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 347915 348110 195 223 348305 348499 348694 348889 349083 349278 349472 349659 349860 350054 194 224 350248 350442 350636 350829 351023 351216 351409 351603 351796 351989 193 225 352183 352375 352568 352761 352954 353147 353339 353532 353724 353916 193 226 354108 354301 354493 354685 354876 355068 355239 355452 345643 355834 192 227 356026 356217 356408 356599 356790 356981 357172 357363 357554 357744 191 228 357935 358125 358316 358506 358696 258886 359076 359266 359456 359646 190 229 359835 360025 360215 360404 360593 360783 360972 361161 361350 361539 189 230 361728 361917 362105 362294 362482 362671 362859 363048 363236 363424 188 231 363612 363799 363988 364176 364363 364551 364739 364926 365113 365301 188 232 365488 365675 365862 366049 366236 366423 366609 366796 366983 367169 187 233 367356 367542 367729 367915 368101 368287 368473 368659 368845 369030 186 234 369216 369401 369587 369772 369958 370143 370328 370513 370698 370882 185 235 371068 371253 371437 371622 371806 371991 372175 372359 372544 372728 184 236 372912 373096 373279 373464 373647 373831 374015 374198 374382 374565 184 237 374748 374932 375115 375298 375481 375664 375846 376029 376212 376394 183 238 376577 376159 376942 377124 377306 377488 377670 377852 378034 378216 182 239 378398 378579 378761 378943 379124 379306 379487 379668 379849 380030 181 240 380211 380392 380573 380754 380934 381115 381296 381476 381656 381837 181 241 382017 382197 382377 382557 382737 382917 383097 383277 383456 383636 180 242 383815 383995 384174 384353 384533 384712 384891 385069 385249 385428 179 243 385606 385785 385964 386142 386321 389499 386677 386856 387034 387212 178 244 387389 387568 387746 387923 388101 388279 388456 388634 388811 388989 178 245 389166 389343 389520 389698 389875 390051 390228 390405 390582 390759 177 246 390935 391112 391288 391464 391641 391817 391993 392169 392345 392521 176 247 392697 392873 393048 393224 393399 393575 393751 393926 394101 394277 176 248 394452 394627 394802 394977 395152 395326 395501 395676 395850 396025 175 249 396199 396374 396548 396722 396896 397071 397245 397419 397592 397766 174 N 0 1 2 3 4 5 6 7 8 9 D 250 397940 398114 398287 398461 398634 398808 398981 399154 399328 399501 173 251 399674 399847 400019 400192 400365 400538 400711 400883 401056 401228 173 252 401401 401573 401745 401917 402089 402261 402433 402605 402777 402949 172 253 403121 403292 403464 403635 403807 403978 404149 404320 404492 404663 171 254 404834 405005 405176 405346 405517 405688 405858 406029 406199 406369 171 255 406540 406710 406881 407051 407221 407391 407561 407731 407901 408070 170 256 408239 408409 408579 408749 408918 409087 409257 409426 409595 409764 169 257 409933 410102 410271 410439 410609 410777 410946 411114 411283 411451 169 258 411619 411788 411956 412124 412293 412461 412629 412796 412964 413132 168 259 413299 413467 413635 413803 413969 414137 413405 414472 414639 414806 167 260 414973 415140 415307 415474 415641 415808 415974 416141 416308 416474 167 261 416641 416807 416973 417139 417306 417472 417638 417804 417969 418135 166 262 418301 418467 418633 418798 418964 419129 419295 419460 419625 419791 165 263 419956 420121 420289 420451 420616 420781 420945 421110 421275 421439 165 264 421604 421768 421933 422097 422261 422426 422589 422754 422918 423082 164 265 423246 423409 423574 423737 423901 424065 424228 424392 424555 424718 164 266 424882 425045 425208 425371 425534 425697 425860 426023 426186 426349 163 267 426511 426674 426836 426999 427161 427324 427486 427648 427811 427973 162 268 428135 428297 428459 428621 428783 428944 429106 429268 429429 429591 162 269 429752 429914 430075 430236 430398 430559 430719 430881 431042 431203 161 270 431369 431525 431685 431846 432007 432167 432328 432488 432649 432809 161 271 