACF or HCD , and their equal angles CAF and HDC , which are equall , because the arch HF is the double measure to them both ; and therefore , as CD , to CA ; so is CH , to CF , which was to be proved . Consectary . Therefore the three sides of a plain oblique angled triangle being given , the reason of the angles is also given . For first , the obliquangled triangle may be resolved into two right angled triangles , by this Axiome , and then the right angled triangles may be resolved by the first Axiome . As in the plain oblique angled triangle , BCD , let the three sides be given , BD 189 paces , BC 156 paces , and DC 75 paces , and let the angle CBA be required . First , by this Axiome , I resolve it into two right angled triangles ; thus : As the true base BD 189 co . ar . 7.7235382 Is to the sum of BC & DC 231 2.3636120 So the difference of BC & DC 81 1.9084850 To the alternate base BG 99 1.9956352 As the the hypothenusal BC , is to Radius : So is the base AB 144 , to the sine of the angle at the perpendicular , whose complement is the angle at the base inquired . In like manner may be found the angle at D , and then the angle BCD is found by consequence , being the complement of the other two to two right angles or 180 degrees . CHAP. VII . Of Sphericall Triangles . A Sphericall Triangle is a figure described upon a Sphericall or round superficies , consisting of three arches of the greatest circles that can be described upon it , every one being lesse then a Semicircle . 2. The greatest circles of a round or Spherical superficies are those which divide the whole Sphere equally into two Hemispheres , and are every where distant from their own centers by a Quadrant , or fourth part of a great circle . 3. A great circle of the Sphere passing through the poles or centers of another great circle , cut one another at right angles . 4. A spherical angle is measured by the arch of a great circle described from the angular point betwixt the sides of the triangle , those sides being continued to quadrants . 5. The sides of a Spherical triangle may be turned into angles , and the angles into sides , the complements of the greatest side or greatest angle to a Semicircle , being taken in each conversion . It will be necessary to demonstrate this , which is of so frequent use in Trigonometry . In the annexed Diagram let ABC be a sphericall triangle , obtuse angled at B , let DE be the measure of the angle at A. Let FG be the measure of the acute angle at B , ( which is the complement of the obtuse angle B , being the greatest angle in the given triangle ) and let HI be the measure of the angle at C , KL is equal to the arch DE , because KD and LE are Quadrants , and their common complement is LD . LM is equall to the arch FG , because LG and FM are Quadrants , and their common complement is LF . KM is equal to the arch HI , because KI and MH are Quadrants , and their common complement is KH . Therefore the sides of the triangle KLM are equal to the angles of the triangle ABC , taking for the greatest angle ABC , the complement thereof FBG . And by the like reason it may be demonstrated , that the sides of the triangle ABC are equal to the angles of the triangle KLM . For the side AC is equall to the arch DI , being the measure of the angle DKI , which is the complement of the obtuse angle MKL. The side AB is equall to the arch OP , being the measure of the angle MLK . And lastly , the side BC is equal to the arch FH , being the measure of the angle LMK , for AD and CI are Quadrants : so are AP and OB , BF and CH. And CD , AO , and CF are the common complements of two of those arches . Therefore the sides of a spherical triangle may be changed into angles , and the angles into sides , which was to be demonstrated . 6. The three sides of any spherical triangle are lesse then two Semicircles . 7. The three angles of a spherical triangle are greater then two right angles , and therefore two angles being known , the third is not known by consequence , as in plain triangles . 8. If a spherical triangle have one or more right angles , it is called a right angled spherical triangle . 9. If a spherical triangle have one or more of his sides quadrants , it is called a quadrantal triangle . 10. If it have neither right angle , nor any side a quadrant , it is called an oblique spherical triangle . 11. Two oblique angles of a spherical triangle are either of them of the same kinde of which their opposite sides are . 12. If any angle of a triangle be neerer to a quadrant then his opposite side : two sides of that triangle shall be of one kinde , and the third lesse then a quadrant . 13. But if any side of a triangle be nearer to a quadrant then his opposite angle , two angles of that triangle shall be of one kinde , and the third greater then a quadrant . 14. If a spherical triangle be both right angled and quadrantal , the sides thereof are equall to the opposite angles . For if it have three right angles , the three sides are quadrants , if it have two right angles , the two sides subtending them are quadrants ; if it have one right angle , and one side a quadrant , it hath two right angles and two quadrantal sides , as is evident by the third Proposition . But if two sides be quadrants , the third measureth their contained angle , by the fourth proposition . Therefore for the solution of these kindes of triangles , there needs no further rule : But for the solution of right angled , quadrantall , and oblique spherical triangles there are other affections proper to them , which are necessary to be known as well as these general affections common to all spherical triangles . The affections proper to right angled and quadrantal triangles we will speak of first . CHAP. VIII . Of the affections of right angled Sphericall Triangles . IN all spherical rectangled Triangles , having the same acute angle at the base : The sines of the hypothenusals are proportional to the sines of their perpēdiculars . As in the annexed diagram , let ADB represent a spherical triangle , right angled at B : so that AD is the sine of the hypothenusal , AB the sine of the base , and DB is the perpendicular . Then is DAB the angle at the base , and IH the sine , and LM the tangent thereof : Also DF is the sine of the perpendicular DB , and KB is the tangent thereof : I say then , As AD , is to FD : So is AI , to IH , by the 16 th . Theoreme of the second Chapter . And because it is all one , whether of the mean proportionals be put in the second place ; therefore I may say : As AD , the sine of the hypothenusal , is in proportion to AI Radius : So is FD , the sine of the perpendicular , to IH the sine of the angle at the base . 2. In all rectangled spherical triangles , having the same acute angle at the base . The sines of the bases , and the tangents of the perpendiculars are proportional . For as AB , to KB ; so is AM , to ML , by the 16 th . Theorem of the second Chapter : or which is all one ; As AB , the sine of the base , is in proportion to AM Radius : so is BK , the tangent of the perpendicular , to ML , the tangent of the angle at the base . 3. If ● circles of the Sphere be so ordered , that the first intersect the second , the second the third , the third the fourth , the fourth the fift , and the fift the fift at right angles : the right angled triangles made by their intersections do all consist of the same circular parts . As in this Scheme , let IGAB be the first circle , BLF the second , FEC the third , GAD the fourth , HLEI the fift . Then do these five circles retain the conditions required . The first intersecting the second in B , the second the third in F , the third the fourth in C , the fourth the fift in H , the fift the angle , we mark or note intersections at B , F , ● to a quadrant . As angles ; therefore I say●nt as the completriangles made by the inte●●or AD we write circles ; namely , ABD , D● write compl . EGI , and GCA do all co●●d AB besame circular parts ; for the circu●●● 〈◊〉 in every of these triangles are , as h●●d by peareth . In ABD are AB BD c BDA c AD c DA● DHL c HLD c LD c LDH DH HL LFE cō ELF LF FE cō FEL c EL EGI IG cō IGE c GE cō GEI IE GCA c GA c AGC GC CA c CAG Where you may observe , that the side AB in the first triangle is equal to compl . HLD in the second , or compl . ELF in the third , or IG in the fourth , or com . GA in the fift ; and so of the rest . To expresse this more plainly , AB in the first triangle is the complement of the angle HLD in the second , or the complement of the angle ELF in the third , or the side IG in the fourth , or the complement of the hypothenusal GA in the fift . And from these premises is deduced this universall proposition . 4. The sine of the middle part and Radius are reciprocally proportional , with the tangents of the extreams conjunct , and with the co-sines of the extreams disjunct . Namely ; As the Radius , to the tangent of one of the extreames conjoyned : so is tangent of the other extream conjoyned , to the sine of the middle part . And also ; As the Radius , to the co-sine of one of the extreams dis-joyned : so the co-sine of the other extream dis-joyned , to the sine of the middle part . Therefore if the middle part be sought , the Radius must be in the first place , if either of the extreams ; the other extream must be in the first place . For the better Demonstration hereof , it is first to be understood , that a right angled Spherical Triangle hath five parts besides the right angle . As the triangle ABD in the former Diagram , right angled at B , hath first , the side AB : secondly , the angle at A : thirdly , the hypothenusal AD : fourthly , the angle ADB : fifthly , the side DB. Three of these parts which are farthest from the right angle , we mark or no●e by their complements to a quadrant . As the angle BAD we account as the complement to the same angle . For AD we write comp . AD , and for ADB we write compl . ADB . But the two sides DB and AB being next to the right angle , 〈…〉 are not noted by their complements . Of these five parts , two are alwayes given to finde a third , and of these three one is in the middle , and the other two are extreams either adjacent to that middle one , or opposite to it . If the parts given and required are all conjoyned together , the middle is the middle part conjunct , and the extreams the extream parts conjunct . If again any of the parts given or required be dis-joyned , that which stands by it self is the middle part dis-joyned , and the extreames are extream parts dis-joyned . Thus , if there were given in the triangle ABD , the side AB , the angle at A , to finde the hypothenusal AD , there the angle at A is in the middle , and the sides AD and AB are adjacent to it ; and therefore the middle part is called the middle conjunct , and the extreames are the extreames conjunct ; but if there were given the side AB , the hypothenusal AD , to finde the angle at D , here AB is the middle part dis-junct , because it is dis-joyned from the side AD by the angle at A , and from the angle at D by the side DB , for the right angle is not reckoned among the circular parts , and here the extreams are extreams dis-junct . These things premised , we come now to demonstrate the proposition it self , consisting of two parts : first , we will prove , that the sine of the middle part and Radius are proportional with the tangents of the extreams conjunct . The middle part is either one of the sides , or one of the oblique angles , or the hypothenusal . CASE 1. Let the middle part be a side , as in the right angled spherical triangle ABD of the last diagram , let the perpendicular AB be the middle part , the base DB and comp . A the extreame conjunct , then I say , that the rectangle of the sine of AB and Radius is equal to the rectangle of the tangent of DB , and the tangent of the complement of DAB : for , by the second proposition of this Chapter , As the sine of AB , is in proportion to Radius : so is the tangent of DB , to the tangent of the angle at A. Therefore if you put the third term in the second place , it will be , as the sine of AB , to the tangent of DB : so is the Radius , to the tangent of the angle at A. But Radius is a mean proportional between the tangent of an arch , and the tangent of the complement of the same arch , by the Corollary of the first reason of the second Axiome of plain Triangles : and therefore as Radius , is to the tangent of the angle at A ; so is the tangent complement of the same angle at A unto Radius : Therefore as the sine of AB is in proportion to the tangent of DB ; so is the co-tangent of the angle at A , to Radius : and therefore the rectangle of AB ▪ Radius , is equall to the rectangle of the tangent of DB , and the co-tangent of the angle at A. CASE 2. Let the middle part be an angle , as in the triangle DHL of the former Diagram , and let compl . HLD be the middle part , HL and compl . LD the extreames conjunct ; then I say , that the rectangle made of the co-sine of HLD and Radius , is equal to the rectangle of the tangent of HL and the co●tangent of LD . For ▪ by the third proposition of this Chapter , compl . HLD is equal to AB , and compl . LD to DB , and HL to compl . DAB ; and here we have proved before , that the rectangle of the sine of AB and Radius , is equal to the rectangle of the tangent of DB , and the co-tangent of the angle at A ; therefore also the rectangle of the co-sine of HLD and Radius , is equal to the rectangle of the co-tangent of LD , and the 〈◊〉 tangent of HL. CASE 3. Let the middle part be the hypothenusal , as in the triangle GCA , let compl . AG be the middle part , compl . AGC , and compl . CAG the extreams conjunct ; then I say , that the rectangle of the co-sine of ▪ AG and Radius , is equal to the rectangle of the co-tangent of AGC , and the co-tangent of CAG : for we have proved before , that the rectangle of the sine of AB and Radius is equal to the rectangle of the tangent of DB and the co-tangent of DAB , but , by the third proposition of this Chapter , compl . AG is equal to AB , compl . AGC to DB , and compl . CAG to compl , DAB ; therefore also the rectangle of the co-sine of AG and Radius , is equal to the rectangle of the co-tangent of AGC and the co-tangent of CAG , which was to be proved . It is further to be proved , that the sine of the middle part and Radius are proportional with the co-sines of the extreams dis-junct . Here also the middle part is either one of the sides , or the hypothenusal , or one of the oblique angles . CASE 1. Let the middle part be a side : as in the triangle ABD , let DB be the middle part , compl . AD and compl . A the opposite extreams : then I say , that the rectangle of the sine of BD and Radius is equal to the rectangle of the sine of AD , and the sine of the angle at A ; for , by the first proposition of this Chapter , as the sine of AD , is to Radius ; so is the sine of DB , to the sine of the angle at A. Therefore , the rectangle of the sine of DB and Radius , is equal to the rectangle of the sine of AD and the sine of the angle at A. CASE 2. Let the hypothenusal be the middle part ; as in the triangle DHL , let compl . LD be the middle part , DH and HL the extreams dis-junct . Then I say , that the rectangle of the co-sine of LD and Radius is equal to the rectangle of the co-sine of DH and the co-sine of HL : for compl . LD is equal to DB , and DH is equall to compl . AD , and HL to compl . DAB , by the third proposition of this Chapter : therefore the rectangle of the co-sine of LD and Radius , is equal to the rectangle of the co-sine of DH and the co-sine of HL. CASE 3. Let one of the oblique angles be the middle part , as in the triangle IEG , let compl . IGE be the middle part : then I say , that the rectangle of the co-sine of IGE and Radius is equal to the rectangle of the sine of GEI and the co-sine of IE : for compl . IGE is equall to DB , and GEI is equal to AD , and EI to compl . DAB . 5. In any Spherical triangle , the sines of the sides are proportional to the sines of their opposite angles . Let ABC be a spherical triangle , right angled at C , then let the sides AB , AC , and CB be continued to make the quadrants AE , AF , and CD , and from the pole of the quadrant AF , to wit , from the point D , let be drawn down the other quadrants DF and DH ; so there is made three new triangles BDE , GDE , and the obliquangled triangle BDG . I say , in the right angled triangle ABC , that the sine of the side AB is in proportion to the sine of his opposite angle ACB : as the sine of the side AC , is to his opposite angle ABC ; or as BC , to BAC : likewise in the obliquangled spherical triangle BDG , I say , that as BG , is to BDG : so is BD , to BGD ; or so is DG , to DBG . For first , in the right angled triangle ABC , the angle ACB and the arch AE are of the same quantity , to wit , quadrants , so likewise the angle BAC and the arch EF are of the same quantity , it being the measure of the said angle . Now then as AB , to AE ; so is BC , to EF , by the first proposition of this Chapter : therefore also as AB , to ACB ; so is BC , to BAC . Then in the obliquangled Triangle BDG , because , by the demonstration of right angled triangles , they are as DB , to DEB ; so is DE , to DBE : and as DG , to DEG ▪ so is DE , to DGE , or to DGB . Therefore changing of the proportional termes , it shall be , as DG , to DB : so is DBE , or DBG , to DGB , which was to be demonstrated . These foundations being thus laid , the businesse of right angled spherical triangles is easily dispatcht . And the proportions to be used in every case may be discovered either by the first , second and fift propositions ; or by the fourth proposition only . The severall cases in a right angled sphericall triangle are sixteen in number , whereof six may be resolved by the first proposition : seven by the second , and three by the fift ; an example in each will suffice . In the triangle ABC , let there be given the hypothenusal AB , and the perpendicular BC , to finde the base AC ; then by the first proposition , the Analogie is , As the co-sine of the perpendicular , is to Radius : so is the co-sine of the hypothenusal , to the co-sine of the base . 2. Let there be given the base AC , and the angle at the base BAC , to finde the perpendicular BC , by the second proposition , the analogie is : As Radius , to the sine of the base ; so is the tangent of the angle at the base , to the tangent of the perpendicular . 3. Let there be given the hypothenusal AB , the angle at the base BAC , to finde the perpendicular BC , by the fifth proposition , the analogie is : As Radius , to the sine of the hypothenusal : so is the sine of the angle at the base , to the sine of the perpendicular : and so of the rest . By the fourth or universall Proposition , the proportions for right angled sphericall triangles may be found two wayes : First , by the equality of the Sines and Tangents of the circular parts of a triangle , that is , of the Logarithmes of the natural , thus by the universal proposition in the aforesaid triangle ABC , the hypothenusal AB , and the angles at AM and B being noted by their complements , I say . 1. The sine of AC added to Radius ; is equal to the sine of AB added to the sine of the angle at ● . 2. The cosine of A added to Radius is equal to the co-sine of BC added to the sine of the angle at B. 3. The co-sine of AB added to Radius , is equall to the co-sine of AC added to to the co-sine of BC. 4. The co-sine of AB added to Radius is equal to the co-tangent of A , added to the co-tangent of the angle at B. 5. The cosine of the angle at B added to Radius is equal to the tangent of BC , added to the cotangent of AB . 6. The sine of BC added to Radius is equal to the co-tangent of the angle at B added to the tangent of AC . And thus he that listeth may set down the equality of the sines and tangents of the other sides and angles , and so there will be ten in all ; but these may here suffice : for to these may the sixteen cases of a right angled spherical triangle be reduced ; namely , three to the first , three to the second , two to the third , two to the fourth , three to the fift , and three to the sixt . As admit there were given the hypothenusal BA , and the angle at B , to finde the base AC ; then , by the first , seeing that the sine of AB added to the sine of the angle at B , is equal to the sine of AC added to Radius . Therefore , if working by natural numbers I multiply the sine of AB by the sine of B , and divide the product by Radius , the remainer will be the sine of AC : and working by Logarithmes , if from the summe of the sines of AB and B I substract Radius , the rest is the sine of AC . Secondly , admit there were given AB and AC , to finde B , then seeing that the sine of AC added to Radius is equal to the sines of AB and B. Therefore , if working by naturall numbers I multiply the sine of AC by Radius , and divide the product by AB , the remainer is the sine of B. Or working by Logarithmes , if from the sum of the sines of AC and Radius , I substract the sine of AB , the remainer will be the sine of B. Or thirdly , if there were given AC and the angle at B , to finde AB : then forasmuch as AC and Radius is equal to the sines of AB and B , therefore if working by natural numbers I multiply AC by the Radius , and divide the product by the sine of B , the remainer is the sine of AB . Or working by Logarithmes , if from the sine of AC and Radius , I substract the sine of B the remainer is the sine of AB : an so of the rest . Which that you may the better perceive , I have here added in expresse words , the Canons or rules of the proportions of the things given and required in every of the sixteen cases of a right angled sphericall triangle , as they are collected from the Catholick Proposition . And here the side subtending the right angle we call the hypothenusal , the other two containing the right angle we may call the sides ; but for further distinction , we call one of these containing sides ( it matters not which ) the base , and the other the perpendicular . The base an angle at the base given , to finde 1. The Perpendicular . ] As Radius , to the sine of the base ; so is the tangent of the angle at the base , to the tangent of the perpendicular . 2. Angle at the perpendicular . ] As Radius , to the co-sine of the base ; so the sine of the angle at the base , to the co-sine of the angle at the perpendicular . 3. Hypothenusal . ] As Radius , to the co-sine of the angle at the base : so the co-tangent of the base , to the co-tangent of the hypothenusal . The perpendicular and angle at the base given , to finde 4. Angle at perpend . ] As the co-sine of the perpendicular , to Radius ; so the co-sine of the angle at the base , to the sine of the angle at the perpendicular . 5. Hypothenusal . ] As the sine of the angle at the base , to Radius ; so the sine of the perpendicular , to the sine of the hypothenusal . 6. The Base . ] As Radius , to the co-tangent of the angle at the base ; so is the tangent of the perpendicular , to the sine of the base . The hypothenusal and angle at the base given , to finde 7. The base . ] As Radius , to the co-sine of the angle at the base ; so the tangent of the hypothenusal , to the tangent of the base . 8. Perpendicular . ] As Radius , to the sine of the hypothenusal , so the sine of the angle at the base , to the sine of the perpendicular . 9. Angle at perpend . ] As Radius , to the co-sine of the hypothenasal ; so the tangent of the angle at the base , to the co-tangent of the angle at the perpendicular . The base and perpendicular given , to finde 10. Hypothenusal . ] As Radius , to the co-sine of the perpendicular : so the co-sine of the base , to the co-sine of the hypothenusal . 11. Angle at the base ] As Radius , to the sine of the base : so is the co-tangent of the perpendicular , to the co-tangent of the angle at the base . The base and hypothenusal given , to finde the 12. Perpendicular . ] As the co-sine of the base , to Radius ; so the co-sine of the hypothenusal , to the co-sine of the perpendicular . 13. Angle at the base . ] As Radius , to the tangent of the base ; so the co-tangent of the hypothenusal , to the co-sine of the angle at the base . 14. Angle at the perpend . ] As the sine of the hypothenusal , to Radius ; so the sine of the base , to the sine of the angle at the perpendicular . The angles at the base and perpendicular given , to finde 15. The perpendicular . ] As the sine of the angle at the perpendicular , is to Radius : so the co-sine of the angle at the base , to the co-sine of the perpendicular . 16. The hypothenusal . ] As Radius , to co-tangent of the angle at the perpendicular ; so the co-tangent of the angle at the base , to the co-sine of the hypothenusal . Secondly , the proportions of all the cases of a right angled spherical triangle , may by the aforesaid Catholick Proposition be known thus : If the middle part be sought , put the Radius in the first place ; if either of the extreams , the other extream put in the first place . And note , that when a complement in the proposition doth chance to concur with a complement in the circular parts , you must take the sine it self , or the tangent it self , because the co-sine of the co-sine is the sine , and the co-tangent of the co-tangent is the tangent . As in the following triangle ABC , let there be given the base AB , and the angle at C , to finde the hypothenusal BC. Here AB is the middle part , BC and C are the opposite extreams , or the extreams disjunct . Now because the extream BC is sought , therefore I must put the other extream , that is , the angle at C , in the first place ; and because that angle , as also the side sought are noted by their complements , therefore I must not say : As the co-sine of the angle at C , is to Radius : so is the sine of the base AB , to the co-sine of the hypothenusal BC : but thus ; As the sine of the angle at the perpendicular ACB , is to Radius ; so is the sine of the base AB , to the sine of the hypothenusal BC. The like is to be understood of the rest . Thus much concerning right angled spherical triangles : as for Quadrantal there needs not much be said , because the circular parts of a quadrantal triangle , are the same with the circular parts of a right angled triangle adjoyning . As let ABC be a triangle , right angled at A , and let one of the sides thereof ; namely , AC be extended , till it become a quadrant , that is to D ; then draw an arch from D to B ; then is DBC a quadrantal triangle , to which there is a right angled triangle adjoyning , as ABC . I say therefore that the circular parts of the quadrantal triangle BCD are the same with the circular parts of the right angled triangle ABC : for the circular parts of either of them are as here appeareth . The five circular parts of the triangle . ABC are AC AB co ABC cō B● cō BCA BCD are com CD CDB DBC cō BC cō BCD Where it is evident , that AD and DB being quadrants , DBA is a right angle , and BA is the measure of the angle at D , so that the side AC in the one is equall to compl . CD in the other : and the side AB in the one is equal to the angle BDC in the other : and compl . ABC in the one is equal to DBC in the other , and compl . BC in the one is the same with BC in the other : and lastly , compl . BCA in the one is the same with compl . DCB in the other ; for the compl . of the acute angle A●● unto a quadrant is also the complement of the obtuse angle BCD , and the circular parts of both triangles being the same , it followes , that that which is here proved touching right angled triangles is also true of quadrantal . And all the sixteen cases thereof may also be resolved by the aforesaid Catholick Proposition . As let there be given the side DC , and the angle at C , to finde the angle at D , then is the side DC the middle , and the angles at D and C are extreams adjacent ; now because the angle at D , one of the extreams is sought , we must put the other extream , to wit , the angle at C in the first place , and that is noted by its complement : and therefore the Analogie is ▪ As the co-tangent of the angle at C , to Radius ; so the co-sine of DC , to the tangent of the angle at D : and so of the rest ; and what is said of the addition of the artificial numbers is to be understood of the rectangles of the natural . CHAP. IX . Of Oblique angled Sphericall Triangles . IN an obliquangled spherical triangle , there are twelve Cases ; two whereof , that is , those wherein the things given and required are opposite , may be resolved by the fift proposition of the last Chapter . CASE 1. Two angles with a side opposite to one of them being given , to finde the side opposite to the other . As in the triangle ABC , let there be given the side BC , with his opposite angle at A , and the angle ABC , to finde the side AC . I say then , by the fift proposition of the last Chapter : As the sine of the angle at A , is to the sine of his opposite side BC : so is the sine of the angle at B , to the sine of his opposite side AC . CASE 2. Two sides with an angle opposite to one of them being given , to finde an angle opposite to the other . As the sine of BC , to the sine of his opposite angle at A : so is the sine of AC , to the sine of his opposite angle B. Other eight cases must be resolved by the aid of two Analogies at the least , and that by reducing the triangle proposed to two right angled triangles , by a perpendicular let fall from one of the angles to his opposite side , which perpendicular falls sometimes within , sometimes without the triangle . If the perpendicular be let fall from an obtuse angle , it falleth within , but if it fall from an acute angle , it falls without the triangle : however it falleth , it must be alwayes opposite to a known angle . For your better direction , in letting fall the perpendicular take this generall rule . From the end of a side given , being adjacent to an angle given , let fall the perpendicular . As in the triangle , ABC , if there were given the side AB , and the angle at A : by this rule the perpendicular must fall from B upon the side AC ; but if there were given the side AC , and the angle at A ; then AB must be produced to D ; and the perpendicular must fall from C upon the side AD. Thus shall we have two right angled triangles , and the side or angle required may easily be resolved by the Catholick Proposition . As suppose there were given the side AB , the angles at A and C , and required the side AC ; then the perpendicular must fall from B upon the side AC , as in the first triangle , and divide the oblique triangle ABC into two right angled triangles , to wit , ABF and BFC . And in the triangle ABF we have given the side AB , and the angle at A , to finde the base AF , for which the analogie , by the Catholick Proposition , is , As the co-tangent of AB , to Radius : so is the co-sine of the angle at A , to the tangent of AF : that is , by the seventh case of right angled triangles . Secondly , by the eighth case , finde the perpendicular BF . Lastly , in the triangle BFC , having the perpendicular BF , and the angle at C , by the sixt case of right angled spherical triangles , you may finde the base FC , which being added to AF , is the side AC . But thus there are three operations required ; whereas it may be done at two : for the obliquangled triangle being reduced into two right angled triangles , by letting fall a perpendicular , as before : the hypothenusal in one of the right angled triangles will be correspondent to the hypothenusal in the other , and the base in the one to the base in the other ; and so the other parts . Then in one of these right angled triangles ( which for distinction sake we call the first ) there is given the hypothenusal and angle at the base , whereby may be found the base or angle at the perpendicular , as occasion requires ; by the seventh or ninth cases of right angled triangles . And this is the first operation . For the second , there must ( of the things thus given and required ) two things in one triangle , be compared to two correspondent things in the other triangle , which two in each with the perpendicular make three things in each triangle , either adjacent , that is , lying together , or opposite of which three the perpendicular is alwayes one of the extreams , and the thing required one of the other extreams . Thus in the triangle ABF , if there were given AF and BF , to finde AB : AB is the middle part , AF and BF are opposite extreams ; and therefore by the Catholick Proposition . Radius added to the co-sine of AB , is equal to the co-sines of AF and BF . Then in the triangle BFC , if there were given BF and FC , to finde BC : BC will be the middle part , BF and FC opposite extreames ; and therefore by the Catholik Proposition . The co-sines of BF and FC are equall to the co-sine of BC and Radius . But if from equal things we take away equal things , the things remaining must needs be equal ; if therefore we take away the Radius , and co-sine BF in both these proportions , it followes , that the co-sine of AB added to the co-sine of FC is equal to the cosine of BC added to the co-sine AF. And therefore , the middle part AB in the first , and the extream FC in the second , is equall to the middle part BC in the second , and the extream AF in the first : or thus ; As the middle part in the first triangle , is in proporion to the middle part in the second : so is the extream in the first , to the extream in the second . Thus by the Catholick Proposition , and the help of this , the eight cases following may be resolved . In the exemplification whereof this sign + signifies addition . By the Catholick Proposition , it is evident that 1 Rad. + cs AB is equal to cs AF + cs FB cs BF + cs FC cs BC + Rad 2 Rad. + s AF is equal to ct A + t FB t FB + ct C s FC + Rad. 3 Rad. + cs A is equal to s ABF + cs FB cs FB + s FBC cs C + Rad 4 Ra. + cs ABF is equal to ct AB + t FB t FB + ct BC cs FBC + Ra. Then taking from either side tangent FB and Radius , or co-sine FB and Radius , it followes , by the former proposition , that 1. cs AB + cs FC is equall to cs BC + cs AF. 2. s AF + ct C is equall to s FC + ct A. 3. cs A + s FBC is equall to cs C + s ABF 4. cs ABF + ct BC is equal to cs FBC + ct AB For seeing that AF and FB are opposite extreams to AB , as CF and FB are to BC : therefore , 1. As cs AF , to cs FC ; so is cs AB , to cs BC : that is , As co-sine the first base , to co-sine the second ; so co-sine the first hypotheriusal , to co-sine the second . And this serves for the third and seventh cases following . And seeing that A and FB are adjacent extreams to AF : as C and FB are to FC : therefore , 2. As s AF , to s FC ; so ct A , to ct C : that is , as the sine of the first base , to the sine of the second ; so co-tangent the first angle at the base , to co-tangent the second , which serves for the fourth and tenth cases . Again , seeing that ABF and FB are opposite extreams to A , as CBF and FB are to C : therefore , 3. As s ABF , to s CBF ; so cs A , to ●s C : that is , as the sine of the first angle at the perpendicular , to the sine of the second ; so co-sine the first angle at the base , to co-sine the second : which serves for the fifth and ninth cases . Lastly , seeing AB and FB are adjacent extreams to ABF , as BC and FB are to CBF : therefore , 4. As cs ABF , to cs CBF ; so ct AB , to ct BC : that is , as co-sine the first angle at the perpendicular , to co-sine the second ; so co-tangent the first hypothenusal , to co-tangent the second : this serves for the sixth and eighth cases following . And this foundation being thus laid , we come now to the severall Cases thereon depending . CASE 3. Two sides and their contained angle given , to finde the third side . First , by the seventh case of right angled triangles , the analogie is : As Radius , to the co-sine of the angle at the base : so is the tangent of the hypothenusal , to the tangent of the base , or first arch . Which being added to or substracted from the base given , according to the following direction , giveth the second arch . If the perpendicular fall Within the triangle , subtract AF the base found from AC the base given , the remainer is EC , the second arch . Without , and the contained angle obtuse , adde the arch found to the arch given , and their aggregate is the second arch . Without , and the contained angle acute , substract the arch given from the arch found , the remainer is the second arch . Then , by the first Consectary aforegoing say : as the co-sine of the first base , to the co-sine of the second ; so the co-sine of the first hypothenusal , to the co-sine of the second : but this we will illustrate by example . Let there be therefore given in the oblique angled spherical triangle ABC , the side or arch AB 38 degrees 47 minutes , the side AC 74 degrees , 84 minutes , and their contained angle BAC 56 degrees , 44 minutes , to finde the side BC. Now then according to the rules given , I let fall the perpendicular BF , and so have I two right angled triangles , the triangle ABF and the triangle BFC . In the triangle ABF , we have the hypothenusal AB 38 degrees , 47 minutes , and the angle at the base BAF 56 degrees 44 minutes , to finde the base AF. First therefore I say , As the Radius 90 , 10.000000 Is to the co-sine of BAC 56.44 . 9.742576 So is the tangent of AB 38 47. 9.900138 To the tangent of AF 23.72 . 9.642714 Now because the perpendicular falls within the triangle , I substract AF 23 degrees , 7● minutes from AC 74 degrees , 84 min. and there remains FC 51 degrees , 1● minutes , the second arch . Hence to finde BC , I say ; As the co-sine of AF 23. 72. co . ar . 0.038331 Is to the co-sine of FC 51.12 . 9.797746 So is the co-sine of AB 38.47 . 9.893725 To the co-sine of BC 57.53 . 9.729802 2. Example . In the same triangle , let there be given the side AB 38 degr . 47 min. the side BC 57 degr . 53 min. and their contained angle ABC 107 deg . 60 min. and let the side AC be sought . First , let fall the perpendicular DC , and continue the side AB to D , then in the right angled triangle BDC , there is given the angle DBC 72 deg . 40 min. the complement of the obtuse angle ABC , and the hypothehusal BC 57 degrees 53 minutes : to finde BD , I say first ; As the Radius 90 , 10.000000 Is to the co-sine of DBC 72.40 . 9.480539 So is the tangent of BC 57.53 . 10.196314 To the tangent of BD 25.42 . 9.676853 Now because the perpendicular falls without the triangle , and the contained angle obtuse , I adde BD 25 degrees , 42 minutes to AB 38 deg . 47 min. and their aggregate is AD 63 deg . 89 min. the second arch : hence to finde AC , I say , As the co-sine of BD , 25.42 . 0.044223 Is to the co-sine of 63.89 . 9.643547 So is the co-sine of BC 57.53 . 9.729859 To the co-sine of AC 74.84 . 9.417629 3 Example . In this triangle , let there be given the side BC 57 deg . 53 min. the side AC 74 deg . 84 min. and their contained angle ACB 37 deg . 92 min. and let the side AB be sought . First , I let fall the perpendicular AE , and the side BC I continue to E , then in the right angled triangle AEC , we have known the angle ACE , and the hypothenus ; al AC , to finde EC , I say then : As the Radius 90 , 10.000000 Is to the co-sine of ACE 37.92 . 9.897005 So is the tangeent of AC 74.84 . 10.567120 To the tangent of EC 71.5 . 10.464125 Now because the perpendicular falls without the tringle , and the contained angle acute , I substract the arch given BC 57 degrees 53 minutes from EC 71 degrees 5 minutes , the arch found , and their difference 13 deg . 52 min. is EB , the second arch . Hence to finde AB , I say : As the co-sine of EC 71.5 . co . ar . 0.488461 Is to the co-sine of EB 13.52 . 9.987795 So is the co-sine of AC 74.84 . 9.417497 To the co-sine of AB 38.47 . 9.893753 CASE 4. Two sides and their contained angle given , to finde one of the other angles . First , by the seventh case of right angled spherical triangles , I say : As Radius , to the co-sine of the angle at the base ; so is the tangent of the hypothenusal , to the tangent of the base , or first arch : which being added to , or substracted from the base given , according to those directions given in the third case , giveth the second arch ; then by the second Consectary of this Chapter , the proportion is : As the sine of the first base , to the sine of the second : so is the co-tangent of the first angle at the base , to the co-tangent of the second . 1 Example . Thus if there were given , as in the first example of the last case , the side AB 38 degrees , 47 minutes , the side AC 74 degrees , 84 minutes , and their conteined angle BAC 56 degrees , 44 min. and ACB , the angle sought , the first operation will in all things be the same , and AF 23 degrees , 72 minutes , the first arch , FC 51 degrees , 12 minutes , the second ; hence to finde the angle ACB , I say : As the sine of AF 23.72 . co . ar . 0.395486 To the sine of FC 51.12 . 9.891237 So is the co-tang . of BAC 56.44 . 9.821771 To the co-tangent of ACB 37.92 . 10.108494 There being no other variation in this case then what hath been shewed in the former , one example will be sufficient . CASE 5. Two angles , and the side between them given , to finde the third angle . First , by the ninth case of right angled spherical triangles , the proportion is ; As Radius , to the co-sine of the hypothenusal ; so the tangent of the angle at the base , to the co-tangent of the angle at the perpendicular , which being added to , or substracted from the other given angle , according to the following direction , giveth the second arch . If the perpendicular fall Within the triangle , substract the angle found from the angle given , the remainer is the second arch . Without , and both the angles given acute , substract the angle given from the angle found , and the remainer is the second arch . Without , and one of the angles given be obtuse , adde the angle found to the angle given , & their aggregate is the second arch . Then , by the third Consectary of this Chapter , the analogie is ; As the sine of the first at the perpendicular , to the sine of the second angle sound : so is the co-sine of the first angle at the base , to the co-sine of the second . 1 Example . In the triangle ABC , let there be given the angles BAC 56 degrees 44 minutes , and ABC 107 degrees , 60 minutes , and the side between them AB 38 degrees 47 minutes , to finde the angle ACB . First , let fall the perpendicular BF , and then in the right angled spherical triangle ABF we have known the angle at the base BAF , and the hypothenusal AB , to finde the angle at the perpendicular ABF . First , then I say : As the Radius 90 , 10.000000 To the co-sine of AB 38.47 . 9.893725 So is the tangent of BAF 56.44 . 10.178229 To the co-tangent of ABF 40.28 . 10.071954 Let there be given , as before , the two angles BAC and ABC , with the side between them AB , to finde the angle ACB , and let the perpendicular EA , and let the side BC be continued to E , then in the right angled triangle AEB we have known the hypothenusal AB 38 degrees , 47 minutes , and the angle at the base ABE 72 degrees , 40 minutes , the complement of the obtuse angle ABC , to finde the angle EAB . First then I say : As the Radius 90. 10,000000 To the co-sine of AB 38.47 . 9.893725 So is the tangent of ABE 72.40 . 10.498641 To the co-tangent of EAB 22.6 . 10.392366 And because the perpendicular falls without the triangle , and one of the angles given obtuse , I adde the angle found EAB 22 degrees 6 minutes to the angle given BAC 56 degrees , 44 minutes , and their aggregate 78 degrees 50 minutes is the angle EAC , the second arch ; and hence to finde the angle at C , I say , as before . As the sine of EAB 22.6 . co . ar . 0.425300 To the sine of EAC 78 . 5● 9.991194 So is the co-sine of ABE 72.40 . 9.480538 To the co-sine of ACB 37.92 . 9.897032 3 Example . Let there be given the angles BAC 56 degrees 44 minutes , and ACB 37 degrees , 92 minutes , with their contained side AC 74 degres , 84 minutes , to finde the angle ABC , let fall the perpendicular CD , and let the side AB be continued to D , then in the right angled triangle ADC , we have known the hypothenusal AC , and the angle at the base DAC , to finde ACD ; first , then I say ; As the Radius 90 10.000000 To the co-sine of AC 74.84 . 9.417497 So is the tangent of DAC 56.44 . 10.178229 To the co-tangent of ACD 68.48 . 9.595726 Now because the perpendicular falls without the triangle , and both the angles given acute , therefore I substract the angle given ACB 37 degrees , 92 minutes from the angle found ACD 68 degrees 48 minutes , and their difference 30 degrees 56 minutes is the angle BCD , the second arch . Hence to finde the angle CBD , I say , as before ; As the sine of ACD 68.48 . co . ar . 0.031382 To the sine of BCD 30.56 . 9.706240 So is the co-sine of DAC 56.44 . 9.742575 To the co-sine of CBD 72.40 . 9.480197 CASE 6. Two angles and the side between them given to finde the other side . First , by the ninth case of right angled triangles , I say , as before ; As Radius , to the co-sine of the hypothenusal ; so the tangent of the angle at the base , to the co-tangent of the angle at the perpendicular . Which being added to or substracted from the other angle given , according to the direction of the fift case , giveth the second arch . Then by the fourth Consectary of this Chapter , As the co-sine of the first angle at the perpendicular , to the co-sine of the second ; so is the co-tangent of the first hypothenusal , to the co-tangent of the second . Example . If there were given , as in the first example of the last case , the angles BAC 56 degrees 44 minutes , and ABC 107 degrees 60 minutes , with the side AB 38 degrees , 47 minutes , to finde the side BC. The first operation will be in all things the same , and the first arch ABF 40 degrees , 28 minutes ; the second arch FBC 67 degrees , 32 minutes . Hence to finde the side BC , I say : As the co-sine of ABF 40.28 . co . ar . 0.117536 To the co-sine of FBC 67.32 . 9 . 5861●9 So is the cotangent of AB 38.47 . 10 . 09986● To the co-tangent of BC 57.53 9.803517 CASE 7. Two sides with an angle opposite to one of them , to finde the third side . First , by the seventh case of right angled sphericall triangles , I say ; As Radius , to the co-sine of the angle at the base ; so is the tangent of the hypothenusal , to the tangent of the base , or first arch . Then , by the first Consectary of this Chapter , the analogie is , As the co-sine of the first hypothenusal , to the co-sine of the second ; so the co-sine of the first arch found , to the co-sine of the second . Which being added to or substracted from the first arch found , according to the direction following , their sum or difference is the third side . If the perpendicular fall Within the triangle , adde the first arch found to the second arch found , and their aggregate is the side required . Without , & the angle given obtuse , substract the first arch found from the second arch found , and what remaineth is the third side . Without , & the given angle acute , substract the second arch found from the first , and what remaineth is the side required . 1 Example . In the oblique angled triangle ABC , let there be given the sides AB 38 degrees , 47 minutes , and BC 57 degrees , 53 minutes , with the angle BAC 56 degrees , 44 minutes , and let the side AC be required . First , I let fall the perpendicular BF , and then in the right angled triangle ABF , we have given the hypothenusal AB , and the angle at the base BAF , to finde the base AF , for which I say : As the Radius 90 10.000000 To the co-sine of BAF 56.44 . 9.742576 So is the tangent of AB 38.47 . 9.900138 To the tangent of AF 23.72 . 9.642714 Secondly , for FC , I say : As the co-sine of AB 38.47 . co . ar . 0.106275 To the co-sine of BC 57.53 . 9.729859 So is the co-sine of AF 23.72 . 9. ●61669 To the co-sine of FC 51.12 . 9.797803 Now because the perpendicular fell within the triangle , therefore I adde the first arch found AF 23 degrees , 72 minutes to the second arch found FC 51 degrees 12 minutes , and their aggregate 74 degrees , 84 minutes is AC the side required . 2 Example . In the same triangle ABC , let there be given the sides AB 38 degrees , 47 minutes and AC 74 degrees 84 minutes , and the angle ABC 107 degrees , 60 minutes , and let BC be required . First then , I let fall the perpendicular AE , and continue the side BC to E , and then in the right angled triangle AEB we have given the side AB 38 degrees , 47 minutes , and the angle ABE 72 degrees , 40 minutes , the complement of ABC , to finde EB : for which I say : As the Radius 90 10.000000 To the co-sine of ABE 72.40 . 9.480538 So is the tangent of AB 38.47 . 9.900138 To the tangent of EB 13.51 . 9.380676 Secondly , to finde EC , I say : As the co-sine of AB 38.47 . co . ar . 9.106275 To the co-sine of AC 74.84 . 9. ●17497 So is the co-sine of EB 13.51 . 9.987813 To the co-sine of EC 71.4 . 9.511585 Now because the perpendicular falls without the triangle , and the given angle obtuse , therefore I substract the first arch found EB 13 degrees 51 minutes , from the second arch EC 71 degrees , 4 minutes , and their difference 57 degrees , 53 minutes is BC , the side required . 3 Example . In the same triangle ABC , let there be given the sides AC 74 degrees , 84 minutes , and BC 57 degrees , 53 minutes , and the angle BAC 56 deg . 44 min. to finde the side AB : I let fall the perpendicular DC , and continue the side AB to D , then in the right angled triangle ADC we have given the hypothenusal AC , and the angle at A , to finde AD. As the Radius 90 10.000000 To the co-sine of BAC 56.44 . 9.742576 So is the tangent of AC 74.84 . 10.567119 To the tangent of AD 63.89 . 10.309695 Secondly , to finde DB , I say : As the co-sine of AC 74.84 . co . ar . 0.582503 To the co-sine of BC 57.53 . 9.729859 So is the co-sine of AD 63.89 . 9.643547 To the co-sine of DB 25.39 . 9 . 95●909 Now because the perpendicular falls without the triangle , and the angle given acute , therefore I substract the second arch found DB 25 degrees , 39 minutes , from the first arch found AD 63 degrees 89 minutes , and their difference 38 degrees 50 minutes is AB , the side required . CASE 8. Two sides with an angle opposite to one of them being given , to finde their contained angle . First , by the ninth case of right angled spherical triangles , I say ; As Radius , to the co-sine of the hypothenusal ; so the tangent of the angle at the base , to the co-tangent of the angle at the perpendicular . Then , by the fourth Consectary of this Chapter , the proportion is : As the co-tangent of the first hypothenusal , to the co-tangent of the second ; so the co-sine of the first angle at the perpendicular , to the co-sine of the second : which being added to , or substracted from the first arch found , according to the direction of the seventh case , giveth the angle sought . Example . If there were given , as in the first example of the last case , the sides AB 38 deg . 47 min. and BC 57 deg . 53 min. with the angle BAC 56 deg . 44 min. to finde the obtuse angle ABC . The perpendicular BF falling within the triangle , then in the right angled triangle ABF , we have knowne the hypothenusal AB , and the angle at A , to finde the angle ABF , I say then , As the Radius 90 , 10.000000 Is to the co-sine of AB 38.47 . 9●93725 So is the tangent of BAF 56.44 . 10.178229 To the co-tang . of ABF 40.28 . 10.071954 Secondly , to finde FBC , I say : As the co-tangent of AB 38.47 . 9.900138 To the co-tangent of BC 57.53 . 9. ●03686 So is the co-sine of ABF 40.28 . 9.882464 To the co-sine of FBC 67.32 . 9.586288 Now because the perpendicular falls within the triangle , I adde the first arch found ABF 40 degrees , 28 minutes , to the second arch found FBC 67 degrees , 32 minutes , and their aggregate is 107 degr . 60 min. the angle ABC required . CASE 9. Two angles and a side opposite to one of them being given , to finde the third angle . First , by the ninth case of right angled spherical triangles , I say : As the Radius , to the co-sine of the hypothenusal ; so the tangent of the angle at the base , to the co-tangent of the angle at the perpendicular . Then by the third Consectary of this Chapter , the proportion is . 〈◊〉 the co-sine of the first angle at the base , to the co-sine of the second ; so is the sine of the first angle at the perpendicular , to the sine of the second : which being added to , or substracted from the first arch found , according to the ●i●ect●on following , their summe or difference is the angle sought . If the perpendicular fall Within the triangle , adde both arches together . Without , and the angle opposite to the given side acute , substract the first from the second arch . Without , and the angle opposite to the given side obtuse , substract the second from the first . 1. Example . In the oblique angled Triangle ABC , let there be given the angle BAC 56 deg . 44 min. and ACB 37 deg . 92 min. and the side AB 38 deg . 47 min. to finde the angle ABC . First , let fall the perpendicular FB , then in the right angled triangle AFB we have known , the hypothenusal AB , and the angle at A , to finde the angle ABF , for which I say , As Radius , 90 deg . 10.000000 To co-sine of AB , 38.47 9 . 893●26 So the tangent of BAF , 56. ●4 10.178229 To the co-tangent of ABF , 40.28 . 10.071955 Secondly , to finde FBC , I say , As the co-sine of BAF , 56.44 0.257424 . To the co-sine of ACB , 37.92 9 . 8970●5 So is the sine of ABF , 40.28 9.810584 To the sine of FBC , 67.32 9.965013 Now because the perpendicular fals within the Triangle , I adde the first arch found ABF 40 deg . 28 min. to the second arch found FBC 67 deg . 32 min. and their aggregate is 107 deg . 60 min. the angle ABC required . 2. Example . In the same Triangle let there be given the angle ACB 37 deg . 92 min. and ABC 107 deg . 60 min. and the side AB 38 deg . 47 min. to finde the angle BAC . First , let fall the perpendicular AE , and let the side BC be continued to E , then in the right angled triangle AEB we have known the Hypothenusal AB , and the angle at B , 72 deg . 40 min. the complement of ABC , to finde EAB , I say then , As the Radius 90 , 10.000000 To the co-sine of AB , 38.47 9.893726 So is the tangent of ABE , 72.40 10.498641 To the co-tangent of EAB , 22. ●6 10,392367 Secondly , to finde EAC , I say , As the co-sine of ABE , 72.40 0.519462 To the co-sine of ACB , 37.92 9.897005 So is the sine of EAB , 22.6 9.574699 To the sine of EAC 78.49 9 . 99●166 Now because the perpendicular falls without the triangle and the angle opposite to the given side acute , I substract the first angle found E. AB 22 deg . 6 min. from the second arch found 78 deg . 49 min. and their difference 56 deg . 43 min. is the angle BAC required . 3 Example . In the same triangle ABC , let there be given the angles ACB 37 deg . 92 min. and ABC 107 deg . 60 min. and the side AC 74 deg . 84 min. to finde the angle BAC . Let fall the perpendicular AE , and then in the right angled triangle AEC , we have known the hypothenusal AC , and the angle ACB , to finde the angle EAC . As the Radius 90 , 10.000000 To the co-sine of AC , 74.84 9.417497 So is the tangent of ACE , 37.92 9.891559 To the co-tangent of EAC , 78.49 9 . 3090●6 Secondly , to finde EAB , I say , As the co-sine of ACE , 37.92 0 . 102●95 To the co-sine of ABE , 72.40 9.480538 So is the sine of EAC , 78.49 9.991177 To the sine of EAB , 22.6 9.574790 Now because the perpendicular falls without the Triangle , and the angle opposite to the given side obtuse , therefore I substract the second arch found EAB , 22 deg . 6 min. from the first arch found , EAC 78 deg . 49 min. and their difference 56 deg . 43 min. is the angle BAC required . CASE 10. Two angles , and a side opposite to one of them being given , to finde the side between them . First , by the 7th . Case of right angled Sphericall Triangles , I say , As Radius , to the co-sine of the angle at the base ; so is the Tangent of the Hypothenusal , to the Tangent of the Base . Then by the second Consectary of this Chapter , the proportion is , As the co-tangent of the first angle at the base , to the co-tangent of the second ; so is the sine of the first base , to the sine of the second : which being added to , or substracted from , the first arch found , according to the direction of the 9th . Case , giveth the side required . Example . In the oblique angled triangle ABC , let there be given the two angles BAC 56 deg . 44 min. and ACB 37 deg . 92 min. with the side BC 57 deg . 53 min. to finde the side AC . Let fall the perpendicular BF , then in the right angled triangle BCF , we have known the Hypothenusal BC , and the angle FCB , to finde the base FC : say then , As the Radius , 90 10.000000 Is to the co-sine of FCB , 37.92 9.897005 So is the tangent of BC , 57.53 10.196314 To the tangent of FC , 51.11 10.093319 Secondly , to finde AF , I say , As co-tangent FCB , 37.92 , co . ar . 9.891559 To co-tangent of BAC . 56.44 9.821771 So is the sine of FC , 51.11 9.891176 To the sine of AF , 23.72 9.604506 Now because the perpendicular falls within the Triangle , I adde the first arch FC 51 deg . 11 min. to the second arch AF , 23 deg . 72 min. and their aggregate is 74 deg . 83 min. the side AC required . CASE 11. The three sides given to finde an angle . The solution of this and the Case following , depends upon the Demonstration of this Proposition . As the Rectangular figure of the sines of the sides comprehending the angle required ; Is to the square of Radius : So is the Rectangular figure of the sines of the difference of each containing side taken from the half summe of the three sides given ; To the square of the sine of half the angle required . Let the sides of the triangle ZPS be known , and let the vertical angle SZP be the angle required , then shall ZS the one be equal ZC . In like manner PS the base of the vertical angle shall be equal to PH or PB , then draw PR the sine of PZ and CK the sine of CZ or ZS . Divide CH into two equal parts in G , draw the Radius AG and let fall the perpendiculars P● and CN which are the sines of the arches PG and CG . The right line EV is the versed sine of a certain arch in a great circle , and SC the versed of the like arch in a less , then if you draw the right line NF parallel to SH bisecting CH in N , it shall also bisect the versed sine SC in F by the 15th . of the second and RM bisecting TP in R , and drawn parallel to TX , shall for the same reason bisect PX in M , and the triangles SCH and FNC shall be like , as also the triangles TPX and RPM are like ; and ZG shall be equal to the half summe of the three sides given , which thus I prove . Of any three unequal quantities given , if the difference of the two lesser be substracted from the greatest , and half the remainer added to the mean quantity , the summe shall be equall to half the summe of the three unequal quantities given . Example . Let the quantities given be 9 , 13 , and 16 , the difference between 9 and 13 is 4 , which being substracted from 16 , there remaineth 12 , the half whereof is 6 , which being added to 13 maketh 19 , the half sum of the three unequal quantities . Now then in this Diagram PC is the difference of the two lesser sides , which taken from PH , the remainer is CH , the half whereof is CG , and CG added to CZ , the mean side , giveth GZ the half summe , and if we substract ZP the lesser containing side of the angle required , from ZG the half sum , their difference will be PG , and if we substract ZC the other side , the difference will be CG . Lastly , let the arch IV be the measure of the vertical angle PZS , and the right line OQ bisect the lines EV and IV , and the right line AQ perpendicular to the right line IV , bisecting the same in Q , I say then . And dividing the two last rectangles by CF , the proportion will be PR × CK And because VO in VA is equal to VQ square ; therefore if you multiply CN by VA , the proportion will be , as PR × CK , to PM × VA ; so is CN × VA , to VO × VA equal to VQ square , which was to be proved . PM × VA CN VO If then the three sides of an oblique angled spherical triangle be given , and an angle inquired ; do thus : 1. Take the sines of the sides comprehending the angle inquired . Or the Logarithmes of those sines . 2. Take also the quadrat of the Radius , or the Logarithme of the Radius doubled . 3. Substract each side comprehending the angle inquired from the half sum of the three sides given , and take the sines of their differences , or the Logarithmes of those sines . 4. If the rectangle of the first divide the rectangle of the second and third , the side of the quotient is the sine of half the angle inquired . Or if the sum of the Logarithme of the first be deducted from the sum of the Logarithmes of the second and third , the half difference is the Logarithme of half the angle sought . Arithmeticall illustration by Naturall Numbers . In the Oblique angled Triangle SZP , having the Sides PS , 42 deg . 15 min.   PZ , 30 00 And SZ , 24 7 To finde the angle PZS .   Sines . The side PZ , 30 deg . 50000 The side SZ , 24 deg . 7 min. 40785     1 The factus of the Sines 2039250000 2 Quadrat of the Radius 10000000000 The summe of the sides 96 deg . 22 min. The half summe , 48 11   Sines . The difference of ZS 24 de . 4 min. 40737 The difference of PZ 18 11 31084     3 Factus of the sines 1266268908 Which being multiplyed by Radids square , 100000.00000 , and divided by 2039250000 , the quotient will be 620●●83●7● , the side whereof is 78802 , the sine of 52 deg . which doubled is 104 , the angle PZS inquired . Arithmeticall illustration by artificiall numbers . The side PS , 42.15 . Logar . Sine . The side PZ , 30 9.698970 The side SZ , 24.7 9.610503     Sum of the sides , 96.22 19.309473     The halfe sum , 48.11   Diff. of ZS and the half sum , 24.4 9.609993 Dif . of PZ & the half sum , 18.11 9.492540 The doubled Radius 20.000000       39.102533     From which substract the sum of the Log. of the sides , ●S . PZ 19.309473 There doth remain , 19.793060 The halfe thereof , 9.896530 is the Logarithm of the sine of 52 deg . whose double 104 is the angle PZ Sinquired as before . Or if instead of the Logarithms of the sines of the sides ●S and PZ , you take their Arithmeticall complements , as was shewed in the 8th . Proposition of the 4th . Chapter , and leave out the doubled Radius , the work may be performed without substraction in this manner . The side PZ , 30 co . ar . 0.301030 The side ZS . 14.7 co . ar . 0.389497 Dif . of ZP and half sum , 18.11 9.492540 Dif . of ZS and half sum , 24.4 9.609993     The summe is 19.793060 The halfe thereof 9.896530 Is the Logarithm of the sine of 52 deg . as before . CASE 12. The three angles of a Sphericall Triangle given , to finde a side . This Case is the converse of the former , and to be resolved after the same manner , if so be we convert the angles into sides , according to the fifth of the sixth Chapter . For the two lesser angles are alwayes equal unto two sides of a Triangle comprehended by the arkes of great Circles drawn from their Poles , and the third angle may be greater then a Quadrant , and therefore the complement thereof to a Semicircle must be taken for the third side . The angle being found , shall be one of the three sides inquired . As in the Triangle ABC , the poles of those arks L , M , K , which connected do make the Triangle LMK , the sides of the former Triangle being equal to the angles of this latter , taking the complement of the greater angle to a semicircle for one . As AB is equal to the angle at L , or the arke EG . The side BC is equal to the angle at M , or the arch FH . And the side AC is equal to the complement of the angle LKM , or the arch DI. Therefore if the angles of the latter triangle LMK be given , the sides of the former triangle AB , BC , and AC are likewise given . And the angles of the triangle LMK being thus converted into sides , if we resolve the triangle ABC , according to the precepts of the last Case , we may finde any of the angles , which is the side inquired . Illustration Arithmetical , by the Artificiall Canon . Let the three angles of the triangle LMK be given . LMK , 104 deg . or the complement of DKI , 76 deg . equal to AC . MLK , or the side AB , 46 deg . 30 min. LMK , or the side BC , 36 deg . 14 min. To finde the side ML , or the angle ABC . The sides AC 76.   AB 46.30 9.859118 BC 36.14 9.770675         Sum of the sides   158.44 19.629893         Halfe sum   79.22           Diff. of AB and the sum   32.92 9.735173 Dif . of BC and half sum   43.08 9.834432 The doubled Radius     20.000000           The summe 39.569605 Sum of the sides   substract 19.629893             The difference 19.939712     Halfe difference 9.969856 The Sine of 68 deg . 90 min. which doubled is 137 deg . 80 min. the quantity of the angle ABC , and the complement thereof to a semicircle 42 deg . 20 min. is the angle FBG , or the arch FG , equal to the side ML which was inquired . Institutio Mathematica : OR , A MATHEMATICALL Institution : The second Part. Containing the application and use of the Naturall and Artificiall SINES and TANGENTS , as also of the LOGARITHMS , IN Astronomie , Dialling , and Navigation . By JOHN NEWTON . LONDON , Printed Anno Domini , 1654. A Mathematicall Institution : The second Part. CHAP. I. Of the Tables of the Suns motion , and of the equation of time for the difference of Meridians . WHereas it is requisite that the Reader should be acquainted with the Sphere , before he enter upon the practise of Spherical Trigonometri , the which is fully explained in Blundeviles Exercises , or Ch●lmades translation of Hues on the Globes , to whom I refer those that are not yet acquainted therewith : that which I here intend is to shew the use of Trigonometrie in the actuall resolution of so me known Triangles of the Sphere . And because the Suns place or distance from the next Equinoctial point is usually one of the three terms given in Astronomical Questions , I will first shew how to compute that by Tables calculated in Decimal numbers according to the Hypothesis of Bullialdus , and for the Meridian of London , whose Longitude reckoned from the Canarie or Fortunate Islands is 21 deg . and the Latitude , North , 51 deg . 57 parts ( min. ) or centesms of a degree . Nor are these Tables so confined to this Meridian , but that they may be reduced to any other : If the place be East of London , adde to the time given , but if it be West make substraction , according to the difference of Longitude , allowing 15 deg . for an houre , and 6 minutes or centesms of an houre to one degree , so will the sum or difference be the time aequated to the Meridian of London , and for the more speedy effecting of the said Reduction , I have added a Catalogue of many of the chiefest Towns and Cities in diverse Regions , with their Latitudes and difference of Meridians from London in time , together with the notes of Addition and Substraction , the use whereof is thus . Suppose the time of the Suns enterance into Taurus were at London Aprill the 10th . 1654 , at 11 of the clock and 16 centesms before noon , and it be required to reduce the same to the Meridian of Vraniburge , I therefore seeke Vraniburge in the Catalogue of Cities and Places , against which I finde 83 with the letter A annexed , therefore I conclude , that the Sun did that day at Uraniburge enter into Taurus at 11 of the clock and 99 min. or centesms before noon , and so of any other . Problem 1. To calculate the Suns true place . THe form of these our Tables of the Suns motion is this , In the first page is had his motion in Julian years compleat , the Epochaes or roots of motions being prefixed , which sheweth the place of the Sun at that time where the Epocha adscribed hath its beginning : the Tables in the following pages serve for Julian Years , Moneths , Dayes , Houres , and Parts , as by their Titles it doth appear . The Years , Moneths , and Dayes , are taken compleat , the Houres and Scruples current . After these Tables followeth another , which contains the Aequations of the Eccentrick to every degree of a Semicircle , by which you may thus compute the Suns place . First , Write out the Epocha next going before the given time , then severally set under those the motions belonging to the years , moneths , and dayes compleat , and to the hours and scruples current , every one under his like , ( onely remember that in the Bissextile year , after the end of February , the dayes must be increased by an unit ) then adding them all together , the summe shall be the Suns mean motion for the time given . Example . Let the given time be 1654 , May 13 , 11 hours , 25 scruples before noon at London , and the Suns place to be sought . The numbers are thus :   Longit. ☉ Aphel . ☉ The Epocha 1640 291.2536 96.2297 Years compl . 13 359.8508 2052 Moneth co . April 118.2775 53 Dayes compl . 12 11.8278 6 Hours 23 9444   Scruples 25 102         Sū or mean motiō 782.1643 96.4308 2. Substract the Aphelium from the mean Longitude , there rests the mean Anomalie , if it exceed not 360 degrees , but if it exceed 360 degr . 360 being taken from their difference , as oft as it can , the rest is the mean Anomalie sought . Example . The ☉ mean Longitude 782.1643 The Aphelium substracted 96.4308 There rests 685.7335 From whence deduct 360. There rests the mean Anomalie . 325.7335 3. With the mean Anomalie enter the Table of the Suns Eccentrick Equation , with the degree descending on the left side , if the number thereof be lesse then 180 ; and ascending on the right side , if it exceed 180 , and in a straight line you have the Equation answering thereunto , using the part proportional , if need require . Lastly , according to the title Add or Substract this Equation found to or from the mean longitude ; so have you the Suns true place . Example . The Suns mean longitude 782.1643 Or deducting two circles , 720. The Suns mean longitude is 62.1643 The Suns mean Anomalie 325.7335 In this Table the Equation answering to 325 degrees is 1.1525 The Equation answering to 326 degrees is 1.1236 And their difference 289. Now then if one degree or 10000 Give 289 What shall 7335 Give , the product of the second and third term is 2119815 , and this divided by 10000 the first term given , the quotient or term required will be 212 fere , which being deducted from 1.1525 , the Equation answering to 325 degr . because the Equation decreased , their difference 1.1313 . is the true Equation of this mean Anomalie , which being added to the Suns mean longitude , their aggregate is the Suns place required . Example . The Suns mean longitude 62.1643 Equation corrected Add 1.1313 The Suns true place or Longitude 63.2956 That is , 2 Signes , 3 degrees , 29 minutes , 56 parts . The Suns Equation in this example corrected by Multiplication and Division may more readily be performed by Addition and Substraction with the help of the Table of Logarithmes : for , As one degree , or 10000 , 4.000000     Is to 289 ; 2.460898 So is 7335 , 3.865400     To 212 fere 2.326298 The Suns mean Motions . Epochae Longitud ☉ Aphelium ☉   ° ′ ″ ° ′ ″ Per. Jul. 242 99 61 355 85 44 M●●di 248 71 08 007 92 42 Christi 278 98 69 010 31 36 An. Do. 1600 290 95 44 095 58 78 An. Do. 1620 291 10 41 095 90 39 An. Do. 1640 291 25 36 096 21 97 An. Do. 1660 291 40 33 096 53 56 1 356 76 11 0 01 58 2 359 52 22 0 18 17 3 359 28 30 0 04 74 B 4 000 03 00 0 06 30 5 359 79 11 0 07 89 6 359 55 19 0 09 47 7 359 31 30 0 11 05 B 8 000 05 97 0 12 64 9 359 82 08 0 14 22 10 359 58 19 0 15 78 11 359 34 30 0 17 36 B 12 000 08 97 0 18 94 13 359 85 08 0 20 52 14 359 00 19 0 22 11 15 359 37 30 0 23 69 B 16 000 11 97 0 25 25 17 359 88 08 0 26 83 18 359 64 19 0 28 41 19 359 40 28 0 30 00 B 20 000 14 97 0 31 61 40 000 29 91 0 63 19 60 000 44 83 0 94 77 80 000 59 83 1 26 39 100 000 74 80 1 57 97 100 00 74 80 01 57 97 200 01 49 58 03 15 94 300 02 24 39 04 73 94 400 02 99 19 06 31 94 500 03 73 97 07 89 91 600 04 48 77 09 47 92 700 05 23 58 11 05 89 800 05 98 36 12 63 89 900 06 73 17 14 21 86 1000 07 47 97 15 79 86 2000 14 95 92 31 59 69 3000 22 43 89 47 39 55 4000 29 91 82 63 19 41 5000 37 39 80 78 99 25               January 030 55 50 0 00 14 February 058 15 30 0 00 25 March 088 70 83 0 00 39 April 118 27 75 0 00 53 May 148 83 28 0 00 67 June 178 40 19 0 00 80 July 208 95 69 0 00 94 August 239 51 22 0 01 06 September 269 08 17 0 01 19 October 299 63 66 0 01 33 November 329 20 61 0 01 44 December 359 76 11 0 01 58 The Suns mean motions in Dayes .   Longit. ☉ Aphel . D ° ′ ″ ′ ″ 1 0 98 55 0 00 2 1 97 14 0 00 3 2 95 69 0 00 4 3 94 25 0 02 5 4 92 83 0 02 6 5 91 39 0 03 7 6 89 94 0 03 8 7 88 52 0 03 9 8 87 08 0 05 10 9 85 63 0 05 11 10 84 22 0 05 12 11 82 78 0 06 13 12 81 33 0 06 14 13 79 91 0 06 15 14 78 47 0 06 16 15 77 03 0 08 17 16 75 61 0 08 18 17 74 16 0 08 19 18 72 72 0 08 20 19 71 30 0 08 21 20 69 86 0 08 22 21 68 41 0 11 23 22 67 00 0 11 24 23 65 56 0 11 25 24 64 11 0 11 26 25 62 94 0 11 27 26 61 25 0 11 28 27 59 80 0 11 29 28 58 36 0 13 30 29 56 94 0 14 31 30 55 50 0 14 32 31 54 05 0 14   Longit. ☉   Long.   Long. H ° ′ ″ M ′ ″ M ′ ″ 1 0 04 11 34 1 39 67 2 75 2 0 08 22 35 1 43 68 2 79 3 0 12 31 36 1 47 69 2 83 4 0 16 42 37 1 51 70 2 87 5 0 20 52 38 1 56 71 2 91 6 0 24 63 39 1 60 72 2 96 7 0 28 75 40 1 64 73 3 00 8 0 32 86 41 1 68 74 3 04 9 0 36 97 42 1 72 75 3 08 10 0 41 06 43 1 76 76 3 12 11 0 45 17 44 1 80 77 3 16 12 0 49 27 45 1 84 78 3 20 13 0 53 39 46 1 88 79 3 24 14 0 52 50 47 1 93 80 3 28 15 0 61 61 48 1 97 81 3 32 16 0 65 72 49 2 01 82 3 37 17 0 69 80 50 2 05 83 3 41 18 0 73 91 51 2 09 84 3 45 19 0 78 03 52 2 13 85 3 49 20 0 82 14 53 2 17 86 3 53 21 0 86 25 54 2 21 87 3 57 22 0 90 36 55 2 25 88 3 61 23 0 94 44 56 2 30 89 3 65 24 0 98 55 57 2 34 90 3 69 25 1 02 66 58 2 38 91 3 74 26 1 06 77 59 2 42 92 3 78 27 1 10 88 60 2 46 93 3 82 28 1 14 99 61 2 50 94 3 86 29 1 19 10 62 2 54 95 3 90 30 1 23 21 63 2 58 96 3 94 31 1 27 32 64 2 62 97 3 98 32 1 31 43 65 2 67 98 4 02 33 1 35 54 66 2 71 99 4 06 ′ ′ ″ ‴ ′ ′ ″ 100 4 11 ″ ″ ‴ ' ' ' ' ″ ″ ‴ ″ ″ ‴ The Equations of the Suns Eccentrick .   Aeq . sub     ° ′ ″   0 0 00 00 360 1 0 03 52 359 2 0 07 03 358 3 0 10 56 357 4 0 14 05 356 5 0 17 53 355 6 0 21 00 354 7 0 24 44 353 8 0 27 89 352 9 0 31 30 351 10 0 34 72 350 11 0 38 17 349 12 0 41 56 348 13 0 44 94 347 14 0 48 30 346 15 0 51 67 345 16 0 55 03 344 17 0 58 36 343 18 0 61 67 342 19 0 64 97 341 20 0 68 24 340 21 0 71 53 339 22 0 74 78 338 23 0 78 03 337 24 0 81 22 336 25 0 84 41 335 26 0 87 56 334 27 0 90 69 333 28 0 94 26 332 29 0 97 30 331 30 1 00 19 330 31 1 03 33 329 32 1 06 41 328 33 1 09 41 327 34 1 12 36 326 35 1 15 25 325 36 1 18 03 324 37 1 20 78 323 38 1 23 50 322 39 1 26 22 321 40 1 28 91 320 41 1 31 58 319 42 1 34 22 318 43 1 36 86 317 44 1 39 50 316 45 1 42 08 315 46 1 44 52 314 47 1 47 05 313 48 1 49 47 312 49 1 51 89 311 50 1 54 16 310 51 1 56 47 309 52 1 58 69 308 53 1 60 86 307 54 1 63 00 306 55 1 65 14 305 56 1 67 25 304 57 1 69 30 303 58 1 71 33 302 59 1 73 28 301 60 1 75 05 300   1 76 92 299 62 1 76 69 298 63 1 80 39 297 64 1 81 97 296 65 1 83 50 295 66 1 85 00 294 67 1 86 44 293 68 1 87 83 292 69 1 89 16 291 70 1 90 44 290 71 1 91 69 289 72 1 92 86 288 73 1 93 96 287 74 1 95 28 286 75 1 96 22 285 76 1 97 14 284 77 1 97 97 283 78 1 98 72 282 79 1 99 61 281 80 2 00 41 280 81 2 01 14 279 82 2 01 72 278 83 2 02 25 277 84 2 02 94 276 85 2 03 14 275 86 2 03 44 274 87 2 03 66 273 88 2 04 05 272 89 2 04 22 271 90 2 04 41 270 91 2 04 47 269 92 2 04 41 268 93 2 04 27 267 94 2 04 11 266 95 2 03 89 265 96 2 03 61 264 97 2 03 33 263 98 2 02 94 262 99 2 02 50 261 100 2 02 03 260 101 2 01 42 259 102 2 00 64 258 103 1 99 83 257 104 1 99 27 256 105 1 98 47 255 106 1 97 64 254 107 1 96 67 253 108 1 95 67 252 109 1 94 55 251 110 1 93 39 250 111 1 92 11 249 112 1 90 89 248 113 1 89 58 247 114 1 88 28 246 115 1 86 89 245 116 1 85 44 244 117 1 83 97 243 118 1 82 39 242 119 1 80 72 241 120 1 79 00 240 121 1 77 19 239 122 1 75 39 238 123 1 73 50 237 124 1 71 50 236 125 1 69 50 235 126 1 67 53 234 127 1 65 39 233 128 1 63 22 232 129 1 61 28 231 130 1 58 77 230 131 1 56 44 229 132 1 54 05 228 133 1 51 64 227 134 1 49 16 226 135 1 46 97 225 136 1 44 16 224 137 1 41 58 223 138 1 38 94 222 139 1 36 31 221 140 1 33 58 220 141 1 30 83 219 142 1 28 08 218 143 1 25 28 217 144 1 22 42 216 145 1 19 55 215 146 1 16 67 214 147 1 13 72 213 148 1 10 61 212 149 1 07 47 211 150 1 04 27 210 151 1 01 00 209 152 0 97 75 208 153 0 94 47 207 154 0 91 19 206 155 0 87 89 205 156 0 84 58 204 157 0 81 28 203 158 0 77 97 202 159 0 74 61 201 160 0 71 25 200 161 0 67 86 199 162 0 64 44 198 163 0 60 97 197 164 0 57 44 196 165 0 53 89 195 166 0 50 33 194 167 0 46 75 193 168 0 43 19 192 169 0 39 64 191 170 0 36 06 190 171 0 32 50 189 172 0 28 91 188 173 0 25 31 187 174 0 21 69 186 175 0 17 08 185 176 0 14 47 184 177 0 10 86 183 178 0 07 25 182 179 0 03 64 181 180 0 00 00 180 A Catalogue of some of the most eminent Cities and Towns in England , Ireland , and other Countreys , wherein is shewed the difference of their Merdians from London , with the height of the Pole Artique . Names of the Places . Diff. in time Pole ABerden in Scotland S 0 12 58 67 S. Albons S 0 02 51 92 Alexandria in Egypt A 2 18 30 97 Amsterdam in Holland A 0 35 52 42 Athens in Greece A 1 87 37 70 Bethelem A 2 77 31 83 Barwick S 0 10 55 82 Bedford S 0 03 52 30 Calice in France   0 00 50 87 Cambridge A 0 03 52 33 Canterbury A 0 08 51 45 Constantinople A 2 30 43 00 Darby S 0 08 53 10 Dublin in Ireland S 0 43 53 18 Dartmouth S 0 25 50 53 Ely A 0 02 52 33 Grantham S 0 03 52 97 Glocester S 0 15 52 00 Hartford S 0 02 51 83 Hierusalem A 3 08 32 17 Huntington S 0 02 52 32 Leicester S 0 07 52 67 Lincolne S 0 02 57 25 Nottingham S 0 07 53 05 Newark S 0 05 53 03 Newcastle S 0 10 54 97 Northampton S 0 07 52 30 Oxford S 0 08 51 90 Peterborough S 0 03 52 38 Richmond S 0 10 54 43 Rochester A 0 05 51 47 Rochel in France S 0 07 45 82 Rome in Italy A 0 83 42 03 Stafford S 0 13 52 92 Stamford S 0 03 52 68 Sbrewsbury S 0 18 54 80 Tredagh in Ireland S 0 45 53 63 Uppingham S 0 05 52 67 Uraniburge A 0 83 55 90 Warwick S 0 10 52 42 Winchester S 0 08 51 17 Waterford in Ireland S 0 45 52 37 Worcester S 0 15 52 33 Yarmouth A 0 10 52 75 York S 0 07 54 00 LONDON   0 00 51 53 Probl. 2. To finde the Suns greatest declination , and the Poles elevation . THe Declination of a Planet or other Star is his distance from the Equator , and as he declines from thence either Northward or Southward , so is the Declination thereof counted either North or South . In the annexed Diagram , GMNB represents the Meridian , LK the Equinoctiall , HP the Zodiac , A the North pole , O the South , MB the Horizon , G the Zenith , N the Nadir , HC a parallel of the Suns diu●nall motion at H , or the Suns greatest declination from the Equator towards the North pole , PQ a parallel of the Suns greatest declination from the Equator towards the South pole . From whence it is apparent , that from M to H is the Suns greatest Meridian altitude , from M to Q his least ; if therefore you deduct MQ , the least Meridian altitude from MH , the greatest , the difference will be HQ , the Suns greatest declination on both sides of the Equator , and because the angles HDL and KDP are equal , by the 9th . of the second , therefore the Suns greatest declination towards the South pole is equall to his greatest declination towards the North ; and consequently , half the distance of the Tropicks , or the arch HQ , that is , the arch HL is the quantity of the Suns greatest declination . And then if you deduct the Suns greatest declination , or the arch HL from the Suns greatest Meridian altitude , or the arch MH , the difference will be ML , or the height of the Equator above the Horizon , the complement whereof to a Quadrant is the arch MO equal to AB , the height of the Pole. Example . The Suns greatest meridian altitude at London about the 11 th . of June was found to be 62 00 00 His least December 10. 14 94 00 Their difference is the distance of the Tropicks 47 06 00 Half that the Suns greatest declin . 23 53 00 Whos 's difference from the greatest Altitude is the height of the Equator 38 47 00 Whos 's complement is the Poles elevation 51 53 00 Probl. 3. The Suns place and greatest declination given to finde the declination of any point of the Ecliptique . IN this figure let DFHG denote the Solsticiall Colure , FBAG the Equator , DAH the Ecliptique , I the Pole of the Ecliptique , E the Pole of the Equator , CEB a Meridian line passing from E through the Sun at C , and falling upon the Equator FAG with right angles in the point B. Then is DAF the angle of the Suns greatest declination , AC the Suns distance from Aries the next Equinoctiall point , BC the declination of the point sought . Now suppose the sun to be in 00 deg . of Gemini , which point is distant from the next Equinoctiall point 60 deg . and his declination be required . In the rectangled spherical triangle we have known , 1 The hypothenusal AC 60 deg . 2 The angle at the base BAC 23 deg . 53 min. Hence to finde the perpendicular BC , by the 8 Case of right angled sphericall triangles , the analogie is , As the Radius , 90 10.000000 To the sine of BAC , 23.53 . 9.601222 So is the sine of AC , 60 9.937531     To the sine of BC , 20.22 9.538753 Probl. 4. The greatest declination of the Sun , and his distance from the next Equinoctial point given , to finde his right ascension . IN the Triangle ABC of the former diagram , having as before , the angle BAC , and the hypothenusal AC , the Right Ascension of the sun AB may be found by the 7 Case of right angled spherical triangles : for As the Radius , 90 10. ●00000 To the Co-sine of CAB , 23.53 9.962299 So is the tangent of AC , 60 10.238561 To the Tangent of AB , 57.80 10.200860 Only note , that if the Right Ascension of the point sought be in the second Quadrant ( as in ♋ ♌ ♍ ) the complement of the arch found to 180 is the arch sought . If in the third Quadrant ( as in ♎ ♏ ♐ ) adde a semicircle to the arch found ; if in the last Quadrant , substract the arch found from 360 , and their difference shall be the Right Ascension sought . Probl. 5. The Latitude of the place , and declination of the Sun given , to finde the Ascensionall difference , or time of the Suns rising before or after the houre of six . THe Ascensionall difference is nothing else but the difference between the Ascension of any point in the Ecliptique in a right Sphere , and the ascension of the same point in an oblique Sphere . As in the annexed Diagram , AGEV represents the Meridian , EMT the Horizon , GMCV the Equator , A the North Pole , VT the complement of the Poles elevation , BC the Suns declination , DB an arch of the Ecliptique , DC the Right Ascension , MC the Ascensionall difference . Then in the right angled triangle BMC , we have limited , 1 The angle BMC , the complement of the Poles elevation , 38 deg . 47 min. 2 The perpendicular BC , the Suns Declination 20 deg . 22 min. Hence to finde MC the Ascensional difference , by the 6 Case of right angled Spherical Triangles , the Proportion is , As the Radius , 90 10.000000 To the tangent of BC , 20.22 9.566231 So is co-tangent of BMC , 38.47 10.099861 To the sine of MC , 27.62 9.666092 Probl. 6. The Latitude of the place , and the Suns Declination given , to finde his Amplitude . THe Suns Amplitude is an arch of the Horizon intercepted between the Equator , and the point of rising , that is , in the preceding Diagram the arch MB , therefore in the right angled Sphericall triangle MBC , having the angle BMC the height of the Equator , 38 deg . 47 min. and BC the Suns declination 20 de . 22 m. given , the hypothenusal MB may be found by the 5 Case of right angled sphericall triangles : for As the sine of BMC , 38.47 9.793863 Is to the Radius , 90 10.000000 So is the sine of BC , 20.22 9.538606 To the sine of MB , 33.75 9.744743 Probl. 7. The Latitude of the place , and the Suns Declination given , to finde the time when he will be East or West . LEt ABCD in the annexed diagram represent the Meridian , BD the Horizon , FG the Equator , HNK an arch of a Meridian , AC the Azimuth of East and West , or first Verticall , EM , a parallel of declination . Then in the right angled sphericall triangle AHN , we have known , 1 The perpendicular AH , the complement of the Poles elevation , 38 deg . 47 mi. 2 The hypothenusal HN , the complement of the Suns declination , 69 deg . 78 m. Hence the angle AHN may be found by the 13 Case of right angled sphericall triangles . As the Radius 90 10,000000 To the tangent of AH 38.47 . 9.900138 So is the co-tangent HN 69.78 . 9.566231 To the co-sine of AHN 72.98 . 9.466369 Whos 's complement NHZ 17 degr . 2 min. being converted into time , giveth one houre , 13 minutes , or centesmes of an hour , and so much is it after six in the morning when the Sun will be due East , and before six at night , when he will be due West . Probl. 8. The Latitude of the place and Declination of the Sun given , to finde his Altitude when he cometh to be due East or west . IN the right angled sphericall triangle NQZ of the last Diagram , we have limited . 1. The perpendicular QN , the Suns declination . 2. The angle at the base NZQ , the Poles elevation 51 degr . 53 min. Hence to finde the hypothenusal NZ , by the fift Case of right angled sphericall Triangles , the proportion is ; As the sine of the ang . NZQ 51.53 . 9.893725 Is to the Radius 90 10.000000 So is the sine of NQ 20.22 . 9.538606 To the sine of NZ 26.20 . 9.644881 Probl. 9. The Latitude of the place , and Declination of the Sun given , to finde the Suns Azimuth at the hour of six . IN the right angled sphericall triangle AIH of the seventh Probleme , we have known : 1. The base AH , the complement of the Poles elevation 38 degr . 47 min. and the perpendicular IH , the complement of the Suns declination 69 degr . 78 min. Hence to finde the angle at the base HAI the suns Azimuth at the houre of six , by the 11 Case of right angled spherical triangles , the proportion is , As the Radius , 90 10.000000 To the sine of AH , 38.47 9.793863 So the co-tangent of HI , 69.78 9.566231 To the co-tangent of HAI , 77. 9.360094 Probl. 10. The Poles elevation , with the Suns Altitude and Declination given , to finde the Suns Azimuth . IN the oblique angled Spherical triangle AHS , in the Diagram of the seventh Probleme , we have known , the side AH , the complement of the Poles elevation , 38 deg . 47 min. HS , the complement of the Suns declination , 74 deg . 83 min. And the side SA , the complement of the Suns altitude , 57 deg . 53 min , to finde the angle SAH : Now then , by the 11 Case of Oblique angled Sphericall Triangles , I work as is there directed .   SH , 74.83     HA , 38,47 9.793863   SA , 57.53 9 . 9●6174       Summe of the sides 170.83 19.720037 Halfe summe 85. 41. 50         Dif . of HA & half sum , 46.91.50 9.863737 Dif . of SA & half sum , 27.88.50 9.669990 The doubled Radius   20.000000         Their summe 39.533727 From whence substract   19.720037 There rests   19.813690 The halfe whereof   9.906845 Is the sine of 53 deg . 80 min. which doubled is 107 deg . 60 min. the Suns Azimuth from the north , and 72 deg . 40 min. the complement thereof to a Semicircle is the Suns Azimuth from the South . CHAP. II. THE ART OF SHADOWS : Commonly called DIALLING . Plainly shewing out of the Sphere , the true ground and reason of making all kinde of Dials that any plain is capable of . Problem 1. How to divide diverse lines , and make a Chord to any proportion given . FOrasmuch as there is continuall use both of Scales and Chords in drawing the Scheams and Dials following , it will be necessary first to shew the making of them , that such as cannot have the benefit of the skilful artificers labour , may by their own pains supply that defect . Draw therefore upon a piece of paper or pastboard a streight line of what length you please , divide this line into 10 equal parts , and each 10 into 10 more , so is your line divided into 100 equal parts , by help where of a line of Chords to any proportion may be thus made . First , prepare a Table , therein set down the degrees , halves , and quarters , if you please , from one to 90. Unto each degree and part of a degree joyn the Chord proper to it , which is the naturall sine of halfe the arch doubled , by the 19th . of the second of the first part : if you double then the naturall sines of 5. 10. 20. 30. degrees , you shall produce the Chords of 10. 20. 40. 60. degrees : Thus 17364 the sine of 10 de . being doubled , the sum will be 34●28 , the Chord of 20 deg . and so of the rest as in the Table following . De Chord 1 17 2 35 3 52 4 70 5 87 6 105 7 122 8 139 9 157 10 175 11 192 12 209 13 226 14 244 15 261 16 278 17 296 18 313 19 330 20 347 21 364 22 382 23 398 24 416 25 432 26 450 27 466 28 384 29 501 30 ●18 31 534 32 551 33 568 34 585 35 601 36 618 37 635 38 651 39 668 40 684 41 700 42 717 43 733 44 749 45 765 44 781 47 797 48 813 49 830 50 845 51 861 52 876 53 892 54 908 55 923 56 939 57 954 58 970 59 984 60 1000 61 1015 62 1030 63 1045 64 1060 65 1074 66 1089 67 1104 68 1118 69 1133 70 1147 71 1161 72 1176 73 1190 74 1204 75 1217 76 1231 77 1245 78 1259 79 1273 80 1286 81 1299 82 1312 83 1325 84 1338 85 1351 86 1364 87 1377 88 1389 89 1402 90 1414 This done , proportion the Radius of a circle to what extent you please , make AB equal thereto , in the middle whereof , as in C , erect the perpendicular CD , and draw the lines AD and BD , equal in length to your line of equal parts , so have you made an equiangled Triangle , by help whereof and the Table aforesaid , the Chord of any arch proportionable to this Radius may speedily be obtained . As for example . Let there be required the Chord of 30 deg . the number in the Table answering to this arke is 518 , or in proportion to this Scale 52 almost , I take therefore 52 from the Scale of equal parts , and set them from D to E and F , and draw the line EF , which is the Chord desired . Thus may you finde the Chord of any other arch agreeable to this Radius . Or if your Radius be either of a greater or lesser extent , if you make the base of your Triangle AB equal thereunto , you may in like manner finde the Chord of any arch agreeable to any Radius given . Only remember that if the Chord of the arch desired exceed 60 deg . the sides of the Triangle AD and DB must be continued from A and B as far as need shall require . In this manner is made the line of Chords adjoyning , answerable to the Radius of the Fundamental Scheme . And in this manner may you finde the Sine , Tangent or Secant of any arch proportionable to any Radius , by help of the Canon of Naturall Sines , Tangents and Secants , and the aforesaid Scale of equall parts , as by example may more plainly appear . Let there be required the sine of 44 degrees in the table of natural sines , the number answering to 44 degrees is 694. I take therefore with my compasses 69 from my Scale of equal parts , and set them from D to G and H ; so is the line GH the sine of 44 degrees , where the Radius of the circle is AB . Again , if there were required the tangent of 44 degrees , the number in the table is 965 ; and therefore 96 set from D to K and L shall give the tangent required ; and so for any other . Your Scales being thus prepared for the Mechanicall part , we will now shew you how to project the Sphere in plano , and so proceed to the arithmeticall work . Probl. 2. The explanation and making of the fundamental Diagram . THis Scheme representeth to the eye the true and natural situation of those circles of the Sphere , whereof we shall have use in the description of such sorts of Dials as any flat or plane is capable of . It is therefore necessary first to explain that , and the making thereof , that the Symetry of the Scheme with the Globe being well understood , the representation of every plane therein may be the better conceived . Suppose then that the Globe elevated to the height of the Pole be prest flat down into the plane of the Horizon , then will the outward circle or limbe of this Scheme NESW represent that Horizon , and all the circles contained in the upper Hemisphere of the Globe may artificially be contrived , and represented thereon , as Azimuths , Almicanters , Meridians , Parallels , Equator , Tropicks , circles of position , and such like , the which in this Diagram are thus distinguished . The letter Z represents the Zenith of the place , and the center of the horizontal circle , NZS represents the meridian , P the pole of the world elevated above the North part of the Horizon N here at London , 51 degrees 53 minutes , or centesmes of a degree , the complement whereof PZ 38 degrees , 47 minutes , and the distance between the Pole and the Zenith ; EZW is the prime vertical , DZG and CZV any other intermediate Azimuths , NOS a circle of position , EKW the Equator , the distance whereof from Z is equal to PN , the height of the Pole , or from S equal to PZ , the complement thereof , HBQX the Tropick or parallel of Cancer , LFM , the Tropique of Capricorn , the rest of the circles intersecting each other in the point P , are the meridians or hour-circles , cutting the Horizon and other circles of this Diagram in such manner as they do in the Globe itself . Amongst these the Azimuths onely in this projection become streight lines , all the rest remain circles , and are greater or lesser , according to their natural situation in the Globe , and may be thus described . Open your compasses to the extent of the line AB in the former Probleme , ( or to any other extent you please ) with that Radius , or Semidiameter describe the horizontal circle NESW , crosse it at right angles in Z with the lines NZS and EZW . That done , seek the place of the Pole at P , through which the hour circles must pass , the Equinoctial point at K , the Tropiques at T and F , the reclining circle at O , and the declining reclining at A ; all which may thus be found . The Zenith in the Globe or Materiall Sphere is the Pole of the Horizon , and Z in the Scheme is the center of the limbe , representing the same , from which point the distance of each circle being given both wayes , as it lyeth in the Sphere , and set upon the Azimuth , or streight line of the Scheme proper thereunto , you may by help of the natural tangents of half their arches give three points to draw each circle by , for if the naturall tangents of both distances from the Zenith be added together , the half thereof shall be the Semidiameters of those circles desired . The reason why the natural tangent of half the arches are here taken , may be made plain by this Diagram following . Wherein making EZ the Radius , SZN is a tangent line thereunto , upon which if you will project the whole Semicircle SWN , it is manifest , by the work , that every part of the lines ZN or ZS can be no more then the tangent of half the arch desired , because the whole line ZN or ZS is the tang . of no more then half the Quadrant , that is , of 45 degrees , by the 19th . of the second Chapter of the first Part ; and therefore WEA is but half the angle WZA and WEB is but half the angle WZB. Now then if EZ or Radius of the fundamental Scheme be 1000 , ZP shal be 349 , the natural tangent of 19 degrees , 23 minutes , 50 seconds , the half of 38 degrees , 47 minutes , the distance between the North pole and the Zenith in our Latitude of 51 degrees , 53 minutes , or centesmes of a degree . And the South pole being as much under the Horizon as the North is above it , the distance thereof from the Zenith must be the complement of 38 degrees , 47 minutes to a Semicircle , that is , 141 degrees , 53 minutes ; and as the half of 38 degrees , 47 minutes , viz. 19 degrees , 23 minutes , 50 seconds is the quantity of the angle PEZ , and the tangent thereof the distance from Z to P , so the half of 141 degrees , 53 minutes , viz. 70 degrees , 76 minutes , 50 seconds must be the measure of the angle in the circumference between the Zenith and the South , the tangent whereof 2866 must be the distance also , and the tangents of these two arches added together 3215 , is the whole diameter of that circle , the half whereof 1607 , that is , one Radius , and neer 61 hundred parts of another is the Semidiameter or distance from P to L in the former Scheme , to which extent open the compasses , and set off the distance PL , and therewith draw the circle WPE for the six of the clock hour . The Semidiameters of the other circles are to be found in the same manner : the distance between the Zenith and the Equinoctiall is alwayes equal to the height of the Pole , which in our Latitude is 51 degr . 53 min. and therefore the half thereof 25 degrees , 76 minutes , 50 seconds is the measure of the angle WEB , and the natural tangent thereof 483 , which being added to the tangent of the complement 2070 , their aggregate 2553 will be the whole diameter of that circle , and 1277 the Radius or Semidiameter by which to draw the Equinoctiall circle EKW . The Tropique of Cancer is 23 degrees , 53 minutes above the Equator , and 66 degrees 47 minutes distant from the Pole , and the Pole in this Latitude is 38 degrees 47 min. distant from the Zenith , which being substracted from 66 degrees 47 minutes , the distance of the Tropique of Cancer from the Zenith , will be 28 , the half thereof is 14 , whose natural tangent 249 being set from Z to T , giveth the point T in the Meridian , by which that parallel must passe ; the distance thereof from the Zenith on the North side is TN 90 degrees , and substracting 23 degrees , 53 minutes , the height of the Tropique above the Equator , from 38 degrees , 47 minutes , the height of the Equator above the Horizon , their difference is 14 degrees , 94 minutes , the distance of the Tropique from N under the Horizon ; and so the whole distance thereof from Z is 104 degrees , 94 minutes , the half whereof is 52 degrees , 47 minutes , and the natural tangent thereof 1302 added to the former tangent 249 , giveth the whole diameter of that circle 1551 , whose half 776 is the Semidiameter desired , and gives the center to draw that circle by . The Tropique of Capricorn is 23 degrees , 53 minutes below the Equator , and therefore 113 degrees 53 minutes from the North pole , from which if you deduct , as before , 38 degrees , 47 minutes , the distance of the Pole from the Zenith , the distance of the Tropique of Capricorn from the Zenith will be 75 degrees , 6 minutes , and the half thereof 37 degrees , 53 minutes , whose natural tangent 768 being set from Z to F , giveth the point F in the Meridian , by which that parallel must pass : the distance thereof from the Zenith on the North side is ZN 90 degrees , as before ; and adding 23 degrees , 53 minutes , the distance of the Tropique from the Equator to 38 degrees , 47 minutes , the distance of the Equator from the Horizon , their aggregate is 62 degrees , the distance of the Tropique from the Horizon , which being added to ZN 90 degrees , their aggregate is 152 degrees , and the half thereof 76 degrees , whose natural tangent 4011 being added to the former tangent 768 , giveth the whole diameter of that circle 4.779 , whose half 2.389 is the Semidiameter desired , and gives the center to draw that circle by . The distance of the reclining circle NOS from Z to O is 40 degrees , the half thereof 20 , whose naturall tangent 3.64 set from Z to O , giveth the point O in the prime vertical EZW , by which that circle must pass ; the distance thereof from the Zenith on the East side is ZE 90 degrees , to which adding 50 degrees , the complement of the former arch , their aggregate 140 degrees is the distance from Z Eastward , and the half thereof 70 degrees , whose natural tangent 2747 being added to the former tangent 364 , their aggregate 3111 is the whole diameter of that circle , and the half thereof 1555 is the Semidiameter desired , and gives the center to draw that circle by . The distance of the declining reclining circle DAG from the Zenith is ZA 35 deg . the half thereof 17 degrees , 50 minutes , whose natural tangent 315 being set from Z to A , giveth the point by which that circle must passe , and the natural tangent of 7● degr . 50 min. the complement thereof 317● being added thereto is 3486 , the whole diameter of the circle , and the half thereof 1743 , the Semidiameter desired , and giveth the center to draw that circle by . The streight lines CZA or DZG are put upon the limbe by help of a line of Chords 30 degrees distant from the Cardinal points NESW , and must crosse each other at right angles in Z , representing two Azimuths equidistant from the Meridian and prime verticall . Last of all , the hour-circles are thus to be drawn ; first , seek the center of the six of clock hour-circle , as formerly directed , making ZE the Radius , and is found at L upon the Meridian line continued from P to L , which cross at right angles in L with the line 8 L 4 , extended far enough to serve the turn , make PL the Radius , then shall 8 L 4 be a tangent line thereunto , and the natural tangents of the Equinoctiall hour arches , that is the tangent of 15 degrees 268 for one hour , of 30 degr . 577 for two , hours , of 45 degrees 1000 for three hours of 60 deg . 1732 for four hours , and 75 deg . 3732 for five hours set upon the line from L both wayes , that is , from L to 5 and 7 , 4 and 8 , and will give the true center of those hour-circles : thus , 5 upon the line 8 L 4 is the center of the hour-circle 5 P 5 , and 7 the center of the hour-circle 7 P 7 ; and so of the rest . The centers of these hour-circles may be also found upon the line 8 L 4 by the naturall secants of the same Equinoctiall arches , because the hypothenuse in a right angled plain triangle is alwayes the secant of the angle at the base , and the perpendicular the tangent of the same angle : if therefore the tangent set from L doth give the center , the secant set from P shall give that center also . The Scheme with the lines and circles thereof being thus made plain , we come now to the Art of Dialling it self . Probl. 3. Of the severall plains , and to finde their scituation . ALL great Circles of the Sphere , projected upon any plain , howsoever situated , do become streight lines , as any one may experiment upon an ordinary bowle thus . If he saw the Bowle in the midst , and joyne the two parts together again , there will remain upon the circumference of the Bowle , some signe of the former partition , in form of a great Circle of the Sphere : now then , if in any part of that Circle the roundnesse of the bowle be taken off with a smoothing plain or otherwise , as the bowle becomes flat , so will the Circle upon the bowle become a streight line ; from whence it follows , that the houre lines of every Diall ( being great Circles of the Sphere ) drawn upon any plain superficies , must also be streight lines . Now the art of Dialling consisteth in the artificiall finding out of these lines , and their distances each from other , which do continually varie according to the situation of the plain on which they are projected . Of these plains there are but three sorts . 1. Parallel to the Horizon , as is the Horizontal only . 2. Perpendicular to the Horizon , as are all erect plains , whether they be such as are direct North , South , East or West , or such as decline from these points of North , South , East , or West . 3. Inclining to the Horizon , or rather Reclining from the Zenith , and these are direct plains reclining and inclining North and South , and reclining and inclining East and West , or Declining-reclining and inclining plains . To contrive the houre lines upon these severall plains , there are certain Spherical arches and angles , in number six , which must of necessity be known , and divers of these are in some Cases given , in others they are sought . 1. The first is an arch of a great Circle perpendicular to the plain , comprehended betwixt the Zenith and the plain , which is the Reclination , as ZT , ZK , and ZF , in the fundamental Diagram . 2. The second is an arch of the Horizon betwixt the Meridian and Azimuth passing by the poles of the plain , as SV or NC in the Scheme . 3. The third is an arch of the plain betwixt the Meridian and the Horizon , prescribing the distance of the 12 a clock houre from the horizontal line , as PB in the Scheme of the 11th . Probl. 4. The fourth is an arch of the plain betwixt the Meridian and the substile , which limits the distance thereof from the 12 a clock houre line , as ZR in the Scheme . 5. The fifth is an arch of a great Circle perpendicular to the plain , comprehended betwixt the Pole of the World ; and the plain , commonly called the height of the stile , as PR in the Scheme . 6. The last is an angle at the Pole betwixt the two Meridians , the one of the place , the other of the plain ( taking the substile in the common sense for the Meridian of the plain ) as the angle ZPR in the fundamental Scheme . The two first of these arches are alwayes given , or may be found by the rules following . To finde the Inclination or Reclination of any plain . If the plain seem to be level with the Horizon , you may try it by laying a ruler thereupon , and applying the side of your Quadrant AB to the upper side of the ruler , so that the center may hang a little over the end of the ruler , and holding up a threed and plummet , so that it may play upon the center , if it shall fall directly upon his level line AC , making no angle therewith , it is an horizontal plain . If the plain seeme to be verticall , like the wall of an upright building , you may try it by holding the Quadrant so that the threed may fall on the plumb line AC , for then if the side of the Quadrant shall lie close to the plain , it is erect , and a line drawn by that side of the Quadrant shall be a Verticall line , as the line DE in the figure . If the plain shall be found to incline to the Horizon , you may finde out the quantity of the inclination after this manner . Apply the side of your Quadrant AC to the plain , so shall the threed upon the limbe give you the inclination required . Suppose the plain to be BGED , and the line FZ to be verticall , to which applying the side of your Quadrant AC , the threed upon the limbe shall make the angle CAH the inclination required , whose complement is the reclination . To finde the declination of a plain . To effect this , there are required two observations , the first is of the horizontal distance of the Sun from the pole of the plain , the second is of the Suns altitude , thereby to get the Azimuth : and these two observations must be made at one instant of time as neer as may be , that the parts of the work may the better agree together . 1. For the horizontal distance of the Sun from the pole of the plain , apply one edge of the Quadrant to the plain , so that the other may be perpendicular to it , and the limbe may be towards the Sun , and hold the whole Quadrant horizontal as neer as you can conjecture , then holding a threed and plummet at full liberty , so that the shadow of the threed may passe through the center and limb of the Quadrant , observe then what degrees of the limb the shadow cuts , counting them from that side of the Quadrant which is perpendicular to the horizontal line , those degrees are called the Horizontal distance . 2. At the same instant observe the Suns altitude , by this altitude you may get the Suns Azimuth from the South , by the 10th . Probleme of the first Chapter hereof . When you make your observation of the Suns horizontal distance , marke whether the shadow of the threed fall between the South , and the perpendicular side of the Quadrant , or not , for , 1. If the shadow fall between them , then the distance and Azimuth added together do make the declination of the plain , and in this case the declination is upon the same coast whereon the Suns Azimuth is . 2. If the shadow fall not between them , then the difference of the distance and Azimuth is the declination of the plain , and if the Azimuth be the greater of the two , then the plain declineth to the same coast whereon the Azimuth is , but if the distance be the greater , then the plain declneth to the contrary coast to that whereon the Suns Azimuth is . Note here further , that the declination so found , is alwayes accounted from the South , and that all declinations are numbered from North or South , towards East or West , and must not exceed 90 deg . 1. If therefore the number of declination exceed 90 , you must take its complement to 180 , and the same shall be the plains declination from the North. 2. If the declination found exceed 180 deg . then the excesse above 180 , gives the plains declination from the North , towards that Coast which is contrary to the Coast whereon the Sun is . By this accounting from North & South , you may alwayes make your plains declination not to exceed a Quadrant or 90 de . And as when it declines nothing , it is a full South or North plain , so if it decline just 90 , it is then a full East or West plain . These precepts are sufficient to finde the declination of any plain howsoever situated , but that there may be no mistake , we will adde an Example . 1 Example . Now because the line of shadow AG , falleth between P the pole of the plains horizontal line , and S the South point , therefore according to the former direction , I adde the horizontal distance PG 24 deg . to the Suns Azimuth GS 40 deg , and their aggregate is PS 64 deg . the declination sought ; and in this case it is upon the same coast with the sun , that is West , according to the rule given , and as the figure it selfe sheweth , the East and North points being hid from our sight by the plain it selfe ; this therefore is a South plain declining West 64 degrees . 2. Example . To finde a Meridian line upon an Horizontal plain . If your plain be levell with your Horizon , draw thereon the Circle BCMP , then holding a threed and plummet , so as the shadow thereof may fall upon the center , and draw in the last diagram the line of shadow HA : then if the Suns Azimuth shall be 50 deg . and the line of shadow taken in the afternoon , set off the 50 deg . from H to S , and the line SN shall be the Meridian line desired . Probl. 4. To draw the houre lines upon the Horizontal plain . THis plane in respect of the Poles thereof , which lie in the Vertex and Nadir of the place may be called vertical , in respect of the plane it self , which is parallel to the Horizon , horizontal , howsoever it be termed , the making of the Dial is the same , and there is but one onely arch of the Meridian betwixt the pole of the world and the plane required to the artificiall projecting of the hour-lines thereof , which being the height of the pole above the horizon ( equal to the height of the stile above the plane ) is alwayes given , by the help whereof we may presently proceed to calculate the hour distances in manner following . This plane is represented in the fundamental Diagram by the outward circle ESWN , in which the diameter SN drawn from the South to the North may go both for the Meridian line , and the Meridian circle , Z for the Zenith , P for the pole of the world , and the circles drawn through P for the hour-circles of 1 , 2 , 3 , 4 , &c. as they are numbred from the Meridian , and limit the distance of each hour line from the Meridian upon the plane , according to the arches of the Horizon , N 11 , N 10 , N 9 , &c. which by the severall Triangles SP 11 , SP 10 , SP 9 , or their verticals NP 11 , NP 10 , NP 9 may thus be found ; because every quarter of the Horizon is alike , you may begin with which you will , and resolve each hours distance , either by the small Triangle NP 11 , or the verticall Triangle KP 11. In the Triangle PN11 , the side PN is alwayes given , and is the height of the pole above the horizon , the which at London is 51 deg . 53 min. and the angle at P is given one hours distance from the Meridian , whose measure in the Equinoctiall is 15 deg . & the angle at N is alwayes right , that is 90 deg . wherefore by the first case of right angled spherical Triangles , the perpendicular N 11 may thus be found . As Radius 90 , 10.000000 To the tangent of NP11 , 15d . 9.428052 So is the sine of PN 51.53 . 9.893725     To the tangent of N11 , 11.85 . 9.321777 Which is the distance of the hours of 1 and 11 , on each side of the Meridian , thus in all respects must you finde the distance of 2 and 10 of clock , by resolving the triangle NP10 , and of 3 and 9 of clock , by resolving the triangle NP9 ; and so of the rest : in which , as the angle at Pincreafeth which for 2 hours is 30 degrees , for 3 hours 45 degr . for 4 hours 60 degr . for 5 hours 75 degr . so will the arches of the Horizon N10 , N9 , N8 , N7 , vary proportionably , and give each hours true distance from the Meridian , which is the thing desired . Probl. 5. To draw the hour-lines upon a direct South or North plane . EVery perpendicular plane , whether direct or declining , lieth in some Azimuth or other ; as here the South wall or plane doth lie in the prime vertical or Azimuth of East and West , represented in the fundamental Diagram by the line EZW , and therefore it cutteth the Meridian of the place at right angles in the Zenith , and hath the two poles of the plane seated in the North and South intersection of the Meridian and Horizon ; and because the plane hideth the North pole from our sight , we may therefore conclude , ( it being a general rule that every plane hath that pole depressed , or raised above it , which lieth open unto it ) that the South pole is elevated thereupon , and the stile of this Diall must look downwards thereunto , erected above the plane the height of the Antartick Pole , which being an arch of the Meridian betwixt the South pole and the Nadir , is equall to the opposite part thereof , betwixt the North pole and the Zenith ; and therefore the complement of the North pole above the horizon . Suppose then that P in the fundamental Scheme , be now the South pole , and N the South part of the Meridian , S the North ; then do all the hour-circles from the pole cut the line EZW , representing the plane unequally , as the hour-lines will do upon the plane it self , and as it doth appear by the figures set at the end of every hour line in the Scheme . Now having already the poles elevation given , as was in the horizontal , there is nothing else to be done , but to calculate the true hour-distances upon the line EZW from the meridian SZN ; and then to proceed , as formerly , and note that because the hours equidistant on both sides the meridian , are equal upon the plane , the one half being found , the other is also had , you may therefore begin with which side you will. In the triangle ZP11 , right angled at Z , I have ZP given , the complement of the height of the pole 38 deg . 47 min. the which is also the height of the stile to this Diall , and the angle at P15 degrees one hours distance from the meridian upon the Equator to finde the side Z11 , for which by the first case of right angled sphericall triangles , the proportion is , as before . As the Radius 90 , 10.000000 To the sine of PZ 38.47 . 9.793863 So is the tangent of ZP11 , 15d . 9.428052     To the tangent of Z11 , 9.47 . 9.221915 And thus in all respects must you finde the distance of 2 and 10 , of 3 and 9 ; and so forward , as was directed for the houres in the horizontal plane . The North plane is but the back side of the South , lying in the same Azimuth with it , & represented in the Scheme by the back part of the same streight line EZW , whatsoever therefore is said of the South plane may be applied to the North ; because as the South pole is above the South plane 38 degr . 47 min. so is the North pole under the North plane as much , and each stile must respect his own pole , onely the meridian upon this plane representeth the midnight , and not the noon , and the hours about it 9 , 10 , 11 , and 1 , 2 , 3 , are altogether uselesse , because the Sun in his greatest northern declination hath but 39 degr . 90 min. of amplitude in this our Latitude ; and therefore riseth but 22 min. before 4. in the morning , and setteth so much after 8 at night ; neither can it shine upon this plane longer then 35 min. past 7 in the morning , and returning to it as much before 5 at night , because then the Sun passeth on the North side of the prime vertical , in which this plane lieth , and cometh upon the South . Now therefore to make this Dial , is but to turn the South Dial upside down , and leave out all the superfluous hours between 5 and 7 , 4 and 8 , and the Diall to the North plane is made to your hand . The Geometricall projection . To project these and the Horizontal Dials , do thus : First , draw the perpendicular line CEB , which is the twelve of clock hour , crosse it at right angles with 6C6 , which is the six of clock hour ; then take with your compasses 60 deg . from a line of Chords , and making C the center draw the circle 6E6 , representing the azimuth in which the plane doth lie ; this done , take from the same Chord all the hour distances , and setting one foot of your compasses in E , with the other mark out those hour distances before found by calculation , both wayes upon the circle 6E6 ; streight lines drawn from the center C to those pricks in the circle are the true hour-lines desired . Having drawn all the hour-lines , take from the same line of Chords the arch of your poles elevation , or stile above the plane , and place it from E to O , draw the prickt line COA representing the axis or heighth of the stile , from any part of the meridian draw a line parallel to 6C6 , as is BA , & it shall make a triāgle , the fittest form to support the stile at the true height ; let the line 6C6 be horizontal , the triangular stile CBA erected at right angles over the 12 of clock line , and then is the Diall perfected either for the Horizontal , or the direct North and South planes . Probl. 6. To draw the hour-lines upon the direct East or West planes . AS the planes of South and North Dials do lie in the Azimuth of East and West , and their poles in the South and North parts of the meridian ; so do the planes of East and West Dials lie in the South and North azimuth , and their poles in the East and West part of the Horizon , from whence these Dials receive their denomination , and because they are parallel to the meridian line in the fundamental Scheme SZN , some call them meridian planes . And because the meridian , in which this plane lieth , is one of the hour-circles , and no plane that lieth in any of the hour circles can cut the axis of the world , but must be parallel thereunto ; therefore the hour lines of all such planes are also parallel each to other , and in the fundamental Scheme may be represented in this manner . Let NESW in this case be supposed to be the Eq●inoctiall divided into 24 equall parts , and let the prickt line E 8. 7. parallel to ZS be a tangent line to that circle in E , straight lines drawn from the center Z thorow the equal divisions of the limbe , intersecting the tangent line , shall give points in 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , thorow which parallels being drawn to the prime vertical , or 6 of clock hour line EZW , you have the hour-lines desired , which may for more certainties sake be found by tangents also ; for making ZE of the former Scheme to be the Radius , and E 8. 7. a tangent line , as before ; then shall the naturall tangent of 15 degr . 268 taken from a diagonal scale equal to the Radius , and set both wayes from E upon the tangent line E 8. 7. gives the distance of the houres of 5 and 7 , the tangent of 30 degr . the distance of the hours of 4 and 8 , and the tangent of 45 degr . the distance of the hours of 3 and 9 , &c. from the six of clock hour , as before ; and is a general rule for all Latitudes whatsoever . The Geometricall projection . The length of the stile being thus proportioned to the plain , make that the Radius of a Circle , and then the Equator DAE shall be a Tangent line thereunto , and therefore , the naturall Tangent of 15 deg . being set upon the Equinoctiall DAE both wayes from A , shall give the points of 5 and 7 : the Tangent of 30 deg . the points of 8 and 4 , &c. through which streight lines being drawn parallel to the six a clock houre , you have at one work made both the East and West Dials , only remember that because the Sun riseth before 4 in Cancer , and setteth after 8 , you must adde two houres before six in the East Diall , and two houres after six in the West , that so the plain may have as many houres as it is capable of . The West Dial is the same in all respect with the East , only the arch BD , or the height of the Equator , must be drawn on the right hand of the center A for the West Dial , and on the left for the East , that so the houre lines crossing it at right angles , may respect the Poles of the world to which they are parallel . Probl. 7. To draw the houre-lines upon a South or North erect plain declining East or West , to any declination given . EVery erect plain lieth under some Azimuth or other , and those only are said to decline which differ from the Meridian and Prime Vertical . The declination therefore being attained by the rules already given , ( or by what other means you like best ) we come to the calculation of the Diall it selfe , represented in the fundamentall Scheme by the right line GZD , the Poles whereof are C and V , the declination from the South Easterly NC , or North Westerly SV , 25 deg . supposing now S to be North , and N South ; W East , and E the West point , the houre circles proper to this plain are the black lines passing through the Pole P , and crossing upon the plain GZD , wherein note generally that where they run neerest together , thereabouts must the sub-stile stand , and alwayes on the contrary side to the declination , as in this example declining East , the stile must stand on the West side ( supposing P to be the South Pole ) between Z and D , the reason whereof doth manifestly appear ; because the Sun rising East , sendeth the shadow of the Axis West , and alwayes to the opposite part of the Meridian wherein he is , wherefore reason enforceth , that the morning houres be put on the West side of the Meridian , as the evening houres are on the East , and from the same ground that the substile of every plain representing the Meridian thereof , must alwayes stand on the contrary side to the declination of the plain and that the houre-lines must there run neerest together , because the Sun in that position is at right angles with the plain . For the making of this Diall three things must be found . 1. The elevation of the Pole above the plain , represented by PR , which is the height of the stile , and is an arch of the Meridian of the plain , between the Pole of the world and the plain . 2. The distance of the substile from the Meridian , represented by ZR , and is an arch of the plain between the Meridian and the substile . 3. The angle ZPR , which is an arch between the substile PR the meridian of the plain , and the line PZ the meridian of the place , and these are thus found . Because the substile is the Meridian of the plain , it must be part of a great circle passing through the pole of the world , and crossing the plain at right angles , therefore in the supposed right angled triangle PRZ , ( for yet the place of R is not found ) you have given the base PZ 38 deg . 47 min. and the angle PZR the complement of the declination 65 deg . and the supposed right angle at R , to finde the side PR , which is the height of the stile as aforesaid , but as yet the place unknown : wherefore by the 8 Case of right angled Spherical Triangles the analogie is , As the Radius , 10.00000 To the sine of PZ , 38.47 9.793863 So the sine of PZR , 65 9.957275 To the sine of PR , 34.32 9.751138 Secondly , you may finde ZR the distance of the substile from the meridian , by the 7 case of right angled Spherical Triangles . As the Radius , 90 10.000000 To the Co-sine of PZR , 65 9.629378 So is the tangent of PZ , 38.47 9.900138     To the tangent of ZR , 18.70 9.529516 These things given , the angle at P between the two meridians may be found by the 9 Case of right angled Sphericall Triangles , for the proportion is , As the Radius , 10.000000 To the Co-sine of PZ , 38.47 9.893725 So the Tangent of PZR , 65 10.331327     To the Co-tang . of RPZ , 30.78 10.225052 Having thus found the angle between the Meridians to be 30 deg . 78 min. you may conclude from thence , that the substile shall fall between the 2d . & third houres distance from the Meridian of the place , and therefore between 9 and 10 of the clock in the morning , because the plain declineth East from us , 9 of the clock being 45 deg . from the Meridian , and 10 of the clock 30 deg . distant , now therefore let fall a perpendicular between 9 and 10 , the better to inform the fancie in the rest of the work , and this shall make up the Triangle PRZ before mentioned and supposed , which being found we may calculate all the houre distances by the first case of right angled sphericall Triangles . For , As the Radius , Is to the sine of the base PR ; S● is the Tangent of the angle at the perpendicular , RP 9 , To the tangent of R 9 the perpendicular The angle at P is alwayes the Equinoctiall distance of the houre line from the substile , and may thus be sound : If the angle between the Meridians be lesse than the houre distance , substract the distance of the substile from the houre distance ; if greater substract the houre distance from that , and their difference shall give you the Equinoctiall distance required . Thus in our Example , the angle between the Meridians was found to be 30 deg . 78 m. and the distance of 9 of the clock from 1● is three houres , or 45 deg . if therefore I substract 30 deg . 78 min. from 45 deg . the remainder will be 14 deg . 22 min. the distance of 9 of the clock from the substile . Again , the distance of 10 of the clock from the Meridian is 30 deg . and therefore if I substract 30 deg . from 30 deg . 78 min. the distance of 10 of the clock from the substile will be 78 centesms or parts of a degree : the rest of the houres and parts are easily found by a continual addition of 15 deg . for every houre , 7 deg . 50 min. for half an houre , 3 deg . 75 min. for a quarter of an houre , as in the Table following you may perceive , the which consists of three columns , the first containeth the houres , the second their Equinoctiall distances from the substile , the third and last the houre arches , computed by the former proportion in this manner . As the Radius , 90 10.000000 Is to the sine of PR , 34.32 9.751136 So is the tang . of RP 9 , 14.22 9.403824 To the tangent of R 9 , 8.13 9.154960 H Aequ . Arches 4 89 22 88 61 5 74 22 63 38 6 59 22 43 43 7 44 22 28 75 8 29 22 17 50 9 14 22 8 13   merid . substil 10 00 78 00 44 11 15 78 9 05 12 30 78 18 56 1 45 78 30 08 2 60 78 45 23 3 75 78 65 80 4 90 78 88 61 The Geometricall Projection . Having calculated the hour distances , you may thus make the Diall ; Draw the Horizontall line ACB , then crosse it at right angles in C , with the line CO 12. Take 60 degrees from a Chord , and making C the Center , draw the Semicircle AOB , representing the azimuth GZD in the Scheme , in which the plane lieth ; upon this circle from O to N set off the distance of the substile from the Meridian , which was found before to be 18.70 . upon the West side of the Meridian , because this plane declineth East , then take off the same Chord the severall hour-distances , as they are ready calculated in the table , viz. 8.13 . for 9 , 17.50 . for 8 : and so of the rest ; and set them from the substile upon the circle RNO , as the Table it self directeth ; draw streight lines from the center C to these several points , so have you the true hour lines , which were desired : and lastly , take from the same Chord the heighth of the stile found to be 34. 32. which being set from N to R , and a streight line drawn from C through R representing the axis , the Diall is finished for use . Nay , thus have you made four Dials in one , viz. a South declining East and West 25 degrees , and a North declining East or West as much ; to make this plainly appear , suppose in the fundamental Scheme if N were again the North part of the horizon , P the North pole , and that GZD were a North declining West 25 degrees , then do all the hour-circles crosse the same plane , as they did the former ; onely DZ which was in the former the East side will now be the West : and consequently the afternoon hours must stand where the forenoon hours did , the stile also , which in the East declining stood between 9 and 10 , must now stand between 2 and 3 of the afternoon hours . And lest there should be yet any doubt conceived , I have drawn to the South declining East 25 , the North declining West as much ; from which to make the South declining West , and North declining East , you need to do no more then prick these hour lines through the paper , and draw them again on the other side , stile and all ; so shall they serve the turn , if you place the morning hours in the one , where the afternoon were in the other . APPENDIX . To draw the hour lines upon any plane declining far East or West , without respect to the Center . THe ordinary way is with a Beamcompasse of 16 , 18 , or 20 foot long , to draw the Diall upon a large floor , and then to cut off the hours , stile and all , at 10 , 12 , or 14 foot distance from the center , but this being too mechanical for them that have any Trigonometrical skill , I omit , and rather commend the way following ; by help whereof you may upon half a sheet of paper make a perfect model of your Diall , to what largenesse you please , without any regard at all to the Center . Suppose the wall or plane DZG , on which you would make a Diall to decline from N to C , that is from the South Easterly 83 degrees , 62 min. set down the Data , and by them seek the Quaesita , according to the former directions . The Data or things given are two . 1. PS the poles elevation 51 degrees , 53 minutes . 2. SA , the planes declination southeast 83 deg . 62 min. The Quaesita or things sought are three , 1. PR the height of the stile 3 degrees 97 minutes . 2. ZR , the distance of the substile from the meridian 38 deg . 30 min. 3. ZPR , the angle of the meridian of the plane with the meridian of the place 85 degrees , which being found , according to the former directions , the substile line must fall within five degrees of six of the clock , because 85 degrees wanteth but 5 of 90 , the distance of 6 from 12. Now therefore make a table , according to this example following , wherein set down the houres from 12 , as they are equidistant from the meridian , and unto them adjoyn their Equinoctial distances , and write Meridian and substile between the hours of 6 and 7 , and write 5 degrees against the hour of 6 , 10 degrees against the hour of 7 , and to the Equinoctial distances of each hour adde the natural tangents of those distances , as here you see . So is the Table prepared for use , by which you may easily frame● the Diall to what greatnesse you will , after this manner . Hours Equ . dist . Tang. 4 8 35 0 700 5 7 20 0 364 6 6 5 0 087     Meridiā Substile 7 5 10 0 176 8 4 25 0 166 9 3 40 0 839 10 2 55 0 1.428 11 1 70 0 2.747 12 12 85 0 11.430 The Geometricall projection . Proportion the plane BCDE , whereon you will draw the Diall to what scantling you think fit . Let VP represent the horizontal line , upon any part thereof , as at P , make choice of a fit place for the perpendicular stile ( though afterwards you may use another forme ) neer about the upper part of the plane , because the great angle between the two Meridians maketh the substile , which must passe thorow the point P , to fall so near the 6 of clock hour , as that there may be but one hour placed above it , if you desire to have the hour of 11 upon the plane , which is more useful then 4 , let P be the center , and with any Chord ( the greater the better ) make two obscure arches ; one above the horizontal line , the other under it , and with the same Chord set off the arch of 51.70 . which is the angle between the substile and horizon , and is the complement of the angle between the substile and meridian , and set it from V to T both wayes , then draw the streight line TPT , which shall be the substile of this Diall . Which is 5 inches , and 66 hundred parts for the distance of 11 a clock from the point H , and will be the same with those points set off by the natural tangents in the Table . Having done with this Equinoctiall , you must do the like with another : to finde the place whereof , it will be necessary first to know the length of the whole line from H the Equinoctial to the center of the Diall in parts of the perpendicular stile PO , if you will work by the scale of inches , or else the length in natural tangents , if you will use a diagonall Scale : first therefore , to finde the length thereof in inch-measure , we have given in the right angled plain triangle HOP , the base OP , and the angle at O to finde HP , and in the triangle OP center . We have given the perpendicular OP , and the angle PO center the complement of the former , to finde H center : wherefore , by the first case of right angled plain triangles : As the Radius 90 10.000000 Is to the base OP 206 ; 2.313867 So is the tang . of HOP 3.97 . 8.841364     To the perpendicular PH14 . 1.155231 Again , As the Radius 90 , 10.000000 Is to the perpend . OP 206 , 2.313867 So is the tang . PO center 86.3 . 11.158636     To the base P center 2972 3.472403 Adde the two lines of 014 and 2972 together , and you have the whole line H center 2986 in parts of the Radius PO , viz. 29 inches , and 86 parts ; out of this line abate what parts you will , suppose 343 , that is , 3 inches and 43 parts , and then the remainer will be 2643. Now if you set 343 from H to I , the triangle IO center will be equiangled with the former , and I center being given , to finde LO , the proportion is ; As H center the first base 2986 , co . ar . 6.524911 Is to HO , the first perpend . 206. 2.313867 So is I center the 2d . base 2643 , 3.422097     To IO the 2d . perpend . 182 , 2.260875 Having thus found the length of IO to be one inch , and 82 parts ; make that the Radius , and then NT4 shall be a tangent line thereunto , upon which , according to this new Radius , set off the hour-distances before found , and so have you 2 pricks , by which you may draw the height of the stile OO , and the hour-lines for the Dial. The length of H center in natural tangents , is thus found , HP 069 is the tangent line of the angle HOP 3 deg . 97 min. and by the same reason P center 14421 is the tangent line of PO center 86.3 . the complement of the other , and therefore these two tangents added together do make 14490 , the length of the substile H center , that is , 14 times the Radius , and 49 parts , out of which substract what number of parts you will , the rest is the distance from the second Equinoctial to the center in natural tangents ; suppose 158 to be substracted , that is , one radius , and 58 parts , which set from H to T , in proportion to the Radius HO , and from the point T draw the line NT4 parallel to the former Equinoctial , and there will remain from T to the center 1291. Now to finde the length of LO , the proportion , by the 16 th . of the second , will be As H center 1449 , co . ar . 6.838932 Is to HO 321 , 2.506505 So is T center 1291 , 3.110926     To TO 286 , 2.456363 Now then if you set 286 from T to O in the same measure , from which you took HO , then may you draw ONO , and the tangents in the Table set upon the line NT in proportion to this new radius TO , you shall have two pricks , by which to draw the hour-lines , as before . Probl. 8. To draw the hour lines upon any direct plane , reclining or inclining East or West . HItherto we have only spoken of such planes , as are either parallel or perpendicular to the horizon , all which except the horizontal , lie in the plane of some azimuth or other . The rest that follow are reclining or inclining planes , according to the respect of the upper or nether faces of the planes , in those that recline , the base is a line in the plane , parallel to the Horizon or Meridian , and alwayes scituate in some azimuth or other : thus the base of the East and West reclining planes lie in the Meridian , or South and North azimuth , and the poles thereof in the prime vertical , but the plane it self in some circle of position ( as it is Astrologically taken ) which is a great circle of the Sphere , passing by the North or South intersections of the meridian and horizon , and falling East or West from the Zenith upon the prime vertical , as much as the poles of the plane are elevated and depressed above and under the horizon . And this kinde of plane rightly conceived and represented in the fundamental Scheme by NOS , is no other but an erect declining plane in any Countrey , where the pole is elevated the complement of ours : for if you consider the Sphere , it is apparent , that as all the azimuths , representing the decliners , do crosse the prime vertical in the Zenith , and fall at right angles upon the horizon , so do all the circles of position , representing the reclining and inclining East or West planes crosse the horizon in the North and South points of the Meridian , and fall at right angles upon the prime vertical . From which analogie it commeth to passe , that making a Diall declining 30 degr . from the Meridian , it shall be the same that a reclining 30 degr . from the Zenith , and contrary , onely changing the poles elevation into the complement thereof , because the prime vertical in this case is supposed to be the horizon , above which the pole is alwayes elevated the complement of the height thereof above the horizon . And therefore the poles elevation and the planes reclination being given , which for the one suppose to be , as before , 51 deg . 53 min. and the other , that is , the reclination 35 degrees towards the West , we must finde ( as in all decliners ) first the height of the pole above the plane , which in the fundamental diagram is PR , part of the meridian of the plane between the Pole of the world and the plane . 2. The distance thereof from the meridian of the place , which is NR part of the plane betwixt the substile and the meridian . 3. The angle betwixt the two meridians NPR , by which you may calculate the hour distances , as in the decliners . First , therefore in the supposed triangle NPR ( because you know not yet where R shall fall ) you have the right angle at R the side opposite PN 51 degr . 53 min. and the angle at N , whose measure is the reclination ZO 35 degr . to finde the side PR , the height of the stile , or poles elevation above the plane , wherefore , by the eighth case of right angled spherical triangles , the analogie is As the Radius 90 , 10.000000 Is to the sine of PN 51.53 . 9.893725 So is the sine PNR 35. 9.758591     To the sine of the side PR 26.69 . 9.652316 Secondly , you may finde the side NR , which is the distance of the substile from the meridian , by the seventh case of right angled spherical triangles ; for As the Radius 90 , 10.000000 Is to the ●●sine of PNR 35. 9.913364 So is the tangent of PN 51.53 . 10.099861     To the tangent of NR 45.87 . 10.013225 Thirdly , the angle at P between the two meridians m●● be found by the ninth case of right angle● spherical triangles . As the Radius ●● 10.000000 Is to the co-sine ●●N , 51.53 . 9.793864 So is the tangent ●f ●NR 35. 9.845227     To the co-tangent of RPN 66.46 . 9.639091 The angle at P being 66 deg . 46 min. the perpendicular PR must needs fall somewhat neer the middle between 7 and 8 of the clock ; if then you deduct the Equinoctial distance of 8 , which is 60 , from 66 deg . 46 min. the Equinoctial distance of 8 of the clock from the substile will be 6 deg . 46 min. again , if you deduct 66 degr . 47 min. from 75 deg . the distance of 7 from the Meridian , the Equinoctiall distance of 7 from the substile will be 08. deg . 53 min. the rest are found by the continual addition of 15 deg . for an hour : thus , 15 degr . and 6 degr . 47 min. do make 21 deg . 47 min. for 9 of the clock ; and so of the rest . And now the hour distances upon the plane may be found by the first case of right angled spherical triangles : for As the Radius 90 10.000000 Is to the sine of PR 26.69 . 9.652404 So is the tangent of RP . 8 , 6.46 . 9.053956     To the tangent of R 8 , 2.91 . 8.706360 These 2 deg . 91 min. are the true distance of 8 of the clock from the substile . And there is no other difference at all in calculating the rest of the hours , but increasing the angle at ● , acccording to each hours Equinoctial distance from the substile . The Geometrical Projection . Having calculated the hour distances , you shall thus make the Diall ; let AD be the base or horizontal line of the plane parallel to NZS , the meridian line of the Scheme . And ADEF the plane reclining 35 degr . from the Zenith , as doth SON of the Scheme ▪ through any part of the plane , but most convenient for the houres , draw a line parallel to the base AD , which shall be GO 12 , the 12 of the clock hour representing NZS of the Scheme ; because the base AD is parallel to the meridian , take 60 degrees from a Chord , and make G the center , and draw the circle PRO , representing the circle of position NOS in the Scheme in which this plane lieth ; from the point O to R Westerly in the East reclining , & Easterly in the West reclining , set off the distance of the substile and meridian formerly found to be 45 degrees , 87 min. and draw the prickt line GR for the substile , agreeable to PR in the Scheme , GO in the Diall representing the arch PN , and OR in the plane the arch NR in the Scheme . From the point R of the substile both wayes set off the hour distances , by help of the Chord , for 8 of the clock 2 degr . 91 min. and so of the rest ; and draw streight lines from the center G through those points , which shall be the true hour lines desired . Last of all , the height of the stile PR 26 degr . 69 min. being set from R to P , draw the streight line GP for the axis of the stile , which must give the shadow on the diall , Erect GP at the angle RGP perpendicularly over the substile line GR , and let the point P be directed to the North pole , GO12 placed in the Meridian , the center G representing the South , and the plane at EF elevated above the horizon 55 degrees ; so have you finished this diall for use , onely remember , because the Sun riseth but a little before 4 , and setteth a little after 8 , to leave out the hours of 3 and 9 , and put on all the rest . And thus you have the projection of four Dials in one ; for that which is the West recliner is also the East incliner 〈◊〉 you take the complements of the recliners ●ours unto 12 , and that but from 3 in the afternoon till 8 at night : again , if you draw the same lines on the other side of your ●●per , and change the houres of 8 , 7 , 6 , &c. into 4 , 5 , 6 , &c. you have the East recliner , and the complement of the East recliners hours from 3 to 8 is the West incliner : onely remember , that as the stile in the West recliner beholds the North , and the plane the Zenith ; so in the East incliner , the stile must behold the South , and the plane the Nadir . Probl. 9. To draw the hour-lines upon any direct South reclining or inclining plane . AS the base of East and West reclining or inclining planes do alwayes lie in the meridian of the place , or parallel thereunto , and the poles in the prime vertical ; so doth the base of South and North reclining or inclining planes lie in the prime vertical or azimuth of East and West , and their poles consequently in the Meridian . Now if you suppose the circle of position , ( which Astrologically taken is fixed in the intersection of the meridian and horizon ) to move about upon the horizon , till it comes into the plane of the prime vertical , and being fixed in the intersection thereof with the horizon , to be let fall either way from the Zenith upon the meridian , it shall truly represent all the South and North reclining and inclining planes also , of which there are six varieties three of South and three of North reclining ; for either the South plane doth recline just to the pole , and then it becommeth an Equinoctial , because the poles of this plane do then lie in the Equinoctiall ; some call it a polar plane , or else it reclineth more and less then the pole , and consequently the poles of the plane above and under the Equinoctiall , somewhat differing from the former . In like manner , the North plane reclineth just to the Equinoctial , and then becometh a polar plane , because the poles of that plane lie in the poles of the world ; some term it an Equinoctiall plane . Or else it reclineth more or lesse then the Equinoctial , and consequently the poles of the plane above and under the poles of the world , somewhat differing from the former . Of the Equinoctiall plane . And because this Diall is no other but the very horizontall of a right Sphere , where the Equinoctial is Zenith , and the Poles of the world in the Horizon ; therefore it is not capable of the six of clock hour ( no more then the East and West are of the 12 a clock hour ) which vanish upon the planes , unto which they are parallel : and the twelve a clock hour is the middle line of this Diall ( because the Meridian cutteth the plane of six a clock at right angles ) which the Sun attaineth not , till he be perpendicular to the plain . And this in my opinion , besides the respect of the poles , is reason enough to call it an Equinoctiall Diall , seeing it is the Diall proper to them that live under the Equinoctiall . This Diall is to be made in all respects as the East and West were , being indeed the very same with them , onely changing the numbers of the hours : for seeing the six of clock hour in which this plane lieth crosseth the twelve of clock hour at right angles , in which the East and West plane lieth , the rest of the hour-lines will have equall respect unto them both : so that the fifth hour from six of the clock is equal to the fift hour from twelve ; the four to the four ; and so of the ●est . These analogies holding , the hour distances from six are to be set off by the natural tangents in these Dials , as they were from twelve in the East and West Dials . The Geometricall Projection . Draw the tangent line DSK , parallel to the line EZW in the Scheme , crosse it at right angles with MSA the Meridian line , make SA the Radius to that tangent line , on which prick down the hours ; and that there may be as many hours upon the plane as it is capable of , you must proportion the stile to the plane ( as in the fifth Problem ) after this manner : let the length of the plane from A be given in known parts , then because the extream hours upon this plane are 5 or 7 , reckoning 15 degrees to every hour from 12 , the arch of the Equator will be 75 degrees : and therefore in the right angled plain triangle SA ♎ , we have given the base A ♎ , the length of the plane from A , and the angle AS ♎ 75 degrees , to finde the perpendicular SA ; for which , as in the fifth Chapter , I say ; As the Radius 90 , 10.000000 Is to the base A ♎ 3.50 . 2.544068 So is the tangent of A ♎ S 15 9,428052     To the perpendicular AS 94 1.972120 At which height a stile being erected over the 12 a clock hour line , and the hours from 12 drawn parallel thereunto through the points made in the tangent line , by setting off the natural tangents thereon , and then the Diall is finished . Let SA 12 be placed in the meridian , and the whole plane at S raised to the height of the pole 51 degr . 53 min. then will the stile shew the hours truly , and the Diall stand in its due position . 2. Of South reclining lesse then the pole . This plane is represented by the prickt circle in the fundamental Diagram ECW , and is intersected by the hour circles from the pole P , as by the Scheme appeareth , and therefore the Diall proper to this plane must have a center , above which the South pole is elevated ; and therefore the stile must look downwards , as in South direct planes ; to calculate which Dials there must be given the Poles elevation , and the quantity of reclination , by which to finde the hour distances from the meridian , and thus in the triangle PC 1 , having the poles elevation 51 degr . 53 min. and the reclination 25 degr . PC is given , by substracting 25 degr . from PZ 38 degr . 47 min. the complement of the poles height , the angle CP 1 is 15 degrees , one hours distance , and the angle at C right , we may finde C 1 , by the first case of right angled spherical triangles : for , As the Radius 90 , 10.000000 Is to the sine of PC 13.47 . 9.367237 So is the tangent of CP 1. 15. 9.428052 To the tangent of C 1 3.57 . 8.795289 And this being all the varieties , save onely increasing the angle at P , I need not reiterate the work . 3. Of South reclining more then the pole . This plane in the fundamental Scheme is represented by the prickt circle EAW , of which in the same latitude let the reclination be 55 degrees , from which if you deduct PZ 38 deg . 47 min. the complement of the poles height , there will remain PA 16 deg . 53 min. the height of the north pole above the plane , and instead of the triangle PC 1 , in the former plane we have the triangle PA 1 , in which there is given as before the angle at P 15 deg . & the height of the pole PA 16 deg . 53 min. and therefore the same proportion holds : for , As the Radius 90 , 10.000000 Is to the sine of PA 16.53 . 9.454108 So is the tangent of A 15. 9.428052 To the tangent of A 1. 4.36 . 8.882160 The rest of the hours , as in the former , are thus computed , varying onely the angle at P. The Geometricall Projection . These arches being thus found , to draw the Dials true , consider the Scheme , wherein so oft as the plane falleth between Z and P , the Zenith and the North pole , the South pole is elevated ; in all the rest the North ; the substile is in them all the meridian , as in the direct North and South Dials ; in which the stile and hours are to be placed , as was for them directed : which being done let the plane reclining lesse then the pole , be raised above the horizon to an angle equal to the complement of reclination , which in our example is to 65 degr . and the axis of the plane point downwards ; and let all planes reclining more then the pole have the hour of 12 elevated above the horizon to an angle equal to the complement of the reclination also , that is in our example , to 35 deg . then shall the axis point up to the North pole , and the Diall-fitted to the plane . Probl. 10. To draw the hour-lines upon any direct North reclining or inclining plane . THe direct north reclining planes have the same variety that the South had ; for either the plane may recline from the Zenith just to the Equinoctial , and then it is a Polar plane , as I called it before , because the poles of the plane lie in the poles of the world ; or else the plane may recline more or lesse then the Equinoctial , and consequently their poles do fall above or under the poles of the world , and the houre lines do likewise differ from the former . Of the Polar plain . This place is well known to be a Circle divided into 24 equall parts , which may be done by drawing a circle with the line of Chords , and then taking the distance of 15 degrees from the same Chord , drawing streight lines from the center through those equall divisions , you have the houre-lines desired . The houre-lines being drawn , erect a streight pin of wier upon the center , of wh●● length you please , and the Diall is finished : yet seeing our Latitude is capable of no more then 16 houres and a halfe , the six houres next the South part of the Meridian , 11 , 10 , 9 , 1 , 2 , and 3 , may be left out as uselesse . Nor can the reclining face serve any longer then during the Suns aboad in the North part of the Zodiac , and the inclining face the rest of the year , because this plain is parallel to the Equinoctial , which the Sun crosseth twice in a year . These things performed to your liking , let the houre of 12 be placed upon the Meridian , and the whole plain raised to an angle equall to the complement of your Latitude , the which in this example is 38 deg . 47 min. so is this Polar plain and Diall rectified to shew the true houre of the day . 2. Of North reclining less then the Equator . The next sort is of such reclining plains as fall between the Zenith and the Equator , and in the Scheme is represented by the pricked circle EFW , supposed to recline 25 degrees from the Zenith , which being added to PZ 38 deg . 47 min. the complement of the poles elevation , the aggregate is PF , 63 deg . 47 min. the height of the Pole or stile above the plane . And therefore in the triangle PF1 , we have given PF , and the angle at P , to finde F1 , the first houres distance from the Meridian upon the plain , for which the proportion is , As the Radius , 90 , 10.000000 Is to the sine of PF , 63.47 9.951677 So is the tangent of FP1 , 15 9.428052 To the Tangent of F1 , 13.48 9.379729 In computing the other houre distances there is no other variety but increasing the angle at P as before we shewed . 3. Of North reclining more then the Equator . The last sort is of such reclining plain ; as fall between the Horizon and Equator , represented in the fundamental Scheme by the prickt circle EBW , supposed to recline 70 deg . And because the Equator cutteth the Axis of the world at right angles , all planes that are parallel thereunto have the height of their stiles full 90 deg . above the plane : and by how much any plane reclineth from the Zenith , more then the Equator , by so much less then 90 is the height of the stile proper to it , and therefore if you adde PZ 38 deg . 47 min. the height of the Equator , unto ZB 70 deg . the reclination of the plain , the totall is PB 108 deg . 47 mi. whose complemenc to 180 is the arch BS , 71 deg . 53 min. the height of the pole above the plain . To calculate the houre lines thereof , we must suppose the Meridian PFB and the houre circles P1 , P2 , P3 , &c. to be continued till they meet in the South pole , then will the proportion be the same as before . As the Radius , 90 , 10.000000 To the sine of PB , 71.53 9.977033 So is the tangent of 1PB , 15 9.428052 To the tangent of B1 , 14.27 9.405085 And so are the other houre distances to be computed , as in all the other planes . The Geometricall Projection . The projection of these planes is but little differing from those in the last Probl. for the placing the hours and erecting the stile , they are the same , and must be elevated to an angle above the horizon equall to the complement of their reclinations , which in the North reclining lesse then the Equator is in our example 65 degrees , and in this plane the houres about the meridian , that is , from 10 in the morning till 2 in the afternoon , can never receive any shadow , by reason of the planes small reclination from the Zenith , and therefore needlesse to put them on . In the North reclining more then the Equator , the plane in our example must be elevated 120 degr . above the horizon , and the stiles of both must point to the North pole . Lastly , as all other planes have two faces respecting the contrary parts of the heavens ; so these recliners have opposite sides , look downwards the Nadir , as those do towards the Zenith , and may be therefore made by the same rules ; or if you will spare that labour , and make the same Dials serve for the opposite sides , turn the centers of the incliners downwards , which were upwards in the recliners ; and those upwards in the incliners which were downwards in the recliners , and after this conversion , let the hours on the right hand of the meridian in the recliner become on the left hand in the incliner , and contrarily ; so have you done what you desired : and this is a general rule for the opposite sides of all planes . Probl. 11. To draw the hour-lines upon a declining reclining , or declining inclining plane . DEclining reclining planes have the same varieties that were in the former reclining North and South ; for either the declination may be such , that the reclining plane will fall just upon the pole , and then it is called a declining Equinoctial ; or it may fall above or under the pole , and then it is called a South declining cast and west recliner : on the other side the declination may be such , that the reclining plane shall fall just upon the intersection of the Meridian and Equator ; and then it is called a declining polar ; or it may fall above or under the said intersection , and then it is called a North declining East and West recliner . The three varieties of South recliners are represented by the three circles , AHB falling between the pole of the world and the Zenith : AGB just upon the pole ; and AEB between the pole and the horizon : and the particular pole of each plane is so much elevated above the horizon , ( upon the azimuth ) DZC , crossing the base at right angles ) as the plane it self reclines from the Zenith , noted in the Scheme , with I , K , and L. 1. Of the Equinoctiall declining and reclining plane . This plane represented by the circle AGB , hath his base AZB declining 30 degrees from the East and West line EZW equal to the declination of the South pole thereof 30 degrees from S the South part of the Meridian Easterly unto D , reclining from the Zenith upon the azimuth CZD the quantity ZG 34 degrees , 53 min. and Passeth through the pole at P. Set off the reclination ZG , from D to K , and K shall represent the pole of the reclining plane so much elevated above the horizon at D , as the circle AGB representing the plane declineth from the Zenith Z , from P the pole of the world , to K the pole of the plane , draw an arch of a great circle PK , thereby the better to informe the fancie in the rest of the work . And if any be desirous , to any declination given , to fit a plane reclining just to the pole : or any reclination being given , to finde the declination proper to it , this Diagram will satisfie them therein : for in the Triangle ZGP , we have limited , First , the hypothenusal PZ 38 degrees , 47 min. Secondly , the angle at the base PZG , the planes declination 30 degrees . Hence to finde the base GZ , by the seventh case of right angled spherical triangles , the proportion is ; As the Radius 90 , 10.000000 To the co-sine of GZP 30 ; 9.937531 So the tangent of PZ 38.47 . 9.900138 To the tangent of GZ 34.53 . 9.837669 the reclination required . If the declination be required to a reclination given , then by the 13 case of right angled spherical triangles , the proportion is As the Radius 90 , 10.000000 To the tangent of ZG 34.53 . 9.837669 So the co-tangent of PZ 38.47 . 10.099861 o the co-sine of GZP 39. 9.937530 And now to calculate the hour-lines of this Diall , you are to finde two things : first , the arch of the plane , or distance of the meridian and substile from the horizontal line , which in this Scheme is PB , the intersection of the reclining plane with the horizon , being at B. And secondly , the distance of the meridian of the place SZPN , from the meridian of the plane PK , which being had , the Diall is easily made . Wherefore in the triangle ZGP , right angled at G , you have the angle GZP given 30 degrees , the declination ; and ZP 38 degr . 47 min. the complement of the Pole ; to finde GP : and therefore , by the eighth case of right angled spherical triangles , the proportion is : As the Radius , 90 10.000000 To the sine of ZP , 38.47 9.793863 So is the sine of GZP , 30 9.698970 To the sine of GP , 18.12 9.492833 Whos 's complement 71 deg . 88 min. is the arch PB desired . The second thing to be found is the distance of the Meridian of the place , which is the houre of 12 from the substile or meridian of the plane , represented by the angle ZPG , which may be found by the 11 Case of right angled sphericall Triangles , for As the Radius , 90 10.000000 Is to the sine of GP , 18.12 9.492833 So is the co-tang . of GZ , 34.53 10.162379 To the co-tang . of GPZ , 65.68 9.655212 Whos 's complement is ZPK 24 deg . 32 min. the arch desired . Now because 24 deg . 32 min. is more then 15 deg . one houres distance from the Meridian , and lesse then 30 deg . two houres distance , I conclude that the stile shall fall between 10 and 11 of the clock on the West side of the Meridian , because the plain declineth East : if then you take 15 deg , from 24 deg . 32 min. there shall remain 9 deg . 32 min. for the Equinoctiall distance of the 11 a clock houre line from the substile , and taking 24 deg . 32 min. out of 30 deg . there shall remain 5 deg . 68 min. for the distance of the houre of 10 from the substile : the rest of the houre distances are easily found by continual addition of 15 deg . Unto these houre distances joyn the naturall tangents as in the East and West Dials , which will give you the true distāces of each houre from the substile , the plane being projected as in the 5 Pro. for the east & west dials , or as in the 8 Prob. for the Equinoctial , according to which rules you may proportion the length of the stile also , which being erected over the substile , and the Diall placed according to the declination 30 deg . easterly , and the whole plain raised to an angle of 55 deg . 47 min. the complement of the reclination , the shadow of the stile shall give the houre of the day desired . 2. To draw the houre lines upon a South reclining plain , declining East or West , which passeth between the Zenith and the Pole. In these kinde of declining reclining plains , the South pole is elevated above the plane , as is clear by the circle AHB representing the same , which falleth between the Zenith and the North pole , and therefore hideth that pole from the eye , and forceth you to seeke the elevation of the contrary pole above the plain , which notwithstanding maketh the like and equall angles upon the South side objected to it , as the North pole doth upon the North side , ( as was shewed in the 7 Prop. ) so that either you may imagine the Scheme to be turned about , and the North and South points changed , or you may calculate the houres as it standeth , remembring to turn the stile upwards or downwards , and change the numbers of the houres , as the nature of the Diall wil direct you . In this sort of declining reclining Dials , there are four things to be sought before you can calculate the houres . 1 The distance of the Meridian from the Horizon . 2 The height of the pole above the plain . 3 The distance of the substile from the Meridian . 4 The angle of inclination between the Meridian of the plane , and the meridian of the place . 1 The distance of the Meridian from the Horizon , is represented by the arch OB , to finde which , in the right angled Triangle HOZ , we have HZ the reclination 20 deg . and the angle HZO the declination , to find HO , the complement of OB , for which , by the first case of right angled sphericall triangles , the analogie is , As the Radius , 90 10.000000 o the sine of HZ , 20 9.534051 o is the tangent of HZO , 30 9.761439 o the tangent of HO , 11.17 9.295490 Whos 's complement 78 deg . 83 min. is OB , the arch desired . 2. To finde the height of the pole above the plane , there is required two operations , the first to finde OP , and the second to finde PR ; OP may be found by the 3 Case of right angled Sphericall Triangles , for , As the Radius , 90 10.000000 Is to the co-sine of HZP. 30 9.937531 So is the co-tang . of HZ , 20. 10.438934 To the co-tangent of ZO , 22.80 10.376465 Which arch being found , and deducted out of , ZP 38 deg . 47 min. there resteth PO 15 deg . 67 min. Then may you finde PR , by the triangles HZO , and PRO both together , because the sines of the hypothenusals and the sines of the perpendiculars are proportional , by the first of the 7 Chap. of Triangles . Therefore , As the sine of ZO , 22.80 9.588289 Is to the sine of ZH , 20 9.534052 So is the sine of PO , 15.67 9.431519 To the sine of PR , 13.79 9.377282 The height of the stile desired . 3 The distance of the substile from the Meridian may be found by the 12 Case of right angled sphericall triangles , for As the co-sine of PR , 13 78 9.987298 Is to the Radius , 90 10.000000 So is the co-sine of PO , 15.67 9.983551 To the co-sine of OR , 7.41 9.996253 The arch desired . 4. The angle of inclination between the Meridians , may be found by the 11 Case of right angled Spherical triangles , for , As the Radius , 90 10.000000 Is to the sine of PR , 13.79 . 9.377241 So is the co-tang . of OR 7.51 10.879985 To the co-tang . of OPR , 28.93 10.257226 Now as in all the former works , the angle P between the two Meridians being 28 deg . 93 min. which is more then one houres distance from the Meridian , and lesse then two , you may conclude that the substile must stand between the first & second hours from the Meridian or 12 of the clock Westerly , because the declination is easterly : and 28 deg . 93 min. being deducted out of 30 deg . there resteth 1 deg . 7 min. for the distance of 10 of the clock from the substile ; again , deducting 15 deg . from 28 deg . 93 min. there resteth 13 deg . 73 min. the distance of the 11 a clock houre line from the substile , the rest are found by continuall addition of 15 deg . as before . And here the true houre distances may be found by the first case of right angled Sphericall triangles , for , As the Radius , 90 10.000000 Is to the sine of PR 13.79 9.377240 So is the tangent of RP , 11.15 9.428052 To the tangent of R 11 , 3.66 8.805292 And so proceed with all the rest . 3. To draw the houre lines upon a South reclining plain , declining East or west , which passeth between the Pole and the Horizon . In this plain represented by the circle of reclination AFB , the North pole is elevated above the plane , as the South pole was above the other , and the same four things that you found for the former Diall must also be sought for this ; in the finding whereof there being no difference , save only deducting ZP from ZO , because ZO is the greatest arch , as by the Scheam appeareth : to calculate the houres of this plane needeth no further instruction . Probl. 12. To draw the houre lines upon a polar plain , declining East or west , being the first variety of North declining reclining planes AS in the South declining recliners , there are three varieties , so are there in the North as many : for either the plane reclining doth passe by the intersection of the Meridian and Equator , and then it is called a declining Polar , which hath the substile alwayes perpendicular to the Meridian ; or else it passeth above or under the intersection of the Meridian and Equator , which somewhat differeth from the former . I will therefore first shew how they lie in the Scheam , and then proceed to the particular making of the Dials proper to them . 1. Of the Polar declining reclining plane . This plane is in this diagram represented by the circle AGB , ZG is the reclination , ZAE the distance of the Equator from the Zenith , the declination NC , K the pole thereof . Here also as in the last Probl. there may be a reclination found to any declination given , and contrary , by which to fit the plane howsoever declining , to passe through the intersection of the Meridian and Equator , by the 7 and 13 Cases of right angled sphericall triangles . As the Radius , 90 10.000000 To the co-sine of GZAE , 60 9.698970 So is the tangent of ZAE , 51.53 10.099861 To the tangent of ZG , 32.18 9.798871 The reclination desired . And , As the Radius , 90 10.000000 To the tangent of GZ , 32.18 9.798831 So is the co-tangent of ZAE , 51.53 9.900138 To the co-sine of GZAE , 60 9.698969 The declination . And now to calculate the houre lines of this Dial , you must finde , first , the distance of the Meridian from the Horizon , by the 8 Case of right angled Spherical triangles . As the Radius , 90 10.000000 Is to the sine of ZAE , 51.53 9.893725 So is the sine of GZAE , 60 9.937531 To the sine of AEG , 42.69 9.831256 Whos 's complement 47 deg . 31 min. is AAE the arch desired . 2. You must finde RP , the height of the pole above the plane , by the 2 Case of right angled Sphericall Triangles , for As the Radius , 90 10.000000 Is to the sine of AEZG , 60 9.937531 So is the co-sine of ZG , 32.11 9.927565 To the co-sine of ZAEG , 42.87 9 . 865●9● Which is the height of the pole above the plane , AER being a Quadrant , PR must needs be the measure of the angle at AE . 3. Because in all decliners ( whose planes passe by the intersection of the Meridian and Equinoctiall ) the substile is perpendicular to the Meridian , therefore you need not seek AER , the distance between the substile and Meridian , which is alwayes 90 deg . and falleth upon the 6 a clock houre . 4. Lastly , the arch AER , which is the distance of the substile from the Meridian : being 90 degrees , the angle at P opposite thereunto must needs be 90 also : from whence it followes , that the houres equidistant from the six of the clock hour in Equinoctial degrees shall also have the like distance of degrees in their arches upon the plane , and so one half of the Diall being calculated , serves for the whole ; these things considered , the true hour-distances may be found , by the first case of right angled spherical triangles : for , As the Radius , 10.000000 Is to the sine of PR 42.87 . 9.832724 So is the tangent of RP 5. 15 d. 9.428052 To the tangent of R 5 ▪ 10.34 . 9.260776 The which 10 degr . 34 min. is the true distance of 5 and 7 from the substile or six of the clock hour , and so of the rest . The Geometrical projection of this plane needs no direction ; those already given are sufficient , according to which this Diall being made and rectified by the declination and reclination given , it is prepared for use . 2. To draw the hou●● lines upon a North reclining plane , declining East or West , which cutteth the meridian between the Zenith and the Equinoctial . All North reclining planes howsoever declining , have the North pole elevated above them , and therefore the center of the Diall must be so placed above the plane , that the stile may look upwards to the pole , neither can it be expected that the plane being elevated above the horizon Southward , should at all times of the year be enlightened by the Sun , except it recline so far from the Zenith , as to intersect the Meridian between the horizon and the Tropique of Capricorn ; this plane therefore reclining but 16 degrees from the Zenith , and declining 60 cannot shew many hours , when the Sun is in his greatest Northern declination , partly by reason of the height of the plane above the horizon , and partly by reason of the great declination thereof , hindring the Sun-beams from all the morning houres , which may be therefore left out as useless . In this second variety , the plane represented by the Circle AMB in the last Diagram , cutteth the Meridian at O between the Zenith and the Equator , ZM being the reclination , 16 deg . ZAE the distance of the Equator from the Zenith , 51 deg . 53 m. and the declination NC 60 as before . As in the former , so in this Diall , the same four things are again to be found before you can calculate the houre distances thereof . The first is the distance of the Meridian from the Horizon , represented in this plain by the arch A● . The second is PR , the height of the pole above the plane . The third is ●R , the distance of the substile or Meridian of the plane , from the Meridian of the place . The fourth is the angle ●PR between the two Meridians : all which , and the houre distances also , being to be found according to the directions of the last Probl. there needeth no further instruction here . 3 To draw the houre lines upon a North reclining plane , declining East or West , which cutteth the Meridian between the Equator and Horizon . The last variety of the six declining recliners , represented by the circle ALB , and cutteth the Meridian at H , between the Equator and the Horizon , ZL being the reclination , 54 deg . the declination NC , 60 deg . as before ; and hence the four things mentioned before must be sought ere you can calculate the houre distances . 1 The distance of the Meridian and Horizon , represented by AH . 2 RH the substile . or Meridian of the plane from the Meridian of the place . 3 PR , the height of the pole above the plane . 4 HPR , the angle between the two Meridians . In finding whereof the proportions are still the same , though the triangles are somewhat altered , for when you have found ZH , it is to be added to ZP to finde PH , both which together do exceed a Quadrant , therefore the sides PN must be continued to X , then is PX the complement of PH to a semicircle , and if RB be continued ●o X also , RX may be found by the 12 Case of right angled spherical triangles as before , whose complement is RH , the distance of the substile from the Meridian ; and hence the angle at P must be found in that triangle also , though the proportion be the same , there being no other variety , I think it needlesse to reiterate the work . The Geometrical Projection . There is so little difference between the South & North declining reclining planes , that the manner of making the Dials for both may be shewed at once : Let the example therefore be a Diall for a South plane declining East 30 deg . reclining 20 deg . And thus have you made four Dials at once , or at least , this Dial thus drawn may be made to serve four sorts of planes , for first , it serves for a South declining East 30 deg . reclining 20 deg . and if you prick the houre lines through the paper , and draw them on the other side stile and all , this Diall will then be fitted for a South plane declining West 30 deg . reclining 20 deg . only remember to change the houres , that is to say , instead of writing 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 1 , 2 , 3 , 4 , from A , the west side of the East declining plane , you must write , 8 , 9 , 10 , 11 , 12 , 1 , 2 , 3 , 4 , 5 , 6 , 7. Again , if you turn the Zenith of your Dial downwards , the South declining East reclining shall in all respects serve for a North declining west inclining as much ; and the South declining West reclining , will likewise serve for a North declining East inclining ; and therefore there needs no further direction either to make the one , or calculate the other . CHAP. III. Of the Art of NAVIGATION . Probl. 1. Of the 32 windes , or Seamans Compass . THe course of a ship upon the Sea dependeth upon the windes : The designation of these depends upon the certain knowledge of one principal ; which considering the situation and condition of the whole Sphere , ought in nature to be North or South , the North to us upon this side of the Line , the South to those in the other hemisphere ; for in making this observation men were to intend themselves towards one fixed part of the heavens or other , and therefore to the one of these . In the South part there is not found any star so notable , and of so neer distance from the Pole , as to make any precise or firm direction of that winde , but in the North we have that of the second magnitude in the tale of the lesser Bear , making so small and incensible a circle about the Pole , that it commeth all to one , as if it were the Pole it self . This pointed out the North winde to the Mariners of old especially , and was therefore called by some the Lead or Lead star ; but this could be only in the night , and not alwayes then . It is now more constantly and surely shewed by the Needle touched with the Magnet , which is therefore called the Load or Lead-stone , for the same reason of the leading and directing their courses to the North and South position of the earth , not in all parts directly , because in following the constitution of the great Magnet of the whole earth , it must needs be here and there led aside towards the East or West by the unequal temper of the Globe ; consisting more of water then of earth in some places , and of earth more or lesse Magnetical in others . This deviation of the Needle , the Mariners call North-easting , & North-westing , as it falleth out to be , otherwise , and more artificially , the Variation of the Compass , which though it pretend uncertainty , yet proveth to be one of the greatest helps the Seaman hath . And the North and South windes being thus assured by the motion either of direction or variation of the needle , the Mariner supposeth his ship to be ( as it alwayes is ) upon some Horizon or other , the center whereof is the place of the ship . The line of North and South found ou● by the Needle , a line crossing this at right angles , sheweth the East and West , and so they have the four Cardinall windes , crosse again each of these lines , and they have the eight whole windes , as they call them . Another division of these maketh eight more which they call halfe windes , a third makeeth 16 , which they call the quarter windes , so they are 32 in all . Every one of these Windes is otherwise termed a several point of the Compasse , and the whole line consisting of two windes , as the line of North and South , or that of East and West is called a Rumb . The Windes and Rumbs thus assigned by an equal division of a great Circle into 32 parts , the angle which each Rumb maketh with the Meridian is easily known , for if you divide a quadrant or 90 degrees in eight parts : you have the angles which the eight windes reckoned from North to East or West do make with the meridian ; and those reckoned from South to the East or West are the same , and for your better direction are here exhibited in the Table following . A Table for the angles which every Rumb● maketh with the Meridian . North South D. part South North     02.8125         05.6250         08.4375     N by E S by E 11.2500 S by W N by W     14.0625         16.8750         19.6875     NNE SSE 22.5000 SSW NNW     25.3125         28.1250         30.9375     NE by N SE by S 33.7500 SW by S NW b N     36.5625         39.3750         42.1875     NE SE 45.0000 SW NW     47.8125         50.6250         53.4375     NE by E SE by S 56.2500 SW b W NW b W     59.0625         61.8650         64.6875     ENE ESE 67.5000 WSW WNW     70.3125         73.1250         75.9375     E by N E by S 78.7500 W by S W by N     81.5625         84.3750         87.1875     East East 90.0000 West West Probl. 