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FIELD ENGINEERING. 
 
 A HAND-BOOK OF THE 
 
 THEORY anp PRACTICE or RAILWAY SURVEYING, 
 LOCATION, AND CONSTRUCTION, 
 
 DESIGNED FOR THE 
 
 CLASS-ROOM, FIELD, AND OFFICE, 
 A LARGH NUMBER OF USHFUL TABLES, 
 ORIGINAL AND SELECTED. 
 
 BY 
 
 WILLIAM H. SEARLES, C.E., 
 
 MoOMBER AMERICAN SOCIETY OF CIVIL ENGINEERS. 
 
 SIXTEENTH EDITION, 
 
 PWEEE TH THOUSAND, 
 
 NEW YORK: 
 JOHN WILEY & SONS, PUBLISHERS, 
 53 East TENTH STREET, 
 1895. 
 
CopyRiGcut, 1880, 
 
 By JoHN WILEY & 50Ns. 
 
 Press of J. J. Little & Co. 
 Astor Place, New York. 
 
PREFACE. 
 
 AuLTHouGH the modern railway system is but about fifty 
 vears old, yet its growth has been so rapid, and the progress 
 in the science of railway construction so great, as to render the 
 earlier technical books on this subject inadequate to the needs 
 of the engineer of to-day. 
 
 In the course of his practical experience as a railway engi- - 
 
 neer, the author was strongly impressed with the want of a 
 more complete hand-book for field use, and finally concluded, 
 at the solicitation of his friends, to undertake the preparation 
 of the present volume. 
 
 The aim in this work has been: 
 
 First—To present the general subject of reilway field work 
 in a progressive and logical order, for the benefit of beginners. 
 
 Second—To classify the various problems presented, so that 
 they may be readily referred to. 
 
 Third—To embrace discussions of all the more important 
 practical questions while avoiding matters non-essential. 
 
 Fourth—To employ throughout the work a uniform and 
 systematic notation, easily understood and remembered, so 
 that after one perusal the formule may be intelligible at a 
 glance wherever referred to. 
 
 Fifth—To express the resulting formula of every problem 
 in the'shape best adapted to convenient numerical compu- 
 tation. 
 
 Siath—To furnish a large variety of useful tables, more com- 
 plete and extended than any heretofore published, especially 
 adapted to the wants of the field engineer. 
 
 An elementary knowledge of algebra, geometry and trigono- 
 metry on the part of the reader has been taken for granted, as 
 a command of these instrumentalities is deemed essential to 
 the education of the civil engineer. The few references to 
 mechanics, analytical geometry, optics and the calculus may 
 be assumed correct by those not conversant with these 
 
 DAN Say 
 
 branches. 
 
iV PREFACE. 
 
 Many of the problems in curves are new, yet there is hardly 
 one that has not presented itself to the author in the course of 
 his practice. The investigation of the valvoid curve is original, 
 and though the mathematical discussion is somewhat difficult, 
 yet the resulting formule, taken in connection with Table X, 
 are exceedingly simple and convenient for the solution of a 
 certain class of problems. 
 
 The treatment of compound curves is novel and exhaustive. 
 A few general equations are established, which, by slight 
 modifications, solve all the problems that can occur. 
 
 No discussion of reversed curves is given, because these are 
 inconsistent with good practice, except in turnouts, under 
 which head they are noticed. 
 
 The chapter on levelling includes a discussion of stadia 
 measurements, with practical formule. The chapter on earth- 
 work contains a review of several methods for calcuiating 
 quantities, and states the conditions under which these suc- 
 ceed or fail in giving correct results. 
 
 Among the tables, numbers 3, 5, 6, 10, 18, 19, 26 and 29 
 are original. The adoption of versed sines and external 
 secants throughout the work, wherever these would simplify 
 the formule, rendered necessary the preparation of tables of 
 these functions. The table of logarithmic versed sines and 
 external secants has been computed from ten-place logarithmic 
 tables of sines and tangents, so that the last decimai is to be 
 relied on, and no pains have been spared to make the table 
 thoroughly accurate. 
 
 Tables numbers 4, 7, 8, 9, 11, 12, 13, 14 and 86 have been 
 recalculated, enlarged, and some of them carried to more deci- 
 mal places than similar tables heretofore published. The 
 intention has been to give one more decimal than usual, so that 
 in any combination of figures the result of calculatiop might 
 be reliable to the last figure usually required. 
 
 The tables which have been’ compiled and rearranged are 
 numbers 1, 2, 15, 16, 17, 24, 25 and 81. The tables of log. 
 sines and tangents here given are the only six-place tables 
 which give the differences correctly for seconds. The table 
 of logarithms of numbers is accompanied by a complete table 
 of proportional parts, which greatly facilitates interpolation 
 for the fifth and sixth figures. 
 
 In all the tables, whether new or old, scrupulous care has 
 
PREFACE. 
 
 been taken to make the last figure correct, and the greatest 
 diligence has been exercised by various checks and compari- 
 sons to eliminate every error. It is, therefore, hoped and 
 believed that a very high degree of accuracy has been ob- 
 tained, and that these tables will be found to stand second to 
 none in this respect. 
 
 The preparation of this work has extended over several 
 years, as time could be spared to it from other engagements. 
 It is, therefore, the expression of deliberate thought, based on 
 experience, and as such is submitted to the judgment of 
 brother engineers. If it shall prove to have even partially 
 met the aim herein announced, and so shall serve to smooth 
 the way of the ambitious student, or to assist the expert in his 
 responsible duties, the labors of the author will not have been 
 in vain. Wa. H. SEARLES, C.E. 
 New Yorg, March ist, 1880, 
 
TABLES. 
 
 I. Geometrical Propositions.........--....- Bate ete rc on 74 
 
 Il. Trigonometrical Formule@.........--.+seses eee ee cece ereeee 275 
 Aiba Curve MOrnati laa: ssh ecreee tare creer lets enero ee otain se Vei ierele ae QU¢ 
 IV. Radii, Offsets, and Ordinates............. arate weet ose wien 280 
 
 V. Corrections for Tangents and Externals..........-.......-. 288 
 
 VI. Tangents and Externals to a One-Degree Curve............ 289 
 VII. Long Chords and Actual Arcs..........-..-20 esters eee eeeee 993 
 VIII. Middle Ordinates to Long Chords...... ......------..--+.+s 298 
 TX.’ Linear Deflectiong. 3 coe vos ec cle o ea oem owen vice eis vem oslo 301 
 X. Curved Defiections; Valvoid Arcs..........-.-.-2+-+--e-00-- 308 
 
 XI. Frog Angles and Switches................. : pita wae eieelebaoeee 303 
 XII. Middle Ordinates for Rails. ........060+-+00-e ens see cee ee es 304 
 XIII. Difference in Elevation of Rails. ......-.....--.---+-----+8- 304 
 XIV.) Grades and ‘Grade Angles te ercc cairns ste cee oat ete ere 305 
 XV. Barometric Heights, in feet..... igies en Facets sees een ie eee 307 
 XVI. Coefficient of Correction for Atmospheric Temperature.... 399 
 XVII. Correction for Earth’s Curvature and Refraction........... 309 
 XVIII. Coefficient for Reducing Stadia Measurements............. 310 
 XIX. Logarithmic Coefficients, Stadia Measurements............ 811 
 xX. (Lengths of CirctlarAreg tere. ce -cne sss eee ee mais emai 8312 
 XXI. Minutes in Decimals of a Degree......... ....- SS. fs eee 313 
 XXII. Inches in Decimals Of a. Hoots. ou: iran ee eine ors oot 814 
 XXIII. Squares, Cubes, Roots and Reciprocals.... .-...-..-.. «- 815 
 XXIV~ Logarithms of NUMDGrs ees <tc ee ene oe Serer 832 
 XXV. Logarithmic Sines, Cosines, Tangents, and Cotangents..... 359 
 XXVI. Logarithmic Versed Sines, and External Secants........... 404 
 XX VII. Natural Sinésiand Cosimesin ease = i-tac 2 siete cto se ereernelote 25. 49 
 XXVIII. Natural Tangents and Cotangents. ..........--.++++-eeeeseee 458 
 XXIX. Natural Versed Sines and External Secants......-........- 40 
 XXX. Cubic Yards per 100 Feet in Level Prismoids............+..- 493 
 
 XXXI. Useful Numbers and Formule............ (Gah de TARGA DIN oats 
 
CITAPPER’ I. 
 
 RECONNUISSANCE. 
 SECTION PAGE 
 2. Topographical considerations. .........--.ceeseeeteeenees voeeseee 1 
 Be Use. OF WORDS oso gec vase Pee sghl cas soos ts See nes es Sem cae aee 3 
 Pe POCKeu MStTUMMCMUS. oe a2. as oe alec + atte etarroniionce wiaaietet tthe ouernre crs sievers 3 
 0 ay ASRS) 00 be i er ene Ra rattan Reon ieee incident aici’ OM AE Cobs9 + FDIC 4 
 10. Pormulss fOr AMET ose a, fs a sae sieveponel oy staraya; a cl ane arama eke pate eae bebeaeich ae 4 
 TG TOCKE AO VE) Sy raise ce eres eis also oie ee oof oae ratenth cmmven ata es ts Seale o ss Stage suesaza ets 7 
 TZ” PrismMatiG: COMPASS <5 ie sjcccse evs ccy Sve oles o,siciere + eee OU Anes s Seis ple 7 
 
 CRAVPTER II. 
 
 PRELIMINARY SURVEY. 
 Le DTI EL ORIG She Neh ees aie ete Te ie Rcted era erat oicconmclolctel obese arn’ oensrenrers 8 
 19. Engineer COTps..-...... sce e cece cece cece rec ewcccsnesreceneccereees 8 
 Oh Ciel Sarina ys. eS ene theo se pein ahbieln was Sal's simwiataTe bio! 9a ete 8 
 D1 ASSIStANt OM IMCS ss cclser » clesie o acclee oo oltre, vinteiaitra dala nieie alaisle sieimcls\tiel8's 9 
 | De CAV ALININV LTS Se ee tae ooh aoe cca Tork via we clorale is el ole ci oRMeKeebeatate miomaue sate he's ole 10 
 | AAAS OUITALT Se ere 5 Nate ie etre Una o nian. eens el atelarerndaretevh oly arakel's skater ote a 11 
 ! S62 TOPORTAPNSL Actos dsc te es ccbras cog be usens vet pmaetparaes sien wees sear 11 
 | EP Tres VOL ee eater et ra tee sae eaten a Sis o a'el eat ems a eis ene eerie laters wigs Marcie e's 13 
 | ys ROTC Oa fee te lees AE oe bn cb Ne UP RE he Bes Pin a wha een ss 3 
 | Fa FRG COMPASS £.c/s des caog dol se se ot hag Ga cums Reet Ree ees ote etree te 14 
 ) EMITS EL ULTA sc tovay tare a ane aroha ea as ah aos Pavol algcat eh re sembeta atatas kere fe etenar at ey avetel erecta ntote 14 
 ; BRE Love 06 55 cus eS) ea ONT ar ee RY cee eeeae ch Baas (Oko 
 SRA THER TOGS alc otc ns aha ceria 4 aisle: © ses bisja! elass aes raretO piel olerpcis Gime tanoue’: ocetersiertys 15 
 SR THE CHMOMELET So... vj, sjeree.0 ojtuie ober ose welelont oie tekegeinte nee cierto ogal fol ge nis 16 
 | BE Te tlAanG-tAble. ccwsceec-s0vpi sa ca >: sae SURE a nea sini te pt 16 
 | AUTO UEATISID A tte: clesalotein cis abelelsie snc Ee te c34 he ee a ee 16 
 | APM TAN SI DOWES cco sie seccmie etn 2, ea Oise cero el apreastn nee sie iceleseminiaieieisis 17 
 | AS es Trangit Gage ast. os 2 haan ols dae orng aieee renee ofeen inset Ue 
 | 44, Obstacles to alignment and zneasurement....... Sane Mery: 18 
 Ae Parallel nes... on. ne co elsccle pallet are ep ole are esialeiann syercle aisha zirpneie’s 18 
 | 46. Lines ata smatl angle. ... ..). 20: .o005 5 on Beeler ad cies wsloe m ne.si6 18 
 | AY, General problem... .. 2... a0 105 eae om sede snes anise noe ole cle Hees ann els 19 
 | 48, Lines at a large Angle... ......- newer cece r etn rce estes meee cers 20 
 49, Selection of angleS:....... 2.2 secececcee ces seer e ete rene ese cceneee 21 
 | 58. Rocky shores; Tie-limes.... 2... .ceseeee eee s eee e reese cere eeeecece 22 
 . System of plotting MapP......-. cece ee eee eee rete eeees Sy oes 23 
 
CONTENTS. 
 
 CHAPTER III. 
 
 THEORY OF Maximum EconoMy IN GRADES AND CURVES, 
 
 SECTION PAGED 
 
 Bor Cnoice Of YOULES. .. « .2.s sa snip eee ae alee nicks 6 gee earn tere 24 
 “Statement ofthe problems. 2... epee tse sie > os eee eels es 2 
 
 Engine traction .....202. 55 deca tes Sten eae g= 4 sees es aac 25 
 ~ Engine expense.....6.. 2065 oe Ses mele oo eens + Seer nee ems + eee 26 
 GQ erd Bu atchtci Fe OLE Om ROW £00615 (0)s PERO G SS hES SOME SROese < spon a sonnaaek 27 
 Resistance’due'to grade =.: Saeet cs see ce > eee ne ele ie eles 20 
 2) Resistance: AUC LO Cul VC tea e wit eee ere eat eine bie mio alas meted om gel 28 
 © Rormuls for resistances: e-ee bee anc clini oe arene lee sieetnwe=ir's 28 
 . Formule for maximum trains........... as pac ani cheese 29 
 Pan PIne=StaLek-\: nse eer ieee ee Peat ate GS eae sicko ee 30 
 6. Graphical solution...... wef tee see et wc eee Sve ER nets oat 2 
 . Train -load-reciprocals =: ag. 5. -elied. a. rte ne « 2s ese sclee olteey mes ae gee 
 . Reduction-of grades ON CULVES..... 2.52.6 cee ieee eee eee eee eee 33 
 J) Behe OS ome ac otc pugs PE ge 2 ROE RES Une aioe Ahise 51 wan Se teleee 33 
 Assisting engines; 2.1ccicc etic dettole Smt ones elt ott ne oie = erate reel 36 
 
 71. Maximum return grades: ... 2... 0.2.8 ee ss eee eee eet cere neon eeaee 36 
 WD. Undulating grades 1.11.21). ahi desieiiekicistet = «sialon eraieieta siete estan te 7 
 75. Comparison Of routes... 2.0... ..e cece cece rece cece tee ceneee ceceres 38 
 VED Value Of distanGe:SAaVCG 2. . cee cies ocr caee ci eo a = aisle weieinseiniclgis oat 39 
 PO CONCLUSION Pos. ss seve bole cee mee eee Oe is ae oe eo oreo rsa spas tats 39 
 
 CHAPTER IV. 
 LOCATION. 
 pCleneral-TeCmarks. d2eccs- caesar oe ee Rm id avace eoreberets 39 
 BY Longe tangents... . 2+. 05 ses smdesinees Wes ses sim aes ais Eis Sista = SO 
 lueveller’s duties; Profiling ss: = sees see MEGS en Sos ti eee 
 . Establishing grade lines..........--..:---.----+-=- et Ores! 
 CHAPTER. V. 
 SIMPLE CURVES. 
 A, Elementary Relations. 
 
 82. Limits to curves and tangents: ....... 6...6..5.: 5 a Bate eaters sae 4)} 
 33) Definition of terms e.5.-.-e eee Cie tates teeta sets Cee MM ASARLSNE 45) 
 84. Radius and degree of curve..... Ss SEDER Ns 3s patios NRE LS 43 
 65>. Measurement of curves....... hie se WH oe bichata ta a Sees wie Ute ARs lols chet Panos. thee 44 
 30. Approximate value-of B...2. 28.22% Aki we wists bine bis tie ee ey Malet eaters 44 
 87. Central angle and length of Curve,........- -.s-eeseee eee e eee eeee 45 
 83. Definition of other elements’. 2. Gaescce.se ss arr RRs Ae Ree 46 
 99, Formula for tangent distance: D7... 2s. cuts emcees oes nee senie's 46 
 ois Kormula for lone chord C@s22: st. escs. 2: oo SPP AER HOST 47 
 92? Formula for middle ordinate M: 2252: co.cc dsc svee eee esos see's 48 
 93. Formula for external distance #:..2:. 22.55. cence cece cee ct ecsses 48 
 95. Formula for radius in terms of Tand A...............- SC ae 49 
 95. Formula for external distanze in terms of Tand A............. 50 
 07. Formula for radius in terms of Hand A............ oat, JE) eee Oe 
 
CONTENTS. 
 
 SECTION PAGE 
 98. Formula for tangent distance in terms of Hand A...... ...... 51 
 99.: To. define the curve ofan old tracks. ..e ete secre cece es 51 
 
 10)2 Other curvesformulay Panis ES See eee cures coos rime 52 
 
 B. Location of Curves by Deflection Angles. 
 
 j[0s Pal BYsva (axeynteyaWt: 4 s faq (s\-jaeeeeD Mrs eR POM RA COT AC CAC ore tage ar” De 
 
 dies Ble Or AENeChONS jasc. stereeie nice 5 Scola tee mene loses en slee fase spei 8 ke = rr 53 
 103, Rule for finding direction of tangent at any poijnt.. ............ 53 
 RU LO CHRONOS oc ierod aha Since ohio one eae lee ay ti sls sore en fae Oe 
 105. WeHeCLLONS 1 OF SUDCNOLGS 4 -.. sijcc «4 sey ois eirpeiaie eet ee biel teh ie hate 54 
 JOGr Correchions fOr SiO CMOTOS a 5 ora jor ysis o osaie et sreue ake iets eee pes eaiejenseln anes chit 55 
 ip RaMOMOL COLTEChION UOrOXCESS OL OTC. oases aclaielorcve slatenietnictet-ieieusiieeis = 55 
 POS Treut Site WOM KOT MOUS VCS ones, creek dw) ans ao, wigs odes lao nll oan carded Ae aT 57 
 PRO OIC IO LES secs 9 oe. peta xea wie = 2,0 snia. euoiehche dig elena siete eee ale are eae te 58 
 110. Central angle in terms of deflections....... 00... .-.d meee +. 6-2 oe 58 
 fii Method Dydenleohons On] Vice .t-yo.0.0i5 « stencil eae eualre eta ae 58 
 
 C. Location of Curves by Offsets. 
 
 1127 Rourmethods seek - ee eae eee ee aera Nike Gaels 59 
 113. By offsets from the chords produced............+--s.seeeeeeee eee 59 
 11 Doshec inning witty a SUDCHOLGS aoc tart. settee tere lelatel a ter-taen 60 
 115. Formula for subchord offsets, approximate....... ST eC 61 
 117. By middle ordinates... seas os soe 0 ies 2h s Seiden anwcas saree e eyes 61 
 118. Do. beginning with a subchord.......-....-----.ee cece eee eee ees 62 
 DLO B yaar Ont Ori SUS ares where bos Netto Sets Stole sroreicing inte inlays seat an ela wiehcas 62 
 120. Do. beginning with a subchord........ 2.26... - see eee eee cece recone 64 
 12t-By ordinates fromalong chord >... .).° 7s eee nee oe 64 
 1298 Dosfor‘an even number of stations... 10" 5. ae essa cee ores scree ee 65 
 1233) Por foran odd number Of Stations. .--c.o. ce cette te aceis ese ses carce 66 
 124, Do. for an even number of half stations......... ..........-2.-.. 67 
 125. Do. beginning with any subchord............-..---..--eeseee sees 67 
 126. Erecting perpendiculars without instrument.................-.-. G9 
 
 D. Obstacles to the Location of Curves. 
 
 Tota ENE VELUOX IN ACCESSIOL Gs. 2.5. so aba sae aaiatne catia pers Cole teicher syste el lot 69 
 128. The point of curve inaccessible...........-- 2.2.2 ee s eee se ne 7 
 129. The.vertex and point of curve inaccessible.................----. 70 
 130, The point of tangent inaccessible... ............-- 0, -0++++-+-+-ees fi 
 131. TO pass. an Obstacle Ona CUIVE: © 6.2 aise cecilia oe? ln eine 2-2 = 72 
 
 EE. Special Problems in Simple Curves. 
 
 132. To find the change in R and # for a given change in T........... (¢ 
 133, To find the change in R and T for a given change in #........... 74 
 134. To find the change in T and E for a given change in f........-.. 79 
 135. General expression for elementary ratios...............+-----0- 75 
 136. To find a new point of curve for a parallel tangent.............. 76 
 137. To find a new radius for a parallel tangent...................... 76 
 
 38 To find new P.C. and new radius for a parallel tangent......... 
 
SECTION 
 139, 
 
 CONTENTS. 
 
 To find new tangent points for two parallel tangents............ %8 
 
 140. To tind new R and P.C. for new tangent at same P.T............ 80 
 141. To find new P.C. for a new tangent from same vertex........... 81 
 142, To find new radius for a new tangent from same vertex......... 81 
 143. To find new R and P.C. for same external distance, butnew A. 82 
 144, To find a curve to pass through a given point............... aactars 83 
 145. To find new radius for a given radial offset.....................- 84 
 146, equation Of the -velVvOld: tees werretemle ter ee oie cca ale ae are re 86 
 147. To find direction of a tangent to the valvoid at any point........ 7 
 148. To find the radius of curvature of the valvoid at any point...... 88 
 149. To find the length of are of the valvoid......... IS fas a's deep eres 88 
 150. To find new position of any stake for a newradius from same 
 
 Pie ee eae ee re aT ee tree eo 89 
 151. To find new radius from same P.C. for new position of any 
 
 batho sic sh fe 26 os Ee eae eee ee area oe enn deo loa caatee ote eters 92 
 152. To find distance on any line between tangent and curve, ........ 93 
 158. To find a tangent to pass through a distant point................ 94 
 jot, To finda line tangent LOU WOlCUrVES as teen. ee bs hie een 96. 
 155. To find a line tangent to two curves reversed..............2.020 98 
 156. Study of location on preliminary map; Templets; Table of con- 
 
 Venient .CUIVES ised cle be EER ee ieee via Irckats fevalectociaenes Rta e ees 100 
 
 CHAPTER VI. 
 CoMPOUND CURVES. 
 A, Theory of Compound Curves, 
 
 i. WDETIDITION . jtesti is acc bons Sace.a.0 DSc Sree Rae ees PA elt Se eee 102 
 158. The circumscribing circle............... adie Gal pterete Mapes ERT ET 102 
 159. The locus of the point of compound Curve.. ©. .¢.0 52.0. <s0%8 ve <0 103 
 160. The inscribed circle of the principal radii.......0.....c.0..00.00- 104 
 
 Cor, 2, Maxima and minima,.o£ thewadiia.) sic ae sesiesee ee 104 
 B, General Equations. 
 161. Formula for radii, central angles, and sides..................... 105 
 162" Given: S,.S,° 4 and Rystotind)7 wee ese and ee ee 106 
 163. Given: AB, VAB, VBA. and R,, tofind A, A,and R,.. ..... 107 
 164. Given: Rk, A, R, Ag, to find the triangle VAB. oo. 0 .. otee ene 108 
 165. Given: A, the radii, and one side to find the other... ........-. 108 
 166. Given: one side, radius and central angle to find the others..... 110 
 o/s Remarks on special Casesijzeex .5c secke cee sche oie eae eee 111 
 163 Obstacles; the.P.C.C.anaccessiblece. 6 ua. hte eee eee 112 
 : C. Special Problems in Compound Curves. 
 169; Todind a new P.C.C. fora, parallel tangent...,.........-..-?sseens 1138 
 170, To find a new P.C.C, and last radius for a parallel tangent...... 115 
 171. To find a new P.C.C. and last radius for the same tangent...... 118 
 172. To find a new P.C.C. and last radius R.’ for new direction of 
 
 tangent throuch sameP. 7) ever ers. en a tlie ete ene aie ot 
 
4) 
 
 CONTENTS. 
 
 SECTION PAGE 
 173. To find a new P.C.C. and last radius FR,’ for new direction of 
 tangent through same P.T.......6. eee eee eee e eee eens agiad Wa! 
 174. To replace a simple curve by a three- centred compound curve 
 between the same tangent points.........-6-- esses eee ee eee 127 
 175. To find the distance between the middle points of a simple curve 
 and three-centred compound CUrVe......-. ss cece eee cee tees 129 
 176. To replace a simple curve by a three-centred compound curve 
 passing through the same middle point........--.---+- ee ie 
 I. The curve flattened at the tangents.........--.... eee seer eee 129 
 Il. The curve sharpened at the tangents. ....-...---4.-++++++eees 1382 
 177. To replace a tangent by a curve compounded with the adjacent 
 CUILEV OR SL ah rae tee crt EE tie dee Pistons rareeevateraiene sieve atalee 134 
 I. When the perpendicular offset p is assumed.........----+-+ 136 
 II. When the angle q or Bp 1S*ASSUIMEG ie ee Crete a iclees caine 137 
 TIl. When the radius R, is assumed ............- ee eee eee eee BT 
 LVes Locus-of the. centre Oj ah ikica ee et sate reece mate ergiaierer=tel 133 
 178. To replace the middle arc of a three-centred compound by an 
 Are Of Ciftorent TAGING .. 6 so. 8 bees ce cepemaincle ewe otic ae orcs 140 
 I. When the radius of the middle arc is the greatest,.......... 140 
 Il. When the radius of the middle are is the least.............. 141 
 III. When the radius of the middle arc is intermediate.......... 142 
 CHAPTER VII. 
 TURNOUTS. 
 
 9. Definitions; Frogs and switches.....:....---+-+seseee sees ece cree 147 
 180. Single turnout from straight track in terms Of from angle ...... 148 
 181. Single turnout from straight track in terms of frog number..... 149 
 182. Double turnout, middle track straight, to calemate Ht... am 151 
 
 183. Double turnout, middle track straight, and three given frogs... 152 
 184. Double turnout on same side of straight track to calculate the 
 
 middie frogs Hie esse ach See Ses conir eaeec city te aut elas 153 
 185. Double turnout on same side of straight track with three given 
 PTOGS 0 ok hone pe sink ote reek eaas os manga asia scl 2p haa ca oie 155 
 a. When the middle track is a simple curve.....-.-++-+-++++++++ 155 
 _ When the middle track is straight beyond F’.........--+-.+-- 158 
 . When the middle track is reversed at P.......+--+-++--++-+++5 159 
 186. scout on the inside of a curved track.......-..-.+--+0+ seeeee- 161 
 187. Turnout on the outside of a curved track....... ..++-+++-+-+0s> 163 
 188. Tougue switches 1.2... opens sawinny uoriitegasAgment meme hems ¢ 164 
 189. Tongue switch turnout from a straight track.......:.2.0sseeee- 164 
 190. Tongue switch double turnout to GheQded ah (pte ee wc odo y ooo enor 165 
 191. Tongue switch double turnout with three given frogs..........-. 166 
 192. Tongue switch double turnout on same side of straight track 
 with three given frogs........--+----+++-- eS Ben SMA hors 167 
 a. The middle track reversed at F........--.-- 22s cee cece ences 167 
 b. The middle track compounded at F’........-.--- Etat & 168 
 
 c. The middle track straight beyond I’.... po Ne ared age aes (ola: 
 
x11 CONTENTS. 
 
 SECTION PAGE 
 
 193. To find the reversed curve for parallel siding in terms of F' and 
 perpendicular: distance sass peeee aa ea eee oe +3, 169 
 
 194. To find the connecting curve from frog to parallel siding on a 
 curve in terms of #'and perpendicular distance p.......... 176 
 a“The siding: outside of main track cf aseiltts dhe bese 171 
 H b-2The siding inside, of mainntrack..ny isewwdcaee eS eee ity 171 
 ! 195. To locate a crossing between parallel tracks. ..:...........0.26- 17% 
 ! 186. To locate a reversed curve crossing between straight tracks.... 178 
 197. To locate a reversed curve crossing between curved tracks..... 174 
 198. To find the middle ordinate m, for one station in terms of D.... 175 
 i 199. To find the middle ordinate m1 for rails, in terms of rail and R.. 1% 
 200, Curving rails; To find m1 in terms of railand m................. 176 
 201, To find elevation of outer rail On CUrveS « s...6. 6.6) ieee el ee 177 
 
 202. To find a chord whose middle ordinate equals the proper eleva- 
 ANON. Ses os aes 6 ois See eo OIE ee a eee ee ee 179 
 203, General remarks on elevation, of Tail! eaiec ws tone eek ee 19 
 204, General remarks on coned: wheels............ sc.eseceee seeees . 180 
 
 CHAPTER. VItt 
 LEV£L1LI.G. 
 205. Use of the engineers’ level. ... 2.2.0.0... c.ccee cece cecees eee ots 181 
 DUOe Le datum, how ASSUNIECc susseincas aes oe ode ce see sbislde eee ee eine 181 
 Avie benchés: how Used $2 BW eee, faa ee acl a Sea 181 
 Ave. FL eie ht, Of  Tstrumrent steal see ee Oe ole os ae aes ee 182 
 209 -Reading Of the Trodscs bes eee eee ee 6 cae owe oe ae eee 182 
 210, -Llevatiom- of mtermediate points: << +.2.s2 . 5a. thawinsheee aeneee 182 
 Bed at ACT SOLUTE See) See Sar stiatewiert ie Se neo eR eee 182 
 Pie. KUle tor DACkKsights and t OPEsi@nts om. -..0 acetate cen Sere 183 
 mis, Horm of field-book: proofot extensions. ......cc.-..2. oe 183 
 ber OE VONIGRY 2. ws. a> 2 oo 07 we'd apo ROR TEE Ub atineeGlt DE ERIE One ee 1&4 
 mio. mibrple leveling; test levels... n1s. need. eset eet sce foe tare 185 
 216. Errors in reading, due to the level; how avoided.=.............. 185 
 217. Errors in reading, due to the rod;. how avoided.................. 185 
 mis; hors GUC LO CUrVALUTe Of THEIeALb hate ee. sete: oe oe See eer 186 
 mig Barors due to refraction -4 > soso. ee nee be es chs Saltese s orate ee 187 
 220.) Wadlus OF Ciryature Of the Garhiits. cose seks seater eee 187 
 Eels evelling by Lransit:or theodolllewunmetcs «nectes cee eee 188 
 22, To find the HJ. by observation of the horizon................... 189 
 223. Stadia measurements; horizontal sights......................... 191 
 224, Stadia measurements; inclined sights, vertical rod.............. 193 
 225, Stadia measurements; inclined sights, inclined rod............. 195 
 j 
 CHAPTER 1X 
 CONSTRUCTION. 
 
 e260. Organization of engineer departments. 2s... 65sul,4-0s Jessie... 196 
 Reve Hearine and Tub bing a5 4 Sk Amsaeherccee ae Ot ae eee 197 
 
 p Lest levelsand:enard. phipgs acu keene at eee ee ae ete ee! 
 
CONTENTS. 
 
 SECTION PAGE 
 990 OrOSs SECHONS* “SlOPOSe si. 22 1s ole nie,0 oe Soleo as Se eiettnieencistare oss wea, hod: 
 230. Cross sections, formule for........ SEE AIG HO Coe Ses Sent ane aie $8 
 931. Cross sections, Staking Out... . 2.2... ccs seccwsre ee Bias anise 200 
 232. Cross sections on irregular ground..... aire he BRS iss oe por Ree 201 
 238. Cross sections on side-hill work.................+--- sgh 3 AOC 201 
 234, Compound Cross sections.............04. AEE PCRS AOL E ree foe 202 
 235. Selection of points for cross sections............ SEaset nes ahha 203 
 DEG)  WOEiICAl, CUEN ES > oe cieia aisle. << apelan oegsuane oalatobenene piel avatoysLoleltarieln ateastayspet 2 203 
 O37 HOT OF CLOSS-SCCLLOM DOOK® =~ clas n.d cele siere sietetate ere erate elataal=)s08\aiatelal a/2 204 
 O38: ixtended Cross profiles. .i32 yt) eatiac grasa «seietet-/-i=10 Meteors cl 205 
 BIO TNACCESSIDIO SOCLIODS « «5 = ais shoe ccyo is ots al eta citielebs eles ato ers Samed iocniatet © 205 
 DANS LSGLADCC TVA SSCS Aa akils.ccc 2 x e-t.c:w «) cue) oro wich teeid etavatel ene tortie eiehah ance csacartes oie 206 
 Ste BOLLOW -pitecnia se yc cleeis.s e's te Lae SS yas aa ee Soca re ea eae ots eis 206 
 POS Sarin kare selIICrEAS’.. sie os «dese dors ol evslela eleieyaiel suet Melee talc) aeaetelete eteiaior= 206 
 WARPED ECE VWoORICS « c scels hase 2 Seas ose 3. pore one A ee te olen eet era ee 20% 
 OAT A ILETAtIOMIOL UTIG., ..c.cr- a ciazeiccie sone elapoteierpie) tie memes ererers ret sams, otras ene 207 
 945 DrainS and Culverts, « <o<.c a. <sjslald -/aa:oeiavelethe Alar alberta Set 2,o ya easke is anaes 208 
 DAG Se ACG Mertsn scans: os ae bicisis shail aero emer taerre Utes g dake Ce eee 209 
 247, Foundation pits; Bridge chords on curves.........--.++++-+--0+5 210 
 O49 Cattle-OUBrds Jo 4 6 cbs ses e-eiee 10:0 + mom nen e.8 bee eee Ral aoa eet a 214 
 DAG IT POSEIG=WOLKK 25 a clcisd < sieve are cise: os -sisbare/e Bi eienege lel Nos Lape emee teeth Ws atiel yn aistay tak . 214 
 250. Tunnels:. Location; Alignment; Shafts; Curves ; Levels ; 
 
 Grades; Sections; Rate of progress; Ventilation; Drain- 
 
 DIES we cieieie e Late 3 a's] nseieis s/sceie jae « ewisn'e sees ae cleieink= ciarey> /e siahiiestammaats 216 
 951. Retracing the lin€.......... ec cee eee cee eee n ee ett teers ey eee: 222 
 959. Side ditches and. drains. 5... x. 2. e106 sisls «slay over starperers]© wimrsoPaererreags 223 
 BES TBallastine so. Feo. ais owe evo Seis tie owinitin ¢ a HN mele Siale 01s mewteresiialeaates Sainte 223 
 254. Track-laying; Expansion of rails; Sidings...........-..-+-++++-- 223 
 
 CHAPTER X. 
 CALCULATION OF HARTHWORK. 
 
 954. Prismoids: Choice of cross sections. .v... 0... 26. coer eee eee eee 225 
 £55. Formule for sectional areas.........-- .--.-+----+-+++6-- tS atone 220 
 256. Prismoidal formule for solid contents. .....-......+--.+---0e-es- 229 
 257. Tables of quantities in cubic yards... .....-...---.e eee eee eee ee 229 
 258. Tables of equivalent depths............ 2... cece eee e eee teen eee. 231 
 259, Formula for equivalent depth in terms of slope angle........... 268 
 260, Conditions necessary for correct results in use of tables........ 25: 
 261. Method of mean areas; correction required... ...............06-. Qee 
 262. Exact calculation of content; examples.................. Shite eins ROO 
 263. Wedges and pyramids. .........0. 22 cece eee acetic ees cece een 286 
 264. Side-hill sections, uniform Slope. ....... 0.522... .050 veecesee eee 226 
 265. Side-hill sections, irregular ground..............--.---+- PS ey 237 
 266. Side-hill secticns in terms of slope angle.............-....-,---- R230 
 267. Systems of diagrams. ...... 6... ccsasene sue dsines se snes ewes wes nobus 238 
 268. Correction for curvature in earthwork................-.222-. eee 239 
 269. Haul; Centre of gravity of prismoid........ 2.2.2... ee ee eee ee 243 
 BiO> Pinal OSUIMALO qca.5 5 clo oe os Lee a ah air ak Waa ae eo wets Gar are ee 245 
 271. Monthly estimates. ,......... rage rages esate ra 246 
 
CONTENTS. 
 
 CHAPTER XI. 
 
 TOPOGRAPHICAL SKETCHING. 
 
 SECTION PAGE 
 ac General remarks 0.0.5. 000 2 aie ae a eka en 247 
 gi) ACU Acial foatures. <. . i, a eee eas ee Ome blag 248 
 274, Natural features; Contours; Hatchinegs. 2.1.7. , 3) EP GE ORS, » 248 
 275. Method of sketching -.. 1g aos te tye een. eee 249 
 CHAPTER XII. 
 ADJUSTMENT OF INSTRUMENTS, 
 eco The transit ....:.-.. 22a aeataan saan enclmeeanne Aiea katie 250 
 Bec abe level... :.2. 7241.5. diame Aeepeaieuet rg cena eae 252 
 gOS Tie theodolite. :-:2-2f5: ots asian 1 aeirad a een 2538 
 
 CHAPTER XIII. 
 EXPLANATION OF THE TABLES, 
 
 wine gy tt FO eh ster ahera! eteNeteratela tates Siete’ Tee cok Ree EN tee 
 
FIELD ENGINEERING. 
 
 CHAPTER I. 
 RECONNOISSANCE. 
 
 1. The engineering operations requisite to and preceding the 
 
 construction of a railroad are in general: 
 THE RECONNOISSANCE, 
 THE PRELIMINARY SURVEY, and 
 THE Location. 
 
 2. The Reconnoissance-is a general and somewhat hasty 
 examination of the country through which the proposed road 
 is to pass, for the purpose of noting its more prominent 
 features, and acquiring a general knowledge of its topography 
 with reference to the selection of a suitable route. The 
 judicious selection of a route may be a very simple or com- 
 plex problem, depending on the character of the topography, 
 and more especially on the direction of the streams and ridges 
 as compared with the general direction of the proposed road. 
 2. A road running along a water-course is most easily 
 located, In this case the choice is to be made merely between 
 the two banks of the stream, or between keeping one bank 
 continuously and making occasional crossings. When the 
 stream is small it will usually be found best to cross it at 
 intervals, the advantage of direct alignement outweighing the 
 cost of bridging; but when the stream is of considerable size 
 the solution of the problem is not so obvious, requiring patient 
 comparison of results in the two cases to determine whether to 
 cross or not, while in the case of the larger rivers crossing 
 may be out of the question. 
 
 When there is a choice of sides, both banks should be 
 traversed by the engineer on reconnoissance, and while exam- 
 ining in detail the one side he should take a general and com- 
 prehensive view of the other. Only thus can he gain a complete 
 knowledge of either side. The points to be considered are the 
 relative value of the property on either side, the number and 
 
FIELD ENGINEERING. 
 
 size of tributary streams, and probable cost of crossing them, 
 the cost of graduation as affected by the amount and character 
 of the material to be removed, and the liability to land slides, 
 the amount and degree of curvature required, and the proba- 
 ple revenues which the road can command. If, in respect to 
 these points, one bank of the stream gives the more favorable 
 result all the way, the question is decided at once; but in 
 case the greater inducements are found on either bank alter- 
 nately, as usually happens, the propriety of bridging the 
 stream, with the costs and advantages, must be considered as 
 an additional element in the problem. 
 4. When no water-course offers along which the road may 
 be located, the difficulties of selecting a route are increased, 
 and these usually become greatest when the streams are found 
 to run about at right angles to the direction of the road. Val- 
 leys and ridges are to be crossed alternately, involving the 
 necessity of ascending and descending grades, diverting the 
 road from a straight line, and increasing the distance and cur- 
 vature. The engineer must now seek the lowest points on the 
 ridges, and the highest banks at the stream crossings, in order 
 to reduce as much as possible the total rise and fall, but these 
 points must be so chosen relatively to each other as to admit 
 of their being connected by a grade not exceeding the maxi- 
 mum which may be allowable. The intervening country 
 between summit and stream must usually be carefully exam- 
 ined, even on reconnoissance, to determine where the assumed 
 grade will find sustaining ground at a reasonable expense for 
 graduation and right of way. 
 
 In selecting stream crossings, regard should be had not only 
 to the height of the bank, but also to the character of the bot- 
 tom, its suitability for foundations, and its liability to be 
 washed by the current. The direction and force of the cur- 
 rent should be observed, and its behavior during freshets, and 
 the extremes of high and low water ascertained, if possible. 
 An approximate estimate of the cost of bridging may be made, 
 
 5. The engineer should not only seek the best ground on the 
 route first assumed, but should have an eye to all other possi 
 ble routes, holding them in consideration pending his accu- 
 mulation of evidence, and being ready, finally, to adopt that 
 one which promises the greatest ultimate economy. He should 
 
 be able to read the face of the country like a map, and to 
 
RECON NOISSANCR,. 3 
 
 carry in his mind a continuous idea or image of any line he is ex- 
 amining, so as to judge with tolerable accuracy of the influence 
 any one portion of the line may have on another as to align. 
 ment and grade, even though many iniles apart. In the success- 
 ful prosecution of a reconnoissance he must depend mainly on 
 his own natural tact and a judgment matured by experience. 
 
 G6. The engineer will bring to his aid in the first place the 
 most reliable maps, and those drawn on the largest scale. The 
 sectional maps of United States surveys will be found very 
 useful when they exist. In addition to these it is often desira- 
 ble to prepare a map on a scale of one or two inches to a mile, 
 on which will be drawn the principal features of the country 
 to be traversed, such as streams, roads, towns, and the princi- 
 pal ridges, if known, but leaving the further details to be filled 
 in by the engineer as he progresses. Such a map furnishes a cor- 
 rect scale for his sketches, and saves much valuable time, as he 
 has only to sketch what the map does not contain, and occa- 
 sionally to make corrections when he finds the map to be in 
 error. He also notes on the map the governing points of the 
 route, such as the best crossings of streams, ridges, or other 
 roads, and any point where the line will evidently be com- 
 pelled to pass. He may then indicate the route by a dotted 
 line on the map drawn through the governing points. Having 
 traversed the route in one direction he should retrace luis steps, 
 verifying or correcting his observations, and making such 
 further notes as seem important. When in a densely wooded 
 country, with but few openings, it may be impossible for him 
 to get a commanding view from any point that will afford him 
 the necessary information as to the general topography. He 
 must then depend largely upon instrumental observations, 
 taking these more frequently, and noting carefully all details 
 likely to prove useful in future surveys. 
 
 ¢. The instruments required on an extended recon- 
 “noissance are the barometer and thermometer, the hand or 
 Locke level, a pocket or prismatic compass, and a telescope or 
 strong field-glass. To these may be added a telemeter for 
 measuring distances at sight, but when good maps are to be 
 had this instrument is seldom needed. o also some portable 
 astronomical instruments are necessary in a new country, for 
 determining latitude and longitude, but would only be a use- 
 Jess incumbrance in a settled district, 
 
FIELD ENGINEERING. 
 
 $. The mercurial barometer has generally been relied upon 
 for the determination of heights, but owing to its inconvenient 
 dimensions and the danger of breaking, it is now discarded by 
 railroad engineers in favor of the more portable aneroid 
 barometer, except in the case of trans-continental surveys, 
 and when astronomical instruments are to be used also. 
 
 9. The best aneroids are designed to be self compen- 
 sating for temperature, so that with a constant atmospheric 
 pressure the reading shall be the same at all temperatures of the 
 instrument. This, however, being a very delicate adjustment, 
 is not always successfully made, so that each instrument is la- 
 ble to have a small error due to temperature peculiar to itself. 
 This error will be found rarely to exceed one hundredth of an 
 inch, plus or minus, per change of ten degrees Fah., and is 
 frequently much less than this. Just what the error is in a 
 particular instrument may be determined by careful compari- 
 son with a standard mercurial barometer at the extremes of 
 temperature, assuming the error found as proportional to the 
 difference of temperature for all intermediate degrees of heat. 
 The error having been determined for any aneroid, it should 
 be applied, with its proper sign, to every reading to obtain 
 the true reading. 
 
 The sizes generally used are 13 and 24 inches in diameter, 
 respectively, and experience seems to prove that there is no 
 advantage in using larger sizes, but rather the contrary. 
 
 10. The ordinary barometric formule and tables have been 
 prepared with reference to the mercurial barometer. In order 
 that they may apply to the aneroid, it is necessary that the 
 latter should be adjusted to read inches of mercury identically 
 with the mercurial column at the sea level at a temperature of 
 39° Fah. But as the aneroid, unlike the mercurial column, 
 requires no correction for latitude, nor for the variation in the 
 force of gravity due to elevation, that portion of the formula 
 which provides for such corrections, as well as that which 
 provides for a correction due to the temperature of the 
 ‘instrument itself, may be omitted when using an aneroid. 
 Thus the general formula is very much simplified, and be 
 comes 
 
 z= log Hf 60384.3 (1 + a 64 ) 
 
RECON NOISSANCE. 5 
 
 in which i, and 7’ are the readings of the aneroid in inches, 
 and ¢, and @ the readings of a Fahrenheit thermometer at the 
 lower and upper of any two stations respectively, and z is the 
 difference in elevation in English feet of those stations. 
 To facilitate the calculation of heights by this formula, we 
 may write 
 Log . 60384.3 = [log 2, — log h'] 60384.3 
 
 and since only the difference of the logs. is required, this will 
 not be affected, if we subtract unity from each. The quan- 
 tities in Table XV. are prepared, therefore, by the formula 
 (log 2 — 1) 60884.3 
 for every ;2,ths of an inch from 19 inches to 31 inches. 
 + t’ — 64° 
 ~ 900 
 
 Table XVI. contains values of f for every de- 
 gree of (¢, + 1’) from 20° to 200° Fah. 
 
 11. To find the difference in elevation of any two stations by 
 the tables : 
 
 Take the difference of the quantities corresponding to /, and 
 h' in Table XY. as an approximation, and for a correction 
 multiply this difference by the coefficient corresponding to 
 (¢, +2, in Table XVI., adding or subtracting the product 
 according to the sign of the coefficient, 
 
 Hrample.— 
 Lower Sta. Upper Sta. 
 in. in. 
 Aneroid h, = 29.92 h' = 23.57 
 Thermometer i tne lien tg 8 ai 
 By Table XV. for 29.92 we have 28741 
 for 23.57 22485 
 
 Difference 6256 
 By Table XVI. for 77.6 + 70.4 = 148 we have -+ .0983 
 
 Then 6256 X .09383 = 583.6848 
 and 6256 + 584 = 6840 ft.= 2.—Ans. 
 
 12. Certain precautions are to be observed in the use of the 
 aneroid. When the index has been adjusted to a correct 
 reading by means of the screw at its back, it should not be 
 meddled with until it can again be compared with a standard 
 mercurial barometer, and even then some engineers prefer to 
 take note of its error, if any, rather than disturb the aneroid. 
 
§ FIELD ENGINEERING. 
 
 Again, since the principle of compensation supposes the 
 aneroid to have a uniform temperature throughout its parts, it 
 must be guarded against sudden changes, as otherwise the 
 metallic case will be considerably heated or cooled before the 
 change can affect the inner chamber, thus inducing very erro- 
 neous results. ‘The aneroid, therefore, should seldom be taken 
 from its leather case, nor exposed to any radiant heat of sun 
 or fire, nor worn so near the person as to increase its tempera- 
 ture above that of the surrounding atmosphere. If removed 
 to an atmosphere of decidedly different temperature, time 
 must be allowed for the aneroid to be thoroughly permeated 
 by the new degree of heat. The aneroid should be held with 
 the face horizontal while being read; it should be handled care- 
 fully, and all concussions avoided, and it should be compared 
 
 with a standard as often as practicable to make sure that it 
 
 has suffered no derangement. Observing these precautions, 
 and having a really good aneroid, the engineer should obtain 
 excellent results in the estimation of heights. It has been 
 found that the slight error in compensation, previously alluded 
 to, is subject to a change during the first year or two after the 
 instrument is made, but subsequently it becomes quite per- 
 manent. 
 
 13. For the purpose of obtaining approximate elevations by 
 a simple inspection of the dial, the modern aneroid is provided 
 with a secondary scale reading hundreds of feet, which is 
 placed outside the scale of inches. It is divided according to 
 the following formula prepared by Prof. Airy: 
 
 é h,—l' t+t— =) 
 Eo MAD core ‘ ; 
 i 550382 i (1 = 1000 
 
 in which it is evident that no correction for temperature is 
 required when the average temperature of the two stations is 
 50°. When the two scales are engraved on the same plate the 
 zero of the scale of feet is coincident with 31 on the scale of 
 inches; but in some aneroids the scales are on two concentric 
 plates, so that the zero of one may be made to coincide with 
 any division of the other, which is in some respects an advan- 
 tage. 
 
 14. The theory of the barometer, as expressed in the above 
 formule, assumes the atmosphere to be at rest, and its pres 
 ure affected only by temperature, whereas, in fact, the pres: 
 
RECON NOISS.A NCE. 7 
 
 sure at any point is liable to sudden changes due to variations 
 in the force of the wind, the amount of humidity, etc. The 
 best way to eliminate errors due to these causes is to take read- 
 ings simultaneously at the points the elevations of which are 
 to be compared. For this purpose an assistant should be 
 stationed at some point of known elevation contiguous to the 
 route to be surveyed, and provided with an aneroid similar to 
 that carried by the engineer. The aneroids, time-pieces, and 
 thermometers having been compared at this point, the assist- 
 ant should record the readings every ten minutes, with the 
 time, temperature, and state of the weather. The engineer 
 will thus have a standard with which to compare his own 
 observations. If the survey is so extended that the same con- 
 ditions of atmosphere are not. likely to be experienced by the 
 two observers, the assistant should be instructed to move for- 
 ward to a new station at a designated time; or two assistants 
 may be employed, one at each of two stations between which 
 the engineer intends to make a reconnoissance. Even with 
 these precautions no attempt should be made to obtain the ele- 
 ration of important points during, or just before, or after a 
 storm of wind or rain. 
 
 15. When but one aneroid is used the observations at the 
 several stations should be taken as nearly together as possible 
 in point of time, and then repeated in inverse order, taking 
 the mean of the observations at each station, and repeating the 
 whole operation if necessary. Only approximate results can 
 be hoped for, however, with a single instrument, unless the 
 atmospheric conditions are very favorable. 
 
 16. The Locke Level is an instrument in which the 
 bubble and the observed object may be seen at the same instant, 
 enabling the operator to keep the instrument horizontal, while 
 holding it in the hand, like an ordinary spy-glass. While 
 very portable, it enables the observer to define rapidly all visi- 
 ble points of the same elevation as his own, and_to estimate 
 from these the relative heights of other points. It may be 
 made useful in a variety of ways which easily suggest them- 
 selves to the engineer in cases where no great precision 18 
 required, and where a more elaborate level is not at hand. 
 
 17. The Prismatic Compass is a portable instrument 
 with folding sights, in using which the bearing to an object 
 may be read at the same instant that the object is observed. 
 
8 FIELD ENGINEERING. 
 
 The bearings are read upon a floating card, graduated and 
 numbered from zero to 360°, so that no error can be made in 
 substituting one quadrant for another. The instrument may be 
 
 held freely in the hand during an observation, though better 
 
 results are obtained by giving it a firm rest. 
 
 CHAPTER II. 
 PRELIMINARY SURVEY. 
 
 18. A preliminary survey consists in an instrumental exam- 
 ination of the country along the proposed route, for the 
 purpose of obtaining such details of distances, elevations, 
 topography, etc., as may be necessary to prepare a map and 
 profile of the route, make an approximate estimate of the cost 
 of constructing the road, and furnish the data from which to 
 definitely locate the line should the route be adopted. The 
 survey is more or less elaborate, according to circumstances. 
 In case the country is new, or the reconnoissance has been 
 incomplete, or if several routes seem to offer almost equal 
 inducements, the survey will partake somewhat of the nature 
 of a reconnoissance, and will be made more hastily than if but 
 one route is to be examined, and that, perhaps, presenting 
 serious engineering difficulties. The survey is made as expe- 
 ditiously as possible, consistent with general accuracy, but 
 should not usually be delayed for the sake of precision in 
 matters of minor detail. 
 
 19. For preliminary survey the Corps of engineers is 
 organized as follows: 
 
 A chief engineer, an assistant engineer, two chainmen, one 
 or two axemen, a stakeman, and. a topographer, these forming 
 the compass (or transit) party, to which a flagman is some- 
 times added; a leveller and one or two rodmen, forming the 
 level party; and to these is sometimes added a cross level party 
 of two or three assistant rodmen. 
 
 20. The chief engineer takes command of the corps, 
 and directs the survey. He ascertains or estimates the value 
 of the lands passed over, the owners’ names, and the boundary 
 lines crossed by the line of survey, He examines all streams, 
 
PRELIMINARY SURVEY. 9 
 
 and estimates the size and character of the culverts and 
 bridges which they will require; he notices existing bridges, 
 and inquires concerning their liability to be carried away by 
 freshet; he selects suitable sites for bridges, examines the 
 character of the foundations, the direction of the current rela- 
 tively to that of the line, and considers any probable change 
 in the direction of the current during freshets; he’ inspects the 
 various soils, rocks, and kinds of timber as they are met with, 
 and takes full notes of all these and kindred items in his field 
 book. He not unfrequently assumes in addition the duties of 
 topographer. He should run his line as nearly as may be over 
 the ground likely to be chosen for location, so that the infor- 
 mation obtained may be pertinent, and so that the length of 
 the line, the shape of the profile, and the estimate based on 
 the survey may approximate to those of the proposed location. 
 To thisend he has due regard to the levels taken, and when 
 they show that the line as run fails to be consistent with 
 allowable grades, he either orders the corps back to some 
 proper point to begin a new line, or makes an offset at once 
 to a better position, or continues the same line with some 
 deflection, simply noting the position and probable elevation 
 of better ground, as in his judgment he thinks best. He 
 should at all times maintain a friendly attitude toward pro- 
 prietors, and by his polite bearing endeavor to secure their 
 cordial support. of the new enterprise. If he is tolerably cer- 
 tain that the location will follow nearly the line of the prelim- 
 inary survey, he should have with him some blank deeds of 
 right of way, and let these be signed by land-owners while 
 they are favorably disposed. When this cannot be done, ¢ 
 blank form of agreement to allow the surveys and construc- 
 tion of the road to proceed until such time as the terms of 
 right of way may be agreed upon may be made very useful. 
 The chief also selects quarters for his men, and in case of 
 camping out he directs the movements of the camp equipage. 
 
 21. The assistant engineer takes the bearings of the 
 courses run, and makes a minute of them, with their lengths, or 
 
 the numbers of the stations where they terminate. He sees that. 
 
 the axemen keep in line while clearing, and the chainmen 
 while measuring; he takes the bearings of the principal roads 
 and streams, and of property lines when met with. In an 
 open country he may save time by selecting some prominent 
 
10 FIELD ENGINEERING. 
 
 distant object toward which the chainmen measure without his 
 assistance, while he goes forward and prepares to take the 
 bearing of the course beyond. In traversing a forest with not 
 too dense undergrowth, when the line is being run to suit the 
 ground according to a given grade, it is a good plan for the 
 assistant to go ahead of the chainmen as far as he can be seen, 
 select his ground, take his bearing by backsight on the last 
 station, and then have the chainmen measure toward him. In 
 this case both he and the head chainman should be provided 
 with a good sized red and white flag, mounted on a straight 
 pole, to be waved at first to call attention, and afterward held 
 vertically for alignement. Otherwise a flagman must be added 
 to the party, who will select the ground ahead, under the in- 
 structions of the chief, and toward whom the survey will pro- 
 ceed in the usual manner. 
 
 22, The head chainman drags the chain, and carries a 
 flag which is put into line at the end of each chain length by the 
 assistant engineer or the rear chainman. It is his duty to 
 know that his flag isin line and that his chain is straight and 
 horizontal before making any measurement, and to*show the 
 stakeman where each stake isto be driven. <A stake is usually 
 driven at the end of each measured chain length, called a 
 station, though in an open and level country the stakes at the 
 odd stations may be omitted, in which case marking pins are 
 used to indicate the odd stations temporarily. In case there 
 is much clearing to be done the head chainman plants his flag 
 in line, and ranging past it, indicates to the axemen what is 
 to be cut, going a little in advance through the bushes so that 
 they may work toward him. The head chainman should be a 
 quick, active and strong man, with a good eye and a taste for 
 his work, as very much of the real progress of the survey 
 depends upon him. 
 
 23. The rear chainman holds his end of the chain firm- 
 ly at the last stake or pin by his own strength, not by means of 
 the stake. He keeps the tally by the pins when they are used, 
 and watches the numbers on the stakes to see that they are cor- 
 rect. The end of a course should always be chosen at the end 
 of achain, if possible, and if not, then at a brass tag indicating 
 tens of feet, as thus the labor of plotting the map will be much 
 lessened. The numbering of stations is not recommenced 
 with each new course, but is continued from the beginning to 
 
PRELIMINARY SURVEY. 1] 
 
 the end of the survey, through all its courses, and if one 
 course ends with a portion of a chain the next course begins 
 with the remainder of it. It is. the rear chainman’s duty to 
 attend_to this, holding the proper link at the compass station, 
 Any fraction of a chain measured on the line is called a plus, 
 and is counted in feet from the previous station. The length 
 of an offset in the line is never included in the length of the 
 line, but if the line should change its course by a right angle, 
 or more, or less, the numbering would go on as usual. 
 
 24. The axemen should be accustomed to chopping and 
 clearing, and are, therefore, to be selected in the country rather 
 than the city. They will cut out so much of the underbrush 
 and overhanging branches as may interfere with the sight of 
 the assistant or leveller; but care must be taken not to cut 
 unnecessarily wide, and no tree of considerable size should be 
 felled, except in rare instances. When running by compass, if 
 the assistant goes ahead of the chain, he can always select a 
 position so that no large tree will interfere; or, if the line must 
 be produced and strikes a tree, the compass may be brought up 
 and set close to the tree on the forward side as nearly in line as 
 can be estimated, the slight error in offset being neglected, 
 since the lin2 will b2 produced parallel to itself by the needle. 
 
 25. The stakeman prepares and marks the stakes, and 
 drives them at the points indicated by the head chainman. 
 
 When no clearing is needed, the axemen keep him supplied 
 
 with stakes, as the rapid progress of the chain will only give 
 him time to drive them. The stakes should be two feet long and 
 pointed evenly so as to drive straight, and are blazed or faced 
 on two opposite sides, one of which is marked in red chalk 
 with the number of the station. The stake must be driven 
 vertically, and with the marked face to the rear, so that it may 
 be read by the rodman as he follows the line. 
 
 26. The topographer makes accurate s<xetches of all 
 features of the country immediately on the line, and extends 
 the sketches as tar each side of the line as he can, in a book 
 prepared for the purpose. He must never sketch in advance 
 of the chain, nor in advance of his own position. His work 
 should be done to scale as nearly as possible, using the same 
 scale for distances on the line and at right angles to it. The 
 scale adopted should never be less than that of the map to be 
 made from the sketches. The ruled lines of a field book are 
 
12 FIELD ENGINEERING. 
 
 usually one quarter of an inch apart, so that a scale of one 
 line to a station equals about four hundred feet to an inch. 
 This is the smallest scale ever used. The scale of two lines to 
 a station is most convenient for general use. Four lines to a 
 station are needed in special cases to show details, as in pass- 
 ing through villages. The scale may be changed from time to 
 time as found necessary, but no two scales should ever be used 
 on the same page. The numbering of the stations up the page 
 indicates the scale of the sketch. 
 
 27. When the contours of the surface are required, the 
 topographer may join the level party in order that his esti- 
 mates of heights and slopes may be corrected by the instru- 
 ment. He should never draw a mass of contours indiscrimi- 
 nately, but should sketch them as they exist at a uniform ver- 
 tical interval. This interval may be assumed at five feet in 
 a gently rolling country, and at twenty feet in a mountainous 
 one, but an interval of ten feet will be found most convenient 
 generally. If the topographer accompanies the level he can 
 assume the contours at the even tens of feet in elevation, and 
 mark them so, noting where a contour crosses the surveyed 
 line, and sketching its direction and shape both ways from 
 that point. He will estimate the rate of slope of the ground 
 at right angles to the line as so many feet per hundred, and 
 record it from time to time, noting ascent from the line on 
 either side by ‘‘ A,” and descent by ‘‘D.” If the slope changes 
 within the limit of the page, the line of change may be 
 sketched and the next slope recorded. When little banks or 
 terraces occur, or bDluffs and rocks, which cannot be suf- 
 ficiently indicated by contours, they should be shown by 
 hatchings, and the height noted. Special care should be 
 taken to sketch roads and houses in their correct positions 
 and dimensions, the latter to be either measured, paced or 
 estimated. The dimensions should also be recorded in num- 
 bers. The outline of forests may be shown by a scalloped 
 line, and the kind of timber, and whether dense or scattered, 
 written within the inclosed space. Correct outlines are essen- 
 tial, but no time should be given to shading up a sketch with 
 conventional signs. A single sign, or the name of the thing 
 intended, is all sufficient. Land-owners’ and residents’ names 
 should be recorded whenever they can be obtained, as well as 
 
 the names of roads, streams and public buildings. 
 
4 
 
 PRELIMINARY SURVEY. 13 
 
 28. The leveller takes charge of the level party and 
 keeps the notes of his work. He reads the rod on benches and 
 at turning points to hundredths of a foot and to tenths at other 
 points. He should direct a bench to be made at least once 
 every half mile, and in a very rough country every quarter of 
 1. mile. The benches need not be far from the line, and, if 
 well chosen, may be used as turning points, thus saving time. 
 
 i 
 The elevation of turning points must be computed when 
 
 tuken, so that the elevation of any one of them may be 
 instantly given when called for, and the other elevations will 
 be filled in as far as may be without delaying the survey. As 
 the levels are usually the most essential part of the survey, 
 much care should be taken to have the instrument well ad- 
 justed and truly level, and the rod held vertically and correctly 
 read on turning points, but the intermediate work should not 
 be so done as to delay the party unnecessarily. The leveller 
 should use every endeavor to follow closely after the survey- 
 ing party, so that the chief and topographer may have the 
 advantage of his notes. 
 
 29. The rodman’s first duty is to hold the rod vertically, 
 and he must learn to do this in calm or windy weather, in level 
 field or on side hill. He may carry a small disk-level, which 
 applied to the edge of the rod will show when it is vertical. 
 The turning points are to be selected for firmness and definite- 
 ness, and so that they will afford a clear view from beyond 
 for a backsight. The rod is held for a reading on the ground 
 at every stake, the number of which is called out to the level- 
 ler as soon as the rodman arrives at it; the rod is also to be 
 held at every prominent change of slope on the line, as the 
 crest and foot of every bank, the rodman calling out its dis- 
 tance from the last stake as plus so many feet, but all gentle 
 undulations and minor irregularities are to be neglected. The 
 rod will always be read at the surface of a stream or pond, 
 and also at its deepest part on the line, when possible; other- 
 wise the depth of the water may be found by sounding, and 
 so recorded. Should the line run along a stream the surface 
 will be taken occasionally, opposite certain stations, and in 
 case of a canal, the elevation of surface above and below cach 
 lock must be noted. The rodman makes inquiry for high- 
 water marks or seeks traces of them himself in an uninhabited 
 district, and holds the rod upon them that their elevation may 
 
14 FIELD ENGINEERING. 
 
 be determined. The rodman carries a small axe or hatchet 
 with which to make benches and to trim out any stray 
 branch that may intercept the leveller’s view. 
 
 80. The assistant rodmen take the slope and elevations 
 of the ground at right angles to the line, using vertical and hori- 
 zontal rods and a pocket level, or a tape line and clinometer. 
 The cross levels are not taken throughout the whole survey, if 
 at all, but only where the roughness of the country seems to 
 demand it. They may be extended to any distance from the 
 centre line required by the chief—not less, however, than fifty 
 feet as a rule. They may be taken at the stations only, or 
 oftener, if necessary, depending upon the roughness of the 
 surface, the object being to define accurately the contours, 
 and so the shape of the ground. The assistant rodmen will 
 also take soundings when they are needed, either on the line 
 or at right angles to it. 
 
 31. In defining the duties of the members of the corps, the 
 instruments used have been incidentally noticed. 
 
 32. The compass is preferable to the transit on prelimi- 
 nary surveys, because it can be operated more rapidly, is lighter, 
 and usually has a better needle. It may have either plain 
 sights or telescope, and be mounted on tripod or jacob staff. 
 The simpler forms are preferred for forest work. Not unfre- 
 quently the engincer’s transit is employed, but using the needle. 
 A preliminary line should not be run by backsights and deflec- 
 tions, unless local attraction is found to exist to such an extent 
 as to destroy confidence in the needle; or, in special cases, 
 where the natural obstacles to a survey are very great. In the 
 latter case the survey partakes of the nature of a location, and 
 should be conducted with similar care and fidelity. 
 
 33. The chain is 100 feet long, and composed of 100 
 links. It should be of steel for lightness, durability, and greater 
 accuracy. Those having rings of hard stecl, unbrazed, are 
 least apt to wear. Five marking pins are needed, each having 
 a piece of red flannel attached, for temporary stations, or for 
 keeping points temporarily while measuring by parts of a 
 chain up or down aslope. A pointed plumb bob, with sev- 
 eral yards of small cord wound on a carpenter’s spool, is use- 
 ful in chaining over steep declivities or bluffs. 
 
 34. The axes should be of. best quality, with hand-made 
 handles, and not too heavy. The axe of the stakeman should 
 
\ 
 
 PRELIMINARY SURVEY. 15 
 
 have a fine edge for dressing and a broad head for driving 
 the stakes. When the stakes are not required to be over two 
 feet long, a stout basket, having.a square, flat bottom, 26x14 
 inches, should be furnished the stakeman. He will then pre- 
 pare a basketful of stakes, ready marked, and place them in 
 the basket regularly, in the reverse order of their numbers, so 
 that they will come to hand as wanted. A small hand-saw 
 no larger than the basket, with rather coarse teeth, wide sct, 
 will be found extremely useful in cutting stakes with square 
 heads and of uniform length, and much more rapidly than can 
 be done with an axe. When not in use, it is to be strapped to 
 the inside of the basket, to prevent its being lost by the way. 
 When the basket is about empty, the stakeman, with the 
 assistance of the axemen, can soon replenish it, and the stakes 
 being all numbered at once, there is less danger of a mistake 
 being made in the tally than when they are marked only as 
 wanted. 
 
 35. The level should be the regular engineer’s level, the 
 same as used on location. 
 
 306. The rod should be self-reading, ¢.¢., to be read by the 
 leveller, as too much time would be consumed in the constant 
 adjustment of a target by the rodman. It should be as long as 
 can be conveniently handied in order to reduce the number of 
 turning points on hill sides. A very convenient rod may be 
 made of thoroughly seasoned clear white pine, sixteen feet 
 long and two inches wide, with a thickness of one inch at the 
 bottom, increasing to one and a quarter inches at six feet from 
 the bottom, and then gradually diminishing to three eighths of 
 an inch at the top. The rodis shod with a stout strap of steel, 
 extending five inches up the edges, and secured by screws. 
 The top is protected for a few inches by a plate of sheet brass 
 on the back. The face of the rod is a plain surface through- 
 out, and is graduated from the lower edge of the steel shoe as 
 zero. The divisions are fine cuts made with the point of a 
 knife. At the foot and half-foot points the cuts extend across 
 the face. For the tenths and half tenths they extend three 
 quarters of an inch from the right hand edge, terminating in a 
 line scribed parallel to the edge of the rod, thus forming rec. 
 tangular blocks half a tenth wide, every other one of which is 
 painted black, the body of the rod being white. The feet are 
 indicated by numerals painted red on the blank part of the 
 
SS iE 
 
 16 FIELD ENGINEERING. 
 
 face, each figure standing exactly on its foot mark, and being 
 exactly one tenth high. For the figure 5 the Roman. \V. is sub- 
 stituted and for 9 the Roman I[X., so that in case a dumpy 
 level is used the 5 may not be mistaken for a 3, nor the 9 fora 
 6. At the half-foot points a red diamond is painted, so that 
 the graduated line bisects it. No other figures nor gradua- 
 tions are required. With this rod the leveller can read quite 
 accurately to hundredths of a foot, and after some practice 
 can estimate the half hundredths. 
 
 o%¢. The horizontal rod for cross-levels may be made 
 of white pine, ten feet long and one inch thick by three wide, 
 tipped with brass, painted white, and graduated to feet and 
 tenths. It must be a straight edge, and is levelled by a pocket 
 level placed upon it when needed, or by a small level embedded 
 permanently in one edge. The vertical rod to be used with it 
 is made of pine eight feet long and one and a quarter inches 
 square, and graduated to feet, tenths, and half tenths. All 
 rods when not in use should be laid on a flat surface to pre- 
 vent their being sprung. Leaning them in acorner soon ruins 
 them for use. . 
 
 28. The clinometer is any small instrument which will 
 measure the slope angle of the surface. The angle is always 
 estimated from the horizon, a vertical being 90°. The rise per 
 100 feet is found by multiplying the nat. tangent of the slope 
 angle by 100. It may often be found more easily by the 
 leveller reading the rod at a station and then 100 feet left or 
 right of the line. If surface measures are taken in connec- 
 tion with a.slope angle they are reduced -to horizontal meas- 
 ures by multiplying them by the cosine of the slope angle. 
 
 39. The plane-table is rarely if ever used on prelimi- 
 nary surveys in the United States. Occasional bearings taken 
 to prominent objects by the assistant engineer, or the use of a 
 prismatic compass by the topographer in connection with his 
 sketches, is found to answer every purpose. 
 
 40. In case a survey is to be made with a tran- 
 sit, it is necessary to add a back flagman to the party, who will 
 hold his flag or rod on the point last occupied by the transit, so 
 that the assistant may take a backsight upon it. The direction 
 of a new course in each case is determined by the deflection 
 angle to the right or left of the preceding course produced. 
 The bearing of one long course near the beginning of the sur- 
 
PRELIMINARY SURVEY. 17 
 
 vey having been carefully ascertained, the bearing of each suc- 
 
 ceeding course is calculated from the BeHestiona, and entered 
 in acolumn of the field book headed Culeulated Beart vngs, from 
 which the line is afterwards plotted. The magnetic bearing 
 of each course should also be taken from the needle, and re- 
 corded as such, but is used only as a check on the transit 
 work. The deflections should be made in degrees, halves, or 
 quarters, if possible, to facilitate the calculation of bearings, 
 and to admit of using a traverse table. 
 
 44. The attached level and vertical arc of the transit are 
 useful in determining approximately the grade of the line run 
 in advance of the level party, or in seeking for one assumed 
 grade to which it is desired that the line shall conform. For 
 this purpose it is only necessary to set the vertical arc to the 
 angle corresponding to the grade as given in Table XIV., and 
 let the head chainman move about until a point on his rod at 
 the same height from the ground as the telescope is covered 
 by the horizontal cross-hair. 
 
 42. The point on the ground where a transit is set up is 
 marked by a good-sized plug, flat headed, and driven down 
 flush, with the ground, with a tack set in the head to show the 
 exact point or centre. This is called a transit point. When 
 a transit point occurs at a regular stetion, the stake bearing 
 the number of that station is set three feet to the left cf the 
 jine opposite the plug and facing it. When a transit point 
 occurs between stations the stake is driven three feet to the 
 left of it, marked with the number of the preceding station 
 + the distance from that station in feet. 
 
 43. As a transit is capable of giving a line with great pre- 
 cision, it is important that the flags used in connection with 
 it should be equally precise in giving points, An excellent flag 
 for this purpose is made of well-seasoned clear white pine ten 
 feet long, two and a half inches wide, and one inch thick. It is 
 tapered for the last four inches to an edge at one end, the edge 
 being formed at the middle of the width, The tapered end is 
 shod with a band of steel covering the edge of the rod, and 
 secured by screws, and the steel is brought to a sharp edge at 
 the point of the rod. The rod is then painted white and 
 tipped with brass at the square or upper end, A centre line 
 on the face is then struck from the point of the steel to the 
 
18 FIELD ENGINEERING. 
 
 middle of the brass tip by means of a piece of sewing silk, 
 and a fine cut made with a knife and steel straight edge. 
 The centre line must not be scribed parallel to one edge of the 
 rod, as this is rarely ever straight. The face of the rod is 
 then divided into one-foot spaces, measured from the head of 
 the rod, and these are painted red on either side of the centre 
 line in alternate blocks. On the back of the rod at three and 
 a half feet from the point is placed a small ground-glass 
 bubble-tube, mounted very simply, and attached to the rod by 
 a brass plate and screws, and guarded by two blocks of wood 
 for protection. The centre line of the rod is made vertical by 
 a plumb-line while the level tube is being attached, which ever 
 after secures a vertical rod. If only two feet of this rod can 
 be seen over any obstruction, a point can be set with great 
 | precision, provided the level tube is in adjustment. This flag 
 | can also be used as a plumb in chaining with much more 
 satisfaction than a cord and weight, especially in windy 
 weather. 
 
 44, A transit survey usually requires more clearing than 
 one made by compass. When a given course is to be produced 
 in a forest, some large trees will inevitably be encountered, but 
 the labor and delay of felling them may be avoided hy the 
 use of auxiliary lines. These may be classified as running 
 parallel to the main line, at a small angle with it, or at a large 
 angle with it. 
 
 45. The parallel line is established by means of two 
 short perpendicular offsets measured with care before reach- 
 ing the obstacle, and the main line is established beyond the 
 obstacle by means of two more equal offsets, But since short 
 
 back-sights are to be avoided, these offsets should be at least 
 i 100 feet apart, so that it may be difficult to find a parallel line 
 of sufficient length which does not strike some other obstacle, 
 1 || or at least require considerable extra clearing. 
 
 46. The auxiliary lines making a small angle 
 with the main line are more convenient, not only on this 
 account, but because they require a less number of transit 
 points. By them an isosceles triangle is formed on the ground, 
 having the intercepted portion of the main line as base, and the 
 vertex near the obstacle. The deflections at the points where 
 
 the lines leave and join the main line are similar and equal, and 
 
bh 
 
 PRELIMINARY SURVEY. 14 
 
 the deflection at the vertex is double in amount and opposite 
 in direction. No calculation is necessary, for the error in 
 measurement due to the deviation is too small to be noticed. 
 and since the main line is immediately resumed, the calculated 
 bearings of the auxiliary lines are unnecessary. Should. the 
 point where the second line joins the main line prove unsuit 
 able for a transit point, the second line may be produced to 
 any convenient point beyond, and so go to form an isosceles 
 triangle on the opposite side of the main line, the triangle 
 being completed by running a third line parallel to the first, 
 and equal to the difference of the first and second. Again, 
 the second line may encounter a serious obstacle before reach- 
 ing the main line. To avoid this a parallel to the main line 
 may be run from the end of the first line for a con- 
 venient distance, and there the second line be put in 
 parallel and equal to its first position, as before de- 
 scribed, thus forming a trapezoid. 
 
 47. The following general solution of this 
 problem allows the engineer to make use of. any 
 number of auxiliary lines, provided that none of 
 them make an angle of much more than one degree 
 with the main line, with a certainty of resuming the 
 main line in position and direction at the extremity 
 of any course desired, and without necessitating 
 any trigonometrical calculation. It is based on the 
 assumption, practically true for small angles, that 
 the sines are proportional to their angles, and is ex- 
 pressed by the following rule: 
 
 Call all deflections to the right plus, and all to 
 the left menus ; multiply the length of each course 
 in feet by the algebraic sum in minutes of all the 
 auxiliary deflections made to reach that course; 
 take the algebraic sum of these products, and 
 when the sum equals zero the extremity of the last 
 course will be on the main line. The deflection 
 required at that point to give the direction of the main line 
 is equal to the algebraic sum of all the preceding deflections, 
 but taken with the contrary sign. 
 
 Thus, if we have left the main line at A, and run by these 
 notes: (Fig. 1.) 
 
 Fie. 1. 
 
FIELD ENGINEERING. 
 
 Sta- Defi. Dist. Factors. Products. 
 A. 16° R 199 2% +16 X 190 = -+- 3040 
 B 31 L 1200 $3. — 15 x 120= — 1800 
 C 18’ R 175 E 5 te BX 1D D0 | 
 D 13' L Le Desai 10 X 265 = = 2650 
 E 15’ R Ba + 5X (?) 
 
 3065 — 4450 
 and their algebraic sum is ~ — 88d 
 
 Therefore to render the sum zero we must add 885 as the pro- 
 duct of the last course. But 5’ is already given as one factor, 
 
 eld 
 
 885 » gent 
 so that the other factor must be ae 177, which is the length 
 
 of the last course, giving some point /’ on the main line. The 
 deflection at / from the last course to give the direction of 
 the main line is 
 
 16 — 31-4 18-1384 15=95' 
 
 and changing the sign we have — 5’; that is, the deflection is 
 to the left. 
 
 The distance on the main line from A to / equals the sum 
 of the courses, or 927 feet, but this we have by the stations, 
 which have been kept by stakes in the ordinary way. All the 
 stakes on the auxiliary lines will be more or less off the main 
 line, but as these offsets are usually very small, they are con- 
 sidered of no consequence on a preliminary survey through a 
 forest. In Fig. 1 the offsets are very much magnified. The 
 field notes of such auxiliary courses should be Kept, not as 
 regular notes, but on the margin or opposite page, and in such 
 a way that, while the line may be retraced by them on the 
 ground, the draughtsman may see that it is not necessary to plot 
 them, when a straight line raled and measured through is suf- 
 ficient. It is obvious that in selecting a closing course either 
 the deflection may be assumed and the length calculated, or 
 vice versa ,; but care should be taken to assume such values as 
 do not involve a fraction in either factor, if possible. 
 
 48. The method of passing an obstacle on the line by 
 auxiliary lines at a large angle with the main line will 
 only be resorted to when circumstances are such that the other 
 methods mentioned cannot be employed, as in passing a build- 
 ing, pond, or densely wooded swamp. In such a case we may 
 
PRELIMINARY SURVEY. 21 
 
 turn a right angle with the transit, and measure accurately one 
 offset, putting a transit point at its extremity, where another 
 right angle will give a parallel line. If the offset prove too short 
 for an accurate backsight, a temporary point at a sufficient 
 distance may be established for that purpose on the offset line 
 produced before the instrument is removed from the main 
 line. If any other angle than 90° is used it should be selected, 
 when circumstances permit, so that the distances on the inter- 
 cepted part of the main line may be in some simple ratio to 
 the distances measured on the auxiliary line. Thus a deflection 
 ot 60° gives a distance on the main line equal to half the 
 length of the auxiliary course, that is, 
 
 60° gives a ratio of += 0.5 
 
 HB IS EH bi es $ 0.6 nearly 
 45° 344" us UT AES 
 mad are es s 0:55 8e 
 25° 50y Shs e 0:9 gars 
 
 the angles being taken to the nearest half minute. 
 
 49. If it be desired that the stakes on the auxiliary line 
 should stand on perpendiculars through the true stations 
 on the main line, a certain correction must be added to each 
 chain length depending on the angle which the auxiliary 
 makes with the main line. If there is a fraction of the chain 
 at either end of the course, a proportional addition must be 
 made for this. Thus, by referring to the table of external 
 secants, we find that we must add a correction as follows: 
 
 2° 834...0.1 ft. per chain. | 6° 453’...0.7 ft. per chain. 
 9 
 vo 
 
 oe De Vi a aga PAI: reiti ta 
 POG AD BEST 7°.302'...09 « 
 Beings i Ga iites -«s 8% (4 pee ce 
 ae © a (0s PD: 6 OO 
 621525 3.20-6.06 -". « TROON igi ah 
 
 These methods of suiting the angle to an even measure are 
 much superior to assuming an even number of degrees deflec- 
 tion, and then calculating the distance by trigonometry. The 
 last table, which may be extended indefinitely by reference to 
 the table of Ex. secants, is perfectly adapted to chaining by 
 surface measure on regular slopes when the slope angle is 
 
22 FIELD ENGINEERING. 
 
 known, the chain being lengthened by the correction corre- 
 sponding to the slope angle. 
 
 50. If the chain is lengthened as per above table on auxil- 
 iary lines, the numbering of the stakes goes on as usual, but 
 they should have an additional mark as X to show that they 
 are off the main line; and they may stand facing the true 
 stations which they represent, and the length of offset, if 
 known, may also be recorded on them. The leveller will then 
 understand that he is to read the rod not only at the stakes as 
 they stand, but also at the true stations, as nearly as may be. 
 The assistant engineer will always make a diagram in his 
 field book, showing exactly the method pursued in reference to 
 auxiliary lines. Having passed the obstacle, it is advisable 
 to return to the main line by a course equal in length to the 
 first auxiliary, and making an equal angle with the main line. 
 If this cannot be done from the end of the first course, a 
 parallel to the main line may be run any convenient distance, 
 and the return line then put in, forming a trapezoid. 
 
 51. When there is no obstruction to sight on the main 
 line, but only to measurement, a transit point should be 
 set in line beyond the obstacle before the transit leaves the 
 main line, as a check on the other operations, and the main 
 line should be afterward produced from this point by back- 
 sight on the main line, rather than by deflection from an 
 auxiliary line. 
 
 52. The main line should always be resumed as soon as 
 practicable, making the auxiliary lines the mere exception. 
 When a number of courses at a large angle are likely to be 
 required before the main line can again be reached, it may be 
 better to consider these as regular courses of the survey, and 
 to note them as such. The sémplest method is always the dest, 
 because least likely to involve mistakes. 
 
 53. When the natural obstacles are so numerous 
 and of such magnitude as to render any continuous line of sur- 
 vey or location extremely difficult, if not impossible, as‘in the 
 case of a bold rocky shore, all the data necessary to a location 
 should be gathered with precision on the preliminary survey, 
 the measurements and angles being taken with the greatest care, 
 and as many checks as possible should be introduced to verify 
 the work. In meandering such a shore it is probable that a large 
 number of short courses will be used, which may be measured 
 
PRELIMINARY SURVEY. 2d 
 
 correctly, but there is liability to error in the angles, To 
 verify the latter the more conspicuous transit stations on 
 prominent points of the shore are selected, and these being 
 named by the letters of the alphabet, the deflections between 
 them are taken by careful observations repeated a number of 
 times, as for a triangulation. These points, joined by tie- 
 lines, then form a survey of themselves, much simpler than 
 the full traverse. To obtain the length of these tie-lines, the 
 angles between them and the courses meeting at the same 
 station are measured. Then since each tie-line forms the 
 closing side of a field, in which all the bearings are known, 
 and all the distances, save one, that one may be calculated by 
 latitude and departures. But the angles should first be tested 
 for error in each complete field, and if the error be large the 
 angles must all be remeasured until the error is found and cor- 
 rected, but if very small it may be distributed among the 
 angles, or among those most probably inaccurate. Before cal- 
 culating the traverse of any of these fields, it will be advanta- 
 geous to assume, for an artificial meridian, a line parallel to 
 the average direction of the shore for several miles, and to 
 refer all courses to this meridian for their bearing. This 
 meridian is called the avis of the survey, and all bearings 
 referred to it are called avial bearings, as distinguished from 
 magnetic bearings. The magnetic bearing of the axis should 
 be some exact number of degrees, so as to facilitate the reduc- 
 tion from one system to the other. 
 
 54. In plotting the map, the axis is first laid down, and then 
 the lettered stations in their respective positions, after which 
 the meandering surveys can be filled in. The map being 
 drawn on a scale of one hundred feet to an inch, and the con- 
 tours constructed from the notes of the level and cross-level 
 parties, the engineer may project the location upon it with 
 great certainty and economy of result. But he should calcu- 
 late the traverse of the location as projected, and compare it 
 with the traverse of the preliminary, to eliminate all errors in 
 drafting, before taking his notes to the field to reproduce the 
 location on the ground. Any point where the location crosses 
 the preliminary should have the same latitude and longitude 
 by the traverse of either line. This system, though laborious, 
 is the only one that will ensure a successful location under the 
 
 circumstances supposed. Advantage may sometimes be taken 
 
24 FIELD ENGINEERING. 
 
 of cold weather to cross bays and inlets on the ice, but there 
 is great liability to error in angles taken upon the ice, due both 
 to its motion and to the sinking of the feet of the tripod into 
 the ice as soon as exposed to the rays of the sun. 
 
 CHAPTER. IIT, 
 THEeorRyY oF MAaxtmumM EconoMy IN GRADES AND CURVES. 
 
 55. Before commencing the field work of location it de- 
 volves upon the engineer to decide as to which of the surveyed 
 routes shall be adopted as being most advantageous in all 
 respects, and also to establish the maximum grade in each 
 direction and the minimum radius of curve on that route. 
 
 The general considerations which guide the engineer in the 
 selection of one of several routes for location are such as were 
 hinted at in the chapter on reconnoissance, but upon the com- 
 pletion of the preliminary surveys he has at hand a large 
 amount of information which enables him to consider this 
 important question much more in detail. Unless his instruc- 
 tions are explicitly to the contrary, he may assume it to be his 
 duty to find the best line, or that one which, for a series of 
 years following the completion of the road, will require the 
 least annual expense, including interest on first cost. The 
 finances of the company may be so limited as not to permit 
 the construction of the best line at once, and it may then be 
 the duty of the engineer to select the cheapest line, or that of 
 least first cost, as a temporary expedient, with the expectation 
 of building the road at its best when the improved credit of 
 the company will permit. But generally he will be able to 
 build the cheaper portions of the best line at once, only making 
 deviations and introducing heavier grades at the expensive 
 points to avoid a cost beyond the present means at his com- 
 mand. The selection of the best line may be a question as 
 between different routes or as between different grades and 
 curves on the same route. We will consider the latter case 
 first. 
 
 56. To solve the problem of true economy we must 
 determine the actual expense both of building and operating 
 
MAXIMUM ECONOMY IN GRADES, ETC. 2D 
 
 the line at a given maximum grade, and also what changes will 
 be made in these expenses by a change in that maximum. We 
 have then, on one hand, the annual interest upon the original 
 cost, and, on the other, the annual expense of operating the road. 
 The best grade ts that which will render the sum of these two a 
 minimum. Both forms or expense consist of two parts: one 
 that is affected by a change in grade, and the other that is not. 
 Clearly the former is the only one we have to consider in either, 
 since when the sum of the variabie portions is a minimum, the 
 sum total will be a minimum also. Tbe varying portions then 
 are functions of the grade, though independent of each other. 
 If, therefore, we let 2’ represent the maximum grade in feet 
 per mile, and let @ represent the corresponding value of that 
 portion of the annual expense which varies with the grade, 
 and establish the relation existing between the two, we shall 
 have z= f(z). Similarly if we let y represent the interest on 
 so much of the first cost as is affected by grade, we shall have 
 y =f' (2). The problem then is to find that value of 2’ which 
 shall render 
 e+y=a minimum. 
 
 Let us now seek the complete expression represented by 
 tees f te). 
 
 The elements that enter into this expression are numerous, 
 and will be considered in succession. 
 
 57. The traction of an engine is the force with which 
 it pulls a train, and is limited by the reaction of the drivers 
 against the rails. It depends on the weight upon each driver, the 
 number of drivers, and the coefficient of friction. The weight 
 on one driver should not exceed 12,000 Ibs., and is usually less 
 than this. If the exact proportions of engine that will be 
 used on the road are not known, the weight per driver may 
 be assumed at 10,000 Ibs., with 4 drivers for ordinary grades 
 and traffic, or at 11,000 lbs. with 6 drivers, if the grades are 
 steep and the tonnage large. For extraordinary grades special 
 engines are required, having 8 or 10 drivers. The coefficient 
 of friction, called also the adhesion, varies from .09 to .37, 
 these being the extremes on record. The lowest is due to 
 extremely unfavorable circumstances, as sleet and frost; the 
 highest doubtless to the use of sand, though not so stated in 
 the record. The more common range of values is from .15 to 
 
26 FIELD ENGINEERING. 
 
 .25. For our present purpose it will be assumed at .20, se 
 that if a 4-driver engine has 10,000 Ibs. on each driver, its 
 traction is 40,000 * .20 = 8000 lbs. when hauling its maximum 
 train. 
 
 08. The expense of running an engine one mile, hauling 
 a train, on the proposed road, can only be estimated from the 
 experience on other roads similarly situated. The expense is 
 composed of the Items of fuel, water, oil and waste, repairs 
 (including renewals), wages of conductor, engineer, and fire- 
 man, engine-house expenses, and_ interest on first cost of 
 engine and engine-stall. The range and approximate average 
 of these items is here given: 
 
 | || : 
 4-DRIVER ENGINE. | 4-DRIVER 6-DRIVER | 8-DRIVER 
 ITEMS. || ae | 
 Lowest. | Highest.'|| Average. Average.; Average. 
 1 EU XE) Wied es ins eis $0.050 $0.210 $0.100 $0.165 $0.213 
 Wa COT cece stems tx .001 010 || .004 006 | .008 
 Oil and waste........ 904 .030 |] .006 .008 -010 
 Repairs and renewals 05 150 .080 104 | 133 
 WIA OS Teeter cine | 050 .100 075 075 O75 
 Engine-house. ......} 025 060 .035 -050 -060 
 Interest).32.00 52% aes .025 .038 030 .038 047 
 OtaIs but .-e scr 205 .598 3830 A46 546 
 
 si a= SE Pee 
 
 In a given case the probable value of each item should be 
 estimated separately, and the sum taken afterwards. In the 
 above averages each.engine is supposed to haul its maximum 
 train. The relative expense of the several classes of engines 
 has not been established conclusively. 
 
 59. The resistance offered to the motion of a railway 
 train is occasioned by a variety of causes, concerning which 
 a great deal of uncertainty exists as to their relative effect. 
 An investigation which should seek to determine the exact 
 amount of each partial resistance, and then by a summation 
 derive the total, would be tedious, and, in the present state of 
 our knowledge, unsatisfactory. We shall therefore simply 
 group the resistances under three general heads, namely: 
 Resistance due to uniform motion on a straight, level track; 
 Resistance due to grade; 
 Resistance due to curvature, 
 
bs 
 
 MAXIMUM ECONOMY IN GRADES, ETC. a7 
 
 60. The first of these, considered as an aggregate of 
 the various items of friction in engine and train, of oscillations 
 and impacts, and of resistance of the atmosphere, is found to 
 vary nearly or quite as the square of the velocity. The frie: 
 tion of an engine is greater in proportion to its weight than 
 that of a car, owing to its many moving parts, so that the 
 resistance of a short train is greater in proportion to its total 
 weight than that of a long train. The resistance of the atmos- 
 
 phere is greater also in proportion to the weight of a short a 
 train than of a long one. An empty train will offer more Mh 
 
 resistance in proportion to its weight than a loaded one. A 
 formula which shall express the resistance of a train to uni- 
 form motion must include at least the velocity and the weight 
 of the train and engine. 
 
 : The following empirical formula is based upon a careful 
 investigation of all such records of experiments on the subject, 
 several hundred in number, as have come to the author’s notice, 
 and is believed to give results agreeing closely with the average 
 experience and practice of the present day. It is designed to 
 give the resistance per ton for all trains, whether freight or 
 
 SEDO LE LOL EDEL LI 
 
 passenger, and at any velocity, under ordinary circumstances. 
 r Accidental circumstances, such as the state of the weather, 
 
 and the condition of the road-bed, rails, and rolling stock, may 
 largely modify the resistance, but these, of course, are not aa 
 taken account of in the formula. ut 
 Let V = velocity of train in miles per hour, | i 
 “ # = weight of engine and tender in tons, AV] 
 ‘““ W = weight of cars in tons, i 
 ‘< T = weight of freight in tons, 
 “q = resistance to uniform motion in Ibs. per ton. 
 We then have the formula 
 
 i 0006? 
 
 61. The second resistance considered is that due to 
 gravity in grades. It varies in the exact ratio of the rise to the 
 length of the grade. 
 
 Let G, = rise of grade in feet per station. 
 
 ‘““ G, = rise of grade in feet per mile. 
 q = resistance in pounds per ton duc to grade. 
 
 ce 
 
FIELD ENGINEERING. 
 
 Then, 
 . re 
 q = 2240 100 4G, 
 (2) 
 G 14 
 ' — 99 a SF 
 gq = 2240 5380 > 387" 
 
 62. The third resistance considered is that due to 
 curvature of the track. This resistance is due to the friction 
 of the wheels upon the top of the rail, and of their flanges upon 
 the side of the rail. The top friction is lateral, due to the 
 oblique position of the wheel on the rail, and longitudinal, due 
 to the greater length of the outer rail, since both wheels are 
 rigidly attached to the axle. The flange friction is due to the 
 reaction of the top friction, which, combined with the parallel- 
 ism of the axles, throws the truck into an oblique position on 
 the track. A forward flange presses the outer rail, while a rear 
 flange is usually in contact with the inner rail. The centri- 
 fugal force of the car will increase the pressure on the outer 
 rail, unless the ties are inclined at an angle sufficient to coun- 
 terbalance this force. But if the ties are inclined too much, 
 or the velocity is less, the pressure’on the inner rail will be 
 increased. An uneven track will cause the truck to pursue a 
 zigzag course, increasing the resistance considerably. 
 
 Experiments for determining the amount of curve reststance 
 have been neither numerous nor very satisfactory, but the 
 generally accepted conclusion is that the resistance isa little 
 less than half a pound per ton on a one-degree curve, and that 
 it varies as the degree of curve. On European roads, how- 
 ever, it is estimated at about one pound per ton per degree of 
 cv-ve, owing largely to the form of rolling stock used. 
 
 63. Let q’ = curve resistance in pounds per ton on any 
 
 curve, 
 and D = degree of curve. ; 
 
 Then, assuming the resistance per ton on a one-degree curve 
 
 at 0.566, we have for any other curve 
 
 g" = 0.56D (3) 
 
 To ascertain what grade upon a straight line will offer the 
 same resistance as a given curve; substitute the value of q’ 
 for q' in eq. (2) and solve for G; whence 
 
 G, = 9.025D 
 
 ‘4 
 Ga 1 B20 ® 
 
MAXIMUM ECONOMY IN GRADES, ETC. 29 
 
 For definition of degree of curve, sce Art. 84. 
 
 GA. It is evident that grades and curves, by their resistances, 
 fix a limit to the weight of a train which a given engine can 
 haul over them. 
 
 A locomotive is usually built with such a surplus of boiler 
 and cylinder capacity that its power, at ordinary velocities, is 
 limited by the adhesion of the drivers, so that the adhesion is 
 the proper measure of the tractive force. | 
 
 Yo find an expression for the maximum train which a gwen 
 engine can haul over a given grade and curve: 
 
 Let P = tractive force of engine in pounds, 
 
 « 7' = weight of paying load in tons per maximum 
 train, 
 
 «« W' = weight in tons of cars carrying the load 7”. 
 Then for uniform motion, at a given velocity, 
 
 (B4+Ww+T)@+¢+o)=h (9) 
 
 Let ¢ = average load of one car in tons 
 ‘w= average weight of one car and load in tons. 
 
 ‘ ae wes nae BB 
 Then W’' =- 7’ = of , substituting which in eq. (5) we derive 
 
 t P ) 
 fyi So. : pS ( 
 CHG, hen wt hae " 
 
 | 
 
 In this equation g = the resistance per ton due to uniform 
 motion, g' = the resistance per ton due to the maximum grade 
 opposed to the direction of the train, and q’’ = the resistance 
 per ton due to the sharpest curve on that grade. 
 
 For accelerated motion the reaction of inertia of the train 
 must be added to the above resistances. This is estimated at 
 4g, in order that a train starting from rest may acquire the 
 requisite maximum velocity, even on a maximum grade, in a 
 reasonable time, say from 8 to 6 minutes. Therefore, for 
 
 accelerated motion, 
 t tes — 
 pr =+(,..-5) (7) 
 
 of TZ and q involve each other, but if we 
 eq. (1) the value of g becomes that used in 
 
 Now, the values 
 accent Wand 7’ in 
 
30 FIELD ENGINEERING. 
 
 eq. (7), and we may eliminate g between these equations, and 
 ilerive the value of 7’; whence 
 
 “(P~ .00098? V7?) 5 
 | F akOtals Gh age A GECDONT yee (8) 
 
 Also, for the weight of maximum train and load, 
 
 a — .00098? V2, 
 gta +81 +0097? 
 
 which is the expression required. 
 When there is no curve on the maximum grade, g" is zero: 
 
 | and when there is no grade, gq’ is zero; hence for a straight level 
 i track eq. (7) becomes 
 
 We 7 
 
 —H# (9) 
 
 and eq. (8 
 Blak ey (10) 
 ~(P — .0009E? V2) 
 (gee pe 
 ° =" 81+ .009 V2 w 
 
 65. An engine-stage is a division of the road to which 
 an engine is limited, and over which it regularly hauls a train. 
 Its length varies, on existing roads, from 50 to 200 miles or 
 more, depending on the grades, on the length of the whole 
 line, and on the distance between points favorable for the loca- 
 tion of shops, etc. The average engine-stage on American 
 roads is not far from 75 miles. If there are to be several 
 engine-stages on the proposed line, the problem of maximum 
 economy of grade must be solved with reference to each of 
 | them separately. 
 
 1 | Let Z = length of engine-stage in miles, 
 
 ““ @ = expense per engine-mile in dollars, 
 
 ‘““ A= average annual paying freight in tons moving in 
 
 one direction, and 
 
 a = average annual paying freight in tons, moving in 
 the vupposite direction; and if these are not equal, let A be 
 greater than a Now TZ’ eq. (8) is the maximum train-load 
 which, at a velocity V, should be hauled up steepest grade 2’ 
 
 tT ge A 
 opposed to the direction of the tonnage A; hence 7 = the 
 
MAXIMUM ECONOMY IN GRADES, ETC. dl 
 
 number of trains per annum; and since each train must go 
 2LA f ae , 
 
 and return, .’. —,,;~- = the total train-mileage per annum. 
 
 If there were no return tonnage, the annual expense charge- 
 
 2A Le : 
 
 able to A would be pr but since some of the cars return 
 loaded with the freight a, these are not chargeable to A, and 
 must be deducted from the above expression. Hence if we 
 denote the annual expense of engine-mileage by 2, i 
 
 ie (2A — a) Le 
 
 ae (11) 
 
 in which the value of the maximum grade 2’ is involved in 
 the value of 7”. 
 
 But we may obtain an expression for 2 in terms of 2’; for, | 
 at any given velocity, the resistance, ¢,, on a level is equal to a | 
 the resistance due to a certain grade z,, the value of which is, ay 
 by eq. (2), for uniform motion, ay 
 
 39 
 o = 74 to 
 
 2 
 
 So the resistance, g, to motion up a grade 2’ is equal to 
 9 We 
 
 € 8 
 = g, the total resistance ie 
 
 being that due to the combined grades z+ 2’. Now, since 
 the gross weight of a maximum train, under a constant engine 
 power, is inversely as the resistances, we have, for conditions 
 of accelerated motion: 
 
 the resistance due to some grade z = 
 
 a 
 
 w D 
 ri TE ees, ik Ti + Hs: 82, : g2+2 
 
 whence 
 
 ion, ee enna eerste (2) 
 
 in which 7”, = maximum train-load on a level line. Substi- ii 
 tuting this value of 7”’ in eq. (11) we have | 
 Wit 
 | 
 By a 
 = = an — (2A —a) Le (18) 
 37", 0,-—H@ +4 (@—2)) 
 
 t 
 w 
 
 which is the complete expression for v = f (z2’) required. 
 
32 FIELD ENGINEERING. 
 
 66. Could we also find a complete expression for y = f' (2"), 
 we might then proceed to find, by analysis, that value of 
 e’ which would render #-+-y =a minimum. But the value 
 of y cannot be formulated, since it depends on the accidental 
 features of the country through which the line passes; it can 
 only be determined for any given value of 2 by an estimate 
 based on the survey. We therefore resort to a graphical 
 solution. 
 
 Equation (13) is the equation of a curve in the plane ZX, 
 Fig. 2. If we assume several values—of 2’, and calculate the 
 corresponding values of 7, we may lay these off by scale on 
 the axes of Z and X respectively, and so obtain several points 
 
 ia. 2. 
 
 through which the curve of annual expense may be drawn. 
 We then make estimates of the cost of constructing the road 
 at the same values of 2’, and taking the annual interest of 
 each estimate as an ordinate y to OZ in the plane ZY, we lay 
 it off to scale at the proper height, thus obtaining a series of 
 points in the plane ZY, through which the curve of annual 
 interest on first cost may be drawn. If now we suppose the 
 plane ZY to be revolved to the left about the axis OZ until 
 it coincides with the plane OX, as in Fig. 2, we shall see 
 that the two curves are convex:to OZ and to each other. The 
 shortest horizontal line intercepted by them indicates the 
 minimum value of («+ y), and the point where this line cuts 
 the axis OZ indicates the corresponding value of 2’, which is 
 the one required. If tangents be drawn to the curves at the 
 points where the shortest horizontal line intersects them, the 
 tangents will be parallel to each other. Any convenient scales 
 may be used to lay off the values of z' and 2, provided that 
 the values of wand y be laid off to the same scale. It is well 
 
MAXIMUM ECONOMY IN GRADES, ETC. Ps) 
 
 to reduce all the values of 2 by an amount common to them 
 all, and the same with respect to values of y, before laying 
 them off to scale. This will bring the two curves nearer 
 together without altering their form. 
 
 6%. To facilitate the calculation of 2, we give on the next 
 
 4 1 . , 
 
 page a table of values of via for several engines, using eq. (8) 
 for this purpose. The value of vis therefore found, eq. (11) 
 or (13), by multiplying (24 — a) Le by the proper tabular 
 number, under conditions assumed as follows: 
 
 t- = 10 tons of freight per car-load; 
 aw = 18 tons per car and load; 
 
 V = 12 miles per hour. 
 
 Fora 4driver engine, # = 42 tons; P= 8100 Ibs. 
 
 Fora 6-driver engine, #= 49.5 ‘“* P= 12600 “ 
 
 For an 8-driver engine, # = 59.4 ‘“* P= 17280 * 
 
 Substituting these values in eq. (8), and making gq” = 0, 
 we find the maximum loads of freight which the several 
 engines can haul up the grade whose resistance is g'. The 
 reciprocals of these loads are given in the table opposite the 
 grades noted in the first and last columns. 
 
 G8. Since qg’ is made zero, the grades in the table are 
 assumed to be on straight lines. In locating a road, the 
 maximum grade should be reduced on a curve by the amount 
 of the equivalent-grade of the curve, eq. (4), so that the resist- 
 ance may be no greater on the curve than elsewhere. But 
 grades less than the maximum need not necessarily be reduced 
 for the curves upon them, unless the sum of the grade and the 
 curve-equivalent exceeds the maximum. 
 
 69. For an example, let us suppose that a certain engine- 
 stage is to be 80 miles long, and that an estimate of the cost of 
 construction has been made, based on a ruling or maximum 
 grade of 52.8 ft. per mile against the heavier traffic, and that 
 the annual interest on the estimate amounts to $168,000. 
 
 Let us further suppose that the average traffic in one direc- 
 tion, is estimated at 375 000 tons per annum, and in the other 
 direction at 125 000 tons, that it is decided to use 6-driver 
 engines, and that the expense per engine-mile is estimated at 
 40 cents; hence (2A — a) Le = 20 000 000. Weare now required 
 to find the most economical maximum grade. 
 
 Wefirst select at least two other maximum grades, and having 
 
FIELD ENGINEERING. 
 
 12 TABLE OF RECIPROCALS OF T’’, b= 104038, 
 
 | | | 
 
 E=2 E=49.5 |-, Fe 280 sae on a ne 
 
 G,. P = 8100 Diff. | P= 12600 | Diff. P = 1780 Diff. ft. per 
 | | | mile. 
 
 "9 842 2 5 D RAW | 
 BAD His OF79 844: | oy anit) CORAL BBS ees Vint ICR Ror berets 
 8.9 | .0457399 | Si a¢9 || -0232 431 | 8 goo || -0157 250 | Pia, | 205.92 
 3.8 | .0436.036 | 59 gay |) 0223739 | Q gig || -0151 786 5 336 | 200.64 
 8.7 | .0415 679 | 49 yoo || -0215 297 | Bong |} -0146.450 | 2 or9 | 195.36 
 8.6 | .0396 259 | 38 x4p || -0207 094 | On4 || .0141 238 | Boos | 190.08 
 8.5 | .0877 712 | 37295 || .0199120 | n/5 || .0186 146 | Fong | 184.80 
 8.4 | .0359 980 | 46 ogg || -0191 867 | + 545 |] -0181 168 | fone | 179.52 
 8.3 | .0343 012 | y¢ ox || -0183 824 | 94) || -0126 802 | Zen» | 174.24 
 8.2 | 0826 759 | 35 Fen || -0176 483 |» o4~ || 0121545 | | peg | 168.96 
 3.1 |- 10311 176 | 15 583'/| ‘o169 336 | “147 '| 0116 s92 | 48 | 463/68 
 
 Bh eta 14 952 || ae. 6 960 | cial 4553 | - 
 
 .0296 | || .0162 876 | amo || .0112 339 nx | 158.40 
 2.9 | ‘028i 864 | 33 So || -o155596 | 6180 1| ‘o1ov sea | 4955 | 153.12 
 2.8 | .0268 061 | 43 oem || -0148 988 | ¢ 445 || -0103524 | A ogg | 147.84 
 2.7 | 0254 784 | 35 mg || -0142546 | g og5 || -0099 255 | 1189 | 142.56 
 2.6 | 0242005 | 35 319 || -0186 264 | g jog || -0095 075 | 4 og4 | 187.28 
 2.5 | .0229 695 | 47 gee || -0180186 | 5 oxo || -0090981 | 741; | 182.00 
 2.4 | .0217 828 | 44 gar || -0124 157 | 2 age || 0086970 | 3 o39 | 126.72 
 2.3 | .0206 881 | 44 o4g || -0118 321 | 5 699 -0083 040 | 3 pr5 | 121.44 
 2.2} .0195 333 | 49 gro || -0112 622 |S 566 -0079 188 | gre | 116.16 
 2.1 | .0184 663 | ‘ 0107 056 | 2 °P) || .CO%5 413 ‘| 110.88 
 
 2.0 | .0174 352 Bee 0101 6 ome C071 712 tie 
 .0174 352 a 20 | C071 712 | on | 105.60 
 1.9 | :0164382 | 9 2%2 || ‘0096308 | 2312 || 0068 02 | 3880 | 100.32 
 1.8} .0154 736 | gaa || .0091115 | Fong || -0064522 | 3493 | 95.04 
 1.7 | .0145 399 | 6 pys || -0086039 | Toe. || .0061 029 | 3 jo» | 89.7 
 1.6 | .0136356 | greg || .0081074 7 Qn8 || .0057 602 3 363 | 84.48 
 1.5 | .0127593 | 3 4oq || -0076 218 4751 || -0054239 | 339, | 79-20 
 1.4 | .0119 099 939 || -0071 467 | 4 gig || -0050 988 | 3 oy49 |+ 13.92 
 1.3} .0110860 | »oox || .0066 818 | gre) || .0047 698 | 37g, | 68.64 
 1.2} .0102865 | srey || .0062267 | 4 gee || .0044517 | 5 yo, | 63.36 
 1.1; .0095104 | ‘‘™ || .0057 810 YF | 0041 393 | | 58.08 
 1.0 | .0087568 | yon, || .0053445 | yor, {| .003882 | ao | 52.8 
 .0 | .0087 a || .0053 445 Es , Lee | 52.80 
 9 | oogozde | F824) ‘0049 101 4277 )| 10085 308 | 20l8") 47.52 
 8 | .0073 123 | 6 923 | .0044 984 4104 |, 2082847 | Soi9 | 42-24 
 .7 | 0066 200 | gigs || -0040880 | 499; -0029437 Speq 36.96 
 -6 | .0059 466 | grx3 || .0086 858 | 3045 || .0026577 Sei, | 31.68 
 -5 | .0052.913 | @ 3a || -0082915 | 3 ga || .0023 766 Orgy | 26.40 
 4) .0046 583 | & 549 || -0029 050 3791 || -0021002 | gaia | 21.12 
 -3 | .0040 320 | @ Ox5 || .0025 259. | Seiq || .0018 284 | 5 Gap | 15.84 
 -2| .0034 268 | £ gog || -0021 540 | 3 ay4g | (0015 612 | 969g | 10.56 
 -1 | 0028870 | Bren || .0017 892 | 3560 || 0012984 | 9 pen 5.28 
 0.0} .0022620 | ° .0014 312 | .0010 399 0.00 
 
 In this table T’ = tons of freight for a maximum train of 
 fully loaded cars hauled up any grade ¢z' at a velocity of 12 
 miles per hour ; the ratio of dead to paying load being assumed 
 at 8to 10. Hence gross load of train behind engine = +5 T’. 
 The track is assumed straight, hence q’ = Qin eq. (8) for 
 
 this table. 
 
MAXIMUM ECONOMY IN GRADES, ETC. 35 
 made an estimate of the cost of constructing the road upon 
 each, take the annual interest of each, as in the first case. Let 
 us suppose the two ruling grades thus selected to be 73.92 ft. 
 and 31.68 ft. per mile, or 1.4 ft. per station and 0.6 ft. per 
 station, and the interest on the estimates to be $145 596 and 
 $204 388 respectively, giving the following statement: 
 
 G;. Y. 1st diff. 2d diff. 
 1.4 145 596 29 404 
 1.0 168 000 a 13 984 
 0.6 204 388 36 35¢ 
 
 Interpolating by second differences, we have the complete 
 statement: 
 
 Gar) Ye. | diff. y. | Cre. | gs ety. z 
 
 1.4 | 145596 ohh ee | 142 934 eatery a2 
 
 1.3 | 149886 Sr ietisron & | 2 Gs ae | 
 
 1.2 | 155 050 6038 | solid 124 534. | 
 
 1.1 161 088 6912 | 8730 115 620 | | 
 
 1.0 168000 | wrge | gig | 106 890 74 890 52.80 
 
 0.9 175786 | ges | gag | 98342 | 274128 47.52 
 
 0.8 184 446 9534. | g0g | 89968 | 274414 42.24 
 
 0.7 193 980 1408.1 e044. 1, SL OO 
 
 0.6 | 204388 | | | @u6 | 31.68 
 | j 
 
 The numbers in the fourth and fifth columns are obtained 
 as follows: the values assumed above give us (2A — a) Le = 
 $20 000 000, and this multiplied by the tabular differences in 
 the preceding table for a 6-driver engine, gives the numbers in 
 the fourth column. We now observe that the differences of 
 z and of y increase in opposite directions, therefore at some 
 point they will be equal; and a simple inspection shows us that 
 this point is at or near the grade of 0.9, which is therefore 
 the grade required. We now multiply the tabular number for 
 0.9, and a 6-driver engine by $20 000 000, for the number in 
 the fifth column, and this added to the value of y on the 
 same line gives the sum of («+ y) for the most economical 
 grade. This of course is not the total annual outlay of the 
 road, or engine-stage, because many items of expense which 
 are independent of a maximum grade have not been con- 
 sidered. 
 
36 FIELD ENGINEERING. 
 
 If an 8-driver engine were to be used, and the expense per 
 engine-mile estimated at 50 cts., then (2A — a) Le = $25 000000; 
 hence 
 
 G,. y. diff. y. diff. a. i woty. z'. 
 ita! 161 088 ae 
 
 6 912 % 6700 4 a 
 ce be oat 7 286 538 95 810 263 810 52.80 
 
 indicating a saving of $10 818 per annum in the case supposed 
 by using 8-driver engines, although on a steeper ruling grade. 
 On the other hand, should we adopt 4-driver engines, and esti- 
 mate the expense per engine-mile at 30 cents, we should find 
 the most economical grade to be 0.7 per station and (# + y) 
 = $2938 280, showing a loss in this case of $19 152 per annum, 
 as compared with the results of 6-driver engines. 
 
 It should be remembered that the table § 67 is prepared on 
 
 : ROT, eordl 
 the assumption that the ratio ea es If cars are to be used 
 giving for full loads any other ratio,. 5 a new table may be 
 LO = snag 
 prepared by multiplying each tabular number by 78 xs = 
 
 The velocity adopted of 12 miles per hour is sufficient a 
 ordinary grades. When the maximum grade is very low, it 
 would be better to use 15 or 18 miles an hour in calculating 
 the value of 2. 
 
 70. Since z, eq. (11), varies directly as Z, it is important 
 that an engine-stage having heavy grades should be short. Its 
 length, however, must be consistent with the economical 
 length of the adjoining engine-stages, and with the amount of 
 work which an engine ought to perform daily. The most 
 favorable condition for a road would be that in which all the 
 
 _ engine-stages were operated at equal expense. But if, to 
 
 secure this result, the engine-stage of heavy grades must be 
 unreasonably reduced in length, it will be better to adapt the 
 grades to the use of two engines per train. 
 
 71. The maximum grade 2’, opposed to the heavier tonnage 
 A, having been determined, we have now to consider what is 
 the limit to grades in the opposite direction. The engines are 
 
MAXIMUM ECONOMY IN GRADES, ETC. 37 
 
 supposed to haul their maximum loads in moving the ton- 
 nage A, and since the return tonnage, a, is less than <A, the 
 engines, in returning, will not be worked to their full capacity 
 if they encounter no grades steeper than ¢’. We therefore 
 have a margin of power in the returning engines which may 
 be taken advantage of to cheapen the cost of construction, or 
 to shorten the line, by introducing grades, steeper than 2’, 
 against the lighter traftic. | . 
 The weight of a maximum train moving up the grade ¢’ is, an a 
 eq. (9), W'-+ 7”; the weight of the train returning will be Wh 
 
 ¢ 
 
 al Diy w—t a ! 
 pen el Tits (a BE 
 
 Substituting this in place of (W’‘ + 7’), eq. (9), and solving for Wi 
 q, we find the resistance due tu a maximum grade opposed to GE 
 the returning train. Whence, by eq. (2), if we let Z=the A) 
 maximum return grade, and make q” = 0, 
 
 Se pea au 
 
 w—t , 24 Wa 
 
 Inasmuch as the value of Z varies with every change made 
 in 2’, the engineer, when estimating the cost of construction 
 upon the basis of any maximum grade 2’, should take care 
 that the return grade Z nowhere exceeds its limit as given by 
 the last equation (14). In the example, $69, 2’ = 47.52; hence 
 T’ = 208.37, eq. (8). Substituting these values, in eq. (14), we 
 find Z = 81.25, which is therefore the limit for return grades 
 in this case. With regard to curves on the maximum grade, 
 see $68. 
 
 72. If ineg. (1) weletz = 
 
 33 
 14 
 offers a resistance equal to the resistance to uniform motion 
 
 on a level, we have 
 
 q be the grade per mile which 
 
 00141422 | 
 = To Son ee 7% 15 | 
 2= 12.784 (01414 + pe) (15) | 
 When V = 20 this becomes | 
 ~ 2 Wy 
 BES 8 BOE); git en AE Me! (153) 
 
 E+W+T 
 
3 FIELD ENGINEERING. 
 
 which is the grade down which a train, whose weight is (H 
 W-+ T), if started at 20 miles an hour, will continue to move 
 at that speed without steam or brakes. As that speed is not 
 objectionable, so the grade 2, which induces it is not, pro- 
 vided it does not exceed the values of 2’ or Z respectively, 
 determined with reference to economy. For the extra work 
 done by the engine in ascending one grade ¢ is utilized in 
 descending the next; and the net result is the same as though 
 the two were replaced by a uniform grade. The engineer 
 therefore is not warranted by economic considerations in 
 reducing undulating grades which do not exceed z to a uni- 
 form grade, when to do this would cause any increase in the 
 cost of construction, unless 2 exceeds the grades z' or Z of 
 maximum economy. . 
 
 73. But when grades exceed 2, eq. (154), the resulting 
 speeds of the maximum train become too great, and the neces- 
 sary application of the brakes absorbs a portion of the power 
 previously expended in gaining the summit, which is thus 
 worse than wasted, since it increases the wear and tear of 
 machinery and track. Therefore the engineer is justified in 
 spending a certain sum of money in reducing grades which 
 exceed z to that limit. A calculation of the loss of power due 
 to the use of brakes on a grade, and of the cost of that lost 
 power, together with the resulting wear and tear per annum, 
 will give the interest on the sum that may be justifiably spent 
 in reducing the grade from its position of cheapest construc- 
 tion. 
 
 74. The limit 2 is not constant, but depends on the weight 
 of the maximum train, which in turn depends on 2’. It will 
 not be the same in both directions unless A =a, giving 2’ = Z. 
 In the example §69, # = 49.5 and W' + 7’ = 366.07; hence, 
 eq. (155), 2 = 21.72 descending in the direction of the traffic A. 
 
 Also W' +4 T' = 230.49, whence z = 23.34 descending in 
 
 the opposite direction. These are the limits in this case at 
 which undulating grades cease to be profitable. 
 
 75. We have finally to consider the method for selecting the 
 best line from several proposed routes. For this purpose we 
 determine the most economical grade on each route thought 
 worthy of consideration, and calculate the interest on the 
 entire cost of constructing the line with that ruling grade, and 
 
S 
 
 LOCATION. 39 
 
 also the annual expense of operating the line, and take the sum 
 of the two. ©That route is best in respect to which this swm is 
 the least. 
 
 7G. The value of saving onc mile in distance on any route 
 is found by dividing the sum of the annual operating expense 
 and the interest on the cost of construction by the rate of 
 interest, and the quotient by the length of the line in miles. 
 
 77. We have now fully discussed the theory and developed 
 the formule necessary to the determination of the most i] 
 economical grades; but the value of the results in a given i 
 case depend upon the correctness of the engineer’s estimates 
 which enter into the formule. These may seldom prove pre- . 
 cisely accurate, yet, if he can bring them within definite Wy 
 limits, he may determine the grades of maximum economy 
 within corresponding limits. In the case of a finished road 
 and in full operation, however, the elements of first cost, of Hh | 
 traffic, and of operating expenses being known, an investiga- 
 tion by means of the foregoing formule becomes a critical test 
 as to the economy of the location and grades; and should the 
 road fail to pay dividends, or be forced to charge high rates 1 
 of toll, we can determine, though perhaps too late, to what | 
 extent the location is chargeable with these results. ii 
 
 CHAPTER IY. ay 
 LOCATION. 
 
 78. A railroad is said to be located when its centre line is 
 established on the ground in the position which it is intended 
 finally to occupy. The location is made by an engineer corps | 
 similar in its organization to that employed on preliminary | | 
 surveys. The instruments used are also the same, except that VW 
 the transit is substituted for the compass, and usually the target | | 
 rod’ for the self-reading rod. The magnetic needle is never Hi 
 used upon the centre line, except as a rough check on the 
 transit work. It is used, however, to obtain the direction of 
 property lines, roads, and other topographical data. 
 
 79. The remarks upon transit work in the preceding 
 chapter apply to the running of straight lines on location. All 1 
 
40 FIELD ENGINEERING. 
 
 field-work on location should be done with accuracy and 
 fidelity. No guesswork, nor rude approximations, are to be 
 tolerated. AJ] transit points are made as secure and permanent 
 as possible, and the more important ones are guarded by other 
 transit points set in safe positions near by, their distances and 
 directions from the main point being recorded. 
 
 The stakes for the stations are made ne atly, and somewhat 
 uniform in size, and they are firmly driven. Sometimes a 
 small plug is driven down flush with the surface of the ground 
 to indicate the station point, and the stake is then set near by 
 as a witness. 
 
 In locating a very long tangent the greatest care is re- 
 quired to make it straight. If the tangent is produced from 
 point to point by backsights and foresights, the observation 
 should be repeated in every instance with reversed instrument, 
 to eliminate any possible lack of adjustment, and to check 
 any accidental error. (Indeed it is proper to observe this rule 
 on curves, as well as on tangents.) When some object in the 
 horizon can be used as a foresight, it is preferable to set the 
 instrument by this rather than by a backsight. For final loca- 
 tion, the line should be cleared to give as continuous a line of 
 sight as possible, but in case of an obstacle which cannot be 
 removed at the time, at least two independent methods of 
 passing it should be employed, so that there may be a check 
 upon the alignment beyond. 
 
 SO. The leveller selects his benches far enough from the 
 line to prevent their being disturbed during the construction of 
 the road. They should be nearly at grade, as a rule, though it 
 is well to leave a bench near a water-course for reference in lay- 
 ing out masonry of trestle-work. The rodman holds the rod 
 at every station, and at every point on the centre line where 
 the slope changes direction, so that these points may be accu- 
 rately defined on the profile. When he uses a target rod, he 
 sets the target as directed by the leveller, and after clamping 
 
 ‘it, takes the reading. He reads to thousandths upon turning 
 
 points and benches, but only to tenths of a foot elsewhere, and 
 announces the readings to the leveller for record. He also 
 records the readings upon turning points and benches in his 
 own book asa check. At the close of each day the leveller 
 and rodman compare notes, and draw a profile of the line sur- 
 veyed. (See also §§ 28, 29, 30.) 
 
LOCATION, 
 
 $1. The fixing of the grade-lines upon the profile is 
 one of the most important operations connected with the loca- 
 tion. It is usually performed by the engineer in charge of 
 the locating party, as being most conversant with the general 
 character and detailed requirements of the line. The maxi- 
 mum gradients will have generally been determined in advance 
 from the preliminary data by the principles laid down in the 
 preceding chapter, but the position of each grade-line, relative 
 to the profile of the surface, must be left to the judgment and 
 skill of the engineer. In general, the grade-line is so placed 
 as to equalize the amounts of excavation and embankment, 
 but there are various exceptions to this rule. Thus, the exca- 
 vation may be in excess: first, when it is necessary to pass 
 under some other road or highway, the grade of which cannot 
 be changed; second, when valuable property is to be avoided, 
 the appropriation of which would cost more than the excava- 
 tion; third, when the grade is at the maximum near a sum- 
 mit, and cannot be raised parallel to itself without incurring 
 too great an expense for masonry, etc., at some other part of 
 the line. The embankment may be in excess, first, when the 
 country is flat and wet, in order to keep the road-bed well 
 drained; (the grade-line should be at least two feet above tke 
 average level of the surface, or above high-water mark, if the 
 district is subject to overflow;) second, in approaching a 
 stream, where it is necessary to raise the grade above the 
 requirements of navigation; third, when the cuttings on the 
 line are largely in solid rock, and a cheaper material for 
 embankments may be conveniently had at other points; 
 fourth, in a district subject to heavy drifts of snow, by which 
 deep cuts would be liable to be obstructed; fifth, in side-hill 
 work, where there is danger of land-slips; sizth, when it is 
 determined to supply the place of a portion of an embankment 
 by a timber trestle-work or other viaduct. 
 
 The apparent equality of cut and fill on the profile does not 
 represent an equality in fact, owing to the different bases and 
 slopes of the sections adopted, and to the various inclinations 
 of the natural surface transversely to the line. This is espe- 
 cially true in side-hill work, where there are both cut and fill 
 at every point, while the profile shows very little of either. In 
 the latter case it is an excellent plan to combine with the pro- 
 file of the centre line the profiles of parallel lines ten 
 
4° FIELD. ENGINEERING. 
 
 or twenty feet either side of the centre, and drawn with differ. 
 ent colored inks, as these will indicate tolerably well the relative 
 amount of cut and fill required. But after the grade has been 
 thus chosen, the only safe method in side-hill work is to 
 actually compute the amounts of excavation and embankment 
 from cross-sections, mark the amount for each cut and fill on 
 the profile, and compare the results. Any changes required in 
 the grade or alignment may then be discovered and effected 
 before the work of construction has begun. 
 
 C. H-A.P TER. .Ve 
 SIMPLE CURVES. 
 A. Hlementary Relations, 
 
 $2. The centre line of a located road is composed alternately 
 of straight lines and curves. 
 
 The straight lines are called tangents because they are laid 
 exactly tangent to the curves. <A tangent may be indefinitely 
 long, but should never, as a rule, be shorter than 200 feet 
 between two curves which deflect in opposite directions, nor 
 shorter than 500 feet between curves which deflect in the same 
 direction. A curve should not be less than 200 feet long. 
 When a tangent is said to be straight, the meaning simply is 
 that it has no deflections to the right or left; for since it fol- 
 lows the surface of the ground, it evidently has as many 
 undulations as the ground. But if we conceive a vertical 
 plane to be passed through the line, a horizontal trace of this 
 plane will accurately represent the line; and so, if we con- 
 ceive a vertical cylinder to be: passed through a curve on the 
 surface of the ground, a horizontal trace of that cylinder will 
 accurately represent the curve, since all distances and angles 
 are measured: horizontally, whatever be the irregularities of 
 the surface. In all problems, therefore, relating to this sub- 
 ject, we may consider the ground to be an absolutely level 
 plain. 
 
 83. A Simple curve isa circular arc joining two tan- 
 gents. It is always considered as limited by the two tangent 
 
SIMPLE CURVES. 
 
 points, and any part of it beyond these points is called the 
 curve produced. The first tangent point, or the point where 
 the curve begins, is called the Point of Curve, and is indicated 
 by the initials P.C. The point where the curve ends, and the 
 next tangent begins, is called the Point of Tangent, and is indi- 
 cated by the initials P.7. When accessible, these points are 
 always occupied by the transit in the course of the survey, 
 and the plug driven to fix the point is guarded, not only by 
 the usual stake bearing the number of the station, but also by 
 another bearing the proper initials, the ‘‘ degree” of the curve, 
 and an ‘‘R” or ‘‘L” to indicate whether the deflection is to 
 the Right or Left. 
 
 S4. A simple curve is designated either by the radius, R, 
 or the degree of curve, D. 
 
 The Degree of Curve, D, is an angle at the centre, sub- 
 tended by a chord of 100 feet. It is expressed by the number 
 of degrees and minutes in that angle, or in the arc of the 
 
 curve limited by the chord of 100 
 b feet. Therefore D equals the num- 
 ber of degrees of are per station. 
 Q The radius R and degree of 
 , curve J) can be expressed in terms 
 of each other. 
 Let ab, Fig. 3, be a chord of 
 
 100 feet subtending an arc de- 
 scribed with a radius a0 = Rk from the centre 0. Then, by 
 definition the angle bea = D. Biscct the angle boa by a line 
 og, and this line will also bisect the chord ad and be perpen- 
 dicular to it; and in the right-angled triangle dgo we have 
 
 Fie. 3. 
 
 bg = ob X sin bog 
 or 
 
 a = sin 4D 
 
 Hence, to find Radius in terms of Degree of Curve: 
 
 50 
 dea a. 16 
 t sin 4D 0 
 
 and to find Degree of Curve in terms of Radius: 
 
 She seers (177) 
 
 I 
 
 & 
 
44 FIELD ENGINEERING. 
 
 It is the practice of English engineers to assume the radius 
 at some round number of feet and calculate the degree of curve, 
 which is therefore fractional. In America, on the contrary, 
 the degree of curve is assumed at some integral number of 
 degrees or minutes, and the radius deduced from this. 
 
 Heample.—W hat is the radius of a 3° 20' curve? 
 
 50 log 1.698970 
 4D = 1° 40’ log sin 8.463665 
 
 Ans. R=1719.12 log 3.235305 
 Thus the second and third columns of Table IV. have beew 
 calculated. 
 
 Example.—W hat is the degree of curve when the radius is 
 600 feet? 
 
 50 log 1.698970 
 R=-600 log 2.778151 
 
 4D = 4° 46' 48".'73 log sin 8.920819 
 Ans, D = 9° 33' 37" .46 
 
 Measurement of Curves. 
 
 85. A railroad curve is always assumed to be measured with 
 a 100-foot chain, and as the chain is stretched straight between 
 stations it cannot coincide with the arc of the curve, but 
 forms a chord to the arc, as in Fig. 3. Consequently the 
 curve as measured from one tangent point to the other is an 
 inscribed polygon of equal sides, each side being 100 feet. 
 The sum of these sides (with any fraction of a side at either 
 end of the curve) is called the Length of curve, L. This length 
 L is evidently a little less than the length of the actual are 
 between the same points, but the latter we very seldom have 
 occasion to consider. 
 
 8G. If the chain lengths were taken on the are instead of as 
 chords of the curve, the degree of curve would be inversely 
 proportional to the radius, and since the are whose length is 
 equal to radius contains 57.3 degrees nearly, we should have 
 
 DS OTF e100 © ie. 
 or 
 _ 5180 
 
 PD 
 
SIMPLE CURVES. 45 
 a convenient formula, but only approximately true when D is 
 small, and seriously at fault when D is large; the error in- 
 volved being proportional to the difference in length of a 
 100-foot chord, and the are which it subtends. 
 
 87. The Central Angle of a simple curve is the angle 
 at the centre included between the radii which pass through the 
 tangent points (P.C.) and (P.7.). It is therefore equal to the 
 number of degrees contained in the entire arc of the curve 
 between those points. The central angle will be designated 
 by the Greek letter A (delta). 
 
 From the definitions of the length and degree of curve we 
 have the proportion, 
 
 LTE. = 100371. 
 
 fence, to find the Length of curve én terms of the central 
 angle: 
 
 A 
 23 18 
 EL = 100 i ( ) 
 
 Example.—W hat is the length of a 4° curve when the cen- 
 tral angle is 29°? 
 D =A’ and A= 20° § 4)2900 
 Ans. I= stations + 25 feet ( 725 feet. 
 To find the Central angle in terms of the length and degree 
 of curve: 
 
 Livample.—What is the central angle of a 5° curve 730 feet 
 long? 
 . 
 Io: cea TOU; ca WV res, 
 Ans. A = 36° 30' 
 
 To find the Degree of curve in terms of the length and 
 central angle: 
 — 400 2 20) 
 D = 100 = ( 
 Example—What is the degree of a curve 8 stations long, 
 and having a central angle of 26° 40'? 
 26°.666 
 
 — = 3°.333 
 
 L = 800, A = 26°.666, 100 800 
 
 Ans. D = 8° 20’ 
 
46 FIELD ENGINEERING. 
 
 88. If two tangents, joined by a simple curve, are produced 
 (one forward and the other backward) until they intersect, the 
 point of intersection, V (Fig. 4), 
 is called the vertex, and the 
 exterior or deflection angle 
 which they make with each 
 other is equal to the central 
 angle, A 
 
 The Tangent-distance, 
 T, is the distance from the 
 vertex to either tangent point; 
 thus in Fig. 4, 7= AV=VB. 
 
 The Long Chord, C, is 
 the line APB joining the two 
 tangent points. 
 
 Fra. 4. The Middle-ordinate, 
 
 M, is the line GH, joining the 
 
 middle point of the long chord with the middle point of the 
 curve. 
 
 The External distance, #, is the line HV, joining the 
 middle point of the curve with the vertex. 
 
 We observe that both the middle-ordinate, J, and the 
 external distance, #, are on the radial line joining the centre, 
 O, with the vertex, V, and that this line is perpendicular to 
 the long chord, C; also, that it bisects the central angle 
 AOB= A, and its supplement AVB. (Tab. 1.14.) We also 
 observe that the angle VAB= VBA =A (Tab. I. 20); and 
 if in the figure we draw the two chords AH and BH, the 
 angle BAH equals one half the angle BOH, or BAH = ABH = 
 4A (Tab. I. 18); also the angle VAH= VBH=4a. 
 
 89. If we have laid out two tangents on the ground, inter- 
 secting at V, and have measured the angle, A, between them, 
 we may then assume any other one of the elements of a 
 simple curve before mentioned, and calculate the rest. If 
 we assume JD, for instance, we then find & by eq. (16) or by 
 Table LY. 
 
 Then, having A and R,we may proceed to. calculate the 
 other elements as they are needed. 
 
 90. To find the Tangent-distance in terms of the 
 Radius and Central Angle: 
 
4 | 
 
 SIMPLE CURVES. 
 
 In the right-angled triangle VOA, Fig. 4, we have 
 
 VA = OA X tan VOA 
 Pl oh ta FA (21) 
 
 Otherwise, approwimately: In Table VI., opposite the central 
 angle, take the value of 7 for a 1° curve and divide it by the 
 degree of curve D. If desirable, add the correction taken 
 from Table V., corresponding to D. 
 
 Hxample.—What is the tangent distance of a 4° curve with 
 a central angle of 30°? 
 
 Diese R(Table IV.) log 3.156151 
 A= BO, 4A = 15° _ log tan 9.428052 
 
 Ans. T' = 3838.89 feet log 2.584208 
 Otherwise: 
 By Table VI. 4)1585.3. 
 Approximate ans. 383 . 82 
 Correction from Table V. .08 
 Ane Tt = 383.90 feet. 
 
 91. To find the Long Chord (C, in terms of Radius and 
 Central Angle: 
 In the right-angled triangle BOG, Fig. 4, we have 
 
 BG = BO X sin BOG 
 or 
 4C0=Rsinta 
 .C=2Rsinta (22) 
 
 But in case A can be divided by D without a remainder, 
 that is, if the curve contains an exact number of stations (not 
 exceeding 12), we may take the long chord at once from 
 Table VII. 
 
 Example.—What is the long chord of a 3° 20’ curve with a 
 
 central angle of 86° 40' ? 
 
 2 log 0.801030 
 =o 20. (lab. LV.) log 3. 2353805 
 A = 36° 40’, 4A = 18° 20' log sin 9.497682 
 
 Ans. C = 1081.48 feet log 3.034017 
 
A8 FIELD ENGINEERING. 
 
 Otherwise: 
 A 863° 
 ie —-- = |{ stations 
 pr BP 
 
 And by Table VII. O = 1081.48. 
 
 92. To find the Middle-ordinate Y, in terms of Radius 
 and Central Angle: — 
 
 It is evident from the figure that if the radius OH were 
 unity, the line GH would be the nat. versed sine of the arc 
 BH. But the arc BH measures the angle BOH= 4A, and 
 Ofek: 
 
 . M= Rversta (23) 
 
 But in case A can be divided by D without a remainder, 
 . that is, if the curve contains an exact number of stations (not 
 i exceeding 12), we may take the middle-ordinate at once from 
 Table VIII. 
 
 Example.—What is the middle-ordinate of a 4° 30’ curve 
 with a central angle of 40° 30’? 
 
 | Dawa R (Tab. IV.) log 3.105022 
 | A = 40° 30, 4A = 20° 15’ log vers 8.791049 
 
 Ans, Ma U8 7 1.896071 
 Otherwise: 
 A> 4075 
 9 Aldo 9 stations 
 
 and by Tab. VIII. M = 78.717 
 93. To find the External Distance £# in terms of 
 Radius and Central Angle. 
 It is evident from the figure that if the radius OA were 
 i unity, the portion HV of the secant line OV would be the 
 i external secant of the arc AH. But the arc AH measures the 
 1 || angle AOH= 4A, and OA=Rf; 
 
 ‘ H= Rex see ga (24) 
 
 Otherwise, approximately: 
 
 In Table VI., opposite the central angle, take the value of 
 E for a 1° curve, and divide it by the degree of curve D. 
 If desirable, add the proper correction corresponding to D, 
 taken from Table V 
 
SIMPLE CURVES. 
 
 Hrample.—W hat is the external distance # of a 7° 30’ curve 
 when the central angle is 60° ? 
 
 DT tau, ?(Tab. IV.) log 2.883371 
 LOU", 4A = 80° log ex sec 9.189492 
 Ans. H = 118.27 feet log 2.072863 
 . Otherwise: 
 By Tab. VI. 7.5)886.38 
 : Approximate ans. 118.184 
 Correction for D = 7° 30' (Tab.. V.) 084 
 Ans. H= 118. 268 
 
 94. But, instead of assuming D or R, we may prefer, or may 
 find it necessary to assume, some other element of the curve, 
 the central angle being given. 
 
 If we assume the tangent distance, then: 
 
 95. To find the Radius and Degree of Curve in terms 
 of the Tangent-distance and Central Angle. 
 
 From eq. (21), and by Table II. 40, we have 
 
 R=T cotia (29) 
 
 Otherwise, approximately: 
 
 Divide the tangent of a 1° curve found opposite the value of 
 A in Table VI., by the assumed tangent distance; the 
 quotient will be the degree of curve in degrees and decimals. 
 
 Kxample,—The exterior angle at the vertex is 54°, and the 
 tangent distance must be about 700 feet. What shall be the 
 degree of curve? 
 
 A = 54°, 4A = 27° log cot 0.292834 
 Rig. 5 (X) 2.845098 
 log B= 3.137932 
 Ans. By Table IV. D= 4° 10’ + 
 
 Otherwise: 
 
 pe 
 
 By Table VI. 700)2919.4 
 Ans, D = 4° 10' 15” 4.170 
 
 =r) 
 
 But as it is difficult to lay out a curve when JD is fractional, 
 we discard the fraction and assume 4° 10’ as the value of D. 
 
50 FIELD ENGINEERING. 
 
 This may require us to recalculate the value of 7, which we 
 do by eq. (21) and find 7’= 700.8 feet log 2.845596. If the 
 other elements are required, they may be calculated by eqs. 
 (22), (23), (24), or directly from 7’ and A, as follows: 
 
 96. To find the External distance LH, in terms of the 
 Tangent-distance and Central Angle. 
 
 In Fig. 5 we have given 
 
 AOB = A. and.AV = 7 stone 
 HV=&. In the diagram draw 
 the chord AH, and through H draw 
 a tangent line to intersect OA pro- 
 duced in J, and join VZ. 
 
 Then HI is parallel to BA, and 
 since HT = AV= 7, and Al= HV 
 — BH, VI is parallel to HA, and 
 Vit = STAB= tet Tababasy 
 In the right-angled triangle VHI we have 
 
 Fe. 5. 
 
 HV = HI x tan VIH 
 or H= Ttan}fa (26) 
 
 Erample.—The angle at the vertex being 54° and the tan- 
 gent-distance 700.80 feet, how far will the curve pass from 
 the vertex ? 
 
 T = 700.80 (from last example) 2.845596 
 A = 54°, 4A = 138° 30’ log tan 9.880354 
 
 foo] 
 
 Ans. H = 168.25 feet log 2.225950 
 
 (For the formule by which to find the long chord and mid- 
 dle-ordinate in terms of the tangent-distance and central angle, 
 see Table III. 12 and 13.) 
 
 97. Again, it may be necessary to assume the eaternal dis- 
 tance in order to determine the proper degree of curve. 
 
 To find the Radius and Degree of Curve in terms of 
 the External distance and Central Angle: 
 
 By eq. (24) 
 
 R238 @) 
 
 ex sec dA 
 
SIMPLE CURVES. 
 Otherwise: 
 In Table VI. divide the external distance of a 1° curve, 
 opposite the given valuc of A, by the assumed -external dis- 
 tance; the quotient is the degree of curve required. 
 Hrample.—The angle at the vertex being 24° 30’, the curve is 
 desired to pass at about 65 feet from the vertex. What is the 
 proper degree of curve ? i 
 
 E=65 log 1.812913 
 
 A = 24°30', 4.4 = 12° 15’ log ex sec 8.367345 | 
 eRe 3.445568 | 
 
 ans, By Table IV. D = 2° 03’ + i 
 
 Otherwise: 
 _ By Table VI. . 65)133.50 
 Ais: D= 2° 03’ 14” 2° .0538 
 
 We may therefore assume a 2° curve, unless required by 
 the circumstances to be more exact, when we might use a 
 2° 03' curve. Assuming a 2° curve, we have by eq. (24) 
 
 E = 68.75 log 1.824460 
 
 Having decided on the degree of curve, we may calculate 
 the remaining elements by eqs. (21), (22), (23), which is always 
 the better way, but we may calculate them directly from # 
 and A. 
 
 98. To find the Tangent-distance in terms of the 
 External distance and Central Angle: 
 
 From eq. (26), and by Table IT. 40, 
 
 T = E cotta (28) | 
 
 t 
 | 
 Example.—The angle at the vertex is 24° 80’, and the curve i 
 passes 66.75 feet from the vertex. How far are the tangent | 
 points from the vertex ? 
 
 i = 66.75 (from last example) log 1.824460 
 
 S 
 
 A = 24° 30', $A =6° 0730" _ log cot 0.969358 
 Ans. T = 622.04 feet 2.793818 
 
 99. Remark.—Kqs. (27) and (28) are particularly useful in 
 defining the curve of a railroad track where all original 
 
on FIELD ENGINEERING. 
 
 points are lost. Produce the centre lines of the tangents of 
 the curve to an intersection V, and there measure the angle A. 
 Bisect its supplement AVA, and measure the distance on the 
 bisecting line from V to the centre line of the track. This 
 will give VH= H. Then 2 and 7 may be calculated, and the 
 distance 7’ laid off from V on the tangents, giving the tangent 
 points A and B. 
 
 (For the formule by which to find the long chord and mid- 
 dle-ordinate in terms of # and A, see Table III. 16 and 17.) 
 
 100. Again, having only the central angle given, we may 
 assume the long chord, or the middle-ordinate, and from either 
 of these and the central angle calculate the remaining ele- 
 ments. Or, finally, the central angle being wnknown, we may 
 suppose any two of the linear elements given, and from these 
 calculate the rest. As such problems have little practical 
 value, their discussion is omitted. The requisite formule for 
 their solution are given in Table III., and the verification of 
 them is suggested as a profitable exercise to the student. 
 
 Bb. Location of Curves by Deflection Angles. 
 
 101. In order that the stakes at the extremities of the 
 100-foot chords, by which the curve is measured, shall be set 
 exactly on the arc of the curve 
 by transit observation, it is neces- 
 sary at the point of curve, A, to 
 deflect certain definite angles 
 from the tangent. AV. Let us 
 suppose that in the curve AB, 
 Fig. 6, the points A, a, 0, c, d, 
 etc., indicate the proper posi- 
 tions of the stakes 100 feet apart, 
 and that OA is the radius of the 
 curve. In the diagram join Oa, 
 Ob, ete.,,and also Aa, ab, be, ete. 
 Then, by definition, the angle AOa=D, and by Geom. 
 (Tab. I. 20 and 11) the angle VAa = 4D. ‘Therefore if 
 we set the transit at A, and deflect from AV the angle 
 4D, we shall get the direction of the chord Aa, on which by 
 measuring 100 feet from A we fix the stake, a, in its true 
 position on the curve. So again, since the angle a@0d, at 
 the centre, = D, the angle qAd, at the circumference, = 1D. 
 
 Fie. 6. 
 
SIMPLE CURVES. 
 If therefore, with the transit at A, we deflect the angle 3D 
 from the chord Aa, we shall get the direction of the chord 
 Ab; and when the stake d is on this chord it will also be on 
 the curve, if 6 is 100 feet distant from a. Thus, in general, 
 we may fix the position of any stake on the curve, by deflect- 
 ing an angle 1D from the preceding stake, and at the same 
 time measuring a chain’s length from it,—the chain giving 
 the distance, while the instrument at A gives the direction of 
 the point. 
 
 1D) is called the Deflection-angle of the curve; so that in 
 any curve, the deflection-angle is equal to one half the degree of 
 curve. 
 
 102. Since each additional station on the curve requires 
 an additional deflection-angle, the proper deflection to be made 
 at the tangent point from the tangent to any stake on the 
 curve is equal to the deflection-angle of the curve multiplied 
 by the number of stations in the curve up to that stake; or it 
 is equal to one half the angle at the centre subtended by the 
 included arc of the curve. 
 
 103. It may happen that all the stations of a curve are not 
 visible from the tangent point, A. When this is the case a 
 new transit-point must be prepared at some point on the 
 curve, by driving a plug and centre in the usual manner, and 
 the transit moved up to it. Let us suppose that the point d, 
 Fig. 6, has been selected for a transit-point, and that the 
 transit has been set up over it. Before the curve can be run 
 any farther, it is necessary to find the direction of a tangent to 
 the curve at the point d. Forthis purpose we deflect from 
 chord dA an angle Adz equal to the angle V-Ad previously 
 deflected to fix the point d. (Tab. 1.16.) Or we may adopt the 
 following 
 
 Rule: To find the direction of the tangent to a 
 curve at the extremity of a given chord, deflect from the chord an 
 angle equal to one half the angle at the centre subtended by the 
 chord. (Tab. 1. 20.) 
 
 Having thus found the direction of the auxiliary tangent 
 zdz, we proceed to deflect from dz, (3D). for the next station e¢, 
 2 (LD) for station f, 3(4D) for station g, etc., as before. When 
 the end of the curve is reached, a transit-point is set at the 
 Point of Tangent, after which it only remains to find the 
 direction of the tangent, by the above rule. Thus if g is to be 
 
AA. FIELD ENGINEERING. 
 
 the point of tangent, we obtain the direction of the tangent by 
 deflecting from the chord gd an angle equal to zdy, or to 
 x 40g. If this tangent VB was already established, the line 
 ge thus obtained should coincide with it; and if it does so, 
 the correctness of our work is proved. 
 
 104. The centre line is measured, and the stations num- 
 
 ered regularly and continuously through tangents and 
 curves from the starting point to the end of the work. It 
 therefore frequently happens that a curve will neither begin 
 nor end at an even station, but at some intermediate point, or 
 plus distance. 
 
 If the Point of Curve occurs a certain number of feet 
 beyond a station, the first chord on the curve is composed of 
 the remaining number of feet required to make 100. 
 
 Any chord less than 100 feet is called a subchord. 
 
 If a curve ends with a subchord, the remainder of the 100 
 feet must be laid off on the tangent from the Point of Tangent 
 to give the position of the next station, so that the stations 
 may everywhere be 100 feet apart. 
 
 105. The deflection to be made for a subchord is equal to one 
 half the arc it subtends, 
 Let c = length of any subchord in feet. 
 “ad = angle at centre subtended by subchord. 
 Then, from eq. (22), by analogy 
 
 ¢ =2R sin id (29) 
 100 
 ; Oy? pt Meee 
 But by eq. (16) EE sinteD 
 sin 1d 
 6 = 100 (30) 
 - sindd ——°~ gin 1D (31) 
 i 100 3 
 
 When D does not exceed 8° or 10°, we may assume without 
 serious error that the angles are to each other as their sines, 
 and the last two equations become 
 
 (approx.) e = 100 (82) 
 
 SI a 
 
s i 
 
 SIMPLE CURVES. 
 
 and id = —— (iD) (33) 
 
 In curves sharper than 10° per station, the error involved in 
 this assumption becomes apparent and must be corrected. 
 
 106. If curves were measured on the actual arc, then 
 eqs. (82) and (33) would be true in all cases; but since a curve 
 is ‘measured by 100-ft. chords, it is evident that if a 100-ft. Hl) 
 chord between any two stations were replaced by two or more | 
 subchords, these taken together would be longer than 100 feet, 
 since they are not in the same straight line. Let us conceive 
 the actual arc of one station to be divided into 100 equal 
 parts; since the arc is longer than the chord, each part will be ni 
 slightly longer than one foot. Now if we take an arc contain- 
 ing any number of these parts (less than 100), the nominal 
 length of the corresponding subchord in feet will egual the 
 number of parts, and the deflection for the subchord will be 
 proportional to the number of parts which the arc contains. Mi il 
 The deflection therefore will be exactly given by eq. (83) if in Ali 
 that equation we let c equal the number of parts in the arc, or 
 the nominal length of the subchord in feet. Having thus 
 obtained the correct value of ($d), we may introduce it into 
 eq. (29) or (80), and obtain the ¢rwe value of the subchord, 
 which will always be a little greater than its nominal value. . 
 
 Suppose, for instance, that the arc of one station is to be wat 
 divided into four equal portions; then each subchord will be , 
 
 29 nia 
 
 nominally 25 feet long; and by eq. (33 
 
 95 | Hl 
 
 $d = GD) =2 GD) a 
 
 which is the correct value of the deflection, whatever be the 
 degree of curve. Substituting this value in eq. (29) or (380) we i 
 obtain the true value of the subchord, ¢, a little greater than AWA 
 25; the eacess is called the correction of the nominal length. 1 
 107. This correction for any given subchord bears an Hy 
 almost constant ratio to the excess of are per station, what- | | 
 ever be the degree of curve. These ratios are shown in the WH 
 following table for a series of subchords, and Table VII. gives | 
 the length of actual are per station for various degrees of 
 curve. Subtracting 100 we have the excess of arc per station, 
 and multiplying this evcess by the ratio corresponding to the 
 
56 FIELD ENGINEERING. 
 
 nominal length of subchord we obtain as a product the proper 
 correction for the subchord. 
 
 TABLE OF THE RATIOS OF CORRECTIONS OF SUBCHORDS TO 
 THE EXCESS OF ARC PER STATION. | 
 
 Nominal | Nominal Nominal ; 
 
 Length of Ratio. || Lengthof| Ratio. || Length of | Ratio 
 
 ’ Subchord. | | Subchord. Subchord. | 
 —_____! a a { 
 
 0 .000 35 .207 70 .3806 
 
 5 .050 40 .oa5 (45) Ry 
 
 10 1099. .o4f) SU4R That oer papeei> | G0 STs Nad [ogy 
 
 15 .147 50 .Ot4 85 .205 
 
 20: .192 55 | .083 90 .169 
 
 25. .234. 60 .883 95 | .092 
 
 30. eels 65 374 100 .000 
 
 We observe that the largest correction is required by a sub- 
 chord between 55 and 60 feet in length. 
 
 Example.—It is proposed to run a 14° curve with a 50-ft. 
 chain. What correction must be added to the chain? 
 
 = ig a — 7° By _ 90 Mo 90 & __ g0 arp 
 D== 14 iD=% Mie (~ = 3°.5 = 8° 80 
 By eq. (30) 
 6100 ee nes 
 sin 7 
 
 Ans. Correction = .093 
 
 Or, by Table VIL., length of arc = 100.249 
 
 excess of arc = 249 
 and by above table, ratio for 50 feet = .Bt4 
 Ans. Correction = product = .093 
 
 Example.—The P.C. of an 18° curve is fixed at + 55 feet 
 beyond a station. What are the nominal and true values of 
 the first subchord, and what the proper deflection? 
 
 Nominal value = 100 — 55 — 45 feet 
 
 ; 45 s 
 Deflection = id= 100 xX 9° = 4°.05 — 4° 03’ 
 and by eq. (80) 
 True value = c = 100 oe ee 
 
 she Uda 45.148 
 
Gq 
 
 SIMPLE 
 
 CURVES. 
 
 Or, by Table VII., excess of are= _ _—-. 412 
 by above table, ratio for 45 feet = _.358 
 Correction = product = __.147 
 
 Ans. True value of subchord = 45.147 
 
 Example.—The last deflection at the end of a 40° curve is 
 found to be 6° 80’. What are the nominal and true values of 
 
 the last subchord? 
 Here 4d = 6° 30’, and by eq. (82) A AT We 
 
 0: _ 38.5 feet | 
 20 
 : ° an i 
 
 Mrugrpabios «100 Tose 89 1008 feat Wi 
 
 sin 20° wip 
 
 Nominal value, ¢ = 100 
 
 Or by Table VILI., excess of arc 40° = 2.060 
 by above table, ratio for 32.5 feet = » .290 
 Correction = product = _ .597 
 
 Nominal value of subchord = 82.5 i i 
 True value = 33.097 Wi 
 
 108. For convenience in making deflections, the zeros of | 
 the instrument should always be together when the line of at 
 collimation coincides with a tangent to the curve. Thus, in 
 beginning a curve, the transit being set at the P.C. zeros | 
 together, and line of collimation on the tangent, the read- iI 
 ing of the limb for any station on the curve has simply to be | 
 made equal to the proper deflection from the tangent for that Hal | 
 station. After the transit is moved forward from the P.C. NG so 
 and set at another point of the curve, the vernier is set toa AVA 
 reading equal to the reading used to establish that point, but | 
 on the opposite side of the zero of the limb, and the line of A | 
 collimation is set on the P.C. just left. Then by simply turn- Hi 
 ing the zeros together again, the line of collimation will be an | 
 made to coincide with a tangent to the curve through the new i 
 point, and the deflections for the succeeding stations can be Pie 
 read off directly, as before. Thus any number of transit | 
 points may be used in locating a curve by finding the direc- Wh | 
 tion of the tangent through each by a deflection from the pre- 1 
 ceding point, until finally the P.7. is reached, where another 
 deflection gives the direction of the located tangent. 
 
FIELD ENGINEERING. 
 
 109. The assistant engineer keeps neat and systematic 
 field-notes of all his operations with the transit in running 
 curves. The numbers of the stations are written in regular 
 order up the first column of the left-hand page of the field- 
 book, using every line, or every other line, as may be pre- 
 ferred. The second column contains the initials of each 
 transit point on the same line as the number of its station, or 
 between lines, if the point occurs between two stations, In 
 the third column, and opposite the initials in the second, is 
 recorded the station and plus distance, if any, of each transit 
 point. The fourth column contains, opposite the ““P.C.,” the 
 degree of curve used, and an R or Z, showing whether the 
 curve deflects to the right or left; the fifth column contains 
 the readings or deflections made from a tangent to set each 
 Station or point, written on the same line as the number of 
 that station or point; and the sixth column contains the cen- 
 tral angle of the whole curve, A, written opposite the ‘‘ P. 7.” 
 The plus distances recorded in 
 the third column are always the 
 nominal lengths of subchords, but 
 if the true lengths have been calcu- 
 lated and laid off on the ground, 
 these should also be recorded in 
 parenthesis. On the right-hand 
 page are recorded the calculated 
 bearings of the tangents and their 
 magnetic bearings; and on the 
 centre line of the page, opposite 
 
 Fia. 7. the record of each transit point, a 
 
 dot is made with a small circle 
 
 around it, to show the relative position of the several points 
 on the ground. Some slight topographical sketches may be 
 made, indicating the more prominent objects, but the full 
 sketches should be taken by the topographer in a separate book. 
 
 110. Since the deflections start from zero at each new 
 transit point, the sum of the deflections by which the transit 
 points are located will be equal to one half the central 
 angle of the curve. 
 
 111. The stations on a curve may be located by deflee- 
 tions only, without linear measurements. For this purpose 
 two transits are set at two transit points on the curve, as A 
 
SIMPLE CURVES. 5Y 
 
 and B, Fig. 7, and the proper deflections for any station are 
 made with both instruments, the station being located by find: 
 ing the intersection of the two lines of collimation. 
 
 This method requires that the two transit points shall have 
 been previously established, that their distance from each 
 other shall be known, that they shall be visible from each 
 other, and that they shail both command a view of the stations 
 to be located. It is not therefore generally useful, but may 
 be resorted to to set stations which fall where chaining cannot 
 be accurately done, as in water or swamps. The chord join- 
 ing the two transit points becomes, in fact, a base-line, and the 
 deflections form a serics of triangulations. 
 
 C. Location of Curves by Offsets. 
 
 112. A curve may be located. by linear measurement only, 
 without angular deflections. There are four general methods, 
 Viz. : 
 
 By offsets from the chords produced, 
 By middle-ordinates, 
 
 By offsets from the tangents, and 
 By ordinates from a long chord. 
 
 To locate a curve by offsets from the chords 
 produced. 
 
 When the curve begins and ends at a station. 
 
 1138. Let A, Fig. 8, be the P. C. of a curve taken at a station, 
 to locate the other stations, a, 0, ¢, 
 etc. The chords Aa, ab, be, etc., 
 each equal 100 feet, and since the 
 angle AOa = D, the angle VAa = 
 4D. (Tab. I. 20.) Taking an off- 
 set ax =t, perpendicular to the 
 tangent, we have in the. right- 
 angled triangle Aza. 
 
 az = Aa X sin }D 
 or 
 
 t =100 sin 4D (34) 
 The offset ¢ is called the tangent 
 offset, and its value is givenfor all 
 degrees of curve in Tab. IV. col. 4. F1a. 8, 
 If the curve were produced 
 backward from A, 100 feet to station 2, the offset zy would 
 
60 FIELD ENGINEERING. 
 
 equal ?t; and if the chord zA were produced 100 feet from A 
 to a’, the offset a’ would also equal t. Therefore the distance 
 aa’ = 2t,and the angle aAa' = D. So if we produce the chord 
 Aa 100 feet to 0’, the distance bd’ = 2¢. 
 
 To lay out the curve, stretch the chain from A, keeping the 
 forward end at a perpendicular distance, t, from the line of the 
 tangent to locate station a. Then find the point 0’ by stretch- 
 ing the chain from @ in line with a and A, and then stretching 
 the chain again from a, fix its forward end at a distance from 
 b’ equal to 2¢. This gives station b. In the same way find 
 other stations. 
 
 When the last station, as d, of the curve is reached, produce 
 the curve one station farther toe’. Then the tangent hee 
 d is parallel to the chord ce", and laying off ¢ from ¢ and e" per- 
 pendicular to this chord, the tangent c’e is found. If the work 
 has been correctly done the tangent c"e will coincide with the 
 given tangent VB. 
 
 When the curve begins or ends with a subchord, 
 
 114. Let ., Fig. 9, be the PC, and Ag the first sub- 
 chord = ¢, and the angle VAa = 24, and let the offset aw = t,. 
 Then 
 
 t; = ¢ sin 4d (35) 
 
 Producing the curve backward to the nearest station 2, we 
 have another subchord Az = (100 — ce), and the angle y.Az = 4 
 (D — d), and putting the offset yz = t, 
 
 ?, = (100 — c) sin } (D— d) (86) 
 
 Laying off the two subchords on the ground, and making 
 the proper offsets, ¢, and ¢,, at the 
 same time, we fix the position of 
 the two stations @ and z on the 
 curve ; after which we may pro- 
 duce the chord 2a 100 feet to b, 
 and proceed as before until the 
 curve is finished. 
 
 If the curve ends with a sub- 
 chord, as dB, produce the curve 
 to the first station beyond B, as 
 - e", then calculate the two offsets 
 for the two subchords Bd and Be", 
 and lay them off from d and e" 
 
SIMPLE CURVES. 
 
 perpendicular to the supposed direction of the tangent. If 
 the line de so obtained coincides with the given tangent, VB, 
 the work is correct. 
 
 115. We may find the values of ¢ and ¢, otherwise than 
 by the formule above, for in Fig. 8 we have shown that the 
 angle aAa' = aOA, and since these triangles are isosceles, 
 they are similar; therefore 
 
 Fig. 8, OA: Aa:: Aa: aa’ | 
 or FR: 100::100 : 2¢ A | 
 _ (100) | 
 5 Fiore en 
 and similarly, Fig. 9, | 
 ce 
 
 ; = —— ( 
 
 iso Bp (38) 
 Hence 
 
 Lee e= (1009 
 roe. et 39 
 = “q00P (39) 
 
 Thus ¢, may be found by multiplying the square of the sub- ae 
 chord by the value of ¢ given in Tab. LV., and dividing the At 
 product by 10000. As c is always less than 100, so ¢, is always 
 less than ¢. 
 
 116. In eqs. (85), (88), and (89) it is customary to use the 
 nominal values of c, and this can produce no error in ¢ or ¢, 
 exceeding -005, when the degree of curve does not exceed ten 
 degrees. In the case of a very sharp curve, the formule eqs. 
 (40) and (41) are preferable. 
 
 To locate a curve by middle-ordinates. 
 
 When the curve begins and ends at a station. 
 
 117. In Fig. 10, let A be the P.C. at a station, and let aand 
 e be the next stations on the curve either way from A. Then, 
 since zy = aw = ft, the chord za is parallel to the tangent A J, 
 and Ag =—¢. Hence, having any two consecutive stations on 
 the curve, as 2 and A, we may lay off the tangent offset ¢ 
 from A to gon the radius, and find the next station, a, 100 feet Pe 
 from A on the line zg produced. Then laying off ai = ¢ on Wt 
 the radius @O, a point on the line Ah produced and 100 feet | 
 from @ will be the next station 4. 
 
62 FIELD ENGINEERING. 
 
 On reaching the end of the curve, the tangent is found 
 precisely as described in the method by chords produced, § 113. 
 In Fig. 10, we observe that if the radius 0A were unity, gA 
 would be the versed sine of the angle a~0A = D.. But GA oat, 
 
 hiss Rovers iD (4()) 
 
 When the curve begins or ends with a subchord, 
 118. Let A, Fig. 11, be the C,, and @ and z the nearest 
 
 Fia. 10. Fie. 11, 
 
 stations. Then Aa = ¢, the first subchord, and a@0A = d, and 
 by analogy, we have from the last equation, if aw = t, and 
 ey = t, 
 
 ¢, = R vers d 
 
 t, = & vers (D—d) § (41) 
 
 or eq. (389) may be used if preferred. 
 
 Having found the two stations, a and z, on the curve, lay 
 I off from the forward station a, ad = t on the radius, and go 
 | continue the curve as described above. 
 
 i When the end of the curve is reached, produce the curve to 
 the next station beyond, and find the tangent by offsets as 
 described in the previous method, § 114. 
 
 To locate a curve by offsets from the tangents. 
 When the curve begins at a station. - 
 
 119. Let A, Fig. 12, be the PC. at a station. Then the 
 next station @ is located by the tangent offset t, taken from 
 
SIMPLE CURVES. 63 
 
 Tab. IV., or calculated by eq. (40). To calculate the distance 
 anc offsets for the following stations, J, c, etc., in the diagram 
 draw lines through the points 4, ¢, etc., parallel to the tangent 
 AJ, intersecting the radius AOing’, g", etc., and draw the 
 lines bz’, cz", etc., perpendicular to the tangent. 
 
 Then 
 Az’ = g'b = Ob sin BOA 
 
 or 
 
 Az’ = Ff sin 2D) 
 
 Az" = Rsin 3D (42) 
 and etc. etc. J 
 Also, 
 
 bz’ = g' A = Ob vers. DOA 
 
 or 
 
 t RR vera 2D 
 
 i" = & vers 3D} (43) 
 and etc. atCiins | 
 
 But these calculations may be avoided, for as twice ag equals 
 the chord of two stations, so twice dg’ equals the chord of four 
 stations, and twice cg" the chord 
 of six stations, etc. So also as Ag 
 is the middle-ordinate of two sta- 
 tion, Ag’ is the middle-ordinate of 
 four, and Ag” the middle-ordinate 
 of six stations, etc. Hence the 
 rule: 
 
 The distance on the tangent from 
 the tangent point to the perpendicu- 
 lar offset for the extremity of any 
 are is equal to one half the long 
 chord for twice that arc; and the 
 offset from the tangent to the, ex- 
 tremity of any are ts equal to the 
 middle-ordinate of twice that are. 
 
 The long chords and middle-ordinates may be taken from 
 Tables VIL. and VIII. for 2, 4, 6, 8, etc., stations, when the 
 P.G. is at a station, or for 1, 3, 5, 7, etc., stations, when the 
 P.O. is at + 50, or half a station. 
 
 If the offsets from the first tangent A V prove inconveniently 
 long, the second half of the curve may be located from the 
 other tangent BV, beginning at the point of tangent B, and 
 closing on a station located from the first tangent. 
 
 Fie, 12. 
 
64 FIELD ENGINEERING. 
 
 When the curve begins with a subchord. 
 120. If d=the angle at centre, subtended by the first 
 subchord, we have for the distances on the tangent (Fig. 13) 
 
 Az = RFR sind 
 Az = Rsin d+ D) 
 
 (44) 
 Ax" = Ff sin (d + 2D) 
 ete, etc. 
 and for the offsets (Fig. 11) 
 t, = vers d 
 ¢ = Rvers d+ D) t (45) 
 t" = FR vers (d + 2D) 
 etc. ete, 
 
 If the first subchord equals 50 feet (nominal), then d =—1D, 
 and the Tables VII. and VIII. may be used as explained 
 
 Fie.13; Fie. 14, 
 
 above. These tables may be used in any case, by adopting a 
 temporary tangent through any station, and laying off the dis- 
 tances on this, and making the offsets from it. 
 
