■n (Vvilliaii.) THE CO.MPLEAT SURVEYOR, Co ole Art of SurveyinR I.ancl. By the Plaiiie Tab! cuiiiferciitor, Peractor, and iithcr instruments. Snn frontispiece of tli.c author at 30 years of age, nunicro ms and crude illustrations, autograph of Tho. Bai contemporary calf. £3 10 0 R. and 11'. Lc) hourne: London 16, nvndes, wlio notes “ Win. Le.vlioiini published nnni.v otl til of which are esteemed.” Tlie fifth edition appeared anmel Ounn, who says in tlio preface, “ Tile author of tl rpieutl.v employed in surveying, inoasiiriug, and mappi es, us evidently ap|iears from the several draughts by Ii met witli in almost every couury in England.” D.N.T!. > t the author “ is said to liave begun life as a printer,” and lat this volume was printed by E. & AV. Leybonriie. T •orks are “ clear and attractive in style.” S&StalcbeCsltiTUmnooo ..zrcL ^ ^ T H E C O M P 1, E A.T Coritainingihe rfbote (iART of jijrvevd'Mi ot BY T H £ vanci and other Inftrumenfs i ; .1 :Y<.)rc l'a'ic . LUid cc^mDcnoiow’ i miiiner, ibcii liatli beenhitberto publilbed by any .- the PlainTdle being f.) c j.itrivcd, tliat it abn= will conveniently perfonn wlutloever may be done by ' any of tbc forc-ra:ntioncd Infirumems , or any oclicr ye; invented) with the lam; cafe and exaft'iela; and in many tafei much better, :r. v';h llw. .ul’ bU nA.uirinr of Heights and Diftances,either acccflible or in-acccirible,thcPlottmg and Protraftini'ofall mannet of Grounds, either fmall Inclofures, Champion Plains, Wood-lands, or any other Mountainous and un-even grounds. Alto, how to take the Plot of a whole Manor, tocaftuptheconteut,and to ! make a perfeft Cliart or Map thereof. All which parti* I culiti arc petfonned three feveral ways,and and by three feveral InfirMnnats, iy hutU; ;vi:\v/ c: Sfill'^Ioi: L vi! by >yhich a man niay be fatisfied whct^icr his Plot will clofe before he be<»in$ to protiaft the fam:,with the manner how to order fuch Water Colourtas arc “necefliry for the beautifying of Mapi and flitt ; Alfo how to know whether fK/iicr may be conveyc i from a Spring-head to any appointed place or not, and how to efleA the fame: With whatfoever elfe is necefliry to the Art of The Second Edition, mthmn) Additions. 'A t: - : \ .uih hhi. h. ' 1 . LONDON, I Printed by and • • for. : • • .atthefigneof tne Bible upon i««/^-gatc-hill; X a il--'-= ±: h.DAlFND HFKGAIE, j 0^ CjraxCshnigEiq. SIR, bLinguniiiv- cd, and ready rolec ■ the light, I e'oiiid not, bethink my lelfc or a ; httcr Patron tircri I your lelr to protcci • it; Yonr knowledge I , in, and air’eclion to the deicnce;di////V-' I maticnl, as allb the civil reipecl yon iiilially | vouchlafe fuch as alFect thole Studies, | arm me with this confidence. I forefee that this my preftimptionin ' expofing thisW’ork to piihlike view,may ‘ meet with fbmc Darnrhi s , bn t Yon r ap- ! probation thereof, will both convince 1 them of their £/7Ye/-, and plentifully latif- | A 2 he ^ Tie I-pjilo Dedicatory, lie me for the pains 1 have taken therein, i Howbeit, vvhatreceptionfoeveritmay : obtain with the Vtdger, my intention (I I doubt not) will give me fiipport and en- ! couragement, my aime therein being I nothing elfe but the piiblike good, and I this my Dedication an evidence to let i You know how much I am, i SIR, Tow mofl humble and obliged Servantj William Leybourn. / TO THE READER. Conneoiis Reader, B Bouc tlircc ycarcs fincc there came into tlic World a little Pamphlet ciuituled PUtometih, or the whole Art of Surteying ot Land, under the name of Lliier fVaUi/.lj, ot vvhichiconfels myfciftobcthe Authour, that name being only the,true letters ot my own name tranfpofcd.I vvas.^indecd \cry unwilling the VVorld fliould | know me to be the Author thereof, it being fo immaterial a Trcatife, and too pankular.for aSubjeftof fo large an extent, but that was occahoneiitby over-much hade, for ( being urged thereunto) it was not abose lix weeks eonceived before it was brought forth j and thcrtfbrc mud needs be little Icfs thenmon- ftrous: yet. the good accep t nice wiiich that Pamphlet receivect, occafioned me toprofecuic mat Subjcid moreat large. Now as the opinions o'Fmcn in tl eWorld are'various, fol know this Work will be. varioufly confuted , and therefore it might (perchance)be ckpcidedby feme, that I Jliould make an apologie for my fclf, as t6 crave pardon cr cxcufcforwhatfocr ver any man fliall be pieafed to object againd, but I mean to make no excufc , for I know of nothing that needs it, neither did I ever know any Book the more favoured for the Authours befpeak- ingit; befides, the fubjeftof theenfuing Tieatife being Gmne- trj, needeth no fuch thing, for [Demaflratio/.] the grand fup- porter thereof, is able tbwith-dand all cppclcrs, and dlcntly with Lines and Figures to mn-fltis the mod malevolent tongue or Pen that (hall either fpeak or write againd it. But to the ju¬ dicious Reader I fhall fay thus much; As I dare not think my doings free from all exception, fo I do not-know of any thing herein contained worthily deferving blame : Some fmall over- fights which, may poflibly have crept in by chance, I mud in- ue^it the friendly Reader to over-lee or wink at , as for the _^_A z under- To the Reader, underftanding Reader 5 1 am furc he will fcorn to cavil at every flight raiftake or literal fault iii the Pfinlitig! as for inatctial faults I know ol none in the whole Work, although 1 hft ve diligently ex* aminfed the Printed (hccts. In the following Trcatifc I have endeavoured to proceed me- thodically, and to infert every particular Chapter, atitoughtto be read and practiled, and have omitted nothing ti at might any way tend to make a man in iRort time become an cxqUilitc profi* cient in the Geometrical part of Surveying. The firft Part of this Book conlifteth of Geometry only, and containeth fuch Problcmes as are meet and neceflary to be known and praftifed by any man that intendeth to exercife himlclf in this employment: by help of thefe Problemcs the Plot of any piece of Land may be inlarged or diminilhed according to any afligned E roportion, and reparation and divifion thereof made, if need be, yRule and Compals only, and alio by Aritlimctick. In the lecond Book, you liave a general del'cription of all the moft neceflary Inftrumentsufed in >urvcyin2,as of the Tljfodil/te, Circumferentor, Plain Talte, and ti.e like, and more particularly of thofc which I make ufe of in the profecuting of this Difeourfe. Alfo I have given fuch direftions for the making of tl;e Plain Talle^ and furnilTied the Index and other parts thereof with divers ntN ceflarie lines for feveral occafions, fo that it being made accord¬ ing to the directions there si vcn,it i' the mofl abfolutc and univeN fai Inftrumentfor all occafions yet ever invented 5 forty it may be performed whatfoever may be done by the Theodolite, Circumfe¬ rentor, or Peri&or, with the fame facility and exadnefs, and in ma¬ ny cafes better, as in the particular ufes thereof will plainly ap¬ pear. And in the fifth Chapter of the fccond Book, I have deferibed the making of an Inftrument which will perform all the ufes of the Theodolite, Semicircle^ Circumferentor, and PeraBor, and for PortabiJi- tyexceeds any of them. In the eighth Chapter of this fccond I Book, there is alfo deferibed a new kinde of ProtraBor, which is 1 I nr more convenient for ufe then that which is ufually made, and as is deferibed in the former part of the faid Chapter. 1 The third Book is o( Trigonometry , or the Doftrine ofthc di- } raenfion of Plain Triangles, by Sines,Tangents, and Legarithmes,by j which the nature&rcafon of the taking of all manner of Heights j and piflancesmay the better appear, and forthatreafoni have i in this third Book added (hort fables of Sines and Ifigaritlmes, • namely, a Table of Sines to every ten minutes of the Quadrant,and a Table of Logarithmesfiom i to 1000, by which more Qjieftions may be refolved in the fpacc of one hour, then by the ufual wayes taught by others can be performed in fix, if thelikecxaancfsbe acquired. And for a further abreviation of thefe Calculations, I have alfo (hewed how to rcfolvc all fuch C^fes in Plain Triangles asiMyatany time comeinufc in the praftice <>f Surveyingliy the Lines of Artificial Sines, aniTangntS', whereby I TotheReadtr, fuch Cafes may be refolvcd without Pen j bke, or Paper, In the fourth Book is (hewn the ufe of all the fore-mentioned Inftruments in the pradiccof futveying, and firlV in taking of all manner of Heights and Diftances, either accelTibleor inacce^ fible, in the pra^ice whereof the young Pra^tionec will take much delight, and receive no fmall (atiswaion. There is alfo tauglit how to take the Plot of any field or other in- clofurc feveral ways,both by the PlainTahle , Tbeodolite, and C;V- ram/rrMfor, by which will appear what congruity and harmony there is between thefe feveral Inftruments, for if you take the Plot of any field by any one of them,and then by another of them, and plot your work by the fame Scale at both your obfervations,you fliall (if you be careful) finde that thefe two Plots will agree to¬ gether as cxadtly as if they had been both taken by one and the iamclnllriiment. And for this rcafon I have made one Scheme or 1 Figure ft rve for tlircc feveral Chapters, which hath muchabre- ] viated the number of Diagrams, and will (Iperfwademy felf) give better (atisfaftion to the Learner, then variety of Figures could have done. In the manner of protraifting, when you have obferved your de¬ grees cut by the Needle in the Circumferentor, or the Index of the PeraP.or, I have (bccaufc the praftice thereof is very ufual and na lefs difficult) in 239 inferted a Figure fo plain and perfpicu- ous, that the very fight thereof will be enough (if there were no words ufed) to explain the ufe thereof. After the Plot of any field is taken and protrafled according to any of the former directions,! come to iTiew how the content tner- of may-be attained feveral ways,that is, to finde how many Acres, Roods and Pcrclicsare contained in any Field thus plotted. Alfo there istaught how to meafurcMountanous and uneven grounds, and to finde the area or content thereof. You arc alfo taught in this fourth Book howto take the true Plot ofa whole Manor, c rof divers feverals, both hj the [lain TaUcf Theodolite, C:rcumfere/.torox PeraBcr, with the manner how to keep account in your Field-Book after the moft fure andexadlcft waj^ And here is alfo added a new way of Surveying of Land by the Circumferentor oT PeraSor, by which when you have gone rounda¬ bout the field & made obfervation of the feveral fidcs and angles, you may with fmall trouble make examination thereof, and be fatisfied whither your plot will clofeornot, before you begin to protrad. Alfo how to reduce your Plot, to any proportion, to I draw a perfed draught thcieof, and to deck and beautifie the fame, with the manner how to order fuch Colours as are necefla- ry thereunto. And in ihclaft place there is an example of Wa¬ ter-leveling, by which you may know whether water may be con-1 veyed irom a Spring-head to any determinate place or not. | Thus have I given you fome general intimation of the principal * heads conteinedin the followingTreatifc, which you may feci more apareni in the following Andjfs, but beft of all in the B^k it ^ felf. To the Reader. fclf, unto which I chiefly refer you, wifhin^^ that you may take the fame delight and plcafure in thepradice of thofc things tlicrcin conteined, as I did in the compoltng of them, fo fliall 1 think my labour well beftowed, and be the more animated to prelcntthee with (bme other MatheraiticalTreatife, who am \ A Friend to all that are iXdatbemticaUy affe&ed. lenionijmt 1653 William Leybourn. ! The following— :G E N A- - R A. L S ¥ R V E Y the whole WORK. ...,# 3A O F r I Raififig and letting fall of Pci pendiculai's,! i 2 The making of equal An^.ts.^ai)d drawing of parallel Lines, 15 ;; The dividing o/right Lines equnllj, 14 4 The con/litnting of right lined Figures, id . y The working of proportions l>/ lines, 17 6 The dividing n/right lines profortio>i.illj^j 2 7 The dividing of Triaiiglcs according to proportion , both Arithmetically and Geo¬ metrically,Ci From an] angle, 19 „ by a line<^2 pointfide,21 qPwbhncs concerning. I, vMo any Me, zv js The power of Lines a,-idSupciRcks,!; 9 The reducing ofC Four-] t Definitions, Fage-^ | flO/Gc ' ometry, which J2 Theoremes, 10 offieth^ . of I s from onejp.^Cfded figures into 0another,] ( Triangles,3!;.(^'f. 'Six ^ I 10 The dividing of any plain Superficies wfa two or more parts , according to any propor- j tion, by lines drawn either from any angle,«r (_ from a point in any Jlde.^i ’ r 1 In general, 41 j 2 O/f/ie Theodolite, 45 5 Of the Circumferentor, 4.] , 4 of the PlainTable, 45 5 Of anew Inftrurnent,49 I 6 Of the Crofs, 50 11 0/ I 7 Of ChTms, and chiefly, of < Inftru- < K^Mr. Rathborns, yt ^Mr. Gunters, ; 1 8 O/fheProtraifior, yy I CF/rf/» J) 1 9 0/Seales,^ and Syg I (Diagonal^ [ lo 0 / 4 Field-book, 60 HI 0/r/jeParallelogram,dI ' CSincs 6 $ III 0 /( 1 1 Of the defeription and life of the Tables of? and ITrigo- ]) ‘ 13 The application andufe of thefe Tables, as ^//sCRight angled,7^ ; trie and (} of the lines of Numbers, Sines and Tangents, a-ad refolving of Plain Triangles, ^Oblique angled 87 ■Jreatife is divided into- f 1 1 Of the Scale /;>«' Of ihcVwMaovwl , ^''/Ar.gh.iSi 0/»>5 \te attic an angle InthefiM :herewhh, 18} ^Circiimfcicntor,j ^a» Hoiizoiual line, or line of level, ^ . 4 Ofth, Label, to ohferve^^^^ ^acceffible 7 ^ CPlain Table, 187 5 0/Mfeo/Diftanees< or Si; fWThcodolitc, 1S9 r ^ linaceeffthleS lCncw:^:t-r.:o jpo.ani I ^ ^ (ropretradth.fonejpil^ S Of the taking LltitfJet hj the Label and Tangent line , ^ ^inacceple,^ 191 (^‘"“^tovrotrall the fame ic^, \j Of taki>igiivcrs\?h\nl3Ae,l^67 Man'ces at ome,l)p I and ‘ ^and to protraR the fame, 199 the ^Theodolite.iyS j 8 To take the plot of a Field 4ff Plain Table, 201 one flat ion taken in the »»/dd/«)Th:odolite, 20 J thereof bp the )Ciicuinfcrcntor,ioy L ** ptotraR the fame, 2c6 9 To taketheplotof aFieldat {Plain Table, icS one flatioH taken in anj jTheodolitc', ibid. thereof, bj the jCircumferentor, no 1 (_ {and to pretraU the fame,'.10 1 10 To take the plot of a jSWdr Plain Table, 212 at tiro ftations taken iwJThcodolite, 214 anj parts thereof , bj tic^Civc-umfereiuor, 216 L {and to protract the fame,zi 6 | 11 Totake the plot of a Field at rn>j. Plain Tabic, 218 I ftations taken inany part thereof, ^Theodolite, 220 only meafnring the fiation.try d;-^Circumterentor,2;o ft ante, by the (_ {and to protraR the fame, 222 IV The ** (fhampion Plains *rC Plain Table, 223 ffjg I ^l'oods,totakePl^titl)ertofbythe^Ti\ettAo\iv :,^^6 Inflrn- ' ^ {andto protraR the famCji^o ^ ments, IFith away to prove the truth thereof, 232 : 1', To take the plot [of ! anjField,tyood,Parif,/'CveumPeremor, 234. to protraR the • Chafe, Forrefi, or j! the ^enAoi, 2;6iand ' to protrahl the fame, 2-14 16 To fiirveigh bj the Chain only, and no other Infirttmcnt, 247 17 To take ‘Plot of a Forreft, Chafe, or fFuod hj the Circumfercn- tor, of which you may know whether your plot will clofe or net be- I fore you begin to protraU, 2^0 iS ToprotraUthefame, 259 fTitY Geomctric.Sqiiarcj2 6i I 7 /jf Long Square, 263 19 Of fndingtheareaor T/jir Triangle,26? content of any piece of Land,theplot<, T"/;# Tiape2ia,264 I therof being firfi taken,& chiefly of t/f”J irregular plot of d I F,eld, 20^ ^The Circle, 2 67 20 the manner of cJlfr.Rathbornsi ^2^7 up the content of any piece ^ > Otain? of LandinAcret,&c,byft_^rr.QimKn ^ C^7® ' 21 To reduce Acres inio Perches, and the contrary, 270 22 TheufeofaSc3lc,of Rodvi Aion necejfary, for finding the fra- Hionparttof an s/icre, 272 13 Divert compendioM Rules for the ready cafiingiipof any plain Superficies, with divert other Coinpendiums in Surveying, bj the Line of Numbcrs,27? 24 Of Statute and Cufiomarymeafire to reduce one to the other at pleafure, 276 25 Of the laying out of common fields into Furlongs, 277 26 How a Lordjhip lying in common fields is to be divided, 279 27 Of Mils and Mountains, how to finde the Lengths oftheHori- stantal Lines on which they /hand fever al ways ,281 28 Of MountanoM and tin-even grounds, how to protrahi or laj the fame down in pla no after the befh manner, giving the area or content thereof, 283 j-Flain Tabler/jr« feveral wayts,2S$ 29 How to tafe the Plot^ of a whole t^anor i7291 ^Peraftor, ^ tVith the ktff'ug an account in your FiclJ-booV, after the befh and mojh certain manner 29'i,it>idto protrat any obfervations fo ta¬ ken, :pi5 I C Semicircles, 30 0/ inlarging or dimini flung of Plots^t^r. Rathborns Ruler, according to any pojftble proportion byyA Line into 100 parts, Cthe Parallelogram, 289 ,31 To draw a perfet draught of a whole Manor, and to furnifh it with allnecefary varieties,‘^00 32 The names of fucb Colours as are uecefary for the wafhingof Maps orPlots, with the manner how to temper^ ufe the fame^oi H Of convey ing of SVater, 305 Note that all Chapters and parts of Chapters that are added in this Edition, arc noted with this mark bc- l^fore the Chapter jy I Or as sftuch ns the rrhok qjit of ^'urveying, of L and is performed hj ii> fir uinents oj federal l^ndsj nm thnt the exaB mi me- jul makttig dKd dividing of all fiich Inftruments is chiefly to be aimed at , I thought good to intimate to fueb as are defrom to ^raBifetbis rt,and do not rradily J^on? where to be fHrniJhedwithnecejfar yhiitummts for the performance thereof j that all j or any of inflrn^ ments ufed or mentioned in this Boo \, or any Ma? thematicaUnftruments wbatfoever are mofl ex- My made by ^K£r. Anthony Thompfon in Bofier Lane neer Smithfield, L endon. , THE COMPLEAT SURVEYOR. The Firft Book, THE AKGVMENT. His firft Boofi confiftcth of j divers Definitions & Tr(n ! blemes GeometricaU, ex- trafted out of the Wri¬ tings of divers ancient and modtmGeometricians, as Euclidi R amus^ Bagiedtne^ Clavim^ cJnc. and arc here fb methodically dif^fed^that any man may gradually proceed from Probleme to Probleme without interruption ^ or being referred to any ether for the Praifticall performance of any of them. Onely the Demnfiration is whol¬ ly omitted; partly, bccanfe thofe Boo^ out of I _ _ B which 2 The ^rgtment, L i b. i. which they were extraded, are very large in that JT particular, and alfo for the avoiding ot nianyo- thtiTropofitiommA Theoremetj which (had the enfuing Probkmes been demonftrated) muft of ne- ccffity have been inferted. Alfo, the figures - would have been fo incumbred with multiplici¬ ty of lines, that the intended Prohlemcs would have been thereby much darkened. And befides it was not my intent in this place to make an ab- folute or entire Treatife of Geometry y and there¬ fore 1 have onely made choice of fuch Problemes as I conceived raoft ulefull for my pn^fent pur- polcj and come moft in ufe in the pradife of Sur- veying, and ought of neceflity to be known by e- very man that intendeth to exercilc hi iifelfc in the Pradice thereof, and thole are chif fly fuch as concern the reducing of Plots from one forme to another, and to inlarge or diminifli them ac¬ cording to any affigned Proportion j alfo divers of the Problemes in this £ooi^ will abundantly Mptht Surveyor in the divifion and feperation of Land, and in the laying out of any afligned quantity, whereby large parcels may be readily divided into divers feverals; and thofe again lub-divideif need be. Alfo for the better latit f adion of the Reader, I have p^^rformed divers of the following Problemes as well Arithmetically as Geometrically. G £- GEOMETRICALL DEFINITIONS. 1 A Point is that which cannot be divided. which is void of all Magni- tude., and is the lead thing chat by minde and underftanding can be imagined and conceived, than which there can be notning lefle, as the Poim or Ptick noted with the A W letter A, which is neither quantity nor part of quantity, but only ctic terms or endsof quantity, and herein a Point in Geometry differech from Unity in Number, 2 A line is a length witJmt breadth or tlnch^efje, A Line is created or made by the moving or drawing out of a a-—1 “-® Point from one place to another, ^ fo the Line A B, is made by mo- ^- vingof a Point from A to B, and according as this motion is, fo is the li/zf thereby created, whe¬ ther freight or crooked. And of the three kindcs of Magnitudes in Ge¬ ometry^ viz. Lengthy Breodthy and Thicknejfej a Lt«eis the firft, confid¬ ing of Length only, and therefore the Line A B,is capable of divifion \n Length onX'jy and may be divided equally inthe/wwtC, orune^ qually in D, and the like, but will admit of no other dimenfion. 9 The ends or bounds ofahmsafe PoinGs. Thisistobe underdoodofafinite Iwf A--- ^ only, as is the line A B, the ends or bounds , whereof ore tlie points A and B; ButinaCirculatZ-iwitisothtt* B 2 Gcometricall Definitions. L i b. i . wife, for there, the Point in its motion rcturneth a;Tain to the place where it firft began, anJfomaketh ti'.c infinite, and the ends or bounds thereof undeterminatc. 4 A Right line is tint ivhxh lieth eqmlly be¬ tween bis pints. ^B As the tine AB lieth freight and cquall between tlic points A and B (which arc the bounds thercot) without bowing, and is the iBortcft of all other lines that can be drawn between tiiolc two points. •y ASuptriidesisthjtwhich Jmbonlylengtb and breadth. 13 As the motion of a point '^r~ l" ~l- l - l-f ! ■ 1 1 1 - 1 1 l -jB prj.J'icetn a Line, the firft ; !! ki.ulc of Magnitude, fo the i •• mitionofa Lineproducctha 5.Yf,//r/V’S,vvhici/isthefccond _■ • kindcol Magnitude, and is ca- ' C T) pabic ofI V 0 dimcnfions,namc- Iv, length and breadth, and fo the S ■l e.pcies A B C D may be divi¬ ded in length from A to B, and alfo in breadth from A to C. 6 Tbeextreams of aSupcsdcitsareLin'^s, As the extreames or ends of a Line arc points, fo the extreams or bounds of a iiu^efficies arc Lines, and lo tic extreams or ends of the Superficies ABC D, arc t)ic lines AB, BD,D C, and C A, which arc the terms or limits tlicrcof. 7 A plain Supfrfides is that which lieth qmlly between his lines. So the Superficies ABC lieth direft and equally between his lines: and whatfoever is faid of a right line, the fame is alio to be underftood of a plain Superficies. 8 A plain A ng1 e »the inclination or bowing of tvro lines the one to the other^ the one tombing the otherj & m being dire&ly joynedtogether. As the two lines A B and B C incline the one to the other, and touch one another in the point B, in which point, byreafonof the ____ *"cli- GeometricaWejimiom.. LiB.f. • inclination of the foi .1 lines, is made th.c AfialeAhC. tilt if thctwolineswliich touch each other be without inclination, A . and be drawn direifily one to the other, / \ I then they make no an^Ie at all, as the / \ lines C D and D E, touch each other in / ^ the point D,an'i yet they make no angle, / . but one continued right line. / And here note , that an Angle ^ i commonly is figned by three Letters, ' i the middle moft'whcreof iTieweth the angular point: As in this fi- I gure, when we fiy tlie angle ABC, you are to underftand the very ' point at B: And note alfo, that the length of the fides containing ' any angle, as tlie fides A B and B C, do nor make the angle ABC ! either greater or lefferi but the angle (fill retaineth the fametquan- i tity be "tlie containing Tides thereof cither longer or fliorter. And if the lines which contain the angle he right lines, then is it called a right lined mgU So the angle A B C is a right lined angle, becaufe the lines A B ^ and B C, which contain the faid angle, arc right ^e?. And of right i lined Angles there arc three forts, whofc Definitions follow. 1 lo When a right line (landing upon a nght line maJ^th the angles on either fine equall , then either of t'’>oft angles is a right angle: and the right line which flandeth eretded, is called a perpendicular line to that whereon it flandeth. As upon the right line C D, fuppofc there do ftand another right liite A B, in dich fort that it maketh the angles on either fide thereof equall, namely, the angle A B D on the one fide, equall to the Angle ABC on the other fide: then arc cither of ■ — the two angles ABC, and C “ " ABD right angles, and the , a-t, riaht line AB, which ftandeth ereaed upon the right lincC U, w?thout inclining to cither part thereof, is a perpendicular to the line C D. Geotnetricall Defnitiom. LlB.l. U An Obtufe angled that n>hich is greater tbm aright angel So the angle C B E is an obtufe angle, becaul'e it is greater than the angle ABC, which is a right angle; for it doth not only contain that right angle, but the angle ABE alio, and therefore is obtufe. 12 Acute angle » D lej[e than a right angle ^ So the angle EBD is an acute angle, for it is leffc than the righ tangleABDfin which it is con¬ tained ) by the other acute angle ABE. 13 A limit or term *r (he end of e’uery thing. As a point is the limit or term of a Lined^ecaufe it is the end thereof, foa Linclikewifeis the limit and term of a Superficies; and a Superficies is the limit and term of a Body. 14 if Figure is that which is contained under As the Figure A is contained under one limit or term, which is the round line. Al- fo the Figure Bis con- rained under three \ \ right lines, which are \ \ 1 ^ / thereof. Likewifc,the ^ I _ f Figure C is contained u r- c j c . under four right lines, the Figure E under five right lines, and fo of all other figures. €[ i^d here note, that in the following work we call any pla'n Suj^cies whofe fides are uncquall, (as the figure E ) a ?Was ofa Field, Wood, Park, Foneft, and the like, L1B. I. Geometrmll Definitions. A Circle is a ^lain Figure contained under one line, which is called a Circumference, unto which all lines drawn from one point within the Figure, and falling upon the Cir¬ cumference thereof are equal one to the other. As the figure A B C D E is a Circlcjcontaiiicd under the crook¬ ed line BCD E,\vliich line is call¬ ed the Circumference: In the mid¬ dle of this Fioure is a point A,from which point all lines drawn to tile Circumference thereof arc cquall, as the lines A Bj A CjAFjAD ; and this point A is called the center of the Circle. D i6 A Diameter of a circle is a right line drawn by the Center thereof, and ending at the Cir¬ cumference-fin eitherfide dividing the Circle into two equall parts. So the line B A D (in the former Figure) is the Diameter there¬ of, bccaufc it palTeth from the pomt B on the one fide of the Cir¬ cumference, to tlie point D on the other fide of the Circumference, andpafleth alfoby the point A, which is the center of the Circle. And morcovcr,it dlvidech the circle ihto two equall parts, namely, BCD being on one fide of the Diameter, equall, to B E D on the other fide of the Diameter. And this obfervation was firft made hy Thales Miletiu!,^oi,(an\ihe, Ifa line drawn by the center of any Circle do not divide it equally, all the lines drawn from the center of that Circle to the Circumference cannot be equal!. %y A Semicircle is a figure contained under the Diameter, and that part of the Circumfe¬ rence, cut off by the Diameter, As in the former Circle, the figure B E D is a Semicircle,becaufe it IS contained of the right line BAD, which is the Diameter, and of the crooked line BED, being that part of the circumference which is cut off by the Diametcr.-alfo the part BCD is a Semicircle. i8 A 8 Geometricall Vefimiom. L i b. i. 18 A Scaion orprtion of a Circle, is a Figure containedunder aright Im, and a fan of the circwnfmnce,greater or lefs then a Semicircle. So the,Figure ABC, which confiftcth of the part of the Cir¬ cumference ABC, and the right line A C is a Sedf ion or portion of a Circle greater than a Semi¬ circle . Aifo the other figure A C Dj which is contained under theright i line AC, and the part of the cir¬ cumference ADC, isa Seftion of a circle Icfle than a Semicircle. 4 [ And here note, that by a Seftion, Segment, Portion, or Part of a Circle, is meant the fame thing, andfignifieth fucha part as is either greater or leffer then a Semicircle, fothata Semicircle cannot properly be called a Seftion, Segment, or part of a Circle. 19 Right linedf^es are fueb as are contain- ed under right lines, 20 Three Jided figures arefucb as are contained j under three right lines, 21 Four fided figures are fiteb as are contained under four right lines, 22 Many fided figures are fitch as have more fides than four. 23 All three fided figures are caEedTnsa^cs’ And fuch arc the Triangles ABC. >4 0 / Geometricall Definitions, 9 Lib.!. 24 Of fourfidedfiguresi a Quadrant or Square is ^that iffhofe fides are equal ; and his angles right. As : the Figure A. 25 ^Long Squares I that n>hich bath right an- I gles blit unequalfides. As I the Figure B. I A Rhombus is a i Figure having four equaU fides but not right angles, ■ As the Figure C. 27 A Rhomboides is a i Figure ipbofe Offofite fides ' are equally andtphofeoffo^ fite angles are alfo equally but it bath neither equaU fides nor equaU angles. As the Figure D. 28 AU other Figures offoure fides (befides tbefe) are called Trapezias, Such are all Figures of four (ides in which is obferved no equality of fides or angles, as the figures A and B, which have neither equal! fides nor equall angles, but are deferibed by all adventures without the ob¬ servation of any order. A B 3 9 Parallel C II GEOMETRICALL PROBLEMES PRQBLEME I. V^on aright line giveny borvtoereU another right linCy which jball he perpendicular to the right linegiven. He right line given is A Bj upon which from the point E iggj itxsrcquiredtoeredttheperpendicularEH. Opening your Compares at pleahire to any convenl- • ui “‘ft^nce, place one foot in the afligned point E, and withtiieotherniakethe marks C and D,EquiSiftanfon each fide the given point E. Then open- ing your Compafles again to G„ j, ..p any other convenient tUftance wider then the former, place F * 'V one foot in C, and with the , ■ other deferibe the arch GG, alfo (the Compafles remaining at the fame diuance) place one foot in the point Djand with the other deferibe the arch FF, ihen^ from the point where X A""' vT * 'jh thele two arches interfeft or ® rot each other (which is at H) draw the right Line HE which (hall be perpendicular to the given right line ABj which was the ^ns required to be done. 12 Geometricall VroUemes, Lib.i, PROB. II. Harp to ereB a Ter^endicular on the end of a right linegiven. L Et O R be a line given, and let it be required to ereft the per¬ pendicular RS. 'Pirft, upon the line OR, with your Coai- palTes poened to any fmall diftance, make five fmall divifions beginning at R,notcd with i,s, 3 , 4 ’ j. Then lake with your Com- paues the diftance from R to 4, and placing one foot , in R, with the other deferibe .. the arch P P. Then take the diftance R j, and placing B one foot of The CompalTes in t, with the other foor de¬ feribe the arch B B, cutting the former arch in the point S, Laftly from the point S, l_ draw the line R S , which fhall be perpendicular to the given line OR. -r-r- PROB. III. Hm to let fall a perpendicular, from any pom af fgned, upon a right line given. T he point given is C , from which point it is required to draw a right line which iliall bp perpendicular to the given right line A B. Firft, from the given point C, to the line AB, draw a line by chance, as C E, which divide into two equall parts in the point D, then placiim one foot of the Coihpalles in the point D, with the di¬ ftance DC, deferibe the Semicircle C F E, cutting the given line A B in the point F. Laftly, if from the PROB. Lib.I. Giometricall Problems^ PROB. IV. Hob? io ma{e an angle eqml! to an angle given. T Et the angle given be A C B. and let it be required to make 1 another analeequall thereunto. JL-/ Firft;'draw . ^ the line EF at plealurCj • then upon the given an- gleat C, (the Com- ■; : pafles opened to any di- fiance) deferibe the ark "ft ^ AB, alfo; upon the point F (the Compares un-altercd ) deferibe the arkc D E: then take with your Compaficsthcdiftance ABjand fet the lamediftance from EtoD. Laftly, drawrhelineDFjfb fl>all the an^le D F E be equall to the given angle A C B. prob. V. A right line being given, howto draw another right line which fhadbepraMtbtbe f^^^ any dtflance required, ^ He line given is AB, unto which it Is required- to dfavit ano- J thg right line parallel thereunto, at the diftance A C, or Firft, Oocn vour Com. C D ^ K paffes to tl le diftance A C or .<•••.X AD, then placing one foot in A, with t lie other deferibe the arke C; alfo place one foot in B, and with the other A B deferibe the. Af ch l>. Laftly, •— Draw the inc-C D fo that it may only touch the arks C andD fo (Rail the lineCDbe parallel to the line A B, and at the diftance required. PROB VI. To divide a right line given into any nmtiber of squadfarts, , T Et A Bbca fight line given, and let it Le required to ivide the *— lame into four equall parts, Firft, Geometricall Defnitions. L i b. i. Firftj from the end of. tlic given line A, draw the line A C,making any ande, then from the other end of the given line, which is at the point B, draw the line \ B D parallel to AC, (or “B make the angle A B D c- quall to the Angle C A B;) then upon the lines AC and B D fet off any three cquall parts which (is one Icfle then the number of parts into which the line A B is to be divided) on each line, as I 2 then draw lines from i to 7, from 2 to j, and from j to 1, which lines, croffing the given line A B, fliall divide it into four c- quall parts as was required. PROB, VII. A right line being given, how to draw another right line farallel thereunto, which jhall alfo < • ' - ■ ■ Et A B be a line given, and let it be required to draw another line parallel thereunto which fliall pafle through the given ■'point C. Firft, Take with your com- .■E pafles the di- _ftance from A toC, andplace- ing one foote thereof in B, with the other . deferibetheark D E; then take in your compaf- An J 'I . • . whole une AB,and placing one foot in the point C,with the other deferibe the arke F G, crofting the former arke D E in the point H. Laftly, tfyou draw the LineC H it fliall be parallel to A B. PROB.1 Lib.1. Geometricall Vrohlemes. 15 P ROB. VIII.. ; Having^ny three pints given ^wbich are mt Ji- i tuate in cl right line , hov? to finde the center of ! an arch of a Circle which fhall pafs diret^ly ; iht OHgb the three given points. He three points given are A Band C, noW it is required to finde the center of a Circlcjwholc circumference mail pafs , through the three points given. I Firft , open your ' Compafi'es to any di- ; fiance greater then halt' ! the diftance between B ! and C, then place one j foot in the point Bj and [ with the other deferibe the arc!) F G > then tlie compalfcs remaining at the lamcdiftanccjplace one foot in C, and with the other turned about make the marks F and G in,the former arch, and draw the line F G at length if need be. Again , opening the Compafles to any di¬ fiance greater then half AB, place one foot in the point A, and with the other deferibe the arch H K,then,the Compalfes remaining at the fame diftance, place one foot in the point B, and turning the o- ther about make the marks H K in the former arch. Laftly, draw the right line H K cutting the line F G in 0 , 0 ) (hall O be the center upon which if you deferibe a circle at the diftance of O A,it fliall pafs direftly through the three given points ABC, which was required. PROBLEME IX. Jny three right lines being given, fo that the two JhorteJl together be longer then the third,to maf{e I thereof a Triangle, j T Et it be required to make a Triangle of the three lines A B and j L C, the two fhorteft whereof, viz. A and B together, are longer then the third line C. Firft, j >f>. Geometricall Vrohkmes, LiB.S.i R- C- rirft. Draw the line D E equall to the given.lineB, then take with your Coni- palTes the line C, and fet- tinr one toot in t, with tiie otlicr deCcribe the arch H G i alio, take tiie given line A in your Compaflc-s, and placing one toot in D, with the other dclcribe the arch H F, cutting ttie for¬ mer arcii H G in the point H. Laftly, it from the point H you draw the lines H E and H D, you (hall confti- tute the Triangle H D E, whofe fides iTiall be equall to the three given lines ABC. : PROB, X. Havingarigk line given, bom to ma 1 \ea Geome- trtcall Squarejupbofe fide jhall be equall to the Right line given, T He lino given is QR, and it is required to makeaGcome* tricall Square whole fide fhall be equall to the line QR. FifRjDraw the line A B, making it equall to the given ‘ » line Q^R, then ( 6 y the jirflorfe- j ^ md Prohleme) upon the point B ! raife the perpendicular B C, ! making the lincBC equall to ' the given line QR alfo. Then ’ taking the line Q^R in your ' CompalTes, place one foot in ' C, and vvitii the other deferibe i the arch D, alfo the Compaf- ' fesfo refting, place one toot in • A, and with the other deferibe ; another arch crofling the for- B mer in the point D. Laftly, „ draw the lines D C and D A. ^ which fhall include the Geo- inetricallfquareABCD. | PROB. j Lib.I. GeometricaU Trohlemes. ! Two right lines being given how to finde atbird right line which fhd be in proportion unto tbem» L Ect t!ic two given A-8 lines be Aand ^ required tofinda" ! third line which Hiall 1 be in jiroportioii unlo \ them. Firft) Draw two right lines making any angle at pleafure, as the lines OPandON, making the angle PON; then taking the line A in your tompaflesjfct the \ - vt Icngtli thereof from O^f ^ ^5 C to Sjalfo? take the line B in your Compaflcs,and fet the length thereof from 0 to R,and alfo from O to D, then draw the right line S D, and from the point R draw tile right line R C parallel to S Dj fo iliall O C be the third proportional required. As OS to OD :: fo OR to OC. 8 12 iz l8 PROB. XII. Three right lines being given, to finde 4 fourth in proportion to them. T H E three a. ----24 lines given n —--2j? arc ABCj c- jfT unto which it is re- — ---42 quired to finde a ^ D fourth proportion- ' all line. This is to perform the rule of , three in lines. \ As in the laft ^ Problem, you muft ^ \ .