Ex Libris C. K. OGDEN THE LIBRARY OF THE UNIVERSITY OF CALIFORNIA LOS ANGELES \\'L 1.1.1 AM \VKS1. Uooksrllrv \ Dubltshi 1 I oi [T\ G E O D^ S I A: OR, THE ART of SURVEY ING A tf D MEASURING LAND madeEafy. SHEWING By Plain and Practical RULES, to Survey, Pro- tract, Caft up, Reduce or Divide any Piece of" Land whatfoever ; with New TABLES for the Eafe of the Surveyor in Reducing the Meafures of Land. MOREOVER A more Facile and Sure Way of Surveying by the CHAIN, than has hitherto been taught. As ALSO To lay out New Lands in AMERICA, or elfewhere : And how to make a Perfeft MAP of a River's Mouth or Harbour; with feveral other Things never yet Publifhed in our Language. BY JOHN LOVE. THE EIGHTH EDITION. CORRECTED and IMPROVED BY SAMUEL CLARK. LONDON: 1 Printed for J. RIVING TON, St. Paul's Church-Yard; G. KEITH, in Gracechurch-Street; and ROBINSON and ROBERTS, in Paternofter-Row. MDCCLXVIII. -rA THE PREFACE T O T H E READER. IT would be ridiculous, to go about to praife an art that all mankind know they cannot live peaceably without, and is near hand as ancient (no doubt on it) as the world : for how could men fet down to plant, with- out knowing fome distinction and bounds of their land ? But (neceflity being the mother of invention) we find the Egyp- tians, by reafon of the Nyle's overflow- A 2 ing, The PREFACE. ing, which either waflied away all their bound-marks, or covered them over with mud, brought this meafuring of land firft into an art, and honoured much the pro- feflbrs of it. The ufefulnefs, as well as the plealant and delightful ftudy, and wholefome exercife thereof, tempted fp many to apply themfelves thereto, that at length in Egypt (as in Bermudas) every ruftic could meafure his own land. From Egypt, this art was brought into Greece by Thales, and was for a long time called Geometry ; but that being too com- prehenfive a name for the menfuration of a fuperficies only, it was afterwards called Geodaefia ; and what honour it ftill has con- tinued to have among the antients, needs no better proof than Plato's aV^sr^' *<5V r 5*V;To. And not only Plato, but moft, if not all the learned men of thofe times, re- fufed to admit any into their fchools, that had not been firft entered in the mathema- tics, efpecially geometry and arithmetic. And we may fee, the great monuments of learn? The PREFACE. learning built on thefe foundations continu- ing unfhaken to this day, Efficiently de- monftrate the wifdom of the defigners in chufing geometry for their ground-ploti Since which, the Romans have had fuch an opinion of this fort of learning, that, they concluded that man to be incapable of commanding a legion, that did not pof- fefs at leaft fo much geometry, as to know how to meafure a field. Nor did they indeed either refpecl: prieft or phyfician, that had not fome infight into the mathematics. Nor can we complain of any failure of refpecl: given to this excellent fcierice by our modern worthies, many noblemen, clergymen, and gentlemen affeding the ftudy thereof ; fo that we may fafely fay, none but unadvifed men ever did, or do now fpeak evil of it Befides the many profits this art brings to man, it is a ftudy fo pleafant, and af- fords fuch wholcfome and innocent exer- cife, that we feldom find a man that has once entered himfelf into the ftudy of A 3 Geo- The PREFACE. Geometry or Geodaefia, can ever after wholly lay it afide : fo natural is it to the minds of men, fo pleafingly infinuating, that the Pythagoreans thought the mathe- matics to be only a reminifcience, or call- ing again to mind things formerly learned. But no longer to light candles to fee the fun by, let me come to my bufinefs, which is to fpeak fomething concerning the fol- lowing book ; and if you afk, why I write a book of this nature, fince we have fo -many very good ones already in our own lan- guage ? I anfwer, Becaufe I cannot find in thofe books many things, of great confe- quence, to be underftood by the furveyor. I have feen young men in America fo often at a lofs, that their books would not help them forward ; (particularly in Carolina,) about laying out lands, when a certain quantity of acres has been given to be laid out five or fix times as broad as long. This I know is regarded as a mere trifle by a ma- thematician ; yet to fuch as have no more of this learning, than to know how to mea- fure The PREFACE. lure a field, it feems a difficult queftion : and to what book of furveying fliall they repair to be refolved ? Alfo concerning the Extraction of the Square Root ; I wonder that it has been fo much neglected by the teachers of this art, it being a rule of fuch abfolute neceffity . for the furveyor to be acquainted with. I have taught it here as plainly as I could de- vife, and that by the beft method now in ufe, ufing fewer figures, and being once well learned, charges the memory lefs than any other way. Moreover, founding .the entrance of a river or harbour is a matter of great im- port, not only to feamen, but to all fuch as feamen live by ; I have therefore done my endeavour to teach the young artift how to do it, and draw a fair draught thereof. Many more things have I added, fuch as I thought to be new, and wanting ; for which I refer you to the book itfelf. As for method, I have chofe that which I thought to be the eafieft for a learn- A 4 er The PREFACE. er ; advifing him firft to learn fome arith- metic, and after, teaching him how to ex- tract the fquare root. But I would not have any neophyte difcouraged; for if he find the firft chapter too hard for him, let him rather flap it, and go to the fecond and third chapters : thofe he will find fo eafy and de- lightful, that I amperfuaded he will be en- couraged to conquer the difficulty of learn- ing that one rule in the firft chapter. From Arithmetic, I have proceeded on to teach fo much Geometry as the art of furveying requires. In the next place, I have fhewed by what meafures land is fur- veyed, and made feveral tables for the re- ducing one fort of meafure into another. From thence I come to the defcription of inftruments, and how to ufe them ; wherein I have chiefly infifted on the fe- micircle, it being the beft that I know of. The fixth chapter teacheth how to ap- ply all the foregoing matters together, in the practical furveying of any field, wood, &c. divers ways, by various inftruments ; and The PREFACE. and how to lay down the fame upon paper. Alfo at the end of this chapter I have large- ly infifted on, and by new and eafy ways, taught furveying by the chain only. The feventh, eighth, ninth, tenth and eleventh chapters, teach how to caft up the contents of any plot of land ; to lay out new lands ; alfo to furvey a manor, county or country ; and, how to reduce and divide lands, 8cc. The twelfth chapter confifts wholly of Trigonometry. The thirteenth chapter treats of heights and diftances, including, amongft other things, how to make a map of a river or harbour, and to convey water from a Ipring- head to any appointed place, or the like. Laftly, at the end of the book, I have added a table of northing or {buthing,eafting or wefting ; or, (if you pleafe to call it fb) a table of difference of latitude and de- parture from the meridian, with directions for the ufe thereof. Alfo a table of fines and tangents, and a table of logarithms. I have The PREFACE. I have followed Mr. Holwell's method, in making the table of fines and tangents but to every fifth minute, that being near enough in practice for the Purveyor's ufe ; for it is not poffible with the beft inftru- ment that ever was yet made, to take an angle in the field nigher, if fo nigh, as to five minutes. All which I commend to the ingenious reader, wifting he may find benefit there- by, and defiring his favourable reception thereof accordingly. I conclude, READER, Your Humble Servant, J. L. H E CONTENTS CHAP. I. OF arithme tick in general* Page i To extratt the ? vulgar arithmetick, 2 fquare root by 5 the logarithms. 7 CHAP. II. Geometrical Definitions. Point, o Aline, ibid yf/z angle * ibid. ^ perpendicular ; i o ydf triang/e, 1 1 A fquare > 1 2 -^ parallelogram, ibid. A rhombus and rhomboldes y ibid. ^ trapezium, ibid. y/w irregular figure, 13 regular polygon, as pent agon, hexagon ^ &c. 14 //6 wte thereto belongs > ibid. Afuperfcie^ 15 I Parallel The CONTENTS. Parallel lines, Page 16 Diagonal lines, ibid* CHAP. III. Geometrical Problems. 1. H^O make a line perpendicular to another two *- ways, 1 7 2. To raife a perpendicular upon the end of aline two ways, I 8 3. From a point aligned, to let fall a perpendi- cular upon a line given, 20 4. To divide a line into any number of equal parts, 21 5. To make an angle equal to any other angle given, 22 6. To make lines parallel to each other, 23 7. To draw a line parallel to another line, and which JJjall alfo pafs through a point affigned, 24 8. To make a triangle with three given lines, ibid. 9. To make a triangle equal to a triangle given, 2 5 10. To make a fquare, 26 1 1 . To make a long fquare or parallelogram, ibid. 12. To make a rhombus or rhomboides, 27 13. To defcribe regular polygons, as pentagons, hex- agons, heptagons, Oft. 28 14. Three points being given, to defcribe a circle, whofe circumference Jhall pafs through thofe points, 3 2 1 5. To defcribe an ellipfis and an oval f ever al ways, 3 3 1 6 To divide a given line into two parts, which Jhall be in fuch proportion to each other 9 as two given lines, 36 17. To The CONTENTS. 1 7. T^ofind a fourth proportional to three given line 's, 37 CHAP. IV. Of Meafures in general. I. /^\ F long meafure, Jhewing by what kind of mea- ^ furesland isfurveyed-, andalj'o how to reduce one fort of long meafure into another , 39 A general table of long meafure, ibid. A table Jhewing how many feet, and parts of a foot j aljo how many perches and parts of a perch, are contained in any number of chains and links from one link to an hundred chains, 41 A table Jhewing how many chains, links, and parts of t a link; alfo how many perches and parts of a perch are contained in any number of feet, froni a foot to 10000, 44. II. Offquare meafure, Jhewing what it is ; and how to reduce one fort into another, 46 A general table of fquare meafure, 47 A table fiewing the length and breadth of an acre, in perches, feet, and parts of a foot y 49 A table to turn perches into acres, roods, and perches, 53 CHAP. The CONTENTS. CHAP. V. Of Inftruments and their Ufe. OF the chain, Page 54 Of inftruments for the taking of an angle in the field, S 6 70 take the quantity of an angle in the field by Plain table, 57 Semicircle, 58 Circumferentor, &c. fever always, ibid. Of the fold- book, 61 Ofthefcale, with feveral ufes thereof; and how to make. a line of chords, 62, &V. Of the protracJor, 68 CHAP, VI. O take the plot of a field, at one Jlation, in any place thereof -, from whence you may fee all the angles by thefemicircle ; and to protract the fame, 7 I 7i? take the plot of the fame field, at one fiation, by the plain table, 74 "To take the plot of the fame field, at one Jlation, by the femicircle, either with the help of the needle and limb together, or by the help of the needle alone, ibid. By the femicircle, to take the plot of afield, at one Jlation, in any angle thereof, from whence the other angles may befeen, and to protract the fame. 76 To The CONTENTS. fo take the plot of afield at two flattens, pro- uidedfrom either ft ation you may fee every angle, and meafuring only the Jlationary diftance, al/b to protrafl the fame, 79, 80, &c. f t*ke the plot of a field at twoftations, when the field is fo irregular, that from one jl at ion you cannot fee all the angles, 83 fo take tfa plot of a field, at one ftation, in an angle, (jo that from that angle you may fee all the other angles) by meafuring round about thefaidfield, 86 fo take the plot of the foregoing field, by mea- furing one line only ; and taking obfervations at every angle, 88 fo take the plot of a large field or wood, by meafuring reund the fatfie, and taking obferva- tions at every angle by the femicircle, 90 When you have furveyed after this manner, to know, before you go out of the field, whether you have wrought true or not, 94 )irettions to meafure parallel to a hedge, ivhm you cannot go in the hedge itfelf ' -, and aifo infucb cafe, how to take your angles, 95 fo take the plot of a field or wood, by obferv- mg near every angle, and meafuring the diftance between the marks of obfervatim, by taking in every line two off-fets to the hedge, 97 An eafier way to do the fame, by taking only one fquare and fever al of -Jets. 99 By the help of the needle to take the plot of a large wood, by going round the fame, and mak- ing ufe of that divj/ion of the card that is num- bered with four 9OJ. or quadrants -, and two ways 3 how The CONTENTS. how to protract the fame, and examine the work, 103, &c. By the chain only ,' 'to take an angle in the field, \ 1 1 By the chain only, to furvey afield by going round the fame, \ \ 3 The common way taught by furveyors, for taking the plot of the foregoing field, 1 1 6 To take the plot of afield, at one jlation, in any part thereof, from whence all the angles may befeen, by the chain only, 1 1 9 CHAP. VII. How to caft up the Contents of a Plot of Land. F a fquarg and parallelogram, 120 Of triangles, 123 To find the content of a trapezium, 1 25 To find the content of an irregular plot, conjift- ing of many fides and angles, \ 2 7 To find the content of a circle, or any portion thereof, J2 g To find the content ofan'oval, 1 3 o To find the content of regular polygons, &c. 13 j CHAP. The CONTENTS. C HA P. VIII. Of laying out new Lands. /] Certain quantity of acres being given t how to -" lay out the fame in a /quare figure ', 1 3 2 To lay, out any given quantity of acres in a right- angled parallelogram, whereof one fide is given, 133 To lay out any given number of acres in a parallelo- gram that Jhall & 4, 5, 6, or 7, &c. times longer than it is broad^ ibid. To make a triangle upon agivcnbafe, that Jhall con- tain any number of acres , 134 To find the length of the diameter of a circle ', which Jhall contain any number of acres required, 136 ', CHAP. IX. Qf REDUCTION. *TO reduce a large plot of land or map into a lefs * compafS) according to any given proportion ; or e contra, bow to enlarge one t 137 To change cujlomary meafure intojlatute, and the con- trary , 141 Knowing the content of a piece of land, tojind whflt fcale it was plotted from, ibid. CHAP. The CONTENTS. CHAP. X, Inftnidlions for furvey ing a Manor, Coun- ty, or whole Country. ULES to be obferved in purveying a manor, 142 T0 take the draught tf a County or country*, 1 44 CHAP. XL v ~. '.'.\ * v v7> " v "vY*\rv.* '-vy?^ 7i\~n i>j^ '( :> ^^ Of dividing Land*; ia&srw VVta TOflA 3 " T i*^ " : " ' N -V -''^'rA^ fc ^iatv\' vil? '0 divide a triangle fever al ways, 146 To The CONTENTS. An eafy 'way of dividing lands, To divide a circle according to any proportion, by a line concentric with the fir ft, 158 CHAP. XII. Trigonometry* CTHE ufe of the table ofjines and tangents, 1 6a ^ To Jind the co-fine or fine- complement ; the co- tangent or tangent complement, of any given number of degrees and minutes, ibid. A fine or tangent, co-fine or co- tangent being given-, to Jind the degrees and minutes belonging thereto^ 1 6 1 Certain theorems for the better under/landing right- lined triangles, 1 63 In a right-angled triangle, the bafe being given, and the acute angle at the bafe, to Jind the bypo- thenufe and perpendicular, 165 To Jind the perpendicular, \ 66 The hypothenufe A C equal to 30 being firjl found, to find the perpendicular, 167 The perpendicular and acute angle A C B being given, to find the bafe and bypothenufe, ibid. The hypothenufe, and one of the acute angles given, to Jind the bafe and perpendicular, 169 The hypothenufe and bafe being given, to find the acute angles, A C B and CAB, 170 The bypothenufe and perpendicular being given, to Jind the acute angles, andalfo the bafe, 171 Of oblique-angled plain triangles, 173 Two angles being given, and a fide oppojite to one of them, to find the other oppojitejide > 175 To The CONTENTS. angle of a triangk tfrg ferpendicular, CHAP. XIII.- -tt~ . Of Heights and Diftances.,^, . W- : ^ -^-^v^. G E O D JE S I A: OR, THE Art of Meafuring Land, CHAR I. Of ARITHMETIC. IT is very neceffary for him that intends to be an artift in the Meafuring of Land, to be well ac- quainted with arithmetic, as being the ground- work and foundation of all arts and fciences mathema- cal ; or at leaft not to be ignorant of the five firft and principal rules thereof, viz. Numeration, Addition, Subt raft ion, Multiplication, and Divifion : which, fuppofing every Perfon that applies himfelf to the ftudy of this art to be fkilled in 5 or if not, refer- ing him to Books or Matters (every where to be found) to learn : 1 (hall name a fixth rule, as ne- ceflary (if not more) to be underftood by the learner ; B which 2 Of Arithmetic. which is the extra&ion of the fquare root ; without which (though feldom mentioned by furveyors in their writings) a man can never attain to a compe- tent knowledge in the art: I iMl not therefore think it unworthy my pains (though perhaps other rnen have better done it before mt) to fhew you eafily and briefly how to do it. To extratt the Square Root of any given Number. In the firft place, it is convenient to tell you what the fquare root is : it is to find out of any number propounded a lefs number, which number being multiplied in itfelf, may produce the number pro- pounded. As for example : fuppofe 8 i be a num- ber given me, I fay 9 is the root of it ; becaufe 9 multiplied in itfelf, .viz. 9 times 9 is 81. Now 8 could not be the root, for 8 times 8 is but 64 : nor could 10, for 10 times ten is joo; therefore, I fay, 9 muft needs be the root, becaufe multiplied in it- felf, it makes neither more nor lefs than the number propounded, viz. 81. Again : fuppofe 1 6 be the number given, I fay the root of it is 4, be- caufe 4 multiplied in it- . felf makes 16. For your 4 better understanding fee this figure, which is a great fquare, containing 1 6 little fquares ; any fide J I. 4* ' of which great fquare contains 4 little fquares : which is called the fquare root. Or, Of Arithmetic. 3 Or, fuppofe a plain fquare figure be given you, as this in the margin, and it be requir- O ed of you to divide it into 9 fmall r^ j | 1 fquares ; your bufinefs is to know in- |_ -4 to how many parts to divide any one ', _9 of the fide lines, which here muft be into 3, and that is the root required. . I < But how to do this readily is the thing I am now going to teach you. The roots of all fquare numbers ender 100, you have in your multipli- cation table ; however, fince it is good for you to keep them in your mind, take this fmall table of them. Roots |i | 2 I 3 | 4 I 5 I 6 | 7 I 8 | 9 Squares I' I 4 I 9 I l6 I 2 5 I 3 6 I 49 I &4 I *** Here you fee the root of 25 is 5, the root of 64 is 8, and fo of the reft. So far as 100 in whole numbers, your memory will ferve you to find the root ; but if the number propounded, whofe root a you are to fearch out, ex- ceed 100, then put a point over the firft figure on the right hand, which is the place of units, and fo proceeding to the left hand, mifs the fecond figure? and put a point over the third ; then miffing the fourth, point the fifth ; and fo (if there be ever fo many figures in the number) proceed on to the end, pointing every other figure, as you may fee here, and fo many points as there are, of fo many figures your root will 1234567 confift, which is very material to re- member : then begin at the firft figure on the left hand that has a point over it, which will always be B 2 ^ the ife L 4 Of Arithmetic. the firft or fecond figure, and fearch out the root of the one figure, or both joined together if there be two ; and when you have found it, or the nigheft lefs to it, which you may eafily do by the table above, or your own memory, draw a little crooked line, as in divifion, and there fet it down ; then fquare that root or quotient, (that is, multiply it by itfelf ) and fet it under the firft fquare ; draw a line and fubtract ; to this remainder bring down the next period, or pair of figures; double the root, and place it on the left hand for a divifor, then fee how often the divifor can be had in the refolvend, (excepting the figure to the right hand) and as often as you find it will go, place it in the root, and alfo in the unit's place of the divifor ; multiply the laft root by the divifor, and place the product under the refolv- end ; continue thus till all the periods or fquare num- bers be brought down, and if any thing remains, add two cyphers, and work as has been taught above ; and for every two cyphers thus added, there will be one decimal place in the root. EXAMPLE. To extracl: the fquare root of 99856. Point it as before (hewn, and it will ftand thus 99856; then feek a number whofe fquare (hall be equal to the firft figure 9, viz. 3, and write it in the root or quotient ; then having fubtracted from 9, 3 by 3, or 9, there will remain o ; to which fet down the fi- gures as far as the next point, viz, 98, for the fol- lowing operation : 2 Then Of Arithmetic. 5 Then taking no notice of the laft figure 8, fay, how many times is the double of 99856(316 3 or 6 contained in. the firft figure 9 ? Anfwer, i* Wherefore having fet i in the root, fubtract the product of i by 61, or 6 1 from 98, and there will remain 37 ; to which connect the laft figures 5 6, and you will have the number 3756, in which the work is next to be carried on ; wherefore, alfo, neglecting the laft figure of this, viz. 6, fay, How often is the double of 31, or 62, contained in 375 (which may be guefled at from the initial figures 6 and 37, by taking notice how many times 6 is contained in 37) ? Anfwer, 6 ; and writing 6 in the quotient, fubtract 6 by 626 or 3756, and there will remain o ; whence it appears that the bufinefs is done, the root coming out 316. Otherwife, with the divifors fet down, it will ftand thus : 99856 (316 2 61)098 61 626)3756 3756 Again, if you were to extract the root out of 22178791 : Firft, having pointed it, feek a num- ber, whofe fquare, (if it cannot be exactly equalled) B 3 fhall l6 6 Of Arithmetic. {hall be the next lefs fquare 22178791 (479>43 6 37> (or neareft) to 22, the fi- gures to the firft point, and you will find it to be 4 : for 5 by 5, or 25 is greater than 22, and 4 by 4, or 16 is lefs : wherefore 4 will be the firft figure of the root. This, therefore, be- ing writ in the quotient, from 22 take the fquare 4 4, or 16, and to the 609 88791 84681 411000 37 6 73 6 3426400 2825649 by 60075100 5651319^ 356190400 282566169 73624231 remainder 6, adjoin the next period of figures 17, and you will have 617 ; from whofe divifion, by the double of 4, you are to obtain the fecond figure of the root, viz. 7, as thus : neglecYmg the laft figure 7, fay, How many times 8 is contained in 6 1 ? An- fwer, 7; wherefore write 7 in the quotient, and from 617 take the product of 7 into 87, or 609, and there will remain 8 j to which join the two - next figures 87, and you will have 887 ; by the di- vifion whereof, by the double of 47, or 94, you are to obtain the third figure ; in order to which fay, How many times is 94 contained in 88 ?-Anfwer, o j wherefore write o in the quotient, and bring down the two laft figures 91, and you will have 88791, by whofe divifion by the double of 470, or 940, you are to obtain the laft figure , thus I fay, How many times 940 in 8879 ? Anfwer 9; where- fore write 9 in the quotient, and you will have the root 4709. But, fince the product 9 by 9409, or 84681, Of Arithmetic. 7 $468 1, fubtracled from 8 8 791, leaves 41 10, the num- ber 4709 is not the compleat root of the number 22178791, but a little lefs. If then it be required to have the root approach nearer, carry on the operation in decimals, by an- nexing to the remainder two cyphers in each ope- ration. Thus the remainder 4110, having two cy- phers added to it, becomes 4110005 the divifion whereof by the double of 4709, or 9418, you will have the firft decimal figure 4. Then having writ 4 in the quotient, fubtradt 4 by 94184, or 376736, from 411000, and there will remain 34264. And fo having added two more cyphers, the work may be carried on at pleafure, the root at length coming out 479>43 6 37 & c - . But when the root is carried half way or above, the reft of the figures may be obtained by divifion alone ; as in this example, if you had a mind to ex- trad the root to nine figures, after the five former 4709,4 are found, the four latter may be found, by dividing the remainder, viz, 342640, by the double of 4704,4. After the fame manner are roots extracted out of decimal numbers, Thus, the root of 329,76 is 18,159; anc ^ ^ e root f 3 2 97 k 1,8159, and the 0,032976 is 0,18159, and fo on. But the root of 3297,6 is 57,4247 ; and the root of 32,976 is 5,74247. And thus the root of 9,9856 is 3,16. There are other ways taught by arithmeticians for finding the fquare root of any number ; but I know no. way fo concife as this, and, after a little practice, fo eafy and ready, or to be wrought with as few figures. To do it indeed by the logarithms, or ar- tificial numbers, is very eafy and pleafant 5 but fur- B 4 veyors 8 Of Arithmetic. veyors have not always books of logarithms about them, when they have occafion to extract the fquare root ; however, I will briefly fliew you how to do it, and give you one example thereof. When you have any number given whofe fquare root you defire, feek for the given number in the table of logarithms under the title numbers ; and right againft it under the title logarithms, you will find the logarithm of the faid number, the half of which is the logarithm of the root defired : which half feek for under the title logarithm, and right againft it under the title number, you will find the root. #: EXAMPLE. Let 625 be the number whofe root is defired. Firft, J feek for it under the title numbers, and right againft it I find this Log. 2795880, which I divide by 2, or take the half of it 1,397640, as you fee : and finding that half under the title Log. right againft it is 25, the root defired. See the fame done by the former way with lefs trouble. * 625 (25 4__ 45) 22 5 225 CHAP. ( 9 ) ' - CHAP. II. GEOMETRICAL DEFINITIONS. A Point is that which has no parts j confequent- ly, of itfelf no magnitude, and may be con- fidered as indivifible. It is commonly marked as a full flop in writing, thus (.) A line has only length, but neither breadth or thicknefs, and may be conceived as generated by the continual motion of a point. There are two forts of lines, viz. ftrait and crooked ; as A B is a ftrait line, B C two crooked lines. An angle is the meeting of two lines in a point \ provided the two lines fo meeting do not make one ftrait line, as the. line A B, an and with the other defcribe the arch GG. C Laffly, 1 8 Geometrical Problems^ Laftly, lay your ruler to the point C, and the in- terfe&ion H of the two arches G G and F F ; then draw the line H C, you have your defire, H C be- ing* perpendicular to A B. See it- here done again after the very fame man- ner, but perhaps plainer for your underftanding. p R o B. n. To raife a perpendicular upon the end of a line. ^> ..0' B Geometrical Problems. 19 A B is the line given, and at B it is required to ereft a perpendicular, as B C. Open your compares to an ordinary extent, and fetting one foot in the point B, let the other fall at adventure, no matter where in reafon, as at the point o ; then without altering the extent of the compafles, fet one foot in the point o, and with the other crofs the line A B as at D : a!fo on the other fide defcribe the arch E E -, then laying your ruler to D and , ^ raw the dotted line D o F. Laftly, from the point B, you began at, through the inter- fedtionat G, draw the line BGC, which will be perpendicular to A B. Another I think rather more ea/y, ilwugb indeed nearly the fame. Let A B be the given line. ti- - H H /f" *-\ -P C 2 Set 20 Geometrical Problems. Set one foot of your compares in B, and with the other at any ordinary extent, defcribe the arch CEFD; then keeping your compaiTes at the fame extent, fet one foot in C, and make a mark upon the arch at E ; and keeping one foot in E, make another mark at F ; then with any extent let one foot in E, and with the other defcribe the arch G G ; alfo fetting one point in F, make the arch H H, then draw a right line, as IB, through the inter- fedion of thofe arches, and the point firft propofed, it: will be the perpendicular required. P ROB. JIT. Prom a point affigned, to let fall a perpendicular upon a line given. The line given is AB,' the point is at C, from which it is deiired to draw a right line down to A B. that may be perpendicular to it. V Firft, Setting one foot of your compares in tha point C, with the other make a mark upon the line A B as at D, and alfo ahother at E 5 then open- ing your companies wider, or (hutting them clofer, either will do; fet one foot in the point of in- terfeclion Geometrical Problems. 21 terfedion at D, and with the other defcribe the arch fg j do the like at E, for the arch h h. Laftly, rom the point afllgned, through the point of inter- feclion of the two arches, gg and hb, draw the per- pendicular line CF. P R O B. IV. To divide a line into any number of equal parts. A B is a line given, and it is re- quired to divide it into 6 equal parts. Make at the point B a line perpendicular to AB, asBC: do the fame at A, the contrary way, as you fee here; o- pen your com- paffes to any con- venient widenefs, and upon the lines BC and AD, mark out five e- qual parts; for it mu,ft be always ond lefs than the number you in- tend to divide the line into : which parts you may number, as you e "S 4 3 22 Geometrical Problems. fee here, thofe upon one line one way, and the other the contrary way ; then laying your ruler from N. i. on the line BC, to N. i. on the line A D, it will interfecT: the lint A B at E, which you may mark with your pen, and the diftance between B and E is one fixth part of the line ; fo proceed on 'till you ccme to N, 5. and then you will find that you have divided the given line into fix equal parts, as required. P R O B. V. To make an angle equal to any other angle givfn. The angle given is A, and you are defired to make one equal to it. Draw the right line B C, then going to the angle A^, fet one foot of your compafles in the point b, and with the other at what diftance you pleafe, defcribe Geometrical Problems. 23 defcribe the arch I K ; then, without altering the extent of the companies, fet one foot in B, and draw the arch fg j after that, meafure with your compares how iar it is from K to I, and the fame diflance fet down upon the arch from g towards f t which will fall at E ; draw the line BED, and the angle DEC will be equal to the given angle A. P R O B. VI. * To draw lines parallel to each other. A B is a line given, and k is required to draw a line parallel to it. c ^ ^N A o- Set one foot of your compafles at or near the end of your given line, as at E, and with the other defcribe the arch a b ; do the fame near the other end of the faid line j then draw th 2 line C D, juft to touch the convexity of thofe arches, and k will be the parallel required. P R O B. 24 Geometrical Problems. P R O B. VII. o draw a line parallel to another line, which Jhatt alfo pafs through a point ajpgned. Let A B be the given line, C the point through which the required parallel line mud pafs. , 7 7 / \ A ' ..