2. Of the description and making of the Sea-chart . THe Sea-mans Chart is a Parallelogram , divided into little rectangled figures , and in the plain Chart are equal Squares , representing the Longitudes and Latitudes of such places , as may be set in the Chart , but the body of the earth being of a Globular form , the degrees of Longitude reckoned in the Equator from the Meridian , are in no place equal to those of the Latitude reckoned in the Meridian from the Equator , save onely in the Equinoctial ; for the degrees of latitude are all equall throughout the whole Globe , and as large as those of the Equinoctial ; but the degrees of Longitude at every parallel of Latitude lessen themselves in such proportion as that parallel is lesse then the Equinoctial : This dis-proportion of longitude and latitude caused for a long time much errour in the practise of Navigation , till at last it was in part reconciled by Mercator , that famous Geographer : and afterwards exactly rectified by our worthy Countreyman Master Edward Wright , in his Book entituled , The Correction of Errours in Navigation : In which he hath demonstrated by what proportion the degrees of Longitude must either increase or decrease in any Latitude , his words are as followeth . Suppose , saith he , a spherical Superficies , with Meridians , Parallels , Rumbes , and the whole Hydrographial description drawn thereupon , to be inscribed into a concave Cylinder , their axes agreeing in one . Let this Spherical superficiees swell like a bladder ( whiles it is in blowing ) equally alwayes in every part thereof ( that is as much in Longitude as in Latitude , til it apply and joyn it self ( round about and all along also towards either pole ) unto the concave superficies of the Cylinder : each parallel upon this spherical superficies increasing successively from the Equinoctial towards either pole , until it come to be of equal diameter with the Cylinder , and consequently the Meridians , stil inclining themselves , till they come to be so far distant every where each from other , as they are at the Equinoctial . Thus it may most easily be understood , how a spherical superficies may by extension be made a Cylindrical , and consequently a plain parallelogram superficies ; because the superficies of a cylinder is nothing else but a plain parallelogram wound about two equal equidistant circles , that have one common axletree perpendicular upon the centers of them both , and the peripheries of each of them equall to the length of the parallelogram , as the distance betwixt those circles , or height of the cylinder is equall to the breadth thereof . So as the Nautical planisphere may be defined to be nothing else but a parallelogram made of the Spherical superficies of an Hydrographical Globe inscribed into a concave cylinder , both their axes concurring in one , and the sphericall superficies swelling in every part equally in longitude and latitude , till every one of the parallels thereupon be inscribed into the cylinder ( each parallel growing as great as the Equinoctial , or till the whole spherical superficies touch and apply it self every where to the concavity of the cylinder . In this Nautical planisphere thus conceived to be made , all places must needs be situate in the same longitudes , latitudes , and directions or courses , and upon the same meridians , parallels and rumbes , that they were in the Globe , because that at every point between the Equinoctial and the Pole we understand the spherical superficies , whereof this planisphere is conceived to be made , to swell equally as much in longitude as in latitude ( till it joyn it self unto the concavity of the cylinder ; so as hereby no part thereof is any way distorted or displaced out of his true and natural situation upon his meridian , parallel or rumbe , but onely dilated and enlarged , the meridians also , parallels , and rumbes , dilating and enlarging themselves likewise at every point of latitude in the same proportion . Now then let us diligently consider of the Geometrical lineaments , that is , the meridians , rumbes , and parallels of this imaginary Nautical planisphere , that we may in like manner expresse the same in the Mariners Chart : for so undoubtedly we shall have therein a true Hydrographical description of all places in their longitudes , latitudes , and directions , or respective situations each from other , according to the points of the compasse in all things correspondent to the Globe , without either sensible or explicable errour . First , therefore in this planisphere , because the parallels are every where equal each to other ( for every one of them is equal to the Equinoctiall or circumference of the circumscribing cylinder ) the meridians also must needs be parallel & streight lines ; and consequently the rumbes , ( making equall angles with every meridian ) must likewise be streight lines . Secondly , because the spherical superficies whereof this planisphere is conceived to be made , swelleth in every part thereof equally , that is as much in Latitude as in Longitude , till it apply it self round about to the concavity of the cylinder : therefore at every point of Latitude in this planisphere , a part of the Meridian keepeth the same proportion to the like part of the parallel that the like parts of the Meridian , and parallel have each to other in the Globe , without any explicable errour . And because like parts of wholes keep the same proportion that their wholes have therefore the like parts of any parallel and Meridian of the Globe , have the same proportion , that the same parallel and meridian have . For example sake , as the meridian is double to the parallel of 60 degrees : so a degree of the meridian is double to a degree of that parallel , or a minute to a minute , and what proportion the parallel hath to the meridian , the same proportion have their diameters and semidiameters each to other . But the sine of the complement of the parallels latitude , or distance from the Equinoctial , is the semidiameter of the parallel . As here you see AE , the sine of AH , the complement of AF , the latitude or distance of the parallel ABCD from the Equinoctial , is the semidiameter of the same parallel . And as the semidiameter of the meridian or whole sine , is to the semidiameter of the parallel ; so is the secant or hypothenusa of the parallels latitude , or of the parallels distance from the Equinoctial , to the semidiameter of the meridian or whole sine ; as FK , ( that is AK ) to AE ( that is GK ) so is LK , to KF . Therefore in this nautical planisphere , the Semidiameter of each parallel being equal to the semidiameter of the Equinoctial , that is , to the whole sine ; the parts of the Meridian at every point of Latitude must needs increase with the same proportion wherewith the secants of the ark , conteined between those points of Latitude and the Equinoctial do increase . Now then we have an easie way laid open for the making of a Table ( by help of the natural Canon of Triangles ) whereby the meridians of the Mariners Chart may most easily and truly be divided into parts , in due proportion , and from the Equinoctial towards either Pole. For ( supposing each distance of each point of latitude , or of each parallel from other , to contein so many parts as the secant of the latitude of each point or parallel conteineth ) by perpetual addition of the secants answerable to the latitudes of each point or parallel unto the summe compounded of all the former secants , beginning with the secant of the first parallels latitude , and thereto adding the secant of the second parallels Latitude , and to the summe of both these adjoyning the secant of the third parallels Latitude ; and so forth in all the rest we may make a Table which shall truly shew the sections and points of latitude in the Meridians of the Nautical Planisphere , by which sections the parallels must be drawn . As in the Table of meridional parts placed at the end of this Discourse , we made the distance of each parallel from other , to be one minute or centesm of a degree : and we supposed the space between any two parallels , next to each other in the Planispere , to contain so many parts as the secant answerable to the distance of the furthest of those two parallels from the Equinoctial ; and so by perpetual addition of the secants of each minute or centesm to the sum compounded of all the former secants , is made the whole Table . As for example , the secant of one centesm in Master Briggs 's Trigonometrica Britannica is 100000.00152 , which also sheweth the section of one minute or centesm of the meridian from the Equinoctial in the Nautical Planisphere ; whereunto adde the secants of two minutes or centesmes , that is 100000. 00609 , the sum is 200000.00761 . which sheweth the section of the second minute of the meridian from the Equinoctial in the planisphere : to this sum adde the secant of three minutes , which is 100000.01371 , the sum will be 3000●0 . 02132 , which sheweth the section of the third minute of the meridian from the Equinoctial , and so ●orth in all the rest ; but after the Table was thus finished , it being too large for so small a Volume , we have contented our selves with every tenth number , and have also cut off eight places towards the right hand , so that in this Table the section of 10 minutes is 100 , of one degree 1000 , and this is sufficient for the making either of the generall or any particular Chart. I call that a general Chart , whose line AE in the following figure represents the Equinoctial , ( as here it doth the parallel of 50 degrees ) and so containeth all the parallels successively from the Equinoctial towards either Pole , but they can never be extended very near the Pole , because the distances of the parallels increase as much as secants do . But notwithstanding this , it may be remed general , because a more general Chart cannot be contrived in plano , except a true projection of the Sphere it self . And I call that a particular Chart which is made properly for one particular Navigation ; as if a man were to sail betwen the Latitude of 50 and 55 degrees , and his difference of Longitude were not to exceed six degrees , then a Chart made , as this figure is for such a Voyage , may be called particular , and is thus to be projected . Probl. 3. The Latitudes of two places being known , to finde the Meridional difference of the same Latitudes . IN this Proposition there are three varieties : First , when one of the places is under the Equinoctial , and the other without ; and in this case the degrees and minutes in the Table answering to the latitude of that other place are the meridional difference of those Latitudes . So if one place propounded were the entrance of the River of the Amazones , which hath no latitude at all , and the other the Lizard , whose latitude is 50 degrees , their difference will be found 57.905 . 2. When both the places have Northerly or Southerly Latitude , in this case if you substract the degrees and minutes in the Table answering to the lesser Latitude , out of those in the same Table answering to the greater Latitude , the remainer will be the Meridional difference required . Example . Admit the Latitude of S. Christophers to be 15 deg . 50 parts or minutes , and the Latitude of the Lizard to be 50 degrees . In the Table of Latitudes , the number answering to 15 deg . 50 min. is 15.692 50 deg . is 57.905 Their difference 42.213 3. When one of the places have Southerly and the other Northerly Latitude ; in this case , the sum of the numbers answering to their Latitudes in the Table , is the meridional difference you look for . So Caput bonae spei , whose latitude is about 36 deg . 50 parts , and Japan in the East Indies , whose latitude is about 30 degrees being propounded , their meridional difference will be found to be 70.724 . For the meridional parts of 36.50 . 39.252 And the meridional parts of 30 d. 31.472 Their sum is the difference required . 70.724 Probl. 4. Two places differing onely in Latitude , to finde their distance . IN this proposition there are two varieties . 1. If the two places propounded lie under the same meridian , and both of them on one side of the Equinoctial , you must substract the lesser latitude from the greater , and the remainer converted into leagues , by allowing 20 leagues to a degree , will be the distance required . 2. If one place lie on the North , and the other on the South side of the Equinoctial ( yet both under the same meridian ) you must then adde both the latitudes together , and the sum converted into leagues , will give their distance . Probl. 5. Two places differing onely in longitude being given , to finde their distance . IN this proposition there are also two varieties . 1. If the two places propounded lie under the Equinoctial , then the difference of their Longitudes reduced into leagues ( by allowing 20 leagues to a degree ) giveth the distance of the places required . 2. But if the two places propounded differ onely in longitude , and lie not under the Equinoctial , but under some other intermediate parallel between the Eqninoctial and one of the poles : then to finde their distance , the proportion is , As the Radius , Is to the co-sine of the common latitude ; So is the sine of half the difference of longitude , To the sine of half their distance . Probl. 6. Two places being given , which differ both in Longitude and Latitude , to finde their distance . IN this Proposition there are three varieties . 1. If one place be under the Equinoctial circle , and the other towards either pole , then the proportion is , As Radius , To the cosine of the difference of longitude ; So is the co-sine of the latitude given , To the co-sine of the distance required . 2. If both the places propounded be without the Equinoctial , and on the Northern or Southern side thereof , then the proportion must be wrought at two operations . 1. Say ; As the Radius , To the cosine of the difference of Longitude So the co-tangent of the lesser latitude , To the tangent of the fourth ark . Which fourth ark substract out of the complement of the greater latitude , and retaining the remaining ark say , As the co-sine of the ark found , Is to the co-sine of the ark remaining ; So is the sine of the lesser latitude , To the co-sine of the distance required . 3. If the two places propounded differ both in Longitude and Latitude , and be both of them without the Equinoctial , and one of them towards the North pole , and the other towards the South pole , the proportion is , As the Radius , Is to the co-sine of the difference of Longit. So is the co-tangent of one of the Latitudes To the tangent of another ark . Which being substracted out of the other Latitude , and 90 degrees added thereto , say : As the co-sine of the ark found , Is to the co-sine of the ark remaining ; So is the co-sine of the Latitude first taken , To the co-sine of the distance . Probl. 7. The Rumbe and distance of two places given , to finde the difference of Latitude . THe proportion is : As the Radius , Is to the co-sine of the rumb from the meridian : So is the distance , To the difference of Latitude . Example . If a ship sail West-north-west , ( that is , upon the sixt rumb from the meridian ) the distance of 90 leagues ; what shall be the difference of Latitude ? First , I seek in the Table of Angles which every Rumb maketh with the Meridian , for the quantity of the angle of the sixt rumb , which is 67 degr . 50 parts , the complement whereof is 22 degr . 50 parts : therefore , As the Radius , 10.000000 Is to the sine of 22.50 . 9.582839 So is the distance in leagues 90 , 1.954242     To the difference of Latitude 34 , and better 1.537081 And by looking the next neerest Logarithm , the difference of latitude will be 34 leagues , and 44 hundred parts of a league . And because 5 centesmes of a degree answereth to one league , therefore if you multiply 3444 by 5 , the product will be 17220 , from which cutting off the four last figures , the difference of latitude will be one degree 72 centesmes of a degree , and somewhat more . Probl. 8. The Rumb and Latitude of two places being given , to finde the difference of Longitude . THe proportion is : As the Radius , Is to the tangent of the rumb from the meridian : So is the proper difference of latitude , To the difference of Longitude . Example . If a ship sail West-north-west ( that is , upon the sixt Rumb from the meridian ) so far , that from the latitude of 51 degrees , 53 centesmes , it cometh to the latitude of 49 degrees , 82 centesmes ; what difference of Longitude hath such a course made ? First , I seek in the Table of Meridional parts what degrees do there answer to each latitude , and to 51 degrees , 53 min. I finde 60. 328 , and to 49 degrees , 82 minutes 57. 629 , which being substracted from 60. 328 their difference is 2. 699 , the proper difference of latitude . Therefore , As the Radius , 10.000000 To the tangent of 67.50 . 10.382775 So is 2.699 . 0.431203     To 6 the difference of Longitude , 0.813978 Or in minuter parts 6. 515 , that is 6 degr . 52 centesmes of a degree fere , which was the thing required . Here followeth the Table of Meridional parts , mentioned in some of the preceeding Problemes , together with other Tables usefull in the Arts of Dialling and Navigation . A Table of Meridional parts . M. Gr. par 0.00 0.000 0.10 0.100 0.20 0.200 0.30 0.300 0.40 0.400 0.50 0.500 0.60 0.600 0.70 0.700 0.80 0.800 0.90 0.900 1.00 1.000 1.10 1.100 1.20 1.200 1.30 1.300 1.40 1.400 1.50 1.500 1.60 1.600 1.70 1.700 1.80 1.800 1.90 1.900 2.00 2.000 2.10 2.100 2.20 2.200 2.30 2.300 2.40 2.400 2.50 2.500 2.61 2.600 2.71 2.700 2.81 2.800 2.91 2.900 3.01 3.000 3.00 3.001 3.10 3.101 3.20 3.201 3.30 3.301 3.40 3.402 3.50 3.502 3.60 3.602 3.70 3.702 3.80 3.803 3.90 3.903 4.00 4.003 4.10 4.103 4.20 4.204 4.30 4.304 4.40 4.404 4.50 4.504 4.60 4.605 4.70 4.705 4.80 4.805 4.90 4.906 5.00 5.006 5.10 5.106 5.20 5.207 5.30 5.307 5.40 5.408 5.50 5.508 5.60 5.609 5.70 5.709 5.80 5.810 5.90 5.910 6.00 6.011 6.10 6.111 6.20 6.212 6.30 6.312 6.40 6.413 6.50 6.514 6.60 6.614 6.70 6.715 6.80 6.816 6.90 6.916 7.00 7.017 7.10 7.118 7.20 7.219 7.30 7.319 7.40 7.420 7.50 7.521 7.60 7.622 7.70 7.723 7.80 7.824 7.90 7.925 8.00 8.026 8.10 8.127 8.20 8.228 8.30 8.329 8.40 8.430 8.50 8.531 8.60 8.632 8.70 8.733 8.80 8.834 8.90 8.936 9.00 9.037 9.10 9.138 9.20 9.239 9.30 9.341 9.40 9.442 9.50 9.543 9.60 9.645 9.70 9.746 9.80 9.848 9.90 9.949 10.00 10.051 10.10 10.152 10.20 10.254 10.30 10.355 10.40 10.457 10.50 10.559 10.60 10.661 10.70 10.762 10.80 10.864 10.90 10.966 11.00 11.068 11.10 11.170 11.20 11.272 11.30 11.374 11.40 11.476 11.50 11.578 11.60 11.680 11.70 11.782 11.80 11.884 11.90 11.986 12.00 12.088 12.10 12.190 12.20 12.293 12.30 12.395 12.40 12.497 12.50 12.600 12.60 12.702 12.70 12.805 12.80 12.907 12.90 13.010 13.00 13.112 13.10 13.215 13.20 13.318 13.30 13.422 13.40 13.523 13.50 13.626 13.60 13.729 13.70 13.832 13.80 13.935 13.90 14.038 14.00 14.141 14.10 14.244 14.20 14.347 14.30 14.450 14.40 14.553 14.50 14.656 14.60 14.760 14.70 14.863 14.80 14.967 14.90 15.070 15.00 15.174 15.10 15.277 15.20 15.381 15.30 15.485 15.40 15.588 15.50 15.692 15.60 15.796 15.70 15.900 15.80 16. ●04 15.90 16.107 16.00 16.211 16.10 16.316 16.20 16 . 42● 16.30 16.524 16.40 16.628 16.50 16.732 16.60 16.836 16.70 16.941 16.80 17.045 16.90 17.150 17.00 17.255 17.10 17.359 17.20 17.464 17.30 17.568 17.40 17.673 17.50 17.778 17.60 17.883 17.70 17.988 17.80 18.093 17.90 18.198 18.00 18.303 18.10 18.408 18.20 18.513 18.30 18.619 18.40 18.724 18.50 18.830 18.60 18.935 18.70 19.041 18.80 19.146 18.90 19.251 19.00 19.356 19.10 19.463 19.20 19.569 19.30 19.675 19.40 19.781 19.50 19.887 19.60 19.993 19.70 20.100 19.80 20.206 19.90 20.312 20.00 20.419 20.10 20.525 20.20 20.632 20.30 20.738 20.40 20.845 20.50 20.952 20.60 21.059 20.70 21.165 20.80 21.272 20.90 21.379 21.00 21.486 21.10 21.593 21.20 21.701 21.30 21.808 21.40 21.915 21.50 21.023 21.60 22.130 21.70 22.238 21.80 22.345 21.90 22.453 22.00 22.561 22.10 22.669 22.20 22.777 22.30 22.885 22.40 22.993 22.50 23.101 22.60 23.210 22.70 23.318 22.80 23.427 22.90 23.535 23.00 23.643 23.10 23.752 23.20 23.861 23.30 23.970 23.40 24.079 23.50 24.188 23.60 24.297 23.70 24.406 23.80 24.515 23.90 24.624 24.00 24.734 24.10 24.844 24.20 24.953 24.30 25.063 24.40 25.173 24.50 25.282 24.60 25.392 24.70 25.502 24.80 25.613 24.90 25.723 25.00 25.833 25.10 25.943 25.20 26.054 25.30 26.164 25.40 26.275 25.50 26.386 25.60 26.497 25.70 26.608 25.80 26.719 25.90 26.830 26.00 26.941 26.10 27.052 26.20 27.164 26.30 27.275 26.40 27.387 26.50 27.499 26.60 27.610 26.70 27.722 26.80 27.834 26.90 27.946 27.00 28.058 27.20 28.283 27.30 28.396 27.40 28.508 27.50 28.621 27.60 28.734 27.70 28.847 27.80 28.959 27.90 29.072 28.00 29.186 28.10 29.299 28.20 29.413 28.30 29.526 28.40 29.640 28.50 29.753 28.60 29.867 28.70 29.981 28.80 30.095 28.90 30.300 29.00 30.324 29.10 30.438 29.20 30.553 29.30 30.667 29.40 30.782 29.50 30.897 29.60 31.012 29.70 31.127 29.80 31.242 29.90 31.357 30.00 31.473 30.10 31.588 30.20 31.704 30.30 31.820 30.40 31.936 30.50 32.052 30.60 32.168 30.70 32.284 30.80 32.409 30.90 32.517 31.00 32.633 31.10 32.750 31.20 32.867 31.30 32.984 31.40 33.101 31.50 33.218 31.60 33.336 31.70 33.453 31.80 33.571 31.90 33.688 32.00 33.806 32.10 33.924 32.20 34.042 32.30 34.161 32.40 34.279 32.50 34.397 32.60 34.516 32.70 34.635 32.80 34.754 32.90 34.873 33.00 34.992 33.10 35.111 33.20 35.231 33.30 35.350 33.40 35.470 33.50 33.590 33.60 35.710 33.70 35.830 33.80 35.950 33.90 36.071 34.00 36.191 34.10 36.312 34.20 36.433 34.30 36.554 34.40 36.675 34.50 36.796 34.60 36.917 34.70 37.039 34.80 37.161 34.90 37.283 35.00 37.405 35.10 37.527 35.20 37.649 35.30 37.771 35.40 37.894 35.50 38.017 35.60 38.140 35.70 38.263 35.80 38.386 35.90 38.509 36.00 38.633 36.10 38.757 36.20 38.880 36.30 39.004 36.40 39.129 36.50 39.253 36.60 39.377 36.70 39.502 36.80 39.627 36.90 39.752 37.00 39.877 37.10 40.002 37.20 40.128 37.30 40.253 37.40 40.379 37.50 40.505 37.60 40 . 63● 37.70 40.757 37.80 40.884 37.90 41.011 38.00 41.137 38.10 41.264 38.20 41.392 38.30 41.519 38.40 41.646 38.50 41.774 38.60 41.902 38.70 42.030 38.80 42.158 38.90 42.287 39.00 42.415 39.10 42.544 39.20 42.673 39.30 42.802 39.40 42.931 39.50 43.061 39.60 43.191 39.70 43.320 39.80 43.451 39.90 43.581 40.00 43.711 40.10 43.842 40.20 43.973 40.30 44.104 40.40 44.235 40.50 44.366 40.60 44.498 40.70 44.630 40.80 44.762 40.90 44.894 41.00 45.026 41.10 45.159 41.20 45.292 41.30 45.425 41.40 45.558 41.50 45.691 41.60 45.825 41.70 45.959 41.80 46.093 41.90 46.227 42.00 46.362 42.10 46.496 42.20 46.631 42.30 46.766 42.40 46.902 42.50 47.037 42.60 47.173 42.70 47.309 42.80 47.445 42.90 47.581 43.00 47.718 43.10 47.855 43.20 47.992 43.30 48.129 43.40 48.267 43.50 48.404 43.60 48.542 43.70 48.681 43.80 48.819 43.90 48.958 44.00 49.097 44.10 49.236 44.20 49.375 44.30 49.515 44.40 49.655 44.50 49.795 44.60 49.935 44.70 50.076 44.80 50.217 44.90 50.358 45.00 50.499 45.10 50 . 64● 45.20 50.783 45.30 50.925 45.40 51.068 45.50 51.210 45.60 51.353 45.70 51.496 45.80 51.639 45.90 51.783 46.00 51.927 46.10 52.071 46.20 52.215 46.30 52.360 46.40 52.505 46.50 52.650 46.60 52.795 46.70 52.941 46.80 53.087 46.90 53.233 47.00 53.380 47.10 53.526 47.20 53.673 47.30 53.821 47.40 53.968 47.50 54.116 47.60 54.264 47.70 54.413 47.80 54.562 47.90 54.711 48.00 54.860 48.10 55.010 48.20 55.160 48.30 55.310 48.40 55.460 48.50 55.611 48.60 55.762 48.70 55.913 48.80 56.065 48.90 56.117 49.00 56.369 49.10 56.522 49.20 56.675 49.30 56.828 49.40 56.981 49.50 57.135 49.60 57.289 49.70 57.444 49.80 57.598 49.90 57.754 50.00 57.909 50.10 58.065 50.20 58.221 50.30 58.377 50.40 58.534 50.50 58.691 50.60 58.848 50.70 59.006 50.80 59.164 50.90 59.322 51.00 59.481 51.10 59.640 51.20 59.800 51.30 59.960 51.40 60.120 51.50 60.280 51.60 60.441 51.70 60.601 51.80 60.763 51.90 60.925 52.00 61.088 52.10 61.250 52.20 61.413 52.30 61.577 52.40 61.740 52.50 61.904 52.60 62.069 52.70 62.234 52.80 62.399 52.90 62.564 53.00 62.730 53.10 62.897 53.20 63.063 53.30 63.231 53.40 63.398 53.50 63.566 53.60 63.734 53.70 63.903 53.80 64.072 53.90 64.242 54.00 64.412 54.10 64.582 54.20 64.753 54.30 64.924 54.40 65.096 54.50 65.268 54.60 65.440 54.70 65.613 54.80 65.786 54.90 65.960 55.00 66.134 55.10 66.308 55.20 66.483 55.30 66.659 55.40 66.835 55.50 67.011 55.60 67.188 55.70 67.365 55.80 67.543 55.90 67.721 56.00 67.900 56.10 68.079 56.20 68 . 25● 56.30 68.438 56.40 68.618 56.50 68.799 56.60 68.981 56.70 69.163 56.80 69.345 56.90 69.528 57.00 69.711 57.10 69.895 57.20 70.080 57.30 70.263 57.40 70.449 57.50 70.635 57.60 70.821 57.70 71.008 57.80 71.195 57.90 71.383 58.00 71.572 58.10 71.761 58.20 71.950 58.30 72.140 55.40 72.331 58.50 72.522 58.60 72.714 58.70 72.906 58.80 73.099 58.90 73.292 59.00 73.486 59.10 73.680 59.20 73.875 59.30 74.071 59.40 74.267 59.50 74.464 59.60 74.661 59.70 74.859 59.80 75.057 59.90 75.256 60.00 75.456 60.10 75.650 60.20 75.857 60.30 76.059 60.40 76.261 60.50 76.464 60.60 76.667 60.70 76.871 60.80 77.076 60.90 77.281 61.00 77.487 61.10 77.694 61.20 77.901 61.30 78.109 61.40 78.317 61.50 78.526 61.60 78.736 61.70 78.947 61.80 79.158 61.90 79.370 62.00 79.583 62.10 79.796 62.20 89.010 62.30 89.225 62.40 89.441 62.50 89.657 62.60 89.874 62.70 81.091 62.80 81.310 62.90 81.529 63.00 81.749 63.10 81.970 63.20 82.191 63.30 82.413 63.40 82.635 63.50 82.860 63.60 83.084 63.70 83.310 63.80 83.536 63.90 83.763 64.00 83.990 64.10 84.219 64.20 84.448 64.30 84.678 64.40 84.909 64.50 85.141 64.60 85.374 64.70 85.607 64.80 85.842 64.90 86.077 65.00 86.313 65.10 86.550 65.20 86.788 65.30 87.027 65.40 87.267 65.50 87.508 65.60 87.749 65.70 87.992 65.80 88.235 65.90 88.480 66.00 88.725 66.10 88.971 66.20 89.219 66.30 89.467 66.40 89.716 66.50 89.967 66.60 90.218 66.70 90.470 66.80 90.723 66.90 90.978 67.00 91.232 67.10 91.489 67.20 91.746 67.30 92.005 67.40 92.264 67.50 92.525 67.60 92.787 67.70 93.050 67.80 93.314 67.90 93.579 68.00 93.846 68.10 94.113 68.20 94.382 68.30 94.652 68.40 94.923 68.50 95.195 68.60 95.468 68.70 95.743 68.80 96.019 68.90 96.296 69.00 96.575 69.10 96.854 69.20 97.135 69.30 97.418 69.40 97.701 69.50 97.986 69.60 98.272 69.70 98.560 69.80 98.849 69.90 99.139 70.00 99.431 70.10 99.724 70.20 100.018 70.30 100.314 70.40 100.612 70.50 100.910 70.60 101.211 70.70 101.513 70.80 101.816 70.90 102.121 71.00 102.427 71.10 102.735 71.20 103.044 71.30 103.356 71.40 103.668 71.50 103.983 71.60 104.299 71.70 104.616 71.80 104.936 71.90 105.257 72.00 105.579 72.10 105 904 72.20 106 230 72.30 106 558 72.40 106 888 72.50 107 220 72.60 107 553 72.70 107 888 72.80 108 226 72.90 108 565 73.00 108 906 73.10 109 249 73.20 109 594 73.30 109 941 73.40 110 290 73.50 110 641 73.60 110 994 73.70 111 349 73.80 111 707 73.90 112 066 74.00 112 428 74.10 112 792 74.20 113 158 74.30 113 526 74.40 113 897 74.50 114 270 74.60 114 645 74.70 115 023 74.80 115 403 74.90 115 786 75.00 116 171 75.10 116 559 75.20 116 949 75.30 117 342 75.40 117 737 75.50 118 135 75.60 118 536 75.70 118 939 75.80 119 345 75.90 119 755 76.00 120 160 76.10 120 581 76 20 121 000 76.30 121 420 76.40 121 843 76.50 12 270 76.60 12 700 76.70 123 133 76.80 123 570 76.90 124 009 77.00 124 452 77.10 124 898 72.20 125 348 77.30 125 801 77.40 126 258 77.50 126 718 77.60 127 182 77.70 127 649 77.80 128 121 77.90 128 596 78.00 129 075 78 10 129 558 78 20 130 045 78 30 130 536 78 40 131 031 78 50 131 530 78 60 132 034 78 70 132 542 78 80 113 055 78 90 113 572 79 00 134 094 79 10 134 620 79 20 135 151 79 30 135 687 79 40 136 228 79 50 136 775 79 60 137 326 79 70 137 883 79 80 138 445 79 90 139 012 80 00 139 585 80 10 140 164 80 20 140 748 80 30 141 339 80 40 141 936 80 50 142 138 80 60 143 147 80 70 143 763 80 80 144 385 80 90 145 014 81 00 145 650 81 10 146 292 81 20 146 942 81 30 147 600 81 40 148 265 81 50 148 937 81 60 149 618 81 70 150 307 81 80 151 003 81 90 151 709 82 00 152 423 82 10 153 147 82 20 153 878 82 30 154 620 82 40 155 372 82 50 156 132 82 60 156 903 82 70 157 685 82 80 158 478 82 90 159 281 83 00 160 096 83 10 160 922 83 20 161 761 83 30 162 612 83 40 163 475 83 50 164 352 83 60 165 242 83 70 166 146 83 80 167 065 83 90 167 999 84 00 168 947 84 10 169 912 84 20 170 893 84 30 171 891 84 40 172 907 84 50 173 941 84 60 174 994 84 70 176 ●67 84 80 177 160 84 90 178 275 85 00 179 411 85 10 180 569 85 20 181 752 85 30 182 960 85 40 184 194 85 50 185 454 85 60 186 743 85 70 188 062 85 80 189 411 85 90 190 793 86 00 192 210 86 10 193 661 86 20 195 151 86 30 196 680 86 40 198 251 86 50 199 867 86 60 201 529 86 70 203 240 86 80 205 005 86 90 206 825 87 00 208 705 87 10 210 649 87 20 212 668 87 30 214 745 87 40 216 909 87 50 219 158 87 60 221 498 87 70 223 938 87 80 226 486 87 90 229 153 88 00 231 95● 88 10 234 891 88 20 237 991 88 30 241 268 88 40 244 744 88 50 248 445 88 60 252 402 88 70 256 652 88 80 261 243 88 90 266 235 89 00 271 705 89 10 277 753 89 20 284 517 89 30 292 191 89 40 301 058 89 50 311 563 89 60 324 455 89 70 341 166 89 80 365 039 89 90 408 011 90 00 Infinite A Table of the Suns Declination , for the years 1654 , 1658 , 1662 , 1666.   Ianu. Febr. Mar Apr. May. June July . Aug. Sep. Octo. Nov Dec. Dayes . south south sout north north north north north nort south south south 1 21 78 13 85 3 48 08 52 18 03 23 18 22 16 15 28 4 50 7 15 17 60 23 13 2 21 62 13 52 3 10 08 88 18 28 23 25 22 ●3 14 98 4 11 7 53 17 86 23 20 3 21 45 13 17 2 70 09 25 18 53 23 30 21 88 14 66 3 73 7 91 18 13 23 26 4 21 27 12 83 2 30 09 60 18 77 23 35 21 73 14 36 3 35 8 28 18 40 23 33 5 21 08 12 50 1 92 09 97 19 00 23 40 21 58 14 05 2 96 8 65 18 66 23 38 6 20 88 12 15 1 52 10 31 19 23 23 43 21 42 13 73 2 56 9 03 18 91 23 43 7 20 68 11 80 1 11 10 67 19 47 23 46 21 25 13 41 2 18 9 40 19 15 23 46 8 20 48 11 43 0 72 11 02 19 68 23 50 21 07 13 08 1 80 9 76 19 40 23 48 9 20 27 11 08 0 33 11 36 19 90 23 51 20 90 12 76 1 40 10 13 19 63 23 50 10 20 05 10 72 0 06 11 70 20 11 23 52 20 71 12 43 1 01 10 48 19 86 23 51 11 19 82 10 37 N 47 12 05 20 31 23 53 20 51 12 10 0 63 10 85 20 08 23 53 12 19 58 09 83 0 85 12 38 20 51 23 52 20 31 11 76 0 23 11 20 20 30 23 51 13 19 35 09 63 1 25 12 72 20 70 23 51 20 11 11 43 0●16 11 57 20 51 23 51 14 19 11 09 25 1 65 13 05 20 88 23 50 19 90 11 08 0 55 11 91 20 71 23 48 15 18 86 08 88 2 03 13 36 21 06 23 46 19 68 10 73 0 95 12 25 20 91 23 46 16 18 61 08 52 2 41 13 68 21 25 23 43 19 47 10 38 1 33 12 60 21 10 23 43 17 18 35 08 13 2 82 14 00 21 41 23 40 19 25 10 03 1 73 12 95 21 28 23 28 18 18 08 07 75 3 20 14 31 21 58 23 35 19 01 09 68 2 11 13 28 21 46 23 33 19 17 81 07 37 3 60 14 63 21 73 23 30 18 78 09 33 2 51 13 61 21 63 23 28 20 17 53 06 98 3 98 14 93 21 88 23 25 18 55 08 96 2 90 13 95 21 80 23 21 21 17 25 06 60 4 37 15 23 22 03 23 18 18 30 08 60 3 30 14 26 21 95 23 13 22 16 96 06 22 4 75 15 53 22 16 23 10 18 05 08 25 3 68 14 60 22 10 23 05 23 16 68 05 83 5 13 15 83 22 30 23 03 17 78 07 88 4 06 14 91 22 25 22 96 24 16 38 05 45 5 51 16 13 22 41 22 95 17 53 07 51 4 46 15 23 22 38 22 86 25 16 08 05 07 5 90 16 41 22 53 22 85 17 26 07 ●5 4 85 15 55 22 51 22 76 26 15 78 04 67 6 28 16 70 22 65 22 75 17 00 06 76 5 23 15 85 22 63 22 65 27 15 46 04 28 6 67 16 97 22 75 22 65 16 71 06 40 5 63 16 15 22 75 22 53 28 15 15 03 88 7 03 17 23 22 85 22 53 16 43 06 01 6 00 16 45 22 85 22 40 29 14 83   7 41 17 50 22 95 22 41 15 86 05 26 6 76 17 03 23 05 22 11 30 14 51   7 78 17 77 23 ●3 22 30 15 86 05 26 6 76 17 03 23 05 22 11 31 14 18   8 15   23 77   15 56 04 88   17 31   21 96 A Table of the Suns Declination , for the years 1655 , 1659 , 1663 , 1667.   Ianu. Febr. Mar Apr. May. June July . Aug. Sep. Octo. Nov Dec. Dayes . south south sout north north north north north nort south south south 1 21 81 13 93 3 58 08 43 17 96 23 16 22 20 15 35 4 58 7 06 17 53 23 10 2 21 65 13 60 3 18 08 80 18 21 23 23 22 06 15 05 4 20 7 43 17 80 23 18 3 21 48 13 26 2 80 09 15 18 46 23 30 21 91 14 75 3 81 7 81 18 06 23 25 4 21 30 12 91 2 40 09 51 18 71 23 35 21 76 14 43 3 43 8 20 18 33 23 31 5 21 11 12 58 2 00 09 88 18 95 23 40 21 61 14 13 3 05 8 56 18 60 23 36 6 20 93 12 23 1 61 10 23 19 18 23 43 21 45 13 81 2 66 8 93 18 85 23 41 7 20 73 11 88 1 21 10 58 19 41 23 46 21 28 13 50 2 28 9 30 19 10 23 45 8 20 53 11 53 0 81 10 93 19 63 23 48 21 11 13 16 1 88 9 66 19 33 23 48 9 20 32 11 16 0 43 11 28 19 85 23 50 20 93 12 85 1 50 10 03 19 56 23 50 10 20 10 10 81 ●●03 11 61 20 06 23 51 20 75 12 51 1 10 10 40 19 80 23 51 11 19 88 10 45 0 36 11 96 20 26 23 53 20 56 12 18 0 71 10 76 20 03 23 53 12 19 65 10 08 0 76 12 30 20 46 23 51 20 36 11 85 0 33 11 11 20 25 23 52 13 19 41 09 71 1 15 12 63 20 66 23 51 20 16 11 51 0●06 11 46 20 46 23 51 14 19 16 09 35 1 55 12 96 20 85 23 50 19 95 11 16 0 46 11 81 20 66 23 50 15 18 92 08 96 1 93 13 28 21 03 23 48 19 75 10 81 0 85 12 16 20 86 23 46 16 18 67 08 60 2 33 13 61 21 20 23 45 19 53 10 65 1 25 12 51 21 06 23 43 17 18 41 08 23 2 71 13 93 21 38 23 41 19 30 10 13 1 63 12 86 21 25 23 40 18 18 15 07 85 3 11 14 25 21 55 23 36 19 06 09 76 2 03 13 20 21 41 23 35 19 17 88 07 46 3 50 14 56 21 70 23 31 18 83 09 41 2 41 13 53 21 60 23 28 20 17 60 07 08 3 88 14 86 21 85 23 26 18 60 09 06 2 80 13 86 21 76 23 21 21 17 31 06 70 4 28 15 16 22 00 23 20 18 35 08 70 3 20 14 20 21 91 23 15 22 17 03 06 31 4 66 15 46 22 13 23 13 18 10 08 33 3 60 14 51 22 06 23 06 23 16 75 05 93 5 05 15 76 22 26 23 05 17 85 07 96 3 98 14 83 22 21 22 98 24 16 45 05 53 5 43 16 05 22 40 22 96 17 58 07 60 4 36 15 15 22 35 22 88 25 16 15 05 15 5 81 16 35 22 51 22 88 17 31 07 23 4 76 15 46 22 48 22 78 26 15 85 04 76 6 20 16 63 22 63 22 78 17 05 06 85 5 15 15 76 22 60 22 68 27 15 53 04 36 6 56 16 90 22 73 22 68 16 78 06 48 5 53 16 08 22 71 22 56 28 15 23 03 98 6 95 17 18 22 83 22 56 16 50 06 11 5 91 16 38 22 83 22 43 29 14 91   7 31 17 45 22 93 22 45 16 21 05 73 6 30 16 68 22 93 22 30 30 14 58   7 70 17 71 23 01 22 33 15 93 05 36 6 68 16 96 23 01 22 16 31 14 26   8 06   23 10   15 65 04 98   17 25   22 01 A Table of the Suns Declination , for the years 1656 , 1660 , 1664 , 1668.   Ianu. Febr. Mar Apr. May. June July Aug. Sep. Octo Nov Dece . Dayes south south sout north north north north north nort south south south 1 21 85 14 01 3 28 08 70 18 16 23 21 22 08 15 11 4 30 7 35 17 73 23 16 2 21 70 13 68 3 90 09 06 18 41 23 28 21 95 14 81 3 91 7 73 18 00 23 23 3 21 53 13 35 2 50 09 43 18 65 23 33 21 80 14 51 3 53 8 10 18 26 23 30 4 21 35 13 00 2 10 09 78 18 90 23 38 21 65 14 20 3 15 8 48 18 53 23 35 5 21 16 12 66 1 66 10 15 19 13 23 41 21 50 13 88 2 75 8 85 18 78 23 40 6 20 98 12 31 1 33 10 50 19 36 23 45 21 33 13 56 2 36 9 21 19 03 23 45 7 20 78 11 96 0 91 10 85 19 58 23 48 21 16 13 25 1 98 9 58 19 28 23 48 8 20 58 11 61 0 53 11 20 19 80 23 50 20 98 12 91 1 58 9 95 19 51 23 50 9 20 36 11 26 0 13 11 53 20 01 23 51 20 80 12 60 1 20 10 31 19 75 23 51 10 20 15 10 90 N26 11 88 20 21 23 52 20 61 12 26 0 81 10 68 19 98 23 52 11 19 93 10 53 0 66 12 21 20 41 23 53 20 41 11 93 0 41 11 03 20 20 23 53 12 19 70 10 16 1 05 12 55 20 61 23 51 20 21 11 60 0 03 11 38 20 41 23 51 13 ●9 46 09 80 1 45 12 88 20 80 23 50 20 00 11 25 0●36 11 73 20 61 23 50 14 19 23 09 43 1 83 13 20 20 98 23 48 19 80 10 90 0 75 12 08 20 81 23 48 15 18 98 09 06 2 23 13 53 21 16 23 45 19 58 10 56 1 15 12 43 21 01 23 45 16 18 73 08 70 2 63 13 85 21 33 23 41 ●9 35 10 21 1 55 12 78 21 20 23 40 17 18 48 08 31 3 03 14 16 21 50 23 38 ●9 13 9 85 1 93 13 11 21 38 23 35 18 18 21 07 93 3 41 14 48 21 66 23 33 18 90 9 50 2 31 13 45 21 55 23 30 19 17 95 07 55 3 80 14 80 21 81 23 26 18 65 9 15 2 71 13 78 21 7● 23 25 20 17 66 07 16 4 18 15 10 21 96 23 21 18 41 8 78 3 10 14 11 21 88 23 18 21 17 40 06 78 4 56 15 40 22 10 23 15 18 16 8 41 3 50 14 43 22 03 23 10 22 17 11 06 40 4 95 15 70 22 23 23 06 17 91 8 05 3 88 14 76 22 18 23 01 23 16 81 06 01 5 33 15 98 22 36 22 98 17 66 7 68 4 28 15 08 22 31 22 91 24 16 53 05 63 5 71 16 28 22 48 22 90 17 40 7 31 4 66 15 40 22 45 22 81 25 16 23 05 25 6 10 16 56 22 60 22 80 17 11 6 95 5 05 15 70 22 58 22 70 26 15 91 04 86 6 48 16 83 22 71 22 70 16 85 6 56 5 43 16 00 22 70 22 58 27 15 61 04 48 6 85 17 11 22 81 22 60 16 56 6 20 5 81 16 30 22 80 22 46 28 15 30 04 06 7 23 17 38 22 90 22 48 16 28 5 83 6 20 16 60 22 90 22 33 29 14 98 03 68 7 60 17 65 22 98 22 35 16 00 5 45 6 58 16 90 23 00 22 20 30 14 66   7 96 17 90 23 06 22 21 15 71 5 06 6 96 17 ●8 23 08 22 05 31 14 35   8 33   23 15   15 41 4 68   17 46   21 90 A Table of the Suns Declination , for the years 1657 , 1661 , 1665 , 1669.   Ianu. Febr. Mar Apr. May. June July Aug. Sep. Octo Nov Dece . Dayes . south south sout north north north north north nort south south south 1 21 73 13 76 3 40 08 60 18 08 23 20 22 13 15 20 4 40 7 25 17 66 23 15 2 21 56 13 43 3 00 08 96 18 33 23 26 22 00 14 90 4 01 7 63 17 93 23 21 3 21 38 13 08 2 61 09 33 18 58 23 31 21 85 14 60 3 63 8 00 18 20 23 28 4 21 21 12 75 2 21 09 70 18 83 23 36 21 70 14 28 3 25 8 36 18 46 23 33 5 21 03 12 41 1 81 10 05 19 06 23 41 21 53 13 96 2 86 8 75 18 71 23 38 6 20 83 12 06 1 41 10 40 19 30 23 45 21 36 13 65 2 48 9 11 18 96 23 43 7 20 63 11 71 1 01 10 75 19 51 23 48 21 20 13 33 2 08 9 48 19 21 23 46 8 20 43 11 35 0 63 11 10 19 73 23 50 21 03 13 01 1 70 9 85 19 45 23 50 9 20 21 11 00 0 23 11 45 11 95 23 51 20 85 12 68 1 31 10 21 19 68 23 51 10 20 00 10 63 N16 11 78 20 16 23 52 20 66 12 35 0 91 10 58 19 91 23 52 11 19 76 10 26 0 55 12 11 20 36 23 53 20 46 12 01 0 53 10 93 20 13 23 53 12 19 53 09 90 0 95 12 46 20 56 23 51 20 26 11 68 0 13 11 30 20 35 23 51 13 19 30 09 53 1 35 12 80 20 75 23 50 20 06 11 35 0●26 11 65 20 56 23 50 14 19 05 09 16 1 73 13 11 20 93 23 48 19 85 11 15 0 65 12 00 20 76 23 48 15 18 80 08 80 2 13 13 45 21 11 23 46 19 63 10 65 1 05 12 35 20 96 23 45 16 18 55 08 41 2 51 13 76 21 28 23 43 19 41 10 30 1 43 12 68 21 15 23 41 17 18 28 08 05 2 90 14 08 21 45 23 38 19 20 9 95 1 83 13 20 21 33 23 36 18 18 03 07 66 3 30 14 40 21 61 23 33 18 96 9 60 2 21 13 36 21 51 23 31 19 17 75 07 28 3 68 14 70 21 76 23 28 18 71 9 25 2 61 13 70 21 68 123 26 20 17 46 06 90 3 08 15 01 21 91 23 23 18 48 8 88 3 00 14 03 21 83 23 20 21 17 18 06 51 4 46 15 31 22 06 23 16 18 23 8 51 3 38 14 35 22 00 23 11 22 16 90 06 13 4 85 15 61 22 20 23 10 17 98 8 15 3 78 14 68 22 15 23 03 23 16 60 05 75 5 23 15 90 22 33 23 01 17 73 7 78 4 16 15 00 22 28 22 95 24 16 30 05 35 5 61 16 20 22 45 22 91 17 46 7 41 4 55 15 31 22 41 22 85 25 16 00 04 96 6 00 16 48 22 56 22 83 17 20 7 05 4 95 15 61 22 55 22 73 26 15 70 04 56 6 36 16 76 22 68 22 73 16 93 6 68 5 33 15 91 22 66 22 61 27 15 38 04 18 6 75 17 03 22 78 22 61 16 65 6 30 5 71 16 21 22 76 22 50 28 15 06 03 78 7 11 17 30 22 88 22 51 16 36 5 93 6 10 16 51 22 86 22 36 29 14 75   7 50 17 56 22 96 22 38 16 08 5 55 6 48 16 81 22 96 22 ●● 30 14 43   7 86 17 83 23 05 22 26 15 80 5 16 6 86 17 10 23 06 22 08 31 14 10   8 23   23 13   15 50 4 78   17 38   21 93 A Table of the Suns right Ascension in hours and minutes .   Janu Febr. Mar. Apr. May Iune Jul Aug. Sept. Octo. Nov. Dece . Dayes H. M. H. M. H. M. H M H. M H. M H M H. M H. M H. M H. M H. M 1 19 53 21 68 23 45 1 33 3 21 5 30 7 36 9 40 11 30 13 10 15 10 17 23 2 19 61 21 75 23 51 1 38 3 28 5 36 7 43 9 46 11 35 13 16 15 16 17 31 3 19 68 21 81 23 56 1 45 3 35 5 43 7 50 9 51 11 41 13 23 15 23 17 38 4 19 75 21 88 23 63 1 51 3 40 5 50 7 56 9 58 11 48 13 28 15 30 17 45 5 19 83 21 95 23 70 1 56 3 46 5 56 7 63 9 65 11 53 13 35 15 38 17 53 6 19 90 22 00 23 75 1 63 3 53 5 63 7 70 9 71 11 60 13 41 15 45 17 60 7 19 96 22 06 23 81 1 70 3 60 5 71 7 76 9 76 11 65 13 48 15 51 17 68 8 20 05 22 13 23 86 1 75 3 66 5 78 7 83 9 83 11 71 13 53 15 58 17 75 9 20 11 22 20 23 93 1 81 3 73 5 85 7 90 9 90 11 78 13 60 15 65 17 83 10 20 18 22 26 00 00 1 88 3 80 5 91 7 96 9 96 11 83 13 66 15 71 17 90 11 20 25 22 33 0 05 1 95 3 86 5 98 8 03 10 01 11 90 13 73 15 80 17 98 12 20 33 22 40 0 11 2 00 3 93 6 05 8 10 10 08 11 95 13 80 15 86 18 05 13 20 40 22 45 0 18 2 06 4 0● 6 13 8 16 10 15 12 01 13 85 15 93 18 11 14 20 46 22 51 0 23 2 13 4 06 6 20 8 23 10 20 12 06 13 91 16 00 18 20 15 20 53 22 58 0 30 2 18 4 13 6 26 8 30 10 26 12 13 13 98 16 08 18 26 16 20 60 22 65 0 35 2 25 4 20 6 33 8 36 10 33 12 20 14 05 16 10 18 35 17 20 66 22 70 0 41 2 31 4 26 6 40 8 43 10 38 12 25 14 11 16 21 18 41 18 20 75 20 76 0 48 2 38 4 33 6 40 8 50 10 45 12 31 14 18 16 28 18 50 19 20 81 22 83 0 53 2 45 4 40 6 53 8 56 10 51 12 38 14 23 16 36 18 56 20 20 88 22 90 0 60 2 50 4 46 6 61 8 63 10 56 12 43 14 30 16 43 18 65 21 20 95 22 95 0 66 2 56 4 55 6 68 8 70 10 63 12 50 14 36 16 50 18 71 22 21 01 23 01 0 71 2 63 4 61 6 75 8 76 10 70 12 55 14 43 16 58 18 80 23 21 08 23 08 0 78 2 70 4 68 6 81 8 81 10 75 12 61 14 50 16 65 18 86 24 21 15 23 13 0 83 2 76 4 75 6 88 8 88 10 81 12 68 14 56 16 71 18 93 25 21 21 23 20 0 90 2 81 4 81 6 95 8 95 10 88 12 73 14 63 16 80 19 01 26 21 28 23 26 0 96 2 88 4 88 7 01 9 01 10 93 12 80 14 70 16 86 19 08 27 21 35 23 33 1 01 2 95 4 95 7 10 9 08 11 00 12 86 14 76 16 93 19 16 28 21 41 23 38 1 08 3 01 5 01 7 16 9 15 11 05 12 91 14 83 17 01 19 23 29 21 48     1 15 3 08 5 08 7 23 9 21 11 11 12 98 14 90 17 08 19 30 30 21 55     1 20 3 15 5 16 7 30 9 26 11 18 13 05 14 96 17 16 19 38 31 21 61     1 26     5 23     9 33 11 23     15 03     19 45 Declination and Right Ascension of the Stars .   Declination Dist . from the pole Right Ascenation The names of the Stars . D. M.   D. M. H. M. Brest of Cassiopeia 54 76 N 35 24 0 35 Pole-star 87 48 N 02 52 0 51 Girdle of Andromeda 33 83 N 56 17 0 83 Knees of Cassiopeia 58 41 N 31 59 1 05 Whales belly 12 00 S 78 00 1 58 South , foot by Andr. 40 65 N 19 35 1 70 Rams head 21 81 N 68 19 1 80 Head of Medusa 39 58 N 50 42 2 76 Perseus right side 48 55 N 41 45 2 98 Buls eye 15 75 N 74 25 4 26 The Goat 45 58 N 44 42 4 85 Orions left foot 08 63 S 81 37 4 96 Orions left shoulder 05 98 N 84 02 5 10 First in Orions girdle 00 58 S 89 42 5 25 Second in Orions gird . 01 45 S 88 55 5 31 Third in Orions girdle 02 15 S 87 85 5 38 Wagoners right shold . 44 86 N 45 14 5 70 Orions right shoulder 07 30 N 82 70 5 60 Bright foot of Twins 16 65 N 73 35 6 30 The great Dog 16 21 S 73 79 6 50 Upper head of Twins 32 50 N 57 50 7 20 The lesse Dog 06 10 N 83 90 7 36 Lower head of Twins 28 80 N 61 20 7 40 Brightest in Hydra 07 16 S 82 84 09 16 Lions heart 13 63 N 76 37 09 83 Lions back 22 06 N 65 94 11 50 Lions tail 16 50 N 73 50 11 51 Great Bears rump . 58 72 N 31 28 10 66 First in the great Bears tail next her rump 57 85 N 32 15 12 63 Virgines spike ●9 32 S 80 68 13 11 Middlemost in the great Bears tail 56 75 N 33 25 13 16 In the end of her tail 51 08 N 38 92 13 56 Between Bootes thighs 21 03 N 68 97 14 00 South Ballance 14 55 S 75 45 14 53 North Ballance 08 05 S 81 95 14 96 Scorpions heart 25 58 S 64 42 16 13 Hercules head 14 85 N 75 15 16 98 Serpentaries head 12 86 N 77 14 17 31 Dragons head 51 60 N 38 40 17 80 Brightest in the Harp 38 50 N 51 50 18 41 Eagles heart 08 02 N 81 98 19 56 Swans tail 44 08 N 45 92 20 50 Pegasus mouth 08 32 N 81 68 21 45 Pegasus shoulder 17 38 N 76 62 23 91 The head of Androm . 27 22 N 62 78 23 85 Rules for finding of the Poles elevation by the meridian altitude of the Sun or stars , and by the Table of their Declinations aforegoing . Case 1. IF the Sun or star be on the meridian to the southwards , and have south declination . Adde the suns declination to his meridian altitude , and taking that total from 90 degrees , what remaineth is the latitude of the place desired . As the 7 of February , 1654 , by the aforegoing Table , the suns decl . south . is 11.80 The suns meridian altitude 15.27 The sum or total is 27.07 Which substracted from 90.00 There remains the North latitude 62.93 But when you have added the suns declination to his meridian altitude , if the total exceed 90 : substract 90 degr . from it , and what remaineth is your latitude to the southwards . As admit the suns declination to be southerly 11.80 And his meridian altitude 87.23 The sum or total is 99.03 From which substracting 90.00 There remains the latitude south . 09.03 Case . 2. If the sun or star be on the meridian to the southwards , and have north declination . Substract the suns declination from his meridian altitude , and that which remains substract from 90 , and then the remainer is the poles elevation northerly . Case 3. If the sun or star be on the meridian to the northwards , and have north declination ▪ Adde the suns declination to his meridian altitude , the total take from 90 , and what remaineth is the poles elevation southerly . But when you have added the suns declination to his meridian altitude , if it exceed 90 , substract 90 from it , and what remaineth is your latitude northerly . Case 4. If the sun be to the northwards at noon , and declination south . Substract the suns declination from his meridian altitude , and that which remains substract from 90 , what then remaineth is your latitude southerly . And what is said of the Sun , is also to be understood of the Stars , being upon the Meridian . Case 5. If you observe when the Sun hath no declination . Substract his meridian altitude from 90 , what remaineth is your latitude . Case 6. If you chance to observe when the Sun or star is in the Zenith , that is 90 degrees above the Horizon . Look in the table for the declination of the Sun or of that star , and the same is your latitude . Case 7. If the Sun come to the meridian under the Pole. If you be within the Artick or Antartick circle , and observe the Sun upon the meridian under the Pole ; substract the Suns declination from 90 , the remainer is the Suns distance from the Pole ; which distance added to his meridian altitude , the sum or total is the latitude sought . And what is here said of the Sun is to be understood of the stars , whose declinations , distances from the pole , and right ascensions we have expressed in the foregoing Table . FINIS . 
A34005	----	The sector on a quadrant, or A treatise containing the description and use of four several quadrants two small ones and two great ones, each rendred many wayes, both general and particular. Each of them accomodated for dyalling; for the resolving of all proportions instrumentally; and for the ready finding the hour and azimuth universally in the equal limbe. Of great use to seamen and practitioners in the mathematicks. Written by John Collins accountant philomath. Also An appendix touching reflected dyalling from a glass placed at any reclination. Collins, John, 1625-1683. 1659 Approx. 798 KB of XML-encoded text transcribed from 209 1-bit group-IV TIFF page images. Text Creation Partnership, Ann Arbor, MI ; Oxford (UK) : 2007-01 (EEBO-TCP Phase 1). A34005 Wing C5382 ESTC R32501 99899660 99899660 66044 This keyboarded and encoded edition of the work described above is co-owned by the institutions providing financial support to the Early English Books Online Text Creation Partnership. This Phase I text is available for reuse, according to the terms of Creative Commons 0 1.0 Universal . The text can be copied, modified, distributed and performed, even for commercial purposes, all without asking permission. Early English books online. (EEBO-TCP ; phase 1, no. A34005) Transcribed from: (Early English Books Online ; image set 66044) Images scanned from microfilm: (Early English books, 1641-1700 ; 1524:9 or 2400:6) The sector on a quadrant, or A treatise containing the description and use of four several quadrants two small ones and two great ones, each rendred many wayes, both general and particular. Each of them accomodated for dyalling; for the resolving of all proportions instrumentally; and for the ready finding the hour and azimuth universally in the equal limbe. Of great use to seamen and practitioners in the mathematicks. Written by John Collins accountant philomath. Also An appendix touching reflected dyalling from a glass placed at any reclination. Collins, John, 1625-1683. Lyon, John, professor of mathematics. Appendix touching reflective dialling. Sutton, Henry, mathematical instrument maker. [16], 284; [2], 54, [10], 26 p., [6] leaves of plates (some folded) : ill. (woodcuts) printed by J.M. for George Hurlock at Magnus Corner, Thomas Pierrepont, at the Sun in Pauls Church-yard; William Fisher, at the Postern near Tower-Hill, book-sellers; and Henry Sutton, mathematical instrument-maker, at his house in Thred-needle street, behind the Exchange. With paper prints of each quadrant, either loose or pasted upon boards; to be sold at the respective places aforesaid, London : 1659. The quadrants described were made and engraved by Henry Sutton, who also calculated some of the tables and drew the projections. "The description and vses of a great universal quadrant" has separate title page dated 1658; pagination and register are continuous. "The description and uses of a general quadrant, with the horizontal projection, upon it inverted" has separate title page dated 1658; pagination, and register are separate. "An appendix touching reflective dialling" by John Lyon has separate title page dated 1658; pagination is separate; register is continuous. A reissue, with cancel title page, of the 1658 edition having "printed by J. Macock" in imprint (Wing C5381). In this issue, the 2 contents leaves, bound after p. 275 in the original issue, are bound with the preliminaries, following title page and "To the reader" (A2). The catchword "The" on the verso of the second contents leaf does not match the first word of the following page (a1). "The description and uses of a general quadrant" filmed separately as Wing C5371 on UMI microfilm set "Early English books, 1641-1700", reel 2400. Reproduction of originals in: Harvard University Library; Henry E. Huntington Library. Created by converting TCP files to TEI P5 using tcp2tei.xsl, TEI @ Oxford. Re-processed by University of Nebraska-Lincoln and Northwestern, with changes to facilitate morpho-syntactic tagging. Gap elements of known extent have been transformed into placeholder characters or elements to simplify the filling in of gaps by user contributors. EEBO-TCP is a partnership between the Universities of Michigan and Oxford and the publisher ProQuest to create accurately transcribed and encoded texts based on the image sets published by ProQuest via their Early English Books Online (EEBO) database (http://eebo.chadwyck.com). 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Keying and markup guidelines are available at the Text Creation Partnership web site . eng Mathematical instruments -- Early works to 1800. Astronomy -- Early works to 1800. Navigation -- Early works to 1800. Dialing -- Early works to 1800. 2005-07 TCP Assigned for keying and markup 2006-03 SPi Global Keyed and coded from ProQuest page images 2006-06 Derek Lee Sampled and proofread 2006-06 Derek Lee Text and markup reviewed and edited 2006-09 pfs Batch review (QC) and XML conversion THE SECTOR ON A QUADRANT , OR A Treatise containing the Description and Use of four several QUADRANTS ; Two small ones and two great ones , each rendred many wayes , both general and particular . Each of them Accomodated for Dyalling ; for the Resolving of all Proportions Instrumentally ; And for the ready finding the Hour and Azimuth Universally in the equal Limbe . Of great use to Seamen and Practitioners in the MATHEMATICKS . Written by JOHN COLLINS Accountant Philomath . Also an Appendix touching Reflected Dyalling from a glass placed at any Reclination . London , Printed by J.M. for George Hurlock at Magnus Corner , Thomas Pierrepont , at the Sun in Pauls Church-yard ; William Fisher , at the Postern near Tower-Hill , Book-sellers ; And Henry Sutton , Mathematical Instrument-Maker , at his House in Thred-needle street , behind the Exchange . With Paper Prints of each Quadrant , either loose or pasted upon boards ; to be sold at the respective places aforesaid . 1659. To the Reader . Courteous Reader , THou hast in this Treatise , the Description and Uses of three several Quadrants , presented to thy View and Acceptance ; and here I am to give thee an account of their Occasion and Original . Being in conference with my loving friend M. Thomas Harvie , he told me , that he had often drawn a Quadrant upon Paper pastboard , &c. derived by himself , and never done by any man before , as to his knowledge , from the Stereographick Projection , which for a particular Latitude , would give the Hour in the equal Limb , and would also perform the Azimuth very well ; and but that it was so particular , was very desirous to have one made in Brass for his own use by an Instrument Maker : whereto replying , that with the access of some other Lines to be used with Compasses , it might be rendred general for finding both the Hour and the Azimuth in the equal Limb : He thereupon intimated his desires to M. Sutton , promising within a fortnight after their conference , to draw up full directions for the making thereof . But M. Sutton having very good practise and experience in drawing Projections , speedily found out the drawing of that Projection , either in a Quadrant or a Semicircle , without the assistance of the promised directions , and accordingly , hath drawn the shape of it for all Latitudes , and also found how the Horizontal Projection might be inverted and contrived into a Quadrant without any confusion , by reason of a reverted tail , and let me further add , that he hath taken much pains in calculating Tables for the accurate making of these and other Instruments , in their construction more difficult then any that ever were before ; and the said M. Sutton conceiving that it would be an advancement to their Trade in general , besides satisfactory to the desires of the studious in the Mathematiques , to have the uses of a good Quadrant published , prevailed with me , in regard M. Harvey was not at leisure ( though willing his Quadrant should be made publique ) to write two or three sheets of the use of it , which I intended to have given M. Sutton ( who very well understood the use as well as the making ) to be published in his own name ; whereto he being unwilling , and finding that therein many of the uses of one Quadrant , much less of more could not be comprized , at his earnest request , I wrote what is here digested , succisivè & horis Antelucanis , having little leisure for that purpose , and all this performed before the Instruments were cut , wherefore the description given of them , may not so nearly agree with the Instruments , as if they had been first made , nor possibly some of the examples about finding the hour of the night by the Stars ; which examples were fitted from Tables of present right Ascension , whereas the Quadrant is fitted to serve the better for the future , the difference notwithstanding will be but small . And thus hoping thou wilt cover my failings with the mantle of love , and kindly accept of my endeavours , tending to the publique advancement and increase of knowledge , I still remain a Wellwiller desirous thereof . John Collins . depiction of a quadrant How the Projections on both the Quadrants may be Demonstrated . TO satisfie the inquisitive Reader herein , I shall only in this Edition quote such Latine Authors and Propositions as will evince the truth thereof , the performance whereof in English , is hereafter intended by my loving friend M. Thomas Harvie , in an elaborate Treatise , concerning all the Projections , with their Demonstration and Application , who is accomplished with singular knowledge in that kind , as in general in the Mathematiques . Now the Demonstration of these two Projections is as much included in the Demonstration of the Stereographick Projection , which by Aguilonius in his 6 Book is largely insisted upon , as a peculiar question in Trigonometry , is included in a general Case , and both the Projections on these Quadrants being derived from the grounds of the said general Projection , are necessarily involved in one and the same Demonstration . Stofler in his Astrolabe supposeth the eye in the South Pole Stereographically projecting upon the Plain of the Equator those Circles between the North Pole Horizon and Troprick of Capricorn , neglecting that part under the Horizon . But the Projection on the Quadrant , considered as it may be derived from his Astrolabe , supposeth the eye in the same Position , and makes use of one half of the Projection of the other part of the Circles intercepted between the Horizon and Tropick of Capricorn , namely , of that space between the Tropick below our Horizon , only changing the names of Cancer for Capricorn in their use , and using the depressed Parallels to the Horizon , instead of the Parallels of Altitude ; so that the Azimuths of the Quadrant made by this inversion , are no other then the Azimuths of Stoflers Projection continued below the Tropick of Capricorn where he breaks them off , and the rule he prescribes to draw the parallel of 18d of Depression for the Twylight serves to draw that , and all the other parallels of Altitude in this Quadrant . In like manner the Horizontal Projection supposeth the eye in the Nadir projecting upon the Plain of the Horizon . That part of the Sphere intercepted between the two Tropicks , neglecting that part thereof under the Horizon . But the Projection on the other great Quadrant , considered as derived therefrom , supposeth the eye there projecting that part of the Sphere which is there neglected with the like change of denomination ; and the Parallels of Declination are no other then the continuance of the said Parallels of the Horizontal Projection round to the Midnight Meridian , and the Hour circles the continuance of the said hours , only the Index of Altitudes is fitted to the Depressed Parallels of the Horizon , in stead of the Parallels of Altitude . Now it is evident , either from the Sphere or Analemma , that that part of either of these Projections which falls under the Horizon , will supply the use of that which hapned above , admitting only a change of denomination ; for in the Horizontal Projection , that Parallel of Declination which was called the Winter Tropick , being no other then the same Circle continued about , now in its use and denomination , becomes the Summer Tropick ; and the reason is , because what ever Altitude the Sun hath in any Sign upon any Hour or Azimuth reckoned from the Noon Meridian , he hath the like Depression on the like Hour and Azimuth in the opposite Signe counted from the Midnight meridian . The terms of Noon and Midnight Meridian are afterwards used in relation to some general Proportions : By the Hour in general is meant the Angle between the Meridian of the Sun or Stars , and the Meridian of the place : By the Hour counted from Noon Meridian , is meant the said Angle counted from that part of the Meridian of the place which falls above the Elevated Pole , continued towards the Depressed Pole : and by the Midnight Meridian , the opposite thereto under the Elevated Pole , continued as before . By the Azimuth counted from the Midnight Meridian , is meant an Angle at the Zenith between the Suns Vertical or Azimuthal circle , and the Meridian of the place , measured by the Horizon , counted from the Intersection of the Horizon with the Meridian under the Elevated Pole ; and by the Azimuth counted from the Noon Meridian , is meant the Complement of the said Angle to a Semicircle , counted from the opposite Intersection of the Horizon with the former Meridian continued above the Elevated , and towards the Depressed Pole , according to which acceptions , the general Proportions are fitted for finding it either way in both Hemispheres , without any restriction to North or South . A more immediate account of these Projections . HItherto we have accommodated our Discourse , to shew how these Projections are derived from Stoflers Astrolabe , and from the Horizontal Projection , of which neither Stofler ( as to my knowledge ) for I have only seen his 8 Book ) nor the learned M. Oughtred , give no peculiar Demonstration , as being particular examples of a general case , largely ( as such ) insisted on ; and this we have done for the accommodation of Instrument makers , to whom this Derivation may seem most suitable ; whereas such a deduction is not at all necessary to the Demonstration of the Projections so derived . For in the Projection derived from the Inversion of Stofler , let the eye be supposed to be placed in the North Pole , projecting upon the Plain of the Equinoctial , such Circles in the Sphere , as are described in the Quadrant between the two Tropicks , a quarter of which Projection will be the same with that on the Quadrant , namely , one of those quarters between the South part of the Meridian and hour of six , which will leave out all the outward part of the Almicanters between it and the Tropick of Cancer , and in stead thereof , there is taken in such a like part of the depressed Parallels to the Horizon between the same Hour of six , and Tropick of Capricorn , which is the reverted tail ; for the Parallels of Depression have the same respect to the Tropick of Capricorn , that the Parallels of Altitude have to the Tropick of Cancer , and will work the same in effect . In like manner , the Eye in the other Projection may be supposed in the Zenith , Stereographically Projecting upon the Plain of the Horizon , that part only of the space between the Tropicks , which falls without the Projection of the Horizontal Circle , save only the reverted tail , which is the Projection of so much of the Parallels of South Declination , as is intercepted between the prime Vertical Circle and the Horizon , and is taken in to serve in stead of that part of the Parallels of North Declination , which will fall without the Quadrant . In any of these Positions of the Eye , all Circles passing through the same , will be projected in right lines by 91 Prop. 6 Book of Aguilonius , such are the Azimuths on the Horizontal Quadrant , and the Hours on the other Quadrant , represented by the thred lying over any Ark in the Limb , so also in this latter Quadrant is the Parallel of Altitude equal to the Latitude of the place , a right line . All Circles parallel to the Horizon and Equinoctial , will be projected in concentrick Circles by 94 Prop. 6. Aguilonius , such are the Parallels of Altitude in the Horizontal Quadrant , and the Parallels of Declination in the other Quadrants , represented by the Bead , when it is rectified to the Index of Altitudes in the one , and to the Ecliptick in the other , carried in a circular trace from one side of the Quadrant to the other . All other Circles in the Sphere , whatsoever and howsoever scituated , being projected according to the supposed position of the Plain and Eye , will be represented by Excentrick Circles . by 96 Prop. 6. Aguilonius , and the hours in the Horizontal Projection will ( if they be produced ) meet with the projected Pole points , so also the Azimuths in the other Projection , which by the like parity of Reason may be denominated The Equinoctial Projection , will ( being produced ) meet with the projected points of the Zenith and Nadir ; and how in particular to project and divide any Circle however scituated in the Sphere , is abundantly shewn in the 6th Book of the aforesaid Author , and amplified with many examples , though none of them agreeing with the particular Draughts of these Quadrants , yet if put in practise according to the proposed Scituation of the Eye , will be found to agree with the prescribed Directions for the making of these Quadrants . See also Clavius his Book of the Astrolabe , Guido Ubaldus his Theorick of the general Planispheres , and M. Oughtreds 2d Scheme B in his late Trigonometry in English . An Appendix to the Description of the Small Quadrant . SInce the Printing of the sheet B , we have thought fit to vary a little from the Description there given of the Small Quadrant . The Dyalling Scale of Hours described in page 9 , near the beginning , which I say in Page 191 may be omitted , is accordingly left out , and instead of it , a line of Versed Sines of 90d put on , the uses whereof are handled in the great Equinoctial Quadrant . Also there is two Scales added to the small Quadrant more then was described ; namely , the Scale of Entrance , the same that was placed upon the Horizontal Quadrant , with a Sine of 51d 32′ put through the whole Limb serving to give the Altitude at six , which the thred will intersect , if it be laid over the Declination in the Limb ; but enough of the uses of these Scales is said in the Horizontal Quadrant . Lastly , Those that like it best , instead of having on the Small Quadrant one loose fitted Scale for the Hour , and another for the Azimuth , may have the Hour-Scale only divided into two parts , serving to give the Hour and Azimuth for the Sun , and all the Stars in the Hemisphere , the one part for South Declinations , the other for North Declinations , in imitation of the Diagonal Scale . An Advertisement . ALL manner of Mathematical Instruments , either for Sea or Land , are exactly made in Wood or Brass , by Henry Sutton , in Thredneedle-street , near Christophers Church , or by William Sutton in Upper Shadwel , a little beyond the Church . Pag. Li. Errors to be thus Corrected . 8 33 apply supply 19 25 76 d 54′ 79 d 54′ 20 7 48 d 45 d 23 3 50 d 41′ 50 d 41′ 23 11 58′ 53′ 24 25 a Letter Character . the Character plus + 28 is lesse if lesse 55 14 difference of one of differences of 55 17 of the Leggs from the Leggs of one of the Leggs The affections in page 57 line 16 , 17. wanting Braces , are expressed at large page 140 , 141. also the last affection in that page having a mistake of lesser for greater in the middle brace is reprinted in page 138 60 14 The last term of the 4th Proportion should be the Sine of the angle sought , and not the Cosecant . 92 29 a Leg and its adjacent angle The Hipotenusal and its adjacent angle 30 to find the other angle by 4 to find the side opposite there to by 8 Case Case . See page 138. 103 34 Acquimultiplex Aequimultiplex 35 therefore by therefore by 18 Prop. 7. Euclid . 121 18 Hour 3¼ Altitude 41 d 31′ Altitude for the hour 3¼ is 43 d 31′ 158 1 Line of Line of Sines 159 21 of 90 d at 90 d 164 ●8 any some 174 15 in the Limb in the lesser Sines 181 25 between as also between 27 would find it would in the other Hemisphere find it 184 21 the common as the common 189 1● either it may be found either 192 24 As the second including side As the Sine of the 2d including side 207 1 great Scale great Quadrant 209 7 60 parts 60 equal parts The Angle C in the Scheam page 52 is wrong cut , and should be 113d 22′ , See it in page 156. Page 98 a wrong Scheam printed , the true one is in page 93. Page 102 in the under Triangle the Angle D should be 108d 37′ See page 201. The first ten lines of calculation p. 53 are somwhat misplac'd , & should stand thus .     D●ffs Log ms     B C 126 Legs 169   2,2278867   A C 194   101   2,0042214 Sum with double Radius A B 270 Base 25 1,3979400 24,232208●   Sum 590           half sum 295   Logarith . 2 4698220 3 8677620 former Rectangle       Residue 20,3644461   The half is the tangent of 56 d , 41′ t.       10,1822230   Which Ark doubled is       113 , 22 A C B the Angle sought .   A Table shewing the Contents of the Book . AN account of the original of the Projections , and their Demonstration . In the Preface . The Description and making of the small Quadrant : Page 1. The Vse of the perpetual Almanack . 12 The Vses of the Projection on the small Quadrant . 15 Of the Stars inscribed thereon . 24 Of the Projection drawn in a Semicircle . 33 Of the Stars and quadrant of Ascensions drawn on the back-side of the quadrant . 34 The Vses of the Quadrat and Shaddows . 35 Divers ways to measure the altitude of a Tower or object . 36 Affections of Plain Triangles 45 Proportions for the Cases of right angled plain Triangles 46 Proportions for the Cases of Oblique Plain Triangles 49 Doubtful Cases manifested , and mistakes about them rectified 50 Four several Proportions for finding an angle when three sides are given , not requiring the help of Perpendiculars 52 Affections of Sphoerical Triangles 56 Six several Proportions suited to each of the 16 Cases of right angled Sphoerical Triangles , with all necessary caution 60 Directions for varying of Proportions . 72 Some Cases of oblique Sphoerical Triangles resolved without the help of Perpendiculars 76 , 83 When some of the Oblique Cases will be doubtful , and when not 80 , 81 Six of those Cases demonstrated to be doubtful 86 , 87 Perpendiculars needless in any of those Cases , and not used 88 Some new Proportions applyed to the Tables 88 And mistakes about Calculating the distances of places noted 89 And Proportions esteemed improper to the artificial Tables applyed for that purpose 96 Proportions both Instrumental and others for finding a Sphoerical angle when 3 sides are given 100 , 101 Those Instrumental Proportions how demonstrated and derived from other Proportions in use for the Tables . 103 New Proportions for finding the Hour and Azimuth demonstrated from the Analemma . 108. And applyed to the Calculating of a Table of Hours . 117 A Table Calculated thereby , showing the Suns altitudes on all hours for some Declinations useful for the trial and construction of some Instruments . 121 A Table of the Suns Altitudes for each 5d of Azimuth fitted to the same Declinations . 122 To Calculate a Table of Azimuths to all Altitudes . 124 And a Table of Altitudes to all Azimuths . 128 And the Proportions used laid down from the Analemma . 132 Affections of some doubtful Cases of Sphoerical Triangles determined . 138 The ground of working Proportions on a quadrant . 144 Proportions is equal Parts resolved instrumentally . 146 , 147 All Proportions in Tangents alone resolveable by the tangent line on the quadrant , or by a tangent of 45d. 148 Proportions in Sines and Tangents wrought on the quadrant . 151 Proportions in equal parts and tangents resolved by the quadrant . 152 Proportions in equal Parts and Sines so resolved , being useful in Navigation . 156 , 157 Of a line of Sines , and how from it or the Limb to take off a Sine , Tangent or Secant . 158 Proportions in Sines wrought on the quadrant . 161 How to proportion out lines to any Radius from lines inscribed in the Limbe . 167 To operate Proportions in Sines and Tangents on the quadrant . 168 A general way for finding the Hour . 173 As also of the Azimuth . 174 Particular Scales fitted for finding the same . 177 Another general way for finding the Hour . 181 And for finding the Suns Altitudes on all Hours . 183 Another general way for finding the Azimuth . 183 And of distances of places by the quadrant . 188 How from a quadrant to take off Chords ▪ 190 The Description of the Diagonal Scale . 193 , 194 The said Scale fitted for the ready finding the Hour and Azimuth for all parts of England , Wales , Ireland , and the uses thereof . 195 , 196 The description and uses of the particular Scales in Dyalling . 197 Also the uses of other Scales fitted for finding the Hour and Azimuth near noon . 200 Hitherto the Uses of the small Quadrant . THe Description of the great Equinoctial Quadrant . 2 The Vses of a line of Versed Sines from the Center . 208 And of Versed Sines in the Limb. 214 The Vses of a fitted particular Scale , with the Scale of entrance , for finding the Hour and Azimuth in the equal Limb. 216 As also for finding the same in the Versed Sines of 90d in the Limb. 220 , 221 And of the Diagonal Scale therewith . 222 , 223 New general Proportions for finding the Hour and Azimuth . 225 Divers Proportions demonstrated from the Analemma . 227 Several general Proportions for finding the Hour applyed . 230 And several Proportions applyed for finding the Altitudes on all Hours . 233 General Proportions for finding the Azimuth 237 A new single Proportion in Sines for Calculating a table of hours . 243 The Stars how put on and the hour found by them . 244 , 245 The Construction of the Graduated Circle . 248 The line of Latitudes demonstrated . 249 Vses of the graduated Circle and Diameter in resolving of Proportions , and in finding the Hour and Azimuth universally . 253 Vses of other Lines within the said Circle . 261 To prick down a Horizontal Dyal in a Square or a Triangle by the Dyalling lines on the Quadrants . 262 Proportions for upright Decliners . 265 To prick down upright Decliners in a Square or rather right angled Parralellogram , is likewise in a Triangle from the Substile . 268 To prick them down in a Triangle or Paralellogram from the Meridian . 272 An Advertisement about observing of altitudes . 275 The description and making of the Horizontal quadrant . Page 1 The perpetual Almanack in another form 12 A tide Table with the use of the Epacts in finding the Moons age . 11 The uses of the Horizontal Projection . 18 The uses of the particular curved line and fitted Scales thereto . 20 Some Proportions demonstrated from the Analemma . 26 A particular Scale for finding the Hour and Azimuth fitted thereto , with its uses . 27 The uses of the Scale of entrance , being another particular fitted Scale for finding the Hour and Azimuth in the Limb. 32 A Chord taken off from the equal Limb and the Hour and Azimuth found universally thereby . 34 A new general Proportion for the Azimuth to find it in the equal Limb. 36 Another for the Converse of the fourth Axiom . 38 The Stars how inscribed , and the hour of the night found by them . 39 Some uses of the Dyalling Lines . 44 , 45 The use of the Line of Superficies . 46 , 47 And of the Line of Solids . 48 And of the line of inscribed Bodies and other Sector Lines . 49 , 50 A Table of the Latitudes of the most eminent places in England , Scotland , Wales , and Ireland . 52 , 53 A Table of the right Ascensions and Declinations of 54 of the most eminent fixed Stars . 53 , 54 Lastly , A Table of the Suns Declination and right Ascension for the year 1666 , with a Table of equation to make it serve sooner or longer . An Errour Page 248 , Line 30 and 31 for so is L N to L F read so is L N to N F. Of the Lines on the foreside of the QUADRANT . ON the right edge from the Center is placed a Line of equal parts , of 5 inches in length , divided into 100 equal parts . On the left edge a Line of Tangents , continued to two Radii , or to 63d 26 m the Radius whereof is 2½ inches . These two Lines make a right Angle in the Center , and between them include the Projection , which is no other then a fourth part of Stoflers particular Astrolabe inverted . Next above this Projection , towards the Center , is put on in the Quadrant of a Circle , the Suns declinations . And above that in four other Quadrants of Circles , the days of the Moneth , respecting the four seasons of the year . Underneath the Projection , towards the Limbe is put on , in one half of a Quadrant , one of the sides of the Geometrical Quadrat , and in the other half the Line of shadows . All which is bounded in by the equal Limbe . There stands moreover on the very edges of the Quadrant , two Dyalling Scales , which do not proceed from the Center ; that on the right edge is called the Line of Latitudes ; and that on the left edge the Scale of Hours ( equal in length to the Sines ) which is no other then a double Tangent , or two Lines of Tangents to 45d each set together in the middle , and so might , if there were need , be continued , ad infinitum . The Construction and making of such of these Lines as are not commonly described in other Treatises . To inscribe the Line of Declinations , there will be given the Suns declination to find his right Ascension , which is the Ark of the Limb , that by help of a Ruler , moving on the Center of the Quadrant , and laid over the same , will in-scribe the Declination proposed . The Canon to find the Suns right Ascension from the nearest Equinoctial point , correspondent to the Declination proposed , is As the Radius To the Cotangent of the Suns greatest Declination : So the Tangent of the Declination given : To the Sine of the Suns right Ascension . The four Quadrants for the days of the moneth are likewise to be graduated from the Limb , by help of a Table of the Suns right Ascensions , made for each day in the year . The Geometrical Quadrat is inscribed in half the Quadrant of a Circle , by finding in the Table of natural Tangents , what Arches answer to every equal Division of the Radius , and so to be graduated from the Limb ; so 300 sought in the Tangents gives the Ark of 16d 42 m of the Limb against which 3 of the Quadrat is to be graduated . The Line of shaddows is no other then the continuance of the Quadrat beyond the Radius , and so the making after the same manner ; thus having the length of the shadow assigned , annex the Ciphers of the Radius thereto , and seek in he natural Tangents , what Ark corresponds thereto ; thus the shadow being assigned thrice as long as the Gnomon , I seek 3000 in the natural Tangents , the Ark answering thereto , is 71d 34′ which being counted from the left edge of the Quadrant , towards the right in the Limb , the Line of shadows may from thence be graduated ; the Complement of this Ark is the Suns Altitude , answering to that length of the right shadow , being 18d 26′ . The Canon to make the Line of Latitudes , will be As the Radius to the Chord of 90d so the Tangents of each respective degree of the Line of Latitudes . To the Tangents of ohter Arks : The natural Sines of which Arks are the numbers that from a Diagonal Scale of equal parts shall graduate the Divisions of the Line of Latitudes to any Radius . To draw the Projection . Those Lines that cross each other , are Arches of Circles , whose Centers fall in two streight Lines . Of the Paralels of Altitude . All those Arks whose Aspect denotes them to be drawn from the right edge of the Quadrant towards the left , are called Paralels of Altitude , and their Centers fall in the right edge of the Quadrant , continued both beyond the Center and Limb so far as is needful . To find the Intersections of the Paralels of Altitude , with the Meridian , that is , Points therein limiting the Semi-diameters of the Paralells . Assume any Point in the right edge of the Quadrant , ( which is called the Meridian Line ) near the Limbe to be the Tropick of Cancer ; the distance of this Point from the Center of the Quadrant , must represent the Tangent of 56d 46′ which is half the Suns greatest Declination more then the Radius ; the distance of the Equator from the Center , shall be equal to the Radius of this Tangent . For the finding the Intersections of the other Paralells of Altitude , it will be best to make a Line of Semi-tangents to the same Radius , that is to number each degree of this Tangent with the double Ark , and so every half degree will become a whole one : Out of this Line of Semi-tangents prick off from the Center of the Quadrant 66d 29′ the Complement of the Suns greatest Declination , which will find the Intersection of the Tropick of Capricorn with the Meridian . Now to fit the Projection to any particular Latitude : Out of the said Line of Semi-tangents from the Center of the Quadrant , prick off the Latitude of the place , and it will find a point in the Meridian Line , where the Horizon , or Paralell of 00d of Altitude will intersect the Meridian ; this Point is called the Horizontal Point , and serves for finding the Centers of all the Paralells . To the Latitude of the place add each degree of Altitude successively till you have included the greatest Meridian Altitude ; these compound Arks are such as being prickt from the Center of the Quadrant out of the Line of Semi-tangents will find points in the Meridian Line , limiting the Semi-diameters of the paralells of Altitude . Above the Horizon , and between the Circle that bounds the Projection falls a portion thereof called the Reverted Tail , which otherwise would if it had not been there reverted , have excurred the limits of a Quadrant . To find the Intersections for those Paralells of Altitude , substract successively each degree out of the Latitude of the place , and the remaining Arks prick from the Center out of the Line of Semi-tangents : The use of this Tail being to find the hour and Azimuth before or after 6 in the Summer time only , it need be continued no further above the Horizon then the Ark of the Suns greatest Altitude at 6 , which at London is 18d 12′ . To finde the Centers of the Paralells of Altitude . These are to be discovered by help of a Line of natural Tangents , not numbred with the double Arks , whose Radius must be equal to the distance of the Equator from the Center of the Quadrant , or which is all one to 90d of the Line of Semi-tangents : Out of this Line of Tangents prick off beyond the Center of the Quadrant the Complement of the Latitude , the distance between the Point thereby found , and the Horizontal Point is the Semi-diameter wherewith the Horizon is to be drawn . To find the Centers of the rest of the Paralells . To the Complement of the Latitude add each degree of Altitude successively till you have included the greatest Meridian Altitudes ; The Tangents of these Arks prick beyond the Center , the distance from the Points so discovered to the Horizontal Point , are the Semi-diameters of the Paralells of Altitude ; the extreamities of which Semi-diameters being limited in the Meridian Line ; these extents thence prict , finds their Centers . Some of these Compound Arks will exceed 90 degrees , as generally where any Meridian Altitude is greater then the Latitude . In this case substract those Arkes from 180d and prick the Tangents of the remaining Arks from the Center of the Quadrant on the Meridian Line continued beyond the Limbe , and then as before the distances between those Points and the Horizontal Point , are the Semi-diameters of those Paralells , whose Extremities are limited in the Meridian Line . To find the Centers of the Paralells of the reverted Tail. From the Complement of the Latitude substract each degree of Altitude in order , till you have included the greatest Altitude of 6 the Tangent of the remaining Arks prick from the Center of the Quadrant , and you will find such Points the distances between which and the Horizontal Point are the Semi-diameters of those Paralells . To find the Centers and Semi-diameters of the Azimuths . All those Portions of Arks which issue from the top of the Projection towards the Limb are called Azimuths , the Centers of them all fall upon that Paralell of Altitude which is equal to the Latitude of the place whereto the Projection is fitted , which will always be a streight Tangent Line . Out of the former Line of Tangents , whose Radius is equal to the distance of the Equator from the Center of the Quadrant , prick down the Latitude of the place on the Meridian Line , and thereto perpendicularly erect the Line for finding the Centers of the Azimuths , which must be continued through and beyond the Projection . Out of the said Line of Tangents and beyond the Center prick down the Tangent of half the Complement of the Latitude at London 19d 14 m and it will discover a Point which is called the Zenith Point , because in it all the Azimuths do meet ; The distance between this Point and the Point where the Center Line of the Azimuths intersects the Meridian make the Radius of a Tangent , out of which Tangent prick down each degree successively , both within and beyond the Projection on the Line of Centers , and you have the Centers for all the Azimuths ; where note , that the Centers of all Azimuths which exceed 90d will fall within the Projection , and of all others without , the distances of these respective Points from the Zenith Point , are the Semi-diameters of the Azimuths , with which extents let them be respectively drawn . To draw the Summer and Winter Ecliptick and to divide them . The Summer Ecliptick is drawn from the Point of the Equator in the left edge of the Quadrant to the Tropick of Cancer , and the Winter thence to the Tropick of Capricorn out of a Line of Tangents to the Radius equal to the distance of the Equator , from the Center prick down the Tangent of 23d 31′ the Suns greatest declination from the Center of the Quadrant on the Meridian Line towards the Limbe , and you shall discover the Center of the Summer Ecliptick with the same extent , being the Semi-diameter thereof , set one foot down at the Tropick of Capricorn , and the other will fall beyond the Center of the Quadrant on the right edge , and discovers the Center for drawing the Winter Ecliptick ; to divide them use this Canon . As the Radius to Tangent of the Suns distance from the nearest Equinoctial Point : So the Cosine of the Suns greatest Declination : To the Tangent of the Suns right Ascension , which must be counted in the Limbe , and from it the Suns true place graduated on both the Eclipticks . To draw the two Horizons , and to divide them . One of the Horizons is the Paralel of 00d of Altitude , which being intersected by the Azimuth Circles , is thereby divided into the degrees of the Suns Amplitude ; this is the upper Horizon , and the drawing hereof was shewed already . The other Horizon is but this inverted , and the Divisions transferred from that , the Center of it is found by pricking the Tangent of the Complement of the Latitude on the Meridian Line from the Center of the Quadrant , the distance of the Equator being Radius . But it may be also done from the Limbe by the Proportion following . As Radius , to Tangent of the Latitude ; So the Tangent of the Suns greatest Declination , to the sine of the greatest Ascensional difference ( which converted into Time , gives the time of the Suns rising or setting before or after 6 ) by which Ark of the Limbe the Horizon is limitted ; Then to divide it say As the Radius , to the Tangent of the assigned Amplitude : So is the Sine of the Latitude : To the Tangent of the Ascensional difference agreeing thereto , which counted in the Limb , from it the Amplitudes may be divided on both the Horizons ; and note , if these Amplitudes be not coincident with those the Azimuths have designed , then are the said Azimuths drawn false . To inscribe the Stars on the Projection . Such only , and no other as fall between the two Tropicks , may be there put on . Set one foot of the Compasses in the Center of the Quadrant , and extend the other to that place of either of the Eclipticks , as corresponds to the given declination of the Star , and therewith sweep an occult Ark : I say then that a Thread from the Center of the Quadrant laid over the Limb to the Stars right Ascension where it intersects , the former occult Ark is the place where the proposed Star must be graduated . Of the Almanack . There is also graved in a Rectangular Square , or Oblong , a perpetual Almanack , which may stand either on the foreside or back of the Quadrant , as room shall best permit . On the Backside of the Quadrant there is , 1. On the right edge a Line of Signs issuing from the Center , the Radius whereof is in length 5 inches . 2. On the left edge a Line of Chords issuing from the Center . 3. On the edges of the Quadrant there are also two Scales for the more ready finding the Hour and Azimuths in one Latitude ; the Hour Scale is no other then 62d of a Line of Sines , whose Radius is made equal to half the Secant of the Latitude being fitted for London ) to the common Radius of the Sines ; the prickt Line of Declination annexed to it , and also continued beyond the other end of it , to the Suns greatest Declination is also a portion of a Line of Sines , the Radius whereof is equal to the Sine of the Latitude taken out of the other part of the Scale , or which is all one the Sine of the Suns greatest declination is made equal to the Sine of the greatest Altitude at the hour of 6 taken out of the other part of the Scale , which at London is 18d 12 m 4. The Azimuth Scale is also 62d of a Line of Sines , whose Radius is made equal to half the Tangent of the Latitude to the common Radius of the Sines , the Line of the Declination annexed to it , and continued beyond it : To the Suns greatest Declination is also a portion of a Line of Sines of such a length whereof the Sine of the Latitude is equal to the Radius of the Sines of the other part of this fitted Scale ; or which is all one , the length of the Suns greatest Declination is made equal to the Suns greatest Vertical Altitude , which in this Latitude is 30d 39′ of the other Sine or Line of Altitudes . The Limbe is numbred both with degrees and time , from the right edge towards the left . Between the Limbe and the Center are put on in Circles , the Scales following . 1 , A Line of Versed Sines to 180 degrees . 2. A Line of Secants to 60d the graduations whereof begin against 30d of the Limbe , to apply which Vacancy , and for other good uses , there is put on a Line of 90 Sines , ending where the former graduations begin ; this is called the lesser Sines . 3. A Line of Tangents graduated to 63d 26′ 4. A Line of Versed Sines to 60d through the whole Limbe , called the Versed Sines quadrupled , because the Radius hereof is quadruple to the Radius of the former Versed Sines . 5. A Line of double Tangents , or Scale of hours , being the same Dyalling Scale as was described on the foreside . 6. A Tangent of 45d or three hours through the whole Limbe for Dyalling , which may also be numbred by the Ark doubled to serve for a Projection Tangent , alias a Semi-tangent . 7. In another Quadrant of a Circle may be inscribed a portion of a Versed Sine to eight times the Radius encreased , of that of 180d called the Occupled Versed Sine , and at the end of this from the other edge , another portion of a Versed Sine to 12 times the Radius encreased may be put on . 8. Lastly , above all these is the Scale of Hours or Nocturnal with Stars names graved within it towards the Center ; this is divided into 12 equal hours and their parts , and the Stars are put on from their right Ascensions , only with their declination figured against them . All the Lines put on in Quadrants of Circles must be inscribed from the Limbe by help of Tables , carefully made for that purpose ; an instance shall be given how the Line of Versed Sines to 180d was inscribed , and after the same manner that was put on , must all the rest : Imagine a Line of Versed Sines to 180d to stand upon the left edge of a Quadrant from the Center with the whole length thereof upon the Center sweep the Arch of a Circle , and then suppose Lines drawn through each graduation or degree thereof continued parralel to the right edge till they intersect the Arch formerly swept which shall be divided in such manner as the Line of Versed Sines on this Quadrant is done . But to do this by Calculation , A Table of natural Versed Sines must first be made , which for all Arks under 90d are found by substracting the Sine Complement from the Radius , so the Sine of 20d is 34202 which substracted from the Radius rests 65798 , which is the Versed Sine of 70d : And for all Arks above 90d are got by adding the Sine of the Arks excess above 90d unto the Radius : thus the Versed Sine of 110d is found by adding the Sine of 20d to the Radius , which will make 134202 for the Versed Sine of the Said Ark. This Table , or the like of another kind , being thus prepared , the proportion for inscribing of it will hold . As the length of the Line supposed to be posited on the left edge , Is to the Radius , So is any part of that length To the Sine of an Arch , which sought in the Tables , gives the Arch of the Limbe against which the degree of the Line proposed must be graduated . But in regard the Versed Sine of 180d is equal to the double of the Radius ; the Table for inscribing it will be easily made by halfing the Versed Sine proposed , and seeking that half in the Table of natural Sines , so the half of the Versed Sine of 70d is 32899 which sought in the Table of natural Sines , gives 19d 13′ fore of the Limb against which the Versed Sine aforesaid is to be graduated , and so the half of the Versed Sine of 110d is 67101 which answers to 42d 9′ of the sines or Limb. So likewise the Table for putting on the lesser Sines was made by halfeing the natural Sines , and then seeking what Arks corresponded thereto in the natural Sines aforesaid ; those that think these Lines to many may very well want the Versed Sines so oft repeated ; And they that will admit of a Radius of 6 or 7 inches , may have the Line of Lines Superficies and Solids , put on in the Limb on the foreside , and the Segments Quadrature , Equated Bodies , Mettalls , and inscribed Bodies , or other Lines at pleasure put on upon the backside , as hath been already done upon some Quadrants . Now to the Use . The Vses of the PROJECTION . Of the Almanack . BEfore the Projection can be used , the day of the Moneth , the Suns place or Declination must be known ; but these are commonly given by the knowledge thereof : Now this Almanack will as much help to the obtaining hereof , as any other common Almanack . It consists of a Rectangular Oblong , or long Square divided into 7 Colums in the breadth to represent 7 days of the week , accounting the Lords day first ; and length ways into 9 Columns , the two uppermost represent the months of the year , accounting March the first , the five middlemost the respective days of each Month , and the two undermost some certain leap years , posited in such Columns , as that thereby may be known by Inspection , what day of the Week the first of March happened upon in the said Leap years ; the contrivance hereof owns its original from my Worthy Friend Mr. Michael Darie , for the due placing of the Months over the Columns of days , take the following Rule in his own words . First having March assign'd to lead the round , The rest o' th Months are easily after found ; If that you take the complement in days To 35 of a plac'd Month always , And count it from its place with due Progression It shews you where the next Month takes possession . Thus placing the Month of March first , then if I would place April , or the second Month , March having 31 days , the Complement thereof to 35 is 4 then counting four Columns from the place of March , it falls upon the 5 Column , where the figure 2 is placed for the 2d Moneth , then April being placed ; if I would place May I take 30 , the number of days in April , from 35 there rests 5 , and counting 5 Columns from the place of April where it ends , which is in the 3d Column , the figure 3 is placed for the 3d Moneth or Moneth of May. The next thing to be known is on what day of the Week the first day of March falleth upon , which is continually to be remembred in using the Almanack . This for some Leap years to come , may be known by counting in what Column the said Leap year is graved , thus in Anno 1660 , the first of March falls upon a Thursday , because 60 is graved in the 5 Column , that being the fift day of the Week : But for a general Rule take it in these words . To the number two add the year of our Lord , and a fourth part thereof , neglecting the odd remainder , when there is any ; the Amount divide by 7 the remainder , when the Division is finished , shews the number of Direction , or day of the week , on which the first day of March falleth , accounting the Lords day the first ; but if nothing remain , it falls on a Saturday .   2   Example for the year 1657 The even fourth thereof 414   2073 ( 296 quotient . 7 ) 1 remaining .   By this Rule there will be found to remain one for the year of our Lord 1657 whence it follows that the first day of March fell on the first day of the Week , alias , the Lords day in that year ; so in Anno 1658 , there remains 2 for Munday : in 1659 , rests 3 for Tuesday ; in 1660 rests 5 for Thursday ; so that hence it may be observed , that every 4 years the first of March proceeds 5 days : Upon which supposition the former Rule is built ; say then As 4 to 5 , or as 1 to 1¼ so is the year of the Lord propounded , to the number of days , the first of March hath proceeded in all that Tract , caused by the odd day in each year , and the Access of the days for the Leap years ; this number divided by 7 , the remainder shews the fractionate part of a Week above whole ones , which the said day hath proceeded , which wil not agree with the day of the Week the first of March falls upon , according to common tradition , unless the number two be added thereto , which argues that the first of March , as we now account the days of the week fell upon Munday , or the second day of the week in the year of our Lords Nativity : This is only for Illustration of the former Rule , being to shew that the adding of the even fourth part of the year of our Lord thereto , works the proportion of 4 to 5. The Vse of this Almanack is to know for ever on what day of the Week any day of the Month falls upon . Remembring on what day of the Week the first day of March fell upon in the year propounded ( which doth then begin in the use of this Almanack , and not sooner or later , as upon New-years day , or Quarter day ) all the figures representing the days of the Month do also represent the same day of the week in the respective Months under which they stand ; and the converse , the Moneth being assigned , all the figures that stand as days under it , inform you what days of the said Month the Week day shall be the same , as it was upon the first day of March , and then by a due Progression it will be easie to find upon what day of the Moneth any day of the week falleth , as well as by a common Almanack , without the trouble of always one , and sometimes two Dominical Letters quite shunned in this Almanack , by beginning the year the first of March , and so the odd day for Leap year is introduced between the end of the old , and the beginning of the new-year . Example . In Anno 1657. looking for the figure 10 in the Column for Months , for the Month of December ; under it I find 6 , 13 , 20 , 27 , now the first of March being the Lords day , I conclude also that these respective days in December , were likewise on the Lords day ; and from hence collect , that Christmas day , which is always the 25 of that Month , happened on a Friday . Vses of the Projection . THis Projection is no other then a fourth part of Stoflers particular Astrolabe , fitted for the Latitude of London inverted , that is , the Summer Tropick and Altitudes , &c. turned downwards towards the Limb , whereas in his Astrolabe they were placed upwards , towards the Center ; thus the Quadrant thereof made , is rendred most useful and accurate when there is most occasion for it ; before the projection can be used , the Bead must be rectified , and because the Thread and Bead may stretch , there may be two Beads , the one set to some Circle concentrick to the Limb , to keep the other at a certainty in stretching , and the other to be rectified for use . To rectifie the Bead. LAy the Thread over the day of the Month in its proper Circle , and if the season wherein the Quadrant is to be used , be in the Winter half year , set the Bead by removing it to the Winter Ecliptick ; but in Summer let it be set to the lower or Summer Ecliptick , and then it is fitted for use , One Caution in rectifying the Bead is to be given ; and that is in Summer time if it be required to find the hour and Azimuth of the Sun by the Projection , before the hour of 6 in the morning , or after it in the evening , or which is all one , when the Sun hath less Altitude then he hath at 6 of the clock ; then must the Bead be rectified to the Winter Ecliptick , and the Parralels above the Horizon in the Reverted Tail , are those which will come in vse . To find what Altitude the Sun shall have at 6 of the clock in the Summer half year . This will be easily performed by bringing the Bead that is rectified to the Summer Ecliptick to the left edge of the Quadrant , and-there among the Paralels of Altitude it shews what Altitude the Sun shall have at 6 of the clock : It also among the Azimuths shews what Azimuth the Sun shall have at the hour of 6. Example , So when the Sun hath 17 degrees of North Declination , as about the 27 of April , his Altitude at the hour of 6 will be found to be 13d 14 m and his Azimuth from the Meridian 79d 14 m whence I may conclude if his observed Altitude be less upon the same day , and the Hour and Azimuth sought , the Bead must be set to the Winter Ecliptick , and the Operation performed in the reverted Tail. Here it may be noted also that the exactest way of rectifying the Bead , will be either from a Table of the Suns Declination , laying the Thread over the same in the graduated Circle , or from his true place , laying it over the same in the proper Ecliptick , or from his right Ascension counted in the Limb. Or Lastly from his Meridian Altitude on the right edge of the Quadrant , for these do mutually give each other the Bead , being rectified to the respective Ecliptick as before . for Example . To find the Suns Declination . The Thread laid over the day of the Moneth , intersects it upon that Circle whereon it is graduated , which in the Summer half year is to be accounted on this side the Equinoctial , North , and in the Winter-half year , South ; so laying the Thread over the 27th . day of April , it intersects the Circle of Declination at 17 degrees , and so much was the Suns Declination . To find the Suns true place . The Thread lying as before , shews it on the respective Ecliptick , So the Thread lying over the 17 of April , will cut the Summer Ecliptick , in 17d 7 m of Taurus ; or in 12d 53 m of Leo , which agrees to the 26 day of July , or thereabouts , the Thread intersecting both these days at once ; and the opposite points of the Ecliptick hereto , are 17d● m in Scorpio , about the 20 of October ; and 12d 53 m of Aquarius , about the 22d of January , all shewed at once by the Threads position . To find the Suns right Ascension . Lay the Thread over the day of the Month as before , and it intersects it in the equal Limb ; whence taking it in degrees and minutes of the Equator , whilst the Sun is departing from the Equator towards the Tropicks , it must be counted as the graduations of the Limb , from the left edge towards the right ; but when the Sun is returning from the right edge towards the left ; the right Ascension thus found , must be estimated according to the season of the year . From June 11 to Sept. 13 It must have 90 degrees added to it . Sep. 13 to Dec. 11 It must have 180 degrees added to it . Dec. 11 to Mar. 10 It must have 270 degrees added to it . But in finding the Hour of the night by the Quadrant , we need no more then 12 hours of Ascension , for either Sun or Star , and the Limb is accordingly numbred from the left edge towards the right , from 1 to 6 in a smaller figure , and thence back again to 12 , and the other figures are the Complements of these to 12. so that when the Sun is departing from the Equator towards the Tropicks ; his right Ascension is always less then 6 hours , and the Complement of it more ; but when he is returning from the Tropicks towards the Equator , it is always more then 6 hours , and the Complement of it less ; the odd minutes are to be taken from the Limb , where each degree being divided into 4 parts , each part signifies a Minute of time , and to know whether the Sun doth depart from , or return towards the Equator , is very visible , by the progress and regress of the days of the month , as they are denominated on the Quadrant . Example . So the Thread laid over 17d of Declination , which will be about the 27 April The Suns right Ascension will be 44d 37 m In time 2 h 58′ 26 July The Suns right Ascension will be 135 23 In time 9 2 20 October The Suns right Ascension will be 224 37 In time 2 58 22 January The Suns right Ascension will be 315 23 In time 9 2 But here the latter 12 hours are omitted . Such Propositions as require the use of the Bead , are , To find the Suns Amplitude , or Coast of rising and setting from the true East or West . Bring the Bead , being rectified to either of the Eclipticks , it matters not which , to either of the Horizons , and the Thread will intersect the Amplitude sought , upon both alike : Example ; The Suns Declination being 17 North , or South , the Suns Amplitude , will be found to be 28● 2m. The Amplitude before found for the Summer half year , is to be accounted from East or West Northwards ; and in the Winter half year from thence Southwards . To find the time of the Suns rising or setting . The Thread lying in the same Position , as in the former Proposition , intersects the Ascensional difference in the Limb , which may there be counted either in degrees or Time. Example . So the Bead lying upon the Horizon , being rectified to 17● of Declination , the Thread intersects the Limb at 22d 38 m , which is 1 h 30 m of time , and so it shews the time of Suns rising in Summer , or setting in Winter , to be at half an hour past 4 ; and his rising in Winter , and setting in Summer , to be at half an hour past 7. To find the length of the Day or Night . The time of the Suns rising and setting are one of them ; the Complement of the other to 12 hours ; so that one of them being known , the other will be found by Substraction ; the time of Suns setting is equal to half the length of the day ; and this doubled gives the whole length of the day ; in reference to the Suns abode above the Horizon , the time of setting converted into degrees , is also called the Semi-diurnal Ark ; the time of Sun rising ( so converted is called the Semi-nocturnal Ark ) doubled gives the whole length of the Night ; so upon the 27th day of April , the Sun having 17d of Declination , the length of the day is 15 hours , and the length of the night 9 hours . To find the Suns Altitude on all Hours ; or at any time proposed . In Summer time , if the hour proposed be before 6 in the morning , or after it in the evening ▪ lay the Thread to the hour in the Limb , the Bead being first rectified to the Winter Ecliptick , and amongst the Paralels of Altitude above the upper Horizon , it shews the Altitude sought . Example . So the Sun having 16d of declination Northwards , as about the 24th of April , laying the Thread over the Declination , I set the Bead to the Winter Ecliptick , and if it were required to find what Altitude the Sun shall have at 36 minutes past 6 in the afternoon , lay the Thread over the same in the Limb , and the Bead among the Parralels of Altitude will fall upon 7d , At all other times the Operation is alike ; the Bead being rectified to that Ecliptick that is proper to the season of the year : Lay the Thread over the proposed hour in the Limb , and the Bead amongst the Parralels of Altitude , sheweth the Altitude sought . Example . So if it were required the same day to find what Altitude the Sun should have at 19 m past 2 in the afternoon ; Lay the Thread in the Limb over the time given , and the Bead among the Parralels of Altitude will fall upon 45d for the Altitude sought . To finde the Suns Altitude on all Azimuths . IN the Summer half year , if the Azimuth propounded be more Northward then the Azimuth of the Sun shall have at the hour of 6 ; The Bead must be rectified to the Winter Ecliptick , and brought to the Azimuth proposed above the upper Horizon , and there among the Parralels of Altitude , it sheweth the Altitude sought . So about the 24th of April , when the Suns Declination is 16d his Azimuth at 6 of the clock will be found to be 76d 54 m from the South ; Then if it were required to find the Suns Altitude upon an Azimuth more remote , as upon 107d from the South , laying the Thread over the Declination , I set the Bead to the Winter Ecliptick , and afterwards carrying it to the Azimuth proposed among the Parralels of Altitude above the upper Horizon , it falleth upon 7d for the Suns Altitude sought . In all other Cases bring the Bead rectified to the Ecliptick proper to the season of the year , to the Azimuth proposed ; and among the Parralels of Altitude it sheweth the Altitude sought ; So far the same day , I set the Bead to the Summer Ecliptick , and if it were required to know what Altitude the Sun shall have when his Azimuth is 50d 48′ from the Meridian carry the Bead to the said Azimuth , and among the Parralels of Altitude it will fall upon 45d for the Altitude sought . The Hour of the night Proposed to find the Suns Depression under the Horizon . IMagine the Sun to have as much Declination on the other side the Equinoctial , as he hath on the side proposed ; and this Case will be co-incident with the former of finding the Suns Altitude for any time proposed ; the reason whereof is because the Sun is always so much below the Horizon at any hour of the night , as his opposite Point in the Ecliptick is above the Horizon at the like hour of the Day . Such Propositions as depend upon the knowledge of the Suns Altititude , are to find the Hour of the Day , and the Azimuth ( or true Coast ) of the Sun. THe Suns Altitude is taken by holding the Quadrant steady , and letting the Sun Beams to pass through both the Sights at once , and the Thread hanging at liberty shews it in the equal Limb , if this be thought unsteady , the Quadrant may rest upon some Concave Dish or Pot , into which the Plummet may have room to play ; but for greate Quadrants there are commonly Pedistalls made . The Altitude supposed to find the Hour of the Day , and the Azimuth of the Sun in Winter . REctifie the Bead to the Winter Ecliptick , and carry it along amongst the Parralels of Altitude till it cut or intersect that Parralel of Altitude on which the Sun was observed , and the Thread in the Limb sheweth the hour of the Day , and the Bead amongst the Azimuths sheweth the Azimuth of the Sun. Example . So about the 18 of October , when the Suns Declination is 13d 20′ South if his observed Altitude were 18d the true time of the day would be found to be either 36 minutes after 9 or 24 minutes past 2 and his Azimuth would be 37 degrees from the South . To finde the Hour of the Day , and the Azimuth of the Sun at any time in the Summer half year . IT was before intimated , That if the question were put when the Sun hath less Altitude then he hath at the hour of 6 of the clock , that then the Operation must be performed among those Parralels above the upper Horizon , in the reverted Tail , the Bead being rectified to the Winter Ecliptick ; and that it might be known what Altitude the Sun shall have at 6 of the clock , by bringing the Bead rectified to the Summer Ecliptick , to the left edge of the Quadrant . So admitting the Sun to have 16d of North Declination , which will be about the 24 April , I might finde his Altitude at 6 of the Clock by bringing the Bead rectified to the Summer Ecliptick to the left edge of the Quadrant ; to be 12d 28 m whence I conclude , if his Altitude be less , the Bead must be rectified to the Winter Ecliptick , and be brought to those Parralels above the upper Horizon ; and it may be noted , that the Suns Altitude at 6 is always less then his declination . Example . Admit the 24th of April aforesaid the Suns observed Altitude were 7d laying the Thread over the Suns Declination , or the day of the month ; I rectifie the Bead to the Winter Ecliptick , and bring it to the said Parralel of Altitude above the upper Horizon ; and the Thread intersects the Limb at 9d 3 m shewing the hour of the day to be 24 minutes past 5 in the morning , or 36′ past 6 in the evening , and the Bead amongst the Azimuths shews the Azimuth or Coast of the Sun to be 107d from the South . Another Example . But admitting the Sun to have more Altitude then he hath at the hour of 6 , the Operation notwithstanding differs not from the former , but only in rectifying the Bead , which must be set to the Summer Ecliptick , and then carried to the Parralel of the Suns observed Altitude , and the Thread will intersect the Limb at the true time of the day , and the Bead amongst the Azimuths sheweth the true Coast of the Sun. So upon the 24th of April aforesaid , the Suns observed Altitude being 45d , I bring the Bead rectified to the Summer Ecliptick , to the said Parralel of Altitude , and the Thread intersects the Limb at 55d 15 m shewing the hour to be either 41 m past 9 in the morning , or 19 m past 2 in the ofternoon ; to be known which by the increasing or decreasing of the Altitude , and the Bead amongst the Azimuths shews the Azimuth or true Coast of the Sun to be 50d 40● from the South : Another Example . Admit when the Sun hath 19d 13 m of North Declination which will be about the 6th of May , his observed Altitude were 56d the Bead being set to the Summer Ecliptick , and brought to that Parralel of Altitude amongst the Azimuths shews the Suns true Coast to be 23d from the South , Eastwards in the forenoon , and Westwards in the afternoon , and the Thread in the Limb sheweth the true time of the day to be either 7′ past 11 in the forenoon , or 5● m past 12. The Depression of the Sun supposed to find the true time of the night with us , or the hour of the day to our Antipodes ; As also the true Coast of the Sun upon that Depression . THis Proposition may be of use to know when the Twilight begins or ends , which is always held to be when the Sun hath 18d of Depression under the Horizon , to perform this , Imagine as much Declination on the contrary side the Equinoctial , as the Declination given , and find the time of the Day , as if the Suns Altitude were 18d So when the Suns Declination is 16d North , as about the 24th of April , laying the Thread over it I rectifie the Bead to the Winter Ecliptick , and bringing it to the Parralel of 18d the Thread in the Limb shew the Twilight to begin at 54 m past 1 in the morning , and ends at 6′ past 10 at night , and the Azimuth of the Sun to be 28d 58′ which in this Case is to be accounted from the North. But if the Suns greatest Depression at night be less then 18d as that it may be in any Latitude where the Meridian Altitude at any time in Winter or the opposite Signe is less then 18d there is no dark night which in our Latitude of London will be from the 12th of May to the 11th of July . Of the Stars graduated on the PROJECTION . SUch Stars as are between the two Tropicks only , are there inscribed , and such haue many things common in their Motion with the Sun when he hath the like Declination , as the same Amplitude , Semidiurnal , Arke , Meridian , Altitude , Ascentional difference , &c. These Stars have Letters set to them to direct to the Circle of Ascensions on the back of the Quadrant , where the quantity of their right Ascension , is expressed from one of the Equinoctial points ; those that have more Ascension then 12 hours from the point of Aries , are known by the Character plus + set to them ; many more Stars might be there inserted , but if they have more then 23d 31′ of Declination , the Propositions to be wrought concerning them are to be performed with Compasses , by the general Lines on the Quadrant . To find the true time of the Day or Night when any Star commeth to the Meridian . In the performing of this Proposition we must make use of the Suns whole right Ascension in time , which how that might be known hath been already treated of , as also of the Stars whole right Ascension , which may be had from the Circle of Ascensions on the back of the Quadrant if 12 hours be added to the right Ascension of a Star taken thence that hath a Letter Character † affixed to it . Substract the Suns whole right Ascension from the Stars whole right Ascension , encreased by 24 hours when Substraction cannot be made without it , the remainder is less then 12 shews the time of the afternoon or night when the Star will be upon the Meridian ; but if there remain more then 12 , reject 12 out of it and the residue shews the time of the next morning when that Star will be upon the Meridian . Example . The 23d of December the Suns whole right Ascension is 18 hours 53′ which substracted from 4 ho : 16′ the right Ascension of the Bulls eye encreased by 24 there remains 9 h 23′ for the time of that Stars comming to the Meridian , and being substracted from 6 ho : 30′ the right Ascension of the great Dogg , there rests 11 ho : 37′ for the time of that Stars coming to the Meridian at night . This Proposition is of good use to Sea-men , who have occasion to observe the Latitude by the Meridian Altitude of a Star , that they may know when will be a fit time for observation . In finding the time of the Night by the Stars , we use but 12 ho : of right Ascension , nor no more in finding the time of their rising or setting , so that when it is found whether it be morning or evening is left to judgement , and may be known by comparing it with the former Proposition , if there be need so to do . To find the Declination of any of these Stars . This is engraven or annexed to the Stars names , yet it may be found on the Projection , by rectifying a Bead to the proposed Star , and bringing the Thread and Bead to that Ecliptick it wil intersect ; and in the same Position the Thread will intersect the said Stars declination in the Quadrant of Declinations ; if the Bead meet with the Summer Ecliptick the Declination is North , if with the Winter South . To find the Amplitude and Ascensional difference of any of the Stars on the Projection . BRing the Bead rectified to the Star to either of the Horizons , the Thread being kept in its due Extent , and where it intersects the same it shews that Stars Amplitude which varies not , and is Northward if the Star have North declination , otherwise Southwards , the Thread likewise intersecting the Limb , sheweth the Stars Ascensional difference . Example . So the Bead being rectified to the Bulls eye , and brought to the lower Horizon , shews the Amplitude of that Star to be 25d 54′ Northwards because the Star hath North Declination ; And the Thread lyeth over 20′ 49′ of the Limb which is this Stars Ascensional difference , which in Time is 1 ho 23 m The Thread in the Limb lyeth over 4 ho 37 m from midnight for the Stars hour of rising , and over 7 ho 23 m from the Meridian for the Stars hour of setting always in this Latitude which with the Amplitude varies not , except with a very small allowance in many years . To find a Stars Diurnal Ark , or the Time of its continuance above the Horizon . When the Star hath North Declination add the Ascensional difference of the Star before found in Time to 6 hours , the Sum is half the time . South Declination Subtract . the Ascensional difference of the Star before found in Time from 6 hours , the Residue is half the time . Of that Stars continuance above the Horizon , which doubled , shews the whole time , the Complement wherof to 24 ho is the time of that Stars durance under the Horizon . Example . So the Ascensional difference of the Bulls eye being in time 1 ho : 23 added to 6 hours , and the Sum doubled makes 14 hours 46 m for the Stars Diurnal Ark or abode above the Horizon , the residue whereof from 24 is 9 ho : 14 m for the time of its durance under the Horizon . To find the true time of the Day or night , when the Star riseth or setteth . THe Stars hour of rising or setting found as before , being no other but the Ascensional difference of the Star added to , or substracted from 6 hours ; which the Thread sheweth in the Limb the Bead being rectified to a Star , and brought to that Horizon it will intersect ; is not the true time of the night ; but by help thereof that may be come by ; this we have denominated to be the Stars hour , and is no other but the Stars horary distance from the Meridian it was last upon ; If a Star have North Declination the Stars hour of rising must be reckoned to be before 6 and the time of its setting after 6 South Declination the Stars hour of rising must be reckoned to be after 6 and the time of its setting before 6 Now the time of the Stars rising or setting found by this and the former Propositions must be turned into common time by this Rule . To the Complement of the Suns Ascension add the Stars Ascension , and the Stars hour from the Meridian it was last upon , the Amount if less then 12 shews the the time of Stars rising or setting accordingly ; but if it be more then 12 reject 12 as oft as may be , and the remaind-sheweth it . So upon the 23d of December for the time of the Bulls eye rising .   h m The Complement of the Suns Ascension found by the foreside of the Quadrant is — 5 7 And the said Stars Ascension on the backside is — 4 16 The Stars hour of rising is — 4 37 14 hours . From which 12 rejected rests 2 hours for the time of that Stars rising , which I conclude to be at 2 in the afternoon , because that Star was found to come to the Meridian at 23 m past 9 at night , the like Operation must be used to get the time of that Stars setting , which will be found to be at 4 ho 46 m past in the morning .   h m Complement ☉ Ascension — 5 7 Stars Ascension — 4 16 Stars hour of setting — 7 23 16 h. 46′ To find what Altitude and Azimuth a Star that hath North Declination shall have when it is 6 hours of Time from the Meridian . REctifie the Bead to the Star , and bring the Bead and Thread to the left edge of the Quadrant , and there among the Parralels of Altitude and Azimuths it sheweth what Altitude and Azimuth the Star shall have . Example . So the Bead being set to the Bulls eye , and brought to the left edge of the Quadrant it will be found to have 12′ 17′ Altitude , and 80d 3′ Azimuth from the South , when it is 6 hours of time from the Meridian , which Proposition is afterwards used to know to which Ecliptick in some Cases to rectifie the Bead as hath likewise been intimated before . The Azimuth of a Star proposed , To find what time of the Night the Star shall be upon that Azimuth , and what Altitude it shall then have . SUpposing the Azimuth proposed to be nearer the South Meridian then that Azimuth the Star shal have when it is 6 hours from the Meridian : Bring the Bead rectified to the Star , to the proposed Azimuth , and among the Parralels of Altitude it shews that Stars Altitude , and the Thread in the Limb shews that Stars hour to be turned into common time to attain the true time sought . Example . If the question were What Altitude the Bulls eye shall have when his Azimuth is 62d 48′ from South , this being less Azimuth then he hath at 6 hours from the Meridian , the rectified Bead being brought to the Azimuth sheweth among the Parralels the Altitude to be 39d and the Stars hour shewn by the Thread in the Limb is either 8 ho : 56′ or 3 ho : 4′ from the Meridian ; then if upon the 23 of December you would know at what time the Star shall have this Altitude on this Azimuth , Change the Stars hour into common time by the former Rule . Decemb. 23 Complement of ☉ Ascension 5 h 7′ 5 h 7′   Stars Ascension — 4 16 4 16   Stars hour — 8 56 3 4     18 19 12 27 And you will find it to be at 19′ past 6 in the evening , or at 27 m past midnight . For Stars of South Declination being they have no Altitude above the Horizon at 6 ho : distance from the Meridian , the operation will be the same , void of Caution . But for Stars of North Declination when the proposed Azimuth is more remote from the South Meridian then the Azimuth of that Star 6 ho from the Meridian , another Bead must be rectified to the Winter Ecliptick , and carried to the Azimuth proposed above the upper Horizon , where amongst the Parralels it shews the Altitude sought ; and the Thread in the Limb sheweth the Stars hour to be converted into common time . Example . The Azimuth of the Bulls eye being 107d 53′ from South , which is more then the Azimuth of 6 hours , the other Bead set to the Winter Ecliptick , and carried to that Azimuth in the Tail , shews the Altitude to be 6d and the Stars hour to be 5 ho : 18′ Or 6 ho : 42′ which converted into common time , as upon the 23d of December , will be either 41 m past 2 in the afternoon , or 5 m past 4 in the morning following . h ' h ' December 23 Complement Suns Ascension 5 7 5 7   Stars Ascension — 4 16 4 16   Stars hour — 5 18 6 42 Rejecting 12 the Total is — 2 41 Or 4 5 The Hour of the night proposed to find what Altitude and Azimuth any of the Stars on the Projection that are above the Horizon shall have at that time . FIrst turn common time into the Stars hour , the Rule to do it is , To the Complement of the Stars Ascension add the Suns Ascension , and the time of the night proposed , the Aggregate if less then 12 is the Stars hour ; if more reject 12 as oft as may be , and the remainder is the Stars hour sought . So the 23 of December , at 8 a Clock 59 minutes past at night what shall be the Horarie distance of the great Dogg from the Meridian Complement of great Doggs h ' Ascension — 5 30 Suns Ascension — 6 53 Time of the night — 8 59 The Sum is , 12 rejected — 9 22 Then for Stars of South Declination , rectifie the Bead to the Star proposed , and lay the Thread over the Stars hour in the Limb , and the Bead amongst the Parralels and Azimuths , shews the Altitude and Azimuth of the Star sought . Example . So the Bead being rectified to the great Dogg , and the Thread laid over 9 ho 22′ in the Limb , the Bead will shew the Altitude of that Star at that time of the night to be 14d and its Azimuth 39d from the South . The Operation is the same for Stars of North Declination when the Stars hour found as before is not more remote from the South Meridian then 6 hours on either side . But if it be more then 6 ho distance from the Meridian as before 6 after its rising , or after it before its setting , then as before suggested , one Bead must be rectified to the Star , and brought to the Summer Ecliptick , where the Thread being duly extended , another must be set to the Winter Ecliptick , and afterwards the Thread laid over the Stars hour in the Limb , this latter Bead will shew the Stars Azimuth and Parralel of Altitude in the reverted Tail above the upper Horizon . Example . So upon the 23 of December , I would know what Azimuth and Altitude the Bulls eye shall have at 4 a Clock 5 minutes past the morning following . Time proposed 4 h 5′ Complement of Bulls eye Ascension — 7 44 Suns Asce●sion 23 of December — 6 53 12 rejected rests — 6h 42′ Proceed then and lay the Thread over 42′ past 6 and the Bead among the other Paralels in the Tail sheweth the Stars Altitude to be 6d and its Azimuth from the Meridian 107d 53′ These two Propositions have a good tendency in them to discover such Stars as are upon the Projection if you know them not , but supposing them known the Proposition of chiefest use is By having the Altitude of a Star given to find out the true Time of the night , and the Azimuth of that Star. If the Stars observed Altitude be less then its Altitude at 6 ho : distance from the Meridian ; Bring the Bead , rectified to the Star , to the Summer Ecliptick and set another Bead to the Winter Ecliptick , Then carry it to the Parralell of Altitude above the upper Horizon in the Reverted taile and there it will shew the Azimuth of that Star ; and the thread in the Limbe shews the houre . Example . So if the observed Altitude of the Bulls eye were 6d its Azimuth would be found to be 107d 53′ from the South , and its hour 42′ past 6 from the Meridian the true time would be found to be 5 minutes past 4 in the morning the 24 of December .   ho :   Complement of ☉ Ascension the 23 of December — 5 7 Stars hour — 6 42 Stars Ascension — 4 16   4 5 But for Stars that have South declination or north , when their Altitude is more then their Altitude being 6 hours from the Meridian , this trouble of rectifying two Beads is shunned ; in this Case only bring the Bead that is rectified to the Star to the Parralel of Altitude , and there among the Azimuths it will shew the Stars Azimuth , and the Thread in the Limb intersects the Stars hour sought . Example . December 11th Bulls eye Altitude 39● Azimuth from the South 62d 48 Hours from the Meridian — 8 h 56′ Complement of ☉ Ascension — 6 00 Ascension of Bulls eye — 4 16 The true time of the night was 12′ past 7 of the Clock 7 12 Another Example . The great Doggs observed Altitude being 14d his Azimuth from the South would be 39d. h m And the Stars hour from the Meridian — 9 22 Stars Ascension — 6 30 If this Observation were upon the 31 of December , the Complement of the Suns Ascension would be — 4 30   8 22 And the true time of the night 22 minutes past eight of the Clock . For varieties sake there is also added to the Book a Draught of the Projection for the Latitude of the Barbados ; in the use whereof the Reader may observe that every day when the Sun comes to the Meridian between the Zenith and the Elevated Pole , he will upon divers Azimuths in the forenoon ( as also in the afternoon ) have two several Altitudes , and so be twice before noon , and twice afternoon , at several times of the day , upon one and the same Azimuth , viz. only upon such as lye between the Suns Coast of rising and setting , and his remotest Azimuth from the Meridian , which causeth the going forward and backward of the shaddow ; but of this more hereafter , when I come to treat of Calculating the Suns Altitude on all Azimuths ; It may also be observed that the Sun for the most part in those Latitudes hath no Vertical Altitude or Depression , and so comes not to the East or West . Moreover there is added a Draught of this Projection for the Latitude of Greenland , in the use whereof it may be observed that the Sun , a good part of the Summer half year comes not to the Horizon , and so neither riseth nor sets . And that no convenient Way that this Projection can be made should be omitted , there is also one drawn in a Semi-Circle for our own Latitude , which in the use will be more facile then a Quadrant , there being no trouble before or after six in the Summer time , with rectifying another Bead to perform the Operation in the reverted Taile , neither doth the Drawing hereof occupy near the Breadth , as in a Quadrant , and so besides the ease in the use is more exact in the performance ; there being no other Rule required for rectifying the Bead , but to lay the Thread over the day of the Month , and to set the Bead to that Ecliptick the Thread intersects . A Semi-Circle is an Instrument commonly used in Surveigh , and then it requires a large Center-hole ; however this Projection may be drawn on a Semi-Circle for Surveigh , but when used at home there must a moveable round Bit of brass be contrived to stop up that great Center-hole , in which must be a small Center-hole for a Thread and Plummet to be fastned , as for a Quadrant and some have been so fitted . The Reader will meet with variety of Lines and furniture in this Book to be put in the Limb , or on other parts of the Semicircle , as he best liketh . The Projection for the Barbados & Greenland , are drawn by the same Rules delivered in the Description of the Quadrant , and so also is the Summer part of this Semi-Circle , and the Winter part by the same Rules that were given for drawing the Reverted Taile . Of the Quadrant of Ascensions . The turning of the Stars hour into the Suns hour and the the converse may be also done by Compasses upon the Quadrant of Ascensions on the back side . To turn the Stars Hour into common time , called the Suns hour . THe Arithmetical Rule formerly given is nothing but an abridgment of the Rule delivered by Mr. Gunter , and others , and the work to be done by Compasses , differeth somewhat from it , though it produce the same Conclusion which is : To get the difference between the Ascension of the Sun and the Star by substracting the less from the greater ; this remainder is to be added to the Stars hour , when the Star is before , or hath more Ascension then the Sun , but otherwise to be substracted from it , and the Sum or remainder is the true time sought . To do this with Compasses , take the distance between the Star and the Suns Ascension , and set the Suns foot to the observed hour of the Star from the Meridian it was last upon , letting the other foot fall the same way it stood before , and it sheweth the time sought , if it doth not fall off the Quadrant . If it doth , the work will be to finde how much it doth excur , and this may be done by bringing it to the end beyond which it falleth , letting the other foot fall inward , the distance then between the place where it now falleth , and where it stood before , which was at the Stars hour , is equal to the said excursion , which being taken , and measured on the other end of the Scale , shews the time sought . This trouble may be prevented in all Cases , by having 12 hours more repeated after the first 12 , or 6 hours more may serve turn if the whole 18 hours be also double numbred , and Stars names being set to the Additional hours , possibly the Suns Ascension and Star do not both fall in the same 12 hours , yet notwithstanding the distance is to be taken in the same 12 hours between the quantity of the Suns Ascension and the Stars , and to proceed therewith as before , and the Compasses will never excur ; in the numbring of these hours , after 12 are numbred they are to begin again , and are numbred as before , and not with 13 , 14 , &c. And this trouble may be shunned when there is but 12 hours by assuming any hour to be the Stars hour , with such condition that the other foot may fall upon the Line ; and the said assumed hour representing the Stars hour ; count from it the time duly in order , till you fall upon the other foot of the Compasses , and you will obtain the true time sought . To turn common time , or the Suns hour into the Stars hour . THis is the Converse of the former ; take the distance between the Star and the quantity of the Suns Ascension , and set the Star foot to the Suns hour , letting the other fall the same way it stood before , and it shews the time sought . Of the Quadrat and Shaddows . Both these as was shewed in the Description of the Quadrant , are no other then a Table of natural Tangents to the Arks of the Limb and may supply the use of such a Cannon , though not with so much exactness , all the part of the Quadrate are to be estimated less then the Radius , till you come against 45● of the Limb , where is set the figure of 1 , and afterwards amongst the shadows is to be accounted more then the Radius , and so where the Tangent is in length 2 Radii as against 63d 26 m of the Limb. 3 Radii as against 71 34 of the Limb. 4 Radii as against 75 58 of the Limb. 5 Radii as against 78 42 of the Limb. are set the figures 2 , 3 , 4 , 5 , and because they are of good use to be repeated on the other side of the Radius in the Quadrat , there they are not figured , but have only full points set to them , falling against the like Arks of the Limb from the right edge towards the left , as they did in the shadows from the left edge towards the right . To find a hight at one Observation . LEt A B represent a Tower , whose Altitude you would take , go so far back from it that looking through the sights of the Quadrant , the Thread may hang upon 45 degrees of the Limb , or upon 1 , or the first prick of the Quadrat , and the distance from the foot of the Tower will be equal to the height of the Tower above the eye , which accordingly measure , and thereto add the height of the eye above the ground , and you will have the Altitude of the Tower. So if I should stand at D and find the Thread to hang over 45d of the Limb , I might conclude the distance between my Station and the Tower to be equal to the height of the Tower above my eye , and thence measuring it find to be 96 yards , so much would be the height of the Tower above the eye . If I remove farther in till the Thread hang upon the second point of the Quadrat , then will the Altitude of the Tower above the level of the eye be third point of the Quadrat , then will the Altitude of the Tower above the level of the eye be fourth point of the Quadrat , then will the Altitude of the Tower above the level of the eye be fifth point of the Quadrat , then will the Altitude of the Tower above the level of the eye be twice thrice four times as much as the distance from the Tower is to the Station . five times as much as the distance from the Tower is to the Station . So removing to C ; I find the Thread to hang upon the second point of the Quadrat , and measuring the distance of that Station from the Tower , I find it to be 48 yards , whence I may conclude the Tower is twice as high above my eye , and that would be 96 yards . So if I should remove so much back that the Thread should hang upon 2 of the shaddows &c. as E the distance between 3 of the shaddows &c. at F the distance between 4 of the shaddows &c. 5 of the shaddows &c. the Station and the foot of the Tower would be twice thrice four times five times as much as the height of the Tower above the eye , and consequently if I should measure the distance between D and E where it hung upon 1 and 2 of the Shaddows , or between E and F , where it hung upon 2 and 3 of the shaddows , &c I should find it to be equal to the Altitude ; but other ways of doing it when inaccessible will afterwards follow . A Second way at one Station . WIth any dimension whatsoever of a competent length , measure off from the foot of the Object , whether Tower or Tree , just 10 or 100 , &c. of the said Dimensions , as suppose from B to K , I measure of an hundred yards ; there look through the sights of the Quadrant to the Top of the Object at A , and what parts the Thread hangs upon in the Quadrat or Shadows , shews the Altitude of the Object in the said measured parts , and so at the said Station at K the Thread will hang upon 96 parts , shewing the Altitude A B to be 96 yards above the Level of the eye , and so if any other parts were measured off , they are to be multiplyed by the Tangent of the Altitude , or parts cut by the Thread , rejecting the Ciphers of the Radius , as in the next Proposition . A third way by a Station at Random . TAke any Station at Random as at L , and looking through the sights observe upon what parts of the Quadrat or Shadows the Thread falls upon , and then measure the distance between the Station and the foot of the object , and the Proportion will hold . As the Radius To the Tangent of the Altitude , or to the parts cut in the Quadrat or shadows , So is the distance between the Station and the Object To the height of the Object above the eye . So standing at L , the Thread hung upon 30d 58 m of the Limb , as also upon 600 of the Quadrat , the Tangent of the said Ark and the measured distance L B was 160 yards , now then to work the former Proportion , multiply the distance by the parts of the Quadrat , and from the right hand of the product cut off three places , and you have the Altitude sought . In this Work the Radius or Tangent of 45d is assumed to be 1000. To measure part of an Altitude , as suppose from a window in a Tower to the top of the Tower , may be inferred from what hath been already said ; first get the top of the Tower by some of the former ways , and then the height of the window which substract from the former Altitude and the remainder is the desired distance between the window and the top of the Tower. The former Proportion may also be inverted for finding of a distance by the height , or apprehending the Tower to lye flat on the ground , and so the height to be changed into a distance and the distance into a height , the same Rules will serve only the height of a Tower being measured , and from the top looking to the Object through the sights of the Quadrant , what Angle the Thread hangs upon is to be accounted from the right edge of the Quadrant towards the left ; but in taking out the Tangent of this Ark , after I have observed it , the Thread must be laid over the like Ark of the Quadrant from the left edge towards the right , and from the Quadrat or Shaddows the Tangent taken out by the Intersection of the Thread , and so to measure part of a distance must be done by getting the distances of both places first , and then substract the lesser from the greater . To find the Altitude of any Perpendicular , by the length of its shadow . THis will be like the first Proposition , with the Quadrant take the Altitude of the Sun , if in so doing the Thread hang over the 1st pricks in the Quadrat , the length of the Shaddow is equal to the height of the Object Tree , or Perpendicular 2d pricks in the Quadrat , the length of the Shaddow is double the height of the Object Tree , or Perpendicular 3d pricks in the Quadrat , the length of the Shaddow is triple the height of the Object Tree , or Perpendicular 4th pricks in the Quadrat , the length of the Shaddow is four times the height of the Object Tree , or Perpendicular 5th pricks in the Quadrat , the length of the Shaddow is five times the height of the Object Tree , or Perpendicular whatever it be ; But if it hang 1 in the shaddows 2 in the shaddows 3 in the shaddows 4 in the shaddows 5 in the shaddows the highth of the object is equal to the length of the double the length of the triple the length of the four times the length of the five times the length of the Shaddow which may happen where the Sun hath much Altitude , as in small Latitudes , and so the length of the Shadow being forthwith measured , the height of the Gnomon may be easily attained . If the Thread in observing the Altitude hang on any odd parts of the Quadrant or Shadows , the Proportion will hold as before . As the Radius To the length of the Shadow , So the Tangent of the Suns Altitude , or the parts cut by the Thread To the height of the Gnomon to be wrought as in the third Proposition ; if the length of the Gnomon , and the length of its shadow were given ; without a Quadrant we might obtain the Suns Altitude , for As the length of the Gnomon Is to the Radius So is the Length of its Shadow To the Tangent of the Complement of the Suns Altitude . And the height of the Sun , and the length of the Gnomon assigned ; We may find the length of the Shadow by inverting the Proportion aforesaid . As the Radius To the length of the Gnomon So the Cotangent of the Suns Altitude To the Length of the Shadow . To find an innaccessible Height at two Stations . To work this with the Pen , Out of the Line of Sines take the Sine of 70d with Compasses , and measure it on the equal parts , where admit it reach to 94 parts , Then multiply the said number by the distance 102 , 1 and the Product will be 95 , 974 from which cutting off three figures to the right hand the residue being 95 , 974 , is the Altitude sought feré , but should be 96 caused by omitting some fractionate parts in the distance which we would not trouble the Reader withall . Another more general way by any two Stations taken at randome . ADmit the first Station to be as before at G where the observed Altitude of the Object was 70d and from thence at pleasure I remove to H , where observing again I find the Object to appear at 48d 29 m of Altitude , and the measured distance between G and H to be ●0 yards , a general Proportion to come by the Altitude in this case will hold . As the difference of the Cotangents of the Arks cut at either Station . Is to the Distance between the two Stations , So is the Radius To the Altitude of the Object or Tower. To save the substracting of the two Arks from 90d to get their Complements , I might have accounted them when they were observed from the right edge of the Quadrant towards the left , and have found them to have been 20d and 41d 31 m ; to work this Proportion lay the Thread over these two Arks in the Limb from the left edge towards the right , and take out their Tangents out of the Quadrat and Shadows , then substract the less from the greater , the remainder is the first tearm of the Proportion , being the Divisor in the Rule of three to be wrought by annexing the Ciphers of the Radius to the Distance ; or as Multiplication in Decimalls ; and then dividing by the first tearm the Quotient shews the Altitude sought , 41d 31′ Tangent from the Shadows is — 885 20d Tangent from the Quadrat is — 364 Difference — 521 By which if I divide 50000 — the distance encreased by annexing the Ciphers of the Radius thereto , the Quotient will be 96 fere the Altitude sought . This may be performed otherwise without the Pen , as shall afterwards be shewn ; If the distance from the Stations either to the foot or the top of the Tower be desired , the Proportions to Calculate them will be As the difference of the Cotangents of the Arks cut at either Station Is to the distance between those Stations , So is the Cotangent of the greater Arke to the lesser lesser Arke to the greater distance from the Station to the foot of the Tower : And so is the Cosecant of the greater Ark to the lesser distance lesser Ark to the greater distance from the top of the Tower to the eye , Or having first obtained the Height , we may shun these Secants , for As the Sine of the Ark of the Towers observed Altitude at the first Station . Is to the height of the Tower or Object above the eye , So is the Radius To the distance between the Eye and the Top of the Tower ; and by the same Proportion using the Ark at the second Station , the distance thence between the Eye and the Top of the Tower may be likewise found . If any one desire to shun this Proportion in a difference , as perhaps wanting natural Tables , it may be done at two Operations : The first to get the distance from the eye to the Top of the Tower at first Station , in regard the difference of two Arks is equal to the difference of their Complements , it will hold As the Sine of the Ark of difference between the Angles observed at each Station Is to the distance between the two Stations , So is the Sine of the Angle observed at the furthest Station , To the distance between the eye and the Top of the Object at the first Station , which being had , it then holds , As the Radius To the said Distance , So the Sine of the Angle observed at the first Station , To the Altitude of the Object . The rest of the Lines on the Quadrant are either for working of Proportions , or for Protractions , and Dyalling depending thereon ; wherefore I thought fit to reduce all the common Cases of Plain and Sphoerical Triangles to setled Cannons , and to let them precede their Application to this or other Instruments , which shall be endeavoured ; it being the cheif aim of this Book to render Calculation facil , in shunning that measure of Triangular knowledge hitherto required , and to keep a check upon it . Some Affections of Plain TRIANGLES . THat any two sides are greater then the third . That in every Triangle the greater side subtendeth the greater Angle , and the Converse . That the three Angles of every Triangle are equal to two right Angles , or to 180. That any side being continued , the outward Angle is equal to the two inward Opposite ones . That a Plain right angled Triangle hath but one right Angle , which is equal to both the other Angles , and therefore those two Angles are necessarily the Complements one of another . By the Complement of an Arch is meant the residue of that Arch taken from 90d unless it be expressed the Complement of it to 180d , to save the rehearsal of these long Words , The Tangent of the Complement , The Secant of the Complement ; others are substituted , namely the Cotangent and the Cosecant . An Obutse angular Triangle hath but one Obtuse angle , an Acutangular Triangle hath none . If an Angle of a Triangle be greater then the rest , it is Obtuse , if less Acute . The Side subtending the right Angle is called the Hipotenusal and the other two the Sides or Leggs . 1. To find a side . Given the Hipotenusal , and one of the Acute Angles , and consequently both . As the Radius , To the Sine of the Angle opposite to the side sought : So is the Hipotenusal , To the side sought . 2. To find a Side . Given the Hipotenusal and the other side . As the Hipotenusal To the given side : So is the Radius to the Sine of the Angle opposite to the given side . Then take the Complement of that Angle for the other Angle . As the Radius , to the Hipotenusal : So is the Sine of the Angle opposite , To the side sought , To the side sought 3. To find a Side . Given a Side , and one Acute Angle , and by consequence both As the Radius , To Tangent of the Angle opposite to the side sought : So is the given side , To the side sought . Or , As the Sine of the Angle opposite to the given side Is to the given side : So is the sine of the other Angle To the Side sought . 4. To find the Hipotenusal . Given one of the Sides , and an Acute Angle consequently both , As the Sine of the Angle opposite to the given side , Is to the Radius : So is the given Side , To the Hypothenusal Or , As the Radius , To Secant of the Angle adjacent to the given Side : So is the given side , To the Hipotenusal . 5. To find the Hipotenusal . Given both the Sides . As one of the given Sides . To the other given Side : So is the Radius , To the Tangent of the Angle opposite to the other side , Then take the Complement of the Angle found for the other Angle . Then , As the Sine of the Angle opposite to one of the given Leggs , or Sides , Is to the given Sides : So is the Radius , To the Hipotenusal . 6. To find an Angle . Given the Hipotenusal , and one of the Sides . As the Hipotenusal , Is to the given Side : So is the Radius , To the Sine of the Angle opposite to the given side ; The Complement of the angle found , is the other Angle . 7. To find an Angle . Given both the Sides . As one of the given sides , To the other given Side : So is the Radius , To the Tangent of the angle opposite to this other Side . The Complement of the angle found is the other Angle . Because the Sum of the Squares of the Leggs of a right angled Triangle is equal to the Square of the Hipotenusal by 47 1 Euclid . Therefore the 2d and 5th Case may be otherwise performed . 2. Case , To find a Side or Legg . Given the Hipotenusal , and other Legg . To do it by Logarithmes , Add the Logarithm of the Sum of the Hipotenusa and Legg , to the Logarithm of their difference , the half Sum is the Logarithm of the Side sought . Which in natural Numbers is to extract the Square root of the Product of the Sum and difference of the two Numbers given , namely , the Hipotenusal and Legg , which said root is equal to the side sought . 5. Case , When both the Sides are given to find the Hipotenusal . To do it by Logarithmes , Substract the Logarithm of the lesser side from the doubled Logarithm of the greater , and to the absolute number answering to the remaining Logarithm add the lesser side the half sum of the Logarithmes of the Sum thus composed , and of the lesser side , is the Logarithm of the Hypotenusa sought . Which work was devised , that the Proposition might be performed in Logarithmes ; The same Operation by natural numbers , would be to Divide the Square of the greater of the Sides by the lesser , and to the Quotient to add the lesser Side , then multiply that Sum by the lesser Side , and extract the Square root of the Product for the Hipotenusal sought . Cases of Oblique Plain Triangles . 1. Data To find an Angle , TWo sides with an Angle opposite to one of them to find the Angle Opposed to the other side . As one of the Sides , To the Sine of its opposite Angle : So the other side ; to the Sine of the angle opposed thereto . If the Angle given be Obtuse , the Side opposite to it will be greater then either of the rest , and the other two Angles shall be Acute ; But if the given Angle be Acute , it will be doubtful whether the angle opposed to the greater side be Acute or Obtuse , yet a true Sine of the 4 ●h Proportional . The Sine found will give the angle opposite to the other given side , if it be Acute , and it will always be Acute when the given angle is Obtuse , But if it be fore known to be Obtuse , the Arch of the Sine found substract from a Semicircle , and there will remain the Angle sought . In this Case the quality or affection of the angle sought must be given , and fore-known , for otherwise it is impossible to give any other then a double answer , the Acute angle found , or its Complement to 180d , yet some in this Case have prescribed Rules to know it , supposing the third Side given which is not , if it were then in any Plain Triangle it would hold , That if the Square of any Side be equal to the Sum of the Squares of the other two Sides , the angle it subtends is a right angle , if less Acute , if greater Obtuse ; but if three Sides are given we may with as little trouble by following Proportions come by the quantity of any angle , as by this Rule to know the affection of it . Two Sides with an Angle opposite to one of them , to find the third Side . In this Case as in the former , the affection of the angle opposite to the other given side , must be fore-known , or else the answer may be double or doubtful . In the Triangle annexed there is given the sides A B and A C with the angle A B C to find the angle opposite to the other given side , which may be either the angle at C or D , and the third side , which may be either B C , or B D , the reason hereof is because two of the given tearms the Side A B , and the angle at B remain the same in both Triangles , and the other given side may be either A C or A D equal to it , the one falling as much without as the other doth within the Perpendicular A E. The quality of the angle opposite to the other given side being known ; by the former Case get the quantity , and then having two angles the Complement of their Sum to 180d is equal to the third angle , the Case will be to find the said side , by having choice of the other sides , and their opposite angles as follows . 2. Two Angles with a Side opposite to one of them given to find the Side opposite to the other . As the Sine of the angle opposed to the given Side , Is to the given Side ; So is the Sine of the angle opposite to the side sought ▪ To the side sought . Ricciolus in the Trigonomerical part of his late Almagestum Novum , suggests in the resolution of this Case , that if the side opposite to the Obtuse angle be sought , it cannot be found under three Operations ; as first to get the quantity of the Perpendicular falling from the Obtuse angle on its Opposite side , and then the quantity of the two Segments of the side , on which it falleth , and so add them together to obtain the side sought ; But this is a mistake , and if it were true by the like reason a side opposite to an Acute angle , could not be found without the like trouble ; for in the Triangle above , As the Sine of the angle at B Is to its Opposite side A C or A D , So is the Sine of the angle B C A , or B D A , To its opposite Side B A , where the Reader may perceive that the same side hath opposite to it both an Acute and an Obtuse angle , the one the Complement of the other to 180d , the same Sine being common to both , for the Acute angle A C E is the Complement of the Obtuse angle B C A ; but the angle at C is equal to the angle at D being subtended by equal Sides , and the Proportion of the Sines of Angles to their opposite Sides is already demonstrated in every Book of Trigonometry . 3. Two Sides with the angle comprehended to find either of the other Angles . Substract the angle given from 180d , and there remains the Sum of the two other Angles , then As the Sum of the Sides given , To tangent of the half sum of the unknown Angles : So is the difference of the said Sides , To Tangent of half the difference of the unknown Angles . If this half difference be added to half sum of the angles , it makes the greater , if substracted from it , it leaves the lesser Angle 4. Two Sides with the Angle comprehended , to find the third side . By the former Proposition one of the Angles must be found , and then As the Sine of the angle found , is to its Opposite side : So is the Sine of the angle given , To the Side opposed thereto , If two angles with a Side opposite to one of them be given , to find the side opposite to the third Angle is no different Case from the former , because the third angle is by consequence given , being the Complement of the two given angles to 180d , and the like if two angles with the side between them were given . 5. Three Sides to find an Angle . The Side subtending the Angle sought is called the Base . As the Rectangle or Product of the half Sum of the three sides , and of the difference of the Base therefrom , Is to the Square of the Radius : So is the Rectangle of the difference of the Leggs , or conteining Sides therefrom , to the Square of the Tangent , to half the Angle sought . And by changing the third tearm into the place of the first . As the Rectangle of the Differences of the Leggs from the half Sum of the 3 sides , Is to the Square of the Radius : So is the Rectangle of the said half sum , and of the difference of the Base there from , To the Square of the Tangent of an angle , which doubled is the Complement of the angle sought to 180d , Or the Complement of this angle doubled is the angle sought . To Operate the former Proportion by the Tables . From the half Sum of the three Sides substract each Side severally then from the Sum of the Logarithmes of the Square of the Radius ( which in Logarithms is the Radius doubled ) and of the differences of the Sides containing the angle sought . Substract the Sum of the Logarithms of the half sum of the three sides , and of the difference of the Base therefrom the half of the remainder is the Logarithm of the Tangent of half the angle sought Example . In the Triangle A B C let the three sides be given to find the Obtuse angle at C. Differences from the half sum . B C 126 Leggs — 101 — 2,0043214 A C 194 Leggs — 169 — 2,2278867 A B 270 Base — 25 Logarithms — —   — 1,3979400 Sum with double Radius 24,2322081 Sum 590     half sum 295 — 2,4698220       — 3,8677620     3,8677620 —     — 20,3644461 Tangent of 56d 41′ which doubled is 113d 22′ the angle sought . 10,1822230 half For the 2d Proportion for finding an Angle the Operat on varies ; to the former sum add the double Radius , and from the Aggregate substract the latter sum , the half of the Relique is the Logarithm of the Cotangent of half the Angle sought . Tangent of 33d 19′ the Complement whereof is 56d 41′ which doubled makes 113d 22′ as before The Tables here made use of are Mr. Gellibrands . It may be also found in Sines . As the Rectangle of the Containing Sides or Leggs Is to the Square of the Radius : So the Rectangle of the differences of the said Leggs from half Sum of the three Sides , To the Square of the Sine of half the Angle sought . Or , We may find the Square of the Cosine of half the Angle sought by this Proportion . As the Rectangle of the containing Sides , Is to the Square of the Radius : So the Rectangle of the half sum of the three Sides , and of the difference of the Base there from , To the Square of the Cosine of half the Angle sought , viz. To the Square of the Sine of such an Ark , as being doubled is the Complement of the Angle sought to 180d , or the Complement of the found Ark doubled is the angle sought . All these Proportions hold also in Spherical Triangles if instead of the bare names of Sides you say the Sines of those Sides , and this Case may also be resolved by a Perpendicular let-fall ; but the Reader need not trouble himself with name nor thing in no Case of Plain or Sphoerical Triangles , but when two of the given Sides or Angles are equal . If the Square of the Radius and the Square of the Sine of an Ark be both divided by Radius the Quotients will be the Radius , and the half Versed Sine of twice that Arch , and in the same Proportion that the two tearms propounded , as will afterwards be shewed , upon this Consideration may any of these Compound Proportions be reduced to single Tearms for Instrumental Operations , by placing the two Tearms of the first Rectangle , as two Divisors in two single Rules of three , and the tearms of the other Rectangle as middle tearms . An Example in that Proportion for finding an Angle in the Sineés : As one of the Leggs or including Sides , Is to the difference thereof from the half sum of the three Sides : So is the Difference of the other Legg therefrom , To a fourth Number . Again . As the other Legg , To that fourth Number : So is the Radius , To the half Versed Sine of the Angle sought ; And so is the Diameter or Versed Sine of 180d Or Secant of 60d To the Versed Sine of the Angle sought . And upon this Consideration , that the fourth Tearm in every direct Proportion bears such Proportion to the first Tearm , as the Rectangle or Product of the two middle Tearms doth to the Square of the first Tearm , as may be demonstrated from 1 Prop. 6. Euclid . We may place the Radius or Diameter in the first place of the first Proportion , and still have the half or the whole Versed Sine in the last place as before , therefore As the Radius , To one of the given Leggs : So the other given Legg , To a fourth Number , which will bear such Proportion to the Radius , as the Rectangle of the two given Leggs doth to the Square of the Radius , then it holds , As that fourth number , To the Radius : So the Rectangle of the difference of 〈…〉 the Leggs from the half sum of the three sides ; To the Square of the Sine of half the Angle sought , and omitting the Raidius , it will hold ; As that fourth Number , To the differences of the Leggs from the Leggs from the half sum of the three Sides : So is the difference of the other Legg therefrom , To the half Versed Sine of the Angle sought . And if the Diameter had been in the first place , then would the whole Versed Sine have been in the last . This Case when 3 Sides are given to find an Angle is commonly resolved by a Perpendicular let-fall , it shall be only supposed and the Cannon will hold , As the Base or greater Side , To the sum of the other Sides : So is the difference of the other Sides , To a fourth Number , which taken out of the Base , half the remainder is the lesser Segment , and the said half added to this fourth Number is the greater Segment . Again . As the greater of the other Sides , To Radius : So is the greater Segment , To the Cosine of the Angle adjacent thereto , Or , As the Lesser of the other Sides , To Radius : So is the lesser Segment , To the Cosine of its adjacent Angle : This for either of the Angles adjacent to the Base or greatest side but for the angle opposite to the greatest side , which may be sometimes Obtuse , sometimes Acute ; not to multiply Directions , the Reader is remitted to the former Cannons , or to find both the Angles at the Base first , and by consequence the third angle is also given , being the residue of the sum of the former from a Semicircle . In right angled Spherical Triangles . By the affection of the Angles to know the Affection of the Hipotenusal , and the Converse . If one of the angles at the Hipotenusal be a right angle The Hipotenusal will be A Quadrant . If both be of the same kind , The Hipotenusal will be lesser then a Quadr. If of a different kind , The Hipotenusal will be greater then a Quadr. By the Hipotenusal to find the affection of the Leggs ▪ and the Converse : If the Hipotenusal be A Quadrant one of the Legs wil be a Quadrant If the Hipotenusal be Less then A Quadrant The Leggs , and their Opposit Angles wil be less then Quadrant If the Hipotenusal be Greater , then A Quadrant One Legg will be greater , & the other less then the Quadrant : The Leggs of a right angled Sphoerical Triangle are of the same Affection as their Opposite Angles and the Converse : If a Legg be a Quadrant the Angle opposite thereto will be a Quadr. If a Legg be less then a Quadrant the Angle opposite thereto will be Acute If a Legg be greater then a Quadrant the Angle opposite thereto will be Obtuse . All these Affections are Demonstrated in Snellius ; I shall add some more from Clavius de Astrolabio , and the Lord Nepair . The three Sides of every Sphoerical Triangle are less then a Whole Circle . In an Oblique angular Triangle , If two Acute Angles be equal the Sides opposite to them shall be lesser then Quadrants . Obtuse Angles be equal the Sides opposite to them shall be greater then Quadrants . Reg. 10 ▪ 11 , 4 If an Oblique angular Triangle , if two Acute Angles be unequal the side opposite to the lesser shall be lesser then a Quadrant Obtuse Angles be unequal the side opposite to the greater shall be greater then a Quadrant Reg. 12 , 13.4 . An Acute angular Triangle hath all its Angles Acute , and each side less then a Quadrant . Two sides of any Spherical Triangle are greater then the third . If a Sphoerical Triangle be both right angled and Quadrantal , the sides thereof are equal to their Opposite Angles . If it hath three right Angles , the three sides of it are Quadrants . If it have two right Angles , the two sides subtending them are Quadrants and the contrary , and if it have one right angle , and one side a Quadrant , it hath two right angles , and two Quadrental sides . Any side of a Sphoerical Triangle being continued , if the other sides together are equal to A Semicircle the outward angle wil be equal to the inward opposit angle on the side continued . lesser then A Semicircle the outward angle wil be lesser then the inward opposit angle on the side continued . greater then A Semicircle the outward angle wil be greater then the inward opposit angle on the side continued . If any Sphoerical Triangle have two Sides equal to lesser then greater then a Semicircle , the two angles at the Base or third Side will be equal to lesser then greater then two right Angles . In every right angled Spherical Triangle having no Quadrantal Side , the angle Opposite to that Side that is less then a Quadrant is Acute , and greater then the said Side ; But that angle which is Opposite to the Side , that is greater then a Quadrant , is Obtuse ; and less then the said Side . In every right angled Spherical Triangle all the three angles are less then 4 right angles , that is the two Oblique angles are less then 3 right angles , or 270d. In a right angled equicrural Triangle , if the two equal angles be Acute , either of them will be greater then 45d , but if Obtuse less then 135d. In every right angled Sphoerical Triangle either of the Oblique angles is greater then the Complement of the other , but less then the difference of the same Complement from a Semicircle . Two angles of any Sphoerical Triangle are greater then the difference between the third angle and a Semicircle , and therefore any side being continued , the outward angle is less then the two inward opposite angles . The sum of the three Angles of a Sphoerical Triangle is greater then two right angles , but less then 6. In Spherical Triangles , that angle which of all the rest is nearest in quantity to a Quadrant , and the side subtending it are doubtful , Whether they be of the same , or of a different affection , unless foreknown , or found by Calculation ; But the other two more Oblique angles are each of them of the same kind as their Opposite Sides , which Mr Norwood thus propounds , Two Angles of a Spherical Triangle , shall be of the same affection as their Opposite Sides , and to this purpose , If any Side of a Triangle be nearer to a Quadrant then its opposite Angle , two Angles of that Triangle ( not universally any two ) shall be of the same kind , and the third greater then a Quadrant . But if any Angle of a Triangle be nearer to a Quadrant then its opposite side , two Sides of that Triangle ( not universally any two ) shall be of the same kind , and the third less then a Quadrant . In any Sphoerical Triangle , if one of the angles be substracted from a Semicircle , and the residue so found substracted from a Whole Circle , the Ark found by this latter Substraction , will be greater then the Sum of the other two Angles . In every Sphoerical Triangle the difference between the sum of two angles howsoever taken , and a whole Circle or 4 right angles is greater then the difference between the other Angle and a Semicircle ; The demonstration of most of these Affections are in Clavins his Comment on Theodosius , or in his Book de Astrolabio , where shewing how to project in Plano all the Cases of Sphoerical Triangles , and so to measure the sides and Angles , he delivers these Theorems to prevent such Fictitious Triangles as cannot exist in the Sphere . The 16 Cases of right Angled Sphoerical Triangles , Translated from Clavius de Astrolabio . 1. To find an Angle . Given the Hipotenusal and side opposite to the Angle sought . As the Sine of the Hipotenusal , To Radius : So the Sine of the given side To the Sine of the Angle sought . Or , As Radius , To sine of the Hipotenusal : So the Cosecant of the side , To the Cosecant of the Angle . As Radius , To sine of the side : So Cosecant of the Hipotenusal , To sine of the angle . As Cosecant of the side , To Radius : So Cosecant of the Hipotenusal , To Cosecant of the angle . As Cosecant Hipotenusal , To Radius : So Cosecant of the side , To Cosecant of the angle . As the Sine of the side to Radius : So sine of the Hipotenusal , To Cosecant of the Angle . The Angle found will be Acute if the Side given be less then a Quadrant , Obtuse if greater . 2. To find an Angle . Given the Hipotenusal and side adjacent to the Angle . As Radius , To Cotangent Hipotenusal : So tangent of the side , To Cosine of the angle . As tangent Hipotenusal . To Radius , So tangent of the Side . To Co-sine of the angle . As the Cotangent of the side , To Radius : So Cotangent Hipotenuse , To Cosine of the angle . As Radius , To Cotangent side : So tangent Hipotenusa , To Secant of the angle . As Cotangent Hipotenusal : To Radius : So Cotangent side , To Secant of the angle . As the tangent of the side , To Radius : So tangent Hipot : To Secant of the angle . The Angle found will be Acute , if both the Hipotenusal and the given side be greater , or less then a Quadrant , but Obtuse if one of them be greater , and the other less . 3. To find an Angle . Given the Hipotenusal , and either of the Oblique Angles As the Radius , To Cosine Hip : ∷ So Tangent of the given angle : To the Cotang of sought angle As the Cotangent of the given angle : To Radius ∷ So Cosine of : the Hipot : To the Cotang of the sought angle . As the Cosine of the Hipot : To Radius ∷ So Cotang : of the given angle : To the tang . of the ang : sought As the Radius , To Secant Hip : ∷ So Cotangent of the given angle : To the tang of the angle sought As Secant of the Hipotenusal : To Radius ∷ So the tangent of the angle given : To the Cotan : of the angle sought As the tangent of the given angle : To Radius ∷ So Secant of the Hipotenusal : To tangent of the angle sought . The angle found will be Acute , if the Hipotenusal be less then a Quadrant , and the given angle Acute ; or if the Hipotenusal be greater then a Quadrant , and the given angle Obtuse ; And the said angle will be Obtuse , if the Hipotenusal be less then a Quadrant , and the given angle Obtuse ; Of if the Hipotenusal be greater then a Quadrant , and the given angle Acute . 4. To find an Angle . Given the Side opposite to the Angle sought , and the other Oblique Angle . As the Radius : To sine of the angle given ∷ So is the cosine of the given side : To the Cosine of the angle sought As the Radius : To Cosecant of the given angle ∷ So Secant of the side given : To Secant of the angle sought As the sine of the given angle : To Radius ∷ So the Secant of the given side : To the Secant of the angle As the Cosine of the given side : To Radius ∷ So the Cosecant of the given angle : To the Secant of the angle sought As the Cosecant of the given angle : To Radius : So Cosine of the given side : To the Cosine of the angle sought As the Secant of the given side : To Radius ∷ So sine of the given angle : To the Cosine of the angle sought The angle found will be Acute , if the side given be less then a Quadrant , Obtuse if greater . 5. To find an Angle . Given a side adjacent to the angle sought , and the other Oblique angle , if it be foreknown whether the angle sought be Acute or Obtuse , or whether the Base or other side not given be greater , or lesser then a Quadrant . As the Cosine of the given side : To Radius ∷ So Cosine of the given angle : To the sine of the angle sought As the Radius : To Secant of the given side ∷ So Cosine of the given angle : To Sine of the angle sought As the Radius : To Secant of the given angle ∷ So Cosine of the given side : To Cosecant of sought angle As the Cosine of the given angle : To Radius ∷ So Cosine of the given side : To the Cosecant of the angle sought As the Secant of the given side : To Radius ∷ So is the Secant of the given angle : To the Cosecant of the angle sought As the Secant of the given angle : To Radius ∷ So is the Secant of the given side : To the Sine of the angle sought The angle found will be Acute , if the side not given be less then a Quadrant , Obtuse if greater . In like manner if the Hipotenusal be less then a Quadrant , and the given angle Acute ; Or if the Hipotenusal be greater then a Quadrant , and the given angle Obtuse , the angle found will be Acute . But if the Hipotenusal be less then a Quadrant , and the given angle Obtuse , or if the Hipotenusal be greater then a Quadrant , and the given angle Acute , the angle found will be Obtuse , 6. To find an Angle . Given both the Leggs . As Radius : To sine of the side adjacent to sought angle ∷ So Cotang of the side opposite to the angle sought : To Cotang of the angle sought As the sine of the side adjacent to sought angle : To Radius ∷ So the tangent of the side opposite to the sought angle : To the tang of the angle sought : As the Tangent of the side opposite to sought angle : To Radius ∷ So the sine of the side adjacent to the sought angle : To Cotang of the angle sought : As the Radius : To Cosecant of the side adjacent to sought angle ∷ So the tangent of the side opposite thereto : To the tang of the angle sought : As the Cosecant of the side adjacent to the angle sought To Radius ∷ So is the Cotang of the side opposite to the angle sought : To Cotang of the sought angle : As the Cotangent side opposite to the sought angle : To Radius ∷ So Cosecant of the side adjacent to the angle sought : To the tang of the angle sought The Angle found will be Acute , if the side opposite to the angle sought be less then a Quadrant , but Obtuse if greater . 7 , To find a Side or Legg . Given the Hipotenusal and the other Legg . As the Cosine of the given side : To Radius ∷ So Cosine of the Hipotenusal : To Cosine of the side sought As Radius : To Secant of the side ∷ So Cosine of the Hipotenusal : To Cosine of the side sought As Cosine Hipotenusal : To Radius ∷ So Cosine of the given side : To Secant of the side sought As Radius : To Secant of the Hipotenusal ∷ So Cosine of the given side : To Secant of the side sought As the Secant of the Hipotenusal : To Radius ∷ So Secant of the given side : To Cosine of the side sought As the Secant of the given side : To Radius ∷ So Secant of the Hipotenusal : To the Secant of the side sought The side sought will be less then a Quadrant , if both the Hipotenusal and given sides be less then Quadrants , but greater then a Quadrant if either the Hipotenusal be greater , and the given side less , Or if the Hipotenusal be less , and the given side greater . 8. To find a Side . Given the Hipotenusal , and an Angle opposite to the side fought . As the Radius : To sine of the Hipotenusal ∷ So sine of the given angle : To the sine of the side sought As Radius : To Cosecant of the Hipotenusal ∷ So the Cosecant of the given angle : To Cosecant of the side sought As the sine of the Hipotenusal : To Radius ∷ So Cosecant of the given angle : To Cosecant of the side sought As the Cosecant of the given angle ▪ To Radius ∷ So the sine of the Hipotenusal : To the sine of the side sought As the sine of the given angle : To Radius ∷ So the Cosecant of the Hipotenusal : To the Cosecant of the side sought The side found will be less then a Quadrant , if the Angle opposite thereto be Acute , but greater if Obtuse . 9. To find a Side . Given the Hipotenusal and Angle adjacent to the Side sought As the Radius : To Cosine of the given angle ∷ So tangent of the Hipotenusal : To the tangent of side sought As the Cosine of the given angle : To Radius ∷ So Cotangent of the Hipotenusal : To Cotang of the side sought As the Cotangent of Hipotenusal : To Radius ∷ So the Cosine of the given angle : To tangent of the side sought As the Radius : To Secant of the angle ∷ So Cotangent of the Hipotenusal : To Cotang of the side sought As the Secant of the given angle : To Radius ∷ So tangent of the Hipotenusal : To the tangent of the side sought As the tangent of the Hipotenusal : To Radius ∷ So Secant of the given angle : To Cotang : of the side sought The Side sought will be less then a Quadrant , if the Hipotenusal be less then a Quadrant , and the given angle Acute : Or if the Hipotenusal be greater then a Quadrant , and the given angle Obtuse ; But it will be greater then a Quadrant , if the Hipotenusal be less then a Quadrant , and the given angle Obtuse : Or if the Hipotenusal be greater then a Quadrant , and the given angle Acute . 10. To find a Side . Given a Side , and an Angle adjacent to the sought side . Provided it be foreknown whether the side sought be greater or less then a Quadrant , or whether the other angle not given be Acute or Obtuse ; or finally whether the Hipotenusal be greater or less then a Quadrant . As the Radius : To the Cotangent of the given angle ∷ So tangent of the given side : To the Sine of the side sought As the tangent of the given angle : To Radius ∷ So tangent of the given side : To the sine of the side sought As the Cotang of the given side : To Radius ∷ So the Cotang of the given angle : To the sine of the side sought As Radius : To Cotangent of the given side : So tangent of the given angle : To Cosecant of the side sought As the tangent of the given side : To Radius ∷ So tangent of the given angle : To the Cosecant of the side sought As the Cotangent of the given angle : To Radius ∷ So Cotang of the given side : To the Cosecant of the side sought The Side found will be less then a Quadrant , if the angle opposite thereto and not given be Acute , but greater if it be Obtuse ; In like manner it will be less , if the Hipotenusal be less then a Quadrant , and the side given also less then a Quadrant : Or if the Hipotenusal be less then a Quadrant , and the given side greater , the side found will be greater then a Quadrant ; Lastly , if both the Hipotenusal , and the side given be greater then a Quadrant , the side found will be less then a Quadrant , but greater if the Hipotenusal be greater then a Quadrant , and the given side less . 11. To find a Side . Given a Side , and an Angle opposite to the Side sought As Radius : To sine given side ∷ So tangent given angle : To tangent of the side sought As the sine of the given side : To Radius ∷ So Cotangent of the given angle : To Cotangent of the side sought As the Cotangent of the given angle : To Radius ∷ So sine of the given side : To the tangent of the side sought As the tangent of the given angle : To Radius ∷ So Cosecant of the given side : To Cotangent of the side sought As the Radius : To Cosecant of the given side ∷ So Cotang of the given angle : To Cotangent of the side sought As the Cosecant of the given side : To Radius ∷ So tangent of the given angle : To tangent of the side sought The side found will be less then a Quadrant , if the given Angle opposite thereto be Acute , but greater if Obtuse . 12. To find a Side . Given both the Oblique Angles . As the sine of the angle adjacent to side sought : Is to Radius ∷ So Cosine of the angle opposite to side sought : To Cosine of the side sought As the Radius : To Secant of the angle opposite to side sought ∷ So sine of the angle adjacent to the side sought : To Secant of the side sought As Radius : To Cosecant of the angle adjacent to side sought ∷ So Cosine of the angle opposite to side sought : To Cosine of the side sought As the Cosine of the angle opposite to side sought : To Radius ∷ So side of the angle adjacent to the side sought : To Secant of the side sought As the Secant of the angle opposite to side sought : To Radius ∷ So Cosecant of the angle adjacent to the side sought : To Cosine of the side sought As Cosecant of the angle adjacent to side sought : To Radius ∷ So Secant of the angle opposite to side sought : To Secant of the side sought The side found will be less then a Quadrant , if the given angle Opposite thereto be Acute , but greater if Obtuse . 13. To find the Hipotenusal . Given a side and an Angle adjacent thereto . As the Radius : To Cosine of the given angle ∷ So Cotangent of the given side To Cotang of the Hipotenusal As the Cosine of the given angle : To Radius ∷ So tangent of the given side : To tangent of the Hipotenusal As the tangent of the given side To Radius ∷ So Cosine given angle : To Cotangent of the Hipotenusal As Radius : To Secant of the given angle ∷ So tangent of the given side ; To the tangent of the Hipotenusal As the Secant of the given angle : To Radius ∷ So Cotang of given side : To the Cotang of the Hipotenusal As the Cotangent of given side : To Radius ∷ So Secant of the given angle : To tangent of the Hipotenusal The Hipotenusal found will be less then a Quadrant , if the given side be less then a Quadrant , and the angle given adjacent thereto Acute ; As also if the given side be greater then a Quadrant , and the given angle adjacent thereto be Obtuse . But it will be greater then a Quadrant , if the given side be greater then a Quadrant , and the given Angle adjacent thereto Acute ; As also when the given side is less then a Quadrant , and the given Angle Obtuse . 14. To find the Hipotenusal Given a Side , and an angle Opposite thereto . If it be fore-known whether the Hipotenusal be greater or less then a Quadrant , or whether the other angle not given be Acute or Obtuse ; Or lastly , whether the other side not given , be greater or less then a Quadrant . As the sine of the given angle : To Radius ∷ So sine of the given side : To the sine of the Hipotenusal As the Radius : To Cosecant of angle given ∷ So sine of the given side : To the sine of the Hopotenusal As Radius : To Cosecant given side ∷ So sine of the given angle : To Cosecant of the Hipotenusal As the sine of the given side : To Radius ∷ So sine of the given angle : To Cosecant of the Hipotenusal As Cosecant of given side : To Radius ∷ So Cosecant of the given angle : To sine of the Hipotenusal As Cosecant of the given angle : To Radius ∷ So the Cosecant of the given side : To Cosecant of the Hipotenusal The Hipotenusal found will be less then a Quadrant , if both the Oblique Angles be Acute or Obtuse , or if both the sides be greater or less then Quadrants . It will also be greater then a Quadrant , if one of the Oblique Angles be Acute , and the other Obtuse , or if one of the sides be less and the other greater then a Quadrant : 15. To find the Hipotenusal , Given both the sides , distinguished by the names of first and second . As Radius : To Cosine 1st side ∷ So Cosine 2d side : To Cosine of the Hipotenusal As Radius : To Secant 1st side ∷ So Secant 2d side : To Secant of the Hipotenusal As Secant 1st side : To Radius ∷ So Cosine of the 2d side : To Cosine Hipotenusal As Secant 2d side : To Radius ∷ So Cosine 1st side : To Cosine of the Hipotenusal As Cosine 1st side : To Radius ∷ So Secant 2d side : To Secant of the Hipotenusal As Cosine 2d side : To Radius ∷ So Secant 1 side : To the Secant of the Hipotenusal The Hipotenusal found will be less then a Quadrant , if both the Sides are less or greater ; But otherwise , it will be greater , if one be less and the other greater . 