The country-survey-book: or Land-meters vade-mecum Wherein the principles and practical rules for surveying of land, are so plainly (though briefly) delivered, that any one of ordinary parts (understanding how to add, substract, multiply and divide,) may by the help of this small treatise alone and a few cheap instruments easy to be procured, measure a parcel of land, and with judgment and expedition plot it, and give up the content thereof. With an appendix, containing twelve problems touching compound interest and annuities; and a method to contract the work of fellowship and alligation alternate, very considerably in many cases. Illustrated with copper plates. By Adam Martindale, a friend to mathematical learning. Martindale, Adam, 1623-1686. 1692 Approx. 245 KB of XML-encoded text transcribed from 96 1-bit group-IV TIFF page images. Text Creation Partnership, Ann Arbor, MI ; Oxford (UK) : 2006-02 (EEBO-TCP Phase 1). A52120 Wing M854A ESTC R217468 99829133 99829133 33569 This keyboarded and encoded edition of the work described above is co-owned by the institutions providing financial support to the Early English Books Online Text Creation Partnership. This Phase I text is available for reuse, according to the terms of Creative Commons 0 1.0 Universal . The text can be copied, modified, distributed and performed, even for commercial purposes, all without asking permission. Early English books online. (EEBO-TCP ; phase 1, no. A52120) Transcribed from: (Early English Books Online ; image set 33569) Images scanned from microfilm: (Early English books, 1641-1700 ; 1917:11) The country-survey-book: or Land-meters vade-mecum Wherein the principles and practical rules for surveying of land, are so plainly (though briefly) delivered, that any one of ordinary parts (understanding how to add, substract, multiply and divide,) may by the help of this small treatise alone and a few cheap instruments easy to be procured, measure a parcel of land, and with judgment and expedition plot it, and give up the content thereof. With an appendix, containing twelve problems touching compound interest and annuities; and a method to contract the work of fellowship and alligation alternate, very considerably in many cases. Illustrated with copper plates. By Adam Martindale, a friend to mathematical learning. Martindale, Adam, 1623-1686. Collins, John, 1625-1683. [12], 225 [i.e. 226], 229-234 p., plates, folding printed for R. Clavel, at the Peacock in St. Pauls Church-yard, and T. Sawbridge, at the Three Flower-de-luces in Little-Britain, London : 1692. "An appendix containing XII. problems touching compound interest & annuities" has separate title page with imprint: London, printed in the year 1692. Register and pagination are continuous. Errata on p. [12]. "To the reader" signed: John Collins. Page 226 misnumbered 225. Reproduction of the original in the British Library. Created by converting TCP files to TEI P5 using tcp2tei.xsl, TEI @ Oxford. Re-processed by University of Nebraska-Lincoln and Northwestern, with changes to facilitate morpho-syntactic tagging. Gap elements of known extent have been transformed into placeholder characters or elements to simplify the filling in of gaps by user contributors. 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Copies of the texts have been issued variously as SGML (TCP schema; ASCII text with mnemonic sdata character entities); displayable XML (TCP schema; characters represented either as UTF-8 Unicode or text strings within braces); or lossless XML (TEI P5, characters represented either as UTF-8 Unicode or TEI g elements). Keying and markup guidelines are available at the Text Creation Partnership web site . eng Surveying -- Early works to 1800. Interest -- Early works to 1800. Mathematics -- Early works to 1800. 2005-07 TCP Assigned for keying and markup 2005-08 Aptara Keyed and coded from ProQuest page images 2005-12 Judith Siefring Sampled and proofread 2005-12 Judith Siefring Text and markup reviewed and edited 2006-01 pfs Batch review (QC) and XML conversion THE Country-Survey-Book : OR LAND-METERS VADE-MECVM . WHEREIN The PRINCIPLES and practical RULES for Surveying of Land , are so plainly ( though briefly ) delivered , that any one of ordinary parts ( understanding how to add , subtract , multiply and divide , ) may by the help of this small Treatise alone , and a few cheap Instruments easy to be procured , Measure a parcel of Land , and with judgment and expedition Plot it , and give up the Content thereof . WITH An APPENDIX , containing Twelve Problems touching Compound Interest and Annuities ; and a Method to Contract the work of Fellowship and Alligation Alternate , very considerably in many Cases . Illustrated with Copper Plates . By ADAM MARTINDALE , A Friend to Mathematical Learning . Frustra sit per plura quod fieri potest per pauciora . London , Printed for R. Clavel , at the Peacock in St. Pauls Church-yard , and T. Sawbridge , at the Three Flower-de-luces in Little-Britain , 1692. TO THE Right Honourable THE LORD DELAMER , Baron of Dunham-Massie , &c. My Lord , THIS small Tract comes to Your Lordship , not as to a Patron to protect its Errors ( if any such there be ) but as to a critical noble Friend , that will be sure faithfully to tell the Author of them . Which favour , together with a chearful acceptance of this poor present , he humbly hopes for , because of its Relation , being writ at Dunham , by Your humble Servant , who besides his domestick dependance , cannot forbear without ingratitude to tell the World that Your Lordship's kindness hath very much encouraged and assisted him in Mathematical Studies , not only by a free Communication of many a choice notion both vivâ voce , and by the loan of Manuscripts . But also by a Considerable number of excellent Books and costly Instruments bountifully bestowed upon him : Who wanting other ways to express his many singular Obligations , and deep sence thereof , humbly offers this Punie Treatise for Your Lordships diversion at spare hours ; and is ambitious to write himself , Oct : 26. 1681. MY LORD , An humble and faithful ( though unworthy ) Servant to all Your Lordship's Noble Family A. M. Mr. COLLINS TO THE READER . Courteous Reader , THE Learned Mr. Adam Martindale formerly Writ two excellent Almanacks , called Country Almanacks , which were Printed , and esteemed by several Members of the Royal Society very useful , especially for Country Affairs , but meeting with some Discouragements from such as knew not how to judge of the Authors worth , he gave over that undertaking , contrary to the desires of many Ingenious Men. And having since Writ a little Treatise of Survey , in which he hath had experience , as well as Theory , and being willing to have the Approbation or Dislike of others , that it might either be Printed or Stifled , imparted the same to some of the Members of the aforesaid Illustrious Body . Upon the Perusal of whom and divers Experienced Artists who make a Livelihood of it , I find it well approved ( as clear and concise ) only the latter were sorry so much was discovered , as detrimental to their Practice , particularly about setting off the outjettings , where it 's inconvenient intirely to measure them . The Book is small , the price ( though not the worth ) mean , no small Encouragements to young Students and the Vulgar , for whom it was chiefly intended : By which , that they may reap Benefit , is the hearty desire of ( a Well-willer to the Author and them ) John Collins . The PREFACE . READER , THe Title sufficiently informs thee in general of my Designs : But I confess I owe thee a a more particular Account , not only of that , but several other things which I shall briefly give thee . 1. I have observed that the Country aboundeth with such , as by their Inclination and Interest are prevailed with to take pains in measuring Land , that for want of better Instruction use ill divided Chains and tedious Methods of Computation , which makes their work intolerable troublesom , if exactness be required . And some for want of skill in the Fundamentals of Geometry , have imbibed prodigiously false Principles , as this for one , viz. That the Content of any Close , of what Figure soever , may be found by Squaring a quarter of the Perimeter . Mathematical-Schools , where better things might be learned , are very rare , and an able Artist to instruct one in private is hard , and charge able to be procured . Excellent Books indeed there are in our English Tongue , Written by our Famous Rathborn , Wing , Leybourn , and Holwell , to which may be added Industrious Mr. Atwells Treatise , and some part of Capt. Sturmy's : But those I rather esteem fit to be read byan able Artist ( towards his perfecting ) than by a new beginner , for in the best of those Books he will find the most useful and plain Rules so intermixed with others that are less necessary , and more intricate , ( though very excellent for their proper ends ) and so many Curiosities touching Trigonometry , Transmtation of Figures , &c. which his business never calls for , that for want of judgment to pick out that which fits his present purpose , and to study higher Speculations afterwards , he is apt to be confounded and discouraged ; whereby it accidentally comes to pass that plenty , makes him poor . Besides , three of these six Books are in Folio , another in Quarto , and the other two ( though Octavo's ) too large for ordinary carrying in a Pocket , and in that regard not so convenient for one that hath much occasion to be out of his own House ; to say nothing of their Price , which to some poor Youths is not the least discouragement . I have therefore made my Book so little , that the Price can neither much empty the Pocket , nor the Bulk overfill it . And yet so plain , that I doubt not to be understood by very ordinary Capacities . My Method is fitted to my Design : Beginning with the Principles of the Art , and so proceeding gradatim till I have shewed how all ordinary Figures may be Measured , Protracted , and Cast up , without any other Instrument of charge but Chains , Compasses and Scales . Afterwards for such as desire higher Attainments , I have endeavoured to speak so fully ( in a little compass ) of the Plain Table , and given such hints , applicable not only to it , but also to the Peractor , Theodolite , and Semicircle , as that an Ingenious Person may make great use of them . But as touching the Doctrine of Triangles , and Transmutation of Figures into others equipollent , with the large Tables ( referring to the former ) of Logarithms of Numbers , Sines , and Tangents , I thought it improper to cumber this small Manual , or the unlearned Reader with them , having ( as I humbly hope ) sufficiently inform'd him how to find all his Sides and Angles by Instruments , and also the Content of any Figure without such transmutation , reserving such Curiosities and many others touching Drawing and Painting of Maps ; Measuring of Ways , and Rivers , &c. to a Second Part , which I may perhaps hereafter Publish upon due encouragement , but if I do not , the Curious may find themselves good store of work in the Authors even now quoted . 2. It may seem strange to some , that in referring to the Figures , I sometimes use Words seeming to imply that the Figure I speak of is in that very Page , and so it was in my Copy , but the Printer and Gravers have otherwise contrived them for convenience in Copper Cuts by themselves . And to give them their due they are generally done with great accuracy , and none of them having any such error as is like to beget trouble or mistake to the Reader , saving only that fig. 19 hath D instead of O at the Center , and the Line OL in the Margin of Fig. 14. should be of the length from L to the uppermost o in the Scale , and the Figures on the side should be made 1 less than they are , viz. 2 should be 1 , 3 made 2 , &c. And lastly , as to the Errata , though I have not been so anxiously careful , as to correct every literal mistake , I have very diligently perused all from p. 1 to p. 224 inclusive , and hope I have sufficiently restored the Sense to the places wronged , when thou hast done them right by the Pen according to the Directions of the Errata following next after the contents , and that you continue the Line in the Margin of p. 34. to the length of the Line OL in fig. 14. THE CONTENTS OF THE CHAPTERS . Chapt. 1. OF Geometrical Definitions , Divisions , and Remarks . p. 1. Chapt. 2. Of Geometrical Problems . p. 6. Chapt. 3. To find the Superficial Content of any right lined Figure , the lines being given . p. 17. Chapt. 4. Concerning Chains , Compasses , and Scales . p. 26. Chapt. 5. How to cast up the Content of a Figure , the lines being given in Chains and Links . p. 35. Chapt. 6. How to measure a Close , or parcel of Land , and to protract it , and give up the Content . p. 41. Chapt. 7. Concerning the measuring of Circles and their parts . p. 48. Chapt. 8. Concerning Customary measure , and how it may be reduced to Statute measure , & e Contra , either by the Rule of Three , or a more compendious may by Multiplication only . p. 52. Chapt. 9. How a Man may become a ready Measurer by Practice in his private Study , without any ones assistance or observation , till he design to practise abroad . p. 65. Chapt. 10. How to measure a piece of Land with any Chain of what length soever , and howsoever divided ; yea with a Cord or Cart-rope ; being a good Expedient when Instruments are not at hand of a more Artificial ●ake . p. 67. Chapt. 11. Concerning dividing of Land Artificially and ●echanically . p. 70 Chapt. 12. Concerning the Boundaries of Land , where the ●ines to be measured must begin and end . p. 80. Chapt. 13. Containing a Description of the Plain-Table , the ●rotractor , and Lines of Chords . p. 82. Chapt. 14. How to take the true Plot of a Field by the ●lain-Table upon the Paper that covers it , at one or ●ore Stations . p. 85. Chapt. 15. Concerning the plotting of many Closes together , ●hether the ground be even or uneven . p. 99. Chapt. 16. Concerning shifting of Paper . p. 102. Chapt. 17. Concerning the plotting of a Town Field , where 〈◊〉 several Lands , Buts , or Doles , are very crooked : ●●th a Note concerning Hypothenusual or sloping ●●undaries , common to this and the fifteenth Chapter . ● 104. Chapt. 18. Concerning taking the plot of a piece of ground 〈◊〉 the Degrees upon the Frame of the Plain-Table se●●●al ways , and protracting the same . p. 108. Chapt. 19. Concerning taking inaccessible Distances by the ●●ain-Table , and accessible Altitudes by the Protractor . ● 121. Chapt. 20. Of casting up the Content of Land by a Table . ● 193. ERRATA . PAge 3 Line 16 Read Trilaterals . p. 4 〈◊〉 Geodates . p. 5 l. 29 Eneagon . p. 10 l. 6 belong as the other two . p. 16 l. 6 Centers at right A●●gles . p. 28 l. 1 forefinger . p. 36 l. 20 Poles or R●● p. 47 l. 15 fourth Diagonal and the sixth side . p. 〈◊〉 l. 28 as in this figure is ABC . p. 56 l. 23 , 〈◊〉 and l. 24 ● . 22 ½ p. 58 l. 11 28. p. 69 l. 5 side 74 l. 25 FG. p. 77 l. 27 138562 , and l. 28 〈◊〉 33 r. 242030. p. 79 l. 33 triangulate . p. 80 l ▪ former . p. 82 l. 24 fitted . p. 86 l. 2 Stationary ●●●stances . p. 90 l. 15 Chart , or Card. p. 95 l. ● Park , Pond . p. 103 l. 22 Line . p. 104. l. 19 〈◊〉 Fig. 1. Fig. 2. Fig. 3. Fig. 4. Fig. 5. Fig. 6. Fig. 7. Fig. 8. Fig. 9. Fig. 10. Fig. 11. Fig. 12. Fig. 13. Fig. 14 Fig. 15. Fig. 16. Fig. 17. Fig. 18. Fig. 19. Fig. 20. Fig. 21. Fig. 22. Fig. 23. Fig. 24. Fig. 25. Fig. 26. Fig. 27. Fig. 28. Fig. 29. Fig. 30. Fig. 31. Fig. 32. Fig. 33. Fig. 34. Fig. 35. Fig. 36. Fig. 37. Fig. 38. Fig. 39. THE Country Survey-Book : OR , LAND-METER's VADE-MECVM . CHAP. I. Of Geometrical Definitions , Divisions , and Remarks . I. A Point is that which hath no parts , either of longitude or latitude , but is indivisible , ordinarily expressed with a small prick , like a period at the end of a sentence . II. A Line hath length , but no bredth nor depth , whose limits or extremities are Points . This is either right or crooked . III. A right Line lies straight , and equal between its extreme points , being the shortest extension between them ; the crocked or circular not so . IV. A Superficies hath length and bredth , but no depth ; of this Lines are the limits . V. A plain Superficies is that which lieth equally ( or evenly ) between its Lines . VI. An Angle is the Meeting or two Lines in one point , so as not to make one straight Line , and if drawn on past that point , they will intersect or cross one another . This is vulgar English may be called a Corner ; of which there be two sorts , one right , the other oblique . VII . A right Angle is that which is made by two right lines , crossing or touching one another perpendicularly , ( or squarely ) like an ordinary Cross , or Carpenters Square . VIII . An oblique Angle is that which is either greater or less than a right Angle , and this is of two sorts , obtuse and acute . IX . An obtuse Angle is greater than a right Angle , like the left and right Corners of a Roman X. X. An acute Angle is less than a right Angle , like the highest and lowest Corners of the same Letter . XI . A Figure is that which is comprehended under one line or many : Of this there are two kinds , a Circle and a right-lined Figure . XII . A Circle is a perfect round Figure , such as is drawn with a pair of Compasses , the one Foot being turned round in a point , and the other wheeled about it . The point in the precise middle is called the Center ; the round line , the Circumference or Peripheri ; a line going through the Center , and divide the Circle into two equal parts , is called the Diameter ; half of that line is a Somidiameter , or Radius ; half the Circle is stiled a Semicircle ; the quarter , a Quadrant ; any portion of it , cut off by a right Line not touching the Center , is called a Segment . XIII . Right-lined Figures are such as are limited by three right Lines or more , and are either Triangles or Triangulate , that is , such as are compounded of , and resolvable into Triangles . XIV . Triangles are Figures comprehended under three right Lines , and ( as Ramus thinks for a Reason that he gives , lib. 6. pr. 6. ) might be better called Trilaterals ; but the name Triangle from the number of the Angles hath obtained . Also from the nature and Quantity of their Angles these Triangles are distinguished into three sorts : 1. Rectangled , having one right Angle ; 2 Obtuse-angled , having one obtuse Angle ; and 3. Acute-angled ; having all acute Angles ; for no Triangle can have more right or obtuse Angles than one , because by by an old Rule ( easie to be demonstrated ) no Triangle upon a plain Superficies can consist of three greater Angles than such , as being jointly taken are equal to two Right . These three sorts of Triangles may , according to the length and proportion of their sides , be subdistinguished into seven ; for each of them may have either two equal Sides or none , and the Acute-angled may have all three Sides or lines equal : To all which kinds , learned Men give distinct Greek Names , which if mine English Reader have a mind to see , they are to be found in 〈…〉 Practices , Book 1. page . 6. for my present purpose the above-mentioned ●rimembred distinction will abundantly suffice ; for be Triangles of what name or kind soever , they are all capable of being exactly measured by one plain Rule as hereafter shall fully appear . XV. Triangulare Figures are such as have more Angles ( and consequently more Sides or Lines ) than three : and these are either Quadrangular or Multangular . XVI . Quadrangular Figures are such as have fo●● Angles ( and as many Side ) and these are either Parallellograms or Trapezia's . XVII . Parallellograms are Figures that are bounded with parallel Lines , that is , such lines as are every where of the same distance one from another , so as if they were infinitely extended they would never meet , like the upright lines of he Roman H. These Parallellograms are either Rectangular or Obliquangular . XVIII . Rectangular Parallellograms are such as have four right Angles . viz. the Square or Quadrat , and the long Square , otherwise called the Oblong . XIX . The Square is that Figure that hath four right Angles , and four equal Sides , like any of the six Faces of a Die. XX. The long Square hath also four right Angles , and the oposite Sides are equal , but the adjoyning Sides meeting at each Angle differ in length . Of this Figure is a well printed Page in a Book , and the Superficies of a well cut Sheet of Paper , or an ordinary Pane of Glass . XXI . Obliquangled Parallellograms are such as have oblique Angles , viz. two acute , and two obtuse . Of these there are two kinds , the Rhombus , and the Rhomboides . XXII . The Rhombus is a Figure that hath equal Sides , but no right Angles , ( like the form of a Diamond on the Cards , or the most ordinary Cut of Glass in Windows ) whose oposite Angles are equal . XXIII . The Rhomboides is ( as it were ) a defective Rbombus , for if from any side of a Rhombus we cut off a part with a parallel Line , the Remainder will be a Rhomboides , which hath neither equal Sides nor Angles , but yet the opposite Sides and Angles are equal . XXIV . The Trapezium is a Figure that is neither parallellogram , nor ( consequently ) hath equal Sides or Angles , but is irregularly quadrangular , as if drawn at adventure . Of this shape most Fields prove , that seem to the Eye to be Squares or Oblongs . XXV . Multangular Figures are such as contain more Sides and Angles than four , and they are either regular or irregular . XXVI . Regular Multangulars take their names from their Number of Angles , so a Pentagon , Hexagon , Heptagon , Octogon , Encagon , Decagon , signifie a multangular Figure of five , six , seven , eight , nine , ten , Angles , and consequently Sides . XXVII . An irregular Polygon or multangular Figure , is that which hath more Angles ( and Sides ) than four , the Sides ( and Angles ) being unequal to one another . CHAP. II. Of Geometrical Problems . I. To draw a Line parallel to another , at any Distance assigned . Fig. 1 OPen your Compasses to the Distance given and chusing two Points conveniently distant in the Line given , as here at A and B describe the Arches C and D , to whose convixity if you apply a Rule , the parallel Line is easily drawn . II. To raise a Perpendicular upon a Line given , or to cross that Line at right Angles in a Point assigned . Fig. 2 Suppose the point C in the Line AB were assigned for the Perpendicular ; open the Compasses to a convenient distance , and mark out the two points E and F in the line AB , then opening them some what wider , you may ( by setting one Foot in E and F severally ) describe the two Arches cutting one another at the point D , from which if you draw a Line to the point C , the work is done for the raising of a Perpendicular ; but if you be to cross the lines at right Angles , you may continue the line from D through C at pleasure . But if the said line AB had been given to be divided in the precise middle , by another Line crossing it at right Angles , the way were to set one Point of the Compasses in A and B severally , and having described two Arches above the line , intersecting one another as at D , do the like below the line AB from the same points and with the same extent of your Compasses , then through the several intersections ( a Rule being laid upon them ) a line may be drawn , cutting the given line exactly in the middle at right Angles . Note , That when one point of your Compasses stand in A , you may make both the Arches belonging to that Center above and below the line , and then removing the Compasses to B , you may cross them both . III. To raise a Perpendicular at the End of a Line . Fig. 3 Let OR be the line given , then to raise a Perpendicular at R , make five little equal divisions , and taking four of them with your Compasses , set one foot of your Compasses in R , and with the other describe the Arch PP ; then take the distance from R to 5 , and placing one Foot in 3 , with the other describe the Arch BB , intersecting the former in the point S ; then shall the line SR ( being drawn by a straight Rule ) be a Perpendicular to the line OR . IV. To let fall a Perpendicular upon a given Line from any Point assigned . Open your Compasses so as one Foot being set in the assigned point the other may go clear over the line given , and thereby describe an Arch cutting the line at two points ; then shall the half distance between those two points be the point to which the Perpendicular may be drawn from the point assigned . But if you think it too much pains to find the point of half distance by trial , you may help your self by the second Problem : For if you describe two Arches intersecting one another on the farther side of the line from the assigned point , placing ( to that purpose ) the Foot of your Compasses first in one of the Intersections of the given line , and then in the other ; you may by laying a Rule upon the assigned Point , and the Intersection of the two Arches , draw a Perpendicular from the said assigned Point , cutting the given Line at right Angles . Note , that all these Problems touching perpendiculars , aim at no greater matter , than what may be performed in a Mechanical way with exactness enough ( and much more neatly by avoiding unhandsome Pricks and Arches ) by the help of a small Square exactly made , ( or for want thereof a Plate Quadrant , or broad Rule , having a right Angle and true Sides ) for if you apply one Leg of such a Square to any Line , so as the Angle of the Square may touch the end of the said Line , or any other Point where the Perpendicular is to be raised , you may by the other Leg draw the Perpendicular . In like sort to let fall a perpendicular from a point assigned you need only to apply one Leg of the Square to the Line , so as the other may touch ( at the same time ) the assigned point whence you may draw the perpendicular , by that Leg that toucheth the Point : If the Angle of your Square be a little blunt either through ill making or long using , you must allow for it when you apply it to the point in a Line . And when you are drawing a Perpendicular , you must stop before you reach the given line , and then by applying the Leg of your Square to that part of the perpendicular already drawn , so as part of that Leg may pass clearly over the given Line , you may draw the rest of your perpendicular as exactly as if the Angle had been true . The like course is to be taken when a line is to be crossed by another drawn quite through it at right Angles . V. An Angle being given , to make another equal to it . Fig. 4 The Angle XAD being given , and a Line drawn at pleasure as is the lowest from the point E , open your Compasses to any convenient distance , and setting one foot in A describe the Arch BC. Then with the same extent setting one foot in E , with the other describe the Arch GH , long enough to equal or exceed the other . Then taking the distance BC between the points of your Compasses , set one in G , and with the other mark the point H in the Arch GH , through which point H a line being drawn from the point E , will make an Angle with the Line EG equal to the Angle given . Note , when we speak of the quantity of Angles , their equality , or unequality , we never regard the length of the Lines ; for if you extend or contract them at pleasure , the Angle is still the same . But that is the greatest Angle whose lines are farthest distant from one another , at the same distance from the Angular Point , or the place where its lines meet . VI. Any three Lines being given ( equal or unequal ) so as no one of them be longer than the other two joyned together to make a Triangle of them . Fig. 5 The Lines A , B , C , being given , set the line A from D to E ; then with your Compasses take the length of the Line B , and setting one Foot in D describe the Arch PO. This being done , take with your Compasses the length of the line C , and setting one foot in E with the other cross the former Arch at F , from which Intersection drawing Lines by a Rule to D and E , the Triangle is finished . Note , that if all the Sides , or two of them , be equal , the method is the same ; but the labour less , because we need not to take the same length twice over with the Compasses . VII . To find the Perpendicular of the Triangle , in order to the measuring of it . Fig. 6 Let the Line AB be accounted the Base , and from the Angle C let fall a perpendicular as was taught , Probl. 