*>:' p Set one foot of your compaffes in C, and clofing them fo that they will juft touch (and no more) the line A B, defcribe the arch a a ; with the fame extent in any part of the given line fet one foot, and defcribe another arch as at D ; then through the affigned point C, draw a right line C D, juft to touch the convexity of the laft defcribed arch, and it will be parallel to A B, as was required. P R O B. VIII. To form a triangle with three given right lines, pro- vided any two of thofe lines, taken together , be longer than the third, Let the three lines given, be thofe marked i, 2, 3, in the next figure. Take Geometrical Problems. 25 Take with your compares- the length of either of the three in this i ___ example : let it be that No. i. viz. 3 thelongeft,andlay it down as hereun- der from A to B ; then taking with your compafTes the length of the line 2, fet one foot in B, and make the arch C j alfo taking the length of the laft line 3, place your compafies at A, and defcribe the arch D, which will interfect the arch C, in the point e j from this point of interfedtion draw right lines to the points A B, which fhall con- ftitute the triangle A e B j the line A B being equal to the line N. i, B,^ to N*. 2, and A e to N. 3. P R O B. IX. To make a triangle equal and equiangular to a tri- angle given. Firft make an angle equal to the angle at A, a* yoa were taught in Prob. 5. Then making the lines A D and A E re- fpe&ively equal to A B and A C, draw the lineDE. Geometrical Problems. P R O B. X. defcribe a fquare figure upon a given right line. A Let A be a line given, and it is required to make a fquare figure thereon. Firft, Lay down the length of your line A, as AB. Secondly, Raife BC perpendicular and equal to B A. Thirdly, Take the length of either of the aforementioned lines with your compaffes, and fet- ting one foot in C, defcribe the arch e e \ do the like at A, and defcribe the arch ff. Fourthly, Draw lines from A and C to the point of inter ieclion, and the fquare is finifhed. P R O B. XL To make a right- angled parallelogram, or long fquare. 2 This is much like the former. Admit two N lines be given, as i, ^2, and it is required to make a parallelo- Q gram of them : what a parallel- Geometrical Problems. 27 a parallelogram is, you may fee in the fecond chap- ter of Definitions. Firft, Lay down your longeft line, as A B, upon the end B of which eredt a perpendicular line, equal in length to your ftiorteft line ; and fo pro- ceed, as you were caught in the foregoing problem. 5 P R O B. XJI. 70 defcribe a rbombus. Firft, Make an angle, A C B, of any magnitude at pleafure with two equal lines, as AC and BC; then taking one of thefe lines in in your compafTes, fet one foot in A, and defcribe the arch b b -, alfo fet one foot in B, and defcribe the arch c c. Laftly, draw lines from A and B to the point of interfe&ion, and it is fmimed. The two equilateral triangles together, form a rhombus. A rhomboides differs from a rhombus, as a right- angled parallelogram does from a fquarc; it being no other than an oblique parallelogram, -and there- fore it is needlefs to give its defcription. P R O B. Geometrical Problems. P R O B. XIII. 50 divide a circle into any number of equal parts > not exceeding ten, and alfo to Jhew how the figures called* Pentagon, Hexagon, Heptagon, Octagon, &c. may be described. Let A B C D be a circle, in which it is required to infcribe an equilateral triangle, being the greateft that can poffibly be made therein. Keeping your compafTes at the fame extent they were at when you made the circle, fet one point of them in any part of the circumfer- ence, as at A, and with the other make a mark at E, and alfo at f; then draw a line for E to f y which will be one fide of the triangle. I need not tell you how to make the other two fides, for as it is an equilateral triangle, all three fides will be of equal length. Geometrical Problems. 29 70 make a pentagon, or jive-feted figure. Draw firft an obfcure circle, as ACBDj then draw a diameter from A C to B ; make another dia- ?f " meter C D perpendicular // * to the firft; then taking with your compaffes the length of the femidiame- ter, fet one point in A, and make the marks E, F, drawing a line be- tween them, as you did to make a triangle. Next, fet one point of your compaffes in the interfeclion at g y and extend the other to C, draw the arch C H : and the chord of this arch, viz. the line CI'H, will be the fide of a pentagon, and the greateft five-fided figure that can be made within that circle. With this extent of your compaffes, you may mark out five points round the circle, and joining thofe points with right lines, the figure will be finifhed. To make a hexagon , or fix-fided Jigure. Draw an obfcure circle, as you fee here, and then, without altering the ex- tent of the compaffes, mark out the hexagon required round the circle ; for the femidiameter of /any circle is the fide of the greateft hexagon that can be made within the fame circle. This jo Geometrical Problems. This is the way coopers ufe to make heads for their cafks. make a heptagon, or figure of f even equal fides and angles. You muft begin and proceed, as if you were going to defcribe an e- quilateral triangle in a circle, 'till you have drawn the line EF; then taking with your compafles the half of that line, viz. from o to E, or from o to F, mark out round the circle your heptagon ; for the half of the line E F is the fide of it. 70 make an ocJagon, commonly called an eight-fquare figure. Firft, Geometrical Problems. 3 1 Eirft, make a circle. Secondly, Divide it into four equal parts by two diameters, the one perpendicular to the other, as AB and CD. Thirdly, Set one foot of the compares in A, and make the arch c e - y alfo with the fame extent fet one foot in C, and make the arch ff\ then through the interfeclion of the two arches draw a line to the center, viz. g h. Laftly, Draw the line I C or I A, either of which is the fide of an odagon. To make a nonagon. Firft, Make a cir- cle, and an equilate- ral triangle in it, as you were taught at Prob. XIII. Then divide the arch AB into three equal parts, ;\ / as A i, 12, and 62. \\ / Laftly, draw the lines A I, 12, 2 B, &C. quite round the cir- cle, and the nona- gon will be corn- plea ted. 10 Geometrical Problems* To make a decagon. You muft work altoge- ther as you did in making a pentagon. See the pen- tagon above, where the di- ftance from the center K, (fee the laft figure but 4) to the point at H, is the fide of a decagon, or ten-fided figure. P R O B. XIV. Three points being given : To defer 'lot a circle^ ivhofe circumference Jhall fafs through thofe given points, provided they are not in a ftrait line. Let A, B, C, be the three points given ; firft fet- ting one foot of your compafles in A, open them to any convenient widenefs, more than half the Diftance Geometrical Problems. . 33 diftance between A and B, and defcribe the arch dd \ then, without altering the extent, fet one point in B, and crofs the firft arch at e and e ; through thofe two interfections draw the line e e. The very fame you mud do between B and C, and draw the line ff\ where thofe two lines in- terfect each other, as at g, there is the center of the circle required j therefore fetting one foot of your compafles in g, extend the other to any of the points given, and defcribe the circle ABC. Note, the cen- tre of a triangle is found the fame way. P R O B. XV. ' T0 defcribe the carpenter's or common oval, federal ways ; alfo to defcribe an eHipfis. 34 Geometrical Problems. Fig. i. Defcribe a circle at pleafure, and through the center thereof draw an indefinite right line, as A B. On the two points where the circumference of this circle cuts the line A B, as centers defcribe circles with the fame radius as before j draw d c per- pendicular to A B, and paffing through the center of the middle circle. From the points C and d t draw C e, C h and dj\ dg : fet one foot of the compares in D, and extend the other to g, defcribe the arch g f' y with the fame extent, and one foot in C, de- . fcribe the arch b e ; thefe arches together, with the circular parts/>, g h, will form the oval AgfB cbA. required. Fig. 2. Draw any trapezium at pleafure, and pro- duce the fides indefinitely towards A, E, F, B, &c. Upon each angular point of this trapezium, as a center, with a radius equal to half the fide thereof, defcribe a circle ; which done, from the centers i, 2, with the radii I F, 2 B, defcribe the arches F C, B H, and the oval is finimed. Fig. 3 . This needs no defcription, it being fo like the two former figures, anji eafier than either of them. Here note, that you may make the ovals i and 3 of any determined length j for in the length of the firft, there is four femidiameters of the fmall cir- cles j and in the laft but three. If therefore any line was given you, of which length an oval was required, you muft take in your compafTes only the fourth part of the line to make the oval Fig. i . but the third part to make the oval Fig. 3, and with that extent you muft defcribe the. fmall circles : the breadth will be always proportional to the length. But if the breadth be given, you muft defcribe the circles Geometrical Problems. circles with one fourth part thereof, to make the oval Fig. 2. Fig. 4. This ellipfis is to be defcribed by having the length and breadth both given. Let A B be the length, C D the breadth of the required ellipfis. Firft, Lay down the line AB equal to the given length, and crofs it in the middle with the perpen- dicular C D, equal to the given breadth. Secondly, Take in half the line A B with your compafTes, viz. A e, or B e j fet one foot in C, and make two marks upon the line AB, viz. f and g ; alfo with the fame extent fet one foot in D, and crofs the former marks at f and g. Thirdly, at the point f and g fix two pins; or if it be a garden-plat, or the like, two ftrong flicks. Then putting a line about them, make faft the two ends at fuch an exact length, that, flretching by the two pins, the bent of the line may exactly touch A or B, or C or D, or h, as in this diagram it does at b ; fo moving the line flill round, it will defcribe a true ellipfis. D 2 P R O B. 36 Geometrical Problems. P R O B. XVI. 70 'divide a given line into two parts, which may be in fuch proportion to each other, as two given lines are. 60 D- 40 Let Geometrical Problems. 37 Let AB be the given line to be divided in fuch proportion as the line C is to the line D. Firft, From A -draw a line at pleafure, as A E ; then taking with your compafTes the line C, fet it off from A towards E, which will fall at F : alfo take the line D, and fet off from F to E. Secondly, Draw the line E B ; and from F draw F G parallel to E B, which (hall interfecl the given line A B in the point required, viz. at G; thus will A G and G B be in like proportion to each other, a$ C and D are. Example by arithmetic. The line C is 60 feet, perches, or any thing elfe ; the line D is 40 ; the line A B is 50 j which is required to be divided in fuch proportion as 60 to 40. Firft add the two lines C and D together, and they make i ao : then fay, If i oo the whole, give 60 for its greater part, what (hall 50, the whole line A B, give for its greater proportional part ? Multiply 50 by 60, it makes 3000 ; which, divided by iocy produces 30 for the greater part ; which 30 taken from 50, leaves 20 for the leffer part : as therefore 60 is to 40, fo is 30 to 20. - ^ , P R O B, XVII. To find a fourth In proportional to three given lines. Let A B C be the three given lines: it is required to find a fourth line, which may be in fuch proportion to C, as B is to A, D, 3 which g 8 Geometrical Problems. which is no more than performing the Rule of Three by lines ; thus, If A 14 give B 18, then what mall C 2 1 give ? Anfwer, 27. But to perform the fame geometrically, work thus : Firft, Make any Angle, as B A C : then take with your compares the firft line A, and fet it from A *o 14. A lib take the fecond line B, and fet it from A to 1 8 j draw the line 14, 18. Then take the third line C with your compares, and fet it from A to 2 1. From 21 draw 21, 27 parallel to 14, 1 8. Then will A, 27 be the length of your fourth line required. And here for a while I {hall leave thefe Problems^ 'till I come to mew you how to divide any piece of land , and to lay out any piece of a given quantity of acres into any form or figure required : and in the mean time I {hall (hew you in the next chapter, what is neceflary to be known relating to meafures of various kinds. CHAP. ( 39 ) C H A P. II. Of M E A S U RE S. AN D firft of long meafures ; which is either inches, feet, yards, perches, chains, &c. Note, that twelve inches make one foot, three feet one yard, five yards and an half one pole or perch, Four perches one chain of Gunter*^ eighty chains one mile. But if you would bring one fort of mea- fure into another, you muft work by Multiplication or Dhi/ion. As for example : Suppofe you would know how many inches are contained in twenty yards ; firft, reduce the yards into feet by multiply- ing them by 3, becaufe 3 feet make one yard, the product is 60 j which, multiplied by 12, the num- ber of inches in one foot, gives 720, and fo many inches are contained in 20 yards length. On the contrary, if you would know how many yards there are in 720 inches, you muft firft divide 720 by 12, the quotient is 60 feet; that again di- vided by 3, the quotient is 20 yards. The like you muft do with any other meafure, as perches, chains, &c. of which more hereafter. Long Link Foot Yard Perch Chain Mile Inches 7.92 12 36 198 792 63360 Link MM 456 2 5 100 8000 Feet 3 16.5 66 5280 Yard 5-5 22 1760 Perch 4 Chain 320 ~~8cT See 40 Of Meafure. See this table of the long meafure annexed, the ufe whereof is very eafy. If you would know how many feet in length go to make one chain, look for chain at the top, and at the left hand for feet ; againft which, in the common angle of meet- ing, is 66 ; fo many feet are contained in one chain. But becaufe Mr. Gunters chain is moft in ufe among furveyors for meafuring of lines, I {hall chiefly infill on that meafiyre, it being the beft in ufe for lands. This chain contains in length 4 pole, or 66 feet, and is divided into i oo links, each link is therefore in length 7 -AV inches : if you would turn any num- ber of chains into feet, you muft multiply them by 66, as 100 chains multiplied by 66, make 660 feet 5 but if you have links to your chains, to be turned into feet, and parts of feet, you muft fet down the chains and links, as if they were one whole num- ber, and after having multiplied that number by 66, cut off from the product the two laft figures to the right hand, which will be the hundredth parts of a foot, and thofe on the left hand the feet required. EXAMPLE. Let it be required to know how many feet there are in 15 chains, 25 links. I fet down thus the Multiplicand I 5> 2 5 The num. of eet in one chain, Multiplicat. 6 6 9150 Produtt 1 006150 feet. The Of Meafure. 41 The />r0 links. In like manner, if it had been afked how many perches had been contained in 15 chains, 25 links? Of Meafure. 43 In the table again/I 10 Perch. Parts. chains (lands 40 5 20 20 Links. QO 80 5 Links. po 20 Anfwer, 60 Percfas. 60 100 Pbferve, that the foregoing table is as large again as ft need to be ; for you fee both the columns are alike in figures, and only differenced by points. I made it fo for your clearer underftanding of it ; which, when you well do, you need ufe no more than one column ; and that, if you ple'afe, you may have placed on a feale, or any. other inftrument. But now, to bring a lefs meafure into a greater, re- quires divifion, and is therefore more troublefome than to bring a greater into a lefs, which requires only multiplication : I have therefore, for your eafe, hereto annexed a large table, with which, by in- fpection only, or at raoft by a little eafy addition, as in the former, you may change any number of feet into chains, links, and parts of a link (Mr. Gunter's chain is here ufed) alfo into perches, and pares of a perch. A TABLE, 44 Of Meafure. Feet. Chain jink J .ofL Per. P. of Per. i c r 5'5 060 2 3 030 121 >~ 3 4 o 4 6 545 ooo o 1*1 242 5 o 7 575 33 D t""* J> ~ c 9 090 363 ? 5* w 7 o 10 606 o 424 o .. ja 8 o 12 121 o 484 9 10 13 636 c o III S'lj? 20 c 3 303 1 212 3 o* S- 3 o 454 1 818 ^ D 40 o 60 606 2 424 ^ MI i-r- 5 o 75 757 3 030 III 60 70 c I 9 06 909 060 3 4 636 424 Its 80 I 21 212 4 848 o ^ ^y 300 t 54 545 lg 181 v. p -. 400 6 c6 060 24 242 tT^ Q-P qro 7 57 575 3 30; M 6oc c 09 090 3^ J6 3 ^ ET- 700 10 60 606 4* i %% & 800 12 12 121 48 484 s o ? 900 13 63 636 54 545 O >-t^ loor Jj K '5' 60 006 3 " S 20CO 3 30 33 121 212 o *0 a- Ort ^ 3 coc 4ooc 60 g 454 606 181 242 8l8 424 ^ ^" ^^ r^ qooo 7 75 757 3^3 OgO M P 6000 90 90 909 363 636 "t o 7000 1 06 06 060 424 2 4 2 8oco 12 21 212 484 848 9000 '3 36 363 0*0 454 060 This Of Meafure. 45 This table is like the former, and needs not much explanation. However, I will give an exam- ple or two. Admit I would know how many chains in length are contained in 500 feet. Firft, in the left-hand column, under title Feet, I look out 500, and right againft it I find 7 chains, 57 links, 575 parts of 1000 of a link, or 7 chains, 57 -^^ links. So likewife under title Perches, I find 30 AVs- perches. But if you would know how many feet the fraction -rrv contains, you muft feek for 303 in the column titled Parts of a Perch, and right againft it you will find 5 feet. So I fay that 500 feet is 30 perches, 5 foot. Again, I would know how many chains and links there are in 15045 feet. Firft, feck for 10000, and write down the chains, links, and parts of a link contained therein. Do the like by 5000 ; alfo by 40 and 5. Laftly, adding them together, you have your defire. Feet. Chain. Link. Parts. 10000 151 51 515 5 ooo 75 75 757 40 o 60 606 5 ~ o 7 575 Added make 227 95 453 Anfwer, 227 chains, 95 links, and 453 parts are contained in 1 5045 feet. One Example more will be fufficient for the ufe this table. How many perches do 10573 feet make ? Fett. 4 6 Of Meafure. Feet. Perches. Parts. icooo 606 060 5 3 70 4 3 o 33 242 181 Add 640 786 The anfwer is, 640 perches, and J- of a perch or 13 feet; a furlong is 40 perchesr in length; 8 furlongs make i mile. Thus much for Long Mea- fure : I (hall now proceed to Square Meafure. Planometry, or the meafuring Superficies or Planes (as Sir Jonas More fays) is done with the fquares of fuch meafures, as a fquare foot, a fquare perch or chain ; that is to fay, by fquares whofe fides are a foot, a perch, or chain ; and the content of any fuperficies is faid to be found, when we know how many fuch fquares it contains. As for example : Suppofe A BCD was a piece o\ land, and the length oj the line A B or CD was 4 perches ; alfo the length of the line A C or B D was 5 perches ; I fay, that piece of land contains 20 fquare perches, as you may fee it here divided ; every little fquare being a perch, having a perch in length * for its fide. If you lay down a fquare figure, 3 whofe Of Meafure. 47 whofe fide is i foot, and at the end of every inch you draw lines croiiing one another, as thefe here, you will divide that fquare foot into 114 little fquares, or fquare inches. Or thus : The line a b is a perch long, or 1 6 feet i, fo is the line b d? and the other 2 lines : the whole fi- gure abed is called a fquare perch. But before we go any farther, take this table fol- lowing of fquare meafure. f > O jq ? < % S ? 5- U r - o fr % D S- 8T ^ fc ON N IM VO N _ ? ** I ^1 *jj O VO ON ^ < _t- ON ON M la OO ^i O\ | <" o O * 8 i ON "o g 8 N wl ~^ O M EH o 8 OO JJi y, 9 2T o H OO 5 l|fe N v^ * fl OS ON rt 00 O VI : a 1 s 00 ?$ "? 4J * * --J ~o sf 1 o z 5 g - V^l "^ * OO B* M O * f> o 8 o 6 * I f o ~ g o ON - r> Os O ' ' "=~ s" r CO 48 Of Meafure. This Table is like the former of long meafure, and the ufe of it is the fame. Example : If you would know how many fquare feet are contained in one chain, look for feet at top, and chain on the fide, and in the common angle of meeting ftands 43 56, fo many fquare feet are con- tained in one fquare chain. The common meafure for land is the acre, which by ftatute is appointed to contain 160 fquare perches, and it matters not what form the acre is of fo it contains juft 1 60 fquare perches -, as in a right angled parallelogram, 10 perches one way, and 16 another, contain an acre : fo does 8 one way, and 20 another ; and 4 one way, and 40 another. If then, having one fide given in perches, you would know how far you muft go on the perpendicular to cut off an acre: you muft divide 160 (the number of fquare perches in an acre) by the given fide, the quotient is your defife. As for example : the given fide is 20 perches, divide 160 by 20, the quotient is 8 : by trut I know, that 20 perches one way, and 8 another, including a right angle, will be the two fides of an acre ; the other two fides muft be parallel to thefe. And here I think it convenient to infert this ne- ceflary table, (hewing the length and breadth of an acre in perches, feet, and parts of afoot : But if your given fide had been in any other fort of meafure ; as for inftance in yards, you muft then have found the fquare yards in an acre, .and that number you muft have divided by the given number of yards, the quoti- ent would have anfwered the queftion. 2 EXAMPLE.- Of Meafure. 49 EXAMPLE. If 44 yards be given for the breadth, how many yards muft there be in length to contain juft one acre. Firft, I find that an acre contains 4840 fquare yards, which I divide by 44, the quotient is no for the length of an acre. And thus knowing well how to take the length and breadth of an acre, you may alfo, by the fame way, know how to lay down any num- ber of acres together ; of which more hereafter. Reducing of one fort of fquare meafure to another, is done, as before taught in long meafure, by multipli- cation and divifion. And becaufe Mr. Gunter's chain is chiefly ufed by fur- veyors, I ihall only inftance in that, and mew you how to turn any number of chains and links into acres, roods, and perches. Note, that a rood is the fourth part of an acre. cc s Length of an CO Length of an C- Acre. Acre. o' n n 1 | 3 if sr y n 2* n "^ Te 28 ~ i M 20 "Y 8 TT 1 2 .13 5i- .-o 5 5^ '3 ~2 5TT 3' J 2 T Li 1 1 7k 3; J ~ '5 1C ! I 33 _4 16 1< O 3-' 4 Il-f 17 _9 ^TT *5 "7 9/r 18 fc '4Tj: 36 _4 ~ 19 8 6 ii 37 4 qi zc 1 & _4 ~ 21 _7 I0f 39 4 if 22 J ~ 40 4 11 6 i<;| 4' 3 4K -i 6 I li _3 '31 2* t ^ 43 3 '-'T "6 ~6 2 T7 ~ 44 ? ioi 7 ~5 4- 1" 2 f And ^o Of Meafure. And firft, obferve that 10 fquare chains makd i acre, that is to fay, I chain in breadth, and 10 in length ; or 2 in breadth, and 5 in length, is an acre, as you may fee by the fmall table here annexed. p Q. ET. S 3 And thus knowing that . TT % 10 chains make an acre^ p ^ if any number of chains P be given to be turned in- to acres, you muft di- g- vide them by 10, and the i 10 Co quotient will be the num- ""2 EP c oo her of acres contained in S/i 2 9 ? -33 -2 fo many chains. But this Crq 3 O. 6 oS JJJ .. .,, J . , , . , S-A. & 2 50 diviiion is abbreviated by o ^ ^ 2 oo nly cutting off the laft ^6 p i 66 666 figure; thus if 1590 ^7 i 42 285 chains were given to be og q x 25 turned into acres, by cut- ? ^ " i ii Hi ting off the laft figure 159(0, there is left 159 acres, which is the fame thing as if you had divided I 59 by 10. If any chains remain after the divifion by 10 is ended, they may be reduced into roods and perches, thus, Multiply the remaining number of fquare chains by 4, and cut off one figure from the right hand towards the left, the remaining fi- gures will denote the number of fquare roods con- tained in the faid chains ; this done, multiply the fi- gure cut off by 40 ; then, from the product, cut off one figure in the fame manner as before, and you will have the number of perches required, exprelTed by the remaining figures towards the left hand. EXAMPLE. Of Meafure. 51 EXAMPLE. In 1599 fquare chains, how many acres, roods, and perehes ? Acres I 59l9 4 Anfwer, 1 59 acres, 3 roods, and 24 perches. Roods 3|6 40 Perches 24(0 On the contrary, if to any number of acres given you annex a cypher, they will be turned into chains. Thus 99 acres are 990 chains, 100 acres 1000 chains, &c. the fame as if you had multiplied the acres by 10. And if you would turn fquare chains into fquare links, annex four cyphers to the end of the chains ; thus 990 chains will be 9900000 links, 1000 chains be 10000000 links ; the very fame as if you had multiplied 990 by 10000, the number of fquare links contained in one chain. And now, whereas in cafling up the content of a piece of land meafured by Mr. Gunfers chain, (viz. multiplying chains and links by chains and links) the 3roduct will be fquare chains and parts ; you muft :herefore from that product cut off five figures to ind the acres; which is the fame as dividing the 3roduc"l by 100000 (the number of fquare links contained in one acre) then multiply the five figures cut off by 45 and from that product cut off five E 2 figures, 52 Of Meafure. figures, you will have the roods. Laftly, multiply by 40, and take away (as before) 5 figures, the reft are perches. EXAMPLE. Admit a right-angled parallelogram, or long fquare, to be one way 5 chains, 55 links ; and the other way 4 chains, 35 links : I demand the content in acres, roods, and perches ? Multiplicand 555 Multiplicator 435 2775 1665 222O Anfwer, 2 Acres ~| Acres 2(41425 4 i Rood J Roods 1)65700 4 And T\\ Parts of a Perch. Perches 26(28000 Laftly, Becaufe fome rather chufe to caft up the content of land in perches, 1 will here briefly mew you how it may be done ; which is only by dividing by 1 60 (the number of fquare perches contained in one acre) the number of perches given. EXAMPLE. Admit a parallelogram to be in length 55 perches, and in breadth 45 perches ; thefe two multiplied together, make 2475 fquare perches: to turn thefe into acres, divide by 160, the quotient is 15 acres, and 75 perches remain j which, to turn into roods, divide Of Meafure. -- divide by 40, the quotient is i rood, and ,r perches remain. So the content of this piece of 'fnd Ti I acres, i rood, and 35 perches 5 and H roodf WS * *"* tO **** The Ufe of the TABLE. In 2475 perches, how many acres and roods ? Perch. Acr. Rood. Per. 2000 12 2 00 40 2 2 00 7 o i 30 To which add the odd 5 perches O O 05 jfnfwer i c j ^ ^ CHAP. [ 54 ] CHAP. V. Of Inftruments, and their Ufes. And frft, of the Chain. THERE are feveral forts of chains, as Mr. Rathborne's of two perches in length ; others of one perch long : fome of i oco feet in length. But that which is moft in ufe among furveyors (as being indeed the beft) is Mr. Gunter's, which is 4 pole long, and contains 100 links, each link being 7 -rsro- inches in length. The defcription of this chain, and how to reduce it into any other meafure, you have at large in the foregoing chapter of mea- fures. In this place, I {hall only give you fome few directions for the ufe of it in measuring land. Take care that they who carry the chain deviate not from a ftrait line j which you may do by (land- ing at your inftrument, and looking through the fights. If you fee them between you and the mark obferved, they are in a ftrait line, otherwife not. But without this trouble, they may carry the chain true enough, if he that follows the chain always caufes him that goeth before to be in a direct line between himfelf and the place they are going to, fo as that the foreman may always cover the mark from him that goes behind. If they fwerve from the Of Inftruments and their Ufes. 55 the line, they will make it longer than it really is, a ftrait line being the neareft diflance that can be between any two places. Be careful that they which carry the chain, miflake not a chain either over or under in their account ; for if they mould, the error would be very confi- derable ; as fuppofe you were to meafure a field that you knew to be exactly fquare, and therefore need only meafure one fide of it; if the chain- carriers fhould miftake but one chain, and tell you the fide was nine chains, when it was really 10, you would make of the field but 8 acres and 1 6 perches, when it mould be 10 acres juft. And if in fo fmall a line fuch a great error may arife, what may be in a greater you may eafily imagine j but the ufual way to prevent fuch miftakes, is to be provided with 10 fmall flicks, fharp at one end, to flick in the ground ; and let him that goes before take all the flicks in his hand at fetting out, and at the end of every chain flick down one, which let him that follows take up : when the i o (licks are ufed, you will be fure they have gone 10 chains: then if the line be longer, let them change the flicks, and proceed as before, keeping in memory how often they change : they may either change at the end of 10 chains, then the hindmoft man muft give the foremofl all his flicks ; or, which is better, at the end of 1 1 chains, and then the lafl man muft give the firfl but nine flicks, keeping one to himfelf. At every change count the flicks, for fear left you have dropt one, which fometimes happens. If you find the chain too long for your ufe, as for fome lands it is, efpecially in America y you may E 4 then 56 Of Inftruments and their Ufes. then take the half of the chain, and meafure as be* fore, remembering ftfll, when you put down the meafure of the lines in your field-book, that you fet down but the half of the chains, and the odd links j as if a line meafured by the little chain be u chains, 25 links, you muft fet down five chains, 75 links ; and then in plotting and cafting up, it will be the fame as if you had meafured by the whole chain. It is ufual at the end of every 10 links to have a ring, or a piece of brafs fixed, for your more ready reckoning the odd links. When you put down in your field-book the length of any line, you may fet it thus, if you pleafe, with a ftop between the chains and links, as 15 chains, 15 links, 15.15; or without, as thus, 1515 links, it will be all one in the cafling up. Of injlruments neceffary for taking of an angle In the feld. There are but two material things (towards the meafuring of a piece of land) to be done in the field ; the one is to meafure the lines (which I have fhewed you how to perform by the chain), and the other is to take the quantity of an angle included by thofe lines j for which there are almoft as many in- ftruments as there are furveyors. Such among the reft as are moft efteemed, are the plain table for fmall inclofures, the femicircle for champaign grounds, die circumferentor, the theodolite, &c. To defcribe thefe to you, with their feveral parts, how to put them together, take them afunder, &c. is like teaching the art of fencing by book ; one hour's ufe of them, or but looking on them in the inflrument* Of Injlntments and their Ufes. 57 jnftrument-maker's (hop, will better defcribe them to you, than the reading one hundred fheets of pa- per concerning them. Let it fuffice that the only ufe of them all is no more (or chiefly at moft) but this, viz. To take the quantity cf an angle. As fuppofe A B and A C are two hedges, or other fences of a field, the chain ferves to mea. fure the length of the fides A B or A C, and thefe inftruments we are fpeaking of, are to take the angle A. And firft, by the Plane Table. Place the table (already fitted for the work, with a meet of paper upon it) as nigh to the. angle A as you can, the north end of the needle hanging di- redtly ever the flower-de-luce-, then make a mark upon the meet of paper at any convenient place for the angle A, and lay the edge of the index to the mark, 5 8 Of Inftruments and their Ufes. mark, turning it about, 'till through the fights you view the point B, then draw the line A B by the edge of the index. Do the fame for the line A C, keeping the index ftiil upon the firfl mark, then will you have upon your table an angle equal to the angle in the field. 