432969 433129 433289 433449 433609 433769 433929 434089 434249 434409 160 272 434569 434729 434888 435048 435207 435366 435526 435685 435844 436004 159 273 436163 436322 436481 436639 436799 436957 437116 437275 437433 437592 159 274 437751 437909 438067 438226 438384 438542 438701 438859 439175 439165 158 275 439333 439491 439648 439806 439964 440122 440279 440437 440594 440752 158 276 440909 441066 441224 441381 441538 441695 441852 442009 442166 442323 157 277 442479 442637 442793 442949 443106 443263 443419 443576 443732 443889 157 278 444045 444201 444357 444513 444669 444825 444981 445137 445293 445449 156 279 445604 445759 445915 446071 446226 446382 446537 446692 446848 447003 155 N 0 1 2 3 4 5 6 7 8 9 D 280 447158 447313 447468 447623 447778 447932 448088 448242 448397 448552 155 281 448706 448861 449015 449169 449324 449478 449633 449787 459941 450095 154 282 452049 450403 450557 450711 450865 451018 451172 451326 451479 451633 154 283 451786 451939 452093 452247 452399 452553 452706 452859 453012 453165 153 284 453318 453471 453624 453777 453929 454082 454235 454387 454539 454692 153 285 454845 454997 455149 455302 455454 455606 455758 455910 456062 456214 152 286 456366 456518 456669 456821 456973 457125 457276 457428 457579 457731 152 287 457889 458033 458184 458336 458487 458638 458789 458939 459091 459242 151 288 459392 459543 459694 459845 459995 460146 460296 460447 460597 460748 151 289 460898 461048 461198 461348 461499 461649 461799 461948 462098 462248 150 290 462398 462548 462647 462847 462997 463146 463296 463445 463594 463744 150 291 463893 464042 464191 464340 464489 464639 464788 464936 465085 465234 149 292 465383 465532 465680 465829 465977 466126 466274 466423 466571 466719 149 293 466868 467016 467164 467312 467460 467608 467756 467904 468052 468199 148 294 468347 468495 468643 468790 468938 969085 469233 469380 469527 469675 147 295 469822 469969 470116 470263 470410 470557 470704 470851 470998 471145 147 296 471292 471438 471585 471732 471878 472025 472171 472318 472464 472610 146 297 472756 472903 473049 473195 473341 473487 473633 473779 473925 474071 146 298 474216 474362 474508 474653 474799 474944 475089 475235 475381 475526 145 299 475671 475816 475962 476107 476252 476397 476542 476687 479832 476976 145 300 477121 477266 477411 477555 477699 477844 477989 478133 478278 478422 145 301 478566 478711 478855 478999 479143 479287 479431 479575 479719 479863 144 302 480007 480151 480294 480438 480582 480725 480869 481012 481156 481299 144 303 481443 481586 481729 481872 482016 482159 482302 482445 482588 482731 143 304 482874 483016 483159 483302 483445 483587 483729 483872 484015 484157 143 305 484299 484442 484585 484727 484869 485011 485153 485295 485437 485579 142 306 485721 485863 486005 486147 486289 486430 486572 486714 486855 486997 142 307 487138 487279 487421 487563 487704 487845 487986 488127 488269 488409 141 308 488551 488692 488333 488974 489114 489255 489396 489537 489677 489818 141 309 489958 490099 490239 490379 490520 490661 490801 490941 491081 491222 140 N 0 1 2 3 4 5 6 7 8 9 D 310 491362 491502 491642 491782 491922 492062 492201 492341 492481 492621 140 311 492760 492900 493039 493179 493319 493458 493597 493737 493876 494015 139 312 494155 494294 494433 494572 494711 494850 494989 495128 495267 495406 139 313 495544 495683 495822 495960 496099 496238 496376 496515 596653 496791 139 314 496929 497068 497206 497344 497483 497921 497759 497897 498035 498173 138 315 498311 498448 498586 498724 498862 498999 495137 499275 499412 499549 138 316 499687 499824 499962 500099 500236 500374 500510 500648 500785 500922 137 317 501059 501196 501333 501470 501607 501744 501880 502017 502154 502291 137 318 502427 502564 502700 502837 502973 503109 503246 503382 503518 503655 136 319 503791 503927 504063 504199 504335 504471 504607 504743 