4. Upon that line at D , which is ready for taking off with Compasses and measuring on a Scale , of which hereafter in the Chapters of measuring the Content of Figures . But if we have no occasion to draw the perpendicular , but only to know the length of it , ( as it most frequently falls out in measuring ) no more is needful but to set one foot of the Compasses in the Angular point C , and extend the other to the Base AB , so as it may touch it , but not go beyond it ; then have we the perpendicular between the points of the Compasses . VIII . One Side being given , how to make a Square . Fig. 7 The Line CD being given , raise a perpendicular at C of the length ( at the least ) of the given line ; then taking the line CD between the feet of your Compasses , set it upon the perpendicular from the Angular Point C to A : With the same distance setting one foot in D describe the Arch OP . Lastly , with the same distance ( or extent ) set one foot in A , and with the other describe the Arch crossing the Arch OP in N , from which intersection a line drawn by a Rule to A , and another to D , finish the Geometrical Square or Quadrant ACDN. IX . To make a long Square , the length and bredth being given . This is so like the former , that a particular Figure is not necessary to conceive of it . Suppose each side of the Square in the last Problem to consist of 8 small equal parts , and you were to make a long Square whose length must be equal to a side thereof , viz. 8. and the bredth half so much given in a line thus — 4 ; then when you had drawn the line CD for the length , and raised the Perpendicular at C , you must take the shorter Line given for the bredth , and set it upon the Perpendicular from C upwards to a Point , which for distinction we shall call the Point E , imagining it so marked : With the same extent of the Compasses describe the Arch , placing ( to that purpose ) one foot in D. Lastly , extending your Compasses to the length of the line CD , set one foot in E , and with the other cross the Arch aforesaid . Then a right Line drawn from that Intersection to E , and another from the same to D , complete the long Square . X. To make a Rbombus , the Sides being given . Fig. 8 If the Angles be not limited draw any oblique Angle at pleasure , either Acute or Obtuse , as here the Angle BAC , which is Acute . Then let the Line OP be the length of a Side , which being taken with your Compasses , set it from the angular point A in both Lines to D and E , in which two points place a foot of your Compasses successively without altering them , viz. in D to describe the Arch FG , and in E to describe the Arch HI , crossing one another in the Point K , from which , right lines drawn to D and E , finish the Rhombus DAEK . Note , if any Angle be given , together with the Side ; to limit the shape and content , begin with that , and proceed as before : For you must know , that to make a Rhombus ( or Rhomboides ) like to another for Figure , or equal to it in Content , it is not sufficient to have the same Sides ; for the more oblique the Angles 〈◊〉 the farther will the Rhombus differ from a 〈◊〉 ( and the Rhomboides from a long 〈◊〉 are ) and the less will be the Content . But 〈◊〉 must have an Angle given , ( which will pro●●●e all the rest ) or else a Diagonal Line , which 〈◊〉 right Line passing through the Rhombus ( or ●omboides ) , from one opposite Angle to another , and dividing the Figure into two equal ●●●angles . If the former ( viz. an Angle ) be ●●n , I have shewed what use is to be made 〈◊〉 . If the latter , ( i. e. a Diagonal ) toget●●● with the length of the Sides , you may by ●●●ng the length of the Sides with your Com●●es , and setting a Foot in the ends of the ●●gonal Line , make a Triangle on the one side ●he Diagonal , by Probl. 6. and then another on the other side by the same problem , the ●●gonal being a common Base to them both ; this will give the Figure exactly . To make a Rbomboides , the Sides being given . Fi. 9 〈◊〉 neither Angle nor Diagonal be given , 〈◊〉 if either of them be limited , the case is ●●en to in the last problem ) make any Angle ●dventures as here ABC . Then supposing 〈◊〉 Lines given to be OP and QR , set the 〈◊〉 of the longer upon the Line BC from B 〈◊〉 , and the shorter on the Line BA to E , 〈◊〉 with the Compasses extended from B to E 〈◊〉 one Foot in D and describe the Arch FG. ●●ewise , with the Compasses extended to the 〈◊〉 of the Line OP , setting one Foot in E 〈◊〉 the other describe the Arch HI , intersecting Fig. 9 the former Arch at K , from which Intersection Lines drawn to D and E finish the Rhomboides . XII . To make a Trapezium , the Diagonal and Lines in order being given . Fi. 10 Let the Line HL be the Diagonal of a Trapezium , whose Sides are the Lines A , B , C , D , the Side A being counted the first , as that which takes its beginning from the point H , and the rest in the order as they are marked Alphabetically . Then with your Compasses set to the length of the Line A , place one Foot in H , and with the other describe the Arch EF. Next taking the length of the Line B , with the one Foot , o● your Compasses placed in L , with the other make the Arch GI intersecting the former at K , from which Point of Intersection , Lines drawn to H and L make the Triangle HKL . Then with the extent of the Line C , set one of the Feet of your Compasses at L , and describe the Arch OP . Lastly , setting them to the length of the Line D , and placing one Foot of your Compasses in H , with the other make the Arch SR intersecting the former at Q ; 〈◊〉 shall Lines drawn from Q to L and H make u● the Triangle LQH , and finish the Trapezium HKLQ . I could have been much briefer in this problem by referring to the sixth ; but this being of very great and frequent use , I desired to be very plain . XIII . To make a regular Polygon , otherwise called a regular multangular , or multilateral Figure , consisting of many equal Sides and Angles , viz. above four apiece . Being satisfied what shall be the distance between the Center and every Angle , with that distance describe a Circle , which being equally divided into as many Parts , as the Figure must have Angles ( or Sides , for they are equal in number ) and Lines drawn from the Points of Division within the Circle from Point to Point , ( ordinarily called Chords ) the Polygon is finished as in this Diagram . Fig. 11 Suppose an Heptagon , or multangular Figure of seven Sides , and as many Angles , be to be described , every Angle being designed to be distant from the Center A , seven Eighths , or three quarters and an half of an Inch ; with that distance describe the Circle BCDEFGH , which being divided into seven equal parts , and Lines drawn from Point to Point , the Heptagon BCDEFGH will be therein included . I shall rather leave my unlearned Reader to find out the Points of Division by many tryals , than to puzzle him with the Geometrical way for finding out Chords to that purpose ; nor shall I busie my self to tell him at large how he may divide 360 by the number of his Angles or Sides , and then finding in his Quotient the Degrees and Parts belonging to every Division , set them readily out by a Protractor , or ( for want thereof ) by a Line of Chords ; for I suppose him yet ignorant of such things . I shall therefore only tell him thus much : A Line drawn through the Circle at the Center divides it into two equal parts , which being crossed in the Center by another Line , the Circle will be parted into four equal Parts or Quadrants , and those by halving them into eight Parts . The extent of the Compasses whereby the Circle is drawn ( usually called the Radius or Semidiameter ) will divide it into six equal parts ; two whereof must be a third part , and half of one a twelfth part ; and these still easily capable of farther Division . XIV . Having the Sides of the Triangles whereof it consisteth , orderly given , to make an irregular multangler , or multilateral Figure . This will be more fully handled hereafter , when I come to shew the method of drawing plots of Ground : In the interim I will give you a Specimen of an irregular Pentagon . Fi. 12 Having the Lines of three Triangles given , ( which by a Rule hereafter to be mentioned are necessary to make up a five Figure ) lay down the greatest of the first , viz. 20 , from A to B for a Base , and by Probl. 6. make a Triangle of it and the other Lines 16 and 10. viz. the Triangle ABO . Secondly , you find by the number o over the first Line of the second Triangle that it is the common Base to them both , and therefore by the same Probl. 6. make the Triangle A B P of the Lines 20 , 14 , 18. Fi. 12 Lastly , finding the Base of the third Triangle to be the same with 18 , one of the Sides of the second , make the Triangle PBQ of the Lines 18 , 11 , 12 : So is the quinquangular Figure finished . How every Line is to be found in its due order in this or any other sort of multangular Figures , so as to give a true and exact account , not only of the superficial Content , but also of the Figure ( or shape ) and situation , is to be taught hereafter in the Doctrine and Practice of protraction . CHAP. III. How to find the Superficial Content of any Right-lined Figure , the Lines being given . AS a Foundation to what I shall say upon this Subject , there are some few Geometrical principles or Theorems out of Enclid and Ramus , which I desire may be remembred ; and because understanding is a mighty help to memory , I design for my Country Reader a kind of ocular Demonstration , which though not so strict and artificial as that which is to be found in the Commentators upon Euclid in the quoted places , will be more serviceable to him , because more easily understood . Theor. 1. Every Parallellogram being of the same length with the Base of a Triangle , and of the same height with the Perpendicular of that Triangle , is double to it , Euclid 41. 1. Fi. 13 Here are two equal Oblongs or long Squares , ABCD and BEFC , and within them two Triangles inscribed , whose Bases are of the same length , and their Perpendiculars ( OP and QR ) of the same height with the Oblongs . Now each of these Triangles being parted into two Right-angled Triangles by their Perpendiculars , then it is plain to the Eye ( and from the nature of Diagonals , which ever divide a Parallellogram into equal parts ) that the two new Triangles OPD and OPC , which make up the first of the given Triangles , are equal to the Triangles DAO and OBC , which make up the remainder of the Parallellogram ABCD. Therefore that Parallellogram is double to that Taiangle , which was to be demonstrated : In like manner it is evident , that the Parallellogram BEFC is double to the Triangle CQF , because CRQ is equal to BQC , and QRF is equal to QEF. Theor. 2. All Triangles having the same Base , and lying between the same Parallels , are equal . Euclid 37. 1. So in our last Diagram , the two given Triangles having Bases of the same length , and lying between the same parallels , are evidently equal , because they are demonstrated to contain each of them the exact half of the Parallellograms , wherein they are inscribed ; and the Parallellograms being equal , their halves must be equal also . Theor. 3. The Sides of a Triangulate ( that is , one that hath four or more Sides ) are ever two more than the Triangles of which it is made . Ram. lib. 10. prop. 2. Fi. 10 , 12. This is plain by inspection , if you view again the Figures of the Trapezium and irregular Pentagon , in the 12th and 14th Problems of the second Chapter . These Theorems being allowed to be found ( as nothing more certain ) the Doctrine concerning the Superficial Content of Right-lined Figures might be reduced to a narrow compass ; for he that knoweth how to husband these three Theorems , may easily take up these Corollaries , ordine inverso . 1. Any quadrangular Figure ( regular or irregular ) may by a Diagonal be parted into two Triangles ; any five-sided Figure by two Diagonals into three Triangles ; and six-sided Figures into four by three Diagonals , &c. by Theor. 3. 2. 'T is no matter of what shape the Triangle is , as to the rule for measuring , for whether it be Right-angled , Acute-angled , or Obtuse-angled , and whether it have three , two , or no Lines equal ; 't is only the length of the Base , and height of the Perpendicular , that is considerable ▪ by Theor. 2. 3. The true measure or content of any Triangle , whether alone , or as part of any triangulate Figure of 4 , 5 , 6 , or more Sides ( and consequently of the whole Figure , by summing up the content of all the several Triangles ) is found by multiplying the whele Base of the Triangle by half the Perpendicular , or the whole Perpendicular by half the Base ; which being a Rule of such infinite use in surveying , I desire it may be remembred ; and that it may be understood , I shall give you a plain Example . Fig. 6 Suppose A B the Base of the Triangle , belonging to the seventh Problem of the foregoing Chapter , to be 44 , and the Perpendicular CD to be 20 : Whether you multiply 44 the whole base by 10 the half of the Perpendicular , or 20 the whole Perpendicular by 22 the half Base , the product gives the Content 440 , as is here apparent . 44 10 440 20 22 40 40 440 Having thus given a general method how all Right lined Figures may be reduced to Triangles , and so their Content found out ; I might pass to the next Head concerning Instruments , and their use ; but because there are nearer ways in measuring particular kinds of triangulate Figures proper to those kinds . I shall briefly touch them . I. To find the Contents of a Square , or long Square . Multiply the length by the bredth , the product gives the Area or Content . Example of a Square 17 Inches length . 17 Inches bredth . 119 17 289 Square Inches . Example of an Oblong 25 Feet long . 13 Feet broad . 75 25 325 Square Feet . II. To find the Area or Content in Measure of a Rhombus . Let fall a perpendicular from one of the obtuse Angles upon the opposite side ; that side multiplied by the perpendicular , gives the Area . 20 Yards the Side . 14 Yards the Perpendicular . 80 20 80 Square Yards . III. To find the Area of a Rhomboides . Divide it into two Triangles by a Diagonal drawn between either pair of the opposite Angles , ( as suppose the Acute ) then from either of the other Angles ( for instance , the Obtuse let fall a perpendicular upon that Diagonal then shall that Diagonal , being multiplied by that perpendicular , give the Area . Example . 19 Rods the Diagonal . 5 Rods the Perpendicular . 95 Square Rods the Content . IV. To find the Content of a Trapezium . F. 10 Divide it by a Diagonal into two parts from Angle to Angle ; as the Trapezium , Ch. 2. pr. 12 is divided by the Diagonal HL , then from the other two Angles ( which in that Figure a●● marked with K and Q ) let fall Perpendicular upon the Diagonal , half the Sum of those Perpendiculars being multiplied by the Diagonal ( or common Base ) gives the Superficial Content . Example . Suppose in the Trapezium before mentioned the Diagonal is 33 , the Perpendicular from 〈◊〉 Fi. 10 13 , and that from Q15 , the Area or Superficial Content is thus computed : 13 Chains the first Perpendicular . 15 The second Perpendicular 28 The Sum of both Perpendiculars . 14 Their half Sum. 33 The Diagonal . 42 42 462 Square Chains the Area . V. To find the Content of a regular Polygonial , or multangular Figure , otherwise called multilateral . Draw a Line from the Center to the middle of any Side ; half of the Perimeter ( or of all the Sides ) being multiplied by that Line before-mentioned , gives the Content . VI. To find the Content of an irregular Polygon , or many sided Figure . Divide it into Trapezia's and Triangles by Diagonals , then find their Content severally , and sum up all together ; which that you may better apprehend , Fi. 12 Suppose the Polygon belonging to the last Problem of the second Chapter were given without the Diagonals AB and BP ; then by drawing those Diagonals , Fi. 12 the Figure is divided into three Triangles , whereof two being upon the same Base AB make up a Trapezium , whose Content may be found , just in the same manner as was taught even now in the fourth Rule , having found the perpendiculars from O and P falling upon the Line AB . Then there remains the Triangle BQP , whose Content may be found by the general Rule concerning Triangles , having found the perpendicular falling from Q on the Line BP ; and then having added the Content of that Triangle to the Content of the Trapezium , you have the Area of the whole polygonial Figure . But now methinks I see ( as it were ) my Country Student scratching his Head , and wishing for an opportunity ) to propound two doubts to me . 1. Why I called the Numbers correspondent to my Lines by divers denominations , as Inches , Feet , Yards , Rods ; and sometimes by none at all but propounding the Numbers abstractly . 2. How I came to know how many of those Measures ( whatever they be ) are represented by the Lines given , and perpendiculars found . To the former I answer , I am not yet teaching how to measure Lines ( that work is presently to follow ) but what Lines of Figures are to be measured , and the measures of those Lines being known ( or supposed ) how the Content upon those real ( or supposed ) Grounds may be found , and to this purpose I might call the Numbers represented by the Lines , Inches , Feet , Yards , or any other Measures , at pleasure , provided I called the Squares , to which the Area is equal , by the same names ; for an Inch in length bears the same proportion to a square Inch ( having length and bredth ) that a Mile in length bears to a square Mile . For this reason I profess not in the Title of this Chapter to teach how to measure Figures , ( much less how to measure the Lines of such Figures ) but how to find the Content , the Lines being given . And then to take away the second doubt , know that the Numbers represented by the Lines , were either given by those Learned Artists from whom I borrowed the Figures , or supposed by my self as grounds to go upon ( as in such cases is ordinary ) or , lastly , found out to be agreeable thereto by some Scale of small equal parts , which he is yet supposed ignorant of . But now I am going to shew him the nature and use of two or three plain and cheap Instruments , by the help whereof he may with much exactness , 1. Measure the length of any Lines bounding Right-lined Figures upon the ground . 2. Draw Lines and Figures upon paper proportionable thereunto ( which we call protracting ) . 3. Find upon his paper-figures the true length of all the desired Perpendiculars , which shall also be proportionable to those on the Ground , but much more easie to be obtained . And withal , I intend to give him such farther Instructions and Cautions for the application of the general and more particular Rules of this Chapter to his peculiar use , as will render them ( especially some of them ) singularly advantagious . CHAP. IV. Concerning Chains , Compasses , and Scales . 1. Amongst the many sorts of Chains used for measuring Land , three are most famous ▪ bearing the Names of their Inventors , Mr. Rathborne , Mr. Gunter , and Mr. Wing , all of them Ingeniously divided , and useful in their kind● but my brievity will give me leave only to describe one , and that shall be Mr. Gunter's , being most in use , and easie to be procured . This Chain contains in length four Statute-Poles or Perches , each Perch containing 1● Feet and a half , or 5 Yards and a half ; so that the whole Chain is 66 Feet , or 22 Yards long . This whole Chain is divided into 100 equal parts or Links , whereof 25 are a just Pole or Perch ; and for ready counting , there is usually a remarkable distinction by some Plate or large Ring at the end of every 25 Links , but especially at the precise middle of the Chain , which should differ from the rest in greatness and conspicuousness . Also at the end of every tenth Link 't is usual to hang a small Curtain-Ring and if there be at every five Links end a piece o● Wire made like the bow of a Link , with a little shank an Inch or less long , ( or some such distinction ) 't is still better . When you are to measure any Line by this Chain , you need to regard no other Denomination but only Chains and Links , set down with a prick of your pen betwixt them , e. g. If you found the side of a Close to be 6 Chains and 35 Links long , it is thus to be put down 6. 35. But if the Links be under 10 , a Cypher must be prefixed ; so 7 Chains 9 Links must be thus set , 7. 09. In the using of this ( or indeed of any ) Chain , care must be taken , both to go strait , and to keep a true account ; for which purpose , it is good that he which goeth before carry in his hand a bundle of Rods , to stick down one at the end of the Chain which leads , having first stretched it well , and that he which follows do not only gather up the Rods to keep the Accompt , but also at every remove , mark whether he see the Leader directly between his Eye and the Angle , or other Mark he aims to measure to ; and if need be , call to the Leader to move towards the right or left hand , till he see him in a direct Line to it . II. Compasses are so well known , that I need not describe them ; only they should be of Brass , with Steel Points small and neatly wrought , nine or ten Inches long from the Joint to the points , turning so truly upon the Rivet that they may be easily opened ; and yet stand so firmly , that an Arch or Circle may be without their shrinking described upon a large Radius . For the form , I would commend above all , those that have large Bows , so contrived , that by pressing them with the hinder part of the Hand , they will gently open , and by the Thumb and fourth Finger be put together ( as others will ) so that they are manageable by one hand , which is a great convenience for one that at the same time should hold his Rule with Scales in the other . These might also be contrived with a Screw to take out one of the points , to place in the room ( upon occasion ) a Black-Lead pen , or any pen to draw Circles , with either black or otherwise coloured . And for a Man that would be an Artist indeed , it were convenient he were furnished with dividing Campasses , beam Compasses , and triangular ones , for several uses not here to be mentioned ; but a Country Surveyor may make a good shift with such a plain pair as I first described ; which Mr. Wynn , over against the Rolls in Chancer●-lane , will help him to , with the Chain and Scales , for a small matter . III. Scales are certain Lines divided into equal parts , upon plates or broad Rules of Brass or Box , and they are of two sorts , 1. Plain ; 2. Diagonal . 1. Plain Scales are made up of two small Lines parallel to one another at a little distance , and these are divided into great equal parts , which signifie Tens , and are noted 10 , 20 , 30 , 40 , 50 , &c. according to the length of the Lines . They may be of any convenient length , but these great divisions are seldom more than Inches , or less than third parts a piece . Again , one of the great Divisions ( or parts ) is subdivided into ten equal parts by short Lines , whereof that in the middle standing for 5 is longer than the rest . According to the Numbers of these little parts contained in an Inch , the Scale is named A Scale of 10 , 11 , 12 , 16 , 20 , 24 , 30 , &c. in an Inch. Fi. 13 That short one which I give you the Figure of at A is of 10 in an Inch , so noted at the top , according as is usual upon Rules , and Indices of plain Tables . The Line marked OQ , separates Unites and Tents ; Unites being taken upward from that Line , and Tens downward ; mixt Numbers both ways . As for Example . 