'To take the quantity of the fame angle by the femi circle. Place the center of your femicircle in the angular point A, and caufe marks to be fet up near B and C, fo far off the hedges, as your inftrument at A ftands j then turn the inftrument about, 'till through the fixed fights you fee the mark at B, there fcrew it faft : next turn the moveable index, 'till through the fights thereof you fee the mark at C, then fee what degree upon the limb is cut by the index; which let be the 45th, then is the angle A B C 45 degrees. To take the fame angle by the circitmferentor. Place your inftrument, as before, at A, with the flower-de-luce towards you, direct your fights to the marks at B, and fee what degree is cut by the fouth end of the needle, which let be the 55th j do the fame to the mark at C, and let the fouth end of the needle there cut i oo j fubtracl the lefTer out of the greater, the remainder is 45, the angle required. If the remainder had been more than 180 degrees, you muft then have fubtracted it out of 360, the lafl remainder would have been the angle de fired. This Of Inftrumsnts and their Ufes. 5 9 77?;; lajl inftrument depends wholly upoa the needle for taking of angles, which often proves er- roneous; the needle yearly of itfelf varying from the true north, if there be no iron mines in the earth, or other accidents to draw it afide, which in moun- tainous lands are often found : it is therefore the beft way for the furveyor, where he poffibly can, to take his angles without the help of the needle, as is be- fore fhewed by the femicircle. But in all lands it cannot be done j but we mull fometimes make ufe of the needle, without exceeding great trouble, as in the thick woods of Jamaica, Carolina, &c. Jt is good, therefore, to have fuch an inftrument as a fe- micircle, with which an angle in the field may be taken either with or without the needle, and there is no better inftrument that I know of, for the fur- veyor's ufe, yet made public ; therefore, as I have before (hewed you, how by the femicircle to take an angle without the help of the needle, I (hall here direct you, How to take the quantity of an angle in the field by the femicircle and needle. Screw faft the inftrument, with the north end of the needle hanging directly over the fawer- de-luce in the card ; turn the index about, till through the fights you fee the mark at B j and note what degree the index cuts, which let be the 4oth j move again the index to the mark at C, and note the degree cut, viz. 85. Subtract the lefs from the greater, remains 45, the quantity of the angle. Cr 60 Of Inftrument $ and their Ufes. Or thus : Turn the whole inftrument, 'till thro' the fixed fights you fee the mark at B ; then obferve what de- gree upon the card is cut by the needle ; which, for example, fuppofe the 31 5th. Turn alfo the inftru- ment, till through the fame fights you efpy C, and note the degree upon the card then cut by the needle, which fuppofe 270; fubtract the lefs from the greater, (as before in the working by the circumfe- rentor) remains 45 degrees for the angle. Note, if you turn the Flower -de- luce towards the mark, you muft look at the north end of the needle for your degrees. Befides the djvifion of the card of the femicircle into 360 equal parts or degrees, it is alfo divided into four quadrants, each containing 90 degrees, begin- ning at the north and fouth point, and proceeding both ways till they end in 90 degrees at the eaft and weft points 5 thofe points are marked contrary, viz. Eaft with a W, and Weft with an E ; becaufe when you turn your inftrument to the eaftward, the end of the needle will hang upon the weft fide, &c. If by this way of divifion of the card, you would take the aforefaid angle, direct the inftrument fo, (the jpr&er-je-Iuce from you) 'till through the fixed fights, you efpy the mark at B ; then fee what de- grees are cut by the north end of the needle, which let be N E 44 ; next direct the inftrument to C, and the north end of the needle will cut N E 89 ; fubtract the one from the other, and there will re- main 45 degrees for the angle. But if at the firft fight the needle had hung over NE 55, and at the fecond SE 80, then take 55 from Of Inftruments and their Ufes. 6 1 from 90, remain 35; take 80 from 90, remains io; which added to 35, makes 45, the quantity of the angle : moreover, if at the firft fight the north- end of the needle had pointed to N W 22, and at the fecond NE 23, thefe two muft have been added together, and they would have made 45, the angle as before. Note, if you had turned the fcuth-part of your inftrument to the marks, then you muft have had refpect to the fouth-end of your needle. Although 1 have been fo long (hewing you how to take an angle by the needle, yet when we come to furvey land by the needle, as you mall fee farther on, we need take but half the pains j for we take not the quantity of the angle included by two lines, but the quantity of the angle each line makes with the meridian ; then drawing meridian-lines upon paper, which reprefent the needle of the inftrument, by the help of a protractor, which reprefents the inftrument, we readily lay down the lines and angles in fuch proportions as there are in the field. This way of dividing the card into four quarters each being 90 degrees is, in my opinion, for any work the beft , but there is a greater ufe yet to be made of it, which (hall hereafter be mewed in its proper place. Of the Field-Book. You muft always have in readinefs in the field a little book, in which fairly to infert your angles and lines ; which book you may divide by lines into columns, as you lhall think convenient in your practice ; leaving always a large column to the right-hand, to put down what remarkable things you meet with in your way, as ponds, brooks, mills, trees, or the like. Thus, for example, if you 62 Of Inftniments and their Ufes. you had taken the angle A, and found it to contain! 45 degrees ; and meafured the line AB, and found it to be 12 chains, 55 links, let it down in your field- book thus : I 1 l^gr. 1 A 1 4 < iviin. oo Chain. 12 Link. 1 i or if at A you had only turned the fixed fights to B, and the needle had then cut 315? in the place of 45, you muft have put down 315. If you fwrvey by Mr. Norwood's way there muft be four columns more for the cardinal points E. W. N. and South. You may alfo make two columns more, if you pleafe, for ofF-fets, to the right and left* Laftly you may chufe whether you will have any lines or not, if you can write {trait, and in good order, the figures directly one under another. For this I leave you chiefly to your own fancy ; for I believe there are fcarce two furveyors in England^ that have exactly the fame method for their field notes. Of the Scale. Having by the inftrument before fpoken of, mea- fured the angles and lines in the field ; the next thing to be done, is to lay down the fame upon paper j for which ufe the fcale ferves. There are feveral forts of fcales, both large and fmall, according as men have occafion to ufe them : but all do princi- pally confift of no more but two forts of lines ; the firft is of equal parts. For the laying down chains and links ; the fecond a line of chords, for laying down or meafuring angles. I cannot better explain the fcale to you, than by {lie wing the figure of fuch a one as are commonly fold in {hops, and teaching how to ufe it. The 11 12 16 M 5 15.344'^rf fl L--L-; ____ JO ^ - ,- ^ - - _ is. .2*. 30 To IO Jo 5o K = i "o F 40 2.0 -2,0 1 30 12. & 60 - 33 41 \ [ 50 4o 100 ~ r Qo pa. 80 o 120 ^ 70 i ~. 5-0 S!L 4o_ So" 6 20 to" ^ = = *- -- -- ? t i 1 j " J ?-4 rt^ '( n 64 Of Injlrumenls and their Ufes. The lines numbered at top with u, 12, 16, Gfc are lines of equal parts, containing u, 12, or 16 equal parts to an inch, foot, yard, &c. If now by the line of 1 1 to an inch, you would lay down i o chains 50 links, look down the line under 1 1, and fet- ting one foot of your compafles in i o, clofe the other till it juft touch 50 links, or half a chain, in the fmall divifions. Then laying your ruler upon the paper, by the lide thereof, with the fame extent of the compafles, ._ _r mark the points A and B, and draw the right line AB, which will mea- fure in length 10 chains, 50 links, by the fcale of 1 1 to .an inch. On the back-fide of the fcale, there is a line of 10 equal parts to an inch, but divided by diagonal lines, for the more readily taking off any required line or number to 100 parts. 70 lay down an angle by the line of chords. Let it be required to make an angle that fhall contain 45 degrees. Of Inftruments and their Ufes. 6 5 Draw a line at pleafure, as A B : then fetting one foot of your compaffes at the beginning of the line of chords, fee that the other fall jufl upon 60 degrees : with that extent fet one foot in A, and defcribe the arch C D. Then take from your line of chords 45 degrees, and fetting one foot in D, make a mark upon the arch at C, through which draw the line A E : fo mall the angle E A B be. 45 degrees. If by the line of chords you would erect a line perpendicular to a given one, it is no more than to make an angle that mall contain 90 degrees. The rcafon why you are to take 60 from the line of chords to make your arch by, is, becaufe the chord of 60 degrees is the femidiameter of a circle, whofe circumference is divided into 360 equal parts. To make a regular Polygon, or a fgure of 5, 6, 7, 8, or more, Jides, by the line of chords. Divide 3 60, the number of degrees contained in a circle, by the number of fides you would have your figure to contain ; the quotient taken from the line of chords mall be one fide of fuch a figure. EXAMPLE. For to make a pentagon, or a figure of five fides : divide 360 by 5, the quotient is 72, one fide of a pentagon. Take 60 degrees from your line of chords, and defcribe an obfcure circle 5 which done, take 72 for your F 66 Of Inflruments and their Ufes. your line of chords, and beginning at any part of the circle, fet off that extent round the circle, as from B to C, and fo round till you come to A again. Then having drawn lines between thofe marks, the pentagon is compleated. The like of any other polygon, be the number of fides what they will. -. Example. For the fide of a heptagon: divide 360 by 7, the quotient will be 51 degr. 25 min. which if you take from the line of chords, and fet off round the circle, you will form a heptagon, D E, EF, FG, &c. being the fides thereof. Of Inflruments and their Ufes. 67 To infcribe an equilateral triangle in a circle by the line of chords. Firft, Take the whole length of your line of chords, or the chord of 90 degrees, with your com- paffes ; this diftance fet off from C to *. Then take 30 degrees from the line of chords, and fet that from * to H. Draw the line C H, which will be one fide of the greateft triangle that can be infcribed in that circle. Or you may make it by fetting off twice the fe- midiameter of the circle -, for 60 and 60 is 120, equal to 90 and 30. To make a line of chords. Firft, Make a quadrant, or the fourth part of a F 2 Circle, 68 Of Inftruments and their Ufes. circle, as A B C j divide the arch thereof, viz. A C, into 90 equal parts ; which you may do by dividing it firft into three equal parts, and every of thofe divifions into three equal' parts more, and every of the laft divifions into ten equal parts. Secondly, continue the femidiameter BC to any convenient length, as to D. Then fetting one foot of your compares in C, let the other fall on 90, and defcribe the arch 90, 90. So likewife 80, 80 j 70, 70, &c. then will C D be the line of chords, and thefe arches cutting it into unequal parts, conftitute the true divifions thereof, as you may fee by the figure : you may, if you pleafe, draw lines parallel to D C, as I have done here, for the better diftmgui(hing every tenth and fifth figure. Of the Profra&or. The protractor is an inftrument with which, with more eafe and expedition you may lay down an angle, than you can by the line of chords : alfo- when you have furveyed by the needle, by placing the diameter of the protractor upon a meridian line made upon your paper, you may readily, with a needle upon the arch of the protradlor, fet off the true fituation of any line from the meridian without fcratching the paper, as may happen if you ufe the line of chords. It is graduated together like the brafs limb of a temicircle, performing the fame upon paper, as that inftrument did in the field : fee here the figure of it, For Of Injiruments and their Ufes. 69 For the ufe of the protradtor, you muft have a fine fewing needle, put into a fmail handle of wood, or ivory, or the like, which is to put through the center of the protractor to any point affigned upon the paper, that the protractor may turn round upon it. To lay down an angle with the protra&or. If it were required by the protractor to lay down an angle of 30 degrees, draw the line AB, then take the protractor, and putting a needle through the center point thereof, place the needle in A, fo that the center of the protractor may lie juft upon the end of the line at A, move the protractor about till you find the diameter thereof lie upon the liae A B $ then at 30 degrees upon the arch, F 3 with yo Of Inftruments and fMr Iffes* with your protracting needle make a mark upon the paper as at C; draw the line C A, which (hall make an angle, as B AC, of 30 degrees. If you furvey according to Mr. Norwood's method before fpoken of, it will be neceffary to have the arch of your protractor divided accordingly, viz. into two quadrants, or twice 90 degrees. I need fay no more of a protractor, any inge- nious man may eafily find the feveral ufes thereof; it being, as it were, but only an epitome of inftru- ments. CHAP. CHAP. VI. 70 tale a plot of a field at one Jlation in any place thereof ^ from whence you can fee all the angles by the femicircle. ADmit A B C D E F to reprefent a field, of which you are to take the plot : Firft, fet your femicircle upon the ftaff in any convenient place thereof, as at o, and caufe marks to be fet up in every angle : Diredt your inftrument, the fower-de- luce being from you, to any one angle : as for ex- ample to A ; and efpying the mark at A through the fixed fights, fcrew faft the inftrument j then turn the 7 2 Divers Ways to take the Plots of Fields. the moveable index about, (the femicircle remaining immoveable) 'till through the fights thereof you efpy the mark at B. See what degrees on the brafs limb are qit by the index, which fuppofe 80 5 write that down in your field book ; fo turn the index round to every one of the other angles, putting down in your field-book what degrees the index points to. As for example, at C 107 degrees, at D 185: But at D, the end of the index will go off the brafs limb, and the other end will come on j you muft therefore look for what degrees the index cuts in the innermoft circle of the limb at the points E and F ; fuppofe 260 and 3 15 degrees refpedlively. Thefe obfervations you muft note down in your field-book thus : > ? o r =** 5' s. rt* en C.O 3 1 I 2. x- 2 O A CO . OO . 8 . 70 B 080 . oo . 10 . 00 C 107 . oo . II . 40 D 185 . oo . 10 . 50 E ato . oo . 12 . OO F 3'? oo . 8_o 7" Secondly, Caufe the diftance between your inftru- ment and every angle to be meafured : thus, from o to A will be found to be 8 chains 70 links ; from G to B, 10 chains oo. All which fet down in or- der in your field-book, as you fee done above ; and then have you got what is neceflary to be known in that field towards meafuring of it. Your next work is to protract or lay it down upon paper. To Divers Ways to take the Plots of Fields. 73 70 protraft the former obfervatiom taken. Firft, draw a line at pleafure, as A a , then take from your fcale with your compaffes, the firft di- flance meafured, viz. from o to A, 8 chains, 70 links ; and fetting one foot in any convenient place of the line, which may reprefent the place where the inltrument ftood, with the other make a mark upon the line as at A, fo mall A be the firfl angle, and o the place where the inftrument ftood. Secondly, Take a protractor, and having laid the center hereof exactly upon o, and the diameter or meridian upon the line A a, the femicircle of the protractor lying upwards j there hold it faft, and with your protracting pen make a mark upon the paper againft 80 deg. 107 deg. &c. as you find them in the field-book. Then for thofe degrees that ex- ceed 1 80, you muft turn the protractor downward, keeping ftill the center upon o, and placing again the diameter upon a A. Mark out by the inner- moft circle of divifions the reft of your obfervations 185, 260, 315. Then applying a fcale to o, and every one of the marks, draw the dotted lines o B, G C, o D, o E, o F. Thirdly, Take with your compafles the length of the line o B, which you find by the field-book to be 10 chains, which fet ofT from o to B. The like to do for o C, o D, and the reft. Laftly, Draw the lines A B, B C, C D, &c. which will inclofe a figure exactly proportionable to the field before furveyed. ft 74 Divers Ways to take the Plots of Fields. To take the plot of the fame field at one Jlation by the plane table. Place your table with a meet of paper upon it at o, and making a mark upon the paper, that {hall fignify where the instrument flands, lay your index to the mark, turning it about 'till you fee through the fights the mark at A; there holding it faft, draw the line A o. Turn the index to B, keep- ing ftill upon the firft mark at o ; and when you fee through the fights the mark at B, draw the line Bo. Do the fame by all the reft of the angles, and having meafured the diftance between the in- ftrument and each angle, fet it off with your fcale and compafles from o to A, from o to B, &c. making marks where, upon the feveral lines, the di- ftances fall. Laftly, Between thofe marks draw lines, as AB, B C, CD, &c . and then have you the true plot of the field ready protracted to your hand. This in- ftrument is fo plain and eafy to be underftood, I mall give no more examples of the ufe of it. The greateft inconveniency that attends it is, that when any rain, or even dew, falls, the paper will be wet, and the inftrument ufelefs. 1o take the Plot of the fame field at one Jlation by the femicircle^ either 'with the help of the needle and limb both together , or by the help of the needle alone. In the beginning of this chapter, I (hewed you how to take the plot of a field at one ftation, by 2 the Divers Ways to take the Plots of Fields. 75 the femicircle, without refpect to the needle, which is the beft way : but that I may not leave you ig- norant of any thing belonging to your inftrument, I (hall here (hew how to perform the fame with the help of the needle two waysj and firft with the needle and limb together. Fix the inftrument as before, in o, making the north point of the needle hang directly over the Flower-de-luce of rhe card ; there fcrew faft the in- ftrument. Then turn the index to all the angles, noting down what degrees are cut thereby at every angle, as at A fuppofc 25, at B 105, at C 132 -, and fo of the reft round ,the field. And when you have meafured the diftances, and are come to pro- traction, you muft firft draw a line crofs your paper, calling it a north and fouth line, which reprefcnts the meridian line of the inftrument. Then applying the protractor to that line, mark round the degrees as they were obferved, viz. 25, 105, 132, &c. and having fet ofT the diftances, and drawn the outward lines agreeable to what you were taught at the begin- ning of the chapter, you will find the figure to be the fame as there defcribed. Now to perform this by the needle only, is in a manner the fame as the former : for inftead of turn- ing the index about the limb, and obferving what degrees are cut thereby, here you muft turn the whole inftrument about, and note at every angle what degrees upon the card the needle hangs over ; which fet down and protract as before. But here obferve, fome cards arc numbered from the north eaft wards io, 20, 30, &c. to 360 deg. others from the north weftward, which indeed are beft for this ufe, 7 6 Divers Ways to take the Plots of Fields. ufe, protractors being made accordingly : for when you turn your inftrument to the eaftward, the needle will hang over the weftward divifion, on the con- trary. As for the ufe of the divifion of the card into four quadrants, I fhall fpeak largely of by and by; therefore for the prefent beg your patience. 20 take the plot of a field, at one jlatlon^ by the femi- circle placed in any angle thereof, from whence the other angles can be feen. Let GABCDEFG be the field, and F the angle at which you would take your obfervations. ing placed your femicircle at F, turn it at Hav- it about the north Divers Ways to take the Plots of Fields. 77 north point of the card from you, 'till through the fixed fights, (Note, that I call them the fixed fights which are on the fixed diameter) you efpy the mark at G. Then fcrew fad the inftrument ; which done, move the index, till, through the fights thereof, you fee the mark at A, and the degrees on the limb there cut by it will be 20. Move again the index to the mark at B, where you will find it to cut 40 deg. Do the fame at C, and it cuts 60 deg. Likewife at D 77, and at E 100 deg. Note down all thefe angles in your field-book : next meafure all the lines, as from F to G 14 chains, 60 links ; from F to A 1 8 chains, 20 links ; from F to B 16 chains, 80 links ; from F to C 2 1 chains, 20 links ; from F to D 1 6 chains, 95 links; from F to E 8 chains 50 links ; and then will your field-book ftand thus: S? $ crc OQ 5. 5" 3 g s F y ? FG 14 60 GFA 20 oo FA 18 20 GFB 40 oo FB 16 ,80 G F C 60 oo F C 2 1 20 GFD 77 oo FD 16 95 GFE 110 oo FE 8 50 jT0 protraft the former obfervations. Draw a line at adventure, as Gg, upon any con- venient place ; on which lay the center of your pro- tractor, as at F, keeping the diameter thereof right upon the line G g. Then make marks round the protractor at every angle, as you find them in the field-book, viz. againft 20, 40, 60, 77, and 100 ; 3 which 7 8 Divers Ways to take the Plots of Fields. which done, take away the protractor, and apply- ing the fcale or ruler to F, and each of the marks, draw the lines F G, FA, FB, FC, F D, andFE. Then fetting off upon thefe lines the true diftances as you find. them in the field-book; thus for the firft lineFG, 14 chains, 60 links; for the fecond FA, 18 chains, 20 links, &c. make marks where the end of theie diftances fall, which let be at G, A, B, C, &c. Laftly, Between thefe marks, drawing the lines GA,^B, BC, CD, DE, EF, FG, you will have compleated the work. When you furvey thus without the help of the needle, you muft remember before you come out of the field to make the meridian line, that you may be able to make a compafs fhewing the true fituation of the land, in refpect of the four quarters of the heavens ; I mean Ea/i, Weft, North > and South : which you may do thus : The inftrument dill {landing at F, turn it about 'till the needle lies directly over the flower-de-luce of the card ; there fcrew it faft. Then turn the move- able indeX) 'till, through the fights, you efpy any one angle. As for example : Let be D : note then what de- grees upon the limb are cut by the index, which let be 10 deg. Mark this down in your field-book, and when you have protracted as before directed, lay the center of your protractor upon any place of the line F D, as at , turning the protractor about 'till to i o deg. lie directly upon the Jine F D. Then againft the end of the diameter of the protractor, make a mark as at N, and draw the line N o, which is a meridian, or north or fouth line, by which you may make a compafs. Nofe, Divers Way 3 to take the Plots of Fields. 7 9 Note, You may take the plot of a field at one ftation, by {landing in any fide thereof, as well as in an angle : for if you had fet your inftrument in a> the work would have been nearly the fame. 70 take the plot of a feld at two Jtations, provided from either flation you can fee every angle - y by mea- furing only the Jlationary dijlance, and obferved angles. Let CDEFGHC be fuppofed a field to be meafured at two (rations: firft, when you come into the field, make choice of two places for your flations, as far afunder as the field will conveni- ently admit of; alfo take care that if the fla- tionary diftance were continued, it would not touch any angle of the field; then fetting the femicir- cle at A, the fir/I ftation, turn it about, the north point from you, 'till, through the fixed fights, you efpy the mirk at your fecond ftalion, which admit to be at B, there fcrew fail the inftrument ; then turn the moveable index to every feveral angle round the whole field, and ice what degrees are cut there - 8o Divers Ways to take the Plots of Fields by at every angle, which note down in your field- book, as followeth : Angles, Divers Ways to take the Plots of Fields. 81 Angles. Degr. Min. BAG 24 30 BAD 97 oo !=?^EE=F Firftftath ^-- H 346 oo Secondly, Meafure the diftance between the two ftations, which let be 20 chains, and fet it down in the field-book thus : Stationary diftance 20 chains, oo links. Thirdly, Placing the inftrument at B, the fecond ftation, look backwards through the fixed fights to the firft ftation at A, (I mean by looking backward, that the fouth part of the inftrument be towards A) and having efpied the mark at A, make faft the in- ftrument, and moving the index as you did at the firft ftation to each angle, fee what degrees are cut by the index, and note them down as followeth ; and then have you done, unlefs you will take a me- ridian line before you move the inftrument ; which you were taught to do in the laft example. Angles. Degr. Min. IBC 84 oo IBD 149 oo \ The fecond ftation. G - 270 --- oo H -- 3 22 --- 00 How 8 2 Divers Ways to take the Plots of Fields. How to protraft or lay down upon paper thefe follow- ing observations. Firft, Draw a line crofs your paper at pleafure, as the line I K ; then take off from the fcale the fta- tionary diftance 20 chains, and let it upon that line, as from A to B, fo (hall A reprefent the firft flation, B the fecond. Secondly, Apply the center of your protractor to the point A, and the diameter lying (Irak upon the line BK j mark out round it the angles, as you find them in the field-book, and through thofe marks from A draw lines of a convenient length. Thirdly, Move your protractor to the fecond ftation B ; and there mark out your angles, and draw lines, as before, at the firft ftation. Laftly, The places where the lines of the firft ftation, and the lines of the fecond interfect each other, are the angles of the field. As for example : At the firft ftation the angle C was 24 degrees 30 minutes, through thofe degrees 1 draw the line AC. At the fecond ftation C was 84 degrees ; ac- cordingly from the fecond ftation I draw the line* B 2 : Now, I fay, where thefe two lines cut each other, as they do at C, there is one angle of the field. So likewife of D, E, and the reft of the an- gles; if therefore between thefe interfedlions you draw ftrait' lines, as CD, D E, E F, &c. you will have a true figure of the field. This may as well be done by taking two angles for your ftations, and meafuring the line between 2 them, Divers Ways to take the Plots of Fields. 8 3 them, as C and D ; from whence you might as well have feen all the angles, and confequently as well have performed the work. T0 take the plot of a field at two Jiations, when the field is fo irregular, that from one Jiation you can- not fee all the angles. G 2 Let 84. Divers Ways to take the Plots of Fields. Let CDEFGHIKLMNOC be a field, in which from no one place thereof all the angles may be feen ; chufe therefore two places for your ftations, as A and B ; and fetting the femicircle in A, direft the diameter to the fecond ftation B ; there making the inftrument fail, with the index take all the an- gles at that end of the field, as CDEFGHIK, and meafure the diftance between your inftrument and each angle j meafure alfo the diftance between the two ftatioos A and B. Secondly, Remove your inftrument to the fecond ftation at B ; and having made it faft, fo as that through the back- fights you may fee the firft ftation A, take the angle at that end of the field, as N O C K L M N, and meafure their diftances alfo as before; all which done, your field-book will ftand thus: Firfl Jlation. Angles. Degr. Min. Chains CAD 25 oo 20 CAE 31 oo 8 CAP 67 oo 9 C A G 101 oo 10 H 137 oo 7 I 262 oo; 6 K 3 1 6 oo 1 3 C 354_oo 24 Links, 75 10 I 5 80 oo 7 70 50 The diftance between the two ftations, 3 1 chains, 60 links. Second Divers Ways to take the Plots of Fields. 8 5 Second Jlation. Angles. Degr. Min . Chains. Links. P BN . 3 30 == 4 2O P BO . ii i . 00 ; 7- OO P BC . HS . oo zr= '5 60 K . 205 . oo - 7 4 8 L . 220 . 00 = '5 OO M . 274 . oo ^ ii . 