 When a curve is located by offsets the chain should be car- 
 ried around the curve, if possible, to prove that the stations 
 are 100 feet apart. , 
 
 To locate a curve by ordinates from a long 
 
 chord. 
 
 When the curve begins and ends at a station. 
 
 121. In Fig. 14 draw the long chord AB, joining the tan- 
 gent points, and from this draw ordinates to all the stations on 
 
i 
 
 SIMPLE CURVES. 65 
 
 the curve. We then require to know the several distances on 
 the long chord Aa’, at’, b'c’, etc., and the length of ordinate 
 at each point. 
 
 Let C =the long chord AB, then eq. (22 
 
 C= 2R sinta 
 
 If a is the second station and 7 next to the last on the curve, 
 join az, and let the chord aa = C’. Then since the arc Aa = 
 tk = D, the angle at the centre subtended by C’ is (A — 2D). 
 
 *. MO=2KR sini (a — 2D) 
 Again, if we join } and / the next stations and let b4= C" 
 C" = 2K sin 4 (A — 4D) 
 and so on for other chords. 
 Since Aa’ = ki, C= C’ + 2Aa’ 
 C—C 
 2 
 
 =o Ag = 
 
 Similarly, : 
 a b= aaa siked 
 2 
 
 Thus we continue to find the distances up to the middle of 
 the curve, after which they repeat themselves in inverse 
 order. 
 
 122. When the long chord C, subtends an even number of 
 stations (as 10 in Fig. 14), the middle ordinate of the chord is 
 the ordinate of the middle station, ase. Since the chords AB 
 and a# are parallel, the ordinate a'a or 7't is evidently equal to 
 the difference of the middle ordinates of these chords. 
 
 Let M, M’, M", etc., be the middle-ordinates of the chords 
 CU, CU’, CO", etc. Then eq. (23) 
 
 Me = vers A 
 M' = Rvers§ (a —2D) 
 M" = Rvers 4 (A —4D) 
 
 etc., etc. 
 And aa=ti=—=M-M 
 
 b'd a h'h = MW — M" 
 
 etc. ete. etc. 
 
 The values of the chords and middle-ordinates may be taken 
 at once from Tables VII. and VIII. 
 
66 FIELD ENGINEERING. 
 
 Ezxample.—It is required to locate a 4 degree curve of ten 
 stations by offsets from the long chord. 
 By Table VIL: 
 
 Diff. VDiff. 
 10 sta. | GC =980.014 | 
 : te Oh ap apa. | 1e0eaid | 95.205 =a 
 eee) 194.059 | 97.080 2 ab = Th! 
 6 « Oi = 595.744 is Ok 
 £8 Crit 298. 7g0 | 196-962 | 98.481 = de = Hg 
 ee rE a Aas 198.904 | 99.452 =e'd' =g'f 
 ee a 199.878 99.939 = d'e' = fre’ 
 0 | OG =000.000 | t 
 From Table VIIL.: 
 Diff. 
 10st. | M =86.402 | | 
 g « | Mi =55.500 | 30.902 =aa=v 
 6 « M* =81.308 | 55.094 —)'b —Wh 
 4 « | Mi = 18.948 | 72.459 —cc =9'g 
 Q « | Miv— 3.490 | 82.912 —da= fF 
 Q ‘ pea = 202000 S| 5/88; 40 hs aie 2 
 
 123. When the long chord C subtends an odd number of 
 stations, the middle ordinate will fall half-way between two 
 stations, and need not be laid off. 
 
 If the ordinates near the middle of the curve prove incon- 
 veniently long, we may subtract 17 — M’, M—M", etc., and so 
 obtain in Fig. 14 @’a, 6"b, ec, etc. We then lay off Aa’, aa, 
 ab", b"b, be", etc., turning a right angle at every point. The 
 chain should be carried along the curve at the same time to 
 make the stations 100 feet apart. 
 
 Example.—It is required to locate a 10-degree curve of nine 
 stations by offsets from the long chord. 
 
 By Table VII. : 
 
 Diff. MDiff. 
 9 sta. 811.814 
 7 «© 658.105 153.209 76.604 = Aa’ 
 5 484 900 173.205 86.603 = ab’ 
 8 «' 996 969 187.938 93.969 ete. 
 1 “ 400.000 196.962 98.481 
 0 9.000... | 100.000 50.000 
 
4 
 
 SIMPLE CURVES. 
 
 By Table VIIL.: 
 
 Diff. 
 @ sta, 168.029 bie tts 
 Saleen 50.000 =U) 
 5 53. 80 | 34.202 = ¢"e 
 Be rae TINORS oy 17.365 ete, 
 1 2.183 | 4 Ha 
 0 0.000 
 
 124. The tables can be used equally well when the curve ill 
 both begins and ends with a half station; also to locate 
 half-station points throughout the curve, but in the latter case 
 the numbers are taken from consecutive columns of the tables . 
 instead of from alternate col- ah 
 umns, as in the above examples. 
 
 When the curve begins or ends 
 with any subchord. 
 
 125. Let A, Fig. 15, be the 
 P.@%. and Aa=e the first sub- 
 chord, and d the angle it sub- 
 tends at the centre. In the dia- 
 gram draw the long chord AB, 
 and the ordinates to each sta- 
 tion, and through each station 
 draw a line parallel to AB, and 
 let AOB= A. 
 
 Since the angle VAB = 4A and Hi 
 VAa = id, theangle aAB=4(A—d). The deflection angle li 
 from the subchord Aa produced to the chord ad is (d+ D), te 
 the deflection angle between any two consecutive chords of i 
 
 i 
 | 
 
 100 feet is }(D+D)=D. Therefore the angle ae 
 bab’ =4 (A — ad) — 4+ D) =F (a —2d-D) 
 che” = 4 (A — 2d —D) — 3 (2D) =4 (a — 2d — 8D) 
 edd’ = 4(A — 2d —8D) — + QD) =4(A — 24—5D) 
 
 ete. 
 
 etc. etc. 
 
FIELD ENGINEERING. 
 
 Solving the several right-angled triangles we have, Fig. 15. 
 
 Ad@=c. cos4(A —d@) ) 
 
 ab" = 100 cos4(A —2d— D) | 
 
 be" = 100 cos (A — 2d— 8D) } | (46) 
 dd" = 100 cos i (A — 2d — 5D) 
 
 etc., etc., | 
 
 And also 
 
 wa=c. sin¢(A —a@) 
 
 b"b = 100 sint¢ (A —2d— D) 
 
 ¢ “ = 100 sin} (A — 2d — 3D) (47) 
 
 = 100 sin 4 (A — 2d — 5D) 
 BAL ete. 
 
 When the middle point of the curve is passed the minus 
 quantities in the parentheses become greater than A, making 
 the parentheses negative, and, therefore, the sines negative, 
 and indicating that such values as are determined by them 
 ii must be laid off toward the long chord AB. 
 | By a proper summation of the quantities determined by eqs. 
 (46) and (47) we obtain the distances Aa’, Ab’, Ac’, etc., and 
 the ordinates a‘a, b'b, c'c, etc., and the curve may be located 
 accordingly. It is well to make all the necessary calculations 
 before beginning to lay down the lines on the ground, thus 
 avoiding confusion and mistakes. 
 
 Example.—The P.C. of a 3° 20' curve is fixed at + 25 feet 
 beyond a station, and the central angle is 16° 24’= a. It is 
 ' required to locate the curve by ordinates from the long chord. 
 i We have c = 100 — 25 = 75 and d= 2° 30’ and D= 3° 20’. 
 Ti Hence, eqs. (46) 
 
 1 Aa'= %5 cos 6° 57 = 74.449 74.449 = Aa’ 
 ir ab" = 100 cos 4° 02' = 99.752 174.201 = Ad’ 
 be" = 100 cos 0° 42’ = 99.993 274.194 = Ae' 
 a'd = 100 cos (— 2° 38’) = 99.894 374.088 = Ad’ 
 e"e = 100 cos (— 5° 58’) = 99.458 473.546 = Ae’ 
 éB= 17 cos (— 7 55’) = 16.8388 490.384 = AB 
 
 By eqs. (47) 
 
 va= Hsin 657 = 9.075 9.075 = wa 
 6"6 = 100 sin 4° 02’ = 7.034 16.109 = b'b 
 e"c = 100 sin 0° 42’ = 1.222 17.331 = ce 
 ca" = 100 sin (— 2° 38) = — 4.594 12.737 = dd 
 de" = 100 sin (— 5° 58’) = — 10.395 2.342 = ee 
 
 ee’ = 17 sin (— Te 55’) ame 2.341 0.000 eo oe e 
 
q 
 
 SIMPLE CURVES. 69 
 
 The same formulz can be used when the curve begins at a 
 station by making ec = 100 and d= D. 
 
 126. The methods of locating curves by linear measure- 
 ments do not require the use of a transit, although one may 
 be used to advantage for giving true lines, turning right 
 ungles, etc. When a transit is not used the alignments should 
 be made across plumb-lines suspended over the exact points 
 previously marked on top of the stakes. A right angle 
 may easily be obtained, without an instrument, by laying off ATR 
 on the ground the three sides of either of the right-angled 1 
 triangles represented in the following table (or any multiples A a 
 of them), always making the base coincide with the given line. i 
 
 TABLE OF RigHt-ANGLED TRIANGLES. Ht 
 
 Base. Hypothenuse. Perpendicular. i 
 4 5 3 He 
 12 13 5 | 
 20 29 21 
 24 25 7 
 40 Al 9 
 60 61 11 
 84 85 13 
 
 D. Obstacles to the Location of Curves. 
 
 127. To locate a curve joining two tangents when the in- 
 tersection Vis inaccessible. Fig. 16. 
 
 From any transit point p on one tangent run a line pq to i 
 intersect the other tangent; measure Hl 
 pq and the angles it makes with the 
 tangents. Then the sum of the de- 
 flections at p and g equals the central 
 angle A. Solve the triangle pyV 
 and find Vp. Having decided on 
 the radius R of the curve, calculate 
 the tangent distance VA by eq. (21), 
 and lay off from p the distance 
 pA == VA — Vp to locate the point 
 of curve. The point p being as- 
 sumed at random, Vp may exceed VA, in which case the differ- itt 
 ence pA is to be laid off toward V. 14 
 
 In case obstacles prevent the direct alignment of any line | 
 pg, a line of several courses may be substituted for it (as 
 
 Fic. 16. 
 
7U0 FIELD ENGINEERING. 
 
 explained in §§ 46, 47, 48,) from which the length of pq will 
 be deduced. The algebraic sum of the several deflections will 
 equal A. 
 
 . 128. To locate a curve when the point of curve is 
 inaccessible. Fig. 17, 
 Assume any distance Ap on the curve which wil] reach to 
 an accessible point p. Then by eq. (19) the angle 
 Dx Ap 
 100 
 Ap' = R sin pOA 
 pp = Rvers pOA 
 Vp' = VA — Ap’ 
 
 pOoA = 
 
 Measure Vp’ and p’'p to locate a transit point at p; and meas. 
 
 ure an equal offset from some transit point on the tangent, as 
 
 q7. This gives a line pq’, parallel 
 
 B to the tangent, from which deflect at 
 
 : p an angle equal to pOA for the 
 
 direction of a tangent through the 
 
 point p. 
 
 f, Instead of measuring the second 
 Ip ' 
 
 offset gq’ we may deflect from pq an 
 
 U 
 
 angle found by tan gpq’ = i and so 
 
 BAS 
 
 obtain the line pg’ parallel to the 
 
 Fic. 17. tangent. Or we may deflect from pV 
 the angle found by tan p Vp’ =F to obtain the line q'p pro- 
 
 duced, from which the tangent to the curve at p is found as 
 above. 
 
 Again, we may lay off from V, the external distance VA 
 found by eq. (24) or Tab. VI on a line bisecting the angle 
 AVB. This gives us h, the middle point of the curve, and a 
 line at right angles to 4V is tangent to the curve at h, from 
 which the curve may be located in either direction. 
 
 129. To locate a curve when both the Vertex and Point 
 of curve are inaccessible. Fig. 18. 
 From any point p on the tangent run a line pq’ to the other 
 
iq 
 
 SIMPLE CURVES. 
 tangent, and so determine pA as in §127. Suppose the curve 
 produced backward to p’ on the perpendicular offset pp’. 
 
 Then 
 
 abe | CT, RIS 5 . 
 sin p'OA = RP and pp’ = R vers p'OA 
 
 Having located the point p’, a parallel chord p’g may be 
 laid off, giving a point g on the curve, since p'g = 2 X pA. Hh | 
 At g deflect from gp’ an angle equal to p’OA for a tangent to | 
 the curve at q.. Wit 
 
 If any obstacle prevents using the chord p’g, any other 
 
 Fia. 19. 
 
 chord as p's may be used, by deflecting from p’g the angle b 
 gp's = 4 (gQs) and laying off its length, vi 
 
 p's= 2R sin (p'OA + qp’s). ft 
 
 At 3 a deflection from the chord sp’ of (p'OA + gp’'s) will give 
 the tangent at s. 
 
 If obstacles prevent the use of any chord, the methods de- 
 scribed in $131 may be resorted to. 
 
 130. To pass from a curve to the forward tangent when the 
 Point of Tangent is inaccessible. Fig. 19. 
 
 From any transit point p on the curve, near the end of the 
 curve, run a chord parallel to the tangent. The middle point 
 g of the chord will be on the radius through the point of tan- 
 gent B. At any convenient point beyond this an offset equal 1 
 to pp’ = R vers pOB may be made to the cangent, and at 
 some other point an equal offset will fix the direction of the 
 tangent. 
 
22 FIELD ENGINEERING. 
 
 Otherwise, if an unobstructed line pq can be found inter. 
 secting the tangent at a reasonable distance from B, measure 
 the angle q'pg = pap’, and lay off the distance 
 
 r 
 
 / Pp 
 
 sin q'pq 
 to fix the point g. Then 
 Ba=p'¢ —p'B= pp’ cot q'pg — R sin pOB. 
 
 Otherwise ; assume an arc of any number of stations from 
 p to q" on the curve produced, and take the length of chord 
 from Tab. VII. Lay off pg’, and from q’ lay off g'g=R 
 vers g"OB, perpendicular to the tangent, to locate g. The 
 angle pg"¢ = 90° — q'pq", and the distance qB = R sin gq" OB. 
 
 131. To pass an obstacle on a curve. Fig. 20. 
 
 From any transit point A’ on the curve take the direction 
 of a long chord which will miss the obstacle, as A'B’. The 
 length of this chord is 2R sin 
 
 V'A'B', V'A' being tangent to the 
 
 curve at A’ (see eq. 22), and by 
 measuring this distance, the point 
 B' on the curve is obtained. If 
 the angle V'A'B' is made equal to 
 the deflection for an exact number 
 of stations, the chord may be taken 
 from Tab. VII. 
 
 If the chord which will clear the 
 obstacles would be too long for con- 
 venience, as A'g’, we may measure 
 a part of it as A’p’, and then, by an 
 ordinate to some station, regain the curve at p. The distance 
 on the curve from A' to p being assumed, the distances A'p’ 
 and p’'p are calculated by the methods given in § 121 to § 125. 
 If p'’p can be made a middle ordinate the work will be much 
 simplified. If more convenient the middle ordinate may first 
 be laid off from A’ to p", and the half chord afterwards 
 measured from p" to locate p. 
 
 Again, we may calculate the auxiliary tangent A’V’ for 
 any assumed length of curve A'B’, and lay off the distance 
 A'V’ and V'B’, deflecting at V’ an angle equal to twice 
 
 Fia. 20. 
 
SIMPLE CURVES. 
 
 V'A'B'. But if the point V’ should prove inaccessible, we 
 may conceive the auxiliary tangents to be revolved about the 
 chord A'B' as an axis, so that V’ will fall at V", and the 
 lines .4'V" and V"B' may be laid out accordingly. If these 
 in turn meet obstructions, we may run a curve from A’ to B' 
 of same radius as the given curve, but tangent to A’V" and 
 VB’. 
 
 Again, the entire curve or any portion of it may be laid out 
 by offsets from the tangents, or by ordinates from a long 
 chord, as already explained, § 119 to $126. 
 
 In case any distance on a curve must be measured by a tri- 
 angulation, as in crossing a stream, a long chord may be 
 chosen, either end of which is accessible, and the triangula- 
 tion is then performed with respect to this chord or a part of 
 it, as upon any other straight line. 
 
 SPECIAL PROBLEMS IN SIMPLE CURVES. 
 
 132. Given: a curve joining twotangents, to find the change 
 required in the radius BR, and external distance EK, for an 
 assumed change in the value of the tangent distance'T. Fig. 21. 
 
 Fia. 21. 
 
 Let.7 =AV= VB and 7’ = A'V= VB' 
 ih = AO “« R'=A'0,' 
 ery == VE. +h SV A 
 
 Then 7’— 7’ = AA' = the given change. 
 
 By eq. (20) Kk =T. cot4a 
 R'= T' cotta 
 
 0G =R—R'=(T—T')cot4a 
 
74. FIELD ENGINEERING. 
 By eq. (26), similarly, 
 HH' = H— E'=(T—T’') tania (49) 
 
 Eqs. (48) (49) give the changes in R and EF for any change 
 in 7. When 7'is increased R and # will be increased also, 
 and vice versa. 
 
 Example.—A 4° curve joins two tangents, making an angle 
 of 388° = A, and it is necessary to shorten the last tangent dis- 
 tance 80 feet. What will be the change in tne radius and in 
 the external distance? 
 
 Eq. (48) T—T'=80 log 1.908090 
 $A 19° log cot 0.463028 
 i | Ans. R —R' 232.34 log. 2.366118 
 } R 1482.69 
 Riss 1200.35 or about 4° 46’ = D’. 
 
 If the tangent distance had been increased 80 feet we should 
 add the above to R. 
 
 ft’ = 1665.03 or about 3° 26’ = D' 
 
 | Eq. (49) 7-7’ =80 log 1.903090 
 | +A 9° 30’ log tan 9.223607 
 | Ans. E—E' 18.387 log 1.126697 
 
 133. Given: a curve joining two tangents, to Jind the change 
 
 required tn the radius R, and tangent distance T, for any 
 | | | assumed change tn the value of the eaternal distance B. Fig. 21. 
 i We suppose HH’ given to find OG and AA’. 
 
 By eq. (24) H =R exsecta 
 E' = R' ex sec4+a 
 OG SR eRe 
 
 ex sec dA 
 
 By eq. (49) 
 AA = T —T' =(H— FE’) cota 
 
SIMPLE CURVES. 75 
 
 Example.—A 4° curve joins two tangents, making an angle 
 of 38° = A, and it is necessary to bring the middle point of 
 the curve 25 feet nearer the vertex V. What changes are re- 
 quired in the radius and point of curve? 
 
 Kq. (50) H-—H'= 25 log 1.397940 
 
 tA 19° log ex sec. 8.760578 
 
 Ans. R—R' 483.87 log 2.637362 
 R 1482.69 
 
 R’' 998.82 or about 5° 44’ = D’ 
 
 Eq. (51) H- £#' 25 log 1.397940 
 tA 9° 30 log cot 0.776393 
 T — T' 149.39 2.174333 
 
 or the P.C. will be moved toward the vertex 149.39 feet. 
 But if the point H, Fig. 21, were to be moved 26 feet 
 further. from the vertex V, then 
 
 R' = 1866.56 or about 38° 04’ = D' 
 
 and the P.C. will be moved 149.39 feet further from the 
 vertex. 
 
 It is preferable to assume some radius from Table IV. near 
 the value of R’ found as above, and from this calculate the 
 value of 7" by eq. (21). 
 
 134. Given: a curve joining two tangents, to find the change 
 made in the tangent distance 'T, and external distance KH, by 
 any assumed change in the value of the radius R. Fig. 21. 
 
 By eq. (48) 
 
 AA'=T—T'=(R—#') tanta (52) 
 By eq. (50) 
 
 HH'=H—H'=(R— R')ex sec4a (53) 
 
 The changes calculated by eqs. (52) (53) will be added to or 
 subtracted from 7 and # respectively, according as the radius 
 ie increased or diminished. 
 
 135. Since for a constant value of the central angle A, 
 
%6 FIELD ENGINEERING, 
 
 the homologous parts of any two curves are proportional to 
 each other, we may write at once 
 
 oon | tee nat ated ane 
 
 SECA’ iabahtt ik yeh 
 ee 2 
 
 a IT ps : ny Uae M' 
 
 Let Reet Lea ace 
 
 etc. etc. ete. 
 
 1356. Given: a curve joining two tangents, to change the 
 position of the Point of curve so that the curve may end 
 in a parallel tangent. Fig. 22. 
 
 Let AB be the given curve, AV, VB the tangents, and 
 V'B' the parallel tangent. Then VV" is the distance from 
 one vertex to the other; and since 
 there is no change in the form or 
 dimensions of the curve, we may 
 conceive it to be moved bodily, 
 parallel to the line AV, until it 
 touches the line V'’B’, when every 
 point of the curve will have moved 
 a distance equal to VV'. Hence 
 AA'=00'= BB'=VV". ‘There- 
 fore, run a line from B parallel to 
 
 Fia. 22. pe a4 Vi intersecting the new tangent in 
 
 B’, measure BB’, and lay off the dis- 
 
 tance from A to find A’. In the figure the new tangent is 
 
 taken outside the curve, and so A’ falls beyond A, but if the 
 
 new tangent were taken inside the curve at V"B", the new 
 P.C. would fall back of A at some point A’, 
 
 If the parallel tangent is defined by a perpendicular offset 
 from B, as Bp; since the angle BB'p= A 
 
 AA' = BB’ =~ (55) 
 
 137. Given: a curve joining two tangents, to find the 
 radius of a curve that, Srom the same Point of curve, will end 
 na parallel tangent. Fig. 28. 
 
 Let AB be the given curve, AV, VB the tangents, and 
 V'B' the parallel tangent; and let AO= Rand AO’ = R’. 
 
mw 
 
 SIMPLE OCURVES. var 
 
 Since the central angle A remains unchanged, the angle 
 4A between the tangent and long chord remains unchanged; 
 therefore V'A B' = VAB, and the new point of tangent is on 
 the long chord AB produced. Find on the ground the inter- 
 section of V'B’ with AB produced 
 and measure BB’. In the diagram 
 draw Be parallel to AO, then Beb' = 
 A, and by eq. (22) 
 
 BB’ = 2Be sin 4A 
 
 but 
 Be = 00' — ee a te 
 BB ~ 
 at ey 2sin tA 8) 0 
 The + sign is used when B’ is be- Fig, 23. 
 
 yond B, as in the figure; but if the 
 parallel tangent is within the given curve it will cut the 
 chord in some point B", and then the — sign must be used, 
 since R' will evidently be less than Lt. 
 
 If the parallel tangent is defined by a perpendicular offset, 
 as Bp = B'f; since Beb' = A 
 
 Bp = Be vers A =(h' — f) vers A 
 a Ri= k-- ——— (57) 
 
 Add or subtract as explained above. 
 If the long chord C= AB is known, then the new long 
 chord C' = AB' or AB" =C + BB’, and by eq. (54) 
 
 pi = pO * BB ne 
 
 138. Given: a curve joining two tangents, to change the 
 radius, and also the Point of curve, so that the new curve 
 may end in a parallel tangent directly opposite 
 the given Point of tangent. Fig. 24. 
 
 Let AB be the given curve, AV, VB the tangents, V'B’' the 
 parallel tangent, and B' the given tangent point on the radius 
 OB produced. 
 
18 FIELD ENGINEERING, 
 
 In the diagram, produce the tangent AV and the radius OB 
 to intersect at K. Then 
 
 BK = R exsec A 
 B'K = R' exsec A 
 
 Subtracting we have 
 
 BB' =(R — R') exsec A 
 
 ae (59) 
 
 Fig. 24, Fig. 25. 
 
 To find the change AA’ of the P’ C., in the diagram draw 
 O'G parallel to _A’A; then 
 O'G = OG tan A . 
 or 
 AA'=(R— RP’) tan a (60) 
 
 Vil By substituting the value of (R — R') from eq. (59) and ob- 
 serving Table II. 42 we have 
 
 AA' = BB' X cot ta (61) 
 Observe that. eqs. (59), (60), and (61) may be derived directly 
 
 from eqs. (50), (52), and (51) respectively by writing A for $A. 
 
 139. Given: a curve Joining two tangents, to find the new 
 tangent points after each tangent has been moved 
 parallel to itself any distance in either direction. Fig. 25. 
 
SIMPLE CURVES. 79 
 
 Let A and B be the given tangent points, and A’ and B’ 
 the new tangent points required. Let the known perpendicu- 
 lar distances Ag=a, and Byp=b. We then require the 
 unknown parallel distances gA’ = @ and pB' = y. 
 
 Since the form and dimensions of the curve remain un- 
 changed we may conceive the curve to be moved bodily 
 into its new position on lines parallel and equal to the. 
 line VV' joining the vertices. Then AA’ 00'= BB'= 
 Be 
 
 In the diagram draw VK parallel and equal to Bp = 6 and 
 
 V'H parallel and equal to A4g=a. Then VH=gA'=2, and 
 V'K=Bp=y. Since VGV' = JA, we have 
 
 VG=- and GH = e 
 
 sin A tan A 
 
 and since 
 VH = VG—GH=z 
 
 * b pe AER 
 
 ~ gin A tan A 
 Similarly (62) 
 
 b a 
 
 Y=Tan A sin A 
 
 When the new tangents are outside of the given curve, the 
 offsets a and } are considered positive; if either new tangent 
 were inside of the given curve its 
 offset would be considered negative. 
 In solving eqs. (62) if 2 and y are 
 found to be positive they are to be 
 laid off forwards from q and 7, as 
 in Fig. 25; if either is found to be 
 negative it is to be laid off in the 
 opposite direction. 
 
 Example.—A certain curve has a 
 central angle of 50° = A, and it is 
 proposed to move the first tangent Fic, 26. 
 in 20 feet and the second tangent 
 out 12 feet. Required, the distances on the tangents from the 
 old tangent points to the new, Fig. 26. 
 
 rs 
 
80 FIELD ENGINEERING. 
 
 Here a = — 20 andd’=-+ 12 
 
 4b 12 1.079181;—a 20 1.301030 
 A 50° —_ log sin 9.884254| A 50° ~_—— dog tan 0.076186 
 
 15.665 1.194927; -— 16.782 1.224844 
 x = 15.665 — (— 16.782) = + 82.450 
 
 +2 12 1.079181 |—a 20 1.301030 
 wie b0° log tan 0.076186 | A 50° log sin 9.884254 
 
 10.069 1.002995 | — 26.108 1.416776 
 
 y = 10.069 — (— 26,108) = + 36.177 
 
 3 (&@ = — 82.450 
 
 For + a and — 0d ly = — 36.177 
 g=— 1,120 
 
 For + a and + 0 ang 
 fh by fee + 1.120 
 
 For — a and — 6 j y = + 15.989 
 
 If we have a and @ given to find d and y: Solving eqs. (62) 
 for 6 and y we obtain 
 
 b=asin A—+-acos ait 
 
 (63) 
 Y¥Y=2cOS A —a@sin A 
 
 In which the algebraic signs of the quantities must be ob- 
 served as above. 
 
 140. Given: a curve joining two tangents, to find a new 
 Radius and new position of the Point of curve, such 
 
 i that the curve may end at the same point as before, but with 
 
 a given change in the direction of .the forward tangent. 
 
 | Fig. 27. 
 
 Let AB be the given curve, AV, VB the given tangents, 
 V’B the new tangent, and VBV' the given change in direc- 
 tion. Let A’= A+ VBYV'". 
 
a | 
 
 SIMPLE CURVES, 81 
 
 In the diagram draw BG perpendicular to AV produced; 
 
 then 
 BG = R vers a 
 <= F' vers A 
 Hence 
 : vers A 
 R= Re (64) 
 vers A 
 and 
 
 AA =AG— A'G=Rsin A—R'sin a’ (65) Hie 
 
 In the figure the change in direction of tangent makes A' 
 greater than A; therefore V' falls beyond V, and A’ beyond 
 
 Fie, 27. Fira. 28. 
 
 A; but if the change made A’ less than A, then VY" and A’ | 
 would fall behind V and A respectively, and R' would be We 
 greater than R. 
 
 The same formule apply to the converse problem in which | 
 B is taken as the point of curve, and A and A’ as points of HN | ig 
 tangent. HW 
 
 | | 
 
 141. Given a curve joining two tangents, to find the change 
 tn the Point of curve when the forward tangent takes a new WW 
 direction from ihe vertex V. Fig. 28. fl 
 
 By eq. (21) 1 
 
 ¥ ; eH 
 
 VA=RianjaA, VA' = Rtan $A’ 
 
 AA' = R (tan 4A — tan $A’ (68) ii 
 
 142. Given: a curve joining two tangenis, to find the new 
 
 f: 
 
82 FIELD ENGINEERING. 
 
 radius, R, when the forward tangent takes a new direc- 
 tion from the vertex, V. Fig. 29. 
 By eqs. (21) (25) 
 
 VA=Rtania, R'=VA cot ta’ 
 K' = R tan 4a cot 4a’ (67) 
 
 143. Given: a curve joining two tangents, and a given 
 change in the direction of the forward tangent from the 
 verted, to find the radius and point of curve of a curte 
 that shall pass at the same distance, VHA, from th. vertex. 
 Fig. 30. : 
 
 Let AB be the given curve, BVB’ the given change in 
 
 i Fie. 29. Fie. 30. 
 
 direction of tangent, and VW’ = VH. Let a’ = a+ BVB', 
 then eq. (24) 
 
 1) VH = Rex sec £4 = VH' = R' ex sec $4’ 
 
 | es R= R. LS8ee eS (68) 
 ii exsec $A 
 i By eq. (28) 
 VA=VHcot4a, VA' = VH' cotta’ 
 ge AA' = VH (cot +4 — cot $4’) (69) 
 
 But in case A’= A — BVB', AA’ becomes negative and 
 must be laid off backward from A, 
 
SIMPLE CURVES. 83 
 
 Example.—Given a 2° curve, A = 80° and BVB' = — 10° 
 
 aie a a ( 
 R log 3.457114 
 tA 40° . log exsec 9.484879 
 VH 874.97 2.941993 
 tA! 35° log exsec 9.348949 
 Re 1° 27’ nearly 3.598044 
 tA 20° cot 2.74748 
 tA’ 17° 30 cot 3.17159 
 
 — 0.42411 
 
 AA' = 874.97 X (— .42411) = — 371.08 
 
 and must be laid off backward from A. 
 
 144. Given: two indefinite tangents, a point situated be- 
 tween them, and the angle A, to find the radius R, and tan- 
 gent distance T of a curve joining the tangents which shall pass 
 through the given point. Fig. 31. 
 
 If the given point is on the bisecting line VO, as H, meas- 
 ure VH = #, and find #& and 7’as in §§ 97, 98. 
 
 When the given point, as P is not on the bisecting line VO; 
 if a line GK is passed through P per- 
 pendicular to VO, it will be parallel 
 to any long chord, as AB, and the 
 angle V@K=+4A. The curve pass- 
 ing through P will intersect GX in 
 some other point P’; the line GK 
 is bisected by the line VO at J, and 
 Wid oeey 2A fh 
 
 If the given point P is located by a 
 perpendicular offset from the tangent, 
 
 RS "as PL; in the triangle PLG, LG = 
 PL cotta. Lay off LG, and at G deflect VGK = 4A, and 
 
 measure GP and PK. Since by Geom. (Tab. I. 24) GA? = 
 GP' x GP, and GP’ = PK; 
 
 GA= VGPX PK (70) 
 
84 FIELD ENGINEERING. 
 
 Lay off GA; and A is the Point of curve, AV= 7" and 
 k= AV cotta. 
 
 If the given point were located by an offset from BV. find 
 B first, and make VA = BV. ' 
 
 If the given point P is located by a perpendicular offset 
 JP from the bisecting line VO; produce JP to intersect the 
 tangent at G and measure PG. Since P'G = GP + 2PI 
 
 GA= VGP(G@P+ 2PT) (71) 
 whence we have the point of curve A, as before. 
 
 145. Given: a curve, AP, and the radial offset PP’ 
 Al to find a curve which shall pass through the point P’, start- 
 Hi ing from the same point of curve A. Fig. 32. 
 
 Fia. 82. 
 
 | Let 6 = PP’, and in the diagram draw P’'@' parallel to the 
 | common tangent 4X, and join AP’. Then 
 
 P'@ =(R + d)sin a 
 
 GA =R—-(R + b)cosa 
 
 1} | rE yom ry: 
 
 i | ig tan $A pie (Rot) A (72) 
 if P' / Ss] 
 
 | Rim 2%, (2 + Deine (73) 
 sin A sin A 
 
 When the offset is outward use R -+ 6, when it is inward 
 use R — 0, 
 
 Haample.—Given: a 8° curve of 16 stations and a radial 
 offset of 205 feet inward from the P. 7. to find the radius of 
 the curve passing through the extremity of the offset, 
 
4 
 
 CURVES. 
 
 SIMPLE 
 
 Here A = 3° X 16 = 48°; and b = 205. 
 R 3°= 1910.08 
 
 R—b 1705.08 log 3.231745 
 A 48° log sin 9.871073 
 Sig day 3.102818 
 RR? log 3.281051 
 
 1.50742 0.178283 
 A 48° cot .90040 
 ZA’ tan .60702 = 31° 153’ 
 
 2 
 
 LS 62° 3L’ log sin 9.947995 
 LP. log 3.102818 
 R' (about 4° 01’). Ans. 3.154823 
 
 If the same offset were made outside of the curve we should 
 find R' log 3.488350, or about a 2° 05° curve. 
 
 This solution is inconveniently long for ordinary field prac- 
 tice. When the offset is small compared with the length of 
 curve, we may use the following 
 
 Approximate Rule: Divide twice the offset b by the 
 length of curve, look for the quotient in the table of nat. 
 sines, and take out the corresponding angle, which multiply by 
 100, and divide by the length of curve. The quotient is the 
 correction for the given degree of curve; to be subtracted when 
 the offset is made owtward, and added when the offset is made 
 inward. 
 
 This rule is expressed by the formula 
 
 100 gna od 
 D'=D*F —F- sin 3 (74) 
 Taking the same example, we have 
 20 = sin 14° 51’ 
 Ty = sin 
 
 : ee acy OU * eal 
 and correction = 14° 51’ < 1600 — = F 0° 56 
 
 Hence D' = 3° 56’ or D’ = 2° 04 
 
FIELD ENGINEERING. 
 
 THE VALVOID. 
 
 146. Given: any number of circular curves of equal length 
 Li, all starting from a common point of curve A, in a@ common 
 tangent AX, to find the equation of the curve joining 
 ther extremities, Fig. 33. 
 
 Let AP be any one of the given curves, 
 
 ‘“ R= its radius AO, 
 
 “ D = its degree of curve, 
 
 “ A = its central angle AOP, 
 
 *“‘ C= its long chord AP. 
 
 Fig. 33. 
 
 By substituting the value of 2 from eq. (16) in eq. (22) we 
 have 
 sin tA 
 
 a ye +D 
 
 (75) 
 
 Substituting in this the value of D from eq. (20) and letting 
 
 C L 
 th t 6 = = —— 9 oe 
 (theta) 4A, (rho) p 700 and VV {00° we have for the 
 polar equation of the required curve 
 sin 9 
 sin NV 
 
 in which p is the radius-vector AP, 6 the variable angle 
 XAP, the unit of measure is one side of the inscribed polygon 
 by which the circular curve AP is measured, and JV the num. 
 ber of these sides in the length of the curve AP. By the 
 
1 
 
 SIMPLE CURVES. 87 
 
 conditions of the problem JV is constant, but 6 may have any 
 value whatever. If we let 9 vary from 0° to + 180° and from 
 0°. to — 180° the point X will describe the curve XP PA 
 shown in the figure, which is called the Valvoid from its re- 
 semblance to the shell of a bivalve. All circular curves tan- 
 gent to AX at A and having a length IL = AX will terminate 
 in the valvoid, and the line PP’ joining the extremities of 
 any two of them is a chord of the valvoid. 
 
 147. To find a tangent to the valvoid at any point 
 P. Fig. 34. See Appendix. 
 Differentiating eq. (76) 
 dp ( if 6 
 ST Ret & cot 6 — +, cot =) (77) 
 which is essentially negative, since p is a decreasing function 
 of 9. 
 Let (phi) p = APG, the angle between the radius vector 
 and the normal PG. 
 1 6 
 tan @ = a cot WoT cot 6 (78) 
 The line PK perpendicular to P@ is tangent to the valvoid 
 at P, and PV perpendicular to PO is tangent to the curve ALF, 
 Then APV =9 and VPG =6 — 9g, and letting 1 = OPK = 
 VPG. 
 i=0-p=}A-— (79) 
 Therefore, to obtain the direction of a tangent to the val- 
 void at any point P, deflect from 
 the radius PO an angle equal to 
 i=(tA — @), on the side of PO 
 farthest from the point ot curve A. 
 The value of 7 may be found by 
 eqs. (78) (79), but we are saved 
 this somewhat tedious calculation 
 by the use of Table X. 1, which K 
 
 Fie. 34. 
 
 wht 
 contains values of the ratio ve U 
 for various values of A, and length of curve L. Multiplying 
 A by the proper tabulated number gives the value of 7 = OPK 
 
 at once; or 
 i=(4A —Q)=UA (80) 
 
88 FIELD ENGINEERING. 
 
 148. To find the radius of curvature of -the valvoid 
 atany point P. See Appendix, 
 Differentiating eq. (77) we have 
 
 dp | 2 6 1 ( 6 
 ae poe tgs ie §80b woot 71.9 Gott 
 ie =P 1 yr Cot G08 sachs cotta; + 1) 
 
 The general formula for the radius 
 
 curves is 
 8 
 2 mer 
 (c T ape 
 ee 
 
 a p? pie 9 dp? ap 
 
 do? ? age 
 
 of curvature of polar 
 
 , 
 
 SUT, witty : d 2 
 Substituting in this the values of p, 7 and fia and putting 
 
 6 
 (7 cot a te cot a) = @ we have after reduction, 
 
 p (1+ a3 
 Y bel —— DY e ne eae ee SS ELE Se 
 
 : (81) 
 1 — a2 4% cot 6 
 This formula bein 
 
 g too complicated for convenient use in 
 ‘he field, its use is 
 
 avoided by referring to Table X. 2, which 
 
 : Pitas : 
 contains values of the ratio Er % for various values of A and 
 
 L. Multiplying the given value of Z 
 ratio, gives the value of the radius of c 
 for a short distance either way from th 
 
 by the proper tabular 
 urvature of the valvoid 
 e given point P; or, 
 
 PEEVE, Tu (82) 
 
 149. To find the length of are of the valooid corre- 
 sponding to a change of one degree in the value of the 
 angle A. Fig, 35, 
 
 From any chord AP suppose a deflection of 4 degree to be 
 made each way to Ap' and Ap"; then the angle p’ Ap" = 4° — 
 the change in 6, and since A — 26, this makes a change of 1° 
 in the value of A. We then require to know the length of 
 
SIMPLE CURVES. 89 
 the arc p'p", and we may, without sensible error, consider it 
 
 to be described by the radius of curvature r= Po for the 
 point P, through an angle p’op". Now 
 
 and since g’ is so nearly equal to m" we may assume uw’ = 
 " ! tw Nie A" ‘ rte 
 Meet RENCE .Q — Dip == re (1 —2w) and p'op" = 
 4 a 
 (A'— Aty (tu). 
 But the condition of the problem requires A'’— A” =1’, 
 hence p'op" = (1 —u)°. 
 Therefore the length of arc p'p" for a change of 1° in the 
 value of A is 
 
 1=r(i—wu) X arc! 
 or (Tab. XVII.) L=r (1 — wu) .0174538 
 and since 7 = voL (Tab. X. 2), 
 1 =v(1 — wu) LZ .0174533 (83) 
 
 By this formula Table X. 3 has been prepared, for various 
 values of A and L. 
 
 150. Given: two curves of the same length L but of 
 different radit, starting from the same point of curve in @ 
 
90 FIELD ENGINEERING. 
 
 common tangent, to determine the direction and length of 
 alne joining their extremities. Fig. 36. 
 
 Let AX be the common tangent, and AP’, AP" the two 
 curves, to determine the direction and length of D a iid 
 
 If we take the point P on the 
 are P'P" determined by the 
 cr! airs ae 
 
 angle A = > and draw 
 
 a tangent PK to the valvoid at 
 P, we may assume without ma- 
 terial error that the chord P'P" 
 will be parallel to PK for any 
 value of P’P" not exceeding 
 4Z, a limit not likely to be ex 
 | ; ceeded in practice. 
 
 | Let O be the centre of the curve AP fixing the point P; 
 
 then AOP — at Pel rae 
 
 OP =t= ae. 
 
 Since P’P" is assumed parallel to PE 
 : PPO = K@0' mata Ka n'— ATA Gg ay) 
 
 P'P'O' — 7 — A Se A (1.46) (84) 
 
 Similarly producing P"P’ to any point H, 
 
 HPO =~ Ma ae rn har (85) 
 
 a 
 
 | whence also 
 t= 2 = At A" (85)' 
 The slight error involved in the above assumption is cor. 
 rected by taking out the value of w (Table X. 1) correspond- 
 
 ing to A", the less of the two given central angles; we have 
 
 therefore written 7% with the double accent in equations (84) 
 and (85). 
 
i 
 SIMPLE CURVES. 91 
 
 When 7’ and 2” are positive, they will be deflected as in 
 Fig. 36, on the side of the radius farthest from A ; should 2" be 
 negative it will of course be deflected from P’O" toward A. 
 
 The arc P’P" corresponds to a change of the central angle 
 from A’ to A" ; hence 
 
 A Ar = A" ite CA BERS 
 br 
 PT ea ie S20 Nn ely (86) 
 
 in which 7, is taken from Table X. 3 for Z= AP, and 
 ee 
 eS 
 
 A em 
 
 As in practice, the distance P’P" is usually small compared 
 with Z, the arc and chord will be almost identical and no 
 further calculation is necessary. If P'P" is large, it will be 
 found that equation (86) gives the “ength of arc very correctly 
 when a does not exceed, 20°, and the length of chord 
 
 A'+ aA" 
 2 
 
 gives a value to P’P", between that of the arc and chord. 
 The arc P'P" may be considered bu be described by the radius 
 
 60° ; tor intermediate mean angles it 
 
 when 
 
 y = vL, v being taken fox tit Eas Ay (Table X. 2), and its total 
 
 curvature is foune »y 4 ea re its length by the degree of 
 curve corres,onding to r (Table IV). 
 
 Example. Given, a 2° 30' curve, and a 1° curve of 12 stations 
 each from the same PC, to determine the distance between 
 
 their extremities. 
 
 A’ = 24° x 12 = 80°, Aaueeaiees a = 21; 
 oe ae BS 1 u" = .83446 
 
 Kq. (84). 2” = 2°.97387 = 2°58'25" 
 
 Eig. (85). ¢7 = 2" + A'—- A= 20°.9737 = 20°58'25" 
 
 Eq. (86). Arc P'P" = 18° X 10.425 = 187.65 ft. Ans. 
 
 Eq. (82). 7 = 1200 x .7479 = 897.48 ft. = (say) a 6°23’ curve. 
 
 Total curvature, P’P” = 6°.883 x 1.8765 = 11°.9777. 
 
 (The distance P’P"” may be found by solving the triangle 
 formed by itself and the long chords of the curves APS 
 
 AP".) 
 
FIELD ENGINEERING, 
 
 151. Given: a curve AP, to Jind a curve starting from the 
 same point A, that shall shift the station P any desired dis- 
 tance PP’ to the right or left. Fig. 36. 
 
 Before we can determine what distance PP’ is desired, we 
 must know (approximately) its direction. We have given, 
 therefore, D, LZ, and A to find the angle OPP’, and (after 
 measuring PP’) to find A’ and D’. 
 
 The solution is necessarily somewhat approximate, yet 
 Close enough for all practical purposes. For if the required 
 value of D’ were obtained precisely, it would probably involve 
 some seconds, and would therefore be discarded in favor of 
 some value in even minutes. 
 
 When P’ is inside the given curve : 
 
 Vi Eq. (80). t= OPK =ua. TableX.1 
 H 
 Eq. (82). r=Po =v. Table X. 2. 
 
 Let 6 (delta) = degree of curve corresponding to 7, by 
 
 Table IV. 
 | ; atese Sieg A ad tied” : 
 | <3 OP sy [00 40 nearly. 
 Eq. (86). A’= A+ ae Table X. 38. 
 
 4 
 
 Instead of taking 2 from Table X. 3 for the exact value 
 
 of A it is well to take it for the estimated value of AA, 
 
 Eq. (20). D= Al 
 
 When P’ is outside of the given curve: 
 
 t= WA, 7 == OTR 
 
 180° — OPP’ = f+ ei - 45 nearly. 
 
 A *100 88 
 
 Heample. Given, a 4° curve of 800 feet, or A = 82° to find 
 
SIMPLE CURVES. 93 
 
 « curve from the same P.@. which shall shift the last station, 
 in, about 55 feet. (Fig. 36.) 
 i = 32° & .8355 = 10°.786 
 7 = 800 X .7450 = 596, -. 6 = 9° 36’ = 9°.6 
 
 geod ono D0 ° +S 9° U 
 OPP' = 10°.736 iain 06 
 
 [) ae ° 19) = % ° 
 
 A' = 82°-+ 53, = 40 
 
 1 = 40 =— 5°. Ans. 
 
 For a 5° curve, the true distance PP’ = 59.53 
 ie RRS 4°59' ‘ce ée¢ ‘ec 6 PP' = 54.60 
 
 which proves this method practically correct. 
 
 152. Given: a tangent and curve, and a straight line 
 inter seotins them, making a given angle with the tangent at 
 a given point, to determine the distance on the line 
 
 from the tangent to the curve. Fig. 3%. 
 
 oO T 
 Fia. 37. 
 
 We have OA, AG, and the angle AGP to find GP. 
 R 
 
 tan AGO = iG PGO = AGO — AGP 
 Og sin PG-O 
 sin OPT = 73 sin PGO = in AGO 
 
 _ 4 sin (OPT— P@O) 
 PG = BR sin PGO- 
 
94 FIELD ENGINEERING, 
 When AGP = AGO, eq. (24), 
 GP = R exsec (90° — AGO) 
 When AGP = 90°, $8 (92), (119), 
 GA 
 
 sin POA‘ —— 
 
 GP= R vers POA, R 
 
 When AGP'> AGO, we have 
 P'GO = AGP' — AGO 
 but the other formule remain unchanged. 
 
 Hxample.—Let R= 955.37, AG = 350, AGP = 40° 
 
 R 955.37 log 2.980170 
 AG 350. log 2.544068 
 AGO 69° 52’ 47” log tan 0.436102 
 AGP. 40° 
 
 PGO 29° 52! 47" 
 
 log sin 9.697387 
 
 AGO 69° 52’ 47” log sin 9.972653 
 
 OPI 32° 02’ 86” log sin 9.724734 
 POG 2° 09' 49” log sin 8.576953 
 
 8.879566 
 R log 2.980170 
 PG 72.40 Ans. 1.859736 
 
 log 1.859736 
 
 This problem may be used in passing from a tangent to a 
 curve when the tangent point is obstructed. The distance 
 AP on the curve is defined by the angle AOP, which is readily 
 found. 
 
 If AGP' > 2AGO the line will not cut the curve. 
 
 153. Given: a curve and a distant point to find a 
 tangent that shall pass through the point. Fig. 38. 
 
 We have the curve adg and the point P visible, but distance 
 unknown, to find the point of tangent B. 
 
SIMPLE CURVES. 
 
 Any chord, as Of, parallel to the required tangent, if pro- 
 
 duced will pass the point Pata perpendicular distance equal 
 to the middle ordinate of that chord. Ranging across every 
 two consecutive stakes on the curve we at first find the 
 range falling outside of the required tangent, as bcG, ca H, 
 etc.; but finally the range falls inside, as dek. We then know 
 that the required point is between ¢ and é. 
 If the range ce falls inside the point Pia , 
 perpendicular distance equal to the middle Ki 
 ordinate of ce, the tangent point is at d. 
 If the perpendicular distance is greater 
 than this, the point B is between c and d. 
 If less, or if the range ce falls outside of 
 P, the point B is between d and ¢. The 
 middle ordinate for ce (200 feet) equals the 
 tangent offset for 100 feet, given in Tab. 
 IV., and it is generally so small that it can 
 be estimated at P without going to lay 
 it off. 
 
 To find the exact point B, when it falls 
 between d and e¢, find by trial a point x 
 on the arc cd in range with ¢e and a point 
 inside of P a perpendicular distance equal 
 to the middle ordinate of ew. The point B 
 is at the middle point of the arc ew If 
 the point B is between ¢ and d, stand at ¢ 
 and find a point 2 on the arc de in the same 
 way. Bis at one half the arc cz. ; 
 
 The middle ordinate of any chord ez is Fig. 38. 
 less than VM for 200 feet, and greater than m for 100 feet. If 
 necessary, its exact value m’ can be found by 
 
 pos eee = = 
 
 Wye PN Non 
 
 = "70000 tt 
 
 7 
 
 ‘and*this equation is nearly true when ea is as great at 300 or 
 400 feet. That is, middle ordinates on the same curve are to 
 each other as the squares of their chords very nearly. 
 
 By this method the point B is found without the use of the 
 transit, so that the plug can be driven at D before the transit 
 
to be near the unknown tangent point B 
 
 96 FIELD ENGINEERING. 
 
 is brought up from the rear. It is therefore preferable to the 
 following solution. Fig. 39. 
 
 From any two points @ and c¢ of the curve measure the 
 angles to the point P, so that with the chord ac asa base, 
 and the measured angles, we may find cP by the formula 
 
 sin caP 
 cP = ac — 
 
 sin 2Pa 
 
 Knowing the angle c that cP makes with a tangent at c, we 
 find the length of the chord cd by cd = 2R sinc. 
 By Geom. Tab. I. 24, 
 
 PB = Pe= V¢eP X dP 
 
 whence we know ce. Opposite e, or on the arc eB described 
 With the radius Pe, we find B. 
 
 Fia. 40, 
 
 154. Given: two curves exterior to cach other, to 
 Jind the tangent points of a line tangent to both and its 
 
 length between tangent points. Fig. 40. 
 
 Let B and A be the required tangent points. Let OB = BR, 
 
 and 0'A = R’, 
 
 On the curve of greater radius R select a point H supposed 
 , and knowing the 
 
~. 
 
 SIMPLE CURVES. 97 
 
 direction of the radius OJZ/, find on the other curve a point 
 having a radius O'# parallel to O/, and measure HK. In 
 the diagram draw Od and O'w perpendicular to Hk. Then 
 the angle KO'a = 90° — HKU’ = KO'A nearly, which is the 
 angle required. We have therefore to find the correction 
 a0'A = 2, and apply it to KO'a. 
 
 Aa = f' vers KO'a; Bb = FR vers KO'a nearly. 
 
 Ka = f' sin K0'a; Hb= ksin KO'a 
 
 Bb — Aa = (R—R') vers KO'a | 
 ab = HK + (#— Lf’) sin KO'a i | 
 (R= R) vers KO'a | 
 lig + (R — R') sin HO'a 
 
 sin. @ == nearly. (88) MM 
 
 KO'A = (K0O'a — x) = HOB 
 
 Observe that HO’a = the angle between the tangent at or il 
 H and the line 7A; and AKO'A = the angle between the aN 
 tangent at 7 or 7 and the required tangent BA. i 
 
 If, instead of H and AH, the points JJ’ and A’ had been Hi 
 selected, then vi 
 (R — R') vers H'Ob qi 
 
 sin. v= 
 
 and 
 H'OB= K'OJA = H'0b-+ 2. 
 The length of BA should be obtained by measurement, but HA 
 it may be calculated by He 
 
 AB = ab — (R— RB’) sina’ (89) A 
 
 KOA = HOB = KO'a —& 
 
 When R = Rk’, x = 0, and HK is parallel to BA. | ue if 
 | | 
 In case the curves are reverse to each other, as tn i | f 
 Fig. 41, i 
 ; (R-+ R') vers KO'a HF 
 in’ = ——_-___—— —-—— nearly. 90 I 
 ane RoR ya KO ae ee ll 
 
 If the points H’ and X’'’ are selected, Fig. 41, 
 
 (R-+ R’) vers H'Ob Ih 
 HK —(RtR) aia Hl Ob nearly. (91) it 
 H'0B = K'0'A = H'0b +. 
 
 sin? = 
 
FIELD ENGINEERING. 
 
 The lines HK, AB, and QO’ all intersect in a common 
 point J, Fig. 41. 
 
 BELGE OE x 
 HI = ELE (92) 
 IB = VHI(HI+2R sin HOB) (98) 
 iS Sea (94) 
 
 These last three equations furnish another method of 
 solving the same problem. They may be applied to Fig. 40 
 by changing the sign of R’. 
 
 In Fig. 41, if R= R’, then HT = 44K and AB = 2/8. 
 
 Fia. 42. 
 
 155. Given: two curves, O and O’, reverse to 
 each other, joined by a tangent BA', and terminating in 
 another tangent, B'F; to change the position of the 
 Point of Tangent B of the first curve, so that the second 
 curve may terminate in a given parallel tangent, B'F'’. 
 Fig. 42. 
 
 Let X be the required new position of B. 
 
 ‘* Q" be the corresponding position of 0’. 
 
 te to =A OLE rand peas ARO 
 
 Since the radii and the connecting tangent are unchanged 
 in length, and all rotate together about O as a centre, O” will 
 be on a circle passing through O', described with a radius 
 OO’, and the required angle BOX = 0'00". 
 
SIMPLE CURVES. 99 
 
 In the diagram, produce 0'A’ and draw the perpendicular 
 OG, and let «=the angle 00'G. Also, draw OF parallel 
 and O"K and O'H perpendicular to B’O'. In the triangle 
 00'G we have 
 
 GO R+&k 
 
 cot OO'G = Goo cota = “eae (95) 
 and 
 00>-= rel (96) 
 COs a 
 
 The angle KOO' = 00'B'’ =a-+ A’. 
 The angle KOO" = OO"B" =a-+ A’ 
 
 KO = 00". cos(a + 'A"),, HO = 00’, cos(a« +. A”. 
 
 HK = 00' [cos(a + A")—cos(a + A’)] = BF" 
 
 cos (a -- A"). "cos (@ -- A Se ee (97) 
 
 OX OOO a ee Be) ey (98) 
 
 If we conceive a line to be drawn through O bisecting the 
 arc 0'O", the angle it makes with B"O" is a mean between 
 B'0'0 and B"0"0; hence the chord 0'O", perpendicular to 
 this line, makes an angle with O'P perpendicular to B'O' of 
 
 POO" =4[(@+ 4) ++] 
 and since 
 O'P = PO" cot PO'O" 
 
 FR’ = B'F' cot t[(«e+t adt(etay] @9 
 
 which gives the distance, measured on the parallel tangent, 
 between the old tangent point and the new. 
 
 This problem occurs in practice when both the connecting 
 tangent and the radius of the last curve are at their minimum 
 limit, and the parallel tangent is énside of the old one, as in 
 the figure. Should the new tangent be outside, the same for- 
 mulse apply, only changing the sign of B’F’” in eq. (97). But 
 in this last case it is usually preferable to employ problem 
 § 136 or § 187. 
 
 Example.—A 1° 40' curve is followed by a tangent of 200 ft., 
 and that by a 4° curve of 10 stations ending in a tangent ; 
 
100 FIELD ENGINEERING. 
 
 and the offset to the given parallel tangent is 80 ft. on the 
 inside. Required, the pasition of the new tangent points X 
 and B", 
 
 Here & = 3487.87, R' == 1482.69, BA’ = 200, B'F’ = 80. 
 
 Iq. (95) R-+ R’ 4870.56 log 8.687579 
 
 BA! 200. log 2.301030 
 
 “Ge Pa} log cot 1.386549 
 Eq. (96) a 2° 21’ log cos 9.999635 
 vs 00' 3.687944 
 Eq. (97) B’F’ 80 1.903090 
 .01641 . 8.115146 
 
 a+ A’ 42° 21’ cos .73904 
 a+ A" 40° 56’ cos .75545 
 
 Wy. (98) BOX): 1°25 .-: BR =—s8b5 tt. Ane 
 Eq. (99) PO'O" 41° 88’ 80” cot 1.12468 x 80 = 89.97 = #"T' 
 
 156. When the tangents of a proposed road are to be in 
 general much longer than the curves, it is desirable to estab- 
 lish the tangents first in making the location, and afterwards 
 determine suitable curves. On the other hand, if the curves 
 necessarily predominate, they should be first selected and 
 adjusted to the ground with reference to grade and easy 
 alignment, and afterwards joined by tangents. In the latter 
 case the field work cannot be successfully accomplished 
 unless the location has been previously worked out upon a 
 correct map constructed from the preliminary surveys. The 
 map Should show contours of the surface, and also the grade 
 contour, or intersection of the surface and plane of the grade. 
 In side-hill work the grade contour indicates approximately 
 the degree and position of the necessary curves. In the work 
 of selecting proper curves upon the map, templets or 
 pattern curves are almost indispensable. The templets are 
 cut to form a series of curves, the radii being taken from 
 Table IV. to a scale corresponding to the scale of the map, 
 which ranges from 400 to 100 feet per inch, according to the 
 difficulty of the location. The templets should represent 
 convenient curves, or those in which the number of minutes 
 
SIMPLE CURVES. 101 
 
 per station bear a simple ratio to 100. Curves of 50’ and 
 multiples of 50° are most convenient; 40° curves and multi- 
 ples standing next in order, and 30° curves and multiples 
 next. 
 
 TABLE OF CONVENIENT CURVES. 
 
 D Ratio of Min. | D \Ratio of Min. D Ratio of Min. 
 : to Feet. || : iP to dmeets 1 ; to Feet. 
 50’ | a bees 40’ 2:5 30’ 3:10 
 1° 40’ 1:1 1° 20’ | 4:5 | 4° QO | 3:5 
 2° 30/ | 3:2 R20 00 | 6:5 | 12° 80’ | 9:10 
 8° 20’ 2:1 De 40’ 8:5 | 2° 00/ 6:5 
 4° 10’ 5:2 |} 3° 207 | 2:1 2° 30/ 3:2 
 5° (0! 3:1 4° 0” | 12:5 3° 00/ 9:5 
 5° 50’ var) | 4° 40/ 14:5 3° 30’ 21:10 
 6° 40’ 4:1 || 5° 207 | 16:5 4° 00’ 12:5 
 7° 30 9:2 6° 00’ | 18:5 4° 30/ 27:10 
 82 20/ 5:1 6° 40’ | 4:1 5° 00’ 3:1 
 g° 10/ 11:2 |! FolOy | . 9222 5 5° 30’ 33 : 10 
 10° 00’ 6:1 8° 00/ | 24:5 ‘| 6° 00’ 18:5 
 
 After drawing the curves and tangents upon the map, the 
 tangent points and central angles are carefully determined, 
 the latter being compared with the lengths of the curves ob- 
 tained by a pair of stepping dividers set precisely by scale to 
 the length of one station. Field notes are then prepared from 
 the map, and if the work has been well done these notes may 
 be followed in the field with scarcely any alterations. 
 
 No ordinary protractor will measure the angles closely 
 enough for this purpose ; it is better to use a radius as large 
 as convenient, of 50 parts. The chord of any arc drawn with 
 this radius equals 100 times the sine of one half the angle 
 subtended. 
 
 The importance of having absolutely straight-edged rulers 
 in such work is obvious. In case a very long line is to be 
 projected upon the map, it is well to use a piece of fine 
 sewing silk for the purpose. See §§ 53, 54. 
 
FIELD ENGINEERING. 
 
 CHAPTER VI. 
 CoMPOUND CURVES. 
 A. Theory. 
 
 157. A compound curve consists of two or more consecu- 
 tive circular arcs of different radii, having their centres on 
 the same side of tbe curve ; but any two consecutive arcs 
 must have a common tangent at their meeting point, or their 
 radii at this point must coincide in direction. The meeting 
 point is called the point of compound curve, or P.C.C. 
 Compound curves are employed to bring the line of the road 
 upon more favorable ground than could be done by the use 
 of any simple curve. 
 
 When a compound curve of two ares connects two tangent 
 lines, the tangent points are at unequal distances from the 
 intersection or vertex, the shorter distance being on the line 
 which is tangent to the arc of shorter radius. 
 
 158, Let VA, VB (Fig. 48) be any two right lines inter- 
 secting at V, and let A be the deflection angle between them. 
 Let A and B be the tangent points of a compound curve (VA 
 less than VS), and let AP, PB be the two arcs of the curve. 
 The centre 0, of the arc AP will be found on AS, drawn per- 
 pendicular to VA; the centre QO, of the arc PB will be found 
 on BS produced perpendicular to VB; and the angle ASB 
 will evidently equal A. Join VS, and on VS as a diameter 
 describe a circle; it will pass through the points A and B, 
 since the angles VAS, VBS are right angles in a semicircle. 
 Draw the chord VQ, bisecting the angle A VB, and join AQ, 
 BQ. Then AQ, BY are equal, since they are chords subtend- 
 ing the equal angles AVQ, BVQ. From Q as.a centre, and 
 with radius QA, describe a circle; it will cut the tangent 
 lines at A and B, and also at two other points G and Y, such 
 that VG = VA, and VY= VB. Hence BG = AY, and the 
 parallel chords AG, BY are perpendicular to VQ. Join AB; 
 then AQB = ASB = A, since both angles are subtended by 
 the same chord AB. 
 
 In the triangle VAB, the sum of the angles at A and B is 
 equal to the exterior angle A between the tangents ; while 
 their difference (A — B) is equal to the angle at the centre Q 
 
COMPOUND CURVES. 103 
 
 subtended by the chord BG, which is the difference of the 
 sides (VB —V.A). For the angle VAB = VAG + GAB, and 
 the angle VBA = VBY — ABY. But VAG = VBY and 
 GAB = ABY, and by subtraction VAB — VBA = 2GAB = 
 GQB, since A is on the circumference and Q at the centre. 
 
 159. Turorem—The circle YAGB, whose centre is Q, % 
 the locus of the point of compound curve P, whatever be the 
 relative lengths of the arcs AP, PB composing the curve. 
 
 Fie. 43. 
 
 On the circle YAGB, and between A and G, take any point 
 P, and on AS find a centre O,, from which a circular arc may 
 be drawn cutting the circle at A and P; also on BS produced 
 find a centre Q., from which a circular are may be drawn 
 cutting the circle at B and P. Join PQ, PO, and POs. 
 Since when two circles intersect, the angles are equal be- 
 tween radii drawn to the points of intersection, QPO:= QAO, 
 
104 FIELD ENGINEERING. 
 
 and @PO, = QBO,. Draw the chord QS and it subtends the 
 equal angles (AO, = QBO.. Hence @PO, = QPO, and the 
 radius PO, coincides in direction with the radius PO., which 
 is the condition essential to a compound curve. 
 
 Now, if we imagine another point P' to be taken on QP or 
 on YP produced, and the arcs AP’ BP', drawn from centres 
 found on AS and BS, it is evident that the equality of angles 
 found in respect to P could not exist in respect to P’. Hence 
 the arcs would intersect in P’ at some angle 0, PO, and would 
 not form a compound curve. Therefore, Q. E. D. 
 
 160. THErorem.—Jn any compound curve the radial lines 
 passing through the three tangent points A, P, and B are all 
 tangent to a cirele having the point Q for its centre, and Jor tts 
 diameter the difference of the sides VB and VA. 
 
 Draw the three lines QM, QN, QL perpendicular to the 
 radial lines BO,, AS, and PO, respectively. Then the three 
 right-angled triangles BQN, PQL, and AQM are equal, since 
 BQ = PQ = AQ =radius of the circle AGB, and the angles 
 at B, P, and A are equal by the last theorem. Hence QM = 
 QL = QN, and if a circle be described with this radius about 
 Q, the three lines BO:, PO2, and AO, produced will be tan- 
 gent to it. Draw QJ perpendicular to VB; it will bisect the 
 chord GB in J; and QV = BI= $BG. Hence the diameter 
 2QN = BG = VB —VA; which was to be proved. 
 
 Corollary 1. The compound curve intersects the circle AGB 
 in the point P, at an angle equal to half the difference of the 
 angles VAB, VBA, For QPL= QBN = BQOT=4BQG. The 
 arc AP is exterior, and the arc PB interior to the circle 
 AGB. 
 
 Cor. 2. Since both centres are on the line PL, the position 
 of the point P fixes the lengths of the radii of a compound 
 curve. As P is moved toward G both radii are increased, 
 until when P reaches G, AO; becomes AK, a maximum, while 
 BO; becomes infinite. As P moves toward A both radii are 
 diminished, but the least value of the are AP depends upon 
 the least radius allowed on the road. If in the diagram we 
 make AQ, equal to the least radius allowed, a right line drawn 
 through the point 0, tangent to the circle ZMW fixes the 
 corresponding minimum value of the arc AP, and also of 
 the radius BO, for given values of VA, VB, and a. Be- 
 
» | 
 COMPOUND CURVES. 105 
 
 tween these limits any desired values of the radii may be em 
 ployed. 
 
 Cor, 3. In the triangle SO,0., the sum of the two central 
 angles A0,P and PO,B is equal to the exterior angle ASB = 
 A; consequently, as the central angle of one arc is increased 
 by any change in the position of the point P, the central 
 angle of the other will be diminished an equal amount. 
 
 or, 4. Only one value of the angle AO, P is consistent with 
 a given value of the radius AQ,, since both depend on the 
 variable position of the line PL; and for the same reason only 
 one value of the angle BO.P is consistent with a given value 
 of the radius BO.. Hence only one radius or one central 
 angle can be assumed at pleasure, the remaining parts being 
 deducible therefrom in terms of the sides VA, VB, and the 
 
 angle A. 
 
 B. General Equations. 
 
 161. Let 8, =the side VA, S, =the side VB 
 Let R, — the radius AN, R, = the radius BO, 
 aye dif VAS = BA; A =the sum VAB+ VBA 
 
 “ a, =central angle AOiP, A. = central angle BOP. 
 In the triangle BQJ, cot BOL = ae But JQ 2eV i xX 
 
 cot 1QV = 4(S2 +S.) cot dA, and Bl = 4(S_ — S1). 
 
 cot tv = = +5 cot 4A (100) 
 
 a) 
 2 
 
 By Cor. 3, AitAr.=A (101) 
 
 In the triangle AQM, AO, = AM— MO,._ But AM = 
 
 MQ cot 4y, and MO, = MQ cot tA. 
 
 Ry = U(S2 — &:) (cot 4v — cot £41) } (102) 
 
 Similarly, Rs = 4(S2—- S:) (cot 4v + cot +A.) \ 
 Subtracting, 
 
 R, — Bi = (Sa — S1) (Cot 4A2 + cot $A1) (108) 
 
FIELD ENGINEERING. 
 
 | RAL aa ora | 
 plO2 — Al 
 From (102), (104, 
 RR 
 cob PAs = 7g gy cota | 
 In the triangle ABG, 
 AB sin BAG 
 coats sin AG@V 
 or 
 3AB sin 4 
 tAiseiQdiceP ae (105) 
 
 sin tA 
 
 by which we find 3(S, — S,), when, instead of the sides and 
 A, we have given AB, and the angles VAB and VBA. 
 
 Rea 
 From (103), 4(S: — 8,) = a roe meee TONE (106) 
 
 R 
 From (102), (107) 
 Re 
 
 cot $v = —.— —; — cotia 
 bi EL ESS Hi 
 
 3(S2 — S,) cot 4 
 From (100) 4/8, + &) = 28s aa By (108) 
 
 S, and S, are found by adding and subtracting the values 
 found by egs. (106), (108). 
 
 $(S_ — 8) sin 2A 
 sin dy 
 
 From (105), 44B= (109) 
 
 ti which may be used instead of (108) when the sides are not re- 
 | quired. VAB=}(A+y)and VBA=i(a — y). 
 
 162. Given: the sides VA = 8, and VB = &, and the 
 angle A; assuming the shorter radius R,, to find Ai, Aa, 
 and Ro. 
 
 Use equations (100), (104), (101), (102), and (18). 
 
 Hzample.—Let VA = 1899.90, VB = 1091.12. a = 74°, and 
 assume A, = 955.37. 
 
COMPOUND CURVES. 107 
 
 (100) 4(S2 + S;) 1495.51 
 1(S, — S,) 404.39 
 
 LA Hf 
 
 a a 
 04), G, (Uv = 6) 
 
 (Sa — 81) 
 ; pA 21° 27’ 
 (101) 4 tA Sis 
 wags Aa 15° 33’ 
 (102) 37 
 
 $(S, — S1) 
 R, (D = 1° 40’) 
 
 log 3.174789 
 2.606800 
 
 < 0.567989 
 cot ‘* 0.122886 
 
 11° 31' 01".5 cot 4.90769 “ “ 0.690875 
 
 2.980170 
 ‘** 2.606800 
 
 2.36249 ‘© 0.378370 
 cot 2. 54520 
 
 3.59370 
 “« 4.90769 
 
 8. 50139 «0.929490 
 
 2.606800 
 3 536290 
 
 (18) Prd tay SS eee 42° 54’, Dy = ilo No 31¢ 06’, Dg — 1866. 
 
 163. Given: the line AB, and the angles VAB, VBA; 
 assuming the longer radius R., to tind A>», Ss, and Ry. 
 
 Example.—Let AB = 2487.82, VAB = 48° 31, VBA = 25° 29, 
 
 and assume R, = 3437.87. 
 
 (105) AB 1218.91 
 
 iy 11° 81’ 
 i 37° 
 - (4 — 81) 404.88 
 (104) Re 
 Wy 11° 81! 
 ie Seas a 15° 33' 
 101) $A oh 
 ey 21° 27 
 i102) 4y 
 3(82 — 81) 
 
 log 8.085972 
 sin “ 9.300276 
 
 ‘2.386248 
 ce 6 9.779468 
 
 «© 9606785 
 3.536289 
 
 8.50166 ‘< 0.929504 
 cot 4.90785 
 
 cot 3. 59381 
 
 cot 2.54516 
 «© 4.90785 
 
 9. 36269 log 0.373407 
 2.606785 
 
 2.980192 
 
108 FIELD ENGINEERING. 
 
 164. Usually a compound curve is fitted by trial to the 
 Shape of the ground, after which it may be desirable to 
 ealculate the sides VA, VB, or the line AB, and the angles 
 VAB, VBA. 
 
 Example.—From the point of curve A, a 6° curve is run 
 715 feet to the P.C.C.; thence a 1° 40’ curve is run 1866 feet 
 to the P.7. Required, the sides VA, VB, and the line AB, 
 and angles VAB, VBA. Here R, = 955.87, A, = 42° 54, 
 fi, = 8487.87, A = 31° 06’. 
 
 (106) R.— Rh, 2482.50 log 3.894889 
 
 $Ai 21° 27 cot 2.54516 
 
 $A2 15° 33! ** 3.59370 
 6.13886 ** 0.788088 
 . 4S. — S,) 404.39 «2.606801 
 (107) A, «2.980170 
 2.36248 ‘< 0.878369 
 
 tar 21° 27’ cot 2.54516 
 
 ey, 11° 81' 01.7 “4.90764  “ 0.690873 
 (108) $(S2 — S;) 2.606801 
 
 ‘* 3.297674 
 
 2A 37° cot “ 0.122886 
 4(Sa + S;) 1495.51 “3.174788 
 Ss 1899.90 
 8, 1091.12 
 VAB 48° 81’ 
 VBA 25° 29' 
 (109) 4(S2 — S,) “2.606801 
 tA 87° sin “ 9.779463 
 2, 386264 
 ly 11° 81’ 01".7 sin “ 9.300294 
 1AB 1218.91 3.085970 
 AB 2437, 82 
 
 165. Given: the radii R,, Ra, the angle A, and one xde, 
 VA, or VB, to find the other side and the central angles Ay, 
 
COMPOUND CURVES. 109 
 
 In the triangle AM/Q, AO, = AM — MO; = IQ — MQ cot 
 MO,Q: or 
 
 R, = 4(S2 + Si) cot $A — $(S2 —Si) cot FA1 
 whence 
 
 4(S, +. 8) = 4(Sq —S,).cot tA) tan 4A + R, tan 4A 
 
 By eq. (106) 
 sin +Ae sin $A 
 sin 4A 
 
 4(Sq — 1) = (Re > R;) 
 Substituting this above, subtracting and reducing 
 Sy a dais — Ry) sin +Ag pais AS ot + R, tan 4A 
 
 But #(A — A:) = $A, and 2 sin? $A. = vers Ae, whence 
 
 i= (R. — fy) vers A: + Ri vers A (110) 
 sin A 
 Transposing, Ht 
 S, sin A — fi, vers A a 
 vers Ace = a eee (111) | 
 Similarly, from the triangle BQO: 7 ad 
 1) 
 
 it —— 4(S, = S;) cot tA ob 4(S2 aad 81) cot +Ag i) H | 
 
 from which and eq. (106) we derive 
 
 2, vers A — (RR. — Ay) vers A | 
 g, Ba vors A’ (Ra— Ri) vers Av aia) | 
 sin A | 
 and 
 R, vets A — S. sin A (112) 
 
 ai 3 i 
 
119 FIELD ENGINEERING. 
 
 Hvample.—Given : VA = 8, = 1091.12, A = 74’, and the 
 radii R, = 955.37, Rz = 3437.87, to find A,, As, and S. 
 
 (111) S, 1091.12 log 3.037873 
 A "4° sin ** 9.982842 
 
 1048.85 “ 3.020715 
 
 Ri © 2.980170 
 
 A 74° vers *£ 9.859956 
 
 692.03 “9 840196 
 
 ! 306.82 «© 2.552449 
 i R, — R, 2482.50 ‘3 394889 
 Hi Ke 81° 06’ vers “ 9.157566 
 1 ASIN 42° 54! «46 9 497954 
 1 (112) R, — R, “3 394889 
 663.96 “9999143 
 
 R, ‘© 3536289 
 
 A vers ‘* 9.859956 
 
 2490.26 “3 396245 
 
 1826.30 “3 961572 
 
 A sin “ 9.982849 
 
 Sa 1899.90 “3 978730 
 
 166. Given : one side, and the radius and central angle of 
 the adjacent arc, to find the other radius and side. 
 From eqs. (111), (118) we have 
 
 S, sin A — R&, vers j 
 
 1 |. y ers A 
 
 Eg Te Se 3 Se sat - 
 vers Ag 
 
 (114) 
 ey vers A — So sin A | 
 vers Ay 
 
 Be Ri = 
 
 by one of which the required radius may be found; the required 
 side is then found by eq. (110) or (112), as in the last problem. 
 
 Lzample.—Given : VA =S; = 1091.12 a = 74°, R, = 955.37 
 and A; = 42°54’; to find R, Aa == 74° — 42° 64’ = $1° 06 
 
COMPOUND CURVES. Lil 
 
 (114) ref 1091.12 log 8.037873 
 
 K "4° gin 9.982842 
 
 1048.85 << 3.020715 
 
 R 955.37 | “ 9.980170 
 
 A 74° “ vers 9.859956 
 
 692.03 “ 2.840126 
 
 356.82 “ 9.552449 
 
 a 31°06 © vers 9.157556 
 
 a. Ry — R, 2482.52 «“ 3.394893 
 SoBe 3437.89 
 
 Fig. 44, 
 
 Otherwise: Fig. 44. If convenient in the field, a tan- 
 gent PV, may be run from the point P to intersect the 
 farther tangent. The -distance PV, multiplied by cot $4; 
 will equal the radius R, by eq. (25). 
 
 167. Remarks.—If the first arc AP be produced to G, 
 Fig. 44, so that A0,@ = A, then Gis the tangent point of a 
 tangent parallel to VB, and by $187, the tangent point B must 
 be on the line PG produced. Conversely, if the point B is 
 assumed, and the arc AG given, the point P must be on 
 the line BG@ produced. The radius R, may be found by 
 
112 FIELD ENGINEERING. 
 
 sho LP being measured on the ground ; 
 similar triangles R,: R, :: BP : GP. 
 
 The distance VD, Fig. 48, from the vertex to the circle 
 AGS is expressed by the formuia 
 
 ite 
 
 or by 
 
 ex'sec ey 
 sin dy 
 
 VD = 3(82—8)) - 
 
 (115) 
 
 Tf the point P falls at D, then VD is also the distance of the 
 curve from the vertex measured on the line VQ. But when 
 P falls at D, the radius PO, is perpendicular to the line AB, 
 and A, = VAB, and A, = VBA. When a, is greater than 
 VAB, the arc AP, being exterior to the circle, cuts the line 
 VD; but when A, is less than VAB, the arc PB cuts the line 
 DQ. 
 
 If the line O.P produced passes through V, we have 
 S.— 
 i+ & 
 giving A: = 4A + QVLand As = $A — QVL. 
 
 When A, is greater than this, we have for the external 
 distance of the vertex 
 
 SiR Wy Da ts Sin (116) 
 
 Hi, = R, ex sec AO, V 
 
 in which the angle AQ, V is found by the formula cot AO, V= 
 
 de, 
 oi? and #, is measured on a line VO,, making the angle 
 £51 
 
 AVVO; = 90° — AO,V. 
 When A, is less than (}A + QVL), we have similar expres- 
 sions with respect to the arc BP and centre Oy. 
 
 168. To locate a compound curve » when the point of com- 
 pound curve is inaccessible. Fig. 45. 
 
 Hach are being in itself a simple curve is located as such. 
 When the P.C.C. is accessible, the transit is placed over it, 
 and the direction of the common tangent found, from which 
 the second arc is then located. 
 
 When the P.C.C. is not accessible, the common tangent 
 Vi V2 may be found by locating the points V,; and V2, which 
 may be easily done, since V:A = ViP= R, tan +Aj, and 
 
COMPOUND CURVES. 113 
 
 V.B = V2P = ft, tan 4 Ae, from which eacl: are may then 
 be located by offsets or otherwise, as in the case of simple 
 curves 
 
 Should the points V,; V2 be obstructed, the common tangent 
 may be found by an offset 7G = LP from any convenient 
 point /7, for knowing the angle HO,P, we have HG = R, 
 vers 110, P, and. GP.= & sin -HO,P. 
 
 If the cutire tangent V, V.is too much obstructed for use, 
 
 the parallel line ZA may be employed, observing that the 
 Ae -gee ey weatdil. 
 angle PO. is found by vers PO.K = Re and the distance 
 v2 
 LE by LK = R, sin PO,K, by which a point # on the second 
 arc is found having a tangent offset KAJ = HG. 
 
 Fig. 46. 
 
 Fic. 45. 
 
 Should the line HK be also obstructed, we may run the in- 
 verted curve HP’ = HP and P'K = PK to find the point K 
 from which so much of the second are as is accessible may be 
 located. 
 
 C. Special Problems in Compound Curves. 
 
 169. Given: a compound curve ending in a tangent; to 
 change the P.C.C. so that the curve may end in a@ given 
 parallel tangent. Fig. 46. 
 
 Let APB be the given curve ending in VB, 
 
 «< V'B' be the given parallel tangent, 
 ‘‘ » = perpendicular distance between tangents. 
 
 It is required to change the point P, and with it the values 
 of A, and Ag, so that with the same radii R, and AR, the new 
 curve AP’B’ may end in the parallel tangent V'B. 
 
114 FIELD ENGINEERING. 
 
 a. When the tangent V'B' is inside of VB: 
 
 Let A, = AGP, Ay = AOLP’, Ar= PO.B, Ad’ = P'0,8B, 
 and in the diagram draw 0G perpendicular to BO,; then 
 GO, = 0:0, cos Ao, KO, = 0,0. cos Ao’. Subtracting, 
 since 0:0. = 0,02' = (Rh, — fa), and K0O;' ~ G0, = GB— 
 AB =p, 
 
 p= (hk; — A) (cos Ay’ — COs Ae) 
 whence 
 pe eee Ay (117) 
 
 cos Ai = 
 dis ee R, : 
 
 PO,P' =(A2 — Ad’) and the point P is zdvanced, 
 
 b. When the tangent V'B' is outside of VB: 
 | p=(R. — R,) (cos Aa — COS Az’) 
 i i whence 
 
 ee S P 
 
 PO, P’ =(Ae’ — As)and the point P is moved back and the 
 arc AP diminished. 
 
 Fie. 47. 
 
 | In case the curve terminates with the arc of shorter 
 ‘| radius, or R, follows R.. Fig. 47%. 
 ec. When V'B is inside of VB: 
 p= (R = Ri) (cos Ay — COs Ai) 
 whence 
 
 cos A, = cos Ai — Peas (119) 
 
 POP’ = (Ay — «A1) and the point P is moved back. 
 
COMPOUND CURVES. 115 
 
 d. When V'B' is outside of VB: 
 p = (fig — Ri) (cos Ay’ — cos A:) 
 whence 
 p 
 
 0S Ai = Os Ai Fiols B (120) 
 bg — 401 
 
 PO,P’ =(A: — 41) and the point P is advanced. 
 Example.—Let R = 9992.01, Ri = 1432.69, Ae= 28°, and 
 p = 20.07 inside of VB; case a. 
 
 p 20.07 log 1.302547 
 (117) R, — R, 859.32 ‘* 2.934155 
 023356 ‘* 8.368392 
 As 28° cos .88205_ 
 A's 25° ‘* 906306 
 POG: 3° 
 
 170. Given: a compound curve terminating mn a tangent, 
 to change the P.C.C. and also the last radius, so that the 
 curve shall end in a parallel tangent at a point on the 
 same radial line as before. Fig. 48. 
 
 Fig, 48. 
 
 Let APB be the given curve ending in the tangent VB; let 
 V'B' be the given parallel tangent; and let 9 = BBA= Hl= 
 tne perpendicular distance between tangents. 
 
 ' Tt is required to change the point P to P’, and also the 
 value of R, to R,', so that the new curve May endin V'B’' at 
 2’ inside of VB on the same radial line BOs. 
 
 In the diagram produce the ‘are AP. to G to meet 0:4 
 drawn parallel to 0.B; then PO,G = Az. Draw the chord 
 PB, and it will pass through G. Lay off the distance p from 
 
116 FIELD ENGINEERING. 
 
 Bon BO; to find B'; draw B'G and produce it to intersect 
 the arc APG in P'.. Then P’ is the P.C. C. required. Join 
 P' 0; and produce it to meet BO: produced in 0... Then 
 P'0.' = B'0,' = Rf,’ the new radius, with which describe the 
 are P'B'. 
 
 By Geom. Tab. I. 18: 
 
 PBV =} P0:B = 44a, and @B'V' =4P'O.B =4A0'. 
 
 PGP’=BGB' = #A2— A?) 
 
 Draw 0,K perpendicular to BO. 
 
 Then OK = BH=s B= 0; O. sin Gy = (Re == R) sin Ag 
 GH 2 GI s2 x 
 
 GI 
 tan tO. = pr tan Ae SBC) EBLE 
 
 p 
 t tAo == te 4 — —— F 
 antAce ané+Ae (Racadhisin Ds (121) 
 
 In the triangle 0:0:02' 
 sin Ae’: in Aat: 0100: O100'::: (Re. —)Ra) 3\(Re = fi) 
 
 and ‘ 
 mae ME |< cla. 
 Ry == (be Ry) Tae: + fy (122) 
 
 If B'V' were outside of VB; 
 
 p 
 
 = tan ? : 9 
 
 tan Ao an 4Aqgt+ (R; Se R,) sin A‘ (1 3) 
 Ri = (Rae hee (122) 
 
 sin Ag 
 
 When the smaller radius R, follows R,: If the given 
 tangent B'V' is inside of BV. Fig. 49. 
 
 tan 4A,’ = tan4Ai + Aiea . (124) 
 1 1 
 j Mpg eT PARIS, ca gee (125) 
 
 sin Ai 
 
COMPOUND CURVES. 
 
 If B'V' is outside of BV: 
 
 pid jhe ye “OS eee 
 tan Ai SS valk 3.28 1 (R; — R) bins Aa (126) 
 
 Rico haa. RD So (125) 
 
 Fia. 49. 
 
 Example 1.—Ig. 48. 
 
 Let R, = 2292.01 p = 20.07 inside. 
 “« Ry, = 1482.69 Ag = 28° 
 
 (121) Ry — R, = 859.32 log 2.984155 Mh 
 As 28° log sin 9.671609 Hh: 
 2.605764 | 
 p 20,07 1.302547 | 
 04975 8.696783 IM 
 tan Ae 24933 | 
 .*. tan $Ae! 19958 ab atg Wh 
 Gea 92° 84’ sin 9.584058 Vn 
 (Ry B,) 2.934155 i 
 3.350097 | 
 i 28° sin 9.671609 
 (Ra’ ~ Ri) 1051.25 3.021706 
 R, 1482.69 
 
 Ans. R,! 9483.94 .*. D = 2° 18' 26" 
 PO P == 128° 22°84) = 5° 26°. PPo = 135.83 ft. 
 
1is 
 
 FIELD 
 
 Keample 2.—Fig. 49. 
 
 Let R, = 2292.01 pp 
 
 ENGINEERING. 
 
 = 20.07 inside. 
 — AG. 
 
 (124) R,— R, 859.82 log 2.934155 
 Ai 46° log sin 9.856934 
 2,.'791089 
 p 20.07 1.302547 
 08247 8.511458 
 tanyA, 42447 23° 
 tan $A,' 45694 24° 333! 
 SNE 49° 07’ log sin 9.878547 
 Peer 9.934155 
 3.055608 
 Nie 46° log sin 9.856934 
 817.60 2.912542 
 Re 2292.01 
 Ans. »| Ry 1474.41 0D =B° 53" 12" 
 PO:P! = Ay! — A: = 8°07.” are PP’ =" = 124.67 ft. 
 
 Observe that in either figure both tangents must be on the 
 same side of the point G, in order to a solution. 
 
 Fra. 50. 
 
 17. Given: a compound curve ending in a tangent, to 
 change the last radius and also the position of the P.C.C., 
 
 so that the curve may end in the same tangent. 
 
 Fig. 50. 
 
y 
 
 COMPOUND CURVES. 119 
 
 I. When the curve ends with the greater radius Re. 
 Let APB be the compound curve in which R; R, A, and 
 A» are known. 
 in the diagram draw the chord PB and produce the first 
 arc AP to meet it in G; draw 0,G, and produce it to meet the 
 tangent in K. Then by § 137 O,K is parallel to 0.B, and by 
 eq. (57) MI 
 GK = (R. — RB.) vers Aa (127) ni 
 
 If we assume P' as the new P.O.C,, we have A2'= P'0,'B,, 
 and the chord P’G produced will intersect the tangent at the 
 new point of tangent B’, and BO,' = R,'. . Similar to eq. (12) | \ 
 we have HH 
 GK = (R;' — R,) vers A: Wi 
 
 and equating the two expressions, we obtain HA) 
 
 rae (R, — Ri) vers Az _ GK 
 Fy! = Fa vers Ae pie vers Ad eo) He 
 
 If we assume Raz, we have vee 
 
 R, — hy GK 
 Fil hts Cate a ere A 2 
 vers Aa = By Bye pied ah Bas Bh , 
 
 In the two right-angled triangles BAG and B'KG, we have 
 
 BK = GK cot tAs i 
 BK = GK cot 4A: Win 
 
 and by subtraction, | 
 | | 
 BB’ = GK (cot 443' — cot 44s) (130) | 
 in which GA is obtained from eq. (127). ti} 
 When BB’ as given by eq. (180) is negative, the point B’ falls He 
 between B and V. 
 If we assume the distance BB’ on the tangent, we have 
 from the last equation, 
 BB' 
 
 cot Aq — cot +Ae 3 GK (131) 
 
120 FIELD ENGINEERING, 
 
 GK being obtained from eq. (127) and FR,’ from eq. (128). In 
 eq. (181) use the + sign when B' is beyond B as in the Fig. 50. ' 
 
 Il. When the given curve ends with the smaller radius 
 R,. Fig. 51. 
 
 Fig. 51. 
 
 We have by a similar reasoning 
 
 GK = (R, — Ri) vers A, (132) 
 2 — Ry 1 
 vers A, vers Ai 
 1 fe fy ~__ GK 
 vers Ai = era, vers: Ay = Drie he (134) 
 BB = GK (cot $41 — cot $A1) (185) 
 BRB 
 
 cot¢A, = cot pA, += (136) 
 
 GK 
 using the — sign when £' is beyond B. 
 Hxanvple.—Fig. 51. 
 Let R. = 2292.01, RA, = 1482.69, aA; = 46°, and let the 
 
 P.C.C. be moved back 200 feet from P to P’; hence PO,’P’ = 
 5° and A,’ = 51°: to find the new radius R,' and the distance 
 
4) 
 
 COMPOUND CURVES. 
 
 log 2.934155 
 
 Kg. (182) R2 — BR, 859.32 
 Ai 46° « vers 9.484786 
 7. Ge log 2.418941 
 eq. (188) Ai’ 51° « vers 9.568999 
 R, — Ri 707.85 2.849942 Hy 
 Ry 2292.01 il 
 syn 1584.16 and D = 3° 87’ ) 
 eq. (185) GK log 2.418941 Hine 
 cot $A, 2.35585 23° A 
 cot $A1' 2.09654 25° 30' | A 
 0.25981 log «9.418819 it 
 - BB 68.04 1.832760 i 
 172. Given: a compound curve ending in a tangent, the last | i} 
 radius being the greater, to change the last radius and AW 
 also the position of the P.C.C. so that the curve may end at the 1 
 
 same tangent point, dui with a given difference in the 
 direction of the tangent. Fig. 92. 
 
 Let APB be the given compound curve, PO, = R, and 
 PO, = R, > fi. Bi 
 Let V’Bbe the new tangent, and the angle VBV' =, the i I 
 given difference in direction: to find BO,’ = R,', BO,’P' = H 
 A,’ and the angle P0,P'. 
 
 g 
 
{22 FIELD: ENGINEERING. 
 
 We have 
 Bo, oe 0,02 = Re Fi (Re <= R,) = Ri 
 Bo,’ a 0, 0, = R, a (Re'— Ry) = Ti; 
 
 From which we see that whatever may be the value of the 
 new radius, the difference of the distances from B and O, to 
 the new centre is constant, and equal to Ri. We therefore 
 conclude that the centres 0, and OQ,’ are on an hyperbola of 
 which B and O, are the foci, and R, the major axis. 
 
 This suggests an easy graphical method of solving the 
 problem. 
 tilt Through B draw a line perpendicular to the new tangent 
 At V'B which will give the direction of the required centre O,', 
 He On this line lay off BK equal to A, and since (y’ — hy) = 
 a 0,0.' = K0,', if we join KOQ,, the triangle KO,'0, is isosceles; 
 i} therefore bisect KO, and erect a perpendicular from the mid- 
 dle point to intersect the line BA produced in 0,'. Draw 0,0, 
 and produce it to intersect the are AP (produced if necessary) 
 in P'.. Then P’ is the new P.O.C. required, and BO,’ = 
 P'0O,' = R,', the new radius. 
 
 The analytical solution 1s as follows: 
 Adopting the usual notation of the hyperbola 
 
 Let 22 = R, = the-fajor axis, 
 ‘© 9 — BO, = the distance between foci. 
 
 Produce the arc AP and through B draw the’ tangent BH, 
 and join HO, = R,. Then im the right-angled triangle BHO, | 
 
 1) BH? = BO, — R,? = 4c° — 4@? 
 Now by Anal. Geom., c? —a@’ = 0’. 
 Therefore 2b = BH = the minor axis. 
 Draw the chord PB and produce the arc AP to cut it in G 
 Then by Geom. (Table I. 24) 
 BH*?= PB x GB = 2R, sin +Ag x 2(Re — R,) sin +Ao 
 
 = 2 sin dae Ratha — Ba) (137) 
 
ny 
 
 COMPOUND CURVES. 123 
 
 Let a = the angle HO,S, then 
 
 BH R, 
 
 tan Q = Pie and Bo, =. Coe a (138) 
 In the triangle BO, 02 let 0,BO. = f ; then 
 igi Ras Bi | i} 
 vg eh Bo, sin Ae (139) iM 
 
 The polar equation of the hyperbola for the branch 10,02’, 
 taking the pole at B and estimating the variable angle » from 
 the line BO,, is Hy 
 c¢.cosv—a ii 
 
 r= 
 
 Wheno =f +4, r= Ry, and substituting the values ot 
 a, b, and ¢ found above, we have 
 
 BH? i 
 2 (BO, cos (6 + 7) — fi) (140) at 
 
 Ry — 
 
 using (+7) when V’ falls between V and A, as in the 
 figure, and ( — 7) when V’ falls beyond V. 
 In the triangle BO,0,', the angle BO.'O, = As’ and 
 
 Ha 
 Bo, } Bi & 
 
 sin As’ = a7 qe sin (f + t) (141) We 
 Finally 
 EE STOR 6 RIE ae | 
 
 Remark.—When V’ falls between V and A, as in Fig. 52, if | 
 the angle é be greater than the angle VBH, the curve ceases to 1 
 be a compound, and becomes reversed. Therefore VBH = | 
 «a — fis the maximum value of 7 possible in this case. When , id 
 V' falls beyond V, the point P’ will fall between Pand A; ll 
 and the largest possible value of 4 will then be that which 
 renders PO, P’ = Ai, and makes the point P’ coincide with A. 
 
124. FIELD ENGINEERING. 
 
 Example.—Fig. 52. Let Ri = 1482.69 Ai = 31° 
 a=. = 2292.01 As = 56° 
 
 2 
 
 (137) R, — RB, 859.82 log 2.984155 
 R, 2292.01 3.360217 
 2) 6.294372 
 3.147186 
 1KY 28° log sin 9.671609 
 2 0.301030 
 a BH 3.119825 
 (138) R, 1432.69 3.156151 
 a 42° 36' 28".7 log tan 9.968674 
 a 43°. 36) 23.7 log cos 9.866889 
 os, BO, 3.289262 
 (139) R, — R, 3 2.934153 
 9.644893 
 Aa 56° log sin 9.91857 
 B 21° 28’ 06".3 log sin 9.563467 
 (140) B+a 27° 98’ 06".3 log cos 9.948053 
 BO, 3.289262 
 1727.09 3.237315 
 R, 1432.69 
 294.40 X 2 = 588.80 2.769968 
 BE? 6.239650 
 Ry’ 2949.05 3.469682 
 (141) .°. Aa’ = 36° 18’ 26" 
 (142) .°. PO,P' = 18° 41' 84" = 342.8 feet. 
 
 Remark—This problem may also be solved by first finding 
 the new sides V'A, V'B, from which and the new central 
 angle (A + 7), and the radius #,, may be found A,’, Az, and 
 
 2.’, as in $162. The new sides are readily found from the 
 old ones by solving the triangle VBV’. If the original sides 
 are not given, they must be calculated as in § 164. 
 
 173. Given: a compound curve ending in a tangent, the 
 last radius being the less, to change the last radius and the 
 position of the P.C.C. 80 that the curve may end at the same 
 tangent point, dut with w given difference im the 
 direction of tangent. Fig. 53. 
 
ny 
 
 COMPOUND CURVES. 125 
 
 Let APB be the given curve, and PQ: = R2,and PO; = 
 R, < R. Let V’Bbe the new tangent, and VBV' =1, the 
 ziven angle; to find BO,’ = R,', BO,'P'= Ay, and PO.P’. 
 
 We have 
 
 Bo, + 0,02 — Ri; + (Re — RF) = R, 
 Bo,’ +. O05 a RS + (R, ae Rs) => R, 
 
 from which we infer that the locus of the centre O,' is an iN 
 ellipse, of which B and OQ, are the foci, and & the major axis, . 
 
 Fie. 53. 
 
 since the sum of the distances BO,’ and 0,0,’ is always equal Bi 
 to Re. 
 
 This suggests an easy graphical solution of the prob- 
 blem, as follows : 
 
 Perpendicular to V'B draw the indefinite line BA, which Hy 
 will contain the required centre 0,', and lay off BK = fy. | 
 Join KOxz, bisect it, and from the middle point erect a perpen- Hi | 
 dicular to intersect BK in 0,'.. Join 020,’ and produce the Hi 
 line to intersect the arc AP (produced if necessary) in P’, | 
 which is the new P.0.0. required. P'O.' = BOS Br abhe 
 required radius, and P! Of Bid 1: 
 
 The analytical solution is as follows: Adopting the 
 usual notation of the ellipse, 
 
 let 24 = R, = the major axis, 
 ‘© 9¢ — BO, — the distance between foci. 
 
 At Berect BH perpendicular to BO; to intersect the arc AP 
 
 w 
 
126 FIELD ENGINEERING. 
 
 (produced if necessary) in H, and join HO, = R,. Then 
 BH? = R,? — BO,’ = 4a — 4¢ 
 
 But by Anal. Geom., a? —. ¢? = 0. 
 
 Hence 2b = BH = the minor axis. 
 
 In the triangle BO,O, we know BO, = R,, and 0,0; = 
 R, — R,, and the included angle BO,O0, = 180° — A,; hence 
 by Trig. (Tab. II. 25) 
 
 tan 3(0,0.B — 0,BO:2) = a tandAi (148) 
 2 
 Wy The angles at B and 0, are then found by (Tab. II. 26). 
 Ail Let 6 = the angle 0,B0.; then 
 
 | sin A, 
 
 | BOs = (ha — fis) sin 6 (144) 
 The value of BH? above may be written 
 BH? = (R, + BO) (R2*— BO2) (145) 
 
 The polar equation of the ellipse, taking the pole at B, and 
 estimating the variable angle v from the axis BOs, is 
 
 }? 
 ~ &@—C. COSY 
 When » = 6 ¥ 7, then r = Ry’, and substituting the values 
 of a, b, and ¢, given above, we have 
 
 BH? 
 
 th = 9(R, — BO, cos (B ¥ 4) (146) 
 
 HA using (# —7) when V" falls between Vand A, as in Fig. 93, 
 ni and (8+ ¢) when V’ falls beyond V. 
 
 | in the triangle BO,'O,, the angle O,'BO, = (fF ¢), and the 
 exterior angle BO;'P’ = Ax‘; hence 
 
 BO ; ; 
 oa Pe sin (f F 7) (147) 
 
 sin Aa = 
 
 Finally PO,P' =(A, #¢)— Al (148) 
 
 When V’ is on AV, then POP’ is negative, showing that 
 it must be laid off from P toward A; but when V’ is beyond 
 
COMPOUND CURVES. 127 
 
 V, then PO,P’ is positive, and P’' will be on AP produced 
 The only limits imposed on the angle ¢ are that the resulting 
 value of PP’ shall not exceed PA, and that R,' shall not be 
 less than a practical minimum. 
 
 Example.— Fig. 538. 
 Let Dy =.3° 20’. Ra=.1719.12. Ar = 28° 20° 
 
 ip Us R, = 955.87 A, = 48 t= 45 
 
 The resulting values are as follows: 
 p 21° 09’ 82".6 
 
 BO, 1572.42 3.196567 
 
 HH? 5.688829 
 
 Ry 1273.65 3.105652 
 Ay 54° 56 
 POP! 14° 41’ 
 
 2 
 PE, 440.5 
 (See also remark at end of $172.) 
 174: Given a simple curve joining two tangents, to re- 
 
 place it by a three-centred compound curve between 
 the same tangent points. Fig. 54. 
 
 Fig. 54. 
 
 Let R = AO = radius of simple curve. 
 R, = P0O,=P'0A,<h A, = PO,P' 
 R, == AO, aa BO; — R INCU = AC a BO;P' 
 AOL 
 Since AQ, is made equal to BO; and VA = VB , AO.P musi 
 equal BO,P', and the compound curve will be symmetrical 
 about the bisecting line VO; and the centre O, will be on the 
 line VO. : 
 We have at once from the figure, 
 2h 4- ets (149) 
 
128 FIELD ENGINEERING. 
 
 In the triangle 00; 02 we have 
 0:0.: O02 :: sin AOV: sin POV 
 whence 
 R, — R)sin $A 
 (By S'B)aingA a0 
 sin 3 Ay’ 3 
 
 which expresses the general relation between the quantities, 
 Rand A being given. 
 
 We may now assume values for Ry and Res subject to the 
 above conditions, viz., Ri < Rand Rk, > Ah; whence 
 
 sin +A — Sa (151) 
 
 In selecting values for A, and fs, the degree of curve D, 
 should be but little greater than D of the simple curve, say 
 from 30 to 60 minutes, while Dz may be taken at 4D to 4D. 
 
 Eeample.—Given: R= 1719.12 D= 3°. 20'. A = 40° 
 
 Let R, = 1432.69 D, = 4° 
 
 «© R, = 5729.65 D, =1° 
 Ry — R 4010.53 
 R, — R, 4296.96 
 
 log 3.603202 
 3 633161 
 
 “« 9.970041 
 tA . 20° log sin 9.584052 
 tA 18° 36°57" ‘* “ 9.504093 
 A 87° 13' 54" 
 Ae 1° 23° 038" 
 AP = P’B : 138.48. 
 Again we may assume A, and fy, whence 
 AN er Aa 7 2Ae 
 and 
 . #sinta — R, sinéAy ve 
 Fees Kinin Gn, <Sstiag Ai eu 
 Example.—Given. R= 1719.12 A= 40° 
 Let R, = 1482.69 A,=1° .*. Ai = 38° 
 Ans. Ry = 7887.24 «+. Dy = 0° 464° AP = 129. 
 
 Finally we may assume Ay and feo, and deduce A, and &; 
 from eqs. (149) (150); but this is the least desirable because 
 
ay 
 
 COMPOUND CURVES. 129 
 the value of R, so found will not usually give a convenient 
 value to the degree of curve D. 
 
 175. To determine the distance HH’ between the middle 
 points of a simple curve and a three-centred compound curve 
 joining the same tangent points AB. Fig. 54. 
 
 In the triangle 00:02, we have 
 
 (Rp, Rye Aa 
 O00, os (R, R,) sin $A 
 HH' = 00, + 0,H' — 0H | 
 HHH = (Bs SR) — (R— By *(158) 
 
 sindA 
 
 in the first example given above HH’ = 14.55, and in the | 
 
 second HH' = 17.05 ft. 
 
 In many instances the distance HH’ is so great as to render 
 this problem practically useless, unless the distance HH, is 
 discounted beforehand by putting the simple curve AHB a 
 sufficient distance inside of the proper location through the 
 
 point H'. But the problem given below is usually preferable. Wi 
 
 176. Given, 4 simple curve joining two tangents to re- Vee 
 place it by a three-centred compound curve which Ha 
 
 shall pass through the same middle point H. 
 I. The curve flattened at the tangents. Fig. 55. 
 
 Let R= AO, the radius, and A = the central angle of the 
 simple curve AHB, and let H be the middle point. 
 
 Midas Ope. 
 Decade Oa. maka GED. 
 
 Let Ti = PO; — HO, 
 Es Fo = BOs = A' 0. — B'O0s 
 
 « A’ and B' be the new tangent points required. 
 We have at once, as in the last problem, 
 
 2ZAe+ Ai= A. 
 
130 FIELD ENGINEERING. 
 
 Since the curve is to be symmetrical about VO, BP = RS 
 PA = P'B, and AA' = BD’. 
 
 Bt 
 
 In the diagram produce the arc HP to G, and draw OG 
 parallel to OA, and produce it to K. Then a tangent line al 
 G will be parallel to VA; and by § 187 the point G will be ov 
 the long chord HA, and on the long chord PA’. GK is the 
 perpendicular distance between parallel tangents, and the 
 problem is similar to that given in $171; whence by eq. (57) 
 we have, in this case, 
 
 GK = (R, — R,) vers Aa = (R — Ri) vers 4A. (159) 
 
 for the general equation in which & and A are given, 
 Analagous to eq. (180) we have 
 
 AA' = KA' — KA = GK cot GA'K — GK cot GAK. 
 - AA' = GK (cot 4A2 — cot $A) (156) 
 in which GK is obtained from (159). 
 We may now assume values for R, and Rz, making RB, < R 
 
 and R, > R, and deduce the values of As, Ai, and AA’, 
 Solving eq. (155) 
 
 _ (R— Ry) versta _ GK 
 Were Ay Cr a ee nly (157) 
 
 Eq. (154) gives A., and eq. (156) gives AA’ 
 
COMPOUND CURVES. 
 
 Example.—Fig. 55. 
 
 Given: R= 764.489 D = TV 30’ 
 
 Let R, = 716.779 ye hl 
 “« R, = 3487.870 Dz = 1° 40’ 
 (155) R — Ay 47.71 
 +A 20° 
 
 GK 
 R,— Ry 2721.091 
 
 Ae (say) 2° 38° 
 ie A'P 158.00 Ai = 34°44 
 (156) tAs 43.5081 = cot 1°19 
 
 +A 5.67138 cot 10° 
 37.8368 
 
 Gk 
 
 AA’ 108.87 
 
 131 
 
 A = 40° 
 
 log 1.678609 
 log vers 8.780370 
 
 log 0.458979 
 ° 3 434743 
 
 log vers 7.024236 
 
 log 1.577914 
 0.458979 
 
 «© 2.036893 
 
 Again, we may assume Az and Ay < R; whence 
 
 A, = A — 2A: 
 
 and 
 eq. (155) GK = (Rk — &) vers tA 
 and 
 GK 
 fis = Briere Ae 
 
 Eq. (156) gives AA’. 
 
 (158) 
 
 Again, we may assume Az and the distance AA’, whence, 
 
 from eq. (156) . 
 AA' 
 
 GK = : 
 cot $A2e— cotzA 
 
 GK 
 
 AACE = R — —_— 
 eq. (155) R= R SAN 
 
 eq. (158) gives Re. 
 
 (159) 
 
 Again, we may assume R, < Rand AA’; then, eq. (155) 
 
 = (R — R,) verstA 
 and eq. (156) 
 
 cot 4A, = cotta + = 
 
 and eq. (158) gives Rs. 
 
 (1603 
 
132 
 
 FIELD ENGINEERING. 
 
 Hxample.—Fig 59. 
 
 Given: R = 764.489 DZ T 3 A = 40° 
 Let) Ah, = 716,779 BD, = 
 ee AA: 
 Hence by last example, 
 GK log 0.458979 
 eq. (160) AA’ 110. 2.041393 
 38.2309 1.582414 _ 
 cot $A 5.6713 10° 
 cot tAe 43.9022 1° 18’ 18” log 1.642486 
 (158) Ae (say) 2° 87 log vers 7.018147 
 GK 0.458979 
 R,— Ri = 2759.5 3.440832 
 R, 3476.3 Deg == lod 
 AP’ 157. Ai = 384° 46 
 
 Il. The curve Sharpened at the tangents. Fig. 56. 
 
 This case will only occur when, with a given external dis 
 tance VH, a simple curve would absorb too much of the tan- 
 gents. 
 
 Fie, 56. 
 
 Let AHB be the simple curve, and 
 “ 4'PHP'B'the required compound curve, 
 hg == PO. = HOF A, POF 
 “ R= PO, = AO. B'0,; 4:2 A OP= P'0;B', 
 We have from the figure, 
 24Ai+ Bo= &. (161) 
 
COMPOUND CURVES. 139 
 
 In the diagram draw 024 parallel to OA cutting the tan- 
 gent at K, and produce the arc HP to G. Draw the chords 
 GH and GP, passing through A und A’ respectively. We 
 have then a discussion similar to the preceding case, and to 
 the problem § 171, Fig. 51, whence we derive the general 
 formule: 
 
 GK = (R, — Rh) vers Ai = (R. — R) vers +A (162) 
 and 
 AA' = GK (cot $A: — cot ¢A) (163) 
 1. Assuming fi < Rand R, > Rk 
 
 Gk Ry — 
 
 R 
 vers: Ai =" = ae, SO vers 4A (164) 
 
 9. Assuming A, < 4A and diy << Se 
 
 ra 1 ae 
 R vers tA — Ry vers Ai (165) 
 
 Ri a 
 vers $A — Vers Ai 
 
 3. Assuming Ai < $A and AA 
 
 AA’ 
 GK = Ta; = cotta ie 
 Gk 
 Ba = Rat vere FA ett 
 GK 
 By = Be ers Bi a 
 
 4, Assuming R, > R and AA’ 
 GK = (R. — R) verszA 
 
 cot.4Ai.= cobtA +. pi (169) 
 
 The third assumption will usually secure most readily the 
 
 desired curve. AA’ should be assumed as small as the nature 
 
 of the case will allow, and A, should not be much smaller 
 than $A. 
 
 It is evidently not necessary that the new curve should be 
 symmetrical; for having laid out the curve A' PH, the simple 
 curve HB may then be used, or, if desirable, some compound 
 curve HP'B' determined by an assumed value of BB’ not 
 
 equal to AA’. 
 
134 FIELD ENGINEERING. 
 
 These formule (154) to (169) are readily adapted to the 
 case of substituting a compound for a simple curve when it 
 is necessary to keep one tangent point fixed, but to move the 
 other a certain distance in either direction on the tangent. 
 For if in Figs. 55, 56, we draw a tangent at H, and make H 
 the fixed point of tangent, it is evident that the central angle 
 of the curve will then be AOH. The only change necessary, 
 therefore, to adopt the formule to this case is to write A in 
 place of +A, and to observe, instead of eqs. (154) (161), that 
 
 Ai 4 Aze= A. 
 Hrample.— Fig. 55. 
 Let it = "TOM. 08 = Be" 
 
 Assume AA’ = 260. Ayes 08" 2S. Age S 
 Kq. (166) AA’ = 260. log 2.414973 
 : cotsA, 2.90421 19° 
 cotta 2.60509 21° 
 .29912 log 9.475846 
 rs GK “ 2.939127 
 Eq. (167) 4A 42° ** vers 9.409688 
 3384.07 3.529439 
 R 1910.08 
 ee R, 5294.15 D = say 1° 05’ 
 Eq. (168) GA log 2.939127 
 Ai 38° *“ vers 9.326314 
 4100.27 3.612818 
 R, 1193.88 D= 4° 48' 
 AP 791.67 PH = 369.23 
 
 177. Given, two curves joined by a common tangent 
 to replace the tangent by a curve compounded with 
 
 _ the given curves. Fig. 57. 
 
 Let R, = BO; the radius of one curve, 
 fits = AO; the radius of the other curve, > R,, 
 
 1 = BA the common tangent, 
 ““ R; = PO, = P'O, the radius of connecting curve. 
 ““ As = POP’ the central angle of “ 4 
 ‘a= AO;P' and 6 = BOP. 
 
COMPOUND CURVES. 135 
 
 In the diagram join 0,0; and draw 0,G parallel to BA. 
 Then in the right-angled triangle 0; GO, we have, 
 
 .. GO,  Bs—k 
 cot 4 = rita be j (170) 
 0:03 = pis preak tC al eS. (171) 
 COS 7 §1nN 2 
 
 which gives the distance between the centres of the given 
 curves. 
 
 ace sena Dewees 
 
 Fra. 57. Wh 
 
 We shall now assume the following geometrical truths, fll} 
 which may be easily demonstrated. it 
 
 If two circles intersect in one point, they intersect in two 
 points; and the line joining the two points is the common 
 chord. 
 
 The common chord is perpendicular to the line joining the 
 centres, and when produced it bisects the common tangents. 
 
 If a third circle is drawn touching the two circles, a tangent 
 to the third cirele, parallel to the common tangent, will have 
 its tangent point on the common chord produced. 
 
 Conversely, therefore, if the tangent BA be bisected at K, HI | 
 and a line, KJ, drawn perpendicular to 0,03, KI will coincide | 
 with the common chord produced, and the angle Anes 
 AO,0, =?. If on KI we assume a point T through which 1 
 it is desirable that the connecting curve should pass, then J is | 
 the tangent point of a tangent parallel to BA; consequently i 
 a line through J perpendicular to BA contains the required | 
 
 eentre Ox. 
 
136 FIELD ENGINEERING. 
 
 All I. Let p = HI = the perpendicular distance between the 
 tangents. 
 
 If in the diagram we join [A and JB, and produce the 
 chords to intersect the given curves in P and P', then P and 
 P’ are the points of compound curvature; and the lines PO, 
 and P'Os produced will intersect JO, in the same point 0,; 
 and the angles P'O,J = « and PO.I = f. 
 
 In the triangle AJB the line KT bisects the base AB, and 
 we have by Geom. Tab. I. 25. 
 
 AI? + BI? = 24K? + 2KT? 
 By eq. (56) AI = 2(R. — Rs) sin 4a 
 i) BI = AR, = R,) sin 38 
 
 AK =H ands /KI 2. 
 S1n 2 
 
 Seley 
 
 . 4a — Rey sin? ta + 4(Ra — Ray sin? 48 = 4P + 
 
 Dividing by 2 and putting vers « = 2 sin? ta and vers paa 
 2 sin? 46 (Tab. IT. 46) 
 
 9 
 (fh, — Rs)? vers « + (Rz — Ri)? vers 6 = 4P +. os 
 But by eq (57) 
 
 { 
 Hi 
 i (R. — Rs) vers a = (R, — R,) vers B = p (172) 
 
  p QRy = (Ry + R))= ye 4 
 
 Hii} sin? 7 
 
 P Dp 
 | | From (172) 
 Pp Pp 
 = ——_ ; So sve a 174 
 vers @ a vers fj jek (174) 
 
 and from the figure 
 
 These formule solve the problem when p is assumed, If 
 desirable we may find a and ( independently of R., for in 
 
COMPOUND CURVES. 
 
 the triangle AJB, JAB = 4a and IBA = 14; and since 
 HK = p cot 2, 
 
 AH 41 — HK l ; ty 
 cot ja = aT = 2p — cot? (176) 
 
 We AK Y ; 
 fe = -— a == op + cot2 (177) 
 
 II 
 
 cot 46 
 
 Il. In case a or f ts assumed, we have from the last equa- 
 tion 
 l l 
 
 = —<~ = 178 
 RES 2(cot +a + cot 7) cot 44 — cot 2) fa79) 
 
 Ill. In case the radius Ry is assumed, then in the triangle 
 0,0.03 we know all three sides ; for 0:02 = (Ry — fh), 
 
 0,03 = (R2 — Ri), and 0,03 = Ba ey 
 COS @ 
 
 By Trig. (Table II. 31.) 
 
 2 ‘ae 9 s— € 
 vers Ae = He oe a ; 0320s) 
 1 2 7 B) 
 
 in which s = + sum of the three sides. 
 Substituting values, and reducing, observing that, 
 
 tai) (itd) maces 
 (om : ern =f SOP AIAN 
 
 and that (R; — R,) tan 7 = J, we have 
 
 [? 
 vers Ac = 5(Ry — Ri) (Ra — Ba) (179) 
 
 In the same triangle. 
 
 : : 030. 
 sin 0,0,0. = sin Ae 60. 
 
 But from the figure 0;0,0. = 7 — #, and taking the value 
 of 0,0, from eq. (171). 
 
FIELD ENGINEERING. 
 
 sin (i — 6) = (A, — Bs) sin As sin (180) 
 
 We then find a@ from eq. (175) and p from (172). 
 
 The angles « and # may be found otherwise, for by Trig 
 “Tab. Il. 27) we have in the triangle 0,0,0; 
 
 sin +(0, Os Oz as 0; 0, Oz) = eal COS Ao 
 1 3 
 
 or 
 
 ‘ ., &€—f\ _ (Rs — R,) cost costAg 
 sin (90 —(@+ ae) a Rate 
 
 *. Cos («+ £ 3") = COS?. Cos 4A, (181) 
 
 which is a convenient formula when ¢ and A; are not too 
 
 a—fp 
 
 small. Having obtained ao 
 ft 
 
 we have 
 
 a— p ie a—p 
 9 PA gre (182) 
 
 a= gAot 
 
 For a constant value of / the Jess the difference of R; — R, 
 the greater will be the value of the angle 7. When R; = R,, 
 cot ¢ = 0 and ¢ = 90° and the tangent point T will be ona per- 
 pendicular to BA drawn through the middle point A; and 
 a= f. On the contrary, as (R,; — ft,) increases, 7 becomes 
 less, and the foot, H, of the perpendicular HZ moves toward 
 B, the tangent point of the curve of smaller radius R;. The 
 distance HK = p cot ¢. The connecting curve is farthest 
 from the tangent BA at J. To find the ordinate from BA to 
 the curve at any other point, subtract from p the tangent 
 offset for the length of curve from J to the ordinate in ques- 
 tion. §115, eq. (89) may be used on flat curves with tolera- 
 ble accuracy, even when the distance equals several hundred 
 feet. 
 
 IV. It is evident that in this problem R, must be greater 
 than either R, or R;. As the centre O, is taken nearer the 
 
a 
 
 a 
 
 COMPOUND CURVES. 139 
 
 linc 0,03, Rz grows less, and is a minimum when OQ, falls on 
 the line 0,03. In this case we have A. = 180°, and 
 
 R, = (Rk; + R, +0,0;); a minimum. (183) 
 
 This limit must be regarded in assuming the value of Ry. 
 Since 
 
 0,0, aps 0:03 — (Lt. rx R,) wy (Re a Rs) == (Rs <7 Rf) 
 
 a constant value, independent of R:, we infer that the centre 
 OQ, is always on a hyperbola of which O,; and Qs are the foci; 
 (R, — R,) equals the diameter on the axis joining the foci; 
 and / equals the diameter at right angles to it, for in the tri- 
 angle 0,4 0s, 
 
 i? =O, 0g he aie (184) 
 
 Example.—Fig. 57. 
 
 Given: R, = 1482.69 Rs; = 1910.08 and / = 400. 
 
 Assume p= 11.4 to find Ry, and £. 
 
 Eq. (170) Rs — A, 477.39 log 2.678873 
 l 400. ‘© 2.602060 
 
 39° 57’ 84" log cot 0.076818 
 
 i 
 Eq. (1°78) i 39° 57' 34" sin 9.807701 
 5 39° 57°34" sin? 9.615402 
 D 11.4 log 1.056905 
 * 217.64 <* 1441508 
 419 «4.602060 
 p “1.056905 
 * - 3508.77 “3.545155 
 R+R, 8842.77 
 2) 6879.18 
 oh R; 3439.59 (say) 8487.87 
 Eq. (174) p 11.4 “© 4.056905 
 R,—R; 1527.79 «3.184064 
 a 7° 00' log vers 7.872841 
 11.4 log 1.056905 
 
 p 
 R,— Rk, 2005.18 «© 3.802153 
 p (nearly) 6° 07’ log vers 7.754752 
 
 13° 07" 
 
140 FIELD ENGINEERING. 
 
 Krample.—Fig 57. 
 
 Given: R, = 1482.69, Rs; = 1910.08, and 7 = 400. 
 Assume fy = 3487.87, to find Aa, 6,a and p. 
 2 
 
 Eq. (179) log 0.301036 
 
 R,—R, 2005.18 © 3 3091538 
 R,— Rs 1527.79 © 3184064 
 “ 6.787247 
 12 5.204120 
 . me 18° 07' 22" log vers 8.416873 
 Hq. (170) Rs— PR, 477.89 log 2.678873 
 eae A00: © 9 602060 
 ve i 39° 57’ 34" log cot 0.076818 
 Eq. (180) j 39° 57' 34" —_ log sin 9.807701 
 es 13°.07' 29” «9 356099 
 R,— Rs 1527.79 log 3.184064 
 log sin 2.347864 
 tae 200. log 2.602060 
 i— 33° 50' 39" —_log sin 9.745804 
 oe B 6° 06’ 55” 
 Eq. (175) a 7° 00! 27" 
 Eq. (172) Ry ~Rs log 3.184064 
 at 7° 00' 27" log vers 7.873309 
 p 11.41 1.057373 
 
 178. Given: a three-centred compound curve to replace 
 the middle are by an arc of different radius. 
 
 I. When the radius of the middle are is the greatest. 
 Fig. 57. 
 
 First find the length and direction of the common tangent 
 AB. Let A». = central angle of the middle arc, k, = its 
 
 radius, and #, and R; the radii of the other arcs. From eq. 
 (179). 
 
 L = V2(R. — Ri) (R2 — Rs) vers Aa (185) 
 
 Then find ¢ by eq. (1'70), a and 7 by eqs. (181) (182), and p by 
 eq. (172). 
 
 For the new arc we may now assume a new value for p, or 
 for Rs, or for a. Indicating the new values by an accent, if 
 we assume p’ we proceed as in the last problem, using eqs. 
 (173), etc. If we assume R., we use eq. (179), etc. If we 
 assume a’, we use eq. (178). 
 
COMPOUND CURVES. 141 
 
 Il. When the radius of the middle arc is the least of the 
 three. Fig. 58. 
 
 In this case the middle arc is within the other two pro- 
 duced; and for the same values of R,R; and 0,03, the locus 
 
 Fia. 58. 
 
 of the centre 0; is the opposite branch of the hyperbola found 
 in §177. When the centre O» falls on the line 0,0:, As = 
 
 180°, and 
 R, = (Rs + Bi — 0102), a maximum. (186) 
 
 Analogous to eq. (185), we have 
 i= V2(R, — Re) (Rs oa Rs) vers Aa (187) 
 which gives the length of the common tangent YZ. 
 
 We then have the values of ¢ and of 0,03 by eqs. (170) (171), 
 and of a and / by egs. (181) (182), and analogous to eq. (172), 
 
 p = (R, — Rs) vers @ = (R;—R:) vers (188) 
 in which p is the perpendicular distance HJ between parallel 
 
 tangents. 
 For the new are we may now assume a new value for p, for 
 
 R,, or for e. Indicating the new values by an accent, if we 
 assume p’, we have, analogous to eq. (178) 
 
FIELD ENGINEERING. 
 
 ia eee VI is (a Baie ae 7) Gem 
 
 and from eq. (188) 
 
 vers = at ; vers-9' = oe (190) 
 If we assume R,’, we have, analogous to eq. (179), 
 
 5 Ue b? 
 vers Ag = 2h — 1B, — Be) (191) 
 
 and we find a and f by eqs. (181) (182), and p’ by eq. (188). 
 
 Ill. When the radius of the middle are has an interaedi- 
 ate value, compared with the other radii. Fig. 59. 
 
 In this case, whatever be the value of hz, we have 
 | 0:0, + 020; = (Rs Ser Re) + (fe Te fi) = (Rs he Rj} 
 
 a constant value independent of R,; hence we infer tha) the 
 locus of Oz is an ellipse, of which O; and Q; are the foci, and 
 (R; — Rf) equal to the transverse axis. 
 
 Let / = QQ = the conjugate axis, and let ¢ = Q0,0, = 
 Q0;,03. 
 
 Produce 0;Q to G, making QG = 0,Q, and join GO, 
 
4 
 
 COMPOUND CURVES. 
 
 Then by similar triangles GO, is perpendicular to 010s, and 
 GO, = 1; and in the right-angled triangle GO30; 
 
 F. agrinstr Oo l 
 phi Od. wpm rd 192 
 sant =o. — Ri Fh, (19%) 
 
 0,0; —(R; — Ri) cost = ¢ cot t (193) 
 
 Analogous to eqs. (189) and (187), we have 
 
 1 = V &R; — Rs) (Ra — Bh) vers Az (194) 
 
 which may also be derived from the triangles 0,0,0; and 
 
 0:03Q. 
 Let « = 0203;0;, and 6 = 020103 
 Then 
 ] 2° oe vi — R ° ° . 
 sin wa = “ a sin Az = aa tan 7. sin Az (195) 
 
 From the figure 6 = 42— & (196) 
 
 In the diagram produce the lime 0;0, and it will intersect 
 ‘e it cuts the inner 
 
 all the arcs. At the points Z and Y, wher 
 and outer arcs, draw tangent lines perpendicular to O30\. 
 Draw the radius O.J parallel to 0:0, and the tangent line 
 
 IL at I. 
 Let g = ZY and p = ZL = HI 
 
 Then by the theory of parallel tangents, $137, the point J is 
 on the chord PZ produced, and it 1s also on the chord P’Y; 
 
 and we have 
 p = ZL =(Rz — R,) vers B. (197) 
 
 q-p= LY = (Rf; — Ry) vers a (198) 
 
 and g equals the swm of these. But g = ZY is the shortest 
 distance between the inner and outer arcs, and has a constant 
 value independent of &:. If we assume R, = (Rk; + #,) the 
 centre QO. will be at Q, anda = fp = i,and p=4q. Making 
 these substitutions above, 
 
 q = (R; — R,) vers 2. (199) 
 
 Also, from the figure, 
 
144 FIELD ENGINEERING. 
 
 ZY = 0;3Y — 0:2 —.0;0s, 
 i or, 
 
 q peed R; ae R, — 0, Os (200) 
 In the triangle ZTY we have by Geom. Tab. I. 26, 
 
 ZI? = IY? + ZY* — 24Y (ZY — ZL) 
 
 or 
 ZY — 2Z2Y.ZL = IY? — Zl? 
 Now, 
 ZI? = 4(f, — Ry)? sin? 46 = 2R, — R,)* vers p 
 ly? a 4(Rs — Re)? sin? 4a = 2(fs = Re) vers @ 
 } Hence 
 
 Hy ZI? = &B— Ry) p and IY? = &R,— Ba) (q—p) 
 
 Substituting these values, and solving for p, we have 
 
 _ Us — Rs — 49) _ (Rs — R. — 49) 
 
 Epa Bie yer 0,0; Kote, 
 Also 
 By = (Ra ~ 4) — p20 (202) 
 For any other value of R,, we have 
 | Bi = (By — 4) — p' 128 
 | i Hence 
 | By — Ry = 228 (p — p) (203) 
 
 Aisi which gives the change in R, for a given change in the value 
 of p 
 
 Observe that as p diminishes R, increases and vice versa. 
 Having determined the value of R,', we find p' by substitut- 
 ing &,' forR, in eq. (201); and from eqs. (197) (198) we have 
 
 vers f' = sy (204) 
 
 (205) 
 
s 
 
 COMPOUND CURVES. 145 
 
 and the change in the points of compound curvature is found 
 
 by (6 — fi’) and (@ — @). 
 
 Remark.—When Rz = +(Rs + Ry), Ae = 2, a minimum, 
 and the long chord PP’ is perpendicular to 0,03. When fz 
 is greater than this, « is greater than /, and vice versa. What- 
 aver be the value of Re, the long chord PP' always cuts the 
 line 0,0; produced in the same point S, at a distance from Zof 
 
 ZS = R, vers 7; 
 
 or from O, of 
 
 OS — R, COS Of 
 
 This item will be found useful in solving the problem 
 
 graphically. 
 
 Hzample. 
 
 781.84 
 1375.40 
 1910.08 
 
 11.30 
 
 Rs — Rs 534.68 
 Ry — fi 593.56 
 
 me 
 Tl Ut 
 
 Ae 
 rae l 458.27 
 (192) Rs — hi 1128.24 
 2 
 (193) 1 
 R; — Rk, 
 
 0,0; 1080.98 
 
 Ae 
 se a 
 (196) fp 
 (203) 0,0; 
 (200) q -97.26 
 0:03 
 q 
 
 9—p 11.30 
 Ra’ — fs 119.78 
 
 1495.18 (say) 1494.95 for 3° 50’ curve. 
 
 agi: 
 =3j47)10'A5) Als. = 482 
 Ds =. 3" 00' 
 log 0.301030 
 «© 2.728094 
 ‘¢ 2.773465 
 48° log vers 9.519657 
 2) 5.322246 
 log 2.661123 
 
 ** 3.052402 
 23° 57’ 55” log sin 9.608721 
 
 23° 57’ 55” log cos 9.960847 
 log 3.052402 
 
 log ¥ 3.013249 
 
 log 2.773465 
 48° log sin 9.871073 
 
 log * 2.644538 
 25° 19! 52" log sin 9.631289 
 22° 40’ 08" 
 
 log 3.013249 
 1.987934 
 
 1.025315 
 log 1.053078 
 ** 2.078393 
 
146 FIELD ENGINEERING. 
 
 (201) R; — Rx’ —4q 366.50 log 2.56407 
 Boe “© 1.025815 
 q et ee 
 23 p 34.57 “1.588759 
 (197) Reis ge T18.41 «2.853157 
 gp 17° 55’ ~— log vers 8 685602 
 (198) roe 62.69 log 1.797198 
 Rees ee alos ke 2.618184 
 a’ 31° 54’ ~—log vers 9.179014 
 na 49° 49’ 
 Hi erat PP = 21689 
 Boe Bs dean eo PP sp eebaald 
 
 The practical difficulty in changing the middle arc of three 
 centred curves lies in the difference of measurement that 
 ensues. Thus, in the last problem, although the total central 
 angle is the same, the new curve is 6.56 feet shorter than the 
 original, making a fractional station at P'’. If the change is 
 made during the location, it is well tu re-run the last arc from 
 P'” to the tangent following, so as to eliminate the fractional 
 station from the curve. 
 
TURNOUTS. 147 
 
 CHAPTER VII. 
 TURNOUTS. 
 
 179. A turnout is a curved track by which a car may 
 leave the main track for another. At the point where the 
 outer rail of the turnout crosses the rail of the main track a 
 frog is introduced which allows the flanges of the wheels to 
 pass the rails. A frog consists essentially of a solid block of 
 iron or steel having two straight channels crossing each other 
 on the upper surface, in which the flanges of 
 the wheels pass. The triangular portion of the 
 upper surface formed by the channels is called 
 the tongue of the frog, and the angle which the 
 channels make with each other is called the frog- 
 ungle. Every railroad is provided with a set of \ 
 frogs of different angles, from which may be i 
 selected one best adapted to any particular case. Fia, 60, 
 
 The frogs may be designated by their angles, 
 but it is customary to designate them by numbers expressing 
 the ratio of the bisecting line FC of the tongue to the base 
 line ad, Fig. 60. Observe that ¥ is at the intersection of the 
 edges produced, ‘and not at the blunt. point of the tongue. 
 
 In the triangle afC, 
 sue. = cot 4aFb 
 
 ad : 
 and if we let 2 = the number of the frog, and # = the frog 
 angle, then 
 
 <b 
 ¢ 
 
 | 
 
 | 
 
 | 
 
 On some roads, however, the frogs are numbered arbitrarily, 
 or according to their length in feet, while on others they are 
 designated by letters of the alphabet. In any case the true 
 number (x) of a frog may be determined by the above for- 
 
 mula. 
 
148 FIELD ENGINEERING. 
 
 The first rail of the turnout is common to both tracks, and 
 is called the switch-rail. It has one end free, so as to be shift- 
 ed from one track to the other as required; the free end, D 
 (Fig. 61), is called the point of switch. The tangent point of 
 the turnout, at A, is called the heel of switch, and the distance, 
 AD, is the length of switch. The switch-rail should be several 
 feet longer than AD, and the excess be spiked down in the 
 line of the main track back of the point A. Then if the point 
 D is thrown over to meet the rail of the turnout at A, the switch 
 rail is sprung into an arc, which coincides with the arc of the 
 turnout, provided that the ,ength of switch AD has been prop- 
 erly taken. The distance DA through which the point moves 
 is called the throw of the switch. It varies on different roads 
 from 44 to 6 inches, but is usually made about 5 inches, or 0.42 
 feet. A turnout should be a simple curve from the heel of the 
 switch to the point of the frog. 
 
 180. Given: a main track, straight, and a frog angle F, to 
 determine the distance BF, on the main track from the heel of 
 switch to point of frog, the radius, ¥, of the centre line of the turn- 
 out, the length of chord af, and the proper length of switch AD. 
 Fig. 61. 
 
 Fie. 61. 
 
 Let C be the centre of the turnout. 
 ““  F= the frog angle, HFT = FCB. 
 ‘* —g = the gauge of track AB. 
 Sry E Oty Se PAGIUS, AG = $C, 
 “«“ DK = the throw of switch. 
 
 Then the radius of the gauge side of the outer rail is (7 + 49), 
 and we have 
 
TURNOUTS. 149 
 
 AB = FC. vers FCB 
 
 or, 
 g = (r+ 4g) vers F 
 whence 
 : oe ee: 
 (7 + 39) = SG (207) 
 The angle AFB=%3Ff 
 and BF = ABcot AFB = g.cot4f (208) 
 
 Again, in the triangle YCB 
 BF = FC: sin FCB = (r+ }y) sin F - (209) 
 The chord af is evidently 
 af = 2rsintF (210) 
 Similar to eq. (207), we have 
 vers ACD = es = ay 
 
 But since the inside rail has the same throw, while its radius 
 is (r — 3g), we may, if convenient, drop the 4g, and hence the 
 length of switch is 
 
 AD =7r.sin ACD (211) 
 
 The degree of curve corresponding to 7 is found from Table 
 IV., or by eq. (17), and the centre line of the turnout may be 
 located by transit deflections from the tangent point a, using 
 chords of 20 or 25 feet -+ the correction found in S$ 106, 107; 
 or the deflection for a 20-foot chord may be calculated at once 
 by 
 
 sin (dso) = —— (212) 
 
 181. Simple as these formule are, they may be rendered 
 still more convenient by introducing the number of the 
 frog, 7. By eq. (206) we have cot 4F = 2n, which substi- 
 tuted in eq. (208) gives 
 
 BF = 2gn (218) 
 
 Drawing the chord AF’ to the outer rail, 
 
 AF = VAB?+ BF? =gV1+4n (214) 
 
150 FIELD ENGINEERING. 
 
 Make BA’ = ABand join #4’, then by similar triangles, 
 AA'F' and AFC, 
 
 CAAA AP SS ale at 
 
 whence 
 AT? 
 fhe = Stat 
 or (7 + 39) = 49 (1 + 4n’) (215) 
 whence Pe 2on = DR Sn (216) 
 
 The chord af to the arc of the centre line is to AF’ as 7 is to 
 
 AR? 
 iO 4 : h _ ES ae 
 (r + 39); hence af 2 iy 
 eqs. (214) (215) we have 
 
 , and substituting values from 
 
 V1 + 4n? 
 Assuming that, for small angles, the tangent offsets vary as 
 
 the squares of their distances from the tangent point, which 
 will lead to no material error in this case; 
 
 (217) 
 
 AB: DK :: BF?: AD? 
 
 whence AD=BFy/ ee 
 AB (218) 
 or AD = y4n? 9. DK = V2. DK 
 
 It is not necessary to determine the degree of curve in order 
 to locate the turnout, for having fixed the position of BF, the 
 position of af is found by laying off Ba, and Ff, each equal to 
 2g. Whatever be the length of the chord af, found by eq. 
 (217) or (210), its middle ordinate is always ig, and the ordin- 
 nates at the quarter points, $. 4g = 3g. Thus for the stan- 
 dard gauge of 4.708 the middle ordinate is 1.177, and the side 
 ordinates 0.883. 
 
 By the preceding formulw Table XI. has been calculated, 
 which gives the required parts of a turnout for various frogs 
 when the gauge is 4 feet 8! inches and the throw 5 inches; 
 uso for a gauge of 3 feet and throw of 4 inches. For any 
 other throw, only AD must be calculated. For a different 
 gauge the engineer will do well to construct a similar table. 
 adapted to the frogs used on the road. 
 
4 
 
 TURNOUTS. La) 
 
 In the table the frog angle 1s given to seconds, in order that 
 the results may agree, whether found by equations in §180 
 $181; but in practice the nearest minute is sufficiently 
 exact. The frogs most used for single turnouts are those 
 from No. 7 to No. 9, inclusive. 
 
 or 
 
 182. In case of a double turnout from the same switch, 
 
 1..ree frogs are required, as at PV F' and I", Fig. 62., and the 
 Ss J ’ 
 
 Fig. 62. 
 
 switch is called a three-throw switch, because its point takes 
 three positions. The frogs #’ and F" are usually alike, and 
 placed exactly opposite each other in the main track. The 
 other frog Fis placed on the centre line of the main track. 
 Its angle 7” and its distance from 4 are now to be determined 
 
 in terms of /. 
 
 In the figure we have vers Ra = ae or 
 vers 32" = 5; a (219) 
 27+ 439) 
 The distance ak" = (r+ 4g) sin 3h" (220) 
 aF" =r. tan3Fh" (221) 
 
 also 
 All the parts of the turnout required to locate the frogs /’ 
 and FF are calculated by the formule in the preceding sec- 
 tions, or are taken from Table XI. | 
 If we let »’” = the number of the frog F''’, then by eq.(206) | 
 
 { 
 
 tan 47" = os which substituted in eq. (221) gives i 
 
 CH Se 
 | 
 
 & 
 
 sy eee ee (222) | 
 
 ee Ont. 
 
%~ 
 
 152 FIELD ENGINEERING, 
 
 Also, in the triangle aF'"C, 
 
 al" = Vr-+ yg? — 7 = Vo(r +49) (223) 
 
 Equating these and replacing r by 2gn?, we obtain 
 
 Nims Mee (224) 
 Qn? +3 
 If we neglect the 4, we have 
 (approx.) n' =" —  wWrin (225) 
 V3 
 
 tii Kxample.—If F = F' =6° 44, or n=7n' = 8.5, then ” = 
 HH 6.0 + or fF” = 9° 32’. 
 
 8 at hand of the angle or number given 
 by eq. (219) or (225), we may select one as nearly like it as pos- 
 sible, and locate the turnout as a compound curv 
 vided that #”” is less than 2F' Fig. 63. 
 
 } 183. In case no frog i: 
 
 ey pro- 
 
 Fig. 63. 
 
 Let r" = O"a, andr = r' = Of = Cf" 
 
 Then analogous to the equations of § 180, 
 
 " Gy 4g 
 BAG od 49 
 * tla 4 exec eam (227) 
 
 ah” = (r' + 39) sin dF" = r” tan $f" (228) 
 
| 
 TURNOUTS. 
 
 The length of the switch, by eq. (218), is 
 AD = V2r" DK 
 
 The curvature of the rail between the frogs 7” and F is 
 F’'CK = (F — $F). 
 
 Draw the chord #"’ fF’ and the perpendicular #’’L; then-the 
 angle LFF" = F— y(h—3P") =F + 4f"); and since 
 LF" = 49; 
 
 ; “up +9 
 ite = sin H+ 4F") (229) 
 LF =%g.cot4(PF+4F") (230) 
 wy Le 6 
 / at a 
 (r + 49) a sin + (F +8 4P") (231) 
 Example.—Let F = 6° 44 F" = 10° 24 
 
 Eq. (226) ig 2.354 log 0.371806 
 4H" 5° 12' log exs 7.616224 
 si r 569.616 2.155582 
 Kq. (228) 4 Pf" 5° 12’ log tan 8.959075 
 aF" 51.839 1.714657 
 
 log 0.371806 
 log sin 9.016824 
 
 Kg. (229) 29 2.354 Pras: 
 MF + 4F") B° 58 
 
 Ss FF 22.645 
 Eq. 81) 17 — 4") 
 
 Ar + 49) 1692.432 
 r 843.862 
 
 0° 46’ log sin 8.126471 
 
 3.228511 
 
 When n” > . 707, 7 will be less than 7’. Should F’ not 
 equal F, (F’" being given), then 7’ and ZL’ F” must be calculated 
 also, by substituting 7’ for Win eqs. (230) and (281). 
 
 184. From the same switch in a straight track tt 1s required 
 to lay two turnouts on the same side. Fig. 64. 
 
 If we assume #”’ = F, and that these two frogs shall be 
 opposite each other, we calculate all the distances of the first 
 turnout for the angle /# (or number 7) by § 180, 181, whence 
 we have the radius 7 = Ca. 
 
 1.354982. 
 
154 FIELD ENGINEERING. 
 
 : | Let r' = C'a, the radius of the centre line of the second 
 , turnout. The angle ACF’ = F, and since #" = F, the angle 
 CF'C'= F, and the triangle CF"C’ is isosceles, and C'F’ = 
 O'C.. But C'F" = C’A = 340A. 
 
 or (7 + 49) = 4(r + 49) - (282) 
 ee r = 47 — Wy) (238) 
 
 C' 
 Fic. 64. 
 
 To calculate the remaining frog at #7", we have from eq. 
 (207) 
 
 owed | eto foe 
 vers #"" — ge (234) 
 4 or from eq. (216) 
 Wp) et 
 | nia (285) 
 i | 2g 
 | BF" = (r' + 49) sin F = 2gn" (236) 
 1) ‘ ek ar" 
 ii Bp OP Osin eh he cee (237) 
 i V1 + 4n'? 
 | and since AO'F" = 2F, ; 
 af = oF sin dt (238) 
 
 The length of switch may be calculated by either 7 or 2", 
 since for 7’, which is about 47, the throw of switch is double, 
 
 thus giving practically identical results. 
 | If we compare the values of F’” as obtained by eqs. (234) 
 and (219), we shall find them almost identical for given values 
 
. 
 
 | 
 
 TURNOUTS. 
 
 of F'and g; and since this may also be proved analytically by 
 assuming that vers +#’" = ¢ vers F'", which is very nearly 
 true for ordinary values of #'", we conclude that a set of frogs 
 (F = F', and #") which is adapted to a double turnout in i) 
 opposite directions from a straight line (as in Fig. 62) is also | 
 adapted to a double turnout on one side (as in Fig. 64), the 
 curves being simple curves in every Case. But this being 
 true, the set is also adapted to a double turnout in opposite 
 directions from any curved track the radius of which is not 
 less than 7 as given for F, since any such case is intermediate 
 between the two cases named. When, ‘therefore, a certain 
 frog, F, is adopted for general use on any road, another frog 
 should also be adopted, whose angle, F", is determined by 
 eq. (219), or whose number is determined by eq. (229). 
 Thus, if 7 = 6° 44, or n = 84, then F" should be 9° 32’, or 
 Re = 0. 
 
 185. In case no frog is at hand of the angle or number given 
 by eqs. (234) (2385), we may select one as near the same angle 
 as possible, and, calling this #’", calculate the distance BF" 
 and the radius CO" F" (Fig. 65) as for a single turnout; $ 180. 
 
 Fia, 65. 
 
 Then assuming any other frog #’’, whether equal to F or not, 
 it is required to find the chord FF’, and the radius 0'F" of 
 the arc F’F’. The point #’ may fall either side of the radius 
 OF. according to the values given to #” and #". 
 
 a. In case F' falls beyond the radius CF, we will assume 
 first, that the entire rail from B to F’ is laid with the same 
 radius BO, and centre 0. (This investigation also applies to 
 the case when F’ falls between B and the line CF’) 
 
 In the diagzam (Fig. 65) draw CF". We then have 
 
FIELD ENGINEERING. 
 
 BF" BR" 
 UAL Besa (ee. 9 
 tan BOP” = Bo = (289) 
 
 and 
 GF" = (r — 49) exsec BCF"’ (240) 
 
 In the triangle #'"CF’. 
 F°C— F'C : F'C+F'C:: tant" F'C— F'F'C) 
 > cot #" CF" 
 Now, since C'’F'C = F', and BC" F"' = F", 
 ot de ee ee 
 
 and 
 Pod Om FR GEO = FORO ms (Fh — BCF") 
 Letting =" C OH Cre Cee BOR) 
 
 and subtracting, we have 
 F'RC-—FF'C=F'+U0 
 Hence the above proportion may be written 
 
 GF" :2BC-+ GF” :: tan 4" + U): cot $F" CF’ 
 whence 
 
 po wes tan (F’ +0) (241) 
 
 (Since BOF" + FCF" = BCF"', and we know the radius 
 BC, the chord or arc BF’ is easily obtained, which fixes the 
 position of the frog #’'; and the problem may end here, 
 frequently, in practice.) 
 
 Now in the same triangle CF", the half sum of F'' F'C 
 and F''F'''C is 90° — 4F'"CF"; while, as we have just seen, 
 the half difference is }(F'' +- U); and by subtracting we have 
 the less, or 
 
 F'r'C=90 — UF’ LU+ POF) (242) 
 
 F'Osin FOF" 
 sin F'P''0 
 
 cot 3#"CF’ = 
 
 Now 4 hie tae 
 
 BC. sin FCF" 
 
 oe Ee son UHL (Lt RY OF 
 
 (243) 
 
“ae 
 
 TURNOUTS. 
 
 To find the angle F'''C' F''; produce the line #''C’' in the dia- 
 gram to intersect the line BO at A. Then the two triangles 
 HO'C' and KOF' have the angle A common, and the sum of 
 the other angles will be equal; that is, 
 
 KOO -- O O'R = KO --— Cr kK 
 
 or FOL ORs BOR hek i 
 and since BCF' = BOF" + FCF’ i 
 BO Gs POR + Fe — 0 (244) Wi 
 
 If we denote the radius /''C’ by » + 49 
 , 4h" F" 
 / 1 — 
 m= SP Or” 
 Example.—Given: the three frogs 7 = 6° 43' 59", #” = | 
 6° 01’ 32", and #" = 8° 47 51" to lay a double turnout on one i, 
 side of a straight track. Fig. 60. il 
 
 (245) 
 
 By Tab. XI. BF = 80.0386 r = 680.806 AD = 23.82 
 BF" = 61.204 7" = 397.826 
 
 Eq. (239) BF" 61.204 log 1.786779 Wall 
 (7 — 39) 677.952 « 2°831199 ui 
 BOF" 5° 09' 88" log tan 8.955580 ih 
 Eq. (240) BOF" 5° 09' 38" log exsec 7.609587 | 
 (ry — 49) 677.952 log 2.831199 | 
 GR’ * 2:760 “0.440786 ul 
 Eq. (241) 2BC-+ GF") 1358.664 «3.133112 H 
 (U = 8° 38’ 13") 2.692326 NW 
 34(#” + U) 4° 49' 52".5 log tan 8.926968 He 
 te Jeane ees Ha 
 (F" OF’) 1°22'35" “© cot 1.619294 I} 
 Eq. (243) F" OF' 9° 45'10" «sin 8.681481 | 
 r—4g 677.952 2.831199 He | 
 Se a oe Se Bia 4 
 1.512680 ) 
 MF’ + U+ F'OF’) 6° 12'27".5 ** cos 9.997446 iH 
 F’F’ 82,752 1.515234 i 
 
 Eq. (245) 4F"C'F' ; 9° 34! 14.5 6“ sin 8 651781 | | | 
 vin Ar’ +49) 780.219 2.863453 i] 
 r 
 
158 FIELD ENGINEERING. 
 
 C 
 
 b. We assume, secondiy, that the middle track is straight 
 beyond F, and tangent to the curve at F. Fig. 66. 
 
 Then whenever the value of /”" 1s dess than that given by 
 eq. (234), the arc AF”, produced with the same radius AC", 
 will intersect the straight rail H#” at some point /#”, and the 
 frog angles f'and fF" will be equal. 
 
 Fia. 66. 
 
 For the straight rail H#” produced backwards, passes 
 through the point A, making an angle #’ with the main 
 track, since the triangles CBF and CHA are equal, and AH 
 = BF. Now any circle, tangent to the main rail at <A, 
 will intersect the line AH in some point /’, and since AF’ is 
 the chord of the are, the angle at #7’ equals the angle at A, 
 which is #. Hence # = #"; and the angle AC" F” = 2F. 
 
 The length of the chord AF" is 
 
 AF’ = 2AC" sin F (246) 
 | | | | The chord #"F" = 2F'"C" sin}(F"C"F’) 
 I, = 2AC" sin }(2F — FP") 
 ti Hence, F'F" = Ar" + 4g) sin(F¥—4F") (247) 
 
 \ | Hrample.—Let fF’ = F = 6° 4359" and F'” = 8° 47' 51” 
 By Table XI. r” = 397.826 _ 
 
 Eq. (247) 27" + 49) = 800.360 log 2.903285 
 Fo—4F" 2° 20'03".5 log sin 8.609915 
 F"F" 32.60 1.513200 
 
 If the frog ¥’ 1s required to be different from F, then the 
 inside curve must be compounded at F', giving other valués 
 to the length and radius of the are #'" F’. 
 
ES 
 
 TURNOUTS. 159 
 
 c. We assume, thirdly, that the curve of the mzddle track is 
 reversed at F. Fig. 67. 
 
 In the diagram, let Q be the centre of the reversed portion, 
 and F" the proper position of the frog F’, and CC’ the centre 
 of the required arc FF". Then Q ison the radial line CF, 
 produced, and 0’ is on the radial line #0" produced. Join il 
 FQ and F'Q, and produce (F'" to intersect these lines in L | 
 and M respectively. Also join FQ, and denote the angle i 
 LF'’Q by U and the angle F'QF” by @. at 
 
 Fig. 67. 
 
 In the triangle F'"Q we know #"F = BF — BF", and the 
 side FQ is given; and the included angle F' FQ =90° + F. Hl 
 Hence we may calculate (Tab. II. 25) the angle F' QF and the | 
 side FQ. 
 
 The triangle CCL gives the angle at L = #” — F’, and the 
 triangle LQ gives LF" G@ = L — Por 
 
 U= F’ —F— F'QF (248) 
 
 In the triangle /' QF" we have | 
 F'Q—F'Q: FQL FQ: tany(F'F"@ — F’'F'Q) i} 
 
 : cot 4(F' QF") | 
 
 But 2’ FQ = F'F"L4+ Uand F’F'Q=Ff 'F'N — F’, and | 
 since FF" N = I'F'L, we have by subtraction, 
 F'P'Q — F’F'Q=U+F' HH 
 
 pO reed aa: 
 Hence cot 49 = F'Q-F'Q tan 4(U + F’) (249) 
 
160 FIELD ENGINEERING. 
 
 (Now the angle FQF’ = Q — F’'' QF, and is subtended by the 
 chord HF", which is therefore easily found, and serves to 
 locate the frog #’’, and frequently this is all that will be 
 required. ) 
 
 In the triangle #”'’'Q/’, thehalf sum of QF’ F’ and QOH Ff" 
 is 90° — 4Q, while, as we have just seen, the half difference is 
 34,\U-+ F’); hence by adding, we have the greater, or 
 
 QF’ F' = 90 4+ KU + FQ 
 
 Pi ae, eee sin Q re 
 ee seed |e © cos (UF — (250) 
 The triangle C’F'M gives F''C'F"’ = F' — M, while the 
 triangle #""MQ gives M= U + Q; hence FP" C'F' = F' — 
 (U + Q); and denoting the radius (''F" by r’ + 4g, 
 
 4" F' 
 
 cb SS cre (Ff — U—@Q) Gey 
 
 Example.—Let F = F" = 6° 43' 59", F’ = 8° 4Y7’ 51", and 
 FQ = 953.012. Then by Tab. XL, BF = 80.036 and BF’ = 
 61.204; hence #’’"# = 18.832; and the included angle is 
 96° 43' 59". 
 
 Solving the triangle 77'"Q we find LOE Or Ae 
 FF"Q = 82° 08' 48", and #"Q = 955.402. Now it © a 
 FQ + 9 = 957.720. 
 
 (249) F’'Q+ FQ 1913.122 log 3.281748 
 TL) See ye 
 
 Q 2,318 © 0 365113 
 
 (U 0° 56’ 34") “2 916630 
 
 KU + F’) 3° 50’ 16".5 log tan 8.826231 
 
 1Q 1° 02° 08".4 “ cot 1.742861 
 
 (250) Q 2° 04’ 16".8 <* sin 8.558033 
 
 HU+ F'— Q) ~ 2° 48’ 08".1  “ cos 9.999480 
 
 8.558553 
 
 F’'Q 957.720 log 2.981289 
 
 ES F'F' — 34.633 1.539892 
 
 (251) 4’ —-U— @Q) 1° 51’ 34".1 log sin 8.511191 
 
 Q(r' + 49) 1068.32 3.028701 
 581.81 
 
TUBNOUTS. 161 
 
 186. Given: a main track, curved, and a frog- -angle F, to 
 locate a turnout on the inside of the curve. Fig. 68. 
 Let R= Oa = radius of main track. 
 « » = Ca = radius of turnout. 
 « H— OFO = the frog angle. 
 
 In the diagram draw the chord AF’ and produce it to inter- 
 sect the outer rail at G; and draw FO and GO. Since the 
 chords AF and AG coincide, and the radii AC and AO 
 
 Fic. 68. 
 
 coincide, the chords subtend equal angles at CO and O respec- 
 tively, and GO is parallel to FC. (See $137.) Hence, FOG 
 — (CFO =F. Let 6 = the angle FOA. 
 
 In the triangle FOA, 9 = GFO — FAO = GFO — FGO,; 
 and in the triangle @FO, GO4+ FO: GO— FO :: tan HGFO 
 
 + FGO) : tan (GFO — FGO), or 2R : g:: cot iF: tan 49 
 : g gn 
 2 fan3o-= oR cot 37 = PB (252) 
 In the triangle CFO, 
 r+ io) = (R—-3 = ein 8 O59) 
 +i) =k - WD FPS (253) 
 
 In the triangle BOF, 
 BF = (R — iy) sin 39 (254) 
 
 In the triangle aCf, 
 af = 2r sin WF+ f) 
 
162 FIELD ENGINEERING. 
 
 The length of switch AD, for a given throw DK, may be 
 found thus: from Table IV. take the tangent offsets, ¢ and f, 
 corresponding to # and 7 respectively, and assuming that the 
 offsets may vary as the squares of their distances from the 
 tangent point, we have 
 
 Bes aad see & Ws Gods OR Ui Ge BE. 
 t—¢ 
 This result is practically the same as that found for length 
 of switch in a turnout from a straight line with the same frog, 
 when J is large. 
 Heample.—Let R = 1482.69 and # = 6° 43’ 59". 
 
 Eq. (252) ig 2.854 log 0.871806 
 iP 
 
 } 3° 21°59".5 log cot 1.230440 
 log 1.602246 
 R (Tab. IV.) ° 3156151 
 19 1° 85'59".8 log tan 8.446095 
 Eq. (254) 8 3°11'59"6  — **_ sin 8.746786 
 P46 9° 55’ 58".6 6 9 936778 
 9.510008 
 R— ig 1480.3386 3.155438 
 r-t4g 462.856 2. 665446 
 
 r 460.502 
 254) 9 log 0.801030 
 (R—4g) 1430.336 “© 3.155438 
 ase 1° 35'59".8 log sin 8.445924 
 : BF 79.872 log 1.902392 
 (255) Qr 921.004 2964269 
 i(F + 6) 4° 57'59".8 log sin 8.937381 
 af 79.734 log 1.901643 
 
 The values of BF and af are found to be so nearly identical 
 in this case with those determined in case of a turnout from a 
 straight line, that the values given in Table XI may be used 
 at once for ordinary values of R; and the degree of curve of the 
 turnout in this problem is approaimately the sum of the degree 
 of curve of the main track and the degree of curve given in 
 Table XI. opposite #. Thus, in the example 4° + 8° 26’ = 
 127526... 7. 7 = 461.7 idarls 
 
TURNOUTS. 163 
 
 187. Given: a main track, curved, and a frog-angle F, to 
 locate a turnout on the outside of the curve. Fig. 69. 
 
 In the diagram draw the chord AF, and produce it to meet 
 the inner rail at G; and draw /O and GO. The triangles 
 OAF and OAG are both isosceles, and have the angles at A 
 equal; hence they are similar, and FCA = AOG. Hence 
 FOG = HFO=F. Let R= Oa, r = Ca, and 6 = FOA, 
 
 Fia. 69. 
 
 In the triangle FOA, 6 = OAG — AFO = FGO — GO; 
 and in the triangle FOG; FO + GO: FO — GO:: tan 
 4(FGO + GO) : tan 4(FGO — GFO), or 2h : g 3: cot 4H 
 
 : tan 46 
 
 : Lae OP Nee 25 
 . tan 40 = oR cot 4+# = R (257) 
 which is identical with (252). 
 In the triangle CFO 
 sin 6 
 (7 + 49) = (Rk + 39) ain (= ® (258) 
 In the triangle BOF, 
 BF = AR -+ ig) sin 46 (259) 
 In the triangle a@f, 
 af = 27. sin 4(F'— 6) (260) 
 
 For a given throw, the length of switch will be 
 
 a an eae Se) 
 
 fay 
 
164 FIELD ENGINEERING. 
 
 | in which ¢ and ¢’ are the tangent offsets (Tab. IV.) corre- 
 | sponding to # and 7, 
 
 In this problem, as in the preceding, we may, for ordinary 
 values of #2, assume the values for BF'and af given in Tab. XT. 
 The degree of curve of this turnout is, approrimately, d — D, 
 taking d from Tab. XI. and D from Tab. IV. corresponding 
 to Rk. Should D = d, this turnout becomes a straight line; 
 
 Fie. 70. 
 
 and when D > d, or when # is less than 7 given in Tab. XI., 
 the centre falls on the same side as O. Fig. 70. In this case, 
 using the same notation, 46 is given by eq. (257). 
 
 sin. 
 (7. — 49) = (kh + ty) sind —F) (262) 
 | Eq. (259) BF = &AR-+ 4g) sin 36 
 | af = 2r sin 44 — F) (263) 
 
 188. A tongue-switeclh is a short, stiff switch which; 
 | when moved, revolves at the heel as on a pivot. When it is 
 HI thrown over to the turnout track, it makes an abrupt angle 
 with the main track, called the switch angle; but in this posi- 
 ii tion it should be tangent to the turnout curve. .The use of 
 this switch is generally confined to yards and warehouses, 
 where but little space can be afforded, and where the motion 
 of the cars is always slow. 
 
 189. Given: a straight track, a frog-angle F, and the length 
 and throw of a tongue-switch, to locate the turnout. 
 
TURNOUTS. 165 
 
 Let AD be the length, and DK the throw of switch, and let 
 S denote the switch-angle DAK. 
 
 DK DIE 
 
 Then sin S = “ap S° = 57.3 AD (264) 
 
 (Compare § 86.) 
 Let @ be the centre of the required turnout, and in the dia- 
 
 gram draw OK and CF; also draw DG perpendicular to the 
 straight track. Then DG = F: and in the triangle KGC, 
 KOF = KGF —GKC, and since CKA isa right-angle, GKC 
 eae OP = Lor a. : 
 
 Draw the chord KF, and since the triangle KCF is isosceles, 
 the angle C/K = 90° — 1(F — 8)... Now, CFI = 90° — F, 
 hence by subtraction, KF = HF -+ 8). 
 
 Fig. 71. 
 
 If g denote the gauge, we know KI = g — DK; and in the 
 
 right-angled triangle KIF, we have 
 f= Ki. ct te+t 8): (265) 
 
 KI As 
 AP = nar er 8) (268) 
 
 KF 
 r + io Sra — 5) C8?) 
 
 These equations are analogous to eqs. (229) (230) (231). 
 
 190. Given: a double turnout with tongue- 
 switch, from a straight track; to find the angle, F', of the 
 
 middle frog. 
 
 Assuming 7” = calculate (r -- 49) by the last equations. 
 n the centre line of 
 
 Since the rails of the turnouts intersect oO 
 
166 FIELD ENGINEERING. 
 
 he straight track, as in Fig 63; if we substitute the value of 
 Lf" F’, eq. (229) in eq. (281), we have 
 
 OF 19) = 2 5in Ea) sin KP EP) 
 
 and by Trig. Table II. 
 
 With off89 7A) Be 
 ia cos 4f'" — cos F 
 
 Be ee 4g 
 whence cos 4" = cos F + —~* 268 
 If the angle of the middle frog to be used does not agree 
 
 it) with #’” found by the last equation, the turnout will be com- 
 Ht) pounded at 7’. 
 
 191. Given: a straight track, the frog-angles I, F' and F", 
 and the switch angle S, to locate a double turnout. 
 Fig. 72. 
 
 | 
 | 
 | 
 | 
 | 
 1 Fia. 72, 
 
 Assuming that #” shall be placed on the centre line of 
 the straight track, let h be a point on the centre line at the 
 point of switch. Then AK = 4g — DK; and since the angle 
 Ff" is bisected by the centre line the necessary formule in this 
 case are obtained from §189 by simply replacing by $F" 
 and AJ by 44; and in the first. members [7 by AF" and r by 
 
 rn’. This is obvious by the similarity of the figures, 
 
TURNOUT. 167% 
 
 LK . cot 44" +8) (269) 
 
 hk 
 = 2 
 — gin 444" +8) (270) 
 
 II 
 
 Hence AF" 
 
 KF “" 
 
 sighs eases Wy 
 
 The location of the remaining frogs is a problem already 
 discussed, § 188, ea. (229), etc. 
 
 192. Given: a straight track, the frog angles F, F', F'", and 
 the switch angle S, to locatea double turnout on one 
 side. Fig. 73. 
 
 Fic. 73. 
 
 The frog Fis located by $189; but for the frog #'” we have 
 evidently a double throw; hence eqs. (265) (266) (267) become 
 
 IF" =(g — 2DK) cot (F'’ + 28) (272) 
 KP =~ y(F -+ 28) oe) 
 IT (274) 
 
 K sido sin 3(#” — 28) 
 To locate the remaining frog F': when F" falls beyond the 
 line CF, there are three cases. 
 a. The middle track reversed beyond F. 
 
 We find the distance #'" #' by subtracting TF", eq. (272) from 
 IF, eq. (265). after which the solution is identical with that 
 
 given $185, @., Fig. 67. 
 
168 FIELD ENGINEERING. 
 
 b. The middle track compounded at F. 
 
 Let Q be the centre of the curve beyond F, and also let ss 
 the angle #’Q/’"; and let U = the angle 0" F'"Q. 
 Then by a course of reasoning analogous to that of case a, 
 we derive . 
 U=F"— F4+ F'QPr (275) 
 
 cot 4Q = re te re tan 3#(U + F’) (276) 
 
 Now since the radius 7’'Q is given, and the angle QF’ = 
 
 Q@ — FQF", we readily determine the distance HF’ ‘, and so 
 locate the frog F”’. 
 
 In the triangle 7" QF", the half sum of QF" F’ and QE'F"’ 
 
 is 90° — 4Q, while the half difference is 4(U -+ #”"); hence by 
 subtraction we have the less, or 
 
 F'F'Q = 90 —KU+ F'4+@ 
 
 ‘Tw ’ sin Q 
 Hence LSS. eae HULEF LO (277) 
 
 Join C'@, and the quadrilateral C’QF"' F" gives 
 f"+Q= U0+ F'"C'F' 
 
 hence #'"0'#”" = F’ — U-+ Q; and denoting the radius C'F" 
 by r’ + 4g, we have 
 
 mig = 
 
 4H" fF" 
 sin (F’— UF Q 
 
 (278) 
 
 Cor. Since the centre Q is assumed at pleasure, it may be 
 made to coincide with the centre (0, and then the compound 
 curve becomes a simple curve. Then also, the above formule 
 will apply when 7” is such that the frog will come on the are 
 JH. But as FQF" will be greater than Q, the difference 
 FQF' will be negative, indicating that the distance HF’ is to 
 be laid off backwards from H. 
 
 c. The middle track straight beyond F, and tan- 
 gent to the curve at F. Fig. 74. ; 
 
 Let #” be the required position of the frog F'. A tangent 
 to the curve at #’ makes an angle (#’ + F) with the main 
 track, and a tangent at #” makes an angle of #’” with the 
 same; hence the angle they make with each other is 
 
; 
 > | 
 TURNOUTS. 169 
 
 (F'+ F— F"), and this is the curvature of the are 1" F", 
 and equals the angle #'"C' F". 
 
 Produce the straight line 7’ H backwards to G, and draw 
 F’G perpendicular to it. Then #"G= FH — FF’. sin F, or 
 
 F'G=g—F'F.sinFf —. 795 
 
 Fig. 74. 
 