7 is the extent of the Compasses upon the Scale A from the Line OQ to K ; 30 is their extent from the Line so marked to OQ ; and 27 is their extent from the Line 20 , to the short Line K aforesaid . Here note , that you must not expect to find the Letters OQ or K upon the Scales which you buy , being only marks used at pleasure , to make my meaning plain ; and likewise that this Scale of 10 in an Inch , and others that are smaller ( all being composed after the same manner ) are usually made for more convenient use , so long as to contain nine or ten ( the more the better ) of the great Divisions , signifying Tens ▪ though the Figure at A being designed for no other use than to help your conceptions , extends but a little beyond 30 , that length being sufficient for my purpose in this place . Fi. 14 These plain Scales , especially the smaller sorts of them ( such as 24 or 30 in an Inch ) are very proper for drawing Figures upon paper , where the Numbers represented by the Lines are not above 100 ( for then every Division may be counted as it is upon the Scale ) or above upon a long Scale . Also in the surveying of Forests , Chases , and great Commons , where the Lines are vastly long , and the mistake of a few Links ( yea , of half a Pole ) is not considerable , they may be conveniently used , accounting the Tens and Unites to signifie so many whole Chains , and so estimating the parts of a Chain with the Compasses upon the small Divisions , which a sagacious Man may do very near upon one of the larger Scales . But it were much better , in my opinion , for ordinary measuring , if the grand Divisions on the Scale were two Inches a piece , as I have one upon the Index of my plain Table ) for then the smaller Divisions being of five in an Inch , would be so large as to be subdivided into five apiece , which represents 20 Links ; and then the half of one of those smaller Divisions signifying 10 Links , and the quarter 5 , a very ordinary judgment may come very near to the truth by estimation . 2. But the Diagonal Scale is so well known to every Mathematical Instrument-maker , so easie to be procured , and every ways so fitted to Gunter's Chain , and our Countryman's use , that I cannot but highly commend it . Of these Diagonal Scales , there are two sorts , the Old and New. Fi. 14 By the Old one , I mean such as is to be found in Mr. Leybourn's Book , whereof I shall present you with a fragment , with such a description as may enable you to understand the whole . 1. It is made ( as appears by the Figure B ) upon eleven parallel Lines equidistant , so as to include ten equal spaces , which are all cut at right Angles by Transwerse Lines dividing them all into four equal parts . 2. One of these Transverse Lines ( viz : PR ) where it toucheth the first and last Lines , separates between the Hundreds ( or whole Chains ) and the Tens , ( representing 10 Links apiece ) the Chains being numbred downwards on the left hand from P only to 3 , but on the Instrument it self they may go on to 9 or 10 , ( the Rule being a Foot long ) but the Tens ( or Decads ) upward from P to 10. 3. From the Points of Division into Tens upon the first Line beginning at P , to the like Points beginning at R in the last Line , are nine Diagonal Lines drawn , the first beginning at P , and ending at the first Division above R. The second beginning at the first Division above P , and ending at the second above R. In a word , they are all drawn from one Division less from P , to one more from R ; by which it comes to pass , that every Diagonal , by that time it hath passed from the first Line to the eleventh is a whole tenth part of an Inch ( which answers to ten Links of the Chain ) farther distant from the Line PR , than at the Point upon the first Line whence it was drawn . Fi. 14 4. Every one of these Diagonals is divided into ten equal parts by the long parallel Lines running through the whole Scale , and numbred on the top from 1 to 9. Whereby it is evident , that the Intersection of any of the nine parallel Lines that are numbred at the head with any Diagonal , must be farther distant from the Line PR , than the Intersection of the Line next before it with the same Diagonal by of that is , by which answereth to a single Link of your Chain . From what hath been said , and inspection of the Figure B , these things plainly follow , which as so many clear instances will help you to understand it fully . 1. The distance from PR to the second Division below it answereth to two Chains . 2. The distance from PR to the eighth Division upward being taken ( with Compasses ) upon the first Line of the eleven from P to 8 , answereth to 80 Links . 3. Consequently the extent of the Compasses from the second grand Division below P to the eighth of the less Divisions upward , is proportionable to 2 Chains 80 Links . 4. The distance from PR to the first Diagonal being taken upon the parallel Line noted with 9 above answereth to 9 Links : Where note , that the first Diagonal is not that which is noted with 1 , but that which is drawn from the point P. 5. The distance upon the same Line from PR to the Diagonal that is marked with 7 , is answerable to 79 Links . Fi. 14 6. The extent of the Compasses from the bottom of the Figure B upon the same Line to the same Diagonal , answereth to 3 Chains 79 Links . Briefly whole Chains may ( by Analogy ) be measured upon any Line from PR to the grand Division noted with the given Number , Decads alone , or Chains and Decads upon the first Line of the eleven where the Diagonals begin . Links alone , Decads with Links , and Chains and De●ads with Links , always upon that Line upon which the number ofodd Links stands at the head of the Scale . And know , that these Directions ( mutatis mutandis ) will as well fit , if half an Inch be only allowed for a Chain , and consequently all the Diagonals drawn within that extent , as it is usual ( and very commodious for longer Lines ) upon the other end of the same Rule , the grand Divisions for Chains going the contrary way , ●nd noted with Numeral Figures in order . It 〈◊〉 good therefore when you furnish your self with Scales , to have Diagonal Scales of both ●imensions on the fore-side of your Rule ; and upon the back-side many plain Scales of equal parts , with a Line of Chords ; all which you may have ( by enquiring only for the Scales described ●n Mr. Leybourn's Book ) of Mr. Wynne aforesaid , ●s likewise all other Mathematical Instruments . Having been so large for my plain Country-man's sake , I shall not proceed to the description of the new Diagonal Scale , of which you may have the Figure and Description in Mr. Wing's Book : For though it be an excellent good one , Fi. 14 as I know by experience ; Mr. Hayes having ( at my desire ) furnished my noble Friend Si● Charles Hoghton with an artificial one of that sort , when I had the honour of assisting him in Mathematical Studies ) yet because 't is pretty cost●● ( if well made ) , and that before described will very well answer its end , I shall at present say no more of it . But my Reader may perhaps object to me , th●● though I have instructed him how he may make a Line of an exact length , to answer to any number of Chains and Links ( given or found by measure ) upon the Diagonal Scales : I have not yet shewed him how to measure a Line ( as suppose a Perpendicular ) whose length is unknown , upon them . To give him therefore all satisfaction ( though what I have writ already , might help him to find this out ) let us suppose , that in some Figure made according to the Diagonal Scale B of 10● in an Inch , we meet in measuring with an unknown Perpendicular equal to the Line in the Margin . Taking it between the Points of my Compasses , I first try whether it be even Chains and finding upon the first view that it is not , 〈◊〉 make a second trial , whether it will prove to be even Decads , or Tens of Links ; to which purpose I set one Foot at 3 Chains in the bottom o● my Scale in the first Line where the Diagonals begin , and the other Foot rests in the same line betwixt 6 and 7 ; whereby I am assured the odd Links above 3 Chains are more than 60 and less than 70. And to find how many above 60 , I remove the Compasses from parallel to Fi. 14 parallel in order , till one Foot in the lowest Line resting in the end of a parallel , the other will touch some Diagonal at the Intersection with that line which falls out to be at L and O in the line marked with 7 , shewing the whole line , being measured by that Scale , to signifie 3 Chains and 67 Links . CHAP. V. How to cast up the Content of a Figure , the Lines being given in Chains and Links . HAving described these plain Instruments , and in some measure shewed the use of them in severals , it were very proper in the next place to teach their joynt use in measuring and protracting ; but because I would have my young Surveyor , before I take him into a Close , able to perform his whole work together , I intend to shew him , 1. How he ought to make his Computations ; 2. The Grounds or Principles that will justifie him in so doing . For the first , take these Rules : 1. Put down your length and bredth of Squares and Oblongs , and your Base and half Perpendicular of Triangles directly under one another , expressed by chains and links with a prick betwixt them , as was taught before , Chap. 4. 2. If the odd links were under ten , put a Cypher before the numeral Figure expressing them , ( as there also was shewed ) and if ther● be no odd links , but all even chains , put tw● Cyphers after the prick . 3. Multiply length by bredth , and Base 〈◊〉 the half Perpendicular , according to the Rul● for finding the Content of Figures , Chap. 3. 4. From their Product cut off 5 Figures ( accoun●ing Cyphers for such ) reckoned from th● right hand backward , with a dash of your 〈◊〉 so shall those to the left hand signifie Acres . 5. If those five cut off were not all Cypher● multiply them by 4 , and cutting off fiv● towar●● the right hand again , the rest will be Roods 〈◊〉 Quarters . 6. If amongst these five Figures towards the right hand that were cut off at the second Multiplication there be any Figures besides Cyphers multiply all the five by 40 , and cutting off fiv● again by a dash of your Pen , those on the left hand signifie square Perches , Poles , or Roods ▪ A few Examples will make all plain . Quest . 1. What is the content of a Square , 〈◊〉 Sides are every one of them 7 Chains , 25 Links ? Length 7.25 Bredth 7.25 3625 1450 5075 525625 525625 4 102500 40 100000 Answ . 5 Acres , 1 Rood , and 1 Perch , as here appears . Quest . 2. In a long Square , whose length is 14 Chains , and the bredth 6 Chains 5 Links , what is contained ? Length 14.00 Bredth 6.05 7000 84000 847000 4 188000 40 3520000 Answ . 8 Acres , 1 Rood , and 35 Perches , as the Work makes it evident . Quest . 3. In a Triangle , whose Base is 3 Chains , and half the Perpendicular 98 Links , what is the Content ? The Base 3.00 Half Perpend . 0.98 2400 2700 29400 4 117600 40 704000 Answ . 0 Acres , 1 Rood , 7 Perches , as here is plain . There be other ways of Computation by Scales , Tables , &c. but that this is sound and demonstrative , I come now to shew by these following Steps . 1. It is evident , that in this way of Multiplication the Product is square Links ; for every Chain being 100 Links , it is all one to multiply 7.25 by 7.25 , or 725 by 725 without pricks , for the pricks signifie something as to Conceptions but nothing at all in Operation . The Product therefore of the first Example was really 525625 Links . 2. Every Chain being 4 Perches long , it follows , that 5 Chains ( or 20 Perches ) in length , and 2 Chains ( or 8 Perches ) in bredth , make an Acre , or 160 square Perches ; for 20 being multiplied by 8 , gives 160. 3. From hence it plainly followeth farther , that there are exactly 100000 square Links in an Acre ; for 5 Chains multiplied by 2 , is the same with 500 Links by 200 , which makes 100000. And he deserveth not the name of an Arithmetician that is ignorant of this old plain Rule , When the Devisor consists of 1 and Cyphers , ( as 10 , 100 , 1000 , 10000 , 100000 , &c. ) cut off from the right hand so many Figures of the Dividend as the Devisor hath Cyphers , accounting them the Remain ; so shall the rest on the left side be the Quotient . It is plain then that 525625 square Links make 5 Acres , and 25625 square Links over . Thus I have made it clear to a very ordinary capacity , that as far as concern Acres , the Rules for Computation are good . Now for Roods and Perches , though I might turn off my Reader with that known Rule in Decimal Arithmetick : Multiplying Decimal Fractions by known Parts , gives those known Parts in Integers , due regard being had to the separation . I shall proceed in my plain way thus : If 25625 square Links , which remain above an Acre , do contain any quarter or quarters of an Acre ; then if they be multiplned by 4 , and divided by 100000 , ( that is , five cut off from the Product ) they will contain so many Acres as now they do quarters ( or Roods ) , for any number of quarters multiplied by 4 , must needs produce the like number of Unites or Integers , and the Division doth only reduce them into the right denomination . Now 25625 being multiplied by 4 , and five Figures being cut off from the product , the result is , 1●02500 , that is an Acre and above ; which shews it was above a quarter before it was multiplied by 4. And to find how much , ( that is , how many square Perches are contained in this last remainder ) you must consider this 2500 , not as square Links remaining above the Rood or Quarter , but as fourth parts or quarters of square Links ; or ( which is all one ) as the true number of square Links multiplied by 4 , and consequently being multiplied by 40 , ( the fourth part of square Perches in an Acre ) it must as often contain 100000 square Links ( or an Acre ) as the quarter of this number 2500 , viz. 625 , signifying square Links , containing square Perches ; and so it doth , for 100000 divided by 160 ( the number of Perches in an Acre ) gives 625 as answerable to 1 Perch ; and 2500 multiplied by 40 , gives 100000 , or 1 Acre ; the five Cyphers being cut off as here is manifest . 2500 40 100000 160 ) 100000 ( 625 960 400 320 800 800 0 Some may perhaps wonder , that in so small a Manual I spend so many words about such ordinary things , as in this and the last Chapter ; but I am most afraid , lest I shall not for all my plainness be sufficiently understood by such as I purposely write for , in things of such necessary and frequent use : And I designed not to make this Treatise small by being obscurely brief in substantial things , but by leaving out such Curiosities as I thought my Country Friend might well spare . CHAP. VI. How to measure a Close , or parcel of Land , and to Protract it , and give up the Content . HItherto we have been like Children learning to spell , now let us set our Syllables together . I mean , let us make use of the Instructions beforegoing to measure a plece of land , to plot it , and to cast up the Content . All Closes , or parcels of land , are either such as need not to be plotted for finding out their true measure , but the Chain alone doth the Work ; or such as cannot be conveniently measured without plotting or protraction . Of the first sort are the Square and long Square , known before-hand to be such , or found so to be by such Instruments as I have not yet described , or by measuring all the Sides and Diagonals . These Squares and long Squares ( I say ) need no protracting , for you need only to multiply the Chains and Links of the length , by the Chains and Links of the bredth , and so proceed as in the first and second Examples of the fifth Chapter : But all others , whether Triangles or Triangulate , are to be protracted . I shall give Examples therefore in the 3 sorts of Figures , triangular , quadrangular , and multangular . But before I proceed to particular Instances , let me advise the young Practitioner thus : Remember , 1. To begin at some notable Angle of the Field , where there is some House , Gate , Stile , Well , or the like ; or if there be none , then to dig up a Clod , drive down a Stake ; or at least , to observe what quarter of the Heavens it pointeth towards , whether East , West , North , or South , and on your Paper mark it with the Letter A , or any other . 2. To go parallel to the side of the Field , 〈◊〉 Pits , Bushes , or the like , hinder not , ( and if they do , to allow for it ) accustoming your self to go either cum Sole , that is , with your left hand towards the Hedges , Walls , or Pales ; or contra Solem , with your right hand towards them ; and when you go contrary to your usual custom , note it on your paper by some mark known to your self . 3. To set down the Chains and Links of every side as you measure them , and not to trust your memory . A Black lead pen will be very proper for this purpose . 4. To take heed ( if you have more Scales than one upon your Rule ) lest you confound your self by taking lines off of several Scales , or measuring perpendiculars upon wrong ones ; for every line of the same Figure must be made by the same Scale , and the perpendiculars measured by it . 5. To make use of a Scale of larger Divisions when you measure small Closes , and of smaller when you measure great ones . 6. To make your lines and points where Angles meet , small , pure , and neat . 7. To set on your Chains and Links at twice , when any line is too long for your Scale . These things being premised , I proceed thus : I. Suppose I measure a triangular Field with my Chain , beginning at the Eastern Angle A , and find the Sides in their order and measures to be severally , thus : ( I going cum Sole ) 2.229 , 3.45 , 4.07 . Fig. 15 Making use of the less Diagonal Scale , because the other would make the Figure too large , otherwise it were more proper for so small a Close ) I first with my Compasses take off the Scale 4 Chains and 7 Links , and setting them from A to C draw that line for the Base , because the longest of the three : Then I take 2 Chains 29 Links off the same Scale , and set them in the Eastern point A where I began , and turning the loose Foot of the Compasses above the line A C , because I went cum Sole , I describe ( at that distance 2. 29. ) the Arch E E. Fi. 15 Next taking with my Compasses upon the same Scale the extent of 3 Chains 45 Links , 〈◊〉 place one Foot in the point C , and with the other make the Arch FF intersecting the former● in the point B ; and drawing the lines A B and B C , ( as was taught in Ch. 2. Probl. 6. ) the Triangle A B C is the Plot of the Triangular Field measured . But before I can give the Content , I must find the length of the Perpendicular , which is done by setting one Foot of the Compasses in B , and extending the other to the Base A C , so as 〈◊〉 touch it and pass not over it , ( according to Ch. ● ▪ Probl. 7. ) for then the length of the Perpendicular is between the Points of the Compasses , and being applied to the same Scale by which the Triangle A B C was made , it appears to be 1 Chain 42 Links . With the half whereof I multiply 4.07 the length of the Base , and proceeding in my Work as was shewed in the last Chapter , the Content appears to be 0 Acres , 1 Rood , 6 Perches , as it is here evident . The Base 4.07 Half Perpend . 0.71 407 2849 28897 4 115588 115588 40 623520 II. Suppose I were to measure a quadrangular or four-corner'd Field , I begin as before at ●ome remarkable Angle ; and going round the ●lose cum Sole , I find the Sides to be 9.04 , 6.72 , ● . 46 , 7.28 , and the Diagonal from that remarkable Angle to the opposite Angle to be 10.02 , I ●●gin therefore to protract it thus . Fi. 16 Having by the help of my Scale and Com●asses drawn my Diagonal 10. 02 from my remarkable Angle A to C the opposite Angle , make a Triangle of it , and the first and second sides 9.04 and 6.72 . according to Ch. 2. Probl. 6. ●nd another after the same method of that Diagonal , and the third and fourth Sides 8.46 and ● . 28 , so have I the Trapezium ABCD. Then by the help of my Scale and Compasses , 〈◊〉 find the Perpendicular of the Triangle A B C 〈◊〉 be 6.02 , and of the other , viz. CDA 6.01 , which added are 12.03 , whereof the half Sum 〈◊〉 6.01 ; by which multiplying the Base 10.02 , and proceeding as formerly hath been shewn , I ●nd the Content of the Field to be 6 Acres , 0 Roods , 3 Perches , as is here apparent . Fi. 16 The Base10 . 02 Half Perpend . 6.01 1002 60120 6●02202 4 ●08808 40 3●52320 Before I pass any further , let me tell you , 1. Any quadrangular Close , or parcel Ground whatsoever , having right Lines , 〈◊〉 be thus measured , protracted , and computed . 2. The odd measure above Perches is 〈◊〉 valuable here , nor in the former Computatio● being always under a square Perch ; but in mu●● angulars where there be many Remainders , th●●● must be summed up , and the Perches contain● in them added to the Content before found . 3. This last , and the following Figures ( wh●● I use any Scale at all ) are made that they mig●● not be too large , by a Scale of 400 in an In●●● i. e. by the less Diagonal Scale , each Chain 〈◊〉 Link being counted two . Fig. 17 III. If this multangular Figure be conceive to represent a Close of seven sides , which is 〈◊〉 be measured , I begin at the remarkable Angle 〈◊〉 and going round the Close ( with the Sun ) find the sides to be in measure 3. 11 , 2. 49 , 2. 4● , 1. 77 , 4. 11 , 2. 29 , 4. 37. Fi. 17 Then I measure the four Diagonals , BD , DF , ●B , and BG , in the order that I named them , ●nd I find them to be 3.45 , 4.97 , 4.13 , and 4.36 , ●hich is as short a way as can be taken , to pre●ent unnecessary Walks . But when I come to protract by the help of ●●y Scale and Compasses , I first make the Trian●●e BCD of the first Diagonal , and the second ●nd third sides . Then the Triangle DEF upon 〈◊〉 second Diagonal , and fourth and fifth sides , 〈◊〉 upon the same Diagonal as a common Base , 〈◊〉 Triangle BDF of the first , second and third Diagonals . Next of the same third Diagonal , together ●ith the fourth and sixth sides I make the Triangle BFG , and upon the fourth Diagonal as ●pon a common Base with the first and last sides 〈◊〉 Triangle ABG , so is the whole Close ●●otted . And now it stands visibly reduced into two ●rapezia's ABFG and BDEF , together with ●he Triangle BCD , which I shall not now cast 〈◊〉 , having so often shewed how such work is 〈◊〉 be done . But I must acknowledge that this sort of plotting of parcels of Land that have many Angles , ●equires not only more care and pains , but better skill and memory than to draw Diago●als upon Paper , when the Plot is already taken by the plain Table , or other standing Instrument . I shall therefore to help my young Practitioner in this case , advertize him of two easie ways to help himself , so as to be out of danger of mistakes . One way is to divide the Multangular Field into two or more parts as the last might have been by the Diagonal BF ; then might each pa●● have been measured severally , as if they had been separated by a Pale , or were sundry Mens Land parted by a Boundary . Another way that much helps both the understanding and memory , is to draw a rude Draugh● of the Figure of the Land you intend to measure , not only as to the sides , but also necessary Diagonals . Then measuring the Lines upo● the Ground correspondent to those on the Paper ( which by the help of the Draught may be easily hit ) set the Lines as you measure them upon the Lines of the Draught , as if it were the true ones , and when you have finished your measuring , protract it truly . Such as you see her● ( but it 's better larger ) will do your business for 't is not a Pin matter how rude or false the Lines or Angles be , resemblance being all that is desired . CHAP. VII . Concerning the measuring of Circles , and their Parts . I Have hitherto abstained of purpose from medling with the Circle and its Parts , that I might lay those things close together without unnecessary mixtures , that are of the greatest use . 'T is wonderful rare , if a Land-meeter ever have occasion to measure any Field or parcel of Land , that will prove either Circle , Semicircle , Quadrant , or Sector . Sometimes indeed there will be a little crook in an old Hedge bowing like an Arch : But I have never seen any offer to measure it as a Segment , but always take it as an Angle or Angles . Yet because it may be expected I should say somewhat of those things , I shall briefly do it . 1. To measere a Circle in the more exact way 〈◊〉 to square the Diameter , and to multiply that Square by . 7854 , so shall the Content be in Integers and Decimals . But the more usual and quick way ( and near enough for any use we shall make of it ) is to multiply the half of the Periphery or Circumference by the Semidiameter . In like manner to find the Content of a Semicircle , Quadrant , or Sector made up of Semidiameters , and arched Lines , multiplying the half Arch by the Semidiameter , gives the Content . Fi. 19 But that which falls out most frequently in Mensuration ( though seldom much regarded ●ave where a curious exactness is required ) is that particular sort of Segment , which we call 〈◊〉 Section , less than a Semicircle , such as this Figure ABC . And to find the Content of it , the center of the Circle whereof this is a Section must be first fou●d out , as here at O , from which Lines drawn to A and B , make up the Sector AOBC ; which being measured according to the last Rule , and from the Content thereof , Fi. 19 the Content of the Triangle AOB sub●racted , the difference or residue is the Content of the Section ABC . But two Questions may be here demanded : 1. How may the Center be found ? 