•■'j \ draw two lines ma- ' ~lTi- ^ Jf king any angle, as '■ tlieanglcDEF Then take the line A in your CompaCfes, and ; let It from E to G, than take the line B in your Gompalfes and le t 1 __ D that i8 Geometricall Froblenas Lib.i. that length from E toH. Then take the third given line in your Compaflesjand fet that from E to Kjand through that point K draw the line K L parallel to G H, lb (hall the line E L be the third pro¬ portional required ; forj As EG to EH :: fo EK to EL. 24 28 36 i)2. Here note that in the performance of this Probicme, that the firft and the third terms (namely 5 the lines A and C) mull be let upon one and the fame lincjas hereupon the line E D, and the fccond term (namely, the line B) mull be fet upon the other line E F, upon which line alfo the fourth propor¬ tional E L will be found. PROBLEME Xin. T0 divide a right line given kto tm pru, which pall have fuch froprtion one to the ether as two given right lines. ^■"■^He line given is A B , and it is required to divide the fame I into two parts, which (hall have fuch fuch proportion one * to the other, as the line C hath to the line D. Firft, from the point ^ ' c A 5 draw fl c line A E, at picafurej making the angle E A B j then take ; in your Cempaftesthe line C 3 and let it from A to Fjalfo take the line D, and fet it from F to E 3 and draw the line __ 2:^:- -1 EB3thcn frem the point A ^^30 FjdrawthelineFGpa- • AT3- . . ^ ^ rallel to EB,cutting the given line A B in the point G; fo is the line AB divide d into two parts in the point G, being in proportion one to the oth.crjas the line C is to the line D; for, As AE to AB :: fo AF to AG. Arithmetically. T Et the line A B contain 40 Pcrches3 and Jet the line C be 20, I and the line D ,o; andjlctit be required to divide the line Firft, •( Geomemcall Problemes. *9 LI B.r. Fir ft, AdJ tlic lines C and D together, their funi is 5 c, then fay by tlic Rule of Proportion: If 50 (which is the fum of two given ! terms) give 40 tlic whole line A B, what (hall 30, the greater given ' term give? Multiply and divide, and you (hall have in the quotient : 24 for the greater part of the line A B, which being taken from 40 1 the whole line, there remains 16 for the other part A G; for, As AE to AB :: fo FE to GB. 50 40 30 24 I PROBLEME XVI. j Horp to divide al ria^gle into two farts j accord* ! ding to any frofortion ajjigned^ by a line drawn I from any Angle thereof, and to lay tbelejfer fart towards any fide affigncd, L Et A B C be a Triangle given, and let it be required to di¬ vide the fame, hyaline drawn from the angle A, into two parts, the one bearing proportion to the other, as the line F doth to the line G, and that the lefTcr part may be towards the fide By the laft Prolleme divide thebafe of Hie Triangle B C in the point D, in proportion as the line F is to the line G, (the leflcr part being fet tromBtoD). Laftly, draw the line A D, which llialJdi¬ vide the Triangle A B C in proper tion as F to G-, for. As the line F, is to the line G j So is the Triangle A D Cj to the Triangle A B D. D 2 PROB. 20 Geometricall Troblemes Lib.i, PROB. XV. T be Bafe of the Tmngle being perform the foregoing Vrcblemeby Aritbmetici^ S Uppofe the Bafe of the Triangle B C to be 40, and let the pro¬ portion into which the Triangle ABC is to be divided, be as a to j. Firft, Add the two proportional terms together, 2 and j, which makes y, then fay by the Rule of Proportion; If 5 (thefumof the proportional terms,) give 40 (the whole bale BC) whatfliall^ (the greater term given) ? Multiply and divide, and tlic quotient will give you 24 for the greater fegment of the Bale D C , which being deduced from the whole B^afe 40, there will remain i d for the leffer fegment B D. PROB. xvr. I How to divide a Triangle, whofe area or content I K l^nown, into two pant, by a line drawn from j an (tAngle offigned, according to any tropor -! iion required, ! L Et the Triangle ABC contain S Acres, and let it be required to divide the lame into two parts, by aline drawn from the Firft 5 Acres;and the other 3 Acres. which (hppofe40, / n Acres (the quantity of the whole f rianale) nit c 40, Sa Sfd 5 AcrescivcVMulti-’ ESi ^ the oreatcr‘ 5 ec.mentof tlier^wuSemar^^^ f (the whole Bafe,) tnerewUlremain 15 for the IclTcr Segment of the BafeBD, then draw Geometricall Prohlemes. 21 Lib.i. draw tlic line A D, whiclr iTiall divide the Triangle ABC accord¬ ing to tlie proportion required. PROB. XVII. Boiv to divide aTriatigle given into tm farts, I according to any po^ortion nffigned ^ by a line draopn from a pint limited in any of the fides thereof: and to lay the greater or lejfer fart to- ivards an Angle ajftgned. He Triangle given is ABC and it is required from the point ^ E,to draw a line that fliall divide the Triangle into two J?- part?, being in proportion one to the other as the line I is to the line Rj and to lay the lelfer part towards B. Fii fl, From the limited point E, draw a line to the oppofite An- elc ac A; then divide the Bafe B.C in proportion as I to K, which pointuf di\ilion will be at D, then draw D F parallel to AE. Laft- Iv, 1 Vom r draw the line F E , which will divide the Triangle into uvo parts being in proportion one to the other,as the line I is to the lineK. PROB. XVIII. Ta ferform the foregoing Vrobkme ArithmetU \ ally, f T is required to divide the Triangle ABC,from the point Edn- to two parts in proportion as 5 to i. Firll,Divide the Bale BC according to the given proportion, j tiicn Cbecaufc the lelfer part is to be laid towards B) mcafurethe dillancc from E to B,which admit 30, then fay by the Rule of Pro- portion; If E B 3c, give D B15 what fliall AG ip (the perpendicu- Geometricdll Problemet L 1 B. 1. larof the Triangle) ;z,ivc? Multiply and divide, the Qiioticnt will bei4;.at whichdillance drawa parallel line to BC, namely F, ti en fr mi F draw the line F.E,wiiicli lliall divide the Triangle ac- cc ding :o the required proportion, . PROB. XIX. Hor^ to divide a Triangle^ whfe ^rea or content is h^orr^n, into two prts, h^ahne drawnjroma pint limited in any fide thereof, according toa- ny nwnher of Acrees^ Roods and Perches. I Nthe foregoing Triangle ABC, whofe area or content is five Acres, iRood, let the limited point beE in thebafe thereof, and let it be required from the point E to draw a right line which fliall divide the Triangle into two parts between Jp and i?, fo that sBmay have 3 Acres, 3 Roods thereof, and Jg may ha\e 1 Acre and 2 Roods thereof. Firft, reduce the quantity ofi? (being the lefTcr) into Perches, which makes 24c, then (confidcring on which fide of the limited point E this part is to be laid,as towards B)mcafuro that part of the Bafc from E to B 30 Perches, whereof take tlie half, which is ly, and thereby divide 240, the part belonging to ^; the quotient will be itf,thelength of the perpendicular FH, atwhich parallel di- ftance from the Bale B C cut the fide A B in F, from whence draw the line FE which lliall cut off the Triangle F BE, containing i Acre, j Roods, the part belonging to jjJ, then will the Trapezia AFEC (which is the part belonging to iP) contain thercliduc, namely, 3 Acres, 3 Roods. PROB. XX. How to divideaTriangleaccording to nnypo- prtiongivtnjhy a line drawn parallel' - one of the fides given, T He follovvi^ Triangle A B C is given, and it divide the fame into two parts by a line drawn para, el the fide A C,which fliall be in proportion ore to the other, the line I is to the line K. Firft(^;7/;f 13 /’/cZ/fw; divide the line BC in E, in porporti- i qnasito Kjthcnf'Z^ //;f 24 P>cLlen:efcIlonh.njfiT.di. prepor- j tional between B E and B C, whicli let be B F from which point F, draw the line FH parallel to AC, which line fliall divide I the Triangle into two parts, liz. the Trapezia A H F C, and the _ Tri- I 'Lib.i. Geoimtncall Trohlemes, 33 Triangle H F B, which arc in proportion one to the other as the line I is to the line K, PROB. XXL I To perform the foregoing Probleme Jritb- m^tically. L Et the Triangle be A BC , and let it be required to divide the fame into two parts, which (hall be in proportion one to the other , as 4 to 5 5 by a line drawn parallel to one of the fides. A j Firft, Lctthc bafeBC containing54be divided accordingto the proportion given, fofliall thelcflcr fcgment BE contain 24, and the greater EC 30; then find out a mean proportionall line ] between B E 24,& the whole bafe B C 54 , by multiplying 54by 24, whofc produft will be 12p6', the fquare root whereof is 36, the mean proportionall fought, which is BF, then, by the rule of pro¬ portion fay; IfB F 36, give BE 24, what AD 35 ? the anlwer isH C 24,atwhichdiftancedrawaparalIcllinetothebare,tocuc the fide A B in H,from whence draw the line H F parallel to A C, which lliall divide the Triangle as was required, PROB. XXII. T 0 (livid eaT riangle of any J^om quantity y into tiro pmsy by a line drarrn parallel to one of the fides ydccordingto any number of Acres,Koods, and Perches, T Hc Triangle given is ABC, whofe quantity is 8 Acres, o Roods, 1 6 Perches, and it is required to divide the fame (by a line drawn parallel to the fide. AC) into two pans, ziz,. 4 Acres, 2 Roods, o Perches, and 3 Acres, 2 Roods, 16 \ Perches. 1 Firft, Geometricall Problemes Lib.!, Firft,Reduce both quantities into perches(as is hereafter taught) and they will be 710, and 576, tlien reduce both thole numbers, by abbreviation, into the lead proportional terms, zh. 5 and 4, and according to that proportion,divide the bale BC 54of tlie gi¬ ven Triangle in E, then feek the mean proportional between B E and B C, wliich proportional is B F je, of which 36 take the halfe, and thereby divide57d5 thelefferquantity of Perches, the Quo¬ tient will be H G 3 2, at which parallel didance from the bafe, cut off the line A B in H,from whence draw the line H F parallel to tlie fide A C , which fliall divide the Triangle given according as was required. PROB. XXIII. From a line given, to cm off any prts required. T He line given is A B, from which it is required to cut off i parts. Fird, draw the line AC, making any angle , as CAB, then from A, fet off any feven equal parts,- as i 3 3 4 5 67, and from 7 draw the line 7 B. Nowbecaule ' istobe jr, • cutoff from the line 'f..''’’ ’ AB, therefore from the point j,draw the i \ line 3 D parallel to \ 7 B cutting the line A B in D, fo fliall A D be 4 of the line A B, and D B fliall -—ji - be -fof the fame linej •* for. As Ay,is to AB :: fois A3,to A D. PROB. XXIV. To find a mean proportional between two lines given, I N the following figure, let the two lines given be A aud B, be¬ tween which it is required to find a mean proportional. Let the two given lines A and B, be joyned together in the pointE,raakingone right line, as CD whicli divide into twoe- qual parts in the point G, upon which point G, with tlie didance G C or G D,dcfcribc the Semicircle C F D,then,from the point E, (v^ere the two lines are joyned together) raife the perpendicular E F, cutting the Perifery of the Semicircle in F, fo fliall the line __EF LIB. I. Gemttricd^ TroblemSs, 35 E F be a wean proportional between the two civen lines A and B; for, As ED to EF :: foEF to CF. 9 li 12 16 PROB. XXV. How to find two lines which togetherJhal be eqnd in pwer to any line given i and in pwerthe one to the other according to any poyortion li N this figure let CD A— be a line given to be B— divided in power as G— the line A is to tncline B. Fir ft. Divide the line C D in the point E, in proportion as A to B> {h^ the 13 ProUeme) then di¬ vide the line CD into two equal parts in the point Gj anci on G, at the diftance G C or G D,de- feribe the femicircle CF, Dj and upon the pobt E, raife the perpendicular E F, cutting the Semicircle in F: LaRJy draw the lines CFand DF, which together in power wall be equall to the power of the given line CD, and yet in power one to the other as A to B. E pROB. Geomet'/mll Troblmef L i b. i, 0“ ' PROB. XXVI. HoiPto divide a line a line in forner according w any fro^ortion given. I divide the line C D in the point E in propottion as A » to Bj then divide the line CD in two parts in the point G> and Upon Gas a center at the diftanccGDj deferibe thefe* micircle C F Djand on the point E raife the perpendicular of E F cutting the fcmicirclc In F, then draw the lines C F and D FjanJ produce the line C F to Hj till F H be equal to F D, and draw the } the line B D, and F K parallel to H D, then (hall the line C D. be divided in K,fo that the fquarc of CK fhall be to the Iquarc of X as C E to E D, or as B to A. PROB. XXVII. Eoyv to enlarge aline in pwer, according to any proportion ajfigned, ITN the Diagram of the 2 jtli. Probleme, let C E be a line givcib ,^tal»enlarged in power as the line B to the line G. ^ Firftjf^jif/ie r jt/;. ProUeme) find a line in proportion to the given line C E, as B is to G, which will be C D, upon which line deferibe the femicirclc C F D,and on the point E, cred tlie per* pedicular E F 5 then draw the line C F, whichlliall be in nower toCE,asGtoB. ^ PROB, jLiB.i. Geometricall Trohlemes. iy PROB. XXVIII. T0 enlarge or diminillj a Plot given, according to any ^roprtion required. L Ec ABC D E be a Plot giveiij tobedihiinifliedinpowcra LtoK. Divide one of tlie fides (as A B)iii power as L toKjin fuch for t, that the power of A F, may be to the power of A B, as L to K. Tl.enfromtheaivj,le A, draw lines to the points Cand D j that done,by F draw a parallel to B C,to cut A C in G,as F G. Again, from G,draw a parallel to D C,tocutin ADin H. Laftly,{rom H, draw a parallel to D Ejto cut A E in I, fb fliall the plot A F G HI be like AB C D E,and in proportion to it, as the line L,to the line K which was requ red- Alfo, if the leffer Plot __ were given, and it were re- quired to make a y reater in ^ proportion to it as K to L. Then from the point A, \ draw the lines A Cand A F" D,atlcnght, alfo increafe AF and AI: that done, in- large AF in power asK toL, which fetfromAto B, then by B draw a paral¬ lel to F G to cut A C in C, as B C. Likewife from C draw a parallel to G H, to cut A D in D, as C D. Laftly, a paral¬ lel from D to H I, as as D E, to cut AI being incrcafcd in E, fo fliall you include the Plot A B C D E, like A F G H I, and in pro¬ portion thcreunto,as the line K is to the L, which was required. PROB. XXIX. How to a Triangle which Jball contain any number of Acre SyRoods and Perches and wbofe bafe fhall be equal to any Qoffthle) number given. I F it be required to make a Triangle which fhall contain S A- cres, two Roods, ? o Perchcs,whofe bafe fhall contain 50 Perch¬ es, you muft firft reduce your 5 Acres, a Roods,30 Perches, all into Perches in this manner. Firft, (bccaule4Roods make one Acre) multiply your 5 Acres by 4 which makes so, to which idd the two odd Roods, fo have Geometricall Troblenm LiB.1. 1 j you 22 Roods in your 5 Acres, 2 Roods. Tlicn (becaufc 40 Perch- I ^ ts make one Rood) multiply your 2 2 Roods by 40, which makes j I 880 Perches,to which add the 30 odd Perches,andyou fhall have 1 * ^lOjandfomanyPcrchesarecontainedinyAcres, zRoods, 30 Perches. Now to make q t. tv a Triangle which iliall contain 5110 pcrches,& whole bafe lhall be jO Perches, do thus, Double the num¬ ber of perches gi¬ ven, namely 910, and they make ] 1820 , then bc- caufe the bafe of ^ the triangle muft contain 50 Perches, divide 1820 by 50, the quotient will be 36^5 which will be the length ot the perpendicular of your Triangle. This done. From any equal Scale lay down the line ABequall to JO Perches, then upon B,raifc the perpendicular BD equal to 36} perches, and draw the line CD parallelto AB: then, from any point in the line C D(as from E)draw the lines E A annd E Bj in¬ cluding the Triangle A E B, which lhall contain 5 Acres,1 Roods, 3 0 Perches which was required. PROB. XXX. How to redttce a T ra^es^ia into a 7 rimgkj by a line drawn from any angle thereof. T He Trapezia gi¬ ven is ABCD, and it is requi¬ red to reduce the fame into a Triangle, Firft, Extend the line DC, and draw the Di- agonallBD, thenfrom B the point A, draw the line A E parallel to B D,extending it till it cut the fide C D in the c T the line B E, conftituting the Tri- angle E B C, which lhall be equal to the Trapezia ABCD. PROB. jLiB.i, Geometricall Trohkmes. 39 PROB. XXXI. Horp to nduce a Trapf ^ia into a Triangle J?y lines ^rom any point in any ofthe fide; thereof Et A B C D be aTrapczia given, and lec H be a point in one of the Tides thereof from which point H let it be rcf)iiired to ■^draw lines which fiiall reduce tlic Trapezia into a Triangle. _ Firft, Extend tlieiide which is oppofite to the gii en point,namclyjthe iide C D, both wayes to E and F , and then from the point H, draw lines to the Angles C andDjastl'.e lines HG and H D ; alio, draw the lines AE and BF ^ parallel to H C and H D, cutting the extend¬ ed line C D in the points E and F. Laftly, If from the point H you draw the lines HE and HF you lliall conftitute the Triangle H E F, which fhall be equal to the Trapezia A B C D. PROB. XXXII. FIon> to reduce an irregular Tlot of five fides into aT riangle, T He irregular Plot given is A B C D E, and it is required to reduce the fame in to a Triangle. Firft, extend the fide A E both wayes to F and G, an from the Angle C, draw the lines C A and C E, to the Angles A and E, Then &om tke point B, draw the line B F paraUel to C A 30 GeOmetricall TroblefUes Lifi.i. cutting the extended fide A E, in F ; alfo from the point D, draw the line DG parallel to CE, cutting alfo the extended fide in G. Laftly, from the angle C, draw the lines C F and C G 3 conflitu- ting the Triangle C F G3 which is equal to the Plot A B C D E. Cl* PROB. XXXIII. Houp to rednee an irregular Flot 0/ 6 3 7, or 8 Jides into a Triangle, A L EcABCDEFGbcan irregular plot given, to be reduced into a Triangle. I have chofen this figure where the angles C and D are ^ without the field, becaufc it often comes in prafticc, and hath not been taught by any to my knowledge. i Firft, draw the lines B t), and parallel thereto the lines C K, j then if you draw the line B K, the two fidcs B C and C D j fhall be reduced to one right line, liz, to the line B K. Alfo ; draw the line G E, and parallel thereto F L, then if you draw ® the line G L, the two fides G F and F E lliall be reduced to oae { ftraight line, to the line G L, and fo the whole plot A B C D i E FG confiding of feven fides, is reduced to a figure of five | fidcs 3 namely, to the figure A B K L G; yet ftill retaining the fame quantity. Now to reduce this plot into a Triangle youmuft work in all refpedts as in the former Probleide. Firfi, produce the fide DE of thegiven figure, on both fidestoH and M, then draw lines A K and A L , and parallel to them, the lines B H and , G M, cutting the line D E being extended in H and M. Lafliy, if; you draw the lines A H and H M, you fliall cenftitute tlic Trian- | gle A H M which fliall be equal to the irregular plot A E C D E : F G which was required. _____,_f And I |L t I. GmmrkaU Vrohlemcs. j ^ Audi here note that the number of fidcs be never fo n;anyj f \«\h.is way ofredudlion will bring them to Triangles> and I more outward angles there are in the Plot, the more I tM^yefome will it be to efte< 5 l:. j PROB. XXXIV. 1 I being gh en , bon^ from any angle I «0/ to divide the fame into tuno parts being j in proprtion one to the other as two given right I line ft and to fet the pan cut off towards an I affgmdfde, j W the Trapezia given be ABC D , and let it be required I I to draw a line from the angle B, wliich iliall divide the Tra- 1 into two parts, being in proportion one to the other, as ! the line G is to the line H, and that the IclTcr part of the figure cut i off? tnay be towards the fide A B. G^--2. 'S\x9it(Mei9Priikme) reduce the Trapezia ABCD into a Trianglcjby drawing the line BF from the afligiied angle, itoe- by conftiiuting the Triangle A B F,equalto the Trapezia ABCD: tins done> divide the bafe of the Triangle A F in proportion as G to H, which will be in the point E. Laftly, draw the line BE,which Biall divide the Trapezia in proportion as GtoH. Now becaule the IclTcr part of the Trapezia was tobe fet towards the hde A B, the other angles. hcreforc the leficr part of the line muftbclettrom Atoc. na lotc that the fame manner of working is tobe obferved, it ithad icon required to divide the Trapezia by a line drawn from any ot PROB. 33 GeOm^iribdl Probkfm LlB.l. PROB. XXXV. A TrapCs^ia hii/ig givenMi^yfrom a point limi¬ ted in any fide thereof 3 to drawn line n>hich jhall divide the fame into two parts m proportion as two given lines. T He Trapezia given is A B C Dj and it is required from the point Hj to draw a line which lliail divide tlx Trapezia in proportion as O to Firft, Prolong the fide C Djand reduce the whole Trapezia into the Triangle H E F by the 30 Fyo- ileme,then divide the line EF in proportion asOtoQ^jwhichwil fall in the point C, therefore draw the f i. ij p line HGjwhich lhall divide the Trapezia into two parts in proportion as 0 to Q, which was required. PROB. XXXVI. Trt^es^ia being given, how to divide the fame into two parts in proportion as two lines — me otner as tne line \ is to the lin partition may be parallel to the fide B D. Confider firft through whichfidesof thcTrapezia thelincof imtion willpafsj as in this figure it will nafs throntri, p^artition wm pafs, as in this figure it will pafsXoighlhe fidS (becaufe parallel to B D5)thereforejextend the fides itill thev concur in F.rli(.n _j_ A K r S ^ O.)therefore,extend the fides A B and C p.tdl thw concur in E;thcn (by the 32 ProMeme Reduce GD thcTriangle BG D, whofe bafeh fo Sat ^ ^ ^ proportion as K to Lj AsKto L;:So DH to HG. jLiB.i. Gemetricall TMemei, This done, findc a mean proportional between E D and E H (bphe 24 ProUeme)a^ E R. Laftly, :hrou:li this point R, draw the line R F parallel to B D, which (liall divide the Trapezia into two parts being in proportion one to the other 5 as the line K is to the lineLjand with a line parallel to the fide BDjwhich was required. But if it had been required to divide thcTrapczia by a line drawn parallel to the fide CDjthcn the lines C A and DBniull have been extended, but the reft of the work rauft be performed as is before taught. PROB. XXXVII. The figure of a V lot heinggiveri,b(m>tO divide the fame into Wo farts being infrofortion on 6 to the other as mo given lines an^ with aiini drrwn from an angle ajfigned. L Et the following figure ABODE repreftnt the Plot of field or fuch like, and let it be required to divide the fame; into twopatts, being in ptoportipa one to the other aith'c lineRisto the line S, by a line drawn from the AnglcB. Firft, Reduce the Plot ABODE into the Triangle BFG, (Ijthe^i ProMeme) fo {hall the line FG be the bale of a Triangle equal to the given Plot, then the ijth Probleme) divide this line F G into two part* in the point H, in proportion one to the other,as the line R is to the line S j fo that, AsRtoS :: foGHtoHF. Ladlyjdraw the line B H, which (liall divide your given Plot into two parts which fliall have fuch proportion one to the otherjts the line R hath to the line S. F PROB. |L I ij. I*. GcOm^rhall Ffoblemer, 5«? i PROS. XX^CIX. j Ho^io divide an irregular Plot of fixfides, into I iiJ?opat'ttj accordingtoanyajpgnedproportionj j h a right line draa?n from a point limited many I of the fide f thereof. T Hc irregular Plot given is A B C D E F, and it is required to divide the fame into two parts, being in proportion one to the otl’.cr, as the line R is to the line S. Firft, Draw the right line H K,and (by the ^oth.ProUemeJ ttdacc theTrapezIa ABF G into the Triangle H GK, then divide the bafethercofinamclyjHK, into two parts in proportion as R toS, whith will be in the point Ojthen draw the line G O, which will divide the Trapezia ABFC into two parts in proportion one to theother, asthelineRistothelineS. ! Secpndlyj From the point Of^yifcr 31 reduce the Tra-1 peiia F C ED into the Triangle 0 L M,and divide the bafe there¬ of j nainclyjL M,in the point Njin proportion as R to S, and draw the line O Nj which will divide the Trapezia F C E D into two parts in proportion as R to S: and by this meant is the whole Plot A B C p E F divided into two parts in proportion as R to Sjby the lines G C) and O N. But it is required to rcfolve the Probleme by one right line only drawn from the point Gj therefore, from tltc j^int G, draw the line C N > and through the point 0 , draw the line O P parallel to G N: and laRly, from G, draw the ri^t line G Pjwhich ihall divide the whole Plot A BCD E F into two parts, beins in pronortion one to the other as the line T isto the line S. F 2 PROB; 36 GeOmetricall Problems Lib.i. PROB. XL Horptodroidem irregular Plotaccordingto any ^roprtiott , by a line dram from any (^Angle thereof. W Et AB C D E F G be an irregular Plot, and let it be requi- 1 red to divide the fame into two equal parts, by a line drawn from the angle A. G- A /" . B N P n -M" I Firft, draw the line H K dividing the Plot into two parts,name¬ ly, into the five fidcd figure ABC FG, and into the Trapezia, FCE^D, then (ly then Prdlme) xeduce the five fided figure A B C F G into theTriangle H A K, the bafe whereof H K divide in¬ to two coual parts in 0,and draw the line O A .which fhall divide the fivended figure A B C F G into two equal parts. Then (lythe loth. Prollme) reduce the Trapezia FCDE into the Triangle O L M., and divide the bafe thereof L M into two equal parts in the point P, and draw the line 0 P, which will divide the Trape¬ zia F C D E into two equal parts by the lines A O and 0 P, but to perform theProblcmc by one right line only, do thus, from the point A, draw the line A P, and parallel thereunto, through the point Ojdraw the line O N. Laftly, if you draw a right line froiri A to Nj it fliall divide the whole Plot into two equal parts. C Here note that whatfoever hath been faid concerning the dividing of figures in proportion to right lines, the fanie may be effedied in. numbers, fo that from any Plot you may cut off any number of Acres,Roods, or Perches. —._ ' _ VK O B. Gimetrmll ILib.I. (JJ» PROS. XLI. Horv to divide a Tra^es^ia inm tmpartshy a line draxpn from anoint mtbouti whicb farts fhall be inpofortion one to the other, as tm given lines. N r s 11* Tit the Trapezia given be ABCD, and let the givtji point y ’.vitiioiu be E, from which it is required to draw a line -fi- ' wliich fliall divide the Trapezia into two parts which {hall be in proportion one to the other, as the line F is to thcline G. Extend the fidcs of die I rapezia B C and AD till they concur in H, then through the point E draw the line EI parallel to A Hj till it meet withthelinc BH being extended to I, then ( hy the 29 Frableme)xQ6Mcc. thcTrapczia ABCD into theTnanglc A BKj and from the point B let fall thcpcrpcndicularBZ 5 then (by the \yh. Probleme) divide thebafe of the Triangle AKintotwo parts in proportion as F to Gjwhich point of Divifion will fall Ln L.This done (by the i ith.PrtblemejUnA a fourth line which fhall be in pro¬ portion to the three linos i.e. H L, and H Bj that is, as IE : to H L :: fo H B: to H M, fo is H M the fourth proportional: then (by the zefh. Prdleme) find a mean proportional between the lines I Hj H M, which is H N, then fet H N perpendicular upon B H^nd di¬ vide HM into two equal parts in Ojthen draw the line ONj which you fliall fet from O toP. Laftly) if you draw the line E P , it fhall divide the Trapezia A B C D in two parts, which fhall be in pro¬ portion one to the otlier, as the lineF is to the line G. 59 LiB.a. ! THE I COMPLEAT SURVEYOR: The Second Book. the ARgVMENT. N this BooJi is contcined both a general and particu¬ lar defcription of all the moftneceffary In^uments belonging to Surveying, as the Theodolite , Circumfe¬ rentor 3 and Plain Table, with all the appurtenances thereunto belonging, astheSw/, Soc}\ets, Screm,Index, Label,and other neceffaries. Now, whereas thelc three Injlruments are the moft convenient for all man¬ ner of pradices in Surveying, 1 have fo ordered the matter, that in this Bool\, after the Theodo¬ lite, and are particularly defcri- bed, as they have ulually been made; I come to the defaiption of the Plain T able, and therein 40 T he ($yfrgnment. L i b. 2 . I have (hewed how that infirumint may be orde¬ red to perform the work of any of the other; fo that whatfoevcr may be done by the Theodolite, Circumferentor,- namely,by Mr. Dtggs, Mr. Hupton-, Mv.RaihUn-, an 11 ill of all in Planonetrla-, yet in this place k will be very necefl lary to give a particular dcfci ipcion of tlicm again, bccaiile if any man ha', c a dcllre to any particular Inftriimcnt, he may givetlic better direftions for the making thereof. I’or thedelcription which I mall make of thefe three Inflruments in particular, it lhall be agreeable to tiiofe Inftnimen ts as-rhey are iilually madepvith forae fmall addition or alteration : But when I come tothedeferiptionofthe/’/.rw 7a^/c, akcrthatlhavcdefcri- bed it according to ti e vulgar way, I will then fiiew you a new ?ne- umrpbcfn oi thatinftrument, making it the moil abfoliiteandu- nivcrlallnftrtimcntyet ever invented, fo that having that one In- ftrtimcnt (made according to the following diredlions) you ikall have need ofno other for the duc,cxa(5l, and fpeedy performance of any thing belonging to the Art of Surveying, For, the frame of tlie Table being graduated according to that defeription, will be an aololute 'r/;r&A///c,.and perform tlievvork thereof with the fame h'.cilityand e>;aancfs, andvvhatfoever maybe done by the limbe V.. eT'Wt •>, the lame the degrees on the frame of the Table will,..-, vu'll perform. Like'vV'fr, the Index and Sights, together with the Box and !'(C-. ” bk.ngtakcn frcijytheTablejandfcrewedtotheStaffeCas in tl c defeription tluredf'itis fo conveniently ordered) will bean abWx\iicChc:imferentor, and in fome refpefts better then the or¬ dinary one licreafter deferibed, becaufe the Sights thereof ftand at a greater diftance, io that thereby the vifual line may be the better dircdled. And this Inftrument (as now contrived) though it be called the Plat.. Aif/fcnly, yet you fee that it contains both the other, and therefore in adviling any man to the ufe thereof chiefly, I do not confine h ra to one, but to all Inflruments, and therefore do not contradidl my former cxprelfron, j Befides.,thcre is another great convenience which doth enfue by the degrees on the Tables frame; for in taking the plot of a field according to tl ie following diredtions by the Plain Table, you may at the fame time perform tne fame work by the degrees on the frame of the Tablcjif at the drawing of every line you obferve the degrees cut by tlie Index , and note them upon the paper. This I fay is a great convenience , for at one obfervatibn you perform two works with the fame labour, as in the ufes of thefc Inltruments feverally a Surveyor by this contrivance, which with praftife will appear of themfclvcs. CHAP. IL I«. 2 . defcri^tion of InJIrntrisnh, CHAP. II. Of the Theodolite, the defer!flion thereof, and thedeteBion of an erronr frequently commit¬ ted in the malting tberdof, with the manner kow> to correU the fame. [He theodolite is an inftrument confifting of Four parts principally. The firft whereof is a Circle di¬ vided into 350 equal partSjcalled degreesjand each degree fub-divided into as many other equal parts as the largenefs ol the Inftrument will bell permit: For the diameter of this Circle, it may be ol any lengthjbut thole ufuallymade in brafs are about twelve or fourteen Inches, andthelimbe thereof divided asaforefaid into36ode- grees,and fub-divided into other parts by diagonal lines drawn from the outmoft and inmoft concentrique Circles of the limbciin the drawing of which concentrique Circlesjthey Ufe to draw them cquidiftant, which is erroneous, as (hall appear hereafter. The fecond part of this Inftrument is the Geometrical Square; whic . is deferioed within the Circle, and the fides thereot divi¬ ded into certain equal parts, but there are few of them made now wit', this Square, for the degrees themfelvcs will better fupply that want, it being only for taking of heights and diftanecs. Yetif py man be defirous to have thisSquare upoh his Inftrument:thcre is a more convenient way to fet it on then tliat which Maftcr Di^s flic'veth, namely, upon the limbe of the Inftrument,, the manner how is scry well known to the Inftrument maker. Tlie third part of this Inftrument is the Box and Needle,fo coa- vcniently contrived to (land upon the Center of the Circle, upon which Center alfo the Index of the Inftrument muft turn about j and fometimes over the Box and Needle there is a Quadrant ere- fted for the taking of heights and diftances. The fourth part of this Inftrument, to fet it upon a ftaff when you make ufe thereof. In the making of this Inftrument, it were' neceflary to have two back-fights fixed at each end of one of the Diameters,for the readier laying out of any angle without moving of the Inftrument. I G z 45 CAAP. defcripion of hftrmrenn. L i b. 2 . CHAP. III. The defcripion of the Circumferentor. iHisInftrt'mcnt hath been much efteemedby many) for portability thereof) it being ufually made to con¬ tain in length about eight inchesjin breadth 4 inches, and in thicknefs about three quarters of an inch; one fide whereof is divided into divers equal parts, moft fitly often or twelve in an inch ; fo thacit may be ufed as the Scale of a Protractor, the inftrument it fclf being fitting to protraft the plot on paper by help of the Needlcjand tl.e degrees of angles, and length of lines taken in the field. On the upper fide of this Inftru¬ ment is turned a round hole, three inches, and a half Diameter, and about halfan inch deep, in which is placed a Card divided commonly into i jo equal parts or degrees, and each of thofe into three, which makes3^o aniwcrablc to the degrees of thcTiieodolitt, in which Card is alfo a Dial drawn to find the hour of the day,and Azimuth of the Sun; within the Box,is hanged a Needle touched with a Load-ftone, and covered over with a clcer glafs to preferve it from the weather. On the upper part of this Inftrument is alfo deferibed a Table of natural Sines, coUeftedanfwerabl* to the Card in the box, that is to fay,if the Card be divided but into 120 parts, the Sines muft be fo alfo; but if into 36o,thc Sines muft be the abfolutc degrees of the Quadrant. To this Inftrument alfo bclongcth two Sights,one double in length to the otherjthc lengeft containing about I'cven inches, being pla¬ ced and divided in all refpcifts,as thofe hereafter mentioned in the defeription of the Plain Table. On the edge of the lliorter fight to¬ ward the upper part thereof, is {riaced a Imall wyer reprefenting the Center ofafuppofed Circle,the Semidiameter whereof is the diftance from theWyer to the edge of the Inftrument underneath the fame,which parts is imaginarilydivided into fixty equal parts, and according to thofe divifions is the right line of divifions on the edge of the Inftrument divided,and numbred by 5,10,15 ,from the perpendicular point to the end thereof: And alio from the fame point on the upper edge of the Inftrument is perfedfed the degrees ofthe Quadrant, fupplying the refidue of thofe which could not be exprelled on the long fight, from 28 to ^o by tens. There is alfo belonging to thefe divifions a little Ruler, atone end whereof is a little hole to put it upon the wyer, on the edge of theftiqrter fight;andat tiie other end of this Ruler is placed a Imal fight,dire6Hy over the fiducial edge thereof; which edge is like- wife divided according to thofe divifions on the edge of the Inftru¬ ment. To this fhort fight is added a plummet to fet the Inftrument horizontal. And this Ihort Ruler, with the divifions thereof, and thole on the edge of the Inftrument ferve for taking of altitudes chiefly, and for the reducing of Hypothennfal to Horizontal lines. __CAAP. jL I B. 2 . / defcri^tioH of Infirnmenti^ 45 j CHAl^. IV. ' ^ Defcripion of the Plain Table, horf it hath I been formerly ma-iCj ard hm it is tonp altered,. i tt being the nioftabfolMte Inflrnmert of any b- ! thcr for a Surveyor to ufe, in that it perform- ■ etb iphjif ')cvcr my be done either by the The- I odolite. Circumferentor, or any other hfrn- ! menty with the fame eafe and exaBnefs. He Tabic it felf is a Parallelogram , containing in length about fourteen inches and a half,& in breadth eleven inches: it is compofed of throe feveral boards, which rrtSiy be taken afundcrjfor eafe and convenience ' carriage. For the binding of thefe three boards faft wl .eii the Table is fet together, there belongcth a joynted frame,, fo contrived jthat it may be taken offhand put on the Table at plea- lure : this frame alfo is to fallen a flicct of paper upon the Table, I wi.tn you arc to deferibe the plot of any field,or other inclofure by I the Table. This frame mull have imon it, neer the inward ledge, i Scalesot equal parts on both fides, for the fpeedy drawing of pa- I rallcl lines upon the paperi and alfd for the lliifting of your paper, w hen one fuecc will not hold your whole work. Unto this Table belongeth a Ruler or Index,containing in lehgth about fixtecn Inches or more, it being full as long as from angle to ansilc of your Table; it ought to be about two inches in bredth,and one third parr of an Inch in thid:nefs.Upon this Ruler or Index two lights mull be placed;one whereof is double in length to the other, the lonacr containing in length about twelve inches, the other fix i on the top of this Ihorter fight is placed a brafs pin,and alfo a thrid and plummet to place your Inllrument horizontal. Through the longer fight muft be made a flit, almoft the whole length thereof, Thefe two fights thus prepared, are to be perpendicularly erefted; upon the Index;in fuch fort,that the wyer on the top of the (hotter fight, and the flit on the longer fight (land precifely over the fidu¬ cial edge of the Index.The fpace or dillanceof thefe two fights one from tlie other, is to be equal to the divided part of the longer fight. Upon the longer fightistobeplaceda Vaneof brafs,to be moved up and down at pleafure, through which a fmall hole is to be made, anfwerable to the flit in the fame fight, and the edge of the Vane. By thefe fights thus placed on the Index there is projeded the GeoinetricarSquare, whofc fide is the divided part of the lon| fight (or the diflance between the two fights.) In tlie middle oT the <^defcri^H(^ of hJlrUmenth L i b. a. the long fight (through tiic whole bicdth thereof there is drawn a I line called the line of Level, dividing the fide of the projeded Square into two equal parts: alfothe fame fide is on this fight divided into a hundred equal parts, which arc numbered up¬ wards and downwards, from the line of Level,by fives and tens to , fifty,on cither fide, which divifions art called the Scale. There is alfo on ti e fame fight another fort of divifionjrcprcfcnt- ing the hypothenufal Lines of the fame Square, as they incrcafe 1 by”UhiKs, and are likcwifc numbered upwards and downwards from the line of Level, from one to twelve, by i, 2,3, &c. fomc- times fignifying loi, 103, 103, &c. thefe divifions fticw how much any hypothenufal or (lope line drawn over the fame Square, exccedcththc dir ciff horizontal line, being the fide of the fame Square On this fight there Is a third fort of divifions, reprefenting the decrees of a Quadrant (or as many as the lame fightiscapapleto receiv'c, which arc about 25) numbered froni the line of Level upward and downward by fives and tens to j 5, which divifions are called the Quadrant. Unto this Inftrument, as unto all others belong thefe nccelTary parts as the Sockct,thcStaffej the Boxi and Needle, &c. |[ Accordingfothis defeription,have Plain Tables formerly been made, but if unto it be added thefe additional parts & alterations (which make it IclTe cumberfome then before) it will be the moft exa£f, abfolute and univerfal Inftrument for a Surveyour that was ever yet invented. Fifft, Let the frame be fo fitted fo the Tables that it may go on cafily, either fide being upwards; fo that as one fide is divided into equal parts(as in the defeription the othet fide mayhave projedfed upon it the 180 degrees of a Semicirclc,from a Center noted in the flipctficies of the Table, which degrees muft be numbered from the left hand towards the ri^ht(whenthc Center is next to you) by fives and tens to 180, ana then beginning again , fee lyo, and fo fucceffively to 360. Thefe degrees thus inferted are of excellent ufeiiTwetor ftormy weather, when you cannot keep a fheet of paper upon your Table, either in refpedt of rain or winde. Alfo thefe degrees will make the Plain "fable to be an abfolute: Theodolite fo that you may work with thefe degrees as if they Were the degrees of a Theodolite. Secondly, Upon the Index or Ruler before fpoken of, (inftcad of ] the fights before defetibed) let there be placed two fights, both of ohe length, and back-fighted; one having a flit below, and a thrid above j and the other, a flit above, and a thrid below, ferving to look backward and forward at pleafure without turning about the Ihftrument, when the Needle is at quiet. The expedition that thefe back-fights will make, will beft appcarc by prafticc; for ufingthefeyoufhallnccd (in going about a field) to plant your Inftrument but at every fecond an gle._Thirdly, IL IB. 2. ($_A dafcri^tion of Injirwnentf, Tliir'JIy, for tl'.e ready caking of heights, and the reducing of ' Hypotlienulal rojdorizontal lines (inftead of the diviiions on the iigiits before mentioned) let there be projefted a Tangent line along tl’C-lide of the Ruler, whofe divifions muft touch the very cdgcla rcof, fo chat a Label or Ruler of Box or Braffe, which is iianged on a pin Ricking infhe fide of one of the back-fights, and having another Imall fight at the end thereof, may move juftly a lon^ tile fide of the Index; thenfehe InRrument ftanding horizon- I tal}if you look through this fmal figlK,and by the Pin on which the Label hangech, mo ting die Label too and fro, till you efpie the mark you look at, then will the labelIh.ew you what Degree of the Tangent line is cut thereb)'. This on line thus projected upon the fide of the Ruler performeth all th.c ufes of thole divided iigl its and IS far bettcr,and leffe cumberfomc tl-en them or a Qua¬ drant (Rich as 1 formerly deferibed in PLi/.ome.ria heeaxikthe degrees arc lar.er. This line of Tangents is projedted on the ’■■'deKfrcm the toot of the fachenmat light, all along the Ruler f:.e'oo'of the nethermoR fight, and up the fide thereof and is rumbred from ito fourtii Book, and is ufed with the Tables of Sines and Loga- rr created of in the third Book, without which Tables, (of fomething equivalent thereunto) this line ot Tangents will be of little life, i aerefore it wi II be con cenient to have upon the Index of your Tab ;■ . o'-- lines ot Artificial Numbers, Sines, and Tangents; by whit..youmay woik any proportion required very fpeedily and c'/. ly, io that if you be deftitute of your Tables, thefe Lines will iutr... .e.'