16. To find the Hipotenusal . Given both the Oblique Angles , distinguisht by the names of the first and second . As Radius : To Cotangent 1st angle ∷ So Cotangent 2d angle : To Cosine of the Hipot : As the tangent 1st angle : To Radius ∷ So Cotangent 2d angle : To Cosine of Hipoten : As tangent of 2d angle : To Radius ∷ So Cotangent 1st angle : To Cosine Hipotenusal As the Radius : To tangent 2d angle ∷ So tangent 1st angle : To Secant of the Hipot : As Cotangent 2d angle : To Radius : So tangent 1st angle : To Secant of the Hipotenus ; As the Cotangent 1st angle : To Radius ∷ So Tangent 2d angle : To Secant Hipotenus : The Hipotenusal found will be less then a Quadrant , if both the Oblique angles be Acute or Obtuse , but greater if one of them be Acute , and the other Obtuse . I shall not spend time to shew Examples of all these Cases , but shall onely instance in an Example or two . In the Right angled Sphoerical Triangle P S N , let the side P N represent the Poles height , the side S P the Complement of the Suns declination , the side S N the Suns Amplitude of rising from the North Meridian , the Angle S P N the time of Suns rising from Midnight , and the angle P S N the angle of the Suns Position ; and in it let there be given the side P N 51d 32′ , and the side P S the Complement of the Suns Declination to find the angle S P N the time of the Suns rising ; then in this Case there is given the Hipotenusal , and the side adjacent to the angle sought to find the said angle ; and this is the 2d Case , whence the Proportion taken is , As Radius , To Cotangent of the Hipotenusal : So the Tangent of the given side , To the Cosine of the angle sought ; and so the Proportion to find the time of Suns rising will be As the Radius , To the tangent of the Suns declination : So the tangent of the Latitude , To the Sine of the time of Suns rising before 6 in Summer or after it in Winter , the Complement whereof is the time of its rising from Midnight . Tangent 13d Suns declination — 936336 Tangent 51d 32′ the Latitude — 1,009922 Sine 16d 53′ — , 946328 the Compl : of which Ark is 73 , 6 , which converted into time shews that the Sun riseth in our Latitude when he hath 13d of North Declination at 52● and a half past 4 in the morning ferè . By the same Data we may find the Side S N the Suns Amplitude of rising or setting , and this will agree with the 7 ●h Case ; for here is given the Hipotenusal , and one of the Leggs to find the other Leg the Proportion will be , As the Cosine of the given side : To Radius : So the Cosine of the Hipotenusal , To the Cosine of the side sought ; that is in this Cass , As the Cosine of the Latitude , To Radius : So is the Sine of the Suns Declination , To the Sine of his Amplitude from the East or West . Example , Logme . Sine 13d † Radius is — 1,935208 Sine 38d 28′ — 979383 Sine 21d 12′ — 955825 the Complement wherof , viz 68d 48′ is the side S N sought , and this Proportion is of good use to obtain the Variation of the Compass at Sea by the Suns Coast of rising ; More Examples need not be given , the Reader may try over all the Cases by the Calculated Triangle annexed . Some may say here are more Proportions then needs , especially seeing there are no Logarithmical Tables of Secants ; but Alterna amant Camenae , they have not hitherto been published in English ; the Instruments to be treated of will have Secants ; besides in some Cannons there are Tables of the Arithmetical Complements of the Logarithmical Sines and Cosines , which augmented by Radius , are the Logarithmical Secants of the Complements of those Arks to which they do belong ; and for Instruments , especially Quadrants , a Proportion having Tangents or Secants many times cannot be Operated on the Quadrant without changing the Proportion , by reason those Scales cannot be wholy brought on , being infinite ; Now the chief Grounds for varying Proportions , are built upon a few Theorems . 1. That the Rectangle or Product of a Tangent , and its Complement is equal to the Square of the Radius , or which is all one , that the Radius is a mean Proportional between the Tangent of an Arch , and the Tangent of its Complement , that is , As the Tangent of an Arch , To Radius : So Radius , To tangent of that Arks Complement , And by Inversion . As the Cotangent of an Arch , To Radius : So Radius , To tangent of that Arch , that is , As the 4 h tearm to 3d , So second to first . 2. That the Radius is a mean Proportional between the Sine of an Arch , and the Secant of that Arks Complement . That is , As the Sine of an Arch , To Radius : So is the Radius , To Secant of that Arks Complement , and the Converse . 3. That the Rectangles of all Tangents and their Complements , being respectively equal to the Square of the Radius , are Reciprocally Proportional , That is , As the Tangent of an Arch or Angle : Is to the Tangent of another Arch or Angle : So is the tangent of the Complement of the latter Arch : To the tangent of the Complement of the former , And by varying the Second Tearm into the place of the Third , we may compare the Tangent of one Ark to the Cotangent of another , &c. that is , As the tangent of an Ark or Angle : Is to the Cotangent of another Ark ∷ So is the tangent of this latter Ark To the Cotangent of the former . 4. That the Sines of Arches , and the Secants of their Complements are reciprocally proportional , that is , As the Sine of an Arch : To the Sine of another Arch or Angle : So is the Cosecant of the latter Arch , To the Cosecant of the former , And by changing the 2 and 3 Tearms , a Sine may be compared with a Secant . Now hence to be directed to vary Proportions , observe that if 4 Tearms or Numbers are Proportional , it is not material which of the two middle Terms be in the second or third place ; for instance if it be , As 2 to 4 ∷ So is 3 to 6 : It will also hold , As 2 to 3 ∷ So 4 to 6. Secondly , that when 4 Tearms are in direct Proportion , if a question be put concerning a fifth Tearm not ingredient in the Proportion , it is not material whether the two former , or the two latter Tearms be taken : As if it should be demanded ; When 2 yards of Linnen cost 4sh . What shall 8 yards ? Answer , 16. It might as well be said , If 3 cost 6 , What 8 ? Answer , 16. Hence then in any Proportion , if the two first Tearms be , As the Tangent of an ark , To Radius , to bring the Radius into the first place , it may be said , As the Radius , Is to the Cotangent of that Ark , because there is the same Proportion between these two latter Tearms , as between two former ; Now in all the former Theorems , the two latter Tearms consist either of the parts , or of the Complements of the parts of the two former , whence it will not be difficult to vary any Proportion propounded . 1. From whence it will follow , that a Proportion wholly in Tangents may be changed into their Complements without altering the Order of the Tearms , and the Converse . If it were As Tangent 10d , To tang 20d : So tang 52d , To tan 69d 15′ It would also be , As tang 80d , To tan 70d : So tan 38d , To tan 20d 45′ 2. That if the two latter Tearms of any Proportion being Tangents are only changed into their Complements , it infers a Transportation of the first Tearm into the second place . That is in the first Example , As Tang 20d , To Tangent 10d : So Tangent 38d , To Tangent 20d 45′ . 3. That if the two former Tearms of a Proportion being Tangents are changed into their Complements , it likewise infers a changing of the third Tearm into the place of the fourth . And then if the fourth Tearm be sought , it will hold , As the second Tearm , To the first : So is the third Tearm , as at first propounded to the fourth . In the first Example , as tangent 70d to tangent 80d : So tang ●2 to tang 69d 15′ . 4. That a Proportion wholly in Secants may be changed into a Proportion wholly in Sines , without altering the Order of the places , only by taking their Complements , and the Converse . If it were , As Secant of 80 To Secant 70d ∷ So Secant 60d To Secant 10d It would also hold in Sines , As the Sine 10 to Sine 20d ∷ So the Sine of 30d To Sine 80d 5. That if the two latter Tearms being Secants , should be changed into Sines , and the Converse , if they were Sines to be turned into Secants , it will be done only by taking their Complements , but then must the second and first Tearms change places one with another . If the Proportion were , As Sine 12d to Sine 42d ∷ So is the Secant of 36d to Secant of 75d 26′ . It would also hold , As Sine 42d to Sine 12d ∷ So Sine of 54d to the Sine of 14d 34′ . 6. That if the two former Tearms of a Proportion in Secants , should be changed into Sines and the Converse ; this would infer a changing of the fourth Term of that Proportion into the place of the Third : But the third Tearm not being that which is sought : The Rule to do it , would be to imagine the two first Tearms to change places , and then to take their Complements . If the Proportion were : As Secant of 39d to Secant of 75d 26′ So is the Sine of 12d to the Sine of 42d. It would also hold , As Sine 14d 34′ , to Sine of 52d. So is the Sine of 12d , to the Sine of 42d. 7 Two Tearms whether the former or latter in any Proportion being as a Sine to a Tangent , may be varied . For , As the Tangent of an Arch , To the Sine of another Arch : So is the Cosecant of the latter Arch , To the Cotangent of the former . And by transposing the Order of the Tearm . As a Sine , To a Tangent : So the Cotangent of the latter Arch , To the Cosecant of the former . This will be afterwards used in working Proportions on the Instrument , and there Instances shall be given of it . 8. Lastly , Observe that if 4 Tearms or Numbers are Proportional , their Order may be so transposed , that each of those Tearms may be the last in Proportion ; and so of any 4 Proportional Tearms , if there be given , the other that is unknown may be found , Thus , As first to second ∷ So third to fourth . As second to the first ∷ So the fourth to the third : As the third to the fourth : , So the first to the second . As the fourth , To the third ∷ So the second , To the first . Cases of Oblique Sphoerical Triangles . 1. TWo Sides together less then a Semicircle with the Angle comprehended given to find one of the other Angles . At two Operations they may be both found by a Proportion demonstrated in the late Trigonometry of the Learned Mr. Oughtred . As the Sine of half the sum of the sides , To Cotangent of half the contained angle : So the sine of half the difference of the sides , To the Tangent of half the difference of the other angles . Again , As the Cosine of half the sum of the sides , To Cotangent of half the contained angle : So the Cosine of half the difference of the sides , To the Tangent of half the sum of the other angles . Add the half difference to the half sum , and you have the greater Angle ; but substracted from it , and there remains the lesser angle . If the sum of the two given Sides exceeds a Semicircle , the Opposite Triangle , must be resolved instead of that propounded . Here note that evey Sphoerical Triangle hath opposite to each angular Point , another Triangle , having the side that subtends the said Angle common to both , and the angle opposite thereto equal , the other parts of it are the Complements of the several parts of the former to a Semicircle . So if in the Triangle B C D there were given the sides B C , and C D with their contained Angle B C D to find the Angle C B D because these two sides are greater then a Semicircle , resolve the opposite Triangle C A D , in which there will be given C A , which may be the Complement of the Latitude 38d 28″ , and C D the Complement of the Altitude 83d with the angle A C D , the Suns Azimuth from the North 73d to find the angle C A D the hour from Noon . C A 38d : 28′ Sides , C D 83 : 00 Sides , 121 : 28′ sum 60 : 44 half sum Logarithms Logarith . 44 : 32 difference 22 : 16 difference , Sine 957854 Cosine — 996634 36 : 30 half the Angle 53 : 30 Complement , Tang : 1013079 Idem — 1013079 1970933 2009713 Sine of 60d 44′ half sum 994069 Cosine 968919 Tang : of 30d 24′ ½ — 976864 tan 68d 39′ 1040794 Sum 99 : 3 hour , in Time 36′ before 6 in the morning , or as much after it in the afternoon , difference 38d 15′ Angle of ☉ position . 2 , Two Angles together less then a Semicircle with the side between them , alias , the Interjacent side , To find one of the other sides . This is but the Converse of the former to be performed at two Operations to get them both , and the Proportion thence applyed by changing the sides into Angles . As the Sine of the half sum of the angles , To the Sine of half their difference : So is the Tangent of half the interjacent side , To the Tangent of half the difference of the other sides . Again . As the Cosine of the half sum of the angles , To the Tangent of half the interjacent side : So the Cosine of half their difference , To the Tangent of the half sum of the other sides . If half the difference of the sides be added to half the sum of the sides , it makes the greater side ; but substracted from it , leaves the lesser . If the Sum of the two given Angles exceeds a Semicircle , then , as in the former Case , resolve the Opposite Triangle . So in the Triangle Z P ☉ if there were given the angle ☉ Z P , the Suns Azimuth from the North 63d 54′ , and the hour from Z P ☉ 105● in time 5 in the morning , or 7 in the evening , and the Complement of the Latitude Z P 38d 28′ , to find the Complement of the Altitude Z ☉ 80d 31′ , or the Complement of the Declination ☉ P 66● 29′ , two Operations finds both , and neither with less . Example . Angle Z P ☉ — 105d , 00d P Z ☉ — 63 : 54 difference — 41d 6′ half difference — 20 : 33 Sine — 954533 Cosine 997144 half the side Z P — 19 : 14 Tang — 954268 Idem 954268 Sum of the 〈…〉 — 168 : 54 1908801 1951412 half sum — 84 : 27 Sine — 999796 Cosine 898549 Tangent — 7 : 1′ 909005 ta 73● 30′ 1052863 73 : 30 Sum — 80 31 the greater side Z ☉ Difference — 66 29 the lesser side ☉ P 3. Two sides with an Angle opposite to one of them given , To find the Angle opposite to the other , its Affection being fore-known . As the Sine of the side opposite to the angle given Is to the Sine of its Opposite angle : So is the sine of the side opposite to the angle sought , To the sine of its opposite angle . Here note , that the same Sine is common to an Arch , and to its Complement to 180 , if the Angle sought be foreknown to be Obtuse , substract the Arch found from 180● and there remains the angle sought . Example . So in the former Triangle , if there were given the side ☉ P 66d 29′ the Complement of the Declination with its opposite angle P Z ☉ 63d 54′ , the Suns Azimuth from the North , and the side Z ☉ , the Complement of the ☉ Altitude 80 31′ , the Angle Z P ☉ the hour from Noon would be found to be 105. Sine 63d 54′ — 995329 80 31 — 999402 1994731 Sine 66d 29′ — 9906234 Sine 75d — 998497 The Complement of 75d is the angle sought , being 105d , and so much is the hour from Noon . In some Cases the Affection of the angle sought cannot be determined from what is given ; Such Cases are , When the given Angle is Acute , and the opposite Side less then a Quadrant , and the adjacent or other Side greater then the opposite Side , and its Complement to a Semicircle also greater then the opposite Side . Also when the given Angle is Obtuse , and the opposite Side greater then a Quadrant , and also greater then the other side , and greater then the Complement of the said other Side to a Semicircle . In all other Cases the Affection of the Angle sought may be determined from what is given ; in these it cannot without the help of the third side ( or something else given ) Where Cases are thus doubtful , there can be but a double answer , and both true ; wherefore find the Acute Angle and its Complement to 180d and the like answer give in Case 4 ●h , 5 ●h , 6th , 7th and 8th following . 4 , Two Angles with a Side opposite to one of them being given , To find the Side opposite to the other , its Affection being foreknown . As the Sine of the angle opposite to the given side , Is to the Sine of the given Side : So is the Sine of the angle opposite to the side sought , To the Sine of the side sought . If the side sought be foreknown to be Obtuse , the Complement of the Ark found to 180 will be the side sought . Example . So in the former Triangle , if there were given the angle at Z the Suns Azimuth from the North 63d 54′ , and the Complement of the Suns Declination ☉ P 66d 29′ with the hour from Noon Z P ☉ to find the Side Z ☉ the Complement of the Suns Altitude , it would be found to be 80d 31′ , and the Altitude it self 9d 29′ . Sine 66d 29′ — 996234 Sine 105 that is of 75d is — 998497 1994731 Sine 63d 54′ is — 995329 Sine 80d 31′ — 999402 In some Cases the Affection of the Side sought cannot be determined from what is given ; Such Cases are , When the given Angle is Acute , and the opposite Side less then a Quadrant , and the other Angle greater then the former Angle , and its Complement to a Semicircle also greater then the said former Angle . Also when the given Angle is Obtuse , and the opposite Side greater then a Quadrant , the other Angle being less then this Angle , and its Complement to a Semicircle also less then this Angle : What Snellius hath spoke concerning these doubts , is in some Cases false , in others impertinent , however I conceive not that Learned Author mistakes , but the Supervisors after his death . In all other Cases the determination is certain , as may be hereafafter shewed . 5. Two sides with an Angle opposite to one of them being given , To find the third side , the kind of the angle opposite to the other side being foreknown . First find the Angle opposite to the other side by 3d Case , and then you have two Sides and their opposite Angles . To find the third side by the Inverse of either of the Proportions used in the 2d Case , the former will be , As the Sine of half the difference of the angles given , To tangent of half the difference of the sides given : So is the sine of half the sum of those angles , To the tangent of half the side required . In the latter Case , if the sum of the given Angles exceed a Semicircle , the opposite Triangle must be resolved . Example . If in the former Triangle there were given the side ☉ P , the Complement of the Declination 66d 29′ and angle ☉ Z P , the Azimuth from the North 63d 54′ with the side Z P , the Complement of the Latitude 38d 28′ , to find the side ☉ Z , the Complement of the Suns Altitude on the Proposed Azimuth : The first Operation will be to find the Suns angle of Position Z ☉ P 37d 32′ , which is always Acute when the Sun or Stars do not come to the Meridian between the Zenith and the elevated Pole. The said angle being found by the former Directions , we proceed to the second Operation . Sides 66 29 difference 28d 1′ half 14d 00′ 30′ Tang — 939705 Sides 38 28 difference 28d 1′ half 14d 00′ 30′ Tang — 939705 Angles 63 54 Sum 101d 26 , half 50d 43 Sine — 988875 Angles 37 32 Sum 101d 26 , half 50d 43 Sine — 988875 1928580 26 22 difference , half 13d 11′ Sine — 935806 Tangent of 40d 15′ 30″ — 992774 doubled is 80d 31′ the side sought being the Complement of the Suns Altitude . 6. Two sides with an angle opposite to one of them being given , To find the angle included , or between them , the species of the opposite to the other side being foreknown . First find the angle Opposite to the other side by 3d Case , and then we have two angles and their opposite sides to find the other angle , by the Inverse of either of the Proportions used in the first Case , the former will be , As the sine of halfe the difference of the sides , To the Tangent of halfe the difference of the angles : So is the sine of halfe the sum of the sides . To the Cotangent of half the angle required ; That is , to the Tangent of an Ark , whose Complement is half the angle inquired . If the sum of the given sides be more then a Semicircle , in the resolution of this latter Case resolve the Opposite Triangle . Example . In the former Triangle given ☉ P Comple : Declination 66d 29′ Z P Comple : Latitude — 38 28 Angle ☉ Z P the Azimuth — 63 54 To find the hour Z P ☉ — 105 The first operation wil find the angle of Position as before 37 d 32′ The second Operation . half difference of the given angles 13 d 11 m Tangent — 936966 half sum of the side — 52 , 28′ , 30″ Sine — 989931 1926897 half difference of the sides 14 d 00′ 30″ Sine — 938393 Tangent 37 d 30′ — 988504 Comple : is 52 30 doubled makes 105 d , the Angle sought . 7. Two Angles with a side opposite to one of them being given . To find the third Angle , the kind of the side opposite to the other Angle being foreknown . First find the side opposite to the other Angle by 4th Case , And then we have two angles , and their opposite sides to find the third angle ; by transposing the order of either of the Proportions used in the first Case , the latter will be , As the Cosine of halfe the difference of the sides , To the Tangent of halfe the sum of the angles : So the Cosine of halfe the sum of the sides , To the Cotangent of half the contained angle . Example . In the Triangle Z ☉ P Data angle ☉ — 37 d 32′ Angle P — 105 00 Side ☉ Z — 80 31 To find the angle Z — 63 54 The first Operation will find Z P — 38 28 The second Operation . half sum of the angles — 71 d 16′ Tangent — 1046963 half sum of the sides — 59 d 29′ 30″ Sine Compl : — 970558 2017521 half difference of the sides 21 d 1′ 30″ Cosine — 997007 Tangent 58d 3′ — 1020514 Compl : 31d 57′ doubled is 63● 54′ the angle sought . 8. Two angles with a side Opposite to one of them being given , To find the Interjacent side , the kind of the side opposite to the other angle being fore known . First find the side opposite to the other angle by 4 Case , And then you have two sides , and their opposite angle given to find the 3 side by , tranposing the Order of either of the Proportions used in the 2d Case , the latter will be , As the Cosine of halfe the difference of the two angles , To the tangent of halfe the sum of the two sides : So the Cosine of halfe the sum of the two given angles , To the Tangent of halfe the third side . Example . In the former Triangle given the Hour angle at P 105d 00 Azimuth angle at Z 63 54 Compl Altitude Side Z ☉ 80 31 To find the Compl. of the Latitude the side Z P 38 28 The first Operation will find the side P ☉ 66● 29′ Second Operation . half the sum of the two sides 73d 30′ Tangent 1052839 half the sum of the two angles 84 27 Cosine 898549 1951388 half the difference of the two angles 20d 33′ Cosine 997144 Tangent of 19d 14′ — 954244 Doubled is 38 28 the side sought These 6 last precedent Cases may be called the Doubtful Cases , because that three given terms are not sufficient Data to find one single answer without the quality of a fourth , which is demonstrated by Clavius , in Theodosium , and seeing it passes without due caution in our English Books , I shall insert it from him : LEt A D and A C be two equal sides including the angle D AC , and both of them less or greater then a Quadrant . Draw through the Points C and D , the arch of a great Circle C D , continue it , and draw thereunto another Arch or Side from A , namely A B , neither through the Poles of the Arch C D , nor through the Poles of the Arch A D , so that the angles B and B A D may not be right angles , nor the angle A D B , if then each of these sides A D A C be less then a Quadrant , the two angles C , and A D C will be Acute ; and if these Arks be greater respectively then a Quadrant , the two angles C and A D C , will be Obtuse , whence it comes to pass that the angle A D B is Obtuse , when the angle A D C is Acute , and the contrary : Now forasmuch as the sides A C and A D are equal to each other , the other Data , viz. the side A B , and the angle at Bare common to both , for in each Triangle A B D , and A B C there is given two sides with the angle at B opposite to one of them ; Now this is not sufficient Data to find the angle opposite to the other side , which may be either the acute angle at C , or the Obtuse angle ADB the Complement thereof to a Semicircle : Nor to find the third side , which may be either B D , or the whole side B C , nor the angle included , which may be either B A D , or B A C , therefore in these 3 Cases we have required the quality of the angle opposite to the other given side A B , and though it be not so much observed ; in the other Trigonometry , by Perpendiculars let fall , without the knowledge of the said angle it could not be determined whether the Perpendicular would fall with in or without the Triangle , nor whether the angle found in the first Case be the thing sought , or its Complement to 180● , nor whether the angles or Segments found by 1st and 2d Operation in the other Cases are to be added together , or substracted from each other , to obtain the side or angle sought . So also two angles with a side opposite to one of them , are not sufficient Data to obtain a fourth thing in the said Triangle , without the affection of the side opposite to the other given angle . LEt A B and A C be two unequal sides containing the angle B A C both together equal to a Semicircle , one being greater , the other less then a Quadrant Draw through the Points B and C , the arch of a great Circle B C , continue it , and draw thereto from A another side AD ; but not through the Poles of A C , nor through the Poles of B C , so that the angles D and C A D may not be right angles , nor the angle A C D a right angle ; for if it were a right angle , the angle A B C whereto it is equal , should be also a right angle , and so the two sides A B and A C , by reason of their right angles at B and C should be equal , and be Quadrants contrary to the Supposition ; Now the angles A C D and A B C being equal , which is thus proved : Suppose the two sides A B and B D to be continued to a Semicircle at E , then will the said angle be equal to its opposite angle at B , the side A C by supposition is equal to the side A E , the Complement of the side A B to a Semicircle , but equal sides subtend equal angles , therefore the angle at C is equal to the angle at B or at E , which being admitted retaining the side A D and angle at D , we have another angle opposite thereto , either C or B , which are equal and common to both Triangles , and so if the side opposite to the given angle at D were sought , a double answer should be given , either the side A C , or the other side A B its Complement to 180 , and the interjacent side might be C D or B D , and the third angle the lesser angle C A D , or the greater B A D , which is not commonly animadverted . Two Sides with the Angle comprehended , to find the third Side . That the former Cases might be resolved without the help of Perpendiculars , hath been long since hinted by Mr Gunter , Mr Speidel , and Mr Gellibrand , but so obscurely that I suppose little notice was taken thereof ; but this Case hath not hitherto been resolved by any man , to my knowledge , under two Operations with a Perpendicular let fall , working by Logarithms , unless by Multiplication and Division in the natural Numbers , which being the onely Case left wherein we are to use Perpendiculars , I shall shew how to shun both , with the joynt use of the Natural and Logarithmical Tables , by a novel Proportion of my own , and illustrate the usefulness thereof by some Examples . Two Sides with the Angle comprehended , to find 3d Side . As the Cube of the Radius , To the Rectangle of the Sines of the comprehending sides : So is the Square of the Sine of half the angle contained , To half the difference of the Versed sines of the third side , and of the Ark of difference between the two including sides , Which half difference doubled , and added to the Versed Sine of the difference of the Leggs or containing sides , gives the Versed Sine of the side sought . And if you will make the third Tearm the Square of the Sine of half the Complement of the contained angle to 180d , you will find the half difference of the Versed Sines of the third side , and of the sum of the two including sides to be doubled and substracted from the Versed Sine of the said sum . But to apply the former to Logarithms . Double the Logarithmical Sine of half the angle given , & thereto adde the Logarithms of the sines of the containing Sides , & from the left hand of the Sum , Substract 3 for the Cube of the Radius , so rests the Logarithm of half the difference of those two Versed Sines above . And if instead of the second Tearm be taken into the Proportion , the double of the Rectangle of the Sines of the containing Sides ; that is , if the Logarithm of the Number 2 be added to the Logm of the other middle Tearms , you will have the Logarithm of the whole Difference in the last place ; having found it , take the Number that stands against it , either in the Natural Sines or Tangents , and accordingly add it to the Natural Versed Sine of the Difference of the Leggs , and the summe is the natural Versed Sine of the side sought . This is the Inverse of the 4 h Axiom , used when 3 sides are given to find an angle , and will be of great use to Calculate the Distances of Stars by having their Declinations and right Ascensions , or Longitudes and Latitudes given , by means whereof the Altitudes of two of them , or of the Sun with the difference of time , or Azimuth , being observed at any time off the Meridian , the Latitude may be found , as also for Calculating the distances of places in the Arch of a great Circle , all of them Propositions of good use in Navigation ; as for the latter it hath hitherto been delivered in our English Books doubtfully , erroneously , or not sufficiently for all Cases , the Rules delivered being only true in some Cases , and doubtful in most , not determining whether the side sought be greater or less then a Quadrant . The Reader may observe how necessary it is to have such Tables , as have the natural Sines and Versed Sines , &c. standing against the Logarithmical Sines , for this and other following Proportions discovered by my self for the easie calculating a Table of hours and Azimuths to all Altitudes , as also a Tables of Altitudes to all hours ; but as yet there are none such made as have the Versed Sines , but will in due time be added to Mr. Gellibrands Tables ; in the interim it may be noted , that the Residue of the Natural Sine of an Ark from Radius called its Arithmetical Complement , is the Versed Sine of that Arks Complement ; thus the natural Sine of 40d is 6427876 substracted from Radius , rests 3572124 , the Versed Sine of 50 d. And for Arks above 90 d we need no natural Versed Sines , because the natural Sine of any Arks excess above 90 d added to the Radius is equal to the Versed Sine of the said Ark , thus the Sine of 40 d augmented by the Radius is equal to the Versed Sine of 130 d and is 16427876 Example of this Case . In the Triangle ☉ Z P let there be given the side ☉ Z , the Complement of the Altitude 70 d 53′ and the side Z P the Complement of the Latitude 38 d 28 n with the angle ☉ Z P 145 d the Suns Azimuth from the North , to find the side ☉ P , the Suns distance from the Elevated Pole. Sine 38 d 28 m — 97938317 Sine 70 53 — 99753646 Sine 72 30 Log m dobled 199588390 Natural Sine against 97280353 it doubled is 10691964 Natural V Sine of 32 d 25 m the difference of the sides — 1558280 The Versed Sine of 103 d the — 12250244 side sought , and therefore the Sun hath 13 d of South declination . Another Example of this Case for Calculating the Suns Altitude on all hours . As the Cube of the Radius , To the double of the Rectangle of the Cosines , both of the Latitude , and of the Suns declination . So is the Square of the Sine of half the hour from noon , To the difference of the Sines of the Suns Meridian Altitude , and of the Altitude sought . This Canon will finde two Altitudes at one Operation , and will have very little trouble in it , the double Rectangle , that is the second tearm of the Proportion , being fixed for that Declination . Add the Logarithms of the Number two , and of the Cosines of the Declination and Latitude together the sum may be called the fixed Logarithm . Double the Logarithm of the Sine of half the hour from noon , and add it to the fixed Logarithm the sum rejecting 3 towards the left hand , for the Cube of the Radius is the Logarithm of the difference : Take the natural Sine that stands against it , and substract it from the natural Sine of the Meridian Altitude , both for the Winter and Summer Declination , and there remains the natural Sines of the Altitudes sought . If this difference cannot be substracted from the Sine of the Meridian Altitude , it argues the Sun hath no Altitude above the Horizon in this Case substract that from this , and there will remain the Natural Sine of the Suns Altitude for the like hour from midnight in Summer . Example . Let it be required to Calculate the Suns Altitude when he hath 23d 31 m both of North and South Declination for our Latitude of London at 2 and 5 a Clock in the afternoon , or which is all one for the hours of 10 and 7 in the morning . Sine 38d 28 m Compl Latitude — 97938317 Sine of 66 29 m Compl Declination — 99623428 Logarithm of Number 2 is — 03010300 Fixed Number — 200572045 Logm of Sine of 15d , doubled — 188259924 Nearest natural Sine against it , 761900 — 88831969 61d 59 m Summer Meridian Altitude Natural Sine — 8828110 Substract — 761900 the difference before found Rests — 8066210 the natural Sine of 53d 46′ the Summer Altitude for the hours of 10 and 2 14d 57 m Winter Meridian Altitude Nat Sine — 2579760 Substract the former difference — 761900 Rests the Natural Sine of 10d 27 m the — 1817860 Winter Altitude for the hours of two and ten . The same day for the Altitude of 5 and 7. Fixed number — 200572045 Sine of 37d 30 m Logm doubled — 195688942 Natural Sine against it 4226183 — 96260987 Winter Meridian Altitude , as before Sine 2579760 Rests — 1646423 the natural Sine of 9d 29 m Summer Altitude for 5 in the morning , or 7 in the evening . Natural Sine . Summer Meridian Altitude , as before — 8828110 The former difference — 4226183 Rests the Natural Sine of 27d 24 m — 4601927 The Summer Altitude for 7 in the morning , or 5 in the afternoon . The former Case may also be performed at two Operations by help of a Perpendicular supposed , without the help of Natural Tables . 1. If both Sides are equal , As the Radius , To the sine of the Common side : So the Sine of half the Angle , To the Sine of half the side sought . 2. If one of the sides be a Quadrant , this by continving the other side to a Quadran ( as shall afterwards be shewed ) wil become a Case of right angled Sphoericala Triangles , in which besides the right angle , instead of the quadrantal side , there will be given a Legg , and its adjacent angle to find the other angle by 4 Case of right angled Sphoerical Triangles ; and so if the angle included were 90d it would be a Case of right angled Sphoerical Triangles , in which besides the right angle , there would be given both the Leggs or Sides to find the Hipotenusal . 3. In all other Cases one or both of the including Sides being less then Quadrants , it will hold , As the Radius , To the Cosine of the angle included : So the tangent of the lesser side , To the tangent of a fourth Ark , If the angle included , be less then 90d substract the 4 ●h Ark from the other side ; but if it be more from the other sides Complement to 180d , The remainder is called the Residual Ark. Then , As the Cosine of the 4th Ark , To the Cosine of the Ark remaining : So the Cosine of the lesser side , To the Cosine of the side sought . The side sought may be greater then a Quadrant , and so be doubtfull , but we may determine , That when the Leggs are of the same kind , and the angle comprehended Acute , the side sought is less then a Quadrant . And when the Leggs or containing Sides are of a different kind , and the angle comprehended Obtuse , the side sought is greater then a Quadrant . Or it may be determined from the affection of the Residual Ark in all Cases . When the contained angle is acute , and the residual Ark more then 90d , or when the said angle is Obtuse , and the residual Ark less then a Quadrant , the side sought is greater then a Quadrant , in all other Cases less . Example . In the Triangle ☉ Z P , let there be given Z P , and ☉ Z with the angle at Z , to find the side ☉ P , the Suns distance from the Elevated-Pole . angle included 145d Logm Or , 35 Compl 55d Sine — 99133645 Tangent of 38 28′ lesser side — 99000865 Tangent 33d , 3′ — 98134510 Compl ☉ P to 180d is 109 7 The Ark remaining or differ : 76d 4 m Cosine — 93816434 Lesser side — 51 32 Cosine — 98937452 192753886 Ark found — 33d 3 m Cosine — 99233450 Sine 13 — 93520436 The Complement hereof 77 d should be the side fought , but because the angle was Obtuse , and the residual Ark less then a Quadrant , the side sought is greater , and therefore 103 d the Complement hereof is the side sought . This Case & the Converse of it being the next Case , I have thus setled to apply the to Logarithmical Tables only , in Case the natural ones were wanting , being all the other Cases are thereto fitted ; and as the trouble about the Cadence of a Perpendicular is here shunned , without so much as the name of it ; so may it be done in all the rest of the Oblique Cases , which I had so fitted up for my own use ; but forbear to trouble the Reader with them , apprehending these to be better , and that he would not willingly Calculate for a portion of an angle , or a Segment of a Side , in order to the finding out the thing sought , when with as little trouble he may come by it , and yet Calculate always either for a side or an angle , one of the six principal parts of the Triangle . Otherwise for Instruments . As the Diameter , To the difference of the Versed Sines of the sum , and of the difference of any two sides , including an Angle . Or , As the Cosecant of one of the including Sides , Is to the Sine of the other side : So is the Versed Sine of the angle included . To the difference of the Versed Sines of the Ark of difference between the two including Sides , and of the third side sought , Which difference added to the Versed Sine of the difference of the Leggs , makes the Versed Sine of the side sought . And so is the Versed Sine of the contained angles Complement to 180d To the difference of the Versed Sines of the sum of the Leggs , and of the side sought , which substracted from the Versed Sine of the said sum , there remains the Versed sine of the side sought , Here note , that the same Versed Sine is common to an Ark greater then 180 d , and to its Complement to 360 d , So the Versed Sine of 200 d is also the Versed Sine of 160 d. The Proportions delivered for Instruments having such Tables as before hinted , will not be so unsuitable to the Logarithms as commonly reputed . Example for Calculating the distance of two places in the Arch of a great Circle , otherwise then according to the general Cannon before delivered . As the Secant of one of the Latitudes , To the Cosine of the other , So the Versed Sine of the difference of Longitude , To the difference of the Versed Sines of these two Arks , The one the Ark of distance sought , the other the Ark of difference between both Latitudes , when in the same Hemisphere , or the sum of both Latitudes when in different Hemispheres , which difference added to the Versed Sine of this latter Ark , the sum is the versed Sine of the distance , By turning the Substraction to be made of the first Tearm into an Addition , the two first Tearms of the Proportion will be , As the Square of the Radius , To the Rectangle of the Cosines of both the Latitudes : Then for the third Tearm being the difference of Longitude , take the natural Versed Sine thereof , and seek that Number in the natural Tangents , and that Logarith Tangent that stands against it take into the Proportion instead of the Logarithm of the Versed Sine proposed . Admit it were required to find the Distance between London and Bantam , in the Arch of a great Circle . Logme Bantam Longitude 140 d Latitude 5 d 40′ South Cosine 9,9978725 London Longitude 25 , 50 Latitude 51 , 32 North Cosine 9,7938317 — difference of Long 114 d 10′ Nat V Sine 14093923 equal to the natural Tangent of 54 d 38′ ½ nearest Logm 10,1489900 Natural Sine 8723538 against it — 2 9940694 Nat Versed Sine of 57 d 12′ the sum of both Latitudes 4582918 — Sum — 13306456 the natural Versed Sine of 109 d 18′ 30″ the Ark of distance sought . And if to the said difference , namely — 8723538 Be added the natural Versed Sine of the difference of both Latitudes , namely the V Sine of 45 d 52′ — 3036695 — The sum being the natural V Sine of 100 d 8′ 30″ is — 11760233 the distance of two places , having the same Latitudes , and difference of Longitude , but are both in the same Hemisphere . Here note , that no two places can have above 180 d difference of Longitude , therefore in differencing the two Longitudes if the remainder be more take its Complement to 360 d. The Complements of these two distances , namely 70 d 41′ 30″ and 79 d 51′ 30″ are the distances of two places of the same Latitudes considered as in different Hemispheres , their difference of Longitude being 65 d 50′ the Complement of the former , and two places in a such Position compared with their former Positions may be apprehended to be Diametrically opposite upon the Globe , as thus , Bantam having 5d 40′ South Latitude , let another place have as much North Latitude , the difference of Longitude between them 180 d and consequently so much their distance ; now whatever be the distance between Bantam and the third place , the Complement of it to 180 d shall be the distance between the two other places . 10. Two angles with the Interjacent side given . To find the 3d angle , the proportion derived from the former Case by changing the angles into sides , and holds without any such change supposed is , As the Cube of the Radius , To the double of the Rectangle of the Sines of the two given angles : So is the Square of the Sine of half the given side , To the difference of the Versed Sines of these two Arks , the one is the angle sought , the other the Ark of difference between one of the including angles , and the Complement of the other to a Semicircle , which difference added to the Versed Sine of this Ark gives the Versed Sine of the angle sought . How to work this by Tables need not be shewed after the Logm of the difference is got , if it be less then the Radius , it may be sought either in the Sines or Tangents , and the natural Sine or Tangent that stands against it and comes nearest taken ; but when it exceeds the Radius always seek it in the Tangents , and take the natural Tangent that stands against it , which difference so found , is to be added to the Versed Sine of the difference of the Leggs to obtain the Versed Sine of the angle sought . Otherwise for Tables the common way by a supposed Perpendicular 1. If both the angles are equal , As the Radius . To the Sine of the angle given : So the Cosine of half the given Side , To the Cosine of half the angle sought . In all other Cases not belonging to right angled Triangles if one or both of the given angles be Acute , it holds , As the Radius , To Cosine of the interjacent side : So the Tangent of the lesser angle , To the Tangent of a 4● h Ark. If the interjacent side be more then 90d substract the 4● h Ark from the other angle ; but if less then 90d , substract the 4● h Ark from the other angles Complement to 180d , noting the residual Ark. Then , As the Cosine of the 4th Ark , To the Cosine of the Ark remainmaining : So the Cosine of the lesser angle , To the Cosine of the angle sought . When the interjacent side is less then a Quadrant , and the residual Ark more , or when the interjacent side is greater then a Quadrant , and the residual Ark less , the angle sought is Obtuse , in all other Cases Acute . In the Triangle ☉ Z P let there be given The angle of Position at ☉ — 21d 28′ The hour from noon angle at P — 33 47 And the side ☉ P the Suns distante from the elevated Pole — 103 00 To find his Azimuth the angle ☉ Z P Sine 13 d the Complement of the interjacent side — 93520880 Tangent 21 d 28′ the lesser angle — 95946561 Tangent of 5 d 3′ — 89467441 The other angle — 33 47 The difference being the residual ark 28 44 Cosine — 99429335 Lesser angle — 21 28 Cosine — 99687773 199117108 Ark first found — 5d 3′ Cosine — 99983109 Sine 55 d — 99133999 The Complement whereof 35 d in this Case is not the angle sought , but the residue hereof from a Semicircle 145 d is the angle sought being Obtuse , because the interjacent side is greater then a Quadrant , and the residual Ark less ; the residual Ark in Operation if greater then a Quadrant , take its Complement to 180 d , because there are no Sines to Arks above a Quadrant , and then the Complement of this Ark to 90d is the Complement of the residual Ark the Sine whereof must be taken for the Cosine of the residual Arke . Otherwise for Instruments . As the Diameter , To the difference of the Versed Sines of the sum and difference of the two including angles , Or , As the Cosecant of one of those angles , Is to the Sine of the other , So the Versed Sine of the interjacent side , To the difference of the Versed Sine of an Ark left by substracting one of the including angles from the Complement of the other to a Semicircle , and of the angle sought , which difference added to the Versed Sine of the said Ark , gives the Versed Sine of the angle sought , And so is the Versed Sine of the interjacent sides Complement to 180 d , To the difference of the Versed Sines of an Ark made by adding one of the including angles to the Complement of the other to a Semicircle , and of the angle sought , which substracted from the Versed Sine of the said Ark , leaves the versed sine of the angle sought . 11. Three Sides to find an Angle . The two sides including the angle sought are called Leggs , and the third side the Base . As the Rectangle or Product of the Sines of the half sum of the three sides and of the difference of the Base therefrom . Is to the Square of the Radius : So is the Rectangle of the sines of the differences of the Leggs from the said half sum , To the Square of the Tangent of half the angle sought . And by changing the third Tearm into the place of the first , As the Rectangle of the Sines of the differences of the Leggs from the half sum of the 3 sides , Is to the Square of the Radius : So the Rectangle of the Sines of the half sum of the three sides , and of the difference of the Base therefrom , To the Square of the Tangent of an Ark , whose Complement doubled is the angle sought , or this Ark doubled is the Complement of the angle sought to 180 d , or it might be expressed , To the Square of the Cotangent of half the angle sought . Otherways in Sines . As the Rectangle of the Sines of the containing Sides or Leggs , Is to the Square of the Radius ; So the Rectangle of the Sines of the differences of the Leggs from the half sum of the three sides , To the Square of the Sine of half the angle sought . Or the Cosine may be found . As the Rectangle of the Sines of the containing sides , Is to the Square of the Radius : So the Rectangle of the Sines of the half sum of the 3 sides , and of the difference of the Base therefrom , To the Square of the Cosine of half the angle sought . These two latter Proportions are demonstrated in the Treatises of the Lord Napier , Mr Oughtred , Mr Norwood , and are those from whence I shall educe the Demonstrations of the rest . To work the third Proportion that finds the Square of the Sine of half the angle . To the Arithmetical Complements of the Logarithms of the sines of the containing Sides or Leggs add the Logarithmical Sines of the differences of the said Leggs from the half sum of the three Sides , the half sum of these four Numbers will be the Logarithm of the sine of half the angle sought . In the Triangle ☉ Z P , Data , the three Sides to find the angle a P the hour from noon . 80 d 31′ Base 66 29 Leggs — Ar comp , 0376572 38 28 Leggs — Ar comp , 2061683 Sum-185 , 28 difference of the Leggs half-92 44 from half sum — 26d 15′ Sine 9,6457058 54 , 16 Sine 9,9094190 Sum — 19,7989503 Sine of 52 d 30′ half — 9,8994751 doubled 105 , the angle at P sought . The Arithmetical Complement of a Logarithm , is the residue of that Logarithm from the next bigger Number , consisting of an Unite and Ciphers . Otherwise for Instruments . As the difference of the Versed Sines of the sum , and of the difference of any two sides including an angle , Is to the Diameter , Or , As the sine of one of the said sides , To the Secant of the Complement of the other . So is the difference of the Versed Sines of the third side , and of the Ark of difference between the two including sides , To the Versed Sine of the angle sought . And so is the difference of the Versed Sines of the third , and of the sum of the two including sides , To the versed Sine of the sought angles Complement to 180d. 12. Three Angles to find a Side . The work here for the Canon or Tables , will be by changing the Angles into Sides , the general Rule for changing all the parts of a Triangle , is to draw a new Triangle , and let the angles be wrot against their Opposite sides , and these against those , only taking the Complements of the greatest Angle , and greatest side opposed thereto to 180 d , this for most convenience that the sides or angles of the new framed Triangle may not be too large , and so cause recourse to the Opposite Triangle , otherwise the Complements of any side and its opposite angle to 180 d ▪ might as well have been taken . But for this Case , seeing there are only angles to be changed into sides , take the Complement of the greatest angle to 180 d and proceed as if there were three sides given to find an angle . But the Proportion in Versed Sines , &c. without any such change will be , As the difference of the Versed Sines of the sum , and of the difference of any two angles adjacent to the side sought . Is to the Diameter , Or , As the Sine of one of the said angles , Is to the Cosecant of the other : So is the difference of the Versed Sines of the third or Opposite angle , and of an Ark left by substracting one of the including angles from the Complement of the other to a Semicircle , To the Versed Sine of the side sought . And so is the difference of the Versed Sines of the third angle , and of an Ark made by adding one of the including Angles to the Complement of the other to a Semicircle . To the Versed Sine of the sought sides Complement to 180d . Thus having finished the Cases , it is to be intimated that the Proportions here used in Versed Sines are variously demonstrated in diverse Writers , but in most the latter part for finding the Complement of an angle to 180d , is quite omitted , those that have demonstrated the former part , do it in these tearms following . As the Rectangle of the sines of the containing sides , Is to the Square of the Radius : So is the difference of the Versed Sines of the Base , or third Side , and of the Ark of difference between the two including sides , To the Versed Sine of the angle sought , which the Reader may see in Lansberg , Regiomantanus , Snellius , Pitiscus , and the learned Clavius , who makes 15 Cases , and twice as many Scheams , to demonstrate this part of it . I shall only shew how it may be inferred from the common Proportions in use fitted to the Tables demonstrated by the Lord Napier , Mr Oughtred , Mr. Norwood . We have two Proportions delivered in Rectangles and Squares the one for finding an angle , the other to find its Complement to 180d. The two first tearms are the Proportion between the Rectangle of the Sines of the containing sides , and the Square of the Radius ; these two tearms being divided by the Sine of one of those sides , the Quotient will be the Sine of the other , if the same Divisor divide the Square of the Radius , the Quotient will be the Secant of the Complement of the Ark belonging to the Divisor , because , As the Sine of an Ark , To Radius , So is the Radius , To the Secant of that Arks Complement ; But if any common Divisor divide any two Tearms of a Proportion , the Dividends will be Acquimultiplex to the Quotients ; and therefore by the Quotients will bear such Proportion each to other as the Dividends , and therefore it holds , As the Rectangle of the Sines of the containing sides , Is to the Square of the Radius : So is the Sine of one of those sides , To the Secant of the Complement of the other . Again , for the third Tearm , to find an angle it is proposed . So is the Restangle of the Sines of the differences of the Leggs from the half sum of the three sides . Or which is all one , So is the Rectangle of the Sines of the half sum , and half difference of the Base or third side , and of the Ark of difference between the two including sides , To the Square of the Sine of half the angle sought , And so to find the Complement of an angle to 180 d. So is the Rectangle of the Sines of the half sum of the three sides , and of the difference of the third side or Base therefrom , Or which is equivalent thereto , So is the Rectangle of the Sines of the half sum , and half difference of the Base or third side , and of the sum of the two including sides , To the Square of the Sine of an Ark , which doubled is the Complement of the angle sought to 180 d , or the Complement of that Arch to a Quadrant doubled , is the angle sought . The former of these two expressions of the third Tearm of the Proportion , as being the more facil for memory is now retained ; but the latter , ( formerly used , and now rejected ) agrees best with the Proportion , as applyed to Versed Sines , for the inferring whereof note , that such Proportion , As the difference of two Versed Sines beareth to another Versed Sine , the same Proportion doth the half difference of those Versed Sines , bear to half the Versed Sine of that other Arch : But that is the same that the Rectangle of the Sines of the half sum and half difference of any two Arks doth bear to the Square of the Sine of half that other Arch , which will be thus inferred , because if the said Rectangle and Square be both divided by Radius , the two Quotients will be the half difference of the versed Sines of the two Arks proposed , and half the versed Sine of the 4 ●h Arch. That the Sines of the half sum and half difference of any two Arks are mean Proportionals between the Radius and the half difference of the Versed Sines of those Arks is demonstrated in Mr Gellibrands Trigonometry in Octavo , that is , As the Radius , To the Sine of half the sum of any two arks : So is the sine of half the difference of those two arks , To half the difference of the versed sines of those two arks , and therefore the said Rectangle divided by Radius , the Quotient is half the difference of the versed sines of the two Arks. And that the Sine of any Arch is a mean Proportional between the Radius and half the versed Sine of twice that Arch , That is , As the Radius , Is to the sine of an Arch : So the sine of that Arch , To half the versed sine of twice that Arch , and therefore the Square of the sine of any Arch divided by Radius , the Quotient is the half versed sine of twice that Arch ; whence the Rule to make a Cannon of whole Logarithmical versed sines is to take half the arch proposed , and to the Logarithm thereof doubled , or twice wrot down , to add the Logarithm of the number two , and from the sum to substract the Radius . We have before inferred , that As the Rectangle of the sines of the containing sides , Is to the Square of the Radius : So is the sine of one of those sides , To the Secant of the Complement of the other , and that by dividing those two Plains by one of those sides ; but if we divide the said two Plains , viz. the Rectangle of the sines of the containing sides , and the Square of the Radius , by the Radius as a common Divisor , the latter Quotient will be the Radius , and the former the half difference of the versed sines of those Arks whereof the two containing sides are the half sum and the half difference ; but those Arks are found by adding the half difference to the half sum to get the greater , and substracting it therefrom to get the lesser ; Which is no other then to get the sum and difference of the two containing sides , it therefore holds , As the Rectangle of the sines of the containing sides , Is to the Square of the Radius , Or , As the sine of one of those sides , To the Secant of the Complement of the other : So is the half difference of the versed sines of the sum and difference of those two sides to the Radius , And by consequence so is the whole difference to the Diameter , and this being admitted the whole Proportion in all its parts may be inferred from Mr Daries Book of the Uses of a Quadrant , where he demonstrates , That , As the difference of the versed Sines of the sum and difference of any two sides including an angle , Is to the Diameter : So is the difference of the versed sines of the third side , and of the Ark of difference between the two including sides , To the versed sine of the angle sought , in that Scheme it lyes , As M S , To G H : So is M P , To H C. And I further add , As M S , To G H : as before , So is P S , To G C. that is , retaining the two first Tearms of the Proportion , the same it holds for the third and fourth Tearm . So is the difference of the versed sines of the third side , and of the sum of the two including sides , To the versed sine of the sought angles Complement to 180d. Now from these Proportions thus Demonstrated , are inferred those others that give the answer in the Squares of Tangents , in order whereto observe , That if 4 Numbers are Proportional , their Squares are also Proportional ( quamvis non in eadem rations ) so that any three of those Squares being given , the Square of the 4th will be found by direct Proportion , and the Proportion for making a Table of Natural Tangents from the Tables of natural sines is , As the Cosine of an Ark , To the sine of the said Ark : So is the Radius , To the Tangent of the said Ark. It will therefore hold by 22 Prop. of 6 h Book of Euclid , As the Square of the Cosine of an Ark , Is to the Square of its sine : So is the Square of the Radius , To the Square of its Tangent , Now from the two Demonstrated Proportions for the Tables , the two first Tearms are common to both , and therefore there is the like Proportion between the two latter Tearms of the first Proportion , and the two latter in the second , as between the two latter , and the two former in each Proportion : Now because the latter Proportion finds the Square of the Cosine , and the former the Square of the Sine of the same Ark , it is inferred that the third tearm in the latter Proportion , bears such Proportion to the third Tearm in the former Proportion , as the Square of the Cosine of an Ark , doth to the Square of its Sine , which is the same that the Square of the Radius bears to the Square of the Tangent of the said Ark , it therefore holds when three sides of a Spherical Triangle , are given to find an angle . As the Rectangle of the Sines of the half sum of the three sides , and of the difference of the Base therefrom , Is to the Rectangle of the Sines of the differences of the Leggs therefrom : So is the Square of the Radius , To the Square of the Tangent of half the angle sought , and by changing the 2d Tearm into the place of the first . As the Rectangle of the sines of the differences of the Leggs from the half sum of the 3 sides , Is to the Rectangle of the sines of the half sum of the three sides , and of the difference of the Base therefrom : So is the Square of the Radius , To the Square of the Cotangent of half the angle sought . These Proportions are published in order to their Application to the Serpentine Line , which will be accomodated for the sudden operating of any of them ; the Axioms to be remembred are not many , the Reader will meet with their Demonstration and Application in Mr Newtons Trigonometry now in the Press , and said to be near finished : The four Proportions in plain Triangles , when three sides are given to find an angle without the Cadence of Perpendiculars are demonstrated in the 27 Section of the late Miscellanies of Francis van Schooten . The Construction of diverse Instruments will require a Table of the Suns Altitudes to the Hour and Azimuth assigned ; And for the Acurate bounding in of the Lines , it may be a Table of Hours and Azimuths to any Altitude assigned ; for the easie Calculating whereof , I am desired for the ease and benefit of the Trade , to render this part of Calculation as facil as I can , and therefore shall handle it the more largely . To Calculate a Table of Hours to all Altitudes in all Latitudes . The 1. Proportion shall be to find the Suns Altitude in Summer , or Depression in Winter at the hour of 6. As the Radius , To the sine of the Latitude : So the sine of the Declination , To the sine of the Altitude or Depression sought This remains fixed for all that day the Suns Declination supposed not to vary , and then it holds , As the Cosine of the Declination , To the Secant of the Latitude : So in Summer is the difference in Winter the sum of the sines of the Suns Altitude proposed , and of his Altitude or Depression at 6 To the sine of the hour from 6 towards noon in Winter , and in Summer also , when the given Altitude is greater then the Altitude of 6 , but when it is less towards midnight . This Proportion also holds for Calculating the Horary distance of any Star from the Meridian . In like manner to Calculate the Azimuth . As the sine of the Latitude , To sine of the Declination : So is the Radius , To the side of the Suns Altitude or Depression in the prime Vertical , that is , being East or West . This remains fixed for one day . Then , As the Cosine of the Altitude , To the Tangent of the Latitude : So in Summer is the difference , and in Winter the sum of the sines of the Suns Altitude proposed , and of his Vertical Altitude or Depression , To the sine of the Azimuth towards noon Meridian in Winter and in Summer also , when the given Altitude is greater then the Vertical Altitude or Depression , but when it is less towards Midnight Meridian . This Proportion is general either for Sun or Stars , when the Declination is less then the Latitude of the place ; But when it is more , say as before , As the sine of the Latitude , To the sine of the Declination : So is the Radius , To a fourth we may call it a Secant . Again . As Cosine Altitude , To the Tangent of the Latitude : So in declinations towards the Depressed Pole is the sum ; but towards the Elevated Pole the difference of this Secant , and of the sine of the Sun or Stars Altitude , To the sine of the Azimuth from the Vertical towards the noon Meridian . Before Application be made , the latter part of these Proportions being of my own peculiar Invention , and of very great use both for Calculation , and Instrumentally , it will be necessary to demonstrate the same . For the Hour from the Analemma . Having in the Scheme annexed drawn the Equator and Horizon , the two prickt Lines passing through the Center , as also the Prime Vertical and Axis , the two streight Lines passing through the same . Let I X and L M represent two Parralells of Declination on each side the Equator , and O X a Parralel of the Suns Altitude in Summer , and P Q of his Depression in Winter , at the hour of 6 , because these Parralells pass through the Intersection of the Parralells of Declination with the Axis . Let R S be a Parralel of Altitude after 6 , and T V a Parralel of Altitude before it ; from the Intersections of these Parralells of Altitude with the Parralels of Declination let fall Perpendiculars on the Parralells of the Suns Altitude or Depression at 6 , and then we shall have divers right Lined right angled Triangles Constituted in which we shall make use of the Proportion of the sines of angles to their opposite sides an Axiom of common demonstration . In the Triangle A F E , As the sine of the angle at F the Radius , To its Opposite side A E , the sine of the Declination : So the sine of the Latitude the angle at E to A F , the sine of the Suns Altitude at 6. Again in the two Opposite Triangles A B C , the smaller before the greater after 6. As the Cosine of the Latitude the sine of the angle at A , To its Opposite side B , C , the difference of the sines of the Suns Altitude at 6 , and of his proposed Altitude : So is the Radius sine of the angle at B , To C A , the sine of the hour from 6 in the Parralel of Declination in the lower Triangle before , in the upper after 6. So in the Winter or lower Triangle A B D C. As Cosine of the Latitude sine of the angle at A , To B C , the sum of the sines of the Suns Depression at 6 B D , and of his given Altitude D C : So is the Radius the sine of the angle at B , To A C , the sine of the hour from 6 towards noon in the Parralel of Declination , The sine of the hour thus found in a Parralel , is to be reduced by another Analogy to the common Radius , and that will be , As the Radius of the Parralel I A , the Cosine of the Declination , Is to the common Radius E AE : So is any other sine in that Parralel . To the sine of the said Arch to the common Radius . Now it rests to be proved that both these Analogies may be reduced into one , and that will be done by bringing the Rectangle of the two middle Tearms of the first Proportion with the first Tearm under them as an improper Fraction to be placed as a single Tearm in the second Proportion , being in value the answer found in the Parralel , and then we have the Rule of three to Operate as it were in whole Numbers and mixt . The Proportion will run , As the Cosine of the Declination , To Radius : So the said Improper Fraction ▪ To the Answer . and so proceeding according to the Rules of Arithmetick . The Divisor will be the Rectangle of the Cosine of the Declination , and of the Cosine of the Latitude , one of the middle Tearms would be the Square of the Radius , and the other the former sum or difference . Now if any two Tearms of a Proportion be divided by a common Divisor , the Dividends being Equimultiplex to the Quotients , the Quotients bear the same mutual Proportion as the Dividends by 18● Propos . 7 Euclid . So in this instance if the Rectangle of the Cosines both of the Latitude and of the Declination be divided by one of those Tearms , the Quotient will be the other , and if the Square of the Radius be divided by the Sine of an Arch , the Quotient will be the Secant of that Arks Complement ; So in the present Example , if the former Rectangle be divided by the Cosine of the Latitude ▪ the Quotient is the Cosine of the Declination , if the Square of the Radius be divided by the same Divisor , the Quotient is the Secant of the Latitude , likewise if both those Plains were divided by the Cosine of the Declination , the Quotients would be the Cosine of the Latitude , and the Secant of the Declination , it therefore holds , As the Cosine of the Declination , To the Secant of the Latitude , Or , As the Cosine of the Latitude , To the Secant of the Declination : So is the former sum or difference of sines , To the sine of the hour from 6 , which was to be proved . Corrollarie . As the Radius , To the sine of an Arch in a lesser Circle or Pararlell : So is the Secant of that Parralell , To the sine of the said Arch , to the common Radius . Hence may be observed a general Canon for the double or compound Rule of three , divide the Tearms into two single Rules , by placing two Tearms of like Denomination in each Rule , and the other remaining Tearm may in most Cases be put among either of these two Tearms of like Denomination , and then by arguing whether like require like , or unlike , the Divisor in each single Rule , may be discovered , and then it will hold in all Cases , As the Rectangle or Product of the two Divisors , Is to the product of any two of the other Tearms : So is the other Tearm left , To the Number sought , For the Azimuth . Having drawn the Horizon and Axis , the two prickt Lines , the Vertical Circle Z N , and the Equinoctial Ae Ae , the Parralels of Declination I K and L M , draw T V a Parralel of lesser Altitude then that in the Vertical , and R S a Parralel of greater Altitude ; Draw also P Q a Parralel of Depression equal to the Vertical Altitude , in the point C aboue the Center the Point A being as much below it b , eing the point where the Parralel of Declination intersects the Vertical Circle , and from the point C in the lesser parralel of Altitude , let fall the perpendicular C B on the parralel of Depression P Q , by this means there will be Constituted divers right lined , right angled Triangles , and through those Points where the parralel of Declination , and parralels of Altitude intersect , are drawn Elipses prickt from the Zenith to represent the Azimuths , and in the three several Triangles thus Constituted , the side A B measureth the quantity of the Azimuth in the parralel of Altitude , and B C in the two upper Triangles is the difference of the sines of the Suns proposed Altitude , and of his Altitude in the prime Vertical : But in the lower Triangle the sum of them , it then holds by the Proportion of the sines of Angles to their opposite sides . In the two upper Triangles , As the Cosine of the Latitude , the sine of the angle at A , To its opposite side B C , the difference of the sines of the Suns Vertical , and of his proposed Altitude : So is the sine of the Latitude , that is the sine of the angle at C , To its opposite side B A , the sine of the Azimuth from the East and West , And the like in the lower Triangle , onely there the third Tearm B C , is the sum of the sines of the Suns Vertical Depression , and of his given Altitude : Such Proportion as as the Cosine of an Ark doth bear to the sine of an Ark , doth the Radius bear to the Tangent of the said Ark , this being the Canon by which the natural Tangents are made from the natural sines , and therefore we may change the former Proportion , and instead thereof say , As the Radius , To the Tangent of the Latitude : So the said sum or difference of Sines , To the Sine of the Azimuth in the Parralel of Altitude : The answer falling in a Parralel or lesser Circle is to be reduced to the common Radius by another Analogy , and that is As the Cosine of the Altitude ( the Radius of the parralel ) To the Radius : So any sine in the said parralel , To the like sine in the common Radius . Now it is to be proved that both these Proportions may be brought into one , and that will be as before , by making an improper Fraction whose Numerator shall be the Rectangle of the two middle Tearms of the former Proportion , the first Tearm , viz. the Radius being the Denominator , and placing this as the third Tearm in the second Proportion , and then those that understand how to operate the Rule of three in whole Numbers and mixt , will find their Divisor to be the Rectangle or Product of the Cosine of the Altitude , and of the Radius , and the Dividend the Product of the three other Tearms , namely , of the Tangent of the Latitude , the Radius , and the former sum or difference of sines , whence it holds , As the Rectangle of the Cosine of the Altitude , and of the Radius , Is to the Rectangle of the Tangent of the Latitude , and of the Radius : So is the former sum or difference of sines , To the sine of the Azimuth . The Reader may presently espy that the two former Tearms of this Proportion may be freed from the Radius by dividing them both thereby , and the Quotients will be the Cosine of the Altitude , and the Tangent of the Latitude , It therefore holds , As the Cosine of the Altitude , To the Tangent of the Latitude : So in Summer is the difference , in Winter the sum of the sines of the Suns Vertical and proposed Altitude , To the sine of the Azimuth from the Vertical . This is general either for Sun or Stars , when their Declination is less then the Latitude of the place ; but when it is more , the Case doth but little vary . In the Scheme annexed fitted to the Latitude of the Barbados having drawn H H the Horizon , P P the Axis , Ae Ae the Equator , Z A the Vertical draw two parralells of Declination F R , K A continued till they intersect the Vertical prolonged , draw the parralel of Altitude B ☉ , and parralel thereto from the Point A draw A E , Then doth the latter part of the Proportion lye as evident as before , In the right angled Triangle C G F right angled at G , As the sine of the Latitude the angle at F , To its Opposite side C G the sine of the Declination , So the Radius the angle at G , To the Secant C Z F. Again in Summer . As the Cosine of the Latitude the angle at ☉ , To its opposite side D Z F , the difference between the former Secant and the sine of the Altitude : So is the sine of the Latitude , the angle at F , To its opposite side D ☉ , the sine of the Azimuth from the Vertical in the Parralel of Altitude . In Winter , As Cosine Latitude angle at A , To B E the sum of the former Secant equal to E M , and of the sine of the Altitude M B : So is the sine of the Latitude the angle B , To A E , equal to B D the sine of the Azimuth in a Parallel as before , to be reduced to the common Radius . From this Schem may be observed the reason why the Sun in those Latitudes upon some Azimuths hath two Altitudes , because the Parralel of his Declination F R intersects , and passeth through the Azimuth , namely , the prickt Ellipsis in the two points S , ☉ . I now proceed to the Vse in Calculating a Table of Hours . For those that have occasion to Calculate a Table of Hours to any assigned Altitude and parralel of Declination , it will be the readiest way to write down all the moveable Tearms first , as the natural sines of the several Altitudes in a ruled sheet of Paper , and then upon a peice of Card to write down the natural sine of the Suns Altitude at 6 and removing to every Altitude , get the sum or difference accordingly , which being had , seek the same in the natural sines , and write down the Log m that stands against it , then upon the other end of the piece of Card get the sum of the Arithmetical Complements of the Logarithmical Cosine of the Declination , and of the Logarithmica Cosine of the Latitude , and add this fixed Number to the Logme before wrote down ; by removing the Card to every one of them , and the sum is the Logme of the sine of the Hour from 6 , if the Logmes be well proportioned out to the differences which may be sufficiently done by guess . Example . Comp Latitude 38d 28′ Ar Comp — 0,2061683 Comp Delinat 66 , 29 Ar Comp — 0,0376572 fixed Number — ,2438255 Let the Altitude be 36● 42′ Nat Sine 5976251 Natural sine Altitude at 6 — 3124174 difference — 2852077 Log against 9 4550441 Sine of 30d the hour from 6 towards noon — 9,6988696 Another Example . N S Altitude at 6 — 3124174 Let the Altitude be 13d 46′ N S 2379684 difference — 744490 Logm — 8,8715646 The former fixed Number — 0,2438255 Sine of 7d 30′ the hour from 6 towards midnight because the Altitude is less then the Altitude of 6. 9,1153901 This method of Calculation will dispatch much faster then the common Canon , when three sides are given to find an angle ; the Azimuth may in like manner be Calculated , but will be more troublesome not having so many fixt Tearms in it , and having got the hour , the Azimuth will be easily found ; in this Case we have two sides and an angle opposite to one of them given , to find the angle opposite to the other , and the Proportion , will hold , As the Cosine of the Altitude , To the sine of the hour from the Meridian : So the Cosine of the Declination , To the sine of the Azimuth from the Meridian . And in this Case the three sides being given , we may determine the affection of any of the angles . If the Sun , or Stars have declination towards the depressed Pole , the Azimuth is always Obtuse , and the hour and angle of position-Acute . If the Sun , &c. have declination towards the Elevated Pole , but less then the Latitude of the place the angle of Position is always acute , the hour before 6 obtuse , the hour and Azimuth between the Altitude of 6 , and the Vertical Altitude both acute , afterwards the hour acute , and the Azimuth obtuse . But when the Sun or Stars come to the Meridian between the Zenith and the Elevated Pole , as when their declination is greater then the Latitude of the place , the Azimuth is always acute , the hour before 6 obtuse , afterwards acute . The angle of position from the time of rising to the remotest Azimuth from the Meridian is acute , afterwards obtuse . Another General Proportion for the Hour . As the Radius , To the Tangent of the Latitude : So the Tangent of the Suns declination , To the sine of the hour of rising from six : Again . To the Rectangle of the Cosine of the Latitude , and of the Cosine of the declination , Is to the Square of the Radius : So is the sine of the Altitude , To the difference of the Versed sines of the Semidiurnal Ark , and of the hour sought . Having got the Logarithm of this difference , take the natural number out of the Sines or Tangents that stands against it , accordingly as the Logme is sought , and in Winter add it to the natural sine of the hour of rising from 6 , the sum is the natural sine of the hour from 6 towards noon . In Summer get the difference between this fourth and the sine of rising from 6 , the said difference is the natural sine of the hour from 6 towards noon , when the Number found by the Proportion is greater then the sine of rising , towards midnight when less . The Canon is the same without Variation as well for South declinations as for North , and therefore we may by help thereof find two hours to the same Altitude . Example . Comp Lat 38d 28′ Ar Comp Sine — , 2061683 Comp declin 77d Ar Comp — , 0112761 fixed number — , 2174444 Let the Altitude be 14d 38′ Sine — 9,4024889 Natural sine against it — 41660 , 00 — 9,6199333 Nat sine of rising from 6 — 29058 , 79 Sum — 70718 , 79 N sine of 45d the hour from six in Winter . Difference — 12601 , 21 Sine of 7d 14′ the hour from 6 in Summer towards noon to the former Altitude , and like declination towards Elevated Pole. Another Example for the same Latitude and Declination . Logme Let the Altitude be 20d 25′ Sine — 9,5426321 The former fixed Logme — 2174444 Natural sine against it — 5754811 Sum — 9,7600765 Natural sine of rising — 2905879 Sum — 9,7600765 Sum — 8660690 sine 60d the ho from 6 in Wint Difference — 2848932 sine 16d 33′ the hour from 6 in Summer towards noon ; And thus may two hours be found at one operation for all Altitudes less then the Winter Meridian Altitude , to be converted into usual Time by allowing 15d to an hour , and 4● to a degree . To Calculate a Table of the Suns Altitudes on all Hours . As the Secant of the Latitude , To the Cosine of the Declination , Or which is all one , As the Square of the Radius : To the Rectangle of the Cosines , both of the Latitude and of the Declination : So is the sine of the hour from 6. To a fourth , namely , in Summer the difference of the sines of the Suns Altitude at 6 , and of the Altitude sought , in Winter the sum of the sines of the Suns Depression at 6 , and of the Altitude sought . Having wrote down the Logarithmical sines of the hour from 6 on the Paper , at one end of a piece of Card may be wrote down the sum of the Logarithmical Cosines of the Latitude and Declination , and add the same to the sine of the hour rejecting the double Radius , and take the natural sine that stands against the sum sought in the Logarithmical sines ; having this natural sine , get the sum and difference of it , and of the natural of the Suns Altitude at 6 , the sum is the natural sine of the Altitude for Summer declinations , and the difference for Winter Declinations when the sine of the Suns Altitude is the lesser : But when it is the greater , the said difference is the natural sine of the Altitude for hours beyond 6 towards midnight . Example . Log m Complement Latitude 38d 28′ Sine — 9,7938317 Compl Declination 66 29 Sine — 9,9623428 fixed Logme — 19,7561745 Let the hour be 30d that is 2 hours before and after six in Summer sine 9,6989700 Sum — 9,4551445 Natural sine against it — 2851308 Nat sine of the Altitude at 6 — 3124174 Sum — 5975482 Sine of 36d 42′ being the two Altitudes for 4 and 8 in the morning or afternoon , in Summer . Difference — 272866 Sine of 1 34 being the two Altitudes for 4 and 8 in the morning or afternoon , in Summer . Another Example . Let the hour be 45d from six Sine — 9,8494850 the former fixed Logme — 19,7561745 N Sine against it — 4032791 Sum — 9,6056595 N Sine of Altitude at 6 — 3124174 Sum — 9,6056595 Sum — 7156965 Sine of 45d 42′ being the two Altitudes for the hours of 9 or 3 in Summer or Winter for Declination 23d 31′ both towards the Elevated and Depressed Pole. Difference — 908617 Sine — 5 13 being the two Altitudes for the hours of 9 or 3 in Summer or Winter for Declination 23d 31′ both towards the Elevated and Depressed Pole. By the former Canon was the following Table of Altitudes calculated , and that with much celerity beyond any other way , it will not be amiss to Calculate the Suns Altitude at 6 by the natural Tables only , however the Logarithms will accurately discover the natural sine of at , if duly Proportioned by the differences . A Table of the Suns Altitudes for each Hour and quarter for the Latitude of London .   North.   South .   Declination . 23d , 31 13 Equator . 13d 23 31   XII . 61d , 59 51 , 28 38 , 28. 25 , 28 14 , 57 XII . . 61 , 49 51 , 21 38 , 22 25 , 20 14 , 52   * 61 , 23 50 , 59 38 , 4 25 , 8 14 , 39   . 60 , 40 50 , 24 37 , 36 24 , 43 14 , 18   I. 59 , 42 49 , 36 36 56 24 , 10 13 , 48 XI .   58 , 29 48 , 35 36 , 5 23 , 26 13 , 9     57 , 4 47 , 24 35 , 4 22 , 34 12 , 23     55 , 29 46 , 1 33 , 55 21 , 33 11 , 29   II. 53 , 45 44 , 3● 32 , 36 20 , 25 10 , 28 X.   51 , 53 42 , 51 31 , 8 19 , 8 9 , 19     49 , 54 41 , 4 29 , 34 17 , 44 8 , 3     47 , 51 39 , 1● 27 , 53 16 , 14 6 41   III. 45 , 42 37 1● 26 , 6 14 , 38 5 13 IX .   43 , 31 35 , 9 24 , 12 12 , 55 3 , 39     41 , 16 33 , 2 22 , 15 11 , 7 1 , 59     38 , 59 30 , 52 20 , 13 9 , 15 0 , 15   IIII. 36 , 42 28 , 37 18. 7 7 , 17     VIII .   34 , 23 26 , 22 15 , 58 5 , 17         32 , 4 24 , 5 13 , 46 3 , 12         29 , 43 21 , 46 11 , 32 1 , 4       V. 27 , 23 19. 27 9 , 16     VII .     25 , 4 17 , 7 6 , 58             22 , 46 14 , 47 4 , 39             20 , 28 12 , 27 2 , 20           VI. 18 , 13 10 , 9 00 , 00 VI.         15 , 58 7 , 51                 13 , 46 5 , 34                 11 , 37 3 , 20               VII . 9 , 30 1 , 7 V.             7 , 25     Declination . Ascensionall difference .         5 , 24             3 , 27     13d   16d , 54′       VIII . 1 , 34 IIII. 23 , 31 33 , 12 R A A Table of the Suns Altitudes for every 5 degrees of Azimuth from the Meridian for the Latitude of London . North. South . Declination . 2● d 31 13d Equator . 13● 23d , 31′ Mer Alt 61 , 59 51 , 28 38d 28′ 25d , 28′ 14d , 57   5 61 , 55 51 , 23 38 , 21 25 , 21 14 , 49   10 61 , 42 51 , 7 38 , 2 24 , 57 14 , 22   15 61 , 21 50 , 40 37 , 30 24 , 20 13 , 39   20 60 , 51 50 , 3 36 , 44 23 , 27 12 , 39   25 60 , 11 49 , 14 35 , 45 22 , 16 11 , 19   30 59 21 48 , 13 34 , 32 20 , 51 9 , 43   35 58 , 20 46 , 59 33 , 3 19 , 7 7 , 46   40 57 , 7 45 , 31 31 , 19 17 , 7 5 , 31   45 55 , 43 43 , 50 29 , 19 14 , 50 2 , 57   50 54 , 3 41 , 53 27 , 3 12 , 13 0 , 03   55 52 , 7 39 , 39 24 , 30 9 , 21       60 49 , 56 37 , 9 21 , 40 6 , 11       65 47 , 27 34 , 22 18 , 34 2 , 46       70 44 , 39 31 , 18 15 , 12           75 41 , 34 27 , 58 11 , 37           80 38 , 10 24 , 23 7 , 51           85 34 , 32 20 , 38 3 , 59           90 30 39 16 , 42               95 26 , 34 12 , 40               100 22 , 28 8 , 41               105 18 , 20 4 , 44 Declinat Amplitude .       110 14 , 15 0 , 54 13d ,   21d , 12′       115 10 , 19     23 , 31′ 39d , 54′       120 6 , 36                   125 3 , 7                 Many Tables may want the naturall Tables standing against the Logarithmicall ; therefore the method of Calculation by the Logarithmicall Tables onely , is not to be omitted , albeit we wave the common Proportions , when three Sides are given to find an Angle . A general Proportion derived from the book of the honorable Baron of Marchist●n , which may bee wrought on a Serpentine Line without the use of Versed Sines , or finding the half distance between the 7th Tearme and the Radius , not encumbred with Rectangles , Squares , or Differences of Sines , or Versed Sines . Three Sides to find an Angle ; two of them or all three being lesse then Quadrants . By a supposed perpendicular , which need not to be named . As the Tangent of half the greater of the containing sides , To the Tangent of the half sum of the other sides : So is the Tangent of half their difference , To the tangent of a fourth Arke . If this Arke be greater then the half of the first assumed side ; namely , then the Arke of the first Tearme , in the Proportion , the ( Supposed perpendicular falls without ) Angles opposite to the two other Sides are of a different Affection , the greatest side subtending the Obtuse angle , and the lesser the Accute . If the Angle Opposite to the greater of the other Sides be sought ▪ Take the difference ; if to the lesser , the Sum of the 4th Arke , and of half the containing Side , which half is the first Tearme in the Proportion , Then , As the Radius , To the Cotangent of the other Containing Side : So the Tangent of the said Sum or difference , To the Cosine of the Angle sought . The first Tearme above needs no Restraint , but when one of the Containing Sides is greater then a quadrant : If the 4th Arke be less then the half of the first assumed Side , the Perpendicular falls within , in this Case , the two Angles , opposite to the two other sides may be found , being both Acute . Get the Sum and difference of half the first assumed Side , and of the 4● Arke , the Sum is the greater Segment , and the Difference or residue , the lesser Segment ; the Perpendicular alwayes falling on the side assumed , first into the Proportion ; Then , As the Radius , To the Cotangent of the lesser of the other Containing Sides ; So is the Tangent of the lesser Segment . To the Cosine of the Angle sought . As the Radius , To the Cotangent of the greater of the other Containing Sides ; So is the Tangent of the greater Segment . To the Cosine of the Angle sought . From these general Directions is derived this Canon for Calculating the Azimuth ; As the Tangent of half the Complement of the Altitude ; To the Tangent of the half sum of the Sun or Stars distance from this elevated pole , and of the Complement of the Latitude : So is the Tangent of half their difference , To the Tangent of a 4 h Arke . If this Ark be less then half the Complement of the Altitude , the Azimuth is Acute ; if more obtuse , in both Cases , get the difference of these two Arkes , if there be no difference , the Azimuth is 90d from the Meridian ; Then , As the Radius , To the Tangent of the Latitude ; So the Tangent of the said Arke of difference , To the Sine of the Azimuth from the prime Verticall . This when the Sun or Stars do not come to the Meridian between the Zenith and the elevated pole : but when they do , Let the sum of the 4th Ark , and of half the Complement of the Altitude , be the third Tearme in the latter Proportion . This is a ready Way to Calculate a Table of Azimuths ; two Tearms in each Proportion being fixed for one Declination ; and the Azimuth being known , the Hour may be found by a single Operation , As the Cosine of the Declination , Is to the Sine of the Azimuth , from the Meridian : So is the Cosine of the Altitude , To the Sine of the Hour , from the Meridian . Example , for 13d of North Declination . 77d Complement of Declination the Polar distance . 38. 28 Complement of Latitude at London . 115 , 28 Sum : half sum 57d , 44′ Tangent . 10,1997231 . 38 , 32 difference : half diff 19 , 16 Tangent 9 ▪ 5434594 fixed for that declination — 19,7432225 Altitude 4d , 44′ Comp. 85d 16′ half 42d 38′ Tang. 9,9640811 Tangent of — 31. 1 — 9,7791414 difference — 11 37 Tangent 9,3129675 Tangent of 51d 32′ the Latitude — 10,0999135 Sine of 15d , the Azimuth 9,4128810 from East or West Northwards , because the Ark found by 1st Operation was less then half the Complement of the Altitude . Another Example for the Altitude 34d 22′   Log ● The fixed Number — 19,7432225 Compl. Altitude 55● 38′ half 27 , 49● Tangent 9,7223147 Tangent of — 46 , 23 — 10,0209078 Difference — 18 , 34 Tangent 9,5261966 Tangent of the Latitude — 10,0969135 The Sine of 25d the Azimuth from — 9,6261101 East to West Southwards because the first Ark was more then half the Coaltitude . The hours to these two Azimuths will be found by the latter Proportion to be — 98d , 54′ 50 , 9 from Noon . To Calculate the Suns Altitude on all Hours and Azimuths . The first operation shall be to find such an Ark as may remain fixed in one Latitude to serve to all Declinations in both Cases : So that but one Operation more need be required . The Proportion to find it is As the Radius to the Cotangent of the Latitude , So is the Sine of any Hour from 6 , or Azimuth from the Vertical To the Tangent of a fourth Ark. This 4th Arke ( if the Azimuth be accounted from the Vertical , that is , from the points of East or West towards noon Meridian ) Is the Altitude that the Sun shall have , being in the Equinoctial , upon that Azumuth , and so one of the Quesita . If the hour from 6 be accounted upward on the Equinoctial , this 4 ●h Ark is the ark or portion of the hour Circle , between the Equinoctial and Horizon . This Ark for Hours and Azimuths beyond 6 , or the Vertical , towards the midnight Meridian , is the Depression under the Horizon , according to the Denominations already given it . For the Altitudes on all Hours . When the Sun is in the Equinoctial , As the Radius , is to the Cosine of the Latitude : So is the Sine of the Hour from 6 To the Sine of his Altitude . In all other Cases , If the Hour from noon be more then 6 Substract the Equinoctial Arks that Correspond to such parts of time as you would Calculate Altitudes for ; out of the Suns distance from the Depressed Pole , and then it will hold , If the Hour from noon be less then 6 Substract the Equinoctial Arks that Correspond to such parts of time as you would Calculate Altitudes for ; out of the Suns distance from the Elevated Pole , and then it will hold , As the Cosine of the Ark found by the first common Proportion , Is to the Cosine of the Ark remaining ; So is the Sine of the Latitude , To the Sine of the Altitude sought ▪ For the more speedy Calculating a Table of the Suns Altitudes for this Latitude to any Declination , there is added a Table of these Equinoctial Arks for every hour and quarter , as also for every 5d of Azimuth ; The use whereof shall be illustrated by an Example or two , observing by the way , that the same Ark belongs to two hours alike remote on each side from six , as also like Arks to two Azimuths equally remote on each side the Vertical . Hours on each side six , Azimuths on each side the Vertical .   d ′ VI 0 , 00 fixed Arks       5 , 55   5d 3 , 59   8 , 49   10 7 , 51 VII 11 , 37 V 15 11 , 37   14 , 19         16 , 55   0 15 , 12   19 , 22   25 18 , 34 VIII 21 , 40 IIII 30 21 , 40   23 , 49         25 , 48   35 24 , 30   27 , 39   40 27 , 3 IX 29 , 19 III 45 29 , 19   30 , 52         32 , 13   50 31 , 19   33 , 27   55 33 , 3 X 34 , 32 II 60 34 , 32   35 , 28         36 , 17   65 35 , 45   36 , 57   70 36 , 44 XI 37 , 30 I 75 37 , 30   37 , 55½         38 , 13½   80 38 , 2   38 , 24½   85 38 , 21 XII 38 , 28   90 38 , 28 Equinoctial Altitudes or Depressions . Example . Admit it were required to Calculate the Suns Altitude for 2 or 10 of the Clock , when his declination is 23d 31′ North. Suns distance from the Elevated Pole — 66 : 29 fixed Ark for 2 or 10 is — 34 : 32 Residue — 31 : 57 Complement of the Residue 58d 3′ Sine — 9,9286571 Sine of — 51 32 the Latitude — 9,8937452 19,8224023 Sine of — 55 28 Com first ark 9,9158200 Sine of — 53 45 the — 9,9065823 Altitude sought — 9,9065823 So if it were required to Calculate the Suns Altitude for the hour ▪ of 5 in the morning when his declination is 20d North ☉ distance from depressed Pole — 110d fixed Ark — 11 : 37 The Residue is 81d , 33′ Or — 98 23 Sine of 8d , 23′ the Compl of the Residue — 9,1637434 Sine of the Latitude — 9,8937452 19,0574886 Sine of 78d 23′ the Compl fixed ark — 9,9910119 Sine of 6d 42′ the Altitude sought — 9,0664767 For the Altitudes on all Azimuths . As the Sine of the Latitude , Is to the Cosine of the Equinoctial Altitude : So is the Sine of the Declination , To the Sine of a fourth Ark. Get the sum and difference of the Equinoctial Altitude , and of this fourth Ark the sum is the Summer Altitude for Azimuths from the Vertical towards noon Meridian . The difference when this 4th Ark is lesser greater then the Equinoctial Altitude is the Winter Summer Altitude for Azimuths from the Vertical towards Noon Midnight Meridian Example . Let it be required to Calculate Altitudes for the Suns Azimuth 30d from the Meridian , that is 60d from the Vertical for Declination 23d 31′ both North and South . This will be speedily done , add the Logme of the Sine of the Declination to the Arithmetical Complement of the Logme of the Sine of the Latitude , this number varies not for that declination , and to the Amount add the Logme of the Cosine of the Equinoctial Altitude , the sum rejecting the Radius is the Logarithmical Sine of the 4th Ark. Sine Latitude Ar Comp — 0,1062548 Sine declination — 9,6009901 Sine of 55d 28′ the Complement of Equinoctial Altitude — 9,7072449 9,9158200 Sine — 24d 49′ — 9,6230649 Equinoct Altitude — 34 32 Sum — 59 21 Summer Difference — 9 43 Winter Altitude for that Azimuth . Another Example : The former Number — 9,7072449 Sine 78d 23′ Comp of Equinoct 9,9910119 Altitude for 15d of Azimuth — Sine of 29d 57′ — 9,6982568 Eq Altitude 11 37 Difference — 18 20 Sum — 41 34 The Summer Altitudes for 15d Azimuth each way from the Vertical to that declination . The Suns Declination 20d North to find his Altitudes for 15d Azimuth on each side the Vertical . Sine of 51d 32′ the Latitude Ar Com — 0,1062548 Sine of 20d the declination — 9,5340517 Sine of 78d 23′ Comp Eq Altitude — 9,9910119 Sine of 25d 20′ — 9,6313184 Sine of — 25d 20′ Eq Altitude — 11 37 Difference — 13 43 Sum — 36 57 The Summer Altitudes for 15d Azimuth on each side the Vertical to that Declination . By the Arithmetical Complement of a Number is meant a residue which makes that first Number equal to the other : And so if from a number or numbers given another Number is to be substracted , and instead thereof a third number added , the totall shall be so much encreased more then it should by the sum made of the number to be substracted , and of that was added : That is to say , in this last Example instead of substracting the Sine of the Latitude from another sum we added the residue thereof , being taken from Radius thereto , and so increased the Total too much each time by the Radius , which is easily rejected . If the Sun or Stars come to the Meridian between the Zenith and the Elevated Pole , as when their Declination is more then the Latitude of the place , the former Rule of Calculation varies not only the sum of the Equinoctial Altitude , alias , the fixed Ark , and of the Ark found by the second Proportion will be more then 90● In this Case the Complement of it to 180 d is the Altitude sought . A double Advertisement . The Declination towards the Elevated Pole supposed more then the Latitude of the place . If the Complement of the Declination be more then the Latitude of the place also , as in this case it always is for the Sun ; the Sun or such Stars shall have two Altitudes on every Azimuth between the Coast of rising or setting , and the remotest Azimuth from the Meridian ; To find what Azimuths those shall be , As the Cosine of the Latitude , To Radius ; So is the Sine of the Declination , To the Sine of the Amplitude , And So is the Cosine of the Declination , To the Sine of the remotest Azimuth from the Meridian . Between the Azimuth of rising , and the remotest Azimuth , the angle of Position is Acute , afterwards Obtuse . The Sun upon the remotest Azimuth , the angle of Position being a right angle , will have but one Altitude to find it . As the Sine of the Declination . To Radius ; So is the Sine of the Latitude , To the Sine of that Altitude . Example . In North Latitude 13d of Barbados . Declination 20 d North , the Suns Amplitude or Coast of rising 69 d 23′ from the North , or 20 d 33′ from the East Northwards , and his remotest Azimuth from the North Meridian 74 d 4●′ . His two Altitudes upon the Azimuth of 74 d from the Meridian 27 d 27′ the lesser and 52 d 27′ the greater , the fixed Ark found by the first Operation , being — 50 d 3′ And by the second Operation the Ark found is — 77 30 Difference being the lesser Altitude is — 27 27 The sum 127d 33′ the Comp to 180 d being the greater Alt is 52 27 Altitude on the remotest Azimuth — 41 07 Upon Azimuths nearer the Meridian then the Coast of rising or setting , it need not be hinted that there will be but one Altitude . The Proportion from the 5th Case of Oblique Sphoerical Triangles to find the Suns Altitude on all Azimuths would be As the Cosine of the Declination , To the Sine of the Azimuth from the Meridian : So is the Cosine of the Latitude , To the Sine of the angle of Position . In such Cases when it will be acute or obtuse is already defined , and where two Altitudes are required it will be both , and being accordingly so made , the Proportion to find the Altitudes would be , As the sine of half the difference of the Azimuth and angle of Position , To the Tangent of half the difference of the Polar distance and Colatitude So the Sine of half the sum of the Azimuth and angle of Position , To the Tangent of half the Complement of the Altitude . The Azimuth being an angle always accounted from the Midnight Meridian ; but the former Proportion derived from the other Trigonometry in this Case is more speedy . Such Stars as have more Declination then the Complement of the Latitude never rise nor set , if their declination be also more then the Latitude of the place , they will have two Altitudes upon every Azimuth , except the remotest from the Meridian , and the Calculation the same as before . Example for the Latitude of London . The middlemost in the great Bears Rump , declination 56d 45′ , The remotest Azimuth will be 61 d 49′ from the Meridian , and the Altitude thereto 69 d 26′ . If that Star have 30 d of Azimuth from the Meridian , The first ark will be 34 d 22′ difference being the lesser Alt 27● 7′ The second ark — 61 39 difference being the lesser Alt 27● 7′ Sum — 96 11 Comp the greater Altitude — 83 49 This will be very evident on a Globe for having rectified it to the Latitude extend a Thread from the Zenith over the Azimuth in the Horizon , then turn the Globe round , and such Stars as have a more utmost remote Azimuth from the Meridian , and do not rise or set will pass twice under the Thread , the Azimuth Latitude and Declination being assigned if it were required to know the time when the Star shall be twice on the same Azimuth it may be found without finding the Altitudes first by 6 ●h Case of Oblique Spherical Triangles , As before get the angle of Position . As the Sine of half the difference of the Complement of the Latitude , and of the Complement of the declination . To the Sine of half their sum : So the Tangent of half the difference of the Azimuth from the Meridian , and of the angle of Position , To the Cotangent of half the hour from the Meridian , to be converted into common time if it relate to Stars : In each Proportion there is two fixed Tearms . The Illustration how these two fixed Arks are obtained , is evident from the Analemma in the Scheme annexed . AE E represents the Equator . H K the Horison , P T the Axis , Z N the Prime , Vertical S D a Parralel of North declination , ☉ another of South , declination , P S ☉ and P A N the Arks of two hours Circles between six and noon , and P D T another before 6. In the Triangle E L B right angled , there is given the angle at E , the Complement of the Latitude , the side E L the hour from 6 , with the right angle at L to find the side L B the ark of the hour Circle contained between the Equinoctial and the Horzion by 11th Case of right angled Sphoerical Triangles , the Proportion will be , As AE E Radius , To AE H Cotangent Latitude : So E L Sine of the hour from 6 , To L B The Tangent of the said Ark. From Z draw the Arches Z S and Z ☉ ( being Ellipses ) through the Points where the Hour Circle , and Parralels of Declination intersect , and they represent the Complements of the Altitudes sought , and let fall a Perpendicular from Z to R , then will P R be equal to L B , because the Proportion above is the same that would Calculate R P ; Which substracted from P S or P ☉ the Suns distance in Summer or Winter from the Elevated Pole rests R S or R ☉ : Now in an Oblique Spherical Triangle reduced to two right angled Triangles by the Demission of a Perpendicular , the Cosines of the Bases are in direct Proportion to the Cosines of the Hipotenusals a consequence derived from the general Axiom of the Lord Napier , and therefore it holds , As the Cosine of the Ark R P , or B L , that we call the common fixed ark , Is to the Sine of the Latitude , the Complement of Z P : So the Cosine of the Ark remaining , that is the Complement of R S or S O , To the Sine of the Altitude , the Complement of Z S or Z ☉ : This for all hours under 90● from the Meridian , but for those before or after 6 in the Summer it may be observed in the opposite Triangles E A N and E A I , counting the hour E A each way from 6 that the Ark of the hour Circle A N equal to A I , as much as it is above the Horizon in Winter , so much is it below the same in Summer , and the Suns distance from the Elevated Pole then equal to his distance from the Depressed Pole now : and the Zenith distance then equal to the Nader distance now , as is evident in the Triangle T N D. So that in this Case the Sun is only supposed to have Winter instead of Summer Declination , and the Rule for Calculating his Altitude the same as for Winter Altitudes . In like manner for the Azimuth . In the Schem following H Q represents the Horizon , AE F the Equator , S G a Parralel of Summer declination , and another passing through M of Winter declination , Z S L and Z N A two Azimuths between the Vertical and Noon Meridian , Z K I N another between it and the Midnight Meridian , from the Points S M I let fall the Perpendiculars S D , M O , I G representing the ☉ Declination in the Ellipses of several hour Circles ; So will L S , M L , A K , represent the Altitudes of these 3 Azimuths respectively , according to the proper Declination , for the finding whereof there is given in the Triangle E L B the side E L the Azimuth from the Vertical the angle B E L , the Complement of the Latitude and the right angle , and the Proportion by 11th Case of right angled Sphoerical Triangle is As AE H the Radius , To H AE the Cotangent of the Latitude : So E L the Sine of the Azimuth from the Vertical , To L B the Tangent of the Equinoctial Altitude to that Azimuth . Thut we see the first Proportions common to both , this Case issuing from the 2d Axiom of Pitiscus , that in many right angled Sphoerical Triangles having the same Acute angle at the Base , the Sines of the Bases and Tangents of the Perpendiculars are proportional . For the second Operation to find B S. Though the Analogy is derivable from the general Proposition of the Lord Napier , yet here I shall take it from Ptitiscus the 1 Axiom . That in many right angled Sphoerical Triangles , having the same acute angle at the Base the Sines of the Perpendiculars and Hipothenusals are in direct Proportion . Therefore in the Triangle AE Z B , and D S B it will hold , As AE Z the Sine of the Latitude , To Z B the Cosine of the Equinoctial Altitude , So is D S , the Sine of the Declination , To the Sine of B S , which is equal to B M : See 29th Prop. of 3d book Regiomontanus , I prove it thus , The opposite angles at B are equal , and the angles at D and O are equal , and the side D S , equal to M O , it will then be evinced by Proportion , As the Sine of the angle at B , To its opposite side M O or D S : So is the Radius , that is the angle at O or D , To its opposite side S B , or M B. This equality being admitted , if unto L B we add B S , the sum is L S the Altitude for Summer Declination , if from B L we take B M equal to B S , the remainder M L is the Altitude for the like Declination towards the Depressed Pole , being the Winter Altitude of that Azimuth . But for Azimuths above 90 d from the Meridian it may be observed in the two Opposite Triangles E A N , and E A I , counting the Azimuth E A each way from the Vertical its Equinoctial Altitude A N in the Winter is equal to its Equinoctial Depression A I in the Summer , and is to be found by the 1 Proportion . The second Proportion varies not . As the Sine of the Latitude N F equal to AE Z , Is to N I equal to Z B the Cosine of the Equinoctial Altitude or Depression : So is the Sine of the Declination , I G equal to D S : To Sine I K , from which taking A I , the Equinoctial Depression rests A K , the Altitude sought . To Calculate a Table of the Suns Altitude for all Azimuths and hours under the Equinoctial . This will be two Cases of a Quadrantal Sphoerical Triangle . 1. For the Altitudes on all Azimuths . There would be given the side A B a Quadrant , the angle at B the Azimuth from the Meridian , and the side A D the Complement of the Suns Declination . If the side B D be continued to a Quadrant , the angle at C will be a right angled , besides which in the Triangle A D C , there would be given A D as before the Complement of the Suns Declination , and A C the measure of the angle at B to find D C the Suns Altitude being the Complement of B D , and so having the Hipotenusal , and one of the Leggs of a right angled Sphoerical Triangle , by the 7th Case we may find the other Legg , the Proportion sutable to this question would be As the Sine of the Azimuth from East or West , Is to the Radius : So is the Sine of the Declination , To the Cosine of the Altitude sought . 2. For the Altitudes on all Hours . There would be given the side A B a Quadrant , A D the Complement of the Suns declination with the contained angle B A D the hour from noon , to find the side B D the Complement of the Suns Altitude . Here again if B D be continued to a Quadrant , the angle at C is a right angle , the side A D remains common , the angle D A C is the Complement of the Angle B A D , See Page 57 where it is delivered , That if a Sphoerical Triangle have one right angle , and one side a Quadrant , it hath two right angles , and two Quadrantal sides , and therefore the angle B A C is a right angle ; this is coincident with the 8th Case of right angled Spherical Triangles , the Proportion thereof is , As the Radius , Is to the Cosine of the Declination , So is the Sine of the hour from six , To the Sine of the Altitude , sought . Affections of Sphoerical Triangles . BEcause the last Affection in page 57 is not Braced in the beginning , and a mistake of lesser for greater , in the last Brace but one , I thought fit to recite it at large . Any side of a Sphoerical Triangle being continued , if the other sides together are equal to a Semicircle , the outward angle on the side continued shall be equal to the inward angle on the said side opposite thereto . If the sides are less then a Semicircle , the outward angle will be greater then the inward opposite angle ; But if the said sides are together greater then a Semicircle , the outward angle will be less then the inward opposite angle . In the Triangle annexed , if the sides A B and A C together are equal to a Semicircle , then is the angle A C D equal to the angle A B C. If less then a Semicircle then is the said angle greater then the angle at B. But if they be greater , then is the said angle A C D less then the angle at B. By reason of the first Affection in page 58 ( which wants a Brace in the first Line ) after the words two sides ) We require in the first , second , and other Cases of Oblique angled Sphoerical Triangles , the sum of the two sides or angles given , to be less then a Semicircle . Before I finish the Trigonometrical part , I think it not amiss to give a Determination of the certain Cases about Opposite sides and Angles in Sphoerical Triangles , having before shewn which are the doubtful , and the rather because that this was never yet spoke to . Two sides with an Angle opposite to one of them , to determine the Affection of the Angle opposite to the other . 1. If the given angle be Acute , and the opposite side less then a Quadrant , and the adjacent side less then the former side . The angle it subtends is acute because subtended by a lesser side , for in all Sphoerical Triangles the lesser side subtends the lesser angle and the Converse . 2. If the given angle be acute , and the opposite side less then a Quadrant , and the other side greater then the former side : This is a doubtfull Case , if it be less then a Quadrant it may subtend either an Acute or an Obtuse angle , and so it may also do if it be greater then a Quadrant , yet we may determine , That when the given angle is acute , and the opposite side less then a Quadrant , but greater then the Complement of the adjacent side to a Semicircle ( which it cannot be unless the adjacent side be greater then a Quadrant ) the angle opposite thereto will be obtuse . 3. The given angle Acute , and the opposite side greater then a Quadrant , and the other side greater . If two sides be greater then Quadrants , if one of them subtends an Acute angle , the other must subtend an Obtuse angle , by the 1st . Affection in pag 58. 4. The given angle Acute , and the opposite side greater then a Quadrant ; The other side cannot be lesser then the former , by what was now spoken . 5. If the given angle be Obtuse , and the opposite side less then Quadrant , the other side less subtends an Acute angle . 6. If the given angle be Obtuse , and the opposite side less then a Quadrant ; The other side greater then the former side must of necessity be also greater then a Quadrant , otherwise two sides less then Quadrants should subtend two Obtuse angles , contrary to the first Affection in p 58. 7. If the given angle be Obtuse , and the Opposite side greater then a Quadrant ; If the other side be greater then the former it will subtend a more Obtuse angle . 8. If the given angle be Obtuse , and the Opposite side greater then a Quadrant , If the other side be less then the former , whether it be lesser or greater then a Quadrant it may either subtend an Acute or Obtuse angle . But we may determine That when the given angle is Obtuse , and the Opposite side greater then a Quadrant ; If the Complement of the adjacent side to a Semicircle be greater then the said opposite side , the angle subtended by the said adjacent side is Acute . Two Angles with a side Opposite to one of them , to determine the Affection of the side opposite to the other . 1. If the given angle be Acute , and the opposite side less then a Quadrant ; If the other angle be less , the side opposite thereto shall be less then a Quadrant because it subtends a lesser angle . 2. If the given angle be Acute , and the Opposite side less then a Quadrant , If the other angle be greater the Case is ambiguous , yet we may determine , If the given angle be Acute , and the opposite side lesser then a Quadrant , if the Complement of the other angle to a Semicircle be less then the Acute angle ( which it cannot be but when the latter angle is Obtuse ) the side subtending it shall be greater then a Quadrant . 3. If the given angle be Acute , and the opposite side grnater then a Quadrant , if the other angle be lesser . Then by 12 of 4 book of Regiomontanus , if two Acute angles be unequal , the side opposite to the lesser of them shall be less then a Quadrant . 4. If the given angle be Acute , and the opposite side greater then a Quadrant , If the other angle be greater , It must of necessity be Obtuse , because otherways two Acute unequal angles , the side opposite to the lesser of them should not be lesser then a Quadrant , contraty the former place of Regiomontanus . 5. If the given angle be Obtuse , and the opposite side lesser then a Quadrant , If the other angle be lesser , It must of necessity be Acute , and the side subtending it less then a Quadrant , otherways two sides less then Quadrants should subtend two Obtuse angles , contrary to 1st Affection in page 58. 6. If the given angle be Obtuse , and the opposite side less then a Quadrant , If the other angle be greater , By 13 Prop of 4 h of Regiomontanus , if a Triangle have two Obtuse unequal angles , the side opposite to the greater of them shall be greater then a Quadrant . 7. If the given angle be Obtuse , and the opposite side greater then a Quadrant , If the other angle be greater or more obtuse then the former ; it is subtended by a greater side . 8. If the given angle be Obtuse , and the opposite side greater then a Quadrant , If the other angle be less then the former , the Case is ambiguous ; yet we may determine , That when the given angle is Obtuse , and the opposite side greater then a Quadrant , if the other angle be less then the Complement of the said Obtuse angle to a Semicircle , the side subtending it shall be less then a Quadrant . The former Cases that are still , and alwais will be doubtful , may be determined when three sides are given . A Sphoerical Triangle having two sides less then Quadrants and one greater , will always have one Obtuse angle opposite to that greater side , and both the other angles Acute . A Sphoerical Triangle having three sides less then Quadrants , can have but one obtuse angle ( and many times none ) and that obtuse angle shall be subtended by the greatest side ; But whether the greatest side subtend an Acute or Obtuse angle cannot be known , unless given or found by Calculation , and that may be found several ways . First by help of the Leggs or Sides including the angle sought by 15● Case of right angled Sphoerical Triangles . As the Radius , To the Cosine of one of those Leggs ; So is the Cosine of the other Legg , To the Sine of a fourth Arch. If the third side be greater then the Complement of the fourth Arch to 90d , the angle included is Obtuse , if equal to it a right angle , if less an Acute angle . Secondly , By help of one of the Leggs and the Base or Side subtending the angle sought by 7th Case of right angled Sphoerical Triangles . As the Cosine of the adjacent side , being one of the lesser sides , Is to the Radius : So the Cosine of the opposite side , To the sine of a fourth Arch. If the third side be greater then the Complement of the 4th Arch to 90d , the angle subtended is Acute , if equal to it a right angle , if less an Obtuse angle . All other Cases need no determination , if a Triangle have two sides given bigger then Quadrants , make recourse to the opposite Triangle and it will agree to these Cases . If three angles were given to determine the Affection of the sides if they were all Acute the three sides subtending them will be all less then Quadrants . But observe that though a Triangle that hath but one side greater then a Quadrant , can and shall always have but one Obtuse angle , yet a Triangle that hath but one Obtuse angle may frequently have two sides greater then Quadrants . In this and all other Cases let the angles be changed into sides , and the former Rules will serue , I should have added a brief Application of all the Axioms that are necessary to be remembred , and have reduced the Oblique Cases to setled Proportions ( with the Cadence of Perpendiculars only to shew how they arise ) whereby they will be rendred very facil ; as also the Demonstration of the Affections , which may be hereafter added to some other Treatise to be bound with this Book . Of working Proportions by the Lines on the Quadrant . BEfore I come to shew how all Proportions may in some measure be performed upon the Lines of this Quadrant , it is to be intimated in general , That the working of a Proportion upon a single natural Line , was the useful invention of the late learned Mathematician , Mr Samuel Foster , and published after his decease as his ; In the use of his Scale , a Book called Posthuma Fosteri , as also by Mr Stirrup in a Treatise of Dyalling ; in which Books , though it be there prescribed , and from thence may be learned , yet I acknowledge I received some light concerning it , from some Manuscripts lent me by Mr Foster , in his life time to Transcribe for his and my own use touching Instrumental Applications ; Yet withal be it here intimated , that there are no ways used upon this Quadrant for the obtaining the Hour and Azimuth with Compasses , and the Converse of the 4th Axiom , but what are wholly my own , and altogether novel , though not worth the owning , for Instrumental Conclusions not being so exact as the Tables are of small esteem with the learned as in Mr Wingates Preface to the Posthuma ; besides the taking off of any Line from the Limb to any Radius ; the Explanation of the reason of Proportions so wrought , the supply of many Defects , and the inscribing of Lines in the Limb , I have not seen any thing of Mr Fosters , or of any other mans , tending thereto . Of the Line of equal Parts . THis Line issueth from the Center of the Quadrant on the right edge of the foreside , and will serve for Mensurations , Protractions and Proportional work . The ground of working Proportions by single natural Lines , is built upon the following grounds . That Equiangled Plain Triangles have the sides about their equal angles Proportional , and this work hath its whole dependance on the likeness of two equiangled Plain Right angled Triangles ; as in the figure annexed , let A B represent a Line of equal parts , Sines or natural Tangents issuing from the Center of the Quadrant supposed at A , and let A C represent the Thread , and the Lines B C , E D making right angles with the Line A C , or with the Thread , the nearest distances to it from the Points B and E. I say then that this Scheme doth represent a Proportion of the greater to the less , and the Converse of the less to the greater . First of the greater to the less , and then it lies , As A B to B C : So A E to A D , whence observe that the length of the second Tearm B C must be taken out of the common Scale A B , and one foot of that extent entred at B the first Tearm , the Thread must be laid to the other foot at C , according to the nearest distance then the nearest distance , from the Point E to the Thread that is from the third Tearm called Lateral entrance , being measured in the Scale A B , gives the quantity of the 4 ●h Proportional . Secondly of the less to the greater . And then it lies , As B C to A B : So E D to A E , Or , As E D to A E : So B C to A B , by which it appears that the first Tearm B C must be taken out of the common Scale , and entred one foot at the second Tearm at B , and the Thread laid to the other at C according to nearest distance then the third Tearm E D must be taken out of the common Scale and entred between the Thread and the Scale , so that one foot may rest upon the Line , as at E , and the other turned about may but just touch the Thread , as at D , so is the distance from the Center to E the quantity of the 4th Proportional ; and this is called Parralel entrance , because the extent E D is entred Parralel to the extent B C : To avoid Circumlocution , it is here suggested , that in the following Treatise , we use these expressions to lay the Thread to the other foot , whereby is meant to lay it so according to nearest distance , that the said foot turned about may but just touch the Thread , and so to enter an extent between the Thread and the Scale is to enter it so that one foot resting upon the Scale , the other turned about may but just touch the Thread . Another chief ground in order to working Proportions by help of Lines in the Limb is , That in any Proportion wherin the Radius is not ingredient the Radius may be introduced by working of two Proportions in each of which the Radius shall be included , and that is done by finding two such midle tearms ( one whereof shall always be the Radius ) as shall make a Rectangle or Product equal to the Rectangle or Product of the two middle Tearms proposed , to find which the Proportion will be . As the Radius , To one of the middle Tearms : So the other middle Tearm , To a fourth , I say then , that the Radius and this fourth Tearm making a Product or Rectangle equal to the Product of the two middle Tearms , these may be assumed into the Proportion instead of those , and the answer or fourth Tearm will be the same without Variation , and therefore holds , As the first Tearm of the Proportion , To the Radius : So the fourth found as above , To the Tearm sought . Or , As the first Tearm of the Proportion , Is to the fourth found as above : So is the Radius , To the Tearm sought ; and here observe , that by changing the places of the second and third Tearm , many times a Parralel entrance may be changed into a Lateral , which is more expedite and certain then the other , having thus laid the foundation of working any Proportion , I now come to Examples . 1. To work Proportions in equal parts alone . If the first Tearm be greater then the second , take the second Tearm out of the Scale , and enter one foot of that extent at the first Tearm , laying the Thread to the other foot , then the nearest distance from the third Tearm to the Thread gives the 4th Proportional sought , to be measured in the Scale from the Center . If the first Tearm be less then the second , still as before keep the greatest Tearm on the Scale , and enter the first Tearm upon it , laying the Thread to the other foot , then enter the third Tearm taken out of the Scale between the Thread and the Scale and it finds the 4th Proportional . Example . Admit the Sun shining , I should measure the length of the Shaddow of a Perpendicular Staff and find it to be 5 yards , the length of the Staff being 4 yards , and at the same time the length of the Shaddow of a Chimny , the Altitude whereof is demanded , and find it to be 22½ yards , the Proportion then to acquire the Altititude would be , As the length of the shaddow of the Staff , To the length of the Staff : So the length of the shadow of the Chimney , To the height thereof , that is As 5 to 4 : So 22 ‑ 5 to 18 yards the Altitude or height of the Chimney sought , Enter 4 or the great divisions upon 5 , laying the Thread to the other foot , then the nearest distance from 22 ‑ 5 to the Thread measured will be 18 , and in this latter part each greater division must be understood to be divided into ten parts . And so if the Sun do not shine , the Altitude might be obtained by removing till the Top of a Staff of known height above the eye upon a level ground be brought into the same Visual Line with the Top of the Chimney , and then it holds , As the distance between the Eye and the Staff , To the height of the Staff above the eye : So the distance between the Eye and the Chimney . To the height of the Chimney above the Eye . Some do this by a Looking Glass , others by a Bowl of Water , by going back till they can see the top of the object therein , and then the former Proportion serves , mutatis mutandis . But Proportions in equal parts will be easily wrought by the Pen , the chief use therefore of this Line will be for Protraction , Mensuration , and to divide a Line of lesser length then the Radius of the Quadrant Proportionally into the like parts the Scale is divided , which may be readily done , and so any Proportional part taken off , to do it Enter the length of the Line proposed at the end of the Scale at 10 , and to the other foot lay the Thread the nearest distances from the several parts of the graduated Scale to the Thread shall be the like Proportional parts to the length of the Line proposed , the Proportion thus wrought is , As the length of the graduated Scale , To any lesser length : So the parts of the Scale , To the Proportional like parts to that other length . Of the Line of Tangents on the left edge of the Quadrant . THe chief Uses of this Scale will be to operate Proportions either in Tangents alone or jointly , either with Sines or equal parts , to prick down Dyals , and to proportion out a Tangent to any lesser Radius ▪ To work Proportions in Tangents alone . 1. Of the greater to the less . Enter the second Tearm taken out of the Scale upon the first , laying the Thread to the other foot , then the nearest distance from the third Tearm to the Thread being taken out and measured from the Center shews the 4th Proportional . But if the Proportion be of the less to the greater , Enter the first Tearm taken out of the Scale upon the second , and lay the Thread to the other foot , then enter the third Tearm taken out of the Scale between the Thread and the Scale , and the foot of Compasses will shew the 4 Proportional . Example . Of the greater to the less , As the Tangent of 50d , To the Tangent of 20d : So the Tangent of 30 , To the Tangent of 10d. To work this take the Tangent of 20● in the Compasses , and entring one foot of that extent at 50d , lay the Thread to the other , according to the nearest distance , then will the nearest distance from the Tangent of 30d to the Thread being measured on the Line of Tangents from the Center be the Tangent of 10d the fourth Proportional . By inverting the Order of the Tearms , it will be , Of the less to the greater . As the Tangent of 20d , To the Tangent of 50d : So the Tangent of 10d , To the Tangent of 30d , to be wrought by a Parralel entrance . This Scale of Tangents is continued but to two Radii , or 63d 26′ whereas in many Cases the Tearms given or sought may out-reach the length of the Scale , in such Cases the Propprtion must be changed according to such Directions as are given for varying of Proportions at the end of the 16 Cases of right angled Sphoerical Triangles . In two Cases all the Rules delivered for varying of Proportions will not so vary a Proportion as that it may be wrought on this Line of Tangents . First when the first Tearm is greater then 63d 26′ the length of the Scale , and the rwo middle Tearms each less then 26d 34′ the Complement of the Scale wanting ; In this Case if any two Tearms of the Proportion be varied according to the Rules for varying of Proportions , there will be either in the given Tearms or Answer such a Tangent as shall exceed the length of the Scale , but it may be remedied by a double Proportion by the reason before delivered for introducing the Radius into a Proportion wherein it is not ingredient . As the Radius , To the Tangent of one of the middle Tearms : So the Tangent of the other middle Tearm , To a fourth Tangent : Again . As the Radius , To that fourth Tangent : So is the Cotangent of the first Tearm , To the Tangent of the fourth Ark sought . The Radius may be otherways introduced into a Proportion then here is done , but not conducing to this present purpose , and therefore not mentioned till there be use of it , which will be upon the backside of a great Quadrant of a different contrivance from this , upon which this trouble with the Tangents will be shunned . An Example for this Case . As the Tangent of 65d , To Tangent of 24d : So the Tangent of 20d , To what Tangent ? the Proportion will find 4d , 10′ . Divided into two Proportions will be , As Radius , To Tangent 24d : So Tangent of 20d , To a fourth , the quantity whereof need not be measured . Again . As Radius , To that fourth : So the Tangent of 25d , the Complement of the first Tearm , To the Tangent of 4d 26′ , the fourth Tangent sought . Operation . First enter the Tangent of 24d on the Radius or Tangent of 45d laying the Thread to the other foot , then take the nearest distance to it from 20d , and enter that extent at 45d , laying the Thread to the other foot , then will the nearest distance from 25d , to the Thread if measured from the Center be the Tangent of 5d 26′ sought . The second Case is when the first Tearm of the Proportion is less then the Complement of the Scale wanting , and the two middle Tearms greater then the length of the Scale . This ariseth from the former , for if the Tearms given were the Complements of those in the former Example , they would be agreeable to this Case , and so no further direction is needful about them , for the Tangent sought would be the Complement of that there found , namely 84d 34′ . Hence it may be observed , that a Table of natural Tangents only to 45d , or a Line of natural Tangents only to 45d may serve to operate any Proportion in Tangents whatsoever . To Proportion on out a Tangent to any Radius . Enter the length of the Radius proposed upon the Tangent of 45d and to the other foot of the Compasses lay the Thread according to the nearest distance , then if the respective nearest distances from each degree of the Tangents to the Thread be taken out they shal be Tangents to the assigned Radius : Because the Tangents run but to 63d 26′ whereas there may be occasion in some declining Dyalls to use them to 75d though seldom further ; to supply this defect , they may be supposed to break off at 60 and be supplied in a Line by themselves not issuing from the Center , or only pricks or full-points made at each quarter of an hour , for the 5th hour , that is , from 60d to 75d , and so these distances prickt again from the Center as here is done , either one way or other , the Proportion will hold , As the common Radius of the Tangents , Is to any other Assigned Radius : So is the difference of any two Tangents to the common Radius . To their Proportional difference in that Assigned Radius , And so having Proportioned out the first four hours , the 5th hour may be likewise Proportioned out and pricked forward in one continued streight Line from the end of the 4th hour . To work Proportions in Sines and Tangents by help of the Limb and Line of Tangents issuing from the Center . THough this work may be better done on the backside where the Tangents lye in the Limb , and the Sines issue from the Center , and where also there is a Secant meet for the varying of some Proportions that may excur , yet they may be also performed here supposing the Radius introduced into any Proportion wherein it is not ingredient , the two middle Tearms not being of the same kind as both Tangents or both Sines . To find the 4th Proportional if it be a Sine . Lay the Thread to the Sine in the Limb being one of the middle Tearms , and from the Tangent being the other middle Tearm take the nearest distance to it , then entring this extent upon the first Tearm being a Tangent lay the Thread to the other foot , and in the equal Limb it shews the Sine sought . So if the Example were , As the Tangent of 50d , To the Sine of 40d : So is the Tangent of 36d , To a Sine , the 4th Proportion would be found to be the Sine of 23d , 4′ . If a Tangent be sought . Lay the Thread to the Sine in the Limb being one of the middle Tearms , and from the Tangent being the other middle Tearm , take the nearest distance to it , then lay the Thread to the first Tearm in the Limb , and the former extent entred between the Scale and the Thread finds the Tangent , being the 4th Proportional sought . If the Example were , As the Sine of 40d , To the Tangent of 50d : So is the Sine of 23d 4′ To a fourth a Tangent , it would be found to be the Tangent of 36. These Directions presuppose the varying of the Proportion , as to the two Tangents , when either of them will excur the length of the Scale , of which more when I come to treat of the joint use of the Sines and Tangents on the backside . If both the middle Tearms be Sines , the 1st Operation will be wholly on the Line of Sines on the backside , by introducing the Radius , and the second upon the Line of Tangents on the foreside , likewise , if both the middle Tearms were Tangents , the first Operation would be on the Line of Tangents on the foreside , and the second on the Line of Sines on the backside ; but this is likewise pretermitted for the present , for such Cases will seldom be reducible to practise . To work Proportions in equal Parts and Tangents . Because the Lines to perform this work do both issue from the Center , the Radius need not be introduced in this Case ; but here it must be known whether the first or second Tearm of the Proportion taken out of its proper Scale be the longer of the two , and accordingly the work to be performed on the Scale of the longer Tearm , which shall be illustrated only by a few Examples , the ground of what can be said being already laid down . Example . To find the Suns Altitude , the length of the Gnomon or Perpendicular being assigned , and the length of its Shadow measured . As the length of the Shadow . Is to the Radius : So is the length of the Gnomon , To the Tangent of the Suns Altitude . Example . If the length of the Shaddow were 8 foot , and the length of the Gnomon but 5 foot , because 8 of the greater divisions of the equal parts are longer then the Tangent of 45d take the said Tangent or Radius , and enter it at 8 , laying the Thread to the other foot , then the nearest distance from 5 of the equal parts to the Thread measured on the Tangents sheweth 32d for the Suns Altitude sought . So the distance from a Tower and its Altitude being observed the Proportion , to get the height of the Tower is , As the Radius , To the measured distance : So the Tangent of the Altitude . To the height of the Tower. So in the Example in Page 38 , the measured distance K B was 100 yards , and the Altitude 43d 50′ to find the height of the Tower take the Tangent of 45 d , and enter it on 10 at the end of the equal Scale , laying the Thread to the other foot , then take the Tangent of 43 d 50′ , and enter it between the Scale and the Thread , and the Compasses will rest at 96 the height of the Tower in yards , sometimes each grand division of the equal Scale must represent but one sometimes 10 , and sometimes 100 , as in Case L B , and the Altitude thereto were given to find the height assigned . Another Example . As the Tangent of 60d , Is to 50 : So is the Tangent of 40d , To 24 ‑ 2 as before . Take 50 equal parts , and enter it upon the Tangent of 60 d laying the Thread to the other foot , then the nearest distance from the Tangent of 40 d to the Thread measured on the equal parts from the Center will be 24 ‑ 2 as before . Otherwise . Enter the Tangent of 40 d upon the Tangent of 60 d laying the Thread to the other foot , then enter 50 equal parts down the Line of Tangents from the Center , and the nearest distance from the termination to the Thread measured in the equal parts will be 24 ‑ 2 as before . If both the Tangents in any Proportion be too long , they may be changed into their Complements if one of them may and the other may not be so changed without excursion , then the Proportion may be wrought by the Pen , taking the Tangents out of the Quadrat and Shaddows , or it may be made two Proportions by introducing the Radius as before shewed ; it will not be needful to speak more to this , only one Example for obtaining the Altitude of a Tower at two Stations . As the difference of the Cotangents of the Arks cut at either Stations : Is to Radius : So the distance between those Stations : To the Altitude of the Tower. In the Diagram for this Case the Complements of the angles observed at the two Stations , viz. at G were 20 d , at H 41 d 31′ . Take the distance on the Line of Tangents between these two Arks , and because equal parts are sought , and the said Extent less then 50 , the measured distance changing the second Tearm of the Proportion into the place of the third , Enter the said Extent upon 50 in the equal parts , laying the Thread to the other foot , then if the Tangent of 45 d be entred between the Scale and the Thread , the Compasses will rest upon 96 for the Altitude sought . To work Proportions in equal Parts and Sines by help of the Limb. TO suppose both the middle Tearms to be either equal Parts or Sines , will not be practical , yet may be performed as before hinted , without introducing the Radius , if it be not ingredient , because both these Lines issue from the Center , and may also be performed by the Pen by measuring the Sines one the Line of equal parts , as was instanced in page 41. But supposing the middle Tearms of a different kind . 1. If a Sine be sought , Operate by introducing the Radius . Lay the Thread to the Sine in the Limb being one of the middle Tearms , and from the other middle Tearm being equal parts , take the nearest distance to it , one foot of this extent enter at the first Tearm , and the Thread laid to the other foot cuts the Limb at the Ark sought . If the Ark sought be above 70 d this work may better be performed with the Line of equal parts and Sines jointly , as issuing from the Center . 2. If a Number be sought ▪ Lay the Thread to the Sine in the Limb being one of the middle Tearms , and from the other middle Tearm being equal parts take the nearest distance to it , Then lay the Thread to the first Tearm in the Limb being a sine , and enter the former extent between the Scale and the Thread , and the foot of the Compasses will on the Line of equal parts shew the fourth Proportional . The Proportion for finding the Altitude of a Tower at one Station by the measured distance , may also be wrought in in equal parts and Sines . For , As the Cosine of the Ark at first Station , To the measured distance thereof from the Tower : So is the Sine of the said Ark , To the Altitude of the Tower. In that former Scheme , the measured distance B H is 85 , and the angle observed at H 48 d 29′ Wherefore I lay the Thread to the Sine of the said Ark in the Limb , counted from the right edge , and from the measured distance in the equal parts take the nearest extent to the Thread , then laying the Thread to the Cosine of the said Ark in the Limb , and entring the former extent between the Thread and the Scale , I shall find the foot of the Compasses to fall upon 96 the Altitude sought . So also in the Triangle A C B , if there were given the side A C 194 , the measured distance between two Stations on the Wall of a Town besieged , and the observed angles at A 25 d 22′ , at C 113 d 22′ , if B were a Battery we might by this work find the distance of it from either A or C , for having two angles given all the three are given , it therefore holds , As the Sine of the angle ot B 41d 16′ , To its opposite side A C 194 , So the Sine of the angle at C 66d , 38′ the Complement , To its Opposite side B A 270 , the distance of the Battery from A Such Proportions as have the Radius in them will be more easily wrought , we shall give some few Examples in Use in Navigation . 1. To find how many Miles or Leagues in each Parralel of Latitude answer to one degree of Longitude . As the Radius , To the Cosine of the Latitude . So the number of Miles in a degree in the Equinoctial , To the Number of Miles in the Parralel . So in 51 d 32′ of Latitude if 60 Miles answer to a degree in the Equinoctial 37 ‑ 3 Miles shall answer to one degree in this Parralel . This is wrought by laying the Thread to 51 d 32′ in the Limb from the left edge towards the right , then take the nearest distance to it from 60 in the equal parts which measured from the Center will be found to reach to 37 ‑ 3 as before . The reason of this facil Operation is because the nearest distance from the end of the Line of equal parts to the Thread is equal to the Cosine of the Latitude , the Scale it self being equal to the Radius , and therefore needs not be taken out of a Scale of Sines and entred upon the first Tearm the Radius as in other Proportions in Sines of of the greater to the less , when wrought upon a single Line only issuing from the Center , where the second Tearm must be taken out of a Scale , and entred upon the first Tearm . 2. The Course and Distance given to find the difference of Latitude in Leagues or Miles . As the Radius , To the Cosine of the Rumb from the Meridian : So the Distance sailed , To the difference of Latitude in like parts . Example . A Ship sailed S W by W , that is on a Rumb 56 d 15′ from the Meridian 60 Miles , the difference of Latitude in Miles will be found to be 33 ‑ 3 the Operation being all one with the former , Lay the Thread to the Rumb in the Limb , and from 60 take the nearest distance to it , which measured in the Scale of equal parts will be found as before , 3. The Course and Distance given to find the Departure from the Meridian , alias the Variation . As the Radius , To the Sine of the Rumb from the Meridian : So the distance Sailed , To the Departure from the Meridian . In the former Example to find the Departure from the Meridian , Lay the Thread to the Rumb counted from the right edge towards the left , that is to 56d 15′ so counted , and from 60 in the equal parts being the Miles Sailed , take the nearest distance to it ; this extent measured in the said Scale will be found to be 49 ‑ 9 Miles , and so if the converse of this were to be wrought , it is evident that the Miles of Departure must be taken out of the Scale of equal parts and entred Parralelly between the Scale and the Thread lying over the Rumb . Many more Examples and Propositions might be illustrated , but these are sufficient , those that would use a Quadrant for this purpose may have the Rumbs traced out or prickt upon the Limb : Now we repair to the backside of the Quadrant . Of the Line of on the right Edge of the Backside . THe Uses of this Line are manifold in Dyalling in drawing Projections in working Proportions , &c. 1. To take of a Proportional Sine to any lesser Radius then the side of the Quadrant , or which is all one , to divide any Line shorter in length then the whole Line of Sines in such manner as the same is divided . Enter the length of the Line proposed at 90 d the end of the Scale of Sines , and to the other foot lay the Thread according to nearest Distance , or measure the length of the Line proposed on the Line of Sines from the Center , and observe to what Sine it is equal , then lay the Thread over the like Arch in the Limb , and the nearest distances to it from each degree of the Line of Sines shall be the Proportional parts sought . And if the Thread be laid over 30 d of the Limbe the nearest distances to it will be Sines to half the Common Radius . 2. From a Line of Sines to take off a Tangent , the Proportion to do it is , As the Cosine of an Arch , To the Radius of the Line proposed : So the Sine of the said Arch , To the Tangent of the said Arch. Enter the Radius of the Tangent proposed at the Cosine of the given Arch , and to the other foot lay the Thread then from the Sine of that Arch take the nearest distance to the Thread , this extent is the length of the Tangent sought ; thus to get the Tangent of 20 d enter the Radius proposed at the Sine of 70 d , then take the nearest distance to the Thread from the Sine of 20 d , this extent is the Tangent of the said Arch in reference to the limited Radius . Otherways by the Limb. Lay the Thread to the Sine of that Arch counted from the right edge whereto you would take out a Tangent , and enter the Radius proposed down the Line of Sines from the Center and take the nearest distance to the Thread then lay the Thread to the like Arch from the left edge , and enter the extent between the Scale and the Thread , the distance of the Foot of the Compasses from the Center shall be the length of the Tangent required . 3. From the Sines to take off a Secant . The Proportion to do it is , At the Cosine of the Arch proposed , To Radius of the Line proposed So the Sine of 90d , the common Radius , To the length of the Secant of that Arch , to the limitted Radius . By the Limb , Lay the Thread to that Arch in the Limb counted from left edge whereto you would take out a Secant , then enter the Limitted Radius between the Scale and the Thread and the distance of the foot of the Compasses from the Center shall be the length of the Secant sought , and the Converse if a Secant and its Radius be given to find the Ark thereto enter the Secant of 90 d then enter the Radius of it between the Thread and the Sines , and the Compasses shews the Ark thereto , if counted from 90 d towards the Center . Otherways . Enter the Radius of the Line you would devide into Secants at the Cosine of that Arch whereto you would take out a Secant , and to the other foot lay the Thread then the nearest distance to the Thread from the Sine of 90 d is the length of the Secant sought : Thus to get the Secant of 20 d enter the Radius limited in the Sine of 70 d then the nearest distance from 90 d to the Thread , is the length of the Secant sought . And here it may be noted , that if you would have the whole length of the Line of Sines to represent the Secant sought , then the Cosine of that Arch which it represents shall be the Radius to it ; so the whole Line of Sines representing a Secant of 70 d , the length of the Sine of 20 d shall be the Radius thereto . It may also be observed , that no Tangent or Secant can be taken of at once larger then the Radius of the Quadrant , nor no Radius entred longer then that is , and that if the Radius entred be in Length ½ ⅓ ¼ of the Sines Tangents to 63 h 26′ 71 34 75 58 Secants to 60d●′ 70 32 75 32 may be taken off by help of the Line of Sines . And here it may be observed , That if the Tangent and Secant of any Arch be added in one streight Line or otherwise in Numbers , the Amount shall be equal to the Tangent of such Ark as shall bisect the remaining part of the Quadrant , as is demonstrated in Pitiscus & Sn●llius . Whence it follows , That if we have a Tangent and Secant no further then to 60 d each , yet a Tangent by the joint use of both Lines may from them be prickt down to 75 d : Wherefore at anty time to lengthen the Tangents double the Arch proposed , and out of the Amount reject 90 d ; The Tangent and Secant of the remainder connected in one streight Sine shall be the Tangent of the Arch sought . Thus to get the Tangent of 70 d the double is 140 d whence 90 d rejected rests 50 d ; the Tangent and Secant of 50 d joined in one streight Line shall be the Tangent of such an Arch as bisects the remaining part of the Quadrant , namely of 70 d. It may also be observed , That the Tangent of an Ark , and the Tangent of half its Complement is equal to the Secant of that Arch as is obvious in drawing of any Projection . A Chord may also be taken off from the Line of Sines , but more facilly by the Line of Chords on the left edge of the Quadrant , and is therefore pretermitted . To work Proportions in Sines alone . Frst , Without the help of the Limb or lesser Sines without introducing the Radius , but upon this Line alone independently . There will be two Cases , 1. If the first Tearm be greater then the second , the entrance is lateral ; Enter the second Tearm upon the first , laying the Thread to the other foot . Then from the third Tearm in the Scale take the nearest distance to the Thread , and measure that Extent from the Center , and it shews the Tearm sought , and so if it were , As Sine 30d , To Sine 10d : So Sine 80d , the fourth Proportional would be found to be 20d. In giving Examples to illustrate the matter , I shall make use of that noted Canon for making the Tables , As the Semiradius , or Sine of 30d , To the Sine of any Arch : So the Cosine of that Arch , To the Sine of that Arch doubled . But when the first Tearm is less then the second , Enter the length of the first Tearm upon the second , laying the Thread to the other foot of that Extent , then enter the third Tearm Parralelly between the Scale and the Thread , and it shews the fourth Proportional sought . So if it were , As the Sine of 10d , To the Sine of 30d : So the Sine of 20d , To a fourth , the 4th Proportional would be found to be 80 d. Another general way will be to do it by help of the Limb , by introducing the Radius in such Proportions wherein it is not Lay the Thread to one of the middle Tearms in the Limbe , and from the other middle Tearm on the Line of Sines take the nearest distance to it , then enter one foot of that extent at the first Tearm on the Line of Sines , and lay the Thread to the other foot , and in the Limbe it shews the 4th Proportional sought . Example . If the three Proportionals were , As Sine 55 d , To Sine 70 d : So Sine 30 d , To a fourth , the fourth Proportional would be found to be 35 d. But if the first Tearm of the Proportion be either a small Ark or the answer above 70 d , the latter part of this general direction for more certainty may be turned into a Parralel entrance , that is to say instead of entring the Extent taken from one of the middle Tearms in the Sines to the Thread laid over the other middle Tearm in the Limb , and entring it at the first Tearm in the Sines finding the Answer in the Limb lay , the Thread to the first Tearm in the Limb , and find the Answer in the Line of Sines by , entring the former extent parralelly between the Scale and the Thread . What hath been spoken concerning the Limb may also be performed by the Line of lesser Sines in the Limb by the same Directions . So if it were , As the Sine of 5d , To Sine 30d : So the Sine of 10d To a fourth , the 4th Proportional would be found to be the Sine of 85d , and the Operation best performed by the joint use of the Line of Sines , and the lesser Sines by making the latter entrance a Parralel entrance . When the Radius is in the third place of a Proportion in Sines of a greater to a less ▪ the Operation is but half so much as when it is not ingredient . Example . As the Cosine of the Latitude , To the Sine of the Declination , So the Radius , To the Sine of the Suns Amplitude . If the Suns declination were 13d , to find his Amplitude in our Latitude for London , take the Sine of 13d and enter one foot of it on the Sine of 38d 28′ and to the other foot lay the Thread , and in the Limb it shews the Amplitude sought to be 21d 12′ . By changing the places of the two middle Tearms , this Example will be turned into a Parralel entrance . Lay the Thread to the Complement of the Latitude in the Limb , and enter the Sine of the Declination between it and the Scale , and you will find the same Ark in the Sines for the Amplitude sought , as was before found in the Limb. Such Proportions of the greater to the less wherein the Radius is not ingredient , that have two fixed or constant Tearms , may be most readily performed by the single Line of Sines without the help of the Limb. An Example for finding the Suns Amplitude . As the Cosine of the Latitude , To the Sine of the Suns greatest declination : So the Sine of the Suns distance from the next Equinoctial Point , To the Sine of the Suns Amplitude . Because the two first Tearms of this Proportion are fixed , the Amplitude answerable to every degree of the Suns place may be found without removing the Thread ; To do it enter the Sine of the Suns greatest Declination 23d 31′ , at the Sine of the Latitudes Complement , and to the other foot lay the Thread , where keep it without alteration , then for every degree of the Suns place counted in the Sines take the nearest distance to the Thread , and measure those extents down the Line of Sines from the Center , and you will find the correspondent Amplitudes . Example . So when the Sun enters ♉ ♍ ♏ ♓ , his Equinoctial distance being 30 d , the Amplitude will be 18 d 41′ , and when he enters ♊ ♌ ♐ ♒ Equinox distance 60 d , the Amplitude will be 33 d 42′ ; and when he enters ♋ ♑ the greatest Amplitude will be 39d 50′ , his distance from the nearest Equinoctial Point being 90 d. But for such Proportions in which there is not two fixed Tearms , the best way to Operate them will be by the joint help of the Limbe and Line of Sines . An Example for finding the Time of the day the Suns Azimuth Declination and Altitude being given . By the Suns Azimuth is meant the angle thereof from the midnight part of the Meridian , the Proportion is As the Cosine of the Declination , To the Sine of the Azimuth : So the Cosine of the Suns Altitude . To the Sine of the hour from the Meridian . Example . So when the Sun hath 18 d 37′ North Declination , if his Azimuth be 69 d from the Meridian , and the Altitude 39 d , the hour will be found to be 49 d 58′ from Noon . So if there were given the Hour , the Declination and Altitude by transposing the Order of the former Proportion , it will hold to find the Azimuth , As the Cosine of the Suns Altitude , To the Sine of the hour from the Meridian : So the Cosine of the Suns Declination , To the Sine of the Azimuth from the Meridian . Commonly in both these Cases the Latitude is also known , and the Affection is to be determined according to Rules formerly given . A Proportion wholly in Secants we have shewed before may be changed wholly into Sines ; but the like mutual conversion of the Sines into Tangents is not yet known , however it may be done in 〈◊〉 of the 16 Cases wherein the Radius is ingredient , for instance , let the Proportion be to find the time of Sun rising . As Radius , To Tangent of Latitude : So the Tangent of the Declination , To the Sine of the hour from 6. Instead of the two first Tearms it may be , As the Cosine of the Latitude , To the Sine of the Latitude , then instead of the Tangent of the Declination say , So is the Sine hereof to a fourth . Again , As the Cosine of the Declination , To that fourth : So Radius , To the Sine of the hour from six : This being derived from the Analemm● by resolving a Triangle , one side whereof is the Arch of a lesser Circle . If a Quadrant want Tangents or Secants in the Limb , but may admit of a Sine from the Center , the Tangent and Secant of the Latitude , &c , may be taken out by what hath been asserted , to half the common Radius , and marked on the Limb , and the Quadrant thereby fitted to perform most of the Propositions of the Sphoere in one Latitude , and how to supply the Defect of a Line of Versed Sines in the Limb shall afterwards be shewne . What hath been spoken concerning a Line of Sines graduated on a Quadrant from the Center , may by help of the equal Limb be performed without it . 1. A Proportional Sine may be taken off to any diminutive Radius . By the Definition of Sines the right Sine of an Arch is a Line falling from the end of that Arch Perpendicularly to the Radius drawn to the other end of the said Arch ; So the Line H K falling Perpendicularly on the Radius F G shall be the Sine of the Arch H G , and by the same Definition the Line G I falling perpendicularly on the Radius F H shall also be the Sine of the said Arch , and whether the Radius be bigger or lesser , this Definition is common , but the Line G I on a Quadrant represents the nearest distance from the Radius to the Thread , therefore a Sine may be taken off from the Limb to any Diminutive Radius , to perform which , Enter the length or Radius proposed down the streight Line that comes from the Center of the Quadrant , and limits the Limb ; observe where the Compasses rests , this I call the fixed Point , because the Compasses must be set down at it , at every taking off , then to take off the Sine of any Arch to that Radius , lay the Thread over the Arch counted in the Limb from the said edge of the Quadrant , and take the nearest distance to it for the length of the Sine sought : But to take out Sines to the Radius of the graduated Limb set down one foot at the Ark in the Limb , and take the nearest distance to the two edge Lines of the Limb , the one shall be the Sine , the other Co-sine of the said Ark. 2. A Proportion in Sines alone may be wrought by help of the Limbe . Take out one of the middle Tearms by the former Prop. and entring it down the right edge from the Center , take the nearest distance to the Thread laid over the other middle Tearm in the Limbe , counted from right edge , then lay the Thread to the first Tearm in the Limb , and enter that extent between the right edge Line and the Thread , the distance of the foot of the Compasses from the Center , is the length of the Sine sought , to be measured in the Limb by entring one foot of that Extent in it : So that the other turned about may but just touch one of the edge or side Lines of the Limb issuing from the Center , or enter that Extent at the concurrence of the Limbe with the said Line , and lay the Thread to the other foot according to the nearest distance , and in the Limbe it shews the Ark sought : Whence may be observed how to prick of an angle by Sines instead of Chords . From this and some other following Propositions I assert the Hour and Azimuth may be found generally by the sole help of the Limb of a Quadrant without Protraction . How from the Lines inscribed in the Limbe to take off a Sine , Tangent , Secant and Versed Sine to any Radius , if less then half the common Radius of the Quadrant . IT hath been asserted , that a Sine may be taken off from the Limb , and by consequence any other Line there put on ; for by being carried thither they are converted into Sines , and put on in the same manner , for by the Definition of Sines , if Lines were carryed Parralel to the right edge of the Quadrant from the equal degrees of the Limb to the left edge they would there constitute a Line of Sines and the Converse . To find the fixed Point enter the Radius proposed twice down the Line of Sines from the Center , or which is all one , Lay the Thread over 30 d of the Limb counted from the right edge towards the left , and enter the limitted Radius between the Thread and the Scale ; so that one foot turned about may just touch the Thread , and the other resting on the Line of Sines , shews the fixed Point , at which if the Compasses be always set down , and the Thread laid over any Ark in the Tangent , Secant or lesser Sines , the nearest distances from the said Point to the Thread shall be the Sine , Tangent , Secant , of the said Ark to the limitted Radius . But for such Lines as are put on to the common Radius , as the Tangent of 45 d , &c. the Radius is to be entred but once down from the Center to find the fixed Point . Of the Line of Secants . This Line singly considered is of small use , but junctim with other Lines of great use for the general finding the Hour and Azimuth : Mr Foster makes use of it in his Posthuma to graduate the Meridian Line of a Mercators Chart , which is done by the perpetual addition of Secants , and the like may be done from this Line lying in the Limb but a better way wil be to do it from a well graduated Meridian Line by doubling or folding the edge of the Chard thereto , and so graduate it by the Pen. Of the Line of Tangents . The joint use of this Line with the Line of Sines is to work Proportions in Sines and Tangents , in any Proportion wrought by help of Lines in the Limb wherein the Radius is not ingredient , the Radius must be introduced according to the general Direction . If the two middle Tearms be Sines there must be one Proportion wrought wholly on the Line of Sines on the Backside , and another on the Line of Tangents on the foreside ; but such Cases are not usual : But if the two middle Tearms be Tangents , the first Operation must be on the line of Tangents on the foreside , and the latter on the line of Sines on this backside , unless the Radius be ingredient . A general Direction to work Proportions when the middle Tearms are of a different Species . If a Sine be sought , Lay the Thread to the Tangent in the Limb being one of the middle Terms , and from the Sine being another of the middle Terms take the nearest distance to it , then lay the Thread to the other Tangent in the Limb , being the first Tearm , and enter the former extent between the Scale and the Thread , and the foot of the Compasses on the Line of Sines will shew the fourth Proportional . Example . If the Proportion were , As the Tangent of 30 d , To the Sine of 25d So is the Tangent of 20 d , To the Sine of 15 d 27′ . Lay the Thread over the Tangent of 20 d in the Limb , and from the Sine of 25 d take the nearest distance to it , then lay the Thread to the Tangent of 30 d , and the former extent so entred that one foot resting on the Sines , the other foot turned about may but just touch the Thread , and the resting foot will shew 15 d 27′ for the Sine sought . 2. If a Tangent be sought . Lay the Thread to the Tangent being one of the middle Tearms , and from the other middle Tearm being a Sine take the nearest distance thereto , then Enter one foot of that extent at the first Tearm being a Sine , and the Thread laid to the other foot shews the fourth Proportional in the Line of Tangents in the Limb. Example . So if the Proportion were , As the Sine of 25 d , To the Tangent of 30 d : So is the Sine of 32 d , To a Tangent , the fourth Proportional would be found to be the Tangent of 35 d 54′ . If the answer fall near the end of the Scale of Tangents , the latter entrance may be made by laying the Thread to the first Tearm in the Limb , and by a Parralel entrance an Ark found on the Line of Sines , then if the Thread be laid over the like Ark in the Limb it will intersect the Tangent sought . These Directions presuppose the varying of the Proportion when the Tangens excur the length of the Scale , according to the Directions in the Trigonometrical part ; but as before suggested , those Directions are insufficient when one of the Tearms or Tangents are less then the Complement of the Scale wanting , and the other greater then the length of the Scale , for two such Arks cannot be changed into their Complements without still incurring the same inconvenience ; in this Case only change the greater Tearm , which may be done by help of the Line of Secants , for , As the Tangent of an Arch , To the Sine of another Arch ; So is the Cosecant of the latter Arch , To the Cotangent of the former . And by Transposing the Order of the Tearms . As a Sine , To a Tangent : So the Cotangent of the latter Arch , To the Cosecant of the former . Example . If the Proportion were , As the Sine of 8d , To the Tangent of 25d So is the Sine of 60d , To the Tangent of 71d : Here we might foreknow by the nature of the Tearms that the Tangent sought would be large or finde by tryal that it cannot be wrought upon the Quadrant : We may therefore vary it thus , As the Tangent of 25d To the Sine of 8d : So the Secant of 30d , To the Tangent of 19d , the Complement of 71d , the Arch sought . Lay the Thread over 8d in the lesser Sines , and set down one foot of the Compasses at the Sine of the same Arch the Thread lyes over in the Limb ; and take the nearest distance to the Thread laid over the Secant of 30 , then lay the Thread to the Tangent of 25d , and enter the former extent between the Thread and the Line of Sines , and the distance of the foot of the Compasses from the Center measured on the Tangents on the foreside sheweth 19d. But a more general Caution in this Case without the help of the Secants , would be by altring the larger Tangent into its Complement by introducing the Radius , and operating the Proportion on the greater Tangent of 45d. If the Proportion were , As the Tangent of 70d , To the Sine of 60d : So the Tangent of 25d , To the Sine of 8d 27′ . By introducing the Radius at two Operations it would be easily wrought , As Radius , To Tangent 25d , So Sine 60d , To a fourth , Again , As the Radius , To the Tangent of 20 d : So that fourth , To the Sine sought . So the former Example wherein a Tangent is sought may be likewise varied . As Radius , To Tangent 25d : So Sine 60d , To a fourth , Again , As that fourth , To the Radius : So is the Sine of 8d , To the Cotangent of the Arch sought , namely to the Tangent of 19d as before . Two Proportions with the Radius in each are as suddenly done as one without the Radius . Operation . Lay the Thread over the Tangent of 25d in the greater Tangents , and from the Sine of 60d take the nearest distance to it , enter that extent at 90 , or the end of the Line of Sines , laying the Thread to the other foot according to the nearest distance , then enter the Sine of 8 parralelly between the Scale and the Thread and the distance of the foot of the Compasses from the Center is the Tangent of the Complement of the Ark sought to be measured in the greater Tangents by setting down one foot at 90d , and the Thread laid to the other , according the nearest-will lye over the Tangent of 19 d. An Example with the Radius ingredient and a Sine sought , Data , Latitude , and Declination , to find the time when the Sun shall be East or West . As the Radius , To the Cotangent of the Latitude : So the Tangent of the Declination , To the Sine of the hour from 6. To be wrought by the help of the lesser Tangents . When the Radius comes first and two Tangents in the middle , change the largest Ark into its Complement to bring it into the first place , and the Radius into the second ; then take out the Tangent of the other middle Ark , either from the foreside from the Scale , or out of the Limb by setting one foot at the Sine of 90 d , and taking the nearest distance to the Thread laid over the Tangent given , then laying the Thread to the Tangent of the first Ark , enter the former extent between the Scale and the Thread , and the foot of the Compasses will shew the Sine sought . Otherways the two middle Tearms being Tangents , as also when the first Tearm and one of the middle Tearms is a Tangent , change the Radius and one of those Tangents into Sines . For , As the Radius , To the Tangent of any Ark : So is the Cosine of the said Ark , To the Sine thereof . And , As the Tangent of any Ark , To Radius : So is the Sine of that Ark , To the Cosine thereof . And so the former Proportion changed will be , As the Sine of the Latitude , To the Cosine of the Latitude : So the Tangent of the Declination , To the Sine of the hour from six , When the Sun shall be East or West . Example . If the Declination were 23d 30′ North , in our Latititude of London 51d 32′ to find the Sine sought , Lay the Thread to the Tangent of the Declination in the Limb , and from the Complement of the Latitude in the Sines take the nearest distance to it , then lay the Thread to the Sine of the Latitude in the lesser Sines and enter the former extent between the Thread and the Scale and the foot of the Compasses sheweth the answer in degrees , if the Thread be laid to the Ark found in the Limb it there sheweth it in Time ; So in this Example the time sought is 20d 14′ , or in Time 1 h 17 h before 6 in the morning or after it in in the Evening . If the Latitude and Declination were given , To find the Suns Azimuth at the Hour of 6. As the Radius , To Cosine of the Latitude : So the Tangent of the Suns Declination , To the Tangent of his Azimuth from the Vertical . In this Case a Tangent being the 4th Tearm sought , the Operation is very facil . Lay the Thread to the Tangent of the Declination in the lesser Tangents , and from the Cosine of the Latitude take the nearest distance to it , and either measure that extent on the Tangents on the foreside , or set one foot of that extent upon the Sine of 90d , and to the other lay the Thread and it will intersect the Tangent sought in the Limb : So in our Latitude when the Sun hath 23d 30′ of declination , his Azimuth at the hour of 6 will be 15 d 9′ from the East or West . Another Example , So if the Suns distance from the nearest Equinoctial Point were 60 d , his right Ascension would be found to be 57 d 48′ . The Proportion to perform this Proposition is , As the Radius , To the Cosine of the Suns greatest Declination : So the Tangent of the Suns distance from the next Equinoctial Point , To the Tangent of the Suns right Ascension , or when the Tangents are large , As the Cosine of the Suns greatest declination , To Radius : So the Cotangent of the Suns distance from the Equinoctial Point . To the Cotangent of his right Ascension . By what hath been said it appears that the working Proportions by the natural Lines is more troublesome then by the Logarithmical , however this trouble wil be shunned in the use of the great Quadrant by help of the Circle on the backside . I now come to shew how the Hour of the Day , and the Azimuth of the Sun may be found universally by the Lines on the Quadrant , which is the principal thing intended . The first Operation for the Hour will be to find what Altitude or Depression the Sun shall have at the hour of 6. The Proportion to find it is , As the Radius , To the Sine of the Latitude : So the sine of the Suns Declination , To the sine of the Altitude sought . Example . So in Latitude 51 d 32′ , the Suns declination being 23 d 31′ , To find his Altitude or Depression at 6 , Lay the Thread to the Sine of the Latitude in the Limb , and from the sine of the Suns Declination take the nearest distance to it , which extent measured from the Center will be found to be 18 d 12′ . This remains fixed for one Day , and therefore must be recorded , or have a mark set to it . Afterwards the Proportion is , As the Cosine of the Declination , To the Secant of the Latitude , Or , As the Cosine of the Latitude , To the Secant of the Declination : So in Summer is the difference , but in Winter the Sum of the sines of the Suns proposed or observed Altitude , and of his Altitude or Depression at 6 , To the Sine of the hour from 6 towards Noon in Winter , as also in Summer when the Altitude is more then the Altitude of 6 , otherways towards Midnight . To Operate this . In Winter to the sine of the Suns Depression at 6 , add the sine of the Altitude proposed , by setting down the extent hereof outward at the end of the former extent ; in Summer take the distance between the sine of the Suns Altitude , and the sine of his Altitude at 6 , and enter either of these extents twice down the Line of sines from the Center , then lay the Thread to the Secant being one of the middle Tearms , and take the nearest distance to it . Lastly , enter one foot of this extent at the first Tearm , being a Sine , and to the other foot lay the Thread , and in the equal Limb it shews the hour from 6 , which is accordingly numbred with hours . But when the Hour is neer Noon , the answer may be found in the Line of Sines with more certainty by laying the Thread to the first Tearm in the Limb , and entring the latter extent Parralelly between the Scale and the Thread . Otherways . Enter the aforesaid sum or difference of sines once down the Line of Sines from the Center , and laying the Thread to the Secant , being one of the middle Tearms , take the nearest distance to it , then lay the Thread to the first Ark in the lesser sines , and enter the former extent between the Thread and the Scale , and the foot on the Compasses on the Line sheweth the Sine of the Hour . Example . If the Altitude were 45d 42′ , take the distance between it and the sine of 18 d 12′ before found enter this extent twice down the Line of sines from the Center , and laying the Thread over the Secant of 51 d 32′ take the nearest to it , then entring one foot of this extent at 66 d 29′ in the Line of Sines the Thread being laid to the other according to nearest distance will lye over 45 d in the Limb shewing the hour to be either 9 in the morning , or 3 in the afternoon , and so it will be found also in the latter Operation by entring the first extent once down the sines , and taking the distance to the Thread lying over the Secant of the Latitude , and then laying the Thread to 66 d 29′ in the 〈◊〉 , and entring that extent between the Scale and the Thread . To find the Suns Azimuth The first Operation will be to get the Suns Altitude in the Vertical Circle , that is , being East ar West . As the sixe of the Latitude , To Radius : So is the Sine of the Declination , To the sine of the Altitud . So in our Latitude of London , when the Sun hath 23d 31′ of declination , his Vertical Altitude in Summer will be found to be 30d 39′ and so much is the Depression when he hath as much South declination . This found either by a Parralel entrance on the Line of Sines by laying the Thread to the sine of the Latitude in the Limb , and entring the sine of the Declination between the Scale and the Thread , or by a Lateral entrance in the Limbe changing the Radius into the third place , and then enter the sine of the Declination on the Sine of the Latitude , laying the Thread to the other foot , and in the Limb it shewes the Altitude sought ; having found this Ark let it be recorded or have a mark set to it , because it remains fixed for one Day , afterwards the Proportion to be wrought is , As the Cosine of the Altitude , To the Tangent of the Latitude : So in Summer is the difference in Winter the sum of the Sines of the Suns Altitude , and of his Vertical Altitude or Depression : To the Sine of the Azimuth from the East or West towards noon Meridian in Winter as also in Summer , when the given Altitude is more then the Vertical Altitude , but if less towards the Midnight Meridian . This Proportion may be wrought divers ways on the Quadrant after the same manner as the former , I shall therefore illustrate it by some Examples . Declination 13d , Latititude 51d 32′ , Vertical Altitude 16d 42′ , Proposed Altitude in Summer — 41d , 53′ . Proposed Altitude in Winter — 12 13 Enter the aforesaid sum or difference of Sines twice down from the Center of the Quadrant , and take the nearest distance to the Thread being laid over the Tangent of the Latitude , this extent set down at the Cosine of the Altitude , and lay the Thread to the other foot and in the Limbe it shews the Azimuth sought . So in this Example the Azimuth will be found to be 40d both in Summer and Winter from East or West towards Noon Meridian . Otherways . Enter the aforesaid sum or difference of sines but down from the Center , and take the nearest distance to the Thread laid over the Tangent of the Latitude , then lay the Thread to the Complement of the Altitude in the lesser sines , and enter the former extent between the Scale and the Thread , and the answer will be given in the Line of Sines , supposing the declination unchanged , if the Altitude were 9d 21′ both for the Winter and the Summer Example , the Azimuth at London would be 9d 22′ from the East or West Northwards in Summer , and 35 d Southwards in Winter . Hitherto we suppose the Latitude not to exceed the length of the Tangents , whether it doth or not this Proportion may be otherways wrought by changing the two first Tearms of it ; Instead of the Co-sine of the Altitude to the Tangent of the Latitude , we may say , As the Cotangent of the Latitude , To the Secant of the Altitude : So when the Sun hath 23 d 31′ of North Declination in our Latitude , and his Altitude 57 d 7′ , take the distance between the sine thereof and the sine of 30 d 39′ the Altitude of East , and enter it once down from the Center , and take the nearest distance to the Thread laid over the Secant of the Altitude , viz. 57 d 7′ , then lay the Thread to 38 d 28′ in the Tangents , and enter the former extent between the Scale and the Thread , and the Compasses on the Line of Sines will rest at 50 d for the Azimuth from East or West Southwards , because the Altitude was more then the Vertical Altitude . Otherways without the Secant in all Cases by help of the greater Tangent of 45d . Enter the aforesaid Sum or difference of the Sines once down from the Center and lay the Thread to the Tangent or Cotangent of the Latitude in the greater Tangents , and take the nearest distance to it . Then for Latitudes under 45d enter the former extent at the Complement of the Altitude in the Line of Sines , and find the answer in the Limb by laying the Thread to the other foot , or if it be more convenient make a Parralel entrance of it , and find the answer in the Sines as before hinted . But for Latitudes above 45 d , first find a fourth by entring the sum or difference of sines between the Scale and the Thread , and then it will hold , As the first Tearm , To that fourth : So Radius , To the Sine of the Azimuth , and may be either a Lateral or Parralel entrance , according as it falls out , and as the Radius is put either in the second or third place , in all these Directions the introducement of the Radius is supposed according to to the general Advertisement . The finding of the Amplitude this way presupposeth the Vertical Altitude known , and then the Proportion derived from the Analemma , not from the 16 Cases is , As the Radius , To the Tangent of the Latitude , So the Sine of the Vertical Altitude , To the Sine of the Amplitude : So also to find the time of Sun rising . As the Cosine of the Declination , To Secant of the Latitude : So the sine of the Suns Altitude at 6 , To the Sine of the hour of rising from six . To find the Suns Azimuth at six of the clock otherways then by the 16 Cases . As the Cosine of the Suns Altitude at 6 , To Tangent of the Latitude , So is the difference of the sines of the Suns Altitude at 6 , and of his Vertical Altitude , To the sine of the Azimuth from the Vertical . To find the time when the Sun shall be due East or West . As Cosine of the Declination , To Secant of the Latitude : So the difference of the Sines of the Suns Altitude at 6 , and of his Vertical Altitude , To the Sine of the hour from 6 , When the Sun shall be due East or West . These Proportions derived from the Analemma , are general both for the Sun and Stars in all Latitudes ; but when the Declination either of Sun or Stars exceed the Latitude of the place , this Proportion for finding the Azimuth cannot be at some times conveniently performed on a Quadrant , but must be supplyed from another Proportion , whereof more hereafter . Of the Hour and Azimuth Scales on the Edges of the Quadrant . These Scales are fitted for the more ready finding the Hour and Azimuth in one Latitude , being only to facilitate the former general Way . The Labour saved hereby is twofold , first the Suns declination is graved against the Suns Altitude of 6 in the Hour scale , and the said Declinations continued at the other end of the said hour Scale to give the quantity of the Suns Depression in Winter equal to his Altitude in Summer ; and secondly they are of a fitted length as was shewed in the Description of the Quadrant , and thereby half the trouble by introducing the Radius shunned . The Vse of the Azimuth Scale . The Altitude and Declination of the Sun given to find his Azimuth . Take the distance between the Suns Altitude in the Scale , and his Declination in Summer time in that Scale that stands adjoyning to the side ; in Winter in that Scale that is continued the other way beyond the beginning , and laying the Thread to the Complement of the Suns Altitude in the lesser sines , which is double numbred , enter this extent between the Scale and the Thread parralelly , and the foot of the Compasses sheweth in the Line of Sines the Azimuth accordingly , Declination 23 d 31′ , Altitude 47 d 27′ , the Azimuth thereto would be 25 d from East or West in Summer , and if the Altitude were 9 d 43′ in Winter the Azimuth thereto would be 30 d either way from the Meridian . And so when the Sun hath no Altitude , lay the Thread over 90 d in the lesser Sines and enter the extent from the beginning of the Azimuth Scale to the Declination , and you will finde the Amplitude which to this Declination will be 39 d 50′ . The Vses of the Hour Scale . To find the Hour of the Day . TAke the distance between the Suns Altitude in the hour Scale , and his Declination proper to the season of the year , then laying the Thread to the Complement of the Suns Declination in the lesser sines enter the former extent between the Scale and the Thread and the foot of the Compasses sheweth the sine of the hour . Example . If the Declination were 13 d North , and the Altitude 37d 13′ take the distance between it in the Scale and 13 d in the prickt Line , then laying the Thread to 77 d in the lesser sine enter that extent between the Scale and the Thread , and the resting foot will shew 45 d for the hour from 6 , that is either 9 in the forenoon , or 3 in the afternoon . The Converse of the former Proposition will be to find the Suns Altitude on all Hours . The Thread lying over the Complement of the Suns Declination in the lesser sines from the sine of the hour , take the nearest distance to it , then set down one foot of that extent in the hour Scale at the Declination , and the other will reach to the Altitude . Example . At London , for these Scales are fitted thereto , I would find the Suns Altitude at the hours of 5 and at 7 in the morning in Summer when the Sun hath 23 d 31′ of Declination . Here laying the Thread to 23 d 31′ the Suns declination from the end of the lesser Sines being double numbred , from the sine of 15 d , taking the nearest distance to it , set down one foot of this extent at 23 d 31′ the declination it reaches downwards to 9 d 30′ , and upwards to 27 d 23′ the Suns Altitude at 5 and 7 a clock in the morning in Summer . Another Example . Let it be required to find the Suns Altitudes at the hours of 10 or 2 when his declination is 23d 31′ both North and South . The Thread lying as before over the lesser sines take the nearest distance to it from 60d in the sines , the said extent set down at 23d 31′ in the prickt Line reaches to 53 d 44′ for the Summer Altitude , and being set down at 23d 31′ on the other or lower continued Line reaches to 10d 28′ for the Winter Altitude . The Hour may be also sound in the Versed sines by help of this fitted hour Scale , Take the distance between the Suns Altitude , admit 36d 42′ , and his Meridian Altitude to that Declination 61 d 59′ , and enter one foot of this extent at the sine of 66 d 29′ , and laying the Thread to the other foot according to nearest distance and it will lye over the hours of 8 in the morning , or 4 in the afternoon in the Versed sines in the Limb , and thereby also may the time of Suns rising be found by taking the distance from 0 to the Meridian Altitude and entring it at the Cosine of the Declination as before and the Converse will find the Suns Altitudes on all hours by taking the distance from the Co-sine of the Declination to the Thread laid over the Versed sine of the hour from Noon , and the said Extent will reach from the Mridian Altitude in the fitted Scale to the Altitude sought . To find the time of Sun rising or setting , Lay the Thread over the Complement of the declination as before , in the lesser sines , and enter the extent between the ☉ Altitude , which is nothing that is from the beginning of the Hour Scale to the Declination between the Scale and the Thread and the foot of the Compasses shews it in the Line of sines , which may be converted into Time by help of the Limb. If these Scales be continued further in length as also the Declinations they will after the same manner find the Stars hour for any Star whatsoever to be converted into common Time , as in the uses of the Projection , as also the Azimuth of any Star that hath less declination then the place hath Latitude , but of this more in the next Quadrant . In Dyalling there will be often use of natural sines , whereas these Scales are continued but to 62 d , if therefore it be desired to take out any sine to the same Radius , the rest of the Scale wanting may be easily supplyed , for the difference of the sines of any two Arks equidistant from 60 d is equal to the sine of their distance . Thus the sine of 20 d is equal to the difference of the sines of 40 d and 80 d Arks of like distance from 60 d on each side , and so may be added either to 40 d forward , or the other way from the end of the Scale . In finding the Hour and Azimuth by these Scales , not in the Versed sines , the Directions altogether prescribe a Parralel entrance , but if the Extent from the Altitude to the Declination be entred at the Cosine of the Altitude or of the Declination in the Line of sines according as the Case is , and the Thread laid to the other foot , the Hour and Azimuth may be found in the lesser sines by a Lateral entrance . Or if the said Extent be doubled and entred as before hinted , the answer will be found in the equal Limb. Example to find the Suns Azimuth . Declination 23 d 31′ North. Altitude — 41 : 34 Having taken the distance between these two Tearms in the Azimuth Scale and doubled it , enter one foot in the Line of sines at 48 d 26′ , the Complement of the Altitude , and laying the Thread to the other according to nearest distance it will lye over 15 d of the equal Limb for the Suns Azimuth from the East or West Southwards . The Vse of the Versed Sin 's in the Limbe . It may be noted in the former general Proportion , I have used the word Azimuth from Noon or Midnight Meridian , though not so proper , because they are more universal and common to both Hemispheres , other expressions besides their Verbosity would be full of Caution for the following Proportion in our Northern Hemispere , without the Tropick that finds it from the South between the Tropick of Cancer and the Equinoctial , when the Sun comes to the Meridian between the Zenith and the Elevated Pole would find it from the North , wherefore it is fit to be retained . A general Proportion for finding the Hour . As the Cosine of the Declination , To the Secant of the Latitude : Or , As the Cosine of the Latitude , To the Secant of the Declination : So is the difference of the Sines of the Suns Altitude proposed , and of his Meridian Altitude , To the Versed Sine of the hour from Noon● And So is the sum of the sines of the Suns proposed Altitude , and of his Midnight Depression , To the Versed sine of the hour from Midnight : And So is the sine of the Suns Meridian Altitude , To the Versed sine of the Semidiurnal Ark : And So is the sine of the Suns Midnight Depression , To the Versed sine of the Seminocturnal Ark. The Operation will be like the former , I shall therefore onely illustrate it by one Example , the Meridian Altitude is got in Winter by differencing , in Summer by adding the Declination to the Complement of the Latitude , if the sum exceed 90 d the Complement thereof to 180 d is the Meridian Altitude . An Example for finding the Hour from Noon . Declination — 23d 31′ North the 11th June . Comp. Latitude — 38 28 London .   