20 To lay this down upon paper, draw at pleafurc the line PBAP; then taking in with the com- paries the diftancc between the two ftations, viz. 3 1 chains, 60 links j fet it upon the line, making marks with the compafles, at A and B ; A being the firft ftation, B the fecond, lay the protractor to A, with the north end of the diameter towards B, and mark out the feveral angles obferved at your firft ftation, drawing lines, and fetting off the diftances, as you were taught in the beginning of this Chapter,* Pig. i. Do the fame at B, the fecond ftation ; and when you have traced out all the diftances between thofe marks, draw the bound-lines. I am rather more brief in this, becaufe it is th& fame as was taught concerning Fig. i j for if you conceive a line to be drawn from C to K, then would there be two diftincl fields to be meafured at one ftation in each. If a field be very irregular, you may, after the fame manner, make three, four, or five ftations, if you pleafe -, but I think it better to go round fuch a field, and meafure the bounding lines thereof: which I (hall mew you how to do farther on. G 3 Note, 8 6 Divers Ways to take the Plots of Fields. Note y In the foregoing figure you might as well have had your ftations in two convenient angles, as D and K, and have wrought as you were taught concerning Fig. 2. the work would have been the fame. To take the plot of a field at one ftation in an angle (provided from that angle you can fee all the other angles) by meafuring round about the faid Jield. ABCDEA is a field, and A the angle appoint- ed for the ftation ; place your femicircle in A, and direct the diameter thereof, 'till, through the fixed fights, you fee the mark at B j then fcrew it faft, and turn the index to C, obferving what degrees are there cut upon the limb ; which fuppofe 68 degrees : turn it further, 'till you efpjr D, and note down the degrees there cut, viz. 76 ; do the like at E, and Divers Ways to take the Plots of Fields, 8 7 do the like at E, and the index will cut 1 24 degrees : this done, meafure round the field, noting down the length of the fide lines between angle and an- gle, as from A to B, 14 chains, oo links; from B to C, 15 chains, oo links j from C to D, 7 chains, oo links; from D to E, 14 chains, 40 links: and from E to A, 14 chains, 5 links. Then will your field-book be as hereunder. Angles. Degr. Min. Links. Chains. Links. BAG 68 . oo AB 14 . oo BAD 76 . oo BC 15 . oo BAE 124 . oo CD 7 . oo DE 14 . 40 EA 14. 5 To protract the figure, draw the right line A B at pleafure ; and applying the center of the protractor to A, (the diameter lying upon the line AB, and the femicircle of it upwards) mark off the angles as 68 : 76 : and 124: thro' thefe marks draw the lines AC, AD, AE, of fufficient length j then take in your compafies, from off the fcale, the length of the Hue AB, viz. 14 chains, and fetting one foot of the compafles in A, with the other crofs the line, as at B ; alfo for B C take off 1 5 chains, and fetting one foot in B, with the other crofs the line A C, which determines the point C ; for the line C D, take off 7 chains, and fetting one foot in C, crofs the line AD, at D j then for D E, take 14 G 4 chains, 8 8 Diver s Ways to take the Plots of Fields. chains, 40 links, and fetting one foot of the com- paffes in D E, with the other crofs the line A E, at E. Laftly, take EA 14 chains 5 links in your compares, and fet one point in E ; then if the other will fall exactly upon A, you have done the work true ; if not, you have erred : between the points of interfeclion draw ftrait lines, which (hall be the bounds of the field, viz. AB, B C, CD, DE, EA. 'To take the pkt of the foregoing feld> by meafuring one line only, and taking obfervations at every angle. Begin as you have been juft before taught, 'till you have taken the angles BAC, BAD and BAE, viz- 68, 76, and 124 degrees j then leaving a good mark at A, which may be feen all round the field, go to B, meafuring as you go the diftance from A to B, which is the only line you need meafure ; and plant- ing your femicircle at B, dired the fouth part there- of toward A, until through the back fixed fights you fee the mark, at A ; there making it faft, turn the index: about 'till you efpy C, and note down the degrees there cut, which fuppofe 129; move your inftru- ment to C, and ftill keeping the fouth part of the diameter to A, turn the index to D, where it will cut 20 degrees : then remove to D, and efpying A through the back fights, turn the index to E, where it will cut 135 degrees. Note all this down in your field-book. Angles Divers Ways to t ah the Plots of Fields. 89 Angles taken at tbe External angles round frft ftation. tbe field. BAG 681 B . 129] BAD 76 {Degrees. C . 20 {'Degrees. BAE 124^ D . 135] Line AB: 14 chains. To protract this, you muft work as you were taught concerning the foregoing figure, until you have drawn the lines AC, AD, A E, and fet off the line A B, 14 chains; then laying the center of your protractor to B, and the fouth end of the dia- meter (or that marked with 180 degrees) towards A, make a mark againft 129 degrees, and through that mark from B, draw the line B C, 'till it inter- feel: the line A C, which it will do at C. Lay alfo the center of the protractor upon C, and the dia- meter thereof along A C; and againft 20 de- grees make a mark, through which from C draw the line C D, 'till it interfect the line A D, which it will do at D. Laftly, Place your protractor at D, with the diameter thereof lying upon the line D A, and make a mark againft 1 3 5 degrees ; through this mark draw the line D E, to interfect the line A E, as at E : draw the line E A, and you have done. This may be other wife performed, thus : After you have, ({landing at A) taken the feveral angles, and meafured the diftance A B, you may only take the quantity of the bounding angles, without re- ject to A ; as the internal angle at B 5 1 degrees, at C (an outward angle, which in your field-book you fliould diftinguifh with a mark -7) 1 1 8, and fo of the reft. And when you come to plot, having found go Divers Ways to take the Plots of Fields. found the place for B, there make an angle of 5 1 degrees, drawing the line 'till it interfedl A C, &c. You may alfo furvey a field after this manner, by fetting up a mark in the middle thereof, and meafuring from that to any one angle ; alfo in the obfervations round the field, having refpedt to that mark, as you had here to the angle A. It is too tedious to give examples of all the varie- ties ; befides, it would rather puzzle than inftrudl a neophyte. 70 take the plot of a large field or wood, by meafuring round the fame, and taking obfervations at every angle thereof by the femicircle. Suppofe Diver. Way 3 to take the Plots of Fields. 9 1 Suppofe ABCDEFGA to be a wood, through which you cannot fee to take the angles, as before di- rected, but mufl be forced to go round the fame ; firft plant the femicircle at A, and turn the north end of the diameter about, 'till through the fixed fights you fee the mark at B ; then move round the index, 'till through the fights thereof you efpy G, the index there cutting upon the limb, fuppofe 146 degrees. 2. Remove to B j and as you go, meafure the diftance A B, viz. 23 chains, 40 links ; and plant- ing the inftrument at B, direct the north end of the diameter to C, and turn the index round to A, it then pointing to 76 degrees. 3. Remove to C, meafuring the line as you go, and fetting your' inftrument at C, direct the north end of the fixed diameter to D, and turn the index till you efpy B, and the index then cutting 205 de- grees ; which, becaufe it is an outward angle, you may mark thus "7 in your field book. 4. Remove to D, and meafure as you go ; then placing the inftrument at D, turn the north end of the diameter to E, and the index to C, the quan- tity of the angle will be 84 degrees. And thus you muft do at every angle round the field, and at E, you will find the quantity of that angle to be 142 degrees, at F 137, at G no : but jthere is no need for taking the laft angle, or mea- furing the two laft fides, unlefs it be to prove the truth of your work ; which is however convenient. When you have thus gone round the wood or field, you will find your field-book to be as folio weth : Angles 9 2 Divers Way 3 to take the Plots of Fields > OTQ Degr, Min. Ch. Link. A . 146 . oo A B .'23 . 40 B . 76 . oo B C . 15 . 20 C . 205 . oo CD . 17 .90 D . 85 . oo "7 DE . 20 . 60 E . 142 : oo E F . 13 . 60 F . 137 . oo GA . 19 . 28 G . no . oo To protract this, draw with a black lead pencil, a line at pleafure, as A B ; upon which fet off the diftance, as you fee in your field-book, 23 chains, 40 links, from A to B ; then laying the center of your protractor upon A, and the diameter along the line A B, with the north end, or that of oo degrees towards B ; on the outfide of the limb make a mark againft 146 degrees, through which from A draw the line A G ; fo have you the firft angle and firft diftance. 2. Place the center of the protractor upon B, and turn it about 'till 76 degrees lie upon the line A B ; there hold it faft, and againft the north end of the diameter make a mark, through which draw a line, and fet off the diftance B C, 15 chains, 20 links. 3. Apply the center of the protractor to C, (the femicircle thereof outward, becaufe you fee by the field- book it is an outward angle) and turn it about till 205 degrees lie upon the line C B ; then againft the upper or fouth end of the diameter make a mark, through which draw a line, and fet off ij chains, 90 links from C to D. 4. Put Divers Ways to take the Plots of Fields. 9 3 4. Put the center of the protractor to D, and make 84 degrees thereof lie upon the line CD; then making a mark at the end of the diameter, or o degt. through that mark draw a line, and fet off 20 chains, 60 links, for D E. 5. Move the protractor to E, and make 142 degr, to lie upon the line E D. Then at the end of the protractor make a mark as before, and fetting off the diftance 18 chains, 85 links, draw the line E F. 6. Lay the center of the protractor upon F, and making 137 degr. lie upon the line EF ; againft the end of the diameter make a mark, through which draw the line F G, which will interfect the line A G as at G : fo you have a true copy of the field or wood. But you may, if you think fit to prove your work, fct off the diftance from F to G ; and at G apply your protractor, making 1 1 o degr. there- of to lie upon the line F G. Then if the end of the diameter point directly to A, and the diftance be 90 chains, 28 links, you may be fure you have done your work true. Whereas I bid you put the north end of the in- ftrument and of the protractor towards B, it was chiefly to mew you the variety of working by one inftrument ; for in the figure before this, I directed you to do it the contrary way ; and in this figure, if you had turned the fouth end of the inftrument to G, and with the index had taken B, and fo of the reft, the work would have been the fame ; re- membering ftill to ufe the protractor the fame way as you did your inftrument in the field Alfo if you had been to have furveyed this field or wood by the help of the needle j after you had planted the femicircle at A, and pofited it, fo that 3 the 94 Divers Ways to take the Plots of Fields. the needle might hang directly over the flower-de- luce in the card, you mould have turned the index to B, and put down in your field-book what degrees upon the brafs limb had then been cut thereby, which, for example, fuppofe 20. Then moving your inftrument to B, make the needle hang over the jhitr -de-luce t and turn the index to C, and note down what degrees are there cut. So do by all the reft of the angles. And when you come to pro- tract, you muft draw lines parallel to one another crofs the paper, not farther diftant than the breadth of the parallelogram of your protractor ; which mall be meridian lines, marking one of them at one end N. for the north, and at the other S. for fouth. This done, chufe any place which you (hall think moft convenient upon one of the meridian lines for your firft angular point, as at A; and laying the diameter of your protractor upon that line, againft 20 degr. make a mark ; through which draw a line, and upon it fet off the diftance from A 19 B. In like manner proceed with the other angles and lines, at every angle laying your protractor parallel to a north and fouth line ; which you may do by the figures graduated thereon, being at either end alike. When you have furwyed after this manner s to difco- ver before you go out of the field \ 'whether you bavf wrought true or not. Add all the obferved angles together, in the pre- ceding example of meafuring the wood, they make 900. Multiply 180 by a number lefs by two than the Divers Ways to take the Plots of Fields. 95 the number of angles j and if the product be equal to the fum of all the angles, then you have wrought true. There were feven angles to that wood, there- fore multiply 180 by 5, and the produdt is 900. If you furvey, by taking the quantity of every angle, and if all be inward angles, you mufl work as before. But if one or more be outward angles, you muft fubtraft them out of 180 degr. and add the remainder only to the reft of the langle. And when you multiply 180 by a fum lefs by 2 than the number of your angles, you are not to account the outward angles into the number. Thus, in the laft example, I find one outward angle, viz. C 205 ; the quantity of which, if it had been taken, would have been but 155 degr. That taken from 1 80 degr. there remains 25 ; which I add to the other angles, and they make in all 720. Now becaufe C was an outward a'ngle, 1 take no notice of it ; but fee how many other angles I have, and I find 6 : a number lefs by two than 6, is 4: by which I multiply 180, and the product is 720, and confequently the work is right. Directions to meafure parallel to a hedge,, (when you cannot go in the hedge itfelf) j and alfo, in fuch cafe t htnv to take your angles. It is impoflible for you, when you have a hedge to meafure, to go on the top of the hedge itfelf; but if you go parallel thereto, either within or without, and make your parallel line of the fame length as the line of your hedge, your work will be the fame. 96 Divers Ways to take the Plots of Fields. fame. Thus, if A B was a bufhy hedge, to which A_ _ ___ B JL (r) ------ ..n . . i you could not conveniently come nigher to plant your inftrument than Q ; let him that goes to fet up your mark at B, take before he goes the diftance A G, which he may do readily with a wand or rod ; and at B let him fet off the fame diftance again, as to +, where let the mark be placed for your ob- fervation ; and when the chain bears, meafure the diftance o -f ; be fure they have refpect to the hedge A B, fo as that they make o + equal to A B, or of the fame length. But to make this more plain : fuppofe A B C to be a field; and for the bufhes, you cannot come nigher than o to plant your inftrument. Let him that fets up the marks take the diftance between t inftrument o, and the hedge A B ; which diftance 1 him fet off again nigh B, and fet up his mark at D j like- s Divers Ways to take the Plots of Fields. 9 7 like wife let him take the diftancc between o and the hedge A C, and accordingly fet up his mark at E. Then taking the angle D o E, it will be the fame as the angle BAG: fo do for the reft of the angles. But when the lines are meafured, they muft be meafured of the fame length as the outfide lines, as the line o D meafured from g to f % &c. The befl way, therefore, is for thofe who meafure the lines, to go round the field on the outfide there- of, although the angles be taken within. 70 take the plot of a field or wood, by obferving near every angle > and measuring the dijlance between the marks of obfervation, by taking, in every line, two of -fet 3 to the hedge. Let A, B, C, D be a wood or field, to be thus meafured. Caufe your afMants to fet up marks in every angle thereof, not regarding the diftance from the hedges, fo much as the convenience for plant- H ing 98 Divers Ways to take the Plots of Fields. ing the inftrument, fo as you may fee from one mark to another. Then beginning at o i , take the quantity of that angle, and meafure the diflance i, 2. But before you begin to meafure the line, take the ofT-fet to the hedge, viz. the diftance o e ; and in taking of it, you muft make that little line o e perpendicular to i, 2 j which is eafily done, when your inftrument ftands with the fixed fights towards 2, by turning the moveable index 'till it lie upon 90 degr. which will then direct to what place of the hedge to meafure, as e, that little line o e, fet down in your field-book under title Off-fet. So like wife when you come to 2, meafure there the off-fet again, viz. o /. Then taking the angle at 2, meafure the line 2, 3, and the off-fets 2g, 3 . The like do by all the reft of the lines and angles in the field. And when you come to lay thefe down upon paper ; firft, as you have been taught before, protraft the figures i, 2, 3, 4. That done, fet off your off-fets as you find them in the field-book, viz. G e, and f, perpendicular to the line i, 2.-, alfo O g> o b, perpendicular to the line 2, 3, mak- ing marks at e y f t g, b y and the reft ; through which draw lines interfering each other at the true angular points, and then defcribe the bound lines of the field or wood. In working after this manner, obferve thefe two things : firft, if the wood be fo thick, that you can- not go on in the infide thereof, you may, after the fame manner, as well perform the work by going on the outfide round the wood. Secondly, If the diftances are fo great, that you cannot fee from angle to angle, caufe your affiftant 2 to Divers Ways to take the Plots of Fields, g 9 to fet up a mark as far from you as you can con- veniently fee it, as at n. Meafure the diftance o i n, and take the oft-fet from n to the hedge. Then at n turn the fixed fights of the inftrumcnt to o j , and by that direction proceed on the line 'till you come to an angle. tti s way of furveying is made eafier (though I cannot fay truer) by taking only a great jquare in the feld -, from the fdes of 'which the of -Jets are taken. I have drawn this following figure fo, that at once you may fee all the variety of this way of work- ing. The beft way, indeed, is to contrive your fquare fo, that, if poffible, you may, from the fides thereof, go upon a perpendicular line to any of the angles. But if that cannot be, then perpendicular H 2 lines ioo Divers Ways to take the Plots of Fields. lines to the fides may do as weil, as you fee here, 15 and 76 are. To begin, therefore, plant your femicircle in any convenient part of the field, as at i, for taking a large fquare; and laying the move- able index upon 90 degr. look through the fights, and caufe a mark to be fet up in that line, as at 4 : looking alfo through the fixed fights, caufe another mark to be fet up, as at 2. Meafure out from your inftrument towards either of thefe marks, any num- ber of chains, as 1 2 from i to 2 ; from 2 to 4, 12 chains. But as you meafure, remember to take the off-fets in a perpendicular line to every angle or fide, if there be occafion, as here 17, which is i chain, 50 links j from my flations I take an off-fet to a fide of the hedge, as 76, and put it down ac- cordingly 5 chains, 40 links. So at 8 I take an off-fet to an angle, viz. 8 B, 6 chains ; which off- fet is at the end of 8 chains, 30 links in my firft line. Then feeing in that line there is no farther occafion for off-fets, I plant my inftrument at 2, and I direcl the fixed fights to my firft ftation ; then laying the index upon 90 degr. I caufe a mark to be fet up, fo that I may fee it through the fights ; and in that direction I meafure out 1 2 chains, taking the off-fets 9, D 10. Proceed in like manner for the other angles, lines, and off-fets. ^ When you have thus laid out your fquare, and taken all your off-fets, you will find in your field- book fuch memorandums as the following, to help you to protract. Divers Ways to take the Plots of Fields, i o r angles 4 right angles. 'The Jides 1 2 chains, oo links each. I went round cum folis, or the hedges being on my left hand. C. L. C. L In the firft r line at i 8 5 30 Off-fet to a fide-line Off-fet to an angle C. In the fecond r 3 line, at I 10 L. 5 70 C. L. Off-fet to an angle 6 Off-fet to an angle 5 In the third c . line, at i IO oo Off-fet to an angle 5 30 C. In the fourth^ line, at L. 33 70 80 C. Off-fet to an angle 4 Off-fet to an angle I Off-fet to a fide 2 20 L. 40 5 Now to lay down upon paper the foregoing work, make firft a fquare figure, as i, 2, 3, 4, whofe fide may be 1 2 chains. Then confidering you went with the fun, take i, 2 for the firft line; and taking from your fcale i chain, 50 links, fet it upon the line from i to 7 ; at 7 raife a perpendicular, as 7, 6, making it according to your field-book, 5 chain?, 40 links long. Alfo tor the fecond off- fet upon the fame line, take from your fcale of equal parts 8 H 3 chains, 103 Divers Way 3 to take the Plots of Fields. chains, 30 links, which fet upon the line from i to 8, and upon 8 make the perpendicular line 8 B, 6 chains in length. For the off-fets of the fecond line, take 3 chains 50 links from the fcale, and fet it from 2 to Q ; at 9 make a perpendicular line 6 chains long, viz. 9 C : alfo for the fecond off-fet of the fame line, take i o chains, 70 links, and fet it from 2 to 10 , at 10 make the perpendicular 10 D, 5 chains, 50 links in length. For the off-fets of the third line, take from your fcale i o chains, and fet it up from 3 to 1 1 ; and at 1 1 make the perpendicular 1 1 E, 5 chains, 30 links long. For the off-fets of the fourth line, take from your fcale 4 chains, 30 links, and fet it from 4 to 12; and at 12 make the perpendicular 12 F, 4 chains 40 links long. Alfo take 6 chains, 70 links, and fet it from 4 to 13; and at 1 3 make the perpen- dicular 13 G, i chain, 50 links long. La/My > Take 10 chains, 80 links, and fet it from 4 to i ; and at I, make -the perpendicular i, 5, chains, 20 links long. Then have you no more to do, but through the ends of thefe perpendiculars to draw the bounding- lines, remembering to make angles where the field- book mentions angles ; and where it mentions fide- lines, there to continue fuch fide-lines 'till they meet in an angle. Although I mention a fquare, yet you are not obliged to take that figure ; for you may with the fame fuccefs life a parallelogram, triangle, or any other figure. Nor are you bound to take the off- fets Divers Ways to take the Plots of Fields, 102 fets in perpendicular lines, notwithftanding it is the beft way; for you may take the angles with the index from any part of the line. This way was chiefly intended for fuch as were not provided with proper inftruments j for with the crofs ftaff only, you may lay out a fquare, the reft of the work being done with a chain. By the help of the needle, to take the plot of a large woody by going round the fame, and making ufe of that divijion of the card that is numbered with four 90' or quadrants. Let A B C D E A reprefent a wood ; fet your in- ftrument at A, and turn it about, 'till, through the fixed fights you efpy B, then obferve what degrees in the divifion before fpoken of, the needle cuts ; which fuppofe 7 degrees from N. towards Weft, meafure AB 28 chains, 20 links; then fetting the inftrument at B, direct the fights to C, and fee what then the needle cuts, which let be 74 degrees from N. towards the E; meafure B C 39 chains, 50 links; in like manner meafure every line, and take every angle, and then your field-book will ftand thus, as followeth hereunder. H 4 Lines 104 Divert Way, to take the Plots of Fields. NNNNNNNNNN I MM j ' N S S pr AB DE EA NW 7 oo 28 20 ~ NE 74 oo 39 50 g. S E 9 oo 38 oo X NW 6 3 20 I 4 rr S W 74 80 28 60 To Divers Ways to take the Plots of Fields. 105 To lay down which upon paper, draw parallel lines, as N. S. N. S. &c. to reprelent meridians, or north and fouth lines, then applying the protractor (which mould be graduated accordingly with twice 90 degrees, beginning at each end of the diameter, and meeting in the middle of the arch) to any con- venient place of one of the lines as to A, lay the meridian line of the protractor to the meridian line on the paper, and again ft 7 dcgr. make a mark, through which draw a line, and fet off thereon the diftance A B 28 chains, 20 links. Secondly ', Ap- ply the center of the protractor to B, and (turning the femicircle thereof the other way, becaufe you fee the courfe tends to the eaftward) make the dia- meter thereof lie parallel to the meridian lines on the paper, (which you may do by the figures at the ends of the parallelogram) and again ft 74 degrees make a mark, and fet off 39 chains, 50 links, and draw the line B C j the like do by the other lines and angles, until you come round to the place where you began. This is the general method of plotting obferva- tions, taken after this manner, ufed by moft furvey- ors in America , where they lay out very large tracts * of land : but there is another way much furer, yet rather tedious, (I think firft made public by Mr. Norwood] whereby you may know before you come out of the field, whether you have taken your an- gles, and meafured the lines truly or not, and is as follows : When you have furveyed the ground as above directed, and find your field-book to ftand as before ; caft up what northing, fouthing, calling, or weft- ing 1 06 Divers Ways to take the Plots of Fields. ing every line makes ; that is to fay, how far at the end of every line you have altered your meridian, and what diftance upon a meridian line you have made. As for example: fuppofe AB, equal to 20 chains, was the fide of a field, N S a meridian line, the angle CAB north 20 deg. eaft. The bufinefs is to find the length of the line A C, which is called the north- ing, or the difference of latitude ; alfo the length of the line C B, which is called the eafting, or differ- ence of longitude j which you may do indifferently true, by laying them down upon paper. By help of the Gunt 'er's fcale, the be ft way is by the tables of fines and logarithms, ufing this proportion : As radius or fine of 90 degrees, viz. the right angle C is to the logarithm of the line AB 20 chains ; So is the fine of the angle C A B 20 degrees to the difference of longitude C B 6 chains, 80 links. Secondly, To find the difference of latitudes, or the line A C, fay, As radius is to the logarithm of the line A B 20 chains, fo is the fine complement of the angle at A, to the logarithm of the line AC 18 chains, 80 links. EXAMPLE. Divers Ways to take the Plots of Fields. 107 EXAMPLE. I find, by my field-book, the firft line (fee the laft figure but one) runs N. W. 7 degrees, 28 chains, 20 links; now to find what northing, and what wefting is hereby made, I (ay thus : As radius Is to the logarithm of the line 287 chains, 20 links. So is the fine of the angle from > the meridian, vix. 7 degrees, 3 To the logarithm of the wcfting ? 3 chains, 43 links, J 10,000000 1,450249 9,085894 Again, As radius, Is to the logarithm 28 chains, 20 links So is the fine complement of 7 degrees To the logarithm of the northing 27 chains, 99 links, 10,000000 1,450249 9,996751 11,447000 And having thus found the northing and wefting of that line, 1 put it down in the field-book againft the line under the proper titles N. W. in like man- ner I find the latitude and longitude of all the reft j and having fet them down, the field-book will ap- pear thus: Lines. 1 o 8 Divers Ways to take the Plots of Fields. s N w A B N W 28 20 27 * OO . B C : N E CD : S E D E N W . uvj 74 : oo 9 : oo 39 : 5 38 : oc */ yy 10 : 86 06 53 37 : 53 37 : 97 >S ; 95 13 oo E A : SW 74 : oo 28 : 6c 07 : 88 2 7 49 ^ : 4' .5 : 41 4.1 : 92 *3 92 This done, add the northings together, alfo all the fouthings, and fee if they agree ; alfo all the eaftings and weftings; and if they agree likewife, then you may be fure you have wrought truly, other- wife not. Thus in the example the fum of the northings is 45 chains, 41 links; fo likewife is the fum of the fouthings ; alfo the fum of the eaftings is 43 chains, 92 links, fo is the fum of the weftings: therefore, I fay, I have furveyed that piece of land true. But becaufe this way of cafting up the northing, fouthing, eafting, and wefting of every line, may feem tedious and troublefome, I have, at the end of this book, added a table ; wherein, by infpection, you may find the longitude and latitude of every line, according to the angle which it makes with the meridian. Another way of plotting the foregoing piece of ground according to the table in the field-book, with regard to the points of the compafs, N. S. E. and W. is as follows : Draw Divers Ways to take the Plots of Fields. 109 Draw an indefinite right line, as n o A S for a meridian line 5 then beginning in any place of that line, as from A to o i, viz. 27 chains, 99 links; then taking with your cornpaiTes the weftings of the fame line, viz. 3 chains, 43 links ; fet one foot in O i, and with the other defcribe the arch a a j next take the length of your firft line, as you find it in the field-book, viz. 28 chains, 20 links; and fett- ing one foot of the compafles in A, with the other crofs HO Divers Ways to take the Plots of Fields. crofs the former arch a a with another arch B b t and in B the interfedion of thefe arches is your fecond angle. Then through B draw another north and fouth.Jine, as NBS, parallel to the fir ft, NAS; then take with your compares the northing of the fecond line, viz. 10 chains, 89 links, and fet it upon that line, from B to o 2 ; take alfo the eafting of the fame line, viz. 37 chains, 07 links ; and fetting one foot of the compares in O 2, with the other fweep the arch c c ; alfo take with your compafles the length of the fecond line, viz. 39 chain!, 50 links ; and fetting one foot in B, crofs the^f(kmer arch with another dd\ the interfedlion C is your third angle. It would be needlefs to go round thus with all the lines 5 for by thefe already drawn, you may 'eafily conceive how all the reft may be done. But obferve when you fweep the arches for the eafting and weft- ing, to turn your companies the right way, and not take eaft for weft, and weft for eaft. Nor can I much recommend this way of plotting the former being as true, and far eafier ; yet when you plot by the former way, it will be proper to prove your work by the table of difference of lati- tude and longitude, before you begin to protract ; and when you find your field-book correct, you ma] hy down your work upon paper, by that method you think the eafieft. To conclude this chapter or fedion, I (hall in the next place {hew you, how to furvey a field by the chain only, ufing no other inftrument, and that after a better manner than hitherto has been taught. Firft, Divers Ways to take the Plots of Fields, in Firft, therefore, I fhall {hew you how to take the quantity of an angle by the chain ; (which being well underflood) there will be no more required : for the bufinefs of a furveyor in the field, is no more than to meafure fides, and take angles, in order to find how many acres any field or piece of land contains. To take an angle in the fold by help of the chain only. Firft, Meafure along the hedge A B, any fmall diftance, as two chains from A to 2 ; alfo meafure along H2 Divers Ways to take the Plots of Fields. along the hedge A C any number of chains you pleafe, (either equal to the former or not) as A 3 two chains: next meafure the diftance 2, 3, equal to i chain, 68 links ; and then have you done in the field. To plot which, draw the line A B at plea- fure, and fet off 2 chains from A to 2 ; then take with your compaffes the diftance A 3, viz. 2 chains, and letting one foot in A, defcribe the arch 2 3 ; take alfo with your compaffes the diftance 2 3, viz. j chain, 68 links j and fetting one foot in 2, with the other crofs the former arch in A, draw the line A 4 3 C ; which, with A B, will make an angle equal to the angle in the field. A more eafy and fpeedy way is to meafure out one chain only along the hedges thus ; I fet a ftrong flick in the angle A, and putting the ring which is at one end of the chain over it, I take the other end in my hand, and ftretch out the chain along the firft hedge A B, and where it ends, as at 5, I ftick down a ftick 5 then I ftretch the chain alfo along the other hedge A C, and at the end thereof fet another ftick, as at 4 ; then loofing my chain from A, I meafure the diftance 4, 5, and find it 74 links, which is all I need note down in my field-book for that angle ; and now coming to plot it, I firft take from my fcale the diftance of one chain, and placing one foot of the compaffes in any part of the paper, as at A, I defcribe the arch 4, 5 ; then I take from the fame fcale 74 links, and fet it off upon that arch, making marks where the ends of the com- paffes fall, as at 4 and 5. Laftly, From A ; through thefe marks I draw the lines A B, and A C, which conftitute the, former angle : always plot your angles with Divers Ways to take thePlots of Fields. 