504878 505014 136 320 505149 505286 505421 505557 505193 505828 505964 506099 506234 506369 136 321 506505 506640 506776 506911 507046 507181 507316 507451 507586 507721 135 322 507856 507991 508126 508260 508395 508529 508664 508799 508934 509068 135 323 509203 509337 509471 509606 509740 509894 510009 510143 510277 510411 134 324 510545 510679 510813 510947 511081 511215 511349 511482 511616 511749 134 325 511883 512017 512151 512284 512418 512551 512684 512818 512951 513084 133 326 513218 513351 513485 513617 513750 513883 514016 514149 514282 514415 133 327 514548 514681 514813 514946 515079 515211 515344 515476 515609 515741 133 328 515874 516006 516139 516271 516403 516535 516668 516796 516932 517064 132 329 517196 517328 517459 517592 517724 517855 517987 518119 518251 518382 132 330 518514 518646 518777 518909 519040 519171 519303 519434 519566 519697 131 331 519828 519959 520090 520221 520353 520484 520615 520745 520876 521007 131 332 521138 521269 521399 521530 521661 521792 521922 522053 522183 522314 131 333 522454 522575 522705 522835 522966 523096 523226 523356 523486 523616 130 334 523746 523876 524006 524136 524266 524396 524526 524656 524785 424915 130 335 520545 525174 525304 525434 525563 525693 525822 525951 526081 526210 129 336 526339 526469 526598 526727 526856 526985 527114 527243 527372 527501 129 337 527629 527759 527888 528016 528145 528274 528402 528531 528659 528788 129 338 528916 529045 529174 529302 529430 525559 529687 529815 529943 530072 128 339 530199 530328 530456 530584 530712 530839 530968 531096 531223 531351 128 N 0 1 2 3 4 5 6 7 8 9 D 340 531479 531607 531734 531862 531989 532117 532245 532372 532499 532627 128 341 532754 532882 533009 533136 533264 533391 533518 533645 533772 533899 127 342 534026 534153 534280 534407 534534 534661 534787 534914 535041 535167 127 343 535294 535421 535547 535674 535800 535927 536053 536179 536304 536432 126 344 536558 536685 536811 536937 537063 537189 537315 537441 537567 537693 126 345 537819 537945 538071 538197 538322 538448 538574 538699 538825 538951 126 346 539076 539202 539327 539452 539578 539703 539829 539954 540079 540204 125 347 540329 540455 540579 540705 540829 540955 541079 541205 541329 541454 125 348 541579 541704 541829 541953 542078 542203 542327 542452 542576 542701 125 349 542825 542949 543074 543199 543323 543447 543571 543696 543819 543944 124 350 545008 544192 544316 544440 544564 544688 544812 544934 545059 545183 124 351 545307 545431 545555 545678 545802 545925 546049 546172 546296 546419 124 352 546543 546666 546789 546913 547036 547159 547282 547405 547529 547652 123 353 547775 547898 548021 548144 548267 548389 548512 548635 548758 548881 123 354 549003 549126 549249 549371 549494 549616 549739 549861 549984 550106 123 355 550228 550351 550473 550595 550717 550839 550962 551084 551206 551328 122 356 551449 551572 551694 551816 551938 552059 552181 552303 552425 552547 122 357 552668 552789 552911 553033 553155 553276 553398 553519 553640 553762 121 358 553883 554004 554126 554457 554368 554489 554610 554731 554852 554973 121 359 555094 555215 555336 554247 555578 555699 555819 555940 556061 556182 121 360 556303 556423 556544 556664 556785 556905 557026 557146 557267 557387 120 361 557057 557627 557748 557868 557988 558108 558228 558349 558469 558589 120 362 558709 558829 558948 559068 559188 559308 559428 559548 559667 559787 120 363 559907 560026 560146 560265 560385 560504 560624 560743 560863 560982 119 364 561101 561221 561339 561459 561578 561698 561817 561936 562055 562174 119 365 562293 562412 562531 562649 562769 562887 563006 563125 563244 563362 119 366 563481 563599 563718 563836 563955 564074 564192 564311 564429 564548 119 367 564666 564784 564903 565021 565139 565257 565376 565494 565612 565729 118 368 565848 565966 566084 566202 566319 566437 566555 566673 566791 