61 59 Meridian Altitude . Proposed Altitude — 36 42 , take the distance between the sines of these two Arks , and enter it once down the Line of sines from the Center , and take the distance to the Thread laid over the Secant , then enter one foot of that extent at the sine being the first Tearm , and to the other lay the Thread , and in the Versed sines in the Limb it will lye over the Versed Sine of the hour from Noon . In this Example , if the Thread be laid over the Secant of 51d 32′ the extent must be entred at the sine of 66d 29′ 23 31 the extent must be entred at the sine of 38 28 either way the answer will fall upon 60 d of the Versed sine shewing the Hour to be either 8 in the forenoon , or 4 in the afternoon . If the hour fall near noon , then the extent of the Compasses may be Quadrupled and entred as before , and look for the answer in the Versed Sines Quadrupled : Or before the distance be took to the Thread the extent of difference may be entred four times down from the Center . The Converse of this Proposition will be to find the Suns Altitude on all Hours universally . As the Secant of the Latitude , To Cosine Declination , Or , As the Secant of the Declination , To Cosine Latitude : So the Versed sine of the hour from Noon , To the difference of the sines of the Suns Meridian Altitude , and of his Altitude sought , to be substracted from the sine of the Meridian Altitude , and there will remain the sine of the Altitude sought . So in Latitude of London , if the Suns Declination were 13 d 00′ , and the hour from noon 75 d , that is either 7 in the morning , or 5 in the afternoon . Lay the Thread over the Versed sine of the hour from noon , namely , 75 d , and from the sine of 77 d the Complement of the Declination , take the nearest distance to it , then lay the Thread to the Secant of the Latitude , and enter the former extent between the Scale and the Thread , and you will find a sine equal to the difference sought , which sine take between the Compasses and setting down one foot at the sine of 51 d 28′ the Meridian Altitude , the other foot turned towards the Center will fall upon the sine of 19 d 27′ the Altiude sought . A General Proportion for the Azimuth . Get the Remainder or Difference between these two Arks , the Suns Altitude and the Complement of the Latitude by Substracting the less from the greater , and then the Proportion will hold , As the Cosine of the Latitude , Is to the Secant of the Altitude , Or , As the Cosine of the Altitude , To the Secant of the Latitude : So is the sum of the sines of the Suns Declination ▪ and of the aforesaid Remainder , To the Versed Sine of the Azimuth from the Noon Meridian in Summer only when the Suns Altitude is less then the Complement of the Latitude . In all other Cases , So is the difference of the said sines , To the Versed sine of the Azimuth , as before from Noon Meridian . Example . The 11th of June aforesaid , the ☉ having 23 d 31′ of North declination , his Altitude was observed to be 18 d 20′ , which substracted from 38 d 28′ the remainder is 20 d 8′ , take out the sine thereof , and set down one foot at the sine of 23 d 31′ , and set the other forwards towards 90 d , then take the nearest to the Thread laid over the Secant of the Latitude 51 d 32′ , enter one foot of this Extent at the Complement of the Altitude by reckoning the Altitude it self from 90 d towards the Center , and the Thread laid to the other foot cuts the Line of Versed sines at 105 d the Azimuth from the South . The same day when the Altitude was more then the Colatitude , suppose 60 d 11′ the Remainder will be found to be 21 d 43′ , take the distance between the sine thereof and of 23 d 31′ , and because the Extent is but small enter it four times down the Line of sines from the Center , and take the nearest distance to the Thread laid over the Secant of the Latitude , which entred at the Cosine of the Altitude , the Thread laid to the other foot shews 25 d in the Quadrupled Versed Sines for the Azimuth from the South . The Proportion hence derived for the Amplitude . As the Cosine of the Latitude , To Secant of the Declination , &c. as before . So in Summer is the sum in Winter , the difference of the sines of the Suns Declination , and of the Complement of the Latitude , To the Versed Sine of the Amplitude from Noon Meridian . The Proportion for the Azimuth will be better exprest by making the difference to be a difference of Versed sines . How the Versed sines in the Limbe may be spared in Case a Quadrant want them . If a Quadrant can only admit of a Line of sines from the Center , the common Quadrant of Mr Gunters very well may , on the right edge above the Margent for the Numbers of the Azimuths , it may be easily fitted for any or many Latitudes by setting Marks or Pricks to the Tangent and Secant of the Latitudes in the Limbe , which may be taken out by help of the Limb , Line of Sines , or by Protraction , and either of these general Proportions wrought upon it , or those which follow , if it be observed that whensoever the Thread lyes over the Versed sine of any Ark in the Limbe , it also at the same Time lyeth over a Sine equal to half that Versed sine to the common Radius : Now because the sine of 30d doubled is equal to the Radius , let it be observed whether the sine cut by the Thread be greater or less then 30 deg . When it is less let the Line of Sines represent the former half of a Line of Versed Sines , and take the sine of the Ark the Thread lay over , and enter it twice forward from the end of the Scale towards the Center , and you will obtain the Versed sine of the angle sought . When it is more take the distance between the sine of 30 d and the said sine , and letting the Line of sines represent the latter half of a Line of Versed sines , enter the said distance twice from the Center , and you will obtain the Versed sine of the Arch sought , namely , the sine of an Arch , whereto 90 d must be added . Three sides to find an Angle , a general Proportion . As the Sine of one of the Sides including an angle , Is to the Secant of the Complement of the other including side : So is the difference of the Versed Sines of the third side , and of the Ark of difference betwen the two including sides , To the Versed Sine of the Angle sought . And So is the difference of the Versed sines of the third side , and of the sum of the two including sides , To the Versed sine of the sought angles Complement to 180d . To repeat the Converse when two sides and the angle comprehended are given to find the third side were needless . If one of the containing sides be greater then a Quadrant , instead of it in referrence to the two first Tearms of the Proportion , take the Complement thereof to 180d for the reputed side , but in differencing or summing the two containing sides alter it not : And further note , that the same Versed sine is common to an Ark less then a Semicircle , and to its Complement to 360 d. The Operation of this Proportion will be wholly like the former , so that there needs no direction but only how to take out a difference of two Versed sines to the common Radius , seeing this Quadrant of so small a Radius is not capable of such a Line from the Center . And here note that the difference of two Versed sines less then a Quadrant , is equal to the difference of the natural sines of the Complements of those Arks. And the difference of two Versed sines greater then a Quadrant is equal to the difference of the natural sines of the excess of those Arks above 90 d. And by consequence the difference of the Versed sines of two Arks the one less , the other greater then a Quadrant is equal to the sum of the natural sines of the lesser Arks Complement to 90d , and the greater Arks excess above it . And so a difference of Versed sines may be taken out of the Line of natural sines considered as such . Or the Line of sines may be considered sometimes to represent the former half of a Line of Versed Sines as it is numbred with the small figures by its Complements from the end of it to 90d at the Center , and sometimes the latter half of it , and then the graduations of it as Sines must be considered as numbred from 90 d to 180 d at the end of it , and so a difference to be taken out of it by taking the distance between the two Tearms , which if the two Arks fall the one to be greater the other less then 90 d will be a sum of two sines , as before hinted , and in this Case the sine of the greater Arks excess above 90 d to be set down outwards if it may be , at the Versed sine of the lesser Ark , or which is all one , at the sine of that Arks Complement , and the distance from the Exterior foot of the Compasses to the Center will be equal to the difference of the Versed sines of the Arks proposed . To measure a difference of Versed sines to the common Radius . In this Case also the Line of sines must sometimes represent the former , sometimes the latter half of a Line of Versed sines , and then one foot of the difference applyed to one Ark , the other will fall in many Cases upon the Ark sought , each Proportion variously exprest , so that possibly either one or the other will serve in all Cases . But if one of the feet of the Compasses falls beyond the Center of the Quadrant : To find how much it falls beyond it , bring the said foot to the Center , and let the other fall backward on the Line , then will the distance between the said other foot , where it now falleth , and the place where it stood before be equal to the excess of the former foot beyond the Center , which accordingly thence measured helps you to the Arch sought , and its Complement both at once with due regard to the representation of the Line ; this should be well observed , for it will be of use on other Instruments . A difference of Versed sines thus taken out to the Common Radius must be entred but once down from the Center . To take out a Difference of Versed sines to half the common Radius . Count both the Arks proposed on the Versed sines in the Limbe , and find what Arks of the equal Limbe answer thereto , then out of the Line of sines take the distance between the said Arks , and you have the extent required , which being but half so large as it should be is to be entred twice down from the Center . To measure a difference of Versed sines to half the common Radius . The Versed sines are largest at that end numbred with 180d , count the given Ark from thence , and laying the Thread over the equal Limb find what Ark answers thereto , then setting down the Compasses at the like Ark in the Line of sines from the end of it towards the Center mind upon what Ark it falls , the Thread laid to the like Ark in the Limb sheweth on the Versed sines the Ark sought . To save the labour of drawing a Triangle , I shall deliver the Proportion for the Azimuth derived from the general Proportion , As the Cosine of the Latitude , To the Secant of the Altitude , Or , As the Cosine of the Altitude , To the Secant of the Latitude : So is the difference of the Versed sines of the Sun or Stars distance from the Elevated Pole , and of the Ark of difference between the Latitude and Altititude . To the Versed sine of the Azimuth sought ; as it falls in the Sphoere that is from the Midnight Meridian . And So is the difference of the Versed sines of the Polar distance , and of the Ark of difference between a Semicircle and the sum of the Latitude and Altitude , To the Versed Sine of the Azimuth from Noon Meridian . A Canon derived from the Inverse of the general Proportion to finde the Distance of places in the Ark of a great Circle . As the Secant of one of the Latitudes , To the Cosine of the other . So the Versed sine of the difference of Longitude , To the difference of the Versed sines of the Ark of distance sought , and of the Ark of difference between both Latitudes , when in the same Hemisphere , or the Ark of the sum of both Latitudes when in both Hemispheres , which difference added to the Versed sine of the said Ark gives the Versed sine of the Ark of distance sought . And So is the Versed sine of the Complement of the difference of Longtitude to 180 d. To the difference of the Versed sines of the Ark of distance sought , and of an Ark being the sum of the Complements of both Latitudes when in one Hemisphere ; Or the sum of the lesser Latitude encreased by 90d , and of the Complement of the greater Latitude when in different Hemispheres , which difference substracted from the Versed sine of the said Ark , there will remain the Versed sine of the Ark of distance sought . This Proportion is to be wrought after the same manner as we found the Suns Altitudes on all hours universally , and the difference to be measured in the Line of sines as representing the former half of a Line of Versed sines , according to the Directions given for measuring of a difference of Versed sines to the common Radius or Radius of the Quadrant . By altring the two first Tearms of the Proportion above , we may work this Proposition by positive entrance . As the Radius , To the Cosine of one of the Latitudes : So the Cosine of the other Latitude , To a fourth . Again . As the Radius , To the Versed sine , as above expressed in both parts , So is that fourth , To the difference as above expressed . An Example for finding the distance between London and Bantum in the Arch of a great Circle the same that was proposed in Page 96. Bantam Longitude 140d Latitude 5d 40′ South . London Longitude 25 , 50′ Latitude 51 32 North. difference of Longitude 114 : 10 Sum 57 12 Lay the Thread to 5d 40′ in the Limb counted from left edge and from 38d 28′ in the sines the Complement of our Latitude take the nearest distance to it , then lay the Thread to 114d 10′ in the Versed sines and entring the former extent down the Line of sines from the Center take the nearest distance to it , then laying the Thread over 57d 12′ in the Versed sine , it cuts the Limbe at 13d 15′ from the right edge , at the like Ark set down one foot of the former extent in the Line of sines , and the other will reach to the Sine of 41d 42′ , then laying the Thread over the like Ark in the Limb , it will intersect the Versed sines at 109d 18′ the Ark of distance sought to be converted into Leagues or Miles according to the number of Leagues or Miles that answer to a degree in each several Country . Thus when we have two sides with the angle comprehended to find the third side , either to half or the whole common Radius without a Line of natural Versed sines from the Center , or by the Proportions in page 93 , or a third way , which I pretermit to the great Quadrant ; and thus the Reader may perceive this small Quadrant to be many ways both Universall and particular , which are of sudden performance though tedious in expression . Three sides to find an angle . Each of the Proportions in Rectangles and Squares before delivered for the Tables , may as before suggested be reduced to single Tearms , an instance shall be given in that which finds the Square of the sine of half the Angle sought . Add the three sides of the Triangle together , and from the half sum substract each of the sides including the angle sought , then it will hold , As the sine of one of the Comprehending sides , ( rather the greater that the entrance may be Lateral . ) Is to the sine of the difference of the same side from the half sum : So is the sine of the difference of the other comprehending side , To a fourth sine , Again . As the Sine of the other comprehending side , Is to that fourth sine : So is the Radius , To half the Versed sine of the angle sought , And So is the Diameter , To the whole Versed sine . To work this on the Quadrant . Upon the first Tearm in the Line of sines being the greatest containing side , enter the extent of the second , and to the other foot lay the Thread , then from the third Tearm in the sines take the nearest distance to it , Which extent enter at the sine of the first Tearm in the second Proportion , and to the other foot lay the Thread , and it will cut the Versed sine at the angle sought . Having shewed how all Proportions may be performed upon the Quadrant : I now proceed to the rest of the Lines . Of the Line of Chords on the left edge of the Backside . I shall not at present speak any thing as to the use thereof , that is intended to be done in a Treatise of the general Scale ; the principal use of a Line of Chords is to prick off readily the quantity of any Arch of a Circle , to do which take the Chord of 60d , and draw a Circle with that Extent then any Arch being to be prict off it is to be taken out of the Line of Chords , and to be transferred into the Circle swept , this supposeth the Radius not to vary . But to do it to any Radius that is lesser take the Semidiameter of the Circle , and enter it at the Chord of 60d laying the Thread to the other foot , and the nearest distances to the Thread will be Chords to the Semidiameter assigned , and the Converse will measure the Chord of any Arch by a Parralel entrance . A Chord may be taken off though no Line of Chords be graduated . The Bead wheresoever it be set carryed from one edge of the Quadrant to the other , the Thread being extended doth describe the Quadrant of a Circle , if therefore extending the Thread down one edge of the Quadrant you set the Bead to the distance of the Radius ( or Semidiameter of the Circle swept ) from the Center , and at it set down one foot of the Compasses , and lay the Thread kept at a certainty in stretching over the Limb to any Arch , & then open the Compasses to the Distance of the Bead , you shall take out the Chord of the said Arch. A Chord may very conveniently be taken off from any Circle swept Concentrick to the Limbe , and divers such there are upon both sides of this Quadrant ; Sweep a Circle of the like Radius on Paper as that on the Quadrant , and then setting one foot to the Intersection of the Concentrick Circle with the right edge , the Thread being laid over any Arch whatsoever in the Limbe , take the distance to the Intersection of the Thread with the Concentrick Circle , and transform the said Extent into the Circle drawn upon Paper . Of the Versed Sines Augmented . These are to be used with fitted Scales thereto , to stand upon a loose Ruler for the ready and more Exact finding the Hour and Azimuth near Noon , or at other times , and shall be treated of in the use of the Diagonal Scale . Of the Line of Latitudes and Scale of Hours on the spare edges of the foreside . The use of these Scales are for the ready pricking down of any Diall that hath a Center in an Equicrural Triangle from the Substile , as shall be shewed in the Use of the great Quadrant , though the Schems be fitted to small Scales . The Scale of Hours standing on the Edge on the foreside may very well be supplyed from thence to another Radius , as shall in due Time be shewed , though it do not proceed from the Center , and therefore may be spared out of the Limbe on the Backside . The Description of the Diagonal Scale . THe particular Scales handled in Page 181 would find the hour and Azimuth in the equal Limbe without doubling the Extent , if laying the Thread over the Cosine of the Declination in the lesser sines when the hour is sought ; or over the Cosine of the Altitude when the Azimuth is sought , it be minded what Ark of the Limbe the Thread intersects , and then make the entrance of one foot of the Extent at the like Ark in the sines laying the Thread to the other foot according to nearest distance . But because these Scales are more convenient being twice as long , there is accordingly a Diagonal Scale fitted to serve for our English Region , and may be accommodated to any 5 or 6 degrees of Latitude , and placed conveniently on any Instrument for Surveigh to give the hour and Azimuth in the Limbe of the Instrument or on the frame of the plain Table . And here I am to intimate that either the hour or Azimuth Scale before described on the small Quadrant , will serve to finde both the Hour and Azimuth , as conveniently as either ; the foundation whereof is , that the same Proportion demonstrated from the Analemma that finds the hour from six being applyed to other sides of the Triangle will also find the Azimuth from the East or West , an instance whereof I give in the Use of a great Quadrant for finding the Azimuth when the Declination of the Sun or Stars exceeds the Latitude of the place . By the like parity of reason the Proportion that found the Azimuth is a general Proportion to find an angle , when the threee sides are given the Canon will be , As the Cosine of one of the including sides , Is to the Radius : So is the Cosine of the side Opposite to the angle sought , To a fourth a Sine or Secant . Again . As the second including side , Is to the Cotangent of the first including side : So when any one of the sides is greater then a Quadrant is the sum , but when all less the difference between the 4th abovesaid , and the Cosine of the second Includer , To the Cosine of the angle sought . Wee do suppose but one of the three sides given to be greater then a Quadrant , if there be any such it subtends an Obtuse angle , and both the other sides being less then Quadrants subtend Acute Angles . When the 4th Sine is less then the Cosine of the second Includer , the Angle sought is Obtuse , other ways Acute . Hence the peculiar Proportion educed for the hour will be , As the Sine of the Latitude , Is to the Radius : So is the Sine of the Altitude , To a 4th a Sine or Secant . Again . As the Cosine of the Declination , Is to the Tangent of the Latitude , So in Summer is the difference , but in Winter the sum of the sines of the Declination of the Sun or Stars , and of the 4th Sine , To the Sine of the hour from six . When the 4th Ark is less then the Declination the hour is Obtuse , when greater Acute , and in Winter always Acute : But of this Proportion I make no use it being liable in some Cases to Excursion , and will not hold backward to find the Suns Altitudes the hour being assigned . This Diagonal Scale is made after the same manner as the Hour Scale described in the small Quadrant , being but several Lines of sines the greater whereof are made equal to the Secant of the Latitude , whereto they are fitted , the Radius of which Secant is 5 Inches long . The lesser Sines continued the other way , their respective Radii are made equal to the sine of the Latitude of that greater sine whereto they are continued . The parralel Lines fitted to the respective Latitudes are not to be equidistant one from another ; but having determined the distance between the two Extream Latitudes , to which they are fitted for the the larger sine it will hold , As the difference of the Secants of the two extream Latitudes , It to the distance between the Lines fitted thereto : So is the difference of the Secants of the lesser extream Latitude , and any other intermediate Latitude , To the distance thereof from the lesser extream . And so for the lesser sine continued the other way , having placed the two Extreams under the two former Extreams , to place the imtermediate Lines , the Canon would be , As the difference of the sines of the two extream Latitudes , Is to the distance between the Lines fitted thereto : So is the difference of the sines of the lesser extream Latitude , and of any other intermediate Latitude , To the distance thereof from the lesser Extream . Having fitted the distances of the greater sine , streight Lines drawn through the two extream sines shall divide the intermediate Parralels also into Lines of sines , proper to the Latitudes to which they are fitted : Now for the lesser sines they are continued the other way at the ends of the former Parralells , the Line proper to each Latitude should be divided into a Line of sines , whose Radius should be equal to the sine of the Latitude of the other sine whereto it is fitted ; and so Lines traced through each degree to the Extreams ; but by reason of the small distance of these Lines , the difference is so exceeding small , that it may not be scrupled to draw Lines Diagonal wise from each degree of the two outward extream Sines , for being drawn true , they will not be perceived to be any other then streight Lines . Whereas these Lines by reason of the latter Proportion should not fall absolutely to be drawn , at the ends of the former Lines whereto they are fitted , and then they would not be so fit for the purpose , yet the difference being as we said , so insensible that it cannot be scaled , they are notwithstanding there placed and crossed with Diagonals drawn through each degree of the Extreams . The Vses of the Diagonal Scale . 1. To find the time of Sun rising or setting . In the Parralel proper to the Latitude take out the Suns Declination out of the lesser continued sines , and enter one foot of this extent at the Complement of the Declination in the Line of sines , and in the equal Limb the Thread being laid to the other foot will shew the time sought . In the Latitude of York , namely 54d , if the Sun have 20d of Declination Northward he rises at 4 and sets at 8 Southward he rises at 8 and sets at 4 2. To find the Hour of the Day or Night for South Declination . In the Parralel proper to the Latitude account the Declination in the lesser continued sine , and the Altitude in the greater sine , and take their distance , which extent apply as before to the Cosine of the Declination in the Line of sines on the Quadrant , and laying the Thread to the other foot according to nearest distance it shews the time sought in the equal Limbe . Thus in the Latitude of York when the Sun hath 20d of South declination , his Altitude being 5d , the hour from noon will be found 45 minutes past 8 in the morning , or 15 minutes past 3 in the afternoon feré . For North Declination . The Declination must be taken out of the lesser sine in the proper Parralel , and turned upward on the greater sine and there it shews the Altitude at six for the Sun or any Stars in the Northern Hemispere , the distance between which Point and the given Altitude must be entred as before at the Cosine of the declination , laying the thread to the other foot and it shews the hour in the Limb from six towards noon or midnight , according as the Sun or Stars Altitude was greater or lesser then its Altitude at six . So in the Latitude of York , when the Sun hath 20d of North declination if his Altitude be 40d , the hour will be 46 minutes past 8 in the morning , or 14 minutes past 3 in the afternoon . 4. The Converse of the former Proposition will be to find the Altitude of the Sun at any hour of the day , or of any Star at any hour of the night . I need not insist on this , having shewn the manner of it on the small quadrant , only for these Scales use the Limb instead of the lesser sines , for Stars the time of the night must first be turned into the Stars hour , and then the Work the same as for the Sun. 5. To find the Amplitude of ehe Sun or Stars . Take out the Declination out of the greater sine in the Parralel proper to the Latitude , and measure it on the Line of sines on the lesser Quadrant , and it shews the Amplitude sought . So in the Latitude of York 54d when the Sun hath 20d of Declination , his Amplitude will be 35d 35′ . 6. To find the Azimuth for the Sun or any Stars in the Hemisphere . For South Declination . Account the Altitude in the lesser sine continued in the proper Parralel , and the Declination in the greater sine , and take their distance enter one foot of this extent at the Cosine of the Altitude on the Quadrant , and lay the Thread to the other according to nearest distance , and in the Limbe it shews the Azimuth from East or West Southwards . So in the Latitude of York , when the Sun hath 20d of South Declination , his Altitude being 5d , the Azimuth will be found to be 44d 47′ to the Southwards of the East or West . For North Declination . Account the Altitude in the lesser sine continued , and apply it upward on the greater sine and it finds a Point thereon , from whence take the distance to the declination in the said greater sine in the Parralel proper to the Latitude of the place , and enter one foot of this Extent at the Cosine of the Altitude on the Line of sines , and the Thread being laid to the other foot according to nearest distance , shews the Azimuth in the Limbe from East or West . So in the Latitude of York , when the Sun hath 20d of North Declination , and 40d of Altitude , his Azimuth will be 23d 16′ to the Southwards of the East or West . When the Hour or Azimuth falls near Noon , for more certainty you may lay the Thread to the Complement of the Declination for the Hour , or the Complement of the Altitude for the Azimuth , in the Limbe , and enter the respective extents Parralelly between the Thread and the Sines , and find the answer in the sines . We might have fitted one Scale on the quadrant to give both the houre and Azimuth in the Equall Limb by a Lateral entrance , and have enlarged upon many more Propositions , which shall be handled in the great Quadrants . Mr Sutton was willing to add a Backside to this Scale , and therefore hath put on particular Scales of his own for giving the requisites of an upright Decliner in this Latitude , which he hath often made upon Rulers for Carpenters and other Artificers and Diallists , and whereof he was willing to afford them a Print ; whereto I have added other Scales for giving the Hour and Azimuth near Noon . On the Backside are drawn these Lines , A large Dyalling Scale of 6 hours or double Tangents , with a Line of Latitudes fitted thereto . A large Chord . A Line for the Substiles distance from the Meridian . A Line for the Stiles height . A Line for the angle of 12 and 6. A Line for the inclination of Meridians . All these Scales relate to Dyalling . An Azimuth Scale being two Lines of natural sines of the same Radius set together at O , and thence numbred with Declinations , this Scale must be made of the same sine that the hour Scale following is made of continued from O one way to 38d 28′ , and the other way to 23d 31′ or further at pleasure ; but numbred from the beginning which is at the end of that 38d 28′ the Complement of the Latitude with 10d 20′ , &c. up to 60d. The Hour Scale is no other then a Line of sines with the declinations set against the Meridian Altitudes in the Latitude of London , the Radius of which sine is equal in length to the Dyalling Scale of hours . Of the Vses of these Scales . The Line of Hours and Latitudes is general for pricking down all Dialls with Centers as will afterwards be shewed in the Use of the great Quadrant , and by help of the Scale of Hours may the Diameter of a such a Circle be graduated as is placed in on the back of the great Quadrant , and the Line of Latitudes will serve as a Chord to divide the upper Quadrant , and the Hour Scale or Line of Sines will serve as a Chord to divide a Semicircle , whose Diameter is equal to the Scale of Hours into 90 equal parts and their Subdivisions , and hereby may Proportions in sines and Tangents , or Tangents alone be wrought by Protraction , and so the necessary Arks in Dyalling found generally as is done by Mr Foster in the three last Schems of his Posthuma , this will easily be understood if the use of the Circle on this Quadrant be well apprehended . The particular Scales give the requisite Arks of upright Decliners in this Latitude by inspection , for count the plaines Declination in the Line of Chords , and a Square laid over it intersects all those Arks or to be found by applying the Declination taken out of the Chords with Compasses to every other Line . Example . So if an upright Plain decline 35d from the Meridian . The Substiles distance from the Meridian will be — 24d 30′ The Stiles height — 30 38 The Inclinations of Meridians — 41 49′ The angle of 12 and 6 — 54 10 These particular Scales also resolve some of the Cases of right angled Sphoerical Triangles , relating to the Motion of the Sun or Stars thus , Of the Line of the Stiles height . Account the Declination in the Line for the Stiles height , and against it in the Chord stands the Amplitude of the Sun or Stars from the Meridian . Example for Amplitude . So when the Sun hath 18d of Declination , his Amplitude will be 67d 13′ from the Meridian , and 29d 47′ from the Vertical . The reason hereof is because the two first fixed Tearms of the Proportion that Calculate the Stiles height are the Radius and the Co-sine of the Latitude , and the two first Tearms that Calculate the Amplitude are the Cosine of the Latitude and the Radius , and therefore must as well serve in this Case as in that . On this Stile Line may be found the Suns Altitudes on all hours , when he is in the Equinoctial by applying the hour from six taken from the Chords to the other end of the Stile Line . Of the Substiler Line . Hereby we may find the time of Sun rising and setting , take the Declination out of the Substilar Line and measure it on the Line of Chords . Example . So when the Sun hath 18 of North Declination , the Ascensional difference is 24d 9′ in time 1 hour 36½ minutes , and so much the Sun rises and sets from six . Hereby may be also found the Equinoctial Altitudes to every Azimuth . Of the Line for the Angle of 12 and 6. Hereby we may find the time when the Sun will be due East or West . Account the Complement of the Declination in this Scale , and against it in the Chords stands the hour from six . Example . So when the Sun hath 18d of North Declination , he will be East or West at 7 in the morning , or 5 in the afternon . By these Scales the requisites of an East or West Reclining or Inclining Diall in this Latitude may be found . 1. The Substiles distance from the Meridian . Account the Complement of the Reclination Inclination in the Chords , and against it in the Line for 12 and 6 stands the Complement of the angle sought . 2. For the Stiles height . Apply the Reclination in the lesser sines on the Diagonal Scale in the Parralel proper to the Latitude to the greater sine and it shewes the Ark sought . 3. For the Inclination of Meridians . This may be also found on the Diagonal Scale when the Substiles distance is not more then the Latitude , By Accounting the Substiles distance on the greater sine , and applying it to the lesser . 4. For the Angle of 12 and six . Account the Complement of the Reclination in the Chords , and against it in the Substilar Line is the Complement of the angle sought . So if an East or West Plain Recline or Incline 35d. The Substiles distance from the Meridian will be — 45d 52′ The Stiles height — 26 41 The Inclination of Meridians — 66 27 And the angle of 12 and 6 — 56 55 Of the Hour and Azimuth Scales . This Scale is fitted to find the Hour from Noon in the Versed sine : augmented , and the Proportion to be wrought by it the same as delivered in the use of the small Quadrant . As the Cosine of the Declination , Is to the Secant of the Latitude : So is the difference of the sines of the Suns proposed and Meridian Altitude , To the Versed sine of the hour from Noon . And of this one Proportion we make two by introducing the Radius . As the Radius , is to the Secant of the Latitude : So is the former distance , To a fourth . By fitting the Radius of the sines equal in length to the Secant of the Latiude ; this first Proportion is removed for the said difference of sines taken out of this fitted Scale is the 4th Proportional , the Proportion that remains to be wrought upon the Quadrant is , As the Cosine of the Declination , Is to the difference of the sines taken out of this fitted Scale : So is the Radius , To the Versed sine of the hour from Noon . By this means if in the same Proportion as we increase the length of the fitted Scale , we also increase the versed sines lying in the Limb , we may find the hour and Azimuth near noon with certainty if the Altitude be well given : These Scales in their Use presuppose the Hour and Azimuth of the Sun to be nearer the noon Meridian then 60d. Operation to find the Hour . Take the distance between the Altitude and the Declination proper to the season of the year out of the Hour Scale , and enter one foot of this Extent at the Cosine of the Declination in the Line of sines , and laying the Thread to the other foot according to nearest distance , it shews the hour from noon in the Versed sines Quadrupled . Example . When the Sun hath 23d 31′ of North Declination , and 60d of Altitude , the hour from noon will be 13d 58′ to be Converted into time . When the hour is found to be less then 40d from Noon , the former extent may be doubled and entred as before , and it shews the hour in the Versed sines Octupled . And when the hour is less then 30d from Noon , the former extent may be tripled and entred as before , and after this manner it is possible to make the whole Limb give the hour next Noon , the Versed Sine Duodecupled , lies on the other side of the Quadrant ; and in this case , an Ark must first be found in the Limb , and the Thread laid over the said Ark , counted from the other edge , will intersect the said Versed Sine at the Ark sought . To find the Suns Azimuth : TAke the distance in the Azimuth Scale , between the Altitude and the Declination , proper to the season of the year , and entring it at the Cosine of the Altitude , laying the Thread to the other foot , according to nearest distance , it will shew the Azimuth in the Versed Sines quadrupled ; or , when the Azimuth is near Noon , according to the former restrictions for the hour , the extent may be doubled , or tripled , and the answer found in the Versed Sines Octupled , or Duodecupled , as was done for the hour . Example . So when the Sun hath 23d 31′ of North Declination , his Altitude being 60d. The Azimuth will be found to be 26d 21′ from the South . By the like-reason , when we found the Hour and Azimuth in the equal Limb by the Diagonal Scale , if those extents had been doubled , the Hour and Azimuth near six , or the Vertical , might have been found in a line of Sines of 30d , put thorow the whole Limb , but that we thought needless . FINIS . THE DESCRPITION AND VSES Of a Great Universal Quadrant : With a Quarter of Stofters particular Projection upon it , Inverted . Contrived and Written by John Collins Accomptant , and Student in the MATHEMATIqUES . LONDON , Printed in the Year , 1658. The DESCRIPTION Of the Great Quadrant . IT hath been hinted before , that though the former contrivance may serve for a small Quadrant , yet there might be a better for a great one . The Description of the Fore-side . On the right edge from the Center is placed a line of Sines . On the left edge from the Center , a line of Versed Sines to 180d. The Limb , the same as in the small Quadrant . Between the Limb and the Center are placed in Circles , a Line of Versed Sines to 180d , another through the whole Limb to 90d. The Line of lesser Sines and Secants . The line of Tangents . The Quadrat and Shaddows . Above them , the Projection , with the Declinations , Days of the moneth , and Almanack . On the left edge is placed the fitted Hour , and Azimuth Scale . Within the Projection abutting against the Sines , is placed a little Scale , called The Scale of Entrance , being graduated to 62d , and is no other but a small line of Sines numbred by the Complements . At the end of the Secant is put on the Versed Sines doubled , that is , to twice the Radius of the Quadrant , and at the end of the Tangents tripled , to some few degrees , to give the Hour and Azimuth near Noon more exactly . The Description of the Back-side . On the right edge from the Center , is placed a Line of equal part being 10 inches precise , decimally subdivided . On the out-side next the edge , is placed a large Chord to 60● , equal in length to the Radius of the Line of Sines . On the left edge is placed a Line of Tangents issuing from the Center , continued to 63d 26′ , and again continued apart from 60d , to 75d The equal Limb. Within it a Quadrant of Ascensions , divided into 24 equal hours and its parts , with Stars affixed , and Letters graved , to refer to their Names . Between it and the Center is placed a Circle , whereof there is but three Quadrants graduated . The Diameter of this Circle is no other then the Dyalling Scale of 6 hours , or double Tangents divided into 90d. Two Quadrants , or the half of this Circle beneath the Diameter , is divided into 90 equal parts or degrees . The upper divided Quadrant , is called the Quadrant of Latitudes . From the extremity of the said Quadrant , and Perpendicular to the Diameter , is graduated a Line of Proportional Sines ; M Foster call it the Line Sol. Diagonal-wise , from one extremity of the Quadrant of Latitude to the other , is graduated a line of Sines ; that end numbred with ●0 d , that is next the Diameter , being of the same Radius with the Tangents . Opposite and parallel thereto from 45d of the Semicircle , to the other extremity of the Diameter , is placed a Line of Sines equal to the former . Diagonal-wise , from the beginning of the Line Sol , to the end of the Diameter , is graduated a Line of 60 Chords . From the beginning of the Diameter , but below it , towards 45d of the Semicircle , is graduated the Projection Tangent , alias , a Semi-tangent , to 90d , being of the same Radius with the Tangents . The other Quadrant of this Circle being only a void Line , there passeth through it from the Center , a Tangent of 45d , for Dyalling , divided into 3 hours , with its quarters and minutes . Below the Diameter is void space left , to graduate any Table at pleasure , and a Line of Chords may be there placed . Most of these Lines , and the Projection , have been already treated upon in the use of the small Quadrant , those that are added , shall here be spoke to . Of the Line of Versed Sines , on the left Edge , issuing from the Center . THis Line , and the uses of it , were invented by the learned Mathematician , M. Samuel Foster , of Gresham Colledge , deceased , from whom I received the uses of it , applyed to a Sector ; I shall , and have added the Proportions to be wrought upon it , and in that , and other respects , diversifie from what I received ; wherein I shall not be tedious , because there are other ways to follow , since found out by my self . The chief uses of it are , to resolve the two cases of the fourth Axiom of Spherical Trigonometry ; as , when three sides are given to find an Angle , or two sides with the Angle comprehended , to find the third side , which are the cases that find the Hour and Azimuth generally , and the Suns Altitudes on all hours . For the Hour , the learned Author thought meet to add a Zodiaque of the Suns-place annexed to it , both in the use of his Sector , as also in the use of his Scale , published since his death , entituled Posthuma Fosteri , that the Suns place being given , which for Instrumental use might be obtained , by knowing on what day of each moneth the Sun enters into any Signe , and allowing a degree for every days motion , come by it prope verum , and being sought in the annexed Zodiaque ( which is no other then two lines of 90d. Sines , each made equal to the Sine of 23d 31′ the Suns greatest Declination ) just against it stands the Suns Declination , if accounted in the Versed Sine , from 90d each way ; but this for want of room , and because the Declination is more easily given by help of the day of the moneth , I thought fit to omit , the rather , because it may also be taken from the Table of Declinations . But from hence I first observed , that if the two first terms of a Proportion were fixed , if two natural Lines proper to those terms , were fitted of an equal length , and posited together , if any third term be given , to find a fourth in the same proportion , it would be given by inspection , as standing against the third ; but if the Lines stand asunder , or a difference be the third term , application must be made from one Line to the other with Compasses , as in the same Scale there is also fitted a Line of 60 parts , equal in length to the Radius of a small Sine , serving to give the Miles in every several Latitude , answerable to one degree of Longitude . Three sides given to find an Angle , the Proportion , As the difference of the Versed Sines of the Sum , and difference of any two Sides including an Angle , Is to the Diameter , So is the difference of the Versed Sines of the third side , And of the Ark of difference between the two including Sides , To the Versed Sine of the Angle sought ; And so is the difference of the Versed Sines of the third side And of the sum of the two including sides , To the Versed Sine of the sought Angles , Complement to a Semicircle . Corollary . And seeing there is such proportion between the latter terms of the fore-going Proportion , as between the former , omitting the two first terms , it also holds , As the difference of the Versed Sines of the third side , and of the Ark of difference between the two including sides Is to the Versed Sine of the Angle sought , So is the difference of the Versed Sines of the third side , And of the sum of the two including sides , To the Versed Sine of the sought Angles , Complement to 180d. And this is the Proportion M. Foster makes use of in his Scale , page 25 and 27. to find the Hour and Azimuth by Protraction , as also in page 68. in Dyalling , when three sides are given to find an Angle , by constituting two right angled equi-angled plain Triangles , the legs whereof consist of the 4 terms of this Proportion . But in that Protraction work , the first and third terms of the Proportion are given together , with the sum of the second and fourth terms , to find out the said terms respectively . The Proportion for the Hour . As the difference of the Versed Sines of the Sum , and difference of the Complement of the Latitude , and of the Sun or Stars distance from the Elevated Pole , Is to the Diameter or Versed Sine of 180d , So is the difference of the Versed Sines of the Complement of the Altitude , and of the Ark of difference between the Complement of the Latitude , and of the Polar distance , To the Versed Sine of the Hour from Noon . And if the latter clause of the third term be the Sum of the Co-latitude and Polar distance , the Proportion will find the Versed Sine of the hour from midnight , And if the sum of any two Arks exceed a Semicircle , take its Complement to 360d , for the same Versed Sine is common to both . When the Declination is towards the Elevated Pole , the Polar distance is the Complement of it to 90d ; and when towards the Depressed Pole , the Polar distance is equal to the Sum of 90d , and of the Declination added together . Example . Let the Suns Declination be 15d 46′ North , Complement , — 74d 14′ The Complement of the Latitude , — 38 28 Sum — 112 : 42 Difference — 35 : 46 And let the Altitude be 20d , Complement — 70 : 00 Operation . Take the distance between the Versed Sines of 35d 46′ , and of 112d 42′ , and entring one foot of that extent at the end of the Versed Scale at 180d , lay the thred to the other foot , according to nearest distance , then take the distance between the Versed Sines of 35d 46′ , and 70d , and entring that extent parallelly , between the Thred and the Scale , and the other foot will rest upon the Versed Sine of 77d 32′ , the quantity of the Hour from the Meridian being either 50′ past 6 in the morning , or 10′ past 5 in the afternoon . The Reader may observe in this work , that the thred lies over a Star , by entring the first extent ; as also , that there is the same Star graduated at 35d 46′ of the Versed Sine , and this no other then the Bulls eye , having 15d 46′ of North Declination , for which Star in this Latitude , there needs be no summing or differencing of Arks , in regard the Stars declination varies not : So to find that Stars hour at any time , having any other Altitude , only lay the thred over that Star in the Quadrant , and take the distance between the Star in the Scale , and the Complement of its Altitude , and enter that extent parallelly between the Thred and the Scale , and it finds the Stars hour from the Meridian : Thus when that Star hath 39d of Altitude , its hour from the Meridian will be found to be 45d 54′ , in time , 3 hours 3½′ , which to get the true time of the night , must be turned into the Suns hour by help of the Nocturnal on the Back-side : But admitting the Suns Declination and Altitude to have been the same with the Stars , the true time of the day thus found , would have been 56½′ past 9 in the morning , or 3½′ past 3 in the afternoon ; and thus the Reader may have what Stars he pleases put on of any Declination , and for any Latitude ; and they may be put on at such a distance from the Center , that the distance from it to the Star , may be a Chord to be measured in the Limb , to give the Stars Ascensional difference , or the like conclusion : And thus the thred being once laid , and the former point found for one example to the Suns Declination , neither of them varies that day ; which is a ready general way for finding the time of the day for the Sun. To find the Semidiurnal , and Seminocturnal Arks. SUppose the Sun to have no Altitude , and the Complement of it to be 90d , and then work by the former precept , and you will find the Semidiurnal Ark from the beginning of the Line , and the Seminocturnal Ark from the end of the Line , which doubled , and turned into time , shews the length of the Day and Night , and the difference between 90d , and either of those Arks is the Ascensional difference , or time of rising and setting from 6. To find the Azimuth generally . The Proportions for this purpose have been delivered before , from which it may be observed , that there are no two terms fixed , and therefore to every Altitude , the containing sides of the Triangle , namely , the Complements both of the Altitude and Latitude must be summed and differenced , when the Proposition is to be performed on this Line solely , and the Operation will be after the same manner , as for the hour , namely , with a Parallel entrance : and this is all I shall say of the Authors general way ; and of any other that he used , I never heard of ; those ways that follow , being of my own supply . By help of this Line to work a Proportion in Sines alone , wherein the Radius leads . As the Radius Is to the Sine of any Ark , So is the Sine of any other Ark To the Sine of a fourth Ark. This fourth Sine , as I have said before , is demonstrated by M. Gellibrand , to be equal to half the difference of the Versed Sines of the Sum , and difference of the two middle terms of the Proportion . Operation . Let the Proportion be , As the Radius Is to the Sine of — 40d So is the Sine of — 27 To a fourth Sine Sum — 67 Difference — 13 Take the distance between the Versed Sines of the said sum and difference , and measure it on the Line of Sines from the Center , and it will reach to 17d , the fourth Sine sought . By help of this Line may the Divisions of the line Sol , or Proportional Sines , be graduated to any Radius less then half the Radius of the Quadrant , the Canon is , As the Versed Sine of any Ark added to a Quadrant , Is to the Radius , or length of the Line Sol , So is the Versed Sine of that Arks Complement to 90d To that length which pricked backward from the end of the Radius of the said Line , shall graduate the Arch proposed . Example . Suppose you would graduate 20 of the Line Sol , enter the Radius of the said Line upon the Versed Sine of 110d , laying the thred to the other foot ; and from the Versed Sine of 70d , take the nearest distance to the thred , which prick from the end of the Line Sol , towards the beginning , and it shall graduate the said 20d. This Line Sol is made use of by M. Foster in his Scale for Dyalling . The Line of Versed Sines was placed on the left edge of the foreside of the Quadrant , for the ready taking out the difference of the Versed Sines of any two Arks , and to measure a difference of two Versed Sines upon it , which are the chief uses I shall make of it ; whereas to Operate singly upon it , it would be more convenient for the hand to have it lie on the right edge of the Quadrant . An example for finding the Azimuth generally , by help of Versed Sines in the Limb , and of other Lines on the Quadrant . I shall rehearse the Proportion , As the Cosine of the Latitude is to the Secant of the Altitude , Or , As the Cosine of the Altitude is to the Secant of the Latitude , So is the difference of the Versed Sines of the Suns distance from the Elevated Pole , and of the Ark of difference between the Latititude and Altitude , To the Versed Sine of the Azimuth from the midnight meridian . And making the latter clause of the third term the Complement of the Sum of the Latitude and Altitude to a Semicircle , the Proportion will find the versed Sine of the Azimuth from the noon Meridian . Example . Altitude , — 51d 32′ Latitude — 34 : 32 Complement 55d 2● Difference 17 : 00 ☉ distance from elevated Pole , 66 : 29 Operation in the first Terms of the Proportion . On the Line of Versed Sines , take the distance between 17d , and 66d 29′ , and entring it twice down the line of Sines , from the Center , take the nearest distance to the thread laid over the Secant of 51d 32′ , the given Altitude , and entring one foot of this Extent at the Sine of 55d 28′ the Complement of the Latitude , lay the thred to the other foot , according to nearest distance , and in the line of Versed Sines in the Limb , it will lie over 95d , for the Suns Azimuth from the midnight meridian . And the Suns declination supposed the same , he shall have the like Azimuth from the North , in our Latitude of London , when his Altitude is 34d 32′ , for the sides of the Triangle are the same . Another Example . To find it in the versed Sine of 90d Latitude — 47d 27′ Altitude — 51 : 32 Sum — 98 : 59 Complement — 81 : 1 Polar distance — 66 : 29 Take the distance in the Line of Sines , as representing the former half of a Line of Versed Sines , between these two Arks counted towards the Center , viz. 66d 29′ , and 81d 01′ , and enter this extent twice down the Line of Sines from the Center , and take the nearest distance to the thred lying over the Secant of the Latitude 47d 27′ , then enter one foot of this extent at 51d 32′ counted from the end of the Sines towards the Center , laying the thred to the other foot , according to nearest distance , and in the Versed Sine of 90d , it shews the Azimuth to be 65d from the South in this our Northern Hemispere . Of the fitted Particular Scale , and the Line of Entrance thereto belonging . THis Scale serves to find both the Hour and Azimuth in the Latitude of London , to which it is fitted , in the equal Limb , by a Lateral or positive Entrance , it consists of two Lines of Sines . The greater is 62d of a Sine , as large as can stand upon the Quadrant , the Radius of the lesser Sine is made equal to 51d 32′ of this greater , being fitted to the Latitude : The Scale of Entrance standing within the Projection , and abutting on the Line of Sines , is no other but a portion of a Line of Sines , whose Radius is made equal to 38d 28′ of the greater Sine of the fitted Scale ; and this Scale of Entrance is numbred by its Complements up to 62d , as much as is the Suns-greatest meridian Altitude in this Latitude . The ground of this Scale is derived from the Diagonal Scale , the length whereof bears such Proportion to the Line of Sines whereto it is fitted , as the Secant of the particular Latitude doth to the Radius , which is the same that the Radius bears to the Cosine of the Latitude , and consequently , making the Line of Sines to represent the fitted Scale , the Radius of that Sine whereto it is fitted , must be equal to the Cosine of the Latitude : and so we needed no particular Scale , but this would remove the particular Scale , or Scale of Entrance , nearer the Center , and would not have been so ready as this fitted Scale ; however , hence I might educe a general method for finding the hour and Azimuth in the Limb , without Tangents or Secants . The first Work would be to proportion out a Sine to a lesser Radius , which would find the point of Entrance , the next would be to finde the Altitude , or Depression , at 6. the third would be to enter the sum , or difference of the Sines of the Altitude , or depression at 6 at the point of Entrance , and to lay the thred to the other foot ; but I shall demonstrate it from other grounds . 1. To find the time of Sun Rising , or Setting . Take the Declination from the lesser Sine , and enter it at the Declination in the Scale of Entrance , laying the thred to the other foot , according to nearest distance , and it shews the time of Rising or setting in the equal Limb. So when the Sun hath 13d of South Declination , he riseth at 8′ past 7 in the morning f●re , and sets at 52′ past 4 in the afternoon . 2. To find the true time of the day . In Summer , or Northwardly Declination , take the distance between the Altitude in the greater Sine , and the Declination in the lesser Sine . In Winter , take the Declination in the lesser Sine , and with your Compasses add it to the Altitude in the greater Sine . These extents enter at the Declination in the Scale of Entrance , and lay the thred to the other foot , according to nearest distance , and in the equal Limb , it will lye over the true time of the day . In Summer , when the Declination in the fitted Scale is above the Altitude , the hour is found from 6 towards midnight , when below it , towards Noon . Example . When the Sun hath 13d of North Declination , his Altitude being 39d 10′ will be a quarter past 9 in the morning , or 3 quarters past 2 in the afternoon ; and when he hath the same South Declination , his Altitude being 16d 14′ the time of the day will be found the same . The Converse will find the Suns Altitudes on all hours by this fitted Scale , which I shall handle the general way . 3. To find the Amplitude . Take the Declination from the greater Sine , and enter it at the beginning of the Scale of Entrance , laying the thred to the other foot , according to nearest distance , and it shews it in the Limb. When the Sun hath 13d of Declination , his Amplitude will be 21d 12′ . 4. To find the Azimuth of the Sun. In Summer ▪ take the distance between the Altitude in the lesser Sine , and the Declination in the greater . In Winter , or South Declinations , take the Declination from the greater Sine , and add it to the Altitude in the lesser Sine with your Compasses . These Extents , enter at the Altitude in the Scale of Entrance , and lay the thred to the other foot , according to nearest distance ; and in the equal Limb , it shews the Azimuth from the East or West . In Summer , when the Altitude falls below the Declination , the Azimuth is found from the East or West , Northwards ; when above it , Southwards . So when the Sun hath 13d of North Declination , his Altitude being 43d 50′ the Azimuth will be found to be 45d from East or West , Southwards ; and when he hath the same South Declination , his Altitude being 14d 50′ he shall have the same Azimuth . These Scales are fitted to give the Altitude at six , and the Vertical Altitude by Inspection . Against the Declination in the greater lesser Sine stands the Vertical Altitude or Depression , Altitude or Depression at six . When the Hour or Azimuth falls near Noon , mind against what Arch of the Line of Sines the point of Entrance falls , the thred may be laid to the like Arch in the Limb , and the respective extents entred parallelly between the Scale and the thred , and the answer found in the Line of Sines . But we have a better remedy by help of the Versed Sine of 90d put thorow the whole Limb. The joynt use of the Fitted Scale , with the Versed Sine of 90d in the Limb. IN the following Propositions , I shall make no use of the lesser Sine of the Fitted Scale . Get the Summer and Winter Meridian Altitude , by summing and differencing the Declination , and the Complement of the Latitude , which may be done with Compasses in the equal Limb , by applying the Chord of the Declination both ways from the Co-latitude . To find the Hour of the Day in Winter . Take the distance between the Meridian Altitude , and the given Altitude , out of the greater Sine of the fitted Scale , and as before , enter it at the Declination in the Scale of Entrance , laying the thred to the other foot , according to nearest distance , and in the Versed Sine of 90d it shews the hour from Noon . So if the Sun have 13d of South Declination , the Meridian Altitude is 25d 28′ , if the given Altitude be 17d 44′ the time of the day will be half an hour past 9 in the morning , or as much after 2 in the afternoon . To find the Hour of the Day in Summer . Take the distance between the Summer Meridian Altitude , and the proposed Altitude , and if this extent be less then the distance of the Declination in the Scale of Entrance from the Center , enter it at the Declination in the said Scale , and laying the thred to the other foot , it will in the Versed Sine of 90d shew the Hour from Noon . If the Sun have 23d 31′ of North Declination , his Meridian Altitude will be 61 59′ , if his given Altitude be 47d 51′ , the time of the day will be a quarter past 9 in the morning , or three quarters of an hour past 2 in the afternoon . If the Extent be larger then the distance of the point of Entrance ▪ to wit , the distance of the Declination in the Scale of Entrance from the Center , the hour must be found from midnight . In this case , with your Compasses add the Sine of the Winter Meridian Altitude , taken from the greater Sine of the fitted Scale , to the Sine of the Altitude in the said Scale , and enter the said whole extent at the point of Entrance , as before ; and in the Versed Sine of 90d , the thred will shew the hour from midnight . When the Sun hath 23 31′ of North Declination , if his Altitude be 5d 24′ , the time of the day will be half an hour past 4 in the morning , or half an hour past 7 in the evening , the Winter Meridian Altitude to this Declination being 14d 57′ . When the hour in these examples falls near Noon , the extent of the Compasses may be doubled , or tripled , and an Ark first found in the Limb , then if the thred be laid over the like Ark from the other edge , it will accordingly in the Versed Sines doubled or tripled , shew the time sought ; and the like may be done for the Azimuth . To find the Azimuth of the Sun in Winter : Get the Ark of difference between the Suns Altitude , and the Complement of the Latitude , and in the greater Sine of the fitted Scale , take the distance between the said Ark , and the Suns Declination , and enter one foot of this Extent at the Altitude in the Scale of Entrance , laying the thred to the other foot , and in the Versed Sine of 90d , it shews the Azimuth from Noon Meridian . Example . Colatitude , — 38d 28′ Altitude , — 12 : 13 Ark of Difference — 26 : 15 Declination — 13 : 00 The Azimuth to this example , will be 50d from the South . In Summer , get the Ark of difference between the Altitude , and the Complement of the Latitude , then when the Suns Altitude is the lesser of the two , take the sum , but when the greater , the difference of the Sines of the Suns Declination , and of the said Ark , and enter it at the Altitude on the Scale of Entrance , and you will find the Azimuth from the noon Meridian , as before ; but when either of those extents are larger then the distance between the point of Entrance and the Center , the Azimuth must be found from the midnight Meridian . In this case , take the difference , that is , the distance of the Sines of the Suns Declination , and of the Ark ▪ being the sum of the Altitude and Colatitude , out of the greater Sine of the fitted Scale , and enter it at the Altitude in the Scale of Entrance , laying the thred to the other foot , and in the Versed Sine it shews the Azimuth from the North. Example for finding the Azimuth from the North. Colatitude — 38d 28′ Altitude — 14 : 15 Sum — 52 : 43 Declination , — 23 : 31 The Azimuth to this example , will be found to be 70d from the North. Of the joynt use of the Diagonal Scale , with the Line of Sines on this Quadrant . If the respective extents that found the Hour and Azimuth in the Limb on the small Quadrant , be doubled , and applyed here to the Line of Sines issuing from the Center , which in this case becomes the Scale of Entrance , the Hour and Azimuth will be also found in the equal Limb of this Quadrant , for all those respective Latitudes to which the Diagonal Scale is accommodated . Of the Hour and Azimuth Scales on the Back-side thereof . THose Scales were fitted to the Versed Sines quadrupled on that small Quadrant , and consequently , are fitted to the Versed Sine of 90d , and the Line of Sines on this Quadrant , which is just double the Radius of that Quadrant . Those Scales are peculiarly fitted for the Latitude of London , and thereby we may alwaies find the Hour and Azimuth in the Versed Sine of 90 , without the trouble of summing or differencing of Arks. 1. By the Hour Scale , to find the Hour of the Day . Take the distance between the Declination , proper to the season of the year , and the Altitude , and entering one foot of that extent at the Complement of the Declination in the Sines , lay the thred to the other foot , according to nearest distance , and it shews the hour from Noon . Example . When the Sun hath 13d of North Declination , his Altitude being 47● 24′ , the Hour will be 30′ past 10 in the morning , or as much past 1 in the afternoon . In Summer , when this extent is greater then the Cosine of the Declination , and that it will be , when the Sun hath less Altitude then he hath at 6. The Declination is graduated against the Meridian Altitudes . In this case , add the Sine of the Altitude given , to the Sine of the Meridian Altitude in Winter , to that Declination , with your Compasses , and enter that whole extent at the Declination counted in the Line of Sines from 90d laying the thred to the other foot , according to nearest distance , and in the Versed Sine of 90d , it will shew the hour from midnight . Declination , — 23d 31′ North , The hour will be found either 4 in the morning , or 8 at night . Altitude — 1 : 34 The hour will be found either 4 in the morning , or 8 at night . 2. By the Azimuth Scale , to find the Azimuth of the Sun. Take the distance between the Declination proper to the season of the year , and the Altitude , and entering one foot of this extent at the Complement of the Altitude in the Lines of Sines issuing from the Center , to the other lay the thred according to nearest distance , and it shews the Azimuth from the noon Meridian in the Versed Sine of 90d. Declination — 23d 31′ North , The Azimuth hereto will be found 65d from the South . Altitude — 47 : 27 The Azimuth hereto will be found 65d from the South . In Summer , when this extent is greater then the Cosine of the Altitude , and that it will be , when the Sun hath less Altitude then he hath in the Vertical , the Azimuth must be found from the midnight Meridian . In this case , because the Azimuth Scale is not continued far enough , the sum of the Altitude and Colatitude must be gotten , and the distance taken between the said Ark and the Declination , counted in the hour-Scale as a Sine , and that extent entred at the Altitude counted from 90d in the Line of Sines , and the thred laid to the other foot , will shew the Azimuth from the North in the Versed Sine of 90d in the Limb. Colatitude , — 38● 28′ The Azimuth to this example will be 65d from the North. Altitude — 10 : 19 The Azimuth to this example will be 65d from the North. Sum — 48 : 47 The Azimuth to this example will be 65d from the North. Declination — 23 : 31 The Azimuth to this example will be 65d from the North. North. General Proportions . It now remains to be shewed , how the Hour , and Azimuth , &c. may be found generally , either in the equal Limb , or in the Versed Sine of 90d , and that without the help of Tangents or Secants , and possibly with more convenience then with them . In page 55. I have asserted , that the fourth term in any direct Proportion , bears such Proportion to the first term , as the Rectangle of the two middle terms doth to the square of the first term . And in page 105. That the Sine of any Arch bears such proportion to the Secant of the Complement of another Ark , as the Rectangle of the Sines of both those Arks , doth to the Square of the Radius . Whence it follows , That , As the Radius , Is to the Sine of one of the sides including an Angle , So is the Sine of the other containing side , To a fourth Sine . I say then , that this fourth Sine bears such Proportion to the Radius , as the Sine of one of those including sides , doth to the secant of the Complement of the other . And therefore , when three sides are given to find an Angle , it will hold , As the Radius , Is to the Sine of one of those including sides , So is the Sine of the other including side , To a fourth sine . Again , As that fourth Sine , Is to the difference of the Versed Sines of the third side , and of the Ark of difference between the two including sides , So is the Radius , To the Versed Sine of the Angle sought . And as that fourth Sine , Is to the difference of the Versed Sines of the third side , and of the sum of the two including sides , So is the Radius , To the Versed Sine of the sought Angles , Complement to 180d , or a Semicircle . Thus we are freed from a Secant in the two first terms of 3 several Proportions that find 〈…〉 Hour and Azimuth : All which I shall further confirm from the Analemma , and then proceed to Application , in the Scheme annexed . Proportions in the Analemma . UPon the Center C , draw a Circle , and let N C be the Axis of the Horizon , and E P the Axis of the World , AE C the Equator , ☉ F and Z Y two Parallels of Declination on each side the Equator , alike equidistant , S G the parallel of Altitude at 6 , and D E F the parallel of Depression at 6 ; draw a parallel of Altitude less then the Altitude at 6 V R , and another greater M N continued ; also a parallel of Depression less then the Depression at 6 W P , and another greater X Y , and there will be constituted diverse right lined , right angled Triangles , relating to the motion of the Sun or Stars , in which it will hold . As Radius is to the Cosine of the Declination , So is the Cosine of the Latitude To a fourth . Namely the difference of the Sines of the Meridian Altitude , and Altitude at 6 in Summer , equal to the Sum of the Sines of the Meridian Altitude and Depression at 6 in Winter , which is equal to the sum of the Sines of the midnight Depression and Altitude at 6 in Summer . A : B ☉ ∷ B : A ☉ . Again the same , As D : S E ∷ E : D S Again the same — G : B F ∷ B : F G. As that fourth is to the Radius , So is the Sine of the Meridian Altitude , To the Versed Sine of the Semidiurnal Ark A ☉ : ☉ B ∷ ☉ I : ☉ K. The two first terms are common to all the rest of the following Proportions . And so is the Sine of the midnight Depression , To the Versed Sine of the Seminocturnal Ark. G F : B F ∷ F L to K F. And so is the Sine of the Altitude , To the difference of the Versed Sines of the Semidiurnal Ark and Hour sought from Noon , ☉ A : B ☉ ∷ I M : K N. And so is the Sine of the Depression , To the difference of the Versed Sines of the Seminocturnal Ark , and of the hour from Midnight , F G : F B ∷ P L : K O And so is the difference of the Sines of the Suns Meridian , and given Altitude , To the Versed Sine of the hour from Noon , ☉ A : ☉ B ∷ ☉ M : ☉ N. If the Sun have Depression , So is the sum of the Sines of the Suns Meridian Altitude , and proposed Depression , To the Versed Sine of the hour from Noon , A ☉ : ☉ B ∷ ☉ Q : ☉ O. And so is the difference of the Sines of the Midnight and propose Depression , To the Versed Sine of the hour from Midnight , F G : F B ∷ P F : O F. But supposing the Sun to have Altitude , retaining still the two first terms , it holds . And so is the sum of the Sines of the Midnight Depression and given Altitude , To the Versed Sine of the hour from Midnight . F G : F B ∷ F R : F S. And so is the Sine of the Altitude , or Depression at six , To the Sine of the Ascensional difference , A ☉ : ☉ B ∷ A I : B-K. In Summer , if the Sun have Altitude , So is the difference of the Sines of the Altitude at six , and of the given Altitude , To the Sine of the hour from six , towards Noon , if the given Altitude be greater then the Altitude at six , otherwise towards Midnight . A ☉ : ☉ B ∷ A M : B N. Also A ☉ : ☉ B ∷ A T : B S. If he have Depression , So is the sum of the Sines of the Altitude at six , and the given Depression , To the Sine of the Hour from six , towards Midnight . A ☉ : ☉ B ∷ A Q : B O. In Winter , if the Sun have Altitude : So is the the sum of the Sines of his Depression at 6 , and of his given Altitude , To the Sine of the hour from 6 toward Noon , S D : S E ∷ D V : Z E. If he have Depression . So is the difference of the Sines of his Depression at 6 , and of has given Depression . To the Sine of the hour from 6 towards Noon , when the Depression is less then the Depression at 6 , otherways towards Midnight . S D : S E ∷ W D : Q E. S D : S E ∷ D X : E Y. When two terms of a Proportion happen in the common Radius , and two in a Parallel , there needs no Reduction . In Latitudes nearer the Poles then the Polar Circles , the Semidiurnal Arks , when the Declination is towards the Elevated Pole , will be more then the Diameters of their Parallels ; in that case , the difference , is the difference of the Versed Sine of the Hour , and of the fourth Proportional , found by the Proportion that finds the Semidiurnal Ark. General Proportions for the Hour . The Proportion selected for the Hour is , As the Radius , Is to the Cosine of the Latitude , So is the Cosine of the Declination , To a fourth : Namely , the difference of the Sines of the Meridian Altitude , and of the Altitude at 6. Again , 1. As that fourth , Is to the Radius , So in Summer , is the difference ; but in VVinter , the sum of the Sines of the Suns Altitude or Depression at 6 , To the Sine of the Hour from 6 towards Noon or Midnight , according as the Altitude or Depression is greater or less then the Altitude or Depression at 6. 2. And so is the difference of the Sines of the Meridian , and proposed Altitude , To the Versed Sine of the Hour from Noon : And so is the sum of the Sines of the Midnight Depression , and given Altitude , To the Versed Sine of the Hour from Midnight . 3. And so is the Sine of the Altitude , To the difference of the Versed Sines of the Semidiurnal Ark , and of the Hour sought . By the first Proportion , the hour may be found generally , either in the equal Limb , or Line of Sines . By the second Proportion , it may be found generally , either in the Versed Sines of 90d , or 180d. By the third Proportion , it may be found in the Line of Versed Sines issuing from the Center in many cases . I shall add a brief Application of all three ways . The first Work will be to find the point of Entrance . Example , For the Latitude of Nottingham , 53d. Lay the thred to the Declination , admit 20 in the Limb , counted from the left edge , and from the Latitude in the Line of Sines , counted towards the Center from 90d ; take the nearest distance to the thred , the said extent measured from the Center , will fall upon 34 25′ , and there will be the point of Entrance ; let it be recorded , or have a mark set to it . If the Suns Declination be North , the Meridian Altitude in that Latitude , will be 57d , the said extent will reach from the Sine thereof , to the Sine of the Suns Altitude , or Depression at 6 , to that Declination , namely , to 15d 51′ : which may also be found without the Meridian Altitude , by taking the distance from 20d in the Sines , to the thred laid over the Arch 53d , counted from the right Edge , and by measuring that extent from the Center , the point thus found , I call the Sine point . Thirdly , If the respective distances between the Sine point , and the Sine of the given Altitude , be taken and entred upon the point of Entrance , laying the thred to the other foot , according to nearest distance , the hour may be found all day for that Declination , when it is North in the equal Limb. Example , For the Latitude of Nottingham , to the former Declination , being North. When the Sun hath 11d 31′ 20 17 of Altitude , the Hour in each Case will be found half an hour from 6 , to the lesser Altitude beyond it , towards Midnight ; to the greater , towards Noon . And when the Altitude is 38d 19′ , the time of the day will be either half an hour past 8 in the morning , or half an hour past 3 in the Afternoon . An Example for the Latitude of Nottingham , when the Declination is as much South . Let the Altitude be 10d 6′ , In this case add the Sine thereof to the Sine of 15d 51′ , the whole extent will be equal to the Sine of 26d 39′ ; Enter this Extent upon the point of Entrance at 34d 25′ laying the thred to the other foot , according to nearest distance , and the time of the day found in the Limb , will be either half an hour past 9 in the morning , or half an hour past 2 in the afternoon . An Example for working the second Proportion . The Summer Meridian Altitude is 57d , if the given Altitude be 46d 11′ , take the distance between the Sines of these two Arks , and entring this extent upon the point of Entrance , lay the thred to the other foot , according to nearest distance , it will in the Versed Sine of 90d , shew the Hour from Noon to be 37d 30′ , that is , either half an hour past 10 in the morning , or half an hour past 1 in the Afternoon . And when the Hour falls near Noon , we may double or triple the extent of the Compasses , and find an Ark in the Limb , which if counted from the other edge , and the thred laid over it ▪ will give answer in the Versed Sines doubled or tripled accordingly . A third Example . If the Altitude were 3d 15′ , in this case the distance between it and the Meridian Altitude being greater then the distance of the point of Entrance from the Center , the hour must be found from Midnight ; add the Sine thereof to the Sine of 17d , the Winter Meridian Altitude , the whole extent will be equal to the Sine of 20 25′ ; Enter the said extent upon the point of Entrance ▪ as before , and in the Versed Sine of 90d , the hour will be found to be either half an hour past 4 in the Morning , or half an hour past 7 in the Evening . Examples for working the third Proportion . Take the Sine of 30d , and enter it upon the point of Entrance , laying the thred to the other foot , according to nearest distance , and there keep it ; then take the nearest distance to it from the Sine of 57 , the Meridian Altitude ; and the said Extent prick upon the Line of Versed Sines on the left edge , and it will reach to 118● 54′ , set a mark to it . Lastly , the nearest distance from the Sine of each respective Altitude to the thred , being pricked from the said mark , will reach to the Versed Sine of the hour from Noon , for North Declinations . So when the Sun hath 24d 48′ of Altitude , the Hour from 7 : 17 Noon will be found to be — 75d 105 A Winter Example for that Declination . The nearest distance from the Sine of 17d , the Winter Meridian Altitude , to the thred , will reach to the Versed Sine of 61d 6′ , the Complement of the former to a Semicircle , at which set a mark ; then if the Altitude were — 12d 30′ 14 : 26 the nearest distances to the thred prickt from the latter mark , would shew the hours to these Altitudes to be 2 hours 1 ½ hour from Noon This last Proportion in some cases will be inconvenient , being liable to excursion in Latitudes more Northwardly . Two sides with the Angle comprehended , to find the third side . As the Radius , Is to the Sine of one of the Includers , So is the Sine of the other Includer , To a fourth . Again , As the Radius , Is to the Versed Sine of the Angle included , So is that fourth , To the difference of the Versed Sines of the third side , and of the Ark of difference between the two including sides , And so is the Versed Sine of the Included Angles Complement to 180. To the difference of the Versed Sines of the third side , and of the sum of the two including sides . Another Proportion for finding it in Sines , elsewhere delivered . By the former Proportion , having the advantage both of lesser and greater Versed Sines , we may find the side sought , either in the line of Sines , or in the line of Versed Sines on the the left edge ▪ issuing from the Center . The Converse of the Proportion that found the Hour , will find the Suns Altitudes on all Hours . As the Radius , Is to the Cosine of the Latitude , So is the Cosine of the Declination , To a fourth Sine . Namely , The difference of the Sines of the Suns Meridian Altitude , and of his Altitude at 6 in Summer , but the sum of the Sines of his Depression at 6 , and Winter Meridian Altitude , hereby we may obtain the point of Entrance and Altitude , or Depression at 6 , as before , and let them be recorded , then it holds , As the Radius , Is to the Sine of the Hour from 6 , So is that fourth Sine , To the difference of the Sines of the Suns Altitude at 6 , and of his Altitude sought ; But in Winter , To the sum of the Sines of his Depression at 6 , and of the Altitude sought . Hereby we may find two Altitudes at a time . Lay the thred to the Hour in the Limb , and from the point of Entrance , take the nearest distance to it , the said Extent being set down at the Altitude at 6 , shall reach upward to the greater Altitude , and downward , to the lesser Altitude . Example . Admit the hour to be 5 and 7 in the morning , the Altitudes thereto for 20 North Declination for the Latitude of Nottingham , will be found to be 7d 17′ , and 24● 48′ If the Hour be more remote from 6 then the time of Rising , we may find a Winter Altitude to as much South Declination , and a Summer Altitude , to the said North Declination . Thus if the Hour be 45d from 6 , that is either 9 in the morning , or 3 in the afternoon , the nearest distance from the point of Entrance to the thred , will reach from the Sine of 15d 51′ , the Altitude at 6 upwards , to the Sine of 42d 18′ , the Summer Altitude to that Declination : But downwards , it reaches beyond the Center : In this case measure , that extent from the Center , and take the distance between the inward foot of the Compasses , and the Altitude at 6 ▪ which measured on the Sines , will be found to be 7d 17′ for the Winter Altitude to that Hour . So if the hour were 60d from 6 , that is either 10 , or 2 , the Summer Altitude would be found to be 49d 42′ , and the Winter Altitude 12d 30′ . And this may be found in the Versed Sines on the left edge , accounted as a Sine each way from the middle , if use be made of the lesser Sines , instead of the Limb , in finding the point of Entrance , as also , in laying it to the Sine of each hour from 6 , in which case the Compasses will alwaies find two Altitudes at once ; for when they fall beyond the midst of the said Line , it shews the Winter Altitudes counted from thence towards the end of the said Versed Sines . Having found the fourth Sine , which gives the point of Entrance as before , the Altitudes on all hours may be found by the Versed Sines of 90d in the Limb , the Proportion will be , As the Radius , Is to the Versed Sine of the Hour from Noon , So is the fourth abovesaid , To the difference of the Sines of the Meridian Altitude , and of the Altitude sought . But for hours beyond 6 , the Proportion will be , As the Radius , Is to the Versed Sine of the Hour from Midnight , So is the fourth abovesaid , To the sum of the Sines of the Suns Depression at Midnight ( equal to his Winter Meridian Altitude , ) and of his Altitude sought , Hereby also we may find two Altitudes at once . Operation . Lay the thred to the Versed Sine of the Hour from Noon , and from the point of Entrance at 34d 25′ , take the nearest distance to it , the said Extent shall reach from the Summer Meridian Altitude , accounted in the Sines to the Altitude sought , also from the Winter Meridian Altitude , to the Altitude sought . Example . Latitude of Nottingham is 53d , Complement — 37● Suns Declination , — 20 Sum being the Summer Meridian Altitude — 57d Difference being Winter Meridian Altitude 17 If it were required to find the Altitudes for the hours of 11 and 1 The Extents so taken out will find the Summer Altitudes to be — 55● 00′ And the Winter Altitudes to the same hours and Declination — 15● 51● 10 and 2 The Extents so taken out will find the Summer Altitudes to be — 49 , 42 And the Winter Altitudes to the same hours and Declination — 12 , 30 9 and 3 The Extents so taken out will find the Summer Altitudes to be — 42 , 18 And the Winter Altitudes to the same hours and Declination — 7 , 17 8 and 4 The Extents so taken out will find the Summer Altitudes to be — 33 , 47 And the Winter Altitudes to the same hours and Declination — 00 , 32 But for hours more remote from the Meridian then 6 , as admit for 5 in the morning , or 7 at night , which is 75d from the North Meridian ; lay the thred to the said Ark in the Versed Sine of 90d , and the distance from the point of Entrance to it , shall reach from the Sine of 57d , the Meridian Altitude , to the Sine of 24d 48′ , the Summer Altitude for the Hour 75d from Noon , and if that Extent be pricked from the Winter Meridian Altitude , it will reach beyond the Center , in which case , enter that Extent upon the Line of Sines , and take the distance between the point of limitation and 17d , which will ( being measured ) be found to be the Sine of 7d 17′ , the Altitude belonging to the hour 105 from Noon . In like manner , the Altitudes for the hours 97d 30 from Noon that is 82d 30′ from Midnight , will be 11d 31′ and for the like hours from Noon 20d 17′ 112 , 30 from Noon that is 67 , 30 from Midnight , will be 3 : 15 and for the like hours from Noon 29 : 19 In like manner , it might have been found in the Versed Sines issuing from the Center , if in finding the point of Entrance , and in laying the thred to the Versed Sine of the Hour , we make use of the lesser Sines , and of the Versed Sine of 180d in the Limb. For the Azimuth . Two of the former Proportions may be conveniently applied to other sides , for finding the Azimuth universally . As the Radius , Is to the Cosine of the Latitude , So is the Cosine of the Altitude , To a fourth Sine . Get the sum of the Altitude and Colatitude ; or , which is all one , the sum of the Latitude and Colatitude ; and if it exceeds a Quadrant , take its Complement to a Semicircle : This fourth Sine is equal to the difference of the Sines of this Compound Ark , and of another Ark to be thereby found , called the latter Ark. Then it holds , As the fourth Sine , Is to the Radius , So in Summer is the difference , but in Winter , the sum of the Sines of this latter Ark , and of the given Declination , To the Sine of the Azimuth from the Vertical . When the latter Ark is more then the Declination , the Azimuth will be found from the Vertical towards the Noon Meridian , otherwise towards the Midnight Meridian , and in winter , always towards the Noon Meridian . For such Stars as come to the Meridian between the Zenith and the elevated Pole , the fourth Ark will never exceed the Stars declination , and their Azimuth will be alwaies found from the Vertical towards the Meridian they come to , above the Horizon . Example for the Latitude of Nottingham . Complement of the Latitude is — 37d Altitude is 40● — 40 Sum — 77 Let the Declination be 20d North. To find the point of Entrance , take the nearest distance to the thred laid over 50 in the Limb , counted from right edge from the Sine of 37d , the said Extent measured from the Center , falls upon the Sine of 27d 26′ , and there will be the point of Entrance ; the said Extent prickt from 77● in the Sines , will reach to the Sine of 30 51′ , where the Sine point falls . Lastly , The distance between the Sine point , and the Sine of 20d being entred at the point of Entrance , and the thred laid to the other foot , the Azimuth will be found in the equal Limb to be 21d 48′ from the East or West Southwards , because the Sine point fell beyond the Declination . Another Example for that Latitude , the Declination being 20d South Altitude . 12d 30′ The point of Entrance will fall at the Sine of 36● The Sine point may be found without summing or differencing of Arks , by taking the nearest distance from the Sine of the Latitude , to the thred laid over the Altitude , counted in the Limb from the right edge ; which Extent being added to the Sine of 20d the Declination , the whole Extent will be equal to the Sine of 31d , this being entred on the point of Entrance , and the thred laid to the other foot , the Azimuth will be found to be 61d 14′ from the East or West Southwards . A third Example for the Latitude of London , 51d 32′ . Let it be required to find the Azimuth of the middlemost Star in the great Bears tail , Declination is 56d 45′ , let the Altitude be 44d 58′ . The nearest distance from the Sine of 38d 28′ to the thred laid over the Altitude counted from the right edge , will find the point of Entrance to be at the Sine of — 26d 6′ . The nearest distance from the Sine of 51d 32′ to the thred laid over the Altitude , counted from the right edge , need not be known , but the distance between that Extent , and the Sine of 56d 45′ , the Stars Declination being entred on the point of Entrance , will find the Azimuth of that Star , by laying the thred to the other foot , to be 40d from the East or West Northwards . Thus we find it the general way , and so it will also be found by the fitted particular Scale ; for the Hour , the point of Entrance , and Sine point , vary not till the Declination change ; but for the Azimuth , they vary to every Altitude . To find the Azimuth in the Versed Sines . As the fourth , found by the former Proportion ; namely , where the point of Entrance hapned , Is to the Radius , So is the difference of the Versed Sines of the Polar distance , and of the Ark of Difference between the Altitude and the Latitude , To the Versed Sine of the Azimuth from Midnight Meridian . This finds the Angle it self in the Sphere . And so is the difference of the Versed Sines of the Polar distance , and of the Ark of residue of the sum of the Latitude and Altitude taken from a Semicircle . To the Versed Sine of the Azimuth from Noon Meridian . This finds the Complement of the Angle in the Sphere to a Semicircle . The Proportion to find it from Midnight Meridian , the third term being express'd in Sines , will be thus . Get the sum of the Altitude and Colatitude , and when it exceeds a Quadrant , take its Complement to a Semicircle , the Ark thus found , is called the Compound Ark. Then it holds , As the fourth found before . Is to the Radius , So in Summer Declinations , is the difference , but in Winter Declinations , the sum of the Sines of the Suns or Stars declination , and of the compound Ark , To the Versed Sine of the Azimuth from the Midnight Meridian of the place . Use this Proportion alwaies for the Sun or Stars , when they come to the Meridian between the Zenith and elevated Pole. And to find it from the Noon Meridian , Get the difference between the Altitude and Colatitude , and then it holds , As the fourth Sine found before , Is to the Radius , So is the sum of the Sines of the said Ark of Difference , and of the Suns Declination , To the Versed Sine of the Azimuth from the Noon Meridian , in Summer only , when the Suns Altitude is less then the Colatitude . In all other cases , So is the difference of the said Sines , To the Versed Sine of the Azimuth , as before , from Noon Meridian . If by the former Proportion it be required to find the Azimuth in the Versed Sine of 90d , a difference of Versed Sines taken out of the Line of Versed Sines on the left edge must be doubled , and being taken out of the Line of Sines , as sometimes representing the former , sometimes the latter half of a Versed Sine , needs not be doubled . Example : Latitude of Nottingham — 53d Altitude of the Sun — 4 Ark of difference — 49 ☉ Declination 20 North , the Polar distance is — 70 The Point of entrance will fall at the Sine of 36d 54′ And the difference of the Versed Sines of 49● and 70● , equal to the distance between the Sines of 41● and 20 being entred at the Point of entrance , and the Tbread laid to the other foot will lye over 61d 30′ of the Versed Sine of 90● , and so much is the Suns Azimuth from the North. Another Example for finding it from the South when the Altitude is more then the Colatitude . Altitude — 47d Colatitude — 37 of Nottingham . difference — 10 The Point of entrance will fall at the sine of 24d 14′ found by taking the nearest extent from sine of 37d to the Thread lying over 43d of the Limb the Coaltitude . Then the distance between the sines of 10d , the Ark of difference as above , and the sine of 20 the Suns North Declination being entred at the Point of entrance , and the Thread laid to the other foot , will shew 53d 55′ in the Versed sine of 90d for the Suns Azimuth from the South . A third Example when the Altitude is less then the Colatitude in Summer . Complement Latitude 37d of Nottingham . Altitude — 34 difference — 3 The Point of entrance will fall at the sine of 29d 55′ , and the sum of the sines of 3d , and of 20d the Suns declination supposed North , is equal to the sine of 23d 13′ : Which Extent entred at 29d 55′ , the Point of entrance , and the Thread laid to the other foot according to nearest distance , it will intersect the Versed sine of 90d at the Ark of 77d 57′ , and so much is the Suns Azimuth from the South . And if there were no Versed sines in the Limbe , find an Ark of the equal Limbe , and enter the sine of the said Ark down the Line of sines from the other end , and you may obtain the Versed sine of the Ark sought . More Examples need not be insisted upon , having found the Point of entrance , the distance between the Versed sines of the Base or side subtending the angle sought , and of the Ark of difference between the two including sides , being taken out of the streight Line of Versed Lines on the left edge , and entred at the Point of entrance , laying the Thread to the other foot shews in the Versed Sine of 180d in the Limb the angle sought ; and if the said distance or Extent be doubled , and there entred it shews the angle sought in the Versed Sine of 90d , when the Angle is less then a Quadrant , when more , the distance between the Versed Sines of the Base and the sum of the Legs , will find the Complement of the angle sought to a Semicircle without doubling in the Versed Sine of 180d in the Limb , with doubling in the Versed Sine of 90d. Lastly , Three sides , viz. all less then Quadrants , or one of them greater , generally to find an angle in the equal Limb , the Proportion will be , As the Radius , Is to the Cosine of one of the including sides : So is the Cosine of the other Includer , To a fourth Sine . Again , As the Sine of one of the Includers , To the Cosecant of the other : So when any one of the sides is greater then a Quadrant is the sum , but when all less , the difference of the fourth Sine , and of the Cosine of the third side , To the Cosine of the angle sought . If any of the three sides be greater then a Quadrant , it subtends an Obtuse angle , the other angles being Acute ; But when they are all less then Quadrants , if the 4th Sine be less then the Cosine of the third side , the angle sought is Acute , if equal thereto , it is a right angle , if greater an Obtuse angle . From the Proportion that finds the Hour from six , we may educe a single Proportion applyable to the Logarithms without natural Tables for Calculating the Hour of the day to all Altitudes , By turning the third Tearm , being a difference of Sines or Versed Sines into a Rectangle , and freeing it from affection . The two first Proportions to be wrought are fixed for one Declination ; The first will be to find the Suns Altitude or Depression at six . The second will be to find half the difference of the Sines of the Suns Meridian Altitude , and Altitude sought , &c. as before defined , the Proportion to find it is , As the Secant of 60d , To the Cosine of the Declination : So is the Cosine of the Latitude , To the Sine of a fourth Arch. Lastly , To find the Hour . Get the sum and difference of half the Suns Zenith distance at the hour of six ; and of half his Zenith distance to any other proposed Altitude or Depression . Then , As the Sine of the fourth Arch , Is to the Sine of the sum : So is the Sine of the difference , To the Sine of the hour from six towards Noon or Midnight , according as the Altitude or Depression was greater or lesser then the Altitude or Depression at six . Observing that the Sine of an Arch greater then a Quadrant , is the Sine of that Arks Complement to a Semicircle . Of the Stars placed upon the Quadrant below the Projection . ALL the Stars placed upon the Projection are such as fall between the Tropicks and the Hour may be found by them with the Projection , as in the Use of the small Quadrant : Which may also be found by the fitted particular Scale , not only for Stars within the Tropicks , but for all others without , when their Altitude is less then 62d , and likewise their Azimuth may be thereby found when their Declination is not more then 62d. For other Stars without the Tropicks , they may be put on below the Projection any where in such an angle that the Thread laid over the Star shall shew an Ark in the Limb , at which in the Sines the Point of entrance will always fall ; And again , the same Star is to be graved at its Altitude or Depression at six in the Sines , and then to find the Stars hour in that Latitude whereto they are fitted , will always for Northern Stars be to take the distance in the Line of Sines between the Star and its given Altitude , and to enter that Extent at the Point of entrance , laying the Thread to the other foot according to nearest distance , and it gives the Stars hour in the equal Limb from six , which may also be found in the Sines by a Parrallel entrance , laying the Thread over the Star. Example . Let the Altitude of the last in the end of the great Bears Tail be 63d , take the distance between it and the Star which is graved at 37d 30′ of the Sines , the said Extent entred at the Sine of 23d , the Ark of the Limb the Thread intersects when it lies over the said Star , and by laying the Thread to the other foot you will find that Stars hour to be 46d 11′ from six towards Noon Meridian , if the Altitude increase , and in finding the true time of the night , the Stars hour must be always reckoned from the Meridian it was last upon ; in this Example it will be 5 minutes past 9 feré . Of the Quadrant of Ascensions on the backside , This Quadrant is divided into 24 Hours with their quarters and subdivisions , and serves to give the right Ascension of a Star , as in the small Quadrant to be cast up by the Pen. It also serves to find the true Hour of the night with Compasses . First having found the Stars hour , take the distance on the Quadrant of Ascensions in the same 12 hours between the Star and the Suns Ascension ( given by the foreside of the Quadrant ) the said Extent shall reach from the Stars hour to the true hour of the night , and the foot of the Compasses always fall upon the Quadrant ; Which Extent must be applyed the same way it was taken , the Suns foot to the Stars hour . Example . If upon the 30th of December the last in the end of the Bears Tail were found to be 9 hours 05′ past the Meridian it was last upon , the true time sought would be 16 minutes past 3 in the morning . Another Example for the Bulls Eye . Admit the Altitude of that Star be 39 d , that Stars hour as we found it by the Line of Versed Sines was 3 ho 3′ from the Meridian , if the Altitude increase , then that Stars hour from the Meridian it was last upon was 57 minutes past 8 — 8 h : 57′ If this Observation were upon the 23d of October , the Complement of the Suns Ascension would be — 9 : 30 The Ascension of that Star is — 4 : 16 The true time of the night would be forty — 10 : 43 three minutes past ten . The distance between the Star and the Suns Ascension being applyed the same way , by setting the Sun foot at the Stars hour will shew the true time sought . When the Star is past the Meridian , having the same Altitude , the Stars hour will be 3′ past 3 , and the true time sought , will be 49′ past 4 in the next morning . The Geometrical Construction of Mr Fosters Circle . THe Circle on the Back side of the Quadrant , whereof one quarter is only a void Line , is derived from M. Foster's Treatise of a Quadrant , by him published in An●o 1638. the foundation and use whereof being concealed , I shall therefore endeavour to explain it . Upon the Center H describe a circle , and draw the Diameter A C , passing through the Center , and perpendicularly thereto , upon the point C , erect a Line of Sines C I , whose Radius shall be equal to the Diameter A C , let 90d of the Sine end at I ; I say then , if from the point A , through each degree of that Line of Sines , there be streight lines drawn , intersecting the Quadrant of the circle C G , as a line from the point D doth intersect it at B the Quadrant C G , which the Author calls the upper Quadrant , or Quadrant of Latitudes shall be constituted , and if C I be continued as a Secant , by the same reason the whole Semicircle C G A may be occupied ; hence it will be necessary to educe a ground of calculation for the accurate dividing of the said Quadrant , and that will be easie ; for A C being Radius , the Sine C D doth also represent the Tangent of the Angle at A , therefore seek the natural Sine of the Ark C D in the Table of Natural Tangents , and the Ark corresponding thereto , will give the quantity of the Angle D A C , then because the point A falls in the circumference of the Circle , where an Angle is but half so much as it is at the Center , by 31 Prop. 3. Euc. double the Angle found , and from a Quadrant divided into 90 equal parts , and their subdivisions , by help of a Table so made , may the Quadrant of Latitudes be accurately divided : but the Author made his Table in page 5. without doubling , to be graduated from a Quadrant divided into 45 equal parts . Again , If upon the Center C , with a pair of Compasses , each degree of the line of Sines be transferred into the Semicircle C G A it shall divide it into 90 equal parts ; the reason whereof is plain , because the Sine of an Arch is half the chord of twice that Arch , and therefore the Sines being made to twice the Radius of this circle , shall being transferred into it , become chords of the like Arch , to divide a Semicircle into 90 equal parts . Again , upon the point A , erect a line of Tangents of the same Radius with the former Sine , which we may suppose to be infinitely continued , here we use a portion of it A E. If from the point C , the other extremity of the Diameter lines be drawn , cutting the lower Semicircle ( as a line drawn from E intersects it at F ) through each degree of the said Tangent , the said lower Semicircle shall be divided into 90 equal parts ; the reason is evident a line of Tangents from the Center shall divide a Quadrant into 90 equal parts , and because an Angle in the circumference is but half so much as it is in the Center , being transferred thither , a whole Semicircle shall be filled with no more parts . The chief use of this Circle , is to operate Proportions in Tangents alone , or in Sines and Tangents joyntly , built upon this foundation , that equiangled plain Triangles have their sides Proportional . In streight lines , it will be evident from the point D to E , draw a streight line intersecting the Diameter at L , and then it lies as C L to C D ; so is A L to A E : it is also true in a Circle , provided it be evinced , that the points B L F fall in a streight line . Hereof I have a Geometrical Demonstration , which would require more Schemes , which by reason of its length and difficulty , I thought fit at present not to insert , possibly an easier may be found hereafter : As also , an Algebraick Demonstration , by the Right Honourable , the Lord Brunkard , whereby after many Algebraick inferences it is euinced , that as L K is to K B ∷ so is L N to ● F : whence it will follow , that the points B , L , F , are in a right line . If a Ruler be laid from 45d of the Semicircle , to every degree of the Quadrant of Latitudes , it will constitute upon the Diameter , the graduations of the Line Sol , whereby Proportions in Sines might be operated without the other supply . From the same Scheme also follows the construction of the streight line of Latitudes , from the point G , at 90● of the Quadrant of Latitudes , draw a streight Line to C , and transfer each degree of the Quadrant of Latitudes with Compasses , one foot resting upon C into the said streight line , and it shall be constituted . To Calculate it . The Line of Latitudes C G bears such Proportion to C A as the Chord of 90d doth to the Diameter , which is the same that the Sine of 45d bears to the Radius , or which is all one , that the Radius bears to the Secant of 45 d , which Secant is equal to the Chord of 90 d ; from the Diagram the nature of the Line of Latitudes may be discovered . Any two Lines being drawn to make a right angle , if any Ark of the Line of Latitudes be pricked off in one of those Lines retaining a constant Hipotenusal A C , called the Line of Hours , equal to the Diameter of that Circle from whence the Line of Latitudes is constituted , if the said Hipotenusal from the Point formerly pricked off , be made the Hipotenusal to the Legs of the right angle formerly pricked off , the said Legs or sides including the right angle shall bear such Proportion one to another , as the Radius doth to the sine of the Ark so prickt off ; and this is evident from the Schem , for such Proportion as A C bears to C D , doth A B bear to B C , for the angle at A is Common to both Triangles , and the angle at B in the circumference is a right angle , and consequently the angle A C B will be equal to the angle A D C , and the Legs A C to C D bears such Proportion by construction , as the Radius doth to the Sine of an Ark , and the same Proportion doth A B bear to B C , in all cases retaining one and the same Hypotenusal A C , the Proportion therefore lies evident . As the Radius , the sine of the angle at B , To its opposite side A C , the Secant of 45d : So is the sine of the angle at A , To its opposite side B C sought . Now the quantity of the angle at A was found by seeking the natural Sine of the Ark proposed in the Table of natural Tangents ; and having found what Ark answers thereto , the Sine of the said Ark is to become the third Tearm in the Proportion . But the Cannon prescribed in the Description of the small Quadrant is more expedite then this , which Mr Sutton had from Mr Dary long since , for whom , and by whose directions he made a Quadrant with the Line Sol , and two Parrallel Lines of Sines upon it , as is here added to the backside of this Quadrant . Of the Line of Hours , alias , the Diameter or Proportional Tangent . This Scale is no other then two Lines of natural Tangents to 45 d , each set together at the Center , and from thence beginning and continued to each end of the Diameter , and from one end thereof numbred with 90 d to the other end . This Line may fitly be called a Proportional Tangent , for whersoever any Ark is assumed in it to be a Tangent , the remaining part of the Diameter is the Radius to the said Tangent . So in the former Schem , if C L be the Tangent of any Ark , the Radius thereto shall be A L. In the Schem annexed , let A B be the Radius of a Line of Tangents equal to C D , and also parralel thereto , and from the Point B to C draw the Line B C , and let it be required to divide the same into a Line of Proportional Tangents : I say , Lines drawn from the Point D to every degree of the Tangent , A B shall divide one half of it as required from the similitude of two right angled equiangled plain Triangles , which will have their sides Proportional , it will therefore hold , As C F , To C D : So F B , To B E , and the Converse , As the second Tearm C D , To the fourth B E : So is the first C F , To the third F B , and therefore C F bears such Proportion to F B , as C D doth to B E , which is the same that the Radius bears to the Tangent of the Ark proposed . If it be doubted whether the Diameter wil be a double Tangent or the Line here described such a Line , a Proportion shall be given to find by Experience or Calculation , what Line it will be ; for there is given the Radius C D , and the Tangent B E , the two first Tearms of the Proportion , with the Line C B the sum of the third and fourth Tearms , to find out the said Tearms respectively ; and it will hold by compounding the Proportion , As the sum of the first and second Tearm , Is to the second Tearm : So is the sum of the third and fourth Tearm , To the fourth Tearm , that is , As C D + B E , Is to B E : So is C F + F B = C B , To F B , see 18 Prop. of 5 of Euclid , or page 18 of the English Clavis Mathematicae , of the famous and learned Mr Oughtred . After the same manner is the Line Sol , or Proportional Sines made , that being also such a Line , that any Ark being assumed in it to be a Sine , the distance from that Ark to the other end of the Diameter , shall be the Radius thereto . A Demonstration to prove that the Line of Hours and Latitudes will jointly prick off the hour Di●tances in the same angles as if they were Calculated and prickt off by Chords . Draw the two Lines A B and C B crossing one another at right angles at B , and prick off B C the quantity of any Ark out of the line of Latitudes , and then fit in the Scale of Hours ; so that one end of it meeting with the Point C , the other may meet with the other Leg of the right angle at A , from whence draw A E parralel to B C ; So A B being become Radius , B C is the Sine of the Arch first prickt down from the line of Latitudes ; from the Point B through any Point in the line of Proportional Tangents , at L draw the Line B L E , and upon B with the Radius B A draw the Arch A D , which measureth the Angle A B E to the same Radius : I say , there will then be a Proportion wrought , and the said Arch measureth the quantity of the fourth Proportional , the Proportion will be , As the Radius , To the Sine of the Ark prickt down from the Line of Latitudes : So is any Tangent accounted in the Scale , beginning at A , To the Tangent of the fourth Proportional ; in the Schem it lies evident in the two opposite Triangles L C B and L A E , by construction equiangled and consequently their sides Proportional . Assuming A L to be the Tangent of any Ark , L C becomes the Radius , according to the prescribed construction of that Line , it then lies evident , As L C the Radius , To C B the Sine of any Ark , So is L A , the Tangent of any Ark , To A E , the Tangent of the fourth Proportional . Namely , of the Angle A B E , and therefore it pricks down the Hour-lines of a Dyal most readily and accurately : the Proportion in pricking from the Substile being alwaies , As the Radius , To the Sine of the Stiles height , So the Tangent of the Angle at the Pole , To the Tangent of the Hour-line from the Substile . Uses of the Graduated Circle . To work Proportions in Tangents alone . In any Proportion wherein the Radius is not ingredient , it is supposed to be introduced by a double Operation , and the Poportion will be , As the first term , To the second , So the Radius to a fourth . Again , As the Radius is to that fourth , So is the third Term given , To the fourth Proportional sought . In illustrating the matter , I shall make use of that Theorem● for varying of Proportions , that the Tangents of Arches , and the Tangents of their Complements are in reciprocal Proportions . As Tangent 23d , to Tangent 35d , So Tangent 55d to the Tangent of 67d. In working of this Proportion , the last term may be found to the equal Semicircle , or on the Diameter . 1. In the Semicircle . Extend the thred through 23d on the Diameter , and through 3● in the Semicircle , and where it intersects the Circle on the opposite side , there hold one end of it , then extend the other part of it over 55 in the Diameter , and in the Semicircle , it will intersect 67d for the term sought . 2. On the Diameter . Extend the thred over 23d in the Semicircle , and 35d on the Diameter , and where it intersects the void circular line on the opposite side , there hold it , then laying the other end of it over 55 d in the Semicircle , and it will cut 67 d on the Diameter . If the Radius had been one of the terms in the Proportion , the operation would have been the same , if the Tangent of 45 d had been taken in stead of it . To work Proportions in Sines and Tangents joyntly . 1. If a Sine be sought , the middle terms being of a different species . Extend the thred through the first term on the Diameter , being a Tangent , and through the Sine , being one of the middle terms , counted in the unequal Quadrant , and where it intersects the Opposite side of the Circle hold it , then extend the thred over the Tangent , being the other middle term counted on the Diameter , and it will intersect the graduated Quadrant at the Sine sought . Example . If the Proportion were as the Tangent of 14d to the Sine of 29d So is the Tangent of 20d to a Sine , the fourth Proportional would be found to be the Sine of 45d. 2. If a Tangent be sought , the middle terms being of several kinds , Extend the thred through the Sine in the upper Quadrant , being the first term , and through the Tangent on the Diameter , being one of the other middle terms , holding it at the Intersection of the Circle on the opposite side , then lay the thred to the other middle term in the upper Quadrant , and on the Diameter , it shews the Tangent sought . Example . If the Suns Amplitude and Vertical Altitude were given , the Proportion from the Analemma to find the Latitude would be , As the Sine of the Amplitude to Radius , So is the Sine of the Vertical Altitude , To the Cotangent of the Latitude Let the Amplitude be — 39d 54′ And the Suns Altitude being East or West — 30 39′ Extend the thred through 39 54′ , the Amplitude counted in the upper Quadrant , and through 45d on the Diameter , holding it at the intersection with the Circle on the Opposite side , then lay the thred over 30d 39′ , the Vertical Altitude , and it will intersect the Diameter at 38d 28′ , the Complement of the Latitude sought . But Proportions derived from the 16 cases of right angled Spherical Triangles , having the Radius ingredient , will be wrought without any motion of the thred . An Example for finding the Suns Azimuth at the Hour of 6. As the Radius to the Cosine of the Latitude , So the Tangent of the Declination , To the Tangent of the Azimuth , from the Vertical towards Midnight Meridian . Extend the thred over the Complement of the Latitude in the upper Quadrant , and over the Declination in the Semicircle , and on the Diameter . it shews the Azimuth sought . So when the Sun hath 15d of Declination , his Azimuth shall be 9d 28′ from the Vertical at the hour of 6 in our Latitude of London . Another Example to find the time when the Sun will be due East or West . Extend the thred over the Latitude in the Semicircle , and over the Declination on the Diameter , and in the Quadrant of Latitudes it shews the Ark sought . The Proportion wrought , is , As the Radius to the Cotangent of the Latitude , So is the Tangent of the Declination , To the Sine of the Hour from 6. Example . So when the Sun hath 15● of North Declination , in our Latitude of London , the Hour will be found 12d 18′ from 6 in time 49⅕′ past 6 in the morning , or before it in the afternoon . Another Example to find the Time of Sun rising . As the Cotangent of the Latitude , to Radius , So is the Tangent of the Declination , To the Sine of the Hour from 6 before or after it . Lay the thred to the Complement of the Latitude in the Semicircle , and over the Declination on the Diameter , and in the Quadrant of Latitudes , it shews the time sought in degrees , to be converted into common time , by allowing 15● to an hour , and 4′ to a degree . So in the Latitude of London , 51d 32′ when the Sun hath 15● of Declination , the ascensional difference or time of rising from 6 , will be 19d 42′ , to be converted into common time , as before . By what hath been said , it appears , that the Hour and Azimuth may be found generally by help of this Circle and Diameter . For the performance whereof , we must have recourse to the Proportions delivered in page 123. whereby we may alwaies find the two Angle adjacent to the side on which the Perpendicular falleth , which may be any side at pleasure ; for after the first Proportion wholly in Tangents is wrought , to find either of those Angles , will be agreeable to the second case of right angled Spherical Triangles , wherein there will be given the Hypotenusal , and one of the Legs , to find the adjacent Angle , only it must be suggested , that when the two sides that subtend the Angle sought , are together greater then a Semicircle , recourse must be had to the Opposite Triangle , if both those Angles are required to be found by this Trigonometry , otherwise one of them , and the third Angle may be found by those directions , by letting fall the perpendicular on another side , provided the sum of the sides subtending those Angles be not also greater then a Semicircle ; or , having first found one Angle , the rest may be found by Proportions in Sines only . IN the Triangle ☉ Z P , if it were required to find the angles at Z and ☉ , because the sum of the sides ☉ P and Z P are less then a Semicircle they might be both found by making the half of the Base ☉ Z the first Tearm in the Proportion , and then because the angles ☉ Z are of a different affection , the Perpendicular would fal without on the side ☉ Z continued towards B , as would be evinced by the Proportiod , for the fourth Ark discovered , would be found greater then the half of ☉ Z ; hence we derived the Cannon in page 124 , for finding the Azimuth ; Whereby might also be found the angle of Position at ☉ ; so if it were required to find the angles at ☉ and P , the sides ☉ Z and Z P being less then a Semicircle the Perpendicular would fall within from Z on the side ☉ P , as would also be discovered by the Proportion , for the fourth Ark would be found less then the half of ☉ P. But if it were required to find both the angles at Z and P , in this Case we must resolve the Opposite Triangle Z B P , because the sum of the sides ☉ Z and ☉ P are together greater then a Semicircle , and this being the most difficult Case , we shall make our present Example . The Proportion will be , As the Tangent of half Z P , Is to the Tangent of the half sum of Z B and P B : So is the Tangent of half their difference , To a fourth Tangent . That is , As Tangent 19d 14′ , Is to the Tangent of 86d 30′ , : So is the Tangent of 9d 30′ , To a fourth . Operation . Extend the Thread through 19 d 14′ on the Semicircle , and 9 d 30′ on the Diameter , and hold it at the Intersection on the opposite side the Semicircle , then lay the Thread to 86 d 30′ in the Semicircle , and it shews 82 d 44′ on the Diameter for the fourth Ark sought . Because this Ark is greater then the half of Z P , we may conclude that the Perpendicular B A falls without on the side Z P continued to A. fourth Ark — 82d 44′ half of Z P is — 19 14 sum — 101 58 is Z A difference — 63 30 is P A Then in the right angled Triangle B P A , right angled at , A we have P A and B P the Hypotenusal , to , find the angle B P A , equal to the angle ☉ Z P. The Proportion is As the Radius , Is to the Tangent of 13d , the Complement of B P : So is the Tangent of P A 63d 30′ , To the Cosine of the angle at P. Extend the Thread through 13 d on the Diameter , and through 63 d 30′ in the Semicircle counted from the other end , and in the upper Quadrant , it shews 27 d 35′ for the Complement of the angle sought . And letting this Example be to find the Hour and Azimuth in our Latitude of London , so much is the hour from six in Winter when the Sun hath 13 d of South Declination , and 6 d of Altitude , in time 1 ho 50⅓ minutes past six in the morning , or as much before it in the afternoon . To find the Azimuth . Again , in the Triangle Z A B right angled at A , there is given the Leg or Side Z A 101 d 58′ , and the Hipotenusal Z B 96 d , to find the angle B Z P ; here noting that the Cosine or Cotangent of an Ark greater then a Quadrant is the Sine or Tangent of that Arks excess above 90 d , and the Sine or Tangent of an Ark greater then a Quadrant , the Sine or Tangent of that Arks Complement to 180 d , it will hold , As the Radius , To the Tangent of 6d : So is the Tangent 78d 2′ , To the Sine of 29 d 44′ , found by extending the Thread through 78 d 2′ on the Semicircle , counted from the other end , alias , in the small figures , and in the Quadaant it will intersect 29 d 44′ ; now by the second Case of right angled Sphoerical Triangles , the angle A Z B will be Acute , wherefore the angle ☉ Z B is 119 d 44′ the Suns Azimuth from the North , the Complement being 60 d 16′ is the angle A Z B , and so much is the Azimuth from the South . To work Proportions in Sines alone . THat this Circle might be capacitated to try any Case of Sphoerical Triangles , there are added Lines to it , namely , the Line Sol falling perpendicularly on the Diameter from the end of the Quadrant of Latitudes , whereto belongs the two Parrallel Lines of Sines in the opposite Quadrants , the upermost being extended cross the Quadrant of Latitudes . The Proportion not having the Radius ingredient , and being of the greater to the less . Account the first Tearm in the line Sol , and the second in the upper Sine extending the Thread through them , and where it intersects the opposite Parrallel hold it ; then lay the Thread to the third Tearm in the line Sol , and it will intersect the fourth Proportional on the upper Parrallel . As the Sine of 30d , To the sine of any Arch : So is the Cosine of that Arch , To the sine of the double Arch and the Converse . By trying this Canon , the use of these Lines will be suddenly attained . Example . As the sine of 30d , To the sine of 20d : So is the sine of 70d , To the sine of 40d. But if it be of the less to the greater , the answer must be found on the Line Sol. Account the first Tearm on the upper Sine , and the second in the Line Sol , and hold the Thread at the Intersection of the opposite Parrallel , then lay the Thread to the third Tearm on the uper Parrallel , and on the line Sol it will intersect the fourth Proportional if it be less then the Radius . But Proportions having the Radius ingredient , will be wrought without any Motion of the Thread . As the Cosine of the Latitude , To Radius : So is the sine of the Declination , To the sine of the Amplitude . So in our Latitude of London , when the Declination is 20d 12′ the Amplitude will be found to be 33d 42′ . Extend the Thread through 38 d 28′ on the line Sol. and through the Declination in the upper Sine , and it will intersect the opposite Parrallel Sine at 33 d 42′ , the Amplitude sought . The use of the Semi-Tangent and Chords are passed by at present . The line Sol is of use in Dyalling , as in Mr Fosters Posthuma , page 70 and 71 , where it is required to divide a Circle into 12 equal parts for the hours , and each part into 4 subdivisions for the quarters , and into such parts may the equal Semicircle be divided ; that if it were required to divide a Circle of like Radius into such parts , it might be readily done by this . Of the Line of Hours on the right edge of the foreside of the Quadrant . This is the very same Scale that is in the Diameter on the Backside , only there it was divided into degrees , and here into time , and placed on the outermost edge ; there needs no line of Latitudes be fitted thereto , for those Extents may be taken off as Chords from the Quadrant of Latitudes , by help of these Scales thus placed on the outward edges of the Quadrants may the hour-lines of Dyals be prickt down without Compasses . To Draw a Horizontal Dyal . FIrst draw the line C E , for the Hour-line of 12 , and cross it with the Perpendicular A B , then out of a Scale or Quadrant of Latitudes set of C B and C A , each equal to the Stiles height , or Latitude of the place , then place the Scale of 6 hours on the edge of the Quadrant , whereto the Line of Latitudes was fitted , one extremity of it at A , and move the Quadrant about , till the other end or extremity of it will meet with the Meridian line C E ; then in regard the said Scale of Hours stands on the very brink or outward most edge of the Quadrant , with a Pin , Pen , or the end of a black-lead pen , make marks or points upon the Paper or Dyal against each hour ( and the like for the quarters , and other lesser parts ) of the graduated Scale , and from those marks draw lines into the Center , and they shall be the hour-lines required , without drawing any other lines on the Plain , the Scale of Hours on the Quadrant is here represented by the lines A E , and E B , the hour lines above the Center , are drawn by continuing them out through the Center . And those that have Paper prints of this line , may make them serve for this purpose , without pricking down the hour points by Compasses , by doubling the paper at the very edge or extremity of the Scale of Hours . Otherwise to prick down the said Dial without the Line of Latitudes and Scale of hours in a right angled Parallellogram . Having drawn C E the Meridian line , and crossed it with the perpendicular C A B , and determining C E to be the Radius of any length , take out the Sine of the Latitude to the same Radius , and prick it from C to A and B , and setting one foot at E , with the said Extent sweep the touch of an Arch at D and F , then take the length of the Radius C E , and setting down one foot at B , sweep the touch of an Ark at D , intersecting the former , also setting down the Compasses at A , make the like Arch at F , and through the points of Intersection , draw the streight lines A F , B D , and F E D , and they will make a right angled Parallellogram , the sides whereof will be Tangent lines . To draw the Hour-lines : Make E F , or E D Radius , and proportion out the Tangents of 15d and prick them down from E to 1 and 11 and draw lines 30 and prick them down from E to 2 and 10 and draw lines through the points thus found , and through the points F and D , and there will be 3 hours drawn on each side the Meridian line . Again , make A F or B D Radius , and proportion out the Tangent of 15d , and prick it down from A to 5 , and from B to 7. Also proportion out the Tangent of 30d , and prick it down from A to 4 , and from B to 8 , and draw lines into the Center , and so the Hour-lines are finished , and for those that fall above the 6 of clock line , they are only the opposite hours continued , after the like manner are the halfs and quarters to be prickt down . Lastly , By chords prick off the Stiles height equal to the Latitude of the place , and let it be placed to its due elevation over the Meridian line . Of Vpright Decliners . DIvers Arks for such plains are to be calculated , and may be found on the Circle before described . 1. The Substiles distance from the Meridian . By the Substilar line is meant , a line over which the Stile or cock of the Dyal directly hangeth in its nearest distance from the Plain , by some termed the line of deflexion , and is the Ark of the plain between the Meridian of the Plain , and the Meridian of the place . The distance thereof from the Hour-line of 12 , is to be found by this Proportion . As the Radius , To the Sine of the Plains Declination , So the Cotangent of the Latitude , To the Tangent of the Substile from the Meridian . 2. For the Angle of 12 and 6. An Ark used when the Hour-lines are pricked down from the Meridian line in a Triangle or Parallellogram , ( and not from the Substile , ) without collecting Angles at the Pole. As the Radius , Is to the Sine of the Plains Declination , So is the Tangent of the Latitude , To the Tangent of an Ark , the Complement whereof is the Angle of 12 and 6. 3. Inclination of Meridians . Is an Ark of the Equinoctial , between the Meridian of the plain , and the Meridian of the place , or it is an Angle or space of time elapsed between the passage of the shaddow of the Stile from the Substilar line into the Meridian line , by some termed the Plains difference of Longitude ; and not improperly , for it shews in what Longitude from the Meridian where the Plain is ; the said Plain would become a Horizontal Dyal , and the Stiles height shews the Latitude , this Ark is used in calculating hour distances by the Tables and in pricking down Dyals by the Line of Latitudes , and hours from the Substile . As the Radius , Is to the Sine of the Latitude , So the Cotangent of the Plains Declination , To the Cotangent of the Inclination of Meridians . Or , As the Sine of the Latitude to Radius , So is the Tangent of the Plains Declination , To the Tangent of Inclination of Meridians . 4. The Stiles height above the Substile . As the Radius , Is to the Cosine of the Latitude , So is the Cosine of the Plains Declination , To the Sine of the Stiles height . Or the Substiles distance being known , As the Radius , To the Sine of the Substiles distance from the Meridian , So is the Cotangent of the Declination , To the Tangent of the Stiles height . Or , The Inclination of Meridians being known . As the Radius , To the Cosine of the Inclination of Meridians , So is the Cotangent of the Latitude , To the Tangent of the Stiles height . 5. Lastly , For the distances of the Hour-lines from the Substilar Line . As the Radius , Is to the Sine of the Stiles height above the Plain , So is the Tangent of the Angle at the Pole , To the Tangent of the Hours distance from the Substilar Line . By the Angle at the Pole , is meant the Ark of difference between the Ark called the Inclination of Meridians , and the distance of any hour from the Meridian , for all hours on the same side the Substile falls , and the sum of these two Arks for all hours on the other side the Substile . These Proportions are sufficient for all Plains to find the like Arks , without having any more , if the manner of referring Declining Reclining Inclining Plains to a new Latitude , and a new Declination in which they shall stand as upright Plains , be but well explained , for East or West Reclining Inclining Plains , their new Latitude is the Complement of their old Latitude , and their new Declination , is the Complement of their Reclination Inclination , which I count always from the Zenith , and upon such a supposition , taking their new Latitude and Declination , those that will try , shall find that these Proportions will calculate all the Arks necessary to such Dials . So if an Upright Plain decline 25d in our Latitude of London from the Meridian . The Substiles distance from the Meridian is — 18d 34′ The Angle of 12 and 6 is — 62 : 00 The Inclination of Meridians is — 30 : 47 The Stiles height is — 34 : 19 To Delineate the same Dial from the Substile by the Line of Latitudes , and Scale of hours in an Equicrutal Triangle . To Draw an Vpright Decliner . An Vpright South Plain for the Latitude of London , Declining 25d Eastwards . TO prick down this Dial by the line of Latitudes , and Scale of Hours in an Isoceles Triangle . Draw C 12 the Meridian Line perpendicular to the Horizontal line of the Plain , and with a line of Chords , make the Angle F C 12 , equal to the Substiles distance from the Meridian , and draw the line F C for the Substile ; Draw the line B A perpendicular thereto , and passing through the Center at C , and out of the line of Latitudes on the other Quadrants , or out of the Quadrant of Latitudes on this Quadrant , set off B C and C A each equal to the Stiles height , then fit in the Scale of 6 hours , proper to those Latitudes , so that one Extremity meeting at A , the other may meet with the Substilar line at F. Then get the difference between 30d 47′ , the inclination of Meridians , and 30d the next hours distance lesser then the said Ark , the difference is 47′ in time , 3′ nearest then fitting in the Scale of hours as was prescribed . Count upon the said Scale , Hour . Min.   0 3 from F to 10 1 3 11 2 3 12 3 3 1 4 3 2 5 3 3 And make points at the terminations with a pin or pen , & draw lines from those points into the Center at C , & they shall be the true hour-lines required on this side the Substile . Again , Fitting in the Scale of Hours from B to F , count from that end at B the former Arks of time . Ho Min   00 , 03 from B to 4 1 , 3 5 2 , 3 6 3 , 3 7 4 , 3 8 5 , 3 9 And make Points at the Terminations , through which draw Lines into the Center , and they shall be the hour Lines required on the other side the Substile . The like must be done for the halfs and quarters , getting the difference between the half hour next lesser ( in this Example 22d 30′ ) under the Ark called the inclination of Meridians , the difference is 1d 17′ in time 33′ nearest to be continually augmented an hour at a time , and so prickt off as before was done for the whole hours . By three facil Proportions , may be found the Stiles height , the Inclination of Meridians , and the Substiles distance from the Plains perpendicular , for all Plains Declining , Reclining , or Inclining , which are sufficient to prick off the Dyal after the manner here described , which must be referred to another place . If the Scale of hours reach above the Plain , as at B , so that B C cannot be pricked down , then may an Angle be prickt off with Chords on the upper side the Substile , equal to the Angle F C A , on the under side , and thereby the Scale of hours laid in its true situation , having first found the point F on the under side . To prick down the former Dyal in a Rectangular ☉ blong , or long square Figure from the Substile . Having set off the Substilar F C , assume any distance in it , as at F to be the Radius , and through the fame at right Angles , draw the line E F D , then having made F C any distance Radius , take out the Sine of the Stiles height to the same Radius , and entring it at the end of the Scale of three hours , make it the Radius of a Tangent , and proportion out Tangents to 3′ and set them off from F to 10 1 hour 3 and set them off from F to G 2 3 and set them off from F to H Again , Take out the Tangents of the Complement of the first Ark , increasing it each time by the augmentation of an hour , namely 57′ and prick them from F to I and from the points 1 ho. 57 and prick them from F to K and from the points 2 57 and prick them from F to E and from the points thus found , draw lines into the Center . Then for the other sides of the Square , make C F the Radius of the Dyalling Tangent of 3 hours , and proportion out Tangents to the former Arks , namely , 3′ and prick them from B to P Also to the latter Arks , 57′ and prick them from A to — N 1 ho. 3 and prick them from B to O Also to the latter Arks. 1 h. 57 and prick them from A to — M 2 3 and prick them from B to L Also to the latter Arks. 2 57 and prick them from A to — D and draw lines from these terminations into the Center , and the Hour-lines are finished ; after the same manner must the halfs and quarters be finished . And how this trouble in Proportioning out the Tangents may be shunned without drawing any lines on the Plain , but the hour-lines , may be spoke to hereafter , whereby this way of Dyalling , and those that follow , will be rendred more commodious . Lastly , the Stile may be prickt off with Chords , or take B C , and setting one foot in F , with that Extent sweep the touch of an occult Arch , and from C , draw a line just touching the outward extremity of the said Arch , and it shall prick off the Angle of the Stiles height above the Substile . To prick off the former Dyal in an Oblique Parallellogram , or Scalenon alias unequal sided Triangle from the Meridian . First , In an Oblique Parallellogram . DRaw CE the Meridian line and with 60d of a line of Chords , draw the prickt Arch , and therein from K , contrary to the Coast of Declination , prick off 62d , the angle of 12 and 6 , and draw the line C D for the said hour line continued on the other side the Center , and out of a line of Sines , make C E equall to 65d the Complement of the Declination ; then take out the sine of 38d 28′ the Complement of the Latitude , and enter it in the line D C , so that one foot resting at D , the other turned about , may but just touch the Meridian line , the point D being thus found , make C F equall to C D , and with the sides C F and C E make the Parallellogram D G F H , namely , F H and G D equal to C E : and E G and E H equal to D C. And where these distances ( sweeping occult arches therewith ) intersect will find the points H and G limiting the Angles of the Parallellogram . Then making E H or C D Radius , proportion out the Tangents of 15d and prick them down from E to 1 and 11 and 30 and prick them down from E to 2 and 10 and draw lines into the Center through those points , and the angular points of the Parallellogram at H and G , and there will be 6 hours drawn , besides the Meridian line or hour line of 12. Then making D G Radius , proportion out the Tangent of 15d , and prick it down from D upwards to 5 , and downward to 7 , also proportion out the tangent of 30d and prick it from D to 8 , and from F to 4 , and draw lines into the Center , and so the hour lines are finished ; after the same manner are the halfs and quarters to be proportioned out and pricked down : and if this Work is to be done upon the Plain it selfe , the Parallel F H will excur above the plain , in that case , because the Parallel distance of F H from the Meridian , is equal to the parallel distance of D G the space G. 8. may be set from H to 4 , and so all the hour lines prickt down . To prick down this Dyal in a Scalenon , or unequal sided triangle from the Meridian , from E to D draw the streight line D E , and from the same point draw another to F , and each of them ( the former hour lines being first drawn ) shall thereby be divided into a line of double tangents , or scale of 6 hours , such a one as is in the Diameter of the Circle on this quadrant , or on the right edge of the foreside ; and therefore by helpe of either of them lines , if it were required to prick down the Dyal , it might be done by Proportioning them out , take the extent D E , and prick it from one extremity of the Diameter in the Semicircle on the quadrant , and from the point of Termination draw a line with black Lead to the other extremity , ( which will easily rub out again either with bread or leather parings ) and take the nearest distance from 15 of the Diameter to the said line , and the said extents 30 of the Diameter to the said line , and the said extents 45 of the Diameter to the said line , and the said extents shall reach from , E to 11 and from D to 7 shall reach from , E to 10 and from D to 8 shall reach from , E to 9 and from D to 9 and the like must be done for the line E F , entring that in the Semicirle as before ; or without drawing lines on the quadrant , if a hole be drilled at one end of the Diameter , and a thred fitted into it , lay the thred over the point in the Diameter , and take the nearest distances thereto . Lastly , from a line of Chords , prick off the substilar line , and the stiles height as we before found it . This way of Dyalling in a Parallellogram , was first invented by John Ferrereus a Spaniard , long since , and afterwards largely handled by Clavius , who demonstrates it , and shews how to fit it into all plains whatsoever , albeit they decline , recline , or incline , without referring them to a new Latitude ; the Triangular way is also built upon the same Demonstration , and is already published by Mr Foster in his Posthuma , for it is no other then Dyalling in a Parallellogram , if the Meridian line C E be continued upwards , and C E set off upwards , and lines drawn from the point , so found to D and E , shall constitute a Parallellogram . An Advertisement about observing of Altitudes . IMagine a line drawn from the beginning of the line Sol , to the end of the Diameter , and therein suppose a pair of sights placed with a thred and bullet hanging from the begining of the said line , as from a Center ; I say the line wherein the sights are placed , makes a right angle with the line of sines on the other side of Sol , and so may represent a quadrant , the equal Limbe whereof is either represented by the 90d of the equal Semicirle , or by the 90d of the Diameter and thereby an Altitude may be taken . Now to make an Isoceles equicrural , or equal legged triangle made of three streight Rulers , the longest whereof will be the Base or Hipotenusal line ; thus to serve for a quadrant to take Altitudes withal , will be much cheaper , and more certain in Wood , then the great Arched wooden framed quadrants . Moreover , the said Diameter line supplies all the uses of the Limbe , from it may be taken off Sines , Tangents , or Secants , as was done from the Limbe ; and therein the Hour and Azimuth , found generally by helpe of the line of Sines on the left edge , as is largely shewed in the uses of this quadrant , besides its uses in Dyalling , onely when such an Instrument is made apart , it will be more convenient to have the line of Sines to be set on the right edge , and the Diameter numbred also by its Complements ; this Diameter or double Tangent , or Hipotenusal line being first divided on , all the other lines may from it , by the same Tables that serve to graduate them from the equal Limbe be likewise inscribed : and here let me put a period to the uses of this quadrant . Gloria Deo. The Description of an Universal small Pocket Quadrant . THis quadrant hath only one face . On the right edge from the Center is placed a line of sines divided into degrees and half degrees up to 60 d. afterwards into whole degrees to 80 d. On the left edge issueth a line of 10 equal parts , from the Center being precise 4 inches long , each part being divided into 10 subdivisions and each subdivision into halfs . These two lines make a right angle at the Center , and between them include a Projection of the Sphere for the Latitude of London . Above the Projection are put on in quadrants of Circles a line of Declinations 4 quadrants for the dayes of the moneth , above them the names of 5 Stars with their right Ascensions graved against them , and a general Almanack . Beneath the Projection are put on in quadrants of Circles a particular sine and secant , so called , because it is particular to the Latitude of London . Below that the quadrat , and shadowes . Below that a line of Tangents to 45 d. Last of all the equal Limbe . On the left edge is placed the Dialling scale of hours 4 Inches long , outwardmost on the right edge a line of Latitudes fitted thereto . Within the line of sines close abutting thereto is placed a small scale called the scale of entrance beginning against 52 d. 35 min. of the sines numbred to 60 d. The line of sines that issueth from the Center should for a particular use have been continued longer to wit to a secant of 28 d. because this could not be admitted , the said secant is placed outward at the end of the scale of entrance towards the Limbe , and as much of the sine as was needful placed at its due distance , at the other end of the scale of entrance . Of the uses of the said quadrant . THe Almanack hath been largely spoke to in pag. 12 , and 13 , also again in the uses of the Horizontal quadrant pag. 11 , 12. The quadrat and shadowes from pag. 35 to 44. The line of Latitudes and scale of hours pag. 250. Again , from page to 262 to 274 , also the line of sines , equal parts , and Tangents , in other parts of the Booke . The use of the Projection . THis projection is only fitted for finding the hour in the limb , and not the Azimuth , all the Circular lines on it are parallels of Altitude or Depression except the Ecliptick and Horizon , the Ecliptick , is known by the Characters of the signes , and the Horizon lyeth beneath it , being numbred with 10 , 20 , 30 , 40. The parallels of Altitude are the Winter parallels of Stofler's Astrolable , and are numbred from the Horizon upwards towards the Center , the parallels of Depression which supply the use of Stoflers Summer Altitudes are numbred downwards from the top of the Projection towards the limb . To find the time of Sun rising , and his Amplitude . LAy the thread over the day of the moneth , and set the Bead to the Ecliptick , then carry the thread and Bead to the Horizon , and the thread in the limb , shewes the time of rising , and the Bead on the Horizon the quantity of the Amplitude . Example . So on the second of August the Suns Declination being 15 d ▪ his Amplitude will be 24 d. 35 min. and the time of rising 41′ past 4 in the morning . To find the Hour of the Day . HAving taken the Suns Altitude and rectified the Bead as before shewed , if the Sun have South Declination bring the Bead to that parallel of Altitude on which the Suns height was observed , amongst those parallels that are numbred upward towards the Center , and the thread in the limb sheweth the time of the day . Eample . So when the Sun hath 15 d. of South Declination , as about 28th of January , if his Altitude be 15 deg . the time of the day will be 39 m. past 2 in the afternoon , or 21 m. past 9 in the morning . But in the Summer half year bring the Bead to lye on those parallels that are numbred downwards to the limb , and the thread sheweth therein the time of the day sought . Example . If on the second of August his Declination being 15 d. his Altitude were 40 d. the true time of the day would be 8 m. past 9 in the morning , or 52 m. past 2 in the afternoon . If the Bead will not meet with the Altitude given amongst those parallels that run donwnwards towards the right edge , then it must be brought to those parallels that lye below the Horizon downward towards the left edge , and the thread in the Limb shewes the time of the day before six in the morning or after it in the evening in Summer . Example . When the Sun hath 15 d. of North Declination as on the second of August if his Altitude be 5 d. the time of the Day will be 44 m. before 6 in the morning , or after it in the evening . Of the general lines on this quadrant . THe line of sines on the right edge is general for finding either the hour or the Azimuth in the equal limb , or in the said line of sines , as I have largely shewed in page 231 for finding the hour , also for finding the Suns Altitudes on all hours , as in page 234 , for finding the Azimuth from page 237 to pag. 2.9 . Though this quadrant hath neither secants nor versed sines as the rest have , yet both may be easily supplyed , let it be required to work this Proportion . As the Co-sine of the declination , Is to the secant of the Latitude , So is the difference of the sines of the Suns Meridian and given Altitude , To the versed sine of the hour from noon , before or after six the hour may be found from midnight by the proportions in page 230. Let the Radius of the sines be assumed to represent the secant of the Latitude , the Radius to that secant will be the cosine of the Latitude , then lay the thread to the complement of the Declination in the limb counted from right edge , and take the nearest distance to it . I say that extent shall be the cosine of the Declination to the Radius of the Secant enter this at 90 d. of the line of sines laying the thread to the other foot according to nearest distance , then in the sines take the distance between the Meridian Altitude and the given Altitude , and enter that extent so upon the sines that one foot resting thereon , the other turned about may just touch the thread the distance between the resting foot and the Center is equal to the versed sine of the ark sought and being measured from the end of the line of sines towards the Center shewes the ark sought . Example . When the Suns Declination is 15 d. North if his Altitude were 35 d. 21 m. the time of the day would be found 45 d. from noon , that is 9 in the morning or 3 in the afternoon . Of finding the Azimuth generally . THough this may be found either by the sines alone in the equal limbe as before mentioned , or by versed sines as was instanced for the hour , see also page 239 , 240 , 241 , yet where the Sun hath vertical Altitude or Depression , as in places without the Tropicks towards either of the Poles , it may be found most easily in the equal limb by the joynt help of sines and tangents by the proportions in page 175. First , find the vertical Altitude as is shewed in page 174. Then for Latitudes under 45 d. Enter in Summer Declinations the difference , but in Winter Declinations the sum , of the sines of the vertical Altitude , and of the proposed Altitude once done the line of sines from the Center , and laying the thread over the Tangent of the Latitude take the nearest distance to it , then enter that Extent at the complement of the Altitude in the Sines , and lay the thread to the other foot , and in the limb it shewes the Azimuth from the East or West . Example . For the Latitude of Rome to witt 42 d. If the Sun have 15 d. of North Declination his vertical Altitude is 22 d. 45 m. If his given Altitude ●e 40 d. the Azimuth of the Sun will be 17 d. 33 m. to the Southward of the West . If his declination were as much South and his proposed Altitude 18 d. his Azimuth would be 41 d. 10 m. to the Southwards of the East or West . For Latitudes above 45. If we assume the Rad. of the quadrant to be the tangent of the Latit . the Rad. to that Tang. shall be the co-tangent of the Latit . wherefore lay the thread to the complement of the Latitude in the line of Tangents in the limb , and from the complement of the Altitude in the Sines take the nearest distance to it , I say that extent shall be cosine of the Altitude to the lesser Radius which measure from the Center , and it finds the Point of entrance whereon enter the former Sum or difference of sines as before directed , and you will find the Azimuth in the equal limb . Or if you would find the answer in the sines , enter the first extent at 90 d. laying the thread to the other foot , then enter the Sum or difference of the Sines of the vertical and given Altitude , so between the scale and the thread , that one foot turned about may but just touch the thread , the other resting on the sines , and you will find the sine of the Azimuth sought . Example . For the Latitude of Edinburg 55 d. 56 m. If the Sun have 15 d. of Declination , his vertical Altitude or depression is 18 d. 14 m. the Declination being North , if his proposed Altitude were 35 d. the Azimuth of the Sun would be 28 d. to the South-wards of the East or West . But if the Declination were as much South , and the Altitude 10 d. the Azimuth thereto would be 46 d. 58 m. to the South-wards of the East or West . The first Operation also works a Proportion to witt . As the Radius Is to the cotangent of the Latitude . So is the cosine of the Altitude , To a fourth sine . I say this 4th sine beares such Proportion to the Radius as the cosine of the Altitude doth to the tangent of the Latitude , for the 4th term of every direct Proportion beares such Proportion to the first terms thereof , as the Rectangle of the two middle terms doth to the square of the first term . But as the rectangle of the co-tangent of the Latitude and of the cosine of the Altitude is to the square of the Radius , So is the cosine of the Altitude is to the tangent of the Latitude , or which is all one , So is the co-tangent of the Latitude , To the secant of the Altitude , as may be found by a common division of the rectangle , and square of the Radius by either of the terms of the said rectangle , by help of which notion I first found out the particular scales upon this quadrant . All Proportions in sines and Tangents may be resolved by the sine of 90 d. and the Tangent of 45 d. on this quadrant if what hath been now wrote , and the varying of Proportions be understood , as in page 72 to 74 it is delivered . Because the Projection is not fitted for finding the Azimuth there are added two particular scales to this quadrant , namely , the particular sine in the limb , and the scale of entrance abutting on the sines fitted for the Latitude of London . Lay the thread to the day of the moneth , and it shewes the Suns Declination in the scales proper thereto . Then count the Declination in the Limbe laying the thread thereto , and in the particular sine , it shewes the Suns Altitude or Depression being East or West . To find the Suns Azimuth . FOr North Declinations take the distance between the sines of the vertical Altitude and given Altitude , but for South Declinations adde with your compass the sine of the given Altitude to the sine of the vertical Altitude , enter the extent thus found , at the Altitude in the scale of entrance laying the thread to the other foot according to nearest distance , and in the equal limb it shewes the Azimuth sought from the East or West , or it may be found in the sines by laying the thread to that arch in the limb that the Altitude in the scale of entrance stands against in the sines , and entring the former extent paralelly between the thread and the sines . Example . So when the Sun hath 13 d. of Declination his vertical Altitude or Depression is 16 d. 42 m. If the Declination were North and his Altitude 8 d. 41 m. his Azimuth would be 10 d. to the North-wards of the East or West . But if it were South and his Altitude 12 d. 13 m. the Azimuth would be 40d to the South-wards of the East or West . By the same particular scales the hour may be also found . To find the time of Sun rising or setting . TAke the sine of the Declination , and enter it at the Declination in the Scale of entrance and it shewes the time sought in the equal lim●e from six . Example . When the hath 10 d. of Declination the Ascensional difference is 49 m. which added to , or substracted from six shewes the time of rising and setting . To find the hour of the day for South Declination . IN taking the Altitude , mind what Ark in the particular Sine the thread cut , adde the Sine of that Ark to the Sine of the Declination , and enter that extent at the Declination in the Scale of entrance laying the thread to the other foot according to nearest distance , and in the equal limb it shewes the hour from six . So if the Declination were 13 d. South and the Suns Altitude 14 d. 38 m. the thread in the particular Sine would cut 18 d. 49 m. and true time of the day would be 9 in the morning or 3 in the morning . To find the time of the day for North Declination . HAving observed what Ark the thread in taking the Altitude hung over in the Particular Sine take the distance between the Sine of the said Arke , and the Sine of the Declination and entring that extent at the Declination in the Scale of entrance the thread in the limbe shewes the hour from six . Example . If the Suns Declination were 23 d. 31 m. North , and his Altitude 39 d. the Arch in the particular Sine would be 53 d. 32 m. and the time of the day would be about 3 quarters past 3 in the afternoon , or a quarter past 8 in the morning . When the Altitude is more then the Latitude the thread will hang over a Secant in the particular Scale , this happens not till the Sun have more then 13 d. of North Declination , in this case take the distance between the Secant before the beginning of the Scale of entrance , and the Sine of the Declination at the end of the same and enter it as before . Example . The Suns declination being 23 d. 31 m. North if his Altitude were 55 d. 29 m. the thread in the particular Scale would hang over the Secant of 18 d. 11 m. and the true time of the day would be a quarter past 10 in the morning , or 3 quarters past 1 in the afternoon . The Proportions here used are expressed in Page 193. The Stars hour is to be found by the projection by rectifying the Bead to the Sar and then proceed as in finding the Suns hour , afterwards the ●ue time of the night is to be found as in page 32. ERRATA . In the Treatise of the Horizontal quadrant , pag. 43 line 6 for the 3 January read the 30th . In the Reflex Dialling pag. 5 , adde to the last line these words , As Kircher sheweth in his Ars Anaclastica . FINIS . THE DESCRIPTION AND USES OF A GENERAL QUADRANT , WITH THE HORIZONTAL PROJECTION , UPON IT INVERTED . Written and Published By JOHN COLLINS Accountant , and Student in the Mathematicks . LONDON , Printed Anno M. DC . LVIII . The Description OF THE HORIZONTAL QUADRANT . THis Denomination is attributed to it because it is derived from the Horizontal projection inverted . Of the Fore-side . On the right edge is a Line of natural Sines . On the left edge a Line of Versed-Sines . Both these Lines issue from the Center where they concurre and make a right Angle , and between them and the Circular Lines in the Limb is the Projection included , which consists of divers portions and Arkes of Circles . Of the Parallels of Declination . THese are portions of Circles that crosse the quadrant obliquely from the left edge , towards the right . To describe them . OBserve that the left edge of the quadrant is called the Meriridian Line , and that every Degree or Parallel of the Suns Declination if continued about would crosse the Meridian in two opposite points , the one below the Center towards the Limbe , and the other above , and beyond the Center of the quadrant , the distance between these two points is the Diameter of the said Parallel , and the Semidiameters would be the Center points . It will be necessary in the first place , to limit the outwardmost Parallel of Declination , which may be done in the Meridian Line at any point assumed . The distance of this assumed point from the Center in any Latitude , must represent the Tangent of a compound Arke , made by adding halfe the greatest Meridian Altitude to 45 Deg. which for London must be the Tangent of 76 Degr. And to the Radius of this Tangent must the following work be fitted . In like manner , the Semidiameters of all other Parallels that fall below the Center , are limited by pricking downe the Tangents of Arkes , framed by adding halfe the Meridian Altitude suitable to each Declination continually to 45 Degr. Now to limit the Semidiameters above or beyond the Center onely prick off the respective Tangents of half the Suns mid-night Depression from the Center the other way , retaining the former Radius , by this meanes there will be found two respective points limiting the Diameters of each Parallel , which had , the Centers will be easily found falling in the middle of each Diameter . But to doe this Arithmetically , first , find the Arke compounded of halfe the Suns meridian Altitude , and 45 Degr. as before , and to the Tangent thereof , adde the Tangent of halfe the Suns mid-night depression , observing that the Suns mid-night depression in Winter-Summer , is equal to his Meridian Altitude in Summer-Winter , his declination being alike in quantity , though in different Hemispheres , the halfe summe of these two Tangents are the respective Semidiameters sought , and being prickt in the meridian line either way from the former points limiting the Diameters , will find the Centers . Or without limiting those Points for the Diameters : first , get the Difference between the Tangents of those Arkes that limit them on either side , and the halfe summe above-said , the said difference prickt from the Center of the quadrant in the meridian line finds the respective Centers of those Parallels , the said halfe summes being the respective Semidiameters wherewith they are to be described . Of the Line or Index of Altitudes . THis is no other then a single prickt line standing next the Meridian line , or left edge of the quadrant , to which the Bead must be continually rectified , when either the houre or Azimuth is found by help of the projection . To graduate it . ADde halfe the Altitudes respectively whereto the Index is to be fitted to 45 Degr. and prick downe the Tangents of these compound arkes from the Center . Example . To graduate the Index for 40 Degr. of Altitude , the halfe thereof is 20 , which added to 45 Degr. makes 65 Degr. which taken from a Tangent to the former Radius , and prickt from the Center , gives the point where the Index is to be graduated with 40 Degrees . Hence it is evident that where the divisions of the Index begin marked ( 0 ) the distance of that point from the Center is equal to the common Radius of the Tangents . Because this quadrant ( as all natural projections ) hath a reverted taile , the graduations of the Index are continued above the Hozontal point ( 0 ) towards the Center to 30 Degr. 40′ as much as is the Sunnes greatest Vertical Altitude in this Latitude , and the graduations of the Index are set off from the Center by pricking downe the Tangents of the arkes of difference between half the proposed Altitude , and 45 Deg. thus to graduate 20 deg . of the Index the halfe thereof is 10 Degrees , which taken from 45 Degrees , the residue is 35 Degrees , the Tangent thereof prickt from the Center gives the point where the Index is to be graduated with 20 Degrees . Of the houre Circles . THese are knowne by the numbers set to them by crossing the Parallels of Declination , and by issuing from the upper part of the quadrant towards the Limbe . To describe them . LEt it be noted that they all meet in a point in the Meridian Line below the Center of the quadrant : the distance whereof from the Center is equal to the Tangent of halfe the Complement of the Latitude taken out of the common Radius , which at London will be the Tangent of 19 Deg. 14′ . The former point which may be called the Pole-point , limits their Semidiameters , to find the Centers prick off the Tangent of the Latitude and through the termination raise a line Perpendicular to the Meridian line , the distance from the Pole-point being equal to the Secant of the Latitude , must be made Radius . And the Tangents of 15 Degrees , 30 Degrees &c. prickt off on the former raised line , gives the respective Centers of the houre Circles , the distances whereof from the Pole point are the Semidiameters wherewith those houre Circles are to be drawne . Of the reverted Tail. THis needs no Rule to describe it , being made by the continuing of the parallels of Declination to the right edge of the quadrant and the houre Circles up to the Winter Tropick or parallel of Declination neerest the Center , however the quantity of it may be knowne by setting one foot of a paire of Compasses in the Center of the quadrant , and the other extend to 00 Degrees of Altitude in the Index ; an Arch with that extent swept over the quadrant as much as it cuts off will be the Reverted Taile , and so much would be the Radius . Of a Quadrant made , of this Projection not inverted . BY what hath been said it will be evident to the judicious that this inversion is no other then the continance of the extents of one quarter of the Horizontal projection . Which otherwise could not with convenience be brought upon a quadrant . Hence it may be observed that . Having assigned the Radius , a quadrant made of the Horizontal Projection without inversion , to know how big a Radius it will require when inverted the proportion will hold . AS the Radius , is to the distance of the intersection of the Aequinoctial point with the Horizon from the Center equall to the Radius of the said Projection when not inverted , in any common measure . So is the Tangent of an Arke compounded of 45 Degrees , and of half the Suns greatest Meridian Altitude . To the distance between the Center and the out-ward Tropick next the Limbe in the said known measure when inverted , whence it followes that between the Tropicks this projection cannot be inverted , but the reverted taile will be but small , and may be drawne with convenience without inversion . Of the Curved Line and Scales belonging to it . BEyond the middle of the Projection stands a Curved or bending Line , numbred from the O or cypher both wayes , one way to 60 Degrees , but divided to 62 Degrees , the other way to 20 Degr. but divided to 23 Deg. 30′ . The Invention of this Line ownes Mr. Dary for the Author thereof , the Use of it being to find the houre or Azimuth in that particular latitude whereto it is fitted by the extension of a threed over it , and the lines belonging to it . The lines belonging to it are two , the one a Line of Altitudes , and Declinations standing on the left edge of the quadrant , being no other but a line of Sines continued both wayes , from the beginning one way to 62 Degrees , the other way to 23. Degrees 30′ . The other line thereto belonging is 130 Deg. of a line of Versed Sines , which stands next without the Projection being parallel to the left edge of the quadrant . To dravv the Curve . DRaw two lines of Versed Sines , it matters not whether of the same Radius or no , nor how posited ; provided they be parallel , let each of them be numbred as a Sine both ways , from the middle at ( 0 ) and so each of them will containe two lines of Sines , to the right end of the uppermost set C , to the left end D , and to the right end of the undermost set A , and to the left end B. First , Note that there is a certaine point in the Curve where the Graduations will begin both upwards and downwards , this is called the Aequinoctial point ; to find it , lay a ruler from A to the Complement of the Latitude counted from ( 0 ) in the upper Scale towards D , and draw a line from A to it , then count it the other way towards C , viz. 38 Degrees 28′ . for the Co-latitude of London , and lay a ruler over it , and the point B , and where it intersects the line before drawn , is the Aequinoctial point to be graduated . Then to graduate the Division on each side of it , requires onely the making in effect of a Table of Meridian altitudes to every degree of Declination ( which because the Curve will also serve for the Azimuth in which case the graduations of the Curve , which in finding the houre were accounted Declinations must be accounted Altitudes ) must be continued to 62 Degrees for this Latitude , and further also if it be intended that the Curve shall find a Stars houre that hath more declination . To make this Table . GEt the Summe and difference of the Complement of the latitude and of the Degrees intended to be graduated , and if the summe exceed 90 Degrees , take its complement to 180 degrees instead of it : being thus prepared the Curve will be readily made . To graduate the under part of the Curve . Account the summe in the upper line from O towards D , and from the point A in the under line draw a line to it . Account the difference in the upper lfne when the degree proposed to be graduated is lesse then the complement of the Latitude from O towards C : but when it is more towards D , and from the point B lay a Ruler over it , and where the Ruler intersects , the line formerly drawn is the point where the degree proposed is to be graduated . Example . Let it be required to find the point where 60 deg . of the Curve is to be graduated . Arke proposed 60 deg . Co-latitude 38 : 28   98 : 28 Summe 81 : 32 Difference 21   32 Count 81 deg . 32′ in the upper line from O towards D , and from the point A draw a line to it . Count the difference 21 degrees 32′ from O towards D , because the co-latitude is lesse then the arke proposed , and lay a Ruler over it , and the point B , and where it intersects the former line is the point where 60 deg . of the Curve is to be graduated on the lower side . Another Example . Let it be proposed to graduate the same way , The arke of 30 degr . 30 degr . Co-latitude 38 : 28 Summe 68 : 28 Difference 8 : 28 Count 68 deg . 28′ from O towards D , and from the point A draw a line to it . Again in the said upper line , count 8 deg . 28′ upwards from O towards C , & from the point B lay a ruler over it , & where it intersects the line last drawn is the point where 30 d. of the curve is to be graduated . To graduate the upper part of the curve requires no other directions , the same arkes serve , if the account be but made the other way , and in accounting the summe the ruler laid over B. in the lower line instead of A , and in counting the difference over A , instead of B , neither is there any Scheme given hereof , the Practitioner need onely let the upper line be the line of altitudes on the left edge of the quadrant continued out to 90 deg . at each end , and to that end next the Center set C , and to the other end D. So likewise let that end of the Versed Scale next the right edge of the quadrant be continued to 180 deg . whereto set A , and at the other end B , and then if these directions be observed , and the same distance and position of the lines retained , it will not be difficult to constitute a Curve in all respects agreeing with that on the fore-side of the quadrant . Of the houre and Azimuth Scale on the right edge of the Quadrant . THis Scale stands outwardmost on the right edge of the quadrant , and consists of two lines , the one a line of 90 sines made equal to the cosine of the Latitude , namely , to the sine of 38 deg . 28′ , and continued the other way to 40 deg . like a Versed sine . The annexed line being the other part of this Scale , is a line of natural Tangents beginning where the former sine began , the Tangent of 38 deg . 28′ being made equal to the sine of 90 deg . this Tangent is continued each way with the sine ; towards the Limbe of the quadrant it should have been continued to 62 deg . but that could not be without excursion , wherefore it is broken off at 40 degrees , and the residue of it graduated below , and next under the Versed sine belonging to the Curve that runnes crosse the quadrant being continued but to halfe the former Radius . Of the Almanack . NExt below the former line stands the Almanack in a regular ob-long with moneths names graved on each side of it . Below the Almanack stands the quadrat● , and shadowes in two Arkes of circles terminating against 45 deg . of the Limbe , below them a line of 90 sines in a Circle equal to 51 deg . 32′ of the Limbe broken off below the streight line , and the rest continued above it . Below these are put on in Circles a line of Tangents to 60 degrees . Also a line of Secants to 60 deg . with a line of lesser sines ending against 30 deg . of the Limbe ( counted from the right edge ) where the graduations of the Secant begins . Last of all the equal Limbe . Prickt with the pricks of the quadrat . Abutting upon the line of sines , and within the Projection stands a portion of a small sine numbred with its Complements beginning against 38 deg . 28′ of the line of sines , this Scale is called the Scale of entrance . Upon the Projection are placed divers Stars , how they are inscribed shall be afterwards shewne . The description of the Back-side . Put on in quarters or Quadrants of Circles . 1 THe equal Limbe divided into degrees , as also into houres and halves , and the quarters prickt to serve for a Nocturnal . 2 A line of Equal parts . 3 A line of Superficies or Squares . 4 A line of Solids or Cubes . 5 A Tangent of 45 degrees double divided to serve for a Dyalling Tangent , and a Semitangent for projections . 6 The line Sol , aliàs a line of Proportional Sines . 7 A Tangent of 51 degrees 32′ through the whole Limbe . 8 A line of Declinations for the Sun to 23 deg . 31′ . Foure quadrants with the days of the Moneth . 9 10 11 12 13 The Suns true place , with the Charecters of the 12 Seignes . 14 The line of Segments , with a Chord before they begin . 15 The line of Metals and Equated bodies . 16 The line of Quadrature . 17 The line of Inscribed bodies . 18 A line of 12 houres of Ascension with Stars names , Declinations , and Ascensional differences . Above all these a Table to know the Epact , and what day of the Weeke , the first day of March , hapned upon , by Inspection continued to the yeare 1700. All these between the Limbe and the Center . ON the right edge a line of equal parts from the Center decimally sub-divided , being a line of 10 inches ; also a Dyalling Tangent or Scale of 6 houres , the whole length of the quadrant not issuing from the Center . On the left edge a Tangent of 63 deg . 26′ from the Center . Also a Scale of Latitudes fitted to the former Scale of houres not issuing from the Center , and below it a small Chord . The Vses of the Quadrant . Lords-day 1657 63 68 74 ☉ 85 91 96 anno 25 1 26 3   4 11 6 epact Monday 58 ☽ 69 75 80 86 ☽ 97 anno 6   7 14 9 15   17 epact Tuesday 59 64 70 ♂ 81 87 92 98 anno 17 12 18   20 26 22 28 epact Wenesday ☿ 65 71 76 82 ☿ 93 99 anno   23 29 25 1   3 9 epact Thursday 60 66 ♃ 77 83 88 94 ♃ anno 28 4   6 12 7 14   epact Friday 61 67 72 78 ♀ 89 95 700 anno 9 15 11 17   18 25 20 epact Saturday 62 ♄ 73 79 84 90 ♄ 701 anno 20   22 28 23 29   1 epact Dayes the same as the first of March. March 1 8 15 22 29 November August 2 9 16 23 30 August May 3 10 17 24 31 Jnuary October 4 11 18 25 0 October April 5 12 19 26 00 July Septem . 6 13 20 27 00 December June 7 14 21 28 00 February Perpetual Almanack . Of the Vses of the Projection . BEfore this Projection can be used , the Suns declination is required , & by consequence the day of the moneth for the ready finding thereof there is repeated the same table that stands on the Back-side of this quadrant in each ruled space , the uppermost figure signifies the yeare of the Lord , and the column it is placed in sheweth upon what day of the Weeke the first day of March hapned upon in that yeare , and the undermost figure in the said ruled space sheweth what was the Epact for that yeare and this continued to the yeare 1701 inclusive . Example . Looking for the yeare 1660 I find the figure 60 standing in Thursday Column , whence I may conclude that the first day of March that yeare will be Thursday , and under it stands 28 for the Epact that yeare . Of the Almanack . HAving as before found what day of the Weeke the first day of March hapned upon , repaire to the Moneth you are in , and those figures that stand against it shewes you what dayes of the said moneth the Weeke day shall be , the same as it was the first day of March. Example For the yeare 1660 , having found that the first day of March hapned upon a Thursday , looke into the column against June , and February , you will find that the 7th , 14th , 21th and 28th dayes of those Moneths were Thursdayes , whence it might be concluded if need were that the quarter day or 24th day of June that yeare hapneth on the Lords day . Of the Epact . THe Epact is a number carried on in account from yeare to yeare towards a new change , and is 11 dayes , and some odde time besides , caused by reason of the Moons motion , which changeth 12 times in a yeare Solar , and runnes also this 11 dayes more towards a new change , the use of it serves to find the Moones age , and thereby the time of high Water . To know the Moons age . ADde to the day of the Moneth the Epact , and so many days more , as are moneths from March to the moneth you are in , including both moneths , the summe ( if lesse then 30 ) is the Moones age , if more , subtract 30 ; and the residue in the Moons age ( prope verum . ) Example . The Epact for the year 1658 is 6 , and let it be required to know the Moons age the 28 of July , being the fift moneth from March both inclusive 6 28 5 The summe of these three numbers is 39 Whence rejecting 30 , the remainder is 9 for the Moons age sought . The former Rule serves when the Moneth hath 31 dayes , but if the Moneth hath but 30 Dayes or lesse , take away but 29 and the residue is her ages To find the time of the Moones comming to South . MUltiply the Moones age by 4 , and divide by 5 , the quotient shewes it , every Unit that remaines is in value twelve minutes of time , and because when the Moon is at the full , or 15 dayes , old shee comes to South at the houre of 12 at midnight , for ease in multiplication and Division when her age exceedes 15 dayes reject 15 from it . Example , So when the Moon is 8 dayes old , she comes to South at 24 minutes past six of the clock , which being knowne , her rising or setting may be rudely guessed at to be six houres more or lesse before her being South , and her setting as much after , but in regard of the varying of her declination no general certaine rule for the memory can be given . Here it may be noted that the first 15 dayes of the Moones age she commeth to the Meridian after the Sun , being to the Eastward of him , and the later 15 dayes , she comes to the Meridian before the Sun , being to the Westward of him . To find the time of high Water . TO the time of the Moones comming to South , adde the time of high water on the change day , proper to the place to which the question is suited , the summe shewes the time of high waters For Example , There is added in a Table of the time of high Water at London , which any one may cast up by memory according to these Rules , it is to be noted , that Spring Tides , high winds , and the Moon in her quarters causes some variation from the time here expressed . Moones age Moon South Tide London Dayes . Ho. mi. Ho. Mi 0 15 12 — 3 00 1 16 12 : 48 3 : 48 2 17 1 : 36 4 : 36 3 18 2 : 24 5 : 24 4 19 3 : 12 6 : 12 5 20 4   7 00 6 21 4 : 48 7 : 48 7 22 5 : 36 8 : 36 8 23 6 : 24 9 : 24 9 24 7 : 12 10 : 12 10 25 8 : 00 11 : 00 11 26 8 : 48 11 : 48 12 27 9 : 36 12 : 36 13 28 10 : 24 1 : 24 14 29 11 : 12 2 : 12 This Rule may in some measure satisfie and serve for vulgar use for such as have occasion to go by water , and but that there was spare roome to grave on the Epacts nothing at all should have been said thereof . A Table shewing the houres and Minutes to be added to the time of the Moons comming to South for the places following being the time of high Water on the change day .   