2. How may such a portion of Land be truly protracted and computed ? To the first I answer , that the most exact and artificial way is by making a Mark any where in the Arch. As for Example . At the Point C ; and then ( by a Problem known not only to every Surveyor , but to ordinary Carpenters and Joyners , for finding the center of a Circle , whose Circumference will pass through 3 given points that are not in a right line as ACB ) to find the Center O. But if you know not how to do it so , cross the Line AB in the middle , as here it is done by the Perpendicular OC , so you may by a few trials find both the due extent of your Compasses , and the point in the Perpendicular that will fit your purpose near enough ; for if a little errour be committed in making up the Sector , the most of it goes off again in the substraction of the Triangle . II. For the latter you may take this ready course : Measure the length of both your Lines , ( the Chord and the Arch ) and their distance at the middle of them both . Then when you come to protract , first take the length of your right Line from the Scale , and having laid it down , cross it in the middle at right Angles with a dry Line as in the last Figure , so shall it intersect the Line AB in the point E ; Fig. 19 then from the same Scale take the measured distance between the two lines in the middle , and set it upon that dry line from the Intersection at E to the point C. Then by trials find a due place in the dry line OEC , and such a distance with your Compasses , that the one Foot resting in that line , the other may describe the Arch ACB , and the Section is protracted . These few hints are as much as I thought necessary for my Country Practitioner concerning circular lines ; but if he think otherwise , there are large Treatises enough , and particularly those I mentioned , whose Rules ( though ingenious , sound , and fit to be known by every one that intends to plunge deep into Mathematical Studies ) will not ( I think ) be of that use to him in ordinary measuring , that I should transcend the intended bounds of brevity to transcribe them . CHAP. VIII . Concerning customary Measure , and how it may be reduced to Statute-Measure , & è contra , either by the Rule of Three , or a more compendious way by Multiplication only . WHereas the Statute-Perch or Pole is 16 Feet and a half , and no more , there be Poles of larger measure used in many places , as of 18 , 20 , 21 , 24 , and 28 Feet , yea in some 22 Feet and a half . It were therefore very convenient , that our young Surveyor were furnished with a Chain fitted to the customary measure of the Country where he lives , as I use to make Chains for my self and Scholars of 21 or 24 Feet to the Pole for Lancashire and Cheshire , where those Measures most obtain . But because these are too large and cumbersome for small Closes , it is very convenient , instead of one Chain of 100 Links , to make two of 2 Poles apiece , each Pole divided into 25 Links as that of 100 is , which two half Chains may in measuring large Fields be tied together by the Loops with Pack-thread , or joyned by a buttoning Keyring for more speedy dispatch ; but in smaller we may use the half Chain of 50 Links , only taking care that we count not half Chains for whole ones . And in these cases where the Poles are large and the Closes small , it were still more convenient if you had a Chain of 2 Poles only , divided into 100 Links : Only you must then take notice , that whereas working by whole Chains and Links , the first Multiplication , after five cut off , gives the Content in Acres and Parts . The like work by half Chains and half Links will give the Content in Roods or quarters of Acres , and parts of such Roods . But though it is no hard matter ( for one that can find out the length of a Link by dividing the number of Feet in a Chain by 100 , and provide himself of good Iron-wyre , and Curtain Rings to make it of , and a sharp edged File , and round nosed Plyers to make it with ) to be furnished with such a Chain ; yet because every one cannot do this , I shall shew you how you may easily and yet very truly reduce Statute-measure into customary , that so the Chain before described may do your business all England over . Know therefore ( for a ground to go upon ) that Acres bear proportion to one another , as the squares of their Poles ; and therefore if you multiply 33 , the number of half Feet in the Statute-pole by it self , which gives 1089 , and also multiply the number of half feet contained in a Pole of that measure you would reduce into ; in the same manner you may by the Rule of Three reverse obtain your desire , making to that purpose 1089 the first number , the Statute-measure the second , and the squared half-feet of the Pole given the third . As for example : 9.33 L. 7.21 B. 933 1866 6531 672693 prod . Suppose of a Close measured by the Statute-Pole , the length , breadth , and their product be as here represented in the Margin . And it is desired that the Content may be cast up according to our large Cheshire measure of eight yards or 24 feet to the Pole or Rood ( as we call it : ) The● before I cut off any Figures , I consider that in the Statute-pole are 33 half-feet , and in the Cheshire-pole 48 , then multiplying 33 by 33 , and 48 by 48 , I have these two square numbers , 1089 and 2304 , which together with the said product may be thus placed : 1089. 672693 ∷ 2304 , and so multiplying 672693 by 1089 , and dividing their product being 732562677 by 2304 , the quotient is 317952 , from which if 5 figures toward the right hand be cut off , and dealt withal as was taught in the fifth Chapter , the Content by our customary measure of 24 feet to the Pole , will be 3 acres , 0 roods , 28 perches as here appears 317952 4 71808 40 2872320 But if the lines on the land had been measured according to our custom here , of 24 feet to the pole , and the Content must have been found according to Statute measure ; then I must have multiplied the product by 2304 , and have divided that latter product by 1089 ▪ And in the same method you may proceed in all or any of the rest . But the truth is , that though this way be very exact , plain and comprehensive , suting all the customary measures before-mentioned without fractions , which for my Learners sake I studiously avoid , and for that reason reduced my poles to half-feet : It is something tedious except he knows how to relieve himself by a large Table of Logarithms , or at least a set of Nepair's-bones , which I cannot stand here to treat of : Therefore to contract the work a little , take notice , that all the customary poles before mentioned , ( saving only those of 20 and 28 feet , which I suppose are somewhat rarely used , because I never heard nor read of them ( to my remembrance ) save only in Mr. Holwel ; all the rest I say , are capable of being divided into half-yards : And therefore if instead of squaring the half-feet you square the half-yards of both poles , and work with them , you will attain the same end without any regardable difference , the small diversity that there is being usually in the useless remainders , not at all affecting the desired Quotient that gives the answer near enough for use . As for Example . If I had squared 11 , the number of half-yards in the Statute-pole , which would make 121 , and also 16 the number of half-yards , in our Cheshire-pole , which would make 256 , as appears in the Margin , and then multiplied the first product 672693 by 121 , the second product would have been 81319853 , which being divided by 256 , the Quotient would have been ( as before ) 317952. And this way is in a manner coincident with Mr. Holwells first Method . 11 11 11 11 121 16 16 96 16 256 Take notice also further once for all , that whether you use either of these or the following Methods , you need not reduce the particular Squares , Triangles , or Trapezia's severally ; but sum up all their products together , and then reduce all at once . But if you would reduce Statute measure into Customary by Multiplication only , take notice of this present Table following . The Content by the Statute-pole being multiplied by . 84027 Gives the Content by the Pole of 18 feet . . 68062 20 . 61734 21 ½ . 53777 22 . 47265 24 . 34725 28 The use of this Table . When you have multiplied Lengths by Bredths , or Bases by half Perpendicalars , multiply these Products by the Decimal Fractions answering to the Customary-measure into which you would reduce Statute-measure , and from that latter product , first cut off five places towards the right-hand as not to be regarded , ( being only parts of a square Link ; ) then cutting off 5 more , and proceeding to multiply by 〈◊〉 , and then by 40 , as hath been often shewed , you will have the Content by that Customary-measure , Example . Suppose once more the length of a Close measured by Gunter's Chain , and multiplied by the breadth measured also by the same , produced 672693 square Links ; and it is desired that the Content may be given in Cheshire-measure of 24 to the Pole : You must multiply 〈◊〉 72693 by .47265 , the decimal fraction answering to 24 feet , and from that product being 31794834645 cut off and cast away 5 places , and the rest being ●17948 , are in the usual way easily ●educible into 3 Acres , 0 Roods , 28 Perches , as here appears , agreeable ●o what it amounted to in the former Method . 317948 4 71792 40 2871680 But if you measured by a Chain of Customary Poles , and desire to know what the Content 〈◊〉 in Statute-measure ; this following Table is fo● your purpose . The Content measured by the Pole of 18 Feet . being multiplied by 1. 19008 gives the Content by the Statute-pole . 20 Feet . 1. 46923 21 Feet . 1. 61983 22 ½ Feet . 1. 85950 24 Feet . 2. 11570 26 Feet . 2. 87970 To understand which , take this Example . Suppose the length and breadth of a lo● Square being measured by a Chain of 24 feet 〈◊〉 the Pole , and multiplied together , make the●● product 317952 , le● this be multiplied b● 2.11570 , which answereth to 24 feet , and th● latter product will be 67269104640 , from which if you cut off and cast away 5 places towards the right-hand , the remainder is 672691 , which in the usual way is easily reduced to 6 Acres , 2 Roods , and 36 Perches , as you see here . 67269 29076 4 363056 One thing more and I have done with th● business of Reduction : If the Content to be r●●duced , be given cast up into Acres , Roods , and Perches , reduce all into Perches , and then i● other respects work as before either by th● Rule of Three , or by this last Method of Multi●●lication only . So shall you have the Content in square Perches according to the Measure desired , which you may reduce into Acres by dividing them by 160 , and if any thing remain , that remainder being divided by 40 , will give you the Roods in the Quotient , and the latter remainder the number of square Perches . For tryal of which Rules , mind the Answer ●o these two following Questions wrought all ●hree ways . Quest . 1. How many Acres , Roods and Perches , according to the Pole of 18 Feet , are contained in 5 Acres , 3 Roods , and 11 Perches , Statute-measure ? Answ . 4 Acres , 3 Roods , and 22 Perches , as ●●ere appears : I. Method . 33 36 5 A : 3 R : 11 P. 33 36 4 — — — 99 216 23 99 108 40 — — — 1089 1296 931 1089 . 931 ∷ 1296 1089 8379 7448 9310 1013859 1296 ) 1013859 ( 782 9072 10665 10368 2979 2592 387 160 ) 782 ( 4 640 40 ) 142 ( 3 120 22 II. Method . 11 11 11 11 121 12 12 24 12 144 121 . 931 ∷ 144 121 931 1862 931 144 ) 112651 ( 782 1008 1185 1152 331 288 43 160 ) 782 ( 4 640 40 ) 142 ( 3 120 22 Or thus : 40 ) 782 ( 19 40 382 360 22 4 ) 19 ( 4 16 3 III. Method . 84027 931 84027 252081 756243 782. 291 37 160 ) 782 ( 4 640 40 ) 142 ( 3 120 22 Quest . 2. How many Acres , Roods and Perches , of Statute-measure are contained in 8 Acres , 3 Roods ( or Quarters ) and 21 Perches of 21 Feet to the Pole ? Answ . 14 Acres , 1 Rood , and 21 Perches , as appears by the three following works in the several Methods . I. Method . 8 A : 3 R : 21 P 4 35 40 1421 42 42 84 168 1764 33 33 99 99 1089 1764 . 1421 ∷ 1089 1764 5684 8526 9947 1421 1089 ) 2506644 ( 2301 2178 3286 3267 1944 1089 855 160 ) 2301 ( 14 160 701 640 40 ) 61 ( 1 40 21 II. Method . 14 14 56 14 196 11 11 11 121 196 . 1421 ∷ 121 196 8526 12789 1421 121 ) 278516 ( 2301 242 365 363 216 121 95 160 ) 2301 ( 14 160 701 640 61 40 ) 61 ( 1 40 21 III. Method . 1.61983 1421 161983 323966 647932 161983 2301.77843 160 ) 2301 ( 14 160 701 640 40 ) 61 ( 1 40 21 CHAP. IX . How a man may become a ready Measurer by Practice in his private Study , without any ones assistance or observation , till he design to practice abroad . THis Art above all parts of the Mathematicks , is burdened with two Inconveniences to the young Practitioner : The necessity of having one to assist him in measuring his Lines with the Chain , as oft as he would practice his skill , or get more , and the exposing of his unreadiness to the view of meddlesom people , while he is yet raw and unexperienced , as every one must needs be at first : Both which may in good measure be avoided by this easie knack . Take a small Packthread , and by knots about half Inch asunder divide it into an 100 parts , as Gunter's Chain is divided ; So shall these small divisions between the knots answer to Links ; and if they be not exactly of one length , the matter is not very weighty , but the more equal and short they are , the better . Having this String thus prepared , and marked with longer and shorter pieces of thread tied in the knots , so as you may readily see where is the middle , and where your divisions of 25 Links , and the smaller divisions of 10 links begin and end ; make all sorts of Figures in your Study or Chamber , by marking places , or sticking Knives or Bodkins at pleasure for Angles , accounting the streight lines betwixt them for sides , and so measure the Figures by your knotted string , and cast up the Content by a Scale . This Work you may manage with your own hands in private , and so make your self very quick and ready when you begin to measure for good and all ; as I once made a full experiment , and it was thus : When I first began to instruct Youths in Mathematical Learning in Warrington , some of my Boys Parents desired a sensible demonstration of their Sons proficiency in somewhat that they themselves could in some measure understand ; and particularly pitched upon measuring a Piece of Land : Whereupon I took four or five of my Scholars to the Heath with me , that had only been exercised within the Walls of the School , and never saw ( that I know of ) so much as a Chain laid on the ground : and to the admiration of the Spectators , and especially of a skilful Surveyor then living in the Town , they went about their work as regularly , and dispatched it with as much expedition and exactness , as if they had been old Land-meters . CHAP. X. How to measure a piece of Land with any Chain of what length soever and howsoever divided ; yea with a Cord or Cart-Rope , being a good Expedient when Instruments are not at hand of a more Artificial make . IF you can procure a Chain , and find it is not divided as before hath been shewed , but into Feet or quarters of Yards , or any such vulgar divisions , make no reckoning of the divisions at all , but measure it as exactly as you can to find out the true length of the whole Chain , and if it fit none of those lengths mentioned in the 8th Chapter , nor any of their halfs , make it to fit , by taking off a Link or two , or piecing it out with a string ; then dividing the length of that Chain by 100 , or the half of it by 50 , find the true length of a Link according to our artificial division , and having got a long stick or rod , set as many of those link-lengths upon it as it will hold ; Then may you measure all the whole Chains by your regulated Chain , and the odd links of every line by your divided stick or rod , as is manifest in this Example following . Being far from mine Instruments , and requested by a Friend to measure him a Close , I procure a pair of Compasses , an ordinary Carpenters Rule of two foot , divided into Inches and quarters , and meeting also with a piece of an old Chain seemingly divided into feet , I measure it by the Rule , and finding it to be 45 feet long , and some odd measure , I piece it out with a pretty strong Cord that will not stretch much , to 48 feet exactly ; then it will serve me for half a Chain of 24 feet to the Pole : This 48 I multiply by 12 ( the number of Inches in a Foot ) and that product being 576 , I divide by 50 , the number of links in half a decimal Chain , and the Quotient is 1166-50 Inches , or 11 Inches and an half , and an trifle over : So then dividing a long stick throughout into such parts , each containing 11 Inches and an half , besides the breadth of the nicks , I am provided of Tools to measure Lines to a Link with exactness enough . In like manner would I proceed with a Cord or Rope , having fitted them to some known length or other . And then for protraction it were easie with the Compasses to make a plain Scale of a large sort , either upon Paper , or an even piece of Wood ; this for once may serve a mans turn well enough . Besides there is a way of measuring the Perpendiculars of Triangles and Trapezia`s upon the ground it self , so as to prevent the necessity of a Scale ; for if you have a little Square with an hole in it , to turn upon the head of a little stick , which you may fix where you please , as you are measuring the Base of a Triangle , or the Diagonal of a Trapezium , you may by a very few trials find the place where the one Leg will be just in the Line which you are measuring , and the other point at the Angle from which the Perpendicular falls on it , and then the space between your Stick and that Angle truly measured , is the Perpendicular . If you have not such a Square , a square Trencher , or any end of a Board that hath one right Angle , and two true sided , will supply the want of it . And now that I am mentioning this way of measuring , I shall make bold to add , that this is a good way , ( and as such ordinarily used by that general Scholar and reverend Minister Mr. Samuel Langley of Tamworth in Staffordshire , whose ancient acquaintance I have long esteemed both mine happiness and honour ) to measure a Trapezium thus , though it be protracted afterwards ; for by measuring the Perpendiculars as aforesaid , and observing at how many Chains and Links end the said Perpendiculars meet the common Base , the whole Trapezium may be truly protracted , without going about it ; this little Square competently supplying the place of an Instrument , which is usually called a Cross or Square , made up ( as it were ) of two small Indices , like those for a Plain-Table ( but much less ) with fore-sights and back-sights , and cutting one another at right Angles , put together , and having an hole at the Center , like those things which here in Cheshire we call Yarndles , being used by Country Housewives in winding of their Yarn . CHAP. XI . Concerning dividing of Land Artificially and Mechanically . WEre it sutable to my Design or Humour to be copious or curious , I had here a fait opportunity ; for four or five modern Survey-Books of the best Accompt lying open before me , would tempt me to transcribe abundance of ingenious things ; but for reasons often hinted before , I shall confine my self to a few plain things that will competently do this business . 1. To divide a Triangle into any parts required ; divide the Base as the Demand imports ; then shall Lines drawn from the Points of Division to the opposite Angle finish the Division of the Triangle . Example . Fi. 20 AC , the Base of the Triangle ABC , being divided into 12 equal parts , a Line drawn from the Angular point B to the point 6 divides the Triangle into two equal parts . 2. Lines drawn to 4 and 8 divide it into three equal parts . 3. Lines drawn to the Points noted with 3 , 6 , and 9 , divide it into four equal parts ; and so Lines drawn to 2 , 4 , 6 , 8 , 10 , divide it into six equal parts . Also it is very obvious , that if the same Triangle were so to be divided , that the one part should be double to the other ; a Line drawn from B to 4 or 8 , doth the work . Or if it be required to divide it into two parts , so as the one shall be triple to the other , a Line drawn from B to 3 or 9 , compleats the Work. So also a Line from B to 2 or 10 divides it into two parts , whereof the one is quintuple ( or five-fold ) to the other , and a Line from B to 1 or 11 , divides it into two parts , whereof the one is 11 times as large as the other . Further yet , if it were required this Triangle should be so divided , that the two parts should in quantity bear proportion , as 5 and 7 , a Line from B to 5 or 7 , doth that feat . But to deal plainly with you , I must confess that sometimes the Division will be a little more intricate than thus , yet not such , but that the seeming difficulty may be easily overcome , by observing the method wherein I shall satisfie the following demand . Suppose a large Triangle of common Land be to be divided amongst three Tenants A , B , and C , according to the quantity of their Tenements , A having 19 Acres of Land to his Tenement , B13 , and C7 , the Base of the Triangle being found by measure to be 17 Chains and 27 Links ; and the Dem and is , where the Points of Division must be placed in the Base , so as Lines drawn from thence to the opposite Angle , shall truly limit each mans part ? To answer this , let us add 13 and 07 to 19 , ( as in the Margin ) and they give 39 : So is the work plainly reduced to the Rule of Fellowship ; and therefore to find every mans distinct portion , we need only to multiply the Base by his 19 13 07 39 particular number , and divide that product by 39 , the sum of all their numbers as here is plain ▪ A 39 . 17. 27 ∷ 19 19 15543 1727 39 ) 32813 ( 841 14 / 39 312 161 156 53 39 14 B 39 . 17. 27 ∷ 13 13 5181 1727 39 ) 22451 ( 575 2● / 3● 195 295 273 221 195 26 C 39 . 17. 27 ∷ 7 7 39 ) 12089 ( 309 38 / 39 117 389 351 38 From these Operations it is plain , that if we set off from the Angular point where the Base begins , 8 Chains 41 Links , and a little above the third part of a Link upon the Base for A , and where that ends , 5 Chains and 75 Links and 2 / 3 of a Link for B , and consequently leave between this second division and the other end of the Base 3 Chains and almost 10 Links for C ; Lines drawn from those points of division to the opposite Angle , will give each man his due . What I have said touching the division of Triangles upon their Bases , will with a little variation serve for the dividing of all sorts of Parallellograms , whether Square , Long-squares , Rhombus's , or Rhomboides's : all the difference is , that in stead of drawing Lines from Points in the Base to the opposite Angle , you must draw parallel Lines from Points in one opposite side to another , as will be sufficiently plain by this one Instance . Fig. 7 Suppose the square Figure in the 8th Prop. of the second Chapter to represent a Close of six Acres , and I am to cut off an Acre at the side AC , having set off the 6th part of the Line CD , from C towards D , and also from A towards N , a Parallel drawn between those Points , takes off exactly a 6th part , or an Acre . If it be not thought convenient ( as in some cases it is not ) to cut off a piece so long and narrow , you may by the Rule of Three find what other length of any greater breadth will limit an equal quantity to it . Or you may multiply the breadth by 2 , 3 , or any other , and divide the length by the same number that you multiplied the breadth by . Or lastly , if you set out a double proportion that is 2 / 6 or 1 / 3 , from C towards D , and from the Point where it falleth , draw a Line to the Angle A , you will have a Triangle equal to 1 / 6 of the Square ACDN. But to return to Triangles , ( the most simple and primitive of all Rectilinears , and therefore the most considerable in this case of partition , as giving Laws often to the rest : ) It may fall out , that a Triangle must be divided ( convenience so requiring ) by a Line from some Point in a side , so as that Line may either be parallel to some other side , or not parallel to any . For the former case take this Example following out of Mr. Wing . Lib. 5. Prob. 5. Fig. 21 Let ABC be a Triangle given , and it is required to cut off 3 / 5 by a Line parallel to AB . First , on the Line AC , describe the Semi-circle , AEC , whose Diameter CA divide into five equal parts according to the greater term , and upon three of those parts ( the lesser term ) erect the Perpendicular DE , which cutteth the arch Line in E ; then set the Line CE from C to F , and from thence draw the Line EG parallel to AB ; so will the Triangle CGF contain 3 / 5 of the Triangle ABC , as was required . Fig. 22 Now for the latter case , when the line of partition goes not parallel with any side , take this Example : Let ABC be a Triangle given to be divided into two parts which shall bear proportion to one another , as 3 and 2 , by a Line drawn from the point D in the Base , or , Line AC . From the limited point D , draw a Line to the Angle B ; then divide the Base AC into five equal parts , and from the third point of Division draw the Line to E , parallel to BD. Lastly , from E draw the Line ED. So shall the Trapezium ABED be in content , as 3 to 2 , to the new Triangle DEC . I have now done with the Division of Triangles , when I have added these three Advertisements . 