 In the right-angled triangle #’GF'", the angle NG Ge 
 F—4(F' 4 F— Ff") =F’ + FF’ — #). 
 
 "a F'G 
 and _ GF = F'F'.. cos 4(F’ + F" — FP) (281) 
 
 Observe that GF" cannot be less than GH = F" F. cos F. 
 
 193. Given: a turnout with a frog angle F, and the perpen. 
 dicular distance p between the centre lines of the main and side 
 
 tracks ; to find the radius x of the curve connecting the 
 turnout with the side track. Fig. 75. 
 
170 TIELD ENGINEERING. 
 
 Let the reversing point be taken at /, and let Q on CF’ pro- 
 duced be the centre of the required curve, and draw QM per- 
 pendicular to the main track. Then QM= QF =r — 4g; the 
 point Wis the point of tangent, and the angle FQM = FL. 
 
 Now JV being the intersection of the rail BF’ with the radius 
 QM, we have MN= QF'vers F, but MN=p—g; hence 
 
 The distance FN is evidently 
 FN = (r — 49) sin & (283) 
 and the chord to the centre line is 
 fm = 2r sin $F (284) 
 
 Should the distance F'N consume too much of the track, it may 
 be lessened by introducing a short tangent at F, denoted by k; 
 then by eq. (48) the radius will be shortened by an amount 
 equal to &. cot 4F, and the distance /N will be shortened by &. 
 
 Since the tangent & reduces the length of the tangent offset 
 of the entire curve by k . sen F, we have for the new radius 7” 
 
 p—g—ksn Pf 
 
 ro Ee “vers # CoH) 
 
 When 2’ is fixed by a limit, we obtain & by resolving eq. (285) 
 
 SPRATT ayyvers a 
 
 é sin 
 
 (286) 
 
 In case the main track is but sghtiy curved, we may at first 
 assume it to be straight, and find 7 as above, eq. (282), and 
 the degree of curve corresponding to 7; but this degree of 
 curve must then be ¢ncreased or dimimshed by the degree of 
 curve of the main track, according as the track is concave or 
 convex toward Q. 
 
 194. Given: the perpendicular distance p between the centre 
 lines of a curved main track and a parallel side track, and the 
 frog angle F of a turnout, to find the radius vr of the connecting 
 curve, and the length FN, or fm, of the curve. Fig. 76, 
 
4) 
 
 TURNOUTS. 
 
 Let FN be the rail of the main track, and GM ihe rail of 
 the siding, adjacent to each other; let O be the centre of the 
 main track, and Q the centre of the connecting curve. Then 
 the connecting curve will terminate at m, on the line OQ pro- 
 duced, 
 
 In the diagram draw MF, and produce it to intersect the 
 rail MG@ at G, and join GO, FO, and FQ. , 
 
 Let R =-radius of centre line of the main track; 7 = radius 
 of centre line of the connecting curve; and @ = the angle wi 
 FOM. © Wy 
 
 Case a.—The siding outside the main track. Fig. ‘76. 
 
 By similarity of the triangles GOM and FQM, GO is paral- 
 lel to FQ, and the angle GOP = F; and by a process similar 
 to that of $186, we have 
 
 AY 
 tan 109 = 55 iP cot 47 (287) I} 
 hie 
 A sin 6 , Hl 
 r— 49 =k 39) in (FL 6) tees) | a 
 FN = 2(R + jg) sin 49 (289) i} 
 fm = 2”. sin 4+ 4) (290) : 
 
 Case b.—The siding inside the main track, Fig. 77. 
 
 By-a process entirely similar to $187, we have Ih 
 
 zt Pp promi , Py 
 tan 49 = aR» cot +# 
 
FIELD ENGINEERING. 
 
 sin 0 
 
 r—wWwa(h — 49) an (P= 8 (292) 
 FPN = AR — 49) sin 46 (293) 
 jm = 2r sin 47 — 9) - - (294) 
 
 When 6 = F’in the last equations, sin (/ — 6) = 0, and r — 
 }g is infinite, and the curve 7M becomes a straight line. 
 
 Fie. 77. Fie. 78. 
 
 When 6 > F, sin ( — 9) is negative, and the centre Q falls 
 
 bf 
 
 on the same side of the track as O, and we have 
 
 sin @ 
 
 r+ig=(hk- 49) Sin 0 — F) (295) 
 fm = 27. sin 46 — Ff) (296) 
 Equations (291) and (293) remain unchanged. 
 
 195. To locate a crossing detween paraliel tracks. 
 Fig. 78. 
 
 When a turnout from one track enters a parallel track by 
 means of another frog and switch, the whole is called a cross- 
 ing. The frogs are alike, and the calculation for one end of 
 the crossing answers for the other. §§180, 181. We have 
 only to find the length of track between the two frogs. 
 
 In the diagram let Af’ be one turnout, and A'/” the other, 
 connected by the straight track #'G@. It is required to deter- 
 mine the length #'G, or the distance NV measured on the 
 main track from F to a perpendicular through 7’. Produc: 
 ing the line #’@ to intersect the rail VF’ at H, we have two 
 
av 
 
 TURNOUTS. 173 
 
 right-angled triangles GFH and F' NH, having the common 
 angle at H = F. Let p= the perpendicular distance between 
 centre lines of main tracks, andg = gauge. Then Gi’= 9g, 
 and fN.= (p.— g-) 
 
 r@=Pru— Gut ~- GFretF 
 sin # 
 1 nike teeidoas i 
 an P@= k= P-L g cot F (297) | 
 
 a - Sk = A hs +a g 9 Al | 
 So FN = NH — FH =(p—g) cot Ff on (298) vi 
 
 When the main tracks are curved the distance F'G may be Hy 
 calculated by the same formula (297) which gives a value only 
 a fraction too small, but in laying the track the rail #’’@ must 
 be curved to a radius which is to R of the main track as 
 
 F'G: NF. 
 
 196. When p is large, or the tracks are very wide apart, it 
 will effect some saving of room to lay the crossing in the form 
 of a reversed curve ; and the frogs being alike, the two 
 arcs will be equal, and the point of reversed curve P will be Hi 
 midway between Fand F"’. Fig. 79. yh 
 
 In the diagram we have aPa' the centre line of the cross- 
 ing, and PL the centre line between tracks; al = 4p, and 
 aC —aC'=r. The radius 7 having been found by § 180 or 
 
 & 181, we have iW 
 
 vers aCP = ne (299) 
 
 and PL=rsin.aCP 
 
174 FIELD ENGINEERING. 
 
 1 The distance between frogs, /“V, measured on the main track 
 is evidently 
 
 FN = APL — BF) (301) 
 
 in which BSF’ is determined by eqs. (209), (218), or by Tab. XI. 
 
 197. To lay a crossing in the form of a reversed 
 curve, when the parallel tracks are on a curve. Fig. 80. 
 
 Let O be the centre of the main curve, Cand C’ the centres 
 of the reversed curve. 
 ut Then in the triangle COC' we know all three sides; for CO 
 a =R+r;CC=r4+r,and0O= R+ p—r';; and the half 
 . sum of the three sides iss = R+7-+ 4p. 
 ill Denoting the angle COC’ by @, we have (Trig. Tab. II. 31) 
 TH p(T +1 — 4p) 
 Wil ves ® = (R48) (RED) 
 | The angle g determines the length of the arc BW described 
 Hil with the radius (& + 4g) and so fixes the position of the point 
 | A’ from A. 
 
 By a formula similar to the above, 
 
 (302) 
 
 10g — Pi — r+ 4p) 
 vers 0'00 = (Rin eer) (303) 
 
TURNOUTS. 175 
 
 The angle O’CO determines the length of the are aP 
 described with the radius 7; the angle (p + C’'CO) = CCA’ 
 determines the length of the arc Pa’, and P is the point of 
 reversed curve. 
 
 In this problem R is known, 7 is found by § 187, and 7 is 
 found by $186, only observing that in this case the value of 
 R must be increased by p. The frog angles /'and F’’ may be 
 equal or otherwise, only taking care that the point P shall be 
 included between the radii C'F" and CF. 
 
 The angle FOC = 6 is given by eq. (257), and the angle 
 F'0C' = 6' is given by eq. (252) (in which the value of £ is 
 to be increased by p); hence the angle FOF’ = mp — (6+ 6), 
 which determines the distance between the frogs, measured on 
 the main track. 
 
 198. To find the middle ordinate m, for 1 sta- 
 tion, or 100 feet, on any curve, in terms of the degree of 
 curve D. 
 
 Referring to Fig. 4 we have in the right triangle AGH 
 
 GH = GA. tan GAH 
 
 But GA = +}AB = 40, and (Tab. L. 18) GAH =4AOB =34; 
 hence 
 M=3C. tan}ta (304) 
 
 a general expression for the middle ordinate of any chord. 
 If in this equation we make C = 100, A becomes D; and 
 denoting the corresponding value of M by m, we have 
 
 m = 4100 tan {1D (305) 
 whence the rule, Multiply the nat. tangent of 4 the degree of 
 
 curve by 100 and divide by 2. Thus the values of m in the 5th 
 column of Tab. IV. have been calculated 
 
 199. To find the middle ordinate for any chord in 
 terms of the chord and radius 
 Referring to Fig. 4 we have 
 
 GH = OF — 0G = OE — VAO?— GA? 
 
 ibe pal / Ge (<) (306) 
 
176 FIELD ENGINEERING. 
 
 When CO = 100 we have for the middle ordinate of one 
 station 
 m= R— VR? — 2500 (307) 
 
 For any subchord ¢, less than 100, we have for the middle 
 ordinate, 
 
 m, = R— j/ (2) | 
 
 2 | pie Secrets 
 m = kR— y/ (e+ =) (x—<) 
 
 By adding se to the quantity under the radical in eq. (808) 
 
 it becomes a perfect square, giving 
 eC 
 
 Mi = se nearly, (809) 
 
 which is a very useful formula, although approximate. The 
 error in m, does not exceed .002 for any subchord ¢ when the 
 radius is greater than 800. On a 20° curve the error will be 
 002 for a chord of 50 feet; and on a 40° curve the error in ™m 
 will be only .003 fora chord of 33 feet. Hquation (809) is 
 therefore practically correct in all cases for finding the middle 
 ordinates of rails. Table XII. is calculated by eq. (808). 
 
 200. Curving Rails. Before any rail is spiked to its 
 place in a curve, it must be evenly bent from end to end, so 
 that it will assume the proper curvature when lying free. 
 The bending may be done by using sledges, but is best accom- 
 plished, especially for turnouts and other sharp curves, by 
 using a bending machine made especially for this purpose. 
 
 The proper curvature of a rail is tested by measuring Its 
 middle ordinate from a small cord stretched from end to 
 end and touching the side of the rail-head, .The cord should 
 also be stretched from the middle point of the rail to either 
 end, and the middle ordinate of each half length measured, 
 to test the wnzformity of curvature. 
 
 From the last equation it appears that, with a given radius, 
 the middie ordinate varies nearly as the square of the chord. 
 
TURNOUTS. 
 
 We may therefore find the middle ordinate of a rail whose 
 length is ¢ by the proportion 
 
 (100)? : @ i: ms Mm 
 
 or Mm, = cm yearly (310) 
 eT } 
 in which m is obtained from Tab. IV., col. 5, for the given 
 radius or degree of curve. 
 
 Example.—What is the middle ordinate of a 30 ft. rail 
 when curved for a 20° curve? 
 
 900 X 4.374 
 Eq. (310 ee ee IN, 
 When a long rail is bent for a sharp. curve, observe that ¢ is 
 the length of the chord of the rail—not of the rail itself. 
 
 For the chord of half a rail the middle ordinate is one-fourth 
 the middle ordinate of the whole rail. Thus, in the above ex- 
 ample it would be .099 or 14% inches. 
 
 Instead of using the chord of the whole rail, it may be.more 
 convenient to assume a chord shorter than the rail, especially 
 when the chord is not an exact number of feet, knotting the 
 string to the length assumed, and applying it to different por- 
 tions of the rail successively. 
 
 201. Elevation of the outer rail on curves. 
 
 When a car passes around a curve, a centrifugal force is 
 developed which presses the flanges of the wheels against the 
 outer rail. This force acts horizontally, and varies as the 
 square of the velocity, and inversely as the radius of the 
 curve. Denoting the centrifugal force by f, we have from the 
 
 wv? 
 
 theory of mechanics f= 39,166 R’ in which 7 = weight of 
 fe ay 
 
 loaded car in pounds, v= velocity in feet per second, and 
 R=radius of curve in feet. 
 
 In Fig. 81, let ab represent a level line at right angles to the 
 track, let @ and ¢ be the tops of rails on a curve, let de = ¢e= 
 elevation of outer rail ¢, and let the point d be the centre of 
 gravity of the car. The force f acts in the direction ad, and 
 if f’ = the component of f in the direction ac, then 
 
 fis fii ab: ae. 
 
178 FIELD ENGINEERING. 
 
 The weight w, resting on the inclined plane ac, developes a 
 component in the direction ca, and denoting this by #', we 
 have by similar triangles, 
 
 wm :w:: bc: ac. 
 
 7 
 Fie. 81. 
 
 Since equilibrium requires that w’ shall equal dir ’, we have after 
 
 dividing one proportion by the other a ithe , or f= —— 
 
 Equating this value of f with that given above we find, 
 
 ook CaS 
 ~ 32.166 R 
 But a = V ae — e’, and ac = distance between rail centres = 
 gauge + one rail head = g + 0.188. Also v= sae V, if V de- 
 
 note the velocity in miles per hour. Making these substitu- 
 tions and reducing, we have 
 
 V2 
 06688— 
 es Ge as <<) enc eet at (311) 
 
 pre 
 fs 
 vam +( 06688 a 
 By this formula Table XIII. is calculated for the standard 
 gauge g = 4' 83", = 4.708. 
 An approximate formula may be obtained by assuming that 
 ab = g for practicable values of ¢e. Substituting this m the 
 
 5280 
 
 first value of e given above, and replacing 7 by 3600 V? we 
 have 
 
 ; 
 {approx.) ei ogess oF (312) 
 
 which is the formula generally employed. 
 
TURNOUTS. 
 
 In laying a new track, the transverse inclination is first 
 given to the ballast by grade pegs driven either side of the 
 centre line at a distance of (g + .188) each side of the centre, 
 the outside peg being set higher, and the inside peg lower 
 than the grade of ballast on the centre line, by the proper 
 elevation selected from Table XIII. But in re-surfacing an 
 old track, the inner rail is taken as grade and the outer rail is 
 raised the necessary amount. 
 
 202. The proper elevation may be found mechan- 
 ically by the following method : 
 
 To find, on a curved track, the length of a chord whose middle 
 ordinate shall equal the proper elevation of the outer rail for any 
 velocity V in miles per hour. 
 
 By the conditions of the problem, we have m, in eq. (809) 
 equal to ¢ in eq. (312), or 
 ec  gV* .06688 
 
 SR R 
 a. c= .73144 VV9 (318) 
 When g = 4.708, 
 ¢ = 1,587V (314) 
 
 Lay off the chord, ¢, upon the rail of the track, stretch a 
 piece of twine between the points so found, and measure the 
 middle ordinate; it will equal the proper elevation. 
 
 203. The velocity assumed in the preceding formule 
 should be that of the fastest regular trains which will pass 
 over the curve in question, since the flanges would be forced 
 against the outer rail were there no centrifugal force devel- 
 oped, by reason of the wheels being rigidly attached to the 
 axles, and the axles being parallel. 
 
 The rails on tangents should be level transversely, except 
 near curves, where for 50 or 100 feet from the curve one rail 
 is gradually raised, so that at the P.C. or P.T. it may have 
 the full elevation due to the curve. Ata P.C.@. the elevation 
 should be an average of the elevations due to the two arcs. 
 Owing to the difficulty of properly adjusting the elevation of, 
 rail, it is objectionable to have arcs of very dissimilar radii 
 join each other; and the objection is much greater in the case 
 of reversed curves unless separated by a short tangent. See 
 
 g 82, 
 
 We 
 
FIELD ENGINEERING. 
 
 On the other hand, a short tangent between arcs which 
 curve in the same direction should be avoided, since it makes 
 a ‘‘flat place” both in line and levels, at once unsightly and 
 injurious to the rolling stock. 
 
 In the case of turnouts, however, no elevation of rail is pos 
 sible (except when both tracks curve in the same direction); 
 hence reversed curves are allowable, the speed of trains being 
 usually quite low also. 
 
 204. The coning of the wheels, by which the 
 wheel on the outer rail gains a diameter enough larger than 
 the other to compensate for the superior length of the outer 
 rail, although a theoretically perfect device, 1s gradually going 
 into disuse. To be effective for the sharpest curves, the coning 
 must be so great as to produce an unsteady motion on tan- 
 gents, very objectionable at high speeds. Moreover, it is un- 
 desirable to seek for an equilibrium of lateral forces in a car 
 on a curve, since the flanges are then sure to strike the inner 
 and outer rails alternately with damaging force, as that equi 
 librium is momentarily disturbed. It is far better that the 
 flange should press steadily against the outer rail, while that 
 pressure is modified and reduced somewhat by the elevation 
 of the rail. For these and other reasons, car-wheels are now 
 made nearly cylindrical. 
 
LEVELLING. 181 
 
 CHAPTER VIII. 
 LEVELLING. 
 
 205. The field operations with the Engincer’s Level are of 
 a more simple character than those performed with the transit, 
 yet require equal skill and nicety of manipulation in order to 
 produce trustworthy results. The transit is used to ascertain 
 the relative horizontal position of points, the level to obtain 
 their relative vertical position. 
 
 206. In order to express the elevation of points, they must 
 be referred to some level surface of known (or assumed) eleva- 
 tion; and in order that the elevations may all be positive up- 
 ward, this surface of reference should be selected below all the 
 points to be considered. The level surface of reference is called 
 the datum. 
 
 The elevation of the datum és always zero. The elevation of any 
 point is its vertical height above the datum. 
 
 Near the coast the sea level is usually adopted as the datum; 
 inland, the low water mark of a river or lake, etc. ; butitis not 
 necessary that the datwm should coincide with a water surface. 
 If any points whose elevations are to be ascertained are below 
 the water surface, the latter may be assumed to have an eleva- 
 tion of 100 or 1000 feet instead of zero; that is, we remove the 
 datum, in imagination, to 100 or 1000 feet below the level of 
 the water surface. 
 
 207. Incase of a survey commencing at a point quite re- 
 mote from any important water surface, any permanent point 
 may be selected as the original point of reference, and its ele- 
 vation may be assumed at 100 or any other number of feet; 
 that is, we fix the datum at the same number of feet below that 
 point. The point of reference is called a bench, or bench- 
 mark, and is designated by the initials B.M. Other benches 
 are established at intervals during a survey, and their eleva- 
 tions determined instrumentally. They are then convenicnt 
 
FIELD ENGINEERING. 
 
 points of known elevation for future reference. We cannot 
 assume the elevation of more than one bench on the same sur- 
 vey, else we should have more than one datum, and all the 
 results would be thrown into confusion. 
 
 208. Having established the first bench and recorded its 
 elevation, the next step is to set up the instrument firmly at a 
 moderate distance from the bench, so that the telescope shall 
 be somewhat higher than the bench, and in full view of a rod 
 held vertically upon it. The instrument having been tested for 
 its several adjustments, and found to be correct, the line of sight 
 through the intersection of the cross-hairsis known to be hori- 
 zontal when the bubble stands at the middle of its tube. Turn- 
 ing the line of sight upon the rod, the point of the rod covered 
 by the horizontal cross-hair is known to be on a level with the 
 cross-hair; and the latter is therefore Aigher than the bench by 
 the distance intercepted on the rod from its lower end. Add- 
 ing this distance to the elevation of the bench, we obtain the 
 clevation of the cross-hair, known technically as the ** Height 
 of Instrument,’’ and designated by the initials ZZ.Z. 
 
 209. The distance intercepted on a rod from its lower end 
 by the line of sight, when the rod is held vertically on any 
 given point, is called the reading of the rod at that point. 
 
 210. Having obtained the height of instrument, the eleva- 
 tion of any point somewhat lower than the cross-hair is easily 
 ascertained by taking a reading of the rod upon it. The read- 
 ing subtracted from the height of instrument gives the eleva- 
 tion of the point above the datum. The elevation of any num- 
 ber of other points may be similarly obtained. But the eleva- 
 tion of points on the ground higher than the cross-hair, or 
 farther below it than the length of the rod, cannot be deter- 
 mined, because in either case the line of sight will not cut the 
 rod, and hence there can be no reading. In order to observe 
 such points, the instrument must be removed to a new posi- 
 tion, higher or lower than before, as the case may require. 
 
 211. Before the instrument is removed to a new position, 
 a temporary bench, called a Turning Point (and designated 
 by 7.P.or ‘‘Peg’”’) must be established, and its elevation ascer- 
 
LEVELLING. 183 
 
 tained as for any other point, but with more care. A turning 
 point must be a firm and definite point whose position cannot 
 readily be altered in the least, nor lost sight of. A small stake 
 firmly driven, or a point of rock projecting upward, is fre- 
 quently used. The reading having been taken on the turning 
 point, the instrument is carried forward to a new position, 
 levelled up properly, and the new Height of Instrument ob- 
 tained by a new reading on the same turning point. Since the 
 cross-hair is higher than the point (otherwise there could be no 
 reading) the reading, added to the elevation of the point, gives 
 the Height of Instrument. 
 
 212. In general, the intersection of the cross-hairs being 
 higher than any point on which a reading is taken: 
 0 find the Height of Instrument, add the reading on a point 
 to the elevation of the point; and :, 
 To find the Elevation of point, subtract the reading on at 
 from the Height of Instrument. 
 
 A reading taken for the purpose of finding the Height of 
 
 Instrument is called a Backsight (B.S). A reading taken 
 for the purpose of finding the elevation of a turning-point (or 
 of a bench used as such) is called a Foresight (F.8). Hence 
 Backsights are always plus, and Foresights always minus. 
 
 213. The form of fieid-book used for the survey of 
 a railroad, or other continuous line, is shown below. The jirst 
 column contains the numbers of the stations on the line and 
 of plus distances to other points on the line where readings are 
 taken—also the initials of benches and turning points, in 
 order, as they occur. The sceond column contains the back- 
 sights, taken on points of known elevation only. The third 
 column contains the height of instrument, recorded on the 
 same line as the elevation of the turning point (or bench) from 
 which it is calculated. The fourth column contains the fore- 
 sights, taken on new turning points, and benches used as such, 
 only. The jifth column contains the readings taken on all 
 other points noted in the first column. The stzth column con- 
 tains the elevations of all points observed. The right-hand 
 page is reserved for remarks, descriptive of the benches and 
 their location—of objects crossed by the line, as roads, streams, 
 swamps, ditches, etc. ; the depths of streams, etc. 
 
184 FIELD ENGINEERING. 
 
 LEVEL BOOK. 
 
 Sta, BES: H.I. E.S. Rod. Elev. Remarks. 
 B.M. | 4.683 | 204.683 200.000 | White oak, 115 R. 
 0 2.1 202.6 
 1 3.4 201.3 | 
 + 50 | 5.2 199.5 
 Peg | 1.791 | 197.260 | 9.214 | 195.469 
 pas Ligon 193.0. 
 + 25 | 7.0 190.3 Brook 5 wide; 1 deep 
 + 50 3.1 194.2 
 3 0.5 196.8 
 Peg | 11.750 | 208.574 | 0.436 | 196.824 
 Peg | 11.938 | 219.528 | 0.979 | 207 595 
 + 90 | | 3.5 S16:01= 9) 
 4 2.6 216.9 
 B.M. | | 2.075 | 217.453 | Maple, 78 L. 
 5 a. 217.8) | 
 o | 0.9 218.6 
 Peg | 9.005 | 227.801 | 0.732 218.796 
 7 | 6.2 221.6 -| 
 39.162 | 11.361 | 
 
 When a bench is not used as a turning point, the reading on 
 it is recorded in the fifth column. 
 
 The numbers in the second, fourth, and fifth columns come 
 directly from the rod, those in the third are obtained by 
 addition, those in the sixth by subtraction, according to the 
 rule given above. The additions and subtractions made on 
 each page should be proved before proceeding to the calcula- 
 tions of the next. When correct, the difference of the sums 
 of the backsights and foresights on the page equals the differ- 
 ence of the first and last elevations on the page. Thus, in the 
 form given 
 
 (39,162 — 11.361) = (227.801 — 200.000) = 27.801 
 
 In this proof we ignore all elevations except those of turn- 
 ing points, and benches used as such, and the height of instru- 
 ment, 
 
 At the end of the survey, as well as at the end of each day’s 
 work, a bench is established from which the survey may be 
 resumed at any future time See &§ 28, 29, and 80. 
 
 214. The object of making such a survey with level and 
 rod is to furnish a profile or vertical section of the entire 
 line, showing in detail the rise and fall of the surface over 
 
a 
 
 LEVELLING. 183 
 
 which it passes. The profile is plotted on profile-paper pub. 
 lished for the purpose, the horizontal scale being usually 400 
 feet to an inch, and the vertical scale 30 feet to an inch. This 
 distortion of scale magnifies the vertical measures so that 
 slight changes in the elevation of the surface may be seen 
 distinctly. 
 
 215. When only the difference of level of two extreme 
 points is required, the survey is more simple. No readings 
 are taken except on turning-points, the backsights and fore- 
 sights being recorded in separate columns. No calculation is 
 required until the survey is finished, when—the first reading 
 having been taken on one of the given points, and the last on 
 the other—the difference of the sums of the backsights and 
 foresights is the difference in elevation of the two points, ac- 
 cording to the method of proof mentioned in § 218. Thus the 
 difference in level of any two benches established on a previ- 
 ous survey may be tested, and, if found correct, all the inter- 
 mediate elevations on the line may be assumed to be correct 
 also. The discrepancy should not exceed one tenth of a foot 
 in any case, and is usually much less. 
 
 216. Any lack of adjustment in the instrument gives 
 the line of sight a slight angle of elevation or depression, 
 causing a slight error in every reading, proportional to the 
 distance of the rod from the instrument. But the errors being 
 equal for equal distances, and the backsights and foresights 
 having opposite signs in our calculations, the errors cancel 
 when the distances are equal. Hence, to avoid errors in ele- 
 vation, each new turnin’g-point should be as nearly as possible 
 at the same distance from the instrument as the point on which 
 the last backsight was taken. For precise reading, the rod 
 should not be more than 400 feet from the instrument. 
 
 217. Another cause of error in readings is want of verti- 
 cality in the rod. This may be avoided by the use of a disk- 
 level, or in the absence of wind, by balancing the rod. The 
 rod may be plumbed one way by the vertical cross-hair of the 
 level, and to ensure a vertical reading in the plane of the line of 
 sight, the rod may be gently waved each side of the vertical 
 toward and from the instrument, the shortest reading being 
 
 Sppremapeiee MPF RE SIO — 
 = SS ae 
 ev 
 
 2 
 
 ae 
 
186 FIELD ENGINEERING. 
 
 the correct one; or in case of a target rod, the target should 
 rise to, but not above the horizontal cross-hair, as the rod is 
 waved. 
 
 218. When very long sights are required to be taken with 
 the level, another source of error must be considered, namely, 
 the curvature of the earth. 
 
 A level line is parallel to a great circle of the earth, and is 
 therefore an arc of a circle, or may be so considered. 
 
 A horizontal line is a straight line parallel to the plane of the 
 horizon. Therefore the line of sight, being a horizontal line, 
 is tangent to the circle of a level line passing through the in- 
 strument. 
 
 To find the correction in elevation due to curvature of the 
 earth for any distant station. Fig. 82. 
 
 Fie. 82. 
 
 Let A be the station of the instrument J, and B the distant 
 station observed. 
 
 Let &, = CI= the radius of curvature of the earth, or of the 
 parallel arc JD. Let Z, = ID = the level distance between 
 Aand B. Then JH, perpendicular to CZ, is the line of sight, 
 BE is the reading of the rod, and DE = EH, = the correction 
 due to curvature. 
 
 By Tab. I., 24, JH? = DH (DE+ 2R.); but since DZ is 
 very small compared with 2f,, it may be omitted from the 
 parenthesis, and since JH = ID = L, very nearly, because 
 the angle ACB is very small, we have Z,? = 2R,£.. 
 
 2 
 B= ae (815) 
 
Ss 
 
 LEVELLING. 18% 
 
 219. Refraction. In observing distant stations the 
 line of sight passing through the atmosphere is refracted from 
 the straight line JH, Fig. 82, and takes the form of a curve, 
 which, for practical purposes, may be considered as the arc of 
 a circle, concave downwards. Its radius, depending on the 
 conditions of the atmosphere, varies from 5} to 74 times the 
 radius of curvature of the earth. 7R, is considered a good 
 average value. 
 
 Refraction causes the observed object to appear too high, 
 while the curvature of the earth causes it to appear too low ;— a] 
 the effects being contrary, the correction for curvature is re- 
 duced by the correction for refraction. If we let H, = the Wi 
 total correction for both curvature and refraction, to be added Hi 
 to the apparent elevation of the observed object, then Hi 
 
 H, =~E,=53° (316) 
 
 Table XVII. is calculated by this formula, assuming a mean Hi 
 value of R, = 20,918,650 feet. 
 
 220. The form of the earth is approximately an el- 
 lipsoid of revolution. Its meridian section at the mean level | 
 of the sea is an ellipse, the semi-axes of which are, according i 
 to Clarke, 
 
 at the equator A = 6378206 metres [6.8046985] | 
 at the poles B= 6356584 ‘“‘ — [6.8082288] ' 
 
 According to the same authority 
 
 1 metre = 3.280869 feet [0.5159889] Wy 
 ae 
 Therefore the semi-axes expressed in feet are | 
 A = 20 926 058 feet ['7.8206874] , | 
 = 20855119 “ ['7.3192127 
 
 Then the radius of curvature of the meridian 
 
 9 
 
 at the equator, es = R, = 20 784 422 ft. [7.3177379) 
 
 A2 
 at the poles, a = R, = 20997 240 ‘ ['7.3221622] 
 
188 FIELD ENGINEERING. 
 
 In latitude 40° the radius of curvature of the meridian is 
 20 871 900, and of a section at right angles to the meridian, 
 20 955 400; the mean valuc,.or 22, = 20 913 650 [7.820430], be- 
 ing adopted for general use. The error in the correction H, 
 eq. (316) due to this assumption will usually be much less than 
 that due to the assumed value of the radius of refraction. 
 
 221. Levelling by Transit or Theodolite. When 
 a transit has a level-tube attached to the telescope, it may 
 be used as a Theodolite for levelling, and for taking vertical 
 angles. If the instrument be in perfect adjustment, the line 
 of sight will be horizontal when the bubble stands at the 
 middle point of the tube, and the reading of the vertical circle 
 will be zero. Should there be a small reading when the line of 
 sight 1s horizontal it is called the ¢rdex error. When the line 
 of sight is not horizontal, the angle which it makes with the 
 plane of the horizon is called an angle of elevation, or of de- 
 pression, according as the object upon which the line of sight 
 is directed is above or below the telescope. This angle is 
 measured on the vertical circle, being the difference of the 
 reading and the index error, when both are on the same side 
 of the zero mark, and their swm, when they are on opposite 
 sides. When the distance to an observed object is known, 
 and its angle of elevation or depression is measured, we may 
 calculate its vertical height above or below the telescope. 
 
 § elevation 
 ’ depression 
 sb I, = the horizontal distance 
 
 Let + a = angle of 
 
 ‘‘ I’ = the distance parallel to 
 line of sight 
 
 & h = difference in elevation of 
 object and instrument. 
 
 Then for short distances, 
 
 hA=Ltane=WL' sina (817) 
 
 Fie, 83. For long distances the curvature of 
 
 the earth and refraction must be considered. Fig. 838. 
 Tet.J be the place of the instrument, and F’ the object 
 observed, 
 
S| 
 
 LEVELLING. 189 
 
 Let Z, = the distance, measured on the chord of the level 
 arc LD, passing through the instrument; and let # = the 
 number of seconds in the arc JD; hence, since for ordinary 
 distances the chord and are are sensibly equal, 
 
 b= = 206264".8 [5.314429] . 
 
 or giving to R, its mean valué, § 220, 
 wy = L, X .0098627 [7.993995] 
 
 or a fraction less than 1” per 100 feet. 
 
 Let IF’ be the arc of the refracted ray, and assuming that its 
 radius is 72,, the arc will contain 4th the number of seconds 
 of the arc LF 
 
 IF’, tangent to IF, is the direction of the telescope; J/ is 
 the chord of the are JF, and JZ is the horizontal. 
 
 Let a — EIF' =observed angle of elevation. Then HIF = 
 true angle of elevation = HIF” — fUIRP=a — + 4b a— 
 
 .O71y. i 
 The angle HID = 4) .°. DIF = 4p + a — .071yp; and Hi 
 
 IDF = 90° + bbe FD = 90° -— We — O71y). | 
 We now solve the triangle [7D for the side DF = h, and He 
 
 find } 
 
 sin (4% + a — .071y) (318) | 
 
 pr L, cos (wb ae ea 0717) 
 
 For an observed angle of depression make a negative in the 
 formula. || 
 
 The coefficient .071 is called the coefficient of refraction, this |) 
 being a fair average value, while its extreme range is from .067 | 
 to .100 under varying conditions of the atmosphere, and values | 
 of the angle a. Wt 
 
 When the difference in elevation of two or more distant 
 objects is required, we obtain the elevation of each separately, 
 and subtract one elevation from another. The elevation of the 
 observed object is given by (7. 1.) + A. 
 
 222. To find the Height of Instrument of a transit or 
 theodolite by an observation of the horizon. Fig. 84. 
 
Sree Be OS 
 
 190 FIELD ENGINEERING. 
 
 Let J be the place of the instrument, and let « = observed 
 angle of depression of the horizon. 
 
 Let # be the point where the refracted ray meets the level 
 surface, and draw the chords JF’ and AF. 
 
 Let w= the angle ACF, let h = AJ, and let & = the coefii- 
 cient of refraction. 
 
 In the triangle JAP, 
 
 IAF = 90° +44, AFT = 4 — kp, AIF = 90° —  — kp) 
 
 Hence FIE = » — kt. But FIH=a-+ ky 
 
 (a 
 
 b= = (319) 
 
 Let F” be the tangent point of a right line drawn through J; 
 
 Fia, 84 
 
 then AI = OF" exsec ACF", but CF" = R,, and, since 7 is 
 
 always very small, ACH" = 4(# + @) very nearly = — a 
 
 Ven hk 
 p= Jey exsec any a (820) 
 
 Giving to R, its mean value, § 220, and assuming k= 7; 
 
 log h = 7.820480 + log exsec 1.0801 a (321) 
 
at 
 
 LEVELLING. 19} 
 
 Otherwise, we may solve the triangle AIF since 
 
 sin (3 — ky) 
 
 : Fae Fee as a sin 4@ (322) 
 Be = 2h, sin 3 — 21° at - 
 COS 7 5h 
 When &k = ze 
 sin 4a@ 
 —~) it ZC 
 h = 2R, sin ca. nae (823) 
 
 Example.—The observed dip of the sea horizon is 24 =a 
 What is the height of the instrument above the sea? 
 
 By eq. (821) 1.0801 x a@ X 60 = 1555".34 3.191825 
 9 
 
 6.383650 
 Table XXVI. q—22 9.070130 
 R 7320430 
 
 ° 
 
 h = 594.58 2.7'74210 
 
 Methods of determining heights by distant observations can- 
 not be relied on for more than approximate results, since they 
 necessarily involve the uncertain element of refraction, and 
 usually a lack of precision in the vertical angle, the are reading 
 only to minutes in ordinary instruments. These methods, how- 
 ever, are useful where no great accuracy is required, as for a 
 temporary purpose until levels can be taken in the regular way, 
 or for interpolating between points of established elevation, 
 
 223. Stadia Measurements. 
 
 It is sometimes convenient to determine distances by instru- 
 mental observation For this purpose two additional cross- 
 hairs may be placed in the telescope parallel to each other and 
 equidistant from the central cross-hair. These are called stadia 
 hairs, and distances determined hy them are called stadia 
 measurements. The stadia hairs are adjusted so as to inter- 
 cept a certain space on a rod held at a certain distance from 
 the instrument and perpendicular to the line of sight. For any 
 
192 FIELD ENGINEERING. 
 
 other place of the rod, the distances and intercepted spaces 
 are nearly proportional. The exact relation is given below. 
 Fig. 88. 
 
 Let / = AB, the distance of the rod from the vertical axis 
 of the instrument; ¢ = the distance from the axis to the ob: 
 ject glass of the telescope; 4 = the distance from the object- 
 
 Fie. 85. 
 
 glass to the rod; ¢ = the space between the stadia hairs; s = 
 CD the space intercepted by them on the rod; and f = the 
 focal distance of the object-glass. We then have by optics, 
 
 Ss a— : . 
 cat oF, whence a — f as and sinceea = 1l—c.°. l— 
 oe 
 
 (f+ o= 78 
 
 f, and the space between the stadia hairs ¢ are constant, while 
 sand c vary with 7. For any other distance lJ’, we then have 
 
 Now in any given instrument the focal distance 
 
 li—(fte)= Ly, and combining the two equations 
 . Seay 
 l= (ft =< +e] (324) 
 
 s'is usually assumed at 1 foot and /' — (f+ c’) at 100 feet. 
 and the stadia hairs are then adjusted accordingly. The focal 
 distance f may be found by removing the object glass and ex- 
 posing it to the rays of the sun and noting at what distance 
 from the surface of the lens the rays form a perfect and min. 
 ute image of the sun on a smooth surface; the distance ¢’ is 
 measured on the telescope when the rod is clearly in focus, 
 at the assumed distance. 
 
 To measure any other distance, the rod is again observed 
 at the desired point, and the space s noted, which, placed in 
 eq. (824), gives 1 — (f+ ce) at once. We then measure ¢ on 
 the telescope, and adding (f+ c), obtain /, the distance re- 
 quired, 
 
LEVELLING. 193 
 
 But inasmuch as ¢ has but a small range of values, it will 
 usually be sufficient to assume for it a mean value, as a con- 
 stant. In this case we may find the value of (f+ ¢) = 1 
 for the instrument used. Making c’ = ¢ in eq, (824), and solv- 
 ing for (f+ c), we have 
 
 2 sl’ — s'l 
 Tees ae a 
 8 
 
 — 
 
 iv) 
 
 =~ 
 iS) 
 ve 
 CU 
 —— 
 
 and by laying off on level ground any two distances from the 
 instrument for 7’ and J, as 100 and 500, and observing the 
 corresponding spaces s’ and s intercepted on a rod, we insert 
 them in eq. (825) and find (f+ o). 
 
 Having found (f+ ¢), lay off (100 + f-+ ¢) from the instru- 
 ment and adjust the stadia hairs to inclose just one foot on 
 the rod at that distance. Any other distance is then found by 
 
 the formula, 
 $= 100s+(f+9 (326) 
 
 Exam ple.—At l’ = 100 we finds’ =: 1.00, and at = 500 we 
 find s = 5.061. 
 
 . 506.1 — 500 
 (QD eat re mete — 9 
 Hence, eq. (825) ft+e= 061 = 1.502 
 
 and eq. (326) 1 =100s-+-1.5; provided the stadia hairs be ad- 
 justed so as to intercept 1 foot at 101.5 fect distance from the 
 centre of the instrument. 
 
 224. The foregoing formule are all that are necessary for 
 horizontal sights, but since the line of collimation is generally 
 inclined more or less to the horizon, 1t follows that the stadia 
 hairs will intercept a larger space on the vertical rod than 
 that due to the true horizontal distance. We therefore require 
 a formula for reducing inclined measurements 
 to the horizontal. Fig. 86. 
 
 Let « = HFG = the angle of inclination of the line of colli- 
 mation JG; 
 “ §@ = CFD = the visual angle defined by the stadia hairs; 
 «< ¢ — OD = space intercepted on a vertical rod. 
 
 Then (Fig. 85), 
 CGH 3 
 re if — 2 face re ee ns — =F 5! 
 tan 34 EF * er Pr) 
 
 a. 
 (SO) 
 © 
 
 ko) 
 
194 FIELD ENGINEERING. 
 Th In Fig. 86 
 ' s= CE — DE = EF [tan (a + 49) — tan (a — 46)] 
 while the true value (for the same distance) would be 
 C'D' = 2EF tan 48 
 Dividing one by the other we derive 
 
 CD 2 tan 40 a 
 $. tan (a + 46) — tan (a — 46) 
 
 By giving to 8s’ and /’—(f+ ec) in eq. (827) their customary 
 
 Fig. 86. 
 
 values, v7z., 1 and 100, we have tan46=.005 .:°. 6 = 34’ 22’.63 
 and by Trig. Table II. 70, 
 
 sin 9 
 aH tan (v@-+-46) — tan (a — 46) = ———_____ —_ 
 
 iia Coase ot ( ) cos (aw + 49) cos (~—44) 
 Since 6 is small, we have sensibly 
 
 Hoi sin 6 = 2 tan 49, and cos (@ + 49) cos (a — 49) = cos? « 
 
 Hii | and the last equation reduces sensibly to 
 
 it C'D 
 
 Hi rae cos? @ (328) 
 
 / 8 
 
 which is the coefficient of reduction required by which 
 i to multiply the observed space s in case of inclined sights. 
 
 Hence the formula for distance (eq. 326) becomes in this case 
 without sensible error 
 
 {| t= 100 s cos? a+ (f+e) (829) 
 
 Tables XVIII. and XIX. have been calculated by the exact 
 formula for the coefficient. 
 
LEVELLING. 19d 
 
 Example.--Given : « = 8° 20' and s = 9,221; what is the 
 horizontal distance to the rod? 
 Eq. (829) 100 log...2. 
 8 9.221 0.964778 | 
 a 8° 20' Tab. XTX. «¢ 9.99078) 
 902.7 2.955558 
 +e 1.5 .. Ans. 904.2 ft. 
 
 The rodman should have a disk level to insure keeping the 
 ro‘ vertical. 
 
 225. Another method of procedure is that in which 
 the rod is always held perpendicular to the line of collimation, 
 however much inclined the latter may be. To secure this posi- 
 tion of the rod, a small brass bar is attached, having sights 
 upon it through which the rodman watches the instrument 
 during an observation, the line of sight being at right angles to 
 the rod. The distance thus obtained is of course parallel to 
 the linc of collimation, and requires to be reduced to the hori- 
 zontal. 
 
 For this purpose, we have (Fig. 87). 
 
 Fia. 87. 
 TH=IGcosa+ LG sin a 
 or IH = (100 s+ f+e)cosa+r sin a (330) 
 
 in which r is the reading of the rod by the line of collimation. 
 For the elevation of the point B above J, Hy 
 
 EB = HG — GBcosa 
 or ERB = (100s+ f+ ¢) sin «~— 7 cos a (3831) 
 
196 FIELD ENGINEERING. 
 
 When the distances are sufficiently great, correction must be 
 made for curvature of the earth and refraction, as already ex- 
 plained. 
 
 This method is employed by the topographical parties of the 
 U. §. Coast Survey in connection with the plane table. Their 
 instruments, however, are so constructed as to give distances 
 in metres, and heights in feet, requiring a modification of the 
 above formule. 
 
 OMAP ih ix. 
 CONSTRUCTION. 
 
 HH 226. The engineer department of a railway com- 
 | pany is usually reorganized for the construction of the road, 
 as follows: Chief engineer, Division engineers, Resident 
 engineers, Assistant engineers. On some roads the division 
 engineers are styled ‘‘ Principal Assistants ;’ the resident 
 engineers, ‘‘ Assistants;” and the assistant engineers are de- 
 ! signated according to their duties, as ‘‘leveller,” ‘ rodman,” 
 Hi etc. | 
 
 A resident engineer has charge of‘a few miles of line, 
 
 i limited to so much as he can personalfy superintend and 
 
 direct. He has one or more assistants and an axman in his 
 i party. All instrumental work is done and all measurements 
 taken by the resident engineer and his assistants. 
 
 A division engineer has charge of several residencies, 
 | and inspects the progress of the work on his division once 
 \'( | or twice a week. In his office, which should be centrally 
 | located, all maps, profiles, plans, and most of the working 
 bial drawings required on his division are prepared. To him the 
 HGH resident engineers make detailed, reports once a month, or 
 
 | oftener if necessary, which he passes upon as to their cor- 
 rectness, and from which he makes up a monthly report, or 
 estimate, of the amount and value of the work done and ma- 
 terials provided by each contractor on his division. The esti- 
 mates are forwarded about the first of each month to the 
 chief engineer, who examines and approves them, returning 
 for modification any that seem to require it. 
 
 9 
 
CONSTRUCTION. 19% 
 
 The chief engineer has charge of the entire work, 
 and directs the general business of the engineer department. 
 He occasionally inspects the work along the line. 
 
 227. Clearing and Grubbing. The first step in 
 the work of construction is to clear off all growth of timber 
 within the limits of the right of way. The resident engineer 
 with his party passes over the line, making offsets to the right 
 and left, and blazing the trees which stand on, or just within, 
 the limits of the company’s property. The blazed spot is 
 marked with a letter 0, as a guide to the contractor. After 
 felling, the valuable timber should* be piled near the boun- 
 dury lines, to be saved as the property of the company. The 
 brushwood 1s burned. 
 
 Where a deep cut is to be made, the stumps are left to be 
 removed as the earth is excavated. In very shallow cuts and 
 fills the contractor will generally prefer to tear up the trees 
 by their roots’ at once, rather than to grub out the stumps 
 after clearing. Where the embankments will be over three 
 feet high, grubbing is not necessary; but the trees require to 
 pe low-chopped, leaving no stump above the roots. The engi- 
 neer should indicate to the contractor the localities where each 
 
 process is suitable. 
 
 clearing is in progress, the engineer should 
 run a line of test Iqvels touching on all the benches to verify 
 their elevations ; Ife may also rerun the centre line, replacing 
 any stakes that may have disappeared, and setting guard plugs 
 to any important transit points which may not have been 
 previously guarded. If any changes in the alignment have 
 been ordered, these may be made at the same time. 
 
 228. While th 
 
 229. Cross Sections. The resident engineer is fur- 
 nished with ‘a profile of the portion of the line in his charge, 
 upon which 1s plainly indicated by line and figures the estab- 
 lished grade, From this he calculates the elevation of grade 
 at each station, and by subtracting this from the elevation of 
 the surface, he derives the depth of cut or fill (+ or —) to be 
 made at each point. The grade given on the profile is that 
 which is subsequently called the subgrade, being the surface 
 of the road-bed. The final or true grade is the upper surface 
 of the ties after the track is laid. 
 
198 FIELD ENGINEERING. 
 
 The base of a cross section is identical with the width of the 
 road-bed. It is made wider in cuts than in fills to allow for 
 the side ditches. Six feet should be allowed in earth, and 
 four feet in rock cuts. The ratio of the side slopes 
 depends upon the material. The usual slope ratio for earth is 
 1} horizontal to 1 vertical for both excavation and embank- 
 ment. Damp clay and solid gravel beds will stand for a time 
 in cuts at 1 to 1, or an angle of 45°, but this cannot be perma- 
 nently depended on. On the other hand, fine sand and very 
 wet clay may require slopes of 12 to 1 or 2 to 1. Exceptional 
 cases require slopes of 3 or 4 to 1. In rock work the slopes are 
 usually made at } to 1 for solid, 4 to 1 for loose, and 1 tol for 
 very loose rock, liable to Tone ceed Rock embankmenis 
 stand at 1 to 1. 
 
 230. All cross sections are taken in vertical planes at 
 right angles to the direction of the centre line. Figs. 88, 89. 
 Formule. 
 
 Let 6 = AB, the base of section, or road-bed. 
 
 Rtas ad # WA Spent % We 
 2 Diy aca the slope ratio 
 
 ““ d = CG = the cut (or fill) at the centre stake. 
 “ h = DH or HN = the cut (or fill) at the side stake. 
 
 x = CD = the ‘‘distance out”’ from centre to side stake. 
 eta teem Cica kD), 
 
 We have at once from the figures the general formula 
 x= 4b-+s8h (3832) 
 When the ground is level transversely, 
 h=d, and # = 40-4 sd, 
 
 For embankment use the same formula, considering d or h as 
 positive in this case also, the figure being simply inverted. 
 
 "Len the ground ts inclined transven ’sely; 
 h=0G+DK=d+y onthe upper side in cuts; 
 w= 4b-+-sd-+ sy (333) 
 
 and h=EN=d—y onthe lower side in cuts 
 
 = 46+ sd — sy (334) 
 
CONSTRUCTION. 199 
 
 For embankments use the same formule, but apply eq. (838) to 
 the lower side and eq. (334) to the upper side, the figure being 
 inverted. The points D and / on the ground are usually found 
 by trial, such that the corresponding values of # and y will 
 verify the formule. 
 
 When the natural slope FD or LE is uniform its ratio 8’ may 
 be found by measuring along the section the horizontal dis- 
 tance necessary to change the reading of the rod 1 foot (or half 
 the distance necessary to change it 2 feet, etc.). Then, having 
 found the depths of cut (or fill) at /’and ZL, distant 4d from the 
 centre C, we have 
 
 BH = sh = s(h — BF) 
 and AN = sh = 8 (AL — h) 
 Yrom these we have, for the upper side in cuts, and lower gide 
 in fills. 
 ' 
 
 SRR ean 
 
 =) 
 
 Vise gees) BF. ..2-= 4) 4 - 
 
 S$. te 
 
 also, for the lower side in cuts, and upper side in fills, 
 
 PTs Fs Be ee Beale awe AL (886) 
 We also have 
 h—- BF= 7 — BR | 
 vag + (337) 
 AL —h= eee AL 
 aereet 
 
 whence the points D and # may be found by the level. 
 
 But points D and # thus calculated should have their post- 
 tions verified by the general formula, eq. (332), lest the slope 
 s may not have been perfectly uniform. 
 
 When the natural surface intersects the base between the 
 points 4A and B, the section is said to be in side hil! work, 
 Fig. 90. Both portions of the section are then determined by 
 eq. (833), or where the slope s' 1s reguiar, by eq. (835) measuring 
 in every case from the centre stake C; but observing that 
 when the centre is in cut and one side in fill, or vice versa, that 
 J must be considered negative for that side, wheuce eq. (883) 
 becomes for this case 
 
 z= 4b—sdt+ ey (833) 
 
200 FIELD ENGINEERING. 
 
 231. Staking out Earthwork. Beginning at a 
 point on the centre line where the grade cuts the natural sur- 
 face, the engineer drives a grade stake (marked 0.0) and notes 
 the point in the cross-section book. If the line of intersection 
 of the road-bed and surface would make an acute angle with 
 the centre line, he also finds the points where the edges of the 
 proposed road-bed will intersect the surface, drives grade 
 stakes, and also stakes out a cross section through each of 
 those points, if necessary. 
 
 Then advancing to the next point on the centre line where 
 a section is required, he finds its elevation with the level (veri- 
 fying or correcting the elevation taken on the location), calcu- 
 lates the depth of cut or fill CG, which 1s then marked upon 
 the back of a stake there driven; a cut being designated by C 
 and a jill by L. 
 
 Lf the ground is level transversely (Fig. 88), he calculates z by 
 
 Fig. 88. 
 
 eq. (382) and lays off this distance at right angles to the centre 
 line, driving slope stakes at the points D and #, marked with 
 the depth of cut or fili. The marked side of slope stakes should 
 face the centre line. 
 
 Lf the ground ws inclined transversely (Fig. 89), he first measures 
 
 Fie. 89. 
 
 the distance, 45, to F’ and finds the depth BF’ for record. He 
 then proceeds to find the point D. If the natural slope be uni- 
 form, D may be found by eq. (335) or (337), verifying the result 
 by eq. (832). The point Hof the other slope may be found 
 similarly, using eq. (336) or eq. (337): verifying by eq. (332). 
 
CONSTRUCTION. 201 
 
 232. If the ground be irregular, the depth of cut or fill is 
 found not only at the centre and edges of the road-bed, but 
 also at every other point along the cross section where the sur- 
 Jace slope changes, all of which depths are recorded, together 
 with their respective distances from the centre. To find the 
 point D: assume a point supposed to be near D, and there 
 take a reading of the rod. The difference of the readings at 
 that point and at Cequals y’ for that point, which inserted in 
 eq. (333) gives a value 2’, If 2’ agrees with the horizontal dis- 
 tance of the assumed point from (C, the true position of D has 
 been found. If 2’ be greater than this, by subtracting the eq. 
 
 ‘= 30-+ sd-+sy’' from eq. (888) we derive 
 
 a= x + sy — ¥) (338) 
 
 the last term of which shows the correction to be added to 2’. 
 Now in advancing from the assumed point to the extremity of 
 
 , the rise of the surface 1s approximately (y — y’), and if, in 
 going the additional distance, s(y — y'), a further rise is en- 
 countered, this last, multiplied by s, must also be added to 2’, 
 and so on until the additional advance makes no change in the 
 value of y. The point thus found, verified by eq. (882), is the 
 point D required. 
 
 But if 2’ be less than the distance of the assumed point from 
 C, we have 
 
 v= a' — y' — Y) (338)' 
 
 the corrections being subtractive. 
 
 The point # on the other slope is found in a similar manner, 
 using eq. (834) for the value of 2’; if 2’ be greater than the as- 
 sumed distance, we have 
 
 2=e' —y—y) (339) 
 
 the corrections being subtractive ; but if @' be less than the as. 
 sumed distance, 
 
 =v + sy —Y) (339)’ 
 
 the corrections being additive. 
 
 233. In side-hill work (Fig. 90) proceed in the same 
 manner, using eqs, (833) or (333)' and (388) in all cases of un- 
 even ground. When the surface slope s’ is uniform, eq. (335) 
 may be used, if preferred, on either side. In addition to the 
 
 ia] 
 
 re 
 
 eee 
 
202 FIELD ENGINEERING. 
 
 centre and side stakes, a grade stake is driven at the point 0, 
 where the surface intersects the grade, the stake facing down 
 hill. 
 
 To find a grade point, set the target to a reading equal to the 
 height of instrument less the elevation of grade, and stand the 
 rod at various points along the given line until the target coin- 
 cides with the line of collimation. 
 
 Fie. 90. 
 
 234. When two materials are found in the same section, 
 as rock overlaid with earth, each material requires its own 
 slope, and a compound section is the result. To stake 
 out work of this description, the depth of earth to the rock must 
 be known, and may be nearly ascertained by reference to an 
 adjacent section already excavated. Fig. 91. 
 
 heal ke Ta ono a ial 
 
 - 4 
 
 Fria. 91. 
 
 Let a; be the depth of earth at C 
 
 “6 is Ct Sha 6c “6 P or Q 
 ‘* s, be the ratio of rock slope 
 Sh Sig oe ** earth slope 
 Then w= 40+ si(d — a) + 91) + 8e(de + Yo) (840) 
 
 in which y; = difference of rod readings on the rock at C, and 
 D,, or C, and #,; and y, = difference of rod readings on the 
 surface at Pand Dz, or at @ and #,. The upper sign applies 
 to the upper side, the lower sign to the lower, 
 
CONSTRUCTION. 203 
 
 It is better, however, to make an indefinite cross profile at 
 first, driving two reference stakes quite beyond the section 
 limits: and when the contractor has removed the earth from 
 between D, and Z,, indicate to him those exact points by 
 marks on the rock, and also set the slope stakes at D. and £. 
 
 235. The frequency with which cross sections should 
 be takén depends entirely upon the form of the surface; where 
 this is regular, a section at each station is sufficient. A cross 
 section should be taken, not only at every point on the centre 
 line where there is an angle in the profile, but also wherever 
 an angle would be found in the profile of a line joining a series 
 of slope stakes on either side, even though the profile of the 
 centre line may be quite regular at the corresponding point :— 
 the object being, not only to indicate the proper outlines of 
 the earthwork, but to furnish the data necessary to calculate 
 correctly the quantities of material removed. Rockwork will 
 generally require more frequent sections than earthwork. 
 
 236. Vertical Curves.—The grades as given on the 
 profile are right lines, which intersect each other with angles 
 more or less abrupt. ‘These angles require to be replaced by 
 vertical curves, slightly changing the grade at and near the 
 point of intersection. A vertical curve rarely need extend 
 more than 200 feet each way from that point. Fig. 92. 
 
 Fig. 92. 
 
 Let AB, BOC, be two grades in profile, intersecting at station 
 B, and let A and C be the adjacent stations. It is required to 
 join the grades by a vertical curve extending from A to @. tt 
 Suppose a chord drawn from A to (;—the elevation of the 
 middle point of the chord will be a mean of the elevations of 
 grade at A and (C; and one half of the difference between this 
 
owe 
 
 204 FIELD ENGINEERING. 
 
 and the elevation of grade at B will be the middle ordinate of 
 the curve. Hence we have 
 
 grade A + grade 0 | 
 
 9 — grade B) (341) 
 
 M=+%4 
 in which M = the correction in grade for the point B. The 
 correction for any other point is proportional to the square of 
 its distance from Aor 0. Thus the correction at A 26 is 
 3M; at A+ 50 it is $M; at A+ 75 it is {M, and the same 
 for corresponding points on the other side of B. The correc- 
 tions in the case shown are subtractive, since M is negative. 
 They are additive when M is positive, and the curve concave 
 upward. 
 
 These corrections are made at the time the cross sections 
 are taken, and the corrected grades are entered in the field- 
 book opposite the numbers of the respective stations. 
 
 237. Form of Field-book.—A complete record of 
 all cross-section work is kept in the cross-section book. 
 On the left-hand page is recorded, in the first column, the 
 numbers of the stations and other points where sections 
 are taken; in the second, the elevations of those points, copied 
 in part from the location level-book, but verified or corrected 
 at the time the section is taken; in the third, the elevation of 
 the grade for the same points; in the fourth, the width of 
 base 0b; in the fifth, the slope ratios, s; and in the sixth, the 
 surface ratio s’ when uniform. The right-hand page has a 
 central column, in which, and opposite the number of the 
 station, is recorded the centre depth of the section, marked 
 + or —, to indicate cut or fill, as the case may require. 
 To the right of this are recorded the notes of that portion of 
 the section which lies on the right of the centre line, as the 
 line was run, and to the left, the notes of the left side. The 
 distance from the centre to each point noted is recorded as 
 the numerator of a fraction, and the cut or fill at the point 
 as the denominator, prefixed by a-+ or — as the case may 
 require. The denominator for a grade point is zero. The 
 numbers of the stations should increase wp the page, as in a 
 transit book, so that there may be no confusion as to the right 
 and left side of the line. The several points being noted in 
 order as they occur from the centre outwards, the notes far: 
 
CONSTRUCTION. 205 
 
 thest from the centre of the page usually appertain to the 
 slope stakes; but in case the cross profile is extended beyond 
 the slope stake, the note of the latter should be surrounded by 
 a circle to distinguish it. The following form is a specimen 
 of a right-hand page, with the first column only of the left- 
 hand page: ; 
 
 Sta. |} Cross | Sections. 
 \| | 
 83 || Hea” MGIC HOGI, SHOR OF SOR SA0 __ 20 CB 56 
 © | + 8.6 + 14 417.7 4 21 | +215) 420.8 +25.6 428.3 +20.4 
 | 
 + 60 || 37.5 _ 10 cod Os exis dO ek 42.6 
 ti 5.0, 410.1192). oe 14 Pid 7 180 1 101 7 
 89 BS 2s AO ah tiie ton 6 10 ; 31.6 
 2 +38 +5.4| 19.4) + 8.5 1117.6 114.4 
 ies fGen Ot IO 19,2 
 S| OT eer ae he 
 + 27 0 
 24 er ane ar fame: 
 Larue! 9.6 aigeeacs Wining 
 SI 25.99 MOVBID 15 
 —12.6-11.2 | —12 | 10.6 — 5.3 
 eg} BBSydi A Frn Orsi hut whier 48! eaGy 
 —17.6 —16.4 | —17.6 | —19.6 —19.1 —12.4 
 
 238. In case there is a liability to land-slips, the profiles 
 of cross sections should be carried beyond the slope stakes, 
 on the upper side of the cut, to any distance thought neces- 
 sary to reach firm ground, and stakes driven for future refer- 
 ence. When a number of consecutive cross profiles are to be 
 considerably extended, it is well to first run, instrumentally, 
 a line parallel to the centre line, and set stakes opposite the 
 stations, taking their elevations. The intermediate surface of 
 the sections may then be taken with cross-section rods if more 
 convenient. See $37, 
 
 239. In case of inaccessible ground, preventing a 
 regular staking out, an indefinite profile of the section may 
 generally be obtained, referred to the datwm for elevation and 
 to the centre line for position, which being plotted on cross- 
 section paper, and the grade line and side slopes added, shows 
 to scale where the slope stakes should be, 
 
206 FIELD ENGINEERING. 
 
 240. Any isolated mass of rock or earth which oc- 
 curs within the limits of the slope stakes, but not included in 
 the regular notes, is separately measured and noted, so that 
 its contents may be computed and added to the sum of the 
 same material found in the cross sections. 
 
 241. Borrow-pits.—When the excavations will not 
 suffice to complete the embankments, material may be taken 
 from other localities, termed borrow-pits. These should be 
 staked out by the engineer and their contents calculated, 
 unless the contractor is to be paid for work by embankment 
 measurements. A number of cross profiles are taken of the 
 original surface, and (on the same lines) of the bottom of the 
 pit after it is excavated, which furnish the depth of cutting 
 at each required point. Borrow-pits should be regularly ex- 
 cavated, so that they may not present an unsightly appear- 
 ance when abandoned. Borrow-pits may be avoided by 
 widening the cut uniformly at the time it is staked out, so 
 that it may furnish sufficient material; provided the material 
 is suitable, the embankment accessible, and the distance not 
 too great. When the excavation is in excess, the surplus ma- 
 terial should be uniformly distributed by widening the adja- 
 cent embankments, if possible; otherwise it 1s deposited at 
 convenient places indicated by the engineer and is said to be 
 wasted, 
 
 242. Shrinkage.—In estimating the relative amounts of 
 excavation and embankment required, allowance must be made 
 for difference in the spaces occupied by the material before ex- 
 cavation and after it is settled in embankment. The various 
 earths will be more compact in embankment, rock less so. The 
 difference in volume is called shrinkage in the one case, and 
 trcrease in the other. 
 
 Shrinkage in 1000 cu. yds. 
 
 Material. -Of excavation. Of settled embkt. 
 Sand:and gravel... /s.5..2. si 80 C. Yds. 87C. Yds. 
 SOLA tecnsouin's face pean nuh cattle 10055 111 35 
 TAU cee ratte Salen see tua ai eae 14 Sa Leh 
 NVGL SULT. OS cae ea ee eee wae es 1508 Ps 200°“ 
 
 Increase in 1000 cu, yds. 
 Rock, large fragments......... 600 C. Yds, 375 C. Yds. 
 «¢ “medium tragments...... TU 413“ 
 
 ‘+ gmall seine, 8 800 i 444 * 
 
CONSTRUCTION. 20% 
 
 Thus, an excavation of sand and gravel measuring 1000 cubic 
 yards will form only about 920 cubic yards of embankment; or 
 an embankment of 1000 cubic yards will require 1087 cubic 
 yards of sand or gravel measured in excavation to fill it; but will 
 require only 587 cubic yards of rock excavation, the rock being 
 broken into medium-sized fragments; while 1000 cubic yards 
 of the latter, measured in excavation, will form 1700 cubic 
 yards of embankment. 
 
 The lineal settlement of an earth embankment will be 
 about in the ratio given above, therefore the contractor should 
 be instructed in setting his poles to guide him as to the height 
 of grade on un earth embankment, to add 10 per cent (average) 
 to the fill marked on the stakes. In rock embankments this 
 is not necessary. The engineer should see that all embank- 
 ments are made full width at first, out to the slope stakes, and 
 by measure at or above grade, so that the whole may settle in 
 a compact mass.. Additions to the width made subsequently 
 are likely to slide off. 
 
 243. The cross-section notes should be traced in ink at the 
 first opportunity to secure their permanence. An office copy 
 should also be made to serve in case of loss or damage to the 
 original. 
 
 244. Alteration of Line.—Inasmuch as the centre line 
 at grade is the base of reference for all measurements and cal- 
 culations in earthwork, any change made in it after the work 
 of grading has begun should be most carefully recorded and 
 explained. The centre stakes of the old line should be left 
 standing until after the new line is established, so that the per- 
 pendicular offset from the old line to the new, at each station, 
 may be measured, as also the distance that the new station may 
 be in advance of, or behind the old one. The date of the change 
 should be recorded. The original cross sections are extended 
 any amount requisite, the distance out being still reckoned from 
 the old centre, while a marginal note states the amount by which 
 the centre has been shifted. 
 
 The difference in length of the lines will make a long or short 
 station at the point of closing. The exact length of such a 
 station should be recorded, so-that 1t may be observed in re- 
 tracing the linc at any time, and in calculating the quantity of 
 
——— = 
 
 208 FIELD ENGINEERING. 
 
 earthwork. The original transit notes of the altered line should 
 be preserved, but marked as ‘‘ abandoned,” with a reference to 
 the notes of the new line on another page. 
 
 245. Drains and Culverts.—The engineer should ex 
 amine the nature and extent of each depression in the profile 
 with reference to the kind of opening required for the passage 
 
 ‘of water. For small springs, and for a limited surface of rain- 
 
 fafl, cement pipes, in sizes varying from 12 to 24 inches diame- 
 ter, serve an excellent purpose as drains. These are easily laid 
 down, and if properly bedded, with the earth tamped about 
 them, are very permanent; but their upper surface should be 
 at least 24 feet below grade. The embankment is protected at 
 the upper end of the drain by a bit of vertical wall, enclosing 
 the end of the pipe. If necessary, a paved gutter may lead to 
 it. ; 
 
 Where stone abounds, the bed of a dry ravine may be partly 
 filled with loose stone, extending beyond the slopes a few feet, 
 which will prevent the accumulation of water. 
 
 When the flow of water is estimated to be too great for two 
 lines of the largest cement pipe, or when the embankment is 
 too shallow to admit them safely, a culvert is required. A 
 pavement is Jaid one foot thick, protected by a curb of stone 
 or wood 8 fect deep at each end, and wide enough to allow the 
 walls to be builtupon it. Itshould have a uniform slope, usu- 
 ally between the hmits of 50 to 1 and 100 to 1 to ensure the 
 ready flow of water. In firm soils the foundation pit is exca- 
 vated one foot below the bed of the stream, butif mud is found 
 this must be removed and the space filled with riprap, the up- 
 per course of which is arranged to form the pavement at the 
 proper level. In a V-shaped ravine, requiring too much ex- 
 cavation at the sides, and where the fall 1s considerable, riprap 
 may be used to advantage, the bed of the stream above the 
 culvert being graded up by the same material to meet the pave- 
 ment. In some cases a curtaim, or cross wall, 1s necessary on 
 the lower end to retain the riprap. 
 
 Culverts should be laid out at right angles to the centre line 
 whenever practicable, the bed of the stream being altered if 
 necessary. The length of an open culvert 1s the entire distance 
 between slope stakes, the walls being parallel throughout, or 
 the length may be taken somewhat less than this, and the walls 
 
? CONSTRUCTION. 2092 
 
 turned at right angles on the upper end, forming a facing to 
 the foot of the slope. The walls are carried up to grade for 
 the width of the road-bed, and are stepped down to suit the 
 siopes. A course is afterwards added to retain the ballast. 
 
 In box culverts the span varies from 2 to 5 feet, the height 
 in the clear from 2 to 6 feet; the thickness of walls from.3 to 
 4 feet; the thickness of cover from 12 to 18 inches, and its 
 length at least 2 feet greater than the span. The walls terminate 
 in short head-walls built parallel to the centre line, the top 
 course being a continuation of the cover. The length of a 
 head-wall, measured on the outer face, 1s equal to the height of 
 the culvert in the clear multiplied by the slope ratio of the 
 embankment. The perpendicular distance from the centre 
 line to the face of a hvad-wall 1s equal to one half the road-bed, 
 plus the depth of the top of the wall below grade multiplied by 
 the slope ratio, or 40 + sk. A coping 1s sometimes added. 
 
 246. Arch culverts are used when the span required is 
 more than 5 feet, and the embankment too high to warrant 
 carrying the walls up to grade as an open culvert. The span 
 varies from 6 to 20 feet; the arch is a semicircle, the thickness 
 varying from 10 or 12 inches to 18 or 20 inches. The height 
 of abutments to the springing line varies. from 2 to 10 feet, the 
 thickness at the springing line from 8 to 5 feet, and at the base 
 from 3 to6 feet, the back of the abutment receiving the batter. 
 The foundations are laid broader and deeper than in box cul- 
 verts, each abutment having its own pit, carried to any depth 
 found necessary. The half length of the culvert is 40 + sh, in 
 which ¥# is the depth of the crown of the arch below grade. 
 The abutments are carried up half way from the spring to the 
 level of the crown of the arch, and thence sloped off toward 
 the crown. ‘The face walls are carried up to the crown, and 
 coped. The wing walls stand at an angle of 30° with the 
 axis of the culvert, they receive a batter on the face, and are 
 stepped (or sloped) down to suit the embankment. Their 
 thickness, at the base, 1s the same as that of the abutment; at 
 the outer end 3 feet. They stop about 3 feet short of the foot 
 of the slope. They need not be curved in plan. 
 
 Any stone structure of dimensions greater than those given 
 above, scarcely comes under the head of culverts, and should 
 be made the subject of a special design by the engineer. 
 
210 FIELD ENGINEERING. 
 
 247. Staking out Foundation Pits.—For box 
 culverts.—The engineer having decided upon the size of cul- 
 vert required, makes a diagram of it in plan, on a page of his 
 masonry book, recording all the dimensions, stating the sta- 
 tion and plus at which its centre is taken, the span and height 
 of the opening, etc. He then sets the transit at the centre A, 
 
 Fig. 93, measures the angle between the centre line and axis, 
 
 Fie. 93. 
 
 (making it 90° if practicable); on the axis he lays off the dis- 
 tances to the ends of the culvert and drives stakes at Band C. 
 Perpendicular to BC he Jays off the half widths of the pit, set- 
 ting stakes at D and #, and laying off DP'and HH = AB; and 
 DGand HI = AC. On IG produced he lays off CJ = CK, and 
 perpendicular to this /M and AZ, and finds the intersections 
 Oand V. A stake is driven at each angle, and upon it is 
 marked the cut required to reach the assumed level for the 
 foundation. These cuts are recorded on the corresponding 
 angles of the diagram. The pit 1s thus no larger than the 
 plan of the proposed masonry, and the sides are vertical, which 
 answers the purpose for shallow pits. 
 
 Kor arch culverts.—The pit for each abutment when 
 shallow may be of the same dimensions as the lower founda- 
 tion course . if more than five feet deep, it should be enlarged 
 by an extra space of one foot all around. In Fig. 94 the inside 
 
CONSTRUCTION. 211 
 
 lines show the plan of the abutments at the neat-lines ; the 
 outside lines represent the pits. Having prepared a plan of 
 the structure suited to the locality, and made a diagram of 
 the same in the masonry book; set the transit at A, and drive 
 stakes at D, H, NV and O on the centre line. Then turning to 
 the axis BC, lay off AC, and set stakes at Gand J. With G 
 as a centre, and a radius equal to 2D#, describe on the ground 
 
 Fie. 94, 
 
 an arc cutting #7 in X or UX = DE. cot 30°) may be calcu- 
 lated; and on XG produced lay off GX, and perpendicular to 
 this, AL. From WN lay off WP, parallel to AC, and measure 
 PL asacheck. Drive a stake at each angle, marked with the 
 proper cutting, and record the same on the diagram. The 
 locality may require the wings to be of different lengths and 
 angles, of which the engineer will judge.. Guard-plugs should 
 be driven in line with the intended face of one or both abut- 
 ments, so that the neat-lines can be readily given when re 
 quired. In case the material 1s not likely to stand vertically, 
 the pit must be staked out with sloping sides, as described 
 below. 
 
 For bridge abutments.—A design for every impor- 
 tant structure is usually prepared in the office after a survey 
 of the site. The foundation pit is then laid out from dimen- 
 sions furnished on a tracing, but a diagram of the pit should be 
 made in the masonry book as usual. When the bredge ison a tan- 
 gent, Fig. 95, set the transit at A on the centre line at its inter- 
 section with the axis BC of the abutment at the level of the scat. 
 
pate FIELD ENGINEERING. 
 
 Deflect from the tangent the angle giving the direction of BC, 
 and lay off AC, AB, setting plugs at B and C, and reference 
 plugs (two on each side) on BC produced. After staking out 
 ihe sides of the pit parallel to BC, set the transit at C, and 
 deflect the angle for the wing, laying off CD, and driving 
 stakes at the corners # and #. Two reference points are 
 then set on the line CD produced. The other wing being 
 
 Fie. 95. 
 
 staked out in the same manner, the cut is found at each stake 
 and marked and recorded. Cross sections are then taken near 
 each corner, perpendicular to each side, and slope stakes 
 (marked ‘‘ slope”) are driven where the slope runs out. Inter- 
 mediate sections are taken when the unevenness of the ground 
 makes it necessary, and the lines joining the slope stakes are 
 produced to intersect, and other stakes are driven at the inter- 
 sections. ‘The position of each stake is shown on the diagram, 
 and the cut recorded. 
 
 A slope of 1 to 1 is usually sufficient for pits. Ifthe material 
 will not stand at 14 to 1, or if space cannot be spared for the 
 slope, the sides may be carried down vertically, supported by 
 sheet piling braced from within. 
 
 The reference points should be so chosen that the points A, 
 Band C may be found by intersection, on any course of the 
 masonry, during the progress of construction. 
 
 When the bridge is on a curve, the bridge-chord 
 should be found and the abutments laid out from this. Fig. 96. 
 The bridge-chord is a line AB, midway between the chord of 
 the curve OD, joining the centres of the abutments, and a tan- 
 gent to the curve at the middle point of the span. Hence 
 
CONSTRUCTION. 
 
 CA = DB=+%4MN, which may be laid off, and A and B are 
 the true centres of the abutments, from which the foundations 
 are staked out as before. 
 
 The distance CH = DF to the points where the bridge-chord 
 cuts the curve is 0.147CD. 
 
 Should an abutment site on a curve be inaccessible, as when. 
 
 Fie. 96. 
 
 under water, from any transit point P on the curve lay off. PX 
 perpendicular to the tangent at MW, observing that 
 
 PX = MQ — AC= KR (wers PM — 4 vers CM) 
 and AX = PQ —tAB= R(sin PM — 4CD) 
 
 The point A may then be found by intersection, or by direct 
 measurement with a steel tape or wire, driving a long stout, 
 stake to show the point above the water. Other points may 
 then be approximately found, sufficient to begin operations. 
 
 In case of a bridge of several spans, the piers are laid out in 
 the same manner, from a centre point and axis. If on a curve, 
 each span has its own bridge-chord, but for convenience, the 
 centre of a pier may be taken on the centre line during its con- 
 struction, and the bridge-chord only found for the purpose of 
 placing the bridge; the piers being long enough to allow of the 
 shift. 
 
 = s 
 
214 FIELE ENGINEERING. 
 
 To locate the centres of piers, a base line is re: 
 quired on one or both shores, and two transits are used to give 
 ihe intersections by calculated angles. When practicable the 
 spans should also be measured with a steel tape or wire. 
 
 The bed of a pit for any sort of structure should 
 receive the closest scrutiny of the engineer, it being his duty 
 to judge whether the material will resist une Joad to be im- 
 posed upon it. A pit may require to be excavated to a greater 
 depth than first ordered, while sometimes a less depth will 
 answer, aS when solid rock is found. When a good material 
 is reached, if any doubt exist as to its thickness, or as to the 
 character of the underlying stratum, borings should be made 
 or sounding rods driven down. Piles may be driven to gain 
 the requisite firmness, and a layer of riprap, of beton, or of 
 timber may be used to afford a uniform bearing. When satis- 
 fied of the stability of the bed, the engineer finds the original 
 centres, and gives points for the courses of masonry. A com- 
 plete record is kept of the amount and kind of excavation, the 
 materials uscd in foundation under the,masonry, and of the 
 size and thickness of each foundation course of masonry; the 
 notes should be taken at the time the work is done, it being 
 generally impossible to take measurements thereafter. 
 
 248. Cattle-guards are shallow pits placed at right, 
 angles across the road at the fence lines to prevent the passage 
 of cattle. They are either entirely open, in which case they 
 should be at least 4 feet deep, or they are covered in part with 
 wooden rails laid a few inches apart. The open guard is 
 preferred. It is built like an open culvert except that no 
 pavement is required. The stringers carrying the rails over 
 any opening should be no longer than the span plus the thick- 
 ness of the walls. 
 
 249. Trestle Work.—No wooden culverts should ever 
 beused. If stone cannot be had at first, two trestle bents may 
 be erected, ,eaving between them a space sufficient to contain 
 the stone structure to be built when the material for it can be 
 brought by rail. The bents may be backed by plank to retain 
 the embankment, and the stringers are then notched down an 
 inch on the caps to receive the pressure of the earth, and 
 render the bents mutually sustaining. The sills are prevented 
 from yielding to the pressure cf the earth by being sunk in 
 
CONSTRUCTION. pple 
 
 a trench, or by sheet piling. Should the span be too long, a 
 central bent may be used, so as not to interfere with building 
 the wall. Sometimes pile-bents may be used with greater ad- 
 vantage, the piles being driven in rows of four each, and cap- 
 ped to receive the stringers. In districts where suitable stone 
 is entirely wanting, pile or trestle abutments and piers are 
 used for the support of bridges, the piles or posts being 
 arranged in groups and capped to receive the direct weight of 
 the trusses. Thcy should not sustain the embankment, but 
 should be connected with it by a short trestle work. 
 
 Trestle work is frequently used as a substitute for embank- 
 ment, either to lessen the first cost, or to hasten the completion 
 of the line, or for lack of suitable material with which to form 
 an embankment. The cost of trestle work, however, is not 
 iess than that of an earth embankment formed from borrow 
 pits, unless its height exceeds about 15 fect, depending on the 
 relative prices of materials and labor. When not exceeding 30 
 fect in height, the dents, for single track, are usually composed 
 of two posts, a Cap and sill, cach 12 X 12, and two batter posts, 
 10 < 12, inclined at 4th to 1, all framed together. Two lengths 
 of 3-inch plank are spiked on diagonally on opposite sides of 
 the bent as braces. The length of the caps should equal the 
 width of the embankment; the posts should be 5 fect from 
 centre to centre, and the batter posts 2 feet from the posts at 
 thecap. The sill should extend about two fect beyond the 
 foot of the batter post. A masonry foundation for the bent is 
 preferable, though pile foundations are not uncommon, and 
 some temporary structures are placed directly on a firm soil, 
 supported only by mudsills laid crosswise under the sill. The 
 spans, or distance between bents, may vary from 12 to 16 feet. 
 The stringers should consist of 4 pieces, 2 under each rail, 
 bolted together, with packing blocks to separate them 2 or 3 
 inches. Over each bent and at the centre of cach span a piece 
 of thick plank about 4 feet long should be placed on edge 
 between the two pair of beams to preserve the proper distance 
 between them, while rods pass through the beams and strain 
 them up to the ends of the plank, to increase the stability of the 
 beams and prevent their buckling under a load. The string- 
 ers should be able to carry safely the heavicst load without 
 bracing against the posts. The bents, however, if high, must 
 be braced against each other. The stringers should be con 
 
216 FIELD ENGINEERING. 
 
 tinuous, the two pieces breaking joints with each other at the 
 bents, to which they are firmly bolted. They may rest directly 
 on the caps, or corbels may intervene. -The spans on a curve 
 should be shorter than on a tangent. The ties should be 
 notched down to fit the stringers closely, and guard rails, either 
 wood or iron, secured to them firmly. Unless the spans are 
 ‘very short, horizontal bracing should be employed consisting 
 of 3-inch plank, extending from the centre of each span to the 
 ends of the caps, which are notched down to receive the plank. 
 For trestles much higher than 80 feet the cluster bent is 
 preferable, so termed because each vertical post is composed of 
 a cluster of four pieces, 8 x 8, standing a little apart to allow 
 the horizontal members to pass between them. The verticals 
 are continuous, breaking joints, two and two, while the hori- 
 zontals pass the posts and are bolted to them at the joints; the 
 framing is accomplished entirely by packing blocks and bolts. 
 The batter posts consist each of two pieces 8 X 8; the horizon- 
 tals may be 4 X10, and extend not only across the bent, but 
 Ht from one bent to another. Proper bracing is also used in every 
 | direction. When very high, a secondary pair of batter posts 
 may be introduced in the lower part of the structure. The 
 ‘| batter need not exceed 1th to1. In some instances two adjoin- 
 1 ing bents are strongly braced together, forming a tower or pier, 
 i and the piers placed from 50 to 100 feet apart, the roadway 
 being carried on trussed bridges. The cluster bent admits of 
 | any piece being removed and a new one inserted when neces- 
 Ht sary. 
 
 Iron trestles are now adopted where a permanent struc- 
 ture is desired. Owing to the expansion of the metal by heat, 
 the bents cannot be continuously connected with each other as 
 in a wooden trestle; hence the pier form is resorted to, having 
 spans varying from 380 to 150 fect, covered by trussed bridges, 
 and the whole structure is more properly styled a viaduct. 
 
 250. Tunnels. Tunnels are adopted in certain cases to 
 avoid excessive excavations, steep grades, high sunimits, and 
 circuitous routes. Their disadvantages are the increased time 
 and cost of their construction compared with an open line, and 
 their lack of light and fresh air when in use. It is desirable 
 | that they should be on a tangent throughout, both for the ad- 
 
 I mission of light and for convenience of alignment. Many 
 
CONSTRUCTION. Q1% 
 
 tunnels, however, have been built with a curve at one or both 
 ends.* 
 
 The location ofa tunnel, other things being equal, should 
 be such as to make not only the tunnel proper, but also its im- 
 Mediate approaches by open cut as short as possible; and the 
 latter should be selected so as not to be subject to overflow, 
 nor liable to land slides. ‘The material to be encountered may 
 frequently be determined with tolerable accuracy by a study 
 of the geological formation in the vicinity, or by actual borings. 
 The most favorable material for tunnelling is a homogeneous 
 self-supporting rock, devoid of springs, which does not disin- 
 tegrate on exposure to the atmosphere. The worst materials 
 are saturated earth and quicksands. The presence of water in 
 any material increases the cost considerably. 
 
 The alignment of a tunnel is made the subject of special 
 survey, after the general location is decided, and this is more 
 or less elaborate according to the length of tunnel. A perma- 
 nent station is established at the highest point crossed by the 
 tunnel tangent, from which, if possible, monuments are set in 
 each direction at points beyond the ends of the tunnel. Ii 
 there are two principal summits, stations on these will define 
 the tangent, which may then be produced. The monuments 
 established beyond the tunnel should be sufficiently distant to 
 afford a perfect backsight from the ends of the tunnel, where 
 other monuments are also established. The first quality of in- 
 struments only should be used, and these perfectly adjusted, 
 and the observations should be repeated many times until it is 
 certain that all perceptible errors are eliminated. Since the 
 line of collimation will be frequently inclined to the horizon 
 at a considerable angle, it is important that it should revolve 
 in a vertical plane; and to secure this, a sensitive bubble tube 
 should be attached to the horizontal axis, at right angles to the 
 telescope of the transit. The distance may be obtained by tri- 
 angulation, though direct measurement is to be preferred. A 
 steel tape is convenient and accurate, providing that allowance 
 be made for variations due to temperature, from an assumed 
 standard. The rods described in § 48 may be uscd instead of 
 
 *The Mont Cenis tunnel, requiring a curve at each end, was first 
 opened on the tangent produced, giving a straight line through, and the 
 curves were excavated subsequently. 
 
218 FIELD ENGINEERING. 
 
 plumb lines, the tape being held at right angles to them, and 
 therefore horizontal. A plug should be driven for each rod to 
 stand on, and a centre set to indicate the line and measure- 
 ment. 
 
 As the excavation of the tunnel proceeds, the centre line is 
 given at short intervals by points either on the floor or roof. 
 Overhead points are generally preferred, from which short 
 plumb lines may be hung, constantly indicating the line, with 
 little danger of being disturbed. When a new transit point is 
 required in the tunnel, it should be established directly under 
 an overhead point, which serves as a check upon its perma. 
 nence, and as a backsight when needed. 
 
 Shafts are sometimes opened to give access to several points 
 of the tunnel at the same time, thus facilitating the work, though 
 at an increased cost. They also serve for ventilation during the 
 progress of the work, though they are worse than useless for 
 this purpose afterward, except possibly in the case of a single 
 shaft near the centre of the tunnel. Some of the longest tun- 
 nels have been formed without shafts, while many shorter ones 
 have had several, which have generally been closed after the 
 tunnel was completed. Shafts are either vertical, inclined, or 
 nearly horizontal; in the latter case they are called adits. In: 
 clined shafts should make an angle of at least 60° with the ver- 
 tical. Vertical shafts may be either rectangular, round, or 
 oval. Their dimensions vary, depending on their depth and 
 the material encountered, between 8 and 25 feet. They are 
 usually sunk on the centre line of the tunnel, though some- 
 times at one side. When over the tunnel the alignment below 
 is obtained directly from two plumb lines of fine wire suspended 
 on opposite sides of the shaft from points very carefully deter- 
 mined at the surface. The plummets are suspended in water 
 to lessen their vibrations, and as soon as the transit can be set 
 up at a sufficient distance to bring the lines into focus, it is 
 shifted by trial into exact line with the mean of their oscilla- 
 tions, the latter being very limited. Permanent points may 
 then be set, but, should be repeatedly verified. As soon as the 
 workings from a shaft communicate with those from either 
 end, or from another shaft, the alignment thus found is 
 tested, and revised if necessary. These operations require the 
 ereatest nicety of observation and delicacy of manipulation to 
 obtain satisfactory results. 
 
CONSTRUCTION, 
 
 From plumb lines in the central shaft of the Hoosac tunnel, 
 the line was produced three tenths of a mile, and met the line 
 produced 2.1 miles from the west end with an error in offset 
 of five sixteenths of an inch. In the Mont Cenis tunnel the 
 lines met from opposite ends with ‘‘no appreciable” crror in 
 alignment, while the error in measurement was about 45 feet 
 in a total length of 7.6 miles. 
 
 When a curve occurs in a tunnel it is usually near one 
 end. The tunnel tangent is produced and established as 
 before described, anda second tangent from some point on the 
 curve outside the tunnel is produced to intersect it, the inter- 
 section being precisely determined and the angle measured 
 with many repetitions. The tangent distances are then calcu- 
 lated, and the position of the tangent points corrected by 
 precise measurements, and permanent monuments are estab- 
 lished. As the tunnel advances, points may be set at short 
 intervals on the curve in the usual manner; but at intervals 
 of 100 feet the regular stations should be defined with finely 
 centred monuments, using a 100-foot steel tape carefully sup- 
 ported in a horizontal position. When it is necessary to use a 
 subchord, its exact length should be calculated as shown in 
 $107. When the curve has advanced so far as to render anew 
 transit point necessary, this should be established at a full 
 station. The subtangents from the two transit points should 
 then be produced to intersect, and measured for equality with 
 each other and with their calculated length. The distance 
 from their intersection to the middle of the long chord should 
 also be measured as a check on the deflections. When no 
 perceptible errors remain, the curve may be produced as 
 before ‘until the P. 7. is reached. It is evident that correct 
 measure is indispensable to correct alignment on curves. 
 Should obstacles on the surface necessitate triangulation, more 
 than ordinary care must be exercised, and as many checks 
 introduced as possible. The triangles should be so arranged 
 that all of the angles and most of the sides may be measured. 
 
 Test levels are carried over the surface with great care, 
 each turning point being made a permanent bench, and its 
 elevation determined with a probable error not exceeding 
 0.005 foot. Levels may be carried down a shaft on a series of 
 bolts or spikes abeat 12 fect apart in the same vertical line, 
 the distances being measured by the same level-rod as that 
 
220 FIELD ENGINEERING. 
 
 with which the benches are determined. The measures should 
 be taken between two graduations of the rod, not using the 
 end of the rod, which may be slightly worn. Fine horizontal 
 lines on the heads of the bolts may be used to mark the exact 
 distances. After the shaft reaches the level of the tunnel, the 
 depth may be measured more directly with a steel tape, the 
 entire length of which has been corrected at the given tem- 
 perature, by comparison with the same rod. 
 
 If the grade of a tunnel is to be continuous, it should 
 be assumed at something less than the maximum of the road, 
 but not less than 0.10 per station, which is required for 
 drainage. If a summit is to be made in the tunnel, the grade 
 from the upper end should not exceed 0.10 per station. 
 Grades are given in the tunnel from day to day, or as often as 
 required by the progress of the work, the marks being made 
 on the sides at some arbitrary distance above grade. Turning 
 points should be taken on permanent benches. 
 
 The least width of a tunnel in the clear should be, for 
 single track about 15 feet, and for double track 26 feet. The 
 least height in the clear above the tie should be 18.5 feet 
 for single track, and 16.5 feet at the outside rails for double 
 track, allowing for tie and ballast; the roof at the centre of the 
 section should be at least 20 feet above subgrade, and with a 
 full centred arch 22 or 23 feet for double track. The form 
 of section depends somewhat on the material traversed. In 
 perfectly solid rock a nearly rectangular section may be used, 
 the roof being slightly rounded. In dry clay, and stratified 
 rock, a flat arch may be used, and in other cases a full-centred 
 arch. .The latter form is rather to be preferred on account of 
 the better ventilation afforded. The sides are made vertical, 
 battered or curved, as necessity or taste may dictate. In wet 
 and infirm soil an invert floor may be required, otherwise it 
 is made level transversely. When a lining is required the 
 original section must of course be made large enough to 
 allow for the masonry, and the temporary timber supports 
 behind it. Hard burned brick is usually adopted for arching, 
 being durable and easily handled. In loose rock the arching 
 may- be from 13 to 26 inches thick, in wet and yielding soil a 
 thickness of from 26 to 89 inches may be necessary. The 
 walls may be from 24 to 6 fect thick. 
 
 In forming a tunnel, @ heading or gallery of smaller 
 
CONSTRUCTION. rae | 
 
 cross section is first driven and afterwards enlarged to the 
 full size required. In firm clay or loose rock which will tem- 
 porarily support itself until the masonry can be put in, it is 
 better to drive the heading along the floor (at subgrade) of the 
 tunnel, the remaining material being then easily thrown down 
 in sections as the archiny is advanced. In solid rock, or wet 
 earth, a top-heading (along the roof) is generally preferred, 
 The dimensions of a heading driven by hand are usually 8 treet 
 high by 8 or 10 feet wide, but in solid rock where drilling 
 machinery is introduced, it is advantageous to make the head- 
 ing as wide as the tunnel at once. By drilling holes into the 
 face at points about five feet each side of the centre, and con-~ 
 verging on the centre line at a depth of about ten feet, a tri- 
 angular mass of rock may be blown out, and the space thus 
 gained facilitates the blasting of the adjacent rock on either 
 side. An advance of about 10 feet in each day of 24 working 
 hours may thus be made, using nitroglycerine in some form 
 as the explosive agent. Owing, however, to unavoidable 
 delays from various causes, this rate of progress cannot 
 always be maintained. At the Hoosac tunnel the greatest 
 advance in one week was 50 feet; in one month 184 feet at 
 one licading. At the Musconetcong tunnel a heading 8 X 22 
 feet in syenitic gneiss was advanced at the average rate of 
 137 feet per month for 6 months, the maximum being 144 feet 
 —the enlargement of the tunnel to full size going on at the 
 same time, a few hundred feet behind. At the St. Gothard 
 tunnel the north heading 2.5 x 8 metres was advanced in 
 mica gneiss, during the year 1875 at the average daily rate of 
 2-71 metres, with a maximum of about 4 metres, but the en- 
 largement was not made. The south heading advanced at 
 the rate of 2 metres a day, timbering being at times necessary. 
 
 In ordinary clay a heading may be driven at from 795 to 180 
 ft. per month, according to circumstances, where timbering is 
 putin. The enlargement, inciuding timbering and masonry, 
 may be advanced at from 20 to 60 ft. per month. Small tun- 
 nels for water conduits are driven through dry clay at the rate 
 of 10 ft. per day, the masonry following at once without tim- 
 bering. 
 
 The compressed air uscd to drive the drilling machinery 
 serves to supply ventilation also. When this is wanting or 
 proves insufficient, exhaust fans are used. At Mont Cenis a 
 
a ee 
 
 pape FIELD ENGINEERING. 
 
 horizontal brattice or partition was built in the tunnel, dividing 
 it so as to secure a circulation of air. When foul gases are en- 
 countered, ventilation becomes a serious question, and in one 
 instance an important work was abandoned for this cause. 
 
 Cross sections of the heading, and also of the tunnel en 
 largement, should be measured at intervals of about 20 fect, as 
 soon as opened, to see that the sides, roof, and floor are taken 
 out to the prescribed lines, at the same time that the latter are 
 exceeded as little as possible. In solid rock, since some ma- 
 terial outside of the true section will necessarily be thrown 
 down, leaving an irregular outline, it 1s well to take two cross 
 sections at the same point, one following the projections and 
 the other the recesses of the rock, from which an average sec- 
 tion may be estimated. A daily, or at least a weekly, record 
 of operations should be kept in tabular form, and the progress 
 indicated by a profile and cross sections drawn on a sufficiently 
 large scale to show dctails. 
 
 The drainage of a tunnel is best secured by a line of 
 stoneware or cement pipe laid in a trench along cach side, and 
 covered with ballast or other loose material. The entire floor 
 is thus made available for the use of the trackmen. - When an 
 invert is used, the drain is placed in the centre between tracks. 
 If the amount of water is large, drain pipe may be laid behind 
 the walls, and the back of the arch may be covered with as- 
 phaltum, or coal tar, to prevent a constant dripping on the 
 track. 
 
 251. Retracing the Line.—<As the grading pro- 
 gresses, in either excavation or embankment, the principal 
 transit points are established on the road-bed from the points 
 of reference, and the centre line is retraced, setting stakes at 
 every 50 fect. Transit points on grade should be fixed upon 
 stout, durable posts firmly set in the ground, and standing 
 high enourh to be easily reached after the ballast is laid. 
 To recover the old line, any discrepancies in measurement 
 must be Icft between the transit points where they occur, 
 and not carried forward. In retracing a curve, if the transit 
 is placed at the forward point, allowing the chain to ad- 
 vance toward it, slight differences in measurement will not 
 affect the position of the curve. If any short or long sta- 
 
CONSTRUCTION. 2Re 
 
 tions have been introduced on the location, their position on 
 the line must not be changed in retracing. The chain may 
 be adjusted so that its measures will agree with the recorded | 
 distances between transit points. Offsets are made right and Hi 
 left from the new stakes to see that the road-bed is of the full | 
 width at all points. The levels are also carried over the i) 
 grade, and any remaining cut or fill found necessary is marked Wh 
 on the back of the stakes, due allowance being made for the i | 
 probable settlement of embankments. MM 
 
 252. As the work approaches completion the contractor 
 voes over the line dressing it to grade and opening the side i 
 ditches if this has not been previously done. | 
 
 Drain-tile should be laid at the bottom of these ditches and 
 lightly covered with earth, particularly if the cut be wet. 
 These not only prevent the water from reé ching the ballast, 
 but by keeping the foot of the slope comparatively dry pre- 
 vent the earth from sliding down and filling up the cut. 
 There is also a marked economy in their use, as the cost 1s 
 trifling, and all further excavation of mud and water from 
 the cut is generally obviated. Should any springs appear in 
 the slope a branch line of smaller tile may be laid to meet it. 
 If the slope is liable to be overflowed from the surface above, 
 an open ditch should be dug a few feet beyond the slope 
 stakes, leading the surface water to discharge elsewhere. 
 
 : 253. The road-bed being prepared, ballast stakes are 
 | driven at every half station, giving the width of the ballast at 
 its base, while the tops of the stakes indicate the proper level 
 of its upper surface, which is the under side of the tie. These 
 stakes should be set so as to give the proper elevation to the 
 outer rails on curves when the ballast is graded to them. The 
 ballast should be about one foot deep before the ties are laid. { 
 Broken stone or a mixture of coarse and fine gravel is the | 
 : best material, affording elasticity and good drainage. The 
 
 side slopes of the ballast are made 1 to 1; its width at the Hl 
 
 wider side of the tie should be one foot greater than the 
 
 length of the tie. 
 
 } 
 : 
 : 
 254. Track-laying.—After the ballast has been laid 
 and graded, the centre line is retraced upon it; short stakes 
 
224 FIELD ENGINEERING. 
 
 are used, each of which is centred. On long tangents, one 
 stake in every 200 fect is sufficient, on ordinary curves one in 
 every 50 feet, and on very sharp curves one in every 25 feet 
 The ties are then spaced evenly according to the number 
 prescribed per mile, or per rail length; but a tie should not be 
 allowed to cover a transit point. ‘Ties for the standard gauge 
 are 8 or 9feet long; they should be sawed off square at the ends 
 and in uniform lengths for appearance sake when laid. 
 Specifications usually call for ties having a thickness of 6 
 inches and a width of from 7 to 10 inches. The ends of the 
 ties are aligned on one side of the road, though if cut into 
 uniform lengths both ends will be equally well aligned. The 
 rails are then laid on, and spiked to gauge. The first spikes 
 are driven in the ties near’ a centre stake, the centre mark of 
 the gauge bar being kept over the centre on the stake. Upon 
 curves the rails must be sprung to the proper arc before they 
 are laid (§199). Ad’ the ties required in a given distance 
 should be laid before the rails are brought upon them. The 
 practice of laying only joint and middle ties at first subjects 
 the rails to the danger of bending from passing loads. 
 
 Owing to the expansion of the rails by heat, a space 
 must be left at the rail-joints. The highest temperature of a 
 rail in the summer sun is about 180° Fah. The expansion of 
 iron or steel per 100° is .0007 per foot; or for a 30-foot rail 
 .021 foot or .262 inch, Therefore when 30-foot rails are laid 
 at a temperature near the freezing point, or 100° below the 
 maximum, the space allowed must be at least «a quarter of an 
 inch. At 80° Fah. or 50° below the maximum, it need be only 
 half as much. The space required is also proportional to the 
 length of rail used. The exact space should be given, as less 
 would result in the rails being forced up by expansion, while 
 more than necessary space gives a rough road; and hastens 
 the destruction of the rail. 
 
 Wherever siding’s ‘are required, the necessary frogs and 
 iong switch-ties should be provided in advance, so that they 
 may be put in place at the time of laying the main track. For 
 every road crossing at grade, heavy oak plank should be pro- 
 vided, and laid upon the ties as soon as the rails are spiked, 
 so that the highway travel may not be impeded. 
 
CALCULATION OF EARTHWORK. 
 
 CHAPTER X. 
 JALCULATION OF HARTHWORK. 
 
 254. The first step toward finding the cubical content of 
 an excavation is to divide it into a number of prismoids by 
 several cross sections. 
 
 A prismoid is a solid having plane parallel bases or ends, 
 and bounded on the sides either by planes, or by such surfaces 
 as may be generated by a right line moving continuously along 
 the edges of the bases as directrices. 
 
 The positions of the cross sections must be so selected 
 that the solid included between any two consecutive sections 
 may be a prismoid as nearly as possible. Upon a tangent the 
 road-bed and side-slopes are planes, so that the prismoidal 
 character of a given solid depends upon the shape of the natu- 
 ral surface. When tlie natural surface is a plane, the sections 
 are taken only at the regular stations, 100 feet apart; when it 
 is curved, warped, irregular, or broken, the sections must be 
 more numerous, so that the surface limited by any two shall 
 be composed substantially of right-lined elements extending 
 from one section to the other. 
 
 If two end sections of a prismoid are somewhat similar, we 
 infer that the corresponding points are connected by right- 
 lined elements, forming in cach case the axis of a ridge or of a 
 hollow. If one section has less breaks than the next, some of 
 these ridges or hollows must vanish; and in order that the 
 solid may be a prismoid, they must vanish in the section of 
 least breaks; therefore a cross section must be taken on the 
 ground through the point where each ridge or hollow vanishes, 
 and the distance of that point from the centre line noted, so 
 that it may be coupled with the proper point in the next section 
 for exact.calculation of content. 
 
 When ridges or hollows run diagonally across the line of 
 road, cross sections must be taken where they are intersected 
 not only by the centre line but also by the side slopes; that is, 
 sections must be taken so that a side stake may stand on top of 
 
226 FIELD ENGINEERING. 
 
 each ridge and at bottom of each hollow. In case the centre 
 line intersects at right angles a retaining wall or other vertical 
 surface, two cross sections are required at the same point, one 
 at top and the other at base of wall, in order to furnish the 
 data necessary to calculate the content each way from the ver- 
 tical surface. (See Art. 235.) 
 
 Every thorough cut terminates in either side-hill cutting, a 
 pyramid, or a wedge; the latter happens only when the con- 
 tour of the natural surface is at right angles to the line of road. 
 Sections should always be taken through the points where the 
 edges of the road-bed meet the surface, as these are the points 
 of separation between thorough and side hill-work. Such sec- 
 tions also serve to define terminal pyramids when they occur 
 as is illustrated by Fig. 97. In side-hill work the foregoing 
 
 Fie. 97%, 
 
 rules apply as well, but sections will generally be more numer- 
 ous than in thorough cuts. The same rules apply also to em- 
 bankment, but as grading is preferably paid for in excavation; 
 the same precision in determining the quantities in embank- 
 ment is not usually necessary. 
 
CALCULATION OF EARTHWORK. 
 
 255. Formule for Sectional Areas. 
 
 Let 6 = base of section or width of road-bed, 
 
 vertical 
 
 : horizontal 
 as — slope ratio = . 
 
 ‘‘ q@ = depth at centre stake. i] 
 ‘« h, k = depths at side stakes. Mi) 
 
 ‘‘m, m= horizontal distances from centre to side stakes. 
 
 | 
 
 | 
 
 | 
 
 I! 
 
 For ground level transversely, the section is a parallelogram, ] 
 
 and the area is evidently Wi 
 
 j 
 
 t 
 
 | 
 
 A = bd-+ sd? (342) 
 or directly from the field notes, 
 A=t+b0+tm+n)d (348) i 
 
 For ground of uniform transverse slope between slope stakes, HHT) 
 
 U ll i 
 
 a ee 1 
 
 See ee Od AV EE OM OES A Ni 
 
 | J Hi 
 
 | Hi 
 h: | 
 i | 
 eh reles OE SHO ARIE SeCnIN tebe oe ii 
 
 N A G 8 i 
 
 Fic. 98. Hh 
 
 Fig. 98, the section consists of the parallelogram 4 20H and | 
 the triangle HOD. Hence | le. 
 
 A= }(AB+ HO)EN + 41EODH — EX) 
 A=}(AB. EN+ LO. DH) | 
 
 or 
 A= 3[bh +k + 2sh) | 
 also (844) 
 A = [dK + hid + sk) | 
 From which also 
 A = tbh + mk ) 
 and (345) 
 
 ee ahh eee 
 
 These formule are independent of the centre depth. They 
 are convenient for calculating the area of a plotted section 
 
228 FIELD ENGINEERING. 
 having an irregular surface after the surface line has been 
 averaged by stretching a silk thread over it. The points 
 where the thread intersects the slope lines determine the 
 values of h, &, m, and n respectively. 
 
 When the ground has uniform slopes transversely from the 
 centre to the side stakes: Fig. 99: If in the diagram we draw 
 
 Qa 
 
 ——— foe ee ee ee 
 
 ro 
 
 Fia. 99. 
 
 EG and DG, the section will be divided into four triangles, 
 two having the common base CG =d and respective heights 
 GN =m and GH=7n, and two having the equal bases AG = 
 GB = and the respective heights HV =h and DH=k. 
 Hence we have for the area of section 
 
 A =4d(m+ n)+ 40h + &) (3846) 
 
 Otherwise, if the slope lines are produced to meet below 
 
 B. l 
 1G = —¥ The area of CHPD is 
 
 8 as 
 
 grade at P, then GP= 
 
 10P xX N= 4 («a -+ °) (m+ n). The area of ABP is AG X 
 aS . 
 
 b? 5 ‘ 
 GP—=— Hence we have for the area of the section 
 
 ° 
 
 9 
 
 b\ b ~ 
 Ae y(a+ =) (m+n) — tig (847) 
 
 Both these formule are convenient, and as the values of the 
 several letters can be substituted directly from the field notes, 
 it ig unnecessary to plot such sections. 
 
 When the surface of the ground ts irregular, verticals are con- 
 ceived to be drawn to the grade line through the slope stakes, 
 
CALCULATION OF EARTHWORK. 229 
 
 and through each break in the surface line, giving a number of 
 trapezoids, the areas of which are severally calculated, and 
 irom their sumis subtracted the area of the two triangles HVA 
 and DiZBb. The remainder is the area of section required. 
 This calculation may be made directly from the data furnished 
 by the field notes without plotting; but if the ground has a 
 number of small breaks, it is generally better to plot the sec: 
 tions and stretch an averaging line over them, finding the areas 
 by eq. (845). Or two averaging lines may be employed extend- 
 ing from the centre stake, cach way, when the area may be cal- 
 culated by eq. (846) or (847). 
 
 256. Prismoidal Formule for Solid Contents. 
 —The content of a prismoid may be exactly calculated by 
 means of the Prismoidal Formula, which is 
 
 Fie aia l ) AR ' 
 8= gy At et 41 (348) 
 
 S = cubic yards, / = length in feet, A, A’ = the areas at the 
 two parallel ends, and MZ = the area of a section midway be- 
 tween the ends. This area is not a mean of the other two, but 
 the linear dimensions of the mid-section are means of the cor- 
 responding dimensions severally of the end sections; from 
 which therefore the area of the mid-section may be computed. 
 
 The labor of calculating the middle area may be avoided in 
 many instances by substituting in the prismoidal formula, eq. 
 (848), for A, A’, and WM, their values as given in eq. (842) for 
 ground level transverscly ; 
 
 if 2 9 i he 
 A=bd-+sd? A'=bd'+sd® M= pt zu d 4 ofl + ag 5.8 
 
 i 
 bt ED (SONY 9 Me Og u agit ! 
 8 =9 27 [2sd ?+ (85 + 2sd')\d + (86 + 28sd')d'] (849) 
 
 in which S is expressed in terms of the end dimensions, 
 
 257. Tabies of cubic vards may be constructed upon this 
 formula which are very convenient in practice. ° The constant 
 values in any one table are / which 1s taken at 100, and 0 and s 
 which are given values corresponding to the road-bed and slope 
 ratio. The variables are d and d’. The columns im the table 
 
FIELD ENGINEERING. 
 
 will be headed by the successive values of d', while each hori: 
 zontal line will be headed by a value of d. For any one 
 column therefore d' is constant, and the only variable is d. 
 Assuming any value for @’, the values of S in that column may 
 be computed, letting d take a series of values differing by unity 
 from zero upwards, and the corresponding values of S will be 
 placed in the column d’ opposite the several values of d. 
 
 But instead of solving the eq. (849) for each value of S re- 
 quired, the process of filling the table may be much abbre- 
 viated by observing that since the equation is of the second 
 degree with respect to the variable d, the second difference of 
 the values of S will be a constant and equal to twice the co- 
 
 Ast 
 6 X 27 
 of first differences of S in the column d’ (é.e. between d = 6 
 and d =1) is expressed by the swm of the coefficients of @ 
 and d; or 
 
 efficient of @’, or 6" = Also the first term in the series 
 
 60 = 
 
 U 9 
 b+ 281-4 d' 
 The first value of S in any column d’ is found by solving 
 eq. (349) after making d= 0; or, 
 Deore go8 bis [ (8b + 28d "\d"] 
 6 X 27 
 Starting with these values we may fill any column d’ simply 
 by successive additions. The values of d’ for the several 
 columns should also differ by unity. The final value of S in 
 each column should be calculated by formula as a check; or 
 since all the final quantities in the same line d of the table 
 form a series of which the second difference is 6", if on taking 
 
 their differences this result is obtained, the quantities are 
 proved to be correct. 
 
 Example.—Given a base of 18 feet and slopes 14 to 1, to fill 
 the column of d' =6 in a table of cubic yards for level cross 
 sections. Here [=100, 5=18, s=3, d'=6. Hence d’= 
 8.70874, by’ = 46.2962, and S, = 266.6666-+. It is not 
 necessary to go beyond the fourth decimal place, since that 
 figure will always be the same as the first decimal (a result 
 
CALCULATION OF EARTHWORK. 20% 
 
 due to dividing by 27), and may be corrected by it after every 
 addition. The process is as follows: 
 
 d S 6' 6” 
 0 266.6666 
 1 312.9629 46.2962 3.7087 
 2 362.9629 ed ake 3.7087 
 3 416.6666 pesdue 3.7037 
 , NYA 57.4074 9 RENE 
 Biers 61.1111 aac 
 6 600.0000 64.8148 3.7037 
 % 668.5185 68.5185 3.7037 
 1 668.5185 72, 9999 3. 
 8 740.7407 eee 3.7037 
 etc. ete. etc. 
 
 In copying into the table, the quantities are taken to the 
 nearest unit only, and the decimals are otherwise neglected. 
 
 The completed table furnishes values of 8 corresponding to 
 any values of d and d@' in even feet. The correction for the 
 decimal parts of the depths, when there are such, is made by 
 adding to S (found opposite the even feet) the product of the 
 half sum of the decimals by the difference between S as 
 found and the next value of S diagonally below to the right. 
 These differences may for convenience he inserted originally 
 in the table under each quantity in small figures. 
 
 If the length of the solid differs from 100 feet, multiply the 
 corrected quantity by the length and divide by 100, since S 
 varies directly as 7. Such tables are published in separate 
 sheets for a variety of bases and slopes, so that usually one 
 may be purchased to suit the case in hand. 
 
 258. These tables may be used to find quantities when 
 the ground is not level transversely, by finding, first, the area 
 of the actual sections, and second, the depths of level sec- 
 tions having equal areas, and then using the depths so found 
 as the values of d and d' in the table of quantities. The 
 depths of equivalent level sections are called equiya. 
 lent depths. They may be calculated by the formula 
 
 b ‘A: pt? 
 SS tebe / emit Ps; (350) 
 which is derived directly from eq. (842). The more convenient 
 
 method, however, is to construct a table on eq. (842), giving to 
 d a series of values varying by one tenth of a foot from zero 
 
FIELD ENGINEERING. 
 
 upward. ‘The values of 8 and s in this table must agree with 
 those of the road and of the table of cubic yards. 
 
 259. When the transverse slope is uniform between slope 
 stakes the equivalent depth may be expressed in 
 terms of the centre depth and slope of surface 
 without reference to the area, Fig. 100. 
 
 D 
 
 Fie. 100. 
 Let HABD be the given section. 
 ‘* RABT be the equivalent level section. 
 Produce the side slopes to mect at P, and let c = CP and 
 Co OrP3 
 Through C draw the horizontal line QZ, and at ZL erect the 
 perpendicular LM = z, and draw LN parallel to PL. 
 The area RPT = area HPD; and QHC= LNC; hence area 
 EPLN = QPL = EPD — NLD. 
 Since VLD is similar to HPD, we have 
 HPD NALD, 33 Cae 
 or HPD — NID: HEPD:: 2-2: ¢ 
 Since YPL and kPT'are similar, 
 
 OPE s BPL tO Hee 
 
 cA 
 oF — 22:02 3: cs ¢,2 or 4? = ——— 
 oe — 2 
 A CL es c? 8? 
 Let s' = slope ratio of surface = —— =—. Then2 = —- 
 ML 2 8 
 
 which substituted gives 
 
 eet raeeen (351) 
 
 eee 
 V3'2 — 3? 
 
CALCULATION OF EARTHWORK. 
 
 If d, = the equivalent depth C,G, then 
 di =d+(Q— ¢) 
 
 A table may be prepared giving (c: — ¢) for various inclina- 
 tions of surface with given base and side slopes. It is then 
 only necessary to add this correction to the centre depth to 
 obtain the equivalent depth. Such tables of correction usually 
 accompany the published tables of cubic yards. This method 
 of obtaining quantities is particularly applicable to preliminary 
 estimates, where the ground has not been cross-sectioned, and 
 only the centre depth and transverse inclination is known. 
 
 260. The use of the earthwork tables described gives 
 correct results ;— 
 
 1st. When the surface of the prismoid is a plane, however 
 much inclined; provided it does not intersect the road-bed 
 within the limits of the prismoid. 
 
 2d. When with regular, or three-level end sections, generally 
 similar to each other, the surface is regularly warped from one 
 end to the other; provided that the side lines and centre line 
 of the surface are straight, and that no two of them are in- 
 clined to grade in opposite directions. 
 
 3d. When the ridges or hollows of an undulating surface 
 are parallel to the line of the road. 
 
 Ath. When a surface of numerous irregularities may be 
 averaged by planes or warped surfaces so as to comply with 
 one of the preceding conditions. 
 
 But the method fails on undulating ground when the 
 ridges or hollows run obliquely to the line of road, even 
 though the sections may appear quite regular. 
 
 In general, the method of equivalent depths holds good 
 when the mid-section of the equivalent level end sections 
 equals in area the actual mid-section or the prismoid; other- 
 wise it fails. 
 
 261. The content of a prismoid may be approximately ob- 
 tained by the method of mean areas, the formula for 
 which is 
 
 y 
 S= - A + A! (352) 
 
 2x 27 
 
 Although approximate, this method is much employed on 
 
ro4 FIELD ENGINEERING. 
 
 account of its convenience. It is approved by statute to be 
 used upon the public works of the State of New York. 
 
 If the values of A and A' derived from eq. (842) be substi- 
 tuted in eq. (852), and then eq. (849) be subtracted from it there 
 remains ' 
 
 3 
 exer 74) 
 Hi which is the correction by which S obtained by eq. (852) must 
 i be diminished to make it equal to S obtained by eq. (349) 
 when the ground is level transversely. 
 Again, for three-level sections, if the values of A, A’, and M 
 derived from eq. (847) be substituted in both eq. (848) and 
 (352), and one subtracted from the other, there remains 
 
 a 1 ; ae , na 
 Av 1237 [(d — d') (m+n —(m'+-n')) ] 
 
 which is the correction by which S ubtained by eq. (352) must 
 be diminished to make it equal to S obtained by eq. (849). 
 Hence we may write at once, for three level-sections, the cor- 
 rect formula: 
 
 Mi 4 bs [m+n — on'-+n 1 | (853) 
 
 & 
 
 s l 
 
 This formula gives results identical with eq. (349), is applica- 
 ble to the same cases, and gives correct results or fails to do 
 so according to the conditions stated in the previous section. 
 
 262. When the conditions of the surface are such that eq. 
 (349) or eq. (853) will not give correct results, the area of 
 the mid-section may be derived from its calculated linear 
 dimensions as stated in § 256. The contents of the prismoid 
 are then given by eq. (848), 
 
 Example 1. (Fig. 101.)—Base 20. Slopes 14 : 1. 
 
 22 0 7.5 
 ae el a A oe ee -°, Aas 443 gn ft 
 i} 84 0 16 
 
 | A a gee: + ig ". A = 200 sq. ft. 
 
CALCULATION OF EARTHWORK. 
 
 If 7 = 100, eq. (848) 
 
 Miccc-sty 161s pe Sele 
 S = Gray (448 + 967 + 200) = 1001 c. yds. 
 
 Had this been calculated by eq. (849) or eq. (853) or by the 
 
 Fie, 101. 
 
 tables, the result would be 1167 c. yds., showing an error of 
 166 cubic yards in excess. 
 
 Heample 2. (Fig. 102.)—Base 20. Slopes 14: 1. 
 
 22 0 19 
 Af ae 7 + re 234 sq, ft. 
 et pe le 
 13 0 16 
 ij tt - v' + 7d 88 sq. ft. 
 17.5 11 0 8 17.5 sad 
 M+ ef : a5 _ + ana can 164.5 sq. ft. 
 
 If 7 = 100, eq. (848) 
 
 y 100 5 > oc (‘Viva ad 
 ‘Pies ora [88 +- 658 + 234] = 605 c. yds. 
 
 Had this been calculated by eq. (849) or eq. (353) or by the 
 
296 FIELD ENGINEERING. 
 
 tables, the result would be 584 c. yds., showing an error of 
 21 cubic yards in deficit. 
 
 263. At the termination of acut or fill we have usually 
 either a wedge or a pyramid. To a wedge the pre- 
 ceding formule and tables based on them apply by makiny 
 
 Fie. 102, 
 
 one end depth equal zero. In the case of a pyramid, the 
 content is equal to the area of the section forming the base 
 multiplied by one third the length of the solid, and divided by 
 24: OF | 
 LA 
 
 Oe KOT 
 
 (354) 
 
 264. Side-hill Work.—When the natural surface has 
 a regular transverse slope and intersects the road-bed, the 
 cross section is reduced toa triangle. If w = the intercepted 
 portion of the road-bed, and & = the side height, then A = 
 tok. Similarly A’ = 40% and 4M = 4w+w’) (A+ hb), 
 which substituted in eq. (848) give 
 
 t l 
 
 S = —— ~~ (Qu + wk 2w' + w) k' (355 
 which is convenient for direct calculation from the field notes. 
 It is not adapted to the construction of tables, since it contains 
 four independent variables, 
 
CALCULATION OF EARTHWORK. 
 
 If the slope of the natural surface is given, let s' be the sur- 
 face slope ratio at one section, and s” that at the other, and s 
 the ratio of the side slope. Then w = k(s' — s) and w' = 
 k'(s" — s), which substituted in eq. (855) give 
 
 Y U 4t im) " ae s' a " ¥] 
 S = 53087 (s'— s)k? + EUs eras kk + (8 ead 8) k? 
 If the surface is a plane, then s" = s', and we have for this 
 case 
 fa ee he tk] (356) 
 PG. 627 
 
 which is a formula of quite limited application; yet it is the 
 one on which tables and diagrams are usually constructed. 
 Consequently the latter will not give correct results, except 
 when the surface is a plane. 
 
 265. When the natural surface is broken the 
 sections may be plotted, and the values of «# and & taken from 
 the points where an averaging line intersects the grade 
 and side slope respectively. Finding values for w' and x’ in 
 the same way, the content may then be obtained by eq. (855) 
 as before. The averaging line should not only cut off the 
 same area as the original section, but should also have in cach 
 case a slope agreeing as nearly as possible with the general 
 slope of the natural surface. The slope is determined simply 
 by inspection of the diagram, but the area may be bad pre- 
 cisely, for, taking w from the averaging line, and knowing A, 
 we may calculate k by the formula k = = or k may be 
 taken from the plot and 2 calculated. 
 
 Otherwise, the actual mid-section may be calculated and 
 the cubic contents determined by the method illustrated in 
 § 262. 
 
 266. To express side-hill areas and cubic yards 
 in terms of the centre depth, d, and transverse 
 slope-ratio s’. Fig. 103. 
 
 5? 
 
 Whend=9, A= 0k = =p 
 
FIELD ENGINEERING. 
 
 For any depth d, add to this area 
 
 ; Reet : b sd 
 sa (e+ 5) = 4a ee 5 ta@ a) 
 and there results, 
 
 _@o+sa" 
 =e pomens it 
 
 _ Wab+e'a? 
 —~ 2X 278" —8) J 
 
 (857) 
 
 Wh Observe that d may be plus or minus, and that its limits are 
 WE b 
 
 a Tables of cubic yards may be constructed on this formula, 
 i HH making d and s’ the variables, which would be extremely con- 
 
 Fie. 108. 
 
 venient for making up estimates upon preliminary lines on 
 which the profile of centre line and angle of transverse slope 
 Hil only are known. Since s’ is the cotangent of the slope angle 
 | the columns of the table may be headed by the angles in a 
 i series of degrees, while the corresponding values of s' are 
 Hi used in the formula. The values of d@ should vary by tenths 
 i] of afoot. The results obtained by eq. (356) and eq. (857) will 
 be identical for the same sections. 
 
 267. Several different systems of diagrams have been 
 devised and published for determining quantities in earthwork 
 by a sort of graphical method. These diagrams, which are 
 substitutes for tables are preferred by some engineers. They 
 
CALCULATION OF EARTHWORK. 239 
 
 are based on the same principles, and are constructed on modi- 
 fications of the same formule. 
 
 268. Correction of Earthwork for Curvature. 
 —The preceding calculations are based on the assumption that 
 the centre line is straight, with cross sections at right angles to 
 it. When an excavation is on a curve, the cross sections, be- Hh 
 ing in radial planes, are inclined to each other, so that the con- Hl 
 | dition of a prismoid is not exactly fulfilled. But by the proper- | 
 
 ty of Guldinus, if any plane area is made to revolve about an 
 axis in the same plane, the volume of a solid generated by the 
 area is equal to that of a prism having a base equal to the given | 
 area, and a height equal to the length of path described by the 
 : centre of gravity of the area. The path, being the arc of a cir- 
 cle, is proportional to the radius drawn to the centre of gravi- Wma 
 ty. If therefore a cross section is symmetrical with respect to t 
 the centre line, the path of the centre of gravity is equal to the Hi | 
 measured length of the centre line, and no correction for cur- 1 
 vature is required. 
 
 But when the ground is inclined transversely, the centre of 
 gravity is one side of the centre line, and its path, if we con- i 
 : ceive it to sweep around the curve, from one end of a prismoid 
 to the other, is longer or shorter than the distance measured on 
 the centre line, according as the centre of gravity is outside or 
 inside of the centre line curve. 
 
 Let C = correction in cubic yards due to curvature. 
 ‘« S = cubic yards as obtained by prismoidal formula. ii 
 ‘*. R= radius of centre line. 
 “* é-= eccentricity of centre of gravity of section. ai 
 
 = horizontal distance from centre line to centre of i] 
 gravity. HI} 
 
 We then have the proportion, | 
 
 S+ Os: Sh te6:R 
 
 As the sections of a solid are seldom similar and equal, we 
 shall usually have a diffcrent value of e for every section, from 
 
240 FIELD ENGINEERING. 
 
 which, however, a mean average value may be deduced, and 
 used in the above formula. But it will be more convenient to 
 correct the areas themselves for eccentricity before finding S, 
 which will then require no correction. For the same result 
 
 will ensue whether we multiply S by 7 or multiply one of 
 
 the component factors of S by the same ratio. 
 If then ¢ = correction of area in square fect due to eccentri- 
 city, we have at once 
 
 Ae 
 R 
 
 and the corrected area equals A + ¢ according as the cut is 
 deeper on the outside or inside of the curve. Each area used 
 in determining the solid contents should, on a curve, be first 
 corrected in this manner, 
 
 To find the value of e for any three-level section, Fig. 104. 
 
 & 
 
 + 
 
 ss Ta ee 0 ws ew anal ee ee 
 
 % DERISION IAL yp 
 
 Fie. 104. 
 
 i 
 | 
 i 
 I 
 ' 
 I 
 \ 
 \ 
 j 
 1 
 t 
 | 
 | 
 I 
 u 
 
 Find the areas either side of the centre line separately, call- 
 ing them Hand X, and take their sum and difference. Using 
 the same notation as in § 255, H = md + ibh, K = ind kik 
 bk, and + K = A, 
 
 K —H = 4d (n — m) + 30 (k —h) 
 In the figure draw CZ’ equal to O#, and the triangle CH’D 
 
 will represent the area (K — #7). Bisect the side E'D, and 
 draw a line from (@ to the middle point. Then the centre of 
 
CALCULATION OF EARTHWORK. 2AI 
 
 gravity of the triangle will be on this line at two thirds its 
 length from @, and the horizontal distance of the centre of 
 gravity from C is ? xX m+n) = 3(m-+n). 'The centre of 
 gravity of the remainder of the section is on the centre line 
 CG, so that the value of ¢ is found from the proportion 
 
 €é:¢t(n+m): K-—H:A 
 
 _n+t+m _,, 
 ein 7 (K — ff) 
 
 | | Ae nm +m 
 : Hence c= Peas a [$d (n —m) +4) (k—h)] (858) 
 
 Sections which are more irregular may be plotted and 
 reduced by averaging lines to three-level sections, in order that 
 the formula may be applied. If the ground is so irregular as 
 to require the computation of the middle section, the correc- 
 tion ¢ should be found and applied to this area (M) also 
 before introducing it into the prismoidal formula. As the 
 correction for curvature is always relatively small, it is usually 
 ignored in practice for thorough cuts, except where deep cut- 
 tings with steep transverse slope occur on sharp curves. 
 
 The correction is of more importance relatively in 
 side-hill work as the centre of gravity of the section is 
 more remote from the centre line. Let the section be reduced 
 
 A ce 
 
 Fre. 105. 
 
 to a triangle by an averaging line (Fig. 105), and w be the base 
 of the triangle formed by the averaging line. The centre of 
 gravity is at one third the horizontal distance from the middle 
 point of 2 to the side stake D, while the distance of this 
 middle point from the centre stake C is evidently 40 — fe. 
 
FIELD ENGINEERING. 
 
 Hence ¢ = 3b — lwo + 3[n — GO — Ww) 
 e=db-+n—w) 
 
 _ Ae b4+n—w, wk f 
 and ¢= R = 3R OK 5 (859) 
 
 The correction ¢ will be plus or minus as before explained. 
 This formula applies to all side-hill triangular sections, 
 whether there be cut or fill at the centre stake 
 
 Example 1.—Thorough cut; base 20; slopes 14: 1. 
 
 1 = 100; 8° curve, left; R = 716.78 
 
 16 12 58 
 Notes. A.+ mn + 6 + 39 
 
 : Powe 8 _ 40 
 Gran ee ba GRIST 
 Then K = +X 58 X 12+4 X 20 X 82 = 508 
 H=4x16X%12+4X20X 4=116... A= 624 
 
 K—-H= 392 
 ye g 
 Kq. (858) ¢ = Ra! awe 5 chee 13.49 
 
 (A+ c) 687.49 
 K'=:i4x40x8tt x 20 X 20 = 260 
 Ho = F318 X8 +X 20X 2= 82. aoe 
 Ki He 198 
 
 2 AR PaO e | 
 © = 330 716.78 = 4:86 
 
 (A’ +c’) = 826.87 
 From which we obtain S = 1758 cub. yds.—Ans. 
 
 Without correction we have 1726 ‘‘ A 
 
 32 66 66 
 
 Showing a difference of 
 
 Had the curve been to the right with same notes, ¢ wouid 
 have been minus, and S would = 1694. 
 
CALCULATION OF EARTHWORK. 245 
 
 Example 2.—Side-hill cut; base 20; slopes 14 :1 
 = 60; 10° curve, right; R = 573.69 
 
 : 6 iPaper fh | 
 tg 0 +28 T 39 i| 
 0 2 ist BY | 
 0.80.01 is i) 
 
 A=} x16 x20 = 160 |i 
 
 2 as i) 
 MT OR oa i ey 7m 3.58 | 
 
 3 X 573.69 
 
 (A —c) = 156.42 
 
 A'=4x8x18= 72 i 
 pein QO OT — Bieks i 
 == ne, Oa 9.05 I 
 © = 3 x 573.69 Ye | 
 
 Hence S = 248 cub. yds. if 
 Without correction Swould = 255 “ « 
 
 Difference © joes y 
 
 269. Haul.—The cost of removing excavated material, I 
 when the distance does not exceed a certain specificd limit, is | 
 included in the price per cubic yard of the material as meas- 
 ured in the cutting. But when the material must be carried 
 beyond this limit, the extra distance is paid for at a stipulated 
 price per cubic yard, per 100 feet. The extra distance is known il 
 by the name of haul, and is to be computed by the engineer 
 with respect to so much of the material as is affected by it. iy 
 
 The contractor is entitled to the benefit of all short hauls 
 (less than the specified limit), and material so moved should not 
 be averaged against that which is carried beyond the limit. 
 Therefore, in all cuts, the material of which is all deposited 
 within the limiting distance, no calculation of haul is to be 
 made. 
 
 On the other hand, the company is entitled, in cases of long 
 haul, to free transportation for that portion of the cutting, no 
 one yard of which is carried beyond the specified limit. . There- Hot 
 fore, this portion is first to be determined in respect to its ex- 
 tent; and the number of cubic yards contained in it is to be de- 
 
R4A+ FIELD ENGINEERING. 
 
 ducted from the total content of the cutting, before estimating 
 the haul upon the remainder. Find on the profile of the line 
 two points, one in excavation, and the other in embankment, 
 such, that while the distance between them equals the specified 
 limit, the included quantities of excavation and embankment 
 shall just balance. These points are easily found by trial, with 
 the aid of the cross sections and calculated quantities, and be- 
 come the starting points from which the haul of the remainder 
 of the material is to be estimated. 
 
 Fic. 106, 
 
 Fig. 106 represents a cut and fill in profile. The distance 
 AB isthe limit of free haul. The materials taken from AO 
 just make the fill OB and without charge for haul; but the haul 
 of every cubic yard taken from AQ, and carried to the fill BD, 
 is subject to charge for the distance it is carried, less AB. It 
 would be impossible to find the distance that each separate yard 
 is carried, but we know from mechanics that the average dis- 
 tance for the entire number of yards is the distance between 
 the centres of gravity of the cut AC, and of the fill BD which 
 is made from it. If, therefore, X and Y represent the centres 
 of gravity, the actual average haul is the sum of the distances 
 (AX-+BY), and this (expressed in stations) multiplied by the 
 number of cubic yards in the cut AC, gives the product to 
 which the price for haul applies. 
 
 But the product of AX by the number of cubic yards in AC 
 is equal to the sum of the products obtained by multiplying the 
 contents of each prismoid in AC by the distance of its own 
 centre of gravity from A. The distance of the centre of gravity 
 of a prismoid from its mid-section is expressed by the formula 
 
 eA A 
 v= 987 8 CY) 
 . l 4 a j . 
 If we replace S by its approximate value, ee which 
 will produce no important error in this case, we have 
 ll A-—A 
 x= (361) 
 
 6 Ae 
 
CALCULATION OF EARTHWORK. 245 
 
 in which A should always represent the more remote end area 
 from the starting point A, fig. 106. Hence, may be + or —, 
 and it must be applied, with its proper sign, to the distance of 
 the mid-section from the starting point A, before multiplying 
 by the contents S. Each partial product is thus obtained. 
 
 By a similar process with respect to the prismoids composing 
 the mass BD, and using the point /as the starting point, we 
 obtain finally a sum of the products representing this portion 
 of the haul. 
 
 If a cut is divided, and parts are carried in opposite direc- 
 tions, the calculation of each part terminates at the dividing 
 line. Ifa portion of the material in AC is wasted, it must be 
 deducted, and the haul calculated only on the remainder. 
 
 The specified limit is sometimes made as low as 100 feet, 
 sometimes as high as 1000 feet. A limit of about 300 feet, how- 
 ever is usually most convenient, as it includes the wheelbarrow 
 work, and a large part of the carting, while it protects the con- 
 tractor on such long hauls as may occur. 
 
 270. The Final Estimate is a complete statement in 
 detail, of the amount of work done and materials provided, in 
 the construction of the road, and is the basis of final settlement 
 between the company and contractor. Its preparation should 
 be begun as soon as possible after the work is in progress, and 
 should be continued, as fast as the necessary data are accumu- 
 lated, while the circumstances are still fresh in mind, and when 
 any omissions in the field notes may be readily supplied. The 
 content of each prismoid, the classification of its material, and 
 the length of haul to which it is subject, should be matters of 
 special record in a book provided for that purpose. These re- 
 sults having been carefully computed by exact methods form 
 a standard of comparison for those approximate results which 
 must be had from time to time during the progress of the work, 
 and furnish a limit to the amounts of the monthly estimates. 
 The same remark applies to all other items of labor and mate- 
 rial. The notes and record of the final estimate should be par- 
 ticularly full and exact in respect to all such items as will be 
 inaccessible to measurement at the completion of the work, 
 such as foundation pits, foundation courses of masonry, cul- 
 verts, and works under water. 
 
246 FIELD ENGINEERING. 
 
 271. Monthly Estimates.—On or before the last day 
 of every month during the progress of construction, measure- 
 ments are taken to determine the total amount of work done 
 and material provided up to that date. The estimates based 
 on these measurements are called Monthly Estimates. It is fre- 
 quently necessary to take measurements for both monthly and 
 final estimates at other times than the end of the month, as in 
 the case of foundations which are not long accessible. With 
 respect to each piece of work satisfactorily completed, the 
 monthly estimate should be exact, and identical in amount 
 with the final estimate. With respect, however, to items of 
 work in progress at the time of measurement, the monthly 
 estimate is only approximate, yet should be as precise as the 
 nature of the case will allow; and the quantities stated should 
 not be in excess of fair proportion of the total quantities given 
 on the final estimate for the same piece of work. 
 
 A special field book is devoted to monthly estimate 
 notes. Each page should be dated with the day on which the 
 notes upon it were taken. The notes consist of measurements 
 of all sorts, principally of cross sections partially excavated. 
 These sections should be at the same points on the line as the 
 original sections, so that comparisons may be made. Where- 
 ever the excavation is finished to grade, it is only necessary to 
 write ‘‘completed” opposite such stations, and the quantities 
 may be taken from the final estimate or computed from the 
 original notes. It is frequently necessary to retrace portions 
 of the centre line in taking estimate notes, so that all the field 
 instruments are required, but a party of three or four men is 
 usually sufficient. 
 
 If the contractor has provided materials, such as stone, lum- 
 ber, etc., which are not as yet put into any structure when the 
 estimate is taken, these should be measured and entered under 
 the head of temporary allowance, an arbitrary price be- 
 ing used somewhat below the actual value of the material as 
 delivered. Such allowances should never be copied from one 
 month’s estimate to the next, but made anew on such material 
 as may be found that seems to require it. But all completed 
 items of contract work, and of extra work when ordered by 
 the engineer, are necessarily copied from one monthly esti- 
 mate to the next during the continuance of the contract. 
 
 A blank form is used by the resident engineer in report 
 
TOPOGRAPHICAL SKETCHING. 24AG 
 
 ing monthly estimates, on which a column is provided for each 
 class of material and work required by the contract, while the 
 several lines, headed by the numbers of the proper stations, are 
 devoted to the different cuttings, structures, etc., in consecu- 
 tive order as they occur on the line of road. The estimates are 
 made out and reported separately for the several sections into 
 which the line of road is divided for letting. 
 
 These reports are reviewed by the division engineer, and 
 the footings copied upon another blank, which is the monthly 
 estimate proper; the prices are attached to the items, and the 
 amounts extended and summed up. This sum indicates ap- 
 proximately the total amount earned by the contractor up to 
 date, fron. which is deducted a certain percentage (usually 15 
 per cent.), which is retained by the company until the comple- 
 tion of the contract. From the remainder is deducted the 
 amount of previous payments, which leaves the amount due 
 the contractor on the present estimate. A blank form of re- 
 ceipt is appended, to be signed by the contractor. 
 
 CHAPTER XI. 
 
 TOPCGRAPHICAL SKETCHING. 
 
 272. Topographical sketches taken on preliminary surveys 
 are usually of great value in projecting a line for location; 
 they should be made therefore as accurate and complete as 
 possible. In too many instances sketches are presented having 
 a picturesque appearance, but conveying little information, if 
 not tending to mislead the map-maker. The aim of the topog- 
 rapher should be to record the topographical features either 
 side of the line with as much precision as those directly upon 
 the line, without taking actual measurements, except in rare 
 instances. The eye and the judgment must be usually depended 
 on for distances and dimensions. The sketch of a tract ex- 
 tending to 400 feet each side of the line ought to be accurate 
 enough to warrant its being copied literally upon the map. If 
 a much wider range is required it may be advisable to use the 
 plane-table; but an approximation to plane-table methods may 
 be employed in ordinary sketching. 
 
FISLD ENGINEERING. 
 
 273. As artificial features are the most readily de- 
 fined and located these should first receive attention in making 
 a sketch. When recorded they form a skeleton upon which 
 the natural features can be drawn with more precision than if 
 the order were reversed. ‘The point where each fence crosses 
 the line and the angle between the two may be sketched exact- 
 ly. The distance along the feuce to any object may be esti- 
 mated, and checked (in case of an oblique angle) by observing 
 where a line from the object perpendicular to the centre line 
 would intersect the latter. The book may be rested on any 
 support, the centre-line of the page coinciding with the line of 
 survey, and the direction of objects defined by a small ruler 
 laid on the page. This operation being repeated from another 
 point gives intersections which locate the several objects on 
 the sketch. If the bearings are taken they may be plotted on 
 the page as well as recorded, giving the same results. The 
 eye may be trained to estimate distances Correctly by observ- 
 ing the appearance of objects along the measured line, the dis- 
 tances to which are therefore known. 
 
 274. After the artificial objects the more distinct natural 
 features are to be sketched, as streams, shores, margins of 
 swamps, forests, etc., ravines, ridges, and bluffs, taking care 
 that all these outlines intersect the features of the sketch 
 already delineated at the proper points. The correct repre- 
 sentation of contours is the most difficult part of sketching, 
 since these lines are quite imaginary, yet for railroad maps 
 they are usually as important as any others. It is desirable to 
 know not only the locality of a hill or slope, but also its shape, 
 steepness, and height. This information is best given by con- 
 tour lines. A contour 1s the intersection of the surface of the 
 ground by an imaginary level surface. When the surface 1s 
 real, like that of a lake, the intersection 1s called ashore. If 
 the water should rise a certain -height a new shore would be 
 defined, and rising double that height still another shore 
 would result, each of which, on the subsidence of the water, 
 would be a contour. <A practiced eye is able to follow on the 
 ground the course of a contour with all its windings; but in 
 sketching them due allowance must be made for the fore- 
 shortening effect of distance. All contours on the same sketch 
 should have the same vertical interval, so that by counting 
 
TOPOGRAPHICAL SKETCHING. 24% 
 
 them the height of the hill may be known. The spaces on the 
 sketch between contours vary as the cotangent of the slope 
 angle, so that the width of the spaces indicates the degree of 
 steepness. ‘The contours nearest the topographer should gene- 
 rally be sketched first, although if there be a shore that is apt to 
 be the best guide to the share of the slopes. Ifthe height of the 
 hillis known and the upper contour located, the other contours 
 can be spaced between with less difficulty, the proper number 
 being ascertained by dividing the height by the assumed verti- 
 cal interval. A special line of levels up an inclined ravine or 
 sloping ridge to fix the contour points is often of the greatest 
 service in obtaining correct results. Other random lines are 
 sometimes run to locate the contours more definitely. These 
 should be made to cross several contours rather than to trace a 
 single one. Old preliminary lines which have proved useless 
 in themselves often furnish by their profiles valuable informa- 
 tion 1n respect to contours. 
 
 The use of hatchings should be avoided in the sketch-book, 
 except to represent precipitous banks, or slight terraces, which 
 would not be sufficiently defined by the contour system. 
 Hatchings frecly used consume too much time, and fail to give 
 an accurate idea of either slope or height, while they obscure 
 the page for the representation of other objects. 
 
 275. The centre line on the page is straight, and for 
 sketching purposes the surveyed line on the ground is assumed 
 to be so also. = Slight deflections in the course of a preliminary 
 line may be ignored 1n the sketch; but if a large angle occurs 
 it is better to terminate the sketch with the course, and begin 
 again, leaving a few blank lines between the two sketches. 
 On a located line with curves, the sketch is continuous. The 
 curved line in the field is represented by the straight line on 
 the page, and the radial lines through the stations are repre- 
 sented by the parallel lines ruled across the page. Al objects 
 vive sketched at the proper offset distance by scale frem the 
 centre line; but longiiudinally the sketch 1s necessarily dimin- 
 ished outside of the curve, and magnified inside of the curve. 
 Consequently topographical lines which are straight in fact ap- 
 pear curved in the sketch, concave to the centre line if inside 
 the curve, and convex if outside of it. Such features are cor- 
 rectly sketched by means of offsets estimated or measured 
 
250 FIELD ENGINEERING. 
 
 from each station of the curve on the radial lines. This kind 
 of distortion creates no confusion if properly done, for in mak- 
 ing the map, after drawing the curve and the radial lines, the 
 same offsets will give the correct positions of the objects delin- 
 eated. This method is preferable to drawing a curved line on 
 the page to represent the centre line, as it is difficult to draw 
 it correctly; it will cross the ruled lines obliquely, rendering 
 them of no service for offsets or scale, and moreover is likely 
 to run off the page altogether. 
 
 CHAPTER XII. 
 
 ADJUSTMENT OF INSTRUMENTS. 
 
 Every adjustment consists of two processes: first the test, and 
 second the correction. Inasmuch as the amount of correction 
 is made by estimation, the test must always be repeated until 
 no further lack of adjustment is observable. 
 
 276. THE TRANSIT. 
 
 The level tubes should be parallel to the 
 vernier plate. 
 
 Test : Place the tubes in position over the levelling screws, 
 and turn the latter till the bubbles are centred; revolve the 
 plate 180°. The bubbles should remain centred; if they have 
 retreated— 
 
 Correction : Bring them half way back to the centre by 
 turning the adjusting screws which attach the tubes to the 
 plate. 
 
 The line of collimation should be perpendi- 
 cular to the horizontal axis. 
 
 West: Clamp the limb, and by the tangent screws bring 
 the intersection of the cross-hairs to cover a well-defined point 
 about on a level with the telescope; plunge the telescope to 
 look in the opposite direction, and note any point about on a 
 level with the telescope and about equidistant with the first 
 point, which the intersection of the cross-hairs now happens to 
 cover. Now unclamp the limb and turn through 180°, and 
 repeat the above operation, using the same first point as before. 
 
ADJUSTMENT OF INSTRUMENTS. 201 
 
 The tiird point obtained should be identical with the 
 second, if not— 
 
 Correction : Move the vertical cross-hair over one fourth 
 of the apparent distance from the third to the second point, by 
 turning the adjusting screws at the side of the telescope. 
 
 The horizontal axis should be parallel to the 
 vernier plate. 
 
 Test; After completing the above adjustments level the 
 limb, clamp it, and bring the intersection of the cross-hairs to 
 cover some high point so that the telescope may be elevated to 
 a large angle; depress the telescope and note some point on the 
 ground now covered by the intersection of the cross-hairs. 
 Now unclamp the limb, turn it through 180°, and repeat the 
 above operation, using the same high point as before. The 
 third point found should be identical with the second; if not— 
 
 Correction: Raise the endof the axis opposite the second 
 point (or lower the other end) by a small amount, by turning 
 the adjusting screws in the standard. The amount of motion 
 required is only determined by repeated trials until the test is 
 satisfied. 
 
 The intersection of the cross-hairs should 
 appear in the centre of the field of view. 
 
 Test ; Bring the cross-hairs into focus and direct the tele- 
 scope toward the sky, or hold a sheet of blank paper in front 
 ofit. If the intersection appear eccentric— 
 
 Correction ; Turn the screws (by pairs) which support 
 the end of the eyepiece until the desired result is obtained. 
 
 If there be a level on the telescope it Should be 
 parallel to the line of collimation. 
 
 Drive two stakes equidistant from the instrument in exactly 
 oposite directions, and having perfected the previous adjust- 
 ments, level the plate carefully, Glamp the telescope in about 
 a horizontal position, and observe a rod placed on each stake. 
 Have the stakes driven by trial until the rod reads alike on 
 both. The heads of the stakes are then on a level.  Re- 
 move the instrument beyond one stake, and set it up in 
 line with the two, level the plate, and elevate or depress the 
 telescope to a position which will again give equal readings 
 on the stakes. The line of collimation is now level— 
 
2D2 FIELD ENGINEERING. 
 
 i) Test: While in this position the bubble of the attached 
 level should stand centred; if not— 
 
 Correction: Bring the bubble to the centre by turning 
 the nuts at one end of the tube, while the cross-hair continues 
 to give equal readings. | 
 
 277. THE Y LEVEL. | 
 
 The line of collimation should coincide with 
 the axis of the telescope. 
 
 Test: Clamp the spindle, and bring the intersection of the 
 cross-hairs to cover a well-defined point by the tangent and 
 levelling screws; revolve the telescope half over in the Ys, so 
 Hil) that thelevel tube is on top. The intersection of the cross- 
 
 HE hairs should still cover the point. If either hair has departed— 
 
 Pi Correction: bring it half way back by means of the pair 
 ' ly oi adjusting screws at the extremities of the other hair. 
 agit The attached level should be parallel to the 
 axis of the telescope. 
 i Test: Bring the telescope over one pair of levelling screws, 
 i, clamp the spindle, open the clips, and bring the bubble to the 
 | centre. Then gently remove the telescope from the Ys, and 
 replace itend forend. Ifthe Ys have not been disturbed, the 
 bubble should return to the centre. If it does not— 
 
 Correction : bring the bubble half way back by turning 
 the nuts at one end of the tube. 
 
 But as now the level tube and telescope may only lie in 
 parallel planes, and yet not be parallel to each other— 
 
 Test: bring the bubble to the centre as before, and turn 
 the telescope on its axis so as to bring the level tube out to one 
 ‘il side. The bubble should remain centred. If it has departed— 
 it Correction : bring it back to the centre by the adjusting 
 ji screws at one end. 
 
 1 The axis of the telescope should be at right 
 i angles to the spindle. * 
 
 Test: Having completed the above adjustments (and not 
 before), fasten down the clips, unclamp the spindle, and bring 
 the bubble to the centre over each pair of levelling screws in 
 succession, then swing the telescope end for end on the spin- 
 dle. The bubble should settle at the centre. If it donot— | 
 
 Correction: bring it half way back by the large nuts at 
 one end of the bar. 
 
EXPLANATION OF TABLES. ROO 
 
 278. THE THEODOLITE. 
 
 This instrument being a combination of Transit and Level, 
 its several adjustments are to be made according to the rules 
 already given for those instruments. 
 
 CHAPTER XIII. 
 EXPLANATION OF TABLES. 
 
 TABLE J.—Contains concise statements of such geometrical 
 truths as are applicable to the various discussions in this volume. 
 References are given to Davies’ Geometry, in which the demon- 
 strations of the propositions may be found. 
 
 TaBLE IJ.—Contains all the formule necessary to the solu- 
 tion of any plane triangle; also, a select list of miscellaneous 
 formule. A few formule with respect to the versed sine and 
 external secant are new. 
 
 TABLE JII.—Contains a complete list of formule expressing 
 the relations between the radius, tangent, chord, versed sine, 
 external secant, and central angle of a railway curve; also, the 
 relations between the radius, degree of curve, length of curve, 
 and central angle. The notation is identical with that used 
 elsewhere in the book. 
 
 Tas.Ee 1V.—Contains the radius, and its logarithm, for every 
 degree of curve to single minutes up’to 10 degrees, and thence 
 by larger intervals up to 50 degrees. With the radius is given 
 also the perpendicular off-set, ¢, from the tangent to a point on 
 the curve at the end of the first 100-foot chord from the tan- 
 gent-point, and the midd!e ordinate, m, of a 100-foot chord. 
 See eqs. (16, 34, 87, 40, and 805). 
 
 TaspLE V.—Contains the corrections to be added to the tan- 
 gents and externals of any railroad curve, as obtained by refe- 
 rence to Table VI., according to the degree of the given curve 
 (found at head of columns), and its central angle, (found in the 
 
254 FIELD ENGINEERING. 
 
 first column.) If the given degree of curve, or central angle, 
 does not appear in the table, the exact value of the correction | 
 may be casily obtained by interpolation. | 
 
 TaBLE VI.—Contains the exact values of the tangents, T, | 
 and externals, E, to a1 degree curve, for every 10 minutes of | 
 central angle, from 1° to 120° 50’. Approximate values of the 
 tangent and external to any other degree of curve may be had 
 by simply dividing the tabular values opposite the given cen- 
 tral angle by the given degree of curve, expressed in degrees. 
 These approximations may be made exact by adding the proper 
 corrections taken from Table V. See eqs. (21) and (24). 
 
 | TABLE VII.—Contains the value of Long Chords of from 2 
 | ib to 12 stations, for every 10 minutes of degrce of curve from 0° 
 
 | to 15°, and of aless number of stations for degrees of curve be- 
 He tween 15° and 80°. As the chord of one station is always 100 
 Ht feet, the column of the first station gives instead the length of 
 i arc subtended by the chord of 100 feet. See §$121, 122, 128, | 
 ia 124, 125. 
 
 TasLE VIII.—Contains the values of Middle Ordinates to 
 long chords of from 2 to 12 stations, for every 10 minutes of 
 aii degree of curve from 0° to 10°, and of from 2 to 6 stations for 
 Hal every curve from 10° to 20°, at 10-minute intervals. The table 
 
 Ml may be used, not only to fix the middle point of an arc, but 
 also, in conjunction with the table of long chords, to locate in- 
 termediate stations. See §$121, 122, 123, 124, 125. 
 
 TABLE 1 X.—Contains the chords of a scries of angles vary- 
 ing by half degrees up to 30° for radii varying by 100 feet up to | 
 1000 feet. It shows, therefore, the linear opening between | 
 Hal} the extremities of two equal lines at any given number of hun- 
 
 i dred feet from their intersection, when the angle does not ex- 
 i) ceed 30°. For any distance exceeding 1000 we have only to 
 add to the value found in that column, the value found in the 
 i column headed by the excess of distance over 1000 feet. Con- 
 versely, the table gives the angular deflection required between 
 two equal lines, in order that at a given distance from the point 
 of intersection they may be separated a given amount. 
 
EXPLANATION OF TABLES. 
 
 " : . a 
 TABLE X.—1. Contains values of the ratio w= ae accord- 
 
 ing to the notation of $147 for finding the angle 7 (Fig. 34) 
 between the radius PO of the curve at any point P, and the 
 tangent P& to the valvoid arc PX by the simple formula 
 eq. (80)¢ =uwA. The table embraces lengths of curve from 
 300 to 2000 feet, and central angles from 10° to 120°. 
 
 LR 4 : : ! 
 = 60° w = 4, and for hasty approximation this 
 value of w may be assumed in any case without consulting 
 the table. 
 
 A F : r : ; 
 2. Contains values of the ratio » = 7 for finding the radius 
 
 4 
 of the valvoid arc at the point P (Fig. 35) in terms of the 
 length of curve £ = AP by the simple formula, eq. (82), 
 mol. 
 
 3. Contains values of the length J, of a valvoid are limited 
 by two curves of equal length laid out from the same tangent 
 and same P.C., but whose central angles differ by 1°. The 
 length Z of each curve is given in the first column, and the 
 
 : : eZ ay eee a een 
 half sum of their central angles (eee 4) is given at the 
 2 
 head of the other columns. 
 
 When the central angles of two curves of equal length 
 differ by @ degrees the length / of the valvoid arc joining 
 their extremities is expressed by the simple formula, Fig. 36, 
 
 eq. (86) Pet Pe PBS CA UAE 
 A'+ A‘ 
 
 in which 7, is taken from the column headed by 
 
 fe 
 and opposite the given value of Z;or 7 is found by inter- 
 polation if necessary. See § 150 and example. 
 
 TABLE XI.—Contains the measurements necessary to lay 
 down a turnout with frogs of given numbers or angles for 
 both a standard and a three-foot gauge. The distance BF is 
 measured on the rail of the given track from the heel of the 
 switch to the point of the frog, while af is the chord of the 
 centre line of the turnout between the same points. The 
 radius 7 applies to the centre line of the turnout. The dis- 
 
 tance af" is measured on the centre line of the straight track 
 
206 FIELD ENGINEERING. 
 
 from the heel of the switch to the point of the middle frog. 
 The length of switch AY should conform to the tabular 
 values unless the throw is to be different from that assumed 
 in the table. Sce §§ 180, 181, 182. 
 
 TABLE XIJ.—Contains the middle ordinates of chords vary- 
 ing in length from 10 to 82 feet, and for degrees of curve vary- 
 ing from 1° to 50°. The use of the table is obvious. See § 199. 
 
 TABLE XIiI.—Gives the proper difference in elevation of 
 rails on curves of various degrees from 1° to 50° for veloci- 
 ties varying from 10 to 60 miles per hour. See § 201. 
 
 Wan TaBLE XIV.—Gives the rise of grades in feet per mile and 
 Hi their angle of inclination corresponding to a rise per station 
 AAT (100 fect) varying from 0.01 foot to 10 feet. 
 
 i TABLE XV.—Contains values of the formula (log 4 — 1) 
 i 60384.3 in which # = reading of the barometer in inches. The 
 inches and tenths of the readings are in the left-hand column, 
 while the hundredths are found at the top of the other columns. 
 The difference of any two values corresponding to two read- 
 ings taken simultaneously at any two stations is the differ- 
 ence in elevation in feet of those stations. But the differ- 
 ence in height so found is subject to a correction for tempera- 
 ture given in the next table. See § 10. 
 
 TABLE XV1.—Contains coeflicients of correction for atmos- 
 pheric temperature, by which the approximate heights ob- 
 tained by Table XY. are to be multiplied for a correction of 
 these heights, which correction is to be added or subtracted 
 according as the coefficient given in the table is marked 
 + or --. See §11. 
 
 i Tapiu XVII.—Contains corrections in feet, required by the 
 
 i curvature of the earth and the refraction of the atmosphere, to 
 be applied to the elevation of a distant object as obtained by a 
 Ievel or theodolite observation for distances ranging from 
 il 300 feet to 10 miles. See § 119. 
 
 TABLE XVIII.—Contains the cvefficients for reducing the 
 space on a vertical rod intercepted by the stadia hairs when 
 
SXPLANATION OF TABLES. 257 
 
 the line of collimation is inclined to the horizon, to the space 
 that would be intercepted were the line of collimation horizon- 
 tal; previded, that the visual angle 9 defined by the stadia hairs 
 is Such that tan 49 = .005 or 9 = 0° 34’ 22.63, which is its 
 customary value in surveying instruments, The angle of in- 
 clination @ is taken at every 10 minutes through half a quad- 
 rant, 
 
 TABLE XIX,—Contains the logarithms of the coefficients 
 given in Table XVIII. 
 
 TABLE XX.—Gives the lengths of circular arcs to a radius 
 =a I 
 
 To find the length of any arc expresscd in degrees, minutes, 
 and seconds, take from the table the lengths of the given num- 
 ber of degrees, minutes, and seconds respecti vely, and multi- 
 ply their sum by the length of the radius, The product is the 
 length of are required. 
 
 TABLE XXI.—-Contains the values of minutes and seconds 
 expresscd in decimals of a degree, for every 10 seconds of arc, 
 and also for quarter minutes up to one degree. 
 
 TABLE XXII.—Contains the values of inches and fractions 
 expressed in decimals of a foot for every 32d of an inch up to 
 one foot. 
 
 TABLE XXIII.—Contains the squares, cubes, square roots, 
 cube roots, and reciprocals of numbers from 1 to 1054. Its 
 use may be greatly extended by observing that if any number 
 is multiplied by 7 its square is multiplied by n’, its cube by 
 
 : ; 1 
 n, and its reciprocal by —. 
 n 
 
 TABLE XXIV.—The logarithm of a number consists of 
 two parts, a whole number called the characteristic, and a dcci- 
 mal called the mantissa. All numbers which consist of the 
 same figures standing in the same order have the same man- 
 tissa, regardless of the position of the decimal point in the 
 number, or of the number of ciphers which precede or follow 
 the significant figures of the number. The value of the char- 
 acteristic depends entirely on the position of the decimal point 
 in the number. It is always one less than the number of 
 
258 FIELD ENGINEERING. 
 
 1 figures in the number to the left of the decimal point. The 
 value is therefore diminished by one every time the decimal 
 point of the number is removed one place to the Jeft, and vce 
 versa. Thus | 
 
 Number. Logarithm. 
 
 13840. 4.141136 
 
 1384.0 3.141136 
 
 188.40 — 2.141186 
 
 13.84 1.141136 
 
 1.384 0.141186 
 1384 —1,141136 : 
 .01384 —2,141136 | 
 .001884 —3,.141136 : 
 
 etc. etc. 
 
 Ht The mantissa is always positive even when the characteristic 
 ET is negative. We may avoid the use of a negative characteristic 
 | by arbitrarily adding 10, which may be neglected at the close 
 i of the calculation. By this rule we have 
 
 Hi Number. Logarithn. 
 ie 1.884 0.141136 
 ae 1384 9.141136 
 . ie .01384 8.141186 
 Hy .001884 7.141136 
 etc, ete. 
 
 | No confusion need arise from this method in finding a number 
 from its logarithm; for although the logarithm 6.141186 repre- 
 
 mil sents either the number 1,384,000, or the decimal .0001384, yet 
 these are so diverse in their values that we can never be uncer- 
 tain in a given problem which to adopt. 
 Halk The table XXIV. contains the mantissas of logarithms, car- 
 Wad ried to six places of decimals, for numbers between 1 and 9999, 
 } Hi inclusive. The first three figures of a number are given in the 
 | first column, the fourth at the top of the other columns. The 
 | i first two figures of the mantissa are given only in the second 
 we column, but these are understood to.apply to the remaining 
 i four figures in either column following, which are comprised 
 between the same horizontal lines with the two. 
 
 If a number (after cutting off the ciphers at either end) con- 
 i) sists of not more than four figures, the mantissa may be taken 
 i direct from the table; but by interpolation the logarithm of a 
 number having six figures may be obtained. The last column 
 contains the average difference of consecutive logarithms on 
 
EXPLANATION OF TABLES. 
 
 the same line, but for a given case the difference needs to be 
 verified by actual subtraction, at least so far as the last figure 
 isconcerned, The lower part of the page contains a complete 
 list of differences, with their multiples divided by 10. 
 
 To find the logarithm of a number having six 
 figures ;—Take out the mantissa for the four superior places 
 directly from the table, and find the difference between this 
 mantissa and the next greater in the table. Add to the man- 
 tissa taken out the quantity found in the table of proportional 
 parts, opposite the difference, and in the column headed by the 
 fifth figure of the number; also add #5 the quantity in the col- 
 umn headed by the sixth figure. The sum is the mantissa 
 required, to which must be prefixed a decimal point and the 
 proper characteristic. 
 
 Hrample.—Find the log of 23.4275. 
 
 For 2342 mantissa is 369587 
 fevdifls 185 coli7 129.5 
 ee ‘é a4 6é 5 9.2, 
 
 Ans. For 23.4275 log is 1.369726 
 The decimals of the corrections are added together to deter- 
 mine the nearest value of the sixth figure of the mantissa. 
 
 To find the number corresponding to a given 
 logarithm.—If the given mantissa is not in the table find the 
 one next less, and take out the four figures corresponding to it; 
 divide the difference between the two mantissas by the tabu- 
 Jar difference in that part of the table, and annex the figures of 
 the quoticnt to the four figures already taken out. Finally, 
 place the decimal point according to the rule for characteristics, 
 prefixing or annexing ciphers if necessary. The division re- 
 quired is facilitated by the table of proportional parts, which 
 furnishes by inspection the figures of the quotient. 
 
 Example.—Find the number of which the logarithm is 
 
 8.268927 8.263927 
 First 4 figures 1836 from 263873 
 Diff. 54.0 
 Tabular diff. = 236 .. oth fig, = 2 47.2 
 6.80 
 6th fig. = 3 7.08 
 
 Ans. No. — .0183623 or 188,623,000. 
 
 259 
 
260 FIELD ENGINEERING. 
 
 reliable beyond the sixth figure. 
 
 table. 
 
 of one second in the arc are given in adjoining columns. 
 
 i angle increases. 
 Erample.—Find the log sin of 9° 28’ 20". 
 ) Log sin of 9° 28’ is 9.216097 
 Hit Add correction 20 x 12.62 252 
 Ans, 9.216349 
 EKxample.—Find the log cot of 9° 28’ 20". 
 
 | Log cotan of 9° 28’ is 10. 777948 
 | Subtract correction 20 x 12.97 259 
 
 Ans. 10777689 
 
 The number derived from a six-place logarithm is not 
 
 At the end of table XXIV. is a small table of logarithms of 
 numbers from 1 to 100, with the characteristic prefixed, for 
 easy reference when the given number does not exceed two 
 digits. But the same mantissas may be tound in the larger 
 
 TaBLE XXV.—The logariinmic gine, tangent, 
 ete. of an arc is the logaritnin oi the natural sine, tangent, 
 etc. of the same arc, but with 10 added to the characteristic to 
 avoid negatives. This table gives log sines, tangents, cosines, 
 
 and cotangents for every minute of the quadrant. Wita the 
 | number of degrees at the lIefé side of the page are to be read 
 Mi the minutes in the left-hand column ; with the degrees on 
 | i the right-hand side are to be read the minutes in the right-hand 
 i column. When the degrees appear at the top of the page the 
 top headings must be observed, when at the bottom those at 
 the bottom. Since the values found for arcs in the first quad- 
 iH rant are duplicated in the second, the degrees are given from 
 0° to 180°. The differences in the logarithms due to a change 
 
 To find the log.sin, cos, tan, or cot of a given 
 arc.; Take out from the proper column of the table the log- 
 arithm corresponding to the given number of degrees and 
 minutes. If there be any seconds multiply them by the ad- 
 joining tabular difference, and apply their product as a cor- 
 
 Ai rection to the logarithm already taken out. The correction 1s 
 . | to be added if the logarithms of the table are increasing with 
 Hii the angle, or subtracted if they are decreasing as the angle in- 
 Hit creases. In the first quadrant the log sines and tangents 1n- 
 al crease, and the log, cosines and cotangents decrease as the 
 
EXPLANATION OF TABLES. 
 
 To find the angle or are corresponding to a 
 given logarithmic sine, tangent, cosine, or co- 
 tangent.—If the given logarithm is found in the proper 
 column take out the degrees and minutes directly; if not, find 
 the two consecutive logarithms between which the given 
 logarithm would fall, and adopt that one which corresponds to 
 the least number of minutes; which minutes take out with the 
 degrees, and divide the difference between this logarithm and 
 the given one by the adjoining tabular difference for a quo- 
 tient, which will be the required number of seconds. 
 
 With logarithms to six places of decimals the quotient is 
 not reliable beyond the tenth of a second. 
 
 Example. —9.383781 is the log tan of what angle? 
 Next less 9.383682 gives 13° 36’ 
 
 Diff. 49.00 -- 9.20 = 05".3 
 
 Ans. 13° 36’ 05".3 
 
 Example, —9.249348 is the log cos of what angle? 
 Next greater 583. gives 79° 46' 
 
 Diff. 200 + 11.67 = 20".1 
 
 Ans. 79° 46° 20"1 
 
 The above rules do not apply to the first two pages of this 
 table (except for the column headed cosine at top) because 
 here the differences vary so rapidly that interpolation made by 
 them in the usual way will not give exact results, 
 
 On the first two pages, the first column contains the number 
 of seconds for every minute from 1’ to 2°; the minutes are 
 given in the second, the log. sin. in the thard, and in the Sourth 
 are the last three figures of a logarithm which is the difference 
 between the Jog sin and the logarithm of the number of sec. 
 onds 1n the first column. The first three figures and the char- 
 acteristic of this logarithm are placed, once for all, at the head 
 of the column. 
 
 To find the log sin of an are less than 2° given 
 to seconds.—Reduce the given arc to seconds, and take the 
 Jogarithm of the number of seconds from the table of loga- 
 rithms, and add to this the logarithm from the fourth column 
 opposite the same number of seconds. The sunr is the log sin 
 required. 
 