.cly help you. T c re is yet another way by which you may take any aItimde,or reduccHypothenufaltoHorizontallines, only by Vulger Arith- metick, withoui; the help of Tables, by having a line of equal parts divided on the edge of the Index, and another line of the fame equal parts on the Label, by wf'ich lines, and Vulgat Aritrmetick, an Altitude may very well be taken. Now becaufe I intend only to Riew in general theufe of thefe equal parts, I will therefore do it in this place, becaufe I fhall have occafion to fpcake no more thereof hereafter; The ufe there¬ of brietely) is thus. Suppofe that the line A B were forae Hi 11 or Tower, whofe Al¬ titude you require; Randing at C, and looking through the fights in your Label till you efpie the top of the Tower at A, tliefe find- the Label to cut 2 30 of the equal parts; then meafuring the di- ftance from your Ration at C, to the foot of the Tower at B, you finde it to contain djo foot, then to finde the altitude A Bj fay. As 48 ^ defcripion of Inflruments. L i b. a. As 330, the parts cut by the Lab^l, Is to too; So is tf50jthemcafurcddiftancc CB, To 282K, ior the altitude AB. Thcreforcjinultiplytfyoby 100,and divide that Produft by ajo, the Quotient will be zSaijjfor the altitude A B. Now for the reducing of Hypothenufal to Horizontal lineSj ha¬ ving raeafured the Hypothenufal line with your Chain, the pro¬ portion will be; As the equal parts cue on the Label, Are to the equal parts cut on the Index; So is the length of the Hypothenufal line meafured, To the length of the Hypothenufal line required. I thought good to give the Reader a view of the feveral wayes there are to perform thefe conclufions, leaving evc^ man at liber¬ ty to ufc that which he bell liketh,or all if he pleale, for all the Ilics may \ cry well be put upon one Inftrument without any con- fufion of lines; but the way which I fliall chiefly infill upon in the profecuting of thisWork,(hall be by the line of Tangcnts,as being (inmyopinion}thebell of all. Now when I come to Ihew you the ufeof rhislu'.eof Tangents, with the Tables of Sines and Loga¬ rithms in therefolving of TrianglesJ will alfo Ihew you how to perform the famePropofii-ion? by the lines of Artificial Numbers, Sines, and Tangents, and tiiereforel would advife every man to havethefe foneceflary lines upon his Index alfo. Fourthly, Unto this Ini'trument alfo belongeth a Box and Nee¬ dle, which is to be fallned ro tlie fide of the Table hy help of two ferews/o that it may be taken off and put on at plcafure.In the bot¬ tom of this Box mull be placed a Card divided into 360 degrees numbred (if you pleafe) after the ufual manner, from the North Eallwardjbut the Card by which all the Examples in this Book ____were LI B.3. ^ defiripm of Inflnmmr, were framed was numbered from tlie North Weftward by i o,io i jOj &c. to 3dOjContrary to the common cuftomc. I There belongeth alio to this Inftrument a Socket of Brafle to be . ft rewed on the back fide of tlic Table, into which muft be put the licadol the three le^’d Staff, this Staff ought to be joynted in the middle, fo that it may be the more portable For the Socket it may be a plain one, but a Ball and Socket with an cndlefs ferew i is thebeftofall, for by help thereof you may place the Table (or ' any other Inftrument) cither Horizontal!, Verticall, or in any other polition, I Note,that this Inftrument(if made according to thefc diredli- I ons) is themoft abfolute Inftrument for a Surveyor to ufc, i I CHAP. V. l.he defer lotion of an Inftrument n^hich mill per¬ forin the wor\ of the Theodolite or Circum¬ ferentor, hut efpecially of the Peraftor, with great eafe and exaHnefs^and for portMity ex- ceedetbanyof tbe forementioned. S Hc Inftrument confifteth of arccfangled PinHeUpipe- ■ t/o;;,of wliat length,breadth and depth you pleafe, but a con\ enient length will be about 15 or 16 Inches, the breadth about 3 Inches,and the depth about .one Inch. On one ofthc broader Superficies, and in the middle tlicreof, let there be cut out a little narrow Box about an Inch and half broad, and in length according to the length you intend yi^r Needle, in the middle'^af this Box'let there be jdrawn a Meridian line marked with a Flome-deluce for thcNorth point,by this means you may have your Needle longer then in the comon Circumferen¬ tor hdioK deferibed; upon the fame fide of the broader fupcrficies of this Parallelepipedon^Xcx. there be at each end thcrof,cut two other places in which to lay the Sights when they are taken out of the Index for convenience in carriage, fo the Sights being laid therein, will be preferved and lie in a little room. ' On one of the narrower fides of the faid PurAtielepivedon^ within three quarters of an Inch of the extream ends thereof, let there be made to Mortizes about half an Inch fquarc, and about an Inch deep. Then prepare a joynted frame confifting of three Tides, fuchas thefiameof a Plain Table to fold in, that it may lie inalittlp room,one of which fides muft be as long as both the other,fo that being folded together, it lie upon the former PtrMefipedon. The two ihorter ends of this franje(whcn you ufc the infttument in the H . (2^ defcripionof hpuments. L i b. a.; field) muft be put into the two former Mortizes, to a certain a;- ! fiance,and there fixed with two ferews, lo will tlic frame repreli. nt j a rcftanglcd Parallelogram, the three lidcs of this frame nniit oe i dividedin all refpc(fts as the frame of a Plain Table, by diviiieiis i transferred. And if you plcafe,you may liave tlie other fide dt\ ided into twoquadrants,each divided into 510 degrees, from 00 to 90, and from 90 to 00, lb may you by this means make the inilrumcnt capable of performing the ufes of a Card divided into 4 nineties, for the North-eafi, South-wcfi,Nortli-eaft,&: South-cafi Quarters of the Heavens, which degrees in this cafe will be very lar. e, and effebt that manner of work with more exafrnefs then by a Imallir Card. ^ ^ Here note, that in the defeription of the former Infm- ments, that when I mention .a Card , every man is at liberty to have what Card liketh him befi, as a Card ct ; 360 degrees, or iso parts, or into four nineties, accord- 1 ing as Matter M)>»w/advifcth j but in my opinion,a Card | which hath all thefc Divifions is the mofi abfoliuc, aild j this may very cafily be done without confufion, it threr Circles be made concentrick, for one and the fame di vifi- ons will very well ferve for the 3 60 degrees, and the tour nineties, and the innermoft for the 120 parts, and to avoid mifiakes it will not be amifs to have themiddlemoficon- ccntrick Circle to be coloured with feme dittinbl tranfpa- rcnuolour, or numbred with red figures. Unto this Inftrument there alfobelongeth an Index and Sights, and if you make two Mortizes in theextream ends of the Idipipedoni wherein to fet the fightSjand caufc a Tangent line to be drawn by the edge of the ParaHelepipedor:, tlicinfirumcnt (with a Label) will be very commodious for the taking of heights. This Inftrument being thus madc,and let upon a ftaffwill be ve¬ ry convenient for all purpofes. ^ CHAP. VI. The definition of an Infirmmt called a Crofs, jHis Inftrument is of good ufe in fmall indofurcs of manyfidcs, the ufewhcrcoffhallbcniewcdhcrc' ■ after in the fourth Book. It is only two Rulers of W*ood, in length about 14 Inches, crotfing one the other in the midft at right , angles, and having at each end of both the Rulers back-fights, which ferves only to fet out right angles in the field it : felf. _ chap. |L I B.a. defcription of InflrumenUc ! CHAP. VII. Of Chains, the feverd forts thereof aircs tl'trc arc clivers ferts , as namely. Foot Chaiiib, cadi link containinga l oot or 12 Inches,and Perch will contain 16i Links or Feet,anfweringto theStatute denomination. I Some Ci ains 1 a\ e each Pole divided into ten equal jvirts, and thcle arc called Decimal Chains, and this grofs divill- on may he const nicin in fome praclifes. The Chain now nfed,and molt eficcmcd amongft Surveyors, arc cipccially two, namely, that generally ufed by Maftcr Rathloui which hath every I’erch divi J.cd into ico Links, and that of Ma¬ iler CT/tA'cr whicii iiath four Poles divided into icoLinks, fothat each Link of Mailer Chain is as long as four of Mailer KA.Ido>,. 5 . ' Now becatile thefe Chains arc moll ellccmed of and ufed by 1 Sure cyorsd wil the rforc make a general dcfcription of them both, kavine every man at liberty to take his choife. i Rathbom ChsAn. T He Chain which Mr. Rahlorn ordinarily ufed (as himfelf iaith) contained in lengh two Statute Poles or Perches, each Pole contatning in length 16I feet, which is ijiS Inches, then each Pole was divided intoio equal part called Pr/wes,every of which contained in length ig| Inches; asain, every of thole e/mns was lub-divided into toother equal parts called S(coi.ih\o thatevery of thefe Seconds contained in le-ngth C’Inch,fo thatthe whole Pole, Perch,Unite, or Commence¬ ment (as he calleth it) was div ided into 130 equal parts or Links called Sccuf.i's. Tl.e Chain (or one Pole tl.crcof) being thus dividedjat tliecnd of ev erv 50 Links or 1 alf Pole, let a large Curtain ring be nrltned folViall you have in a whole Chain oltvvo Perches long, three of thefe Rins^, the middle moll being the divifion of the two Poles. Then at the end of every Pr/wcjthat is at the end of every ten Links let a I'mrller Curtaine Ring be fallcned. By this dillindlion of Rings, th.e Chain is divided into thefe three denominations Unites ?,//rr>,aud .?(Y«//v'swhofe Charadters are thefe• • •,fo that it you would exiueflc qs Uni'.es^ 8 ytimes and 7 Scruff's, they arc thus to be written, 4087, by which you may perceive chat thofc Figures which have no pricks over them are /'/,;.vsor/«/cw/.s, andthe fitiurc under the firfl point Primes, and under the next Seconds-, fo alfo, ilircc ILius , feven Primes, and two 5 c>ro/;'/y,will (land th.us, 378. I H 2 Bcfides ^ defcripioH of InftrHmeras. L i b. a. j Befides thefe divifions Maftcr Rathborn for his own ufc/ewcd at i the end of every two MWranda lialfe (which is a i]uarcer ot a I PoIe)a (mail red clothjan J at every feven Primes and a lialfe(bcing I threeqiiartcrsofaPolc)the like of yellow: or other difcernable colour, which much helped him in the ready reckoning of the feveral Rings upon the Chain, remembring this Rule: That if it be thencxtRingihorcottheRed,itistwo if the next over, three, if the next iTiort of the yellow feven Primes ifthe next over cightj if the next fliort of the great halfe Ring it is four, the next over fix: and if the next (hortofthe middle great Ring, it is nine, and if the next over one. 4 [ But here is to be noted, that if you ufe this diftinflion by co¬ lours, you muft alwayes work with one end of the Chain from you. This Chain being thus divided and marked, you have every whole Pole equal to ten Primes^or too every three quatters of a Pole equal to feven Primes and a half, or 7, SecorJsi every half Pole equal to fi ve/’r/wf.f, or ',0 Seconds: andlaftly, euery quarter of a Pole equal to two Primes and a halfe, or 25 Seconds. And here is to be noted,that in the ordinary ufe of this Chain,for meafurin^ and platting, you need take notice only of Unites and Primes, which is exaft enough for ordinary ufe, but in cafe that fe- paration or divifion ofLands into feveral parts, you may make ufe oi Seconds. Of Mafter Gmters Chain. A 5 every Pole ofMafter^at/j^orw Chain was divided into the 1 /» Links, fo Mafter Gunters whole Chain (which is alwayes 1 made to contain four Poles) is divided into 100 Links one of ; thefe Links bein? four times the length of the other. Now if this I Ci.ainbe madeaccjrdingtotheStatute, each Perch to contain 167 Feet, tnen each Link of this Chain will contain 7 Inches, and ! 74Tofan Inch, and the whole Chain 72^ Inches, or 68 Foot. j In meafuring with this Chain, you arc to take notice only a ' Chains and Links, as faying fuch a line mcafured Iw the Chain j contains 72 Chain,45 Links, which you may expreftc more brief- | lythus, 7 3^8, and thefe arc all the Denominations which are neceflary to be taken notice ofinSutveying of Land. For the ready counting of the Links of this Chain, there ought robe thefe diftin(ftionsnameIy3ln the middle thereof, which is at orbrafte like a Rhombus, fo is the whole Cliain (by this plate) divided into two equal parts. jL I B.3. ^ defcrqtion of Inflrumenu, 5 g I Secondly) Let each of thefc two parts be divided into two I othel: equal parts by fm.'iller Rings or Circular plates of braflc) I fofliall the whole Chain be divided into four equal Parts or ' Perches, each Perch containing 25 Links. ^ Thirdly, At every ten Linkslctbe tallencd a lelTer Ring then i the former, or elle a Plate of fome other falliion, as a Semicircle i or the like. And laftly, at every lift link (if you pleafe) may be I faftencdothcrmarks,fo by this means you fliall nloft eafily and ; exaftly count the Links of your Chain without any trouble. The ; Chain being thus diftinguillied, it mattereth not which end ! thereof be carried forward , becaufe the notes of diftinftion • j proceed alike on both fides from the middle of the Chain. i ^ Herenotc, thatin all the examples in this Bock, the lines j are fuppofed to be meafured by this fdur Pole Chain of I Mailer it being the bell of any other: the manner ! how to cad up the content of any plot mcalurcd therewith, i iliall be hereafter taught in its due place. i I Cautions to be obferved in the ofe of any Chain. I N meafuring a large dillance with your Chain, you may cal'ually miliakeormilTcaChainortwo in keeping your ac- counnt, fri'm whence w ill enfuc a conliderablc errour: Alfo in meafuring of diftances(whenyougonotalongby a hedge fide) you can hardly keepe your Inftrument, Chain and Marke, irt- right line, which if you do not, muft ncceflanly make your mea- fured diftance greater then in reality it is. For the avoyding of either of thefe miflakes,you ought to pro\ ide ten fmall flicks or ar¬ rows, which let him that leadeth the Chain carry in his hand be fore, and at the end of every Chaine, flick on of thefc Arrows into the ground, which let him that followeth the Chain take up, fo going on till the whole number of Arrows be fpent and then you may conclude that you have m.eafured ten Chains without any further trouble, and thefe ten Chains (if the diftance you arc to mcafurc be large) you may call a Change, and fo you may denominate every large diftance by Changes, Chains and Links. Or you may at the end of every ten Chains fet up another kind of flick, by which (Handing at the Inftrument) you may fee whe- tl cr vour eye, the flick, and the Mark to which you are to mca- f' i-cbein a right line or not, and accordingly guide'thofe that car¬ ry the Chain, with the more exaftnefle to direft it to the Mark intended. How 54 defcripion of InftrumCrJs. L i b. a- ! How to reduce any number of Chains and j jnto Feet. I N tl'.c pradticc of many Geometrical Conclufions, as in the ta¬ king of' Heights and Diftanccs, htreafeer taugiu, it it. rcquilite togivcyourmeaiurc (infuch cafes) in Feet or Yards, and not in Poles or Perches; yet bccaufc yciir Ciiain is tiie moll necclFiry Inftrumcnt to mcalurc withal,! thougiit it contcnicncin tliisplacc toflicwyouhowtoreduccanynundierof Chains and Links into Feet, which is thus. Multiply your number of Chains and Links together as one whole number, by 66, cutting off from the produ^l the two laid figures towards the right hand, fo fliall the reft of the prodiid be Feet, and th.c two figures cut off fliall be hundred parts of a Foot. i EXAMPLE. Let it be required to know how many Feet are contained in five Chains, 3 2 Links. Firft, Set down your five Chains,3 2 Links as is before taught, and as you fee in tlie firft Examplc,with a Cemma between the Chains and Links, then multiplying this five Chains, 3a Links by 66,theprodu6lwill be 35112, from which, cutoff the two laft figures toward the right hand svitli a Comma,tl.cn vvil the number be 33 1,12, which is 351 Feet and -Vi parts of a foot, and fo many Feetare contained in five Chains, 32 Links. , Example /. £.v.w;y;/f 11, 5^32 9^% 66 66 3192 5430 3192 5430 3SI5I2 557=30 But let the number of Chains be what they will, if the number' of Links be lefs then io,as in the fccond Example it is nine Chains five Linksjyou muft place a Cypher before the five Links as there you fee,and then multiplying that number ('c/i.9.05.) by 66, tlie produift wilbe 597J0 from which taking the two Lll figures there will remain j 97 Feet, and ~ parts of a Foot. Tiie like may be done for any other number of Chains and Links whatfoever. According to thefe Examples is made thcTable following,whi ch ftieweth how many Feet arc contained in any number of Chains Links from five Links to eight Chains, for every lift Link,which is fufficient for ordinary ufe, by which Table you may Ice tliar in fix Chains 40 Links, is contained 422 Feet, and -ri^of a Foot, Alio in five Chains 5s Links is contained 360 Feet .-and -,!) parts of a Foot: and fo of any other. _____ ; LI B.3. defcription of Inflrnrkents, 55 i A Table fliewing how many Feet, and parts of j a Toot arc contained in any number cf Chains and Links ' ! between five Links and eight Chains. 6,60 9i9°\ 15 ) 10 ! 66,oo| 96 ) 30 , 7- 75 ) 90 | 79 )’-o J I 5 82,so ip)8o 85,80^ 15.10 8 p,io' 26,40! 5»2,40j i9)7o; 95.70; 53,00; sr9,oo 36,5^1102,30 39 ) 6 o!io 5 , 6 o 4 :,Sioj(o 3 ,po 46.10 112,20 49,50 115,50 52,8o'ii8,8o! 56.10 122,10] 59,40115,40 62,704 28,701 4 ! 5 132,00 ip8.,oo 264,00 330,oo'3p6,oo'462,oo 135,30 201,30 167,301333,301395,30465,30 138,60 204,60 270,6 o'336,6o‘40236o'468,6o 141,90 207,90 273,90 339390 405,9o!47i590 145,20 211,2C 277,20,343.20;409,2o!475,20 148,50214,50 28o,5oi346,5o'4i2,5o‘478,5o i 5 i,. 502 I 7 , 8 o 283 , 8 o' 349 , 8 o 4 I 5 , 8 o' 48 i,So 155,10221,10287,10 353,io4i9,io'4S5,10 158,40,224,40 290.40356,40422,401488,40 161,70227,70 293,70'359,70 425,70491,70 165,00 231,00 297,00 363,00 429,00495,00 234.30 3 oo,3o!366,30 432,301498,30 237,60303,60369,60 435,6o|50i,6o '-40,90 306,5o'372,90|438,90 504,90 44,20 210,20 376,20 442,20 508,20 i6S,3o| 171,6; 7-h90 7 «,ic 181,50 184,80 188,10 1244,20310,20' , 47,503155501 250,80:316,80 |254,ioj320,io 191,40257,40323,40 379 ,50I445,501511,50 382,80448,80514,80 386,10452,10518,10 389,40^4555405^1,40 i94, 7o;26o,7o‘326,7o|392,7c!458,7c'524,79 CHAP. VIII. Of the Protraftor. Protractor is an inftrumentby whicli you may protraft K m or lay down upon paper or otherwife, the true fyme- try or proportion of any field, having made obferva- tion of tW fides and angles thereof by fomc of the In- Itremcnts before dcfcribcd. Thisinftrument cofifteth of two parts, the one is a Semicircle divided into degrees, as is the frairic of the Table, and the other is a Seale divided into equal parts, the Semi¬ circle being to lay down the angles, and the Scale to plot the fides. This Inllrnment ought to be made of a piece ot thin brafs well po- lifiicd, the ed',cs thereof being wry fmooth,and the Scale thereof, ninicly,ti’criglitanElcd Parallelogram, or long fquare containing in lens til l i om Ato Babcut four Inches and three quarters, and in breadth, irom A toC about oncanda half. Let the two ends of the Scale,n.imelv. tiie fides, A C and B D be divided intocqual parts of i6cr ;o in anincli, and letthefide CD be divided according to a Scale cf I cor 12 in an Inch. 5 « defirfftmoflttfirmentr. Lib. 3.1 I The Scale being thuis divided, on the middle of the line A B, j 3 SatH, deferibe the Semicircle EGF, which divide into two I C^adrants in the point G, by help of the perpendicular HG: then ; divide each of thofe Quadrants into 90 equal parts called * degrees, fo fhall the whole Semicircle contain 180 degrees, I whichmuftbenumberedby 10,20,30,40, &c. to 180, from E by ] G toF, and the fame way alfo from 180 to 3 60, as you fee done i in the Figure, the numbers of the firft Semicircle from 00 to 180 . beingfortheEaftftdeofthe Protradlor, and the other numbers : from 180 to 360 for the Weft fide. Now you are to note, that the line AB always reprefenteth the Meridian line, and is fomtimes noted with the letters S and N for South and North, but then it is necclTary that the Protraftor be divided on either fide the plate, which this double numbering avoideth,fqr the line AB being taken for the Meridian in general, the Semicircle oftheProtraftor may be turned anyway( either upward or downward)andfo one Semicircle being divided will I be fufficicnt;yet if any man be defirous, he may have it made I according to his own fancie, but this manner of ,.numbcring(in my opinion ) is the bell, it being mod asrccable to your Infiru- ments. |Li B. 2 , defcripion of Injlpmem, IHT The Protraftor here deferibed is that which is commonly j j made and ufed, but this (thedefeription whcl-eof followeth) in ) my judgment was the beft contrivance I have I’cen, the firftj ! whereof being made for my loving friend ivitl.Forfter now in/re-j J Und which is asiolloweth. I A B The Prptraftor confiftethof a piece of thin bralTc in forme ' of a re(ftan:led Parallelogram, upon which at the diftance of haU’e an inch draw the line E F parallel to C D, which line di¬ vide into two equal rpatts in the point G for the center, from which center let the fides E A , A B, and B F be divided by lines ifluing from the center G into i8o degrees, and nUmbred by 10, io, 30,40, &c. to 180, and back again from »80 to 360, in aU refpefts as thefemicirclein the other Protraftor was numbred. Now becaufe that in protrafting (many times)the parallels will fall off of your paper or parchment fo that you mult addea piece thereto for the prefent, there is in this Protraftor, thePa- I rallelogram IM O L cut quite out, that you may fe^our worke through, and either fide of the narrow ilip of bralie which is cut out, namely 1 L and M O mull be divided into the fame parts with the fides E A and B F, by a ruler laid from fide to fidcj and thefe two fides mull be numbredby 10, 20, 30, &c. as farre as they will extend. To this Protraftor (if you plcafe ) you may have added upon thccdgeCDa Scale of any equal part asof lO, 12, 20, 24, or 3c in an inch,this Protraflor thus made, is very convenient for ufe and much cxcecdeth the other before deferibed. To ufc with your Protraftor in protrjfting, you mull provide a fine needle, put into a piece of Box or Ivory neatly turned,this will ferveto fix in your center, note your degrees, and for otherufes in drawing your Ploq ahd is called a Pro- ttaftingpin. 1 CHAP, CHAP. IX; ^/Scales. B Or the ready layin;' down of Lines and Argics according to any aiTigned quanti¬ ty, you nuift provide cii\CTs Scales. The Scales now ordinari¬ ly I'fcd by Surveyors, arc principally two: Firlijof equal parts, for the protradf ing of Lines; and Seconilly, of Chords, for the protrading of An¬ gles. Unto thefe may DC added. Thirdly, a Diagonall Seal , which is (indeed) no ether then a Seale of equal! parts more Icrupuloufly divided. If you defire a conve¬ nient Seale, let it be made in this manner, to contain in Itn -th : about 8 cr 9 Inches,; and in breadth one , Inch and a quarter: ■ on ore fide thereof j let be placed divers Scales, as of ir, u, ' II, 1 6,20,2q, and 30 in an Inch, C Here is to be no¬ ted , that wi en I fay a Seale of rj in ah Inch,you arc to undtrftand a part of a line divi¬ ded into loenuall parts, ji ofwhich pares would make an |Ll B.3. dejcri^tionof Infirumentf, j an Incli, and the like is to be underftood of any other number • ofcqual parts whatfoever. i ' On the fame fide of the Ruler let be placed a line of Chords ex^ i tended up to go , aud numbred as you fee in the figure by lo, lo, ! 30j &c. to po. This Scale will be of good ufe for many purpofesjos ! to divide the circumference of a circle, and to protraft angles in I lome cafesbetter then the Protradlor. \ ' On the other fide of the Ruler let be drawn a Diagonal Scalejof I 1 o in an Inch, which will be an excellent Scale for large Plots, out { of which you may very well take the hundred part ol an Inch, and I this Scale will agree with your four Pole Chain exceeding wclli for as your whole Cham contains loo Links, foeach Inch of this ! Scale contains ico parts,fo that out of it you may take any number ■ mcafuredby vourChain,toaLink,and lay it down upon paper. You may alfo have half an Inch divided into too parts, which i S calc will be of good ufe alfo to lay down a fmallcr Plot; j Gil* Rut if you would have your Scales to be anfwcrablc to your j Cliain,and to agree with the divifions thereof, then you arc to take j notice that o'! TioV faf) ij Ini' j 112 |Poleinaninch]ix| i A Scalcof.' 16 Imuft be a i6 > in 4 inches:;4 '201 Scale of laoj or jj Hi [50J |H L30J L 7 t. Then will each of thofe parts reprefent a Chain, and if you fub- dividc the lad fingle part into ten,thofe parts fliall reprefent links. Thcfe Scales, alfo the lines of numbers Signesand Tangents, and reducing Scale hereafter mentioned , may very conveniently be placed on the Index of the plain Table. To ufe with this Scalc,you muft provide a pair of neat Compaf- fesofBrafs, with ftcel points, filed vcryfmall, and alfo a neat pair of Cl mpalfcs with three points,& Screws to alter the points, lb that you may draw lines of Circles with black lead, flr any co^ , loured Inke, which will be very necelfary and convenient in beau¬ tifying of your Plots after Protradlion. I2 char: 6o Adefcri^tionof hfrmenn. C U \ V Y Of a Field Brn'k: iT wil be fufficicnt in this plac'c only to dcfcribc the man- 'nerhmva Field-Book ought to be ruled ; Let the Book 'contain any quantity of paper, more or lefTe, andin what volume you pleafc, but a long folio is the belt. Let it be ruled,towards the left Margine of every page,with fi ve incs in red inkc, fo (hall you have four Columns,in the firft wliercof you rtiuft note down the degrees cut either by the Index on the frame of the Table, of elfe by the Needle on the Card of the Circumferentor, or degrees of the Peraftof, at every angle you obferve, and the fccond Column is to note the minute? or parts of a Degree, for you arc to note i that every degree on the frame ol the Tabic, or in the Card of the Circumferentor, is fuppofed to be divided in¬ to do other parts called Minutes, which cannot be expreffedby rcafonof thcfmabcfsof thelnftruments, and therefore mull on¬ ly be eftimated, yet if your Inflrument be large enough, you may have each degree divided into three equal parts, folhall every part contain ao minutes. The other two Columns fervetonote down the lengths meafured by your Chain, as the Chains and Links. Now fuppofe that making any obfervation in the Field cither with the Degrees on thefrariieof theTablc, or with the Circum- ferentofi and that obferving any angle, (as is hereafter taught) you findc the Index of the Plain Table, ortheNeedleinthcCir- cumfetentor, to cut 326 degrees j 45 minutes, thefejao degrees mull be fet down in the firft Column of your Field-Book, and the 45 minutes in the fccond Column, as you fee licrc done. Alfo if youmcafure any length in the Field with your Chain, as fup- —__pofe :Lib.2. A defcnpttdn of Injirments 6i i pofelomedirtancemcafurccltdcontain id Chains, 87 Links, the | ! idChainshiurthefccinthc thirdCoIi;rnn, aniithe87 Links in: the fourth Coluihn, undcrthci'rrcfpcftixcTJtlcs, as you fee here done. ■ j ; gj" hut if you bfe a Card of four nineties according to yi.il\stK" tant wherein they are at large/o that thole who delircto make a further Icrutiny intoTV/gwowe- try, may perufe the forementioned Authours. In this BooJi 1 have only infilled upon fuch Ca¬ fes as may come in ufe in Surveying, and there¬ fore have omitted divers,yet thole which I have infilled on, are performed both by the Tables^ following in this Bool\i and allb by th&Lines of\ ArtificiiSNumbers,Sines sindTangents before fpoken of in the defeription of the Index of the Plain Table in the lad Bool{. I TRii 6'5 i TRIGONOMETR[E. CHAP. L 7 be Ex^lamtio/i and Vfe of the f die of Si tiES. Efcrc I ccmc to tV.c nicnfuration of ; Triangles, it will be ncceffary to explain J and lliewthc ufeof the Tables of ^/V.« ' and Loga/itlms following, by which : Tables the fides or angles of right lined ; Triangles may be readily and exaftly i meafured, fo that in any plain Triangle, ; if there be any three parts thereof gi¬ ven , a fourth may be eafily difeove- , red. The Table of Sines confifteth of two Rows or Columns, the j firftwhereoffheweth the Degrees and Minutes of the Qnadrant, , having over the head thcreol thefe two letters, D. M. ftanding | for Degrees and Minutes: Inxhe fccond Column is the Artifici¬ al! an fwering to every Degree and lO''' Minute of the Qua¬ drant,having the word Sine over the head thereof. The ufe of this Table will appear by the following Propofitions, PROP. I. Degree and Minute king given, to find the Sine thereof pirft. Seek the Degree and Minute in the firft Column of the Table, under D. M. and right againft it, in the next Column towards the right liand,under the word Sine, you fliall have your degree. K EX~ Trigonometrie. Lib. HI. EXAMPLE. I. Suppofeit were required to find the Sine of 20 degrees, Firft, youmuft feekio in the firft Column of tlie Table under D.M. and right again 20 in the feeond Column under the word Si/ie^yon iTiall find 95534052, which is the Sine of 20 degrees. In the fame manner you fliall find the Sine of 50 degrees to be 9,884254, and the Sine of 76 degrees to be 9,986904. EXAMPLE. II. Let it be required to find the Sine ot 40 degrees, jo minutes. Firft, you muft find 40, 30 (which is 40 degrees 30 minntes) in the firft Column, under the letters D. M. andagainftityou fliall find9,812544,which is the Sine of 40 dcgrccs,30 minutes. Alfothe Sine of 6 i degrees 10 minutes, will be found to be 9,946<504,and theSincof86 degrees 30 minutes will be9,999189, and in this manner may you find the artificial Sine of any number of degrees and minute* exprefled in the Table. PROP. II. Any Sine being given , to find the number of de¬ example. T Et 9,866470 be a Sine given, and let it be required to find the ■^degree and minute of the Quadrant anfwering thereunto. Firft, feek in the feeond Column amongft the Sines for 9,866470 , and againftit (on thclefthand) you fliall find 47 degrees 20 minutes, which is the arch of the Quadrant anfwering thereunto. Again, Letit be required to find the arch anlwering tothis Sine9,821264, having found 9,821264 in the feeond Column un¬ der the word. Sine , againft it you fliall find 41 degrees 30 min. aod that is the arch or degree anlwering thereunto. |[ But in cafe you have a number given which you cannot ex- adly find in the Table,you muft then in dead thereof,take the nee- reftin the Tabic. As ifyour number given were 9,675 859, ifyou look in the Table for this number it c.annot be found there, but the neereft thereunto is 9,676328, which is the Sine of 28 degrees 30 minutes,which you muft take inftead thereof. THE |l. B.11I. T rigmometrie <5? The Table of Sines. h . At. Sines D. M. Sines 1 D . M . Sines o o 0,OOOOOC 8 0 S’,>43 555 16 0 9,440338 10 7,463726 10 9,152451 lo 9,4447:0 20 7.764754 20 1 9,161164 10 9,449054 5 o; 7'94'’842 3o| 9,169702 30 9,45334: 40 8,065776 40 1 9,178072 40 9,457584 5° 8,162681 50 9,186280 50 9,461782 I 8,241855 9 0 9,194332 >7 0 9,465935 20 8,308794 9,202234 10 9,460446 20 8,366777 20 j 9,209972 10 9,474“ 5 ;o 8yf'79'9| 30 9,217609 9,478142 40 8,463665 9,225092 40 9,482128 50 8,505043 50 91:3:444 50 9,486075 2 8,54:819 10 0 19.:39670 18 0 9,489981 10 8,577566 lo 1 19.146795 lo 9,493851 8,6097 34 20 1 9,’-5 376 i 20 9,49768* ;o ; V'S'679; 30 1 9,:6 c 633 :o 9,501476 40 8,667685,; 40I 9,:67395 401 9.505*34 501 3998 50 9,:74049 50 9,505955 3 0 i8.7>S8oc] II 0 ] 9,280599 >9 9,51*64: 10 8,74'-:59 10 9,:87048 10 9,516194 20 8,764511 20,9>*95399 20 9,519911 lO ! 1^8)675 30; 9,:99655 30 9.5:3495 AO 1 18,803652 40 19,305819 40 9,5:7046. 501 [8,825130 50 ‘9,511899 1 9,3 >7879 5° 9,530565 4 0 8,"43584: 11 0 20 Oi 9’534 o 5: 10 1 8,861283] 10 9,323780 10 9,537507 1 8,878:85 2° 9,329599 20 9,54°93». 1 8,89464:1 30 [9,3353371 30 9,54431s 1 4*^ ^,710404 40 9,340996 4°, 9,547689 5= 8,92.007 5° 9,34657s 5° 9,551024 1 S 8,94°:96; 13 0 9,35208s 2l 9.554329 10] ' ,954499 10 9,3575:4 I0| 9,557606 20 8,968:47 10 9,362889 20 9,560855 18,98157; 30 0,368185 50 9)564075 40 8,994497 40 9,3734:4 40 9,567269 i 50] 9,007044] 50 9,378577 501 9,570435 i ^ 0 9.6«9J35 14 0 9,383675 22 9,573575 lo 9,031089 lo 19,388711 io| 9,576689 20 9,042625 20 19,393685 20 1 9)579777 i 30 9;053859 50 1 0.30660c 30 9,582840 1 40 9,064806 40 9’4o3455 40 1 9,585877 1 50 9,075480 5° 19,408254 50 9,588890 I 7 0 10 9,065694 9,'"960621 1 15 0 10 9,412996: 9,417684 *5 10 19,591878 19,594*4: 20 1 9.10599:] 1 20 9.4::3I7 20 9,597783 30 9,115696, 1 30 0.426809 30 9,600700 40 9,125157 40] 9,43*4*9 40 9.603594 1 50 9.‘!447'’ 1 50 ?.4359>8 1 i 5° 9,606465 Ki 68 The Table of Sines. H ‘S 6 ^7 28 * 9 . 30 51 0 9,609313 10 9,612148 20 9,614944 30 9,617727 40 9,620488 50 9,623225, 9,625948 10 9,628647 20 9,631326 30 9,633984 40 9,636625 50 9,63924’- 0 9,641842 lo 9,644423! 20 9,646984 30 9,649527 40 9,652052 50 9,654558 9,657047 20 9/5595*7 20 9,661970 50 9,664406 40 9,666824 50 9,669225 0 9,671609 10 95673977 20 9,676328 5°| 19,678663 401 9,680982 50 9,683284 0 9,68517* 10 9,687842 20 9,690098 30 9,692339 40 95694564 9,696774 0 9,698970 10 9,70*151 20 9,7033*7 30, 9,705469 40 9,707606 50 9,709730 0 S»,711839 lo 9,7*3935 20] 9,7*6013 30 9,718085 40 9,720140 5° 9,722181' D. Sines 32 0 9,724210 10 9,726225 20 9,728227 30 9,730216 40 9,732193 50 33 0 9,736109 10 9,738048 20 9->n99T) 30 9,741889 40 9,743792 50 9,745683 34 0 9,747562 10 9,749429 20 9,751284 30 9.753128 40 9,754960 so 9,75678* 35 0 9,35859* 10 9,760390 26 9,762177 30 9,763954 40 9.765720 50 9,767474 36 0 9,769219 101 9,770952 20 9.772675 301 9,774388 40 9,776090 So 9,77778* 37 ° 9,779463 10 9,781134 20 9,782796 30 I 9.784447 40 9,786088 5° 9,787720 38 0 9,789342 xo| 9,790954 20 9>792S57 30 1 40 9,794*49 9,795733 so 9,797307 39 0 [9,79887: lo 1 9.800427 201 9,80197; 30 9,803510 40 J 9,805038 5°^ 9,806557! 9,808067 9,809569 9,8icc6i 9,812544 9,814019 9,815485 9,816943 9,818392 9,819832 9,821264 9,822688 9,824104 9,8^511 9,826910 9,828301 9,829683 9,831058 9,832425 9,8357«3, 9,835154 9,85^477 9,837812 9 , 839 I 4 < 5 ; 9,8404591 9,841771 9,845079' 9,844372 9,845662 9,846944 9,848218 9,849485 9,850745 9,85*997 9,853242 9,954480 9,8557*° 9,85<5954 9,858150 9,859360 9,860562 9,861757! 9,862546 9,864117 9,865302 9,866470 9,867651 9,868785 9,869933 The Table of Sines. D, /W. Sines D. Sines 48 0 9,871075 56 0 9,918574 10 9,872208 10 9,9194*4 ao 9,873335 20 9,916268 30' 9,874456 30 9,9*1107 40 9,87 5 57 • 40 9,911940 so 9,876678 5^ 9,912768 49 ° 9,877780 57 . ® 9,9*3591 10 91878875 10 9,9*4409 20 9,87996: 20 9,915122 30 9,881045 30 9,926029 40 9,88212! 40 9,926831 5° [^883191 50 9,927628 50 0 9,884254 58 0 9,928420 10 9,885511 10 9,9*9*07 20 9,886561 20 9,929989 50 9,887406 30 9,930766 40 9,888444 40 9,93i537\ SO 9,889476 50 9,932304 SI 0 9,890505 59 ° 9,933066 10 9,891521 10 9,933822 20 9,892556 20 9>934574 30 9,89354'] 30 9,935320 40 9,894546 40 9,93606: 5° 9,89554* 9,936799 S* 0 9,89653* 60 0 9,937531 101 9,897516 10 9,938*57 ' 201 ; 9,8984941 20 9,938980 ■ 30 9,8994671 30 9,939697 9,900455 i 40 9,940409 ! 