566909 118 369 567026 567144 567262 567379 567497 567614 567732 567849 567967 568084 118 N 0 1 2 3 4 5 6 7 8 9 D 370 568202 568319 568436 568554 568671 568788 568905 569023 596139 569257 117 371 569374 569491 569608 569725 569842 569959 570076 570193 570309 570426 117 372 570543 570659 570776 570893 571009 571126 571243 571359 571476 571592 117 373 571709 571825 571942 572058 572274 572291 572407 572523 572639 572755 116 374 572872 572988 573104 573219 573336 573452 573568 573684 573799 573915 116 375 574031 574147 574263 574379 574494 574609 574726 574841 574957 575072 116 376 575188 575303 575419 575534 575649 575765 575880 575996 576111 576226 115 377 576341 576457 576572 576687 576802 576917 577032 577147 577262 577377 115 378 577492 577607 577722 577836 577951 578066 578181 578295 578409 578525 115 379 578639 578754 578868 578983 579097 579212 579326 579441 579555 579669 114 380 579784 579898 580012 580126 580241 580355 580469 580583 580697 580811 114 381 580925 581039 581153 581267 581381 581495 581608 581722 581836 581949 114 382 582063 582177 582291 582404 582518 582631 582745 582858 582972 583085 114 383 583199 583312 583426 583539 583652 583765 583879 583992 584105 584218 113 384 584331 584444 584587 584670 584783 584896 585009 585122 585235 585348 113 385 585461 585574 585686 585799 585912 586024 586137 586249 586362 586475 113 386 586587 586699 586812 586925 587037 587149 587262 587374 587486 587599 112 387 587712 587823 587935 588047 588159 588272 588384 588496 588608 588719 112 388 588832 588944 589056 589167 589279 589391 589503 589615 589726 589838 112 389 589949 599061 590173 590284 590396 590507 590619 590730 590842 590953 112 390 591065 591176 591287 591399 591509 591621 591732 591843 591955 592066 111 391 592177 592288 592399 592509 592621 592732 592843 592954 593064 593175 111 392 593286 593397 593508 593618 593729 593839 593950 594061 594171 594282 111 393 594393 594503 594614 594724 594834 594945 595055 595165 595276 595386 110 394 595496 595606 595717 595827 595937 596047 596157 596267 596377 596487 110 395 596597 596707 596817 596927 597037 597146 597256 597366 597476 597586 110 396 597695 597805 597914 598024 598134 598243 598353 598462 598572 598681 110 397 598790 598899 599009 599119 599228 599337 599446 599556 599665 599774 109 398 599883 599992 600101 600210 600319 600428 600537 600646 600755 600864 109 399 690073 601082 601191 601299 601408 601517 601625 601734 601843 601951 109 N 0 1 2 3 4 5 6 7 8 9 D 400 602059 602169 602277 602386 602494 602603 602817 602819 602928 603036 108 401 603144 603253 603361 603469 603577 603686 603794 603902 604009 604118 108 402 604226 604334 604442 604550 604658 604766 604874 604982 605089 605197 108 403 605305 605413 605521 605628 605736 605844 605951 606059 606166 606274 108 404 606381 606489 606596 606704 606811 606919 607026 607133 607241 607348 107 405 607455 607562 607669 607777 607884 607991 608098 608205 608312 608419 107 406 608526 608633 608739 608847 608954 609061 609167 609274 609381 609488 107 407 609594 609701 609808 609914 610021 610128 610234 610341 610447 610555 107 408 610660 610767 610873 610979 〈◊〉 611192 611298 611405 611511 611617 106 409 611723 611829 611936 612042 612148 612254 612359 612466 612572 612678 106 410 612784 612889 612996 613102 613207 613313 613419 613525 613630 613736 106 411 613842 613947 614053 614159 614264 614369 614475 614581 614686 614792 106 412 614897 615003 615108 615213 615319 615424 615529 615634 615739 615845 105 413 615950 616055 616160 616265 616370 616476 616581 616686 616790 616895 105 414 617000 617105 617210 617315 617419 617525 617629 617734 617839 617943 105 415 618048 618153 618257 618362 618466 616571 618676 618780 618889 618989 105 416 619093 619198 619302 619406 619511 619615 619719 619824 619928 620032 104 417 620136 620240 620344 620448 620552 620656 