 The logarithm in the fourth column may need a slight inter. 
 
262 FIELD ENGINEERING. 
 polation of the last figure, to make it correspond closcly to the 
 given number of seconds. 
 
 Example.—Find the log sin of 1° 39° 14".4. 
 
 1° 39’ 14".4 = 5954".4 log 3.774838 
 add (q —l) 4.685515 
 
 i Ans. log sin 8.460353 
 
 Log tangents of small arcs are found in the same way, only 
 taking the last four figures of (¢ — @) from the fifth column. 
 
 Exampile.—Find the log tan of 0° 52’ 35". 
 
 HAGE 52’ 35” = (8120” + 385”) = 8158’ log 3.498999 
 Hi add (¢ — l) 4.685609 
 
 | Ans. log tan 8.184608 
 
 i if To find the log cotangent of an angle less than 
 ae 2° riven te seconds.—Take from the column headed ( g++ 2) 
 it the logarithm corresponding to the given angle, interpolating 
 for the last figure 1f necessary, and from this subtract the loga- 
 rithm of the number of scconds in the given angle. 
 
 Example.—Find the log cotan of 1° 44° 227.5. 
 
 i 5 qg + ! 15.314292 
 i 6240" +- 22".5 = 6262.5 log 3.796748 
 
 | Ans. 11.517544 
 
 aa These two pages may be used in the same way when the 
 \ given angle lics between 88° and 92°, or between 178° and 180°; 
 } but if the number of degrees be found at the dottom of the paye, 
 the title of each column will be found there also; and if the 
 . number of degrees be found on the wight hand side of the page, 
 ni the number of minutes must be found in the nght hand col- 
 umn, and since here the minutes increase upward, the number 
 i of seconds on the same line in the first column must be dimin- 
 if ished by the odd seconds in the given angle to obtain the num- 
 | _ber whose logarithm is to be used with (7+/) taken from the 
 . ! table. 
 
 Ezample.—Find the log cos of 88° 41° 12".5 
 
 (¢g — 1) 4.685537 
 4740" —12".5 = 4727.5 log 3.674631 
 
 Ans. 8.360168 
 
EXPLANATION OF TABLES. 
 
 Ezrample.—Find the log tan of 90° 30’ 50”. 
 q+ 15.814413 
 1800" + 50” = 1850" log 3.267172 
 Ans. 12.047241 
 
 To find the are corresponding to a given log 
 Sin, cos, tan, or cotan which falls within the 
 limits of the first two pages of Tabie XXV. 
 Find in the proper column two consecutive logarithms be- 
 tween which the given logarithm falls. If the title of the 
 given function is found at the top of that column read the 
 degrees from the top of the page; if at the bottom read from 
 the bottom. 
 
 Find the value of (¢ — ) or (+1), as the case may require, 
 corresponding to the given log (interpolating for the last figure 
 if necessary). Then if g= given log and / = log of number of 
 seconds, 7, in the required arc, we have at once J = g — (q — J) 
 or = (¢-+-l) — g, whence n is easily found. 
 
 Find in the first column two consccutive quantities between 
 which the number 7 falls, and if the degrees are read from 
 the left hand side of the page, adopt the Jess, take out the 
 minutes from the second column, and take for the seconds 
 the difference between the quantity adopted and the number 
 nm. But if the degrees are read from the right hand side of the 
 page, adopt the greater quantity, take out the minutes on the 
 same line from the right-hand column, and for the seconds 
 take the difference between the number adopted and the num- 
 ber n. 
 
 Example. —11.734268 is the log cot of what arc? 
 
 qtl 15.314376 
 q 1.734268 
 
 n= 3802.8 3.580108 
 
 For 1° adopt 3780. giving 03’ 
 
 Difference 22".8 
 
 Ans. 1° 03' 22".8 or 178° 56’ 37".2. 
 
 Hxrample.—8.201795 is the log cos of what arc? 
 
 7 sil 4.685556 
 q 8.201795 
 nos 3282".8 d. 916239 
 
 For 89° adopt 3300. giving 05’ 
 
 Difference 17’.2 
 
 Ans. 89° 05' 17".2 or 90° 54’ 42".8. 
 
264 FIELD ENGINEERING. 
 
 TABLE XXVI.—Contains logarithmic versed sines and ex- 
 ternal secants for every minute of the quadrant, with the 
 differences of the same corresponding to a change of 1 second 
 in the arc or angle. Interpolation for seconds is made in the 
 same manner as with log sines of the preceding table, except 
 onthe first two pages. For angles less than 4° the differences 
 vary so rapidly that interpolation by direct proportion will not 
 give exact values. 
 
 On the first two pages the column headed g — 2/ contains 
 the difference between the log versed sine (or log ex secant) of 
 an are and twice the logarithm of the number of seconds in the 
 same are. The characteristic, and first three decimals (9.070) 
 are common to all the logaritlims in these columns up to 3° 19’ 
 for log vers sines, where it changes to (9.069), as shown at the 
 foot of the column; and up to 4° for log ex secants, where it 
 changes to (9.071). At the point of change a cipher is replaced 
 by the mark ¢ to call attention. 
 
 To find the log vers sin, or log ex sec of an 
 angle given to seconds.—Reduce the angle to seconds, 
 take the logarithm of this number, multiply it by 2, and add 
 the product to the logarithm in the column (q¢ — 2!) found op: 
 posite the given angle. The log (g — 2/) should be corrected 
 by interpolation for the fractional part of a minute in the given 
 angle. 
 
 Hreample.—W hat is the log ex secant of 2° 14’ 48".77 
 2° 14’ 43".7 == 8040” + 43.7 = 8083".7 log 3.907610 
 9 
 
 QI 7.815220 
 (@—2)° 9.070064 
 ngs MOM, 6.885284 
 
 To find the are corresponding to a given log 
 vers, or log ex sec.—Find in the column of log vers, or 
 log ex sec the two values between which the given log falls, 
 and take out from the column (qg— 2/) the logarithm corres- 
 ponding to the given log (interpolating for the value of the last 
 figure if necessary). Subtract this from the given logarithm 
 and divide by 2. The quotient is the logarithm of the num- 
 ber of seconds in the required are. 
 
EXPLANATION OF TABLES. 26d 
 
 Example.—7.344728 is the log vers of what arc? 
 
 q 7.344728 
 3° 49’ + (q — 21) 9.069960 
 
 2)8.274 274768 
 13720".9 ee 4.1373 
 13680. 
 
 Ans. 3° 48' 40".9 
 
 To find the log ex sec of an are greater than 
 88° given to seconds.—Take from the column (¢-+J) 
 the logarithm corresponding to the given arc, interpolating for 
 the fraction of aminute. From this subtract the logarithm of 
 the number of seconds in the complement of the given arc. 
 
 Hrample.—W hat is the log ex sec of 88° 24' 20".5? 
 
 For 88° 24' g+/ 15.302188 
 Correction 129 x a 44 
 
 g+l 15.302227 
 Comp. 88° 24’ 20".5 = 5789".5 log 3.758874 
 
 Ans. log ex sec 11.548353 
 
 To find the angie corresponding to a given 
 log ex sec when the angle is greater than 88°,— 
 Find in the table the two consecutive log ex secants between 
 which the given one falls, and then find by interpolation the 
 value of the log (¢-+-/) corresponding to the given log ex sec 
 and subtract the latter from it. The difference will be the 
 logarithm of the number of seconds in the complement of the 
 required angle, which is then easily found. 
 
 Hxample.—11.924368 is the log ex sec of what arc? 
 
 Given log ex sec 11.9243868 
 Next less 11.918290 g+l 15.809225 
 Diff. 6078 
 ; .  _--- 9852 — 9225 Be ¥4 
 Correction = 99141 — 18290 xX 6078 = i1 
 q+l — 15.309296 
 Given log ex sec 11.924368 
 0° 40’ 26".2 = 2426.2 .. log 8.384928 
 
 Ans. 89° 19 83".8, 
 
266 
 
 FIELD ENGINEERING. 
 
 TABLE XXVII.—Contains natural sines and cosines, to five 
 
 places of decimals for every minute of the quadrant. 
 
 Correc- 
 
 tions for fractions of a minute are made directly proportional 
 to the difference of consecutive values in the table; positive 
 for sines, negative for cosines. 
 
 TABLE XX VIII.—Contains natural tangents and cotangents 
 to five places of decimals for every minute of the quadrant. 
 Corrections for fractions of a minute are made directly propor- 
 tional to the difference of consecutive values in the table ; 
 positive for tangents, negative for cotangents. 
 
 TABLE XXIX.—Contains natural versed sines and external 
 secants to five places of decimals for every minute of the 
 
 quadrant. 
 
 Corrections for fractions of a minute are made 
 
 directly proportional to the difference of consecutive values. 
 They are positive in every case. 
 
 LAR LB 
 
 Contains the number of cubic yards con- 
 
 tained in prismoids of various side slopes, bases, and depths, 
 as indicated by the title and the numbers i. the first column. 
 Each prismoid is supposed to have a uniform level cross sec- 
 
 tion throughout. 
 
 These tables are chiefly useful in making up 
 
 preliminary estimates from the profile, or in other cases where 
 
 only approximate results are required. 
 
 For monthly and final 
 
 estimates more elaborate tables are required, such as are des- 
 
 cribed in § 257. 
 
 Oo 
 e 
 
 To make an approximate estimate of quanti- 
 ties from a profile by use of Table XX X.—Select the 
 proper column for base and slopes, and. if the outline of a cut 
 on the profile is roughly a four-sided figure, stretch a fine silk 
 thread over the surface line to average it, note the depth from 
 thread to grade line midway of the cutting, and multiply the 
 tabular number opposite this depth by the average length of 
 
 the cutting in stations of 199 fect. 
 
 (By average length is meant 
 
 the half sum of the length of the grade line in the cutting and 
 of so much of the surface line as is covered by the thread.) If 
 the area of a cutting as seen on the profile is approximately 
 triangular, stretch an averaging line over each incline, and 
 note the depth from the intersection of these lines to grade, 
 
 and multiply the tabular number opposite this depth by one- 
 
meen 
 
 EXPLANATION OF TABLES. 267 
 
 half the length of the cut measured on the grade line in sta- 
 tions. The resulting quantities will be slightly in excess if the 
 ground is level transversely, but-may be too small if the trans- 
 verse slope is steep, and cutting on the centre line is small. 
 In general they furnish a good approximation. Quantities in 
 embankments may, of course, be found similarly. <A cut or 
 fill may be divided on the profile into several portions, and the 
 contents of each portion found senarately if preferred. 
 
 The content of a prismoid, level transversely, but having 
 different end depths, may be found correctly by this table thus: 
 add together the quantities opposite each end-depth and 4 times 
 the quantity opposite the half sum of the depths; multiply the 
 sum by the length in feet, and divide by 600. 
 
 TABLE XX XI.—Contains a variety of useful numbers and 
 formule. The logarithms are here given to seven places of 
 decimals. 
 
mN 
 ea 
 an 
 aa 
 <q 
 Ss 
 
TABLE I.—GEOMETRICAL PROPOSITIONS. 
 
 The References are to Davies’ Legendre, Revised Edition. 
 
 ~I 
 
 (oo) 
 
 9) 
 
 10 | 
 
 11 | 
 
 15 
 
 16 | 
 
 | from the vertex 
 of an _ isosceles 
 triangle bisects 
 the base, 
 
 | If one side of a tri- 
 
 Leo Vi Cor 
 
 He OQ WERE Pome 
 
 eX 
 
 LDL bles ees 
 
 (BTR OYE Bah ees 6 
 
 Vein Conie.. 
 NA no G WW IRS oe Satoac 
 
 EEE, Vilise Saeco 
 
 Mio tis). wks | 
 
 IIl., XIV., Cor... 
 
 is 6 Oe 
 
 angle is pro- 
 duced, 
 
 | If two triangles are 
 
 mutually equian- 
 
 | gular, 
 
 | If the sides of a 
 
 | polygon are pro- 
 
 | duced in the 
 same order, 
 
 |If a figure is a 
 quadrilateral, 
 
 If a figure is a 
 | parallelogram, 
 
 | : 
 If three points are | 
 | not in the same | 
 
 | straight line, 
 | 
 | If two ares are in- 
 
 tercepted on the | 
 
 same circle, 
 
 If two arcs 
 similar, 
 
 are 
 
 | If two areas are 
 circles, 
 
 If a radius is per- 
 chord, 
 
 If a straight line is 
 
 | tangent to .@ 
 circle, 
 
 | If from a point 
 tangents 
 drawn 
 the circle, 
 
 pendicular to a | 
 
 to touch | 
 
 | The 
 
 without a circle | 
 are | 
 
 | 
 1 
 
 No REFERENCE. HYPOTHESES. CONSEQUENCES, 
 1” GRD. Ea 5 Ayo Ifatriangleisright | The square on the hypothe- 
 angled, nuse is equal to the sum of 
 the squares on the other 
 two sides. 
 2) L, XI, Cor.1....|If a triangle is | It is equiangular. 
 | equilateral, | 
 ie Bg @ hes eo |If a triangle is | The angles at the base are 
 | isosceles, | equal. 
 AG TeX ly WOT. ee as. |If a straight line | It bisects the vertical angle. 
 
 And is perpendicular to the 
 base. 
 
 | The exterior angle is equal to 
 
 the sum of the two interior 
 and opposite angles. 
 
 And their 
 sides are 
 
 They are similar. 
 corresponding 
 proportional. 
 
 of the exterior 
 right 
 
 sum 
 angles equals four 
 angles. 
 
 The sum of the interior angles 
 equals four right angles. 
 
 | The opposite sides are equal. 
 
 The opposite. angles are 
 equal. It is bisected by its 
 diagonal. And its diagonals 
 bisect each other. 
 
 A circle may be passed 
 
 through them. 
 
 They are proportional to the 
 corresponding angles at the 
 centre. 
 
 They are proportional to their 
 radii. 
 
 They are proportional to the 
 squares on their radii. 
 
 It bisects the chord. And it 
 bisects the are subtended 
 by the chord. 
 
 It meets itin only one point. 
 
 And it is perpendicular to 
 
 the radius drawn to that 
 point. 
 There are but two. They are 
 
 equal. And they make 
 equal angles with the chord 
 joining the tangent points. 
 
18 
 
 19 
 
 20 
 
 oO 
 wt 
 
 25 
 
 TABLE I.—GEOMETRICAL PROPOSITIONS 
 
 aN. 
 
 The References are to Davies’ Legendre, Revised Edition. 
 
 REFERENCE. 
 
 HYPOTHESES. CONSEQUENCES. 
 
 iw en 
 
 DT AVL. 2X 
 
 I1., XVIIL., Cor.2 
 eae oa eh, geo 
 
 IV.,XXVIII.,Cor. | 
 
 LV JAA. Coro 
 
 Ve ec LX Cor, 
 
 PV IER: SE 
 
 Eee L Vicaccsensssichoss 
 
 TV Sse Xl Dee ee hd 
 
 If two lines are | They intercept equal ares of 
 parallel chords a circle. 
 
 ora tangent and | 
 
 parallel chord, 
 
 If an angle at the |The angle at the circumfer- 
 
 circumference of ence is equal to half the 
 a circle is sub-| angle at the centre, 
 
 tended by the 
 same are as an 
 angle at the cen- | 
 tre, 
 
 If an angle is in- | Itisa right angle. 
 scribed ina semi- 
 circle, 
 
 if an angle is | It is measured by one half of 
 formed by a tan- the intercepted are. 
 gene and chord, 
 
 If two chords in- | The rectangle of the seg- 
 tersect each oth- | ments of the one, equals the 
 er in a circle, rectangle of the segments 
 
 of the other. 
 
 And if onechord is | The rectangle of the seg- 
 a diameter, and ments of the diameter is 
 the other per- equal to the square on half 
 pendicular to it, the other chord. ind the 
 
 half chord is a mean pro- 
 
 | portional between the seg- 
 ments of the diameter. 
 
 If two secants | The rectangles of each secant 
 meet without the and its external segment 
 circle, | are equal. 
 
 If a secant and | The rectangle of the secant 
 
 tangent meet, | and its external segment is 
 equal to the square on the 
 tangent. And the tangent 
 is a mean proportional be- 
 tween the secant and its 
 external segment, 
 
 If a straight line | The sum of the squares on 
 
 from the vertex the two sides is equal to 
 of a triangle bi- twice the square ot half the 
 secis its base, base increased by twice the 
 square of the bisecting line. 
 
 | If a perpendicular | The square of a side opposite 
 
 be drawn from an acute angle is equal to 
 
 the vertex of a the sum of the squares of 
 triangle to the | the other :ide and the base, 
 base, diminished by twice the 
 
 rectangle of the base and 
 the distance from the ver- 
 tex of the acute angle to 
 the foot of the perpendicu- 
 
 eda 
 
TABLE II.—TRIGONOMETRIC FORMULAL. 
 
 Let A (Fig. 107) = 
 
 AH => ie 
 We then have 
 
 sin A 
 cos A 
 tan A 
 cot A 
 sec A 
 
 cosee A 
 versin A 
 covers A 
 exsec A 
 coexsec A 
 chord A 
 chord 2 A 
 
 In the right-angled triangle ABC (Fig. 107) 
 het Ao = c, 40 = 0,andtbC = a 
 We then have: 
 
 1. sin A == 
 De COSTA = 
 3. tan A = 
 4. cota = 
 5.. sec A ce 
 
 6. cosec A = 
 
 8, exsec4 = -- 
 
 9, covers A = 
 
 10. coexsec 4 = 
 
 angle BAC = arc BF, and let the radius AW’ = AB == 
 
 = [39 
 = B= 2BC 
 
 vers A oe 
 
 TRIGONOMETRIC FUNCTiONS. 
 
 H_ ik G 
 
 AG 
 CRFrasBE 
 BK=AL 
 
 Fie. 107. 
 = cos B 11. a=csinA=dOtanaA 
 =isiy 8 29 b=ccosA=acotaA 
 = cot B 13 C= “ = 
 =i, C 2 oO. ye E = = == 
 sin A cos A 
 = tan i ,o3=i\c.cos B= Peo 
 = cosec B 15). 2 OeSa6 Si =O taets 
 = sec B 48 tel pes : 
 IE ‘ = | \Coseb ia sinwe. 
 Les COVELS 2 17. &@= V(e+b)(e— b) 
 = coexsec B 18 b= ¥(eta)(c—a) 
 = versin B 19. ¢ = Vq2 + p2 
 = exsec B 20°" C= 90° = A+B 
 ab 
 QB: = —_— 
 rea 3 
 
 ae 
 
TABLE IIl.—TRIGONOMETRIC FORMULAE. 
 
 Pe ys yee 
 SOLUTION OF OBLIQUE TRIANGLES. 
 B 
 t 
 | 
 } 
 | | 
 } 
 } it 
 } i 1 
 ft 
 Fie. 108. 1 
 | 
 ay | i 
 | GIVEN. SOUGHT, FORMULA. | 
 22 | A, Boa | C, b,c | C = 180° — (A4-B), b= ing gc in B, | 
 i 
 | i a tf 
 | -= ——- sin(A+B li, 
 | - slu A. ‘ ni 7) Hi 
 | t i 
 Rene r sin A : | 
 | Bers, O. | BO. es bein ie a . C= 180°—-(A+ B), iW 
 } i.) 
 | a : | i 
 ! a iC, ih 
 | | sin 4 ; 
 | 24 ; Gab |4A+B)|/%(44 B)=9°-—YKC i 
 { | 4 
 ae ly, Ka : aes Pa ee 
 25 134(A — B)| tang (4 — B) = ——~“ tan 144(A+ B) if 
 atb iF 
 | 26 i ES A es Ca) ea Ry ha 
 ingee ee R \ 
 | | | B=K(4+L)—-%U4-=B) 
 > 4 Ci OS V4 (u A 1. B) sin V6(A 4 aime B) | 
 27 c le = b)- =(a—b 
 igs | Tana tare) TS sin gee ie 
 i | | } ! 
 28 | | area K=Y¥absin C. } 
 ) ia /(s~b) (s—c) . 
 29 | a,b,c A Lets =le(a+b-+e);sinkgA =4/ pies: a i] 
 | ’ b) } 
 4 
 | pate =o: (s- ~b)(s—c) 
 i 3 2 A= ; tang A= 
 89 | Cos ¥ A= V % /* s (s—a) _ 
 a] | a ee 
 a8 en 4 eV 3 ts a) (s- b)(s — ec). 
 | 31 | |sin A = a : 
 | | 2(s— b) (s—c) |} 
 | | | vers A = — a 
 bc 1] 
 32 | area K = Vs(s — a) (8 — 5) (§—oO) | 
 | 
 ie 407 sin Bio sin C 
 | A, B,C,a | area ea we — | 
 i 
 
TABLE IL.—TRIGONOMETRIC FORMULA. 
 
 GENERAL FORMULA, 
 
 41 
 
 2) 
 ~ 
 
 43 
 
 df 
 
 50 
 
 51 
 
 Or 
 ce) 
 
 Si 4 
 
 Sin 4 p= 
 
 sin A= 
 
 cos A = 
 
 COs A= 
 
 cos A = 
 
 1 sin A - : 
 tan 4 = —-> eae = 4/ sec? A— 1 
 x cot A cos 4 4 
 Yb Wakes /1— cos? A 
 tad hy Bs pean) Soe ee ae 
 / cos? A cos A 
 1— cos? A vers 2.4 
 tan A = ae Lope eee 
 sin 2 4 sin 2 A 
 1 cos A —— 
 @ot A => eS te, 4 CcOser Fis 
 tan A sin A v 
 | 
 sin 2 A sin 2 A 1-+ co 
 cot Ay =~ = = St 
 1 — cos2 # vers 2 4 
 tan 4 A 
 cot A = — 7 
 
 vers A 
 vers A 
 
 exsec A 
 
 cos 4A 
 
 COS 2 x 
 
 1 
 cosec A 
 
 2Qsinlkg AcosZA = 
 
 VW % vers2 
 
 exsec A 
 
 = 1—cosA = 
 
 V¥1—cos?A = 
 
 A= ¥Wi6(1—cos2A) 
 
 4-i-—sin?-_ A —= 
 
 sin Atanigd = 
 
 = exsec 4 cos A 
 
 = gsecA-—1 = 
 
 ie mi 
 / vers A 
 
 ves: +e 
 att ear oa 
 
 = 2sinAc 
 
 os A 
 
 cos? 4 — sin? A 
 
 vers A cot 4 A 
 
 cot Asin A 
 
 ti 
 
 tan AtankwA = 
 
 tan A cos A 
 
 sin? 14 A 
 
 sin 2 2 
 
 1 cos 2.4 
 
 = exsec Acotlg A 
 
 A 
 
 sin 2 A 
 
 2Qsint7 lg Aa 
 
 {1 = 2 sin! 
 
TABLE 
 
 II.—TRIGONOMETRIC FORMULZ. 
 
 | 70, 
 
 . tanA-+tan B= 
 
 GENERAL FORMULA. 
 
 6 CE eel 
 
 a i 
 53, tan}g A =~ we cited A = cat 4a 
 1+ sec A sin 4 
 2tan A 
 54, 2 = soe 
 eres 1 — tan? A 
 ) 
 | sin A 1+ cos A 1 
 Raerpeie fe a UE Con Ae 
 
 vers4 sinA  cosec A—cot A 
 
 cot? A —1 
 aodacot 2 A= —— 
 ae cot 2 Pcot wd 
 ly , Mes 
 Bi, vers A= — 9 Le Sot f= cog 4 
 | 1+ V1—MversA 2+ V2(1+ cos A) 
 58. vers 2. A = 2 sin? A =2sin A cos A tan A 
 59. exsecl4 A = - ace coe & — 
 (1+ cos 4) + ¥2(1 + cos A) 
 60. exsec 2. A = _2 tan? 4 
 1 — tan? A 
 61. sin (4 + B) = sin A.cos B+ sin B.cos A 
 | 62. cos CA’ .5) = cos A eos Basin 42sin PB 
 63. sin A + sin B = 2sin14(A+ B)cosl4(A — B) 
 
 . sin A —sin B= 2cos(A+ 2)sin'4(4 — B) 
 . cos A+ cos B= 2cos (A + B) ccs l4(A — B) 
 . cos B— cos A = 2sin 4(A+ DP)sin 4(A — B) 
 . sin? ys — sin? B = cos? B — cos? A = sin (A + B) sin (4 — B) 
 
 8. cos? A -— sin? B = cos (A + B) cos (A — DB) 
 
 sin (A+ B) 
 cos 4.cos B 
 
 sin (A — B) 
 
 3 4—tan B= 
 ae : cos A .cos B 
 
TABLE III.—CURVE FORMULAE. 
 
 vo 
 
 = 
 
 FORMULA. | 
 ve G0 Hae? 2 
 foe sin 144 D 
 ‘ 50 
 oO D ee re 
 sin 4% R 
 A 1 
 L = 100 —— 
 E D 
 Ate LF 
 or TOO 
 A 
 D = 100 — 
 ” ic 
 
 T=Rtan&a 
 C=2R sin’ a 
 M=Rvers\% aA 
 E= Rexsec 4% a 
 R= Tcot’%a 
 = Ttanl a 
 C=2T cos%a 
 M=T cot a.vers%a 
 
 Oo alee 
 exsec 4 a 
 
 T= Heot%a 
 
 Cu2p Sw 4 
 exsec 4 b 
 M= HcosWYa 
 
 C 
 2sinlg a 
 
 M=%Ctan a 
 
 Cc 
 T = ———~ 
 2cos% a 
 exsec 4 A 
 H-= 1410 ——= = 
 sat yi sin lg a 
 M 
 Prat vers lon 
 C=2Mcot%a 
 r= yw BBB 
 vers 14 A : 
 ee ert 
 | 
 
 cos % A 
 
TABLE IJI.—CURVE FORMULA. 
 
 | 
 | GIVEN. SOUGHT. 
 
 FORMULA. 
 
 R,T | A 
 
 VT? +R 
 C 
 2R 
 
 LY (e+ $) Ge 
 
 Ets 
 
 > | 
 26 || tanga = 
 27 ; 2. [sin W™a = 
 | 
 | | 
 > | ’ | | = 
 28 Ire Oe Ts A |sin 4 a = 
 { | | 
 oC e | ‘ | aos A 
 “| | | cos aA 
 | 
 | | | 
 | H { 
 | US ew «See Ses A | vers 4 A 
 | | | 
 | 
 | 
 31 | * ; |cos ea = 
 | | | 
 82 | R, K | A | A 
 
 | 
 33 | nS - | cos % 
 ye 84 eC a4 A ; cos % 
 385 | : | tan 14 
 
 36 T, i ..| A | tan 14 
 
 cos 4 
 
 | cos 4% 
 | tan 14 
 
 GC. |c= 
 
 ky 
 i 
 
 ae 
 Ce 
 BR 
 Q 
 S 
 
 cos 4 4 
 
 | exsec 14 
 
 ray 
 
 tan 4 a 
 
 7s 
 
 97 
 4) 
 
 M=R— 
 
 | | VT?2+-R2 
 
 ER 
 
 STO 
 / 2TLE 
 
 - 
 Gy A A a ane 
 2M 
 ie 
 C2 — 4M? 
 C2+4M? 
 M 
 ae 
 ‘wai 
 E+M 
 
 ~ V(R+% 0) (R—16C) — 
 
 Orv 
 
 8 
 
 v8 
 
 C 
 
 9 
 a 
 
 ) 
 
TABLE UI.—CURVE FORMULA 
 
 | 
 GIVEN. | SOUGHT. FORMULA. 
 | | 
 tere : | _RyM@R-M) 
 48 R,M | T T = fe ol, 
 49 ue | 24 C= 2VM2R —M) 
 RM 
 4 se | Ph _ == — 
 | i eam ae pag iees 
 | 51 R, E T T= VEQR+E) 
 } | 
 9 R V E2R any 
 KO 66 1 ee 
 fal | . ‘Es R+E 
 Bé | ‘ I M= 
 53 M RLE 
 | er 
 id } R = — —_——--—---— a 
 if} 54 T C | R Vv 27 4 C) Br C) 
 } 
 ! Jor 
 Mei Sie CO 
 Mil) 55 fs M |\M=16C 
 eaatealti 2 | x 2T4 
 i | |  sric 
 Het ee | 5 Oy eae oe 
 Hg So ‘ E |#=1 ‘4/ STAC. 
 Hl | | T+E)(P 
 i Bi mle oe ey ie re eed : ne ae 
 i | | | ol 22 (r? — Ey | 
 es aie 58 6 | Y a / 
 ai : be 24 72 
 : Hi | E ( jaws Ls E?) 
 \ BC “ mi LM == 
 59 | M | T2462 
 Hi | 
 th | M2 14 C)2 
 \) 60 C, M ee Ns a) 
 Hi 2. AL 
 i} c(c2+ 4M?) 
 i 61 “ T eens 
 | s BBs (C2 — 4M?) 
 | 2 
 | 62 “ E |= ar 
 RO 7 > ee es EM 
 | 03 M, Hf R tay = Per ae M 
 a | | /E+M 
 ii | 64 6é | 7 = E 3 Van 
 i il} | | S | V H-M 
 vi | | “E+M | 
 bal | 65 aS | C CO =2M 4/ Py 
 i | | A 2 19 
 ] e664 Pa. R R3 — FR? seg et + RT? —14 MT2=0 
 HM 67 “ E E84. 1? M— ET? -4.MT2=0 
 68 “i C 1Cc:+270214M2C0—8M2T=0 
 4H2%—C? C7 Wor 
 | 28 i pet te eR = = 
 69 OF Fe | R R3+ Bh “e R= . 0 
 | #0 is fe | 273 — T? C—2 TE?—CE2=0 
 | | C2. ©23 
 at} M | ee 
 
 | 3 M2H# .- TI ce 
 Met B+ Me, j 
 
 279 
 
TABLE IV.—RADII, LOGARITHMS, OFFSETS, ETC. 
 
 ‘oe = eal 
 
 | | ; | 
 rid adine | Woga- | Tan. | Mid. || i : Loga- | Tang. | Mid. 
 Deg. Ber gheae rithm. | Off. | Ord. || Deg. | Radius. | jithm. Of | Ord. 
 ; D. R. log. R. t. m. D. R. log. R. t. | m. | 
 0° 0’ | Infinite | Infinite | .000 | .000 | 1° 0’) 5729.65 | 3.758128 | .873} .218 - 
 1 | 343775. | 5 5386274 | .015 | .001 || 1 | 5685.72 750950 887 | .222 | 
 2 | 171887. | 5.235244 | .029 | .007 2 | 5544.83 . 743888 902 | 225 
 3 | 114592. | 5.059153 | .044 | .011 3) 5456.82 | 1736939 | .916 | .229 | 
 4 | 85943.7 | 4.934214 | .058 | .015 4 | 5371.56 . 730100 931 | .233 
 5 | 68754.9 837304 | .073 | .018 5 | 5288.92 (23367 .945 | 236 HW 
 6 | 57295.8 | .753123 | .087 | .022 6 | 5208.7 716787 .960 | .240 Ht 
 7 | 49110.7 .691176 | .102 | .025 7 | 5181.05 . 710206 974 | .244 | Ht 
 8 | 42971.8 .633184 | .116 | .029 8 | 5055.59 708772 .989 | .247 Wii 
 9 | 38197.2 | .582031 | .131 | .033 9 | 4982.33 | 697432 | 1.004 | .251 It 
 10 | 34377.5 | 4.586274 | .145 | .036 10 | 4911.15 | 3.691183 | 1.018 | .255 | i 
 11 | 31252.3 | 4.494881 | .160 | .040 || 11 | 4841.98 | 3.685023 | 1.033 | .258 
 12 | 28647.8 .457093 | .175 | .044 || 12 | 4774.7 .678949 | 1.047 | .262 
 18 | 26444.2 .422331 | .189 | .047 || 13 | 4709.33 .672959 | 1.062 | .265 i 
 14 | 24555.4 | .390146 | .204 | .051 14 | 4645.69 .667051 | 1.076 | .269 i 
 15 | 22918.3 |  .360183 | «R18 | .055 15 | 4583.75 .661221 | 1.091 | .273 i 
 16 | 21485.9 332154 | .233 | .658 16 | 4523.44 .655469 | 1.105 | .27 | 
 17 | 20222.1 . 805825 | .247 | .062 17 | 4464.70 .649792 | 1.120 | .280 i 
 18 | 19098.6 -281002 | .262 | .065 18 | 4407.46 .644189 |-1.184 | .284 i 
 19 | 18093.4 .257521-| .276 | .069 19 | 4351.67 .638656 | 1.149 | .287 i 
 20 | 17188.8 | 4.235244 | .201 | .07 20 | 4297.28 | 3.633194 | 1.164 | .291 i 
 21-| 16370.2 | 4.214055 | .3805 | .076 21 | 4244.23 | 3.627799 | 1.178 | .295 i 
 22 | 15626.1 .193852 | .320 | .080 2 | 4192.47 .622470 | 1.193 | .298 i 
 23 | 14946.7 | .174547 | .335 | .084 23 | 4141.96 .617206 | 1.207 | .302 it 
 24 | 14323.6 .156064 | .3849 | .087 || 24 | 4092.66 .612005 | 1.222 | .305 i 
 25 | 18751.0 | .138335 | .3864 | .091 || 25 | 4044.51 .606866 | 1.2386 | .309 i 
 26 | 13222.1 .121302 | .378 | .095 26 | 3997.49 .601787 | 1.251 | .313 i 
 27 | 127382.4 .104911 | .393 | .098 || 27 | 3951.54 .596766 | 1.265 | .316 i 
 28 | 12277.7 .089117 | .407 | .102 || 28 | 3906.54 .591803 | 1.280 | .320 
 29 |! 11854.3 073877 |. .422 | .105 || 29 | 3862.7 .586K96 | 1.294 | 824 j 
 80 | 11459.2 | 4.059154 | .436 | .109 || 30 | 3819.83 | 3.582044 | 1.309 | .327 
 31 | 11089.6 | 4.044914 | .451 | .118 || 31 | 8777.85 | 3.577245 | 1.824 | .331 
 32 | 10743.0 .031125 | .465 | .116 32 | 3736.7 .572499 | 1.338 | .335 
 33 | 10417.5 .017762 | .480 | .120 || 33 | 3696.61 .567804 | 1.353 | .338 
 84 | 10111.1 | 4.004797 | .495 | .124 34 | 3657.29 .563160 | 1.367 | .342 | 
 85 | 9822.18 | 3.992208 | .509 | .127 | 35 | 8618.80 .508564 | 1.3882 | .345 
 36 | 9549.34 979973 | .524 | .181 | 36 | 3581.10 .554017 | 1.3896 | .349 
 37 | 9291.29 .968074 | .588 | .135 37 | 3544.19 .549517 | 1.411 | .353 ) 
 38 | 9046.75 956493 | .553 | 1138 38 | 3508.02 .545063 | 1.425 | .356 
 39 | 8814.7 945212 | .567 | .142 || 39 | 3472.59 .540654 | 1.440 | .360 
 40 | 8594.42 | 3.934216 | .582 | .145 4G | 3487.87 | 3.536289 | 1.454 | .364 ig 
 41 384.80 | 3.923493 | .596 | .149 i | 3403.83 | 3.531968 | 1.469 | .367 
 42 | 8185.16 | .913027 | .611 | .153 42 | 3370.46 .527690 | 1.483 | .3871 
 43 | 7994.81 .902808 |. .625 | .156 43 | 3337.7 .523453 | 1.498 | .875 
 44 | 7813.11 .892824 |. .640 | .160 || 44 | 3305.65 .519257 | 1.513; .37 
 45 | 7639.49 .883065 | .654 | .164 45 | 3274.17 .515101 | 1.527 | .382 
 46 | 7473.42 | .873519 | .669 | .167 46 | 8243.25 .510985 | 1.542 | .385 
 47 | (314.41 .864179 | .684 | .171 47 | 3212.98 .506908 | 1.556 | .3889 
 48 | 7162.03 | .855036 | .698 | .174 48 | 3183.23 502868 | 1.571 | .393 | 
 49 | 7015.87 | .846082.| .713 | <178 | 49 | 3154.03 .498866 | 1.585 | .396 
 50 | 6875.55 | 3.887308 | .727 | .182 | 50 | 3125.36 | 3.494900 | 1.600 | .400 . 
 51 | 6740.74 | 3.828708 | .742 | .185 51 | 3097.20 | 3.490970 | 1.614 | -.404 
 52 |#$611.12 .820275 | .756 | .189 52 | 3069.55 487075 | 1.629 | .407 
 53 | 6486.38 .812002 | .771 | .198 53 | 3042.39 .483215 | 1.643 | .411 
 54 | 6366.26 .803885 | .785 | .196 54 | 3015.7 .479389 | 1.658 | .414 
 55 | 6250.51 .795916 | .800 | .200 55 | 2989.48 .475596 | 1.673 | .418 
 56 | 6138.90 788091 | .814 | .204 56 | 2963.7 .471836 | 1.687 | .422 
 57 | 6031.20 | .780404 | .829 | .207 57 | 2938.39 -468109 | 1.702 | .425 
 58 | 5927.22 772851 | .844 | .211 58 | 2913.49 .464413 | 1.716 | .429 ? 
 59 | 5826.76 765427 | .B58 | .215 59 | 2889.01 460749 | 1.731 | .433 ! 
 60 | 5729.65 | 3.758128 | .873 | .218 60 | 2864.93 | 3.457115 | 1.745 | .4386 
 
| 2728.52 
 
 Radius. 
 
 R. 
 
 2864.93 
 2841.2 
 2817. 
 2795. 
 2772.8 
 
 2750. ¢ 
 
 2707. 
 2685. 
 2665. 
 2644. 
 2624. 
 2604. 
 2584. 9: 
 2565. 
 2546. € 
 2527. 
 2509. 
 2491.2 
 2473.: 
 2455.7 
 2438, 
 2421. 
 2404. 
 2387. 
 2371. 
 2854.8 
 2338.7) 
 
 2322.98 
 2307.39 
 2292. 01 
 
 1910.08 
 
 ee 
 
 Loga- 
 rithm. 
 
 log. R. 
 
 3 457115 
 453511 
 449937 
 . 446392 
 -442876 
 -439388 
 485928 
 .482495 
 -429089 
 -425710 
 
 3.422356 
 
 3.419029 
 415727 
 412449 
 -409197 
 
 -405968 . 
 
 -402763 
 399582 
 -896424 
 393289 
 390176 
 3887085 
 384016 
 - 080969 
 3877943 
 -874938 
 .3(1954 
 - 368990 
 . 366046 
 . 3863122 
 860217 
 8573382 
 354466 
 .851618 
 348789 
 845979 
 843187 
 .840412 
 .887655 
 .334916 
 3.332193 
 
 3.329488 
 326799 
 324127 
 821471 
 318832 
 . 316208 
 -313600 
 -811008 
 808431 
 
 3.305869 
 
 3.303323 
 -800791 
 298274 
 -295771 
 293283 
 290809 
 
 co cw 
 
 Co co 
 
 "288349 
 
 285902 
 283470 
 8.281051 
 
 WWW*W WWW Ree Prk ek pe ek ee ee ee pe pe 
 Jeo) 
 se 
 Ve) 
 
 0 0 09 09 0 
 — 
 Or 
 © 
 
 2.487 
 
 TH WW WWW WWW 
 orore * 
 oS 
 oO 
 
 : Loga- 
 Deg. | Radius. | rithm: 
 D. R. log. R. | 
 | | | 
 | 3° 0’! 1910.08 | 3.281051 
 1 | 1899.53 218646 
 2 | 1889.09 206253 
 3 | 1878.77 278804 
 4 | 1868.56 | 271508 | 
 5 | 1858.47 | .269155 
 6 | 1848.48 | .266814 
 7 | 1888.59 - 204486 
 8 | 1828.82 .262170 
 9 | 1819.14 .259867 
 10 | 1809.57 | 3.257576 
 11 | 1800.10 | 8.255296 
 12 | 1790.7 .253029 
 13 | 1781.45 20077 
 14 | 1772.27 - 2485380 
 15 | 1763.18 246297 
 16 | 1754.19 - 244077 
 17 | 1745.26 -241867 
 18 | 1736.48 239669 
 19 | 1727.75 .237481 
 20 | 1719.12 | 3.235805 
 21 | 1710.56 | 3.233140 
 22 | 1702.10 230985 
 23 | 1693.75 . 228841 
 24 | 1685 .42 .226707 
 25 | 1677.2 . 224584 
 26 | 1669.06 222472 
 27 | 1661.00 - 220369 
 28 | 1653.01 218277 
 29 | 1645.11 -216195 
 80 | 1637.28 | 3.214122 
 31 | 1629.52 | 8.212060 
 32 | 1621.84 .210007 
 33 | 1614.22 - 207964 
 34 | 1606.68 . 205930 
 85 | 1599.21 . 203906 
 86 | 1591.81 .201892 
 87 | 1584.48 . 199886 
 88 | 1577.21 . 197890 
 89 | 1570.01 . 195903 
 40 | 1562.88 | 3.193925 
 41 | 1555.81 | 8.191956 
 42 | 1548.80 . 189996 
 43 | 1541.86 . 188045 
 44 | 1534.98 . 186103 
 45 | 1528.16 . 184169 
 46 | 1521.40 182244 
 47 | 1514.7 - 180327 
 48 | 1508.06 .178419 
 49 | 1501.48 .176519 
 50 | 1494.95 | 3.174627 
 51 | 1488.48 | 8.172744 
 52 | 1482.07 . 170868 
 53 | 1475.7 . 169001 
 54 | 1469.41 . 167142 
 55 | 1463.16 . 165291 
 56 | 1456.96 .163447 
 7 | 1450.81 . 161612 
 58 | 1444.72 . 159784 
 59 | 1438.68 . 157963 
 60 | 1432.69 | 8.156151 
 
 TABLE IV.—RADII, LOGARITHMS, OFFSETS, ETC. 
 
 ‘ | 
 Tang. | Mid. 
 
 | 
 | 
 
 Off. | Ord. | 
 tit at 
 2.618 | .654 
 2.652 | 658 
 2.647 | 662 
 2.661 | 665 
 2.676 | 669 | 
 2.690 | .673 
 2.705 | .676 
 2.7719 | .680 
 2.734 | .684 
 2.749 | .687 
 2.763 | 691 | 
 2.778 | .694 
 2.792 | .698 
 2.807 | .702 
 2 821 | .705 
 2.836 | .709 
 2.850 | .713 
 2.865 | .716 
 2.879 .| .720 
 2.894 | .723 
 2. 727 
 
 2. 
 2.938 | .734 
 2.952 | .738 
 2.967 | .742 
 2.981 | .745 
 2.996 | .749 
 3.010 | .%53 
 3.025 | .756 
 3.039 | .760 
 3.054 | .763 
 3.068 | .767 
 3.083 | .77 
 
 3.127 | |782 
 3.141 | 1785 
 3.156 | .789 
 3.170 | 793 
 3.185 | _796 
 3.199 | 800 
 3.214 | .808 
 3.228 | 807 
 3.243 | “811 
 3.657 | .814 
 3.972 | 818 
 3.286 | 822 
 3.301 | .825 
 3316 | _829 
 3.330 | _832 | 
 3.345 | _836 
 3.359 | .840 
 3.374 | 848 
 3.388 | 1847 
 3.403 | 1851 
 8.417 | .854 
 3.432 | “858 
 3.446 | “962 
 3.461 | |865 
 3.475 | _869 
 3.490 | |872 
 
~~ 
 
 LIL, 
 
 IS ) 3 
 TABLE IV,—RADI, LOGARITH 
 
 OFFSETS, ETC. 
 
 WCMOIOor war cS 
 
 Pasa <p Sica iasameente ai aan icbeccs heh ts eoiibiicablaiiaceee rare ME Soe eon a 
 
 5 | 1348, 
 
 03 | 1173.65 
 
 5¢ | 1157.8 
 
 ar by Loga- 
 g. _ Radius. rithm. -| 
 R. | log. R. | 
 1432.69 3.156151 
 1426.74 | .154346 ! § 
 1420.85 + .152548 
 } 1415.01 | .150758 | ¢ 
 1409.21 | .148975.) § 
 1403.46 , .147200 | : 
 1397.76 | .145431 | 
 | 1892.10 | .143670 | § 
 1386.49 | .141916 
 1380.92 | 140170 
 | 1875.40 |3,138430 
 1369.92 {3.136697 
 1364.49 | .184971 
 1359.10 | .138251 
 1353 .7 . 181539 
 45 | 129833 
 1843.15 | .128134 
 1337.65 | .126442 
 1332.77 | .124756 
 1327.63 | .123077 
 1322.53 |3 121404 
 1317.46 |8.119738 
 1312.43 | .118078 
 1307.45 | .116424 
 1302.50 | .114777 
 1297.58 | .113136 
 1292.7 .111501 
 1287.87 | .109872 
 1283.07 | .108249 
 1278.30 | .106632 
 1273.57 |3.105022 
 1268.87 |3.103417 
 1264.21 | .101818 
 1259.58 | .100225 
 1254.98 | .098638 
 1250.42 | .097057 
 1245.89 | .095481 
 1241.40 | .093912 
 1236.94 | .092347 
 1282.51 | .090789 
 1228.11 |3.089236 
 1223.74 |3.087689 
 1219.40 | .086147 
 1215.30 | .084610 } 
 1210.82 | .083079 
 1206.57 | .081553 
 1202.36 | .080033 
 1198.17 | .078518 
 | 1194.01 | .077008 
 9 | 1189.88 | .075504 
 | 1185.78 |8.074005 
 1181.71 |3.072514 
 66 | .071022 
 | 65 | .069538 
 | 1169.66 | .068059 
 | 1165.7 . 066585 
 1161.76 | .065116 
 85 | 063653 
 1158.97 } .062194. 
 1150.11 | .060740 
 1146.28 |3.059290 
 
 ES 
 
 bet et ee hh pp Ph fed fk fe fk feed 
 
 28 
 
 | Mid. || H : ! Loga- | Tan 2 
 | Ord. °|| Deg. | Radius. | rithm. Of 
 ma. R. | log. R. t. 
 872 || 1146 . 28 I3. 059290 | 4.362 
 O76 || | 1142.47 | .057846 | 4.276 
 . 880 | 1188.69 | .056407 | 4.391 
 .883 i 1184.94 | .054972 | 4.405 
 .887 | 1181.21 | ,053542 | 4.490 
 891 1177.50 | .052116 | 4.435 
 894 ' 1123.82 | .050696 | 4.449 
 .898 1120.16 | .049280 | 4.464 
 . 902 1116.52 | .047868 | 4.478 
 . 905 | 1112.91 | .046462 | 4.493 
 . 909 1109.83 3: 045059 | 4.507 
 .912 | 1105.76 3.043662 4,522 
 .916 2 | 1102.22 | .042268 | 4.536 
 920 1098.7 .040880 | 4.551 
 923 1095.20 | .039495 | 4.565 
 927 1091.73 | .038115 | 4.580 
 931 | 1088.28 | .036740 | 4.594 
 934 1084.85 | .035368 | 4.609 
 988 j; 1081.44 | .034002 | 4.623 
 942 1078.05 | .032639 | 4.638 
 945 1074.68 '3.031281 | 4.653 
 .949 1071.34 {3.029927 | 4.667 
 952 1068.01 | .028577 | 4.682 
 956 1064.71 | .027231 | 4.696 
 .960 1061.43 | .025890 | 4.711 
 963 | 1058.16 | .024552 | 4.725 
 967 1054.92 | .028219 | 4.740 
 971 1051.7 .021890 | 4.754 
 974 1048.48 | .020565 | 4.769 
 978 1045.31 | .019244 | 4.783 
 982 1042.14 |3.017927 | 4.798 
 985 1039.00 |3.016614 | 4.812 
 . 989 1085.87 | .015305 | 4.827 
 993 1032.76 | .018999 | 4.841 
 . 996 1029.67 | .012698 | 4.856 
 .000 1026.60 | .011401 | 4.87 
 003 1023.55 | .010107 | 4.885 
 007 1020.51 | .008818 | 4.900 
 O11 | 1017.49 | .007532 | 4.914 
 014 1014.50 | .006250 | 4.929 
 .018 1011.51 |8.004972 | 4.943 
 22 1008.55 {3.003698 | 4.958 
 025 1005.60 | .002427 | 4.972 
 029 1002.67 |3.001160 | 4.987 
 032 999.762 {2.999897 | 5.001 
 036 996.867 | .998637 | 5.016 
 .040 993.988 | .997381 | 5.03 
 .043 991.126 | .996129 | 5.045 
 047 988.280 | .994880 | 5.059 
 051 | 985.451 | .993635 | 5.07 
 054 | 982.638 |2.992393 | 5.088 
 .058 | 979.840 |2.991155 | 5.103 
 062 | 977.060 | .989921 | 5.117 
 065 | 974.294 | .988690 | 5.132 
 .069 | 971.544 | .987463 | 5.146 
 073 968.810 | .986238 | 5.161 
 076 | 966.091 | .985018 | 5.175 
 .080 | 963.387 | .988801 | 5.190 
 083 | 960.698 | .9825 87 5.205 
 .088 | 958. 025 | .98137 5.219 
 091 | 955.366 2: 980170 5.234 
 
 Dora 
 
TABLE IV.—RADII, 
 
 LOGARITHMS, OFFSETS, ETC. 
 
 283 
 
 ; | | 
 [eg Ay ‘nee 
 Heke Loga- | Tang.| Mid. || : Loga- | Tang.) Mid 
 | Deg. | Radius.) yithm. | Off | Ord. | Pes: | Radius. | rithm. | “Off.” | Ord 
 | D: R. log. BR. |. | mas |] DL | RL) PogORR. | | me 
 | | | | 
 G° 0’! 955.366 2.980170 | 5.234 | 1.309 || *7°-0 | 819.020 |2.918295 | 6.105 | 1.528 
 1 | 952.722 | .979066 | 5.248 | 1.318 | 1 | 817.077 | .912263 | 6.119 | 1.531 
 2 | 950.093 | .977766 | 5.263 | 1.317 2 | 815.144 | .911234 | 6.134 | 1.535 
 3 | 947.478 | 976569 | 5.277 | 1.820 3 | 813.238 | .910208 | 6.148 | 1.539 
 4 | 944.877 | 975375 | 5.202 | 1.824 || 4 | 811.803 | .909183 | 6.163 | 1.548 
 5 | 942.291 | 974185 | 5.206 | 1.327 || 5 | 809.397 | 908162 | 6.177 | 1.546 
 6 | 939.719 | .972998 | 5.321 | 1.331 6 | 807.499 | .907142 | 6.192 | 1.550 
 7 | 937.161 | .971814 | 5.385 | 1.385 7 | 805.611 | .906125 | 6.206 | 1.553 
 8 | 934.616 | .970633 | 5.850 | 1.358 8 | 803.731 | 905111 | 6.221 | 1.557 
 9 | 932.086 | 969456 | 5.364 | 1.342 |) —-9 | 801.860 | 904098 | 6.236 | 1.561 
 10 | 929.569 |2.968282 | 5.379 | 1.346 || — 10 | 799.997 2.903089 | 6.250 | 1.564 
 11 | 927.066 2.967111 | 5.393 | 1.349 || 11 | 798.144 [2.902081 | 6.265 | 1.568 
 12 | 924.576 | 965943 | 5.408 | 1.358 || 12 | 796.299 | .901076 | 6.279 | 1.572 
 13 | 922.100 | .964778 | 5.422 | 1.356 || 13 | 794.462 | .900073 | 6.204 | 1.575 
 14 | 919.637 | .963616 | 5.437 | 1.360 || 14 | 792.634 | .899073 | 6.308 | 1.579 
 | 15 | 917.187 | .962458 | 5.451 | 1.364 || 15 | 790.814 | .898074 | 6.823 | 1.582 
 a 16 | 914.750 | .961303 | 5.466 | 1.368 16 | 789.003 | .x97078 6.887 | 1.586 
 i. 17 | 912.326 | .960150 | 5.480 | 1.37 17 | 787.210 | .896085 | 6.352 | 1-590 
 Wei 18 | 909.915 | .959001 | 5.495 | 1.875 || 18 | 785.405 | .895094 | 6.366 | 1.593 
 tu 19 | 907.517 | _957855 | 5.510 | 1.378 || 19 | 783.618 | .804105 | 6.881 | 1.597 
 Re 20 | 905.131 2.956711 | 5.524 | 1.382 | 20 | 781.840 [2.893118 | 6.395 | 1.600 
 i oi | 902.758 2.955571 | 5.539 | 1.386 || 21 | 780.069 |2.892133 | 6.410 | 1.604 
 | 92 | 900.397 | .954434 | 5.553 | 1.389 92 | 778.307 | .891151 | 6.424 | 1.608 
 Hh, 93 | 898.048 | .953300 | 5.568 | 1.393 23 | 776.552 | .890171 | 6.439 | 1.611 
 li 94 | 895.712 | .952168 | 5.582 | 1.397 || 24 | 774.806 | .889193 | 6.453 | 1.615 
 ie 95 | 893.388 | 951040 | 5.597 | 1.400 || 25 | 773.067 | .888217 | 6.468 | 1.619 
 il 96 | 891.076 | .949915 | 5.611 | 1.404 || 26 | 771.336 | .887244 | 6.482 | 1.628 
 | a7 | 888.776 | 948792 | 5.626 | 1.407 27 | 769.613 | .886272 | 6.497 | 1.626 
 ih 98 | 886.488 | .947673 | 5.640 | 1.411 28 | 767.897 | .885303 | 6.511 | 1.630 
 ial! 99 | 884.211 | .946556 | 5.655 | 1.415 29 | 766.190 | .884336 | 6.526 | 1.638 
 Vi 30 | 881.946 2.945442 | 5.669 | 1.418 30 | 764.489 2.888871 | 6.540 | 1.637 
 i 31 | 879.693 (2.944331 | 5.684 | 1.422 || 31 | 762.797 |2.882409 | 6.555 | 1.641 
 Hil 32 | 877.451 | .943223 | 5.698 | 1.426 || 32 | 761.112 | .881448 | 6.569 | 1.644 
 | 33 | 875.221 | .942118 | 5.713 | 1.420 33 | 759.434 | .880490 | 6.584 | 1.648 
 | 34 | 873.002 | 941015 | 5.727 | 1.438 || 34. | 757.764 | .879534 | 6.598 | 1.651 
 35 | 870.795 | .939916 | 6.742 | 1.437 35 | 756.101 | .878580 | 6.613 | 1.655 
 36 | 868.598 | .938819 | 5.756 | 1.440 36 | 754.445 | .877627 | 6.627 | 1.659 
 37 | 866.412 | .937725 | 5.771 | 1.444 7 | 752.796 | 876678 | 6.642 | 1.662 
 38 | 864.238 | 936633 | 5.785 | 1.447 38 | 751.155 | .8757B0 | 6.656 | 1.666 
 39 | 862.075 | .935545 | 5.800 | 1.451 39 | 749.521 | .874784 | 6.671 | 1.67 
 40 | 859.922 (2.934459 | 5.814 | 1.455 40 | 747.804 [2.873840 | 6.685 | 1.673 
 41 | 857.780 '2.983876 | 5.829 | 1.458 41 | 746,274 |2.872898 | 6.700 | 1.677 
 42 | 855.648 | .932295 | 5.844 | 1.462 42 | 744.661 | .871959 | 6.714 | 1.680 
 | 43 | 853.527 | .981218 | 5.858 | 1.466 43 | 743.055 | .871021 | 6.729 | 1.684 
 44 | 851.417 | .930142 | 5.873 | 1.469 44 | 741.456 | .870086 | 6.743 | 1.68 
 45 | 849.317 | .929070 | 5.887 | 1.473 45 | 739.864 | 869152 | 6.758 | 1.601 
 | 46 | 847.228 | .928000 | 5.902 | 1.476 46 | 738.979 | .868221 | 6.773 | 1.695 
 | 47 | 845.148 | .926933 | 5.916 | 1.480 47 | 736.701 | .867291 | 6.787 | 1.699 
 48 | 843.080 | 925869 | 5.931 | 1.484 48 | 735.129 | 1866363 | 6.802 | 1.702 
 49 | 841.021 | 924807 | 5.945 | 1.487 49 | 733.564 | 1865438 | 6.816 | 1.706 
 50 | 888.972 [2.928747 | 5.960 | 1.491 || 50 | 732.005 [2.864514 | 6.831 | 1.710 
 | 51 | 836.933 2.922691 | 5.974 | 1.495 Bi | 730.454 [2.863593 | 6.845 | 1.718 
 52 | 834.904 | .921637 | 5.989 | 1.498 || 52 | 728.909 | 862673 | 6.860 | 1.717 
 53 | 832.885 | .920585 | 6.003 | 1.502 || 53 | 727.870 | .861755 | 6.874 | 1 720 
 54 | 830.876 | .919536 | 6.018 | 1.505 54 | 725.838 | 860840 | 6.889 | 1.72. 
 55 | 828.876 | .918489 | 6.082 | 1.510 Bb | 724.312 | .859926 | 6.903 | 1.728 
 56 | 896.886 | 917446 | 6.047 | 1.513 || 56 | 722.793 | 859014 | 6.918 | 1.731 
 57 ) 824.905 | .916404 | 6.061 | 1.517 BY | 721.280 | /858104 | 6.932 | 1.735 
 58 | 822.934 | .915365 | 6.076 | 1.520 || 58 | 719.774 | .857196 | 6.947 | 1.739 
 59 | 820.973 | .914329 | 6.090 | 1.524 59 | 718.273 | 1856290 | 6.961 | 1.742 
 i “020 |2.913295 | 6.105 | 1.528 60 | 716.779 121855385 | 6.976 | 1.746 
 
Loga- 
 | rithm 
 
 log. R 
 
 TABLE IV.—RADI, LOGARITHMS, OFFSETS, ETC. 
 
 Tang | Mid 
 Ord 
 
 m. 
 
 8° 0 | 716.779 |2.855385 
 1°} 715.291 “854483 
 2 | 713.810 | .853583 | 
 3 | 712.335 | “Boek | 
 4 | 710.865 | .851787 
 5 709.402 | .850892 
 6 | 707.945°>| .849999 | 
 7 | 706.493 | 849108 | 
 8 | 705.048 | .848219 
 9 | 703.609 | .847331 | 
 : 10 | 702,175 |2.846445 | 
 11 | 700.748 |2.845562 
 : 12 | 699.326 | .844679 
 ) 18 | 697.910 | .843799 
 : 14 | 696.499 | .842991 
 : 15 | 695.095 | .842044 
 16 | 693.696 | .841169 
 17 | 692.302 | .840296 
 : 18 | 690.914 | .839424 
 19 | 689.532 | .838555 
 : 20 | 688.156 |2.837687 
 : 21 | 686.785 |2.836821 
 92 | 685.419 -835956 
 23 | 684.059 | .83 
 24 | 682.704 | 834232 
 25 | 681.354 | .83337 
 : 26 | 680.010 "9305415 
 : 27 | 678.671 | .831G660 
 : 28 | 677.338 | .S80805 
 : 29 | 676.008 | .829953 
 30 | 674.686 /2.829102 
 31 | 673.369 |2.828253 
 32 | 672.056 | .827405 
 
 83 | 670.748 | 826560 
 
 -825715 
 
 35, | 668.148 | .82487: 
 36 | 666.856 | .824032 
 37 | 665.568 | .823193 
 38 | 664.286 | 829355 
 89 | 663.008 | .821519 
 40 | 661.736 |2.820685 
 41 | 660.468 |2.819852 
 42 | 659.205 | .819021 
 43 | 657.947 | .818191 
 44 | 656.694 | .817363 
 45 | 655.446 | 816537 
 46 | 654.202 | .815712 
 47 | 652.963 | .814889 
 48 | 651.729 | .814067 
 49 | 650.499 | .813247 
 
 U., $e | 
 50 | 649.274 {2.812428 
 | 2.811611 
 
 92 | 646.838 | .810796 
 53 | 645.627 | .809982 
 54 | 644.420 | .809169 
 55 | 643.218 | .808358 
 56 | 642.021 | .807549 
 57 | 640.828 | .806741 
 58 | 639.639 | .805935 
 59 | 638.455 | .805130 
 
 60 | 687.275 |2.804327 
 
 WNWNWNWWWNWWWd Sip ees oe 
 
 WNWWNWNWNWWWWWD 
 
 ND W W ®W ® ® W 
 
 Tang.| Mid. | Loga- 
 On Ord. | Deg. | ‘Radius rithm Oft 
 | 
 leikte || Gms) ap. || aan.) ietdetes. |cate, 
 6.976 | 1.746 | -9° 0'| 637.275 |2.804827 | 7.846 
 6.990 | 1.%49 | 1 |-636.099 | .803525 | 7.860 
 7.005 | 1.753 | 2 | 634.928 | .802724 | 7.875 
 7.019 |1.756 3 | 633.761 | .801926 | 7.889 
 7.084 | 1.761 4 | 632.599 | .801128 | 7.904 | 
 7.048 | 1.764 | 5 | 631.440 | .800382 | 7.918 
 7.063 | 1.768 | 6 | 630.286 | .799538 | 7.933 | 
 007 | 1.771 7 | 629.186 | .798745 | 7.947 | 
 7.092 | 1.775 8 | 627.991 | .797953 | 7.962 | 
 7.106 | 1.778 9 | 626.849 | .797163 | 7.976 
 7.121 | 1.782 10 | 625.712 |2.796374 | 7.991 | 
 7.135 | 1.786 11 | 624.579 |2.795587 | 8.005 
 7.150 | 1.790 12 | 623.450 | .794801 | 8.020 
 7.164 | 1.793 | 13 | 622.325 | .794017 | 8.084 
 7.179 | 1.797 | 14 | 621.203 | .793234 | 8.049 | 
 7.193 | 1.801 15 | 620.087 | .792453 | 8.063 | 
 7.208 | 1.804 | 16 | 618.974 | .791673 | 8.078 
 7.222 | 1.807 | 17 | 617.865 | .790894 | 8.092 
 7.237 | 1.811 | 18 | 616.760 | .790117 | 8.107 | 
 7.251 | 1.815 | 19 | 615.660 | .789341 | 8.121 
 7.266 | 1.819 | 20 | 614.563 |2.788566 | 8.136 
 7.280 | 1.822 | 21 | 613.470 |2.787793 | 8.150 
 7.295 | 1.826 | 22 | 612.380 | .787021 | 8.165 | 
 7.309 | 1.829 | 23. | 611.295 | .786251 | 8.179 | 
 7.824 | 1.833 | 24 | 610.214'| .785482 | 8.194 | 
 7.888 | 1.837 | 25 | 609.136 | .784714 | 8.208 | 
 7.353 | 1.840 | 26 | 608.062 | .783948 | 8.223 
 7.867 | 1.844 27 | 606.992.| .783183 | 8.237 | 
 7.882 | 1.848 28 | 605.926 | .782420 | 8.252 
 7.896 | 1.851 29 | 604.864 | .781657 | 8.266 
 7.411 | 1.855 30 | 603.805 |2.780897 | 8.281 
 7.425 | 1.858 31 | 602.750 |2.780137 | 8.295 
 7.440 | 1.862 | 32 | 601.698 | .7'79 379 8.310 
 7.454 | 1.866 | 33 | 600.651 | .778622 | 8.324 
 7.469 | 1.869 34 | 599.607 | .777867 | 8.389 | 
 7.483 | 1.873 | 35 | 598.567 | .777112 | 8.353. | 
 7.598 | 1.877 | 36 | 597.530 | .776360 | 8.368 | 
 7.512 | 1.880 | 37 | 596.497 | .775608 | 8.382 
 7.527 | 1.884 | 38 | 595.467 | .774858 | 8.397 
 7.541 | 1.887 | 39 | 594.441 | .774109 | 8.411 | 
 7.556 | 1.892 40 | 593.419 /2.773361 | 8.426 
 7.570 | 1.895 41 | 592.400 |2.772615 | 8.440 
 7.585 | 1.899 42 | 591.884 | .771870 | 8.455 
 7.599 | 1.908 43 | 590.872 | .771126 | 8.469 
 7.614 | 1.906 44 | 589.364 | .770383 | 8.484 
 7.628 | 1.910 45 | 588.359 | .769642 | 8.498 
 7.643 | 1.914 46 | 587.357 | .768902 | 8.513 
 7.657 | 1.918 47 | 586.359 | .768164 | 8.527 
 7.672 | 1.921 48 | 585.364 | .767426 | 8.542 
 7.686 | 1.924 | 49 | 584.373 | .766690 | 8.556 | 
 7.701 | 1.928 50 | 583.385 |2.765955.| 8.571 
 7.715 | 1.932 | 51 | 582.400 |2.'765221 | 8.585 
 7.780 | 1.935 | 52 | 581.419 | /764489 | 8.600 
 7.744 | 1.939 | 53 | 580.441 | .763758 | 8.614 
 7.759 | 1.943 | 54 | 579.466 | .763028 | 8.629 | 
 7.773 | 1.946 | 55 | 578.494 | .762299 | 8.643 | 
 7.788 | 1.950 | 56 | 577.526 | .761572 | 8.658 
 7.802 | 1.953 | 57 | 576.561 | .760845 | 8.672 | 
 7.817 | 1.957 | 58 | 575.599 | .760120 | 8.687 
 7.831 | 1.961 || 59 | 574.641.| {759307 | 8.701 
 7.846 | 1.965 || 60 | 573.686 |2.758674 | 8.716 
 
 ROS Ff AE ee eat bet et 
 
 G 
 
 WwW 
 
 WW NWWWYD 
 
 965 
 
 968 
 9G . 
 SON 
 
 : 9479 
 .983 
 . 987 
 .990 
 994. 
 .998 
 .001 
 .005 
 .008 
 
 .012 
 .016 
 019 
 023 
 .026 
 030 
 034 
 037 
 041 
 045 
 
 .048 
 .052 
 
 056 
 
 .060 
 .063 
 .066 
 070 
 O%4 
 
 077 
 081 
 084 
 .088 
 .092 
 .096 
 .099 
 .108 
 .106 
 .110 
 
 113 
 S17 
 
 121 
 
 125 
 .128 
 .182 
 oe 
 
Radius. 
 
 TABLE IV.—RADII, LOGARITHMS, OFFSETS, ETC. 
 
 Loga- 
 
 562.466 
 10.638 
 
 3.823 
 ’,019 
 
 52227 
 3.447 
 
 678 
 920 
 174 
 3.438 
 714 
 
 3.001 
 
 .298 
 
 39. 606 
 37.924 | 
 
 ).253 
 593 
 
 32.943 
 
 803 
 
 29.673 
 28.053 
 26.443 
 
 843 
 
 3.252 
 
 671 
 100 
 589 
 ).986 
 
 5.443 
 3.909 
 2.385 
 
 .869 
 .863 
 
 JT. 865 
 
 376 
 .896 
 
 3.425 
 
 . 962 
 507 
 061 
 
 97.624 
 
 5.195 
 
 hin 
 odd 
 
 3. 801 
 956 
 
 .559 
 SAW 
 ’.790 
 5.417 
 
 5 051 
 33.694 
 344 
 
 .001 
 .666 
 
 60 | 478.839 
 
 “) 
 
 () 
 
 ww 
 
 rithm. 
 R. log. R. 
 3.686 |2.75867 
 (1.784 | .757282 
 39.896 | . 755796 
 38.020 | .754364 
 6.156 | .752937 
 44.305 | .751514 
 
 750096 
 
 4274 
 745870 
 144471 
 .743076 
 741686 
 .'740300 
 738918 
 737541 
 736169 
 734800 
 738436 
 732077 
 . 730721 
 . 729870 
 . 728023 
 . 726681 
 (253842 
 - 724008 
 
 722677 
 
 (21351 
 720029 
 18711 
 2.717397 
 (16087 
 “14781 
 713479 
 712181 
 710887 
 109596 
 708310 
 107027 
 105748 
 704473 
 103202 
 701934 
 100671 
 699410 
 698154 
 696901 
 695652 
 694407 
 693165 
 
 691926 
 .690692 
 
 WWW WNW WWNWWW 
 
 WWW WWWWWWYW 
 
 a WWW 
 
 . 689460 
 . 688283 
 . 687008 
 685788 
 . 684570 
 683357 
 682146 
 . 680989 
 
 '2.679735 
 
 Radius. 
 
 R. 
 
 442.814 
 
 441 . 684 
 440.559 
 
 | 489.440 
 
 438.326 
 437.219 
 436.117 
 435.020 
 433.929 
 432.844 
 431.764 
 430.690 
 429. 620 
 428.557 
 427.498 
 426.445 
 425.396 
 424.354 
 423.316 
 422.283 
 421.256 
 420.233 
 419.215 
 418.203 
 417.195 
 
 | 416.192 
 
 415.194 
 414.201 
 413.212 
 412.229 
 411.250 
 
 | 410.275 
 
 rw) 
 
 co] 
 
 a 
 
 .676145 |10.! 
 674954 |10U. 
 673767 |10.! 
 672584 |10.62 
 .671403 |10.6: 
 670226 |10.6 
 2.669052 |10.71: 
 
 2.667881 |10. 
 666713 |10. 
 665549 |10. 
 
 | .664387 10. 
 663229 |10. 
 
 662074 |10.8 
 
 660922 |10.5 
 
 659773. |10.4 
 
 658628 |10. 
 
 657485 |11. 
 
 656345 
 
 655208 |11. 
 
 654075 |11. 
 
 652044 
 
 651816 |11. 
 
 650691 |11. 
 
 649570 |11. 
 
 648451 |11. 
 
 647835 |11. 
 
 646221 /11. 
 
 645111 |11. 
 
 644004 |11. 
 
 642899 |11. 
 
 641798 |11. 
 
 640699 |11. 
 
 639603 |11. 
 
 638510 |11. 
 
 637419 |11. 
 
 636331 |11. 
 
 635246 |11. 
 
 634164 |11. 
 
 633085 |11.6 
 
 682008 |11.6 
 
 630934 |11.6 
 
 629863 |11 
 
 628794 |11.7 
 
 627728 |11. 
 
 626665 |11.£ 
 
 625604 |11.£ 
 
 624546 11.8 
 
 2.623490 11. 
 
 622437 |11. 
 
 621387 |11 
 
 620889 |11 
 
 619294 
 
 618251 12 
 
 617211 |12. 
 
 616173 12 
 
 615138 |12 
 
 614106 |12 
 
 613075 |12 
 
 = 
 eo 
 
 | 
 —" 
 
 Loga- | Tang. | 
 
 rithm. Off. 
 log. R. | t. 
 9 |2.679735 |10.453 
 | 678535 (10.482 | 
 .677388 |10.511 
 
 . 956 
 985 
 014 
 
 .043 | & 
 
 O71 
 100 
 129 
 158 
 187 
 
 3% *~— 
 
 9% ~~ 
 
 ~ WW 
 
 WWW WWNHWNYW WW WW WNWWNWWYW 
 0 OO = See aS spedis 
 
 927 | 2.$ 
 
EEE EEO a ee 
 
 Loga- 
 rithm, 
 
 log. R. 
 
 275 |2.613075 
 
 - 612048 
 .611028 
 .610000 
 . 608980 
 .607962 
 . 606946 
 605933 
 .604923 
 .603914 
 
 . 602908 
 
 57| .601905 | 
 
 609904 
 -999905 
 .598908 
 .097914 
 
 .596922 | 
 
 .995933 
 594945 
 
 .593960 | 
 
 592978 
 
 -991997 
 .991019 
 .990043 
 .589069 
 988097 
 .987128 | 12 
 .086161 | 
 
 985196 
 
 984233 
 
 2.58327: 
 “582314 
 
 -5813858 
 .580403. | 
 .579451 
 
 978501 
 977553 
 .976608 
 .975664 
 
 944722 
 
 .573783 
 
 972845 
 .971910 
 
 |) .070977 
 970045 
 
 .569116 
 . 568189 
 
 | .567264 
 
 566340 
 565419 
 564500 
 
 .563582 
 .562667 
 .561754 
 . 560843 
 559933 
 .559026 
 .558120 
 .557216 
 .556315 
 2.555415 
 
 ic=>) 
 Co Oo 8 GO OO 2 9 
 
 Coon cu co CD Co CD 
 De 
 
 Loga- 
 ‘| rithm, 
 log. R. 
 
 2.555415 
 004517 
 .553621 
 .552727 
 .551834 
 550944 
 590055 
 .549169 
 548284 
 547401 
 
 2.546519 
 545640 
 544762 
 543887 
 543013 
 542140 
 
 752 | .541270 
 
 540401 
 
 | 539585 
 
 | 588670 
 
 537806 
 
 | 536945 
 
 | .B36085 
 
 535227 
 
 584370 
 
 533516 
 
 532663 
 
 531811 
 
 530962 
 
 530114 
 
 |2.529268 
 528424 
 527581 
 526740 
 | .525900 
 525062 
 524226 
 523392 
 BR255$ 
 521728 
 2.520898 
 520070 
 519244 
 518419 
 517596 
 516774 
 515954 
 | 515136 
 | .514319 
 | 513504 
 
 .512690 
 5| .511878 
 .511067 
 | .510258 
 509451 
 5 | .508645 
 9} .507840| 
 | 507037 
 506236 
 .505436 
 
 | 319.623 2.504638) 12 
 
 13.917 
 18.946 
 13.975 
 14.004 
 14,033 
 14.061 
 14,090 
 14.119 
 
 14,148 | & 
 
 14.177 
 14.205 
 14.234 
 14.263 
 14.292 
 14.320 
 14.349 
 14.378 
 14.407 
 14.436 
 14.464 
 
 14.493 
 14.522 
 14.551 
 14.580 
 14.608 
 14.637 
 14.666 
 14.695 
 14.723 
 14.752 
 14.781 
 14.810 
 14.838 
 14.867 
 14.896 
 14.925 
 14.954 
 14.982 
 15.011 
 15.040 
 
 15.069 
 15.097 
 15.126 
 15.155 
 15.184 
 15.212 
 15.241 
 15.270 
 
ae 
 oO 
 
 re) 
 e 
 
 60 | 287. 
 
 939 2.459300 
 
 TABLE IV.—RADII, LOGARITHMS, OFFSETS, ETC. 
 
 : Loga- 
 Radius. rithm. 
 
 R. log. R. 
 
 | 319.623 |2.504638 
 319.037 | .503841 
 318.453 | .503045 
 317.871 | .502251 
 317.292 | .501459 
 316.715 | .500668 
 316.139 | .499879 
 315.566 | .499091 
 314.993 | .498304 
 314.426 | .497519 
 313.860 |2.496736 
 313.295 | .495953 
 312.732 | .495173 
 312.172 | .494393 
 311.613 | .493616 
 311.056 | .492839 
 310.502 | .492064 
 309.949 | .491291 
 309.399 | .490518 
 
 | 308.850 | .489748 
 308.303 |2.488978 
 307.759 | .488210 
 
 1 | 307.216} .487444 
 
 > | 306.675 | .486679 
 306.136 | .485915 
 305.599 | .485152 
 
 | 305.064 | .484391 
 304.531 | .483632 
 
 6 | 304.000 | .482873 
 
 | 303.470) .482116 
 
 ‘| 302.943 |2.481361 
 
 | 302.417 | .480607 
 
 | 301.893 | .479854 
 301.371 | .479102 
 
 | 300.851 | .478352 
 
 | 300.833 | .477603 
 
 | 299.816 | .476855 
 
 | 299.302 | .476109 
 
 | 298.789 | .475364 
 298 .278| .474621 
 
 | 297.768 |2.473878 
 
 | 297.260 | .473137 
 296.755 | .472398 
 
 | 296.250 .471659 
 
 | 295.748 | .470922 
 
 | 295.247 | 470186 
 
 2 | 294.748 | .469452 
 34 | 294.251] .468718 
 3 | 203.756 | .467986 
 
 | 998.262 | .467256 
 
 | 292.770 |2.466526 
 
 42 | 292.279| .465798 
 44 | 291.790} .465071 
 291.803 | .464345 
 
 48 | 290.818| .463621 
 50 | 290.384 | .462897 
 52 | 289.851} .462175 
 | 289.371 | .461455 
 288 .892| .460735 
 
 58 | 288.414| .460017 
 
 —_ 
 
 Pe ek ek ek ek ek ek Pk Rk 
 
 | BRRQVQVRMIAINVWWIRN NRO 
 
 Mid. 
 Ord. | 
 
 | 4.089 
 | 4.096 
 | 4.103 
 eT 
 118 | 
 AP} 
 4.188 || 
 4.140 | 
 147 || 
 5155.1) 
 162 || 
 .169 
 sa leerg 
 184 | 
 191 || 
 .199 || 
 206 
 
 9 | 4.352 || 
 | 4.360 || 
 4.367 || 
 365 | 4.3874 || 830° 0/| 193.185 
 
 | 287. 
 285. 
 | 283. 
 30 | 280. 
 | 278. 
 | 276: 
 | 274. 
 272. 
 270. 
 | 268. 
 | 266. 
 0) | 264. 
 
 262. 
 260. 
 | 258. 
 | 256. 
 254. 
 252. 
 | 250. 
 | 249. 
 | 247. 
 30 | 245. 
 40 | 243 
 50 | 242. 
 
 | 24° 0’| 240. 
 10 | 238 
 
 20 | 237 
 
 30 | 235. 
 
 40 | 234 
 
 50 | 232. 
 25° 0’| 231 
 10 | 229 
 
 20 | 228 
 
 30 | 226. 
 
 40 | 225. 
 
 50 | 2238 
 
 | 26° 0’ 222. 
 10 | 220. 
 20 | 219. 
 30 | 218. 
 40 | 216 
 50 | 215. 
 || 27° 0’| 214. 
 10 | 212. 
 20 | 211. 
 30 | 210. 
 40 | 209. 
 50 | 207. 
 | 206. 
 10 | 205 
 20 | 204. 
 30 | 208. 
 40 | 201. 
 50 | 200. 
 | 29° 0’, 199. 
 10 | 198. 
 
 20 | 197. 
 30 | 196. 
 40 | 195. 
 50 | 194. 
 
 . |Radius. 
 
 R. 
 
 939 | 
 583 
 267 
 988 
 746 
 541 
 370 
 234 
 132 
 062 
 024 
 018 
 
 042 
 098 
 180 
 292 
 431 
 599 
 793 
 013 
 258 
 
 529 
 
 825 
 
 144 
 487 | 
 
 853 
 241 
 
 652 
 
 084|. 
 
 537 
 
 O11 |2 
 506 
 020 | 
 
 4-4-4 
 
 555 
 108 
 
 .680 
 271 2. 
 
 879 
 506 
 150 
 
 811 
 
 489 
 183 
 893 
 620 
 362 
 119 
 891 
 
 78 | 
 
 480 
 296 
 125 | 
 969 
 826 
 696 
 580 
 476 
 B85 
 306 
 240 
 
 Crore Ore 
 
 C2 > SD OVOT ON OT OL OT OTOH 
 
 > DD DADAIADAHI AIS 
 or or E 
 
Deg. 
 
 D. 
 
 Radius. 
 
 TABLE IV.—RADI, 
 
 Loga- | Tang. 
 rithm, | 
 
 R. |log R.| t. 
 
 ~ 
 ~~ 
 fore 
 ans 
 
 40 
 
 30° 20’) 191.111 2.281286 | 26.163 
 189.083 | .276652. | 26.443 
 
 : 31° (| 187.099} 272071 | 26.724 
 
 32° 
 
 : 20 
 
 4) 
 
 20 | 185.158! .267541 | 27:004 
 183.258} .268062 | 27.284 
 0’ | 181.898 | .258632 : 
 179.577 | .254250 | 27.843 
 
 40 | 177.794 .249916 | 28.123 
 88° 0/| 176.047 | 245628 | 28.462 
 
 \) 
 ~ 
 on 
 ror) 
 nse 
 
 20 | 174.336] .241386 | 28.680 
 0 | 172.659] 287188 | 28.959 
 
 4 
 0! | 171.015 
 169.404] .228924 | 99. 
 
 167.825 | .224855 | 29.793 
 
 233035 | 29.237 
 
 ci) 
 
 ~~] 
 oS 
 Or 
 pm, 
 on 
 
 0’| 166.275 | .220828 | 30.07 
 
 164.756 | .216842 | 30.348 
 163.266 | .212895 | 80.625. 
 
 0’ | 161.803 |} .208988 | 30.902 
 
 160.368 | .205119 | 31.178 
 158.960 | .201288 | 31.454 
 
 Q’| 157.577 | .197494 | 31.730 
 
 156.220 | .193736 | 32 006 
 154.887 | .190014 | 32.282 
 
 0’ | 153.578 |2.186328 | 82.557 
 
 TABLE V.—CORRECTIONS FOR TANGENTS AND EXTERNALS. 
 
 LOGARITHMS, OFFSETS, ETC. 
 
 2. ae 42.262 
 
 | 2 
 Mid. se |Radina | Loga- | Tang. | Mid. ; 
 Ord. | Deg. Radius. rithm. Of | Ord. | 
 
 m. || D. R log. R.| t. m. 
 | 
 
 | 6.657 || 38°30’) 151.657/2.180863 32.969 | 8.479 
 6.731 || 39° 0’) 149.787] .175475 33.381 | 8.592 
 6.805 || 30 | 147.965) .170160 33.792! 8.704 
 6.879 || 40° 0’| 146.190) .164918 34.202! 8.816 
 6.958 || 144.460) .159747 34.612] 8.929 
 7.027 || 41° 0’, 142.778] .154645 35.021! 9.041 
 7.101 || 141.127) .149610 35.429) 9.154 
 7.175 || 42° 0’ 139.521] .144641) 35.887 | 9.967 
 7.250 30 | 187.955} .189736 36.244] 9.380 
 7.324 || 43° 0’ 136.425] .134895 36.650! 9.493 
 4.398 134.932) .130114 37.056! 9.606 
 7.473 || 44° 0’ 133.478/2 .125395 37.461 | 9.719 
 7.547 30 | 182.049} .120734 37.865 | 9.832 
 7.621 || 45° 0’ 130.656} .116130 38.268] 9.946 
 7.696 30 | 129.296) .111584 38.671 110.059 
 7.770 || 46° 0’ 127.965} .107092 39.073 }10.173 
 7.845 80 126.664} .102655 39.474 |10.286 
 7.919 || 4'7° 0’ 125.392 -098270 39.875 |10.400 
 7.994 30 124.148) .093938, 40.275 110.516 
 8 068 || 48° 0’ 122.930} .089657, 40.674 110.628 
 8.148 30. 121.738} .085425. 41.072 110.742 
 8.218 || 49° 0% 120.571) .081243 41,469 110.856 
 8.292 30 | 119.429} .077109, 41.866 |10.970 
 8.367 || 50° 0’, 118.310 11.085 
 
 For TANGENTS, ADD 
 10° | 15° | 20° | 25° | goe ||Ang 
 Cur.| Cur.| Cur.| Cur.| Cur.|| © 
 06 Oe 1S Peas 19 || 10° 
 13 19| .26| .82 39 || 20 
 19 .29| .89) .49] .59]} 30 
 26 40 Bia 12267 80)! 40 
 .o4 .51/ .68] .85/)1.021) 50 
 .42 .63| .84/ 1.05 11.271! 60 
 .51 76 | 1.02 | 1.28 | 1.541) 7 
 61 .91 | 1.22 | 1.53) 1.84]| 80 
 72 | 1.09 | 1.45 | 1.83 | 2.20]! 90 
 .86 | 1.380) 1.74 | 2.18 | 2.62 \100 
 1.03 | 1.56 | 2 08 | 2.61 | 3.141 /110 
 1.25 | 1.93 | 2.52 | 3.16 | 3.81 ||120 
 
 For EXTERNALS, ADD 
 
 5° | 10° 
 Cur.| Cur. 
 
 001 | .003 
 .006 | .011 
 .013 | .025 
 .023 | .046 
 .087 | .075 
 .056 | .112 
 .080 | .159 
 .110 | .220 
 .149 | .299 
 .200 | .401 
 .268 | .536 
 .360 | .721 
 
 15° 
 
 | Cur. 
 
 004 
 O17 
 "038 
 
 290° 
 Cur. 
 
 006 
 
 25° 
 
 007 
 028 
 .065 
 ers 
 .189 
 .283 
 -403 
 .558 
 756 
 1.015 
 )1.355 
 \1.825 
 
 Cur. | 
 
f 
 
 TABLE VIL—TANGENTS AND EXTERNALS TO A 1° CURVE. 
 
 | | | | 
 S Tan- | Exter- || Tan- | Exter- || Tan- | Exter- 
 Angle. gent nal. | Angle. gent nal. | Angle. gent. | nal, 
 i T. E.— 4+ 2 T. E. A To) Ble 
 1° 50.09 218 || 11 551.70 | 26.500 || Qy°¢ 1061.9 | 97.57% 
 10’| 58.34 297 10’| 560.11 | 27.813 || 10’| 1070.6 | 99.155 
 20 | 66.67 .888 20 | 568.53 | 28.137 || 20 | 1079.2 | 100.75 
 30 | 75.01 491 30 | 576.95 | 28.974 || 30 | 1087.8 | 102.35 
 40 | 83.84 .606 40 | 585.36 | 29.824 || 40 | 1096.4 | 103.97 
 50 | 91.68 733 50 | 593.79 | 30.686 50 | 1105.1 | 105.60 
 2 100.01 873 || 12 602.21 | 31.561 | 99 1118.7 | 107.24 
 19 | 108.35 | 1.024 10 | 610.64 | 32.447 || 10 | 1122.4 | 108.90 
 20 | 116.68 | 1.188 20 | 619.07 | 33.347 90 | 1181.0 | 110.57 
 30 | 125.02 | 1.864 30 | 627.50 | 34.259 30 | 1139.7 | 112.25 
 40 | 133.36 | 1.552 40 | 635.93 | 35.188 40 | 1148.4 | 113.95 
 50 | 141.7 1.752 50 | 644.37 | 36.120 || 50 | 1157.0 | 115.66 
 3 150.04 | 1.964 || 13 652.81 | 37.070 || 93 1165.7 | 117.38 
 10 | 158.38 | 2.188 10 | 661.25-! 38.031 || 10 | 1174.4 | 119.12 
 90 | 166.72 | 2.425 20 | 669.70 | 39.006 || 20 | 1183.1 | 120.87 
 30 | 175.06 | 2.674 30 78.15 | 39.993 || 30 | 1191.8 | 122.63 
 40 | 183.40 | 2.934 40 | 686.60 | 40.992 40 | 1200.5 | 124.41 
 50 | 191.7 3.207 50 | 695.06 | 42.004 | 50 | 1209.2 | 126.20 
 4 200.08 | 3.492 || 14 703.51 | 43.029 | 94 1217.9 | 128.00 
 10 | 208.43 | 3.790 10 | 711.97 | 44.066 || 10 | 1226.6 | 129.82 
 20 | 216.77 | 4:099 20 | 720.44 | 45.116 | 20 | 1235.3 | 131.65 
 30 | 225.12 | 4.421 30 | 728.90 | 46.178 30 | 1244.0 | 183.50 
 40 | 233.47 | 4.755 40 | %87.37 | 47.253 40 | 1252.8 | 185.35 
 50 | 241.81 | 5.100 50 | 745.85 | 48.841 || 50 | 1261.5 | 137.23 
 5 950.16 | 5.459 || 15 754.32 | 49.441 || 95 1270.2 | 189.11 
 10 | 258.51 | 5.829 10 | 62.80 | 50.554 | 10 | 1279.0 | 141.01 
 20 | 266.86 | 6.211 20 | 771.99 | 51.679 || 20 | 1287.7 | 142.93 
 30 | 275.21 | 6.606 30 | 779.77 | 52.818 || 30 | 1296.5 | 144.85 
 40 | 283.57 | 7.018 40 | 788.26 | 53.969 40 | 1305.3 | 146.79 
 50 | 291.92 | 7.482 50 | 796.75 | 55.132 || 50 | 1314.0 | 148.7 
 6 300.28 | 7.863 || 16 805.25 | 56.309 | 96 1822.8 | 150.71 
 10 | 308.64 | 8.807 | 10 | 818.75 | 57.498 || 10 | 1331.6 | 152.69 
 20 | 316.99 | 8.762 || 20 | 822.25 | 58.699 || 20 | 1340.4 | 154.69 
 30 | 825.35 | 9.230 30 | 830.76 | 59.914 | 30 | 1849.2 | 156.7 
 40 | 333.7 9.710 40 | 839.27 | 61.141 || 40 | 1858.0 | 158.72 
 50 | 342.08 | 10.202 || 50 | 847.78 | 62.381 | 50 | 1366.8 | 160.76 
 uf 350.44 | 10.707 || 17 856.80 | 63.634 | 27 1375.6 | 162.81 
 10 | 358.81 | 11.224 10 | 864.82 | 64.900 || 10 | 1884.4 | 164.86 
 20. |.367.17 | 11.753 20 | 873.35 | 66.178 || 20 |.1393.2 | 166.95 
 30 | 375.54 | 12.294 30 | 881.88 | 67.470 | 30.| 1402.0 | 169.04 
 40 | 383.91 | 12.847 40 | 890.41 | 68.774 || 40 | 1410.9 | 171.15 
 50 | 392.28 | 13.413 50 | 898.95 | 70.091 || 50.| 1419.7 | 173.27 
 8 400.66 | 13.991 || 18 907.49 | 71.421 || 28 1428.6 | 175.41 
 10 | 409.03 | 14.582 10 | 916.03 | 72.764 | 10 | 1437.4 | 177.55 
 20 | 417.41 | 15.184 20 | 924.58 | 74.119 | 20.| 1446.3 | 179.72 
 30 | 425.79 | 15.799 30 |. 933.18 | 75.488 | 30 | 1455.1 | 181.89 
 40 | 434.17 | 16.426 40 | 941.69 | 76.869 | 40 | 1464.0 | 184.08 
 50 | 442.55 | 17.065 50 | 950.25 | 78.264 || 50 | 1472.9 | 186.29 
 9 450.98 | 17.717 || 19 958.81 | 79.671 || 29 1481.8 | 188.51 
 10 | 459.32 | 18.381 || 10 | 967.38 | 81.092 || 10 | 1490.7 | 190.7 
 20 | 467.71 | 19.058 20 | 975.96 | 82.525 || 20 | 1499.6 | 192.99 
 30 | 476.10 | 19.746 || 30 | 984.53 | 83.972 || 30 | 1508.5 | 195.25 
 40 | 484.49 | 20.447 || 40 | 993.12 | 85.431 | 40 | 1517.4 | 197.53 
 50 | 492.88 | 21.161 50 | 1001.7 | 86.904 | 50 | 1526.3 | 199.82 
 10 501.28 | 21.887 || 20 / 1010.3 | 88.389 || 30 1535.3 | 202.12 
 10 | 509.68 | 22.624 10 | 1018.9 | 89.888 || 10 | 1544.2 | 204.44 
 20 | 518.08 | 23.375 20 | 1027.5 | 91.399 20 | 1553.1 | 206.77 
 30 | 526.48°| 24.188 |! 30 | 1036.1 | 92.924 30 | 1562.1 | 209.12 
 40 | 534.89 | 24.918 || 40 | 1044.7 | 94.462 | 40 | 1571.0 | 211.48 
 543.29 | 25.700 || 50 | 1053.3 | 96.0138 50 | 1580.0 | 213.86 
 
TABLE VI.—TANGENTS AND EXTERNALS TO A 1° CURVE. 
 
 ‘pes ee ha ee ee ee ae 
 Tan- | Exter- | Tan- | Exter- Tan- | Exter- 
 Angle. gent nal. {| 4ngle.|- cont. | nal. || Angele. gent. | nal. 
 T. KE. A T. E. A ae EK. 
 : 3ie 1589,0 | 216.25 || 41° | 2142.2 | 387.38 || 51° | 2732.91 618.39 
 | 10 | 1598.0 | 218.66 10’) 2151.7. | 390.71 10’| 2748.1 622.81 
 20 : 1606.9 | 221.08 | 20 | 2161.2 | 394.06 | 20 | 2753.4 627.24 
 : 30 | 1615.9 | 223.51 | 30 | 2170.8 | 897.43 | 380 | 2763.7 631.69 
 : 40 | 1624.9 | 225.96 40 | 2180.3 | 400.82 40 | 2773.9 636.17 
 : 50 | 1633.9 | 228.42 | 50.| 2189.9 | 404.22 | 50. | 2784.2 640.66 
 : 32 1643.0 | 230.90 || 42 2199.4 | 407.64 52 2794.5 645.17 
 | 10 | 1652.0 | 233.39 | 10 | 2209.0 | 411.07 | 10 | 2804.9 649.7 
 : 20 | 1661.0 | 235.90 | 20 | 2218.6 | 414.52 20. | 2815.2 654.25 
 : 30 | 1670.0 | 238.43 80 | 2228.1 | 417.99 30 | 2825.6 658 .83 
 : 40-| 1679.1 | 240.96 | 40 | 2237.7 | 421.48 40 | 2885.9 663 .42 
 : 50 | 1688.1 | 243.52 | 50 | 2247.3 | 424.98 50 | 2846.3 668 .03 
 | 33 1697.2 | 246.08 || 43 2257.0. | 428.50 || 53 2856.7 | 672.66 
 : 10 | 1706.3 | 248.66 10 | 2266.6 | 432.04 10 | 2867.1 677.382 
 : 20: | 171523: | 251.26 20. | 2276.2 | 485.59 20 | 2877.5 681.99 
 : 30 | 1724.4 | 253.87 30. | 2285.9 | 439.16 30 | 2888.0 686.68 
 40 | 1733.5 | 256.50 40 | 2295.6 | 442.95 40 | 2898.4 691.40 
 50 | 1742.6 | 259.14 | 50. | 2805.2 | 446.35 | 50 | 2908.9 696.13 
 : 34 1751.7 | 261.80 || 44 2314.9 | 449.98 || 54 2919.4 700.89 
 ) 10 | 1760.8 | 264.47 | 10 | 2324.6 | 458.62 10 | 2929.9 705.66 
 | 20 770.0 | 267.16 | 20 | 2334.3 | 457.27 | 20 | 2940.4 710.46 
 : 30-1 1779.1 | 269.86 80 | 2344.1 | 460.95 30 | 2951.0 715.28 
 40 | 1788.2 | 272.58 | 40 | 2353.8 | 464.64 40 | 2961.5 720.11 
 50 | 1797.4 | 275.31 50 | 2363.5 | 468 .35 50 | 2972.1 (24.97 
 35 1806.6 | 278.05 | 45 2373.3 | 472.08 || 55 2982.7 729.85 
 10 | 1815.7 | 280.82 10 | 2883.1 | 475.82 10 | 2993.38 734.76 
 20 | 1824.9 | 283.60 20 | 2392.8 | 479.59 20 | 8003.9 739.68 
 30,! 1834.1 | 286.39 30 | 2402.6 | 483.37 30 | 3014.5 744.62 
 40 | 1843.3 | 289.20 40 | 2412.4 | 487.17 40 | 3025.2 749.59 
 50 | 1852.5 | 292.02 50 | 2422.3 | 490.98 50 | 3035.8 754.57 
 36 1861.7 | 294.86 || 46 2482.1 | 494.82 || 56 3046.5 759.58 
 10 | 1870.9 | 297.72 | 10 | 2441.9 | 498.67 10 | 3057.2 764.61 
 | an 2 1880.1 | 300.59 | 20 | 2451.8 | 502.54 20 | 8067.9 769.66 
 : 30 | 1889.4 | 803.47 | 30 | 2461.7 | 506.42 80 | 8078.7 | 774.78 
 40 | 1898.6 | 306 .37 40 | 2471.5 | 510.33 40 | 3089.4 779.83 
 50 | 1907.9 | 309.29 50 | 2481.4 | 514.25 50 | 3100.2 | 784.94 
 37 1917.1 | 312.22 || 47 2491.3 | 518.20 || 57 8110.9 790.08 
 10 | 1926.4 | 815.17 10 | 2501.2 | 522.16 | 10 | 3121.7 | 795.24 
 20 | 1935.7 | 318.18 20 | 2511.2 | 526.13 20 | 3182.6 800.42 
 30 | 1945.0 | 821.11 30. | 2521.1, | 530.18 | 30 | 3148.4 | 805.62 
 40 | 1954.3 | 824.11 40 | 2531.1 | 584.15 | 40 | 3154.2 | 810.85 
 50 | 1963.6 | 327.12 50. | 2541.0 | 5388.18 50 | 8165.1 | 816.10 
 38 | 1972.9 | 380.15 || 48 2551.0 | 542.23 || 58 3176.0 821.37 
 10 | 1982.2 | 333.19 10 | 2561.0 | 546.30 10 | 4186.9 826.66 
 20 | 1991.5 | 336.25 20 | 2571.0 | 550.39 20 | 3197.8 831.98 
 30 | 2000.9 | 339.32 30 | 2581.0 | 554.50 30 | 3208.8 | 837.31 
 40 | 2010.2 | 342.41 40 | 2591.1 | /58.63 40 | 8219.7 | 842.67 
 50 | 2019.6 | 345.52 50 | 2601.1 | 562.77 50 | 8280.7 848.06 
 39 2029.0 | 348.64 || 49 2611.2 | 566.94 || 59 8241.7 853.46 
 10 | 2038.4 | 351.78 10. | 2621.2 | 571.12 10 | 8252.7 858.89 
 20 | 2047.8 | 354.94 20 | 2631.3 | 575.32 20 | 3263.7 864.34 
 30 | 2057.2 | 358.11 | 30 | 2641.4 | 579.54 30 | 8274.8 869.82 
 | 40 | 2066.6 | 361.29 | 40 | 2651.5 | 583.78 | 40 | 3285.8 875.82 
 50 | 2076.0 | 364.50 | 50 | 2661.6 | 588.04 | 50 | 3296.9 880.84 
 40 2085.4 | 367.72 || 50 2671.8 | 592.32 || 60 3308.0 886.38 
 10 | 2094.9 | 370.95 10 | 2681.9 | 596.62 10 | 3319.1 891.95 
 20 | 2104.3 | 374.20 20. | 2692.1 | 600.93 | 20 | 3330.3 897.54 
 { 80 | 2118.8 | 877.47 | 3 2702.3 | 605.27 80 | 3341.4 908.15 
 | 40 | 2123.3 | 380.76 | 40 | 2712.5 | 609.62 | 40 | 3352.6 908 . 7 
 50 | 2132.7 | 384.06 50 | 2722.7 | 614.00 50 | 3363.3 914.45 
 ey 
 
 290 
 
TABLE VIL—TANGENTS AND EXTERNALS TO A 1° CURVE. 
 
 Tan- 
 gent. 
 
 4 ke 
 
 3375.0 
 
 3386.3 
 3397.5 
 3408.8 
 3420.1 
 3431.4 
 3442.7 
 3454.1 
 3465.4 
 3476.8 
 3488 .3 
 3499.7 
 3511.1 
 3522. 
 3584 
 3545. 
 3557. 
 3568. 
 3580. 
 3591. 
 | 8603 
 3615 
 3626. 
 3638. 
 
 SOR RICO] NOH MOWIWAHM 
 
 1002.3 
 1008.3 
 1014.4 
 1020.5 
 1026.6 
 1082.8 
 1039.0 
 1045.2 
 1051.4 
 1057.7 
 
 1063.9 
 1070. 
 1076. 
 1082. 
 1089. 
 1095. 
 1102. 
 1108. 
 1115: 
 1121. 
 1128 
 1134. 
 1141 
 1148 
 1154. 
 1161. 
 1168. 
 1174. 
 1181 
 1188 
 1195 
 1202. 
 1208. 
 1215. 
 
 1222. 
 1229 
 1286. 
 1243. 
 1250. 
 1257. 
 1265. 
 1272 
 127 
 
 1286. 
 1293.6 
 1300.9 
 
 DOW PODHWIOH DWVHARWIWOARW 
 
 orp et OO CO-IVIWIAVI’ WD 
 
 920.14 
 925 .85 
 931.58 
 937.34 
 943.12 
 948 . 92 
 954.75 
 960.60 
 
 966 .48 
 
 972.38 
 
 978.31 
 984.27 | 
 990.24 | 
 996 .24 
 
 | 1890. 
 
 | 1413. 
 
 COO WHO MWWIAIH OUCH QS ~~ 
 
 | 1583. 
 
 i 
 DO wWIRADOWUDWA CHOROCUR RAH 
 
 eo 
 
 WDNWOVAP www 
 
 1308 .2 
 1315.6 
 1822. 
 1330. 
 1337 
 1345 
 1352. 
 1360. 
 1367. 
 1375. 
 1882. 
 
 1398. 
 1405. 
 
 1421. 
 1429. 
 14236. 
 1444. 
 1452. 
 1460. 
 1468. 
 1476. 
 1484. 
 1492. 
 1500. 
 1508 
 1516. 
 1524. 
 1533. 
 1541 
 1549. 
 1558. 
 1566. 
 1574. 
 
 1591. 
 1600. 
 1608. 
 1617. 
 1625 
 1634. 
 1643. 
 1651 
 1660. 
 1669 
 1678 
 1686. 
 
 DH WAIORH 
 
 | 
 
 SOHOHUWOS 
 
 rt Sort Od MRI WONRH ORNATE PP POTHOOW IOS HOC 
 
 fa 
 aS; 
 co) 
 aE 
 JIN DW 
 
 Tan- | Exter- 
 gent. nal. 
 | se EK. 
 4893.6 | 1805.3 
 10'| 4908.0 | 1814.7 
 20 | 4923.5 | 1824.1 
 30 | 4937.0 | 1833.6 
 40 | 4951.5 | 1843.1 
 50 | 4966.1 | 1852.6 
 4980.7 | 1862.2 
 10 | 4995.4 | 1871.8 
 20 | 5010.0 | 1881.5 
 30 | 5024.8 | 1891.2 
 40 | 5039.5 | 1900.9 
 50 | 5054.38 | 1910.7 
 | 5069.2 | 1920.5 
 10 | 5084.0 | 1980.4 
 20 | 5099.0 | 1940.3 
 30 | 5113.9 | 1950.3 
 40 | 5128.9 | 1960.2 
 50 | 5143.9 | 1970.3 
 | 5159.0 | 1980.4 
 10 ; 5174.1 | 1990.5 
 20 | 5189.3 | 2000.6 
 30 | 5204.4 | 2010.8 
 40 | 5219.7 | 2021.1 
 50 | 5284.9 | 2081.4 
 | 5250.3 | 2041.7 
 10 | 5265.6 | 2052.1 
 20 | 5281.0 | 2062.5 
 30 | 5296.4 | 2073.0 
 40 | 5811.9 | 2088.5 
 50 | 5327.4 | 2094.1 
 | 5843.0 | 2104.7 
 10 | 5858.6 | 2115.3 
 20 | 5874.2 | 2126.0 
 30 | 5389.9 | 2136.7 
 40 | 5405.6 | 2147.5 
 50 | 5421.4 | 2158.4 
 | 5437.2 | 2169. 
 10 | 5453.1 | 2180. 
 20 | 5469.0 | 2191 
 0 | 5484.9 | 2202. 
 “40 | 5500.9 | 22138. 
 50 | 5517.0 | 2224. 
 5533.1 | 2235. 
 10 | 5549.2 | 2246. 
 20 | 5565.4 | 2258. 
 80 | 5581.6 | 2269 
 40 | 5597.8 | 2280 
 50 | 5614.2 | 2292 
 5630.5 | 2303. 
 10 | 5646.9 | 2315. 
 20 | 5663.4 | 2326. 
 30 | 5679.9 | 2388. 
 40 | 5696.4 | 2849. 
 50 | 5718.0 | 2361 
 | 5729.7 | 2378. 
 10 | 5746.3 | 2885. 
 20 | 5763.1 | 23897. 
 30 | 577! 2408. 
 40 | 5796.7 | 2420. 
 50 | 5813.6 | 2482. 
 
 SDD OM WN DNAOM CHWONMWNMNWH WE 
 
TABLE VI.—TANGENTS AND EXTERNALS TO A 1° CURVE 
 
 i | | | 
 Tan- |. Bx | | or | | 
 | Angle, : 2s Angle Tan- Ex- | ¥ x 
 | ent. | ternal. ngle. | oo iter Ste pa eee 
 | | ‘ | Be | 2 | | gent. | ternal. | Angle. | gent. .| ternal 
 P| ics ale eee dn aedeinsl ick, ae e e e ie E. 
 | ; ‘ 
 : 91° | 5830.5 | | | | 3 
 5830.5 | 2444.9 | ° | 6950.6 | 3978 
 10’| 5847.5 | 3457 ’ on 10’) pate | 3278.1 || L11° | 8336.7 | 4386.1 
 Ba idooe ee Bab? tH 971.3 | 3204-1 || ~~ 10"| 8362.7 | 4407.6 
 20 | 5864.6 | 2469.3 || 20 | 6992 | 99 pose 4407.6 
 | gies ee et 20 | 6092.0 | 3310.1 20 | 8388.9 | 4429.2 
 | 40 | 5898.8 | 2493.8 || 30 | Role. | 8326.1 | 30 | 8415.1 | 4450.9 
 i | 50 | 5916.0 | 2506.1 || 50 | t05d'5 | aces | Boe oe mele 
 : 99 | 5933.2 | 95185 || 102 eign 3358.5 50 | 8468.0 | 4494.6 
 | | 10 | 5950.5 | 2531.0 || 10 | 7006:6 | B301°2 || 22? 49 | SA94-8 | 4518.6 
 | | ~~ 20 | 5967.9 | 2543'5 || 20 | 11718 | sare Se eer = aoa 
 ih | ~~ 80} 5985.3 | 9556.0. 80] ri39.0 | Sakae Site 
 = Sapien 30 | 7139.0 | 3424.3 30 | 8575.0 | 4583.4 
 | sy | 0002-7 | 2568.6 |) | 7160.3 | 3440.9 40 | 8602.1 | 4606 
 6020.2 ROSL.o | 50 181.7 | 8457 @ J 6.0 
 eens | (181.7 | 3457.6 50 | 8629.3 | 4628.6 
 93 6037.8 | 2594.0 || 198 7203.2 | ar | 
 | 10 | 6055.4 | 2606'8 || ~~ 10 | waee:7 | 3401-3 |] 228 49 | 8638-6 | 4651-8 
 | 20 | 6073.1 | 2619-7 | oat eerie) aco 10 | 8684.0 | 4674.2 
 | 30 | 6090.8 | 2632'6 | 30 | 796801 sree’ || a0 | 8711.5 | 4697.2 
 | 40 | 6108.6 | 2645 5 40 | 2080°8 | 8025.2 || 380 | 8739.2 | 4720.3 
 | Ey ised Pee. | 40 | 7289.8 | 3542.4 || 40 | 8767.0 | 4743.6 
 7 94 6144.3 | 2671.6 || 104 ae | 8559.6 50 | 8794.9 | 4766.9 
 | 10 | 6162.2 | 9684.7 | 10 Ente | 8576.8 | 114 8822.9 | 4790.4 
 | 20 | 6180.2 | 2697.9 || 20 | 7377.8 | 3611.7 Si eee ae 
 | 30 | 6198.3 | 2711.2 | BOT anarad Seed eed Seen 
 : 40 | 6216.4 | ovoq's | 30 | 7299-9 | 3629.2 30 | 8907.7 | 4861.7 
 | 50 | 6234.6 | 2737.9 || 50 | vaaa'e | soau'e cabernet sce 
 Maree ose e tart gdiace’ | | dots tel PO seEPOS. 0: 20002 
 10 | Ga7i.i | areas || 105 ie ie 3682.3 || 115 8993.8 | 4934.1 
 | 20 | 6289.4 | 9779's || ee eS 9.6 | 3700.2 || 10 | 9022.7 | 4958.6 
 : 30\16307.0 | ste's 20 | 512.2 | 8718.2 20 | 9051.7 | 4983.1 
 | 40 | 6326.3 | 2805.6 40 | 1587-7 | Bea 0 Wi eaie | cae 
 : 50-| 6344.8 | S819 4 il Perse 3754.4 || 40 | 9110.3 | 5032.6 
 96 | 6363.4 | 9833 9 | 106 7603.5 | 3072.6 | 50 9139.8 5057.6 
 | 10 | 6382.1 | 2847.0 | 10 | 7626-6 | song. || 248 44 | $183-4 | p08: 
 at mea toe 4 || 10 | 7626.6 | 3809.4 | 10 | 9199.1 | 5107.9 
 Seba eee 3 il 20 | 7649.7 | 3827.9 20 | 9229.0 | 5133.3 
 40 | 6438.4 | 2889°0 ye cone ea he eugeeere ai eS 
 siesta: 40 | (606.3 | 3865.2 40 | 9289.2 | 5184.5 
 A nae 50 | 7719.7 | 3884.0 | 50 | 9319.5 | 5210.3 
 : 10 | 6495.2 | 29316 |; 107 10 (143.2 | 3902.9 || 117 9349.9 | 5236.2 
 | Gipsy rare asian 10 | 7766.8 | 3921.9 10 | 9380.5 | 5262.3 
 30 | 6533.4 | 2960.3 30 ‘i483 oo Sn otis shee 
 40 | 6552.6 | der4'2 | ee Ay 2060.1 30 | 9442.2 | 5315.0 
 50 | 6571.9 | 2989.2 50 | va6e.1 | swe. Fo ieesne ¢ heer 
 96 5012 | sees.8 || 10g.” | seed | 2998:7 50 | 9504.4 | 5368.2 
 : 10 | 6610.6 | 3018.4 10 | 7910.4 | doar || 228, | 9535-7 | 5895.1 
 20 | 6630.1 | 3033"1 90 | 7934°6 | aoe s 10 | 9567.2 | 5422.1 
 30 | 6649.6 | 3047.9 30 | 7950.0 | dory'a 30 pegs ee 
 gree a ge 830 | 7959.0 | 4077.2 30 | 9630.7 | 5476.5 
 sotltene eee S| 40 | 7983.5 | 4097 1 40 | 9662.6 | 5504.0 
 90 ane ea 7 117.0 50 | 9694.7 | 5531.7 
 10 | 6728/4 | 310777 || 29° 40) B04 | ater a | te ty hoes | no 
 20 | 6748.2 | 3122.9 | 20 | 8082.3 | 4178 20 | 9ro20 | bee's 
 ay bres 4 t cee | 0 | S082. 4177-6 20 | 9792.0 | 5615.5 
 40 | 6788.1 | 3153.3 | 40 8130°3 met: fe see cpteonde 
 50 | 6808.2 | 3168.7 || | Seeeeeeml oon See 9857.7 | 5672.3 
 LOO hecae stares lags 50 8157.5 | 4239.0 50 | 9890.8 | 5700.9 
 0 | 6848.5 | 3199.6 || ~~ 10 | g208/9 gaoo-7 |, 120 peed 0 | 6120.7 
 eee | | 10 | $208.2 | 4280.5 | 10 | 9957.5 | 5758.6 
 Big hee A 20 | 8288.7 | 4901-4 20 | 9991.0 | 5787.7 
 Doe. | 30 | 8259.3 | 4302.4 30 |10025.0 | 5817.0 
 50 | 6930.1 | 3262.3 50 | 8310.8 | 4960°¢ at dee ae Me Ee 
 | 262.3 | 50 | 8310.8 | 4364.8 50 |10093.0 | 5876.1 
 
TABLE 
 
 VII.—LONG CHORDS. 
 
 i= el 
 Actual Lone CHORDS. 
 Degree | _ Are, 
 of One | 
 eee 2 | 3 A. 5 6 
 CoE . Station. Stations. Stations. | Stations. | Stations. | Stations. 
 | | 
 O° 10’ 100.000 200.000 299.999 399.998 499.996 599.993 
 20 000 199.999 299.997 399.992 499 . 983. 599.970 
 30 000 199.998 299 992 399.981 499 . 962 599.933 
 40 001 199.997 299.986 399.966 499 . 932 599. 882 
 50 001 199.995 299.979 399.947 499 . 894 599.815 
 1 100.001 199.992 299.970 399.924 499.848 599.733 
 10 002 199.990 299.959 399.896 499.793 599.637 
 20 002 199.986 299 .946 399.865 499 .729 599.526 
 30 003 199.983 299 . 932 399.829 499.657 599.401 
 40 003 199.979 299.915 899.789 499.577 599.260 
 50 .004 199.974 299 .898 399.744 499.488 599.105 
 2 100.005 199.970 299 .878 399.695 499.391 598 .934 
 10 006 199.964 299.857 399.643 499 285 598.750 
 20 007 199.959 299.834 3998586 499.171 598.550 
 30 008 199.952 299.810 399 524 499 .049 598.336 
 40 009 199.946 299.783 399 .459 498.918 598.106 
 50 .010 199.939 299.756 399.389 498.778 597.862 
 3 100.011 199.931 299.726 399.315 498 . 630 597.604 
 10 S013 199 . 924 299 .695 399 . 237 498 .474 597.33 
 20 014 199.915 299.662 399.154 498 .309 597.043 
 | 30 015 199.907 299 .627 399 .068 498.136 596.740 
 40 “017 199.898 299.591 398.977 497 .955 596.423 
 50 019 199.888 299.553 398 . 887, | 497.765 596.091 
 4 100.026 199.878 299.513 398.782 497 .566 595.7 
 10 022 199.868 299.471 398 .679 497.360 595.383 
 20 .024 199.857 299 428 398.571 497.145 595.007 
 30 | -026 199.846 299.388 | 398.459 | 496.921 594.617 
 40 028 199.834 299 .337 398.343 | 496.689 594.212 
 50 030 199.822 299 .289 398 .223 496 .449 593.792 
 5 100.032 199.810 299 .239 398.099 | 496.201 593.358 
 10 034 199.797 299.187 397 .970 495.944 592.909 
 20 .036 199.783 299 134 397.837 495.678 592.446 
 30 .038 199.770 299.079 397.700 495.405 591.968 
 40 041 199.756 299 .023 397.559 495.123 591.476 
 50 043 199.741 298.964 397.418 494 832 590.97 
 6 100.046 199.726 298.904 397 .264 494.534 590.449 
 10 .048 199.710 298 843 397.110 494 227 589.913 
 20 051 199.695 298.77 396.952 493.912 589.364. 
 30 054 199.678 298.714 396.790 493 588 588 .800 
 40 .056 199.662 298 .648 396 .623 493.257 588 221 
 50 059 199.644 298.579 396 .453 492.917 587.628 
 ui 100.062 199.627 298.509 396.278 492.568 587.021 
 10 065 199.609 298 .438 396 .099 A92 212 586.400 
 20 068 199.591 298 . 364 395.916 491.847 585.765 
 30 071 199.57 298 .289 395.729 491.474 585.115 
 40 075 199.553 298 .212 395 .538 491 .093 584.451 
 50 078 199.533 298.134 395 .342 490.704 583.773 
 8 100.081 199.513 298 . 054 395.142 490.306 583.081 
 10 .085 199.492 297 .972 394.938 489 .900 582.375 
 20 088 199.471 297 .888 394.731 489 .486 581.654 
 30 092 199.450 297 803 394.518 489 .064 580.920 
 40 095 199.428 297 .'716 394.302 488 . 634 580.172 
 50 099 199.406 297 .628 394 .082 488.196 | 579.409 
 b 9 100.103 199.383 297.538 393.857 487 .749 578.633 
 10 107 199.360 297 .446 398.629 87 294. 577.843 
 20 A111 199.337 297 352 393.396 486. 832 577.039 
 30 115 199.313 297 257 393.159 486 .361 576 . 222 
 40 119 199.289 297.160 392.918 485.882 575.390 
 50 123 199.264 297 .062 392.678 485.395 54.545 
 10 100.127 199.239 296.962 | 892.424 484.900 573.686 
 
 293 
 
 sca tea tS SEL EO 
 
Degree 
 
 of 
 
 Curve. 
 
 O° 
 
 10 
 
 10/ 
 20 
 30 
 40 
 
 TABLE VII.—LONG CHORDS. 
 
 Lone CHoRDSs. 
 
 8 
 
 vi 9 10 11 12 
 
 Stations. | Stations, | Stations. | Stations. | Stations. | Stations. 
 699. 988 799. 982 899.974 999. 965 1099.95 1199.94 
 699 .953 799.929 899.899 999.860 1099.81 1199.76 
 699.893 799.840 899.772 999. 686 1099.58 1199.46 
 699.810 799.716 899.594 999.442 1099.25 1199.08 
 699.704 799.556 899.365 999. 128 1098 . 84 1198.49 
 699.574 799 .360 899.086 998.744 1098 .33 1197 .82 
 699. 420 799.180 898.757 998 . 290 1097.72 1f97 .04 
 699.242 798 . 863 898.376 997. 768 1097 .02 1196.13 
 699.041 798.562 897.945 | 997.175 1096 .23 1195.11 
 698.816 798 .224 897 .464 996.518 1095.35 1193.96 
 698 .567 797 .852 896.931 995.782 | 1094.38 1192.69 
 698 .295 797 444 896.349 994. 981 1093.31 1191.31 
 698 . 000 797 000 895.716 994.112 1092.15 1189.80 
 697.680 196 .522 895.033 993.17 1090.90 1188.18 
 697.338 796.008 894.299 992.165 1089.56 1186.43 
 696.971 195.459 893.515 991.088 1088.12 1184.57 
 696.581 794.874 892.681 989.943 1086.60 1182.59 
 696.168 794.255 891.798 988 . 729 1084.98 1180.49 
 695.731 793. 600 890.864 987.447 1083.28 1178.28 
 695.271 792.911 889.880 986 096 1081.48 1175.94 
 694.787 792.186 888 . 846 984.677 1079.59 1173.49 
 694.280 791.427 887.763 | 983.190 1077.61 1170.93 
 693. 750 790. 632 886.630 | 981.636 1075.54 1168.25 
 693.196 789 .803 885.448 980.014 1073.3 1165.45 
 692.619 788 . 939 884.217 978.325 1071.14 1162.54 
 692.018 788 .040 882.936 976.569 1068.81 1159.51 
 691.395 787.108 881 . 606 974.746 1066.38 1156.37 
 690.748 786 . 140 880 . 228 972.856 1063.87 1153.12 
 690.079 785.188 878.800 970. 900 1061.27 1149.7 
 689.386 784.101 77 824 968.877 1058.59 1146.28 
 688 . 670 783 .030 75.800 966.788 1055.81 1142.69 
 687.930 781 . 925 874.227 964. 634 1052.95 1138.99 
 687.169 780.786 872.605 962.415 1050.01 1135.18 
 686 . 384 779.612 870.936 960.130 1046.97 1131.26 
 685.576 778.406 869.219 957.780 1045.86 1127.24 
 684.745 77.165 867.454 955 . 366 1040.66 1123.10 
 683.892 775.890 865.642 952.888 1037.37 1118.86 
 683 .016 774.582 863.782 950.345 1034.01 1114.51 
 682.117 773.240 861.875 947.739 1080.55 1110.05 
 681.195 771.864 859. 922 945.069 1027.02 1105.49 
 680.251 770.455 857.921 942.337 1023.40 1100.83 
 679.285 769.014 855.874 939.542 1019.7 1096.06 
 678 296° 767 .539 853.780 936 . 684 1015.93 1091.19 
 677 .284 766.03 851.640 933 . 764 1012.07 1086 .22 
 676.250 764.490 849.455 930.783 1008.13 1081 .15 
 675.194 762.916 347 224 27.741 1004.11 1075.98 
 674.116 761.309 844.947 924.638 1000.01 1070.71 
 673.015 759.670 842.625 921.47 995.834 | 1065.34 
 671.892 757.999 840.258 918.250 991.580 | 1059.88 
 670.748 756.295 837.845 914.966 987.250 | 1054.82 
 669.581 754.560 835.389 911.623 982.844 | 1048.66 
 668 .393 752.792 832.888 908.221 978.362 | 1042.91 
 667 .182 750 . 993 830.342 904.761 973.806 | 1037.06 
 665.950 749.161 827.754 901 . 242 969.175 | 1031.13 
 664.697 747.299 825.121 897 66% 964.471 1025.11 
 663.421 745.404 822.445 894 .033 959.694 | 1018.99 
 662.124 743.479 819.726 890.343 954.844 1012.79 
 660.806 TAL 522 816.965 886 .597 949.924 | 1006.49 
 659. 466 739.585 814.160 882.795 944 , 933 1000.12 
 658.105 737 68 811.314 878.938 939.871 993.653 
 
 204 
 
 eee re 
 
Curve. 
 
 10° 10’ 
 20 
 
 11 
 
 12 
 
 13 
 
 14 
 
 15 
 
 16; 
 
 17 
 
 18 
 
 19 
 
 20 
 
 Actual 
 Are, 
 One 
 
 Station. 
 
 100.131 
 136 
 
 100.183 
 
 Lona CHORDS, 
 
 TABLE VII.—LONG CHORDS. 
 
 2 
 Stations. 
 
 199.213 
 199.187 
 199.161 
 199.134 
 199.107 
 199.079 
 199.051 
 199.023 
 198 . 994 
 198.964 
 198.935 
 
 198.904 
 198.874 
 198.843 
 198.811 
 198.779 
 198.747 
 198.714 
 198.681 
 198 .648 
 198.614 
 198.579 
 198 .544 
 198.509 
 198.474 
 198.437 
 198.401 
 198.364 
 198.327 
 198 .289 
 198 .251 
 198.212 
 198.173 
 198.134 
 198.094 
 
 198.054 
 198.013 
 197.972 
 197.930 
 197.888 
 197.846 
 197.803 
 197.760 
 197.716 
 197.672 
 197.628 
 197.583 
 197.538 
 197.492 
 197.446 
 197.399 
 197.352 
 197.305 
 197.256 
 197.209 
 197.160 
 197.111 
 197.062 
 197.012 
 196.962 
 
 3 | 4 5 
 Stations. | Stations. | Stations. 
 296.860 392.171 484.397 
 296.756 391.914 483.886 
 296.651 391.652 483 .367 
 296.544 391.387 482.840 
 296 .436 891.117 482.305 
 296 825 390.843 481 .'762 
 296.214 390.565 481.211 
 296.100 390.284 480.6538 
 295 .985 389.998 480.086 
 295.868 389.708 479 511 
 295 .750 389.414 478 .929 
 295 .629 389.116 478.838 
 295 .508 888.814 77.740 
 295 . 384 888 .508 77.135 
 295 .259 388.197 476 521 
 295 .182 387.883 475 .899 
 295 .004 887.565 475.27 
 294.874. 387 243 474.633 
 294.742 386 .916 473.988 
 294 .609 386.586 473.336 
 294 474 386 .252 472.675 
 294 337 885.914 472.007 
 294.199 885.572 471 .382 
 294.059 385 . 225 470.649 
 293.918 884.875 469.958 
 293.77. 384.521 469 .260 
 293 .629 884.163 468 .554 
 293 .483 883.801 467.840 
 293 3835 383 .435 467.119 
 293.185 383.065 466.3890 
 293 .034. 382.691 465 . 654 
 292.881 882.313 464.911 
 292 .'726 881.931 464.160 
 292.57 381.546 463.401 
 292.412 881.156 462.635 
 292 . 252 380.763 461.862 
 292.091 880.365 461.081 
 291.928 79.964 460.293 
 291 . 764 79.559 459.498 
 291.598 379.150 458 .695 
 291.430 378.737 457.886 
 291.261 378.3820 457.069 
 291.090 377.900 456.244 
 290.918 3877 47 455.413 
 290.743 77.047 454.57. 
 290.568 376.615 453.728 
 290.390 876.179 452.875 
 290: 211 875.739 452.015 
 290.031 875.295 451.147 
 289 .849 874.848 450.373 
 289.665 374.397 449 392 
 289.479 373.942 448 504 
 289 292 373.483 447.608 
 289 . 104 373.021 446.706 
 288 .913 372 554 445.797 
 288 .722 872.084 444.881 
 288 .528 71.610 443 957 
 288 .3833 371.1383 443 .028 
 288 . 137 370.652 442.091 
 287 .939 370.167 441.147 
 
 295 
 
 6 
 Stations, 
 
 572.818 
 
 71.926 
 571.027 
 570.113 
 569.186 
 568.245 
 567.292 
 566.3824 
 565.343 
 564.349 
 563.341 
 
 562.321 
 561.287 
 560.240 
 559.180 
 558.107 
 557.020 
 555.921 
 554 °809 
 553.684 
 552.546 
 551.395 
 550.232 
 
 549.056 
 547.867 
 546.666 
 545.452 
 544.226 
 542.987 
 541.736 
 540.472 
 539.196 
 537.908 
 536.608 
 535.296 
 583.972 
 582.635 
 531.287 
 529.927 
 528.555 
 af eal 
 
 525.77 
 
 524.369 
 522.950 
 521.519 
 520. 
 518. 6% 
 
 517.16 
 515.685 
 514. 
 512. 
 511. 
 509.67 
 508. 1: 
 506. 5S 
 505. 
 5038. 
 501. { 
 500. ¢ 
 
 498.7 
 
TABLE VII.—LONG CHORDS. 
 
 — 
 / Lone CHorDs. 
 Degree | | 
 | of 
 Curve, s 9 | 10 11 12 
 ; Stations.| Stations. Stations. | Stations.| Stations. | Stations. 
 . | | | 
 10° 10 | 656.728 735.467 808 .426 875 .025 934.741 987.105 
 : 20 | 655.320 733.887 805.495 371.058 | 929.542 980.47% 
 30 | 653.895 731.277 802.524 867.038 924.276 973.760 
 40 | 652.450 729.137 799.512 862.963 918.943 966. 967 } 
 50 650.983 726. 967 796.458 858 . 836 913.544 960.0938 i 
 : 11 649.496 724.767 793.364 854.656 908 . 080 953.141 | 
 | 10 | 647.989 722.537 790 . 280 850.425 902.550 946.112 Wt 
 | fs 20| 646.460 720.278 787.056 846.140 896.957 939.007 Mi 
 80 | 644.911 717.990 783.843 841.808 891 .303 931.828 fi 
 40 | 643.342 715.672 780.590 837.424 885.586 924.575 i} 
 : 50 | 641.752 713.825 777.298 832.990 879.807 917.250 i 
 | 12 640.142 710.950 773.968 828 .507 73.968 909 .854. Hl 
 | 10 | 638.512 708.546 770.600 823.974 868.070 902.389 i 
 : 20 | 636.862 706.113 767.193 819.394 862.1138 894.855 
 : 30 | 635.191 703 .653 763.749 814.766 856.099 887.254 | 
 : 40 | 683.501 701.164 760.268 810.092 850.028 879.588 } 
 50 | 631.792 698.647 756.749 805 .370 843.900 871.857 i 
 : 13 630.062 696.103 753.194 800.602 837.718 864.063 i 
 : 10 328.313 693 .531 749.603 795.790 831.482 856.208 | 
 20 | 626.544 690 . 932 745.976 790 .932 825.192 848 .293 it 
 : 380 | 624.756 688.306 742.313 786 .030 818.850 840.318 | Wa 
 : 40 | 622.949 685 . 653 738.616 781.085 812.457 832.286 nai 
 : 50 | 621.123 682 . 974 734 883 776.096 806.013 824.198 , 
 14 619.27 680.268 731.116 771.066 799 .520 816.056 i 
 10 | 617.413 677.535 727.815 765.993 792.979 807.860 i 
 20 | 615.530 674.777 723.480 760.879 786.389 799 612 th 
 30 | 613.628 671.993 719.612 (55.725 779 753 791.313 i 
 40 | 611.708 669.183 (abnareel 750.531 (73.072 782.966 i 
 59 | 609.769 666.348 LOBE 745.297 766 .345 774.571 1 
 15 607 .812 663.488 707.811 740.024 159.575 766.130 i 
 10 | 605.836 660.603 703.814 734.714 752.763 
 2 603.842 657.693 699.785 729 366 745 .908 . 
 30 | 601.831 654.758 695.725 723 . 982 739.014 } 
 40 | -599.801 651.799 691 . 634 718.561 732.078 
 50 | 597.753 648.817 687.512 713.105 725 . 104. | 
 16 595.688 645.810 683 .362 707.614 718 .092 |} 
 10 | 593.605 642.780 679.182 702.088 711.043 1 
 20 | 591.505 639.727 674.973 696.529 703.959 | 
 30 | 589.388 636 .650 370.735 690.938 | 
 40 587.253 633 .550 666.469 685 .314 im 
 50 585.101 630.428 662.175 679.659 
 17 582.933 627.283 657.854 73.972 
 10 | 580.747 624.117 653.506 668 .256 ; 
 20} 578.545 620.928 649.131 662.510 
 30 | 576.326 617.717 644.730 656 735 
 40 574.091 614.485 640.304 650.9338 
 50 | 571.839 611.232 635.852 645.103 | 
 18 569.571 607.958 631.375 | 639.245 
 10 | 567.287 604.664 626.874 
 20 | 564.988 601.3849 622.349 
 30 | 562.673 598.013 617.801 
 40 | 560.342 594.658 613.229 
 50 | 557.996 591.283 608 . 635 
 19 555 . 634 587.888 604.018 
 10-4558. 257 | 584.475 599 .37$ 
 20 550.864 581.042 594.720 (1 
 30 548.457 577.591 590.039 
 40 546 . 035 574.121 585.339 
 50 543.599 570.634 580.618 | 
 541.147 567.128 75.8% j : : 
 
Actual 
 Are, 
 One 
 
 Station. 
 
 100.562 
 100.617 
 100.675 
 100.735 
 100.798 
 100.8638 
 100.931 
 101.002 
 101.075 
 101.152 
 
 TABLE VII.—LONG CHORDS. 
 
 Lone CHORDS. 
 
 2 
 Stations, 
 
 196.651 
 196.825 
 195.985 
 195.630 
 195.259 
 194.874 
 194.474 
 194,059 
 193.630 
 193, 185 
 
 3 
 Stations. 
 
 286.716 
 285.487 
 284.101 
 282.709 
 281 .262 
 
 79.759 
 278.201 
 276.589 
 274, 924 
 278.205 
 
 pe RR RT WS 
 
 4 
 Stations. 
 
 367.179 
 364.060 
 360.810 
 357.483 
 353 . 930 
 350.303 
 346 .555 
 342,688 
 338 .'704 
 334.607 
 
 5 
 Stations. 
 
 435.345 
 429 305 
 423.033 
 416.5385 
 409.819 
 402.891 
 395 . 758 
 388 .428 
 380.908 
 373.205 
 
 6 
 Stations. 
 
 488 . 931 
 478.705 
 468.270 
 457 .433 
 446.280 
 434.827 
 423 .092 
 411.092 
 398.846 
 386.370 
 
 — 
 
fora 
 ry 
 Fat 
 =< 
 Ps 
 — 
 fo 
 eS 
 © 
 me 
 — 
 (exn) 
 jam) 
 jn 
 i 
 — 
 — 
 — 
 me 
 bol 
 a) 
 =< 
 es 
 
TABLE VII.—MIDDLE ORDINATES. 
 
 oe 
 | 
 YEE | 1 | 2 8 
 Curve, | Station. | Stations. | Stations. 
 .036 .145 232 
 078 291 654 
 .109 .436 .982 
 145 .582 1.309 
 .182 127 1.636 
 218 .873 1.963 
 255 1.018 2.291 
 291 1.164 2.618 
 327 1.309 2.945 
 .364 1.454 | «827 
 Tot e400 1.600 3.599 
 .436 1.745 3.926 
 Gate oc Sea ee ey 4.253 
 | 509 «| 2.086 4.580 
 545 2.181 4.907 
 .582 2.827 5.234 
 .618 2.472 5.561 
 654 | 2.618 5.888 
 694 2.763 6.215 | 
 (27 2.908 6.542 
 163 3.054 6.868 
 .800 3.199 | 7.195 
 .836 3.345 7.522 
 .872 3.490 7.848 
 |  .909 3. 635 8.175 
 te. 945 3.781 8.501 
 .982 3.926 8.828 
 | 1,018 4.071 9.154 
 | 1.054 4.217 9.480 
 1.091 4.362 9.807 
 1.127 4.507 10.188 
 1.164 4.653 10.459 
 1.200 4.798 10.785 
 1.287 4.943 yb es ot Peas 
 1.273 5.088 11.486 | 
 1.309 5.284 11.762 
 | 1.846 5.379 12.088 
 1.382 5.524 12.413 | 
 1.418 5.669 12.739 
 1.455 5.814 13.064 
 1.491 5.960 13.389 
 1.528 6.105 13.715 
 1.564 6.250 14.040 
 1.600 6.395 14.365 
 1.637 6.540 | 14.689 
 1.673 6.685 | 15.014 
 1.710 6.831 | 15.339 
 1.746 6.976 15.663 | 
 | 1.782 7.12 15.988 | 
 1.819 7.266 16.312 
 1.855 7.411 16.636 
 1.892 7.556 16.960 
 1.928 7.701 | 17.284 
 | 1.965 7.846 17.608 | 
 | 2.001 7.991 | 17.982 
 2.037 8.136 18.255 
 2.074 8.281 18.578 
 2.110 8.426 18.902 
 2.147 8.57 19.225 
 2.183 19.548 
 
 | 
 
 es) 
 Nl 9 
 — 
 oS 
 
 4 5 6 
 Stations. | Stations. |Stations. 
 582 .909 1.309 
 1..164 1.818 2.618 
 1.745 2.727 | 3.926 
 2.327 8.686 | 5.285 
 2.909 4.545 | 6.544 
 3.490 5.4538 | 7,852 
 4.072 6.362 9.160 
 4.654 7 270 10.468 
 5.235 8.179 11.775 
 5.816 9.087 13.082 
 6.398 9.994 14.389 
 6.979 10.902 | 15.694 
 7.560 11.809 | 17.000 
 8.141 12.716 | 18.304 
 8.722 13.623 | 19.608 
 9.303 14.529 | 20.912 
 9.883 15.485 | 22.214 
 10.464 16.341 | 23.516 
 11.044 17.246 24 817 
 11.624 18.151 | 26.117 
 12.204 19.055 | 27.416 
 2.784 19.959 | 28.714 
 13.363 20.863 30.012 
 13.943 21.766 31.308 
 14.522 22.668 32.603 
 15.101 23 57 33.896 
 15.680 24.471 35.189 
 16.258 25.372 36.480 
 16.837 26.272 | 37.770 
 17.415 27.171 | 39.059 
 7992 28.070 | 40.346 
 18.570 28.968 | 41.631 
 19.147 29.866 | 42.916 
 19.724 80.762 | 44.198 
 20.301 31.658 | 45.479 
 20.877 82.553 46.759 
 21.453 33.448 | 48.037 
 22.029 84.341 | 49.315 
 22.604 35.234 | 50.587 
 23.17 36.126 51.860 
 23.754 7.017 | 58.180 
 24.328 37.907 | 54.399 
 24.902 38.796 55.666 
 25 .476 39.684 56.931 
 26.049 40.57 58.193 
 26.622 41.458 59.454 
 27.195 42.343 60.712 
 27 767 a3 227 61.969 
 28 338 44.110 63 . 228 
 28.910 44.992 64.475 
 29.481 45.873 65.72 
 30.051 46.753 66.972 
 30.621 47 632 68.216 | 
 31.190 48.510 | 69.459 
 31.759 49.386 70.699 
 32.328 50.261 71.936 
 32.896 51.135 "3 171 
 33.464 52.008 74.403 
 34.031 52.880 75.632 
 34.597 53.750 76.859 
 
 298 
 
Degree 
 of 
 Curve. 
 
 O° 10’ 
 20 
 30 
 40 
 50 
 
 10 
 20 
 30 
 40 
 50 
 
 10 
 20 
 
 TABLE VUI.—MIDDLE ORDINATES. 
 
 Tn el a 
 
 7 
 
 Stations. 
 
 72.037 
 
 Data otot= 
 
 YO WF OUT 2 
 or 00 
 = 
 oS 
 
 ® 
 ~) 
 
 poh 
 
 CO 
 
 82.2 
 
 12 
 
 Stations. 
 
 103.675 
 108.747 
 113.808 
 118.841 
 
 123.862 
 128. 864 
 133.847 
 138.810 
 143.753 
 148.674 
 153.572 
 158.448 
 163.300 
 168.128 
 172.931 
 177.708 
 182.459 
 187.182 
 191.878 
 196.545 
 201.183 
 205.792 
 210.370 
 214.916 
 219.431 
 223.914 
 228 .363 
 232.7 
 
 237.160 
 241.507 
 245.818 
 250.093 
 254.331 
 258.531 
 262.694 
 266.818 
 270.904 
 274.949 
 278.955 
 282.919 
 286.843 
 
11 
 
 12 
 
 13 
 
 14 
 
 15 
 
 16 
 
 17 
 
 18 
 
 19 
 
 Degree 
 of 
 Curve. 
 
 | 
 
 Station. 
 
 4.045 
 
 cw Co Ww rw) 
 © 
 CO 
 
 2 
 Stations. 
 
 3 6 
 Stations. | Stations. | Stations. | Stations. 
 19.870 35.164. 54.619 78.083 
 20.193 35.729 55.486 79.805 | 
 20.516 36.294. 56.3538 80.523 
 20.888 36.859 57.218 81.739 
 21.160 7.423 58.081 82.951 
 21.483 37.986 | 58.943 84.161 
 21.804 88.549 | 59.804 85.368 
 22.126 39.111 | 60.663 86.571 
 22.448 39.67% 61.521 87.705 
 22.769 40 .284 62.377 88 . 969 
 23.090 40.795 63 232 90.164 
 23.412 41.355 | 64.085 91.3855 
 23.782 41.914 64.9387 92.542 
 24.053 42.473 65.787 93.727 
 24.374 43.031 66. 656 94.908 
 24.694 43 588 67.482 96 .086 
 25.014 44.145 | 68.32 7.260 
 25 .3834 44.701 69.17 | 98.431 
 25.654 45.256 | 70.018 | 99.598 
 25.974 45 .811 70.854 | 100.762 
 26.298 46 .365 (12092, = 101Ge2 
 26.612 46.919 72 .529 103.079 
 26.931 47 472 73.364 104.232 
 27.250 48 .024 74.197 105.381 
 27.569 48 .575 75 .029 106.527 
 27.887 49.126 75 . 859 107.669 
 28 . 206 49.676 76.687 108 . 807 
 28.524 50.225 77.518 109.941 
 28.841 50.7% 78 3837 111.071 
 29.159 51.32 79.159 112.197 
 29.476 51.868 79.979 113.319 
 29.794 52.414 80.798 114.488 
 30.111 52.959 81.614 115.552 
 30.427 53.504 82.429 116. 662 
 30.744 54.048 83 . 241 117.768 
 31.060 54.591 84.052 118.870 
 31.37 55.133 84.861 119.967 
 31.692 55.675 85 . 667 121.061 
 82.008 56.215 86.471 122.15 
 82.3823 56.755 8.274 128 .23 
 32.638 57.294 88 . 074. 124.315 
 32.953 57.882 88.872 25.891 
 33 . 267 58 .369 89. 668 126.463 
 33.582 58. 906 90 . 462 127.680 
 33.896 59.441 91.254 128.593 
 34.210 59.976 92.043 129.651 
 84.523 60.510 2.83 180.704 
 34.837 61.042 93.616 131.758 
 85.150 61.57 94 398 132.797 
 35.463 62.106 ae Yd 133 .&37 
 35.775 62.686 95 957 134.872 
 36.088 63.165 96.783 135 . 902 
 36.400 63.693 97 .506 186 .$28 
 36.712 64.221 98.278 137.848 
 37.023 64.747 99.047 188.964 
 37.3084 65.273 99.813 139.975 
 87.645 65.797 100.577 140.981 
 37.956 66.321 101.389 141 . 982 
 38 . 266 66.843 102.098 142.978 
 38.576 67.365 102.855 143.969 
 
 TABLE VIII.—MIDDLE ORDINATES. 
 
 aoe | 
 
TABLE IX.—LINEAR DEFLECTION TAPLE. 
 
 Q ( 
 | aoe | 900. | 1000. 
 } / | | 
 | sy | 0.87! 1.75] 2.62| 8.49] 4.386) 5.24) 6.11] 6.98) 7.85) 8.73 
 ph cape 1.75| 8.49] 5.24} 6.98] 8.73) 10.47] 12.22] 18.96) 15.71) 17,45 | 
 3) | 2.62| 5.24/ 7.85] 10.47] 13.09] 15.71) 18.38] 20.94) 23.56] 26.18 i 
 2 3.49) 6.98) 10.47] 13.96] 17.45] 20.94] 24.43) 27.92) 31.41) 34.90 | 
 39 | 4.36! 8.72| 13.09/ 17.45] 21.81] 26.18) 30.54| 34.90) 39.27) 43.63 i 
 ee eo Ag 5.24| 10.47] 15.71| 20.94] 26.18] 31.41} 36.65/ 41.88) 47.12| 52.85 a 
 30 | 6.11| 12.22] 18.32] 24.43] 30.54] 86.65) 42.75) 48.86) 54.97) 61.08 Hi 
 4 6.98] 13.96] 20.94) 27.92] 34.90} 41.88) 48.86] 55.84) 66.82) 69.80 I 
 30 | 7.85) 15.70) 23.56] 31.41) 39.26] 47.11] 54.96] 62.82) 70.67) 78.52 1A 
 rl: ob 8.73) 17.45) 26.17) 84.80} 43.62! 52.34] 61.07) 69.79] 78.51) 87.24 | 
 30 | 9.60! 19.19! 28.79] 38.33] 47.98] 57.57] 67.17| 76.76] 86.36] 95.96 Hi 
 6 10.47| 20.93' 31.40] 41.87] 52.34] 62.80] 73.27| 83.74] 94.20) 104.67 1H 
 i 30 | 11.34] 22.68] 34.02} 45.35] 56.67] 68.03} 79.37) 90.71] 102.05) 113.39 | 
 7 12.21) 24.42! 36.63] 48.84] 61.05] 73.26) 85.47| 97.68] 109.89} 122.10 i 
 | 30 | 13.08) 26.16) 39.24) 52.32] 65.40] 78.48] 91.56| 104.64] 117.73} 130.81 i 
 8 3.95| 27.90) 41.85) 55.80} 69.76] 83.71) 97.66 | 111.61} 125.56) 139.51 i 
 | 30 | 14.82| 29.64) 44.47) 59.29] 74.11] 88.93) 103.75 | 118.57| 133.40 | 148.22 il 
 | 9 | 15.69} 31.38) 47.08) 62.77} 78.46] 94.15} 109.84 125.53} 141.23) 156.92 i 
 | 30 | 16.56) 33.12) 49.68) 66.25} 82.81) 99.37 | 115.93) 182.49] 149.05) 165.62 i 
 | 10. =|: 17.43} 34.86) 52.29) 69.72] 87.16] 104.59 | 122.02) 189.45] 156.88 | 174.31 ! 
 30 | 18.30] 36.60] 54.90) 73.20] 91.50] 109.80) 128.10 | 146.40) 164.70) 183.00 |} 
 | 11 19.17|. 38.34] 57.51) 76.68] 95.85 | 115.01 | 184.18 | 153.35] 172.52] 191.69 i 
 | 30 | 20.04) 40.08) 60.11] 80.15] 100.19] 120.28 | 140.26 | 160.30] 180.34) 200.38 | 
 12 =| 20.91} 41.81] 62.72} 83.62] 104.53 | 125.43 | 146.34 | 167.25/ 188.15 | 209.06 i 
 30 | 21.77! 43.55] 65.32) 87.09! 108.87 | 130.64] 152.41 | 174.19] 195.96 | 217.73 I 
 13 22.64) 45.23) 67.92) 90.56] 113.20] 185.84] 158.48 | 181.13) 203.77 | 226.41 H 
 39 | 23.51| 47.01] 70.52| 94.03] 117.54 | 141.04 | 164.55 | 188.06] 211.57 | 235.07 i 
 14. =| 24.387} 48.75] 73.12} 97.50] 121.87 | 146.24 | 170.62} 194.99) 219.36 | 243.74 I 
 30,| 25.24) 50.48) 75.72) 100.96) 126 201 151.44 | 176.68 | 201.92] 227.16 | 252.40 i 
 | 15 | 26.11] 52.21) 78.32) 104.42] 130.53 | 156.63 182.74 | 208.84) 234.95 | 261.05 } 
 30 | 26.97) 53.94] 80.91) 107.88} 134.85 | 161.82 | 188.79 | 215 76) 242.73) 269.70 
 | 16 | 27.83] 55.67) 83.50) 111.34] 139.17] 167.01 | 194.84 | 222.68) 250.51 | 278.35 
 30 | 28.70| 57.40} 86.10) 114.79| 143.49 | 172.19 | 200.89 | 229.£9 | 258.29 | 286.99 
 17 29.56) 59.12] 88.69] 118.25 | 147.81 | 177.37 | 206. 93 | 236 .50| 266. 06| 295.62 
 30 | 30.42) 60.85] 91.27/ 121.70| 152.12 | 182.55 | 212.97 | 243.40) 273.82 | 304.25 1] 
 18 31.29} 62.57] 93.86) 125.15] 156.43 | 187.72 | 219.01 | 250.30] 281.58 | 312.87 
 30 | 32.15] 64.30] 96.45] 128.59] 160.74 | 192.89 | 225.04 | 257.19 | 289.34 | 321.49 
 19 33.01) 66.02) 99.03) 132.04] 165.05 | 198.06 | 231.07 | 264.08 | 297.08 | 830.09 | 
 30 | 33.87) 67.74 101.61] 135.48] 169.35 | 203.22 | 237.09 | 270.96 | 304.83 | 338.7 
 20 34.73 69.46 1104.19 138.92] 173.65 | 208.38 | 243.11] 277.84 | 812.57 | 347.30 al 
 30 | 35.59) 71.18 /106.77| 142.35] 177.94 | 213.53 | 249,12 | 284.71 | 320.30 | 355.89 | 
 21 |: 36.45) 72.89 109.34) 145. 79| 182.24) 218.68 | 255.13 | 291.58] 828.02 | 364.47 \ 
 30 | 37.30] 74.61 /111.91] 149.22] 186.52 | 223.83 | 261.18 | 298.44 ) 835.74 | 873.05 |} 
 22 38.16] 76.32 |114.49| 152.65| 190.81 | 228.97 | 267.13 | 305.29 | 343. 46 | 381 62 | 
 30 | 39.02] 78.04/117.05| 156.07] 195.09 | 234.11 | 273.13 | 312.14|351.16 | 390.18 . 
 93 39.87| 79.75 |119.62) 159.49] 199.37 | 239.24 | 279.12 | 818.99 | 858.86 | 398.74 | 
 30 | 40.73) 81.46 122.19) 162.91} 203. 64 | 244.87 | 285.10 | 825.83 | 866.56, 407.28 
 94 41.53) 83.16 124.75) 166.33] 207.91 | 249.49 | 291 .08 | 832.66 | 874.24 | 415.82 
 30 | 42.44) 84.87 |127.31 169.74| 212.18 | 254.61 | 297.05 | 839.48) 881.92] 424.36 [| 
 25 43.29 86.53 |129.86) 173.15] 216. 44 | 259.73 | 803 .02| 864.80 | 889.59 | 432.88 | 
 | 80 | 44.14) 88.23/132.42) 176.56] 220.70 | 264.84 | 808.98 | 853. 12|897.26 | 441.39 
 26 44.99| 89.98 134.97 179.96) 224.95 269.94 814.93 |359.92| 404.91 | 449.90 
 30 | 45.84! 91.68 /137.52) 183.36] 229.20 | 275.04 | 820.88 | 866.72 | 412.56 | 458.40 
 27 46.69} 93.38 /140.07) 186.76] 233.45 | 280.14 | 826.82 | 873.51| 420.20 | 466.89 
 30 | 47.541 95.07 |142.61| 190.15) 237.69 285.22 | 332.76) 380.30 | 427.83 | 475.37 . 
 28 48.38| 96.77 /145.15/ 193.54| 241.92 | 290.81 | 838. 69| 887.08 | 435.46 | 483.84 
 30 | 49.23| 98.46 /147.69 | 196.92) 246.15 295.33 | 844.62) 893.85 | 443.08 492.31 . 
 29 50.08 1100.15 150.23, 200.80] 250.38 800.46 | 350.53] 400.16 | 450.68 | 500.7 
 30 | 50.92/101 .84/152.76 203.68) 254.60. 305.52 | 356.44 | 407.36 | 458.28 | 509.20 
 30 51.76|103.53 155 .29| 207 .06| 258.82. 310.59 | 8362.35] 414.11 1465.87 | 517.64 
 
TABLE X.—COEFFICIENTS FOR VALVOID ARCS. 
 
 Oy 
 
 Z 
 I.—RaTIO OF u = —: 
 A 
 
 —— 
 
 3337 | 38336 
 
 3343 
 8339 
 
 | 3366 | .8364 
 . 8358 | .8356 
 . 3352 | 3350 
 - 3348 | 3346 
 
 3341 
 
 . 3337 | .3335 
 
 .3333 | 3331 
 
 L 10° | 20° | 30° | 40° | 50° | 60° | «0° | 80° | 90° | 100° | 110° | 120° 
 | | posts. | | 
 
 300 | .8518 | .3516! .3514|.3510| 3506) .3500 .8493} 3485 | 3476 | 3466 | .3455| .3444 
 
 400 | .3437 | 3436 | .3433 | .3430| .3426] .3421 | .3415) 8408} .3399 | .3390| .3380) .3368 
 
 500 | .3400 | .3398 | .3396 | .3393,] 8389) . 3383 | .3379] :3372| 3364 | .3856 8845} 3335 
 
 600 | .3879 | .38878 | .3376 |.3373 | 8369] .38365 | .38359) 8353] .8345 | .38337 | .8827| .3317 
 
 8361 
 8359 
 8348 
 3344 
 .8339 
 
 3357 
 83849 
 3844 
 8340 
 .3336 
 3831 
 3328 
 
 8353 | 3347) . 
 3345 | .38840 
 
 8340 | .3384! . 
 .8336 | 3331) . 
 .8931 | 3326} . 
 8827 | 8822) . 
 
 . b024 | 8319 
 
 3334 
 83826 
 3821 
 
 8326 
 8318 
 .8318 
 .3317 | .38310 
 .3313 | .8805 
 .3309 | .3301 
 . 3306 | 38298 
 
 .3316 | 
 . 3809 
 . 3304 
 83801 
 8296 
 8292 
 
 . 38289 
 
 8306 | 
 8299 | 
 8294 
 8291 
 .3286 
 8283 
 3280 
 
 co 
 2 
 Ww 
 Co 
 
 I.—RATIO oF 
 
 L 
 
 10° 
 
 tide 
 | 
 
 30° 
 
 90° | 100° | 110° | 120° 
 
 80° 
 
 400 
 500 
 600 
 700 
 800 
 900 
 1000 
 1200 
 1500 
 2009 
 
 300 
 
 7706 | 
 7611 |. 
 
 z 
 ‘ 
 
 F 
 
 |. (545 | 7506 | 7452 | .7384 | .'7 
 F 
 
 a 
 
 ‘ 
 
 . 1683 | 7648 | .7588 | .'7518) . 
 7588 | .7549 | .'7495 | 7425 | .7 
 7522! 7483 | 7430] 72 
 | 7508 |. 7 7 
 
 22 | 7499 | .'7461 | 7407) . 7 
 | 7492! 7454 | .7401 | . 7338 
 
 2} 7489 | 7450 | 7397 | .7329 | .'7% 
 05 | .'7483 | .7444 | 7391 |. 732 
 W501 | 17478 | 7440 | 7387 | 731! 
 bak Me be 7436 piaga ta 
 
 | 7218 | 7090 | .6949 | .6795 
 7130 | 7004 | .6865 | .6714 | 
 2). 7091 | .6966 | 6828) .6678 | 
 7070 | .6946 | .6808] . 6659 | 
 .7057 | .6933 | .6797'| .6648 
 .7049 | 6926 | .6789) 6640 | 
 7044 | 6920 | 6784 | 6635 | 
 (040.6917 6780) . 6632 | 
 5| 7035 | 6912) .6775 | . 6627 | 
 |. 031 | .6908 | .67'72 | . 6624 | 
 028 |-6904) .6760) .6621 
 
 F 2: a | | 
 
 . 6630 
 6551 
 .6516 
 6498 
 .6487 
 .6480 
 6475 
 6472 
 .6468 
 6464 
 .6461 
 
 469 | .7416| . 7 
 
 Tii.—Ratrio'or §, = —— = 
 A’—A 
 TO A CHANGE OF ONE DEGREE IN THE 
 
 l 
 — = LENGTH OF VALVOID ARC CORRESPONDING 
 
 ANGLE A. 
 
 L 
 
 re 
 f=) 
 fe} 
 
 100° 
 
 300 
 400 
 500 
 600 
 700 
 800 
 900 
 1000 
 1100 
 120) 
 1300 
 1400 
 1500 
 1600 
 1700 
 
 1800 
 1900 
 
 . 2000 
 
 CO OO 3 Od CI OT PB GY 0 
 
 SOOO 
 .46 |10. 
 .69/11.: 
 .21)12. 
 .08 13. 
 9513-5 
 182 |14.7 
 .69 |15.65 
 sO 10605 ; 
 4417.39.17. 
 
 62 | 61) 
 
 | 3.48 
 36} 35 
 23) 5 2 | 
 10| 6 
 
 rey 
 es 
 3D SD OT HR CO 09 
 
 97) 6 
 85| 7 .82| 
 72| 8.69| 
 
 =) 
 a) 
 © WH 38 OD SI O1 ® 09 09 i 
 5 lo} 
 | 
 bss | > 
 on 6 a 
 a => 
 o or 
 j=) Co Go GS GoW 
 mat ° Rie 0 ww co 
 *; OO GIO 
 IS? OT OU GY 0 ior) J 9 9-9 3 
 oe B W WW WW 
 wg Go CO HS HS OT Or 
 CO OI mae ses 
 co J J I 
 “ S peer et et 
 ee < Sas 2 OOH PA OUOT DD if : 
 y ROH ReTORS DG 
 e0) 
 Oo 
 a ° 
 oO 
 2} le) 
 (<5) 
 ~ fo) 
 el 
 _ 
 a 
 toh Eilers 
 (—p) 
 a ° 
 aa 
 oO 2 
 co 
 Qo ° 
 neg 
 i 
 
 ery 
 co 
 
 ror) 
 
 O DFO DOr 09 
 OMI ODOT ROO e 
 oS 
 oo 
 COWDNIOUP KW 
 
 ~ | 
 
 oie 
 ROO OID wr 
 CM) 
 i) 
 WOO WOIO DOB OCD 
 
 11 |12.03)|11.93 }11.80|11.65 11.48 111.29 86| 10.61 
 98 |12.89 }12.78 /12.64|12.48 12.30 112.10) 11.88/11.63! 11.37 
 84 }13.75 |13.63 }13.49 |13.32|13.12 12.91} 12.67 12.41} 12.13 | 
 71 |14.61)14.48 |14.33}14.15 13.94 18.71 | 13.46/13.18| 12.88 
 57 15.47/15 88 |15.17|14.98 |14.76 14.52/14.25 113.96) 13.64 
 
 ).01 15.81 
 3.86 16.65 
 
 115.58 15.3: 
 16.40 16. 
 
 30 |17.19}17.04| 
 
 302 
 
TRACK. 
 
 TABLE XI.—TURNOUTS AND SWITCHES FROM 
 §§ 180, 181, 182. 
 
 A STRAIGHT 
 
 GAUGE, 4 FEET i SyorEnS- = 4, iOS: 
 
 THRow, 5 INCHES = 0.417: 
 
 No Angle Dist. | Chord | | Switch | Radius | Log’thm.| Degree 
 N. ole BF. | af. AD De log. r. | of Curve. 
 4 | 14° 15/ 00" | 387.664; 387.8738) 11:209) - 150.656 2.177986 | 38° 45/5 Die 
 | 414 |12 40 49| 42.372) 42.113 12.610, 190.674 | 2.280292 | 30 24 09 
 : 5 11 2% 16 7.080 | 46.846| 14.012] 235.400 | 2.371806 | 24 31 36 
 544 | 10 23 20 51.788 | -51.575 | 15.413 284.834 2.454592 | 20 18 13 
 
 : Gy eed. 31.539 56.496 | 56.301 16.814 | 338.976 | 2.580169 | 16. 57 52 
 | 6146 | 8 47 51 61.204) 61.024 18.215 | 397.826 2.599693 | 14 26 25 
 ii 8 10 16 65.912 | 65.744 19.616 461.384 2.664063 | 12 26 34 
 : 16 | 7 37 41 | 70.620) 70.464) 21.017 | 529.650 | 2.723989 | 10 50 02 
 : 8 7 09°10 |. 75.828 | 75.181} 22.418 602.624 | 2.780046} 9 31 07 
 ) 8144 | 6 43 59 80.086 | 79.898} 28.820 680.3806 2.882704 8 25 47 
 : 9 6 21 35 84.744 | 84.613] 25.221 762.696 2.882352 7 31 04 
 : 916 | 6 01 32] 89.452) 89.328) 26.622 849.794 | 2.929814 | 6 44 46 
 : 10 5 43 29 94.160} 94.048} 28.023 941 .600 2.973866 6 05 16 
 : 104 | 5 27 09 98.868 | 98.756 29.424 10388 .114 3.016245 5 381 17 
 11 5 12 18 | 108.576 | 103.469 30.825 1139 .336 3.056652 5 01 50 
 
 : 11% | 4 58 45 | 108.284} 108.182 | 32.227 1245 ..266 8.095262 4 36 08 
 : 12 4 46 19 | 112.992 | 112.894) 33.628] 1855.904 | 3.182229) 4 138 36 
 : | z 
 
 GAUGE, 3 Feet. Turow, 4 INCHES = == 0) 338, 
 
 No Angle Dist. | Chord , Switch! Radius | Log’thm. | Degree 
 n. | Ff. BF, 55 2 OL (a log. r. of Curve. 
 4 | 14°15’ 00" 24 «| 23.815; 8 96.0 1.982271 | 62° 46/ 24” 
 
 4t46 | 12 40 49 27 =| 26.835 | ) 121.5 2.084576 48 36. 04 
 
 5 11 25 16 30 =| 29.851 10 150.0 2.176091 38 56 35 
 
 516 | 10 23 20 3: 32.865 | 11 181.5 2.258877 | 31 58 55 
 
 6 9 31 39 36 SOOO le 216.0 2.884454 26 46 07 
 
 616 8 47 51 389 38.885 13 258.5 2.403978 22 45 04 
 
 4 8 10 16 42 41 893 14 294.0 2.468347 19° 35. Of 
 
 76 tot 4) 45 44.900 15 307.5 2 -ER8274 17 02) 21 
 
 Teo! eOOE LT 48 47.906 16 384.0 2.584331 14 57 48 
 
 86 | 6 43 59 51 50.912 17 433.5 2.686989 3 14 47 
 
 9 6 21-35 54 53.917 18 486.0 2.65 6686 11 48 3% 
 
 944 OxO1- 32 Sf 56.921 19 541.5 2.736 10 35 46 
 
 10 5. 48 29 60 59.925 20 600.0 2. 9 383 38 
 
 101, 5 27 09 3 62.929 21 661.5 2. 8 40 12 
 
 11 5 12 18 66 65.932 22 726.0 2.860937 (ents 37 
 
 114% 4 58-45 69 68.935 23 193.5 2.899547 % 13 32 
 
 12 4 46 19 72 71.988 24 864.0 2.986514 6 &8 06 
 
 ANGLE AND DisTANCE OF MIDDLE FROG, F"" 
 
 | Gauge Gauge | Gauge | Gauge 
 
 No.| No. | Angle | 4,84. 3. || No.| No. Angle | 4,8%.| _ 3. 
 
 nd. nv" ys Dist. Dist. || 7, qu". he Dist Dist. 
 
 GE. \aie*s 4 aF'' ak; 
 
 4 | 2.817} 20° 07’ 36"| 26.786 | 17.037 || 8 5.651 | 10° 06! 44"| 53.817 | 33.97 
 
 416; 3.172) 17 54 52) 30.054 | 19.151 |} 814) 6.005| 9 31 08) 56.643 | 86.094 
 
 5 8.021| 16 08° 19) 33.374) 21-266 9 6.359 | 8 59 80] 59.969 | 88.213 
 
 5 6) 3.881; 14 40 58] 36.695 | 23.383 || 9146) 6.713| 8 81 10] 63:296 | 40.333 
 
 G6 | 4.235) 138 27°57 | 40.018 | 25.500 |} 10 | 7.067 | 8 05 40] 66.623 | 42.45: 
 
 614| 4.589) 12 26 07} 43.342.| 27.618 || 1014) 7.420) 7 42 35| 69.950 | 44.57 
 % | 4.943) 11 83-04] 46.666 | 29.736 i Gee a ae 7 21 36) 73.277 | 46.693 
 746) 5.297| 10 47 02) 49.991 | 31.855 || 1116) 8.128] V 02 26] 76.605 | 48.813 
 
 8 | 5.651) 10 06 44/1 58.317 | 33 974 2 | 8.482 | 6 44 51} 79.982 | 50.984 
 
 ape ee — 
 2 
 
TABLE XII.—MIDDLE ORDINATES FOR CURVING RAILS. 
 