5c 9,90'39I 5°! 9,941116 1 S3 o' 19,902349 61 0 9,941819 1 10 9,903*98 10 9,941517 1 9,904*41, 20 9.943210 30 9,905179 30 9,943898 i 4® 9,906111 40 9,944581 5°i 9,907037 j 501 9,945261 ; 54 0 9.907958' 62 0 9,945935 9,908873 10 9,946604 20 9,909782 20' 9,947269 30, 9.910686 30 9,9479*9 4oi 9,911584 40 9,948584 ! 9,91*477 501 9,949*35 1 SS ® 9,913364 63 o| 9,94988, 10 9,914*46 10 9,950522 9,915123 20 9.951159 1 9,915994 30 9,951791 1 40 9,916859 4° 9,952419 50 9J517719 50 9,95304* I). M.. _ Sines '54 o 9>953^6 o ' 10 9>S> 54^74 2o!P>P5488? 9>9S5483 9>956o88 9>95tf684 95957^7^ 9>95786i 91958445 9>959oz3 91959596 91960165 9,960730 9,961290 9,961846 9,962398 9,962945 9,963488 9,964026 9,964560 9,965090 9,965615 9,966136 9,966653 9,967166 ®,967674 9,968171 •S',968678 '*>1969173 9,969665 9,970152 9,97°i534 9,97>iii 9>97M88 9,97*858 9,972524 9,97*986 *'•973443 91973897 9,974;4<* 9>97479* 9,975283 .9,975670 19,976103 ■9,976532 9,977956 9>977377 9,977794 70 Trigonometrie. LibIII. ■ The Table of Sines. D. M. Sinti |D. yi/ ■ 5/W 75. // } Sinet T - 9,9782od 78 0 9,990404 8"4~^ 1 9,997614 10 9,97861 1 10 9,990671 ’,9,997732 20 9,979019 20 ' ‘;>99°934 .'9,99787? 30 9>9794i5'i 30 9,991193 30 i 9,997996 40 9,979816; 40 9>99M4' 4019.998106 50 9,980208; 50 9,9916,;9 50 : 9,99813 2 73 ° 9,980596, 79 0 9.99-'947 85 0 j 9,998344 10 9,980980! 10 9,992190 10 19,998453 20 9,981361! 20 9.992430 20 1 9.998558 3° 9^981737 • 30 9,992666 3019,998659 40 9,982109; 40 93992898 4019,998757 50 9,982477! 9.993127 50 9. .'98851 74 0 9,982842 n 3o 0 9.993351 86 0 9,598941 10 9,983202 10 9.993572 10 ' 9,999927 20 9^983558 20 9.993789 20 9.999110 30 9,983910 3° 9,994003 30 9,999189 40 9,984259 40 9,994212 , 40 9.999265 50 9,984603 50 9.994418 5° 9.999336 75 0 9.984943 81 0 9.994620 87 0 9.999404 10 9,985280 10 9.994818 1019,999469 1 20 9,985613 20 9.995012 20 9,999529 30 9,985942 30 9.995205 30 9,999586 40 9,986266 40 9.995390 40 9.999640 50 9,986587 0 50 9.995573 50 9,999689 76 0 9,98690.1 32 0 9.995753 88 0 9.999735 10 9,987217 10 9,995918 10 9,999778 20 9,987526 20 p,996ioo 20 9,999816 . 30 1 9,987832 30 9^196269 30 9.999851 • 40 9,988133 40 9.996433 40 9.999882 50 9>98843o 50 9.996594 50 9,999910 77 . 0 9,988724 83 0 9.996751 80 ° 9.999934 10 9,989014 10 9,996904 ^ 10 9.999954 20 9.989299 20 9.997053 20 9 > 99997 ‘ 30 9,989581 30 9.997199 30 9.999983 40 9,989860 40 9.997341 40 9.999993 50 9,990134 50 9,99748© 50 9.999998 c AP. 1 Lib.111. T rigonometrie 7 * ; CHAP. IL j Tbe Explanation and Vfe of tbeTahle | i ! 0/Logarithms. I F'/ P A i.v -ff Hc Tabic of Logarithms following confiftcth of two Rows or Columns, the firft of which (namely that towards the left hand, having the word ATaw, ! at the head thereof) containcthall abfolutc num- I increafing by a Unite in continual proportion from 1, to looo. In the otlicr Column is placed the Logarithms of thofc abfolutc number?; which Logarithms arc numbers fo fitted to proportio¬ nal iu:inhers,that thcrafelves retain equal differences. BythisTable, tlacLogarithmcof any abfolutc number under 10.0, may be readily found: Orifany Logarirhme, whofeabfo- !;;tc number cxcceJeth not looo, be given, thisTable will plainly difeoverwhat abfolutc number anfwcrcth thereunto. Theufeof ; this Tabic will appear by the Propofitions following. PROP. I. A number being given t to find the Logarhhme thereof, L Etitbe requiredto find the Logarithm of 223, Firft, feck 223 in the firft Column of the Table under the word Num. and againft it in the fecond Column you lliall find 2,348305, which is the Logarithm thereof. Alfo, Let it tic required to find the Logarithm of 629, if you fcck 619 in the firft Column, againft it in the fccond you (hall find ICCK 019 mu ... ' 2,798651, which IS the Logarithm thereof. PROP. II. i A Logarithme beinggiveny bow to find theahfo- \ lute number thereunto belonging. L Et 1,731589 bea Logarithm given, whofcabfolutc number you require: you muft firft feek this number in the fccond Co- ' lumn of tlie Table,under the word Low, againft which you lhall • find 539, which is the abfolutc numW anfwcring to that Loga- ! rithme. i But in this Table, as in the Table of Sints , if you cannot • ^ fiiiJ the direft Logarithm which you look for in the Table, I you muft take the neareft thereunto. 7^ Trigonometrie, Lt 6 .111.1 The Table of Logarithms. Logarithm Nttm . Logarith. Num. L'gmtk. I OjOOOOOO 51 1,707570 lOI 2,004521 2 0,501030 51 1,726003 102 2,008600 i 0,477121 53 1,714176 103 2,012857 4 0,602060 54 i> 73 i 394 104 2,017053 J 0,698970 55 1,740365 105 2,021189 6 0.77&I5I 56 1,748188 106 2,025506 7 0,835098 57 IC7 2,029584 8 0,903090 58 1,763418 108 2,035424 9 0,954242 59 1,770852 109 2,057426 10 IjOOOO 00 60 2,778151 no 2,041595 n I5O41595 61 1,785330 111 1,045313 1,079181 6 z i>79i39i 112 2,049218 1,11594: 63 1,799341 ii^ 2,055078 14 1,146128 64 1,806180 114 2,036905 iJ 1,176091 65 1,812012 115 2,060698 itf 1,204120 66 1.819544 116 2,064458 17 1,230449 67 1,826075 117 2,068186 18 ',255171 68 1,852509 iiS 2,071882 19 ',178754 69 1,838849 119 i,°75 547 zo 1,301030 70 1.845098 120 1,079181 41 1,511219 71 1,851258 III 2,082785 32 ',341413 71 '>857331 122 2,086339 *3 1,361718 73 1,863313 I>3 2,089905 24 14380211 74 1,869232 124 1,093411 *5 1,397940 75 1,875061 115 2,096910 16 '>414973 76 1,880814 126^ 2,100571 27 ',431364 77 1,886491 117 2,103804 28 1.447158 7« 1,892095 128 2,107209 29 1,462598 79 1,897627 119 2,110389 30 M77121 80 1,903089 130 1,113943 31 1,491362 81 1,908483 131 1,117171 3J 1,505150 81 1,913814 131 2,120574 35 i> 5 i 8 ji 4 83 ! 1,919078 153 2,123851 34 '>531479 84 1,924279 134 2,117105 35 1,544068 85 1 1,919419 135 2,130334 36 1,556302 86 1.934498 136 1>13 35.39 57 1,568202 87 ! ',939519 137 2,156721 38 *>579783 88 1,944483 138 1,139879 39 1,591065 89 1,949390 139 1,143015 40 1,602060 90 1,954141 140 2,146128 41 1,612784 91 1,959041 141 2,149219 4’- 1,623249 91 1,963788! 142 2,152288 43 1^533468 93 1,968483 143 1,15533^ 44 1,643455 94 '>973118 '44 2,158362 45 1,653212 95 '>977714 '45 2 ,i6ij68 45 1.662758 96 1,982271 146 2,164553 47 1,672098 97 *,986772) '47 2,167317 48 1,681241 98 1,991226 148 [ 2,170261 49 1,690196 99 1,995635 '49 '1,173186 __ 50 1,698970' 100 1,00000 0 *50 2,176091 L IB. 5. Trigonomitrie , 75 The Table of Logarithms. N’nm J^ 3 g 4 rnh. 1 1 Num, j Logurith 1 iV«w LogAnth '51 I5I78977I lor 2,30319 *51 *,396'674 2,181844 101 *,30555 *52 2,401401 2,184691, 203 *,30749 *53 2,405121 1 i»i!^7)i 204 1 *,309*53 *54 2,404854 2,19053 205 1 *,31175* *55 2,406540 ijfi i>i95'2 3. I06 *,31386 *J6 2,408259 ^ ) / 1 h' 9)^99 207 1 2,31597c *57 2,409935 158 *>' 98657 j 20 C> 2,318065 : 256 *,411619 159 160 2 >ioi 397 i 209 2,320146 1 *59 2,413299 2,204119' 210 2,3 22219 1 260 M14973 i6l 2,2068261 2II 2,324282 1 ^6l 2,416641 1&2 21209515! 212 2,326536 262 *,418301 16} 252I21S7I 213 2,328579 *63 *,419956 2,214844 214 2,330414 264 *,421604 2,217484. 215 2,532438 *65 *,412246 1® 2,220108 H6 2,534454 265 1,4:4881 2,222716^ -17 *,356459 *67 *,426511 2,225509, 2l8 2,338456 268 *1428.35 iCi, 2,227887 2 i 9 2,340444 169 *,4*9751 2,230449 220 2,342*27 *70 *,431364 17^ 2,23299^ 22 I 2,34439: *71 *,432969 l/ w •2,13 5 5 *22 *534*5533 272 M34569 2,238046 213 2,348305 *73 *^436163 2,24054c 1*4 3,55024s *74 *,437751 775 2,245038 215 2,352183 *75 *>439333 j .*7^ 2,24551.- z^6 2,354108 176 *,440909 177 i 7 S 2,-4797' 227 2-556016 *77 *>44*479 1/0 2,25042c 228 2,357935 *78 *,444045 *79 180 2,252855 129 2,559833 *,445604 1 2,255273 230 2,561711 280 *,447158 1 2,2 57*579 231 2,365611 iSi *,448706 1 2,260071 252 2,365488 282 *,450*49 / 2,26245. *33 2.36735*5 *83 *,451786 184 2,264818 234 2,569216 *84 *,453318 . 185 1^6 2,267172 *35 2,371068 285 *,454^45 2,269513 236 2,372912 286 *,456565 '®7 2,271842 *37 2,574748 287 *,459889 2,27415'* 238 2,376577 188 M59391 1^9 2,276462 *39 2,378398 189 *,460898 190 2,27875c) *40 1,580211 290 *462398 191 2,281083 *41 2,582017 191 *,463893 iji 2,183301 242 2,383815 291 M65383 19} 2,285557 *43 2,385606 293 *^66868 1P4 2,187802 244 *,387389 294 *,468347 '$>5 2,290035 *45 2,389166 *95 2,4698** 196 2,2921^ 246 2,390935 296 *,471291 'S7 2,194466 *47 2,392697 *97 *,471756 I98 2,296665 I 248 2.39445* 298 *,474216 199 2,298853 1 *49 2,596195 299 *,475671 200 1,301029 *50 2,397940 300 *,4771*1 L 74 Trigonometrie. L i b. 5. The Table of Logarithms. j Logarith 1 Ntim. j Log,nith. 1 7V«w. 1 L/garitl. IC\ :,478s6d 351 2,543507' 401 2 , 6 o.M 44 J02 2,48000} 552 2,546543! 402 2,604226 303 2^48144 353 2,547775’ 405 2,605305 304 2,48287^ 354 2,5490031 404 2,606381 38J ^)484-95 355 2,350228! 405 2,607455 jO(5 2,48572 356 2,15,449 406 2,608526 307 2,487135 357 2.552668 407 2,609594 ;o8 2,48855 35ii 2,553883 408 2j6Jo66o 509 2,489958 359 2,355094! 409 2,6**723 310 2,491363 360 2,3563031 410 2,612784 3” -,49276c 361 2,357507 4'i 2,613842 M94>y5 362 2,538705 412 2,614897 313 ^495544 ; 363 3,559907 4*3 2,615950 314 2,496915, ; 364 2,^61101 4*4 2,617000 315 2,498311 365 2,562293 4*5 2,618048 5i479 390 2,591065 440 21^43453 341 2,532754! 39« 2,592177 44* 2,644439 342 2,5340.6' 392 2,59328c 442 2,645422 343 2,535294 393 2,594393 443 2,646404 ■ 344 2,536558 394 2,593496 444 2,647383 345 2,5 3 78'9; 395 2,596597 445 2,648360 345 2,5390761 396 2,397695! 446 2,649335 347 2,540329] 397 2,598790, 447 2 650308 34S 2,541579 598 2,5998831 448 2,651278 549 2,542825 399 2,60097^! 449 2,652246 350 2,5440681 400 2,602059) 450 2,653213 g. Trigonometrie, The Table of Logarithms. Lararith . J Mow. . Ltjr .- trith . Nnm //;• j 45^ ^«54'77! 501 j 1 2,699358 \ 2,741132! 45^ 2/55158 502 ! i »700704 531 2,74*939 1 45' 2,656098 505 2,701568 1 555 2,742725 454 2,657056 504 2,702450 j 554 2,74! 509 455 2,658011 1 505 1 2,705291 ; 555 1 2,744295 45*5 2,658965' 506 2,704151 i 556 2,745273 457 2>'5599>«5i 507 2,705008 ! 557 2,743^33 458 J 2,660865] 508 2,7058641 5581 2.746634 459 1 2,661815 509 22706718 559 2,747412 460 2,662758; 510 2,707570 5601 ' 2,748188 461 :,6657oi| 511 2,708421 5611 2,748963 461 I 664642 512 2,700260 562 1 2,749736 46? ( 2,6^5581 515 2,7i“'i7 i 2,750508 464 2,666518 514 1,71096! 1564 1 2,75,279' 46V 2.66745!: 5I5 2,711807 i 565 j 2,751048 466 2,668586, 516 2,712649 5661 i 2,751816 467 2>(5695I7, 517 2,715491 567 j ^753585 •’6S 2,670246! 518 2.714329 I 568 2,73454s 469 2,67 "75 519 2,715167 569 1 2,735**2 470 2,672098 510 2,716003 j 157° 1 2,735875 47« 2,675021 5ii 2,71685!' 1571 1 2,736636 471 2,<573942 522 2,717671 572 2,757396 475 , 2,674861 2,718502 573 1 2,75815 5 474 1 2 s <575778 524 2>7‘5'33 i 574 2,738912 475 2,676694 525 2,720159 575 2,759668 476 1 2,67760; 526 2,720986 576 2,760422 477 1 2,678518 527 2,721811 577 2,761176 478 2,679428 528 2,722634 578 2,761918 479 2,680556 529 2,725456 579 1 2,762679 480 2,681241 55“ 2,724276 580 2,765428 481 2,682145 551 2,725095 581 2,764176 481 2,685047 552 1 2,725912 582 2,764923 48? 2,685947 5J5 2.726727 583 2,763669 4'-4 1 2,684845 554 2,727541 584 2,76641? 485 2,685742 535 21728354! 585 2,767136 1 48^ 1 2,686656 55753999 392 2,772322 495 2,692847 543 1 2-754799 393 2,773°55 494 2,695727 544 1 2,735599 594 2,773786 495 1,694605 545 2,736397 595 2,7745*7 496 2,695482 54<5 2,737192 596 2,775246 : 497 2,696556 547 2,757987 597 2,775974 1 498 2,697129 548 2,738781 398 1,776701 ! 499 2,69810! ! 549 2,759572 399 2,777427 ■ 500 1 i/98970' I 550 2,740363 600 2,778131 L i 7^ Trigonometrie, Lib. 3 . The Table of Logarithms. Nnm 6qi 6oi 603 60^ 605 6c6 607 (5c8 dop 610 611 dl2 di3 614 613 did 6i'j di8 d!^ 611 dj2 dij d^' djd 617 d2& djp d’o 631 ^32 <^34 ^35 d36 ^'7 d38 d39 d4o d4l d4z 643 d44 <543 646 647 648 d49 djo Logaritf 2,77887 2,77959 2,78031 i 2,78103 h,78175 ^,78247 12,78318 2,78350 2,784^1 2,78532 2,78do4 2,78d73 2,787461 2,78887 2,78958 2,790283 2,790988 2,791691 2,793092 2,793791 2,794488 2,795283 2,795880 2,797268 2,797959 2,798631 2,799342 2,800029 2,800717 2,801404 2,802080 2,802774 2,803437 2,804139 2,804821 2,803301 2,806179 2,806338 2,807535 2,808211 2,808886 2,809559 2,810233 2,810904 2.811373 2,812243 2,812913 • 4 ' j Logarith. 631 632 2,81424 ^33 1 2,81491 f54 2,813378 ^55 2,816241 ‘^56 2,816902 937 2,817365 ^58 2,818226 ^S9 2,818883 660 ,,819343 6dl 2,820201 ^62 2,820838 ^63 2,8215,4 “64 2,822168 °65 2,822822 2,823474 ^68 2,824776 2,8^426 ^7° 2,826075 ^7' 2,826723 ^7* 2,827569 ”73 2,820015 ^74 2,828659 ^75 2,829304 f7^ 2,829947 ”77 2,830589 678 2;83.h9 ^79 2,851869 2,834421 ”85 2 835691 686 2,836324 ^836957 688 ,837588 689 2,8382,9 690 2,838849 69« ,,8;947« ”92 2,840,06 693 2,840733 694 2,841339 695 2,841983 M 2,842609 1 698 1 2,843833 699 1 2,844477 700 1 2,843098 7Vi/w., L garitk, 701 1 2,843718 702' 2,846337 703 12,846933 704 1 2,847373 705 2,848189 706 1 2,848803 707 1 2,849419 70S 1 2,S 3003 3 709 2,830646 7'o 2,831238 7* I 2,831869 712 2,832479 7*3 2,833089 7*4 2,833698 7*5 2,834306 7*6 2,834915 7*7 2,833319 7>S 2,836124 7*9 2,836729 7’-° 2,857332 7»* 2,837933 7*- 2,858337 7»3 2,839138 7*4 2,859739 7*5 2,860338 726 2,860937 7*7 2,861334 7*8 2,362131 7*9 2,862728 750 2,>-63323 73* 2,863917 7 ^- 2,864511 733 2,865104 734 2,863 696 735 2,866287 736 2,866878 737 2,867467 738 2,868036 739 *.868643 740 2,869231 741 2,869818 742 2,870404 743 *,870989 744 *,87*573 745 2,872156 746 2,8727jp 747 2,873321 748 2,873903 749 2,874482 750 12,873061 jLiB. 5. Trigonometrie, 77 The Table of Logarithms. LigAritb. . Mum. Loganth. Nnm. Logarith. 7SJ 2,8756391 801 2,90•.6^3 851 2,929929 7)» 2,876218 801 2,904174 852 2,935439 yn 2,87679) 803 2,9047,6 853 2,930949 m 2,877371. S04 2,905256 8S4 2,93145s 755 ^>877947, 805 2,905796 855 2,931966 70 2,8785221 806 2,906335 856 2)932474 757 2,879095; 807 2,906874 857 2,932981 7)'^ 2,87069; 808 2,907411 858 2,933487 759 2,88->242] 809 2^907949 1 859 2,933993 760 2,8So8i4 Sio 2,908485 g6o 2,934498 761 2,881385; Sii 2,909021 «6i 2,935003 761 2,881955 812 2,909556 862 2,935507 76? 2,882525' 813 2,910051 863 2,936311 7(54 2,883093! S14 2,910624 86in 2,936514 765 2883661' 815 2,911150; 865 2,937016 76(5 2,8842:9' 816 2,911690 866 2,937518 767 2.884795I 817 2,912222 867 2,938019 768 2,885561] S18 2.912773 868 1,938519 769 2,885926! 819 2,913284 S69 2,939019 77° 2,386491 820 2,915814 870 2,939519 771 2.07054 821 2>9>4'43 871 2,940018 77- 2,887617 822 2,914872 872 2,940516 775 ,888179 823 i> 9I5399 873 2,941014 774 2,888741 824 2>9iS9»7 874 2,941511 775 2,88920: 825 21916454 875 2,942008 776 2,88986: 8:6 2,916980 876 2,942504 777 2,8904,! 827 2.917506 877 2,942999 778 2,89'9 ■ 9 828 2,918030 878 2,943493 779 2,891 <(57 S29 2.918555 879 2,945989 780 1 2,892095 830 2)9'9078 :8o 2,944483 781 2,89:651 831 2,919601 8S1 2,944976 782 2,893207 8;: 2.920125 881 2,945468 785 2,8937'^= 1 2.920645 885 2,945961 784 2,894316 834 2,921166 884 2,946452 1 78s 2,894^9 835 2,921680 885 2,94694? ! 7M6 2,8954-5 836 2,922206 886 11947434 787 2,895975 =^’37 2.922725 ^‘■7 2,947924 788 2,8965:6 838 2,923244 888 2,948413 7‘’9 2,897077 839 2,923762 S89 2 948902 79D 2,897627 840 2,724279 890 2,949590 791 2,898136 84. 2,924796 891 2,949878 791 2,898725 842 2,925312 891 2,950365 79^ 2,899273 843 2,925825, 893 2,950851 794 2,899821 844 2,9263421 894 2,951338 1 795 2,900567 845 2,926857 895 2,95182? 1 796 2,900913 846 2,927370 896 2,952308 i 797 2,901458 847 2,927883 897 2,952792 1 798 2,902003 S48 2,928396 898 2,953276 1 799 2,902547 849 2,928908 899 1,953759 / 800 2,903089 1 '850 2,919419 900 2,954*43 78 Trigommetrie . Lib. 3, The Table of Logarithms. N ’ fim . Logarith . Nnm. j Log , xrith . Ltgixrifh , 901 2,954725 935 2,070812 968 2,985875 902 2,955207 936 2,971276 969 2,986324 903 2,955688 937 2,971739 97° 2,986771 904 j 2,956168 1 938 2,972203 971 2,987219 905 2,956640 , 939 2,972666 972 2,987666 906 2,957128 940 2,973128 973 2,988113 9O7 I 2,957607 ! 941 2)973589 974 2,988559 908 j 3,py8o85 942 2,974050 975 2,980009 909 2,958564 943 2,974512 976 2,989449 910 2,959041 944 2,974972 977 2,989895 911 2,959518 945 2,9754-2 978 2,990339 912 2)959995 9 i ^\ 2,975891 979 2,990783 9^3 2,960471 947 2,976349 980 2,991226 914 2,960946 j 948 2,976808 981 2,991669 915 ! 2,961421 949 2,977266 982 2,992111 916 2,961895 950 2,977724 983 2)992 554 917 2,962369 95* 2,978181 ! 984 2)992995 918 2,962842 952 2,978637 985 2)993436 919 2,963315! 953 2,979093 986 2.993877 920 2,963788 954 2,979548 987 2,994317 9:1 2,964259 955 2,980003 988 2,99475<5 9 i 2 2,964731 956 2,980458 989 2,995196 923 2,965202 957 2,980912 990 2,995635 924 2,965672 958 2,981366 991 2,996074 9^5 2,966142 959 2 .g8i8iQ 2,996512 916 2,966611 960 2,982271 993 21956949 927 2,967079 961 2,982723 994 2,997386 928 1,967548 962 2,983175 995 2,997823 929 2,968016 963 2.983626 996 2,998259 950 2,968483 964 2,984077 997 2,598695 931 2,968949 965 2,9845271 99S 2,999133 932 2,969416 966 2,984977 1 999 2,959565 933 2,969882 967 ’ 2,985426 1000 3,ooooco 934 2,97o?47 1 Chap., 19 T Lib. 5. CHAP. III. ths ufe of the Tablet of Sines and Logarithms in the revolving of Plain Triangles. ■ Eforelconv: toflicw how the quantity of the Tides and angles of any Triangle may b: found by help of the former Tables, it will be convenient firft to de¬ liver thcfe following confiderations and Theoremes, as necelfaries thereunto. I A Triangle is a figure confiftingofthreefit*esand C three angles, as is the fi¬ gure D B"C. i 2 Any two Tides of a Triangle are callcdthe Tides of the angle comprehended by them, as the Tides C B j and D B ate the Tides con- f raining the angle C B D. I 3 The mcafure of an An¬ gle, is the quantity ofan arch of aCircledefcribedontheanguUr I pointjand cutting bo:’,; the containing fides of the fame angle,as in i the Triangle following, the arch CB, isthemeafureof the angle j at A; thearchKDistnemeaTureoftheangleat E; and the arch I F G is the meafureof the angle at H; each of thefe arches are de- ' feribed on the angular points A,H,E, and cut the containing Tides. 1 ^ 5 A Semicircle containeth 180 degrees. 6 A Quadrant containeth po degrees. 5 The complement of an angle leflc then a Quadrantds To much as that angle wanteth of 90 degrees, as if the angle H A E fiiould contain 5 0 degrees,the complement thereof woiUd be 40 degrees, for if you take jofrom 90 there will remain 40. 8 The Trigommetrie. Lib. 3. 8 The complement of an angle to a S emicirclejis the remainder thereof to 180 degrees. 9 An angle is either Right, Acute, or Obtiile. 10 A Right angle is that whofe meafure is a Quadrant. 11 An Acute angle is lelTe then a right angle. 12 An Obtufc angle is greater then a Quadrant. 13 A Triangle is cither Right angled, or Oblique angled. 14 A Right angled Triangle is that which hath one right angle, as the Triangle A H E is right angled at E. 15 In every right angled Triangle, that fide which fubtendeth or lieth oppofite to the rigiit angle, is called the Hypothenufaljand ot the other two fides,the one is called the Perpendicular, and the other the Bafc,at plcafurcjbut moll commonly the iliorteft is called the Perpendicular;and the longer is called the Bafe. So in the for¬ mer Triagle, the fide A His (l.'eHypothcnufil, HE the Bafe, and AE the Perpendicular. 1 6 In every right angled Triang’c, if you have one of tlvc acute angles given,the other is alfo eivendt being the ccmplement there¬ of to 90 degrees. As in the Triangle A H E, fuppofe there were given the angle A H E 40 degrees, tht n by conlequcncc the angle H A E muft be 50 degrees, which is the ccmplement of the other to ,0 desrees to the Sine of3o degrees, the lame cxtcntwill reach Iroin 4^2 toi3t: and that is the fide C A. CASE IV. The Hypotbe'/mfall anda^gle at the Bafebei'/tg given , to finds the Baje, ■t" Et there be given in the former Triangle the Hypothenufal B C, and the angle at the bafe C B A, and by confequence the anelcBCA the complement of the other to 90; then to firidc BA, the proportion is, _ ' As 85 T L i b. 5. | ; As the Sine of the angle C A B, 90 degrees 10,000000 ! Is to the Hypothenufall 30,462 2,664641 j So is the Sine ofthc angle BOA, 60 degrees, 9,937531 j To the Logatithm of the Bafe B A, ^2,602173 Thenccreft number anfwering to 2,602173, is the Logarithm of 400, and fo long is the Bafe B A. By thelines of Sines and Numbers. As before,Extend the CompalTes from the Sincofpo,to4623 the fame extent will reach from the fine of 60 degrees, to40oin the line of Numbers. Or, extend the Compafles from the fine of 90, to the Sine of 6o, the fame extent will reach from 462 to 400, which is the length of the Bafe B A. CASE V. The Perpendicular, and angle at the Bafe being given, to finde the Hypcthenufall, I F the Perpendicular C A be given 231, and the angle at the BaleC B A 30 degrees, the HypothcnufallBC may be found thus; for. As the fine of the angle C B A, 30 degrees, 9,698970 Is to the Logarithm of the perpendicular C A 231 12,363612 So is the Sine of the angle CAB, sodegrees, 10,000000 To the Logarithm of the Hypothenufoll B C 2,664642 If Hcre,liecaufe the angle C A B is a right angle,or 90 degrees and comes in the third place, 1 therefore only put an unite before the fecond term, and from thatfecond terra fubftraft the firft terme, and the remainder is, 2,664642, the abfolutc number anfwering thereunto is 462, the fide B C. By the lines of Sines and Numbers.’ Extend the Compafles from the fine of 30 degrees, to 231 the fame extent will reach from the fine of 90 degrees to 462. Or,the diftance between the Sine of 30 degrees and 90 degrees SScrequireV^^ diftance between a3i,atid462,whichgiveth cAse CASE VI. The Hjfothennfall and Perpendicular being gi- *ien tojinde the angle at the Bafe. I N tlie forcgoins Triangle tliere isgivcn the Hypothenuftll B C 462 lcct,andtncpcrpendicuIarC A,231 feet, and it is requi¬ red to finde the angle C B A, the proportion is, As the Logarithm of the Hypothenufall B C 461 5,66464a Is to the right angle B A C, 90 degrees, io,ooo3co So is the Logarithm of the perpendicular C A, 231, 12,36 {612 To the fine of the angle C B A, 30 degrees. 9.698^70 Ey the Lines of Sines and Numbers.- Extend the Compafles from 461, to the fine of 90, the fame ex¬ tent wi 1 reach from 2 31 to the fine of 30 degrees. Or, Extend the Ctmpafics from .62 to 23 , the fame extent will reach from the fine of 90 degrees, to the fine of 30 degrees, which is the quantity of the enquired angle C B A. Cf oblique angled plain Triangles^ CASE VIL Having two angles and a fide oppofite to one of tberngiveny to finde the fide oppofite to the other, I N the Triangle Q.RS, there is given the angle QS R 24 deg. 20 min. and the angleQ,RS4;"' degrees 10 minutes,and the fide Q.S 303 feet, and it is required to finde the fide Q^R. ^|[ Here note, that in c blique angled plain Triangles,as well as in Right angled, the fides are in proportion one to the other, as tiiC Sines of the angles oppofite to thofc fides. Therefore, AsthefineofthcangleQRS45deg.iomin. 9,8 50745 , Is to the Logarithm of the fide QS 303 feet, 2,48144? So is the fine of the angle Q^S R 24 degrees 20 min, 9,614944 thcfumofthefccondand third terms-i2,o96,'87 the firft terme fubftraiSed—9,850745 j To the Logarithm cf the fide Q.R, 2.245642 j The ncereft abfolute number anfwcring to this Logarithm is i76, and fo many feet is tl.c fide QJ{. M4 By 8o Tri^nometrie, By the lines of Sines and Numbers, The lines of Sines and Humbers will rcfolve thefe Trianolcs by the fame manner of work as in the other before. For, If you extend the Compalfes from the Sine of 45 deg. 10 min. to 303, the fame extent will reach from the Sine of 14 degrees jo minutes, to 176,and fo much is the fide QR. Or, Extend the Compaffes from the Sine of 45 degrees 10 min. to 24 degrees 20 minutes, the fame Extent will reach from 303, to 175, the length of the inquired fide. In like manner, if the Angle R QS 110 degrees 30 minutes, and the Angle S 45 degrees^ 10 minutes, and the fide QS 303 feet, had been given, and the fide RS required, the manner of work had been the fame j for, As the fine of the Angle QRS 45 degrees 10 min. ^5850745 Is to the Logarithm of the fide Q^S 303 feet, a 3481443 56 is the Sine of R QS i iodeg.3omin.(or 69 deg.^o m.)?,97i588 the fum of the fccond and third terms -12,45 3031 thefirft terme fubftrafted—9,850745 To the Logarithm of the fide R S, 2,602286 The abfolutc number anfwering to this Logarithm is 400, and fo much is the fide RS. C In this cife,bccau(e the angle R Q S is more then 90 degrees, you rauft therefore take the complement thereof toi8o degrees, fo 110 degrees 30 minutes, being taken from 180 degrees, there remains 69 degrees 30 min. whofcSinc is the fame with no deg. 30 min. and being ufed in Ac-ad thereof, will effed the fame thing. By the lines of Sines and Numbers. Extend the Compaffes from the Sine of 45 degrees 10 min. to 303, the fame extent will reach from the fine of 69 deg. 30 min. to 400, which is the fide R S required. Or 8 ? j Li B. 9. TrigDr^trie. Or the Compafles being opened to the diftancebenveefitht I fincof45 deg.iomin. and6ydcg.3omin. the famediftance will I reachirom 303 1040035 before. > I • ■ CASE VIII. I Tm fidetaftd an anglhfpdfite to one of them he^ I ' ing giveuy to find the angle o^fofiteto the other, I N the fame Triangle, let there be given, the fide QS 30J, dnd QR 176, togetlicr with the angle QSR^degites 20 minutts, I and let it be rcijurred to find the angle Q.R Sj the proportion is, ,■ AstheLogarithnkof thcfideQR 176, *3245^13 Istochefinc of thcan'ilcQS R, 24^eg.2omin. 9,£14^44 So is the Logarithm of the fide QS ^03. 2,481443 i The fum of tlie fccond and third numbcrs-i^-i 2,096387 j The firft number fubtradled from the fum—2,145513 To the fine of the angle QRS, 9,850374 The necrcfl degree anfwering to this fine is 45 degrees 10 min, which is the quantity of the angle QJR S, required. By the lines oTSines and Numbers. r, 'tend the Compafles from 175, to the fine of 24 degrees 20 rri . : ,utes, the fame extent will reach from 303 to 45 d^. loinin. t'-.e angle QRS. Or. the diftancebetwten 17^ and 303, will be equal to the di- fiaiict between 24 degrees 20 minutes, and 45 deg. 10 min. CASE IX. Having tm fidesjOnd the angle containMy them given,, to find either of the other angles, T His Cafe will fcldome come in ufc in Surveying, becaufe the thing required is an angle, which arc moft commonly given, they being obfervedby inftrument, and therefore in this place may be omitted, partly becaufe the propofition is not wrought by Sines and Logarithms,but by Tangents and Logarithmsjand there Is no Tables of Tangents in this Book, to work the proportion by.- I Yet thofc that are defirous to rcfolve all kinde of Triangles by the proporrionall lines, may have added to the lines of artificial fines and Numbers, a line of artificial Tangents, and thefc three lines I N to- Trigonometrie, Lib. 3. j together, will refolveall Cafes ki Sphrtical-, as well as in plain Triangles. . ■ ' For the performance of this Probleme, fuppefe there were gi¬ ven the fide QS 30jjand tlic fide R Q^iyejahd the angle compre¬ hended by them ;namely,the angle R QS 11 o degrees 30 minutes, and it were required to find titlttr of the other angles. Firftj Take the fum and difference of the two given fidcs, their fum 15479, 'and their'diffcrenceiS'ii7. Thcn kr(o\iving that the three angles of all rightlined Triangles arc equal to two t^btan- gfesjof 180 deg.fij the i 7 T/.w.«/'C/w;>.3.)therefore the angle R QS fcing 110 deg. 3 3 minutes, if you fubtraft this angle from 180 de¬ grees , the remainder will be 69 deg. 30 min. which is the fum of the two unknown angles at R and S, the half whereof is 34 deg. 4$ min. The fide QS, *303 The fide QR 17^ The fum of the fides, 471^ The difference of the fides, 117 The half fum of the two unknown angles 34 deg. 45 min. Q- Thcfunland difference of the fides being thus found, andalfd the half fum of the two unknown angles, the proportion by which you muft find the angles fcverally iS, As the Logarithm of the fum of the fides,,^79, J>68o335 Is to the Logarithm of the difference of the fides, 127,2,103804 SoistheTangcntofthchalffumofthetwounknovVii? . , *• angles 34 degrees, 45 minutes, ^"^304* * •/ the fum of the fecond and third HJifebers-i 1,944991 the firft number fi^ltra&ed-s,^8o33y to theTangenit of 10 degrees 25 minitcs, Thcfe ILi "frigomnetrle. j Thclc ten degrees 2 5 minutes, bging added to iJie lialf (urn of ; tbcM’-.iaDkirowii’andcf) namely, to 34.degrees 45 minutes; thb ; fum will be 45 degrc-cs 10 minutes, tl.e quanrity.'of the angle i QJl S, whicli is the greatef angle of tlic two; Alfo, thefe ten de -1 j gi ec5 2 5 minutes, being fubftrailed frorn the lame half fum, there j .1 remainerh 24 degrees 20 minutes for the angle Rjwhich is the j I Icflcrottljc unknown aaglcs'.fand thus are either of the enquired '! angles eafily found. j By the lines of Tangents and Numbers. i Extend the CompaiTcs from the fum of the lidbs 479, to the I difference of the fides 127, the fame extent upon the line of taii- ; gbnts wil I each from the Tangent of 34 degrees 4y.mrnuies(which ■ is the half fum of the two unknown angles) to the Tangent of ten [ degrees 2 5 niiniKcs, and thefe ten degrees 25 pinutes, added to, ' and fubradedfrom the half fonijas before is fliewed,vvill give the quantity of either of the twdunknown angles. CASE X. 7 hd three fides of a right lined Tim Triangle I being given, bom to finde the Area, or the fu -1 ferficiali content thereof ■ff^rlljAdd the three fides together, and from the half fumme ^ fubftrail^each fide fcverally,to the end you may hare the dif- -ib fercncc betwixt that halfe lummeand each fida- this done i aJJc A _ li C the Logarithms of thefaid halfc lummc, and of thole differences together: and laftly, dividing the fumme of thofe Lo»arithms by a, you have the L^arithmof the fuperficiall conteiit or area of the Triangle. EXAMPLE, LctthcTriangle given be A B C , fides thereof being lo, 13,11, how mivh IS the fupcrfkiaU Content thereof? N a The Thefum of the fidcsis44,thclialf fummeis w,the differences betwixtcach fide and that half arc which numbers rank m this order following. The half fiutij 22 r2 0,301030 The differences, Pp 0,^54243 x, 04 i^ The fum of the Logarithms 3,63908^ The Area or Content required, 65 . 1,8 19544 And this Area, or fupcrficial Content thus found, is alwayes of the lame nature with the fides of the Triangle,that is to fay,if the tides of the Triangle, be given in feet,then is the content found in f^t) alfo, it the fides be Perches, you (hall have the content in Perches, and fo of any other mcafure whatfdcver. I might add hereunto divers other Cafes, but ip this place at prcfentletthefc fuffice, The end of the third Bool{l *77 Ili B.4. [THE compleat SURVEYOR: The Fourth Book. the ARgVMENT. hath hitherto been our lU:Pri^ bufinefs to provide neccf- fary Inftrumcnts, and to learn fuch things , which of neccflity ought to be known before we enter the Fields to Survey. Be¬ ing thus providedwee come now to apply them (everal wayes: Firft, in taking of Heights and Diftances whether ac- ceflibleor in-acceffiblcj and then in Surveying of Land. In this Book every kinde of work is performed three Iweral ways j by three feveral InlhumentSiVi^.^t PlainTable, the Theodo¬ lite^ and Circumferentorhy which the congruity and harmony of the feveral maybe ____A a_ eafily The Argument, 17^ Lib. ‘1 eafily di(cerned,and the truth of every Example may the better appear. Here is aifo divers ways of Surveying by one and the fame InftTument^ that is, to take the Plot of a Field fcvcral ways, and to meafurc all kind of Cf rounds whatfoever, v/h^thGrPVoodland or other. Here is alfo fliewn how to take the Plot of a whole c^Mannor^ and to keep your account in your Field Boo\^ after the beft and moft eafieftmanner: with divers ^les y Cautions and Dire^ionsy throughout the whole ^oa^inferted. I i TH E jAPPLIGATION AND VSE 1 of the feveral Inftruments (before delcribed) in the praSice of I SV‘^VEXI3^G. CHAP. I. The life of the Scale, Aving before defcribed the fevcrall In¬ ftruments belonging to Surveying, I will now fhewtheufeof them: and firft, of the Scale. The Scale is principally in¬ tended for the laying out of lines, for which purpofe the feveral Scales of equal parts arc there divided, fome of greater and fome of lefter quantities: the ufes qfall the lines being the fame, for each line is divided into certain equal parts rc- prefenting ii Chains, & thefe grand divifions are numbred with ArithmeticallFigurcsby 1,2,3, &c. to lo, then the uppermoft large di\ifion is again divided into ten other fmaller parts, each part containing lo links of your chain,cach of which fmaller parts younuy luppolctobc again divided into ten other lellcr parts reprclcining fingic Links of your Chain. ’ 1 (^.Any length being meafured by your Chain, bom to Uydomn the fame difiance uyon Fofer, a ofcjthat mcafuring a long a hedge with your Chain, you e length thereof to contain 5 Chains 60 Links; Now to take this diftance from your Scale, andlay it down upon paper do thm. Firft, Draw a line as AB, then place one foot of your Com- paffes upon your Scale at the figure 5. for your five Chains, and A a 2 ex- T be ufe of feveral L i b. 4 I ^ extend the other foot to fix of the fmal divifions(\v!iich S rcprefcntsthc6o Links)thcn fet this dillanccupon the line drawn from A to B, fo fliall the line A B contain 5 Chains eo Links 5 if you take the diftance from the ■ 'C Scale of 10 in an Inch, But if you would have your line flaortcr, and yet to contain 5’Chains 60 linksjthcu take your diftance from afmallcrScalcjasof n, 16, iCjor 2if inan Inch , fo fliall the 5 Chains 60 Links end at Cjif taken from the Scale of 12 in an iuch, or at Djby the Scale of 16, or at E by the Scale of 24: either of which lines will con¬ tain 5 Chains 60 Linksjand be in proporcion one to tl.e other as. the Scales from whence they were taken. And ■' ^ in this manner may any number of Chains and Links be taken from any of the Scales, 2 A right line being given ^ to find ho)P many Chains and Lkl\s are therein - - E contained, according to any Scale af figned, a ofe A B were a line givemand it were required how many Chains and Links are contained therein,according totheScaleof loinanlnch. Take in your CompaiTes the length of the line A B, and ap¬ plying it to your Scale of 1 o in an Inch, you fliall finde the extent of the Compaffes to reach from 5 of the great divifions, to fix of the leffcr divifious, wherefore the line A B contains 5 chains and 60 Links; The like muft be done for any linc,and alfo by any of the other Scales, Upon the Ruler there is (befides the fevtral Scales ' ofequal parts)a line or Scale of Chords,which is gum- j bred by 10,20,30, &c, to yo, and this line ferveth to j protradl or lay down angles; butinallthepradiceof Surveying j a Protradlor is much more convenient, yet for other ufes this line I may be very ferviccable, and when a Protradlor is wanting, it may fupp'y that defed: the manner how to ufe it is thus. taining any number of degrees and minutes by the line of Chords, Draw a line as plcafure, as A B, and from the point A, let it be required to piotraiSl an angle of 40 degrees 20 minutes, Firft, ex¬ tend your Compaffes upon the line of Chords, from the beginning thereofto 60 degrees alwayes, and with this diftance, fetting one _ ' foot I foot upon the point Ajwith the other dcfcribe the pricked archB Cj then with your Compaffes take 40 degrees 20 minutes( which is the quantity of the inquired anglc)outof the line of Chords, from the beginning thereof to/j.o degrees 20 minutes then (the CompaiTcs fo refting ) if you fet one foot thereof upon B the other will reach upon the arch to C. Laftly, draw the line A Cjfo the angle CAB fhall contain 40 degrees jo minutes ^ angle being given to finde what number of degrees and minutes are contained therein* Suppofe CAB were an angle given, and that it were retired to finde the q uantity thereof. Open your Compaffes (as before to 60 degrees of your Chord, and placing one foot in A, with the other'deferibe tlie arch C B, then take in your Compaffes the di- ftanceCB,and meafuring that extent upon the line of Chords from the beginning thereof, you (ball nnde it to reach to 40 degrees 20 minutes,which is the quantity of the required angle. ■ Ifany angle given or required (hall contain above 90 degrees, you murt then protraft it at twice,by taking firft the whole line, and tiien the remainder. CHAP. II. Of theujeof the FrotraBor, •yS^Lthough the chief ufesof theProtraftor may beper- l^i^formcdby the line of Chords laft fpokenof, yet tor "^'^avoiding luperfluous lines and arches (which muft otherwilc be drawn all over your Plot) the Protractor is far more convenient, iheufe whereof is, I. To The nfe of fever al Lib.4.