620760 620864 620968 621072 104 418 621176 621280 621384 621448 621592 621695 621799 621902 622007 622110 104 419 622214 622318 622421 622525 622628 622722 622835 622939 623042 623146 104 420 623249 623353 623456 623559 623663 623766 623869 623973 624076 624179 103 421 624282 624385 624488 625591 624695 624798 624901 625004 625107 625209 103 422 625312 625415 625518 625621 625724 625827 625929 626032 626135 626237 103 423 626340 626443 626546 626648 626751 626853 626956 627058 627161 627263 103 424 627366 627468 627571 627673 627775 627878 627979 628082 628185 628287 102 425 628389 628491 628593 628695 628797 628899 629002 629104 629206 629308 102 426 629409 629512 629613 629715 629817 629919 620021 630123 630224 630326 102 427 630428 630529 630631 630733 630835 630936 631038 631139 631241 631342 102 428 631444 631545 631647 631746 631849 631951 632051 632153 632255 632356 101 429 632457 632559 632659 632761 632862 632963 633064 633165 633266 633367 101 N 0 1 2 3 4 5 6 7 8 9 D 430 633468 633569 633670 633771 633872 633973 634075 634075 634276 634376 100 431 634477 634578 634679 634779 634880 634981 635081 635182 635283 635383 100 432 635484 635584 635685 635785 635886 635986 636087 636187 636288 636388 100 433 636488 636588 636688 636789 636889 636989 637089 637189 637289 637389 100 434 637489 637589 637689 637789 637889 637989 638089 638189 638289 638389 99 435 638489 638589 638689 638789 638888 638988 639088 639188 639287 639387 99 436 639486 639586 639686 639785 639885 639984 640084 640183 640283 640382 99 437 640481 640581 640680 640779 640879 640978 641077 641177 641276 641375 99 438 641475 641573 641672 641771 641871 641969 642069 642168 642267 642366 99 439 642465 642563 642662 642761 642860 642959 643058 643156 643255 643354 99 440 643453 643551 643650 643749 643847 643946 644044 644143 644242 644340 98 441 644439 644537 644636 644734 644833 644931 645029 645127 645226 645324 98 442 645422 645521 645619 645717 645815 645913 646011 646109 646208 646306 98 443 646404 646502 646599 646698 646796 646894 646992 647089 647187 647285 98 444 647383 647481 647579 647676 647774 647872 647969 648067 648165 648262 98 445 648360 648458 648555 648653 648750 648848 648945 649043 649140 649237 97 446 649335 649432 649529 649627 649724 649821 649919 650016 650113 650210 97 447 650308 650405 650502 650599 650696 650793 650890 650987 651084 651181 97 448 651278 651375 651472 651569 651666 651762 651859 651956 652053 652149 97 449 652246 652343 652439 652536 652633 652729 652826 652923 653019 653116 97 450 653213 653309 653405 653502 653598 653695 653791 653888 653984 654080 96 451 654177 654273 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678427 91 477 678518 678609 678700 678791 678882 678973 679064 679155 679246 679337 91 478 679428 679519 679609 679700 679791 679882 679972 680063 680154 680245 91 479 680336 680426 680517 680607 680698 680789 680879 680969 681060 681151 91 480 681241 681332 681422 681513 681603 681693 681784 681874 681964 682055 90 481 682145 682235 682326 682416 682506 682596 682686 682777 682867 682957 90 482 683047 683137 683227 683317 683407 683497 683587 683677 683767 683857 90 483 683947 684037 684127 684217 684307 684396 684486 684576 684666 684756 90 484 684845 684935 685025 685114 685204 685294 685383 685473 685563 685652 90 485 685742 685831 685921 686010 686099 682189 686279 686368 686458 686547 89 486 686636 686326 686815 686904 686994 687083 687172 687211 687351 687439 89 487 687529 687618 687707 687796 687885 687975 688054 688153 688242 688331 89 488 688419 688509 688598 688687 688776 688865 688953 689042 689131 689220 89 489 689309 689398 689486 689575 687664 689753 689841 689930 690019 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 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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.