 LENGTH OF RAIL-CHORD. 
 D : a D 
 a2 | so { 28 | 26 | 24 | 22.| 20 | 48 | 16 | 14 | 12 | 10 
 1°} .022| .020| .017} .015| .013| .011| .009 .007 |.006 |.004|.003 | .002| 1° 
 2} .045 | .039] .034) .030| .025 .021 | .017| .014 |.011 | .009 | .006 | 004 2 
 3 | .067| .059} .051) .044} 038) .032| .026} .021 |.017 |.013 |.009 | .007| 3 
 4 | .089| .079| .068| .059| .050 .042 | .035 | .028 |.022 |.017 |.013 | .009| 4 
 5 | .112} .098| .086| .074| .063| .053) .044) .035 |.028 | .021 /.016 | 011] 5 
 6 | .134| .118} .103 | .088| .075 | .063| .052) .042 | .034 | .026 | .019 | 013] 6 
 7 | -156| .137| .120 | .103| .088 | .074 | .061 | .049 | .039 |.030 |.022 | 015) 7 
 8 | .179| 157] .187 |) .118) .100] .084| .070] .057 | 045 | .034 |.025 | .017| 8 
 9 | 201} .177) .154] .133) .113] .095 | .078| .064 |.050 |.038 |.028 | .020] 9 
 10 | .223| .196 | 171 | .147 | .126] 105) .087| .071 |.056 | .043 | 031 | .022 | 10 
 11 | .245} .216| .188| .162] .138 | .116 | .096 | .078 |.061 |.047 | 035 | 024 |. 11 
 12 | .268) .235| .205| .177| .151| .127| .105 | .085 |.067 |.051:| 038 | .026 | 12 
 14 | B12} .274| .238 | .206) .175 | .147 | .122 | .099 |.078 |.(60 |.044 | .030| 14 
 16 | .856| .313 | .273 | .235 | .200| .168 | .139| .113 |.089 |.068 | .050 | .035 | 16 
 18 | .400} .352| .307 | .264) .225 | .189 | .156| .127 |.100 | .077 |.056 | .039 | 18 
 20 | 445) .3891 | .3840) .293| .250 | .210 | .174| .141 |.111 |.085 |.063 | .043 | 20 
 24] 531) .467| .407) .851 | .299 | .251 | .207| .168 |.133 |.102 | .075 | .052 | 24 
 28 | 618) .543| .473| .408 | .347| .292 | .241 | .195 |.154 |.118 |.087 | .060| 28 
 2 | .705| .619| .539| .465 | .396 | .833 | .275 | .223 |.176 |.135 |.099 | .069 | 32 
 36 | .791| .696| .606 | .522! .445 | .873 | .309| .250|.197 |.151 |.111 | .077 | 36 
 40 | .878| .772 | .672 | .579 | .493 | .414 | .3842 | .277 |.219 |.168 |.123 | .086 | 40 
 45 | .983| .863| .752| .648) .552| .463 | .883 | .305 |.245 |.188 |.137 | .096 | 45 
 50 (1.087 | .955| .831 | .716) .610| .512 | .423] .343 |.271 |.207 |.152 | .106 | 50 
 TABLE XUI.—DIFFERENCE IN ELEVATION OF RAILS ON 
 CURVES, §201. . 
 VELOCITY IN MILES PER Hour. 
 D D 
 10; 15 | 20 | 2 | 30 | 3 40 45 50 60 
 
 1 | .006 | .013 | .023 | .036 | .051 | .070| .091 | .116| .143| .206| 1 
 2| .011 | .026 | .046 | .O71 | .103 | .140 | .183 | .281 | .285 | .410] 2 
 3 | .017 | .039 | .069 | .107 | 154.) .210) .274] 846] .427) 612] 3 
 4 | .023 | .051 | .091 | .143 | .206 | .280| .865] .461 |] .568| 811] 4 
 5 | .029 | .064 | .114 | .179 | .257 | .849 | .455 | .574 | .707 | 1.006 | 5 
 6 | .034 | .077 | .137 | .214 | .808 | .418 | .545 | .687 | .844| 1.196 | 6 
 7 | .040 | .090 | .160 | .250 | .859 | .487 | 1634 | .798 | .97 
 
 8 | .046 | :103 | .183 | .285 | .410 | .556 | .723 | .908 | 1.112 
 
 9 | C51 | .116 | .206 | .820 | .460 | .624 ; .811 | 1.017 
 
 10 | .057 | .129 | .228 | .856 | .511 | .692 | .898 | 1.124 
 
 11 | .063 | .142 | .251 | .891 | .561 | .760 | .984 
 
 12 | .069 | .154 | .274 | .427 | .611 | .826 | 1.069 
 
 30 | . 818 ‘11 | .959 
 : /1.088 
 
io) 
 cal 
 — 
 Zi 
 re 
 — 
 << 
 =} 
 i 
 om) 
 = 
 <— 
 iva) 
 i=) 
 — 
 — 
 os 
 T 
 — 
 amma 
 bet 
 <= 
 <e 
 2Q 
 =< 
 & 
 
TABLE XIV.—GRADES AND GRADE ANGLES. 
 
 | | 
 
 ae | Feet | Feet 
 per | Feet per | Inclina-|| per | Feet per | Inclina- || per | Feet per |Inclin- 
 Sta-| Mile. tion. || Sta-| Mile. tion. || Sta-| Mile. | ation. 
 
 tion tion tion 
 lo se So e/ ” } (oe URL 
 01 528 21 .51 | 26.928 17 32 1.01 | 53.328 34 48 
 2 1.056 41 .52 | 27.456 17 53 1.02 | 53.856 35 04 
 03 1.584 1 02 .538 | 27.984 18 18 || 1.08 | 54.384 35 24 
 04 2.112 1 23 .54 | 28.512 18 34 || 1.04 | 54.912 35 45 
 .05 2.640 1 43 .55 | 29.040 18 54 || 1.05 | 55.440 36 05 
 .06 3.168 2 04 56 | 29.568 19 15 || 1.06 | 55.968 36 26 
 07 3.696 2 24 57 | 30.096 19 86 || 1.07 | 56.496 36 47 
 .08 4.224 2 45 58 | 30.624 19 56 |' 1.08 | 57.024 37 08 
 09 4.752 3 06 59 | 31.152 20 17 1.09 | 57.552 87 28 
 10 5.280 3 26 60 | 31.680 20 88 || 1.10] 58.080 37°49 
 11 5.808 3 47 61 | 382.208 20 58 1.11 | 58.608 38 09 
 12 6.336 4 08 62 | 32.736 21.19 || 1.12 | 59.136 388 30 
 13 6.864 4 28 63 | 33.264 21 389 |) 1.138 | 59.664 38 51 
 14 7.392 4 49 64 | 33.792 22 00 || 1.14 | 60.192 39 11 
 15 7.920 5 09 65 | 34.320 22:21) || 1:15") °60:720 39 32 
 .16 8.448 5 30 66 | 34.848 22 41 1.16 | 61.248 39 53 
 mls 8.97 Ns Se BSE 25°02 SHAT bie 1G 40 13 
 .18 9.504 6 11 68 | 35.904 23 23 || 1.18 | 62.304 40 34 
 .19 | 10.032 6 32 69 | 386.4382 23 43 || 1.19 | 62.882 40 54 
 20 | 10.560 6 53 70 | 36.960 24 04 1.20 | 63.360 41 15 
 21 | 11.088 7 13 71 | 387.488 24 24 1.21 | 68.888 41 35 
 22 | 11.616 7 34 72 | 388,016 24 45 || 1.22] 64.416 41 56 
 23 | 12.144 7 54 7 38.544 25 06 1.23 | 64.944 42 17 
 24 12.672 8 15 74 39.072 25 26 1.24 65.472 42 38 
 25 | 13.200 8 36 7 39.600 25 47 1.25 | 66.000 42 58 
 26 | 13.728 8 56 7 40.128 26 08 1.26 | 66.528 43 19 
 R27 | 14.256 9 17 77 | 40.656 26 28 1.27 7.056 43 39 
 28 | 14.784 9 38 7 41.184 26 49 1.28 7.584 44 00 
 29) 15.812 9 58 7 41.712 27 09 1.29 | 68.112 44 21 
 30 | 15.840 1019 || .80 | 42.240 7 30 1.30 | 68.640 44 41 
 31 | 16.368 10 39 .81 | 42.768 27.51 || 1.81 | 69.168 45 02 
 82 | 16.896 11 00 .82 | 43.296 28 11 || 1.82) 69.696 45 23 
 33 7.424 ib hepa .83 | 43.824 28 32 1.83 | 70.224 45 43 
 34 | 17.952 11 41 .84 | 44.352 28 53 1.34 | 70.752 46 04 
 85 | 18.480 12 02 .85 | 44.880 29 13 1.85 |. 712280 | 4624 
 36 | 19.008 12 23 .86 | 45.408 29 34 || 1.386] 71.808 46 45 
 7 | 19.536 12 43 .87 | 45.936 20 54 iad 87% leiecsae 47 06 
 38 | 20.064 13 04 88 | 46.464 80 15 || 1.38]. 72.864 7 26 
 39 20.592 13 24 89 46.992 30 36 1.39 73.392 7 47 
 40 | 21.120 13 45 90 7.520 30 57 || 1.40 | 73.920 48 08 
 41 | 21.648 14 06 .91 | 48.048 31 17 1.41 | 74.448 48 28 
 2 22.176 14 26 .92 48.576 81 38 || 1.42 74.976 48 49 
 43 22.704 14 47 93 49.104 31 58 1.43 75.504 49 09 
 44 | 23.232 15 08 94 49 .632 32 19 || 1.44 76.032 49 30 
 45 | 23.760 | 15 28 -95 | -50.160 382 39 || 1.45 | 76.560 49 51 
 46 | 24.288 15 49 .96 50.688 33. 00 1.46 77.088 50 11 
 47 | 24.816 16 09 977) 51, 216 33 21 1.47 | 77.616 50 382 
 48 | 25.344 16 30 .98 | 51.744 83 41 1.48 | 78.144 50 52 
 49 | 25.87 16 51 .99 | 52.272 34 02 1.49 | 78.672 51 13 
 50 | 26.400 uci a 1.00 | 52.800 34 23 || 1.50 | 79.200 51 34 
 
TABLE X1IV.—GRADES aND GRADE ANSLES, 
 
 Feet per Inclina- || per | Feet per 
 
 WNWNWWNKTWWWWew 
 
 D9 aA 0 
 Sete BAT 
 
 wopee wapapaoraprw 
 2g Oe a3 
 
 OPAC 
 
 Co 09 09 C8 
 1) 
 fan io) 
 
 SOODDYIRIBMONM 
 STMONSMAS WAS & 
 
 woe 
 on) 
 
 frek fe pk peek ek pe ek pe pe ek ek ek ek ek ek ek ek pk pk pe ek eek fk pe 
 
 CORRES RRR RR He C9 C2 CO CO 09 GO GH 09 2 
 
 DD et ek et et ed et ek et ek ek ek dk ek ek ek ek ek ek ek tk et ek ek at eed. Pek Pek Ret et ek et Bk ek Ped Ped Rt 
 Ne ee ae os : . pele Ey Ear aui, ae, ah -@ ° eae ee we eh ees ° aie ca eee es Neots ie Sap 
 - m Ae We a SiG 3 5 sone a 2A 4 > he xy se, he 
 
 Mile. 
 
 | 
 
 108.240 | 
 110.880 
 
 118.520 | 
 116.160 
 118.800 
 121.440. | 
 124.080 
 126.720 | 
 
 129.360 
 
 7 
 (3) 
 NN 
 for) 
 (—) 
 
 — 
 =, 
 or) 
 [ey] 
 rp) 
 © 
 
 et 
 ~3 
 
 io 2) 
 forge) 
 [=Jer 
 oom 
 
 © 29 
 OW 
 oo =~2 
 
 > o9 D> 
 SS WS 
 SS 
 
 258 ..720 
 
 264.000 
 
 OVO OT OT OTT 
 BOR WWE 
 SSSSS6Sso 
 
 WM WNWNWNWNWNWNWWWD WDHB HHH eee Yee ee Bee eee ee ee Oo 
 a ere S rN?) SS Par Cm 
 
 wa 
 
 we 
 
 2D 0O 
 
 VW WW We 
 
 we 
 
 2, 
 
 Feet per | 
 
 Mile. 
 
 269.280 | 
 274.560 
 279.840 | 
 285.120 
 290. 400 
 295. 680 
 300.960 
 306.240 
 311.520 
 316.800 
 
 322.080 
 327.860 
 352.640 
 337.920 
 843.200 | 
 348 .4&0 | 
 353.760 | 
 359 .040 
 
 364.820 | ¢ 
 
 369.600 
 
 374.80 | ¢ 
 
 380.160 
 8&5 .440 
 360.720 
 306 .C00 
 401.280 
 406 .5€0 
 411.840 
 417.120 
 422.400 
 
 306 
 
 | 
 
 gcc cco ce to CCV cmH CU CU CHE CUCU WW Y 
 
 mb eww WMeRH He OOo 
 MPWOMWO BDBMWOOUW MW Mor > 
 rc 
 
 OT OT OTe 
 
 Or OV OT Or OF OF Ot OF OF O1 
 
TABLE XV.—FOR OBTAINING BAROMETRIC HEIGHTS IN FEED. 
 
 Barom- Dur 
 eter. 23 0 ' | Diir. per 
 Traits 0.00 0.02 0.04 0.06 0.08 “002 fe 
 ay (i 
 ae aes : 
 19°.0 16832 16860 16888 16915 16943 2.8 
 a1 16970 16997 17025 17052 17080 2.8 
 ae 17107 17134 17162 17189 7216 - 2 
 33 17248 47270 17298 17325 17352 23% 
 4 17379 17406 17483 17460 17487 2.7 
 5) 17514 17540 17567 17594 17621 230 
 .6 17648 T674 17701 17728 17755 22% 
 =F 17781 17808 17834 17861 17887 enki 
 8 17914 17940 17967 7993 18020 2.7 
 9 18046 18072 18099 18125 18151 2.6 
 20°.0 1817 18204 18230 18256 18282 2.6 
 1 18308 18334 18360 18386 18413 2.6 
 32 18438 18464 18490 18516 18542 2.6 
 iS 18568 18594 18620 18645 18671 2.6 
 \ 4 18697 18723 18748 1877 18799 236 
 ati 5 18825 18851 18376 18902 18927 2.6 
 Hit .6 18953 18978 19004 19029 19054 rates 
 i 7 19080 19105 19130 19156 19181 2.5 
 | 8 19206 19231 19256 19282 19307 2.5. 
 i 9 19332 19357 19382 19407 19432 2.5 
 21°.0 19457 19482 19507 195382 19557 Pia) 
 a! 19582 19606 19631 19656 19681 2.5 
 2 19706 19730 19755 19780 19804 2.5 
 aa 19829 19854 19878 19903 19927 2.5 
 4 19952 19976 20001 20025 20050 2.5 
 35 20074. 20098 20123 20147 20172 250 
 6 20196 20220 20244. 20269 20293 2.4 
 are 20317 20341 20365 20389 20413 2.4 
 8 20438 20462 20486 20510 20534. 2.4 
 Hi 9 20558 20581 20605 20629 20653 2.4 
 } | 222°0 20677 20701 20725 20748 20772 2.4 
 tk 1 20795 20820 20843 20867 20891 2.4 
 i 12 20914. 20938 20952 20985 21009 aA: 
 iit 3 21032 21056 21079 21103 21126 2.4 
 iy i 4 21150 21173 21196 21220 =| ~— 21243 2.3 
 Wig 5 21266 21290 21313 21336)- ..| 2521859 2.3 
 | 6 21383 21408 21429 21452 21475 Bae 
 Ha ae 21498 21522 21545 21568 21591 2.3 
 Wit 8 21614 21637 21660 21683 21706 Pe 
 Palin| 9 21728 21751 2177: 21797 21820 es 
 Hil 23°.0 21843 21866 21888 21911 21934. Te 
 Ht 1 21957 21979 22002 22025 22047 2.3 
 VHT 2 22070 92022 92115 22138 | 22160 2.3 
 init io 22183 22205 22228 22250 22972 ue 
 ij 4 22295 92317 22340 22362 22384 Dae 
 Wi 5 22407 92429 22451 22474 22496 9.2 
 i | 6 22518 22540 22562 22585 22607 Pe 
 sai ai 22629 292651 22673 22695 22717 2.2 
 8 22739 22761 22783 22805 99907 2.2 
 i) 9 22849 92871 22898 92915 22937 2.2 
 24°.0 22959 22981 23003 23024 23046 272 
 1 23068 23090 23111 23133 23155 Psy 
 Hit 2 23176 23198 93220 93941 23263 9.2 
 iy il 3 23285 23306 93328 23349 93371 2.2 
 4 23392 93414 23435 23457 23478 2.2 
 . 5 23500 23521 23542 23564 23585 2.1 
 i 6 23606 23628 23649 23670 23692 24 
 i} “fe 23713 23734 93755 23776 | 23798 Ost 
 WA | | 8 23819 23840 23861 23882 23903 2.1 
 9 23924 23945 23966 23987 24008 ot 
 
 307 
 
TABLE XV. 
 
 —FOR OBTAINING BAROMETRIC HEIGHTS IN FEET. 
 
 Barom- 
 eter. 
 Inches 
 nai (Ie 
 
 25° 
 
 26°. 
 
 . 
 [ry 
 
 27°. 
 
 “rs 
 
 ~ 
 
 28°. 
 
 29°. 
 
 30°. 
 
 las) 
 
 pean) 
 
 0.00 
 
 0.02 
 
 0.04 
 
 0.06 
 
 0.08 
 
 WOIRAMIPWwWHO | 
 
 DIOP WWHOS 
 
 DOIN WWH OS OWVIOURWWHO 
 
 CS OO =2 C2 OTH CO CO ES 
 
 WHO 
 
 DIR OAC 
 
 24029 
 24134 
 24238 
 24342 
 24446 
 24549 
 24651 
 24754 
 24855 
 24957 
 25058 
 25159 
 25259 
 25359 
 25458 
 20557 
 25656 
 25755 
 25853 
 25950 
 26048 
 26145 
 26241 
 26337 
 26433 
 26529 
 26624 
 26719 
 26813 
 26908 
 
 27001 
 27095 
 27188 
 27281 
 27373 
 27466 
 27557 
 
 27649 
 27740 
 27831 
 27922 
 28012 
 28102 
 28192 
 28281 
 28370 
 28459 
 28547 
 28635 
 28723 
 98811 
 28898 
 QB9R5 
 29072 
 29158 
 29244 
 29330 
 29416 
 29501 
 29586 
 
 24363 
 
 24466 
 
 24569 
 24672 
 2477. 
 
 24876 
 24977 
 25078 
 25179 
 25279 
 25379 
 25478 
 25577 
 25676 
 
 Sei 
 257°C¢ 
 
 25872 
 25970 
 
 26067 
 26164 
 26260 
 26357 
 26452 
 26548 
 26643 
 26738 
 26882 
 
 26926 
 
 27020 
 27114 
 27207 
 27299 
 27392 
 27484 
 27576 
 27667 
 27758 
 27849 
 27940 
 28030 
 28120 
 28209 
 28299 
 28388 
 28476 
 28565 
 98653 
 28741 
 28828 
 28915 
 29002 
 29089 
 29175 
 29261 
 29347 
 29483 
 29518 
 
 29603 
 
 24071 
 24176 
 24280 
 24384 
 24487 
 24590 
 24692 
 24794 
 24896 
 24997 
 
 25098 
 25199 
 25299 
 25399 
 25498 
 25597 
 25696 
 25794 
 25892 
 25989 
 
 26086 
 26183 
 26280 
 26376 
 26472 
 26567 
 26662 
 26757 
 26851 
 26945 
 
 27039 
 27132 
 27225 
 27318 
 27410 
 27502 
 27594 
 27685 
 orrent 
 
 wlidd 
 27867 
 27958 
 28048 
 28138 
 28227 
 28317 
 28405 
 28494 
 28582 
 28670 
 28758 
 98846 
 28933 
 99020 
 29106 
 99192 
 29278 
 29364 
 29450 
 29535 
 
 £9620 
 
 24092 
 24197 
 24301 
 24404 
 24508 
 24610 
 247138 
 24815 
 24916 
 25018 
 
 25118 
 25219 
 25319 
 25419 
 25518 
 25617 
 25715 
 25813 
 25911 
 26009 
 
 26106 
 26203 
 26299 
 26395 
 26491 
 26586 
 26681 
 26776 
 25870 
 26964 
 
 27058 
 27151 
 27244 
 27386 
 27429 
 27521 
 27612 
 27704 
 27795 
 27885 
 27976 
 28066 
 28156 
 28245 
 283384 
 28423 
 28512 
 28600 
 28688 
 28776 
 28863 
 28950 
 29037 
 29124 
 29210 
 29296 
 29381 
 29467 
 99552 
 
 29637 
 
 24113 
 RAR? 
 24321 
 24425 
 24528 
 24631 
 24733 
 24835 
 24937 
 25038 
 
 25138 
 25239 
 25339 
 
 25931 
 
 26028 
 
 26125 
 26222 
 26318 
 26414 
 26510 
 26605 
 26700 
 26795 
 26889 
 £6983 
 
 TO76 
 27169 
 27262 
 27355 
 27447 
 27589 
 27631 
 alae 
 27813 
 27904 
 27994 
 28084. 
 28174 
 28263 
 28352 
 28441 
 28529 
 28618 
 28706 
 28793 
 
 28881 
 88968 
 29054 
 29141 
 29227 
 29313 
 29398 
 29484 
 99569 
 
 29654 
 
 308 
 
 ) 
 
 WOWOWOHOHNHDH SDOOSOSOOOSCSS COSCOHH HEHEHE 
 
 G0 00 G0 G0 G0 G6 GD 0D DD OM MMO MDMDMDwO 
 
 | aS Beek peek pr eek fed frat freak fk peek ek bak fk rk fr fk fk eek fem foek fk fk peek fed fred fk Peek Pek ee ee ek DO VDMV AD WO AMMA WWAW WWM WW WWwiwe 
 MINIT WII OW To ooo: oof ¢ ae 
 
TABLE XVI.—COEFFICIENT OF CORRECTION FOR TEMPERATURE. 
 
 ! | 
 t+ 4 — 64° t-+¢t/ ~ 64°]! ttt — 64°]| l¢-+¢ — 64° | 
 / /) / — = / ae = 
 
 Bed 900 at 900 bat 900 . Sa 900 
 
 = 4. | 2 | 
 20° 0489 65° | + .0011 110° | + .0511 || 155° 1011 
 
 21 = .0478 || 66 0022 || 111 0522 ||. 156 .1022 : 
 
 22 0467 7 0033 112 0533 157 .1033 | 
 
 23 0456 68 0044 113 0544 || 158 1044 : 
 
 24 . 0444 69 0056 114 0556 |) 159 . 1056 | 
 
 25 0433 "0 0067 115 .0567 | 160- 1067 | 
 
 26 . 0422 v4 0078 116 .0578 || 161 1078 | 
 
 Q7 .0411 72 0089 117 .0589 |) 162 1089 
 
 | 28 ~ .0400 ve 0100 118 0600 | 163 1100 | 
 29 .0389 7. 0111 119 0611 164 1111 
 
 30 0378 "5 .0122 120 + 0622 165 .1122 | | 
 
 { 31 — .0367 Vi + 0133 121 0633 || 166 + .1133 | 
 
 uh 32 0356 U7 0144 122 0644 |) 167 1144 | 
 
 in 33 0344 78 0156 123 .0656 |) 168 1156 | 
 ih 34 .0333 ” 0167 124 .0667 || 169 1167 
 
 35s .0322 || 80 0178 125 .0678 || 170 1178 : 
 
 | 36 0311 81 .0189 126 .0689 || 171 1189 | 
 Hail 37 .0300 82 .0200 127 0700 || 1% 1200 
 Sa 38 0289 83 0211 128 0711 1% 1211 
 ne | 39 0278 84 0222 129 .0722 74 . 1222 
 hale 40 0267 85 0233 130 + .0733 || 1%5 .1233 
 Ma 41 — .0256 86 4. (0244 131 0744 || 17 +. 1244 
 A a 2 0244 87 0256 132 0756 || 177 .1256 
 Hi a | 43 0233 88 .0267 133 0767 ted? 1267 
 LT 44 0292.6 1 89 0278 134 0778 7 1278 
 a 45 0211 90 0289 135 .0789 | 180 1289 
 
 lH 46 -0200 91 -0300 || 136 0800 || 181 1300 | 
 
 Ay .0189 92 0311 137 .0811 | 182 1311 
 48 0178 93 . 0322 138 (822 |) 1838 .1322 
 49 .0167 || 94 .0333 139 0833 |! 184 1333 
 50 — .0156 95 0344 140 + .0844 || 185 1344 
 51 0144 96 + .0856 141 0856 || 186 + .1356 
 52 0133 97 0367 142 .0867 || 187 .1367 
 53 .0122 98 037 143 .0878 |) 188 .1378 
 54 0111 99 0889 144 .0889 || 189 .1389 
 55 .0100 || 100 .0400 145 .0900 |; 190 1400 
 
 56 0089 101 0411 146 0911 191 1411 | 
 
 57 0078 102 0422 147 0922 |; 192 1422 | 
 
 58 0067 103 0433 148 0933 || 193 1433 | 
 59 0056 104 0444 149 .0944 || 194 1444 
 
 60 0044 105 0456 150 + .0956 195 1456 | 
 
 61 — .0033 106 + .0467 151 .0967 || 196 + .1467 | 
 
 62 .0022 107 0478 152 .0978 || 197 1478 | 
 63 .0011 || 108 0489 153 0989 198 1489 
 
 64 .0000 — || 109 .0500 154 .1000 |} 199 1500 | 
 
 Hii TABLE XVII.—CORRECTION FOR EARTH’S CURVATURE AND 
 REFRACTION. §119. 
 
 . L° | H°|| L° | He |] Lo | wo || Le | B° || Lo | we ||Miles| He 
 i| 800 | .002 |; 1300} .035 || 2300] .108 || 3300| .223 || 4300! .379 || 1 ye 
 400 | .003 |, 1400] :040 || 2400) '118 || 3400] 1237 || 4400| {397 |} 2 | 2/285 | 
 tj 500 | .005 |) 1500; .046 '| 2500] 128 || 8500) .251 |/ 4500] .415 || 3 | 5.142 | 
 | 600 | .007 |} 1600) .052 ;| 2600] .139 || 3600| |266 || 4600] 1434 || 4 | 9.141 
 i] 700 | .010 || 1700; .059 }} 2700] .149 || 3700) .281 || 4700! .453 5 14.282 
 800 ] .013 || 1800} .066 | 2800} .161 || 3800) .296 || 4800| 472 6 | 20.567 
 900 | .017 || 1900] .074 || 2900) :172 || 3900) [312 || 4900| 1492 || 7 | 27.994 
 Wii 1000 020 |} 2000} 082 || 8000) .184 || 4000} .328 15000] .512 | 8 36.563 
 Wit 1100 | .025 | 2100} .090 |} 3100} .197 || 4100] .345 ||5100} .533 9 46.275 
 | 1200 | .030 || 2200] .099 |].3200} |210 || 4200] 362 |} 5200] 554 || 10. | 57.130 | 
 
TABLE XVIIl.—COEF FICIENT FOR REDUCING INCLINED STADIA 
 MENTS TO THE HORIZONTAL. § 224. 
 
 ° 
 
 10 
 
 MEASURE 
 
 0’ 
 
 1.000000 
 . 999696 
 . 998782 
 997261 
 
 995134 
 
 992404 
 
 989074 
 -985148 
 . 980631 
 975528 
 . 969846 
 -963591 
 . 956772 
 .949396 
 
 941473 - 
 
 .933011 
 924022 
 .914517 
 
 .904507 - 
 .894003. - 
 883020 - ; 
 
 .871569 ~ 
 
 .859667 
 847326 
 
 .834561 °° 
 
 821390 
 807826 
 . 793888 
 T9591 
 764954 
 - 749994 
 TBA729 
 . 719179 
 . 703361 
 687296 
 
 .671002 ° 
 
 .654500 
 .637810 
 . 620952 
 . 603946 
 586814 
 
 .569576 
 992253 
 .534867 
 .517438 
 
 .499988 © 
 
 10’ 
 
 . 999992 
 999586 
 998571 
 . 996949 
 . 994721 
 .991891 
 . 988461 
 . 984436 
 .979821 
 .974621 
 . 968843 
 . 962494 
 . 955581 
 .948113 
 . 940100 
 .931550 
 922474 
 . 912883 
 . 902790 
 .892206 
 .881143 
 .869617 
 .857640 
 .845227 
 . 832394 
 .819156 
 .805529 
 . 79152 
 
 helio 
 . 762483 
 T4 7471 
 (82157 
 .716561 
 . 700700 
 . 684595 
 . 668266 
 .651731 
 .635011 
 .618127 
 .601099 
 .583948 
 
 566694 
 .549359 
 .531964 
 .514530 
 .497079 
 
 20" 
 
 999967 
 . 999459 
 . 998343 
 . 996619 
 994291 
 .991360 
 987831 
 . 983708 
 . 978995 
 .973698 
 967824 
 . 961380 
 954375 
 946815 
 938711 
 . 9380073 
 920911 
 911236 
 .901060 
 .890395 
 879254 
 867652 
 855601 
 843117 
 .830215 
 .816911 
 803221 
 789161 
 TATAD 
 . 760002 
 144939 
 72957 
 . 718935 
 .698033 
 .681889 
 665524 
 . 648957 
 . 632208 
 .615299 
 598248 
 .581079 
 
 .5638810 
 546464 
 529061 
 .511622 
 
 -494170 
 
 30’ 40’ 
 . 999924 .999865 
 9993815 . 999154 
 . 998098 997886 
 99627¢ .995910 
 993844 993381 
 . 990814 . 990250 
 987185 . 986522 
 982963 982202 
 978152 977294 
 972759 . 971804 
 . 966790 . 965739 
 . 960252 .959107 
 953153 .951916 
 945502 944174 
 .937309 .935891 
 928582 927077 
 919334 917742 
 90957. . 907899 
 899316 897558 
 .888571 886733 
 877352 8754387 
 865674 . 863684 
 853550 851487 
 .840996 838862 
 828025 825825 
 814656 812890 
 800903 (98575 
 786783 784396 
 772814 . 769870 
 157518 755015 
 142399 . 789850 
 . 726989 . 724893 
 . 7113802 . 708662 
 695858 692677 
 679176 676457 
 662776 . 660023 
 646177 . 643893 
 629401 626588 
 -612466 - . 609630 
 595395 592537 
 978207 575332 
 .560924 558036 
 943567 . 540668 
 -526156 528251 
 508714 505805 
 .491261 488353 
 
 50’ 
 
 999789 
 998977 
 ‘997557 
 . 995531 
 992901 
 989670 
 985843 
 981424 
 976419 
 970833 
 964673 
 957948 
 950664 
 942831 
 934459 
 9255577 
 916137 
 906209 
 895787 
 884883 
 873510 
 
 861681 
 849412 
 836718 
 823613 
 .810113 
 196236 
 781998 
 67416 
 152509 
 T37294 
 
 721790 
 
 706015 
 689990 
 .673733 
 657264 
 640604 
 623772 
 .606790 
 589677 
 572455 
 
 555145 
 537768 
 520345 
 502897 
 485445 
 
TABLE XIX.—LOGARITHM OF COEFFICIENT FOR REDUCING _IN- 
 CLINED STADIA MEASUREMENTS TO THE HORIZONTAL. § 224. 
 
 a 0/ 10/ 20/ 30/ 47 | 50 
 0° | 0.000000 | 9.999996 | 9.999985 | 9.999967 | 9.999941 | 9.999908 
 1 9.999868 | .999820 | .999765 | .999702 | 999633 | .999555 
 2 999471 | .999379 | .999280 | .999173 | .999059 | 998938 
 3 .998809 | .998673 | .998529 | .998879 | 998220 | ‘998055 
 4 997882 | .997701 | .997514 | .997318 | .997116 | 996906 
 5 .996689 | .996464 | .996232 | .995992 | (995745 | 995491 
 6° | 9.995229 | 9.994959 | 9.994683 | 9.994899 | 9.994107 | 9.993808 
 @ .993501 | .993187 | .992866 | .9925387 | .992201 | 1991857 
 8 .991506 | .991147 | .990780 | .990406 | .990025 | .989636 
 9 .989240 | .988836 | .988424 | 988005 | 987579 | ‘987144 
 10 -986703 | 986253 | 985797 | .985332 | 984860 | 1984380 
 11°. | 9.983893 | 9.983398 | 9.982895 | 9.982885. | 9.981867 -| 9.981342 
 12 .980808 | .980268 | .979719 | .979163 | 978599 | ‘978027 
 13 977447 | 976860 | .976265 | .975663 | 975052 | ‘974434 
 14 .973808 | .973174 | .972532 | .971888 | 971225 -| _970560 
 15 .969887 | .969206 | .968517 | .967820 | (967116 | (966403 
 AS 16° | 9.965683 | 9.964954 | 9.964218 | 9.963473 |. 9-962721 | 9.961960 
 ig 17 961192 | .960415 | 959631 .958838 | 958037 | 957229 
 Hi 18 .956412 | 955587 | .954753- | .9538912 | 953063 | |952205 
 Lacie 19 .951339 | 950465 | .949583 | 948692 | ‘947793 | |946886 
 i 20 945970 | .945047 | .944114 | 1943174 | [942025 | “941968 
 ta 21° | 9.940802 | 9.939828 | 9.938345 | 9.987354 | 9.936355 | 9.935347 
 Ne 22 .934330 | .9883805 | .932271 .931229 | 930178 | 929119 
 HI 23 928050 | .926974 | .925888 | .924794 | [993691 | ‘go257 
 Hil 24 .921458 | .920329 | .919191 .918044 | 916888 | 915723 
 a 25 914549 | .913366 | .912175 | 910974 909764 | 908546 
 Hd 26° | 9.907318 | 9.906081 | 9.904835 | 9.903580 | 9.902316 | 9.901042 
 HT 27 .899759 | .898467 | .897166 | 895855 | 894535 | 893206 
 | 28 .891867 | .890519 | .889161 | .ss7794 | ‘secd17 | ‘885031 
 Hl 29 .883635 | .882230 | .880815 |. .879390 | 877956 | (876512 
 pil 30 .875058 | .873594 | 87212 .870637 | .869144 | 867641 
 i 31° | 9.866127 | 9.864604 | 9.868071 | 9.861528 | 9.959974 | 9.858411 
 i 32 .856837 | .855253 | .853659 | .852054 | .850439 | _84ssi4 
 can 33 847178 | .845532 | 1843876 | .842209 | 840531 | _838843 
 ch 34 .887144 | 835434 | .833714 | .831982 | 830040 | “g2s498 
 35 .826724 | .824949 | .823163 | .821367 | :819559 | _817740 
 36° | 9.815910 | 9.814068 | 9.812216 | 9.810352 | 9.808476 | 9.806589 
 37 .804691 | .802781 | .800860 | .798927 | 796982 | .795026 
 38 .793058 | .791078 | .789086 | .787082 | (735066 | .78380388 
 39 (80998 | .778946 | .776882 | .774805 | (772716 | ..770614 
 40 768500 | .766374 | .764235 | 762083 | .759919 | *. 757742 
 ANE 41° | 9.755552 | 9.753349 | 9.751133 | 9.748904 | 9.746662 | 9.744407 
 att 42 742138 | .739857 | . 737561 (35253 | .732931 730595 
 va 43 728246 | 2725883 | 1723506 | 1721115 | .718710 | 7162901 
 at 44 713858 | 711411 .708950 | .706474 | .708988 | .701479 
 Hi 45 9.698959 | 9.696425 | 9.693876 | 9.691313 | 9.688734 | 9.686140 
 
TABLE XX.—LENGTHS OF CIRCULAR ARCS: RADIUS = 1. 
 
 Length, | 
 
 0000048 
 0000097 
 0000145 
 0000194 
 0000242 
 0000291 
 0000339 
 0000388 
 0000436 
 0000485 
 
 .0000533 
 .0000582 
 .0000630 
 .0000679 
 .0000727 
 .0000776 
 . 0000824 
 .0000873 
 .0000921 
 .0000970 
 
 .0001018 
 .0001067 
 .0001115 
 .0001164 
 0001212 
 0001261 
 .0001309 
 -0001357 
 .0001406 
 .0001454 
 
 .0001503 
 .0001551 
 . 0001600 
 .0001648 
 .0001697 
 .0001745 
 .0001794 
 .0001842 
 .0001891 
 .0001939 
 
 .0001988 
 . 0002036 
 .0002085 
 .0002133 
 0002182 
 .0002230 
 0002279 
 .0002327 
 .0002376 
 .0002424 
 
 .00024'7% 
 
 .0002521 
 .0002570 
 .0002618 
 .0002666 
 0002715 
 0002763 
 0002812 
 
 . 0002860 
 . 0002909 
 
 92 
 
 SO OFS OC 
 
 Length. || Deg 
 . 0002909 il 
 .0005818 2 
 .0008727 3 
 .0011636 4 
 .0014544 5 
 0017453 6 
 .0020862 ff 
 .0023271 8 
 . 0026180 9 
 .0029089 10 
 .0081998 11 
 .0034907 ile 
 .00387815 13 
 .0040724 14 
 .0043633 15 
 .0046542 16 
 .0049451 % 
 .0052360 18 
 . 0055269 19 
 .0058178 20 
 .0061087 21 
 . 0063995 22 
 . 0066904 23 
 .0069813 24. 
 0072722 25 
 .0075631 26 
 .0078540 Wi 
 .0081449 28 
 .0084358 29 
 . 0087266 30 
 .0090175 al 
 .0098084 || 32 
 .0095993 || 83 
 .0098902 || 34 
 .0101811 || 35 
 .0104720 36 
 .0107629 || 37 
 0110588 |} 3 
 
 .0113446 || 39 
 .0116855 40 
 .0119264 41 
 OL2e173 «rede 
 .0125082 43 
 .0127991 44 
 .0130900 45 
 .0138809 46 
 .0186717 {i 
 .0139626 48 
 .0142535 49 
 .0145444 50 
 .0148353 51 
 .0151262 52 
 0154171 53 
 .0157080 54 
 .0159989 55 
 .0162897 56 
 0165806 57 
 .0168715 58 
 .0171624 ieeog 
 ies | 60 
 
 pie pits 
 
 Length. | Deg Length. 
 0174533 61 | 1.0646508 
 0349066 62 1.0821041 
 0523599 63 | 1.0995574 
 0698132 64 1.1170107 
 0872665 |} 65 1.1844640 
 1047198 66 1.1519173 
 1221730 37 1.1693706 
 1396263 68 1.1868239 
 1570796 69 12042772 
 1745329 " 1. 2217305 
 1919862 "1 1. 2391838 
 2094395 12 1. 2566371 
 . 2268928 "3 1.2740904 
 2443461 "4 12915436 
 2617994 15 1.3089969 
 2792527 "6 1.8264502 
 2967060 rit 13489035 
 | .8141593 78 1.8613568 
 | 8316126 vi 1.8788101 
 | .8490659 80 1.8962634 
 | .8665191 81 1.4137167 
 | .8889724 || 82 1.4311700 
 .4014257 83° | 1.4486233 
 4188790 84 1.4660766 
 4362323 85 14835299 
 | .4537856 86 1.5009832 
 | 4712389 87 1.5184364 
 | 4886922 88 | 1.5358897 
 5061455 || 89 1.5533480 
 5285988 || 90 1.5707963 ° 
 | 5410521 91 1.5882496 
 BB85054 92 1.6057029 
 | .57%59587 93 1.6231562 
 | 5934119 94 1.64060°5 
 6108652 95 16580628 
 6283185 96 1.6755161 
 6457718 97 1.6929694 
 6632251 98 1.7104227 
 6806784 99 1.7278760 
 6981317 100 1.7453293 
 7155850 || 101 1. 7627825 
 7880883 || 102 | 41.7809358 
 7504916 103 1.7976891 
 7679449 || 104 1.8151424 
 TR5 5B9R2 | 105 1.8825957 
 8028515 106 1.85 500490 
 8202047 107 | 1.8675028 
 8377580 || 108 | 41.8849556 
 .8552113 || 109 | 4.9024089 
 8126646 || 110 | 1.919862 
 8901179 111 | 1,9373155 
 B0T5112 || 112 1.9547688 
 J posbods | 113 1 9722224 
 9424778 {| 114 1.9896753 
 9599311 | 115 20071286 
 97738844 || 116 | 9 9945819 
 9948377 | 117 9 04203852 
 1.0122910 | 118 | 2.0594885 
 | 1.0297443 || 119 | 2.0769418 
 | 120 | 2.0943951 
 
TABLE XXI.--MINUTES IN DECIMALS OF A DEGREE. 
 
 , 0” 10” 15" 20" 30" 40" | 45" Bo" | 
 0 | .oo000 | 00278 | .00417 | .00556 |) .00833 | .01111 | .01250 | .01389 | 0 
 1 | 01667 | .01944 | .02083.} .02222 || .02500 | 02778 | .02917 | .03055 | 1 
 9 | |03333 | .03611 | .03750 | .03889 || .04167 | .04444 | .04583 | .04722 | 2 
 3 | 05000 | .05278 | .05417 | .05556 || .05833 | .06111 | .06250 | 06389 | 3 
 4 | (06367 | .06944 | .07083 | 07222 || .07500 | .07778 | .07917 | .08056 | 4 
 5 | (08333 | 08611 | .08750 | .08889 || .09167 | 09444 | .09583°| .09722 | 5 
 s | 10000 | .10278 | .10417 | .10556 || .10833 | .11111 | .11250! 111389 | 6 
 ~ | ‘41667 | .11944 | .12083 | .12222 || .12500 | .12778 | .12917 | .13056 | 7 
 g | (13333 | 18611 | .13750 | .18889 || 14167 | .14444 | 114588.) 14722 | 8 
 9 | 15000 | 15278 | .15417 | .15556 |} .15838 | .16111 | .16250 | .16389 | 9 
 10 | 16667 | .16944 | .17083 | .17222 |) .17500 | .17778 | .17917 | .18056 | 10 
 i 11 | 18383 | .18611 | .18750 | .18889 |) .19167 | 19444 | .19583 | .19722 | 11 
 12 | .20000 | |20278 | .20417 | 20556 || .20833 | .21111 | .21250 | .21389 | 12 
 13 | (21667 | .21944 | .22083 | .22222 || 22500 | 22778 | .22917 | 123056 | 13 
 
 14 23333 | .23611 | .28750 | .23889 || .24167 | .24444 | .24583 | 124722 | 14 
 15 25000 | .25278 | .25417 | .25556 || .25833 | .26111 | .26250 | .26889 | 15 
 | 16 26667 | .26944 | .27083 | .27222 || .27500 | .27778 | .27917 | .28056 | 16 
 A 17 28333 | .28611 | .28750 | .28889 || .29167 | .29444 |-.29583 | .29722 | 17 
 
 Hh 18 -30000 | .80278.| .380417 | .380556 || .80833 | .381111 | .381250 | .31389 | 18 
 
 ‘tid 19 | .31667 | .381944 | .32083 | .32222 || .82500 | .82778 | .82917 | .83056 | 19 
 | it 90 | .33333 | .33611 | .383750 | .383889 || .84167 | .34444 | .34583 | .34722 | 20 
 Hae | 21 | .35000 | .35278 | .85417 | .85556 || .35833 | .36111 | .36250 | .36389 | 21 
 
 HI 99 | (36667 | .36944,| .37083 | .37222 || .37500 | .87778 | .387917 | 138056 | 22 
 i 93 | °39333 | 138611 | .38750 | .38889 || .39167 | .39444 | .39583 | .39722 | 23 
 | 24 | |40000-| .40278 | .40417 | .40556 |) .40833 | .41111 | .41250 | 141389 | 24 
 
 li a5 | 41667 | .41944 | .42083 | .42222 || .42500 | .42778 | .42917 | 43056 | 25 
 iis | 96 | 43388 | |43611 | .43750 | .43889 || 44167 | .44444 | 144583 | 144722 | 26 
 Hy 97 | '45000 | (45278 | .45417 | .45556 || .45838 | .46111 | .46250 | .46389 | 27 
 i 98 | |46667 | .46944 | .47083 | 47222 || .47500 | .47778 | .47917 | .48056 | 28 
 
 29 48333 | .48611 | .48750 | .48889 || .49167 | .49444 | .49583 | .49722 | 29 
 30 .50000 | .50278 | .50417 | .54556 || .50833 | .51111 | .51250 | .51389 | 30 
 
 <8 .51667 | .51944°} .52083 | .52222 || .52500 | .5277 .£2917 | .538056 | 31 
 32 53333 | .53611 | .538750 | .53889 || .54167 | .54444 | .54583 | .54722 | 382 
 33 .55000 | .55278 | .55417 | .55556 || .55833 | .56111 | .56250 | .56389 | 33 
 34 56667 | .56944 | .57083 | .57222 || .57500 | .57778 | .57917 | .58056 | 34 
 35 58333 | .58611 | .58750 | .58889 || .59167 | .59444 | .59583 | .50722 | 35 
 36 .60000 | .6027 .60417 | .60556 || .60883 | .61111 | .61250 | .61389 | 36 
 
 37 | .61667 | .61944 | .62083 | .62222 || .62500 | .62778 | .62917 | .68056 | 37 
 
 38 63333 | .63611 | .63750 | .63889 || .64167 | .64444 | .64583 | .64722 | 38 
 
 39 .65000 | .65278 | .65417 | .65556 || .65833 | .66111 | .66250 | .66389 | 39 
 
 40 | .66667 | .66944 | .67083 | .67222 || .67500 | .67778 | .67917 | .68056 | 40 
 
 41 68333 | .68611 | .68750 | .68889 || .69167 | .69444 | .69583 | .69722 | 41 
 
 2 .70000 |. .70278, | .70417 | .70556 || .70833 | .71111 | .71250 | .713889 | 42 
 
 Tne 43 71667 | .71944 | .72083 | .72222 || .72500 | .72778 | . 72917 | .73056 | 43 
 | } 44 73333 | .73611 | .73750 | .73889 || .74167 | .'74444 | .74588 | .'74722 | 44 
 Palit 45 75000 |. 7527 5417 | .75556 || .75888 | .76111 | .76250 | . 76389 | 45 
 Aa 46 .76667 | .76944.] 77083 | .77222 |) .77500 | 27777 JTI9TT | £78056 | 46 
 
 i] 7 78333 | .78611 ; .78750 | .78889 || .79167 | .79444 | .79588 | .79722 | 47 
 
 it 48 .80000 | .80278 | .80417 | .80556 || .80833 | .81111 | .81250 | .81389 | 43 
 Mi 49 81667 | .81944 | .82083 | .82222 || .82500 | .8277 .82917 | .83056 | 49 
 
 50 .83333 | .83611 | .83750 | .83889 || .84167 | .84444 | .84583 | .84722 | 50 | 
 51 .85000 | .8527 .85417 | .85556 || .858383 | 86111 | .8625 .86389 | 51 
 52 86667 | .86944 | .87083 | .87222 || .87500 | .87778 | .87917 | .88056 | 52 
 53 88333 | .88611 | .88750 | .88889 || .89167 | .80444 | .89583 | .89722 | 53 
 i 54 .90000 |. .90278 | .90417 | .90556 || .908383 | .91111 | .91250 | .91389 | 54 
 Hi 55 .91667 | .91944 | .92083 | .92222 |) .92500 | .92778 | .92917 | .93056 | 55 
 
 Hy 56 .93333 | .93611 | .93750 | .93889 || .94167 | .94444 | .94583 | .94722 | 56 
 HH | 57 .95000 | .95278 | .95417 | .95556 || .95833 | .96111 | .96250 | .96389 | 57 
 | 58 .96667 | .96944 | .97085 | .97222 || .97500 | .97778 | .97917 | .98056 | 58 
 
 59 .98333 | .93611 | .98750 | .98889 || .99167 | .99444 | .99583 | .99722 | 59 
 
 AR OF) |EL08 15°-¢|, 20° || GesOTN - dome] ) 50a OR pos tare 
 31d 
 
| | 
 In. hag | oP tase 6 TSF O-eledOy yh 11 
 
 \ | | | 
 
 | eons BBS baie 
 
 i Agneta Saas eee aoe a | : aig | ma! | | 
 | | 0 _ |Foot! .0833) 1667} .2500) 3333 ..4167| 5000) .5833! 6667} .7500 8333] 9167 
 1-32 | .0026) .0859! . 1693] . 2526) .38859) .4193) .5026) .5859! . 6693} .'7526| 8859) .9193} 
 | 4-16 |.0052! .0885).1719| 2552) .3385) 4219] 5052) .5885/ 6719] 7552) .8385/ 9219) 
 | 3-32 |.0078 .0911| 1745) 2578) .3411) 4245) .5078) 5911 .6745| .7578) .8411) 9245 
 1-8 |.0104) 0938) 1771] .2604) 3438) 4271) .5104| .5938) .6771| .7604| 8438) 9271 
 | 5-32 |.0130 .0964) 1797) .2630) 8464) 4297) 5130 5964! .6797| 7630, 8464) .9297 
 3-16 |.0156! .0990) 1823] .2656) .3490! .4323| .5156| 5990) .6823) .7656) .8490) 9323 
 7-32 | 0182). 1016) 1849] .2682) .3516) .4849] 5182) 6016) .6849] .7682| .8516) .9349 
 Tt .0208| .1042) .1875| .2708) .3542! .4375| 5208) .6042| .6875! 7708] 8542! 9875 
 9-32 | 0234 1068] 1901] .2734| .3568) .4401|.5234) .6068] .6001| .7’734/ .8568) .9401 
 5-16 | .0260) 1094 1927] 2760) 8594) .4427/ 5260) . 6094) .6927) 7760) .8594 9427 
 11-32 | 0286) .1120) 1953] .2786) 8620] .4453) .5286) .6120) .6953) . 7786 .8620) .9453 
 8-8 |.0313) .1146) 1979] 2813] .3646) 4479! 5813) .6146] 6979] .7813| .8646) 9479 
 13-32 | .0339| 1172) .2005].2839) .3672! 4505] .5339/ 6172) .7005| .7839) .8672! 8505 
 7-16 | .0365} 1198) 2031) .2865] 8698) .4531] 5365) .6198) .7081| .7865) .8698) 9531 
 15-82 |.0391| 1224) 2057) .2891| 3724) .4557|. 5391] 6224) .7057| .7891| .8724| .9557 
 1-2 |.0417|.1250! 2083] .2917| 3750] .4583] .541’7| . 6250) .7083] .791'7| .8750| .9583 
 17-82 | 0443) .1276] 2109] 2943) .3776) .4609] .5443) .6276] .7109] .7943) .8776| .9609 
 9.16 |.0469) 1302) 2135] 2969) .3802) .4635} .5469) . 6302] .'7135) .'7969| .8802/ .9635 
 19-32 | 0495) .1328) .2161) 2995) 8828) .4661] .5495) 6828) .7161] 7995) 8828) 9661 
 5-8 | .0521) .1354| .2188] 8021) .8854) .4688} .5521| .6354/ .7188) .8021! .8854) .968= 
 21-32 | 0547) .1380) .2214] .3047| 8880) .4714| .5547| .6380} .7214| .8047| .8880] .9714 
 11-16 | 0573) .1406) 2240) 3073} .8906) .4740} .5573) .6406] .7240) .8073 .8906] .9740 
 23-32 | 0599) 1432 .2266] 3099) 3932 .4766) 5599 6432) .7266) 8099 8932. 9766 
 0625} .1458} . 2292} 3125) .3958) .4792) .5625) 6458) . 7292! .8125) 8958! .9792 
 
 0807 
 
 .0651) .1484) .2318) 
 -0677'| .1510} .2344 
 0703} . 1536} .2370) 
 0729) .1563) .2396 
 0755) . 1589) . 2422) 
 0781) 1615) 2448) 
 
 1641} 2474) 
 
 TABLE XXII.--INCHES IN DECIMALS OF A FOOT. 
 
 <olol} 
 OL77 
 .8203} . 
 .8229| 
 . 8255 
 .O201). 
 3807) 
 
 3984 
 .4010). 
 
 4036! 
 4063! 
 
 4089} 
 
 4115) 
 
 4141} 
 
 .4818 | 
 4844) 
 4870} .f 
 .4896) 
 .4922 
 .4948 
 4974 
 
 5651 | 
 5677 
 
 703 
 
 5129 
 
 5055 
 
 5781 
 5807 
 
 .6484 
 .6510 
 6536 
 . 6563 
 6589 
 .6615 
 . 6641 
 
 7318 | 
 7344 
 7370 
 
 7396) . 
 
 - (422 
 (448 | 
 TATE 
 
 8151 
 8177 
 8208 
 8229) 
 8255 
 8281} .9115) 
 8307 
 
 8984 | 
 “9010 
 "9036 
 9063) 
 9089 
 
 .9818 
 
 .9896 
 . 9922} 
 | 9948) 
 9141) .9974 
 
 . 9844) 1: 
 9870) 27 
 
 15-8; 
 
 1-2 
 17-82 
 
 9-16 
 19-82 
 
 5-8 
 21-32 
 11-16 
 23-32 
 
 3-4 
 
 25-32 
 
 29-32 
 15-16 
 31-32 
 
 8 
 
 9 | 10 | 11 
 
TAPLE XXIII.—SQUARES, CUBES, SQUARE ROOTS. 
 
 | [ | 
 | i 
 No. |Squares.| Cubes. BS eae y | Cube Roots. 
 1 1 1 1.0000000 1.0000000 
 2 4 8 1.4142136 1.2599210 
 3 9 Q7 1.'7320508 1.4422496 
 4 16 64 20000000 1.5874011 
 5 25 125 2. 2360680 1.7099759 
 6 36 216 2 4494897 1.8171206 
 v 49 | 343 °2. 6457513 1.9129312 
 8 64.1 512 2. 8284271 20000000 
 9 81 729 3.0000000 2.0800837 
 10 - 100 1000 3.1622777 2.1544347 
 11 121 1331 3.3166248 2. 2239801 
 12 144 1728 3.4641016 2 2894286 
 13 169 2197 8.6055513 2.3513347 
 14 196 2744 8.741657 2.4101422 
 15 225 3375 3.87:29833 24662121 
 16 256 4096 4 0000000 2.5198421 
 17 289 4913 4. 1231056 2.5712816 
 18 324 5832 4 2426407 2 6207414 
 19 361 6859 4. 3588989 2. 6684016 
 20 400 8000 4. 4721360 2.'7144177 
 g 44} 9261 4 5825757 2.'7589243 
 99 484 10648 4.6904158 2. 8020393 
 23 529 12167 4.958315 2. 8438670 
 24 576 1382 4. 8989795 2.8844991 
 25 625 15625 5 0000000 2. 9240177 
 26 376 17576 5 0990195 2. $624960 
 27 729 19683 5. 1961524 3.0000000 
 28 784 21952 5 2915026 3 .0365889 
 29 841 24389 5.3851648 3.0723168 
 30 900 27000 5. 4772256 3. 1072325 
 31 961 29791 55677644 3. 1413806 
 32 1024 32768 5. 6568542 3.1748021 
 33 1089 35937 57445626 8.2075343 
 34 1156 39304 5. 8309519 3. 2396118 
 85 1225 42875 59160798 3.2710663 
 at! 36 1296 46656 6.0000000 3.3019272 
 ait | °* 37 1369 50653 6. 0827625 3. 3322218 
 att | 38 1444 54872 6.1644140 3.8619754 
 a 39 1521 59319 6.2449980 3.3912114 
 a 40 1600 64000 6.3245553 3.4199519 
 a 41 1681 68921 64031242 3.4482172 
 ii} 42 1764 74088 6 .4807407 3.4760266 
 al 43 1849 79507 6.5574385 3.5033981 
 Ht 44 1936 85184 6. 6332496 . 8.5808483 
 i 45 2025 91125 6..7082039 35568933 
 il 46 2116 97336 6..7823300 3.5830479 
 Wii " 2209 103823 68556546 3.6088261 
 48 2304 110592 6. 9282022 3 6342411 
 ' 49 2401 117649 7 0000000 3.6593057 
 | 50 , 2500 125000 70710678 8.6840314 
 51 2601 132651 % 1414284 3.7084298 
 i 52 2704 140608 72111026 3.7325111 
 | 53 2809 148877 72801099 8.7562858 
 | 54 2916 157464 % 3484692 3.7797631 
 55 3025 166375 74161985 3. 8029525 
 ) 56 3136 175616 74833148 3. 8258624 
 i BY 3249 185193 7 5498344 3.8485011 
 i 58 3364 195112 "6157731 38708766 
 iF 59 3481 205379 76811457 3. 8929965 
 60 3600 216000 % V459667 8.9148676 
 61 3721 226981 78102497 3.9364972 
 62 3844 238328 78740079 3.9578915 
 OP ew . ee ee ee sa a = 
 
 1 
 
 | 
 Reciprocals. 
 
 1.000000000 
 - 500000000 
 8333838333 
 . 250000000 
 
 - 200060000 
 . 166666667 
 - 142857148 
 - 125000000 
 Meh mabol 
 
 . 100000000 
 -090909091 
 083383833 
 076923077 
 071428571 
 . 066666667 
 . 062500000 
 . 058823529 
 055555556 
 052631579 
 
 - 050000000 
 047619048 
 045454545 
 043478261 
 041666667 
 .040000000 
 038461538 
 -037037037 
 085714286 
 034482759 
 . 033333333 
 082258065 
 031250000 
 .030303030 
 029411765 
 628571429 
 
 OTT 77 
 
 027027027 
 .026315789 
 025641026 
 025000000 
 024390244 
 023809524 
 023255814 
 022727273 
 022222229 
 021739130 
 021276600 
 020833333 
 020408163 
 020000000 
 019607843 
 019230769 
 018867 925 
 018518519 
 .018181818 
 017857143 
 017545860 
 .017'241379 
 .016949153 
 016666667 
 016303443 
 016129032 
 
 31d 
 
CUBE ROOTS, AND RECIPROCALS. 
 
 bot 
 oO 
 
 He Co OO Ret 
 
 mJ AF HJ AF IF FI 
 
 OO O22 OF 
 
 | Squares. 
 
 3969 
 4096 
 4225 
 4356 
 4489 
 4624 
 4761 
 4900 
 5041 
 5184 
 5329 
 5476 
 5625 
 5776 
 5929 
 6084 
 6241 
 
 10404 
 10609 
 10816 
 11025 
 11236 
 11449 
 11664 
 11881 
 12100 
 12321 
 12544 
 12769 
 12996 
 13225 
 13456 
 13689 
 13924 
 14161 
 
 14400 
 14641 
 14884 
 15129 
 15376 
 
 Cubes. 
 
 250047 
 262144 
 274625 
 287496 
 300763 
 3144382 
 328509 
 343000 
 857911 
 373248 
 3889017 
 405224 
 421875 
 438976 
 456533 
 474552 
 493039 
 512000 
 531441 
 551368 
 571787 
 592704 
 614125 
 636056 
 658503 
 681472 
 704969 
 729000 
 753571 
 773688 
 804357 
 830584 
 857375 
 884736 
 912673 
 941192 
 970299 
 
 100000 
 
 1030301 
 1061208 
 1092727 
 1124864 
 1157625 
 1191016 
 
 1225043" 
 
 1259712 
 1295029 
 1331000 
 1367631 
 1404928 
 1442897 
 1481544 
 1520875 
 1560896 
 1601613 
 1643032 
 1685159 
 
 728000 
 1771561 
 1815848 
 1860867 
 1906624 
 
 Square 
 koots. 
 
 9372539 
 0000000 
 0622577 
 1240384 
 1853528 
 2462113 
 .8066239 
 3666003 
 4261498 
 4852814 
 5440037 
 6028253 
 6602540 
 T177979 
 7749644 
 8317609 
 8881944 
 9442719 
 0000000 
 0553851 
 1104336 
 1651514 
 2195445 
 2736185 
 38273791 
 .8808315 
 .4339811 
 .4868330 
 5393920 
 .5916630 
 .6436508 
 6953597 
 7467943 
 7979590 
 8488578 
 8994949 
 9.9498744 
 
 10.0000000 
 10.0498756 
 10.0995049 
 10.1488916 
 10.1980390 
 10.2469508 
 10.2956301 
 10.3440804 
 10.3923048 
 10.4403065 
 
 10. 4886385 
 10.5356538 
 10.5830052 
 10.6801458 
 10.¢770783 
 10.7238053 
 10.7'703296 
 10.8166538 
 10.8627805 
 10.9087121 
 
 10. 9544512 
 11.00 10000 
 11.0453610 
 11.0905365 
 11.1855287 
 
 Mmmm anm- 
 
 316 
 
 Cube Roots. 
 
 3.979057 
 4.0000000 
 4.0207256 
 4.0412401 
 4.0615480 
 4,0816551 
 4,1015661 
 4.1212853 
 4.1408178 
 4.1601676 
 4.1798390 
 4.1988364 
 4.2171633 
 4 2358236 
 4 2543210 
 4 2726586 
 4.2908404 
 4 3088695 
 43267487 
 8444815 
 3620707 
 8795191 
 3968296 
 4140049 
 4310476 
 79602 
 4647451 
 4814047 
 .4979414 
 .5143574 
 .5306549 
 5468359 
 . 5629026 
 5788570 
 .5947009 
 .6104363 
 .6260650 
 .6415888 
 6570095 
 .6723287 
 .6875482 
 . 7026694 
 . 7176940 
 . 7326235 
 ~7474594 
 . (622032 
 . 7768562 
 .'7914199 
 .8058955 
 .8202845 
 .$345881 
 .8488076 
 .8629442 
 .8769990 
 .8909732 
 9048681 
 9186847 
 9324242 
 9460874 
 9596757 
 9731898 
 9866310 
 
 Reciprocals. 
 
 015873016 
 .015625000 
 .015384615 
 .015151515 
 .014925373 
 .014705882 
 .01449275 
 
 .014285714 
 .014084507 
 013888889 
 .013698630 
 .013518514 
 013333333 
 .013157895 
 012987013 
 012820513 
 01265822: 
 
 012500000 
 012345679 
 -012195122 
 .612048193 
 .011904762 
 .011764706 
 .011627907 
 .011494253 
 .011363636 
 011235955 
 
 011111111 
 .010989011 
 .010869565 
 .010752688 
 . 010638298 
 .010526316 
 .010416667 
 .010309278 
 .010204082 
 .010101010 
 
 .010000000 
 009900990 
 009803922 
 009708738 
 009615385 
 
 .009523810 
 .009433962 
 .009345794 
 005259259 
 .0091743812 
 .009090909 
 _ .009009009 
 .0089285; 1 
 .008849558 
 .008771930 
 . 008695652 
 .008620690 
 .008547009 
 .008474576 
 .008403361 
 008333385 
 .008264463 
 .008196721 
 .008130081 
 .008064516 
 
 aa eee 
 
Squares, 
 
 15625 
 15876 
 16129 
 16384 
 16641 
 16900 
 17161 
 17424 
 17689 
 
 7956 
 18225 
 18496 
 18769 
 19044 
 19321 
 
 19600 
 19881 
 20164 
 20449 
 20736 
 21025 
 21316 
 21609 
 21904 
 22201 
 
 22500 
 22801 
 23104 
 23409 
 23716 
 24025 
 24336 
 24649 
 24964 
 25281 
 25600 
 25921 
 26244 
 26569 
 26896 
 27225 
 27556 
 27889 
 28224 
 28561 
 28900 
 29241 
 29584 
 29929 
 30276 
 30625 
 30976 
 31329 
 31684 
 32041 
 
 32400 
 32761 
 
 33124 
 33489 
 
 33856 
 84225 
 34596 
 
 Cubes. 
 
 1953125 
 2000376 
 2048383 
 2097152 
 2146689 
 2197000 
 2248091 
 2299968 
 2352637 
 2406104 
 2460875 
 2515456 
 2571353 
 2628072 
 2685619 
 2744000 
 2803221 
 2863288 
 2924207 
 2985984 
 8048625 
 8112136 
 3176523 
 8241792 
 8307949 
 8375000 
 3442951 
 8511808 
 3581577 
 8652264 
 8723875 
 3796416 
 38869893 
 3944312 
 4019679 
 
 4096000 
 4173281 
 4251528 
 4330747 
 4410944 
 4492125 
 4574296 
 4657463 
 4741682 
 4826809 
 4913000 
 5000211 
 5088448 
 S1LTTT17 
 5268024 
 5859375 
 5451776 
 5545233 
 5639752 
 5735339 
 5832000 
 5929741 
 6028568 
 6128487 
 6229504 
 6331625 
 6434856" 
 
 Square 
 Roots. 
 
 11.1803399 
 112249722 
 11.2694277 
 11.8137085 
 11.8578167 
 11.4017543 
 11. 4455231 
 11.4891253 
 115325626 
 115758369 
 11.6189500 
 11.6619038 
 11. 7046999 
 11.7473401 
 11.7898261 
 11.8321596 
 11.874342 
 
 119163753 
 11. 9582607 
 12.0000000 
 12.0415946 
 12.0830460 
 12.1248557 
 121655251 
 22065556 
 2. 2474487 
 12.2882057 
 12 38288280 
 12.3693169 
 124096736 
 12.4498996 
 12.4899960 
 12..5299641 
 12.5698051 
 12. 6095202 
 
 2.6491106 
 12.6885775 
 12. 7279221 
 12.7671453 
 128062485 
 128452326 
 
 2.8840987 
 12.9228480 
 
 2.9614814 
 13 0000000 
 
 13..0334048 
 13.0766968 
 13.114877 
 
 13. 1529464 
 13. 1909060 
 13. 2287566 
 132664992 
 13. 3041347 
 13.3416641 
 133790882 
 13.4164079 
 13. 4536240 
 13.4907376 
 13.5277493 
 13.5646600 
 136014705 
 136381817 
 
 TABLE XXIII—SQUARES, CUBES, SQUARE ROOTS. 
 
 Cube Roots. 
 
 5.0000000 
 5.0132979 
 5 .0265257 
 5.0396842 
 5.0527743 
 5.0657970 
 5.0787531 
 5. 0916434 
 1044687 
 1172299 
 1299278 
 . 1425632 
 1551867 
 . 1676493 
 1801015 
 
 1924941 
 2048279 
 5. 2171034 
 52293215 
 5.2414828 
 5.2585879 
 5.2656374 
 5. 2776821 
 5. 2895725 
 3014592 
 
 8132928 
 .8250740 
 . 83868033 
 . 38484812 
 . 8601084 
 53716854 
 5, 8882126 
 5 3946907 
 54061202 
 5.4175015 
 5. 4288352 
 5. 4401218 
 5.4513618 
 5 .4625556 
 5.4737037 
 5. 4848066 
 5.4958647 
 5 5068784 
 55178484 
 5 5287748 
 5.5396583 
 .5504991 
 .5612978 
 5720546 
 5827702 
 .59384447 
 6040787 
 5.6146724 
 6 .6252263 
 5. 6357408 
 5 6462162 
 5.656652 
 
 5.6670511 
 56774114 
 5.6877340 
 56980192 
 
 Or 
 
 Crorororoer 
 
 OU OT Or Or OT OC 
 
 10817 
 
 5. 7082675 
 
 Reciprocals. | 
 
 .008000000 
 -007936508 
 007874016 
 007812500 
 ; 7751938 
 007692308 
 7633588 
 007575758 
 -007518797 
 007462687 
 007407407 
 -007352941 
 -007299270 
 007246377 
 007194245 
 007142857 
 007092199 
 007042254 
 006993007 
 006944444 
 - 006896552 
 006849315 
 - 006802721 
 006756757 
 .006711409 
 006666667 
 -006622517 
 006578947 
 006535948 
 - 006493506 
 -006451613 
 .006410256 
 -006369427 
 006329114 
 006289308 
 
 - 006250000 
 -€06211180 
 -006172840 
 006134969 
 -006097561 
 - 006060606 
 . 006024096 
 - 005988024 
 -005952381 
 -005917160 
 
 .005882353 
 005847953 
 005818953 
 005780347 
 005747126 
 005714286 
 .005681818 
 -005649718 
 005617978 
 005586592 
 
 005555556 
 005524862 
 005494505 
 005464481 
 005434783 
 -005405405 
 -005376344 
 
CUBE ROOTS, AND RECIPROCALS. 
 
 | _ 
 No. | Squares. Cubes. eee Cube Roots, | Reciprocals. 
 —$——— |—_—__—_———_- -_—_ —— — —- 
 187 34969 6539203 13.6747943 5.7184791 005847594 
 188 35344 6644672 13.7113092 5.7286543 .005319149 
 189 85721 3751269 13. 7477271 5.7387936 005291005 
 190 36100 6859000 13.7840488 5.7488971 .005263158 
 | 191 36481 6967871 13.8202750 5.7589652 005235602 i 
 192 36864 (0778 13 .8564065 5. 7689982 005208333 
 | 193 37249 7189057 -13.8924440 5.7788966 .005181347 | 
 194 376386 7301384 13. 9283883 5.'7889604 .005154639 } 
 195 | 388025 TA14875 13.9642400 5.'7988900 .005128205 i og 
 | 196 38416 7529536 14.0000000 5. 8087857 005102041 Will 
 197 38809 7645373 14.0356688 5.81 ee 005076142 i 
 198 | 39204 7762392 14.0712473 5 828476 .005050505 
 199 39601 7880599 14, 1067360 5.83827 25 . 005025126 
 200 40000 8000000 14.1421356 58480355 005000000 
 : 201 40401 8120601 14.1774469 5. 8577660 004975124 i 
 202 40804 8242408 14.2126704 5.8674643 604950495 i 
 203 41209 8365427 14. 2478068 5.8771307 004926108 | 
 204 41616 8489664 14. 2828569 5. 8867653 004901961 Mi 
 205 42025 8615125 14.3178211 5 8963685 004878049 i 
 | 206 42436 741816 =| 14.3527001 5.9059406 004854369 I 
 207 42849 8869743 14. 3874946 59154817 004830918 i 
 208 43264 8998912 14. 4222051 5. 9249921 004807692 i} 
 209 43681 9129829 14. 4568323 5. 9344721 004784689 \ 
 210 | 44100 9261000 14.4913767 5 9439220 .004761905 
 211 44521 9393931 14.5258390 5.95388418 .004739336 i 
 212 44944 9528128 14.5602198 5. 9627320 004716981 i 
 213 45369 9663597 14.5945195 | 5.9720926 .004694836 i 
 214 45796 9800344 14.6287888 | 5.9814240 .004672897 i 
 215 46225 9938875 14. 6628783 5.9907264 .004651163 i 
 i 216 46656 10077696 14.69693885 6.0000000 . 004629630 ; 
 217 47089 10218313 14.7309199 6.0092450 .004608295 ! 
 218 | 7524 10360232 14.764828 6.0184617 .004587156 i 
 219 47961 10503459 14.7986486 6 .0276502 004566210 ' 
 22 48400 10648000 14.8823970 |  6.0868107 004545455 
 eal 48841 10798861 14.8660687 |  6.0459435 004524887 
 222 49284 10941048 14.8996644 | 6.0550489 .004504505 
 e2% 49729 11089567 14.9831845 | 6.064120 004484305 
 224 50176 11239424 14.9666295 | 6.0731779 -004464286 | 
 225. | 50625 11390625 15. 0000000 6 .C8E2020 004444444 Wa 
 226 51076 11548176 15 .0882964 6.0911994 -004424779 | 
 227 51529 11697083 150665192 6.1001702 .004405286 
 22 51984. 11852852 15 .0996689 6.1091147 .004825965 3 
 229 52441 12008989 15.1827460 61160832 004866812 
 | 
 230 52900 12167000 15.1657509 6. 1269257 .004 34 7826 | 
 231 53361 12326391 15.1986842 6.1857924 004229004 | 
 232 53824 12487168 15. 2315462 6.1446837 "004310845 
 233 54289 12649337 15 .2648375 6.184495 004291845 | 
 234 54756 12812904 15.29705 85 6.1622401 .00427 8504 
 2395 55225 12977875 15.8297097 6.1710058 : 04D5E 319 
 236 55696 13144256 15.3622915 6.1797466 004237288 
 237 56169 13312053 15 38948043 6.1&84628 .004219409 | 
 238 56644 13481272 15 4272486 6.1971544 -004201681 
 239 57121 18651919 15 .4596248 6.2058218 .C04184100 | 
 : 240 57600 13824000 | 15.491933: 6.2144650 004166667 
 2A1 58081 18997521 =| 15.5241747 6. 2280843 .C04149878 
 242 58564 14172488 15 5563492 6.2316797 -004182281 
 243 59049 14348907 15.5884573 6. 2402515 004115226 a 
 244 59536 | 14526784 15 6204994 6 .2487998 . 004098361 
 245 60025 14706125 15 .6524758 6 .2573248 .004081683 
 246 60516 14886936 15.6843871 6. 2658266 .004065041 
 247 61009 15069223 15.7162836 6.2743054 004048583 
 | 248 61504 15252992 15.7480157 6.2827613 .004032258 
 
 318 
 
No, Squares. Cubes. 
 —— oe. hese 
 249 62001 15438249 
 250 62500 15625000 
 251 63001 15813251 
 252 63504 16003008 
 253 64009 16194277 
 254 64516 16387064 
 255 65025 16581375 
 255 65536 16777216 
 257 66049 16974593 
 258 66564 17173512 
 259 67081 17373979 
 260 67600 17576000 
 261 68121 17779581 
 262 68644 17984728 
 263 69169 18191447 
 264 69696 18399744 
 265 70225 18609625 
 266 10756 18821096 
 | 267 71289 19034163 
 268 71824 19248832 
 At hg 269 72361 19465109 
 iil 270 72900 19683000 
 Wet 271 73441 19902511 
 Ne 272 73984 20123648 
 Hy 273 %4529 20346417 
 Hi Q74 75076 20570824 
 iii! Q75 75625 20796875 
 276 "6176 21024576 
 277 76729 21252933 
 278 77284 21484952 
 279 V7841 21717639 
 280 78400 21952000 
 281 78961 22188041 
 282 9524 22425768 
 283 80089 22665187 
 284 80656 22906304 
 285 81225 23149125 
 286 81796 23393656 
 287 82369 23639903 
 288 82944 23887872 
 289 83521 24137569 
 290 84100 24389000 
 291 84681 24642171 
 292 85264 24897088 
 293 85849 25153757 
 294 86435 25412184 
 295 87025 25672375 
 296 87616 25934336 
 297 | 88209 26198073 
 298 88804 26463592 
 299 89401 26730899 
 300 90000 27000000 
 301 90601 27270901 
 302 91204 27543608 
 303 91809 27818127 
 304 92416 28094464 
 805 93025 28372625 
 306 93636 28652616 
 307 94249 28934443 
 308 94864 29218112 
 309 95481 29503629 
 310 96100 29791000 
 
 7 
 q 
 
 ‘17.0880075 
 ‘ 
 
 Square 
 Roots. 
 
 15.779733 
 15.8113883 
 158429795 
 158745079 
 159059737 
 15.9373775 
 15.9687194 
 160000000 
 16.0312195 
 16, 0623784 
 160934769 
 
 16. 1245155 
 16. 1554944 
 16. 1864141 
 16..2172747 
 16.2480768 
 16.2788206 
 16 .3095064 
 163401346 
 16.3707055 
 16.4012195 
 164316767 
 16.4620776 
 164924225 
 16.5227116 
 16. 5529454 
 165831240 
 166132477 
 16. 6433170 
 16..6733320 
 16. 7032931 
 167332005 
 16.7630546 
 16. 7928556 
 168226038 
 168522995 
 168819430 
 16.9115345 
 16. 9410743 
 169705627 
 70000000 
 0293864 
 0587221 
 
 1172428 
 171464282 
 7.1755640 
 172046505 
 7,2336879 
 72626765 
 @.2916165 
 78205081 
 7 .8493516 
 7.3781472 
 7 .4068952 
 7 .4855958 
 74642492 
 74928557 
 75214155 
 75499288 
 7.5788958 
 7. 6068169 
 
 319 
 
 TABLE XXIII.—SQUARES, CUBES, SQUARE ROOTS, 
 SS ee ae eee 
 
 Cube Roots, 
 
 62911946 
 
 6.2996053 
 6.3079935 
 6.3163596 
 6.38247035 
 6. 3330256 
 6.3413257 
 6 .3496042 
 68578611 
 68660968 
 6.3743111 
 6.3825043 
 6.3906765 
 6.3988279 
 6. 4069585 
 6.4150687 
 6.4231583 
 6 .4312276 
 6.4392767 
 6.4473057 
 64553148 
 
 6.4633041 
 6.4712736 
 64792236 
 6.4871541 
 6.4950653 
 5029572 
 .5108300 
 5186839 
 5265189 
 .9343351 
 . 0421326 
 9499116 
 .55716722 
 .9654144 
 5731885 
 . 5808443 
 5885323 
 5962023 
 6038545 
 .6114890 
 .6191060 
 . 6267054 
 6342874 
 6418522 
 6493998 
 6569302 
 6644437 
 6719403 
 6794200 
 . 6868831 
 6943295 
 7017593 
 7091729 
 7165700 
 7239508 
 7318155 
 7386641 
 7459967 
 6 .7583134 
 6.7606143 
 6.7678995 
 
 De D2 ID 
 
 Reciprocals, 
 
 .004016064 
 
 004000000 
 .003984064 
 003968254 
 .003952569 
 .003937008 
 003921569 
 003906250 
 .003891051 
 003875969 
 . 003861004 
 
 003846154 
 . 003831418 
 003816794 
 003802281 
 .003787879 
 .003773585 
 .003759398 
 003745818 
 0037313843 
 008717472 
 003703704 
 0038690037 
 .003676471 
 .003663004 
 . 003649635 
 . 003636364 
 . 003623188 
 . 003610108 
 003597122 
 - 003584229 
 
 .003571429 
 .003558719 
 . 003546099 
 . 003533569 
 .008521127 
 .0035087'7% 
 .0034965038 
 .0038484321 
 .003472222 
 - 003460208 
 .003844827' 
 
 . 003436426 
 - 003424658 
 .003412969 
 .003401361 
 .003389831 
 .003378378 
 . 003367003 
 .003355705 
 . 003344482 
 
 .0033338333 
 . 003322259 
 .003311258 
 . 0038800330 
 . 008289474 
 . 0038278689 
 . 0038267974 
 . 0038257329 
 . 003246753 
 .098286246 
 . 003225806 
 
} 
 CUBE ROOTS, AND RECIPROCALS, 
 | - 
 | | | | 
 No. | Squares, Cubes. = aoa | Cube Roots. : Reciprocals, 
 Atal | i Been, (ORE OE 
 Goat: aes 96721 | 80080231 | 176351921 6.751690 Seed ios, 
 312 97344 30371828 17..6635217 6. 7824229 . 003205128 
 313 97969 30664297 176918060 6.7896613 . 003194888 
 314 98596 30959144 177200451 6.7968 844 003184713 i 
 815 99225 31255875 17 . 7482393 68040921 .003174603 | 
 316 99856 31554496 17.7763888 6.811247 | .008164557 
 317 100489 | 81855013 17 .8044938 6.8184620 . 008154574 } 
 318 101124 82157432 =| 178325545 6.8256242 Oates 4. | 
 319 101761 32461759 7.8605711 6.832714 .003134796 
 f 
 320 102400 32768000 178885438 6.8399037 Sotto Hit: 
 321 103041 33076161 | 17.9164729 6.8470213 | 003115265 
 322 103684 33386248 7.9443584 6.8541240 . 063105590 | 
 323 104329 33698267 179722008 68612120 .002095975 
 S24 104976 34012224 | 18.0000000 6. 8682855 003 B64: 20 ) 
 825. | 105625 34328125 18.0277564 6.8753443 003071692 i 
 826 | 106276 34645976 18 0554701 6. 8823888 003067485 
 B27 | ~—:106929 34965783 18.0831413 |  6.8894188 .003058104. \ 
 828 | 107584 85287552 18. 1107703 6.8964345 .003048780 | 
 329 108241 35611289 18, 1383571 6. 9034359 “003039514 i 
 330 108900 35937000 18. 1659021 6.910422 .003030303 | 
 33 109561 36264691 181934054 6.9173964 .003021148 i 
 332 | 11022 36594368 18 . 2208672 6.9243556 | 003012048 i 
 333 | 110889 86526037 18. 2482876 6.9318008 002003003 i 
 834 111556 37259704 18. 2756669 6 9382321 . 002994012 i 
 330 112225 875953875 18.8030052 6.9451496 .C02985075 | 
 | 336 112896 37933056 18 .3303028 6 .9520533 .002976190 
 337 113569 88272753 18.3575598 6.9589484 | _002967359 i 
 338 114244 38614472 18 .3847763 6.9658198 | .002958580 
 83! 114921 38958219 18.4119526 6. 9726826 .002949853 | 
 115600 39304000 18.4390889 6.9795321 . 002941176 
 116281 | 39651821 18466185: 6. 9863681 002982551 
 116964 | 40001688 18. 4932420 6. 9931906 .002923977 
 117649 | 40353607 18.5202592 7 0000000 .0029154 452 
 : 118336 | 40707584 185472870 7 0067962 . 00290697’ 
 119025 | 41063625 18.5741756 7 0035791 "002898551 
 119716 | 41421736 18. 6010752 70203490 .002890173 
 120409 41781923 18. 6279360 #0271058 002881844 
 121104 | 42144192 18.6547581 7.0338497 .002873563 
 121801 |. 42508549 18.6815417 70405806 .002865830 
 122500 42875000 18.7082869 7 .0472987 0088571438 i 
 123201 43243551 18.7349940 7.054004] . 002849003 ( 
 123904 | 43614208 18. 7616630 70606967 . 002840909 | 
 124609 | 43986977 | 18.7882942 70673767 | 002832861 tH 
 35 125316 44361864 18.8148877 7.0740440 | 002824859 | 
 3855 126025 44738875 18.8414437 70806988 .002816901 i 
 356 126736 45118016 18.8679623 70873411 . 002808989 
 357 127449 45499293 18.8944436 7.0939709 . 002801120 | 
 358 128164 45882712 18.9208879 71005885 .002793296 
 359 128881 46268279 18.9472953 71071937 002785515 
 360 129600 46656000 18. 9736660 71137866 0027777 78 
 361 130321 47045881 19.0000000 V. 1203674 "00277008 
 362 131044 47437928 19 .0262976 71269360 .002 + ODA3t 
 | 363 131769 | 47882147 190525589 7. 1334925 002754821 
 364 182496 | 48228544 19.0787840 7. 1400370 002747253 
 | 865 133225 48627125 19.1049732 71465695 .002739726 
 | 366 33956 49027806 19.1311265 7.1580901 "002732240 
 367 | 134689 49430863 19.1572441 7’ .1595988 002724796 
 868 | 135424 49836032 19. 1833261 71660957 002717391 
 369 136161 50243409 192093727 7.1725809 .002710027 
 370 136900 50653000 192353841 71790544 .002702703 
 37 37641 51064811 19.2613603 71855162 .002695418 
 372 138384 51478848 19.2873015 7.1919663 .002688172 
 
 L bees Ae get et a 
 520 
 
873 | 189129 
 374 | 139876 
 
 | 75 140625 
 
 | 346 | 141876 
 
 377 | 142129 
 
 378 | 142884 
 
 379 | 143641 
 
 380 | 144400 
 
 381 | 145161 
 
 p> 882 | «145924 
 
 ; 883 | 146689 
 
 | 884 | 147456 
 
 | 885 148225 
 
 | 386 148996 
 
 | 387 | 149769 
 
 888 | 150544 
 
 889 | 151821 
 
 390 | 152100 
 
 vii 391 | . 152881 
 
 | 892 .| 153664 
 
 li | 398 | 154449 
 
 lit | 394 155236 
 
 Hn | 895 | 156025 
 
 HH 396 | 156816 
 
 Hi | 397 | 157609 
 
 iy } 398 158404 
 
 Na | 399 159201 
 fen | 
 
 | 400 | 160000 
 
 401 160801 
 
 402 161604 
 
 403 162409 
 
 | 404 163216 
 
 405 | 164025 
 
 | 406 164836 
 
 407 165649 
 408 166464 
 409 167281 
 
 | 410 | 168100 
 | 411 | 168921 
 WW 412 | 169744 
 vit 413 | 170569 
 i 414 | 171396 
 Alt 415 | 172225 
 A 416 | 173056 
 a 417 | 173889 
 ii 418 | 174724 
 
 419 | 175561 
 420 | 176400 
 177241 
 422 | 178084 
 423 | 178929 
 424 | 179776 
 425 | 180625 
 
 426 181476 
 42% 182529 
 
 428 183184 
 429 184041 
 430 184900 
 185761 
 432 186624 
 
 | 187489 
 434 | 188856 
 
 | 
 
 | Squares. | 
 
 Cubes. 
 
 51895117 
 52313624 
 52734375 
 58157376 
 53582633 
 54010152 
 54439939 
 54872000 
 55306341 
 55742968 
 56181887 
 56623104 
 57066625 
 7512456 
 7960603 
 58411072 
 58863869 
 59319000 
 59776471 
 60236288 
 60698457 
 61162984 
 61629875 
 62099136 
 32570773 
 63044792 
 63521199 
 64000000 
 64481201 
 64964808 
 65450827 
 65939264 
 66430125 
 66923416 
 67419143 
 67917312 
 68417929 
 68921000 
 69426531 
 69934528 
 70444997 
 70957944 
 1473875 
 71991296 
 2511713 
 73034632 
 #3560059 
 4088000 
 74618461 
 75151448 
 "5686967 
 36225024 
 "6765625 
 77308776 
 7854483 
 78402752 
 78953589 
 79507000 
 80062991 
 80621568 
 81182737 
 81746504 
 
 Square 
 xn te | Cube Roots, 
 19.3132079 |  %.1984050 
 19.3390796 | 7.204882 
 19.8649167 | 7.2112479 
 19.3907194 |  7.2176522 
 19.4164878 |  7.2240450 
 194422221 |. 7.2304268 
 19.4679223 | 72367972 
 19.4935887 | '%.2431565 
 19.5192218 | 7.2495045 
 19.5448203 | 7.2558415 
 195703858 72621675 
 195959179 % 2684824 
 19 .6214169 7 2747864 
 19.6468827 7 2810794 
 196723156 1 2873617 
 19.6977156 72936330 
 19. 7230829 |  7.2998986 
 197484177 | 7.3061436 
 19.7737199 | %.3123828 
 19.7989899 | 7%.3186114 
 19.8242276 |  %.3248295 
 19. 8494332 7 .8310369 
 19.8746069 % 3372339 
 19.8997487 | 7,3484205 
 19.9248588 | 17.3495966 
 19.9499373 | %.8557624 
 19.9749844 | %.8619178 
 20.0000000 | %.8680630 
 20.0249844 | 7. 8741976 
 20.0499377 | 7.8808227 
 20.0748599 % 2864373 
 20.0997512 7.3925418 
 20. 1246118 7 3986363 
 20.1494417 | 7.4047206 
 20.1742410 |  7.4107950 
 20.1990099 |  7.4168595 
 20. 2237484 7 4229142 
 20. 2484567 7 4289589 
 202731349 7 4349938 
 20.2977831 74410189 
 203224014 4470342 
 20. 3469899 74580399 
 203715488 74590359 
 20.3960781 7 4650223 
 20.4205779 7.4709991 
 20.4450483 7 4769664 
 20.4694895 74829242 
 204939015 7. 4888724 
 20 5182845 74948113 
 205426386 75007406 
 20.5669638 |  7.5066607 
 20 5912603 75125715 
 20.6155281 75184730 
 20. 6397674 7 5243652 
 20.6639783 75302482 
 20. 6881609 7 5361221 
 20.7123152 75419867 
 20.7364414 7 5478423 
 207605395 7 5536888 
 20.7846097 75595263 
 20. 8086520 75653548 
 7.5711743 
 
 20. 8326667 
 
 321 
 
 TABLE XXIII.—SQUARES, CUBES, SQUARE ROOTS, 
 
 | 
 
 Reciprocals. 
 
 002680965 
 002673797 
 002666667 
 002659574 
 002652520 
 002645503 
 002638522 
 
 .002681579 
 . 002624672 
 .002617801 
 .002610966 
 .002604167 
 .002597403 
 . 002590674 
 002583979 
 002577320 
 002570894 
 002564108 
 .002557545 
 .002551020 
 002544529 
 002588071 
 . 002531646 
 . 002525253 
 .002518892 
 002512563 
 002506266 
 
 002500000 
 .002493766 
 .002487562 
 -002481390 
 002475248 
 002469136 
 . 002463054 
 002457002 
 002450980 
 .002444988 
 
 00243902 
 002433090 
 002427184 
 - 002421308 
 002415459 
 002409639 
 002403846 
 002398082 
 002392344 
 002386635 
 002380952 
 002375297 
 002369668 
 002364066 
 002358491 
 002352941 
 002347418 
 - 002341920 
 002336449 
 002331002 
 002325581 
 002320186 
 002314815 
 .0023809469 
 002304147 
 
CUBE ROOTS, AND RECIPROCALS. 
 
 Sta ae Werke “Be pee Ralaeen ae 
 | | | ] 
 | a 4 = t; Square > | 
 : No. (Squares. | Cubes, | Sheil | Cube Roots. | Reciprocals. 
 435 | 189225 82312875. |  20.8566536 7.576984: 20885) 
 436 | 190096 | 82881856 | 20. 8806130 apes | Prgsee 
 |. 437 | 790969 | 83453453 |  20.9045450 | 7.5885793 “002258330 
 438 | 191844 84027672 | — 20.9284495 75043633 | 7002985105 
 439 | 19z721 | 84604519 | 20.9523268 | 7.600185 | 02277904 | 
 | 440 | 193600 | 85184000 | 20.9761770 | 7.6059049 |  .o02272727 | 
 | 440 | 133600 | Beresisn | 2120000000 | y-6146626 | |. onaabzarG | 
 : 442 | 195364 | 86350888 | 21.0237960 | 7.6174116 | “026243 | 
 443 | 196249 | 86038307 | 21.0475652 | 716231519 | [002257336 | 
 : 444 |< 197136 Wepsse, | 21.0718075 | 7.688887 |  “oopasvane 
 | 445 | 198025 88121125 21 095023 76346067 gu Sate i 
 | 446 | 198916 | 88716536 21.1187121 76403213 rinserkts fi 
 ; 447 | 199809 | 89314623 | 21.1423745 | 76460272 “002287136 | 
 418 | 200v0e | sogis392 | 21.1660105 | 7.GorTed? “DoRDae LA | 
 449 | 201601 | 90518819 | 21.1896201 | 716574133 |  ‘oo22RQ7171 
 | | 450 | 202500 | 91125000 | 21.2132034 | 7.6630943 | 229295 
 | 4 | Sesto, | grass: | St cbaeve0s | Feosrees “0B21 7208 
 452 | 20130 | gesiodng | 212602016 | 6741303 | 002212389 i 
 53 | 205209 | 92059677 | 21.28387967 | 7.s800857 | — 1002207506 i 
 454 | 206116 | 93576664 S 3072758 | ressraes |. looBa0ao4s 
 | 455 | 207025 | 94196375 | 21.8807200 |  7.6013717 “003197802 ii 
 456 | 207936 94818816 21 3541565 76970023 “002109083 | 
 457 | 208849 | 95443003 | 21.8775583 | 7. 7026246 “02188184 \ 
 458 | 209764 | 96071912 | 21.4009316 7. @082388 “00218306 | 
 | 459 | 210681 | 96702579 | 21.4242853 | 7.713848 002178649 " 
 460 | 211600 | 9733600 21.4476 77194426 021734 ) 
 te. | Sige | gyoreisy | aiaveoios | fracas |. <oopieig? 
 462 | 213444 | 98611128 |  21.4941853 irposiat | “002164502 
 | ~ 463 | 214369 | go2see7 | 21 5174348 | 7. 7361877 “02150827 | 
 | 464 | 2t506 | goggrada | 1.540592 | 77417532 | [002 eae | 
 465 | oieoas | 10nsi462s | 2icsesessy | ‘tira7siog | oozi50588 | 
 466 | 217156 | 101194696 21.5870331 |  7.7528606 “002115923 
 467 | 218089 | 101847563 21.6101828 | 7. 7584023 “pooidieer 
 468 | 219024 | 102503232 | 21.6333077 | 7.760361 | “OO3136752 
 $ p : | ; Ate as aoe A IIo ~UURLO0 (08 
 469 | 219961 | 103161709 | 21.6564073 | 7. 7604620 “002132196 
 470 | 220900 | 103823000 |  21.679483 7.77498 212766 
 | Spies | qouernit | sicroessie | f-yeodood “posiaa142 | 
 va | Seared | 105154048 | 217255610 | 7.759928 “D021 S64 | 
 | 473 | aeare9 | t0sea3817 | 21.7485632 | 7.7914875 “DOR114165 
 | aca | geaeve | 105496424 | oucevisaii | 7969745 “02100705 | 
 4v5 | 905605 | 107iTI875 | 21 vo4i947 | 7.024538 “002105263 | 
 476 | 206576 | 107850176 | 21 8174242 | Y_80r9e54 ‘002100810 
 | 477 4 927529 | 108531333 |  21.8403207 |  7.8133892 002096436 [ 
 | 473 | 228484 | 109215352 | 21 86B2L1 78188456 “02002080 
 lt oor ODE Qe NBR f 5616 Te OnOsbee 
 | 479 | 220441 | 109002289 "| 21.8860686 | 7.8242942 ‘002087683 
 | 480 | 230400 | 110592000 |  21.9089023 | 7.820735: 208833: 
 fa | Seat | itiseiet | sisirise | 4 8351068 “(02070002 
 | 482 | 239304 | 111980168 |  21.9544984 #8403049 “002074680 
 | 483 | 233989 | 1izvens7 | 21 9772610 | 718460134 “902070303 
 | 484 | 231255 | 113379904 | 92/0000000 |  7l8514e44 “002066116 
 485. | 235295 | 114081125 | 2210227155 | 78568281 “002061856 
 | 486 | 2361968 | 114791256 | 22.0454077 | 7 8622242 “002057613 
 487 | 237169 | 115501303 |  22.0680765 | — 7.8676130 “002058388 
 ty | past | iioatiere | 22.000 | rsr2004 | “ooe0dgTs0 
 43) | 239121 | 116930169 | 221133444 | 7 8783084 "002041999 
 490 | 240100 | 117649000 |  22.1350436 | 7.8837352 | — .002040816 
 tr | sires, | qissvorrt. | gecdsesies | 7.ss90016 | [902036660 
 42 | 242064 | 119005188 | a2.1810730 | F180L4463 | “002032520 
 493 | 243049 | 119823157 |  22.2036033 | esogzg17 | 109202838 
 494 | 244036 | 120553784 | 22.2261108 |  7.9051204 | “902021201 
 495 | 245025 | 121997375 | 2.248505 |  7_910450% “003020203 
 | re Wiper > | sect io) 22 2485955 7. 9104599 002020202 
 | 495 | e4oo16 | Teepesea6 | Re.zrIOsiD | T.91DTERR ‘oozsisize — | 
 | 
 
 9 
 322 
 
Or 
 
 | 
 | 
 | 
 | 
 | 
 
 Squares. Cubes. 
 217009 12276347 
 248004 123505992 
 249001 | 124251499 
 250000 125000000 
 251001 125751501 
 25204 126506008 
 253009 127263527 
 254016 128024064 
 255025 128787625 
 256036 129554216 
 257049 1303238843 
 258064 1381096512 
 259081 131872229 
 260100 132651000 
 261121 183482831 
 262144 134217728 
 263169 185005697 
 264196 135796744 
 265225 136590875 
 266256 137388096 
 267289 188188413 
 268324 188991832 
 269361 ° 139798359 
 270400 140608000 
 271441 141420761 
 272484 142236648 
 273529 143055667 
 274576 143877824 
 275625 144703125 
 276676 145581576 
 277729 146363183 
 278784 147197952 
 279841 148035889 
 280900 148877000 
 281961 149721291 
 283024. 150568768 
 284089 151419437 
 285156 152273304 
 286225 1531380375 
 287296 153990656 
 288369 154854153 
 289444 155720872 
 290521 156590819 
 291600 157464000 
 292681 158340421 
 293764 159220088 
 294849 160103007 
 295936 160989184 
 297025 161878625 
 298116 162771336 
 299209 163667323 
 300304 164566592 
 301401 165469149 
 802500 166375000 
 303601 167284151 
 804704 168196603 
 805809 169112377 
 3806916 170031464 
 808025 1709538875 
 809136 171879616 
 3810249 172298693, 
 
 173741112 
 
 311364 
 
 TABLE XXIII.—SQUARES, CUBES, SQUARE ROOTS, 
 
 Square 
 Roots, 
 
 222934968 
 22.3159136 
 22. 3383079 
 223606798 
 223830293 
 22. 4053565 
 22. 4276615 
 22. 4499443 
 22.4722051 
 22. 4944438 
 225166605 
 
 25388553 
 225610283 
 225831796 
 22. 6053091 
 226274170 
 
 2. 6495033 
 
 2. 6715681 
 22. 6936114 
 227156334 
 227376340 
 23.7506134 
 22. 7815715 
 228035085 
 
 2. 8254244 
 22.8473193 
 22. 8691933 
 228910463 
 229128785 
 229346899 
 229564806 
 22. 782506 
 23. 0000000 
 
 28 0217289 
 23 . 0434372 
 23 .0651252 
 23 . 0867923 
 23. 1084400 
 23. 1300670 
 23 .1516788 
 23 .17382605 
 23 1948270 
 23.2168735 
 23.23879001 
 23. 2594067 
 23 .2808935 
 23 38023604 
 23 . 3238076 
 23 .8452351 
 23 8666429 
 23 .3880311 
 23 .4093998 
 23 .4807490 
 
 23. 4520788 
 23.4733892 
 23.4946802 
 23 5159520 
 23 5372046 
 23 .5584380 
 23 5796522 
 23.6008474 
 23. 6220236 
 
 BOD OMMDMMDODOWDW WHWMOMWOOMWHMM-s-F aaa asa eda sett 
 
 © cp co 
 
 9210994 
 . 9264085 
 9317104 
 9370053 
 9422931 
 9475739 
 9528477 
 9581144 
 96383743 
 9686271 
 9738731 
 9791122 
 9843444 
 
 . 9895697 
 . 9947883 
 0000000 
 0052049 
 .0104032 
 .0155946 
 0207794 
 20259574 
 .0811287 
 0362935 
 .0414515 
 .0466030 
 0517479 
 0568862 
 .0620180 
 .0671482 
 0722620 
 0773743 
 .0824800 
 0875794 
 .0926728 
 .097'7589 
 . 1028390 
 .1079128 
 1129803 
 1180414 
 1230962 
 1281447 
 1331870 
 1882280 
 
 1482529 
 1452765 
 1532939 
 1583051 
 1633102 
 1683092 
 1733020 
 1782888 
 1832695 
 1882441 
 1932127 
 1981752 
 .20313819 
 
 8. 2080825 
 
 8.21380271 
 8.2179657 
 8 .2228985 
 8. 2278254 
 8. 2327465 
 
 323 
 
 Reciprocals. 
 
 | Cube Roots. 
 
 002012072 
 :002008032 
 . 002004003 
 
 002000000 
 001996098 
 .001992032 
 .001988072 
 001984127 
 .001980198 
 .001976255 
 .001972387 
 .001968504 
 .001964637 
 
 001960784 
 .001956947 
 .001953125 
 001949318 
 .001945525 
 .001941748 
 001937984 
 . 601934236 - 
 .001930502 
 . 001926782 
 001923077 
 .001919386 
 .001915709 
 .001912046 
 .001908397 
 .001904762 
 .001907141 
 .001897533 
 .001893939 
 .001890359 
 .001886792 
 . 001883239 
 .001879699 
 .001876173 
 001872659 
 .001869159 
 001865672 
 .001862197 
 .001858736 
 . 001855288 
 001851852 
 001848429 
 .001845018 
 .001841621 
 001838235 
 .001834862 
 .001831502 
 .001828154 
 .001824818 
 .001821494 
 
 001818182 
 .001814882 
 .001811594 
 .001808318 
 - 001805054 
 .001801802 
 .001798561 
 001795832 
 .001792115 
 
CUBE ROOTS, AND RECIPROCALS. 
 
 No. | Squares.| Cubes. star oi 
 559 | 312481 74676879 |  23.6431808 
 560 313600 175616000 23 6643191 
 561 314721 176558481 23 6854386 
 562 315844 177504328 23 7065392 
 563 316969 178453547 23 7276210 
 564 318096 179406144 23. 7486842 
 565 319225 180362125 23 7697286 
 566 320356 181321496 23 7907545 
 567 321489 182284263 23 .8117618 
 568 | 322624 183250432 23 8327506 
 569 323761 184220000 25 .8537209 
 BYE 324900 185193000 23. 8746728 
 571. | 326041 186169411 23 8956063 
 572 327184 187149248 23,.9165215 
 Aye 328329 188132517 239374184 
 57 329476 189119224 23.9582971 
 575 | 330625 190109375 23 9791576 
 | 576 331776 191102976 24 0000000 
 | 577 332929 192100033 24 0208243 
 57 334084 193100552 24 0416306 
 | 57 335241 194104529 24 0624188 
 580 336400 195112000 24 0831891 
 581 337561 196122941 24 1039416 
 582 338724 197137368 24 1246762 
 583 339889 198155287 24. 1453929 
 584 341056 199176704 |  24.1660919 
 585 342225 200201625 | 24.1867'732 
 | 586 343396 201230056 24 2074369 
 | 587 344569 202262003 24 2280829 
 588 3457 903297472 | 24.2487113 
 589 346921 204336469 24 2698222 
 590 348100 205379000 24 2899156 
 591 349281 206425071 | — 24.8104916 
 592 350464 207474688 | 24.3310501 
 593 351649 208527857 | 24.38515913 
 594 352836 209584584 | 24.8721152 
 595 354025 210644875 |  24.3926218 
 596 355216 211708736 24 4131112 
 597 356409 212776173 24 4335834 
 598 357604 213847192 24 4540285 
 599 358801 214921799 24 4744765 
 600 360000 216000000 244948074 
 601 361201 217081801 245153013 
 602 | 362404 218167208 24 5356883 
 | 603 | 363609 219256227 24 5560583 
 604 364816 220348864 245764115 
 | 605 366025 221445125 245967478 
 606 | 367236 222545016 | 24.6170673 
 607 368449 223648543 24 6373.00 
 | 608 369664 294755712 246576560 
 : 609 370881 295866529 24 6779254 
 610 372100 226981000 24 .6981781 
 G11 373321 228099131 24 7184142 
 | 612 37444 229220928 | 24.7386338 
 | | 613 375769 230346397 | 24.7588368 
 | 614 76996 231475544 | 24.7790234 
 | 615 378225 232608375 | 24 7991935 
 616 | 3879456 933744896 | 24.8193473 
 617 380689 934885113 |  24.8394847 
 618 381924 236029032 24 8596058 
 619 383161 237176659 24.8797106 
 620 384400 238328000 248997992 
 
 cots 
 324 
 
 Bsace" 
 } 
 | Cube Roots. | Reciprocals. | 
 / 
 | §,2376614 001788909 
 8.242506 001785714 
 82474740 001782531 
 8. 2523715 001779359 
 | 8. 2572633 .001776199 
 | §.2621492 001773050 
 8.2670294 + .001769912 | 
 82719039 001766784 
 82767726 001763668 
 82816355 001760563 
 82864928 001757469 
 8.2913444 001754386 
 | 8.2961903 001751313 
 | 8.2010804 001748252 
 8.8058651 001745201 
 8.3106941 001742160 
 83155175 001739130 
 8 3203353 001736111 
 8.82514"5 .001733102 
 8 8299542 001730104 
 8.8847553 001727116 
 8.3895509 001724138 
 83443410 001721170 
 8.3491256 001718213 
 8.2539047 001715266 
 8.586784 .001712329 
 | 8.3624466 001709402 
 83682095 001706485 
 8.8729668 001708578 
 8.3777188 001700680 
 8. 2824653 .001697793 
 |. 8.3872065 001694915 
 83919423 .001692047 
 88966729 .001689189 
 8.4013981 .001686341 
 8.4061180 001688502 
 8. 4108326 001680672 
 8.4155419 |. .001677852 
 8. 4202460 .001675042 
 8.4249448 001672241 
 8. 4296383 001669449 
 8 4343267 .001666667 
 8.4390098 001663894 
 8. 4436877 .001661130 
 84483605 001658375 
 84530281 . 001655629 
 8. 4576906 001652893 
 8.462347 .001650165 
 8. 4670601 001647446 
 84716471 001644737 
 8. 4762892 .001642036 
 8 4809261 .001639344 
 84855579 -001636661 
 8. 4901848 001633987 
 8.4948065 001631321 
 8 4994233 .601628664 
 85040350 . 001626016 
 8. 5086417 001623377 
 85132435 001620746 
 8.5178403 00161812: 
 85224321 001615509 
 8.5270189 .0C01612908 
 
TABLE XXIII.—SQUARES, CUBES, SQUARE ROOTS, 
 
 Squares, | 
 
 3885641 
 886384 
 388129 
 3889376 
 890625 
 891876 
 393129 
 394384 
 395641 
 896900 
 398161 
 399424 
 40689 
 401956 
 403225 
 404496 
 405769 
 407044 
 408321 
 409600 
 410881 
 412164 
 413449 
 414736 
 416025 
 417316 
 418609 
 419904 
 421201 
 422500 
 423801 
 425104 
 426409 
 427716 
 429025 
 430336 
 431649 
 432964 
 43,4281 
 435600 
 436921 
 438244. 
 439569 
 440896 
 442225 
 443556 
 444889 
 446224 
 447561 
 
 448900 
 450241 
 
 451584 
 452929 
 454276 
 455625 
 
 456976 
 
 458329 
 459684 
 
 461041 
 462400 
 463761 
 
 465124 
 
 ade 
 
 Square | Cube Roots. | Reciprocals. 
 
 hoots. 
 
 24.9198716 8.5316009 .001610306 
 24.9399278 8.53861780 -OOL607717 
 24 9599679 8.5407501 .001605136 
 24.9799920 | 8.5453173 .001602564 
 25 . 0000000 8.5498797 .001600000 
 25 .0199920 8.5544872 .001597444 
 25 .0399681 8.5589899 .001594896 
 25 0599282 8.56385377 .001592357 
 25. 0798724 8. 5680807 . 001589825 
 85 .0998008 8.5726189 .001587302 
 25 1197134 85771523 . 001584786 
 95 1396102 8.5816809 .001582278 
 25. 1594913 8.5862047 001579779 
 25 .1798566 8 .5907238 001577287 
 25 .1992063 8.5952380 .001574803 
 25.2190404 |  8.5997476 001572327 
 25. 2388589 — | 8. 6042525 .001569859 
 25 2586619 8. 6087526 001567398 
 25.2784493 | 8.6132480 001564945 
 25.29822138. | 8.6177388 .001562500 
 25.38179778 |  8.6222248 001560062 
 25 .33877189 | 8.6267063 . 001557632 
 25.3574447 | © §.63118380 .001555210 
 25.38771551 | °8.6356551 .001552795 
 25 38968502 | 8 .6401226 .001550388 
 25.4165301 8.6445855 .001547988 
 25 .4861947 |  8.6490437 .001545595 
 25.4558441 | 8.65384974 .001543210 
 25.4754784 | 8.6579465 .001540832 
 25.4950976 | 8.6623911 .001538462 
 25.5147016 8. 6668310 .001536098 
 25 .5842907 8.6712665 001533742 
 25 .5588647 | 8.6756974 -001531894 
 20D (o42a7 | 8.6801237 -001529052 
 25.5929678 | 8.6845456 -001526718 
 25.6124969 | 8.6889630 001524390 
 25 .6320112 8.6933759 .001522070 
 25..6515107 8.6977843 .001519757 
 25.6709953 |  8.7021882 .001517451 
 25.6904652 | 8.7065877 .001515152 
 25.7099203 |  8.7109827 - 001512859 
 25.7298607 | 8.7158734 -001510574 
 25 . 7487864. 8.7197596 -001508296 
 25 .7681975 8.7241414 -001506024 
 25 ..7875939 8. 7285187 . 001503759 
 25. 8069758 8.7328918 .001501502 
 25.8263431 8.7372604 .001499250 
 25.8456960 | 8.7416246 001497006 
 25 .8650343 | 8 .7459846 .001494768 
 25 . 8848582 8.503401 .001492537 
 25 9036677 8.546913 .001490313 
 25 . 9229628 8 .7590883 .001488095 
 25 . 9422485 §.7638809 001485884 
 25.9615100 8.7677192 001483680 
 25 . 9807621 8.720532 .001481481 
 26 .0000000 8.7'763830 .001479290 
 26 0192237 8. 7807084 .001477105 
 26.0384331 8.'7850296 001474926 
 25 0576284 8.7893466 .001472754 
 26 0768096 8.7936593 .001470588 
 26 .0959767 8.7979679 001468429 
 26 .1151297 8 .8022721 .001466276 
 
 90K 
 3825 
 
CUBE ROOTS, 
 
 | 
 | 
 
 AND RECIPROCALS. 
 
 | 
 
 No. eauareey Cubes. | ee: | Cube Roots, | Reciprocals. 
 683 | 466489 318611987 26 . 1342687 8.8065722 001464129 
 681 | 467856 320013504 | 26.1588937 8. 8108681 001461988 
 685 | 469225 321419125 | 26.1725047 8.8151598 001459854 
 686 470596 322828856 | 26.1916017 8.8194474 001457726 
 687 | 471969 824242703 26 2106848 8. 8237307 001455604 
 G88 | 473344 325660672 26 2297541 8. 8280099 001453488 
 689 | 474721 327082769 26 2488095 8 .8322850 001451379 
 690 | 476100 998509000 | 26.2678511 |  8.8865559 001449275 
 691 477481 329939371 | 26.2868789 | 8. 8408227 001447178 
 692 478854 331373888 |  26.3058929 88450854 001445087 
 693 | 480249 | 332812557 | 263248932 | 8.8493440 001443001 
 694 | 481636 | 334255384 | 26.8488797 | 8.8535985 | 001440922 
 695 483025 335702375 | 26.8628527 | 8.8578489 | 001438849 
 696 481416 | 387152536 | 26.3818119 | 8 .8620952 001436782 
 697 | 485809 | 338608873 | 26 4007576 8.8663375 001434720 
 698 | 487204 | 340068392 | — 26.4196896 3.8705757 =|, .001482665 
 699 | 488601 | 341532099 | 26 4886081 8.8748099 | .001480615 
 700 | 490000 | 343000000 | 26 4575131 8.8790400 |  .001428571 
 VOL | 491401 344472101 | 26.4764046 | —8.8832661 001426534 
 702 | 492804 345048408 | 26.4952826 | 8.8874882 | 001424501 
 "03 | 494209 | 347428927 965141472 | 8.8917063 | = 001422475 
 704 | 495616 | 3848918664 265329983 | 8.8959204 |  .001420455 
 705 497025 | 350402625 | 26.5518361 | 8.9001304 |  .001418440 
 706 498436 | 351895816 |  26.5706605 89043366 | 001416431 
 707 499849 353393243 |  26.5894716 3.9085387 | .001414427 
 708 | 501264 | 354894912 | 26.6082694 8.9127369 001412429 
 709 | 502681 356400829 | 26 6270539 | 8.9169311 001410437 
 "10 | 504100 357911000 |  26.6458252 8.9211214 , 001408451 
 711 505521 | 359425431 26 6645833 | 8.9253078 | 001406470 
 712 | 506944 | 360944128 26 6833281 8.9294902. | 001404494 
 #13 | 508369 | 362467097 26. 7020598 8.9336687 001402525 
 ee 509796 863994344 26. 72077 89378433 | 001400560 
 15 | 511225 365525875 26 .7394839 8.9420140 | .001398601 
 716 512656 367061696 26 . 7581763 8.9461809 001396648 
 717 514089 368601813 26 .7768557 8.9503438 | 001394700 
 W718 .| 515524 3701462382 26 .7 955220 89545029 | .001892758 
 519 | 516961 371694959 268141754 8.9586581 001390821 
 "20 | 518400 373248000 |  26.8328157 8. 9628095 001388889 
 m21 | 519841 374805361 |  26.8514432 8. 9669570 401886963 
 (22 521284 | 876367048 26 8700570 8.9711007 001885042 
 723 522729 377933067 |  26.8886593 8. 9752406 .001383126 
 724 | 524176 379503424 26 . 9072481 89793766 .001881215 
 125 525625 381078125 |  26.9258240 8 .9835089 001379310 
 726 527076 382657176 26 . 9443872 8.9876373 .001377410 
 1270 528529 | 384240583 26 . 9629375 8. 9917620 .001375516 
 728 529984. | 385828352 26 .9814751 8 . 9958829 001873626 
 729 531441 387420489 27 .0000000 9 .0000000 001371742 
 730 532900 389017000 27 0185122 90041184 001369863 
 731 | 534361 390617891 27 0370117 90082229 | 001367989 
 732 | 535824 392222168 27 0554985 9 0128288 .001366120 
 733 | 587289 393832837 27 07389727 9 .0164309 001364256 
 734 | 538756 395446904 27 .0924344 9 0205293 001362898 
 "735 | 640225 397065375 | 27.1108834 9 0246239 .001860544 
 726 | 541696 398688256 | 27.1293199 9 0287149 001358696 
 737 | 543169 400315553 27 1477439 9 0328021 | .001356852 
 mga | 544644 | 401947272 | 27.1661554 9 0368857 |  .001855014 
 nag | 546121 | 403583419 |  27.1845544 90409655 |  .001353180 
 "40 | 547600 | 405224000 | 97.2029410 | 9.0450419 .001351351 
 TAL 549081 406869021 | 27.2218152. | 9.0491142 001849528 
 742 550564 408518488 | 27.23896769 | 9.0531831 | 001347709 
 743 552049 | 410172407 | 27. 2580263 90572482 | .001845895 
 744 553536 | 411830784 | 27.2763634 9,0613098 {|  .001844086 
 
 326 
 
 =e 
 
hI 
 i 
 | 
 } 
 
 Squares. 
 
 TC) 
 
 Oo 
 — 
 
 e 
 
 ~~ ~> =] ~t =} ~ 
 OTOL UL OT Or 
 
 “2 Oo OUHS Co 0 
 
 OV oror 
 Laie 2) 
 
 J 3-3 5 
 
 ~ 
 
 > SS 
 i=) 
 
 J 3 3-5 5 
 VNININS 
 et 
 
 OU CO a 
 
 2-2 +) 
 COIS 
 
 i ae ee 
 
 555025 
 556516 
 558009 
 559504 
 561001 
 562500 
 564001 
 565504 
 567009 
 568516 
 570025 
 571586 
 573049 
 574564 
 576081 
 577600 
 579121 
 580644 
 582169 
 583696 
 5852 
 586756 
 588289 
 589824 
 
 591361 
 
 592900 
 594441 
 595984 
 597529 
 599076 
 600625 
 602176 
 603729 
 605284 
 606841 
 608400 
 609961 
 611524 
 613089 
 614656 
 616225 
 617796 
 619369 
 620944 
 62252 
 
 I 
 
 624100 
 625681 
 627264 
 628849 
 630436 
 632025 
 633616 
 635209 
 636804. 
 
 338401 
 640000 
 641601 
 648204 
 644809 
 646416 
 648025 
 649636 
 
 Cubes. 
 
 413493625 
 415160936 
 416832723 
 418508992 
 420189749 
 421875000 
 423564751 
 425259008 
 426957777 
 428661064 
 430368875 
 432081216 
 433798093 
 435519512 
 437245476 
 
 438976000 
 440711081 
 442450728 
 444194947 
 445943744 
 447697125 
 449455096 
 451217663 
 452984832 
 454756609 
 456533000 
 458314011 
 460099648 
 461889917 
 463684824 
 465484375 
 467288576 
 469097433 
 470910952 
 472729139 
 
 474552000 
 476379541 
 478211768 
 480048687 
 481890304. 
 483736625 
 485587656 
 487443403 
 489303872 
 491169069 
 
 493039000 
 494913671 
 496793088 
 498677257 
 500566184 
 502459875 
 504358336 
 506261573 
 508169592 
 510082399 
 
 512000000 
 513922401 
 515849608 
 517781627 
 519718464 
 521660125 
 523606616 
 
 TABLE XXTII.—SQUARES, CUBES, SQUARE ROOTS, 
 
 7 
 7.3678644 
 7 
 7 
 
 Square 
 Roots. 
 
 Cube Roots. 
 
 Reciprocals. 
 
 27 2946881 
 273180006 
 7 .8813007 
 
 "8495887 
 
 38861279 
 "4043792 
 
 27. 4226184 
 27. 4408455 
 274590604 
 27 47726828 
 27 .4954542 
 (51863830 
 % 5917998 
 27 5499546 
 27 5680975 
 275862284 
 27 60438475 
 27.62.4546 
 7 6405499 
 
 ", 6586334 
 
 276767050 
 27. 6947648 
 27 .7'128129 
 
 ", 7808492 
 
 27. 7488739 
 27. 7668868 
 27.7848880 
 27 8028775 
 27 8208555 
 
 8388218 
 
 27 8567766 
 «8747197 
 
 ’, 8926514 
 
 279105715 
 27 9284801 
 27 . 9463772 
 9642629 
 ', 9821372 
 8. 0000000 
 8.0178515 
 28.0356915 
 8 .0535203 
 280718377 
 28. 0891438 
 
 . 1069886 
 
 28 . 1247222 
 3. 1424946 
 8. 1602557 
 28. 1780056 
 28.1957444 
 28 2134720 
 28.2311884 
 28 .2488938 
 28 .2665881 
 
 2842712 
 
 28 3019434 
 
 9196045 
 3372546 
 .8048938 
 
 28.3725219 
 
 .3901391 
 
 9.0653677 
 9 .0694220 
 9.0734726 
 9.0775197 
 9.0815631 
 9 .0856030 
 9. 0896392 
 9.0936719 
 90977010 
 9.1017265 
 9.1057485 
 9.1097669 
 9.1187818 
 9.1177931 
 9.1218010 
 
 9.1258053 
 9.1298061 
 9.1388034 
 9.137°7971 
 9.141787 
 
 9.145742 
 9.1497576 
 9.15387375 
 9.157'7139 
 9.1616869 
 9.1656565 
 9. 1666225 
 91785852 
 9.1775445 
 9.181E003 
 9. 1854527 
 9.1894018 
 91988474 
 9.1972897 
 9.201226 
 
 9.2051641 
 9 2090962 
 9 2180250 
 9.2169505 
 9.2208726 
 92247914 
 9 .2287068 
 9.2526189 
 9.286527 
 92404383 
 9. 2443355 
 9. 2462344 
 9. 2521300 
 9 .2560224 
 9 .2599114 
 9.2637973 
 9 .2676798 
 9.2715592 
 9.2754352 
 9.2798081 
 
 9. 2881777 
 9 2870440 
 9.2909072 
 9.2947671 
 9 2986239 
 9 8024775 
 9 .3063278 
 
 327 
 
 .001342282 
 .0013840483 
 .00133886&8 
 .-001886898 
 .001835113 
 . 001333333 
 -0U13881558 
 -001829787 
 . 0013828021 
 - 001826260 
 . 001824503 
 .001322751 
 . 001821004 
 -001319261 
 .C01317523 
 
 .001815789 
 .C01314060 
 -C01812336 
 -001310616 
 -001808901 
 .001307190 
 .001805483 
 .001803781 
 .0013802083 
 - 001300390 
 
 001298701 
 .001297017 
 001295337 
 -001293661 
 .001291990 
 001280323 
 .001288660 
 .001287001 
 001285347 
 .001283697 
 001282051 
 . 001280410 
 .001278772 
 001277139 
 001275510 
 001273885 
 001272265 
 .001270648 
 .C01269036 
 001267427 
 
 001265823 
 .C01264223 
 001262626 
 .001261034 
 001259446 
 .001257862 
 001256281 
 .001254705 
 .001258133 
 . 001251564 
 
 001250000 
 .001248489 
 .001246883 
 001245330 
 .001243781 
 001242236 
 001240695 | 
 
CUBE ROOTS, AND RECIPROCALS. 
 
 \ | Cc 7Tra 
 No. | Squares. Cubes. Ph otan 
 807 651249 525557943 28 .4077454 
 808 652864 527514112 28 . 4258408 
 809 | 654481 529475129 28 .4429253 
 810 656100 531441000 28 .4604989 
 811 657721 5383411731 28 .4780617 
 812 659344 535387328 28 .4956137 
 813 660969 537867797 28.5131549 
 814 662596 589353144 28 .5306852 
 815 654225 541343375 23.5482048 
 816 665856 543338496 28 .5657137 
 817 | 667489 545338513 28 .58382119 
 818 | 669124 547343432 28 . 6006995 
 819 670761 5493538259 28,6181760 
 82 672400 551368000 28 .635642 
 821. | 674041 553387661 286530976 
 | 822 675684 555412248 28 6705424 
 82% 677329 557441767 288 6879766 
 | 82 678976 55947622 28 .'7054002 
 | 825 680625 561515625 28.72281382 
 826 682276 563559976 287402157 
 | 827 683929 565609288 | 28.7576077 
 ) 828 685584 567663552 28.7749891 
 | 829 687241 569722789 >| 23.'7923601 
 830 688900 571787000 28 .8097206 
 831 690561 573856191 288270706 
 832 692224 575930368 28 .8444102 
 : 833 693889 578009537 28 .8617394 
 ) 834 695556 580093704. 28.8790582 
 : 835 697225 582182875 28.8963666 
 : 836 698896 584277056 28 .9186646 
 837’ | 700569 586376253 28 9309523 
 838 702244 588480472 28 . 9482297 
 839 703921 590589719 28.9654967 
 840 705600 592704000 28 .9827535 
 841 707281 594823321 29 . 0000000 
 842 708964 596947688 29.0172363 
 843 | 710649 599077107 29. 034462: 
 844, | 712386 601211584 29.0516781 
 845 714025 603351125 29. 0688837 
 846 | 715716 605495736 29 .0860791 
 347 | =. 717409 607645423 29 .1032644. 
 848 719104 609800192 29 . 1204396 
 849 720801 611960049 29.1376046 
 850 722500 614125000 29.1547595 
 351 | . 724201 616295051 29.1719043 
 852 72590 618470208 29.1890390 
 853 727609 620650477 29.2061637 
 854 | 729316 22835864. 29 2232784 
 | 855 731025 625026375 29 .2403830 
 | 856 732736 627222016 29. 2574777 
 857 734449 629422793 29.2745623 
 | 858 736164 631628712 29 .291687 
 | rola\ Pima ies UE ifoteil 6338397769 29.38087018 
 860 739600 636056000 29 3257566 
 861 741321 638277381 29.3428015 
 862 743044 640503928 29 .38598365 
 863 744769 642735647 29.3768616 
 864 | 746496 644972544 29 3938769 
 865 | | 748225 647214625 29 4108823 
 : 866 | . 749956 649461896 294278779 
 : 867. | | 751689 651714363 29 4448637 
 868 | | 7538424 653972032 29 .4618397 
 
 3 
 
 28 
 
 Cube Roots. 
 
 | Reciprocals. 
 
 .8101750 
 .8140190 
 
 .B178599 
 
 e 304 
 . 38381916 
 .38370167 
 . 38408386 
 .38446575 
 38484731 
 8522857 
 .0- 60952 
 .8599016 
 .8537049 
 . 8675051 
 
 9 .38718022 
 9.3750963 
 
 3788873 
 
 9. 38826752 
 9 .3864600 
 9 .3902419 
 9. 3940206 
 
 8977964 
 
 9 
 9.4015691 
 ") 
 
 4058387 
 
 9 .4091054 
 9 .4128690 
 
 Ow 
 
 SODOOO OOsewssd 
 
 la) 
 
 .4166297 
 .4203873 
 .4241420 
 .4278936 
 .4816423 
 .4358880 
 .43913807 
 .4428704. 
 .4466072 
 .4503410 
 .4540719 
 .4577999 
 .4615249 
 .4652470 
 .4689661 
 4726824 
 .4763957 
 .4801061 
 .4838136 
 .4875182 
 .4912200 
 .4949188 
 .4986147 
 .5023078 
 _ 5059980 
 
 5096854 
 .5138699 
 .5170515 
 . 5207308 
 5244063 
 5280794. 
 9817497 
 5354172 
 .5390818 
 
 .001289157 
 .00123762 
 
 .001286094 
 .001234568 , 
 .001283046 
 .001231527 
 .001230012 
 .001228501 
 . 001226994. 
 . 001225490 
 .001223990 
 .001222494 
 - 001221001 
 
 .001219512 
 .001218027 
 . 001216545 
 .001215067 
 .001218592 
 .001212121 
 .001210654 
 .001209190 
 .001207729 
 .001206273 
 .001204819 
 .001203369 
 .001201923 
 .001200480 
 .001199041 
 001197605 
 .001196172 
 .001194743 
 .001198317 
 .001191895 
 
 .001190476 
 .001189061 
 .001187648 
 001186240 
 .001184834 
 .001183432 
 .001182033 
 .001180638 
 001179245 
 .001177856 
 001176471 
 .001175088 
 .001173709 
 .001172833 
 .001170960 
 .001169591 
 .001166224 
 .001166861 
 .001165501 
 ; .001164144 
 
 .001162791 
 .001161440 
 .001160093 
 .001158749 
 .001157407 
 .001156069 
 .001154734 
 .001158403 
 , 001152074 
 
TABLE XXIII.—SQUARES, CUBES, 
 
 SQUARE ROOTS. 
 
 | 
 
 80.4959014 
 
 ae 
 
 | No. Squares. Cubes. Tks | Cube Roots. Reciprocals. 
 a : = 
 869 755161 656234909 | 29.4788059 9 5427437 001150748 
 870 756900 658503000 | 294957624 9.5464027 | 001149425 
 aff 758641 660776311 | 29.5127091 9.5500589 | .001148106 | 
 872 760384 663054848 | 29.5296461 | = 9.5537123 | _ .001146789 
 873 762129 665338617 | 29.5465784 | 9.5573630 001145475 
 87 768876 667627624 | 29.5634910 | 9.5610108 | .001144165 
 875 765625 669921875 | 29.5803989 |  9.5646559 001142857 
 87 767376 6722213876 | 29.5972972 1} 9.5682982 | .001141553 
 877 769129 6745261338 | 29.6141858 9.57193877 | .001140251 
 78 770884 | 676836152 | 29.6310648 9.5755745 | 001188952 
 879 772641 79151439 | 29.6479342 9.579208 | .001137656 
 880 774400 | 681472000 | 29.66479389 | 9.5828397 | .001136364 
 Sot a  COLOk 683797841 | 29.6816442 | 9.5864682 | .001135074 
 882 777924 686128968 | 29.6984848 | 9.5900939 | .001133787 
 883 | 779689 688465387 | 29.7153159 | 9.59387169 | .001182503 
 884 | 781456 690807104 | 29.7321875 | 9.5973873 | 001131222 
 a 885 | 783225 693154125 | 29.7489496 | 9.6009548 | .001129944 
 Hi 886 784996 | 695506456 | 29.7657521 | 9.6045696 | 001128668 
 Hil: 887 786769 697864103 | 29.7825452 | 9.6081817 | .001127896 
 Heat, 888 | 788544 700227072 =| 29.7998289 | 9.6117911 | .001126126 
 Ma 889 | 7903821 | 702595369 / 29.81610380 | 9.6153977 | .001124859 
 Wie 890 | 792100 | 704969000 | 29.8828678 | 9.6190017 | .001123596 
 891 | 793881 | TO7347971 | 29.8496281 | 9.6226080 | .001122334 
 i 892 795664 709732288 | 29.8663690 | 9.6262016 | .001121076 
 ist 893 797449 712121957 | 29.8831056 9.62979%5 001119821 
 i 894 799236 714516984 | 29.8998328 | 9.6338907 .001118568 
 ih 895 801025 716917375 | 29.9165506 | 9.6369812 .001117318 
 Hi 896 802816 719323186 | 29.9882591 | 9.6405690 .001116071 
 897 804609 721734278 | 29.94995838 | 9.6441542 .001114827 
 898 806404 724150792 | 29.9666481 9.647367 .001113586 
 899 808201 726572699 = -29.9838287 | 9.6513166 .001112847 
 | | 
 
 900 810000 729000000 | 30.0000000 | 9.6548938 .001111111 
 901 811801 | 731482701 | 30.0166620 | 9.6584684 .001109878 
 902 813604 7188870808 30.0383148 | 9.6620403 .001108647 
 903 815409 736314327 80 .0499584 9. 6656096 .001107420 
 904. 817216 788763264 30 .0665928 96691762 .001106195 
 905 819025 741217625 30.08382179 9 .6727403 001104972 
 906 820836 743677416 | 30.0998339 9.67638017 .001103753 
 907 822649 746142643 | 30.1164407 9.6798604 .001102536 
 908 824464 748613312 | 30.1330383 9 .6834166 .001101322 
 909 826281 751089429 | 30.1496269 9 .6869701 .001100110 
 910 828100 753571000 | 30. 1662063 9.6905211 .001098901 
 911 829921 756058031 80. 1827765 9.6940694 .001097695 
 912 831744 758550528 | 30.199337'7 9.6976151 | .001096491 
 all 913 833569 761048497 80.2158899 9.011583 .001095290 
 914 835396 | 763551944 380 . 2824329 9. 7046989 .001094092 
 ait 915 837225 766060875 380. 2489669 9 7082869 .001092896 
 4 916 839056 | 768575296 30.2654919 9.7117728 .001091703 
 917 840889 | 771095213 30. 2820079 9.7158051 .001090513 
 918 842724 | 7773620632 80 .2985148 9 7188354 .001089325 
 919 844561 | 776151559 80.3150128 9 7223631 . 0010881389 
 
 920 846400 | ee 30 .38315018 9 7258883 001086957 
 
 921 | 848241 | 781229961 30.34'79818 9 .7294109 .001085776 
 922 850084 | 783777448 80 .3644529 9 7329309 .001084599 
 923 851929 | 786330467 80.3809151 9.7364484 | .001083423 
 924 853776 | 788889024 80 .8978683 9.7399634 | .001082251 
 925 855625 | 791453125 | 80.4138127 9 .'7434758 .001081081 
 926 857476 | 794022776 | 30.4302481 97469857 .001079914 
 | 927 859329 796597983 30 .4466747 9. 7504930 .001078749 
 | 928 861184 T99178752 80 .4630924 9.7539979 .001077586 
 929 863041 801765089 30.4795013 9.7575002 .001076426 
 930 864900 804357000 9.7610001 .001075269 
 
CUBE ROOTS, AND RECIPROCALS. 
 
 ! 
 
 | 
 
 } | 
 No. Squares, | Cubes. Pete | Cube Roots. | Reciprocals. 
 931 866761 806954491 80.5122926 97644974 .001074114 
 932 868624 | 809557568 30 .5286750 9.7679922 .001072961 
 933 870489 | 812166287 | 30.5450487 9.7'714845 .001071811 
 934 872356 814780504 | 380.5614136 9.749743 .001070664 
 935 | 874225 817400375 30.5777697 9.7784616 .001069519 
 936 | 876096 820025856 380.5941171 9.7819466 .001068576 
 937 =| 877969 322656953 30.6104557 97854288 001067236 | 
 9388. | 879844 | Sati 3672 80. 6267857 9.7889087 .001066098 | 
 9389 881721 | 27936019 380.6481069 | 9.7928861 .001064663 
 940 | 883600 830584000 30.6594194 | 9.7958611 . 0010688380 
 941 | 885481 833257621 30.6757233 | 9.7993236 | .001062699 
 942 | 887364 835896888 30.6920185 | ee | 001061571 
 943 | 889249 | 838561807 | 380.7083051 | 9.8062711 .001060445 
 944. |. 891136 | 841282384 40.7245880 | 9.80973862 .001059822 
 945. | 898025 843908625 | 30.7408523 | 98131989 .001058201 
 946 | 894916 846590536 30.7571130 | 9.8166591 .001057082 
 947 | 896809 849278123 30.7788651 | 9.8201169 .001055966 
 948 | 898704 851971392 30.7896086 | 9.8285723 001054852 
 $49 | 900601 | 854670349 | 30.8058436 9 8270252 -001053741 
 950 | 902500 | 857375000 30.8220700 | 9.8304757 | .001052632 
 951 904401 | 860085851 30.8882879 | 9.838389288 .001051525 
 952 906304 862801408 80.8544972 9.8378695 .001050420 
 953 | = 908209 865523177 380.8706981 9. 8408127 .001049318 
 954. | 910116 868250664 | 80.8868904 9 8442536 .001048218 
 95 | 912025 | 870983875 30. 9030743 98476920 .001047120 
 956 | 913936 873722816 80. 9192497 9.8511280 .001046025 
 957 915849 876467493 30.9354166 | 9.8545617 | .001044932 
 958 | 917764 879217912 30.9515 751- | = 9.8579929 .0010438841 
 959 | 919681 881974079 | 380.9677251 , 9.8614218 | .001042753 
 960 | 921600 | 884736000 80. 9828668 9.€648483 | .001041667 
 961 | 928521 | 887508681 | 81.0000C00 9.68 2724 001040583 
 962 925444 890277128 | 31.0161248 9.8716941 001039501 
 963 927369 898056847 | 381.0822413 9 8751135 001088422 
 964 929296 895841344 ; 81.0488494 9.6785805 | .001087344 
 965 931225 898632125 | 31.0644491 9.819451 | .001086269 
 966 933156 901428696 | 381.0805405 9. €85357 001085197 
 967 985089 904231063 | 81.0966286 9 .E887673 .001084126 
 968 937024 907039232 31. 1126984 9.8921749 .001083058 
 969 938961 909858209 381. 1287648 9.8955€01 |  .001081992 
 970 940900 912673000 | 381.1448230 9.8959830 | .001030928 
 971 942841 915498611 381.1608729 9. C028&85 .001029866 
 972 944784 918380048 81 .1769145 9. seth .001 028807 
 973 946729 921167317 81 .1929479 9.909177 .001027'749 
 97. 948676 924010424 | 81.2089731 | 99125419 001026694. 
 975 950625 926859375 | 31.2249900 |! 9.9159624 . 001025641 
 76 952576 929714176 | 381.2409967 {| 9.9198513 .001024590 
 O77 | 954529 9825748838 | 81.2569992 | 9.9227379 .001023541 
 978 956484 935441352 | 381.2729915 9 .9261222 001022495 
 979 958441 938318739 | 31.2889757 | 9.9295042 .001021450 
 980 960400 941192000 | 381.8049517 | 9.9828839 .C01020408 
 981 962361 944076141 | 31.8209195 |; 9.9862613 .001019368 
 982 964324 946966168 31 .3868792 | ~ 9.9896868 .001018330 
 983 966289 949862087 31.3528308 | 9 9430092 .001017294 
 984 968256 952763904 81.38687743 9 .94638797 .001016260 
 985 970225 955671625 31.3847097 | 9.9497479 .001015228 
 986 972196 958585256 | 81.4006369 | 9 .9581138 .001014199 
 987 974169 961504803 81 .4165561 9.9564775 | .0010138171 
 988 | 976144 964480272 31 .4324673 9 . 95983889 .001012146 
 989 978121 967361669 31.4483 3704 | 9.9631981 .€01011122 
 990 980100 970299000 81 .4642654 9 9665549, .001010101 
 991 982081 973242271 81 .4801525 | 9.9699095 .001009082 
 992 | 984064 976191488 381.4960315 | 9.9782619 . 001008065 | 
 —-———~d 
 
 380 
 
| 
 
 i 1049 | 1100401 
 Hit 1050 1102500 
 
 1051 1104601 
 1052 1106704 
 1053 1108809 
 1054 1110916 
 
 1154320649 
 1157625000 
 1160935651 
 1164252608 
 1167575877 
 1170905464 
 
 No. Squares. Cubes. ee 
 | 
 | 
 
 993 986049 979146657 | 31.5119025 
 994 988036 982107784. 315277655 
 995 990025 985074875 | 31.5436206 
 996 992016 988047936 315594677 
 997 994009 99102697: 31.5753068 
 | = 998 996004 994011992 31.5911380 
 
 999 998001 997002999 31 6069613 
 
 1000 1000000 1000000000 | 31.6227766 
 
 1001 1002001 1003003001 | 31.6385840 
 
 1002 1004004 1006012008 | 31.6543836 
 
 me 1003 1006009 1009027027 | 81.6701752 
 Aan 1004 1008016 1012.\48064 | 31.6859590 
 Be | 1005 1010025 1015075125 | 381.7017349 
 1096 1012036 1018108216 | 81.7175030 
 
 1007 1014049 1021147343 | 31.7332633 
 
 : 1098 1016064 1024192312 | 31.7490157 
 aah 1009 1018081 1027243729 ) 31.7647603 
 ia 1010 1020100 103)301000 | 31.7804972 
 
 HAA | 1011 1022121 1033361331 | 31.7962262 
 iN 1012 1024144 1036433728 | 31,8119474 
 He | 1013 1026169 1039509197 318276609 
 Hil 1014 1028196 1042590744 : 31.8433666 
 We ae 1015 1030225 1045678375 318590646 
 Bee a 1016 1032256 1048772096 31.8747549 
 ah ~ 4017 1034289 1051871913 31. 8904374 
 
 Wit 1018 1036324 1054977832 319061123 
 
 na 1019 1038361 1058089859 | 31.9217794 
 Hi 1020 1040400 1061208000 | 31.9374388 
 Ht 1021 1042441 1081332261 319530906 
 
 all 1022 1044484 1067462648 31. 9687347 
 in| 102 1046529 1070599167 | 31.9843712 
 Bani 1021 1048576 1073741824 | 32.0000000 
 | 1025 1050825 1076890625 32.0156212 
 
 i} 1026 1052876 1030045576 32.0312348 
 1027 1054729 1083206683 32.0468407 
 
 102: 1056784 1036373952 | 82.0624391 
 
 1029 1058841 1039547339 | 32.0780298 
 
 1030 1080900 1092727009 | 82.0936131 
 
 1031 1062961 1095912791 | 32.1091887 
 
 1032 1085024 1099104768 32. 1217568 
 
 1033 1067089 1102302937 32. 1403173 
 
 1034 1089156 1105507304 32.1558704 
 
 1035 1071225 1108717875 32.1714159 
 
 1036 1073296 1111934656 | 32.1869539 
 
 1037 1075369 1115157653 32. 2024844 
 
 me | 1038 1077444 1118386872 | 382.2180074 
 i 1039 1079521 1121622319 32 2335229 
 il 1040 1081600 1124864000 322490310 
 aul 1044 1083681 1128111921 322645316 
 siti 1042 1085764 1131366088 32. 2800248 
 1043 1087849 1134626507 32.2955105 
 
 1044 1089936 1137893184 32.3109888 
 
 1045 1092025 1141166125 32. 3264598 
 
 HT 1046 1094116 1144445336 32. 3419233 
 Wi 1047 1096209 1147730823 32.3573794 
 it | 1048 1098304 1151022592 32. 3728281 
 
 2. 3882695 
 
 4037035 
 
 .4191301 
 4345495 
 .4499615 
 4653662 
 
 TABLE XXIII.—_SQUARES, CUBES, ETC. 
 
 | ee 
 } | 
 
 | { 
 
 | Cube Roots. 'Reciprocals. 
 
 | 
 | | 
 | | 
 
 | 9.9766120 
 
 .001007049 
 | 9,9799599 .001006036 
 99833055 | .001005025 
 9.986488 _; .001004016 
 9.9899900 | .001003009 
 
 | 
 
 9 .99338289 .001002004 
 |  9,9966656 .001001001 
 10.0000000 .001000000 
 10.0033322 | .0009990010 
 | 10.0066622 .0009980040 
 10.0099899 .0009970090 
 10.0133155 | .0009960159 
 
 10.0166389 | .0009950249 
 10.0199601 | .0009940358 
 10.0232791 | .00099380487 
 10.0265958° | 0009920635 
 
 10.0299104 .0009910808 
 10.0332228 0909900990 
 10.0365330 0009891197 
 10 .0398410 . 0009881423 
 10.0431469 .0009871668 
 10.0464506 | .0009861933 
 10.0497521 .0009852217 
 10.0580514 - 0009842520 
 100563485 . 0009832842 
 10.0596485 . 0009823183 
 10.0629364 .0009813543 
 10.066227 . 0009803922 
 
 10.0695156 | .0009794319 
 10. 0728020 0009784736 
 10.0760863 .0009775171 
 10.0793684 0009765625 
 10.0826484 .0009756098 
 
 10.0859262 .0009746589 
 10.0892019 .0009737098 
 10.0924755 .0009727626 
 10.0957469 .0009718173 
 10.0990163 0009708738 
 
 10. 1022835 .0009699321 
 
 10. 1055487 -00.9689922 
 10.1088117 . 0009680542 
 10. 1120726 .0009671180 
 
 10. 1153314 . 0009661836 
 10.1185882 .0009652510 
 10. 1218428 .0009643202 
 10. 1250953 .0009633911 
 10. 1283457 -0009624639 
 10.13815941 .0009615385 
 
 10. 1348403 0009606148 
 
 * 10.1880845 .0009596929 
 
 10. 14138266 .0009587'738 
 10. 1445667 .0009578544 
 10.1478047 0009569378 
 10.1510406 .0009560229 
 10. 1542744 .0009551098 
 10. 1575062 0009541 985 
 10.1607359 .0009532888 
 10. 1639636 .0009523810 
 10.1671893 .0009514748 
 10.1704129 .0009505703 
 10.1736344 .0009496676 
 
 10.1768539 .0009487666 
 
 301 
 
—— 
 
 | Wo. 100 L. 000.] 
 
 [No. 109 L. 040. | 
 
 TABLE XXTV.—LOGARITHMS OF 
 
 NUMBERS. 
 
 igo eee a ee | 2 3 4 5 6 7 | 8.94 9.) pitt 
 aes | ns Se | | 
 100 | 000000 0434 | 0868 1301 | 1734 || 2166 2598 3029 | 3461 | 3891 | 432 
 1 4321 | 4751 | 5181 | 5609 | 60388 | 6466 | 6804 T7321 | 7748 | 8174 | 428 ! 
 2 8600 9026 | 9451 9876 | | | : | 
 = os | 0300 || 0724 | 1147 | 1570 | 1993 | 2415 | 424 
 8 | 012887 | 3259 | 3680 | 4100 | 4521 || 4940 | 5360 | 8779 | 6197 | 6616 | 420 
 4 7033 | 7451 | 7868 | 8284 | 8700 || 9116 | 9532 | 9947 | | 
 | — ——}| 0361 | 0775 | 416 
 5 | 021189 | 1603 | 2016 | 2428 | 2841 || 8252 | 3664 | 4075 | 4486 | 4896 | 412 
 6 | 5306 | 5715 | 6125 | 6583 | 6942 || 7350 | 7757 | 8164 | 8571 | 8978 | 408 
 7 | 9384 | 9789 | | | | | = 
 0195 | 0600 | 1004 || 1408 | 1812 | 2216 | 2619 | 8021 | 404 
 8 | 033424 | 3826 | 4227 | 4628 | 5029 || 5430 | 5830 | 6230 | 6629 | 7028 | .400 
 9 | 7426'| 7825 | 8223 | 8620 | 9017 || 9414 | 9811 | | 
 | 04 | | i | | o207 | 0602 | 0998 | 397 
 PROPORTIONAL PARTS. 
 ree | | | 
 dy 2 els Ge Cane 2g a ON | 5 ify SM ea 7p 8 9 
 | | | i | | 
 434 | 43.4) &6.8 | 130.2 | 173.6 | 217.0 | 260.4 | 303.8 | 347.2 | 390.6 
 433. | 43.3| 86.6.) 129.91 173.2} 216.5) 259.8 | 3803.1 | 346.4 | 389.7 
 432 3.2| 86.4 | 129.6 | 172.8 | 216.0 | 259.2 | 302.4 | 345.6 | 388.8 
 431 | 43.1} 86.2 | 129.3! 172.4% 215.5 | 258.6) 301.7 | 344.8 387.9 
 43 3.0 | 86.0. | 129.0] 172.0 | 215.0 | 258.0 |) 301.0 | 344.0) 387.0 
 4299 | 42.9} 85.8 28.71 171.6 }- 214.5) 257.4 | 300.3) 348.2% 386.1 
 428 9 § |. 85.6 128.4 | 171.21) 214.0 | 256.8 299.6 | 342.4 | 385.2 
 427 | 42.71 85. 128.1.|° 170.8 | 218.5 | 256.2) 298.9 | 841.6) 384.3 
 426 | 42.6} 85.2 | 127.8| 170.4) 218.0! 2556); 293.2 | 340.8 | 883.4 
 425 |142.5 | 85.0 127.5 | 170.0} 212.5 | 255.0) 297.5 | 340.0 | 382.5 | 
 494 | 42.4) 84.8 127.2 | 169.6 | 212.0 | 254.4 | 296.8 | 389.2 | 381.6 
 423 | 42.3 | 84.6 126.6 169.2 | 211.5 | 258.8! 296.1; 838.4 | 380.7 
 92 | 42.2) 84.4 126.6 | 168.8 | 211.0°| 258.2} 295.4 | 337.6 | 379.8 
 421. | 42.1) 84.2 126.3 | 168.4 | 210.5 | 252.6 | 294.7 | 336.8) 378.9 
 420 | 42.0 | 84.0 126.0 | 168.0 | 210.0 | 252.0} 294.0 | 336.0 | 378.0 
 419 | 41.9 | 83.8 125.7 | 167.6 | 209.5 | 251.4! 293.3 | 335.2 | 377.1 
 418 | 41.8; 83.6 125.4 | 167.2} 209.0 | 250.8 | 292.6 | 334.4 | 376.2 
 41V>| 44.7.) 83:4 125.1 | 166.8 | 208.5 | 250.2 | 291.9 | 333.6 | 375.3 
 416 | 41.6 | 83.2 124.8 | 166.4 | 208.0 | 249.6 | 294.2 | 382.8 | 374.4 
 4145. | 41.5 | 83.0 124.5 | 166.0} 207.5 , 249.0, 290.5 | 3832.0 | 373.5 
 414 | 41.4] 82.8 124.2| 165.6 | 207.0 | 248.4 | 289.8 331.2 | 372.6 
 413° | 41.3) 82.6 123.9 | 165.2} 206.5 | 247.8 | 289.1 | 330.4 | 371.7 
 412 | 41.2 | 82.4 123.6 | 164.8 | 206.0 | 247.2 | 288.4 | 3829.6 | 370.8 
 Ailes "44 A | 2.2 123.3 | 164.4 | 205.5 | 246.6 | 287.7 | 328.8 | 369.9 
 410 | 41.0} 82.0 123.0 | 164.0! 205.0 | 246.0 | 287.0 | 828.0 | 369.0 
 409 | 40.9| 81.8 {| 122.7 | 163.6 | 204.5} 245.4 286.3 | 327.2 | 368.1 
 408 | 408] 81.6 122.4 | 163.2} 204.0 | 244.8 | 285.6 | 326.4 | 867.2 
 407. | 40.7 | 81.4 122.1 | 162.8! 208.5 |} 244.2 | 284.9 | 325.6 | 366.3 
 406°} 40.6 | 81.2 121.8 | 162.4 | 203.0) 24836) 284.2 | 824.8 | 365.4 
 405 | 40.5} 81.0 121.5 | 162.0 | 202.5 | 243.0! 288.5 | 3824.0 | 364.5 
 404 | 40.4 | 80.8 421.2 | 161.6 | 202.0 | 242.4 | 282.8 | 323.2 | 363.6 
 403 |.40.3 | 80.6 420.9 | 161.2 | 201.5 | 247.8] 282.1 | 822.4 | 3862.7 
 402 } 40.2! 80.4 120.6 | 160.8 | 201.0] 2412] 281.41 3821.6 | 361. 
 401 | 40.1 | 80.2 120.3 | 160.4 | 200.5 | 240.6 | 280.7 | 820.8 | 360.9 | 
 400 | 40.0 | 80-0 120.0 | 160.0} 200.0} 240.0: 280.0 | 820.0 | 360.9 | 
 399 |.39.9 | 79.8 119.7} 159.6 | 199.5} 230.4 | 279.3 | 319.2 | 359.1 
 398 | 39.8 | "9.6 | 119.4| 159.2 | 199.6) 288.8 | 278.6} 318.4 | 358.2 | 
 397 | 39.7 | 79.4 | 119.1} 158.8 | 198.5.) 238.2) 277.9 | 317.6 | 357.3 
 396 | 39.6 | 79.2) | 118.8! 158.4 | 198.0 | 287.6 | 277.2 316.8 | 356.4 
 395 | 39.5 | 79.0 118.5 | 158.0 5 | 237.0 | 276.5 ' 316 0! 355.5 
 
 | 
 
TABLE XXTV.—LOGARITHMS OF NUMBERS, 
 
 RR STR aa ce cr 
 
 | No. 110 L, 041.] [No. 119 L. 078, 
 
 9 | Diff. 
 
 a] 
 ie 2) 
 
 6 
 
 or 
 
 | N. | 0 pee 2 3 | 4 
 | | Recaecacar | 
 
 | | 
 
 | 110 | 041393 | 1787 | 2182 | 2576 |, 2969 |] aga | 
 | 1) 5323 | 5714 | 6105 | 6495 | 6885 || 7275 | weed | 8053 2 | 820! 590 
 | 2%) 9218 | 9606 | 9993 i| - 
 
 | 0380 | 0766 || 1153 | 1538 | 1924 | 2209 | c@o4 | 886 
 3 | 053078 | 3463 | 3846 | 4230 | 4613 || 4996 5378 | 5760 | 6142 | 6524 | $83 
 4 6905 | 7286 | 7666 | 8046 | 8426 || 8805 9185 | 9563 | 9942 
 
 3755 | 4148 | 4540 | a9se | agg 
 2 ») 
 
 | 0820 | 3879 
 4083 | 376 
 
 5 | 060698 | 1075 | 1452 | 1829 | 2906 || e582 | eos | £333 | 3709 | 
 : 7815 | 373 
 
 | 
 6 | 4458 | 4832 | 5206 | 5580 | 5953 || 6326 | 6699 OTL | 7448 | 
 © | 8186 | 8557 | 8928 | 9298 | 9668 . — 
 | ass 0088 | 0407 | 0776 | 1145 | 1514] 370 
 \\ S*) 071882 | 2250 | 2617 | 2985 | 3352 || 3718 | 40gs 4451 | 4816 | 5182 | 366 
 ee 9) 5547 | 5912 | 6276 | 6640 | 7004 |] 7368 | 731 | €094 | | 
 
 ee 
 Si 
 or 
 ee 
 
 8819 | 363 
 | 
 
 PROPORTIONAL Parts. 
 
 Dah =e ae <= 
 
 | 
 | Diff. 1 2 3 4 5 6 vi 8 9 
 \| joe i ‘ ee ae 
 395 | 39.5] 79.0 118.5 | 158.0 | 197.5 | 237.0 | 9765 316.0 | 355.5 
 | 304-| 39.4] 78:8 118.2 | 157.6 | 197.0] 236.4 | 9275/8 | 815.2 | 354.6 | 
 393. | 39.3 | 78.6 7.9) 157.2 | 166.5 | 235.8] on5 4 | 814.4 | 853.7 | 
 892 | 39.2 | 78.4 117.6 | 156.8) 196.0} 235.9] 2974/4 | 313.6 | 352.8 
 391 | 89.1 | 78:9 17.3 | 156.4 | 195.5 | 234.6] 93°77 | 312.8 | 251.9 
 390 | 39.0! 8.0 V7.0 | 156.0; 165.0 | 984.0] 2973/0 | 812.0 | 351.0 
 389 | 88.9] 77.8 116.7 | 155.6 | 194.5} 9&3 4 | ere | 811.2 | 850.4 
 388. | 88.8 | 77.8 116.4 | 155.2! 194.0] F298] 9717/6 | 210.4 | 349.2 | 
 887 | 88.7% | 274 116.1 | 154s 103.5 | 222.2 | 270.91 309.6 | 348.3 
 386 | 88.6 | "7.9 115.8 | 154.4 | 198.0} 231.6 | 940.2 808.8 | 347.4 
 885 "| 88.5 1° 77-0 115.5 | 154.0 | 152.5 | 231.0] 269.5 | 808.0 | 846.5 
 384 | 38.41 76.8 115.2 | 158.6 | 162.0] 220.4] 268.8 | 207.2 | 345.6 
 883 :| 38.3°| 76.6 114.9 | 158.2] 191.5 | 229.81 9684 306.4'| 344.7 
 882 | 38.2-| 76.4 114.6 | 152.8 | 191.0} 209.9] 967.4 305.6 | 343.8 
 381. | 88.1} 76.2 114.3 | 152.4] 190.5] 998.6 266.7 | 804.8 | 342.9 
 880 | 88.0 | 46.0 114.0 | 152.0} 190.0] 9228 0 266.0 | 804.0 | 342.0 
 379 | 87.9 | 75.8 113.7 | 151.6 | 189.5 | 2297.4] 965.3 | 808.2 | 841.1 
 318 937,821) G56 113.4 | 151.2 | 189.0 | 226.8] 964'6 | 802.4 | 340.2 
 Othe! 3%. Zale 724 113.1 | 150.8 | 188.5 | 226.2 | 963.9 | 301.6 | 339.3 
 316 | 87.6.1 75.2 112.8 | 150.4 | 188.0] 22.6] 963.9 | 800.8 | 388.4 
 375. | 87:5 | 75.0 112.5 | 150.0 | 187.5 | 225.0} 9262/5 | $00.0 337.5 
 3ov4_ | 87.4 | 74:8 112.2 | 149.6 | 187.0 | 224.4} 961.8 | 299.2 | 336.6 
 i 8784) S784 (7456 111.9 | 149.2 | 186.5 | 223.8 261.1 | 298.4 | 335.7 
 | 87201 88.24 P4c4 111.6 | 148.8] 186.0} 223.2 | 960.4 | 297.6 | 334.8 
 |: 3871-1 87.11. 74.2 111.3 | 148.4 | 185.5] 229.6 259.7 | 296.8 | 233.9 
 870 | 387.0 | 74:0 111.0 | 148.0 | 185.0 | 222.0 | 9599 266.0 | 883.0 
 | 369 | 36.94 73.8 110.7 | 147.6 | 184.5 | 201.4] $583 295.2 | 882.1 
 | 868 186.8} 73.6 110.4 | 147.2} 184.0} 220.81 957/6 204.4 | 331.2 
 i | 867 | 36.71 73:4 110.1 | 146.8} 183.5 | 980.9 | 256.9 | 293.6 | 330.3 
 I 366 | 86.6 | 73.9 109.8 | 146.4 | 183.0] 219.6 | 956.2 | 292.8 | 829.4 
 | 565 | 36.51 73.0 109.5 | 146.0} 182.5} 219.0] 955.7 } 292.0 | 328.5 
 1] 364 | 86.4] 72.8 109.2} 145.6) 182.0] 218.4] 954.8 291.2 | 327.6 
 Hil 363. -| 36.3.) 72-6 108.9 | 145.2 | 181.5) 217.8] 25471 | ‘990.4 | 306.7 
 362 | 36.2] 72.4 108.6 | 144.8} 181.0] 217.9 | 953 4 | 289.6 | 825.8 
 361 | 36.1 | 472.2 108.3 | 144.4 | 180.5] 216.61] 2959.7 288.8 | 824.9 | 
 360 | 36.0! 72.0 108.0 | 144.0} 180.0] 216.0 959.0 | 228.0 | 324.0 | 
 359 | 35.9 | 71.8 107.7 | 148.6} 179.5] 215.4] 251.3 287.2 | 223.4 
 358 | 85.8 | 71.6 107.4 | 143.2 | 179.0} 214.8) 950.6 | 286.4 | 322.2 
 OO7.31° 85.7 Vid 107.1 | 142 8) 178.5] 9214.9 | 949°9 285.6 | 221.2 | 
 556 | 35.6 | 71.2 | 106.8 | 4 
 
 142.4 178.0 213.6 249.2 284.8 | &€0. 
 
TABLE XXIV. 
 
 No. 120 L. 079.) 
 
 LOGARITHMS OF NUMBERS. 
 
 [No. 134 L. 130, 
 
 | Diff. 
 
 Nokyt@e tthete & [8 +) 4 eS iin 9 
 | | | | 
 | : 
 9 97918 954! C ~~, ———|] I 
 120 079181 | 9543 | 9904 | O565 | o¢a6 || ooST | 1347 | 1707 | 2067 | 2426 | 360 
 1 082785 | 3144 | 3503 8861 4219 | 4576 | 4934 | 5291 | 5647 | 6004 | 35% 
 3| 6360 | 6716 | 7071 | 7426 | 7781 | 8136 | 8490 | 8845 | 9198 | 9552) 35d 
 3| 9905 | — |——— | | | 
 — 0258 | 0611 | 0963 | 1315 | 1667 | 2018 | 2370 | 2721 | 8071 | 852 
 4 | 093422 | 3772 | 4122 | 4471 4820 || 5169 | 5518 | 5866 | 6215 6562) 349 
 5 | 6910 | 7257 | 7604 | 7951 | 8298 |) 8644 | 8990 | 9385 | 9681 \———_| 
 \- - 0026 | 346 
 6 | 100371 | 0715 | 1059 | 1403 | 1747 |, 2091 | 2434 | 2777 | B119 | 8462 | 34s 
 | 3804 | 4146 | 4487 4828 | 5169 || 5510 | 5851 | 6191 | 653i | 6871 | S4l 
 8! 7210 | 7549 | 7888 S227 | 8565 || 8903 | 9241 | 9579 | 9916 |— 
 .| | | eee| 0253 | 3838 
 9 | 110590 | 0926 | 1263 | 1599 | 1934 |) 2270 | 2605 | 2940 | 8275 | 8609 | 335 
 130 3943 | 4277 | 4611 | 4944 | 5278 || 5611 | 5943 | 6276 | 6608 | 6940 | 338 
 1 | 771 | 7603 | 7934 8265 | 8595 || 8926 | 9256 | 9586 | 9915 | | 
 — | | | 0245 | 330 
 | | 120574 | 0903 | 1231 | 1560 | 1888 || 2216 | 2544 | 2871 | 8198 | 8525 | 328 
 | 31} 3852 | 4178 | 4504 | 4830 | 5156 || 5451 | 5806 6131 | 6456 | 6781 | 825 
 4| 7105 | 7429 | 7758 | 8076 | 8399 || 8722 | 9045 | 9368 | 9690 
 | 13 | | | ooi2 | 82 
 | 
 PROPORTIONAL PARTS. 
 | 
 | Diff.| 1 2 3 4 5 se eee 8 9 
 55. | 35.5 | 71.0 | 106.5 | 142.0 | 177.5 | 213.0) 248.5) 284.0 | 319.5 
 a4 | 35.4| 70.8 | 106.2} 141.6] 177.0 | 212.4 | 247.8) 283.2 | 318.6 
 gna! | 35.3 | 70.6 | 105.9 | 141.2 | 176.5 | 211.8 | 247.1) 262.4 | 3iv.% 
 359 | 35.21 70.4 | 105.6 | 140.8] 176.0 | 211.2 | 246.4 | 281.6 | 316.8 
 ost | 35.1 70.2 | 105.3} 140.4 | 175.5 | 210.6) 245.7 | 280.8 | 315.9 
 350 | 35.0! 70.0 | 105.0} 140.0] 175.0 | 210.0 | 245.0 | 280.0 | 315.0 
 | 349 | 34.91 69.8 | 104.7 | 139.6] 174.5 | 209.4) 244.3) 279.2 | 314.1 
 | 348 | 34.8] 69.6 | 104.4] 139.2] 174.0 | 208.8) 243.6) 278.4 | 818.2 
 a4z | 34.7 | 69.4 | 104.1 | 188.8) 173.5) 208.2 | 242.9) 277.6 | 312.3 
 346 | 34.6 | 69.2 | 103.8 | 188.4 | 173.0) 207.6 | 242.2 | 206.5 | 311.4 
 | | | 
 ain | 34.5 | 69.0 | 108.5 | 138.0] 172.5] 207.0) 241.5 | 276.0 | 310.5 
 } 344 | 84.4 | 68.8 | 108.2} 137.6) 172.0 206.4 | 240.8 | 275.2 | 309.6 
 343 | 34.3 | 68.6 | 102.9| 137.2 | 171.5 | 205.8) 240.1 | 274.4 | 308.7 
 342 | 34.2 | 68.4 | 102.6| 136.8] 171.0} 205.2 | 239.4) 273.6 | 307.8 
 B41 | 34.1 | 68.2 | 102.3 | 186.4 | 170.5 | 204.6 | 238.7 | 212.8 | 806.9 
 310 | 34.0 | 68.0 | 102.0| 136.0 | 170.0 | 204.0 | 238.0 | 272.0 | 306.0 
 939 | 33.9 | 67.8 | 101.7 | 135.6 | 169.5 | 203.4 | 287.3 | 271.2 | 306.2 
 398 | 33.8 | 67.6 | 101.4| 135.2| 169.0] 202.8} 236.6) 270.4 | 304.2 
 337 | 33.7) 67.4 | 101.1} 134.8 | 168.5 | 202.2) 235.9) 269.6 | 308.3 
 335. | 33.6 | 67.2 | 100.8} 134.4] 168.0 | 201.6 | 235.2) 268.8 | 302.4 
 | 385 | 98.5 | 67.0 | 100:5:| 134.0) 167.5.) 201.0 | 230.5 268.0 | 301.5 
 | 334 | 83.4 | 66.8 | 100.2) 133.6 167.0 | 200.41 233.8) 267.2 | 300.6 
 | 333 | 33.3 | 66.6 99.9 | 133.2 | 166.5 | 199.8 | 2838.1 | 266.4 | 299.7 
 339 | 33.2! 66.4 | 99.6| 182.8 | 166.0| 199.2 | 232.4 | 265.6 | 298.8 
 331 | 33.1 | 66.2 | 99.3| 182.4] 165.5 | 198.6 | 281.7 | 264.8 | 297.9 
 339 | 33.0! 66.0 | 99.0} 132.0! 165.0 | 198.0) 231.0 | 264.0 | 297.0 
 329 | 32.9 | 65.8 98.7 | 131.6 | 164.5) 197.4 | 230.3 | 263.2 | 296.1 
 308 | 32.8| 65.6 | 98.4 | 1381.2 | 164.0 | 196.8 | 220.6 | 262.4 | 295.2 
 327 2.7 | 65.4 98.1 130.8 163.5 | 196.2} 228.9 | 261.6 | 294.3 
 326 | 32.6 | 65.2 97.8 | 130.4 | 163.0 | 195.6 | 228.2} 260.8 | 293.4 
 2.5 | 65.0 97.5 | 130.0 | 162.5 | 195.0 | 227.5 | 260.0 | 292.5 
 241 64.8 | 97.2 | 129.6] 162.0] 194.4 | 226.8 | 259.2 | 201.6 
 2.3 | 64.6 96.9 | 129.2} 161.5, 193.8 926.1 | 258.4 | 290.7 
 2.2 | 64.4 96.6 | 128.8! 161.0! 193.2 | 225.4] 257.6 | 289.8 
 
No. 135 L. 130.] 
 
 TABLE XXTV.—LOGARITHMS OF NUMBERS. 
 
 [No. 149 L. 1%, 
 
 1 
 
 2 
 
 | 0655 
 3858 
 
 | 7037. | 
 
 | 0 
 41% 
 7 
 
 f 
 ‘ | 
 B54. 
 
 97 
 
 1298 | 1619 || 1939 | 2260 
 | 4496 | 4814 1} 5133 | 5451 
 7671 | 7987 || 8303 | 8618 | 
 
 2900 
 6086 
 
 9249 | 
 
 | 0508. | 
 | 8639 | 
 | 6748 | 
 | 9885 
 
 0822 | 11 
 8951 | 42 
 7058 | 73 
 
 36 || 1450 | 1763 
 63 || 4574 | 4885 
 367 || 7676 | 
 
 | 2389 
 | 5507 
 
 | 8603 
 
 2594 
 | 5640 
 | 8664 
 
 2900 | 
 
 5943 
 8965 
 
 9266 | 9567 || 9868 | 
 
 | 0142 | 0449 || 0756 | 1063 
 3205 | 3510 | 
 6246 | 6549 || 6862 | 7154 
 
 8815 
 
 | 1676 
 | 4728 
 
 C159 | 
 
 | 1667 
 
 | 4650 
 7613 
 
 1967 
 4947 
 7908 
 
 | 2266 | 2564 
 5244 | 5541 || 5888 | 6134 
 8203 | 8497 || 8792 
 
 2863 | 3161 
 
 9086 
 
 | 0769 | 
 | 38758 | 
 | 6726 
 
 | 9674 
 
 0555 
 3478 
 
 0848 
 3769 
 
 | 1141 | 1434 || 1726 
 4060 | 4351 || 4641 | 4932 
 
 2019 
 
 | 
 
 26083 
 
 5512 
 
 PROPORTIONAL PARTS, 
 
 i a) 
 
 1D 
 
 Cos 
 
 t 
 
 =) 
 
 et ec CoH 
 
 DDS 
 
 Sr 2 
 
 He O 
 
 - 
 
 ww Oo 
 
 oor 
 
 COM HE -3 O to 
 
 oO 
 
 L950 Hwa 
 
 ee 
 
 CO he = 
 
 w on 
 
 © 
 
 Oo 
 
 WHIP IOW HDowordueshwoO es 
 
 2X 
 
 © & 
 = & 
 wos 
 
 o] 
 2] 
 ~ 
 
 pare 
 Noi) 
 
 ~J iH ooo 
 
 S 
 Su -2 
 Cow © 
 
 rw) 
 Nort Le Cora 
 
 rar) 
 He rt GO OTAICO OC 
 
 2 © 
 Parco 
 eo CO OO 
 
 o- 
 
 i] 
 DAs 
 
 4 5 6 
 128.4 | 160.5 |] 192 
 128.0 160.0 192. 
 127.6 159.5 191. 
 127.2 159.0 | 190 
 126.8 158 5a 19 
 126.4 58.0} 189 
 126.0 | 157.5 | 189 
 125.6 | 157.0 | 188 
 125.2 156.5.) 18% 
 124.8 156.0 187 
 124.4 | 155.5 186 
 124.0°| 155.0}. 186 
 123.6 | 154.5.| 185 
 122.8 | 158.5 84. 
 122.40) T58.0ab 183° 
 122.0 | 152.5} 183. 
 121.6 | 152.0 } 182. 
 12132 15155 181 
 120.8 152.-Oe}) 187 
 120.4} 150.51 180 
 120.0 150.0 80 
 119.6,| 149.5.) 179. 
 119.2 1490+ 178. 
 118.8; 148.5 i 78. 
 118.4 148.0 17%. 
 118.0 147.5 Lik. 
 117.6 TAG Ont R622 
 117.2 | 146.5 175: 
 116.8 146.0 175. 
 116.4 145.5 174. 
 T1650 145.05) 77a 
 Aid Gay” 144.5 13. 
 115.2 | 144.0} 179, 
 114.8} 1438.5 | 179. 
 114.4.) 143.0) (174. 
 