H. m. Quinborough , Southampton , Portsmouth , Isle of Wight , Beachie , the Spits , Kentish Knocke , half tide at Dunkirke . 00 :  00 Rochester , Maulden , Aberdeen , Redban , West end of the Nowre , Black taile . 00 :  45 Gravesend , Downes , Rumney , Silly half tide , Blackness , Ramkins , Semhead . 1 :  30 Dundee , St. Andrewes , Lixborne , St. Lucas , Bel Isle ; Holy Isle . 2 :  15 London , Tinmouth , Hartlepoole , Whitby , Amsterdam , Gascoigne , Brittaine , Galizia . 3 :  00 Barwick , Flamborough head , Bridlington bay , Ostend , Flushing , Bourdeaux , Fountnesse . 3 :  45 Scarborough quarter tide , Lawrenas , Mountsbay , Severne , King sale , Corke-haven , Baltamoor , Dungarvan , Calice , Creeke , Bloy seven Isles . 4 :  30 Falmouth , Foy , Humber , Moonles , New-castle , Dartmouth , Torbay , Caldy Garnesey , St. Mallowes , Abrowrath , Lizard . 5 :  15 Plymouth , Weymouth , Hull , Lin , Lundy , Antwerpe , Holmes of Bristol , St. Davids head , Concalo , Saint Malo. 6 :  00 Bristol , foulnes at the Start. 6 :  45 Milford , Bridg-water , Exwater , Lands end , Waterford , Cape cleer , Abermorick Texel . 7 :  20 Portland , Peterperpont , Harflew , Hague , St. Magnus Sound , Dublin , Lambay , Mackuels Castle . 8 :  15 Poole , S. Helen , Man Isle , Catnes , Orkney , Faire Isles , Dunbar , Kildien , Basse Islands , the Casquers , Deepe at halfe tide . 9 :   Needles , Oxford , Laysto , South and North Fore-lands . 9 :  45 Yarmouth , Dover , Harwich , in the frith Bullen , Saint John de luce , Calice road . 10 :  30 Rye , Winchelsea , Gorend , Rivers mouth of Thames , Faire Isle Rhodes . 11 :  15 To find the Epact for ever . IN Order hereto , first , find out the Prime Number divide the yeare of the Lord by 19 the residue after the Division is finished being augmented by an Unit is the Prime sought , and if nothing remaine the Prime is an Unit. To find the Epact . MUltiply the Prime by 11 , the product is the Epact sought if lesse then 30 , but if it be more , the residue of the Product divided by 30 is the Epact sought , there note that the Prime changeth the first of January , and the Epact the first of March. Otherwise . Having once obtained the Epact adde 11 so it the Summe if lesse then 30 is the Epact for the next yeare if more reject 30 , and the residue is the Epact sought . Caution . When the Epact is found to be 29 for any yeare , the next yeare following it will be 11 and not 10 , as the Rule would suggest . A Table of the Epacts belonging to the respective Primes . Pr. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 Ep. 11 22 3 14 25 6 17 28 9 20 1 12 23 4 15 26 7 18 29 The Prime number called the Golden Number , is the number of 19 years in which space the Moone makes all variety of her changes , as if she change on a certain day of the month on a certain yeare she shall not change the same day of the moneth again till 19 yeares after : and then it doth not happen upon the same houre of the day , yet the difference doth not cause one dayes variation in 300 yeares , as is observed by Mr. Philips . The Vses of the Quadrant . WIthout rectifying the Bead nothing can be performed by this Projection , except finding the Suns Meridian Altitude being shewn upon the Index , by the intersection of the Parallel of declination therewith . Also the time when the Sun will be due East or West . TRace the Parallel of Declination to the right edge of the Projection , and the houre it there intersects ( in most cases to be duly estimated ) shewes the time sought , thus when the Sun hath 21 deg . of North declination , we shall find that he will be due East or West , about three quarters of an houre past 4 in the afternoon , or a quarter past 7 in the morning . The declination is to be found on the Back-side of the quadrant by laying the thread over the day of the moneth . To rectifie the Bead. LAy the thread upon the graduated Index , and set the Bead to the observed or given Altitude , and when the Altitude is nothing or when the Sun is in the Horizon set the Bead to the Cypher on the graduated Index , which afterwards being carried without stretching to the parallel of Declination the threed in the Limbe shewes the Amplitude or Azimuth , and the Bead amongst the houres shewes the true time of the day . Example . Upon the 24th of April the Suns declination will be found to 16 deg . North. Now to find his Amplitude and the time of his rising , laying the threed over the graduated Index , set the Bead to the beginning of the graduations of the Index , and bring it without stretching to the parallel of declination above being 16 d ▪ and the threed in the limbe will lye over 26 deg . 18′ for the Suns Amplitude or Coast of rising to the Northward of the East , and the Bead amongst the houres sheweth 24 minutes past 4 for the time of Sun rising . Which doubled gives the length of the night 8 houres 49 min. In like manner the time of setting doubled gives the length of the day . The same to find the houre and Azimuth let the given Altitude be 45 degrees . HAving rectified the Bead to the said Altitude on the Index and brought it to the intersect , the parallel of declination the thread lyes over 50 degrees 48′ . For the Suns Azimuth from the South . And the Bead among the houres shewes the time of the day to be 41 minutes past 9 in the morning , or 19 minutes past two in the afternoon . Another Example wherein the operation will be upon the Reverted taile . Let the altitude be 3 deg . 30′ And the declination 16 deg . North as before . TO know when to rectify the Bead to the upper or neather Altitude will be no matter of difficulty , for if the Bead being set to the neather Altitude will not meet with the parallel of declination , then set it to the upper Altitude , and it will meet with Winter parallel of like declination , which in this case supplyes the turn . So in this Example , the Bead being set to the upper Altitude of 3 deg . 30′ and carried to the Winter parallel of declination . The thread in the Limbe will fall upon 68 deg . 28′ for the Suns Azimuth from the North , and the Bead among the houres shewes the time of the day to be either 5 in the morning or 7 at night . Another Example . Admit the Sun have 20 degr . of North Declination ( as about the 9th of May ) and his observed altitude were 56 deg . 20′ having rectified the Bead thereto , and brought it to intersect the parallel of 20 deg . among the houres it shewes the time of the day to be 11 in the morning or 1 in the afternoon , and the Azimuth of the Sun to be 26 deg . from the South The Vses of the Projection . TO find the Suns Altitude on all houres or Azimuths will be but the converse of what is already said , therefore one Example shall serve . When the Sun hath 45 deg . of Azimuth from the South . And his Declination 13 deg . Northwards . Lay the threed over 45 deg . in the Limbe , and where the threed intersects the Parallel of Declination thereto remove the Bead which carried to the Index without stretching , shewes 43 deg . 50′ for the Altitude sought . Likewise to the same Declination if it were required to find the Suns Altitude for the houres of 2 or 10. Lay the threed over the intersection of the houre proposed with the parallel of Declination , and thereto set the bead which carried to the Index shewes the Altitude sought namely 44 deg . 31′ . The same Altitude also belongs to that Azimuth the threed in the former Position lay over in the Limbe . This Projection is of worst performance early in the morning or late in the evening , about which time Mr. Daries Curve is of best performance whereto we now addresse our selves . Of the curved line and Scales thereto fitted . This as we have said before was the ingenious invention of M. Michael Dary derived from the proportionalty of two like equiangled plain Triangles accommodated to the latitude of London , for the ready working of these two Proportions . 1. For the Houre . As the Cosine of the Latitude , is to the secant of the Declination , So is the difference between the sine of the Suns proposed and Meridian Altitude . To the versed sine of the houre from noone , and the converse , and so is the sine of the Suns Meridian Altitude , to the versed sine of the semidiurnal Arke . 2. For the Azimuth . The Curve is fitted to find it from the South and not from the North , and the Proportion wrought upon it will be . As the cosine of the Latitude , is to the Secant of the Altitude . So is the difference of the versed sines of the Suns ( or Stars ) distance from the elevated Pole , and of the summe of the Complements both of the Latitude and Altitude , to the versed sine of the Azimuth from the noon Meridian . Which will not hold backward to find the Altitude on all Azimuths , because the altitude is a term involved , both in the second and third termes of the former proportion . If the third terme of the former Proportion had not been a difference of Sines , or Versed sines , the Curved line would have been a straight-line , and the third term always counted from one point , which though in the use it may seem to be so here , yet in effect the third term for the houre is always counted from the Meridian altitude . Here observe that the threed lying over 12 or the end of the Versed Scale , and over the Suns meridian altitude in the line of altitudes , it will also upon the curve shew the Suns declination , which by construction is so framed that if the distance from that point to the meridian altitude , be made the cosine of Latitude , the distance of the said point from the end of the versed Scale numbred with 12 shall be the secant of the declination to the same Radius , being both in one straight-line by the former constitution of the threed , and instead of the threed you may imagine a line drawn over the quadrant , then by placing the threed as hereafter directed it will with this line & the fitted scales constitute two equiangled plaine triangles , upon which basis the whole work is built . In the three first Proportions following relating to time , the Altitude must alwayes be counted upwards from O in the line of Altitude , and the Declination in the Curve upwards in Summer , downwards in Winter . 1 To find the time of the Suns rising and setting by the Curve . WE have before intimated that the suns Declination is to be found on the back of the quadrant , having found it , lay one part of the thread over 0 deg . in the Line of Altitude , and extending it , lay the other part of it over the Suns Declination counted from O in the Curve , and the thread upon the Versed scale shewes the time of Suns rising and setting , which being as much from six towards noon in Winter as towards mid-night in Summer , the quantity of Declination supposed alike both wayes on each side the Equinoctial , the thread may be layd either way from O in the Curve to the Declination . Example . When the Sun hath 20 deg . of Declination , the thread being laid over 20 deg . in the Curve and O in the Altitude on the left edge shewes that the Sun riseth setteth 1 houre 49′ before after six in the Summer and riseth setteth as much after before six in the Winter . 2 The Altitude and Declination of the Sun being given to find the houre of the day . COunt the Altitude from O in the Scale of Altitudes towards the Center , and thereto lay the thread , then count the Declination from O in the Curve , if North upwards towards the Center , if South downwards towards the Limbe . And lay the thread extended over it , and in the Versed Scale it shewes the time of the day sought . Example . The Altitude being 24 d. 46′ and the Declination 20 d. North counting that upwards in the Scale of Altitudes , and this upward in the curve , and extending the through thread , it will intersect the Versed Scale at 7 and 5 , shewing the houre to be either 7 in the morning , or 5 in the afternoon . Another Example for finding when twliight begins . Let the Suns Declination be 13 deg . North , the Depression supposed 18 degr . under the Horizon . In stead of the case propounded , suppose the Sun to have 13 deg . of South Declination , and Altitude 18 deg . above the Horizon accordingly extending the thread through 18 in the Altitudes counted upward from O in the line of Altitudes and through 13 deg . counted downward in the Curve from O , and upon the Versed Scale , the thread will shew that the Twilight begins at 28 minutes past 2 in the morning , and at 32 min. past 9 at night . 3 The Converse of the last Proposition is to find the Suns altitudes on all houres . EXtend the thread over the houre proposed in the versed Scale and also over the Declination in the Curve counted upward if North , downward if South . And in the Scale of Altitudes it shewes the Altitude sought . Example . If the Sun have 13 deg . of North Declination his Altitude for the houre of 7 in the morning , or 5 in the after-noon will be found to be 19 deg . 27′ . In the following Propositions the altitude must alwayes be counted from O in the Curve downwards , and the Declination in the line of altitudes , if North downward , if South upwards . 4 To find the Suns Amplitude or coast of rising and setting . Example . If the Sun had 20 deg . of Declination the thread being laid to O in the Curve , and to 20 in the line of altitudes or Declinations , either upwards or downwards the thread will lye 33 deg . 21′ from 90 in the Versed Scale , for the quantity of the Suns Coast of rising or setting from the true East or West in Winter Southward , in Summer Norhward . 5 The Suns altitude and Declination being proposed to find his Azimuth . COunt the altitude from O in the Curve downward , and the declination in the Winter upon the line of Declinations from O upwards , in Summer downwards , and the thread extended sheweth the Azimuth sought , on the Versed Scale . Example . So when the Sun hath 18 deg . 37′ of North Declination , as as about 19 July , if his altitude were 39 deg . the Suns Azimuth would be found to be 69 deg . from the South . 6 The Converse of the former Proposition will be to find the Suns Altitude on all Azimuths . THe Instrument will perform this Proposition though the Porportion for finding the Azimuth cannot be inverted . Lay the thread to the azimuth in the Versed Scale , and to the Declination in the Scale on the left edge , and upon the Curve it will intersect the altitude sought . Example . If the Sun had 16 deg . 13′ of South Declination , as about the 27th of October , if his Azimuth were 39 deg . from the South the altitude agreeable thereto would be found to be 14 deg . These Uses being understood ▪ if the houre and altitude or the azimuth and altitude were given to find the Declination , the manner of performance cannot lurke . Of the Houre and Azimuth Scale on the right Edge of the Quadrant . THis Scale being added by my selfe , and derived from Proportions in the Analemma , I shall first lay them down , and then apply them . In the former Scheme draw F C V the Horizon , Z C the Axis of the Horizon , C P the axis of the Spheare G C continued to N the Equator , O L a parallel of North , and E I a parallel of South Declination , W X a parallel of winter altitude , S L a parallel of altitude lesse then the Complement of the latitude , N Z P a parallel of greater altitude , and from the points E and B. let fall the perpendiculars E F and B H , and from the points B G and N let fall the perpendiculars B G , G M , and N O which will be the sines of the Suns declination , by this meanes there will be divers right lined right angled plaine Triangles constituted from whence are educed , the Proportions following to calculate the Suns houre or Azimuth . Note , first , that T V is the Versed sine of the Semidiurnal arke in Summer , and E I in Winter , and Y V the sine of the houre of rising before six in Summer , equal to the distance of I from the Axis continued in Winter , which may be found in the Triangle C Y V , but the Proportion is . As the Cotangent of the latitude , To Radius . So the Tangent of the Suns Declination , To the sine of his ascentional difference , being the time of his rising from six , thus we may attaine the Semidiurnal arke . Then for the houre in the Triangle B H I it holds . As the Cosine of the latitude , to the sine of the altitude . So is the Secant of the Declination . To the difference of the Versed sines of the Semidiurnals arke , and of the houre sought . In the Triangle B H I it leys . As the Cosine of the Latitude the sine of the angle at I. To its opposite side B H the sine of the altitude . So is the Radius or the angle at H. To B I h difference of the Versed sine of the Semidiurnal arke , and of the houre sought , in the parallel of declination and by consequence , so is the secant of the Declination , to the said difference in the Common Radius as we have else where noted , if this difference be subtracted from the Versed sine of the semidiurnal arke there will remaine E B the versed sine of the houre from noon , the like holds , if perpendiculars be let fall from any other parallel of Declination , from the same Scheme it also followes . As the Cosine of the Latitude , Is to the secant of the Declination . So is the sine of the Meridian Altitude . To the versed sine of the semidiurnal arke . Here observe the like Proportion between the two latter terms , as between the two former which may be of use on a Sector . If the Scheme be considered not as fitted to a peculiar question for finding the houre , but as having three sides to find an angle , it will be found upon such a consideration in relation to the change of sides , that the Proportion for the Azimuth following is no other then the same Proportion applyed , to other sides of the Triangle , and so we need have no other trouble to come by a Proportion for the Azimuth , but it also followes from the same Scheme . In the Triangles C D A and C K G , and C Z N the first operation will be to find A D , and G K , and N ● in all which the Proportion will hold . As the Radius to the Tangent of the Latitude . Or as the Cotangent of the Latitude to Radius . So is the Tangent of the Altitude , to the said respective quantities , which when the Altitude is lesse then the Complement of the Latitude , are the sins of the Suns Azimuth from the Vertical belonging to the proposed Altitudes when the Sun is in the Equinoctial , or hath no declination . The next proportion will be . As the Cosine of the Latitude , Is to the Secant of the Altitude . So is the Sine of the declination . To the difference sought being a 4 Proportional . Hereby we may find A B in the Winter Triangle A G B which added to A D , the summe is the sine of the Azimuth from the Vertical consequently W B , is the Versed sine of the Azimuth , from the noon Meridian . Also we find G L in the Summer triangle L M G , when the Altitude is lesse then the Complement of the Latitude , which added to S G the summe S L is the Versed sine of the Azimuth from the South . Likewise we may find N R in the Triangle R O N , and by subtracting it from N Z , there will remaine R Z , and consequently Q R the versed sine of the Azimuth from the Meridian in Summer when the Altitude is greater then the Co-latitude . And for Stars that come to the Meridian between the Zenith , and the Elevated Pole , we may find N c , in the Triangle N c d where it holds , as the sine of the Angle at N , the complement of the Latitude , to its opposite sides c d , the prickt line , the sine of the Declination : so is the Radius to N c , the parallel of altitude the Azimuth sought . The latter Proportion lyes so evident , it need not be spoken to , if what was said before for the houre be regarded , and the former Proportion lyes . As the Cosine of the Latitude , the sine of the Angle at A. To its Opposite side D C , the sine of the altitude . So is the sine of the Latitude , the angle at C. To its opposite side A D in the parallel of altitude . And in stead of the Cosine , and sine of the Latitude . We may take the Radius , and the Tangent of the Latitude . Another Analogy will be required to reduce it to the common Radius . As the Cosine of the Altitude to Radius . So the fourth before found in a parallel . To the like quantity to the Common Radius . These Analogies or Proportions being reduced into one , by multiplying the termes of each Proportion , and then freed from needlesse affection will produce the Proportion at first delivered . The Vses of the said Scale . WE have before noted , that if two termes of a Proportion be fixed , and naturall lines thereto fitted of an equal length , that if any third term be sought in the former line , the fourth term will be found in the other line by inspection , as standing against the third . So here , in this Scale which consists of two lines , the one an annexed Tangent , the other a line of Sines continued both Wayes , the Radius of the Sines being first fittted , the Tangent annexed must be of such a Radius , as that 38 deg . 28′ , of it may be equall in length to the Radius of the Sine to which it is adjoyned , and then looking for the Declination in the Tangent just against it stands the time of rising , from six or ascentional difference , or the Semidiurnal arke , if the same be accounted from the other end as a Versed Sine . So if the Suns Altitude be given , and accounted in the Tangent , just against it stands the Suns Azimuth , when he is in the Equinoctial upon the like altitude , and thus the point N will be found in the Tangent at the altitude , when it is more then the Colatitude . 1 An Example for finding the time of the Sun rising . If the Declination be 13 deg . looke for it in the annexed Tangent , and just against it in the houre Scale stands 16 deg . 53′ the ascentional difference in time 1 houre 7½ min. shewing that the Sun riseth so much before , and setteth so much after 6 in Summer , and in Winter riseth so much after , and setteth before 6 , for this arke may be found on either side of six where the declination begins each way . 2 To find the time of the day . To perform this Proposition wee divide the other Proportion into two , by introducing the Radius in the Middle . As the Radius is to the Secant of the Declination . So is the sine of the altitude to a fourth . Again . As the Cosine of the Latitude to Radius . So the fourth before found . To the difference of the Versed Sines of the Semidiurnal arke , and of the houre sought . The former of these Proportions must be wrought upon the quadrant , the latter is removed by fitting the Radius of the Sines that gives the answer , equal in length to the Cosine of the latitude . Wherefore to find the time of the day , lay the thread to the Secant of the declination in the limbe , and from the sine of the altitude take the nearest distance to it , and because the Secant is made , but to halfe the Common Radius , set downe one foot of this extent at the Declination in the annexed Tangent , and enter the said extent twice forward , and it will shew the time of the Day . Example . Let the Declination be supposed 23 deg . 31′ North , and the Altitude 38 deg . 59′ the nearest distance from the Sine thereof , to the thread laid over the Secant of 3● deg . 31′ will reach being turned twice over from 3● d. 31′ in the annexed Tangent neerest the Center to 33 deg . 45′ in the Sines , aliàs to 56 d. 15′ counted as a Versed Sine shewing the time of the day to be a quarter past 8 in the morning , or three quarters past three in the afternoon . 3 To find the Suns Altitude on all houres . Take the distance between the houre and the Declination in the fitted Scale , and enter it downe , the line of Sines from the Center , then laying the thread over the Cosine of the Declination in the Limbe , the nearest distance to it shall be the sine of the Altitude sought . Example . Thus whee the Sun hath 13 deg . of South Declination , count it in that part of the annexed Tangent nearest the Limbe , if then it were required to find the Suns Altit . for the houres of 10 or 2 by the former Prescriptions the Altitude would be found 10 d. 25′ 4 To find the Suns Amplitude . Take the Sine of the Declination from the line of the Sines , and apply it to the fitted Scale where the annexed Tangent begins , and either way it will reach to the Sine of the Amplitude . Example . So when the Sun hath 15 deg : of Declination his Amplitude will be found to be 24 deg . 35′ . 5 To find the Azimuth or true Coast of the Sun. Here we likewise introduce the Radius in the latter Proportion . 1 In Winter lay the thread to the Secant of the Altitude in the Limbe , and from the sine of the Declination , take the nearest distance to it , the said extent enter twice forward from the Altitude in the annexed Tangent , and it will reach to the Versed Sine of the Azimuth from the South . Example . So when the Sun hath 15 deg . of South Declination , if his Altitude be 15 deg . the nearest distance from the sine thereof to the thread laid over the Secant of 15 degrees , shall reach in the fitted Scale from the annexed Tangent of 15 deg . being twice repeated forward to the Versed sine of 39 deg . 50′ for the Suns Azimuth from the South . 2 In Summer when the Altitude is lesse then 40 deg . enter the former extent from the sine of the Declination to the thread laid over the Secant of the Altitude twice backward from the Altitude in the annexed Tangent , and it will reach to the Versed sine of the Azimuth from the South . Example . So if the Sun have 15 deg . of North Declination , and his Altitude be 30 deg . the prescribed extent doubled shall reach from the annexed Tangent of 30 deg . to the Versed sine of 75 deg . 44′ for the Suns Azimuth from the South . 3 In Summer when the Altitude is more then 40 deg . and lesse then 60 deg . apply the extent from the sine of the Declination to the thread , laid over the Secant of the Altitude once to the Discontinued Tangent placed a Crosse the quadrant from the Altitude backwards minding how farre it reaches , just against the like arke in the annexed Tangent stands the Versed sine of the Azimuth from the South . 4 When the Altitude is more then 60 deg . this fitted Scale is of worst performance , however the defect of the Secant might be supplyed by Varying the Proportion . 6 To find the Suns Altitude on all Azimuths . JUst against the Azimuth proposed stands the Suns altitude in the Equator suitable thereto , which was the first Arke found by Calculation when we treated of this subject , and the second arke is to be found by a Proportion in sines wrought upon the quadrant . This quadrant is also particularly fitted for giving the houre , and Azimuth in the equal limbe . The sine of 90 deg . made equal to the sine if 51 deg . 32′ gives the altitude of the Sun or Stars at six , for if the thread be laid over the Declination counted in the said sine , it shewes the Altitude sought in the limbe , so when the Sun hath 13 deg . of Declination his Altitude or Depression at 6 is 10 deg . 9′ . It also gives the Vertical Altitude if the Declination be counted in the limbe , seeke what arke it cuts in that particular sine , when the Sun hath 13 deg . of Declination , his Vertical Altitude or Depression is 16 deg . 42′ . To find the houre of the Day . HAving found the Altitude of the Sun or Stars at six , take the distance between the sine thereof in the line Sines , and the Altitude given , and entring one foot of that extent at the Declination in the Scale of entrance laying the thread to the other foot according to nearest distance , it will shew the houre from six in the limbe . Example . When the Sun hath 13 deg . of Declination his Altitude , or Depression at six will be 10 deg . 9′ if the Declination be North , and the Altitude of the Sun be 24 deg . 5′ the time of the day will be halfe an houre past 7 in the morning , or as much past 4 in the afternoon . In winter when the Sun hath South Declination as also for such Stars as have South Declination , the sine of their Altitude must be added to the sine of their Depression at six , and that whole extent entred as before . When the Sun hath the same South Declination , if his Altitude be 11 deg . 7′ the time of the day will be half an houre past 8 in the morning , or 30 min. past 3 in the afternoon . To find the Azimuth of the Sun or Stars . LAy the thread over their Altitude in the particular sine fitted to the Latitude , and in the equal Limbe it shewes a fourth Arke . When the Declination is North , take the distance in the line of Sines between that fourth Arke and the Declination , and enter one foot of that extent at the Altitude in the Scale of entrance , laying the thread to the other foot , and in the equal Limbe it shewes the Azimuth from the East or West . Example . When the Altitude is 44 deg . 39′ the Arch found in the equal Limbe will be 33 deg . 20′ then if the Declination be 23 deg 31′ North , the distance in the line of sines between it and the said Arke being entred at 44 deg . 39′ in the Scale of entrance the thread being laid to the other foot will shew the Azimuth to be 20 deg . from the East or West . But if the Declination be South , adde with your Compasses the sine thereof to the sine of the fourth Arke , and enter that whole extent as before , and the thread will shew the Azimuth in the equal limbe . Example . When the Altitude is 12d . 13′ the fourth Arch will be found to be 9 degrees 32 minutes , then admit the Declination to be 13 degrees South , whereto adding the Sine of the fourth Arke , the whole will be equall to the sine of 22 deg . 41 minutes , and this whole extent being entred at 12 deg . 13′ in the Scale of entrance lay the thread to the other foot according to nearest distance , and it will intersect the equal . Limbe at 40 deg . and so much is the Suns Azimuth from the East or West . Because the Scale of entrance could not be continued by reason of the Projection , the residue of it is put on an little Line neare the Amanack the use whereof is to lay the thread to the Altitude in it when the Azimuth is sought , and in the Limbe it shewes at what Arke of the Sines the point of entrance will happen which may likewise be found by pricking downe the Co-altitude on the line of Sines out of the fitted houre Scale on the right edge . How to find the houre and Azimuth generally in the equal limb either with or without Tangents or Secants hath been also shewed , and how that those two points for any Latitude might be there prickt and might be taken off , either from the Limbe , or from a line of Sines , or best of all by Tables , for halfe the natural Tangent of the Latitude of London , is equal to the sine of 〈◊〉 39 deg . And half the Secant thereof equal to the sine of 〈◊〉 53d . 30 Against which Arkes of the Limbe the Tangent and Secant of the Latitude are graduated , but of this enough hath been said in the Description of the small quadrant . Of the Quadrat and Shadowes . THe use thereof is the same as in the small quadrant onely if the thread hang over any degree of the Limb lesse then 45d . to take out the Tangent thereof out of the quadrat count the Arch from the right edge of the quadrant towards the left , and lay the thread over it , the pricks are repeated in the Limbe to save this trouble for those eminent parts . Of the equal Limbe . WE have before shewed that a Sine , Tangent and Secant may be taken off from it , and that having a Sine or Secant with the Radius thereof the correspondent Arke thereto might be found , & that a Chord might be taken off from Concentrick Circles or by helpe of a Bead , but if both be wanting enter the Semidiameter or Radius whereto you would take out a Chord twice downe the right edge from the Center , and laying the thread over halfe the and laying the thread over halfe the Arch proposed , take the nearest distance to it , and thus may a chord be taken out to any number of degrees lesse then a Semicircle . It hath been asserted also that the houre and Azimuth might be found generally without Protraction by the sole helpe of the Limb with Compasses and a thread . Example for finding the houre . THe first work will be to find the point of entrance take out the Cosine of the Latitude by taking the nearest distance to the thread laid over the said Arke from the concurrence of the Limbe with the right edge , and enter it down the right edge line and take the nearest distance to the thread laid over the complement of the Declination counted from the right edge , this extent entred down the right edge finds the point of entrance , let it be noted with a mark . Next to find the sine point take out the sine of the Declin . & enter it dowh the right edge , & from the point of termination , take the nearest distance to the thread laid over the ark of the Latit . counted from the right edge , this extent enter from the Center and it finds the sine point , let it be noted with a marke . Thirdly , take out the sine of the Altitude & in Winter add it in lenght to the sine point , in Summer enter it from the Center & take the distance between it & the sine point which extent entred upon the point of entrance , if the thread be laid to the other foot shewes the the houre from 6 in the equal limb before or after it , as the Sine of the Altitude fell short or beyond the sine point . Example . In the latitude of 39 d. the Sun having 23d . 31′ of North Declination , and Altitude 51 deg . 32′ the houre will be found to be 33 deg . 45′ from six towards noon . Note the point of entrance and sine point Vary not , till the Declination Vary . After the same manner may the Azimuth be found in the limb , by proportions delivered in the other great quadrant . Also both or any angle when three sides are given may be found by the last general Proportion in the small quadrant which finds the halfe Versed sine of the Arke fought , which would be too tedious to insist upon & are more proper to be Protracted with a line of Chords . To find the Azimuth universally . THe Proportion used on the smal quadrant for finding it in the equal limbe ( wherein the first Operation for the Vertical Altitude was fixed for one day , ) by reason of its Excursions will not serve on a quadrant , for the Sun or Stars when they come to the Meridian between the Zenith and the elevated Pole , but the Proportion there used for finding the houre applyed to other sides will serve for the Azimuth Universally , and that is As the Radius , Is to the sine of the Latitude , So is the sine of the Altitude , To a fourth sine . Again . As the Cosine of the Altitude , Is the Secant of the Latitude . Or , As the Cosine of the Latitude , Is the Secant of the Altitude . So In Declinations towards the Elevated Pole is the difference , but towards the Depressed Pole the summe of the fourth sine , and of the sine sine of the Declination . To the sine of the Azimuth from the Vertical . In Declinations towards the Depressed Pole , the Azimuth is alwayes obtuse , towards the elevated Pole if the Declination be more then the fourth Arch it is acute , if lesse obtuse . Example for the Latitude of the Barbados 13 deg . Altitude 27 deg . 27′ . Declination 20 deg . North. Lay the thread to 27 deg . 27′ in the Limbe , and from the sine of 13 deg . tahe nearest distance to it which enter on the line of Sines from the Center , and take the distance between the limited point , and the sine of 20 deg . the Declination , this latter extent enter twice downe the line of the Sines from the Center , and take the nearest distance to the thread laid over the Secant of 27 deg . 27′ this extent enter at the sine of 77 deg . the Complement of the Latitude , and laying the thread to the other foot it will lye over 16 deg . in the equal Limbe , the Suns Azimuth to the Northwards of the East or West . Otherwaies . Another Example for the same Latitude and Declination , the Altitude being 52 deg . 27′ lay the thread to it in the Limbe , and take the nearest distance to it from the sine of 13 deg . as before , and enter it downe the line of sines from the Center , and from the point of the limitation take the distance to the sine of 20 deg . the Suns Declination , this latter extent enter once downe the line of sines from the Center , and take the nearest distance to the Thread laid over the Secant of the Altitude 52 deg . 27′ then lay the thread to 77 deg . the Complement of the Latitude in the lesser sines , and enter the former extent between the Scale and the thread , and the foot of the Compasses sheweth 16 deg . as before , for the Suns Azimuth to the Northward of the Vertical , that the Sun may have the same Azimuth , upon two several Altitudes hath been spoken to before , and how to do this without Secants hath been shewne . Two sides with the Angle comprehended to find the third side . DIvers wayes have been shewed for doing of this before , I shall adde one more requiring no Versed sines nor Tangents . 1 If both the sides be lesser then quadrants , and the Angle at liberty . Or , 2 If one of the sides be greater then a quadrant , and the Angle included acute , it will hold . As the Radius , To the Cosine of one of the including sines . So is the Cosine of the other , To a fourth sine . Again . As the Cosecant of one of the including Sides ● is the Sine of the other , So is the Cosine of the angle included , To a seventh Sine . The difference between the fourth and the seventh Sine , is the Cosine of the Side sought . 1 In the first case if the angle given be obtuse , and the seventh Sine greater then the fourth Sine , the Side sought is greater then a quadrant in other cases lesse . If in the second case the seventh Sine be lesse then the fourth , the side sought is greater then a quadrant in other cases lesse . In this second case when one of the includers is greater then a quadrant , and the angle obtuse resolve the opposite Triangle by the former Rules , or the summe of the fourth and seventh Sine shall be the Cosine of the side sought in this case greater then a quadrant . We have before noted that the Cosine of an Arke greater then a quadrant is the Sine of that Arkes excesse above 90 deg . this no other then the converse of the Proportion for the houre demonstrated from the Analemma , in the Triangle O Z P. Let there be given the Sides O P 113 deg . 31′ the side Z P 38 deg . 28′ and the angle comprehended Z P O 75 to find the Side O Z. Operation . Lay the thread to 51 deg . 32′ in the Limbe , and from 13 deg . 31′ in the Sines take the nearest distance to it which measured from the Center will reach to the sine of 18 deg . 12 minutes the fourth Sine . Again , laying the thread to 23 deg . 31′ in the Limbe , from the Sine of 15 deg . take the nearest distance to it , then lay the thread to the Secant of 51 deg . 32′ and enter the said extent between the Scale and the thread , the distance between the resting foot , and the Sine of 18 deg . 12 minutes before found measured from the Center is equal to the Sine of 9 deg . 32′ being the Cosine of the side sought which in this instance because the seventh Sine is lesse then the fourth sine is greater then a quadrant , and consequently must have 90 deg . added thereto , therefore the side O Z is 99 deg . 28 minutes if the question had been put in this Latitude what depression the Sun should have had under the Horizon at the houres of 5 or 7 in the Winter Tropick it would have been found 9 deg . 28′ and this is such a Triangle as hath but one obtuse Angle yet two sides greater then quadrants , and how to shunne a Secant , and a parallel entrance hath been shewed els-where . Of the Stars on the Projection , and in other places of the fore-side of the quadrant . SUch only are placed on the Projection as fall between the Tropicks being put an according to their true Declinations , and in that respect might have stood any where in the parallel of Declination , but in regard we shall also find the time of the night by them with Compasses , they are also put on in a certain Angle from the right edge of the quadrant , to find the quantity of the Angle for Stars of Northerly declination , get the difference of the Sines of the Stars Altitude six houres from the Meridian , and of its Meridian Altitude , and find to the Sine of what Arch the said difference is equal , against that Arch in the Limbe , let the Star be graduated in its proper declination , but for Stars of Southwardly Declination , get the summe of the Sines of their Depression at six and of their Meridian Altitude , and find what Arke in the Sines corresponds thereto as before . We have put on no Stars of Southwardly Declination that will fall beyond the Winter Tropick , but some of Northerly Declination falling without the Summer Tropick , are put on that are placed without the Projection towards the Limbe . All these Stars must be graduated against the line of Sines at their respective Altitudes or Depressions at the Stars houre of Six from the Meridian , and must have the same letter set to them in both places , as also upon the quadrant of 12 houres of Ascension on the Back-side where they are put on according to their true Ascension with their Declinations and Ascensional differences graved against them with the former Letter , and such of them as have more then 12 houres of right Ascension have the Character plus ✚ affixed , denoting that if there be 12 houres of Ascension added to that Ascension they stand against , the summe is their whole true right Ascension . To find the quantity of a Stars houre from the Meridian by the Projection . SEt the Bead upon the Index of Altitude to the Stars observed Altitude , and bring it to the parallel of Declination the Star is graved in , so will it shew among the houre lines , that Stars houre from the Meridian , and the thread in the Limbe will shew the Stars Azimuth . Example . Admit the Altitude of Arcturus be 52 deg . the houre of that Star from midnight , if the Altitude increase will be 7′ past 10 ferè , and the Azimuth of that Star will be 47 deg . 43′ to the Eastwards of the South . The houre and Azimuth of any Star within the Tropicks , may be also found by the fitted Scale on the right edge of the quadrant , or by the Curve , after the same manner as for the Sun , using the Stars Declination as was done for the Suns , or in the equall limb as we shewed for the Sun , which may well serve for most of the Stars in the Hemisphere . Otherwise with Compasses according to the late suggested placing of them . To find the houre of any Star from the Meridian that hath North Declination . TAke the distance between the Star point in the line of Sines , and it s observed Altitude , and laying the thread over the Star where it is graved on or below the Projection , enter the former extent paralelly between the thread and the Scale , and it shewes the Stars houre from six in the sines towards noone , if the Altitude fell beyond the Star point , otherwise towards midnight . Example . For the Goat Star let its Altitude be 40 deg . and past the Meridian , the houre of that Star will be 44′ from six , for the Compasses fall upon the sine of 11 deg . 4′ the houre is towards noon Meridian , because the Altitude is greater then 34 deg . the point where the Star is graved , the thread lying over the Star intersects , the Limbe at 25 deg . 47′ if the distance between the Star , and its Altitude be entred at the sine of that Arke , and the thread laid to the other foot , the houre will be found in the equal Limbe the same as before . For Stars of Southwardly Declination . BEcause the Star point cannot fall the other way beyond the Center of the quadrant , therefore the distance between the Star point , and the Center must be increasing by adding the sine of the Stars Altitude thereto , which will fall more outwards towards the Limbe , and then that whole extent is to be entred as before . Example . The Virgins Spike hath 9 deg 19′ of South Declination the Depression of that Star at six will be found by help of the particular sine to be 7 deg . 17′ and at that Arke in the sines the Star is graved , if the Altitude of that Star were 20 deg . the sine thereof added to the Star will be equal to the sine of 29 deg . 6′ this whole extent entred at the sine of 37 deg . 52′ the Arke of the Limbe against which the Star is graved , and the thread laid to the other foot , the houre of that Star if the Altitude increase will be 19′ past 9. To find the true time of the right . THis must be done by turning the Stars houre into the Suns houre or common time , either by the Pen as hath been shewed before , which may be also conveniently performed by the back of this quadrant , for the thread lying over the day of the moneth sheweth the Complement of the Suns Ascension in the Limbe . Or with Compasses on the said quadrant of Ascensions . THe thread lying over the day of the moneth , take the distance between it and the Star on the said quadrant , the said extent being applyed , the same way as it was taken the Suns foot to the Stars houre shall reach from the Stars houre to the true houre of the night , and if one of the feet of the Compasses fall off the quadrant , a double remedy is els-where prescribed . Example . If on the 12th of January the houre of the Goat Star was 16′ past 5 from the Meridian , the true time sought would be 49′ past 1 in the morning . Example . If upon the third of January , the houre of the Virgins Spike , were observed to be 19′ past 9 , the true time sought would be 45′ past 2 in the morning . To find the time of a Stars rising and setting . THe Ascentional difference is graved against the Star , the Virgins Spike hath 48′ of Ascentional difference , that is to say , that Stars houre of rising is at 48′ past 6 , and setting at 12′ past 5 , And the true time of that Stars rising upon the third of January , will be at 22′ past 10 at night , and of its setting at 47′ past 8 in the morning , found by the former directions . Of the rest of the lines on the back of this quadrant . THey are either such as relate to the motion of the Sun or Stars , or to Dialling , or such as are derived from Mr Gunters Sector . The Tangent of 51 deg . 32′ put through the whole Limbe is peculiarly fitted to the Latitude of London , and will serve to find the time when the Sun will be East or West , as also for any of the Stars that have lesse Declination then the place hath Latitude . Lay the thread to the Declination counted in the said Tangent , and in the Limbe it shewes the houre from 6 if reckoned from the right edge . Example . When the Sun hath 15 deg . of North Declination the time of his being East or West will be 12 deg . 17′ in time about 49′ before or after six , ferè . The Suns place is given in the Ecliptick line by laying the thread over the day of the moneth in the quadrant of Ascensions , of which see page 16 & 17 of the small quadrant . Of the lines relating to Dialling . SUch are the Line of Latitudes , and Scale of houres , of which before , and the line Sol in the Limbe , of which I shall say nothing at present , it is onely placed there in readinesse to take off any Arke from it , according to the accustomed manner of taking off lines from the Limbe to any assigned Radius . The requisite Arkes of an upright Decliner will be given by the particular lines on the Quadrant for the Latitude without the trouble of Proportionall worke . 1 The substiles distance from the Meridian . ACcount the Plaines declination as a sine in the fitted hour Scale on the right edge of the fore-side , and just against it in the annexed Tangent , stands the substiles distance from the meridian . If an upright Plaine decline 30 deg . the substiles distance will be 21 deg . 41 minutes . 2 The Stiles height . Count the Complement of the Plaines Declination in the said fitted houre scale as a sine and apply it with Compasses to the line of sines issuing from the Center , for the former Plaine the stiles height will be found 32 deg . 37′ . 3 The Inclination of Meridians . Account the stilts height in the annexed tangent of the fitted hour Scale , and just against it in the sine stands the Complement of the Inclination of meridians which for the former plaine will be found to be 36 deg . 25′ 4 The Angle of 12 and 6. Account the Plaines Declination in the Limbe on the Backside from the right edge , and lay the thread over it , and in the particular Tangent it shewes the Angle between the Horizon and six 32 deg . 9′ in this Example the Complement whereof is the Angle of 12 and 6 , namely 57 deg . 51 min. Also the requisite Arkes of a direct East or West , reclining or inclining Dial may be found after the same manner for this Latit . 1 The substiles distance . ACcount the Plaines Reinclination in the Limbe on the Backside from the left edge , and in there lay the thread , and in the particular Tangent it shewes the Arke sought . So if an East or West plain recline or incline 60 deg . the substiles distance will be found to be 32 deg . 12′ . 2 The stiles height . Account the Reinclination in the particular Sine on the foreside and in the Limbe it shewes the stiles height , which for the former Example will be found to be 42 deg . 41′ . 3 The inclination of Meridians . The Proportion is , As the Sine of the Latitude , to Radius . So is the sine of the substiles distance . To the sine of the inclination of Meridians , when the substiles distance is lesse then the Latitude of the place it may be found in the particular sine on the foreside , by the intersection of the thread , and for this Example will be 42 deg . 53′ . 4 The Angle of 12 and 6. Account the Complement of the Reinclination in the peculiar hour Scale as a sine , and just against it in the annexed Tangent stands the Complement of the Angle sought , in this Example the Angle of 12 and 6 is 68 deg . 20′ . In other Latitudes the Operations must be performed by Proportional worke with the Compasses . Of the Lines derived from Mr. Gunters Sector . Such are the Lines of superficies Solids , &c. Of the Line of Superficies or Squares . THe chiefe uses of this Line joyntly with the Line of Lines in the Limbe , is when a square number is given to find the Root thereof , or a Root given to find the square number thereto , these Lines placed on a quadrant will perform this some what better then a Sector , because it is given by the Intersection of the thread without Compasses , the properties of the quadrant casting these lines large where on a Sector they would be narrow . To find the square Root of a number . The Root being given to find the Square Number of that Root . IN extracting the square Root pricks must be set under the first third , fift , and seventh figure , and so forward and as many pricks as fall to be under the square number given , so many figures shall be in the Root , and accordingly the line of lines , and superficies must vary in the number they represent , I am very unwilling to spend any time about these kind of Lines , as being of small performance , and by my self and almost by all men accounted meere toyes . If a number be given in the superficies , the thread in the lines sheweth the Root of it , and the contrary , if a number be given in the lines the thread laid over it intersects the Square thereof . The performance thereof by these lines is so deficient that I shall give no Example of it . When a number is given to find the square thereof , if not to large the Reader may correct the last figure of it by multiplying it in his memory . To three numbers given to find a fourth in a Duplicated Proportion . That is to worke a Proportion between Numbers and Squares . Example . If the Diameter of a Circle whose Area is 154 be 14 , what shall the Diameter of that Circle be whose Area is 616. Example . Lay the thread over 616 in the superficies , and from 14 in the equal parts , take the nearest distance to it , then lay the thread to 154 in the superficies , and enter the former extent between the thread and the Scale , and the foot of the Compasses will rest upon 28 the diameter sought . To find a Proportion between two or more like superficies . ADmit there be two Circles , and I would know what Proportion their Areas bear to each other , in this case the proper use of a Line of superficies would be to have it on a ruler , and to measure the lengths of their like sides , for Circles the lengths of their Diameters upon it , and then I say , the numbers found on the superficies beare such Proportion each to other as the Areas or superficial contents , and for small quantities may be done on the quadrant by entring downe the larger extent of the Compasses on the Line of Lines from the Center , and mind the point of limitation , enter then the other extent on the point of limitation , and lay the thread to the other foot , find what number it cuts in the superficies , and the greater shall beare such Proportion to the lesser as 100 , &c. the length of the whole line doth to the parts cut . The Proportion that two superficies beare each to other is the same that the squares of their like sides , and therefore their sides may be measured either in foot or inch measure , and then the Squares taken out as before shewed . The line of superficies serves for the reducing of Plots to any proportion . ADmit a Plot of a piece of ground being cast up containes 364 Acres , and it were required to draw another Plot which being cast up by the same Scale should containe but a quarter so much , and let one side of the said Plot be 60 inches , against 60 in the lines , the square of it will be found to be 3600 , and the fourth part hereof would be 900 , which account in the superficies and you will find the Square Root of it to be 30 , and so many inches must be the like side of the lesser Plot if being cast up by the same Scale it should containe but ¼ of what it did before . If the line of Superficies were on a streight ruler , then to perform such a Proposition as this , would be to measure therewith the side of the Plot given , minding what number it reaches to in the Superficies , the fourth part of the said Number being reckoned on the Superficies , and thence taken shall be the length of the side in the Proportion required . Of the Line of Solids . IF a number be duly estimated in the said line , and the thread laid over it , it will in the line of lines shew the cube Root of that number , and the converse the Root being assigned , the Cube may be found , but by reason of the sorry performance of these Lines I shall spend no time about it , if this line be placed on a loose Ruler , and the like sides of two like Solids be measured therewith , those Solids shall beare such Proportion in their contents each to other as the measured lengths on the Solids . Three Numbers being given to find the fourth in a Duplicated Proportion . Example . IF a Bullet of 4 inches Diameter weigh 9 pound , what shall a Bullet of 8 inches Diameter weigh ? Answer 72 pounds . In this case let the whole line of Solids represent 100 , alwayes the Solid content whether given or sought , must be accounted in the line of Solids , and the Sides or Diameters in the Equall parts . Lay the thread to 9 in the line of Solids , and from 8 in the inches take the nearest distance to it , enter one foot of that extent at 4 in the inches , and lay the thread to the other foot : and it will lye over 72 in the Solids for the weight of the Bullet sought . An Example of the Converse . If a Bullet whose Diameter is 4 Inches weigh 9 pound , another Bullet whose weight is 40 pound , what shall be the Diameter of it . Lay the thread to 40 in the Solids , and from 4 Inches in the lines take the nearest distance to it . Then lay the thread to 9 in the Solids , and enter the said extent at the equal Scale , so that the other foot turned about may but just touch the thread , and it it will rest at 6½ Inches nearest , which is the Diameter sought . Of the Line of inscribed Bodies . This Line hath these letters set to it . D Signifying the Sides of a Dodecahedron S Signifying the Sides of a _____ I Signifying the Sides of a Icosahedron C Signifying the Sides of a Cube O Signifying the Sides of a Octohedron T Signifying the Sides of a Tetrahedron And the Letter S Signifieth the Semidiameter of a Sphere , the use whereof are to find the Sides of the five Regular Bodies that may be inscribed in a Sphere . Example . A joyner being to cut the 5 Regular Bodies desires to know the lengths of the sides of the said 5 Regular Bodies that may be inscribed in a Sphere where Diameter is 6 inches . Lay the thread over S ▪ and take 3 inches out of the line of equal parts or Inches , and enter that extent so that one foot resting on the said Scale of inches , the other turned about may but just touch the thread , the resting point thus found , I call the point of entrance , from the said point take the nearest distances to the thread laid over the Letters .   Inch. Dec. parts D And measure those Extents on the Line of Inches , and you will find them to reach to 2.13 I 3.15 C 3.45 O 4.23 T 4.86 Which are the Dimensions of the respective sides of those Bodies to which the Letters belong . The uses of the Lines of quadrature , Segments , Mettals and Equated Bodies , I leave to the Disquisition of the Reader , when he shall have occasion to put them in practice , which I think will be seldome or never ; and wherein the assistance of the Pen will be more commendable . These lines were added to this quadrant to fill up spare room , and to shew that what ever can be done on the Sector , may be performed by them on a quadrant . A TABLE Of the Latitude of the most eminent Places in England , Wales , Scotland and Ireland .   d. m. Bedford 52 8 Barwick 55 54 Bristol 51 27 Buckingham 52   Cambridge 52 12 Canterbury 51 17 Carlisle 55   Chichester 50 48 Chester 53 16 Colchester 51 58 Derby 52 58 Dorchester 50 40 Durham 54 50 Exceter 50 43 Gilford 51 12 Gloucester 51 53 Hartford 51 49 Hereford 52 7 Huntington 52 19 Ipswich 52 8 Kendal 54 23 Lancaster 54 10 Leicester 52 40 Lincolne 53 14 London 51 32 Northampton 52 14 Norwich 52 42 Nottingham 53   Oxford 51 46 Reading 51 28 Salisbury 51 4 Shrewsbery 52 47 Stafford 52 52 Stamford 52 38 Truero 50 30 Warwick 52 20 Winchester 51 3 Worcester 52 14 Yorke . 53 58 WALES d. m. Anglezey 53 28 Barmouth 52 50 Brecknock 52 1 Cardigan 52 12 Carmarthen 51 56 Carnarvan 53 16 Denbigh 53 13 Flint 53 17 Llandaffe 51 35 Monmouth 51 51 Montgomeroy 51 56 Pembrooke 51 46 Radnor 52 19 St. David 52 00 The ISLANDS . d. m Garnzey 49 30 Jersey 49 12 Lundy 51 22 Man 54 24 Portland 50 33 Wight Isle . 50 39 SCOTLAND . d. m. Aberdean 57 32 Dunblain 56 21 Dunkel 56 48 Edinburgh 55 56 Glascow 55 52 Kintaile 57 44 Orkney Isle 60 6 St. Andrewes 56 39 Skirassin 58 36 Sterling . 56 12 IRELAND . d. m. Autrim 54 38 Arglas 54 10 Armach 54 14 Caterlagh 52 41 Clare 52 34 Corke 51 53 Droghedah 53 38 Dublin 53 13 Dundalk 53 52 Galloway 53 2 Youghal 51 53 Kenny 52 27 Kildare 53 00 Kings towne 53 8 Knock fergus 54 37 Kynsale 51 41 Lymerick 52 30 Queens towne 52 52 Waterford 52 9 Wexford . 52 18 A Table of the right Ascensions and Declinations of some of the most principal fixed Stars for some yeares to come .   R. Ascension . Declination . Magnitude   H m D. m.   Pole Star 00 31 87 34 N 2 Andromedas Girdle 00 50 33 50 N 2 Whales Belly 01 35 12 S 3 Rams head 1 48 21 49 N 3 Whales mouth 2 44 2 42 N 2 Medusas head 2 46 39 35 N 3 Perseus right side 2 59 48 33 N 2 Buls eye 4 16 15 46 N 1 Goat 4 52 45 37 N 1 Orions left foot 4 58 8 38 S 1 Orions left shoulder 5 6 5 59 N 3 First , in Orions girdle 5 15 00 35 S 3 Second , in Orions girdle 5 19 1 27 S 3 Third , in Orions girdle 5 23 2 9 S 3 Orions right shoulder 5 36 7 18 N 2 The Wagoner 5 39 44 56 N 2 Bright foot of the Twins 6 18 16 39 N 3 Great Dog 6 30 16 13 S 1 Castor or Apollo 7 12 32 30 N 2 The little Dog 7 22 6 6 N 2 Pollux or Hercules 7 24 28 48 N 2 Hidra's heart 9 10 7 10 S 1 Lions heart 9 50 13 39 N 1 Lions Neck 9 50 21 41 N 3 Great Beares rump 10 40 58 43 N 2 Lions back 11 30 22 4 N 2 Lions tail 11 31 16 30 N 1 The Virgins girdle 12 38 5 20 N 3 First in the great Bears taile next the rump 12 38 57 51 N 2 Vindemiatrix 12 44 15 51 N 3 Virgins Spike 13 7 9 19 S 1 Middlemost in the Great Beares tail 13 10 56 45 N 2 Last in the end of the Great Beares tail 13 34 51 05 N 2 Arcturus 14 00 21 03 N 1 South Ballance 14 32 14 33 S 2 Brightest in the Crown 15 24 27 43 N 3 North Ballance 14 58 08 03 S 3 Serpentaries left hand 15 56 02 46 S 3 Scorpions heart 16 08 25 35 S 1 Serpentaries left knee 16 18 09 46 S 3 Serpentaries right knee 16 49 15 12 S 3 Hercules head 16 59 14 51 N 3 Serpentaries head 17 19 12 52 N 3 Dragons head 17 48 51 36 N 3 Brightest in the Harp 18 25 38 30 N 1 Eagle or Vultures heart 19 34 08 00 N 2 Upper horn of Capricorn 19 58 13 32 S 3 Swans tail 20 30 44 05 N 2 Left shoulder of Aquarius 21 13 07 02 S 3 Pegasus mouth 21 27 08 19 N 3 Right shoulder of Aquarius 21 48 01 58 S 3 Fomahant 22 39 31 17 S 1 Pegasus upper Wing , or Marchab 22 48 13 21 N 2 Pegasus Lower Wing . 23 55 33 25 N 2 Mr. Sutton knowing that some of the Tables of Declination and Right Ascension in our English Books are antiquated and removed forward , took the pains to Calculate a new Table of Right Ascensions and Declinations to serve for the future , in regard I was not at leisure to accomplish it ; which followeth . Dayes . January ☉ R A. ☉ Decl. H. M. D. M. 1 19 35 21 46 2 19 39 21 36 3 19 43 21 25 4 19 47 21 14 5 19 51 21 03 6 19 56 20 52 7 20 00 20 40 8 20 04 20 27 9 20 09 20 15 10 20 13 20 01 11 20 17 19 48 12 20 22 19 34 13 20 26 19 20 14 20 30 19 05 15 20 34 18 50 16 20 38 18 35 17 20 42 18 19 18 20 46 18 03 19 20 50 17 47 20 20 54 17. 30 21 20 58 17 13 22 21 03 16 56 23 21 07 16 39 24 21 11 16 21 25 21 15 16 03 26 21 19 15 44 27 21 23 15 26 28 21 27 15 07 29 21 31 14 48 30 21 35 14 28 31 21 38 14 09 Dayes . February ☉ R A. ☉ Decl. H. M. D. M. 1 21 42 13 49 2 21 46 13 29 3 21 50 13 08 4 21 54 12 48 5 21 58 12 28 6 22 02 12 06 7 22 06 11 45 8 22 10 11 24 9 22 14 11 03 10 22 17 10 41 11 22 21 10 19 12 22 25 9 57 13 22 29 9 35 14 22 33 9 13 15 22 36 8 51 16 22 40 8 26 17 22 44 8 06 18 22 48 7 43 19 22 52 7 20 20 22 55 6 57 21 22 59 6 34 22 23 03 6 11 23 23 06 5 48 24 23 10 5 24 25 23 13 5 01 26 23 17 4 37 27 23 21 4 14 28 23 25 3 51 29         30         31         Dayes . March ☉ R. A. ☉ Decl. H. M. D. M. 1 23 28 3 27 2 23 32 3 03 3 23 36 2 39 4 23 39 2 16 5 23 43 1 52 6 23 46 1 29 7 23 50 1 05 8 23 53 0 41 9 23 57 0 18 10 0 01 North 6 11 0 05 0 30 12 0 08 0 53 13 0 12 1 17 14 0 15 1 41 15 0 19 2 04 16 0 23 2 28 17 0 26 2 51 18 0 30 3 15 19 0 33 3 38 20 0 37 4 01 21 0 41 4 24 22 0 44 4 48 23 0 48 5 11 24 0 52 5 34 25 0 55 5 57 26 0 59 6 19 27 1 03 6 42 28 1 06 7 04 29 1 10 7 27 30 1 14 7 49 31 1 17 8 11 Dayes . April . ☉ R. A. ☉ Decl. H. M. D. M. 1 1 21 8 33 2 1 25 8 55 3 1 29 9 17 4 1 33 9 38 5 1 36 9 51 6 1 40 10 21 7 1 44 10 42 8 1 47 11 03 9 1 51 11 24 10 1 54 11 44 11 1 58 12 05 12 2 02 12 24 13 2 06 12 45 14 2 10 13 04 15 2 13 13 24 16 2 17 13 43 17 2 21 14 02 18 2 25 14 21 19 2 29 14 40 20 2 32 14 58 21 2 36 15 16 22 2 40 15 34 23 2 44 15 52 24 2 48 16 09 25 2 51 16 27 26 2 55 16 43 27 2 59 17 00 28 3 03 17 16 29 3 07 17 32 30 3 10 17 48 31         Dayes . May ☉ R. A. ☉ Decl. H. M. D. M. 1 3 14 18 03 2 3 18 18 18 3 3 22 18 33 4 3 26 18 48 5 3 30 19 02 6 3 34 19 16 7 3 38 19 29 8 3 42 19 42 9 3 46 19 55 10 3 50 20 08 11 3 54 20 20 12 3 58 20 32 13 4 02 20 44 14 4 06 20 55 15 4 10 21 05 16 4 14 21 16 17 4 18 21 26 18 4 22 21 36 19 4 26 21 45 20 4 30 21 54 21 4 34 22 02 22 4 38 22 11 23 4 42 22 19 24 4 46 22 26 25 4 50 22 33 26 4 54 22 40 27 4 58 22 46 28 5 02 22 52 29 5 06 22 57 30 5 11 23 02 31 5 15 23 07 Dayes . June . ☉ R. A. ☉ Decl. H. M. D. M. 1 5 19 23 11 2 5 23 23 15 3 5 27 23 19 4 5 31 23 22 5 5 36 23 24 6 5 40 23 26 7 5 44 23 28 8 5 48 23 29 9 5 52 23 30 10 5 56 23 31 11 6 00 23 31½ 12 6 04 23 31 13 6 08 23 30 14 6 12 23 29 15 6 17 23 28 16 6 21 23 26 17 6 25 23 24 18 6 29 23 21 19 6 33 23 18 20 6 38 23 14 21 6 42 23 11 22 6 46 23 06 23 6 50 23 01 24 6 54 22 56 25 6 58 22 51 26 7 02 22 45 27 7 06 22 39 28 7 10 22 32 29 7 14 22 25 30 7 19 22 17 31         Dayes . July ☉ R. A. ☉ Decl. H. M. D. M. 1 7 23 22 09 2 7 27 22 01 3 7 31 21 52 4 7 35 21 43 5 7 39 21 34 6 7 43 21 24 7 7 47 21 14 8 7 51 21 04 9 7 55 20 53 10 7 59 20 42 11 8 03 20 30 12 8 07 20 18 13 8 11 20 06 14 8 15 19 54 15 8 19 19 41 16 8 23 19 28 17 8 27 19 14 18 8 31 19 00 19 8 35 18 46 20 8 39 18 32 21 8 43 18 17 22 8 47 18 02 23 8 51 17 46 24 8 55 17 31 25 8 58 17 15 26 9 02 16 59 27 9 06 16 42 28 9 10 16 25 29 9 14 16 08 30 9 17 15 51 31 9 21 15 33 Dayes . August ☉ R. A. ☉ Decl. H. M. D. M. 1 9 25 15 16 2 9 29 14 58 3 9 33 14 39 4 9 37 14 21 5 9 40 14 02 6 9 44 13 43 7 9 48 13 24 8 9 51 13 04 9 9 55 12 45 10 9 58 12 25 11 10 02 12 05 12 10 06 11 45 13 10 10 11 25 14 10 14 11 04 15 10 17 10 43 16 10 21 10 22 17 10 25 10 01 18 10 28 9 40 19 10 32 9 18 20 10 35 8 57 21 10 39 8 35 22 10 43 8 14 23 10 46 7 52 24 10 50 7 30 25 10 53 7 07 26 10 57 6 45 27 11 01 6 22 28 11 04 6 00 29 11 08 5 37 30 11 11 5 14 31 11 15 4 51 Dayes . September ☉ R. A. ☉ Decl. H. M. D. M. 1 11 19 4 28 2 11 23 4 6 3 11 26 3 42 4 11 30 3 19 5 11 33 2 56 6 11 37 2 33 7 11 41 2 10 8 11 44 1 46 9 11 48 1 23 10 11 51 0 59 11 11 55 0 3● 12 11 59 0 12 13 12 02 South 11 14 12 06 0 35 15 12 09 0 58 16 12 13 1 22 17 12 17 1 46 18 12 20 2 09 19 12 24 2 33 20 12 27 2 56 21 12 31 3 19 22 12 35 3 43 23 12 38 4 06 24 12 42 4 30 25 12 45 4 53 26 12 49 5 16 27 12 53 5 39 28 12 57 6 02 29 13 01 6 26 30 13 04 6 49 31         Dayes . October ☉ R. A. ☉ Decl. H. M. D. M. 1 13 08 7 11 2 13 12 7 34 3 13 15 7 57 4 13 19 8 19 5 13 22 8 42 6 13 26 9 04 7 13 30 9 26 8 13 34 9 48 9 13 38 10 10 10 13 41 10 31 11 13 45 10 53 12 13 49 11 14 13 13 53 11 36 14 13 57 11 57 15 14 00 12 18 16 14 04 12 38 17 14 08 12 59 18 14 12 13 19 19 14 16 13 39 20 14 20 13 59 21 14 24 14 19 22 14 28 14 38 23 14 32 14 57 24 14 36 15 16 25 14 39 15 35 26 14 43 15 5● 27 14 47 16 1● 28 14 51 16 29 29 14 55 16 47 30 14 59 17 04 31 15 03 17 21 Dayes . November ☉ R. A. ☉ Decl. H. M. D. M. 1 15 07 17 38 2 15 11 17 54 3 15 15 18 10 4 15 19 18 26 5 15 23 18 41 6 15 27 18 56 7 15 31 19 11 8 15 36 19 26 9 15 40 19 40 10 15 45 19 53 11 15 49 20 07 12 15 53 20 19 13 15 58 20 32 14 16 02 20 44 15 16 07 20 56 16 16 11 21 08 17 16 15 21 19 18 16 19 21 29 19 16 23 21 39 20 16 28 21 49 21 16 32 21 58 22 16 36 22 08 23 16 40 22 16 24 16 44 22 24 25 16 49 22 32 26 16 53 22 39 27 16 57 22 46 28 17 02 22 52 29 17 06 22 58 30 17 11 23 03 31         Dayes . December . ☉ R. A. ☉ Decl. H. M. D. M. 1 17 15 23 08 2 17 20 23 13 3 17 25 23 17 4 17 29 23 20 5 17 34 23 23 6 17 38 23 26 7 17 42 23 28 8 17 47 23 29 9 17 51 23 30 10 17 56 23 31 11 18 00 23 31½ 12 18 05 23 31 13 18 09 23 30 14 18 14 23 29 15 18 19 23 27 16 18 24 23 25 17 18 28 23 22 18 18 33 23 19 19 18 37 23 15 20 18 41 23 11 21 18 45 23 07 22 18 49 23 02 23 18 54 22 56 24 18 58 22 50 25 19 03 22 43 26 19 07 22 36 27 19 11 22 29 28 19 16 22 21 29 19 20 22 13 30 19 25 22 04 31 19 30 21 55 A Rectifying Table for the Suns Declination .   Years Years Years   1657 1661 1665 1669 1673 1659 1663 1667 1671 1675 1660 1664 1668 1672 1676 Moneths min. min. min. January 3 s 2 a 5 a 4 s 3 a 7 a 5 s 4 a 9 a February 5 s 5 a 10 a 5 s 5 a 11 a 6 s 5 a 11 a March 6 s 5 a 13 s 5 a 5 s 12 a 5 a 5 s 12 a April 5 a 5 s 11 a 5 a 5 s 10 a 4 a 4 s 9 a May 4 a 4 s 8 a 3 a 3 s 6 a 2 a 2 s 4 a June 1 a 1 s 2 a 0 s 0 a 0 s 1 s 1 a 3 s July 2 s 2 a 5 s 3 s 3 a 7 s 4 s 4 a 9 s August . 5 s 5 a 10 s 5 s 5 a 11 s 6 s 5 a 12 s Septēber 6 s 5 a 13 s 6 a 5 s 13 a 6 a 5 s 12 a October 6 a 5 s 12 a 5 a 5 s 11 a 4 a 5 s 9 a Novem. 3 a 4 s 7 a 2 a 3 s 5 a 1 a 2 s 3 a Decemb. 0 a 1 s 1 a 1 s 0 a 1 s 2 s 1 a 3 s The use of the Rectifying Table . NOte that the minutes under the respective years is to be added or substracted to or from the Suns Declination in the former Table , as is noted with the letter a or s : and also note that the first figure in each moneth stands for the first 10 dayes of the moneth , and the second for the second 10 days , & the third for the last 10 dayes , except in March or September , which in March will be the first 9 dayes only , and in September the first 12 dayes . Example . I would know the Suns Declination the 15 day of May 1668. Now because this day of the moneth falls in the second 10 dayes , I look in the Table under the year 1663 , and right against May you shall finde that in the second place of the moneth stands 6 a , which shews me that I must adde 6 minutes to the Suns Declination in the former Table 21 degrees 5 min. that stands against the 15 day of May , and then I find that the Sun will have 21 deg . 11 min. of North Declination , and so for the rest , which will never differ above two minutes from the truth , but seldome so much , and for the most part true . Note that the former Table of the Suns Declination is fitted exactly for the year 1666. by the Rules Mr. Wright gives in his Correction of Errours , and from his Tables , and may indifferently serve for the years 1658 , 1662. 1670 , 1674 , without any sensible errour , and the Table of Right Ascensions will not vary a minute of time in many years . FINIS . Errours in the Horizontal Quadrant . PAge 5 line 6 in an Italian letter should not have been distinct , nor in another letter from the former line . page 5. line 9. for quarter , read half . p. 5. l. 13. r. of a quadrant . p. 11. l. 7. r. 63 d. 26′ . p. 19. l. 7. r. the same day to . p. 23. l. 17. r. and ends at 32′ past 9. p. 27. l. 7. for N R , r. N Z. p. 28. l. 4. r. in the parallel . p. 30. l. 9 , & l. 10. r. 23 d. 31′ . p. 38. l. 4. r. Is to the sine . p. 50. l. 5. r. whereof the Diameter . AN APPENDIX Touching REFLECTIVE DIALLING . By JOHN LYON : Professor of this , or any other part of the Mathematicks , neer Sommerset House in the Strand . LONDON , Printed Anno Domini , 1658. DIRECT DIALLING By a Hole or Nodus . To draw a Dial under any window that the Sun shines upon by help of a thread fastened in any point of the direct Axis found in the Ceiling , and a hole in any pane of glasse , or a knob or Nodus upon any side of the window or window-post . CONSTRUCTIO . FIrst , draw on pastboard or other material , an Horizontal Dial for the Latitude proposed . Then by help of the Suns Azimuth , which may be found by help of a general Quadrant , at any time , or by knowing the true hour of the day with the help of the said Horizontal Dial : and draw that true Meridian from the hole or Nodus proposed , both above on the Cieling , and below on the walls and floor of the Room ; so that if a right line were extended from the said hole or Nodus by any point in any of those lines , it would be in the meridian Circle of the World. To finde a point in the direct Axis of the world , which will ever fall to be in the said Meridian , in which point the end of a thread is to be fastened . FIrst , fix the end of a thread or small silk in the center of the Hole or Nodus , and move the other end thereof up or down in the said meridian formerly drawn on the Cieling or wall , untill by applying the side of a Quadrant to that thread , it is found to be elevated equal to the Latitude of the place ; so is that thread directly scituated parallel to the Axis of the world , and the point where the end of that thread toucheth the meridian either on the Cieling or wall , is that point in the direct Axis sought for , wherein fix one end of a thread , ( which thread will be of present use in projecting of hour-points in any place proposed , then : To find the Hour-points either under the window , or any other convenient place in the Room . Place the center of the said Horizontal Dial in the Center of the Hole or Nodus ; also scituate the said Dial exactly parallel to the Horizon , and the meridian of the said Dial in the meridian of the world , which ( as before ) may easily be done , if at that instant you know the true hour of the day . ) Then take the thread whose end is fixed in a point in the direct Axis , and move it to and fro , until the said thread doth interpose between your eye , and the hour-line on the said Horizontal Dial which you intend to draw , and then keeping your eye at that scituation , make a point or mark in any place where you please , or under the window , so that the said thread or string may interpose between that point or mark so made , and your eye , as aforesaid ; which said point so sound will shew the true time of the day at that hour all the year long , the Sun shining thereon , so will that point , together with the said thread , serve to shew the hour , instead of an hour-line . In like manner , the said thread fixed in the Axis may be again moved to and fro , until the said thread doth interpose between the eye and any other hour-line desired on the said Horizontal Dial and then ( as before ) make another point or mark in any place at pleasure , or under the said window , by projecting a point from the eye , so that the said thread also interpose between that point to be made and the eye , so will that point so found shew the true time of the day for the same hour that did the hour line on the said Horizontal Dial , which was shadowed by the said thread . In like manner may be proceeded ( by help of that thread , and the several hour-lines on the said Horizontal Dial ) to finde the other hour-points which must have the same numbers set to them as have the hour-lines on the said Horizontal Dial. Otherwise to make a Dial from a hole in any pane of glasse in a window , and to graduate the hour-lines below on the Sell , or Beam , or on the ground , that hole is supposed to be the center of the Horizontal Dial , and being true placed , the stile thereof , if supposed continued , will run into the point in the Meridian of the Cieling before found , where a thread is to be fixed ; then let one extend a thread fastned in the center of the Horizontal Dial parallelly to the Horizon , over each respective hour-line , and holding it steady , let another extend the thread fastened in the Meridian , in the Cieling along by the edges of the former Horizontal thread , and so this latter thread will finde divers points on the ground , through which if hour-lines be drawn , and the Sun shine through the hole in the pane of Glasse before made , the spot of the Sun on the ground shall shew the time of the day . For the points that will be thus found on the Beam or Transome , the thread fixed in the Cieling , or instead of it a piece of tape there fixed must be moved so up and down , that the spot of the Sun may shine upon it , and being extended to the Transome or Beam graduated with the hour-lines , as before directed , it there shews the time of the day . Here note , that it will be convenient to have that pane of Glasse darkened through which that spot is to shine . In like manner may a Dial be made from a nail head , a knot in a string tied any where a crosse , or from any point driven into the bar of a window , and the hour-lines graduated upon the Transome or board underneath . To make a Reflected Dial on the Ceiling of the Room is onely the contrary of this , by supposing the Horizontal Diall with its stile to be turned downwards , and run into the true meridian on the ground , where the thread is to be fixed , and to be extended along by the former Horizontal thread ( held over the respective hours as before ) upward , to find divers points in the Cieling , as shall afterwards be shewed . Of Dials to stand in the Weather . These may be also made by help of an Horizontal Dial. DRive two nails or pins into the wall , on which the edge of a Board of competent breadth may rest , then to hold up the other side of the Board , drive two hooks into the wall above , whereto with cord or line the outside of the Board may be sustained , and this Board being Horizontal , place the Horizontal Dial its Meridian-line in the true Meridian of the world . If a Plain look towards the South , the stile of the Horizontal Dial continued by a thread from the center will run into the Plain , which note to be the center of the new Dial , as also that line is the new stile , which must be supported with stayes , when you fix it up . By a thread from the center laid over every hour-line on the Horizontal Dial , cross the Horizontal line of the Plain , which note with the same hours the Horizontal Dial hath . The hour-lines on the Plain are to be drawn from the center before found through those points , and so cut off by the Dial , or continued at pleasure . If the Center of the Dial be assigned before you begin the work , in such Cases you may remove the Horizontal Dial up and down , keeping it still to the true position or hour , till you finde the Axis or stile run into the Center . But if the Plain look into the East or West , then possibly the Axis of the Horizontal Dial will not meet with the Plain : in such Cases you must fix a board so , that it may receive the Axis , ( the board being perpendicular to the Plain ) this stile or Axis is to be fastened to the Plain by two Rests , the hour-lines may be drawn by the eye , or shadowed out by a Light : Bring the thread that represents the Axis or stile into any hour-point ( on the Horizontal Dial ) by your eye or shadow ; at the same time the thread or shadow making marks on the Plain , shews where the hour-lines are to passe . After the same manner any hour-line is to be drawn over any irregular or crooked Plain . Further observe , that any point in the middle , or neer the end of the stile will as well shew the hour of the Day , as the whole stile . Of Refracted Dials . IF you stick up a pin or stick , or assign any point in any concave Boul or Dish , to shew the hour , and make that the center of the Horizontal Dial , assigning the meridian-line on the edges of the Boul , point out the rest of the hour-lines also on the edges of the Boul , and taking away the Horizontal Dial , elevate a string or thread from the end of the said pin fastned thereto over the Meridian-line equal to the Elevation of the Pole or the Latitude of the place ; then with a candle , or if you bring the thread to shade upon any hour-point formerly marked out on the edges of the Boul , at the same time the shade in the Boul is the hour line . And if the Boul be full of water , or any other liquor , you may draw the hour-lines , which will never shew the true hour , unlesse filled with the said Liquor again . Reflected Dialling . To draw a Reflected Dial on any Plain or Plains , be they never so Gibous , and Concave , or Convex , or any irregularity whatsoever , the Glass being fixed at any Reclination at pleasure , ( provided it may cast its Reflex upon the places proposed . ) Together with all other necessary lines or furniture thereon , viz. the Parallels of Declination , the Azimuth lines , the Parallels of Altitude ( or proportions of shadows ) the Planetary Hour-lines , and the Cuspis of those Houses which are above the Horizon , &c. 1. If the Glasse be placed Horizontal upon the Transome of a window , or other convenient place : How upon the Wall or Cieling whereon that Glasse doth reflect to draw the Hour-lines thereon , although it be never so irregular , or in any form whatsoever . CONSTRUCTIO . FIrst , draw on Pastboard or other Material an Horizontal Dial for the Latitude proposed . Then by help of the Azimuth , or at the time when the Sun is in the Meridian ; or by knowing the true hour of Day , whereby may be drawn several lines on the Cieling , Floor , and Walls of the Room : so as in respect of the center of the Glasse they may be in the true Meridian-circle of the World : For if right lines were extended from the center of the said Glasse by any point , though elevated in any of those lines so drawn , it would be directly in the Meridian Circle of the World. Now all Reflective Dialling is performed from that principle in Opticks , which is , That the angle of Incidence is equal to the angle of Reflection . And as any direct Dial may be made by help of a point found in the direct Axis , so may any Reflected Dial be also made by help of any point found in the Reflected Axis . And in regard the reflected Axis for the most part will fall above the Horizon of the Glasse without the window , so that no point there can be fixed , therefore a point must be found in the said Reflected Axis continued below the Horizontal of the said Glasse , until it touch the ground or floor of the Room in some part of the Meridian formerly drawn , which point will be the point in the reversed Axis desired , and may be found , as followeth . One end of the thread , being fixed at or in the center of the said Glasse , move the other end thereof in the meridian formerly drawn below the said Glasse , until the said reversed Axis be depressed below the Horizon , as the direct Axis was elevated above the Horizon , which may be done by applying the side or edge of a Quadrant to the said thread , and moving the end thereof to and fro in the said meridian , until the thread with a plummet cut the same degree as the Pole is above the Horizontal Glasse , and then that point where the end of the thread toucheth the Meridian either on the floor or wall of the room , is the point in the reflected reversed Axis sought for . Now if the Reversed Axis cannot be drawn from the Glasse by reason of the jetting of the window or other impediment , that point in the reverse Axis may be found by a line parallel thereto , by fixing one end of it on the Glasse , and the other end in the meridian , so as that it may be parallel to the floor or wall in which the reversed Axis-point will fall , and finde the Axis point from that other end of the lath : so if the same Distance be set from that point backward in the Meridian on the floor , as is the Lath , the point will be found in the Reversed Axis desired . Thus having found a point in the reflected reversed Axis ; it is not hard , by help whereof and the Horizontal Dial , to draw the reflected hour-lines on any Cieling or Wall , be it never so concave or convex . To do which : First note , that all straight lines in any projection on any Plain , do always represent great Circles in the Sphere , such are all the hour-lines . Place the center of this Horizontal Dial in the center of the Glasse , the hour-lines of the said Dial being horizontal , and the Meridian of the said Dial in the Meridian of the world , which may be done by plumb lines let fall from the meridian on the Cieling : Then fix the end of a thread or silk in the said center of the Dial or Glasse , and draw it directly over any hour-line on the Dial which you intend to draw , and at the further side of the room , and there let one hold or fasten that thread with a small nail . Then in the point formerly found on the reversed Axis on the ●oor , fix another thread there ( as formerly was done in the center of the Diall ) then take that thread , and make it just touch the thread ( on the hour-line of the Horizontal Dial extended ) in any point thereof , it matters not whereabouts , and mark where the end of that thread toucheth the Wall or Cieling , and there make some mark or point . Then again move the same thread higher or lower at pleasure , till it , as formerly touch the said same hour thread , and mark again whereabouts on the wall or Cieling , the end of the said thread also toucheth . In like manner may be found more points at pleasure , but any two will be sufficient for the projecting or drawing any hour-line on any plain , how irregular soever . For if you move a thread , and also your eye to and fro , until you bring the said thread directly between your eye and the points formerly found , you may project thereby as many points as you please at every angle of the Wall or Cieling , whereby the reflected hour-line may be exactly drawn . Again , in like manner remove the said thread fastned in the center of the Horizontal Dial , ( which also is the center of the Glasse ) on any other hour-line desired to be drawn , and as before fasten the other end of the thread , by a small nail , or otherwise at the further side of the room , but so that the said thread may lie just on the hour-line proposed to be drawn on the Horizontal Dial. Then ( as before ) take the thread fastened in the point on the reflected Axis , and bring it to touch the thread of the hour-line in any part thereof , and mark where the end of that thread toucheth the said Wall or Cieling : Then again ( as before ) move the said thread so , as that it only touch the said thread of the hour-line in any other part thereof , and also mark where the end of that thread toucheth the said Wall or Cieling : So is there found two points on the Wall or Cieling , being in the reflected hour-line desired , by help of which two points the whole hour-line may be drawn ; for if ( as before ) a thread be so scituated , that it may interpose between the eye and the said two points found , you may make many points at pleasure , whereunto the said thread may also interpose , which for more conveniency may be made at every angle or bending of the Wall or Cieling , be they never so many : So that if lines be drawn from point to point , that said reflected hour-line will be also exactly drawn . In like manner may the other hour-lines be drawn so , that the Reflex or spot of the Sun from the said Horizontal Glasse scituated in the said window ( as before ) shining amongst the said reflected hour-lines drawn on the wall or Cieling , will exactly shew the hour of the day desired . Now if lines be drawn round about the said Room , equal to the Horizon of the said Glasse , it will shew when the Sun is in or neer the Horizon . To draw the Aequator and Tropicks on any Wall or Cieling to any Horizontal reflecting Glasse . 1 To draw the Reflected Aequator or Equinoctial-line on the Wall or Cieling , which represents a great Circle . TAke the thread fixed in the Center of the Glasse , and move the end thereof to and fro in the meridian line drawn on the Cieling , untill by help of a Quadrant the said thread be elevated equal to the complement of the Latitude , ( which will be alwayes perpendicular to the reversed Axis ) marking in the Meridian where the end of that thread falls , then on that point and the said meridian line on the Cieling erect a perpendicular line , which line may be continued on any plane whatsoever , and is the reflected Equinoctial line desired . Note that all great Circles are right lines , & are alwayes drawn or projected from a right line . 2. To draw the Tropicks . Note , that all Parallels of Declination are lesser Circles , and are Conick Sections . FIrst , make or take out of some Book a Table of the Suns Altitude for each hour of the day , calculated for the place or Latitude proposed , when the Sun is in either of the Tropicks . Then take the thread fixed in the center of the Glasse , and by applying one side of a quadrant to the said thread , and moving one end of it to and fro in the hour-line proposed , elevate the said thread answerable to the Suns height in that hour , when he is in that Tropick you desire to draw , and mark where the end of that thread so elevated toucheth in that hour-line proposed . So may you in like manner finde a several point in each hour-line for the Suns height in that Tropick , whereby a line may be drawn on the Wall or Cieling from point to point formerly made in the said hour-lines , which the Tropick desired . In like manner may any parallel of Declination be drawn : If there be first calculated a Table of the Suns altitude at all hours of the day , when the Sun hath any Declination proposed , whereby may be drawn either the Parallels of the Suns place , or the parallels of the length of the day . To draw the parallels of Declination to any Reflected Glasse most easily , by help of a Trigon first made on past board or other material . FIx the Trigon to the reflected roversed Axis , so that the center of the Trigon may be in the center of the Glasse , then will the Equinoctial on the Trigon be perpendicular to the said Axis : then take the thread fixed in the center of the Glasse , and lay it along either of the Tropicks or other parallels of Declination required , which is drawn on the said Trigon , which thread must be continued so , that the end thereof may touch any hour-line , and on that hour-line mark the point of touch , the thread being still laid on the same parallel of declination on the Trigon : in the same manner finde a point in each hour-line . Lastly , draw a line by those points so found , which will be the Tropick-line or other parallel of declination , as the thread was laid on , on the Trigon . To draw the Azimuth-lines on any Wall or Cieling to any Horizontal reflecting Glasse . Note that all Azimuths are great Circles . FIrst , find a vertical point , either above to the Zenith , or below to the Nadir of the Glasse ( by some called a perpendicular or plumb line ) and mark in what point it cuts the floor of the room , which point I call the reflected vertical point , wherein the end of a thread is to be fixed : For by a point found in the reflected Axis of the Horizon the Azimuths may be drawn , as by a point found in the reflected Axis of the Equinoctial the hour-lines may be drawn . Then on pastboard or other material draw the points of the Compasse or other degrees , and fix the center thereof in the center of the Glasse , and the meridian thereof in the meridian of the world , as was shewn in drawing the hour-lines , being careful to place it horizontal . Then take the thread fixed in the place of the glasse , and draw it over any Azimuth , which is desired to be drawn , and at the further side of the Room fasten that thread with a small nail as it was in drawing the reflected hour-lines : Then take the thread whose end is fastened in the said reflect vertical point , and bring that thread so as just to touch the said horizontal thread , and augment it , until the end thereof touch the wall or Cieling , and there make a mark or point . In like manner , move the said thread , whose end is fastened in the said vertical point , higher or lower at pleasure , till as formerly it touch the said horizontal thread , and mark again whereabouts the end thereof toucheth the said Wall or Cieling : Now by help of these two points found in the reflected Azimuth line , the whole Azimuth line may be drawn ; for if ( as before in drawing the Hour-lines ) a thread be so scituated , that it may interpose between the eye and the said two points , you may make many points at pleasure , to which the said thread so situated may also interpose , which may be made at every angle or bending of the wall or Cieling ( as before ) whereby the reflected Azimuth-line desired may be drawn . In like manner may the other reflected Azimuth lines be drawn . Also there may be lines drawn parallel to the Horizon round about the room , by help of the thread fixed in the center of the Glasse , and a Quadrant for the elevation thereof , which will shew the Suns altitude at any appearance thereof . Thus have I shewed the drawing of a Reflected Dial from an Horizontal Glasse , with all the usual furniture thereon , though the wall or place on which it is to be drawn be never so gibous or irregular , or in what shape soever . Now the Glasse may be exactly situated Horizontal , if you draw a reflected parallel for the present day , and know also the true hour , and so place the Glasse , that the spot or reflex of the Sun may fall thereon on the Cieling , for there is no way by an Instrument to do it , the Glasse is so small . Of Reclining Reflecting Glasses . Reflected Dialling from any Reclining Glasse . I shall now shew how to draw any Reflected Dial , with all the Furniture ( that possible may be ) the Glass being set at any possible Reclination . In the drawing of which there is principally to be considered , 1 The Reflected Horizon . 2 The Reflected Meridian . Note , the Horizon & Meridian are two great circles . 1 To draw the Reflected Horizon according to the situation of any reclining Glasse whatsoever . FIrst , let two pieces of nealed wire be fastened on the window on each side of the said Glasse , the ends thereof being without the room in the air , at whose ends let there be fastned a thread which may be pulled straight at pleasure , by bending of the wire , then bend those wires upward or downward , until the thread fastened at the end of each wire be exactly horizontal with the center of the Glasse , which may be tried by a quadrant : Then I tie a string or thread cross the room , in such sort that I may from most part of the thread see the reflecting glass , and therein the said horizontal thread without the room : Then on the said thread cross the room , I tie a slipping knot to move to and fro at pleasure , which knot I move to and fro on the said thread , until by looking in the said Glasse I finde from my eye the said knot and part of the horizontal thread without , all as it were in a right line , the one interposing the sight of the other . Then being careful to keep the knot in that position , fasten one end of a thread in the place of the center of the reclining reflecting glasse , and bring that thread so , as just to touch the aforesaid knot , augmenting that thread , until the end thereof touch the wall or Cieling , and there make a mark or point , so is there one point found on the Wall or Cieling in the Reflected Horizon of the World. Then I begin again , and remove the position of that thread ( which went overthwart the Room ) either higher or lower at pleasure , still having regard that I may from the most part of the said thread see the Reflecting Glasse , and therein the same horizontal thread without the room . Then , as before , I move the said knot on the said thread to & fro , until ( as before ) by looking in the said Glasse I find from my eye the said knot , and part of the Horizontal thread both in one right line , the one interposing the sight of the other ; and by the said knot I bring that thread , whose end is fastened in the center of the said glasse , and keeping it just to touch the said knot , I continue it , until the end thereof touch the Wall or Cieling , as before , and there I make another mark or point ; so is there two points found in the said reflected Horizon on the wall or Cieling . By which said two points , if a thread ( as before ) be so scituated , that it may interpose between the eye and the said two points , there may be many points made to be in the same interposition of the thread , which ( as before ) may be made at every bending or angle of the Wall or Cieling , whereby the reflected Horizon desired may be drawn , by drawing a line from point to point round about the Room ; Which wil be the true reflected Horizon according to the situation of the glasse . 2 To draw the Reflected Meridian , according to the situation of any Reclining Glasse whatsoever . FIrst , take a lath or thin piece of wood of any convenient length at pleasure , as some one and an half , or two foot long , and at each end thereof make a hole , the one to hang a thread and plummet , and the other is to put a small nail therin to fasten it in some part of the window over the center of the Glasse , so that the thread and plummet may hang without the room : then by help of the Suns Azimuth you may draw the meridian line , ( as before ) as if the Glasse were horizontal , and move the lath with the thread and plummet at the end of it to and fro , until the thread and plummet be in the direct meridian of the world with the center of the Glasse . Then ( as before ) tie a thread crosse the room , in such sort that from or by some part of the said thread both the Reclining glasse and the thread to which the plummet is fastened may be seen at one time . Then ( as before ) on the said thread , which crosses the room , I tie a slipping knot , which I move to and fro on the said string , until by looking in the said Glasse I find from my eye the said knot and some part of the perpendicular thread without , all as it were in one right line , the one shadowing or interposing the sight of the other , being then very careful to keep that knot in the same position , then take the thread ( whose end whereof being fastened in the said center of the Glasse ) and bringing it just to touch the said knot , I augment that thread , until the end thereof touch the said wall or Cieling , and the said thread also touch the knot , as before : then in that place where the end of the said thread toucheth the wall or Cieling , I make a mark , which mark or point will be directly in the reflected meridian of the world , according to the situation of that Glasse . Then again I remove that thread ( overthwart the room ) on which the said knot is , either higher or lower then it formerly was at pleasure , still having regard that from some part of the said thread within , you may see both the Reclining Glasse , and the perpendicular thread without at one time ; and ( as before ) move the said slipping knot on the said thread , until by looking in the said Reclining Glasse , you see the said knot and some part of the perpendicular thread without in one right line , so as the one shadows or hinders the sight of the other , ( as before ) which knot then must not be removed from its situation , then take that thread ( whose end is fastened in the Glasse ) and bring it to touch that knot , the end of the said thread being continued to touch the wall or Cieling : so is that point of touch on the Cieling another point found in the Reflected Meridian of the world . So is there two points found in the said Reflected Meridian , on the wall or Cieling ; by which , if a thread ( as before ) be so situated , that it may interpose between the eye and the said two points , many points thereby in the said reflected Meridian may be made at every bending or angle of the wall or Cieling , whereby the Reflected meridian desired may be drawn , by drawing a line from point to point obliquely in the Room , which will be the true Reflected Meridian of the world , according to the situation of that Glasse . Now this Reflected Horizon and Meridian being first drawn , they will be of great use in drawing the Hour-lines , together with all the furniture that possibly can be drawn on any Diall . To draw the Reflected Hour-lines to any Reclining Glasse on any plane whatsoever , that the Sun will be reflected on : By help of an ordinary Horizontal Dial for that Latitude . FIrst , extend several threads from the center of the Glasse to the extremity of the Reflected Horizon in the Room ( which for more conveniency and use may be the several hour-lines , and may also serve as a bed to situate the Horizontal Diall on the Reflected Horizon ) having regard to situate the center of the Dial on the center of the Glasse , and the Meridian of that Dial on the Reflected Meridian of the World : Then to finde the point in the Reflected reversed Axis on the floor of the Room ; Take a thread , one end thereof being fastened in the center of the Glasse , and move the other end thereof to and fro in the reflected meridian under the Reflected Horizon , until by help of a Quadrant the said thread is found to be depressed under the reflected Horizon , equal to the latitude of the place , and where the end of the said thread intersects or meets the Reflected Meridian either on the floor or wall , that point is the reflected reversed Axis , as was required . In which point fasten one end of a thread , which thread will be of great use in drawing the reflected hour-lines on any wall or Cieling whatsoever . Now if this thread , whose end is fastened in a point on the reflected reversed Axis , be taken and brought to touch any part of any one of the threads of the hour-lines ( produced to and fastened in the reflected Horizon ) the said thread being continued so , as the end thereof may touch the wall or Cieling , and also any part of the said thread touch the hour-line or thread proposed ; that point on the wall or Cieling is in the reflected hour-line desired to be drawn : Also the other point in the same reflected hour-line may be found ; If the said thread , whose end is fastened in the Reflected Axis , be brought to touch some other part of the same hour-thread proposed ; so that when ( as before ) the end of the said thread toucheth the wall or Cieling , some part of that thread may also touch the hour-line desired , which point of touch on the wall or Cieling , is also another point in the said reflected hour-line desired . By which two points so found ( as before ) the reflected hour-line may be drawn by a thread , projecting by those points from the eye , as it was formerly directed in drawing the reflected hour-lines to an Horizontal Glasse . To draw the Reflected Equinoctial line , and also the Tropicks on any wall or Cieling , to any Reclining Reflecting glasse . 1 To draw the reflected Equinoctial line on the Wall or Cieling . TAke that thread , whose end is fastened in the center of the reclining glasse , and move the other end thereof to and fro in the said Reflected meridian formerly drawn , until ( by help of a quadrant ) the said thread is elevated above the reflected Horizon formerly drawn , equal to the Complement of the Latitude , ( which as before will be alwayes perpendicular to the reversed Axis ) and make a point in the said reflected meridian , where the end of the said thread toucheth ; then on that point and the said reflected meridian on the Cieling , raise a perpendicular line , which is the Reflected Equinoctial line desired . 2. To draw the reflected Tropicks , or other Parallels of Declination . FIrst , ( as before ) make or take out of some Book a Table of the Suns Altitude for each hour of the day , calculated for the place or Latitude proposed , when the Sun is in either of the Tropicks , or other parallel of Declination : then take that thread , whose end is fastened in the center of the Glasse , move the other end thereof to and fro in the hour-line proposed , until by applying one side of a quadrant to the said thread you find the said thread elevated above the reflected Horizon answerable to the Suns height in that hour proposed , when he is in that Tropick or degree of Declination proposed . Which altitude required will be found in the foresaid Table for that end calculated , which said thread being of the elevation above the reflected Horizon , as the said Table directeth : then mark where the end of the thread ( so elevated ) toucheth the Wall or Cieling in that hour-line : so is one point found in the reflected parallel of Declination desired to be drawn . In like manner , find in the said Table in the same parallel or degree of declination what altitude the Sun hath at the next hour , and elevate the said thread , whose end is fastened in the center of the Glasse , equal to the Suns altitude in that hour above the said reflected Horizon , by help of the said Quadrant , and where the other end of the said thread falleth in the hour-line proposed , make another mark or point . And so in like manner make the points ( belonging to that parallel of Declination ) in the remaining hour-lines , according to the several Altitudes found in the said Table of Altitudes : Then drawing by hand a line to passe through those several points so found , as before , which line is the reflected parallel of the Suns declination desired . In like manner may be drawn all or any other parallel of Declination , which may have respect to the Suns place , or the length of the day , as shall be desired . Or , To draw the said reflected Tropicks , or other parallels of Declination , without any Tables calculated , only , by help of a Trigon first made on pastboard or other material . Note that all Parallels are lesser Circles . FIrst ( as formerly is shewd in drawing the parallels of Delination to a Reflecting Horizontal Glasse ) fasten the Trigon on the reflected reversed Axis , so that the center of the Trigon may be in the center of the Glasse , then also will the Equinoctial on the Trigon be perpendicular to the said reflected reversed Axis : then take the thread fixed in the center of the said Glasse , ( which is also in the center of the Trigon ) and lay it upon that parallel of Declination , drawn on the said Trigon , whose reflected parallel is required to be drawn on the plane or Cieling : then move the Trigon , the thread lying on the said parallel , until the end of the said thread touch any hour-line on the said wall or Cieling , in which point of touch on that hour-line make a mark , so will that point be in the reflected parallel of Declination desired . In like manner , move the said Trigon , still keeping the thread on the same parallel , until the end of that thread touch another hour-line on the said plane or Cieling , and there also make another mark . And so in like manner find a point in each hour-line through which that reflected parallel must passe ; then drawing a line to passe through those several points on the said plane or Cieling , which line is the reflected parallel of the Suns Declination desired . In like manner may be drawn any other reflected parallel of Declination required . To draw the reflected Azimuth-lines to any reclining Glasse , on any plane whatsoever that the Sun-beams will be reflected on . Here note that Azimuths are great Circles . FIrst , know that the reflected vertical point in the Axis of the Reflected Horizon , will alwayes be found in the reflected meridian . And look how many degrees the reflected Horizon differs from the direct Horizon , so many must the reflected Axis of the Horizon differ from the direct Axis of the Horizon : Hence the reflected vertical point , whereby the reflected Azimuth-lines are drawn , may be thus found . Take that thread whose end is fixed in the center of the Glasse , and move the other end thereof to & fro in the reflected meridian , until by applying one side of a quadrant thereto , you find the said thread depressed just 90 degrees , or perpendicular under the reflected Horizon ; then make a mark or point where the other end of the said thread toucheth the said reflected Meridian on the Wall , Ground , or Floor of the Room , which point so found is the reflected vertical point desired , in which point fasten one end of a thread : Then on pastboard or other material draw the points of the Compasse or other degrees , placing the center thereof in the center of the Glasse , and the meridian thereof in the reflected meridian of the world , which said pastboard must be also situated in the reflected Horizon just as the Horizontal Dial was formerly directed to be situated for drawing the reflected hour-lines : And as the threads from the center fastened in the reflected Horizon were also the hour-lines on the Horizontal Diall , whereby the reflected hour-lines were drawn . So now the threads from the center fastened in the Reflected Horizon may be the Horizontal Azimuth lines , whereby the reflected Azimuth-lines may be drawn : Or if that thread which fastned in the center of the glass be drawn exactly over any Azimuth-line , the end whereof being fastened by a nail or other means in the reflected Horizon on the other side of the Room , there may several points be found in the wall or Cieling , through which the reflected Azimuth line must passe , as followeth : Take that thread , one end of which is fastened in the said vertical point , and bring it just to touch the Azimuth thread formerly fastened , and continue it until the end thereof touch the wall or Cieling , ( and also the thread it self touch the said Azimuth it self , as before ) in which point of touch on the wall or Cieling make a mark , through which point that reflected Azimuth-line must passe . Then move the said string fastened in the said vertical point , so that it may just touch the said thread again , but in another place : then as before continue that thread , untill the end thereof touch the wall or Cieling again , as before , and there make another mark , through which the said reflected Azimuth line must also passe ; In like manner may more points be found for your further guide , in drawing that Azimuth-line . But two points being found will be sufficient . To draw any reflected line by any two points given over any plane whatsoever , without projecting by the eye . FAsten two threads in the place of the center of the said reclining Glasse , drawing the said threads straight , fastening each of the other ends in the two reflected Azimuth-points formerly found on the wall or Cieling . Then situate a thread cross or thwart the room , so as it may crosse those other threads from the center , neer at right angles , and also just touch both of them in that situation . By which said thread crosse the room may any number of points in the said reflected Azimuth-line to be drawn , be found at pleasure : For if the end of another thread be also fastened in the center of the said Glasse , making the other end thereof to touch the wall or Cieling , but so that it may also just touch the said thread , which is fastened crosse the room , which point of touch on the said wall or Cieling is another point in the said reflected Azimuth line required to be drawn . In like manner may more points be found at every angle or bending of the wall or Cieling for the exacter drawing the reflected Azimuth line required , which doth find points , whereby is drawn the same reflected Azimuth line ( or other lines ) as was formerly done by a thread so situated , that it may interpose between the eye and any two points assigned on the wall or Cieling . In like manner , if the thread fastened on the further side of the room were removed on another Azimuth line on the said pastboard , and then fasten it again on the further side of the room ( as before ) you may by help of the said thread fastened in the said vertical point find several points on the wall or Cieling , through which that Azimuth-line will passe ; So may you either by this or the former way draw what Azimuth lines you please , either in points of the Mariners Compasse or degrees , as you please , by drawing it first on pastboard , as before is directed . And note generally , that such relation the point found on the floor or ground in the reflected reversed Axis , hath to the hour-lines drawn on the Horizontal Dial , in drawing the reflected hour-lines ; The same hath the Reflected vertical point found on the floor or ground , to the Azimuths drawn on the pastboard in drawing the reflected Azimuth-lines . To draw the reflected parallels of the Suns altitude , or proportions of shadows to any reclining Glasse on any Plane whatsoever , that the Sun-beams will be reflected on . Here note , that parallels of Altitude are lesser Circles , therefore are not represented by a right line . FIrst , know generally that what respect the parallels of Declination have to the hour-lines , such have the parallels of Altitude to the Azimuths . For if one end of a thread be fastened in the place of the center of the reclining Glasse , and the other end moved to and fro in any reflected Azimuth line , until the said thread be elevated any number of degrees proposed above the reflected Horizon ( the Elevation of which thread being found , by applying a Quadrant thereto , and making a mark or point where the end of the said thread toucheth the said reflected Azimuth drawn on the wall or Cieling , that point so found is the point through which that Almican●er or reflected parallel of the Suns altitude must passe . In like manner , remove the other end of the said thread fastned in the center of the Glasse to another reflected Azimuth-line , and ( as before ) move it higher or lower , untill by applying the edge of a quadrant to that thread , you find the said thread above the reflected Horizon the same number of degrees first proposed , and at the end of the said thread in that Reflected Azimuth-line drawn on the wall or Cieling I make another mark or point , through which the same Reflected Almicanter or parallel of Altitude must also passe : And so in like manner I find a point on each reflected Azimuth-line , through which the same parallel of Altitude must passe . Then drawing by hand a line to passe through these several points so found , as before , that line is the Reflected parallel of the Suns Altitude proposed . In like manner may be drawn all the other parallels of Altitude desired , which will shew the Suns altitude or the Proportion of any shadow to its altitude , at any appearance of the Suns reflex thereon . To draw the Jewish or old unequal hour-lines to any Reclining Glasse on any plane whatsoever that the Sun-beams will be reflected on . Here note that the Jewish hour-lines are great Circles . FIrst , ( by the Rules formerly given ) draw two reflected parallels of Declination of 16 d. 55′ , the one being neer the Summer , and the other neer the Winter-Tropick : for when the Sun hath that Declination , the day is 15 hours long in the Summer , and 9 in the winter : Then ( as is formerly directed ) situate a thread just between the eye , and those three points in the said Reflected Dial , as is expressed in the insuing Table , so may you thereby draw all or any of those Jewish hour-lines desired , which will at any appearance of the spot by the reflex of the Glasse amongst those hour-lines , shew how many of the equal hours is past since Sun-rising , as was desired . Now in this Latitude of 51 deg . 30′ , If the parallels of the Suns declination be drawn , both when the day is 9 and 15 hours long , that is , when it is 16 d. 55′ , any of those Jewish hour-lines will intersect the common hour-lines , either upon the hours , half hours , or quarters . And such a declination may be found , that it shall so do in any Latitude desired . Unequal Hours . 15 H. M Equ . H. 9 H. M. 0 4 30 6 7 30 1 5 45 7 8 15 2 7 00 8 9 00 3 8 15 9 9 45 4 9 30 10 10 30 5 10 45 11 11 15 6 12 00 12 12 00 Unequ . hours 15 H. M. Equ . H. 9 H. M.             7 1 15 1 0 45 8 2 30 2 1 30 9 3 45 3 2 15 10 5 00 4 3 00 11 6 15 5 3 45 12 7 30 6 4 30 To draw the Circles of Position to any reclining Glasse on any plane whatsoever , that the Sun-beams will be reflected on . NOte that all Circles of Position are great Circles of the Sphere , and do alwayes intersect each other in that point of the Reflected meridian which toucheth the Reflected Horizon , which may be called the common intersection ; which said Circles of Position are reckoned upon the Reflected Equinoctial both wayes from the said meridian down to the said Horizon : The Horizon Eastward being the Cuspis of the first House , and the Horizon Westward being the Cuspis of the seventh House ; and the Reflected meridian the cuspis of the tenth House . So that those Meridian-planes , whose Reclination is 60 degrees Westwards , ( being measured from the meridian in the Equinoctial ) lies in the Cuspis of the eighth House , and 30 degr . Westward lies in the Cuspis of the ninth house , and 30 deg . Eastward in the Cuspis of the eleventh House , and 60 deg . Eastward in the Cuspis of the twelfth House : which are all the Houses above the Horizon . Now to draw any Circle of Position , or the Cuspis of any House on any Cieling or wall to any reclining Glasse is done as followeth : First , fasten a thread , in such sort , within the Room , as that it may interpose between the eye and the said common point of intersection on the wall or Cieling , and also between that point where the reflected hour-line of 4 ( being 60 deg . Westward from the said Meridian ) intersects the reflected Equinoctial also on the Cieling , whereby points may be made at every bending or angle of the wall or Cieling , to which the thread so situated may also interpose , by which points the Reflected Cuspis of the eighth House may be drawn . In like manner may the Cuspis of any other House above the Horizon , as the 9 th . or 10 th . which is the Meridian ( or Medium Coell ) or 11 th . or 12 th . be drawn also . For if ( as before ) the said thread be again so fastened within the Room , as that it may also interpose between the eye and the said common point of intersection , and also those points where the reflected hour-line of 2 ( being 30 deg . Westward from the said meridian ) do cut the reflected Equinoctial , whereby may be drawn the reflected Cuspis of the ninth House . Or where the Reflected hour-line of 10 ( being also 30 deg . Eastward from the meridian ) do also cut the said reflected Equinoctial , whereby may be drawn the Cuspis of the 11 th . House . Or where the reflected hour-line of 8 ( being 60 deg . Eastward from the meridian ) do also cut the said reflected Equinoctial , whereby may be drawn the Cuspis of the 12 th . House . The Horizon alwayes being the Cuspis of the first and seventh Houses , and the meridian the Cuspis of the tenth house or Medium Coeli : wherein generally it is to be noted , That in all planes which cut the common Intersection of the meridian and Horizon , ( as doth the Horizontal , and also all meridian planes both Direct and Reclining ) these Circles of Position are all parallel to the meridian , and therefore parallel each to other . For look what respect the hour-lines in all Direct or Reclining Polar Planes , or Direct meridian Planes have to the Axis of the World : Such respect have the Circles of Position , in all Horizontal , or Direct meridian or Reclining meridian Planes , to the Axis of the Prime vertical : For as the hour-lines in the first are all parallel to the Axis of the Equinoctial , in whose Poles they meet : So the Circles of Position in the second are all parallel to the Axis of the Prime Vertical , in whose Poles they also meet . The reason why Glasses reflect a double Spot , is because they are polisht on both sides , which may be remedied with a Pumex-stone . Those that desire to read more of this Subject may see what is written by Kircher , in primitiis Gnomicae Catoptricae , and since him by Magnan and others , VALE . FINIS . 
A64224	----	Thesaurarium mathematicae, or, The treasury of mathematicks containing variety of usefull practices in arithmetick, geometry, trigonometry, astronomy, geography, navigation and surveying ... to which is annexed a table of 10000 logarithms, log-sines, and log-tangents / by John Taylor. Taylor, John, mathematician. 1687 Approx. 1270 KB of XML-encoded text transcribed from 283 1-bit group-IV TIFF page images. Text Creation Partnership, Ann Arbor, MI ; Oxford (UK) : 2003-01 (EEBO-TCP Phase 1). A64224 Wing T534 ESTC R23734 07887917 ocm 07887917 40282 This keyboarded and encoded edition of the work described above is co-owned by the institutions providing financial support to the Early English Books Online Text Creation Partnership. This Phase I text is available for reuse, according to the terms of Creative Commons 0 1.0 Universal . The text can be copied, modified, distributed and performed, even for commercial purposes, all without asking permission. Early English books online. (EEBO-TCP ; phase 1, no. A64224) Transcribed from: (Early English Books Online ; image set 40282) Images scanned from microfilm: (Early English books, 1641-1700 ; 1215:4) Thesaurarium mathematicae, or, The treasury of mathematicks containing variety of usefull practices in arithmetick, geometry, trigonometry, astronomy, geography, navigation and surveying ... to which is annexed a table of 10000 logarithms, log-sines, and log-tangents / by John Taylor. Taylor, John, mathematician. [15], 507 p., 8 leaves of plates : ill., port. Printed by J.H. for W. Freeman, London : 1687. The logarithm tables have separate t.ps. Reproduction of original in the Cambridge University Library. Created by converting TCP files to TEI P5 using tcp2tei.xsl, TEI @ Oxford. Re-processed by University of Nebraska-Lincoln and Northwestern, with changes to facilitate morpho-syntactic tagging. 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Keying and markup guidelines are available at the Text Creation Partnership web site . eng Mathematics -- Early works to 1800. 2000-00 TCP Assigned for keying and markup 2001-08 Apex CoVantage Keyed and coded from ProQuest page images 2002-11 Kirk Davis Sampled and proofread 2002-11 Kirk Davis Text and markup reviewed and edited 2002-12 pfs Batch review (QC) and XML conversion Vera Effigies Johannis Tayl●r Thesaurarium Mathematicae , OR THE TREASURY OF THE MATHEMATICKS . CONTAINING Variety of usefull Practices in Arithmetick , Geometry , Trigonometry , Astronomy , Geography , Navigation and Surveying . AS ALSO The Mensuration of Board , Glass , Tiling , Paving , Timber , Stone , and Irregular Solids . LIKEWISE It teacheth the Art of Gauging , Dialling , Fortification , Military-Orders , and Gunnery ; Explains the Logarithms , Sines , Tangents and Secants : Sheweth their use in Arithmetick , &c. To which is Annexed a Table of 10000 Logarithms , Log-Sines and Log-Tangents . Illustrated with several Mathematical Sculptures on Copper Plates . By JOHN TAYLOR , Gent. — Deus regit Astra ; feruntur Illius arbitrio Sydera , Terra , Fretum . LICENSED , June 26. 1686. Rob. Midgley . LONDON , Printed by J. H. for W. Freeman at the Artichoke next St. Dunstan's Church in Fleetstreet . 1687. To the Right Honourable GEORGE Lord DARTMOUTH , Master of the Horse to K. James II. Master General of His Majesty's Ordnance and Armories , One of His Majesty's most Honourable Privy Council , &c. This small Mathematical Treasury is humbly Dedicated and presented by My LORD , Your Honour 's most humble and obedient Servant John Taylor THE PREFACE TO THE READER . HOW admirably profitable the study of the Mathematicks hath been to these British Islands , and to all other parts of the Universe in which any kind of good Learning hath been esteemed and practised , is well known to all wise and judicious men . And indeed it is an undeniable truth , that among all humane Arts and Sciences whatsoever , the Noble Science Mathematical hath obtained the greatest evidence of certainty , as being the Queen of Truth that imposeth nothing on her Subjects but what she proves by most infallible Demonstrations . Now this prerogative results from the verity and perspicuity of its Principles , which consist of Definitions , Postulats and Axioms . Hence comes it to pass that all Propositions that are proved by those most infallible Precepts are called certain demonstrative truths ; for which cause it hath been the endeavour of sundry Philosophers to make the force of their Arguments ( as far as the quality of their discourse woùld admit ) amount unto Mathematical Demonstrations , as being the most convincing proof of a Proposition that by humane reasoning can be given . Now having for divers years ( amongst my other Studies ) been conversant in the study of the Mathematicks , and for my own private use compiled this Treatise , never in the least intending it should have appeared in publick in this nice and critical Age ; but it by chance falling into the hands of some of my Mathematical Friends and Acquaintance , I have at their requests condescended to publish it , though not without a great aversion in my own mind to expose my self in any publick thing . But this difficulty being overcome , I shall give the Impartial Reader to understand that I have faithfully compiled this Treatise from the best of Authors ( and my own Experience ) that I have contracted their various Works into this little Cabinet or choice Compendium of the Mathematicks , in which thou shalt find the whole Subject clearly and intelligibly handled : I have used a plain and easie method : I have laboured to be as plain and perspicuous as possible : I have applied such Examples to each as may best demonstrate their Operation , be most easie for memory , and applicable to practice ; here is indeed Multum in Parvo , the whole Marrow of the Mathematicks is in this Tract afforded thee , which is as a true and Golden Key to unlock the choicest mysteries in those Arts contained . Thus , Reader , I have laid my labours before thee , and must intreat thee to use me as thou wouldst be done by , which is the Spontaneous act of every good man. But if this shall chance to fall into the hands of any curious conceited person , who thinks himself wiser than the rest of the world , and so he beginneth enviously to carp hereat , and like a Countrey Cur bark at my backside , to him I shall in modesty onely say , — Facilius est unicuivis nostrum aliena curiosè observare : quam propria negotia rectè agere . 'T is much easier for those captious Readers to carp than to copy . If the Collections of Authors shall offend any , and so procure the same censure with the Jackdaw , as our learned Poet hath long since cautioned against — Ne , si fortè suas repetitum venerit olim Grex avium plumas , moveat cornicularisum , Furtivis nudata coloribus . I would have such contentious Readers know that I have robbed no man of the honour of his Works , but have given to each his due ; only I have borrowed some choice things of them , which is no more than what the most learned have always done . Thus , Courteous and Impartial Reader , 't is only for thee that I have taken these pains , and have submitted to the publication hereof , and it is to thee the future parts of my study shall be serviceable , hoping that thou wilt find success in all thy Studies according to thy desire and endeavour ; which are and shall be the hearty wishes of him who is London , July 12. 1686. Thine and Urania's Servant , John Taylor . To the READER . BEing desired to peruse this Mathematical Treasury , accordingly to gratifie the Request of my Friend I did , and I must confess with no small satisfaction to my self to see so much Practical Matter of usefull Mathematical Arts so neatly and compendiously digested into this Portable Volume ; 't will be usefull not only to Learners and meer Tyro's , but to others also who have made some considerable progress in these Studies . 'T is well Methodiz'd , very Concise , yet Plain and Perspicuous , so that any person of a pregnant fancy , may without a Tutour ( in some reasonable time ) wade through the whole , or any part thereof , and such as would be more expeditious may take the assistance of a Teacher to instruct them . The Author is wholly a stranger to me , but to give him his due , in my opinion he has discharg'd himself like a Master in these Arts , and an Ingenious Mathematician , to whom I return thanks for this his generous offer in presenting his Mathematical Treasury to the Publick , and remain October 25. 1686. A true Lover of the Mathematical Sciences , and all such that really delight in those pleasing ( but usefull ) Speculations , Henry Coley . Courteous Reader , I Have perused this Treatise , and find that the Author has in every respect discharged himself like an Artist ; the Work throughout the whole , is very plain and easie , nothing being omitted that might render it Intelligible to the meanest capacity : And indeed , I know not any Treatise of this nature extant that is more Practically handled , so that I doubt not but that it will be very serviceable to the Publick ; and that thou in particular mayst find incouragement in the perusal thereof , is the hearty wish of him who is Thine and Truths Servant , John Hawkins , Philomath . Octob. 30. 1686. 2h . 40 ' P. M. To his learned and ingenious Friend Mr. John Taylor , in the deserved Praise of his Excellent Book intituled The saurarium Mathematicae . ATlas and Hercules whom Poets feign , The heavy load of the Earth to sustain : If so ? great Toyl and Labour then they took , Yet not so much as thou hast in thy Book . Not like to them thy Labours , fictions are , Thy works so true ingenious and so rare , That seldom yet such works from man did flow , For thou by them dost teach us all to know The secrets of all Sciences and Art , Which freely unto us thou dost impart . Thou shew'st us how Numbers to understand , And how the Speech of Numbers to command . Geometry , the Queen of truth , did cease The Egyptian trouble , and did cause a peace ; When proudly Nile had overflown their ground , And all their Bounds and Land-marks did confound : By it , each man his proper Right did gain , And Peace by it great Egypt did obtain . This Art so conspicuous thou hast made , That to thy Glory it can never fade . By Sines , Tangents and Secants thou dost show Us all the parts of Triangles to know . Thy lofty Genius viewed the Stars on high , So that full well thou know'st Astronomy . All motions of the Sun and Planets thou Dost understand , thy works do shew it now . For thou such Rules and Precepts dost apply , In this thy Book unto Astronomy ; Demonstrated by rules so rare and plain , That he 's a Dunce that can't it now obtain . Thus having view'd the Spheres of Heaven well , Then on our Mother Earth thy Genius fell . Thou viewd'st her round , ev'n by inspecting all The known parts of the Cosometick Ball. And here in this thy Book thou let'st us see , How Nations all most disagreeing be . The Seaman he adores thee as his Friend , So liberal unto him thy Art doth lend ; And thou from him wilt not thy Talent hide , Thy Book 's a Light-house Mariners to guide . Surveying thou dost teach and that so plain , That any one that Art may well obtain : And by that means Injustice to disband , Attending Lord and Tenant of the Land. To thee the Brooks and Springs do all submit , And they will glide to that place thou think'st fit . Thou shew'st Mechanick well to apprehend , To measure Board or Glass , nay as a Friend , Teachest them how both Timber round and square , And Stones to measure of what kind so e're . Therefore to thee they praises still will give , And tho ' thy Body's dead thy Fame shall live . The Art of Gauging thou dost plainly teach , And farther far than worthy Oughtred reach Into the Mystery of that curious part , And noble Branch of Mathematick Art. Thou measurest the course of times short stay , Thus Dials shew us how time flies away ; That thereby we may mind our fading breath , And preparation make for certain death . Thy Book 's also prepar'd Mars to withstand , In raising Forts for to defend the Land. In ordering Armies in Battail aray , And them Encamping when they make a stay . The Gunner's Magazine lies in this Tract , From whence directions he may have to sack , Or storm a Town , or batter down a wall , Or make a Breach and at the joyfull fall Of Turrets high Huzza's to make for joy , And entring in , his Enemies destroy . These curious Arts with more than here are nam'd , In this rich treasury so neatly fram'd , Our friendly Authour doth to all impart , Wishing success , and that with all his Heart . But stop my Muse let us not be so rude , We 'll only wish him well and so conclude . Maist thou , O Authour of this Treasury , Reap to thy self profit and praise thereby ; And maist thou ever , ever happy be , That we more of thy Learned works may see . Live thou in splendid comfort to thy end , So prays thy humble Servant , and thy Friend , July 20. 1686. Geo. Barrow . To the Learned Authour my much respected Friend Mr. John Taylor on his Herculean labours in the Composure of this Excellent Mathematical Treasury . I ncrease our Muse , rouze up ye Sisters nine , O n us bestow your Art that we may praise H is works , his worth and his real design ; N ot honour vain , but skill aloft to raise . Vain Glory 's to him but a trifling Toy , 'T is Art alone , 'T is that which is his Joy. T he Earth he hath trac'd , the Spheres of Heav'n view'd A nd Stars and Seas whose billows loud do roar ; Y et is he not nor can he be so rude , L ike many others to lock up his store , O h he doth not ! his Treasure ope ' doth stand : R eceive it as a Jewel from his Hand . Our noble Friend and Authour what 's thy due ? Honour thou slight'st , Treasure 's too vain for you . Thy mind is fixt on Sciences above , Thou art Urania's favourite and love . Thou knowst her ways , her Art 's at thy command , She smiles upon thee , guides thee by the hand : For which thy Name we will extoll and praise , As far as Phoebus sends his golden raies . Therefore in happiness let thy time run , And rest in Peace when that thy Period's come . Sept. 27. 1686. Tho. Robinson . An Acrostick on the Name of my much respected and ingenious Friend Mr. John Taylor . I f Mathematicks be the Art to teach , O by thy Book the Learned then may reach H eavens Poles , and Circles without doubt or fear , Not to find out each Star it's Hemisphere . T hough Archimedes hath much glory got A mongst the Syracusians , why not , Y ea Statues be erected to thy Name ? L et Eagles wings towre and soar thy fame . O happy maist thou be , and this thy Book R eaders instruct when e're they in it look . London , July 29. 1686. Fran. Pierce . ADVERTISEMENTS . A LL Gentlemen , or other Persons that shall have occasion for any sort of Mathematical Instruments , either for Sea or Land , may be furnished by John Worgan , Mathema●ical Instrument-maker ; under St. Dunstan's Church ●n Fleet-street , London . At St. George's Church in Southwark are taught Writing , Arithmetick , Merchants Accounts , Geometry , Trigonometry , Astronomy , Navi●ation , Surveying , Dialling , Gauging and Gun●ery by John Hawkins , Philomath . Arts and Sciences Mathematical , profess'd and taught by HENRY COLEY , Philomath . at his House in Baldwins Court over against the Old Hole in the Wall , in Baldwins Gardens near Grays-Inn-Lane . ARITHMETICK in Whole Numbers and Vulgar Fractions . Decimal , and by Logarithms . GEOMETRY . The Rudiments thereof , also the Demonstration and Practice , according to the best Authours . ASTRONOMY . The use of the Globes Coelestial , and Terrestrial . To project the Sphere in Plano to any Latitude several ways . To calculate the Longitude and Latitude of the Planets , with their Declination and Ascension . Also the true Time , Quantity , and Duration of Eclipses of the Luminaries for any time past or to come . TRIGONOMETRY . Or the Doctrine and Calculation of Triangles , both . — Plain and Spherical . With the Application of the several Cases thereof in the most useful Questions in . — Geometry . Astronomy . Geography . Navigation . Dyalling , &c NAVIGATION . In either of the three principal kinds of Sayling , viz. by the Plain and Mercator's Chart Great Circle . DYALLING . Geometrically Instrumentally Arithmetically by The Sector , and other convenient Scales . The Logarithms , Sines & Tan. SURVEYING . Several ready ways to measure a Plat , and divide Land , &c. also the taking of Altitudes , Profundities , Distances , &c. together with the Mensuration of all manner of Superficies , as Boards , Glass and Pavement : also all Solids , viz. Timber , Stone , &c. Regular and Irregular . GAGING . To find the just quantity of Liquor in any Cask , whether full or partly empty . Also the content or solidity of Brewers Vessels , &c. Tuns , Coppers , Backs , Coolers , &c. ASTROLOGY . In all its parts , and according to the best Authors , with several varieties therein , not known to every Professor . Non nobis nati sumus . The Contents . CHAP. I. OF Arithmetick Page 1. CHAP. II. The Explanation and Use of the Table of Logarithms . p. 18. CHAP. III. The Explanation of the Sines , Tangents and Secants . p. 28. CHAP. IV. Of Geometry . p. 32. CHAP. V. Of Trigonometry , or the Doctrine of Triangles . p. 59. CHAP. VI. Of Astronomy . p. 96. CHAP. VII . Of Geography , with a Geographical Description of the Earthly Globe . p. 122. CHAP. VIII Of Navigation . p. 186. CHAP. IX . Of Surveying . p. 214. CHAP. X. Of Measuring Boards , Glass , Tiling , Paving , Timber , Stones and Irregular Solids , such as Geometry can give no Rule for the Measuring thereof . p. 242. CHAP. XI . Of Gauging . p. 250. CHAP. XII . Of Dialling . p. 255. CHAP. XIII . Of Fortification , according to the modern and best ways now used by the Italian , Dutch , French and English Inginiers . p. 279 CHAP. XIV . Of Military Ordèrs , or the Embattelling and Encamping of Soldiers . p. 301. CHAP. XV. Of Gunnery . p. 306. A Table of Logarithms . p. 337. A Table of Proportional Parts . p. 401. A Table of Artificial Sines and Tangents . p. 417. Arithmetick . CHAP. I. Of ARITHMETICK . ARITHMETICK is an Art of numbring well , for as magnitude , or greatness , is the subject of Geometry , so is multitude , or number , that of Arithmetick . I shall not in this place trouble you with the first Rudiments of Arithmetick , as Numeration , Addition , Substraction , Multiplication , and Division : because they are already largely handled by many , as Mr. Leybourn , Mr. Wingate , and divers others , and also that then this Book would swell to too big a bulk for the Pocket , and so my design would be frustrated ; I shall therefore only propose and operate some principal Propositions , that are of Special moment in Arithmetick , and which most immediately concern the other following parts of this Treatise . SECTION I. The Explication of some Arithmetical Propositions . PROPOSITION I. To three numbers given , to find a fourth in a Direct proportion . To operate this proportion Multiply thē third term , by the second term , and their product divide by the first term , the Quotient shall be a fourth term required . Examp. 1. Admit the Circumference of a Circle whose Diameter is 14 parts be 44 parts , what is the Circumference of that Circle , whose Diameter is 21 parts ? Now according to the Rule if you multiply the third term 21 , by the second term 44 , it produceth 924 ; which divided by the first Term 14 , the Quotient is 66 , and so the Circumference of the Circle , whose Diameter is 21 , will be 66 parts , and so for any other in a direct proportion . PROP. II. To three numbers given , to find a fourth in an Inversed proportion . To operate this proportion , Multiply the first term , by the second term , and their product divide by the third term , the Quotient is the fourth term required : Examp. Admit that 100 Pioneers , be able in 12 hours to cast a More of a certain length , breadth , and depth ; in what time shall 60 Pioneers do the same ? Now if according to the Rule , you Multiply the first term 100 , by the second term 12 , their product is 1200 ; which divided by the third term 60 , the Quotient is 20 , so I say that in 20 hours , 60 Pioneers shall do the same , and so for any other in an Inversed proportion . PROP. III. To three numbers given , to find out a fourth in a Duplicate proportion . The nature of this proposition is to discover the proportion of Lines , to Superficies , and Superficies , to Lines ; for like Plains are in a duplicate Ratio ; that is as the Quadret of their Homologal sides ; therefore to Operate any Example in this proportion , Square the third term , and its square multiply by the second Term , their product divide by the square of the first Term , the Quotient is the 4th . term sought ; Examp. Admit there be two Geometrical squares ; now if the side of the greater square be 50 feet , and require 3000 Tiles to pave it ; what number shall the lesser square require , whose side is 30 feet ? To operate this according to the Rule , I square the third Term 30 , whose square is 900 : then I multiply it by the second Term 3000 , its product is 2700000 , which divided by 2500 , the square of the first Term 50 , the Quotient is 1080 , and so many Tiles will pave the lesser square , whose side is 30 feet . PROP. IV. To three numbers given , to find a fourth in a Triplicate proportion . THE nature of this proposition is to discover the proportion of Lines to Solids , and Solids to Lines ; for like Solids , are in a Triplicate Ratio , that is to the Cubes , of their Homologal sides : Therefore to operate any Question in this proportion , Cube the third Term , and his Cube multiply by the second Term , and their product divide by the Cube of the first Term ; the Quotient is the fourth Term sought . Examp. Admit an Iron Bullet whose diameter is 4 Inches , weigh 9 pounds ; what is the weight of that Bullet whose Diameter is 6 Inches ? Now to operate this proportion ; first according to the Rule I Cube the third Term 6 whose Cube is 216 , then I multiply its Cube by the second Term 9 , the product is , 1944 , which divided by 64 , the Cube of the first Term ; the Quotient is 30 24 / 64 pounds which is equal unto 30l . 6 ounces : which is the weight of the propounded shot ; and so for any other . PROP. V. To two numbers given , to find out a third , fourth , fifth , sixth , &c. Numbers in a continual proportion . To operate this proportion , you must multiply the second number by it self , and that product divide by the first Term , the Quotient is a third proportional : Again you must multiply the third Term by it self , and its Quadret divide by the second Term , the Quotient is a fourth proportional , and so after this manner a fifth , sixth ; or as many more proportionals as you please may be found : Examp. Let it be required to find six numbers in a continual proportion to one another ; as 4 to 8. To operate this first according to the Rule , I multiply the second Term 8 by it self the product is 64 , which divided by the first Term 4 , the Quotient is 16 : so is 4 , 8 , and 16 in a continual proportion ; And so observing the Rules prescribed , proceed in your operation untill you have found your six numbers in a continual proportion ; which in this Example will be 4 , 8 , 16 , ●2 , 64 , and 128 , and so will you have form'd six numbers in a continual proportion . PROP. VI. Between two numbers given , to find out a mean Arithmetical proportional . THIS proposition might be performed without the help of the rule of proportion : nevertheless because it conduceth to the Resolution of the next ensuing proposition , I insert it in this place ; To operate it this is the Rule : add half the difference of the given Terms , to the lesser Term , so that Agragate , is the Arithmetical mean required : Examp. Admit 20 and 50 to be the two numbers propounded : Now to operate this proposition , first according to the Rule , I find that the difference of the two given Terms 20 , and 50 , is 30 , whose half is 15 , which being added to the lesser Term 20 , it makes 35 , so is 35 , a mean Arithmetical proportion betwixt 20 , and 50 , given . PROP. VII . Between two numbers given , to find out a mean Musical Proportional . BOETIUS hath this Rule for it , wherefore take his own words : * saith he , Differentiam terminorum in minorem terminum multiplica , & post junge terminos , & juxta cum qui inde confectus est ; committe illum numerum , qui ex differentiis & termino minore productus est , cujus cum latitudinem inveneris , addas eam minori termino , & quod inde colligitur medium terminum pones . That is , Multiply the difference of the Terms , by the lesser term , and add likewise the same Terms together : this done if you divide the product , by the sum of the Terms , and to the Quotient thereof , add the lesser Term ; the last Sum is the Musical mean desired : Examp. Admit the two numbers given be 6 , and 12. I say that if the difference of the Terms which is 6 , were Multiplied by the lesser Term 6 , it would produce 36 ; then if you add the two terms 6 , and 12 , together : their sum would be 18 , now if you divide 36 , by 18 , the Quotient is 2 ; lastly if to the Quotient 2 , you add the lesser Term 6 , the sum thereof will be 8 , which is a Mean Musical proportional required . PROP. VIII . How to find the Square-Root of any whole number , or Fraction . To Extract the Root of any Square number propounded , is to find out another number , which being Multiplied by it self , produceth the Number propounded . Now for the more easie and ready Extraction of the Square-Root of any number given , This Table here under annexed will be usefull ; which at first sight giveth all single Square numbers , with their respective Roots . ROOT . 1 2 3 4 5 6 7 8 9 SQUAR . 1 4 9 16 25 36 49 64 81 The Explication of the Table . In the uppermost rank of this Table , is placed the respective root of every single Square-number , and in the other the single Square-numbers themselves ; so that if the Root of 25 were demanded , the Answer would be 5 , so the Square root of 49 , is 7 , of 81 is 9 ; and so for the Rest , and so contrarily the Square of the Root 5 is 25 , of 7 is 49 , of 9 is 81 , &c. Example : If the Square root of 20736 , were required , first they being wrote down in order as you see , draw the Crooked-line , * then to prepare this or any other number for Extraction , make a point over the place of Unites ; and so on every other figure towards the Left-hand ; as you see in the Margent . Then find the Root of the first Square 2 , which is 1 ; place it in the Quotient , and also under 2 ; then draw a line , and substract 1 from 2 , there remains 1 ; which place under the line , then to the last remainder 1 , bring down the next Square 07 ; and then there will be this number 107 , which number I call a Resolvend : Then double the Root in the Quotient 1 , whose double is 2 , which 2 place under the place of tens in the Resolvend , under 0 ; so is this 2 called a Divisor ; and 10 called a Dividend . Then demand how often the Divisor 2 , can be had in the Dividend 10 , it permitteth but of 4 , which place in the Quotient , and under 7 the place of Unites in the Resolvend , and there will appear this number 24 ; Then Multiply this 24 , by 4 , ( the last Square placed in the Quotient ) it produceth 96 , which place orderly under 24 , as you see , and this 96 is called a Ablatitium ; ( but some calleth it a Gnomon : ) then draw a line under it , and substract 96 , the Ablatitium , out of the Resolvend 107 , there remains 11 , which place orderly under the last drawn line , then thereunto bring down the next Square 36 , so will there be a new Resolvend 1136 ; then double the whole Root 14 in the Quotient , whose double is 28 ; place it under the Resolvend 1136 as was afore directed ; so shall 28 be a new Divisor , and 113 be a Dividend ; then I find the Divisor 28 can be had in the Dividend 113 , 4 times , which four place in the Quotient , and under the place of Unites in the Resolvend , so there appeareth this number 284 , which number , multiplyed by 4 , the last figure in the Quotient , produceth a new Ablatitium 1136 ; which place orderly under the Resolvend 1136 , and then draw a line , then substract the Ablatitium 1136 , from the Resolvend 1136 ; and the remainder is 00 , or nothing : and thus the work of Extraction being finished , I find the Root of the Square number 20736 , to be 144 ; and so must you have proceeded gradually step by step , if the number propounded , had consisted of some 4 , 5 , 6 , or more Squares ; still observing the aforegoing Rules and Directions . NOTE . BUT when a whole number , hath not a Root exactly expressible by any rational or true Number , then to find the fractional part of the Root very near ; To the given whole number annex pairs of Cyphers , as 00 , 0000 , or 000000 , then esteem the whole number , with the Cyphers both annexed thereunto , as one intire whole number : and Extract the Root thereof according to the foregoing Directions , then as many points as were placed over the Integers , so many of the first figures in the Quotient must be taken for Integers ; and the remainder for the Roots fractional part in Decimal parts , and so you may proceed infinitely ne●r the true Root of a Number . To Extract the Square-Root of a Vulgar or Decimal Fraction , and a Mixt-number . First if the Fraction propounded be not in its least Ter●● , reduce it , and then by the Rules aforegoing , find the Root of the Numerator for a new Numerator ; and of the Denominato● for a new Denominator ; so shall this n●w Fraction be the Square-root of the Vulgar Fraction propounded , so the Square-root of 16 / ● is 4 / 〈…〉 But many times the Numerator and Denominator of a Vulgar Fraction hath not a perfect Square-root ; to find whose Root infinitely near , you must reduce it into a Decimal Fraction , whose Numerator must consist of an equal number of places , to wit , 2 , 4 , 6 , &c. Then Extra●● the Square-root of that Decimal , as if i● were a whole number , and the Root that procee●eth from it is a Decimal Fraction , pre●●ing the Square-root of the Fraction proposed , infinitely near : so the Root of 13 / 16 ( whose De●●ma is , 81250000 ) will be found to be 〈…〉 which is very near , for it wanteth not 1 / 10000 of an Unite of the exact Square-root , of 13 / 16 propounded . Now having a Mixt Number propounded whose Ro●● is required , ●o find which reduce it into an improper Fraction , and then Extract the Root thereof as before . Suppose the Number propounded be 75 24 / 54 ; its improper Fraction is 679 / 9 , whose Square-root I find to be 26 / 3. or 8 ⅔ , very near , &c. But if it had not an Exact Square-root , then reduce the Fractional part of the given Mixt-number into a Decimal Fraction , of an even number of places , and then annex this Decimal to the Integers , and so Extract the same , as a whole number ; and observe that so many points as were set over the Integers , so many of the first figures in the Quotient must be esteemed Integers ; and the Remainder for the Roots Fractional part . PROP. IX . How to find the Cube-Root of any whole Number , or Fraction . To Extract the Cube-Root of any Number propounded , is to find out another Number , which being multiplied by it self , and that product by the number again , shall produce the number propounded ; Now for the more easie and ready Extraction of the Cube-root of any number propounded , this Table hereafter annexed will be usefull , which at first sight giveth the Cube-root of any whole number under 1000 ; which are called single Cube-numbers . ROOT . 1 2 3 4 5 6 7 8 9 CUBE . 1 8 27 64 125 216 343 512 729 The Explication of the Table . In the uppermost rank of the Table is placed the respective Roots of every single Cube , and in the other the respective single CubeNumbers ; for if the Cube-root of 512 were desired , the Answer would be 8 , of 64 is 4 ; and so of the rest : and if the Cube of the Root 7 were desired , it would be found 343 ; of 9 it would be 729 , &c. Examp. Admit the Cube root of the Number 262144 , were required , first they being wrote down in order as you see , draw the Crooked-line . Then place a point over * the place of Unites , and another over the place of Thousands ; and so on still intermitting two places between every adjacent point ; and observe that as many points , as in that order are placed over any number propounded , of so many figures doth the Root consist of : so that in this Example , there being two points , therefore the Root consisteth of two places as you see in the Quotient ; Now first find the Root of the first Cube 262 ; which permitteth but of 6 , place 6 in the Quotient , and subscribe its Cube 216 , under 262 , and then draw a line under it , and substract 216 , out of 262 , and the remainder is 46 , which place in order under the last drawn line as you see . Then to the Remainder 46 , bring down the next Cube-number 144 , so will there appear 46144 , which I call a Resolvend : then draw a Line under it , and square the Number in the Quotient 6 , whose square is 36 ; Then Triple it and it will be 108 , Then subscribe this Triple square 108 , under the Resolvend , so that the place of Unites in the Triple Square 8 , may stand under 1 the place of Hundreds in the Resolvend : Then Triple the Root in the Quotient 6 , whose Triple is 18 , Then subscribe the Triple 18 , under the Resolvend , so that the place of Unites 8 in the Triple , may stand under 4 the place of Tens in the Resolvend , and so draw a Line under neath it , and add the Triple Square 108 , and the Triple 18 together in such order as they stand , their Sum is 1098 , which may be called a Divisor , and the whole Resolvend 46144 , except 4 the place of Unites a Dividend ; then draw another line . Then seek how many times 1098 the Divisor , can be had in 4614 the Dividend , it permitteth but of 4 , which subscribe in the Quotient ; Now Multiply the Triple square 108 , by 4 , it produceth 432 , which in order subscribe under the Triple square 108 : Then square 4 , the figure last placed in the Quotient , whose square is 16 ; and Multiply it by 18 the Triple , it produceth 288 , which subscribe under the Triple orderly , then subscribe the Cube of 4 ( last placed in the Quotient ) which is 64 , in Order under the Resolvend . Then draw a ●ine underneath it , then add the three numbers , viz. 432 , 288 , and 64 , together in such order as they are placed , their sum is 46144 : Then draw another line under the Work , subtracting the said total 46144 , from the Resol●end 46144 , there remains 00 , or nothing , which remainder subscribe under the last drawn ●ine , thus the work being finished I find the Cube root of 262144 the number propounded , to be 64 : And thus you must have proceeded orderly step by step , if the number propounded ●ad arisen to some 3 , 4 , 8 , 10 , or more places , observing the direction prescribed untill all had ●bserved compleated . NOTE . BUT when a whole number , hath not a Cube-root expressible by any true or Rational number , then to proceed infinitely near the Exact truth annex to the number Tenaries of Cyphers as 000 , 000000 , 000000000 , &c. then esteeming the whole number with the Cyphers annexed as one intire whole Number , Extract the root thereof , as is afore taught . Then as many points as were placed over the Whole Number , so many places of Integers will there be in the Root , and the rest expresseth the Root his Fractional part very near . To Extract the Cube-Root , of any Vulgar or Decimal or Mixt Fraction consisting of a Whole Number and a Fraction . To Extract the Cube-root of any Vulgar Fraction , you must first reduce it into his least terms , and then according to the former directions Extract the Cube-root of the Numerator , the Root found shall be a new Numerator so likewise the Root of the Denominator shall become a new Denominator ; so shall this new Fraction be the Cube-root of the Fraction propounded , so I find the Cube-root of 8 / 125 to be 2 / 51 and so for any other Vulgar Fraction . But many times the Numerator , and Denominator , hath not a true Root : Then to find the Root thereof infinitely near , you must reduce the Fraction given , into a Decimal , whose numerator is Tenaries of places , and then Extract the Root according to the former Directions , so shall the Root found , be a Decima● Fraction expressing near the Cube-root of th● Fraction propounded , so I find the Root of 8 / 12 or ⅔ , whose Decimal is , 666666666 , to be , 873 / 1000 very near the Root of 8 / 12 or ⅔ propounded . Now having a Mixt-number propounded , whose Root is required , first reduce it into an Improper Fraction , and then Extract the Cube-root thereof , as is afore directed , so the Cube-root of 12 10 / 27 , Improper 343 / 27 , will be found to be 7 / 3 or 2 ⅓ . But if it hath not an Exact Cube root , Then Reduce the Fractional part of the given Mixt-number into a Decimal Fraction , which shall consist of Tenaries of places , Then to the whole number annex the Decimal Fraction , and Extract the Cube-root of the Whole , and observe that so many points as are over the Integers , so many of the first places in the Quotient must be Esteemed Integers , and the rest Expresseth the Fractional part of the Root in Decimal parts of a Fraction , so the Cube-root of 2 ⅜ , Decimal 2 , 375000000 &c. will be found to be 1 , 334 , or 1 334 / 1000 , and is very near the true Root , and so for any other Mixt-number of this nature . CHAP. II. The Explication , and use of the Tables of LOGARITHMS . SECT . I. The Explication of the Tables of the Logarithms , and of parts proportional . THE Logarithms , were first invented , found out and framed , by that never to be forgotten and thrice Honourable Lord , the Lord Nepeir : which Numbers , so found out and framed by his diligent industry he was pleased to call Logarithms ; which in the Greek signifies the Speech of Numbers , I shall not here trouble you with the manner or the Construction of those Tables of Logarithms but shall first lay down some brief and general Rules , that thereby the better you ma● Understand those Tables , and then I shall e●plain their manifold uses , in sundry Examples Arithmetical , &c. PROP. I. Any Number given under 10000 , or 100000 , to find the Logarithm corresponding thereunto . 1. If the number propounded consist of one place whose Logarithm is required to be found , as suppose ( 5 , ) look for 5 , in the top of the left hand Column under the Letter * N , and right against 5 , and in the next Column under LOG . * you will find this number or rank of figures , 0698970 , which is the Logarithm of the number 5 required . 2. If the number consisteth of two places as if it were 57 , look 57 under N , and opposite to it and under LOG . you will find this number 1. 755875 , which is the Logarithm of 57 , the number propounded . 3. If the number propounded consist of three places as 972 , look for 972 , under N , and opposite to 972 , and under ●o ) the Column , you shall find this number 2. 987666 , which is the Logarithm of 972 , the number which was propounded . 4. But if the number consist of four places as 685 , look the three first figures 168 , under the Column N , and opposite to that , and un●er 5 at the top of the page , you will find this number 3. 226599 , which is the Logarithm of 1685 , the number propounded . 5. But if the number given be above 10000 , and under 100000 , you may find its Logarithm by the Table of parts proportional , printed at the latter end of this Book . Thus if the Logarithm of 35786 , be sought , first seek the Log. of 3578 , which will be 553649 , and the common Difference under D is 121 ; with this difference 121 , Enter the Table of parts proportional , and finding 121 in the first Column under D , you may then lineally under 6 , find the number 72 , which add to the Log. of 3578 , that is 553649 , it produceth , 553712 , which is the Log. of 35786 the number propounded : now because the number propounded 35786 , ariseth to the place of X. M. therefore there must be the figure 4 prefixed before its Logarithm , and then it will be thus 4 , 553712 , which 4 , is called the Index , as shall be hereafter shewed . Now before we proceed to find numbers corresponding to Logarithms , it will be necessary to explain the meaning of the first figure to the left hand of any Logarithm placed , Mr. Briggs calleth it a Characteristick or Index , which doth represent the distance of any the first figure of any whole number from Unity , whose Index is 0 , a Cypher ; so the Index o● 10 is 1 , and so to 100 whose Index is 2 , and s● to 1000 whose Index is 3 , and so to 10000 whose Index is 4 , and so if you persist furthe● the Characteristick is always one less in dignity than the places or figures os the number propounded . PROP. II. To find the Logarithm belonging to a Vulgar Fraction , and a Mixt number . First as is before shewed if it be a Vulgar Fraction , find the Log. of the Numerator , and the Log : of the Denominator , then substract the Log : of the Numerator , from the Log : of the Denominator , the remainder is the Log : of the Fraction propounded : Now if you would find the Logarithm of 5 / 7 , do as is prescribed whose Log. I find to be 0. 146121 , Now to find the Log. of a Mixt Number , reduce it into an Improper Fraction , and then do as before , so the Log of 15 ⅖ , Improper 77 / 5 ; , is 1 , 187 , 52 , and so do for any other Mixt number . PROP III. A Logarithm propounded to find the whole , or Mixt number , corresponding thereunto . For the more speedy finding the number , answering unto the Logarithm propounded , observe that if the Index be 0 , then the Number sought may be found between 1 and 10 ; If 1 , between 10 and 100 ; if 2 , between 100 , and 1000 ; if 3 between 1000 and 10000 , and so on still observing the Rules of the Characteristick , or Index , therefore loo , in the Table untill you find the Logarithm proposed , and against it in the Margent according to the aforegoing directions under N , you shall find the number belonging thereunto . This Rule holds in force in Mixt Numbers also . Thus. 0. 845098 1. 556302 2. 130334 3. 980276 Are the Logarithms of , 7 36 135 9556 NOTE . But if you cannot find the Logarithm exactly in the Table , as in many operations it so hapneth , you must then take the nearest Logarithm Number to the Logarithm propounded , and so take the number belonging thereto ●or the desired number . SECT . II. Of the Admirable use of the Logarithms in Arithmetick . PROP. I. To Multiply one number by another . Admit 90 , be to be multiplied by 42 , what is the product ? To find which first find the Log. of the Multiplicand 90 , whos 's Log. is 1. 95424 : Then find the Log. of the Multiplier 42 , whose Log : is 1 62324 , then add these two Log : together , viz. the Log : of the Multiplicand , and Multiplier , their sum is 3 , 57748 , which is the Log : of 3780 , the product of 90 ; and 42 , Multiplied together . PROP. II. To Divide one number by another . Admit the Dividend ( or number to be divided ) be 648 , and the Divisor 72 , what is the number that the Quotient shall consist off ? To find which , first write down the Logarithm of the Dividend 648 , which is 2. 81157 and also write down the Logarithm of the Divisor 72 , which is 1. 85733. Now substract the Log : of the Divisor , out of the Log : of the Dividend , the remainder is 0. 95424 , which is the Logarithm of 9 , so I conclude that the Divisor 72 , is contained in the Dividend 648 , 9 times , and so do for any other . PROP. III. To find the Square-Root of a Number . Admit it be required to Extract the Square-Root of the Number 144 , to perform which first write down the Log : of 144 which is 2. 15836. Then take the half thereof which is 1. 07918 which number 1. 07918 , is the Log : of 12 , the Root of 144 propounded , and so do for any other . NOTE . Now on the Contrary by doubling the Log. of any number , you have the Geometrical Square thereof . PROP. IV. To find the Cube Root of any Number . Admit it be required to Extract the Cube Root of 1728 , to perform which , First write down the Log of 1728 which is 3. 23754 , then take the third part thereof which is 1. 07918 , which is the Log. of 12 ; which is the Cube-root of the Number propounded 1728 , and so for any other . Note on the contrary if you multiply the Log. of any Number propounded by 3 , it produceth the Log. of the Cube thereof . PROP. V. A Summ of Money being forborn for any number of years , to find how much it will amount unto , reckoning Interest on Interest , according to any Rate propounded . Admit 300 pounds Sterling , be put out for 4 years , for Compound Interest at 6 l. per Cent. what will it amount to when the four years are expired ? To find which substract the Log of ●00 l. the principal , whose Log. is 2. 477121 , out of the Log. of 318 l. Principal and Interest for a year whose Log. is 2. 502427 , the remainder is 0. 025306 , which being multiplyed by 4 , the number of years of its continuance , produceth 0. 101224 , which added to the Log. of the principal 300l . to wit , to 2. 477121 , makes ● , 578345 , which is the Log. of 378 l. 14 s. 10d . 2q . very near , and so much will 300 l. amount to . PROP. VI. A Summ of Money being to be paid hereafter , to find what it is worth in ●eady Money . Admit 100 pounds Sterling , to be paid at 30 years end ; I demand how much it is worth in ready Money ? after the rate of Interest of 6 l. per Cent. To find which substract the Logarithm of 100 the principal , whose Log. is 2. 000000 from the Log. of 106 Principal and Interest , whos 's Log. is 2. 025306 , the remainder is 0 025306 , which Multiplyed by 30 the number of years to succeed , produceth 0. 759180 , which substracted out of 2. 000000 , leaveth 1. 240820 , which is the Log. of 17 411 / 1000 , which sheweth the said 100 l. is worth but 17 l. 8s . 2d 3q . fere . PROP. VII . A yearly rent , or Annuity to continue any number of years , to find what it is worth in ready Money , at any Rate of Interest propounded . What is 100 pound per annum to continue 30 years , worth in ready money at 6 l. per Cent. To find which first substract the Log of 100 l. the principal , which is 2. 000000 from the Log. of 106 l principal and interest for a year , whose Log. is 2. 025306 the remainder is 0. 025306 : Then Multiply 0. 025306 , by 30 the number of years of its continuance , it produceth the number 0. 759180 ; Then Divide 100 l. by 6 the rate of interest and the Quotient is 16 6667 / 10000 , &c. which : 16 6667 / 10000 , is the proportional parts of 100 l. the principal , then add the Log. thereof which is 1. 221829 to the former Log. 0. 759180 it produceth 1. 981009 , which is the Log. of 95 7215 / 10000 parts the Arrearages with the said some for that Time , then from those Arrearages 95 7215 / 10000 , substract the parts proportional of 100 , to wit 16 6667 / 10000 , the remainder is 79 ●●48 / 10000 , which is the bare Arrearages for that proportional part ; Then take the Log. of 79 0548 / 10000 , which is 1. 897929 , out of the which take the Log. found by Multiplication of years , to wit 0. 759180 , there remains 1. 138749 , which is the Log. of the value of the Arrearages in ready money , Then to the Log. 1. 138749 , add the Log. of 100 l. principal , 2. 000000 , it produceth this number 3. 138749 ; the Log. of 137 6 48 / 100 , reduced is 1376 l. 9. sh. 7d . 80 / 100 or ⅘ fere : and so much is the said Annuity worth in ready money . CHAP. III. The Explication of the SINES , TANGENTS , and SECANTS . SECT . I. Of Right Signs , Tangents , Secants , Cosines , Tangents , and Secants : Of any Arch , or Angle of a Triangle . PROP. I. To find the right Sine , or Tangent of any Arch or Angle of a Triangle containing any number of Degrees and Minutes . IF the Angle or Arch of the Triangle propounded beless than 45 Deg. the Sine , or Tangent belonging thereunto , is found in the Column under the Title SINE , or TANGENT , at the top of the Table ; and if there be any Minutes annexēd unto the Degrees , you must find them out in the first Column under M. signifying Minutes , and opposite to those Minutes , and under the title aforesaid , you shall have the Logarithm of the Sine or Tangent , of the Arch or Angle required . But if the Arch or Angle of a Triangle exceed 45 Degrees , you must then look for the Sine or Tangent belonging thereunto , in the bottom of the said Table , and if thereunto are Minutes annexed , you must look for them in the first Column to the Right hand under M. and so opposite to those Minutes in the Column above the Title , Sine , or Tang ; there have you the Log. of the Sine , or Tangent , of the Arch or Angle , of the Triangle propounded . Examp. Suppose it were required to find the Log-Sine or Log-Tangent ; of an Angle of 25 D. 37 M. whose Log Sine , whereof according to the former directions I find to be 9. 635833. and Tangent thereof to be 9. 680768. and so for any other under 45 degrees . Again , suppose it were required to find the Log-Sine or Log-Tangent , of an Angle of 64D . 23M the Sine whereof , I find to be this number'9 , 955065 , and the Tangent thereof , 10. 319231 , and so for any other Arch , or Angle of a Triangle , above 45 degrees . PROP. II. To find the Co-Sine or Co-Tangent of any Arch , or Angle propounded . The Co-sine or Co tangent , of an Angle or Arch , is the remaining part of the Angle propounded , to a Quadrent or 90 Degrees ; and is by some called the Complement of an Angle , thus the Arch or Angle of 64D . 23M . taken out of 90D . leaves 25D . 27M . for its Complement , on the contrary if 25D . 37M . were taken out of 90 Degrees , there would remain 64D . 23M . for its Complement . So you see that these two Angles , are the Complements of each other , because they two are equal to a Quadrent or 90 Degrees . Now the Logarithm of the Complement , may be exactly found with ease , for the Sines and Tangents of every degree , and Minute of the Quadrent in one Column is joyned with his Complement in the next Column , so that without substracting the Angle from 90D . you may readily find the Complement thereof either the Arch in Degrees and Minutes , or the Log. Sine , or Tangent thereof , as you have occasion : Thus the Log. of the Sines Complement before mentioned , to wit , 64D . 23M . Comp. is 25D . 37M . is 9. 635833 , Tang. is 9. 680768 ; so 64D . 23M . is the others Compl. whose Sine is 9. 955065 , and his Tang. is 10 , 319231 ; so for any other . PROP. III. To find the Secant of any Arch or Angle propounded . In this little Book I have not room to set down the Tables of Artificial Secants at large , as I have done with the Sines and Tangents : Nevertheless I will not here omit to shew how they may be easily found out , by the Tables of Sines . The method is thus , substract the Logarithm Sine , of the Sines compl of an Angle , from the double Radius of the Tables , and the remainder shall be the Secant required : As if I desire the Secant of 25D . 37M . I find the Logarithm-sine of his complement to be 9. 955065 , which substracted from the double Radius , that is 20. 000000 : there remains 10 , 044935 which is the Secant of it , and so the Secant of 64D . 23M . is 9. 955065 ; which is the Complement of the former , because they both are Equal to 20. 000000 , the double Radius ; and so may any other be found out . CHAP. IV. Of GEOMETRY . THE End and Scope of Geometry is to measure well : for as Number or Multitude , is the Subject of Arithmetick : so is Magnitude that of Geometry : to measure well is therefore to consider the Nature of every thing that is to be measured ; to compare such like things one with another : and to understand their Reason and proportion , and also their similitude : And this is the End and Scope of Geometry * . I shall not trouble you with the Definitions of Geometry , they being largely handled by many , and herein every one meanly conversant in the study of the Mathematicks is acquainted , but shall immediately fall in hand with the principal Propositions , which chiefly concern the other following parts of this treatise . SECT . I. The Explication of some Geometrical Propositions . PROP. I. To erect a perpendicular on any part of a line assigned . LET the Line be A , B , and on the point D , 't is required to raise a perpendicular to A , B , To operate which first open your Compasses to any convenient distance , and placing one foot thereof in D , with the other make the two marks C , and E , equidistant from D ; then open the Compasses to some other convenient distance , and set one foot in E , and describe the Arch FF ; then likewise in C , describe the Arch GG , then through the Intersections of these two Arches , and to the point D , draw H D , perpendicular to A B ; as was required . PROP. II. To Erect a Perpendicular , on the End of a Line . Let the given line be A B , and on the End thereof at B , 't is required to raise a Perpendicular line : To perform which open your Compasses to the distance B D , then on B as a Center , describe the Arch D , E , F , then from D , to E , place BD ; then placing one foot in E , describe the Arch CF , then remove your Compasses to F , and draw the Arch CE ; Lastly through their Intersection draw C B , which is a Perpendicular to AB , on the end B ; as required . PROP. III. From a Point above to let fall a Perpendicular on a Line . Let the line given be B A , and 't is required from the point above at C ; to let fall a Perpendicular to the said Line : To perform which place one foot of your Compasses in C , and open them beyond the given line A B , and describe the Arch EF ; divide EF , into two parts in D ; Lastly draw CD , which shall be perpendicular unto AB , falling from the point above at C , as was so required . PROP. IV. To draw a right line Parallel to a right line , at any distance assigned . Let the distance assigned be O E , and the Lime given be A B , and 't is required to draw C D , Parallel to A B ; at the distance O E : To perform which , take in your Compasses the distance O E , and on A , describe the Arch H , and on B , the Arch K ; then draw C D , so as it may justly touch the two Arches , but cut them not , so shall C D ; be parallel to A B , at the assigned distance O E , as was required . PROP. V. To Protract an Angle of any Quantity of Degrees propounded . Let it be required to Protract , or lay down an Angle , of 40 degrees : To perform which first draw a right line as A B , then open y●●r Compasses to 60 degrees , in your line of Chords : and with that Distance on A , describe the Arch E F , then take 40 degrees in your Compasses out of your line of Chords , and place it on the Arch , from F , to H ; Lastly through the point H , and from A draw A C ; so shall the Angle CAB contain 40 degrees as required . PROP. VI. To measure an Angle already protracted . Let the Angle given be C A B , and 't is ●equired to find the Quantity thereof : To ●erform which take in your Compasses 60 de●rees from your line of Chords ; and on A , ●escribe the Arch EF ; then take in your Com●asses the Distance FH , and apply it to your 〈…〉 ne of Chords ; and you will find the Angle , 〈…〉 AB to contain 40 degrees . PROP. VII . To divide an Angle into two Equal parts . Let the Angle given be BAC , and 't is required to divide it into two equal parts : To perform which do thus : first take in your Compasses any convenient distance , and placing one foot in A , describe the Arch FKHE , then on H , describe the Arch KK , and on K , the Arch HH ; lastly through the Intersections of these two Arches , draw the line AD , to the Angular point A ; so shall the Angle BAC , be divided into two equal parts , viz. BA● , and DAC ; as required . PROP. VIII . To divide a right line into any Number of Equal or Unequal parts ; or like to any divided line propounded . Let the line A B , be given to be divided into 5 equal parts ; as the line CD . To perform which do thus : first on the point C , draw out a line making an Angle with CD at pleasure : then make CF , equal to AB ; and joyn their Extremities FD , then draw Parallel lines to FD , through all the 5 points of CD , ( by the 4 prop. aforegoing ) which shall divide AB , into 5 equal parts ; as required : This way is to be observed , when the line given to be divided , is greater than the divided line propounded . CASE II. But if AB , be shorter than the given divided line CD ; take the line AB , in your Compasses , and on D strike the Arch F , then draw the Tangent CF , then take the nearest distance from the first division of CD , to the Tangent-line CF , which distance shall divide AB into 5 equal parts , as the given divided line CD ; as required . PROP. IX . How to Protract or lay down any of the Regular Figures , called Polygons . To perform which divide 360 degrees , ( the number of degrees in a Circle ) by the number of the Poligon his sides : as if it be a Pentagon by 5 , if a Hexagon by 6 , &c. the Quotient is the Angle of the Center ; its Complement to 180D . ( or a Semi circle ) is the Angle at the Figure , half whereof is the Angle of the Triangle at the Figure : Now I will shew how to delineate any Poligon three ways , viz. 1 by the Angle at the Center , 2. by the Angle at the Figure , 3. by the Angle of the Triangle at the Figure : I have hereunto annexed a Table , which gives at the first sight , ( without the trouble of Division ) 1. the quantity of the Angle at the Center ; 2. the quantity of the Angle at the Figure ; and 3 the Quantity of the Angle at the Triangle of the Figure , from a Triangle to a Decigon . Names of the Poligons . Sides Angles at the Center Angles at the Figure Angles at the Trian . D M D M D M Triangle 3 120 00 60 00 30 00 Square 4 90 00 90 00 45 00 Pentagon 5 72 00 108 00 54 00 Hexagon 6 60 00 120 00 60 00 Heptagon 7 51 43½ 128 34½ 64 17¼ Octogon 8 45 00 135 00 67 30 Nonigon 9 40 00 140 00 70 00 Decigon 10 36 00 144 00 72 00 CONSTRUCTION I. First by the Angle at the Center , to delineate a Hexagon , whose Angle at the Center is 60 degrees , first lay down an Angle of 60 deg . ( by prop. the 5. aforegoing ) making its sides of a convenient length at pleasure , then take such a distance from O the Center of the figure , equally on both sides , as may make the third side equal to the side of the Poligon given ; which here is 100 parts : * Then divide the third side equally into two equal parts , and draw a line through it , from ☉ the Center : set each half of the side of the Poligon 100 , to wit 50 , on each from the middle of the third line . † thus having placed the side of the Hexagon PP , 100 parts , in order ; describe the whole Hexagon PPPPPP , as was required . CONSTRUCTION II. Now by the Angle of the Figure , to delineate any regular Poligon , Let it be required to protract a Hexagon , whose side as afore is 100 parts ; first I draw a line and make it 100 of those parts , then I sind in the precedent Table the Angle of a Hexagon at the figure to be 120 degrees : Then on each side of the drawn line , I lay down an Angle of 120 deg . ( according to the 5 precedent propositions ) and so work 6 times , ( or as many times as your Poligon hath sides ) making each side 100 parts , and each Angle 120 degrees ; so shall you have enclosed the Poligon PPPPPP , as required . CONSTRUCTION . III. To Protract or lay down a Hexagon , or any other regular Poligon , by the Angle of the Triangle , do thus ; First draw the side of the Hexagon P P , make it 100 parts . I find in the precedent Table that the Angle of the Triangle is 60 deg ; then at each end of the line P P , I lay down an Angle of 60 deg . ( by prop. 5. precedent ) and continue the two lines PO , and PO ; untill they intersect each other in O : then on O , as a Center ( OP : being Radius ) describe a Circle , and within it describe the Hexagon PPPPPP , as you see in the figure : and so may you delineate any other Poligon : whose Angels from a Triangle , to a Decigon , are all specified in the precedent Table . PROP. X. To divide a line according to any assigned proportion . Admit the right line given to be AB , and 't is required to divide the same into two parts , bearing proportion the one to the other as the lines E , and F doth : To perform which , first draw the line CD , equal to the given line AB : Then draw the line HC , from C , to contain an Angle at pleasure . Then from C to G , place the line F , and from G , to H , place the line E : Then draw the line HD . And lastly , draw GK parallel to HD , ( by the 4 prop. precedent ) so is the line DC , equal to AB , and divided into two parts , bearing such proportion to each other , as the two given lines E , and F , as was required . PROP. XI . To two lines given , to find a third proportional to each of them . Admit the two given lines be A and B , and 't is required to find a third proportional to A , as A , to B : First make an Angle at pleasure ; as HIK . Then place the line B , from I , unto P ; and the line A , from I , unto L ; and draw PL. then also place the line A , from I unto M , and draw QM , parallel unto LP , ( by 4 prop. ) so shall the line IQ , be a third proportional unto the two given lines A , and B , as was required . For as B , is to A , so is A , unto the proportional found IQ . PROP. XII . To three lines given to find out a fourth proportional unto them . Admit the three given lines to be A , B , and C ; and 't is required to find a third proportional to them , which shall have such proportion unto A , as B , hath unto C. To perform which , first make an Angle at pleasure as DKG , now seeing the line C , hath such proportion to B , as the line A , unto the line sought : Therefore place the line C , from K , unto H and B , from K , to F , and draw FH . Again , place the line A , from K , to I , and draw IE , parallel unto FH , ( by 4 prop. ) until it cutteth DK , in E ; so have you the line KE , a fourth proportional , as was required . For as C , is unto B , so is A , unto the found line KE . PROP. XIII . To find a mean proportional Line between any two right lines given . Let the two given lines be A , and B , between which it is required to find a mean proportional line . To perform which , first joyn the two lines A , and B together , so as they make the right line CED : Then describe thereon a Semicircle CFD . Then on the point E , erect the perpendicular EF , ( by 1 prop. ) to cut the limb of the Semi-circle in F , so shall EF , be a mean proportional line , between the two given lines A , and B , as required . PROP. XIV . To find two mean proportional Lines between any two right Lines given . Let the two given lines be A , and B ; between which 't is required to find two mean proportionals . To perform which , first make an Angle containing 90 deg . making the sides CD , and CE of a convenient length : then from C , place the line B , unto F , and the line A , from C , unto G ; and draw FG , which divide equally in H , and describe the Semi-circle F K G. Then take the line B in your Compasses , and placeing one soot in G , with the other make a mark in the limb of the Semi-circle in K , then draw ST , in such sort that it may justly touch the Semi-circle in K , and may cut through the two sides of the Angle , equidistant from the Center of the Semi-circle H ; so shall SF , and TG , be two mean proportionals , betwixt the two given lines A , and B , as required . PROP. XV. To make a Geometrical square equal to divers Geometrical squares . Let there be given the 5 sides of five Geometrical Squares , viz. A , B , C , D , E ; and 't is required to make one Geometrical Square , equal to the said five Sqares : To perform which first make a Right Angle as ABC , making its contained sides of a convenient length . Then from B , place A , to D , and from B , place B , to E , and draw Ed. Then place Ed , from B , to F , and C , from B , to G ; and draw GF . Then place GF , from B , to H , and D , from B , to I ; and draw ●H . Lastly from B , unto K , place IH , and from B , unto L , place the line E ; and draw LK . So shall LK , be the side of a Square , equal to the five Squares propounded . PROP. XVI . To make a Circle equal to divers Circles propounded . Let the two Circles propounded be A , and B , and 't is required to make a third Circle , ●e●ual to the said Circles propounded . To perform which , first take the Diameter , of the lesser Circle A , and place it as a Tangent , on the Diameter of the greater Circle B , at right Angles ; as ECD . Then draw the Diagonal ED , which divide equally in F , on which as a Center describe the Circle K , making E D , the Diameter of which Circle K shall be equal unto the two given Circles A , and B , as required * SECT . II. Of Planometry , or the way to measure any plain Superfice . PLanometry is that part of the Mathematicks , derived from that Noble Science Geometry , by which the Superficies or Planes of things are measured , and by which their Superficial Content is found , which is done most commonly by the Squares of such Measures , Viz. a Square Inch , Square Foot , Square Yard , Square Pace , Square Perch , &c. That is whose side is an Inch , Foot , Yard , Pace , or Pearch Square . So that the Content of any Figure is said to be found , when you know how many such Inches , Feet , Yards , Paces , &c. are contained therein : Thus the End and Scope of Geometry is to measure well . PROP. I. To find the superficial Content of a Geometrical square . Let the side of the Square AA be 4 Perch , what is the Area , or superficial content thereof ? To find which multiply its side 4 , by its self , it produceth 16 , which is the content of that Square AAAA , propounded . PROP. II. To find the superficial content of a Parallelogram , or long Square . Multiply the length in parts , by the breadth in parts ; the product is the content thereof . So in the Parallelogram , or long Square ABCD , the length of the side AB , or CD is 20 Paces , and the breadth AC , or BD is 10 paces , and his superficial content is required . I say therefore if according unto the Rule , you multiply the length 20 , by the breadth 10 , it produceth 200 Paces ; which is the content of the Parallelogram or long Square ABCD. PROP. III. To find the superficial Content of any Right-lined Triangle . Although right-lined Triangles are of several kinds , and forms ; as first in respect unto their Angles , they are either Right-angled , or Oblique-angled , i. e. Acute-angled , or Obtuse-angled . Secondly in respect of their sides , they are either an Equilateral , Isosceles , or Scalenium Triangle : But now seeing they are all measur'd by one and the same manner , I shall therefore add but one Example for all ; which take for a general Rule : which is , Multiply the length of the Base , by the length of the Perpendicular , half their product is the Area or superficial content thereof . So if the content of the Triangle ABC , be required . To find which first from the Angle B , let fall the Perpendicular DB , on the Base AC , ( by prop. 3. § . 