1. You must be sure to take very exactly the distance of every point , where a dividing line cutteth any side , to one of the ends of the same side , as in this last Figure , the distances BE and AD , which distances being applied to the Scale by which the Triangle was protracted , will shew at how many Chains and Links-end you are to make your dividing Line on the Field it self . 2. The proportions by which you are to divide , are not always so formally given as in the former examples , but are sometimes to be found out by Arithmetical working , as in this case . Suppose a Triangular Field of 6 Acres , 2 Roods , and 31 Perches , must be divided , so as the one of the two parts shall be 4 Acres , 3 Roods , and 5 Perches , and the other ( consequently ) 1 Acre , 3 Roods , and 26 Perches , reduce both measures into Perches , and the one will be 765 , and the other 306. Their Sum is 1071 , which by their common measure being reduced into their lowest terms of proportion in whole Numbers , will be 5 , 2 , and 7 , which shews that the Triangle being divided into 7 equal parts , the one must have 5 of those 7 parts , and the other 2. And observe , that it will be sufficient to find the common measure between the Sum of the terms and either of the terms ; the method whereof is shewed in every Arithmetick Book for reducing Fractions into their lowest terms . But if my unlearned Reader cannot skill of that work , he may multiply either of the parts ; ( as suppose 765 ) by the length of the Base , which we will suppose to be 8 Chains and 75 Links , or 875 Links : and that product divided by 1071 , ( the Content of the whole Close in Perches ) gives by the Rule of Three direct , 625 Links , or 6 Chains and 1 Pole , the true distance from either end of the Base , that his mind or occasions may direct him to begin with , to the point of Division ; for the Division must be not only for proportion or quantity , but also as to position or situation of parts upon the Paper , as it is required to be on the Ground . 3. In these and all other divisions of Land , where a strict proportion in quantity is to be observed , you must have respect to the Rules hereafter following , concerning measuring of uneven Ground , Chapter 15. especially if one part prove much more uneven than another : and if there be any useful Pond or Well to draw your Line of Division through it ; but if it be an unuseful Pond , Lake or Puddle ; or if there be an boggy or barren ground , that must be cast out in the divisions ; measure that first , and substract it from the Content of the whole Close , and then lay the just proportion of the remainder on that side that is free from it , that the other may have its just part also , besides that which is useless . What hath been said , with an ordinary measure of discretion , may sufficiently instruct a young Artist to divide Triangles , Parallellograms , and regular Polygonials , in an artificial way : but because many Closes and open Grounds are Trapezia's , and many irregular Polygons , and even those that are regular enough , may fall under an irregular division , in regard of the quality of the Land , Woods upon it , or Quarries in it : Or the conveniences of ways , Currents of Water , situation in respect of adjacent Lands , &c. I shall propose a Method , which though it hath somewhat of the Mechanick in it , will be singularly useful in such cases . Fig. 23 Let ABCD be a Trapezium to be divided betwixt a young Heir and his Mother , so as his part may be double to hers . Having by the Diagonal BC divided it into two Triangles , I find the Content of the Triangle ABC to be 138550 square Links , and the Triangle BCD to contain 103468 , in all 242018 square Links , which if reduced , as hath been formerly taught , would amount to 2 Acres , 1 Rood , and 27 Perches : ( for which see Chap. 5. ) But as to my present Work , they are in a better Order already . Dividing then 242018 into three parts , each of them is 80676 and 2 / 3 , Fig. 23 two therefore of those third parts must contain 161353 and 1 / 3 , which 1 / 3 being inconsiderable , I regard not . Then resolving to lay out the double part towards the Line BD , I strike at adventures the Line EF , and measuring the Trapezium bounded by that Line , and the opposite side BD , together with the Interjacent parts of the Lines AB and CD , I find it to contain 119140 square Links , but because it should have been 161353 , I substract 119140 , out of 161353 , and their difference is 42213 , and perceiving that the Lines AB and CD are very near parallel , and finding their distance where they are cut by the Line EF , to be 326 Links , or 3 Chains and 26 Links , I divide 42213 by 326 , and the quotient is 129 Links and almost half , at which distance I draw the Line GH parallel to EF ; so shall the Trapezium GBDH be the Heirs part . Another way whereby this may be preformed , is thus : Finding the Triangle ABC , to contain 138550 square Links , substract it out of the Heirs part , viz. 161353 , the difference 22803 , shews how many square Links must be taken out of the Triangle BCD , and added to the Triangle ABC : Which to perform with all necessary exactness ; suppose the side or line BD to be the Base , which by measure proves to be 344 Links , or 3 Chains and 44 Links . Say by the Rule of Three direct , If the whole Content of the lesser Triangle , viz. 103468 , give 344 ; what shall 22803 give ? so will the result be 75 links , and somewhat more than 4 / 5 of a Link ; for 22803 multiplied by 344 , gives 7844232 , which being divided by 103468 , the Quotient is 75 84132 / ●03468 ; or ( according to decimal Division ) 75.8131 , which is ( as I said before ) somewhat more than 75 Links and 4 / 5 , wherefore extending your Compasses upon the Scale to almost ' 76 Links , set that distance upon the Line BD , from B to I , and draw the Line CI : so shall the Trapezium ABIC be double to the Triangle ICD , within so small a matter as is not worth regarding , though the Land were a rich Meadow . I hope I need not stand to tell any man of sense , that if he please he may begin with the less part , and take out that : or if there be many Partners , he may divide betwixt any one and all the rest , ( putting their parts together ) and then by the same method subdivide amongst them till each hath his due share ; nor to spend many words in telling him he must substract where he hath by a separating Line at adventures , or by choosing , out a Triangle , taken too much . as I added , when I took too little . Nor lastly , that these methods are not only applicable to Trapezia's , but to any triangular Figure whatsoever , whether regular or irregular . CHAP. XII . Concerning the Boundaries of Land , where the Lines to be measured must begin and end . IF there be no agreement between the Parties concerned , ( for if there be , that must be observed ) Reason and Custom are the Surveyor's Guide . The Farmer speaks loudly , that when a piece of Arable or Meadow-land is let for a year to be sown or mown , no more should be measured nor expected to be paid for , either to the Letter or Workmen , than the Plow or Sythe can go over . So also when a parcel of Land is let for Pasture by measure to a Farmer , it seems very reasonable , that all and only so much should be measured as is useful to that purpose . But Commons to be enclosed are usually measured ( except it be otherwise agreed ) to the uttermost bounds of every mans particular proportion , without any allowance for Ditch or Fence ; every man being to make them upon his own of what breadth he pleases : Nor is this unreasonable , for 't is as good for one as another , and the rate paid to the Lord is usually very little , sometimes nothing . It is also very usual in measuring betwixt Lord and Tenant , in case of Leases for Lives , and long terms of years , to extend the Lines to the utmost bounds of the Tenants claim , taking in the very Walls , Hedges , and Ditches : but this is accounted very hard , and oft proves very unequal among the Tenants of a Lordship ; some being forced to make much more waste of their Ground this way , than others that hold as much or more . But where the Custom obtains , the Survey or must observe it : For it is others work to appoint what must be measured , and his only to measure truly what is so appointed . A good Landlord may ( and will be apt ) to consider it in his Rates , and a bad one 't is like will be tenaC●●us of a Custom to his own advantage . Lastly , In case of Sale by measure at a rate agreed upon per Acre ( no Boundaries being specified in the Bargain ) the Rule ( as I had it from an old famous Surveyor many years ago ) is to extend the Lines to the quick Wood-row , that is , as Reason prompts me to understand him , to the place where the quick-wood actually groweth , or where according to custom it ought to be set . I confess these things are trivial , but yet more necessary to be known , than many artificial things much stood upon , for a young Artist , whom ignorance of these things may expose to considerable mistakes in practice . CHAP. XIII . Containing a description of the Plain-Table , the Protractor , and Lines of Chords . THough what hath been already said , may competently suffice to instruct my young Artist in measuring a Close of Land , yet to advance him a degree higher in useful knowledge , I shall take occasion to describe unto him the Plain-Table , which with Mr. Wing I account the best of all fixed Instruments : This Instrument consists of several parts . 1. The Table it self , which is a Parallellogram of Wood fourteen Inches and an half long , and eleven Inches broad , or thereabouts , and for necessity may be made by an ordinary Country-workman of one Board : but for neatness , convenience of carriage , and freedom from warping , it is usually made of three little Boards joyned together side-ways , with a ledge at each end to hold them fast together , and upon the middle Board a Socket of Brass fixed with three Screws , and with a fourth to be fastned on the head of a three-legged Staff ; of which anon . 2. A Frame of Wood fixed to it , so as a sheet of Paper being laid on the Table , the Frame being forced down upon it , squeezeth in all the edges , and makes it lie firm and even , so as a Plot may be very conveniently drawn upon it : this is usually made with Joynts for more easie carriage , but a plain one may suffice . Upon one side of this Frame should be equal divisions , for drawing parallel Lines both longwise and cross-wise ( as occasion may require ) over your Paper , and on the other side the 360 degrees of a Circle projected from a Center of Brass conveniently placed in the Table . 3. A Box with a Needle and Card , to be fixed with two Screws to the Table , very useful for placing the Instrument in the same Position upon every remove . 4. A three-legged Staff to support it , the Head being made so as to fill the Socket of the Table , yet so as the Table may be easily turned round upon it , when 't is not fixed by the Screw . 5. An Index , which is a large Ruler of wood , ( or rather brass ) at the least sixteen Inches long , and two Inches broad , and so thick as to make it strong and firm , having a sloaped edge , ( by which we draw the Lines ) called usually the fiducial edge , and two sights of one height , ( whereof the one hath a slit above , and a thread below , and the other a thread above and a slit below ) so set in the Ruler , as to be perfectly of the same distance from the fiducial edge . Upon this Index 't is usual to have many Scales of equal parts , and there might be a Diagonal Scale if the Instrument-maker please , and Lines of Chords of sundry lengths : but if you have such a Scale as I before described ; you need not to have them here . The Protractor is an Instrument so well known , and so easie to be made and procured , that I shall be very brief in the description of it . As it is usually made , it consists of two parts , a Scale and a Semicircle , but the Scale is no necessary part of it , but serving ( if you be not otherwise provided ) for other uses before mentioned in the case of Plain-Scales . But the Semicircle is more essential , and it may be made of Brass or other Metal of any convenient size , as four Inches ( more or less ) for the straight sid , this Semicircle being bounded as all others are by two Lines , the one right or strait , the other circular . The right Line is divided in the precise middle by a Point which is in the Center , upon which the Circular boundary is drawn , and two other Arches concentrial with it . The Center , when the Semicircle goes alone without the Scale , should be guarded with two little lips , on each side one , or a little loop , for more convenient turning of the Instrument about upon a Pin fixed in a Paper . The arched or circular edge is divided into 180 Degrees or equal Parts numbred by Tens upon the upper Concentrick Arch , from 0 to 180 , and in the lower from 180 to 360. So that by applying the straight edge of the Protractor twice to any Line , ( keeping the Center right● upon a Pin fixed in the Line ) that is , with the Semicircle first above it , and then below it , or contrarily , you may draw a whole Circle by the guidance of the Arch , or set out any number of Degrees , as will appear more plainly hereafter . A Line of Chords is a Line divided into 90 unequal parts , whereof 60 , and the Radius upon which the Circle was drawn , are equal , and the Divisions upon that Line are equal to the next Extent in a right line , of so many Degrees from the beginning of the Quadrant as answer thereunto . When lines of Chords are cut upon wood , 't is both usual and necessary that there be two Studs of Brass , the one at the beginning , and the other at 60 Deg. with little holes for the feet of the Compasses , when you take the extent of the Radius , to preserve the line from being wounded by the Compasses ; and being thus fenced , it will for need do the work of a Protractor , but not altogether so commodiously . CHAP. XIV . How to take the true Plot of a Field by the Plain-Table , upon the Paper that covers it , at one or more Stations . THere are three ways or methods for doing this work , two more usual and ordinary , the third more unusual and extraordinary , though now pretty well known to most Surveyors , and in late Books published . The first performs the work by measuring every line from the Instrument to every Angle , and is a very sure substantial way where it can be done , as it ordinarily may in most Closes . The second doth it by measuring only the Station or the Distances , and is very quick , but not so sure and exact as the other ; yet if managed by a skilful Artist , that knows how to plant his Instrument , so as to avoid making acute Angles unnecessarily , it will come near enough the matter in many cases ; as in measuring for Workmen , that take the Mowing or Reaping of Fields by the Acre , or when Tythes are let at a small value per Acre , as in poor barren Parishes they usually are . The third is the way of Circuition or Perambulation , the Instrument being oft to be planted , and the Plot to be measured about , by which not only difficult Closes , but even the thickest Woods , yea Bogs , Meres , and Pools of Water , may be plotted , which by neither of the other methods can be performed . In all these Methods , two things are to be performed : 1. At every Angle where there is no perspicuous Mark already , as a Tree , Bush , Stile , &c. one must be placed , as a white Paper , or such like ; or else some one must go from Angle to Angle , and remain there as your Mark to look at , till you bid him remove to another : only when Angles are very near you , this labour may be spared . 2. When ever you have occasion to plant your Instrument more than once , ( as it will often fall out in the first Method , and ever in in the two latter ) you must be sure it stand just as it did the first time for situation , for which your Needle if well touched and hung , will be good direction , but is not thought sufficient without back sight and fore-sight , ( by some practical Surveyors : ) I shall therefore in due season shew you that knack . Now for the first Method . I. When you go about to plot your parcel of Land , find such a place in it if possible , from whence you can see all the Angles , and in that place plant your Instrument covered with a sheet of Paper , and turning it about till the Needle playing at liberty , hang over the Flower-de-luce , ( or any other notable place that you make choice of ) screw it fast . Then choosing any convenient place in your Paper for a Center , and to represent your station ( or place where you fix your Instrument ) make a prick with the small point of your Compasses , to which prick applying the fiducial edge of the Index , ( which is easily done if you keep the Point of the Compasses resting in it ) direct the Index by the sight to all the Angles , and when through the slit or long sight , you see the opposite Thread cut the Mark in the Angle , draw a neat dry line along the fiducial edge to or from the Center : then measuring from the Instrument to every Angle , set the measure by a Scale and Compasses from the Center towards the Angle , upon the line that points at it , making a prick in the line where the Chains and Links ( reckoned from the Center ) do truly end : then shall Lines drawn by a straight Rule from prick to prick , give you the perfect plot upon your paper , which you may divide ( as hath been before shewed ) into Trapezla's and Triangles , and to find the true Content . To make which plain , mark this Example . Fig. 24 Suppose ABCDEFGH to be a Field ; having planted my Plain-Table as before directed at a convenient advantage , so as to see all the Angles , ( as at I ) I make a prick to represent my station in the little Circle ☉ marked with I , upon which laying the fiducial edge of my Index , and directing the Sights to all the Angles , I draw dry lines toward A , and all the rest of the Angles in order from the Center ; and then measuring upon the ground from the Instrument to the Angle A , I find it to be 3. 45 , which I set ( by the help of my Scale ) from the Center to the Point A , and so upon all the rest according to their due measures , and then black lines drawn from Point to Point , as from A to B , from B to C , &c. limit the true Figure of the Field according to the Scale I used , viz. of 400 in an Inch. And now before I pass to further varieties , let my Reader take notice of these following things . 1. From henceforth I shall forbear ( for brevities sake ) to take any notice of the measures of lines measured from the Instrument to the several Angles , having so often shewed how to measure by a Scale . 2. When I speak of measuring from or to the Instrument , I always mean from or to that part of ground that is perpendicularly under the head of the Instrument , where you are to draw your Plot , which will ever be enclosed with the three Legs of your Staff. 3. That it 's usually the quickest way to measure first from the Instrument to the first Angle , and then back from the second Angle to the Instrument , and so the rest in order , still one from the Instrument , and the other to it . 4. It is no matter at all whether your Plain-Table be placed towards the middle of a Field , as was represented in this Figure , or at an Angle , as will appear anon . 5. In all workings by this Instrument , you must have a care that the Instrument be not moved out of its due place , till you have finished the work of the present Station ; for which purpose , cast your eye now and then upon your Needle , observing whether it continue to hang directly over the same point you set it at when you began your work , and to rectifie your Instrument if you see cause . But because all Tables have not Needles , and where Needles are , they are not accounted over-trusty , make use of the following help . When you have planted your Instrument , and made a point or prick in your paper , representing your station , set the fiducial edge of your Index to it , and turning it softly about till you find one remarkable thing or other upon one side of the Close , and another on the opposite side as you look through the sights of your Index ( which we call fore-sight and back-sight ) draw a remarkable Line with Ink , or rather with a Black-lead Pen quite over your Paper , which in this Figure is represented by the black line KL : and then if you suspect that by any accidental jog , or other casuality , the Instrument is any thing removed , you may easily try and rectifie it , by applying the fiducial edge to the same Line , and making use of fore-sights and back-sights again , upon the same Marks which you before observed upon the opposite sides of the Close . But if there be no convenient place for the placing of your Instrument , whence you may see all the Angles of the Field , more stations must be made use of thus : Fig. 25 Let ABCDEFGHIKLMN be a Field whose Angles cannot be all seen from any one Angle , or other place in it : I plant my Instrument at the Angle A , and if it have a Needle , I mark what Degree of the Chord it cuts , or turn about the Table on the head of the Staff , till the Needle hang over over some remarkable place ; ( as suppose the Flower-de-luce ) and screw it fast ; then setting up a Stick with a white paper or cloath on the head of it , where I intend my second Sation ( as here at Q. ) I make a prick or point in my paper , to signifie the point A upon the paper on the Table : to which point I apply the fiducial edge of the Index , and when I see the white at Q , so as looking through the slit , I see the thread , cut it , I draw the Line OP quite through my Paper with a Black-lead Pen , and then keeping the fiducial edge still upon the same point , and turning it round by degrees , I look at the Angles BCDEFLMN , still drawing dry Lines with the points of my Compasses , and setting on the Measures from the Station A to every Angle measured to or from , as I did in the last Example . Then I remove the Instrument to the place of my second station , having set up a mark at A , and laying the fiducial edge to the Line OP , I turn about the Table upon the head of the Staff , till through the slit of the back-sight , I see the thread cutting the mark at A , and then screw it fast , so will my Needle , if a good one , hang directly over the same point that it did at the first station ; but however that be , fore-sight and back-sight will do the business ; for which purpose it is good to take back-marks as well as fore-marks at every station , as was taught in the Example of a single station , only taking notice that the back-mark when the Instrument is planted in an Angle , must needs be out of the Field ; as suppose here at O. But to proceed . Having measured the distance between the first and second station , and finding it to be 7. 10 , I set it upon the Line OP , from A to Q , where I make another point to represent the second station , and turning about my Index with the fiducial edge upon that point , and so looking through the sights at the Angles GHIK , I draw Lines towards them on my Paper , and having measured between every one of those four Angles , and the Instrument , I set those Measures as I did the other , with my Scale and Compasses , from Q towards every Angle upon his proper Line : and then having drawn the black bounding-lines from A to B , from B to C and so round about the Close , the Protraction is finished . But here to make this Figure yet more advantageous , let me ( according to my usual method ) add some Advertisements . 1. Sometimes a Station is so taken , that you may measure towards two Angles at once , ( as here from Q to G and H ) in which case you are to set down the Chains and Links where the first Angle falleth , but still be proceeding to the further Angle , causing the remainder of the Chains at the fore-end to advance beyond the former Angle , so going on with whole Chains so far as you can , to which the odd Links at the end are to be added . 2. If at any of your Stations ( as suppose A ) you can see an Angle ( for example E ) to which you cannot measure in a direct Line without passing the boundaries of your parcel of Land given to be measured ; you may notwithstanding take in that Angle by a strait measured Line , as I have done , provided it may be lawfully done without trespass , and conveniently without troublesom passing of Fences , otherwise it must be taken from another station . 3. I here took one of my Stations at an Angle , and the other within the body of the Field , to shew the variety of working taught by other Authors , and that 't is no great matter where you make your Stations , so you can see the Angles : else it had been full as convenient to have taken my first Station also within the body of the Field , as suppose at R. 4. Though this Figure representeth to your eye only two Stations , A and Q , your fancy may multiply them at pleasure , for suppose the Angle H could not have been seen from A or Q , how easie had it been to have set up a Mark at S , and then to have removed the Instrument thither , observing the same directions ●hat were given at the removal from A to Q. II. In the second Method the Instrument is to be planted twice , or oftener as occasion is , the Rules for removal of the Instrument fore-sight and back-sight , and measuring the distance of Stations , being the same as formerly was ●aught : but instead of measuring to and from ●very Angle , we only view each Angle through ●he Sights from two Stations , having applied ●he fiducial edge to the Points representing ●hose Stations , and having drawn Lines with the point of the Compasses , or a protracting Needle , the Interfections represent the Angles , from which the boundary-lines may be drawn , so is the Field protracted : Which that my Reader may understand , let him note these three Figures . Fig. 26 Here in these three Figures the Angles are marked Alphabetically , ABCDEF , &c. and ●he Stations by a point in a small Circle numbred , 1 , 2 , or 1 , 2 , 3 , according to their number and order . Fig. 26 The first of these Figures represents the Plot●ing of a Field at two Stations within it , from both which all the Angles may be seen . Fig. 27 The second , performs the same work by two Stations taken without the Field , by which 〈◊〉 a Close may be measured , though the present Possessor will not give us leave to come ●nto it . Fig. 28 The third , shews how the Work may be performed at three Stations or more , when two such places cannot be found whence to view al● the Angles ; which last having more of difficulty than the two former , ( though indeed no● very much ) and the Explanation of that will sufficiently help to the understanding of them ▪ I shall a little explicate the meaning of it in these particulars . 