| 1 To lay down upn p^er an angle of any quan-- tity» Firftj draw a right line at length as AB, then on any part thereof, as on C, place the center of the Protrador, in which point alfo Sk yourprotrafting pin, and turn the Protraftor about upon the center, till the Meridian line of the Protraftor (noted in the defeription thereof with E F) lie diredly on this line A B, the S emicircle of the Protraftor lying upwards (or from you)then clofe to the edge of the Semicircle, at the divifionof5o degrees, mark the point D with your protraftingpinj and draw the line CD,fo fliall the angle D C A, contain 50 degrees. a (^ny angle being given, to find the quanti ty thereof by the ProtraBor. SuppofeDCBwcreananglegiven, and that it were required to finde the quantity thereof by the Protraftor. Firft, you muft apply the center of the Protraftor to the point C, and the Meridi¬ an line thereof dirciftly upon the line DC, then fhall you finde the line CB to lie direftly under 130 degrees of the Protrador, and fnch is the quantity of the angle D C B required. CHAP. III. Of the.Plain Table^ how to fet the parts there^ of togetherj and mal^ it fit for the Field, Hen you would make your Table fit for the field, lay the three boards thereof together, and alfo the ledges at each end thereof in their due places, according as they are marked. Then Jay a fhect of white paper all over the Table which muft be ftretched over all the boards by putting on the Framc,which bindcs both the paper to the boards,and the boards one to another. Then ;Li B.4. Infimmenihf Surv^kg. ! Then fcre\v the Socket on the back fide of th’eTable, and a!fo the-.. : Box and Needle in its due place, the Meridian line of the Card j (which is in the Box) lying parallel to the Meridian or Diameter : ; of the Table; which diameter is a right line drawn upon the Table ^ ; froiri the beginning of the degrees through the center, and loto | I the end of the degrees. Then put the Socket upon the head of| I the Staffe,and there fcrew it. Alfo, put the fights into theindex, | ; and lay the Index on the Table, fo is your Inftrumeiit prepared for j ufcasa Plain Table or Theodolite, the difference only being in placing of the Index, for when you ufe your Inftrument as a Plain Table, you may pitch your center in any part of the Table, which you fliall think moft convenient for the bringing on of the worke which you intend: But if you ufe youiTnft'rumcnt as a TiieoJo- • lite, then the Index muft be turned about upon the Center of the Table, for which purpofe there is a piece of wicr which goes through a fmall hole of brafs fattened to the Index,and fo into the center, by which means the Index keeps his conftant place, on¬ ly moving upon the center. Your Inftrument being thus ordered, you may ufe it either as a Plaid Table or a Theodolite,but if you would ufe it as a Circum¬ ferentor, you need only fcrew the Box and Needle to the Index, and both of them to the head of the StafFe,witha braffc ferew-pin fitted for that purpofe, fb that the S taffe being fixed in any place, theindex and Sights may turn about at pleafure without moving of the Staff, and now is your Inftrument a good Circumferentoi^ nay better then that before deferibed in the fecond Book. Alfo, when you have occafion to mealure any Altitudejhang tlic Labell upon the farther Sight,. and thus are you exaffly fitted for all occafions. / CHAP. IV. Hom’ to meafme the quantity of any angle in the field, by the PlainTable, 1 heodoiite, and Circumferentor: and alfo to obferve an angle of Altitude, Ou muft underftand that when I mention the Plain Table, or perform any work thereby, that I mean the Table when it is covered with a flieet of paper upon which,alobfervations of angles that are taken upon the Table in the field do agree exaftly in pro¬ portion with thofe of the field itfelfe, but arc not denominated by their quantities, but by their fymetry or pro¬ portion. Secondly, when I mention the Theodolite, or, work by that ___ ___ 184 Theufeof fever al Inftrumcnt, I do not mean the Theodolite before deferibed in the 1 Chapter oithe z Book, but I mean the degrees deferibed on the frame of the Table, which fupplies the ulc thereof. Tliirdly 5 When I mention or make ufe of the Circumferentor, I mean the Index with the Box and Needle ferewed to the Staff. Having thus given you a fufficienc defeription ofthcfevc- rall Inuruments and their parts, I como now to theufeof them, fhewinghow any angle in the field may be mcafured by any of them. And, f. Hoa? to ohftrve an angle in the Field by the Plain Table. SuppofeEKandKGtobetwohedges, or two fides ofa field including the angle EKG, andti.at it were required to draw upon your Table, an angle equal thereun .o.Firftjplace your Inftru¬ mcnt as ncer the angular point Kjas convenicncie will permit,turn¬ ing it about rill the North end of the Needle hang direfflyovcr the plg\yer-dc-lucc in the Box, and then ferew the Table laft. Tnen upon your Table, with your protrafting pin or Compaffc pointjaffignc any point at pleafure upon the Table,and to that point apply the edge of the Index, turning the Index about upon that point,till through the fights thereof you cfpic a mark fet* up at E, j or parallel to the line E K, and then, with your protrafting pin, or j Compaffe point, or Black-lead, draw a line by the fide of the In¬ dex to the affigned point upon the Table.Thcn (the Tabic remain¬ ing immoveable) turn the Index about upon the fame point, and direff the fights to a mark fetup at G, or parallel thereto,that is, fo far diftant from G, as your Inftrumcnt is placed from K, and * then, by the fide of the Index, draw another line to the affigned point, fo (hall you have drawn upon your Table two lines, wliich (hall reprefent the two hedges EKand KG,and thofc lines (hall include an angle equal to the angle,E K G, and although you know j not the quantity of this angle yet you may (by the lotzChspters I of this Book) findc the quantity thereof if there were any need, for j in working by this Inftrumcnt, itisfufficient only to give thefy- j merry or proportion ofanglcs and not their quantities, as in work¬ ing by the Theodolite or Circumferentor it is. Alfo, in working by the Plain Table, there needeth noprotraftion at all, for you (hall have upon your Table the true figure of any angle or angles which you obfer ve in the field, in their true pofitions”, without a- ny farther trouble. ' 2 Hok? to find the quantity ofan angle in the field by the Theodolite. Let it be required to find the quantity of the angle EKGby ___ the 1 ;Ll B.4. 191 the ThcoJolitc; place your Iiiftrutncnt at K, laying the Index oii j the diamccer thereof j then turn the whole Iiiftrumcnt about (the ;i Index ftill refting on the Diameter ) till through the fights you el'pic the mark at li, then ferewing the Inftrumcnt fall there, turn the Index about upon the center > till through the fights you efpie the mark at Gjthen note what dcgrees(on the frame of the Table) are cut by the Index, which you will findc to be i I4dcgrec5jand that IS the quantity of the angle E K G. 3 Hon? find th quantity of any angle in the field by the Circumferentor. If it were required to findc the quantity of the former angle EKGjby the Circumferentor; Firftjplace your Inftruihent (as before )at K, with the Flowcr-dc-luccj in the Card j towards you; then direft your fights to Ejand oBferve what degrees in the Card arc cut by the South end of the Needle, which let be 296, then turning the Inftrumcnt about the (faff (the Flowcr-de-Iucc always towards you) direft the fights to G,noting then alfo what degrees arc cut by the South end of the Ntedle, which fuppofe 182, this done (alwayes)fubftra(ft the leflcr number of degrees oiit of th.c j greater, as in tliis Example 182 from 296 , and the remainder is ■ 114 degrees, which is the true quantity of tlie angle E K G. I Again, the Inftruhicnt ftanding at Kjand the fights being direft-' ed mE, as before, fuppofe that the South end of the Needle had ! cut 79 degrees; and then direfting the fights to G, the'fame end of ( the Needle had cut 325 degrees, now, if from 3 2 5, you fubftra(ft • 79,thcremainder is 246, but becaufethis remainder 246 is great- j er then 186,you muft therefore fiibftrad 246 the remainder, from ; 3603and there will remain 114, the true quantity of the inquired | angle, and thus you muft alwayes do, when the remainder excccd- eth I ifo degrees. C This adding and fiibftraifting for the finding of angles, may feem tedious to fonu-jbut here the Reader is dcfired to take __ B b no- i86 Tbeufeof feveral L i b. 4, notice, that for quick difoatch the Circumferchtof is as good an Inftrument as the belt, for in going round a field, or in S ur- veyingofawhole Manner, you are not to take notice of the quantity of any angle, but only to obferve what degrees the Needle cutteth, which in thofe cafes is fufficient, as will ap¬ pear hereafter, bnt in taking of Diftances by the Circumfe¬ rentor it is altogether neceflary, as may appear by the 7 Chap, ter following, and for that rcafbn I have here flicwed how to find an angle by the Circumferentor, and alfo that you might thereby perceive wha^t congruity and harmony there is in all the three Inftruments'';.but the Circumferentor is not a fitting Inftrument for "the taking of Diftrnces. 4 Horp to fet the Index and Labal Horis^ontal H^ontbeStaf: When you have ferewed the Index and fights to the Staff as a' Circumferentor, before you put the Label uponthebrafspinor wicr,’you muft hang a line and plummet upon that pin, and then put on.thc Label,! then move the Index up and down till the thrid and |l^*q:^t hang direftly upon a line which is gaged from under thepin 'a;llraIong,the Sight, and then doth the Inftrument ftand, horizontaljir level, which it muft always do when "you take an aU titude therewith. " , 5 How to obferve an (^ngk of Altitude, The Label which is to be hanged on one of the Sights of the Circumferentor (as was intimated in the defeription thereof) and j the Tangent line on the edge of the Index, is only for the finding of angles of altitudcjand is therefore only ufeflil in taking of heights, and in furveying of mountanoui and un-even grounds. The manner now to obferve an angle of Altitude by this Label, •nd thc.TdiKcnt line on the Index, is thus. Supppfc C A to be a Tree, Tower, or Hill, whofc height were requited. Your Inftrument being placed at B,exa(ftly level, diredl the fights thereof towards C A, and there fix it, hanging the La¬ bel onthefartheft~fight,uponapinfor that purpofej then move the Label too iidi^oi along the fide of the Index, till through the I fight at the end of thcJL^ql, and by the Pin orij which,the Label i hangeth, you efpy the v^y.top of the objeft tq.b|:.meafured at C, j then note what d^ree of the Tangent linci'is cut by the Label, j which fuppofejo, and that is the quantity of the Angle of Alti-, tude, it being equal to the angle C BA, ' Thus i ;Lib.4. InjhrmentsmSHfv^ing. 187 i Thus by the Rules in this Chapter deliveredjmay the true quan- i tity ofanyan’lebeeafily taken, and this is the njoft convenient ■ ufe to be firft placed, I will now Ihew how by your fevcral Inftru- ! ments you may take all manner of heights and diftances,whether ! accclTible or inaccefliblc, fevcral wayes, with divers other nccel- i fa ryconclufioiis incident thereunto. Horp to an inaccejfble Dijiance at tm flatu | ons by the three fore-mentioned InfirumentSiaid by the Plain Table. Ou arc taught in the lad Chapter how to make ob- fervation of any angle in the held by the levcral In- ftruments before mentioned, as the Plain Table, Thcodo!itc,and Circumferentor, and alfoan angle of Altitude by the Index, and the Label thereunto annexed. I conceive it now convenient to (hew how all manrer of heights and diftaticcs may be readily aridexaftly mcalurcd, fevcral wayes, whether they be accclfiblc or inaccem- blc: and firft of diftances. You may remember that I formerly intimated,that the mea- furing ( r a Hciglit or Diftance is only to refolve a Triangle, fo that when you make any obfervation cither of Height or j[)iftancc, the obfervation of angles which you make are the angles of (cme Triangle, and the lints which you m'eafure on the sreun j, arc tlic iidcs of the I'anieTriangle, and thele are the yiveil pares of theTrianalc. Bb 2 the Bb 2 i88 The ufe of fe veral L i b. .1 The manner how to take a diftance by the Plain Table is thus, I Suppofeyou were (landing in a field at R,andthac at S wcrclbmc eminent mark, (asaTrcc,Church,Hou(b,orfuchlike) and that it were required to finde the diftance between R and S. Firft, place your Table at R, and thereon affigne any point at pleafure , unto which point apply the edge of your Index, turning it about upon that point, till through the fights you cipic the mark at S,and draw a line by the fide of the In icXjas R S. Tnen in fomc other convenient place of the field (as at Q^)lct a ftaftc or other mark becrefted, and the Table remaining asbe- fore,turn the Index about, till through the fights you cfpie the mark at Q^, drawing a line by the fide thereof, asRC^., lohavc yeu Jeferibed upon your Table an angle equal to the angle S, Then (with your Chain) meafure the diftance Q^R, which let be 176 loot, then take with your Compafles 176 out of any Scale, and fet it upon your Table from C to Q , fo (hall this point Qjip- on your Table, reprefent the mark at ^in the field. This donc,{ct up a Half at R,and remove your Table to lay¬ ing the Index upon the line and holding it faft there, turn the whole Table about till through the fights you cfpic tlie mark let up atyour former place of (landing at R: thenferew the Tabic fall, and lay the Index on the point turning it about, till through the fights you efpic your mark at S, then draw a line by the fide of the Index, which will cut the line R S ^firft draw) in the point S. CL. I By this means (hall you have upon your Table a Triangle equal j j to the Triangle ^RS, the correfpondent (Ides and angles thereof j being proportionally equal with thofc in the field ; therefore if , with your Compalles you take the length of the fide RS, & apply ; that diftance to the fame Scale from wh ccc you took the fide^gR, ! you (hall find it to contain 400 foot, and that is the diftance be- j tween Rand S. Likewife, if you take with your Compalfesthc j length of the line ^,and apply it to the fame Scale,you (hall find j it to contain almoft 303,and fo many foot is the diftance ! In this manner (liall the diftance between any two places be ^ meafurcd,although they be fo fituated, that by rcafon of' , water or other impediments you cannot approach necrun- • ‘ to. And here note, that when you take your fecond ftation, ■ I that you take it as large as the ground will permit, fo (hall | _ _yp'i'’. :Lib.4. Injirmentsin Surveying. 189 1 j your work be fo much the truer,by now much the diftance j taken is the larger. ; CHAP. VI. Hoip 10 taJ^ean mccejpble difiance at two Jlations by Theodolite* N the former Diagramdet R and two ftations/roth either of which it is required to finde the diftance to S. Inftrument at,Rjlaying the Index and fights upon the Diameter thereof turning the whole Jn- firument about, till throligh the fights you cfpie your fecond fta- tion at and there fcKwitfaft, then turn the Index about upon ■ the center, till through the fights ybiicfpy the mark at S , noting ; the degrees cut by the Index, which fuppofc45dcgreesiomi- nutes.Then remove your inftrument to laying the Index on the Diameter thereof, and holding it there turn the whole inftrument about,till through the fights ycu cfpie your mark at S, and fixing the Inftrument thcre,turn the Index about till through the fights you fee the riiark fet up at yoiir former ftation at R, noting the de¬ grees there cut, which let be 110 degrees 30 minutes. This done, mcafurc the diftance of your two ftations which Ictbciytf fcct,fo in the Oblique angled Triangle ^ R,you have given, (i ) : the angles R.Sj45 degrees to minutes, the angle obferved at ' your firft ftation. (j) the angle R^S , no degrees 50 minutes, which was the angle obferved at your fecond ftation. And (3) you have given the fide R ^ 176 foot, which is the diftance ol your two ftations: artd you arc to find the t\vo other fides R S, and ^S which you niay find by the 7 Cafe of the 4th Chapter of the 3 Bock, in this manner: for. Having the two angles R^and R^ given you have alfo the third angle RS,g_given, 24 degrees 20 minutes, it being the complement of the other two to 180 degrees tby the 17th Chap, of the 3d, Then to find the other t\vo fides, theproportion is j I. For the fide ^S. As the fine of the angle R S.^, 24 degrees 20 minutes. Is to the Logarithm of the fide R ,^175 foot, So is the fine ofihcangle^R S 4J degrees,iominutcs. To the Logarithm of the fide .^,303 iooiferh i 1 . For the fide R S. As the fine of the angle ^R Sj 45 degrees 10 tnimites, Is to the Logarithm of the fide ^S, 303 foot. So is the fine of the anglcR i to d.30 min. (or 69 d.36.m.) To the Logarithm of the fide R S, 400 foot. vVhich is the diftance required. '____ €.1 Tbenfeof fever al f[ I have been larger upoiulus particular then I intended, ha¬ ving fufficicntly infilled thereon before in the dinienfion of plain Triangles) but that tlie Reader may fully underRand thefc neceflary conclufionsd have in this example iifed all th.e perfpicuity I could imagine, fo that in the fubicquent Chap¬ ters I may be the briefer, for this being well underllood, i.e may eafily apprehend any ot the other at the firft view. I CHAP. VII. Hon^ to taJ^ aniri'accejfible difiance at mo flam ons by the Circumferentor. - (Et it be required to find the diftance from R and Qjo ‘S. Firft, place your inftrument at R, anddiredlti’.e fights to S,obfcrving what degrees the South end of the Needle cutteth, which let be 315 degrees 30 min. then turning the Inftrument about, direft the fights to Qpbfcrving what degrees the Needle there cutteth, which let be lyodegrccs 20minutcs,therfore from 315 degrees 30 minutes, fubftradl 270 degrees zominutes, and there will remain ^5 de¬ grees 10 minutes, which is the quantity of the angle S R Then remove the Inftrument toQ^, and direift the fights to R, the Needle cutting 91 degrees 00 minutes, alfo, direft the fights toS,theNecdIccutting 340degrees30minutes, now ifyou lub- ftraftpi degrees 00 minutes,from 340 degrees 30 minutes, the re¬ mainder is 249 degrees 30 minutes, which becaufe it exccedcth 180 degrees) fubftradt from 360 degrees, and there remains no degrees 30 min. the true quantity of the angle R QS. Having thus obtained the two angles RQS and SRQ^, you muft meafurc the ftationary diftance Q^R 176 foot, fo have you given in the Tringle C^R S , (1) the angle R QS 110 degrees 50 minutes, (2) the angle ^R $,45 degrees 10 minutes, (3) the angle QS R, 20 degrees 10 minutes, (the complement of the other) two to iSo degrees, and (4) the ftationary diftance QR 176 foot, whereby you may find the other fides and RS, according to the doftrine delivered in tlw foregoing Chapter. deg. min. ^315 30 Firft ftation at R, degrees cut/ tiyo 20 The quantity of the angle QJlS 45 10 Second jLi B.4. Inftrwneats of SuTv^ing. 191 ^ deg, min,. Cj40 30 Second Nation at Q^j degrees our< t 91 00 , 249 30 450 00 The quantity of the angle RQi loo 30 The ftationary diftance 176 footi Having thefe things given, if you rcfolve the Triangle Q^S, ; you fliall find the fide R S to contain 400 footjand the fide QS 303 1 foot/^re, as in the laft Chapter. CHAP. VIII. How tofrotraB or lay down a DiHance ac¬ cording to the direBions of the two lajl Chapters, upon faper, by help of your FrotraBor or line of Chords, Hen you 'make any obfervations in Theodolite or Circumferentor, youaretonotCMVvm the quantities of the feveral lines and angles obferve&' in the field, in a Book or paper, fo that they may Re: ready at hand when you come to protraftion^andi^- istheiifualway. Suppofc it were required to draw upon paper the uuefymctrycr proportion ol the diftance taken inme lalfChap- ter. •* Firft, upon your paper draw a line atlengtqSsi^<^,^n,upon one end thereof, as at R, place the center of your Protract j and lay the Meridian line E F of the Protraftor, d^wfy'jpon the line CVR; then, (bccaufe the angle ^R S is 45 degrees lo minutes, therrorc,againft 45 degrees lOminutes ofyour Protra(ftor,makea mark upon your paper with your protrtdling pin (a^ is before taught C/m^. 2.) and draw the line RS.This done, frora/Sny Scale, ■__ __take i86 T beUfe of fei)^d L i b. 4.| I take your flationary diftanccR 0,176 foot, and fet it from Rto Qj^Then upon the point Q,(bccaufe the angle RQ,S contains 110 degrees 30 minutes) place the center of the Protra6for,and turn it about till the line U QJie direftly under i to degrees, then (at the pointEofehe Protradfor) make a mark with your protrading l)in, and through tiiat point draw the line QS , which will cut the line R S in the point S : then it you mcafurc the length of the lines QS and R S, by the farn'e Scale from whence you took 176 for the line QR,you lliall find the line QS to contain t05,and the line R S to contain 40o,c\adly agreeing with the numbers found in the laft Chapter. Hon^ totaJ^e the altitude of my fower^ Tree, Stceph, of the (being accejjibleJ by the S Av'ihg in the fifth SOfioit of the fourth Chapter of this Hook, (hewn how to obferve an angle of Altitude by the Label and Tangent Lincjwc now come to the fur¬ ther ufc thereof. Suppofe therefore that the line C A were a Tree, ' Tbwfcj Steeple,'of other thing, whofe height were required. JFirft, placeyoiirlnffruinent at any convenient diftance from : thbBafeor foot of thedbjbft to bemcafured, asatB, and there looking through the fights of the Label, till you efpy the top of the j Altitude at C, note what degrees of the Tangent Line is cut by j the Label, for that is the quantity of the angle of Altitude, name- j ly, theangleCBA,whichfuppofc30degrees: then istheothcr 1 angle B C A 60 degrees, it being the complement of the former i to 50 degrees. Then ! |Li B.4. InfirUmentsin Sutveying, 199 Thcn(\vith yonr Chain orothenvife) mcafurc the diftance from B(thc place of yourftanding) to A 5 the foot of the thing to be nicafured,) which (uppofc 400 foot: Then in the Triangle ABC, there is given())tlie angle C BA 30 degrees, (z)che angle B C A, 60 dcgrces,and(,3)the Siftance B A 400 foot, and it is required to ^ findc the fide C A,by the t Cafe of right angled plain Triangles: ; For, i As the fine of the angle B C A, 60 degrees, I Is to the Logarithm of the fide B A 40c foot; So is the fine ofthe angle C B A 30 degrees, , To the Logarithm of the fide C A. } This proportion being wrought according to the former dirc- (ftions, the fide C A will be found to contain alraoft 231 foot, and I that is the height of C A required. CHAP, X. Hon^tofrotra& or lay downu^on fa^cr, tbeoh^ fervation made in thelaji Chapter. Aving drawn a line upon your paper as B A, place the ^center of the Protradfor upon B, now (becaufe wh.en ® y®”*" obfervation at B,the di^rces cut were ^^^^3c)turn the Protradlor about till the line BA lie juft ^^^^r®^undcr 30degrees, then (withyourprotradfing pin) make a mark by the edge of your Protradfor againft 00 degrees,& draw tlie lineBC.folTiall the angle C B A contein 3odegf. Then (becaufe the mcafurcd diftanccB A was 4C0 foot,)take 400 from any of your S calcs of equal parts,and fet that diftance from B to A and from tl'.c point AjCrcdf the perpendicular AC,which perpen¬ dicular being taken in your Compaffes, and mcafurcd upon the fame Scale from whence the 400 foot was taken, you idiall find it tocontcinalmoft23ifoot,and fomuch is the altitude C A as be¬ fore. i Cc CHAP, 194 Ths ufe of fe veral ^ CHAP. XL Horn to take an inaccejfihle Altitude^by the Label and Tangent line, M Or the cffcain;;; hereof you muft make two obfervations with your Infirument. ^t the line B C in this figure re- prefent fome Objeft whofe. height is required: Firft, place your Inftrument at/^ricl direft the fights to Bjthe top of the objc(ft, noting what dqg^’s of the Tangent line are cut by the Label, which let be yo.de^ees, the quantity of the angle BAC. Now, bccaufe you;cannot comctomearurcthcdiftance from Ato Cjbvr^alpnof f9me iRiver or other impedin^ent lying between A and,C:(tl>^«f^e,with your Chain,meafure outfrom A towards C, 4 hy nutijli^^.Qf tecc, according as the ground will per- raic,as from A to Djwl^e^ fuppofe to be 20 o footjand at D, place 'vr*. 3^A yotir Inftrument again, au J there pblef y'c tjio’cjuaqdtypf the aiigle BDC , which fuppofe tobetfVdtgi^e^ thef(:ctvcif4lji|ts b^^ known,the two oppofite angles ar^ alfo knqW’n,' for't^‘d^?^BAC i being 5odegrecs,thewholeapgifi'A;Bd. muftbe 4b^^^®,"thc 1 complement of the fornier to 90 efegreos; againj the D C j being 64 degrees, the angle DflC muft be the compl^tncnc there¬ of, namely 25 de^rces,t 6 en ifybp'ltibfti'aA’the angle'DBC i6 de¬ grees,from the whole angle A frC 49 degrees, there will remain ; 14 degrees for the angle A B E)i t»y the knowledge whereof you | _ ma . Lib.4- Injlrumeiitsm Surveying. may atccin the .ildcL'.-ic; DC; for, iiul'.c Triangle ADD you have ' eiven, 1 Tlic angle D A Djjo degrees, i 2 Ti'ic angle A B D, 14 degrees, j 3 Tile diiiance A D i 00 foot, i VVliith (hy ti.e former diicAions) will help you to finde the I fcivathoftlicri.'e DB,eitlicrbytlicTablcs in the 3 Soc^,orby the Lilies ofartificial Numbers, .Sines andTangents on the Index of . your Table, as is formerly taught, the proportiou being, i As tiie fineol tiieair.le A B D, 14 degrees, 1 Is to tl'.e Louarithm ot the fide A D, 2 co foot; ! So is the line of the angle B A D, 50 degrees, ' To tie Louarithm of the fide D B. ; Which by woikingaceording to the former diredion^ will be ) ; found to be d3 3 loot! i i Tiicn mull you m.ikc a fecond work in the Triangle B C D,in which you liave gh cn, . 1 The angle B DC, 64 degrees, 2 The angle D B C 2 6 degrees, 3 Tl'.e fide DB, 033 foot. And you arc to find the fide of BC, the altitude required, wherefore fay again, As the fine of the angle BCD, 90 degrees, I, / ■ Is to the Lorarithm of the fide D B <533 footj ' 'i.j So is the fine of the angle B D C 64 degrees, To the Logarithm of the Altitude BC ; VVhich according to the former Doftrinc will be found to be 569 foot. ^ CHAP. XII. Hoip to VioiraB the chjervation taJ^n in the lafi Chapter, ! J Hep you have made your obfervationasinthelaft Chap- ; ter, and noted downinaBook orotherwife, that the de- I .igreescutyourfirftftationat Awerc50, and the degree I cutatthcftcond ftationatD were 64, and that your ftationaric , diftance-AD was 200 foot, you may immediately find tlic Alti- ! tude B C by protradion, thus, ' Firft, draw a line as A C, in which line let A reprefent your firft flation,whcrcon lay the center of your Protraflor, and make , the aiwlc B A C to coincin 50 dee. (as hatli been leveral times be- i C c fore \^6 I'beufeof feveral L1B.4. fore fhewnOaikl draw the line AB. Then upon the line AC fee oft thediftanceof your two ftacions z^o foot from A toDj then bring your Protractor to D (which reprefents your fccon J ftation) and placing the center of your Protractor thereupon, fet oft'an an¬ gle of 64 degrees, as B DC, and draw the line DB then where tnefe two lines A Band D B inierlcft or meet, which is in the point B, from that point let fall the perpendicular B C, the lenght vvhcreofbcingmealured upon the fame Scale from whence you tookc the diftance AD, will give you 569 loot, and that is the altitude of A B, which was required. CHAP. XIII. nojp to tal\e the dijiance ofdivers f luces one from anotherj according to their true femationy in PianOj and to mal^C^ M ' ; ofj by the Plain Table. IC'ff His Propolition is ofgoodufeto deferibe in/'/a/so the moft eminent places in a Town or City, and to Wn Kw (as it were) a Map thereof. Let A B C D E BfiCT E G. be certain eminent places Icituate in fome Town or City, and let it be required to defenbe ®®®*®*all thofe places upon paper, by which diftance of any of them one from another, may be readily found. r At fome convenient diftance horn the City, Town, or Field, make choice of two other convenient places as K and L, from either ofwhich you may plainly difeern all the marks whicii you intend to deferibe in your Map. Then, at one of thefe places, (as i at K) place your Table and neere one of the lides thereof draw a I line parallel to the edge of the Table; In this line aftigne any point as K for your firft ftation, and laying the Indeinipon this line, turn the Table about, till through the fights you efpie the other place which you intend for your fecond ftation, which found, fcrevvthe j Table fall there. Then laying the Index to the point K, turn it about, till through ‘ the fights you hfpie your firft mark at A, and by the fide ot tKe ! Index draw the line A K. Secondly, turn the Index to the Iccoad ' mark at B, and draw the line BK. Thirdly, direftyour fights to j C, and draw the line CK. Fourthly, direft your fights to D, and ) I draw the line DK. Fifthly, dired the fights to E, and draw the j j lincEK. Sixthly, direft the fights to F, and draw the lineKF. i Laftly,dire6l the fights to G,and draw the line KG, fo have you • finifticd your work at yonr firft ftation. ___Thu; 1L1B.4. Injirmentsin Surveying. iy; This done, with your Cliainj meafurerhc diftanct ofyourtwo ftationsK and L, which fuppoie to contain Soo foot) aad remo¬ ving yci I TaLIetoL,]ay thelndcxuponthcIincKL, turning the Tablc-cltrt rillthroughthcfightyoufec yourfirft ftation acK, and there ferew it faft fo that i t alter not fo long as yout \Vork con- tinueth. Tlicn laying the Index to the point Lj direft your fights to the feverall marks asbefore,namcly)toACBF DEG, and from each ofthofe marks draw lines by the fide of the Indfeii, as A Lj C L, B L, F L) D Lj E L, and G L, fo is your work ffnirtied at your fccond ftation alfo. Having thus done, firftobferve where thclinelvAcroffeth the line L A,which is at Ajat which point you may draw the figure or write the name of the thing which it reprefenteth. Sccondlyjob- ferv e where the line KB crofleth the lincL B, which is at B,at which point write the name of the place as before. Thirdly, oblcrve where the linesKC and LCinterfeft, whicii is at C, at which point alfo note the place, Fourthly, at the interfeftion ol K D and L D which is at D, write the name of the place as before. Do thus with all the reft of the places he they never fo many, fo j fhallthe feveral points of inrerfeftion A B C D E F G upon your Table, reprefent tiic refpedive places in the Town or City. Now to know the diftanceofany ol thefe places one from ano -1 thcr,youmufttakethcdiftancerequiredijiyourGompa(res, andi j apply it to the fame Scale by which the ftationarie diftanceKL* was laid down, and it will there fticw you the diftancc required, j I CHAP.' '/Tier ufe of feveral L i b. 4 . i i CHAP. XIV. I Hoivio I erf me themrl^ofih, lajiChd^terly ! the Theodolite. S intl'.c lap. CiiptcTj niakcdioia' of two places, from “ cither ol whicti you may coiivenicnrly fa-ailchofc Marks which you intend to delcribc, winch two places _, _ let beKanJ L. Then placing thclnftrumcnt at K, lay thclndcxonche Diameter thereof, and turn the wholelnllrumen'c about till through the fights you dpic your lecond liation at L; then fixini the Inftriimcnt there, dired your fights to the fevcral marks ABC D E F G,obfcrving what degrees tiie Indexcuttech when direded to any of the marks intended. As.luppofi, your inftrument being fixed at K,and the fights direded tc<-A, the Index cuts 83 degrees 50 minutesjat F, 97 degrees 5) minutes; at C, 114 degrees lominures; at D, 113 degrees 4c minutes; at E, I34de2rees35 minutes;atF3i38degrecs30minutes ;and at G, 155 degrees,20 minutes. Then removing your Inftrument to L, lay theindexon iheDi- ' ameter thereof, and turn it about till through the fights you cfpie your former ftationatK, as is before taught: Then diredting the fights to your firft mark A, the Index cuts 33 degrees 30 minutes; at C,43 degrees 40 minutes; at B, 54 degrees 10 minutes; at F, 64 degrees;atD,73 degrees20minutes;at E 87 degrees ij mi¬ nutes; and at G, 113 degrees 40 minutes. Thefc feveralobfervations of the degrees cut by the Index at both ftations, Bught to be noted in a Book or paper, together with the ftationarie diftance, as in this example. rA dtg: 83 50 IB 97 55 |c iM 10 Firft ftatio^D 123 40 134 35 138 30 [g 1 J 5 20 The Stationarie diftance 800 Foot. fA 33 50 \n 43 40 |c 54 10 Second ftation reprefentingyour two ftatiops, then up¬ on yoiirfirft ftation at Kj lay the Center of your Protraftor, with the Meridian line therof (which is noted with EF) diredlly upon the IjncTCL. Then lay the Table of your obfervations before you, and feeing that at yourfirft obfervation the Index cut Xj degrees 50minutes, you muft therefore with your protrafting pin make a mark againft 8 ? donees 50 minutes ol your Protraftor. Again, feeing that at your fecond obfervation the Indek cut 97 degrees 5 5 minutesjthcrefore, with your protrafting pin, make a mark upon your paperjagainft 97 degrees 5 5 minutes of your Protraftor. And thirdly/ccing that at your third obfervatioii your Index cut 114 degrees 10 minutes. you mufl; likewife make a mark againft 114 degrees lominutes, 200 T be nfe of feverd L ^ b, 4 J and thus mud you do with all the reft ol your obRrvatiorvs, be i thev never lb many. Which bein^ done, I'rom the point or fta- j tion K, you muft draw th.c ftraightlines K A? K B, K C,K DjJkc. j Then remove your Procraftor to Lj which lignifies your lecond : ftation, laying tlie Meridian line thereof upon the lincKL^ and then by your Table, note the angles of your obfervations made at your fecond ftation in all refpefts as you did thole of your ' firft ftation: fothall you find that atthc'firft obfervatiou at your i fecond ftation 5 theIndeK cut 3 5 degrees 50 minutes, theieiorc, with your protrafting pin make a mark upon tiie paper againft 3 3 ; degrees 50 minutes of the Protraftor. Again, the degrees cut at your fecond obfervatiou were 43 degrees 40 minutes, therefore make a mark againft 43 degrees 40 minutes of your Protractor, Alio,the degrees cut at your third obfervation were 5 4 degrees i o minutes,againft which likewife make a mark,dcaling with all t!ic reft of your obfervations in the fame manner: then through tliefc fevcral points, from your ftation L,draw ftraight lines till they in- ferfed thofe lines before drawn from K, which will be the poiiits ABC DEF andG, which points bear a juft proportion to the Marks which you obferved. Now to find the diftanceof any of thefc marks one from ano¬ ther, you muft divide a line into luch equal parts, fo that your fta- tionarie diftance K L may contain 800 of them. Your Scale be¬ ing thus made, take in your Compaffesthe diftance between any two marks or places here deferibed, and apply it to your Scale fo fliall it exadly fliew you the true diftance between the two places fo taken,in the fame parts as the line K L was divided. In this manner may you with fpcedand exadnelfe attein the true diftance and fituation of anv Mark or Marks far remote, without approaching neer any of them: and thus in over-grown land,where you can neither go about it, nor mcafure witl.in it, this Chapter will be of excellent ufe. f[ I might here infert divers other Cafes concerning the taking ofHeights andDiftanccsjasjdivcrs places lying in the fume right line to find thair diftance; or, part of a Diftance or Altitude being given, to find the whole, with infinite o- therof that nature, but feeing that thefc are but pans or branches ofvvhat is here delivered, and arc rather Pro- blcmcs of curiofity then ufe, I will therefore paffc them o- ver, and the rather, becaufc thefc being rightly undcr- ftood, the performance of any other will be very cafie. But remember alwayesin taking of inacccflible Heights and Dirtances, as alfo in the plotting of un-paficable grounds, that you take this your ftationaric diftance as as may be. And if at any time you be required to the alti¬ tude of a Caftle, Church or Tree, ftanding on a Hill, you you muft perform it at two operations, firft, by taking the altitude of the Caftle and Hill together as one altitude, andfecondly, by taking the height of the Hill alone; then _ __ _ _ by fiiliihaftini tlic Iici^hc of tlie Hill from the whole heights the ri mainJer lhall be the hciiiht of the Caftle. And here notealfo, that in the taking ofall manner of Altitudes, whe¬ ther actellible or inacceifible, you muft alvvayes adde the height foundjihe height of your inftrument from the ground. CHAP. XVI. Howtotah^ethetrue plot of afield at one Ration fallen mthki th famefieldy fo that from thence you may fee all the angles of the fame field by the Plain Table, ■ Hen you enter any field to furvey, your firft work muft be to fet up fomc vifible mark at each angle thereof, or let one go continually before you to every angle, holdingupa white cloth, or the like, to direft bcingdone,makechoiccoffomeconvcnicntpIacea- boutthemiddlc'ofthe field, from whence you may behold all your Marks, and there place your Table covered with a fticet of paper, the needle hanging direftly over the Meridiau line of the Card (which you mufi alwayes have regard unto, efpccially when you arc to furvey many fields together). Then make a marke 1 about themiddleofyourpapcr,which iliall reprefent that part I of the field where your Table ftanJeth, and laying the Index unto ■ I this point, dirccft your fights to the fcverall angles where you 1 I before placed your marks, and draw lines by the fide of the Index j Dd upon The uje of fe veral L i b. 4J upon chc paper-, then meafnre the Jiftance of every of thefe marks from your Table, anvi by your Scale let the lame diftanccs upon the lines drawn upon the Table, makiivr Imall marks with your Protraftino pin or CompalTc point at the end of every of them; then lines being drawn from one to another of thefe points, you lliall have upon your Table the exaft plot of your Field, all the lines and angles upon the Table being proportional to thole of the Field. Suppolcyou wore to take the Plot of the Field ABC DEF. Having placed marks in the fcvcral angles thereof, make choife of lomc convenient place about themiddle of the Field, asat L, Irom whence you may behold all the marks before placed in the fcveral angles, and there place your Table, then turn your Inftru- mentabout, till the needle hang over the Meridian line of the Card,the North end of which line is noted with a Flower-de-luce, and is reprefented in this figure by rhe line N S. Your Table being thus placed, with a llicet of paper thereup¬ on, make a mark about the middled your Table which fliallre- prefent rhat place in the field where your Table ftandoth : then, applying your Index to thispoint,dirc£l the fights to the firft mark j at A, and the Index refting there, draw a line by the fide thereof j to the point L, then with your Chain meafure the difiance from I L, the place where your Table ftandeth, to A your firll mark, which luppofe to be 8 Chains lo Links, then take 8 Chains 10 Links from any Scale, andfet that diftance upon your Table from L to A, and at A make a mark, [ Then direding the fights to B the fecond mark, draw a line by [ the fide of your Index as before, and meafure the diftance from your Table atL, toyour markatB, which fuppofe 8Chains75 links, this diftance muftbc taken from your Scale, andfet upon your Tabic from L to B, and at B make another mark. Then direft the fights to the third markC, and draw a line by ! the fide of the Index, meafuring the diftance from L to C, which ^ fuppofe 10 Chains65 links; thisdiftan.cbeing taken fromyour Scale and applycd to your Table from LtoC, iBall give you the ; point C, reprefenting your third mark, ' In this manner youmuft deale with the reft of the marks at D E and F, and more, if the field had confifted of more angles. Laftly,when you have made obfervation of all the marks round . the Field,and found the points A B C D E and F upon yourTable, you muft draw lines from one point to another till you conclude ■ where you firft began: as draw a line from A to B, from BtoC, j from C to D, from D to E, from E to F, and fromF to A, where | you begamthen will A B CD E F be the exaft figure of your field, j , the fides and angles of the faid figure bearing anexaft proporti- ' on to thofe in the Field, and the line N S, in this andtl»e follow- i ing figures,alwaycs reprefenteth the Meridian line. I CHAP. I i CHAP. XVII. Horp to the^lotof afieldat one Ration taf^o in the middle thereof by the Tlicodolitt ^Lacc marks at die feveral angles ofthc field as before " and make choice of fomc convenient place about die middle xhercof,as L,from whence you may fee all thcj marics, and there place your Inftrumchtj the Needle hanging dire(aiy over the Meridian line in tlie Card. Thisdoncj dired your fights to the firft mark at A, noting what degrees the Index cutteth, which let be 56 degrees 45 mi¬ nutes, tl.elcj^ degrees 45 minutes mufl be noted down in your field-book in the firft and fecond Columns tlicrcof. Tiicn meafure the diftance from L the place ofyour !hftrument,to A your firft mark, which let contain 6 Chains to Links, thefe 8 Chains toj Links muft be placed in the third and fourth Column of your field -1 book, as hath been direiftcd in the defeription thereof. Then dircft the figlns to B your fecond mark, and note tlie de- g-eescutby the Index, whichlct be 99 degrees ij minutes, and the diftance LBS chains 75 links, the 99 drerecs 15 minutes muft be noted in the firft and fecond Columns of ^our field-book, and ] the 8 Chains 75 Links in the third and fourth Columns. Dd 2 Then 2'g4 7 be ufe of feveral L i b. 4^ Then direct your fights to C, your third mark, and note the de¬ grees cut by the Index, which let be 16 ? degrees 1 5 minutes, and JecthcdiftanceLCbeioChains65LinkSithe i 6 j degrees 15 minutes muft be noted in the firft and (econd columns of your field book,and the 1 o Chains 65 Links in the third and fourth columns thereof. Then dired your fights to D,your fourth mark, and note the degrees cut by the Index; which let be a i i degrees: |[ And here you mull note chat in ufing the degrees on the frame of the Table, that after the Index hath palfed 180 de¬ grees, which is at the line N S (reprefenting alwayes the Meridian line) you muft then count the degrees backward; according as they are numbered on tlie frame of the Table from 1^0103^0. Then meafure the diftance L D, which let be 8 Cliains 5 3 Links; the a I z degrees muft be noted in the firft Columns of your field- book, and the 8 Chains 5 3 Links in the third and fourth Columns thereof. Then direft your fights to E, the Index cutting 287 degrees 15 minutes, and the diftance L E being 8 Chains 15 Links, tbefe • muft be noted in your field-book as before, the 287 degrees xy minutes in the firft and fecond columns, and the 8 Chains ij Links in the third and fourth. Laftly, direft the fights to F, your laft mark, the Index cutting 342 degrees, and the diftance L F being 9 Chains 5 s Links, thefe muft be noted down in your field-book in all refpefts as the for¬ mer,!