113 with a very large fcale ; but your lines may be fet off with a fmaller. I fhall give you two "Examples of this way of mea- furing, and then leave you to your own pra&ice. Firft, To furvcy a feld with the chain only, by going round the infide of it. Let ABCDEFA be the field ; and beginning at the angle A in that point, ftick down a ftafF through I the H4 Divers Ways to tale the Plots of Fields. the great ring at one of the ends of your chain, and taking the other end in your hand, ftretch out the chain in length, and fee in what part of the hedge A F the other end falls j as fuppofe at a, there let up a ftick ; and do the like by the hedge A B, and let the chain end at (a) alfo : meafure the neareft di- ftance between a and a, which admit to be I chain, 60 links ; this note down in your field-book : meafure next the length of the hedge A B, 12 chains, 50 links ; note this down alfo in your field-book. Next, coming to B, take that angle in like manner as you did the angle A, and meafure the diftance B C : after this manner you muft take all the angles, and meafure all the fides round the field. But left you be at a nonplus at D, becaufe it is an inward angle, thus you muft do : ftick a ftaff down with the ring of the chain round it in the angular point D, then taking the other end of the chain in your hand, and ftretch- ing it at length, move yourfelf to and fro, till you perceive yourfelf in a direct line with the hedge DC, which will be at G ; where ftick down an arrow, or one of your furveying-fticks ; then move round 'till you find yourfelf in a direct line with the hedge D E ; and there the chain being ftill ftretched out to its full length, plant another ftick, as at H j then meafure the neareft diftance, from H to G, which let be i chain, 43 links j and note it down in your field-book ; proceed on to meafure the line D E 5 but in your field-book make fome markagainft D, to fignify it is an inward angle, as "/% or the like. And when you come to plot this, you muft plot the fame angle outward that you took inward ; for the angle G D H is the fame as the angle d D d. If you furvey a wood, Divers Ways to take the Plots of Fields. 115 wood, by going round it on the outfide, then the angle d D d is an outward angle. Having thus taken all the angles, and meafured all the fides ; the next thing to .be done, is to lay them down upon paper, according to your field-book ; which you will find to (land thus : Crofs Lines or Chords. 1 as 9 r- i- i- A* i 60 AB 12 5 Eb i 84 B C 2 3 37 Cc i 06 CD 19 3 Dd i 43 "7 DE 20 oo E e o 80 EF 29 00 F/ i 52 FA 3 1 5 It being more convenient that the angles fhould be made by a greater fcale than the lines are laid down with j I have therefore, in this figure, made the angles by a fcale of one chain in an inch, and laid down the lines by a fcale of ten chains in one inch. Now to begin to plot, take from your large fcale one chain ; with that diftance, in any convenient place of your paper, as at A, fweep the arch a, a - t then from the fame fcale, take off i chain, 60 links, and fet it upon that arch, as from a to a; and draw through the point, a and #, the right lines A B, I 2 AF: 1 1 6 Divers Ways to take the Plots of Fields. A F : then repairing to your (hotter fcale, take from thence the firft diftance, viz. 12 chains, 50 links j and fetting it from A to B, draw the line A B. Secondly, Repairing to B, take from your large fcale one chain, and fetting one foot of the com- pafles in B, with the other defcribe the arch b b ; alfo from the fame fcale take your chord line, viz. i chain, 84 links, and fet it upon the arch b b, hav- ing one foot of the compafTes in the point where the arch interfecls A B, the other will fall at b ; then through b draw the line B C ; and from your fmaller fcale fet off the diftance 23 chains, 37 links, from B to C, where the next angle muft be made. After this manner proceed on according to your field-book, till you have done. And here note, That you need neither in the field, nor in plotting upon the paper, take any no- tice of the angle F, nor yet meafure the lines E F or A F ; for if you draw thofe two lines already drawn, they will interfect each other at the true an- gular point F : however, for proof of the work, it is proper to meafure them, and likewife to take the angle in the field. I muft not omit in this place to mew the ufual way taught by furveyors, for meafuring a field by the chain only ; as true indeed as the former, but more tedious, which is as follows : The common way taught by furveyors for taking the plot of the foregoing, or any other field. In order that you may not be confufed with too many lines in one figure, I have here again placed the 3 Divers Ways to take thePlots of Fields. 117 the fame. Fir ft, Meafure round the field, and note down in your field book every line thereof, as in this field has been before done. Secondly, Divide the whole field into triangles, by drawing the diagonals AC, AD, A E, and note them down in your field-book, thus : I 7 AC 1 1 8 Divers Ways to tafo the Plots of Fields. e n A C 33 70 AD 25 70 A E 45 40 To plot which, firft draw a right line A C at pleafure, and fet off thereon 33 chains, 70 links, ac- cording to your field book, for the diagonals ; then taking with your compafles the length of the line A B, viz. 1 2 chains, 50 links, fet one foot in A, and with the other deicribe the arch a a ; alfo take the line B C, viz. 23 chains, 37 links, and fetting one foot in C, with the other defcribe the arch c c, cutting the arch a a in the point B ; then draw the lines A B, C B, which {hall be the two bound- lines of the field. Secondly, Take with your compafles the length of the diagonal A D, viz. 25 chains, 70 links, and fetting one foot of the compafles in A, with the other defcribe the arch d d ; alfo taking the line C D, viz. 1 9 chains, 30 links, fet one foot in C, and with the other defcribe the arch e e, cutting the arch dd in the point D, to this interfection draw the line CD. Thirdly, Take with your compafles the length of the diagonal A E, viz. 45 chains, 40 links ; and fetting one foot in A, with the other defcribe an arch, zsff; alfo take the line DE, 20 chains, and therewith crofs the former arch in the point E, to which draw the line D E. Laftly, Divers Ways to tale the Plots of Fields. 119 Laftly, Take with your compafles the length of the line A F, viz. 3 1 chains, 50 links ; and fetting orte foot in A, defcribe an arch, as 1 1. Allb take' the length of the line E F, viz. 2 9 chains, oo links ; and therewith defcribe the arch h h, which cuts the arch 1 1 in the point F } to which point draw the lines A F and E F, and fo will you have a true figure of the field. I have (hewed you both ways, that you may take your choice. . And now I proceed to my fecond example promifed. To take the plot of a field at one Jlation^ near the middle thereof, by the chain only. Let ABCDEA be the field, o the appointed place, from whence by the chain to take the plot thereof. Stick a ftake down at o through one ring of the chain, and make your afiiftant take the other end, and ftretch it out. Then caufe him to move up and down, till you fee him exactly in a line be- tween the ftick and the angle A ; there let him fet down a ftick, as at #, and be fure that the ftick a be in a direct line between o and A ; which you may eafily perceive by ftanding at o, and looking to A. This done, caufe him to move round towards B; and at the chain's end, let him there ftick down another ftick exactly in the line between o and B, as at b. Afterwards let him do the fame at c y at the content. The fquare of the diameter is K 40000, 130 How to caft up the 40000, which, multiplied by 11, makes 440000 ; which, divided by 14, gives 31428, or i rood, 14 perches, and fomething more for the content. 70 meafure the fuperficial content cfafetforofa circle. Multiply half the compafs thereof by the femi- diameter of the circle, the product will anfwer your defire. In the foregoing circle, I would know the con- tent of that little piece DCB; the arch D B is 78 links i ; the half of it 39 T which multiplied by i chain, o links, the femidiameter gives 3925 fquare links, or 6 ^ perches. 70 find the content of a fegment of a circle, without knowing the diameter. Let E F G Be the fegment, the chord E F is i chain, 70 links, or 170 links > the perpendicular G H 50 links j now multiply ^ of the one by the whole of the other, the product will be the con- tent nearly ; the two thirds of 170 is the neareft 1 13, which multiplied by 50, produces 5650 fqnare links, or 9 perches. To find the fuperficial content ofvn oval. The common way is to multiply the long diame- ter by the fhorter, and obferve the product; and then, as if you were meafuring a circle, fay, As Contents of a Plot of Land. 131 As 14 to u, fo the faid product to the content of the oval, but this is not exact. A better way is, As 1,-rw is to the length of the oval, fo is the breadth to* the content j or nearer, as 1,27324. to the length, fo is the breadth to the content. '.'"VVJr * V -'. ' '- .'\ .'';? "''u ; ;A ' 0* To fnd the fuperfaial content of regular polygons*, as pentagons, hexagons, Jeptagons* &c. Multiply half the fum of the fides by a perpen- dicular let fall from the center upon one of the fides, the product will, be the area or fuperficial content of the polygon. In the following penta- gon the fide BC is 84 links, the whole fum of the five fides, therefore, muft be 420, the half of which is 2io; which, multiplied by the perpendicular A D, 56 links gives 1 1760 fquare links for the content, or 18 perches T V of a perch, al- i9P erches - I have been fhorter about thefe three lafl figures than my ufual method, becaufe they very rarely fall into the furveyor's way to meafure them in land, though indeed in broad meafure, paving, &c. often. K CHAP^ ( 132 ) CHAP. vnr. ;^v; ] Of laying out new lands ; very ufeful for furveyors y in his Majeftys plantations in America. A certain quantity of acres being given, bow to lay out the fame in a fquare figure. ANNEX to the number of acres given 5 cyphers, which will turn the acres into links; then from the number thus increafed, extract the fquare root, which (hall be the fide of the propofed fquare'/; 5 \& EXAMPLE. Suppofe the number given be 100 acres, which I atn to lay out in a fquare figure ; 1 join to the 100 5 cyphers, and then it is 100,00000 fquare links; the root of which is 3 162 neareft, or 3 i chains, 62 links, the length of one fide of the fquare. Again : If I were to cut out of a corn-field one fquare acre, I add to i five cyphers, and then is it 100000; the root of which is 3 chains, 1 6 links, and fome- thing more for the fide of that acre. To Of laying out New Lands. 133 Te> Jay out any given quantify of acres in a right-angled parallelogram^ 'whereof one fide is given. Turn firft the acres into links, by adding, as before, five cyphers ; that number thus , increafed divide by the given fide, the quotient will be the other fide. EXAMPLE. It is required to lay out 100 acres in a parallelo- gram, one fide of which (hall be 20 chains ; firft, to the 100 acres 1 add five cyphers, and it is ico,ooooo-j which divide by 20 chains, or 2000 links; the quotient is 50 chains, o links, for the other fide of the parallelogram. To lay out any given number of acres in a parallelo- gram that JJ:all be 4, 5, 6, or 7, &V. times longer than it is broad. In Carolina, ail lands lying by the fides of rivers, except feigniories or baronies, are (or ought, by order of the lords proprietors, to be) thus laid out. To do which, firft, as above taught, turn the given quan- tity of acres into links, by annexing five cyphers; which fum divide by the number given for the pro- portion between the length and breadth, as 4, 5, 6, 7, &c. the fquare root of the quotient will fhew the fhorteft fide of fuch a parallelogram. K 3 EXAM- 134- Of laying out New Lands. EXAMPLE. Admit it were required of me to lay out 100 acres in a parallelogram, that mould be five times as long as broad : firft, to the i oo acres I add five cyphers, and it makes 100,00000 ; which fum I di- vide by 5, the quotient is 2000000; the root of which is neareft 14 chains, 14 links; and that, I fay, mail be the mort fide of fuch a parallelogram ; and by multiplying that 1414 by 5, mews me the longeft fide thereof to be 70 chains, 76 links. 70 make a triangle upon a given bafe, that JJiall con- tain any number of acres. To double the given number of acres, annex five cyphers, and divide by the bafe, the quotient will be the length of the perpendicular required. EXAMPLE. Upon a given bafe whofe length is 40 chains, I am to make a triangle that (hall contain 100 acres, Firft, I double the i oo acres, and annexing five cy- phers thereto, it makes 200,00000 ; which 1 divide by 40 chains, the given bafe ; the quotient is 50 chains, o links, for the height of the perpendicular. As in this figure, A B is the given bafe 40 ; on any part of it, as C, I make the perpendicular C D, equal Of laying out New Lands. 135 equal to 50 chains, and I draw the lines DA, D B, which makes the triangle DAB, containing juft 100 as required. If I had made the perpendicu- acres. lar E F equal to 50, and drawn the lines FA, FB, I mould have made the triangle FAB, containing 100 acres, the fame as DAB. If you confider this well when you are laying out a new piece of land, of any given content, in Ame- rica, or elfewhere, although you meet in your way with many lines and angles ; yet you may, by mak- ing a triangle to the firft ftation you began at, cut off any quantity required. K 4 out New Lands. To find the length of the diameter of a circle, which Jhall contain any number of acres required. Say, As 1 1 is to 14, fo will the number of acres given be to the fquare of the diameter of the circle required. EXAMPLE. What is the length of the diameter of a circle, whofe fuperficial content (hall be 100 acres? Add five cyphers to the 100, and it makes 100,00000 links; which multiply by i^facit 140000000; which divided by u, gives for quotient 12727272; the root of which is 35 chains, 67 links, and bet- ter, almoft 68 links : and fo much {hall be the dia- .meter of the required circle. I might add many more examples of this na- ture ; fuch as to make ovals, regular polygons, &c. that mould contain any affigned quantity of land. But becaufe fuch things are merely fpeculative, and feldom or never come in practice, J here omit jthem. CHAP. { 137 ) CHAP. IX. Of RE DUCTION. To reduce a large plot of land or map into a lefe compafs, according to any given proportion j or, e contra, bow to enlarge one. TH E beft way to do this, is, if your plot be - not over large, to plot it over again by a fmaller fcale : but if it be large, as the map of a county, or the like, the only way is to compafs in the plot firft with one great fquare ; and afterwards lo divide that into as many little fquares as you fhall fee convenient. Alfo make the fame number of little fquares upon a fair piece of paper by a lifs fcale, according to the proportion given. This done, obferve in what fquare, and part of -the fame fquare, any thing remarkable happens to be, and accordingly put it down in your leffer fquares ; and that you may not miftake, it will be proper to num- ber your fquares. I cannot make this plainer, than by giving you the following example, where the plot A B C D, made by a fcale of i o chains in an inch, is reduced into the plot E F G H, of 30 chains in an inch* There 138 Reduction of Land. There Redu&ion of "Land. 139 There are feveral other ways taught by furveyors for reducing plots or maps. Mr. Rathborn, and af- ter him Mr. Hclwell, advifeth to make ufe of a fcale or ruler, (having a center hole at one end, through which to pin it down on a table, ib that it may play freely round) numbered from the center en4 to the other with lines of equal parts. The life of which is thus : lay down upon a fmooth table the map or plot that you would reduce, and glew it with mouth glew faft to the table at the four corners thereof. Then take a fair piece of paper, of fufficient bignefs to contain your reduced plot, and lay that down upon the other ; the middle of the laft about the middle of the firft. This done, lay the center of your reducing fcale near the center of the white pa- per, and there with a needle through the center make it faft ; yet fo, that it may play eafily round the needle. Then moving your fcale to any remark- able thing of the firft plot, as an angle, a houfe, the bend of a river, or the like ; fee againft how many equal parts of the fcale it ftands, as fuppofe 100; then taking the i, the >, the T, or any other part thereof, according to the proportion you would have the reduced plot to bear, and make a mark upon the white paper againft 50, 25, 33 T, &c. of the fame fcale : and thus turning the fcale about, you may firft reduce all the outermoft parts of the plot : which done, you muft double or treble, &c. the lines in the lefler plot, in order to reduce the innermoft part near the center. But 1 advife rather to have a long fcale, made with a center hole for fixing it to the table, at the diftance of about one third part of the fcale, fo that t of the 140 Reduction of Land. the fcale may be one way numbered with equal parts from the center hole to the end ; and T part thereof numbered the other way to the end, with the fame number of equal parts, though lefs. Upon this fcale may be feveral lines of equal parts, the lefs to the greater, according to different proportions. Being thus provided with a fcale, glew down upon a fmooth table your greater plot to be reduced ; and clofe to it, upon the fame table, a paper, about the bignefs whereof you would have your fmaller plot Fix with a ftrong needle the center of .your fcale be- tween both ; then turning the longer end of your fcale to any remarkable place in your plot ; to be re- duced, fee what number of equal parts it cuts, as fuppofe ioo; there holding faft the fctle, againft 100 upon the fmaller end of your fcale, make a mark upon the white paper. Proceed thus round all the plot, drawing lines, and putting down all other particulars as you proceed, for fear of confu- fion through many marks in the end ; and when you have done, although at firft the reduced plot will feem to be quite contrary to the other, yet when you have unglewed it from the table, and turned it about, yov will find it to be an exact epi- tome of the firft. You may have for this work divers centers made in one fcale, with equal parts proceed- ing from them accordingly , or you may have feve- ral fcales, according to the different proportions, which is better. What has been hitherto faid concerning the re- ducing of a plot from a greater volume to a lefs, the fame is to be underftood, vice verja, of enlarging a plot from a lefs to a greater. But this lafi fcldorn comes in practice. 70 ReduElion of Land. 14* 70 dange ciiftomary meafure into flat tit e t and the coti* trary. In fome parts of England, for wood-lands j and in moft parts- of Ireland, for all forts of lands; they account 18 feet to a perch, and 160 fuch perches to an acre, which is called cuftomary meafure : whereas our true meafure for land, by aft of par- Uament, is but 160 perches for one acre, at 16 feet and an half to the perch. Therefore, to reduce the one from the other, the rule is, As the fquare of one fort of meafure, is to the fquare of the other : So is the content of the one, to the content of the other. Thus, if a field meafured by a perch of 1 8 feet, accounting 160 perches, to the acre, contain joo acres; how many acres mall. the fame field contain by a perch of j 6 feet f ? Say, If the fquare of 16 feet v, viz. 272. 2 r give the fquare of 18 feet, viz. 324, What mail 100 acres cuflomary give? Anfwer, 119 A of an acre Jlatute meafure. Knowing the content of a piece of land, to fnd ivhaf J'cale it 'was plotted from. Firft, By any fcale meafure the content of the plot ; which done, fay : As the content thus found, is to the fquare of the fcale I tried by ; So is the true content, to the fquare of the true fcale it was plotted by. Admit 142 InftruEiions for furveylng a Manor. Admit there is a plot of a piece of land contain- ing 10 acres, and meafuring it by a fcale of 11 equal parts to an inch, I find it contains 1 2 acres T V of an acre: then I fay, If 12 A give 121, (the fquare of the number of equal parts in the length of the fcale I ufed) what fhall 10 give? Anfwer, 100, whofe fquare root 10 is the number of equal parts in the length of the fcale the plot was pro- tracted by. Therefore, if the fcale I meafured by contained 1 1 equal parts to an inch, the plot was laid down from a fcale of 10 equal parts to an inch. v CHAP. X. InJlruElions for furveying a manor ^ county -, or whole country. To furvey a manor > obferve the following rules. I. TT TALK or ride over the manor once or VV twice, that you may have, as it were, a map of it in your head ; by which means you may the better know where to begin, and proceed on with your work. 2. If you can conveniently furvey round the whole manor with your chain and inftrument, tak- ing all the angles, and meafuring all the lines thereof; taking notice of roads, lanes, or commons, as you crofs InftruSlions for furveying a Manor. 143 crofs them : alfo minding well the ends of all di- viding hedges, where they butt upon your bound- hedges, in this manner. 3. Take a true draught of all the roads and bye- lanes in the manor, putting down alfo the true buttings of all the field-fences to the road. If the road be broad, or goes through fome common or wafte ground, the beft way is to meafure, and take the angles on both fides thereof; but if it be a narrow lane, you may only meafure along the midft thereof, taking the angles and off- fets to the hedges, and meafure your diftances truly: alfo if there be any confiderable river either bounds or runs through the manor, furvey that alfo truly, as is hereafter taught. 4. Make a true plot upon paper of all the fore- going work, and then will you have a refemblance of the manor, though not compleat ; which to make fo, go to all the buttings of the hedges, and there furvey every field diftinclly, plotting it accordingly every nighr, or rather twice a day, 'till you have perfected the whole manor. 5. -When thus you have plotted all the fields, ac- cording to the buttings of the hedges found in your firft furveys, you will find that you have very nigh, if not quite, done the whole work t but if there be any fields which lie fo within others, that they are not bounded on either fide by a road, lane, or river, then you mud alfo furvey them, and place them in your plot, accordingly as they are bounded by other fields* 2 6. Draw 144 InJtruStions forfurveying a Manor. 6. Draw a fair draught of the whole, putting down therein the manor-houfe, and every other con- fiderable houfe, wind-mill, water-mill, bridge, wood, coppice, crofs paths, rills, runs of water, ponds, and any other matters notable therein. Alfo in the fair draught, let the arms of the lord of the manor be neatly drawn, and a compafs in fome wafte part of the paper ; alfo a fcale, the fame by which it was plotted. You muft a4fo beautify your map with co- lours and cuts, according as you fhall fee conve- nient. Write down alfo in every field the true content thereof; and if it be required, the names of the pre.fent poflefTors, and their tenures, by which they hold i! of the lord of the manor. The quality alfo of the land you may take no- tice of, as yo x u pafs over it, if you have judgment therein, and it be required of you. 70 take the draught of a county or country. j. If the county or country is in any place there- of bounded with the fea, furvey firft the fea-coaft thereof, meafuring it all along with the chain, and taking all the angles thereof truly. 2. Which done, and plotted by a large fcale, furvey next all rocks, fands, or other obftacles that lie at the entrance of every river, harbour, bay, or road upon the coaft of that county or country ; which plot down accordingly, as (hall be (hewn farther on. 3. Sur- InftruftionsforfurueyingaManor. 145 3. Survey all the roads, taking notice as you go along, of all towns, villages, great houfes, rivers, bridges, mills, crofs- ways, &c; Alfo take the bear- ing at two ftations of fuch remarks as you fee out of the road, or by the fide thereof. 4. Alfo furvey all the rivers, taking notice how far they are navigable, what branches run into them, and where they begin j what fords they have, bridges, &c. 5. All this being exactly plotted, will give you a truer map of the country, than any that I know of hath yet been made in England. However, you may look upon old maps, and if you find therein any thing worth notice, that you have not yet put down, you may go and furvey it ; and thus by de- grees you may finim a country, fo as not to leave out even one gentleman's houfe ; for it will fcarcely happen but fome very remarkable thing will come into your view, either from the roads, the rivers, or fea-coaft. 6. Laftly, With a large quadrant take the true latitude of the place, in three or four places of the county j which put down upon the edge of your map accordingly. CHAP. [ '46 ] CHAP. xr. Of dividing lands. 70 divide a triangle feveral ways. SUppofe ABC a triangular piece of land, contain- ing 60 acres, to be divided between two men, one to have 40 acres cut off towards A, and the* other 20 acres to- wards C ; and the line ofdivificn to pro- ceed from the angle B. A^ ., 1 T/rA > C Firft, meafurethe bafe 3$ 'jf 2) -lo'O A A> i A C, viz. 50 chains ; then fay by the Rule of Three, If the whole con- tent 60 acres give 50 chains for its bafe, what (hall 40 acres give ? Multiply and divide, the quotient will be 33 chains, 33 links; which fet off upon the bafe from A to D, and draw the line B D, which fhall divide the triangle as was required. If it had been required to have divided the fame into 3, 4, 5, or more unequal parts in given proportions, you muft, as before, by the P.ule of Three, have found the length of each feverai bafe ; much after the fame manner as merchants calculate their lofs or gain by the rule of Fellowfhip. There are feveral ways of doing this geometrically, without the help of arithmetic -, but my bufinefs is Of Dividing Lands. 147 is not fo much to (hew you what may be done, as how to do it by the moll eafy and practicable way. To divide a triangular piece cf land into any number of equal and unequal parts y by lines proceeding from any point ajpgned in any Jide thereof. Let ABC, the triangular piece of land, contain- ing 60 acres, be divided between three men, the firft to have 1 5 acres, the fecond 20, and the third 25 acres, and the lines of divifion to proceed from D : firtl:, meafure the bafe, which is 50 chains ; then divide the bafe into three parts, as you have been before taught, by faying, If -60 give 50, wj^at (hall 15 give? Anfwer, 12 chains, 50 Ijpks fbr!th>e E D firft man's bafe ; which fet off from A to E. Again, fay, If 60 give 50, what (hall 20 give? Anfwer, 16 chains, 66 links for the fccond man's bafe; which fet ofT from E to F ; then confequently the third man's bafe, viz. from F to C, muft be 20 chains, 84 links. This done, draw an obfqure line from the point afligned D, to the oppofite angle B ; and from E to F draw the lines EH, FG parallel to BD. Laftly, From D draw the lines D H, D G, which (hall divide the triangle into three uch parts as were required. L 2 % 148 Of Dividing Lands. To divide a triangular piece of land, according to any proportion given by a line parallel to one of the Jides. ABC is the triangular piece of land, containing 60 acres, the bafe A C is 50 chains. This piece of land is to be divided between two men, by a line parallel to BC, in fuch proportion that the one have 40 acres, the other 20. Firft, Divide the bafe, as has been before taught, and let the point of divifion fall in D, AD being 33 chains, 33 links; and D C 16 chains, 67 links. Secondly, Find a mean proportion between A D and AC; by multiplying the whole bafe 50 by AD 33, 33, the product is 16665000; of which the fquare root being extracted, gives 40 chains, 82 links ; fet this off from A to E. Laftly, From E draw E F parallel to B C, which divides the trian- gle as demanded. Of dividing four-Jided figures or trapezia. Before I begin to teach you how to divide pieces of land of four fides, it is convenient firft to mew you how to change any four-fided figure into a tri- 3 angle > Of Dividing Lands. 149 angle j which done, the work will be the fame as in dividing triangles, 30 reduce a trapezium into a triangle^ by lines drawn from any angle thereof. Let A BCD be the trapezium to be reduced into a triangle, and B the angle afiigned : draw the obfcure line BD, and from C make a line parallel thereto, as CE 5 extend alfo the bafe AD, 'till it meet CE in E : then draw the line B E, which (hall make the triangle BAE equal to the trapezium ABCD. Now to divide this trapezium according to any afiigned proportion, is no more than to divide the triangle ABE, as before taught j which will alfo divide the trapezium. EXAMPLE. Suppofe the trapezium ABCD, containing 1 24 acres, 3 roods, and 8 perches, is to be divided be- L 3 tweeu I 5 Of Dividing Lands* tween two men, the firft to have 50 acres, 2 roods, and 3 perches j the other 74 acres, i rood, and 5 perches, and the line of divifion to proceed from B. Firft, Reduce all acres and roods into perches, then will the content of the trapezium be 19968 perches; the firft man's {hare 8083 perches, the fecond 1 1885. Secondly, Meafure the bafe of the triangle, viz. AE 78 chains, oo links: Thenfay.If 19968 the whole > 8 cha - OQ , inkSj content give for it s bale 3 What fcall 8083, the firft, cha ; Hnks: man s part, give r Anfwer ) which fet off from A to F, and drawing the line F B, you divide the trapezium as deiired -, the triangle ABF being the firft man's portion, and the trapezium BCFDthefecond's. How to reduce a trapezium into a triangle, by lines drawn from a point aj/igned in anyjide thereof. A BCD A the trapezium, E the point afligned, from whence to reduce into a triangle, and run jhe divifion line j the trapezium is of the fame con- H tent Of Dividing Lands. \ $ I tent as the former, viz. 19968 perches; and it is to be divided as before, viz. one man to have 8083 perches, and the other 11885. Firft, For to re- duce it into a triangle, draw the lines ED, EC, and from A and B make lines parallel to them, as A F, B G ; then draw the lines EG, E F, and the triangle E F G will be equal to the trapezium A B C D A, which is to be divided as before ; therefore find, by the rule of proportion, what the firft man's bafe muft be, viz. 3 i chains, 52 links, fet it from F to H, and draw the line H E ; fo (hall you divide the trapezium according to the former proportion. 