1. ) let therefore the length of the Perpendicular BD be 24 , and the Base AC 44 parts . Now if the Base AC 44 , were multiplyed by BD 24 , the product is 1056 , half whereof is 528 , the Content of the Triangle ABC , propounded . PROP IV. To find the superficial Content of a Rhombus . First let fall a Perpendicular from one of the Obtuse-angles , unto its opposite side , ( by prop. 3. § . 1. ) and then Multiply the length of the side thereof , by the length of the Perpendicular , their product is the Content thereof . So in the Rhombus ABCD , the side AC , or BD is 16 Inches , and the Perpendicular KC is 14 Inches , which multiplyed into the side 16 , produceth 224 Inches ; which is the Area , or superficial Content , of the Rhombus ABCD , propounded . PROP. V. To find the superficial content of a Rhomboides . Frst let fall a Perpendicular , as in the former proposition , then the length thereof multiply by the length of the Perpendicular ; the product is the Area , or superficial content thereof . For in the Rhomboides EDAH , whose length AH , or ED is 32 Feet , and the length of the Perpendicular HK is 16 Feet , which multiplyed together produceth 512 Feet , which is the Area or superficial content of the Rhomboides AHED , propounded . PROP. VI. Te find the superficial Content of any Poligon , or many equal sided Superficies . First from the Center unto the middle of either of the sides of the Poligon , let fall a Perpendicular , ( by 3. prop § . 1. ) Then multiply the length of half the Perifery , by the Perpendicular , the product shall be the Superficial Content of the Poligon . Admit the Poligon to be an Hexagon AAAA AA , whose side AA is 22 Feet , and the Per●end●cular BE 19 Feet ; now , if 66 half the Perifery , be multiplyed by 19 it produceth 1254 Feet ; which is the Content of the Poligon AA , &c. as required . PROP. VII . To find the superficial Content of a Circle . Multiply half the Circumference , by one half of the Diameter , their product is the superficial Content thereof . Admit the Circumference of a Circle ACBD , be 44 Inches , what is the Area or Content thereof . ( by the 9. prop. § . 2. ) I find the Diameter to be 14 Inches , therefore I say if 22 , half the Circumference , be multiplied by 7 , half the Diameter , it shall produce 154 Inches ; which is the superficial Content of the Circle ACDB , as required . PROP. VIII . By the Diameter of a Circle given , to find the Circumference . Suppose the Diameter be 14 , what is the Circumference ? The Analogy or Proportion holds thus , as 7 , to 22 , so is 14 , unto 44 , the Circumference required . PROP. IX . By the Circumference of a Circle given , to find the Diameter . Suppose the Circumference of a Circle be 44 what is the Diameter ? the Analogy or Proportion is , as 22 , to 7 , so is 44 , unto 14 , the Diameter required . Now the proportion of the Diameter , unto the Circumference is as 7 , unto-22 ; or as 113 , to 355 ; or as 1 , unto 3 , 1415926 , &c. so is the Diameter to the Circumference . PROP. X. By the Content of a Circle given , to find the Circumference . Suppose the Content of a Circle be 154 , what is the Circumference , the Analogy or Proportion ? As 7 , unto 4 times 22 , which is 88 , so is 154 the Content of the given Circle ; to the square of the Circumference 1936 , whose root being Extracted , as is taught ( in prop. 8. § . 1. chap. 1. ) gives the Circumference 44 , as required . PROP. XI . By the Content of a Circle given , to find the Diameter . Suppose the Superficial Content of a Circle be 154 parts , what is the Diameter thereof ? to find which this is the Analogy or Proportion . As 22 , To 4 times 7 , which is 28 , So is 154 , the given Content , To the Square of the Diameter 196 , whose Root being Extracted ( by 8 prop chap. 1. § . 1. ) ●iveth the Diameter 14 , as required . PROP. XII . By the Diameter of a Circle given to find the side of a square equal thereto . To find which this is the Analogy or Proportion . As 1 , 000000 , To 0 , 886227. So is the Diameter of the Circle propounded . To the side of a Square , whose superficial Content , is equal unto the superficial Content , of the Circle propounded . PROP. XIII . By the Circumference of a Circle given , to find the side of a square equal to it . This is the Analogy or Proportion . As 1. 000000 , To 0. 282093. So is the Circumference of the Circle propounded , to the side of a Square equal to the Circle . PROP. XIV . By the Content of a Circle given to find the side a square equal to it . To do which , Extract the Square-Root o● the Content propounded , ( by prop. 8 chap. ●● § . 1. ) so is the Root , the side of a Geometrica● Square , equal thereunto . PROP. XV. By the Diameter of a Circle given , to find the side of an Inscribed square . This is the Analogy or Proportion . As 1. 000000 , To 0. 707107 , So is the Diameter of the Circle propounded , To the side of the inscribed Square . PROP. XVI . By the Circumference of a Circle given , to find the side of an Inscribed Square . This is the Analogy , or Proportion . As 1. 000000 , To 0. 225079. So is the Circumference of the Circle propounded , To the side of the inscribed Square . PROP. XVII . ●o find the Superficial Content of an Oval , or Elleipsis . Let the Oval given be ABCD , and 't is re●uired to find the Area or Superficial Content 〈…〉 ereof ? To do which multiply the length A 〈…〉 40 Inches , by the breadth CD 30 Inches , the ●● is 1200. Which divide by 1. 27324 ; 〈…〉 e Quotient is 942 48 / 100 parts . Which is the 〈…〉 ea or Superficial Content of the Oval ABCD 〈…〉 opounded PROP. XVIII . To find the Superficial Content of any Section , or Portion of a Circle . Multiply half the Circute of the Section , by the Semidiameter of the whole Circle , and the product thence arising is the Area or superficial Content thereof . Suppose there be a Circle whose Diameter is 14 parts , and the Circute of the Quadrent ABC is 11 parts , and the Content of the said Quadrent is desired ? To find which multiply 5 ½ or , 5. , 5 half the Circute of the Quadrent , by 7 the Semidiameter , the product is 38 5 / 10 , which is the Content of the Quadrent ABC propounded . SECT . III. Of STEREOMETRY , or the way how to measure any Regular Solid . STereometry is that part of the Mathematicks , springing from Geometry , by which the Content of all Solid Bodies are discovered by two Multiplications , or three Dimention and is valued by the Cube of some famous Mea sure ; as an Inch-Cube , a Foot-Cube , a Yard Cube , or a Perch-Cube , &c. PROP. I. To find the solid Content of a Cube . Multiply the side into its self , and that product by its side again ; their product is the solid Content thereof . Suppose there be a Cube A , whose side is 2 Feet ; and his solid Content is required ? I say if his side 2 , be multiplyed by its self , it produceth 4 , which again multiplyed by 2 , it produceth 8 Feet , which is the solid Content of the Cube propounded . PROP. II. To find the solid Content of a Parallelepipedon . First get the Superficial Content of the End , ( by prop. 1 , or 2 , § . 2. ) which multiply into the length , the product is the solid Content . Suppose there be a Parallelepipedon B , whose sides of the Base is 40 , and 30 Inches , and length 120 Inches , and his Solid Content is demanded ? I say if you multiply 30 , by 40 , the product is 1 , 200 , which is the superficial Content at the Base . Which multiplyed by the length 120 Inches produceth 144000 Inches , which is the solid Content of the Parallelepipedon B , propounded . PROP. III. To find the solid Content of a Cylinder . First get the superficial Content of the Circle at the Base , ( by prop. 7. § . 2. ) and by it multiply its length , their product is the solid Content thereof . Suppose there be a Cylinder as D , whose Diameter of the Circle at the Base is 7 parts , and the length of the Cylinder is 14 parts , and 't is required to find the solid Content thereof ? First I find the superficial Content of the Base to be 38. 5 , which multiplied into 14 the length , giveth 539 parts , which is the solid Content of the Cylinder propounded . PROP. IV. To find the solid Content of a Pyramid . First get the superficial Content of the Base of the Pyramid , ( by some of the aforegoing propositions in Planometria ) and then multiply that into ⅓ of his Altitude , the product is the solid Content thereof . Suppose there be a Pyramid H , whose side of the Base is 4 ½ parts , or 4 5 / 10 , and his Altitude 12 parts , and his solid Content is required ? First I find , ( by prop. 1. § . 2. ) the superficial Content of the Base to be 20 25 / 100 or 20 ¼ , which multiplyed by 4 , ( which is ⅓ of the Altitude 12 ) produceth 81 parts , for the solid Content of the Pyramid propounded . PROP. V. To find the solid content of a Cone . First find the superficial Content of the Circle at the Base , ( by prop. 7. § . 2. ) then multiply it by ⅓ of its Altitude or Heighth , the product is the solid Content thereof . Suppose there be a Cone as B , whose Diameter of the Base is 7 , and his Altitude or Heighth is 15 parts , and his solid Content is required ? First I find the superficial Content of the Base to be 38½ or 38. 5 ; which multiplyed into 5 , ⅓ of its Altitude or Heighth ) produceth 192. 5 , or ½ , which is the solid Content of the Cone propounded . PROP. VI. By the Diameter of a Globe to find his solid Content . This is the Analogy or Proportion . As 6 times 7 , which is 42. Is to 22 , So is the Cube of the Diameter of the Sphere , or Globe propounded . To the solid Content thereof . Suppose there be a Sphere or Globe , whose Diameter is 12 Inches ; what is the solid Content thereof ? say , ( see the Globe R. ) As 42 , Is to 22 , So is 1728 , the Cube of the Diameter , To the solid Content 905 6 / 42 or 1 / 7 of the Globe , or Sphere propounded : This and all other such like Propositions , are performed by the help of the first Proposition , of the first Chapter of this Book . PROP. VII . By the Circumference of a Sphere , or Globe , to find his solid Content . This is the Analogy or Proportion . As 1. 000000 , To 0. 016887 , So is the Cube of the Circumference of the Globe or Sphere propounded To the solid Content thereof . PROP. VIII . By the Axis of a Globe , to make a Cube equal thereunto . This is the Analogy or Proportion . As 1. 00000 , To 0. 80604 , So is the Axis of the Sphere propounded , To the ●u●●-Root , which shall be equal to it . PROP. IX . By the Circumference of a Globe , to make a Cube equal thereunto . This is the Analogy or Proportion . As 1. 000000 , To 0 256556. So is the Circumference of the Globe propounded , To the Cube-Root , which shall be equal to the Sphere , or Globe , propounded . PROP. X. By the solid Content of a Sphere or Globe , to make a Cube equal thereunto . Extract the Cube-root of the solid Content of the Sphere or Globe , ( by prop. 9. § 1. chap. 1. ) so shall the Root , so found , be the side of a Cube , equal unto the Globe or Sphere propounded . PROP. XI . A Segment of a Sphere being given to find the solid Content thereof . To find which first say , As the Altitude of the other Segment , is to the Altitude of the Segment given : so is that Altitude of the other Segment increased by half the Axis , unto a fourth : Then say , As 1 , to 1 , 0472 , so is the product of the Quadrant of half the Chord of the Circumference of that Segment , multiplyed by that fourth , To the solid Content of the Segment propounded . CHAP. V. Of TRIGONOMETRY . Or the Doctrine of Triangles . SECT . I. Some general Maxims , belonging to plain or Right-lined Triangles . TRIGONOMETRY is necessary in most parts of the Mathematicks , and herein indeed consisteth the most frequent use of the Logarithms , Sines , Tangents , and Secants : It is conversant in the measuring of Triangles , Plain or Spherical , comparing their Sides , and Angles together ; according unto their known Analogies , or Proportions : So that any three parts of a Triangle being given , the other parts may be found out , and known : Now in the Doctrine of Right-lined Triangles , it will be necessary to know these Maxims following . 1. That a Right-lined Triangle , is a Figure constituted , by the Conjunction , or Intersection , of the three Right , or Streight-lines thereof ; in their Angles or Meeting-places . So that every Triangle hath six distinct parts , Viz. Three Sides , and three Angles . 2. That all Right-lined Triangles , are either Right-angled , That is , which hath one Right-Angle , as ABC Fig. 34. Or Oblique-angled , whose three Angles are all Acute ; that is , less than a Quadrant , or 90 deg ; or else they have One Angle Obtuse , or greater than a Quadrent : So all Triangles , that have not one Right-angle , are called Oblique-Triangles ; as Fig. 36. to wit , the Triangle ABC . 3. That the three Angles , of any Right-lined Triangle , are equal unto two Right-angles ; or 180 Degrees . So that any two of their Angles being known , the third Angle is also found , being the Complement of the other two ; unto 180 Degrees : But this is more readily found in a Rectangled Triangle , for the Rectangle being a Quadrent , or 90 degrees , one of the acute Angles therefore being given , the other is readily known , being the Complement thereof unto 90 Degrees . 4 That the three sides , comprehending the Triangle , some call Leggs , others Sides , but in Rectangled Triangles , as in the Triangle ABC , I call AB , the Base , BC the Cathetus or Perpendicular ; and AC the Hypothenuse . 5. That the Sines , of the Angles are proportional unto their opposite Sides ; and their Sides , to their opposite Angles . So that if the Side of a Triangle were desired , put the Sine of the opposite Angle in the first place . Also if an Angle be required , put the Logarithm of his opposite side in the first place . 6. That the sides of any Rectangled Triangle may be measured by any Scale of equal parts , as Inches , Feet , Yards , Poles , Miles , Leagues , &c. 7. That if an Angle propounded , be greater than 90 deg . and so not to be found in the Tables , take the Complement thereof , unto 180 deg . and work by the Sine , or Tangent thereof , and the work will be the same . And here for the more short , and speedy performance of these conclusions in Trigonometry ; I have annexed , and used , these following Symbols ; which I would have you take notice of . = Equal , or Equal to . + More . - Less . × Multiplyed by . ° Degrees as 15° . ' Minutes as . 40 ' . cr . A Side . cr s , Sides . V An Angle . VV Angles . Z Sum. X Difference . S Sine . Sc Co-sine . T Tangent . Tc Co-tangent . Se Secant . Sec Co-secant . Co. Ar. Compl. Arithmetic . R A Right-angle . 2R Two Right-angles . Q Square . SECT . II. Of Plain Rectangled Triangles . PROP. I. Two Angles and the Base of a Rectangled Triangle given , to find the other parts . ADmit the Triangle given be ABC : Now the Angle at B , is an Angle of 90° , or a Right-angle ; And the Angle at C is 57° 35 ' , and the Base AB ; is 736 parts . Now first I find the Angle at A , to be 32° 25 ' : it being the Complement of the Angle at C , unto 90° : Secondly , to find the Cathetus , or Perpendicular , this is the analogy or proportion . Add the Log. of the third and second Terms together , and from their Sum , deduct the Log. of the first number , so is the Remainder , the Log. of the fourth Term , or Number sought , as you see in the aforegoing Example . Thirdly to find the Hypothenuse AC , the Analogy or Proportion hold thus . As S. V , C 57° 35 ' , To Log. Base AB 736 ●arts . So Radius or S. 90° , To Log. Hypothenuse AC 871 8 / 10 parts required : Thus are the three required parts , of the given Triangle ABC found , viz. the Angle A to be 32° 25 ' , the Cathetus BC to be 467 4 / 10 parts , and the Hypothenuse AC to be 871 8 / 10 parts , as was so required to be found . PROP. II. The Hypothenuse , Base , and one of the Angles Of a Rectangled Triangle given , to find the other parts thereof . In the Triangle ABC , the Hypothenuse AC is 871 8 / 10 parts , the Base AB is 736 parts , and the Angle at B , is known to be a Right-angle ; or 90° : First to find the Angle at the Cathetus C , the analogy or proportion holds thus . As Log. Hypothen : AC 871 8 / 10 parts To Radius or S. 90° . So Log. Base AB 736 parts , To the S. V. at Cathetus C 57° 35 ' . Secondly , now having found the Angle at the Cathetus C , to be 57° 35 ' ; I say the Angle of the Base A is 32° 25 ' , being the Compl. of the Angle C , unto 90° . Thirdly to find the Cathetus BC , this is the ●nalogy , or proportion . As Radius or S. 90° , To Log. Hypothen . AC 871 8 / 10 parts , So S. V. at Base A 32° 25 ' . To Log. Cathetus BC , 467 4 / 10 parts required . It may also be found , as in the former Proposition . PROP. III. In a Rectangled Triangle , the Base , and Cathetus given to find the other parts thereof . In the Triangle ABC , the Base AB is 736 parts , and the Cathetus BC is 467 4 / 10 parts , and the Angle B , between them is a right angle o● 90° : And here you may make either side of the Triangle , Radius , but I shall make BC the Cathetus Radius , and then to find the Angle at the Cathetus C , this is the Analogy or ●●●portion . As Log. Cathet , BC 467 4 / 10 parts , To Radius or S 90° . So Log. Base AB 736 parts , To T. V. Cathe C 57° 35 ' , as required . Secondly , I find the other Angle , at A to be 32° 25 ' , it being the Complement , to C 57 35 ' , unto 90° . Thirdly , To find out the Hypothenuse AC this is the analogy or proportion . As S. V. Cathe C. 57° 35 ' , To Log. Base AB 736 parts , So Radius or S. 90° , To Log. Hypothenuse AC 871 8 / 10 parts , 〈…〉 quired . But making the Base AB Radius , yo● may find the Hypothenuse AC , by this anal 〈…〉 or proportion . Plate 1 Page 65 As Radius or S. 90° , To Log. Base AB 736 parts . So Sc. V. Base A 32° 25 ' , To Log. Hypothenuse AE 871 8 / 10 parts required , and thus you have all the parts of the Triangle propounded . PROP. IV. The Base , and Hypothenuse , with the Angle between them given , to find the other parts of a Rect-angled Triangle . In the Triangle ABC , the Base AB is 736 parts , and the Hypothenuse AC is 871 8 / 10 parts , and the Angle A included between them is 32° 25 ' . First to find the Angles , and first remember that the Angle B is a right Angle ; or 90° . Secondly , that the Angle at C , is the Complement to the Angle at A 32° 25 ' unto 90° : and therefore is 57° 35 ' : Now these being known , you may find the Cathetus , by this analogy or proportion . As S. V. Cathe . C. 57° 35 ' , To Log. Base AB 736 parts . So S. V. Base A 32° 25 ' , To Log. Cathe . BC 467 4 / 10 parts required . Thus I have sufficiently explained all the Cases of Plain Rect-angle Triangles , for to these rules they may be all reduced . SECT . III. Of Oblique-Angled Plain Triangles . PROP. I. Two Angles , and a side opposite , in an Oblique-Angled Triangle given , to find the other parts thereof . IN the Triangle ABC , the Angleat A is 50° , and at C is 37° , and the side AB is 30 parts , and opposite to the Angle C First , to find the Angle B , remember that ( as 't is said , in the third Maxim aforegoing ) 't is the Complement , to the Angles A 50° , and C 37° , to 180° , and therefore is the Angle at B 93° . Secondly , having thus found the Angles , the two unknown sides , may be found by the proportion they bear to their opposite Angles , for that proportion holds also in these ; thus to find the side BC , this is the analogy or proportion . As S. V. C 37° 00 ' , To Log. side AB 30 parts . So S. V. A. 50° 00 ' , To Log. side BC 38 19 / 100 parts required to be found . But it may be more readily found , and performed in such case as this , where you have a Sine , or Tangent , in the first place , by the Arithmetical Complement thereof , and so save the Substraction . Now the readiest way to find the Arithmetical Complement is that of Mr. Norwood , in his Doctrine of Triangles ; which is thus : begin with the first Figure towards the left hand of any Number and write down the Complement , or the remainder thereof , unto 9 : And so do with all the rest of the Figures , as you see here done . Saying 9 , wants of 9 , 0 : and again 9 , wants 0 : 6 , wants 3 ; 2 , wants 7 : 3 , wants 6 ; 9 , wants 0 : only when you come to the last Figure to the right hand , take it out of 10 , so 8 , wants 2 ; of 10 : Thus you may readily find the Co-Ar . of any Sine , almost as soon as the Sine it self . But if you want the Complement Arithmetical of any Tangent , you may take the Co-tang . which is exactly the Co-Arith . of the double Radius , so that the Tangent , and Co-tangent , of an Arch makes exactly 20. 000000. Now if the Radius be in the first place , then there is no need of taking the Co-Arith . of the first Number , only you must cut off , the first I , to the left hand thus X , and you will have the Logarithm of the Number desired . Thirdly , now to find the side AC , by the opposite Angle B ; which is 93° 00 ' : And see 〈…〉 ng the Angle B , exceeds 90° , you must work 〈…〉 y the Complement to 180° ) as in the seventh 〈…〉 ork in page 61 is taught . Thus having found all the parts of the Triangle propounded , Viz. The Angle B , to be 93° 00 ' , the side AC to be 49 78 / 100 parts , and the side BC to be 38 19 / 100 parts , as was required to be found . PROP. II. Two sides , and an Angle opposite to one of them in an Oblique-angled Triangle given , to find the other parts thereof . In the Triangle ABC , the side AB is 30 parts , and the side AC , is 49 78 / 100 parts , and the opposite Angle C , is 37° 00 ' . First , To find the Angle at B , this is the Analogy or Proportion . As Log. cr . AB 30 parts , To S. V. at C 37° 00 ' . So Log. cr . AC 49 78 / 100 parts , To Sc. V. B 93° 00 ' , as was required to be found . Now seeing that the Angle C , is 37° 00 ' , and the Angle B , is 93° 00 ' , which makes 120° 00 ' , therefore must the Angle A be 50° 00 ' ; the Complement to 180° : so having found all the three Angles , you may find the other side CB , 38 19 / 100 parts , as afore in the first proposition , by his opposite Angle . PROP. III. Two Sides of an Oblique-angled Triangle , with the Angle included between them given , to find the other parts thereof . In the Triangle ABC , the side AC is 49 78 / 100 parts , the side DB is 30 parts , and the Angle A between them is 50° 00 ' ; and 't is required to find the other parts of the Triangle propounded . To resolve this Conclusion , let fall a Perpendicular DB , from the Angle B , on the side AC ; ( by prop. 3. § . 1. chap. 4 ) and then proceed thus . First , Seeing the Oblique-angled Triangle , ABC is divided into two Rectangled Triangles , Viz. ADB , and BDC : Now I will begin with the Triangle ADB , in which is given the Angle A 50° 00 ' , and the Angle D is a right Angle , or 90° , and the side AB 30 parts , and the sides AD , and DB , and the Angle at B , are required . First to find the Angle at B , remember that it is the Complement unto the Angle A 50° 00 ' , unto 90° 00 ' , and therefore must the Angle B be 40° 00 ' ; Now for to find the Cathetus BD , ( as in prop. 1. and 2 § . 2. chap. 5. ) by the Rule of opposition , the Analogy or Proportion holds thus . As Radius or S. 90° , To Log. Hypoth . AB 30 parts . So S. V. at A 50° 00 ' , To Log. Cath. BD 22 98 / 100 parts sought . And AGAIN , say . As Radius or S. 90° , To Log. Hypoth . AB 30 parts . So S. V. at B 40° 00 ' , To Log. Base AD 19 28 / 100 parts sought . Thus in the Triangle ADB , you have found the Angle B , to be 40° 00 ' , the Cathetus BD , to be 22 98 / 100 parts ; and the Base AD to be 19 28 / 100 parts , as was so required . Now for the other Triangle which is BDC , in which there is given the side BD , 22 98 / 100 parts , and the Angle at D , is a Right-angle , or 90° , and the sides DC , and CB , and the Angles B , and C , are required . First to find the side DC , substract AD , 19 28 / 100 parts , out of AC , 49 78 / 100 parts ; there remains the Base DC ; 30 50 / 100 parts : Thus have you the two sides of the Triangle , to wit the Base DC , 30 50 / 100 parts , and the Cathetus BD , 22 98 / 100 parts , and the Angle D between them is a Right-angle or 90° . Now you may find the Angle at B , by the Tangent ( as in prop. 3. § . 2. chap. 5. ) thus . As Log. Cath. BD , 22 98 / 100 parts , To Radius or S. 90° . So Log. Base CD 30 50 / 100 parts . To T. V. B. 53° 00 ' . Secondly , For the Angle C , remember 't is the Complement of the Angle B , 53° , to 90° ; and therefore is the Angle C , 37° 00 ' , required . Thirdly , To find the Hypoth . BC , this is the Analogy or Proportion . As S. V. B. 53° 00 ' , To Log. Base DC 30 50 / 100 parts . So Radius or S. 90° , To Log. Hypoth . BC 38 19 / 100 parts : Thus have you found all the required parts of the Triangle ABC propounded , viz. the Angle C to be 37° 00 ' , the Angle B , to be 93° 00 ' , * and the Side BC , 38 19 / 10● parts , as required to be found . Another way to perform the same . Take the Sum of the two sides , and the difference of the two sides ; and work as followeth . Now to find the two Angles B , and C , this is the Manner , and by this Analogy or Proportion , they are found out and known . As Log. Z. cr s. AB , and CA , 79 78 / 100 parts , To Log. X. cr s. AB , and CA ; 19 78 / 100 parts , So T. of ½ VV unknown , 65° 00 ' , To T. ½X . of VV , 28° 00 ' . This difference of Angles 28° 00 ' , add unto 65° 00 ' , ( half the difference of the unknown Angles ) and it shall produce 93° 00 ' , which is the greater Angle , and substracted from it , leaves 37° 00 ' , which is the lesser Angle C : so have you the required Angles . PROP. IV. The three sides of an Oblique-angled Triangle given , to find the Angles . In the Triangle ABC , the side AC , is 49 78 / 100 parts , the side AB , is 30 parts , and the side BC , is 38 19 / 100 parts ; and the three Angles of the Triangle are required . The resolution of this Conclusion is thus . Take the Summ and Differ . of the two sides AB , and BC ; And then work as follows : To find a Segment of the Base AC , to wit CE ; say : As Log. Base AC , 49 78 / 100 parts , To Z. cr s. AB , and BC ; 68 19 / 100 parts , So X. cr s. AB , and BC ; 8 19 / 100 parts , To Log of a Segment of the Base AC , to wit C E 11 22 / 100 parts . This Segment of the Base CE , 11 22 / 100 parts , being substracted from the whole Base AC , 49 78 / 100 parts , the remainder is EA 38 56 / 100 parts , in the middle of which as at D , the Perpendicular DB , will fall from the Angle B ; and so divide it into two Rectangled Triangles , to wit , ADB , and CDB , whose Base DA is 19 28 / 100 parts , which taken from AC 49 78 / 100 parts , leaves the Base of the greater Triangle CD 30 50 / 100 parts . Now having the two Bases of these two Triangles , and their Hypothenuses ; to wit CD 30 50 / 100 parts , DA 19 28 / 100 parts , CB 38 19 / 100 parts , and BA 30 parts ; you may find all their Angles , by the Rule of Opposite sides , to their Angles as afore . I. In the Triangle CDB . To find the Angles , this is the Analogy or Proportion . As Log. BC 38 19 / 100 parts , To Radius or S. 90° . So Log. DC 30 50 / 100 parts , To S. V. B 53° 00 ' : whose Complement is the Angle at C 37° 00 ' unto 90 : or a Quadrant . II. In the Triangle ADB . To find the Angles , this is the Analogy or Proportion . As Log. AB , 30 parts , To Radius or S. 90° . So Log. AD 19 28 / 100 parts , To S. V. B , 40° 00 ' . The Complement whereof , unto 90° 00 ' , is the Angle at A 50° 00 ' . Now in the first Triangle CDB , there is found the Angle C , to be 37° 00 ' , and the Angle B , to be 53° 00 ' . In the second Triangle ADB , there is found the Angle A ; to be 50° 00 ' , and the Angle B , to be 40° 00 ' . Now the two Angles at B , to wit 53° 00 ' ; and 40° 00 ' ; makes 93° 00 ' , which is the Angle of the Oblique-angled Triangle ABC , at B : Thus the three Angles of the said given Triangle ABC , are found as was required , viz. the Angle A to be 50° 00 ' , the Angle B to be 93° 00 ' , and the Angle C to be 37° 00 ' , as sought . Thus I have sufficiently , fully and plainly explained all the Cases of Plain Right-lined Triangles , both Right and Oblique-angled : I shall now fall in hand with Spherical Triangles , both Right and Oblique-angled . SECT . IV. Of Spherical Rectangled Triangles . And here first it will be necessary also to understand those few general Maxims or Rules , that are of special Moment , in the Doctrine of Spherical Triangles . 1. THat a Spherical Triangle is comprehended and formed , by the Conjunction and Intersection of three Arches of a Circle , described on the Surface of the Sphere or Globe . 2. That those Spherical Triangles , consisteth of six distinct parts , viz. three Sides and three Angles , any of which being known , the other is also found out and known . 3. That the three Sides of a Spherical Triangle , are parts or Arches of three great Circles of a Sphere , mutually intersection each other : and as plain or Right-lined Triangles , are measured by a Measure , or Scale of equal parts : So these are measured , by a Scale or Arch of equal Deg●ees . 4. That a Great Circle is such a Circle that doth bessect the Sphere , dividing it into two equal parts ; as the Equinoctial , the Ecliptick , the Meridians , the Horizon , &c. 5. That in a Right-angled Spherical Triangle , the Side subtending the Right-angle we call the Hypothenuse , the other two containing the Right-angle we may simply call the Sides , and for distinction either of them may be called the Base or Perpendicular . 6. That the Summ of the Sides of a Spherical Triangle are less than two Semicircles or 360° . 7. That if two Sides of a Spherical Triangle be equal to a Semicircle ; then the two Angles at the Base shall be equal to two Right-angles ; but if they be less , then the two Angles shall be less ; but if greater , then shall the two Angles be greater than a Semicircle . 8. That the Summ of the Angles of a Spherical Triangle , is greater than two Right-angles . 9. That every spherical Triangle is either a Right , or Oblique-angled Triangle . 10. That the Sines of the Angles , are in proportion , unto the Sines of their opposite Sides ; and the Sines of their opposite Sides , are in proportion unto the Sines of their opposite Angles . 11. That in a Right-angled Spherical Triangle , either of the Oblique-angles , is greater than the Complement of the other , but less than the Difference of the same Complement unto a Semicircle . 12. That a Perpendicular is part of the Arch of a great Circle , which , being let fall from any Angle of a spherical Triangle , cutteth the opposite Side of the Triangle at Right-angles , and so divideth the Triangle into two Right-angled Triangles , and these two parts ( either of the Sides or Angles ) so divided must be sometimes added together , and sometimes substracted from each other , according as the Perpendicular falls within or without the Triangle . PROP. I. Case 1. A Side and an Angle adjacent thereunto being given , to find the other Side . In the Triangle ABC , there is given the Side AB 27° 54 ' ; and the Angle A 23° 30 ' , and the Side BC is required , to find which this is the Analogy or Proportion . PROP. II. Case 2. A Side and an Angle adjacent thereunto being given , to find the other Oblique-angle . In the Triangle ABC , there is given the Side AB 27° 54 ' , and the Angle A 23° 30 ' , and the Angle at C is required , to find which say by this Analogy or Proportion . As the Radius or S 90° 00 ' , To Sc. of cr . AB 27 , 54. So is S. V. at A 23 , 30 , To Sc. V. at c 69 , 22 required . PROP. III. Case 3. A Side and an Angle adjacent thereunto being given , to find the Hypothenuse . In the Triangle ABC , there is given the Side AB 27° 54 ' , and the Angle at A 23° 30 ' , and the Hypothenuse AC , is required ; which may be found by this Analogy or Proportion . As the Radius or S. 90° 00 ' , To Sc. of V. at A 23 , 30. So is Tc cr . AB , 27 , 54. To Tc. Hypothenuse AC , 30 , 00 required . PROP. IV. Case 4. A Side and an Angle opposite thereunto being given , to find the other Oblique-angle . In the Triangle ABC , there is given the Side BC 11° 30 ' , and the Angle A 23° 30 ' , and the Angle C is required , to find which , say by this Analogy or Proportion . As Sc. cr . BC , 11° 30 ' , To Radius or S. 90 , 00. So is Sc. V. at A , 23 , 30 , To S. V. at C. 69 , 22 , as required . PROP. V. Case 5. A Side and the opposite Angle given , to find the Hypothenuse . In the Triangle ABC , there is given the side BC 11° 30 ' , and the Angle at A 23° 30 ' , and the Hypothenuse AC , is required , which may be found by this Analogy or Proportion . As S. V. at A 23° 30 ' , To Radius or S. 90 , 00. So is Ser. BC 11. 30 , To S. Hypothenuse AC 30 , 00. as required . PROP. VI. Case 6. A side and the opposite Angle given , to find the other side . In the Triangle ABC , there is given the side BC 11° 30 ' , and the Angle at A 23° 30 ' , and the side AB is required , to find which this is the Analogy or Proportion . As Radius or S 90° 00 ' , To Tc. of V. at A. 23. 30 , So is T. cr . BC 11 , 30 , To S. of cr . AB 27. 54 as was required . PROP. VII . Case 7. The Hypothenuse , and an Oblique Angle given , to find the side adjacent thereunto . In the Triangle ABC , there is given the Hypothenuse AC , 30° 00 ' , and the Angle A 23° 30 ' , and the side AB , is required , which is found by this Analogy or Proportion . As the Radius or S. 90° 00 ' , To Sc. V. at A , 23 , 30. So is T. Hypoth . AC , 30 , 00 , To T. cr . AB , 27 , 54 , as was required . PROP. VIII . Case 8. The Hypothenuse , and an Oblique-angle given , to find the opposite Side . In the Triangle ABC , there is given the Hypothenuse AC , 30° 00 ' , and the Angle at A 23° 30 ' , and the Side BC , is required , which is found by this Analogy or Proportion . As the Radius or S. 90° 00 ' , To S. Hypoth . AC , 30 , 00. So is S. V. at A , 23 , 30 , To the S. cr . BC , 11 , 30. which was required . PROP. IX . Case 9. The Hypothenuse , and an Oblique-angle given , to find the other Oblique-angle . In the Triangle ABC , there is given the Hypothenuse AC 30° 00 ' , and the Angle A , 23° 30 ' , now the Angle at C , is required , which may be found by this Analogy or Proportion . As the Radius or S. 90° 00 ' , To Sc. Hypoth . AC , 30 , 00. So is T. of V. at A , 23 , 30 , To Tc. of V. at C. 69 , 22 , as was required . PROP. X. Case 10. The sides given , to find the Hypothenuse . In the Triangle ABC , there is given the side AB 27° 54 ' , and the side BC 11° 30 ' , and the Hypothenuse AC is required , to find which say by this Analogy or Proportion . As the Radius or S. 90° 00 ' , To Sc. cr . BC. 11 , 30. So is Sc. cr . AB 27 , 54 , To Sc. Hypothenuse AC 30 , 00. required . PROP. XI . Case 11. The sides given , to find an Angle . In the Triangle ABC , there is given , the side AB 27° 54 ' , and the side BC 11° 30 ' , and the Angle at A , is required , which may be found by this Analogy or Proportion . As the Radius or S. 90° 00 ' , To S. cr . AB . 27 , 54. So is Tc. cr . BC. 11 , 30 , To Tc. of V. at A. 23. 30. as required . PROP. XII . Case 12. The Hypothenuse , and a side given , to find the other side . In the Triangle ABC , there is given , the Hypothenuse AC 30° 00 ' , and the side AB 27° 54 ' and the side BC is required , which may be found by this Analogy or Proportion . As Sc. cr . AB . 27° 54 ' , To Radius or S. 90 00. So is Sc. Hypothenuse AC . 30° 00 ' , To Sc. cr . BC. 11° 30 ' as required . PROP. XIII . Case 13. The Hypothenuse , and a Side given , to find the contained Angle . In the Triangle ABC , there is given the Hypothenuse AC 30° 00 ' , and the side AB 27° 54 ' , and the Angle at A is required , which may be found by this Analogy or Proportion . As the Radius or S. 90° 00 ' , To T. cr . AB . 27° 54 ' So is Tc. Hypoth . AC 30° 00 ' , To Sc. of V. at A , 23° 30 ' , as required . PROP. XIV . Case 14. The Hypothenuse , and a Side given , to find the opposite Angle . In the Triangle ABC , there is given the Hypothenuse AC 30° 00 ' , and the side AB 27° 54 ' , ●ow the Angle C , is required , which may be 〈…〉 ound by this Analogy or Proportion . As the S. Hypoth . C , 30° 00 ' , To Radius or S. 90° 00 ' . So is S. of cr . AB , 27° 54 ' , To S of V. at C. 69 22 , as required . PROP. XV. Case 15. The Oblique Angles given , to find either Side . In the Triangle ABC , there is given the Angle A 23° 30 ' , and the Angle at C 69° 22 ' , and the side BC , is required , which may be found by this Analogy or Proportion . As the S. of V. at C , 69° 22 ' , To the Radius or S. 90° 00 ' . So is the Sc. of V. at A , 23° 30 ' , To the Sc. of cr . BC , 11° 30 ' , as required . PROP. XVI . Case 16. The Oblique-angles given , to find the Hypothenuse . In the Triangle ABC , there is given the Angle A 23° 30 ' , the Angle C , 69° 22 ' , and the Hypothenuse AC , is required , which may be found by this Analogy or Proportion . As the Radius or S. 90° 00 ' , To Tc. of V. at C. 69° , 22 ' , So is Tc. of V. at A , 23 30 , To Sc. Hypoth . AC , 30 00 , as required . SECT . V. Of Oblique-angled Spherical Triangles . PROP. I. Case 1. Two Sides , and an Angle opposite to one of them given , to find the other opposite Angle . IN the Triangle ADE , there is given the Side AE , 70° 00 ' , the Side ED , 38° 30 ' , and the Angle A , 30° 28 ' , now the Angle at D , is required , to find which this is the Analogy or Proportion . As S. cr . DE , 38° 30 ' , To S. V. at A , 30 28. So is S. cr . AE , 70 00 , To S. V. at D , 130 03 , required . PROP. II. Case 2. Two Angles and a Side opposite to one of them given , to find the Side opposite to the other . In the Triangle ADE , there is given the Angle at D , 130° 03 ' , the Angle E , 31° 34 ' , and the Side AE , 70° 00 ' , now the Side AD , is required , which may be found by this Analogy or Proportion . As S. V. at D , 130° 03 ' , To S. cr . AE , 70 00. So is S. V. at E , 31 34 , To S. cr . AD. 40 00 , required . PROP. III. Case 3. Two Sides and an Angle included between them being known , to find the other Angles . In the Triangle ADE , there is given the Side AE , 70° 00 ' , the Side AD , 40° 00 ' , and the Angle A 30° 28 ' , Now the Angles D , and E , are required , which is thus found : take the Sum and Difference of the two Sides , and work as followeth , saying . As S. ½ Z. cr s. AE and AD , 55° co ' , To S. ½ X. cr s. AE and AD , 15 00. So is Tc. ½ V. at A , 15 14 , To T. ½ X. VV. D and E. 49 1430 " . AGAIN . As Sc. ½ Z. cr s. AE and AD , 55° 00 ' , To Sc. ½ X. cr s. AE and AD , 15° 00 ' . So is Tc. ½ V. at A , 15° 14 ' , To T. ½ Z. VV. D and E , 80 48 30 " . This difference of the Angles unknown D and E , 49° 14 ' 30 " , being added unto the half Sum of the Angles 80° 48 ' 30 " , ( unknown ) produceth the Greater Angle D 130° 03 ' , and substracted from it , leaves the Lesser Angle E , to wit 31° 34 ' . PROP. IV. Case 4. Two Angles , and their Interjacent side being known , to find the other sides . In the Triangle ADE , there is given the Angl● at A 30° 28 ' , and the Angle at D 130° 03 ' , and their Interjacent-side AD 40° 00 ' , and the Sides DE and EA , are required : Which is thus found . Take the Sum and Diffference of the two Angles , and work as followeth , saying . As S. ½ Z. of VV. A and D , 80° 15 ' 30 " , To S. ½ X. of VV. A and D , 49 47 30. So is T. ½ cr . AD , 20 00 00 , To T. ½ X. cr s. DE and EA , 15 45 00. AGAIN Say. As Sc. ½ Z. of VV. A and D , 80° 15 ' 30 " , To Sc. ½ X. of VV. A and D , 49 47 30. So is T. ½ cr . AD , 20 00 00 , To T. ½ cr s. Z. DE and AE . 54 15 00. Add the half Difference of the Sides DE and AE , 15° 45 ' , unto half the Sum of the Sides DE and AE , 54° 15 ' . It produceth the greater Side , the Side AE 70° 00 ' , but if deducted from it , leaves the lesser Side ED , which is 38° 30 ' , as was required . PROP. V. Case 5. Two Sides and an Angle opposite to one of them given , to find the third side . In the Triangle ADE , there is given the Side AE 70° 00 ' , the Side DE 38° 30 ' , and the Angle A 30° 28 ' , the Side AD is required . First by Case 1. Prop. 1. I find the Angle at D to be 130° 03 ' , and then proceed thus First take the Sum and Difference of the two Angles ; then also find the Difference of the two Sides given , and then work as followeth . Now say , As S. ½ X. VV. D and A , 49° 47 ' 30 " , To S. ½ Z. VV. D and 〈…〉 , 80 15 30. So is T. ½ X. cr s. AE and ED , 15 45 00 , To T. ½ cr . AD. 20° 00 ' 00 " : which doubled giveth the Side AD , 40 00 00 , as was required . PROP. VI. Case 6. Two Angles and a Side opposite to one of them given , to find the third Angle . In the Triangle ADE , there is given the Angle A 30° 28 ' , the Angle D 130° 03 ' , and his opposite Side AE 70° 00 ' , and 't is required to find the Angle at E. First by Prop. 2. Case 2. I find the Side DE , opposed to the Angle A ; to be 38° 30 ' , then proceed thus . Fi●st find the Sum and Difference of the Sides . Then find the Difference of the Angles . Now say , As S. ½ X. cr s. DE and AE 15° 45 ' , To S. ½ Z. cr s. EA and DE 54 15. So is T. ½ X. VV. D and A 49 47 30 " , To Tc. ½ V. at E 15° 47 ' 00 " . which doubled giveth the Angle at E 31 34 , as required . PROP. VII . Case 7. Two Sides and an Angle opposite to one of them given , to find the Included Angle . In the Triangle ADE , there is given the Side AE 70° 00 ' , the Side ED 38° 30 ' , and the Angle opposite thereunto at A 30° 28 ' , and the Angle E is required . First by Prop. 1. Case 1. I find the Angle D , opposite to AE , to be 130° 03 ' , then proceed thus . First find the Difference of the Angles , then find the Sum and Difference of the Sides . Now say , As S. ½ X. cr s. AE and ED 15° 45 ' , To S ½ Z. cr s. AE and ED 54 15. So is T. ½ X. of VV. D and A 49 47 30 " , To Tc. ½ V. at E 15° 47 ' . Which doubled is the Angle at E 31° 34 ' , as was required . PROP. VIII . Case 8. Two Angles and a Side opposite to one of them being known , to find the Interjacent Side . In the Triangle ADE , there is given the Angle E 31° 34 ' , the Angle D 130° 03 ' , and his opposite Side AE 70° 00 ' , Now the Side ED is required . First by Prop. 2. Case 2. I find AD opposed to E , to be 40° 00 ' , and then work thus . Take the Sum and Difference of the Angles , then also find the Difference of the two Sides : Now say , As S. ½ X. VV D and E 49° 14 ' 30 " , To S. ½ Z VV D and E 80 48 30. So is T ½ X cr s. AD and AE 15 00 00 " , To T. ½ cr s. ED , 19° 15 ' 00 " , which being doubled is the Side ED 38° 30 ' , as required . PROP. IX . Case 9. Two Sides and their Included Angle being known , to find the third Side . In the Triangle APZ , there is given the Side ZP 38° 30 ' , the Side PA 70° , and the Angle P , let be 31° 34 , and the Side AZ is required . The Resolution of this Case depends on the Catholike proposition of the Lord of Marchiston , by supposing the Oblique-Triangle to be divided ( by a supposed Perpendicular falling either within or without the Triangle ) into two Rectangulars . Now in the Triangle AZP , let fall the Perperpendicular ZR ; so is the Triangle AZP divided into two Rectangulars ARZ and ZRP . Now the Side AZ may be found at two Operations thus : say , As the Radius or S. of 90° 00 ' To Sc. of the included V , P. 31 34. So is T. of the lesser Side PZ . 38 30 , To T. of a fourth Arch. 34 08. If the contained Angle be less than 90° , take this fourth Arch from the greater Side ; but if it be greater than 90° , from its Complement unto 180° , the Remainder is the Residual Arch : Now again say , As Sc. of the fourth Arch. 34° 08 ' To Sc. Residual Arch. 35 52 So Sc. of the lesser Side PZ . 38 30 To Sc. AZ the Side required . 40 00 ☞ But note that many times the Perpendicular will fall without the Triangle , as it doth now within ; in such case the Sides of the Triangle must be continued , so will there be two Rectangulars , the one included within the other : as in the Triangle HIK , the Perpendicular let fall is KM , falling on the Side HE , and so the two Rectangulars found thereby will be IM K , and KMH , and so by the directions in the former proposition find out the Side IK , if required to be found . PROP. X. Case 10. Two Angles and their Interjacent Side known , to find the third Angle . In the Triangle AZP , there is given the Side ZP 38° 30 ' , the Angle P 31° 34 ' , and the Angle Z 130° 03 ' , and the Angle at A is required . First the Oblique-Triangle AZP , being reduced into two Rectangulars ARZ , and ZRP , by Case 9 aforegoing , I find the Angle RZP , to be 64° 19 ' , ( in the Triangle ZRP . ) which taken out of Angle AZP 130° 03 ' , leaves the Angle AZR 65° 44 ' : Now the Angle A is found by this Analogy or Proportion . As S. V. PZR , 64° 19 ' , To S. V. AZR 65 44 , So is Sc. V. at P 31 34 , To Sc. V. at A 30 28 : which was required to be found out and known . PROP. XI . Case 11. Three Sides given , to find an Angle . In the Triangle APZ , the Side AZ is 40° 00 ' , the Side ZP is 38° 30 ' , the Side AP is 70° 00 ' , and the Angle Z is required . To find which do thus . Add the three Sides : together , and from half their Sum , deduct the Side opposite , to the required Angle : and then proceed as you see in the Operation following . ½Sum is 65° 07 ' 30 " , the Sc. ½ V. at Z which doubled is 130° 03 ' 12 " ; the Angle at Z required . PROP. XII . Case 12. Three Angles given , to find a Side . In the Triangle AZP , the Angle A is 30° 28 ' 11 " , the Angle Z 130° 03 ' 12 " , the Angle P is 31° 34 ' 26 " , and the Side AZ , opposite to P , is required . This Case is likewise performed as the former Case or Proposition , the Angles being converted into Sides , and the Sides into Angles , by taking the Complement of the greatest Angle unto 180° : see the work . which being doubled , gives the Side AZ 40° 00 required to be found out and known ☞ But if the greater Side AP were required the Operation would produce the Complem 〈…〉 thereof unto a Semicircle or 180° ; therfo 〈…〉 substract it from 180° , it leaves the remaining required Side sought . Thus I have laid down all the Cases of Triangles , both Right-lined and Spherical ; either Right , or Oblique-angled ; I might hereunto have annexed many Varieties unto each Case , and some fundamental Axioms , which somewhat more would have Illustrated and Demonstrated those Cases , and Proportions ; but because of the smallness of this Treatise , which is intended more for Practice than Theory , I have for brevity sake omitted them , and refer you for those things to larger Authors , who have largely discoursed thereon to good purpose . CHAP. VI. Of ASTRONOMY . ASTRONOMY is an Art Mathematical , which measureth the distinct course of Times , Days , Years , &c. It sheweth the Distance , Magnitude , Natural Motions , Appearances and Passions , proper unto the Planets , and fixed Stars , for any time past , present and to come ; by this we are certified of the Distance of the starry Sky , and of each Planet , from the Center of the Earth , and the Magnitude of any fixed Star or Planet , in respect of the Earth's Magnitude . SECT . I. Of Astronomical Definitions . 1. ASphere or Globe is a solid Body , containing onely one Superficies , in whose middle there is a point ( called the Center , ) from which all right or streight lines drawn unto the Circumference or Superficies , are Equal . 2. The Poles of the World , are two fixed points in the Heavens Diametrically opposite the one to the other , the one called the Artick or North-Pole ; noted in the Scheme by P. The other is called the Antartick , or South-Pole ; as S. and is not to be seen of us , being in the lower Hemisphere . 3. The Axis of the World , is an imaginary line drawn from the North-Pole , through the Center of the Earth , unto the South-Pole , about which the Diurnal motion is performed , from the East to the West ; as the line PS . 4. The Meridians are great Circles , concurring and intersecting one another , in the Poles of the World , as PES , and Pc S. 5. The Equinoctial , or Equator , is a great Circle , 90 deg . distant from the Poles of the World , cutting the Meridians at Right-angles , and divideth the World into two Equal parts , called the Northern , and Southern Hemispheres , as E ♎ Q. in Scheme 42. 6. The Ecliptick is a great Circle , crossing the Equinoctial , in the two opposite points Aries and Libra , and maketh an Angle therewith ( called , its Obliquity ) of 23° 30 ' , represented by ♋ ♎ ♑ . This Circle is divided into 12 Sines , each containing 30° 00 ' : As Aries ♈ , Taurus ♉ , Gemini ♊ , Cancer ♋ , Leo ♌ , Virgo ♍ , ( which are called Northern Sines ) Libra ♎ ; Scorpio ♏ , Sagitarius ♐ ; Capricornus ♑ , Aquarius ♒ , and Pisces ♓ ; these are called Southern Sines . 7. The Zodiack is a Zone or Girdle , having 8 deg . of Latitude on either side the Ecliptick , in which space the Planets make their revolution . This Circle is a Circle which regulates the Years , Months , and Seasons , * and is distinguished in the Scheme by the 12 Sines . 8. The Colures are two Meridians , dividing the Ecliptick , and the Equinoctial , into four equal parts ; one of which passeth by the Equinoctial points Aries , and Libra , and is called the Equinoctial Colure , as P ♎ S. The other by the beginning of Cancer , and Capricorn , and is called the Solstitial Colure , as P ♋ , S ♑ . 9. The Poles of the Ecliptick are two points , 23° 30 ' distant from the Poles of the World , as I and K. 10. The Tropicks are two small Circles , Parallel unto the Equinoctial , and distant therefrom 23° 30 ' , limiting the Sun's greatest declination . The Northern Tropick passeth by the beginning of Cancer , and is therefore called the Tropick of Cancer , as ♋ a D. The Southern Tropick passeth by the beginning of Capricorn , and is therefore called the Tropick of Capricorn ; as B b ♑ . 11. The Polar Circles , are two small Circles parrallel to the Equinoctial , and distant therefrom 66° 30 ' ; and from the Poles of the World 23° . 30 ' . That which is adjacent unto the North Pole , is called the Artick Circle , as G d I. and the other the Antartick Circle , as Kd M. 12. The Zenith , and the Nadir , are two points , Diametrically opposite the one to the other : the Zenith is the Vertical point , or the point over our heads , as Z , The Nadir , is opposite thereto as the point N. 13. The Azimuths or Vertical Circles are great Circles of the Sphere , concurring and intersecting each other , in the Zenith , and Nadir , as Z f N. 14 The Horizon , is a great Circle , 90 deg . distant from the Zenith , and Nadir ; cutting all the Azimuths , at Rightangles , and dividing the World into two equal parts , the upper and visible Hemisphere , and the lower and invisible Hemisphere , represented by H ♎ R. 15. The Meridian of a Place , is that Meridian , which passeth by the Zenith , and Nadir , of the place as P Z S N. 16. The Alinicanthars , or Parallels of Altitude , are small Circles , parrallel unto the Horizon , ( imagined to ●pass through every degree and minute of the Meridian , between the Zenith , and Horizon , B a F. 17. Parallels of Latitude , or Declination , are small Circles parallel unto the Equinoctial ; they are called Parallels of Latitude , in respect to any place on the Earth , and Parallels of Declination , in respect of the Sun , or Stars , in the Heavens . 18. The Latitude of a place , is the height of the Pole above the , Horizon ; or the distance between the Zenith and the Equinoctial . 19. The Latitude of a Star , is the Arch of a Circle , contained betwixt the Center of a Star , and the Ecliptick line : this Circle making Right-angles , with the Ecliptick , is accounted either Northward or Southward ; according to the Scituation of the Star. 20. Longitude on Earth is measured by an Arch of the Equinoctial , contained between the Primary Meridian , ( or Meridian of that place where Longitude is assigned to begin ) and the Meridian of any other place , counted always Easterly . 21. The Longitude of a Star , is that part of the Ecliptick , which is contained between the Star's place in the Ecliptick , and the beginning of Aries , counting them according unto the succession of Sines . 22. The Altitude of the Sun or Stars , is the Arch of an Azimuth , contained betwixt the Center of the Sun , or Star , and the Horizon . 23. Ascension is the rising of any Star , or part of the Equinoctial , to any degree above the Horizon ; and Descension is the setting of it . 24. Right Ascension , is the number of Degrees and Minutes of the Equinoctial ; ( i. e. from the beginning of Aries ) which cometh unto the Meridian , with the Sun or Stars ; or with any portion of the Ecliptick . 25. Oblique-Ascension , is an Arch of the Equinoctial , between the beginning of Aries , and that part of the Equinoctial which riseth with the Center of a Star ; or with any portion of the Ecliptick in an Oblique Sphere : and Oblique Descention , is that part of the Equinoctial , tha● setteth therewith . 26. The Ascentional difference , is an Arch of the Equinoctial , being the difference betwixt the Right and Oblique-Ascension . 27. The Amplitude , of the Sun or Stars , is the distance of the rising or setting thereof , from the East or West point of the Horizon . 28. The Parallax , is the difference between the true and apparent place of the Sun or Star ; so the true place in respect of Altitude , is in the line ACE , or ADG , the Sun or Star being at C , or D , and the apparent place in the Line BCF , and BDH , so likewise the Angles of the Parallax are ACB , or ECF ; and ADB , or GDB : also in the said Scheme , ABK representeth a Quadrent ( of the Globe or Earth , ) on the Earth's Superficies : A the Center of the Earth , and B any point of the Earth's Surface . 29. The Refraction of a Star , is caused by the Atmosphere , or Vapourous thickness of the Air near the Earth's Superficies , whereby the Sun and Stars seem always to rise sooner , and and set later than really they do . SECT . II. Of Astronomical Propositions . PROP. I. The Distance of the Sun from the next Equinoctial point ( either Aries or Libra ) being known , to find his Declination . THE Analogy or Proportion . As Radius or S. 90° , To S. of the Sun's distance from the next Equinoctial point , So it S. of the Sun 's greatest Declination , To the S. of the Sun 's present Declination sought . PROP. II. The Sun's place given , to find his Right-Ascension . This is the Analogy or Proportion . As Radius or S. 90° , To T. of the Sun's Longitude from the next Equinoctial point , So is the Sc. of his greatest Declination , To T. of his Right-Ascension from the next Equinoctial point . PROP. III. To find the Sun's place or longitude from Aries , his Declination being given . This is the Analogy or Proportion . As S. of the Suns greatest Declination , To Radius or S. 90° 00 ' , So is S. of his present Declination , To S. of the Suns Place or Longitude from Aries * PROP. IV. By knowing the Suns Declination , to find his Right Ascension . This is the Analogy or Proportion . As Radius or S. 90° , To Tc. of the Suns greatest Declination , So is T. of the Declination given , To S. of the Suns right Ascension required † . PROP. V. By knowing the Latitude of a Place , and the Suns Declination , to find the Ascensional Difference . This is the Analogy or Proportion . As Radius or S. 90° , To Tc. of the Latitude given , So is T. of the Suns Declination given , To the S. of the Ascensional difference required . ☞ Note that if you reduce the degrees , &c. of the Ascensional difference into hours , it will shew you how much the Sun riseth , or setteth before , or after six a Clock , in that Latitude . PROP. VI. To find the Suns Oblique Ascension or Descension . First find the Ascensional Difference by the 5th Proposition , and his Right-ascension by the fourth : Now if the Suns Declination be Northerly , deduct the Ascentional Difference out of his Right Ascension , from the beginning of ♈ , ( for the six Northern Signs ♈ ♉ ♊ ♋ ♌ ♍ ) it leaves the Oblique Ascension ; and added unto the Right-ascension , giveth the Oblique-descension . But if the Suns Declination be Southerly , the Ascentional Difference , added to the Right-ascension , ( for the six Southern Signs ♎ ♏ ♐ ♑ ♒ ♓ ) giveth the Right-ascension , and substracted there from leaves the Oblique-descension . Plate 11 Page 105 PROP. VII . By knowing the Suns Declination , and the Latitude of a Place , to find the Suns Amplitude . This is the Analogy or Proportion . As Sc. of the Latitude , To the Radius or S. 90° . So is the S. of the Suns Declination , To the S. of the Amplitude from the East or West Points of the Horizon . PROP. VIII . By knowing the Suns Declination and Amplitude , from the North part of the Horizon , to find the Latitude . This is the Analogy or Proportion . As Sc. of the Amplitude from the North , To Radius or S. 90° 00 ' So is S. of his Declination given , To Sc. of the required Latitude . PROP. IX . By knowing the Latitude of a place , and the Sun's Declination , to find at what time the Sun will be on the true East or West Points . The Analogy or Proportion is . As T. of the given Latitude , To T. of the Sun's Declination propounded , So is Radius or S. 90° 00 ' , To , Sc. of the Hour from Noon . PROP. X. By knowing the Sun's Declination , and Latitude of a place , to find his Altitude at six a Clock . This is the Analogy or Proportion . As Radius or S. 90° 00 ' , To S. of the Sun's Declination , So is S. of the Latitude of the place , To S. of the Sun's Altitude at six a Clock . PROP. XI . By knowing the Latitude of a place , and the Sun's Declination , to find the Azimuth at six . This is the Analogy or Proportion . As Radius or S. 90° 00 ' , To the T. of the Sun's Declination , So is Sc. of the Latitude of the place , To the T. of the Azimuth sought . PROP. XII . By knowing the Latitude of a place , and the Sun's Declination , to find the Sun's Altitude when he i● on the true East or West points . This is the Analogy or Proportion . As S. of the Latitude , To the Radius or S. 90° 00 ' , So is the S. of the Declination , To the S. of the Sun's Altitude being due Ea●● or West . PROP. XIII . To find the Sun's Altitude at any time of the day . The Analogy or Proportion is . As Radius or S. 90° 00 ' , To Tc. of the Poles height , So is S. of the Sun's Distance , From the Hour of Six , To the T. of an Arch : which being substracted from the Sun's Distance from the Pole ; say , As Sc. of the Arch found , To Sc. of the remaining Arch of the Sun's Distance from the Pole , So is S. of the Poles height , To the S. of the Sun's Altitude at the Hour required . PROP. XIV . By knowing the Latitude of a Place , with the Sun's Declination , and Altitude , to find the Hour of the Day . To solve this Conclusion do thus : get the Sum of the Complements of the Latitude , Declination and Altitude given * , Then find the Difference betwixt their half Sum , and the Complement of the Altitude ; then say , As Radius or S. 90° 00 ' , To Sc. of the Sun's Altitude , So is Sc. of the Latitude of the Place , To a fourth Sine : then again say , As the fourth S. To the S. of ½ Z. of the Lat. Declin . and Alt. So is the S. of X. of the Altitude to the ½ Z , To a fifth S. unto which Sine , if you add the Radius or 90° 00 ' , half that Sum shall be the Sine of an Arch , whose double Complement is the Hour from the Meridian . PROP. XV. To find the Time of the Sun 's Rising or Setting , and consequently the Length of the Day or Night . To resolve this Conclusion , first by prop. the 5. find the Ascensional Difference , which reduced into Hours , and Minutes of Time , by allowing for every 15 Deg. one Hour , and for every Deg. less than 15° , 4 ' , of Time , and for every 15 Min. one Minute of Time. Secondly , If the Sun's Declination be Northerly , the Ascentional Difference added unto 6 Hours , gives the Time of Sun-setting , and substracted therefrom , leaves the Time of Sun-Rising : On the contrary , if the Sun's Declination be Southerly , the Ascentional Difference added unto 6 Hours , gives the Time of Sun-Rising , and deducted therefrom , the Time of Sun-setting . Thirdly , If you double the Time of Sun-Rising , it gives you the length of the Night ; and the Time of Sun-setting , the length of the Day . PROP. XVI . The Sun's Declination , Altitude and Azimuth known , to find the Hour of the Day . The Analogy or Proportion is . As the Sc. of the Sun's Declination , To the S. of the Azimuth , So is the Sc. of the Altitude , To the S. of the Hour from Noon : which converted into Time , will shew the Hour of the Day . PROP. XVII . By knowing the Sun's Declination , Altitude , and Hour from Noon , to find the Azimuth . The Analogy or Proportion is . As Sc. of the Sun's Altitude , To S. of the Hour from Noon , So is Sc. of the Sun's Declination , To the S. of the Azimuth , required . PROP. XVIII . By knowing the Latitude of a place , the Altitude of the Sun , and the Hour from Noon , to find the Angle of the Sun's Position . This is the Analogy or Proportion . As Sc. of the Sun's Altitude , To S. of the Hour from Noon , So is Sc. of the Latitude , To S. of the V. of the Sun's Position , at the time of the Question . PROP. XIX . By knowing the Sun's Altitude , Declination , and Azimuth ; to find the Latitude . The Analogy or Proportion is . As S. of the Sun's Azimuth , To S. of his Distance from the North-pole , So is S. of V of the Sun's Position , To Sc. of the Latitude required . PROP. XX. To find the length of the Crepusculum , or Twilight The Crepusculum or Twilight , is nothing else but the Refraction of the Sun's Beams in the Density of the Air. Which the Learned Pet. Nonnius found the length of the Crepusculum ( by his many strict observations * ) to continue from the time of the S 〈…〉 passing below the Horizon of a place , untill the Sun had run below the said Horizon 18° 00 ' , and then followed the shutting in of the Twilight , and untill the Sun was departed so low the Twilight continued . — To find which observe this Analogy or Proportion . As Radius or S. 90° , To Sc. of the Sun's Declination , So is Sc. of the Poles-height , To a fourth Sine : which keep . Then out of the Sun's Distance from the South-Pole , subduct the Complement of the Pole ; and of that remains and the degrees 62 , being added to it , their Sum and Difference found , say again . As the fourth Sine found , To S. ½ Z of the remainder and 62° 00 ' , So is S. ½ X. of the remainder and 62° 00 ' , To a Number , which being multiplyed by the Radius is equal unto the Quadrat of the Sine of the ½ Angle of the Sun's Distance at the Ending of the Twilight , from Noon next ensuing . Then from the Sun of the whole Angle converted into Hours , substract the Hour of the Sun 's setting * , it gives you the length of the Crepusculum , or Twilight . But the Sun being in the Winter Tropick , makes the Twilight longest of any Twilight , the whole Winter half year : Now in a certain Parallel , betwixt that Tropick , and the Equinoctial is the shortest Crepusculum : the Declination of which Parallel , is thus found . As the Tc. of the Pole , To the S of the Pole , So is the T. of 99° 00 ' , To S. of the Declination of the Parallel , in which the Sun maketh the shortest Crepusculum of the Year . But before the Crepusculum come to be shortest , there is another Parallel , in which the Crepusculum is equal to that of the Equinoctial : the Declination of which is found thus . As the Radius or S. 90° 00 ' , To S. of the Poles Elevation or Altitude , So i● S. of 18° 00 ' , To S. of the Declination of the Parallel , in which the Sun maketh the Crepusculum equal to that in the Equinoctial . PROP. XXI . To find the Quantity of the Angles , which the Circles of the 12 Houses make with the Meridian . This is the Analogy or Proportion . As the Radius or S. 90° , To T. of 60° : for the 11th , 9th , 5th and 3d House , Or to the T. 30° for the 12th , 8th , 6th , and 2d House , So is the Sc. of the Pole , To the Tc. of any House with the Meridian . PROP. XXII . To find the Right Ascension of the Point in the Equinoctial : and also the Point in the Ecliptick ; called Medium Coeli or Cor Coeli . First , To find the Right Ascension of the Point of the Equinoctial ; called Medium Coeli , vel Cor Coeli , find out the Sun 's Right Ascension , by prop. 2. Then reduce the whole Time from Noon last past into degrees , which add unto the right Ascension of the Sun , so shall their Agragat , be the right Ascension of the point , which in the Equinoctial , is called Medium Coeli , vel Cor Caeli , required to be found . Secondly , By the 2 propositions aforegoing , you may find the right Ascension of the point in the Ecliptick Culminant in the Meridian , called Medium Coeli vel Cor Coeli , which is the Cuspis of the tenth House : and his Declination by prop. the first . PROP. XXIII . To find the Angle of the Ecliptick with the Meridian . The Analogy or Proportion is . As the Radius or S. 90° , To S. of the Sun 's Greatest Declination , So is Sc. of the Sun 's right Ascension , from the next Equinoctial point , To Sc. of the V. of the Ecliptick , with the Meridian . PROP. XXIV . To find the Angle of the Ecliptick with the Horizon . The Analogy or Proportion is . As Radius or S. 90° , To Sc. of the Altitude of Cor Coeli , So is S. of the V. Ecliptick with the Meridian , To Sc. of the V. of the Ecliptick and Horizon sought . PROP. XXV . To find the Amplitude Ortive of the Ascendent , or Horoscopus . This is the Analogy or Proportion . As Radius or S. 90° , To S. of Altitude of Med. Coeli , So is T. of V. Ecliptick with the Meridian , To Tc. of the Amplitude Ortive of the Ascendent , or the distance of the Azimuth from the Meridian . PROP. XXVI . To find the Ascendent degree of the Ecliptick , or the Cuspis of the first House . The Amplitude Ortive of the Ascendent , is equal to the Distance of the Azimuth of 90° , from the Meridian , wherefore the Cuspis of the first House , or Ascendent Degree of the Ecliptick , may be found thus . As Radius or S. 90° , To Sc. of the V. Ecliptick with the Meridian , So is Tc. of the Altitude of Med. Coeli , To T. of the Distance of Med. Coeli , from the Ascendent Degrees . PROP. XXVII . To find the Distance of the Cuspis of any House , from Med. Coeli . This is the Analogy or Proportion . As Sc. of the remaining part of V. of the Ecliptick with the Meridian , ( found by prop. 28. ) To Sc of the former part of the V , So is T. of the Altitude of Med. Coeli , To T. of the Distance of the Cuspis of that House sought , from Med. Coeli . PROP. XXVIII . To find the parts of the Angle of the Ecliptick with the Meridian , cut with an Arch perpendicular to the Circle of any of the Houses . The Analogy or Proportion is : As Radius or S. 90° , To Sc. Altitude of Med. Coeli , So is T. of the Circle of any House with the Meridian , To Tc. of that part of that Angle which is next the Meridian : Then substract that part found , out of the whole Angle , for the remaining or latter part PROP. XXIX . To find the Pole's Altitude , above any of the Circles of the Houses . The Analogy or Proportion is . As the Radius or S 90° , To S. of V. of the Circle of the House with the Meridian : ( found by the 21 prop. ) So is the S of the Poles Elevation , above the Horizon of the Place , To S. of the Altitude of the Pole , above the Circle of Position . PROP. XXX . By knowing the Latitude and Longitude of any fixed Star , to find his Right Ascension and Declination . The Analogy or Proportion is . 1. As Radius or S. 90° , To S. of the Stars Longitude from the next Equinoctial point , So is Tc. of the Stars Latitude , To T. of a fourth Arch. Page 216 Plate III AGAIN , say . 2. As S. of the fourth Arch , To S. of the fifth Arch , So is T. of the Stars Longitude , To T. of the Stars Right-ascension from the next Equinoctial point . 3. As Sc of the fourth Arch , To Sc. of the fifth Arch , So is S of the Stars Latitude , To S. of the Stars Declination . I might also shew how by having the Latitude and Longitude of any two fixed Stars , to find their Distance : but because 't is the very same with finding the Distance of any two Places on Earth , I refer you to the Directions of Prop 1 , 2 and 3. of Chap. 7 , ensuing , where you will see the plain Demonstration thereof . PROP. XXXI . By knowing the Pole's Altitude , to find when any fixed Star shall be due East or West . This is the Analogy or Proportion . As Radius or S. 90° , To T. of the Stars Declination , So is Tc. of the Pole , To Sc. of the Stars Horary Distance from the Meridian . PROP. XXXII . By knowing the Poles Altitude , to find the Elevation of any fixed Star above the Horizon , being due East or West . This is the Analogy or Proportion . As S. of the Poles Altitude , To Radius or S. 90° , So is S. of the Stars Declination , To S. of the Stars Elevation , above the Horizon , at due East or West . PROP. XXXIII . To find out the Horizontal Parallax of the Moon . The Analogy or Proportion . As the Moons Distance from the Center of the Earth , To the Earth's Semidiameter , So is Radius or S. 90° , So S. of the Moon 's Horizontal Parallax in that Distance . PROP. XXXIV . The Horizontal Parallax of the Moon being known , to find her Parallax in any apparent Latitude . This is the Analogy or Proportion . As Radius or S. 90° , To S. of the Moon 's Altitude , So is S. of the Moon 's Horizontal Parallax , To S. of the Parallax in that Altitude . PROP. XXXV . By knowing the Moon 's Place in the Ecliptick , ( having little or no Latitude ) and her Parallax of Altitude , to find the Parallaxes of her Longitude and Latitude . First , If the Moon be in the 90° of the Ecliptick , she hath then no Parallax of Longitude , and the Parallax of the Latitude , is the very Parallax in that Altitude . Secondly , But if the Moon be not in the 90th . Degree of the Ecliptick , to find the Parallaxes of the Latitude and Longitude , the Analogy or Proportion is , 1. As Radius or S. 90° , To T. of the V. of the Ecliptick and Horizon , So is Sc. of the Moon 's Distance from the Ascendent , or Descendent deg . of the Ecliptick , To Tc. of the Ecliptick's V , with the Azimuth of the Moon . AGAIN say , 2. As the Radius or S. 90° , To S. of that V. found , So is the Parallax of the Moon 's Altitude , To the Parallax of her Latitude sought . LASTLY say , 3. As the Radius or S. 90° , 00 ' To Sc. of the former V. found , So is the Parallax of the Moon 's Altitude , To the Parallax of his Longitude sought , which being added to the true Motion of the Moon , if she be on the East part of the 90° of the Ecliptick . Or from it to be deducted if she be on the West part of the 90° of the Ecliptick . PROP. XXXVI . How by knowing the Refraction of a Star , to find his true Altitude . For the speedy performance of which I have annexed this Table of Refractions of the Stars observed by Tycho Brabe a Nobleman of Denmark , and a most famous Astronomer . A Table of the Refraction of the Stars observed by Tycho Brabe . Altitude . Refraction . o° 30 ' 00 " 1 21 30 2 15 30 3 12 30 4 11 00 5 10 00 6 9 00 7 8 15 8 6 45 9 6 00 10 5 30 11 5 00 12 4 30 13 4 00 14 3 30 15 3 00 16 2 30 17 2 00 18 1 15 19 0 30 20 0 00 The USE of which Table is thus . EXAMPLE . Suppose the Altitude of a Star were found by Observation to be 13° ; the correspondent Refraction is 4 ' 00 " , which substracted from 13° leaves 12° , 56 ' , which is the true Altitude CHAP. VII . Of GEOGRAPHY . GEOGRAPHY is an art Mathematical , which sheweth how the Situations of Kingdoms , Provinces , Cities , Towns , Villages , Forts , Castles , Mountains , Woods , Havens , Rivers , Creeks , &c. being on the Surface of the Terrestrial Globe , may be described , and designed , in commensuration Analogical to Nature , and Verity : and most aptly to our view may be represented . Ptolomy saith of Geography , 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 . That it is a description of all the known Earth , imitated by writing and delineation : with all other things belonging thereunto . Of all which I shall say somewhat , as to its Situation , Commodity , Customs , &c. concerning which Ovid saith , Met. lib. 2. Terra , viros , Urbesque gerit , frugesque , ferasque , Fluminaque ; haec super est Caeli fulgentis imago . In English Thus. The Earth , Men , Towers , Fruits , Beasts , and Rivers bears , And over these are place'd the Heavenly Spheres . SECT . I. Of GEOGRAPHICAL Definitions . 1. THE Globe of the Earth is a Spherical Body composed of Earth , and Water , and is divided into Continents , Islands and Seas . 2. A Continent is a great Quantity of Land not separated , interlaced or divided by the Sea , wherein are Kingdoms , Principalities and Nations , as EUROPE , ASIA and AFRICA , are one Continent : and AMERICA is another . 3. An Island is such a part of the Earth that is environed round with Water on every Side , as the Isle of Great Britain , Java , Wight , &c. 4. A Peninsula is such a Tract of Land which being almost cut off from the Main Land , and encompassed round with Water , yet nevertheless is joyned unto the firm Land , by some little Isthmus , as Peloponesus , Peruviana , Taurica , Cymtryca and Morea in the Levant . 5. An Isthmus is a little narrow Neck of Land which joyneth the Peninsula unto the Continent . 6. A Promontory is some high Mountain , which shooteth it self into the Sea , the utmost end of which is called a Cape : as Cape-boon , Esperance , Cape d'Verde , and Cape d'Coquibocao . 7. The Ocean is a general Collection of Waters , which environeth the World on every side , and produceth Seas , Straits , Bays , Lakes , and Rivers : Of which and other Waters Ovid thus speaks in his Metamorphosis . Tum Freta diffudīt , rapidisque tumescere ventis Jussit , & ambitae circundare littora terrae . He spread the Seas , which then he did command To swell with Winds , and compass round the Land. 8. The Sea is part of the Ocean , to which we cannot come but through some Strait , as the Mediterranean , or Baltick Sea. 9. A Strait is a part of the Ocean restrained within narrow bounds , yet openeth a way to the Sea , as the Straits of Gibralter , Helespont , &c 10. A Creek is a crooked Shoar thrusting , as . it were , two Armes forth to hold the Sea ; as the Adriatick , Persian , and Corinthian Creeks : from whence are produced Rivers , Brooks and Fountains : which are engendred of Congealed Air in the Earths Concavity , and seconded by Sea-water creeping through the hidden Cranies of the Earth . 11. A Bay is a great Inlet of Land , as the Bay of Mexico , and Biscay . 12. A Gulph is a greater Inlet of Land and deeper than a Bay , as the Gulph of Venice , and Florida . 13. A Climate is a certain space of Earth and Sea , included within the space of two Parallels ; and there have been anciently accounted these seven : viz. 1. Dia Meros , 2. Dia Syenes , 3. Dia Alexandria , 4. Dia Rhodes , 5. Dia Rhomes , 6. Dia Boristhenes , and 7. Dia Ripheos . 14. A Zone is a certain space of Earth contained betwixt certain Circles of the Sphere , of which there are five : viz. The Torrid or Burning Zone , two Temperate , and two Frigid or Frozen Zones . The Torrid Zone is that which lieth on each side the Equinoctial , whose bounds are the two Tropicks of ♋ and ♑ . The two Temperate Zones are those which lieth betwixt the two Tropicks of ♋ and ♑ , and the Palar Circles . The two Frigid Zones lieth between the Artick and Antartick Circles , and their respective Poles : Of which Ovid thus speaks . Metam . 1. Utque duae dextrâ Coelum , totidemque sinistrâ Parte secant Zonae , quinta est ardientior illis : Sic onus inclusum munero distinxit eodem Cura Dei , totidemque plaga tellure premuntur : Quarum quae media est , non est habitalis aestu ; Nix tegit alta duas : totidem inter utramque locavit Temperiemque dedit mistâ cum Frigore Flammâ . SECT . II. Geographical Descriptions of the Earth . THE whole Earth is divided into four parts . VIZ. EUROPE , ASIA , AFRICA and AMERICA . EUROPE , the first part of the World , is divided from ASIA by the Mediterranean Sea ; bounded on the West with the Western Ocean ; East with the River Tanais . It is lesser than ASIA , or AFRICA , yet doth excell all the other parts , in Worthiness , Fame , Power , multitudes of well builded Cities , strong Fortifications , full of a Wity and Learned People , Courageous Wariours , and the knowledge of God , better than all the Riches of the World. It once had the dominion of ASIA and AFRICA , and in it were fourteen Mother Tongues , and doth contain these Provinces : Viz. Italy , Spain , Alps , France , Britain , Belgia , Germany , Denmark , Sweden , Russia , Poland , Hungary , S●lavonia , Dacia , and Greece , with its several Islands , which shall be mentioned in their due places . Italy * the Mother of Latine learning , is bounded East with the Adriatick and Tuscan Seas , West with France , North with Germany , and South with the River Varus , and the Alps. It hath had seven kinds of Governments : First Kings , Dictators , Consuls , Decimivires , Tribunes , Emperours , and lastly Popes . It far excelleth all the other Lands in EUROPE in fruitfulness and pleasantness . The Inhabitants are witty and frugal , yet hot and lascivious , and very jealous of their Wives ; they are of the Popish Religion , and its chief Commodities are Rice , Silk , Velvets , Sattins , Taffeties , Grogerams , Arras , Gold and Silver , Threed , Venetian Glasses , &c. Italy at this day contains the Kingdom of Naples , Sicily , Sardina , the Lands of the Pope , now Innocent the XI . the Dukedom of Tuscany , Urbin , the Republick of Venice , Genoa , and Luca. The Estates of Lumbardy , being the Dukedom of Millain , Mantua , Modena , Parma , Mountferrat , and the Principality of Piemont , of all which we shall treat in their order . The Kingdom of Naples is environed with the Adriatick , Ionian , and Tuscan Seas ; except where it is joyned to the Lands of the Church , from which 't is separated by a line drawn from the mouth of the River Tronto , falling into the Adriatick , and to the spring-head of Axofenus , taking in all the East of Italy , 1468 miles . It is very fertile , abounding with all things necessary for the life of Man , delight , and Physick : from hence come the Neopolitan Horses . It hath had 13 Princes , 24 Dukes , 25 Marquesses , 19 Earls , and 900 Baronets , and 26 Kings of several Countries of the Norman and Spanish race , whom 't is now under : here the Disease called the French-Pox derived its Original : the Arms are Azure Seme of Flower-d'-lices , or a File of three Lables Gules ; its revenues are 2500000 Crowns , 20000 of which belongs to the Pope , and the rest are imployed to maintain the Garisons against the Turks ; so that scarcely 60000 Crowns falls to the King of Spain s share ; it hath 20 Archbishops , and 124 Bishops Sees . Sicily is situated under the fourth Climate , it shoots forth into the Seas with three Promontories ; the Inhabitants are Eloquent , Ingenious , and Pleasant , but very unconstant , and Talkative ; the first Inventors of Oratory . It 's a fruitfull Soil , it yields Wine , Grain , Oyl , Hony , Gold , and Silver , Agats , Emeralds , Allom , Salt , Sugar , and Silks . Here is the Hill Aetna , supposed to be Hell , and by the Papist Purgatory , because of its vomiting Smoak and Fire : it hath many Cities , Rivers , Lakes , whose descriptions must here be omitted ; it hath had eight Kings ; the first were of the Arragon Family , and began Rule Anno 1281. But it 's now united to the Crown of Spain ; its Revenues are 800000 , or a 1000000 of Ducats , which is disburst on the Account of the Vice-Roy , and Defence of the Countrey ; the Arms are four Pallets Gules , Sable for Aragon , between two Flanches Argent , charged with as many Eagles Sable beaked Gules . It hath had seven Princes , four Dukes , thirteen Marquesses , fourteen Earls , one Viscount and forty-eight Barons , they are of the Romish Religion , and have three Archbishops , and nine Bishops . The Kingdom and Isle of Sardina , lieth West from Sicily and Cap Bara , whose length is 180 , and breadth 90 Miles ; the People are low of Stature , and of a swarthy Complection , rude , slothfull , and rebellious , their diet mean , yet rich in their Apparel , they are of the Romish Religion ; but have an ignorant and illiterate Clergy . It belongs to the King of Spain , and governed by a Vice-Roy , under whom are two Deputy Spaniards : but other inferiour Officers may be Natives . It hath neither Wolf , nor Serpent , nor venemous Beast , but the Fox only , and a little Spider , which cannot endure the light of the Sun ; they are destitute of Water , and are therefore forced to keep the Rain that falls in Summer for their Use in Winter , the Air is unhealthfull and Pestilential ; the Soyl Fertile , but ill manured ; it hath plenty of Cattel , their Horses will last very long , the Natives ride on their Bullocks as we on our Horses , here is also a Beast called Mufrones , resembling a Stagg , whose Hide is used as Armour , and an Herb which eaten produces Death with excessive 〈…〉 aughter , it yields to the King of Spain but a small Revenue . The Arms are Or , a Cross Gules betwixt four Sarazens Heads , Sabled Curled Argent , it hath several Isles belonging thereunto , it hath three Archbishops , and fifteen Bishops . The Lands of the Church or the Pope's Dominion in Italy , lieth West of Naples , extended North and South from the Adriatick to the Tuscan Seas , bounded on the North with the River Trontus ; on the South-east with Axofenes : hy the Rivers Poe and Frore , separated from the Republique of Venice , on the South-west by the River Piscio ; by which 't is divided by the Modern Tuscany , it is in the middle of Italy , its breadth is 115 , and length 300 Miles , it 's most exceeding fruitfull , very Populous , there have been 15 Exarches of Ravena in Romandiola , 17 Dukes and Marquesses of Ferata ; the Revenue thereof to the Pope is 250000 Crowns , there hath also been 6 Dukes of Urbin , its Revenue are 100000 Crowns , but the most splendid Glory of Italy is the City of Rome , sometimes the Empress of the World , and was the Seat of the past Popes , and the now present Pope Innocent the XI . the inferior spiritual Governours , are these , Viz. Cardinals , Friers of the Order of St. Basil , Austin , Jerome , Carmelites , Crouchedfriers , Dominicans , Benedictines , Franciscans , Jesuits and Oratorians ; and of Nuns , the Order of St. Clear and Bridget , which to name wholly doth deserve a particular Treatise , here are 44 Archbishops , and 57 Bishops . The Republique of Venice , lieth Northward of the Popes Dominions , from Romandiola to the Alps , limited on the South with the Territories of Ferrata , and Romandiola ; on the West with the Dukedom of Millain ; on the North with the Alps ; and on the East with the Adriatiok Sea , and the River Arsia . It is a very fruitfull Countrey , well peopled , their Government Aristocratical , and popular , their Religion Popish , they baptize the Sea yearly ; they have had a hundred Dukes , they have two Principal Orders of Knighthood of St. Mark the Patron of the famous City of which the Poet speaks . Viderat Adriacis venetiam Neptunus inundis Stare Urbem , & toto ponere jurae mari : Nunc mihi Tarpeias , quantumvis Jupiter , arces Objice , & illa tui moenia martis ait . Sic Pelago Tibrim praefers , urbem aspice utramque Illam bomines dices , hanc posuisse Deos. Instituted 1330 , and renewed 1562 , they are to be all of Noble Blood : their Motto is Pax tibi Marce. The other is of the Glorious Virgin , iristituted 1222 , their Duty is to be a refuge to Widows and Orphans , and to procure the peace of Italy ; their Habit a White Surcoat over a Russet Cloak , representing Religion , as well as Belliarcity , there are two Patriarchs , and sixteen Bishops . The Dukedom of Florence , being the Seat of the Great Duke of Tuscany , is bounded on the East by the River Pisca ; on the West by the River Macra , and the Fort Sarzana ; on the North by the Appenuine Hills ; and on the South with the Tuscan Seas . It s length is 261 Miles , and breadth not known ; the Order of Knighthood is that of St. Stephen , instituted 1567 , they are to be Nobly born , and in lawfull Wedlock , without Insamy : their Robe is of White Chamblet , with a Red-Cross on the left Side of their Midway Garment , their Number I cannot certainly know , the Grand Duke is their Sovereign ; the Revenue of this Countrey is great , their Duke is also a Merchant , and receiveth Excise of all Commodities , the Arms is Or , five Tortecax Gules , two , two , and one , one , on chief Azure charged with three Flower-de-Luces , of the first . They are of the Popish Religion , and they have three Archbishops , and twenty six Bishops . The Estate of Luca , lieth betwixt the Estate of the Grand Duke , and the Republique of Genoa : The Government is Aristocratical , and Democracy , their Principal Magistrate is called , Gon Fatinere , and is changed every second Month : being assisted by a certain Number of Citizens , which are changed every six Months , during which time they lie together in the Common Hall ; their Protector is Elective from some Neighbouring Prince : they are a very generous People , good Merchants , they sell rich Cloths of Gold and Silver ; the Revenues yearly are 80000 Crowns , it can raise for War 15000 Foot , and 3000 Horse , they are of the Popish Religion , and have two Bishops , and acknowledge the Bishop of Florence for their Metropolitan . The Republique of Genoa , lieth West of Tuscany , from whence 't is divided by the River Macra , it was formerly a large State but have now only Liguria , and the Isle of Corsica ; the Inhabitants are good Warriours , Merchants , and subtle Userers ; here the Women have the most ●iberty of any in all Italy , so that they may convense with whom they will , either publiquely , or privately ; from hence ariseth a Proverb , That Genoa is a Country of Mountains without Woods ; Seas without Fish ; Men without Faith ; and Women without shame . They have a Duke , with Eight Assistants , all subject to the General Council of 400 Men ; which hold but two years , they are of the Popish Religion , and have one Archbishop , and fourteen Bishops . The Estates of Lumbardy is bounded on the East with Romandiola , and Ferrata ; on the West and North with the Alps ; and on the South with the Apenuine hills : Now as Italy is the Garden of EUROPE , so is Lumbardy of Italy , for its exceeding Fruitfulness . The Dukedom of Millain hath on the East the State of Mantua , and Parma ; on the West Piemont , and part of Switzerland ; on the North Marca Trevigana ; and on the South the Apenuine , parting it from Liguria : it was once the chief Dukedom in Christendom , and is now in the Spanish Territories ; its Revenues are 8000 Ducat's , their Arms are Argent , a Serpent Azure Crown'd , Or , in his George an Infant Gules , their Religion is Popish , they have one Archbishop , and six Bishops . The Dukedom of Mantua , is bounded West with Millain ; East with Romandiola ; North with Marca Triugiana ; and South with the Dukedom of Parma . The Countrey yields good store of Corn , Fruit and Wine , the Inhabitants are rustick , foolish in their Apparel , it is a free state and hath had many Dukes , the Order of Knighthood is that of the Blood of Christ : instituted 1608 , it consisteth of twenty Knights , the Mantuan Duke is their Sovereign ; the Coller hath threads of Gold , layed on with Fire , with this Motto Domine probasti , to the Collar are pendent two Angels , supporting three drops of Blood , and circumscribed with this Motto , Nihil ista triste receptô . It s Revenue is 500000 Ducats ; Its Arms are Argent , a Cross Patere Gules , between four Eagles , Sable membred of the second , under an Eschucheon in Fife , charged quarterly with Gules , a Lion Or , and Or , three barrs Sable : Their Religion is Popish , here is one Archbishop , with four Bishops . I shall pass by the Dukedoms of Modena , Parma and Mountferrat , they being but small Estates of Italy , having but four Bishops : they are of the Popish Religion , the Arms of Modena and Parma are as Ferrata ; and the Arms of Mountferrat , a chief Argent . And here we should describe Piemont the last part of Italy , but being but part in Italy , and the Alps belonging to the Duke of Savoy , I shall defer it to the Alpian Descriptions . Now Italy hath these most famous Cities , viz. Genoa , Milain , Venice , Florence , Rome , Bologne and Naples , the Rivers most famous are Arnus , Po and Tiber , and so much for Italy . The Alps begin about the Ligustick Seas , and crosseth all along the Borders of France and Germany , and extend as far as the Gulph of Cornero ; It hath these Provinces , viz. the Dukedom of Savoy ( to which Piemont belongs ) Geneva , Wallisland , Switzerland , and the Countrey of the Grizons , of all which I shall give a short and plain Description . Piemont is part of the Alps , situated at the Foot of the Mountains , bounded North with the Switzes ; East with Millain and Mountferrat ; West with Savoy ; and on the South it runneth into a Narrow Vally to the Mediterranean , having Mountferrat on one side , and Province and part of the Alps on the other : it 's very fruitfull compar'd with Savoy , but yet inferiour to any part of Italy : The Arms are Gules , a Cross Argent , charged with a Lebel of three points Azure . Savoy is bounded East with Wallisland and Piemont ; West and South with Daulphin , and La-Bress ; and North with Switzerland , and the Lake of Geneva : this is a Mountainous Countrey , very healthfull , but not very fruitfull . The Inhabitants are dull and slothfull , it hath had thirty Dukes and Earls , it is a place of Natural strength ; its Revenues is yearly 1000000 of Crowns . The order of Knighthood is that of Anunciado , instituted 1480 , their Coller hath 50 links , ( to shew the Mystery of the Virgin ) appendent to it is her Effigies , and instead of a Motto these Letters F. E. R. T. i. e. Fortitudo ejus Rhodum tenuit , which is engraven on each link of the Chain , interwoven like a True-lovers-knot . The Number fourteen , besides the Duke Soveraign of the Order , their Arms are G. a Cross A. Geneva was a City of the Dukedom of Savoy , but now a free State : having both cast off the Duke and his Holiness the Pope , with all the Clergy . They are now Calvanist Protestants : their Government Presbyterial ; their Language the worst of French , they are an industrious People , and good Merchants . Wallisland reacheth from the Mount De Burken , to the Town of St. Maurice , where the Hills do shut up the Valleys , so that a Bridge is lain from one Hill to 'tother , under which passeth the River Rosue , which Bridge is defended by a Castle and two strong Gates ; on the other side 't is surrounded with steep and horrid Mountains ; covered with a Crust of Ice not passable by Armies ; the Inhabitants are courteous to Strangers , but unnatural to each other : they are of the Romish Religion , and subject to the Bishop of Sion ; the Deputies of the seven Resorts , have voices in his Election , and joyn with him in Diets , for chusing Magistrates ; desiding Grievancies , and determining matters of State. The Valleys of this Countrey is very fruitfull in Saffron , Corn , Wine and Delicate Fruits , they have a Fountain of Salt , many hot Baths , and Spaw-Waters , they have plenty of Cattle , with a wild Stag footed as a Goat , and horned as a fallow Deer : who in Summer is blind with heat . Switzerland is bounded East with the Grisons ; West with Mount-Jove and the Lake of Geneva ; North with Suevia ; and South with Wallisland , and part of the Alps ; this Land is a very Mountainous Countrey , but yet hath some rich Meddows . It is 240 in length and 180 Miles in breadth , the Inhabitants are rich , but rugged like their Soyl : good Souldiers : they are some Papists and some Protestants , others Zwinglians , yet have they toleration under a Popular Government . The Countrey of the Grisons is bounded East with Tyrol , North with Switzerland , South with Suevia , Switzerland , and Lumbardy ; it is a very Mountainous and Barren Land , their Religion Protestant , their Government Popular ; there are in this Alpin Provinces two Archbishops , and thirteen Bishops . It s chief Cities are Turin , Geneva , Basil and Zurich , in all of which are Universities . France is bounded East with Germany ; and South and East with the Mediterranean Seas and Alps ; North with the British Seas ; It hath been esteemed the worthiest Kingdom in Christendom , it yields plenty of Grains and Wines , wherewith it supporteth other Lands , it consisteth of many great Dukedoms and Provinces . It hath great and mighty Cities , the People are Ingenious and good Warriers , the Government is Monarchial , their Religion Popish , but intermixt with Protestants , which of late hath endured grievous Persecutions . Their Orders of Knighthood are that of St. Michael instituted 1409 , consisting of 300 Persons , their Habit is a long Cloak of White Damask down to the Ground , with a Border interwoven with Cockleshels of Gold , interlaced and furred with Ermins , with a Hood of Crimson-velvet , and a long Tippit about their Necks , and a Coller woven with Cockleshells , with this Motto , Immensitremor Oceani , to it hangs appendent the Effigies of St. Michael conquering the Dragon . Their Seat is St. Michael's Mount in Normandy . 2dly the Order of the Holy Ghost instituted 1579 , so that whosoever was admitted to the Order of St. Michael , must and was first dignified with this ; proving their Nobility by three Descents ; and be bound by Oath to maintain the Romish Religion ; and persecute all Dissenters thereunto . Their Robe is a Black Velvet Mantle , portrayed with Lillies and Flames of Gold : the Coller of Flower-de-Luces , and Flowers of Gold , with a Dove and Cross appendent to it . The Arms of France , are Azure three Flower-de-Luces , Or ; It hath seventeen Archbishops , 107 Bishops , 132000 Parishes , and hath these Magnisicent Cities , viz. Amiens , Rouen , Paris , Troys , Nants , Orleans , Diion , Lyons , Burdeaux , Toulose , Marsailles , Grenoble and Anverse ; the Rivers of most Note are the Loyre , Garone , Rhone and the Seyne . The Pirenean-hills lyeth betwixt France and Spain , and are two Potent Kingdoms , esteemed 240 Miles long , the People are barbarous and scarce of no Religion at all . Spain is separated from France by the Pirenean-hills ; on all other sides environed with the Sea ; this Land yieldeth all sorts of Wine , Oyl , Sugars , Grains , Metals , as Gold and Silver , and it is Fertile ; the Inhabitants are Ambitious , Proud , Superstitious , Hypocrites and Lascivious , yet good Souldiers ; by enduring Hunger , Thirst , Labour , &c. It containeth divers Kingdoms . 1. Goths . 2. Navars , it hath had 41 Kings . Their Arms are Gules , a Carbuncle Nowed Or , their Order of Knighthood was of the Lilly , their Blazon a Pot of Lillys , with the Effigies of the Virgin on it , their Duty is to defend the Faith , and daily to repeat a certain Number of Ave-Maries . 3dly Biscay and Empascon , hath had nineteen Lords , their Arms Argent , two Wolves Sable , each in his Mouth a Lamb of the second . 4ly Leon and Oviedo hath had thirty Kings , the Arms Argent , a Lion Passant crowned Or. 5ly . Galicia hath had ten Kings , the Arms Azure Sema of Cressels siched a Chalice crowned Or. 6ly . Corduva hath had twenty Kings , the Arms Or , a Lion Gules armed and crowned of the first , a Border Azure charged with eight Towers Argent . 7ly . Granado hath had twenty Kings , the Arms Or , a Pomgranet slipped Vert. 8ly . Marcia . 9ly . Tolledo hath had eleven Morish Kings . Ioly . Castile hath had twenty Kings , the Order of Mercia is his chief Order , here the Armes is a Cross Argent , and four Beads , Gules in a Field Or , their Habit white , the Rule of their Order that of St. Augustine , they are to redeem Captives from Turky . 11ly . Portugal ( the Native soyl of the most serene Catharine Queen Dowager ) hath had 21 Kings , the Orders of Knight here is first Avis , wearing a Green Cross , 2dly of Christ instituted 1321 , their Robe is a black Cassock under a white Surcoat with a Red Cross hanging in the midst a white Line , and their Duty is to expell Mores out of Boetica , the Arms are Argent , on five Escucheons Azure , as many Befants in saltire of the first pointed , Sable within a powder Gules , charged with seven Towers Or. 12ly . Majorica hath had four Kings . 13ly . Arragon hath had twenty Kings , their Order of Knighthood is of Mintesa , their Robe a red Cross on their Breast , the Arms Or , four Pallets Gules , all which Kingdoms are now united into one Monarchy , under the King of Spain , their Religion Popish : the King is not rich by reason of his great Expences to keep his Dominions , in which are eleven Archbishops , and 52 Bishops , and hath these most notable Cities , viz. Toledo , Madrid , Leon , Fax , Siville , Grenado , Mursy , Saragosa , Bracelon , Pamphelune , Bilbo , Priede , St. James of Compostella , and Lesbone , and Rivers famous are the Dower , Tagus , Gadian and Guadalguinr . Great Britain consisteth of England and Scotland , and is the Biggest Isle in EUROPE , and the Glory thereof ; it is a temperate Soyl , a sound Air , and yieldeth all manner of good things , 't is environed all round with the Seas ; I shall begin first with England . England hath many pleasant Rivers well stored with Fish , excellent Havens , commodious Mines of Silver , Lead , Iron and Tinn , abundance of Woods , good Timber , plentifull in Cattle , good Wool of which is made fine Cloath , which serves not only themselves but vended into other Countreys , the chief City is London , in which are two of the Wonders of the World , viz. the Monument and Bridge over the Thames , the People are brave Warriers , both by Sea and Land , as Europe has felt and can testifie to their Grief , they are learned in all manner of noble Sciences ; the Order of Knighthood is that of St. George , or the Garter , there are 26 Knights of it , whereof the King is the Soveraign , their Ensign is a blue Garter buckled on the left Leg , with this Motto — Hony Soit Qui Mall 〈…〉 Pense , and about their Necks they do wear a blue Ribbon , at the End of which hangs the Image of St. George , upon which day this Order is Celebrated : secondly of the Bath , instituted 1009 , they use to be Created at the Coronation of Kings and Queens , and at the Enstalling of the Prince of Wales . The Knights thereof distinguished by a red Ribbon , which they wear about their Necks , their Duty is to defend Religion , Widows , Maids and Orphans , with the Kings right . Thirdly of Barronets and Hereditary Honour , the Arms are Mars , three Lions Passant , Gardant Sol , their Religion is the Protestant ; they have two Archbishops and twenty Bishops . The length of England is 320 , and breadth 250 Miles , it hath 857 famous Bridges , 325 Rivers , it 's defended and invironed with Turbulent Seas ; guarded by unaccessible Cleves and Rocks ; and defended by a strong and Puissant Navy ; so that of it may well be said , Insula praedives , que toto vix eget Orbe ; Et cujus totus indiget Orbis ope . Insula praedives , cujus miretur & optet Delicias Solomon octavianus opes . It s chief Cities are London , York , Bristol , and Rivers are the Thames , Severn , Humber , and the Ouze . Wales is bounded on all sides with the Sea , except towards England on the East ; it is a barren and mountainous Countrey : It s chief Commodities are their Freeze , and Cottons . The Inhabitants are faithfull in their promises to all men , but yet much enclined to Choler , and subject to Passion , which Aristotle calleth 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 . It contains 14 Shires , 13 Forests , 36 Parks , 230 Rivers , and 1016 Parishes . They are so resolute and valiant ( saith Henry III. writing to Emanuel then Emperour of Constantinople ) That they dare encounter Naked with armed Men , being ready to spend their Blood for their Countrey , and pawn their Life for Praise . They are Protestants , and have four Bishops but no Towns of Note . Scotland is the Northern part of Britain , environed all round with the Sea , unless where it is joyned to England . Polydore saith it is 480 in length , and 60 miles in breadth , divided into Highlands and Lowlands : the Highlands are Irish-Scots , and the Lowlands English-Scots . It is not so fruitfull as England ; the chief City is Edenbrough . Its Commodities is course Wool , and Cloth , Malt , Hides , and Fish. The Order of Knighthood is that of St. Andrew , the Knights did wear about their Necks a Coller interlaced with Thistles , with the Picture of St. Andrew appendant thereunto , having this Motto — Nemo me impune lacessit . 2. Of Nova Scotia instituted by King James , Anno 1622 , hereditary the Knight hereof distinguished by a Ribbond of Orange-tawny . The Arms of Scotland is Sol , a Lion Rampant , Mars within a double Tressure counterflowered ; they are Protestants , and have 2 Archbishops and 12 Bishops . The Cities most Famous are Edenbrough , Sterlin , Aberdeen and St. Andrews : and they have the Famous Rivers Tay and Tweed . Ireland is on all sides environed round with the Irish Seas , and St. Georges Channel . In length is 300 and breadth 120 miles . The Natives are strong and nimble , haughty , careless , hardy , bearing cold and hunger with patience and in a word , if they are bad you shall neve● find worse ; but if good scarcely find better . The Wild Irish have a custom to kneel down to the New Moon , praying it to leave them in as good health as it found them . They received the Christian Faith 435. The Soil is fruitfull and it hath good Pasture , yet full of Boggs and Woods , and multitude of Fowls , and in it will dwell no venomous Creature . The Revenues yearly have been 40000 li. The Air is Temperate , cooler in Summer , and hotter in the Winter than in England . Their Arms are Azure an Harp or stringed Argent , they are some Protestants and Papists mixt : they have 4 Archbishops , and 19 Bishops ; the chief City is Dublin . The Islands belonging to Great Britain , are 1. Wight ( the place where I first drew my Breath , and the Land of my Nativity ) 2. Surlings , 3. Garnsey , 4. Jersey , 5. Anglesey , 6. Man , 7. Hebrides , 8. Orcades , 9. Portland , 10. Sunderland , 11. Holy Island . And thus I have done with the British Empire ; all these Parts described belong to it , and are under the Royal Sceptre of his Sacred Majesty JAMES the Second ( whom God long preserve . ) Thus I have finished the description of Great Britain having this only to say — Quae Deus conjunxit nemo separet . Belgia , or the Low Countreys , consisteth of several wealthy Provinces : viz. The Dukedom of Brabant , Guelderland , Lymburge , Flanders , Artois , Henault , Holland , Zeland , Mamen , Zukfen , the Marquisate of the Holy Empire , Freezeland , Michlen , Ouserisen , and Graving . All which Lands are very fertile and populous , having 208 Cities , and 6300 Villages , with Parish-Churches , Castles , and Forts ; and is watered with the Rhine and the Mose , the Mara and the Sheld . It hath commodious Havens , the Inhabitants are brave Warriours , good Mechanicks , their chief Commodity is Rhenish Wine , Linnen and Woollen Cloth , Camericks , Lace of Gold and Silver , Silk , Taffatys , Velvet , Grogerams and Sayes , all manner of Twined threds , refined Sugars , Buff , Ox-hides , Spanish-leather , Pictures , Books , Cables , Ropes and Herrings . Now Belgia is bounded East with Westphalen , Gulick , Cleve , and the Isle of Triers ; West with the Main Ocean ; North with the River Ems ; and South with Picardy and Champagne . The People are of the reformed Religion all except Flaunders and Artoise , and they have the Popish Tenents , here are three Archbishops , and fifteen Bishops . The Order of Knighthood is that of the Golden Fleece , instituted 1439. their Habit is a Coller of Gold , interlaced with Iron , Or. Ex ferre Flammam , at the end thereof hangs a Golden Fleece . Their chief Cities are Mentz , Antwerp , Amsterdam , Roterdam , and Rivers are the Sheild and Mosa . Germany is the greatest Province in all EUROPE , and is bounded East with Russia , Poland and Hungary ; West with France , Switzerland and Belgia ; North with the Baltick Seas , and part of Denmark ; and South with the Alps and parted from Italy : it contains Bohemia and Pragu , it is adorned with Magnificent Towers , strong Fortifications , Castles and Villages , very Popular ; the Soyl is Fertile ; many Navigable Rivers do to it belong , Good Spaws , Hot Baths , Mines of Gold and Silver , Tinn , Copper , Lead and Iron ; they are some Papists , others Protestants , Zwinglians , Calvinists , and Lutherans . The Arms is Sol , an Eagle displayed with two Heads , Saturn armed , and Crowned Mars . There are six Archbishops , and 34 Bishops . They are a People much given to drinking ; which made the Poet say — Germani possunt cunctos tolerare labores , O utinam possent tam bene ferre sitim . The chief Cities of Germany are these , viz. Strasborough , Cologn , Munster , Norimbergh , Ausburg , Numick , Vienna , Prague , Dresda , Berlin , Stetin , Lubeck ; It s chief Rivers the Rhine , Weser , Elbe , Oder , and the Danow , and Cities of Bohemia , are Cutenberg , and Budrozu . Denmark and Norway , are two great Regions and bounded South with Germany ; they have North Latitude 71° 30 ' , toward the East they border on Sweden ; and elsewhere environed with the Sea. Their Commodities are Oxen , Grain , Fish , Tallow , Sand , Nuts , Oyl , Hides , Goat-skins , Fir-trees for Masts , Boards , &c. Pitch , Tarr and Brimstone : they are Lutherans . The Order of Knighthood is that of the Elephant , their Badge a Coller powdered with Elephants Towred , supporting the Kings Arms ; having appendent the Effigies of the Virgin Mary ; the Arms of the Land are Quarterly . 1. Or , three Lions passant Vert , Crowned of the first , for the Kingdom of Denmark . 2dly , Gules a Lion Rampant , Or , Crowned and Armed of the first , in his Paws a Dansk hatchet ; Argent , for the Kingdom of Norway . There are two Archbishops , and 13 Bishops ; its chief City is Coppenhagne . Sweden is a mighty Kingdom , is bounded East with Muscovia , West with the Dorfirin hills , North with the Frozen Ocean , and South with Denmark , Liesland and Mare Balticum ; the Commodities are Copper , Iron , Lead , Furr , Buff , &c. They are brave Warriers , their Religion is Lutherans . The Arms Azure three Crowns Or , it hath two Archbishops , and eight Bishops . Russia is bounded East with Tartaria , West with Livonia and Finland , North with the Frozen Ocean , and South with Lituania , and Mare Caspium , This Countrey is extreme cold : but yet Nature hath counterpoized it by supplying the Land plentifully with the best of Furrs , viz. Sable , White-fox , Martin , &c. It 's subject to the Emperour of Russia ; a vast Tract , and as wild a Government . The Inhabitants are Base and Ignorant , Contentious and Foolish , they deny the proceeding of the Holy-Ghost , they bury their Dead upright , with many other foolish Ceremonies ; Muscovia is the Seat of the Empire . Its Commodities are Furrs , Flax , Ropes , Hides , Fish , and Whales-grease . The Arms are Sable , a Portal open of two Leaves , and as many degrees Or , they are of a mixt Romish Religion , not observing Learning as any thing : They have one Patriarch , two Archbishops , and eighteen Bishops . It s chief Cities are Mucon , Wolodimax , St. Michael , Cazan , and Astracan , it 's chief Rivers are the Dwine , Volaga , and the Tana . Poland is bounded South of Moldavia and Hungary ; East with Moscovia , and Tartaria ; West with Germany ; and North with the Baltick Seas . The Commodities are Spruce-Beer , Amber , Wheat , Rye , Hony , Wax , Hemp , Flax , Pitch , Tarr ; it hath Mines of Tinn and Copper ; their Religion is partly Romish , and partly of the Greek-Church , and so there are of the Greek Church , two Archbishops , and six Bishops , and of the Romish Church three Archbishops , and nineteen Bishops : The Arms are Quarterly . 1. Gules an Eagle Argent Crowned and Armed Or , for Poland , and two Gules a Chevalier armed Cap-a-peid , advancing his Sword Argent , Mounted on a Barbed Course of the Second , for the Dukedom of Latuania . It s chief Cities are Cracovia , Warsovia , Damzerk , Vilna , Kion , Cameneca , and Smolensco ; and Rivers are Vistula , Niemen , Dunae and the Boristhenes . Hungary is bounded East with Transilvania , and Walachia , West with Stiria , Austria and Moravia , North with the Carpathian Mountains , and South with Sclavonia and part of Dacia . The People are valiant , and shew their Antiquity to be Scythians by their barbarous Manners , and neglect of Learning Their Sons equally inherit without Priviledge of Birthright , and their Daughters portion is only a New Attire . Its Commodities are Colours , Wheat , Beef , Salt , Wine and Fish , the German Emperour and Turk hath it between them . The Arms is eight Barrs Gules , and Argent , they are some of the Romish , and others Mahometans . There are two Archbishops , and thirteen Bishops , and its chief Cities are Transilvania , Valastia , Moldavia , Buda , Presbrough , Hermonstada , Tergoguis , Czuchan , Craffa and Bargos . Its Rivers are the Drin , Oxfeus , Peneus , Vardax , Marize and the Danubus . Sclavonia is bounded South with the Adriatick Seas , East with Greece , North with Hungary , and West with Carniola . It is fruitfull of all those Commodities found in Italy , and is under several Governments , viz. Turks , Venetians , Hungarians and Austrians . The Arms are Argent , a Cardinals Hat , the strings Pendant , and Pleated in a True-lovers-knot , meeting in the Base Gules . They are some Christians , and some Mahometans . There are four Archbishops , and twenty six Bishops . It s chiefest Cities are Nova , Zara , Nonigrad , Tinu , Sebenico , S. Nicolo , Trau , Spalato , Salona , Almisse , Starigrad , Vesicchio , Catara , and Doleigne . Dacia is bounded East with the Euxine Seas , on the West with Hungary and Sclavonia , North with Podolia , and South with Thrace , and Macedonia . The Soyl is fruitfull in Corn and Wine . It yieldeth medicinal Plants , they have plenty of Fowls , both wild and tame , very Populous and of Nature like the Hungarians ; they are all Mahometans ; It s most famous Cities are Triste and Pedena . Greece is bounded East with Propontick Hellespont , and Aegean Seas , West with the Adriatick , North with Mount Haemus , and South with the Ionian Seas . It was once the Mother 〈…〉 Arts and Sciences , but now the very Den 〈…〉 the Turkish Empire . The Soyl is very fruitfull 〈…〉 well manur'd , which made the Poet say — Impius haec tam culta novalia miles habebit ? Barbarus has segetes ? en queis consevimus arva Its Commodities are Gold , Silver , Copper , 〈…〉 Wines , Velvets , Damask ; here is the Mount of Parnassus : Here was the Temple of Delphos , consecrated to Apollo ; where the Devil through the Oracle did deceive the People , but after the Crucifixion of Christ the Oracle ceased . Augustus ( saith Suidas , in whose time Christ was born ) consulting with the Oracle , received this Answer — 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 An Hebrew Child , whom the blest Gods adore , Hath bid me leave these Shrines , and pack to Hell , So that of Oracle I can no more : In silence leave our Altar , and farewell . Their Religion is mixt but they are chiefly Mahometans . The Arms of this Empire were Mars a Cross , Sol , between four Greek Beta's of the second ; Bodin saith the four Beta's signified 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 . The most famous Cities in Greece are Buda , Salonique , Adrianopolis , Scutary , Durazzo , La Valone , L'Armiro , Brevezza , Larta , Lepanto , Setines or Athens , Thebes , Corinth , Patras , Misira or Lacaedomia . I shall pass over the Islands of Sicily , Sardinia , Candia and Corsica : and thus we have finished the description of the first part of the World , called by the Name of EUROPE . ASIA . ASIA the second part of the World is bounded on the North with the Northern Ocean ; South , with the Red Sea ; East , with the East Indian Ocean ; and on the West with the Flood Tunais . It is bigger than EUROPE , or AFRICA , and is far more rich , Viz. in Pretious Stones , and Spices , and hath been renowned by the first and second Monarchs of the World. Here Man was Created , placed in Eden , seduced by Satan , and redeemed by our Blessed Saviour . In it was done most of the History mentioned in the Old Testament . It hath been Ruled by the Kings of China , of Persia , the Great Turk , and the Emperour of Rushia , and contains these Provinces . Viz. Anatolia , Cyprus , Syria , Palestine , Arabia , Chaldea , Assyria , Mesopotamia , Turcomania , Media , Persia , Tartaria , China , India and the Oriental Isles . Anatolia is bounded West with the Thracian-Bosphorus , Helespont and the Aegean Seas ; East with Euphrates ; North with the Black Sea ; and South with the Rhodian , Lydian and Pamphylian Seas . It s length is 630 , and breadth 210 miles ; the Air is sound , the Soil fruitfull , but in some places desolate : it is inhabited by Greeks and Turks . It hath these Cities of note . Viz. Anatolia , Bruce , Chiontai , Augoure , Trebisond , Sattalie : and Rivers are Alie , Jordan , Euphrates , and Tygris . Syprus is bounded all round with the Syrian and Sicilian Seas ; whose length is 200 , and compass 550 miles . It is stored with plenty of all things , so that it wanteth no help of other Nations . It s Commodity is Wine , Oyl , Corn , Sugar , Cotton , Honey , &c. for which plenty of all things 't was Consecrated to Venus , as Ovid saith : Festa dies Veneris , tota celeberrima Cypro , Venerat ; Ipsa suis aderat Venus aurea festis . The People are Warlike , Strong and Nimble , and very hospitable to Strangers Their Arms are Quarterly . 1. Argent , a Cross , Patent , betwixt 4 Crosses Or. 2. Cross-wise of 8 pieces Argent , and Azure , supporting a Lion Passant , Azure Crown'd Or. 3. A Lion Gules . 4 Argent a Lion Gules : they are of the Popish Religion , and have 2 Archbishops and 6 Bishops . Syria is bounded East with Euphrates ; West with the Mediterranean Seas ; North with Cilicia ; and South with Palestine and Arabia . It s length is 525 miles , and breadth 470. They are inhabited by Mahometans , Christians , and Pagans . They are a stout and warlike people . In this Countrey there are said to be Sheep whose Tails weigh some 30 , and some 40 pounds ; the People are also gluttenous ; it is almost overrun by the Turks : It s most famous Cities are Aleppo , Te , Tripoly and Damal . Palestine is bounded West with the Mediterranean Seas ; East with the Arabian Desarts ; North with the Anti-Libianus ; and South with Arabia . The Inhabitants are of a middle stature , strong constitution , yet a stiffe necked and murmuring People and Idolaters . In this is the Land of Canaan , and the famous City Hierusalem , tho' now a Den of Idolatrous Mahometans . It abounds with all good things . Arabia is bounded East with Chaldea and the Gulph of Persia ; West with Palestine , Aegypt , and the Red Sea ; North with Euphrates ; and South with the Southern Ocean . The Inhabitants are Mahometans . Job's Habitation was here . It yields Frankincense , Pretious Stones , &c. It is now under the Great Turk's Sceptre . The most famous Cities are Herac , Ava , Medina , and Mectar : and it hath the famous River Cayban . Chaldea is bounded East with Susiana ; West with Arabia Deserta ; North with Mesopotamia ; and South with the Persian Bay. The Country is exceeding fruitfull ; in it is supposed to have been the Garden of Eden ; they were great Southsayers , and therefore flouted by the Satyrist . — Chaldeis sed major erit fiducia , quicquid Dixerit Astrologus , credent à fonte relatum Ammonis , &c. The Inhabitants are stout and valiant ; they are Mahometans . Here Julian the Apostate breathed his Soul out to Satan , in these dying words , — Vicisti tandem Galileae : the chief Cities are Babylon , Bagdad , Balfora , and Sipparum , with the famous River Fazze . Assyria is bounded East with Media ; West with Mesopotamia ; South with Susiana ; and North with Turcomania and Chaldea . This is a very plain and level Countrey , and very fruitfull , having good Rivers : the Natives are brave stout Warriours , formal in their Habit. It is under the Turk's command , and governed by one of his Bassa's ; who is able to bring into the Field at any time 100000 Souldiers : here are also a Sect of the Nestorians , and fifteen Christian Churches : it s most famous Cities are Calach , Cittace , and Arbela . Mesopotamia is bounded East with the River Tigris ; West with Euphrates ; North with Mount Taurus ; and South with Chaldea , and Arabia Desertae . It aboundeth with all good things necessary for the life of Man ; they are Mahometans , and a people unable to defend themselves but by the assistance of their Neighbours : It belongs to the Mahometan Empire . It s chief Cities are Edessa , and Cologenbar . I shall not describe Mount Taurus , because it is of no moment . Turcomana is bounded East with Media and Mare Caspium ; West with the Euxine Seas , Cappadocia , and Armenia major ; North with Tartaria ; and South with Mesopotamia and Assyria . It is a very mountainous Countrey ; the people are handsome , stout and brave Warriours : the Women are good Archers . It hath Gold and Silver Mines : It yields Grain , Fruit and Wine ; and in Colchis ( a part thereof , and in Assyria ) they sell their Children : The Arms are the Half Moon Or. It is inhabited by Mahometans , and under the Turkish Empire . It s chief Cities are Musol , Bagded , Batfora , Sanatopdy , and Derbent ; with the famous River Arais . Media is bounded East with Parthia ; West with Aremenia ; North with Mare Caspium ; and South with Persia. The Countrey is of a large extent and very different , even to a Miracle , for in the North part it is cold and barren , their Bread is dryed Almonds , and Drink the Juice of Herbs and Fruits . Their Food is Venison , and other Wild Beasts , which they catch by hunting . And in the South side the Country is of a rich Soil , plentifull in Corn , Wine , &c. They have been brave Warriours , and it was a custom with them to poyson their Arrows , in an Oyl called Oleum Mediacum : they are Mahometans . Persia is bounded East with India ; West with Media , Assyria , and Chaldea ; North with Tartaria ; and South with the Southern Ocean . This is a mighty rich Countrey governed by the Sophy , the people are strong and valiant , and though Mahometans , yet they War with the Turks for the Mahometan Religion in expounding the Alcoran . From hence comes Bezoars and other pretious Stones , Pearls , and Silk Works . It hath these famous Cities with Media : Viz Taurus , Gorgia , Cogsolama , Hysphan , Erat , Sus , Schiras , and Ormutz : and these Rivers Tiriditiri , and Bendimuz . Tartary bounded East with China , the Oriental Ocean , and the Straits of Anian ; West with Russia and Podolia ; North with the Frozen Ocean ; and South with China . Now the Tartarians are divided into certain Collonies , and differ in manners and Trade of Living , and are Men of a Square Stature , broad Faces , and look a Sq●int ; they are hardy and valiant ; they will eat either Horse-flesh or Man's Flesh. They drink Blood and Mares-milk ; their Habit is very homely ; they are some Mahometans , and some Pagans ; their chief Commodity is rich Furrs , and they are governed by the great Cham of Tartary , and hath these famous Cities , viz. Zahasp , Samarcanda , Thibet , Cambalu and Tatur ; and Rivers famous are Joniscoy , Oby , Chezel and Albiamu . China is bounded East with the Oriental Ocean ; West with India and Cathay ; North with Altay and the Eastern Tartaries ; and South with Canchin-China . It hath 591 Provinces , 1593 Walled Towns , 1154 Castles , 4200 unwalled Towns , and such an infinite Number of Villages , that the whole Countrey seems as one Town . It is reported that the Prince can bring into the Field 300000 Foot and 200000 Horse . The Land is fruitfull in Grain , full of wild and tame Beasts , it yields Silk , Pretious Stones , Gold , Copper , &c. The People are ingenious and great Artists , Witness their Wagon made to sail over the Land driven by the Wind : and Historians tells us , that the Art of Printing and of making Guns , is more Ancient with them than with us . They are Idolaters and worship the Sun , Moon and Stars , also they worship the Devil himself , that he may not hurt them . And it hath these most famous Cities , viz. Paguin , Quinjay , Caneun , Macao , Mancian and Magaia , with the great River Quinam . India is bounded East with the Oriental Ocean , and part of China ; West with the Persian Empire ; North with Mount Taurus ; and South with the Indian Ocean . This Countrey hath an Exact temperature of Air ; two Summers and a double encrease , blest with all things necessary for the Life of Man. It hath Mines of Gold and Silver , Pretious Stones , Spices , and Medicinal Druggs , abundance of Cattle , and Cammels , Apes , Dragons , Serpents , also multitude of Elephants ; a Creature of a vast Bigness , some of which are said to be nine Cubits high , and as many long , and five Cubits thick . It is a Creature of wonderfull Sence : for 't is reported of the Elephant on which King Phorus sate in the Warrs of Alexander , finding his Master strong and lusty , rushed boldly into the thickest of the Enemies Army : But when he once perceived him to be faint and weary , he withdrew himself out of the Battel , kneel'd down , and into his own Trunck received all the Arrows , directed at his Master . It also is of a most prodigious strength , for it is reported to carry a Wooden Tower on his Back , with thirty fighting men besides the Indian that Rules him . The Sea yields variety of Pearls and Fish ; here is also the Leviathan or Whale , of which Pliny says there are some of 960 Foot long ; here is the Rhinoceros also found : ( such as hath of late been publickly shewed at the Bell-savage Inn on Ludgate-hill in London ) a deadly and cruel Enemy to the Elephant , for though he be less , yet he will whet his horn against the Rocks , and then therewith strive to rip up the Elephants Belly , and is by many Naturalists supposed to be the Unicorn , for all the parts of his Body , especially his Horn , is a soveraign Antidote against Poyson . This Countrey is inhabited by Indians , Moors , Arabians , Jews , Tartars and Portugeze . The Natives are Tawny , tall and strong , and very punctual to their word . They eat no Fish nor Flesh , but live on things without life ; being Pythagoreans . It is also reported that when the Husband dies , and is burning on the Funeral Pile , that then the Wife leaps into the Fire , and so the living and the dead burn together , which made the Poet say — Et certamen habent lethi , quae viva sequatur , Conjugium ; pudor est non licuisse mori . Ardent victrices , & praebent pectora flammae , Imponuntque suis ora perusta viris . In India these are the chief Cities , viz. Amedabur , Cambaia , Gouro , Diu , Bengala , Pangab , Agra , Goa , Calicut , Visnagor , Pegu , Arracan , Malaca , Camboge , and Faefo . The fairest Rivers are Indus , Ganges and Mecon . The Oriental Islands are these , viz. 1. Japan , 2. The Phillepinae Isles , 3. The Maluccose , 4. Bantam , 5. The Selebes , 6. Borneo , 7. The Isles of Java , 8. Sumatra , 9. Zeiland , and other lesser Isles of which we shall not treat . 1. Japan is a rich Island abounding with Gold : So that Paulus Ventius saith , that in his time the King's Palace was covered therewith . It is a Mountainous Countrey , a healthfull Air ; here the Wheat is ripe in May. It 's full of Woods of tall Cedars , abundance of Beasts , Wild and Tame ; and also Fowls . The Inhabitants are strong , and witty , and have but one Language . They are Christians , and Idolaters , and the chief Cities are these , viz. Bungo , Meaco and Sacay . The Phillipine Isles are in Number 40 , called so in honour to Philip II. King of Spain , and are now inhabited by the Natives , and Spaniards , they are in a good Air and stored with rich Commodities ; and in them are these Cities , Lusor , Manille , and Mindanao . The Moluccoes Islands are many in Number , their Commodity is Cinnamon , ( which grows in whole Woods ; it is the Bark of a Tree , stript and laid in the Sun till it looks red ; and in three years time the Tree receives his Bark again . ) Ginger , Nutmegs , Mastick , Aloes , Pepper and Cloves : now the Clove groweth on a Tree like a Bay Tree ; yielding blossoms first white , then green , and at last red and hard , and then are Cloves . In it is also found the Bird of Paradice , and no where else , which for the strangeness and fairness of Feathers exceeds all the Birds in the World. The People are Pagans . Here is a Mountain of a prodigious height , above the Clouds , and agreeing to the Element of Fire , which it seems to mount unto , through Flames , wherewith , a dreadfull Thunder , and a dark Smoak it sends forth continually . The Isles of Bantam are in Number seven , one of which is continually burning , the Inhabi●ants are Barbarous , Weak of Bodies , Slothfull , Dull , and lying most confusedly together , without Rule , and are Mahometans . Its Commodities are Nutmegs , and both the yellow and white Saunderses . Now the Nutmeg grows on a Tree like a Peach Tree , the innermost part of whose Fruit is the Nutmeg , and is covered over with a Coat which ripe is called Mace ; they yield their Fruit thrice in the Year , to wit , at April , August and December . The Selebes are a Number of Isles full of Barbarous People , and Man-Eaters , they have abundance of strange Birds : It yields Sugar , Cocanuts , Cloves , Oranges , &c. In some of these Isles they make Bread of the Pith , and Drink of the Juice of the Tree called Sagu : It hath these chief Towns , viz. Senderem and Macassar . Borneo lieth West of the Celebes , and is in compass 2200 Miles , the Countrey yields Asses , Oxen , Herbs of Cattle and Horses . It yields Camphire , Agarick , and Mines of Adamants : They think the Sun and Moon to be Husband and Wife , and the Stars their Children , they reverently salute the Sun at his first rising . Their Affairs of State they Treat of in the Night , at which time the Councellor of State meets , and ascends some Tree , viewing the Heavens till the Moon ariseth , and then they go to their House of State. In it are these Towns , viz. Borneo , Taiopura , Tamaoratas , Malno and Sagadana . It is under the Government of the Kings of Borneo and Laus ; the People are Idolaters . Java Major , and Java Minor , are two Islands opposite to Borneo . They have plenty of Fruits , Grains , Beasts , Fish and Fowls , Gold and Pretious Stones . The Natives are of a middle Stature , broad faced and tawny , their Religion Mahometans , and they will eat their nearest of kin : the chief Town is Panarucan near a burning Hill , which in 1586 broke forth , and cast huge Stones into the City for three Days together , and destroyed much People . From the top of this vast high Mountain the Devil environed with a white and shining Cloud , doth sometimes shew himself unto his Worshippers , which live about those Hills . Sumatra lieth North of Java Major , betwixt it and the straits of Sincapura , its length is 900 Miles , and breadth 200 ; it is full of Fenns and Rivers , with thick Woods , and hath a very hot Air ; it is not fruitfull in Grain . Its Commodities are Ginger , Pepper , Agarick , Cassia , Wax , Honey , Silk , Cotten , Iron , Tinn and Sulphur . It hath also Mines of Gold , and is supposed to be Solomon's Ophyr . The King's Furniture of his House , and Trapping for his Elephants was beaten Gold , and he intituleth himself King of the Golden Mountains . Here is the notable Mountain Balalvanus , said to burn continually ; out of which or not far off do arise two Fountains , the one is said to run pure Oyle , and the other Balsamum Sumatra ; the People are Mahometans . The chief City and Seat of the King is Achen , beautifyed with the Royal Pallace , to which you pass through seven Gates one after another , with green Courts betwixt the two outermost ; which are guarded with Women , that are expert at their Weapons , and use both Sword and Guns with great dexterity , and are the only Guard the King hath for his Person . The Government is Absolute and Arbitrary , merely at the King's pleasure . Zeiland lies West of Sumatra , it is a good Soyl , and yields these Commodities , viz. Cinnamon , Oranges , Lemmons , most delicate fruit , Gold , Silver and Pretious Stones , it 's full of wild and tame Beasts , Fish and Fowls , yet destitute of the Vine : the People are strong and tall , given to Ease and Pleasure , and are in general Mahometans . The chief Towns are Candia , Ventane , and Janasipata . They have Fish-shells passing currant for money , there are other lesser Isles which we do for brevity sake omit , and thus we have done with the description of the second part of the World called ASIA . AFRICA . AFRICA the third part of the World is bounded East with the Red-Sea ; West with the Atlantick-Ocean ; South with the Southern-Ocean ; and North with the Mediterranean Sea ; and contains these Provinces , viz. Egypt , Barbary , Numidia , Lybia , Terra-Nigritarum , Aethiopia-superior and Aethiopia-inferior , with the Islands thereto belonging . Its Commodities are Balm , Ivory , Ebony , Sugar , Ginger , Dates , Myrrh , Feathers , &c. Egypt is bounded East with Idumea , and the Bay of Arabia ; on the West with Barbary , Numidia , and Lybia ; North with the Mediterranean Sea ; and South with Aethiopia-superior . It s length is 562 Miles , and breadth 160. The Natives are of a Tawny Complection , their Wives are the Merchants , whilst the Husband attends the Houshold Affairs . They were the Inventers of Mathematical Sciences ; they were also Magicians , and are still endued with a special Dexterity of Wit : They worship in every Town a particular God , but the God by them most adored was Apis. This Land is very fruitfull in all manner of Cattle , Cammels , and abundance of Goats ; they have plenty of Fowls both wild and tame : It hath Metals and Pretious Stones , Good Wines and rare Fruits , as Oranges , Lemmons , Cittons , Pomgranets , Figgs , Cherries , &c. Here also groweth the Palm-Tree , which grow the Male and Female together ; both put out Cods of Seeds , but the Female is not fruitfull unless she grow by the Male , and have her Seed mixt with his . The Pith of this Tree is good for Sallads , of the Wood they make Bedsteads , of the Leaves Baskets , Mats , and Fanns , of the outward husk of the fruit Cordage , of the inward brushes . It s fruit is the Dates , good for food , and finally 't is said to produce all things necessary for the Life of Man , and its Branches are worn in token of Victory , as saith Horace . — Palmaque nobilis , Terrarum Dominos evebit ad Deos. It hath many other Rarities which I am forc'd to omit . In it are these famous Cities , viz. Sabod , Cairo , Alexandria , Rascha , Damietta , Cosir and Surs , with the famous River Nilus , which by its overflowing makes the Land fertile , according to that of Lucia . — Terra suis contenta bonis , non indiga mercis , Aut Jovis ; in solo tanta est fiducia Nilo . Barbary is bounded East with Cyrenaica ; West with the Atlantick-Ocean ; North with the Mediterranean , the Straits of Gibralter , and part of the Atlantick-Ocean ; and on the South by Mount Atlas . It is full of Hills and Woods , stored with Wild Beasts : as Lyons , Bears , &c. Large Herds of Cattle ; it hath Dragons , Leopards , and Elephants ; beautifull , swift , and strong Horses ; it is the fruitfullest Countrey ●● the World in some parts of it ; for ●liny saith that not far from the City Tacape , you shall see a great Palm-Tree overshadowing an Olive ; under that a Figg-Tree ; under that a Pomgranat ; under that a Vine ; and under all Pease , Wheat and Herbs ; all growing and flourishing at one time , which the Earth produceth of it self : Its length is 1500 Miles , and breadth 300 Miles , the Natives are of a Tawnyish Colour , rare Horsemen , crafty and unfaithfull , and above measure Jealous of their Wives . It contains these Kingdoms , Viz. Tunis , Algiers , Morocco and Feze , and it hath these Isles , Viz. Pantalaria , Carchana , Zerby , Gaulos and Malta , the two latter of which Isles are inhabited by Christians , and are of the Romish Religion ; but for the other parts of Barbary , they are either Mahometans or Pagans . The most famous Cities are Morocco , Feze , Tangier , ( which formerly was a Principal City of Barbary ; but is now demolisht and lain level with the Ground , by the Command of His late Majesty Carolus II. of blessed Memory , and performed by the indefatigable skill and industry of the right Honourable George Lord Darmouth Anno , 1683. ) Teleusin , Oran , Algi●r , Constantine , Tunis , Tripoly and Barca , with these famous Rivers , Ommiraby and Magrida . Mount Atlas is a ridge of Hills of no small length , but of an exceeding heighth , above the Clouds , and is always covered with Snow , Summer and Winter , full of thick Woods , and against ASIA so fruitfull , that it affords excellent Fruit of it's natural growth ; it received it's Name from Atlas a King of Mauritania , fain . ed by the Poets to be turned into that Hill , by the Head of Medusa ; he was seigned to be so high that his Head touched Heaven : The ground of this Fiction I suppose was from his extraordinary knowledge in Astronomy , which Virgil seems to intimate — Jamque volans apicem & latera ardua cernit Atlantis duri , Coelum qui vertice foluit . N●vidia is bounded East with Egypt ; West with the Atlantick-Ocean ; North with Mount Atlas ; and South with Libia Deserta . The Natives are a wandring and unstable People , for they spend their Lives in Hunting , and continue not above four or five Days in one place , but so long as it will graze their Camels . Here grow abundance of Dates , with which they feed themselves , and with the Stones fat their Goats . The Air here is so sound that it will cure the Fr●nch-Pox without any Course of Physick . They are Mahometans : its chief Provinces are Dara , Pescara , Fighig , Tegorarin and Biledulgerid ; and its chief Cities are Taradath , Dara and Zev ; they belong to the Scepter of M●rocco . Lybia is bounded North with Numidia ; East with Nuba ; South with Terra-Nig 〈…〉 tarum ; and West with Gualata . This is well termed a Desart , for in it may a man travel eight or ten Days and not see any Water , no 〈…〉 Trees , nor Grass . So that Merchants are forced to carry their Provision with them on Camels , which if it fails they kill their Camels , and drink the Juice of their Entrails It contains these Provinces , Viz. Zahaga , Zv●nz●ga , Targa , Lembta and Bordea . They are governed by the chief of the Clans , and are a People only differing from Brute-Beasts , by their Shape and their Speech . Terra Nigritarum is bounded East with Ethiopia-superior ; West with the Atlantick-Ocean ; North with Lybia ; and South with the Ethiopick-Ocean . The Countrey is under the Torrid-Zone , full of People , and most excessive hot ; the soyl is exceeding fruitfull , brave Woods , Multitudes of Elephants and other Beasts : they have Mines both of Silver and Gold , very fine and pure ; the Natives are Cole-Black , or very Tawny , and are now some of them Mahometans , but most of them Pagans . It hath now these Provinces , Viz. Ora , Anterosa , Gualata , Agadez , Cano , Ca●●na , Sanaga , Gambra , Tombrutum , Melli , Gheneoa , G●ber , Gialofi , Guinea , Benin , Guangara , Bornum and Goaga , ( in which groweth a Poyson , which if any eateth but the tenth part of a Grain it will end his Days ) Bito , Temiano , Zegzeg , Zanfara , Gothan , Medra and Daum . And in it are these most remarkab'e Cities : Gue , Eata , Gueneha , Tomta , Agad●s , Cu 〈…〉 a , Tuta , Waver and Sanfara . The Rivers here that are most famous are Sernoga , Cambua and Ri●-Degrand . Aethiopia-superior is bounded East with the Red-Sea , and Sinus Barbaricus ; West with Lybia-inferior , Nubia , and Congo ; North with Egypt , and Lybia Marmarica ; and South with Monta-Luna . Now its length is said to be 1500 Miles , and Circute 4300. It is under the Command of the Abassine Emperour : here the Air and Earth is so hot and pieircing , that if the Inhabitants go out of their Doors without Shoes they lose their Feet ; here they also roast their Meat with the Sun : they have some grain , their Rivers are almost choaked up with Fish , their Woods stuffed with Deer , yet they will not trouble themselves to catch them . The Inhabitants are Lazy and destitute of all Learning , ●hey are of an Olive Tawny : here is also a Fountain , that if a man drinks thereof he either falleth mad , or else for a long time is troubled with a continual Drowsiness , of which Ovid thus speaks — Aethiopesque Lacus ; quos si quis faucibus hausit , Aut furit , aut patitur mirum gravitate saporem . And it contains these Provinces , Viz. Guagere , Tigremaon , Angote , Amara , Damut , Gojamy , Bagamedrum , Barnagasse , Dancali , Dobas , Adel , Adea , Fatigar , Xoa and Barus . Now as for the Government of these Empires'tis merely Regal : here is the Order of St. Anthony , to which every Father that is a Gentleman , is to give one of his Sons : out of which they raise about 12000 Horse , which are to be a standing Guard of the Emperour's Person : their Oath is to defend the Frontiers of their Kingdom , to preserve Religion , and to root out the Enemies of their Faith ; the Principals of their Religion are these . First , they circumcise their Children both Males and Females . Secondly , they Baptize the Males at 40 and Females at 80 Days after Circumcision . Thirdly , after the Eucharist they are not to spit till Sun set . Fourthly , they profess but one Nature and one Will in Christ. Fifthly , they accept but of the three first general Councils . Sixthly , the Priests live by the own labour of their hands , and are not to beg . Seventhly , they baptize themselves every Epiphany in Lakes or Ponds , because that Day they say Christ was baptized by John in Jordan . Eighthly , they eat not of those Beasts which Moses pronounced unclean , keeping the Jews Sabbath , with the Lord's Day . Tenthly , they administer the Lord's Supper to Infants presently after Baptism . Eleventhly , they teach the Reasonable Soul of Man comes by Seminal Propagation . Twelfthly , that Infants dying unbaptized are saved , being sanctifyed by the Eucharist in the Womb , and finally they produce a Book of Eight Volumes , writ as they say by the Apostles at Jerulalem for that purpose , the Contents whereof they observe most solemnly , and thus they differ from the Papists . Now the chief Cities in this Empire are these , Viz Barone , Caxumo , Amarar , Damont , ●●●●tes , Narre , Goyame , and Adeghena with the famous Rivers Zaire and Quilm●nei . Aethiopia-inferior is bounded East with the Red-Sea ; West with the Aethiolick Ocean ; North with Terra-Negritarum , and the higher aethipy ; and South with the Main Ocean And it contains these Provinces , Viz. Zanzibar , M●nomotapa , Cafravia , and Manigongo . The Natives are Black , with curled Hair , and are Pagans . In it are great Herds of Cattle , abundance of Deere , Antelopes , Baboons , Foxes , Hares , Ostriches , Pelicans and Herons , and in a Word what else is necessary for the Life of Man. In it are these most famous Cities , Viz. Banza , Loanga , S. Salvador , Cabazze , Sabula , Simbaos , Butua , and Tete . The Rivers are Cuama , Spiritus Sancto , and Dos Infante . The Islands in AFRICA are these , Viz. the Aethiopick-Isles , Madagascar , Socofara , Mohelia , Mauritius , St. Helens , the Isles of Ascention , St. Thomas-Isles , the Princes-Isles , the Isles of Annibon , the Isles of Cape d'Verd , the Canaries , Madera , Holyport and the Hesperides . The Description of all which I am forced to omit because I have been so very large in the Description of the third part of the World called AFRICA . AMERICA . AMERICA , the fourth part of the World , was first discovered by Christopher Columbus , Anno 1492 , but it hath received its Name from Americus Vesputius , who in the year of Christ 1597 did fail about it . Now this fourth part of the World is bounded East with the Atlantick Ocean ; West with the West-Indian Ocean ; South with the Magellanick Sea ; and on the North with the Northern Ocean . When first the Spaniards had entred on America they found the people without Apparel , and their Bread was made of the Jucca-Root , whose Juice is a strong poyson : but it being squeezed out and dried it makes Bread. They worshipped Devillish Spirits , which they call Zema ; in remembrance of which they keep Images made of Cotton Wool , to which they did great reverence , supposing the Spirits of their Gods were there ; and to blind them the more , the Devil would cause these Puppets to seem to move and to make a noise , so that they feared them so greatly that they durst not offend them ; which if they did , then the Devil would come and destroy their Children . They were so ignorant that they thought the Spaniards to be immortal ; but the doubt continued not long , for having taken some of them Prisoners , they put them under Water untill they were dead , and then they knew them to be mortal like other Men. They were quite destitute of all good Learning , reckoning their Time by a confused knowledge of the course of the Moon ; they were honest and kind in their Entertainments , encouraged thereunto by an Opinion that there was a certain place to which the Souls of those that so lived , and dyed for the defence of their Countrey , should go to , and there be for ever happy . So natural is the knowledge of the Soul's Immortality , and of some Ubi , for its future reception , that we find some tract of it in the most Barbarous Nations . The Americans were of a fair and clear Complexion . This Countrey is very plentifull in Spices , and Fruits ; and such Creatures which the other parts never knew : So fu l of Cows and Bulls , that the Spaniards kill thousands of them yearly only for the Hides and Tallow . Blest with abundance of Gold , that in some Mines they have found more Gold than Earth . They have Grey Lyons , their Dogs snowted like Foxes , neither can they bark ; their Swine hath Talons sharp as Rozors , and their Navel on the ridges of their Backs ; the Stags and Deer without Horns ; their Sheep are so strong that they make them carry burthens of 150 pound weight ; they have a Creature with the forepart as a Fox , and hinder as an Ape , except the Feet which are like a Man's ; beneath their Belly is placed a Receptacle like a Purse , in which their young remains till they can shift for themselves , never coming thence but when they suck and then go in again . The Armadilla is like a barbed Horse , armed all over with Scales that seem to shut and open . The Vieugue resembling a Goat , but bigger , in whose Belly is found the Bezoar , good against Poyson . A Hare having a Tail like a Cat , under whose Skin nature hath placed a Bagg , which she useth as a Store-house : for having filled her self she putteth the residue of her provision therein . Pigritia , a little Beast that can go no further in fourteen days than a Man will cast a Stone . For their Birds they are of such variety of Colours and Notes , which are so rare and charming , that they surpass all other Birds in any other parts . Now America is divided into two parts , viz. Mexicana , whose compass is said to be 13000 miles , and that other part called Peruana , whose Circumnavigation is esteemed 17000 miles . The Provinces of Mexicana are these : Viz Estotilant , Canada , Virginia , Florida , Califormia , Nova-Gallicia , Nova-Hispania , and Guatimala . Peruana contains these Provinces : viz. Castella-Aurea , Nova-Granado , Peru , Chile , paragnay , Brasila , Guiana , and Paria . To Peruana belongs these principal Isles : viz Hispaniola , Cuba , famaica , Porto-Rict , Barbadoes , the Charibe-Isles , Insula-Margaretta , Molaque-Isles , Remora , Insula Solamnis , and some other small Isles . But first of Mexicana . Estotilant hath on the East the Main Ocean ; South Canada ; West Terra Incognita ; and North Hudson's Bay. It comprehends Estotilant , so principally called , Terra Corterialis , New-found-land , and the Isles of Baccala●s . It is well stockt with all things necessary for the life of Man : the Natives are barbarous , fair , swift of Foot , and good Archers . They are Pagans . Canada is bounded North with Cortelialas ; South with New England ; East with the Main Ocean ; and West with Terra Incognita . It contains these several Regions : viz. Nova Francia , Nova Scotia , Norumbegne , and four small Isles adjoyning thereto . The people when first discovered were very rude and barbarous , going Naked only a piece of Fishes Skin to cover their private parts , and had two or three Wives a piece , which never Marry after the death of their Husbands The Soil is fruitfull , and yields all manner of good things Here groweth the Sea Horse whose Teeth is an Antidote against Poyson It hath these principal Cities : viz. Hochelaga and Quebeque . Virginia hath North Canada ; South Florida ; East Mare-del-Noo 〈…〉 ; West with Terra Incognita . And it is now divided into New England , New Belgium , and Virginia strictly so called . It is in some parts ( yea most parts ) Mountainous , Wooddy and Barren , and full of Wild Beasts . It yields plenty of Cattle , wild and tame Fowls . Its Commodities are Furrs , Amber , Iron , Rop●s , Tobaco , Sturgecn , &c. The Natives are but few in number , and those very different both in Speech and Size , to a Miracle : those whom they call Sasques Honoxi , are to the English as Giants clad in Bears Skins ; those whom they call Wig●ocomici , are as Dwarfs ; for the most part without Beards ; they hide their nakedness with a Skin , the rest of their Body they paint over in the figures of horrid Creatures The chief Towns are `fames's , and Plimouth , and Isle of Bermoodus , which I here omit . Florida is bounded North-East with Virginia ; East with Mare-del-Noort ; South with the Gulph of Mexico . It was first discovered by the English , Anno 1497. The Soil is very fertile in Grain and Fruit , Beast wild and tame , and so also for Fowls : It yields lofty Cedars , and Sassafras : It hath Gold and Silver Mines , and also Pearls . The Natives are of an Olive-Colour , strong and fierce , and are clad like the former Natives of America . The Women when their Husbands are dead cut off their Hair , and cannot Marry till their Hair is grown out again . To it belongs these Islands : viz. the Isles of Tortugas , Martyres , and Lucaios : there are also about 24 small ones more which are insignificant . The Women here are most extreamly beautifull ; the Natives are Pagans . It s chief Towns are St. H●elens , Ax Carolina , and Port-Royall . Califormia is an Island having on the West New Spain , and New Gallicia ; and so unto those undiscovered parts which lie furthest North , to the Straits of Anian ; and 't is divided into these four parts : viz. Quivira , Cibola , Califormia , specially so called , and Nova Albion . All which Countreys are indifferent fruitfull , full of Woods , and both wild and tame Beasts ; plenty both of Fish and Fowl wild and tame : They worship the Sun as their chief God : They go naked both Men and Women in some parts , others are half way cloathed ; and so very various that I cannot in this small Tract describe them . It s chief Town is Chichilticala . And here I cannot chuse but remark that in Quivira their Beasts are of strange forms , and are to them both Meat , Drink and Cloathes . For the Hides yields them Houses ; their Bones and Hair , Bodkins and Threed ; their Sinews , Cords ; their Horns , Guts and Bladders , Vessels ; their Dung , Fire ; their Calveskins , Buckets to draw and keep Water in ; their Blood , Drink ; and their Flesh , Meat ; and so much for Califormia . Nova Gallicia is bounded East and South with Nova Hispania ; West with the River Buena , Guia , and the Gulph Califormia ; and North with Terra Incognita . It comprehendeth these Provinces : viz. Chialoa , Contiacan , Xalisco , Guadalajara , Zacatecas , New Biscay , and Nova Mexicana . In which Provinces the Air is indifferently temperate , yet sometimes given to Thunder , Storms , and Rain . It is full of Mountains , yields Brass , Iron , &c. They have plenty of Fish , Beast , Fowls , Fruit , and abundance of Honey . The Natives are wavering , crafty and lazy , given to singing and dancing . They go not naked : they are subject to the King of Spain . It s chief Cities are Guadalajara , and St. Johns . Nova-Hispania is bounded East with the Gulph of Mexico , and the Bay of New-Spain ; West with Nova Gallicia , and Mare-del-Zur ; on the North with part of Nova Gallicia , and part of Florida ; and on the South with the South Sea. It comprehendeth these Provinces : Viz. Mexicana , Mechoacan , Panuco , Trascala , Guaxata , Chiapa and Jucutan . In all which the Air is healthfull and temperate , rich in Mines of Gold and Silver , Cassia , Coccineel , which grows on a shrub called Tuna , yields grain , and delicate Fruit , Birds and Beasts both Wild and Tame : their Harvest is in October and in May. The Natives are witty and hardy , yet so ignorant that they thought the Spanish-horse and Man to have been but one Creature , and thought when the Horses Neighed they had spoken . The Spaniards whose Cruelties will never be forgotten , did in less than 17 years kill of the Natives 6000000 ; here is a Tree called Meto , it bears 40 kinds of Leaves , of which they make Conserves , Paper , Flax , Mantles , Matts , Shoes , Girdles ; it yields a Juce like Syrup , which boyled becomes Hony , if purified Sugar ; the Bark roasted is a good Emplaisture for Punctures or Contusions ; and it yields a Gum Sovereign against Poyson : here is also a Burning Mountain called Propaeampeche , which sends forth two streams the one of Red and the other of Black Pitch : the Inhabitants are Pagans . Guatimala is bounded North with Jacuta , and the Gulph Honduras ; South with Mare-del-Zur ; East with Castella-Aurea ; and West with New Spain . The Soyl and People are as in Nova Hispania : it contains these Provinces , Viz. Chiapa , Verapaz , Guatimala , Hondarus , Niceragna and Teragna . And Towns of most Note are Cutrinidao and St. Michael's , the People are Pagans . And so much for Mexicana . Peruana the Second Part of AMERICA , so called from Peru a Place of Note therein , and it doth contain these Provinces , Viz. Castella-Aurea , Nova-Granada , Peru , Chile , Paragnay , Brasile , Guyana , and Paria and its Isles . But such Isles that fall not properly under some of these must be referred to the general Heads of the American Islands . Castella-del Oro , is bounded East and North with Mare-del-Noort ; West with Mare-del-Zur ; and South with Granada . And it containeth these Provinces , Viz. Panama , Darien , Nova-Andaluzia , St. Martha and De-la-Hacha . In all which Provinces the Air is very hot and unhealthfull : the Soyl either Mountainous and Barren , or low and Miry : plenty of Beast and Fowls . Here is said to be a Tree which if one touch he is poysoned to death : the old Natives are now almost quite rooted out . It s chief City is Carthagena , which Sir Francis Drake in 1585 took by Assault . This Land hath abundance of Gold. Nova-Granada is bounded North with Castella Aurea ; West with Mare-del-Zur ; East with Venez●●la ; and South with Terra Incognita . It s length is 390 Miles , and as much in breadth . It doth consist of these two parts , Viz. Granada , specially so called , and Popayan , both which hath a temperate Air , brave Woods , well stored with Cattle , and Fowls both wild and tame , plenty of Emeralds and Guacum : the People tall and strong ; the Women handsome and better drest than their Neighbours : The chief Towns are S. Toy d'Bagota and Popayan . Peru is bounded East with the Andes ; West with Mare-del Zur ; North with Popayan ; and South with Chile . It is 2100 Miles in length , and its breadth is 300 Miles : it is a Mountainous Country : And here 't is to be noted that in the Plains it never raineth ; and that on the Hills it continually raineth from September to April , and then breaks up . In the Hilly Countreys the Summer begins in April , and endeth in September . In the Plains the Summer beginneth in October and endeth in April , So that a man may travel from Summer to Winter both in one Day ; be frozen in the Morning when he setteth out , and scorched with heat before the dawning of the Day . It is not very plentifull of Corn nor Fruits , but they have a kind of Sheep which they call Pacos as bigg as an Ass , profitable both for fleece and burthen , but in tast as pleasant as our Mutten : So subtile that if it be overladen it will not for blows move a foot till the burthen be lessened , and it is a very hardy Creature . Here is a Figg-tree , the North part of which looketh towards the Mountains , and yieldeth its Fruit in Summer only , and the Part facing the Sea in Winter only . They have another Plant , that if put into the hands of the Sick and the Patient looks merry , they will recover ; but if sad , die . It yieldeth also Multitudes of Rarities more . It 's chief Commodities are Gold , Silver , Tobacco , Sarsaparilla and Balsamum d'Peru , and many other rich Drugs . The Natives are almost now rooted out of the Country . They are fierce and Barbarous . Now it contains these Provinces : viz. Quito , Los Quinxos , Lima , Cusco , Charcos and Colla● . Chile is bounded North with Deserta Alacama ; West with Mare del Zuz ; South with the Straits of Magellan ; and East with Rio de la Plata . It s length is 1500 miles , and breadth uncertain . The Soil hereof in the Mid-land is mountainous and unfruitfull ; towards the Sea-side level and fertile ; with products of Maize and Wheat , plenty of Gold and Silver , Cattle and Wine . The Natives are very tall and warlike , some of them affirmed to be eleven foot high ; their Garments of the Skins of Beasts ; they are of a white Complexion ; their Armes Bows and Arrows . It is divided into Chile ( especially so called ) and Magellanica . Here Sir Walter Rawleigh planted two Collenies , who for want of timely Succors were either starved at home , or eaten by the Salvages , as they ranged the Countrey for food . Paraguay is bounded South with Magellanica ; East with the main Atlantick ; North with Brazila ; and West with Terra Incognita . It is said to be of a fruitfull Soyl , well stored with Sugar-Canes , Fraught with Mines of Gold , Brass , and Iron : great plenty of Amathyses , and Monkeys , Lyons and Tigers , the People are as the other Salvages , and it contains these Provinces , viz. Rio de la Plata , Tucaman and La Crux de Sierra , and it 's chief Towns are Puenas Agrees , and Chividad . Brazila is bounded East with Mare del Noort ; West with Terra Incognita ; North with Guiana ; and South with Paraguay . It s said to be 1500 Mi'es long and 500 broad . The Countrey is full of Mountains , Rivers and Forests , the Air sound and healthfull ; the Soyl is indifferent fruitfull : It s chief Commodities are Sugar and Brazele-wood . There is a Plant called Copiba which yields Balsam , soveraign for Poyson . An Herb called Viva , which if touched will shut up and not open till the Toucher is out of fight . A Creature which hath the Head of an Ape , the Foot of a Lyon , and the rest of a Man. The Ox-Fish with Arms , Fingers and Duggs , the rest as a Cow. So that it may be said of Brasila — Semper aliquid apportat novi . The people are witty as appears by their sayings to the Christians ( holding up a Wedge of Gold ) say'd they , Behold your God oh ye Christians ! on their Festival-days they go Naked , both Men and Women ; and are able Swimmers , staying under water an hour and half : the Women are delivered without great pain : some of the Natives are all over Hairy , like Beasts : it containeth not Provinces , but these Captain-Ships : viz. Vincent , Rio de Juneiro , Holy Ghost , Porto-Seguro , Des Ilheos , Todos los Santos , Fernambuck , Tamaraca , Paraiba , Rio Grande , Saiara , Maragnon , and Para. Its chief Cities are , Meranhan , Tamaracai , and Olinda , and the great River Zoyal . Guiana is bounded East with the Atlantick ; West with Mount-Peru ; North with the Flood Orenoque ; and South with the Amazones . The Air here is indifferently good : near the Sea it is plain and level , up in the Countrey Mountainous ; here the Trees keep their leaves all the year , with their fruit always ripe , and growing . The Inhabitants are under no settled Government : they punish only Murder , Theft , and Adultery ; their Wives are their Slaves , and they may have as many as they please ; they are without Religion or Notion of a Deity . It doth contain these Provinces : viz. Rio de las Amazones , Wiapoce , Orenoque , and the Isles of Guiana . Its Comodities are Sugar , and Cotton : in it are plenty of Beast , Fish , and Fowles ; they are Swarthy in Complection , and great Idolaters ; as for Cities it hath none of note . Paria is bounded on the East with Guiana ; West with the Bay of Venezuela ; North with the Atlantick Ocean ; and South with Terra Incognita : and contains these Provinces : Viz. Cumana , Venezuela , S. Margarita , Cutagana , and its Isles . All which are not very fruitfull ; it is well stored with Pearls ; the People paint their Teeth and Bodies with Colour : The Women are trained up to ride , run , leap and swim : and also to Till the Land. In it are these most noted Cities : Viz. St. Jago , St. Michael de Nevery , and Mahanao . As for the Descriptions of the American Isles I must beg the favour to omit : I shall therefore only name them having been so very large already ; and they are these : Viz. Los . Ladrones , Fernandes , the Caribes ; as Granada , S. Vincent , Barbados , Matinino , Dominica , Desrada ; Guadalupe , Antego , S. Christopher , Nieves , Sancta Crux , and some lesser Isles belonging to them : As also Portorico , Monico Hispaniola , Cuba and Jamaica . Thus I have finished the Description of the known Earth . Now the Names of the Seas are these : Viz. the Ocean Sea , Narrow Sea , Mediterranean Sea , Mare Major , Mare Pacificum , Mare Caspium , the East-Indian Sea , Perfian Sea , Red Sea , and Mare-del-Zuz , which are all the Principal-Seas . Thus through the Blessing of God I have given you a brief , tho'true Description of all the known Earth and Seas , and have thus finished my Geographical Descriptions of the Division of the Earthly Globe . The Author on the Difficulties in the Collection of his 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 , or little Description of the great World. Oh thou Urania ! Thou that hast now brought Our Ship to Harbour sound , and richly fraught . Tho'Aeolus his blustring Gales did send , And foaming Billows high , the Skies did rend : Tho'Blustring storms , and Thunder loud did roar , And darkness Grim , opprest our Souls all'ore ; So that we could not view the Stars , nor Sky , Nor Sun , nor Moon , nay Earth , could not espy . Yet by thy Art , such safety we did find , Safely to pass both raging Seas , and Wind. And at the last a Harbour , safe did gain : Rejecting fears ; we quite cast off our pain . When Seas are calm , and Winds more serene be , Then we again will put our Ship to Sea ; That when refresht we farther may descry , And search into this Noble Treasury . 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 SECT . III. Of Geographical Propositions . PROP. I. How to find the Distance of any two Cities or Places , which differ onely in Latitude . IN this Proposition there are two Varieties which are these . 1. If both the Places lie under one and the same Meridian , and on one and the same side the Equinoctial , either on the North or South side thereof ; then substract the lesser Latitude from the greater , and convert their difference into Miles ( by allowing 60 Miles for a Degree ) so have you the distance of the two Places propounded . 2. But if the two Places lie under one and the same Meridian , but the one on the South side of the Equinoctial , and the other on the North side , then add both their Latitudes together their Sum is their Distance . PROP. II. To find the Distance of any two Places which differ only in Longitude . There are also in this Proposition two Varieties . 1. The two Places may both lie under the Eqinoctial , and so have no Latitude : and if so , substract the lesser Longitude out of the greater , and convert the remainder into Miles ; so have you the distance of any two Places so posuited . 2. But if the two Places differ only in Longitude , and lieth not under the Equinoctial , but under some other Intermediate Parallel of Latitude , between the Equinoctial , and one of the Poles : Then to find their distance , this is the Analogy or Proportion . As Radius or S. 90° , To Sc. of the Latitude : So is S. of ½ X. of Longitude , To S. of ½ their distance , which being doubled , and converted into Miles , giveth the required distance . PROP. III To find the Distance of any two Places , which differ both in Latitude , and in Longitude . In this Proposition three Varieties do present themselves to our View . 1. One of the Places may lie under the Equinoctial and have no Latitude , and the other under some Parallel of Latitude between the Equinoctial and one of the Poles . In such case observe this Analogy or Proportion . As Radius or S. of 90° , To Sc. of their X. of Longitude : So is Sc. of their Latitude , To Sc. of their Distance required . 2. But if both the Places proposed shall be without the Equinoctial , but on the one side either both towards the North , or both towards the South , then their Distance may be found , by this Analogy or Proportion . As Radius or S. 90° 00 ' , To Sc. of their X. of Longitude : So is To. of the greater Latitude , To the T. of a fourth Arch , which substracted from the Complement of the lesser Latitude , the remainder must be the fifth Arch ; Then say , As Sc. of the fourth Arch , To Sc. of the fifth Arch : So is S. of the greater Latitudes , To Sc. of the Distance of the two proposed Places . 3. The two Places propounded may be so situated , that one of them may lie on the North , and the other on the South side the Equinoctial : the Distance of Places so situated may be obtained , by this Analogy or Proportion . As Radius or S. 90° , To Sc. of X. of Longitude : So is Tc. of the greater Latitude , To T. of a fourth Arch , which being substracted out of the Summ of the other Latitude , and the Radius or 90° Deg. the remainder is a fifth Arch ; Then say , As Sc. of the fourth Arch , To Sc. of the fifth Arch : So is S. of the Latitude first taken , To Sc. of the Distance required . These are all the Varieties of the Positions of Places on the Terrestrial Globe : For if the Distance of any two Places be required , they must fall under one or other of these Varieties , and may be obtained by one or other of the Proportions , mentioned in the three aforegoing Propositions . Also if you know the Latitude and Longitude of any two fixed Stars , or their Right Ascension and Declination , then by these Rules their Distance may be found , which is of good use to Astronomy . It may also be applyed to Circular Sailing ; of all other ways the most perfect : which is treated of in its due Place . CHAP. VIII . of NAVIGATION . NAVIGATION So called from Navis a Ship , is an Art Mathematical , which sheweth how by the shortest good Way , by the aptest Direction , and in the shortest Time , to conduct a Ship from any one place unto any other place assigned : it hath been highly esteemed by the Ancients ; it is the Glory , Beauty , Bullwark , Wall and Wealth of Britain , and the Bridge that joyns it to the Universe . Navigation is commonly divided into three sorts of Sailing : viz. Plain sailing , Mercator's sailing , and Circular sailing : Of all which three Parts I shall treat in their Order . SECT . I. Of Plain sailing , or sailing by the Plain Chart. PLain Sailing , or sailing by the Plain Chart , is the most plainest way , and the Foundation of all the Rest : and although the Ground and Projection of the Plain Chart is erroneous , yet seeing it is more facile to the Learner , and may serve indifferently near the Equinoctial , because there the Degrees of Longitude , as well as the Degrees of Latitude , are Equal : Each Degree being divided into 60 Minutes , or Milles , though they are somewhat more than English Miles , Each Minute or Mile containing about 6000 Feet . PROPI . The Rumb , and Distance sailed thereon being given , to find the Difference of Latitude , and the Departure from the Meridian . Admit a Ship sails N. W. by N. 372 ' or Miles , or 124 Leagues , I demand her Difference of Latitude and departure from the Meridian ? In the Triangle ABC , the Hypothenuse AC representeth the distance sailed , or Rumb-line , BC the departure from the Meridian , and AB the difference of Latitude . 1. To find which say , As Radius or S. 90° , To Log. distance sailed 372 ' . So is Sc. V. A of the Course 56° 15 ' , To Log. cr . AB 309 3 / 10 Minutes , which being divided by 60 ' giveth 5° . 9 ' . 18 " for the Difference of Latitude . 2. To find the Departure from the Meridian say , As Radius or S 90° , To Log. Rumb-line AC 372 ' . So is S. of V. of the Course A 33° 45 ' , To Log. cr . BC 206 6 / 10 Minutes , the departure from the Meridian , which divided by 60 giveth 3° . 26 ' . 36 " for the difference of Longitude . Note that by this Proposition you may keep an Account how much you have sailed either East or West , North or South . PROP. II. By the Rumb and Difference of Latitude given , To find the Distance , and the Departure from the Meridian . Admit a Ship sail N. W. by W. untill her difference of Latitude be 309 3 / 10 Minutes , I demand her distance sailed , and her departure from the Meridian ? 1. To find the distance , say , As Sc. of V. of the Course A 56° 15 ' , To Log. cr . AB the X. of Lat. 309 3 / 10 Minutes . So is the Radius or S. 90° , To Log. AC 372 the distance sailed . 2. For the Departure , say , As Sc. of A V. of the Course 56° 15 " , To Log. cr . AB . X of Lat. 309 3 / 10 Minutes . So is S. of V. of the Course A 33° 45 ' , To Log. cr . AB 206 6 / 10 Minutes , the Departure required . By the help of this Proposition , when your Latitude by Observation doth not agree with your dead reckoning , ( kept by the former Proposition ) Then according to this Rule , you may make your way saild agree with your Observed Latitude , and so correct your Account or dead Reckoning . PROP. III. By knowing the Distance of the Meridians of two Places , and their Difference of Latitude , to find the Rumb , and Distance . Admit A , to represent the Lizard , AB the Parallel thereof , C. St. Mary's Islands , being one of the Azores , and CB the Meridian thereof . In the Triangle ABC , there is given the side AB 816 Minutes , the Distance of the Lizard , from the Meridian of St. Marys , and the side CB their difference of Latitude 768 Minutes , I demand the Rumb : i.e. the Angle at C , and the Distance of the Lizard from St. Marys ? 1. For the Rumb or Angle at C , say , As Log. cr . CB 768 ' , To radius ●● S 90° . So is Log. cr . AB 816 ' , To T. of the B 〈…〉 , or Angle at C 46° 44 ' , and is from the Lizard unto St. Marys to the fourth Rumb of the Meridian , and 1° 44 ' more , Viz. S. W. and ● 44 ' . Westerly , or from St. Marys , to the Lizard , N. E. and 1° 44 ' Easterly : and thus it shall be by the Plain Chart. 2. For their Distance AC , say , As S. Rumb . or V. at C. 46° 44 ' , To Log. cr . AB 816 ' Minutes . So is Radius or S. 90° , To Log. Hypoth . AC 1120 ' 1 / 10 which is the Distance of the Lizard , unto St. Marys Istand , and such should be the distance by the Plain Chart. PROP. IV. Admit two Ships to set sail from one Port , one Ship sails W. S. W. 40 ' , the other W. by N. so far untill she finds the first Ship to bear from her S. E. by E. I demand the second Ships distance from the Port , and their Distance asunder ? In the Triangle ADE , let A represent the Port , AD the W. S. W. course , and AE the Course W. by North. 1. To find the second Ships distance from the Port , say , As S. of V. at E. 22° 30 ' , To Log. cr . AD 40 ' Minutes . So is S. of V. at D 123° 45 ' , To Log. cr . AE 86 98 / 100 Minutes , which is the distance required . 2. To find the two Ships their distance Asunder , say , As S. of V. at E 22° 30 ' , To Log. cr . AD 40 Minutes . So is S. of V. at A 33° 45 ' , To Log. cr . DE 58 12 / 100 Minutes , which is the Distance required . PROP. V. Two Ships sets sail from two Ports , which lie N. and South of each other , the one sails from the Northermost Port 72 29 ' / 100 , and then meets she other Ship , which came from the Southermost Port , on a N. W. Course , and had sailed from thence 56 80 ' / 100 I demand the Rumb on which the first ship made her way , and also the Distance between the two Ports ? In the Triangle ADE , let A be the Southermost Port , AD the Course and way of the second Ship N. W. 56 80 ' / 100 , let E be the Northermost Port , ED the Course and Way of the other Ship 72 29 ' / 100 , and D the Place where they both meet . 1. To find the Rumb on which the first Ship sailed , say , As Log. cr . DE 72 29 / 100 Minutes , To S. of V. at A 45° 00 ' . So is Log. cr . DA 56 80 / 100 Minutes , To S. of V. at E 33° 45 ' , which sheweth the Course of the first Ship to be S. W. by South . 2. To find the Distance between the two Ports A and E , say , As S. of V. at A 45° 00 ' , To Log. cr . DE 72 29 / 100 Minutes , So is S. of V. at D 101° 15 ' , To Log. cr . EA 100 ' , which is the required Distance . PROP. VI. Admit a Ship coming off the Main Ocean and I had sight of a Promontory or Cape , by which it is my desire to sail , I find it to bear from me S. S. E. and distant by Estimation 33 ' , or Miles : But keeping still on my Course S. untill the Evening , having sailed 36 ' or Miles , I would then know how the Cape bears , and its distance from the Ship ? In the Triangle ADE , admit that at A , I do observe the Cape D , to bear from me S. S. E. 33 ' , and having sail'd from A , to E 36 ' South , I desire to know its Distance , and bearing . In the Triangle , there is therefore given , AD 33 ' , AE . 36 ' , and the Angle at A 22° 30 ' . 1. To find the Angle at E , say , As Z. cr s. AE , and AD 69 ' , To X. cr s. AD , and AE 03 ' . So is T. ½ VV unknown D and E 78° 45 ' , To the T. of 12° 20 ' , which taken from 78° 45 ' , leaves the Angle at E 66° 25 ' , so that the Cape D then bears from me E. N. E. and 01° 05 ' Northerly . 2. To find the Distance of the Cape ED from the Ship , say , As S. V. at E 66° 25 ' , To Log. cr . AD 33 ' . So is S. V. at A 22° 30 ' , To Log. cr . ED 13 78 / 100 Miles distant , so that the Cape is then distant from the Ship 13 78 / 100 Miles . PROP. VII . Two Ports both lying in one Latitude , distant 64 ' or Miles , the Westermost of those Ports lieth opposite to an Island , more Northerly distant therefrom 47 ' or Miles , which Island is also distant from the Eastermost Port , 34 ' or Miles , I demand the Course from the Westermost Port to that Island ? In the Triangle ADE , let A be the Westermost Port , and E , the Eastermost Port , distant Asunder 64 ' ; and let D be the Island , distant from A 47 ' , and from E 34 ' : Then is the Angle at A required , which is the Course or Rumb , from the Westermost Port , unto the Island : To find which , say , As Log. cr . AE 64 ' , To Log. Z. cr s. AD , and ED 81 ' . So is Log. X. cr s. AD , and ED 13 ' , To Log. os a certain line AO 16 454 / 〈…〉 . Which added to AE 64 , is 80 454 / 1000 , The ½ whereof is AB , 40 227 / 1000 Then again say , As Log. cr . AD 47 ' , To Radius or S. 90° . So is Log. AB 40 227 / 1000 , To Sc. V. at A , 58° 51 ' , that is N. E. by E. 2° 36 ' Easterly , which is the Course from the Westermost Port A , unto the Island D. SECT . II. Of sailing by the true Sea Chart , commonly called MERCATOR'S Chart. THE true Sea Chart , commonly called MERCATOR'S Chart * , performs the same Conclusions as the Plain Chart , and almost as speedily , but far more exactly : Because all Places may be laid down hereon , with the same truth as on the Globe it self : both to their Latitudes , Longitudes , Bearing and Distance from each other . And here it will be necessary to have a Table of Meridional Ports , which I have extracted out of Mr. Wright's Tables , to every tenth Minute of Latitude ; accounting it in single Miles , or Minutes of the Equinoctial , and have hereunto annexed the said Table . A Table of Meridional Miles The Deg. of Lat. The Minutes of each Degree . The Difference . 0 10 20 30 40 50 The Meridional Miles . 0 0 10 20 30 40 50 10 1 60 70 80 90 100 110 10 2 120 130 140 150 160 170 10 3 180 190 200 210 220 230 10 4 240 250 260 270 280 290 10 5 300 310 320 330 340 350 10 6 360 370 380 390 400 410 10 7 421 431 441 451 461 471 10 8 481 491 501 511 521 532 10 9 542 552 562 572 582 592 10 10 603 613 623 633 643 653 10 11 664 674 684 694 704 715 10 12 725 735 745 755 766 776 10 13 786 797 807 817 827 838 10 14 848 858 869 879 889 900 10 15 910 920 931 941 951 962 10 16 972 983 993 1004 1014 1024 10 17 1035 1045 1056 1066 1077 1087 10 18 1098 1108 1119 1129 1140 1150 10 19 1161 1172 1182 1193 1203 1214 10 20 1225 1235 1246 1257 1267 1278 11 21 1289 1299 1310 1321 1332 1342 11 22 1353 1364 1375 1386 1396 1407 11 23 1418 1429 1440 1451 1462 1473 11 24 1484 1499 1505 1516 1527 1538 11 25 1549 1561 1572 1583 1594 1605 11 26 1616 1627 1638 1649 1661 1672 11 27 1683 1694 1705 1717 1728 1738 11 28 1751 1761 1773 1785 1796 1808 11 29 1819 1830 1842 1853 1865 1867 11 The Deg. of Lat. The Minutes of each Degree . The Difference . 0 10 20 30 40 50 The Meridional Miles . 30 1888 1899 1911 1923 1934 1946 12 31 1958 1969 1981 1993 2004 2016 12 32 2028 2040 2052 2063 2075 2087 12 33 2099 2111 2123 2135 2147 2159 12 34 2171 2183 2195 2207 2219 2231 12 35 2244 2256 2268 2281 2293 2305 12 36 2318 2330 2342 2355 2367 2380 12 37 2392 2405 2417 2430 2442 2455 12 38 2468 2481 2493 2506 2519 2532 13 39 2544 2557 2570 2583 2596 2609 13 40 2622 2635 2648 2662 2675 2668 13 41 2701 2714 2718 2741 2754 2768 13 42 2781 2795 2808 2822 2835 2849 13 43 2863 2876 2890 2904 2918 2931 14 44 2945 2959 2973 2987 3001 3015 14 45 3030 3044 3050 3072 3086 3101 14 46 3115 3130 3144 3159 3173 3188 14 47 3202 3217 3232 3247 3261 3276 15 48 3291 3306 3321 3336 3351 3366 15 49 3382 3397 3412 3428 3443 3459 15 50 3474 3490 3505 3521 3537 3553 16 51 3568 3584 3600 3616 3632 3649 16 52 3665 3681 3697 3714 3730 3747 16 53 3763 3780 3797 3814 3830 3847 17 54 3864 3881 3899 3616 3933 3950 17 55 3968 3985 4003 4020 4038 4056 18 56 4074 4092 4110 4128 4146 4164 19 57 4182 4201 4219 4238 4257 4275 19 58 4294 4313 4331 4351 4370 4390 20 59 4409 4428 4448 4468 4487 4507 20 The Deg. of Lat. The Minutes of each Degree . The Difference . 0 10 20 30 40 50 The Meridional Miles . 60 4527 4547 4567 4588 4608 4629 20 61 4643 4670 4691 4711 4733 4754 21 62 4775 4796 4818 4839 4861 4883 22 63 4905 4927 4949 4972 4994 5017 23 64 5039 5062 5085 5018 5132 5155 23 65 5179 5203 5226 5250 5275 5299 24 66 5324 5348 5373 5390 5423 5449 25 67 5474 5500 5520 5552 5678 5704 26 68 5631 5658 5685 5712 5739 5767 27 69 5795 5823 6021 5879 5908 5937 28 70 5966 5996 6125 6055 6085 6115 30 71 〈…〉 6177 6208 6239 6271 6303 31 72 6335 6368 6401 6431 6468 6501 33 73 6535 6570 6605 6640 6675 6718 35 74 6747 6783 6620 6857 6895 6933 37 75 6972 7010 7050 〈…〉 7130 7170 40 76 7211 7253 7295 7338 7381 7424 43 77 7469 7513 7559 7605 7651 7651 7698 46 78 7746 7795 7844 7894 7944 7996 50 79 8048 8100 8154 8209 8264 8320 55 80 8377 8435 8495 8555 8616 8678 60 81 8742 8806 8872 8939 9007 9077 68 82 9148 9221 9295 9371 9449 9523 77 83 9609 9692 9778 9865 9954 〈…〉 88 84 10141 10238 〈…〉 10441 10547 10656 105 85 10770 10887 11007 11133 〈…〉 〈…〉 128 86 11539 11686 11839 11999 12168 12344 165 87 12521 12718 〈…〉 13150 13388 〈…〉 〈◊〉 88 13920 14221 14550 14914 15321 15783 386 89 16318 16950 17726 18729 20152 22623 PROP. I. To find by the Table , what Meridional parts are contained in any Difference of Latitude . The Use of the Table is demonstrated by the several Examples following , after this Manner . In this Proposition three Varieties present themselves unto our View . 