 WOOWO 
 
 © OIA Emod 
 
 261. 
 260. 
 259. 
 258. 
 a 
 
 ony 
 
 | && 
 
 DBI wuUP OW 
 
 Hoo 
 
 S 
 Nw 
 
 =) 
 
 i OD dD 
 
TABLE XXIV.—LOGARITHMS OF NUMBERS. 
 
 ae 
 
 | No. 150 L. 176.] [No. 169 L. 230, 
 
 Oo] wl a] a | 4] s 6) 7 | s.| 9 | pis 
 
 176091 | 6381 ; 6.70 | 6959 | 7248 || 75386 | 7825 | 8113 | 8401 | 8689 | 289 
 | 8977 | 9264 | 9552 | 9839 = | ee Pe 
 
 : 0126 || 0413 | 0699 | 0986 | 1272 | 1558 | 287 
 181844 | 2129 | 2415 | 2700 | 2985 || 8270 | 8555 | 38389 | 4123 | 4407 | 285 
 | 4691 | 4975 | 5259 | 5542 | 5825 || 6108 | 6591 | 6674 | 6956 | 7239 | 283 
 7521 | 78uB | 8084 | 8366 + $647 || 8928 | 9209 | 9490 | 9771 
 
 | 4 | 0051 | 281 
 | 190332°| 0612 | 0892 | 1171 | 1451 || 1780 | 2010 | 2289 | 2567 | 2846 | 279 
 | 8125 | 3403 | B681 | 3959 | 4237 || 4514 | 4792 | 5069 | 5346 | 5623 | 278 i 
 | 5900 | 6176 | 6458 | 6729 | 7005 || 7281 | 7556 | 7882 | 8107 | 8882 | 276 i 
 8657 | 8932 | 9206 | 9481 | 9755 || | HH 
 | || 0029 | 0303 | 0577 | 0850 | 1124 | 274 | 
 201397 | 1670 | 1943 | 2216 | 2488 || 2761 | 8033 | 8805 | 3577 | 3848 | 272 
 4120 | 4301 | 4663 | 4934 | 5204 || 5475 | 5746 | 6016 | 6286 | 6556 | 271 
 6826 | 7096 | 7365 | 7634 | 7904 || 8173 | 8441 | 8710 | 8979 | 9247 | 269 
 9515 | 9783 : | ‘ i 
 | 0051 | 0319 | 0586 || 0853 | 1121 | 1888 |. 1654 | 1921 | 267 | 
 212188 | 2454 | 2720 ; 2986 | 3252 || 2518 | 3783 | 4049 | 4314 | 4579 | 266 | 
 4844 | 5109 | 5873 | 5638 | 5902 || 6166 | 6420 | 
 7484 | 7747 | 8010 | 8273 | 3798 | 9060 
 | | - — 
 220108 | 0370 | 0631 | 0892 | 1153 |) 1414 | 1675 | 1936 | 2196 | 2456 | 261 i 
 2716 | 2976 | 3236 | 3496 | 8755 |) 4015 | 4274 | 4533 | 4792 | 5051 | 259 
 5309 | 5568 | 5826 | 6084 | 6342 || 6600 | 6858 | 7115 | 7372 | 7630] 258 
 
 6694 | 6957 | 7221 264 i 
 9323 | 9585 |. 9846 | 262 Hl 
 
 ® 
 Or 
 Co 
 C3 
 ee 
 ~> 
 OO 
 
 om es 
 ' We, Soe = ict a} 2 
 COOID TRO WHOS © DIM Bww HS] = 
 
 | I 
 7887 | 8144 | 8400 | 8657 | 8913 |) 9170 | 9426 | 9682 | 9938 | —— I 
 | Seay | | | | 0193 | 256 | 
 PROPORTIONAL PARTS, | 
 Bestia: 4] x ie zB Aim 
 BE Aad 1 testes Cadet ok ee (atehs PR aie ie! 
 | | A r | Set UE | | i 
 | | 285 | 28.5] 57.0.| 85.5 | 114.0] 142.5] 171.0] 199.5 | 228.0 | 256.5 
 | | 284- | 28.4 | 86.8 | 85.2 | 113.6 | 142.0} 170.4} 198.8 | 227.2 | 255.6 
 | 283 | 28.3} 56.6 | 84.9 | 118.2] 141.5] 169.8 | 198.1 | 226.4 | 254.7 
 | 282 | 28.2 | 56.4 | 84.6 | 112.6| 141.01 169.2 | 197.4 | 225.6 | 258.8 ' 
 | 281 | 28.1 | 56.2 | 84.3 | 1124] 140.5 | 168.6 | 196.7} 224.8 | 252.9 | 
 | 280 | 28.0} 56.0 | 84.0 | 112.0] 140.0| 168.0] 196.0 | 224.0 | 252.0 et 
 279 | 27.9) 55.8 | 83.7 | 111.6 | 139.5 | 167.4 | 195.3 | 223.2 | 251.1 
 | 278 | 27.8 | 55.6 | 88.4 | 111.2] 139.0} 166.8) 194.6 | 222.4 | 250.2 } 
 a7 | 27.7 | 55.4 | 88.1 | 110.8 | 138.5} 166.2] 193.9 | 221.6 | 249.3 , 
 : 276 | 27.6) 55.2 | 82.8 | 110.4] 138.0°| 165.6 | 193.2 | 220.8 | 248.4 I 
 : | 975 | 97.5 | 55.0 | 82.5 | 110.0| 187.5} 165.0! 192.5 | 220.0 | 247.5 } 
 "ora | 27.41 54.8 | 82.2 | 109.6 | 137.0} 164.4} 191.8 | 219.2 | 246.6 } 
 973 | 27.3) 54.6 |) 81.5 109.2 | 136.5 | 163.8! 191.1 | 218.4 | 245.7 | i 
 | 272 | 27.2; 54.4 | 81.6 | 108.8] 186.0} 163.2} 190.4; 217.6 | 244.8 ; 
 | | 271 | 27.1) 54.2 | 81.8 | 108.4 | 185.5) 162.6! 180.7 | 216.8 | 248.9 
 | | 270 | 27.0] 54.0 | 81.0 | 108.0! 185.0] 162.0) 189.0) 216.0 | 2438.0 
 | 1 269 | 26.9! 53.8 | 80.7 | 107.6) 134.5 | 161.4] 188.3! 215.2 | 242.1 
 | 268 | 26.8! 53.6 | 80.4 | 107.2| 184.0! 160.8) 187.6 214.4 | 241.2 
 267 | 26.7) 53.4 | 80.1 | 106.8] 188.51 160.2} 186.9} 218.6 | 240.% 
 | 266 | 26.6 | 53.2 | 79.8 | 106.4 | 133.0 | 159.6 | 186.2) 212.8 | 220.4 | 
 | 265 | 26.5 | 53.0 | 79.5 | 106.0| 132.5) 159.0} 185.5 ' 212.0 | 238.5 | | 
 : 264 | 26.4| 52.8 | 79.2 | 105.6] 182.0) 158.4 | 184.8) 211.2 | 237.6 | 
 ) 263' | 26.3 | 52.6 | 78.9 | 105.2) 181.5 | 157.8) 184.1 210.4 | 236.7 
 | 262 | 26.21 52.4 | 78.6 | 104.8] 181.0] 157.2 | 183.4 | -209.6 | 285.8 
 | 261 | 26.1| 52.2 | 78.3 | 104.4 | 130.5) 156.6 | 182.7 | 208.8) 2384.9 
 | 260 | 26.0; 52.0 | 78.0 | 104.0 | 120.0 156.0 | 182.0 | 208.0 | 234.0 
 | | 259 | 25.9; 51.8 | 77.7 103.6 | 129.5 155.4} 181.38 | 207.2 | 288.1 
 , 258 | 25.8) 51.6 | 77.4 | 108.2 | 129.0) 154.8 | 180.6 | 206.4 | 282.2 
 DT | 95.9% S40 F714 102.8 | 128.5) 154.2 | 179.9 | 205.6 | 231.8 
 P56 @)25.671 251.2 76.8 |-102.4 | 128.0; 153.6) 179.2 | 204.8) 230.4 
 255 125.51 51.0 | 76.5 | 102.0) 127.5) 153.0} 178.5 | 204.0 | 229.5 | 
 
No. 170 L. 230.] 
 
 TABLE XXIV.—LOGARITHMS OF 
 
 NUMBERS. 
 
 [No. 189 L. 278. 
 
 0 
 
 1 
 
 =i 
 ~) 
 
 2 
 
 3. | 64 
 
 5 
 
 6 
 
 8 
 
 | 
 | 
 | 
 } 
 
 9 
 
 Diff. 
 
 ‘0 | 230449 | 0704 | 0960 | 1215 | 1470 || 1724 | 1979 | 2284 | 2488 | 2742 | 255 
 1} 2996 | 3250 | 8504 | 8757 | 4011 || 4264 | 4517 | 4770 | 5023 | 5276 | 253 
 2] 5528 | 5781 | 6083 | 6285 | 6537 || 6789 | 7041 | 7292 | 7544 | 7795.| 252 
 3 | 8046 | 8297 | 8548 | 8799 | 049 || 9299 | 9550 | 9800 |———| 
 | | | 0050 | 0300 | 250 
 4 | 240549 | 0799 | 1048 | 1297 | 1546 || 1795 | 2044 | 2293 | 2541 | 2790 | 249 
 5 | 3038 | 8286 | 3534 | 8782 | 4030 || 4277 | 4525 | 4772 | 5019 | 5266 | 248 
 6 | 5513 | 5759 | 6006 | 6252 | 6499 || 6745 | 6991 37 | 7482 | 7728 | 246 
 7 | 7973 | 8219 | 8464 | 8709 | 8054- || 9198 | 94438 7 | 99382 ar 
 2 —| 0176-! 245 
 8 | 250420 | 0664 | 0908 | 1151 | 1395 || 1638 | 1881 | 2125 | 2368 | 2610 | 243 
 9 | 2853 | 3096 | 3338 | 3580 | 3822 || 4064 | 4306 | 4548 | 4790 | 5031 | 242 
 180 | 5273 | 5514 | 5755 | 5996 | 6237 || 6477 | 6718 | 6958 | 7198 | 7439 | 241 
 1 7679 | 7918 | 8158 | 8398 | 8637 || 8877 | 9116 | 9355 | 9594 | 98383 | 239 
 2 | 260071 | 0310 | 0548 | 0787 | 1025 || 1263 | 1501 | 1739 | 1976 | 2214 | 288 
 3 | 2451 | 2688 | 2925 | 3162 | 3399 | 3636 | 3873 | 4109 | 4346 | 4582 | 237 
 4| 4818 | 5054 | 5290 | 5525 | 5761 || 5996 | 6232 | 6467 | 6702 | 6937 | 235 
 5 | 7172 | 7406 | 7641 | 7875 | 8110 || 9344 | 8578 | 8812 | 9046 | 9279 | 234 
 6 9513 | 9746 | 9980 | | 
 0213 | 0446 || 0679 | 0912 | 1144 | 1377 | 1609 | 233 
 7 | 271842 | 2074 | 2306 | 2538 | 2770 || 8001 | 3233 | 3464 | 3696 | 3927 | 232 
 8 | 4158 | 4389 | 4620 | 4850 | 5081 || 5311 | 5542 | 5772 | 6002 | 6232 | 280 
 9 | 6462 | 6692 | 6921 | 7151 | 7380 '| 7609 | 7828 | 8067 | 8296 | 8525-| 229 
 PROPORTIONAL PARTS. 
 Diff.| 1 2 3 4 5 6 7 8 9 
 255 | 25.5] 51.0 | 76.5 | 102.0] 127.5] 158.0 | 178.5 | 204.0 | 229.5 
 254 | 25.4.) 50:8} 76.2 9) 101.6) 127.0 | 152.4:) 17728 |> 208.9 228.6 
 QS) 25. Bc 5O.6ech Hah TORQ: -126.5)|) 161 7Bsle 177 dd Ge 202. aah, Saraem. 
 252 | 25.2) 50.4 | 75.6 100.8 | 126.0 | 151.2} 176.4 | 201.6 | 226.8 
 251 | 25.15). 50.2 4 58>) 100.4 + 125.5:| 150.6.) 175.7 |, 200.8 | 225.9 
 250 | 250| 50.0 | %5.0 100.0 | 125.0} 150.0 | 175.0 | 200.0 | 225.0 
 219 | 24.9) 49.8 | 74.7 99.6} 124.5 | 149.4] 174.3 | 199.2 | 224.1 
 248 | 24.8| 49.6 | 74.4 99.2 | 124.0] 148.8} 173.6 | 198.4 | 223.2 
 Bay Ay) AGA | aed 98.8 | 128.5 | 148.2 | 172.9 | 197.6 | 222.8 
 246 | 24.6 | 49.2 | 73.8 98.4 | 123.0] 147.6] 172.2 | 196.8 | 221.4 
 245 | 24.5 | 49.0 | 8:5 98.0 | 122.5 | 147.0} 171.5 | 196.0 | 220.5 
 244 | 24.4] 48.8 | 73.2 97.6 | 122.0] 146.4} 170.8 | 195.2 | 219.6 
 243 | 24.8 | 48.6 72.9 97.2) 121.5) 145.8}. 170.1.| 194.4 | 218.% 
 242 | 24.2] 48.4 | 72.6 96.8 | 121.0] 145.2] 169.4 | 193.6 | 217.8 
 BAdhe) 24.49): 489-2) 7253 96.4 | 120.5 | 144.6] 168.7 | 192.8 | 216.9 
 240 | 24.0 | 48.0 | 72.0 96.0 | 120.0} 144.0] 168.0] 192.0 | 216.0 
 239 | 23.9 | 47.89] 71:7 95.6). 119.5) 143.4 | 167.8 | 191.2 | 215.1 
 938 | 23.8! 47.6 | 71.4 95.2 | 119.0], 142.8] 166.6 | 190.4 | 214.2 
 i EG ev ae ev Se Sa ae | 94.8 | 118.5 | 142.2] 165.9 | 189.6 | 213.3 
 236 | 23.6] 47.2 70.8 94.4] 118.0] 141.6 | 165.2 | 188.8 | 212.4 
 tee ea: 5 Tt 47.0" f 67085 94.0 | 117.5 | 141.0) 164.5 | 188.0 | 211.5 
 234 | 23.4] 46.8 | 70.2 93.6 | 117.0 | 140.4] 163.8 | 187.2 | 210.6 
 9 3 ale eS Was 46.6 69.9 93.2 116.5: . 18928 163.1 | 186.4 | 209.17 
 232 | 23.2} 46.4 | 69.6 92.8 | 116.0 | 139.2 | 162.4 | 185.6 | 208.8 
 231 | 23.1 | 46.2 | 69.3 92.4 | 115.5 | 138.6 | 161.7 | 184.8 | 207.9 
 230 | 23.0] 46.0 | 69.0 92.0 | 115.0] 138.0] 161.0] 184.0 | 207.0 
 229 | 22.9] 45.8 | 68.7 91.6] 114.5 | 187.4} 160.3 |. 183.2 | 206.1 
 228 | 22.8| 45.6 | 68.4 91.2 | 114.0] 186.8] 159.6 | 182.4 | 205.2 
 227 22.7 45.4 68.1 90.8) 118.5 136.2 158.9 181.6 | 204.3 
 226 | 22.6 | 45.2 | 67.8 90.4 | 113.0] 185.6 | 1582} 180.8 | 203.4 
 
TABLE XXTV.—LOGARITHMS CF NUMBERS. 
 
 | No. 190 L. 278.] 
 
 Sa a 
 | 
 
 owe | 
 199 | 278754 | 8982 
 
 ase 
 U 
 Y 
 > 
 
 | 1261 | 1488 | 
 
 [No, 214 L, 232, | 
 + hee | 
 
 9 
 
 2 3201 | 3527 
 3 5557 | 5782 
 4 | [7802 | 8026 
 5 | 290085 | 0257 | 
 6 2256 | 2478 | 
 v 4466 | 4687 | 
 
 8 | 6665 | 6884_| 
 8853 | 9071 | 
 
 9507 
 
 200 | 301030 | 1247 
 1 | 3196 | 38412 
 
 | 2} 5351 | 5566 | 
 | 3 | 7496 | 7710 
 4} 9630 | 9843 | 
 
 | 
 | 5 | 811754 | 1966 | 
 |} 3867 | 4078 
 | 5970 | 6180 | 
 | | 8 | 8063 | 8272 | 
 
 ID 
 
 | 1681 
 | a04d4 
 
 5996 
 
 $1387 
 
 83851 
 
 | O268 
 | 2589 | 
 | 4499 
 
 6599 
 8689 
 
 | 0481 
 
 2600 
 4710 
 6809 
 
 Z 
 8898 
 
 5760 | 
 W854 
 
 Q2: 
 9958 
 
 | 320146 
 
 =) 
 co 
 Ol 
 a 
 
 2219 | 2426 
 | 1| 4282 | 4488 
 | | 2 | 6386 | 6541 | 
 
 3 |  8880-| 8583 | § 
 
 | O769 | 
 
 IQS 
 WO 
 
 4899 | 
 
 6950 
 8991 
 
 0977 | 
 8046 
 
 5105 
 
 1155 | 
 
 9194 
 
 4 | 330414 | C617 | 0819 | 
 
 1022 
 
 1225 
 
 PROPORTIONAL PARTS. 
 
 2012 | 
 
 4nhnw 
 
 407% 
 
 6131 
 
 | 8176 
 
 0211 | 
 2230 | 
 
 en. | b>} 42 
 
 oo 
 
 Rey 
 
 | z | was 
 
 | 205 | 22.5 | 45.0 
 
 | 924 | 224] 44.8 
 
 | 993 |92.3| 44.6 
 
 | 999 | 999) 44.4 
 294 | 2.1) 44.2 
 220 | 22.0] 44.0 
 219 | 21.9| 43.8 
 
 rw 
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 215 1 2e 
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 | | 218 21.3 | 2.6 
 212 21.2 | 2.4 
 211 el MA Ale 
 | 210 ALO 42 .( 
 | 209 20. 41.8 
 i} 208 20.8 41.6 
 207 20.7 41.4 
 206 20.6 41.2 
 205 20.5 4d .0 
 £04 20.4 40.8 
 2038 20.3 40.6 
 202 | 20.2 40.4 
 
 Cwo°iwore W-2Oco 
 
 5 eS 
 OWS 
 
 owce 
 Oooce 
 
 t 
 
 oo 
 CHOW OW ROT 
 
 OROOWBORO 
 
 Wwe wwwww 
 
 It ..St 
 
 CU 
 
 at 
 
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 WwWwww 
 
 ~j =} 
 
 Wo LO OO 
 
 Oo 
 
 ee ee on oe on eed 
 
 S Cr Coa 
 
 SOAS % 
 
 It 
 
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 } PARED CD COWS HE Cr 
 
 Bek fk fk ed Ped Pe Pe 
 
 | 
 
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 9S OG KW 
 
 = 
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 Or 
 
 SONG 
 
 2 
 
 Go 
 
 He CIM HO 
 
 OO wo oO 
 
 0 
 
 bo 
 
 3044 | 32 
 
 TABLE XXTV.—LOGARITHMS OF NUMBERS. 
 
 pas cS See Fe ae 
 No, 215 L, 832.] 
 
 | 8417 
 
 8649 
 
 [No. 239 L. 880. 
 
 7 | 8 
 
 | 4051 
 
 2028 | 2225 
 
 332438 3850 
 4454 | 50ET | || 5458 | 5658 | 5859 | 6059 
 6460 7060 7459 | 7659 | 7858 | 8058 
 8456 | 9054 9451 | 9650 | 9849 |——— 
 | | 5: 41 | 0047 
 
 340444 *| 1039 | '| 1435 | 1682 | 1830 
 2423 3014 3409 | 3606 | 8802 | 8999 | - 
 4392 4931 | 5374 | 5570 | 5766 | 5962 
 6353. | 6939 7330 | 7525 | 7720 | 7915 
 8305. | | 8889 9278 | 9472 | 9666 | 9860 
 
 350248 0829 | 1216 | 1410 | 1603 | 1796 | 
 £188 | 2761 | 8147 | 3839 | 3532 | 3724 
 4108 4685 5068 | 5260 | 5452 | 5643 
 6026 | | 6599 | 6981 | 7172 | 7363 | 755 
 7935 8506 8886 | 9076 | 9266 | 9456 
 9835 | / 
 
 —— 0404 || 0783 | 0972 | 1161 | 1850 
 
 361728 | 2294 | || 2671 | 2859 | 3048 | 3236 
 
 3612 | 88 
 5488 | 56 
 7356 | % 
 9216 
 
 4176 
 
 + 6049 | 
 
 “915 
 
 9772 
 
 lin | 
 
 4551 
 
 8287 
 
 | 4739 
 6423 
 
 6610 
 8473 
 
 4926 
 6796 
 8659 
 
 5113 
 6983 
 8845 
 
 371068 
 2912 
 4748 
 6577 
 
 8398 
 
 38 
 
 | 1622 
 
 3464 
 
 5298 
 
 7124 
 8943 
 
 0148 
 1991 
 8831 
 
 5664 | 
 7488 | 
 | 9306 
 
 0328 
 
 2175 
 
 4015 
 5846 
 7670 
 
 9487 
 
 PROPORTIONAL PARTS. 
 
 0513 
 2360 
 4198 
 6029 
 7852 
 
 9668 
 
 0698 
 
 2544 | 272 
 
 4382 
 6212 | 
 8034 
 9849 
 
 J dD 
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 SDOSSS | a 
 
 MAM BAD 
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 Nard raIIW 
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 WNL KK oror 
 
 STE Wd 3 3 SS 
 
 rt 0o 
 
 101. 
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TABLE XXIV.—LOGARITHMS OF NUMBERS, 
 
 No, 240 L. 380.] [No. 269 L, 431. 
 
 Mein oe | boli a {owl ae lbs 6 | a] 8 | 9 [pie 
 
 240 | 380211 | 0392 | 0573 | 0754 | 0934 |) 1115 | 1296 | 1476 | 1656 | 1887] 181 
 
 Ae) 2017 | 2197 | 2377 | 2557 | 2787 || 2917 | 3097 | 38277 | 8456 | 3636 | 180 
 
 ; 3815 | 3995 | 4174 | 43853 | 4533 || 4712 | 4891 | 5070 | 5249 | 5428 | 179 
 
 3) 5606 | 5785 | 5964.) 6142 | 6321 || 6499 | 6677 | 6856 | 7034 | 7212 178 
 
 | 4 7390 | 7568 | 7746 | 7924 | 8101 || 8279 | 8456 | 8634 | 8811 | 8989 178 
 | 5 9166 | 9343 | 9520 | 9698 | 9875 || 
 
 6 | 390935 | 1112 | 1288 
 7 | 2697 | 2873 | 3048 | 3224 | 3400 || 3575 | 3751 | 3926 | 4101 | 4277 | 176 
 
 8 | 4452 | 4627 | 4802 | 4977 | 5152 || 5326 | 5501 | 5676 | 5850 | 6025 | 175 Ht! 
 9 | 6199 | 6374 | 6548 | 6722 | 6896 || 7071 | 7245 | 7419 | 7592 | 7766 | 174 
 
 250 | 7940 | 8114 | 8287 | 8461 | 8634 |! 8808 | 8981 | 9154 | 9328 | 9501 | 17% 
 | 9674 | 9847 | = 
 
 i} 
 
 0051 | 0228 | 0405 | 0582 | 0759 | 177 i 
 1464 | 1641 || 1817 | 1993 | 2169 | 2345 | 2521 | 176 
 
 _—_—. 0020 | 0192 | 0365 || 0538 | 0711 | 0883 | 1056 | 1228 173 
 2 | 401401 | 1578 | 1745 | 1917 | 2089 || 2261 | 2438 | 2605 | 28777 | 2949 172 | 
 3 | 8121 | 8292 | 3464 | 3635 |.3807 || 38978 | 4149 | 4320 | 4492 | 4663 171 f| 
 £ 4834 | 5005 | 5176 | 5346 | 5517 |) 5688 | 5858 | 6029 | 6199 | 6370 171 i} 
 5 6540 | 6710.| 6881 | 7051 | 7221 || 7291 | 7561 | 7731 | 7901 | 8070 170 il 
 6 | 8240 | 8410 | 8579 | 8749 | 8918 || 9087 | 9257 | 9426 | 9595 | 9764 169 i} 
 th 9933 ii 
 
 | 0102 | os71 | 0440 | 0609 |) 0777 | 0946 | 1144 | 1288 | 1451 i 
 8 | 411620 | 1788 | 1956 | 2124 | 2293 |) 2461 | 2629 | 2796 | 2964 | 3132 | 168 \ 
 
 art 
 o> 
 vo) 
 
 9 3300 | 3467 | 3685 | 8803 | 3970 || 4187 | 4305 | 4472 | 4639 | 4806 167 
 260 4973 | 5140°| 5307 | 5474 | 5641-1; 5808 | 5974 | 6141 | 6308 | 6474 | 167 
 1 6641 | 6807 | 6973 | 7189 | 7306.) 7472 | 76388 | 7804 | 7970 | 8135 166 i 
 2 8301 | 8467 | 8633 | 8793 | 8964 |) 9129 | 9295 | 9460 | 9625 | 9791 165 i 
 3 9956 | ee | 
 \ 
 
 | 
 
 | 0121 | 0286 | 0451 | 0616 781 | 0945 | 1110 | 1275 | 1489 165 
 4 | 421604 | 1768 | 1933 | 2097 | 2261 || 2426 | 2590 | 2754 | 2918 | 3082 164 
 5 3246 | 3410 | 3574 | 3737 | 3901 || 4065 | 4228 | 4392 | 4555 | 4718 164 
 5371 | 5534 
 
 6 4882 | 5045 | 5208 
 
 5697 | 5860 | 6028 | 6186 | 6349 163 | 
 f 
 
 7 | 6511 | 6674 | 6836 | 6999 | 7161 || 7324 | 7486 | 7648 | 7811 | 7973 | 162 
 8 | 8135 | 8297 | 8459 | 8621 | 8783 |) 8944 | 9106 | 9268 | 9429 | 9591 | 162 
 
 9| 9752 | 9914 i | 
 43 | 0075 | 0236 | 0398 | 0559 | 0720 | 0881 | 1042 | 1203.| 161 
 i 
 PROPORTIONAL PARTS, I 
 hi 
 | 
 : - le 
 | pie.| 1 | 2 3 4 5 6 " ea a: I 
 | | } 
 178 | 17.8| 35.6 | 53.4 | 71.2 | 89.0 | 106.8) 124.6} 142.4 | 160.2 I 
 177 | 17.7| 35.4 | 53.1 | 70.8 | 88.5 | 106.2] 123.9 | 141.6 | 159.8 | 
 176 | 17.6] 35.2 | 52.8 | 70.4 | 88.0 | 105.6] 128.2] 140.8 | 158.4 } 
 Pe 17 5 | 85.04") 52.5.8) 70.0 7.5 | 105.0) 122.5 | 140.0 | 157.5 | 
 174 | 17.4| 34.8 | 52.2 | 69.6 | 87.0 | 104.4] 121.8 |. 139.2 | 156.6 
 173 | 17.3] 34.6 | 51.9 | 69.2 | 86.5 | 108.8) 121.1 | 138.4 | 135.7 
 172 | 17.2] 34.4 | 51.6 | 68.8 | 860 | 103.2] 120.4] 187.6'| 154.8 
 | 171 | 17.1] 34.2 | 51.8 } 68.4-} 8.5 | 102.6] 119.7] 186.8 | 153.9 
 : 170 | 17.0} 34.0 | 51.0 | 68.0) 85.0 | 102.0} 119.0] 186.0.| 153.0 
 169 |,16.9°) 33.8 | 50.7 | 67.6-4 81.5 | 101.4] 118.3] 185.2} 15214 
 163 | 16.8} 33.6 | 50.4 | 67.2 + 84.0 | 100.8} 117.6 | 134.4 | 151.2 
 167 | 16.7] 383.4 | 50.1 | 66.8 + 83.5 | 100.2] 116.9 |. 183.6] 150.3 
 166 | 16.6] 33.2 | 49.8 | 66.4 } 83.0 99.6 | 116.2 | 132.8 | 149.4 
 165 | 16.5] 33.0 | 49.5 | 66.0 | 82.5 99.0] 115.5 | 132.0 | 148.5 
 164 | 16.4] 32.8 | 49.2 | 65.6 4 82.0 98.4} 114.8] 131.2 | 147.6 
 163 | 16.3} 32.6 | 48.9 | 65.2 |} 81.5 97.8| 114.1] 130.4 | 146.7 
 162 | 16.2) 32.4 | 48.5 | 64.8-'F 81.0 97.2 | 118.4] 129.6 | 145.8 
 161 | 16.1] 32.2 | 48.3 | 64.4 } 80.5 96.6 | 112.7 | 128.8 | 144.9 
 
TABLE XXTV.—lLOGARITHMS OF NUMBERS. 
 
 No. 270 L. 431.] [No. 299 L. 476. | 
 
 N.| 0 1 2 3 4 || 6 6 7 8 9 | Lift. 
 
 | 431364 | 1505 | 1685 | 1846 | 2007 
 
 2167 | 2328 | 2488 | 2649 | 2809 | 161 
 
 270 | 
 1 2969 | 81380 } 8290 | 3450 | 3610 || 3770 | 3930 | 4090 | 4249 | 4409 160 
 2 4569 | 4729 1 4888 | 5048 | 5207 || 53867 | 5526 | 5685 | 5844 | 6004 159 
 3 6163 ‘ 6322 | 6481 | 6640 | 6799 || 6957 | 7116 | 7275 | 7483. | 7592 159 
 4 7751 | 7909 | 8067 ' 8226 | 83884 || 8542 | 8701 | 8859. | 9017 | 9175 158 
 
 9333 | 9491 | 9648 | 9806 | 9964 |' —-— |- 
 } 0122 | 0279-| 0487 | 0594 | 0752 
 
 1695. | 1852 | 2009 | 2166 | 2823 
 
 SP EE ay ae 15 
 440909 | 1066 | 1224 | 1381 15 
 2480 | 2687 | 2793 | 2950 | 8106 || 8268 | 3419 | 38576 | 3732 | 8889 | 15 
 4045 | 4201 | 4857 | 4513 4669 || 4825 | 4981 | 5137 | 5293 | 5449 | 15 
 15 
 
 15 
 
 1588 
 
 5604 | 5760 | 5915 | 6071 | 6226 || 6382 | 6537 | 6692 | 6848 | 7003 
 | 158 | 7313 | 7468 | 7623 | 7778 || 7938 | 8088 | 8242 | 8397 | 8552 
 706 | 8861 | 9015 | 9170 | 9324 || 9478 | 9633 | 9787 | 9941 
 
 — — =| OOOR at 
 450249 | 0403 | 0557 | 0711 | 0865 || 1018 | 1172 | 1326 | 1479 | 1633 | 154 
 1786 | 1940 | 2093 | 2247 | 2400 || 2553 | 2706 | 2859 | 3012 | 3165 | 15 
 3318 | 3471 | 8624 , 8777 | 3930 || 4082 | 4235 | 4387 | 4540 | 4692 | 153 
 | 4845 | 4997 | 5150 | 5302 | 5454 || 5606 | 5758 | 5910 | 6062 | 6214 152 
 6366 | 6518 | 6670 | 6821 | 6973 || 7125 | 7276 | 7428 | 7579 | 7731 152 
 7882 | 8033 | 8184 | 8336 | 8487 || 8638 | 8789 | 8940 | 9091 | $242 | 151 
 9392 | 9543 | 9694 | 9845 | 9995 
 
 pa St 0146 | 0296 | 0447 | 0597 | 0748 | 151 
 460898 | 1048 | 1198 | 1348 | 1499 | 1649 | 1799 | 1948 | 2098 | 2248 | 150 
 
 ORWDOE DS OW DODNIOOUIRWMO HO OBOMNS 
 
 29 2398 | 2548 | 2697 | 2847 | 2997 || 8146 | 8296 | 3445 | 3594 44 150 
 8893 | 4042 | 4191 | 4840 | 4490 || 4639 | 4788 | 4986 | 5085 | 5234 149 
 53883 | 5582 | 5680: | 5829 | 5977 || 6126 | 6274-| 6423 | 6571 | 6719 149 
 6868 | 7016 | 7164 | 7312 | 7460 |! 7608 | 7756 | 7904 | 8052 | 8200 148 
 8847 | 8495 | 8648 | 8790 | 8988 7; 9085 ; 92383 | 9380 | 9527 | 9675 148 
 9822 | 9969 . |. | 
 es ——/ 0116 | 02638 | 0410 0557 | O704 | 0851 | 0998 | 1145 147% 
 6 71292 | 1488 | 1585 | 1732 | 1878 2025 | 2171 | 2318 | 2464 | 2610 146 
 df 2756 | 2903 | 8049 | 3195 | 3841 8487 | 3633 | 8776 8925 | 4071 146 
 8 4216 | 4862 | 4508 | 46538 | 4799 4944 | 5090 | 5285 | 5881 | 5526 146 
 9 5671 | 5816 | 5962 | 6107 | 6252 397 | GE42 | C687 | 6882 | 6976 145 
 PROPORTIONAL PARTS. | 
 f 
 Diff 1 2 | 3 4 5 6 i 8 9 
 161 | 16.1} 32.2 | 48.8 | 644 | 80.5 | 96.6. | 112.7] 128.8 | 144.9 | 
 160 | 16.0 | 32.0 | 48.0 | 64.0 | 80.0 | 96.0 | 112.0] 128.0 | 144.0 
 159 15.9 31.8 Movil 63.6 79.5 95 .4. 111.3 127.2 | 1438.1 
 158 | 15.8 31.6 ad 63.2 79.0 94.8 110.6 126.4; | 142.2 
 157 151.7 31.4 {ion 62.8 78.5 94.2 109.9 125.6 | 141.38 
 156. «| 15.6 Slee 46.8 62.4 78.0 93.6 109.2 124.8 | 140.4 
 155 15.5 31.0 46.5 62.0 (G5) 93.0 108.5 124.0.) 1389.5 
 154 15.4. 30.8 46.2 61.6 (00 92.4 107.8 123.2 | 138.6 | 
 i laser ira Rages’ 80.6 45.9 61.2 G55 91.8 107.1 122.45) 1st er 
 152.-| 15.2 | 380.4 45.6 60.8 76.0 91.2 106.4 121.6 | 136.8 
 iSiay) 15.291 SOre 45.3 60.4 RD .5 90.6 105.7 120.8-| 185.9 
 150 15.0 + 30:0 45.0 60.0 75.0 90.0 105.0 120.0 | 185.0 
 149 | 14.9} 29.8 44.7 59.6 44.5 89.4 104.3 410 Se) Sarr 
 148 14.8 29.6 44.4 59.2 .| %.0 88.8 103.6 118.4°) 133.2 
 147 14.70) 29.4 44.1 58.8 73.5 88.2 102.9 117.67) A3sa3 
 146 14,62) 29%2 43.8 58.4 fo: 0 7.6 oh T0269 116.8 | 131.4 
 145 14.5 | 29.0 43.5 58.0 425 87.0 oh 10165 116.0 | 180.5 
 144 14.4 28.8 43.2 57.6 72.0 86.4 100.8 115.24 12946 
 143 ie Os) 28 .6 42.9 57.2 1.5 85-8 ah 10031 114.4 | 128.7 
 142. | 14.2 28.4 42. ¢ 56.8 71.0 85.2 99.4 113.6) 127.8 
 144. 4 14.1 28.2 42.3 56.4 70.5 84.6 98.7. | 112.8 | 126.9 
 140 | 14.0 | 28.0 2.0 56.0 70.0 84.0 98.0 112.0.) 426.0 
 
 ou) 
 awe 
 a 
 
| No. 300 L. 477.) 
 
 TABLE XXIV.—LOGARITHMS OF NUMBERS. 
 
 [No. 339 L. 581. 
 
 0099 | 
 
 N.| 0 | 1 2/ sila | 5 | 6 | 7 | 8 | 9 | Diff. 
 || | | | 
 
 800 : 477121 | 7266 | 7411 | 7555 | 7700 || 7844 | 7989 | 8138 | 8278 | 8422 | 145 
 1 8566 | 8711 | 8855 | 8999 | 9143 i 9287 | 943 | 9575 | 9719 | 9863 | 144 
 
 | | a | | 
 
 | 2) 480007 | 0151 | 0294 | 0438 | 0582 |! 0725 | 0869 | 1012 | 1156 | 1289 | 144 
 
 as 1443 | 1586 | 1729 | 1872 | 2016 || 2159 | 2802 | 2445 | 2588 2731 143 
 
 | 4 29874 | 3016 | 38159 | 8802 | 2445 || 8587 | 87380 | S872 | 4015 | 4157 148 
 
 ie ey 4300 | 4442 | 4585 | 4727 | 4869 || 5011 | 5158 | E295 437 | 5579 | 142 
 6 5721_-| 5863 | 6005 | 6147 | 6289 || 6480 | 6572 |- 6714 6597 142 
 7 7138 | 7280 | 7421 | 7563 | “704 || 7845 | 7986 | 8127 | 8410 | 141 
 8 | 8551 |. 8692 | 8833 | 8474 | 9114 9255 | 9896 | 9587 9818 | 141 
 9| 9958 | =| | 
 
 9203 
 
 9337 
 
 947 
 
 9606 
 
 | 510545 
 
 1883 
 | 38218 
 ' | 4548 
 5874 
 7196 
 
 UW 
 BHO OMNI C9 
 
 oe 8514 
 | 9828 | 
 2 | 5211388 
 3| 2444 
 4). 3746 
 ‘ae 5045 
 6 | 633! 
 fg 7630 | 
 8 | 8917 
 
 0679 
 2017 
 38351 
 4681 
 6006 
 7328 
 8646 
 9959 
 
 1269 
 2575 
 
 | 3876 
 5 | 5174 | 
 | 6469 | 
 
 WG59 
 
 | 9045 
 
 | 0009 
 
 0143 | 
 
 0411 124 
 1 
 
 | | 0239 | 0880 | 0520 || 0661 801 | 6841 | 1081 | 1222 | 140 
 | 
 
 310 | 491362 | 1502 | 1642 | 1782 | 1922 2062 | 2201 | 2341 | 2481 | 262 140 
 1 2760 | 2900 | 3040 | 8179 | 3319 || 3458 | 8597 | 5787 | 8676 | 4015 139 
 9) 4155 | 4294 | 4438 | 4572 | 4711 || 4850 | 4989 | 5128 | 5267 | 5406 | 189 
 3 | 5544 | 5683 | 5822 | 5960 | 6099 || 6238 | 63876 | 6515 €€58 | 6791 189 
 4 6930 | 7068 |.7206 | 7344 | 7483 || 7621 | 7759 | T8S7 | EC25 | 8173 | 188 
 5 8311 | 8448 | 8586 | 8724 | 8862 || 8999 | 9187 | S275 | 412 | CEEO | 188 
 
 6 9687 | 9824 | 9962 If _—. = 
 | | 0099 | 0286 || 0874 | 0511 | 0648 | OCES | 0922 | 187 
 % | 501059 | 1196 | 1333 | 1476 | 1607 | 1744 | 18€0 | £017 | S1E4 |} 2291 1G 
 8 | 2497 | 9564 | 2700 | 2887 | 2973 || 8109 | 3246 | 8882 | E518 | E655 126 
 9 3791 | 3927-| 4063 | 4199 | 4335 || 4471 | 4607 | 4743 | 46.8 | £014 | 1286 
 320 | 5150 | 5086 | 5421 | 5557 | 5693 || 5628 | 5964 | 6099 | 6224 | E870 | 186 
 | 6505 | 6640 | 6776 | 6911 | 7046 || 7181 | 7316 | 7451 | EEG | 7721 135 
 #256 | 7991 | 8126 | 8260 | 8895 || 8530 | 8664 | 8799 | E924 | C068 | 185 
 
 0813 | 0947 | 1081 || 1215 | 13849 | 1482 | 1616 750 8 
 2151 | 2284 | 2418 || 2551 | 2684 | 2818 | 2051 | ced 18: 
 3484 | 3617 | 8750 || 8883 | 4016 | 4149 | 42€2 | 4415 182 
 4818 | 4946 | 5079 || 5211 | 5844 | 5476 | ECCO | 5741 183 
 6139 | 6271 | 6403 || 5 | 6668 | GECO | CGe2 | 764 132 
 7460 | 7592 | 7724 || 7987 | 8119 | $251 | G82 | 182 
 8777 | 8909 | 9040 | 9171 | £303 | 9484 | S5C6 | SEO" 131 
 
 0090 
 1400 
 9705 
 4006. | 
 53804 
 
 6598 
 7888 
 
 0221 | 0358 |) 0484 
 
 1530 | 1661 || 1792 | 1922 
 2835 | 2966 || 8096 | 3226 
 4136 | 4266 || 4896 | 4526 
 5434 | 5563 || 5693 | 5822 
 
 6727 | 6856 || 6985 | 7114 
 8016 | 8145 || 8274 | 8402 
 | 9480 || 9559 | 9687 
 
 0615 
 
 | QOF 
 
 | 9815 | 
 
 0745 
 2053 
 uve 
 
 4€E6 
 5951 
 (243 
 
 631 
 
 47&5 | 4916 180 
 €c81 | 6210 129 
 Gate NBO aes 
 FECO | S788 129 
 GC 
 
 |. 580200 | 0328 
 
 sy: 
 At 
 
 2 || O€ 
 
 PROPORTIONAL PARTS. 
 
 968 
 
 1C96 | 
 
 43 
 0072 128 
 223 | 1351 128 
 
 39 
 a0 | ) 
 38 8 
 | Qn rn 
 af ( 
 
 | MCVBwwwwwwwnw 
 
 Ten os 
 
 Pm be pk Re tek ek ek eek Re ek peek ek PL 
 
 OO OO CO GO CO CO GH C9 GO GO CO GO 
 
 IDOOrWO KE OL 
 
 ee ee 
 
 ; MWOOrwwror 
 
 \ 
 | 
 | 
 | 
 
 27.8 
 27.6 
 27 4 
 Ohoe 
 27.0 
 26.8 
 26.6 
 26.4 
 26.2 
 26.0 
 95.8 
 Ro .6 
 25.4 
 
 3 4 5 6 
 ret 6 69.5 83 .4 
 41.4 £3 69.0 32 .£ 
 41-1 8 68.5 82.2 
 40.8 | A 68.0 81.6 
 40.5 0 67.5 81.0 
 
 ORS D tI 6 67.0 | 80.4 
 | 39.9 | 2 oh 66.5. 7958 
 } 39.6 | 8 66.0 79.2 
 39.3 2.4 65.5 [8.6 
 | 89.0 0 65.0 "8.0 
 38.7 1.6 64.5 } 77.4 
 38.4 2 64.0 76.8 
 | 38.1 VE 63.5 76.2 
 
 WAWSEVBAHDEUWONDW 
 
 a pew 
 S > 
 = Or 
 OoMo 
 
 — 
 oo) 
 rus) 
 
 111.2 | 195-4 
 110.4 | 124-2 
 109.6 | 123.3 
 108.8 | 1£2.4 
 108.0 | 121.5 
 107.2 | 120.6 
 106.4 | 119.47 
 
 ~ Aa N-I CS 
 
 G9 OH 
 
TABLE XXIV.—LOGARITHMS OF NUMBERS, 
 
 =a 
 
 | { No. 840 L, 881.] [No. 879 1. 579. 
 1 oe = ===> 
 
 . | 
 N.| 0 1 21a) al 51 8 |e 8 | 9 | pitr. 
 
 340 | 531479 | 1607 | 1734 | 128 
 1 2754 | 2882 | 3009 | 3136 | 3264 3391 | 8518 | 3645 | 3772 | 3899 | 127 
 R 4026 | 4153 | 4280 | 4407 | 4534 |! 4661 | 4787 | 4914 | 5041 | 5167 127 
 3 5294 | 5421 | 5547 | 5674 | 5800 || 5927 | 6053 | 6180 | 6306 | 6432 126 
 
 1862 | 1990 || 2117 | 2245 | 2372 
 
 2500 | 2627 
 
 4 6558 | 6685 | 6811 | 6937 | 7063 || 7189 | 7315 : 7441 | 7567. | 7693 126 
 5 7819 | 7945 | 8071 | 8197 | 8322 || 8448 | 8574 | 8699 | 8825 | 8951 
 6 
 7 
 
 9076 | 9202 | 9327 | 9452 | 9578 || 9703 | 9829 | 9954 a 
 | ———,— —| 0079 | 0204 | 125 
 640329 | 0455 | 0580 | 0705 | 0830 || 0955 | 1080 | 1205 | 1330 | 1454 | os 
 8; 1579 | 1704 | 1829 | 1953 | 2078 | 2203 | 2827 | 2152 | 2576 | aroq | 
 9] 2825 | 2950. | 8074 | 8199 | 3323 3447 | 3571 | 8696 | 3820 | 3944 | jo4 
 350 | 4068 | 4192 | 4316 | 4440 | 4564 | 4688 | 4812 | 4936 | 5060 | 5188 | 124 
 Wn 1 530% | 5431 | 5555 | 5678 | 5802 | 5925 | 6049 | 6172 | 6296 | E419 | 2 
 Ne i 2 | 6543 | 6666 | 6789 | 6913 | 7036 || 7159 | 7282 | 7405 | 7529 | 7659 | jO3 
 ct 3 7¢S | 7898 | 8021 | 8144 | 8267 | 8389 | 8512 | 8635 | 8758 | 8881 23 
 4 9003 | 9126 | 9249 | 9371 | 9494 | 9616 | 9739 | 9861 | 9984 | 
 gpa = : | 0106 | 128 
 5 | 550228 | 0351 | 0473 | 0595 | 0717 | 0840 | 0962 1084 | 1206 | 1328 | 122 
 6 1450 | 1572 | 1694 | 1816 | 1938 | 2060 | 2181 | 2303 | a49% 2547 | 122 
 7 
 8 
 9 
 
 2668 | 2790 | 2911 | 8033 | 3155 || 8276 | 3398 | 3519 | 3640 | 2g | Jo 
 
 8883 | 4004 | 4126 | 4247 | 4368 || 4489 | 4610 | 4731 | 4959 | 4973 | 121 
 
 5094 | 5215 | 5336 | 5457 | 5578 | 569 | 5820 | 5940 | Go61 | Gige | jot 
 360 | 6303 | 6423 | 6544 | 6664 | 6785 |/ 6905 | 7026 | 7146 | 7267 | e397 | 490 
 Py) 7507 | 7627 | 7748 | 7868 | 7988 || 8108 | 8228 | 8349 | g169 | gaeq | 490 
 2) 8709 | 8829 | 8948 | 9068 | 9188 || 9308 | 9428 | 9548 | 9667 | 9787 | 120 
 3] 9907 | —— | 
 ~——~| 0026.| 0146. | 0265 | 0385 | 0504 | 0624 | 0743 | 0863 | cose | 119 
 SULL0L | 1221 ) 1340 | 1459 | 1578 || 1698 | 1817 | 1936 | 2055 | oye | 179 
 2293 | 2412 | 2531 | 2650 | 2769 8006 | 3125 | 3244 | 3362 | 119 
 8481 | 3600 | 8718 | 8837 | 3955 || 4074 | 4192 | 4311 | 4429 | gs4g | 119 
 4666 | 4784 | 4903 | 5021 | 5139 || 5257 | 5376 | 5494 | 5612 | 5730 | 148 
 5848 | 5966 | 6084 | 6202 | 6320 || 6437 | 6555 | 6673 | 6791 | 6909 | 118 
 W026 | 144 | 7262 | 7379 | 7497 || 7614 | 7732 | 7849 | 7967 | gos, | 118 
 8202 | 8319 | 8436 | 8554 | 8671 || 8788 | 8905 | 9028 | 9140 | gos | 447 
 9374 | 9491 | 9608 | 9725 | 9842 |} 9959 a 
 . | = —| 0076 | 0193 | 0309 | 0426 | 4117 
 570543 | 0680 | 0776 | 0893 | 1010 |, 1126 | 1243 | 1359 | 1476 | 1599 | 117 
 1109 | 1825 | 1942 | 2058 | 2174 || 2291 | 2407 | 2523 | 2639 | aras | 146 
 2872 | 2988 | 3104 | 3220 | 3336 || 3568 | 3684 | 3800 | 3915 | 116 
 A031 | 4147 | 4263 | 4879 | 4494 || 4610 | 4726 | 4841 | 4957 | 5072} 116 
 5188 | 5303 | 5419 | 5534 | 5650 || 5765 | 5880 | 5996 | 6111 | 6296 | 115 
 Gaal | 6457 | 6572 | 6687 | 6802 || 6917 | 7032 | 7147 | 7262 | 3771 715 
 (492 | 7607 | 7722 | 7836 | 7951 || 8066 | 8181 | 8295 | 8110 | 85951 i7s 
 8639 | 8754 | 8868 | 8983 | 9097 | 9212 | 9326 | 9441 | 9555 | 9¢60 | 14 
 
 2» 
 @ 
 oa 
 NS 
 
 is) 
 
 COImOMAwWs HS CMOMOOR 
 
 Oo 
 NS 
 Or 
 rw) 
 
 es 
 Qa 
 on 
 ror 
 
 PROPORTIONAL Parts. 
 | — 
 ee Oe ae oh a ge! 7 8 9 
 | | | 
 5 Sam | / | aD ae Moy Ts 
 128 | 12.8] 25.6 | 38.4 51.2 | 64.0 “6.8 | 89.6 | 102 115.2 
 127 | 12.7] 25.4 B81) 50.8 63.5 76.2 | 88.9 | 101.6 | 114.3 
 126 | 12.6] 25.2 37.8 | 50.4 63.0 15.6 “| 88/2 2100-8 418.4 
 125 | 12.5 | 25.0 87.5 | 50.0 | 62.5 75.0 | 87.5 | 100.0 | 112.5 
 12 12.4] 24.8 37.2 | 49.6 | 62.0 74.4 | 86.8 99.2 | 111.6 
 . 123- | 12.3} 24.6 36.9: |. 49.9 \) 61.5 73:8 | 86.1 98.4 -| 110.7 
 eth Iie) 2440+ 26.6 | 48.8 | 61.0 73.2 | 85.4 | 97.6 | 109.8 
 ane) 124 24.0% 368 8.4 60.5 (2.6 | 84.7 | 96.8 | 108.9 
 120 12.0 24.0 | 36.0 48.0 60.0 72.0 84.0 96.0 | 108.0 
 | TGS) TL 23.84 35M ape 59.5 71.4 
 
 | 
 | 
 | 
 | 
 | 
 
 83.5 | 95.9 LO tele 
 
TABLE XXIV.—LOGARITHMS 
 
 OF NUMBERS. 
 
 aye 
 
 No. 380. L. 
 
 9.] 
 
 [No. 414 L. 617. 
 
 | 
 
 6 
 
 So 7.9 |) Der 
 
 579784 | 9898 | ~ a) = 
 SOS ee | 0S. 0355 0469 | 0583 | 0697 | 0811 | 114 
 1 580925 | 1039 1153 | 1495 | 1608 | 1722 336 | 1950 
 2} 2063 | 2177 | 2291 2631 | 2745 | 2858 | 2972 | 3085 
 3 3199 | 3312 | 3426 | § 3765 | 8879 | 3992 | 4105 | 4218 
 4} 4331 | 4444 | 4557 | 4896 | 5009 | 5122 | 5235 | 5348 | 113 
 5 | 5461 | 5574 | 5686 | 6024 | 6137 | 6250 | 6362 | 6475 
 6 6587 | 6700 | 6812 | 7149 | 7262 | 7374 | 7486 | 7596 
 7 G711 | 7823 | 7935 | || 8272 | 8384 | 8496 | 8608 | 8720 | 112 
 8 8832 | 8944 | 9056 || 9391 | 9503 | 9615 | 9726 | 9836 
 9 9950 | | | 
 — | 0061 | 0173 0507 | 0619 | 0730 | 0842 | 0953 
 9 | 591065 | 1176 | 1287 | 1621 | 1732 | 1843 | 1955 | 2066 
 1} 2177 | 2288 | 2399 | 2732 | 2843 | 2954 | 8064 | 3175 | 111 
 2} 3286 | 3397 | 3508 3840 | 3950 | 4061 | 4171 | 4282 
 3 | 4893 | 4503 | 4614 | 4945 | 5055 | 5165 | 5276 | 5386 
 4| 5496 | 5606 | 5717 | 6047 | 6157 | 6267 | 6577 | 6487 
 5 | 6597 | 6707 | 6817 | "146 | 7256 | 7366 | 7476 | 7586 | 110 
 6 "695 | 7805 | 7914 8243 | 8353 | 8462 | 8572 | 8681 
 4 8791 | 8900 | 9009 9337 | 9446 | 9556 | 9665 | 9774 
 8 9883 | 9992 | — 109 
 Si 2 eee Hotel 0428 | 0537 | 0646 | 0755 | 0864 
 9 | 600973 | 1082 | 1191 1517 | 1625 | 1734 | 1843 | 1951 
 () 2060 | 2169 | 2277 | § | 2603 | 2711 | 2819 | 2928 | 3036 
 1 3144 | 3253 | 2361 3686 | 8794 | 8902 | 4010 | 4118 | 408 
 Q 4226 | 4334 | 4442 | | | 4766 | 4874 | 4982 | 5089 | 5197 
 3 | 5305 | 5413 | 5521 | | 5844 | 5951 | 6059 | 6166 | 6274 
 4 6381 | 6429 | 6596 | 6919 | 7026 | 7133 | 7241 | 7348 
 5 7455 | 7562 | 7669 | || 7991 | 8098 | 8205 | 8312 | 8419 | 407 
 6} 826 | 8633 | 8740 | 9061 | 9167 | 9274 | 9381 | 9488 
 “i 9594 | 9701 | 9808 | eS 
 ee eats 0128 | 0234 | 0341 | 0447 | 0554 
 8 | 610660 | 0767 | 0873 | |} 1192 | 1298 | 1405 | 1511 | 1617 
 1723 | 1829 | 1936 | 2254 | 2860 | 2466 | 2572 | 2678 | 106 
 2784 | 2890 | 2996 8313 | 3419 | 3525 | 3630 | 3736 
 8842 | 3947 | 4053 | 4370 | 4475 | 4581 | 4686 | 4792 
 4897 | 5003 | 5108 | 5424 | 5529 5634 | 5740 | 5845 
 5950 | 6055 | 6160 | 6476 | 6581 | 6686 | 6790 | 6895 | 105 
 7000 | 7105 | 7210 "525 | 7629 | 7734 | 7839 | 7943 
 PROPORTIONAL PARTS. 
 | | 
 11.8 | 28.6 | 35.4 “3 59.0 "0.8 82.6 94.4 | 106.2 
 10.7 1593.47" 35.1 8 58.5 70.8 81.9 | 93.6 | 105.3 
 IG a es 2 of 34-R 4 58.0 69 81.2 | 92.8 | 104.4 
 1115. | 23.0 34.5 46.0 | 57.5 69. 80.5 | 92.6 | 1038.5 
 | 11.4 |' 22.8 24.2 45.6 | 57.0 68 "9.8 | 91.2 | 102.6 
 | 11.34 22.6 33.9 45.2 | 56.5 64 79.1 90.4 | 101.7 
 | 11.2] 22.4 | 33.6 | 44.8 | 56.0 | 67 78.4 | 89.6 | 100.8 
 | | 
 pete ADS. 33.3 44.4 | ° 55.5 66. fay? 88.8 | 99.9 
 111.0! 22.0 33.0 44.0 55.0 66. 77.0 88.0 99.0 
 10.9 | 21.8 32.7 43.6 54.5 65. "6.3 8732 | 5a 
 10.8 | 21.6 32.4 43.2 54.0 64. "5.6 86.4 97.2 
 102% | 2.4 32.1 42.8 53.5 64 | AOI) 85.26 96.3 
 1028 |) 21.27 | 4 8H8 42.4 53.0 3, 74.2 | 84.8 | 95.4 
 10.5] 21.0 | 31.5 42.0 52.5 33.0 | %B.5 84.0 94.5 
 10.5.| 21.0 | 31.5 42.0 52.5 63.0 | 73.5 84.0 94.5 
 10.4| 20.8 | 31.2 | 41.6 | 52.0 72.8 | 83.2 | 93.6 
 
TABLE XXIV.—LOGARITHMS 
 
 OF NUMBERS. 
 
 { 
 | 
 | 
 
 No, 415 L, 618.] [No. 459 L. 662 | 
 1| 
 N.| 0 122% | 4 || 56 | 6 ro s Diff. 
 | | | 
 | | | | | | 
 415 | 618048 | 8153 | 857 | 8362 | 8466 || 8571 | 8676 | 8780 | gas4 | 8989 || 105 
 6 | 9093 | 9198 | 9302 | 9406 | 9511 || 9615 | 9719 | 9824 | 9928 |-——— 
 ——- 0082 | 
 7 | 620136 | 0240 | 0344 | 0448 | 0552 || 0656 | 0760 | 0864 | 0968 | 1072 | 104 
 8 | 1176 | 1280 | 1384 | 1488 | 1592 || 1695 | 1799 | 1903 | 2C07 | 2110 
 9 | 2214 | 2318 | 2421 | 2525 | 2628 || 2732 | 2835 | 2939 | 3042 | 3146 
 420 3249 | 3353 | 3456 | 8559 | 3663 || 38766 | 3869 | 3973 | 4076 | 4179 
 1 | 4282 | 4385 | 4488 | 4591 | 4695 || 4798 | 4901 | 5004 | 5107 | 5210 | 103 
 2 | 5812 | 5415 | 5518 | 5621 | 5724 || 5827 | 5929 | 6082 | 6185 | 6238 
 3 | 6340 | 6443 | 6546 | 6648 | 6751 || 6853 | 6956 | 7058 | 7161 | 7263 
 4 7366 | 7468 | 7571 | 7673 | 7775 || 7878 | 7980 8082 | 8185 | 8287 
 i 5 | 83889 | 8491 | 8593 | 8695 | 8797 || 8900 | 9002 | 9104 | 9206 | 9308 | 402 
 6 | 9410 | 9512 | 9613 | 9715 | 9817 || 9919 ; 
 | | 0021 | 0123 | 0224 | 0526 
 % | 630428 | 0530 | 0631 | 0733 | 0835 || 0936 | 1088 | 1189 | 1241 | 1342 | 
 8 1444 | 1545 | 1647 | 1748 | 1849 || 1951 | 2052 | 2158 | 2255 | 2356 | 
 i 9 | 2457 | 2559 | 2660 | 2761 | 2862 |) 2963 | 8064 | 3165 | 8266 | 3867 | 
 Hi 430 | 3468 | 3569 | 38670 | 3771 | 3872 |) 8973 | 4074 | 4175 | 4276 | 4376 | 101 
 i | 1 4477 | 4578 | 4679 | 4779 | 4880 || 4981 | 5081 | 5182 | 5283 | 5388 
 2| 5484 | 5584 | 5685 | 5785 | 5886 || 5986 | 6087 | 6187 | 6287 | 6388 
 3 | 6488 | 6588 | 6688 | 6789 | G889 || E9&9 | 7089 | 7189 | 7290 | 7390 | 
 4 | 7490 | 7590 | 7690 | 7790 | 7200 || 7990 | 8C90 | 8190 | 8290 | 83889 | 499 
 5 | 8489 | 8589 | 8689 | 8789 | 8888 || 8988 | 9088 | 9188 | 9287 | 9387 
 6 | 9486 | 9586 | 9686 | 9785 | 9885 || 9984 + _| 
 — | | 0084 | 0183 | 0283 | 0382 
 7 | 640481 | 0581 | 0680 oe | C879 |; 0978 | 1077 | 1177 | 1276 | 1875 
 8 1474 | 1573 | 1672 | 1771 | 1871 || 1970 | 2069 | 2168 | 2267 | 2366 
 9 | 2465 | 2563 | 2662 | 2761 | 2s6v | 2059 | 3058 | 3156 | 8255 | 3854 99 
 440 | 3453 | 8551 | 3650 | 3749 | 8847 || 3946 | 4044 | 4143 | 4242 | 4340 
 1 | 4489 | 4587 | 4636 | 4734 | 4282 || 4931 | 5029 | 5127 | 5226 | 5824 
 2 | 5422 | 5521 | 5619 | 5717 | 5815 || 5913 | 6011 | 6110 | 6208 | 6806 
 3 | 6404 | 6502 | 6600 | 6698 | 6796 || 6804 | 6992 | 7089 | 7187 | 7285 98 
 4) 4 | 7883 | 7481 | 7579 | 7676 | 7774 || 7872 | 7969 | 8067 | 8165 | 8262 
 5 | 8860 | 8458 | 8555 | 8653 | 8750 || 8848 | 8945 | 9043 | 9140 | 9287 
 6 | 9335 | 9432 | 9530 | 9627 | 9724 || 9821 | 9919 i 
 _——— | | 0016 | 0113 | 0210 
 7 | 650308 | 0405 | 0502 | 0599 | 0696 |) 0793 | 0890 | 0987 | 1084 | 1181 2 
 8} 1278 | 1375 | 1472 | 1569 | 1666 || 1762 | 1859 | 1956 | 2053 | 2150 97 
 9 | 2246 | 2348 | 2440 | 2536 | 2633 | 2730 | 2826 | 2923 | 3019 | 3116 
 | 450 | 3213 | 3809 | 3405 | 3502 | 85598 || 2695 | 8791 | 3888 | 3984 | 4080 
 | 1 4177 | 4273 | 4369 | 4465 | 4562 || 4658 | 4754 | 4850 | 4946 | 5042 
 2} 5138 | 5235 | 5331 | 5427 | 5523 || 5619 | 5715 | 5810 | 5906 | 6002 | 96 
 3 | 6098 | 6194 | 6290 | 6386 | 6482 || 6577 | 6673 | 6769 | 6864 | 6960 
 4| 7056 | 7152 | 7247 | 7343 | 7488 || 7534 | 7629 | 7725 | 7820 | 7916 
 5 | 8011 | 8107 | 8202 | 8298 | 8393 || 8488 | 8584 | 8679 | 8774 | S870 
 hie 6 | 8965 | 9060 | 9155 | 9250 | 9346 || 9441 | 9536 | 9631 9726 | 9821 
 HT 7 | 9916 | | 
 i} ————} 0011 | 0106 | 0201 | 0296 || c391 | 0486 |/0581 | o676 | O77 95 
 i 8 | 660865 | 0960 | 1055 | 1150 | 1245 |) 1339 | 1434 | 1529 | 1623 | 1718 
 } 9 1813 | 1907 | 2002 | 2096 | 2191 |r 2286 | 2380 | 2475 | 2569 | 2663 
 iil PROPORTION AL Parts. 
 | Diff.| 1 2 3 4 5 6 7 8 9 
 105) 10.5! )- 21:0 | 88181) 42.0], 52.52 15 68.0) | 738.5 1 S41 pees 
 104 | 10.4] 20.8 31.2 41.6 52.0 62.4 72 8 83.2 93.6 
 103.1103) |-5.20:6 | 80:98). 42) SRB] 6h Be) F2a4 | “Roa a oes 
 102 | 10.2 | 20.4 | 30.6 | 40.8 | 51.0 | 61.2 | 71.4 | 81.6 91.8 
 101 | 10.1] 20.2 30.3 40.4 50.5 60.6 707 | 
 { 100 | 10.0] 20.0 30.0 40.0 50.0 60.0 70 0 
 89h OD (19: 8)> | 296% 1.6 | 49.5 | 59.4 | 69.3 
 
TABLE XXIV.—LOGARITHMS OF NUMBERS. 
 
 | 
 
 I No, 460 L. 662.1 
 
 Ate 
 . 499 L. 698. | 
 
 | i 
 N | 0O 2 8 4 5 @ 8 Diff. 
 = ‘ ah | 3 = ———— 
 460 | 662758 | 2947 | 3041 | 3135 || 3220 3418 | 3512 
 1 3701 3889 | 8983 | 4078 ||. 4172 | 4360 | 4454 
 2| 4642 | 4830 | 4924 | 5018 || 5112 | 5299 | 5393 94 
 8 | 5581 5769 | 5862 | 5956 || 6050 | 6143 | 6287 | 6331 
 4 6518 | 6705 | 6799 | 6892 || 6986 | 7079 | 7173 | 7266 
 5 7453 7640 | 7733 | 7826 || 7920 | 8013 | 8106 | 8199 | 
 6 8386 | 8572 | 8665 | 8759 || 8852 | 9038 | 9131 
 ‘i 9317 | 9503 | 9596 | 9689 || 9782 | 9875 | 9967 |— 
 : ie iy | - 0060 | 98 
 8 | 670246 | 0431 | 0524 | 0617 || 0710 | 0895 | 0988 
 9 1173 | | 1358 | 1451 | 1543 || 1636 | 1728 | 1821 | 1918 
 470 | 2098 | 2283 | 2375 | 2467 || 2560 2 | 2744 | 2836 
 1} 3021 | | 3205 | 3297 | 3390 || 3482 3666 | 3758 
 2 3942 4126 | 4218 | 4310 |) 4402 4586 | 4677 | 92 
 3 | 4861 | 5045 | 5137 | 5228 || 5820 5508 | 5595 | 
 4| 5778 | | 5962 | 6053 | 6145 | 6236 6419 | 6511 
 5 | 6694 | | 6876 | 6968 | 7059 | 7151 | 7242 | 7388 | W424 | 
 6 | 7607 7789 | 7881 | 7972 || 8063 | 8154 | 8245 | 8336 
 | 7] 8518 | 8700 | 8791 | 8882 | 8973 | 9064 | 9155 | 9246 91 
 8 | 9428 | | 9610 | 9700 | 9791 | 9882 | 9973 | - | 
 | ee | 0063 | 0154 
 9 | 680336 | 0517 | 0607 | 0698 || 0789 | 0879 | 0970 | 1060 | 
 480 | 1241 | | 1422 | 1513 | 1603 || 1693 1874 | 1964 
 1 2145 | % 2326 | 2416 | 2506 || 2596 | 2686 | 2777 | 2867 
 2| 3047 37 | 8227 | 8817 | 3407 || 3497 3677 | 3767 90 
 8} 3947 | | 4127) 4217 | 4307 || 4396 4576 | 4666 
 4 | 4845 | 5025 | 5114 | 5204 | 5294 | 5473 | 5563 
 5 | 5742 | 5831 | 5921 | 6010 | 6100 || 6189 | 6368 | 6458 
 6 6636 6815 | 6904 | 6994 | 7083 | 7261 | 7851 
 Earle 7529 | 7618 | 7707 | 7796 | 7886 | 7975 | 8064 | 8153 | 8242 89 
 | 8 |) 8420 | 8509 | 8598 | 8687 | 8776 | 8865 | 8953 | 9042 | 9181 
 Big 9309 | 9486 | 9575 | 9664 | 9753 | | 9930. | 
 — | | ——) 0019 
 490 | 690196 0373 | 0462 | 0550 | 0639 | 0728 | 0816 | 0905 
 1 1031 | 1258 | 1847 | 1435 || 1524 | 1700 | 1789 
 2 1965 | 2142 | 2280 | 2318 || 2406 | 2583 | 2671 
 3 | 2847 | 3023 | 3111 | 3199 || 3287 3463, | 3551 88 
 | 4 | 3727 | 3903 | 3991 | 4078 || 4166 4342 | 4430 
 5 | 4605 4731 | 4868 | 4956 || 5044 | 5219 | 5307 | 
 6 5482 | | 5657 |. 5744 | 5882 || 5919 6094 | 6182 
 7 | 6356 | 6531 | 6618 | 6706 || 6793 6968 | 7055 | 
 8 7229 | 7404 | 7491 | 7578 || 7665 7839 | 7926 
 9 | 8100 | 8275 | 8362 | 8449 || 8535 | 8709 | 8796 87 
 | } 
 PROPORTIONAL PARTS. 
 | 
 lela] 2 fs] 4s 
 98 | 9.1 | 29.4 | 80.2 | 49.0 | 58.8 | 68.6 8. 88.2 | 
 i ae ey 29.1 38.8 48.5 58.2 67.9 Wop 87.3 | 
 96 *; 9.6 | 28.8 38.4 48.0 57.6 67.2 6. 86.4 | 
 95. | 9.5 28.5 38.0.| 47.5 57.0 | 66.5 6 85.5 
 94 9 23.2 37.6 47.0 56.4 65.8 $4.0 | 
 , 93 9.3 L279 37.2 -|—46.5 55.8 65.1 83.4 
 92 9.2 | 27.6 | 36.8 46.0 55.2 64.4 82.8 
 91 | 9.1 Pee 8) Ao. 4 ous 4o.b 54.6 63.7 81.9 
 90 | 9. | 27.0'-| 86.0 | 45.0 | 54.0 | 63.0 81.0 
 89 | 8.9 | 26.7 | 35.6 | 44.5 53.4 32.3 80.1 
 88 8.8 26.4 | 35.2 44.0 52.8 61.6 79.2 
 | | 94:8 3.5 | 52.2 60.9 78-3 
 \evd4ia 0 51.6 f 0% 4 
 
 | & 
 vo 
 
TABLE XXIV.—LOGARITHMS OF NUMBERS. 
 
 : 
 
 No, 500 L. 698.] [No. 544 L, 736. 
 
 or 
 
 Nala 0% |) ig | 8 | | aed 6 | 7 | 8 | 9 | pi. 
 | | | 
 
 500 | 698970 | 9057 | 9144 | 9231 | 9817 || 9404 | 9491 | 957 | 9664-| 9751 
 1 | 9888 | 9924 | | | | _| | 
 | 0011 | 0098 | 0184 | 0271 | 0358 | 0444 | 0531 | 0617 | 
 
 (0 6) 
 
 2 | 700704 | 0790 | 0877 | 0963 | 1050 || 1136 | 1222 13809 | 1895. | 1482 | 
 
 3} 1568 | 1654 | 1741 | 1827 | 1913 || 1999 | 2086 R172 | 2258 | 2344 | 
 
 4 | 2431 | 2517 | 2603 | 2689 | 2775 || 2861 | 2947 | 3083 | 3119 | 8205 | 
 
 S| 38291 | 3377 | 3463 | 3549 | 3635 || 3721 | 3807 3893 | 3979 | 4065 86 
 6 4151 | 4236 | 4822 | 4408 | 4494 || 4579 | 4665 | 4751 | 4837 | 4922 
 
 7 5008 | 5094 | 5179 | 5265 | 5350 || 5436 | 5522 | 5607 | 5693 | 5778 
 
 8 5864 | 5949 | 6035 | 6120 |- 6266 || 6291 | 6376 | 6462 | 6547 | 6632 
 
 9 6718 | 6803 | 6888 | 6974 | 7059 || 7144 | 7229 | 7315 | 7400 | 7485 
 
 HB 510 7570 | 7655 | 7740 | 7826 | T9114 | 7996 | 8081 | 8166 | 8251 | 8336 85 
 } 1 8421 | 8506 | 8591 | 8676 8761 || 8846 | 8931 | 9015 | 9100 9185 
 i 2 9270 | 9855 | 9440 | 9524 | 9609 || 9694 | 9779 | 9863 | 9948 | — 
 | | _|| | 0033 
 3 | 710117 | 0202 | 0287 | 0871 | 0456 | 0540 | 0625 | 0710 | 0794 | 0879 
 i 4 0963 | 1048 | 1132 | 1217 | 1301 1885 | 1470 | 1554 | 1639 | 1793 
 j 5 1807 | 1892 | 1976 | 2060 | 2144 2229 | 2313 | 2397 | 2481 | 2566 
 6 2650 | 2734 | 2818 | 2902 | 2986 8070 | 3154 | 8288 | 3323 | 3407 84 
 ff 3491 | 3575 | 3659 | 3742 | 3896 | 3910 | 3994 | 4078 | 4162 | 4246 : 
 8 4330 | 4414 | 4497 | 4581 | 4665 4749 | 4883 | 4916 | 5000 | 5084 
 9 5167 | 5251 | 5335 | 5418 | 5502 || 5586 5669 753 | 5886 | 5920 
 520 6003 | 6087 | 61% 6254 | 6337 || 6421 | 6504 | 6588 | 6671 754 
 1 6838 | 6921 | 1004 | 7088 | 7171 | 7254 | 73888 | 7421 | 7504 | 7587 
 2 7671 | 7754 | 7837 | 7920 | 8003 || 8086 | 8169 | 8253 | 8336 | 8419 83 
 3 8502 | 8585 | 8668 | 8751 | 8834 || 8917 | 9000 | 9083 | 9165 | 9248 : 
 4 9331 | 9414 | 9497 | 9580 | 9663 || 9745 | 9828 | 9911 | 9994 | 
 | | | | 0077 
 5 | 720159 | 0242 | 0325 0407 | 0490 || 057% | 0655 | 0788 | 0821 | 0903 
 6 0986 | 1068 | 1151 | 123% 1316 || 1898 | 1481 | 1563 | 1646 1728 
 ff 1811} 1893 | 1975. | 2058 | 2140 | 2222 | 23805 | 2387 | 2469 | 9559 
 8 2634 | 2716 | 2798 | 2881 | 2963 || 8045 | 3127 | 3209 | 3291 | 3974 
 9 3456 | 2588 | 3620 | 3702 | 3784 || 3866 | 3948 | 4080 | 4112 | 4194 82 
 
 530 4276 | 4358 | 4440 | 4522 | 4604 || 4685 | 4767 | 4849 4951 | 5013 
 5095 | 5176 | 5258 | 5340 | 5422 || 5503 | 5585 | 5667 | 5748 | 5830 
 2 5912 | 5993 | 6075 | 6156 | 6238 || 6320 | 6401 | 6483 6564 | 6646 
 3 6727 | 6809 | 6890 | 6972 | 7053 || 7134 | 7916 7297 | 7379 | 7460 
 4 (41 | 7623 | 7704 | 7785 | 7866 || 7948 | 8029 | 8110 8191 | 827 
 5 8354 | 8435 | 8516 | 8597 | 8678 || 8759 | 8841 | So29 9003 | 9084 
 6 9165 | 9246 | 9327 | 9408 | £489 || 9570 | 9651 | 9732 .9813 | 9898 81 
 7 9974 / 
 
 = 
 ~) 
 S) 
 
 0055 | 0186 | 0217 | 0298 || 0378 | 0459 | 0540 | 062 “02 
 8 | 780782 | 0863 | 0944 | 1024 | 1105 || 1186 | 1266 1347 | 1428 | 1508 
 9 1589 | 1669 | 1750 | 1830 | 1911 {| 1991 | 2072 | 2152 | 2238 | 2313 
 
 iH 540 | =. 2394 | S474 | 2555 | 2635 | 2715 || 2796 | 2876 | 2956 | 3037 | 3117 
 ana To eae 3197 | 8278 | 3358 | 3438 | 8518 || 3598 | 3679 | 3759 | 3889 3919 
 
 2 3999 | 4079 | 4160 | 4240 | 4320 || 4400 | 4480 | 4560 | 4640 | 4720 80 
 3 4800 | 4880 | 4960 | 5040 5120 || 5209 | 5279 | 5359 | 5489 | 5519 
 Til 4 5599. | 5679 | 5759 | 5838 | 5918 | 5998 | 6078 | 6157 | 6237 | 6317 
 
 PROPORTIONAL PARTS. 
 
TABLE XXIV.—LOGARITHMS OF NUMBERS. 
 
 r a a a a 
 No. 545 L. 786.] LNo, 584 L. 767. 
 
 mi Of Fe paps |e] s pe pa) s | ov pig 
 
 == | 
 
 545 | 736397 | 6476 | 6556 | 6635 | 6715 | 6795 | 6874 | 6954 | 7034 | 7113 | 
 | 6) 7193 | 7272 | 7352 | 7431 | 7511 || 7590 | 7670 | 7749 | 7829 | 7908 | 
 7987 | 8067 | 8146 | 8225 | 8305 || 8384 | 8463 | 8543 | 8622 | S101 | 
 8 8781 | 8860 | 8939 | 9018 | 9097 || 9177 | 9256 | 9335 | 9414 | 9493 | 
 9572 | 9651 | 9731 | 9810 | 9889 || 9968 | oe a 
 | | 0047 | 0126 | 0205 | 0284} 79 
 | 550 740363 | 0442 0521 | 0600 | 0678 || 0757 | 0886 | 0915 | 0994 | 1073 
 1 | 11527] 1230 | 1309 | 1388 | 1467 || 1546 | 1624 | 1703 | 1782 | 1860 | 
 2) 1939 | 2096 | 2175 | 2254 || 2332 | 2411 | 2489 | 2568 | 2647 | 
 3| 2725 | 2804 | 2882 | 2961 | 3039 || 3118 | 3196 | 3275 | 3353 | 3431 iM 
 4 3510 | 3588 | 3667 | 3745 | 3823 || 3902 | 38980 | 4058 | 4136 | 4215 t 
 5 | 4293 | 4371 | 4449 | 4528 | 4606 || 4684 | 4762 | 4840 | 4919 | 4997 | 
 6 | 5075 | 5153 | 5231 | 5809 | 5387 || 5465 | 5543 | 5621 | 5699 | 5777 | 78 | 
 7 | 5855") 5933 | 6011 | 6089 | 6167 |) 6245 | 6323 | 6401 | 6479 | 6556 . 
 8 | 6634 | 6712 | 6790 | 6868 | 6945 || 7023 | 7101 | 7179 | 7256 | 7334 
 9 | 7412 | 7489 | 7567 | 7645 | 7722 || 7800 | 7878 | 7955 | 8038 | 8110 
 560 | 8188 | 8266 | 8343 | 8421 | 8498 || 8576 | 8653 | 8731 | 8808 | 8885 
 1} 8963 | 9040 | 9118 | 9195 | 9272 || 9350 | 9427 | 9504 | 9582 | 9659 | 
 2| 9736 | 9814 | 9891 | 9968 | 
 
 2 
 = 
 (e 6) 
 
 | ee | 0045 || 0123 | 0200 | 0277 | 0354 | 0431 
 3 | 750508 | 0586 | 0663 | 0740 | 0817 || 0894 | 0971 | 1048 | 1125 | 1202 | Hi 
 4} 1279 | 1856 | 1433 | 1510 | 1587 |, 1664 | 1741 | 1818 | 1895 | 1972 | os i 
 5 2048 | 2125 | 2202 | 2279 | 2356 || 2483 | 2509 | 2586 | 2663 | 2740 ‘ . 
 6 | 2816 | 2893 | 2970 | 8047 | 3123 |! 8200 | 8277 | 8853 | 3430 | 3506 { 
 7 | 3583 | 3660 | 3736 | 3813 | 8889 |, 8966 | 4042 | 4119 | 4195 | 4272 ; 
 8 | 4848 | 4425 | 4501 | 4578 | 4654 || 4730 | 4807 | 4883 | 4960 | 5036 | 
 9 5112 | 5189 | 5265 | 5841 | 5417 || 5494 | 5570 | 5646 | 5722 | 5799 i 
 
 570 | 5875 | 5951 | 6027 | 6103 | 6180 |; 6256 | 6882 | 6408 | 6484 | 6560 i 
 1 | 6636 | 6712 | 6788 | 6864 | 6940 || 7016 | 7092 | 7168 | 7244 | 7320 16 i 
 2| 7896 | 7472 | 7548 | 7624 | 7700 || 7775 | 7851 | 7927 | 8003 | 8079 t 
 3 8155 | 8230 | 8306 | 8382 | 8458 || 8533 | 8609 | 8685 | 8761 | 8836 i 
 4 8912 | 8988 | 9063 | 9139 | 9214 || 9290 | 9866 | 9441 | 9517 | 9592 i 
 5 | 9668 | 9743 | 9819 | 9894 | 9970 | 
 
 | 0045 | 0121 | 0196 | 0272 | 03847 
 
 6 | 760422 | 0498 | 0573 | 0649 | 0724 | 0799 0875 | 0950 | 1025 | 1101 , 
 7 | 4176 | 1251 | 1826 | 1402 | 1477 || 1552 | 1627 | 1702 | 1778 | 1853 | 
 8 1928 | 2003 | 2078 | 2153 | 2228 |! 2808 | 2878 | 2458 | 2529 | 2604 | a» 
 9} 2679 | 2754 |.2829 | 2904 | 2978 || 3053 | 3128 | 3203 | 3278 | 3358, 
 580 | 8428 | 3503 | 8578 | 3653 | 3727 || 3802 | 3877 | 3952 | 4027 | 4101 
 1| 4176 | 4251 | 4326 | 4400 | 4475 || 4550 | 4624 | 4699 | 4774 | 4848 i 
 2 | 4923 | 4998 | 5072 | 5147 | 5221 || 5296 | 5870 | 5445 | 5520 | 5594 l 
 3 | 5669 | 5743 | 5818 | 5892 | 5966 || 6041 | 6115 | 6190 | 6264 | 6838 i 
 4| 6413 | 6487 | 6562 | 6636 | 6710 || 6785 | 6859 | 6933 | 7007 | 082 H 
 | | | 
 PROPORTIONAL PARTS, 
 pi,| i | 2 3 | 4 5 6 4 8 9 
 | | | 
 §3 83 16.6 | (24.9 | 33.2 41.5 49.8 58.1 66.4 (4.7 
 R2 Spor 16.4. |t-24.Gi al sone 41.0 49.2 57.4 65.6 73.8 | 
 81 Ae oy eS SP B24 40.5 48.6 56.4 64.8 2.9 
 80 8.0| 16.0 24.0 32.0 40.0 48.0 56.0 64.0 72.0 
 Bee Or) 158812) 28; Ar |i" Blues! a9eGen | 47,4 5D. 63.2° | 74:1 
 7 | 7.8) 1516 23.4 31.2 | 39.0 46.8 54.6 62.4 (0.2 
 me | 7.7 | 15.4 23.1 30.8 | 38.5 46.2 53.9 61.6 | 69.3 
 “6 | 7.6-| 15.2 | 22.8 | 30.4 | 388.0 | 45.6 53.2 | 60.8 | 68.4 
 fo-75 |. %.5a|. 15.0 22.5. | 30.0 | 37.5 | 45.0 52.5 60.0 v5 
 | 7% | v4 | 14.8 22.2 | 29.6 37.0 14.4 51.8 | 59.2 | 66.6 
 
or 
 
 or 
 
 for) 
 
 62 
 
 TABLE XXTV. 
 
 LOGARITHMS OF NUMBERS. 
 
 8651 | 
 
 8720 
 
 8996 
 
 No. 585 L. 767.] 
 prueioe | 2.) ab e*] sa oe wi wer 
 | | | 
 85 | 767156 | 7230 | 7304 | 7379 | 7453 || 7527 | 7601 | 7675 | 7749 
 6 | 7898 | 7972 | 8046 8120 | 8194 8268 8342 | 8416 | 8490 
 1 8638 | 8712 8786 8860 | 8934 || 9008 | 9082 | 9156 | 9230 
 8 | 9877 | 9451 | 9525 9599 | 9673 || 9746 | 9820 | 9894 | 9968 
 9 | 770115 | 0189 0263 | 0336 | 0410 | 0484 0557 | 0631 0705 
 90 0852 | 0926 | 0999 | 1073 | 1146 1220 1293 | 1367 | 1440 
 1 1587 | 1661 | 1734 | 1808 | 1881 || 1955 | 2028-) 2102 | 2175 
 2 2399 | 2395 | 2468 | 2542 | 2615 | 2688 | 2762 | 2835 | 2908 
 3 8055 | 3128 | 320i | 3274 | 3348 || 3421 | 3494 | 3567 | 3640 
 4! 8786 | 3860 | 3933 | 4006 | 4079 || 4152 | 4225 | 4298 | 4371 | 
 5 | 4517 | 4590 | 4663 | 4736 | 4809 | 4882 | 4955 | 5028 | 5100 | 
 6 | 5246 | 5819 | 5392 | 5465 | 5538 | 5610 | 5683 | 5756 | 5829 | 
 7| 5974 | 6047 | 6120 | 6193 | 6265 | 6388 | 6411 | 6483 | 6556 | 
 8 | 6701 | 6774 | 6846 | 6919 | 6992 || 7064 | 7187 | 7209 | 7282 
 9| 7427 | 7499 | 7572 | 7644 | 7717 | 7789 | 7862 | 7934 | 8006 
 00 | 8151 | 8224 | 8296 | 83868 | 8441 | 8513 | 8585 | 8658 | 8730 
 1| 8874 | 8947 | 9019 | 9091 | 9163 | 9286 | 9308 | 9380 | 9452 | 
 2! 9596 | 9669 | 9741 | 9813 | 9885 |) 9957 |- : 
 . | 0029 | 0101 | 0173 | 
 3 | 780317 | 0389 | 0461 | 0533 | 0605 || 0677 | 0749 | 0821 | 0893 
 4 1037 | 1109 | 1181 | 1253 | 1324 || 1396 | 1468 | 1540 | 1612 
 5 | 1755 | 1827 | 1899 | 1971 | 2042 || 2114 | 2186 | 2258 | 2329 
 6 | 2473 | 2544 | 2616 | 2688 | 2759 || 2831 | 2902 | 2974 | 3046 
 7 | 8189 | 8260 | 3332 | 3403 | 3475 || 3546 | 3618 | 3689 | 3761 
 8 | 3904 | 8975 | 4046 | 4118 | 4189 |! 4261 | 4332 | 4408 | 4475 
 9 {| 4617 | 4689 | 760 | 4831 | 4902 | 4974 | 5045 5116 | 5187 
 10 | 5380 | 5401 | 5472 | 5543 | 5615 || 5686 | 5757 | 5828 | 5899 | 
 1 6041 | 6112 | 6183 | 6254 | 6325 || 6396 | 6467 | 6538 | 6609 | 
 2| 6751 | 6822 | 6893 | 6964 | 7035 || 7106 | 7177 | 7248 | 7319 | 
 8 | 7460 | 7531 | 7602 | 7673 | 7744 |) 7815 | 7885 | 7956 | 8027 | 
 4| 8168 | 8239 | 8810 | 8881 | 8451 || 8522 | 8598 | 8663 | 8734 | 
 5 8875-| 8946) 9016 | 9087 | 9157 || 9228 | 9299 | 9369 | 9440 
 6 | 9581 | 9651 | 9722 | 9792 | 9863 || 9933. | | | 
 | | | | | 0004 | 0074 | 0144 
 7 | 790285 | 0356 | 0426 | 0496 | 0567 || 0637.) 0707 | 0778 | 0848 | 
 8 | 0988 | 1059 | 1129 | 1199 | 1269 || 1840 | 1410 | 1480 | 1550 
 9 1691 | 1761 | 1831 | 1901 | 1971 || 2041 | 2111 | 2181 | 2252 | 
 20 | 2392 | 2462 | 2582 | 2602 | 2672 | 2742 | 2812 | 2882 | 2952 | 
 1| 3092 | 3162 | 3231 | 3801 | 3371 || 8441 | 3511 | 3581 | 3651 | 
 2] 3790 | 3860 | 3930 | 4000 | 4070 || 4189 | 4209 | 4279 | 4349 
 3] 4488°| 4558 | 4627 | 4697 | 4767 || 4886 4906 | 4976 | 5045 | 
 4| 5185 | 5254 | 5824 | 5893 | 5463 || 5532 | 5602 | 5672 | 5741 | 
 5 | 5880 | 5949 | 6019 | 6088 | 6158 || 6227 | 6297 | 6366 | 6436 | 
 6 | 6574 | 6644 | 6713 | 6782 | 6852 || 6921 | 6990 | 7060 | 7129 | 
 7 | 7268 | 7337 | 7406 | 7475 | 7545 || 7614 | 7683 | 7752 | 7821 
 8 | 7960 | 8029 | 8098 | 8167 | 8236 | 8305 | 8374 | 8443 | 8513 | 
 9 | | 8858 | 8927 | 9065 | 9134 9203 | 
 
 9 
 
 7823 
 8564 
 9303 
 
 0042 
 
 0778 
 
 1514 
 2248 
 2981 
 
 | 8713 
 
 4444 
 5173 
 5902 
 6629 
 7354 
 8079 
 8802 
 9524 
 
 9245 
 0965 
 1684 
 2401 
 3117 
 3832 
 4546 
 5259 
 5970 
 6680 
 7390 
 8098 
 8804 
 9510 
 
 0215 
 0918 
 
 | 1620 
 
 2322 
 3022 
 
 721 
 4418 
 5115 
 5811 
 6505 
 7198 
 7890 
 8582 
 
 9272 
 
 | 
 | 
 | 
 | 
 
 73 
 
 oI 
 
 70 
 
 PROPORTIONAL PARTS. 
 
 | DOMwOm ET 
 
 | ROWROWEX 
 
 ee BR OTOTort or 
 DODO 
 WOH OL 
 
 [No. 629 L. 799. | 
 
TABE 
 
 EK 
 
 | No. 630 L. 799.] 
 
 XXIV.—LOGARITHMS OF 
 
 NUMBERS. 
 
 [No. 674 L. 829. 
 
 [N-| 0 | a" as 5 fae fa | ep. sal pim 
 630 | 799341 | 9409 | 9478 | 9547 | 9616 || 9685 | 9754 | 9823 | 9892 | 9961 | 
 1 | 800029 | 0098 | 0167 | 0236 | 0305 || 0373 | 0442 | 0511 | 0580 | 0648 | 
 2 0717 | 0786 | 0854 | 0923 | 0992 |) 1061 | 1129 | 1198 | 1266 | 1335 | 
 3 | 1404 | 1472 | 1541 | 1609 | 1678 ||'1747 | 1815 | 1884 | 1952 | 2021 | 
 ee fate | Y 1 Ole OC | JON | 2021 | 
 | 2089 pe | att | oe nite | 2432 | 2500 2568 | 2637 | 2705 
 G | Bis7 | 35e5 | asgd | S002 | 3730 |) avas.| ssor | 3085 | 4008 | sock | 
 ; | 4139 | 4208 | 1976 | 4344 41412 || ri rit 3935 | 4003 | 4071 | 
 | Oc Ad wid | € rs) } ¢ C | 5 8 ) > | 4685 | V5E i 
 § | 4821 | 4880 | 4057 | 5025 | 5093 |) 5161 | 229 | 5207 | 5365 | 54133 | 68 
 9 | 5501 | 5569 | 5637 | 5705 | 5773 || 5841 | 5908 | 5976 | 6044 | 6112 | 
 640 | 806180 | 6248 | 6316 | 6384 | 6451 || 6519-| 6587 | 6655 | 6723 | 6790 | 
 OP] 1 Gia | 6025 OUD: | TORE, | TAG: || AE) rank | RL | TAO | 7a | 
 3 | Gort | S270 | B10 | Sd | SiBt || S540 | 8616 | Sost | Brot | sHi8 | 
 Ke Owtle oO xe | : Ax 2 
 4 | 8886 | 8953 | 9021 | 9088 | 9156 || 9223 | 9290 0358 495 592 
 5 | 9560 | 9627 | 9694 | 9762 | 9829 || 9896 | 9964 eal 
 -——| - -——— | —| 0031 | 0098 | 0165 
 6 | 810233 | 0300 0367 | 0434 | 0501 || 0569 | 0636 | 0703 | 0770 | 0837 
 7 | 0904 | 0971 | 1039 | 1106 | 1173 || 1240 | 1307 | 1374 | 1441 | 1508 | @7 
 8 | 1575 | 1642 | 1709 | 1776 | 1843 |) 1910 1977 | 2044 | 2111 | 2178 
 9} 2245 | 2312 | 279 | 245 | 212 |) V9 | 2646 | 2713 | 2780 | 247 
 | | 650 | 2918 | 2980 | 3047 | 3114 | 3181 || go47 | 3314 | 3881 | 3448 | 2514 
 | a) eee) aoe | aber | daar | dei || aoe | aoee | aria | 4700 | 4817. 
 | 2 248 | 4314 | 4381 7 | 4514 || 4581 | 4647 | 4714 | 4780 | 4847 
 | 3 4913 | 4980 | 5046 | 5113 | 5179 || 5246 | 5312 | 5378 | 5445 | 5511 
 | $ Poe pi ae aR oa || 5910 596 6042 | 6109 | 6175 
 | 32 DBU0 é | : ID | 657% 3638 705 | 6771-1 68 
 6 |. 6904 | 6970 | 7036 | 7102 | 7169 || ot 7301 fae? | 7433 | 7409 
 | Bye nae 0070: | FONG: 1s 10a | CGR Pee ra ORE alls 
 | : | = ee ree ie — : 2: ee ae | 8094 | 8160 
 8 | 8226 | 8292 | 8358.| 8424 | 8490 || 8556 | 8622 | 8688 | 8754 | 8820 | 
 | 9 | 8885 | 8951 | 9017 | 9083 | 9149 || 9215 | 9281 | 9346 o4t2 | 9478. |. % 
 | | 660 | 9544 | 9610 | 9676 | 9741 | 9807 || 9873 | 9939 gar ti 
 ¥ | _ —|———| 0004 | 0070 | 0136 
 | 1 | 820201 | 0267 0383 | 0399 | 0464 || 0530 | 0595 | 0661 | 0727 | o702 
 2| 0858 | 0924 | 0989 | 1055 | 1120 || 1186 | 125 317 | 138% 18 
 a] asd | 1570| 1048 vn0-| 1775 || 1841. | 1906 jo73-| 2087 | 2108 
 4 | 2168 | 2233 | 2299 | 2364 | 2430 || 2495 one La DO ehee 
 | £ | SES5 | Sat | dose | Sore | B00 || 3148 | aera | der9 | 33d | S400 
 6 3474 | 3539 ae be Z JK BU69 3 Lat ¢ 213 8279 3344 | 3409 
 | 6 3474 | 3538 3605 3670 | 3735 || 3800 | 3865 | 3930 | 3996 | 4061 
 I | Z| 4126 | 4191 | 4256 | 4821 | 4386 |) 4451 | 4516 | 4581 4646 | 4711 | 
 | 8 | 4776 | 4841 | 4906 | 4971 | 5036 |) 5101 | 5166 | 5231 5296 | 5361 | © 
 9 | 5426 5491 | 5556 | 5621 | 5686 |) 5751 | 5815 | 5880 | 5945 | 6010 
 | 670 | 6075 | 6140 | 6204 | 6269 | 6334 || 6399 | 6464 | 6528 | 6593 | 6658 
 1| 6723 | 6787 | 6852 | 6917 | G9BL |) 7046 | 7111 | 7175 | 7240 | 7305 
 2| 7369 | 7434 | 7499 | 7563 | 7628 || 7692 | 7757 | 7821 | 7886 | 7951 
 3B} S015 | 8080 | 8144 | 8209 | 8273 | 8338 | 8402 | 8467 | 8581-) 8595 
 4 | 8660 | 8724 | 8789 | 8853 | 8918 |) 8982 | 9046 | 9111 | 9175 | 9239 
 | PROPORTIONAL PARTS. 
 | Diff. | 1 2 a am ee 5 6 q 8 9 
 se = 5 ; saya as 
 | 68 | 6.8} 18.6 | 20.4 | 27.2 34.0 | 40.1 47.6 | 54.4 | 61.2 
 / 6% O.4 13.4 20.1 | 26.8 33.5 40.2 46.9 58.6 60.3 
 | | 66 | 6.6 13.2 | 19.8 | 26.4 | 33.0 30.6 | 46.2 | 52.8 | 59.4 
 | 6.5 3. 9.5} 26.0 32.5 389.0 Vo 52 58.5 
 | 64 | 64] 12.8 | 19.2 | 25.6 32.0 | 38.4 4.8) 51.2 | 57.6 | 
 
 © 
 
 cr 
 
 fom) 
 
TABLE XXIV.—-LOGARITHMS OF NUMBERS. 
 
 No. 675 L. 829.] [No. 719 L. 857. 
 pe 
 woe Pop ita fee al ot ere ig AP ome 
 . | bree | H | | 
 | | | 
 675 | 829304 | 9368 | 9432 | 9497 | 9561 || 9625 | 9690 | 9754 | 9818 | 9882 
 6 9947 | = | Wecricdineel am icad Stee Saree 
 ————| 0011 | 0075 | 0189 | 0204 |} 0268 | 0382 | 0896 | 0460 | 0525 | 
 @ | 8380589 | 0653 | 0717 | 0781 | 0845 |) 0909 | 0973 | 1087 | 1102 | 1166 | 
 8 1230 | 1294 | 1358 | 1422 | 1486 || 1550 | 1614 | 1678 | 1742 | 1806 | 64 
 9 1870 | 1984.) 1998 | 2062 | 2126 || 2189 | 2253 | 2817 | 2381 | 2445 | 
 380 2509 | 2573") 2637 | 2700 | 2764 || 2828 | 2892 | 2956 |. 3020 | 3083 | 
 1 8147 | 3211 | 3275 | 3338 | 3402 || 3466 | 3530 | 3593 | 3657 | 3721 | 
 2 8/84 | 3848 | 3912 | 3975 | 4039 |; 4103 | 41€6 | 4230 | 4294 | 4357 | 
 2 4421 | 4484 | 4548 | 4611 | 4675 |) 4739 | 4802 | 4866 | 4929 | 4993 | 
 4 5056 | 5120 | 5183 | 5247 | 5310 |) 5873 | 5487 | 5500 | 5564 | 5627 | 
 5 5691 | 5754 | 5817 | 5881 | 5944 || 6007 | 6071 | 6134 | 6197 | 6261 | 
 6 6324 | 6887 | 6451 | 6514 | 6577 || 6641 | 6704 | 6767 | 6830 | 6894 | 
 ie 6957 | 7020 | 7083 | 7146 | 7210 |) 7273 | 7336 | 7399 | 7462 | 7525 | 
 8 88 | 7652 | 7715 | 7778 | 7841 || 7904 | 7967 | 8030 | 8093 | 8156 | 
 9] 8219 | 8282 | 83845 | 8408 | 8471 || 8534 | 8597 | 8660 | 8723 | 8786 | 68 
 690 8849 | 8912 | 8975 | 9038 | 9101 || 9164 | 9227 | 9289 | 9352 | 9415 
 1 | 9478 | 9541 | 9604 | 9667 | 9729 || 9792 | 9855 | 9918 ) 9981 beer 
 | | | | . | 0043 
 2 | 840106 | 0169 | 0232 | 0294 | 0357 || 0420 | 0482 | 0545 | O608 | 0671 | 
 3 0733 | 0796 | 0859 | 0921 | 0984 || 1046 | 1109 | 1172 | 1284 | 1297 
 4} 1359 | 1422 | 1485 | 1547 | 1610 || 1672 | 1735 | 1797 | 1860 | 1922 
 5 | 1985 | 2047 | 2110 | 2172 | 2235 || 2297 | 2360 | 2422 | 2484 | 2547 
 6 | 2609 | 2672 | 2734 | 2796 | 2859 || 2921 | 2983 | 3046 | 3108 | 3170 
 7 | 82383 | 8295 | 3357 | 3420 | 3482 || 3544 | 3606 | 3669 | 3731 | 3793 
 8 | 8855 | 3918 | 3980 | 4042 | 4104 |) 4166 | 4229 | 4291 | 4353 | 4415 
 9 | 77 | 4539 | 4601 | 4664 | 4726 || 4788 | 4850 | 4912 | 4974 | 5036 
 700 | 5098 | 5160 | 5222 | 5284 | 5346 || 5408 | 5470. | 5532 | 5594 | 5656 62 
 1 5718 | 5780 | 5842 | 5904 | 5966 || 6028 | 6090 | 6151 | 6213 | 6275 
 2 | 6837 | 6399 | 6461 | 6523 | 6585 | 6646 | 6708 | 6770 | 6832 | 6894 
 3 | - 6955 | 7017 | 7079 | 7141 | 7202 || 7264 | 7326 | 7388 | 7449 | 7511 
 4 1573 | 7634 | 7696 | 7758 | 7819 | 7881 | 7943 | 8004 | 8066 | 8128 
 5 | 8189 | 8251 | 8312 | 8374 | 8435 | 8497 | 8559 | 8620 | 8682 | 8743 
 6 | 8805 | 8866 | 8928 | 8989 | 9051 || 9112 | 9174 | 9285-| 9297.) 9858 
 7 | 9419 | 9481 | 9542 | 9604 | 9665 | 9726 | 9788 | 9849 | 9911 | 9972 
 8 | 850033 | 0095 | 0456 | 0217 | 0279 || 0340 | 0401 | 0462 | 0524 | 0585 
 9 | 0646 | 0707 | 0769 | 0880 | 0891 | 0952 | 1014 | 1075 | 1136 | 1197 
 710 | 1258 | 1820 | 1881 | 1442 | 1503 || 1564 | 1625 | 1686 | 1747 | 1809 
 1 | 1870 | 1931 | 1992 | 2053 | 2114 || 2175 | 2286 | 2297 | 2358 | 2419 
 2 | 2480 | 2541 | 2602 | 2663 | 2724 || 2785 | 2846 | 2907 | 2968 | 3029 61 
 3; 8090 | 3150 | 8211 | 3272 | 3333 || 3394 | 3455 | 3516 | 3577 | 3637 
 4 3698 | 3759 | 3820 | 3881 | 3941 || 4002 | 4063 | 4124 | 4185 | 4245 
 5 | 4306 | 4367 | 4428 4488 | 4549 | 4610 | 4670 | 4731 | 4792 | 4852 
 6 | 4913 | 4974 | 5034 | 5095 | 5156 || 5216 | 5977 | 5337 | 5398 | 5459 | 
 7 5519 | 5580 | 5640 | 5701 | 5761 | 5822 | 5882 | 5943 | 6003 | 6064 
 8 6124 | 6185 | 6245 | 6806 | 6366 | 6427 | 6487 | 6548 | 6608 | 6668 
 9 | 6729 | 6789 | 6850 | 6910 | 6970 || 7031 | 7091 | 7152 | 7212 | 7272 
 | | | | 
 PROPORTIONAL PARTS. 
 - Nl j 
 Diff. | 1 2 3 4 | 5) 6 q 8 9 
 65 | 6.5 | 13.0 19.5 | 26.0 | 32.5 39.0 45.5 52.0 58.5 
 64. | 6.4] 12.8 19.2 25.6 | 32.0 38.4 44.8 51.2 57.6 | 
 63 6.3 | 12.6 | 18.9 25.2 | 81.5 | 387.8 44.1 50.4 56.7 
 62 6.2 | 12.4 18.6 24.8 | 31.0 | 37.2 | 43.4 49.6 55.8 
 61 | 6.1] 12.2 18.3 24.4 | 30.5 36.6 2.7 48.8 54.9 
 60 6.0} 12.0 | 18.0 | 24.0 | 30.0 36.0 | 42.0 48.0 | 54.0 
 
 a a ee ee ee 
 
 ee 
 
 | 
 | 
 
 Tee 
 
 ———<—— 
 
 L 
 
2 |2| 
 
 TABLE XXIV.—LOGARITHMS OF 
 
 NUMBERS. 
 
 [No. 764 L. 883. 
 
 No, 720 L. 857.] 
 
 6 
 
 9 | Diff. 
 
 8297 
 
 9499 
 
 7694 
 
 8417 | 8477 
 9018 | 9078 
 9619 | $679 60 
 
 8898 
 
 5 | (875 
 
 0098 
 0697 
 1295 
 1893 
 2489 
 3085 
 3680 
 4274 
 4867 
 5459 
 6051 
 6642 
 7282 
 7821 
 8409 
 8997 
 9584 
 
 0170 
 
 15D 
 1339 
 1923 
 2506 
 3088 
 3669 
 4250 
 4830 
 5409 
 5987 
 6564 
 7141 
 
 0218 | 0278 
 0817 | 0877 
 1415 | 4475 
 2012 | 2072 
 2608 | 2668 
 3204 | 3263 
 99 | 3858 
 4392 | 4452 
 4985 | 5045 
 5578. | 5687 
 6169 | 6228 
 6760 | 6819 
 350 | 7409 | 99 
 939 | 7998 
 8527 | 8586 | 
 9114 | 9113 
 
 01 ; 9760 
 
 287 | 0345 
 72 | 09380 
 1456 | 1515 
 2040 | 2008 
 2622 | 2681 
 8204 | 38262 
 85 | 3844 
 
 | 5524 | 5582 
 6102 | 6160 
 6680 | 6737 
 D6 | 7314 
 832 | 7889 
 8407 | 8464 
 8981 | 9039 
 9555 | 9612 
 
 0127 | 0185° 
 
 4366 | 4424 58 
 4945 | 5003 
 
 0699 | 0756 
 1271 | 1328 
 1841 | 1898 2 
 2411 | 2468 | 5% 
 2980 | 3037 
 3548 | 3605 
 
 0 1 | 2 
 
 857332 | 7393 | 7453 
 7935 | 7995 | 8056 
 8537 | 8597 | 8557 
 9138 | 9198 | 9258 
 
 9739 | 9799 | 9859 | 
 5 | 860338 | 0398 | 0458 
 6 0937 | 0996 | 1056 
 i 1534 | 1594 | 1654 
 8 2131 | 2191 | 2251 
 9 |. 2728 | 2787 | 2847 
 730 | 3323 | 8382 | 3442 
 1} 8917 | 38977 | 4036 
 21 4511 | 4570 | 4630 
 3 5104 | 5163 | 5222 
 4| 5696 | 5755 | 5814 
 5 6287 | 6346 | 6405 
 6 | 6878 | 69387 | 6996 
 %| 7467 | 7526 | 7585 
 8 | 8056 | 8115 | 8174 
 9 | 8644 | 8703 | 8762 
 740 | 9232 | 9290 | 9349 
 1| 9818 | 9877 | 9935 
 2 | 870404 | 0462 | 0521 
 3 0989 | 1047 | 1106 
 4 1573 | 1631. | 1690 
 5 2156 | 2215. | 2273 
 6 9739 | 2797 | 2855 
 va 3321 | 3379 | 3437 
 8 3902 | 3960 | 4018 
 9 | 4482 4540 | 4598 
 750 5061 | 5119 | 5177 
 1 | 5640 | 5698 | 5756 
 Q 6218 | 6276 | 6333 
 3 6795 | 6853 | 6910 
 4 7371 | 7429 | 7487 
 5 7947 | 8004 | 8062 
 6 8522 | 857 8637 
 7% | 9096 | 9153 | 9211 
 8 | 9669 | 9726 | 9784 
 9 | 880242 | 0299 | 0356 
 760 0814 | 0871 | 0928 
 1 | 1385 | 1442 | 1499 
 2 1955 | 2012 | 2069 
 B} 9525 | 2581 | 2638 
 4 3093 | 3150 | 3207 
 
 1 2 
 
 GO G0 6 
 Sao | 
 
 0  W%* 
 
 8 9 
 47.2 53.1 
 46.4 52.2 
 45 .6 51.3 
 44.8 50.4 
 
 $$$ 
 
TABLE XXIV.—LOGARITHMS OF NUMBERS. 
 
 No. 765 L. 883.] [No. 809 L. 908. 
 
 N.| 0 i;/2)}% /4 6 | ® | & | 8) 9) | pif. 
 765 | 883661 | 3718 | 3775 | 3832 | 3888 | 38945 | 4002 | 4059 | 4115 | 4172 | | 
 6 422 4285 | 4842 | 4899 | 4455 4512 | 4569 | 4625 | 4682 | 4739 | | 
 ton 4°95 | 4852 | 4909 | 4965 | 5022 || 5078 | 5135 | 5192 | 5248 | 5305 
 8 5861 | 5418 | 5474 | 5531 || 5587 5644 | 5700 | 5757 | 5813 | 5870 
 9 5926 | 5983 | 6039 | 6096 | 6152 || 6209 | 6265-| 6321 | 6878 | 6434 | 
 770 6491 | 6547 | 6604 | 6660 | 6716 || 6773 | 6829 | 6885 | 6942 | 6998 
 hea} 7054 | 7111 | 7167 | 7223 | 7280 73386 | 73892 | 7449 | 7505 | 7561 
 2 7617 | 7674 | 7730 | 7786 | 7842 7898 | 7955 | 8011 | 8067 | 8123 
 3 8179 | 8236 | 8292 | 8348 | 8404 8460 | 8516 | 857% 8629 | 8685 
 4 8741 | 8797 | 8853 | 8909 | 8965 9021 | 9077 |. 9184 | 9190 | 9246 
 5 9302 | 9358 | 9414 | 9470 | 9526 || 9582 | 9688 | 9694 | 9750 | 9806 56 
 6 9862 | 9918 | 9974 . —— — 
 0030 | 0086 0141 | 0197 | 0253 | 0309 | 0365 
 7 | 890421 | 0477 | 05383 | 0589 | 0645 0700 | 0756 | 0812 | 0868 | 0924 
 ih 8 0980 | 1035 | 1091 | 1147 | 1203 1259 | 1314 |-13870 | 1426 | 1482 
 9 1537 | 1593 | 1649 | 1705 | 1760 1816 | 1872 | 1928 | 1983 | 2039 
 fa 780 2095 | 2150 | 2206 | 2262 | 2317 23873 | 2429 | 2484 | 2540 | 2595 
 i i 2651 | 2707 | 2762 | 2818 | 2873 || 2929 | 2985 | 8040 | 8096 | 3151 
 nt | 2 8207 | 3262 | 3318 | 3373 | 3429 || 3484 | 3540 | 3595 | 3651 706 
 (rea 3 3762 | 3817 | 38873 | 38928 |. 3984 || 4089 | 4094 | 4150 205 | 4261 
 Hy ie || 4 4316 | 4871 | 4427 | 4482 | 4538 4593 | 4648 | 4704 | 4759 | 4814 
 Cal 5 4870 | 4925 | 4980 | 5036 | 5091 5146 | 5201 | 5257 | 53812 | 5367 
 u 6 5423 | 5478 | 5533 | 5588 | 5644 5699 | 5754 | 5809 | 5864 | 5920 
 { | ff 5975 | 6030 | 6085 | 6140 | 6195 6251 | 6806 | 63861 | 6416 | 6471 
 i 8 6526 | 6581 | 6686 | 6692 | 6747 6802 | 6857 | 6912 | 6967 | 7022 
 i| 9 7077 | 7182 | 7187 | 7242 | 7297 || 73852 | 7407 | 7462 | 7517 | 7572 55 
 Hat 790 7627 | 7682 | 7737 | 7792 | 7847 7902 | 7957 | 8012 | 8067 | 8122 
 ips | 1 8176 | 8231 | 8286 | 8341 | 8396 8451 | 8506 | 8561 | 8615 | 8670 
 Hh 2 3725 | 8780 | 8835 | 8890 | 8944 8999 | 9054 | 9109 | 9164 | 9218 
 et | 3 9273 | 9828 | 9383 | 9487 |. 9492 9547 | 9602 | 9656, | 9711 | 9766 
 Rint | 4 9821 | 9875 | 99380 | 9985 —_—— | —_ =e _——|——_—— 
 at — 00389 0094 | 0149 | 0208 | 0258 | 0312 
 | 5 | 900867 | 0422 | 0476 | 0531 | 0586 || 0640 | 0695 749 | 0804 | 0859 
 i i \\ 6 0913 | 0968 | 1022 | 1077 | 1181 1186 | 1240 | 1295 | 1349 | 1404 
 i i t 1458 | 1518 | 1567 | 1622 | 1676 1731 | 1785 | 1840 | 1894 | 1948 
 8 2003 | 2057 | 2112 | 2166 | 2221 2275 | 2829 | 2384 | 2488 | 2492 
 H, | 9 2547 | 2601 | 2655 | 2710 | 2764 2818 | 2873 | 2927 | 2981 | 30386, 
 id i 800 8090 | 3144 | 3199 | 8258 | 3307 38361 | 3416 | 3470 | 8524 | 3578 
 tt Ia | 3633 | 3687 | 38741 | 3795 | 3849 8904 | 3958 | 4012 | 4066 | 4120 
 if i 2 4174 | 4229 | 4283 | 4337 | 4891 4445 | 4499 | 4558 | 4607 | 4661 
 | 3 4716 | 477 4824 | 487 4932 4986 | 5040 | 5094 | 5148 | 5202 54 
 al 4 5256 | 5810 | 5864 | 5418 | 5472 5526 | 5580 | 5634 | 5688 | 5742 
 Ut a 5 796 | 5850 | 5904 | 5958 | 6012 6066 | 6119 | 617% 6227 | 6281 
 ana | 6 63835 | 6389 | 6443 | 6497 | 6551 6604 | 6658 | 6712 | 6766 | 6820 
 | 74 6874 | 6927 | 6981 | 7035 | 7089 || 7143 | 7196 | 7250 | 73804 | 7358 
 Ai 8 7411 | 7465 | 7519 | 7578 | 7626 ||: 7680 | 7734 | 7787 | 7841 | 7895 
 9 7949 | 8002 | 8056 | 8110 | 8163 || 8217 | 8270 | 8324 | 837, 8431 
 i | 
 i 
 PROPORTIONAL PARTS. | 
 | | 
 Diff 1 2 3 4 5 6 ve 8 9 | 
 2 BN fe Sat — ies kittie ah Sa, Shee aan 
 I} BY 5.7 11.4 Ved 22.8 28.5 34.2 39.9 45.6 51.3 
 | 56 os0 ee 16.8 22.4 28.0 33.6 39.2 44.8 50.4 | 
 5.5 . 16.5 22.0 27.5 44.0 49.5 
 5.4 16.2 21.6 7.0 43 .2 48.6 
 
TABLE XXTV.—lIOGARITHMS 
 
 OF 
 
 NUMBERS. 
 
 [No. 854 L. 981. | 
 
 No. 810 L. 908.] 
 
 N.| 0 1 2 3 4 | Bil 8 q 8 
 810 | 908485 | 8539 | 8592 | 8646 | 8699 || 8753 | 8807 | 8860 | 8O14 
 ai 9021 | 9074 | 9128 | 9181 | 92385 {| 9289 | 9342 | 9396 | 9449 
 
 2} 9556 | 9610 | 9663 | 9716 | 9770 || 9823 | 9877 | 9930 | 9984 
 3 | 910091: | 0144 | 0197 | 0251 | 0304 |) 0358 | 0411 | 0464 | 0518 
 4 | 0624 | 0678 | 0731 | 0784 | 0838 || 0891 | 0944 | 0998 | 1051 
 5 | 1158] 1211 | 1264 | 1317 | 1371 || 1424 | 1477 | 1530 | 1584 
 6 | 1690 | 1743 | 1797 | 1850 | 1903 || 1956 | 2009 | 2063 | 2116 
 7 | 2aR2 | 2275 | 2328 | 2381 | 2435 || 2488 | 2541 | 2594 | 2647 
 8 | 2753 | 2806 | 2859 | 2913 | 2966 || 3U19 | 8072 | 8125 | 3178 
 9 | 3284 | 3337 | 3390 | 3443 | 8496 || 3549 | 8602 | 3655 | 3708 
 g20 | 3814 | 3867 | 3920 | 3973 | 4026 || 4079 | 4132 | 4184 | 4287 
 1| 4343 | 4396 | 4449 | 4502 | 4555 |! 4608 | 4660 | 4713 | 4766 
 2| 4872 | 4925 | 4977 | 5030 | 5083 || 5136 | 5189 | 5241 | 5294 
 3.| 5400 | 5458 | 5505 | 5558 | 5611 || 5664 | 5710 | 5769 | 5822 
 4| 5927 | 5980 |.6033 | 6085 | 6138 || 6191 | 6243 | 6296 | 6349 
 5 | 6454 | 6507 | 6559 | 6612 | 6664 || 6717 | 6770 | C822 | 6875 
 6 | 6980 | 7033 | 7085 | 7138 | 7190 || 7243 | 7295 | 7348 | 7400 
 7 | 7506 | 7558 | 7611-| 7663 | 7716 || 7768 | 7820 | 7873 | 7925 
 8 | 8030 | 8083 | 8135 | 8188 | 8240 || 8293 | 8345 | 8897 | 8450 
 9 | 8555 | 8607 | 8659 | 8712 | 8764 || 8816 | 8869 | S921 | 8973 
 830 | 9078 | 9130 | 9188 | 9235 | 9287 || 9340 | 9392 | 9444 | 9496 
 1| 9601 | 9653 | 9706 | 9758 | 9810 || 9862 | 9914 | 9967 os 
 ACRE | pert elgg Pe cag ph) Sa 0015 
 
 2 | 920123 | 0176 | 0228 | 0280 | 0382 || 0384 | 0436 | 0489 | 0541 
 3| 0645 | 0697 | 0749 | 0801 | 0853 || 0906 | 0958 | 1010 | 1062 
 4| 1166 | 1218 | 1270 | 1322 | 1374 || 1426 | 1478 | 1580 | 1582 
 | 5 | 1686 | 1738 | 1790 | 1842 | 1894 |} 1946 | 1998 | 2050 | 2102 
 | 6 | 2206 | 2258 | 2310 | 2362 | 2414 || 2466 | 2518 | 2570 | 2622 
 | v | 9795 | 9777 | 2829 | 2881 | 2933 || 2985 | 3087 | 3089 | 3140 
 g | 3244 | 3296 | 3348 | 3399 | 3451 || 3503 | 3555 | 3607 | 3658 
 9 | 38762 | 8814 | 8865 | 8917 | 3969 || 4021 | 4072 | 4124 | 4176 
 840 | 4279 | 4331 | 4383 | 4434 | 44s6 || 4538 | 4589 | 4641 | 4693 
 1| 4796 | 4848 | 4899 | 4951 | 5008 || 5054 | 5106 | 5157 | 5209 
 21 5312 | 5364 | 5415 | 5467 | 5518 || 5570 | 5621 | 5673 | 5725 
 3 | 5828 | 5879 | 5931 | 5982 | 6034 || 6085 | 6137 | 6188 | 6240 
 4 | 6342 | 6394 | 6445 | 6497 | 6548 || 6600 | 6651 | 6702 | 6754 
 5 | 6857 | 6908 | 6959 | 7o11 | 7062 || 7114 | 7165 | 7216 | 7268 
 6 | 7370 | 7422 | 7473 | 7524 | w576 |) 7627 | Te78 | 7730 | 7781 
 7 |  7g93 | 7935 | 7986 | 8037 | 8088 || 8140 | 8191 | 8242 | 8293 
 8 | 8396 | 8447 | 8498 | 8549 | 8601 || 8652 | 8703 | 8754 8805 
 9 | 8908 | 8959 | 9010 | 9061 | 9112 || 9163 | 9215 | 9266 | 9317 
 850 | 9419 | 9470 | 9521 | 9572 | 9623 || 9674 | 9725 | 9776 | 982% 
 1 | 9930 | 9981 18 = Et | ie 
 —__|____| 9032 | 0083 | 0134 || 0185 | 0236 | 0287 | 0338 
 2 | 930440 | 0491 | 0542 | 0592 | 0643 || 0694 | 0745 | 0796 | 0847 
 31 0949 | 1000 | 1051 | 1102 | 1153 || 1204 | 1254 | 1805 | 1356 
 4| 1458 | 1509 | 1560 | 1610 | 1661 || 1712 | 1763 | 1814 | 1865 
 
 Diff. 
 
 53 
 
TABLE XXIV.—LOGARITHMS OF NUMBERS. 
 | No. 855 L. 931.] [No, 899 L. 954, | 
 N. | 0 2 5 6 7 8 Diff. 
 855 | 931966 | 2017 | 2068 1 2220 | 2822 | 2372 | 
 6 | 2474 | 2524 | 2575 | 2727 2829 | 2879 
 7 | 2981 | 3031 | 3082 | 32 3335 | 3386 | 3437 
 8 | 3487 | 3538 | 3589 | 3740 | 3841 | 3892 
 9 3993 | 4044 | 4094 (4246 | 4347 | 4397 | 
 860 | 4498 | 4549 | 4599 | 4051 | 4852 | 4902 
 1 | 5003 | 5054 | 5104 || 5255 5356 | 5406 | 
 2| 5507 | 5558 | 5608 || 5759 5860 | 5910 
 3| 6011 | 6061 | 6111 6262 3363 | 6413 
 4| 6514 | 6564 | 6614 6765 6865 | 6916 
 5 | 7016 | 7066 | 7116 7267 | 7367 | 7418 | 
 6 7518 | 7568 | 7618 7769 | 7869 | 7919 BO 
 7 | 8019 | 8069 | 8119 | 8269 83870 | 8420 
 8} 8520 | 8570 | 8620 877 | 8870 | 8920 
 9} 9020 | 9070 | 9120 927 9869 | 9419 
 870 | 9519 | 9569 | 9619 9769 | 9869 | 9918 
 1 | 940018 | 0068 | 0118 | 0267 0367 | 0417 
 2| 0516 | 0566-| 0616 | 0765 | 0865 | 0915 
 3 1014 | 1064 | 1114 1263 | 1362 | 1412 
 4| 1511 | 1561 | 1611 1760 | 1859 | 1909 
 5 | 2008 | 2058 | 2107 | 2256 | 2355 | 2405 
 6 | 2504 | 2554 | 2603 || 2752 2851 | 2901 | 
 7 | 3000 | 3049 | 3099 3247 | 3346 | 3396 
 8 | 3495 | 3544 | 3593 3742 3841 | 3890 
 9 | 3989 | 4038 | 4088 | 4236 4335 | 4384 
 880 | 4483 | 4582 | 4581 | 4729 | 4828 | 4877 
 1| 4976 | 5025 | 5074 || 5222 5821 | 5370 
 2| 5469 |.5518 | 5567 | 5715 5813 | 5862 
 3| 5961 | 6010 | 6059 6207 | 68305 | 6354 
 4| 6452 | 6501 | 6551 6698 | 6796 | 6845 
 5 | 6943 | 6992 | 7041 7189 | 7287 | 7336 49 
 6| 7434 | 7483 | 7582 | 7679 | 7777 | 7826 
 7| 7924 | 7973 | 8022 8168 | | 8266 | 8315 
 8} 8413 | 8462 | 8511 8657 | | 8755 | 8804 
 9} 8902 | 8951 | 8999 9146 | 9244 | 9292 
 ; 890 | 9390 | 9439 | 9488 | 9634 | 9731 | 9780 
 1]. 9878 | 9926 | 9975 | [aes | 
 / | || 0121 | 0219 | 0267 
 2 | 950365 | 0414 | 0462 0608 0706 | 0754 
 3} 0851 | 0900 | 0949 1095 | 1192 | 1240 
 4} 1888 | 1386 | 1435 | 1580 1677 | 1726 
 5 | 1823 | 1872 | 1920 | 2066 | 2163 | 2211 
 G6 | .2308 | 2356 | 2405 2550 2647 | 2696 
 7 | 2792 | 2841 | 2889 | 3034 | 3131 | 3180 
 8 | 8276 | 3325 | 3373 3518 | 3615 | 3663 
 9 | 3760 | 3808 | 3856 4001 | 4098 | 4146 
 
TABLE XXTIV.—LOGARITHMS OF NUMBERS. 
 
 i] 
 |No 900 1. 954.1] [No. 944 L. 975. 
 | | 
 IN. | 0 Pi Pie | ai ae | ep 3 8 | 9 Diff. 
 | | 
 | | | 
 | 900 | 954243 | 4291 | 4339 | 4387 | 4435 || 4484 | 4582 | 4580 | 4628 | 4677 
 rod 4725 | 4773 | 4821 | 4869 | 4918 || 4966 | 5014 | 5062 | 5110 | 5158 
 We 2 5207 | 5255-| 5303 | 5351 | 5399 || 5447 | 5495 | 5543, 5592 | 5640 
 | 31 5688 | 5736 | 5784 | 5882 | 5880 || 5928) 5976 | 6024 | 6072 , 6120 
 | 4 | 6168 | 6216 | 6265 | 6313 | 6361 || 6409 | 6457 | 6505 | 6553 | 6601 48 
 | 5 | 6649 | 6697 | 6745 | 6793 | 6840 :| 6888 | 6936 | 6984 | 7032 | 7080 
 | 6! 7128 | 7176 | 7224 | 7272 | 7820 || 7368 | 7416 | 7464 | 7512 | 7559 
 rf 7607 | 7655 | 7703 | 7751 | 7799 | 7847 7894 | 7942 | 7990 | 8038 
 8! goxG | 8134 | 8181 | 8229 | 827% || 8825 | 83873 | 8421 | 8468 | 8516 
 9 | 8564 | 8612 | 8659 | 8707 | 8755 || 8803 | 8850 | 8898 | 8946 | 8994 
 910 | 9041 | 9089 | 9137 | 9185 | 9232 | 9280 | 9828 | 9375 | 9423 | 9471 
 | 1] 9518 | 9566 | 9614 | 9661 | 9709 || 9757 | 9804 | 9852 | 9900 | 9947 
 | 2} 9995 | = | — = 
 | | 0042 | 0090 | 0188 | 0185 || 0283 | 0280 | 0328 | 0876 | 0423 
 3 | 960471 | 0518 | 0566 | 0613 | 0661 || 0709 | 0756 | 0804 | 0851 | 0899 
 4 | 0946 | 0994 | 1041 | 1089 | 1136 || 1184 | 1231 | 1279 1826 | 1374 
 5 | 1421 | 1469 | 1516 | 1563 | 1611 || 1658 | 1706 | 1753 | 1801 | 1848 
 | 61 1895 | 1943 | 1990 | 2088 | 2085 || 2182 | 2180 | 2227 | 2275 | 23822 
 7 | 2369 | 2417 | 2464 | 2511 | 2559 || 2606 | 2653 | 2701 | 2748 | 2795 
 8 | 9948 | 9890 | 2937 | 2985 | 3082 || 3079 | 3126 | 3174 | 3221 | 8268 
 | 9 | 3316 | 3863 | 3410 | 3457 | 3504 |) 3552 | 3599 | 3646 | 3693 | 3741 
 920 3788 | 3835 | 3882 | 3929 | 2977 || 4024-) 4071 | 4118 | 4165 | 4212 
 1| 4260 | 4307 | 4354 | 4401 | 4448 || 4495 | 4542 | 4590 | 4637. | 4684 
 2 4731 | 4778 | 4895 | 4872 | 4919 || 4966 | 5013 | 5061 . 5108 | 5155 
 | 3 | 5202 | 5249 | 5296 | 5343 | 5390 || 5437 | 5484 | 5531 5578 | 5625 
 | 4 5672 | 5719 | 5766 | 5813 | 5860 || 5907 | 5954 | 6001 | 6048 | 6095 4g 
 5 | 6142 | 6189 | 6236 | 6283 | 6329 |) 6376 | 6423 | 6470 | 6517 | 6564 
 6 | 6611 | 6658 | 6705 | 6752 | 6799 || 6845 | 6892 | 6939 | 6986 | 7033 
 7 "ogo | 7127 | 71'73.| 7220 | 7267. || 7314 | 7361 | 7408 | 7454 | 7501 
 8 rag | 7595 | 7642 1 7688 | 7735 || 7782. | 7829 | 7875 | 7922 | 7969 
 a 8016 | 8062 | 8109 | 8156 | 8203 || 8249 | 8296 | 8343 | 8390 | 8436 
 930 8483 | 8530 | 8576 | 8623 | 8670 || 8716 | 8763 | 8810 | 8856 | 8903 
 1 | 3950 | 8996 | 9043 | 9090 | 9136 || 9183 | 9229 | 9276 | 9323 | 9369 
 2} 9416 | 9463 | 9509 | 9556 | 9602 || 9649 | 9695 | 9742 | 9789 | 9885 
 3 | 9882 | 9928 | 9975 | La 
 | | | 0021 | 0068 |} O114 | 0161 | 0207 | 0254 | 0300 
 4 | 970347 | 0393 | 0440 | 0486 | 0533 || 0579 | 0626 | 0672 | 0719 | 0765 
 | 5 0812 | O858 | 0904 | 0951 | 0997 || 1044 | 1090 | 1137 | 1183 | 1229 
 6| 1276 | 1322 | 1869 | 1415 | 1461 || 1508 | 1554 | 1601 | 1647 | 1693 
 | 7 | 1740 | 1786 | 1882 | 1879 | 1925 || 1971 | 2018 | 2064 | 2110 | 2157 
 8 2903 | 2249 | 2295 | 2342 | 2388 || 2434 | 2481 | 2527 | 2573 | 2619 
 9 | 2666 | 2712 | 2758 | 2804 | 2851 |} 2897 | 2943 | 2989 | 8035 | 3082 
 | | | | 
 1940 | 3128 | 3174 | 3220 | 3266 | 3313 || 3359 | 3405 | 3451 | 3497 | 3543 
 iP ot 3590 | 3636 | 3682 | 3728 | 3774 || 3820 | 3866 | 3913 | 3959 | 4005 
 | 2 4051 | 409% | 4143 | 4189 | 4235 || 4281 | 4827 | 4874 | 4420 | 4466 
 3! 4512 | 4558 | 4604 | 4650 | 4696 || 4742 | 4788 | 4834 | 4880 | 4926 
 | 4} 4972 | 5018 | 5064 | 5110 | 5156 |} 5202 | 5248 | 5294 | 5340 | 5386 46 | 
 | if 
 PROPORTIONAL PARTS. 
 | | | poe 
 | Diff | 1 | 2 | 3 sat 15.5 6 Fe 8 9 
 } | 
 el aa 2a als ead wed wk | 
 | AG | ANG OM | 14.1 18.8 | 23.5 28 2 32.9 37.6 42.3 | 
 | 4 | 40] 92 | 18 | 184 | 23.0 | a6 | a2 | 36.8 41.4 | 
 
{ 
 
 so 
 or 
 
 CRS SONI PWWHS S® 
 
 OOS Ot Co 
 
 BS 
 S 
 
 COOIMopomwHS CO VNIOounrwmore 
 
 TABLE XXTIV.—LOGARITHMS OF NUMBERS. 
 a ee 
 No. 945 L. 975.] 
 
 0 1 2 3 4 5 6 v4 
 975432 | 5478 | 5524 | 5570 | 5616 || 5662 | 5707 | 5753 
 5891 | 6937 | 5983 | 6029 | 6075 || 6121 | 6167 | 6212 
 6350 | 6896 | 6442 | 6488 | 6523 || 6579 | 6625 | 6671 
 6808 | 6854 | 6900 | 6946 | 6992 || 7037 | 7088 | 7129 
 7266 | 7312 | 73858 | 7403 | 7449 || 7495 | 7541 | 7586 
 TRA | 7769 | 7815 | 7861 | 7906 || 7952 | 7998 | 8043 
 8181 | 8226 | 8272 | 8317 | 8363 || 8409 | 8454 | 8500 
 8637 | 8683 | 8728 | 8774 | 8819 || 8865 | 8911 | 8956 
 9093 | 9188 | 9184 | 9230 | 9275 || 9321 | 9366 | 9412 
 9548 | 9594 | 9639 | 9685 | 9730 || 9776 | 9821 | 9867 
 980003 | 0049 | 0094 | 0140 | 0185 || 0231 | 0276 | 0322 
 (458 | 0503 | 0549 | 0594 | 0640 || 0685 | 0730 | 0776 
 0912 | 0957 | 1003 | 1048 | 1093 || 1189 | 1184 | 1229 
 1366 | 1411 | 1456 | 1501 | 1547 || 1592 | 1637 | 1683 
 1819 | 1864 | 1909 | 1954 | 2000 '| 2045 | 2090 | 2135 
 2271 | 2316 | 23862 | 2407 | 2452 || 2497 | 2543 | 2588 
 2723 | 2769 | 2814 | 2859 | 2904 || 2949 | 2994 | 3040 
 3175 | 3220 | 8265 | 3310 | 3856 || 3401 | 3446 | 3491 
 3626 | 3671 | 3716 | 3762 | 8807 || 3852 | 3897 | 3942 
 4077 | 4122 | 4167 | 4212 | 4257 |) 4802 | 4347 | 4392 
 4527 | 4572 | 4617 | 4662 | 4707 752 | 4797 | 4842 
 4977 | 5022 | 5067 | 5112 | 5157 || 5202 | 5247 | 5292 
 5426 | 5471 | 5516 | 5561 | 5606 || 5651 | 5696 | 5741 
 5875 | 5920 | 5965 | 6010 | 6055 || 6100 | 6144 | 6189 
 6324 | 6369 | 6413 | 6458 | 6503 || 6548 | 6593 | 6637 
 6772 | 6817 | 6861 | 6906 | 6951 || 6996 | 7040 | 7085 
 7219 | 7264 | 7209 | 7353 | 7398 || 7443 | 7488 | 7582 
 7666 | 7711 | 7756 | 7800 | 7845 || 7890 | 7934 | 7979 
 8113 | 8157 | 8202 | 8247 | 8291 || 8336 | 8381 | 8425 
 8559 | 8604 | 8648 | 8693 | 8737 || 8782 | 8826 | 8871 
 9005 | 9049 | 9094 | 9138 | 9183 || 9227 | 9272 | 9316 
 9450 | 9494 | 9539 | 9583 | 9628 || 9672 | 9717 | 9761 
 9895 | 9939 | 9983 _ 
 0028 | OO72 |} 0117 | 0161 | 0206 
 990329 | 0383 | 0428 | 0472 | 0516 || 0561 | 0605 | 0650 
 0783 | 0827 | 0871 | 0916 | 0960 || 1004 | 1049 | 1098 
 1226 | 1270 | 1315 | 13859 | 1403 || 1448 | 1492 | 1536 
 1669 | 1713 | 1758 | 1802 | 1846 || 1890 | 1935 | 1979 
 
 [No. 989 L. 995. | 
 
 Diff. 
 
 45 
 
 ~J 
 (=r) 
 22 
 ris) 
 
| No. 990 L. 995.] 
 
 TABLE XXTV.—LOGARITHMS OF NUMBERS. 
 
 [No. 999 LL. 999. 
 
 Nei 8 1 2 3 4 5 6 3 8 9 | Diff. 
 | eS —_——_—- | oo - —————- ———— 
 990 | 995635 | 5679 W283 | 5767 | 5811 |, 5854 | 5898 | 5942 | 5986 | 6030 
 1 6074 | 6117 | 6161 | 6205 | 6249 |. 6293 | 6337 | 6880 | 6424 | 6468 44 
 2| 6512 | 6555 | 6599 | 6643 | 6687 | 6731 | 6774 | 6818 | 6862 | 6906 
 3 | 6949 | 6993 | 7037 | 7080 | 7124 | 7168 | 7212 | 7255 | 7299 | 7343 
 4 | 386 | 7430 | 7474 | V51T | 7561 | 7605 | 7648 | (692 | 7736 | TTT 
 5 | 7823 | 7867 | 7910 | 7954 | 7998. 8041 | 8085 | 8129 | 8172 | 8216 
 6 | 8259 | 8803 | 83847 | 8390 | 8434 | 8477 | 8521 | 8564 | 8608 | 8652 
 7 | 8695 | 8739 | 8782 | 8826 | 8869 | 8913 | 8956 | 9000 | 9048 9087 
 8 | 9131 | 9174 | 9218 | 9261 | 9305 | 9848 | 9392 | 9485 | 9479 | 9522 
 9 | 9565 | 9609 | 9652 | 9696 | 9739 | 9783 | 9826 | 9870 | 9913 | 9957 | 4g 
 | | | | 
 LoGARITHMS OF NUMBERS FROM 1 To 100. 
 \| | | 
 N. Log. || N.| Log. Wot Logo N.. ik Log. <1 Nye Log 
 1 | 0.000000 || 21 | 1.322219 || 41 | 1.612784 || 61 | 1.785330 || 81 | 1.908485 
 2 | 0.301030 || 22 | 1.342428 || 42 | 1.623249 || 62 | 1.792392 || 82 | 1.918814 
 3| 0.477121 || 23 | 1.861728 || 43 | 1.633468 || 63 | 1.799341 || 83 | 1.919078 
 4 0.602060 || 24 | 1.880211 || 44 | 1.643453 || 64 1.806180 |) 84 | 1.924279 
 5 | 0.698970 || 25 | 1.897940 |} 45 | 1.658218 || 65 1.812918 || 85 | 1.929419 
 6 | 0.778151 || 26 | 1.414973 || 46 | 1.662758 || 66 | 1.819544 || 86 | 1.934498 
 7 | 0.845098 || 27 | 1.431864 || 47 | 1.672098 || 67 | 1.826075 || 87 | 1.989519 
 8 | 0.903090 || 28 | 1.447158 || 48 | 1.681241 || 68 | 1.882509 |; 88 | 1.944483 
 9 |, 0.954243 || 29 | 1.462398 || 49 | 1.690196 || 69 | 1.838849 || 89 | 1.949390 
 10 | 1.000000 || 30 | 1.477121 || 50 | 1.698970 || 70 | 1.845098 |; 90 | 1.954243 
 11 | 1.041893 || 31 | 1.491362 || 51 | 1.707570 || 71 | 1.851258 || 91 | 1.959041 
 12 | 1.079181 || 32} 1.505150 || 52 | 1.716003 || 72 | 1.857332 || 92 | 1.963788 
 13 1.113943 || 83 | 1.518514 || 53 | 1.724276 || 73 1.863323 || 93 | 1.968483 
 14 | 1.146128 || 34 | 1.531479 || 54 | 1.782894 || 74 | 1.869282 || 94 | 1.973128 
 15 | 1.176091 || 35 | 1.544068 || 55 | 1.740363 || 75 | 1.875061 || 95 | 1.977724 
 16 | 1.204120 || 36 | 1.556303 || 56 | 1.748188 || 76 | 1.880814 || 96 | 1.98227 
 17 | . 1.230449 || 37 | 1.568202 || 57 | 1.755875 i 77 | 1.886491 || 97 | 1.986772 
 18 1.255272 38 | 1.579784 || 58 | 1.768428 || 78 | 1.892095 || 98 | 1.991226 
 19 1.278754 || 39 | 1.591065 || 59 | 1.770852 || 79 | 1.897627 || 99 | 1.995635 
 20 | 1.301030 || 40 | 1.602060 || 60 | 1.778151 || 80 | 1.903090 ||100 | 2.000000 
 { ! 
 | Sign Si Val Si Val Si Val 
 | a7 Sig r Sign | Value} Sign alue| Sign alue 
 Jaton. in ist Veins in 2d at in 3d at j|in4th} at 
 Lac Fd Quad. |" *| Quad.| 180°. | Quad.! 270° | Quad.} 360°. 
 ——— | ae | =| | (Ee 
 Binv?. 02 Peep pe ee FMM oes ey |* Ne reed) 
 Tan Oo oe) _ O a oa) — O 
 Sent PSR R co coc to ts RH ae oO Le R 
 Versin....| O aE R + 2R {| + R -- O 
 COS tS tacts | R O _ R | — O +. R 
 Oban Jee ae |} @ -++- O -- oa +f O ~ 00 
 Oosec. ..<.. | io) + R -{- a0 _ R _— ora) 
 —— — — — — t —_ — ——$_—___—_— = iy 
 
 R signifies equal to rad; © signifies infinite; O signifies evanescent. | 
 
 308 
 
60 
 
 120 
 
 180 | 
 240 | 
 300 | 
 360 | 
 
 420 
 480 
 
 AO | 
 600 
 660 | 
 720 
 780 
 
 840 
 
 900 
 960 
 1020 | 
 1080 
 | 1140 
 
 200 
 
 1260 | 
 1320 | 22 
 1380 | 2% 
 1440 | 
 1500 | 2 
 
 1560 
 1620 
 1680 
 1740 
 1800 
 1860 
 1920 
 
 1980 | 3: 
 2040 | < 
 2160 | 3: 
 2160 | 3 
 
 2226 
 2280 
 2340 
 
 2400 | 4( 
 
 2460 
 
 2520 
 
 Ow) | 
 2580 | 
 
 2640 
 
 2700 | 4 
 2760 | 
 
 2820 
 2880 
 2940 
 
 3000 | 5 
 
 3060 
 3120 
 i 3180 
 
 | $240 | 
 2300 | ! 
 
 3360 
 ' 8120 
 
 | 8180 Ie 
 | 8340 |! 
 
 3600 
 
 bat or) 
 
 ~3 
 
 ~ 
 
 CO ~3 
 
 (o-2) 
 
 TABLE XXV.—LOGARITHMIC SINES. 
 
 Sine. 
 
 Inf. neg. 
 | 6.463726 
 |. .764756 
 
 . 940847 
 065786 
 . 162696 
 
 241877 | 
 
 . 808824 
 .3866816 
 .417968 
 .463726 
 
 542906 
 
 577668 | 
 609853 | 2 
 
 639816 
 
 667845 
 694173 | 
 718997 
 742478 | 
 
 164754 
 785943 
 .806146 
 825451 
 843934 
 .861662 
 878695 
 895085 
 . 910879 
 .926119 
 . 940842 
 
 . 955082 
 . 968870 
 982233 
 . 995198 
 007787 
 .020021 
 .031919 
 043501 
 054781 
 065776 
 
 J 
 
 | 8.076500 
 
 086965 
 097183 
 107167 
 . 116926 
 126471 
 . 135810 
 . 144953 
 . 153907 
 - 162681 
 .171280 
 179713 
 187985 
 . 196102 
 204070 
 
 .211895 | 556 
 
 219581 
 227134 
 234557 
 
 | 8.241855 
 
 | 
 ~N 
 
 Tang. 
 
 1 Q 
 2) 
 
 Or 
 
 — O> 
 
 =? 
 
 Ovorororororor 
 PO Is oe 
 Ovoror or or 
 IS 
 
 ry Or Or Or Or Or Or 
 
 Ia 3 I I I 
 
 It 
 
 He ee OT OT OT 
 
 orer Oror1roror 
 {I QR 
 
 ha 
 
 505118 | 
 
 tJ 
 tS 
 ~ 
 
 -z 
 
 Cosine. 
 
 f. neg. 
 463726 
 164756 
 . 940847 
 .065786 
 . 162696 
 241878 
 808825 
 .866817 
 .417970 
 .463727 
 7.505120 
 .542909 
 577672 
 .609857 
 639820 
 667849 
 694179 
 719008 
 . 742484 
 (64761 
 7.785951 
 .806155 
 825460 
 843944 
 .861674 
 878708 
 895099 
 910894 
 926134 
 . 940858 
 
 . 955100 
 . 968889 
 982253 
 .995219 
 007809 
 -020044 
 .031945 
 048527 
 .054809 
 .065806 
 
 95 8.076531 
 
 3} 086997 
 097217 
 107208 
 116963 
 126510 
 135851 
 144996 
 153952 
 162727 
 
 171328 
 .179763 
 . 188036 
 . 196156 
 204126 
 -211953 
 219641 
 220195 
 .234621 
 8.241921 
 
 Cotang. 
 
 13.536274 
 235244 
 13.059153 
 12.934214 
 837304 
 758122 
 691175 
 633183 
 582030 
 53627 
 12.494880 
 457091 
 422398 
 390143 
 . 360180 
 332151 
 305821 
 280997 
 257516 
 235239 
 12.214049 
 193845 
 174540 
 “156056 
 138326 
 121292 
 104901 
 089106 
 073866 
 059142 
 12.044900 
 031111 
 017747 
 
 12.004781 
 
 . 979956 
 . 968055 
 .956473 
 .945191 
 . 984194 
 . 923469 
 . 913003 
 902783 
 892797 
 
 883037 
 
 1 
 
 are 
 
 .864149 
 . 855604 
 . 846048 
 837273 
 .828672 
 ~ 820237 
 .811964 
 . 803844 
 795874 
 . 788047 
 780359 
 .77 2805 
 . 765379 
 
 11.758079 
 
 1 
 
 are 
 
 Tang. 
 
 Cotang. 
 
 Inf. pos. | 
 
 We) 
 
 =) 
 
 ive} 
 
 | 11.992191 
 
 ie) 
 
 .873490 
 
 Sine: 
 
 ID 1" Cosine, 
 
 ten 
 ten 
 ten 
 ten 
 ten 
 ten 
 
 . 999999 | f 
 . 999999 
 . 999999 
 . 999999 
 . 999998 
 
 999998 
 999997 
 -999997 | 
 999996 
 
 999996 
 999995 
 999995. | 
 “999994 
 “999993 
 999993 
 
 9.999992 | 
 -999991 |" 
 . 999990 
 . 999989 
 . 999989 
 . 999988 
 .999987 
 . 999986 
 . 999985 
 . 999983 
 
 .999982 
 . 999981 
 .999980 
 .999979 
 989977 
 .999976 
 .999975 
 . 999973 
 . 999972 
 .999971 
 
 . 999969 
 . 999968 
 . 999966 
 999964. 
 . 999963 
 . 999961 
 999959 
 . 999958 
 999956 | 
 . 999954 
 | 
 999952 
 . 999950 
 999948 
 . 999946 
 . 899944 
 . 999942 
 . 999940 
 . 999938 
 
 999936 
 
 -03 | 9’ 999934 
 
3960 
 4020 
 4080 
 4140 
 4200 
 
 4260 
 4320 
 4380 
 4440 
 4500 
 4560 
 4620 
 4680 
 4740 
 4800 
 
 4860 
 4920 
 4980 
 5040 
 5100 
 5160 
 5220 
 5280 
 5340 
 5400 
 5460 
 5520 
 5580 
 5640 
 5700 
 5760 
 5820 
 5880 
 5940 
 6000 
 
 6060 
 6120 
 6180 
 6240 
 6300 
 6360 
 6420 
 6480 
 6540 
 6600 
 
 | 
 6660 
 
 6720 
 6780 
 6840 
 6900 
 6960 
 7020 
 7080 
 7140 
 7200 
 
 4 Sine. q—t 
 | 4.685 
 0 | 8.241855 | 553 | 619 
 1) .249033 | 552 | 620 | 
 2) 256094 | 551 | 622 
 3 | 263042 | 551 | 623 | 
 4 | .269881 | 550 ,| 625 | 
 5 | .276614 | 549 | 627 
 6 | 288243 | 548 | 628 
 7 | ».289778 | 547 || 630| 
 8 | .296207 | 546 | 632 
 9 | .302546 | 546 | 633 | 
 10 | .308794 | 545 :| 635 | 
 11 | 8.314954 | 544 | 637 
 12.| 321027 | 543 | 638 
 13 | .827016 | 542 | 640 
 14| .332924 | 541 | 642 
 15| .338753 | 540 | 644 
 16| .844504 | 5389 | 646 
 * | 850181 | 589. | 648 
 18 | .355783 | 588 || 649 
 19 | .3613815 | 5387.,| 651 
 20 | .366777 | 536 || 653 
 21 | 8.372171 | 585 | 655 
 22 | .377499 | 534 | 657 
 23 | .382762 | 533 || 659 
 24 | .387962 | 532 | 661 
 25 | .893101 | 531 | 663 
 26 | .398179 | 530 | 666 
 27 | .403199 | 529 | 668 
 28 | .408161 | 527 | 670 
 29 | .413068 | 526 | 672 
 30) 417919 | 525, 67 
 31 | 8.422717 | 524 || 67 
 32 | .427462 | 528 || 67 
 33 | .482156 | 522 || 681 
 34.| 436800 | 521 || 683 
 35 | 441894 | 520 || 685 
 36 | .445941 | 518 || 688 
 87 | .450440 | 517 || 690 | 
 38. | .454898 | 516 | 693 
 39 | .459301 | 515 || 695 
 40 | .463665 | 514 | 697 
 41 | 8.467985 | 512 || 700 
 42 | .472263 | 511 || 702 
 43 |  .476498 | 510 || 705 
 44 | .480693 | 509 || 707 
 45 | .484848 | 507 || 710 
 46 |. .488963 | 506 || 713 
 47 | .493040 | 505 || 715 
 48 | .497078 | 503 || 718 
 49 | 501080 | 502 | 720 
 50 | .505045 | 501 || 723 
 51 | 8.508974 | 499 | 726 
 52 | 512867 | 498 | 729 
 58 | .516726 | 497 || 731 
 54. | .520551 | 495 || 734 
 55 | | 524343 | 494 || 737 
 56 | .528102 | 492 740 
 57 | .531828 | 491 || 748 
 58 | .5355283 | 490 | 745 
 59 | .539186 | 488 | 748. 
 60 | 8.542819 | 487) | 751 
 4.685 
 ’ | Cosine. q—l 
 
 TABLE XXV.—LOGARITHMIC SINES, 
 
 Tang. 
 
 8.241921 
 249102 
 .256165 
 .263115 
 . 269956 
 -276691 
 283323 
 289856 
 296292 
 802634 
 . 808884 
 
 | 8.315046 
 
 -321122 
 .827114 
 833025 
 . 388856 
 .844610 
 850289 
 800895 
 .861430 
 . 866895 
 
 8.372292 
 3877622 
 882889 
 . 888092 
 38932 
 3898315 
 403838 
 -408304 
 413218 
 .418068 
 
 8.422869 
 427618 
 432315 
 -486962 
 -441560 
 .446110 
 -450618 
 455070 
 -459481 
 463849 
 
 8.468172 
 472454 
 476693 
 480892 
 485050 
 489170 
 493250 
 497293 
 501298 
 505267 
 
 8.509200 
 513098 
 516961 
 520790 
 
 524586 
 528349 
 532080 
 535779 
 539447 
 
 8.543084 
 
 11. 758079 
 750898 
 748885 
 . 736885 
 730044 
 .(23309 
 16677 
 710144 
 103708 
 . 6973866 
 .691116 
 
 684954 
 678878 
 . 672886 
 666975 
 .661144 
 .655390 
 649711 
 .644105 
 688570 
 633105 
 . 627708 
 622378 
 617111 
 .611908 
 . 606766 
 601685 
 .596662 
 .591696 
 586787 
 581982 
 
 577181 
 572882 
 . 567685 
 .568038 
 558440 
 .558590 
 .549887 
 
 .544980 
 
 11 
 
 11 
 
 11 
 
 .586151 
 11 
 027546 
 .528807 
 .519108 
 .514950 
 .510880 
 
 498702 
 
 1 
 
 bak 
 
 .486902 
 .488039 
 .479210 
 475414 
 .471651 
 .467920 
 .464221 
 .460553 
 11.456916 
 
 Tang. 
 
 Cotang. 
 
 .540519 | 
 
 581828 
 
 .£06750 
 
 502707 | 
 
 494733 
 .490800 | 
 
 | 
 
 gti ie at | 
 15,314'| | [ere 
 381 |! gg | 9.999934 | 60 
 380 | ‘gs 999982 | 59 
 378 | "pg  -999929 | 5 
 ce | "03 .999927 | 57 
 75 |) "on .999925 | 56 | 
 73 || "ga | .999922 | 55 
 72 || "FS | 999920 | 54 
 70 | “ox | .999918 | 53 
 368 “Og 999915 | 52 
 867 | ‘os | -999913 | 51 
 365 || “~? | ..999910 | 50 
 363 | “03 9.999907 | 49 
 362 | "os | - -999905 | 48 
 860 | ‘op . -999902 | 47 | 
 358 “og — .999899 | 46 
 856 ‘ps, -999897 | 45 
 854 "on | .999804 | 44 
 852 ‘pp | .999801 | 43 
 851 | “pe |. 999888 | 42 
 3849 | ‘on | .999885 | 41 
 847 |" 9.999882 | 40 
 345 | “0? 9.999879 | 39 
 343 “gs | .999876 | 38 
 841 || ‘92 | .999873 | 37 
 839 | “op |. .999870 | 36 
 B37 | "op | .999867 | 85 
 834 | ‘or |  .999864 | 34 
 832 | “oe | .999861 | 33 
 830 |/.‘97 | .999858 | 82 
 328 || “os 999854 | 31 
 326 || ° .999851 | 30 
 aed | +03] 9.999848 | 29 
 321 “05 999844 | 28 
 819 || "op | .999841 | 27 
 817 || “p» | -999888 | 26 
 | 815 |] ‘os | -999884 | 25 
 312 || 72 | .999831 | 24 
 810 | 05 999827 | 23 
 B07 |) “ow | . 999824 | 22 
 | 805 || ‘o | . 999820 | 21 
 303 |} °** | . .999816 | 20 
 | 300 || 03 | 9.990813 | 19 
 | 298 || a. | ..999809 | 18 
 | 295 |) "oy |. 999805 | 17 
 | 293 || ‘oy | . 999801 | 16 
 290 || or | .. 969797 | 15 
 287 Il ae]. .999794 | 14 
 285 || “g» | .999790 | 18 
 282 || “G, | . 999786 | 12 
 280 | -o | ..999782 | 11 
 | 277 |)" | 999778 | 10 
 ar Eis 9.999774 | 9 
 271 | ‘on | 999769 | 8 
 269 || “oy | ..999765 | 7 
 266 || ‘oy | ..999761 | 6 
 263 || a4) ..999757 | 5 
 266 “OR | ..9997538 | 4 
 257 || “ay | ..999748 | 38 
 255 || 741 ..999744 | -2 
 252 || ‘og | ..999740 | 1 
 249 y |) 9.999735 | 0 
 15.314| 
 | 
 wey Peay PERMA Levee 
 q+1||D1*| Sine. [aes 
 
Dale. 
 
 ! 
 4 Sine. Dek. Cosine. | D. 1’. Tang. 
 0 | 8 542819 9.999735 | 8.543094 
 1 | .546422 ote | ‘999731 | “Oy || .546691 
 2 .549995 59. 07 . 999726 OF. Web 550268 
 3 | .553539 | 59" 5g || -999722 08 553817 
 4| .557054 | se'y9 || -999717 07 557336 
 5 | 560540 | 2 ae || .999718 “08 560828 
 6 | .563999 | 24-09 || 999708 | ‘oe 564291 
 7 | ‘pezasr | 26-22 |) 999704 | “og 567727 
 8 | (570886 | 26-49 || 999699 | og || -571187 
 | ipra2i4 | 56:39 || “999604 | “og || 574520 
 10 577566 | 58:82 || (999680 | gy || -577877 
 11 | 8.580892 | ~» no || 9.999685 og || 8.581208 
 12| .584193 | 24'¢9 || -999680 08 584514 
 13 | .587469 | Pa'o9 || -9996%5 “08 587795 
 14| .59072t | Za'ng || -999670 “Og 591051 
 15 | .593948 | P3749 || -999665 | “og 594283 
 16| .597152 | > “00 . 999660 “08 597492 
 17 | 7600332 | 23-09 || ‘999655 | “og || .600677 
 18 | 603489 | 52-08 || 999650 | ‘og ||. 603889 
 19 | .60662% =. || 999645 “68 606978 
 20 | ‘609734 | 51-85 |) “900640 | gg |] | -610004 
 21 | 8.612823 | ~ | 9.999635 8.613189 
 S| 615801 | 51-8 || ~ ‘990629 40 || “e1626e 
 23 | .618937 ce “ho || 999624 “08 619313 
 24 | .621962 | sng || -999619 “08 622343 
 25 | .624965 | Gong || -999614 | “49 . 625352 
 26 | .627948 | 4o'ag || -999608 | ‘og 628340 
 27 | .630911 | 4o‘g5 || 299603 <6 631308 
 28 | .633854 48.70 999597 08 634256 
 29 | 636776 “ty || -999592 | “40 637184 
 30 | 639680 | 48-40 || ‘909586 | “og. || - 640098 
 31 | 8.642563 | ye we || 9.999581 | 8.642982 
 a2} 645428 | 40-12 || .9995%5 “40. || 645858 
 83} .648274 | “4eyg || -99957 "46 648704 
 84| 651102 | 4e'gs || -999564 40 651537 
 35 | .653911 | Je '55 || 999558 6 654352 
 36 | .656702 | Je'o9 || -999553 46 657149 
 387 | .659475 | Je‘go || -999547 10 659928 
 38-| .662230 | 4p'gg || 999541 “40 .662689 
 39 | .664968 0-09 |) 999535 30 665433 
 40 | ‘667689 | 45-25 || 999529 | og || -668160 
 41 | 8.670893 | 4, -a || 9.999524 8.670870 
 42 | .673080 | at || 999518 “10° || 2673568 
 43 | 675751 | 4493 || -999512 “10 676239 
 44| .678405 | 43’g7 ||  -999506 16 678900 
 45 | .681043 | 43mg || -999500 es 681544 
 48 | 683665 | 43°45 || 999493 a3 684172 
 7 | .686272 | 13.18 | 999487 10 686784 
 48 | 688863 | 45%o9 || -999481 40 689381 
 49 | 691438 | 45"pe || -9994% “10 691963 
 50 | .693998 | 45" 4o || -999469 i0 694529 | 
 B1 |. 8.696543 |< 9.999468 8.697081 
 52 | (699073 | 42-44 || 990456 22 || 1699617 
 53 | 701589 | 41 7¢g || 999450 12 702139 
 54] 704090 | 45 74x || -999448 10 7104646 
 Bs | .706577 | 4y‘o9 || -999487 | “10 107140 
 56 | .709049 | dog || -999431 49 . 709618 
 7 | .711507 | go's || -999424 45 712088 
 58 | .713952 | 40 53 .999418 "yo. (|| 14584 
 59 | .716383 | 4n'9g || - 999411 19 || ,- 716072 
 60 | 8.718800 | 42-8 || 9.999404 2 || 8.719396 
 fi Cos*~e. | D1”. |i. Sine. Dat Cotang. 
 
 ee 
 
 Cotang. | f 
 | 
 
 11.456916 | 
 
 453309 | 59 
 .449732 | 58 
 -446183 | 57 
 .442664 | 56 
 .489172 | 55 
 .485709 | 54 
 .482278 | 53 
 428863 | 52 
 .425480 | 51 
 .422123 | 50 
 11.418792 | 49 
 .415486 | 48 
 .412205 | 47 
 .408949 | 46 
 405717 | 45 
 "402508 | 44 
 .899823 | 438 
 .896161 | 42 
 .398022 | 41 
 889906 | 40 
 11.386811 | 39 
 .883738 | 38 
 .880687 | 37 
 .877657 | 36 
 .874648 | 35 
 .3871660 | 34 
 .868692 | 33 
 .3865744 | 382 
 .862816 | 31 
 .3809907 | 30 
 11.357018 | 29 
 .854147 | 28 
 .3891296 | 27 
 348463 | 26 
 .345648 | 25 
 . 842851 | 24 
 .840072 | 23 
 .8dlell | 22 
 .034567 | 21 
 331840 | 20 
 11.329180 | 19 
 .826487 | 18 
 823761 7 
 .821100 | 16 
 "318456 | 15 
 .815828 | 14 
 .313216 | 18 
 .3810619 z 
 .8080387 | 11 
 .205471 | 10 
 11.302919 9 
 . 800383 8 
 297861 cs 
 . 295354 6 
 . 292860 5 
 .290882 | 4 
 287917 | 3 | 
 (285466 | 2 | 
 "283028 | 1 | 
 11.280604 | 0 
 Tange 3 
 
 87° 
 
| 
 
 | 
 
 Cosine. | D. 1’. 
 
 PP ELAS « 
 
 | 
 g fotanzs | 
 Tang. Didi © g | 
 Sine. Ds We: ft 
 | per of ieee 11.280604 | 69 
 | ges VW |" “O7gi94 | 5 
 ; 7 || 9-999404 | 49 || 8 721806 eS oy | <2iS1ML 09 
 0 | 8.718800 | 40.0% .999398 "19 | oe 29.9% i 3 
 1 | .721204 | 90° e6 999391 13: || | ete 20.73 Bate | 
 2 | .423595 | 39 69 “999384 “35 728088 20.52 | Panes 
 Pe load ee Tre -999878 | “15 728059 39.30 le: 
 4} .728337 39.18 || “go93°4 12 7aa663 | 39-10| 08s 
 ‘ 2878 BO8887 12 735996 38.68 | "961683 | 52 
 ic) visas | ‘Boon | ie BBY | Sotelo 259374 | Bl 
 | Bat) Ba ‘gua | 2 740626 | 38°57 "957078 | 50 
 gal Dipsgoen | Bae 999348 eS | ae 
 10'|  fraeesa | Son! nee e Sasa ae eae 
 "999; 12 | ravaro | 3087 Be 48 
 11 | 8.744536 | ge gggaee | +12 ep | Bi . 
 2 - (46802 37.55 999315 aR fare 7.48 | peu ‘a 
 13 | .749055 | 3°3> . 999308 39 edie 3730 28 | 
 14] .151297 | eg 999301 | “12 Tee | 87.10 pag 
 dott Udamcone 1320308 Sees hasta ||| + eae so) cause | 
 16 | 55747 86.80 ||  “ gq999% by a | 8.7 Bis i 
 17 U5T955 36.60 999279 “12 fosois | 28-25 Ss | 4 
 18 | 760151 36 43 fond ee ‘ants a8 | | 
 202) | WEASEL | Bee dl | - ; : cma | 2 ss | 
 a od=4 9.999257 12 e058 a ane : : 
 21 | 8.706875 | 55 9g 999250 | 22 fu | | Bat ei 
 BE) OBO | BR ‘goa | 18 Tt | 208 
 a imo ann || See ees -€05995 | 935" 39 "991886 | 34 
 baal wecags (> BUIBRS | eee iB | Fae Boe: 
 25) .775223 | ae "ye 999220 || es | 35.38 ie: 
 26 | .777333 | 9x" 02 999212 "12 eee 3497 21th : 
 ign, if Nate os : = j Savane “4a ‘feuos | 2480 216 3 
 mR 36C : .999197 ‘ “ 3 34.63 > 
 : weber, | 24:50 999189 — 786490 tu eee 
 30 | {785675 coment it com) he aap 
 ‘30e Pooorra | 22. |] PFapets 3415 | 200887 | 28 
 mie | 38 ‘soos | 28 || “toeg62 33.98 | “Sos999 | 96 
 Ss | tomes | 0 |) poo ee || | zed | apie 203269 | 25 
 x rss 33.70 te 13 ; (96031 33.68 "001248 24 
 sr | Bag | gun AD || | goeene | Bee 199237 | 28 
 6 : 804 33.38 oa 13 || 800763 33.37 "197235 | 22 
 aq) aie | a |) | Boeees | ears 195242 | 21 
 88 | .801892 | 33 o7 rch io 13 wie | Big) oi 
 39 803876 | 82.93 “999110 a ‘$2 | ee : 
 ass | 3 | Sa | spay 89317 | 18 
 40] | = : : ; : 
 ie vant | | ‘Bipoa | 32-68 187359 | 17 
 41 | 8.807819 | 99 ¢3 999004 | “43 1008 | 22.08 a | 
 42 809777 32.48 “999086 i. eel | eat i | 
 43 811726 32.35 .999077 13 ‘lame | 233 tera | 15 
 44} 813667 32.20 999069 13 sisioi | 32.20 1815 | 1 
 45 ‘ 815599 82.05 999061 “43 ; yee s 05 ra c 
 46 | .817522 | 31°99 ee 13 0 | 58 ms 
 47 | 819436 | 3) "ng 999044 43 sims | L78 135 1 
 49 | ‘ss3040 | 31-62 || “po9034 a5 |) Se | ares) 
 49 | 828240 | 355 999086 i * a aa : 
 50] j625i80 | Sten || |: . comm | os ras 
 ai ri 9.999019 | 4. Oey atan | fam | 3 
 RE OE en eget tts 8 |) Seg] aa] i 
 oe) eet | arenas, eee 15 | “33613 | 31:08 160387 | 4 
 pt | ‘sscor | 32-82) “ooeags 1h. |) garam | Bhar “360670 | 4 
 BS | isouise | 30-82 || ones ieee Sa 160837 | 3 
 BS) | 3s “goss a 839163 30.58 "159002 | 2 
 pr | isgsizo | 30-55 . 998967 15 || “g4g98 30.45 "157175 | 1 
 me ese | 3) || ankton |. Sapam. |S oete 1:155356 | 0 
 Bea eevee |" Bree 998950 | Te Wri Bay 1 
 col Ripiages 4¢: B18 9.998941 | 8.844644 by vo 3 
 7 | Sine. D. 1”. || Cotang. | D. 1 é 
 i | D 1" Sine. 5 ae. 
 ‘~} Cosine. 
 