1. From the first Station taken acording to former directions , I see the Angles ABCD FGK , and acordingly draw Lines upon my Paper towards them from the point representing that Station , by the fiducial edge of my Inde● with the point of my Compasses . 2. Having removed my Instrument to the second Station , ( and in so doing , observed the Rules before given , touching such removals ) I thence see the Angles ABDEFGHIK ▪ and draw Lines upon my Paper towards them from the point representing the second Station . And now viewing my Work , I find upon my Paper Interfections for the Angles ABDF GK , but only single Lines toward the Angles CEHI : therefore . 3. Removing the Instrument regularly as before , to a third Station , I thence see those four Angles CEHI , and drawing Lines towards them , I have interfections for them also ; so that having drawn the Lines AB , BC , &c. from one Interfection to another , I have the Field perfectly protracted . For these bounding-lines from Angle to Angle , do not only signifie the Boundaries of a piece of Land given to be measured , limiting the Figure or shape thereof , ( and are to that purpose given in this and all other Survey-books ) but also are the true distance by a Scale from Angle to Angle for the Plot upon the Paper : I mean by the same Scale by which the stationary distances were laid down upon their own lines . And this holds true in all kind of true plotting , whether in this Method or any other . III. The third Method is that of Circuition , and this hath several varieties , according to these three following Cases . 1. When the distance from Angle to Angle , ( without any exception ) is measured quite round the Plot , either within or without . 2. When the distance is taken only between some more notable Angles , and the Perpendiculars of the rest measured as you pass along their Bases , within the Plot , proper for plain solid ground . 3. When the like is done without the Plot , as in the case of Plotting thick Woods , Meres , Pools , Bogs , &c. The first of these is very easie , consisting in nothing but planting the Instrument at every Angle ( either within or without , as necessity and convenience determine it ▪ ) observing the former directions for planting and removing the instrument ; and also for measuring the stationary-lines on the ground , and protracting them on the paper , as is manifest in this Example . Fig. 29 Let ABCDEF be a Park-Pond or Close to be protracted , I first plant my Instrument at A , and direct the sights to a Mark in the Angle B , drawing a dry Line from a convenient Poin● on my Paper towards B on the ground , and having measured by my Chain the distance AB I set it by a Scale upon the Correspondent Line from A to B , drawing a black-line between them with Ink or a Lead-pen , the Extremities whereof are the Points A and B on my Paper , and the little pricked Line that goes beyond B , represents the remainder of the dry Line drawn at random ( as to length ) with the point of the Compasses . Then setting up a Mark at A ( if there was none before ) I remove the Instrument to B , and laying the fiducial edge to the Line AB , I turn about the Instrument upon the Saff , till through the sight I perceive the thread cutting the Mark at A , and my Needle ( if I have one ) directly over the same point , that it was when it was planted at A , and so screw it fast . My next work is to lay the fiducial edge to the point B , and direct the sights to C , drawing a dry Line towards it , and setting the distance BC measured by the Chain from B to C. In this manner I proceed , surrounding the Close till I come at last to A , where I began , by planting the Instrument at every Angle , using the help of Back-sight and my Needle , as I did at B , and then from the point representing my present station , directing the sight to the next Angle , as I did from B to C. In the second Case , we do not plant the Instrument at every Angle , but at the more considerable , taking in the smaller by their Perpendiculars from the Base as we pass along , of which this following Figure may be an Instance . Fig. 30 Let ABCDEFGHIKL be a Close to be measured ; by planting the Instrument only at ACF and K , I have the main substance of the Close in the Trapezium ACFK , and for the five small Triangles which must be added to the Trapezium , they may be easily protracted by the help of such a little Square as was mentioned towards the latter end of the Tenth Chapter ; for thereby finding at how many Chains and Links distance from A upon the ground , the Perpendicular B b falleth upon the Line AC ; and having measured the length of that Perpendicular , and taken it between my Compass-points off my Scale , I erect a Perpendicular of that length at b , which is the point upon the Paper , where so many Chains and Links determine , as were measured upon the Ground , from the Angles A , to the place where the Perpendicular fell on Ac , viz. at b. Just in the same manner I raise the other Perpendiculars , Dd , Gg , Ii , and Ll : and then by the help of the Perpendiculars , I draw ( from and to the proper Angles ) the Boundary-lines AB , BC , CD , DE , FG , GH , HI , IK , K L , LA ; which together with the Line EF between the Angles E and F , give the true Plot of the Field in one large Trapezium , and five small Triangles ready for casting up . The third case is so like the second , that there needeth no new Direction concerning it , but to annex one plain Diagram ; all the difference consisting in this , that because we cannot go within it , ( being supposed to be some Pool , Bog , or Thicket ) we must of necessity go on the out-side , and consequently all the Triangles made by inward Angles , and their Lines upon the measured Bases , must be excluded by the boundary-lines from being any parts of the Plot , as here is manifest . Fig. 31 Supposing ABCDEFGHIKLMN OP , to be a great Pool , though here be fifteen Angles , I plant my Table only five times , viz. at AEFH and M , and upon the dry Lines AE , FH , HM , and MA , I raise their Perpendiculars in due places , ( according to measure ) and also of a right height : by which and my five stationary Angles , I draw the bounding-lines of the Plot , excluding all the Triangles as foreign to it , they being no resemblances of any part of the Pool , but of Land adjacent . Where note , 1. That both in this and the former Case , such a little Square as I mentioned in the Second Chapter , will be very useful for speedy raising of Perdendiculars ; but where the Triangles are very small , it needs not be used , not the other mentioned Chap. 10. 2. That if by reason of troublesom Brush-wood , Gorse , or Bogs , &c. I could not have measured close to the sides EF , HI , or LM , it would be the same thing if I went parallel to them . And this is a shift that the practical Surveyor will oft be put to make use of , in other cases as well as this . CHAP. XV. Concerning the Plotting of many Closes together , whether the Ground be even or uneven . THough I design not so high in this Manual as to make my Reader able to Survey Lordships and Forests , much less to draw Maps of Countries , but to measure a parcel of Land with truth and judgment ; yet I would have him so expert , as not to be puzzled , if any should desire him to draw ( as it were ) a true Map of a Tenement or small Demesn , consisting of several Closes ; for which purpose , let him that knows no better observe this Method . Fig. 32 Suppose ABCDEFGHIKLM to be a Tenement or small Demesn divided into fourteen Closes , to be measured and protracted according to their several shapes and situation . I first draw the Plot of the whole by the Method of Circuition , planting mine Instrument either at every Angle , or only the most considerable either within or without , as I find most convenient . This being done , a Line from B to M gives the Triangle ABM from the first Close : In the next place , I go round the second Close beginning at M , then to B , and so about ( cum scle ) to M again : And then for the third Close I plant my Table at C and go round to B , ( the Line BC being protracted already ) and so of all the rest , still observing which are common Lines belonging to several Closes ( representing the Fences ) that I may avoid the trouble of measuring ●hose Lines oftener than once , and lay ever● part of ev●●y Close in its due place ; and that I be su●● to keep the Instrument throughout the whole Work to its true Position by Needle , fore-sight and back-sight . There are I confess divers other ways of doing this Work , but none more sure or plain , especially if the ground be uneven ; for in that case , if you protract according to the length of Lines measured from your Station to the Angles , you will put your Closes into unproportionable shapes , except you reduce Hypothenusal Lines to Horizontal , by instruments or otherwise ( which is somewhat troublesom : ) and the like may be said when you Plot with the Chain only . Indeed the Method of measuring only the stationary distances were very proper for setting out the Figure of each particular Close , provided the distance of the stations be large , and taken ( if possible ) upon pretty even ground , ( which sometimes may be done , though most of the Close be uneven ) and the Work so ordered , as not to make too acute Angles : but because this requireth skill and care , I rather advise my young Artist to use the circling way , as ordinarily most commodious in my poor judgment , ( but not prejudicing other men's that may differ from me in opinion ) and where need requires let him observe the directions in the 17 Chapter . But which way soever you go to work , there is one very necessary Rule to be observed . If the Ground be uneven considerably , you must not give up the Content by measuring the Bases and Perpendiculars of the Triangles on the Paper by your Scale ; but you must measure the Lines correspondent to them on the Ground , and cast up the Content according to that measure . And if it be desired that you should adjoyn to your Plot ( as is usual ) a Scale of Chains to measure distances by ; you must either by making the Forms of Hills erect and reverse , or some other Note in writing , mark out your uneven Ground , lest those that try it by the Scale , judge your work erroneous : for though you make that Scale exactly correspondent to that you protracted by ( as you ought to do ) the Hills and Dales in the ground truly measured , may make a considerable alteration . It is convenient when you plant your Table that the Needle hang just over the North-point of the Compass under it in the Box ; then may you by the Lines overthwart the Frame of the Table , easily draw two Lines quite through the Plot , cutting one another at right Angles , the one pointing at North and South , and the other at East and West . And if your skill serve you to make the Two and thirty Points of the Compass upon the place where they intersect , and to draw the Forms of the Houses , Woods , and other remarkable things upon the Demesn , and the course of Brooks and Rivers running through it , it will add to your commendation . And so it will also , if you take in such parcels of Land bounding it , whether common or peculiar to other men , as will make your Plot to look handsomly , like a perfect Square or Oblong . But however that be , you must be sure to protract truly all Lanes going into it , or through it , and all Closes of other mens mixed with it ; and also all considerable Ponds , Ways and Outlets , with the Names of the Closes and quality of the Ground , whether Meadow , Pasture , Arable , &c. CHAP. XVI . Concerning shifting of Paper . IN such work as that of the last Chapter , it may sometimes fall out ( through the multitude and largeness of Fields ) that one sheet will not hold your whole Plot , in which case you may help your self by shifting Paper ( as we call it ) thus . Fig. 33 Let ABCD represent our sheet of paper that covereth the Table , upon which the Plot of the large piece of Land EFGHIK should be drawn ; having made my first station at E , and the second at F , I find my paper will not receive the Line FG : but however I draw so far as it will go to the edge of the paper , and planting my Table again at E , proceed in my Circuition the contrary way to K and I , where I find my self again at a loss for my Line IH , but draw it also to the edge of the Paper ; Then with the Point of my Compasses striking the Line PO , parallel to the edge of the Paper BC , and the Line QO parallel to DC , and cutting PO in O , I throw aside that paper for a while , covering the Instrument with a new one , which I mark with the figure ( 2 ) for my second sheet . Fig. 34 Upon which second sheet ( the leading part whereof is represented by the three Lines meeting in the Angular points A and B ) I draw PO parallel to AB the leading edge of the paper , and crossing it at right Angles in the point O , by a parallel to BC , viz. the Line OR , being of the same distance from BC , that QO in the former sheet was from DC . Then with a Rule and a sharp Pen-knife I cut off the end of the first sheet at the Line PO , and applying the edge of it to the Line PO of the second sheet , so as it may touch that Line all along , and the Line QO of the former , touch the Line OR in the latter , so as to make one Line with it : I draw the Lines PG , being the Remainder of the Line FG , and the Line OH being the remainder of the Line IH , and from their extremities the Line GH . And if the Plot required it , you might proceed on in the second sheet , and annex a third and a fourth , &c. as there is occasion . These sheets may be pieced together with Mouth-Glew or fine Paste , applying the edge of the former ( as you did upon the Table ) to the Line PO of the latter . And note here once for all , that when I speak of applying the edge of the paper to a Line , I mean the precise edge cut by the Line PO ; but when I speak of drawing Lines to the edge of the Paper upon the Table , I hope none will think me so absurd as to mean the edge that is couched under the Frame , but that my meaning is , that the Lines must be continued on the paper till they touch the Frame . CHAP. XVII . Concerning the Plotting of a Town-Field , where the several Lands , Butts , or Doles , are very crooked : VVith a Note concerning Hypothenusal , or sloaping Boundaries , common to this and the Fift●enth Chapter . Fig. 35 SUppose ABCDE divided in the manner of a common Field , into seven parts or Doles , belonging to seven several men : First Plot the whole as before hath been taught , then measuring from A to B upon the Land : set one Note down ( as you go along ) at how many Chains or Links ( or both ) the Division is between Dole and Dole , and accordingly mark them out by the help of Scale and Compasses in the Line AB on the Paper plot . In the very same manner you must measure and mark out the Lines OC and ED ; which being done , take the Paper off the Instrument , and laying it before you on a Table , with the side AE towards you ; the Compasses must be so opened and placed ( as by a few tryals they may ) that one foot resting upon the Table , the other may pass through the Points of Division upon all the three Lines , viz. AB , OC , and ED , as in this Figure they do . If the Content of any one or more of these Parts , Butts or Doles , be desired without Plotting ; it may easily be done without your Plain-Table thus : Take the breadth by your Chain at the head , middle , and lower end , and adding these Numbers together , the third part of their Sum is the equated breadth : by which multiplying the length measured down the ridge ( or middle ) the Product gives the Content . But both in this case , and that mentioned in the 15th Chapter , the Figure of a Plot may be somewhat disordered , not only by the unevenness of the ground within , for which I have given due caution already ( that being both the more common and more considerable case ) but also by the great diclivity of the Ground where the boundary-lines go , either of the whole Plot or particular parcels . For whereas in Plotting , every Line is presumed to be Horizontal ( or level ) that it may pass from Angle to Angle the shortest way , and that every part may be duly situated , and none thrust another out of its right place : If it be not level , but falling down towards a Valley , or rising up Hill , or compounded of both ; a Line over such Ground ( though true for the measure , and for giving up the Content ) will be false as to the Plot , and therefore must be reduced to a level , and so taken off the Scale and protracted . For the doing of this there are several Instruments very proper , especially Mr. Rathbourne's Quadrant upon the head of his Peractor ( though it were better to have a Semicircle than a Quadrant so placed ) and divers others . But supposing my Country friend to have no other but such as I have already described ; I shall shew him a plain easie way much used by practical Surveyors , especially in Ireland , as some of themselves have told me , being the very same that he may meet with in Mr. Leybourn's Book , Intituled , The Compleat Surveyor ; I mean the second way by him discovered . Fig. 36 Suppose ABC to be part of an Hill falling within my plot , my Boundary-line going crookedly from A to B , following the Surface of the Ground . To find the Horizontal Line ( equal to AC ; ) I cause one to stand at the point A ( the foot of the Hill ) and to hold up the end of the Chain to a convenient height , and gently ascending the Hill , I draw it level and make a mark where it toucheth the Hill , observing the number of Links betwixt mine assistant's hand and that place , where he must take his second standing , and hold it up as before , and so I draw it out level again till it touch the Place , where he must take his third standing , noting the Links as before , and so proceed , till at last from his fifth standing I draw the Chain level to the highest point within my plot , viz. the point B. And now as the pricked Lines of this Figure put together , and evidently equal to the Line AC : So are the Links noted down at every Station , when summed up , equal to the Horizontal Line of that part of the Hill. In the very same manner , only inverting the Order , you may find the Horizontal Line going down-hill , where that is most convenient : And if there be both Ascents and Descents in one Line betwixt two Angles , the Horizontal Lines of both must be found and joyned together in Protraction . All this concerning Declivities of rising or falling Ground , is to be understood when they are considerable , and a very exact plot required : for small ones , especially when much exactness is not expected , are not regardable . CHAP. XVIII . Concerning Plotting a piece of Ground by the Degrees upon the Frame of the Plain-Table several ways , and Protracting the same . HItherto I have shewed the use of the Plain-Table as such , and I think my Directions have been near as plain as the Instrument it self : At which some quarrel for its over-plainness , exposing the Art to proud ignorant people , who judging the rest of the Surveyors work to be as easie as looking through sights at a Mark , and drawing lines by a Rule , are apt to undertake to use it , or slight the skill of such as do . Others say , ( and that truly ) that for vast things , such as Forests , Chases , &c. the Circumferentor is more proper : And every one must grant , that in wet Weather , either that a Peractor , a Theodolite , or Semicircle , must needs be better than the Plain-Table covered with the paper which cannot endure wet . Hence it is that some Artists have to good purpose shewed how the Box screwed to the Index , and that made to turn on the head of the three-legged Staff become a Circumferentor . And if these thus fixed be turned about upon the Center of the Table ; they will ( say some with good reason , Mr. Leybourn for one ) perform the work of the Peractor , much better than the Peractor it self . Others shew , ( as I shall briefly ) that taking the Instrument as it is without the charge of further fitting it , or trouble of removing the Box , the Index turned upon the Center will by help of the Degrees on the Frame , perform the work of the Theodolite , to which the Semicircle is near of kin . And though I might easily answer all these Objections , by saying the first is frivolous ; such foolish Arrogance being easily contemned or checkt ( if worth the while ) by putting such conceited Fools upon the harder part of the work . The second impertinent to our purpose , who design not to plot such vast parcels of Land : And the third concerning only an extraordinary case , and that well provided for otherways , for sure no man that hath not a Body of the same Metal with his Instruments , will ordinarily measure Land in continual Rain , ( a sudden shower may be fenced against by a cover : ) and if any be so eager upon his work , I have shewed ●how it may be done in the former Chapters of this Book , without planting any Instrument at all , by Chain , Scale , and Compasses alone : Yet I shall shew how the plot of a Field may be ta●en by the Degrees on the Frame not every way that I could imagine , nor that I could transcribe , ( for that would be tedious ) but two ways only , whereof the one is proper for an ordinary Close , where all the Angles may be seen from 〈◊〉 Station within it , the other fitting any par●●l of Land though much larger , whatever be the Figure of it . For the former take this Example Fig. 37 Let ABCDE represent the Figure of Field to be plotted by the Plain-Table in rain Weather : I put on the Frame without a Pap●● the graduated side upwards , and plant it i● some convenient place , whence I can see a●● the Angles , as at O ; then placing the Index upon th● Table , so as the fiducial edge do 〈◊〉 the same time go through the Center upon the Table , and the Lines upon the Frame of th● Table cutting it perpendicularly at 360 , ( whe●● the Degrees begin and end ) and 180 ( the 〈◊〉 act half ) I turn about the Table upon the Sta●●●head , till through the Sights ( the Side marke● with 180 being next mine eye ) I see th●● Angle A , and then screw it fast , observin● where my Needle cutteth , and by back-sigh● causing a mark to be set up in the Line CD 〈◊〉 the point F , that the Instrument may be ke●● firm from moving ( or be rectified if it be moved ) during the Work. And now the Li●● AOF passing upon the Land from the Ang●● A , directly under the Sights of the Instrume●● to the Mark at F , is ( as it were ) the pri●● Diameter whence the Degrees of the Angles 〈◊〉 to be numbred , and accordingly I mark th● Angle A in my Table hereafter to be exempl●●fied with 360 Degrees . But to proceed , turning my Index with the fiducial edge upon 〈◊〉 Center , till I see the Thread cutting the Ma●●● at B , the said edge cuts upon the Frame at 〈◊〉 Deg. 15. Min. which I note down for that Angle : The like work I do , turning the Sights to CD and D , ( but not to F , for there is no Angle , but only a Mark in the Boundary ) and I find mine Index to cut for every Angle as I have marked them within the pricked Circle of the last Figure , viz. 157 Deg. 35 Min. for C , 225 Deg. 20 Min. for D , and 278 Deg. and 50 Min. for E. Then I measure ( or cause to be strictly measured by others ) the Distances betwixt the Place where the Instrument stands , and every Angle , and find them to be as I have set them upon the pricked lines in the little Circle , viz. A4 Chains ●0 Links , B4 Chains 3 Links , C3 Ch. 84 Li. D5 Ch. 35 Li. E5 Cha. 6 L. And now my Table both for Lines and Angles is thus perfected , and the Work is ready for Protraction within Doors .   