1. When one Place is under the Equinoctial , the other having North , or South Latitude , his Meridional parts corresponding , is to be esteemed for the Meridional Difference of Latitude . 2. When both Places are towards one of the Poles , then the Meridional parts of the lesser , taken from the Meridional parts of the greater Latitude , the remainder is the Meridional difference required . 3. When one Place hath North , and the other South Latitude , their corresponding Meridional parts added together gives the Meridional difference of Latitude sought : thus having sound them out they may thus be applyed . PROP. II By knowing the Latitudes , and the difference of Longitude of any two Places , to find the Rumb , and Distance . Admit there be a Port in the Latitude of 50° 00 ' North , and another in the Latitude of 13° 12 ' North , and their Difference of Longitude is 52° 57 ' West , I demand the Rumb and Distance ? In the Triangle A b c , let A b represent the proper difference of Latitude , bc the Departure , Ac the distance sailed , A the Angle of the Course , c the Complement of the Course . In the Triangle ABC , AB is the Meridional difference of Latitude , BC the Difference of Longitude , A the Angle of the Rumb , C the Compl. of the Angle of the Rumb : These things being understood the work evidently appears to be the same as in Rightangled Plain Triangles . There is then required first the Difference of Latitude , and this falls under the second Variety . 1. To find the Rumb or Course say , As Merid. X. Lat. 2676 ' , To Radius or S. 90° . So is X. of Longitude 3177 ' , To T. of the Rumb 49° 53 ' , the Course there fore is S. W. ½ W , &c. 2. To find the Distance , As Sc. Course 40° 07 ' , To proper X. of Lat. 2208 ' . So is Radius or S. 90° , To the Distance 3426 Minutes as required . PROP. III. By knowing the Latitudes , and distance of two Places , to find the Rumb , and Difference of Longitude . 1. To find the Rumb or Course say , As the Distance sailed , To Radius or S. 90° . So is the X. of Latitude , To Sc. of the Rumb required . 2. To find the Difference of Longitude say , As Radius or S. 90° , To the X. of Latitude in Merid. Parts . So is T. of the Rumb , To the X. of Longitude required . PROP. IV. By knowing the Latitudes , and Rumb of two Places , to find their Distance , and Difference of Longitude . 1. To find the Distance say , As Sc of the Rumb , To the X of Latitude . So is Radius or S. 90° , To the Distance required . 2. To find the Difference of Longitude say , As Radius or S. 90° , To the X. of Latitude in M. Parts . So is T. of the Rumb , To the X. of Longitude required . PROP. V. By knowing the Rumb , Difference of Longitude , and one Latitude , to find the other Latitude , and the Distance . 1. To find the other Latitude say , As T. of the Rumb , To the X of Longitude in parts . So is Radius or S. 90° , To the Merid. X. of Latitude required . 2. To find the Distance say , As Sc. of the Rumb , To the X. of Latitude . So is Radius or S. 90° , To the required Distance , PROP. VI. By knowing the Distance , one Latitude , and Rumb , to find the other Latitude , and Difference of Longitude . 1. To find the Difference of Latitude say , As Radius or S. 90° , To the Distance . So is Sc. of the Rumb , To the X. of Latitude required . 2. To find the Difference of Longitude say , As Radius or S. 90° , To the Merid. X. of Latitude . So is T. of the Rumb , To the X. of the Longitude required . SECT . III. Of Circular Sailing , or Sailing by the Arch of a Great Circle . THIS is of all other the most exact way of sailing , and above all other most perfect , shewing the nearest way , and distance between any two Places : and although it is hardly possible to keep close unto the Arch of a great Circle , yet it is of great advantage to keep conveniently near it , especially in an East or West Course : In the former Propositions of sailing , we used Meridians , Parallels and Rumbs , as the Sides of every Triangle , whether by the Plain or Mercator's Chart : but in Circular sailing the Rumbs are not used so , because they are Helispherical-lines , and not Circles ; nor the Parallels , because they are not great Circles : Whereas the sides comprehending every Spherical Triangle are Arches of great Circles : Therefore here we use Arches of the Meridians , of the Equinoctial , and of other great Circles described , or so imagined to be described , from one Place unto another , on the Spherical Superficies of the Earth and Sea. Therefore here ariseth two things observable : and , 1. If the two places lie under the Equinoctial , then is their Position East and West , and their distance is their Difference of Longitude , converted into Miles : or , 2. If the two Places proposed be in one and the same Meridian , then is their Position North and South , and their Distance is their Difference of Latitude converted into Miles . And thus far doth Circular sailing agree with the former ; their difference will evidently appear by these following Propositions . PROP. I. Two Places , the one under the Equinoctial , the other in any Latitude given ; also their difference of Longitude given , to find . 1. Their Distance in the Arch of a great Circle . 2. The direct Position of the first place from the second . 3. And of the second Place from the first . Here we call the Angles which the Arch makes with the Meridians of the places propounded , the Angles of the Direct Positions of those places one from the other : because the Arch of a great Circle . drawn between two places is the nearest distance ; and the most direct way of the one , to the other Place . Now I shall not here demonstrate it by Schemes , as I have done in the other two Sections , but shall only lay down the proportions , whereby the required parts may be found ; and so leave the ingenious Seamen to practice it with Schemes at his leasure : and , 1. To find the nearest distance from Place to place , in the Arch of a Great Circle : Say according to the 10 Case of Rectangled Spherical Triangles . As the Radius , To Sc. of X. of Longitude . So is Sc. of X. of Latitude ; To Sc. of the Distance in the Arch required . 2. For the Direct Position , say by the 11 Case thus , As the Radius , To S. of X. of Latitude . So is Tc. of X. of Longitude , To Tc. of V. of Position required . 3. For the Direct Position of the second Place from the first , say by the 11 Case thus , As the Radius , To S. of the X of Longitude . So is Tc. of X. of Latitude , To Tc. of V. of Position required . PROP. II. Two Places proposed , the one lying under the Equinoctial , the other in any Latitude given ; with their distance in a great Circle of the same Places being also known , to find . 1. Their Difference of Longitude . 2. The direct Position from the first to the second Place . 3. And from the second to the first Place . 1. For their Difference of Longitude , say by Case 12 , As Sc. of the Latitude , To Radius . So is Sc. of their Distance in the Arch , To Sc. of their Difference of Longitude required . 2. Now to find the Direct Position from the first place to the second , say by the 13 Case ; and thirdly , for the Direct Position from the second place to the first , say by the 14 Case of Rectangulars . PROP. III. Two Places lying in one Latitude given , their difference of Longitude being also known , to find . 1. The nearest distance of those two Places . 2. The direct Position of one Place from the other . The Resolution of this Proposition depends on the 9 Case of Oblique Spherical Triangles : by supposing the Oblique Triangle , to be transfigured or converted into two Rectangulars , by a supposed Perpendicular : and then , 1. To find the nearest distance in the Arch , say by the 8 Case of Rectangulars . As the Radius , To Sc. of the Latitude . So is S. of half X. of Longitude , To S. of half the required distance , which doubled giveth the distance of the two places in the Arches , as sought . 2. For the Direct Position , say by the 9 Case . As the Radius , To S. of the Latitude . So is T. of half X. of Longitude , To Tc. of V. of Position required . PROP. IV. Two Places lying both in one Latitude given , and the nearest distance being also known , to find . 1. Their Difference of Longitude . 2. The direct Position of the one Place from the other . The Resolution of this Proposition falls under the 11 Case of Oblique Spherical Triangles : for here you have the three sides of the Triangle given , viz. the Arch of Distance , and the other two sides ( are both equal ) being the Complements of the places Latitude : and here seeing the two sides are equal , therefore the two Angles of Position are also equal : now there is required the three Angles , 1. To find their Difference of Longitude , add the double of the Complement of Latitude to the Arch of Distance ; then from half this Sum , deduct the Arch of Distance , and then proceed in all points as you see in Case the 11th . So shall their Difference of Longitude be obtained . 2. To find their direct Position : First , to the double Complement of Latitude , add the Arch of Distance , then from half that agragate , deduct the Complement of Latitude , and then work as before , so shall the direct Position be attained . PROP. V. Two Places proposed lying in one Latitude , and the distance of those Places in their Parallel given ; to find . 1. Their Difference of Longitude , 2. Their distance in the Arch of a great Circle , 3. The direct Position of the one from the other . Now you must understand , that as the Semidiamiter of a Parallel , is in proportion to the Semidiamiter of the Equinoctial : so is any number of Miles in that Parallel , to the Minutes of Longitude answering to those Miles : fo that if we suppose the Semidiameter of the Equinoctial to be Radius , then the Semidiameter of any Parallel is the Sine of that Parallel's distance from the Pole , that is the Sc. of the Latitude of that Parallel : Therefore , 1. To find the Diff. of Longitude say , As Sc. of the Latitude , To the Radius , So is the Distance in that Parallel , To the Diff. of Longitude required . 2. Now the Difference of Latitude being obtained , the nearest distance may be found , as in the third proposition aforegoing : 3. so likewise may the Angles of Position also . PROP. VI. By knowing the nearest Distance of two Places , their Difference of Longitude , and one of their Latitudes ; to find the Direct Position thereof from the other . This Proposition falls under the first Case of Oblique Spherical Triangles , and is thus resolved : therefore , As S. of the Distance of the two Places , To S. of their X. of Longitude . So is Sc. of the Latitude of the one Place given , To S. of the Direct Position from the other as was so required . PROP. VII . By knowing the Latitudes of two places , and likewise their Difference of Longitude ; to find , 1. The distance in the Arch. 2. The direct Position from the first to the second place . 3. The direct Position from the second to the first place . 4. The Latitudes and Longitudes by which the Arch passeth . 5. The Course and Distance from Place to Place through those Latitudes and Longitudes according to Mercator . I shall here make use of M. Norwood's example of a Voyage from the Summer-Islands , unto the Lizard : now because the work is various I have therefore illustrated it with a Scheme , and shall be as brief and facile as possible . Therefore , In the Triangle ADE , let A be the Summer-Islands , whose Latitude is 32° 25 ' , AD the Complement thereof 57° 35 ' , let E represent the Lizard whose Latitude is 50° 00 ' , and ED the Complement thereof 40° 00 ' , and let their Difference of Longitude , namely the Angle ADE be 70° 00 ' , now Drepresenteth the North-Pole , and AE an Arch of a great Circle passing by these two Places : now see the operation . 1. By having the Complements of the Latitudes of the two Places , viz. AD 57° 35 ' , and ED 40° 00 ' , and their Difference of Longitude , namely the Angle EDA 70° 00 ' : you may find the nearest distance EA to be 53° 24 ' ; by Case the 9. § 5. chap. 5. 2. Then having found the nearest distance in the Arch EA to be 53° 24 ' , ( or 3204 Miles ) the Angle of Position from the Summer Islands to the Lizard , namely the Angle DAE , may be found by Case the 1. § 5. chap. 5. to be 48° 48 ' , that is N. E. and 03° 48 ' Easterly . 3. And also by the same Case , may the Direct Position from the Lizard , to the Summer-Islands , namely the Angle AED befound to be 81° 10 ' , that is W. by N. and 2° 25 ' Westerly . 4. In order to the finding the Latitudes and Longitudes by which the Arch passeth , first let fall the Perpendicular DB , so is the Oblique Triangle ADE converted into two Rectangulars , viz. ABD , and DBE : secondly , by Case the 8. § 4. chap. 5. you may find the length of the Perpendicular DB to be 39° 26 ' , whose Complement is 50° 34 ' , which is the greatest Latitude by which the Arch ABE passeth , so the greatest Obliquity BDc 48° 31 BDf 38 31 BDg 28 31 BDh 18 31 BDj 08 31 of the Equinoctial from that Circle is 50° 34 ' . — Thirdly , by Case the 9. § 4 chap. 5. you must find the vertical Angles , viz. ADB , and BDE , which will appear , the Angle ADB to be 58° 31 ' , and EDB to be 11° 29 ' : now these things being obtained , the Latitudes by which the Arch passeth at every tenth degree of Longitude from A , may be found by resolving the several Right-Angled Triangles , viz. BDc , BDf , &c. substracting 10° from ADB 58° 31 ' , there remains BDc 48° 31 ' , and so for the rest as in the Table . Now by knowing these Angles last found , and the Perpendicular BD before found to be 39° 26 ' , you may by Case the 3. § . 4. chap. 5. find the Latitudes of the several points A. c. f. g. h. i. B. and E. to be as in the subsequent Table . 5. Thus having Latitude . Longitude . A. 32° 25 ' 00 00 ' c. 38 51 10 00 f. 43 34 20 00 g. 46 54 30 00 h. 49 04 40 00 i. 50 15 50 00 B. 50 34 60 00 E. 50 00 70 00 found the Latitudes and Longitudes of the Arch , and the other required parts aforementioned , we now come to shew how the Course , and the Distance from place to place according to Mercator may be found . So to find , first the Course and Distance Ac. now there is given the Latitude of A 32° 25 ' , and of c 38° 51 ' , and their Difference of Longitude is 10° 00 ' , now the Proper Difference of Latitude is 6° 26 ' , or 386 ' , and Meridional Difference of Latitude is 475 ' . Now knowing these things by proposition 2. § 2. chap. 8. yon may find the Course from A to c , to be N. E. 51° 38 ' ; and the Distance Ac to be 622 ' , and so those Rules prosecuted will shew the course and distance from c to f ; from f to g ; from g to h , &c. So of the rest , which for brevity sake I shall omit , and leave the Ingenious Seaman to Calculate at his Pleasure . I might hereunto annex many more propositions of Circular Sailing , but because of the smallness of this Treatise , and that those Propositions already handled , being by the Ingenious Seaman well understood , will be sufficient to enable him to perform any other Conclusion in Circular Sailing whatsoever , I therefore here omit , and hasten forwards unto the other parts of this Mathematical Treasury . A Table of Angles , which every Rumb makethwith the Meridian . These on this side the W. incline towards the N. end of the Meridian Angles of Inclination with the Meridian . These on this side the E. incline to the N. end of the Meridian . Rumbs . North Rumbs . N. by W. 11° 15 ' N. by E. N. N. W. 22 30 N. N. E. N. W. by N 33 45 N. E. by N. North West 45 00 North East N. W. by W. 56 15 N. E. by E. W. N. W. 67 30 E. N. E. W. by N. 78 45 E. by N. West 90 00 East W. by S. 78 45 E. by S. W. S. W. 67 30 E. S. E. S. W. by W. 56 15 S. E. by E. South West 45 00 South East S. W. by S. 33 45 S. E. by S. S. S. W. 22 30 S. S. E. S. and by W. 11 15 S. and by E. Rumbs South Rumbs These on this side the W. incline unto the S. end of the Meridian . These on this side the E. incline towards the S. end of the Meridian . Note that if you account in quarter of Points , add for one quarter 2° 48 ' , for one half 5° 37 ' , for three quarters 8° 26 ' , ( not regarding the Seconds in Navigation . ) CHAP. IX . Of SURVEYING . IT hath been a custom among Modern Authors , that have treated on this Subject , that before they entred on the Work it self , to give the Description of the Instruments , used in ; and chiefly appertaining to the Art of Surveying : viz. the Circumferentor , the Theodolite , the Plain-Table , and the Semicircle : concerning the descriptions of which Instruments I shall not here treat , but refer you unto those Authors that have largely and amply described them . I shall in this place onely demonstrate the Use of the Semicircle in taking the Plots of Enclosures , Champain-Plains , Woods and Mountains divers ways * ; and also in taking of Accessible , and Inaccessible Heights and Distances ; and also I shall shew the use of a little Instrument called a Protractor , in the delineating on Paper the Plot of a Field , &c. which Instrument being so commonly known , and so generally used makes me omit the description thereof as superfluous . As for your Chain , I would have you , have it made of good round Wyre ; to contain in length four Poles , or Perch , to be divided into an hundred equal parts called Links . And here before we enter on the Work it self , it will be necessary to understand how by the Protractor to lay down an Angle of any quantity of degrees propounded , or to find the quantity of an Angle given . SECT . I. Of the use of the Protractor . PROP. I. By the Protractor , to Protract an Angle of any quantity of degrees propounded . AN Angle may be laid down easily according to the directions of Prop. 5. § . 1. Ch. 4. but because this is more usefull in Surveying , Know that if it be required to protract an Angle of 50 deg . having drawn the line A B at pleasure , place the Centre of your Protractor on C , and moving it by your Protracting Pinn , untill the Meridional line thereof be directly on the line A B , then make a Mark by the division of 50° on the limb of the Protractor as at D , and draw the line CD , so shall the Angle DCB , be an Angle of 50 degrees . PROP. II. By the Protractor given , to measure an Angle given . This is performed by the line of Chords also , according to prop. 6. § . 1. chap 4. and by the Protractor is found thus : Suppose DCB were an Angle whose Quantity were desired , to find which , first the Center of the Protractor applyed unto the Angular point C , and its Meridional line lying justly with CB ; you shall perceive the Point D , to touch the limb of the Circle at 50 deg . Therefore I conclude the Measure of the Angle DCB , to be 50 degrees . SECT . II. Of the Manifold Use of the Semicircle , in taking the Plots of small Enclosures , Plains , Woods , or Mountains divers Ways . PROP. I. How to take the Plot of a Field , by the Semicircle at one Station taken in any part thereof , from whence all the Angles may be seen , and measuring from the Station unto every Angle thereof . SUppose ABCDEF were a Field , and 't is required to take the Plot thereof : Having placed marks at all the Angles thereof , and made choice of your Station , which let be K ; at which , place your Instrument , and turning it about untill the Needle hang over the Meridian Line of the Chart , there screw it fast : Then directing your sight to A , you 'l find the Degree out by the Index to be 40° 15 ' : Then measuring KA with your Chain it appears to be 5 Chains and 20 Links , which note down in your Field-book : and so do by all the rest untill you have found all the Angles and Distances from your Station K , to each respective Angle , which finished your work will stand thus . Angles . D. M. C. L. A. 40 15 5 20 B. 88 00 6 10 C. 130 00 5 50 D. 200 00 7 00 E. 250 00 5 00 F. 310 00 5 20 PROP. II. How to delineate on Paper any Observation taken according to the Doctrine of the last Proposition . Upon your Paper draw a Line to represent the Meridian line as M , H , then Placing the Center of your Protractor on the point K , laying the Meridian line of the Protractor on the Meridian line M , H , then seeing the Angle at A was 40° 15 ' , make a Mark against 40° 15 ' of the Protractor , as at A , and so do with all the other Angles , as you find them in your Table : Then remove your Protractor , and draw the Lines KA , KB , &c. This done lay down on each line his respective Measure , as it appeareth in the Table . Lastly draw the Lines AB , BC , &c. So have you on the Paper the exact Figure of the Field . PROP. III. How by the Semicircle to take the Plot of a Field at one Station in any Angle thereof , from whence you may view all the other Angles , by measuring from the Stationary-Angle , unto all the other Angles . Admit A , B , C , D , E , F , G , to be a Field , whose Plot is required : Place your Semicircle at G , and turning it about untill the Needle hang over the Meridian line of the Chart , and there screw it fast : Then direct your sights to the several Angles , viz. B , C , D , &c. in order one after the other , and so shall eace respective Angle be found , as in the subsequent Table : Then with your Chain , measure from your Stationary-Angle G , to all the other respective Angles , which done you have finished , and the work standeth thus . Angles . D. M. C. L. B. 40 00 5 00 C. 88 00 6 00 D. 120 15 6 40 E. 165 00 6 30 F. 193 00 3 40 A. 348 07 4 00 PROP. IV. How to delineate any Observation taken according to the Doctrine of the last Proposition . Upon your Paper draw a streight line as M , N , then take a point therein as G , to represent the Stationary-Angle , to which point apply the Center of your Protractor , ( in all respects as is before taught ) then according to the Notes in the Table , prick off all the Angles , viz. B , C , &c. according to their due quantity , then draw all the lines , viz. GB , GC , GD , &c. and on them place their respective measure ( as appeareth in your Notes ) lastly draw the lines AB , BC , CD , &c. So is there on the Paper the exact Figure of the Field , as was required . PROP. V. How by the Semicircle to take the Plot of a Field at two Stations , by measuring from each Station to the visible Angles : the Field being so Irregular that from no one Place thereof , all the Angles can be seen . Admit A , B , C , D , E , F , G , H , I , K , to be the Figure of a Field , whose Plot is required : having made choice of your two Stations , viz. Q , and P , and placed Marks in all the Angles : Then place your Semicircle at Q , and there six it with the Needle hanging over the Meridian of the Chart , represented by R , Q , X , and direct your sights unto all the visible Angles , viz. A , B , C , D , E , and F , and note down the Quantity of each Angle in your Field-book : Then measure with your Chain from your Station Q , to the Angles A , B , C , D , E , and F , and their length so found , note down in your Field-book also . This done direct your sight unto your second Station P , and note down in your Field-book the degree of Declination , of your second-station P , from the Meridian . Then measure the Stationary Distance PQ with your Chain , and note it down in your Field-book also . Then remove the Instrument unto P , your second-station , and there fix it with the Needle hanging over the Meridian line of the Chart represented by TPB , then direct your sights to the several visible Angles at this second Station , viz. F , G , H , I , and K , in order one after another , and note down the Quantity of each Angle in your Field-book : Then with your Chain measure from your Station P , to these several Angles G , H , I , and K , ( in all respects as at the first station Q. ) and their length so found note down in your Field-book likewise : So have you finished your Observation , and your work standeth thus . The Observation taken at the first Station Q. Angles . D M C. L A 50 00 6 60 B 80 00 7 65 C 140 12 12 00 D 220 07 11 10 E 270 05 12 60 F 330 00 6 00 The Declination of the Station P , from the Meridian R Q X , is 30° 00 ' , and the Stationary distance Q P is 9 Chains . The Observation taken at the second Station P. Angles . D M C L F 227 11 00 00 G 297 00 12 00 H 347 16 9 90 I 60 00 6 00 K 90 00 6 26 ☞ Note that the manner of taking the Plot of a large Champain Field , at many Stations , is almost the same with this Proposition ; for he that can do the one , can also perform the other : therefore for brevity sake I here omit it as superfluous . PROP. VI. How to delineate any Observation taken according to the Doctrine of the last Proposition . Upon your Paper draw the Meridian-line R Q X , then place the Center of your Protractor on Q , ( representing your first Station ) and its Meridional-line lay equal to R Q X , then prick off the Angles visible at your first Station Q , viz. A , B , C , D , E , and F , Of their due quantity , then draw Q A , Q B , &c. laying on them their corresponding measure , noted in your Field-book . Now because your second Station P , doth decline 30° 00 ' , from the Meridian RQX , prick off 30° 00 ' , and draw PQ , making it 9 Chains as in your Field-book appeareth , so doth P represent your second Station . Then in all respects as before , place your Protractor at P your second Station , and draw the Meridian T P B parallel to R Q X , then prick off the several Angles , viz. F , G , H , I , and K , Of their due quantity , and then draw PF , PH , PI , &c. of their due length . Lastly draw the lines AB , BC , CD , &c. and so shall you have on your Paper the exact Figure of the Field as required . PROP. VII . How by the Semicircle , to take the Plot of a Field at t 〈…〉 Stations , which lieth remote from you , when either by opposition of Enemies you may not , or by some other Impediment you cannot come into the same . Admit the Figure A , B , C , D , E , F , to be a Field into which by no means you can possibly enter , and yet of necessity the Plot thereof must be had , for the obtaining of which chuse any two Stations , it mattereth not whether near at hand or far off , so that all the Angles may be seen . Let your two Stations be H and L , ( the full length of the Field if possible ) then place your Instrument at H , and fixing it as is afore shewed , direct your sights to the several Angles of the Field , viz. A , B , C , &c. orderly one after another , observing their degrees as is afore taught , noting it down in your Field-book : then take up your Instrument , leaving a mark in its room at H , And measure with your Chain from Hunto L , your second Station , which note down in your Field-book ; Then placing your Instrument at L , your second Station , and as is before taught , fixing it there , make the like Observation to the several Angles , viz. A , B , C , D , &c. as at the first Station H , and note it down in your Field-book also , And having so done you have finished , and your Work standeth thus . Observations at the first Station H , are The Angle from H the first Station , unto L the second Station , is 180° 00 ' , the Stationary distance HL , is 60 Chains . Observations at the second Station L , are 1 Angles D M A 104 00 B 88 07 C 59 00 D 48 00 E 26 00 F 21 30 2Angles D M A 16 00 B 39 00 C 50 09 D 74 00 E 100 00 F 29 15 PROP. VIII . How to delineate any Observation taken according to the Doctrine of the last Proposition . Upon your Paper draw a Line as HL , which make equal to 60 Chains , then placing the Center of your Protractor on H , your first Station , prick off all the Angles A , B , C , &c. as you find them in your Field-book , and draw HA , HB , HC , &c. at pleasure : then remove your Protractor unto your second Station L , placing it as before , and prick off all the Angles A , B , C , D , &c. as you find them in your Field notes ; and draw the lines LA , LB , LC , &c. at length untill they intersect the former lines , HA , HB , &c. in the Points A , B , C , &c. which Points of Intersection are the Angles of the Field . Lastly draw AB , BC , CD , &c. So shall you have on your Paper the Figure of your Field , required . PROP. IX . How by the Semicircle , to take the Plot of a great Champain-Plain , Wood , or other overgrown Ground , by measuring round about the same , and making Observation at every Angle thereof . Admit A , B , C , D , be the figure of a Large overgrown Champain-Field ; whose Plot is required . First Place your Instrument at A , laying the Index on the Diameter ; and turn it about , untill you espy the Angle at D , and there fix it fast : and direct your sights to B , and note the Degree cut by your Index , in your Field-book , ( as afore is taught ) then remove your Instrument to B , and there make the like observation , and so to C , and D , noting it down in your Field-book , as asore . Then with your Chain , measure the Sides AB , BC , CD , and DA , whose length note down in your Field book , and so you have finished and your work standeth thus . Angles . D M C L DAB 100 00 12 20 ABC 117 15 10 00 BCD 71 30 19 20 CDA 71 15 12 20 PROP. X. How to delineate any Observation taken according unto the Doctrine of the last Proposition . Upon your Paper draw the line AB , at Pleasure , and placing the Center of your Protractor on the Point A , prick off an Angle of 100° , and draw AD , setting on it , and also on AB , their corresponding measure , in your notes : Then on B , protract an Angle of 117° 15 ' , draw BC of its due length : Then draw the line CD , so have you the exact figure of the Field , on your Paper . PROP. XI . How to take the Plot of any Field , by the help of the Chain only . Admit the Figure A , B , C , D , E , to represent a Field whose Plot is required . To obtain the which , first measure the sides CD , CB , and BD , and note their due length down in your Field-book , and then measure the Sides CA , and BA , and then note down their Length in your Field-book . Then measure the sides BE , and ED , for the sides BC , and BD , were before known ) which note down in your Field-book . So is your Field A , B , C , D , E , reduced into three Triangles , viz. CBD , CAB , and BED , the length of whose sides are all known , thus you have finished , and the works stands as you see . PROP. XII . How to delineate any Observation , taken according to the Doctrine of the last Proposition . Upon your Paper , draw a streight line , as CD , make it 5 Chains , 97 / 100 , take CB in your Compasses , and strike an Obscure Arch ; then take BD , and with that extent in D , cross the former Arch in B , and draw BC , and BD. Then take in your Compasses BE , and on B , strike an Obscure Arch , then take DE , and also cross the former Arch in E , and draw BE , and ED. Lastly take the line CA , and on C strike an Obscure Arch , then take AB , and on B , intersect the former Arch in A , then draw CA , and AB , so have you on your Paper the exact figure of the Field A , B , C , D , E , as was required . SECT . III. Of finding the Area or superficial Content of any Field , lying in any Regular or Irregular Form : by reducing the Irregular Fields into Regular Forms . HAving already shewed how to take the Plot of any Field divers ways , by the Semicircle and Chain , and also by the Protractor how to delineate the Draught thereof on Paper , &c. I now come to shew how the Area or superficial Content of a Field may be attained , i. e. how many Acres , Roods and Perches are therein contained . To which end know ; That a Statute Pole or Perch contains 16½ Feet ; that 40 of those Perches in length , and 4 in breadth makes an Acre . So that an Acre contains 160 Perches , and a Rood 40 Perches ; according to the Statute 33 , of Edward the First . Now the Original of the Mensuration of Land , and all other Superficies , depends on the Mensuration of certain Geometrical Figures ; as a Triangle , Square , &c. which may be measured according to the directions of § . 2. chap. 4 of Geometry : It would therefore here be superfluous to make a repetition of things already handled : I shall therefore omit it , and come to shew how any Field lying in any Irregular Form , may be measured by converting it into Regular Figures ; for it seldom happeneth , but that the Plot of a Field , is either a Trapezium * , or a many-sided Irregular Figure : therefore I shall first shew how to find the Content of a Trapezium . Secondly , of any many sided Irregular Figure ; and thirdly , how to reduce any number of Perches into Acres , &c. and on the contrary any number of Acres , into Roods and Perches . PROP. I. How to find the Area or superficial Content of a Trapezium . Trapeziums are Quadrangles of sundry forms : yet take this as a general Rule , whereby their Content may be found . Admit it be required to find the Area or superficial Content of the Trapezium ABCD , to find which , first by drawing the Diagonal AD , you reduceth it into two Triangles , ABD , and ADC . Then by prop. 3. § . 1. of Chap. 4 let fall the two Perpendiculars on AD , from B , and C , Then by prop 3. § . 2 Ch. 4. find the superficial Content of the two Trianangles ABD , and ADC , which added together , is the Content os the Trapezium ; by which Rule the Content of the Trapezium , A , B , C , D , is found to be 630 Perches . PROP. II. To find the Area or superficial Content of a many-sided Irregular Figure . Admit A , B , C , D , E , F , G , to be an Irregular many-sided Figure , representing a Field whose Content is required : now in regard the Field is Irregular , therefore reduce it into Triangles , viz. ABC , ACG , EDG , DEG , and DFG , and then find the Content of all the said Triangles , by prop. 3. § . 2. Chap. 4 and add their Contents together ; so shall that Sum be the Content of the said Figure ; and so do for any other . PROP. III. How to reduce any Number of Perches into Acres , and on the contrary , Acres into Perches . To find how many Acres are contained in any Number of Perches given , you must consider that 160 Perches do make a Statute Acre , therefore if you divide the Number of Perches propounded , by 160 , the Quotient is the number of Acres contained therein ; and if there be a remainder which exceed 40 , then divide it by 40 , the Quotient shall be Roods , and the remainder Perches . But on the contrary , if it were required to find how many Perches are contained in a certain Number of Acres propounded . You must multiply the Number of Acres , by 160 : the product shall be the Perches contained therein . It may be here expected , that I should shew how to reduce customary Measure to statute Measure ; and also that I should treat of the Division and Separation of Land. But because Mr. Rathborne , and of late Mr. Holwell , hath sufficiently explained the same , by many varieties , I shall for brevity sake omit it , and leave you to consult those Authors . SECT . IV. Of the Use of the Semicircle in taking Altitudes , Distances , &c. PROP. I. How by the Semicircle to take an Accessible Altitude . ADmit AB , be the Height of a Tower , which is required to be known . First placing your Semicircle at D , ( with the Arch downwards and the two sights fixed ) place it Horizontal * and screw it fast ; Then move your Index , till through the sights thereof , you espy the top of the Tower at B , and observe what degree the lower part of the Index cutteth and that will be equal unto the Angle at D 50 deg Then measure the distance DA , which let be 299 Feet . Now the heighth of the Tower AB , is found , according to prop. 1. § . 2. Chap. 5. thus , As Sc. V. at A 50° 00 ' , To Log. cr . DA 299 Feet . So is S. V. at A 50 00 , To Log. AB 356 3 / 10 Feet the height of the Tower AB required . PROP. II. How by the Semicircle to take an Inaccessible Altitude , at two Stations . Let AB be a Tower whose height is required ; having placed your Instrument at E , as before direct your sights unto the Top of the Tower at B , and finding the Degree cut by the Index , to be 23° 43 ' , I say it is the Quantity of the Angle at E : Now by reason of Water , or such like Impediment , you can approach no nearer the Base of the Tower , than D , Therefore measure ED , which is found to be 512 Feet , then at D , make the like Observation , and the Angle at D , appeareth to be 50° 00 ' , whose Complement is the Angle DBA , 40° 00 ' , and the Complement of the Angle E 23° 43 ' , is the Angle EBA 66° 17 ' : Now if the lesser Angle at B , be taken out of the greater , the remainder is 26° 17 ' , the Angle EBD : Now first to find the side BD , of the Trangle EBD , say according to prop. 1. § . 3. chap. 5. thus . As S. of V. EBD , 26° 17 ' , To Log. cr . ED 512 Feet . So is S. of V. at E 23° 43 ' , To Log. cr . BD 465 2 / 10 Feet required . Now to find the Height of the Tower AB , say according to prop. 2. § . 2. chap. 5. thus . As Radius or S. 90° , To Log. cr . DB 465 2 / 10 Feet found . So is S. of V. BDA 50° 00 ' , To Log. cr . BA 356 3 / 10 Feet , which is the height of the Tower required . ☞ Note that in taking any manner of Altitude the height of your Instrument must be added unto the height found , and that will give you the True Altitude required . PROP. III. How by the Semicircle to take an Inaccessible Distance at two Stations . Admit A , and B , be the two Stations , from either of which it is required to find the distance unto the Church at C ; placing your Instrument at B , the Index lying on the Diameter , and direct your sights unto the Church at C , fasten your Instrument , and turn your sights about untill you see through your sights , your second Station at A , so will you find your Index to cut 30° 00 ' , which is the Quantity of the Angle ABC . Then measure the distance AB , which is found to be 250 Yards , then with your Instrument at A , make the like Observation as before , and you will find the Angle BAC to contain 50° 00 ' . Now by the third Maxim of Plain Triangles § . 1. Chap. 5 you find also the Angle ACB , to be 100° 00 ' : now to find the distance AC , and BC , you may by their opposite proportion according to prop. 1. § . 3. chap. 5. find the distance of AC , thus . As S. of V. at C 100° 00 ' , To Log. cr . AB 250 yards . So is S. of V. B 30° 00 ' , To Log. cr . AC 127 yards . Which is the distance of the Church from A. Now to find the distance BC , say , As S. of V. at A 100° 00 ' , To Log. cr . AB 250 yards . So S. is of V. at A 50° 00 ' , To Log. cr . BC 194 4 / 10 yards , which is the distance of the Station B , from the Church at C. PROP. IV. How to find the Horizontal line of any Hill or Mountain , by the Semicircle . Let Figure 63 be a Mountain , whose Horizontal-line AB is required to be found : to find which , place your Instrument at A , and having caused a Mark to be placed on the Top of the Mountain at C ; ( of the just height of your Instrument ) then move your Index , untill through the sights thereof you espy the Mark at C , so will you find the Quantity of the Angle CAD , to be 50° 00 ' , and by consequence the Angle ACD to be 40° 00 ' , then measure up the Hill AC , which is 346 yards . Now having obtained these several things , 't is required to find the length of AD part of AB ; to find which say , As Radius or S. 90° , To Log. cr . AC 346 Feet . So is Sc. of V. at A 50° 00 ' , To Log. cr . AD 222 4 / 10 Feet . Now seeing the Hill or Mountain descendeth on the other side , you must place your Instrument at C , and direct your sights unto the Bottom at B , and the Angle DCB will be found 50° 00 ' , and the Angle CBD 40° 00 ' . Then measuring down the Mountain as CB , it appeareth Plate IV Page 237 To find DB , part of AB , say , As Radius or S. 90° , To Log. cr . CB 415 Feet . So is S. of V. BCD 50° 00 ' , To Log. cr . DB , 318 Feet : Now AD 222 4 / 10 Feet added thereunto produceth AB 540 4 / 10 Feet , which is the Horizontal line required of the Mountain ACBD . ☞ Note that when you come to delineate a Field wherein are Hills , you must protract the line AB , instead of the Hypothenusal Lines AC , and CB , and 't will be necessary to distinguish those kind of Fields , by shadowing them off with Hills and Dales . SECT . V. How to find whether Water may be conveyed from a Spring-Head unto any appointed Place . THE Art of conveying of Water from a Spring-Head unto any appointed Place , hath a special respect unto measuring , and therefore I think it not amiss to assert it in this place , and enroll it under the Title of Surveying . In the performance of which we make use of a Water-level , the Construction and making whereof is sufficiently known to those who make Mathematical Instruments : Now if it were required to find whether Water may be conveyed in Pipes , &c. to any Place assigned : to perform which observe these Rules . First at some 10 , 20 , 30 , 40 , 60 , or 100 yards distant from the Spring-head in a right-line towards the Place unto which your Water is to be conveyed . Place your Water-level , being prepared of two Station Staves with moveable Vanes on each of them , graduated also after the usual Manner : Cause your first Assistant to set up one of them at the Spring-Head ; Perpendicular unto the Horizon , and your second Assistant to erect another , as far from your Water-level towards the Place to which the Water is to be conveyed , as your Water-level is distant from the Spring-head : Now the Stationstaves in this order erected , and your Water-level placed precisely Horizontal , go unto the end of the Level , and looking through the sights , cause your first Assistant to move a Leaf of Paper , up or down your Station staff , untill through the sights you espy the very edge thereof , and then by some known sign or sound , intimate to your Assistant that the Paper is then in its true position , then let the first Assistant note against what Number of Feet , Inches , and parts of an Inch the edge of the Paper resteth ; which he must note down in a Paper . Then your Water-level remaining immoveable , go to the other end thereof , and looking through the sights towards your other Station-staff , cause your second Assistant to move a Leaf of Paper along the Staff , till you see the very edge thereof through the sights , and then cause him by some known sign or sound , to take notice what number of Feet , &c. are cut by the said Paper , which let him keep , as your first Assistant did . This done let your first Assistant bring his Station-staff from the Spring-head , and cause your second Assistant to take that Staff , and carry it forwards towards the Place , unto which the Water is to be conveyed ; some 30 , 40 , 60 , or 100 yards , and there to erect it Perpendicular as before , letting your second Assistant's staff stand immoveable , and your first Assistant to stand by it : Then in the Midway between your two Assistants , place your Water-level exactly Horizontal , and looking through the sights thereof , cause your first Assistant , and after that your second , to make their several observations in all respects as before . In this manner you must go along from the Spring-head , to the place unto which you would have the Water conveyed , and if there be never so many several Stations , you must in all of them observe this manner of work precisely ; so that by comparing the notes of your two Assistants together , you may easily know whether the Water may be conveyed from the Spring-head , or not , by calling your two Assistants together , and causing them to give in their notes of observation at each Station , which add together severally : Then if the Notes of the second Assistant , exceed the Notes of the first Assistant , take the lesser out of the greater , and the remainder will shew you how much the appointed Place , to which the Water is to be conveyed , is lower than the Spring-head . The first Assistant's Note . Station . Feet . Inch. Parts . 1 15 3 . 50 2 2 1 . 25 3 1 6 . 00 Sum 18 10 . 75 The second Assistant's Note . Station . Feet . Inch. Parts . 1 3 2 . 75 2 14 0 . 25 3 3 11 . 00 Sum 21 2 . 00 By these two Tables you may perceive that the Notes of the first Assistant collected at his several Stations , being added together , amounts unto 18 Feet , 10 Inches , and 75 / 100 or ¾ of an Inch : and the Notes of your second Assistant collected at his several Stations , amounts unto 21 Feet , 2 Inches : So that the number of the first Assistant's Observations , being taken from the second 's , there will remain 2 Feet , 3 Inches , and 25 / 100 or ¼ of an Inch. And so much is the place unto which the Water is to be brought , lower than the Spring-Head , according to the sleight Water-Level , and therefore the Water may easily be conveyed thither . And here observe these Notes . 1. In your Passage between the Spring head , and the appointed Place , from Station to Station , you must observe this order , that your first Assistant at every Station must stand between the Spring-head , and your Water-level : otherwise great Errours will ensue . 2. That if the Notes of your first Assistant , exceed the Notes of the second Assistant , then 't is impossible to bring the Water from that Spring-head unto the appointed place , but if their Notes are equal , it may be done , if the distance be but short . 3. That the most approved Authors concerning this particular do aver , that at every Mile's end there ought to be allowed 4½ Inches more than the Streight-level , for the current of the Water . 4. That if there be any Mountains lying in the way betwixt the Spring-head and the Place to which the Water is to be conveyed , you must then cut a Trench by the side of the Mountain , in which you must lay your Pipes equal with the streight Water-level , with the former allowance : and in case there be a Valley , you must then make a Trunk of strong wood , well under-propped with strong pieces of Timber , well Pitched , or Leaded , as is done in divers places between Ware and London . 5. That when the Spring will have too violent a Current , you must then convey your Water to the place assigned , by a Crooked or Winding line , and you also ought to lay the Pipes , the one up , and the other down , that thereby the Violence of the Current may be stopped . CHAP. X. Of MEASURING , Of Board , Glass , Tiling , Paving , Timber , Stone , and Irregular Solids , such as Geometry can give no Rule for the Measuring thereof . SECT . I. Of the Measuring of Board , Glass , Paving , Tiling , &c. I Have already in the fourth Chapter of this Book , and the second Section thereof , applyed Geometry to the finding out of the Superficial Content of all Regular Superficies . I have also in the ninth Chapter , and the third Section thereof , shewed how the Superficial Content of any Irregular Superficies may be found , by reducing them into Regular Forms : which I have explained amply in that Section , I shall therefore here be as plain and brief as is possible . PROP. I. To Measure a Piece of Board , Plank , Glass , &c. In Measuring of Board , Glass , &c. Carpenters and other Mechanicks measure by the Foot , 12 Inches unto the Foot ; so that a Foot of Board , or Glass , contains 144 Square Inches . Now if a Piece of Board , Plank , or Glass , be required to be measured , let it be either a Parallelogram , or Tapering Piece : first by the Rules aforegoing find the Content thereof in Inches , and that Product divide by 144 , the Quotient is the Content of that Superficies in Feet . PROP. II. To measure Tiling , Flooring , Roofing , and Partitioning-works . In Tiling , Flooring , Roofing , and Partitioningwork , Carpenters , and other Workmen , reckon by the Square , which is 10 Feet every way ; so that a Square containeth 100 Feet : Example . There is a Roof 14 Feet broad , what length thereof shall make a Square ? Divide 100 by 14 , it yields 7 1 / 7 Feet . Now if you have any Number of Feet given , and the Number of Squares therein contained are required , divide that Number by 100 , the product is Squares . PROP. III. To measure Paving , Plaistering , Wainscotting , and Painting-work . In Paving , Plaistering , Wainscotting , and Painting-work , Mechanicks reckon by the Yard Square , so each Yard is equal unto 9 Square Feet . By the Rules aforegoing find the Superficial Content of the Court , Alley , &c. in Feet : which divide by 9 , the Quotient is the Number of Yards in that work contained . SECT . II. Of the Measuring of Timber , Stone , and Irregular Solids . IN Superficial Measure a Superficial Foot contains 144 Square Inches ; but in Solid Measure a Foot contains 1728 Cubick Inches . Now having already in the fourth Chapter of this Book , and the third Section thereof , largely applyed Geometry unto the Measuring of all Regular Solids , I shall therefore in this Place be as brief as possible , only I shall be somewhat larger in the Mensuration of Irregular Solids , which is of special Moment in sundry parts of the Mathematical Practices . PROP. I. How to Measure any kind of Timber , or Stone , whether Three-square , Four-square , Many-square , Round , or of any other fashion , provided it be streight and equal all along . To perform which first by the Rules aforegoing in Chap. 4. § . 2. get the Superficial Content at the End , and then say , As 144 , the Inches of the Superficial Content of the End of a Cubick Foot , To a Cubick Foot containing 1000 parts ; So is the Superficial Content of the End of any piece of Timber , To the Solid Content of one Foot length of the said piece of Timber . According to which Mr. Phillips calculated the ensuing Table , which I have thought fit hereunto to annex . Case 2 Or the solid Content in Feet , &c. may be found otherwise thus . By the Rules aforegoing find the Content of the End of the piece of Timber in Inches , which Content multiply by the length of the said piece of Timber , or Stone in Inches , and that Product divide by 1728 , it produceth the Solid Content of that Piece of Timber , or Stone , in Feet , and parts of a Foot. A Table shewing the Solid Content of one Foot-length of any Piece of Timber , according to the Superficial Content at the End thereof . Feet . Parts . Feet . Parts . The Inches of the Content at the End. 1 0 007 The Inches of the Content at the End. 200 1 398 2 0 014 300 2 083 3 0 021 400 2 778 4 0 028 500 3 472 5 0 035 600 4 167 6 0 042 700 4 861 7 0 049 800 5 556 8 0 056 900 6 250 9 0 062 1000 6 944 10 0 069 2000 13 888 20 0 139 3000 20 833 30 0 208 4000 27 778 40 0 278 5000 34 722 50 0 347 6000 41 666 60 0 417 7000 48 711 70 0 485 8000 55 555 80 0 556 9000 62 500 90 0 625 10000 69 444 100 0 694 20000 138 888 PROP. II. To measure Round Timber which is Hollow : or any other Hollow Body . If Hollow Timber be to be measured , first measure the Stick as though it were not Hollow , then find the Solidity of the Concavity , as though it were Massie Timber , then substract this last found Content , out of the whole Content before found , the remainder is the Content of that Hollow Body . PROP. III. To Measure Tapering Timber , or Stone . Those Tapering Bodies are either Segments of Cones , or Pyramids : now the way to measure such bodies , is demonstrated in Prop. the 4. and 5. § . 3. Chap. 4 : But now to find the Content of these Segments do thus : measure the Solidity of the whole Cone , or Pyramid , and then find the Content of the Top part thereof cut off , ( as if it were a Cone , or Pyramid of it self ) and the Content thereof , deduct from the Content of the whole Cone , or Pyramid : so shall the remainder be the Content of the Segment required : which reduced into Feet gives the Solid Content of that Piece of Timber in Feet . Now to find the length of the Top part cut off , from the Cone , or Pyramid , say , As the Difference of the breadth of the two Ends , To the length between them : So is the breadth of the greater End , To the whole length of the Cone , or Pyramid . PROP. IV. How to find the Solid Content of any Solid Body , in any strange form , such as Geometry can given : no Rule for the measuring thereof . These strange forms are either Branches in Metal , Crowns , Cups , Bowles , Pots , Screws , or Twisted Ballisters , * or any other Irregular-Solid , that keep not in thickness one Quantity , but are thicker in one place , than in another , so that no man by Geometry , is possible to measure their Solidity . Now for the finding the Content of any such like Irregular Body in Inches or Feet , do thus : Cause to be made a Hollow Cube , or Parallelepipedon , so that you may measure it with an Inch-Rule without Difficulty , and so to know the true Content of the whole , or any part thereof at pleasure within the Concavity : Then take some other convenient Vessel , and put pure Spring-water therein ; then having filled the Vessel to a known Measure , make a Mark precisely round the very edge of the Water , then take the solid body and put it therein , then take out as much of the Water ( as by means of the body put therein ) is arisen above the Mark , untill the Water do justly touch at the Mark again : then put the Water taken forth into the Hollow Cube , and find the solid Content thereof ( being transformed into a Cubick Body ) in Feet , Inches , and parts of an Inch : Which Content is the just solidity of the Body put into the Water . ( Archimedes by this Proposition found the deceit of the Crown of Gold which Gelo the Son of Hiero had vowed unto his Gods : now the Workmen had mixed Silver with the Gold , which Theft was discovered by the great skill of Archimedes ) * And herein you must be very curious not to spill any of the Water , or take out of the Vessel , or put into the Hollow Cube , any more than the just quantity arisen above the Mark , for if you do it will produce infinite Errours , and thus may the Solidity of any Irregular Body be found . CHAP. XI . Of GAUGING . IN GAUGING there are two things chiefly necessary to be noted , yet both controverted . First , that seeing all manner of Casks , made to hold Liquor in , are for the most part the Trunk of a Sphereroid , cut off with two Circles , at Rightangles with the Base , and therefore Irregular , Therefore they must , first be reduced into a Regular Proportion . — And the second thing necessary to be noted , is to find the true quantity of an Ale , or Wine-Gallon in Cubick-Inches or parts of a Foot , that thereby the Content of the Vessel or Cask in Gallons may be known . SECT . I. Of Gauging any Beer , Ale , or Wine-Cask , also any manner of Brewers Tuns . PROP. I. To find the Solid Content in Inches of any Cask . I Shall follow Mr. Oughthred's method , which is , Take the Diameter of the Cask both at Head and Bung , by which find the Area's of their Circles , which done , then take two thirds of the Area of the Bung , and one third of the Area at the Head , which added together , shall be the Mean Area of the Cask ; which multiplyed into the length of the Vessel , it will shew how many solid Inches are contained therein . Example : Suppose the Diameter at the Head of a Vessel be 18 , and at the Bung 32 , and length is 40 Inches . Now I find the Aggregate of the two Circles to be 620 , and 989 , Cubick Inches : which multiplyed by 40 , the length , produceth 24839 , 56 / 100 Cubick Inches , for the whole Content of that Cask in Cubick Inches . PROP. II. To find the Content of a Vessel in Wine , or Ale Gallons . The Wine Gallon is established by the Consent of Artists , in these and other Nations , to contain 231 Cubick Inches * . Yet Dr. Wybard affirms it to be somewhat less , to wit 225 , at most : The Ale Gallon contains 282 Cubick Inches , according to the Establishment of Excise . Herein Artists differ somewhat in their Experiments . Now having already shewed how to find the Content in Inches of any Cask , I now come to shew how to find the Content in Gallons , of any Beer , Ale , or Wine Cask , which is thus : Divide the Number of Inches given by 231 , for Wine Measure , and 282 , for Ale Measure . In the former Example I find the said Cask to contain 107 , 53 Wine Gallons , and 88 , 8 , &c. Gallons in Ale Measure . PROP. III. How to Gauge or Measure Brewers Tuns , &c. Those Tuns are most commonly Segments of Cones or Pyramid , whose Basis is either a Square Parallelogram , Circle , or Oval ; to measure which , let their form be what it will you must do thus . By the former Rules of Measuring such Segments or Bodies , you must find their Solid Content in Cubick Inches , ( as in prop. 3. § . 2. chap. 10. ) which Content divide by 282 Inches , ( the Inches in one Gallon ) it sheweth the Content in Gallons , and dividing the Gallons by 36 , ( the number of Gallons in a Barrel ) it shews the Content in Barrels . SECT . II. Of Gauging or Measuring , and the Moulding of Ships . PROP. I. To Gauge a Ship , thereby to find how many Tuns her burthen is . IN the Gauging or Measuring of Ships , Naupegers , or Ship-Wrights , observe these three Particular Rules : First , that if you measure the Ship within , you shall find the Content , or the Burthen the Ship will hold or take in . Secondly , if the Ship be measured on the outside , to her light mark as she swims being unladen , you shall have the Content of the Empty Ship. Thirdly , but if you measure from the light mark , to her full draught of Water being laden , you shall have the true Burthen of the Ship. Now to find the Content of the King 's Royal Ships : Measure the length of the Keel , the breadth of the Mid ship Beam , and the depth of the Hold : which three multiply into one another , and divide their Product by 100 ; so shall you find how many Tuns her Burthen is . But for Merchant's Ships , which give no allowance for Ordnance , Masts , Sails , Cables , Anchors , &c. which are all a Burthen , but no Tonnage , you must divide the product by 95 , so shall their true Burthen be found . PROP. II. By knowing the Measure of a Ship , of one Burthen , to make another Ship , of the same Mould , which shall be double , or triple , or in any proportion , either more or less than the said Ship. First you shall multiply the Keel Cubically ; and in like manner every Beam ; the Mid ship Beams multiply them Cubically ; and also the Reaking of the Ship , both at Stem , and Stem-Post , multiply them Cubically ; likewise the principal Timbers , that doth mould the Ship , multiply them Cubically ; and the depth of the Hold , multiply it Cubically ; and so consequently every Place , or Places , which doth lead any work , multiply them Cubically ; then if it be required to have a Ship as big again , or thrice as big ; double , or triple each respective Cubical number ; then by prop. 9. § . 1. chap. 1 : Or by prop. 4. § . 2. chap. 2. find the Cuberoots thereunto belonging ; then according unto these respective Numbers , make your Keel , your Timbers , Beams , &c. which being done , you shall make a Ship of the Mould and Proportion desired . CHAP. XII . Of DIALLING . HOROLOGIOGRAPHIA , or the Art of DIALLING , is an Art Mathematical , which demonstrateth the precise Distinction of Times , by the Sun , Moon and Stars , whereby the Time of the Day , or Night , may be known * . Now the Demonstrative delineation of Dials , consisteth chiefly in the finding out the Hour-lines , and their true distance one from the other : which lines are great Circles of a Sphere , which being projected on a plain Superficies , become streight-lines ; which lines do continually vary , according as the Planes on which they are described , or projected , do lie situated in respect of the Horizon of the Place . Now a Dial may be made on any Plain Superficies , for all Plain Superficies are Posited either Perpendicular , Parallel , or Oblique , to the Horizon of the Place , in which the Plane is seated . In the delineation of all which Dials in this Chapter described , ( which are the most Eminent , and usefull Dials now used ) I have used this Method : First , I have shewed how to delineate them by Geometrical Projection , by Scale , and Compass only : and secondly how they may be described by Arithmetical Calculation , of both which I have been very plain and large . SECT . I. Of the Delineation and Projection of sundry most usefull Dials . PROP. I. How to draw the Hour-lines on an Equinoctial Plain . AN Equinoctial Plane , is such which lieth Parallel unto the Equinoctial , and is an Horizontal Plane , under the Pole. This is the first and plainest kind of Dials , and is made after this manner : First describe the Circle AE , W , E , R , for your Planes , then Cross it with the two Diameters EW , and AER . Then divide the Semicircle E , W , R , into 12 equal parts in the points ☉ , ☉ , ☉ , &c. Then from the Center Q , and through the said points draw streight lines , which shall be the true Hour-lines belonging unto this Equinoctial Plane . Now because these Planes are capable of receiving all the Hour-lines from Sun-rising unto the Sun-setting , in Summer ; therefore the Hour-lines of 4 , and 5 , in the Morning ; and 7 , and 8 , in the Evening ; must be delineated as you see done in the Figure : These Hours may be sub-divided into half Hours , and Quarters : The Stile of this Dial , must be a streight Pin , or Wyre set Perpendicular , to the Plain , on the Center Q. and of any convenient length . This Dial may be made for any Latitude , and is of good use for Seamen , and others . PROP. II. How to draw the Hour-lines on a Polar Plane . A Polar Plane is one that lies Parallel unto the Pole , and under the Equinoctial is an Horizontal Dial : the way to make this Dial is thus . First draw the line AB , for the Horizontal line of the Plane ; and cross it at the Middle at right angles , with the line 12 , Q , 12 , which is the Meridian or Hour line of 12 ; Then upon the line 12 , Q 12 , either above or below the point Q , assume any point as S , then setting one foot of your Compasses in S , describe the Semicircle CED , which divide into 12 Equal parts , in the points ☉ , ☉ , ☉ , &c. Then lay a Ruler unto S , and unto the several points ☉ , ☉ , ☉ , &c. and it will cross the line AB , in the points x , x , x , &c. Then through those points draw ( by prop. 4. § . 1. chap. 4. ) right lines all Parallel unto 12 Q 12 , and so is your Dial finished . Then according unto the breadth of the Plane , you may proportion your Stile , * Whose height must be equal to the distance between the two Hour-lines 12 , and 9 , or 12 , and 3 , and then will the shadow of the upper edge thereof shew the Hour of the day : The height of the Stile , is also found thus . As the Tangent of the Hour-line 4 or 5 , To the Distance hereof from the Meridian . So is the Radius , To the Height of the Stile . Then for the other Hour-line , say , As the Radius , To the Height of the Stile . So is the Tangent of any other Hour-line , To the Distance thereof from the Meridian line . PROP. III. How to draw the Hour-lines on a Meridian Plane , which is an East , or West Dial. A Meridian Plane stands upright directly in the Meridian , and hath two Faces , one towards the East , and the other towards the West . Now admit it be required to make a direct East Dial , in the Latitude of 51° 32 ' : let A , B , C , D , be a Dial-plane , on which you would describe a Direct East Dial , on the point D , describe an obscure Arch HG , with the Radius of ●our line of Chords , then take 38° 28 ' , the Complement of your Latitude , place it from G to L ; then draw DL quite through the Plane ; Then to proportion your Stile unto your Plane , so that all the Hours may be placed thereon , from Sun-rising to 11 a Clock . Assume two points in the line LD , as K , for 11 ; and I for the 6 a Clock Hour lines ; then draw 6 , 16 , and 11 , K 11 , Perpendiculur to LD . This done , with the Radius of your line of Chords on L , strike the Arch OP , and from P , to O , place 15° 00 ' ; and draw OK , to cut 6 I 6 , in M , so shall IM be the height of the Stile proportioned unto this Plane ; which may be a Plate of Brass , whose breadth must be equal to the distance between the Hour-lines of 6 , and 9 , which must be placed Perpendicular to the Plane , on the line 6 , I 6 , whose shadow of the upper edge , shall shew the Hour of the day . Now to draw the Hour-lines , with the Radius of your line of Chords , on M strike the Arch QN , which divide into 5 equal parts in the points ● , ● , ● , &c. Then lay a Ruler from M unto each of those points , and it will cut the line JK in the points * , * , * , &c. through which points ( by prop. 4. § 1. chap. 4. ) draw Parallels to 6 I 6 , as the lines 77 , 88 , &c. which shall be the true Hour-lines of an East Plane , from 6 in the Morning , till 11 before Noon . Then for the Hour-lines of 4 , and 5 , you must prick off 5 as far from 6 , as 6 is from 7 ; and 4 , as far as 6 is from 8 ; and draw the Hour-lines 55 , and 44 , as before . Thus is your Dial compleated , and in the forming of which , you have made both an East , and a West Dial ; which is the same in all respects , only whereas the Arch H G , through which the Equinoctial passed in the East Dial , was described on the right hand of the Plane , in the West it must be drawn on the left hand , and the Hour-lines 4 , 5 , 6 , 7 , 8 , 9 , 10 , and 11 , in the Forenoon in the East Dial , must be 8 , 7 , 6 , 5 , 4 , 3 , 2 , and 1 , in the West in the Afternoon ; as in the Figure plainly appeareth : Now you may find the distance of the Hour-lines from the Substile , by this Analogy or Proportion . As the Radius , To the Height of the Stile . So is the Tangent of any Hours distance from 6 , To the distance thereof from the Substile . PROP. IV. How to draw the Hour-lines on a direct South , and North Plane , This Plane or Dial must stand upright , having his face or Plane , if it be a South Dial , directly opposite unto the South ; but if a North Plane , directly opposite unto the North ; now admit it be required to make a Direct South Dial , for the Latitude of 51° 32 ' : To make which first describe the Circle ABCD , to represent an E●ect direct South Plane , cross it with the Diameters CB , and AD , then out of your Line of Chords take 38° 28 ' , the Complement of the Latitude , and set it from A , unto a , and from B , unto b , Then lay a Ruler from C unto a , and it will cut the Meridian ARD , in P , the Poles of Plate V Page 261 As Radius or S. 90° , To Sc. of the Latitude . So is T. of the Hour from Noon , To T. of the Hour-line from the Meridian . PROP. V. How to draw the Hour-lines on an Horizontal Plane . This Horizontal Plane , or Dial , is one of the best and most usefull Dials in our Oblique Hemisphere : Admit it be required to make an Horizontal Dial , for the Latitude of 51° 32 ' : To make which , first describe the Circle AB CD , which representeth your Horizontal Plane , Then cross it with the two Diameters ARC , and BRD , Then take 51° 32 ' out of your Line of Chords , and set it from B , to a , and from C , to b , Then lay a Ruler from A , unto a , and it will cut the Meridian BD , in P , the Pole of the World , Then lay a Ruler from A , unto b , and it will cut ABD the Meridian , in the point AE , where the Equinoctial cutteth the Meridian , then through the three points A , AE , and C , draw the Equinoctial Circle , whose Center is at H ; ( and found as in the former proposition ) Then divide the Semicircle ADC into 12 equal parts , in the points ● , ● , ● , &c. Then lay a Ruler to R the Center of the Plane , and on those points , so shall the Equinoctial Circle AAeC , be by it divided into 12 unequal parts in the points * , * , * , * , &c. Then a Ruler laid unto P the Pole of the World , and those Points , shall cut the Semicircle CDA in those Points I , I , I , &c. Lastly , from the Center R , and through those Points , let there be drawn right lines , which shall be the true Hour-lines of such an Horizontal Plane , from 6 in the Morning , untill 6 at Night ; but for the Hours of 4 and 5 in the Morning ; and 7 and 8 in the Evening ; they are delineated by producing 4 and 5 in the Evening , through the Center R , and 7 and 8 in the Morning ; extending them out , unto the other side of the Plane , so shall you have those Hour-lines also on your Plane delineated as you see in the Figure . The Stile of this Plane may be a thin Plate of Brass , cut exactly unto the Quantity of an Angle of 51° 32 ' , and set Perpendicular on the Meridian line , for the forming of this Stile take out of your Line of Chords 51° 32 ' , and set it from D , unto e , and draw Re , which shall be the Axis of the Stile , you may also prefix the Halves , and Quarters of Hours , in the very same manner as the Hours themselves were drawn . Now to find out the distance of the Hour-lines from the Meridian , say , As the Radius or S. 90° , To the S. of the Latitude . So is the T. of the Hour from Noon , To the T. of the Hour-line , from the Meridian Line . These kinds of Dials being so frequently used with us , in this Oblique Sphere , for the help of the speedy delineating of them , I have annexed hereunto the Table of Longomontanus , wherein the Hour-lines , for many Latitudes , are calculated . A Table shewing the Distance of the Hour-lines from the Meridian , in these Degrees of Latitude . An Horizontal Dial , Latitude . The Hours from the Meridian . A South Erect Dial , Latitude . xi . i. x. ii . ix . iii viii . iv . vii v. vi . D M D M D M D M D M D M 30 7 38 16 6 26 34 40 54 61 49 90 00 60 31 7 51 16 34 27 14 41 42 62 28 90 00 59 32 8 4 17 1 27 53 42 30 63 6 90 00 58 33 8 17 17 27 28 34 43 17 63 45 90 00 57 34 8 30 17 54 29 13 44 5 64 42 90 00 56 35 8 43 18 20 29 49 44 46 64 56 90 00 55 36 8 56 18 45 30 25 45 28 65 27 90 00 54 37 9 9 19 9 31 1 46 9 65 58 90 00 53 38 9 21 19 34 31 37 46 50 66 29 90 00 52 39 9 33 19 57 32 9 47 26 66 55 90 00 51 40 9 46 20 20 32 40 48 1 67 20 90 00 50 41 9 58 20 43 33 14 48 37 67 45 90 00 49 42 10 10 21 7 33 47 49 13 68 11 90 00 48 43 10 22 21 29 34 17 49 44 68 32 90 00 47 44 10 24 21 50 34 46 50 14 68 52 90 00 46 45 10 43 22 12 35 15 50 45 69 14 90 00 45 46 10 54 22 33 35 44 51 16 69 37 90 00 44 47 11 5 22 33 36 10 51 43 69 53 90 00 43 48 11 16 23 12 36 35 52 9 70 10 90 00 42 49 11 26 23 32 37 1 52 35 70 28 90 00 41 50 11 36 23 51 37 27 53 1 70 43 90 00 40 51 11 46 24 9 37 50 53 24 70 58 90 00 39 52 11 56 24 26 38 13 53 46 71 12 90 00 38 53 12 5 24 44 38 36 54 8 71 27 90 00 37 54 12 14 25 2 38 59 54 30 71 41 90 00 36 55 12 23 25 18 39 18 54 50 71 53 90 00 35 56 12 32 25 33 39 38 55 9 72 4 90 00 34 57 12 46 25 49 39 58 55 28 72 16 90 00 33 58 12 48 26 5 40 18 55 46 72 27 90 00 32 59 13 56 26 19 40 36 56 1 72 38 90 00 31 60 13 58 26 30 40 53 56 15 72 47 90 00 30 PROP. VI. How to draw the Hour-lines , on an Erect declining Plane . These Planes are made to set on the sides of Houses , wherein the Meridian is always a Perpendicular , drawn on the Plane , in whose top is the Center , where the Substile , and the Hour-lines all meet . Now before we can delineate the Hour-lines on any such Planes , two things must be given : As the Latitude of the Place , and the Planes Declination ; by having which we must find these three things : viz. The Poles height above the Plane . The distance of the substile from the Meridian . And the Plane's difference of Longitude . For the finding of which Requisites , by Geometrical Projection , we describe on the Dial Plane , these Circles of the Sphere , viz. The Horizon , Meridian , and Equinoctial , which being described in their true Position , on the Plane , we proceed thus . Admit it be required to make a Direct South Dial , on an Erect , Direct South Plane , Declining Westward 24° 20 ' , in the Latitude of 51° 32 ' . Now in order to find the requisites before mentioned , describe the Circle ZHNO , and cross it with the two Diameters ZQN , and H QO : now Z is the Zenith , N the Nadir , ZQN the Hour-line of 12 , HQO the Horizon . Now seeing the Plane declines S. W. 34° 20 ' : make Na , and Ob , each equal to 34 20 : Then a Ruler layed from Z , to a , will cut the Horizon in S , the South point of the Horizon , through which draw the Meridian ZSN , whose Center is at Y , found as in the fourth Proposition aforegoing : Then a Ruler laid from Z to b , will cut the Horizon in W , the West point thereof . Now the Horizon and the Meridian being projected on the Plane , take out of your line of Chords 51° 32 ' , which place from H , unto c , and from N , unto d ; then lay a Ruler from W , unto c , and it cutteth the Meridian in P , the Pole of the World. Then through P and Q , draw the line PQD , which representeth the Axis of the World , and the Substilar line of the Dial , then lay a Ruler from W , to d , it cutteth the Meridian in AE , so is W AE two points through which the Equinoctial must pass , whose Center is found as afore to be at M , ( being always in the Axis of the World ) so have you on your Plane the Horizon HQO , the Meridian ZPSAe N , and the Equinoctial LAeKWG , described on the Plane as required . Now first to find the Poles height above the Plane , which in this Scheme is represented by BP , Lay a Ruler from G , unto P , and it shall cut the Plane in V , then measure the distance BV , on your line of Chords , and you will find it to contain 34° 33 ' , which is the Poles height above the Plane . Secondly , To find the distance of the Substile from the Meridian represented in the Scheme by the Arch ZB , or ND , which measured as afore will appear to be 18° 08 ' , the distance of the Substile from the Meridian . Thirdly , To find the Plane's Difference of Longitude , which in the Scheme is represented by the Angle AEPK , lay a Ruler from P , unto AE , and it cutteth the Plane in X , then measure the Arch DX , as afore , and so will you find the Planes Difference of Longitude , to be 30° 00 ' : Thus by Geometrical Projection have we found all the three Requisites : Now to find them by Arithmetical Calculation observe these Analogies or Proportions . 1. For the Poles height above the Plane , say , As Radius or S. 90° , To Sc. of the Latitude 38° 28 ' . So is Sc. of the Declination 65° 40 ' , To S. of the Poles height above the Plane 34° 33 ' . 2. For the Distance of the Substile , from the Meridian , say , As the Radius or S. 90° 00 ' , To the S. of the Plane's Declination 24° 20 ' . So is Tc. of the Latitude 38° 28 ' , To the T. of the Substilar Distance from the Meridian 18° 10 ' . 3. For the Plane's Difference of Longitude , say , As the Sc. of the Latitude 38° 28 ' , To the Radius or S. 90° 00 ' . So is S. of the Substilar Distance 18° 10 ' , To the S. of the Difference of Longitude 30 Deg. Or , it may be found thus . As the S. of the Latitude , To the Radius . So is the T. of the Declination , To the T. of the Difference of Longitude required . These things found , we come now to shew how the Hour-lines may be projected . To project which observe , First , to lay a Ruler from P the Pole of the World , to AE the Intersection of the Equinoctial with the Meridian , and it will cut the Plane in x , where begin to divide the Semicircle L x G , into 12 Equal parts in the Points ● , ● , ● , ● , &c. Then lay a Ruler from Q , to every of those parts , and it shall cut the Equinoctial ; and divide it into 12 unequal parts , in the points * , * , * , * , &c. Then a Ruler laid from P the Pole of the World unto each of these points , it will divide the Plane into 12 unequal parts in the Points I , I , I , I , &c. Then by a Ruler laid from the Center Q , to those points , draw right lines , which shall be the true Hour-lines proper unto such a Declining Plane , as you see plainly demonstrated by the Scheme . Now the Substilar line falleth in this Dial , just on the Hour-line of 2 , in the Afternoon , because the Plane declineth Westerly . The Angle of the Stile is DQR 34° 33 ' . which may be either a Plate or Wyre , brought into such an Angle , which must be placed Perpendicular to the Plane , and directly over the Substilar line QD 2. Now the distance of the Hour-lines , from the Substilar line , may also be found by this Analogy or Proportion . As the Radius , To the S. height of the Pole above the Plane . So is the T. of the Hour-line from the Meridian of the Plane , To the T. of the Hour-line from the Substile . Thus have you compleated your Dial ; as you see in the Scheme , and here you may take notice that having finished a West Decliner , you have also made an East Decliner ; if you only convert the Hour-lines of the West Decliner , in such manner as you see in Fig. 72. on the East Decliner , and compleat all as you see in that Scheme . Thus I have explained the making and delineating of the best and most usefull Dials both by Geometrical Projection , and also by Arithmetical Calculations , in as brief and compendious a manner as possible . There are sundry other kind of Dials , as Incliners , Decliners , and Recliners , which being not so usefull , for brevity sake , they are here omitted : As for Instrumental Dials , as Quadrants , Rings , Cylinders , &c. Which depend on the Sun's height , I refer you to Mr. Edm. Gunter's Book , wherein they are largely described . As for the Beautifying and Adorning of those Dials , &c. by describing on them the Equinoctial , Tropicks , Parallels of Declination , Parallels of the Sun's Place , Length of Days , the Sun 's Rising and Setting , Jewish , Italian , and Babylonish Hours , Almicanthars , Azimuths , Circles of Position , the Signs Right Ascending , Descending , Culminating , &c. I do advise you to consult Mr. Gunter , Mr. Foster , Mr. Wells , and Mr. Holwel's Works , all which Authors have very learnedly shewed the describing of them , by several large Schemes , and Figures , for the plainer Illustration thereof . Now seeing the Latitude of a Place must be first known , before a Dial can be made to it , Plate VI P 〈…〉 A Table of the Names and Latitudes of all the Principal Cities , Towns , and Islands , in and about Great Britain and Ireland . ENGLAND . D. M. ARundel 51 00 Bedford 52 15 Barwick 55 54 Bristol 51 35 Buckingham 52 10 Cambridge 52 20 Canterbury 51 25 Carlisle 55 20 Chichester 50 48 Chester 53 18 Colchester 52 08 Dover 51 20 Derby 53 00 Dorchester 50 50 Durham 54 56 Exeter 50 48 D. M. Falmouth 55 22 Glocester 51 57 Guilford 51 12 Hartford 51 54 Hereford 52 17 Huntington 52 30 Ipswich 52 20 London 51 30 Lincoln 53 20 Leicester 52 45 Lancaster 54 15 Northampton 52 24 Norwich 52 45 Nottingham 53 00 Newcastle 55 12 Oxford 51 50 Portsmouth 51 08 Plimouth 50 36 Reding 51 40 D. M. Salsbury 51 12 Stafford 52 50 Stanford 54 44 Shrewsbury 52 50 Truero 50 30 Winchester 51 03 Worcester 52 25 Warwick 52 30 York 54 00 WALES . D. M. ANglesey 53 28 Barmonth 52 50 Brecknock 52 01 Cardigan 52 12 Caermarthen 51 56 Carnarvan 53 16 Denbigh 53 13 Flint 53 17 Landaffe 51 35 Monmouth 51 51 Montgomery 51 56 Pembroke 51 46 Radnor 52 19 St. David 52 00 SCOTLAND . D. M. ABerdeen 57 30 Dunblain 56 21 Dunkel 56 48 Edenbrough 56 00 Glascow 55 58 Kinsaile 57 44 Orkney 60 06 D. M. St. Andrews 56 40 Skyrassin 58 38 Sterling 56 12 IRELAND . D. M. ANtrim 54 38 Arglas 54 10 Armagh 54 14 Carterlagh 52 41 Clare 52 34 Corke 51 55 Droghedagh 53 58 Dublin 53 55 Dundalke 53 52 Galloway 53 02 Kenney 52 30 Kildare 53 00 Kings Town 53 08 Knockfergns 54 40 Kynsale 51 41 Lymerick 52 30 Queens Town 52 52 Waterford 52 09 Wexford 52 18 Youhall 51 53 ISLANDS . D. M. WIght 50 48 Portland 50 30 Man 54 24 Limdey 51 22 Jerzey 49 12 Garnzey 49 〈◊〉 CHAP. XIII . Of FORTIFICATION . THE Utility of this Mathematical Art called Fortification , or Military Architecture , is so well known , that it needs not my commendation , and therefore to speak any thing thereto , were but to light a Candle before the Sun. In the handling of this part of the Mathematicks , I shall be as brief as possible , yet as plain as can be desired : In the prosecution of which , I shall use this Method . As First , I shall give you the most principal Definitions or Terms belonging to this Art. Secondly , I shall prescribe the most conducing Maxims or Rules herein observed . Thirdly , I shall shew how to delineate the Ground-line of any Fortification , according to the several Proportions , used by the best and most experienc'd Inginiers of Italy , France , Holland and England ; Fourthly , I shall describe the Construction of the chief and principal Out-works now in use ; and Lastly , lay down some general Maxims or Rules , by most Modern Authors observed in Irregular Fortifications . SECT . I. Of the Definitions of the Lines , and Angles , belonging to the Principal Ground work of any Regular Fortification . 1. THE Exterior or outward Line , which boundeth the Rampart , at the Foot next the Ditch , is the principal and only Line to be regarded 〈…〉 all Regular , or Irregular Fortifications , being the Basis on which all the other Lines , and parts of the Fortification doth depend . 2. The Exterior Polygon , is the outward side of any Regular Figure , as in the Hexagon ( which Figure I shall make use of through this Tract ) the side AA , is the Exterior Polygon . 3. The Interior Polygon , is the inward side of any Regular Figure , as in the Hexagon is noted by any of the sides between P and P. 4. The Bastion or Bulwork , is that great work of any Fort , that advanceth its self towards the Campaigne , and here are six all marked with B , the lines which terminate them , are two Gorges , two Flanks , and two Faces . 5. The Demi-Gorge or Gorge-line PC , is half the Entrance into the Bastion , and terminates the point C , whereby the Flank shall be raised . 6 The Flank is another Out-line of the Bastion as CF , which terminateth the Curtain , and Face . 7. The Face is the utmost line of the Bastion , as FA , two lines thereof doth form the Angle of the Bastion A , or the Flanked Angle . 8. The line forming the Flank FF , is a pricked line , made use of by the Dutch Inginiers , and others . 9. The Capital is AP , part of the line coming from the Center ☉ , terminated at the point of the Bastion A. 10. The Curtain is that part of the Interior Polygon CC , which lieth betwixt the two Bastions B , and B. 11. The line of Defence is AC , passing from A , the point of the Bastion , to C the Angle of the Flank , and Curtain , and ought never to exceed 800 English Feet * . 12. The line Stringent , is the line coming from the point of the Bastion A , and prolonged on the Face AF , to the Curtain D , which sheweth that DC , the part of the Curtain , ( by some called the second Flank will scour the Face . 13. The Diameter of the Interior Polygon , is the line ☉ P , coming from the Center thereof ☉ . 14. The shortest line from the Center unto the Curtain , is ☉ m. These are the Definitions of the principal lines , appertaining to the Ground-work of any Regular Fortification , the Angles followeth . 15. The Angle of the Center of the Polygon is P ☉ P 16. The Angle of the Polygon PPP , is always the Complement of the Angle at the Center , or remainer unto 180 Degrees . 17. The Angle of the Triangle PPO is always the one half of the Angle of the Polygon PPP . 18. The Angle of the Bastion , or the Flanked Angle FAF , is exposed unto the Batteries of the Besiegers , and formed by the two Faces , FA , and FA , which ought never to be less than 60 , nor much above 100 Degrees . 19. The Angle of the Espaule , or Shoulder , is formed by the Face , and Flank , as AFC . 20. The Angle of the Flank CCF , is formed by the Curtain , and the Flank , and is most commonly a Right Angle , but by some later Inginiers , is made Obtuse , or more than a Right Angle , or 90 Degrees . 21. The Angle made by the two lines Sitchant , At A is called the Angle of the Tenaile . 22. The Angle forming the Flank , is CPF , which Angle is made use of by most of the Dutch Inginiers . SECT . II. Of General Maxims or Rules observed in Fortifications . 1. THat all the parts of the Place , be of Cannon Proof flanked , i. e. defended from another place , which place is no farther distant than the reach of a Musket-shoot , from the place to be Flanked or defended . * 2. That in all the Place , there may be no part of the Wall , or outside of the Rampire , that is not seen from the top to the bottome of the Mote , or Ditch . 3. That the Bastions are large , and full of Earth , and not empty ; the bigger they are , they are the more to be esteemed , there being the more room to intrench , in case of necessity : whose Gorge let be at least 35 fadoms , and their Flank at least 18 fadoms . 4. That the Angle of the Bastion , or Flanked Angle , be not much above 90 , nor much less than 60 Degrees , for in the former it would lie too very Obtuse , and open , at the Point ; and in the latter it would be too slender , and so easily to be battered down , by the Enemies Cannon . 5. That the Angle of the Flank may be somewhat Obtuse ; neither is there any more virtue in a Right-angle , than in any other , for the defence of the Fort. 6. That the length of an extended Curtain be not above 135 Fadoms , nor the single above 80 Fadoms , nor be less than 40 Fadoms , to be well defended from two Flanks . 7. That the Rampire be so wide , that so a Parapet of Earth Cannon proof may be erected thereon , and a Teraplane left , full wide for the Ordnance to be recoiled . 8. That the Mote or Ditch be at least 20 Fadoms broad , and as deep as possible . Now dry Motes in great Cities are to be preferred before others , that are full of Water , to facilitate the Sallies , the relief , and retreat of the Besieged ; and in small Fortifications the Motes full of Water are the most Esteemable , because in such Sallies are not necessary , and Surprises are very much to be feared . 9. that the Parts that are most remote from the Center , be commanded by those which are nearest to it . 10. That the Defence of a Face is much stronger , when the Angle made by the Face , and Exterior Polygon is a great Angle ; this Maxim is so very essential , that it will try the goodness of any Fortification whatsoever : Thus I have described the 10 chiefest Maxims , necessary for good Fortifications . SECT . III. Of the Construction and making of the principal Ground-line of a Fort , according to the most Modern ways , used by the Italian , Dutch , French , or English Inginiers . I. Of the Italian Fortifications . GEnnaro Maria , Mathematician to the Catholick King , wrote at Florence , his Elements of Military Architecture entituled , Breve Trattato delle Moderne Fortificazioni . This Italian Author was a very Learned and Skilfull Mathematician , and famous in his Nation . In his said Book Printed 1665 , he makes the Interior Polygon 800 , and not less than 600 Feet , his Demi-Gorge , he makes ⅛ of it , and so for the Flank of the Quadrangle . But for the Pentagon , and all Figures above , he makes the Flanks 1 / 10 part of the Gorge more , and he placeth his Flank at Right Angles with the Curtain . Supposing his Interior Polygon 1000 parts , his Gorges will be 125 , and in the Quadrangle the Flanks will be 125 , but of the Pentagon , and all above , 138 parts . For the Faces , he makes them to fall on the third part of the Curtain , unless in the Square , which he allows no second Flank . PROP. I. To fortifie a Hexagon according to this Author's Proportion . First describe the Hexagon PPP , &c , then divide the Interior Polygon PP , into 1000 equal parts , take 125 for the Gorges , and set it from P to C. Then on C raise a Perpendicular , make it equal to 138 parts , for your Flanks CF , then draw the Face AF , falling on the third part of the Curtain CC , at D , and so do on every Bastion , untill the work is compleated . II. Of the French Fortifications . Monsieur De la Mont , in his Fortifications Offensive , and Defensive , printed 1671 : And Monsieur Manesson Mallet in his late work , intituled Travaux de Mars , printed 1672 , assigneth these proportions for the laying down the Ground-line of a Fort. Both these Authors make the Interior Polygon 768 English Feet , which they divide into 5 parts , and taking one for the Gorge 153½ Feet . Both divides it into 3 parts , and takes one for the Capital , that is 256 Feet . Now our first Author De la Mont , makes the Flank to stand at Right-angles and takes 115½ Feet for it , which is ¼ of the Curtain , and so draws the Bastions , in all save the Quadrangle , and Pentagon , which he makes to have no second Flank . PROP. II. To fortifie a Hexagon according to the Proportion of De la Mont. First describe your Hexagon P , P , P , &c. Now supposing your Interior Polygon PP , 1000 parts , the Capital 333 , the Gorge 200 , and the Flank 150 parts , take out of your Triangular Scale Fig. 75 , ( which is made for the more speedy delineation according to this proportion of De la Mont ) PA for the Capital , and prick it off from PA , on all the Bastions . Then take PC , and prick off all the Gorges from P to C. Then take FC and prick it off at Right Angles , from C to F. Lastly draw all the Faces AF , AF , &c. so is your Hexagon compleat , as required . PROP. III. To fortifie a Hexagon according to Manesson Mallet's Proportion . Now our Authour Monsieur Manesson Mallet , in his Works intituled Travaux de Mars , deviates from our former Authour , only in this : that as De la Mont did place his Flanks at Right Angles , he places them at 98 Degrees with the Curtains , and leaves no second Flank in all his Fortifications . Therefore having described the Polygon PP , &c. divide PP into 1000 parts , prick off the Capitals PA 333 , and the Gorges PC 200 , then lay off the Flanks CF , 150 parts , at an Angle of 98 deg . with the Curtain CC ( by prop 5. § . 1. chap. 4. ) and draw all the Faces , AF , AF , &c. Falling on C the point of the Flank and Curtain , so shall your Hexagon be fortified as was required . III. Of the Dutch Fortifications . The Emperour Ferdinand III. hath learnedly altered the Method of Fritach , Dogen , Goldman , and Faulhaberus , all which were Dutch Inginiers , and wrote large Volumes on this Subject ; in his Works intituled Amussis Ferdinandea , published 1654 ; by turning their way of working by Angles , into working by Sides . Thus he setteth down a Catholick way of delineating the Sides , or Lines of any Fort by his 60 prop. thus , the Interior Polygon to be 66 , the Capital 24 , the Gorge 15 , and the Flank 12. Or in making the Interior Polygon 22 , the Capital 8 , the Gorge 5 , and the Flank 4. Or yet making the Interior Polygon 1000 , the Capital 363 , the Gorge 227 , and the Flank 181 , this is an Epitome of all the Dutch Fortifications , and is general excepting for the Square , which must have no second Flank . PROP. IV. To fortifie a Hexagon according to the Emperour's Proportion . First describe the Polygon PPP , &c. divide P P , &c. into 22 parts , take 8 for the Capitals PA , which prick off all round from P to A , take 5 for the Gorges ; which prick off all round from C to P , then take 4 for the Flank CF , which prick off all round at Right-angles from C to F , lastly draw the Faces AF , AF , AF , &c. So is the Hexagon compleated as was required . IV. Of the English Fortifications . His late Majesty of Great Britain Carolus II. of ever blessed Memory , hath much facilitated the Method of Count Pagan , who in his Fortifications printed at Paris 1645 , did place the Flanks at Right-angles with the Line of Defence , and he works by the Exterior Polygon . Now His Majesty places the Flank , at Right-angles with the line of defence of the Interior Polygon , and works after another manner : Count Pagan makes the proportion of the Grand Royal Fort. Supposing the Exterior Polygon to be 1000 parts , will make the Perpendicular MT to be 150 , and the Complement of the line of Defence TC to be 185 , which may serve for a general proportion be the length what it will , only in a Square the proportions must thus be altered in the Grand Royal Fort , the Perpendicular MT must be 162 , in the Mean R 144 , and in the Petty Royal 126 , the Complement of the Line of Defence for the Grand Royal Fort is 228 , and for the Mean Royal Fort 198 , also for the Petty Royal Fort 198. PROP. V. How to fortifie a Hexagon according to Count Pagan's Proportion . To delineate this Work draw a line , about the middle whereof as at M , set off MA , the half of the Exterior Polygon 500 parts , which makes the Exterior Polygon 1000 , then on M ( by prop. 1. § . 1. chap. 4. ) raise the Perpendicular Mm , which make Mt , MT Equal to 150 , then draw ATC , and ATC , then take 185 , and place it from T to C , and to C , and draw CC for the Curtain , then on the points C raise Perpendiculars CF , to the line of defence CA , for the Flanks , so have you also the Faces FA. Then on the Points A set off half the Angle of the Figure , to wit 60° ( as you see in the Table in page 38 ) and draw the lines OA and O A , so shall O be the Center of the Figure , and PC the Gorge , and AP the Capitals : then finish each Bastion at your own discretion , and the Work is finished as required . PROP. VI. To fortifie a Hexagon according to the way prescribed by His Majesty Carolus II. His late Majesty C. II. hath much facilitated this Work , as will appear in this following Example , by making the line of Defence , stand at Right-angles with the Flank of the Interior Polygon , by this Table , which supposes the Interior Polygon to be 1000. Then Polygons 4 5 6 7 8 9 10 Strait-lines . Capital 398 437 367 333 312 300 291 233 Gorge-line 155 196 203 242 252 260 263 300 Now describe the Hexagon PP , &c. Then divide the Interior Poligon PP , into 1000 parts , take 367 and prick off all the Capitals PA ; Then take 203 and prick off all the Gorges from P to C. Now draw the lines of defence AC and AC , &c. Then at C , set the Flanks at Right Angles with the line of Defence AC , so shall FC be the Flank , and FA the Faces , then finish every Bastion , and your Hexagon is fortified as was required . ☞ Thus have I set down the several Ways and Rules , for laying the fundamental Ground-line , from the most considerable Inginiers of this last Age , out of all which it's most agreeable to those Authors , and to practice , to take ⅓ of the Interior Polygon for the Capital , ⅕ for the Gorge , and Flank , which leaves 6 / 10 for the Curtain , and let this be taken for a general Rule , where the Flank , and Curtain , stand at Right Angles . PROP. VII . By the Semicircle to lay down on the Ground , any of the former Fortifications . Having drawn the Plot of your Fort on Imperial paper , or Vellom , and if it be a Regular Fort you need not describe it but two half Bastions from the Center , for that will be sufficient . Having such a Plate whose length is set down on each respective line , and all proper Angles expressed , will not only be usefull for laying down the Work , but for finding the Solidity of the Ramparts , Parapets , and the other Earth Works See Fig 76. If it be in such a Place , that from the Center of the Fort , all the Angles may be seen , place your Semicircle at Z , and lay off all the Angles of the Center , which here is 60° ; then mark out the Diametrical lines , and making them their due length , as by your Plate they appear to be , set Piquets , on all the P , P' s upright with the Plane , Then take up your Instrument and place a Piquet at Z. Then lock-spit out all the Polygons PP . Then mark out the Gorges CP , then set out the Flanks CF , either at Right Angles , or as otherwise required . Then lock-spit out the Flanks CF , and the Faces AF , having first set off the Capital PA , so is the Fort lined out for the Ground-line . But if there be Houses and Obstacles in the way , that from the Center all may not be seen , then must you mark out any one side and measure it , and at each End set off the Angles of the Polygon , ( which here is 120° ) and draw side after side , untill all be finished : Then finish the Bastions as before , and here great care must be had , or else you will run into infinite Errours . ☞ But you have liberty Experimentally to alter any of the former proportions , as you have occasion , and as will best serve the Place ; as you see by the fortifying a streight lined Figure : Fig. 77. wherein Count Pagan's or in Manesson's way it may not be allowed without some alterations . SECT . IV. Of the Dimensions , and Measures of the Rampires , Parapets , Mote , Coridor , or Covert-way , and its Esplanade , or Breast-work . THE Rampire's thickness and height , must receive its Determination from the Judgment of the Inginiers , and Purse of the Prince . The Height T S , must not exceed 18 Feet , not be less than Ten ; the thickness may be from 50 , to 80 RA , in all Royal Works , and according as Earth is to be had . The slope of the inward side of the Rampire TR , is commonly a foot for a foot , therefore RS , the Talu , will be equal to the Height T S , so if T S be 18 , RS will be 18 , if 15 then 15 feet . The outward Slope QA , is generally proportioned ½ a Foot for a Foot , so if the Height OQ be 18 , the Talu OA , will be 9 Feet , &c. The Height of the Parapet ZD , must always be 6 Feet , the Exterior Height PM must be 4 Feet , the thickness of the Parapet DQ , in light Earth must be 20 Feet , in stiff Earth 16 , and in Solid Rough Clay 14 Feet ; suppose it be 18 Feet , PM will be 4 Feet , MQ 2 Feet , LD 1 Foot , so will the lower thickness LQ be 21 Feet . The Height of the Banquet VX is 1½ Foot , and thickness VL , 3 Feet . The Lizier must be made so wide , as to support the Rampire from slipping into the Ditch , and is taken from 3 , to 10 Feet ; the Mote or Ditch may be from 70 , to 130 Feet broad , that is , from E to G , and the depth IF may be 8 , 10 , or 12 Foot deep , the little Ditch at the bottom of the Mote represented by c q g , must be as large and deep as the Earth and Work will give leave . The Coridor and the Esplanade or Breast-work on it , is left about 18 Feet wide , from G to C ; on which is placed a Parapet , and Banquet , like that on the Rampire , which Parapet or Esplanade , must slope so into the Campaigne , that a streight line drawn from Z , the Top of the Rampire , may terminate OFd , the Slope thereof . PROP. VIII . How to lay down the Profile of the Work , according to this Table . Feet . The Base of the Rampart RA 70 Height T S and QO 16 Interior Talu RS 16 Exterior Talu OA 8 Base of the Parapet LQ 21 Interior Height ZD 6 Exterior Height MP 4 Exterior Talu MQ 2 Interior Talu QD 1 Breadth of the Banquet V , L 3 Height of it V , X 1½ The Terra Plana TV 25 The Lizier AE 3 The Mote's breadth EG 112 The Depth of it IF 12 Breadth at its bottom FH 88 The Talus EI , or KG 12 The Breadth of the little Ditch c g. 18 The Depth of it 5 The Coridor GC 18 The Seat of the Esplanade 60 The Height CF 6 Now to lay down this Profile draw a line of a convenient length as RSOACGD for the level or Ground-line , then by your Scale of 20 , o● 30 , at most in an Inch , representing Feet . Take out of it 70 for RA , 16 for R S , 8 for OA , 3 for AE , 112 for EG , 18 for GC , and 60 for CD , and mark them off on your Paper ( as in Fig. 78. ) at S , O , I , K , C , raise or let fall Perpendiculars ( by prop. 1 , 2 , or 3. § . 1. chap. 4. ) then take 16 for ST , and OQ , 12 for IF and KH , and 6 for Cf , and draw RT , TQ , QA , EF , HG , cf , fd : Then from Q set off QL 12 , LV 3 , QM 2 , and LD 1 , and raise the Perpendiculars MP 4 , DZ6 , and VX 1½ Then draw VX , XY , YZ , ZP , and PQ , and make the little Ditch by its measure , so is th● Profile perfected : as for the Faus-Bray , they ar● now out of use , therefore I omit them . The Solid Content of those Earth-Works may easily be attained by the former Rules which Content being got in Feet , divide that product by 324 , the Quotient shall be the Soli● Flores contained therein , a Flore being 18 foo● square and 1 Foot deep . SECT . V. Of the Dimensions and Construction of Pla 〈…〉 forms , Caveleers and Cazemats in t 〈…〉 Flanks . 1. PLatforms are Plantations where G 〈…〉 are to be placed , and are common 〈…〉 made of Plank , and Sleepers , there neede● for one Gun , to be but one Platform , whi 〈…〉 must be 8 Feet broad next the Parapet , and 14 Feet wide at the other End , and their length should be 18 Feet . 2. An Embrasure is the Port-Hole made in the Parapet , which towards the Gun must be 4 Feet wide , and towards the Campaigne 8 Feet wide , whose height must be proportioned unto the Wheel of the Carriage ; and are 16 , 18 or 20 Feet assunder . 3. Cavaleers or Mounts are Massy pieces of Earth raised on or near the Rampart , above the Parapet , on which Ordnance and small shot may be planted . As to their Construction I shall follow the Method of Manesson , who places them in the Gorge of the Bastion , and gives this Rule for it , [ saith he ] . Lengthen out the line of defence to E , untill it cut the Capital , the Center of your Cavaleer shall be the middle point betwixt P and E , to wit at F , then with the distance of 84 Feet on the Center F strike a Circle , which shall be the Base of your Cavaleer : Now its height ought to be at least 20 Feet ; and if the Work be to be faced with Stone , or Brick it needs not not have a Talu above 3½ Feet , so that the Diameter at the top will be about 153 Feet , whereon , set a Parapet of 20 Seat , and high , and other Demensions as aforesaid in the Rampire , and there will be a Terra-plana at the top of above 100 Feet , whereon six pieces of Ordnance may be planted , making Embrasures and Platforms as was last directed . 4. Cazemats are made in the Corners of the Flanks , and are several Platforms for Guns to be planted on , thereby to be hid from the Battery of the Enemy : As to the Construction I shall follow Manesson's Directions , first as to the form , and also to the measure : [ saith he ] The Caremate shall take up one half of the Flank , and no more ; The Grand Caremate D B is about 7 , 8 or 9 Feet from the Level of the Plane of the Fort , and hath a passage into it from within the Fort A , C is its Parapet of 20 or 22 Foot Seat , and in it let there be 3 or 4 Embrasures ; D is the part thereof most hid from the Enemies Cannon ; F is the Magazine for this Battery ; H is the second Caremate , G the Ladder , and L the Magazine , and M the Parapet ; this is to hold but one Gun ; M is the third Caremate on the level of the Bast. which let be all firm , in which let there be no void place . The Dimensions and Construction according to the Method of this our Authour are thus [ saith he ] Lengthen the Line of Defence from C to G some 40 Feet , then draw CD , parallel to Cf , ( by prop. 4. § . 1. chap. 4. ) let CF be half of cf , so that cF may be equal to Ff , then from the middle of the Face opposed , draw KF , and let it cut GD , in I , then make I L , and FM equal to 6 Feet , then make MN 66 Feet , and draw NO parallel to the Flank , which let be 24 Feet : Lastly [ saith he ] for the Orillon or Blind , prolong the Face FT 36 Feet , and also FV 36 Feet , then joyn TV , and make that part all solid : So is your Caremate finished : Let the height of the lower Cazemat , be 6 Feet as before , and let all the rest be compleated as you see in the Figure . SECT . VI. Of the Dimensions , and Constructions of those Out-Works , called Ravelins , Horn , Crown-works , &c. THE Ravelin is a certain Work lying beyond the Mote , or Ditch , for the covering the Curtain , Bridge , and Gate ; the Angle of the Ravelin must not be less than 60 , nor much above 100 deg . the manner of delineating it is thus . Lengthen out the middle line of the Curtain OM unto a convenient length , then take with your Compasses the length of the Curtain CC , and setting on Foot in F , the point of the Face and Flank , cross the middle line in q ; then laying your Ruler at q , and to the points F , draw the lines of the Ravelin q R and qS , which shall be the Ground-lines of the Ravelin : The M●te surrounding it must be half the breadth of the Great Mote ; the Rampert may be 30 Feet thick , and some 6 , 7 or 8 Feet high , on which may stand a Parapet equal to that of the Rampire . Now if from the points F you raise streight lines into the Campaign , at Right Angles to the Curtain , and from the points F set off FE , and FE 720 or 750 Feet , then may you joyn EE either with , A ; Single Tenaile : which is done by joyning EE , and dividing it into four equal parts , take one and place from D to N , and so draw EN and EN , so have you a Single Tenail IENEL , which must have a Mote Rampire , and Parapet like the Ravelin . Secondly it may if occasion require be fortified with , A ; Horn-Work : which is done by joyning the points E E , and fortifying the Exterior Polygon EE as is afore taught : Or divide EE into three parts ; make ME , and EN equal to MO ; then draw N M , which divide so likewise at O and P ; then draw E O and E P ; then at P and O raise Perpendiculars O Q and P R , so shall M , E , Q , O , P , R , E , N , be the Horn-Work which was desired : which must likewise have small Rampires and Parapets , as afore . For the Crown-Work : From the Center of the Fort O draw O M B of a convenient length , then from the middle of the Ravelin set off 1000 or 800 Feet to B , then on q , strike the Arch D B E , set off the Curtain , and Demi-Gorge P C C , from B to F , and G both ways , then draw C F and C E , to terminate the points I and H on the Counterscerp ; then toke ⅓ part of B F or B E , and set it from B to M ; and srom F to L , and from E to M ; then draw L M , and M N ; then for your Demi-Bastions make N P and L O equal to N E , &c. Then for the Demi-Gorges of the whole Bastion in the middle , let them be equal to ⅕ of the Interiour Poligon L M or M N , viz. M Y or M X ; then finish the Bastions by drawing the lines of Defence , and raising Perpendiculars , or making Angles of 98° at O , X , Y , and P , then the Crown work is finished as desired . You may make Ravelins and other Works ( beforementioned ) before these Curtains if occasion require . There are some other Works which are used ; as Half Moons , Bonnets , Double Tenails , Counter-gards , Horseshoes , Priests-Caps , &c. which would be superfluous to speak of in this place . 5. Cittadels , are Castles or Forts of the least sort , and are the Out-works lastly used , which are * commonly of 4 or 5 Bastions , and are placed in such Order , that there may be two Faces , and a Curtain towards the Town : the Construction whereof is after this manner . Lengthen out the line OM , and therein find the Center of the Cittadel , the Interior Polygon of the Pentagon may be ¾ of the Curtain adjoyning , or a little more ; the Center of the Square may be on P the point of the Interior Polygon , the Center of the Hexagon may be near the outward point of the Bastion of the Town , taken away to make the Cittadel in , which may be delineated as afore : The Motes and other Works in proportion accordingly , and the Rampires as high as those of the City or Town . SECT . VII . Of some Maxims or Rules necessary to be known in Irregular Fortification . IRregular Fortifications is when any Town or Place is to be fortified , which lieth in an Irregular form ; i. e. whose Sides and Angles are unequal in the fortifying of Irregular Figures * . I shall here say very little , only I shall lay down some Precepts that are of immediate concern in fortifying of Irregular Figures , and shall refer you to peruse Marlois , Dogen , Fritach , Taurnier , Dilichius , &c. which will greatly satisfie and help you : To this end know , 1. That the same Laws and Maxims for Regular Fortifications stand and be in force for Irregular ; i. e. that the line of Defence must not exceed the Port of a Musquet , nor the Angles of the Bastion be less than 60° , nor much above 90° , &c. 2. That no inward Angle of the Place be less than 90° , if it be so it must be altered , and that point may be made the outward point of a Bastion . 3. That between Regular and Irregular Fortifications , there is no other difference , but by rectifying the sides that are too short , or too long , and altering the Angles that are too little ; as for the sides , if they be above 500 , and under 1000 Feet , they may be fortified by Bastions placed according to the usual manner , at the extreme points thereof ; But if the sides be between 1000 and 1700 Feet , then in the midst you may place a Plat Bastion , and at the Extreme Points , place two Bastions , as before : But if the line be less than 500 Feet , you may lengthen it , by producing it into the Plane : As for the Angles , they are made greater or lesser according as occasion requireth . For the Raising the Rampires , Parapets , and other Out-works , they are to be as in the Regular , and the Out-work may be placed before the Curtains as was before mentioned . 4. That the Capital , in any Regular or Irregular Bastion , is found by dividing the Angle of the Polygon into two equal parts ( by prop. 7. § . 1. chap. 4. ) and by producing the line of Angular Division or Separation , on which the due length of the Capital must be placed , which observe for a general Rule . SECT . VIII . Of the Dimensions and Construction of small Forts , or Scones , which are built for the Defence of some Pass , River , or other place . WHEN they are made Regular , of 4 , 5 , or 6 Bastions , then they may be fortified by the precedent Rules , but there are others of smaller Dimensions fit for the same purpose : viz. Triangle with Demi-Bastions , Square with Demi-Bastions , Parallelograms with Demi-Bastions and Tong , Star Redoubts of four , five or six points , and Plain Redoubts . PROP. IX . To fortifie a Triangle , with Demi-Bastions . This Triangle may consist and be comprehended of three equal or unequal sides in this Example : let it be an Equilaterial Triangle PPP Now divide PP into three parts , then take 1 , and prick off the Capitals PA , &c. and the Gorges make equal thereunto , as PC , PC , &c. then make the Flanks FC to stand at Right Angles , and to be ½ of PC or PA , then draw the Faces AF , AF , &c. and the Work is finished as required . PROP. X. To fortifie a Square with Demi-Bastions . The sides of the Square may be from 100 to 200 Feet , let PP be 180 Feet , which divide into 3 parts , take one for the Gorges PC , and for the Capitals PA , and prick them off all round as you see , then take ⅙ of PP , and at Right Angles prick off the Flanks CF , then draw the Faces AF , AF , &c. and the Figure is compleated . PROP. XI . To fortifie a Parallelogram with Demi-Bastions , and Tong. First describe the Parallelogram , or LongSquare , PPPP , then divide PP into 6 parts ( the side on which the Tong , or Tenaile , is placed ) and make MC equal unto ⅙ thereof , and also MG , and MH . Draw CG , GC , and CH , HC , then finish the Demi-Bastions as before , so shall the Work be compleated as was required . A Long Square may also be fortified as Fig. the 77. PROP. XII . To fortifie a Star Redoubt of 4 , 5 , or 6 Points 1. A Star Redoubt of four points may have his side from 40 to 60 Feet : First describe the Square PPPP , then divide PP into two parts at M , take ¼ of PM , ( and by prop. 1. § . 1. ch . 4. ) raise Perpendiculars round at M , make MA equal to ¼ of PM , and draw all as in the Figure . 2. A Star Redoubt of five points is thus fortified . Describe the Pentagon PP , &c. then divide PP into halves at M , raise the Perpendiculars MA , make MA equal to ⅓ of PM , and draw the Fort in all respects as the Figure representeth . 3. A Star Redoubt of six points is thus fortified . Describe the Hexagon PPP , &c. divide PP into two equal parts at M , then raise Perpendiculars at the M' s , then make MA equal to ½ of PM , or ¼ of PP , and draw every respective line as you see in the Figure . PROP. XIII . To Delineate a Plain Redoubt Plain Redoubts are called Grand Redoubts , which are used as Batteries in Approaches , whose side may be from 60 to 80 Feet , or Petit Redoubt , which are used for a Court of Guards in the Trenches , and may be from 20 to 50 Feet , and are framed and delineated in all respects as you see in Fig 90. The Profile's to be set on these several Works , and the Motes , are alterable and uncertain , for they being sometimes used in Approaches ; then they do require the Breast-work at the Bottom to be 7 or 8 Foot wide , and the Interior Height 6 , and the Exterior 5 Feet , and the Mote to be either 8 , 10 or 12 Feet , sometimes 14 or 20 Feet wide at the bottom , and the height of 7 , 8 or 9 Feet , to have two , or three ascents to rise to the Parapet . There are many other things belonging to this Art , which the limitation I am bound to , will not permit here to be treated of . CHAP. XIV . Of Military Orders , or the Embattelling and Encamping of Souldiers . SECT . I. Of the Embattelling and Ordering of Souldiers . BATTAILS are considered either in respect of the number of Men , or in respect of the form of Ground . In the respect of the number of Men , it is either a Square Battail , a Double Battail , a Battail of the Grand Front , or a Battail of any proportion , of the number in Rank to the number in File . In respect of the form of the Ground , the Battail is either a Geometrical Square of Ground , or a long Square of Ground . For the Distance , or Order of Souldiers , martialled in Array , is distinguished either into Open Order , which is when the Centers of their places are 7 Feet distant assuunder , both in Rank and File , or Order ; which is when the Centers of the places are 3 ½ Feet distant both in Rank and File ; or else 3 ½ Feet in Rank , and 7 Feet in File . PROP. I. To Order any number of Souldiers into a Square Battail of Men. Admit it were required to Martial into a Square Battail 16129 Men : To doe which extract the Square Root of 16129 ( by prop. 8. § . 1. chap. 1. ) which is 127 , therefore you are to place 127 Men in Rank , and also in File . PROP. II. To Order any number of Souldiers into a Double Battail . Admit 16928 Men were to be Martialled into a Double Battail , extract the Root of half the number of Men ; i. e. of 8464 , whose Root is 92 , therefore I say that 92 Men must be placed in File , and 184 in Rank , to order that number of Men propounded into a Double Battail . Plate VII Page 302 PROP. III. To Order any number of Souldiers into a Battail of the Grand Front. Admit 16900 Souldiers were to be Martialled into a Battail of the Grand Front , that is Quadruple . Extract the Square Root of 4225 ( that is ¼ of the Men ) the Root is 65 ; therefore I say 65 must be placed in File , and 260 in Rank , to form a Battail of the Grand Front. PROP. IV. Any number of Men , together with their distance in Rank and File , being propounded , to Order them into a Square Battail of Ground . Admit 2500 Souldiers were to be Martialled into a Square Battail of Ground , in such sort that their distance in File should be 7 feet , and in Rank 3 feet , and 't is required to know how many Men must be placed in Rank and in File to draw up 2500 Men into Square Battail of the Ground . According to prop. 1. § . 1. ch . 1. say , As — 7 to 3 , So is 2500 to 1071 , &c. whose Square Root is 32 , &c. Therefore I say 32 Men are to be placed in File . Now to find how many Men are to be placed in Rank , divide 2500 by 32 , the Quotient is 78 , which are the number of Men to be placed in Rank , and 4 Men to be disposed elsewhere . PROP. V. Any number of Souldiers propounded , to Order them in Rank and File , according to the reason of any two Numbers given . Admit 6400 Souldiers are to be Martialled into Array , in such Order that the number of Men placed in File , shall bear such proportion to the number in Rank as 7 to 13 ; ( according to prop. 1. § . 1. chap. 1. ) say as 7 to 13 , so is 6400 to 11885 , &c. whose Square Root is 109 , &c. the number of Men to be placed in Rank , by which divide 6400 , it produces , 58 , &c. the number of Men to be placed in File , and 78 Men to be employed elsewhere . SECT . II. Of Castermetation , or Quartering and Encamping of Souldiers . IN Quartering and Encamping of Souldiers , it is requisite , the Quarter-Master General , and all other under Quarter-Masters , be skilled at Foot measure , that so they may lay out their Quarters as directed . The common allowance for the depth of Ground , that a Regiment of Horse or Foot will take up , the wideness must be answerable to the Number of Men 200 Feet for the Huts in length , and 100 for the Commanders , and Sutlers , before them ; every two Souldiers to a Hut , 8 Feet broad , and 8 Feet deep , 2 Feet Hut , from Hut , so that there may stand 20 Huts in the 200 Feet , the Ally betwixt Hut , and Hut , may be 8 Feet , that is 16 Feet in width , and 200 in length for 40 Men , which is 3200 Feet , and for the 100 Feet more , 1600 Feet , in all 4800 Feet , and there must be 25 Rows of Huts , for 1000 Men ; so that for a Regiment of Foot containing 1000 Men , with Officers , and Sutlers , will take up 120000 Feet , which is 2 Acres and 3 Roods , which because of Ways may be allowed 3 Acres of Ground , for every Regiment , which may be 350 Feet deep , and 370 Feet wide , or near 360 Feet square : Now if 1000 Men , Officers , Sutlers , High-ways and all take up a Square of 360 Feet , how many Feet shall the Side of a Square be wherein 10000 Footmen , &c. may be encamped ? say ( by prop. 1. chap. 1. ) as 1000 , to 10000 , so is the Square of 360 , viz. 129600 , to 1296000 , the Square of 1138 Feet , which is very near 30 Acres of Ground . For the Quartering of Horse , you must keep the same depth of 300 Feet for all , and take 200 Feet for the Huts , the Horse Huts must be 10 Feet deep , and 4 wide ; so that 12 Horses may stand in one Hut together , which is 48 Feet long , and 10 wide , and 6 Feet a Street ; The Huts for the Troops , will be 6 , for 12 Troops ; now conceive a Regiment to consist of 8 Troops , 50 to a Troop , it will take up leaving 20 Feet Streets , and Cross-ways , very near as much Ground as a Regiment of Foot , Ways and all must be allowed 3 Acres , near 360 Feet square , so that 10 Regiments of Horse will take up 30 Acres : Moreover , it will be needfull and you may very well allow , as much ground as both Horse and Foot will take , for the Train of Artillery , Victuallers , Parade Places , &c. From these considerations the young beginner , nay even the better practised Souldier may receive help , and thereby be enabled to Encamp an Army if required . CHAP. XV. Of GUNNERY . SECT . I. Of the Names of the Principal Members of a Piece of Ordnance . 1. ACANNON is a long round Body , either of Brass , or Iron , formed and made hollow by Art , and proportion , to offend afar off , with a Ball of Iron , Stone , or any Artificial Substance , charged with Gun-Powder , in its charged Cilinder , which being fired , in an instant performs its desired Effect . This Machine was invented by an Englishman , and first put in practice by the Venetians against the Genoveses at Chiezza , Anno 1376. 2. The Superficies of the Mettal , is the outside round about the Piece . 3. The Body is the Substance of the whole Mass of Mettal . 4. The Chase is the Concavity of the Piece , in which they put the Charge . 5. the Muzzel is the Extremity of the Chase by which you load , and unload the Piece . 6. The Calibre is AB the Diameter of the Muzzel or Mouth . 7. The Touch-hole , is that little vent , which passeth from the Convex Superficies , to the very Chamber of the Piece , made to give fire to the Powder within as C , that which encloseth the Extremity of the Chase about the Touch-hole is called the Breech or Coyl . 8. The Cascabel is the Pammel at the Breech or Coyl as D. The Trunnions , are pieces of Metal fixed unto the Exterior Superficies of the Gun on which he moves in the Carriage as E , E. The Body of the Piece , is that which is comprehended betwixt the Center of the Trunnions and the Cascable EG . The Vacant Cylinder , is comprehended betwixt the Cent : of the Trunnions & the Muzzel as EB . The Frees , or Muzzel Ring is that thick Cornish which , incompasseth the Convex Superficies of the Piece at I , The Base Ring is KLG , The Reinforced Ring is M , The Trunnion Ring is N , and the Cornish Ring is O. The Line of the Cylinder , is a direct line imagined to be described along the Chase Parallel unto the middle of the Chase as XZ . The Line of Metal , is a line touching both Cornishes , as MNI. The Dispart line of the Piece , is the difference betwixt the Semidiameter of the Muzzel , and Base Ring as the line IH . The Vent of the Piece is the difference betwixt the Diameter of the Shot , and the Mouth of the Piece , as e d. The Chamber , or Charged Cylinder , is that part of the Chase towards the Touch-hole equally large , nor narrower in one place than in another , and doth contain the Powder and Ball. SECT . II. Of the Dimension of our Usal English Cannon , and other Ordnance , &c. IN the following Table I have set down the length and weight of our most usual English Ordnance , the Diameters and Weight of their Bullets , the length and breadth of their Ladles , the Weight of Powder to Charge them , &c. The Names of the several Pieces of Ordnance . Guns length Guns weight Guns bore Bullets diamet . Bullets weight Ladles length Ladles breadth Powder weight Shoots Level Utmost Random Feet Inches Pounds Inches 8 parts Inches 8 parts Pounds Ounces Inches 8 parts Inches 8 parts Pounds Ounces Paces Paces A Base . 4 6 200 1 2 1 1 0 5 4 0 2 0 0 8 60 600 A Rabinet . 5 6 300 1 4 1 3 0 8 4 1 2 4 0 2 70 700 A Falconet . 6 0 400 2 2 2 1 1 5 7 4 4 0 1 4 90 900 A Falconi 7 0 750 2 6 2 5 2 8 8 2 4 4 2 4 130 1300 Minion ordinary . 7 0 800 3 0 3 7 3 4 8 4 5 0 2 8 120 1200 Minion largest . 8 0 1000 3 2 3 0 3 12 9 0 5 0 3 4 125 1250 Saker leaft . 8 0 1400 3 4 3 2 4 12 9 6 6 4 3 6 150 1500 Saker ordinary . 9 0 1500 3 6 3 4 6 0 10 4 6 6 4 0 160 1600 Saker old sort . 10 0 1800 4 0 3 6 7 5 11 0 7 2 5 0 163 1630 Demiculver least . 10 0 2000 4 2 4 0 9 0 12 0 8 0 6 4 174 1740 Demiculver ordinary . 11 0 2700 4 4 4 2 10 11 12 6 8 0 7 4 175 1750 Demiculver old sort . 11 0 3000 4 6 4 4 12 11 13 4 8 4 8 8 178 1780 Culverin least . 11 0 4000 5 0 4 6 15 0 14 2 9 0 10 0 180 1800 Culverin ordinary . 12 0 4500 5 2 5 0 17 5 16 0 9 4 11 6 181 1810 Culverin largest . 12 0 4800 5 4 5 2 20 0 16 0 10 0 11 8 183 1830 Demicannon least 11 0 5400 6 2 6 0 30 0 20 0 11 4 14 0 156 1560 Demicannon ordin . 12 0 5600 6 4 6 1 32 0 22 0 12 0 17 8 162 1620 Demicannon large 12 0 6000 6 6 6 3 36 0 22 6 12 6 18 0 180 1800 Cannon Royal 12 0 8000 8 0 7 4 58 0 24 0 14 0 32 8 185 1850 PROP. I. How to know the different Fortification of a Piece of Ordnance . In fortifying any Piece of Ordnance there are three degrees observed , as first Legitimate Pieces , which are those that are ordinarily fortified ; secondly Bastard Pieces , which are such whose Fortification is lessened ; thirdly Double fortified Pieces , or extraordinary Pieces . The Fortification of any Piece of Ordnance , is accounted by the thickness of the Metal at the Touch-hole , Trunnions , and at the Muzzel , in proportion to the Diameter of the Bore . The Legitimate Pieces , or the ordinary fortified Cannons , have ⅞ at the Touch hole , ⅝ at the Trunnions , and ⅜ at the Muzzel of the thickness of the bore , in thickness of Metal . Bastard Cannons , or lesned Cannons , have ¾ at their Touch-hole , or 12 / 16 , and 9 / 16 at their Trunnions , and 7 / 16 at their Muzzel : the Double fortified Cannons have full one Diameter of the Bore in thickness of Metal at the Touch-hole , and 11 / 16 at the Trunnions ; and 7 / 16 at their Muzzel . Now all double fortified Culverins , &c. are 1 ⅛ at the Touch-hole , 15 / 16 at the Trunnions , and 9 / 16 at the Muzzel , and the Ordinary fortified Culverins , are fortified every way as double fortified Cannons , and lesned Culverins as Ordinary Cannons in all respects . PROP. II. How to know how much Powder is fit for proof , and what for service , for any Piece of Ordnance . For Cannons take ⅘ of the weight of their Iron Bullet of good Corn Powder for Proof , and for service ½ the weight of the Iron Bullet is sufficient , especially for Iron Ordnance , which will not endure so much Powder , as Brass ones will receive by ¼ in Weight , for Culverins allow the whole Weight of the Shoot for Proof , and ⅔ for Service . For Sakers , and Falcons , take ⅘ of the Weight of the Shoot , and for lesser Pieces the whole weight may be used in service , untill they grow hot , but then there must be some abatement made at discretion , and take 1 ⅓ of the weight of their Iron Bullet for Proof . PROP. III. To know what Bullet is fit to be used in any Piece of Ordnance . The Bullet must be somewhat less than the Bore of the Gun , that so it may have vent in the discharge , some Authors affirm ¼ of an Inch less than the Bore will serve , all Ordnance , but this vent is too much for a Falcon , &c. and too little for a Cannon : therefore I approve them not , but commend Mr. Phillipes's proportion * to your Use , which is to divide the Bore of the Gun into 20 equal parts ; and let the Diameter of the Bullet be 19 / 20 thereof , according to which proportion the precedent Table is calculated . PROP. IV. By knowing the proportion of Metals one to another , and by knowing the Weight of one Ball , to know what any other shall weigh . The common received proportions for Metals are these . Lead is to Iron as 2 , to 3. Lead is to Brass as 24 , to 19. Lead is to Stone as 4 , to 1. Iron is to Lead as 3 , to 2. Iron is to Brass as 16 , to 18. Iron is to Stone as 3 , to 8. The more exact proportion betwixt Metals are thus known . Admita Cube , or Ball of Gold , weigh 100 l. A Cube of any of those Metals ensuing of the same bigness , shall bear such proportion , as followeth , to the said Cube of Gold. li. pts . li. pts . Gold. 100 00 Iron . 42 10 Quicksilver . 71 43 Tinn . 38 95 Lead . 60 53 Stone . 15 80 Silver . 54 39 Water . 05 68 Brass . 47 37 It is the opinion of Dr. Wybard in his Tactometria , that a Bullet of Cast Iron , whose Diameter is 4 Inches , doth weigh 9 l. Averdupoize weight . Now to find what any other Bullet , or Cube shall weigh ; say ( as in prop. 4. chap. 1. ) As the Cube of the Bullet propounded , is to his weight , so is the Cube of another Bullet given , to his weight , and so observe still this proportion . SECT . III. Of the Qualification of an able Gunner , and necessary Operations before shooting , and in shooting . A Gunner ought to be a Man of Courage , Experience , and Vigilant ; he ought to have good skill in Arithmetick , to know the Extraction of the Roots , &c. He ought to have skill in Geometry , to take heights , distances , &c. to know the Divisions and Use of his Circle , Quadrant , and Quadret ; to know how to level , and to lay Platforms , and to raise Batteries . He must know the Names of all sorts of Ordnance , their Weight , the Height of their Bore , the Height and Weight of their Shot , the length and breadth of their Ladles , how much Powder to use for proof , and action ; The Shoots Level , and the Shoots Random ; He must know the Names of all the Members of a Piece of Ordnance , he must also know the length , thickness and breadth of all manner of Carriages , and must know all the parts thereof : Viz. the Cheeks or Sides , the Axtree , Spokes , Nave , Hoops , Transomes , Bolts , Plates , Drawing-Hooks , the Clout , the Hole for the Linspin , the Shafts , the Thill and Thill-bolt , the Fore-lock , and Fore-lockkeys , Capsquares , the Fore-lock-pins and Chain , the Pintle and Bolt-hole , Fellows , Nayles , Fellow-bars , Stirropes , the Ruts of the Wheel , Dowledges , Beds , Coines , Leveres , Hand screws , &c. He must also know how to make his Ladles , Spunges , Cartridges , whether of Paper or Canvas , and to have by him Formers of all sorts , Sheep-skins undrest to make Spunges , Powder , Shot , Needles , Thread , Paste and Starch , Marlin , Twine , Nails , Handspikes , Crows of Iron , Granado-shells , and Materials for Composition , Fasces , Budg Barrels , Cannon-Baskets , &c. These being general things he is to know , and at all times to have ready by him , and he is more particularly to know these following parts of his Art : As , PROP. I. How to Tertiate , Quadrate , and to Dispart a Piece of Ordinance . 1. To Tertiate a Piece , is to find whether it hath its due thickness at the Trunnions , Touch-hole , and Neck ; if the Trunnions , and the Neck are in its due order , and the Chase streight . 2. To Quadrate a Piece mounted , is to see whether it be directly placed , and equally poized in the Carriage ; which is known by finding in the Convex Superficies of the Base , and Muzzel Ring ; the point which is Perpendicular , over the Soul of the Piece which may be found by the Gunners Instrument , called a Level ; an Instrument whose use is so vulgarly known , that it needeth not my Explanation . 3. To Dispart a Piece , is to fix , or elevate on the Convex point of the Muzzel Ring , a Mark , as far distant from the Cylinder , or Soul of the Piece , as is the point of the Base-Ring ; to the end , that the Visail-ray which passeth by these marks , may be Parallel to the Chase , Soul , or Cylinder of the Piece . Now the Dispart , i. e. the difference of the Semidiameters of the Cornishes , may be by a pair of Calliper Compasses attained . Which found , place on the Top of the Cornish-Ring , near the Muzzel , over the middle of the Inferior Cylinder . PROP. II. To know how far any Piece of Ordnance will shoot , &c. As to the several shootings in Artillery , Authors differ much in their Judgments , and Opinions , but they all unanimously agree that the Ball being shot forth flies through the Air , with a Violent , Mixt and Natural motion ; describing a Parabolical line , in whose beginning and ending are lines sensibly streight , and in the middle carved : In the beginning the Imprest force driving forward by the Fire , the Natural gravity of the Ball doth describe a Right-line , called the Direct line , or Rangs of the Ball 's Circute . In the middle that force diminisheth , and the Natural Gravity prevaileth , so that it describeth a curved line , called the Ball 's middle Helical or Conical Arch * ; In the End the Natural Gravity overcoming the Imprest violence , ( which becomes altogether weak and faint ) describes a new right line , called the Ball 's declining line , in which the Ball tends towards the Center of the Earth , as towards a Place natural unto all heavy bodies : See Figure the 92. These motions are somewhat longer , according as the Piece is mounted from the level unto the Angle of 45 deg . which is called the Utmost Random : The Elevation of which , is regulated by the Gunners Quadrant , the Use of which Instrument is so generally known , and by so many Authors fully explained , that I here crave leave to omit it : But take these for General Rules . 1. That a Shoot at Right Angle , strikes more violent and furiously than at Oblique Angles , therefore Gunners use when they are to bàtter down a Tower , Wall , or Earth-work , to shoot point blank at the Object , Tire by Tire ; by discharging all the Pieces in Battery against the self same . Object , in the same ; Instant , holding it for a Maxim , that ten Cannons discharged together , do fan more Execution than discharged one after another . Now at Oblique Angles they shoot either Cross ways or by rebounding . 2. That the speediest way to make a B●●a●h in a Wall , &c. Is by shooting at the Object from two Batteries , which ruins for more speedily than by striking the Object , with one Battery , at Right Angles , although that one Battery , hath as many Cannon as the other two hath . 3. That if you were to batter a Flank , covered with an Orillion , ( which because you cannot possibly batter it right forward ) you must therefore of necessity batter it Obliquely , by way of Rebounding , thus : Chuse a fit place in the Curtain to be your Object , on which you may play with your Battery obliquely , so that by a rebound the shoot may leap 〈…〉 the Flanks , holding for a Maxim , in this operation , * That the Angles of Incidence and Reflection are Equal . Now we come to shew the length of the Right Range , of all our Common English Ordnance , which is set down in the precedent Table , in which the Cannon exceed not 185 Paces , &c. Esteeming the Pace 5 English Feet , nor his utmost Random above 1850 Paces , which Table so sheweth for all other Natures . As for the Ranges , and Randoms , to the several Degrees and points of Mounture of the Quadrant , I have hereunto annexed the Tables , calculated by the Experiments of sundry most Eminent Artists , whose Works will perpetuate their Worth and Name to succeding Generations . A Table of Ranges , and Randoms , to the several Degrees of Mounture of the Quadrent . A TABLE OF Right Ranges or Points Blanks . Randoms or the First Graze . The Degrees of the Pieces Mounture . 0 The Right Range in Paces of 5 Feet . 192 The Degrees of Mounture . 0 The Paces of the Random 5 Feet a Pace . 192 1 209 1 298 2 227 2 404 3 249 3 510 4 261 4 610 5 278 5 722 6 285 6 828 7 302 7 934 8 320 8 1044 9 337 9 1129 10 354 10 1214 20 454 20 1917 30 693 30 2185 40 855 40 2289 50 1000 50 2283 60 1140 60 1792 70 1220 70 1214 80 1300 80 1000 90 1350 90 The Use of the Table of Randoms . This Table is most agreeing to Cannons , and Culverins ; and the greatest sort of Ordnance , the Use thereof is thus . Admit a Saker to be mounted to 3 deg . shoots the Bullet 323 Paces , how far will it shoot being mounted unto 7 deg . Say ( by prop. 1. chap. 1. ) As 510 the Tabular distance for 3 deg . of Mounture , to 323 , the distance found , So is 934 the Tabular distance for 7 deg . of Mounture , to 591 272 / 510 , the distance required , which the Saker according to this Experiment shall shoot at 7 deg . of Mounture . Mr. NYE in his Book of Gunnery printed Anno , 1647 , saith he made an Experiment by a Saker of 8 Feet long , which he loaded with three pounds of Powder , of an exact weight , both Powder and Wad at every charge , every time ramming it down with three equal stroaks , as near as possible ; but on the Bullet he put no Wad , because the Saker was mounted ; And thus he made four Shoots , each of them half an Hour after the other , that so the Piece might be of equal temper , and mounted his Piece to these 4 degrees of Mounture , viz. 1 deg . 5 deg . 7 deg . 10 deg . and found these Randoms . At 1 Deg. the Random was 225 Paces . At 5 Deg. the Random was 416 Paces . At 7 Deg. the Random was 505 Paces . At 10 Deg. the Random was 630 Paces . According to which Experiment , he framed this Table of Randoms . Deg. Paces Deg. Paces 0 206 6 461 1 225 7 505 2 274 8 548 3 323 9 589 4 370 10 630 5 416 Captain HEXAM in his Book of Gunnery , shews how by finding out the Random of a Cannon , for the first Degree of Mounture , thereby to find the Random for every Degree to 45 deg . or utmost Random , and this is his Rule to perform it . First find how many Paces the Cannon will shoot being laid level by the Metal , ( which by him is accounted 1 deg . ) Then divide the distance found , by 50 , then multiply the Quotient by 11 , so shall the product be the greatest Digression , or Difference betwixt Rangs , and Rang ; which being divided by 44 , the Quotient giveth the Number of Paces , which the Bullet will lose in the other Rangs , from Degree , unto Degree ; according to this Rule , this Table is calculated . A Table of Randoms to 45 Degrees , accounting a Pace 2 ½ Foot. D. Moun Paces . Diff. D. Moun. Paces . Diff. 0 0775 225 23 4685 110 1 1000 220 24 4795 105 2 1220 215 25 4900 100 3 1435 210 26 5000 95 4 1645 205 27 5095 90 5 1850 200 28 5185 85 6 2050 195 29 5270 80 7 2245 190 30 5350 75 8 2435 185 31 5425 70 9 2620 180 32 5595 65 10 2800 175 33 5560 60 11 2975 170 34 5620 55 12 3145 165 35 5675 50 13 3310 160 36 5725 45 14 3470 155 37 5770 40 15 3625 150 38 5810 35 16 3775 145 39 5845 30 17 3920 140 40 5875 25 18 4060 135 41 5900 20 19 4595 130 42 5920 15 20 4325 125 43 5935 10 21 4450 120 44 5945 5 22 4570 115 45 5950 I have hereunto also annexed the Table calculated by Alexander Bianco , for all sort of Ordnance , ( which Table I account one of the best that was ever yet found Extant ) In his Work printed 1648. A Table of Randoms for the first six Points of the Gunner's Quadrant . Points . 1 2 3 4 5 6 Falconet . 375 637 795 885 892 900 Falcon. 550 935 1166 1254 1309 1320 Minion . 450 765 954 1026 1071 1080 Saker . 625 1062 1325 1125 1487 1500 Demi-culver . 725 1232 1537 1653 1725 1740 Culverin . 750 1275 1590 1710 1785 1800 Demi-cannon . 625 1062 1325 1425 1487 1500 Cannon of 7. 675 1147 1431 1489 1606 1620 Double Cannon . 750 1275 1590 1710 1785 1800 SECT . IV. Of Shooting in Mortar-Pieces . A Mortar-Piece is a short Piece , with which they shoot Bombs , Granado-Shells , Stone-Balls , &c. not by a Right line but from a Curved , from on high ; so that it may fall where it should be desired : Now this Mortar is placed in the Carriage , in all respects as you see in Fig. 93. in which A signifies the Carriage , B the Mortar , C the Course the shoot flies , and D the Place on which it falls . Bombs are great hollow Balls of Iron , or Brass , in which are put fine Sifted Gun-Powder , which by a Fuse , they proportion to them a due Fire , that so they may break assoon as they fall amongst the Enemies . These Fuses are small Trunks of Wood , Tinn , or Iron , filled with a prepared Composition for that purpose . Granadoes are of the same form with Bombs , only smaller , and many times are cast by hand , and are made of Iron , Brass , Glass , or Earth . Now in Order to the well shooting in those kind of Machines called Mortars , 't is requisite to observe these following Rules : as , 1. That before you make a shoot at any Place , you find the distance thereof from your Mortar , which may be obtained by Prop. 3. § . 4. Chap. 9. 2. That the Bombs , or other Bodies that are to be shot , be of equal weight , otherwise the shoots will vary 3 That the Carriage in breadth be always on a Level , and without any descent , that so it may not leap in discharging . 4. That the Powder with which the Mortar is loaded , be always of the same force and weight . 5. That the Charge of the Mortar , as well in Powder as in Wadding , be always rammed in with blows equally heavy , and of equal number . 6 That the Wadds be always either of Wood , or Tampeons , or else of Okam , for the strongest drives it farthest . 7. That the Fuses be newly made , in those days that they are to be used , and that they be made of a Composition proportionable to the Range that the shoot shall make in the Air , so that the Bomb may break in the very moment of its fall ; which Composition must be such , that though it fall in the Water , yet not to extinguish , but the Bomb there to break . Now before we proceed any farther , I think it necessary , to shew how to compose your Ingredients for your Fuse . PROP. I. To make Fuses for Bombs , &c. The Composition for Bombs must be of a slow motion , that so time enough may be given to throw either Bombs , Granadoes , Fire-balls , Thundring-Barrels , &c. They are compounded of these Ingredients , thus : Take a pound of Gun-Powder , 4 / 16 of Sulphur 4 / 16 of Salt-Peter , well beaten , dry , and sifted separately , then mix it , and make up your Fuse thereof : Or take Powder of Benjamin , and Small-Coles , all well beaten and mixed together with some Oyl of Piter , and so fill your Fuse therewith . Now the use of Mortar-Pieces , being for the most part to shoot up at Random , therefore the Randoms of these Pieces is very necessary to be known : Therefore hereunto I have annexed the Tables of Randoms , calculated by the Experience of the best of Authors , which have wrote on this Subject ; most of which do agree in their Randoms , although they are in a several dress . Diego-Uffano-Zutphen in his Works printed 1621 , hath calculated these two following Tables , the one for the 12 points of the Quadrant , the other for every Degree , taking the one Half of each Number , and so 't is reduced into our English Paces of 5 Feet , which Tables were esteemed and made use of , both by Captain Hexam , and Mr. Norton , and are as followeth . A Table of Randoms for Mortar-Pieces , to the 12 Points of the Gunner's Quadrant , calculated by Diego-Uffano-Zutphen . 583 570 534 468 377 248 100 6 5 4 3 2 1 0 . . . . . . . ☉ 6 7 8 9 10 11 12 583 570 534 468 377 248 000 Now suppose the Mortar to be placed at ☉ , the Pricks in the middle line representeth the several Randoms , numbred with the Degrees of the Quadrant , forward and backward , unto which the several Randoms are set ; so you see that the Mortar being levelled point blank , throweth the Bomb 100 Paces , if the Mortar be mounted one Point , it throws the Bomb 248 Paces , &c. untill 't is mounted to the 6th . point , 583 Paces , which is the utmost Random : Now if the Mortar be mounted higher to 7 , 8 , 9 , &c. Points , the Randoms decrease again as before they did increase : as you see in the Table . But in those latter Randoms there lieth a great mistake , as shall be made palpably appear . For if as they are distant from the sixth Point you make them equal to one another , then the Random of the 12 points , must be equal to the Random of 0 point , or the Level Random , which is 100 Paces from the Mortar . Now it is contrary to all Art and Reason , to think that if the Mortar be elevated to the 12th . point , i. e. bolt upright , it should shoot the Bomb 100 Paces from the Mortar ; no , it cannot be ; but according to all Reason the Bomb must fall down either on , or near the Mortar , and not 100 Paces distant , as is most erroneously conceived ; the like errour is in the following Table of our said Author ; but because Mr. Phillipps in his Mathematical Manual hath amply demonstrated their Errours , I therefore shall say no more to the Errours that have been a long time generally conceived and embraced as a truth , but now are removed . A Table of Randoms for Mortar-Pieces , to every Degree of the Quadrant . The Degrees of Mounture . 0 The Paces of the Random . 100 The Degrees of Mounture . 89 The Degrees of Mounture . 23 The Paces of the Random . 480 The Degrees of Mounture . 66 1 122 88 24 490 65 2 143 87 25 500 64 3 164 86 26 510 63 4 185 85 27 518 62 5 204 84 28 525 61 6 224 83 29 531 60 7 243 82 30 536 59 8 263 81 31 540 58 9 280 80 32 543 57 10 297 79 33 549 56 11 315 78 34 552 55 12 331 77 35 558 54 13 347 76 36 562 53 14 362 75 37 568 52 15 377 74 38 573 51 16 393 73 39 577 50 17 406 72 40 580 49 18 419 71 41 581 48 19 432 70 42 582 47 20 445 69 43 583 46 21 457 68 44 584 22 460 67 45 585 The most exact Tables of Randoms for the Mortar , that I have seen or can find in any Ancient , or Modern Author , is this following Table , calculated by the experience and trial of that Famous Inginier Tomaso Moretii of Brescia , Inginier to the most serene Republique of Venice , in his Works Intituled , Trattatu delle Artiglieria , printed 1665. Where he supposeth the utmost Random , equal to 10000 , according to which proportion he framed this following Table . A Table of the several Randoms of each Degree of the Quadrant , the greatest Equal to 10000. Elev . Elev . Elev . Elev . 1° 349 89° 23° 7193 67° 2 698 88 24 7431 66 3 1045 87 25 7660 65 4 1392 86 26 7880 64 5 1736 85 27 8090 63 6 2079 84 28 8290 62 7 2419 83 29 8480 61 8 2756 82 30 8660 60 9 3090 81 31 8829 59 10 3420 80 32 8988 58 11 3746 79 33 9135 57 12 4067 78 34 9272 56 13 4384 77 35 9397 55 14 4695 76 36 9511 54 15 5000 75 37 9613 53 16 5299 74 38 9703 52 17 5592 73 39 9781 51 18 5870 72 40 9848 50 19 6157 71 41 9903 49 20 6428 70 42 9945 48 21 6691 69 43 9976 47 22 6947 68 44 9994 46 45 10000 45 The Use of the Precedent Table is explained by these following Propositions . PROP. II. Finding that a Mortar of 300 , with a Tampeon of Wood , being elevated 45° , or 6 Points of the Quadrant , sends a Bomb 800 Paces , how many Paces shall the same shoot , at the Elevation of 54° ? Look at the said 54° of the Table , and you Demonstration . will find thē proportional Number 9511 , to correspond thereunto . Now you find the proportional Number belonging to 45° is 10000 , then by Prop. 1. Chap. 1. Say as 10000 , to 800 , so is 9511 , to 760 88 / 100 , which are the Paces , the Mortar will send the Bomb at the Elevation of 54 Degrees . PROP. III. Finding that a Mortar of 300 , being elevated 54° , sends his Bomb 760 88 / 100 Paces , what Degree of Elevation must that Mortar have , to shoot the Bomb 555 Paces ? This is but the Converse of the former , therefore ( according to Prop. 1. Chap. 1. ) say , as 760 88 / 160 Paces , gives the proportional part or number 9511 ; so doth 555 Paces , give the proportional part 6945. Which number sought among the proportional Numbers , in the Table , you will find 68 Degrees to correspond to that proportional Number 6945 , so that the Mortar must be elevated to 68 Degrees to shoot the Bomb 555 Paces , which was required to be known . These Rules and Precepts here delivered , I esteem necessary to be known by every Gunner , who intends to be serviceable for his Prince and Countrey . Vive , vale : Siquid novisti rectius istis , Candidus imperti : Si non his utere mecum . Hora. lib. 1. Epist FINIS . Plate VIII A TABLE OF Logarithm Numbers , From One to Ten Thousand : Whereby the LOGARITHM OF ANY NUMBER Under Four Hundred Thousand may be readily discovered . LONDON , Printed by J. Heptinstall for W. Freeman , at the Artichoke next St. Dunstan's Church in Fleet street . MDCLXXXVII . N Log. N Log. N Log. 1 0. 000000 34 1. 531479 67 1. 826075 2 0. 301030 35 1. 544068 68 1. 832509 3 0. 477121 36 1. 556303 69 1. 838849 4 0. 602060 37 1. 568202 70 1. 845098 5 0. 698970 38 1. 579783 71 1. 851258 6 0. 778151 39 1. 591064 72 1. 857332 7 0. 845098 40 1. 602060 73 1. 863323 8 0. 903090 41 1. 612784 74 1. 869232 9 0. 954242 42 1. 623249 75 1. 875061 10 1. 000000 43 1. 633468 76 1. 880813 11 1. 041393 44 1. 643452 77 1. 886491 12 1. 079181 45 1. 653212 78 1. 892094 13 1. 113943 46 1. 662758 79 1. 897627 14 1. 146128 47 1. 672098 80 1. 903090 15 1. 176091 48 1. 681241 81 1. 908485 16 1. 204120 49 1. 690196 82 1. 913814 17 1. 230449 50 1 698970 83 1. 919078 18 1. 255272 51 1. 707570 84 1. 924279 19 1. 278753 52 1. 716003 85 1. 929419 20 1. 301030 53 1. 724276 86 1. 934498 21 1. 322219 54 1. 732394 87 1. 939519 22 1. 342422 55 1. 740362 88 1. 944482 23 1. 361728 56 1. 748188 89 1. 949390 24 1. 380211 57 1. 755875 90 1. 954242 25 1. 397940 58 1. 763428 91 1. 959041 26 1. 414973 59 1. 770852 92 1. 963788 27 1. 431364 60 1. 778151 93 1. 968483 28 1. 447158 61 1. 785330 94 1. 973128 29 1. 462398 62 1. 792391 95 1. 977723 30 1. 477121 63 1. 799340 96 1. 982271 31 1. 491361 64 1. 806180 97 1. 986772 32 1. 505150 65 1. 812913 98 1 991226 33 1. 518514 66 1 819544 99 1. 995635 N 0 1 2 3 4 5 6 7 8 9 D 100 000000 000434 000868 001301 001734 002166 002598 003029 003461 003891 432 101 004321 004751 005181 005609 006038 006466 006894 007321 007748 008174 428 102 008600 009026 009451 009876 010299 010724 011147 011570 011993 012415 424 103 012837 013259 013679 014100 014521 014940 015359 015779 016197 016616 416 104 017033 017451 017898 018284 018700 019116 019532 019947 020361 020775 416 105 021189 021603 022016 022428 022841 023252 023664 024075 024486 024896 412 106 025306 025715 026125 026533 026942 027349 027757 028164 028371 028978 408 107 029384 029789 030195 030599 031004 031408 031812 032216 032619 033021 404 108 033424 033826 034227 034628 035029 035429 035829 036229 036629 037028 400 109 037426 037825 038223 038620 039017 039414 039811 040207 040602 040998 396 110 041393 041787 042182 042576 042969 043362 043755 044148 044539 044932 393 111 045323 045714 046105 046495 046885 047275 047664 048053 048442 048830 389 112 049218 049603 049993 050379 050766 051153 051538 〈◊〉 052309 052694 386 113 053078 053463 053846 054229 054613 054996 055378 055760 056142 056524 382 114 056905 057286 057666 058046 058426 058805 059185 059563 059942 060320 379 115 060698 061075 061452 061829 062206 062582 062958 063333 063709 064083 376 116 064458 064832 065206 065579 065953 066326 066699 067071 067443 067815 372 117 068186 068557 068928 069298 069668 070038 070407 070776 071145 071514 369 118 071882 072249 072617 072985 073352 073718 074085 074451 074816 075182 366 119 075547 075912 076276 076640 077004 077368 077731 078094 078457 078819 363 120 079181 079543 079904 080266 080626 080987 081347 081707 082067 082426 360 121 082785 083144 083503 083861 084219 084576 084934 085291 085647 086004 357 122 086359 086716 087071 087426 087781 088136 088490 088845 089198 089552 355 123 089905 090258 090610 090963 091315 091667 092018 092369 092721 093071 351 124 093422 093772 094122 094471 094820 095169 095518 095866 096215 096562 349 125 096910 097257 097604 097951 098298 098644 098989 099335 099681 100026 346 126 100371 100715 101059 101403 101747 102091 102434 102777 103119 103462 343 127 103804 104146 104487 104828 105169 105510 105851 106191 106531 106871 340 128 107209 107549 107888 108227 108565 108903 109241 109579 109916 110253 338 129 100589 100926 111263 111599 111934 112269 112605 112939 113275 113609 335 N 0 1 2 3 4 5 6 7 8 9 D 130 113943 114277 114611 114944 115278 115611 115943 116276 116608 116939 333 131 117271 117603 117934 118265 118595 118926 119256 119586 119915 120245 330 132 120574 120903 121231 121559 121888 122216 122544 122871 123198 123525 328 133 123852 124178 124504 124830 125156 125481 125806 126131 126456 126781 325 134 127105 127429 127753 128076 128399 128722 129045 129368 129689 130012 323 135 130334 130655 130977 131298 131619 131939 132259 132579 132899 133219 321 136 133539 133858 134177 134496 134814 135133 135451 135769 136086 136403 318 137 136721 137037 137354 137671 137987 138303 138618 138934 139249 139564 315 138 139879 140194 140508 140822 141136 141449 141763 142076 142389 142702 314 139 143015 143327 143639 143951 144263 144574 144885 145196 145507 145818 311 140 146128 146438 146748 147058 147367 147676 147985 148294 148603 148911 309 141 149219 149527 149835 150142 150449 150756 151063 151369 151676 151982 307 142 152288 152594 152899 153205 153509 153815 154119 154423 154728 155032 305 143 155336 155639 155943 156246 156549 156852 157154 157457 157759 158061 303 144 158362 158664 158965 159266 159567 159868 160168 160469 160769 161068 301 145 161368 161667 161967 162266 162564 162863 163161 163459 163758 164055 299 146 164353 164650 164947 165244 165541 165838 166134 166430 166726 167022 297 147 167317 167613 167908 168203 168497 168792 169086 169380 169674 169968 295 148 170262 170555 170848 171141 171434 171726 172019 172311 172603 172895 293 149 173186 173478 173769 174059 174351 174641 174932 175222 175512 175802 291 150 176091 176381 176669 176959 177248 177536 177825 178113 178401 178689 289 151 178977 179264 179552 179839 180126 180413 180699 180986 181272 181558 287 152 181844 182129 182415 182699 182985 183269 183555 183839 184123 184407 285 153 184691 184975 185259 185542 185825 186108 186391 186674 186956 187239 283 154 187521 187803 188084 188316 188647 188928 189209 189490 189771 190051 281 155 190332 190612 190892 191171 191451 191730 192009 192289 192567 192846 279 156 193125 193403 193681 193959 194237 194514 194792 195069 195346 195623 278 157 195899 196176 196453 196729 197005 197281 197556 197832 198107 198382 276 158 198657 198932 199206 199481 199755 200029 200303 200577 200850 201124 274 159 201397 201670 201943 202216 202488 202761 203033 203303 203577 203848 272 N 0 1 2 3 4 5 6 7 8 9 D 160 204119 204391 204663 204934 205204 205475 205746 206016 206286 206556 271 161 206826 207096 207365 207364 207904 208173 208441 208710 208978 209247 269 162 209515 209783 210051 210319 210586 210853 211121 211388 211654 211921 267 163 212187 212454 212720 212986 213252 213518 213783 214049 214314 214579 266 164 214844 215109 215373 215638 215902 216166 216429 216694 216957 217221 264 165 217484 217747 218010 218273 218536 218798 219060 219323 219585 219846 262 166 220108 220369 220631 220892 221153 221414 221675 221936 222196 222456 261 167 222716 222676 223236 223496 223755 224015 224274 224533 224791 225051 259 168 225309 225568 225827 226084 226342 226599 226858 227115 221372 227629 258 169 227887 228142 228400 228657 228913 229169 229426 229682 229938 230193 256 170 230449 230704 230959 231215 231469 231724 231979 232234 232488 232742 254 171 232996 233250 233504 233752 234011 234264 234517 234770 235023 235276 253 172 235528 235781 236033 236285 236537 236789 237041 237292 237544 237795 252 173 238046 238297 238548 238799 239049 239299 239549 239799 240049 240299 250 174 240549 240799 241048 241297 241546 241795 242044 242293 242541 242789 249 175 243038 243286 243534 243782 244029 244177 244525 244772 245019 245266 248 176 245513 245759 246006 246252 246499 246745 246991 247237 247482 247728 246 177 247973 248219 248464 248709 248954 245198 249443 249687 249932 250176 245 178 250420 250664 250908 251151 251395 251638 251881 252125 252368 252610 243 179 252853 253096 253334 253580 253822 254064 254306 254548 254789 255031 242 180 255273 255514 255755 255996 256237 256477 256718 256958 257198 257438 241 181 257679 257918 258158 258398 258637 258877 259116 250355 259594 259833 239 182 260071 260309 260548 260787 261025 261263 261501 261739 261976 262214 238 183 262451 262688 262925 263162 263399 263636 263873 264109 264346 264582 237 184 264818 265054 265289 265525 265761 265996 266232 266467 266702 266937 235 185 267172 267406 267641 267875 268109 268344 268578 268812 269046 269279 234 186 269513 269746 269979 270213 270446 270679 270912 271144 271377 271609 233 187 271842 272074 272306 272538 272769 273001 273233 273464 273696 273927 232 188 274158 274389 274619 274850 275081 275311 275542 275772 276002 276232 230 189 276462 276692 296921 277151 277379 277609 277838 278067 278296 278525 229 N 0 1 2 3 4 5 6 7 8 9 D 190 278754 278982 279211 279439 276667 279895 280123 280351 280578 280806 228 191 281033 281261 281488 281714 281942 282169 282396 282622 282849 283075 227 192 283301 283527 283753 283979 284205 284431 284656 284882 285107 285332 226 193 285557 285782 286007 286232 286456 286681 286905 287129 287354 287578 225 194 287802 288026 288249 288473 288696 288919 289143 289366 289589 289812 223 195 290035 290257 290479 290702 290925 291147 291369 291591 291813 292034 222 196 292256 292478 292699 292920 293141 293362 293584 293804 294025 294246 221 197 294466 294687 294907 295127 295347 295567 295787 296007 296226 296446 220 198 296665 296884 297104 297323 297542 297761 297979 298198 298416 298635 219 199 298853 299071 299289 299507 299725 299943 200161 300378 200595 300813 218 200 301030 301247 301464 301681 301898 302114 302331 302547 302764 302979 217 201 303196 303412 303628 303844 304059 304275 304491 304706 304921 305136 216 202 305351 305566 305781 305996 306211 306425 306639 306854 307068 307282 215 203 307496 307709 307924 308137 308351 308564 308778 308991 309204 309417 213 204 309630 309843 310056 310268 310481 310693 310906 311118 311329 311542 212 205 311754 311966 312177 312389 312600 312812 313023 313234 313445 313656 211 206 313867 314078 314289 314499 314709 314920 315130 315340 315551 315760 210 207 315970 316180 316389 316599 316809 317018 317227 317436 317646 317854 209 208 318063 318272 318481 318689 318898 319106 319314 319522 319730 319938 208 209 320146 320354 320562 320769 320977 321184 321391 321598 321805 322012 207 210 322219 322426 322633 323839 323046 323252 323458 323665 323871 324077 206 211 324282 324488 324694 324899 325105 325310 325516 325721 325926 326131 205 212 326336 326541 326745 326949 327155 327359 327563 327767 327972 328176 204 213 328379 328583 328787 328991 329194 329398 329601 329805 330008 330211 203 214 330414 330617 330819 331022 331225 331427 331629 331832 332034 332236 202 215 332438 332640 332842 333044 333246 333447 333649 333859 334051 334253 202 216 334454 334655 334856 335057 335257 335458 335658 335859 336059 336259 201 217 336459 336659 336859 337059 337259 337459 337659 337859 338058 338257 200 218 338456 338656 338856 339054 339253 339453 339650 339849 340047 340246 199 219 340444 340642 340841 341039 341237 341435 341632 341830 342028 342225 198 N 0 1 2 3 4 5 6 7 8 9 D 220 342227 342620 342817 343014 343212 343409 343606 343802 343999 344196 197 221 344392 344589 344785 344981 345178 345373 345569 345766 345962 346157 196 222 346353 346549 346744 346939 347135 347330 347525 347720 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690007 89 N 0 1 2 3 4 5 6 7 8 9 D 490 690196 690285 690373 690462 690550 690639 690728 690816 690905 690993 89 491 691081 691169 691258 691347 691435 691524 691612 691700 691789 691877 88 492 691965 692053 692142 692229 692318 692406 692494 692583 692671 692759 88 493 692847 692935 693023 693111 693199 693287 693375 693463 693551 693639 88 494 693727 693815 693903 693991 694078 694166 694254 694342 694429 694517 88 495 694605 694693 694781 694868 694956 695044 695131 695219 695307 695394 88 496 695482 695569 695657 695744 695832 695919 696007 696094 696182 696269 87 497 696356 696444 696531 696618 696706 696793 696880 696968 697055 697142 87 498 697229 697317 697404 697491 697578 697665 697752 697839 697926 698014 87 499 698101 698188 698275 698362 698449 698535 698622 698709 698706 698883 87 500 698970 699057 699144 699281 699317 699404 699491 699528 699664 699751 87 501 699838 699924 700011 700098 700184 700271 700358 700444 705031 700617 87 502 700704 700790 700877 700963 701049 701136 701222 701309 701395 701482 86 503 701568 701654 701741 701827 701913 701999 702086 702172 702258 702344 86 504 702430 702517 702603 702689 702775 702861 702947 703033 703119 703205 86 505 703291 793377 703463 703549 703635 703721 703807 703893 703979 704065 86 506 704151 704236 704322 704408 704494 704579 704665 704751 704837 704922 86 507 705008 705094 705179 705265 705350 705436 705522 705607 705693 705778 86 508 705863 705949 706035 706130 706206 706291 706376 706462 706547 706632 85 509 706718 706803 706888 706974 707059 707144 707219 707315 707399 707485 85 510 707570 707655 707740 707826 707911 707996 708081 708166 708251 708336 85 511 708421 708506 708591 708676 708761 708846 708931 709015 709100 709185 85 512 709269 709355 709439 709524 709609 709694 709789 709863 709948 710033 85 513 710117 710202 710287 710371 710456 710540 710625 710709 710794 710879 85 514 710963 711048 711132 711217 711301 711385 711469 711554 711639 711723 84 515 711807 711891 711976 712060 712144 712229 712313 712397 712481 712566 84 516 712649 712734 712818 712902 712986 713070 713154 713238 713223 713407 84 517 713491 713575 713659 713742 713826 713910 713994 714078 714162 714246 84 518 714329 714414 714497 714581 714665 714749 714833 714916 714999 715084 84 519 715167 715251 715335 715418 715501 715586 715669 715753 715836 715919 84 N 0 1 2 3 4 5 6 7 8 9 D 520 716003 716087 716170 716254 716337 716421 716504 716588 716671 716754 83 521 716838 716921 717004 717088 717171 717254 717338 717421 717509 717587 83 522 717671 717754 717837 717920 718003 718086 718169 718253 718336 718419 83 523 718502 718585 718668 718751 718834 718917 718999 719083 719165 719248 83 524 719331 719414 719497 719579 719663 719745 719828 719911 719994 720077 83 525 720159 720242 720325 720407 720490 720573 720655 720738 720821 720903 83 526 720986 721068 721151 721233 721316 721398 721481 721563 721646 721728 82 527 721811 721893 721975 722058 722140 722222 722305 722387 722469 722552 82 528 722634 722716 722798 722881 722963 723045 723127 723209 723291 723374 82 529 723456 723538 723619 723702 723784 723866 723948 724029 724112 724194 82 530 724276 724358 724439 724522 724604 724685 724767 724849 724931 725013 82 531 725095 725176 725258 725339 725422 725503 725585 725667 725748 725829 82 532 725912 725993 726075 726156 726238 726319 726401 726483 726564 726646 82 533 726727 726809 726890 726972 727053 727134 727216 727297 727379 727459 81 534 727541 727623 727704 727785 727866 727948 728029 728110 728191 728273 81 535 728354 728435 728516 728597 728678 728759 728841 728922 729003 729084 81 536 729165 729246 729327 729408 729489 729569 729651 729732 729813 〈◊〉 81 537 729974 730055 730136 730117 730298 730378 730459 730540 730621 730702 81 538 730782 730863 730944 731024 731105 731186 731266 731347 731423 731508 81 539 731589 731669 731749 731830 731911 731991 732072 732152 732233 732313 81 540 732394 732474 732555 732635 732715 732796 732876 732956 733037 733117 80 541 733197 733278 733358 733438 733518 733598 733679 733759 733839 733919 80 542 733999 734079 734159 734239 734319 734399 734479 734559 734639 734719 80 543 734799 734879 734959 735039 735119 735199 735279 735359 735439 735519 80 544 735599 735679 735759 735838 735918 735998 736078 736157 736237 736317 80 545 736397 736476 736556 736635 736715 736795 736874 736954 737034 737113 80 546 737191 737272 737352 737431 737511 737590 737669 737749 737829 737908 79 547 737987 738067 738146 738225 738305 738384 738463 738543 738622 738701 79 548 738781 738859 738939 739018 739097 739177 739259 739335 739414 739493 79 549 739572 739651 739731 739809 739889 739968 740047 740126 740205 740284 79 N 0 1 2 3 4 5 6 7 8 9 D 550 740363 740442 740521 740599 740678 740757 740836 740915 740994 741073 79 551 741152 741230 741309 741388 741467 741546 741624 741703 741782 741860 79 552 741939 742018 742096 742175 742254 742332 742411 742489 742568 742647 79 553 742725 742802 742882 742961 743039 743118 743196 743275 743353 743431 78 554 743509 743588 743667 743745 743823 743902 743979 744058 744136 744215 78 555 744293 744371 744449 744528 744606 744684 744764 744840 744919 744997 78 556 745075 745153 745231 745309 745387 745465 745543 745621 745699 745777 78 557 745855 745933 746011 746089 746167 746245 746323 746401 746479 746556 78 558 746634 746712 746789 746868 746945 747023 747101 747179 747256 747334 78 559 747412 747489 747567 747645 747722 747800 747878 747955 748033 748110 78 560 748188 748266 748343 748421 748498 748576 748653 748731 748808 748885 77 561 748963 749040 749118 749195 749272 749349 749427 749504 749582 749659 77 562 749736 749814 749891 749968 750045 750123 750199 750277 750354 750431 77 563 750508 750586 750663 750739 750817 750894 750971 751048 751125 751202 77 564 751279 751356 751433 751510 751587 751664 751741 751818 751895 751972 77 565 752048 752125 752202 752279 752356 752433 752509 752586 752663 752739 77 566 〈◊〉 752893 752969 753047 753123 753199 753277 753353 753429 753506 77 567 753583 753659 753736 753813 753889 753966 754042 754119 754195 754272 77 568 754348 755425 754501 754578 754654 754730 754800 754883 754959 755036 76 569 755112 755189 755265 755341 755417 755494 755569 755646 755722 755799 76 570 755875 755951 756027 756103 756179 756256 756332 756408 756484 756560 76 571 756636 756712 756788 756864 756940 757016 757092 757167 757244 757320 76 572 757396 757472 757548 757627 757699 757775 757851 757927 758003 758079 76 573 758155 758230 758306 758382 758458 758533 758609 758685 758761 758836 76 574 758912 758988 759063 759139 759214 759290 759366 759411 759517 759592 76 575 759668 759743 759819 759894 759969 760045 760121 760196 760272 760347 75 576 760422 760498 760573 760649 760723 760799 760875 760949 761025 〈◊〉 75 577 761176 761251 761326 761402 761477 761552 761627 761702 761778 761853 75 578 761928 762003 762078 762153 762228 762303 762378 762453 762529 762604 75 579 762679 762754 762829 762904 762978 763053 763128 763203 763279 763353 75 N 0 1 2 3 4 5 6 7 8 9 D 580 763428 763503 763578 763653 763727 763802 763877 763952 764027 764101 75 581 764176 764251 764326 764400 764475 764549 764624 764699 764774 764848 75 582 764923 764998 765072 765147 765221 765296 765370 765445 765519 765594 75 583 765669 765743 765818 765892 765966 766041 766115 766189 766264 766338 74 584 766413 766487 766562 766636 766710 766785 766859 766933 767007 767082 74 585 767156 767230 767304 767379 767453 767527 767601 767675 767749 767823 74 586 767898 767972 768046 768119 768194 768268 768342 768416 768490 〈◊〉 74 587 768638 768712 768786 768860 768934 769008 769082 769156 769229 769303 74 588 769377 769451 769525 769599 769673 769746 799820 769894 769968 770042 74 589 770115 770189 770263 770336 770410 770484 770557 770631 770705 770778 74 590 770852 770926 770999 771073 771146 771219 771293 771367 771440 771514 74 591 771587 761661 771734 771808 771881 771955 772028 772102 772175 772248 73 592 772322 772395 772468 772542 772615 772688 772762 772835 772908 772981 73 593 773055 773128 773201 773274 773348 773421 773494 773567 773640 773713 73 594 773786 773859 773933 774006 774079 774152 774225 774298 774371 774444 73 595 774517 774589 774663 774736 774809 774882 774955 775028 775100 775173 73 596 775246 775319 775392 775465 775538 775610 775683 775756 775829 775902 73 597 775974 776047 776119 776193 776265 776338 776411 776483 776556 776629 73 598 776701 776774 776846 776919 776992 777064 777137 777209 777282 777354 73 599 777427 777499 777572 777644 777717 777789 777862 777934 778006 778079 72 600 778151 778224 778296 778368 778441 778513 778585 778658 778729 778802 72 601 778874 778947 779019 779091 779163 779236 779308 779380 779452 779524 72 602 779596 779669 779741 779813 779884 779957 780029 780101 780173 780245 72 603 780317 780389 780461 780533 780805 780677 780749 780821 780893 780965 72 604 781037 781109 781181 781253 781324 781396 781468 781539 781612 781684 72 605 781755 781827 781899 781971 782042 782114 782186 782258 782329 782401 72 606 782473 782544 782616 782688 782759 782831 782902 782974 783046 783117 72 607 783189 783260 783332 783403 783475 783546 783618 783689 783761 783832 71 608 783904 783975 784046 784118 784189 784261 784332 784403 784475 784546 71 609 784617 784689 784759 784831 784902 784974 785045 785116 785187 785259 71 N 0 1 2 3 4 5 6 7 8 9 D 610 785329 785401 785472 785543 785615 785686 785757 785828 785899 785970 71 611 786041 786112 786183 786254 786325 786396 786467 786538 786609 786680 71 612 786751 786822 786893 786964 787035 787106 787177 787248 787319 787389 71 613 787460 787531 787602 787673 787744 787815 787885 787956 788027 788098 71 614 788164 788239 788309 788381 788451 788522 788593 788663 788734 788804 71 615 788875 788946 789016 789087 789157 789228 789299 789369 789439 789510 71 616 789581 789651 789722 789792 789863 789933 790004 790074 790144 790215 70 617 790285 790356 790426 790496 790567 790637 790707 790778 790848 790918 70 618 790988 791059 791129 791199 791269 791339 791409 791480 791550 791620 70 619 791691 791761 791831 791901 791971 792041 792111 792181 792252 792322 70 620 792392 792462 792532 792602 792672 792742 792812 792882 792952 793022 70 621 793092 793162 793231 793301 793371 793441 793511 793581 793651 793721 70 622 793791 793860 793930 793999 794069 794139 794209 794279 794349 794418 70 623 794488 794558 794627 794697 794767 794836 794906 〈◊〉 795045 795115 70 624 795185 795254 795324 795393 795463 795532 795602 795672 795741 795810 70 625 795880 795949 796019 796088 796158 796227 796297 796366 796436 796505 69 626 796574 796644 796713 796782 796852 796921 796990 797059 797129 797198 69 627 797268 797337 797406 797475 797545 797614 797683 797752 797821 797890 69 628 797959 798029 798098 798167 798236 798305 798374 798443 798513 798582 69 629 798651 798719 798789 798858 798927 798996 799065 799134 799203 799272 69 630 799341 799409 799478 799547 799616 799685 799754 799823 799892 799961 69 631 800029 800098 800167 800236 800305 800373 800442 800511 800579 800648 69 632 800717 800786 800854 800923 800992 801061 801129 801198 801266 801335 69 633 801404 801472 801541 801609 801678 801747 801815 801884 801952 802021 69 634 802089 802158 802226 802295 802363 802432 802500 802568 802637 802705 69 635 802774 802842 802910 802979 803047 803116 803184 803252 803321 803389 68 636 803457 803525 803594 803662 803730 803798 803867 803935 804003 804071 68 637 804139 804208 804276 804344 804412 804480 804548 804616 804685 804753 68 638 804821 804889 804957 805025 805093 805161 805229 805297 805365 805433 68 639 805501 805569 805637 805705 805773 805841 805908 805976 806044 806112 68 N 0 1 2 3 4 5 6 7 8 9 D 640 806179 806248 806316 806384 806451 806519 806587 806655 806723 806790 68 641 806858 806926 806994 807061 807129 807197 807264 807332 807399 807467 68 642 807535 807603 807670 807738 807806 807873 807941 808008 808078 808143 68 643 808211 808279 808346 808414 808481 808549 〈◊〉 808684 808751 808818 67 644 808886 808953 809021 809088 809156 809223 809298 809358 809425 809492 67 645 809559 809627 809694 809762 809829 809866 809964 810031 810098 〈◊〉 67 646 810233 810299 810367 810434 810501 810569 810636 810703 810778 810837 67 647 810904 810971 811039 811106 811173 811239 811307 811374 811441 811508 67 648 811575 811642 811709 811776 811843 811909 811977 812044 812111 812178 67 649 812245 812312 812379 812445 812512 812579 812646 812713 812779 812847 67 650 812913 812980 813047 813114 813181 813247 813314 813381 813448 813514 67 651 813581 813648 813714 813781 813848 813914 813981 814048 814114 814181 67 652 814248 814314 814381 814447 814514 814581 814647 814714 814780 814847 67 653 814913 814979 815046 815113 815129 〈◊〉 815312 815378 815445 815511 66 654 815578 815644 815711 815777 815843 815909 815978 816042 816109 816175 66 655 816241 816308 816374 816440 816506 816573 816639 816705 816771 816838 66 656 816904 816910 817036 817102 817169 817235 817301 817367 817433 817499 66 657 817565 817631 817698 817764 817829 817896 817962 818028 818094 818159 66 658 818226 818292 818358 818424 818489 818556 818622 818688 818754 818819 66 659 818885 818951 819017 819083 819149 819215 819281 819346 819412 819478 66 660 819543 819609 819676 819741 819807 819873 819939 820004 820070 820136 66 661 820201 820267 820333 820399 820464 820529 820595 820661 820727 820792 66 662 820858 820924 820989 821055 821120 821186 821251 821317 821382 〈◊〉 66 663 821514 821509 821645 821709 821775 821841 821906 821972 822037 822103 65 664 822168 822233 822299 822364 822429 822495 822560 822626 822691 822756 65 665 822822 822887 822952 823018 823083 823148 823213 823279 823344 823409 65 666 823474 823539 823605 823669 823735 823800 823865 823930 823996 824061 65 667 824126 824191 824256 824321 824386 824451 824516 824581 824646 824711 65 668 824776 824841 824906 824971 825036 825101 825166 825231 825296 825361 65 669 825426 825491 825556 825021 825686 825751 825815 825880 825945 826009 65 N 0 1 2 3 4 5 6 7 8 9 D 670 826075 826139 826204 826269 826334 826399 826464 826528 826593 826658 65 671 826723 826787 826852 826917 826981 827046 827111 827175 827239 827305 65 672 827369 827434 827499 827563 827628 827692 827757 827822 827886 827951 65 673 828015 828079 828144 828209 828273 828338 828402 828467 828531 828595 64 674 828659 828724 828789 828853 828918 828982 829046 829111 829175 829239 64 675 829304 829368 829432 829497 829561 829625 829689 829754 829818 829882 64 676 829947 830011 830075 830139 830204 830268 830332 830396 830460 830525 64 677 830589 830653 830717 830781 830845 830909 830973 831037 831102 831166 64 678 831229 831294 831358 831422 831486 831549 831614 831678 831742 831806 64 679 831869 831934 831998 832062 832126 833189 832253 832317 832381 832445 64 680 832509 832573 832637 832700 832764 832828 832892 832956 833019 833083 64 681 833147 833211 833275 833338 833402 833466 833529 833593 833657 833721 64 682 833784 833848 833912 833975 834039 834103 834166 834229 834294 834357 64 683 834421 834484 834548 834611 834675 834739 834802 834866 834929 834993 64 684 835056 835119 835183 835247 835310 835373 835437 835500 835564 835627 63 685 835691 835754 835817 835881 835944 836007 836071 836134 836197 836261 63 686 836324 836387 836451 836514 836577 836641 836704 836767 836830 836894 63 687 836957 837019 837083 837146 837209 837273 837339 837399 837462 837525 63 688 837588 837652 837715 837777 837841 837904 837967 838030 838093 838156 63 689 838219 838282 838345 838408 838471 838534 838597 838660 838723 838786 63 690 838849 838912 838975 839038 839101 839164 839227 839289 839352 839415 63 691 839478 839541 839604 839667 839729 839792 839855 839918 839981 840043 63 692 840106 840169 840232 840394 840357 840419 840482 840545 840608 840671 63 693 840733 840796 840859 840921 840984 841049 841109 841172 841234 841297 63 694 841359 841423 841485 841547 841609 841672 841735 841797 841859 841922 63 695 841985 842047 842109 842172 842235 842297 842359 842432 842484 842547 62 696 842609 842672 842734 842796 842859 842921 842983 843046 843108 843170 62 697 843233 843295 843357 843419 843482 843544 843606 843669 843731 843293 62 698 843855 843918 843979 844042 844104 844166 844229 844291 844353 844415 62 699 844477 844539 844601 844664 844726 844788 844849 844912 844974 845036 62 N 0 1 2 3 4 5 6 7 8 9 D 700 845098 845160 845222 845284 845346 845408 845470 845532 845594 845656 62 701 845718 845779 845842 845904 845666 846028 846089 846151 846213 846275 62 702 846337 846399 846461 846523 846585 846646 846708 846769 846832 846894 62 703 846955 847017 847079 847141 847202 847264 847326 847388 847449 847511 62 704 847573 847634 847696 847758 847819 847881 847943 848004 848067 848128 62 705 848189 848251 848312 848374 848435 848497 848559 848620 848682 848743 62 706 848805 848866 848928 848989 849051 849112 849174 849235 849297 849358 61 707 849419 849481 849542 849604 849665 849726 849788 849849 849911 849972 61 708 850033 850095 850156 850217 850279 850339 850401 850462 850524 850585 61 709 850646 850707 850769 850829 850891 850952 851014 851075 851136 851197 61 710 851258 851319 851381 851442 851503 851564 851625 851686 851747 851809 61 711 851869 851931 851992 851053 852114 852175 852236 852297 852358 852419 61 712 852479 852541 852602 852663 852724 852785 852846 852907 852968 853029 61 713 853089 853150 853211 853272 853339 853394 853455 853516 853577 853637 61 714 853698 853759 853819 853881 853941 854002 854063 854124 854185 851245 61 715 854306 854367 854428 854488 854549 854609 854670 854731 854792 854852 61 716 853913 854974 855034 855095 855156 855216 855277 855337 855398 855459 61 717 855519 855579 855640 855701 855761 855822 855882 855943 856003 856064 61 718 856124 856185 856245 856306 856366 856427 856487 856548 856608 856668 60 719 856729 856789 856849 856910 856970 857031 857091 857152 857212 857272 60 720 857332 857393 857453 857513 857574 857634 857694 857755 857815 857875 60 721 857935 857995 858056 858116 858176 858236 858297 858357 858417 858477 60 722 858537 858597 858657 858718 858778 858838 858898 858958 859018 859078 60 723 859138 859198 859258 859318 859379 859439 859499 859559 859619 859679 60 724 859739 859799 859859 859918 859978 860038 860098 860158 860218 860278 60 725 860338 860398 860458 860518 860578 860637 860697 860757 860817 860877 60 726 860037 860996 861056 861116 861176 861236 861295 860355 861415 861475 60 727 861534 861594 861654 861714 861773 861833 861893 861952 862012 862072 60 728 862131 862191 862251 862310 862369 862429 862489 862549 862608 862668 60 729 862728 862787 862847 862906 862966 863025 863085 863144 863204 863623 60 N 0 1 2 3 4 5 6 7 8 9 D 730 863323 863382 863442 863501 863561 863620 863679 863739 863799 863858 59 731 863917 863977 864036 864096 864155 864214 864274 864333 864392 864452 59 732 864511 864570 864629 864689 864748 864808 864867 864926 864985 864045 59 733 865104 865163 865222 865282 865341 865400 865459 865519 865578 865637 59 734 865696 865755 865814 865874 865933 865992 866051 866110 866169 866228 59 735 866187 866346 866405 866465 866524 866583 866642 866701 866759 866819 59 736 866878 866937 866996 867055 867114 867173 867232 867291 867349 867409 59 737 867467 867526 867585 867644 867703 867762 867821 867879 867939 867998 59 738 868056 868115 868174 868233 868292 868350 868409 868468 868527 868586 59 739 868643 868703 868762 868821 868879 868938 868997 869056 869114 869173 59 740 869232 869290 869349 869408 869466 869525 869584 869642 869701 869759 59 741 869818 869877 869935 869994 870053 870111 870169 870228 870287 870345 59 742 870404 870462 870521 870579 870638 870696 870755 870813 870872 870930 59 743 870989 871047 871106 871164 871223 871281 871339 871398 871456 871515 58 744 871573 871631 871689 871748 871806 871865 871923 871981 871039 872098 58 745 872156 872215 872273 872331 872389 872448 872506 872564 872622 872681 58 746 872739 872797 872855 872913 872972 873029 873088 873146 873204 873262 58 747 873321 873379 873437 873495 873553 873611 873669 873727 873785 873844 58 748 873902 873959 874018 874076 874134 874192 874249 874308 874366 874424 58 749 874482 874539 874598 874656 874714 874772 874829 874888 874945 875003 58 750 875061 875119 875177 875235 875293 875351 875409 875466 875524 875582 58 751 875639 875698 875756 875813 875871 875929 875987 876045 876102 876160 58 752 876218 876276 876333 876391 876449 876507 876564 876622 876679 876737 58 753 876795 876853 876910 876968 877026 877083 877141 877199 877256 877314 58 754 877371 877429 877487 877544 877602 877659 877717 877774 877832 877889 58 755 877947 878004 878062 878119 878177 878234 878292 878349 878407 878464 57 756 878522 878579 878637 878694 878752 878808 878866 878924 878981 879039 57 757 879096 879153 879211 879268 879325 879382 879459 879497 879555 879612 57 758 879669 879726 879784 879841 879898 879955 880013 880070 880127 880185 57 759 870242 880299 880356 880413 880471 880527 880585 880642 880699 880756 57 N 0 1 2 3 4 5 6 7 8 9 D 760 880814 880871 880928 880985 881042 881099 881156 881213 881271 881328 57 761 881385 881442 881499 881556 881613 881669 881727 881784 881841 881898 57 762 881955 882012 882069 882126 882103 882239 882297 882354 882411 882468 57 763 882525 882581 882638 882695 882752 882809 882866 882923 882979 883037 57 764 883093 883050 883207 883264 883321 883377 883434 883491 883548 883605 57 765 883661 883718 883775 883832 883888 883945 884002 884059 884115 884172 57 766 884229 884285 884342 884399 〈◊〉 884512 884569 884625 884682 884739 57 767 884795 884852 884909 884965 885022 885078 885135 885192 885248 885305 57 768 885361 885418 885474 885531 885587 885644 885700 885757 885813 885869 57 769 885926 885983 886039 886096 886152 886209 886265 886321 886378 886434 56 770 886491 886547 886604 886659 〈◊〉 886773 886829 886885 886941 886998 56 771 887054 887111 887167 887223 887279 887336 887392 887449 887505 887561 56 772 887617 887674 887720 887786 887842 887898 887955 888011 888067 888123 56 773 888179 888236 888292 888348 888404 888460 888516 888573 888629 888685 56 774 888741 888797 888853 888909 888965 889021 889077 889134 889189 889246 56 775 889302 889358 889414 889469 889523 889582 889638 889694 889749 889806 56 776 889862 889918 889974 890029 890036 890141 890197 890253 890309 890365 56 777 890421 890477 890533 890589 890645 890700 890756 890812 890868 890924 56 778 890979 891035 891091 891147 891203 891259 891314 891370 891426 891482 56 779 891537 891593 891649 891705 891760 891816 891872 891928 891983 892039 56 780 892095 892150 892206 892262 892317 892373 892429 892484 892539 892595 56 781 898651 892707 892762 892818 892818 892929 892985 893040 893096 893151 56 782 893207 893262 893318 893373 893429 893484 893539 893595 893651 893706 56 783 893762 893817 893873 893928 893984 894039 894094 894149 894205 894261 55 784 894316 894371 894427 894482 894538 894593 894648 894704 894759 894814 55 785 894869 894925 894980 895036 895091 895146 895201 895257 895312 895367 55 786 895423 895478 895533 895588 895644 895699 895754 895809 895864 895919 55 787 895975 896029 896085 896140 896195 896251 896306 896361 896416 896471 55 788 896526 896581 896636 896692 896747 896802 896857 896912 896967 897022 55 789 897077 897132 897184 897242 897297 897352 897407 897462 897517 897572 55 N 0 1 2 3 4 5 6 7 8 9 D 790 897627 897682 897737 897792 897847 897902 897957 898012 898067 898122 55 791 898176 898231 898286 898341 898396 898451 898506 898561 898615 898670 55 792 898725 898780 898835 898889 898944 898999 899054 899109 899164 899218 55 793 899273 899328 899383 899437 899492 899547 899602 899656 899711 899766 55 794 899821 899875 899929 899985 900039 900094 900149 900203 900258 900312 55 795 900367 900422 900476 900531 900586 900640 900695 900749 900804 900859 55 796 900913 900968 901022 901077 901131 901186 901240 901295 901349 901404 55 797 901458 901513 901567 901622 901676 901731 901785 901839 〈◊〉 901948 54 798 902003 902057 902112 902166 902221 902275 902329 902384 902438 902492 54 799 902547 902601 902655 902709 902764 902818 902873 902927 902981 903036 54 800 903089 903144 〈◊〉 903253 903307 903361 903416 903469 903524 903578 54 801 903633 903687 903741 903795 903849 903904 903956 904012 904066 904120 54 802 904174 904229 904283 904337 904391 904445 904499 904553 904607 904661 54 803 904716 904769 904824 904878 904932 904986 905039 905094 905148 905202 54 804 905256 905310 905364 905418 905472 905526 905580 905634 905688 905742 54 805 905796 〈◊〉 905904 905958 906012 906066 906119 906173 906227 906281 54 806 906335 906389 906443 906497 906551 906604 906658 906712 906766 906819 54 807 〈◊〉 906927 907981 907035 907089 907143 907196 907250 907304 907358 54 808 907411 907465 907519 907573 907626 907680 907734 907787 907841 907895 54 809 907949 908002 908056 908109 908163 908217 908270 908324 908378 908431 54 810 908485 908539 908592 908646 908699 908753 908807 908860 908914 908967 54 811 909021 909074 909128 909181 909235 909289 909341 909396 909449 909503 54 812 905556 909609 909663 909716 909769 909823 909877 909930 909984 910037 53 813 910091 910144 910197 910251 910304 910358 910411 910464 910518 910571 53 814 910624 910678 910731 910784 910838 910891 910944 910998 911051 911104 53 815 911158 911211 911263 911317 911371 911424 911477 911530 911584 911637 53 816 911690 911743 