'To reduce an irregular frue-fded figure into a tri- angle, and to divide the fame in any given propor- tion. Let ABCDEA be the five-fided figure; to re- duce this into a triangle, firft draw the lines A C, A D ; and parallel thereto B F, EG, extending the bafe from C to F, and from D to G ; then draw the lines A F, A G, which will make the triangle A F G equal to the five-fided figure. If this wer e L 4 t 152 Of Dividing Lands, to be divided into two equal parts, you muft take F H equal to half of the bafe of the triangle ; and from H draw the line H A, which divides the fi- gure ABC D E into two equal parts. Jf in dividing the plot of a fold, there be outward angles, you may change them after the following manner. Suppofe A BCD E A be the plot of a field, and B the outward angle. Draw the line C A ; parallel thereunto draw alfo B F ; join the points C F, and the five-fided figure, having one angle, is now reduced Jnto a four-fided figure, or trapezium ; which you may again reduce into a triangle, as has been before taught. Of Dividing Lands. 153 To divide an Irregular plot of any number ofjides t ac- cording to any given proportion, by a Jlrait line through it. AB *54 Of Dividing Lands. ABCDEFGHIA is a field to be divided e- qually between two men, by a flrait line proceeding from A. Firft, Confider how to divide the field into five- fided figures and trapezia, that you may the better reduce it into triangles : as by drawing the line K L, you cut off the five-fided figure A B C H I ; which reduce into the triangle A K L, and meafuring half the bale thereof, which will fall at Q, draw the line OA. Secondly, Draw the line MN, and from the point Q^reduce the trapezium C D G H into the triangle MNQ^; which again divide into halfs, and draw the line QR. Thirdly, From the point R reduce the trapezium D E F G into the triangle R O P ; and taking half the bafe thereof, draw the line R S ; and then have you divided this irregular figure into two equal parts by the three lines AQ^.QJl, R S. Fourthly, Draw the line A R, alfo QJT parallel thereto. Draw alfo A T, and then have you turned two of the lines into one. Fifthly, From T draw the lines T S, and parallel thereto the line R V : draw alfo T V. Then is your figure divided into two equal parts by the two lines AT and TV. Laftly, Draw the line A V, and parallel thereto T W. Draw alfo A W, which will cut the figure in- to two equal parts by a flrait line, as was required. You may, if you pleafe, divide the whole of fuch a figure into triangles ; and then divide each triangle from the point where the divifion of the laft fell, and then will your figure be divided by a crooked line, which you may bring into a ftrait one, as above. The Of Dividing Lands. The above is a good way of dividing land's- but furveyors feldom take fo much pains about it ; I (hall therefore (hew you how they commonly abbre- viate their work, as is indeed An eafy way of dividing lands. Admit the following figure A B C D E contains 46 acres, to be divided equally between two men, by a line proceeding from A. Draw firft a Hne at pleafure through the figure, as the line A F. Then caft up the content of either part, and fee what it wants, or what it is more than the true half mould be. As for example : I caft up the content of A E G, and find it to be but 1 5 acres j whereas the true half is 23 acres j 8 acres, being in the part ABCDG more than AEG. Therefore I make a triangle containing 8 acres, and add it to A E G, as the tri- angle A G I ; then the line A I parts the figure into two equal quantities. But more plainly how to make this triangle : meafure firft the line A G, which is 23 chains, 60 links. Double the 8 acres, they make 16 ; to which add five cyphers to turn them into chains and links, and then they make 16000005 which divide by AG 2360, the quotient is 6 chains, 77 links - y for the perpendicular H I, take from your fcale 6 chains, 77 links, and fet it fo from the bafe A G F, that the .end of the perpendicular may juft touch the line E D, which fuppofe at I. Then draw the line A I, which makes the triangle ACI juft 8 acres, and divides the whole figure as defired. If 156 Of Dividing Lands. If it had been required to have fet off the perpen- dicular the other way, you muft (till have made the end of it but juft touch the line ED, as LK does : for the triangle A K G is equal to the triangle A G I, each 8 acres. And thus you may divide any piece of land of any number of fides, according to any proportion, by ilrait lines through it, with equal certainty,* and more cafe, than the former way. Note, You might alfo have drawn the line A D, and meafured the triangle A G D, and afterwards have divided the bafe G D in I, according to the given Of Dividing Lands. 157 given proportion ; this will appear plainer by the following example : Suppofe a field, containing 27 acres, is to be di- vided between three men, each to have 9 acres, and the lines of divifion to run from a pond in the field, fo that each perfon may have the benefit of the water, without going over another's land. Firft, From the pond o draw lines to every angle, as o A, o B, o C, oX>, O E ; and then is the figure divided into five triangles ; meafure each of thefe, and put their contents down feverally, ^Perches. The whole content is 4320 perches, or 27 acres, and each man's proportional part 1440 perches. From which reduce into , rA I B Triangle < C D U per< o o o .rllCb Bl C D E A , and [ n you \* 39 1238 911 158 Of Dividing Lands. From O to any angle draw a line for the firft divi- fion-line, as p A. Then confider that the firft tri- angle A o B' is but 674 perches, and the fecond B o C 390, both together but 1064 perches, lefs by 376 than 1440, one man's portion. You muft therefore cut off from the third triangle C ' D 376 perches for the firft man's dividing line : which you may do thus : the bale D C is \ 8 chains, the con- tent of the triangle 1238 perches: fay then, If 1238 perches give the bafe 18 chains, oo links ; What (hall 376 parches give? Anfwer, 5 chains, 45 links ; which fet from C to F, and drawing the line o F, you have th\2 firft man's part, viz. A o F, CBA. Secondly, Obferve the remaining part of the tri- angle C o D, after 37 6 is taken out, and you will find it to be 86 1 perches, which is lefs than 1440 by 578. Therefore fro m the triangle D E cut off 578 perches, and the point of divifion will fall in G. Draw the line o O, which with o A and o F, divides the figure into three equal parts.- To divide a circle according to any proportion^ by a line concentric with the firft. Th<; areas of circles are in proportion to one ano- ther a s the fquares of their femi-diameters ; there- fore, if you divide the fquare of the femi-diameter by the proportion given, and extract the fquare root, you w ill have your defire. EXAMPLE. Let A B C E) be a circle to be equally divided be- tweea two men. The Trignometry. The diameter thereof is 2 chains. The femidiameter i chain, or 100 links. Thefquare thereof 10000. Half the fquare 5000* The root of the half 7 1 links ; which take from your fcale, and upon the fame center defcribe the circle G E H F, which will divide the circle A B C D into two equal parts. CHAP. XII. Trigonometry : or y 'fbe menfuration of right-lined triangles. TH E ufe of the table of logarithmn num- bers I have fhewed in chap. I. concerning the extraction of the fquare root. Here follows 22* 160 Trigonometry, The life of the table of fines and tangents. Any angle being given in degrees and minutes, how to find the fine or tangent thereof. Let 27 degrees, 10 minutes be given, to find the line and tangent thereof. Firft, in the table of fines and tangents, at the head thereof feek for 27 ; and having found it, look down the firft column on the left-hand under M, for the 10 minutes, and right againft it under the title Sin. ftands the fine required, viz. 9,659517 , alfo in the fame line under the title 'Tang, ftands the tangent of 27. 10'. viz. 9,710282 : but if the degrees exceed 45, then look at the foot of the tables for the degrees, and upon the right-hand column for the minutes 5 and right againft it you will find the fine and tangent above the title Sine. fan. Thus, the fine of 64 degrees, 50 minutes, is 9,956684 ; the tangent thereof is 10,328037. Tofndthe co-Jine> or fine-complement ; the co- tangent, or tangent-complement, of any given number of degrees and minutes. The co-fine, or co-tangent, of an angle, is the fine and tangent of the remaining degrees and minutes, after fubftra&ing it from 905 thus, take 25 degrees 10 minutes from 90 degrees, oo min. there remains 64 degrees, 50 minutes ; the fine of which is, as before, 9,9566843 and that is alfo the fine-comple- ment of 25 degrees, 10 minutes. But Trigonometry. 1 6 1 But a more ready wdy to find the co-fine, or co- tangent of any number of degrees given, is to look for the degrees and minutes as before taught, for fines and tangents ; and right againft it under titles co-fine and co-tangent , or above, if the degrees exceed 45, you will find the co-fine or co- tangent required : thus the co-fine of 50 degrees, 15 minutes, is 9,936431. the co-tangent of 58 degrees, 10 minutes is 9,792974. A fine or tangent ', co-fine or co tangent being given^ to Jind the degrees and minutes belonging thereto. This is the converfe of the former ; for you muft feek in the tables for the fine, Gfc. given, or the nigheft that can be found thereto, and right againft it you will find the minutes and degrees over-head. Let the fine 8,742259 be given, right againft it ftands 3 degrees, 10 minutes. Remember that multiplication is performed with thefe logarithmic tables by addition, and divifion by fubt.r action. If I were to multiply 5 by 4, firft I look for the logarithm of 5, which is 0,698970 The logarithm of 4 is 0,602060 Added together, they make 1,301030 which 1,301030! feek for in the table of logarithms ; and right againft, under title num. ftands 20, the product of 5 multiplied by 4. M Jf 1 6 2 'Trigonometry. If I were to divide 20 by 5, firft I look for the logarithm of 20, which, as above, is 1,301030 the logarithm of 5 is 0,698970 After fubftradtion remains 0,602060 and the number anfwering to this logarithm you will find to be 4. And thus by addition and fubtradtion the rule of three is performed with the logarithms, viz. by adding the fecond and third terms together, and from their fum fubtradting the firft. EXAMPLE. If 15 give 32, what will 45 give ? The logarithm of 1 5 is 1,176091 The logarithm of 45 is 1,653212 The logarithm of 3 2 is i ,505 1 co The two laft added together make 3 , 1 5 8 3 6 2 From that fum I fubtradl the firft term ) and there remains - - J 1,9*2271 Againft 1,982271 in the table I find the number 96. 1 anfwer therefore, If 15 gives 32, 45 will give 96. This you muft obferve to do in the following cafes of triangles, always to add the fecond and third numbers together, and from their product to fubtradt the firft, the remainder will be the logarithm number, fine, or tangent of your required line or angle. Certain Trigonometry. 163 Certain theorems for the better undemanding right-lined triangles. 1 . A right-lined triangle is a figure comprehended within three flrait lines. 2. It is either right-an- gled, as A, having one right angle, which contains juft 90 degrees, viz. that at b\ or elfe oblique as B, which confifts of three acute an- gles, neither of them fo great as 90 degrees ; or which confifts of two a- cute angles and one obtufe, viz. as that C d. 3. All the three angles of any triangle are equal to two right angles, or 180 degrees; fo that one angle being known, the fum of the other two becomes known alfo ; or two being known, the third may be found by fubtrafting the two known angles out of 180 degrees, the remainder is the third angle. To underftand well what the quantity of an angle is, take this following iliuftration. Let A BCD be a circle, whofe circumference is divided (as all circles are fuppofed to be) into 360 equal parts, called degrees, and each of thefe again divided into 60 equal parts, which are called minutes: now a right-angled triangle is that which cuts off one M 2 fourth 164 'Trigonometry. fourth part of this circle, viz. 90 degrees fuch is the triangle PEG. An angle cut off being lefs than 90 degrees, is a- cute as the angle H E F. The angle G E I is obtufe becaufe the two lines that proceed from E, take in betwixt them more than a quarter of the circle. A 5. Every trigonometry. 165 5. Every triangle hath fix parts, viz. three fides and three angles; the fides are fometimes called legs ; but moft commonly in right-angled triangles the bottom line, as B C, is called the bafe, then is A C the perpendicular ; but the longeft line A B is al- ways called the hypothenufe. The fides are pro- portional to the lines of their oppofite angles ; fo that any three parts of the fix being known, (except the three angles) the reft may eaiily be found. 6. When an angle exceeds 90 degrees, fubtraft it out of 1 80, and work by the remainder, which is called the fupplement thereof. CASE I. .7 r.,'i.', f . ' * J JlU"I,\! 3f:J 3'1 O'J In a right-angled tri angle t the bafe being given > and the acute angle at the bafe j to find the hypothenufe and perpendicular. J->i S-flj '-"Tl.'r jf r J r>.'IJJ Vii i./."! In a right-angled triangle ABC, there is given the bafe A B, 26 equal parts, as yards, perches, &c. and the angle at A 30 degrees ; to find the length of the hypothenufe A C ; As the fine complement of the angle at A is to the logarithm of the bafe 26, So is the radius or fine of 90 to the logarithm of the hypothenufe A C ecjual 3' witt* ^ M 3 1 66 Trigonometry. The fine complement of 30 degrees is 9,937531 The logarithm of 26 is J >4H973 The radius, or fine of 90* 10,000000 The two laft added together 1 1,414973 Remains, after fubtradling the firft numb. 1,477442 Which if you look for in your table of logarithms, you will find the neareft number anfwering thereto to be 30, the length of the hypothenufe required. Note, in your table, when you cannot find ex- actly the logarithm you look for, you muft take the neareft thereto ; as in this example, I find 1,477121 to be the neareft to 1,477442. Note alfo, that the fine complement of the angle at A, is the right fine of that at C. For the angle at A in a right- angled triangle being given, you may, by fubtraclion, cafily find the angle at C j becaufe, by the rule above, all the three angles of a triangle are equal to two right angles, or 180 degrees; confequently, if you take the fum of the right angle at B 90', and that at A 30 out of 180, there will remain the angle at C equal to 60. But to purfue our quef- tion, 70 jind the perpendicular. As the fine of the angle A C B 60 is to the log. of the bafe A B 26 ; So is the fine of the angle CAB 30* to the log. of the perpendicular C B 15. . t Note, Trigonometry. 167 Note, When three letters are put to exprefs an an- gle, the middle letter denotes the angular point, The fine 60 degr. is 993753 I The log. of the bafe A B 26 is I >4*4973 The fine of 30 degr. is 9,698970 The fum of the two laft 11,113 943 From which fubtracT: the firft, remains 1,176413 The neareft number anfwering to which is 15, equal to the length of the perpendicular Ifcie C B. Otberwfe ; the bypotbenufe A C equal to 30 being Jirjl found, you may find the perpendicular thus : As the fine of the right-an. C B A or rad. 10,000000 is to the log. of thehypoth. A 30 1,477121 So is the fine of the angle C A B 30 deg. 9,698970 to the log. of the perpendicular 15, 1 1 , 1 7609 1 CASE II. The perpendicular and acute angle A C B being given, to find the bafe and bypothenufe. Let the perpendicular C B be 1 5, as before the angle A C B 60 deg. to find the bafe. Cf As 1 6 8 'Trigonometry. As the co-fine of the angle A CB is to the logarithm of the perpendicular B C 15 j So is the fine of the angle A C B to the logarithm of the bafe A B 26. The co-fine of the angle A C B 60, is 9,698970 The logarithm of CB 15 is 1,176091 The fine of the angle A CB 60, is 9)93753* 11,1 13622 The neareft log. anfwering 1026, is' 1,414652 For the bypothenufe* -,' v As the fine complement of the angle A C B 6o q is to the log. of the perpendicular C B 15, So is the fine of the angle A B C, or 90, to the log. of the hypothenufe 30. The co-fine of the angle A C B is 9,698970 The log. of the perpend. C B 15 is 1,176091 The radius 10,000000 The log. of the hypothenufe 30 -1,477121 Or the bafe being firjl found, find the hypothenufe thus. As the fine of the angle A C B 60 9>93753 r is to the log. of the bafe 26 1,414973 So is radius 10,000000 to the log. of the hypothenufe (30) 1,487442 CASE Trigwometry. 169 CASE III. Tie hypothenufe, and one of the acute angles given, to . Jind the baje and perpendicular. s S A '.o..^ol -?: r i ri to sm t.Jj Let the hypothenufe be A 30 5 The angle CAB 30. W*vv^ 'A\ \rib x*itf'ia t tt "^ A'-'^ V-^:, '7.vvi'> Tofadthekafe A B, work thus : As radius or the fine of the right angle / CBA 9 o% tj 1 is to the log. of thehypoth. A 030 1,477121 So is the co-fine of the angle CAB 300 9,93753 1 to the log. of ttte bafe A.B (26) 11,414652 C, wcr To find the 'perpendicular B C, ie^r^ thus : As radius or the fine of the right angle r CBA 9 o ;j 10,000000 is to the log of the hypoth. AC 3.0 1,477121 So is the fine of the angle CAB 3-0 9,698970 Jo the log. of the perpend. (15) 11,176091 Or 170 Trigonometry. Or the bafe being firft found, find the perpendicular thus: As the co-fine of the angle C A B 30 9>93753 1 is to the log. of the bafe A B 26 1,414973 So is the fine of the angle CAB (30) 9,698970 II > I *3943 to the neareft log. of the perpend. (15) 1,176412 CASE IV. hypotbenufe and bafe being given, to find the acute angles, ACBWCAB. Let AC, the hypothenufe, be 30 equal parts A B the bafe 26 j required the angle A C B, 36 As the logarithm of the hypothenufe A C 30, is to radius, or the fine of the angle CB A 90 So is the logarithm of the bafe A B 26, to the fine of the angle A C B 60. Vbe Trigonometry. 171 tte Operation. The log. of the hypothenufe AC 30 is 1,477121 The radius 10,000000 The logarithm of the bafe A B 26 i ,41 4973 The line of A C B the angle required, 60 9,937852 For the angle CAB, work thus : As the log. of the hypothenufe A C 30 1,477121 is to the radius or fine of 90. 10,000000 So is the logarithm of the bafe A B 26 1,414973 to the co-line of the angle required 30* 9,937852 CASE V. The bypotbenufe and perpendicular being given, to fad the acute angles > andalfo the bafe. The hypothenufel is 30. The perpendicular 15. I g A B C a right angle. 172 Trigonometry. Tojindthe angle at A, work thus : As the log. of the hypothenufe AC 30 1,477121 to the radius 10,000000 So is the log. of the perpendicular C B 15 1,17609 1 to the fine of the angle at A 30*, n 9,698970 To jind the angle at C, 'work thus : As the logarithm of the hypothenufe AC 30 is to the radius or fine of 90. So is the logarithm of the perpendicular B C 15 to the co-fine of the angle A, or the right fine of the angle C 60. To find the bafe, work as you were taught in Cafe 2. Here note, that any two fides of a right-angled triangle being given, the third fide may be found by extraction of the fquare root. EXAMPLE. \ ilL, In the right- angled triangle A , let the given bafe be 20, the perpendicular 15, and. the hy- pothenufe required. The fquare of the bafe 20, or 20 multiplied by itfelf, is 400 : fquare alfo the per- pendicular 15, it gives 225 j add thefe fquares toge- ther, and we have 6255 the fquare root, of this trigonometry. 173 this fum viz. 25 is the length of the hy- pothenufe; but if the hypothenufe and 20 either of the other fides be given to find 625(25 the third, you muft fubtra<5t the lefs fquare 45 out of the greater, and the fquare root of the remainder is the fide required. As for example ; fuppofe the hypothenufe 25 and the bafe 20, to find the perpendicular multiply the hypothenufe by itfelf, the product is 625 multiply the bafe by itfelf, it makes 400 Subtract 400 from 625, there remains 225 The root of which is 15, the perpendicular requir'd. C A S E VI. Of oblique-angled plain triangles. Two fides of an oblique triangle being given, and an angle oppofite to either of the fides, to find the other angles and alfo the third fide. 174- Trigonometry. In the triangle ABC there is given the fide A B 40, the fide BC 32 ; The angle at A 40 degrees, Required the angle at C Note, that in all oblique triangles the fides are in fuch proportion one to another, as the fines of their oppofite angles. As the logarithm of the fide B C 32 i, 505 150 is to the fine of the angle A 40 9,808067 So is the logarithm of the fide A B 40 1,602060 11,410127 To the fine of the angle atC 53*. 28 ; 9,904977 To find the angle at B. Add the two known angles together, viz. that at A 40, and that at C 53.28, and they make 93 degrees, 28 minutes ; which fubtradled from 180 degrees, leaves 86 degrees, 32 minutes, for the angle atB. Lqflfyy to fad the line A C, fay, As the fine of the angle A 40 9,808067 is to the logarithm of the fide B C 32 1,505150 So is the fine of the angle 86* : 32 9,999204 to the log. of fide AC required 50 Trigonometry. 175 Note, the neareft whole number anfwering to the logarithm 1,696287 is 50 ; but if you admit fractions, the length of the line A C will be only 49 -AV. CASE VII. Two angles being given, and a Jide oppojite to one of tbem t to find the other oppofitejtde. In the foregoing triangle there is given the angle A 40 degrees, the angle 53 degrees, 28 minutes 5 and the fide A B 40 , to find the fide B C. As the fine of the angle C 53 : 28 9,904992 is to the log. of the fide A B 40 1,602060 So is the fine of the angle A 40 9,808067 11,410127 to the log. of the fide B C, 3 2 nearly i , 505 1 3 5 CASE VIII. Twojtdes and the contained angle of a triangle being given \ to find either of the other angles. 500 In 1 7 6 Trigonometry. In the triangle ABC there is given the fide A B 1 97, The fide A 500, The angle at A 40 To find either of the other angles As the log. of the furn of the two fides 627 2,843233 is to the log. of thei- difference 303 2, 48 144 3 So is the tang, of the half fum of the} Q g two oppofite angles 70 degrees 5 J 43 934 12,920377 to the tangent of half their difference 50 degr. 4 min. >77 I 44 Which 50. 4'. added to the half fum of the two unknown angles, viz. 70 gives 120. 4'. for the quantity of the angle at B ; but taken from 70, leaves 19. 56'. the angle at C, CASE IX. of an oblique triangle being given, tofnd the angles. . You muft divide your oblique triangle into two right-angled triangles, thus : In Trigonometry. In the triangle ABC The fide A C is The fide A B The fide 6 C The fum of the two fides AB, BC Their difference As the log. of the fide A C 50 is to the log. of the fum of the 7 other two (ides 56 S So is their diff. 16 177 5 36 20 E l6 1,698970 I,748l88 1,204120 2,952308 othe log. of a fourth number 18 J > 2 53338 Subtract this 1 8 out of the greateft fide A C 50, and there remains 32; the half of which, viz. 1 6, is the bafe of the right-angled triangle B D C, and the remainder of the line A C, viz. A D 34, is the bafe of the remaining right-angled triangle, B D A. Now in each of thefe right-angled triangles, having the bafe and hypothenufe given, you may find the angles ; by Cafe IV. Note, It is eafier to find the 4th number, for divid- ing an oblique-angled triangle into two right-angled triangles by vulgar arithmetic, than by the tables of logarithms, thus : N Square 1 7 8 Trigonometry. Square the three given fides, add the two greater fquares together, and from that fum fubtract the lelTer; half the remainder divide by the greater fide, the quotient will be the bafe of the greater right-angled triangle. EXAMPLE. In the foregoing triangle, the fquare of the greateft fide A C 50, is 2500 The fquare of the fide A B 36, is 1 296 Added together, make 3796 From which fubtracl: the fquare of the 7 leaft fide 4< Remains 3396 The half 1698 Which 1698 divide by 50 the longeft fide, the quo- tient is 33pr, the bafe of the greater right-angled triangle, viz. A D ; and that being fubtracled from 50, leaves 16 T V for the bafe DC of the lefler right- angled triangle. CASE c Trigo?2ometry. 1 79 CASE X. three Jides of an oblique triangle being given, haw to find the fuperficial content, without knowing the perpendicular. From half the fum of the three fides fubtract each particular fide. Add the logarithms of the three differences, alfo the logarithm of half the fum of the three fides together. Half the 'total is the loga- rithm of the content required. In the foregoing triangle, the fides are 50, 36, 20, their fum is 106 j the half fum 53. The differences between the half fum and each par- ticular fide, are 3 log. 0.477121 17 1-230449 33 1.518514 The half fum 53 1.724276 Total added 4.950360 The half 2.475180 The number anfwering to that log. is 298, which is the content of the triangle required. By vulgar arithmetic thus : Multiply the firft difference by the fecond, that product by the third, that product by the half fum. Laftly, Extract the fquare root, and you have the N 2 fuperficial i8o Heights and Diftances. fuperficial content. So 3 multiplied by 17 makes 515 which multiplied by 33, makes 1683; that, multiplied by 53, the half fum makes 891995 the fquare root of which is 298, the content required. C H A P. XIII. Of Heights and Diftances. To take tie height of a tower, Jteep/e, tree, or any fuch thing. LET A B be a tower, whofe height you would know. Firft, At any convenient diftance, as at C, place your femicircle, or fuch other inftrument you judge moft fit for the taking an angle of altitude, us a large quadrant, or the like, and there obferve the angle A C B. But to be more plain, place your fe- micircle horizontally at C, by making a plummet- line fixed to the center fall juft upon 90 deg. (in fome femicircles there is a line on the back-fide of the brafs limb, on purpofe for the fetting it hori- zontal.) Then (firil fere wing the inftrument fafl) move the index up and down, 'till through the fights you efpy the top of the tower at A ; fee then what degree upon the limb is cut by the index, fuppofe Heights and Diftances. 181 fuppofe 58, fo much is your angle of altitude. Mea- fure next the diftance Cd t between your inftru- inent and the foot of the tower, which let be 25 yards ; then have you all the angles known, (admit- ting the angle the tower makes with the ground, viz. d to be a right angle) and the bafe to find the perpendicular A B ; which you may do, by Cafe I. >f Trigonometry j thus, N As 182 Heights and Diftances. As the fine of the angle A at 32 9,724210 is to the log. of the bafe Cd 25 1,397940 So is the fine of the angle at C 58 9,928420 to the log. height of the tower A B, i , , or rather A W 2 ? 6 11,705464 to the height of the tower 54 yards, log. i ,73 1 1 1 8 To take this at two ftations, without approaching the foot of the tower, you mufl proceed as before directed ; for if you take the angles at C, and then meafure to F, and there in like manner as before, take your angles again, thereby you may find all the angles, and the line A F ; then fay, As the fine of the angle A B F is to the logarithm of the line F A, So is the fine of the angle A F B to the log of the height of the tower A B, Of Diftance , Having before mewed how to take the proper fliftances in furveying a field at two ftations, yet as it naturally occurs in this place I will give you one ex- thereof: fuppofe the following figure to be part 1 86 Heights and Diflances. part of a river, and you being rneafuring along one fide of it, all defirous to know the breadth of it, as alfo to make a true plot thereof, by putting down what remarkable things are feen on the other fide. Beginning at o i, the firft ftation, caufe one of your afiiftants to go to the next bend of the river, as o 2, and there fet up a mark then obferve what angle from the meridian o J, o 2 makes, which let be N. 6 deg. W. : alfo feeing feveral marks on the other fide of the river, take their bearings, as the houfe A, which ftands upon the bank, the breadth of the river bears N. W. 52 deg. the wind-mill B up in the land bears N. W. 40 deg. the tree C by the water-fide bears N. W. 17 deg. All this note down in your field-book, and meafure the diftance O i, o 2, 18 chains, 20 links. After this, coming to o 2, fee how the next bend of the river bears from you, viz. o 3; which is NE 15 deg. fee alfo how the houfe A there bears from you, viz. S. W. 20 deg. the wind-mill S. W. 50 deg. the tree N. W. 77. Alfo as you are going forward, if you fee any thing more at this fecond ftation, taking the bear- ing thereof, as a noted houfe D up in the land bears N. W. 28. and a church E clofe by the river's brink N. W. 4. Meafure the diftance 2, 3, and placing your inftrument at 3, the church bears from you N. W. 88 deg. the houfe up in the land D you cannot fee for the church, therefore let it alone for the next ftation. But here you may fee forward a little village F, the firft houfe whereof bears from you N. W. 32 deg. Meafure the diftance 3, 4, and planting your inftrument in 4, the firft houfe of the village F bears from you S. W. 30 deg. and the houfe 1 8 8 Heights and Diftances. 32 deg. and the houfe D, which you could not fee at the third ftation, S. W. 24. Having put down all thefe things in your field-book, it will appear thus: G i N. W. 6. and 18 chains, 20 links, equal to o 2. g r A tree upon the brink of the river bears *i \ N. W. 17. oo'. J 1 A windmill up in the land N. W. 40. oo'. t A houfe upon the river bank N. W. 52. oo'. G 2 N . E. 1 5. and 1 8 chains, i o links, equal to o 3. f The tree N. W. 77.] Thefe look back to < The houfe S. W. 20. > the obfervation of 1 The windmill S. W. 50) o i. r A noted houfe far up in the-j \ land N. W. 28. /Forward obferva- ") A church upon the river's f tions. I bank N. W. 40. 3 o 3 N. W. 1 5. and 20 chains, 50 links, equal to o 4. r The church bears N. W. 88. , Thefe look back to \ The noted H. cannot be feen. J theobf. of o 2. yThe end of a little village. 7 A forward obfer- L N. W. 32. i vatien. 4""~ * " The end of the little village, -j S. W. 32. /Thefe refpect o 3 The houfe refpectjng o 2 in (" and o 2. the land. S. W. 24. J ! To protract this, draw the line N. S. for a meri- dian, and laying your protractor upon it, the center thereof to o i j again ft N. W. 6. make a mark for the line that goes to o 2 : alfo againft N. W. 17. a mark for the tree, and againft 40. and 52, 2 ' for Heights and Diftances. 