D. M. C. L A 360 00 4 20 B 76 15 4 03 C 157 35 3 84 D 225 20 5 35 E 278 50 6 06 Your judgment will easily inform you , that in such weather we shall hardly stand to make our Table neat and formal , but any thing ( how rude soever ) that we can understand , doth the feat . A Welsh Slate with a sharp Style , ( or for want thereof , a Black-lead Pen and a smooth end of an hard Board like a Trencher ) is more convenient at such a season than Pen , Ink and Paper . But of all I would commend for expedition a Red-lead Pen , whereby you may mark out every Angle neatly with one touch upon the Table it self , just where it toucheth the Frame , by help of the fiducial edge , and close by it the length of the Line from the Center to that Angle : All which may be easily cleared off by a wet Sponge or Cloath so soon as you have protracted . Or if through the sponginess of the wood , the head of the Table ( which we use to cover with paper ) were made a little reddish , what great harm were that ? We are forced to do it more real wrong by the points of the Compasses in the ordinary way . Now to protract our Observations : I draw upon a paper the Line AF at adventures , so it be long enough , and stick a Pin in it at pleasure for the Center O , upon which I place the Center of the Protractor , so as the straight side ( or Diameter ) of the Protractor may just lie upon the Line AF , the Limb or Arched-side being upwards towards B , by help whereof I make a prick or point on the paper at 76 Deg. 15 Min. for B , and at 157 Deg. 35 Min. for C , according to the numbers nearest to the Limb. Then turning the Protractor about on the Pin with the Arch or Limb down towards D and E , till the Diameter lie again just upon the Line AF , I number downwards from the right hand towards the left , by that rank of Figures that are nearer to the Center , beginning 190 , 200 , &c. and over against the places where 225 Deg. 20 Min. and 278 Deg. 50 Min. fall , I prick the Paper at the side of the Limb , and through those four points I draw so many several Lines , ( having laid aside the Protractor ) upon which and also upon the Line AO , I mark out by Points the true measure of every Line ( by a Scale ) from the Center , and from those points drawing the Lines AB , BC , CD , DE , and EA , I have the true Plot of the Field . Where note by the way , that we estimate Minutes as well as we can both upon the Frame of the Plain-Table , and the Protractor , accounting half a Degree , 30 Minutes ; a third part , ●0 ; a fourth part , 15 , &c. And though by this means it is impossible to avoid small errours , 't is easie to avoid sensible ones ; and the like may be said when we protract by a Line of Chords , of which I now come to treat . Having proceeded in the Field as before , and made my Table for Lines and Angles , or done that which is equivolent by a Red-lead Pen , I draw the Line AF , and having extended my Compasses to the Radius ( or 60 Degrees ) on a Line of Chords , I set one Foot towards the middle of the Line AF , and with the other I describe a Circle like that in this Figure of a five-angled Field , but much larger , according to the length of the Radius : Then extending the Compasses from the beginning of the Line to 76 Deg. 15 Min. I set one foot in the Intersection of the Circle by the Line A , and with the other foot make a mark in the Circumference of the Circle upwards towards the right-hand , and through it draw the dry Line BO . In the next place I substract the angle 76. 15 from 157. 35 , where the Index cut for the Angle C , and there resteth 81 Deg. 20 Min. which I take off the Line as before , and set it upon the Circumference from the Intersection by BO , towards the end of the Diameter marked with F , and through the Point were it falleth , draw the dry line CO. In like manner I subtract 157 Deg. 35 Min. from 225 Deg. 20 Min. and the difference is 67 Deg. 45 Min. which I set from the Intersection by the Line CO downwards past the prime Diameter AF , and through the point where it falleth , draw the Line DO . Lastly , Having subtracted 225 Deg. 20 Min. from 278 Deg. 50 Min. there resteth 53 Deg. 30 Min. which must be set downward towards the left-hand from the Intersection by DO , and through the point where that falleth , I draw the Line EO . And now when I have set the particular Measures upon every Line , and drawn the Boundary lines , as I must have done if I had used a Protractor , the Plot is finished . But for better assurance that I have done my Work well , I take the measure of the remaining Angle AOE upon its proper Arch , viz. from the Intersection of the Circumference by AF , to the Intersection by EO , and applying it to the Line of Chords , I find it to be 81 Deg. 10 Min as it ought to be , for it should be the Complement of 276 Deg. 50 Min. to 360 , and so it is , And for further satisfaction , I sum up the Degrees and Minutes of all the five Angles , which for plainness sake I have noted in every one of them on the outside of the Circle in the Figure so often referred to , and their sum is 360 , as it ought to be , and as here is evident . 76. 1● 81. 20 67. 4● 53. 3● 81. 1● 360. 00 My Reader may now perhaps expect that I teach him how to take a Plot at two or more Stations , when all the Angles cannot be seen from one : But because this is so easie from the grounds already laid , to any that is Ingenious , and in part rend●red unnecessary by the Method presently following , I shall content my self to give this general hint . When you have from one Station taken in all the Angles you can see from thence , and then are to remove to your second Station , do just as you would do if the Table were covered with a Paper ; only it is at your choice , whether you would guide your self for back-sight by a Line that may be rubbed off , drawn upon the Table it self from the Center to the Degrees on the Frame along the fiducial edge , or by noting only what Degrees it cuts on either side of the Center , the edge passing through it , that by the help thereof and the Needle , the Instrument may be placed in the same Line and Situation as before , for taking in the rest of the Angles , if it can be , if not , another Station must be taken after the same manner . But now to my second Method ; Fig. 38 Let ABCDE be the Figure of a Field to be plotted , the Weather being bad ▪ I send mine Assistants to find the length of every side , measuring it about , cum sole , beginning at A , who return me such an account of every side , in Chains , and Links , as I have noted them upon the Figure and in the Table following , viz. AB 3 Chains , 73 Links , BC 4 Chains , 91 Links , &c. In the mean season , I make haste to find the Angles , and without curiosity plant the Instrument at B , and laying the Index on the Center , I look at C , and find the Index cutting 10 Deg. 15 Min. and looking at A , it cuts 126 Deg. 45 Min. out of which if I subtract 10 Deg. 15. Min. there resteth 116 Deg 30 Min. for the Angle A : but because I like not my Quarters so well as to subtract there , I set them down thus ; B A 126. 45 C 10. 15. the meaning whereof is , that B notes the Angle , and CA the Lines meeting there , cutting such Degrees on the Frame , and the reason why I set A above , is for more ready subtracting afterwards : then removing to the Angle C , and thence looking at B and D , I find the Index to cut as here expressed , C B 153. 10 C 15. 40. In like manner I find at D thus , D C 96. 05 E 28. 50 At E thus , E D 141. 20 A 11. 45. And lastly , at A I find them thus , A E 98. 30 B. 9. 20 An. D. M Sides . Ch. L. A 89 10 A B 3 73 B 116 30 B C 4 91 C 137 30 C D 1 88 D 67 15 D E 6 64 E 129 35 E A 2 29 This being done , I hast under Covert , and by Subtraction find 116 Deg. 30 M. for the Angle B. 137 Deg. 30 M. for C. 67 Deg. 15 Min. for D. 129 Deg. 35 Min. for E. and 89. Deg. 10 Min. for A , as you find them on the Figure , and in this Table together with the length of the Lines . Note , that there is a way to find the Angles without Subtraction , if at every Station you lay the fiducial edge over the Center , and the Divisions 180 and 360 , turning about the head of the Instrument upon the Staff , till through the Sights you see one of the Neighbouring Angles , for the Index turned upon the Center to the other Angle , will give you the quantity of the Angle you are at , but this exact planting at every Angle is more tedious than the other , and therefore not so fit for wet weather . But now to protract this Plot : First , by my Scale , Rule , and Compass , I draw the Line AB in length 3 Chains , 73 Links , ending at the pont B : then laying the Center of my Protractor upon the Line AB , so as the Center of it be upon the Point A , and that end of the Diameter from which the Numbers are reckoned on the Arch or Limb towards B , I make a point for the Angle A at 89 Deg. 10 Min. by the guidance whereof and the point A , I draw the Line AE , which according to my Scale , must be 2 Chains , 29 Links . In like manner placing the Diameter upon AE , just as it was upon AB , and the Center upon the point E , I mark out by the Limb ( for the Angle E ) 129 Deg. 20 Min. by which I draw the Line ED , 6 Chains , 64 Links . In the next place , I bring the Center of the Protractor to the point D its Diameter , lying on the Line ED , and its Limb towards A , by which I prick out 67 Deg. 15 Min. for the Angle at D and draw the Line 1 Chain , 88 Links . Lastly , the Center being at C , and the Diameter upon the Line DC , in such manner as before at other Angles , I prick out by the Limb or Arch 137 Deg. 30 Min. and draw the Line CB , for at B my Plot should close , and if rightly done , the Angle at B will be 116 Deg. 30 Min. and the side BC 4 Chains 91 Links , which by measure I find so to be . But if I plot by a Line of Chords , I am not bound to this Order , but may go from A to B , and so round that way if I please , which I could not so well do with a Protractor , without reckoning my Numbers backward , yet it must be granted that a Line of Chords neither doth the work so quickly nor conveniently , for this is the way . When I have drawn the Line AB of a right length , I set the Compasses to the Radius , and placing one Foot of the Compasses in the point B , and with the other describe an Arch of a competent length , beginning at that side of the Line AB , that is designed to be the inward-side , and upon this Arch , 116 Deg. 30 Min. must be set , but because my Line of Chords gives me only 90 , I set them first on from the Line AB , and then take off the remainder 26 Deg. 30 Min. I joyn them to the 90 upon the Arch , making a Point , through which the Line B must be drawn of a due length . In the like manner must I do at CE , but the Angles at A and D need no such piecing , being capable of being measured out by a Line of Chords at once . Nor do your Angles only give you trouble in this kind of work , but oft-times your Lines will be found too short to receive the touch of an Arch upon the Radius , especially if the Line of Chords be large and your Scale little , and so it may often fall out when you use the Protractor upon such short lines as AE and CD of this last Figure : In which case a Rule must be applied to them , and they must be extended to a due length that the Arches may meet them without the Figure . And if those Extensions of lines and describing of Arches spoil the beauty of your Plot , the matter is not weighty , 't is so easie to be retrieved , for if you lay it on a clean paper and prick through every Angle , lines drawn between those points will give you the Plot neat and perfect . One thing more before I close this long Chapter ; the Artist sometimes loseth his labour of Protraction through some error in the Field , so as his Plot will not close : 't is therefore good to know before we begin that work , whether it will or no ; for which purpose if we take a Number less by two than the number of Angles in the Plot , and thereby-multiply 180 , that Product being found to be equal to all the Angles , the Plot will close , and so it appears by our Plot in this present work : the Multiplier being 3 , because the Angles are 5 , and the Multiplier must be two less than the number of Angles . 180 3 540 Deg. Min. 89. 10 116. 30 137. 30 67. 15 129. 35 540   This kind of tryal is grounded upon two principles of Euclid and Ramus mentioned in the first and third Chapters of this Book , shewing that in all plain Triangles , all the Angles taken together , are equal to two right Angles , and that the sides ( consequently the Angle also ) of every triangulate Figure , are more by two than the Triangles of which it consisteth : But I think it not proper to be large in such things whereof my young Artist is like to make but little use ; for when all is done , I confess with Mr. Wing in his Art of Surveying , Lib. 6. Chap. 15 , that this way of Plotting by the Degrees , is far more troublesome , tedious , and liable to errour , than the other ways upon a sheet of paper , and therefore not ordinarily to be used , but when necessity compelleth us . CHAP. XIX . Concerning taking inaccessible Distances by the Plain-Table , and accessible Altitudes by the Protractor . Fig. 39 THe Substance of what is to be said for the first of these , is gathered from the Instructions given for Plotting a Field , by measuring only the Stationary-distance ; but to make the case more plain to an ordinary capacity : Suppose the Line AC to be the unknown breadth of a River , over which a Bridge of Boats is to be laid , and the General ( that he may inform himself what store of Boats and Planks is necessary to be brought down ) commands me to tell him the true distance from A where he is at present , to C a little Boat-house on the other side the water . To satisfie his demand , I plant my Table covered with a paper at A , causing one to set me up a mark at B at a good distance from me , along the Bank of the River ( the further the better , if distance do not hinder sight : ) Then having chosen a point to represent A , and laid the fiducial edge upon it , I direct my sights towards C and B , and strike lines towards them . Which done , I set up a mark at A , and from thence measure to B , ( 6 Chains , 32 Links ) and so plant my Instrument at B , laying the fiducial edge to the line AB , and turning about the head of the Instrument upon the Staff , till through the sights I spy the Mark at A , and then screw it fast . In the last place , I take 6 Chains 32 Links off my Scale , and set it on the Line AB , from A to B , and laying the fiducial edge to the point B , from thence direct the sights to C , and draw the Line BC , meeting or cutting the Line AC in C : So shall the space AC measured on the Scale ( viz. 8 Chains , 29 Links be the distance desired : and because the Chain is 22 Yards long , if I multiply 8. 29 by 22 , the Product is 182 Yards and 38 / 100 of a Yard , which by reduction is some little more than 13 Inches and 3 / 5 of an Inch. Now to take the height of a Tree , Tower , or Steeple by a Protractor , without any Arithmetical operation , hang a Plumet with a fine Silk Thread at the Center of it , and hold it stedfastly with that end to your Eye where the Numbers begin , then look streight along the Diameter , as if you were to shoot in a Cross-bow without a Sight ( still removing backward and forward as there is occasion ) till you see the top of the Tree , Tower , or Steeple , and the Thread at the same time fall upon 45 degrees ; 10 shall the distance from your Eye to the Tree , Tower , or Steeple , measured in an horizontal or level Line , together with the height of your Eye above the bottom of it , be equal to the height thereof . If either for convenience of sight , or any other reason , you think good to set the other end of the Diameter to your eye , then the Thread for the tryal aforesaid , must fall upon 135 Deg ▪ instead of 45. Other ways of doing this work by this Instrument ( or a Quadrant ) with the help of Trigonometry , and by other Instruments , I forbear at present ( till I write a second Part ) considering whose benefit is here intended . CHAP. XX. Of casting up the Content of Land by a Table . TO make up the number of my Chapters to an even score , I shall add one at the desire of my worthy Friend Mr. S. L. before mentioned ( to whose Experience and Communicative●ess I acknowledge my self indebted for the notion of measuring crooked Lands or Doles at the middle and both ends , marking every where how the Divisions fall , as is mentioned in the beginning of Chapter 17. ) concerning the use of a Table borrowed out of the 46th Chapter of Mr. Leybourns Compleat Surveyor , second Edition , Page 271 , which with the use take as followeth . Links R. P. 1     00000 4 00 90000 3 24 80000 3 08 70000 2 32 60000 2 16 50000 2 00 40000 1 24 30000 1 08 20000 0 32 10000 0 16 8750 0 14 8125 0 13 7500 0 12 6875 0 11 6250 0 10 5625 0 09 5000 0 08 4375 0 07 3750 0 06 3125 0 05 2500 0 04 1875 0 03 1250 0 02 625 0 01 This Table consists of three Columns , the first containing Links , the second Roods ( or Quarters of Acres ) the third Perches : and the use of it is thus : 7. 25 5. 50 36250 3625 3625 3. 98750 Suppose a Field to be 7 Chains and 25 Links long , and 5 Chains 50 Links broad , these by multiplication make 398750 ( as here is evident ) whereof five Figures being cut off towards the right-hand , the Figure 3 signifies Acres , and the rest , viz. 98750 denote parts , and to reduce them into Roods and A R P 3 3 24     14 3 3 38 Perches , I first subtract from 98750 the greatest number of Links in my Table that can be subtracted from it , viz. 90000 ( and put down for it 3 Roods , 24 Perches which I find over against it in the annexed Columns ) and the remain being 8750 , I look in the Table , and find over against it 14 Perches , which by addition makes 3 Roods 38 Perches : So is the whole Content of the Field 3 Acres , 3 Roods , 38 Perches . But not here , that if the Remainder after the first Subtraction cannot be found in the Table , you may take the nearest to it , so the error will be but part of a Perch . As for Example : 7. 35 , being the half Perpendicular , and 9. 23 , the Base , give for their Product 6.78405 . The 6 signifies Acres and from the rest 70000 being subducted , ( to which 2 Rood , 32 Perches answer ) there resteth 8405 , which because I cannot find in my Table , I take the nearest , which is 8125 , to which 13 Perches answer : So the whole Content of that Triangular Close is 6 Acres , 3 Roods and 5 square Perches , and a little better , A R P 6 2 32     13 6 3 05 But Manum de Tabula — I am at present taking leave of my Country-Man , and supposing he brought with him any competent stock of Natural Capacity , and so much Arithm●tick as enabled him to add , Subtract , Multiply , and Divide ; I dare make him ●udge , after he hath as faithfully laboured to understand me , as I to be understood ; whether I have not performed what I undertook in my Title or elsewhere . FINIS . ADVERTISEMENT . The Author hereof useth in Winter and Spring Seasons to board young Gentlemen and others at his Habitation near Dun●am in Cheshire , and to instruct them in these parts of the Mathematicks , viz. Arithmetick . Vulgar concerning Whole Numbers . Fractions Balancing Accounts . Artificial by Decimals Logarithms Instruments Symbols or Algebra . Geometry and therein The Doctrine of plain Triangles . Measuring Superficials , and particularly Land , by all usual Instruments . Measuring of Solid Bodies . Gauging of Casks . The Doctrine and use of Globes and Spherical Triangle● . The Principles of Astronomy and Navigation . The Art of Dialling , by Logarithms , Scales , and Geometrical Projection , &c. Vivat Rex . Floreat Regnum . AN APPENDIX CONTAINIMG XII . PROBLEMS TOUCHING Compound Interest & Annuities . BEING Part of a Letter sent by the Author to his worthy Friend Mr. John Collins Fellow of the Royal Society . Together with A quick and easie Method to contract the Works of Fellowship and Alligation alternate . LONDON Printed in the Year . 1692. XII . PROBLEMS Touching Compound Interest and Annuities , expressed in Symbols , to be resolved by Logarithms ; first presented in Twelve short Lines , to the Right Honourable the Lord Delamer ; afterwards Explained by the Inventor Adam Martindale , and by his Consent presented to the Royal Society by Mr. Collins , and now applied to pertinent Questions in a Practical way , to make them more plain and useful . THese Problems are distinguished into three Ranks , whose Symbols are thus to be understood . p Principal , r Rate , viz. 1 l. with its Rate , t Time , common to all the three Ranks . a Amount or aggregate , proper to the first Rank . s Sum of Principal and Arrearages , proper to the second Rank . d Difference of Principal and Worth , proper to the third Rank . Their Capitals stand for the Logarithms 〈◊〉 the Numbers signified by the small Symbolic● Letters before mentioned . D. signifies Data , Q. Quaesita , Prob. Problem Res . Resolution . The first Rank , concerning Compound Interest for a single Sum of Money . 1. Prob. D p , r , t. Q. a ? Res . Rt + P = ● 2. Prob. D. a , r , t. Q. p ? Res . A − Rt = ● 3. Prob. D. p , a , t. Q. r ? Res . A − P / t = ● 4. Prob. D. p , a , r. Q. t ? Res . A − P / R = ● Examples relating to the Four Problems in order . Quest . 1. What will 15 l. 10 s. amount to i● 12 yeaps at 6 per Cent. Compound Interest ? Answ . 31 l. 3 s. 9 d. 1 q. as appears by the work . R viz. of 1.06 = 0.25306 t = 12 050612 025306 Rt = 0.303672 P 15.5 = 1.190332 A 31.188 = 1.494004 Quest . 2. What is 31 l. 3 s. 9 d. 1 q. due ●●elve years hence worth in ready money , abating ●●er 6 per Cent. Compound Interest . Answ . 15 l. 10 s. as here appears . A 31.182 = 1.494004 R = 0.025306 t 12 050612 025306 Rt = 0.303672 A − Rt = P = 15.5 = 1.190322 Quest . 3. At what Rate of Compound Interest 〈◊〉 15 l. 10 s. amount to 31 l. 3 s. 9 d. 1 q. in ●●elve years ? Answ . At 6 per Cent. as here . A = 1.494004 P = 1.190382 0.303672 12 ) 0.303672 ( 0.025306 = 1 . 0● 0 Quest . 4. In wha● time will 15 l. 10 s. amount to 31 l. 3 s. 9 d. 1 q. at 6 per Cent. Compound Interest . Answ . In 12 years . A = 1.494004 P = 1.190332 R. 025306 ) 0.303672 ( 12 = t 0 The second Rank , touching Annuities in arrear , grounded upon these two Axioms . 1. The Annuity and Rate of Interest bein● given , the Principal correspondent to that Annuity is in effect given also , being easily found by the Rule of Three , thus ; As the Interest of any Principal ( ex gr . of 1● 10 , 100 , 1000 , &c. ) is to that Principal : S● the Annuity of Pension , to its Principal . 2. The Sum of the Principal and Arrearage● of all the Payments being found , the Arrearage● alone may be obtained by subtracting the Principal from that Sum. 1 Prob. D. p , r , t : Q. s ? Res . Rt + P = S 2 Prob. D. s , r , t. Q. p ? Res . S − Rt = P 3 Prob. D. p , s , t. Q. r ? Res . S − P / t = R 4 Prob. D. p , s , r. Q. t ? Res . S − P / R = t Examples suted to the first and last of these Four Problems . Quest . 1. What will 33 l. per annum amount to in 14 years at 6 per Cent. Compound Interest ? Answ . 693 l. 10 s. as here . 6 . 100 ∷ 33 33 300 300 6 ) 3300 ( .550 30 30 30 00 R = 0.025306 t = 14 0.101224 0.25306 0.354284 Rt = 0.354284 P = 2.740363 S = 3.094647 = 12435 550 6935 Quest . 2 In what time will 33 l. per annu●● raise a stock of 693 l. 10 s. at 6 per cent . Compound Interest ? Answ . In 14 years , as here is evident . 6 . 100 ∷ 33 . 550 550 + 693.5 = 1243.5 = s● S = 3.094647 P = 2.740263 25306 ) 0.354285 ( 14 25306 101224 101224 0 The third Rank of Problems touching Annuities anticipated or bought for a Sum in hand ( o● that which is equivalent ) at compound Interes● discounted , is bottomed upon the former of th●● two Axioms above mentioned , and this that followeth : Axiom . If the difference of the Principal and Worth be once found , the Worth is easily obtained by subtracting that difference from the Principal , which is ever greater , being the Worth of the Annuity at that Rate for ever . 1 Prob. D. p , r , t. Q. d ? Res . P − Rt = D 2 Prob. D. d , r , t. Q. p ? Res . D + Rt = P 3 Prob. D. p , d , t. Q. r ? Res . P − D / t = R 4 Prob. D. p , d , r. Q. t ? Res . P − / R = t Examples fitted to the first and fourth of these Problems . Quest . 1. What is 17 l. 10 s. per annum to continue for 11 years , worth in present Money , at 8 per cent . Compound Interest allowed to the Pur●●aser ? Answ . 124 l. 18 s. 8 d. as here is shewed . 8 . 100 ∷ 17.5 ? 17.5 500 700 100 8 ) 1750.0 ( 218.75 0 〈◊〉 R = 0.033424 11 033424 033424 Rt = 0.367664 P = 2. 339948 Rt = 0.367664 — 21875 D = 1. 972284 = 93817 124933 Note , that if in stead of subducting Rt from P , I had turned Rt into the Arithmetical Complement 9632335 , and added that to the P 2. 339948 , it would have done the same thing in a more convenient manner , ( save that it is not so suited to the letter of the Problems ) as here is evident , 10 being rejected ( as in this case it must ever be ) from the Index . 0.033424 11 033424 033424 0.367664 9.632335 2.339948 1.972283 Quest . 2. In what time will 17l . 10s . pay off a Debt of 124 l. 18 s. 8 d. allowing the Creditor after 8 per cent . Compound Interest ? Answ . In 11 years , thus manifest : 8 . 100 ∷ 17.5 . 218.75 = p 124.933 93 817 = d P = 2.339948 D = 1.972284 R = 33424 ) 0.367664 ( 11 0 Any one that understands the very Elements of Algebra , may contract these Twelve Problems into Three ; for the First of any Rank will by Reduction , Application , and Transposition , produce the rest . Then the work of the second Rank may be performed by the first , if one but understand , that instead of the Pension , he must take the principal correspondent to it , and work with it till he have found the Amount , from which the principal must be subducted when the Arrearages of an Annuity are sought , and so proportionably in the rest of the Problems . My Noble Lord Delamer , only noted down for his own use Rt + P = A , and P − Rt = D : by which and a small Canon of Logarithms he will quickly answer any Question of this kind . But for the help of young Mathematicians , I have set them forth thus explicitely . The only inconvenience of any importance that I can yet discover in this Method , is , that in both Ranks concerning Annuities , the second and third Problems are rather for demonstration of the other , and to compleat the Rounds than for any other great use , proceeding upon such . Data as are seldom given for finding the pensions and rate , ( or if they were , the Work would be rendred useless ) yet we are not left without sufficient help to find out pensions or annuities by plain and proper Data . For in reference to the second Rank , if the Arrearages ( or Stock to be raised ) with the rate of Interest and time , be given for finding out such an Annuity , as at such a rate , and in so much time , will raise a given Stock by its Arrearages ; 't is easily found thus : Suppose an Annuity at pleasure , and by the first Problem of the second Rank , find out its Arrearages : Then say by the Rule of Three . As the Arrearages found , to the supposed Annuity : So the Arrearages given , to the Annuity required . For clearing whereof , let this be the Example . Quest . What Annuity will at 6 per cent . comp . Interest , raise a Stock of 693 l. 10 s. in fourteen years . Answ . 33 l. per annum , as here is manifest . Suppose 30 l. then the Work goeth on thus : 6 . 