//^, the 341 degrees in the firft column, and the 6 Chains S5 Links in the third and fourth; then will your obfervations noted in your field-book Hand as in this Table following. Degrees Minutes 1 Chains. Links 36 45 8 10 99 IS 8 75 163 15 lO 65 212 1 00 8 53 287 1 8 IS 34 » 1 00 9 55 Chap. |Li B.4. Injimmentsm Surv^ing, loj' CHAP. XVIII. Hoiv to tal^ the f lot of1 Field at one fiation ta^fn in the middle thereof by the Circumferentor. ^^assaiat Here is little difference between the work of this and the ^ Chapter: for, the marks being placed in the feve- ^ ^ ral angles of the field, and the ftation appointed at L> place mere the Inftrument, and turning it about, direft thefights to A(theFlowcr-de-luceof the Card being alwayes to wardsyou)the South end of the Needle cutting j6 degrees 45 minutes the fame which the Index of the Theodolite did in the laftChaptcr,thenmeafuringthediftance fromL to A, you will findeittocontain,asbefore,8Chains 10Links,which you muft note down in your field-book as m the laft Chapter. Then turning the whole Inftrument about (as before) dire^ the fights toB,thc South end oftheNccdlccuttngp^degrees 15 mi¬ nutes,and the diftanceLB will contain 8 Chains 75 Links,which note down in your Book alfo. In this manner muft you direft the fights to all the other angles | CD EandF,and7ou fhall finds the South end of the I^ce^e ^ • al- IC^ "Iheitfeof fiveral LiB.4| alwaycs to cut the fame degrees in tlie Card as the Index of the Theodolite did, and the meafured lines L C, L D, L E, and L F, will be likewife the fame, fothat the Table ol obfervations in the laft Chapter will ferveto protraft cither tliis or the other work, as is taught in the next Chapter. CHAP. XIX. Horp to frotra^ any obfer vations taf\en according to the direUions in the laft Chapter. ■ Irft, draw upon your paper or parchment a line at length, which niallreprefcnt the Meridian lines NS in the fi¬ gure, then makcnoicc of fome point or other in tliat line which fhallreprefent your ftation or place of ftandingin the field,as K: upon this point place the center of your Protrador, lo that the Meridian line E F of the Protraftor, may lie diredlly upon the Meridian line N S ol this figure. Then laying your field-book before you; feeing chat at your firftobfervationat A,theIndcxofthcThcodolitc,or the Needle of the Circumferentor, cut 36 degrees 45 minutes, you muft therefore againft 36 degrees 45 minutes ol your Protrador make a marke upon your paper. 2 Seeing the degrees cut at your fecond obfervation were gg degrees 15 minutes, you muft mark upon your paper againft gg degrees 15 minutes of your Protraftor. ^ The degrees cut at your third obfervation were i6y degrees i^ imnutes, therefore againft 163 degrees 15 minutes make a m ^ upon your paper. ' 4 The degrees cut by the Index or Needle at your fourth obfer¬ vation being 212 degrees —^ |[ Now becaufe 212 degrees is greater then 180 degrees you muft therefore turn the Semicircle of the Protraiftor downwards, yet the line E F thereof muft lie direftly upon the Meridian line N S, as before. j ==you muft againft 212 degrees of the Protraftor make a mark upon your paper. ) Seeing the degrees cut at your fifth obfervation were 287 deg. i-j minutes : therefore make a mark againft 287 degrees ij mi¬ nutes of the Protraftor. Laltly, the degrees cut at your laft obf rvation were 342, tliere- lore againft 342 degrees ofjour Protraftor make a mark with your Protrading pin, as before. .. proiradcd all the degrcccs of your feveral obferva- lions, take away your Protrador, and laying a ruler to the point!, - _ __ draw |Lib.4. Inflrumentsin Purveying, draw obfaire lines from L through thofe points, which lines will be L A, L B, L C, L D, L E, and L F. This done, youmuftobferveby your field-book the length ol every line As the line L A at your firft obfervation was 8 Chains lo Links, therefore, 8 Chains 10 Links being taken from your Scale and fet upon your paper from L to A, it fliall give you the point A upon your paper. 2 The length of your fccond line being 8 Chains 7 j Links, you muft take8Chains75 Links from your Scale, and fet it upon your paper from L to B. 3 The line LC being loChainsdy Links, you muft therefore take 10Chains 65 Links from your Scale, and fet it upon your paper from L to C. And thus muft you dealc with all the reft of the lines, as L D, LE,andLF. Laftly 3 draw the lines A B, B G, C D, D E, E F, and F A, fo (hall you have the exaft figure of the field upon your paper. C In thefe four laft C hapters you are taught how to take the plot ofany field at one ftation taken in the midft thereofrboth ^ the Plain Table, Theodolite, and Circumferentor, andaHo how to protraft the fame. This way of plotting of a field is feldome, or never, ufed in furveying of divers parcels, but for i hii nfe of feverd for one particular field k is as good as any, but divers other var.eries vvillappearein tdefollowing Cnaptcrs. CHAP. XX. Hoa? to ta\e the f lot of a Field at onejiation in any an^le thereof from whence all the other anglesmiiy be feen^ by the Plain 1 able. t Lace your Table in feme convenient angle in the field to be meafuredjand turn it about till tiie Needle i.ang direiSly over the Meridian line in the Card,and there fix it: then draw a line parallel to the fide cf your Table, as N S: in which lineafllgne any point at plealurc, as H, which ikallrcprelent your ftation or place of ftanding, unto this pointapply theindex, and direct the lights to A and draw a line upon your paper as H A; and meafure the difiance H A (as was dire£led before in Ck-i/). 16.) Then direft the fights to B, your (e- cond mark, and there likewife draw aline HB, mealuring the diftance A B, as was taught in the fore-mentioned Chapter. In like manner direft the fights to C D E F and G,drawing lines by the fide of your Index at every obfervation, and meafure witli your Cliain the diftance from H (the place where your Inftru- ment ftandeth) to the feverall angles of the field A,B,C,D,E,F, and G j which diftanccs being taken in your Compaffes, from any Scale, and fet upon your Table from H upon the feveral lines H A H B, H C, H D, H E, H F, and H G, fo Ikall you have upon your Table the points A,B,C,D,E,F, and G, by which marks draw the lines HA,AB,BC,CD,DE,EF,FG,andGH,vvhich lines will , include the exaft figure of the field upon your Table. . 1 CHAP. XXI. How to tal\e the p/o^ of a Field at one fation ta- in any angle thereof by the Theodolite. the fame figure following, having placed your Inftru- ment at H, as is taught in the fore-going Chapter direft the fights to A, your firft mark, noting the de- ^ grees cut by the Index, which fuppofe la degrees 15 minutes, thefe degrees and minutes muft be noted in the firft and fecond columns of your field-book (as hath been before fufficientlytaught.)Then with your Chain meafure the diftance fromyourftationatHtotheangleA, which let be 8 Chains 46 Links, which you muft place in the third and fourth columns of your field-book, according to the former diredtions. 2. Direft Li B.4. Injlrumemsin Surveywg. I 2 Direa yourfights to B noting, the degrees/there cut, which 1 fuppofe 4idegrees,45 minutes, thefe degrees and minutes place Columns 0^ field-book, and me^re tne diftance H B, 15 Chains 21 Link s, and note them down in the third and fourth Columns thereof, 3 Direayom fights to C, the degtees cut being Thc firftftarioq 17 atO. L 28a 00 9 95 M 304 30 8 05j The ftationary diftance O Qjs 8 Chains 8p Links, and die an- gleO^N iSd^rees 15 minutes, the inclination or diderence of Meridians. E 5» 15 3 lol F 99 50 7 ^4 G 14S 30 6 40 ^The fecond fta- H 232 30 5 tionatQj. I 275 CO 6 ,j K 321 30 7 tfij 7 he ufe of fe veral L i b, 4. CHAP. XXVI. Howtouh^e the Tiot of a hieId at Wo (iatioas tiJ^nin any farts tbtreof by the Circumfe¬ rentor. He life ol this Inftrumcnt in taking the plot of a field by obfervini^ the lines and angles in tnc niidft thereof, is fuffiiicntly llicwn already in Chap.iS. & the work ot cliis Chapter difterctli notliin37 30 8 85 lao 30 >3 4 121 30 7 70 126 30 3 61 >43 30 7 87 Gg 5 chap. 230 Tbe ttfe of fe veral L1 b. 4 Chap, xxxiii. • j How to ^rotraBo'^hy down any ohfer vatiom ta * ! h^n according to the doBrine of the Uji Chapter. Onfidcr which way your Plot will extend, and accor¬ dingly upon the paper that you would have the Plot of your field deferibed, draw a line at pleafure, as the line G A. Then place the center of your Protradlor up:n the poiutAjandfbecaufe the angle at your firft obfer- vation at A, was 130 degrees 00 miniues)turn it about till the line AG liedireftly under 130degrees,and then at the beginning of the Protrador (which is at 00 degrees,noted (in the figure there- , of/vi|.5i.)with the letter E,) make a mark, and through it draw • the line A B, letting 12 Chains 5 Links (the length of the fame line) from A to B. 2 Lay the center of your Protraftor upon the point B, and fee¬ ing the degrees cut at B were 120 degrees 30 minutes, therefore turn I he Protraftor about rill the line A B lies direC^ly under 120 degrees 30 minutes, and then at the beginning o( the degrees make a mark,and through it draw the line BC,ihe len'Jththerc- i ofbeing4Chains 4J Links. ° : 3 Lay the center of the Protraftor on the point C, turning it I about till the line BC lies direftly under ijy.de^recs 30 minutes 1 (which were the degrees cut at your obfervatiori at C,)and then making a mark at the beginning or 00 degrees of your Protraftor, thr ugh it draw the linc‘CD,fetting 8 Chains 85 Links thereon from C to D. 4Bifc the center of your Protraftor to the point D, turning itabaut'tillthelineCDliesdireiftly under 120 degrees 30 trir- nutes, and then making a mark at the beginning of the Protraftor, through it draw the line DE, and upon ufetn Chains 4. links ! from D to E. [ In this manner muft you dealc with all the reft of the angles, and when you come to protraft the angle at F, which is thc°laft angle, and have drawn the line FG, youlhallfindcit to cut the line AG firft drawn in the point G, leaving the line A G to con¬ tain 7 Chains 8j Links, and the line F G j Chains 67 Links-, and in this, praftife is better then many words,and the fight of the fi¬ gure better then a whole Chapter of information, in which figure, you may fee the Protraiftor lie at every angle in its true pofition. This work may be performed other wife by protraiftincr your laft obfervation firft fo having drawn the line A G, lay the center of the Protraiftoron G,and the Meridian line thercoRnamcly E F) on the line G A, then (becaufe the degrees cut at your obfervation at G,wcro 143 degrees 30 minutes)make a mark with your pro- traftmgpinagainft 143 degrees 30 minutes and through it draw ; the line G F, upon which line from G to F, fee 5 Chains 67 Links. ' _____ Then j L I B.4. hprumentsin Surveying. 2}i Then placing the center of your Protraftor on the point F, and the Meridian line thereof upon the line F G," making a mark by the edge of the Protraftor^gainft la 6 degrees 30 minutesfwhicn were the degrees cut by the Index at your obfervation a^F, and through that point draw the line F E> fetting 7 Chains 70 Links thereupon from F to E. And in this manner muft you proceed with the reft of the lines and angles, and at laftyou fhall ftnde the plot of your field toclofe at A, as before it did at G, and if the fidcs and angles were never fo many, the manner of the work would be the fame. Here note that if ingoing^abouta field, and mcafuring the angles thereof with the Theodolite or degrees on the frame ofthcTablefas in Chap. 31.) that it you meet with any angle that bendeth inwards in the field, you muft reckon that angle to be fo much above 180 degrees as the bending is, and" when you note the degrees ot Tuch an angle in your j field-book, you may make this > or the like marka- gainft them for a remembrance when you come to pro- trad , and in protrading you muft turn the Semicircle of the Protraftor the contrary way to what yon do in pro- rtading of other angles. CHAp Z'yi Jbenfeof feveral LiB^j ! CHAP. XXXIV. Ho 7 p to J^omivbcther JOH have tal^nibe angles of a Field truly j as in ' AvingmaJcobfcrvationofallthcaujlcs in the field h with your I.nllrumcnt, and noted them down in your field-book as is done in tl'c later end of 31. colleft the quantity of all the angle sfiundat your Ic- verall obfervations into one liimj and naiiltiply i8o deirces by a number lefle by two then the number of angles in the field, and if the product of this multiplication be equa'll to the to- tall Inmmc of your angles, then is your work true, other wife not. EXAMPLE. den, min. In the work of the 3 2 Chap, the angles found .130 00 were as in the margine, the funime of them 120 30 being 500 degrees 00 minutes. Now, bc- 137 30 cattle the field confided of 7 angles, you niiult 120 30 therefore multiply t bo degrees by 5, f which lai 30 is a number lefle by two then the number of 11(5 50 angles in the fieldj and th.e produft will be *43 30 $00, which exadtly agreeing with the fumme ioo 00 of all the angles in the field as you found them byobfervation, you may conclude that your work is exaftly performed. ST This rule is general when all the angles of the field be in¬ ward angles,but if any of them be outward angles, then you are to 1 take the complements of fuch angles to 180 degrees, which num¬ ber of degrees is to be added to the reft of theangleSjand their acr- gregate fliall be equal to theproduft of the multiplication of i8o, by a number Jefs by two then the angles of the field, but thefe out¬ ward angles (be there never fo many in one field) muft not be ac¬ counted as angles, but wholly rejefted. CHAP. XXXV. How to tal\e the Plot of any H 'oody Pay\, or other large Champion plaifti by going about the fame^ and mab^ng ohfervation at every angle thereof Circumferentor. have before fliewn the ufe of the Circumferentor in *?^‘*’gtlicplotofanyfmallinclofurefcvcral wayc 5 ,but fprtliofc kind ofpraftifes the Circumferentor is no con- venicntlnftrumentj the ufe thereof in thofc works was j only intimated, that the agreementof the fcv’ral Inftruments in the performance of the fame thing, might the better appear. Now the !Lib.4. hfinments in Surveying. 2,31 A ! the Circumferentor is a moft abfoluteinftrument for thefurvey- ‘ ingof any large and fpacK us bufmeiTc, as a Park, Wood, oro- i thcr large Common field or Champion plain, the ufe thereof i differing from all that hath hitl.crro been delivered. ! Suppofe then that AB C D E FGH Kvveie a large field or ; other inclofure to be plotted by the Circumferentor. 1 Placingyourlnlfrumcntat A (ti e Flower-de-luce towards you)direct the fights to B, the South end of the Needle cutting ’ ij>i degrees, and the ditch, wall or hedge *A B contabing lO Chains 75 Links, the degrees cut, and the line meafured, mimbe noted down in your field-book as in the foregoing examples. ’ 2 Place your inftrument at B, and dirctS the fights to C, the South end of the Needle cutting 279 degrees and the lineB C con- : lining 6 Chains 6j Links, which note down b your field-book j as before. 3 Place the inftrument at C, and diredf the fights to D, the ' Needle cutting 216 degrees 30 minutesjand the line C D contab- • ing 7 Chains 82 Links. - j 4 Place the Inflrument at D, and diredf the fights to E, the I needle cutting 327 degrees, and the line DE contabing 9 Chabs : 96 Links. i 5 Place the Inftrument at E and direft the fights to F, the j Needle cuting 12 degrees 3c minutes, and the linc£ F containing | 9Cliains7i Links. ' 6 Place the Inftrument at F, and dircift the fights to G, the Needle cutting 342 degrees 30 minutes, and the line F G, con- | tabing 7 Chains 54 Links. 1 7 Place the Inftrument at G, and dired the fights to H; the : 1 Needle cutting 98 degrees 30 minutes, and the line G H contain- i j ing 7 Chabs 5 2 Links. i _ 8 Place z^z T be nfe of fever al L i b. 4. 8 Place tlic Inftrument at and direft the fights to K, the Needle cutting 71 degrees, and the line HK containing 7 Chains 78 Links, 9 Place the luftrumcnt at K, and direft the fights to A ( where you began) the Needle cutting 161 degrees jo minutes, and the line K A containing 8 Chains 2 a Links. A Having gon round the field in this manner, and coUeded tbc degrees cut, and the lines meafured in the fitvcrall columns of your field-book according to former dire&ions, you (hall finde them to Hand as followeth, by which you may protra^ and draw the plot of your field as in the next Chapter. VegrteS. Mimites, Chsins. Links. 191 00 10 75 ^19 00 6 83 216 30 7 82 3*5 00 6 96 12 30 9 71 341 30 7 54 9S 30 7 54 71 00 7 78 161 30 8 22 In I Lib. 4* Inflnments in Surveying;^. 23^ ' In going about a field in this manner, you may perceive a won- j ' derful quick difpatch, for you arc only to take notice of the de- I i gres cut once at every angle, and not to ufe any back-fight as fit 1 i tlie fore going work of thcTlicodolite:but to ul'c back-fights with j I the Circumferentor is tell for to confirm your work; for when j ! you Itand at any oiulc of a field, and direct your fights to the next, ! and obftrve what degrees the South end of the needle cutteih, if I you rcnio\ e yonr Inftrument from this angle to the next, and look I to the mark or angle where it lall flood, with your back- fightsthe ; Needle will tlierc alfo cut the fame degree as before, which ' ought to be done, and may be, without much lode of time.- j Soti elnflriimentbeingplacedat Aifyoudiredl the fights to ; B,you lhall finde the Needle to cut 191 degrees, then removing ■ I yourIn(lrrmt-nttoB,ifyou direct theback-figlns to A, the Needle ; will then alio cut 191 degrees. Now for difpatch and exadtnelTe (if the Needle be good, the Card well divided, and the degrees»by a good eye) truly eftimat- ; cd)tl.e Circi mlerentor, for large and fpacious grounds is as good i as any, and therefore obferve well the manner of protradting. Notwithftandingthcquickdifpatchthis Inftrumentma- keth, there isonc Compendium more whidalwillhere inferr, whereby yif one be taken) the one half of the work will be faved, for whereas (by the diredions ip this Cha¬ pter) you are to place the Inftrument at every angle,it will, be fulncient now to place it, but at every fecond angle, I will inftance in the foregoing example. 1 Placing your Inftrument at A, and diredting the fights to B, you find the Needle to cut 191 degrees; Then j 2 placing tl e Inftrument at B, and ciredling the fights to C, I you find the Needle to cut 279 degrees. And 3 PlaciugthcInftrumentatC, anddiredling the fights toD’ you find the Needle to cut 216 degrees. New, having placedycurlnftnmcntat A, and noted down! the degrees cut by the Needle, which wcrei9i, youncednotgo; to the angle Bat all, but go next to the angle G, and there place! your Inftrument, and dire dime your fights backwards to B, you fh.all find the Needletoeut 279 decrees, which was the famede-' _^_ greesJ z '^6 Tbeufeof fiveral L i B.4, grecs as were before cut when the Inflrunlent was ptaced at B, fo that the labour of placing the inftrument is wholly faved. Then (the Inftrment ftill ftanding at C ) direft the fights to D, and the Needle will cut a 16 degrees, as before 5 which note in your field- book. This done remove your Inftrument to E, and make obferva- tion according to this laftdireftion, and you (hall find your work tobe the fame as beforcjthen remove the Inftrument from E to G, and G toK, and fo to every fecond angle, be there never fo many, and here you fee that half the labour is cleeily faved,and the work the fame, as ii theinfttumenthad been placedac either angle. CHAP. XXXVI. How to ^romU any ohfervations fallen hy the Gircumfcrentor^ according to the doSrtne of the lafl Chapter, ^ Ccording to thclargenefleofyour Plot provide a ftieet ^ of paper or skin of parchment, as L M N 0,upon which draw the line L M,and parallel thereto, draw divers o- ® tiler lines,quitc through the whole paper or parchment, as the pricked lines in the figure drawn between L M and NO,and lecthediftanceofeach of thefe parallels one from another be fomewhatlelTe then the breadth of the Scale of your Protradlor. Thefe parallel lines thus drawn do reprefent Meridians, and are hereafter fo called, upon one or other of thefe linesfor parallel to orcofthem) the Meridian line of ycurProtraftor noted in the fi¬ gure thereof i, with E F)muft alwayes be laid when you pro- trad any obfervations taken by the Circumferentor as in the Chapter before going. Hh Your Zj 8 Jbetifeoffeveral LiB. 4 j i Your paper or parchment being thus pre pared^affigne any point upon any of the Meridians, as A, upon which point place the cen¬ ter of your Protraaor, laying the Meridian line thereot juft upon the Meridian line^rawn upon your paper, as you fee it lie in the fiaure annexed.Then lookc in your field-book what degrees the needle cut at A which were 191 degrees, now,becaulc the degrees were more then 180, you muft therefore lay the fcmicircle ot the Protraaor downwards and holding it there,with your protraaing pin make a mark againft 191 degrees, through which point, trom A, draw the line A B, which contains 10 Chains 75 Links. 1 Lay the center of the Protraaor on the point B, with the i meridian line thereof parallel to one of the pricked Meridians 1 drawn upon the paper, and feeing the decrees cut at B were more then 180,279, therefore the Semicircle muft he downwards, ' andfolioldingit,makeamark 3 gainft 279 degrccs, and through ' it draw the line B C, containing 6 Chains 8 j Links. 3 Place the center of the Protraaor on the point C, the Me¬ ridian line thereof lying parallel to one of the pricked Meridians ! drawn on the paper, then the degrees cut by the Needle at your ’ third obfervation at C being above 180, namely 216 degrees 30 miniuf s, therefore muft the Semicircle lie downwards, then ma- 1 king a mark againft 2 16 degrees 30 minutes,through itdraw the line”c D, containing I7 Chains 82 Links. 4 Lay the center of the Protraaor upon the point D, the de¬ grees cut by the Needle at that angle being 325, which, being a- bove 180, lay the Semicircle of the Protraaor downwards, and againft 325 degrees make a mark with your protraaing pm, I through which point, and the angle D,draw the lincDE,makmg it to contain 6Chains 96 links. .■ 5 Remove your Protraaor to E,laying the Meridian Imc there¬ of upon for parellel tojoneof the Meridians drawn upon your pa- I per, and becaufc the degrees cutby the Needle at this angle were ‘ lefle then 180, namely,'12 degrees 30 minutes, therefore, lay the ' Semicircle of the Protraaor upwards, and make a mark againft ! j jdeorees 30 minutes, throuih which draw the line E F, con- ! t3ining9 Chains 71 Links, i 6 Lay the center of tlicProtraflor upon the point F, and be- caufc the degrees to be protraSed are above i So, t/i*. 341 degrees 30 minutes, lay the Semicircle of the Protraftor dovrawards, and make a mark againft 341 degrees 30minutes, drawing the Imc F G which conteins 7 Chains 5 4 Links. And in this manner muft you motraft all the other angles G, H, and K, and more, if the field had confifted of more angles, al- wayes obferving this tor a generall rule,to lay the meridian line of the Protraiflor upon (or parallel*) one of tne Meridians drawn upon your paper (which the Imall divifions at each end of the Scale of the Protraftor will help you to do,)and if the degrees you are to protraft be lefie then 180 (as thofe at G H and K are) to lay the Semicircle of the Protraftor upwards,or from youjand jfthey be above 180 degrees(as thole at A B C and D are)to lay the Se¬ micircle downwards-, as you lee done in the figure. O* Notwithftanding the Compendium in the laft Chapter, where the Inftrument was placed but at every fecond angle, yet the manner of protradling is the fame as in this Chapter without any alteration. Hh 2 CHAP. 240 Tbenfeof fever al L1B.4. CHAP. XXXVII. tloup to taf^ the Plot of any Par\i Forrejlfihafe, I Wood, or other large Champon flain, by the j Index md Needle, together mth the degrees on the frame of the Table, mu^ commodionfly finlying the nfe of the Perador. |He ufe of the Plain Tabic, Theodolite and Circum¬ ferentor, hath been fufHciently taught m the prece¬ ding Chapters, and their agreement in all kind of praiVifes lully intimated, fo that you may perceive by what hath been hitherto delivered,that for fome ktndofwcrks one Inftrument is better then ano¬ ther, and for large and fpacious bufineffes, the Circumferentor is the bcft(thc Needle being good, no impediment necre to hinder the playing or vertue thereof)thcrc being only this objeflion to be made againft it, "^/^.tliat the degrees in the C ard arc(for the moft part)fo fmalthat they cannot be truly eftimated and fo may occa- fion the greater errour in protradlion For the halving of this grand inconvenience Mafter^tWerw hath a contrivance in his Book of Surveyiilgjby an Inflrumcnt which he callcth a Peradlor which is no other then a Thcodolite,only the Box and Needle is fo fitted to the center of the Inftrument that when the Inftrument is fixed in any pofitionvvhatfocvcr, the Index may be turned about, and yet theBoxandNecdle remain immoveable The benefit of this con¬ trivance is,that wheras in the Circumferentor the degrees are cut by the Needle, here the fame degreesarc cut by the Index, and therefore are larger, the ufe whereof is thus. Place the Per aftor at any angle of a field, and turn it about till the Needle hang direftly over the Meridian line in the Card then fix the Inftfument there, and turn the Index about till through the fights you efpie the mark or angle you would lookeat, then (hall the Index cut the fame degrees and minutes upon the Limbe of the Peraftor, as the Needle would have cut upon the Card of the Circumferentor, if ufed as is before taught; yet notwitliftand- ing this contrivance you fee you muft be beholding to the Needle, the convenience only being,that the degrees which you arc to note in your field-book, are larger upon the limb of the Inftrument then in the Card, which fl confeftejis fomthing confidcrable. Nowifanyman haveadefire to make ufe of this Inftrument, thinking nonebetter, he is much deceived, for the Box and Nee¬ dle being ferewed to the index of the Plain Tabic,and faftened to the center of the degrees upon the fame frame of the Table, per- formeth the work ofthcPcradlor much better then the Peraftor it feff; for,whereas in the ufe of Pcractor,you always let the needle _ liang I I hang over the meridian line, and let the Index cut the deg, in this you (hall fee tliat in going round a field, the Needle in the Card, and the Index on the frame of the Table will cut like degrees, fo that you have a double teftimonie for every obfervation with the fame facility,which is no fmall fatisfaftion. Now becaufo(I know) there arc fome which are wedded to the ufe of this inftrument, and induce all men whom they can perfwade to the ufe thereof, thinking none fo good, cr at lead better,! will here in one example briefely fhewiheufcthereof,asitisto be performed by the d^ greesprojefted on the fra me of the Plain-Table,and thereby make the Plain Tabic more gencrall. , Let A B C D E be a field to be meafured by the Index and Needle on the Plain Table, fupplying the ufe of ch6 Peractor. I Place your Inftrument at A, laying the Indexed fights widi the Box and Needle ferewed thereto upon the Diameter of the Table then theindex fo lyingjturn the whole Inftrumem about ti Z4Z T be nfe of fever al L i b. 4. I I in the Boxjabout 218 c!e2rccs,&; the IndexCat the fame time)upon I : theTable will cut 218 degrees lominntcs, which muft be noted i ; down in your field book as hath been fcverall times before taught, : and mcalure the diftance A B 9 Chains 65 Links, which you muft note down in your field- book alfo. ^ By this you may fee th.e convenience of counting the degrees cut by the Index rather tlien by the Needle, as here you lee i lominntesarelcftineftimation, which the Index giveth j more precifely,nay, fometimes you may polfibly miue haif > I or a whole degree by the Needle. I - I I 2 Place your Inftrument at Bj laying the Index on the diameter ; ' thereof, and turn the Inftrument about till the Needle hang over I i thoMcridian line in the Card, then fixing the Inftrument there, I I turn the Index and fights to C,fo (hall both the Needle in the Box i I and the Index on the frame oi theTable cut 298 decrees 30 mi- I 1 nutes aud mcaluring the diftance B C,you fliall find iuo contain 7 1 j Chains 28 Links, the degrees and minutes, and the length of the j line meafured, muft be noted down in your field-book as before. 3 Place your Inftrument at C,and lay the Index and fightsup- on the diameter thereof, then turn the Inftrument about'till the Needle hangover the Meridian line, then fixing it there, turn the Index about till through the fights you efpie the fourth angle at D, then will both the Needle and Index cut 15 degrees 40 minutes thefe degrees and minutes, with the meafured diftance CD 5, Chains 70 Links,muft be fet down in your field-book. 4 Your Inftrument being placed at D, with the Index on the diameter thereof, turn it about till the Needle hang over the Me¬ ridian line, and there fixing it, turn the Index about till through the fights you fee the next angle at E, then will both the Needle and Index cut 68 degrees, and the diftance G E will be 8 Chains 72 Links, which note in your field-book as before. 5 Laftly, place your Inftrument at E,(obfcrving all the former cautions)and direft the fights to A, where you fhall finde both the Needle and Index to cut 142 degrees 45 minutes, and the meafu¬ red diftance E A to be 7 Chains nLinks,which note down in vour field-book. And thus mav you go about any field, let it confift of never fo many fides and angles , obferving alwayes this general rule to lay the Index with the Box and Needle, on the diameter of the Table, and to turn the Table about rilhhcNecdle hangs direiftly over the meridian line in the Card, and then fixing tlic Table, turn theindex about till through the fights you efpie ti.e mark you looke fonthen will both the Index and'the Needle cut tlic degrees which you muft note in your ficld-book,fo will the collcited notes . of this example Hand as followeth. I . Dea, ees Having thus colleftedyour feverall obfcrvations, you may pro¬ ceed to protraft your work as is taught in the next Chapterjwnich difFerem nothing from that in the ieCfjap. ^ It will be here objeded by the affedors of the Peraftor, that here it is required that the Needle (hould plajr twice at i eacli obfer vation,to which I anfwetjit is true but if you n^- Jedl the later of them; itisbothasfpeedyandas exaft as thePera(ftor,andif you have opportunity to obferve both (which you may conveniently do) it will then be better. ’ CHAP. ILIB. 4. in^rumetiif in STtrviying. 147 2. Rfmo\c your Protraftor to the point B, which reprefents your fccon J liition or anile,laying die Meridian line tl'.creof upon : (or parallel to^one of the Meridians drawn upon tlie paper, and bc- caule the degrees cut at B are above i Sojiay the Semicircle down¬ wards as beiorc,and againft 298 degrees 50 minutes make a mark, and ilirouih it draw tlie line B C containing 9 Chains 28 Links. 3. Bring your Protractor to C, and lay it parallel to fome one of your Meridians,and bccanle the de.recs obferved at C were un¬ der 18 -',namcly 15 degrees 40 minutes,lay the femicirclc upwards, and againft 15 degrees 40 minutes make a mark, drawing the line C D containing 5 Chaines 70 Links. 4. Place your Protraftcr as be fore upon the pointD, with the Semicircle upwards,and againft c 8 degrees thereof make a mark, and draw the line DE containing 8 Chains 7 2 Links. Laftly, Remove your Protradt'or to E, placing it as before, and againft 142 dengrees 45 minutes (which were the degrees obferved at your Itation at E) make a mark, and through it and the point E ; draw the line E A, which (if your work be true) will pafle through the point A,and will contain 7 Chains ii Links. CJ* CHAP. XXXIX. How to take th plot of nn Irregular field without /// other Injirinimiitlut the Chain: andIjpur Scale His way of taking the plot of a field (if it be carefully ■ praflifed) is inferior to none of the ways formerly taught, though it be fomething tedious .-Neither can it bepiaftifcdupon Wood-lands, large Commons, Chafes, Forrefts, or Mountainous lands, but upon fmall inclofuresonly, the manner of dfeifting it is as followeth; Let A B C D E F,bc a field to be plotted, Firft, meafurc every fidethereofbeginningattheangleA, fo fliall you finde, { The fide A B to Contain 2 Chains 5 Links ! B C 1 48 CD 2 6 D E 4 75 E F 3 15 FA 2 52 I Now bccaufc the field confiftethoffixfides it may therefore be - I reduced into two Trapezias, and beft (in this figure) hyaline | j fuppofedtobe drawn from the angle F to the angle C. by which j I fuppofed line which being meafured will contain 2 Chains 89 ' ; Links, tlie whole field is divided into the two Trapezias A B C F, j , andF C E D. Ii Then ' The nfe of feveral •248 Lib, 4* ' Then again, if you meafurc with your Chain from Cto A, which wUl contain a Chains 70 Links, and from Cto E, which conteins 3 Chains 60 Links, you fhall divide tliofc two Trapezias each into two Triangles, viz. the Trapezia A B C F into the Tri¬ angles A B Cj and A C F: And the Tapezia F C E D into the Tri¬ angles F E C, and C D E. In every of whichTriangles you have all the fides given,by help whereof you may upon paper or parch¬ ment, draw the exaft figure of your field according to what Scale you pleafe, in manner following. Firft (taking into confidcration the fituation of your field) up¬ on your paper draw the line AC contcining 2 Chains 70 Links of any Scale, Then,bccaufc the fide A B conteins 2 Chains 5 links, take 2 Chains 5 Links from the fame Scale, and placing one foot in Awith the other deferibe the arch ak Likewife, bccaufc the I fide B C conteins i Chain 48 Links take one Chain 48 Links from the Scale, and placing one foot of the CompafTes in C, with the other deferibe the arch c t/, croffing the former arch ^ in the point B, then drawing two lines from C and A, to tl.c point B, you have protrafted the Triangle ABC. Secondly, For the Triangle AF C, bccaufc the fide A F con¬ teins 2 C. 52 L. take 2 C. 52 L. out of your Scale, and placing one foot in A, with the other deferibe the arch f/. Likewife, becaufe the fide F C conteins 2 Chains 89 L. take i C, 89. L out of your Scale, and placing one foot of your CompafTes in C, with thco- _____ther I i LIB. /}. Inifrumcnts in Surveying, ; tiler dcfcribc the arch ff/;, cutting the former arch c fin the point i F, then drawing two line; from A and C to the point F, you iliall enclofethe Triangle AFC. I Thirdly. For the Triangle F E C, becaiifc the fide F E conteins I a Chains 15 Links, take2 Chains 15 LinksoutofyourScale.and placing one foot in F, with the other deferibe the arclu'/t. Like- wifcj becaufe the fide C E conteins 3 Chains 60 Links, take 3 ! Chains 60 Lienks out of your Scale, and placing one foot in C, ' with the other deferibe the arch / m crofling the former arch t k in the point E, then drawing two lines from the points F andC, to ’ the point E, you lhall inclofe, tlie Triangle F E C, I Lallly, Fortl.e TriangleC E D, becaufe the fide E D conteins 4 Chains 75 Links take 4 Chains 75 Links out of your Scale, and placine one foot in E, with the other deferibe the arch;; 0; likc- ! wifcj becaufe the fide CD, conteins 1 Chains 6Links, take 2 Chains 6 Links out of your Scale,and placing one foot in C, with I the other deferibe the archp q cutting the former arch «oin the 1 point D, then drawing two lines from the points C and E to the ' point D, you fhall inclofe the Triangle C E D, and fo is your i workefininud, ! Altiiough this way be fomewhat tedious, yet if it be but care- I fully performed it will effeft the thing intended with muchexaft- neflc, and the chiefe caution to be obfen ed in the performance hereof is, tliat w’ncn you mcafutc any of the diagonals, asC A, C F, or C E, you carry your Chain in a dired line, which you may eafily do if you caufe a marketo be fetupat each oppofite analc, and a flick or fucli like in the mid-way between the two an¬ gles, as (in this Example) if 3 flicks or marks werecreded in the Diagonals at the letters O, P, and Q. 51 * CHAP. XL. Tbc i/fiof the Iiijlnimnt defcrikdiiitbe fifth Chapter of the fimd Booke, asalfo of tbc Crofie mentionedin thefixth Chapter. 