189 for the windmill and houfe. Then from o I through thefe marks, draw the lines o A, o B, o C, Q 2. Secondly, Take from your fcale 18 ch. 20 \m, and fet it off upon the line o 2, which will reach from o to 2. There lay again the center of your protractor, the diameter thereof parallel to the line N. S. ; and make marks, as you fee in the field- book, againft N.E. 1 5. N. W. 77. S. W. 20*. S. W. -50. N.W. 28. N. W. 4. and through thefe marks draw lines. The firft line directs to your third flation, the fecond line N.W. 70. directs you to the tree C upon the river's bank ; for that line cut- ting the line i C, (hews you by the interfection where the tree flood, and alfo the breadth of the river. Alfo the line S. W. 20. cuts the line from the firft (lation N.W. 52*. in the place where the houfe A (lands upon the bank of the river. If therefore you draw a line from A to C, it will re- prefent the farther bank of the river. And fo you may proceed on plotting, according to the notes in your field-book; and you will not only have true plot of the river, but alfo know how far the wind- mill B, and the houfe D, &c. (land from the water- fide. Plow to take the horizontal line of a kill. When you jneafure a .hill, you mud meafure the fuperficies thereof, and accordingly cad up the con- tents. But when you plot it down, becaufe you cannot make a convex fuperficies upon the paper, you muft only plot the horizontal or bafe thereof j which you muft (hadow over with the refemblance of a hill, that other furveyors, when examining your 1 90 Heights and Diftances. your work, may not fay you were miftaken. This iorizontal or bafe line, you may find after the fame manner as you have been taught in taking heights. For fuppofe ABC A a hill, whofe bafe you would know. Place your femicircle at A, and caufe a mark to be fet up at B, as high above the top of the hill, as the inftrument ftands from the ground at A; and with your inftrument horizontal, take the angle B A C 58 deg. Meafure the diftance AB 1 6 chains, 80 links ; then fay, As radius 10,000000 is to the line A B 16 ch. Solin. 3,225309 So is the fine complement of A 58 9,7242 10 to A D (made by the perpendicu- lar B D) 8 ch. 90 lin. 12,949519 But if you have occafion to meafure the whole hill, place again your inftrument at B, and take the angle C B D, which fuppofe 46 deg. Meafure alfo the diftance B C 2 1 chains ; then fay, As Heights and Diftances. 191 As Radius 10,000000 is to the line BC 21 ch. (log.) 1,322219 Sp is the fine of the angle C BD 46 9^56934 to the remaining part of the bafe D C 15 ch. i, L. ".I79IS3 Which added to 8.90 makes 24 chains, i link, for the whole bafe AC; which is to be plotted, inftead of A B or B C ; although they were uied in finding the content of the land. 1 mentioned this way, for your better underftand- ing how to take the bafe of part of a hill ; for it often happens your furvey ends upon the fide of a hill. But if you find you are to take in the whole hill, you may facilitate the work by the former way. Thus : take, as before, the angle at A 58 deg. Meafure alfo A B. Then at B take the whole angle ABC 78 deg. Subtract the fum of thefe angles from 1 80 deg. remains 44 for the angle at C; then fay, As the fine of the angle C is to the log. of the fide A B, So is the fine of the angle ABC to the log. of the bafe A C. To take the Jhoah of a rivers mouth, and plot the fame. Meafure firft the fea-coaft on both fides of the river's mouth, as far as you think you (hall have occafion to make ufe thereof; and make a fair draught thereof, putting down every remarkable thing in it's true fituation, as trees, houfes, towns, wind-mills, &V. Then going out in a boat to fuch i fands 192 Heights and Diftances. fands or rocks as make the entrance difficult, at every confiderable bend of the fands, take with a fea-com- pafs the bearing thereof to two known marks upon the more ; and having thus furveyed round all the fands and rock, you may eafily, upon the plot before taken, draw lines which (hall interfect each other at every considerable point of the fands, whereby you may truly point out the fands, and give directions either for laying buoys, or making marks upon the more for the direction of {hipping. EXAMPLE. Suppofe the following figure reprefent a part of fome fea-coaft. Firft, I make a fair draught of it, with the mouth of the river as far up as there is oc- cafion, putting down every thing remarkable as you fee in the figure except the rocks and fand, which I {hall now (hew you how to take. Go in a boat down the river, till you find the beginning of the firft fand A as at #, and there take a fight to the red houfe, which bears S. W. 86 deg. alfo to the tree which lies S. E. 6 deg. To plot which, draw lines in oppofite directions to your obfervations; as from the red-houfe draw a line N. E. 86. and from the tree a line N. W. 6 deg. thofe lines will interfect each other in the point a, which (hews the beginning of the fand A. Row along this fand, founding as you go, 'till you find it bends confiderably, and there take again two obfervations, and protract them as before. The like do at the bending of every fand, 'till either you come round Heights and Diftances. mnd the fand, or come to the place where it joins ith the {hore. O I: 194 Heights and Difta?ices. It would be needlefs to give you all the remaining obfervations, having already in this treatife fo often defcribed the manner of making them ; therefore I will mention only one place of obfervation more, which I judge fufficient. In the fand C, I find the bend (2), and there, as I mould do at all the reft, I take two obfervations to fuch. things on the fhore as appear moft confpicuous, viz. firft, to the beacon, which bears from me S. W. 25. deg. fecondly to the wind- mill, which bears from me N. W. 40. deg. Now after I have taken the other angles at the bends of that fand, and am come home, I draw a line from the beacon oppolite to my obfervation S. W. 25 deg. viz. N. E. 25 deg. Alfo from the wind-mill I draw a line S. E. 40 deg. Now where thefe two lines interfect each other, as at 2, I mark for one point of the fand C. In like manner as I did this, I obferve and protract every other line of the fand C, and of all the other fands and rocks, fo will you have a fair map, fitting for feamen's ufe. Now to give directions for feamen's coming in here, draw a line through the middle of the fouth channel, which line will' cut both the church and wind-mill > fo that if a (hip be coming from the fouthward, bring the church and wind- mill both into one, and keep them fo, then fhe may fafely run in, 'till (he brings the river's mouth fair open, and then fail up the river. Likewife if coming from the northward, bring the tree and beacon bothlnto one line, and keep them fo 'till the river's mouth is fair open. But left they fhould miftake, and run upon the ends of the fands A or B, it would be necefTary that a mark was fet up behind the red-houfe, in a ftrait line with the middle of the river, as 4^. Then Heights and Diftances. 195 Then a fhip coming from the fouthward, or northward, muit keep her forrner marks both ^ one, 'till me bring the red-houfe an4 j^ both in one line ; and keeping them fo, me may run boldly up the^river, till all danger is part. I haye put do\#n this wind-mill and beacon, not as if fuch good mar^s would always happen ; but to (hew you how to place marks, or lay buoys if it be required. You muft, after having taken all the fands, alfo take the founding quite crofs the channels, all up and down, and put them down accordingly ; the b^fc time pf doing which, is at low- water in fpring- tides. To know whether water may be made to run from ~a fpring head to any appointed place. For this work, the diameter of the femjcircle is a little too fhort; however, an indifferent mift may be made therewith ; but it is better to get a water- level, fuch as you may buy at the inftrument-makers; with which being provided, as alfo with two affiftants, and each of them with a flaff divided into feet, inches, and parts of an inch, go to the fpring- head; and caufing your firft affiftant to (land there with his ftaff perpendicular, make the other go in a right line towards the place deflgned for bringing the water any convenient diftance, as 100, 150, or 200 yards, and there let him ftand, and hold his ftaff perpen- dicular alfo. Then fet your inftrument nigh the mid-way between 'em, making it (land level or horizontal ; and look through the fights thereof to your firft afliftant's ftaff, he moving a piece of white paper up and down the ftaff, according to the iigns you make to him, 'till through the fights you fee juft the edge of the paper. Then by a fign give him . O2 to 196 Heights and Diftances. to underftand that you have done with him ; and let him Write it down how many feet, inches, and parts, the paper refted upon. Alfo going to the other end of your level, do the fame by the fecond affiftant, and let him write down alfo what number of feet, &c. the paper was from the ground. This done, let your firft affiftant come to the fecond affi ft ant's place, and there let him again ftand with his ftaff; and let the fecond affiftant go forward 100, 200 yards, as before ; and placing yourfelf and inftrument in the midft between them, take your obfervations altogether as before, and let them be put down as before. And fo muft you do till you come to the place to which the water is to be conveyed. Then examine the notes of both your affiftants, and if the notes of -the fecond affiftant exceed that of the firft, you may be fure the place is lower than the fpring- head, and that therefore water may be conveyed thence. But if the firft's notes exceed the fecbnd's, you may conclude it impoffible without being raifed by fome engine for that purpofe. The firft Affiftant's Note. The fecond Affiftant's Note. Scat. Feet. Inch. Parts. Stat. Feet. Inch. Parts. 6 i 435 o i H 5 i 021242 Oz 463 3 3 5 O3 9 2 4 20 o 8 28 i 8 Here the fecond affiftant's note exceeds the firft, 8 feet, one inch ; which is enough to bring the water with a ftrong current, and to make it alfo rife up 6 or 7 feet t in the houfe, if occafion be ; for fuch as have written of this matter, allow but 4 inches and v fall in a mile to make the water run. A TABLE TABLE O F T H E ; ^>jwi j; c .. jJ.oi ;oi o.o las' ..,. . C ? Northing or Southing, Eafting or Wefting, of every Degree from the Meridian, ac- cording to the Number of Chains run upon any DEGREE. O 3 ATall* A fable of Northing and Southing, i Deg. 2 Deg. 3 Deg. , e P. O B N S EW i? N S EW ? N S EW n D B n o P P i I.O .0 1 I.O .0 I I.O .j 2 2.O .0 2 2.0 .1 2 2.0 .1 3 3- .0 3 3-0 .1 3 3-0 .1 4 4.0 . 4 4.0 .1 4 4-0. .2 5.0 ij 5.0 .2 5 5.0 .2 6 6.0- . 6 6.0 .2 6.0 3 7 7.0 . 7 7.0 .2 7 7.0 4 8 8.0 . 8 8.0 3 8 8.0 4 9 9 3 P 9,^0 > -3 9 9.0 5 10 10.0 .2 10 10.0 3 10 IO.O 5 20 2O.O 4 20 2O.O 7 20 20.0 I.O 3 30.0 5 30 3O.O I.O 30 30.0 1.6 40 40,0 t 'jf' 40 4O.'oj 1.4 40 40.0 2.1 fo 50.0 60.0 ;J r.r 50.0 60.0 i. 7 : 50' 60 5O.O 59-9 2.6 3- 1 76 70.0 :-f.2- 70 70.0: 2.4 70 69.9 3-7 80 '80.0 4 80 ^'o.o 2.8 So 79-9 4.2 90 90.0 .6 96 ^9-9 3.1 90 4-7 IOO 1 00.0 .8 IOO 99-9 3-5 IOO 99.9 5.2 d EW. N S p EW N S S EW N S I 89 Deg. 88 Deg. 87 Deg. Rafting or Wejling. 4 Deg. 5 De g . 6 Deg. ~ "71 a T i H N S EW 1 N S' few B N S EW i ! ' 1 I.O .1 > I.O i I.O .1 2 2.O .1 2 2.0' .d 2 2.0 S 3' .2 3 3 J 3 ( 3-' f 4 4.0 3 4 4.0 3 4 4.0' A 5 5-o ( 3 5 f.o 4 5 5.0 '5 6" 6.0 4 6 6.o< 5 6 6.0 .6 7 7.0] 5 7 7.0 .6 7 : 7.0 * / 8 $.0 .6 8 ^8.0 7 8 9; 9-! .6 9 9.0^ .8 9 8^ .c 10; IO.O 7 10 10.0' 9 IO 1 9-9 I.O 20' 20.0 i .4 20 20. o l,y 20 i-o.o 2.1 3 29.9 2.1 3 29.9 2.6 30 29.8 3-1 40^ 39-9 2.8 40 39-o 3-5 4) 39-8 5tt 49-9 3-5 50 49^ 4.4 TO 49-7 " 5-a oo 59-9 4.2 60' 59- g; 5.2 60 59-7- 6.3 7o : 69.8 4-9 70 69.7 6.1 7g. 60 1 Dcg. A Unable of Northing or Southing* 31 Deg. 32 Deg. 33 Deg. B 5 5 I N S EW D N S EW f N S EW S P I 9 5 I .8 . / i .8 5 2 1.7 I.O 2 1.7 i.i 2 1.7 i.i 3 2.6 J -5 3 2.5 1.6 3 2.5 1.6 4 3-4 2.1 4 3-4 2.1 4 3-4 2.2 5 4-3 2.6 5 4.2 2.6 5 4.2 2.7 6 5-* 3- 1 6 5- 1 3-2 6 5.0 3-3 7 6.0 3.6 7 5-9 3-7 7 5-9 8 6.9 4.1 8 6.8 4.2 8 6.7 4.4 9 7-7 4.6 9 7.6 4.8 9 7.6 4-9 10 8.6 5.1 10 8-5 5-3 10 8.4 5-4 20 17.1 10.3 20 17.0 io.e> 20 1 6.8 10.9 30 25-7 15.4 3 25-4 15.9 30 25.2 16.3 40 34-3 20.6 40 33-9 21.2 40 33-5 2J.8 50 32-9 25.7 50-. 42.4 26.5 50 41.9 27.2 60 5^-4 30-9 60 50.9 31.8 60 50-3 32.7 7 60.0 36.0 70 59-4 70 58.7 38.1 80 68.6 41.2 3o 67.8 42.4 80 67., 43-6 9 77- 1 46.3 90 76.3 47-7 9 75-5 49.0 IOO 85.7 51.5 IOO 84,8 53-0 IOO 83-9 54-5 g EW N S 2 EW N S q EW N S 59 Deg. 58 Deg. 57 Deg. Ea/ling or Wejling. 34 Deg. 35 Deg. 36 Deg. g a a r? 1 3 N S EW 5 N S EW | ; NS EW O n S P I .8 .6 i .8 .6 . .8 .6 2 1.7 i.i 2 1.7 i.i 2 1.6 1.2 3 2-5 i-7 3 2-5 J-7 3 2.4 1.8 4 5 3-3 4.1 2.2 2.8 i * 3-3 4.1 2.3 2.9 4 5 3-2 4.0 2.3 2.9 t> 5.0 3-4 6 4-9 3-4 6 4.8 3-5 7 5-8 3-9 7 5-7 4.0 7 5-6 4.1 8 6.6 4-5 8 6.6 4.6 8 6. 4 4-7 9 7-5 5.0 9 7-4 5-2 9 7.2 5-3 10 8-3 5.6 10 8.2 5-7 10 8.1 5-9 20 1 6.6 II. 2 20 16.4 1 1.5 20 1 6.* n.8 30 24.9 16.8 30 24.6 17.2 3 24-3 17.6 40 33-2 22.4 40 32.8 22.9 40 3 2 -4 23-5 50 41.4 28.0 50 41.0 28.7 50 40.4 29.4 60 49-7 33-5 60 49.1 344 60 48.5 35-3 70 58.0 39- i 70 57-3 40.2 70 56.6 41.1 80 66.3 44-7 80 65-5 45-9 80 64.7 47-0 90 74-6 503 9 737 51-6 9 7^.8 52.9 IOO 82.9 55-9 ICO 81.9 57-4 100 80.9 58.8 g EW N S 5 EW N S O EW N S 7.7 55 Deg. 54 Deg. A tfable of Northing or Southing, 37 Deg. J / O 38 Deg. i S N S EW C I N S EW a D N S EW n O a r* n> 1 .8 .6 i .8 .6 I .8 .6 2 1.6 1. 2 2 1,6 1.2 2 1.6 J *3 /j 2.4 1.8 j 2.4 1.8 3 2-3 1.9 4 3- 2 2.4 4 3- 1 2.5 4 3- 1 2-5 5 4.0 3- 5 3-9 3- 1 5 3-9 3.1 6 4.8 3.6 6 4-7 3-7 6 4-7 3-8 5.6 4.2 ^ 5-5 43 7 5-4 4.4 8 6.4 4.8 8 6*3 4.9 8 6.2 5.0 9 7-2 5-4 9 7- 1 5-5 9 7.0 5-7 10 80 6.0 10 7-9 6.2 10 7.8 6-3 20 1 6.0 12. 20 15.8 12.3 20 15-5 12.6 30 24.0 180 3 23.6 185 30 23-3 18.9 40 31.9 24.1 40 31.5 24.6 .40 jl'.f 25-2 i 50 39-9 30.1 50 39-4 30.8 50 ^8.8 60 47-9 36.1 60 47-3 36.9 60 46.6 37-8 7 55-9 42.1 70 55- 2 43- 1 70 54-4 44.0 80 63.9 48.1 8c 63.0 49-3 80 62.2 50-3 90 71.9 54.2 90 70.9 55-4 90 69.9 56.6 IOO 79-9 60.2 IOC 7 *.8 61,6 IOO 77-7 62.9 g EW N S e EW N S O EW N S p P g 53 De g- 52 Deg. MJ Rafting or Wefting; 40 Deg. 41 Deg. 42 Deg. g D g 9 N S EW ? N S EW B N S EW 3 3 3 a P ft j ,8 .6 I .8 7 I 7 7 2 i-5 '3 2 l -5 1.3 2 1.5 '3 3 2 *3 1.9 3 2.3 2.0 3 2.2 2.0 4 3- 1 2.6 4 2-6 4 3 2.7 5 3-8 3-2 5 3-8 3-3 5 3 7 3-3 6 4.6 3-8 6 4-5 3-9 6 4.4 4.0 7 5-4 4-5 7 5-3 4.6 7 5.2 5-7 8 6.1 5-i 8 6.0 5-2 8 5-9 5-3 9 6.9 5-8 9 6.8 5-9 9 6.7 6.0 10 7-7 6.4 10 7-5 6.6 10 7-4 6.7 20 15-3 12.9 20 15., I 3- 1 20 14.9 l 3-4 30 23.0 30 22.6 19.7 30 22.3 20. i 40 30.6 25-7 40 30.2 26.2 40 29.7 26.8 5 38.3 32.1 50 37-7 32.8 50 37-2 33-5 60 46.0 38.6 60 45-3 39-4 60 44.6 40.1 70 p 45-0 70 52.8 45-9 7 52.0 46.8 80 80 to.4 ,52.5 80 59-4 53-5 90 68.9 57-9 90 67.9 59.0 9 66.9 60.2 IOO 76.6 64-3 IOO 75-5 65-6 IOO 74-3 66.9 g EW N S b EW N S P EW N S 50 Deg. 49 Deg. 48 Deg. A fable of Northing or Southing, 1 43 Deg. 4, P., 45 Deg. 2 N S EW C N S EW 3 N S EW 3 ^ S a j 7 7 j 7 7 j 7 7 2 '5 H 2 1.4 i .j 2 1.4 1.4 3 2.2 2.o| 3 2.2 2. 1 3 2.1 2.1 4 2.9 2.7 4 2.9 2.b 4 2.8 2.8 5 3 .6 3-4 3-6 3-5 ^ 3-5 3-5 6 4-4 4.1 6 4-3 4.2 6 4.2 4.2 7 5- 1 4.8 7 5.0 4.9 7 4-9 4-9 8 5.8 5-4 b fc 8 5- c 8 5-6 5-6 9 6.6 61 < 6.2 9 6.4 6.4 10 7-3 6.8 10 7.2 6.9 10 7- 1 7- 1 20 14.6 13-6 20 14.4 13.9 20 14.1 3 21.9 20.5 3 21.6 20.8 30 21.2 21.2 40 29.2 2 7-3 4 28.8 27.8 40 28.3 28-3 5 36.6 34.1 36.0 34-7 50 35-3 353 60 43-9 40.9 60 43 - 2 41.7 60 42.4 42.4 70 51.2 47-7 7 .50-3 48.6 70 49-- 49-5 80 5-5 H-6 8 57-5 55-^ 80 566 56.6 90 65.8 61.4 9 62.5 9 63.6 63.6 100 73- 68.2 10 71.9 69. 100 70.7 70.7 tjj EW N S | EW N 3 EW N S 47 Deg. 46 Deg 45 Deg. The USE of the foregoing TABLE. HAVE already fufficiently, in the (ixth c ^ a P ter f ^ is book, explained the ufe of this table -, however, becaufe it is made fomewhat different from fuch of this kind as have been made by others, I will briefly, by an example or two, explain it to you. Admit in furveying a wood, or the like, you run a line N. E. 40 degrees, 10 chains : or, which is the fame thing, a line jo chains in length, that makes an angle with the meridian of 40 degrees to the eaftward ; and you would pat down in your Field-Book the northing and easing of this line, under their proper titles N. and E. accord ing to Mr. Norwood's way of furveying, taught in the fixth chapter of this book. Firft, at the head of the table find 40 degrees, then in the column of diftances feek for 10 chains : which being had, you will find to fland right againft at, under the title N. 7. 7. for the northing, which P 3 is is 7 chains, _i_ of a chain : and for the eafling, un- der the title E. 6. 4. which is fix chains, T V of a chain, as nigh as may be expreffed in tenths of a chain : but if you would know to one link, join a cypher to the diflance, fo will 10 be 100; which'feek for in the fame page of the table, and right againft it you will find under title N. 76.6, or 7 chains, 66 links for your northing ; and under title E. 64 3, or 6 chains, 43 links for your eafling : which found, put down in your Field- Book accordingly ; and hav- ing done fo by all your lines, if you find the north- ing and fouthing the fame, alfo the eafling and weft- ing, you may be fure you have wrought true, other- wife not. If the diflance confjils of odd chains and links, as mofl commonly happens, then take them feverally out of the table, and by adding all together you will have your defire. As for example : Suppofe my diflance run upon any line be N. W. 35 degrees, 15 chains, 20 links : N. Ch. Ch. Lin. Firft in the table I find the northing i Q of ip chains to be - - - - P J 9 5 4 10 20 L. o i6 T 4 ~ 12 Which added together, makes 12 chains, 45 links T \- for the northing of that diflance run. In like manner ( 3 ) manner under 35 degrees and title W. I find the wefting of the lame line, as here : Ch. 10 Ch. 5 5 2 20 Lin. o 8 Lin. 74 87 By which I conclude the northing of that line to be 12 chains, 45 links T 4 -, and the wefting 8 chains, 72 links T J - : which thus you may prove by the lo- garithms. As radius 10,000000 is to the diftance 1*5.26 - '- ' 3,181844 So fs the fine of the courfe 3 5 degrees 9*75859 1 To the wefting 8 chains, 72 links 12,940435 And, as radius - - 10,000000 to the diftance 15 chains, 20 links 3,181844 So co-fine of the courfe 55 - - 9>9 J 33^4 to the northing 1 2 chains, 45 links 1 3,095208 Nate, If your courfe had been S. E. it would have been the fame thing as N. W. for you fee in the ta- bles N. and S. E, and W. are joined together. If P 4 your ( 4 ) your degrees exceed 45, then feek for them at the foot of the table : and over the titles N. S. E. W. find out the northing, fouthing, eafting, or wefting. This 1 apprehend to be as much as need be faid concerning the preceding table : and as for finding the horizontal line of a hill, and fuch other things by the table, you will find fufficient directions in the chapter of trigonometry; A TABLE TABLE O F SINES and TANGENTS To every Fifth Minute of the QJJ A D R A N T. The Table of Sines and Tangents. o M Sine Co-fine. Tangent. 0.000000 Co-tang. o.oooooo IO.COOOCO Infinita. 60 4 10 *5 20 25 3 7.162696 7.463726 7.639816 7-7 6 4754 7.861662 7.940842 9-999999 9.999998 9.999996 9-999993 9.999989 9-9999 8 3 7.162696 7-4637 2 7 7.639820 7.764761 7.861674 7.940858 ^.837304 1-2.5362-7^ 12.360180 12.235239 12.138326 12.059142 55 50 45 40 35 3 35 40 45 5 55 60 8.007767 8.065776 8:116926 8.162681 8.204070 8.241855 9-999977 9.999971 9.999963 9-999954 9 999944 9-999934 8.007809 8.065806 8.116963 8.162737 .204126 8.241921 M. 992191 u-9.24194 11.883037 11.837273 l! -795 8 74 11.758079 25 20 15 10 5 Co- fine. Sine. Co-tang. Tangent. M *9- i. ,VI 'Sine. Co fine. Tangent. 8.241921 Co- tang. o 8.241855 9-99S934 11.758079 60 5 10 '5 20 2/V 32 35 40 45 5 b 60 8.276614 8.308794 8-338753 8.366777 8.393101 !.4i79L9 9.999922 9.9999 10 9.999897 9.999882 9.999867 9.9998/51 8.276691 8.308884 8.338856 8.366895 8 -393 2 34 8.418068 11.723309 1 1.691 1 1 6 1 1.661 144 1 1.630105 11.606766 1 1.581932 55 50 45 40 35 30 8.4413Q4 8.463665 8.484848 8.505045 8.5 2 4343 8.542810 9.999834 9.999816 9-999797 9.999778 9-999757 9-9W35 8.441560 8.463849 8.485050 8 505267 8.524586 8.543084 11.558440 n-53 6l 5 ! 11.514950 11-494733 11.475414 1 1.456916 25 20 15 IO c O Co -fine. Sine. Co-tang. Tangent. M 88. 'I he Table of Sines and Tangents. 2. M bine. Cofine. 1 angcnt. Co-tang. o 5 10 5 20 25 3 8.542810 9-999736 8.543084 1 1.456916 60 8.560540 8.577566 8.593948 8.609734 8.624965 8.639680 9-9997 * 3 9.999689 9.999665 9.999640 9.999614 9-9995 86 9.999558 9999529 4.999500 9.999469 9-999437 9.999404 8.560828 8.577877 8.594283 8.610094 8.625352 8.640093 11.439172 11.422123 11.405717 11.389906 11.374648 11.359907 53 50 45 40 35 3 35 40 45 5 55 60 8.65391 1 8.667689 8 681043 8.693998 8.706577 8.718800 8.654352 8.668160 8.681544 8.694529 8.707140 8.719396 11.345648 11.331840 11.318456 i -3547 ' 11.292860 11.280604 2 5 20 15 IO 5 o Cofine. Sine. Co-tang. Tangent. M 87. 3- iVi inne. Co line. 1 angent. Co tang. o 8.718800 9.999404 8.719396 1 1.280604 60 5 10 '5 20 2 5 30 8.730688 8.742250 8.753528 8.764511 8.775223 8.785675 9-99937 9.999336 9.699301 9.999265 9.999227 9.999189 8 -73 I 3'7 8.742922 8.754227 8.765246 8-775995 8.786486 11.268683 1 1.257078 11.245773 11.234754 11.224005 11.213514 55 5 45 40 35 3 35 40 45 50 55 60 cS. 795881 8.805852 8.815599 8.825130 8.834456 8.843585 9.999150 9.999110 9.999069 9.999027 9.998984 9.998941 8.796731 8.806742 8.816529 8.826103 8.835471 8.844644 11.203269 11.103258 11.183471 11.173897 1 1.164529 H.I55356 2-5 20 15 IO 5 o Co-fine. Sine. Co-tang. Tangent. M 86. The Table of Sines and Tangents. ivi 5 10 5 20 25 |p bine. Co-line. Tangent. Co-tang 8.843585 9-9994 8.844644 n.155356 60 8.65252;, 8.86? 28^ 8.869868 8.878285 8.886^42 8 89; 643 9.998896 9.998851 9.998804 9-99 8 757 9.998708 9.998659 8.853628 8.852433 8.871064 8.879529 8.887833 8 895984 8*^07987 8.9118+6 8.919568 8. 9 2 7 ,-5~ 8.934616 8 941952 1.146372 1.137567 1. 12 '^9,'! 1.120471 I.I 12167 I.IO4.OI6 55 50 45 40 35 3 35 40 45 5 55 60 8.9025^6 8.910104 8.918073 8.925609 8-9330I5 8.040296 9.998609 '.^.998558 9.998506 9-99^453 9.998399 0.008314 1.096013 1.088154 I.O8O432 1.07^844 1.065.584 1.058048 2 5 20 '5 10 5 o line. bi'nrr. Co-tan^ Tangent. M ? 5. R Sine. Co- line. 9.9.^8344- Tangent. CD tang. o 8.940296 8.941952 i 1.058048 60 5 10 >5 20 25 3^ 8.947456 8.954499 8.961429 8.968249 8.97^962 S-9 8 '573 9.9982*9 9.9,98232 9.998174 9.9981 16 4.998056 9.007996 8.949168 8.956267 8.963255 8.970133 8.976906 8.9 8 3577 .050832 43733 036745 .029867 .023094 .016423 55 50 45 40 35 3 35 40 45 50 55 60 8.98^083 8 994497 9.000816 9.007044 9.03182 90192/55 9-vS>7935 9.997872 9.997809 9-997745 9.997680 9.997614 Sine. 8.990149 8 996624 9.003007 9.009298 9.015502 9.021620 .009851 .003376 10.996993 10.990702 10.984498 10 978380 ^5 20 15 10 5 o Co line. Co tang. Tangent. M 84. "The "Table of Sine sand Tangents. 6. M Sine. Co-fine. i angent. Co tang. o 5 10 '5 20 25 3 9.019235 9.997614 9-997547 9.997480 9.997411 9-997341 9997271 9.997199 9.021620 10.978380 60 55 50 45 40 35 30 9.025203 9.031089 9.036X96 9.042625 9.048279 953 8 59 9.027655 9.033609 9394^5 9.045284 9.051008 9.056659 10.972345 10.966^91 10.9605 1 5 10.954716 10.948992 IO.Q43341 35 40 45 50 55 60 9-593 6 7 9.064806 9.070176 9.075480 9.080719 9.085894. 9.997127 9-997053 9.9969-9 9.996904 9.996828 9.996751 9.062240 9.067752 9.073197 9.078576 9.083891 9.089144 10 937760 10.93224.8 10.926803 10.921424. 10.916109 10.910856 25 20 { 5 10 5 Co-line. Sine. Co-tang. I angent. M 3- 7- M Sine. Co-line. Tangent. 9.089144 ^.o-tang. o 9.085894 9.996751 10.910^50 60 5 10 5 20 25 30 9.091008 9.096062 9.101056 9.105992 9.110873 9.115698 9.996673 9.996594 9.996514 9-99 6 433 9-99 6 35 I 9.996269 9-9433^> 9.099468 9.104542 9- I0 9559 9.114521 9.119429 9.124284 9.129087 9- I 33 8 39 9.138542 9.143196 9.147803 10.905664 10.900532 10.895458 10.890411 10.885479 10 880571 55 5 45 40 35 3 35 40 45 50 55 60 9.120469 9.125187 9.129854 9.134470 9- I 3937 9- ' *3555 9.996185 9.996100 9.996015 9.995928 9.995841 9-995753 10.875716 10.870913 10.866161 10.861458 10.856804 10.852197 25 20 '5 10 5 o Co- fine. Sine. Co- tang. Tangent. M 82. The Table of Sines and Tangents. 8. M Sine. 9-H3555 Co-fine. 9-995753 Tangent. 9.147803 Co- tang. 10.852197 60 5 10 '5 20 ^5 9.148026 9.152451 9.156830 9.161164 9.165454 0.169702 9.995664 9-995573 9.9954^2 9-99539 9-995297 9-995203 9- 1 5 2 3 6 3 9.156877 9.161347 ?-^5774 9.170157 9.174499 10.847637 10.843123 10.838653 10.834226 10.829843 /o 825501 55 50 45 40 35 30 ;5 40 45 50 55 >o 0.173908 9.178072 9.182196 9.180280 9.190325 9^9433 Co-line. 9.995108 9-995 J 3 9.994916 9.994818 9.994720 9.994620 Sine. 9.17^799 9.183059 9.187280 9.191462 9.195606 9iL9973 Lo-tang. 10.821201 10.816941 10.812720 10.808538 10.804394 10.800287 Tangent. 25 20 *5 10 5 M 81. 9- M Sine. Co-fine. Tangent. Co-tang. o 5 10 '5 20 2 5 12 35 40 45 50 55 60 9- '94332 9^94620 9-994I59 9 994418 9.994316 9.994212 9.994108 9.994003 9-99.^97 9-9937*9 9.993681 9.99'572 9.993462 9-99?35' Sme. 9- I 997 I 3 10.800287 60 55 50 45 40 35 30 25 20 15 10 5 M 9.198302 9.202234 9.206131 9.209992 9.213818 9.217609 9.221367 9.225092 9.228784 9.232444 9.236073 i32rZ2 Co- fine. 9.203782 9.207817 9.211815 9.215780 9 219710 9.223607 9.227471 9.231302 9.235103 9.238872 9.242610 9246319 Co-tane. 10.796218 10.792183 10 788185 10.784220 10.780290 to.77^393 10.772529 10.768698 10.764897 10.761 128 0.757390 10.753681 Tangent. 80. 7 he Table of Sines and Tangents. 10. M Sine. Co- fine. Tangent. Co-tang. 9.239670 9-993351 9 246319 10.753681 60 5 10 J 5 20 25 3 9-243237 9.246:75 9.250282 9.253761 9.257211 9.260633 9.993240 9.993127 9-9930I3 9-99Z898 9.992783 9.992666 92499^8 9.253648 9.257269 9.260863 9.2644.28 9.267967 10.750002 10.746352 10.742-731 IO -739 I 37 10.735572 10.732033 55 5 45 40 35 30 35 4 45 5 55 60 9.264027 9- 26 7395 9-270735 9.274049 9- 2 77337 9.280599 Co-fine. 9.992549 9-992430 9.992311 9.992190 9.992069 9.991947 9.271479 9.274964 9.278424 9.281858 9.285268 9.288652 10.728521 10.725036 10.721576 10.718142 10.714732 10.71 1348 25 20 '5 10 5 Sine. Co-tang. 1 angent. M 79- 1 1. M Sine. Co- fine. Tangent. Co-tang. 60 55 50 45 40 35 3 o 1 10 15 20 25 22 9.2^0599 9.283836 9.287048 9.290236 9-293399 9- 2 9 6 539 9-299 6 55 9.991974 9.288652 10.711348 9.991823 9.991699 9-99*574 9.991448 9.991321 9.991193 9.292013 9- 2 95349 9.298662 9.301951 9.305218 9.308463 10.707987 10.704651 10.701338 10.698049 10.694782 10.691537 35 40 45 5 55 60 9.302748 9.305819 9.308867 9.311893 9.314897 9.317879 9.991064 9.990934 9.990803 9.990671 9.990538 9.990404 9.311685 9.314885 9.318064 9.321222 9-32435 8 9-327475 10.688315 10.685115 10.681936 10.678778 10.675642 10.672525 25 20 15 10 5 o Co-fine. Sine. Co- tang. Tangent. M 78. The Table of Sinfs and Tangents. 12. M ~5 10 '5 20 25 30 35 40 45 5 55 60 Sine. Co-fine. Tangent. Co-tang. 9- 3 '7 8 79 9.990404 9-3 2 7475 10.672525 60 9.320840 9.323780 9.326700 9-329599 9-33 2 47 8 9-335337 9.990270 9.990134 9.989997 9.989860 9.989721 9.989582 9-33 570 9-333646 9.336702 9-339739 9-34^757 9-345755 10.669430 10.666354 10.663298 10.660261 10.657243 10.654245 55 50 45 40 35 3 25 20 15 IO 5 M 9.338176 9.340996 9-343797 9-34 6 579 9-349343 9.352088 Co-fine-.' 9.989441 9.989300 9.989157 9.989014 9.9^8869 9.988721 Sine. 9-34*735 9-35' 6 97 9.354640 9-357566 9.360474 9-3 6 33 6 4 Co-tang' 10.651265 10.648303 10.645360 10.642434 10.639526 10.636636 Tangent. 77- 13- M Sine. Co-fine. Tangent. Co-tang. 9.3^2088 9.988724 9-363364 10.636636 60 55 5 45 40 35 30 5 10 J 5 20 25 3 9.354815 9'3^7524 9.3*60215 9.362809 9.365546 9.368185 9.988578 9.988430 9.988282 9.988133 9.987983 9.987831 9.366237 9.369094 9-37'933 9-374756 9-377563 9.380354 10.633763 10.630906 10.628067 10.625244 10.622437 1 0.619646 35 40 45 50 60 9 370808 9-3734*4 9.376003 9-378577 9-38ii34 9-38^675 9.987679 9.987526 9.987372 9987217 9 987061 9.986901 9.383129 9.385888 9 388631 9391360 9-39^73 9.396771 io.6i6b'7i 10.614112 10.6113^69 10.608640 10.605927 10.603229 Tangent. 25 20 15 1 Co- fine. Sine. Co- tang, M 76. "The Table of Sines and Tangents, 14. M Sine. Co-fine. Tangent. Co-tang. o 5 IO 15 20 2 5 30 9-3 8 3675 9 986904 9.396771 10.603229 60 55 50 45 40 35 30 9.386201 9.388711 9.381206 9.396150 9.398600 9.986746 9.986587 9.986427 9.986266 9.986104 9.985942 9-399455 9.402124 9.404778 9.407419 9.410045 9.412658 10.600545 10.597876 10.595222 10.592/581 10.589955 10.587342 35 40 45 50 55 |6o 9.401035 9 403455 9.405862 9.408254 9.410632 9.412996 9.985778 9.985613 9.985447 9.985280 9.985113 9.984944 9-4I5257 9.417842 9.420415 9.422974 9.425519 0.428052 10.584743 10.582158 IO -5795 8 5 10.577026 10.574481 10.571948 25 20 15 IO 5 o Co-fine. bine. Co tang. Tangent. M 75- 5- M Sine. Co- fine. 1 an gent. Co- tang. o 5 10 20 25 3 35 40 45 50 55 60 9.412996 9.984944 9.426052 10.571948 60 9-4I5347 9.417684 9.420007 9.422318 9.424615 9.426899 9.429170 9.431429 9-433675 9.435908 9.438129 9 -4403 3 8 9.984774 9-9 8 4 f 03 9.984432 9.984259 9.984085 9.983911 9-430573 9.43^,080 9-435576 9-43 8 59 9.440529 9.4429^.8 9-445435 9.447870 9.450294 9.45 706 9-455I07 9-457496 Co tang. 10.569427 10.566920 10.56-1424 10.561941 JO-55947 i 10.557012 10.554565 10.552130 10.549706 10.547294 10.54^893 10.542504 55 50 45 40 35 3 25 20 15 IO M 99 8 3735 9-9 8 355 9.983381 9 983202 9.983022 9.982842 Sine. Co-fine. Tangent. 74. The Table of Sines and Tangents. \ 1 6. M o Sine. 9~4403^8 Co- fine. 9.982842 Tangent. 9.457496 Co -tan hi;. 10.5; 251,4 60 ! 5 10 T 5 20 25 }O 9'442535 9.441720 9.446893 9.449054 9.451204 9-453342 9.982660 9,982477 9.982294 9.9821,9 9.981924 v-9 8l 737 9-459875 9.462242 9.464599 9.466945 9.469280 9.471605 10.540125 10.537758 10.535401 10.533055 10.530720 10.528395 55 50 45 40 35 3 35 40 45 50 1 9.455469 9-457584 9.459688 9.461782 9.463864 9.465935 Co- fine. 9.981549 9 981361 9.981171 9.980981 9.980789 9-98059J Sine. S)-4739*9 9.476223 9.478517 9.480801 9.483075 9-4^5339 Co-tang. 10 526081 IO -5 2 3777 10.521483 10.519199 10.516925 10.514661 Tangent. 25 20 15 10 5 o M 73- '7- M Sine. Co-line. Tangent. Co-tang. o 9-465935 9.980596 9-485339 10.514661 60 5 10 '5 20 g 9.467996 9.470446 9.472086 9-474i5 9-47 6l 33 9.478142 9980404 9.980208 9.980012 9.979816 9.979618 0.979420 9.487593 9.489838 9.492073 9.494299 9-49 6 5 J 5 9.498722 10.512407 10.510162 10.507927 10.505701 10.503485 10.501278 55! 50 45 40 35" 3 25 20 ^5 S 1 IVJ 35 1-0 *5 50 5 60 .9.48014.0 9,4*2128 9484107 9.486075 9. .< 8034 9.48098 >. Co- fine. 9.979220 9.979019 9.978817 9.978615 9.978411 9 978206 Sine. 9.500920 9503.09 9.505289 9.507460 9.509622 9^ I[ 776 Co- tang. 10.499080 10.496^91 10.494711 10.492540 10.490378 10.4^224 Tangent. 72. .. ''The Table of bines and 'Tangents. 18. M Sine. Co-line. Tangent. Co-tang. 9.489982 9.978206 9.511776 10.488222 60 5 10 '5 20 25 30 9.491922 9.493851 9 495772 9.497682 94995 8 4 9.501476 9.978001 9-977794 9.977586 9-977377 9977167 9-976957 9-5<39 21 9.516057 9.518186 9-520305 9522417 9.524520 10.486076 10.483943 10.481814 10.479695 IO -47753 10.475480 55 50 45 40 35 30 35 40 45 50 55 60 9.503360 9-505 2 34 9.507099 9.508950 9.510803 9.512642 Co- fine. 9-97745 9-976532 9.976318 9.9/6103 9.975887 MTS^ZS Sine. 9.526615 9.528702 9.530781 9-53 28 53 9-5349 16 2l3&972_ Co-tang. IO -473385 12.471298 10.469219 10.467147 10.465084 10 463028 Tangent. 