COSINES, TANGENTS, AND COTANGENTS. 
 
 iar 
 ’ | Sine. /D, ae Cosine. | D. 1". Tang. D. 1". | Cotang. Oy 
 
 | 
 
 | Se a 
 | 11.155356 | 60 
 
 | 8.843585 | , || 9.998941 8.844644 
 
 3751 ©, 905736 | 52: .092853 | 23 
 38 | .907297 | *&?- 
 39 | .908853 | 2: 
 | 910404 
 
 41 | 8.911949 
 2 . 913488 
 43 . 915022 
 44 | .916550 
 45 . 918073 
 
 998589 
 998578 
 “998568 
 -©2 |) 998558 
 
 907147 | $2: 
 "908719 i 091281 | 22 
 910285 | Se'go | . 080715 | 21 
 -911846 | S5"99 | 088154 | 20 
 
 © 
 <= 
 Di 
 
 Oro 
 (9 2) 
 or 
 
 ~~ 
 | 0 | 8.848585 | 35 og tes 15 || S-BasOe* | e998 
 1 api | 29.93 || -998982 "15 || -846455 | g'9g | — .158545 | 59 
 Btemerano |OROS00 Ihccapsee, | OLAS It) eee Minton | epdet ad 158 
 3 | .848971 | SoG || .998014 15 |; -850057 | So'go | -149948 | 5% 
 4) .850751 | 99'5> || -998905 | “48 851846 | Sony | -148154 | 56 
 5 | -R5w25 | a9 43 |) 908606 | 75 | -soBoNs | DOS | 14632 | 5b 
 | &) S549) Soca5 || -2abesz | 18 || -soed03 | Soh | 14s5a7 | 54 
 g} -c20he) | 99.20 || 22888 | a5 || 857171 | -Oo°g5 | — .142829 | 53 
 j 8 | .857801°'| 5998 .998869 45 858932 | $9’ o9 141068 | 52 
 ‘a sarees | 28 95 eae "45 860686 | 9942 |  -189814 | 51 
 | O85 | tae || O98R5 a eke ENS a6 wd.in | errer 
 | a 28-85 || spe | Mea 862433 99/09 |  - 180564 | 50 
 11 | 8.863014 | 96 7g || 9.998841 15 || 8.864173 | og ge | 11.185827 | 49 
 a | Leny Hos | 98°69 || fod ae 865906 poor .184094 | 48 
 t ¢ Oe Ty || . B2¢ Pe 567635 | ao 3236 | 47 Wi 
 14| ‘g68ie5 | 28-50 || ‘ooggi3 | -17 "860851 eres Lae 46 
 ae Eececonés | 228928 \rseacaod | oI. |bateermes |obeeb5 | norapeees p86 
 Die cmtan | 282R8 jlosiegaron | 15. Par gigoee | Mg as |) .xfeeese fae 
 16 | 871565 | 98°5~ 998795 0 82770 | Saag | 127280 | 44 
 : 17 | 878255 | So'oe ||. 998785 er 874469 | So°35 | .125531 | 43 
 18 | 874938 28.05 | C0876 Sait 4 Ly 28.22 | 99e< | 
 | mv | eects 2795 || easoag 17 eee 98 19 . 123838 42 
 : 9 | 2O10615 | nee ee | .IIO(6 cans R7784! Prien OTT | 
 20 | ie7eess | 27-83 || ‘gggtsz | 15 |] “erasag | 28:00) -Fourt | Go 
 i oscs, DON ace Sas | Rea aaa se ek 
 21 | 8.879949 | oe po || 9.998747 ~ || 8.881202 | og» we | 11.118798 | 39 
 : 22 | 881607 | Spo || 908738 | 13 || “gence | 20-78 | "117131 | 38 | 
 : 23 | 888258 | 92745 || 998728 , 884530 | Soke .115470 | 37 i 
 24 | 884903 | So°gq || .998718 | 4 886185 | Ger |  .118815 | 36 
 Bees near | REPO loi eeeee | A |i, BREE Abe as | TIMtOT 85 t 
 | 26 | 8881 (4) oe Fo . 998699 er 289476 oy on |  .110524 | 34 i 
 BE Pee oy) SPO liigieeegey | O17 llaxmeanien Coed? | tutes dees 
 28 | .891421 | S445 |] .99867% a 892742) | Sy 7258 | 3: 
 99 | 893035 | 26.90 || ae i pithas 27.0% 107258 32 i 
 Gr tawinsers | 2000 ockenecc:| Celt |feadteence | 26-07 | er ckeeOes fel i 
 | 80 | 894643 | 36 79 || 998655 ty -895984 | Se g~ |  .104016 | 30 
 B1 | 8'896246 | 96 gg || 9.998649 | 17 || 8.897596 | 96 ng | 1.102404 | 29 | 
 82 | 897842 | Seiry ||  -998639 1” -899203 | $6'gy |  -100797 | 28 
 | ss 890432 | 26.42 || -998629 | “yy .900803 | Seng |  -099197 | 27 
 $ ( t | aU.26 |! R6 . QN9202 wd. led Vays a 
 Be ratiercs 020-82 les taeceie 1 Sci? |lesneesees | oe'as | on Uez60e y26 
 De Mahecrics | P2022 Ih wlgueeon) | Tale: llestareven | 2008S | ryepnaeee ieee 
 36 904169 | 56°45 . 998595 47 . 905570 9698 .094430 | 24 
 18 
 17 
 17 
 1? 
 rf 
 
 . 918401 
 
 = 
 =) 
 
 65 || 9-998548 OF 83 | 
 ar || 998537 914951 | Orme | 085049 | 18 
 “47 | 998527 |} -916495 | 95° no |  .088505 | 17 
 ‘oo ||. 998516 BS “018034 | aoa 081966 | 16 
 ; 998506 wos 919568 | S2°°S | 080432 | 15 
 
 gg | 11.086599 
 
 ID 
 co 
 
 TOL OU OL OTOL 
 iY) 
 CO 
 
 OW WWW PW IW WW 
 
 _ 
 ° 
 
 46 | 919591 | $59 || -998495 | “yn -921096 | Ge°ag |  -078904 | 14 
 
 47 | .921103 | G2 "45 || 998485 | “42 -922619'| Sr "So 077881 | 13° | | 
 
 48 | .922610 | orgs || -998474 | 4 924136 | Se°So | 075864 | 12 | | 
 | 49 | .924112 | Oy'gn ||  .998464 "ig || -925649 | eo. | eed! ay aes 
 
 998453 QRTIEB.| “SecRs .072844 | 10 
 9.998442 1g || 8-926658 | x | 11.071342 
 . 52 928587 | 37: SOBABT | ates tS */O80155 1) Agata bi SY OG9845 
 
 53 | .930068 | Sigg || 998421 | Hl 981647 | os ates 068353 
 54 | 931544 | O)'r5 || .998410 a8 933134 066866 
 a | » 1 
 
 5U .925609 | 9S," 
 
 51 | 8.927100} 
 
 ra) 
 x 
 SO 
 Qo 
 
 55 | .933015 | S¢-28 998399 | ; 984616 | S7rnc 065884 
 
 =o be ps 24.43 boa | tS Anas 4 eae ohne 
 
 56 | .934481 | Siia~ || .998888 | : . 986093 | iS 063907 
 
 By | 935942 | 24.35 | 998307 18 02" C5 | 24. 162435 
 
 of JOIAIAK 2 27 j JI | 18 ~JO(O0D | 94 45 002435 
 | | 
 
 58 | .987398 || .998866 . 9850382. | 
 
 O49 060968 
 
 | 94,909 | | |), Aes 4 
 59 | 938850 | 31:9? || ‘998355 | 48 || ‘940494 Sr ay |... .059506 
 60 | 8.940206 | **:10 || 9 oogaa4 | - 8.941952 | “4-9 | 14 "o58048 
 | 
 
 ‘ | Cosine. | D. 1’. 
 
 Sine. | D.1’. || Cotang.| D.1". | Tang. 
 
 co ~| COR wwhonwMHO.e 
 qn | 
 ° 
 
 94° 
 
TABLE XXV.—LOGARITHMIC. SINES, 
 
 3 Sine. D. 1", || Cosine, | D. 1’. 
 ene: ARON See Desay 
 | pares | 2403 | ogesss | 18 
 2| 943174 | 23°25 || .gogsee | -48 
 € A ane | we. | O82 oA 
 4} “p1o3t | 23-80 |) “goss00 | 18 
 | MARC wO.dU | QgR9R 16 
 6 | coseera | 23-63 || ‘gepaey | 20 
 7 | 1950287 eae || -pgeece | trie 
 FRED ae teed) | Giant | fe Bp 
 9 | 1953100 | 33-29 || ‘oge24: By 
 10 | 954499 | 53-82 || ‘oogzae | -28 
 11 | 8.955804 | 45 "4p 9.998220 ie 
 ‘ 9575 3. QgR9(x : 
 13 | lasso70 | 23-10) “ggsior | 20 
 14 | 960052 | $3°p5 || -298185 | “2p 
 Mae 2.87 cs lle .18 
 17 | -oeivo | 22-82 || copsist | 2° 
 18 | 965534 | 99°65 || 998189] “4g 
 19 | 966803 | 22-65 || ‘oogi28.|  -38 
 20 | .968249 | $3-89 || ‘oosiie | 30 
 21 | 8.969600 | 5 yx || 9.998104 m5 
 i- rngan | reed 992099 . 
 53 | ‘o7e089 | 22-37 || “booq | -20 
 24 | 973628 | 22-82 |] “gogogg | -20 
 25 | 974962 | 22-23 |) ‘oggos6 | -20 
 Be eens | Hoed0 [I beoaananll| Re 20 
 35 | coreoat | 22-03 || “gogpnn | -20 
 20 | 980259 | 21-97 || ‘oogoo8 | -22 
 30 | .981573 | 31:39 || .997006 | «30 
 98288; rw || 9.99798 
 32 | cossisg | 24-27 || “gorore | -20 
 Qe FAQ @lwla oar BC oe 
 at | ‘gsorsa | 21-63 |] “gyrauy | 20 
 35 | 988083 | 9150 |) -907085 | op 
 37 | 990660 | 21-43 || “ggroio | -20 
 38 | 1991943 | 21-88 || ‘ogegor | -22 
 39 | 993222 13a. || eacoenesb || ate 
 40 | .994497 a ie || 907B72,| S25, 
 41 | 8.995768 | 5, 4» || 9.997860 ae 
 ¢ 907028 ra e 9978 ~ «ee 
 43 | vgog209 | 21-05 |] “Boreas | 20 
 44 | 8.999560 | 21-02 || “gg7gog | +22 
 45 | 9.000816 | 20:93 || ‘gore | -22 
 lee epee (ereo288 ll aaenaecoy | mee 
 ae eaanara (re20-82 ete 18422 
 48 | 1004563 | 20-75 || “ggvmay |  .22 
 49 | 005805 ores | lgorr5a |  -22 
 50 | .007044 | 55° 57 | 997745 39 
 51 | 9.008278 | | 9.997732 
 2 20.53 cit 22 
 52 1.009510 | 99" 45 oe | aaeee 
 BP Ve tetany [ede (a eeaens | base 
 bs | loisiea | 20-83 || “orega | 22 
 56 | .014400 | 20-80 || “oovee7 | -22 
 »& =< € ae | ner 7 
 pe oMaiercese | 2018 || Seed pee 
 BoP hiere, (20d? aeeeent | Maps 
 60 | 9.019235 | 9.997614 
 4 Cosine. | D. 1" Dsl" 
 
 Sine. 
 
 174° 
 
 Tang. D. 1". | Cotang. | ’ | 
 | 8.941952 | 94 99 | 11.058048 | 60 
 943404 | Sits .056596 | 59 
 || -944852 | Sips | -055148.| 58 
 || -946295 | 93°98 053705 | 57 
 leew 947734 oS “90 052266 | 56 
 | 949168 | 55° o5 "050832 | 55 
 950597 | G3" mn .049403 | 54 
 952021 | 95" 67 047979 | 53 
 953441 | 53"59 |  -046559 | 52 
 954856 | 99'55-|  -045144 | 51 
 956267 | 93" 4r 043733 | 50 
 | 8.957674 | oe a | 11.042326 | 49 
 | .959075 aah "040925 | 48 
 || -960473 | $355 039527 | 47 
 -961866 | $9°T5 .038134 | 46 
 . 963255 93 07 086745 | 45 
 964639 93 00 .035361 | 44 
 .966019 99 “99 .033981 | 43 
 967394 | 55" ge 032606 | 42 
 968766 ‘ay 2 031234 | 44 
 970133 99.79 .029867 | 40 
 8.971496 | 55 gx | 11.028504 | 39 
 1972855 | es 0 027145 | 38 
 -974209 | 55°29 .025791 | 37 
 975560 | 55°43 .024440 | 36 
 -976906 | 99’ 30 . 023094 | 35 
 978248 | 55°39 021752 | 3 
 979586. | SS-Se 020414 | 33 
 -986921 | 55°4r .019079 | 32 
 982251 | 95° 0 017749 | 31 
 983577 203 .016423 | 30 
 8.984899 | 9, gy | 11.015101 | 29 
 .986217 | 51 "99 | 018783 | 28 
 987532 | 51 gs 012468 | 97 
 . 988842 | 5; ne .011158 | 26 
 -990149 | 54", 009851 | 25 
 991451 | 9) 65 008549 } 24 
 992750 | 61 "58 007250 | 23 
 994045 | 94 "R3 005935 | 22 
 995337 | 94°45 .004663 | 2 
 996624 | 54°45 003376 | 20 
 8.997908 | 94 99 | 11.002092 | 19 
 8.999188 | 9,58 | 11.000812 | 18 
 | 9.000465 | 94°59 | 10.999535 | 17 
 001788 | 94°45 .998262 | 16 
 -003007 | 519g 996993 | 15 
 004272 | 51°93 995728 | 14 
 005584 | “Sa°e8 .994466 | 13 
 -006792 | 30"¢ Mi .993208 | 12 
 -008047 | 55 gx .991953 | 114 
 -009298 | 59°gq .990702 | 10 
 9.010546 | 99 we | 10.989454 | 9 
 -011790 | 50‘ 68 . 988210 | 8 
 -013031 | 59° go 986969 | 7 
 -014268 | 5p ‘ae 985732 | 6 
 015502 30) 50 .984498 | 5 
 016732 | 9945 983268 | 4 
 -017959 | 59°49 .982041 | 3 
 -019183 | $9‘33 .980817 | 2 
 -020403 | 59 ‘g 979597 | 1 
 9.021620 *“° | 10.978380 | 0 
 Cotang. | D. 1’ Tang. d 
 
 \ 
 
 oo 
 a 
 
COSINES, TANGENTS, AND COTANGENTS. 
 
 Cotang. 
 
 911902 
 
 | 10.910856 
 
 Sine. | D.1". || Cosine. | Tang. 
 | 
 #019285 | 99.00 || 9-997614 | 9.021620 | 
 020435 | 39 ‘95 . 997601 022834 
 -021632 | 49" g9 997588 024044 
 -022825 | 49 gs 990574 .025251 
 -024016 | 397m 997561 026455 
 -025203 | 39/0 997547 027655 
 -026386 | 39°49 || . 997534 028852 
 027567 | 19.62 997520 | .030046 
 O28744-| 39" 5> 997507 | .031237 | 
 .029918 19 52 . 997493 | 032425 
 -031089 | 49 47 997480 | .0383609 
 QOOrM | ree OUP 
 rary panto. | 097 06 Neekin 
 - me ‘Ad | 19.35 ee th ow 085969 | 
 | -084582 | 39°90 .997439 .037144 | 
 | -085741 | 49'ox || 997425 .038316 | 
 | .036896 | 59 ‘5p 997411 .089485 
 | .038048 | 39°45 .997897 .040651 
 | 039197 | 49'o8 997383 || .041813 
 | .040342) 39°p5 || .997369 |} .042973 | 
 -041485 ] 49°99 || -997355 || .044130 | 
 -042625 | 49’ 95 || .997341 | 045284. | 
 9.043762 | 4 g¢ || 9.997327 9.046434 | 
 044895 | 38° gx .997313 | .047582 
 .046026 18.80 997299 048727 | 
 | 047154 | 53h. 997285 || 049869 | 
 | -048279 | jo' 63 || .997271 | .051008 | 
 -049400 | 39'e5 || .997257 |} .052144 
 -050519 | 38° @q || .997242 [| 053277 
 051635 | je’ 5m |) .997228 || 054407 | 
 052749 | 3Q°59 || 997214 || .055535 
 -058859 | 39°45 || .997199 | .056659 
 91054966 | 49 49 || 9.997185 || 9.057781 | 
 056071 | 18.35 .997170 “69 ||. - 058900 
 057172} 38°35 997156 “ox || 060016 | 
 -O58271 | 42°57 997141 con {t= eobd Le 
 | .059367 | je°So .997127 Sor? || 2062240 
 | 060460 | 48°58 997112 "93. «|| ~~ - 068348 
 | .061551 | 36°43 .997098 ee .064453 
 | -062639 | j6'og || -997083 Seb? .065556 
 063724 | 49°93 || .997068 | “5? 066655 
 | 064806 | jr'99 || .997053 93 067752 
 9.065885 | 17.95 || 9.997039 | 95, || 9.068846 
 | 066962 wn 90 || . 997024 On .069938 
 | 068056 | je"p5 || .997009 ase 071027 
 | 069107 | jn" 25 . 996994 : 072113 
 OVO176 | sme || 996979 | 073197 
 OT12A2 | sn 'ns . 996964 | .074278 
 072306 | 44" 62 . 996949 | .075856 
 .073366 63 || 996934 | .076432 
 cen 7 60 || -996919 | 077505 
 07548 ee :996904 078576 
 (.00 
 9.076533 750 || 9.996889 9.079644 
 077583 | je ge |) .996874 | 080710 | 
 .078631 LE pil .996858 08177 
 079676 1738 || .996843 | 082833 
 | -080719 | 57°39 || .996828 | .083891 
 | 081759 | 1730 996812 | .084947 
 | 082797 | anos || .996797 |; .086000 
 083832 | 44°95 || .996782 |} 087050 
 | 084864 | ye'Fy || .996766 .088098 
 | 9.085894 | -~‘:** || 9.996751 | 9.089144 
 4 | 
 Gositre. | Dr1% Sine. || Cotang. | 
 
 Tang. 
 
 | 10.978380 
 
 .977166 
 
 975956 
 974749 
 973545 
 972345 
 .971148 
 969954 
 968763 
 967575 |! 
 966391 
 965209 
 .964031 
 . 962856 
 . 961684 
 .960515 
 959349 
 .958187 
 957027 
 955870 
 954716 
 
 10.953566 
 .952418 
 . 951273 
 . 9501381 
 . 948992 
 . 947556. 
 . 946723 
 . 945593 
 . 944465 
 . 943341 
 
 942219 
 . 941100 
 . 939984 
 . 938870 
 . 937760 
 . 936652 
 985547 
 - 934444 
 . 938345 
 - 932248 
 
 .931154 
 . 980062 
 928978 
 927887 
 . 926803 
 925722 
 924644 
 923568 
 922495 
 921424 
 
 . 920356 
 . 919290 
 918227 
 . 917167 
 .916109 | 
 . 915058 
 . 914000 
 
 912950 
 
 CHUWWRORMUIDS 
 
 ~ 
 
TABLE XXV.—LOGARITHMIC SINES, 
 
 Ro 
 
 co 
 
 | 9.995753 
 
 4 Sine. 
 0 | 9.085894 ” 13 
 1 | .086922 a 3 
 21 087947 tee 
 3 | .088970 be 0 
 4 | 039990 | Te-p) 
 5 | (091008 16 93 
 6 | .092024 Ba 
 7 | .093037 Oe 
 8 094047 16.82 
 9 |. 095056 | 48"rr 
 10 | .096062 16. “9 
 11 | 9.097065 es 
 2) 098066 | ane 
 13 | .099065 16 62 
 14 | .100062 | j¢°ee 
 15 | .101056 | 16 53 
 16 | .102048 16.48 
 17 3308087 | “Fea 
 18 | .104025 16. Hs 
 19 ||; *.105010 | “Fe *5e 
 590¢ =< 
 20 | 105092 | 36°35 
 21 | 9.106973 | 16 5 
 22 | .107951 | i¢°5, 
 23 | .108927 16 23 
 24 | 3109901 | As 
 25 | 110873 | 16-20 
 26 | .t11842 | 18-18 
 27 | 112809 | 46°08 
 28 | 118774 | 46'o 
 29) .114737 16 02 
 30 .115698 15.97 
 31 | 9.116656 
 BONO | aa 
 33 | (118567 veg 
 34 | .119519 is 
 35 | 120469 | 15.88 
 86 | (121417 Hee 
 7 | .122362 15 ae 
 38 |  .123306 | 15 
 89 | .124248 | 465 
 41 | 9.126125 
 42 | .127060 Hes 
 43 )x-doggo8 | (20x82 
 44 | 198925 | 18-58 
 45 | 129854 oo 
 46 | .130781 15 49 
 7 | .181706 | 33°40 
 48 | .132630 
 - pas 15.35 
 49 . 1383551 15.32 | 
 50 | .134470 | 35°53 
 51 | 9.135387 > 
 p2 | .136303 | 15-27 
 53 | .187216 | 42°65 
 B4 | 188128 | “42°F 
 55 | .139037 15 40 
 B6 | 189944 | 45°49 
 7 | .140850 | 32"o» 
 58 | .141754 nop 
 59 | .142655 we 
 60 | 9.148555 | °°: 
 / | Cosine. | D, 1” 
 
 Dy 1s | Cosine. 
 
 9.996751 
 . 996735 
 . 996720 
 . 996704 
 . 996688 
 . 996673 
 . 996657 
 . 996641 
 . 996625 
 . 996610 
 . 996594 
 
 . 996578 
 
 . 996562 
 . 996546 
 . 996530 
 . 996514 
 . 996498 
 
 ive} 
 
 996482 
 
 . 996465 
 . 996449 
 . 996433 
 
 9.996417 
 
 . 896400 
 .996884 
 
 .§96368 
 
 . 996351 
 
 996335 
 
 . 996318 
 
 -986302 | 
 
 $9285 
 996269 
 996252 
 996235 
 996219 
 . 996202 
 996185 
 996168 
 996151 
 996134 
 996117 
 996100 
 996083 
 996066 
 996049 
 996032 
 996015 
 995998 
 995980 
 995963 
 995946 
 995928 
 
 9.995911 
 
 .995894. | 
 
 995876 
 . 995859 
 . 995841 
 995823 
 . 995806 
 . 995788 
 99577 
 
 Sine. 
 
 D. 1". || Tang. Doi. | Cotang. ? 
 a7 || 9-089144 | 4» 99 | 10.910856 | 60 
 95 || -090187 | jn er 909813 | 59 
 o7 091228 | 4729 .§08%72-} 58 
 OF 092266 | 305m GUT%84 | 57 
 “On 092802 | yn os .§06698 | 56 
 On 094586 | 4°59 .90E664 | 55 
 "ov || -095867 | jaya |  .804638 | 54 
 “Sy || 096895 | 1712 902605 | 53 
 $25 097422 | jn’ go 802578 | 52 
 “Or 098446 | yo" p9 901554 | 51 
 OW .099468 16.98 . 900532 | 50 
 oy || 9.100487 | 46 os | 10.899513 | 49 
 OW 101504 | 3695 .§98496 | 48 
 joe .102519 | 3996 697481 | 47 
 “On 108582 | jg'g3 | 896468 | 46 
 oy || 204542 | 679 | -£95458 | 45 
 pas -105550 | 46° re 694450 | 44 
 “ae -106556 | 3¢ "ro 893444 | 43 
 97 || -107559 | 36" 68 .€92441 | 42 
 On | .108560 | 36 '¢5 891440 | 41 
 a | ORAC . 48 
 ‘oy || shea 16 62 .890441 |} 40 
 
 || 9.110556 ro | 10.889444 | 39 
 28 || aat551 | 18-58 | eeeaao | 38 
 Sa 112543 | 3659 | 887457 | 87 
 | “og -113583 | 36° gr .886467 | 36 
 ae 114521 16 43 885479 | 35 
 “38 -115507 | 36°49 884498 | 2 
 | oe -116491 | 3¢"95 .€83509 | 33 
 98 117472 | 4633 .€82528 | 82 
 oy || -218452 | 36°58 .881548 | 31 
 So «|| («119429 16.25 880571 |; 80 
 og || 9-120404 | 46 a | 10.879596 | 29 
 ‘or || 121877 | 648 878623 | 28 
 “9g || -122848 | 4645 877652 | 27 
 og || 128817 |. 36745 876683 | 26 
 | ‘og || -124284 | 368 '9g 875716 | 25 
 og || -125249 | 36°03 874751 | 24 
 oR :126211-| 36°90 873789 | 23 
 "og || 12792 | 55" ge .872828 | 22 
 9g ||, -128130 15 95 .871870 | 21 
 
 2 &O | € lo st FOOTE € 
 3 scam | | eae 
 28 || 1ag0coa | 19-88 |“ “eego06 | 18 
 “38 131944 | 32" g0 868056 | 17 
 “58 182893 | 32m .867107 | 16 
 | ‘og || 188889 | 92° p5 866161 | 15 
 "30 (|| +184%84 | 45 '29 865216 | 14 
 ‘og || +185726 | 55 '68 864274 | 13 
 38 126667 | 35" ¢3 .863333 | 12 
 “30 187605 | 45°69 .862395 | 11 
 "og || -188542 | 55's .861458 | 10 
 9.189476 10.860524 | 9 
 88 || 1240409 | 18-85 |" “esoso1 | 8 
 "98 -141840 | 35°48 .858660 | 7 
 “30 142269 | 35°45 857731 | 6 
 30 148196 | 45°49 .256804 | 5 
 "OR 144121 | 35°99 855879 | 4 
 “30 145044 | 45°37 854956 | 3 
 “38 145966 15 32 854034 | 2 
 30 146885 | 45°39 858115 | 1 
 os 9.147803 | | 10.852197 | 0 
 Deg Cotang. | D. 1’ Tang. 
 
Sine. 
 
 | 9.143555 
 144453 
 . 145349 
 . 146243 
 147136 
 . 148026 
 148915 
 . 149802 
 . 150686 
 . 151569 
 152451 
 
 9.153330 
 154208 
 155083 
 155957 
 . 156830 
 157700 
 158569 
 159435 
 160301 
 .161164 
 
 9.162025 
 162885 
 163748 
 164600 
 . 165454 
 . 166307 
 167159 
 . 168008 
 168856 
 169702 
 
 9.170547 
 171389 
 172230 
 173070 
 173903 
 174744 
 175578 
 176411 
 1V7 ate 
 
 .178072 
 
 | 9.178900 
 2179726 
 . 180551 
 .181374 
 . 182196 
 . 183016 
 . 183834 
 .184651 
 
 185466 | 
 | . 186280 
 | 9.187092 
 | .187903 
 .188712 
 » 189519 
 
 . 190825 
 .191180 
 
 .191933 
 .192734 
 * 193584 
 9.194332 
 
 D2". | Cosine. 
 
 COSINES, TANGENTS, AND COTANGENTS. 
 
 Tang. 
 
 | 9. 995753 
 995735 | 
 995717 
 . 995699 | 
 . 995681 
 . 995664 
 . 995646 
 b 99; 5628 
 . 995610 
 . 995591 
 . 995573 
 
 995555 
 . 995537 
 - 995519 
 995501 
 995482 
 995464 
 . 995446 
 - 995427 
 995409 
 . 995390 
 
 | 9. 995372 
 995353 
 . 995334 
 . 995316 
 . 995297 
 .995278 
 . 995260 
 995241 
 995222 
 
 995203 
 
 995184 
 . 995165 
 . 995146 
 995127 
 .995108 
 .995089 
 995070 
 .995051 
 . 995032 
 . 995013 
 
 9. 994995 
 994974 
 994955 
 994935 
 . 994926 
 .994856 
 994877 
 . 994857 
 .994838 
 .994818 
 
 994798 
 99477 
 994759 
 994739 
 994720 
 994700 
 994680 
 994660 
 | .994640 
 
 9.994620 
 
 io) 
 
 | 9.147803 
 || (148718 
 - 149682 
 .150544 
 .151454 
 . 152363 
 .158269 
 154174 
 .155077 
 .155978 
 . 156877 
 
 1 wre 
 
 Cbbe 
 
 158671 
 . 159565 
 160457 
 161347 
 162286 
 163128 
 . 164008 
 .164892 
 165774 | 
 | 9.166654 
 
 167582 
 
 . 168409 
 . 169284. 
 .170157 
 .171029 
 .171899 
 
 172767 
 . 178634 
 . 174499 
 .175362 
 . 176224 
 177084. 
 .177942 
 .178799 
 .179655 
 . 180508 
 . 181360 
 . 182211 
 . 183059 
 
 .188907 
 .18475 52 
 185597 
 
 . 186439 
 187280 
 
 .188120 
 . 188958 
 .189794 
 . 190629 
 . 191462 | 
 
 192294 
 193124 
 .193953 
 .194780 
 . 195606 
 .196430 
 .197253 
 198074 
 . 198894 
 9.199713 
 
 Cotang. 
 
 | 10.852197 
 
 851282 
 850368 
 849456 
 848546 
 847637 
 846731 
 845826 
 844923 
 844022 | 
 843123 
 
 10. 842225 
 841329 
 (840435 
 839543 
 838653 
 837764 
 836877 
 835992 
 835108 
 834226 
 
 10.833346 
 832468 
 831591 
 830716 
 829843 
 828971 
 828101 | 
 827233 
 826366 
 825501 
 
 10. 824638 
 823776 | 
 822916 
 822058 
 
 .821201 | 2 
 
 820345 
 819492 | 
 818640 
 817789 
 816941 
 10.816098 
 815248 
 814403 
 813561 
 812720 
 811880 
 811042 
 810206 
 809371 
 - 808538 
 
 | -10.807706 
 
 806876 
 
 806047 
 . 805220 
 . 804894 
 ~ 808570 
 802747 
 "801926 
 .801106 
 
 10.800287 
 
 Cosine. 
 
 | 
 |} 
 
 Sine. 
 
 Cotang. 
 
 Tang. 
 
 fone whe sie 
 
 Or DO CoP OF 
 
 ~ 
 
eS 
 CODNOTR WWE OS | 
 
 Sine. 
 
 9.194832 
 195129 
 1195925 
 .196719 
 197511 
 . 198802 
 .199091 
 .199879 
 200666 
 .201451 
 
 (202234 | 
 
 9.203017 
 253797 
 204577 
 205854 
 -206131 
 . 206906 
 207679 
 
 208452 
 
 209222 
 
 . 209992 
 
 210760 
 .211526 
 .212291 
 .218055 
 213818 
 214579 
 .215838 
 216097 
 216854 
 217609 
 9.218363 
 .219116 
 . 219868 
 220618 
 221867 
 .222115 
 222861 
 
 2238606 
 
 i=) 
 
 224349 | 
 
 225092 
 9.225833 
 226573 
 220311 
 228048 
 228784 
 
 2229518 | 
 
 280252 
 230984 
 231715 
 232444 
 .233172 
 . 233899 
 234625 
 .235349 
 .236073 
 
 ve) 
 
 .286795 
 
 237515 
 
 238235 
 
 238958 
 
 9.239670 
 
 o . . . . . . . 
 
 Jes} 
 
 TABLE XXV.—LOGARITHMIC SINKS, 
 
 Tang. 
 
 9.199713 
 200529 
 201345 
 
 | , 202159 
 202971 
 203782 
 204592 
 205400 
 206207 
 207013 
 207817 
 
 9.208619 
 
 | .209420 
 
 210220 
 
 211018 
 
 211815 
 
 212611 
 
 213405 
 
 214198 
 
 214989 
 
 | 215780 
 
 9.216568 
 217356 
 218142 
 218926 
 219710 
 220492 
 221272 
 222052 
 222830 
 223607 
 
 | 9224382 
 
 225156 
 
 225929 
 
 226700 
 
 227471 
 
 228239 
 
 . 229007 
 
 22977 
 
 . 230539 
 
 .231302 
 
 9.282065 
 232826 
 233586 
 234345 
 235108 
 235859 
 2086614 
 237368 
 2388120 
 238872 
 
 | 9.289622 
 
 | .240871 
 
 241118 
 
 248354 
 . 244097 
 244839 
 245579 
 9.246319 
 
 241865 
 242610 
 
 fre re pe fred fh fee fed peek eek 
 
 WWW WWWWWHO 
 
 Cosine. 
 
 Cotang. 
 
 o 
 
 | 10.800287 
 | 
 
 799471 
 T9BG55 
 797841 
 797029 
 796218 
 795408 
 794600 
 793793 
 792987 | 
 792183 
 
 10.791381 
 "790580 
 “789780 
 788982 
 “788185 
 787389 
 786595 
 785802 
 "785011 
 784220 
 
 10.7834382 
 
 10. 
 
 10.767935 | 
 
 . 768386 
 7162652 
 61550 
 761128 | 
 
 10.760378 
 
 158629 
 758882 
 758185 
 757890 
 756646 
 755908 
 755161 
 154421 
 10.753681 
 
 Tang. 
 
Sine. 
 
 Cosine. 
 
 DrIDvwk WOH OS 
 
 =) 
 
 ao) 
 
 . 239670 
 
 . 240386 
 .241101 
 
 .241814 
 
 242526 
 
 . 243237 
 -243947 
 244656 
 . 245363 
 . 246069 
 246775 
 247478 
 248181 
 . 248883 
 
 249583 
 250282 
 250980 | 
 
 201677 
 .252373 
 .252067 
 208761 
 254453 
 
 255144 | 
 
 250884 
 - 200523 
 ania 
 201898 
 208583 
 259268 
 .209951 
 .260633 
 . 261314 
 .261994 
 262673 
 
 .263351 
 
 264027 
 264708 
 265377 
 
 266051 
 . 266723 
 
 207395 
 
 268065 
 
 . 268734 
 . 269402 
 . 270069 
 
 .270735 
 .271400 
 .272064 
 212126 
 
 . 273888 
 . 274049 
 .274708 
 275867 
 
 276025 
 
 276681 | 
 
 277339 
 
 wh 
 
 277991 | 
 278645 | 
 
 219297 
 
 . 279948 
 
 9.280599 
 
 DNF AP ay 7 
 
 =) 
 
 Oo 
 
 Ve) 
 
 Je) 
 
 9.993351 
 993329 
 993307 
 993284 
 993262 
 993240 
 993217 
 993195 
 993172 
 993149 | 
 993127 
 993104 
 993081 
 993059 
 993036 
 992013 
 || .992990 
 = || 992067 
 992944 
 992921 
 992898 
 
 . 992875 
 . 992852 
 992829 
 . 992806 
 ', 992783 
 . 992759 
 992736 
 992713 
 - 992690 
 
 . 992666 
 
 992648 
 . 992619 
 . 992596 
 992572 
 . 992549 
 . 992525 
 . 992501 
 . 992478 
 . 992454 
 . 992430 
 992406 
 . 992382 
 
 . 992859 
 - 992335 
 992311 | 
 - 992287 
 . 992268 
 992239 
 . 992214 
 . 992190 
 
 992166 
 992142 
 992118 
 .992093. | 
 992069 
 992044 
 992020 
 991996 | 
 ~ || .991971 
 
 || 9.991947 
 
 | Cosine. | D. 1". | 
 
 Sine. 
 
 COSINES, TANGENTS, AND COTANGENTS, 
 
 Tang. 
 
 9.246319 
 .RAT057 
 -RATT94 
 . 248530 
 249264 
 . 249998 
 . 250730 
 .251461 
 
 .252191 
 
 252920 
 203648 
 9.254374 
 
 .255100 
 255824 | 
 
 206547 
 257269 
 .297990 
 208710 
 259429 
 . 260146 
 . 260868 
 9.261578 
 202292 
 . 268005 
 268717 
 
 . 264428 
 
 .265138 
 . 265847 | 
 
 . 266555 
 .267261 
 .267967 
 9.268671 
 . 269375 
 270077 
 210079 
 201479 
 202178 
 
 212876 | 
 
 27857 
 244269 
 274964 
 
 9.275658 
 . 276351 
 .277043 
 277784 
 . 278424 
 .279113 
 .279801 
 280488 
 281174 
 .281858 
 
 9 282542 
 . 283225 
 .283907 
 . 284588 
 285268 
 . 285947 
 286624 
 .287301 
 287977 
 
 9.288652 
 
 Cotang. 
 
 10.%53681 
 
 152943 
 752206 
 
 751470 | 2 
 750786 
 
 . 750002 
 
 .T49270 
 
 748539 
 
 747809 
 747080 
 746852 
 745626 
 
 . 744900 
 T4416 
 
 7438453 
 742731 | 
 742010 
 741230 
 740571 
 . (89854 
 739137 
 «738422 
 737708 
 
 786995 
 736283 
 135572 
 . 134862 
 (84153 
 . 188445 
 132739 
 732033 
 731329 
 730625 
 (29923 
 (29221 
 
 728521 
 
 nr 
 1271822 
 
 (a 
 
 727194 
 726427 
 725731 
 725036 
 724842 
 7123649 
 722957 
 - 122266 
 21576 
 20887 
 720199 
 19512 
 T1886 
 
 718142 
 
 717458 
 
 T1675 
 
 (40 
 
 - 716093 
 
 712699 
 712023 
 | 10.711348 
 
 | Cotang. 
 
 Tang. 
 
 CHW W ROOM =300 
 
TABLH AXV.—LOGARITHMIC SINES, 
 
 WOMOIAIP WWE © 
 
 ile) 
 
 =) 
 
 We) 
 
 Sine. 
 
 Cosine. 
 
 9.280599 
 
 281248 
 281897 
 282544 
 .283190 
 283836 
 
 284480 
 
 285124 
 285766 
 286408 
 287048 
 287688 
 288326 
 288964 
 .289600 
 290236 
 . 290870 
 291504 
 .292137 
 292768 
 293399 
 294029 
 294658 
 295286 
 .295913 
 .296539 
 297164 
 
 297788 
 
 .298412 
 299034 
 .299655 
 ae 
 . 80089! 
 
 "301514 
 802182 
 802748 
 803364 
 808979 
 .804593 
 805207 
 .805819 
 806430 
 807041 
 .807650 
 
 808259 | 
 . 808867 
 
 . 809474 
 .810080 
 
 .3 10685 | 
 
 .811289 
 f° 811893 
 
 812495 
 .313097 
 313698 
 
 14297 | 
 
 .314897 
 .815495 
 
 .816092 
 .3816689 
 2317 ae 
 
 9.31787 
 
 Cosine. 
 
 ive) 
 
 v=) 
 
 9.991947 
 991922 
 .991897 
 .991873 
 .991848 
 .991823 
 .991799 
 99177 
 .991749 
 99172 
 .991699 
 
 | 9.991674 
 .991649 
 . 991624 
 .991599 
 991574 
 . 991549 
 . 991524 
 
 991498 
 991473 
 991448 
 
 | 9 991422 
 
 .9913897 
 991372 
 . 991346 
 . 991321 
 .991295 
 .991270 
 
 .991244 
 
 ,991218 
 .9911938 
 991167 
 .991141 
 
 . 991115 
 
 . 991090 
 . 991064 
 . 991088 
 
 .991012 
 . 990986 
 . 990960 
 . 990934 
 
 .990908 
 . 990882 
 . 990855 
 . 990829 
 . 990803 | 
 990777 | 
 . 990750 
 . 990724 
 . 990697 
 . 990671 
 990645 
 . 990618 
 . 990591 
 990565. | 
 990538 | 
 . 990511 
 . 990485 
 . 990458 
 990431 
 9.990404 
 
 Tang. 
 
 289826 
 289999 
 290671 
 . 291342 
 . 292013 
 292682 
 . 293350 
 .294017 
 294684 
 .295349 
 
 296013 
 296677 
 297389 
 298001 
 . 298662 
 299322 
 .299980 
 . 800638 
 801295 
 .801951 
 
 802607 
 .808261 
 .803914 
 304567 
 .805218 
 .805869 
 . 806519 
 .807168 
 .807816 
 .808463 
 .809109 
 809754 
 .810399 
 .811042 
 .811685 
 -812827 
 .812968 
 .813608 
 -3014247 
 .814885 
 
 815523 
 .3816159 
 . 816795 
 817480 
 .818064 
 318697 
 819330 
 
 xo 
 
 ie) 
 
 Je) 
 
 © 
 
 .819961 
 
 «820592 
 
 821222 
 
 9.821851 
 
 322479 | 
 
 -823106 
 .823783 
 B24 358 
 
 824983 
 "305607 
 326231 
 .826853 
 
 || 9.827475 
 
 | 9.288652 
 
 Cotang. 
 
 08 3393 
 
 y 0p 661 
 .701999 
 . 701838 
 .700678 
 700020 
 . 699362 
 .698705 
 .698049 
 .697393 
 . 6967389 
 .696086 
 . 695433 
 .694782 
 ,694131 
 .693481 
 . 692832 
 .692184 
 .6915387 
 | 10.690891 
 690246 
 . 689601 
 .688958 
 
 688315 
 
 687673 
 . 687032 
 686392 
 .685753 
 . 685115 
 684477 
 -683841 
 683205 
 682570 
 .681936 
 .681303 
 .680670 
 
 680039 
 
 679408 
 678778 
 | 10.678149 | 
 .677521 
 676894 | 
 676267 | 
 675642 | 
 .675017 
 .674893 
 .673769 
 673147 
 10.672525 
 
 Sine. 
 
 Cotang. | 
 
 Tang. 
 
 10.711348 
 - 710674 
 -710001 
 - 709829 
 708658 
 TOV 987 
 707318 
 106650 
 705983 
 
 705316 
 
 704651 
 : f (08987 ( 
 
COSINES, TANGENTS, AND COTANGENTS. 
 
 9.317879 
 
 o 
 —) 
 
 - 
 
 = 
 SO OID OS CO 2 
 
 i=) 
 
 =) 
 
 Sine. 
 
 Cosine. 
 
 Tang. 
 
 Cotang. 
 
 .318473 
 .819066 
 .3819658 
 820249 
 .820840 
 .821480 
 .822019 
 
 ‘ .322607 
 
 823194 
 
 323780 | 
 
 . 824366 
 . 824950 
 .825534 
 .826117 
 .828700 
 827281 
 .3827862 
 20442 
 .829021 
 -829599 
 
 .330176 
 
 3380753 
 .831329 
 .331903 
 
 .832478 
 
 .833051 | 
 
 .333624 
 ~334195 
 .834767 
 
 835337 
 
 .835906 
 .836475 
 .887048 
 .337610 
 .838176 
 
 888742. | 
 . 889307 | 
 .839871 
 
 .3840484 
 . 840996 
 841558 
 842119 
 842679 
 
 .843239 | 
 
 843797 
 844355 
 
 .844912 
 
 845469 
 
 .846024 | 
 
 346579 
 847134 
 347687 
 348240 
 .348792 
 
 3349343 | 
 
 . 849893 
 .8504438 
 .850992 
 .851540 
 
 | 9.352088 
 
 We} 
 
 Ve) 
 
 eo) 
 
 | 9.990404 
 990378 | 
 990351 
 . 990824 
 
 990297 
 .990270 
 . 990243 
 .990215 
 .990188 
 
 .990161 
 . 990134 
 
 9.990107 
 .990079 
 “990052 
 990025 
 989997 
 989970 
 989942 
 989915 
 989887 
 989860 
 989832 
 . 989804 
 989777 
 989749 
 989721 
 989693 
 989665 
 989637 
 989610 
 989582 | 
 
 .989553 
 . 989525 
 .989497 
 . 989469 
 . 989441 
 .989413 
 . 989385 
 . 989356 
 . 989328 
 . 989300 
 
 989271 
 . 989243 
 . 989214 
 989186 
 
 . 989157 
 
 . 989128 
 989100 
 989071 
 . 989042 
 . 989014 
 . 988985 
 988956 
 988927 
 988898 
 988869 
 
 . 988840 
 98881 1 
 988782 
 988753 
 
 . 988724 
 
 io!) 
 
 Je) 
 
 ile) 
 
 || 9.827445 
 
 o28095 
 Oso 1d 
 O2Id384 
 .8 29953 
 .830570 
 .081187 
 .831803 
 .082418 
 .0830383 
 .883646 
 804259 
 .384871 
 . 800482 
 .8386093 
 886702 
 837311 
 .837919 
 .o80027 
 .3889133 
 .8097389 
 .840344 
 .840948 
 -041552 
 842155 
 042757 
 .343358 
 .343958 
 .844558 
 -340157 
 . 845755 
 . 846353 
 346949 
 .047545 
 
 (Ten) 
 len) 
 
 848141 | 
 
 348735 
 349829 
 349922 
 350514 
 351106 
 351697 
 352287 
 352876 
 353465 
 354053 
 .854640 
 355227 
 855813 
 . 3806398 
 356982 
 357566 
 358149 
 858731 
 .009313 
 359893 
 360474 
 361053 
 361632 
 362210 
 362787 
 
 i=) 
 
 || 9.363364 
 
 a vm 
 
 PJ FFF IIS 
 
 | 10.672525 
 671905 
 671285 
 670666 
 670047 
 669430 
 668813 
 668197 
 667582 
 666967 
 666354 
 
 10. 665741 
 665129 
 664518 
 663907 
 663298 
 662689 
 662681 
 661473 
 660867 
 660261 | 
 
 10.659656 
 659052 | § 
 658448 | : 
 657845 
 
 .658647 | 
 .658051 
 .652455 | 
 .651859 
 . 651265 
 . 650671 
 . 650078 
 .649486 
 . 648894. 
 . 648303 
 | 10.647713 
 .647124 
 . 646535 | 
 . 645947 
 . 645360 
 -644773 
 .644187 
 . 648602 
 .648018 
 . 642434 
 
 .641851 
 
 .641269 
 .640687 
 .640107 
 . 6389526 
 .6388947 
 . 688368 
 
 .6387790 
 
 637213 
 636636 
 
 e NaCosimes. |.) 1. 
 
 Sine, 
 
 Cotang. 
 
 Tang. | 
 
TABLE XXV.—LOGARITHMIC SINES, 
 
 - Sine. D. 1". |} Cosine. | D. 1’. Tang. D. 1°. | Cotang. / 
 0 | 9.352088 | 4 45 |) 9.989724 4g || 9-363364 | 9 ,, | 10.636636 | 60 
 
 1) 352685 | 9°79 |, 988695 | -48 ||” “363940 958 |. -636060 | 59 
 
 2| 353181 | 9: 988666; 28 || “364515 635485 | 58 
 
 3 | 1353726 , a 988636.) a | 1365090 : » 634910 | 57 
 
 4] .354271 oi . 988607 0 |. 865664 “Of | 684836 | 5G 
 
 5 | 1354815 | 9-07 .988578 | “25 || "366937 | 9-55 | “gagren | me 
 
 6 | 355358 | 9-05 988548 | -0 .366810 | 9-53 | “gssig9 | 54 
 
 7 | 1355901 | 9-95 ‘gsss19 | -48 -367382 | 9-53 |  “gg0618 | 53 
 
 8 | .366443 | 9-03 || ‘osgigg BO || 1867958 oe | 682047 | 52 
 
 9 | .356984 | 3-02 988460 | -48 368524 | 9-7. | 631476 | 51 
 
 10 | .857524 9/00 988430 “48 || .369094 9.48 .680906 | 50 
 
 | 11 / 9.358064 | ¢ og || 9.988401} |", || 9.869663 | 9.4 | 10.630837 | 49 
 Hh 12 | 358603 | 8-88 988371 | .*:50 870282 | 9-42 | 1629768 | 48 
 Hn 13 | 5941 | 8.97 988342} «38 370799 | 3-42 | 629201 | 47 
 14 | .359678 8195 .988312 | , 50 871367 : 3 . 628633 | 46 
 
 15 | .360215 | ‘8.95 988282 | © -> .371933 | 9-4 628067 | 45 
 
 HW 16 | 360752 Sieg 988252 | 80 || “372499 3-42 | 627501 | 44 
 7 |. .Beien7 | 8.82 -988223°| «48 373064 | 9-4 | “620936 | 43 
 
 uh 18 | 361822 | 8.92 988193 | -P? 873629 | 9°45 | .626871 | 42 
 Hi | 19 | 362856 | 8.90 988163} -P0 374193 | 3-38 | 625807 | 41 
 He 20 | 362889 | 8-88 988133 | - 0 874756 | 8°38 | 625244 | 40 
 AV ig 21 | 9.363422 | 2°. || 9.988103 50 || 9-875819 | 9 » | 10.624681 | 39 
 vil ,22 | 363954 | 8.87 -988073'| +0 875881 | 9°34 | 624119 | 38 
 23 | .364485 885 . 988043 50 3876442 9.35 623558 | 87 
 24 | 365016 | 8-8 988013 | <P? 3770038 | 9°39 | 622997 | 36 
 
 25 | 365546 | 8.83 987983} :P0 377563 | 9°35 | 1622437 | 35 
 
 26 | 266075 | 8-82 987953. | 80 878122 | 8°35 | 621878 | 34 
 
 27 | 366604 | 8-89 -op7ge2 | <b -378681 | 9°55 | 621319 | 33 
 
 28 | 367131 | 8.78 987892. | -p0 379289 | 9°39 | .620761 | 32 
 
 Hk 29 | 367659 | 8-80 987862 | 20 879797 | 9°58 | 1620208 | 31 
 il 30 | 868185 | 8.7 987832 | <7 380354 | 9°59 | 1619646 | 30 
 A 81 | 9.368711 | gos || 9.987801 | ~',) |) 9.s80910 9.97 | 10.619090 | 29 
 ol 32 | .869236 ue 987771 "Po «|| «6881466 ee .618534 | 28 
 i 33 | 369761 | 8-75 -987740 | >? || 1389000 | 9-23 | “gizago | oy 
 Hil B4| 870285 | 8-2 || ‘osrrto PO |) 1882575 O33 | 1617425 | 26 
 85 | .870808 | §.72 987679 | -B 883129 | 6°53 | 1616871 | 95 
 
 36 | 371330 | 8-7 987649 | -20 "383682 “616318 | 24 
 
 37 | 371852 | 8.70 "987618 |  -52 884234 | 9-20 | “i566 | og 
 38 | 1372373 oon 987588 | «0 "384786 ae "615214 | 29 
 
 39 | 372804 | 8.68 987557 | Fp || -ae5aaz | 8-18] “614663 | 21 
 
 AQ, : AOR .t KC . ¢ 
 
 40] .avsai4 | 8-67 987526 | Fo || 885888 | 9-18) 614112 | 20 
 
 41 | 9.373933 | .'g. || 9.987496 se || 9386438 | 9 4. | 10.618562 | 19 
 
 42 | 374452 863 987465 =) .886987 9°15 613013 | 18 
 43 | .374970 8 62 987434 “B0 887536 9°13 .612464 | 17 
 
 el weeeeee | geo || e8C40B: | “2 -888084 | g'i9 | 611916 | 16 
 
 45 | .376003 | §-60 ‘perei2 |. 62 888631 | 9°15 | 1611869 | 15 
 
 46 | .876519 8 60 987341 “BO .889178 9°10 .610822 | 14 
 
 7| .3770a5 | 8.60 987310 | - 389724 | 9°49 |  .610276 | 13 
 
 48] ‘8rrsz9 | S87 987279 ae | "390270 a 609730 | 12 
 
 49 | .378063 g RY . 987248 “30 890815 908 609185 | 11 
 
 50} .s7s577 | 8.5% 987217 | “Pe 391360 | gps | .608640 | 10 
 
 51 | 9°379089 | . +. | 9.987186 > _ || 9.891903 ~ | 10.608097 | 9 
 
 52) .879601 | pss || 987155 | -P2 || ‘390447 | 9:07 | “ “Gomnca | 
 
 53} .880113 | §-°8 | ‘ogvioa | -52 || “3959g9 9.03: “go70i1 | 7 
 
 54] (880624 | 8-52 || “oproga | — -58 393531 | 2-93 |  “eog469 | 6 
 
 55 | .38iis4 | 8-90 || “oszoei | 52 |! “304073 303) 1605927 | 5 
 
 56 | .881643 | gv4s || -9svo30 | -82 |) “Baaeia 900 |  -605886 | 4 
 
 b7 | 882152 | ee || .986998 as , 395154 9°00 604846 | 3 
 
 58) .382661 | oye || .986967 “25 «|| «| «6895694 8/98 604306 | 2 
 
 59 | 883168 | B42 || “986936 22 396283 | Bor | .603767 | 1 
 
 60 | 9.383675 | °-4° || g gseg04 | 58 |) g Bggen °-3¢ | 10603229 | 0 
 
 ’ | Cosine. | D.1”. || Sine. D. 1". 4| Cotang: | D. 1°. Tang. 2 
 
- 
 
 Sine. 
 
 SO OI OTR OO DH OS 
 
 9.388675 
 .304182 
 .884687 
 .885192 
 .385697 
 
 | - 886201 
 
 .886704 
 
 .881207 
 
 .887709 
 
 .3888210 
 
 .o88711 
 
 9.389211 
 889711 
 .890210 
 .390708 
 .891206 
 .3891703 
 .3892199 
 .3892695 
 
 .893191 | 
 
 .393685 
 | 9.394179 
 | .394673 
 395166 
 895658 
 396150 
 396644 
 897132 
 897621 
 
 898111 
 
 .3898600 
 9.399088 
 
 899575 
 .400062 
 
 .400549 
 .401035 
 .401520 
 .402005 
 .402489 
 .402972 
 403455 
 9.403938 
 .404420 
 .404901 
 .405382 
 .405862 
 .406341 
 .406820 
 
 407299 
 
 40777 
 408254 
 9.408731 
 .409207 
 .409682 
 .410157 
 .410632 
 .411106 
 .411579 
 .412052 
 412524 
 9.412996 
 
 | 
 
 10 GD 00 CO OO 
 
 CO 
 
 wiowwwwwe a aye ee 
 
 z| Cosine. ! D. 
 
 COSINES, TANGENTS, AND COTANGENTS, 
 
 Cosine. | D, 1". i Tang. Cotang. 
 = es | 
 9.986904 | 9.896771 | 2 10603229 
 . 986873 .397309 |g” 602691 
 :. 986841 897846 | 9° 602154 
 986809 898883 |g" 601617 
 1.986778 |} .898919 |g" 60081 
 -986746 || 899455 | 9° 600545 
 986714 |} 899990 | 9" .600010 
 . 986683 |} -400524 | ¢" 599476 
 (986651 |} .401058 | 9°¢ 598942 
 .986619 || .401591 | 9’ 598409 
 . 986587 || -402124 | 9° 597876 
 | 9.986555 || 9.402656 | 10.597344 
 986523 || .408187 | 9° .596813 
 . 986491 |} 408718 | Qos 596282 
 . 986459 || .404249 | 9° 595751 
 -. 986427 || 404778 | 9’. 595222 
 986395 |} -405808 | g'¢ 594692 
 :986363 || .405836 } 9'¢ 594164 
 986331 .406364 | 9° .593636 
 986299 || .406892 | ¢° 593108 
 986266 || .407419 | 9’ 592581 
 9.986234 | 9.407945 |g mm | 10.592055 
 986202 || -408471 | g'ns 591529 
 .986169 .408996 | g'nx .591004 
 .986137 409521 | g"hg 590479 
 . 986104 .410045 | 9'r3 589955 
 . 986072 | .410569 | Q"r5 589431 
 .986039 | .411092 | p'n5 588908 
 .986007 || .411615 | 9° 588385 
 . 985974 .412137 | 9'¢ 587863 
 ,985942 || 412658 | 9" 587342 
 9.985909 || 9.413179 10.526821 
 985876 .413699 586301 
 985843 .414219 | 9 585781 
 985811 .414738 : 585262 
 .985778 BB .415257 | 0’ ¢ 584743 
 :985745 fee 415775 | 9" Ge 584225 
 985712 | ‘ee 416293 | 9’ g¢ 583707 
 985679 Bee .416810 | o'¢ 583190 
 .985646 “3B 417826 | 9° 582074 
 | 1985613 “BB A1T842 |g” 582158 
 9. 985580 9.418358 | 10.581642 
 ges5a7 | -P5 || 4ise73 | 3° 581127 
 985514 os 419387 | 580613 | 17 
 . 985480 5S .419901 -580099 | 16 
 985447 55 420415 -519585 15 
 985414 Bs || | -420927-| 919073 | 14 
 985381 "no || 421440 578560 |, 13 
 . 985314 “my || 422468 5775: (| il 
 . 985280 — || 422074 577026 | 10 
 9.985247 | nn || 9.423484 | 10.576516 | 9 
 985213 “a5 || 428998 576007 | 8 
 .985180 By ae oe 424503 5 4 5494 ¢ 
 985146 "rs || -425011 | 574989 | 6 
 985113 | “py || .425519 | 574481 | 5 
 985079 | “py || .426027 573973 | 4 
 985045 “ny || 420584 | 573466 | 8 
 985011 “a5 || «427041 572959 | 2 
 | .984978 oe || | Saeed? 572453 | 1 
 |) 9.984944 | °°" |) 9.428052 10.571948 | 0 
 Sine. . || Cotang. Tang. / 
 
 & 
 
TA y 
 \BLE XXV.—LOGARITHMIC SINES 
 
 / . 
 Si " | 
 ne, D. rt 4 | Cosine. D. 1’. Park : 
 tk 3 Oe i | g. D. 1". | Cotang. vey. | 
 0 | 9.412996 | 
 1} 413467 | ig Oiuee (i Me: {eee | Lb | 
 2 7.85 5 ¢ Py eae te ae i As | 
 3 Siri 783 “981876 Br. |i ARSE ors ee | 6 | 
 asiava | | 7488 984845 57 .429062: | 9° enn trie 
 5 | <l4tead7 782 speaoa |) Bll gaunne oe “br0as4 | pe 
 Pel Ghagseis | TSO mean, ‘57 || -420070 | 3°38 “560980 | e6 
 7 | 416983 | 7.80 984740 | +e 430573 | 8-38 ga rae 
 ol sttginey | TURD AH) eeeaten py || -431078 | B37 "B6R0D5 | bad 
 10 | 417684. | 7:78 "984638 | -3¢ 432079 | 8-30 ERS. | oe 
 | --417684 | ote 984688 | i5g || | - 482580 8.35 | -567921 | 52 | 
 11| saisiso| || 9. ‘py || «483080 | 8-38 ‘GOtA2) | Od 
 te | 418615 | 0:5 ee ma ie sanege 8.33 .566920 | 50 
 3 re ” - 984535 teks : 
 | wro | 28 |) $880) ) 38 |) es) sm 05000 | 48 
 ig \ aseoor |) 782° deeaaee | 434579 | 8-82 | “565401 | 47 
 16 sas yh . 984432 157 .435078 32 Spec f 
 17 | 1420983 ie 984397 58 ee 828 “56LRt | 45 
 18 94205 Vaid . 984363 : bec vs : Or 
 18 | 421395 | fy || 984828 |» Ft aosi0 | 828 | ‘563180 | 43 
 oy tuessis (7 MOC Te Geers 4 Sette’, po seenden a 
 ry RF As i 40 (ODE . F. ra ae . 
 21 | 9.422778 (.67 . 984259 “58 ‘438059 | 8-2¢ 562437 | 41 
 oe "493038 | 7.67 ae fire » || 9.438554 re aati 
 93 92RQ7 ” 65 . 90 mY ae 55 ¢ ‘ : 
 gy gases | ia osiibs | 3S 439048 | 835 ae Ie 
 95 | “494615 | 7-65 . 984120 : ne 89s 56045" 
 ised ) : 26 122 560457 Wg 
 26 | .425073 | 1-68 984085 | Pe aoe tt Siae 559964 | 3¢ 
 2 | 425580 | tgp |] 984015 | Ep ware ioe “ppe078 | 34 
 oa) aogtes || estes (ic fee seen a0 | ree tae 
 30 | capos09 | 7 epit||) woese4s |) Gee "442006 | 820 | “psraod | Sp 
 426899 6 983911 .58 -442497 8.18 ; 55% 994 | 82 
 ap | oiommad Ili grease “eo. || 3442988 | 8-38 557508 | 31 
 32 | 427809 | 7-98 9.983875 ine 8.18 .557012 | 30 
 82) 427809 | 757 || -983840 | G8 nei 10.5565 
 gets | Lee ll| perce 8 Vaca Bo eee 
 35 | .499170 | 7-55 9837 58 444458 | ooy 5 SeeSEta {7 
 Fe) if en Vv 15 .55dDd42 or 
 36 42962 7 5b . 9837385 .58 444947 RRRAre 5! 
 pote tbe f “ganas | 8148%| eeteee | oe 
 37 | 430075 | 708. || “ogenee eet atbi85 | lig | -554565 | 2 
 38 | .4305e7 | 1-58 983664 .60 -445923 | oy: 554077 A 
 39 | ‘430078 | 7-5 ieeseco |) 286. 1 caaeeae Age | Oe ee 
 Bl eee | (rie 983620 | 53 || -440898 | 8°75 “553102 | 28 
 a) Sr od : 2 6 ; > z . € ‘ pepe 
 a1 | 9.4zis79 | gases | +82 || <aa7e70 | 8-10 552616 | 24 
 . 432329 | 7.50 9. oe fo || 9.448858 8.10 .552130 | 20 
 5 90nhh Lo A, wa RS Nya V0 y O50 
 “4 | 433096 nay Spee || abe lt gates 7 LOG 
 | dass, | TBO | pees “60 || 449826 | Boe pers al ae 
 ae 3 ¢ 07 550674 | 17 
 aa) Bias | eee nee 58 449810 | 8° gy Se reme 
 47 As peng KY 5 983345 .60 450294 07 oO Le J 16 
 | Bin | 18 | cies | 2 | ia | Bo igees | 1d 
 49. |  A3b4g0 | 7-4 reece (200 tl eepneee 1 8 an ane 
 40 | -aioaca | F-23 || cosas | BS || “apzoos 8.05 | -S4sos7 | Te 
 ) by Cag 2951)¢ : - 452225 Us -0F0e0¢4 | 12 
 | 51 | 9.486353 fee re “60 “450706 | 8-02 547775 | 11 
 52 | .436793 | 7.42 9.983166 | 9459187 8.02 .547294 | 10 
 ba | 436798 | 7°79 || 988130 | “Gp Seen 5 | 10.5468 
 54 | ‘43763 | 7-40 -83004 | “6° ee oe “BAG839 5 
 55 | .438129 | 7.38 “ossosg | -89 |] “graeos | 8.00 BA5R5O | 
 56 | 438572 | 7! (983022 | 80 454628 | 2-08 ee Le 
 Br | 14a9014 (88 |} ‘oegs6 | — -60 “155586 098 ‘514808 | 
 BY | 480014 | 757 |] -982950] “69 || 455586 | 7-38 |  paddi4 
 59 139807 | 7-3 “98291 60 -456064 ‘. a7 082" 4 
 60 | 91440585 | 7 35 speereeg ft) 200. Name eee Be: ieee |e 
 Tait Wiccan ae pigeaeia (1.800 fl cee 7,05 5| | Soieee eg 
 Te AEE : peer 9.457496 | @:9° .542981 | 1 
 | Cosine. D. 1’ cee ap ae I 10.542504 0 
 ; real : oe 
 otang. | D. 1 Tang. / 
 
 =I 
 TS 
 oO 
 
COSINHS, TANGENTS, AND COTANGENTS. 
 
 Cosine. | D. 1". |) Tang. D. 1”. | Cotang. | u 
 
 Sine. Dales 
 
 ———— - —— | 
 
 | 9.440388 | » 95 || 9.982842! | 9.457496 
 
 ; | 
 10.542504 | 60 
 
 | 0 Pe 
 
 Pa) lagers | 4:83. || ““osesos | 62 || ““aszora | 5-88 542027 | 59 
 2 | 441218 | 7:33 || Toseveo | 69. |) “assaag | 1:98 "541551 | 58 
 Si} <4aress | 2:33 || ‘oseraa | 89° || “aseoan | 1-98 "541075 | 57 
 Bat Serene | | Tepes | Berens |) BO |i cepa 7 ae PAGO) + O8 
 5 | °,442535 3 . 982660 60 || .459875 90 ~b40125 . 55 
 6 | .442973 og ||  - 982624 eat .460349 90 .5389651 | 54 
 7| 443410 | 7°58 || -osess7 | -8 || ‘46ose3 | 1:00 ‘539177 | 53 
 B | cft8847-| 4'5q! ||. oee5bt | Bo |) Vaeigor | 7280 "538703 52 
 g| 444984] 2:25. || “gosta | 82 |) Cagi77 & "538230 51 
 0 
 
 444720 
 
 11 | 9.445155 
 12 | 445590 
 13 | .446025 
 14} .446459 
 15.| 446893 
 16| 447326 
 17 | 447759 
 18 | 448191 
 19 | 448623 
 20 | .449054 
 449485 
 22 | .449915 
 23 | .450845 
 
 te 537758 , 50 
 
 10.537285 | 49 
 85 "536814 | 48 
 536342 | 
 585872 | 46 
 585401 | 45 
 534931 
 534461 | 43 
 538992 | 42 
 583523 | 41 
 “80 .533055 | 40 
 , 10.532587 | 39 
 bd 531653 | 
 
 — 
 
 Os 982477 | 65 462242 
 on || 9.982441 gg || 9.462715 
 3: 982404 “69 .463186 
 | asa | 8 || aes 
 23 Nae 62 eine 
 4 982294 a 62 : 464599 
 39 982257 eg 465069 
 30 982220 "62° || -465539 
 50) . 982183 an .466008 
 18 || -982146 Tay 466477 
 18 .982109 "gg || -466945 
 » || 9.982072 gg || 9-467413 
 ~ || 982085 .467880 
 1” .981998 "69 468347 
 
 aN 
 ~ 
 
 iS 
 ne 
 
 ne) 
 are 
 ile) 
 
 J 
 
 WEAF AF QT APPIN AAI AHH NI WANN HAIN NAN 
 2 é 
 \ 
 
 24] 1450775 | #75 .981961 | *¢5 .468814 ne .531186 | 36 
 Q5 | 451204 | »'y5 /]| .981924) “63 .469280 wy 530720 | 35 
 26 | .451632 13 981886 | “¢5 469746 5 530254 | 84 
 27 | .452060 3 981849 | “65 ATO2A1 ne 529789 | 83 
 28 . 452488 {27.| . 981812 63 .47(0676 a5 529824 82 
 29 | .452915 19 98177 “ga || 4c114 73 528859 | 31 
 30 |  .453342 10 981737 | G5 || 471605 °3 -528395 | 30 
 31 | 9.453768 10 || 9-981700 gg || 9.472069 wo | 10.527931 | & 
 82.| .454194 | »°p9 981662 | “65 472532 | nny 527468 | 28 
 83.| 454619 08 981625 | “gg || .472005 | ne 527005 | 27 
 4 | 455044 | 7'og |] 981587 | gg |] 48457 | aig | -BRODKB | 26 
 85 | .455469 07 981549 go || -473919 Ff 526081 | 25 
 86 | .455893 981512 | ‘eg || .474881 | »'¢ 525619 | 24 
 
 525158 | 23 
 .524697 | 22 
 524237 | 21 
 
 ~ 
 my, RoaKrin 20 
 
 37 .456316 
 38 .456739 
 39 457162 
 
 ‘on || 981474 | 788 |] lazagae 
 05. || -981486 | *g5 ||  .475308 
 03 | ,981399 - 63 475763 
 
 : 
 > O> 
 eM) 
 
 > FFF HF BPD I VIII 
 S 
 Ww 
 
 40 | 1457584 OeteEL |, Ses oll cateaes.| Oe08 | 1 ances 
 | 41 | 9.458006 » || 9.981823 | gs || 9.476683 | » gs | 10.528317 | 19 
 42.| 458427 | «*oo || .981285 | “ga || 477142 | 9 "¢e 522858 | 18 a 
 43-| 1458848 | 4-72 || ‘ogioa7 | = 83 || larre01 3 | .592899 | 17 
 44) 1459268 | 4-29 || ‘osi209 | -63 || -aze050 | f:a5 521941 | 16 
 45 | 450688 | fy) || cosiizi | 68 || :4rssi7 | “fgg | 1521488 | 15 | 
 46} .460108 | d-o3 || .981138| “83 || c4vsovs | #05 | ‘521025 | 14 
 47 | .460527 | G93. || .981095 | “B38 | oS | 1520568 | 13 
 48 | .460946 | 8-8" || ‘osios7 |  -B3 vanes : 12 
 49 | 461364 | Poy || .981019| “es a 11 
 
 “80 519199 | 10 
 
 BU | .461182 | gos 1} .980981 
 10.518743 | 
 
 .462199 6 |} 9.980942 | 63. || 9.481257 
 
 | of | 9 704 .95 Han 2 Ur 24449 58 290Q 
 52 |  .462616 | 9°93 -980904 63 || 481712 58 518288 
 | 538] .468032 | 698 . 980866 “65 482167 57 .517833 
 54 18 | 698 .980827 ot 482621 5 514379 | 
 Bb ant Owe 980789 ie 483075 ts 516925 | 
 vo Qo . ( i. fr 
 56 | Cane "980750 | ° 483529 of 516471 
 
 oo 516018 | 
 “8g 515565 
 
 Bz | :4646oa | 8-22 || ‘ogovie | -B8 || -4gs982 
 | 58 | ‘465108 | 9: 980673 | 63 || 484435 
 
 Sc 
 « ~ 
 * 
 ee 
 
 aJ  a a NOM SS SS SON SS  S O S  O ed 
 
 ORD WR OLD -IO0O 
 
 | 50 | 1465522 | 8-9 || ‘ogosa5 | -68 || apager | 2:2: "515113 
 
 | 60 | 9.465935 | ©-88 |) o’9s0596 | -® || 9.485839 | 4:3 | 10:514661 
 
 | ‘| Cosine. | D.1”". || Sine. Dal" sai} Cotanes|\3Ds1" Tang. Z 
 10° 720 
 
é Sine. | D.1". || Cosine. 
 0 | 9.465935 | @ gg || 9.980596 
 1 | .466348 | gga || 980558 
 2 | .466761 | ¢'oo .980519 
 3 | 467173 6 87 980480 
 4 | .467585 6 ' 35 980442 
 5 | .467996 | gs 980403 
 6 | .468407 | @'g3 .980364 
 7 | .468817 | 6 ’39 980325 
 8 | .469227 | g’ga 980286 
 9} .469637 | 6'g5 980247 
 10 | .470016 | @"g5 980208 | 
 11 | 9.470155 | 4 ay 9.980169 
 12] .470863 | @g9 .980130 
 13 | .471271 | bgp 980091 
 14 | .471679 | Gr 980052 
 15 | .472086°)  ¢ bad -980012 
 16 | .4724192 67 979973 
 17 | .472898 697 979934 
 18 | .473304 | @'te 979895 
 19 | .473710 | g"hs 979855 
 20 | (474115 6.73 979816 
 21 | 9.474519 | og 9.97977 
 2) .474923 | @° "3 979737 
 23 | 475827 |g "9 979697 
 24] .475730 | p'n5 .979658 
 25 | .476133 | g'n5 .979618 
 26 | .476536 6 70 979579 
 297 | .476938 6.7 979539 
 28 | .477340 | ¢ 68 979499 
 29 | 477741 | 66g 979459 
 30 | .478142 | @ 7 979420 
 31 | 9.478542 | ¢ gn 9.979380 
 32 | .478942 | ¢° 67 979340 
 33 | 479342 | 6g 979300 
 84 | 479741 |g 65 979260 
 35 | .480140 6 65 979220 
 36 | .480539 | ¢" 63 979180 
 37 | 480937 6 62 979140 
 38 | 481334 | 6 '¢5 979100 
 39 | .481731 | 6 '¢5 979059 
 40 | .482128 | 669 979019 
 41 | 9.482525 | & 9.978979 
 42 | 482021 | 2: 58 978939 
 43 | .483316 6 60 978898 
 44 | .488712 | re 978858 
 45 | .484107 6 87 978817 
 46 | .484501 6 57 97807 
 47 | .484895 6 BY 978737 
 48 | .485289 | pe 978696 
 49 | .485682 6 bE 978655 
 50 | .486075 6 53 978615 
 51 | 9.486467 | ¢ 5x 9.978574 
 52 | 486860 | p> 978533 
 53 | .487251 | @ “5g . 978493 
 54 | 487643 | 655 978452 
 55 | .488034 | ¢54 978411 
 56 | .488424 6 50 978370 
 5Y | .488814 650 978329 
 58 | .489204 | ¢ “48 978288 
 59 | .489593 6 48 978247 
 60 | 9.489982 om tL 9.9%3206 
 4 Cosine. | D 1”. Sine. 
 
 TABLE XXV.—LOGARITHMIC SINES, 
 
 | 
 DD a** Tang. D, 1". | Cotang. | fi 
 nf nae Sh | Aes ee 
 bs 9.485839 » 5g | 10.514661 | 60 
 63 ‘ag5vo1 | 2-58 "514209 | 59 
 65 | 48022 | pp | 518758 | 8 
 63 | casriss | 750 | cSrpepr | 56 
 . JD AQUOS f.o S ony xe 
 65 || 4agoa3 | 7:50 | °-Biigne | 
 65 || “gggag0 | 2-48 "511508 | 53 
 ee (|| 488941 | 248° | “511059 | 52 
 "65 489390 a 7 510610 | 51 
 te ||| teeeoees |; bee "510162 | 50 
 QO? Ww 
 | me | ray | sso | a 
 ee || -491d80 | 2-45 | “s08se0 | 47 
 + Ue On . 972 | “AR 
 67 7 ||) satenek |! reat woes al We 
 65 ‘4go519 | 7-43 "507481 | 44 
 ‘65 || "499965 | 1-48 507035 | 43 
 65 || agaaio | 7-42 506590 | 42 
 . 92Q6 : 50614 | 
 a eee cea 
 7 |, sae. solo ee Ae 
 65 || 9-494743 | gq | 10.505257 | 39 
 + Oe 51 504s | 9 
 ai | 2 | Pi | 
 OS || .496073 | 2-88 | ‘503927 | 36 
 a | 496515 | 03% | 508485 | 35 
 . ie AQ mOkNh -t . € . 
 |) a | Far | See | 
 67 |) “4g7g41 | 7-37 "502159 | $9 
 .67 “agora || Tee ieaieia teke 
 8) BES) eB | Sek a 
 £67 ||| PCE otis souhee 
 tr Wt Seana || eee |e ee 
 67 Ap x ec 7 39 0) C Oe sf 28 
 “|| 50002 | 4-82 499958 | 27 
 . 2 : am ie oF 
 S67. |} Gengasy (eae aaeeaa bene 
 67 || “501359 | 7-82 “498641 | 24 
 67 | “sor707 | 7-30 | “aoapo3 | 93 
 -67 || “502035 | 1-30 497765 | 22 
 oy || poze | 1-28 | ‘ag7goe | 91 
 “ey . 503109 98 .496891 | 20 
 ~ || 9.508546] » 4. | 10.496454 | 19 
 S| Rs | 2 | ae | 
 67 Meagaxa {0 Viet “hy doo pl ef 
 | a) ie | aie 
 67 BO5724 | 0-20 494276 | 14 
 a “506159 | 1:29 "493841 | 13 
 08 |) soroer | 223) jours | ti 
 “68 507460 | 9°55 .492540 | 10 
 Weer? y 6) nw 
 ‘68. |) OP etOs | ibe Boe A 
 6% || “s0a759 | 7-22 491241 | 4 
 “ba (| -500191 | £-22 | 7490609 | 6 
 OC NOR. peas ary 
 | gee | Bons 
 ea. {| .310485 | 2-48 | lago515 | 8 
 "fg || - 510016 | 7:18 |  ‘apoosa | 2 | 
 . K1794R8 . Nevada | 
 $68. |) gesaeeca| (THEY AB88DE | | 
 9.51177 10. 488224 0 
 D. 1". || Cotang. | D: 1°. Tang. i 
 
BOIS OR G0 
 
 OD Y Se * G 
 
 Sine. 
 
 .490371 
 
 .490759 
 
 .491147 
 .491535 
 
 .491922 
 492308 
 
 .492695 
 .493081 
 .493466 
 493851 
 
 9.494236 
 .494621 
 -495005 
 - 4953888 
 495772 
 .496154 
 .496537 
 .496919 
 .497301 
 . 497682 
 
 | 9.498064 
 
 |  .498444 
 
 498825 
 
 . 499204 
 
 499584 
 
 .499963 
 
 .500342 
 
 .500721 
 
 501099 
 
 501476 
 
 | 9.501854 
 
 502231 
 502607 
 502984 
 “503360 
 503735 
 504110 
 504485 
 “504860 
 505234 
 505608 
 505981 
 506354 
 506727 
 .507099 
 OVE 
 507843 
 508214 
 BOSS85 
 508956 
 
 | 9.509326 
 | 509696 
 510065 
 510434 
 510803 
 
 © 
 
 .511172 | 
 
 .511540 
 .511907 
 
 512275 
 
 | 9.512642 
 
 0 | 9.489982 | 
 
 eo} 
 
 S Sd G2 ¢ > S> S> > > 
 ee ee ek ek eek ek Re A 
 WO WO 0 GY CLOT OT OT z 
 
 | 9.978206 
 
 .978165 
 978124 
 .978083 
 978042 
 . 978001 
 977959 
 
 977918 
 ITBTT 
 977835 
 877794 
 
 ioleds4 
 . 97 ( (52 
 
 =) 
 
 ITT 
 
 . 977669 
 977628 
 977586 
 977544 
 977503 
 
 977461 
 
 977419 
 77377 
 9.977335 
 977293 
 
 IV7251 
 
 977209 
 977167 
 977125 
 977083 
 
 977041 
 
 . 976999 
 . 976957 
 976914 
 976872 
 976830 
 976787 
 976745 
 . 976702 
 . 976660 
 976617 
 . 976574 
 .976E32 
 . 976489 
 . 976446 
 . 976404 
 
 io) 
 
 .976361 
 
 .976318 
 
 Or 
 
 976275 
 . 976282 
 . 976189 
 - 976146 
 
 976103 
 
 || 9.976060 
 
 975974 
 . 975930 
 . 975887 
 . 975844 
 
 . 975800 
 ON 5 S57 
 dIlodiodd 
 
 975714 
 
 9.975670 
 
 Cosine. | 
 
 A a a a a a a a a a 8 a a 
 
 WWNHHWNWWNWDN YWWWNWNWHWWSOD WOWwWowsd 
 
 Tang. 
 
 9.511776 
 || 512206 
 512635 
 513064 
 | 513493 
 | 513921 
 | 1514349 
 514777 
 515204 
 515631 
 | 516057 
 
 9.516484 
 516910 
 517335 
 517761 
 518186 
 518610 
 519034 
 519458 
 519882 
 520305 
 
 9.520728 
 521151 
 521573 
 521995 
 522417 
 522838 
 523259 
 523680 
 
 | 524100 
 
 | 524520 
 
 9.524940 
 .5253859 
 525778 
 .526197 
 .526615 
 .527083 
 
 | 527451 
 
 .527868 
 528285 
 528702 
 
 9.529119 
 .529535 
 .529951 
 .580866 
 530781 
 .531196 
 .531611 
 
 | 582025 
 
 .532439 
 
 See 
 | .582853 
 
 | 9.533266 
 .538679 
 .534092 
 584504 
 .534916 
 585828 
 .535739 
 .536150 
 5386561 
 
 9.586972 
 
 Cotang. 
 
 WE AE AE AEA EVE APE EEE PEI IE I IM N AIAN 
 
 DAIMDMDADAA ADA SD 
 
 oS 
 
 APOIAS 
 
 DS SH SG. Sd 
 
 487794 
 487365 
 .486956 
 .486507 
 .486079 
 .485651 
 485223 
 .484796 
 .484369 
 483943 
 
 | 10.483516 
 
 483090 
 482665 
 "482239 
 481814 
 481390 
 480966 
 480542 
 480118 
 479695 
 
 10.479272 
 478849 
 478427 
 -478005 
 477583 
 477162 
 476741 
 476820 
 475900 
 475480 
 
 10.475060 
 474641 
 474222 
 -473803 
 473885 
 472967 
 472549 
 472182 
 41715 
 .471298 
 
 | 10.470881 
 
 470465 
 -470049 
 .469634 
 .469219 
 468804 
 -468389 
 467975 
 467561 
 467147 
 
 | 10.466734 
 
 466321 
 -465908 
 .465496 
 465084 
 -464672 
 .464261 
 -463850 
 .463439 
 10468028 
 
 Cosine, | 
 
 Sine. 
 
 Cotang. 
 
 Tang. 
 
 | 10.488224 
 
 Or dD CO O10 FF OC 
 
 ~ 
 
 ~t | 
 = | 
 ° 
 
TABLE XXV.—LOGARITHMIC SINES, 
 
 ~ 
 
 Sine. D. 1", || Cosine. | D. 1’. || Tang. D. 1". | Cotang. | é 
 
 i 
 | 
 1 
 
 | 9.512642 
 
 0 | > || 9.975670] ». || 9.586972 463028 | 60 | 
 1| .513009 | ©-12 || “‘ovsgo7 |. -72 Roreae |) 08880 || VRE eed 
 3 | ‘513375 | 6-10. || “Ooeess | 78 || “pares? | gigg, | | +462618)) 59 
 | -513375 | g°49 975583 73 || -5B0792 | Bes .462208 | 58 | 
 3 | 513741 | 6-1) 975539 mp || 388202 | 6-88 1461798 | 57 
 4! .5iai07 | 8-18 975496 | “8 5as611 | 6-82 "461389 | 56 | 
 5) 514472 |g og 975452 73 539020 | B95 460980 | 55 | 
 6) 1514837 | Bp3 || 975408.) -23 suodeg | 6682 ‘460571 | 54: | 
 al sper 6:07 =a “ns 539837 | go .460163 | 53 | 
 .51556 at By . 975382 Seo. it rode oe 459755 | 52 
 Peon | 6.0 ua ore E | 2 oO 409 (00 ve 
 9 515930 | mae 915207 oh ‘| .540653 | oe "459347 | 51 
 10 | 516204 | 6-07 975283 | +3 541061 | 6-8 “458939 | 50 
 11 | 9.516657 | .,~ || 9.975189; “.. || g.saiaeg| 458532 | 49 | 
 12 | 5170290 | 8-05 ‘ovis | 73 || OReere | 6:78 yrereee 48 
 18 | 1517882 | 6-938 || “opsioy | -73 |] ‘54008] | 6.77 457719 | 47 | 
 | 14 | 517745 | 8-0 |) lov5057 | 73 || ‘sageng | 6:78 | “asraie | 46 | 
 i 15 | 1518107 | 6-03 || ‘975018 | 73 |) “543004 | 8-7 |. “azgooe | 45_| 
 i 16 | .518468 | 8-92 || “g749g9 | 48 ‘543499 | 6.75 “4a56501 | 4a. | 
 7 | .518829 | 6-9 || “oragos 13 |] “543905 | 6.77 456095 | 43 
 | 18 | 1519190 | 8-08 ‘g7asgo | 45 544319 | 6.75 “455690 | 42 
 i 19 | 1519551 | 8-02 974936 |  -73 544715 | 6.45 "455285 | 41 
 i, 90] “teipo11:|, 82007 || “Bean ||, ca || CAPER | gong, || TUdbBeBE A Mt 
 a 51S au 974792 bs 545119 | 6-23 -454881 | 40. | 
 id 21 | 9.520271 9.974748 | ,. || 9.545504 | . |, 54476 | § 
 Hn 22 | '520631 | 9:20 || “‘ozazo3 | -45 pépoee | eta) | eee | e 
 | 23 | 520990 | 5-98 974659 | 43 546321 | 6.72 453669 | 37 
 | 24 | ‘521349 | 5-98 |] “oraeia 05 "E4675 | 8-73 deriomertde Fe 
 i 95 | _Bo1” Bu? || Oa and 13 marian | O.t2 pret E> 
 Hi | uB@l707 |, Soa: || 97454 hi 547138 wis 452862 | 35 
 | 26 | 522066 | 5-95 || “ovanes | -75 |] “Byenag | 6.70 452460 | 34 
 Q7 | ‘5ee404 | 5.9% 974481 | +48 547943 | 6.72 “452057 | 3s 
 28) 1522781 | 2-9 || ‘oraaag | «75 548345 | 8.70 “ae16e6 | 88 
 29 | 1523138 | 5-95 || “ov4zo1 | -75 | sagyar | 6.70 | “gators | St 
 fF 30] 523495. | 595 || 974347 |B || oaongg | 8-7 | asoas1 | 30 
 | 31 | 9.523852 | 9.974302 ‘we || 9.54955 | 29 
 Hi 32 | "1524208 | 5-93 || ““oyao57 | 75 || "pagern | 6.68 | ASOD | a8 
 | 33 | 524564 | 2-28 974212 | ‘550352 | 6-68 449648 | 27 
 | Fy ROAOs 5.9% Sea ay "5 ee iv tet kovihe Bs. 
 Mt SA | £24920 | bop |] -9r4t0v | +f || “B50782 by | «449248 | 26 
 i B | 2525271 me tT gvaig2 | «4 BB115: 68 | 47 
 36 | 2525630 | 5-92 || ‘orgory | 73 || “perare | 6.65 prin 54 
 Wt 37 | 1525984 | 5-90 974032 | +5 551952 | 8-67 "448048 | 23 
 a, a8 | .526339 | 2-78 || covzos7 | -15 |) “aseas1 | 8-85 |  “agreag | 20 
 an 39 | .526593 | 5.90 ‘973042 | «05 552750 | 6.65 447250 | 21 
 Bilas | ornare 0.0 raoqr 45 ap Bon ; 5 - . 
 a 40 | 527046 | 3°99. || .-978807 | -f2 |] 558149 | 6-65 | (446851 | 20 
 iii 41 | 9.527400 : 9.973852 | *,, BBS 16452 
 a 42 | “sav753 | 5-88 ovsco? | &73. || eeatan | oxeay | tOMaaRG ae 
 Hi a5 | yes 5 97 | 91 380% ria -558946 6 63 .446054 | 18 
 i a Lost! : 5 88 | aie “as 554344 6 62 .445656 | 17 
 a 45 | ‘52esi0 | 8?) || ‘oreri | -75 || “pesigg | 6-68 | “Saagor | de 
 ih 46 | ‘529161 | 9-85 973625 | fe 55°536 | 6-62 prrity * 
 ii Aq | 529513 5.87 || 973580 | ay A; ap pled od 6.62 geo 14 
 pe) of! 520018 5 8h - 973560 pe 55593838 oe .444067 | 13 
 48 | 529864 BO || g7g585 | sf 556399 | 6.60 3671 | 12 
 49 | ‘530215 | 5-85 || “o734e9 | 217 grid | 6.60 AA3071 12 
 50 53065 | 5-88 “993 144 5 aD een tt AaOO 443275 | 11 
 | . e 5.83 «dle | v4 I .557121 | 6 60 442879 Hl 10 
 51 | 9.530915 | . o 9.973398 | 51 é 
 52-| 53193 | 5-88 fens Pore pasts oh 6.60 peri : 
 | 53.| 531614 | 3-82 "973307 | 23 |] “xsgaqgg | 6.58 441602 | 7 
 54.| (531963 | 5-82 973261 | ff | ‘asgeqg| 6.58 441297 | 6 
 55 | 532312 | 5-82 973215 | ee “s59097 | 6-54 4400081 LB 
 56 | 539661 | 5-82 -g73ic9 | fe ‘50491 | 8-54 sip 4 
 57 533009 5.80 || 973 wie! "5 1a ML |B By , 009 | 4 
 Bg | 283008 | igo || files | Te 559885 | BP 440115 | 3 
 : .05d500 | ny | .9f8018 | Arche 56027 “ts 43975 1 «ee 
 59 | 1583704 | 5-48 || ‘orgose | 7% || “Beogeg | 6.5% ‘faeser | 4 
 Mii 60 | 9.534052 | °-°? || 9'972986 | 77 || g'61066 | 6-55. | 10‘438934 | 0 
 igocc coe «ee ree erro > | — 
 t | Cosine. |.D.4".) ||) Sinew» } Dia*er| hea Tang. } 
 
 | Cotang. 
 
 ps | 
 O° | 
 ive) 
 ° 
 ~ 
 © 
 ° 
 
 378 
 
Sine. | D. 1". 
 
 Cosine. 
 
 COSINES, TANGENTS, AND COTANGENTS. 
 
 Tang. 
 
 ISO Wwe © 
 
 9.534052 | 
 
 ely, 
 534399 | 5.78 
 534745 | PR 
 535092 | 248 
 535438 | 5 pe 
 535783 s 
 
 .586129 
 536474 | 
 .536818 
 5387163 
 537507 
 
 9.537851 
 
 C9 OLED OT 2 OF 
 
 III I-A 
 
 3 
 ina) 
 
 .538194 | ~3 
 538538 - 
 538880 | 2 "ho 
 (539223 | 2746 
 589565 | 2-4 
 
 539907 | 
 £40249 
 540590 
 | .540931 
 | 9.541272 | 
 | 541613 
 
 541953 oe 
 542293 | O¢ 
 542632 | 65 
 542971 | 65 
 
 .543310 
 .543649 
 .543987 
 .544325 
 | 9.544663 
 .545000 | 
 545838 
 545674 | 
 .546011 | 
 546347 
 .546683 
 .547019 | 
 5473854 
 | 1547689 
 9.548024 
 
 + 
 © 
 
 548359 | 2-28 
 548693 | ee 
 549027 | 22s 
 549360 | Pu 
 549693 | 2-22 
 550026 | 2:22 
 550359 ay 
 
 vd 
 
 550692 
 .551024 
 
 | 9.551356 
 
 Orgor 
 
 TON NNT OOO TOT OT OUT TTT OTT TOTO T TTT OTT LOTT TUN TUT TUTTE, SUITE OT OT IIIT UIT AN 
 er : = ; She Mr Po a gu Cure thee 
 oo (ou) 
 
 eet 52 
 551687 5 
 a8 | See 
 ee ope ¢ ve 
 552849 5) 
 552680 pe 
 553010 =9 
 553841 48 
 553670 : 
 ~vv00 t 50 
 .5_B4000 ae 
 
 | 915 48 
 
 54329 
 
 9.972986 
 972940 
 972894 
 972848 
 972802 
 x OT 275 = 
 
 10d 
 
 972709 
 . 972663 
 .972617 
 .972570 
 972524 
 972478 
 972431 | 
 .97 2385 
 972338 | 
 972291 
 972245 
 .972198 
 .972151 
 972105 | 
 . 972058 
 972011 | 
 971964 | 
 971917 
 .971870 
 971823 
 971776 
 971729 | 
 971682 
 .971635 
 .971588 
 ) 971540 
 .971493 
 .971446 
 .971898 
 .971851 
 .9718038 
 .971256 
 .971208 
 .971161 
 .971113 
 971066 
 .971018 
 970970 | 
 970922 | 
 970874 | 
 970827 
 077 
 .970731 
 970683 
 . 970635 
 .970586 
 970588 
 970490 | 
 970442 | 
 970394 
 .970845 
 970297 
 970249 | 
 .970200 | 
 9.970152 | 
 
 INQ D-III W-I-I-2-3 
 
 By So a eo st a 
 
 OW W-3 WOW O-32D 
 
 @ 
 
 Sag at) IES: 
 
 | 9.561066 
 
 .561459 
 .561851 
 562244 
 .562636 
 563028 
 
 563419 
 
 .563811 
 564202 
 564598 
 564983 
 
 | 9.565873 
 
 565763 
 .966158 
 566542 
 566982 
 
 .567320 
 
 .567709 
 .568098 
 .568486 
 568873 
 
 9.569261 
 | .569648 
 
 .5700385 
 .570422 
 570809 
 571195 
 .571581 
 571967 
 .5 72352 
 572738 
 
 | 9.573128 
 
 578507 
 575892 
 514276 
 574660 
 575044 
 ST5AR7 
 .575810 
 576198 
 576576 
 9.576959 
 OV 7841 
 57128 
 
 578104 
 .578486 
 
 579629 
 
 .580009 
 . 580889 
 
 9.580769 
 .581149 
 
 .582665 
 
 5838044 
 583422 
 .5838800 
 
 9.584177 
 
 anne? 90909 OD Ww o kegs . 7 
 
 > SD 
 
 Cotang. 
 
 10.488934 
 .438541 
 .488149 
 437756 
 .437864 
 .436972 
 .436581 
 .436189 
 .435798 
 .4385407 
 
 435017 | 
 
 10.484627 
 
 .434237 
 
 433847 
 433458 
 433068 
 432680 
 .432291 
 .431902 
 .431514 
 .431127 
 10.480739 
 430352 
 429965 
 
 429578 | 
 429191 
 
 428805 
 
 428419 
 428033 
 
 .427648 
 427262 
 10.426877 
 426493 
 
 426108 
 
 425724 
 
 425340 | 
 424956 | 
 
 424578 
 .424190 
 423807 
 
 423424 | 
 10.423041 
 
 .422659 
 
 422277 
 
 21896 
 
 “491514 
 
 .421138 
 420752 
 .420371 
 
 .419991 
 
 419611 
 
 10.419231 
 418851 
 
 .418472 
 .418093 | 
 ALTT14 | 
 
 417385 
 .416956 
 .416578 
 
 .416200 | 
 
 10.415828 
 
 Het 
 > “CO 
 
 a 
 
 ardradieed 
 Go He Or 
 
 +] 
 
 we 
 
 et 
 _ 
 
 10 
 
 oahio) 
 
 CS mt 09 CO HB OT S2 3 
 
 ‘ | Cosine. 
 
 oO 
 
 bets 
 
 Sine. 
 
 Cotang. 
 
 =) 
 
 Tang. 
 
Sine. Dl" | Cosine. | D. 1". | Tang. | D. 1". | Cotang. | / 
 | 
 | ; | | i a PM ae eae ek ee 
 RRA Qe m 5 pyre | 
 phate | 5.48 || 9-P0152 | gg || 9.584177 | aq | 10.415823 | 60 
 “ -00FIG ¢ mk oun Ot De a 0049382 ne 41506 5 
 3} .655315 | 3-4" | ‘970006 Bo || 885809 | 6:28 “4i4eo1 | 27 
 4 |) 55568 | E47 || 969957 | 8° || pabese | 6-28 | . “ayant, | 56 
 5) -So5071 | E47 || 969909) -89 || 586062 | 8-27 413938 | 55 
 6 | 556209 | 545 || 969860 | 65 || 5864a9 | 8-28 | “args: | 54 
 i) BES 28 | Me) S| Be) ge | tee 
 eases igs -JOIIO 5 008 LE sated .41281 By’ 
 oe | eee bo wae |] Soeeria | 8) Us eerbeg | 6-28 $1243 | Bi 
 557606 | 5°43 || 969665 | 82 |) ‘oyioai | 8-25 412059 | 50 
 Wi 11 | 9.557982 | 5 43 || 9.960616 | “4, || 9.588816 Foe | to-4tiess | 49 
 Whey fe) .BOSB53 | Sp || -960567 | 82 || paseo | 6-25 ‘411309 | 48 
 14 | 558009 | f'4o || 960469 | 88 || “beguz0 | 8-23 .410560 | 46 
 15 | -Bpe2 | 5.40 || 969420) -g5 |] -580814 | 6-23 | latorg6 | 45 
 | o} peeps | 5.42 || 269870) “> |) -s0188 | 6-33 | laoggie | 44 
 i) Ti | epegees | Bug || peemet | Aes || Go0bee | 688 409438 | 43 
 jit 18) -p60207 | 540 || 960272) +85 || -Bo0095 | 8-22 | ‘409065 | 42 
 1 49) 560531 | 5'49 || 969223 | -g3 | -b91808 | 6-22 |  ‘4osege | 41 
 Hi “D60855 | 5138 || 960173 |g) |} -doues1 | 33 | .408819 | 40 
 i] vay om 5 ; 
 "ie 21 | 9.581176 | 555 | 9. 969124 | go || 9-592054 | og | 10407946 | 2 
 Ae He 23 | “561824 | 5-38 |] “gggo95 | 83 || “pyered | 6.22 pe 
 vi 33 | -po1s824 | 537 || 969025 | --g) |] -b9ev00 | 6-22 | ‘aoreo1 | 87 
 i) a ero | me? i a "99 598542 6.20 -406458 | 35 
 4 36) b6200 | 537 || 968877 | -g5 |] -bosoa | 6-20 | ~dogose | 34 
 i 2 | 5612 | Fy 968827 | G3 |] 594285 || 6-18 405715 | 83 
 aM A Tee epee | Biay loeeeeere | tee || Spates || Bute 405344 | 82 
 6) 2BBTS | Sieg |] 960728 |B [lc ponoe7 | B18 404973 | 31 
 a 564075 | 5’a5 || -966678 | gs || 595808 | 6-18 | ‘4os6oe | 80 
 il BI | 9.564306 | 5 a9 | 9. 968628 | gg || 9.595768] ¢ 47 | 10.404282 | 29 
 f G2'| . ROATIG |. Brag || 968578 | 83 | pogias | 8.1? 408862 | 28 
 Me 34. | (pebase | BeBB || 7eeebe8 | olga |] -896508 | g-te .403492 | 27 
 H 35 | 565676 | 5-33 || “eggion | 188 || 7BP6STB | g'a5 | 408122 | 26 
 Z . x ‘£4 5.35 ie Se) ae | ‘ SOU 12 Ry .402753 | 25 
 6 ) 565905 | 255 || 90870 | 83 || ‘sore16 | 6-15 | “aopsea | 21 
 eh lowipeeens 8oad Hae oe | 195 || -597985 | g48 .402015 | 28 
 567269 | 5139 || 968178 | 7g 599091 | 743 -400909 | 20 
 A swastowa OQ | : rm ; 
 42 | eofee, | 5.28 || 9-968128 | “gy || 9.599450 | 1 | 10.400541 | 19 
 42 | -p61904 | 5°59 || 268078 | G2 |] -sa9827 | 6-18 | «~“aoor7s | 1a 
 ill] BB) pebee2 | peg ||: 908027 | (88 | “enon, | Bel2 899806 | 17 
 Hi to | ceesee | Bier || 967927 | +g |) | 600020 | 6-12 |  “agg0rt | 15 
 | 46 | -B6O172 | Siar |) 260876 | 58 || o1a0s | 6-12 | © “sonz04 | 14 
 | 44 | 369488 | Sor || 967826] 23 || ‘601663 | 6-12 "898887 | 18 
 | By | Uses | Bee 1 gee || cree 1] Sb0R0e8 | a8 397971 | 12 
 iil 49 | 970120 | oe 967725 "Br 602395 | @” 897605 | 11 
 | | 50 | -BM0S85 | 5:7 |] 907674 | 83 || Leon761 | 6-10 | “397099 | 30 
 Bo Brows 3m Be Ps ; ; 
 so | ceriogg | 5-25 || 9-2982624) gs |] 9.60818 | ¢ 49 | 10.396873 | 9 
 53 | 1571380 | 5-28 |] “ogeton | 85 |] 808493) | g'og |  -896507 | 8 
 a - os oes 595 : IOTSR2 g5 | 608858 | 6.08 . 896142 ff 
 55 | 1572009 | 3-23 || 7pol8"1 | vg || -604228-1 ggg |  .B05z77 | 6 
 fe cel” | Bias 967421 "a5 || 604588 | @'ng .895412 | 5 
 Bo | 722828) 5ia0 || -967370| -g? || ecaona | 8-98 |  “Rosod7 | 4 
 Bi 72686 | sag || 907819) 2 || “ecs3i7 | 8-07 |  “aqaess | 3 
 Be] 7pie850 | pian || 967268] ge |] -eoscsz | 6-08 | “squgi8 | 2 
 op | gpueee3 | 5.20 || 967217 "gs -606046 | 6 o> .893954 | 1 
 0 | 9.57857 9.967166 “|| 9.606410 ™" | 10.893590 | 0 
 Pine al Gaaeeke i are ee we; 
 Rees Cosine. | D. 1 Sine. |! D. 1". |! Cotang.! D.1". | Tang. | / 
 
 111° 
 
— 
 
 D. 1", || Cosine. | D. 1". || Tang. 
 = = es eG ae 
 0 | 9.573575 | 5 99 || 9.967166 | es | 9-606410 
 1| (573888 | 359 || -967115 | “ge || .606773 
 2| .574200 | F'59 ||  -967064 "gx || -607187 
 3 | .574512 | Foo || -967013 | “or || 607500 
 4) .574824 | 559 || -966961 "Br 607863 
 5 | :5751386 | - jg || .966910 "35 608225 
 6 | 57447 | 54g ||  -966859 “85 608588 
 7 | .575758 | £49 || 966808 | gn . 608950 
 B | .576069 | P'y> || -966756 | “on .609312 
 9} .576379 | Fyn || 966705 a 609674 
 10 | 576689 | Fyy || 966653 "Bs .610036 
 11 | 9.576999 | ~ 47 || 9.966602 ~ || 9.610897 
 2 | .577309 5 15 .966550 "8s 610059 
 13 | 577618 | 245 . 966499 “Be .611120 
 14) BY7927 | P45 966447 eee .611480 
 15 | 4578236 | 245 966395 "gs || -611841 
 16 | .578545 | "43 . 966344 “2 ||. 812201 | 
 17 | 578853 | 2-72 966292 | “So || .612561 
 18 | 1579162 alee . 966240 pee . 612921 
 19 | 579470 8 12 .966188 ee 618281 
 20 | 579777 5 13 . 966136 "gs 613641 
 21 | 9.580085 | ~ 49 || 9.966085 gy || 9-614000 
 22-| 580892 | 5745 || -266033 rats 614859 
 23 | .580699 | -'39 || -965981 “OF 1614718 | 
 24 | 1581005 | 245 || 265929 “ag || -615077 | 
 25 | 1581312] 2739 || -965876 |“, 615435 
 26 | 1581618 | S39 ||  -965824 we 615793 
 27 | .581924 | Fog || .965772 o 616151 
 28 | 1582229] £49 || 965720 | || 616509 
 29 | 582535 | F'og || 965668 “eg || 616867 
 30 | .582840 | F*og ||  -965615 "gr || 617224 
 31 | 9.583145 | ~ gy || 9.965563 | » || 9.61%582 
 32 | 583449 | Pog || .965511 | “68 617939 | 
 83 | .588754 | B'o~ || -965458 | > || .618295 
 34 | 584058 | pion || -965406 | “GQ || .918652 
 35 | 584361 | pp» || -965353 | “oe |) .619008 | 
 36 | 584665 | 5" 05 965301 “oR 619364 
 387 | .584968 | =o 965248 | “G8 619720 
 88 | 585272 | z'o3 || -965195 | “gn 620076 | 
 89 | 585574 | 5o5 || 965143 | ogg || -Sa0dse 
 40 | a 585874 ( 5 ‘ 03 ie 965090 , 88 s 620 ( 8 ( 
 4i | 9.586179 | x ox || 9.965037 | gg || 9.621142 
 42 | .586482 | p'o9 || -964984 “ag || «| 621497 | 
 43 | .586783 5 08 || 964931 ees! 621852 | 
 44} 587085 | poo ||  -964879 | “6 622207 
 45 | .587386 | p'o9 || -964826 “og 622561 
 46 | .587688 | Foo 964773 “og 622915 | 
 47 | 587989 | 2" aq .964720 | “gq || - 628269 | 
 48 | .588289 | Py og || -964666 "oq || -623623 
 49 -588590 5 00 -964613 | “a8 | 623976 
 50 | 588890 | 2"o9 ||  -964560 “gg || -624330 
 51 | 9.589190 , || 9.964507 | 19 ||: 9.624688 
 52 | .5eode9 | 4-98 || “‘o64454 | “88 || -625036 
 53 | .589789 | frog || -964400 | “ge .625388 
 54} .590088 4.98 | -964847 | “ge 625741 
 55 | .590887 | 4igg || -964294 “99 || + - 626093 
 56 | .590686 | 4'g, || -964240 | “gg ||. 626445 
 57 | .590984 | 4igy || -964187 | “99 || 626797 
 58 | .591282 | 4ign || -964183 | “gg || .627149 
 59 | 591580 | 4.97 . 964080 90 627501 
 60 | 9.591878 Ai 9.964026 “|| 9.627852 
 Y + Cosine: |°Dr1"7 ||; --Sine. Dra F11 Cotang. 
 
 COSINES, TANGENTS, AND COVANGENTS. 
 
 Dae 
 
 6.05 
 6.07 
 6.05 
 6.05 
 6.03 
 6.05 
 6.03 
 6.03 
 6.03 
 6.03 
 6.02 
 6.03 
 6.02 
 6.00 
 6.02 
 6.00 
 6.00 
 6.00 
 6.0 
 6.00 
 5.98 
 5.98 
 5.98 
 5.98 
 5.97 
 5.97 
 5.9 
 5.97 
 5.97 
 5.95 
 5.97 
 5.95 
 5.93 
 5.93 
 5.98 
 
 CS CS <O 
 Co wo Oo 
 
 OroTvOwor ar Ororororororororordr Ororororor 
 
 i) 
 ~) 
 
 Or or or 
 3 
 
 Cotang. | / 
 10.393590 | 60 
 ,393227 | 59 
 .892863 | 58 
 .392500 | 57 
 .892137 | 56 
 B917%5 | 55 
 891412 | 54 
 .891050 | 53 
 .890688 | 52 
 .890326 | 51 
 .389964 | 50 
 10.389603 | 49 
 .889241 | 48 
 888880 | 47 
 .888520 | 46 
 .888159 | 45 
 .B87799 | 44 
 887439 | 43 
 .387079 | 42 
 .886719 | 41 
 
 | .886359 | 40 
 10.386000 | 39 
 885641 | 38 
 .885282 | 37 
 884923 | 36 
 .884565 | 35 
 .884207 | 34 
 .883849 | 33 
 883491 | 32 
 .883133 | 31 
 382776 | 30 
 0.382418 | 29 
 382061 | 28 
 881705 | 27 
 .881348 | 26 
 -380992 | 25 
 .880636 | 24 
 380280 | 23 
 379924 | 22 
 .879568 | 21 
 .379213 | 20 
 10.378858 | 19 
 .878503 | 18 
 .378148 | 17 
 377793 | 16 
 877489 | 15 
 BU7085 | 14 
 LOT Fol ako 
 876377 | 12 
 876024 | 11 
 .375670 | 10 
 10.375317 | 9 
 874964 | §& 
 874612 | 
 374259 | 6 
 373907 | 5 
 BIB555 | 4 
 .873203 | 3 
 872851 | 2 
 372499 | 1 
 10.372148 | 0 
 Tang. 
 
TABLE XXV.—LOGARITHMIC SINES, 
 
 | ! 
 Sine. D. 1". | Cosine. | D. 1”. || Tang. | D.1". | Cotang. | ’ 
 | | | 
 | 
 0 91878 ~ || 9.964026 || 9.627852 ~ | 10.872148 | 60 
 1| 592176 | 4-95 || 963072 | -29 || gase03 | 5:85 | “arr707 | 59 
 2| .592473 | 49s . 983919 "90 |; 628554 | Pgs .871446 | 58 
 3 | 592770 | 493 963865 “90 628905 | 239 371095 | 57 
 4 | 593067 | 793 963811 "99 ||. 629255 | Boge 310745 | 56 
 5 | .593363 | 793 963757 "gg || | 629806 | F "98 .870394 | 55 
 6 | .593659 | 493 963704 "90 || 620956 | B*p5 .3870044 | 54 
 7 | 593035 | 493 963650 "99 || 630806 | 5°35 .869694 | 53 
 8 | .594251 | 493 || .963596 "99 || -680856 | 535 .869344 | 52 
 9 | .591547 | 4io5 || 963542 "99 | 631005 | 2°38 .368995 | 51 
 10] .594842 | 4g || .963488 "99 |, -631855 | S65 .868645 | 50 
 i1 | 9.595137 | 4 99 || 9.963434 gg | 9-631704 | ~ g5 | 10.868296 | 49 
 2| 595432) 495 || .963379 "99 || -682053 | ees .867947 | 48 
 B} 595727 | 4°95 963325 ‘9p || 682402 5 80 867598 | 47 
 14 | 1596021 | 4°95 963271 "99 || 682750 | 5 "65 .367250 | 46 
 15 | .596315 | 4°90 963217 "99 || -688099 | 3 °p5 .366901 | 45 
 16 | .596609 4g _«|| 963163 "92 || 688447 | 5°65 .366553 | 44 
 7 | .596903 | 4°65 .963108 “90 633795 | 5 "6p .366205 | 43 
 al 18 | 597196 | 4"o9 || -963054 ‘92 || -684143 | Bp .365857 | 42 
 i) 19 | 5974901 4°93 . 962999 “90 -634490 | "Gq .865510 | 41 | 
 Li 20 | 597783 gg || 962945 "92 || 684838 | She .865162 | 40 
 Be 21 | 9.598075 4 go || 9.962890 go || 9-685185 | ~ no | 10.364815 | 39 
 it 22 | .598368 ; 4'g- || .962836 "gn || 635582 | Bs .364468 | 38 
 Nth 23 | .593660 “oy || .962781 "99 ||. -685879 | Ac .364121 | 37 
 Hi 24] 598052 | 4:8" || 962727 oo || .636296 | 5-78 363774 | 36 
 nt 25 | 599244 | 4°64 |) 962672 "99 || 686572 | Reng . 363428 | 35 
 thi 26 | .599536 | Gig. || 962617 a 4680019 4) ore .363081 | 34 
 27 | .599827 | 4 ‘oe 962562 “90 637265 | "mw 362735 | 83 
 Hit 28] .600118 | 4g. || 962503 too || 687811 |) Ripe . 862389 | 32 
 hal 29 | 600409 | 4g. || .962453 "99 || -687956 | oan .362044 | 31 
 yt 30 | .600709 | 4'g3 || 962398 ‘92 || -638302 | Pes . 361698 | 80 
 1H | 81 | 9.609999 | 4 gg || 9.962343 gg || 9.638647 | ~~ | 10.361353 | 29 
 Baal | 32 | 601280 | 4°99 . 962288 aS -638992 | 5 ’ne .361008 | 28 
 Hit 83 | .601570-| 4gq ||  .962233 i 639337 | 2 'np .360663 | 27 
 ii 84] .601860 | Jigq || 962178 “99 639682 | 2 "ne .360318 | 26 
 nt 85 | .602150 | 499 || 96212: “93 «|| «=~ 640027 B73 859973 | 25 
 ei 86 | .602439 | fgg || 962067 | +05 || 640871 5 5 .359629 | 24 
 Ha 37 | 602728 | 4'gg || .962012] 95 || 640716 5 3 .859284 | 23 
 Pune: 38 | 608017 | 4°35 -961957 | 95 || .641060 | Pens .858940 | 22 
 cial 89 | .603305 | 4'g5 || -961902 | 93 || 641404 5 72 .858596 | 21 | 
 40 | 603594 | 4°35 . 961846 "gg || 641747 | rng - 858253 | 20 | 
 41 | 9.603882 | 4 gy || 9.961791 gg |, 9-642091 | 5 ag | 10.3857909 | 19 | 
 42] .604170 | 4p, .961735 "gn || 642434 | png 857566 | 18 | 
 i 43) 604457 | 785 . 961680 “93 C4277 | Bing .857223 | 17 
 i 44) .604745 | 4p 961624 “99 643120 | 2°15 .356880 | 16 
 45 | 605032 | 4'n .961569 “98 643462 | 2°45 356537 | 15 
 a 46} 605319 | 7+ 961513 a5 .643806 | 2-6 .856194 | 14 
 | 7 | .605606 | 7"+- 961458 “93 644148 5 40 355852 | 13 
 Ha 43 | -605892 | ging || -961402] “98 644490 | Pn5 .3855510 | 12 
 Wnt 49 | 608179 | gin || -961846 | -95 || 1644832 | ?-f 355168 | 11 | | 
 Hil! 50 | .600465 | 4's -961290 | “99 || .645174 | 2-6 .3854826 | 10 | 
 51 | 9.606751 > 9.961235 2 || 9.645516 10.354484 | 9 
 52 607036 | 4-72 || ‘961179 oa || 1645857 | 5.88 "354143 | 8 
 53 | 07322 | Fins -961123 | “93 || 646199 |e "ha .353801 | 7 
 b4} corer | 7-2 961067 | 93 || 646540 | 2-68 _353460 | 6 
 55 | .607892 | 7 "ne 961011 "93_|| 646881 | 5 G0 -B53119 | 5 
 | 56 | © .608177 493 960955 “93 |, 647222 5 67 352778 | 4 
 57 | .608461 | 7°25 . 960899 "93 || 647562. ] Fe eos | 3 
 BS} 608745 | 4-73 || 960843 | 92 || 647903 | 5-88 097 | 2 
 59 | 609029 | fons 960786 | “93 648243 | Pope ‘35r73" | 1 
 60 | 9.609313 “1 9.960730 | | 9.648583 | °°?" | 10.351417} 0 
 7 Gosime.-| D241 Sine. PPE RAS 4 Cotang. | D. 1” Tang. | 1) 
 
Bine..-,|. Di", | Cosine. | D..1 | Tang. | D.1". ; Cotang | , 
 aS 
 = a 
 0 | 0931: | ve 58: 
 0 9.600818. 4-3. || 9.960780 | 93 9.648583. |, @ | 10.351417 | 60 
 @ | asBOe80 | aia" <PEOGTS “gy |; 649263 | pgn | 800737 | 58 
 B estaitse | aoae iL is 60561 | ‘93 |) .649602 | 5 67 .850398 | 57 
 5} ceiory | 420 || “oeousa | 28 | Rees 5.65 oe |e 
 Bl ernie | 40% |) CQeaaos |) f8r |! Seen || 5-85 BageT9 | 85 
 6 012 | Zr. |] -960892 3 || [650620 |. 2: 349380 | 54 
 | .611294 ‘ 960335 | “22 59 |, 9-63 “B49041 | 53 
 8| ‘611576 | 449 "960279 ot ‘eoteo7 | 583 | “Sasros | Be 
 Sears | oa | See | cae. | ih Se | aes 
 | : 2 . 93 Vo ( rR € . 348026 50 
 11 | 9.612421 | | o.e5312 | 
 12 | venroe | 4:53 || °Sooos9 | 93. |) epans0 | 8-63 | 7-Betaso Fe: 
 33 | ce1a9es | 4-68 |] “osgo95 | 95 |! “oppose | 8-88 | cBatore | a7 
 | cerae6s | 453 |) “gpg | 93 || cesaaen | 8-63 | “Stgora | 46 
 15 | 18545 4-68 || ‘ososse | 783° || 658663 | 5.62 | “Bieeee | 45 
 613825 , 4°00 59825 | “22 |! <7654000 |; 5-82 
 17 | cori | 48% || cSsomos | “23 | epaaar 882 | “Bases | 43 
 18 | “614385 | 4°8% || “osovi1 | °9 |) ‘esaerg | 5-62 pret 
 9 | me lt aie tate be = || -654674 ee .345326 | 42 
 o | 614065 | 65 |] 939054 sey || Eeoeenit st Bake "344980 | 41 
 oi eto 4.65 ae 05 655348 | 560 .044652 | 40 
 S| Grae | 4.65. || 2ENES | 95 || *bep5090 | 5:60 | 7° Ses9a0 | Bs 
 93 | eisrs1 | 4-8 || ‘osq405) -25 || “@5635g | 5-60 Serie | ay 
 S| ce1eoeo | 455 || gsyses | 225. |! “eseone | 5:80 | “Biss08 | 86 
 | ceiosay | 4-63 || “gagso | 9% |] ceoroas | 5:60 | “Sisore | Bs 
 96 | 616616 | 4:63 || ‘959953! 25 || “G57gea | 5-80 "Baeese | BA 
 a | ceiesoe | 463 || cgpyros | 22 |) ce8tosa. B88 | “brpsor | 83 
 98 | ¢i7i72| 4:83 || ‘osgisg| <2 | ‘gosoat | 5-58 Bito06 | 32 
 29 | 617450 4.63 | “g59080 | -2% |] ° \assaco | 3-58 Su1031 | 31 
 ‘ C17797 -0# Kans oot Re, : , 
 eure | 4:82 || “959023 | -29 || lessvos | 2°33 |  -841296 | 30 
 a 9.618004 3 an 9.958065 95 || 9-659039) 5 x | 10.340961 | 29 
 2| 61881 | 4° "958908 | 25 ||° 1659373 | 2 340627 
 33 | 619558 | 4:52 || “95885 ‘97 |) “E5q708 | 5-58 Sites | 
 Bt | “eres | 4-60 |) “deeroa | -9% |! “eeoos | 9-32 eaeee (28 
 35 | 619110 4.60 || 958734 | <2f || 660376 | b,0% ees | 35 
 5 | “e19386 | 4: ‘958677 | 22 || ‘eeovi0 | 2-52 |  ‘3sgg00 | 2 
 37 | 619662 | 4:80 5 ‘OT | “Be 1o43 | 5-55 ener te 
 | Giooss | 4-50 |) capssor | 2% |) copier | 5:57 | “Sapo | 22 
 BF | cebeers | 458 || “pores | 2% |} ceerrio | -5:55 | ‘Baso00 | 31 
 [ -oandss | 53 |) 2585 | oor 662043 | >>? 37957 | 20 
 41 | 9.620763 | 5g |) 9.958387 | g, |) 9.662876 | “~~ | 10.337624 | 19 
 2| 1621038 | 4° "958329 | 2% ||. ‘62709 | 5-55 | © ‘aava9 
 43 | 1621313 | 4°oo || coseevl | <4 pean | 5:55 | “B3g099 | 47 
 44} 621587 | 472 ‘osseia | 2% |) “eusa7s | 5-55 Sets tee 
 Biker | 4:57 |] cQeeise | 98 |) cGoaror | 8-23 | “Bsoe08 | 15 
 46 | .6e2ias | 4-27 || ‘958006 | 2 || “664039 | 5-33 ‘S35001 | 14 
 7 | 1622409 | Gree 58035 OF 664371 | 2:38 "33562 
 Br (ieee | Ace i tiers || 208 || criers | 5 :b8 | Bate, Ee 
 PP ieee | iis, leet | ibe |i cceeayOS | pbs | cater me 
 ey cere | 4a5b [oe ieee | ct || -665035.) 559 834905 | 17 
 4.55 ~dd (ODD 98 . 665866 | kK 53 .304634 10 
 51 | 9.628502 9.9578 || 9.665 oe. 
 Bl | 9.628502 | 4g nq ||,9.057804 |g, || 9.085608 | 5 59 10.334302 | 9 
 52 | .oaur7d | fps |) 980746 | <3 || 600% | bio | -BaBOTL | 8 
 BS | 624047 | 43 || -957687 | “gg || - 666360 | es .333640 | 7 
 Bd |r s624319 | 453 || ee “OF 666691 | 2°25 .333309 | 6 
 56 | 624863 | 258 || ‘osvai| +28 “gorse | 8-52 8 | a4 
 57 | 605185 | 4°28 || opvene : «28 “porees | 3-00 | “bammg |B 
 58 | ‘62540 | 4-22 |] “osvao3 | 28 || “eeg013 | 5-52 aiiay | Se 
 59 | 625677 | 4°22 || ‘os7Ba5 | -24 || “osasaa | 3-30 spiosy | 
 2 age vn yrrore . : nisi 2 y | 
 60 | 9.625048 |" 9.957276 9.668673 | 5-59 | 49'331397 | 0 
 Gosine. |. D; 17; || ~Sines } Dil Cotang. | D.1". Tang. ear 
 
 COSINES, TANGENTS, AND COTANGENTS. 
 
10 
 
 Sine. 
 
 9.625948 
 626219 
 . 626490 
 . 626760 
 627030 
 627300 
 627570 
 .627840 
 628109 
 .628378 
 628647 
 
 9.628916 
 .629185 
 629453 
 629721 
 .629989 
 .630257 
 630524 
 .630792 
 -631059 
 6313826 
 
 9.631593 
 .631859 
 632125 
 . 632392 
 . 632658 
 632923 
 . 633189 
 633454 
 633719 
 . 633984 
 
 9.634249 
 -634514 
 63477 
 .635042 
 . 635306 
 . 635570 
 .635834 
 . 636097 
 . 636360 
 636623 
 
 9.636886 
 .637148 
 637411 
 .637673 
 637935 
 638197 
 638458 
 638720 
 638981 
 639242 
 
 9.639503 
 .639764 
 . 640024 
 . 640284 
 .640544 
 . 640804 
 . 641064 
 .641324 
 .641583 
 9.641842 
 
 Cosine. 
 
 i 
 
 1 
 
 HB Orororor 
 
 wwwwoesweasS GRGRARASLS SBRRSSSSSSIR 
 
 SP RP 
 
 ¢ Ww CIt co CDWWKWHRA HR AHH 
 GLOUNESITNINENIDON DODDOOOSOCOW 
 
 co GY Co 
 
 ALA AA AL AAA AAA ADA AAA AAA AD AAA RAAA AA AAA AAA RADA AAAAAARARAA 
 Gt 6559 Or Or 
 
 aad 
 Co Cs 
 02% 
 
 Cosine. 
 
 9.957276 
 957217 
 957158 
 957099 
 957040 
 . 956981 
 956921 
 . 956862 
 .956803 
 956744 
 . 956684 
 
 9.956625 
 956566 
 956506 
 956447 
 956387 
 956327 
 956268 
 . 956208 
 . 956148 
 . 956089 
 
 9.956029 
 . 955969 
 . 955909 
 . 955849 
 955789 
 . 955729 
 955669 
 . 955609 
 . 955548 
 . 955488 
 
 9.955428 
 955368 
 . 955307 
 955247 
 . 955186 
 955126 
 955065 
 . 955005 
 954944 
 954883 
 
 9.954823 
 . 954762 
 954701 
 . 954640 
 . 954579 
 954518 
 954457 
 - 954396 
 954335 
 954274 
 
 9.954213 
 - 954152 
 -954090 
 954029 
 . 953968 
 . 953906 
 953845 
 . 953783 
 . 953722 
 
 9.953660 
 
 Sine. 
 
 Dias 
 
 bet eR eh ee pe ee ee pe pe pp | ell eel geet eel eel eee ee ed beh peek | tl eel and 
 he one ape are ore he. He ime ate th see Ais may a ee pdt oale ai, Sori She ote ak yak oe yas Tye Se Sal ee GA ce e 2 ys 20 Ces Sea OA 
 
 > 
 
 i=) 
 
 p| 
 | 
 
 | 9.668673 
 
 TABLE XXV.—LOGARITHMIC SINES, 
 
 Tang. 
 
 . 669002 
 . 669332 
 -669661 
 .669991 
 670320 
 670649 
 670977 
 671806 
 .6716385 
 671963 
 
 9.672291 
 672619 
 672947 
 673274 
 673602 
 673929 
 674257 
 674584 
 674911 
 675237 
 
 9.675564 
 675890 
 676217 
 676543 
 676869 
 677194 
 677520 
 677846 
 678171 
 678496 
 
 9.678821 
 679146 
 679471 
 679795 
 680120 
 680444 
 680768 
 681092 
 681416 
 681740 
 
 9.682063 
 682387 
 682710 
 . 683033 
 . 683356 
 . 683679 
 684001 
 684324 
 .684646 
 . 684968 
 
 9.685290 
 .685612 
 - 685934 
 . 686255 
 686577 
 . 686898 
 687219 
 .687540 
 .687861 
 
 9.688182 
 
 Cotang. | 
 
 D. 1’. 
 
 5.48 - 
 
 LPL PPR PP PP Sh PP PP 
 SSSSSSERS5E5 fo fo to Ge Oo wo Go to Or Go Ot OW OF I ON SS3ea55 
 
 HAAN NNO OOOO HOON TON TOT OUT, CLOT OT ON OT OT OT OT OTT. OTOTOTOTOTON 
 PLLA DLP PPA . 8 Pa SE te Te War Ban 
 
 TOLOTOVOUN OTSI AE AE 
 
 OCeOLOvr gor orgorvororor O1ror 
 a7 re aera ee Co OO 
 
 o 
 
 Cotang. 
 
 10.331327 
 . 880998 
 . 830668 
 330339 
 
 . 830009 
 . 829680 
 829351 
 
 829023 
 828694 
 828365 
 828037 
 
 10.827709 
 .327381 
 .3827053 
 326726 
 826898 
 .826071 
 820743 
 820416 
 . 325089 
 824763 
 
 10.324436 
 -824110 
 828783 
 823457 
 .823131 
 
 822806 
 
 822480 
 - 822154 
 -821829 
 821504 
 
 10.321179 
 820854 
 820529 
 820205 
 319880 
 319556 
 819282 
 . 318908 
 . 318584 
 .318260 
 
 10.317937 
 .3817613 
 317290 
 3816967 
 .316644 
 £816321 
 . 3815999 
 -815676 
 -815854 
 -815082 
 
 10.314710 
 .314388 
 .314066 
 .818745 
 813423 
 . 813102 
 312781 
 312460 
 .8121389 
 
 10.311818 
 
 Tang. 
 
 | 
 
et 
 
 ~ 
 
 COnt Co Ot te C52 
 
 COSINES, TANGENTS, AND COTANGENTS, 
 
 Sine. Diss 
 
 9.641842 
 "642101 reo 
 642360 | 4-32 
 642618 q's3 
 642877 | 4-32 
 "643135 e+ 
 643393 | 4°30 
 613650 | 4°55 
 .643908 4 28 
 
 644165 , 
 RA AADE 4.30 
 644423 4°28 
 
 | 9.644680} 4 o» 
 
 "644936 | 4 
 "645193 ye 
 "615450 
 
 6550: 
 
 . 655805 
 
 . 656054 
 
 656302 
 
 .656551 
 
 .656799 | 
 9.657047 
 
 or 
 645706 a 
 645962 | 7°56 
 646218 427 
 646474 1.9% 
 646729 | 4°95 
 646984 | 4°5> 
 Garad | 423 
 647749 het 
 648004 | 4°53 
 648258 | 4°53 
 648512 | 45. 
 648760 | 4°5: 
 649020 | 4°53 
 649274 | 7°55 
 649527 | 4°55 
 9.649781 A 
 “650031 | 4°32 
 650287 | 7°59 
 650539 | 4°55 
 650792 | 4°55 
 651014 | 4°55 
 651297 | 4°59 
 651549 | 4°75 
 651800 | 4"op 
 “652052 | 4° 
 9.652304 mi 
 .652¢ 
 658555 a 
 652806 | 4°48 
 653057 | 449 
 653308 | 4‘d> 
 653558 | 417 
 653808 418 
 "654059 | @:18 
 654309 reel 
 654558 | "4p 
 4 
 4 
 4 
 4 
 4 
 4 
 4 
 4 
 
 Cosine. | D. 1’. 
 
 Cosine. 
 
 9.953660 
 . 958599 
 -958537 
 . 953475 
 .953413 
 - 953352 
 - 958290 
 . 953228 
 - 953166 
 .958104 
 . 953042 
 
 | 9.952980 
 
 952918 
 952855 
 952793 
 952731 
 952669 
 952606 
 952544 
 952481 
 952419 
 952356 
 952294 
 952231 
 952168 
 952106 
 952043 
 951980 
 951917 
 951854 
 951791 
 
 9.951728 
 951665 
 . 951602 
 .95158 
 951476 
 .951412 
 951849 
 - 951286 
 951222 
 
 .951159 
 
 co 
 
 . 951032 
 950968 
 950905 
 950841 
 95077 
 950714 
 950650 
 . 950586 
 950522 
 
 9950458 
 
 . 950138 
 . 950074 
 . 950010 
 .949945 
 9.949881 
 
 Sine. 
 
 9.951096 
 
 950894 
 
 950330 | 
 950266. | 
 950202 | 
 
 D1"; 
 
 Tang. 
 
 9.688182 
 . 688502 
 . 688823 
 .689143 
 . 689463 
 .689783 
 .690103 
 .690423 
 .690742 
 .691062 
 .6913881 
 
 9.691700 
 . 692019 
 . 692358 
 . 692656 
 .692975 
 698293 
 .693612 
 693930 
 694248 
 .694566 
 
 | 9.694883 
 .695201 
 .695518 
 .695836 
 .696153 
 .696470 
 696787 
 .697103 
 .697420 
 .697736 
 
 9.698053 
 | .698369 
 . 698685 
 .699001 
 .699316 
 . 699632 
 .699947 
 . 700263 
 700578 
 . 700893 
 701208 
 .701523 
 .701837 
 702152 
 702466 
 . 102781 
 . 708095 
 - 703409 
 103722 
 704036 
 | 9.704350 
 | .704663 
 . 704976 
 | .7'05290 
 . 405603 
 ~ 705916 
 706228 
 706541 
 | .'706854 
 | 9.707166 
 
 © 
 
 | Cotang. 
 
 Dsl"; 
 
 Orororor 
 ) OO 
 © 
 
 CrOLCOvrOrorwore 
 eo a 
 i) 
 
 OCvrOT Or or or 
 ry) 
 So 
 
 5.382 
 
 CUNO UOT OT OT Oot OT OT OU OU St OF OF OF OF OF 
 ra) 
 (SC) 
 
 | 
 | 
 
 Cotang. 
 
 10.311818 
 
 | 
 
 -811498 | 
 
 811177 
 810857 
 
 810537 | 
 310217 | 
 309897 
 
 809577 
 -809258 
 
 808938 | 
 
 . 808619 
 
 10.308300 
 
 807981 
 -807662 
 807344 
 807025 
 
 306707 
 
 - 806388 
 806070 
 805752 
 805434 
 
 10.305117 
 . 804799 
 . 804482 
 . 804164 
 . 803847 
 .008530 
 et BUGR IS 
 - 802597 
 . 802580 
 .002264 
 
 10.301947 
 -801631 
 .801315 
 
 * 300999 
 
 .800684 | 
 300368 | 
 800053 | 
 299737 
 
 299422 
 299107 
 
 10.298792 
 298477 
 298163 
 297848 
 297534 
 291219 
 296905 
 
 296591 
 
 296278 | 
 
 295964 
 
 10. 295650 
 337 
 
 ee 
 
 a 
 
 istatoa 
 .294339 ¢ 
 
 294084 | 
 
 2932 
 -298459 
 
 .293146 
 
 10.292834 
 
 Tang. 
 
 , 
 
Sine. Cosine. 
 0 | 9.657047 | 4 43 9.949881 
 1} .657295 | 449 949816 
 2| .657542 | 445 949752 
 8 | .657790 | a9 949688 
 4 | .6580387 | 4745 949623 
 5 | .658284 | 445 949558 
 6 | 658531 | 4745 949494 
 7 | 658778 | 445 949429 
 8} .659025 | 4°49 949364 
 9 | .65927 410 . 949300 
 10 | 659517 | 4°49 949235 
 11 | 9.659763 | 4 49 || 9.949170 
 12 | 660009 | 4°49 949105 
 13 | .660255 | 4°49 949040 
 14] .660501 | Jog 948975 
 15 | 660746 | {og 948910 
 16 | 660991 | 4p. 948845 
 17 | 661236 | 49g 948780 
 18 | .661481 | 4 "og 948715 
 19 | 661726 | Gop . 948650 
 Q 661970 407 . 948584 
 21 | 9.062214 | 4 og 9.948519 
 22 | 662459 | 4" o 948454 
 23. | 662703 4.05 948388 
 24 | 662946 | 4 oe 948323 
 25 | .663190 | 4"ox 948257 
 26 | .663433 1.07 948192 
 27 | .663677 1.05 948126 
 28 | .663920 4. 05 948060 
 29 | 664163 | 7p: 947995 
 30 | .664406 4.08 947929 
 31 | 9.664648 4.05 9.947863 
 32 | .664891 4.08 947797 
 33 | 665133 | 7 "o3 947731 
 B4 | .665375 | 49g 947665 
 35 | 665617 | 4°93 947600 
 36 | 665859 | 4'o5 947533 
 37 | .666100 | 7 "oa 947467 
 38 | .666342 402 947401 
 39 | 666583 | 4 "95 947335 
 40 | 666824 | 4'o9 947269 
 41 | 9.667065 | 4 9 9.947203 
 2 | 667205 | 495 947136 
 43 | .667546 | 4"p5 947070 
 44 | 667786 | 7° 947004 
 45 | 668027 | 4'o9 946937 
 46 |  .668267 | 3*9g || -946871 
 7 | .668506 | 4'o9 || 946804 
 48 | .668746 | 4'o9 || -946738 | 
 49 | 668986 | S'o3 || .946671 
 50 | .669225 | 3°98 || .946604 
 51 | 9.669464 | ,, 98 || 9.946538 
 52} .669703 | 5°99 || .946471 | 
 53 | 669942 | 5*98 946404 | 
 54 | 670181 | 3 '" 946337 | 
 55 | .670419 | 39g 946270 | 
 56 | .670658 3°97 . 946203 | 
 57 | 670896 | 3 *o. . 946136 
 58 | .671134 3°97 946069 | 
 59 | 671372 | 3 'o¥ 946002 | 
 60 | 9.671609 “e 9.945935. | 
 ’ | Cosine. Dslr Sine. 
 
 Dx? Tange 18D,A". Cotang. 
 Hog || BARES br ogtog | ieee 
 1.07 || crorzg0 | 5-2 292210 | 
 tog. || -70st02 | 8°20. | © “Sareog | 
 1.08 || -/0Stlt | 5.20 eee 
 1.07 || “pooa7 | 518° | “Soa63 | 
 1.08 "709349 | 3-20 290651 
 1.08 || “vo9e60 | 5-18 "290340 
 pe || Lemmag7t |. Bate "290029 
 poe -7ion62 | 2-18 989718 | 
 hank 5) » 
 08 || Sipongs | Onde | “aeseeang | 
 1.08 “711215 | 5-18 “O88785 | 
 1.08 711595 | 2-17 “ORRATS 
 1.08 "711836 | 2-18 "288164 
 1.08 “712146 | 2-17 "987854 | 
 1.08 “712456 | 5-17 O87 44 | 
 1 ° 08 ; ay 2786 5 . WV 3 987234 
 10° || cvago7e | 2-27 |  Topegeg 
 108 713386 | 5°49 286614 | 
 1.10 eyagia | 5-15 "O85 686 | 
 1.08 14604 | 5-17 "985876 | 
 1.10 "714933 | 5-15 "O85 067 | 
 1.08 || cised2 | 3-15 | | lopaiss | 
 1.10 715551 | 5-15 "984449 | 
 ioe W15860 pe "984140 
 08 716168 | 5-1: "983839 | 
 10 "716477 ee 1983593 | 
 716785 9715 
 rio! || Catiebee | Bele | eens | 
 340. || Satan | ae "989599 
 1.10 “7iv709 | 2-2 "989291 | 
 1.08 m1goi7 | 5-13 281983 | 
 t8for || asztees | Baas 981675 
 1-10 118038 | pos .281367 
 J "719940 | 9-12 "981060 
 ager || Setaeas | B38 "980752 
 110 719555 | 2°45 .280445 
 9.71986 28018 
 1.12 || “stooieg | 5-12. | 77Seoest 
 1.10. || “0476 | 5-18 "970524 
 1 . 10 : Pores | 5 ° 12 ‘ 979917 
 1.12 || “rer0a9 | 5-10 | “oregon 
 1.10" || “igeia0e | 5-12, | tepgeed 
 1.02" || o7et702 | 5-18. | “Te7eeos 
 1.10 aeeeans |i Boke dbeies 
 tag || ezeeatd | 8-49. | lopreas | 
 110 T2221 | yg | 207879 | 
 toe! | Teese | bape) Aas 
 112 |] pass | Pay | 376462 | 
 = 2 1] gee o. Weare Es. 
 132) |) ae | 508 see 
 The “Od A | 5.08 | "OWBB46 
 1.12 "94760 | 5-10.) ‘oenoan 
 1.12 ‘wonogs | 5-08 “O74935 
 1.12 Liste “0 5.08 974830 
 1.12 caeuel 5.07 peer 
 ‘12 || go 7osera | 5: 1027432 
 D: 1", {I Cotang. | D. 1’. | Tang. 
 
 vo) 
 
 OM dD wR OIE 30 
 
 ~ 
 
 for) 
 9 | 
 ° 
 
COSINES, TANGENTS, AND COTANGENTS. 
 
 , Sine. D. 1”. 
 Je le a 
 0 | 9.671609 | 4 o, 
 1| 67187 | 3.97 
 2| 672084 | 3-9 
 3 | 672821 | 3.95 
 4| 672558 | 3-95 
 5.| .672795 | 3-95 
 6 | .673032 | 3-95 
 7 | .673268.| 3-98 
 8 673505 3 3 
 | 673741 | 3-83 
 10 | pane 393 
 11 | 9.67421: 
 12| .674448 ae 
 13) 674684 | 3-88 
 14 | 674919 | 3-92 
 15 | 675155 | 3-98 
 16 | .675390 | 3:92 
 7 | 675624 | 3-90 
 18 |, .675859 | 3:92 
 19 | 676094 | 3.92 
 20 | .676328 | 3-5 
 * % 6763 : 3.90 
 
 21 | 9.676562 
 22 | 67796 | 3-90 . 
 23. | (677030 oor 
 24 | 677264 | 3:20 
 25 | .677498 | 3:00 
 26] .e777si | 3-88 
 27 | .677064 | 3-88 
 28 | .67si97 | 3-88 
 29 | .678430 | 3-88 
 30 | 678663 | 5°68 
 31 | 9.678895 ie 
 30 | B79128 3.88 
 33 | .679360 | 3.8” 
 34} 679502 | 3°» 
 35 | .670se4 | 3-87 
 86 | .680058 | 3°20 
 37" .680288 | 3-87 
 38 | .680519 3 Bs 
 39 | .680750 | 3-8 
 2AnARD | OY. 
 40 pene 3.85 
 41 | 9.681218 
 42 | 681443 | 3.53 
 43} .681674 385 
 44] .681905 | 3°93 
 45 | .682135 | 3/83 
 46 | .682365 | 3-83 
 47 | 682595 | 3-82 
 48 | .682825 | 3-82 
 49 | .683055 | 3°85 
 = 209909 -¢ 
 50 | 688284 | 3-88 
 51 | 9.683514 | 2 a, 
 52 | 683743 | 3-05 
 53 | 683072 | 3-82 
 54] 684201 | 3-88 
 55 . 684430 3 ; 80) 
 56 | .684658 | 3°35 
 by | .684887 | 3°05 
 58 | 685115 | 3°85) 
 59 | 685343 | 3°5) 
 60 | 9.685571 | 2 
 / Dit 
 
 Cosine. 
 
 Cosine. | D. 1’. Tang. D. 1".-| Cotang. 
 9.945935 ; 9.725674 10. 274326 
 945868 | 1-12 || l725979 oe 274021 
 $1500 | ire || “BOM | sir |) 2reH16 
 . 945733 1 : 12 ; (26583 5 07 213412 
 945666 | 4°43 726892 | 2" 6s .273108 
 945598 | 3°45 27197 | "op .272803 | 
 945531 | 5°55 20501 | 3 *oe .272499 
 945464 | 375! 727805 | Pye 272195 
 945396 | 343 728109 | 5 "ox .271891 
 945828 | 3775 (28412 | 2° op 271588 
 945261 | 3°33 . 728716 5 Oy 271284 
 9.945193 | , ,, || 9.729020] _ 10.270980 
 945125 | 7-13 || ° "729393 Pee 270677 
 |} 945058 | 3°33 ..729626 5 05 270874 
 944990 | "45 . 729929 5 07 270071 
 Qoo ote KO MONE . POVRYY 
 944786 13 |) crgoses | 5.5 969162 
 944718 | 3°45 sata || ete 268859 
 -944650 | 4°33 731444 | Poo9 268556 
 944582 | 3745 731746 | "og .268254 
 9.944514. | | 9.782048 | _ 10. 267952 
 stauie | 148 |) rs | 3.05 | reso 
 toga | 1d8t ||) Oeoeee (2 Saas || Saat 
 “O44: 143+ || Shaper | 5208 itis 
 944241 115 (88257 5 02 266743 
 WAIT? | S49 733558 | "93 -266442 
 -944104 | 3+43 || .738860 | F°o3 .266140 
 944086 | 45 || .784162 | P95 .265838 
 948967 | 3°73 (34463 | o5 .265537 
 943899 | "45 734764 | P'o3 £ 265236 
 9.943830 ‘i: 5 || 9.735066 | o9 | 10.264934 
 943761 | 3°45 735367 | 299 264633 
 943693 | 55 735668 | 299 .264332 
 943624 | 3773 735969 | 299 264031 
 .948555 115 736269 | Foo 268731 
 -$43486.| 4°95 |] «786510 | 5G |. - 263180 
 angie |e AD || hgeke || Bs02e PE gener 
 eeeee> Hy a, | eee | 5 60 ee 
 948279 {| 15 . (37471 5 00 -2OROR9 
 943210 | "75 BM | 269 262229 
 | 9.943141 | | 9.738071 10.261929 
 oin07e 4 14E || “zagazt | 2-00 261629 
 . 943003 145 |) -88671 | Fog .261329 
 .942934 147 | -¥38971 | 509 -261029 
 Soon i iae ¢\|, Seeneen (oe | Pee 
 942795 1,13 | (395700 5.00 me ‘ 
 . 942726 1 “47 | . (39870 4.98 -260130 
 942656 | "ys || -740169 | 4'98 259831 
 942587 | "77 || 740468 | Fog .259532 
 M2517 | 45 140767 | 49g . 259233 
 9.942448 | 1 yn || 9.741066] 4 og | 10.258934 
 942378 | yyy || -741865 4.98 258635 
 -942308 | 3 "15 741664 | 4 "9p 258336 
 |} 942239 | "4° 741962 | 498 . 258038 
 942169 | 5 "y0 742261 | 4 ge 257739 
 942099 | "yn || -742559 | 49g 257441 
 -942029 | syn || -742858 | 4 gn 257142 
 || -941959 | 44 743156 | 4» ee 
 Whee Serer ieee brea | Pe ee a 256546 
 9.941819 9.743752 10. 256248 
 | | | | | 
 Sine. Dey Cotang. | D. 1". Tang. 
 
 Dr COP O13 OO 
 
 ~ 
 
 { 
 
 top] 
 _ 
 ° 
 
Sine. | 18 A hase | 
 
 9.685571 
 685799 
 686027 
 686254 
 - 686482 
 686709 
 . 686936 
 687163 
 687389 
 .687616 
 687843 
 
 9. 688069 
 688295 
 . 688521 
 688747 
 688972 
 .689198 
 689423 
 .689648 
 .689873 
 .690098 
 
 9.690323 
 690548 
 690772 
 690996 
 691220 
 691444 
 691668 
 691892 
 “692115 
 692339 
 
 . 692562 
 692785 
 .693008 
 .693231 
 .693453 
 693676 
 
 ive) 
 
 . 693898 »}: 
 
 694120 
 . 694342 
 694564 
 
 9.694786 
 .695007 
 695229 
 .695450 
 695671 
 695892 
 .696113 
 696334 
 .696554 
 696775 
 
 9.696995 
 697215 
 697435 
 .697654 
 697874 
 . 698094 
 .698313 
 698532 
 698751 
 
 9.698970 
 
 Cosine. 
 
 3. 
 
 Fates 
 
 -~3 +3 3 
 OLOTOre Or 
 
 STAFF a st st ss 
 
 Wo Wewowowwwwo 
 
 68 
 
 OCTOCOHN OTR NIEOT SIA RIO ND 
 
 Cosine. 
 
 9.941819 
 941749 
 941679 
 .941609 
 941539 
 941469 
 - 941398 
 941328 
 . 941258 
 - 941187 
 941117 
 
 9.941046 
 940975 
 . 940905 
 940834 
 .940763 
 940693 
 940622 
 940551 
 940480 
 . 940409 
 
 9.940338 
 - 940267 
 . 940196 
 . 940125 
 . 940054 
 . 939982 
 939911 
 . 939840 
 939768 
 . 939697 
 939625 
 939554 
 939482 
 . 939410 
 939339 
 939267 
 939195 
 . 939123 
 - 939052 
 . 938980 
 
 9.938908 
 938836 
 928763 
 938691 
 . 938619 
 938547 
 938475 
 933402 
 938330 
 .938258 
 
 .938185 
 . 938113 
 . 938040 
 937967 
 937895 
 . 937822 
 937749 
 937676 
 . 937604 
 9.937531 
 
 Js) 
 
 oO 
 
 D. 1". 
 
 oo 
 
 o 
 
 =| ARAAAAGIAGS Agora 
 
 Sine. 
 
 TABLE XXV.—LOGARITHMIC SINES, 
 
 Tang. 
 
 9.743752 
 744050 
 744348 
 744645 
 (44943 
 745240 
 745538 
 145835 
 (45182 
 146429 
 - (46726 
 
 9.747028 
 
 747319 
 
 147616 
 
 T4918 
 
 748209 
 
 (48505 
 
 748801 
 
 449097 
 
 . 149393 
 
 . 749689 
 
 (49985 
 
 750281 
 
 150576 
 
 . T5087 
 
 (51167 
 
 . 751462 
 
 01757 
 
 (92052 
 
 192347 
 
 . 152642 
 
 9.752937 
 158231 
 (03026 
 153820 
 
 =) 
 
 Jo) 
 or 
 » SO 
 es) 
 a 
 S 
 
 (59687 
 159979 
 160272 
 760564 
 . (60856 
 761148 
 9.761489 
 
 Cotan ge. 
 
 D. 1". | Cotang. 
 10. 256248 
 rp 955950 
 ae "955652 
 ye "955355 
 re "955057 
 = “254760 
 re "954462 
 ae 954165 
 ae 953868 
 ae "958571 
 10 .252977 
 pe "852681 
 re "252087 
 be "951791 
 4th "951495 
 a 251199 
 cae 950908 
 ‘ik “250607 
 ri "950811 
 10. 250015 
 bit "249719 
 dike "949494 
 ri "249198 
 aes 248833 | 
 re "948538 
 Ae 248243 
 re 247948 | 
 oe "247653 
 10.247063 
 re 246769 
 re "246180 
 ye "945885 
 br "945591 
 re "945207 
 pe 245008 
 7a "241709 
 it "O44415 
 10.244199 
 ie "343998 
 re "843535 
 ie 243241 
 4.88 242948 
 4 7 88 3 242055 
 re "949369 
 ‘ee 242069 
 re ‘Ont 
 re "941483 
 ~ | 10.241190 
 ‘e 240898 
 ie "240605 
 4 "940313 
 4a "240021 
 4 4 87 . 939728 
 ee 939436 
 aa "939144 
 rs "238852 
 ‘85 | 40'938561 
 D: 1 Tang. 
 
 / 
 
 SrKHwmOWhROIOIMHO 
 
 ~ 
 
oO 
 
 BO A3.C> OT C9 2D 
 
 Sine. D. 1”. || Cosine, | D, 1’, Tang. 1D ge Cotang. | / 
 9.698970 ; || 9.937581 9.761439 0.2385 
 60189 | 3-83 |) oazasg | 1-22 |] “verzai | 4.82 rears 
 699407 | 3"Re 937885 | 7°38 “7En023 | 4:84 237977 | 58 
 090626 | $"ga || .9a7312 | 1:22 || cyeeaia | 4-85 237686 | 57 
 699844 | 343 || .937238 | 7-33 || 762600 | 4-87 "237304 | 5G 
 700062 | 3°63 937165 | j'oo | || seape7 | 4785 237103 | 55 
 700280 | 3°85 || <o3zo92 | 1-22 || ‘7esigg | 4-85 936812 | 54 
 700498 | 3°43 || .937019 | j-e5 || zeaav9 | 4-85 236521 | 53 
 -700716 | 3"¢5 || .936946 | j-Sq || .zea770 | 4-85 | “aaeps0 | 52 
 700983 | 3°95 || .936872 | j-S || :ve4o61 | 4:85 235939 | 51 
 701151 | 3'g9_ || 936799 | 393 ||) .7es352 | 4-85 | “235648 | 50 ! 
 9.701368] 5 go || 9.936725 | 5 o9. || 9.764643 | 4 95 | 10.235357 | 49 
 W1585 | 3°go || .936652 | j'55 ||. 763933 | 4-83 "235067 | 48 
 701802 | 3'g5 || 936578 | 565 ||, 765224 | 4:85 | loaarve | a7 
 702019 | 3"B5 936505 | 795 ||’ 765514 | 4-88 234486 | 46 
 702286 | 5°gy || 936431 | 4-53 || i765805 | 4-85 "834195 | 45 
 102452 | 3'go || .936857 | 1-53 || ‘766005 | 4:88 | lessons | 44 
 702669 | 3°go || 936284 | j-55 || “76s285 | 488 |  losseis | 43 
 702885 | 3'gq ||’ .936210 | }-53 || <vese7 | 4-88 "933305 | 49 
 703101 | 3 Eq 936136 | "55 7166965 | 4°G8 "233035 | 41 
 708317 | 3°69, || -936062 | 4°53: || .767255 | 4-83 232745 | 40 
 9.703583 | 3 gq || 9.985988. | 1 93 || 9.767545:| 4 49 | 10.282455 | 39 
 703749 | 3'zg° || .980014 | 1-58 || “veresa | 4-82 "232166 | 38 
 703064 | 3'5g || .985840 | j'p3 || .768ie4 | 4-83 |  ‘o3i876 | 37 
 704179 | 3°69 || 935766 | t-53 || -ves4id | 4-88 "931586 | 36 
 -704395 | 3°53 || .985692 | j'53 ||° .768703 | 4-82 231297 | 35 
 701610 | 32g || .935618 | jd, || 768002 | 4-82 | ‘931008 | 34 
 701825 | 3"pe' || 995543 | 1-62 ||: czeoz8i | 4-82 "230719 | 33 
 205040 | 3'57 || 985469 | jog || 769571 | 7-83 | ‘230490 | 32 
 705254 | 35g || .985895 | j-5e || .7e0860 | 4-82 "930140 | 31 
 705469 | 3°p7 || 935820 | 1-53 || .770148 | 4-80 '229852 | 30 
 9.705683 9.935246. | 4 o~ || 9.770437 , | 10.229563 | 29 
 “705808 | 3'Py || .oai7t | 1-28 ||“ \vrora6 | 4-82 | ‘gpoerd | 28 
 706112 | 3'57 || .935097 | ‘ox P01 | 2 4'e5 | 828985 | a7 
 706826 | "px || .935022 |. 3-53 rene | ee 228697 | 26 
 “706589 | 3"po || 934948 | 755 Tribee.| | 788 "228408 | 25 
 906753 | | 57 || 1934873 |. } Se 771880 | | 4:80 "228120 | 24 
 706967 | "Px || 934798 | J "oe 772163 | 7°80 227832 | 23 
 707180 | g:pe || 2984723 |. 1-53 TI2457 | | Fos 227543 | 29 
 707893 | 3 pe || 1984649 | j-S5 || <7re74s |. 4-20 "227255 | Qt 
 .707606 | 3°55 934574 | 5 "on 173033 | 4" 99 .226967 | 2 
 9.707819 | 2 xx || 9.934499 ~ || 9.178821 10.226679 
 oso | 3°72 || .934aza | 7-28 || “l773co8 | 4-%8 "|" lep6a02 | 18 
 08245 | 522 || 934849 | joe || = ar7ae0e | 3-8 "226104 | 17 
 708458 | 3'px || .934274 | jon ||| 774184 | 4°23 225816 | 16 
 708670 | Sigg || 1984199 | J°6> || rama | 8 "225529 | 15 
 108882 | 3°53 984123 | 4°98 WATS | 4 ng 1225241 | 14 
 709001 | 3'r3 || 934048 | joe || 775046 | 4° "224954 | 13 
 209806 | ‘sq || .983073 | joe *||: aves | 4-23 "924667 | 12 
 709518 | 5"p3 || .933898 | j76> || .7zee | 4-3 "924379 | 11 
 a sige) ||" 080822) joe: ||. srese08 | ae 224092 | 10 
 9. 709941 9.933747 | 1, 9.776195 |, ng | 10.223805 | 9 
 10158 | 3-P3'\) fe TTL crease | Ske | | aap | 8 
 710364 | 359 197 TUGT68 | Ang 223232 | 7 
 710375 | 3°59 93352 195 171055 | Ang 222945 | 6 
 -710786 | 3°55 933445 |} o9 T7342 | om .222658 | 5 
 -710997 | 3°55 933369 | 3 '9* TUE | Ging 222372 | 4 
 ePM208 |. oo en 933293 | 46 a UUT915 497 222085 | 3 
 711419 3 50 933217 | oy 78201 | 4'ng 221799 | 2 
 .711629 | 350 933141 — . 778488 ae 221512 |} t 
 9.711839 | 3-59 || 9 932066 | 1-25 || givzerza | 4-7 | 10/221226 | 0 
 Cosine. ' D.1". Sine. Dyer” Cotang. | D. 1” Tang. nie 
 
 COSINES, TANGENTS, AND COTANGENTS. 
 
 aA 
 Oo 
 ° 
 
Sine. Dri". Cosine. | D. 1”. 
 9.711889 xo || 9.983066 a 
 712050 | 3-35 || 92990 | 2.27 
 seen): | | Sa 932914 | 1.37 
 712469 | 3°55 932838 197 
 piee7 |) 2 932762 | 1.27 
 “719689 | 3-50 932685 | 1-38 
 713098 | 3-58 982609 | 1-3¢ 
 713308 | 3:88 992533 | 1-37 
 earsatT | | oop 932457 | 1.37 
 KhyAQKhO . < é 
 -713935 | 3°48 veal 
 9.714144 9.93222 . 
 "714352 a "932151 es 
 714561 | 3-48 932075 | 1-32 
 714769 3 48 -931998 | "58 
 mags | 3-48 -osi921 | 1.58 
 Aaa. || 3.28 hilt Seeeeee (11 18s 
 ~é 5394. 3 47 931 4 68 1 98 
 “715602 | 3-42 931691 | 1-38 
 715809 | 3-45 931614 | 1-28 
 M6017 | 3-46 931537 15 
 9.716224 ~ || 9.981460 
 716432 | 3-47 || ° “931383 i 
 716639 | 3-48 931306 | 1-58 
 “716846 | 3-43 931229 | 1-58 
 a1708a | (3°03 931152 | 1-38 
 tea |. |e 931075 | 4-58 
 717466 | 3-4 980998 | 1-58 
 gers |. 1303 930921 | 1-35 
 717879 | 3-3 930843 | 1-58 
 718085 | 3:43 930766 | 1-35 
 9.718291 9.930688 
 718497 | 3-48 || ‘osoe1i | 1-28 
 “718703 | 3-43 coponas | 1-8 
 718909 ie 930456 eo 
 “M1911: "93037 
 “719820 | 3-43 || 1930300 | 1-30 
 719595 | 3.2 930228 | 1-55 
 visza0 | 3-2 930145 | 1-3) 
 prigees | 12-8 930067 | 2-3) 
 720140 | 3-3 929989 | 3°39 
 9.720845 9.929911 
 720549 | 3-49 || 920833 | 1-3) 
 mania | 3-48 929755 | 1-35 
 720958 | 3-40 929677 | 4-35 
 yoiiee |’ (3-2 929599 | 1°39 
 721366 | 3-40 929521 | 1-39 
 Siero | 12, 40sttl | Wee 
 woes |\ oS 929207 | +35 
 9.722885 9.929129 
 7ee5es | 3-38 -|| ‘920050 | 1-38 
 T2201 | 33-2 928972 | 1-35 
 vezoo, | 3-38 928893 | 1-55 
 (23197 3 3 38 . 928815 { 4 29 
 me -t HQKHHe vw 
 423400 | (3°38 928736 | 4°39 
 ‘ 23603 3 37 § 92865 ( f 329 
 723805 | 3-37 928578 | 1-35 
 724007 | 3-32 928499 | 1-35 
 9.724210 | 3" 9.928420 | 1-9 
 Cosine. Sine. D, I. 
 
 HY Cotane. 
 
 TABLE XXV.—LOGARITHMIC SINES, 
 
 Tang. 
 
 9.778774 
 779060 
 779346 
 779632 
 779918 
 780203 
 780489 
 18077 
 781060 
 781346 
 781631 
 
 9.781916 
 782201 
 782486 
 E271 
 782056 
 783341 
 783626 
 783910 
 184195 
 784479 
 
 9.784764 
 785048 
 785832 
 785616 
 785900 
 786184 
 786468 
 186752 
 787036 
 787319 
 
 9.787603 
 787886 
 788170 
 788453 
 788736 
 789019 
 789302 
 789585 
 789868 
 790151 
 
 9.790434 
 
 (90716 
 
 . (90999 
 
 791281 
 
 791563 
 ~ .791846 
 .792128 
 792410 
 . 792692 
 19297 
 . 193256 
 . 193538 
 . ($3819 
 94101 
 794883 
 794664 
 1949 16 
 795227 
 795508 
 9.79578) 
 
 © 
 
 D, 1". 
 
 | 
 | 
 
 mF BES ae eg ee tg gets 
 
 NW OW NOC OTST ST OLOT-SIONt ISAT IAas 
 
 IIIS 
 C9 09 09 OD 
 
 4.68 
 
 Cotang. 
 
 10.221226 
 220940 
 220654 
 220368 
 220082 
 219797 
 219511 
 219225 
 218940 
 218654 
 218369 
 
 10.218084 
 217799 
 217514 
 217229 
 216944 
 216659 
 216374 
 216090 
 215805 
 215521 
 
 10.215236 
 -214952 
 214668 
 214884 
 -214100 
 -213816 
 213532 
 213248 
 212964 
 212681 
 
 10.2123897 
 212114 
 -211830 
 .211547 
 -211264 
 . 210981 
 210698 
 .210415 
 .2101382 
 - 209849 
 
 10. 209566 
 . 209284 
 209001 
 208719 
 . 2084387 
 208154 
 207872 
 .207590 
 207308 
 207026 
 
 10.206744 
 206462 
 .206181 
 . 205899 
 . 205617 
 - 205336 
 205054 
 204773 
 204492 
 
 10. 204211 
 
Sine. 
 
 9.724210 
 724412 
 124614 
 
 724816 | 
 
 725017 
 .725219 
 725420 
 (25622 
 T2582 
 726024 
 - (26225 
 9.726426 
 .126626 
 
 730217 
 
 | 9 730415 
 “730613 
 “730814 
 "731009 
 "731206 
 "731404 
 "731602 
 "731799 
 "731998 
 732193 
 | 9.732390 
 9 | .732587 
 3 | 739784 
 | "1732980 
 “733177 
 "733373 
 "733569 
 733765 
 "733961 
 "734157 
 9.734353 
 734549 
 
 | i734744 
 "734939 
 "735135 
 "735830 
 
 . 135525 
 
 735719 | 
 | .735914 
 | 9.736109 
 
 Cosine. 
 
 COSINES, TANGENTS, AND COTANGENTS. 
 
 Tang. 
 
 Cotang. 
 
 je) 
 Se) 
 
 ee 
 
 oO 
 
 WW WNWWWW2 ra ra IWweOwwe worwwowowwc Goo? 29 ¢ 9 09 C OC > oo co O9 09 OH CY OH 
 
 os 0) 09 Co 
 D2 Ww? 
 
 9 928420 
 928342 
 928263 
 928183 
 “928104 
 “928025 
 “927946 | 
 “927867 
 927787 
 “927708 
 927629 
 927549 
 927470 
 -927390 
 927310 | 
 (927231 
 “927151 
 927071 
 926991 
 926911 
 926831 
 926751 
 926671 
 926591 
 926511 
 926431 
 926351 
 926270 
 926190 
 -926110 
 926029 
 925949 
 925868 
 925788 
 925707 
 925626 
 925545 
 925465 
 925384 
 925303 
 
 9252 
 
 Todi 
 
 925141 
 925060 
 924979 
 924897 
 
 924816 
 924735 
 924654 
 . 924572 
 . 924491 
 . 924409 
 
 924328 
 924246 
 924164 
 . 924083 
 924001 
 9238919 
 . 928837 
 
 or 
 
 .923759 | 
 923673 
 
 923591 
 
 | 9.812517 
 
 9.795789 
 796070 
 796351 
 . 796632 
 796913 
 -T9T19A 
 T9T4AT4 
 19TTSS 
 798036 
 .798316 
 798596 
 198877 
 799157 
 799437 
 st OOe17 
 W99997 
 800277 
 800557 
 800836 
 .801116 
 801896 
 801675 
 801955 
 802234 
 802518 
 802792 
 . 808072 
 803351 
 803630 
 803909 
 804187 
 
 9.804466 
 804745 
 805023 
 805302 | 
 .805580 
 805859 
 .8061387 
 806415 
 806693 
 806971 
 
 9.807249 
 807527 
 807805 
 808083 
 808361 
 .808538 
 8038916 
 .809193 
 809471 
 809748 
 
 9.810025 
 .810302 
 810580 
 810857 
 811134 
 .811410 
 .811687 
 811964 
 812241 
 
 is) 
 
 ie) 
 
 10.204211 
 
 203930 | 
 203649 | 5! 
 203368 | 5 
 
 . 203087 ) 
 202806 
 202526 
 202245 
 201964 
 201684 
 201404 
 201123 
 200843 
 200563 
 200283 
 200003 
 199723 
 199443 
 199164 
 198884 
 198604 
 198325 
 198045 
 197766 
 197487 
 197208 
 196928 
 196649 | ; 
 .196370 | 32 
 .196091 | ; 
 
 195813 
 
 195584 
 .195255 | % 
 194977 
 . 194698 
 . 194420 
 194141 
 193863 
 193585 
 193307 
 198029 
 192751 
 192473 
 .192195 
 .191917 
 191639 
 .191362 
 191084 
 190807 
 190529 
 190252 
 189975 
 189698 
 . 189420 
 189143 
 188866 
 188590 
 188313 
 . 188036 
 187759 
 
 10.187483 
 
 | Cosine. | 
 
 Sine. 
 