'Xving in the 37 th. Chapter of this Book Hicwcd how to make the degrees on the j frame of the Plain Table to fupply the ufe , of the infirument which Mafker Rathhourn j fo highly commendeth, and calleth by the ■ name of the Perador, I thought good in this place to intimate that the Infirument mentioned in the fifth Chapter of the fe- cond Book will efifed this Workc aswell as the degrees on the frame of the Table, and for portability ex- ccedeth any of the forementioned, Theulcof this Infirument is 250 The ufe of feverd L1 b. 4. fb obvious, that I lliallnot need to give any particular example concerning the ufc thereof, it being the Fame that is already tauf-ht in the ufe ot thcTlicoJolitc, and Circumlcrcncor only you have this advantage, that your needle is much longerj and the degrees much larger tlicn can be in any ordinary Card. ‘ i Intheiixth Chapter ofthefccond Book there is mention made i of an Inilrumcnt called a CrofTcjwhichlnftrumcntmay be of good ufein fma'il inciolures, which conlift of few fides and angles, the Inilrumcnt hath no graduations upon it, biit is only to lay out right angles in the field it felt without protraft ng, and fo to call up the content of any fiich field by right angled parallelograms, or Ion® fc|uaies, and Irianiles, the Inilrumcnt at the firft reducing the field into the largeil right angled parallelogram'that may be. CHAP. XLI. Row to tab tktiiie plot of a hrreji, Chafe, Wood, Vark, or other large inclofiire, hj tk Circumferentor, hj a mofe exaU wap then the form', hp which pou map how kfore poukginto protrahl, whether pour?lot willckfe or not, and alfo difeover wherempou have er¬ red, fo thatpoii map go over that part again, and exa- ime tije truth of pour work before pu go out oftk field. e Or the pcrfoiming of this Worke, the Card of the Circumierentor, ( or the degrees on the frame of the plain Table) muft be divided into four Quadrants or Quarters, each conteining 90 degrees, andnumbred by 10} 2C, o, Scc.topo, beginning at the Meridian, and fo reckoning to the Eall and Weil points, as in this figure. Your Inil'umcnt being provided with fucha Card as this fi¬ gure reprefenteth, you may proceed to take the plot of the ground you intend in manner followin':, but firft provide your field-book 1 which I in this cafe) muft be ruled with red inke into fix Co- j lumns, the firft of which muft be fobroad, artoconteinthenum- ! ber of degrees, and the quarter of the Compalfc in which the de- I grecs are cut by the South end of the Needle, the fecond Column I I is for the Chains and Links which every fide of theficldcontcin- eth, the four other Columns which have written at the head thor- j ; o\ North, South, are to hold a certain number of Chains i , and Lmks,by the addition of which,yoii may examine your worke i and difeover wlicthcrit will clofeor not before you proceed to I i protraftion. ! THE) LIB. 4. In^rtmentS' in z^% The figure ofthc Card of the Circumferentor. ! The manner of working. I Let ABCDEFGH K be a field to be mcafured,' having pla- i cedmarkes at every angle thereof, Firfl:,place.yourInftrument at ; A,(the flower deluce towards you) and direA tlie-fights to B, the ; South end of the Needle cutting 54 degrees in the South-weft ■ Quadrant,and let the length ofthe line ABjbe 5 Chains laLinks, Now you muft (in the firft column of your ficld-book-)fec down S VV 54de^.omin. rcprefcntingSo«t/;-Wr/i 54degrees o minutes, and inthefccond Column fetdown 5.12 rcprelcnting j Chains 12 Links. Secondly, place your Inftrumcnt at B, and direct the Sights to C, the needle cutting A/o/vZ; 45 degrees, and the line B C con- teining 2 Chains 8p Links. Thirdly, place your inftrument atCanddircdl the Sights to D tile needle cutting North Weft 76 degrees, and the line C D j conte nini 3 Chauis 35 Links. | Fourthly ,place your Inftrument at D & dircfft the fight to E, the needle cutting North 31 degrees; and the line DE contcining 4 ‘ Chains 55 Links. I Fifthly, place your InftrumentatEandeireftyourfightstoF, the needle cutting North Eaft 56 degrees, and the lincE F contcin¬ ing 2 Chains 67 Links. __ li 3 _ Sixthly, Sixthly, place your InftrumcntatFj and dircft the fights to G, the needle cutting ^orth Eafi 2 1 degrees and the line F G contein- ing 2 Chains J4Links. Seventhly, place your Tnflrument at G, and dirc(fV the fights to H, the needle cutting Sauth Eaji 51 degrees, and the line G H con- teiningi Chains 95 Links. Eightly, place your Inftrumcnt at H, and direft the fights to K. the needle cutting Siuth Eafi 34 degrees, and the line H K contcin- ing 3 Chains 25 Links. Laftly, place your Inftrumcnt at K, anddirc(ft the fights to A, the Needle cutting 5o»t/;tf'<’/f4degrces,andthelineK A contein- ing 2 Chains 95 Links. Having thus made obfervation of all the angles and meafured all the fides with your Chain and let them down in your field book, you fhall findc them to ftand as followeth. The j LIB. 4. In^rumeits in Surveying, 253 The figure of your field Bock A s w eUg._^ 54 min. 00 C.L. 5.12 North j South Eafi '^'4 1 B NW' 45 CO a.8p 1 1 1 C 'n w 76 00 3-35 1 1 1 D 'N E 3' 00 4-55 1 1 1 pi N E 5^ 00 ^. 6 ^ 1 1 F N E 2l 00 2.24 1 1 g| ,S E 51 001 2.py H S E 34 00 |3-25 Ik Is w 4 00 1 12-95 11*1 How to Examine your VVorkc, whether you have truly wrought, ornot. Your workc being finilhed, the degrees cut by theNecdleand the lengths of the Tides meafured by the Chain fet down in the firft antifecond columns of your field book, you may proceed to protrafting, but if you defire to be fatisfied of the truth of your work before you go out of the field, you may by help of the lines of Sines and Numbers very fpeedily make tryal, and difeover wherein you have erred, and amend that error before you proceed [further. The proportion is this. 1 Firft, As thcRadius, or Sine of 90 degrees is to the length of the fide of the field in Chains and Links So is the fineof the degrees cut by the Needle to the length in the P:irallel in Chains and Links. Wherefore, Extend the CompafTes, from the Sine of po, to the i length of the fide of the field, in the line of Numbers, the fame cx- [ lentwill reach from the fine of the degrees cut by the Needle, to , the length in the parallel. Secondly, As the Radius, or figneof 90 degrees is to thclength of the fide of the field in Chains and Links; So is the fine of the Complement of the degrees cut by the Needle to the length in the Meridia/j in Chains and Links; Wherefore, Extend the CompafTes from the fine of po, to the j length of the fide of the field in the line of Numbers, the fame ex- tent will reach from the fine of the complement of the degrees cur j by the Needle, to the length in the Meridian. fi Note that the two columns in your field book which are noted with North and South, are the Meridian columns, and the two columns noted with Eaft and Weft atzihc Parallel columns, _ _____ that 254 f^^verd L i b. 4* i tint istoC.iy, the line of North and South, noted in the following j hyure withN S, is called the and the line noted with I b \'\' in the fame fi;^ure is called ti.e P.milkl. > EXAMPLE. j Let it be required toexamine whether the degrees be rightly ’ taken, and the (ides truly iiiealurcdin tins figure^ before you be¬ gin ro piotraft. I 'I'our deg. being noted and your lengths of lines orderly placed in j your Held book, we proceed to examine the trutli thcreoi tnus. i f irll, Tl'.e degrees cut by the Needle when th.e Inflrument was | , placed at A being S. \V. 5 4 degrees, and the length ot tlie line A B, j being 5 Chains 12 Links, it you extend the compafles from the 1 line of 90 degrees, to J Chains 12 Links in line of Numbers, that : extent will reach from the Sine of 54 degrees (whicii were the ! degrees tutby theNcedleat A) to 4 Chains 14 Links in the line I of Numbers, which is the diftance in the paralk-l. And alfo th.e I fame extent will reach from the line of 36 de-.rees (which is t!.e j ccmplcment of the degrees cut by the needle at A ) to 2 Chains I is thodiftancein.the parallel. And alfo the fame, ext^t.will rd^ph from the Sip? pf 39, which is the complement of*, tlie degrees cut,by.theNcc;dle atG, to i Chain 83 Links in the ^ lir)§;of Numbdfs,; which j^tfie diftance in the Meridian. Where, fore,, (bccap|e the quarty{^>pf.^he.C.ompaffe.was^« ins 2 CliainsSs Links, if you extend the Compades from the Sine I or'90 ck'itccstoz Chains 95 Links inthelineof Numbers, that I extent will reach from the fine of^, degrees which were the dc- I grees cut by the Needle at K, too Chains 6 Links in the lii;e of ! Numbers, which is the dillancc in the jiarallel. And alfo die fame ' extent will reach fromthc Sine of 86, which is thecomplcmeiu of die degrees cut by tltcNeedlc at K, to 2 Chains p2 Links in the line ol Numbers, which is the dilbnee in the Meridian. Where¬ fore, (becaulethc quarterof thcCompaflevvas 5 o«.-/;wj?, feto Chains 6 Links (which is the dillancc in the parallel) in die Co¬ lumn of Weil, and alfo fee 2 Chains 92 Links ( which is the diftance in the Meridian) in the Columne of South, and fo have vou done with your ninth Station at K. Having thus goncot cr every of your angles and fidcs,as you fee here done, and noted them down in your field book under their refpeftive Titles, of N’orth, Soinh, EaJI, and ii-'t]?, you iliall finde them to llandasfollowcth.- Thefigureof your field Bock mia. C.L. A S W 54 00 5.12 B Inw 45 ' 00 2.89 c N W 76 00 3-35 N E 3 ' 00 4-55 E N E 5 ^ 00 2.67 F iNE 2t 00 2.24 G S E 51 00] 2-95 S E 34 00 l 3-25 IK S W 4 00 1 12-95 No,ih South Enft 2.97 4.14 2.C4 2.04 0.83 3-25 3-93 2-35 i.jo 1 2.22 1 2.to 0.80 1 1 1-83 2.30 1 1 2.68 1.82 1 ' 2.92 1 0.06 Sum.- 10.40 1 10.40 I p.49 I 9.49 1 This done, addc all the figures in the North Column together, and you iLall finde that they make 10 Chains 40 Link's, and addc the South Column, and they make 10 Chains 40 Links alfo. Then adde the Half Column, and they make 9 Chains 49 Linksj Laftly, addctlie Weft column and they alfo make 9 Chains 49 Links. Nowbecaufe the Sum of the North column, and the Sura of the South column arc all one wi thout any difference; and alfo the fum of the Eaft column, and the fum of the Weft column alfo c- qual, youmaybeaftiiredthatyourworkcis true, bucilthcfums ^ ofihc North and South columns, had diffcrcdyouranulcsor fidcs ; had been falfly obferved and alfo if the Sums of the Eaft & Weft columns had differed, it fad difeovered an errour, butbeinci they ; agree you may beaffured your Workc is true, and therefore i may with confidence proceed to protradlion, according to the di- 1 rcdlions in the Chapter following. CHAP. Honp to p' dny ohf^rvatiom taJ^n accord ing to th .lircUton of the laji Chapter. Papcror Parchment drawfirfta hue a'' Nj raprefenting tlie Meridian^ and at S '^ riehtan . Ics thereto another line E V/V rcprefcnc- in m 8 to A. where it will alfo cxadlly lall if you have truly wrong!'t. ^ ■ . Thus have you found the points i.i.3,4.5'.6,7.8. upon the Parallel E. W: threwsh every oE . which pTiniis draw obfeure lines with black Icad^ fuch likfei ;parallcl;ta:the MeridianS Ni asthe lines, I B;; C. 3l).4>Eii5F. d G^yHrS K. : This ddrte,y6u fliall filtdfc'th'at thd Jifid i Bwhich is draivn pa¬ rallel to the Parallel E. W. and the lineT B which is drawn pi- fallcl to the Meridian S.N. will crolTe one another in the point B, wherefore a line drawn from A to B fhall reprefent the fiaeof the field A E, and if you have wrought truly you fliall findc it to con- tcin 5 Chains 12 Links. . . Likewife 26 z The ttfe of feverd L i b. 4* Likcwifcj the lines 2 C and 2 C crofle one another in the point C Likcwife, the lines 3 D and 3 D crofle one another in the point D Likewife, the lines 4 Hand 4E crofle one another in the point E Likewife, thelines5 FandjF crofleone another in the point F Likewife, the lines 6 G and 6 G crofle one another in the point G Likewife, the lines 7 H and 7 H crofle one another in the point H Likewife, the lines 8 K and 8 K crofle one another in the point K Now if you draw the lines A B. B C. C D. D E,E F. F G. G H. H K. and K A. you fliall have upon your paper or parchment, the true and exaft figure of the land you Surveyed, and this way for cxa£tnefle,cxceeaeth any other that I know off. CHAP. XLIII. Hovp to finde hon^ many Acres, R oods and Ter cbes, are contained in any piece of Land,the plot thereof being firft tal^en by any Injlrmiem, ^^^^Aving (hewn how to take the plot of any field or other inclofure feveral ways, and alfo to protraft the fame O ISHCT psper, it is now ncceflary to ftiew how the con- tent tnereofmay be attained, that is to fay, how raa-' ny Acres, Roods and Perches, any fiela fo plotted doth con ein •• In the performance hereof you mnft confider that the original of the menfuration of all fuperficial figures, fuch as Land, Board, Glafle or the like, doth depend upon the exaft mea- furing of certain regular figures, as the CtometrmlSr^uarr, the Long SquareQt Payallelogram ^ the 7 V/Perches. 2 aF GHS ^23 iS Thefe three uumbers being added together produce 618 exaft- ly agreeing with the former. Q Here note, that at any time when you reduce any irregular plot into Triangles, your number of Triangles will belefs by two then the number of the fides of your plot 5 as in this figure, the plot confifted of 8 fides, and you fee it is redu¬ ced into 6 Triangles. Of\ Lib. 4 . InfrmentsinSurv^ing, i 6 j Of the Circle, THc proportion of the Circutn- ' fcrcncc of any Circle is to its diameterjasytozz. Nowcofindethc area or con¬ tent of any Circle, you muft mul¬ tiply the diameter there ofin it fclf,and multiply that fum by 11, which produa being ,divided by 14, (hall give you the area of the ■ Circle. EXAMPLE. In this Circle A B C D, let the : In tins Circle A iJ c 1^, ict tne { diameter thereof D B be 28,which multiplied in it felf giveth 784, I this number multiplyed by 11 giveth 8624, which being divided I by 14, the quotient will be 616, and that is the area of the^-"’-' le Circle fhe Circumference of a Circle being given^ 1 "N/fUltipIy the Circumference by yjand divide theprodufl by za.j ) ^ the Quotient iliall be the length of the Diameter. I EXAMPLE. I Let the Circumference of the Circle A B,C D be 88 , this mul- i tiplycd by 7, giveth 616, which being divided by 22,giveth 28 for I the Diameter D B. i _ CHAP. XL IV. Of the manner of cafing the content of any pece of Land in Acresj Roods and Perches, by Ma- fierRsithhotnsCbain, the fifth Chapter of the fecond Book, you have X^'^^^adefcriptionol Chains in general, and more parti- cularly of Matter KathkrKSandNlafker Cumers.In the /^x^meafuringof Landby Matter Xathhras Cham, you j.gjj every Pole or Perch thereof (which is divided into a 100 Links) a Unite, and every ten of thofe Links you call a 1 Prime, and every (ingle Linke’you callaSfrw7>/2fs; and in breadth i Unitte, 3 Prmes, 2 Seconds:, and that --- it were required to finde the area or content thereof in Perches, whichtoeffedyoumuft multiply the length by the breadth as is taught in the laftChap- : terc tl icrcforc, the length being 16 /te/trs, 2 Prfwes, and the breadth ' ■ I lUiu-:, 3 P/irnes, 2 Seconds:, thefe two numbers multiplied together fliall produce the a rea. Sec your numbers down as you are taught in the 5 th Chapter of the fecond Book,or as you fee them Rand in this Example , with a prick over the head of_ every fradion : under thefe numbers draw a line, 3:4 and multiply them together in all refpeds as if they 4S6 were whole numbers, and then the work will ftand 162 ' thus, the prodtid of your multiplication being 2:6 m 21384. Now becaufe in your two numbers,nzz..your multiplicand and your multiplyetjthere are three fradions, namely, one in your multiplicand,and two in your multiplyer,you muft thercfore(wich a dalBofyourpen) cutoff the three laft figures of the produd towards your right hand, and then will 21'384 your produd ftand thus, the three laft figures vvher- of are the numerator of a fradion, whofe denominator is looc, and the other two figures towards your left hand arc Integers of vour multiplication 3 fo that the fum of this multiplication is ’ I perches, parts of a perch, which is fomethingmore then a ciurdpartof aperch. But to exprefs the exad c^uantity of thefe fradions in a bufinefs cf this nature were fuperfluous,onely obferve this one Rule for all, namely, that if the figures cut off come neer to a Unite , that is, vvhen the figures cut off are neer as much as thofe underneath them, or the firft figure cut off is either 7,8, or 9, you may then increafeyour whole number by a Unite, and not at all regard'the fradion. But for your further pfadice take another Example, which let be a a piece of Land containing in breadth 5 Unites , 6 Primes^ Sf- and in length, 15 Unites, 4 Primes, and 2 Seconds, which place as before, . . ■ . Now Lib. 4. h^rumentsmSmejmg* z6^ 1 Now if youmulciply thefe numbers one by ano¬ ther as if they were whole numbers, then will they ftand as in the margin, the produft being 868146,_ 563 , from whence take the 4 laft figures (becaufe there i ard four fratlions in your two numbers) there re- 9 * 51 i mains86perches,and parts of a perch; now - . I becaufc8i46isneerto ioooo,Iaddito86, ma- ^<518146 I king it 87 perches, dif-regarding thcexcefs as immaterial, i In like manner, fuppole the perpendicular of a Triangle fhauld I contain i U/tite, 3 Primes, 2 Secmds, aiidhalfthe Length of the bafe j l'houldcontaini6K«tfr,2Pr/w«, thefe numbers being placed as I thole before, and multiplied one by another,will produce this pro- I duft 213843 fro™ whence cut off the three M .figures (becaufe i there were three Iraftious in your numbers multip'lyed) and there j will remain 21 perches, and parts ofa perch, which being but of fmall value you may rejeft. . CHAP. XLV. to reduce any number of Perches into Roods and Acres^ or any number of Acres and Roods into perches. . Y a Statute made the 33 of Edtv. i. ah Acre of ground ought to contain 160 fquare Perches,and every Rood of Land 40 fquare Perches, and every Perch was to contain 16 foot and a half. Now, if any number of Perches be given to be turned into Acres, you muft divide the number given by 16o(the number of perches contained in one Acre) and the quotient iliall Hiew you how many Acres are contained in that number of Perches, and if any thing remain fif it be under 40) it is Perches; but if the remainder exceed 40, then you muft divi .le it by 40 (the number of perches contained in one Rood) and the quotient lliall be Roods, and the remainder Per¬ ches. EXAMPLE. perches be given to be reduced into Acres,firft,divide 5267by :6'o,and the quotient vvilbe 32,and iqyremainmgjwhich divide by 40, the quotient will be 3, and 27 remaining, fo that the whoif amounteth to 3 2 Acres 3 Roods and 27 Perches. Aiiandec 5456 Perches be given to be reduced into Acres,ffrft, dinidc 3496by 160, the quotientwill be34,_and 56 remaining, which •; 6 being divided by 40, the quotient wUl bei, and i6re- maiivug, fo that the whole will be 34, Acres i Rood and 16 Perches. To 2'JO The life of feverd Lib. 4. T0 reduce Acres into Percbes. THis is but the converfe of the formerj for (as before) to reduce * perches into Acres, you divided by 160,you muft now, to re¬ duce Acres into Perches, multiply by 160. EXAMPLE. Let 32 Acres 3 Roods and 27 Perches, be given to be reduced into Perches: firft, multiply the 32 Acres by 160, andtheproduft will be 5120, then multiply the 5 5120 Roods by 40, the produft is 120, thefc two pro- 120 dudts, and the 27 Perches being added together, 27 thefum will be 5267, 2nd fo many Perches are contained in the forefaid number of Acres, Roods and Perches: and thus much concerning the ufc of Mafter Rath- Lon.s Chain. CHAP. XLVI. Heup to cajl up the content of any piece of Land in Acresy Roods and Perchesy by Majier Gmt' tisCbain. N meafuringby Mafter Gunters Chain,you arc in your account only to take notice of Cl ains and Lnksjaswas before intimated in the deferipti- 1^3 ^ on thereof,0/».7.Z,/^2. Suppofc then that the figure B were a piece of Land lying in along Square, and that being meafured by Mafter Cun. ferJ Chain ftiould contain in length 9 Chains 50 Links, & in breadth 6 Chains 25 Links. Set your numbers down as before is taught -- & as in this Example, drawing a line under _ them, then multiplying them together, you ^ ftiall finde the Produft to be 593 7 5 o, from which Prqduft you muft: always cut off the * ———J fiyp figures towards the right band withadafli of your pen, thcnwillthcPro- duAftand thus, 5193750, fo is the 5 towards the left handcom- pleat Acres, and the 93750 hundred thoufand parts of an Acre, which 93750 being multiplycd by 4, the number of Roods in one Acrc,the ProduA will be 3750co,froin which produA cutting oft five figures towards the right hand as before, it will ftandthus, 3l 7 5000, fo is the 3 towards the left hand complcat Roods, and the ! the .75C00 hundred thoufand parts of a I Rood, which being multiplyed by 40, the ! number of Perches in a Rood, the produd i will be 3 0000c o, from which cutting of the I five laft figures towards the right hand, the ‘ Produd will ftajid thus,30looooo, and the ' 30 towards the left hand is the number of ; Perches, and fo the Area or content of the whole Oicce will bey Acres, three Roods and 3 perches. Or the P3750 hundred ■ thouf ; i parts of an Acre may be reduced i into’ cods and Perches by help of thcTa- bie owing. ; T , it you look for poooo, under the title : z (which is the firft %ure with Cy- I added)you lhal find againft it 3 JJWr, Vnficr, then look for 3750, and againft ; ii V u iRaU finde 6 Perches, all which being ; adaed together as here you fee, the area or i content of the whole piece will be y Acres,3 Roods and 3oBcrches. ! J. ,R. T. ; 5 00 00 (Another Example. ■ Suppofe the bafe of a Triangle fhould j contain 16 Chains 55 Links, and half the i Perpendicular of fame Triangle 4 1 Chains 52 Lmks, thefe being multiplyed i one in the other will produce the area or 5 conseasof the whole Triangle, is done, ar. ;* multiply one by the otherfo will the Pr -ludtbe 715392, from which ctirtingoff /e five laft; figures towards the . righthan.. • .ere will be left before the line I ofpanitic . ■ ' hich is n compleat Acres, “andbehir, ■ n there will be 153923 which are . /fnoufand parts of thatis,theTc. lumnefor vcoi 950 475 ° 1900 1700 519375 ° 3I75000 4 ° 30I0C000 r: p. 10000° 4 0 90000 3 24 Soooo 3 8 70000 2 32 60000 3 16 50000 2 0 Ijoooo I 24 30000 I 8 20000 0 3 ^ 10000 0 9375 0 15 S750 0 H 8125 0 13 7500 0 12 6875 e II <5250 , 0 10 5 ^ 2 y}o 9 1 cOOOi 0 8 4575! 0 7 3750, d 6 Jl25 0 5 25 °° 0 4 1875 0 3 125c 0 2 1 <524 c J. I < 5 ) 5(5 4)31 6624 7I15392 .houfand parts of an Acre, and how much e?.:uy Ihewjforifyoulooke in the firft Co- ___ .rdV'cvou(hallfinde00Roods 16 Perches-, rfien for y.cr you’finditnot, but the neereft thereto is Mm 5625 The ufe of federal L i b. 4.! A’, f. 56253ag.'iinft which there ftandcch pPer- 7 CO 00 ci’.cs, all thefe number;; being added toge- therwill produce 7 Acres, 00 Roods, 27 __Perches, which is the Area ot the Tri- 7 CO 27 angle, Tims may you finde the area of any ' 1 ; langie or Parallelogram U'l^’ cafily by one mnlciplication and ac w'-.icii is much ca- fier then the way of calling up bv Maicu A’.i, liLr,-.s Chain. By this manner of work it cr .■ icn.tn and breadth ot a long Square or Parallelogram given li 'n.id be 9 Chains 75 Links, and 6 Chains 25 Links,the area ot Uich a long Square would be found tobe6 Acres, 00 Roods 15 Perc ies. Or the length and breadth being 12 Chains,42 Links, and i Chain ^6 Links,tlic area or con¬ tent will be found to be one Acre, two Roods, ^oPerches. Alfo, the length and breadtl I being 12 Chains 86 Links, and 5 Chains, 25 Links, the area will be found to be fix Acres, three Roods, 00 Perches. But left you fhould be deftitute of this Table when you have need thereof, you may have it put upon feme tparc place ot your Infttument, or rather (inftead of this Table) aScale, which 1 will now fhew you the ufe of, which performeth that work far better and more eafily then the Table, and may conveniently be gradua¬ ted upon the Index of your Table, the dividing and numbering whereof is well known to the Inftrumcnt maker. The Scale confifteth of two parts, one whereof is fquare Perch, es, theothcrfquarcLinks, the Seale, of fquare perches procccd- cth gradually from i to 40 with fub-divifions, and is numbred by 5,10,15,20, &c. to 40. The Scale of fquare Links proceedeth gra¬ dually from I to 25000, and is alfo fub-divided and nnmbred by 1000,2000,&c.to 2 jooo, equal to 1 Rood or 40 Perches. The ufe of the Scale of RcdvBion. We will inftance in the fecond example beforc-going,whcrc the length and breadth of the long Square was 16 Chains 36 Links and4Cbains3J Links, thefe being multipycd together produce 7 1 3 392 j'lnd the fi VC laft figures being cut off, thcre^is 7 Acres and 15392 remaining, now to finde how many Roods and Perches this is,look in the Seale of Square Links for fifteen thoufand three hun¬ dred ninety two,and againftit, in the Seale of fquare Perches you lhall finde 24 Perches, and above half a Perch. (^Another Example, . Let us take the firft Example before-going, where the numbers I mul.tiplycd were 9.30,and 6.23, thefe being raultiplycd one by an- ' other produce 5937 JO, and the five laft figurcsbeing cut off, there will be 5 Acres,and 93730 remaining ; now to know howmany ■ Roods and Perches are contained therein by the Scale. C You •Lib. 4, Jn^rmenU in Surveying* ^73 ^ Youmiift confidcr that 25000-rquare Links are equal to one Rood or 40 Perches, as appeareth by the Scale it fclfj and alfo by the Table, then 50000 equal to two I Roods, and 75000 equal to 3 Roodsj therefore, if your ! number remaining exceed 25000, and be under 50000, I you may conclude i Rood and odd Perches to be con¬ tained thcrein.If it exceed 50000, and be under 75000J • you may conclude two Roods and fome odd Perches to be therein. If above -,5000, youmay then concludej ' Roods and odd Perches to be therein. I Now in this Example, the number remaining is 93750, which ' becaufc it exccedcth 75000 ,1 conclude there is 3 Roods contained therein, which I fet to the 5 Acresjand fubftraft 7)00ofrom9375o, the remainder being 18750, A. R. F. this number , eighteen thoufand feven hundred 5 3 30 ^ and fifty. Keek in the Scale of Square Links, and rightagainftit ! I find 30 Perches, which added to the former,^vetlP5 Acres, 3 ! Rood s, and 50 Perches, which is the area pr content required, i Thus you fee with what celerity and exaftnefs the Scale cfFcd- : ethyour’defire, and therefore let it be graduated upon the Index i of your Table that it may always be rea'dy at hand when you Imvc ; need thereof. The conftruftion of this Reducing Scale I received i of my honour’d Friend 5 . F.dcceafed. j CHAP. XL VII. j Comaining divert compendious Rules for the rea* I dy casing up of the Content af aty plain Si^r^ fetes J and ether necejjary. Gonclujions incident to Surveying} hy the Line of Numbers. A i iCa He line of Numbers is of Angular ufein catting up of the Content of any Superficies i and for Land V^iv] meafuring efpecially Matter GtwteifEaihfevefal Propofitions, uke unto which, I will infertftveno- HkS j ther Propofitions which Will be of Angular ufc in JB thepradfice or Surveying, \7 he length and breadth ofi right an^tfPdral -. lelosram or longSquarebeing given^.f erchos, to jinde the content thereof in Perches* , ^ As 1 Perch, is to thebreadth of the Parallelograna in Perches j S 0 is the length in Perches, to the content in Perches. Mm 2 _ fa - The ufeof feveral m LI B. 4. D In tins long Square or Paral¬ lelogram ABC Djifthe breadth thereof C B be 3 6y Perches, and the length there¬ of A B 50 perch¬ es , the content will be found to be 1820 Perches: for, If you extend the Compaffes from i to 361 the length, the fame extent will reach from 50 the breath , to 1820 , the area or content in Per¬ ches which you may reduce into Acres as is taught in the 41 Chap. z The length and breadth of along Square be inggiven in Perchesj tofinde the content in A< cres. As i'5o, to the breadth in Perches; So the length in Perches, to the content in Acres. So in the former figure, if the length thereof A B be 50 Pcrclies, and the breadth thereof 364, the content will be bound tobe 11 A- cres 40 partSj which is i Rood 20 Perches 5 for. If you extend the Compaffes from idp to 36 4 , the fame extent w^U reach from j0 to n Acres 40 parts. As I o, the! breadth in Chains; So the length in Chains, to the content in Acres. _ ■ Soitfrclength of thefong Square A Bbeing 12 Chtfiris jo tinfcfji ahd the breadth B G y Chains 10 Links,the area will be ftunclfii bail Acres>;3^!paBt?,Qr ;i,Rood 20 Ptrches,for, ' rlfryou.dxterfd theCiompaffesfrom totoji Chainsi iaLinks,thfi; fame extent will reach frqm xa Chains 5 0 Links, to i i Acres 37' 'AaJbOjUBthe Pebpotidicular; ' 1 So thei^gthofi ihe)Bafe^ to the content in Acres. LI B. 4. Infruments in StirT^ing, So in the Triangle LAB, ii the line BD be taken for the Per- i pendicularof the Triangle, then the length of the bafe being 50 Perches, and the perpendicular 36 ’, the area will be found to be 5 Acres zz parts, which is 1 Roods 30 Perches, then, j Ifyou extend the CompalTes from 320to 367 the Perpendicu- 1 lar, the fame extent will reach from 50 the length of the bafe, to I : 5 Acres 2 2 parts. ; 5 1 be Bjfe and Perpendicular of aT rianglebe^ ; Cham, to finde the content in A- i cres, I As 20, to the Perpendicular, So the Bafe, to the content in Area. So i n the former figure, If A B12 Chains 50 Links be taken for the Bafe, andBD 4 Chains 55 Links for the Perpendicular of the Triangle A LB; the area (by this proportion) will be found to be 5 Acres <58 parts, that is, 5 Acres 2 Roods 30 Perches, therefore, If you extend the CompafTcs from 20, to 4 Chains j 5 Links, the I fame extent will reach from 12 Chains 50 Links, to 5 Acres 68 j parts, which is 2 Roods 30 Perches. 6 he Area or fuperficial content of any piece of Land being given according to one b^nde of perch, to fade the content thereofaccording to another f^nde of perch. As the length of the fccond Pei'ch, To the length of the firft Perch; So the content in Acres, To a fourth number. And that fourth number to the edntent in Acres required. Suppofe the figure B were a piece of Land, which being plot¬ ted and caftup by a Chain of 16 foot and: an naif to the Perch, lliould contain 8 Acres, and that it were required to finde liow much' the fame piece would contain if it were meafurca with a Chain of 18 foot to the Perch, if youworkaccordinipto the proportion here delivered, you fhall finde it to contain 6 Aefes 7 2 parts 5 for. If you extend the Cotripafles from 18 to i6t> that extent will reach from 8 to 7.30, and from 7.30 to 6.7 , and fo would the figure B contain if it were meafuredbyaPerchor lo The ufe of fever d L i b. 4* j 7 Idavingthe lergth cf tie Fvrlcig to find the \ breadth of the Acre, ^ As the length ofthe Furlong in Perches, to i6o; So is I Acre to the breadth in Perches. So if the length of the furlong be 50 Perchesjthe breadth for one Acre will be 3.20 : for, If you extend the Compaffes from 5 0, the length of the Furlong in Perches, the fame extent will reach frem i Acre to 3.20 Per¬ ches. But if the length of the Furlong ieghen in Chains^ then, As the length of the Furlong in Chains, is to 10; So is I Acre, to the breadth of the Furlongin Chains. So the length of the Furlong being 12 Chains 50 Links , the breadth thereof will be found to be 00 Chains 80 Links; for, Ifyou extend the Compaffes from 12 Chainsjo Links, toio, that extent will reach from i Acre to So Links, which is the breadth of the Furlong required. CHAP. XLVIII. Horn to reduce one f^jnde of meafm ir.to another^ as Statute Meafure to Cufiomary Meafure, and the 6 Profojition of the laft Chapter you may per- ® form this work by the Line of Numbers as is there ^ taught, bqt however, it will notbeamifsin this 1 ^ P perform the fame Arithme- I ^ tically,thatthc reafon thereof may the better ap- i pgar. Now whereas (by the fore-mentioned Sta- . tute) an Acre ofground was to contain 160 fquare Perches, mea- ; fured by the Pole or Perch of 16 foot and a half, but in many pla- ! ces of this Nation (through long cuftomc)thcre hath been received I other quantities called Cuftomary,as namely, «f 18,20,24, and I a8 foot to the Pole or Perch. ; It is therefore ncceflary for a Surveyor to know how readily to : reduce Cuftomary meafure to Statute meafure, and the contrary, , Snppofe then, that it were required to reduce 5 Acres,2;Roods, i aoPprches,meafurcd by thei 8 footPole into Statute meafure,you j muft feck out the leaft proportional terms between 18 foot, and I 1 6 foot and a half, which to perform do thus. Becaufe 16 and a ' half ^77 L1 B. 4. in^rments in Svruejing, i half btarech a fraftion, reduce 16 and a half into halves, and that ' both your num'ocrs may be ot one denomination, ycumuft reduce ! 18 (the cullomary Pole) into halves alfo, then will your numbers i Hand thus which abreviated by 3, by faying how many times ; 3 in 33 ? the quotient will be 11,and a^ain, how many times 3' 3 in 36? the quotient will be iz , fowill two proportional terms between 16 and a half and 18, be n and 12. This dene, reduce your given quantity (5 Acres, 2 Roods, and 20 perches) into Perches, which makes 900 Perches: Now confi- dcring that what proportion the fquareof 11, which is 121, bears to the fquare ot 12, which is iij4, the fame proportion doth the ' Acre of 16 foot and a half to the Perch, bear to the Acre of 18 foot to the Perch. Nowfbccaufethc greater mcafure is tobe reduced into the let fer) multiply the given quantity gzo Perches by i44,the greater fquare,and the produd will be 129600,which divided by 121,the ^ quotient will be J 07I-^^Pcrchcs,which being reduced into Acres, ' giveth 6 Acres, 2 Roods, 31 Perches, and .Ti’parts of a Perch,ac- ' cording to Statute meafure j But on the contrary, fuppofe it had been required to reduce Sta- ; tute mcalure into Culicmary meafure, then you muft havemulti- I plyed 900 perches (your given quantity) by 121 theleffer Square, j (bccaulc the lefTcr meafure is to be reduced into the greater) the I produd will be 108900, which divided by the great Square 144, ] the quotient will be 7565 perches, which reduced into Acresis4 ! Acres, two Roods 36 perches and a quarter. The fame manner of work is to be obferved in the reducing of i anyCuftomaryquautitywhatfoevcr. CHAP. XLIX. out fever d Furlongs in Common fields divers I euants, „ AvingplottedthewholcField,Common,orotherIn- clofure, with its particular bounds, as you obferve them in the furvey of the whole Manor, or if you only . V lurvey that particular, you muft take fpecial notice of all (he bounds thereof, then provide a Book or paper which muft be ruled or divided into 8 columns, in the firft where¬ of towards tl e left hand is to be written theTcnants namcjahd the tenor by which he holds the fame Land, the pwo next columns are to contain the length of every mans Furlong in Chains and Links, inthetwonexteoiumnsis expreffed the breadth of every mans Furlong in Chains and Links, as by the Letters over the head of each Column doth appear. In the three laft Columns is tobe exprefled the quantity of each Tenants Furlong; in Acres, Roods and Perches. In The ufe of feverd L i b. 4. In the laying out of feveral parcclsin this kind, you will have ufc onely of your Chainj then when you begin your woi k,you mult tirlt write the name of the held, and in the firft column of your Book or paper, you mult write the Tenants name, and tlic tenour bv which he holds the fame, from what place you bcain to mca- furc,and upon what point of the Compalle you pafl'e from tl-.eiice, ! and obferving this diredlion in all the reft, you may (il need re- i ouirc)bound every parcel. j This being noted in your Book, obferve the fpecics or lliape of j the furlong, wliether it be all of one length or not, if of one iencth, 1 then you need take the length thereof but once for all, but il it be i irregular, that is, in fomc places lliorter and in others longe r, then i you mult take the length thereof at eveiy Iccond or third breadth, j andexpreftc the fame in your Book, under the title of length. As j for the exprefling of the feveral breadths, you need but to crofl'e o- I ver the whole Furlong, raking every mans breadth by the middle j thereof, and entering the fame as you pafle along, but in cafe there * be a conftdcrable difference at either end, then ! would advife you ; to take thcbreadih at either end, and find a line which fliallbea ' proportion between them, for your mean breadth, andenter this ' in your Book or paper under tjie title of breadth. i In this manner you may proceed from one Furlong to another till you have gone through the whole field, which wficn you have i done and noted down the feveral lengths and breadths in your i book, you may multiply the length and breadth ol every parcel to- getherjas is taught before,and fo fhal you have the quantity of eve¬ ry parcel by it fclfjwhich quantity muft be noted down in the three laft columns ofyour Book;as in thefollowing example appears. Mordon Field ! The Tenants names i Le/igth. Ij Breadth. 1 Comer.:. j .natcour, |c.|L.|jajL: A.|R-P. i i 1327611 3'4 j! i i l3oii2|| 216311 7330 1 j 1 1 1 Robert Dmon.,ht l28|6o | 8|l2| |23'o36 1 i||i 23;|| 3i,o| 2' CHAP, i LIB. Z], Injlrnments in Siirvaying. d* cha;p. L. How a Lordjljif lyi/ig in Common Field is to be inclofed. Clioptcr was wlioily omitted in the former Editi- '■ Ki.'ok, and indeed had not been thought of fecond hditioiijhad not my worthy friend Mailer " ^,*’‘■^*■‘"2niyBook was in thcPrefsjthe fi tend time) lent me word that this Chapter would be ntcelTary to be infeTtcdjand vichall fcntnie in writins, theway which , ei’fetu toeffed tl.e lame , wiiich method I iTking well, !i.’, e ma jir bold to inlert .fiijitm, as ix- lent it to me j which take asiolloweth. S/.'O /' DireP.ior.s fir;; u.n /.ar a o i,i open field , is’.u Le inclofed. f T moll commonly hapneth wlx-n a Lordihip is to be improved, * 'vlierein aie many Frce-holders, tiiat tlieir ground(confillingof didere.it Qualities) lies for tlie moll parr dilperled,and intermixt oneamonJlanoti.erinallpartsandquartersof the Field, tliere- I lore to linde rlie ju'i Quantity of every mans 2round,both Arrable, Le\ - . round and Medow, the Siirt eyor is to prepare a Field-book, wi:i reiiuowards ti;e right hand of every Page, let there be three Imall Columns diitinguiilicd one from another by a black line mad.e witii a pen or penfilc,and one greater Columne towards the lelt liand, wliich lliall contain the butting, bounding and number of every mans particular Lands, Leys, Dolesof Medow, or the like, vvj.icli being ti.iis fitted for ufe, the Surveyor when he comes nuon.e i leld, istobeeinin lomc corner thereof, ashcniallfinde m.vrcGi'.venitntfot taking the Field in order , and then entring i p e 111 'on !,he fna 1 firll fet down the name of the Furlong, and uii.'