25 20 15 10 5 M 7 1 - 19. M Sine. Co-line. .Tangent. Co-tang. o 9.51264.1 9.975670 9-53 6 972 10.463028 60 5 10 '5 20 25 32 35 40 45 50 1 9.514472 9.516294 9.518107 9.519911 9521707 9-523495 9-525i75 9.527046 9.528810 9-53 565 9-532J' 2 9-534Q5* Co-fine. 9-97545 * 9-975233 9-9750I3 9.974792 9-97457 9-97,4347 9974122 9.973897 9.973671 9-973444 9-9732I5 9-972986 Sine. 9 539020 9.541061 9-543094 9-5451*9 9.547138 9-549H9 9-55^53 9-553H9 9-555139 9.557121 9-559 97 9.551066 Co-tang- 10*460980 10.458939 10.456906 10.454881 10.452862 10.450851 10,448847 10.446851 10.444861 10.442879 10.440903 10.438934 Tangent. 55 5 45 40 35 3 2 5 20 15 IO O "M 70. Q.2 The Table of Sines and Tangents. 20. M bine. Co- fine. Tangent. Co -tang. o 9.534052 9.972986 9.561066 10.438934 60 5 10 '5 20 25 3 9-5357 8 3 9-53757 9-539 22 3 9-54Q93' 1 9.542632 9-5443 2 5 9-972755 9.972524 9.972291 9.972058 9.971823 9-97'583 9.563028 9.564983 9.566932 9.568873 9.570809 9-572738 9.574660 9.576576 9.578486 9.580389 9.582286 9.584177 10.436972 10.435017 10.433068 10.431 127 10.429191 10.427262 55 50 45 40 35 3 35 40 45 5o 55 60 9.54601 1 9.547689 9.549360 9.551024 9.552680 9-554329 9-97 I 35* 9.971113 9.970874 9-97 6 35 9-970394 9.970152 Sine." l o-425340 10.423424 10.421514 10.41961 i 10.417714 10.415823 25 20 15 10 5 o M Co-fine. Co tang. Tangent. 69. 21. M Sine. Co-fine. Tangent. Co-tang. ' B 10 15 20 25 30 9-5543 2 9 9-55597' 9.557606 9-559 2 34 9.560855 9.562468 9-5 6 4075 9.970152 9.584177 10.415^23 60 9 969909 9.969665 9.969420 9.969173 9.968926 9.968678 9.586062 9.587941 9.589814 9.591681 9-593542 9-595398 10.413938 10.412059 10.410186 10.4083 19 10.406458 10.404602 55 50 45 40 35 3 35 40 45 50 55 60 9.565676 9.567269 9.568856 9-570435 9.572009 9-573575 9.968429 9.968178 9.967927 9.967674 9.967421 9.967166 9-597 2 47 9.599091 9.600929 9.602761 9.604588 9 606410 10.402753 10.400909 10.399071 10.397239 10.395412 10.393590 2 5 20 15 10 5 o Co-fine. Sine. Co- tang. Tangent. M 68. The "Table of Sines and Tangets. 22. M Sine. Co- fine. Tangent. Co-tang. o 9-573575 9.967166 9.606410 I0 -393590 60 55 5o 45 40 35 30 5 10 '5 20 25 30 9-575I36 9.576689 9.578236 9-579777 9.581312 9.582840 9.966910 9.966653 9-966395 9.966136 9.965876 9-965615 9.608225 9.610036 9.611841 9.613641 9-6i5435 9.617224 IO -39 I 775 10.389964 10.388159 10.386359 10.384565 10.382776 35 40 45 50 55 60 9.584361 9-585*77 9.587386 9.588890 9-590387 9.591878 Co-fine. 9-9 6 5353 9.965090 9.964826 9.964560 9.964294 9.964026 Sine. 9.619008 9.620787 Q. 622561 9.624330 9.626093 9.627852 10.380992 10.379213 IO -377439 10.375670 IO -37397 10.372148 25 20 15 10 5 Co-tang. Tangent. M 67. 23- M Sine. Co- fine. Tangent. Co-tang. 5 10 15 20 25 3 35 40 45 50 55 60 9-59*87^ 9-593363 9.594842 9-59 6 3'5 9-597783 9.599244 9.600700 9.602150 9- 6 03594 9.605032 9.606465 9 607892 9.609313 9.964026 9-9 6 37~57 9.963488 9.963217 9.962945 9.962672 9.962398 9.627852 9.629606 9- 6 3!355 9.633098 9.634838 9.636572 9.638302. 10.372148 10.3703"^. 10.368645 10.366902- 10.365162 10.363428 10.36.1698 ^o-35~9973 10.358253 10.356537 10.354826 10.353119 10.351417 60 55 50 45 40 35 1 25 20 15 10 5 o 9.962123 9.961846 9.961569 9.961290 9.96.1011 9.9607^0 9.640027 9.641747 9643463 9.645174 9.640881 9.648583 Co- fine. Sine. Co-tang. Tangent. M 66. 'The Table of Sines and Tangents. 24. M Sine. Co-fine. Tangent.; Co-tang. o 9.604313 9.960730 9.64.8583, 10.351417 60 5 10 15 20 is 30 9,6107 9 9.612140 9- 6l 3545 9.614944 9.616338 9.617/27 9,960448 9.960165 9.959882 9.959596 9-9593' 9-959023 9.650281 9.651974 9-653663 9- 6 55348 9.657028 9.658704 10.34971,, 10.34 020 '0.346337 10.344654 10.34297- 10.341296 55 50 45 40 35 3 2 5 20 '5 10 5 35 40 45 50 II 9.619110 9.620488 9.621861 9.623229 9624591 9.625948 9-95*734 9-958445 9-958i54 9-957 8 63 9-957570 9.957276 9.660376! 10.339624 9.662043:10.337957 9.663707^0.336493 9.665266 10.334634 9.667021 | 10.332979 9.668673! 10.331327 Co- line. bine. Co-tang. Tangent. M 65. 25. M' o sine. 9.625948 Co-fine. 9-957276 1 angent. 9.668673 Co- tang. i0 -33 I 3 2 7 60 5 1C 15 20 25 30 9 627300 9.628647 9.629989 9.631326 9.632658 9- 6 33984 9.956981 9.956684 9-956387 9.956089 9-9557 8 9 9.955488 9.670320 9.671963 9.673602 9-675237 9.676869 9.678496 10.329680 10.328037 10.326398 10.324763 10.323131 10.322504 55 50 45 40 35 3 25 20 15 10 5 o 35 40 45 50 55 60 9'S353 6 9.636623 9-6:5793,5 9.639242 9.640544. 9.64184.2 9.955186 9'954883 9-954579 9-954*74 9-9539^ 9.053660 9.6 o 120 9.681740 9-683356 9.684968 9.686577 9.688182 10.319880 10.318260 10.316644 10.315032 10.313423 10.31 1818 Co fine. Sine. Co- tang. Tangent. M 64. 'The Table of Sims and Tangents. 26. M| Sine. Co- line. Tangent. 9.688iI 9.689^"^ 9,691381 9.692975 9.694566 9.69^153 9.697736 Co-tang. 5 10 '5 20 25 JO 9.641842 9- t) 43 I 35 9.644423 9.645706 9,646984- 9.648258 9.649527 9.953660 9-953352 9.953042 9-95273I 9.952419- 9.952106 9-95I79I 10.311818 10.310217 10.308619 10.307025 10.305434 10.303847 10.302264 60 55 50 45 40 35 30 35 40 45 5 1 9.650792 9,652052 9-6$338 9- 6 54558 9.655805 9.657047 9.951476 9-95 "59 9-950841 9.950522 9.950202 9.949881 9.699316 9.700893 9.702466 9.704036 9.705603 9.707166 10.300084 10.299107 10.297534 10.295964 10.294397 10.292834 .25 20 15 10 5 Co-fine. Sine. Co-tang. Tangent. M 63- 27. M Sine. Co-fine. Tangent. Co-tang. 60 55 50 45 40 35 30 o 5 10 5 20 2 5 3 9.657047 9.658284 9.659517 9.660746 9.661970 9.663190 9.664406 9.9498*** 9-949558 9-949 2 35 9,948910 9.948584 9.948257 9.947929 9.707166 9.708726 9.710282 9.71 1836 9.713386 9-7*4933 9.716477 10.29283^; 10.291274 10.289718 10.288104 10.286614 10.285067 10.283523 35 40 45 5 55 60 9.665617 9.666824 9.668027 9.669225 9.670419 9.671609 9.947600 9.947269 9-946937 9.946604 9.946270 9-945935 Sine. 9.718017 9-7 I 9555 9.721089 9.722621 9.724149 9.725674 10.281983 10.280445 10.278911 10.277379 10.275851 10.274326 25 20 *5 10 5 Co fine. Co tang. Tangent. M 62. , The Table cf Sines and Tangents. 28. LVJ Sine. 9.671609 9.672795 9- 6 73977 9- 6 75 I 55 9.676328 9.677498 9.678663 Co- line. 9-945935 Tangent. 725674- Co-tang. 10.274326 60 55 50 45 40 35 30 5 10 *5 20. 25 3 9-94559 h 9.945261 9.944922 9.944582 9.944241 9-94]99 9.727197 9.728716 9-73 2 33 9-73 1 746 9-733 2 57 9-7347 6 4 10.272803 10.271284 10.269767 10.268254 10.266743 10.265236 35 40 45 50 55 60 9.679824 9.680982 9.682135 9.683284 9.684430 9.685571 Co-fine. 9-943555 9.943210 9.942864 9.942517 9.942169 2i 9 il 8 Ji Sine. 9.736269 9-73777 1 9-739 2 7 I 9.740767 9.742261 9-743752 Co-tang. 10.263731 10.262229 10.260729 10.259233 'Q-257739 10.256248 Tangent. 25 20 15 IO c J M 61. 29. M bine. Co-fine. Tangent. Co-tang. 60 o 9.68^571 9.941819 9-74375' 10.256248 5 9.6^6709 10 9.687843 15 9.688972 20 9.690098 25 9.691220 30 9.692339! 9.941469 9.941117 9.940763 9.940409 9.940054 9-939697 9.745240 9.746726 9.748209 9.749689 9.751167 q. 752642 10.254760 10.253274 10.251791 10.250311 10.248833 10.247358 55 50 45 40 35 30 35 9- 6 93453 40 9.694564 45 ; 9- 6 95 6 7 I 50 9.696775 55 9- 6 97 8 74 60 9.698970 9-939319 9.938980 9.938619 9.938258 9-9.-i7 8 95 9-93753' 9-7541 '4 9-7555 8 6 9-757052 9'7585i7 9-759979 9.761439 10.245885 10.244415 10.242948 10.241483 10.240021 10.238561 25 20 *5 10 5 o Co- fine. bine. Co-tan^. Tangent. iVJ to. The Table of Sines and Tangents. 30. M Sine. 9.698970 Co-fine. 937531" Tangent. 9-76i439 Co- tang. 10.27856! 60 5 10 '5 20 25 30 9.700002 Q.70U5I 9.702236 9-7033 i 7 9-704395 9.705469 9-7o 6 539 9.707606 9.708670 9.709730 9.710786 9.711839 9-937 I &5 9.936799 9.936431 9.936062 9-935 6 9 2 9-935320 9.762897 9-7 6 4352 9.765805 9-7 6 7255 9.768703 9.770148 10.237103 10.235648 10.234195 10.232745 10.23129-^ ^0.229852 55 50 45 40 35 30 35 40 45 5 55 60 9.934948 9-934574 9-934*99 9.933822 9-933445 9.933066 9-77 I 59 2 9-773033 9-774471 9.775908 9-777342 9.778774 10.228408 10.226967 10.225529 10.224092 10.222658 10.221226 25 20 1 5 1C 5 o Co- fine. Sine. Co-tang. Tangent. M 59- 3i- M Sine. Co- fine. Tangent. Co-tang. 9.711839 9.933066 9-778774 10.221226 60 5 10 '5 20 25 30 9.712889 9-7'3935 9.714978 9.716017 9-7 I 753 9.7.8085 9.932685 9-932304 9.931921 9-931537 9.931152 9.930766 9.780203 9.781631 9.783056 9.784479 9-7859 9.787319 10.219797 10.218369 10.216944 10.215521 10.214100 10.212681 55 5 45 40 35 30 35 40 45 50 55 60 9.719114 9.72OI4C 9.721 162 9.722181 9.723197 9.724210 Co- fine. 9-950378 9.929989 9-9 2 9599 9.929207 9.92881.5 9 928420 Sine. 9.788736 9.790151. 9-79*563 9 792974 9-7943 8 3 g957?g Co-tane. 10.211264 10.209849 10.208437 10.207026 10.205617 10.204211 Tangent. 2 5 20 15 10 5 M 58. Ibe Table of Sines and tangent 3. 32- M v in?. 9.724210 Co-iine. Tangent. Co- tang. 10.20421 1 o 1 IO ''5 20 25 3 35 40 45 50 55 60 9.928420 9.7957^9 60 9.725219 9.726225 9.727228 9. 28227 9.729223 9.720217 9.928025 9.927029 9.92723 9.926831 9.^.26431 9. 92 60 -'9 9.797194 9.798596 9.799997 9.801396 9.802792 9.^04187 10.202806 10.201404 10.200003 10.198604 10.197208 10.195813 55 50 45 40 35 30 9.731206 9'732'I93 9-733'77 9-734-I57 9-735*35 9.736109 9.925626 9.925222 9.924816 9.924409 9.924001 9-92359 1 9.805580 9.806971 9.808361 9.809748 9.811134 9.812517 10.194420 10.193029 10.191639 10.190252 10.188866 10.187483 2 5 20 15 10 5 o Co-iine. Sine. Co tang. Tangent. 1 M 57- 1VJ ome. Co-line. i'angcnt. Co-tang. 9.736109 9-92359I 9.812571 10.187483 60 5 10 15 20 2 5 30 9.737080 9.73804^- 9.739013 9-739975 9-740934 9.741889 9.923181 9.922769 9-9 22 355 9.921940 9.921524 9.921 107 9.813899 9.815280 9.816658 9.818035 9.819410 9.820783 i o. 186101 10.184720 10.183342 10.181965 10.180590 10.179217 55 50 45 40 35 30 35 40 45 5 55 60 9.742842 9-74379 2 9-744739 9-745603 9.746624 9-475 62 9.920688 9.920268 9.919846 9.919424 9 919000 9.918574 9.822154 9.823524 9.824893 9.826259 9.827624 9.828987 10.177846 10.176476 10.175107 10.173741 10.172376 10.171013 25 20 15 10 5 Lo-fine. Sine. Co- tang. Tangent. M 56. The "Table of Sines and Tangents. 34- M o Sine. Co- fine. 99' 8 574 Tangent. 9.828987 Co- tang. 60 9.747562 10.171013 5 10 IA 20 25 3 9-74^497 9.749429 9-750358 9.751284 9.752208 9-753128 9.918147 9.917719 9.917290 9.916859 9.916427 9-9I5994 9-830349 9.831709 9.833068 9-834425 9.835780 9-837!34 10.169651 10.168291 10.166932 10.165575 10.164220 10.162866 55 50 45 40 35 30 35 40 45 5 55 60 9.754046 9 754960 9-755872 9.756782 9.757688 9-75859I 9-915559 9.915123 9.914685 9.914246 9.913806 9-913365 9.838487 9.839838 9.841187 9-842535 9.843882 9-845227 10.161513 10.160162 10.158813 10.157465 10.156118 'O.J54773 25 20 15 10 5 o Co-fine. Sine. Co-tang. Tangent. M 55- 35- M Sine, 9-75859 1 Co fine. Tangent. 9-845 2 27 Co-tang. 9-9!33 6 5 i-i54773 60 5 10 1 5 20 25 12 35 40 45 50 55 60 9-759492 9.760390 9.761285 9.762177 9.763067 9-763954 9 764838 9.765720 9.766598 9-767475 9.768348 9-769219 Co-line. 9.9129:2 9.912477 9.912031 9.911584 9.911136 9.910686 9-9 I02 35 9.909782 9.909328 9.908873 9.908416 9.907958 9.1*46570 9.847913 9.849254 9-850593 9.851931 9.853268 9.854603 9-85593 s 9.857270 9.858602 9-859932 9.861261 10.153430 10.152087 10.150746 10.149407 10.148069 10.146732 LO. (45397 10.144062 10.142730 10.141398 10.140068 10.138739 55 50 45 40 35 3. 2 5 20 15 10 5 Sine. Co-tang. Tangent. LM 54- 'I be iable of Sines and Tangents. 36- M Sine. Co- fine. Tangent. Co-tang. 9.769219 9.907758 9.861261 10.138739 60 5 10 '5 20 25 30 9.770087 9,770952 9.771815 9.772675 9-773533 9.774388 9907498 9907037 9.906575 9.906111 9.905645 9.905179 9.862589 9.863915 9.865240 9.866564 9.867887 9 869209 9.870529 9.871849 9.873167 9.874484 9.875800 .9.877114 10.137411 10.136085 10.134760 10.133436 10.132133 10.130791 55 50 45 40 35 30 35 40 45 50 1 9.775240 9.776090 9.776937 9.777781 9.778624 9-779463 9.904711 9.904241 9.903770 9.903298 9.902824 9.902349 10.129471 10.128151 10.126833 10.125561 10.124200 10.122886 25 20 15 10 5 o M" Co-fine. Sine. Co-tang. 1 angent. 53- 37- M o Sine. Co -fine. ^"902349 Tangent. Co-tang. 60 9-779463 9.877114 10.122880 5 10 *5 20 25 3 35 40 45 5 55 60 9.780300 9.781134 9.781966 9.782796 9.783623 9.784447 9.785269 9.786089 9.786906 9.787720 9.788532 q. 789342 9.901872 9.901394 9.900914 9.900433 9.899951 9-S9946'/ 9.898981 9.898494 9.898006 9.897516 9.897025 9.896530 9.878428 9.879741 9.881052 9.882363 9.883672 9.884980 9186288 9.887594 9.888900 9.890204 9.891507 9.892810 10.121572 10.120259 10.118948 10.117637 10.116328 i o.i 15020 10.113712 10.112406 10. JI 1 100 10.109796 10.108493 10.107190 55 50 45 40 35 30 25 20 15 10 5 Co-fine. Sine. Co- tang. Tangent. M 52. *lhe Table of Sines and Tangents. 38. IVl Sine. Co line. Tangenc. 9.892810 9.894111 9.895412 9.8^6712 9.898010 9.899308 9.900605 Co-tang. o 9.789342 9.896532 10.107190 60 55 50 45 40 35 3 5 10 '5 20 25 3 9.790149 9.790954 9-79 T 757 9-79 2 557 9-793354 9-794I5 9.896038 9.895542 9.895045 9.894546 9.894046 9- 8 93544 10.105889 10.104588 10.103288 10.101990 10.100692 10.099395 35 40 45 5 55 60 9.794942 9-795733 9.796521 9.797307 9.798091 9.798872 9.893041 9.892536 9.892030 9.891523 9.891013 9.890503 9.902901 9.903197 9.904491 9.905785 9.907077 9.908369 10.098099 10.096803 10.095509 10.094215 10.092923 10.091631 2 5 20 15 10 5 o Co-fine. Sine. Co-tang. Tangent. M 5'- 39- LVl o bine. Co -line. Tangent. Lo-tang. 9.798872 9.890503 9.908369 10.091631 60 5 10 15 20 25 30 9.799651 9.800427 9.801201 9.801973 9.802743 9.803^1 1 9.889990 9.889477 9.888961 9.888444 9.887926 9.887406 9.909660 9.910951 9.912240 9-9'35 2 9 9.914817 9.916104 10.090340 10.089049 10.0^7760 10.086471 10.085183 10.083895 55 5 45 40 35 30 35 40 45 50 55 60 9.804276 9.805039 9.805799 9806557 9.807314 9.808067 9.8^6885 9.886362 9.885837 9.885311 9.884783 9.884254 9.917391 9.918677 9.919962 9.921247 6.922530 9.923814 10.082609 10.081323 10080038 10.078753 10.077470 10.0761 86 25 20 15 IO c O Co-fine. Sine. Co-tang. Tangent. M 0. 2.809559 2.810233 2.810904 2.811^75 2 8 122*45 2.812913 2.813581 2.814248 2.814913 <>77 678 679 680 68 1 "682 6*3 684 685 686 2.830589 2 831229 2.831869 2.83250-, 2.833147 2.833704 2.834421 2.83 ',056 2.835691 2.836324 "621 622 623 624 625 626 627 628 629 630 2.793092 2 79379' 2.794488 2.795185 2.795880 2.796574 2.797268 2 '797.959 2.798651 2.799341 654 655 T 65? 6 5 8 6 59 66c 661 662 663 2.815578 2.816241 2.816904 2. 8 I 756 r, 2.818226 27^18885 2.819543 2.82O2O1 2.820858 2.821514 687 688 689 690 691 692 6 93 694 %5 696 2.83^957 2.8,7588 2.838219 2.838849 2.839478 2.840106 2.840/33 2.841359 2.841985 2.842609 I 3 ' 632 ; *33 '634 i.8ocrC20 2.8po7i7 2.801404 2.802089 66j 66 5 t>66 667 2.822168 2.822822 2.823474 2.824126 697 698 699 700 2.843233 2.843855 2.844477 2.845098 700 yoo A Table of Logarithms. N. Logarith. N. Logarith. N. Logarith. 2.884795 2.885361 2.885926 2.886491 2.887054 2.887617 2.888179 2.888741 2.889302 2.889862 701 702 73 704 705 2.845718 2.846337 2.846955 2.847573 2.848189 734 735 736 737 738 739 740 74i 742 743 744 745 746 747 748 2.865696 2.866287 2.866878 2.867467 2.868056 2.868643 2.869232 2.869818 2.870404 2.870989 2.871573 2.872156 2.872739 2.873321 2 873902 2.874482 2.875061 2.875639 2.876218 2.876795 767 768 769 770 771 772 773 774 775 776 706 707 708 709 _Zi 711 712 7*3 7'4 7^5 2.848805 2.849419 2.850033 2.850646 2.8-1258 2.851869 2.852479 2.853089 2.853698 2.854306 2.854913 2-855519 2.^56124 2.856729 2-857332 777 778 779 780 781 782 783 784 785 786 2.890421 2.890979 2.891537 2.892095 2.892651 2.893207 2.893762 2.894316 2.894869 2-895423 716 717 718 719 720 749 750 75i 752 753 721 2.857935 722 2.858537 723 2.859138 724:2.859739 725 2.860338 754 755 756 757 75 2.877371 2.877947 2.878522 2.879096 2.879669 77 788 789' 790 -791 2-895975 2.896526 2.897077 2.897677 2.898176 7 26 727 728 729 _Z-1 731 732 733 734 2.8co937 2.861534 2.862131 2.862728 2.863323 2.863917 2.86451 1 2.865104 2,865696 759 760 761 762 J*l 704 7 6 5 766 1.7^7 2.880242 2.880814 2.881385 2.881955 2.882525 2.8830^3 2.883661 2.884229 2.881795 792 793 794 79^ 79 6 797 798 799 .800 2.898725 2.899273 2.899821 2.900367 2.900913 2.901458 2.902003 2.902547 2.903089 800 800 A Table of Logarithms. N. Logarith. N. Logarith. N. Logarith. 801 802 803 804 805 2.903633 2.904174 2.904716 2.905256 2.905796 834 835 836 837 838 2.921166 2.921686 2.922206 2.922725 2-923244 867 868 869 870 871 872 873 874 875 876 2.938019 2.938519 2.939019 2-9395I9 2.940018 806 807 808 809 810 2-906335 2 906874 2.907411 2.907949 2.908485 839 840 841 842 843 2.923762 2.924279 2.924796 2.925312 2.925828 2.940516 2.941014 2.941511 2.942008 2.942504 811 812 813 814 815 2.909021 2.909556 2.91009! 2.910624 2.911158 844 845 846 847 848 2.926342 2.926857 2.927370 2.927883 2.928396 2.928908 2.929419 2.929929 2.930949 2.931458 2.931966 2.932474 2.932981 2.933487 877 878 879 880 881 2.942999 2 -943495 2.943989 2.944483 2.944970 816 817 818 819 820 2.91 1690 2.912222 2.912753 2.913284 2.913814 849 850 851 852 853 882 883 884 885 886 2.945468 2.945961 2.946452 2.946943 2-947434 821 822 823 824 825 2.9H343 2.914872 2.915927 2.916454 854 855 856 857 858 887 888 889 890 891 2 9479M- 2.948413 2.948902 2.949390 2.940878 826 827 828 829 830 2.916980 2.917506 2.918030 2.918555 2.919078 859 860 861 862 863 2-933993 2.934498 2-935003 2.935507 2.936011 2.973128 2-973589 2.974050 2.974^12 972 973 974 975 976 2.98/666 2.988113 2.988559 2.989005 2.989449 911 912 9*3 914 9'5 2.959518 2-959995 2.960471 2.960946 2.961421 944 945 946 947 948 2.974972 2-975*32 2.975891 2.976349 2.976808 977 97 979 980 981- 2.989895 2 -99339 2.990783 2.991226 2.991669 916 '917 918 919 920 ^.961895 2.962369 2.962843 2- 9 6 33I5 2.963788 "949 950 95 l 952 053 2. 977 z66 2.977724 2.978181 2-97 s>6 37 2.979093 982 9"3 984 9^5 986 2.992111 2.992554 2.992995 2.993436 2.993877 92i 922 9 2 3 924 9 2 5 9.6 927 928 9 2 9 93 2.964259 2.964731 2.965202 2.965672 2 966142 2.96661 ] 2.967079 2.967548 2.968016 2.968483 954 955 956 957 958 959 960 961 962 9 6 3 2.979548 2.980003 2.980458 2.98091 2 2.981366 2.981819 2.982271 2.982723 2.983175 2.983626 987 988 989 990 99* | 992 1 993 994 995 996 997 998 999 1000 2.994317 2.994756 2.995196 2 -995635 2.996074 2.996512 2.996949 2.997386 2.997823 2.998250 93 1 93 2 933 934 2.968949 2.969416 2.969882 2-97?47 964 9 6 5 966 967 2.984077 2.984527 2.984977 2.985426 2.998695 2.999130 2.999560 3.000000 The TH E ufe of thefe TABLES hath been already at large (hewed in the firft and twelfth chapters; therefore 1 {hall fay no more of them here. A N A N APPENDIX, ;, ,.i>ifi&l$mii&f&fiH.fo >> Shewing farther How to furvey by the Chain only : with an ufeful Table to that Purpofe. HAVING, in the fixth chapter of the foregoing Treatife, (hewn a ready and eafy way for taking the quantity of an angle in the field by the chain only ; and undemanding it has been approved of by fqrveyors and others : I think it not improper to fay fomething more upon that fubjecl, in this place. And that this way of working may be practifed with as much eafe as by ufmg the mod coflly inftruments, there are two feeming difficulties mud be removed. The firft is, when the angle becomes very obtufe, or, contains 170 degrees or more, then the fubtending or chord-line will hardly be diftinguifhable between five or fix degrees, there being but 4 part of a link difference between 170 degrees and 171 degrees, and not above -^ part, between 178 and i 79 degrees. To remedy which, you need not take the quantity of that angle at all, efpecially if it be an in- ward angle, but meafure directly from B to C ; and when you come right againft A, take an off-let (which i you 2 'An APPENDIX, &c. you may do with a rod or line only, as true as with a crofs or other instrument) this off-let, being put down in your field-book, will do the bufmefs when you come to protract, as well as if you had taken the angle in the field : but if that is not JL^ fufficient, or any other reafon necefii- tate you to take the angle A, there place a ftrong flick in the angular point A, and putting the ring of the chain over it, ftretch it out at full length, both in the line A B and ^A C ; and where the end of the chain falls, there place flicks alfo, as at D and G. Remove your chain from A, and put the ring over the flick at D, and ftretch it out to- wards E. Now you mould have ano ther chain, or a fmall line, (which you may carry in your pocket) exactly of the length of a chain, with a loop at each end ; which put over the flick at A, and taking the other loop of the line in one hand, and the loofe end of the chain in the other hand, go back- ward 'till both being flretched flrait meet at E, then will L) A E be an equi- lateral triangle, to which add another equilateral trian:le by loofing the chain at D, and putting it over the flick at E, Jetting the line remain as it was fattened at A, and taking the loofe ends again of the chain and line in your hands, go backwards as before, 'till both being flretched flrait, meet in F. So have you formed two equilateral triangles, and DAE will contain 120 degrees. Laftly, with your chain meafure the nearefl diftance EG, which fuppofe to be 84 links and a half; which fum look for in the following table, and right againfl it you will find An APPENDIX, See. 3 50 degrees, which added to 120, make 170 degrees, the quantity of the angle fought , or if you have not a mind to ufe the table, you may note it down in your field-book, thus, A A 847, fignifying that angle con- fifts of 1 20 degrees and 84-'- links for its fiibtending line ; and you may plot it, by performing with your compafles upon the paper, what you did in the field with the chain. But to take the quantity of an obtufe angle with a fingle chain only, let EAI, in the following figure, be an an- gle of about 170 degrees; meafure from A towards B and C, half a chain on each fide, as to D and E, where ftick down flicks, and one at A ; then put the ring at one end of the chain over the ftick at A, and the other end over the ftick at D, and taking the chain in the middle by the ring that is commonly at the end of 50 links, go backwards 'till both parts are ftrait, and there ftick down a ftick -, as at F. Then loofe the ring from D, and put it over the ftick at F, and taking the very middle of the chain, make both parts ftrait, which they will be at C, where ftick down a ftick, from which mea- fure to E, noting it down in your Field-Book, A A 4275 and when you plot it, remember to make the fides of your equilateral triangles but of 50 links each, for in this cafe you cannot have recourfe to the following TABLE, that being made to the radius of 100 links, unlefs you double the number of links found between C and E j or which is better, when you have finilhed your two equi- lateral 4 An APPENDIX, &c. lateral triangles, one end of the chain hanging at A, itretch the other at full length over the ftick at G, which will fall at H ; then meaiuring the ncareft diftance be- tween H and C, you will find it to be 84-^ links, againft which in the table (land 50 degrees, which added to the angle DAG 120 degrees, make 1 70 degrees for the angle BAG. [See this figure.} \^f * 2 i5- i_ * But if you had rather meafure this angle, by firft tak- ing out a right one from it ; proceed thus : [See the figure on the other fide. ~\ put one ring of your chain over the ftick at the angular point A, and ftretching out the chain, let the other end fall any where at pleafure, as at B or C , where ftick a ftick through the ring, and loofing that end at A, take it -in your hand, and ftretching it ftrait, fee in what part it will juft touch the hedge A E ; which will be at D, if the other end be at C ; or at E, if the other end be at B ; and there make a mark , which done, keeping the end of your chain in your hand, go backward from B or C, towards G or F, 'till your chain is ftrait -, then moving yourfelf tideways to and fro, 'till you perceive your chain to lie in a ftrait line with B E or C D, at the end of it place a ftick, at F or G, from whence to A will be aline perpendicular to A E -, where- fore from A fet oft' one chain in that line, which will fall at H ; and one chain upon the line A I, which falls at I, and meafuring the diftance H I, you will find it 1 28 links -iVpafts of a ^ n k, or 80 degrees ; which added to the right angle, makes 1 70 degrees, which was the angle required. Otherwife you may form a right angle, by fixing one end of tlje chain in the angular point, and the other end at An APPENDIX, &c. .j at 40 links diftance in the hedge , then take 50 links in one hand, and 30 in the other, and ftretch both parts ftrait, their meeting will conftitute a right angle, it be- ing well known that three lines in the proportion of 3, 4, and 5, will form a right-angled triangle. Several other ways might be mewn, to take a right angle in the field by the chain only, and alfo to meafure the quantity of an .obtufe angle ; but omitting thofe, I (hall only mention one way more, to take the quantity of an obtufe angle, which is as follows : Let A be the angle required to be taken in the field ; by the chain firft from A, fet off two chains, one to B, the other to C ; then fixing one end of the chain inB, itretch the other directly in ailraitline towards C, making a mark where the end falls, as at 7 ; meafure the dif- tance from 7 to A, which fuppofe 8 links /-<>- parts of a link ; look in the following TABLE, for this number, and right againft it, you will find 5 degrees j which dou- bled (the angles A C 7 and A B 7 being equaj, becaufe the fides AB and AC are equal) makes 10 degrees; and fubtradted from 180, leaves 170 for the angle at A required. If the angle be an outward one, continue one of the lines, as D A to C, making A C one chain, alfo let off one chain upon the other line from A to B ; then meafure the diftance B C, which fuppofe to be 17 links T V parts of a link, and anfwers in the table to 10 degrees, which is the complement of the angle A to 180 degrees ; there- fore take 10 from 180, remains 170 for the outward AB. Thefc t 'An. APPEND IX, &c. Thefe examples, I apprehend, will remove the difficulty of measuring obtufe angles, and make the matter both plain and eaiy : as for acute angles, and fuch obtufe ones as do not much exceed 90 degrees, you have the way to meafure them already in the fixth chapter of the foregoing treatife, with fundry ways to meafure a field with the chain only, to which I refer you. Jt remains now to fpeak of the fecond feeming diffi- culty, which lies in the trouble of plotting after this way : to remove which, you may have a protractor made with links on it inftead of degrees, or both, if you pleafe ; which the inftrument-maker may foon do by the help of this table. Or you may very well ufe the common pro- tractors i for fcuving a copy of this table in the field with you, you may at once note down the degrees of every angle, without mentioning the fubtendents \ or if you only note down the fubtendents in your field-book when you come home, you may at once take all the angles in degrees anfwerable to them, and fo plot with an ordinary protra&or, as at other times. I have extended the table only to 140 degrees; for, when an angle exceeds that number, your beft way of meafuring it, is by the method juft now taught. What has been already faid, I prefume, will fufficiently explain the following table, and the ule thereof, there- fore fhall not trouble you with repetitions , only defire you to remember, that the table is made for the radius of one chain, or too links ; and the fubtendents, or chord lines, are in links, and decimal parts of a link : fo that when you would ufe this table, you muft fet off but one chain from the angle (you defire to know the quantity of) on either hedge, and rneafuring the neareft diirance be- tween the two ends of the chains a-crois from hedge to hedge, look for the number of Jinks in the table that neareft diftance contains, and right againft it you will find the quantity of the angle as true, as if taken by the beft Semicircle, Circumfercntor, or Theodolite. EXAMPLE. An APPENDIX, &c. 7 EXAMPLE. I would know the quantity in degrees of an angle whofe fubtendent is (accounting one chain radius) 80 Jinks : accordingly I look for 80 links in the table, and the neareft number to it is 79 links -^ parts of a link, and right againft it (lands 47 degrees : wherefore I fay that angle confifts of 47 degrees, and fomething more ; and if you defire to know how much the remaining ^ is you may fee by the table, that in an angle of this magni- tude one link and half anfwers to a degree-, fo that ^ parts of a link is juft 12 minutes. The exact angle therefore is 47 y 12'. What has been faid concerning meafuring a field, or taking an angle by the chain only, either in the Appen- dix, or fixth chapter, may as well be applied to a pole or ftrait rod divided into 100 equal parts 5 every divifion of the rod anfwering to a link of the chain : and the table ferves as well for a rod fo divided as the chain, only ob- ferve in meafuring the length of the lines, to call every 4 poles i chain, and every 4 divifions of the pole i lir}k ; then you may caft it up as if it had been meafured by the chain. You may provide a rod made to moot one part into another like a fifhing-rod, to be ufed as a cane, in the head whereof place a fmall compafs , and this inftru- ment will befufficient tofurveyany piece of ground with, without a horfe-load of brafs circles and femicircles, heavy ball-fockets, wooden tables and frames, and three- legged ftaffs, &c. which only ferve to amufe the igno- rant countryman, and make him more freely pay the furveyor. The TABLE of Chords ', or Subtendents to t-:e Radius of one Chain of GunterV w ico Links. 3 ^ 1 j t < ^ -4 > 8J to O O J^ cr. i J ! g> .s 1 1 u 5 " "e -3 60 .S : to .S = -~ p-i JP J H ' Q tn ___^j j I 7 j 3 / 6 1 ' ' h 71 116 i 106 159 7 2 3 5 137 .63 4 72 117 5 107 i 60 i 3 5 2 3^ 65 1 73 n 9 o 108 161 8 470 39 ^ 8 74 i.o 4 109 162 8 587 40 68 4 7 ,- 121 8 i 10 163 8 6 10 ; 41 70 o 76 123 i in 164 8 7 I2 2 -42-71 7 77 124 5 112 16? 8 8140 43 73 3 78 J2 5 9 113 i 66 8 9 J 5 7 44 74 9 79 127 2 114 167 7 10 |7 ^ 45 r 6 5 80 128 5 115 168 7 ii 192 4 6 78 2 81 129 9 116 169 6 12 26 p 47 79 7 82 13! 2 117 170 5 J3 22 6 48 81 3 83 132 5 118 171 4 '4 2 4 4 I ^ 20 .1 16 27 8 49 8 2 9 5 o 84 5 51 86 i 84 133 8 8 5 135 ' 86 136 4 119 172 3 I2O 173 2 121 174 I 17 29 6 5' 87 7' 87 137 7 122 174 9 <3 31 3 53 8 9 * 88 139 o 123 175 7 19 33 54 90 8 '9 140 2 124 176 6 .20 3* T J5 9 2 3 90 141 4 '25 J77 4 5 4 5 6 93 9 91 142 6 126 178 2 -i2 $3 2 57 95 4 92 143 8 27. 179 o '3 39 9 58 97 o 93 14? o 28 i?9 8 24 41 6 59 9? 5 94 146 2 29 j 80 5 2 5 43 3 60 ico o 95 J 47 4 30 181 3 26 44 9 61 ioi 5 96 ij8 6 31 182 o 27 46 7 6j 103 o 97 HV 8 32 182 7 28 48 A 63 104 5 V8 151 o 33 I8 3 4 29 50 i 64 106 o O'J. '52 1 34 1 8 4 "o 5 r 8 3' 53 4 5 107 4 66 108 9 100 153 2 ioi 154 3 35 l8 4 7 6 185 4 32 5? ' 67 no 4 102 i s -5 4 7 186 i 53 5 6 8 8in8 03 '5 6 5 8 186 7 34 5* 5 9 "3 3 04 157 6 9 l8 7 3 .*< 60 j o j 14. 7 o 187 9 FINIS. This book is DUE on the last date stam ed below _ ..-TO 11 ft oa, !m-6,'52(A1855)47( a S&:,?2ffifACILITV