100 ∷ 30 30 6 ) 3000 ( 500 30 000. R. 0.025306 14 0.101224 0.25306 Rt = 0.354284 2. 698970 S = 3. 053254 = 1130●46 500● 630●46 630.46.30 ∷ 693 . 5● 30 630.46 ) 20805.0 ( 32.999 44046 Here we account 32. 999 for 33 , being so much within an inconsiderable trifle . Likewise in relation to the third Rank , if the Worth ( or Price ) Rate and Time be given , and the Pension to be purchased be required ; find the worth of any supposed Pension by the first Problem of the third Rank , then the proportion stands thus : As the Worth found , to the supposed Annuity : So the Worth ( or price ) given , to the Pension required ; as in the Example follow 〈◊〉 : Quest . What Annuity to continue eleven years will be purchased for 124 l. 18 s. 8 d. at 8 per cent . Comp. Interest . Answ . 17l . 10s . as here . Suppose 12 l. Then 8 . 100 ∷ 12 ? 12 200 100 8 ) 1200 ( 150 8 40 40 00 R = 0.033424 11 0.033424 0.33424 Rt = 0.367664 P = 2. 176091 Rt = 0.367664 — 150 1. 808427 = 64332 85668 85.668.12 ∷ 124.933 ? 12 249866 124933 85.668 ) 1499.196 ( 17. 5 60 If the Interval betwixt every two payments 〈◊〉 less than a year , consider what part it is ; ●hether ½ ¼ 1 / 12 1 / 13 1 / 32 1 / 365 , &c. And by the ●enominator of the Fraction signifying that part , ●ivide the Logarithm of 1 l. and its Rate of In●rest , ( usually called the Logarithm of the Rate ) ●en the absolute Number answering to that ●●otient , being made less by an Unite , will be 〈◊〉 new Divisor , whereby dividing the given Pen●on so payable , half , yearly , quarterly , &c. the ●orrespondent Principal will appear in the Quo●ent , with which you may proceed , as if the payments were so many yearly ones ; as this ●ext Example shews . Quest . 1. What is a quarterly Rent of 3 l. 15 s. 〈◊〉 continue Thirteen Years and one Quarter , worth in ●●dy Money at 6 per cent . Compound Interest ? Answ . 137 l. 10 s : 1 d : as appears here . R = 4 ) 0.025306 0.006326 = 1●01467 - 1● ●01467 .01467 ) 3.75000 ( 235.623 1059 1 / 4 R = 0.006326 Number of Quarters 53 0.018978 0.31630 1 / 4 Rt = 0.335278 Ar. C. 9●664721 P = 2.407600 — 255623 2.072321 = 118-119 137504 Whereas I multiplied the Fourth part of the Log. of the Rate , viz. 0.006326 by 53 ( the Number of Quarters ) I might as well have multiplied the whole Log. of the Rate , viz. 0.025306 ▪ by 13.25 , i. e. 13 years and 1 quarter , and it would have been full as exact , or rather more , ( two places being cut off from the Logarithm so multiplied towards the right hand : ) I say rather more exact , because there remained 2 when I quartered the Log. of the Rate . Quest . 2. What is 5 s. per Week to continue 7 ●ears and three quarters worth in ready Money at 8 per cent . Compound Interest ? Answ . 75 l. 17 s. 8 d. 2 q. 52 ) 0.033424 ( 0.000642 = .00148 40 .00148 ) .25000 ( 168.918 136 R = 0.033424 t = 7.75 0 167120 2 33968 2 33268 Rt = 0.259036.00 Ar. ●9 . 740960 P = 2.227675 — 168918 1.968638 = 913033 75885 Here I multiplied ( not by the Number of Weeks ) but by the Number of Years and Quarters , cutting off two Cyphers from the Product . And I also took the Arithmetical Complement of 0.259036 , viz. 9.740963 , and added it to the Log. of the Principal in stead of substracting the Log. 0.259036 from it . Lastly , As to the Rate of Interest , according to the sense of Dr. Newton , Mr. Dary , and others , I see not why this Method may not b● made use of as well as any other . This universal Rule being first understood : In Questions belonging to the Second Rank , the greater Rate 〈◊〉 Interest , the greater Arrearages ; and the less Rate the less Arrearages . But in Questions belonging to the third Rank , the greater the Rate of Interest the less the present Worth ( or price ) and the less the Rate of Interest , the greater the Worth , & vice versa . I shall shew this in answering the Question following , concerning a yearly payment of 1 l. because that by bare Multiplication may easily be applied to any other . A Field worth 1 l. per annum clearly , is offered to be let 11 years for 15 l. to be paid at the end of those years : What is the Rate of Interest demanded for the several payments ? Answ . First , I suppose 6 per cent . and trying it by the first Problem of the second Rank , it find the Arrearages to be but 19 l. 19 s. 5 d. which is 6 d. 3 q. too little . Then trying what the Arrearages will amount to at 6. 2 ( or 6 l. 4 s. ) per cent . I find 15. 13 , or 15 l. 2 s. 7 d. 1 q. Perceiving then I have over-shot my Mark , I make my third tryal at 6.1 ( or 6 l. 2 s. ) per cent . and the Result is 15.05 , or 15 l. 1 s. which is pretty near , being but 〈◊〉 above the truth . But that I may yet bring it nearer , I first subtract the Result at 6 per cent . which is 14.971 from the Result of the Arrearages at 6.1 . viz. 15.05 ( because these are the Results that come nearest to the truth ) and their difference is .079 . 2. I subtract the Rate 6 from the Rate 6.1 , and there resteth . 1. 3. I substract the nearest Result to 15 , viz. 14. 971 , from 15 the given Arrearages , and their difference is .029 . Then working by the Rule of Three , I find ( as followeth ) .0367 .079 .1 ∷ 029 1 .079 ) .0290 ( .0367 7 Which .0367 being added to 6 , gives 6.0367 , or 6 l. 0 s , 8 d. 3 q. per cent . Compound Interest . And that this Rate exactly fitteth , is plain by the following work . 6.0367.100 ∷ 1 1 6.0367 ) 100.0000 ( 16.565 20245 R. 0.025456 11 0.025456 0.25456 0.280016 1.219191 1.499207 = 31 / 565 16 / 565 15 / 000 But perhaps it will be said , that though in this and all other matters foregoing , I agree well enough with the learned Authors before mentioned , and others that have writ of these Subjects ; yet in my Country-Almanack for the Year 1677 , the Fourth Problem agrees neither with them , nor what I have here written , but apparently clasheth with them . I confess this Charge had been very just , had that Problem been designed for the same end , and produced a different effect : ( for in such a case to differ is to contradict . ) But forasmuch as in my very entrance upon it , I began thus . There is a Problem in some Learned mens Works , seemingly of the same importance with this , but indeed much different both in the Design and Effect : And after in answering Objections show wherein the Design differs ; I cannot but admire the Undertakings of that able Artist , who was so much at leisure as to prove with a great deal of Pomp , that mine will not attain the end for which I declared it was never designed , as clearly as words can utter it . A Sword may be a very good one , yet a very bad Instrument to fell Trees with : But I hear he is dead , and I shall rather lament the loss of him , and divers other famous Mathematicians ( which Death hath of late deprived us of ) than unnecessarily to expose the impertinence of his Paper to the Publick View ; or so much as name him to the prejudice of his Memory . But from henceforth I expect from all ingenious persons , that they neither take for granted what I professedly deny and disprove ; nor urge mine Objections against my self , without taking notice of mine Answers . For understanding of what follows , it were very convenient to have inserted herein the whole Discourse in the Country-Almanack about this business ; but because this little Paper cannot afford it room , I shall only point briefly at the Design and Management ; and answering all the Objections I ever met with of any seeming Importance , make things as clear as I can in so narrow a compass . The Design was to discuss in plain language ( suitable to Country-mens understanding ) this practical Question , wherein they are oft concerned ; viz. Whether it be more advantage for the Lender to receive for 100 l. in hand , Compound Interest at 8 per cent . viz. 71 l. 7 s. 7 d. 3 q. at the end of seven years above the stock of 100 l. supposing it can be legally assured ( as in Ireland it may , and in England by equivalence in Goods or Lands , without mentioning Interest ) Or to enjoy a Farm for seven years in consideration of 100 l. that will clear him just 20 l. per annum ? My Answer was , that the former was the better Bargain ( and consequently the higher Rate of Interest for the whole Stock during the whole term ) for which I gave in these grounds under the feigned persons of A and B. 1. A is to receive ( as is said ) 171 l. 7 s. 7 d. 3 q. and if B , who takes the Farm , receive not equivalent , his rate of Interests is lower . 2. B receives only 140 l. by 20 l. per annum ; only he hath the advantage to improve the several payments from the time they grow due , to the end of the term . 3. These must be computed at some certain rate of Compound Interest : For to Compute them at no rate of Interests , or at Simple Interest , or one taken up arbitrarily , will not sute the case . 4. The Rate tolerated by the Statute , viz. 6 per cent . under which none will take , and above which none dare expresly bind any to give , and at which any responsible man may be fitted , is to be preferred before any other . 5. At this Rate the 7 payments will amount but to 167 l. 17 s. 6 d. q. Of these 5 Pillars , the 132 , and 5 th , were never attempted to be shaken by any that I know of : What assaults have been made against the other two , or design of the whole Fabrick , I shall briefly consider ; and I find them ( besides those answered when I first published that Problem ) to be these three : 1. Ob. The Question is not at all , how the Receiver improves his Payments , but what Improvements was made in the Debtors hands , and the same Rate carried on ? Ans . We are agreed that the Rate is not to be computed according to what the Usurer actually makes of it . If he gives away , loses , or lends freely all the several Payments : Or if he make new Bargains more oppressive than the Original one ; the true value of the loan of so much Mony for such terms , is one and the same . But I utterly deny that it is to be reckoned , as if the same Rate must be carried on , and that for unanswerable Reasons ( as I suppose them to be ) laid down in my former discourse , or easily colligible from it . 1. The Laws of our Nation prohibit upon severe Penalties , the taking of more than 6 per cent therefore that is the utmost of the legal worth , which he that exceeds , runs ( as I take it ) the hazard of all , and a great fine . 2. If he can evade the penalty , no responsible man needs to give more than 6 per cent . and few are willing to give more than they need . 3. If he get over both these blocks , and make ●ew oppressive Bargains , this is nothing to the Question as I propounded it ; for that was , whether A. or B. received the highest Interest for his Money , by vertue of the above mentioned Bargains , not by vertue of occasional after Bargains . When the Debtor hath paid in a years Pension , he hath done with it ; and if he have ●t not ready , he may take it up at ordinary Interest , and the proportion is broken off for so much . 2. Object . If we regard Laws in the case , what need we any Rules concerning Compound Interest , seeing the Laws of the Land allow only simple Interest : Ans . 1. The Laws are not against Compound Interest , as it may be managed : that is , the Usurer may receive his simple Interest at the years end , and put out that as a new Stock , and so undeterminately from time to time ; or if you will call this simple Interest , it 's the same to the purse . 2. I am told they have ways in London for putting out very small Sums to mean Traders upon sufficient security ; and then it is both legal and practical , though I say not how lawful before God ; especially as I hear some use it . 3. Object . It ought to be a Question only o● Art , without dependance upon Laws and Usages ; it being the nature of Art in these kind 〈◊〉 Questions , not to shew so much what ought 〈◊〉 ●e done , as what is really done . Ans . 1. It is one Question , what is the natur● of Art , and another how far Art is concerned that is , whether nothing else save the Rules o● Art be regardable in the case ; which I deny , because Laws and Usages have a great influenc● upon it , hindring the continuance of the proportion . 2. I consider not the Laws as pinching the U●surers Conscience ( as to what is lawful ) but 〈◊〉 tying up his hands , and so obstructing the proportionable increase of his gain . 3. Whatsoever may b● said of Art ( as Art i● the strict notion of it ) the Artist must not be such a Slave to the Rules of it , as not to allow for unavoidable Obstructions and irregularities I hope I may be allowed to tell my Scholars that learn Navigation , that though the direct cours●● from one Port to another be upon such a Point of the Compass ; yet other Courses must be steered sometimes in regard of Rocks , Shallows● crooked Chanels , Currents , Trade-winds 〈◊〉 convenience of Harbour , and fresh Supplies 〈◊〉 Or to avoid Pirats , Enemies , Forts , and Places where great Customs or Payments will be exacted ; and many such things which the experienced Sea-man is better acquainted with , than I with their Names . And to say to such as learn Merchants Accompts , that though it were more Artificial and Rational , that Rebates of Interest for Money paid before ●t be due , should be computed at Compound Interest ( as certainly it is : ) yet forasmuch as it is usual with Merchants to allow no more than simple Interest ( as appears by printed Accompt-Books ) they must submit to the Laws of the great Tyrant Custom . An Artist hath not the same liberty of supposition in answering Questions ( especially such such as are real and practical ) that he hath in proposing Questions , or receiving them from others , when tryal of skill is only or chiefly intended . I may without absurdity in this latter case demand or find the Amount or present worth of any single Sum of Money , or yearly Pension for any terms propounded at any rate of Interest given , though such as is never likely to come in practice : ( ex . gr . at 2 , 3 , 14 , or per cent . ) But if I be put by the nature of my Work , to compute the Amount or present worth of Payments , no rate of Interest being named , I ought prudently to weigh all Circumstances , and pitch upon that which is possible and rational . And this is not a work of Art , but of discreet Judgment , ( wherein great respect is to be had to penal Laws , Usages , and other obstructions ) after which Art takes it own Province , in computing after such a Rate resolved upon . To make all this plainer ( if possible ) than I made it at the first Publication ; Let us for once suppose a rare case , viz. That B a Mathematician turns Usurer , and for 100 l. ready Money , takes a Farm for 7 years , that he lets to another for 30 l. per annum clearly , knowing before hand he could so let it : and A his familiar Friend thus accosts him ; I wonder to bear you are grown such an Extortioner , as to receive 20 l. per annum , seven years together for 100 l. B answers , Before you find fault with the Mote in mine Eye , take the Beam out of your own : You have bought the Reversion of a piece of Land after seven years for 100 l. for which I will bona fide give you 172 l. at the time of your Entrance upon it ; which I find by the Rules of mine Art to be 12 s. 4 d. 1 q. above Compound Interest at 8 per cent . A replies , Learned men , say such a Bargain as yours will clear 9 l. 3 s. 4 d. per cent . and better . B rejoyns , 'T is true they do so , but then they suppose the same Rate to be continued , ( for none will say , that if the Payments lie by unimproved , or be let out at an under Rate , that in such cases that Rate of Interest can be answered ) and this supposition is really impracticable ; or ( to say the least ) that which a rational man cannot depend upon . I did accidentally meet with this Bargain from the hands of a weak man , which yet I durst not have accepted ( the Laws are so strict ) if it had been a Rent-charge in Money . And will you undertake to find me Fools that shall at every years end , take off the several Payments , yea and all the increase of them as they grow due upon the same terms that I put out the 100 l. upon , and to let me Land for it ? If you think you can , I will make you a great Bargain , ( because I know you to be a punctual responsible man ) you shall receive the first yearly Payment , and at the end of the second year , pay me simple Interest for it at 6 per cent . as the Statute allows , and I will instantly return it you , together with the second 20 l. which we will joyn into one Sum , and you shall take it at the same rate : and thus we will do every year to the end of the term , still adding that years growth to what was before , and so keep up all the Accounts to 6 per cent . and if you can make any more profitable use of it , much good may it do you , and I will heartily thank you to boot , for helping me so readily to place out my small parcels without loss . But if you dare not do this , let ingenuity mollifie your charge , and I shall not quarrel with you about your rate of 9 l. 3 s. 4 d. which perhaps you borrowed ( or some one for you : ) from Dr. Newton , his Trigonometria Britannica , P. 37. for I acknowledge it is that and somewhat more , though not much ; but I say ( which I desire you to observe ) that I receive no compound Interest after that rate , but only simple : and that not for the whole Sum during the whole term ( as you do compound Interest for all your Stock ) but only for the whole 100 l. for the first year : for you know that if at the end of that year 20 l. be subtracted from 109 l. 3 s. 4 d. all the Interest is paid , and so much of the Stock as brings it down to 89 l. 3 s. 4 d. This 89 l. 3 s. 4 d. being computed at the same rate of Interest , 20 l. abated from it , clears the Interest again , and brings down the Stock much lower : and so year by year the Stock is dwindled away , till at last the seventh payment ( if the Rate were absolutely exact , as it is near ) clears off 〈◊〉 Stock and Interest . And this is all I receive , save only the benefit of the severa● yearly payments , which I offer you ( or any other solvent man ) at 6 per cent . as aforesaid 〈◊〉 At which rate ( abating not a Farthing for loss of time , but supposing good places ready for eve●ry parcel as it becomes due ) it will but amount● to 167 l. 17 s. 6 d. q. which being the whole Aggregate of Stock and Interest , if we substract from its Logarithm , the Log. of 100 , and divide their difference by 7 , as we are directed by Mr. Wingan's 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 , Chap 5. Prov. 13. the rate of Interest will appear to be 7 l. 13 s. 7 d. 2 q. I profess freely , Sir , I cannot withstand B's Arguments ; but if any other can , I shall neither envy his happiness , nor despise , ( if fairly and ingeniously offered ) his Animadversions ; in that case you shall command a Sheet from me at any time , in consideration of any thing so objected ; but for such injurious and passionate discourses as are apter to provoke than convince , I confess , 't is such a piece of drudgery to answer them , that ( if I could not ease my self by contempt ) I should think it hard measure that my silence should be interpreted as their Victory . Sir , You will p●rdon this Prolix Appendix , when 't is considered that it is not intended for such as your self , but for such to whom nothing can be too plain , by SIR , Your Friend and Servant , Adam Martindale . A Compendious Method for working out many Conclusions in Arithmetick , wherein the Rule of Three is often repeated ; long since invented , practised and taught by the Author , but never till now published . IN all Books of Arithmetick that I have had the happiness to peruse , the works of both Rules ●f Fellowship and Alligation alternate are performed by the Rule of Three , so oft to be repeat●d , as there ere particular proportionable parts ●o be found ; which proving intolerable tedious , when the Partners in Fellowship , or the parcels 〈◊〉 Alligation are many , caused me to think of ●is following course , every way as plain and useful , and much more compendious , which to ●e short is this : The Rule . Divide that which is usually made the second Number in the Rule of Three , by that which is as ●sually made the first and the common Divisor ; That Quotient multiplied severally by the respe●●ive third Numbers , gives the particular results . ●ut here it will be necessary if you use Decimal Division ( which I judge most convenient ) that you continue your Division till either nothing remain , or you have six places of Decimals in the Quotient , accounting Cyphers ( if there be any ) into the number . And if upon the Addition of your Parcels , the Total amount not just to the Sum expected , but an Unite short in the Integers , & three more Figures of 9 immediately following , take it for exact , for so it is within a matter of nothing . All this will be plain by three Examples whereof the first shall be wrought ( for plainness ) both in the old and in this newer way ; the other two ( for brevity ) this latter way only . 1. Example in Fellowship without time . A B C Put into the Common Stock 15 24 33 72 The Sum whereof is 72 l. and their gain 63 l. what is each Mans Part ? The Resolution in the old Method is thus . A 72.63 ∷ 15 15 315 63 72 ) 945 ( 13.125 Or 13l . 2 s. 6 l. B 72.63 ∷ 24 24 252 126 72 ) 1512 ( 21 l. 0 C 72.63 ∷ 33 33 189 189 72 ) 2079 ( 28.875 0 Or 28l . 17 s. 6 d. 13.125 21. 28.875 63.000 Or thus , 13-02-06 21-00-00 28-17-06 60-00-00 In the new way the work is thus done . 72 ) 63.0 ( .875 A .875 — 15 0 — 4375 875 13.125 B. 875 24 3500 1750 21.000 C. 875 33 2625 2625 8.875 I hope by this time mine intelligent Reader is aware , that if the Partners had been many ( as in Voyages and Adventures it oft falls out ) the difference betwixt the two Methods would have been yet more signally conspicuous ; for one single Division sufficeth , be the Partners never so many : Though I confess there is yet a nearer way by the help of Logarithms ; of which I shall present this Specimen . 63 = 1.79934055 72 = 1.85732250 .875 = 1.94201805 .875 = 1.94201805 24 = 1.38021124 21 = 1.32222929 .875 = 1.94201805 15 = 1.17609126 13.125 = 1.11810931 .875 = 1.94201805 33 = 1.51851394 28.875 = 1.46053199 The second Example in Fellowship with time . A laid in 43 for 8 Months They gained 73l . Q. each Mans part ? B 52 5 C 63 7 The Resolution by this new Method is thus . A 43 8 344 B 52 5 260 C 63 7 441 344 260 441 1045 1045 ) 73.00 ( .069856 480 A. .069856 344 279424 279424 209568 24.030464 or 24 l. — 0 s. — 7d . — 2 q. B .069856 260 4191360 139712 18.162560 or 18 l. — 3 s. — 3d. — 0 q. C .069856 441 069856 279424 279424 30.806496 l. s. d. q. or 39 l. — 16 s. — 1 d. — 2 q. Proof 24.030464 18.162560 30.806469 72.999520 or thus 24 l. — 00 s. — 7 d. — 2 q. 18 l. — 03 s. — 30 d. 30 l. — 16 s. — 1 d. — 2 q. 73 l. — 00 s. — 0 d. — 0 q. The third Example in Alligation alternate . Suppose a mixture of Wine of 119 Quarts be required , that must be made up of these several prices 7 d. 8 d. 14 d. and 15 d. so as the whole may be afforded at 12 per Quart , the parts may be found out in thir method ( without decimals ) thus . Having linked 8 to 14 , and 7 to 15 , and Counterchanged their differences from the Common price 12d , I find the Sum of their differences to be 14 , by which dividing 119 the Quotient is 8 / 14 or 8 ½ , which for convenience of Multiplication we shall change into the improper Fraction so ● 2 the Resolution● will be thus . 8 2 14 4 7 3 15 5 14 Quarts . 17 / 2 X 2 = 34 / 2 = 17 17 / 2 X 4 = 68 / 2 = 34 17 / 2 X 3 = 51 / 2 = 25 ½ 17 / 1 X 5 = 85 / 2 = 42 ½ 119 Having our just Measure of Wine , let us try the Prices how they suit our purpose by Alligation medial , for considerable errors may be caused by misapplication of Prices , when the parts were truly taken ; but here-under it is apparent that each parcel multiplied by its price , the Sum of the Products is 1428 pence , which divided by 119 give 12 for the common price . 17 8 136 34 14 136 34 476 136 476 178 ½ 637 ½ 1428 25 ½ 7 178 ½ 42 ½ 15 217 ½ 42 637 ½ 119 ) 1428 ( 12 119 238 238 0 This Proof by Alligation medial I do not account a needless curiosity ; but very useful to be throughly understood , for experience informs me , that young Men being defective in Skill , Care and Memory , are apt to mistake in several points , but especially one : That is so as to esteem the several parcels to be of the several prices from which the differences ( by which they are found ) were originally taken , and not ( as the truth is ) of the prices to whom in the counterchange they were annexed . As to that which seems to look like an unnece●●sary affectation of Novelty , in linking the pric● and differences by separate couples ; I designed 〈◊〉 no higher thing than to free the Printer ( if th●● pass 〈◊〉 hand ) from the trouble of lookin● up his dusty Cuts of Semilunes intersecting or enclosing one another , as in our usual Books of Arithmetic●● Though the truth is , in such cases as this before us , no linking at all is needful ; but when the common Quotient is multiplied severally by all th● differences , any price above the common may b● assigned to any product made by the difference o● riginally belonging to an under price , and contr●●rily , so as true couples be observed . So here 〈◊〉 might have assigned the price 15 to the parcel 3● found out by the difference 4 originally belongi●● to 8 , and the price 14 to the parcel 42 ½ , found b● multiplying the common Quotient by 5 the difference of 7 from the common price . But then 〈◊〉 must be sure to do justice , so as to assign the pri●● 8 to the parcel 25 ½ found by 3 the difference o● 15 from the common price , and the price 7 to the parcel 17 arising from 2 the difference of 14● us here is plain . 17 X 7 = 119 25 ½ X 8 = 204 34 X 15 = 510 42 ½ X 14 = 595 1428 119 ) 1428 ( 12 119 238 238 0 FINIS . Notes, typically marginal, from the original text Notes for div A52120-e2310 ☞ ☜ ☞ ☞ ☞ ☞ ☜ ☜ ☜ ☜ ☞ ☞ ☜