.nw. at point of the Compafs he begins; next put down ti t name of toe Free-i older,rl at tirll begins it,with the number of hiQ 1 ,1 u if which in the firll of the three lc(rcrcoImns,write tt.e leniiriud the lands in the fecond put tlie bredth,&in the third i and lad, tile Qiiantity. VN'iiich done, let down the nameof the j 1 ree-nolder, that lies next, and the numbir of his Lands, together with tlie lengtli, breadtii, and ouantity as before; and foproceed inerdtr til! you have finillK-d the Furlong. Then go to tlx-next Furlong, writing the name thereof, and wliere you begin, and proceed as before,and fo on from Fiirlons to Furlong, till you ha\ c finifned the Field. But to explain it further; I lliall here give you a more particular draught of the Field- book. N>i The i.____ z8o Jheufe of feveral Lib. 4 . TheSuriejof ibeLordjhip of PILTON the Countj | of RuclandjwWf wOdob. i 6 j 5 . 1 I Micldle-Hill Furlong begin. South. mfi Furlong begin. Fiji. i Loni!.\Int'it. Qjututilj. I) I Long.: Q^umitj i \P- I’. \P. P. ! \ ]obnFalkner'i\^nAs ’51.18 7.60393.968c Tho.Tomhlmfon ihadf.zo 90.5000 1 Th.Tontlinfon ^ 9-44 3.19150.6150 ThePniioiiage4land i 34 ’ 8 o 8.10 281.8800 Peter Bitcklcj%\mAi ’47.10 :0.16478.5360 JohnFdkjier 6\inA 34.00 u.c 274. i ^iM/3rtF;!/^«c!-6linds46.7c 7,12332.5040 Peter Blackley ^\mA 3400 4.' 136. I 7'fc.7'(;w/;/i’H/w; 1 lands 45.ooj 1.30 58.5000 \ Henry Swift <; Finds 34.00 9.2c 312.80CO \ fobnFalkneriuhnds :44.15 12.00 5:9.8300 AbrakdFalk>,er<\\i^s 7.1234.5000 I rW;3«f6lands :44.0c 8 .ooi 35 i.oOoo ]1 Th:Paironage4lcys 33 - 0 7.0231. ; Andrew Coo^hnds r^.ii\ij6.^-^6o \\pknFalkner iky 35-02.50] 82.5000 ! South Me-tdoiv begin. E />jl. Red-hill Furlong begin. South. Pfto-?!/.(cyt7 one doll 13 6.10' 2.5oj 90.50CO jj Jo/w F 4 .%ot 3 lands. j 45 .oo|i 2 .o’ 540 John Fancier Wtdok Uc.oo; 8. oj32o || y 7 Wj.F/j/t»«’lo l.inds45.oo:20.oj poo >4^-.':/i/’.»»Fii%.")Mdok'!4...coio. 0:420 |1 1 ho.Tomblinfin ■^hndsU-.^.odyz.Ql 540 T be Par [on on: dd-: j 4 i .00 8.0328 Tbe T arfonage ■^hndi 50 .ooji 2 ,o 600 Tbo.Tomblwfon idd: i4C.3o| 6.50)263 Peter Bkckjcy 6 kys so.oo 20.0 1000 j T'fo.Frf/I^i one dole ''40.006.00240 Henry Swift ikys 50,005,0 250 Andrew Cook^ on: dok 140.00] 6.00.240 I bom at Falks r^hys 50.0010.0 500 Peter Blackley iw’odoksM.oo n.'^dl^do | land 50.00 a.oj lOO I phnFalner OIK dok 139.50' 3.001118.5000 | i>. 8 lands 50.00'4.0I 200 1 Having finildied your rough Book (after this manner) you are next to make a particular ot every Mans Arable, Leys, andMe- dow ground feverally, that fo you may be ready to give a juft ac¬ count of what every man holds diftindlyjthat by help thereof you may be enabled (with the help of Arbitrators chofen to alTift you) to give every man, not onely the true quantity in his Plot, but alfo confidcration, for the Quality of his ground, as necr as may bc.To which end in drawing your Particular, you are to make fo many Columns as there are Frce-holders, every one whereof is to be fubdivided into three, fo fhall you have one for Arable Land, an¬ other for Ley-ground, and a third for Medow. Then turning to the ! Field-book, I begin with John fafe£r,and write in the particular in ) its proper Column, under Arable 393 P.968 then Thomett Tomblin- f /b.:? lyo.p.eiyo, next P.P/«r^/g,478.536o.whichIplacelikewife5 j under their names, and in their due place, and fo I proceed till I j have finiflied the Book, placing every mans Arable Leys, and Me- j dow in their order, which being effefted, then make your Summu \ totiilis} a.s you may fee in the following S ynopfis. j _ A LIB. zj; Injiruments in Surveying, i8 (?.A Particular of all the Arrahle^ LeySj and fSAdeadow (f round in the Lordjhif of Pilton, Com, Rutland. JolinF-ilh" II Tho.TcmblinfenW Peter Bl.ic l(fey H Abraha F.tikrer Arable Leys, Med- ij Arab. Leys Med. || ArabJLcysj Med. || Arab. leysMed. ??; Sij 510II150 j j 163 li ^ySiooj 90 11352 ; 23 .ij 4 ’o r -9 1 118 11 58 1 II 136) bdo ijpco i 374 II 9° 1 II =°'’| 11 I I 11^ I . .. I „ I I 1836 I J58 II 858 I 6 1165 II 8i4|rooo l 550 I I 1252 12_y 4 1836, II I 838, 11 ' icoo )i ( 123 V ;; . ,H ii Il 5 «»|i 3 j ycur Inftrument at D, obferving the angle A D C to contain 41 ; I i degrees, and the angle D A C 49 degrees, and the fide D 80 ; 1 ! feet: now to finde the fide C D the proportion will be, j j AstheSincofthcangleAC D 90(fegrecs, I j Isto'hefide AD, .'ofect; j i So is the Sine of the angle C A 0,49 degrees, ! ! To the fide CD to; feet. ' j j W liichi added !o the line B Cjgivctli 109 feet, which vou may | ' __ . reJu- 1 Qyfmthermj. There is another way alfo ufed by fomc for tlie meafurin? of hc- rizoncal Jincs, which is without the taking ot the Hils altitude, or muig of any Arithmetical proportion, but by mcafuring with ihc Cham only, the manner whereof is thus. Suppofc A B C were a hill or mountaiiijand that it were requi' red to findc the length of the Horizontal line thereof A C. At the foot of the Hill or Mountain, as at A, let one hold the Chain up, then let another take the end thcrof,and carry it up the HHljhold- ing it level, fo lhal the Chain meet witli the Hill at D,the length A D being 6 o Links,then at D let the Chain be held up agaiujand | let another carry it along level til it meet with the fide of the hil at | E, the Length being 5 4 Links-.then again let one (land at E,& hold up the Cliain, anotlier going before To the top of the Hil at B, the length bcins 48 Links,thefc three numbers being added together make t6i Links or 1 Chain 62 Links, which is thelengthof the horizontal Line A C. This way of mcafuring is by fomc praifiifcd, but the other (in my opinion) is far to be preferr’d before it, only when youarc deftitute of better hclpsyou may make ufchcreof. ^ But if the Hil or Mountain iliould have a defeent back again on the other fidc,you muft then ufe the lame way of working as before, and add all together tor the Horizontal Line. I CHAP. LIi. ; Hon? to f lot Montdnm and uneven grounds^unitb j the befi way to finde the content thereof. H ' Or the plotting of anv mounianous or uneven piece of 2round,asABCDEFG , you muft lira place your fn- ftrument at A,Sz direft the fights to B, mcafuring the line I A B,tl-cn in regard that frorii B to C there is an afeent or hill, you I muft findc the horizontal Line thereof, and draw that upon your | z84 The ufe of feveral L i b. 4 . Table, accounting thereon the length of the hypothenufal Line, then mcafurc round the field according to former diredions , and having the figure thcreol upon your table reduce it into Trapczias, asintochcTrapczias ABEG, BCDE,andthcTrianglc GEF; then from the angles ACE and F let fall the perpendiculars AK, C H,E I, and F M. New in regard there arc many Hilsand Val¬ leys ail over the field, you muft meafure with your Chainin the field over Hill and Dale from B to D, and to the Line B D fee the number of chains and Links as you find them by meafuringjwhich wil be much longer then the ftraight line BD meafuredon your S cale, then by help of your Inftrument finde the point H in the line BD‘ and meafure with your Chain from C to Hjover hill and dale as before,and to this perpendicular CH fee the number asyou find it by the Chain;then find the perpendicular IE, and meafure that ' with your Chain alfo, all which Lines (in refped of the Hils and 1 VaUeys)will be found much longer then if they were meafured by your Scale: then by the meafured Lines BD,C HandlE,caftup the content of the Trapezia BCD E.In this manner you muft caft up the content ofthc Trapezia A BEG, and the Triangle G E F, , and this is the exafleft way I can preferibe for the meniuration of I uneven grounds, which being well and carefully performed, will not vary much of thetrue content: Fork is apparent that if fuch i mountanous grounds were plotted truly according to their area in piano, the figure thcreofwould not be contained within its proper limits,and being laid down amongft other srounds would fwcll be- ■ yond the bounds, and force the adjoyning grounds out of their j places; now for diftin£fion in your Plot you may fliadow them off • with Hils as in this figure, left any man feeing your Plot fhculd j meafurebv your Scale, and finde your work to differ. i _;_^_ CHAP, j LIB. J, Injiruments in Surveying. CHAP, LIII, Horp'o taJ^e the Plot of a whole , or of divers fa'cels of Land lying together jwbetber Wood lands or Champion Plains, by the Phin Table. Lthough pradiccjin the performance hereof, be better then many words, and that the rules already delivered li'fficicnt extent to perform the work of thisChap ter, yet (for farther fatisfaftion in this particular) I will herein deliver the moft fure ar.d compendious way I can imagine. Siippole tl'crefore that the following figure A L M N P QS TY X G H and K were part of a Manor, or divers parcels of land ly¬ ing together, and that it were required to take the plot thereof up¬ on your Plain Table. Now thebeft way (in my opinion) is firft togoround about the whole quantity to be meafured, and draw upon your Table a per- feft Plot thereof,as if it were one entire field (which you may do by the 51 Chapter of this Book) and then to make feparationand divillon thereof in an orderly way, as is taught in this Chapter ; But before you begin your work,it will be very neccflary to ride Or walk about the whole Manor, or at Icaft fo much as you are to fur- vey, that you may be the better acquainted with the feveral boun- ders,and in your paffage you ought to take fpecial notice of all e- minent things lying in your way, as Churches,Houfes,MiIs,High- way 5 ,Ri\ ers, &c. winch will much help vou, alfo in this your paf fagcic wcrcncccfTary to take notice of fome convenient place to begin your work as followeth. Having made choice of fome convcnicntplace in the periphery or outward part of the Manor,as at A,place there yourTable,turn- 1 ingi: about till the Needle hang over the Meridian line in the Car Jrand there fix it,thcn upon the Tablefwith moft convenience) affigne any point at plcafurc,as A, unto which point lay the Index,' and turn it about till through the fightsyou "fee a mark fet up at the next angle atL,thcn by the fide of the Index draw the line AL, which fuppofe to contain 8 Chains 68 links, take thefe 8 Chains 68 links from any Scale, and place that length upon your Table from A to L, 2 BringyourInftrumenttoL, and lay the Index upon the line L A,turning the whole Table about till through the fights you fee a mark fet up at A where your Table laft flood, and there fix it, fo will the Needle hang direftly over the Meridian line in the Card as before,then lay the Index upon the point L,and turn it about till through the fights you fee a mark fet up at the next angleat M,and draw a line by the fide of the Index, which fuppofe to contam 6 Chains 55 links, this length being taken from the fame Scale as the farmer line was, will reach upon your Table from the point L unto M - 3 Remove your Tabic to M, and lay tlic Index upon the line ML, turnin" the Table about till through thcTightsyou cfpica mark fet up at the angle L, where your Table laft Rood, and there fixing it,you Riall Rill finde the Needle to hangdircaiy over the Meridian line, if you proceed truly in your work; then laying the Index to the point M, turn ir about tillthroughthcfightsyoue- Ipy foinc mark let up at the next angle atN, and drawalincby the fide of the Index, then raealuring with your Chain from M to N, youidiallfind itro contain 7 Chains 27 lii ks, wliich take from the lame Scale as before, and place tlie length tliereof upon your Table from M unto N. 4 Place your InRruracnt at N, laying the Index upon the line N M,and turn the Table about till tliiouah the ficiusyoufeca mark let up at your former Ration at M and there fix the Table,fo will the Necolc hang over the Meridian iinc as before, then turn the Index about upon the point N, till through the fights you efpy the next angle at P, and draw a line by tlie fide thereof, then mca- fure the diRance N P 9Chains 32 links, which take from the Scale, and fee it upon your Tabic from N unto P, _ In this manner muli you go round about the whole Manor, ma¬ king obfervation at every angle, thereof, as at P QS T Y X G H andK^andfettingdownthelcngthof every line uponyourTable as you find it by meafuring witli your Chain, you Riall have upon — ____ y our 1 LIB. /j; IttflrHments in Surveying, your Tabic the figure of one large Plain; which mud iiiclud c all thereft of the workjand in thus going about you iball (if you have truly wrought all the way) find your plot to clofc cxaftly in the point Ajwherc you bcganjbut if it do notjgo over your work again> for otherwife, all that you do alterwards within the fame willbc falfc. ^ Here note, that if one fhect of paper will not contain your whole Plot, you miift then lliift your paper in tliis man- nertwhen any Line fallcth off of your Table, draw two lines at right angles crofs your paper,wliich tl'.c cijual divifions on the frame wil help you to dojthcn lay anotl:Cr clean llicet of paper upon your Table, andbytlic fame parallel divifions at the contrary end of thcTabIc, draw two other Lines at right an¬ gles, and upon them note what part of your Plot crofled the two other Lines before drawn, and at thofc points begin to go forward with the reft of your work : and thus may you fliift divers Papers one after another, if need be. Having thus drawn the true plot of the outward bounds or peri¬ phery of the whole Manor upon your Table, as the figure A L M NPQS TYXGHandK; and cxatSIy clofcd your plot at A where you b^an, you may proceed now to lay out the fcyeral Clofcs therein contained, in this manner. 1 Placcyour Table at A, laying the Index and fights upon the Line AL before drawn,and turn it about till through the fights you efpy the angle L, and there fixing it, the Needle will hang direft- ly over the Meridian line in the Card; then turn the Index about upon the point A, till through the fightsyou efpy a mark fet up at the angle B,and by tlie fide of tlie Index draw tlie line AB contain¬ ing d Chains 45 Links 2 Rdmove the Table to B,laying the Index on tlic Line B A,and turnthe Table about till through tlie fights you fee the angle A, then fix it, and turn tlie Index about upon B, till you fee the next angle at C, drawing tl.c line BC by the fide of the Index, wliich fuppofero contain S Chains 5 Links. 3 Place the Table at C, laying the Index upon the Line C B,& turn it about till tlirough the figlits you fee your fornicr flation at B,and there fixing it, mrn the Index about upon the point C, till through the fightsyou fee the angle at E, and draw the line C E containing 10 Chains aa Links,which fet frem C to E, and again (before you move your Tablc)dire6l the figh'S to O, and draw the lincOC containing 6 cliains 64 Links,which takefom your Scale I and fet from C to 6,and^becaulc 0 is the next angle to the bound- i cr) voumav (withoutplacingyourlnhrumcntatOjOrmcaluring the diftance 0 N) draw the tine 0 N upon your Table, which (if I the reft of the work betrue) will contain 4 Chains 45 Links. 1 4RC010VC your Table to E,laying the Index upon the Line EC, ' and turn the Table .about till throngh the fights you fee the angle ■ at C then fix it, and turn ilic Index about upon the point E, till I ’ Oo you' you efpy tlic next angle at F, and draw the line E F containing 5 Chains jo Links, which fet from E to F, now (becaulc the angle at F is the next angle to the bounder) you may draw the Line F G upon your Table without any further trouble, which (ifthe reftof your work be true) will contain 6 Chains 68 Links. 5 Remove your Inftrument. to T,laying the Index upon the line TS,and turn it about till through the luhts you efpy the angle at S, and there fixing it, turn the Index about upon the point T, till through the fights you efpy the next angle at V, and by the iideof ' the Index draw the LineTV containing 6 Chains 15 Links,which ; fet upon the Table from T to V: now (becaulc V is the angle next I the bounder) you may only draw the Line V G, without placing j your Inftrument at V, or mealuring the diftance VG, upon the I ground, which vif the reft of the work be true) will contain 6 I Chains 38 Links. 1 6 Bring your Inftrument to Q? and lay the Index, upon the Line PQj turning the Table above till through the fights you fee > the angle at P, then fixing the Table there, turn the index about ■ upon the point ^ till through the fights you efpy theangleatR, ' and by the fide of the Index draw the Line containing ten ! Chains 75 Links, which fet from R. " j Laftly.Bring your Table to R, and laying the Index on the Line .^R,turn the Table about till through the fights you fee theanglc .^)and there fix it,then turn the Index about upon the point R, I till through the fights you efpy the angle atD , and draw the ■ I _ Line ' LIB. , /njlrwnents in Surveying . Line R Dj vvhicli (if the reft of ihc work be rue) will contain j Chains} Link'. Thus lave you an cxaift and perfeft draught of the whole Ma¬ nor, or of fcvcTallnclofures, in the performance whereof Have been fonithing large, bccaufc I would fliew the mod natural way firft :but the fame thing maybe performed with m.'re brevity as followeth, wlicrein (if you mark it well) you Hall plainly perceive rhat hall tlie w ork will be abreviated, and the fame thingeffefted with almoft lialf the ineafuiing. Having made choice of the angle A to begin your work, place your Table there,turning it about till the Needle hang direftlyo- ver the Meridian Line in the Card,and there fix it, then aflignany point upon th.eTablc,for your beginning ftation,as the point A,and laying the Index to this point, turn it aMut till through the fights you efpy the next angle at L, then draw the Line A t containing 8 Chains 68 Links, Which take from your Seale and fet from A to L: and alfo (before you move your Table) direft the fights to B, andby tlicfideof the Index draw the Line AB, but you need not meafiire the length thereof. 2 Ti.cn go forward with your work as in the former part of this Chaptcr,placing your Table at the angles LM and_N, and when you come to N, and 1 ave drawn the Line N P, you may (before you move your Table) draw the Line N O, but not meafure it. } Alio when you come to the angle Q,and have drawn the Line Qj,you may draw the Line alfo,at once placing of the Table, 4 VVhen you come to oberve at the angle T,and have drawn the Line T Y, you may at the fame time alfo draw the Line T V, but need not meafure it. 5 VV'hcn you come to ti c angle G, and have drawn the Line G H,you may alfo draw the Line GV,which will cut the Line TV in the pome V; and at th.c fame time alfo you may draw the LineGF containing 6 Chains 6S Links. Has ing thus gon round the whole Manor, and made a Plot of the outward part or Periphery thereof, and alfo drawn the Lines A B,N O, ^ R,T V,G V, and G F,as you went along the bounder, thertmaindcrof the work will (by this means) be much abrevia¬ ted, for you have no more to do, but 1 To place your Tabic at F, laying the Index upon the Line F G, and to turn it about till through the fights you efpy the angle at G, and fixing it there diredl the fights to E , and draw the Line E F containing 5 Chains jc Links. 2 Place the Table at E,and lay the Index on the Line E F,turn- ing thcTable aboiit,rilI you fee through the fights the angle F,then fix it, and turn the Index about upon the point E till through the fishtsyou efpy thcan3f your Protfador make a mark, and through it draw the Line L M containing 6 Chams 55 Links. Then place the center of the Protraftor upon the point Mj and look in your field-book what degrees were cut at M,protrad thofe degrees (as before) and draw the Line M N containing 7 cWns 27 Links. Then place the center of theProtraftor upon the point Nj the degrees cut being 35s degrees 40 minutes, and the Line N 0 con¬ taining 4 Chains45 Links, and becaufe againft thefe35j degrees 40 minutes you finde in your Field-book thismark-|-there placed, you muft therefore ( with Black Lead or the like) make the fame mark at the point N upon your paper, to fignifie that you muft there begin to protraCl fome other Clofe. In this manner muft you proceed with all the other Lines and Angles, as you finde them noted in your Field-book, till you have gone over your firft Clofe, and doled your Plot at A. Having thus finiflied your firft Inclofure,you muft deal in the fame manner with the fecond, third and fourth, and fo on, were there never fo many. And to know where to begin to protraft your fe¬ cond Inclofure.you muft have recourfe to your Field-book, where you (ball find this mark 0 at which you muft begin your fecond Inclofurc, which is and the like mark upon your Paper at the point C, which is your remembrancer to put vou in minde that at the point C you muft begin toprotraft your fecond Inclo- fure, as you did your firft Clofe. ^ In this manner of protrafting there is no difference nor cau¬ tions to beobferved, more then thofe already hinted in Chapter 36 and 38 of this Book uiz. that if the degrees to be protraded be under 180, to lay the Semicircle of the Protrador upwards or from youj and if they be above 180, to lay the Semicircle downwards. Pp» . CHAP. , ^ , A' Theufe of feveral CHAP. LVI. Tie figure of any flat being given^ hom to inlarge or diminifb the fame according to any ajfgned ^rofortion may fo fall out that when you have taken the Plot of whole Manor upon your PlainTablcjin clivers ihcets of Paper, or obferved the angles, and afterwards protraded them, as in the two laft Chapters, it may Wb Jo fall out that your Plot may be either bigicr or lelTcr then is defired, now if atany time it be required to inlarge or di- minilTn any Plot according to any proportion,this Chapter willac* compliiTi your defire. The inftruments for the performance hereof are divers, as was intimated in the ninth Chapter of the fecond Book. Now for ge¬ nerality and exaftnefs, the two Indexes there ipoken of, having at each end thereof a Semicircle, is inferiour to none, but the In- llrument being very chargeable,and the ufe thereof vciy intricate and tedious, I lhall wholly omit to fpeak any more of it. There is another way alfo which Maftcr RAthborn ufed, which vyas with a Ruler by him iaventedior that purpofe, which would indifferent well reduce a Plot from one bignefs to another accord¬ ing to fome particular proportions. The making of this Ruler is fo well known, and the ufe thereof fo apparent, tbatlfhall not need to fay any thing concerning the defeription or ufe of it: I only intiiTWte that there is fuch a Ruler, that tnofe which pleafe may have it made; . Another way is by one Line, divided into loooriooo equal parts only, which by the help of Arithmetick will perform this werk very well, but this (as being very tedious) I negledf. To pafsby ,thefe and divers others which I could name, I fliall fay Ibmethirig of.the Parallelogram,which for generality, exadl- nefs, and difpatclTj furpaffech all the reft, unto which (in my opi¬ nion) there is none comparable. Of Parallelograms there arc di¬ vers fojrts, hut that which I (hall inftance in,confifteth but of four Rulers only, the making whereof is well known to the Inftruracnt- maker, and the manner of ufing it as followeth. Take the Plot which you would reduce, and fallen it to a Table with Mouth-glew,thcn by it, upon the fame Table,faften your fair Paper or Parchment, upon which you would have your new Plot; then having fitted your Parallelogram according to the proportion infoWKch you would have your Plot reduced,fix the Parallelogra to the Tablc,by a point for that purpofe; then put-your drawing f^en into fome one hole on one of the fides of the Parallelogram, and upon it a plummet of lead or brafs to keep the pen down clofe Lib Jnfimnejits in Surveying, igg to the paper, when it is moved thereupon: and here note, that at any time when the Parallelogram is thus fitted, the point that fticketh in the Table, the Pen which is to draw, and the Tracer • whichyoumuftmovealong the lines oi your old Plot , will lieal- waycsinarightLinc, but this by the way : Your Parallelogram being fixed to the Table, and the Pen in its true place fitted to draw , take thcTracer in your right hand, and with it, lightly go o\ er all the Lines ol your old Plot, fo fliall the motion thereof oc- cafion the Pen to draw upon your clean paper or parchment, the trucand exadf figure of your former Plot, though of another bi | r Akc Gum Arabcckwlut quantity you plcafc, of the whitcft and clecrcft you can get, vvliich bruife into Imall pieces, and j' tic tlicm up lool'cly in a fine linnen ^rag, then take of the clecrcft ; vva:cr you can get.cC put it into a clean veftehas a potingtr^or fuch ! likc)then hang your Cloth in which you put yoiu: Gum in this wa- ! ler, letting it liang till all tlic Gum be diftblvcdj then when ' you put your fingers into this Water, if you finde them to ftick to- i gctlicr, as if they were clewed, your water is to ftiff of the Gum, which you may remedy by putting thereto more fair water, and if you finde it too weak you may add more Gum, with this VVatcr moft Colours are to be tempered, z Howto ma^ c^ttumWater. JAke a pound of Allum,andbcatit to powder, then take a Gal¬ lon of clean Water,and fee it on a fire, letting it boil till all the Allum be melted, then take it off the fire, and when it is cold, you may put it into a veffel and keep it for yourufe with this water, if you wet your Paper before you lay on your Colours, it will keep them from finking into the paper, andwillalfoaddaluftreand beauty to the Colours laid thereon. 9 The names of fuch Colours as are neceffary for the Wajbing of c5Vi ays jP lots, or Charts, Reds Vermiion Lake Roflet Brafil RedLcad Blews Yellows Greens Bice Gumbuege Green Bice Indigo Yellow Berries Vcrdcercecc Smalt Orpiment diftllled Verditer Sap-green Umber, Lamp-black Being thus provided of thefc fcveral Colours here named, which you may have in divers plaoes in London, as alfo of a grind¬ ing ftone and Muller, which any Mafon in London will furnifh you with, alfo divers pencils of fcveral fizes and Gaily potsj. Gar | Glaffes, or HorfeMufclc-lhels, to put your Colours in when they ; arc grounded and tempered, you are then ready at any time to | makeufeof them, and now will I fhew you how all the fore-men- j tioned Colours are to be ground and tempered. I Hm to grinde Vermillion:, La\e^ and Red Lead. YOur grinding ftone and Muller being very clean, take either Vermillion,Lake, cr Red Lead,and lay it upon your Stone,then __ take 1 iiiB 4- InftrHtneminSurwying. take fo miicli ,Gum-vva:cras\vill wet the fame, tlicn with your Muller grinde the Vermillion, Lake, or Red Lead with the Gum- water very well together,till there be no grit left, then with a knife i ora piece of Lanthorn horn take it off the (lone and put itintoa Gaily pot or lliell, and keep it for your ufe: if it be too thick for Vour life, you may at any time remedy that by putting in of more Gum water, and ftirring it about with a (licit or pencil, but it is mo 11 convenient to prindc no more of any Colour at a time then you (liall have occafion to ufe. How to prepare Brajil. Y Akc two Ounces of Brafil ground, and put thereto a pint of * finall Becr,and as much Beer Vineger fee tnen on the fire, and let them boil well together , then take iialfan ounce of Allom and beat it very fine, alfo a little bit of Gum Arabeck, and put tiiem in while it is on the fire, letting all boil to ether till the Allom and Gum be quite diffoU cd • then take it off and drain it through a Cloth, into a Glafs or other Vcffel, this liquour is of it fclf a very fair red Colour, and is ufed to rule Books with red Lines, and is commonly called Red Inkc. If you take Logwood and boil it in all refpedtsas you did your Brafil, it will make a very curious Purple Colour. Hon? to grinde and temper Rojfet, ^Akc Roffet and lay it on your grinding done,and grinde it with a little Gum-water, fo that it be very diff, then when it is ground fine enough, take it & pur it into a Galley Pot,5c when you would ufe it, ternper it with Brafil water, this Colour differeth not much from the Colour of Lake, but it is much cheaper, and will not keep colour half fo long. • Hon? to grinde and temper Blew or (jreen Bice, yAkc Blew or Green Bice, and grinde it upon your done with * fair water, then take it and put it into a Galley pot, and fill the pot with fiir water,and fo letit refttwo or three hours, often ftir¬ ring it about with a dkk, then let it (land till the Bice be all fetlcd totlicbottcmc , then poureaway the water from the Bice, and put in more clean water, letting it (land, and ftirring itas before, thcnpoiirc that water away, and put in more clean water, thus continuing 4 or y times, then laftly, when you would life it temper it up with Gum-water. j ! 1 Hon? to grinde Indico. TAkc Indico,and grinde it with Gum-water upon your ftonc ve¬ ry fine, then take it and put it intoapotorfhell, and it is fit for ! 3°4 Tbevfe offewrd Lib. 4. ufc, this Colour of it felf is a very fad blew, but you may make it I lighter at pleafurcjby adding thereto fome white Lead ground al- | fo with Gum-water. How to temper Smalt, | -yAke fine Smalt, and put it into a Galley pot, th en put tiicreto ! -*■ ibme Gum-water, and temper it up with a pencil. This Colour i neeJeth no grinding, it being fine enough of it felf. | How to grinde & temper Bine or (freen Verditer, ■ TAkc Blue or Green Verditer,and grinde it with Gum-water on ' your ftonc, when it is fine enough, take and put it into a pot or i fliellforyourufe. How to life Gum-boogd. XAke Gumboogd, and break i. in fmall pieces, then put it into a ^ clean pot, and put thereto clean watt r, letting it there he till it 1 I be diffolved, this is a fine tranfparcnt Colour, and excellent good j lowafliwithall. ‘ j How tonfeT edow Berries. j ^ A’ " vcflow Berries, and brulfethcm a little in a Mortcr, then ; put Cl., ni iiKoa por, and put thereto lomeAllom-wattr,letting j them ftctp tiicrein and in half anhourcs time tl.c Allom-watcr will be a very curious tranfparcnt Colour, it the Colour be too ' faint you may help it by putting in more Berries, but the longer ; the Berries lieinftcep the better tic colour will be, you may (if • yv'u pleafe) when the Colour is as you would have it flr.'in the ' waicr from the Berries, and referve it for your ufc, or you ni.’.y life ' it without Braining. J i How to grinde and temper Orpimmt. ' 'J'Here is two forts of Orpiment, throne is a Light Yellow, and the other, is an Orient or Marigold Colour commonly called i i Orange Tawny. Take of either of tl e!c forts and grinde them upon ' your ftonc with Gum-watcr,kccp them in pots cr iLels for your ufc i How to life Verdigreafe, | I ^He beft Verdigreafe is that which is dilfolved, take therefore | I diftilled Verdigreafe, and'ftcepc it ten or twelve hours in j Rood Mufcadinc, and it wil be a very fine tranfparcnt Green Co¬ lour. ' I '__ Horv. ■ LIB 4* ^ nflrments in Skrveyi^, Hojp to hfe Sap prten* TAkeSap-jrccnrandputhintoapotorfliellj and put thereto white Wine Vineoer, and a little bit oi Allohi, this will make an excellent Green lor Trees, Hedges, &c.- if at any time it grovy too thick add more Vineger or Water, or rather half Water and half Vineger. Tlicre are fome other ordinary Colours,which will be very ufe- lul for you efpecially when you do to exprefs Houres,Trccs,Rivers, Mils, Mines, Gates, Cottages, Brooks, Pales, Barns,and fuch like; all which are to be ground and tempered with Gum-water, the names ot which Colours arc thefe following:White Lcad,Laml3C black,Umber,SpanifliBrown; Almegum, and Bolcarmoniack : thefe Colours are all very cheap, and arc all to be graund with with Gum-waier,andby mingling two of thefe.or other Colours to2cthcr,youmay triakc almoftwhat Colour you plcafe of any Blew and Yellow mixed together make a Gree n, &c. of thefe lall mentioned Colours, mod of which arc very fad Colours, they ought to be ground very fine, and good ftorcof Gum-waltcrufed in the tempering of them. CHAP. LIX. now to find whether watermay he conveyed from a Spring heady to any appointed placid Here is an inftrument called a Water-level, for the ‘ performance hereof, the making whereof is fufficient- ly known. Nowit it were required to know whether water may be conveyed in Pipes or Trenches from a Spring-head to any determinate place, obferve the following dirtdiions. Place your Water-level at fome convenient diftance from the Spring-i tadjin a right line towards the place to which the water istobe conveyed; as at 30,4c, 60, or 100 yards diftant from the iprins-i'.cad. Then having inareadinels two long ftraight poles (which you may call your ftation ftaves) divided into Feet,Inches, and parts of Inches from the bottom upwards: being thus provi¬ ded, caufc one(whom you riiay call your aJsifiant)to fet up one of the faid ftaveS at the Spring-head, and require another (which you may call your fecond afi^ant) tocxcdithc other ftaft beyend iyoui-Inftrument at 30,40;ti0,or too yards forward, towards the place to which the water fliould be conveyed. Thefe ftauon ftaves king trefted perpendicular, and your Water-level in the;mid- prccifcly horizontaljgo to the end of the Level, and lookinj thromh the fights, caufc putfrjl apfiara to movea leaf ofoapei up and down your ftation ftaff, rill through the . fights you fee the edge thcrcof,and then by fome knovyn figne or Ibmndjiorimate ^c6 -.Ihgitfroffevird • Lib. 4 . to him that the paper is then in its true DQfition , then lee this note asainu what number of Ficr, Incl'es, and parts of an i Inch the edge of the paper reftech, which he muft note down in a paper. Then yout V\ ater-level remaining immoveable, go to the other end thereof, and looking through the fights towards your o- ther ftation ftaff, caufeyour ]ireo/ie/ 4 y/V]?crc:bcmanyHils between. • Here note, that in your paffage between the Spring head and - tbeappoihtcd place, from ftation to ftatloni you muft obferve : . 1 i this! order,etherw ife great errour wil enfuc,i«J that yovrfirfi ■ every iiztion, (land between the Spring head " andyour Water-level: and your/reW 4/f/}I(Mfmuftalwaycs Hand between youi-Water-level and the place to which'the 1 ■ water is tobe conveyed, thus by obferving thisordcr in your !__- work Lib./}. Injirments in Siiwi^ing, 307 work you rtial have no confulion, neither llial one of your Af fillancs take more pains then the other. Havin^^ tints orderly proceeded from the Spring head to the place appointed,call both your Alfiftants togetherjand caufe them to give in their notes of the obfervations at each iVation, and add them together feverally: then if the note of the feW ex¬ ceed (or be greater then) the note of the firfi aflflant, take the lef- fer out of the greater,and the remainder will iliew you how much the appointed place to which the water is to be brought is lower then the Spring head. neMlAfifla«ts^o:e. \ St.ithK Feet P^t. I If , 3 j 2 2 , I I _ 4 _ T I 0 Sum ~s 10 111 jThc fccond Ajfijhms Note. \srath>i Feet .Inch \PaTn ^Fans I 3-2 i4!_0| 31 3 ii By this Tableyou may perceive that the notes of the firfiaffidm conceded at h.s Icveral dations being added together, aLinteth 'Vl In^li^j^nd I of an Inch: and the notes of your femd ttpd* ‘t at Ins fevcral ftationsbeing added together amounteth to 21 Feet and two Inches, fo the number ohhc firfi afiSants obkrvT tions being taken from die number of, the fccond, there will fe mam 2 Feet, 3 lncr.es and and i ofan Inch, & fo much is the place' to which the water is to b^rought, lolvcr then the Spring Ld, according to the (Iraight Water-level,^and therefore the water may eafily be conveyed. Having exprefled as plainly as I can in words the manner of con¬ veying of water from a Spring-head to any appointed place, it will not be amifs to infcrt a figure by the fight Whereof you may plain¬ ly perceive how it is effefted, in which figure note that A repre joS T be ufe of feveral L i b. 4. i i 1 ptcfents the Inftrurticnt or Watcr-level, and the ftation ftaves; C the Spring-head, and D the place to which the water is to be con¬ veyed. f[ Here note that when you have called your two AifiHants toge¬ ther, and examined tiieir feveral Notes, and added them to^ gether, if then you fliall finde the lumme of your frjh/iflms Note to be greater then the iumof yom feco/.J that then it is impoflible to bring the water irom tliat Spring¬ head to the intended place: but if the Sums of th.e Notes of your two Afliftants do exaftly agree 5 there is ihc^ a poffibility of effe 4 >ingit, if the diftance be but iTiort, though with more charge and difficulty. |[ Note : , That ihemoft approved Authours concerning his particular do aver, that at every miles end there ought to be allowed Inches more then the ftraight Level for the cur. . rent of the water. ^ Note 3 , If there be any Hill lying in the way between the Spring-head and the place to which tlie water is to be convey¬ ed you muft then cut a Trench by the fide of the Hill in which you muft lay your pipes equal with the ftraighr Water-level, with the former allowance. And in this cale there be a V alley you mull then make a trunk of firong wood well under-prep- ped with ftrong pieces of timber, and well pitched or leaded, as is done in divers places between mre and Loadon. CNote 4, That in conveying of water to an appointed place, itis not ednvenientto bring it from 'he Sprrng-headby the ■ ncereft diftance or in a ftraight Line, but by a crooked or - winding way; and,you ought alfo to lay the Pipes one up and another down, but this is to be obferved but in fomc cafes on¬ ly, where the water will have too violent a current. Thus 1 have finiflied my intended difeourfe of Surveying of land, in which I have ftudied rather to make every particular therein j come ined plain and perfpicuous to the meane ft capacity , then with too much brevity to obfcuic that which I chiefly aimed at, namely, to inftruft the ignorant: Iconfcfs, Imaybejuftly blamed by thole who ate Mafters of the Art, or have a confidcrable know¬ ledge thereof already, for ufing too many circumlocutions, but I anfwer, it was not written for tficir fakes, yet I hope it will not be rejefted by them ; and although I do not attempt to teach luch more then they know already, yet (poflibly) they may herein find fomethini worth their peiulal & pra£lice,or(at leaft) it may be a Remembrancer unto them to bring to minde what otherwife they may have forgotten. But ccafing to apologize any more iermy Book, let it now fpeak for it felf. finis.