911797 911849 911903 911956 912009 912063 912116 912169 53 817 912222 912275 912323 912381 912435 912488 912541 912594 912647 912700 53 818 912753 912806 912859 912913 912966 913019 913072 913125 913178 913231 53 819 913284 913337 913380 913443 913496 913549 913602 913655 913708 913761 53 N 0 1 2 3 4 5 6 7 8 9 D 820 913814 913867 913919 913973 914026 914079 914132 914184 914237 914290 53 821 914343 914396 914449 914502 914555 914608 914660 914713 914766 914819 53 822 914872 914925 914977 915030 915083 915136 915189 915241 915294 915347 53 823 915399 915453 915505 915558 915611 915664 915716 915769 915822 915875 53 824 915927 915979 916033 916085 916138 916191 916243 916296 916349 916401 53 825 916454 916507 916559 916612 916664 916717 916769 916822 916875 916927 53 826 916980 917033 917085 917138 917190 917243 917295 917348 917400 917453 53 827 917506 917558 917611 917663 917716 917768 917820 917873 917925 917978 52 828 918030 918083 918135 918188 918240 918293 918345 918397 918449 918502 52 829 918555 918607 918659 918712 918764 918816 918869 918921 918973 919026 52 830 919078 919130 919183 919235 919287 919339 919392 919444 919496 919549 52 831 919601 919653 919706 919758 919810 919862 919914 919967 920019 920071 52 832 920123 920176 920228 920279 920332 920384 920436 920489 920541 920593 52 833 920645 920697 920749 920801 920853 920906 920958 921009 921062 921114 52 834 921166 921218 921270 921322 921374 921426 921478 921530 921582 921634 52 835 921686 921738 921790 921842 921894 921946 921998 922050 922102 922154 52 836 922206 922258 922310 922362 922414 922466 922518 922569 922622 922674 52 837 922725 922777 922829 922881 922933 922985 923037 923089 923140 923192 52 838 923244 923296 923348 923399 923451 923503 923555 923607 923658 923710 52 839 923762 923814 923865 923917 923969 924021 924072 924124 924176 924228 52 840 924279 924331 924383 924434 924486 924538 924589 924641 924693 924744 52 841 924796 924848 924899 924951 925003 925054 925106 925157 925209 925261 52 842 925312 925364 925415 925461 925518 925569 925621 925673 925725 925776 52 843 925828 925879 925931 925982 926034 926085 926137 926188 926239 926291 51 844 926342 926394 926445 926497 926548 926599 926651 926702 926754 〈◊〉 51 845 926857 926908 926959 927011 927062 927114 927165 927216 927268 927319 51 846 927370 927422 927473 927524 927576 927627 927678 927729 927781 927832 51 847 927883 927935 927986 928037 928088 928139 928191 928242 928293 928345 51 848 928396 928447 928498 928549 928601 928652 928703 928754 928805 928857 51 849 928908 928959 929009 929061 929112 929163 929215 929266 929317 929368 51 N 0 1 2 3 4 5 6 7 8 9 D 850 929419 929470 929521 929572 929623 929674 929725 929776 929827 929879 51 851 929929 929981 930032 930083 930134 930185 930236 930287 930338 930389 51 852 930439 930491 930542 930592 930643 930694 930745 930796 930847 930898 51 853 930949 930999 931051 931102 931153 931204 931254 931305 931356 931407 51 854 931458 931509 931559 931610 931661 931712 931763 931814 931865 931915 51 855 931966 932017 932068 932118 932169 932220 932271 932322 932372 932423 51 856 932474 932524 932575 932626 932677 932727 932778 932829 932879 932930 51 857 932981 933031 933082 933133 933183 933234 933284 933335 933386 933437 51 858 933487 933538 933589 933639 933689 933740 933791 933841 933892 933943 51 859 933993 934044 934094 934145 934195 934246 934296 934347 934397 934448 51 860 934498 934549 934599 934649 934700 934751 934801 934852 934902 934953 50 861 935003 935056 935104 935154 935205 935255 935306 935356 935406 935457 50 862 935507 935558 935608 935658 935709 935759 935809 935859 935910 935960 50 863 936011 936061 936111 936162 936212 936262 936313 936363 〈◊〉 〈◊〉 50 864 936514 936564 936614 936665 936715 936765 936815 636865 936916 936966 50 865 937016 937066 937117 937167 937217 937267 937317 937367 937418 937468 50 866 937518 937568 937618 937668 937718 937769 937819 937869 937919 937969 50 867 938019 938069 938119 938169 938219 938269 938319 938369 938419 938469 50 868 938519 938569 938619 938669 938719 938769 938819 938869 938919 938969 50 869 939019 939069 939119 939169 939219 939269 939319 939369 939419 939469 50 870 939519 939569 939619 939669 939719 939769 939819 939869 939918 939968 50 871 940018 940068 940118 940168 940218 940267 940317 940367 940417 940467 50 872 940516 940566 940616 940666 940716 940765 940815 940865 940915 940964 50 873 941014 941064 941114 941163 941213 941263 941313 941362 941412 941462 50 874 941511 941561 941611 941660 941710 941759 941809 946859 941909 941958 50 875 942008 942058 942107 942157 942207 942256 942306 942355 942405 942455 50 876 942504 942554 942603 942653 942702 942752 942801 942851 942901 942950 50 877 942999 943049 943099 943148 943198 943247 943297 943346 943496 943445 49 878 943495 943544 943594 943643 943692 943742 943791 943841 943890 943939 49 879 943989 944038 944088 944137 944186 944236 944285 944335 944384 944433 49 N 0 1 2 3 4 5 6 7 8 9 D 880 944483 944532 944581 944631 944680 944729 944779 944828 944877 944927 49 881 944976 945025 945074 945124 945873 945222 945272 945321 945370 945419 49 882 945468 945518 945567 945616 945665 945715 945764 945813 945862 945912 49 883 945961 946009 946059 946108 946157 946207 946256 946305 946354 946403 49 884 946452 946501 946551 946599 946649 946698 946747 946796 946845 946894 49 885 946943 946992 947041 947090 947139 947189 947238 947287 947336 947385 49 886 947434 947483 947532 947581 947629 947679 947728 947777 947826 947875 49 887 947924 947973 948022 948070 948119 948168 948217 948266 948315 948364 49 888 948413 948462 948511 948559 948609 948657 948706 948755 948804 948853 49 889 948902 948951 948999 949048 949097 949146 949195 949244 949292 949341 49 890 949390 949439 949488 949536 949585 949633 949683 949731 949780 949829 49 891 949878 949926 949975 950024 950073 950121 950170 950219 950267 950316 49 892 950365 950414 950462 950511 950559 950608 950657 950706 950754 950803 49 893 950851 950900 950949 950997 951046 951095 951143 951192 951240 951289 49 894 951338 951386 951435 951483 951532 951580 951629 951677 951729 951775 49 895 951823 951872 951920 951969 952017 952066 952114 952163 952210 952259 49 896 952308 952356 952405 952453 952502 952550 952399 952647 952696 952744 48 897 952792 952841 952889 952938 952986 953034 953083 953131 953179 953228 48 898 953276 953325 953373 953421 953469 953518 953566 953615 953663 953711 48 899 953759 953808 953856 953905 953953 954001 954049 954099 954146 954194 48 900 954243 954292 954339 954387 954435 954484 954532 954580 954628 954677 48 901 954725 954773 954821 954869 954918 954966 955014 955062 955110 955158 48 902 955207 955255 955303 955351 955399 955447 955495 955543 955592 955639 48 903 955688 955736 955784 955832 955880 955928 955976 956024 956075 956120 48 904 956168 956216 956265 956313 956361 956409 956457 956505 956533 956601 48 905 956649 956697 956745 956793 956840 956888 956936 956984 957032 957080 48 906 957128 957176 957224 955272 957319 957368 957416 957464 957512 957559 48 907 957607 957655 957703 957751 957799 957847 957894 957942 957900 958038 48 908 958086 958134 958181 958229 958277 958325 958373 958421 958468 958516 48 909 958564 958612 958659 958707 958755 958803 958850 958898 958946 958994 48 N 0 1 2 3 4 5 6 7 8 9 D 910 959041 959089 959137 959185 959231 959279 959328 959375 959423 959471 48 911 959518 959566 959614 959661 959709 959757 959804 959852 959899 959947 48 912 959995 960042 960090 960138 960185 960233 960280 960328 960376 960423 48 913 960471 960518 960566 960613 960661 960709 960756 960804 960851 960899 48 914 960946 960994 961041 961089 961136 961184 961231 961279 961326 961374 47 915 961421 961469 961516 961563 961611 961658 961706 961753 961801 961848 47 916 961895 961943 961990 962038 962085 962132 962179 962227 962275 962322 47 917 962369 962417 962464 962511 962559 962606 962653 962701 969748 962795 47 918 962842 962886 962937 962985 963032 963079 963126 963174 963221 963268 47 919 963315 963363 963410 963457 963504 963552 963599 963646 963693 963741 47 920 963788 963835 963882 963929 963977 964024 964071 964118 964165 964212 47 921 964259 964307 964354 964401 964448 964495 964542 964589 964637 964684 47 922 964731 964778 964825 964872 964919 964966 965013 965061 965108 965155 47 923 965202 965249 965296 965343 965389 965437 965484 965531 965578 965624 47 924 965672 965719 965766 965813 965859 965906 965954 966001 966048 966095 47 925 966142 966189 966239 966283 966329 966376 966423 966470 966517 966564 47 926 966611 966658 966705 966752 966799 966845 966892 966939 966986 967033 47 927 967079 967127 967173 967220 967267 967314 967361 967408 967454 967501 47 928 967548 967595 967642 967688 967735 967782 967829 967875 967922 967969 47 929 968016 968062 968109 968156 968202 968249 968296 968343 968389 968436 47 930 968483 968529 968576 968623 968669 968716 968763 968809 968856 968902 47 931 968949 968996 969043 969089 969136 969183 969229 969276 969323 969369 47 932 969416 969463 969509 969556 969602 969649 969695 969741 969789 969835 47 933 969882 969928 969975 970021 970068 970114 970161 970207 970254 970300 47 934 970347 970393 970439 970486 970533 970579 970626 970672 970719 970765 46 935 970812 970858 970904 970951 970997 971044 971090 971137 971183 971229 46 936 971286 971322 971369 971415 971461 971508 971554 971601 971647 971693 46 937 971739 971786 971832 971879 971925 971971 972018 972064 972110 972157 46 938 972203 972249 972295 972342 972388 972434 972481 972527 972573 972619 46 939 972666 972712 972758 972804 972851 972897 972943 972989 973035 973082 46 N 0 1 2 3 4 5 6 7 8 9 D 940 973128 973174 973220 973266 973313 973359 973405 973451 973497 973543 46 941 973589 973636 973682 973728 973774 973820 973866 973913 973959 974005 46 942 974050 974097 974143 974189 974235 974281 974327 974374 974419 974466 46 943 974512 974558 974604 974649 974695 974742 974788 974834 974819 974926 46 944 974972 975018 975064 975109 975156 975202 975248 975294 975339 975386 46 945 975432 975478 975524 975569 975616 975662 975707 975753 975799 975845 46 946 975891 975937 975983 976029 976075 976121 976167 976212 976258 976304 46 947 976349 976396 976442 976488 976533 976579 976625 976671 976717 976763 46 948 976808 976854 976899 976946 976992 977037 977083 977129 977175 977220 46 949 977266 977312 977358 977403 977449 977495 977541 977586 977632 977678 46 950 977724 977769 977815 977861 977906 977952 977998 978042 978089 978135 46 951 978181 978226 978272 978317 978363 978409 978454 978500 978546 978591 46 952 978637 978683 978728 978774 978819 978865 978911 978956 979002 979047 46 953 979093 979138 979184 979229 979275 979321 979366 979412 979457 979503 46 954 979548 979594 979639 979685 979730 979776 979821 979867 979912 979958 46 955 980003 980049 980094 980139 980185 980231 980276 980322 980367 980412 45 956 980458 980503 980549 980594 980639 980685 980730 980776 980821 980867 45 957 980912 980957 981003 981048 981093 981139 981184 981229 981275 981320 45 958 981366 981411 981456 981501 981547 981592 981637 981683 981728 981773 45 959 981819 981864 981909 981954 981999 982045 982090 982135 982181 982226 45 960 982271 982316 982362 982407 982452 982497 982543 982588 982633 982678 45 961 982723 982769 982814 982859 982904 982949 982994 983039 983085 983129 45 962 983175 983220 983265 983310 983356 983401 983446 983490 983536 983581 45 963 983626 983671 983716 983762 983807 983852 983897 983942 983987 984032 45 964 984077 984122 984167 984212 984257 984302 984347 984392 984437 984482 45 965 984527 984572 984617 984662 984707 984752 984797 984842 984887 984932 45 966 984977 985022 985067 985112 985157 985202 985247 985292 985337 985382 45 967 985426 985471 985516 985561 985606 985651 985696 985741 985786 985830 45 968 985875 985920 985965 986009 986055 986099 986144 986189 986234 986279 45 969 986324 986369 986413 986458 986504 986548 986593 986637 986682 986727 45 N 0 1 2 3 4 5 6 7 8 9 D 970 986772 986817 986861 986906 986951 986996 987040 987085 987129 987175 45 971 987219 987264 987309 987353 987398 987443 987488 987532 987577 987622 45 972 987666 987711 987756 987800 987845 987889 987934 987979 988024 988068 45 973 988113 988157 988202 988247 988291 988336 988381 988425 988469 988514 45 974 988559 988604 988748 988693 988737 988782 988826 988871 988916 988960 45 975 989005 989049 989094 989138 989183 989227 989272 989316 989361 989405 45 976 989449 989494 989539 989584 989628 989672 989717 989761 989806 989850 44 977 989895 989939 989983 990028 990072 990117 990161 990206 980150 990294 44 978 990339 990383 990428 990472 990516 990561 990605 990649 990694 990738 44 979 990783 990827 990871 990916 990960 991004 991049 991093 991137 991182 44 980 991226 991270 991315 991359 991403 991448 991492 991536 991580 991625 44 981 991669 991713 991758 991802 991846 991890 991935 991979 992023 992067 44 982 992111 992156 992199 992244 992288 992333 992377 992421 992465 992509 44 983 992554 992598 992642 992686 992730 992774 992819 992863 992907 992951 44 984 992995 993039 993083 993127 993172 993216 993259 993304 993348 993392 44 985 993436 993480 993524 993568 993613 993657 993701 993745 993789 993833 44 986 993877 993921 993965 994009 994053 994097 994141 994185 994229 994273 44 987 994317 994361 994405 994449 994493 994537 994581 994625 994669 994713 44 988 994756 994801 994845 994889 994933 994977 995021 995065 995108 995152 44 989 995196 995240 995284 995328 995372 995416 995459 995504 995547 995591 44 990 995635 995679 995723 995764 995811 995854 995898 995942 995986 996029 44 991 996074 996117 996161 996205 996249 996293 996337 996380 996424 996468 44 992 996512 996555 996599 996643 996687 996731 996774 996818 996862 996906 44 993 996949 996993 997037 997080 997124 997168 997212 997255 997299 997343 44 994 997386 997430 997474 997517 997561 997605 997648 997692 997736 997779 44 995 997823 997867 997910 997954 997998 998041 998085 998129 998170 998216 44 996 998259 998303 998347 998390 998434 998477 998521 998564 998608 998652 44 997 998695 998739 998783 998826 998869 998913 998956 998999 999043 999087 44 998 999133 999174 999218 999261 999305 999348 999392 999435 999479 999522 44 999 999565 999609 999652 999696 999739 999783 999826 999869 999913 999957 43 A TABLE OF PROPORTIONAL PARTS , WHEREBY The Intermediate Logarithms of all Numbers , AND The Numbers of all Logarithms from 10000 to 100000 may more readily be found out by the foregoing Table of Logarithms . LONDON , Printed by J. Heptinstall for W. Freeman , at the Artichoke next St. Dunstan's Church in Fleetstreet . MDCLXXXVII . A TABLE OF Proportional Parts . D 1 2 3 4 5 6 7 8 9 43 4 8 12 17 21 25 30 34 38 44 4 8 13 17 22 26 30 35 39 45 4 9 13 18 22 27 31 36 40 46 4 9 13 18 23 27 32 36 41 47 4 9 14 18 23 28 32 37 42 48 4 9 14 19 24 28 33 38 43 49 4 9 14 19 24 29 34 39 44 50 5 10 15 20 25 30 35 40 45 51 5 10 15 20 25 30 35 40 45 52 5 10 15 20 26 31 36 41 46 53 5 10 15 21 26 31 37 42 47 54 5 10 16 21 27 32 37 43 48 55 5 11 16 22 27 33 38 44 49 56 5 11 16 22 28 33 39 44 50 57 5 11 17 22 28 34 39 45 51 58 5 11 17 23 29 34 40 46 52 59 5 11 17 23 29 35 41 47 53 60 6 12 18 24 30 36 42 48 54 61 6 12 18 24 30 36 42 48 54 62 6 12 18 24 31 37 43 49 55 D 1 2 3 4 5 6 7 8 9 63 6 12 18 25 31 37 44 50 56 64 6 12 19 25 32 38 44 51 57 65 6 13 19 26 32 39 45 52 58 66 6 13 19 26 33 39 46 52 59 67 6 13 20 26 33 40 46 53 60 68 6 13 20 27 34 40 47 54 61 69 6 13 20 27 34 41 48 55 62 70 7 14 21 28 35 42 49 56 63 71 7 14 21 28 35 42 49 56 63 72 7 14 21 28 36 43 50 57 64 73 7 14 21 29 36 43 51 58 65 74 7 14 22 29 37 44 51 59 66 75 7 15 22 30 37 45 52 60 67 76 7 15 22 30 38 45 53 60 68 77 7 15 23 30 38 46 53 61 69 78 7 15 23 31 39 46 54 62 70 79 7 15 23 31 39 47 55 63 71 80 8 16 24 32 40 48 56 64 72 81 8 16 24 32 40 48 56 64 72 82 8 16 24 32 41 49 57 65 73 83 8 16 24 33 41 49 58 66 74 84 8 16 25 33 42 50 58 67 75 85 8 17 25 34 42 51 59 68 76 86 8 17 25 34 43 51 60 68 77 87 8 17 26 34 43 52 60 69 78 88 8 17 26 35 44 52 61 70 79 89 8 17 26 35 44 53 62 71 80 90 9 18 27 36 45 54 63 72 81 91 9 18 27 36 45 54 63 72 81 92 9 18 27 36 46 55 64 73 82 D 1 2 3 4 5 6 7 8 9 93 9 18 27 37 46 55 65 74 83 94 9 18 28 37 47 56 〈◊〉 75 84 95 9 19 28 38 47 57 66 76 85 96 9 19 28 38 48 57 67 76 86 97 9 19 29 38 48 58 67 77 87 98 9 19 29 39 49 58 68 78 88 99 9 19 29 39 49 59 69 79 89 100 10 20 30 40 50 60 70 80 90 101 10 20 30 40 50 60 70 80 90 102 10 20 30 40 51 61 71 81 91 103 10 20 30 41 51 61 72 82 92 104 10 20 31 41 52 62 72 83 93 105 10 21 31 42 52 63 73 84 94 106 10 21 31 42 53 63 74 84 95 107 10 21 32 42 53 64 74 85 96 108 10 21 32 43 54 64 75 86 97 109 10 21 32 43 54 65 76 87 98 110 11 22 33 44 55 66 77 88 99 111 11 22 33 44 55 66 77 88 99 112 11 22 33 44 56 67 78 89 100 113 11 22 33 45 57 67 78 90 101 114 11 22 34 45 57 68 79 91 102 115 11 23 34 46 57 69 80 92 103 116 11 23 34 46 58 69 81 92 104 117 11 23 35 46 58 70 81 73 105 118 11 23 35 47 59 70 82 94 106 119 11 23 35 47 59 71 83 95 107 120 12 24 36 48 60 72 84 96 108 121 12 24 36 48 60 72 84 96 108 122 12 24 36 48 61 73 85 97 109 D 1 2 3 4 5 6 7 8 9 123 12 24 36 48 61 73 86 98 110 124 12 24 37 49 62 74 86 99 111 125 12 25 37 50 62 75 87 100 112 126 12 25 37 50 63 75 88 100 113 127 12 25 38 50 63 76 88 101 114 128 12 25 38 51 64 76 89 102 115 129 12 25 38 51 64 77 90 103 116 130 13 26 39 52 65 78 91 104 117 131 13 26 39 52 65 78 91 104 117 132 13 26 39 52 66 79 92 105 118 133 13 26 39 53 66 79 93 106 119 134 13 26 40 53 67 80 93 107 120 135 13 27 40 54 67 81 94 108 121 136 13 27 40 54 68 81 95 108 122 137 13 27 41 54 68 82 95 109 123 〈◊〉 13 27 41 55 69 82 96 110 124 139 13 27 41 55 69 83 97 111 125 140 14 28 42 56 70 84 98 112 126 141 14 28 42 56 70 84 99 112 126 142 14 28 42 56 71 85 99 113 127 143 14 28 42 57 71 85 100 114 128 144 14 28 43 57 72 86 100 115 129 145 14 28 43 58 72 87 101 116 130 146 14 29 43 58 73 87 102 116 131 147 14 29 44 58 73 88 102 117 132 148 14 29 44 59 74 88 103 118 133 149 14 29 44 59 74 89 104 119 134 150 15 30 45 60 75 90 105 120 135 151 15 30 45 60 75 90 105 120 135 152 15 30 45 60 76 91 106 121 136 D 1 2 3 4 5 6 7 8 9 153 15 30 45 60 76 91 107 122 137 154 15 30 46 61 77 92 107 123 138 155 15 31 46 62 77 93 108 124 139 156 15 31 46 62 78 93 109 124 140 157 15 31 47 62 78 94 109 125 141 158 15 31 47 63 79 94 110 126 142 159 15 31 47 63 79 95 111 127 143 160 16 32 48 64 80 96 112 128 144 161 16 32 48 64 80 96 112 128 144 162 16 32 48 64 81 97 113 129 145 163 16 32 48 65 82 98 114 130 146 164 16 32 49 66 82 98 114 131 147 165 16 33 49 66 82 99 115 132 148 166 16 33 49 66 83 99 116 132 149 167 16 33 50 66 83 100 116 133 150 168 16 33 50 67 84 100 〈◊〉 134 151 169 17 33 50 67 84 101 118 135 152 170 17 34 51 68 85 102 119 136 153 171 17 34 51 68 85 102 119 136 153 172 17 34 51 68 86 103 120 137 154 173 17 34 51 69 86 103 121 138 155 174 17 34 52 69 87 104 121 139 156 175 17 34 52 70 87 105 122 140 157 176 17 35 52 70 88 105 123 140 158 177 17 35 53 70 88 106 123 141 159 178 17 35 53 71 89 106 124 142 160 179 17 35 53 71 89 107 125 143 161 180 18 36 54 72 90 108 126 144 162 181 18 36 54 72 90 108 126 144 162 182 18 36 54 72 91 109 127 145 163 D 1 2 3 4 5 6 7 8 9 183 18 36 54 73 91 109 128 146 164 184 18 36 55 73 92 110 128 147 165 185 18 37 55 74 92 111 129 148 166 186 18 37 55 74 93 111 130 148 167 187 18 37 56 74 83 112 130 149 168 188 18 37 56 75 94 112 131 150 169 189 18 37 56 75 94 113 132 151 170 190 19 38 57 76 95 114 133 152 171 191 19 38 57 76 95 114 133 152 171 192 19 38 57 76 96 115 134 153 172 193 19 38 57 77 96 115 135 154 173 194 19 38 58 77 97 116 135 155 174 195 19 39 58 78 97 117 136 156 175 196 19 39 59 78 98 117 136 156 176 197 19 39 59 78 98 118 137 157 177 198 19 39 59 79 99 118 138 158 178 199 19 39 59 79 99 119 139 159 179 200 20 40 60 80 100 120 140 160 180 201 20 40 60 80 100 120 140 160 180 202 20 40 60 80 101 121 141 161 181 203 20 40 60 81 101 121 142 162 182 204 20 40 61 81 102 122 142 163 183 205 20 41 61 82 102 123 143 164 184 206 20 41 61 82 103 123 144 164 185 207 20 41 62 82 103 124 144 165 186 208 20 41 62 83 104 124 145 166 187 209 20 41 62 83 104 125 146 167 188 210 21 42 63 84 105 126 147 168 189 211 21 42 63 84 105 126 147 168 189 212 21 42 63 84 106 127 148 169 190 D 1 2 3 4 5 6 7 8 9 213 21 42 63 85 106 127 149 170 191 214 21 42 64 85 107 128 149 171 192 215 21 43 64 86 107 129 150 172 193 216 21 43 64 86 108 129 151 172 194 217 21 43 65 86 108 130 151 173 195 218 21 43 65 87 109 130 152 174 196 219 21 43 65 87 109 131 153 175 197 220 22 44 66 88 110 132 154 176 198 221 22 44 66 88 110 132 154 176 198 222 22 44 66 88 111 133 155 177 199 223 22 44 66 89 111 133 156 178 200 224 22 44 67 89 112 134 156 179 201 225 22 45 67 90 112 135 157 180 202 226 22 45 67 90 113 135 158 180 203 227 22 45 68 90 113 136 158 181 204 228 22 45 68 91 114 136 159 182 205 229 22 45 68 91 114 137 160 183 206 230 23 46 69 92 115 138 161 184 207 231 23 46 69 92 115 138 161 184 207 232 23 46 69 92 116 139 162 185 208 233 23 46 69 93 116 139 163 186 209 234 23 46 70 93 117 140 163 187 210 235 23 47 70 94 117 141 164 188 211 236 23 47 70 94 118 141 165 188 212 237 23 47 71 94 118 142 165 189 213 238 23 47 71 95 119 142 166 190 214 239 23 47 71 95 119 143 167 191 215 240 24 48 72 96 120 144 168 192 216 241 24 48 72 96 120 144 168 192 216 242 24 48 72 96 121 145 169 193 217 D 1 2 3 4 5 6 7 8 9 243 24 48 72 97 121 145 170 194 218 244 24 48 73 97 122 146 170 195 219 245 24 49 73 98 122 147 171 196 220 246 24 49 73 98 123 147 172 196 221 247 24 49 74 98 123 148 172 197 222 248 24 49 74 99 124 148 173 198 223 249 24 49 74 99 124 149 174 199 224 250 25 50 75 100 125 150 175 200 225 251 25 50 75 100 125 150 175 200 225 252 25 50 75 100 126 151 176 201 226 253 25 50 75 101 126 151 177 202 227 254 25 50 76 101 127 152 177 203 228 255 25 50 76 102 127 153 178 204 229 256 25 51 76 102 128 153 179 204 230 257 25 51 77 102 128 154 179 205 231 258 25 51 77 103 129 154 180 206 232 259 25 51 77 103 129 155 181 207 233 260 26 52 78 104 130 156 182 208 234 261 26 52 78 104 130 156 182 208 234 262 26 52 78 104 131 156 183 209 235 263 26 52 78 105 131 157 184 210 236 264 26 52 79 105 132 158 184 211 237 265 26 53 79 106 132 159 185 212 238 266 26 53 79 106 133 159 186 212 239 267 26 53 80 106 133 160 186 213 240 268 26 53 80 107 134 160 187 214 241 269 26 53 80 107 134 161 188 215 242 270 27 54 81 108 135 162 189 216 243 271 27 54 81 108 135 162 189 216 243 272 27 54 81 108 136 163 190 217 244 D 1 2 3 4 5 6 7 8 9 273 27 54 81 109 136 163 191 218 245 274 27 54 82 109 137 164 191 219 246 275 27 55 82 110 137 165 192 220 247 276 27 55 82 110 138 165 193 220 248 277 27 55 83 110 138 166 193 221 249 278 27 55 83 111 139 166 194 222 250 279 27 55 83 111 139 167 195 223 251 280 28 56 84 112 140 168 196 224 252 281 28 56 84 112 140 168 196 224 252 282 28 56 84 112 141 169 197 225 253 283 28 56 84 113 141 169 198 226 254 284 28 56 85 113 142 170 198 227 255 285 28 57 85 114 142 171 199 228 256 286 28 57 85 114 143 171 200 228 257 287 28 57 86 114 143 172 200 229 258 288 28 57 86 115 144 172 201 230 259 289 28 57 86 115 144 173 202 231 260 290 29 58 87 116 145 174 203 232 261 291 29 58 87 116 145 174 203 232 261 292 29 58 87 116 146 175 204 233 262 293 29 58 87 117 146 175 205 234 263 294 29 58 88 117 147 176 205 235 264 295 29 59 88 118 147 177 206 236 265 296 29 59 88 118 148 177 207 236 266 297 29 59 88 118 148 178 207 237 267 298 29 59 89 119 149 178 208 238 268 299 29 59 89 119 149 179 209 239 269 300 30 60 90 120 150 180 210 240 270 301 30 60 90 120 150 180 210 240 270 302 30 60 90 120 151 181 211 241 271 D 1 2 3 4 5 6 7 8 9 303 30 60 90 121 151 181 212 242 272 304 30 60 91 121 152 182 212 243 273 305 30 61 91 122 152 183 213 244 274 306 30 61 91 122 153 183 214 244 275 307 30 61 92 122 153 184 214 245 276 308 30 61 92 123 154 184 215 246 277 309 30 61 92 123 154 185 216 247 278 310 31 62 93 124 155 186 217 248 279 311 31 62 93 124 155 186 217 248 279 312 31 62 93 124 156 187 218 249 280 313 31 62 93 125 156 187 219 250 281 314 31 62 94 125 157 183 219 251 282 315 31 63 94 126 157 189 220 252 283 316 31 63 94 126 158 189 221 252 284 317 31 63 95 126 158 190 221 253 285 318 31 63 95 127 159 190 222 254 286 319 31 63 95 127 159 191 223 255 287 320 32 64 96 128 160 192 224 256 288 321 32 64 96 128 160 192 224 256 288 322 32 64 96 128 161 193 225 257 289 323 32 64 96 129 161 193 226 258 290 324 32 64 97 129 162 194 226 259 291 325 32 65 97 130 162 195 227 260 292 326 32 65 97 130 163 195 228 260 293 327 32 65 98 130 163 196 228 261 294 328 32 65 98 131 163 196 229 262 295 329 32 65 98 131 164 197 230 263 296 330 33 66 99 132 165 198 231 264 297 331 33 66 99 132 165 198 231 264 297 332 33 66 99 132 166 199 232 265 298 D 1 2 3 4 5 6 7 8 9 333 33 66 99 133 166 199 233 266 299 334 33 66 100 133 167 200 233 267 300 335 33 67 100 134 167 201 234 268 301 336 33 67 100 134 168 201 235 268 302 337 33 67 101 134 168 202 235 269 303 338 33 67 101 135 169 202 236 270 304 339 33 67 101 135 169 203 237 271 305 340 34 68 102 136 170 204 238 272 306 341 34 68 102 136 170 204 238 272 306 342 34 68 102 136 171 205 239 273 307 343 34 68 102 137 171 205 240 274 308 344 34 68 103 137 172 206 240 275 309 345 34 69 103 138 172 207 241 276 310 346 34 69 103 138 173 207 242 276 311 347 34 69 104 138 173 208 242 277 312 348 34 69 104 139 174 208 243 278 313 349 34 69 104 139 174 209 244 279 314 350 35 70 105 140 175 210 245 280 315 351 35 70 105 140 175 210 245 280 315 352 35 70 105 140 176 211 246 281 316 353 35 70 105 141 176 211 247 282 317 354 35 70 106 141 177 212 247 283 318 355 35 71 106 142 177 213 248 284 319 356 35 71 106 142 178 213 249 284 320 357 35 71 107 142 178 214 249 285 321 358 35 71 107 143 179 214 250 286 322 359 35 71 107 143 179 215 251 287 323 360 36 72 108 144 180 216 252 288 324 361 36 72 108 144 180 216 252 288 324 362 36 72 108 144 181 217 253 289 325 D 1 2 3 4 5 6 7 8 9 363 36 72 108 145 181 217 254 290 326 364 36 72 109 145 182 218 254 291 327 365 36 73 109 146 182 219 255 292 328 366 36 73 109 146 182 219 256 292 329 367 36 73 110 146 183 220 256 293 330 368 36 73 110 147 184 220 257 294 331 369 36 73 110 147 184 221 258 295 332 370 37 74 111 148 185 222 259 296 333 371 37 74 111 148 185 222 259 296 333 372 37 74 111 148 186 223 260 297 334 373 37 74 111 149 186 223 261 298 335 374 37 74 112 149 187 224 261 299 336 375 37 75 112 150 187 225 262 300 337 376 37 75 112 150 188 225 263 300 338 377 37 75 113 150 188 226 263 301 339 378 37 75 113 151 189 226 264 302 340 379 37 75 113 151 189 227 265 303 341 380 38 76 114 152 190 228 266 304 342 381 38 76 114 152 190 228 266 304 342 382 38 76 114 152 191 229 267 305 343 383 38 76 114 153 191 229 268 306 344 384 38 76 115 153 192 230 268 307 345 385 38 77 115 154 192 231 269 308 346 386 38 77 115 154 193 231 270 308 347 387 38 77 116 154 193 232 270 309 348 388 38 77 116 155 194 232 271 310 349 389 38 77 116 155 194 233 272 311 350 390 39 78 117 156 195 233 273 312 351 391 39 78 117 156 195 233 273 312 351 392 39 78 117 156 196 234 274 313 352 D 1 2 3 4 5 6 7 8 9 393 39 78 117 157 196 235 275 314 353 394 39 78 118 157 197 236 275 315 354 395 39 79 118 158 197 237 276 316 355 396 39 79 118 158 198 237 277 316 356 397 39 79 119 158 198 238 277 317 357 398 39 79 119 159 199 238 278 318 358 399 39 79 119 159 199 239 279 319 359 400 40 80 120 160 200 240 280 320 360 401 40 80 120 160 200 240 280 320 360 402 40 80 120 160 201 241 281 321 361 403 40 80 120 161 201 241 282 322 362 404 40 80 121 161 202 242 282 323 363 405 40 81 121 162 202 243 283 324 364 406 40 81 121 162 203 243 284 324 365 407 40 81 122 162 203 244 284 325 366 408 40 81 122 163 204 244 285 326 367 409 40 81 122 163 204 245 286 327 368 410 41 82 123 164 205 246 287 328 369 411 41 82 123 164 205 246 287 328 369 412 41 82 123 164 206 247 288 329 370 413 41 82 123 165 206 247 289 330 371 414 41 82 124 165 207 248 289 331 372 415 41 83 124 166 207 249 290 332 373 416 41 83 124 166 208 249 291 332 374 417 41 83 125 166 208 250 291 333 375 418 41 83 125 167 209 250 292 334 376 419 41 83 125 167 209 251 293 335 377 420 42 84 126 168 210 252 294 336 378 421 42 84 126 168 210 252 294 336 378 422 42 84 126 168 211 253 295 337 379 D 1 2 3 4 5 6 7 8 9 423 42 84 126 169 211 253 296 338 380 424 42 84 127 169 212 254 296 339 381 425 42 85 127 170 212 255 297 340 382 426 42 85 127 170 213 255 298 340 383 427 42 85 128 170 213 256 298 341 384 428 42 85 128 171 214 256 299 342 385 429 42 85 128 171 214 257 300 343 386 430 43 86 129 172 215 258 301 344 387 431 43 86 129 172 215 258 301 344 387 432 43 86 129 172 216 259 302 345 388 433 43 86 129 173 216 259 303 346 389 434 43 86 130 173 217 260 304 347 390 435 43 87 130 174 217 261 304 348 391 A TABLE OF ARTIFICIAL SINES AND TANGENTS To every DEGREE and MINUTE OF THE QUADRANT . LONDON , Printed by J. Heptinstall for W. Freeman , at the Artichoke next St. Dunstan'S Church in Fleet street . MDCLXXXVII . Degree 0. M Sine Co-sine Tangent Co-tang . 0 0. 000000 10. 000000 0. 000000 Infinita . 60 1 6. 463726 9. 999999 6. 463726 13. 536274 59 2 6. 764756 9. 999999 6. 764756 13. 235244 58 3 6. 940847 9. 999999 6. 940847 13. 059153 57 4 7. 065786 9. 999999 7. 065786 12. 934214 56 5 7. 162696 9. 999999 7. 162696 12. 837304 55 6 7. 241877 9. 999999 7. 241878 12. 758122 54 7 7. 308824 9. 999999 7. 308825 12. 691175 53 8 7. 366816 9. 999999 7. 366817 12. 633183 52 9 7. 〈◊〉 9. 999999 7. 417970 12. 582030 51 10 7. 463726 9. 999998 7. 463727 12. 536273 50 11 7. 505118 9. 999998 7. 505120 12. 494880 49 12 7. 542906 9. 999997 7. 542909 12. 457091 48 13 7. 577668 9. 999997 7. 577272 12. 422328 47 14 7. 609853 9. 999996 7. 609857 12. 390143 46 15 7. 639816 9. 999996 7. 639826 12. 360180 45 16 7. 667844 9. 999995 7. 667849 12. 332151 44 17 7. 694173 9. 999995 7. 694179 12. 305821 43 18 7. 718977 9. 999994 7. 719003 12. 281997 42 19 7. 742477 9. 999993 7. 742484 12. 257516 41 20 7. 764754 9. 999993 7. 764761 12. 235239 40 21 7. 785943 9. 999992 7. 785951 12. 214049 39 22 7. 806146 〈◊〉 999991 7. 806145 12. 193845 38 23 7. 825451 9. 999990 7. 825460 12. 174540 37 24 7. 843034 9. 999989 7. 843944 12. 156056 36 25 7. 861662 9. 999989 7. 861674 12. 138326 35 26 7. 878695 9. 999988 7. 878708 12. 121292 34 27 7. 895085 9. 999987 7. 895099 12. 104901 33 28 7. 910879 9. 999986 7. 910894 12. 089106 32 29 7. 926119 9. 999985 7. 926134 12. 073866 31 30 7. 940842 9. 999983 7. 940858 12. 059142 30 Co-sine Sine Co-tang . Tangent M Degree 89. Degree 0. M Sine Co-sine Tangent Co-tang . 30 7. 940842 9. 999983 7. 940858 12. 059142 30 31 7. 955082 9. 999982 7. 955100 12. 044900 29 32 7. 968870 9. 999981 7. 968889 12. 031111 28 33 7. 982233 9. 999980 7. 982253 12. 017747 27 34 7. 995198 9. 999978 7. 995215 12. 004781 26 35 8. 007787 9. 999978 8. 007810 11. 992191 25 36 8. 020021 9. 999976 8. 020044 11. 979956 24 37 8. 031919 9. 999975 8. 031945 11. 968055 23 38 8. 043601 9. 999973 8. 043527 11. 956473 22 39 8. 054781 9. 999972 8. 054809 11. 945181 21 40 8. 065776 9. 999971 8. 065806 11. 934194 20 41 8. 076500 9. 999969 8. 076531 11. 923469 19 42 8. 086965 9. 999968 8. 086997 11. 913003 18 43 8. 097183 9. 999966 8. 097217 11. 902783 17 44 8. 107167 9. 999964 8. 107203 11. 892797 16 45 8. 116926 9. 999963 8. 116963 11. 883037 15 46 8. 126471 9. 999961 8. 126510 11. 873490 14 47 8. 135810 9. 999959 8. 135851 11. 864149 13 48 8. 144953 9. 999958 8. 144996 11. 855004 12 49 8. 153907 9. 999956 8. 153952 11. 846048 11 50 8. 162681 9. 999954 8. 162737 11. 837273 10 51 8. 171280 9. 999952 8. 171328 11. 828672 9 52 8. 179713 9. 999950 8. 179763 11. 820237 8 53 8. 187985 9. 999948 8. 188036 11. 811964 7 54 8. 196102 9. 999946 8. 196156 11. 803844 6 55 8. 204070 9. 999944 8. 204126 11. 795674 5 56 8. 211895 9. 999942 8. 211953 11. 788047 4 57 8. 219581 9. 999940 8. 219641 11. 780359 3 58 8. 227134 9. 999938 8. 227195 11. 772805 2 59 8. 234557 9. 999936 8. 234621 11. 765379 1 60 8. 241855 9. 999934 8. 241921 11. 758079 0 Co-sine Sine Co-tang . Tangent M Degree 89. Degree 1. M Sine Co-sine Tangent Co-tang . 0 8. 241855 9. 999934 8. 241921 11. 758079 60 1 8. 249033 9. 999932 8. 249102 11. 750898 59 2 8. 256094 9. 999929 8. 256165 11. 743835 58 3 8. 263042 9. 999927 8. 263115 11. 736885 57 4 8. 269881 9. 999925 8. 269956 11. 730044 56 5 8. 276614 9. 999922 8. 276691 11. 723309 55 6 8. 283243 9. 999920 8. 283323 11. 716677 54 7 8. 289773 9. 999918 8. 289856 11. 716144 53 8 8. 296207 9. 999915 8. 296292 11. 703708 52 9 8. 302546 9. 999913 8. 302634 11. 697366 51 10 8. 308794 9. 999910 8. 308884 11. 691116 50 11 8. 314954 9. 999907 8. 315046 11. 684954 49 12 8. 321027 9. 999905 8. 321122 11. 678878 48 13 8. 327016 9. 999902 8. 327114 11. 672886 47 14 8. 332924 9. 999899 8. 333025 11. 666975 46 15 8. 338753 9. 999897 8. 338856 11. 661144 45 16 8. 344504 9. 999894 8. 344610 11. 655390 44 17 8. 350180 9. 999891 8. 350289 11. 649711 43 18 8. 355783 9. 999888 8. 355895 11. 644105 42 19 8. 361315 9. 999885 8. 361430 11. 638570 41 20 8. 366777 9. 999882 8. 366895 11. 633105 40 21 8. 372171 9. 999879 8. 372292 11. 627708 39 22 8. 377499 9. 999876 8. 377622 11. 622378 38 23 8. 387762 9. 999873 8. 382889 11. 617111 37 24 8. 387962 9. 999870 8. 388092 11. 611908 36 25 8. 393101 9. 999867 8. 393234 11. 606766 35 26 8. 398179 9. 999864 8. 398315 11. 601685 34 27 8. 403199 9. 999861 8. 403338 11. 596662 33 28 8. 408161 9. 999858 8. 408304 11. 591696 32 29 8. 413068 9. 999854 8. 413213 11. 586787 31 30 〈◊〉 9. 999851 8. 418068 11. 581932 30 Co-sine Sine Co-tang . Tangent M Degree 88. Degree 1. M Sine Co-sine Tangent Co-tang . 30 8. 417919 9. 999851 8. 418068 11. 581932 30 31 8. 422717 9. 999848 8. 422869 11. 577131 29 32 8. 427462 9. 999844 8. 427618 11. 572382 28 33 8. 432156 9. 999841 8. 432315 11. 567685 27 34 8. 436800 9. 999838 8. 436962 11. 563038 26 35 8. 441394 9. 999834 8. 441560 11. 558440 25 36 8. 445941 9. 999831 8. 446110 11. 553990 24 37 8. 450440 9. 999827 8. 450613 11. 549387 23 38 8. 454893 9. 999824 8. 455070 11. 544930 22 39 8. 459301 9. 999820 8. 459481 11. 540519 21 40 8. 463665 9. 999816 8. 463849 11. 536151 20 41 8. 467985 9. 999812 8. 468172 11. 531828 19 42 8. 472263 9. 999809 8. 472454 11. 527546 18 43 8. 476498 9. 999805 8. 476693 11. 523307 17 44 8. 480693 9. 999801 8. 480892 11. 519108 16 45 8. 484848 9. 999797 8. 485050 11. 514950 15 46 8. 488963 9. 999794 8. 486170 11. 510830 14 47 8. 493040 9. 999790 8. 483250 11. 506750 13 48 8. 497078 9. 999786 8. 497293 11. 502707 12 49 8. 501080 9. 999782 8. 501298 11. 498702 11 50 8. 505045 9. 999778 8. 505267 11. 494733 10 51 8. 508974 9. 999774 8. 509200 11. 490800 9 52 8. 512867 9. 999769 8. 513098 11. 486902 8 53 8. 516726 9. 999765 8. 516961 11. 483039 7 54 8. 520551 9. 999761 8. 520790 11. 479210 6 55 8. 524343 9. 999756 8. 524586 11. 475414 5 56 8. 528102 9. 999753 8. 528349 11. 471651 4 57 8. 531828 9. 999748 8. 532080 11. 467620 3 58 8. 535523 9. 999744 8. 535779 11. 464221 2 59 8. 539186 9. 999740 8. 539447 11. 460553 1 60 8. 542819 9. 999735 8. 543084 11. 456916 0 Co-sine Sine Co-tang . Tangent M Degree 88. Degree 2. M Sine Co-sine Tangent Co-tang . 0 8. 542819 9. 999735 8. 543084 11. 456916 60 1 8. 546422 9. 999731 8. 546691 11. 453309 59 2 8. 549995 9. 999726 8. 550268 11. 449732 58 3 8. 553558 9. 999722 8. 553817 11. 446183 57 4 8. 557054 9. 999717 8. 557336 11. 442664 56 5 8. 560540 9. 999713 8. 560827 11. 439172 55 6 8. 563999 9. 999708 8. 564291 11. 435709 54 7 8. 567431 9. 999703 8. 567727 11. 432272 53 8 8. 570836 9. 999699 8. 571137 11. 428863 52 9 8. 574214 9. 999694 8. 574520 11. 425480 51 10 8. 577566 9. 999689 8. 577877 11. 422123 50 11 8. 580892 9. 999685 8. 581208 11. 418792 49 12 8. 584193 9. 999680 8. 584514 11. 415486 48 13 8. 587469 9. 999675 8. 587795 11. 412205 47 14 8. 590721 9. 999670 8. 591051 11. 408949 46 15 8. 593948 9. 999665 8. 594283 11. 405717 45 16 8. 597152 9. 999660 8. 597492 11. 402508 44 17 8. 600332 9. 999655 8. 600677 11. 399323 43 18 8. 603488 9. 999650 8. 603838 11. 396161 42 19 8. 606622 9. 999645 8. 606978 11. 393022 41 20 8. 609734 9. 999640 8. 610094 11. 389906 40 21 8. 612823 9. 999635 8. 613189 11. 386811 39 22 8. 615891 9. 999629 8. 616262 11. 383738 38 23 8. 618937 9. 999624 8. 619313 11. 380687 37 24 8. 621967 9. 999619 8. 622343 11. 377657 36 25 8. 624965 9. 999614 8. 625352 11. 374648 35 26 8. 627948 9. 999608 8. 628340 11. 371660 34 27 8. 630911 9. 999603 8. 631308 11. 368692 33 28 8. 633854 9. 999597 8. 634456 11. 365744 32 29 8. 636776 9. 999592 8. 637184 11. 362816 31 30 8. 639679 9. 999586 8. 640093 11. 359907 30 Co-sine Sine Co-tang . Tangent M Degree 87. Degree 2. M Sine Co-sine Tangent Co-tang . 30 8. 639679 9. 999586 8. 640093 11. 359907 30 31 8. 642563 9. 999581 8. 642982 11. 357017 29 32 8. 645428 9. 999575 8. 645853 11. 354147 28 33 8. 648274 9. 999570 8. 648704 11. 351296 27 34 8. 651102 9. 999564 8. 651538 11. 348463 26 35 8. 653911 9. 999558 8. 654352 11. 345648 25 36 8. 656702 9. 999553 8. 657149 11. 342851 24 37 8. 659475 9. 999547 8. 659928 11. 340072 23 38 8. 662230 9. 999541 8. 662689 11. 337311 22 39 8. 664968 9. 999535 8. 665433 11. 334567 21 40 8. 667689 9. 999529 8. 668160 11. 331840 20 41 8. 670393 9. 999523 8. 670869 11. 329130 19 42 8. 673080 9. 999518 8. 673563 11. 326437 18 43 8. 675751 9. 999512 8. 676239 11. 323761 17 44 8. 678405 9. 999506 8. 678899 11. 321100 16 45 8. 681043 9. 999499 8. 681544 11. 318456 15 46 8. 683665 9. 999493 8. 684172 11. 315828 14 47 8. 686272 9. 999487 8. 686784 11. 313216 13 48 8. 688892 9. 999481 8. 689381 11. 310619 12 49 8. 691438 9. 999475 8. 691963 11. 308037 11 50 8. 693998 9. 999469 8. 694529 11. 305471 10 51 8. 696543 9. 999462 8. 697081 11. 302919 9 52 8. 699073 9. 999456 8. 699617 11. 300383 8 53 8. 701589 9. 999450 8. 702139 11. 297861 7 54 8. 704090 9. 999443 8. 704646 11. 295354 6 55 8. 706576 9. 999437 8. 707130 11. 292860 5 56 8. 709049 9. 999431 8. 709618 11. 290381 4 57 8. 711507 9. 999424 8. 712083 11. 287917 3 58 8. 713952 9. 999418 8. 714534 11. 285466 2 59 8. 716383 9. 999411 8. 716972 11. 283028 1 60 8. 718800 9. 999404 8. 719396 11. 280604 0 Co-sine Sine Co-tang . Tangent M Degree 87. Degree 3. M Sine Co-sine Tangent Co-tang . 0 8. 718800 9. 999404 8. 719396 11. 280604 60 1 8. 721204 9. 999398 8. 721806 11. 278194 59 2 8. 723595 9. 999391 8. 724254 11. 275796 58 3 8. 725972 9. 999384 8. 726588 11. 273412 57 4 8. 728336 9. 999378 8. 728959 11. 271041 56 5 8. 730688 9. 999371 8. 731317 11. 268683 55 6 8. 733027 9. 999364 8. 733663 11. 266337 54 7 8. 735354 9. 999357 8. 735996 11. 264004 53 8 8. 737667 9 999350 8. 738317 11. 261683 52 9 8. 739969 9. 999343 8. 740626 11. 259374 51 10 8. 742259 9. 999336 8. 742922 11. 257078 50 11 8. 744536 9. 999329 8. 745007 11. 254793 49 12 8. 746801 9. 999322 8 747479 11. 252521 48 13 8. 745955 9. 999315 8. 749740 11. 250240 47 14 8. 751297 9. 999308 8. 751989 11. 248011 46 15 8. 753528 9. 999301 8. 754227 11. 245773 45 16 8. 755747 9. 999294 8. 756453 11. 243542 44 17 8. 757955 9. 999286 8. 758668 11. 241332 43 18 8. 760151 9. 999279 8. 760872 11. 239128 42 19 8. 762337 9. 999272 8. 763065 11. 236935 41 20 8. 764511 9. 999265 8. 765246 11. 234754 40 21 8. 766675 9. 999257 8. 767417 11. 232583 39 22 8. 768828 9. 999250 8. 769578 11. 230422 38 23 8. 770970 9. 999242 8. 771727 11. 228273 37 24 8. 773101 9. 999235 8. 773866 11. 229134 36 25 8. 775223 9. 999227 8. 775995 11. 224005 35 26 8. 777333 9. 999220 8. 778114 11. 221886 34 27 8. 779434 9. 999212 8. 783222 11. 219778 33 28 8. 781524 9. 999204 8. 782320 11. 217680 32 29 8. 783605 9. 999197 8. 784404 11. 215592 31 30 8. 785675 9. 999189 8. 786486 11. 213514 30 Co-sine Sine Co-tang . Tangent M Degree 86. Degree 3. M Sine Co-sine Tangent Co-tang . 30 8. 785675 9. 999189 8. 786486 11. 213514 30 31 8. 787736 9. 999181 8. 788554 11. 211446 29 32 8. 789787 9. 999174 8. 790613 11. 209387 28 33 8. 791828 9. 999166 8. 792662 11. 207338 27 34 8. 793859 9 999158 8. 794701 11. 205299 26 35 8. 795881 9. 999150 8. 796731 11. 203269 25 36 8. 797894 9. 999142 8 798752 11 201248 24 37 8. 799897 9. 999134 8. 800763 11 199237 23 38 8. 801891 9. 999126 8. 802765 11. 197235 22 39 8 803876 9. 999118 8. 807458 11. 195242 21 40 8. 805852 9 999110 8. 806742 11. 193258 20 41 8. 807819 9. 999102 8. 808717 11. 191283 19 42 8. 809777 9. 999094 8. 812683 11. 189317 18 43 8. 811726 9 999086 8. 812641 11. 187359 17 44 8. 813667 9. 999077 8. 814589 11. 185411 16 45 8. 815598 9. 999069 8. 816529 11. 183471 15 46 8. 817522 9. 999061 8. 818461 11. 181539 14 47 8. 819436 9. 999052 8. 〈◊〉 〈◊〉 13 48 8. 821342 9. 999044 8. 822298 11. 177702 12 49 8. 823240 9. 999036 8. 824205 11. 175795 11 50 8. 825130 9. 999027 8 826103 11. 173897 10 51 8. 827011 9. 999019 8. 827992 11. 172003 9 52 8. 828884 9. 999010 8. 829874 11. 170126 8 53 8 830749 9. 999002 8. 831748 11. 168252 7 54 8. 832106 9. 998993 8. 833613 11. 166387 6 55 8. 834456 9. 998984 8. 835471 11. 164529 5 56 8. 836297 9. 998976 8. 837321 11. 162679 4 57 8. 838130 9. 998967 8. 839163 11. 160837 3 58 8. 839956 9 998958 8. 840998 11. 159002 2 59 8. 841774 9 998940 8. 842825 11. 157175 1 60 8. 843585 9. 998941 8. 844644 11. 155356 0 Co-sine Sine Co tang . Tangent . M Degree 86. Degree 4. M Sine Co-sine Tangent Co-tang . 0 8. 843584 9. 998941 8. 844644 11. 155356 60 1 8. 845387 9. 998931 8. 846455 11. 153545 59 2 8. 847183 9 998923 8. 848240 11. 151740 58 3 8. 848971 9. 998914 8. 850057 11. 149943 57 4 8. 850751 9. 998905 8. 851846 11. 148154 56 5 8. 852525 9. 998896 8. 853628 11. 146372 55 6 8. 854291 9. 998887 8. 855403 11. 144597 54 7 8. 856049 9. 998878 8. 857171 11. 142829 53 8 8. 857801 9. 998869 8. 858932 11. 141068 52 9 8. 859546 9. 998860 8. 860686 11. 139314 51 10 8. 861283 9. 998851 8. 862433 11. 137567 50 11 8. 863014 9. 998841 8. 864173 11. 135827 49 12 8. 864738 9. 998832 8. 865906 11. 134094 48 13 8. 866454 9. 998823 8. 867632 11. 132368 47 14 8 868165 9. 998813 8. 869351 11. 130649 46 15 8. 869868 9. 998804 8. 871064 11. 128936 45 16 8. 871565 9. 998795 8. 872750 11. 127230 44 17 8. 873255 9. 998785 8. 874469 11. 125531 43 18 8. 874938 9. 998776 8. 876162 11. 123838 42 19 8. 876615 9. 998766 8. 897849 11. 122151 41 20 8. 878285 9 998757 8. 879529 11. 120471 40 21 8. 879949 9. 998747 8. 881202 11. 118798 39 22 8. 881607 9. 998738 8. 882869 11. 117131 38 23 8. 883258 9. 998728 8. 884530 11. 115470 37 24 8. 884903 9. 998718 8. 886185 11. 113815 36 25 8. 886542 9. 998708 8. 887833 11. 112167 35 26 8. 888174 9. 998699 8. 889476 11. 110524 34 27 8. 889801 9. 998689 8. 891112 11. 108888 33 28 8. 891421 9. 998679 8. 892742 11. 107258 32 29 8. 893035 9. 998669 8. 894366 11. 105634 31 30 8. 894643 9. 998659 8. 845984 11. 104016 30 Co-sine . Sine Co-tang . Tangent . M Degree 85 Degree 4 M Sine Co-sine Tangent Co-tang . 30 8. 894643 9. 998659 8. 895984 11. 104016 30 31 8. 896246 9. 998649 8. 897596 11. 102404 29 32 8. 897842 9. 998639 8. 899203 11. 100797 28 33 8. 899432 9. 998629 8. 900803 11. 099197 27 34 8. 901017 9. 998619 8. 902398 11. 097602 26 35 8. 902596 9. 998609 8. 903987 11. 096013 25 36 8. 904169 9. 998599 8. 905570 11. 094430 24 37 8. 905736 9. 998589 8. 907147 11. 092853 23 38 8. 907297 9. 998577 8. 908719 11. 091281 22 39 8. 908853 9. 998568 8. 910285 11. 089715 21 40 8. 910404 9. 998558 8. 911846 11. 088154 20 41 8. 911949 9. 998548 8. 913401 11. 086599 19 42 8. 913488 9. 998537 8. 914951 11. 085049 18 43 8. 915022 9. 998527 8. 916495 11. 083505 17 44 8. 916550 9. 998516 8. 918034 11. 081966 16 45 8. 918073 9. 998506 8. 919568 11. 080432 15 46 8. 919591 9. 998495 8. 921096 11. 078904 14 47 8. 921103 9. 998485 8. 922619 11. 077381 13 48 8. 922610 9. 998474 8. 924136 11. 075864 12 49 8. 924112 9. 998464 8. 925649 11. 074351 11 50 8. 925609 9. 998453 8. 927156 11. 072844 10 51 8. 927100 9 998442 8. 928658 11. 071342 9 52 8. 928587 9. 998431 8. 930155 11. 069845 8 53 8. 930068 9. 998421 8. 931647 11. 068353 7 54 8. 931544 9. 998410 8. 933134 11. 066866 6 55 8. 933015 9. 998399 8. 934616 11. 065384 5 56 8. 934481 9. 998388 8. 936093 11. 063907 4 57 8. 935942 9. 998377 8. 937565 11. 062435 3 58 8. 937398 9. 998366 8. 939032 11. 060968 2 59 8. 938850 9. 998355 8. 940494 11. 059506 1 60 8. 940296 9. 998344 8. 941952 11. 058048 0 Co-sine Sine Co-tang . Tangent . M Degree 85. Degree 5. M Sine Co-sine Tangent Co-tang . 0 8. 940296 9. 998344 8. 941952 11. 058048 60 1 8. 941738 9. 998333 8. 943404 11. 056596 59 2 8. 943174 9. 998322 8. 944852 11. 055148 58 3 8. 944606 9. 998311 8. 946295 11. 053705 57 4 8. 946034 9. 998300 8. 947734 11. 052266 56 5 8. 957456 9. 998289 8. 949168 11. 050832 55 6 8. 958814 9. 998277 8. 950597 11. 049403 54 7 8. 950287 9. 998266 8. 952021 11. 047979 53 8 8. 951696 9. 998255 8. 953441 11. 046559 52 9 8. 953099 9. 998243 8. 954856 11. 045144 51 10 8. 954499 9. 998232 8. 956267 11. 043733 50 11 8. 955894 9. 998220 8. 957674 11. 042326 49 12 8. 957284 9. 998209 8. 959075 11. 040925 48 13 8. 958670 9. 998197 8. 960473 11. 039527 47 14 8. 960052 9. 998186 8. 961866 11. 038134 46 15 8. 961429 9. 998174 8. 963254 11. 036746 45 16 8. 962801 9. 998163 8. 964639 11. 035361 44 17 8. 964170 9. 998151 8 966019 11. 033981 43 18 8. 965534 9. 998139 8. 967394 11. 032606 42 19 8. 966893 9. 998128 8. 968766 11. 031234 41 20 8. 968249 9. 998106 8. 970133 11. 029867 40 21 8. 969600 9. 998104 8. 971495 11. 028505 39 22 8. 970947 9 998092 8 972855 11. 027145 38 23 8. 972289 9. 998080 8. 974209 11. 025791 37 24 8. 973626 9. 998068 8. 975560 11. 024440 36 25 8. 974962 9. 998056 8. 976906 11. 023094 35 26 8. 976293 9. 998044 8. 978248 11. 021752 34 27 8. 977619 9. 998032 8. 979586 11 020414 33 28 8 978941 9. 998020 8. 980921 11. 019079 32 29 8. 980259 9. 998008 8 982251 11. 017749 31 30 8. 981573 9. 997996 8. 983577 11. 016423 30 Co-sine . Sine Co-tang . Tangent . M Degree 84. Degree 5. M Sine Co-sine Tangent Co-tang . 30 8. 981573 9. 997996 8. 983577 11. 016423 30 31 8. 982883 9. 997984 8. 984899 11. 015101 29 32 8. 984189 9. 997971 8. 986217 11. 013783 28 33 8. 985491 9. 997959 8 987532 11. 012468 27 34 8. 986789 9. 997947 8. 988842 11. 011158 26 35 8. 988083 9. 997935 8. 990149 11. 009851 25 36 8. 989374 9. 997922 8. 991451 11. 008549 24 37 8. 990660 9. 997910 8 992750 11. 007250 23 38 8. 991943 9 997897 8 994045 11. 005955 22 39 8. 993228 9. 997885 8. 995337 11. 004663 21 40 8. 994497 9. 997873 8. 996624 11. 003376 20 41 8. 995768 9. 997860 8 997908 11. 002092 19 42 8. 997036 9 997847 8. 999188 11. 000812 18 43 8. 998299 9. 997835 9. 000465 10. 999535 17 44 8. 999560 9. 997822 9. 001738 10. 998262 16 45 9. 000816 9. 997809 9. 003007 10. 996993 15 46 9. 002069 9. 997797 9. 004272 10. 995728 14 47 9. 003318 9. 997784 9. 005534 10. 994466 13 48 9. 004563 9. 997771 9. 006792 10. 993208 12 49 9. 005805 9. 997758 9. 008047 10. 991953 11 50 9. 007044 9. 997730 9. 009298 10. 990708 10 51 9. 008278 9. 997732 9. 010546 10. 989454 9 52 9. 009510 9. 997719 9 011790 10. 988210 8 53 9. 010737 9. 997706 9. 013031 10. 986969 7 54 9. 011962 9. 997693 9. 014268 10. 985732 6 55 9. 013182 9. 997680 9. 015502 10. 984498 5 56 9. 014399 9. 997667 9. 016732 10. 983268 4 57 9. 015613 9. 997654 9. 017959 10. 982041 3 58 9. 016824 9. 997641 9. 019183 10. 980817 2 59 9. 018031 9. 997628 9. 020403 10. 979597 1 60 9. 019235 9. 997612 9. 021620 10. 978380 0 Co-sine Sine Co-tang . Tangent M Degree 84. Degree 6 M Sine Co-sine Tangent Co-tang . 0 9. 019235 9. 997614 9. 021620 10. 978380 60 1 9. 020435 9. 997601 9. 022834 10. 977166 59 2 9. 021632 9. 997588 9. 024044 10. 975956 58 3 9. 022825 9. 997574 9. 025251 10. 974749 57 4 9. 024016 9. 997561 9. 026455 10. 973545 56 5 9. 025203 9. 997548 9. 027655 10. 972345 55 6 9. 026386 9. 997534 9. 028852 10. 971148 54 7 9. 027567 9. 997520 9. 030046 10. 969954 53 8 9. 028744 9. 997507 9. 031237 10. 968763 52 9 9. 029918 9. 997493 9. 032425 10. 967575 51 10 9. 031089 9. 997480 9. 033609 10. 966391 50 11 9. 032257 9. 997466 9. 034791 10. 965209 49 12 9. 033421 9. 997452 9. 035969 10. 964031 48 13 9. 034582 9. 997439 9. 036144 10. 962856 47 14 9. 035741 9. 997425 9. 038316 10. 961684 46 15 9. 036896 9. 997411 9. 039485 10. 960515 45 16 9. 038048 9. 997397 9. 040651 10. 959349 44 17 9. 039197 9. 997383 9. 041813 10. 958187 43 18 9. 040342 9. 997369 9. 042973 10. 957027 42 19 9. 041485 9. 997355 9. 044130 10. 955870 41 20 9. 042625 9. 997341 9. 045284 10. 954716 40 21 9. 043762 9. 997327 9. 046434 10. 953566 39 22 9. 044895 9. 997313 9. 047582 10. 952418 38 23 9. 046026 9. 997299 9. 048727 10. 951273 37 24 9. 047154 9. 997285 9. 049869 10. 950131 36 25 9. 049279 9. 997271 9. 051008 10. 948992 35 26 9. 049400 9. 997256 9. 052144 10. 947856 34 27 9. 050519 9. 997242 9. 043277 10. 946723 33 28 9. 051635 9. 997228 9. 054408 10. 945592 32 29 9. 052749 9. 997214 9. 055535 10. 944465 31 30 9. 053859 9. 997199 9. 056640 10. 943340 30 Co-sine Sine Co-tang . Tangent M Degree 83. Degree 6. M Sine Co-sine Tangent Co-tang . 30 9. 053859 9. 997199 9. 056640 10. 943340 30 31 9. 054966 9. 997185 9. 057781 10. 942219 29 32 9. 056071 9. 997170 9. 058900 10. 941100 28 33 9. 057172 9. 997156 9. 060016 10. 939984 27 34 9. 058271 9. 997141 9. 061130 10. 938870 26 35 9. 059367 9. 997127 9. 062240 10. 937760 25 36 9. 060460 9. 997112 9. 063348 10. 936652 24 37 9. 061551 9. 997098 9. 064453 10. 935547 23 38 9. 062638 9. 997083 9. 065556 10. 934444 22 39 9. 063723 9. 997068 9. 066655 10. 933345 21 40 9. 064806 9. 997053 9. 067752 10. 932248 20 41 9. 065885 9. 997039 9. 068847 10. 931153 19 42 9. 066962 9. 997024 9. 069938 10. 930062 18 43 9. 063036 9. 997009 9. 071027 10. 928973 17 44 9. 069107 9. 996994 9. 072113 10. 927887 16 45 9. 070176 9. 996979 9. 073197 10. 926803 15 46 9. 071242 9. 996964 9. 074278 10. 925722 14 47 9. 072306 9. 996949 9. 075356 10. 924644 13 48 9. 073366 9. 996934 9. 076432 10. 923568 12 49 9. 074424 9. 996919 9. 077505 10. 922495 11 50 9. 075480 9. 996904 9. 078576 10. 921424 10 51 9. 076533 9. 996889 9. 079644 10. 920356 9 52 9. 077583 9. 996874 9. 080710 10. 919290 8 53 9. 078631 9. 996858 9. 081773 10. 918227 7 54 9. 079676 9. 996843 9. 082833 10. 917167 6 55 9. 080719 9. 996828 9. 083891 10. 916109 5 56 9. 081759 9. 996812 9. 084947 10. 915053 4 57 9. 082797 9. 996797 9. 085999 10. 914100 3 58 9. 083832 9. 996782 9. 087050 10. 912950 2 59 9. 084864 9. 996766 9. 088098 10. 911902 1 60 9. 085894 9. 996751 9. 089144 10. 910856 0 Co-sine Sine Co-tang . Tangent M Degree 83. Degree 7. M Sine Co-sine Tangent Co-tang . 0 9. 085894 9. 996751 9. 089144 10. 910856 60 1 9. 086922 9. 996735 9. 090187 10. 909813 59 2 9. 087947 9. 996720 9. 091228 10. 908772 58 3 9. 088970 9. 996704 9. 092266 10. 907734 57 4 9. 089990 9. 996688 9. 093302 10. 906698 56 5 9. 091088 9. 996673 9. 094336 10. 905664 55 6 9. 092024 9. 996657 9. 095367 10. 904633 54 7 9. 093037 9. 996641 9. 096395 10. 903604 53 8 9. 094047 9. 996625 9. 097422 10. 902578 52 9 9. 095056 9. 996610 9. 098446 10. 901554 51 10 9. 096062 9. 996594 9. 099468 10. 900532 50 11 9. 097065 9. 996578 9. 100487 10. 899513 49 12 9 098066 9 996562 9 101504 10. 898496 48 13 9. 099065 9. 996546 9. 102519 10. 897481 47 14 9. 100062 9. 996530 9 103532 10. 896468 46 15 9. 101056 9. 996514 9. 104542 10. 895458 45 16 9102048 9 996498 9. 105550 10. 894450 44 17 9 103037 9. 996482 9. 106556 10. 893444 43 18 9. 104025 9 996465 9. 107559 10. 892441 42 19 9. 105010 9 996449 9. 108560 10. 891440 41 20 9. 105992 9. 996433 9. 109559 10. 890441 40 21 9. 106973 9. 996417 9. 110556 10. 889444 39 22 9. 107951 9. 996400 9. 111551 10. 888449 38 23 9. 108927 9. 996384 9. 112543 10. 887457 37 24 9. 109901 9. 996368 9. 113533 10. 886467 36 25 9. 110873 9. 996351 9. 114521 10. 885478 35 26 9. 111842 9. 996335 9. 115507 10. 884493 34 27 9. 112809 9. 996318 9. 116491 10. 883509 33 28 9. 113774 9 996302 9. 117472 10. 882528 32 29 9. 114737 9. 996235 9. 118452 10. 881548 31 30 9. 115698 9. 996269 9. 119429 10. 880571 30 Co sine Sine Co-tang . Tangent M Degree 82. Degree 7. M Sine Co-sine Tangent Co-tang . 30 9. 115698 9. 996269 9. 119429 10. 880571 30 31 9. 116656 9. 996252 9. 120404 10. 879596 29 32 9. 117612 9. 996235 9. 121377 10. 878623 28 33 9. 118567 9. 996218 9. 122348 10. 877652 27 34 9. 119519 9. 996202 9. 123317 10. 876683 26 35 9. 120469 9. 996185 9. 124284 10. 875716 25 36 9. 121417 9. 996168 9. 125248 10. 874751 24 37 9. 122362 9. 996151 9. 126211 10. 873789 23 38 9. 123306 9. 996134 9. 127172 10. 872828 22 39 9. 124248 9. 996117 9. 128130 10. 871870 21 40 9. 125187 9. 996100 9. 129087 10. 870913 20 41 9. 126125 9. 996083 9. 130041 10. 869959 19 42 9. 127060 9. 996066 9. 130994 10. 869006 18 43 9. 127993 9. 996049 9. 131944 10. 868056 17 44 9. 128925 9. 996032 9. 132893 10. 867107 16 45 9. 129854 9. 996015 9. 133839 10. 869161 15 46 9. 130781 9. 995998 9. 134784 10. 865216 14 47 9. 131706 9 995980 9. 135726 10. 864274 13 48 9. 132630 9. 995963 9. 136666 10. 863334 12 49 9. 133551 9 995946 9. 137605 10. 862395 11 50 9. 134470 9. 995928 9. 138542 10. 861458 10 51 9. 135387 9. 995911 9. 139476 10. 860524 9 52 9. 136303 9. 995894 9. 140409 10. 859591 8 53 9. 137216 9. 995876 9. 141340 10. 858660 7 54 9. 138127 9. 995859 9. 142269 10. 857731 6 55 9. 139037 9. 995841 9. 143196 10. 856804 5 56 9. 139944 9. 995825 9. 144121 10. 855879 4 57 9. 140850 9. 995806 9. 145044 10. 854956 3 58 9. 141754 9. 995788 9. 145965 10. 854035 2 59 9 142655 9. 995770 9. 146885 10. 853115 1 60 9. 142555 9. 995753 9. 147803 10. 852197 0 Co-sine Sine Co-tang . Tangent M Degree 82. Degree 8. M Sine Co-sine Tangent Co-tang . 0 9. 143555 9. 995753 9. 147803 10. 852197 60 1 9. 144453 9. 995735 9. 148718 10. 851282 59 2 9. 145349 9. 995717 9. 149632 10. 850368 58 3 9. 146243 9. 995699 9. 159544 10. 849456 57 4 9. 147136 9. 995681 9. 151454 10. 848546 56 5 9. 148026 9. 995664 9. 152363 10. 847637 55 6 9. 148919 9 995646 9. 153269 10. 846731 54 7 9. 149881 9. 995628 9. 154174 10. 845825 53 8 9. 150686 9. 995610 9. 155077 10. 844923 52 9 9. 151569 9. 995591 9. 155978 10. 844022 51 10 9. 152451 9. 995573 9. 156877 10. 843123 50 11 9. 153330 9. 995555 9. 157775 10. 842225 49 12 9. 154208 9. 995537 9. 158671 10. 841329 48 13 9. 155082 9. 995519 9. 159565 10. 840435 47 14 9. 155957 9. 995501 9 160457 10. 839543 46 15 9. 156830 9. 995482 9. 161347 10. 838653 45 16 9. 157700 9. 995464 9. 162236 10. 837764 44 17 9. 158569 9. 995446 9. 163123 10. 836877 43 18 9. 159436 9. 995427 9. 164008 10. 835992 42 19 9. 160301 9. 995409 9. 164892 10. 835108 41 20 9. 161164 9. 995390 9. 165773 10. 834226 40 21 9. 162025 9. 995372 9. 166654 10. 833346 39 22 9. 162885 9. 995353 9. 167532 10. 832468 38 23 9. 163743 9. 995334 9. 168409 10. 831591 37 24 9. 164600 9. 995316 9. 169284 10. 830716 36 25 9. 165454 9. 995297 9. 160157 10. 829843 35 26 9. 166307 9. 995278 9. 171029 10. 828971 34 27 9. 167158 9. 995260 9. 171899 10. 828101 33 28 9. 168008 9. 995241 9. 172767 10. 827233 32 29 9. 168856 9 995222 9 173634 10. 826366 31 30 9. 169702 9. 995203 9. 174499 10. 825501 30 Co-sine Sine Co-tang . Tangent M Degree 81. Degree 8. M Sine Co-sine Tangent Co-tang . 30 9. 169702 9. 995203 9. 174499 10. 825501 30 31 9. 170546 9. 995184 9. 175362 10. 824638 29 32 9. 171389 9. 995165 9. 176224 10. 823776 28 33 9. 172230 9. 995146 9. 177084 10. 822916 27 34 9. 173070 9. 995127 9. 177942 10. 822057 26 35 9. 173908 9. 995108 9. 178799 10. 821201 25 36 9. 174744 9. 995089 9. 179655 10. 820345 24 37 9. 175578 9. 995070 9. 180508 10. 819492 23 38 9. 176411 9. 995061 9. 181360 10. 818640 22 39 9. 177242 9. 995032 9. 182211 10. 817789 21 40 9. 178072 9. 995012 9. 183060 10. 816940 20 41 9. 178900 9. 994993 9. 183907 10. 816093 19 42 9. 179726 9. 994974 9. 184752 10. 815248 18 43 9. 180551 9. 994955 9. 185597 10. 814403 17 44 9. 181374 9. 994935 9. 186439 10. 813561 16 45 9. 182196 9. 994916 9. 187280 10. 812720 15 46 9. 183016 9. 994896 9. 188120 10. 811880 14 47 9. 183834 9. 994876 9. 188957 10. 811042 13 48 9. 184651 9. 994857 9. 189794 10. 810206 12 49 9. 185466 9 994838 9. 190629 10. 809371 11 50 9. 186280 9. 994818 9. 191462 10. 808538 10 51 9. 187092 9. 994798 9. 192294 10. 807706 9 52 9. 187903 9. 994779 9. 193124 10. 806876 8 53 9. 188712 9. 994759 9. 193953 10. 806047 7 54 9. 189519 9. 994739 9. 194780 10. 805220 6 55 9. 190325 9. 994719 9. 195606 10. 804394 5 56 9. 191130 9. 994699 9. 196440 10. 803569 4 57 9. 191933 9. 994680 9. 197253 10. 802747 3 58 9. 192734 9. 994660 9. 198674 10. 801926 2 59 9. 193534 9. 994640 9. 198894 10. 801106 1 60 9. 194332 9. 994620 9. 199712 10. 800287 0 Co-sine Sine Co-tang . Tangent M Degree 81. Degree 9. M Sine Co-sine Tangent Co-tang . 0 9. 194332 9. 994620 9. 199712 10. 800287 60 1 9. 195129 9. 994600 9. 200529 10. 799470 59 2 9. 195925 9. 994580 9. 201345 10. 798655 58 3 9. 196718 9. 994560 9. 202159 10. 797841 57 4 9. 197511 9. 994540 9. 202971 10. 797029 56 5 9. 198302 9. 994519 9. 203782 10. 796218 55 6 9. 199091 9. 994499 9. 204592 10. 795408 54 7 9. 199879 9. 994479 9. 205400 10. 794600 53 8 9. 200666 9. 994459 9. 206207 10. 793793 52 9 9. 201451 9. 994438 9. 207013 10. 792987 51 10 9. 202234 9. 994418 9. 207817 10. 792183 50 11 9. 203017 9. 994398 9. 208619 10. 791381 49 12 9 203797 9. 994377 9. 209420 10. 790580 48 13 9. 204577 9. 994357 9. 210220 10. 789780 47 14 9. 205354 9. 994336 9. 211018 10. 788982 46 15 9. 206131 9. 994316 9. 211815 10. 788185 45 16 9. 206906 9. 994195 9. 212611 10. 787385 44 17 9. 207679 9. 994174 9. 213405 10. 786595 43 18 9. 208452 9. 994154 9. 214198 10. 785802 42 19 9. 209222 9. 994133 9. 214989 10. 785011 41 20 9. 209992 9. 994112 9. 215780 10. 784220 40 21 9. 210760 9. 994191 9. 216568 10. 783432 39 22 9. 211526 9. 994171 9. 217356 10. 782644 38 23 9. 212291 9. 994150 9. 218142 10. 781858 37 24 9. 213055 9. 994129 9. 218926 10. 781070 36 25 9. 213818 9. 994108 9. 219710 10. 780294 35 26 9. 214579 9. 994087 9. 220491 10. 779508 34 27 9. 215338 9. 994066 9. 221272 10. 778728 33 28 9. 216097 9. 994044 9. 222052 10. 777948 32 29 9. 216854 9. 994024 9. 222830 10. 777170 31 30 9. 217609 9. 994003 9. 223607 10. 776393 30 Co-sine Sine Co-tang . Tangent M Degree 80. Degree 9. M Sine Co-sine Tangent Co-tang . 30 9. 217609 9. 994003 9. 223607 10. 716393 30 31 9. 218363 9. 993982 9. 224382 10. 775618 29 32 9. 219116 9. 993960 9. 225156 10. 774844 28 33 9. 219868 9. 993939 9. 225929 10. 774071 27 34 9. 220618 9. 993918 9. 226704 10. 773300 26 35 9. 221367 9. 993897 9 227471 10. 772529 25 36 9. 222115 9 993875 9 228240 10. 771760 24 37 9. 222861 9. 993854 9. 229007 10. 770993 23 38 9. 223606 9. 993832 9. 229774 10. 770226 22 39 9. 224349 9. 993811 9. 230539 10. 769461 21 40 9. 225092 9. 993789 9. 231302 10. 768698 20 41 9. 225833 9. 993768 9. 232065 10. 767935 19 42 9. 226573 9. 993746 9. 232826 10. 767174 18 43 9 227311 9. 993725 9. 233586 10. 766414 17 44 9. 228048 9. 993703 9. 234345 10. 765655 16 45 9 228784 9. 993681 9 235103 10. 764897 15 46 9. 239518 9. 993660 9. 235859 10. 764141 14 47 9. 230252 9. 993638 9. 236614 10. 763386 13 48 9. 230984 9. 993616 9. 237368 10. 762632 12 49 9. 231715 9. 993594 9. 238120 10. 761880 11 50 9. 232444 9. 993572 9. 238872 10. 761128 10 51 9. 233172 9. 993550 9. 239622 10. 760378 9 52 9. 233899 9. 993528 9. 240371 10. 759629 8 53 9 234625 9. 993506 9. 241118 10. 758882 7 54 9. 235349 9. 993484 9. 241865 10. 758135 6 55 9. 236073 9. 993462 9 242610 10 757390 5 56 9. 236795 9. 993440 9. 243354 10. 756646 4 57 9. 237515 9. 993418 9. 244097 10. 755903 3 58 9. 238835 9. 993396 9. 244839 10. 755161 2 59 9. 238952 9. 993374 9. 245579 10. 754421 1 60 9. 239670 9. 993351 9. 246319 10. 753681 0 Co-sine Sine Co-tang . Tangent M Degree 80. Degree 10. M Sine Co-sine Tangent Co-tang . 0 9. 239670 9. 993351 9. 246319 10. 753681 60 1 9. 240386 9. 993329 9. 247057 10. 752943 59 2 9. 241101 9. 993307 9. 247794 10. 752206 58 3 9. 241814 9. 993284 9. 248530 10. 751470 57 4 9. 242526 9. 993262 9. 249264 10. 750736 56 5 9. 243237 9. 993240 9. 249998 10. 750002 55 6 9. 243947 9. 993117 9. 250730 10. 749270 54 7 9 244656 9. 993195 9. 251461 10. 748539 53 8 9. 245363 9. 993172 9. 252191 10. 747809 52 9 9. 246070 9. 993149 9. 252920 10. 747080 51 10 9. 246775 9 993127 9. 253648 10. 746352 50 11 9. 247478 9. 993104 9. 254374 10. 745626 49 12 9. 248181 9. 993011 9. 255200 10. 744900 48 13 9. 248883 9. 993059 9. 255824 10. 744176 47 14 9. 249583 9. 993036 9. 256547 10. 743453 46 15 9. 250282 9. 993013 9. 257269 10. 742731 45 16 9. 250980 9. 992990 9. 257990 10. 742010 44 17 9. 251677 9. 992967 9. 258710 10. 741290 43 18 9. 252373 9. 992944 9. 259429 10. 740571 42 19 9. 253067 9. 992921 9. 260146 10. 739854 41 20 9. 253761 9. 992898 9. 260863 10. 739137 40 21 9 254453 9. 992875 9. 261578 10. 738422 39 22 9. 255144 9. 992852 9. 262292 10. 737708 38 23 9. 255834 9. 992829 9. 263005 10. 736995 37 24 9. 256523 9. 992806 9. 263717 10. 736283 36 25 9. 257211 9. 992783 9. 264428 10. 735572 35 26 9. 257898 9. 992759 9. 265138 10. 734862 34 27 9. 258583 9. 992736 9. 265847 10. 734153 33 28 9. 259268 9. 992713 9. 266555 10. 733445 32 29 9. 259951 9 992690 9. 267261 10. 732739 31 30 9 260633 9. 992666 9. 267967 10. 732033 30 Co-sine Sine Co-tang . Tangent M Degree 79. Degree 10. M Sine Co-sine Tangent Co-tang . 30 9. 260633 9. 992666 9. 267967 10. 732033 30 31 9. 261314 9. 992643 9. 268671 10. 731329 29 32 9. 261994 9. 992619 9. 269375 10. 730625 28 33 9. 262673 9. 992596 9. 270778 10. 729923 27 34 9. 263351 9. 992572 9. 271479 10. 729221 26 35 9. 264027 9. 992549 9. 271479 10. 728521 25 36 9. 264703 9. 992525 9. 272178 10. 727822 24 37 9. 265378 9. 992501 9. 〈◊〉 10. 727124 23 38 9. 266051 9. 992478 9. 273573 10. 726427 22 39 9. 266723 9. 992454 9. 274269 10. 725731 21 40 9. 267395 9. 992430 9. 274964 10. 725036 20 41 9. 268065 9. 992406 9. 275658 10. 724342 19 42 9. 268734 9. 992382 9. 276351 10. 723649 18 43 9. 269402 9. 992362 9. 277043 10. 722957 17 44 9. 270069 9. 992335 9. 277734 10. 722267 16 45 9. 270735 9. 992311 9. 278424 10. 721576 15 46 9. 271400 9. 992287 9. 279113 10. 720887 14 47 9. 272063 9. 992263 9. 279801 10. 720199 13 48 9. 272726 9. 992239 9 280488 10. 719512 12 49 9. 273388 9. 992214 9. 281174 10. 718826 11 50 9. 274049 9. 992190 9. 281858 10. 718142 10 51 9. 274708 9 992166 9. 282542 10. 717458 9 52 9. 275367 9. 992142 9. 283225 10. 716775 8 53 9. 276025 9. 992118 9283907 10. 716093 7 54 9. 276681 9. 992093 9. 284588 10. 715412 6 55 9. 277337 9. 992069 9. 285268 10. 714732 5 56 9. 277991 9. 992045 9. 285946 10. 714053 4 57 9. 278685 9. 992020 9 286624 10. 713376 3 58 9. 279297 9. 991996 9. 287301 10. 712699 2 59 9 279948 9. 991971 9. 287977 10. 712023 1 60 9. 280599 9. 991947 9. 288652 10. 711348 0 Co-sine Sine Co-tang . Tangent M Degree 79. Degree 11. M Sine Co-sine Tangent Co-tang . 0 9. 280599 9. 991947 9. 288652 10. 711348 60 1 9. 281229 9. 991922 9. 289326 10. 710674 59 2 9. 281897 9. 991897 9. 289999 10. 710001 58 3 9. 282544 9. 991873 9. 290671 10. 709329 57 4 9. 283190 9. 991848 9. 291342 10. 708658 56 5 9. 283836 9. 991823 9 292013 10. 707987 55 6 9. 284480 9. 991799 9. 292682 10. 707318 54 7 9. 285124 9. 991774 9. 293350 10. 706650 53 8 9. 285766 9. 991749 9. 294017 10. 705983 52 9 9. 286408 9. 991724 9. 294684 10. 705316 51 10 9. 287048 9. 991699 9. 295349 10. 704651 50 11 9. 287688 9. 991674 9. 296013 10. 703987 49 12 9. 288326 9. 991649 9. 296677 10. 703323 48 13 9. 288964 9. 991624 9. 297339 10. 702661 47 14 9. 289600 9. 991599 9. 298001 10. 701999 46 15 9. 290236 9. 991574 9. 298662 10. 701338 45 16 9. 290870 9. 991549 9. 299322 10. 700678 44 17 9. 291504 9. 991524 9. 299980 10. 700020 43 18 9. 292137 9. 991498 9. 300638 10. 699362 42 19 9. 292768 9. 991473 9. 301295 10. 698705 41 20 9. 293399 9. 991448 9. 301951 10. 698049 40 21 9. 294029 9. 991422 9. 302607 10. 697393 39 22 9. 294658 9. 991397 9. 303261 10. 696739 38 23 9. 295286 9. 991372 9. 303914 10. 696086 37 24 9. 295913 9. 991346 9. 304567 10. 695433 36 25 9. 296539 9. 991321 9. 305218 10. 694782 35 26 9. 297164 9. 991295 9. 305867 10. 694131 34 27 9. 297788 9. 991270 9. 306519 10. 693481 33 28 9. 298412 9. 991244 9. 307168 10. 692832 32 29 9. 299034 9. 991218 9. 307816 10. 692184 31 30 9. 299655 9 991193 9. 308463 10. 691537 30 Co-sine Sine Co-tang . Tangent M Degree 78. Degree 11. M Sine Co-sine Tangent Co-tang . 30 9. 299655 9 991193 9. 308463 10. 691537 30 31 9. 300276 9. 991167 9. 309109 10. 690891 29 32 9. 300895 9. 991141 9. 309754 10. 690246 28 33 9. 301514 9. 991115 9. 310399 10. 689601 27 34 9. 302132 9. 991090 9. 311042 10. 688958 26 35 9. 302749 9. 991064 9. 311685 10. 688315 25 36 9. 303364 9. 991038 9. 312327 10. 687673 24 37 9. 303979 9. 991012 9. 312968 10. 687032 23 38 9. 304593 9. 990986 9. 313608 10. 686392 22 39 9. 305207 9. 990960 9. 314247 10. 685753 21 40 9. 305819 9. 990934 9. 314885 10. 685115 20 41 9. 306430 9. 990908 9. 315523 10. 684477 19 42 9. 307041 9. 990882 9 316159 10. 683841 18 43 9. 307650 9. 990855 9. 316795 10. 683205 17 44 9. 308259 9. 990829 9. 317430 10. 682570 16 45 9. 308867 9. 990803 9. 318064 10. 681936 15 46 9. 309474 9. 990777 9. 318647 10. 681303 14 47 9. 310080 9. 990750 9. 319330 10. 680670 13 48 9. 310685 9. 990724 9. 319961 10 680039 12 49 9. 311289 9. 990697 9. 320592 10. 679408 11 50 9. 311899 9. 990671 9. 321222 10. 678778 10 51 9. 312495 9. 990645 9. 321851 10. 678149 9 52 9. 313097 9. 990618 9. 322479 10. 677521 8 53 9. 313698 9. 990591 9 323106 10. 676894 7 54 9. 314297 9. 990565 9. 323733 10. 676267 6 55 9. 314897 9. 990538 9. 324358 10. 675642 5 56 9. 315495 9. 990512 9. 324983 10. 675017 4 57 9. 316092 9. 990485 9. 325607 10. 674393 3 58 9. 316689 9. 990458 9. 326231 10. 673769 2 59 9. 317284 9. 990431 9. 326853 10. 673147 1 60 9. 317879 9. 990404 9. 327475 10. 672525 0 Co-sine . Sine Co-tang . Tangent M Degree 78. Degree 12. M Sine Co-sine Tangent Co-tang . 0 9. 317879 9. 990404 9. 327475 10. 672525 60 1 9. 318473 9. 990377 9. 328095 10. 671905 59 2 9. 319066 9. 990351 9. 328715 10. 671285 58 3 9. 319658 9. 990324 9. 329334 10. 670666 57 4 9. 320250 9. 990297 9. 329953 10. 670047 56 5 9. 320840 9. 990270 9. 320570 10. 669430 55 6 9. 321430 9. 990242 9. 331187 10. 668813 54 7 9. 322019 9. 990215 9. 331803 10 668197 53 8 9. 322607 9. 990188 9. 332418 10. 667582 52 9 9. 323194 9. 990161 9. 333033 10. 666967 51 10 9. 323780 9. 990134 9. 333646 10. 666354 50 11 9. 324366 9. 990107 9. 334259 10. 665741 49 12 9. 324950 9. 990079 9. 334871 10. 665129 48 13 9. 325534 9. 990052 9. 335482 10. 664518 47 14 9. 326117 9. 990025 9. 336093 10. 663907 46 15 9. 326699 9. 989997 9. 336700 10. 663298 45 16 9. 327281 9 989970 9. 337311 10. 662689 44 17 9. 327862 9. 989942 9. 337919 10. 662081 43 18 9. 328441 9. 989915 9. 338527 10. 661473 42 19 9. 329020 9. 989887 9. 339133 10. 660867 41 20 9. 329599 9. 989860 9. 339739 10. 660261 40 21 9. 330176 9. 989832 9. 340344 10. 659656 39 22 9. 330753 9. 989804 9. 340948 10. 659052 38 23 9. 331328 9. 989777 9. 341552 10. 658448 37 24 9. 331903 9. 989749 9. 342155 10. 657845 36 25 9. 332478 9 989721 9. 342757 10. 657243 35 26 9 333051 9. 989693 9. 343358 10. 656642 34 27 9. 333624 9. 989665 9. 343958 10. 656042 33 28 9. 334195 9. 989637 9. 344558 10. 655442 32 29 9. 334766 9 989609 9. 345157 10. 654843 31 30 9. 335337 9. 989581 9 345755 10. 654245 30 Co-sine Sine Co-tang . Tangent . M Degree 77. Degree 12. M Sine Co-sine Tangent Co-tang . 30 9. 335337 9. 989581 9. 345755 10. 654245 30 31 9. 335906 9. 989553 9. 346353 10. 653647 29 32 9. 336475 9. 989525 9. 346949 10. 653051 28 33 9. 337043 9. 989597 9. 347545 10. 652455 27 34 9. 337610 9. 989469 9. 348141 10. 651859 26 35 9. 338176 9. 989441 9. 348735 10. 651265 25 36 9. 338742 9. 989413 9. 349329 10. 650671 24 37 9. 339306 9. 989384 9. 349922 10. 650078 23 38 9. 339870 9. 989356 9. 350514 10. 649486 22 39 9 340434 9. 989328 9. 351106 10. 648894 21 40 9. 340996 9. 989299 9. 351697 10. 648303 20 41 9. 341558 9. 989271 9. 352287 10. 647713 19 42 9. 342119 9. 989243 9. 352876 10. 647124 18 43 9. 342679 9. 989214 9. 353465 10. 646535 17 44 9. 343239 9. 989186 9. 354053 10. 645947 16 45 9. 343797 9. 989157 9. 354640 10. 645360 15 46 9. 344355 9. 989128 9. 355227 10. 644773 14 47 9. 344912 9. 989100 9 355812 10. 644187 13 48 9. 345469 9. 989071 9. 356398 10. 643602 12 49 9. 346024 9. 989042 9. 356982 10. 643018 11 50 9. 346579 9. 989014 9. 357566 10. 642434 10 51 9. 347134 9. 988985 9. 358149 10. 641851 9 52 9. 347687 9. 988956 9. 358731 10. 641269 8 53 9. 348240 9. 988927 9. 359313 10. 640687 7 54 9. 348792 9. 988898 9. 359893 10. 640107 6 55 9 349343 9. 988869 9. 360474 10. 639526 5 56 9. 349893 9. 988840 9. 361053 10. 638947 4 57 9. 350443 9. 988811 9. 361632 10. 638368 3 58 9. 350992 9. 988782 9. 362210 10. 637790 2 59 9. 351540 9. 988754 9. 362787 10. 637213 1 60 9. 352088 9. 988724 9. 363364 10. 636636 0 Co-sine . Sine Co-tang . Tangent . M Degree 77. Degree 13. M Sine Co-sine Tangent Co-tang . 0 9. 352088 9. 988724 9. 363364 10. 636636 60 1 9. 352635 9. 988695 9. 363940 10. 636060 59 2 9. 353181 9. 988666 9. 364515 10. 635485 58 3 9. 353726 9. 988636 9. 365090 10. 634910 57 4 9. 354271 9. 988607 9. 365664 10. 634336 56 5 9. 354185 9. 988578 9. 366237 10. 633763 55 6 9. 355358 9. 988548 9. 366810 10. 633190 54 7 9. 355901 9. 988519 9. 367382 10. 632618 53 8 9 356443 9. 988489 9. 367953 10. 632047 52 9 9. 356984 9. 988460 9. 368524 10. 631476 51 10 9. 357524 9. 988430 9. 369094 10. 630906 50 11 9. 358064 9. 988401 9. 369663 10. 630337 49 12 9. 358603 9. 988371 9. 370232 10. 629768 48 13 9. 359141 9. 988341 9. 370799 10. 629201 47 14 9. 359679 9. 988312 9. 371367 10. 628633 46 15 9. 350215 9 988282 9. 371933 10. 628067 45 16 9. 360752 9 988252 9. 372499 10. 627501 44 17 9. 361287 9. 988223 9. 373064 10. 626936 43 18 9. 361822 9. 988193 9 373629 10. 626371 42 19 9 362356 9. 988163 9. 374193 10. 625807 41 20 9. 362889 9. 988133 9. 374756 10. 625244 40 21 9. 363422 9. 988103 9. 375319 10. 624681 39 22 9 363954 9. 988073 9 375881 10. 624119 38 23 9. 364485 9. 988043 9. 376442 10. 623558 37 24 9. 365016 9. 988013 9 377003 10. 622997 36 25 9 365546 9. 987983 9 377563 10. 622437 35 26 9 366075 9. 987953 9. 378122 10. 621878 34 27 9. 366604 9. 987922 9. 378681 10. 621319 33 28 9 367132 9. 987892 9. 379239 10. 620761 32 29 9. 367659 9. 987862 9. 379797 10. 620203 31 30 9. 368185 9. 987832 9. 380354 10. 619646 30 Co-sine Sine Co-tang . Tangent M Degree 76. Degree 13. M Sine Co-sine Tangent Co-tang . 30 9. 368185 9. 987832 9. 380354 10. 619646 30 31 9. 368711 9. 987801 9. 380910 10. 619090 29 32 9. 369236 9. 987771 9. 381466 10. 618534 28 33 9. 369761 9. 987740 9. 382021 10. 617980 27 34 9. 370285 9. 987710 9. 382575 10. 617425 26 35 9. 370808 9. 987679 9. 383129 10. 616871 25 36 9. 371330 9. 987649 9. 383682 10. 616318 24 37 9. 371852 9. 987618 9. 384234 10. 615766 23 38 9. 372373 9. 987588 9. 384786 10. 615214 22 39 9. 372894 9. 987557 9. 385337 10. 614663 21 40 9. 373414 9. 987526 9. 385888 10. 614112 20 41 9. 373933 9. 987496 9. 386438 10. 613562 19 42 9. 374452 9. 987465 9. 386987 10. 613013 18 43 9. 374970 9. 987434 9. 387536 10. 612464 17 44 9. 375487 9. 987403 9. 388084 10. 611916 16 45 9. 376003 9. 987372 9. 388631 10. 611369 15 46 9. 376519 9. 987341 9. 389178 10. 610822 14 47 9. 377035 9. 987310 9. 389724 10. 610276 13 48 9. 377549 9. 987279 9. 390270 10. 609730 12 49 9. 378063 9. 987248 9. 390815 10. 609185 11 50 9. 378577 9. 987217 9. 391360 10. 608640 10 51 9. 379089 9. 987186 9. 391907 10. 608097 9 52 9. 379601 9. 987155 9. 392467 10. 607553 8 53 9. 380113 9. 987124 9. 392989 10. 607011 7 54 9. 380624 9. 987092 9. 393531 10. 606469 6 55 9. 381134 9. 987061 9 394074 10. 605927 5 56 9. 381643 9. 987030 9. 394614 10. 605386 4 57 9. 382152 9. 986998 9. 395154 10. 604846 3 58 9 382661 9. 986967 9. 395694 10. 604306 2 59 9. 383168 9. 986936 9. 396233 10. 603767 1 60 9. 383675 9. 986904 9. 396770 10. 603229 0 Co-sine Sine Co-tang . Tangent M Degree 76. Degree 14 M Sine Co-sine Tangent Co-tang . 0 9. 383675 9 986904 9. 396771 10. 603229 60 1 9. 384181 9. 986873 9. 397309 10. 602694 59 2 9. 384687 9. 986841 9. 397846 10. 602154 58 3 9. 385192 9. 986809 9. 398383 10. 601617 57 4 9. 385697 9 986778 9. 398919 10. 601081 56 5 9. 386201 9. 986746 9. 399455 10. 600545 55 6 9. 386704 9. 986714 9. 399990 10. 600010 54 7 9. 387207 9. 986683 9. 400524 10. 599476 53 8 9. 387709 9. 986651 9. 401058 10. 598942 52 9 9. 388210 9. 986619 9. 401591 10. 598409 51 10 9. 388711 9. 986587 9. 402124 10. 597876 50 11 9. 389211 9. 986555 9. 402656 10. 597344 49 12 9. 389711 9. 986523 9. 403187 10. 596813 48 13 9. 390210 9. 986491 9. 403718 10. 596282 47 14 9. 390708 9. 986459 9. 404249 10. 595751 46 15 9. 391206 9. 986427 9. 404778 10. 595222 45 16 9. 391703 9. 986395 9. 405306 10. 594692 44 17 9. 392199 9. 986363 9. 405836 10. 594164 43 18 9. 392695 9. 986331 9. 406364 10. 593636 42 19 9. 393190 9. 986299 9. 406892 10. 593608 41 20 9. 393685 9. 986266 9. 407419 10. 592581 40 21 9. 394179 9. 986234 9. 407945 10. 592055 39 22 9. 394673 9. 986201 9. 408471 10. 591529 38 23 9. 395166 9. 986869 9. 408996 10. 591001 37 24 9. 395654 9. 986137 9. 409521 10. 590479 36 25 9. 396150 9. 986104 9. 410045 10. 589954 35 26 9. 396641 9. 986072 9. 410569 10. 589431 34 27 9. 397131 9. 986039 9. 411097 10. 588908 33 28 9. 397621 9. 986007 9. 411615 10. 588385 32 29 9. 398111 9. 985974 9. 412137 10. 587863 31 30 9. 398600 9. 986942 9. 412658 10. 587342 30 Co-sine Sine Co-tang . Tangent M Degree 75. Degree 14. M Sine Co-sine Tangent Co-tang . 30 9. 398600 9. 985942 9. 412658 10. 587342 30 31 9. 399087 9. 985909 9. 413179 10. 586821 29 32 9. 399575 9. 985876 9. 413699 10. 586301 28 33 9. 400062 9. 985843 9. 414219 10. 585781 27 34 9. 400549 9. 985811 9. 414738 10. 585262 26 35 9. 401035 9. 985778 9. 415257 10. 584742 25 36 9. 401520 9. 985745 9. 415775 10. 584225 24 37 9. 402005 9. 985712 9. 416293 10. 583707 23 38 9. 402489 9. 985679 9. 416810 10. 583190 22 39 9. 402972 9. 985646 9. 417326 10. 582674 21 40 9. 403455 9. 985613 9. 417842 10. 582157 20 41 9. 403938 9. 985580 9. 418357 10 581642 19 42 9. 404420 9. 985547 9. 418873 10. 581127 18 43 9. 404901 9. 985513 9. 419387 10 580613 17 44 9. 405382 9. 985480 9. 419901 10. 580099 16 45 9. 405862 9. 985447 9. 420415 10. 579585 15 46 9. 406341 9. 985414 9. 420927 10. 579072 14 47 9. 406820 9. 985380 9. 421440 10. 578560 13 48 9. 407299 9. 985347 9. 421951 10. 578048 12 49 9. 407776 9. 985314 9. 422463 10. 577537 11 50 9. 408254 9. 985280 9. 422973 10. 577026 10 51 9. 408731 9. 985247 9. 423484 10. 576516 9 52 9. 409207 9. 985213 9. 423993 10. 576007 8 53 9. 409682 9. 985180 9. 424503 10. 575497 7 54 9. 410157 9. 985146 9. 425011 10. 574989 6 55 9. 410632 9. 985112 9. 425518 10. 574480 5 56 9. 411106 9. 985079 9. 426027 10. 573973 4 57 9. 411579 9. 985045 9. 426534 10. 573466 3 58 9. 412052 9. 985011 9. 427041 10. 572959 2 59 9. 412524 9. 984977 9. 427547 10. 572453 1 60 9. 412996 9. 984943 9. 428052 10. 571947 0 Co-sine Sine Co-tang . Tangent M Degree 75. Degree 15. M Sine Co-sine Tangent Co-tang . 0 9. 412996 9. 984944 9. 428052 10. 571947 60 1 9. 413467 9. 984910 9. 428557 10. 571442 59 2 9. 413938 9. 984876 9. 429067 10. 570938 58 3 9. 414408 9. 984842 9. 429566 10. 570434 57 4 9. 414878 9. 984808 9. 430070 10. 569930 56 5 9. 415347 9. 984774 9. 430573 10. 569427 55 6 9. 415815 9. 984740 9. 431075 10. 568925 54 7 9. 416283 9. 984706 9. 431577 10. 568423 53 8 9. 416850 9 984672 9. 432079 10. 567921 52 9 9. 417217 9. 984637 9. 432580 10. 567420 51 10 9. 417684 9. 984603 9. 433080 10. 566920 50 11 9. 418149 9. 984569 9. 433580 10. 566419 49 12 9. 418615 9. 984535 9. 434080 10. 565920 48 13 9. 419079 9. 984500 9. 434579 10. 565421 47 14 9. 419544 9. 984466 9. 435078 10. 564922 46 15 9. 420007 9. 984431 9. 435576 10. 564424 45 16 9. 420470 9. 984397 9. 436073 10. 563927 44 17 9. 420933 9. 984363 9. 436570 10. 563430 43 18 9. 421395 9. 984328 9. 437067 10. 562933 42 19 9. 421856 9. 984293 9. 437563 10. 562437 41 20 9. 422317 9. 984259 9. 438059 10. 561941 40 21 9. 422778 9. 984224 9. 438554 10. 561446 39 22 9. 423238 9. 984189 9. 439548 10. 560952 38 23 9 423697 9. 984155 9. 439543 10. 560457 37 24 9. 424156 9. 984120 9. 440036 10. 559964 36 25 9. 424615 9. 984085 9. 440529 10. 559471 35 26 9. 425072 9. 984050 9. 441022 10. 558978 34 27 9 425530 9 984015 9. 441514 10. 558486 33 28 9. 425987 9. 983980 9. 442006 10. 557994 32 29 9. 426443 9. 983945 9. 442497 10. 557503 31 30 9. 426899 9. 983910 9. 442988 10. 557011 30 Co-sine Sine Co-tang . Tangent M Degree 74. Degree 15. M Sine Co-sine Tangent Co-tang . 30 9. 426899 9. 983910 9. 442988 10. 557011 30 31 9. 427354 9. 983875 9. 443479 10. 556521 29 32 9. 427809 9. 983840 9. 443968 10. 556031 28 33 9. 428264 9. 983805 9. 444458 10. 555542 27 34 9. 428717 9. 983770 9. 444947 10. 555035 26 35 9. 429170 9. 983735 9. 445435 10. 554565 25 36 9. 429623 9. 983699 9. 445923 10. 554077 24 37 9. 430075 9. 983664 9. 446411 10. 553589 23 38 9. 430507 9 983629 9. 446898 10. 553102 22 39 9. 430978 9. 983593 9. 447384 10. 552616 21 40 9. 431429 9. 983558 9. 447870 10. 552129 20 41 9. 431879 9. 983523 9. 448356 10. 551644 19 42 9. 432328 9. 983487 9. 448841 10. 551159 18 43 9. 432778 9. 983452 9. 449326 10. 550674 17 44 9. 433206 9. 983416 9. 449810 10. 550181 16 45 9. 433674 9. 983380 9. 450294 10. 559706 15 46 9. 434122 9. 983345 9. 450777 10. 549223 14 47 9. 434569 9. 983309 9. 451260 10. 548740 13 48 9. 435016 9. 983273 9. 451743 10. 548257 12 49 9. 435462 9. 983238 9. 452225 10. 547775 11 50 9. 435918 9. 983202 9. 452706 10. 547294 10 51 9. 436353 9. 983166 9. 453187 10. 546813 9 52 9. 436798 9. 983130 9. 453668 10. 546332 8 53 9. 437242 9. 983094 9. 454148 10. 545852 7 54 9. 437686 9. 983058 9 454629 10. 545372 6 55 9. 438129 9. 983022 9 455107 10. 544893 5 56 9. 438572 9. 982986 9. 455586 10. 544414 4 57 9. 439014 9. 982950 9. 456064 10. 543936 3 58 9. 439456 9. 982914 9. 456542 10. 543458 2 59 9. 439897 9. 982878 9. 457019 10. 542980 1 60 9. 440338 9. 982842 9. 457496 10. 542503 0 Co-sine Sine Co-tang . Tangent M Degree 74. Degree 16. M Sine Co-sine Tangent Co-tang . 0 9. 440338 9. 982842 9. 457496 10. 542503 60 1 9. 440778 9. 982805 9. 457973 10. 542027 59 2 9. 441218 9. 982769 9 458449 10. 541551 58 3 9. 441658 9. 982733 9. 458925 10. 541075 57 4 9. 442096 9. 982696 9. 459400 10. 540600 56 5 9. 442535 9 982660 9. 459875 10. 540125 55 6 9. 442973 9. 982623 9. 460349 10. 539651 54 7 9. 443416 9. 982587 9 460829 10. 539177 53 8 9. 443848 9. 982550 9. 461297 10. 538703 52 9 9. 444284 9. 982514 9. 461770 10. 538230 51 10 9. 444720 9. 982477 9. 462242 10. 537758 50 11 9. 445155 9. 982441 9. 462714 10. 537285 49 12 9. 445590 9. 982404 9. 463186 10. 536814 48 13 9. 446025 9. 982367 9. 463658 10. 536342 47 14 9. 446459 9. 982330 9. 464129 10. 535871 46 15 9. 446893 9. 982294 9 464599 10. 535401 45 16 9. 447326 9. 982257 9 465069 10. 534931 44 17 9. 447759 9. 982220 9. 465539 10. 534461 43 18 9. 448191 9. 982183 9. 466008 10. 533992 42 19 9. 448623 9. 982146 9. 466476 10. 533523 41 20 9. 449054 9. 982109 9. 466945 10. 533055 40 21 9. 449485 9. 982072 9. 467413 10. 532587 39 22 9. 449915 9. 982035 9. 467880 10. 532120 38 23 9. 450345 9. 981998 9. 468347 10. 531653 37 24 9. 450775 9. 981961 9. 468814 10. 531186 36 25 9. 451203 9 981923 9. 469280 10. 530720 35 26 9. 451632 9 981886 9. 469746 10. 530254 34 27 9. 452060 9. 981849 9. 470211 10. 529789 33 28 9. 452488 9. 981812 9. 470676 10. 529324 32 29 9. 452915 9. 981774 9. 471141 10. 528859 31 30 9. 453342 9. 981737 9. 471605 10. 528395 30 Co-sine Sine Co-tang . Tangent M Degree 73. Degree 16. M Sine Co-sine Tangent Co-tang . 30 9. 453342 9. 981737 9. 471605 10. 528395 30 31 9. 453768 9. 981699 9. 472068 10. 527931 29 32 9. 454194 9. 981662 9. 472532 10. 527468 28 33 9. 454619 9. 981624 9 472995 10. 527005 27 34 9 455044 9. 981587 9. 473457 10. 526543 26 35 9. 455469 9. 981549 9. 473919 10. 526081 25 36 9. 455892 9. 981512 9. 474381 10. 525619 24 37 9. 456316 9. 981474 9 474842 10. 525158 23 38 9. 456739 9. 981436 9. 475303 10. 524695 22 39 9. 457162 9. 981398 9. 475763 10. 524237 21 40 9. 457584 9. 981361 9. 476223 10. 523777 20 41 9 458006 9. 981323 9. 476683 10. 523317 19 42 9. 458427 9 981285 9. 477142 10. 522858 18 43 9. 458848 9. 981247 9. 477601 10. 522399 17 44 9. 459268 9. 981209 9. 478059 10. 521941 16 45 9. 459684 9. 981171 9. 478517 10. 521483 15 46 9. 460108 9. 981133 9. 478975 10. 521025 14 47 9. 460527 9. 981095 9. 479432 10. 520168 13 48 9. 460946 9. 981057 9. 479886 10. 520111 12 49 9. 461364 9. 981019 9. 480345 10. 519655 11 50 9. 461782 9. 980980 9. 480801 10. 519199 10 51 9. 462199 9. 980942 9. 481257 10. 518743 9 52 9. 462616 9. 980904 9. 481712 10. 518288 8 53 9. 463032 9. 980866 9 482167 10. 517833 7 54 9. 463448 9. 980827 9. 482621 10. 517379 6 55 9 463864 9. 980789 9. 483075 10. 516925 5 56 9. 464279 9. 980750 9 483528 10. 516471 4 57 9. 464694 9. 980712 9. 483982 10. 516018 3 58 9 465108 9. 980672 9. 484434 10. 515565 2 59 9. 465522 9. 980635 9. 484887 10. 515113 1 60 9. 465935 9. 980596 9. 485339 10. 514661 0 Co-sine Sine Co-tang . Tangent M Degree 73. Degree 17. M Sine Co-sine Tangent Co-tang . 0 9 465935 9. 980596 9. 485339 10. 514661 60 1 9. 466348 9. 980558 9. 485791 10. 514209 59 2 9. 466761 9. 980519 9. 486272 10. 513758 58 3 9 467173 9. 980480 9. 486693 10. 513307 57 4 9. 467585 9. 980441 9. 487143 10. 512857 56 5 9. 467996 9. 980403 9. 487593 10. 512407 55 6 9 468407 9. 980364 9. 488043 10. 511957 54 7 9. 468817 9. 980325 9. 488493 10. 511507 53 8 9. 469227 9. 980286 9 488941 10. 511059 52 9 9. 469637 9. 980247 9. 489390 10. 510610 51 10 9. 460446 9. 980208 9. 489838 10. 510162 50 11. 9 470455 9. 980169 9. 490286 10. 509714 49 12 9. 471863 9. 980130 9. 490733 10. 509267 48 13 9. 471071 9. 980091 9. 491180 10. 508820 47 14 9. 471678 9. 980052 9. 491627 10. 508373 46 15 9. 472086 9. 980012 9. 492073 10. 507928 45 16 9. 472492 9. 979973 9. 492519 10. 507481 44 17 9. 472898 9. 979934 9. 492964 10. 507035 43 18 9. 473304 9. 979894 9. 493410 10. 506590 42 19 9. 473710 9. 979855 9. 493854 10. 506145 41 20 9. 474115 9. 979816 9. 494299 10. 505701 40 21 9. 474519 9. 979776 9 494743 10. 505257 39 22 9. 474923 9. 979737 9. 495186 10. 504813 38 23 9. 475327 9. 979697 9. 495630 10. 504370 37 24 9. 475730 9. 979658 9. 496073 10. 503928 36 25 9. 476133 9. 979618 9. 496515 10. 503485 35 26 9. 476539 9. 979578 9. 496957 10. 503043 34 27 9 476938 9. 979539 9. 497399 10. 502601 33 28 9. 477340 9. 979499 9. 497840 10. 502160 32 29 9. 477741 9 979459 9. 〈◊〉 10. 501718 31 30 9. 478142 9. 979419 9. 498722 10. 501278 30 Co-sine Sine Co-tang . Tangent M Degree 72. Degree 17. M Sine Co-sine Tangent Co-tang . 30 9. 478142 9. 979419 9. 498722 10. 501278 30 31 9. 478542 9. 979380 9. 499163 10. 500837 29 32 9 478942 9. 979340 9. 499602 10. 500398 28 33 9. 479342 9 979300 9. 500042 10. 499958 27 34 9. 479741 9. 979260 9. 500481 10. 499519 26 35 9. 480140 9. 979220 9. 500920 10. 499080 25 36 9. 480538 9. 979180 9. 501359 10. 498641 24 37 9. 480936 9. 979140 9. 501797 10. 498203 23 38 9. 481334 9. 979099 9. 502234 10. 497765 22 39 9. 481731 9. 979059 9. 502672 10. 497328 21 40 9. 482128 9 979019 9. 503109 10. 496891 20 41 9. 482525 9. 978980 9. 503546 10. 496454 19 42 9. 482921 9. 978939 9. 503982 10. 496018 18 43 9. 483316 9. 978898 9. 504418 10. 495582 17 44 9. 483711 9. 978858 9. 504854 10. 495146 16 45 9. 484106 9. 978817 9. 505289 10. 494711 15 46 9. 484501 9. 978777 9. 505724 10. 494216 14 47 9. 484895 9. 978736 9. 506158 10. 493841 13 48 9. 485289 9. 978696 9. 506593 10. 493407 12 49 9. 485682 9. 978655 9. 507026 10. 492973 11 50 9. 486075 9 978615 9. 507459 10. 492540 10 51 9. 486467 9. 978574 9. 507892 10. 492107 9 52 9. 486859 9. 978533 9. 508326 10. 491674 8 53 9. 487251 9. 978493 9. 508759 10. 491241 7 54 9. 487642 9. 978452 9. 509181 10. 490809 6 55 9. 488033 9. 978411 9. 509622 10 490377 5 56 9. 488424 9. 978370 9. 510044 10 489916 4 57 9. 488814 9. 978329 9. 510486 10. 489515 3 58 9. 489204 9. 978288 9. 510916 10. 489084 2 59 9. 489593 9. 978247 9. 511346 10 488654 1 60 9. 489982 9. 978206 9. 511776 10. 488225 0 Co-sine Sine Co-tang . Tangent M Degree 72. Degree 18. M Sine Co-sine Tangent Co-tang . 0 9. 489982 9. 978206 9. 511776 10. 488224 60 1 9. 490371 9. 978165 9. 512206 10. 487794 59 2 9. 490759 9. 978124 9. 512635 10. 487365 58 3 9. 491147 9. 978083 9. 513064 10. 486936 57 4 9. 491534 9. 978042 9. 513493 10. 486507 56 5 9. 491922 9. 978000 9. 513921 10. 486079 55 6 9. 492308 9. 977959 9. 514349 10. 485651 54 7 9. 492695 9. 977918 9 514777 10. 485223 53 8 9. 493080 9 977877 9. 515204 10. 484796 52 9 9. 493466 9. 977835 9. 515631 10. 484369 51 10 9. 493851 9. 977794 9. 516057 10 483942 50 11 9. 494236 9 977752 9. 516484 10. 483516 49 12 9. 494620 9. 977711 9. 516910 10. 483090 48 13 9. 495005 9. 977669 9. 517335 40. 482665 47 14 9. 495388 9. 977628 9. 517761 10. 482239 46 15 9. 495771 9. 977586 9. 518185 10. 481814 45 16 9. 496154 9. 977544 9. 518610 10. 481390 44 17 9. 496537 9. 977503 9. 519034 10. 480966 43 18 9. 496919 9. 977461 9. 519458 10. 480542 42 19 9. 497301 9. 977419 9. 519882 10. 480118 41 20 9. 497682 9. 977377 9. 520305 10. 489695 40 21 9. 498063 9. 977335 9. 520728 10. 479272 39 22 9. 498444 9 977293 9. 521151 10. 478849 38 23 9. 498824 9 977251 9. 521573 10. 478427 37 24 9. 499204 9. 977209 9. 521995 10. 478005 36 25 9. 499584 9. 977167 9. 522417 10. 477583 35 26 9. 499963 9. 977125 9. 522838 10. 477162 34 27 9. 500342 9 977083 9. 523259 10. 476741 33 28 9. 500720 9. 977041 9. 523679 10. 476320 32 29 9. 501099 9 977999 9. 524109 10. 475900 31 30 9. 501476 9. 977956 9. 524520 10. 475480 30 Co-sine Sine Co-tang . Tangent M Degree 71. Degree 18. M Sine Co-sine Tangent Co-tang . 30 9. 501476 9. 976956 9. 524520 10. 475480 30 31 9. 501854 9. 976914 9. 524939 10. 475060 29 32 9. 502231 9. 976872 9. 525359 10. 474641 28 33 9. 502607 9. 976830 9. 525778 10. 474222 27 34 9. 502984 9. 976787 9. 526197 10. 473803 26 35 9. 503360 9. 976745 9. 526615 10. 473385 25 36 9. 503735 9. 976702 9. 527033 10. 472967 24 37 9. 504110 9. 976660 9. 527451 10. 472549 23 38 9. 504485 9. 976617 9. 527868 10. 472132 22 39 9. 504840 9. 976574 9. 528285 10. 471715 21 40 9. 505234 9. 976532 9. 528702 10. 471298 20 41 9. 505608 9. 976489 9. 529118 10. 470881 19 42 9. 505981 9. 976446 9. 529535 10. 470465 18 43 9. 506354 9. 976404 9. 529950 10. 470049 17 44 9. 506727 9. 976361 9. 530366 10. 469634 16 45 9. 507099 9. 976318 9. 530781 10. 469219 15 46 9. 507471 9. 976275 9. 531196 10 468804 14 47 9. 507843 9. 976232 9. 531611 10. 468389 13 48 9. 508214 9. 976185 9. 532025 10. 467975 12 49 9. 508585 9. 976146 9. 532436 10. 467561 11 50 9. 508955 9. 976103 9. 532852 10. 467147 10 51 9. 509326 9. 976060 9. 533266 10. 466734 9 52 9. 509696 9. 976017 9. 533679 10. 466321 8 53 9. 510065 9. 975973 9. 534092 10. 465908 7 54 9. 510434 9. 975930 9. 534504 10. 465496 6 55 9. 510803 9. 975887 9. 534916 10. 465084 5 56 9. 511171 9. 975844 9. 535328 10. 464672 4 57 9. 511540 9. 975800 9. 535739 10. 464261 3 58 9. 511907 9. 975757 9. 536150 10. 463849 2 59 9. 512275 9. 975713 9. 536561 10. 463439 1 60 9. 512642 9. 975670 9. 536972 10. 463028 0 Co-sine Sine Co-tang . Tangent M Degree 71. Degree 19. M Sine Co-sine Tangent Co-tang . 0 9. 512642 9. 975670 9. 536972 10. 463028 60 1 9. 513009 9. 975626 9. 537382 10. 462618 59 2 9. 513375 9. 975583 9. 537792 10. 462208 58 3 9. 513741 9. 975539 9. 538202 10. 461798 57 4 9. 514107 9. 975496 9. 538610 10. 461389 56 5 9. 514472 9. 975452 9. 539020 10. 460980 55 6 9. 514837 9. 975408 9. 539429 10. 460571 54 7 9. 515202 9. 975364 9. 539837 10. 460163 53 8 9. 515566 9. 975321 9. 540245 10. 459755 52 9 9. 515930 9. 975277 9. 540653 10. 459347 51 10 9. 516294 9. 975233 9. 541061 10. 458939 50 11 9. 516657 9. 975189 9. 541468 10. 458532 49 12 9. 517020 9. 975145 9. 541875 10. 458125 48 13 9. 517382 9. 975101 9. 542281 10. 457719 47 14 9. 517745 9. 975057 9. 542688 10. 457312 46 15 9. 518107 9. 975013 9. 543094 10. 456906 45 16 9. 518468 9. 974969 9. 543499 10. 456501 44 17 9. 518829 9. 974925 9. 543905 10. 456095 43 18 9. 519190 9. 974880 9. 544310 10. 455690 42 19 9. 519551 9. 974836 9. 544715 10. 455285 41 20 9. 519911 9. 974792 9. 545119 10. 454881 40 21 9. 520271 9. 974747 9. 545524 10. 454476 39 22 9. 520631 9. 974703 9. 545927 10. 454072 38 23 9. 520990 9. 974659 9. 546331 10. 453669 37 24 9. 521349 9. 974614 9. 546735 10. 453265 36 25 9. 521707 9. 974570 9. 547138 10. 452862 35 26 9. 522065 9. 974525 9. 547540 10. 452459 34 27 9. 522423 9. 974480 9. 547943 10. 452057 33 28 9. 522781 9. 974436 9. 548345 10. 451655 32 29 9. 523138 9. 974391 9. 548747 10. 451253 31 30 9. 523495 9. 974346 9. 549149 10. 450851 30 Co-sine Sine Co-tang . Tangent M Degree 70. Degree 19. M Sine Co-sine Tangent Co-tang . 30 9 523495 9. 974346 9. 549149 10. 450851 30 31 9. 523851 9. 974302 9. 549550 10. 450450 29 32 9. 524208 9. 974257 9. 549951 10. 450049 28 33 9. 524564 9. 974212 9. 550352 10. 449648 27 34 9. 524920 9. 974167 9. 550752 10. 449248 26 35 9. 525275 9. 974122 9. 551152 10. 448848 25 36 9. 525630 9. 974077 9. 551552 10. 448448 24 37 9. 525984 9. 974032 9. 551952 10. 448048 23 38 9. 526339 9. 973987 9. 552351 10. 447649 22 39 9. 526693 9. 973942 9. 552750 10. 447250 21 40 9. 527046 9. 973897 9. 553149 10. 446851 20 41 9. 527400 9. 973852 9. 553548 10. 446452 19 42 9. 527753 9. 973807 9. 553946 10. 446054 18 43 9. 528105 9. 973761 9. 554344 10. 445656 17 44 9. 528458 9. 973716 9 554741 10. 445259 16 45 9. 528810 9. 973671 9. 555139 10. 444861 15 46 9. 529161 9. 973615 9. 555536 10. 444464 14 47 9. 529513 9. 973580 9. 555932 10. 444068 13 48 9. 529864 9. 973535 9. 556329 10. 443671 12 49 9. 530214 9 973489 9. 556727 10. 443275 11 50 9. 530565 9. 973443 9. 557121 10. 442879 10 51 9. 530915 9. 973398 9. 557517 10. 442483 9 52 9. 531265 9. 973352 9. 557912 10. 442088 8 53 9. 531614 9. 973307 9. 558308 10. 441693 7 54 9. 531963 9. 973261 9. 558702 10. 441298 6 55 9. 532312 9. 973215 9. 559097 10. 440903 5 56 9. 532661 9. 973169 9. 559491 10. 440509 4 57 9. 533009 9. 973123 9. 559885 10. 440115 3 58 9. 533357 9. 973078 9 560279 10. 439721 2 59 9. 533704 9. 973032 9. 560673 10. 439327 1 60 9. 534052 9. 972986 9. 561066 10. 438934 0 Co-sine . Sine Co-tang . Tangent M Degree 70. Degree 20. M Sine Co-sine Tangent Co-tang . 0 9. 534052 9. 972986 9. 561066 10. 438934 60 1 9. 534399 9. 972940 9. 561459 10. 438541 59 2 9. 534746 9. 972894 9. 561851 10. 438148 58 3 9. 535091 9. 972848 9. 562244 10. 437756 57 4 9. 535437 9. 972801 9. 562636 10. 437364 56 5 9. 535782 9. 972755 9. 563028 10. 436972 55 6 9. 536129 9. 972709 9. 563419 10. 436580 54 7 9 536474 9. 972663 9. 563811 10. 436189 53 8 9. 536818 9. 972617 9. 564202 10. 435798 52 9 9. 537163 9. 972570 9. 564592 10. 435407 51 10 9. 537507 9. 972524 9. 564983 10. 435017 50 11 9. 537851 9. 972477 9. 565373 10. 434627 49 12 9. 538194 9. 972431 9. 565763 10. 434237 48 13 9. 538537 9. 972384 9 566153 10. 433847 47 14 9. 538880 9. 972338 9. 566542 10. 433457 46 15 9. 539222 9. 972291 9. 566932 10. 433068 45 16 9. 539565 9. 972245 9. 567320 10. 432679 44 17 9. 539907 9. 972198 9. 567709 10. 432291 43 18 9. 540249 9. 972151 9. 568097 10. 431902 42 19 9. 540590 9. 972105 9. 568486 10. 431514 41 20 9. 540931 9. 972058 9. 569873 10. 431126 40 21 9. 541272 9. 972011 9. 569261 10. 430739 39 22 9. 541612 9 971964 9. 569648 10. 430351 38 23 9. 541953 9. 971917 9. 570035 10. 429964 37 24 9. 542292 9. 971870 9 570422 10. 429578 36 25 9. 542632 9. 971823 9. 570809 10. 429191 35 26 9. 542971 9. 971776 9. 571195 10. 428805 34 27 9. 543310 9. 971729 9. 571581 10. 428419 33 28 9. 543649 9. 971682 9. 571967 10. 428033 32 29 9 543987 9. 971635 9. 572352 10. 427648 31 30 9. 544325 9. 971588 9. 572738 10. 427262 30 Co-sine . Sine Co-tang . Tangent M Degree 69. Degree 20. M Sine Co-sine Tangent Co-tang . 30 9. 544325 9. 971588 9. 572738 10. 427262 30 31 9. 544663 9. 971540 9. 573123 10. 426877 29 32 9. 545000 9. 971493 9. 573507 10. 426492 28 33 9. 545338 9. 971446 9. 573892 10. 426108 27 34 9. 545674 9. 971398 9. 574276 10. 425724 26 35 9. 546011 9. 971351 9. 574660 10. 425340 25 36 9. 546347 9. 971303 9. 575044 10. 424956 24 37 9. 546683 9. 971256 9. 575427 10. 424573 23 38 9. 547019 9. 971208 9. 575810 10. 424189 22 39 9. 547354 9. 971161 9. 576193 10. 423807 21 40 9. 547689 9. 971112 9. 576576 10. 423424 20 41 9. 548024 9. 971065 9. 576958 10. 423041 19 42 9. 548358 9. 971018 9. 577341 10. 422659 18 43 9. 548693 9. 970970 9. 577723 10. 422277 17 44 9. 549026 9. 970922 9. 578104 10. 421896 16 45 9. 549360 9. 970874 9. 578486 10. 421514 15 46 9. 549693 9. 970826 9. 578867 10. 421133 14 47 9. 550026 9. 970779 9. 579248 10. 420752 13 48 9. 550359 9. 970731 9. 579628 10. 420371 12 49 9. 550692 9. 970683 9. 580009 10. 419991 11 50 9. 551024 9. 970634 9. 580389 10. 419611 10 51 9. 551355 9. 970586 9. 580769 10. 419231 9 52 9. 551687 9. 970538 9. 581149 10. 418851 8 53 9. 552018 9. 970490 9 581528 10. 418472 7 54 9. 552349 9. 970442 9. 581907 10. 418092 6 55 9. 552680 9. 970394 9. 582286 10. 417713 5 56 9. 553010 9. 970345 9. 582665 10. 417335 4 57 9. 553340 9. 970297 9. 583043 10. 416956 3 58 9. 553670 9. 970249 9. 583422 10. 416578 2 59 9. 554000 9. 970200 9. 583800 10. 416200 1 60 9. 554329 9. 970152 9. 584177 10. 415823 0 Co-sine Sine Co-tang . Tangent M Degree 69. Degree 21. M Sine Co-sine Tangent Co-tang . 0 9. 554329 9. 970152 9. 584177 10. 415822 60 1 9. 554658 9. 970103 9. 584555 10. 415445 59 2 9. 554987 9. 970055 9. 584932 10. 415068 58 3 9. 555315 9. 970006 9. 585308 10. 414691 57 4 9. 555643 9. 969957 9 585686 10. 414314 56 5 9. 555971 9. 969909 9. 586062 10. 413938 55 6 9. 556299 9. 969860 9. 586439 10. 413561 54 7 9. 556626 9. 969811 9. 586815 10. 413185 53 8 9. 556953 9. 969762 9. 587190 10 412800 52 9 9. 557279 9. 969713 9. 587566 10. 412434 51 10 9. 557606 9. 969665 9. 587941 10. 412059 50 11 9. 557932 9. 969616 9. 588316 10. 411684 49 12 9. 558258 9. 969567 9. 588691 10. 411309 48 13 9. 558583 9. 969518 9. 589066 10. 410934 47 14 9. 558909 9. 969469 9. 589440 10. 410560 46 15 9. 559234 9. 969419 9. 589814 10. 410185 45 16 9. 559558 9. 969370 9. 590188 10. 409812 44 17 9. 559883 9. 969321 9. 590561 10. 409438 43 18 9. 560207 9. 969272 9. 590935 10. 409065 42 19 9. 560531 9. 969223 9. 591308 10. 408692 41 20 9. 560855 9. 969173 9. 591681 10. 408319 40 21 9. 561178 9. 969124 9. 592054 10. 407946 39 22 9. 561501 9. 969075 9. 592426 10. 407574 38 23 9. 561824 9. 969025 9. 592798 10. 407201 37 24 9. 562146 9. 968976 9. 593170 10. 406829 36 25 9. 562468 9. 968926 9. 593542 10. 406457 35 26 9. 562790 9. 968877 9. 593914 10. 406086 34 27 9. 563112 9. 968827 9. 594285 10. 405715 33 28 9. 563433 9. 968777 9. 594656 10. 405344 32 29 9. 563754 9. 968728 9. 595027 10. 405073 31 30 9. 564075 9. 968678 9. 595397 10. 404602 30 Co-sine Sine Co-tang . Tangent M Degree 68. Degree 21. M Sine Co-sine Tangent Co-tang . 30 9. 564075 9. 968678 9. 595397 10. 404602 30 31 9. 564396 9. 968628 9. 595768 10. 404232 29 32 9. 564716 9. 698578 9. 596138 10. 403862 28 33 9. 565036 9. 968528 9. 596508 10. 403492 27 34 9. 565356 9. 968478 9. 596878 10. 403122 26 35 9. 565675 6. 968428 9. 597247 10. 402753 25 36 9. 565995 9. 968378 9. 597616 10. 402384 24 37 9. 566314 9. 968328 9. 597985 10. 402015 23 38 9. 566632 9. 968278 9. 598354 10. 401646 22 39 9. 566951 9. 968228 9. 598722 10. 401277 21 40 9. 567269 9. 968178 9. 599091 10. 400909 20 41 9. 567587 9. 968128 9. 599459 10 400541 19 42 9. 567904 9. 968078 9. 599827 10. 400173 18 43 9. 568222 9. 968027 9. 600194 10. 399806 17 44 9. 568539 9. 967977 9. 600562 10. 399438 16 45 9. 568855 9. 967927 9. 600929 10. 399071 15 46 9. 569172 9. 967876 9. 601296 10. 398704 14 47 9. 569488 9. 967826 9. 601662 10. 398337 13 48 9. 569804 9 967775 9. 602029 10. 397971 12 49 9. 570120 9. 967725 9. 602395 10. 397605 11 50 9. 570435 9. 967674 9. 602761 10. 397239 10 51 9. 570751 9. 967623 9. 603127 10. 396873 9 52 9. 571065 9. 967573 9. 603493 10. 396507 8 53 9. 571380 9. 967522 9. 603858 10. 396142 7 54 9. 571695 9. 967471 9. 604223 10. 395777 6 55 9. 572009 9. 967420 9. 604588 10. 395412 5 56 9. 572322 9. 967370 9. 604953 10. 395047 4 57 9. 572636 9. 967319 9. 605317 10. 394683 3 58 9. 572949 9. 967268 9. 605681 10. 394318 2 59 9. 573263 9. 967217 9. 606046 10. 393954 1 60 9. 573575 9. 967166 9. 606409 10. 393590 0 Co-sine Sine Co-tang . Tangent M Degree 68. Degree 22. M Sine Co-sine Tangent Co-tang . 0 9. 573575 9. 967166 9. 606409 10. 393590 60 1 9. 573888 9. 967115 9. 606773 10. 393227 59 2 9. 574200 9. 967064 9. 607136 10. 392863 58 3 9. 574512 9. 967012 9. 607500 10. 392500 57 4 9. 574824 9. 966961 9. 607862 10. 392137 56 5 9. 575135 9. 966910 9. 608225 10. 391774 55 6 9. 575447 9. 966859 9. 608588 10. 391412 54 7 9. 575758 9. 966807 9. 608950 10. 391050 53 8 9. 576068 9. 966756 9. 609312 10. 390688 52 9 9. 576379 9. 966705 9. 609674 10. 390326 51 10 9. 576689 9. 966653 9. 600036 10. 389964 50 11 9. 576999 9. 966602 9. 610397 10. 389603 49 12 9. 577309 9. 966550 9. 610758 10. 389241 48 13 9. 577618 9. 966499 9. 611119 10. 388880 47 14 9. 577927 9 966447 9. 611480 10. 388520 46 15 9. 578236 9. 966395 9. 611841 10. 388159 45 16 9. 578545 9. 966344 9. 612201 10. 387799 44 17 9. 578853 9. 966292 9. 612561 10. 387438 43 18 9. 579161 9. 966240 9. 612921 10. 387078 42 19 9. 579469 9. 966188 9. 613281 10. 386719 41 20 9. 579777 9. 966136 9. 613641 10. 386359 40 21 9. 580084 9. 966084 9. 614000 10. 386000 39 22 9. 580392 9. 966032 9. 614359 10. 385641 38 23 9 580698 9. 965980 9. 614718 10. 385282 37 24 9. 581005 9. 965928 9 615077 10. 384923 36 25 9. 〈◊〉 9. 965876 9. 615435 10. 384565 35 26 9. 581618 9 965824 9. 615793 10. 384207 34 27 9 581923 9. 965772 9. 616151 10. 383448 33 28 9. 582229 9. 965720 9. 616509 10. 383491 32 29 9. 582534 9. 965668 9. 616867 10. 383133 31 30 9. 582840 9. 965615 9. 617224 10. 382776 30 Co sine Sine Co-tang . Tangent M Degree 67. Degree 22. M Sine Co-sine Tangent Co-tang . 30 9. 582840 9. 965615 9. 617224 10. 382776 30 31 9. 583144 9. 965563 9. 617581 10. 382418 29 32 9. 583449 9. 965511 9. 617938 10. 382061 28 33 9. 583753 9. 965458 9. 618295 10. 381705 27 34 9. 584058 9. 965406 9. 618652 10. 381348 26 35 9. 584361 9. 965353 9. 619008 10. 380992 25 36 9. 584665 9. 965301 9. 619364 10. 380635 24 37 9. 584968 9. 965248 9. 619720 10. 380279 23 38 9. 585271 9. 965195 9. 620076 10. 379924 22 39 9. 585574 9. 965143 9. 620432 10. 379568 21 40 9. 585877 9. 965090 9. 620787 10. 379213 20 41 9. 586179 9. 965037 9. 621142 10. 378858 19 42 9. 586481 9. 964984 9. 621497 10. 378503 18 43 9. 586783 9. 964931 9. 621852 10. 378148 17 44 9. 587085 9. 964878 9. 622206 10. 377793 16 45 9. 587386 9. 964825 9. 622561 10. 377439 15 46 9. 587687 9. 964772 9. 622915 10. 377085 14 47 9. 587988 9. 964719 9. 623269 10. 376731 13 48 9. 588289 9. 964666 9. 623623 10. 376377 12 49 9. 588589 9. 964613 9. 623976 10. 376024 11 50 9. 588890 9. 964560 9. 624330 10. 375670 10 51 9. 589190 9. 964507 9. 624683 10. 375317 9 52 9. 589489 9. 964454 9. 625036 10. 374964 8 53 9. 589789 9. 964400 9. 625388 10. 374612 7 54 9. 590088 9. 964347 9. 625741 10. 374259 6 55 9. 590387 9. 964294 9. 626093 10. 373907 5 56 9. 590686 9. 964240 9. 626445 10. 373555 4 57 9. 590984 9. 964187 9. 626797 10. 373203 3 58 9. 591282 9. 964133 9. 627149 10. 372850 2 59 9. 591580 9. 964080 9. 627501 10. 372499 1 60 9. 591878 9. 964026 9. 627852 10. 372148 0 Co-sine Sine Co-tang . Tangent M Degree 67. Degree 23. M Sine Co-sine Tangent Co-tang . 0 9. 591878 9. 964026 9. 627852 10. 372148 60 1 9. 592175 9. 963972 9. 628203 10. 371797 59 2 9. 592473 9. 963919 9. 628554 10. 371446 58 3 9. 592770 9. 963865 9. 628905 10. 371095 57 4 9. 593067 9. 963811 9. 629255 10. 370744 56 5 9. 593363 9. 963757 9. 629606 10. 370394 55 6 9. 593659 9 963703 9. 629956 10. 370044 54 7 9. 593955 9. 963650 9. 630306 10. 369694 53 8 9. 594251 9. 963596 9. 630655 10. 369344 52 9 9. 594547 9. 963542 9. 631005 10. 368995 51 10 9. 594842 9. 963488 9. 631354 10. 368645 50 11 9. 595137 9. 963433 9. 631704 10. 368296 49 12 9. 595432 9. 963379 9. 632053 10. 367947 48 13 9. 595727 9. 963325 9. 632401 10. 367598 47 14 9. 596021 9. 963271 9. 632750 10. 367250 46 15 9. 596315 9. 963217 9. 633098 10. 366901 45 16 9. 596610 9. 963102 9. 633447 10. 366553 44 17 9. 596903 9. 963108 9. 633795 10. 366205 43 18 9. 597196 9. 963054 9. 634143 10. 365857 42 19 9. 597490 9. 962999 9. 634490 10. 365510 41 20 9. 597783 9. 962945 9. 634838 10. 365162 40 21 9 598075 9. 962892 9. 635185 10. 364815 39 22 9 598368 9. 962836 9. 635530 10. 364468 38 23 9. 598660 9. 962781 9. 635879 10. 364121 37 24 9. 598952 9 962726 9. 636226 10. 363774 36 25 9 599244 9. 962672 9. 636571 10. 363428 35 26 9. 599536 9. 962617 9. 636918 10. 363081 34 27 9. 599827 9. 962562 9. 637205 10. 362735 33 28 9. 600118 9. 962507 9. 637610 10. 362389 32 29 9. 600409 9. 962453 9. 637956 10. 362044 31 30 9. 600700 9. 962398 9. 638302 10. 361698 30 Co-sine Sine Co-tang . Tangent M Degree 66. Degree 23. M Sine Co-sine Tangent Co-tang . 30 9. 600700 9. 967398 9. 638302 10. 361698 30 31 9. 600990 9. 962343 9. 638647 10. 361353 29 32 9. 601280 9. 962288 9. 638992 10. 361007 28 33 9. 601570 9. 962233 9. 639337 10. 360662 27 34 9. 601860 9 962178 9. 639682 10. 360318 26 35 9. 602149 9. 962122 9. 640027 10. 359973 25 36 9. 602439 9. 962067 9. 640371 10. 359629 24 37 9. 602728 9. 962012 9. 640716 10. 359284 23 38 9 603017 9. 961957 9. 641060 10. 358940 22 39 9. 603305 9. 961902 9. 641404 10. 358596 21 40 9. 603594 9. 961846 9. 641747 10. 358253 20 41 9. 603882 9. 961791 9. 642091 10. 357909 19 42 9. 604170 9. 961735 9. 642434 10. 357566 18 43 9. 604457 9. 961680 9. 642777 10. 357223 17 44 9. 604745 9. 961624 9. 643120 10. 356980 16 45 9. 605032 9. 961569 9. 643463 10. 356537 15 46 9. 605319 9. 961513 9. 643806 10. 356194 14 47 9. 605606 9. 961458 9. 644148 10. 355852 13 48 9. 605892 9. 961402 9. 644490 10. 355510 12 49 9. 606179 9. 961346 9. 644832 10. 355168 11 50 9. 606465 9. 961290 9. 645174 10. 354826 10 51 9. 606750 9. 961235 9. 645516 10. 354484 9 52 9. 607036 9. 961179 9. 645857 10. 354142 8 53 9. 607322 9. 961123 9. 646199 10. 353801 7 54 9. 607607 9. 961067 9. 646540 10. 353460 6 55 9. 607892 9. 961011 9. 646881 10. 353119 5 56 9. 608176 9. 960955 9. 647222 10. 352778 4 57 9. 608461 9. 960899 9. 647562 10. 352438 3 58 9. 608745 9. 960842 9. 647903 10. 352097 2 59 9. 609029 9. 960786 9. 648243 10. 351757 1 60 9. 609313 9. 960730 9. 648583 10. 351417 0 Co-sine Sine Co-tang . Tangent M Degree 66. Degree 24. M Sine Co-sine Tangent Co-tang . 0 9. 609313 9. 960730 9. 648583 10. 351417 60 1 9. 609597 9. 960674 9. 648923 10. 351077 59 2 9. 609880 9. 960617 9. 649263 10. 350737 58 3 9. 610163 9. 960561 9. 649602 10. 350398 57 4 9. 610446 9. 960505 9. 649942 10. 350058 56 5 9. 610729 9. 960448 9. 650281 10. 349319 55 6 9. 611012 9. 960392 9. 650620 10. 349380 54 7 9. 611294 9. 960335 9. 650959 10. 349041 53 8 9. 611576 9. 960279 9. 651297 10. 348703 52 9 9. 611858 9. 960222 9. 651636 10. 348364 51 10 9. 612140 9. 960165 9. 651974 10. 348026 50 11 9. 612421 9. 960109 9. 652312 10. 347688 49 12 9. 612702 9. 960052 9. 652650 10. 347350 48 13 9. 612983 9. 959995 9. 652988 10. 347012 47 14 9. 613264 9. 959938 9. 653326 10. 346674 46 15 9. 613545 9. 959881 9. 653663 10. 346337 45 16 9. 613825 9. 959824 9. 654000 10. 345999 44 17 9. 614105 9. 959768 9. 654337 10. 345662 43 18 9. 614385 9. 959710 9. 654674 10. 345325 42 19 9. 614665 9. 959653 9. 655011 10. 344989 41 20 9. 614944 9. 959596 9. 655348 10. 344652 40 21 9. 615223 9. 959539 9. 655684 10. 344316 39 22 9. 615502 9. 959482 9. 656020 10. 343980 38 23 9. 615781 9. 959425 9. 656356 10. 343643 37 24 9. 616060 9. 959367 9. 656692 10. 343308 36 25 9. 616338 9. 959310 9. 657028 10. 342972 35 26 9. 616616 9. 959253 9. 657363 10. 342636 34 27 9. 616894 9. 959195 9. 657699 10. 342301 33 28 9. 617172 9. 959138 9. 658034 10. 341966 32 29 9. 617450 9. 959080 9. 658369 10. 341531 31 30 9 617727 9. 959023 9. 658704 10. 341296 30 Co-sine Sine Co-tang . Tangent M Degree 65. Degree 24. M Sine Co-sine Tangent Co-tang . 30 9. 617727 9. 959023 9. 658704 10. 341296 30 31 9. 618004 9. 958965 9. 659039 10. 340926 29 32 9. 618281 9. 958908 9659373 10. 340627 28 33 9. 618558 9. 958850 9. 659708 10. 340292 27 34 9. 618834 9. 958792 9. 660042 10. 339958 26 35 9. 619110 9. 958734 9. 660376 10. 339624 25 36 9. 619386 9. 958677 9. 660710 10. 339290 24 37 9. 619662 9. 958619 9. 661043 10. 338957 23 38 9. 619938 9. 958561 9. 661377 10. 338623 22 39 9. 620213 9. 958503 9. 661710 10. 338290 21 40 9. 620488 9. 958445 9. 662043 10. 337956 20 41 9. 620763 9. 958387 9. 662376 10. 337623 19 42 9. 621038 9. 958329 9. 662709 10. 337291 18 43 9. 621313 9. 958271 9. 663042 10. 336958 17 44 9. 621587 9. 958212 9. 663374 10. 336625 16 45 9. 621861 9. 958154 9. 663707 10. 336293 15 46 9. 622135 9. 958096 9. 664039 10. 335961 14 47 9. 622409 9. 958038 9. 664371 10. 335629 13 48 9. 622682 9. 957979 9. 664703 10. 335297 12 49 9. 622956 9. 957921 9. 665035 10. 334965 11 50 9. 623229 9. 957862 9. 665366 10. 334634 10 51 9. 623502 9. 957804 9. 665697 10. 334302 9 52 9. 623774 9. 957745 9. 666029 10. 333971 8 53 9. 624047 9. 957687 9. 666360 10. 333640 7 54 9. 624319 9. 957628 9. 666691 10. 333309 6 55 9. 624591 9. 957570 9. 667021 10. 332979 5 56 9. 624863 9. 957511 9. 667352 10. 332648 4 57 9. 625134 9. 957452 9. 667682 10. 332318 3 58 9. 625406 9. 957393 9. 668012 10. 331987 2 59 9. 625677 9. 957334 9. 668343 10. 331657 1 60 9. 625948 9. 957276 9. 668672 10. 331327 0 Co-sine Sine Co - 〈…〉 . Tangent M Degree 65. Degree 25. M Sine Co-sine Tangent Co-tang . 0 9. 625948 9. 957276 9. 668672 10. 331327 60 1 9. 626219 9. 957217 9. 669002 10. 330998 59 2 9. 626490 9. 957158 9. 669332 10. 330668 58 3 9. 626760 9 957099 9. 669661 10. 330339 57 4 9. 627030 9. 957040 9. 699990 10. 330009 56 5 9. 627300 9. 956981 9. 670320 10. 329680 55 6 9. 627570 9. 956922 9. 670649 10. 329351 54 7 9 627840 9. 956862 9. 670977 10. 329022 53 8 9. 628109 9. 956803 9. 671306 10. 328694 52 9 9. 628378 9. 956744 9. 671634 10. 328365 51 10 9. 628647 9. 956684 9. 671963 10. 328037 50 11 9. 628916 9 956625 9. 672291 10. 327709 49 12 9. 629184 9. 956565 9. 672619 10. 327381 48 13 9. 629453 9. 956208 9. 672947 10. 327053 47 14 9. 629721 9. 956446 9. 673274 10. 326725 46 15 9. 629989 9. 956387 9 673603 10. 326398 45 16 9. 630257 9. 956327 9 673929 10. 326070 44 17 9. 630524 9. 956267 9. 674256 10. 325743 43 18 9. 630792 9. 956208 9. 674584 10. 325416 42 19 9. 631059 9. 956148 9. 674910 10. 325089 41 20 9. 631326 6. 956088 9. 675237 10. 324763 40 21 9. 631592 9. 956029 9. 675564 10. 324436 39 22 9. 631859 9. 955969 9. 675890 10. 324110 38 23 9 632125 9. 955909 9. 676216 10. 323783 37 24 9. 632392 9. 955849 9. 676543 10. 323457 36 25 9. 632657 9. 955789 9. 676869 10. 323131 35 26 9. 632923 9. 955739 9. 677194 10. 322805 34 27 9. 633189 9. 955669 9. 677520 10. 322480 33 28 9 633454 9 955609 9. 677845 10. 322154 32 29 9. 633719 9. 955548 9. 678171 10. 321829 31 30 9. 633984 9. 955488 9. 678496 10. 321504 30 Co-sine Sine Co-tang . Tangent M Degree 64. Degree 25. M Sine Co-sine Tangent Co-tang . 30 9. 633984 9. 955488 9. 678496 10. 321504 30 31 9. 634249 9. 955428 9. 678821 10. 321179 29 32 9. 634514 9. 955367 9. 679146 10. 320854 28 33 9. 634778 9 955307 9. 679471 10. 320529 27 34 9. 635042 9. 955246 9. 679795 10. 320205 26 35 9. 635306 9. 955186 9. 680120 10. 319880 25 36 9. 635570 9. 955125 9. 680444 10. 319556 24 37 9. 635833 9. 955065 9. 680768 10. 319232 23 38 9. 636097 9. 955004 9. 681092 10. 318908 22 39 9. 636360 9. 954944 9. 681416 10. 318584 21 40 9. 636623 9. 954883 9. 681740 10. 318260 20 41 9. 636886 9. 954823 9. 682063 10. 317937 19 42 9. 637148 9. 954762 9. 682386 10. 317613 18 43 9. 637411 9. 954701 9. 682710 10. 317290 17 44 9. 637673 9. 954640 9. 683033 10. 316967 16 45 9. 637935 9. 954579 9. 683356 10. 316644 15 46 9. 638197 9. 954518 9. 683678 10. 316321 14 47 9. 638458 9. 954457 9. 684001 10. 315999 13 48 9. 638720 9. 954396 9. 684324 10. 315676 12 49 9. 638981 9. 954335 9. 684646 10. 315354 11 50 9. 639242 9. 954274 9. 684968 10. 315032 10 51 9. 639503 9. 954213 9. 685290 10. 314710 9 52 9. 639764 9. 954152 9. 685612 10. 313388 8 53 9. 640024 9. 954090 9. 685934 10 314066 7 54 9. 640284 9. 954029 9. 686255 10. 313745 6 55 9. 640544 9. 954968 9. 686577 10. 313423 5 56 9. 640804 9. 953906 9. 686898 10. 313102 4 57 9. 641064 9. 953845 9. 687219 10 312781 3 58 9. 641323 9. 953783 9. 687540 10. 312460 2 59 9. 641583 9. 953722 9. 687861 10. 〈◊〉 1 60 9. 641842 9. 953660 9. 688182 10 311818 0 Co-sine Sine Co-tang . Tangent M Degree 64. Degree 26. M Sine Co-sine Tangent Co-tang . 0 9. 641842 9. 953660 9. 688182 10. 311818 60 1 9. 642101 9. 953598 9. 688502 10. 311498 59 2 9. 642360 9. 953537 9. 688823 10. 311177 58 3 9. 642618 9. 953475 9. 689143 10. 310857 57 4 9. 642876 9. 953413 9. 689463 10. 310537 56 5 9. 643135 9. 953351 9. 689783 10. 310237 55 6 9. 643393 9. 953290 9. 690103 10. 309897 54 7 9. 643650 9. 953228 9. 690423 10. 309577 53 8 9. 643908 9. 953166 9. 690742 10. 309258 52 9 9. 644165 9. 953104 9. 691063 10. 308938 51 10 9. 644423 9. 953042 9. 691381 10. 308619 50 11 9. 644680 9. 952980 9. 691700 10. 308300 49 12 9. 644936 9. 952917 9. 692019 10. 307981 48 13 9. 645193 9. 952855 9. 692338 10. 307662 47 14 9. 645449 9. 952793 9. 692656 10. 307343 46 15 9. 645706 9. 952731 9. 692975 10. 307025 45 16 9. 645962 9. 952668 9. 693293 10. 306706 44 17 9. 646218 9. 952606 9. 693612 10. 306388 43 18 9. 646473 9. 952544 9. 693930 10. 306070 42 19 9. 646729 9. 952481 9. 694248 10. 305752 41 20 9. 646984 9. 952419 9. 694566 10. 305434 40 21 9. 647239 9. 952356 9. 694883 10. 305117 39 22 9. 647494 9. 952294 9. 695201 10. 304799 38 23 9. 647749 9. 952231 9. 695518 10. 304482 37 24 9. 648004 9. 952168 9. 695835 10. 304164 36 25 9. 648258 9. 952105 9. 696153 10. 303847 35 26 9. 648512 9. 952043 9 696470 10. 303530 34 27 9. 648766 9. 951980 9. 696786 10. 303213 33 28 9. 648020 9. 951917 9. 697103 10. 302897 32 29 9. 649274 9. 951854 9. 697420 10. 302580 31 30 9. 649527 9. 051791 9. 697738 10. 302264 30 Co-sine Sine Co-tang . Tangent M Degree 63. Degree 26. M Sine Co-sine Tangent Co-tang . 30 9. 649527 9. 951791 9. 697738 10. 302264 30 31 9. 649781 9. 951728 9. 698052 10. 301947 29 32 9. 650034 9. 951665 9. 698369 10. 301631 28 33 9. 650287 9. 951602 9. 698685 10. 301315 27 34 9. 650519 9. 951539 9. 699001 10. 300999 26 35 9. 650798 9. 951476 9. 699316 10 300684 25 36 9. 651044 9. 951412 9 699632 10. 300368 24 37 9. 651296 9. 951349 9. 699947 10. 300052 23 38 9. 651648 9. 951286 9. 700263 10. 299737 22 39 9. 651800 9. 951222 9. 700578 10. 299422 21 40 9. 652052 9. 951159 9. 700893 10. 299107 20 41 9. 652303 9. 951095 9. 701208 10. 298792 19 42 9. 652555 9. 951032 9. 701522 10. 298477 18 43 9. 652806 9. 950968 9. 701837 10. 298163 17 44 9. 653057 9. 950905 9. 702152 10. 297848 16 45 9. 〈◊〉 9. 950841 9. 702466 10. 297534 15 46 9. 653558 9. 950777 9. 702780 10. 297219 14 47 9. 653808 9. 950714 9. 703095 10. 296905 13 48 9. 654059 9. 950650 9. 703409 10. 296591 12 49 9. 654309 9. 950586 9. 703722 10. 296277 11 50 9. 654558 9. 950522 9. 704036 10. 295964 10 51 9. 654808 9. 950458 9. 704350 10. 295656 9 52 9. 655057 9. 950394 9. 704663 10. 295337 8 53 9. 655307 9. 950330 9. 704976 10. 295023 7 54 9. 655556 9. 950266 9. 705290 10. 294710 6 55 9. 655805 9. 950202 9. 705603 10. 294397 5 56 9. 656053 9. 950138 9. 705915 10. 294084 4 57 9. 656302 9. 950074 9. 706228 10. 293771 3 58 9. 656550 9. 950009 9. 706541 10. 293459 2 59 9. 656799 9. 949945 9. 706853 10. 293146 1 9. 656347 9. 〈◊〉 9. 707166 10. 292834 0 Co-sine Sine Co-tang . Tangent M Degree 63. Degree 27. M Sine Co-sine Tangent Co-tang . 0 9. 657047 9. 949880 9. 707166 10. 292834 60 1 9. 657295 9. 949816 9. 707478 10. 292523 59 2 9. 657542 9. 949752 9. 707790 10. 292210 58 3 9. 657790 9. 949687 9. 708102 10. 291897 57 4 9. 658037 9. 949623 9. 708414 10. 291586 56 5 9. 658284 9. 949598 9. 708726 10. 291274 55 6 9. 658531 9. 949494 9. 709037 10. 290962 54 7 9. 658777 9. 949429 9. 709349 10. 290651 53 8 9. 659024 9. 949364 9. 709660 10. 290340 52 9 9. 659271 9. 949300 9. 709971 10. 290029 51 10 9. 659517 9. 949235 9. 710282 10. 289718 50 11 9. 659763 9. 949170 9. 710593 10. 289407 49 12 9. 660009 9. 949105 9. 710904 10. 289096 48 13 9. 660255 9. 949040 9. 711214 10. 288785 47 14 9. 660500 9. 948976 9. 711525 10. 288475 46 15 9. 660746 9. 948910 9. 711836 10. 288164 45 16 9. 660991 9. 948845 9. 712146 10. 287854 44 17 9. 661036 9. 948760 9. 712456 10. 287544 43 18 9. 661481 9. 948715 9. 712766 10. 287234 42 19 9. 661726 9. 948650 9. 713076 10. 286924 41 20 9. 661970 9. 948584 9. 713386 10. 286614 40 21 9. 662214 9. 948519 9. 713695 10. 286305 39 22 9. 662459 9. 948453 9. 714005 10. 285995 38 23 9. 662702 9. 948388 9. 714314 10. 285686 37 24 9. 662947 9. 948323 9. 714624 10. 285376 36 25 9. 663190 9. 948257 9. 714933 10. 285067 35 26 9. 663433 9. 948191 9. 715241 10. 284758 34 27 9. 663677 9. 948126 9. 715550 10. 284449 33 28 9. 663920 9. 948060 9. 715859 10. 284140 32 29 9. 664163 9. 947995 9. 716168 10. 283832 31 30 9. 664406 9. 947929 9. 716477 10. 283523 30 Co-sine Sine Co-tang . Tangent M Degree 62. Degree 27. M Sine Co-sine Tangent Co-tang . 30 9. 664406 9. 947929 9. 716477 10. 283523 30 31 9. 664648 9. 947863 9. 716785 10. 283215 29 32 9. 664891 9. 947797 9. 717093 10. 282907 28 33 9. 665133 9. 947731 9. 717401 10. 282598 27 34 9. 665375 9. 947665 9. 717709 10. 282290 26 35 9. 665617 9. 947599 9. 718017 10. 281983 25 36 9. 665858 9. 947533 9. 718325 10. 281675 24 37 9. 666100 9. 947467 9. 718633 10. 281367 23 38 9. 666341 9. 947401 9. 718940 10. 281060 22 39 9. 666583 9. 947335 9. 719248 10. 280752 21 40 9. 666824 9. 947269 9. 719555 10. 280445 20 41 9. 667065 9. 947203 9. 719862 10. 280138 19 42 9. 667305 9. 947136 9. 720169 10. 279831 18 43 9. 667546 9. 947070 9. 720476 10. 279524 17 44 9. 667786 9. 947004 9. 720783 10. 279217 16 45 9. 668026 9. 946937 9. 721089 10. 278911 15 46 9. 668266 9. 946871 9. 721395 10. 278604 14 47 9. 668506 9. 946804 9. 721702 10. 278298 13 48 9. 668746 9. 946738 9. 722008 10. 277991 12 49 9. 668986 9. 946671 9. 722315 10. 277685 11 50 9. 669225 9. 946604 9. 722621 10. 277379 10 51 9. 669464 9. 946537 9. 722927 10. 277073 9 52 9. 669703 9. 946471 9. 723232 10. 276768 8 53 9. 669942 9. 946404 9. 723538 10. 276462 7 54 9. 670181 9. 946337 9. 723843 10. 276156 6 55 9. 670419 9. 946270 9. 724149 10. 275851 5 56 9. 670657 9. 946203 9. 724454 10. 275546 4 57 9. 670896 9. 946136 9. 724759 10. 275240 3 58 9. 671134 9. 946069 9. 725065 10. 274935 2 59 9. 671372 9. 946002 9. 725369 10. 274630 1 60 9. 671609 9. 945935 9. 725674 10. 274326 0 Co-sine . Sine Co-tang . Tangent M Degree 62. Degree 28. M Sine Co-sine Tangent Co-tang . 0 9. 671609 9. 945935 9. 725674 10. 274326 60 1 9. 671847 9. 945868 9. 725979 10. 274021 59 2 9. 672084 9. 945800 9. 726284 10. 273816 58 3 9. 672321 9. 945733 9. 726588 10. 273412 57 4 9. 672558 9. 945666 9. 726892 10. 273107 56 5 9. 672795 9. 945598 9. 727197 10. 272803 55 6 9. 673032 9. 945531 9. 727501 10. 272499 54 7 9. 673268 9. 945463 9. 727805 10. 272195 53 8 9. 673505 9. 945396 9. 728109 10. 271891 52 9 9. 673741 9. 945328 9. 728412 10. 271587 51 10 9. 673977 9. 945261 9. 728716 10. 271284 50 11 9. 674213 9. 945193 9. 729020 10. 270980 49 12 9. 674448 9. 945125 9. 729323 10. 270677 48 13 9. 674684 9. 945058 9. 729626 10. 270374 47 14 9. 674919 9. 944990 9. 729929 10. 270070 46 15 9. 675154 9. 944922 9. 730232 10. 269767 45 16 9. 675389 9. 944854 9. 730535 10. 269464 44 17 9. 675623 9. 944786 9. 730838 10. 269162 43 18 9. 675859 9. 944718 9. 731141 10. 268859 42 19 9. 676094 9. 944650 9. 731443 10. 268559 41 20 9. 676328 9. 944582 9. 731746 10. 268254 40 21 9. 676562 9. 944514 9. 732048 10. 267952 39 22 9. 676796 9. 944446 9. 732351 10. 267649 38 23 9. 677030 9. 944377 9. 732653 10. 267347 37 24 9. 677264 9. 944309 9. 732955 10. 267045 36 25 9. 677497 9. 944241 9. 733257 10. 266743 35 26 9. 677731 9. 944172 9. 733558 10. 266441 34 27 9. 677964 9. 944104 9. 733860 10. 266140 33 28 9. 678197 9. 944016 9. 734162 10. 265838 32 29 9. 678430 9. 943967 9. 734463 10. 265537 31 30 9. 678663 9. 943898 9. 734764 10. 265236 30 Co-sine . Sine Co-tang . Tangent M Degree 61. Degree 28. M Sine Co-sine Tangent Co-tang . 30 9. 678663 9. 943898 9734764 10. 265236 30 31 9. 678895 9. 943830 9. 735666 10. 264934 29 32 9. 679128 9. 943761 9. 735362 10. 264633 〈◊〉 33 9. 670360 9. 943692 9. 735668 10. 264332 27 34 9. 679592 9. 943624 9. 735968 10. 264031 26 35 9. 679824 9. 943555 9. 736269 10. 263731 25 36 9. 680056 9. 943486 9. 736570 10. 263430 24 37 9. 680288 9. 943417 9. 736870 10. 263130 23 38 9. 680519 9. 943348 9. 737171 10. 262829 22 39 9. 680750 9. 943279 9. 737471 10. 262529 21 40 9. 680982 9. 943210 9. 737771 10. 262229 20 41 9. 681213 9. 943141 9. 738071 10. 261929 19 42 9. 681443 9. 943072 9. 738371 10. 261629 18 43 9. 681674 9. 943003 9. 738671 10. 261329 17 44 9. 681904 9. 942933 9. 738971 10. 261029 16 45 9. 682135 9. 942864 9. 739271 10. 260729 15 46 9. 682365 9. 942795 9. 739570 10. 260430 14 47 9. 682595 9. 942725 9. 739870 10. 260130 13 48 9. 682825 9. 942656 9. 740169 10. 259831 12 49 9. 683055 9. 942587 9. 740468 10. 259532 11 50 9. 683284 9. 942517 9. 740767 10. 259233 10 51 9. 683514 9. 942448 9. 741066 10. 258934 9 52 9. 683743 9. 942378 9. 741365 10. 258635 8 53 9. 683972 9. 942308 9. 741664 10. 258336 7 54 9. 684201 9. 942239 9. 741962 10. 258038 6 55 9. 684430 9. 942169 9. 742261 10. 257739 5 56 9. 684658 9. 942099 9. 742559 10. 257441 4 57 9 684887 9. 942029 9. 742858 10. 257142 3 58 9. 685115 9. 941059 9. 743156 10. 256844 2 59 9. 685343 9. 941889 9 743454 10. 256546 1 60 9. 685571 9. 941819 9. 743751 10. 256248 0 Co-sine Sine Co-tang . Tangent M Degree 61. Degree 29. M Sine Co-sine Tangent Co-tang . 0 9. 685571 9. 941819 9743752 10. 256248 60 1 9. 685799 9. 941749 9. 744050 10. 255950 59 2 9. 686027 9. 941679 9. 744348 10. 255652 58 3 9. 686254 9. 941609 9. 744645 10. 255355 57 4 9. 686482 9. 941539 9. 744943 10. 255057 56 5 9. 686709 9. 941468 9. 745240 10. 254760 55 6 9. 686936 9. 941398 9. 745538 10. 254462 54 7 9. 687163 9. 941328 9. 745835 10. 254165 53 8 9. 687389 9. 941257 9. 746132 10. 253868 52 9 9. 687616 9. 941187 9. 746429 10. 253571 51 10 9. 687842 9. 941116 9. 746726 10. 253274 50 11 9. 688069 9. 941046 9. 747023 10. 252977 49 12 9. 688295 9. 940975 9. 747319 10. 252680 48 13 9. 688523 9. 940905 9. 747616 10. 252384 47 14 9. 688747 9. 940834 9. 747912 10. 252087 46 15 9. 688972 9. 940763 9. 748209 10. 251791 45 16 9. 689198 9. 940693 9. 748505 10. 251495 44 17 9. 689421 9. 940622 9. 748801 10. 251199 43 18 9. 689648 7. 940551 9. 749097 10. 250902 42 19 9. 689873 9. 940480 9. 749393 10. 250607 41 20 9. 690098 9. 940409 9 749689 10. 250311 40 21 9. 690323 9. 940338 9. 749985 10. 250015 39 22 9. 690548 9. 940267 9. 750281 10. 249719 38 23 9. 690772 9. 940196 9. 750576 10. 249424 37 24 9. 690996 9. 940125 9. 750872 10. 249128 36 25 9. 691220 9. 940053 9. 751167 10. 248833 35 26 9. 691444 9. 939982 9. 751462 10. 248538 34 27 9. 691668 9. 939911 9. 751757 10. 248243 33 28 9. 691892 9. 939840 9. 752052 10. 247948 32 29 9. 692115 9. 939768 9. 752347 10. 247653 31 30 9. 692339 9. 939697 9. 752642 10. 247358 30 Co-sine Sine Co-tang . Tangent M Degree 60. Degree 29. M Sine Co-sine Tangent Co-tang . 30 9. 692339 9. 939697 9. 752642 10. 247358 30 31 9. 692562 9. 939625 9. 752937 10. 247063 29 32 9. 692785 9 939554 9. 753231 10. 246769 28 33 9. 693008 9. 939482 9. 753526 10. 246474 27 34 9. 693231 9. 939410 9. 753820 10. 246180 26 35 9. 693453 9. 939339 9. 754115 10. 245885 25 36 9. 693676 9. 939267 9. 754409 10. 245591 24 37 9. 693898 9. 939195 9. 754703 10. 245297 23 38 9. 694120 9. 939123 9. 754997 10. 245003 22 39 9. 694342 9. 939051 9. 755291 10. 244709 21 40 9. 694564 9. 938980 9. 755584 10. 244415 20 41 9. 694786 9. 938908 9. 755878 10. 244122 19 42 9. 695007 9. 938835 9. 756172 10. 243828 18 43 9. 695229 9. 938763 9 756465 10. 243535 17 44 9. 695450 9. 938691 9. 756759 10. 243241 16 45 9. 695671 9. 938619 9. 757052 10. 242948 15 46 9. 695892 9. 938547 9. 757345 10. 242655 14 47 9. 696113 9. 938475 9. 757638 10. 242362 13 48 9. 696334 9 938402 9. 757931 10. 242069 12 49 9. 696554 9. 938330 9. 758224 10. 241776 11 50 9. 696774 9. 938257 9. 758517 10. 241483 10 51 9. 696995 9. 938185 9. 758810 10. 241190 9 52 9. 697215 9. 938112 9. 759102 10. 240898 8 53 9. 697435 9. 938040 9. 759395 10. 240605 7 54 9. 697654 9. 937967 9. 759687 10. 240313 6 55 9. 697874 9. 937895 9. 759979 10. 240021 5 56 9. 698093 9. 937822 9. 760271 10. 239728 4 57 9. 698313 9. 937749 9. 760564 10. 239436 3 58 9. 698532 9. 937676 9. 760856 10. 239144 2 59 9. 698751 9. 937603 9. 761147 10. 238852 1 60 9. 698970 9. 937531 9. 761439 10. 238561 0 Co-sine Sine Co-tang . Tangent M Degree 60. Degree 30. M Sine Co-sine Tangent Co-tang . 0 9. 698970 9. 937531 9. 761439 10. 238561 60 1 9. 699189 9. 937458 9. 761731 10. 238269 59 2 9. 699407 9. 937385 9. 762023 10. 237977 58 3 9. 699626 9. 937312 9. 762314 10. 237686 57 4 9. 699844 9. 937238 9. 762606 10. 237394 56 5 9. 700062 9. 937165 9 762897 10. 237103 55 6 7. 700280 9 937092 9. 763188 10. 236812 54 7 9. 700498 9. 937019 9. 763479 10. 236521 53 8 9. 700716 9. 936945 9. 763770 10. 236230 52 9 9. 700933 9. 936872 9. 764061 10. 235939 51 10 9. 701151 9. 936799 9. 764352 10. 235648 50 11 9. 701568 9. 936725 9. 764643 10. 235357 49 12 9. 701585 9. 936652 9. 764933 10. 235067 48 13 9. 701802 9. 936578 9. 765224 10. 234776 47 14 9. 702019 9. 936505 9. 765514 10. 234486 46 15 9. 702236 9. 936431 9. 765805 10. 234195 45 16 9. 702452 9. 936357 9. 766095 10. 233905 44 17 9. 702669 9. 936284 9. 766385 10. 233615 43 18 9. 702885 9. 936210 9. 766675 10. 233325 42 19 9. 703101 9. 936136 9. 766965 10. 233035 41 20 9. 703317 9. 936062 9. 767255 10. 232745 40 21 9. 703533 9. 935988 9. 767545 10. 232455 39 22 9. 703748 9. 935914 9. 767834 10. 232166 38 23 9. 703964 9. 935840 9. 768124 10. 231876 37 24 9. 704179 9. 935766 9. 768413 10. 231587 36 25 9. 704395 9. 935692 9. 768703 10. 231297 35 26 9. 704610 9. 935618 9. 768992 10. 231008 34 27 9. 704820 9. 935543 9. 769281 10. 230719 33 28 9. 705040 9. 935469 9. 769570 10. 230430 32 29 9. 705254 9. 935395 9. 769859 10. 230141 31 30 9. 705469 9 935320 9. 770148 10. 229852 30 Co-sine Sine Co-tang . Tangent M Degree 59. Degree 30. M Sine Co-sine Tangent Co-tang . 30 9. 705469 9. 935320 9. 770148 10. 229852 30 31 9. 705683 9. 935246 9. 770437 10. 229563 29 32 9. 705897 9. 935171 9. 770726 10. 229274 28 33 9. 706112 9. 935097 9. 771015 10. 228985 27 34 9. 706327 9. 935022 9. 771303 10. 228696 26 35 9. 706539 9. 934948 9. 771592 10. 228408 25 36 9. 706753 9. 934873 9. 771880 10. 228120 24 37 9. 706967 9. 934798 9. 772168 10. 227832 23 38 9. 707180 9. 934723 9. 772456 10. 227543 22 39 9. 707393 9. 934649 9. 772745 10. 227255 21 40 9. 707606 9. 934574 9. 773033 10. 226967 20 41 9. 707819 9. 934499 9. 773321 10. 226679 19 42 9. 708032 9. 934424 9. 773608 10. 226391 18 43 9. 708245 9. 934349 9. 773896 10. 226104 17 44 9. 708457 9. 934274 9. 774184 10. 225816 16 45 9. 708670 9. 934199 9. 774471 10. 225529 15 46 9. 708882 9. 934123 9. 774759 10. 225241 14 47 9. 709094 9. 934048 9. 775046 10. 224954 13 48 9. 709306 9. 933973 9. 775333 10. 224666 12 49 9. 709518 9. 933897 9. 775621 10. 224379 11 50 9. 709730 9. 933822 9. 775908 10. 224092 10 51 9. 709941 9. 933747 9. 776195 10. 223805 9 52 9. 710153 9. 933671 9. 776482 10. 223518 8 53 9. 710364 9. 933596 9. 776768 10. 223232 7 54 9. 710575 9. 933520 9. 777055 10. 222945 6 55 9. 710786 9. 933444 9. 777342 10. 222658 5 56 9. 710997 9. 933369 9. 777628 10. 222372 4 57 9. 711208 9. 933293 9. 777915 10. 222085 3 58 9. 711418 9. 933217 9. 778201 10. 221799 2 59 9. 711629 9. 933141 9. 778487 10. 221513 1 60 9. 711839 9. 933066 9. 778774 10. 221226 0 Co-sine Sine Co-tang . Tangent M Degree 59. Degree 31. M Sine Co-sine Tangent Co-tang . 0 9. 711839 9. 933066 9. 778774 10. 221226 60 1 9. 712049 9. 932990 9. 779060 10. 220940 59 2 9. 712259 9. 932914 9. 779346 10. 220654 58 3 9. 712469 9. 932838 9. 779632 10. 220368 57 4 9. 712679 9. 932761 9. 779918 10. 220082 56 5 9. 712889 9. 932685 9. 780203 10. 219796 55 6 9. 713098 9. 932609 9. 780489 10. 219511 54 7 9. 713308 9. 932533 9. 780775 10. 219225 53 8 9. 713517 9. 932457 9. 781060 10. 218940 52 9 9. 713726 9. 932380 9. 781346 10. 218654 51 10 9. 713935 9. 932304 9. 781631 10. 218369 50 11 9. 714144 9. 932227 9. 781916 10. 218084 49 12 9. 714352 9. 932151 9. 782202 10. 217799 48 13 9. 714561 9. 932074 9. 782480 10. 217514 47 14 9. 714769 9. 931998 9. 782771 10. 217229 46 15 9. 714977 9. 931921 9. 783056 10. 216944 45 16 9. 715186 9. 931845 9. 783341 10. 216659 44 17 9. 715394 9. 931768 9. 783626 10. 216374 43 18 9. 715601 9. 931691 9. 783910 10. 216090 42 19 9. 715809 9. 931614 9. 784195 10. 215805 41 20 9. 716017 9. 931537 9. 784479 10. 215520 40 21 9. 716224 9. 931460 9. 784764 10. 215236 39 22 9. 716431 9. 931383 9. 785048 10. 214952 38 23 9. 716639 9. 931306 9. 785332 10. 214668 37 24 9. 716846 9. 931229 9. 785616 10. 214384 36 25 9. 717053 9. 931152 9. 785900 10. 214099 35 26 9. 717259 9. 931075 9. 786184 10. 213816 34 27 9. 717466 9. 930998 9. 786468 10. 213532 33 28 9. 717672 9. 930920 9. 786752 10. 213248 32 29 9. 717879 9. 930843 9. 787036 10. 212964 31 30 9. 718085 9. 930766 9. 787319 10. 212681 30 Co-sine Sine Co-tang . Tangent M Degree 58. Degree 31. M Sine Co-sine Tangent Co-tang . 30 9. 718085 9. 930766 9. 787319 10. 212681 30 31 9. 718291 9. 930688 9. 787603 10. 212397 29 32 9. 718497 9. 930611 9. 787886 10. 212114 28 33 9. 718703 9. 930533 9. 788170 10. 211830 27 34 9. 718909 9. 930456 9. 788453 10. 211547 26 35 9. 719114 9. 930378 9. 788736 10. 211264 25 36 9. 719320 9. 930300 9. 789019 10. 210981 24 37 9. 719525 9. 930223 9. 789302 10. 210698 23 38 9. 719730 9. 930145 9. 789585 10. 210415 22 39 9. 719935 9. 930067 9. 789868 10. 210132 21 40 9. 720140 9. 929989 9. 790151 10. 209849 20 41 9. 720345 9. 929911 9. 790433 10. 209566 19 42 9. 720549 9. 929833 9. 790716 10. 209284 18 43 9. 720754 9. 929755 9. 790999 10. 209001 17 44 9. 720958 9. 929677 9. 790281 10. 208719 16 45 9. 721162 9. 929599 9. 791563 10. 208436 15 46 9. 721366 9. 929521 9. 791846 10. 208154 14 47 9. 721570 9. 929442 9. 792128 10. 207872 13 48 9. 721774 9. 929364 9. 792410 10. 207590 12 49 9. 721978 9. 929286 9. 792692 10. 207308 11 50 9. 722181 9. 929207 9. 792974 10. 207024 10 51 9. 722385 9. 929129 9. 793256 10. 206744 9 52 9. 722588 9. 929050 9. 793538 10. 206462 8 53 9. 722791 9. 928972 9. 793819 10. 206180 7 54 9. 722994 9. 928893 9. 794101 10. 205899 6 55 9. 723197 9. 928814 9. 794383 10. 205617 5 56 9. 723400 9. 928736 9. 794664 10. 205336 4 57 9. 723603 9. 928657 9. 794945 10. 205054 3 58 9. 723805 9. 928578 9. 795227 10. 204773 2 59 9. 724007 9. 928499 9. 795508 10. 204492 1 60 9. 724210 9. 928420 9. 795789 10. 204211 0 Co-sine Sine Co-tang . Tangent M Degree 58. Degree 32. M Sine Co-sine Tangent Co-tang . 0 9. 724210 9. 928420 9. 795789 10. 204211 60 1 9. 724412 9. 928341 9. 796070 10. 203930 59 2 9. 724614 9. 928262 9. 796351 10. 203649 58 3 9. 724816 9. 928183 9. 796632 10. 203368 57 4 9. 725017 9. 928104 9. 796913 10. 203087 56 5 9. 725219 9. 928025 9. 797194 10. 202806 55 6 9. 725420 9. 927946 9. 797474 10. 202522 54 7 9. 725622 9. 921867 9. 797755 10. 202245 53 8 9. 725823 9. 927787 9. 798036 10. 201964 52 9 9. 726024 9. 927708 9. 798316 10. 201684 51 10 9. 726225 9. 927628 9. 798596 10. 201404 50 11 9. 726426 9. 927549 9. 798877 10. 201123 49 12 9. 726626 9. 927469 9. 799157 10. 200843 48 13 9. 726827 9. 927390 9. 799437 10. 200563 47 14 9. 727027 9. 927310 9. 799717 10. 200283 46 15 9. 727228 9. 927231 9. 799997 10. 200003 45 16 9. 727428 9. 927151 9. 800277 10. 199723 44 17 9. 727628 9. 927071 9. 800557 10. 199443 43 18 9. 727828 9. 926991 9. 800836 10. 199163 42 19 9. 728027 9. 926911 9. 801116 10. 198884 41 20 9. 728227 9. 926831 9. 801396 10. 198604 40 21 9. 728427 9. 926751 9. 801675 10. 198325 39 22 9. 728626 9. 926641 9. 801955 10. 198045 38 23 9. 728825 9. 926591 9. 802234 10. 197766 37 24 9. 729024 9. 926511 9. 802513 10. 197487 36 25 9. 729223 9. 926431 9. 802792 10. 197207 35 26 9. 729422 9. 926351 9. 803072 10. 196928 34 27 9. 729621 9. 926270 9. 803351 10. 196649 33 28 9. 729820 9. 926190 9. 803630 10. 196370 32 29 9. 730018 9. 926110 9. 803908 10. 196091 31 30 9. 730216 9. 926029 9. 804187 10. 195813 30 Co-sine Sine Co-tang . Tangent M Degree 57. Degree 32. M Sine Co-sine Tangent Co-tang . 30 9. 730216 9. 926029 9. 804187 10. 195813 30 31 9. 730415 9. 925949 9. 804466 10. 195534 29 32 9. 730613 9. 925868 9. 804745 10. 195255 28 33 9. 730811 9. 925787 9. 805023 10. 194977 27 34 9. 731009 9. 925707 9. 805302 10. 194698 26 35 9. 731206 9. 9. 25626 9. 805580 10. 194420 25 36 9. 731404 9. 925545 9. 805859 10. 194141 24 37 9. 731601 9. 925464 9. 806137 10. 193863 23 38 9. 731799 9. 925384 9. 806415 10. 193585 22 39 9. 731996 9. 925303 9. 806693 10. 193309 21 40 9. 732193 9. 925222 9. 806971 10. 193028 20 41 9. 732390 9. 925141 9. 807249 10. 192751 19 42 9. 732587 9. 925060 9. 807527 10. 192433 18 43 9. 732784 9. 924978 9. 807805 10. 192195 17 44 9. 732980 9. 924897 9. 808083 10. 191917 16 45 9. 733177 9. 924816 9. 808361 10. 191639 15 46 9. 733373 9. 924735 9. 808638 10. 191362 14 47 9. 733569 9. 924653 9. 808916 10. 191084 13 48 9. 733765 9. 924572 9. 809193 10. 190807 12 49 9. 733961 9. 924491 9. 809471 10. 190529 11 50 9. 734157 9. 924409 9. 809748 10. 190252 10 51 9. 734353 9. 924328 9. 810025 10. 189975 9 52 9. 734548 9. 924246 9. 810302 10. 189697 8 53 9. 734744 9. 924164 9. 810580 10. 189420 7 54 9. 734939 9. 924083 9. 810857 10. 189143 6 55 9. 735134 9. 924001 9. 811134 10. 188866 5 56 9. 735330 9. 923919 9. 811410 10. 188589 4 57 9. 735525 9. 923837 9. 811687 10. 188313 3 58 9. 735719 9. 923755 9. 811964 10. 188036 2 59 9. 735914 9. 923673 9. 812241 10. 187759 1 60 9. 736109 9. 923591 9. 812517 10. 187483 0 Co-sine Sine Co-tang . Tangent M Degree 57. Degree 33. M Sine Co-sine Tangent Co-tang . 0 9. 736109 9. 923591 9. 812517 10. 187483 60 1 9. 736309 9. 923509 9. 812794 10. 187206 59 2 9. 736497 9. 923427 9. 813070 10. 186930 58 3 9. 736692 9. 923345 9. 813347 10. 186653 57 4 9. 736886 9. 923263 9. 813623 10. 186377 56 5 9. 737080 9. 923180 9. 813899 10. 186101 55 6 9. 737274 9. 923098 9. 814175 10. 185824 54 7 9. 737467 9. 923016 9. 814452 10. 185548 53 8 9. 737661 9. 922933 9. 814728 10. 185272 52 9 9. 737854 9. 922851 9. 815004 10. 184996 51 10 9. 738048 9. 922768 9 815279 10. 184720 50 11 9. 738241 9. 922686 9. 815555 10. 184445 49 12 9. 738434 9. 922603 9. 815831 10. 184169 48 13 9. 738627 9. 922520 9. 816107 10. 183893 47 14 9. 738820 9. 922438 9. 816382 10. 183617 46 15 9. 739013 9. 922355 9. 816658 10. 183342 45 16 9. 739205 9. 922272 9. 816933 10. 183066 44 17 9. 739398 9. 922189 9. 817209 10. 182791 43 18 9. 739590 9. 922106 9. 817484 10. 182516 42 19 9. 739783 9. 922023 9. 817759 10. 182240 41 20 9. 739975 9. 921940 9. 818035 10. 181965 40 21 9. 740167 9. 921857 9 818310 10. 181690 39 22 9. 740359 9. 921774 9. 818585 10. 181415 38 23 9. 740550 9. 921691 9. 818860 10. 181140 37 24 9. 740742 9. 921607 9. 819135 10. 180865 36 25 9. 740934 9. 921524 9. 819410 10. 180590 35 26 9. 741125 9. 921441 9. 819684 10. 180315 34 27 9. 741316 9. 921357 9. 819959 10. 180041 33 28 9. 741507 9. 921274 9. 820234 10. 179766 32 29 9. 741698 9. 921190 9. 820508 10. 179492 31 30 9. 741889 9. 921107 9. 820783 10. 179217 30 Co-sine Sine Co-tang . Tangent M Degree 56. Degree 33. M Sine Co-sine Tangent Co-tang . 30 9. 741889 9. 921107 9. 820783 10. 179217 30 31 9. 742080 9. 921023 9. 821057 10. 178943 29 32 9. 742271 9. 920939 9. 821332 10. 178668 28 33 9. 742461 9. 920855 9. 821606 10. 178394 27 34 9. 742652 9. 920772 9. 821880 10. 178120 26 35 9. 742842 9. 920688 9. 822154 10. 177846 25 36 9. 743032 9. 920604 9. 822429 10. 177571 24 37 9. 743223 9. 920520 9. 822703 10. 177297 23 38 9. 743412 9. 920436 9. 822977 10. 177023 22 39 9. 743602 9. 920352 9. 823250 10. 176739 21 40 9. 743792 9. 920268 9. 823524 10. 176476 20 41 9. 743982 9. 920184 9. 823798 10. 176202 19 42 9. 744171 9. 920099 9. 824072 10. 175928 18 43 9. 744361 9. 920015 9. 824345 10. 175655 17 44 9. 744550 9. 919931 9. 824619 10. 175381 16 45 9. 744739 9. 919846 9. 824892 10. 175108 15 46 9. 744928 9. 919762 9. 825166 10. 174834 14 47 9. 745117 9. 919677 9. 825439 10. 174560 13 48 9. 745306 9. 919593 9. 825713 10. 174287 12 49 9. 745494 9. 919508 9. 825986 10. 174014 11 50 9. 745683 9. 919424 9. 826259 10. 173741 10 51 9. 745871 9. 919339 9. 826532 10. 173468 9 52 9. 746059 9. 919254 9. 826805 10. 173195 8 53 9. 746248 9. 919169 9. 827078 10. 172922 7 54 9. 746436 9. 919084 9. 827351 10. 172649 6 55 9. 746624 9. 918999 9. 827624 10. 172376 5 56 9. 746811 9. 918915 9. 827897 10. 172103 4 57 9. 746999 9. 918830 9. 828170 10. 171830 3 58 9. 747187 9. 918744 9. 828442 10. 171558 2 59 9. 747374 9. 918659 9. 828715 10. 171285 1 60 9. 747562 9. 918574 9. 828987 10. 171012 0 Co-sine Sine Co-tang . Tangent M Degree 56. Degree 34 M Sine Co-sine Tangent Co-tang . 0 9. 747562 9. 918574 9. 828987 10. 171012 60 1 9. 747749 9. 918489 9. 829260 10. 170740 59 2 9. 747936 9. 918404 9. 829532 10. 170468 58 3 9. 748123 9. 918318 9. 829805 10. 170195 57 4 9. 748310 9. 918233 9. 830077 10. 169923 56 5 9. 748497 9. 918147 9. 830349 10. 169651 55 6 9. 748683 9. 918062 9. 830621 10. 169379 54 7 9. 748870 9. 917976 9. 830891 10. 166106 53 8 9. 749056 9. 917891 9. 831165 10. 168834 52 9 9. 749242 9. 917805 9. 831437 10. 168563 51 10 9. 749429 9. 917719 9. 831709 10. 168291 50 11 9. 749615 9. 917634 9. 831981 10. 168019 49 12 9. 749801 9. 917548 9. 832253 10. 167747 48 13 9. 749986 9. 917462 9. 832525 10. 167475 47 14 9. 750172 9. 917376 9. 832796 10. 167204 46 15 9. 750358 9. 917290 9. 833068 10. 166932 45 16 9. 750543 9. 917204 9. 833339 10. 166660 44 17 9. 750729 9. 917118 9. 833621 10. 166389 43 18 9. 750914 9. 917032 9. 833882 10. 166118 42 19 9. 751099 9. 916945 9. 834154 10. 165846 41 20 9. 751284 9. 916819 9. 834425 10. 165575 40 21 9. 751469 9. 916773 9. 834696 10. 165304 39 22 9. 751654 9. 916686 9. 834967 10. 165033 38 23 9. 751838 9. 916600 9. 835238 10. 164762 37 24 9. 752023 9. 916514 9. 835509 10. 164491 36 25 9. 752207 9. 916427 9. 835780 10. 164220 35 26 9. 752392 9. 916340 9. 836051 10. 163949 34 27 9. 752576 9. 916254 9. 836322 10. 163678 33 28 9. 752760 9. 916167 9. 836593 10. 163407 32 29 9. 752944 9. 916080 9. 836864 10. 163136 31 30 9. 753128 9. 915994 9. 837134 10. 162866 30 Co-sine Sine Co-tang . Tangent M Degree 55. Degree 34. M Sine Co-sine Tangent Co-tang . 30 9. 753128 9. 915994 9. 837134 10. 162866 30 31 9. 753312 9. 915907 9. 837405 10. 162595 29 32 9. 753495 9. 915820 9. 837675 10. 162325 28 33 9. 753679 9. 915733 9. 837946 10. 162054 27 34 9. 753862 9. 915646 9. 838216 10. 161784 26 35 9. 754046 9. 915559 9. 838487 10. 161513 25 36 9. 754229 9. 915472 9. 838757 10. 161243 24 37 9. 754412 9. 915385 9. 839027 10. 160973 23 38 9. 754595 9. 915297 9. 839297 10. 160702 22 39 9. 754778 9. 915210 9. 839568 10. 160432 21 40 9. 754960 9. 915123 9. 839838 10. 160162 20 41 9. 755143 9. 915035 9. 840108 10. 159892 19 42 9. 755325 9. 914948 9. 840378 10. 159622 18 43 9. 755508 9. 914860 9. 840647 10. 159352 17 44 9. 755690 9. 914773 9. 840917 10. 159083 16 45 9. 755872 9. 914685 9. 841187 10. 158813 15 46 9. 756054 9. 914597 9. 841457 10. 158543 14 47 9. 756236 9. 914510 9. 841726 10. 158273 13 48 9. 756418 9. 914422 9. 841996 10. 158004 12 49 9. 756600 9. 914334 9. 842206 10. 157734 11 50 9. 756781 9. 914146 9. 842505 10. 157465 10 51 9. 756963 9. 914158 9. 842804 10. 157195 9 52 9. 757144 9. 914070 9. 843074 10. 156926 8 53 9. 757316 9. 913982 9. 843343 10. 156657 7 54 9. 757507 9. 913894 9. 843612 10. 156387 6 55 9. 757688 9. 913806 9. 843882 10. 156118 5 56 9. 757869 9. 913718 9. 844151 10. 155849 4 57 9. 758049 9. 913630 9. 844420 10. 155580 3 58 9. 758230 9. 913541 9. 844689 10. 155311 2 59 9. 758411 9. 913453 9. 844958 10. 155041 1 60 9. 758591 9. 913361 9. 845227 10. 154773 0 Co-sine Sine Co-tang . Tangent M Degree 55. Degree 35. M Sine Co-sine Tangent Co-tang . 0 9. 758591 9. 913364 9. 845227 10. 154774 60 1 9. 758772 9. 913276 9. 845496 10. 154504 59 2 9. 758952 9. 913187 9. 845764 10. 154235 58 3 9. 759132 9. 913099 9. 846033 10. 153967 57 4 9. 759312 9. 913010 9. 846302 10. 153698 56 5 9. 759492 9. 912921 9. 846570 10. 153429 55 6 9. 759672 9. 912833 9. 846839 10. 153161 54 7 9. 759851 9. 912744 9. 847107 10. 152892 53 8 9. 760031 9. 912655 9. 847376 10. 152624 52 9 9. 760210 9. 912566 9. 847644 10. 152356 51 10 9. 760390 9. 912477 9. 847913 10. 152087 50 11 9. 760569 9. 912388 9. 848181 10. 151819 49 12 9. 760748 9. 912299 9. 848449 10. 151551 48 13 9. 760927 9. 912210 9. 848717 10. 151283 47 14 9. 761106 9. 912121 9. 848985 10. 151015 46 15 9. 761285 9. 912031 9. 849254 10. 150746 45 16 9. 761464 9. 911942 9. 849522 10. 150478 44 17 9. 761642 9. 911853 9. 849789 10. 150214 43 18 9. 761821 9. 911763 9. 850057 10. 149943 42 19 9. 761999 9. 911674 9. 850325 10. 149675 41 20 9. 762177 9. 911584 9. 850593 10. 149407 40 21 9. 762356 9. 911495 9. 850861 10. 149139 39 22 9. 762534 9. 911405 9. 851128 10. 148872 38 23 9. 762712 9. 911315 9. 851396 10. 148604 37 24 9. 762889 9. 911226 9. 851664 10. 148336 36 25 9. 763067 9. 911136 9. 851931 10. 148069 35 26 9. 763245 9. 911046 9. 852199 10. 147801 34 27 9. 763422 9. 910956 9. 852466 10. 147534 33 28 9. 763599 9. 910866 9. 852731 10. 147267 32 29 9. 763777 9. 910776 9. 853001 10. 146999 31 30 9. 763954 9. 910686 9. 853268 10. 146732 30 Co-sine Sine Co-tang . Tangent M Degree 54 Degree 35. M Sine Co-sine Tangent Co-tang . 30 9. 763954 9. 910686 9. 853208 10. 146732 30 31 9. 764131 9. 910596 9. 853532 10. 146465 29 32 9. 764308 9. 910506 9. 853802 10. 146198 28 33 9. 764485 9. 910415 9. 854069 10. 145930 27 34 9. 764662 9. 910325 9. 854336 10. 145664 26 35 9. 764838 9. 910235 9. 854603 10. 145397 25 36 9. 765015 9. 910144 9. 854870 10. 145130 24 37 9. 765191 9. 910054 9. 855137 10. 144863 23 38 9. 765367 9. 909963 9. 855404 10. 144596 22 39 9. 765544 9. 909873 9. 855671 10. 144329 21 40 9. 765720 9. 909782 9. 855937 10. 144063 20 41 9. 765896 9. 909691 9. 856204 10. 143796 19 42 9. 766071 9. 909601 9. 856471 10. 143529 18 43 9. 766247 9. 909510 9. 856737 10. 143263 17 44 9. 766423 9. 909419 9. 857004 10. 142996 16 45 9. 766598 9. 909328 9. 857270 10. 142730 15 46 9. 766774 9. 909237 9. 857537 10. 142463 14 47 9. 766949 9. 909146 9. 857803 10. 142197 13 48 9. 767124 9. 909055 9. 858069 10. 141931 12 49 9. 767299 9. 908964 9. 858336 10. 141664 11 50 9. 767474 9. 908873 9. 858602 10. 141398 10 51 9. 767649 9. 908781 9. 858868 10. 141132 9 52 9. 767824 9. 908690 9. 859134 10. 140866 8 53 9. 767997 9. 908599 9859400 10. 140600 7 54 9. 768173 9. 908507 9. 859666 10. 140334 6 55 9. 768348 9. 908416 9. 859932 10. 140068 5 56 9. 768522 9. 908324 9. 860198 10. 139802 4 57 9. 768696 9. 908233 9. 860464 10. 139536 3 58 9. 768871 9. 908141 9860730 10. 139270 2 59 9. 769045 9. 908049 9. 860995 10. 139005 1 60 9. 769219 9. 907958 9. 861261 10. 138739 0 Co-sine Sine Co-tang . Tangent M Degree 54. Degree 36. M Sine Co-sine Tangent Co-tang . 0 9. 769219 9. 907958 9. 861261 10. 138739 60 1 9. 769392 9. 907866 9. 861527 10. 138473 59 2 9. 769566 9. 907774 9. 861792 10. 138208 58 3 9. 769740 9. 907682 9. 862058 10. 137942 57 4 9. 769913 9. 907590 9. 862323 10. 137677 56 5 9. 770087 9. 907498 9. 862589 10. 137411 55 6 9. 770260 9. 907406 9. 862854 10. 137146 54 7 9. 770433 9. 907314 9. 863119 10. 136880 53 8 9. 770606 9. 907221 9. 863385 10. 136615 52 9 9. 770779 9. 907129 9. 863650 10. 136350 51 10 9. 770952 9. 907037 9. 863915 10. 136085 50 11 9. 771125 9. 906945 9. 864180 10. 135820 49 12 9. 771298 9. 906852 9. 864445 10. 135554 48 13 9. 771470 9. 906760 9. 864710 10. 135289 47 14 9. 771643 9. 906667 9. 864975 10. 135024 46 15 9. 771815 9. 906574 9. 865240 10. 134759 45 16 9. 771987 9. 906482 9. 865505 10. 134495 44 17 9. 772159 9. 906389 9. 865770 10. 134230 43 18 9. 772331 9. 906296 9. 866035 10. 133965 42 19 9. 772503 9. 906203 9. 866300 10. 133700 41 20 9. 772675 9. 906111 9. 866564 10. 133436 40 21 9. 772847 9. 906018 9. 866829 10. 133171 39 22 9. 773018 9. 905925 9. 867094 10. 132906 38 23 9. 773190 9. 905832 9. 867358 10. 132642 37 24 9. 773361 9. 905738 9. 867623 10. 132377 36 25 9. 773533 9. 905645 9. 867887 10. 132113 35 26 9. 773704 9. 905552 9. 868152 10. 131848 34 27 9. 773875 9. 905459 9. 868416 10. 131584 33 28 9. 774046 9. 905365 9. 868680 10. 131320 32 29 9. 774217 9. 905272 9. 868945 10. 131055 31 30 9. 774388 9. 905179 9. 869209 10. 130791 30 Co-sine Sine Co-tang . Tangent M Degree 53. Degree 36. M Sine Co-sine Tangent Co-tang . 30 9. 774388 9. 905179 9. 869209 10. 130791 30 31 9. 774558 9. 905085 9. 864773 10. 130527 29 32 9. 774729 9. 904992 9. 867337 10. 130263 28 33 9. 774899 9. 904898 9. 870001 10. 129999 27 34 9. 775070 9. 904804 9. 870265 10. 129735 26 35 9. 775240 9. 904711 9. 870529 10. 129471 25 36 9. 775410 9. 904617 9. 870793 10. 129207 24 37 9. 775580 9. 904523 9. 871057 10. 128943 23 38 9. 775750 9. 904429 9. 871321 10. 128679 22 39 9. 775920 9. 904335 9. 871585 10. 128415 21 40 9. 776090 9. 904241 9. 871849 10. 128151 20 41 9. 776259 9. 904147 9. 872112 10. 127888 19 42 9. 776429 9. 904053 9. 872376 10. 127624 18 43 9. 776598 9. 903959 9. 872640 10. 127360 17 44 9. 776768 9. 903864 9. 872903 10. 127097 16 45 9. 776937 9. 903770 9. 873167 10. 126833 15 46 9. 777106 9. 903676 9. 873430 10. 126570 14 47 9. 777275 9. 903581 9. 873694 10. 126306 13 48 9. 777444 9. 903486 9. 873957 10. 126043 12 49 9. 777613 9. 903392 9. 874220 10. 125780 11 50 9. 777781 9. 903298 9. 874484 10. 125516 10 51 9. 777950 9. 903203 9. 874747 10. 125253 9 52 9. 778119 9. 903108 9. 875010 10. 124990 8 53 9. 778287 9. 903013 9. 875273 10. 124727 7 54 9. 778455 9. 902919 9. 875536 10. 124464 6 55 9. 778623 9. 902824 9. 875799 10. 124201 5 56 9. 778792 9. 902729 9. 876063 10. 123937 4 57 9. 778960 9. 902634 9. 876326 10. 123674 3 58 9. 779129 9. 902539 9. 876589 10. 123411 2 59 9. 779295 9. 902444 9. 876851 10. 123149 1 60 9. 779463 9. 902349 9. 877114 10. 122886 0 Co-sine Sine Co-tang . Tangent M Degree 53. Degree 37. M Sine Co-sine Tangent Co-tang . 0 9 779463 9. 902349 9. 877114 10. 122885 60 1 9. 779631 9. 902253 9. 877377 10. 122623 59 2 9. 779798 9. 902158 9. 877640 10. 122360 58 3 9. 779965 9. 902063 9. 877903 10. 122097 57 4 9. 780133 9. 901967 9. 878165 10. 121834 56 5 9. 780300 9. 901872 9. 878428 10. 121572 55 6 9. 780467 9. 901776 9. 878691 10. 121309 54 7 9. 780634 9. 901681 9. 878953 10. 121047 53 8 9. 780801 9. 901585 9. 879216 10. 120784 52 9 9. 780968 9. 901488 9. 879478 10. 120522 51 10 9. 781134 9. 901391 9. 879741 10. 120259 50 11 9. 781301 9. 901298 9. 880003 10. 119997 49 12 9. 781467 9. 901202 9. 880265 10. 119734 48 13 9. 781634 9. 901106 9. 880528 10. 119472 47 14 9. 781800 9. 901010 9. 880790 10. 119210 46 15 9. 781966 9. 900914 9. 881052 10. 118948 45 16 9. 782132 9. 900828 9. 881314 10. 118686 44 17 9. 782298 9. 900722 9. 881576 10. 118424 43 18 9. 782464 9. 900626 9. 881839 10. 118161 42 19 9. 782690 9. 900529 9. 882101 10. 117899 41 20 9. 782796 9. 900433 9. 882363 10. 117637 40 21 9. 782961 9. 900337 9. 882625 10. 117375 39 22 9. 783127 9. 900240 9. 882886 10. 117114 38 23 9. 783292 9. 900144 9. 883148 10. 116852 37 24 9. 783457 9. 900047 9. 883410 10. 116590 36 25 9. 783623 9. 899951 9. 883672 10. 116328 35 26 9. 783788 9. 899854 9. 883934 10. 116066 34 27 9. 783953 9. 899757 9. 884195 10. 115805 33 28 9. 784118 9. 899660 9. 884457 10. 115543 32 29 9. 784282 9. 899563 9. 884719 10. 115281 31 30 9. 784447 9. 899467 9. 884980 10. 115020 30 Co-sine Sine Co-tang . Tangent M Degree 52. Degree 37. M Sine Co-sine Tangent Co-tang . 30 9. 784447 9. 899467 9. 884980 10. 115025 30 31 9. 784616 9. 899370 9. 885242 10. 114758 29 32 9. 784776 9. 899273 9. 885503 10. 114497 28 33 9. 784941 9. 899175 9. 885765 10. 114235 27 34 9. 785105 9. 899078 9. 886026 10. 113974 26 35 9. 785269 9. 898981 9. 886288 10. 113712 25 36 9. 785433 9. 898884 9. 886549 10. 113451 24 37 9. 785591 9. 898787 9. 886810 10. 113190 23 38 9. 785761 9. 898689 9. 887072 10. 112928 22 39 9. 785925 9. 898592 9. 887333 10. 112667 21 40 9. 786088 9. 898494 9. 887594 10. 112406 20 41 9. 786252 9. 898397 9. 887855 10. 112145 19 42 9. 786416 9. 898299 9. 888116 10. 111884 18 43 9. 786579 9. 898201 9. 888377 10. 111623 17 44 9. 786742 9. 898104 9. 888638 10. 111362 16 45 9. 786909 9. 898006 9. 888899 10. 111101 15 46 9. 787069 9. 897908 9. 889160 10. 110840 14 47 9. 787232 9. 897810 9. 889421 10. 110579 13 48 9. 787395 9. 897112 9. 889682 10. 110318 12 49 9. 787557 9. 897614 9. 889943 10. 110057 11 50 9. 787720 9. 897516 9. 890204 10. 109796 10 51 9. 787883 9. 897418 9. 890465 10. 109535 9 52 9. 788045 9. 897320 9. 890725 10. 109275 8 53 9. 788208 9. 897222 9. 890986 10. 109014 7 54 9. 788370 9. 897123 9. 891248 10. 108753 6 55 9. 788532 9. 897025 9. 891507 10. 108493 5 56 9. 788694 9. 896926 9. 891768 10. 108232 4 57 9. 788856 9. 896828 9. 892028 10. 107972 3 58 9. 789018 9. 896729 9. 892289 10. 107711 2 59 9. 789180 9. 896631 9. 892549 10. 107451 1 60 9. 789342 9. 896532 9. 892810 10. 107190 0 Co-sine Sine Co-tang . Tangent M Degree 52. Degree 38. M Sine Co-sine Tangent Co-tang . 0 9. 789342 9. 896532 9. 892810 10. 107190 60 1 9. 789504 9. 896433 9. 893070 10. 106930 59 2 9. 789665 9. 896335 〈◊〉 10. 106669 58 3 9. 789827 9. 896236 9. 893591 10. 106409 57 4 9. 789988 9. 896137 9. 893851 10. 106149 56 5 9. 790149 9. 896038 9. 894111 10. 105889 55 6 9. 790310 9. 895939 9. 894371 10. 105628 54 7 9. 790471 9. 895840 9. 894632 10. 105368 53 8 9. 790632 9. 895741 9. 894892 10. 105108 52 9 9. 790793 9. 895641 9. 895152 10. 104844 51 10 9. 790954 9. 895542 9. 895412 10. 104588 50 11 9. 791115 9. 895443 9. 895672 10. 104328 49 12 9. 791275 9. 895343 9. 895932 10. 104068 48 13 9. 791436 9. 895244 9. 896192 10. 103808 47 14 9. 791596 9. 895144 9. 896452 10. 103548 46 15 9. 791756 9. 895045 9. 896712 10. 103288 45 16 9. 791917 9. 894945 9. 896971 10. 103028 44 17 9. 792077 9. 894846 9. 897231 10. 102769 43 18 9. 792237 9. 894746 9. 897491 10. 102509 42 19 9. 792397 9. 894646 9. 897751 10. 102249 41 20 9. 792557 9. 894546 9. 898010 10. 101990 40 21 9. 792716 9. 894446 9. 898270 10. 101730 39 22 9. 792876 9. 894346 9. 898530 10. 101470 38 23 9. 793035 9. 894246 9. 898789 10. 101211 37 24 9. 793195 9. 894146 9. 899049 10. 100951 36 25 9. 793354 9. 894046 9 899308 10. 100692 35 26 9. 793513 9. 893946 9. 899568 10. 100432 34 27 9. 793673 9. 893845 9. 899827 10. 100173 33 28 9. 793832 9. 893745 9. 900086 10. 099913 32 29 9. 793991 9. 893645 9. 900346 10. 099654 31 30 9. 794149 9. 893544 9. 900605 10. 099395 30 Co-sine Sine Co-tang . Tangent M Degree 51. Degree 38. M Sine Co-sine Tangent Co-tang . 30 9. 794149 9. 893544 9. 900605 10. 099395 30 31 9. 794308 9. 893444 9. 900864 10. 099135 29 32 9. 794467 9. 893343 9. 901124 10. 098876 28 33 9. 794626 9. 893243 9. 901383 10. 098617 27 34 9. 794784 9. 893142 9. 901642 10. 098358 26 35 9. 794942 9. 893041 9. 901901 10. 098099 25 36 9. 795101 9. 892940 9. 902160 10. 097839 24 37 9. 795259 9. 892839 9. 902419 10. 097580 23 38 9. 795417 9. 892738 9. 902678 10. 097321 22 39 9. 795575 9. 892637 9. 902937 10. 097062 21 40 9. 795733 9. 892536 9. 903196 10. 096803 20 41 9. 795891 9. 892435 9. 903455 10. 096544 19 42 9. 796049 9. 892334 9. 903714 10. 096285 18 43 9. 796206 9. 892233 9. 903973 10. 096027 17 44 9. 796364 9. 892132 9. 904232 10. 095768 16 45 9. 796521 9. 892030 9. 904491 10. 095509 15 46 9. 796678 9. 891929 9. 904750 10. 095250 14 47 9. 796836 9. 891827 9. 905008 10. 094991 13 48 9. 796993 9. 891726 9. 905267 10. 094733 12 49 9. 797150 9. 891624 9. 905526 10. 094474 11 50 9. 797307 9. 891522 9. 905784 10. 094215 10 51 9. 797464 9. 891421 9. 906043 10. 093957 9 52 9. 797621 9. 891319 9. 906302 10. 093698 8 53 9. 797777 9. 891217 9. 906560 10. 093440 7 54 9. 797934 9. 891115 9. 906819 10. 093181 6 55 9. 798091 9. 891013 9. 907077 10. 092923 5 56 9. 798247 9. 890911 9. 907336 10. 092664 4 57 9. 798403 9. 890809 9. 907594 10. 092406 3 58 9. 798560 9. 890707 9. 907852 10. 092147 2 59 9. 798716 9. 890605 9. 908111 10. 091889 1 60 9. 798872 9. 890503 9. 908369 10. 091631 0 Co-sine . Sine Co-tang . Tangent M Degree 51. Degree 39. M Sine Co-sine Tangent Co-tang . 0 9. 798872 9. 890503 9. 908369 10. 091631 60 1 9. 799028 9. 890400 9. 908627 10. 091373 59 2 9. 799184 9 890298 9. 908886 10. 091114 58 3 9. 799339 9. 890195 9. 909144 10. 090856 57 4 9. 799495 9. 890093 9. 909402 10. 090598 56 5 9. 799651 9. 889990 9. 909660 10. 090340 55 6 9. 799806 9. 889888 9. 909918 10. 090081 54 7 9. 799961 9. 889785 9. 910176 10. 089823 53 8 9. 800117 9. 889682 9. 910435 10. 089565 52 9 9. 800272 9. 889579 9. 910693 10. 089307 51 10 9. 800427 9. 889476 9. 910951 10. 089049 50 11 9. 800582 9. 889374 9. 911209 10. 088791 49 12 9. 800737 9. 889271 9. 911467 10. 088533 48 13 9. 800892 9. 889167 9. 911724 10. 088275 47 14 9. 801047 9. 889064 9. 911982 10. 088017 46 15 9. 801201 9. 888961 9. 912240 10. 087760 45 16 9. 801356 9. 888858 9. 912498 10. 087502 44 17 9. 801510 9. 888755 9. 912756 10. 087244 43 18 9. 801665 9. 888651 9. 913014 10. 086986 42 19 9. 801819 9. 888548 9. 913271 10 086729 41 20 9. 801973 9. 888444 9. 913529 10. 086471 40 21 9. 802127 9. 888341 9. 913787 10. 086213 39 22 9. 802282 9. 888237 9. 914044 10. 085956 38 23 9. 802435 9. 888133 9 914302 10. 085698 37 24 9. 802589 9. 888030 9. 914560 〈◊〉 36 25 9. 802743 9. 887926 9. 914817 10. 085183 35 26 9. 802897 9. 887822 9. 915075 10. 084925 34 27 9. 803050 9. 887718 9. 915332 10. 084668 33 28 9. 803204 9. 887614 9. 915590 10. 084410 32 29 9. 803357 9. 887510 9. 915847 10. 084153 31 30 9. 803510 9. 887406 9. 916104 10. 083895 30 Co-sine . Sine Co-tang . Tangent M Degree 50. Degree 39. M Sine Co-sine Tangent Co-tang . 30 9. 803510 9. 887406 9. 916104 10. 083895 30 31 9. 803664 9. 887302 9. 916362 10. 083638 29 32 9. 803817 9. 887198 9. 916619 10. 083381 28 33 9. 803970 9. 887093 9. 916876 10. 083123 27 34 9. 804123 9. 887989 9. 917134 10. 082866 26 35 9. 804276 9. 886884 9. 917391 10. 082609 25 36 9. 804428 9. 886780 9. 917648 10. 082352 24 37 9. 804581 9. 886675 9. 917905 10. 082094 23 38 9. 804734 9. 886571 9. 918162 10. 081837 22 39 9. 804886 9. 886466 9. 918420 10. 081580 21 40 9. 805038 9. 886361 9. 918677 10. 081323 20 41 9. 805191 9. 886257 9. 918934 10. 081066 19 42 9. 805343 9. 886152 9. 919191 10. 080809 18 43 9. 805495 9. 886047 9. 919448 10. 080552 17 44 9. 805647 9. 885942 9. 919705 10. 080295 16 45 9. 805799 9. 885837 9. 919962 10. 080038 15 46 9. 805951 9. 885732 9. 920219 10. 079781 14 47 〈◊〉 9. 885627 9. 920476 10. 079524 13 48 9. 806254 9. 885521 9. 920733 10. 079267 12 49 9. 806406 9. 885416 9. 920990 10. 079010 11 50 9. 806557 9. 885311 9. 921247 10. 078753 10 51 9. 806709 9. 885205 9. 921503 10. 078496 9 52 9. 806860 9. 885100 9. 921760 10. 078240 8 53 9. 807011 9. 884994 9. 922017 10. 077983 7 54 9. 807162 9. 884889 9. 922274 10. 077726 6 55 9. 807314 9. 884783 9. 922530 10. 077469 5 56 9. 807464 9. 884677 9. 922787 10. 077213 4 57 9. 807615 9. 884572 9. 923044 10. 076956 3 58 9. 807766 9. 884466 9. 923300 10. 076699 2 59 9. 807917 9. 884360 9. 923557 10. 076443 1 60 9. 808067 9. 884254 9. 923813 10. 076186 0 Co-sine Sine Co-tang Tangent M Degree 50 Degree 40. M Sine Co-sine Tangent Co-tang . 0 9. 808067 9. 884254 9. 923813 10. 076180 60 1 9. 808218 9. 884148 9. 924070 10. 075930 59 2 9. 808368 9. 884042 9. 924327 10. 075673 58 3 9. 808519 9. 883936 9. 924583 10. 075417 57 4 9. 808669 9. 883829 9. 924839 10. 075160 56 5 9. 808819 9. 883723 9. 925096 10. 074904 55 6 9. 808969 9. 883617 9. 925352 10. 074647 54 7 9. 809119 9. 883510 9. 925609 10. 074391 53 8 9. 809269 9. 883404 9. 925865 10. 074135 52 9 9. 809419 9. 883297 9. 926121 10. 073878 51 10 9. 809569 9. 883191 9. 926378 10. 073622 50 11 9. 809718 9. 883084 9. 926634 10. 073366 49 12 9. 809868 9. 882977 9. 926890 10. 073110 48 13 9. 810017 9. 882871 9. 927147 10. 072853 47 14 9. 810166 9. 882764 9. 927403 10. 072597 46 15 9. 810316 9. 882657 9. 927659 10. 072341 45 16 9. 810465 9. 882550 9. 927915 10. 072085 44 17 9. 810614 9. 882443 9. 928171 10. 071829 43 18 9. 810763 9. 882336 9. 928427 10. 071573 42 19 9. 810912 9. 882228 9. 928683 10. 071317 41 20 9. 810061 9. 882121 9. 928940 10. 071060 40 21 9. 811210 9. 882014 9. 929196 10. 070804 39 22 9. 811358 9. 881907 9. 929452 10. 070548 38 23 9. 811506 9. 881799 9. 929708 10. 070292 37 24 9. 811655 9. 881692 9. 929964 10. 070036 36 25 9. 811804 9. 881584 9. 930219 10. 069781 35 26 9. 811952 9. 881477 9. 930475 10. 069525 34 27 9. 812100 9. 881369 9. 930731 10. 069269 33 28 9. 812248 9. 881261 9. 930987 10. 069013 32 29 9. 812396 9. 881153 9. 931243 10. 068757 31 30 9. 812544 9. 881045 9. 931499 10. 068501 30 Co-sine Sine Co-tang . Tangent M Degree 55. Degree 40. M Sine Co-sine Tangent Co-tang . 30 9. 812544 9. 881045 9. 931499 10. 068501 30 31 9. 812692 9. 880937 9. 931755 10. 068245 29 32 9. 812840 9. 880829 9. 932010 10. 067989 28 33 9. 812988 9. 880721 9. 932266 10. 067734 27 34 9. 813135 9. 880613 9. 932522 10. 067478 26 35 9. 813283 9. 880505 9. 932778 10. 067222 25 36 9. 813430 9. 880397 9. 933033 10. 066967 24 37 9. 813578 9. 880289 9. 933289 10. 066711 23 38 9. 813725 9. 880180 9. 933545 10. 066455 22 39 9. 813872 9. 880072 9. 933800 10. 066200 21 40 9. 814019 9. 879963 9. 934056 10. 065944 20 41 9. 814166 9. 879855 9. 934311 10. 065688 19 42 9. 814313 9. 879746 9. 934567 10. 065433 18 43 9. 814460 9. 879637 9. 934822 10. 065177 17 44 9. 814607 9. 879529 9. 935078 10. 064922 16 45 9. 814753 9. 879420 9. 935333 10. 064666 15 46 9. 814900 9. 879311 9. 935589 10. 064411 14 47 9. 815046 9. 879202 9. 935844 10. 064156 13 48 9. 815193 9. 879093 9. 936100 10. 063900 12 49 9. 815339 9. 878984 9. 936355 10. 063645 11 50 9. 815485 9. 878875 9. 936610 10. 063389 10 51 9. 815631 9. 878766 9. 936866 10. 063134 9 52 9. 815777 9. 878656 9. 937121 10. 062879 8 53 9. 815923 9. 878547 9. 937376 10. 062623 7 54 9. 816069 9. 878438 9. 937632 10. 062368 6 55 9. 816215 9. 878328 9. 937887 10. 062113 5 56 9. 816361 9. 878219 9. 938142 10. 061858 4 57 9. 816506 9. 878109 9. 938397 10. 061602 3 58 9. 816652 9. 877999 9. 938653 10. 061347 2 59 9. 816797 9. 877890 9. 938908 10. 061092 1 60 9. 816943 9. 877780 9. 939163 10. 060837 0 Co-sine Sine Co-tang . Tangent M Degree 49. Degree 41. M Sine Co-sine Tangent Co-tang . 0 9. 816943 9. 877780 9. 939163 10. 060837 60 1 9. 817088 9. 877670 9. 939418 10. 060582 59 2 9. 817233 9. 877560 9. 939673 10. 060327 58 3 9. 817378 9. 877450 〈◊〉 10. 060072 57 4 9. 817523 9. 877340 9. 940183 10. 059816 56 5 9. 817668 9. 877230 9. 940438 10. 059562 55 6 9. 817813 9. 877120 9. 940693 10. 059307 54 7 9. 817958 9. 877009 9. 940948 10. 059052 53 8 9. 818103 9. 876899 9. 941203 10. 058797 52 9 9. 818247 9. 876789 9. 941458 10. 058542 51 10 9. 818392 9. 876678 9 941713 10. 058287 50 11 9. 818536 9. 876568 9. 941968 10. 058032 49 12 9. 818681 9. 876457 9. 942223 10. 057777 48 13 9. 818825 9. 876347 9. 942478 10. 057522 47 14 9. 818969 9. 876236 9. 942733 10. 057267 46 15 9. 818113 9. 876125 9. 942988 10. 057012 45 16 9. 819257 9. 876014 9. 943243 10. 056757 44 17 9. 819401 9. 875904 9. 943498 10. 056502 43 18 9. 819545 9. 875793 9. 943752 10. 056248 42 19 9. 819689 9. 875682 9. 944007 10. 055993 41 20 9. 819832 9. 875571 9. 944262 10. 055728 40 21 9. 819976 9. 875459 9. 944517 10. 055483 39 22 9. 820119 9. 875348 9. 944771 10. 055229 38 23 9. 820263 9. 875237 9. 945026 10. 054974 37 24 9. 820406 9. 875125 9. 945281 10. 054719 36 25 9. 820549 9. 875014 9. 945535 10. 054464 35 26 9. 820693 9. 874903 9. 945790 10. 054210 34 27 9. 820836 9. 874791 9. 946045 10. 053955 33 28 9. 820979 9. 874679 9. 946299 10. 053701 32 29 9. 821122 9. 874568 9. 946554 10. 053446 31 30 9. 821264 9. 874456 9. 946808 10. 053192 30 Co-sine Sine Co-tang . Tangent M Degree 48. Degree 41. M Sine Co-sine Tangent Co-tang . 30 9. 821264 9. 874456 9. 946808 10. 053192 30 31 9. 821407 9. 874344 9 947063 10. 052937 29 32 9. 821550 9. 874232 9. 947317 10. 052682 28 33 9. 821692 9. 874120 9. 947572 10. 052128 27 34 9. 821835 9 874008 9. 947826 10. 052173 26 35 9. 821977 9. 873896 9. 948081 10. 051919 25 36 9. 822120 9. 873784 9. 948335 10. 051664 24 37 9. 822262 9. 873672 9. 948590 10. 051410 23 38 9 822404 9 873560 9. 948844 10. 051156 22 39 9. 822546 9. 873447 9. 949099 10. 050901 21 40 9. 822688 9. 873335 9. 949353 10. 050647 20 41 9. 822830 9 873223 9. 949607 10. 050393 19 42 9. 822972 9. 873110 9 949862 10. 〈◊〉 18 43 9. 823114 9. 872998 9. 950116 10. 049884 17 44 9. 823255 9. 872885 9 950370 10. 049630 16 45 9. 823397 9. 872772 9. 950625 10 049375 15 46 9. 823538 9. 872659 9. 950879 10. 049121 14 47 9. 823680 9. 872546 9. 951133 10. 048867 13 48 9. 823821 9. 872434 9. 951388 10. 048612 12 49 9. 823962 9. 872321 9. 951642 10. 048358 11 50 9. 824104 9. 872208 9. 951896 10. 048104 10 51 9. 824245 9. 872094 9. 952150 10. 047850 9 52 9. 824386 9. 871981 9. 952404 10. 047575 8 53 9. 824527 9. 871868 9. 952659 10. 047341 7 54 9. 824667 9. 871755 9. 952913 10. 047087 6 55 9. 824808 9. 871641 9. 953167 10. 046833 5 56 9. 824949 9. 871528 9. 953421 10. 046579 4 57 9. 825090 9. 871414 9. 953675 10. 046325 3 58 9. 825230 9. 871301 9. 953929 10. 046071 2 59 9. 825370 9. 871187 9. 954183 10. 045817 1 60 9. 825511 9. 871073 9. 954437 10. 045563 0 Co-sine Sine Co-tang . Tangent M Degree 48. Degree 42. M Sine Co-sine Tangent Co-tang . 0 9. 825511 9. 871073 9. 954437 10. 045562 60 1 9. 825651 9. 870960 9. 954691 10. 045308 59 2 9. 825791 9. 870846 9. 954945 10. 045054 58 3 9. 825931 9. 870732 9. 955199 10. 044800 57 4 9. 826071 9. 870618 9. 955453 10. 044546 56 5 9. 826211 9. 870504 9. 955707 10. 044292 55 6 9. 826351 9. 870390 9. 955961 10. 043038 54 7 9. 826491 9 870275 9. 956215 10. 043784 53 8 9. 826631 9. 870161 9. 956469 10. 043531 52 9 9. 826770 9. 870047 9. 956723 10. 043276 51 10 9. 826910 9. 869933 9. 956977 10. 043023 50 11 9. 827049 9. 869818 9. 957231 10. 042769 49 12 9. 827189 9. 869704 9. 957485 10. 042515 48 13 9. 827328 9. 869589 9. 957739 10. 042261 47 14 9. 827467 9. 869474 9. 957993 10. 042007 46 15 9. 827606 9. 869360 9. 958246 10. 041753 45 16 9. 827745 9. 869245 9 958500 10. 041500 44 17 9. 827884 9. 869130 9. 958754 10. 041246 43 18 9. 828023 9. 869015 9. 959008 10. 040992 42 19 9. 828162 9. 868900 9. 959262 10. 040738 41 20 9. 828301 9. 868785 9. 959515 10. 040485 40 21 9. 828439 9. 868670 9. 959769 10. 040231 39 22 9. 828578 9. 868555 9. 960023 10. 039977 38 23 9. 828716 9. 868439 9. 960277 10. 039723 37 24 9. 828855 9. 868324 9. 960530 10. 039469 36 25 9. 828993 9. 868209 9. 960784 10. 039216 35 26 9. 829131 9. 868093 9. 961038 10. 038962 34 27 9. 829269 9. 867978 9. 961291 10. 038608 33 28 9. 829407 9. 867862 9. 961545 10. 038451 32 29 9. 829545 9. 867747 9. 961799 10. 038201 31 30 9. 829683 9. 867631 9. 962052 10. 037947 30 Co-sine Sine Co-tang . Tangent M Degree 47. Degree 42. M Sine Co-sine Tangent Co-tang . 30 9. 829683 9. 867631 9. 962052 10. 037947 30 31 9. 829821 9. 867515 9. 962306 10. 037694 29 32 9. 829959 9. 867399 9. 962560 10. 037440 28 33 9. 830096 9. 867283 9. 962813 10. 037187 27 34 9. 830234 9. 867167 9. 963067 10. 036933 26 35 9. 830372 9. 867051 9. 963320 10. 036680 25 36 9 830509 9. 866935 9. 963574 10. 036426 24 37 9. 830646 9. 866819 9. 963827 10. 036173 23 38 7. 830784 9. 866703 9. 964081 10. 035919 22 39 9. 830921 9. 866586 9. 964335 10. 035665 21 40 9. 831058 9. 866470 9. 964588 10. 035412 20 41 9. 831195 9. 866353 9. 964842 10. 035158 19 42 9. 831332 9. 866237 9. 965095 10. 034905 18 43 9. 831469 9. 866120 9. 965348 10. 034652 17 44 9. 831606 9. 866004 9. 965602 10. 034398 16 45 9. 831742 9. 865887 9. 965855 10. 034144 15 46 9. 831879 9. 865770 9. 966109 10. 033891 14 47 9. 832015 9. 865653 9 966362 10. 033638 13 48 9. 832152 9. 865536 9. 966616 10. 033384 12 49 9. 832288 9. 865419 9. 966869 10. 033131 11 50 9. 832425 9. 865302 9. 967122 10. 032878 10 51 9. 832561 9. 865185 9. 967376 10. 032624 9 52 9. 832697 9. 865068 9. 967629 10. 032371 8 53 9. 832833 9. 864950 9. 967883 10. 032117 7 54 9. 832969 9. 864833 9. 968136 10. 031864 6 55 9. 833105 9. 864716 9. 968389 10. 031611 5 56 9. 833241 9. 864598 9. 968643 10. 031357 4 57 9. 833376 9. 864480 9. 968896 10. 031104 3 58 9. 833512 9. 864363 9. 969149 10. 030851 2 59 9. 833648 9. 864245 9 969403 10. 030597 1 60 9. 833783 9. 864127 9. 969656 10. 030344 0 Co-sine . Sine Co-tang . Tangent M Degree 47. Degree 43. M Sine Co-sine Tangent Co-tang . 0 9. 833783 9. 864127 9. 969656 10. 030344 60 1 9. 833919 9. 864010 9. 969909 10. 030091 59 2 9. 834054 9. 863892 9. 970162 10. 029838 58 3 9. 834189 9. 863774 9. 970416 10. 029584 57 4 9. 834324 9. 863656 9 970669 10. 029331 56 5 9. 834460 9. 863537 9. 970922 10. 029078 55 6 9 834595 9 863419 9.971175 10. 028827 54 7 9. 834730 9. 863301 9. 971428 10. 028571 53 8 9. 834865 9. 863183 9. 971682 10. 028318 52 9 9. 834999 9. 863064 9. 971935 10. 028065 51 10 9. 835134 9. 862946 9. 972188 10. 027812 50 11 9. 835269 9. 862827 9. 972441 10. 027559 49 12 9. 835503 9. 862709 9. 972694 10. 027306 48 13 9. 835538 9. 862590 9. 972948 10. 027052 47 14 9. 835672 9. 862471 9. 973201 10. 026799 46 15 9. 835806 9. 862353 9. 973454 10. 026546 45 16 9. 835941 9. 862234 9. 973707 10. 026293 44 17 9. 836075 9. 862115 9. 973960 10. 026040 43 18 9. 836209 9. 861996 9. 974213 10. 025787 42 19 9. 836343 9. 861877 9. 974466 10. 025533 41 20 9. 836477 9. 861757 9. 974719 10. 025280 40 21 9. 836611 9. 861638 9 974973 10. 025027 39 22 9. 836745 9 861519 9 975229 10. 024774 38 23 9. 836878 9 861399 9 975479 10. 024521 37 24 9. 837012 9. 861280 9. 975732 10. 024268 36 25 9. 837146 9. 861161 9. 975985 10. 024015 35 26 9. 837279 9. 861041 9. 976238 10. 023762 34 27 9. 837412 9. 860921 9. 976491 10. 023509 33 28 9 837546 9. 860802 9. 976744 10. 023256 32 29 9. 837679 9. 860682 9. 976997 10. 023003 31 30 9 837812 9. 860562 9. 977250 10 022750 30 Co-sine . Sine Co-tang . Tangent M Degree 46. Degree 43. M Sine Co-sine Tangent Co-tang . 30 9. 837812 9. 860562 9. 977250 10. 022750 30 31 9. 837945 9. 860442 9. 977503 10. 022497 29 32 9. 838078 9. 860322 9. 977756 10. 022244 28 33 9. 838211 9. 860202 9. 978009 10. 021991 27 34 9. 838344 9. 860082 9. 978262 10. 021738 26 35 9. 838477 9. 859962 9. 978515 10. 021485 25 36 9. 838009 9. 859842 9. 978768 10. 021232 24 37 9. 838742 9. 859721 9. 979021 10. 020979 23 38 9. 838875 9. 859601 9. 979274 10. 020726 22 39 9. 839007 9. 859480 9. 979527 10. 020473 21 40 9. 839140 9. 859360 9. 979780 10. 020220 20 41 9. 839272 9. 859239 9. 980033 10. 019967 19 42 9. 839484 9. 859118 9. 980285 10. 019714 18 43 9. 839536 9. 858998 9. 980538 10. 019461 17 44 9. 839668 9. 858877 9. 980791 10. 019209 16 45 9. 839800 9. 858756 9. 981044 10. 018956 15 46 9. 839932 9. 858639 9. 981297 10. 018703 14 47 9. 840064 9. 858514 9. 981550 10. 018450 13 48 9. 840196 9. 858398 9. 981803 10. 018197 12 49 9. 840428 9. 858272 9 982056 10. 017944 11 50 9. 840459 9. 858150 9. 982309 10 017691 10 51 9. 840591 9. 858029 9. 982562 10. 017438 9 52 9. 840722 9. 857908 9. 982814 10. 017185 8 53 9. 840854 9. 857786 9 983067 10. 016933 7 54 9. 840985 9. 857665 9. 983320 10. 016683 6 55 9. 841116 9. 857543 9. 983573 10. 016427 5 56 9. 841247 9. 857421 9. 983826 10. 016174 4 57 9. 841378 9. 857300 9. 984079 10. 015921 3 58 9. 841509 9. 857178 9. 984331 10. 015668 2 59 9. 841640 9. 857056 9. 984584 10. 015416 1 60 9. 841771 9. 856934 9. 984837 10. 015163 0 Co-sine Sine Co-tang . Tangent M Degree 46. Degree 44. M Sine Co-sine Tangent Co-tang . 0 9. 841771 9. 856934 9. 984837 10. 015162 60 1 9. 841902 9. 856812 9. 985090 10. 014910 59 2 9. 842033 9. 856690 9. 985343 10. 014657 58 3 9. 842163 9. 856568 9. 985596 10. 014404 57 4 9. 842294 9. 856445 9. 985848 10. 014151 56 5 9. 842424 9. 856323 9. 986101 10. 013899 55 6 9. 842555 9 856201 9. 986354 10. 013646 54 7 9. 842685 9. 856078 9. 986607 10. 013393 53 8 9. 842815 9. 855956 9. 986859 10. 013140 52 9 9. 842945 9. 855833 9. 987112 10. 012888 51 10 9. 843076 9. 855710 9. 987365 10. 012635 50 11 9. 843206 9. 855588 9. 987618 10. 012382 49 12 9. 843336 9. 855465 9. 987871 10. 012129 48 13 9. 843465 9. 855342 9. 988123 10. 011877 47 14 9. 843595 9. 855219 9. 988376 10. 011624 46 15 9. 843725 9. 855096 9. 988629 10. 011371 45 16 9. 843855 9. 854973 9. 988882 10. 011118 44 17 9. 843934 9. 854850 9. 〈◊〉 10. 010866 43 18 9. 844114 9. 854727 9. 989387 10. 010613 42 19 9. 844243 9. 854603 9. 989640 10. 010360 41 20 9. 844372 9. 854480 9. 989893 10. 010107 40 21 9. 844502 9. 854356 9. 990145 10. 009855 39 22 9. 844631 9. 854233 9. 990398 10. 009602 38 23 9 844760 9 854109 9. 990651 10. 009349 37 24 9. 844889 9. 853986 9. 990903 10. 009096 36 25 9. 845018 9. 853862 9. 991156 10. 008844 35 26 9. 845147 9. 853738 9. 991409 10. 008591 34 27 9. 845276 9. 853614 9. 991662 10. 008338 33 28 9. 845404 9. 853490 9. 991914 10. 008086 32 29 9. 845533 9. 853366 9. 992167 10. 007833 31 30 9. 845662 9 853242 9. 992420 10. 007580 30 Co-sine Sine Co-tang . Tangent M Degree 45. Degree 44. M Sine Co-sine Tangent Co-tang . 30 9. 845662 9. 853242 9. 992420 10. 007580 30 31 9. 845790 9. 853118 9. 992672 10. 007328 29 32 9. 845919 9. 852994 9. 992925 10. 007075 28 33 9. 846047 9. 852869 9. 993178 10. 006822 27 34 9. 846175 9. 852745 9. 993430 10. 006569 26 35 9. 846304 9. 852620 9. 993683 10. 006317 25 36 9. 846432 9. 852496 9. 993936 10. 006064 24 37 9. 846560 9. 852371 9. 994189 10. 005811 23 38 9. 846688 9. 852246 9. 994441 10. 005559 22 39 9. 846816 9. 852122 9. 994694 10. 005306 21 40 9. 846944 9. 851997 9. 994947 10. 005053 20 41 9. 847071 9. 851872 9. 995199 10. 004801 19 42 9. 847199 9. 851747 9. 995452 10. 004548 18 43 9. 847327 9. 851622 9. 995701 10. 004295 17 44 9. 847454 9. 851497 9. 995957 10. 004043 16 45 9. 847582 9. 851372 9. 996210 10. 003790 15 46 9. 847709 9. 851246 9. 996463 10. 003537 14 47 9. 847836 9. 851121 9. 996715 10. 003285 13 48 9. 847964 9. 850996 9. 996968 10. 003032 12 49 9. 848091 9. 850870 9. 997220 10. 002779 11 50 9. 848218 9. 850745 9. 997473 10. 002527 10 51 9. 848345 9. 850619 9. 997726 10. 002274 9 52 9. 848472 9. 850493 9. 997979 10. 002021 8 53 9. 848599 9. 850367 9. 998231 10. 001769 7 54 9. 848726 9. 850242 9. 998484 10. 001516 6 55 9. 848852 9. 850116 9. 998737 10. 001263 5 56 9. 848979 9. 849990 9. 998989 10. 001011 4 57 9. 849106 9. 849864 9. 999242 10. 001758 3 58 9. 849232 9. 849737 9. 999495 10. 000505 2 59 9. 849359 9. 849611 9. 999747 10. 000253 1 60 9. 849485 9. 849485 10. 000000 10. 000000 0 Co-sine Sine Co-tang Tangent M Degree 45. FINIS . Some Books sold by W. Freeman at the Artichoke next St. Dunstan's Church in Fleet-street . REports in the Court of King's-Bench from the 12th . to the 30th . Year of the Reign of our late Sovereign Lord King Charles II. in 3 Vol. Taken by J. Keble of Grey's-Inn , Esquire ; with new and usefull Tables to all the 3 Vol. Sheppard's President of Presidents . 8 o. Zenophon's History of the Affairs of Greece , inseven Books , being a Continuation of the Peloponnesian War , from the time where Thucydides ends , to the Battel at Mantinea ; to which is prefixed an Abstract of Thucydides , and a brief Account of the Land and Naval Forces of the ancient Greeks . Translated from the Greek by John Newman . The Institution and Life of Cyrus the Great , written by the famous Philosopher and General , Zenophon of Athens . Translated by F. Digby and J. Norris . Holy Devotions with Directions to Pray : Also a Brief Exposition on the Lord's Prayer , Creed , Ten Commandments , seven Penitential Psalms , and the seven Psalms of Thanksgiving , by the Right Reverend Father in God Lancelot Andrews late Bishop of Winchester . Daily Exercise for a Christian , or a Manual of Private Devotions , as well for every day in the Week , as upon Particular Occasions , by J. Bridal , Senior , Esq late of the Rolls . A New Years Gift composed of Prayers and Meditations with Devotions for several Occasions ; the whole six parts compleat . A Catalogue of Books . The Old Religion : A Treatise wherein is laid down the true State of the Difference betwixt the Reformed and Roman Church . Serving for the Vindication of Our Innocence , for the setling of Wavering Minds , for a Preservation against Popish Insinuations . By the Reverend Father in God Jos. Hall late Lord Bishop of Excester , and afterwards of Norwich . The Manners of the Israelites in 3 parts . 1. Of the Patriarchs . 2. Of the Israelites , after their coming out of Egypt , untill the Captivity of Babylon . 3. Of the Jews , after their Return from the Captivity , untill the Preaching of the Gospel , in 12 o. The Means to preserve Peace in Marriage ; being an ingenious Treatise written ( Originally in French ) by the Authour of the Rules of Civility . The Penitent Pardoned , or a Discourse of the Nature of Sin and Efficacy of Repentance , under the Parable of the Prodigal Son , by Dr. Goodman . An infallible way to Contentment in the midst of Publick and Personal Calamities , together with the Christians Courage and Encouragement against evil Tidings and the fear of Death . Argalus and Parthenia , by F. Quarles . Nanis History of Venice , Fol. History of the Government of Venice , 8 o. Policy of the Venetians , 12 o. The Mistaken Beauty , a Comedy . The Dutchess of Malphey . The Empress of Morocco , a Farce . The Court of the Gentiles in 4 parts compleat , by Theophilus Gale. Notes, typically marginal, from the original text Notes for div A64224-e630 From my House in Baldwin's Court in Baldwin's Gardens , over against the Old Hole in the Wall. Notes for div A64224-e3770 Omnia quaeeunque à primaeva rer●●m natura constructa sunt , Numcrorum vid●ntur ratione formata . Hoc enim fuit principale in animo conditor is exemplar . Boetiu● Arith. lib. 1. cap. 2. * In his second Book of his Arithmetick and the 38 Chapter ; where he saith that this proportion hath , Magnam vim in Musici modulaminis temperamentis , & in Speculatione naturalium questionum : i. e. Great force in Musical composition , ( or in the Composure of Musick ) and in the discovery of the Secrets of Nature . Defin. * As you use to doe in Division to represent the Quotient . The manner of Extracting the Square-root of a Decimal Fraction . The manner of Extracting the Square root of a M●x●-Number . Defin. * As you use to do in Division , to represent the Quotient . The manner of Extracting the Cube-Root of a De●imal Fraction . The manner of Extracting the Cube-Root of a Mixt-number . Defin. * Signifies the Number , or figure sought . * Signifies the Logarithm answering to the number Opposite . ☞ Note this Rule well , for this explains the use of the Table of proportional parts , printed at the end of this Book . Definition . The Rule to find the Characteristick or Index appertaining to any Logarithm . And here'tis necessary to understand , that every Circle is supposed to be divided into 360 Equal parts which are called Degrees , and every of those Degrees into 60 Minutes , and every Minute into 60 Seconds , and every Second into 60 Thirds , &c. so that a Semi-Circle contains 180 Degrees , and a Quadrant 90 Degrees ; Now an Arch or Angle of a Triangle , is the Intersection of its two sides , and the measure thereof , is an Arch of a Circle , which cutteth each of the two sides equidistant from the Angular point , ( which is the Center . ) Now the Logarithm Sine , or Tangof any such Arch of a Triangle , containing any Number of Degrees or Minutes of the Quadrant , may be found in the Tables , printed at the End of this Book , where they are plainly expressed , and are found as directed in the precedent Rules . Defin. Defin. * Quod quaeritur cognoscendi illius gratia , quod semper est , non & ejus quod oritur , quandoque & interit . Geometria , ejus quod est semper , Cognitio est . Ac tollet igitur ( ô Generose vir ) ad veritatem , animum : atque ita , ad Philosophandum prepar●vit cogitationem , ut ad supera convertamus : quae nunc , contra quàm decet , ad inferiora dejicimus , &c. Plato lib. 7. de Rap. Fig. 1 , Fig. 2 , Fig. 2. Fig. 3. Fig. 4. Fig. 5. Fig. ● . Fig. 6. Fig. 7. Fig. 8. Those two propositions well understood , doth demonstrate many other propositions , and thereon is grounded the vse of the Sector . 〈…〉 9. * But this in a Hexagon need not be done , because the 3 sides of the Triangle are equal , but in all other Poligons it must be done . † But if the third line do exceed or be short of the side of the Poligon propounded , then by parallels on each side , cut the sides of the Triangle , till you have found by those Intersections where to set the line proposed , in any Poligon , &c. Fig. 10. Fig. 11 ☞ Observe these Rules well , for you will find them of infinite use in Fortification , &c. Fig. 11. Fig. 12. Fig. 13. Fig. 14. Fig. 15. Fig. 16. And this of all other the Inventions of Plato , Apollonius , Sporus , Architas , Diocles , Nicomedus ; & many other famous Geometricians and Philosophers , I like best for the ready performance of this Conclusion , whose several Methods I could here describe , but for brevity sake do omit them . Fig. 17. Fig. 18. Fig. 18. * So after the same manner , may divers Circles be added into one by the help of the former proposition well understood . Fig. 19. Fig. 20. The Rule . Fig. 21. Fig. 22. Fig. 23. Fig. 24. Fig. 25. Fig. 26. Fig. 27. Fig. 28. Fig. 29. Fig. 30. Fig. 31. Fig. 32. Fig. 33. ☞ Note that every Sphere is equal unto two Cones , whose Height and Diameter of the Base is the same with the Axis of the Sphere . And a Sphere is two thirds of a Cylinder , whose Height and Diameter of the Base is the same with the Axis of the Sphere ; according unto the 9th . Manifestation of the first Book of Archimedes of the Sphere and Cylinder Fig. 34. Fig. 36. Fig. 35. Fig. 34. Observe this for a general Rule in Trigonometry . Fig. 34 Fig. 34. Fig. 34. Fig. 35. 〈…〉 Fig. 35. Fig. 35. Fig. 36. ☞ The Rule to find the Complement Arithmetical , of any Logarithm Number . 9. 962398. 0. 037602. [ ] Fig. 36. Fig. 36. Fig. 36. Fig. 37. Fig. 37. * That is equal unto the two Angles B 40° and B. 53° as afore in the former Proposition . [ ] Fig. 37. Fig. 37. Fig. 37. Fig. 37. Fig. 38. Fig. 38. Fig. 38. Fig. 38. Fig. 38. Fig. 38. Fig. 38. Fig. 38. Fig. 38. Fig. 38. Fig. 38 : Fig. 38. Fig. 38. Fig. 38. Fig. 38. Fig. 38. Fig. 39. Note that in thes● Operations , for the more facility of the learner , I omit Seconds , which doth belong unto the Angles , &c. Fig. 39. Fig. 39. ☞ And here observe that if the Sum of the two contained Sides exceed a Semicircle , then substract each side severally from 180° , and proceed with those Complements , as with the sides given , the Operation produceth the Complements of the Angles sought , unto a Semicircle or 180 Degrees . Fig. 39. ☞ And here observe that if the Sum of the two given Angles excede a Semicircle or 180° , substract them from a Semicircle , and proceed with the Residues , the Operation will produce each side 's Complement to a Semicircle , or 180 Degrees . Fig. 39. Fig. 39. Fig. 39. Fig. 39. Fig. 39. Fig. 39. Fig. 40. ☞ Note that if the Angles at the Base be both of one kind , that then the Perpendicular falls within the Triangle : if of diverse kinds , without the Triangle . Fig. 41. Fig. 40. Fig. 40. Fig. 40. Fig. 40. Fig. 40. Defin. Defin. Fig. 42. Fig. 42. Fig. 42. Fig. 42. Fig. 42. * Which is by the Greeks called 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 , i e. bring Life ; because the life of all Creatures depend on the cause of that Circle , for the Sun ascending in it and moving towards us , brings the Generation of things , and in descending the Corruption of all things sensible , and insensible , which are below the concavity of the Moon , &c. Fig. 42. Fig. 42. Fig. 42. Fig. 42 Fig. 42. Fig. 42. Fig. 42. Fig. 42. Fig. 43. Fig. 43. * If the Sun's Declination be North , and increasing , this Proportion finds the Sun's distance from ♈ ; but if decreasing from ♎ in the northern Sines . But if the Sun's Declination be South , and increasing from ♎ ; if decreasing from ♈ , among the Southern Sines . † From the next Equinoctial point either ♈ , or ♎ . * As in Case 11 of Oblique Spherical Triangles . * Watched the Time after Sun-setting when the Twilight in the West was shut in , so that no more Twilight than in any other part of the Skie near the Horizon appeared there : then by one of the known fixed Stars , having found the true Hour of the Night , he found the length of the Twilight , to be as in the rule is mentioned . * Or ½ Diurnal Arch. To find the length of the least Crepusculum or Twilight . Defin. Defin. Europe . * Which Pliny hath adorned in these words ( saith he ) Italia terrarum omnium alumna , eadem & parens , numine Deûm electa , qua Coelum ipsum clariùs faceret , sparsa congregaret imperia , ritus molliret , tot populorum discordes linguas sermonis commercio ad Colloquia distraheret , & humanitati hominem daret , i. e. Italy ( saith Pliny ) is the Nurse and the Parent of all Religion , was elected by the Providence of the Gods , to make ( if possible ) the Heavens more famous ; to gather the scattered Empires of the World into one Body , to temper the Barbarous rites of the Nations , to unite so many disagreeing Languages of Men by the benefit of one common Tongue : and in a word to restore Man to his Humanity . Europe . Europe . Europe . Europe . Europe . Europe . Europe . Europe . Europe . Europe . Europe . Europe . Europe . Europe . Europe . Europe . Europe . Europe . Europe . Europe . Europe . Europe . Europe . Asia . Asia . Asia . Asia . Asia . Asia . Asia . Asia . Asia . Asia . Asia . Asia . Africa . Africa . Africa . Africa . Africa . Africa . Africa . Africa . America . America . America . America . America . America . America . America . America . America . America . America . The Doctrine of Rightlined Triangles , both Right , and Oblique-Angled , applied to Propositions of Plain Sailing . Fig. 44. Fig. 44. Fig. 44. Fig. 44. Fig. 45. Fig. 45. Fig. 46. Fig. 46. Fig. 47. Fig. 47. Fig. 48. Fig. 49 ' . * But is indeed the Invention of our Learned Countryman Mr. Edw. Wright , although this Stranger hath almost got the Name and Praise thereof . Fig. 50. Fig. 50. Fig. 51. Fig. 51. Fig. 51. Fig. 51. Fig. 51. Fig. 51. * Which Instrument , and the Plain Table , I esteem as the two aptest Instruments for Surveying of Land ; i. e. the Plain Table for small Enclosures , and the Semicircle for Champain Plains , Woods , and Mountains . Fig. 52. Fig. 52. Fig. 53. Fig. 53. Fig. 54. Fig. 54. Fig. 55. Fig. 55. Fig. 55. Fig. 55. Fig. 55. Fig. 55. Fig. 56. Fig. 56. Fig. 56. Fig. 56. Fig. 57. Fig. 57. Fig. 58. Fig. 58. Fig. 58. Fig. 58. * Is a Quadrangle , whose sides are not Parallel , nor equal . Euclides postulat hant fabricam'Trapezium , tanquam mensulam vocari : & sanè nominis ejus ratio Geometrica nulla est : P. Rami lib. 14 pag. 94. Fig. 59. Fig. 60. Fig. 61. * Which to do is no more than thus ; with a Thread and Plummet fastened at the Center of the Semicircle , so that it hath liberty to play , move the Semicircle until the Thread playeth against 90 deg . then screw it fast , and it is Horizontal . Fig. 61. Fig. 61. Fig. 62. Fig. 63 Fig. 63. * Whose Surface is bounded by a Line called by Proclus a Helicoides , but it may also be called a Helix , a Twist or Wreath , &c. * See Procl . lib. 2. cap. 3. & Viturvius lib. 9. cap 3. * See Mr. Oughthred in his Book of the Circles of Proportion , page the 57. and Mr. Edm. Gunter in his Book of the Cross-staff , part 21. chap. the 4. * Which doth appear to have been in use above this 2400 Years , for King Achaz had a Dial : This Art requireth good skill in Geometry , and Astronomy : Now Cresibius that famous Philosopher measured the Hours and Times by the orderly running of Water . Then by Sand was the Hours measured . After that by Trochilike with Weights , and of late with Trochilike with Springs . Fig. 64. Fig. 64. Fig. 65. Fig. 65. * Which may be either a Pin of the length of Q S , placed on Q , and Perpendicular unto the Plane , or it may be a piece of brass or elsewhat of the breadth of 12 , to 3 , or 9. Fig. 66. Fig. 66. Fig. 66. Fig. 67. Fig. 68. Fig. 70. Fig. 70. Fig. 70. Fig. 70. Fig. 71. Fig. 71. Fig. 71. Fig. 71. Fig. 71. Fig. 71. Fig. 71. Fig. 71. Fig. 72. Defin. Fig. 73. Fig. 73. Fig. 73. Fig. 73. * Because the length of the part of a Musket doth not much exceed that Mèasure . Fig. 73. Fig. 73. Fig. 73. * Because the Defence ought to be easie , quick , certain , and of little charge , all which qualities the Musket hath and the Cannon hath not , therefore the Defence of Fortification ought to be measured by the Port of a Musket , and not by that of a Cannon . Fig. 74. Fig. 74. Fig. 74. Fig. 74. Fig. 74. Fig. 74. Fig. 74. Fig. 74. Fig. 74. Fig. 74. Fig. 74. Fig. 74. Observe this for a general Rule in Regular Fortification . Fig. 76. Case 1. Fig. 76. Case 2. Fig. 77. Fig. 78. Fig. 78. Fig. 78. Fig. 78. Fig. 78. Fig. 79. Fig. 79. Fig. 79. Fig. 80. Fig. 81. Fig. 83. Fig. 84. * Built to bridle the Town or the Place , left the Burghers should be rebellious , and to be the last refuge or place of retreat . * The Inginier must first form a Map of the Town or Place , with all the Ways , Passages , Old Walls , Rivers , Pools , Enclosures , and all other matters fit to be known in the draught , and then he is to design what Works he findeth most agreeing to the place to be Fortified . Fig. 84. Fig. 85. Fig. 86. Fig. 87. Fig. 88. Fig. 89. Fig. 90. Defin. This Military Engine Bombarda , Gun , Cannon , &c. So called from Bombo , a resounding Noise , Cannone , or Cannon , from the likeness it holds with his Canna , Bore , or Concavity ; Artigleria , from Artiglio , the Talons , or Claws of Ravenous Fowls , because its shot flying afar off tears and defaces all that it doth meet ; from whence some Natures of this Machine are called Smeriglii , long winged Hawks , Falconi , Falconets ; Passa volanti , swift flying Arrows , &c. Fig. 91. Fig. 91. Fig. 91. * In his Mathematical Manual . page 165. Fig. 91. * See Mr. Diggs in his Pantometria , page 179. General Rules to be observed in the battering down of a Place , or making of Breaches . * According to learned D'Chales , on the 4th Prop. of the first Book of Euclid . Fig. 93.