 Cotang. | 
 
 Tang. 
 
 o 
 
 09 He OT =F COW 
 
 ~ 
 
 Ord 
 
 ~ 
 
~ 
 
 10 
 
 SOANQIDUR WMHS 
 
 TABLE XXV.—LOGARITHMIC SINES, 
 
 Sine. Cosine. Tang. Cotang. | 
 ie | 
 736109 9.923591 9.812517 87483 | 
 736303 "923509 “Siar | 4-620] TORR 
 736498 "993497 ‘813070 | 4-6 186930 | 
 136692 923345 "813347 | 4-6 "186653. | 
 736886 923263 "813623 | 4-8 "486377 | 
 737080 (923181 “813899 | 4:6 136101 | 
 37244 923098 “814176 | 4-6 "185824 | 
 7346 "923016 'gi4452 | 4-6 "185548 | 5 
 731661 922983 814728 | 4-8 “js53r2 
 “T3785 922851 815004 ‘4 “TR008 | 
 ss he 922768 815280 | 7 184720 | 
 1382 922686 815555 5 
 "7B8434 "929603 "815831 | 4:6 ae his | 
 738627 "922590 ‘816107 | +: "183893 | 
 738820 922438 916882 | 4-5 "183618 
 739013 "929355 “816658 | 4-9 483342 
 “7739206 "929972 "816983 | 4-5 483067 | 
 739308 922189 | "917209 | 4: "182791 
 (39590 922106 ‘gi7484 | 4: "182516 | 
 739783 922023 ‘g17759 | 4:5 182241 
 139015 921940 818035 | 4° 181965: | 
 9.740167 9.921857 818310 -¢ | 10.181690 | ; 
 740850 | 3- 92177 "g18585 | 4:4 181415 | 
 740550 | 3. "921601 818860 | 4: "181140 | 
 140742 | 3: 921607 (819135 | 4: "180865 | 
 740984 | 3. “921524 ‘819410 | 4: 180590 
 iatie5 | 3. 921441 ‘819684 | 4: "180316 | 
 741316 | 3: 921357 819959 | 4+ 180041 
 741508 | 3: “921274 820234 | 4 179766 
 741699 | 3° “991190 "820508 | 4+ 179492 
 741889 | 3-4 921107 820783 = 179217 
 9.742080] 5 9.921023 9821057 10, 178943 
 ar |B "920989 | 921332 | 4: "178668 
 742 mage | , 3: "920856 821606 | 4: 178394 
 742652 | 3. "920772 "g21880 | 4: 478120 
 74e842 | 3: "920688 320154 | 4: 177846 
 743083 (3. 920604 822429 | 4. ATT 
 743223 | 3. "920520 “822708 4. 97297 
 743413 | 3-4 920436 ‘go0977 | 4: 177023 
 43602 | 3-7 "920352 “gazes, | 4! 176749 
 TABI | 3 "920268 823504 ; "176476 
 743982 | 9 4 920184 9 823798 ; 10.176202 
 mai | 3-5 "920099 | 1824072 | 4:2 "175928 
 744361 | 3-1 920015 824345 | 4 175655 
 744550 | 3-42 || 919931 | 824619 | 4: “175881 
 744739 | 3° 919846 "824803 | 4 "15107 
 744928 | 3-1, 919762 925166 | 4: 174834 
 TABI? | 374 919677 825439 | 4: "174561 
 745806 | 3-4 "919593 "925713.| 4: "174287 
 T45494 | 8: “919508 “925986 | 4:5 T4014 
 745683 | 3- 919424 26259 | 4:55 1737 
 45083 | 49 26259 | 4-22 “173741 
 T4871 | 9 4x || 9.919339 826582 | 4 xx | 10.173468 | 
 746060 3°13 919254 “926805 | 4:59 "173195 | 
 746248 | 3.13 || ‘oro169 “27078 | 4:25 |. “172922 | 
 746486 | 3-18 “919085 827351 | 4:58 "472649 
 746624 | 3:43 || “919000 827624 | 455° | ' Ci7a3v6 
 796812 | 3-13 || “orgots 27807 | 4:25 172108 | 
 746999 | 3-12 || 918830 828170 | 4-55 |” 7171880 
 TAS; | 3-18 “918745 828449 | 4-53 171558 | 
 “47874 | 3745 "918659 “gos715 | 4:99 “T1285 
 9.747562 | 3-13. |) 9 oresey | gsosoe7 | 4°23 | y07 174023 | 
 Cosine, | D. 1’. Sine. -Cotang, er DA Be Tang. 
 
a 
 
 CosI 
 NES, TANGENTS, AND COTANGENTS 
 
 / ° 
 Sine 
 . D. Cosine. | D. 1" | Ta; 
 —- | ng. D. i"; }-Cotang. | 7 
 4 | 9.747562 | 9 yo. || 9.918574 | 
 TATT49 SfO® |My ced | | = eee ees 
 2 | narperes = a 9 918489 1.42 | 9.828987 4 
 BY tater Ser i) Nereis | 1-2 929260 | 4-23 ee ee 
 4{ -v4gsi0 | 3-12 meieta (Lek! Mi Mpanene i eaten | ee 
 SB haaeeene |S aida Ih aeptere | 1142 || -829805 ao \T ote unige | Bs 
 G6 | :7486s3 | 3-10 ‘918147 | 1-48 Be, ae ee Uden | bs 
 7| ‘ragg70 | 3-22 ‘918062 | 1-42 -830349 4.53 ease | 56 
 8 | 749056 | 3-10 ‘917976 | 1-48 “g30621 | 4°23 Pe eeme 
 Bh teas | Saba? Ni ceteane 14s || -830803 | 4783 ere, | Be 
 10 | {749429 | 3-10 oir | 1-43. |] ceatase ae ieeeas | Be 
 a “y f°) 8.40 ‘gizzi9 | 1-43 831437 | {PS - 168685. | 02 
 11 | 9.749615 1.42 "831709 | 4:22 .168563 | 51 
 12 -T49801 3.10 9.917634 . : 4.53 .168291 | 50 
 Bot Shager | 8.10 |l venues ifdat || Seer hs 4 eeeeaie (eds 
 14} “750{72 | 3-08 ‘g17462 | 1-43 882253 | 7° 53 peat 49 
 Me Bicpees 1 Sal0* Ill cae tag || 892525 | 4iee | ee | a 
 16 150543 3.08 917290 1.43 8327 96 4: = 167 ’, ! 
 Bul Gress || S10) 1) Gama 1:43 || 838068 | 4-25 Mees | ae 
 i8| 750914 | 3-08 ‘oi7tig | 1-43 833339) oe byt | 3 
 19 | 751099 | 3-08 ‘917032 | 1-43 eooe! Re oe a 
 90 | “751284 | 3-98 ‘916946 | 1-43 933882 | “4-33 1664 sg Be 
 21 | 9.751469 ; 1.43 "ga44on | 4.52 165846 | 41 
 | TOI! | 3 0p Maier | 1a |ll See Puc nee | 
 wm | 751054 | Bog |] 916687 | tgp || 83404 
 ae aes 3.08 "|| “o1e800 | 1-45 834967 | 7b 1 ere | ae 
 Be serach | | 8508" ||| Seteer 143 || “S828 | Gop ees | ee 
 26. | “752392 | 3-00 "916497 | 1-4 "885509 | 4°R6 ei | oe 
 97 | l75e5t6 | 3-0¢ ‘916341 | 1-43 - 835780 yo teres oe 
 el Warsece (18M! |) Satetey 145 || 886051 | 4-35 seaniy ‘LSee 
 99 | ‘sa944 | 3-07 ‘916167 | 1-49 836322 4.52 % Sees | ea 
 3) | ‘ssajog | 8-07 ‘g16081 | 1-43 836593 | 4:25 pete 33 
 . 23 | 3 'or “915994 | 1-45 "g36a64 | 4:52 .163407 | 382 
 31 | 9.753312 1.45 "e37q34 | 4.90 .163136 | 31 
 39 | "753195 | 3-99 9.915907 ue | 4.52 "162866 | 30 
 33 753679 3.07 . 915820 1.45 9.837405 | 4.5 10.162595 | 2 
 S| 753862 3.07 || ante aaa tee Rage iid Bite 162325 | 2 
 35.1 95 aT} 91564 45 001920 peed aaRe aoe 
 Onn ages | | 3.05 915648 | G5 || 838210 | bp 162054 | 27 
 7 |. 754412 | 3-0 “ieire | 1-45 |] “Shersy 4.52 | 1161518 | a 
 38 ws4ng5 | 8-05 |] "915385 | 1-45 -83875% oO "46 ; 3 1 
 39 | RAT > ||, Zorpeo7 | 1-47 939027 | 4:20 -161243 | 24 
 to | 751060 303 || peewee 1H 1 aS 1) peu 4i608. | cigars | ge 
 Suen (i) comes [i age 930568 | 4:52 -160703 | 22 
 41 | 9.755145 : 1:47 "g3ggag | 4-90 160432 | 21 
 42 | 755326 | 3-9 || 9.915035 4.50 .160162 | 20 
 43.| “755508 | 3-93 || "914948 | 1-45 9.840108 | 4 5 | 10.159892 
 ts | -ra5om0 | 33 914860 | 445 810873} 45) | «150622 19 
 5 | ‘yssgr2 | 3-03 || (ee eRe ¢ 409; Aiag "159852 
 46 | 756054 |. 3-08 "914685 | 1-46 4017 ee aoaae | de 
 Bit adeee bi 8208? ||| agraeee tag ||) cBatte7 | Diape | pa ede 
 43.| ‘75641g | 3-03 ‘914510 | 1-46 “841457 | 4-2) eee ae 
 49 | 756600 | 3-03 ‘914492 | 1-42 SALT2¢ oe oes Bs 
 Bel Beets | 1308 ||| Sopris Loeyt |i) peaaeoe era ean de 
 82 | 3'o9 || «914246 1.47 842266 4.50 -158004 | 12 
 51 | 9.756963 ant 1.47 "gy9535 | 4-48 157734 | 11 
 53 957326 3 03 .914070 1.47 -¢ 42805 78 10.157195 
 Br iameere f82 * ||| capeeas 147 i pions | 448 | 156: aie 
 55 | 757688 | 3-02 ‘gizso4 | 1-47 949313 | 4.48 fee 8 
 56 | “757869 | 3-02 "913806 | 1-44 813612 4.48 efuee sf 
 57 | 7sgos0 | 3-02 ‘913718 | 1-42 gt3eee | 4.38 eon 6 
 58 | 758290 | 3-00 ‘913630 | 1-40 | 844151 ‘oe perce 5 
 59 | 2758411 | 3-02 |, ores | IMB Il! sages fue Ween ag 
 ed cde ok. (8.00: || ‘oigtaass a || 844689 | 24g Rey ey 
 ee  oapegeral |. 150 * It eines 4.38 55811 | 2 
 7 | Cosi ; meres || g g45an7 | 4:48 ..155042 | 1 
 aceak D. 1 Sine eo ee C | 10.154773 | O 
 ; Joi. ota S= 
 ang. | Dek: Tang. 
 
TABLE XXV.—LOGARITHMIC SINES, 
 
 r | 
 Ab Sine. ih Del" ett!) Casime: 1.8") 1 Tang. | D.1". | Cotang. | / 
 | pee =< ie 
 3365 520% | 10.154773 | 60 
 0 | 9.758591 | 5 gp || 9.913365 | 1 4g || 9.45201 aes 10. 154773 | 6 
 | eypegse | 8:00. ||. Seaeene |: 1ea8s |) Teateeey |e olay) | eee ee 
 S| Swsgice | 8006 || Soteney | tel cl Pagers (dpi 9 dee 
 3 | 759182 | 3-00 913099 | 3-48 816033 | 948 153067 | Br 
 B | coepadon | (82000 |] pigs | tear ll Babee? | aide || ge eee 
 8| “tore | 3-00 || “Oioegs | 1-48 |] Bienen | 4.48 | tBRIR0 | Be 
 7 | cesasse | 2-00 |] “orerat | 1-48 |] Ramos | 4:48 | 1e8np0 | Bs 
 b | Sreopet || 2e98-||) Serages | 1eaB |): Geatee? | Gaede Ses 
 6 | Sopeoat (8-00 ||) SBipeee |. aBy ||| Beet Tia ay ell) aameeee ae 
 3 | “heosso | 298+|| coir | 1aB ct Seyeee | 4aeo| | Rete 
 10 760390 2.98 912477 1.48 . ( 4,47 | i i e 
 Vet t os hE 8 
 11 | 9.760569 } 9 9g || 9.912388 | 4g 9.818181 gel ae 151819 49 
 12} -.760748 | 3‘o9 912209 | 4°48 S189 |p “151551 | 48 
 14 | 761106 | $38 912121 | 185 848086 | fr 151014 | 46 
 ie | “4e1ie4 | 2-98 ‘pitode | 1-48 "849502 ae 150478 | 44 
 of 5 ) 2 97 2 e a 1.48 Ts ed ‘ "150210 8B 
 18 | ‘reir | 2-98 || “utes | 1.00 "850057 re 149943 | 43 
 | Pf 4 (9) 97 2d ioe 1.48 . xd ; ned ; iy . ; ar 
 19 | .761999 et -o11674 | +58 850325 2 149675 at 
 21 | 9.762356 | 9 97 || 9.911495 | 4 59 9.850861 que | Gs 149139 39 
 ba |. ree5a4 | | 2:87 911405 | 1-20 S51129 | 4-45 “14881 | 38 
 3 Eee 2.95 “pitase (18 "851664 rp "148336 | 36 
 - (06000 ¢ Y . we 1.50 eae A , 3. 
 | 25 | .763067 4 dl 911186 | 3-39 81931 ae 148069 35 
 26: |" 2768045 | S:af 911046 | 1-39 852199 | 445 147901 | 3 
 7 | -68422 | 9" oe -910956 | 4 "59 Poemee |! 4d Soe lee 
 i 28 | 763600 | 3-94 910866 | 3:20 852133 | 7 141267 | 82 
 29 163777 2 95 91077 : 1 50 eos 38 4.45 146732 30 
 30 . 763954 9 ‘ 95 . 910686 1 7 50 . 85826 4 § 45 ° rf a 
 . =e =4 35 ahd 46! € 
 31 | 9.764131 | 9 ox || 9.910596 | 1 '., 9.853535 dasa 40465 29 
 a2 | .764308 | §:22 910508 | 3-78 858802 | 445 146198 | 28 
 33 | . 764485 | 3-02 o104i5 | 1-8 854069 | 4-42 15081 | 
 I 764662 igy 910325 2 854336 > 145664 | 26 
 34) .764662 | 9°93 terre |i Bey Sorin es ore hae 
 35 | .7e4sa8 | 5:08 910085 || .1+9) 854003 | 445 145897 | 2 
 Be HBetgH 2.98 BOREL 1.50 "855137 re "144863 | 9: 
 3” T6515 3 9: 91005 159 -855 a ate : 
 38 | 765867 | 2:98 909963 | T"pe $5408 aa “144596 2 
 89] . 765544 | 9°98 -909873 | 4759 eer | 04 ia ie. 
 40 | .765720| 5-33 909782 | -}:D5 855938 | 4-23 1410 
 Hil 41 | 9.765896 | 9 93 || 9.909601 | 45 9.856204 ye 10. 148796 19 
 qa 42] .766072 | 9°99 909601 | 3°55 856471 | 4°43 ie lee 
 aaa As 766247 909510 ‘ 856737 i 143265 7 
 A ri Lives 2.93 909419 | 1-52 857004 hoe 142996 | 16 
 i caer. 1o TG Byes 2 1.52 Saath 4 149730 | 15 
 wa) 45 | 766598 | 3:92 909828 | 1 °p8 857270 | 45 142730 | 15 
 46) .(66774 | 999 909237 | 459 ete | dake reteal es. 
 47 | 766049 | 5-05 909146 | 1°28 851803 | 443 142197 | 13 
 49 | .767300 | 9'99 -908964 | 3 "59 Seas | 4s alana 1a 
 50 ° T67475 9 } 90 . 908873 1 ; 53 O00 i 4 : 43 . ae 
 51 | 9.767649 | o9 || 9.908781 | 4 ‘5 9.858808 “shy 10. 141122 9 
 ef ihatenn’ | eee | dese 1.58 360400 4.43 "140600 | % 
 53 767998 290 -t 00) 1.53 . 2 = 4.438 1 10334 6 
 Bd |. 208173 |~/3 80 908507 | 5 "p3 859608 | 7°43 40334 | 6 
 55 | 768348 | 2-9 908416 |} ps “850932 | 4-48 140068 | 5 
 56 || q7esne2 | 2:20 908324 Bes 800198 ie ‘ 139802 4 
 Bz | 768607 | 3-92 9082! es 860464 | 4-43 139536 | 3 
 58 | .7esevt | 2 60 908141 | 7:23 860730 | 4-43 139210 | 2 
 59 | 760045 | 2: ‘gos049 | 1-5 860995 139005 | 1 
 60 | grees | 2.90 9.907958 | 1-52 |! 9'sero6r | 4:43 .| 407438739 | 0 
 | / | Cosine. | D. 1”. Sine. | D. 1". Cotang. | D. 1" Tang. : 
 
| 
 , Sine. 
 | 
 | 0 | 9.769219 
 Bet 769393 
 2 . 769566 
 3 769740 
 Gav! 769913 
 PD |.> 770087 
 6 | .770260 
 | @ | .770433 
 | 8 | .770606 
 9 T0779 
 10 770952 
 11 | 9.771125 
 2 | .%'71298 
 13 | .771470 
 14 | .771643 
 15 | .771815 
 16 |° .771987 
 17 772159 
 18 772331 
 19 |! 7725038 
 2 ~ 772675 
 21 | 9.772847 
 22 . 773018 
 23 (73190 
 24 .773861 
 25 STLOOOD 
 26 773704 
 27 ~TI8875 
 28 774046 
 29 Ne42de 
 30 .7743888 
 ; 31] 9.774558 
 | 82 | 774729 
 33 | .774899 
 34 775070 
 35 ~ 175240 
 36 . 775410 
 30 775580 
 38 TIS T50 
 39 775920 
 40 .776090 
 41 | 9.776259 
 42 716429 
 43 |  .776598 
 44 776768 
 45 776937 
 46 ~777106 
 47 UTD 
 48 1071444 
 49 77613 
 50 T7781 
 51 | 9.777950 
 52 | .778119 
 BS cane? 
 54 78455 
 DD | 778624 
 56 778792 
 57 | 778960 
 | 58} .779128 
 59 779295 
 60 779463 
 
 | 
 Ds 
 
 cM) 
 
 |’ 1 Cosine. 
 
 o 
 
 Wwrwrmwrwwwwwn wnwnwnwnnwnnwnwnnwwna 
 
 BW W WO WW W WWW WWW WWW W WWW WWW WWNWWNWW)DW 
 
 8 
 
 MAE AQVOU SPST MEI AEE ak 
 
 Cosine. 
 
 9.907958 
 | 907866 
 IOVT7T4 
 . 907682 
 . 907590 
 1}. 907498 
 || .907406 
 .907314 
 907222 
 907129 
 907037 
 9.906945 
 . 906852 
 . 906760 
 . 906667 
 . 906575 
 . 906482 
 . 906389 
 . 906296 
 . 906204 
 .906111 
 
 9.906018 
 
 905925 
 || 905832 
 || .905739 
 || 905645 
 || 905552 
 
 905459 
 
 905366 
 
 905272 
 
 905179 
 9.905085 
 || .904992 
 904898 
 904804 
 904711 
 904617 
 904523 
 904429 
 904335 
 (904241 
 
 9.904147 
 904053 
 903959 
 903864 
 90377 
 . 903676 
 903581 
 
 i| .903487 
 
 | 903392 
 
 || .908298 
 
 | 9.903203 
 
 | .903108 
 
 ‘| 903014. 
 
 | ,902919 
 
 902824 
 
 i} ,902729 
 
 l| . 902634 
 
 || 902589 
 
 | 902444 
 
 || 9.902349 
 
 | Sine. 
 
 Dei", 
 
 BB | 
 BB || 
 
 ie} 
 
 Pek eh fre fh fee fk fed fe fk fe fed fame fk peck eh fr free foemch fore fre fame fore fom prc feel fem fmeh feed foe fom farm bere feel fremh prem peak fom fem fem fom prem fre fem fmm from fom free from fommh meh. fem fom famh fem fem free fed feed peek fed 
 Pe Nc Co ati, ae ED CL ers ee ek re SOS ATMO ile Sle tic. OLE ett ee te Cite Sa eel ee Na OM te ate yan ea Me Dare Aire) he 
 On 
 NX 
 
 Tang. 
 
 | 9.861261 
 
 .861527 
 861792 
 862058 
 862323 
 862589 
 862854 
 .863119 
 863385 
 863650 
 863915 
 
 9.864180 
 864445 
 .864710 
 .864975 
 865240 
 865505 
 86577 
 866035 
 866300 
 866564 
 
 9.866829 
 867094 
 867358 
 867623 
 867887 
 868152 
 868416 
 .868680 
 868945 
 869209 
 
 9.869473 
 .869737 
 870001 
 870265 
 870529 
 870793 
 871057 
 871821 
 871585 
 871849 
 
 .872112 
 872376 
 .872640 
 .872903 
 .873167 
 .873430 
 . 873694 
 . 873957 
 . 874220 
 874484 
 9.874747 
 .875010 
 .875273 
 SOLOD3 « 
 .875800 
 
 . 876063 
 876326 
 .876589 
 876852 
 9.877114 
 
 Cotang. 
 
 COSINES, TANGENTS, AND COTANGENTS. 
 
 D, 1". 
 
 HSS 
 
 SMW WWOWWWW 
 
 nC 
 
 PEP LEB 
 OOOCWwonwow 
 
 He 
 Ps 
 
 4,40 
 
 geo, Seid ma oo ian - g 
 io.2) 
 
 Cotang. 
 
 10.138739 
 .138473 
 . 188208 
 .187942 
 .137677 
 137411 
 .137146 
 .136881 
 .136615 
 . 136350 
 .136085 
 
 10. 135820 
 . 135555 
 . 135290 
 .135025 
 . 134760 
 , 134495 
 134230 
 . 133965 
 . 133700 
 .183486 
 
 10.13317 
 . 132906 
 . 182642 
 . 182377 
 . 182113 
 . 1381848 
 .181584 
 . 131320 
 .131055 
 .180791 
 
 10.130527 
 . 1302638 
 . 129999 
 .129%85 
 129471 
 .129207 
 . 128943 
 . 128679 
 .128415 
 128151 
 
 10.127888 
 127624 
 127360 
 127097. 
 
 126833 
 
 126570 
 126306 
 
 "426043 
 "125780 
 
 . 125516 
 
 10. 125253 
 
 . 124990 | 
 £124727 | 
 124463 | 
 124200 
 
 1238937 
 123674 
 123411 
 123148 
 10. 122886 
 
 TNIGS 
 
 Cc 
 
 Ort Cor ot 
 
 ~ 
 
TABLE XXV.—LOGARITHMIC SINES, 
 
 Det Cosine: | D, 1’. Tang. D. 1”. | Cotang. 
 
 9.877114 
 | Oat? 
 .877640 
 877603 
 .878165 
 . 878428 
 .878691 
 .878953 
 ,879216 
 .879478 
 879741 
 9.880003 
 880265 
 . 880528 
 .880790 
 .881052 
 .881314 
 881577 
 . 881839 
 . 882101 
 .882863 
 9.882625 
 . 882887 
 .883148 
 883410 
 . 883672 
 883934 
 .884196 
 . 884457 
 . 884719 
 .884980 
 9.885242 
 . 885504 
 . 885765 
 886026 
 .886288 
 . 886549 
 886811 
 .887072 
 . 887333 
 887594 
 9.887855 
 . 888116 
 . 888378 
 .888639 
 888900 
 . 889161 
 . 889421 
 . 859682 
 .889943 
 . 890204 
 
 9.890465 
 890725 | 
 .890986 
 .891247 
 891507 
 891768 
 892028 
 . 892289 
 
 | ,892549 
 9.892810 
 
 Cotang. | D. 1’. Tang. 
 
 10122886 
 122623 
 122360 
 122097 
 .121835 
 121572 
 121809 
 121047 
 120784 
 . 120522 
 .120259 
 119997 
 119735 
 119472 
 119210 
 118948 
 118686 
 118423 
 .118%61 
 117899 
 117637 
 
 117875 
 117118 
 116852 
 . 116590 
 116328 
 116066 
 . 115804 
 115548 
 115281 
 115020 
 
 114758 
 .114496 
 114235 
 118974 
 1138712 
 118451 
 118189 
 112928 
 112667 
 112406 
 10.112145 
 111884 
 .111622 
 111861 
 -111100 
 110839 | 
 110579 
 .110318 
 -110057 
 . 109796 
 
 109585 
 .109275 | 
 .109014 | 
 . 108753 
 .108493 | 
 . 108232 
 107972 | 
 107711 
 .107451 
 10.107190 
 
 779463 9.902349 
 479681 | <*5 902253 
 W79798 | 5° 902158 
 7799668 | 5" 902063 
 780133 | 574 901967 
 780300 | 5°4 901872 
 780467 | 5° 901776 
 780634 | 5° 901681 
 730801 | 4" 901585 
 780968 | 5° 901490 
 781134 | 5° 901394 
 781301 we || 9.901298 
 781468 | 5° 901202 
 781634 | 5° 901106 
 781800 | 5° 901010 
 781966 900914 
 7182132 900818 
 782298 900722 
 782464 900626 
 782630 900529 
 782796 900433 
 782961 900337 
 783127 900240 
 783292 900144 
 783458 900047 
 783623 899951 
 783788 899854 
 783953 899757 
 784118 899660 
 784282 899564 
 784447 899467 
 784612 9.899370 
 784776 899273 
 784941 .899176 
 785105 899078 
 785269 898981 
 785433 898884 
 785597 898787 
 785761 898689 
 785925 898592 
 786089 898494 
 
 786252 9.898397 
 786416 898299 
 786579 898202 
 786742 898104 
 786906 898006 
 787069 897908 
 787232 897810 
 187395 897712 
 BUST 897614 
 787720 897516 
 487883 9.897418 
 188045 897320 
 7838208 897222 
 788370 897123 
 788532 | of 897025 
 788694 | =f 896926 
 788856 | 896828 
 789018 | >°f 896729 
 789180 | 5-f 896631 
 | 9.789342|  *° 9.896532 
 
 9 G2 0° 
 
 BSMISU FL COWH SO | 
 
 BS EAP a ee eg gg + 
 
 WW CWC OCI OD OTD GLOTRS COTOTOTOTR OI OTS NEN 
 
 Si dnt Boas Jn Teas 
 
 WwwWwwoOWwte CWO WWW CO CU CU wt tt) CHC a eg 
 
 ESSA! Ov Ov OT OU 09 OU OF OU 3 OF RRRRSRSRES POTS OU SETS OT SSE SE AE AAI AE = HAS BO 3 CO =2 0 GOOD 
 
 Fare Se Sa 
 
 SWS VWOCWNNVWWWWS 
 
 ~ 7 
 
 WRPWWWwWwWwww wwwwwwwwww 
 
 Cosine. F Sine. 
 
 oy eT RAE ee ee Cte test y. >> 2S > > 2S > SD ie ee i 
 
COSINES, TANGENTS, AND COTANGENTS, 
 
 wo © 
 = © 
 
 ivy) 
 iva 
 
 ~ 
 i) 
 
 Jeo) 
 
 Le 
 
 / 
 0 
 14 
 2 
 3 
 4 | 
 5 
 6 | 
 ii 
 8 
 9 
 10 | 
 11 | 
 oe 
 13 | 
 14 | 
 15 | 
 16 | 
 q 
 18 
 19 
 22 
 23 
 24 | 
 25 
 26 
 
 | 
 60 | 
 
 Ae 
 
 Sine. 
 
 9.789342 
 789504 
 789665 
 - 789827 
 789988 
 .790149 
 . 790310 
 -790471 
 . 790632 
 790793 
 - 790954 
 
 9.791115 
 791275 
 .791436 
 ~ 791596 
 T9157 
 -T91917 
 192077 
 792237 
 - 792397 
 - 192557 
 . 792716 
 792876 
 .793035 
 .793195 
 793354 
 . 798514 
 . 793673 
 798832 
 .793991 
 - 794150 
 
 794308 
 (94467 
 794626 
 794784 
 (94942 
 795101 
 - €95259 
 95417 
 95575 
 
 195733 
 
 ve} 
 
 | 9.795891 
 
 . 796049 
 796206 
 
 . 796364 
 
 796521 
 796679 
 796836 
 796993 
 
 797150 
 
 797307 | 
 | 9.797464 
 
 797621 
 GIG 
 
 TIT 9B4 
 
 .798091 
 798247 
 
 798403 
 . 798560 
 . 798716 
 9.798872 
 
 Cosine. 
 
 2. 
 
 TWWWWWD 
 
 gS 1920} 
 
 IWWNWNWWWWWW 
 
 G9 WD 09 WD LO 
 
 WW WWW WWW 
 
 WWW NWWWWWW WW WWW WWW WW 
 
 65 
 
 | 
 | 
 | 
 | 
 
 Cosine. 
 
 || 9.896532 
 
 . 896433 
 .896335 
 . 896236 
 .896137 
 .896038 
 .895939 
 .895840 
 .895741 
 .895641 
 .895542 
 
 9.895443 
 .895343 
 . 895244 
 .895145 
 .895045 
 .894945 
 .894846 
 894746 
 .894646 
 894546 
 9.894446 
 .894346 
 .894246 
 .894146 
 .894046 
 .893946 
 .893846 
 .893745 
 .893645 
 . 893544 
 
 9.893444 
 .893343 
 -893243 
 .893142 
 893041 
 .892940 
 892839 
 .892739 
 .892638 
 .892536 
 
 9.892435 
 . 892334 
 892233 
 .892132 
 892030 
 .891929 
 891827 
 891726 
 891624 
 891523 
 
 9.891421 
 
 .891319 
 
 .891217 
 .891115 
 .891013 
 .890911 
 
 .890809 
 . 890707 
 .890605 
 
 9.890503 | 
 
 Sine. 
 
 1);,.1?. 
 
 Be ek ek ek ek ek ek el ek el ee pe ee pe he ep Re eh ee ee ee ee ek ee ee ep 
 fer) 
 2 
 
 o 
 
 Cotang. | / | 
 
 141° 
 
 Tang. De ehz, 
 | 9.892810 | 4 9, | 10.107190 | 60 
 893070 | 4°33 "106930 | 59 
 .893331 4.33 .106669 | 58 
 893501 | 4°33 106409 | 57 
 mall ae eee 
 yn bape 4.35 espe 
 894372 | 4-23 "405628 | 54 
 804632 | 4°33 "105368 | 53 
 g4gg2 | 4-3 5 2 
 Oe is | dees 
 -895152 | 4°33 -104848 | 
 .895412 4.33 .104588 | 50 
 9.895072 | 4 95 | 10.104388 | 49 
 895932 | 4°33 104068 | 48 
 eopien | 38 | aatisese lugs 
 "go67i2 | 4:33 "103288 | 45 
 ‘so6o71 | 4-32 "403029 | 44 
 ‘g97231 | 4-33 402769 | 43 
 "gorag1 | 4-33 402509 | 4: 
 ve Nake 4.33 Bl 2009 42 
 S| 18 | ea 
 -89E 4.33 -1019¢ 
 9.898270 | 4 9 | 10.101730 | 39 
 898530 | 4°35 101470 | 38 
 aes a 
 } 4 32 3 5 36 
 899308 | 7°38 "100692 | 35 
 899568 | 4°33 "400432 | 34 
 899827 | 4°38 "400173 | 33 
 900087 | 4°33 "099913 | 32 
 900346 | 4°38 099654 | 31 
 900605 | 4°35 "099395 | 30 
 9.900864 | 4 9, | 10.099186 | 29 
 901124 | 4°33 "098876 | 28 
 901383 | 4°35 098617 | 27 
 901642 | 4739 098358 | 26 
 901901 | 4°35 “098099 | 25 
 902160 | 4°3 097840 | 24 
 902420 | 4°35 097580 | 28 
 fon | 13 | wan |e 
 Porcgeg wer ere Tae 
 903197 4 39 .096803 290 
 9.903456 | 4 9, | 10.096544 | 19 
 OOSTI4 |: ae 096286 | 18 
 “goiese | 4-82 | cogszes | 4g 
 “go4491 | 4:32 "095509 | 15 
 ‘904750 | 4:32 “095250 | 14 
 “905008 | 4-30 "091992 | 4: 
 "905267 | 4:32 “094733 | 13 
 905526 | 4-32 “o0ss74 | 41 
 “905785 | 4:82 DOS Ss 
 ot D185 4.30 094215 10 
 9.906043 > | 10.093957 | 9 
 906302 | 4-32 | ““‘o93698 | 8 
 906560 va 093440 | 7 
 emer rik ape eenea Wine 
 ett yee O92923 | 5 
 . 907336 4 ps .092664 4 
 tm ( oes QI406 § 
 Seog | 48 | ee 3 
 908111 | 4°29 | 091889 | 4 
 9.908369 | 4:39 | 40091631 | 0 
 Cotang. | D. 1". Tang. Z 
 
— 
 POUR WWEH COMMA RPWWOH SC 
 
 Sine. 
 
 9.798872 
 - 799028 
 199184 
 . 799339 
 - 799495 
 799651 
 - (99806 
 799962 
 800117 
 -800272 
 800427 
 
 9.800582 
 800737 
 800892 
 801047 
 -801201 
 801356 
 -801511 
 801665 
 -801819 
 .801973 
 
 9.802128 
 802282 
 802436 
 802589 
 802748 
 802897 
 . 803050 
 803204 
 . 803357 
 803511 
 
 9.803664 
 .803817 
 .803970 
 804123 
 804276 
 804428 
 804581 
 804734 
 -804886 
 805039 
 
 9.805191 
 -805343 
 805495 
 805647 
 805799 
 -805951 
 .806103 
 806254 
 .806406 
 806557 
 .806709 
 .806860 
 807011 
 807163 
 807314 
 807465 
 807615 
 807766 
 
 807917 
 
 | 9.808067 
 
 Cosine. 
 
 TABLE XXV,—LOGARITHMIC SINES, 
 
 Or OF OF OF Or Or 
 CO =2 2 =F CO CO “2 
 
 ~Ovrorgorororororor O14 
 
 OUR OURS OT 3 2 OL 43 23 
 
 WWWWNWWNWWWW WWW WW WW YW? 
 
 B® 0 
 a cron or: 
 
 ris) 
 
 5% WW 2 WWWWYD 
 
 DW We 
 
 ris) 
 
 22929 
 
 WWNwWWW 
 Creer er ore 
 
 “Delt 
 
 Cosine. 
 
 9.890503 
 .890400 
 .890298 
 .890195 
 
 - ,890093 
 .889990 
 .889888 
 .889785 
 .889682 
 .889579 
 .889477 
 .889374 
 .889271 
 .889168 
 889064 
 .888961 
 .888858 
 . 888755 
 .888651 
 .888548 
 .888444 
 
 9.888341 
 888237 
 888134 
 888030 
 887926 
 887822 
 887718 
 887614 
 887510 
 
 887406 © 
 
 9.887302 
 887198 
 887093 
 886989 
 886885 
 886780 
 886676 
 .886571 
 .886466 
 886362 
 
 9.886257 
 886152 
 886047 
 885942 
 885837 
 885732 
 .885627 
 885522 
 885416 
 885311 
 
 9.885205 | 
 885100 | 
 .884994. 
 .884889 
 .884783 | 
 .884677 
 
 884572 
 
 884466 
 
 .884360 
 9.884254 
 
 Sine. 
 
 s2orsze 
 
 PIF I-73 
 
 Beh ee ee ee eR pp Ph ek ek ek et ek pe ek ek ee ee ep ee pp 
 z 2-6 ° ’ z ay 25 a Mtis ‘ 7 om os x x . + 2 
 AB 3 3 OF 3-3 I-35 OF e 5 ah Oe Ra G as : - + 
 
 3+ 53 a I+ 
 
 | 
 | 
 
 9 
 
 Tang. 
 
 | 9.908369 
 
 | 
 
 . 908628 
 . 908886 
 . 909144 
 . 909402 
 . 909660 
 .909918 
 910177 
 910435 
 
 -916693 
 
 -910951 
 
 9.911209 
 . 911467 
 911725 
 911982 
 912240 
 912498 
 912756 
 . 913014 
 918271 
 913529 
 
 9.913787 
 914044 
 .914302 
 .914560 
 914817 
 .915075 
 915332 
 915590 
 915847 
 .916104 
 
 9.916362 
 . 916619 
 -916877 
 . 917134 
 .917391 
 .917648 
 . 917906 
 . 918163 
 . 918420 
 -918677 
 
 9.918934 
 -919191 
 .919448 
 -919705 
 . 919962 
 . 920219 
 . 920476 
 . 920733 
 .920990 
 . 921247 
 921503 
 . 921760 
 . 922017 
 922274 
 . 922530 
 922787 
 . 923044 
 . 923300 
 923557 
 
 9.923814 
 
 Dp... 
 
 oo oo 
 SSS38 
 
 oo 0 O89 CO GO Ww Ww 
 conan ask a) 
 
 SSRRS 
 
 do w wird 
 DHDD 
 
 Cotang. | D. 1”. 
 
 WwWwnd Wwnwwwwnwww 
 
 Cotang. 
 
 10.091631 
 
 - 090856 
 
 -090598 
 
 .090340 
 -090082 
 -089823 
 089565 
 .089307 
 
 089049 
 
 10.088791 
 -088533 
 
 088275 | 
 
 .088018 
 .087760 
 087502 
 087244 
 
 086986 
 
 086729 
 086471 
 10.086213 
 085956 
 .085698 
 .085440 
 085183 
 084925 
 -084668 
 084410 
 .084153 
 083896 
 
 10.083638 
 083381 
 0838123 
 082866 
 082609 
 082352 
 082094 
 
 -081837 
 
 .081580 
 081323 
 10.081066 
 080809 
 .080552 
 080295 
 .080038 
 079781 
 
 .079524 
 
 079267 | 
 
 .079010 
 
 078753 * | 
 10.078497 
 
 . 78240 } 
 
 077983 
 077726 
 017470 
 077213 
 076956 
 076700 
 076443 
 10.076186 
 
 Tang. 
 
 | 60 
 091372 | 
 “091114 
 
COSINES, TANGENTS, AND COTANGENTS. 
 
 " | Sine. | D. 1°. | Cosine. | D. 1’. 
 | 
 0 | 9.808067 | 2 59 } 9.884254 | Ly 
 Ba aeyeoes k 20d: ll eresasen |: Lc 
 3} isossi9 | 2-2 || ‘aesgag | 1-77 
 4 | “gogg69 | 2-29 || “ggagog | 1-78 
 | ~OUC 69 5) 50 | 883829 1 v7 
 SS ecrees il BBO | gh 
 Elicia | 20d dives tie 
 | amo: | sp | ee a 
 sai ~OUe y OF f Oe) ais 
 } : 92 2.50 | SOK B ni 
 S| -amost | 30 |) SO] TR 
 CULE eos 2.50 Taek £27 
 11 | 9.809718 | 9 59 || 9.883084 17 
 2| .809868 | $"4g 882977 | yay 
 ‘ > Pied aw. OQr’ 4 
 | eur | a0 | eet | 
 15 | 1810316 wid "882657 ie 
 > ’ v4 We RRO } 4 
 17 | isioera | 2-48 || cgqias | 17 
 18.| ‘siov63 | 2-48 "982336 | 1-78 
 19 | Sioa | 343 |] 882229) 16) 
 20 | .811061 | 9'48 882121 | 5"p 
 21 | 9.811210 | 9 yo |] 9.882014 | | » 
 2 | > 811358 | 3 74o -881907'| +745 
 23 | ..811507 | "jo 881799 | 5 "p 
 OA 2 << BOO . 
 | eine 2 48 -881692 1.80 
 26 | .811952 | 9 47 881407 | "a9 
 S| oan) ae | Bae) is 
 29 | (812396 | 2-44 || “geri53 | 1-80 
 30 | .gi2544 | 2-49 || ‘agiogg | 1-48 
 upp 2.47 . 1.80 
 31 | 9.812692 | 5 yw || 9.880938 
 aa | 812810 | 347 || 880880 | 1p 
 33 | .812988 | 9°45 880722 | 3 "65 
 36 | ceisis0 | 2-45 || “Bogaye | 1:80 
 me Ol wagers 224% “agqado |’ 1-80 
 peli cameos |; 24m {| prgepecs || 1:82 
 Bel betters, |) 2240p || or ORO") 7280 
 39 . 813872 | 9 45 -880072 1 82 
 40 .814019 | ° 45 .879963 1 80 
 41 | 9.814166 | , || 9.879855 
 ¢ n O46 2.45 | LOK AR 1.82 
 42 ing |! Sap | pie 1.82 
 pul Roaeeacr |) Qedee- |) peateee | 7 Be 
 44 | 814607 } >) 43 ~ 879529 1 go 
 45 | .814753 | 2 45 -879420 | 5 "g5 
 Ak fe lot . 
 | som) oa || aa | TB 
 48 .815193 | 343 || ‘gr9093 | 1-82 
 49 | .815339 | 9°43 878984 | 5°65 
 50 .815485 2.45 878875 1.82 
 51 | 9.815632 | 9 43 || 9.878766 | 1 go 
 Bel dicitaes || 48> It Gece |. one 
 Bi | “sioc9 | 2-42 |) ‘Sram | 1-8 
 55 | .816215 | S77. .878328 | 3:3 
 56 | 1816361 | non | [878219 | +t 
 Bel Feber: || 22" |) eens | 188 
 59 | “ster | 2-43 || “eraoo | 18 
 60 | 9.816943 | ~-4* || 9.877780 
 ‘ | Cosine. | D. 1". | Sine. Derk": 
 
 Tang. 
 
 9, 923814 
 - 924070 
 - 924327 
 924583 
 924840 
 925096 
 925852 
 . 925609 
 925865 
 926122 
 . 926378 
 
 9.926634 
 . 926890 
 927147 
 927403 
 927659 
 927915 
 928171 
 928427 
 . 928684 
 . 928940 
 
 9.929196 
 929452 
 -929708 
 929964 
 - 930220 
 930475 
 . 930781 
 . 980987 
 . 931243 
 . 931499 
 
 9.931755 
 932010 
 932266 
 932522 
 932778 
 933033 
 933289 
 983545 
 . 933800 
 934056 | 
 
 9.934311 
 984567 
 934822 
 935078 
 935333 
 935589 
 935844 
 936100 
 936355 
 936611 
 
 9.936866 
 987121 
 IBI377 
 . 937632 
 937887 
 . 988142 
 . 938398 
 . 938653 
 . 938908 
 
 9.939163 
 
 Cotang. 
 
 Del”. 
 
 SWWMWMNWNWNWNWWW WW 
 
 Wrewrwaraorw wa 
 
 MEO ADMIT AE AEA a A AAI OA AA 
 
 CX) 
 
 We 
 
 Wd 2 
 Or » 3 
 
 I 29 29 29 29 29 2 2929 
 SKRLSRRLRSB 
 
 | 
 | 
 
 9 
 _ 
 2 
 
 | Cotang. 
 
 10.076186 
 075930 
 075673 
 075417 
 075160 
 074904 
 074648 
 074391 
 074135 
 073878 
 073622 
 
 10.073366 
 073110 
 072853 
 072597 
 072341 
 072085 
 071829 
 071573 
 071316 
 071060 
 
 10.070804 
 070548 
 070292 
 070036 
 069780 
 069525 
 069269 
 069013 
 068757 
 068501 
 
 10.068245 
 067990 
 067734 
 067478 
 067222 
 066967 
 066711 
 066455 
 066200 
 065944 
 
 10.065689 
 .065433 
 .065178 
 -064922 
 . 064667 
 064411 
 .064156 
 .063900 
 .063645 
 063389 
 
 10.063134 
 062879 
 062623 
 062368 
 .062113 
 .061858 
 .061602 
 .0613847 
 .061092 
 
 10.060837 
 
 Tang. 
 
El | 
 
 —_ 
 
 fol pes et 
 wre 
 
 SOMDIAMTIP OD 
 
 Sine. 
 
 9.816943 
 .817088 
 817233 
 817379 
 817524 
 .817668 
 .817813 
 .817958 
 .818103 
 818247 
 .818392 
 9.818536 
 .818681 
 818825 
 818969 
 819113 
 819257 
 .819401 
 .819545 
 .819689 
 819832 
 9.819976 
 820120 
 820263 
 820406 
 820550 
 820693 
 820836 
 820979 
 821122 
 821265 
 
 9.821407 
 .821550 
 .821693 
 .821835 
 .82197'7 
 .822120 
 .822262 
 .822404 
 ,822546 
 . 822688 
 
 9.822830 
 822972 
 823114 
 823255 
 . 823397 
 . 823539 
 .823680 
 823821 
 .823963 
 .824104 
 
 | 9.824245 
 824386 
 824527 
 824668 
 824808 
 824949 
 825090 
 825230 
 825371 
 | 9.825511 
 
 Cosine. 
 
 D..1". 
 
 | 
 | 
 
 ITWWNWNWWW 
 
 SESS See eee SERee ESE EEE 
 NAHDHHDSDHS SHSSSSSSOW SHNSOHWHNSHHNO 
 
 WNWNWWWWWNWNVMY WNWNVNNNWVNYW WHNVYW 
 
 CQO QO WW BWW 
 
 WWNWNWNWNWWWWM WWNWNWNWNWWNWWW 
 WOWWWWWWWD WWWWWWWWtt WWwWWWwWWWKWRWWwWWw 
 G2 OV 09 CLOT OS OTOUCOT OT OTS OLOUTSE NI OTSA AIRINNWNID VRNDD 
 
 wrwwrwwwrwre 
 
 Cosine. 
 
 9.877780 
 877670 
 877560 
 877450 
 877340 
 877/230 
 877120 
 877010 
 876899 
 876789 
 876678 
 
 9.876568 
 
 id ded 
 
 -Ol006 
 
 9.875459 
 .875348 
 875237 
 875126 
 875014 
 874903 
 
 874680 
 874568 
 874456 
 9.874344 
 874232 
 874121 
 874009 
 873896 
 873784 
 87367 
 873560 
 873448 
 873335 
 9.873223 
 873110 
 872998 
 872885 
 87272 
 872659 
 872547 
 1872434 
 872321 
 872208 
 9.872095 
 871981 
 .871868 
 | ..871755 
 | .871641 
 | .871528 
 871414 
 871301 
 | .871187 
 9.871073 
 
 Sine. 
 
 874791 | 
 
 Didl'’s 
 
 ia a a a Drm mck mk fr fh feed fed fend fred fh mech fel forme fom feemh frrmh femed fered feed fod Pr pre pre pr fee femal femme frm fk fom fm fmt fom fmt fc fc fom fumed fom femme mh fom frm fem emma form fol frome fork feel from 
 3 
 
 TABLE XXV.—LOGARITHMIC SINES, 
 
 Tang. 
 
 9.939163 
 . 939418 
 . 939673 
 . 939928 
 . 940183 
 . 940439 
 940694 
 . 940949 
 941204 
 941459 
 941713 
 
 9.941968 
 942223 
 942478 
 -942733 
 942988 
 - 943243 
 - 943498 
 943752 
 944007 
 944262 
 
 -944517 
 944771 
 - 945026 
 - 945281 
 - 945535 
 945790 
 -946045 
 946299 
 946554 
 946808 
 
 co 
 
 | 9.947063 
 
 .947318 
 947572 
 947827 
 . 948081 
 . 948335 
 . 948590 
 . .948844 
 . 949099 
 949353 
 
 9.949608 
 
 . 949862 
 . 950116 
 . 950371 
 - 950625 
 - 950879 
 .951133 
 .951388 
 . 951642 
 . 951896 
 9.952150 
 - 952405 
 952659 
 952913 
 . 953167 
 . 953421 
 953675 
 - 953929 
 . 954183 
 9.954437 
 
 Cotang. 
 
 529 29 
 
 OL OLCOTT OURS OT OT OU OT 
 
 WWW WWW 
 
 4.25 
 
 WMNWNW WWNONwwnwnw 
 CO OLd9 Ot OOS O19 OF OL 09 
 
 Wrwwmwwwwwwyw w ds 
 WOW ow we sreaee Ot Sie 
 
 2 9 29 
 
 SSSR 
 
 WWWWW 
 Co G8 Oo Go OO 
 
 Cotang. 
 
 10.060837 
 .060582 
 0603827 
 -060072 
 
 .059817 | 
 
 059561 
 059306 
 059051 
 
 058796 
 058541 
 
 058287 
 
 10.058032 
 
 057777 
 
 VOC 
 -057522 
 
 057267 | 
 
 .057012 
 056757 
 . 056502 
 .056248 
 .055993 
 055738 
 10 .055483 
 -055229 
 054974 
 054719 
 
 -054465 | 
 -054210 - 
 
 058955 
 
 053701 | 
 053446 | 
 053192 | 
 
 10.052937 
 052682 
 052428 
 -052173 
 .051919 
 -051665 
 .051410 
 .051156 
 050901 
 -050647 
 
 10.050392 
 .050138 
 049884 
 
 .049629 | 
 
 049375 
 -049121 
 048867 
 .048612 
 048358 
 048104 
 
 10.047850 
 
 047595 
 -047341 
 047087 
 
 046833 
 046579 
 046325 
 046071 
 
 045817 
 
 10. 045563 
 
 Tang, 
 
COMA WWH OS 
 
 —" 
 
 or 
 
 rN 
 CO 
 
 Sine. 
 
 9.825511 
 .825651 
 825791 
 825981 
 826071 
 -826211 
 .826351 
 826491 
 826631 
 826770 
 .826910 
 
 9.827049 
 -827189 
 827328 
 827467 
 827606 
 827745 
 827884 
 828023 
 .828162 
 828301 
 
 9.828439 
 82857; 
 828716 
 828855 
 828993 
 829131 
 829269 
 829407 
 829545 
 1829683 
 
 29821 
 
 829959 
 
 830097 
 
 880234 
 
 .830372 
 
 . 830509 
 
 830646 
 
 830784 
 
 .830921 
 
 831058 
 
 | 9.831195 
 
 831332 
 831469 
 .831606 
 831742 
 831879 
 832015 
 832152 
 832288 
 832425 
 
 | 9°832561 
 
 832697 
 832833 
 
 832969 
 
 833105 | 
 
 833241 
 833377 
 833512 
 
 983648 
 9.833783 
 
 Cosine. 
 
 | 
 | 
 
 wd 
 icy) 
 Oo 
 
 WWWWW WWNWWOWNWWNWWW 
 LOW wCWwWwWWwWwWWwWoN WC 
 Ooo SCOoCocooowonw 
 
 2.30 
 
 2 0 Ww 
 22 
 @ 
 
 BODO 20 20 
 % 
 
 De i | Cosine. 
 
 9.871073 
 
 | .870960 
 
 870846 
 870732 
 870618 
 870504 
 870390 
 870276 
 «870161 
 870047 
 .869933 
 
 9.869818 
 869704 
 869589 
 869474 
 869360 
 869245 
 .869130 
 .869015 
 .868900 
 868785 
 
 9.868670 
 . 868555 
 .868440 
 868324 
 868209 
 868093 
 .§07978 
 . 867862 
 867747 
 8676381 
 
 9.867515 
 .867399 
 867283 
 .867167 
 .867051 
 
 || .866935 
 
 || .866819 
 
 .866703 
 
 || .866586 
 
 .866470 
 
 |] 9.866353 
 || 866237 
 || 866120 
 |) 866004 
 865887 
 86577 
 865653 
 865536 
 865419 
 865302 
 9.865185 
 865068 
 864950 
 864833 
 864716 
 864598 
 864481 
 864363 
 864245 
 9.864127 
 
 Sine. 
 
 | 
 
 DL: 
 
 Tang. 
 
 =) 
 (ou) 
 
 9.954487 
 954691 
 954946 
 . 955200 
 955454 
 . 955708 
 955961 
 956215 
 - 956469 
 956723 
 956977 
 
 9.957281 
 957485 
 957739 
 957993 
 958247 
 . 958500 
 958754 
 . 959008 
 959262 
 - 959516 
 
 9.959769 
 . 960023 
 960277 
 960530 
 960784 
 . 961038 
 . 961292 
 . 961545 
 .961799 
 . 962052 
 
 9.962806 
 962560 
 . 962813 
 . 962067 
 . 963820 
 . 963574 
 . 963828 
 . 964081 
 . 964335 
 964588 
 
 . 964842 
 . 965095 
 965349 
 . 965602 
 . 965855 
 . 966109 
 . 966362 
 - 966616 
 . 966869 
 967123 
 
 9.967376 
 967629 
 .967883 
 . 968186 
 . 968359 
 968643 
 . 968896 
 . 969149 
 969403 
 
 9.969656 
 
 Cotang. I'D: 137 | 
 
 SAA ALAA A AAA AA AAA AHL AAD ALAA A AAA DAE PAAAALRA BARBRA 
 WMWWWDW WW NWNW NNN NWYW WNWNWNWNWNVWNWNYW WNHWwNwwwwwn»n wwrwr 
 Wwe HBSVSSSSBVES SYRSRERRRY GRGRVVRVRLY CRBS 
 
 oc 
 
 Cotang. 
 
 10.045563 
 .045309 
 .045054 
 .044800 
 044546 
 044292 
 .044039 
 043785 
 .043531 
 043277 
 048023 
 
 10.042769 
 042515 
 042261 
 042007 
 041753 
 .041500 
 041246 
 .040992 
 .0407 -8 
 040484 
 
 10.040281 
 .039977 
 .039723 
 039470 
 .039216 
 038962 
 .038708 
 038455 
 038201 
 037948 
 
 10.037694 
 .037440 
 037187 
 -036933 
 .036680 
 036426 
 036172 
 035919 
 035665 
 035412 
 
 10.035158 
 .034905 
 
 034651 | 
 
 034398 
 034145 
 033891 
 033638 
 033384 
 033131 
 032877 
 10.032624 
 
 0323711 
 032117 
 
 031864 
 
 031611 
 031357 
 
 .031104 
 .030851 
 0380597 
 
 10.030344 
 
 Tang. 
 
CMOIRDUIP WMHS | 
 
 Sine. 
 
 9.833783 
 833919 
 
 834054 
 
 834189 
 834325 
 834460 
 834595 
 .834730 
 834865 
 .834999 
 835134 
 9.835269 
 .835403 
 835538 
 835672 
 835807 
 835941 
 .836075 
 836209 
 836343 
 836477 
 9.836611 
 836745 
 836878 
 837012 
 837146 
 837279 
 887412 
 837546 
 837679 
 837812 
 9.837945 
 838078 
 838211 
 838344 
 8388477 
 . 838610 
 838742 
 838875 
 839007 
 839140 
 
 9.839272 
 839404 
 * 839536 
 839663 
 839800 
 839932 
 840064 
 840196 
 840328 
 840459 
 
 9.840591 
 840722 
 840854 
 840985 
 841116 
 841247 
 841378 
 .841509 
 .841640 
 
 9.841771 
 
 Cosine. 
 
 Dai". 
 
 SIOWUVNNWY VYNVNVNVNVNYYNY VVVNVVNVNLYNNY WKVNVNMNYYNVKVVY 
 BS OD OND HOOD ODD ICD TCD OTD OUT OO MANION ST 
 
 2% 
 SOSCS SCHWOCHWOCKWNWNWNWYHW 
 
 wWwWwN 
 
 ~) 
 
 20 
 
 DWWWWWW WWMWNWW WWW 
 
 Cosine. 
 
 9.864127 
 .864010 
 863892 
 863774 
 863656 
 863538 
 .863419 
 .863301 
 .863183 
 863064 
 862946 
 
 | 9.862827 
 
 862709 
 .862590 
 
 862471 
 
 862353 
 862284 
 862115 
 .861996 
 -861877 
 861758 
 
 9.861638 
 -861519 
 .861400 
 861280 
 .861161 
 861041 
 860922 
 860802 
 860682 
 860562 
 
 9.860442 
 860322 
 860202 
 860082 
 859962 
 859842 
 859721 
 859601 
 859480 
 859360 
 
 9.859239 
 859119 
 858998 
 858877 
 858756 
 858635 
 858514 
 858393 
 858272 
 858151 
 
 9.858029 
 857908 
 857786 
 857605 
 
 857543 
 857422 
 
 857300 
 85717 
 857056 
 
 | 9.856934 
 
 Sine. 
 
 Dr". 
 
 98 
 
 Seoooeoe 
 lane oan ie oes) 
 
 WWWWW WNWWBDEHDWBEHVWHH WEP eH HH Eee 
 2 
 Sssé S 
 
 eooooo 
 YOOOoOooO 
 
 eosscoo 
 SD DWNWWD 
 
 CWO WWMOMOIM Wa 
 
 09, 02.09 8 WW WW HW 0 WW 
 
 Seesegeos9o 
 
 o 
 
 TABLE XXV.—LOGARITHMIC SINES, 
 
 Tang. 
 
 9.969656 
 . 969909 
 . 970162 
 970416 
 . 970669 
 - 970922 
 971175 
 -971429 
 . 971682 
 972188 
 
 9.972441 
 972695 
 972948 
 
 973454 
 .I3707 
 973960 
 974218 
 .974466 
 974720 
 9.974973 
 975226 
 975479 
 975732 
 975985 
 976238 
 .976491 
 976744 
 .976997 
 
 9.977508 
 977756 
 978009 
 . 978262 
 978515 
 
 979021 
 979274 
 979527 
 979780 
 
 . 980286 
 . 9805388 
 980791 
 
 981297 
 .981550 
 .981803 
 . 982056 
 . 982309 
 
 9.982562 
 . 982814 
 . 983067 
 . 983320 
 . 988573 
 . 983826 
 . 984079 
 . 984332 
 . 984584 
 9.984837 
 
 Cotang. 
 
 971985 | 
 
 .973201 
 
 977250 | 
 
 978768 | 
 
 9.980033 
 
 -981044 | 
 
 D. 1”. 
 
 | 
 | 
 
 SWWNWNWNNNWNWWW 
 SESSSSSLCSIS 
 
 Mow 
 0 0 09 
 
 wWwmnnney»y 
 GARGS PR) 
 
 IMWMW WOWNHWWOWYM 
 
 ww wo 
 
 4.22 
 
 ri 
 g 
 
 2 
 ] 
 
 ri 
 PR 
 
 wWrmdwwwwwnwwye ww 
 RRS SSRBSSSRSE RE 
 
 4 
 
 WHWwND 
 
 Cotang. 
 
 | 10.030344 | 
 030091 | 
 .029838 
 029584 | 
 .029331 | 
 
 .029078 
 
 028825 
 
 028571 
 
 028318 
 
 028065 
 
 027812 
 10.027559 | 
 
 027805 | 
 027052 
 .026799 
 
 .026546 
 026293 
 026040 
 025787 
 025534 
 
 -025280- 
 10.025027 
 
 02477 
 
 024521 | 
 024268 | 
 024015 | 
 023762 | 
 023509 | 
 
 023256 
 
 .028003 | 
 
 022750 
 10.022497 
 022244 
 .021991 
 021738 
 
 .021485 
 .021232 | 
 020979 
 
 020726 
 020478 
 020220 
 10.019967 
 
 .019714 | 
 .019462 
 .019209 | 
 
 018956 
 018703 
 018450 
 
 018197 
 017944 
 017691 
 
 10.017438 
 
 .017186 
 
 016933 | 
 .016680 
 
 .016427 
 .016174 
 015921 
 .015668 
 
 015416 
 
 10.015163 
 
 or 
 or 
 
 g 
 
 ~ 
 lor) 
 
 (SV) 
 ies} 
 
 coq te 
 He OUS> 
 
 [sy] 
 So 
 
 ta) 
 © 
 
 nil emul aed 
 
| 
 
 ie =) 
 
 Sine. 
 
 .841902 
 .842033 
 .842163 
 .842294 
 842424 
 . 842555. 
 
 .842815 
 .842946 
 .843076 
 9.843206 
 
 COIS) Or 
 
 | 
 
 . 843466 
 . 843595 
 843725 
 . 843855 
 .843984 
 . 844114 
 844243 
 844372 
 9.844502 
 844631 
 .844760 
 .844889 
 .845018 
 .845147 
 845276 
 .845405 
 . 845533 
 /845662 
 9.845790 
 .845919 
 .846047 
 .846175 
 . 846304 
 846432 
 846560 
 .846688 
 .846816 
 846944 
 847071 
 .847199 
 847327 
 
 =) 
 
 847582 
 
 847964 
 848091 
 
 © 
 
 848345 
 848472 
 “848599 
 848726 
 “848852 
 
 849232 
 849359 
 
 | 9.841771 | 
 
 . 842685 | 
 
 .843336 | 
 
 847454 
 847709 | 
 847336 | 
 
 848218 | 
 
 09 0 WW WNWWWWNHWNKNWW WWWNWWNWWNWWW WWNUWNWNWNMNWMWNWW WNWWNWWWNWWwW®D wi ri 
 
 : 848979 | 
 57 | .849106 
 
 ® WW 
 
 9.849485 | 
 
 Cosine. 
 
 Cosine. Tang. Cotang. 
 9.856934 9.984837 4.29 10.015163 
 .856812 . 985090 ¥ 90 .014910 
 .856690 . 985343 4 "99 014657 
 .856568 . 985596 a 20 014404 
 .856446 . 985848 rg ae 014152 
 .856323 .986101 ry 99 .013899 
 .856201 986354 4 ee .013646 
 .856078 . 986607 ry Re .013393 
 .855956 .986860 A. S .0138140 
 855833 987112 | 4°55 .012888 
 .855711 - 987365 4 ; 29 .012635 
 9.855588 9.987618 | 4 99 | 10.012382 
 855465 987871 4 20 .012129 
 855342 . 988123 vy 29 .O11877 
 855219 . 988376 4.99 .011624 
 855096 . 988629 a. 99 .0113871 
 854973 . 988882 ia 20 .011118 
 -854850 . 989134 4 "99 .010866 
 .854727 . 989387 4 Pie .0106138 
 854603 -989640 | 7°55 010360 
 854480 .989893 490 010107 
 9.854356 9.990145 4.99 10.009855 
 854233 . 990398 ra pa .009602 
 .854109 . 990651 re 20 .009349 
 .853986 -990903 | 4°55 009097 
 853862 . 991156 ae as . 008844. 
 .853738 . 991409 ae 99 008591 
 853614 .991662 4.90 008338 
 853490 “91914 | 7°55 .008086 
 . 853366 . 992167 4.99 007833 
 853242 . 992420 4.20 007580 
 9.853118 9.992672 | 4 gq | 10.007828 
 .852994 | * 992025 | 45s 007075 
 . 852869 3. .993178 ‘As 39 .006822 
 852745 3: 993431 4. 50 .006569 
 852620 oy . 993683 vig 99 .006317 
 .852496 sy . 993936 4 99 . 006064. 
 852371 a . 994189 a 20 .005811 
 852247 on .994441 4. Bes .005559 
 852122 9 : . 994694 ay 29 005306 
 .851997 0. . 994947 4 "30 . 005053 
 9.851872 | -995199 | 4 99 | 10.004801 
 SSOLV afar: 995452 | 459 004548 
 .851622 ee . 995705 a an . 004295 
 .851497 9 ; . 995957 4 "99 .004043 
 .851372 9. . 996210 4. 99 .003790 
 .851246 3. . 996463 re 20 .003537 
 .851121 o .996715 4 ‘99 . 003285 
 .850996 pa . 996968 re Se .0038032 
 .850870 . 997221 as 20) 00277 
 850745 oe . 997473 a 99 002527 
 9.850619 | 5 9.997726 | 4 59 | 10.002274 
 . 850493 9 : | 997979 4. 350 002021 
 850368 3 . 998231 Ps 96 .001769 
 850242 | %: .998484 | 7°55 001516 
 .850116 | 5 998787 | 7 °S5 001263 
 . 849990 Ss . 998989 4 sese .001011 
 849864 | %° -999242"| 7°55 000758 
 . 849738 ae . 999495 4 30 .000505 
 849611 | 5° 999747 | 4°50 000253 
 9.849485 | ; 10 .000000 a 10.000000 
 Sine, Cotang. ! D. 1” Tang. 
 
 SCHWWRPUAMNIMDLO 
 
 - 
 
TABLE XXVI—LOGARITHMIC VERSED SINES. 
 
0° 
 
 TABLE XXVI.—LOGARITHMIC VERSED SINES 
 
 ° 
 
 COMNIAIP WMHS 
 
 OU 
 
 or 
 
 So Or 
 
 Vers. 
 
 \Inf. neg. 
 2.626422 
 13228482 
 
 .580665 
 
 3.830542 
 
 024362 
 182725 
 316618 
 .432602 
 534907 
 626422 
 709207 
 784784 
 854308 
 918678 
 978604 
 034661 
 087319 
 136966 
 183928 
 228481 
 250859 
 311266 
 349876 
 - 386843 
 422300 
 456367 
 489148 
 520736 
 551216 
 580662 
 
 .609143 
 .636719 
 663447 
 .689377 
 ~714555 
 739023 
 . 762821 
 . 785985 
 . 808547 
 .830537 
 .851985 
 .872915 
 . 893353 
 . 913322 
 . 982841 
 ~9519381 
 .970611 
 . 988898 
 .006807 
 . 024355 
 
 041555 | 
 
 058421 
 .074965 
 .991201 
 107138 
 
 5 | .122789 
 = 
 
 . 138162 
 153268 
 168116 
 
 6.182714 
 
 q — 2l | Ex. sec. 
 9.070 
 120 || 120 Inf. neg. 
 120 | 120 2.626422 
 | 120 || 120 8.228482 
 | 120 || 120 .580665 
 | 120 || 120 3.880542 
 120 || 120 |4.024363 
 120 || 120 | .182725 
 120 || 120} .816619 
 120 || 121 | .432603 
 119 |} 121 | .584908 
 119 || 121] .626424 
 119 || 122 4.709209 
 119 || 122 | .784787 
 119 |} 122 | .854812 
 119 || 123} .918681 
 119 || 123 '4.978608 
 119 || 124 |5.0384666 
 119 |) 124} .087825 
 119 || 125] .136972 
 119 || 125} .183935 
 119 || 126 | .228488 
 118 || 126 5.270868 
 118 || 127 | .3811275 
 118 || 128} .349886 
 118.|/ 129} .886854 
 118}| 129 | .422312 
 118 || 180} .456379 
 118 || 181} .489161 
 117 || 182 | .520750 
 117 || 188} .5512381 
 117 || 184] .580679 
 117 || 134 5.609160 
 117 || 185 | .636738 
 116 || 186} .663467 
 116 || 187 | .689398 
 116 || 188) .714577 
 116 || 140} .7389047 
 116 ||} 141 | .762847 
 115 || 142 | .786012 
 115 || 143 | 80857 
 115 || 144 | .830567 
 115 || 145 |5.852016 
 114 || 147 | .872948 
 114 || 148) .893387 
 114 || 149] 913357 | 
 114 || 151} .98287 
 113 || 152 | .951970 
 113 || 154] 970652. 
 113 || 155 |5 988940 
 112 || 157 6.006851 
 112 || 158) .024401 || 
 | 112 || 160 6.041602 
 } 111 || 161} .058470 
 /111]/ 163} .075017 
 111 || 164} .091254 
 110 || 166| .107194 
 1 110]/ 168) .122846 
 110 || 169 | .1388222 
 109 || 171 | .153880 
 109 || 173 | .168180 
 109 |) 175 |}6.182780 
 
 | 6480 
 
 | 6840 
 |, 6900 
 
 4} 
 
 3600 
 3660 
 3720 
 3780 
 3840 
 8900 
 3960 
 4020 
 4080 
 4140 
 4200 
 4260 
 4320 
 4380 
 4440 
 4500 
 4560 
 4620 
 4680 
 4740 
 
 4800 
 4860 
 4920 
 4980 
 5040 
 5100 
 5160 
 5220 
 5280 
 5340 
 5400 
 
 5460 
 5520 
 5580 
 5640 
 5700 
 5760 
 5820 
 5880 
 5940 
 6000 | 
 
 6060 
 6120 
 6150 
 6240 
 6300 
 6360 
 6420 
 
 6540 
 6600 
 6660 
 6720 
 6780 
 
 6960 
 7020 
 7080 
 7140 
 
 _ 
 ar CSCOMDIMHDOUPWWH © 
 ao 
 
 a 
 
 SUR) 
 
 (or) for) 
 
 a 
 
 7200 
 0 
 
 Vers. 
 
 .182714 
 .197071 
 .211194 
 .225091 
 28877 
 
 . 252236 
 . 265497 
 .278558 
 -291426 
 .3804106 
 .3816603 
 . 828923 
 841071 
 . 803052 
 . 864869 
 .316528 
 . 3888032 
 .399386 
 .410593 
 .421657 
 .482583 
 6.448372 
 .454029 
 .464557 
 .474959 
 -485238 
 .495396 
 .505438 
 .515364 
 .525178 
 - 5384882 
 .544480 
 .553972 
 . 563362 
 .572651 
 .581842 
 . 590936 
 .599937 
 
 . 608845 
 .6176638 
 . 626392 
 . 6850384 
 .643591 
 
 . 65264 | 
 . 660456 
 . 668767 
 . 677000 
 .685155 
 . 693234 
 . 101239 
 709171 
 \6.717030 
 . (24820 
 . 732540 
 .740192 
 CATT 
 155297 
 . 762752 
 .770144 
 107403 
 
 6.784741 
 
 | 175 | 
 
 lor) 
 
 EX. sec. 
 
 6.182780 
 197139 
 211264 
 225164 
 288845 
 252314 
 265577 
 278641 
 291511 
 304193 
 316693 
 
 6.329016 
 341167 
 853150 
 364970 
 376631 
 388138 
 399494 
 
 410705 
 421772 
 432700 
 
 6.443498 
 454158 
 464684 
 475089 
 48537 
 495532 
 505577 
 515506 
 525324 
 535031 
 
 6.544632 
 .554128 
 563521 
 572813 
 .582008 
 .591106 
 .600110 
 .609021 
 .617843 
 626575 
 
 6.635221 
 643782 
 652259 
 . 660655 
 . 668970 
 677206 
 685365 
 . 693448 
 . 701457 
 709393 
 
 T7257 
 
 725050 
 7382775 
 740431 
 . 748020 
 . 755544 
 . 763004 
 . 770400 
 777733 
 
 6.785005 
 
AND EXTERNAL SECANTS. 
 
 ny 
 
 i SS 
 
 Wan 
 
 } 
 
 7260 
 7320 
 
 9 
 ~ 
 
 7380} 3} 
 
 7440| 4 
 
 7500) 5} . 
 7560| 6) . 
 
 7620| 7 
 
 7680| 8| . 
 
 7740| 9 
 7800 10 
 7860) 11 
 
 792012) . 
 
 7980 13) °. 
 
 8040 | 14 
 8100 | 15 
 8160 | 16 
 8220| 17 
 8280/18 
 
 10080 | 48 
 10140) 49 
 10200) 50 
 
 10260) 51 | 
 
 10820 | 52 
 
 10380) 53] 
 
 10440) 54 
 10500. 55 
 10560 56 
 
 10620 57} 
 
 10680: 58 
 10740 59 
 10800 
 
 u 
 
 SEE 
 
 7200) 0 6.784741 | 
 hy 
 
 60) 
 é 
 
 | | Ex. sec. 
 
 16.'785005 
 792217 
 .799370 
 . 806464 
 813501 
 .820482 
 . 827406 
 83427 
 .841093 
 847857 
 854568 
 | .861228 
 .867837 
 874396 
 .880907 
 .887369 
 .893783 
 .900151 
 . 906472 
 .912748 
 .918979 
 
 925165 
 . 931308 
 . 937408 
 943465 
 .949480 
 955455 
 .961388 
 967281 
 9731385 
 978949 
 984725 
 990463 
 996164 
 001827 
 007454 
 013044 
 .018599 
 024119 
 029604 
 035054 
 
 040471 
 045854 
 051204 
 056522 
 .061807 
 067061 
 .072282 
 077473 
 082633 
 087763 
 3 | 7.092862 
 3} .097932 
 102978 
 107985 
 112968 
 117922 
 . 122849 
 
 1277 
 
 132619 
 
 Abas 
 7.137464 
 
 WOOsIOR WMH © 
 
 ba 3 
 
 ~ 
 
 Vers. 
 
 7.136868 
 .141679 
 . 146464 
 151222 
 155954 
 . 160661 
 . 165342 
 . 169998 
 174630 
 . 179286 
 . 183819 
 188377 
 192912 
 197423 
 .201910 
 -206375 
 210817 
 215286 
 
 219683 
 
 224007 
 
 228360 
 7.232691 
 237000 
 241288 
 "245555 
 249801 
 254027 
 258232 
 262416 
 266581 
 210726 
 274851 
 278056 
 283043 
 287110 
 291158 
 295187 
 299197 
 303190 
 307164 
 311119 
 315057 
 318977 
 322880 
 326705 
 330632 
 334483 
 338316 
 342133 
 345933 
 349716 
 853483 
 357233 
 360968 
 364686 
 368389 
 372076 
 
 30 |7.386668 
 
 ~t 
 
 =I 
 
 S84 
 
 =) 
 
 = 
 
 Ex. sec. 
 
 7.137464 
 142281 
 147072 
 151837 
 156577 
 . 161290 
 .165978 
 170641 
 175279 
 179893 
 . 184483 
 
 7.189048 
 193589 
 198108 
 202602 
 200074 
 211523 
 215949 
 220853 
 224785 
 229095 
 
 238433 
 237750 
 242046 
 . 246320 
 250574 
 254807 
 259019 
 - 263212 
 267384 
 201587 
 
 275669 
 279783 
 283877 
 287952 
 292007 
 296045 
 800063 
 804068 
 808045 
 .312009 
 
 815955 
 .319883 
 .823794 
 327687 
 .3831563 
 .890422 
 839263 
 843089 
 846897 
 850689 
 
 "354464 
 | , 808228 | 
 .861966 
 
 9 | .365693 
 >| .869404 
 .3873100 
 .3876780 
 880444 
 . 884094 
 ’ 887728 
 
TABLE XXVI.—LOGARITHMIC VERSED SINES 
 
 4° 5° 
 Vers. | D. 1",.| Ex. see. | D. 1" / Vers. D. 1°. | Ex. see.|D, 1” 
 0 | 7.386668 | 60.17 | 7.387728 | 60.32 0 | 7.580389 | 48.15 |7.582045 | 48.33 
 1 .890278 | 59.938 .3891847 | 60.07 1 .583278 | 47.98 | .584945 | 48.17 
 2 .398874 | 59.67 .394951 | 59.82 2 .586157 | 47.82 | .587835 | 48.00 
 3 .897454 | 59.48 .898540 | 59.57 3 -589026 | 47.67 | .590715 | 47.87 
 4 -401020 ; 59.18 402114 | 59.33 4 .591886 7.52 | .598587 oul | 
 5 .404571 | 58.93 .405674 | 59.10 5 594737 | 47.35 | .596449 | 47.53 | 
 6 .408107 | 58.70 .409220 | 58.85 6 .597578 | 47.20 | .599301 | 47.38 
 ve -411629 | 58.47 .412751 | 58.62 v6 . 600410 7.05 | .602144 7.25 
 8 .415187 | 58.23 .416268 | 58.38 8 . 603233 | 46.90 | . 604979 7.08 
 9 .418631 | 58.00 .419771 | 58.15 9 -606047 | 47.7: .607804 | 46.92 
 L077 42208Y 15077 .423260 | 57.92 || 10 .608851 | 46.60 | .610619 | 46.78 
 11 | 7.425577 | 57.53 | 7.426735 | 57.70 || 11 | 7.611647 | 46.43 |7.613426 | 46.63 
 12 .429029 | 57:30 .430197 | 57.45 || 12 .614433 | 46.30 | .616224 | 46.48 
 13 .4382467 | 57.08 .433644 | 57.25 || 18 -617211 | 46.15 | .619013 | 46.35 
 14 .435892 | 56.85 .437079 | 57.00 || 14 .619980 | 45.98 | .621794 | 46.18 
 15 .439303 | 56.63 .440499 | 56.80 || 15 -622739 | 45.87 | .624565 | 46.05 
 16 .442701 | 56.42 .443907 | 56.57 16 -625491 | 45.7 .627328 | 45.90 
 17 .446086 | 56:20 -447301 | 56.35 17 .628233 | 45.57 | .630082 | 45.75 
 18 .449458 | 55.97 .450682 | 56.13 18 .630967 | 45.42 | .632827 | 45.62 
 19 -452816 | 55.77 .454050 | 55.92 19 .633692 | 45.28 | .635564 | 45.48 
 20 .456162 | 55.55 .457405 | 55.7% 20 .636409 | 45.13 | .638293 | 45.33 
 21 | 7.459495 | 55.33 | 7.460748 | 55.48 || 21 | 7.639117 | 44.98 |7.641013 45.18 
 22 .462815 | 55.42 .464077 | 55.28 || 22 .641816 | 44.87 | .648724 | 45.07 
 23 .466122 | 54.92 .467394 | 55.08 || 23 .644508 | 44.72 | .646428 | 44.90 
 24 .469417 | 54.70 -470699 | 54.87 || 24 -647191 | 44.57 | .649122 | 44.78 
 25 .472699 | 54.50 .473991 | 54.65 || 25 -649865 | 44.45 | .651809 | 44.65 
 26 .475969 | 54.28 477270 | 54.47 |) 26 -652532 | 44.30 | .654488 | 44.50 
 27 .479226 | 54.10 .480538 | 54.25 || 27 .655190 | 44.17 | .657158 | 44.37 
 28 .482472 | 53.88 .4838793 | 54.05 28 .657840 | 44.05 | .659820 | 44.93 
 29 -485705 | 53.7 .487036 | 53.85 |} 29 .660483 | 43.90 | .662474 | 44.12 
 30 .488927 | 53.48 -490267 | 58.67 || 30 .663117 | 48.77 | .665121 | 43.97 
 31 | 7.492136 | 53.28 | 7.493487 | 53.45 || 81 | 7.665743 | 43.63 |'7.667759 43 .83 
 82 495333 | 53.10 .496694 | 53.27 || 32 .668361 | 43.50 | .670889 | 43.7: 
 33 498519 | 52.90 .499890 | 53.07 || 33 .670971 | 43.38 | .673012 | 43.57 
 34 501693 |} 52.7% .503074 | 52.88 || 34 .678574 | 43.23 | .675626 | 43.45 
 35 504856 | 52.52 006247 | 52.68 385 .676168 | 43.12 | .678233 | 43.33 
 36 508007 | 52.33 .509408 | 52.50 || 36 .678755 | 42.98 | .680833 | 43.18 
 37 511147 | 52.13 .512558 | 52.32 || 37 .681834 | 42.87 | .683424 | 43.07 
 38 -614275 | 51.95 .015697 | 52.12 || 38 .683906 | 42.73 | .686008 | 42.95 
 39 517392 | 51.7 .618824 | 51.93 389 .686470 | 42.60 | .688585 | 42.89 
 40 -620498 | 51.58 -621940 | 51:77 40 .689026 | 42.48 | .691154 | 42.68 
 41 | 7.523593 | 51.40 | 7.525046 | 51.57 || 41 | 7.691575 | 42.35 7.698715 | 42.57 
 42 526677 | 51.22 .528140 | 51.38 || 42 .694116 | 42.%3 | .696269 | 42.43 
 43 529750 | 51.03 081220 |°51.22°'| 43 .696650 | 42.12 | .698815 | 49.33 
 44 532812 | 50.85 .534296 | 51.02 44 .699177 | 41.98 | .701855 | 42.90 
 45 .5385863 | 50.68 .537857 | 50.85 || 45 |. .701696 | 41.87 | .'703887 42.07 
 46 .538904 | 50.50 .540408 | 50.68 || 46 - 704208 | 41.73 | .706411 | 41.97 
 ik .541934 | 50.32 .543449 | 50.50°|| 47 | .706712 | 41.63 | . 708929 | 41.83 
 48 .5449538 | 50:15 .546479 | 50.33 48 - 709210 | 41.50 | .711439 | 44.72 
 49 .547962 | 49.98 .549499 | 50.15 || 49 .711700 | 41.88 | .718942 | 41.60 
 50 .550961 | 49.80 .552508 | 49.98 || 50 714183 | 41.27 | .716438 | 41.48 
 51 | %.553949 | 49.63 | 7.555507 | 49.80 || 51 | 7.716659 | 41.15 |'7. 718997 41.37 
 52 .556927 | 49.48 .558495 | 49.65 52 .719128 | 41.03 | .721409 | 41.95 
 53 .559895 | 49.28 .561474 | 49.47 || 53 .721590 | 40.92 | .723884 | 41.13 
 54 .562852 | 49.13 .564442 | 49.32 || 54 . 724045 | 40.80 | .726352 | 41.02 
 55 -565800 | 48.95 .567401 | 49.13 55 -726493 | 40.68 | .728813 | 40.90 
 56 .568737 | 48.80 .5703849 | 48.98 56 728934 | 40.57 | .731267 | 40.7 
 57 .571665 | 48.63 .5738288 | 48.82 || 57 . 731368 | 40.47 | .733714 | 40.68 
 58 5745838 | 48.47 .576217 | 48.65 || 58 . 733796 | 40.388 | .786155 | 40.57 
 59 .577491 | 48.30 .579136 | 48.48 59 -736216 | 40.23 | .738589 | 40.45 
 7.580389 | 48.15 | 7.582045 | 48.33 || 60 | 7.738630 | 40.13 |'7.741016 40.33 
 
| 
 
 DAS mMeWMS | ~ 
 
 Pee a I en i ee a ae emcee oa ERE a al SE ca EN RAMI ERRATA DRA a 
 CWJCW ICH OCH an Cru a ce g 
 CPG 02 = t oe tS Se ”) < 
 
 Vers. 
 
 7.738630 
 . 741038 
 . 748488 
 . 745832 
 . 748219 
 750600 
 - 752974 
 (55842 
 757703 
 -'760058 
 762406 
 7.764749 
 767084 
 . 769414 
 ART 
 774054 
 776365 
 - 778670 
 - 780968 
 783261 
 785547 
 . 787828 
 . 790102 
 792371 
 .794633 
 .796890 
 .799141 
 .801885 
 . 803625 
 .805858 
 808086 
 
 .810308 
 .812524 
 .814734 
 .816939 
 .819139 
 .8213382 
 . 823521 
 .8257038 
 .8278380 
 . 8380052 
 
 | 7.932218 
 
 . 
 
 “J 
 
 ~I 
 
 “851475 
 
 7.853589 
 .855697 
 .857800 
 .859898 
 .861991 
 864079 
 866162 
 . 868240 
 .870313 
 
 % 872381 
 
 Ex. sec. 
 
 | 40.13 | 7.741016 
 
 748486 
 . 745850 
 . 748258 
 ~750658 
 "753052 
 
 . 755440 
 
 (57821 
 760196 
 762565 
 764927 
 T67282 
 - 769632 
 T1975 
 774312 
 . 776643 
 778968 
 . 781286 
 . 783599 
 785905 
 ~ 788206 
 7.790500 
 792789 
 795071 
 797348 
 799619 
 801884 
 804143 
 806397 
 808644 
 810886 
 
 7.813123 
 .815353 
 .817578 
 .819798 
 . 822012 
 . 824220 
 826423 
 . 828620 
 . 830812 
 .882999 
 
 7.885180 
 . 837356 
 .839526 
 .841691 
 .843851 
 . 846005 
 .848155 
 .850299 
 .852437 
 .854571 
 
 856700 
 858823 
 .860942 
 .863055 
 865163 
 867266 
 869365 
 871458 
 873546 
 7.875630 
 
 =z 
 
 ~ 
 
 AND EXTERNAL SEUANTS. 
 
 Vers. 
 
 | 7.872881 
 
 874444 
 876502 
 878555 
 880603 
 882647 
 . 884686 
 886720 
 888749 
 89077 
 892793 
 
 7.894808 
 896818 
 898824 
 - 900825 
 902821 
 .904813 
 - 906800 
 908783 
 
 -910761 | 
 
 912735 
 
 7.914704 
 . 916668 
 . 918629 
 . 920584 
 922536 
 924483 
 926425 
 . 928364 
 930297 
 982227 
 
 7.984152 
 936073 
 . 937990 
 . 939903 
 941811 
 948715 
 . 945615 
 947511 
 -949402 
 951290 
 
 7.953173 
 955052 
 .956928 
 958799 
 . 960666 
 - 962529 
 . 964388 
 966243 
 . 968094 
 969941 
 
 7.971785 
 973624 
 975459 
 977291 
 .979118 
 .980942 
 982762 
 984578 
 . 986391 
 
 7.988199 
 
 Ex, sec. | 
 
 % 875630, | 
 877708 
 879782 
 881851 
 883915 
 885974 
 888029 
 890078 
 892123 
 894164 
 896199 
 
 7.898280 
 . 900256 
 902278 
 904295 
 . 906307 
 -908315 
 .910319 
 912317 
 .914312 
 . 916802 
 
 7.918287 
 920268 
 922245 
 924217 
 926184 
 . 928148 
 .930107 
 . 932062 
 . 934012 
 .935958 
 
 7.937900 
 939838 
 941772 
 .948701 
 . 945626 
 947547 
 949464 
 .951376 
 958285 
 955189 
 
 7.957090 
 . 958986 
 . 960878 
 962767 
 . 964651 
 . 966531 
 - 968408 
 - 970280 
 972148 
 .974018 
 
 7. 975874 
 977730 
 979583 
 .981432 
 9838277 
 . 985119 
 . 986956 
 988790 
 . 990620 
 
 30.08 17.992446 
 
TABLE XXVI.—LOGARITHMIC VERSED SINES 
 
 <—lS 
 8° 9° 
 Vers. | D.1’. | Ex. sec. | D. 1’. ||’ Vers. | D.-1". | Ex. see./D. 1°, 
 7.988199 | 30.08 | 7.992446 | 80.388 || 0 | 8.090317 | 26.72 |8.095697 | 27.05 
 .990004 | 380.02 - 994269 | 30.82 1 -091920 | 26.68 | .097320 | 27.02 
 .991805 | 29.95 .996088 | 30.25 2 .093521 | 26.63 | .098941 | 26.97 
 . 993602 | 29.88 .997903 | 30.18 3 .095119 | 26.58 | .100559 | 26.92 
 995395 | 29.83 | 7.999714 | 30.13 4 .096714 | 26.52 | .102174 | 26.87 
 997185 |.29.7°7 | 8.001522 | 30.07 5 .098305 | 26.48 | .103786 | 26.82 
 7.998971 | 29,72 -003326 | 30.00 6 .099894 | 26.43 | .105395 | 26.77 
 8.000754 | 29.63 .005126 | 29.95 vg -101480 | 26.40 | .107001 | 26.7% 
 .002582 | 29.60 -006923 | 29.88 8 . 103064 | 26.33 | .108605 | 26 67 
 -004308 | 29.52 .008716 | 29.83 9 -104644 | 26.28 | .110205 | 26.63 
 .006079 | 29.47 .010506 | 29.7% 10 106221 -| 26.25 | .111803 |. 26.58 
 8.007847 | 29.40 | 8.012292 | 29.% j 11 | 8.107796 | 26.18 |8.113398 | 26.53 
 .009611 | 29.35 .014074 | 29.65 |} 12 .109867 | 26.15 | .114990 |. 26.48 
 .011872 | 29.28 .015853 | 29.58 || 13 - 110936 -|| 26.10 | .116579 |. 26.45 
 .013129 | 29.22 .017628 | 29.53 || 14 112502 | 26.05 | .118166'|. 26.38 
 .014882 | 29.17 .019400 | 29.47 || 15 -114065 |.26.00 | .119749 | 26.35 
 .016632 | 29.10 .021168 | 29.42 || 16 115625 |-25.95 | .121330 | 26.30 
 .018878 | 29.05 022933 | 29.85 || 17 117182 || 25.92 | .122908 | 26.25 
 .020121 | 29.00 024694 | 29.30 || 18 .118787 | 25.87 | .124483 | 26.2% 
 .021861 | 28.93 -026452 | 29.23 || 19 120289 | 25.82 | .126056 | 26.17 
 028597 | 28.87 | 028206 | 29.18 || 20 | .121888 | 25.77 | .127626 | 26.12 
 8025329 | 28.82 | 8.029957 | 29.13 || 21 | 8.128384 | 25.72 |8.129193 | 26.07 
 -027058 | 28.75 .081705 | 29.07. || 22 124927 | 25.68 | .130757 | 26.02 
 .028783 | 28.70 033449 | 29.00 || 23 126468 | 25.65 | .1382318 | 25.98 
 .030505 | 28.65 .035189 | 28.97 || 24 128006 | 25.58 | .188877 | 25.93 
 -032224 | 28.58 036927 | 28.90 || 25 129541 | 25.55 | .185433 |. 25.90 
 -033939 | 28.53 .038661 | 28.88 || 26 .181074 | 25.50 | .186987 |. 25.85 
 .035651 | 28.47 .040391 | 28.78 7 . 182604 | 25.45 | 2188588 |.25.80 
 .037359 | 28.42 -042118 | 28.7 28 .184131 | 25.40 | .140086 |. 25.75 
 -039064 | 28.37 .043842 | 28.68 || 29 - 185655: | 25.37 | .141681 |. 25.7% 
 -040766 | 28.30 | .045563 | 28.62 || 830 | 137177 | 25.82 | .148174 |.95 67 
 8.042464 | 28.25 | 8.047280 | 28.57 || 31 | 8.138696 | 25.97 |8. 144714 | 25.63 
 .044159 | 28.20 048994 | 28.50 || 32 - 140212 | 25.23 | .146252 | 25.58 
 .045851 | 28.13 .050704 | 28.47 || 33 141726 | 25.18 | .147787 | 25.53 
 .047539 | 28.08 052412 | 28.40 || 34 148237 | 25.13 | .1493819 | 25.50 
 .049224 | 28.03 .054116 | 28.35 || 385 -144745 | 25.10 | .150849 | 25.45 
 -050906 | 27.98 -055817 | 28.28 || 36 . 146251 | 25.05 | .152876 | 25.40 
 052585 | 27.92 .057514 | 28.25 || 37 147754 | 25.02 | .158900 | 25.37 
 054260 | 27.87 -059209 | 28.18 || 38 . 149255 | 24.95 | .155422 | 25.33 
 055932 | 27.82 .060900 | 28.13 || 39 -150752 | 24.93 | .156942 | 25.27 
 057601 | 27.7 .062588 | 28.08 || 40 152248 | 24.88 | .158458 | 25.25 
 8.059266 | 27.72 | 8.064273 | 28.03 || 41 | 8.153741 | 24.88 |8.159973 | 95.18 
 -060929 | 27.65 065955 7.97 || 42 .155281 | 24.7 .161484 | 25.17 
 062588 7.60 067638 7.93 || 43 .156718 | 24.75 | .162994 | 25.10 
 .064244 | 27.55 . 069809 7.87 || 44 . 158203 4.72 | .164500 | 25.07 
 .065897 | 27.48 .070981 | 27.82 || 45 .159686 | 24.67 | .166004 | 25.08 
 -067546 | 27.45 072650 7.77 |) 46 .161166 | 24.62 | .167506 | 24.98 
 -069193 | 27.38 .074316 | 27.4% 47 -162643 | 24.58 | .169005 | 24.95 
 -070836 | 27.33 075979 | 27.67 || 48 .164118 | 24.53 | .170502 | 24.90 
 072476 | 27.30 077639 | 27.60 || 49 165590 | 24.50 | .171996 | 24.87 
 -OV4114 | 27.23 | .079295 | 27.57 || 50 | 1167060 | 24.45 | 173488 | 24.82 
 8.075748 | 27.18 | 8.080949 | 27.52 51 | 8.168527 | 24.42 (8.174977 | 24.78 
 -OT7379 | 27.18 -082600 | 27.45 || 52 .169992 | 24.37 | .176464 | 24.7% 
 .079007 | 27.07 -084247 | 27.42 3 171454 | 24.83 | .177948 | 24.7 
 -080631 | 27.03 085892 | 27.37 || 54 172914 | 24.80 | .179430 | 24.65 
 082253 | 26.98 087584 | 27.30 || 55 174872 | 24.25 | .180909 | 24.62 
 083872 | 26.93 089172 | 27.27 || 56 175827 | 24.20] .182386 | .24.58 
 .085488 | 26.87 .090808 | 27.20 v 177279 | 24.17 | .183861 | 24.53 
 -087100 | 26.83 -092440 | 27.17 || 58 178729 | 24.13 | .185883 | 24.50 
 .088710 | 26.78 -094070 | 27.12 || 59 -180177 | 24.08 | .186803 | 24.47 
 8.090317 | 26.72 | 8.095697 | 27.05 || 60 8.181622 | 24.05 18.188271 | 24.42 
 
~ 
 
 = 
 So MIM URwWDWKo| 
 
 ao 
 
 Z 
 
 Vers. 
 
 . | Ex. sec. | 
 
 | 8.181622 | 2 
 
 .183065 
 . 184505 
 .185943 
 .187379 
 .188812 
 
 190243 | 23.8 
 191671 
 -193097 | 2 
 .194521 | 2 
 195942 
 
 197361 
 “DOe78 
 
 "200192 | 2 
 “201604 
 
 . 203014 
 .204421 
 .205826 
 . 207229 
 - 208630 
 .210028 
 -211424 
 .212318 
 .214209 
 -215599 
 .216936 
 .218371 
 .219753 
 .221183 
 222512 
 223888 
 
 225261 
 
 226633 
 . 228002 
 . 229369 
 
 -230735. | 
 
 232097 
 233458 
 .284317 
 236173 
 
 -20102 27 j 
 
 238830 
 . 240230 
 
 241578 
 242924 
 
 244267 
 
 . 245609 
 
 . 246948 
 
 218286 
 219621 | 
 250955 
 
 § 252286 
 
 . 253615 
 . 204942 
 2562 68 
 201) 591 
 . 258912 
 260231 
 . 261548 
 
 - 262863 
 
 60 | $.261176 
 
 ) Ww d 
 
 8.188271. | 
 
 1897 36 
 . 191198 
 -192659 
 194117 
 195572 
 197025 
 198476 
 . 199925 
 -201371 
 202815 
 
 8.204257 
 205696 
 .207133 
 208568 
 .210001 
 
 211451 
 
 6212859 
 .214285 
 215708 
 217130 
 8.218549 
 -219966 
 221i BRO 
 222793 
 
 word 
 
 .224203 
 .225611 
 . 227017 
 
 222424 
 . 229822 
 . 231221 
 8.232619 
 .234014 
 . 235407 
 .236797 
 . 238186 
 . 239572 
 anand 
 2423: 
 3449 
 
 .245097 
 
 8.246473 
 . 247847 
 ~ 249219 
 . 250589 
 
 . 251957 
 
 . 258822 
 . 254686 
 .206047 
 .257407 
 .208764 
 8.260120 
 .261473 
 . 262825 
 .264174 
 . 265522 
 . 266867 
 .268211 
 .269552 
 .270892 
 8.272229 
 
 AND EXTERNAL SECANTS. 
 
 eanerzaer > 
 
 (o 2) 
 
 Oacan Meshes 
 RA RRB LHKS 
 
 Ex, Sec. | 
 
 '8 .272229 
 . 273565 
 | .274898 
 276230 | 
 .277560 
 
 . 278888 
 . 280213 
 .2815387 
 . 282859 | 
 . 284179 
 . 285498 
 
 '8.286814 
 288128 
 289441 
 290751 
 
 292060 
 293367 
 294672 
 295905 
 
 297276 
 
 "29857 
 
 j0.299873 
 .801169 
 802463 
 3803755 
 3805045 
 806334 
 .807620 
 .3808905 
 .310188 
 .311469 
 
 812749 
 .814026 
 .815802 
 .316576 
 .317849 
 .819119 
 820388 
 .821655 
 .322920 
 .824183 
 
 325445 
 329705 
 327964 
 820220 
 830475 
 881728 
 332980 
 334229 
 835477 
 836724 
 '8.337968 
 339211 
 840453 
 841692 
 842030 
 344166 
 343401 
 846634 
 347865 | 20. 
 2 8.349095 | 20.47 
 
 ore 2>HAS 33 ( ( 
 z 5 = - G 
 OO 
 
12° 
 
 y Vers. | D. 1’ 
 0 | 8.339499 | 20.02 
 1 .840700 | 20.00 
 2} 3841900 | 19.95 
 3 .343097 | 19.95 
 4 .844294 | 19.90 
 5 .3845488 | 19.88 
 6 .846681 | 19.85 
 & .847872 | 19.82 
 8 .849061.| 19.80 
 9 .850249 | 19.7 
 10 .301485 | 19.7 
 | 11 | 8.352620 | 19.72 
 12 .853803..; 19.68 
 ! 13 004984 | 19.67 
 14 .806164 | 19.63 
 15 .857342 | 19.60 
 16 -3808518 | 19.58 
 7 .809693 | 19.55 
 18 .360866.} 19.53 
 19 .3862083 | 19.50 
 20 .868208 | 19.48 
 21 | 8.364377 | 19.438 
 22 .3805548 | 19.48 
 2é .866709 | 19.38 
 2! 067872 | 19.37 
 25 .869034 | 19.35 
 26 .870195 | 19.382 
 27 .38113854 | 19.28 
 28 872511 | 19.27 
 29 310067. | 19.25 
 380 .874822 | 19.20 
 31 | 8.875974 | 19.18 
 32 804125 | 19.17 
 28 o1e270 | 19.13 
 3¢ .379423.| 19.12 
 35 .880570 | 19.08 
 36 .881715 | 19.05 
 37 .882858 | 19.03 
 38 .884000 | 19.02 
 39 .885141 | 18.98 
 40 .886280 | 18.95 
 41-| 8.387417 | 18.93 
 42 .888553.| 18.92 
 43 .889688 | 18.88 
 44 -890821 | 18.85 
 45 .891952. | 18.838 
 46 .893082 | 18.82 
 47 .894211.| 18.4 
 48 .895888 | 18.75 
 4s -096463 .| 18.7% 
 50 397587 | 18.72 
 51 | 8.898710 | 18.68 
 52 .38998381 | 18.67 
 53 .400951 | 18.63 
 54 .402069 | 18.62 
 55 .408186 | 18.58 
 56 -404301 | 18.57 
 57 .405415 | 18.53 
 58 .406527 | 18.52 
 59 .407688 | 18.50 
 60 | 8.408748 | 18.47 
 
 Ex. sec. | SOPs ih a 
 
 | 
 
 TABLE XXVI.—LOGARITHMIC VERSID SINES 
 
 13° 
 
 f Vers. | D. 1". | Ex. sec.jD. 1”. 
 | | | | 1 
 
 8.849095 | 20.47 || 0 | 8.408748 | 18.47 ‘8.420024 | 18.95 
 .350323 | 20.43 || 1] .409856 | 18.43 | .421161 | 18.93 
 
 851549 | 20.42 || 2] .410962 | 18.42 | .422297.) 18.90 
 
 852774 | 20.38 || 3] 412067 | 18.40 | .423431 | 18.88 
 
 .853997 | 20.35 |} 4 | .413171 | 18.88 | .424564.| 18.87 
 
 .855218 | 20.33 || 5 | .414274 | 18.35 | .425696 | 18.83 
 
 856438 | 20.30 || 6 | .415375 | 18.32.| .426926 | 18.82 
 
 857656 | 20.28 || 71 .416474 | 18.30.) .427955.| 18.80 
 
 858873 | 20.25 || 8 | .417572 | 18.28-| .429083.! 18.77 
 
 860088 | 20.22 || 9 | .418669 | 18.25 | .430209 | 18.75 
 
 .861301 | 20.20 |! 10 | .419764 | 18.23 | .431334 | 18.73 
 
 8.862513 | 20.18 || 11 | 8.420858 | 18.22 |8:432458 | 18.70 
 863724 | 20.13 || 12 | 1421951 | 18.18 | .423580.| 18.67 
 
 264982 | 20.12 || 13 | 1423042 | 18.17 | .484700.| 18.67 
 
 .366139 | 20.10 || 14 | .424132 | 18.13 | .485820 | 18.63 
 
 867345 | 20.07 || 15 | 1425220 | 18.12 | .436938.| 18.62 
 
 .868549 | 20.03 || 16 | .426307 | 18.10°; .438055 | 18.58 
 
 869751 | 20.02 || 17 | 1427393 | 18.07 | .439170 | 18.57 
 
 870952 | 19.98 ||.18 | 1428477 | 18.05 | .440284 | 18.55 
 
 872151 | 19.95 || 19 | .429360 | 18.02 | .441397 | 18.53 
 
 873348 | 19.95 || 20 | .430641 | 18.02 | .442509 | 18.50 
 
 8.374545 | 19.90 |! 21 | 8.431722 | 17.97 18.443619 | 18.47 
 875739 | 19.88 || 22] 1432800 | 17.97 | .444797 | 18.47 
 
 .376982 | 19.85 || 23} .433878 | 17.93 | .445835 | 18.43 
 
 878123 | 19.83 || 24 | .434954 | 17.92 | .446941.| 18.42 
 
 .879813 | 19.82 ;; 25 | .436029 | 17.68 | .448046 | 18.88 
 
 880502 | 19.78 || 23 | .437102 | 17.87 | .449149 | 18.38 
 
 881689 | 19.75 || 27] .438174 | 17.85 | .450252 | 18.35 
 
 882874 | 19.73 || 28] .439245 | 17.82 | .451353 | 18.32 
 
 .884058 | 19.70 || 29} .440814 | 17.80 | 452452 | 18.32 
 
 885240 | 19.68 |! 8 441382 | 17.78 | .453551 | 18.28 
 
 8.386421 | 19.65 || 81 | 8.442449 | 17.75 8.454648 | 18.95 
 .887600 | 19.63 |; 82 | 1448514 |-17.73 | .455748.| 18.25 
 
 883778 | 19.60 || 83} 1444578 | 17.72 | .456838 | 18.22 
 
 .889954 | 19.58 || 84] 445641 | 17.68 | .457931.| 18.20 
 
 .891129 | 19.55 || 25 | .446702 | 17.68 | .459023.| 18.18 
 
 892302 | 19.53 || 86] .447763 | 17.63 | .460114.| 18.15 
 
 893474 | 19.50 || 87 | .448821 | 17.63 | .461203 | 18.13 
 
 .894644 | 19.48 |) 88°] .449879 | 17.62 | .462201 | 18.12 
 
 895813 | 19.45 |; 39 | .4509385 | 17.58 | .463378 | 18.10 
 
 .3896980 | 19.43 |} 40 | .451990 | 17.55 | .464464 | 18.07 
 
 8.398146 | 19.42 || 41 | 8.453048 | 17.55 |8.465548 | 18.05 
 .899311 | 19.38 || 42] .454096 | 17.52 | .466631 | 18.03 
 
 .400474 | 19.85 |} 43 | .455147 | 17.48 | .467713 | 18.00 
 
 .401635 | 19.83 || 44] .456196 | 17.48 | .468793 | 18.00 
 
 .402795 | 19.82 || 45 | .457245 | 17.45 | .469873 | 17.97 
 
 .403954 | 19.28 || 46 |. .458292 | 17.438 | .470951 | 17.95 
 
 .405111 | 19.27 || 47 | .459388 | 17.40 | .472028 | 17.92 
 
 .406267 | 19.23 || 48 | .460382 | 17.40 | .473103 | 17.90 
 
 .407421 | 19.22 || 49 | .461426 | 17.37 | .474177 | 17.90 
 
 .408574 | 19.18 |} 50 | .462468 | 17.35 | .475251 | 17.85 
 
 8.409725 | 19.17 || 51 | 8.463509 | 17.32 8.476822 | 17.85 
 410875 | 19.13 || 52) .464548 | 17.30 | .477393 | 17.88 
 
 .412023 | 19.13 || 58] .465586 | 17.28 | .478463 | 17.80 
 
 .418171 | 19.08 || 54 | .466623 | 17.27 | .479531 | 17.78 
 
 .414316 | 19.08 || 55 | .467659.| 17.23 | .480598 | 17.77 
 
 415461 | 19.03 || 56 | .468693 | 17.23 | .481664 | 17.73 
 
 .416603 | 19.03 |! 57 | .469727 | 17.20] .482728 | 17.73 
 
 417745 | 19.00 || 58 | .470759 | 17.17 | .483792 | 17.70 
 
 .418885 | 18.98 || 59 | .471789 | 17.17 | .484854 | 17.68 
 
 8.420024 | 18.05 || GO | 8.472819 | 17.13 |8.485915 | 17267 
 
AND EXTERNAL SECANTS. 
 
 ~ 
 
 Vers. 
 
 . | Ex, sec. 
 
 Vers. 
 
 PaielUx. SEC 
 
 COMRIAOUFRWWH OS 
 
 —" 
 
 op oves crew enepenee WIWIWIWIWIWIVISI® Wr : 
 
 | 8.472819 
 
 473847 
 474874 
 .475900 
 
 476925 
 477948 
 
 .478970 
 -479991 
 481011 
 .482029 
 483046 
 
 | 8.484062 
 
 .485077 
 .486091 
 .487103 
 .488115 
 .489125 
 
 490134. | 
 
 .491141 
 .492148 
 .493153 
 494157 
 .495160 
 .496162 
 497162 
 .498162 
 .499160 
 .500157 
 
 5011538 
 502148 | 
 
 . 0038142 
 
 504134 
 
 505125 
 
 .506116 
 .507108 
 
 .50§092 |! 
 
 .509079 
 .510065 
 .511049 
 .512033 
 .518015 
 .513996 
 
 514976 | 
 
 .515955 
 .516932 
 ~517909 
 
 .518684 || 
 
 .519859 
 
 5203832 
 
 .521804 
 Sots 
 
 beetle 
 523745 
 Ov4s714 
 020682 
 
 526648 
 
 .527'614 
 .528578 
 .D29542 
 .580504 
 .531465 
 
 8.532425 
 
 Ww 09 Cl GO MW Oo 
 
 ie) 
 
 Say AZALI +3 FH MHDMHOO 
 
 Lr 
 
 8.485915 
 .486975 
 .488033 
 .489091 
 .490147 
 .491202 
 .492256 
 .493308 
 .494360 
 .495410 
 .496459 
 21 8.497507 
 498554 
 .499600 
 .500644 
 .501687 
 .502730 
 .5038771 
 .501810 
 .505849 
 .506887 
 . 507923 
 .508958 
 ~5099938 
 .511026 
 .512057 
 .518088 
 .514118 
 .515146 
 .516174 
 .517200 
 518225. | 
 ~D19249 
 
 »520272 
 
 .527402 
 528416 
 
 529429 
 .530441 
 .581452 
 .532462 
 583471 
 .pd4478 
 580485 
 . 536490 
 .531495 
 .5388498 
 .589501 
 ~)46502 
 ~541502 
 
 .542501 
 .548499 
 544497 
 
 545493 
 
 546488 
 
 8.547482 
 
 YAPAZ PAY APY -I-I-V- 
 
 Pm ek ek ek et ek fel fet Pk tp 
 
 > 
 fa ee pe ee et ep 
 Oe Re eRHEHwtDw wwwsd 
 
 SAPARD a ne - AAAES 
 
 ‘yal: 
 oO 
 cS 
 
 Qn | 
 
 we 
 
 Co 
 
 OF Ee) 
 
 —" 
 
 Peek ped Pd et 
 > OUH CO OD mt OOD 
 
 DOWWwWRDWwwowt 
 
 or a Or & OT cy 
 FDO SO 
 
 oro 
 B= 
 
 8.582425 | 
 533384 
 . 84342 
 .085299 
 .586255 
 .5387210 
 .588163 
 .539116 
 .540068 
 .541018 | 
 .541968 
 .542916 
 .548863 
 .544810 
 
 od-4-4 
 
 .545755 
 .546699 
 .547642 
 548584 | 
 549525 
 .550465 
 .551404 
 
 55887 
 
 .559809 
 
 .560738 
 8.561666 
 . 562592 
 .563518 
 .064443 
 .565367 
 .566289 
 2567211 
 .5681382 
 .569052 | 
 .569970 | 
 .5T0888 
 571805 
 het 
 573636 
 
 579104 | 
 580012 
 580919 
 581825 
 
 582730 
 
 533634 
 
 B3I537 | 
 585440 
 586341 
 STH 
 8.538141 | 
 
 ee 
 
 OrOvorgorororvorw orc 
 
 = 
 — 
 
 ae) 
 
 Orr ot ur Ui or OF OF UT OU. OT OT Ore 
 
 Fe OTOL. OT OF ST. Gr vor 
 
 8.547482 | 
 548474 | 
 “549466 
 550457 
 .BD1447 | 
 552436 
 553424 
 554410 | 
 555396 
 556381 
 557364 
 558347 
 “559329 
 560209 
 561289 
 
 562267 
 “563245 
 564222 
 565197 
 566172 
 567145 
 568118 
 569090 
 570060 
 571030 
 571999 
 572966 
 573933 
 .574899 
 _BT5864 
 576827 
 577790 
 
 578752 
 
 (0A 
 
 579713 | 
 .580673 
 .581682 
 .582590 
 588547 
 .584503 
 .585458 
 .586412 
 .5873865 
 .588318 
 + 2589269 
 
 .590219 
 .591169 
 .592117 
 598065. | 
 ,594012 
 .594957 
 595902 
 
 8.596846 
 | 599789 | 
 .598731 
 599672 
 600612 
 601551 
 602490 
 
 603427 
 
 604363 
 97 |B.605299 | 
 
 WWW RIOOWWw 
 
 = 
 29) 
 
 > Cd Dd Cd Dd Sd G2 SH? OI S 
 CO ODA pe ee Ct OF OT OF 
 
 WITRINVIO WN 
 
 > D> 3 2 > > > > 7D 
 hk pak ee DD BA wy 009 G9 09 
 
 wo 
 
 TOCRIDONW OT 
 
 Or =3-2 7 =2- 
 
 > D 
 RS) 
 
| TABLE XXVI.—LOGARITHMIC VERSED SINES 
 
 16° i 
 
 / Vers. | DA ol Exe sec.) Dei? af Vers. | D. 1’. | Ex. sec.|D. 1’ 
 | } | 
 
 0 | 8.588141 | 14.97 | 8.605299 | 15.58 0 | 8.640434 | 14.08 '8.659838 | 14.72 
 if .589039 | 14.95 .606234 | 15.55 1 .641279 | 14.07 | .660721 | 14.72 
 2 .589936 | 14.95 .607167 | 15.55 2 642123 | 14.05.} .661604 | 14.70 
 3 .590833 | 14.93 .608100 | 15.53 3 642966 | 14.05 | .662486 | 14.68 
 4 .591729 | 14.90 .609032 | 15.52 4 .648809 | 14.02 | .663367 | 14.68 
 5 . 592623 | 14.90 .609963 | 15.50 5 .644650 | 14.02 | .664248 | 14.65 
 6 .593517 | 14.88 .610893 | 15.50 6 .645491° | 14.00 | .665127 | 14.65 
 a .594410 | 14.87 .611823 | 15.47 7 .646331 | 18.98 | .666006 | 14.63 
 8 .595302 | 14.83 612751) 15.45 8 .647170 | 18.97 | .666884 ; 14.62 
 : 9 .596192 | 14.83 .613678 | 15.45 9 .648008 | 18.95 | .667761 | 14.60 
 iH 10 .597082 | 14.82 .614605 | 15.438 10 .648845 | 18.95 | .668637 | 14.60 
 11-| 8.597971 | 14.82 | 8.615531 | 15.42 |] 11 | 8.649682 | 13.93 |8.669513 | 14.58 
 12 .598860 | 14.78 .616456 | 15.38 12 .650518 | 13.92 | .6703888 | 14.57 
 13 | .599747 | 14.7 .617379 | 15.38 || 13 .651353 | 13.90 | .671262 | 14.55 
 14 .600633 | 14.75 .618302 | 15.38 ||} 14 .652187 | 13.88 | .672185 | 14.55 
 15 .601518 | 14.75 .619225 | 15.35 15 .653020 | 138.87 | .673008 | 14.52 
 Pat (nt | 16 .602403 | 14.72 .620146 | 15.33 || 16 .653852 | 18.87 | .6738879 | 14.52 
 Fah 17 .603286 | 14.72 .621066 | 15.33 17 .654684 | 18.85 | .674750 | 14.50 
 jaan 13 .604169 | 14.70 .621986 | 15.30 18 6655515 | 138.83 | .675620-| 14.50 
 ME 19 .605051 | 14.67 .622904 | 15.380 || 19 .656345 | 138.82 | .676490 | 14.47 
 aaa AS .605931 | 14.67 .623822 | 15.28 || 20 .657174 | 13.82 | .677358 | 14.47 
 21 | 8.606811 | 14.65 | 8.624739 | 15.27 |) 21 | 8.658003 | 13.78 |8.678226 | 14.45 
 Aaa 22 .607690 | 14.63 .625655 | 15.235 || 22 .658839 | 13.7 .679098 | 14.45 
 a i 23 .603563 | 14.62 62657 15.23 |} 23 .659657 | 13.77 | .679960 | 14.42 
 ta reas .609 445 | 14.60 627484 | 15.23 || 2k .660483 | 18.75 | .680525 ) 14.42 
 25; .610321 | 14.60 .628398 | 15.20 || 25 .661308 | 13.7: .681690 || 14.40 
 26 | .611197 | 14.57 .629310 | 15.20 || 26 .662132' | 13.73 | .682554 | 14.38 
 27 .612071 | 14.57 .630222 | 15.18 27 .662956 | 18.72 | .683417 | 14.38 
 23 .6129145 | 14.53 .631183 | 15.17 2 .66377 13.70 | .684280°) 14.35 
 29 613317 | 14.53 .639943 | 15.15 29 .664601 | 18.68 | .685141 | 14.35 
 i; 30 .614639 | 14.52 .632952 | 15.13 30 .665422 | 13.67 | .686002 | 14.35 
 ind 31 | 8.615560 | 14.50 | 8.633360 | 15.13 || 31 | 8.666242 | 13.67 |8.686863 | 14.32 
 H 32 .616439 | 14.48 .63£753 | 15.10 || 32 .667032 | 13.65 | .687722 | 14.32 
 33 .617299 | 14.47 635674 | 15.10 33 .657881 | 13.63 | .688581 | 14.30 
 ed (et| Be! .618167 | 14.45 .636530 | 15.08 || Bt .668699 | 18.62 | .689439 | 14.28 
 Bay Wei] 3) .619034 | 14.45 .6374385 | 15.07 35 .669516 | 13.66 | .690296 | 14.28 
 i it 33 619991 | 14.42 .638339 | 15.05 35 .670332 | 13.60 | .691153 | 14.25 
 a 37 .620765 | 14.42 .639292 | 15.05 || 37 .671148 | 18.58 | .692008 | 14.25 
 bin 33 .621631 | 14.40 .640195 | 15.02 || 38 .671953 | 13.57 | .692863°) 14.25 
 a 30 .622495 | 14.38 641095 | 15.02 39 672777 | 13.55 | .6938718 | 14.22 
 j | 4) .623358 | 14.37 .641997 | 15.00 40 .673590 | 18.55 | .694571 °| 14.22 
 Hed 41 | 8.624220 | 14.35 | 8.642897 | 14.98 || 41 | 8.674403 | 13.53 |8.695424 | 14.20 
 al 42 .625081 | 14.33 .643796 | 14.97 42 .675215 | 138.52 | .69627 14.18 
 13} 43 .625941 | 14.33 644694 | 14.95 43! .676026 | 13.50 | .697127 | 14.18 
 NATE 44 .626801 | 14.30 .645591 | 14.95 44 .676836 | 13.48 | .697978 | 14.17 
 45 | .627659 14.30 .646488 | 14.93 45 .677645 | 138.48 | .698828 | 14.15 
 46 .628517 | 14.28 .647334 | 14.92 || 46 .678454 | 18.47 | .699677 | 14.13 
 47 | .629374 14.27 .648279 | 14.90 47 .679262 | 18.45 | .700525 |) 14.13 
 43 .6302380 | 14.25 .649173 | 14.88 48 .680069 | 18.43 | .701373 | 14.12 
 49 .631085 | 14.23 .650065 | 14.87 49 .680875 | 18.43 | .702220 | 14.10 
 50, .631939 ; 14.22 .650958 ; 14.87 50 .681681 ; 18.42 ; .703066 | 14.10 
 51 | 8.632792 | 14.22 | 8.651850 | 14.85 || 51 | 8.682486 | 13.40 |8.703912 | 14.07 
 52 .633645 | 14.18 .652741 | 14.83 || 52 .683290 | 18.38 | .704756 | 14.07 
 53 .634496 | 14.18 -653631 | 14.82 || 53 .684093 | 13.38 | .7O05600°| 14.07 
 54 . 6385347 | 14.17 .654520 | 14.80 54 684896 | 13.35 | .706444°! 14.03 
 55 .636197 | 14.15 .655408 | 14.80 55 | .685697 | 13.35 | .707286'| 14.03 
 56 .637046 | 14.13 656296 | 14.77 56 .686498 | 18.385 | .708128 | 14.02 
 57 .637894 | 14.18 .657182 | 14.77 || 57 | .687299.| 138.32 | .708969 | 14.02 
 58 .638742 | 14.10 .658068 | 14.7 58 | .688098 | 13.32 | .709810°} 14.00 
 59 .639588 | 14.1 10 | .658954 | 14.73 59 .688897 | 13.30 | .710650 | 18.98 
 60 | 8.640434 | 14. 08 | 8.659838 | 14.72 |( 49’ 8.689695 | 13.28 '8.711489 | 13.97 
 
AND EXTERNAL SECANTS. 
 
 COIR erm e | =] 
 
 . | Ex. sec.) D. 1’. 
 
 Ex. sec. 
 
 | 8.689695 
 690492 
 691289 | 
 . 692084 
 
 . 692879 
 693674 
 694467 
 . 695260 
 696052 
 
 .696843 | 
 
 697634 
 698424 
 .699213 
 700001 
 .700789 
 701576 
 . 702362 
 
 . 703147 
 
 . 703932 
 . 704716 
 705499 
 . 706282 
 . 707063 
 
 107844 
 - 708625 
 709404 | 
 
 710183 
 
 710961 
 
 711789 
 . 712516 
 713292 
 .714067 
 
 714842 
 715616 
 
 . 716389 
 
 717161 
 717983 
 
 718704 
 
 719475 | 
 720214 | 
 721013 
 
 . 721782 
 
 722549 
 
 . 723316 | 
 . 724083 
 724848 | 
 
 . 725613 
 726377 
 ~ 127140 
 .727903 
 . 728665 
 .'729427 
 720187 
 730947 
 
 . 731707 
 
 132465 
 . 7338223 
 . 733981 
 134737 
 . 735493 
 
 } 8.736248 
 
 woe 3 QV 
 
 (92) 
 
 WWWNWW 9 WW WW WWW WWW 
 
 pose frm poe frm frm fame frre fem ee fe feck Prk peek fomek feed. 
 
 711489 
 712327 
 
 713164 | 
 “714001 | 
 “714838 
 115673 
 -716508 
 717342 
 18175 
 
 -719008 
 
 -719840 
 
 420671 
 . 721502 
 722332 
 SOBRE! 
 - 723989 | 
 724817 
 (25644 
 726471 
 127297 
 728122 
 728946 
 729770 
 ~730593 
 731415 
 71382237 
 - 738058 
 738878 
 . 734698 
 735517 
 . 7363385 
 737158 
 737970 
 738786 
 739602 
 740417 
 7412381 
 . 742045 
 «742858 
 . 748670 
 . 744482 
 
 745293 
 . 746103 
 
 pea Ate 
 
 on 
 
 OC | 
 3.99 
 
 WOIOWIPWWH OS 
 
 739263 
 
 751214 
 51955 
 752696 
 "753436 
 54175 
 
 762266 
 762998 
 763729 
 . 764459 
 (65189 
 .%65918 
 
 . 766647 
 67374 
 . 768102 
 168828 
 169554 
 
 Crovrergr grag 
 
 It 
 SSSRRGAS 
 
 re 
 
 WOwwnwe 
 
 ~ 
 
 eee ee eee 
 5 
 
 MDW WO 0b 
 
 “50 
 
 18760578 
 
 .765358 
 .766152 
 .766946 
 167739 
 768531 
 8.769328 
 770114 
 770905 
 771695 
 772484 
 173273 
 
 8.785031 
 785810 
 . 786588 
 787366 
 788144 
 788920 
 789696 
 790472 
 791247 
 792021 
 8.792795 
 793568 
 794340 
 795112 
 795884 
 196654 
 OTA 
 798194 
 798963 
 199732 
 8.800500 
 801267 
 802034 
 802800 
 803565 
 804330 
 805095 
 805859 
 
 806622 
 8.807385 
 
~ 
 
 TABLE XXVI.—LOGARITHMIC VERSED SINES 
 
 21° 
 
 Ex, sec. D. 1”. 
 
 COIAMAwWWHS | 
 
 Vers. . | Hx. sec. 4 Vers. Dial 
 | 8.780370 5 | 8.807385 | 0 |. 8.822296 | 11.385 |8.852144 | 
 .781087 .808147 | 1) Oh .822977 | 11.35 | .852874 
 - 781802 .808908 2 .823658 | 11.3838 | .8538604 
 782517 .809669 3 .8243838 | 11.38 | 854832 
 783231 .810430 4 .825018 | 11.82 | .855061 
 783945 .811190 5 .825697 | 11.32 | .855789 
 . 784658 .811949 6 .8263873 | 11.80 | .856516 
 785371 .812708 7 827054 | 11.28 | .857243 
 . 786083 .813466 8 827781 | 11.28 | .857969 
 (86794. 814224 9 .828408.| 11.28 | .858695 
 - 787505 .814981 10 .829085 | 11.27 | .859420 
 788215 8.815737 11 | 8.829761 | 11.25 |8.860145 
 788924 .816493 12 .8304386 | 11.25 | .860869 
 . 789633 .817249 13 .8381111.| 11.23 | .861593 
 - 790342 .818004 14 .881785. | 11.23 | .862816 
 -791049 mG .818758 1-15 .8382459 | 11.22 | .863039 
 - 791756 Se .819512 16 .8331382 | 11.2 .863761 
 792463 3 . 820265 17 .888804 | 11.20 | .864483 
 .793169 ai .821018 18 .834476 | 11.20 | .865204 
 798874 Be .821770 19 .835148 | 11.18 | .865925 | 
 - 794579 we 822521 20 .835819 | 11.17 | .866646 | 
 - 795283 .%3 | 8.823272 21 | 8.836489 | 11.17 |8.867365 
 . 795987 ory .824023 22 .887159.| 11.17 | .868085. | 
 . 796690 ai 824773 23 .887829.| 11.15 | . 868804 | 
 (97392 f . 825522 | 24 .888498 | 11.13 | .869522 
 798094 826271 | 125 .839166 | 11.13 | .870240 
 . 798795 .827019 | | 26 .839834.] 11.12 | .870957 
 - 799496 . 827767 27 .840501 | 11.12 | .871674 
 .800196 . 828514 28 .841168 | 11.10 | .872390 
 . 800896 822261 | 29 .844834 | 11.10 | .873106 
 . 801594 .880007 30 .842500. | 11.08 | .878822 
 802293 8.830752 | 81 | 8.843165.| 11.07 (8.874537 
 .802991 .831497 2 .843829.| 11.07 | .875251 | 
 .803688 . 832242 | 38 .844493 | 11.07 | .875965 | 
 . 804384 832986 | | 34 .845157 | 11.05 | .876678 | 
 .805080 . 833729. | | 85 .845820. | 11.05 | .877391 
 80577 . 884472 86 .846483. | 11.03 | .878104 
 .806471 .835215 37 .847145. | 11.02 | .878816. | 
 .807165 .835957 38 .847806 | 11.02 | .879528 
 .807859 .836698 39 .848467. | 11.00 | .880239 
 -808552 .837439 40 .849127 | 11.00 | .880949 
 8.809244 8.838179 41 | 8.849787 | 11.00 {8.881659 
 .809936 .838919 | 42 .850447 | 10.98 | .882369 
 .810628 . 839658 43 .851106 | 10.97 | .888078 
 .811319 .840396 44 .851764. | 10.97 | .888787 
 .812009 .841135 | 45 , 852422. | 10.95 | .884495 
 .812699 .841872 || 46 .853079.| 10.95 | .885203 
 .815388 . 842609 || 47 .853736. | 10.93 | .885910 
 .814077 .843346 | 12. 48 854392 | 10.93 | .886617 | 
 .814765 | (844082. ) 12.% 49 .855048 | 10.92 | .887323 | 
 .815452 .844817 | 12. 50 .855703. | 10.92 | .888029 
 .816139 8.845552 | 12. | 51 | 8.856358 | 10.90 |8.888754 
 .816825 .846287 | 12. 11° 52 .857012. | 10.90 | .889489 
 817511 .847021."| 12. it 5a .857666 | 10.88 | .890144 
 818196 847754 | 12. || 54 .858319 | 10.88 | .890848 
 818881 . 848487 | 12. || 55 ~858972 | 10.87 | .891551 | 
 .819565 . 849220 | 12.20 || 56 859624 | 10.87 | .892254 
 . 820249 .849952 | 12. | 57 .860276.| 10.85 | .892957 
 . 820932 .850683 | 12. 58 .860927 | 10.85 | .893659 
 .821614 .851414 | 12. 59 .86157 10.83 | .894361 
 8.822296 8 852144 | 12. 60 | 8.862228 | 10.82 |8.895062 | 
 
 . . . -_ . ae . . . . . We ~ = 2 é 7 ~ 
 WWD CCITT o) I We 5 S ox Ww 0 C9 OV GIR 
 
 CONWWNWWOTO TURN 
 
 BMS oe ap od 5 ag ey oo a 
 
: ie 
 93° 
 4 Vers Dig 
 0 | 8.862228 | 10.82 
 1 .862877 | 10.83 
 2 .863527 | 10.80 
 3 .864175 | 10.80 
 | 4] .864823 | 10.80 
 fap | .865471| 10°78); 
 6 .866118 | 10.78 | 
 7 .86676> | 10.77 | 
 8 .86741l | 10.77 | 
 9 .868057 | 10.75 | 
 10 .868702 | 10.73 
 11 | 8.869346 | 10.75 
 12 .869991 | 10.72 
 13 .870634 | 10.72 
 : 14 871277 | 10.72 
 15 .871920 | 10.70 
 16 .872562 | 10.70 
 17 .8738204 | 10.68 
 18 .873845 | 10.68 
 | 19 .874486 | 10.67 
 20 .875126 | 10.67 | 
 21 | 8.875766 | 10.65 
 22 .876405 | 10.65 | 
 | 23 .877044 | 10.63 
 24 .877682 | 10.63 
 25 | .878320 | 10.62 
 26 | 878957 | 10.62 | 
 27 879594 | 10.60 | 
 | 23 . 880230 | 10.60 | 
 29 .880866 | 10.58 | 
 | 30] .881501 | 10.58 | 
 31 | 8.882136 | 10.58 
 32 .882771 | 10.57 
 | 33 883405 | 10.55 
 34 884038 | 10.55 
 35 .884671 | 10.53 | 
 36 885303 | 10.53 
 | ov .885935 | 10.53 
 38 .886567 | 10.52 
 39 .887198 | 10.52 
 || 40} .887829.| 10.50 
 | 41 | 8.888459 | 10.48 
 42 . 889088 | 10.48 
 43 | 889717 | 10.48 
 44 .890346 | 10.47 
 45 .890974 | 10.47 
 | 46 .891602 | 10.45 
 47 892229 | 10.45 
 48 .892856 | 10.43 | 
 AS .893482 | 10.43 | 
 50 | .894108 | 10.42 
 | 
 P| 51 | 8.894733 | 10.42 | 
 52 | .895358 | 10.42 | 
 538. | .895983 | 10.40 | 
 54 | .896607 | 10.38 | 
 55 .897230 | 10.38 
 56 | .897853 | 10.38.} 
 57 | .898476 | 10.37 
 58 | .899098 | 10.35 | 
 59 | .899719 | 10.37 | 
 60 | 8.900341 | 10.33 | 
 
 q _ 
 
 AND EXTERNAL SECANTS. 
 
 Taxa sec.) DO 1". i 4 Vers. 
 | 
 8.895062 | 11.68 || 0 | 8.900841 
 .895763 |; 11.67 || 1 | .900961 
 896463 | 11.67 |; 2 | .901582 
 897163 | 11.65 |} 3 | .902201 
 .897862 | 11.65 || 4] .902821 
 .893561 | 11.63 | 5'| .903440 
 899259 | 11.63 || 6; .904058 
 .899957 | 11.63 || 7 . 904676 
 .900655 | 11.62 |i § . 905293 
 .901352 | 11.60 9 | 905910 | 
 .902048 | 11.62 || 10 . 906527 
 8.902745 | 11.58 || 11 | 8.907143 
 .903440 | 11.60 2 . 907759 
 .904136 | 11.57 || 13 . 908374 
 .904830 | 11.58 |; 14 . 908989 
 pQ0d0eD | ALIS (Wels . 909603 
 .906219 | 11.55 || 16 -910217 
 .906912 | 11.55 || 17 .910830 
 .907605 | 11.53 || 18 .911443 
 -908298 | 11.53 |} 19 . 912056 
 .908990 | 11.52 || 20 . 912668 
 8.909681 | 11.52 || 21 | 8.913279 
 -9103872 | 11:52 |} 22 .913891 
 911063 | 11.52 || 23 .914501 
 917754 | 11:48 || 24 .915111 
 .912443 | 11.50 |} 25 915721 
 .913133 | 11.48 || 26 . 916331 
 .913822 | 11.47 || 27 | .916940 
 .914510 | 11.47 || 28 .917548 
 :915198 | 11747 |; 29 .918156 
 .915888 | 11.45 30 | .918764 
 8.916573 | 11.45 |! 31 | 8.919371 
 .917260 | 11.48 || 82 .919977 
 917946 | 11.43 || 83 | .920584 
 .918682 | 11.43 |} 34 . 921190 
 .919318 | 11.42 || 335 | .921795. 
 .920003 | 11.40 || 383 . 922400 
 920687 | 11.42 || 37 .923094 
 .9213872 | 11.38 || 88} .923608 
 .922055 | 11.40 || 89 | .924212 
 .922739 | 11.37 || 40 . 924815 
 8.923421 | 11.88 || 41 | 8.925418 
 924104 | 11.37 || 42 . 926020 
 924786 | 11.35 || 43 . 926622 
 .925467 | 11.37 || 44 927224 
 926149 | 11.33 || 45 | .927825 
 926829 | 11.35 |; 46 | .928425 
 927510. | 11.32 || 47 . 929025 
 .928189 | 11.83 || 48 . 929625 
 928869 | 11.82 || 49 | .930224 
 929548 | 11.30 || 50 | .980823 
 8.930226 | 11.82 || 51 | 8.931421 
 .930905 | 11.28 |} 52 | .932019 
 .931582 | 11.30 |) 53 | = .982617 
 . 9382260.) 11.27 || 54 933214 
 . 932936 | 11.28 || 55 | .933811 
 933613 | 11.27 || 56 | .934407 
 .934289 | 11.27 || 57 . 935003 
 .934965 | 11.25 |; 58 . 935598 
 .985640 | 11.25 || 59 . 936193 
 8.936315 | 11.23 || 60 | 8.936788 
 
 Dax see. D. 17 
 10.33 '8.936815 | 11.23 
 10.35 | .936989 | 11.23 
 10.32 | .937663 | 11.22 
 10.33 | .938336 | 11.22 
 10.32 | .939009 | 11.22 
 10.30 | .939682 | 11.20 
 10.30 | .940354 | 11.20 
 10.28 | .941026 | 11.20 
 10.28 | .941698 | 11.18 
 10.28 | .942369 | 11.17 
 10.27 | .943039 | 11.18 
 10.27 |8.943710 | 11.15 
 10.25 | .944379-| 11.17 
 10.25 | .945049 | 11.15 
 10.23 | .945718.| 11.13 
 10.23 | .946386 | 11.13 
 10.22 | .947054 | 11.18 
 10.22 | .947722 | 11.12 
 10.22 | .948389 | 11.12 
 | 10.20 | .949056 | 11.12 
 10.18 | .949723 | 11.10 
 10.20 |8.950389 | 11.10 
 10.17 | .951055 | 11.08 
 10.17 | .951720 | 11.08 
 10.17 | .952385 | 11.07 
 10.17 | .953049 | 11.07 
 10.15 | .953713 | 11.07 
 10.13 | .954377-| 11.05 
 10.13 | .955040 | 11.05 
 10.13 | .955703 | 11.05 
 10.12 | .956366 | 11.03 
 10.10 8.957028 | 11.03 
 | 10.12 | .957690 | 11.02 
 10.10 | .958351 | 11.02 
 10.08 | .959012 | 11.00 
 10.08 | .959672 | 11.00 
 10.07 | .960332 | 11.00 
 10.07 | .960992 | 10.98 
 10.07 | .961651 | 10.98 
 10.05 | .962310 | 10.98 
 10.05 | .962969 | 10.97 
 10.03 8.963627 | 10.97 
 10.03 | .964285 | 10.95 
 10.03 | .964942 | 10.95 
 10.02 | .965599 | 10.95 
 10.00 | .966256 | 10.98 
 10.00 | .966912 | 10.93 
 10.00 | .967568 | 10.92 
 9.98 | .968223 | 10.92 
 9.98 | .968878 | 10.92 
 9.97 | .969533 | 10.90 
 9.97 18.970187 | 10.90 
 9.97 | .970841 | 10.88 
 9.95 | .971494 | 10.88 
 9.95 | .972147 | 10.88 
 9.93 972800 | 10.87 
 9.93 | .973452 | 10.87 
 9.92 | .974104 | 10.87 
 9.92 | .974756 | 10.85 
 9.92 | .975407 | 10.85 
 9.90 18.976058 | 10.83 
 
 —— 
 
, Vers. | D: 1". | Ex. sec. | D. 1° / 
 0 | 8.936788 | 9.90 | 8.976058 | 10.83 || 0 
 1 | .937382 | 9.90] .976708 | 10.83 || 1 
 2| .937976 | 9.88) .977358 | 10.83 || 2 
 3} 988569 | 9.88 | .978008 | 10.82 3 
 4} .939162 | 9.87 | .978657 | 10.82 |} 4 
 5 | .989754 | 9.87] .979306 | 10.80 || 5 
 6 | .940346 | 9.87 | .979954 | 10.80 6 
 7 | .940938 | 9.85] .980602 | 10.80 7 
 8 | .941529 | 9.85] .981250 | 10.80 || 8 
 9} .942120 | 9.83 - .981898 | 10."8 9 
 10 | .942710 | 9.83] .982545 | 10.77 1} 10 
 11 | 8.943300 | 9.82 | 8.983191 | 10.77 |] 14 
 12 | .943889 | 9.83 | .983837 | 10.77 || 12 
 13 | .944479 | 9.80 | .984483 | 10.77 || 13 
 14 | .945067 | 9.80 | .985129 | 10.75 || 14 
 15.| .945655 | 9.80] .985774 | 10.75 || 15 
 16} .946243 | 9.80; .986419 | 10.73 || 16 
 17 | .946831 | 9.78} .987063 | 10.7 17 
 18 | .947418 | 9.77) .987707 | 10.73 || 18 
 19 | .948004 | 9.77 | 988351 | 10.72 || 19 
 
 20 | .948590 | 9.77 | .988994 | 10,72 || 2 
 21 | 8.949176 | 9.75 | 8.989637 | 10.70 || 21 
 22 | .949761 | 9.75 |} .990279 | 10.72 || 22 
 23 | .950346 | 9.75 | .990922 | 10.68 || 23 
 24 | (950931 | 9.73 | .991563 | 10.70 || 24 
 25} .951515 | 9.73 | .992205 | 10.68 || 25 
 26 | .952099 | 9.72 | .992846 | 10.68 || 26 
 27 | .952682 | 9.72] .993487 | 10.67 || Q7 
 28 | .953265 | 9.70} .994127 | 10.67 || 28 
 29 | .958847 | 9.701 .994767 | 10.65 || 29 
 30 | .954429 | 9.70 | .995406 | 10.67 || 30 
 31 | 8.955011 | 9.68 | 8.996046 | 10.65 || 31 
 2} .955592 | 9.68 |} .996685 | 10.63 || 32 
 83 | .956173 | 9.67 | .997323 | 10.63 || 33 
 
 34 | .956753 | 9.68] .997961 | 10.63 || 3 
 35 | .957334 | 9.65] .998599 | 10.62 || 35 
 36 | .957913 | 9.65 | .999236 | 10.62 || 36 
 37 | .958492 | 9.65 | 8.999873 | 10.62 || 37 
 38 | .959071 | 9.65 | 9.000510 | 10.60 || 38 
 89 | .959650 | 9.63} .001146 | 10.62 || 39 
 40 | .960228 | 9.62] .001783 | 10.58 || 40 
 41 | 8.960805 | 9.62 | 9.002418 | 10.58 || 44 
 42 | .9613882 | 9.62] .003053 | 10.58 || 42 
 43 | .961959 | 9.60] .003688 | 10.58 || 43 
 44 | .962535 | 9.60} .004323 | 10.57 || 44 
 45 | .963111 | 9.60 | .004957 | 10.57 || 45 
 .963687 | 9.58 | _.005591 | 10.55 || 46 
 7 | .964262 | 9.58] .006224 | 10.57 || 47 
 48 | .964837 | 9.57 | .006858 | 10.53 || 48 
 ¢ .965411 | 9.57 | .007490 | 10.55 || 49 
 .965985 | 9.57 | .008123 | 10.53 || 50 
 | 8.966559 | 9.55 | 9.008755 | 10.53 || 51 
 ; .967132 | 9.55 | °009887 | 10.52 || 52 
 53| .967705 | 9.53] .010018 | 10.52 || 53 
 54.| .968277 | 9.53] .010649 | 10.52 || 54 
 5 | .968849 | 9.53] .011280 | 10.50 || 55 
 .969421 | 9.52 | .011910 | 10.50 || 56 
 7 | .969992 | 9.52] .012540 | 10.50 || 57 
 .970563 | 9.50 | .013170 | 10.48 || 58 
 59 | .971133 | 9.501] .013799 | 10.48 || 59 
 0 | 8.971703 | 9.50} 9.014428 | 10.47 || 60 
 
 TABLE XXVI.—LOGARITHMIC VERSED SINES 
 
 Versett) Dt" 
 8.971703 | 9.50 |. 9.014428 
 . 972273 | 9.48 .015056 
 . 972842 | 9.48 .015685 
 .973411 | 9.48 .016312 
 . 973980 | 9.47 .016940 | 
 .974548 | 9.47 . 017567 || 
 975116 | 9.45 .018194 | 
 .9756838 | 9.45 .018821 | 
 . 976250 | 9.43 .019447 
 .976816 | 9.43 .020073 
 . 9773882 | 9.43 . 020698 | 
 8.977948 | 9.43 | 9.021323 
 .978514 | 9.42 .021948 
 979079 | 9.40 | 022572 | 
 .979643 | 9.40 .023197 
 .980207 | 9.40 . 023820 
 . 98077 9.40 .024444 | 
 .981335 | 9.88 | .025067 
 .981898 | 9.37 | .025690 | 
 . 982460 | 9.38 .026312 | 
 . 983023 | 9.37 . 0269384 
 8.983585 | 9.35 | 9.027556 
 .984146 | 9.35 028177 
 984707 | 9.35 .028798 
 .985268 | 9.33 . 029419 
 . 985828 | 9.33 . 030039 
 .986388 | 9.33 .030659 
 .986948 | 9.32 .031279 
 987507 | 9.32 .031899 
 -988066 | 9.32 .032518 
 .988625 | 9.30 | .033136 
 8.989183 | 9.28 | 9.033755 
 .989749 | 9.30 . 034873 
 .990298 | 9.28 . 034991 
 .990855 | 9.27 . 035608 
 .991411 | 9.28 .036225 
 .991968 | 9.25 . 0386842 
 . 992523 | 9.27 037458 | 
 . 993079 | 9.25 .038074 
 . 993684 | 9.25 . 038690 
 .994189 | 9.23 .039805 
 8.994743 | 9.23 | 9.039920 
 .995297 | 9.23 040535 
 .995851 | 9.22 | .041150 | 
 . 996404 | 9.22 .041764 
 . 996957 | 9.20 . 042378 
 -997509 | 9.22.71 .042991 
 . 998062 | 9.18 048604 
 .998613 | 9.20 044217 
 -999165 | 9.18 . 044830 
 8.999716 | 9.17 045442 
 9.000266 | 9.18 | 9.046054 
 000817 | 9.17 046665 
 .001867 | 9 15 047276 
 .001916 | 9.17 047887 
 .002466 | 9.138 .048498 
 0038014 | 9.15 .049108 
 .003563 | 9.13 .049718 
 .004111 | 9.13 050828 
 .004659 | 9.12 . 050937 
 9.005206 | 9.12 | 9.051546 
 
 ) | 
 Ex. see. D. 1’. 
 
 10.47 
 10.48 
 10.45 
 10.47 
 10.45 | 
 10.45 |} 
 10.45 
 10.43 
 10.43 
 
 | 10.42 
 
 10.42 
 
 10.42 
 10.40 
 10.42 
 10.38 
 10.40 
 10.38 
 10.38 
 
 10.15 
 
AND EXTERNAL SECANTS. 
 
 / Vers. | D..1". | Ex, see. |‘D, 1" / Vers, |D.1".| Ex. sec. |D. 1’. 
 0 | 9.005206 | 9.12 | 9.051546 | 10.15 || 0 | 9.037401 | 8.77 | 9.087520 | 9.83 
 j) -005753 | 9.12) .052155 | 10.13 || 1] .037997 | 8.75 | (088110 | 9/83 
 2; .006300 | 9.10} .052763 | 10.13 || 2] .088452 | 8.77 | [088700 | 9.83 
 3} .006846 | 9.10 058371, | 10:13 || 3 .038978 | 8.75 .089290 | -9.83 
 4| .007392 | 9.10] .053979 | 10.12 || 4] .039503 | 8.73 | 089880 | 9.89 
 5) -007988 | 9.08 | .054586 | 10.12 || 5 | .040027.| 8.75 | .090469 | 9.82 
 6 | .008483 | 9.08} .055193 | 10.12 || 6] .040552 | 8.73 | 091058 | 989 
 7 | ..009028 | 9.07 | .055800.| 10.10 || 7] .041076 | 8.721 091647 | 9/80 
 8 | .009572 | 9.07 | .056406 | 10.10 || 8 | .041599 | 8.73 | ‘092985 | 9/80 
 9 | .010116 | 9.07 | .057012 | 10.10 || 9} .042123 | 8.72 | 1092823 | 9.80 
 10; .010660 | 9.05 | .057618 | 10.10 || 10 | .042646 | 8.70 | .098411 | 9.%8 
 11 | 9.011203 | 9.05 | 9.058224 | 10.08 || 11 | 9.043168 | 8.72 | 9.093998 | 9.80 
 12 | .011746 | 9.05 | .058829,/ 10.08 || 12 | .043691 | 8.7 094586 | 9.78 
 13 | .012289 | 9.03 | .059434 | 10.07 41 13 | .044213 | 8.70 | .095173 | 9.77 
 14 | .012831 | 9.03 | .060038 | 10.08 || 14 | .044735 | 8.68 | .095759 | 9.78 
 15 | .013373 | 9.03 | .060643 | 10.07 || 15 | .045256 | 8.68] .096346 | 9.77 
 16 | .013915 | 9.02 | .061247 | 10.05 || 16 | .045777 | 8.68] .o96932! 9.77 
 17 | .014456 | 9.02 | .061850 | 10.07 || 17 | 046298. | 8.67 | 097518 | 9.75 
 18} .014997 | 9.02} .062454 | 10.05 || 18 | .046818 | 8.67 | .098103:| 9.77 
 19 | .015538 | 9.00 | .063057 | 10.03 || 19 | .047388 | 8.67 | 098689 | 9.75 
 20} .016078 | 9.00} .063659 | 10.05 || 20 | .047858 | 8.65 | _og9e74 | 9.78 
 21 | 9.016618 | 8.98 | 9.064262 | 10.03 || 21 | 9.048377 | 8.65 | 9.099858 | 9.75 
 224 .017157 | 9.00 | .064864 | 10.03 || 22} .048896 | 8.65 | .100443 | 9.73 
 Qs 917697 | 8.97 | .065466 | 10.02 || 23! .049415 | 8.63 | 1010287 | 9.73 
 24 | .018235 | 8.98 | .066067 | 10.02 || 24 | .049933 | 8.631 .101611 | 9.72 
 25 | .018774 | 8.97 | .066668 | 10.02 || 25.| .050451 | 8.63 | 102194 | 9.73 
 26 | .019312 | 8.97 |; .067269 | 10.02 || 26 | .050969 | 8.68 | 10277 9.72 
 27 | .019850 | 8.95 | .067870 | 10.00 || 27} 051487 | 8.62 | .103361°| 9.70 
 28 | .020887 | 8.95 | .068470 | 10.00 || 28} .052004 | 8.60 | .103943| 9.7% 
 29 | .020924 | 8.95 | .069070 | 10.00 || 29 | 052520 | 8.62 | 104526 | 9.70 
 30 | .021461 | 8.93] .069670 | 9.98 || 30 | .053037 | 8.60 | .105108 9.7% 
 31 | 9.021997 | 8.93 | 9.070269 | 9.98 || 31 | 9.053553 | 8 60 | 9.105690 | 9.68 
 82 | .022533 | 8.93 | .070868 | 9.98 || 32] .054069 | 8.581 .106271 | 9.7 
 33 | .023069 | 8.92 | .071467 | 9.97 || 83] 054584 | 8.58 | 106853 | 9.68 
 34 | 1.923604 | 8.92 | .072065 | 9.971; 34 | .055099 | 8.58 | .107434 | 9.68 
 35} .024189 | 8.90 | .072663 | 9.97 || 85 | .055614 | 8.58 | .108015 | 9.67 
 36 | .024673 | 8.92 | .073261 | 9.97 || 86 | .056129 | 8.57 | .108595 | 9.67 
 87 | .025208 | 8.90 | .073859 | 9.95 || 37 | .056643 | 8.57 | .109175 | 9 67 
 88} .025742 | 8.88 | .074456 | 9.95 || 38} .057157 | 8.55 | .109755 | 9.67 
 3° 026275 | 8.88 | .075053 | 9.93 | 39 | .057670 | 8.55 | .110885 | 9.65 
 4) | .026808 | 8.88} .075649 | 9.95 | 40 | .058183 | 8.55 | .110914 | 9.67 
 41 | 9.027341 | 8.88 | 9.076246 | 9.93 || 41 | 9.058696 | 8.55 | 9.111494 | 9.63 
 42 | .027874 | 8.87 | .076842 | 9.92 || 42} 059909 | 8.53 | .112072 | 9.65 
 43 | .028406 | 8.87 | .077437 | 9.93 || 43 | .059721 | 8.53 | 112651 | 9.63 
 44; .028938 ; 8.85 | .078033 | 9.92 || 44] .060233 | 8.53 | .113229 | 9.68 
 45 | .029469 | 8.85 | .078628 | 9.92 || 45 | .060745 | 8.52 | .113807 | 9.63 
 46 | .030000 | 8.85 | .079223 | ‘9.90 | 46] .061256 | 8.52 | .114385 | 9.63 
 7 | .030531 | 8.85 | .079817 | 9.92 || 47 | .061767 | 8.50 | .114963 | 9.62 
 43 | .031062| 8.83} .080412 | 9.90 || 48 | .062977 | 8.52] 1155401 9.62 
 49 | .031592 | 8.83] .081006| 9.88 || 49 | .o62788 | 8.50 | .116117 | 9.60 
 50.; 032122 , 8.82} .081599 ; 9.90 || 50; .063298 | 8.48, .116693 | 9.62 
 51 | 9.032651 | 8.82 | 9.082193 | 9.88 |) 51 | 9.063807 | 8.50 | 9.117270 | 9.60 
 52 | .033180} 8.82| .082786 | 9.87 || 52 | .064317 | 8.48] .117846| 9.60 
 53 | .033709 | 8.80] .083378 | 9.88 || 53! 064826 | 8.48] 1184929] 9.58 
 54 | .034237 | 8.80] .083971 | 9.87 || 54! .065335 | 8.47 | 118997 | 9/60 
 55 | .034765 | 8.80 | .084563 | 9.87 |] 55 | .065843 | §.47 | 119573 | 9.58 
 56} .035293 | 8.78 | .085155 | 9.87 || 56 | .066351 | 8.47 | .120148| 9.58 
 57 | .035820 | 8.78 | .085747 |. 9.85 || 57 | 066859 | 8.45 | 120798 | 9/57 
 58 | .036347 | 8.7 .086338 | 9.85 || 58 | .067366 | 8.47 | .121297 | 9.57 
 59 | .036874 | 8.7 -086929 | 9.85 || 59 | 067874 | 8.43 | 121871 | 9.57 
 60 | 9.037401 | 8.77 9.83 || 60 8.45 9.57 
 
 9.087520 
 
 9.068380 
 
 9.122445 
 
 417 
 
TABLE XXVI.—LOGARITHMIC VERSED SINES 
 
 .| Ex. sec. |D. 
 
 / Vers, | D..1"..). Ex. sec. | D. 1’. 4 Vers. | D. 1’.| ID. 1 
 | i | 
 
 0 | 9.068380 8.45 | 9.122445 9.57 0 | 9.098229 | 8.15 | 9.156410 9.30 
 
 1 . 068887 8.43 .1238019 OES (i ekd 098718 | 8.13- . 156968 9.32 
 
 2 .069393 8.43 . 1238593 9.55 2 .099206 | 8.12 | .157527 9.28 
 
 3 . 069899 8.43 . 124166 9.55) |e t3 .099693 | 8.13 | .158084 9.380 
 
 4 .070405 8.42 . 124739 9.53 || 4 .100181 | 8.12 . 158642 9.30 
 
 5 .070910 8.42 .125311 9.55 || 5 .100668 | 8.12 . 159200 9.28 
 
 6 .071415 8.40 . 125884 9.53 6 101155-|- 8.12 159757 9.28 
 
 # .071919 8.42 . 126456 9:53. ||) 7 .101642.} 8.10 . 160314 9.27 
 
 8 .072424 8.40 .127028 9.52 8 .102128.| 8.10 . 160870 9.28 
 
 9 072928 8.40 .127599 9.53 9 .102614 | 8.10 .161427 9.27 
 
 10 078432 8.38 128171 9.52 || 10 .103100 | 8.08 .161983 9.27 
 
 11 | 9.673935 | 8.88 | 9.128742 | 9.52 || 11 | 9.103585 | 8.08 9.162539 ; 9.27 
 
 12 .074438 8.38 . 129313 9.50 || 12 .104070 | 8.08 . 163095 9.25 
 
 13 074941 8.37 . 129883 9.50 +) 13 .104555 | 8.08 . 163650 9.25 
 
 eal 14 075448 8.38 . 130453 9.50 || 14 .105040 | 8.07 . 164205 9.25 
 ave it | 15 .075946 §.385 . 181028 9.50 || 15 . 105524 | 8.07 .164760 9.25 
 iit 16 .076447 8.37 .1815938 9.50 || 16 .106008 | 8.05 . 165315 9.25 
 ea 7 .076949 8.35 .1382163 9.48 7 .106491 | 8.07 . 165870 9.23 
 ee Ca 18 .077450 8.35 . 1827832 9.48 || 18 .106975 | 8.05 . 166424 9.23 
 ii] a 19 077951 8.35 138301 9.48 |; 19 .107458 | 8.05 .166978 9.23 
 Wy 20 078452 8.33 . 1383870 9.47 |\.2 .107941 | 8.08 .167582 9.22 
 21 | 9.078952 | 8.33 | 9.134438 9.47 || 21 | 9.108423 | 8.05 | 9.168085 | 9.23 
 
 ile 22 .079452 8.33 . 185006 9.47 || 22 .108906 | 8.03 . 168639 9.22 
 th) 23 .079952 8.32 135574 9.47 || 23 .109888 | 8.02 . 169192 9.22 
 
 We hi} 24 .080451 8.32 . 186142 9.45 || 24 .109869 | 8.03 .169745 9.2 
 et i 25 080950 8.32 . 186709 9.47 || 25 .110351 | 8.02 .170297 9.22 
 
 Hi 26 .081449 8.32 187277 9.45 || 26 .110882 | 8.02 . 170850 9.20 
 
 2¢ .081948 8.30 . 137844 9.43 || 27 .111313 | 8.00 .171402 9.20 
 
 ty 28 082446 8.30 . 188410 9.45 || 28 .111793 | 8.CO .171954 9.18 
 ep | 29 082944 8.28 .1388977 9.43 || 29 .112273 | 8.06 .172505 9.20 
 Hh | 30 .083441 8.30 . 1389543 9.43 || 80 .112753 | 8.00 .178057 9.18 
 HE 31 | 9.083939 | 8.28 | 9.140109 9.42 || 81 | 9.113233 | 8.00 | 9.173608 | 9.18 
 ay Net 32 .084436 8.27 . 140674 9.43 ||: 382 118713 | 7.98 174159 9.18 
 33 .084932 8.28 .141240 9.42 || 33 .114192 | 7.98 .174710 9.17 
 
 aa ti} 34 .085429 8227 .141805 9.42 || 34 114671 .| 7.97 .175260 9.17 
 patie) 35 .085925 8.25 - 142370 9.40 || 35 .115149 | 7.97 .175810 9.17 
 Tuan | 36 .086420 8.27 . 142934 9.42 || 36 115620 - (87 . 176860 9.17 
 i 37 .086916 8.25 .1431499 9.40 || 37 .116105 | 7.97 .176910 9.1% 
 a | 38 087411 8.25 .144063 9.40 || 88 .116588 | 7.97 .177460 9.15 
 bit} 39 .087905 8.23 144627 9.38 || 39 .117061 | 7.95 | .178009 9.15 
 40 .088400 8.25 . 145190 9.40 || 40 .117538 | 7.95 .178558 9.15 
 
 uy 41 | 9.088895 8.23 | 9.145754 9.38 || 41 | 9.118015 , 7.93 | 9.179107 9.15 
 HG 42\ 089389 | 8.22 | .146317 | 9.38 || 42] .118491 | 7.95 | .179656 | 9.13 
 Vea 43 . 089882 8.23 . 146880 9.37 || 48 .118968 | 7.93 .180204 | 9.13} 
 44 .090376 8.22 147442 9.38 || 44 119444.) 7.92 180752 | 9.13) 
 45 .090869 8.22 .148005 9.37 45 119919 | 7.90 | .181800 9.13 | 
 46 .0913862 8.20 . 148567 9.37 || 46 .120895 | 7.92 ; .181848 | 9.12 
 7 .091854 8.20 . 149129 9.35 fi 120870 | 7.92! .182895 9.13 
 
 48 .092846 8.20 . 149690 9.35 || 48 121845. | 7.92 | .182943 | 9.12 
 
 49 .092838 8.20 150251 9.37 || 49 .121820 | 7.90 . 183490 9.10 
 
 50 .093330 8.18 . 150813 9.33 |} 50 122294 | 7.90 , .184036 9.12 
 
 51-| 9.093821 | 8.18 | 9.15137 9.35 || 51 | 9.122768 | 7.90 | 9.184583 9.10 
 
 52 .094312 8.18 151934 9.33 1) 52 123242 | 7.88 . 185129 9.10 
 
 53 094803 | 8.17 152494 9.35 || 53 e315 -| ‘7.90 . 185675 9.10 
 
 54 .095293 | 8.17 . 153055 9.32 || 54 .124189 | 7.88 . 186221 9.10 
 
 55 095783 | 8.17 . 152614 9.33 || 55 124662 | 7.87 . 186767 9.08 
 
 56 096273 ; 8.17 .154174 9.32 || 56 .1251384 | 7.88 .187312 9.10 
 
 57 .096763 | 8.15 154733 9.33 || 57 .125607 | 7.87 .187858 | 9.08 
 
 58 097252 | 8.15 | .155293 9.30 || 58 .126079 | 7.87 . 188403 9.07 
 
 59 | .097741 8.13 | .155851 9.32 || 59 .126551 | 7.85 .188947 9.08 
 
 60 | 9.098229 | 8.15 | 9.156410 9.30 || 60 | 9.127022 | 7.87 | 9.189492 9.07 
 
AND EXTERNAL SECANTS. 
 
 | 9.146126 
 
 | 9.150717 
 
 141971 
 142434 
 . 142896 
 143358 
 . 143820 
 144282 
 144743 
 145204 
 145665 
 
 146586 
 147046 
 147506 
 147966 
 148425 
 “148884 | 
 "149343 | 
 149801 
 
 150259 
 
 ~151175 
 .151633 
 152090 
 . 152547 
 . 153003 
 . 153460 
 153916 
 154872 | 
 9.15 4828 | 
 
 WE AE AY AP APA FAYVAFAY PAF VrFQ VIII INN NNN NNN NN NNN NANN NNNNNVUYNANN NNINQNVNINNVAINN 
 
 CoCow 
 
 IoI-F 
 
 | 9.216971 
 
 . 206799 
 .207337 
 207874 
 . 208410 
 208947 
 209483 
 . 210020 
 . 210556 
 .211091 
 
 9.211627 
 212162 
 . 212697 
 213232 
 213767 
 .214301 
 . 214836 
 -2153870 
 215904 
 .216437 
 
 217504 
 218037 
 218570 | 
 219102 
 219635 
 220167 
 220699 
 221231 
 
 9.221762 
 
 ie 8) 
 
 . 169275 
 . 169722 
 .170169 
 .170616 
 . 171062 
 Bikes 509 
 
 .175070 
 .175514 
 175958 
 1 Me ne 
 
 .179058 | 
 
 .179500 
 . 179942 
 . 180388 
 .180825 
 . 181265 
 
 | 9.181706 
 
 ESS Ay a EE a a NS BHI INN MN NINN NINN 
 
 FY Vers: 2D. 1.) Exec! | D. 1". })°7 Vers. |D.1".| Ex. sec. 
 0 | 9.127022 | 7.87 | 9.189492 | 9.07 || 0 | 9.154828 | 7.58 | 9.221762 
 1} .127494 85 | .190086 | 9.07 || 1] .155288 58 | .222293 
 9| .127965 85 | .190580.; 9.07 || 2 155738 58 | 222825 
 3 | .128436 83 | .191124 | 9.07 || 3] .156193 58 | .228355 
 4 | .128906 83 | .191668 | 9.05 || 4 | .156648 | 7.57 | .223886 
 5 | .129376 83; .192211 | 9.05 || 5! 1157102.) 7.57 | 224417 
 6 | "129846 83 | .192754 | 9.05 || 6 157556 | 7.57 | .224947 
 7 |. .180316 82} .198297 | 9.05 || ¥ 158010 BY | 225477 
 8 | .130785 83 | .193840 | 9.03 || 8 158464 55 | .226007 | 
 9 | .181255 82 | .194882 | 9.05 || 9 158917 BB | .226587 | 
 10} .131724 80. | .194925 | 9.03 |} 10 15937 55 |. .227066 
 11 | 9.132192 80 | 9.195467 | 9.03 |) 11 in 55 | 9.227595 
 12 | .132660 82°} .196009 | 9.02 || 12 | .16027¢ 53 | .228125 
 18 | .133129 |} 7.784 .196550 | 9.03 || 13 "160728 | 7.53 |. .228653 
 14.| .133596 8) | .197092 | 9.02 || 14) .161180 53 | 229182 
 15 | .134064 73 | .197633.| 9.02 11 15] .161632 52 | : 229711 
 16 | .18453 7 | .198174 | 9.02 || 16 | .162083 53 | .230289 
 17 —_ 78 | .198715.| 9.00 |/-17 | .162585 52 | .230767 
 18 | .3354( W@ | .199255 | 9.00 || 18 | .162986 52 | .231295 
 19 031 ‘7 | .199795 | 9.00 || 19 163437 50 | .231822 
 20 | .1363 7% | .200335 | 9.00 || 2 163887 52 |. 282850 
 21 | Altea 77 | 9.200875 | 9.00 || 21 | 9.164338 50 | 9.232877 
 93 | .137329 | 7.75 | .201415 | 8.98 || 22] .164788 4§ 233404 
 23 | 137794 77.| 201954} 9.00 || 93} 165237 50 | .238981 
 24 | .138260 73 | .202494 | 8.97 || 24°] .165687 48 | .284458 
 95 | .138724 75 | .203032 | 8.98 || 25 |} .166136 48 | .234984 
 26 | .139189 43 | .20857 8.98 || 26 | .166585 48 | .235510 
 97 | .139653 73 | .204110 | 8.97 || 27 | .167034 48 | .236036 
 98 | .140117 3 | .204648 | 8.97-|| 28 | .167488 ” | .286562 
 29 | .140581 43 | .205186 | 8.97 || 29 | .167931 7% | .237088 
 30 | .141045 72 | .205724 | 8.97 || 80 | .168379 7 | 237613 
 31 | 9.141508 2 | 9.206262 95 || 31 | 9.168827 v | 9.238139 
 
 | 9.248602 
 
 238664 
 .239189 
 .239718 
 . 240238 
 240762 
 . 241286 
 . 241810 
 . 242333 
 242857 
 9.243380 
 . 243903 
 . 244426 
 . 244949 
 245471 | 
 .245994 | 
 . 246516 
 247088 
 .247559 | 
 248081 
 
 249123 
 
 249644 | 
 250165 | 
 250686 | 
 251206 | 
 251726 
 
 252246 | 
 252766 | 
 
 9.253286 
 
 8. 
 
 4 
 
 83 
 
 78 
 
 wJotoFoIssst-g 
 NOVI 
 
 CO OV 09 OVOT 
 
 ev) 
 
 ~F =F J -JIYF +I 
 
32° 
 
 Vers. D: 1": 
 
 9.181706 | 7.35 
 182147 | 7.33 
 .182587 | 7.33 
 .188027 | 7.32 
 .183466 | 7.33 
 .183906 | 7.32 
 .184845 | 7.32 
 .184784 | 7.32 
 185223 ; 7.32 
 .185662 | 7.30 
 
 jak 
 _ SODIRORWWHOS | ~ 
 
 .186100 | 7.30 
 9.186538 | 7.30 
 
 12 | 1186976 | 7.28 
 13 | .187413 | 7.30 
 14 | 1187851 | 7.28 
 15 | .188288 | 7.27 
 16 | 188724 | 7.28 
 17 | .180161 | 7.27 
 18 | .189597 | 7.28 
 19 | .190034 | 7.95 
 20 | .190469 | 7.27 
 21 | 9.190905 | 7.27 
 22 | 1191341 | 7.25 
 23) 1191776 | 7.25 
 24 1192211 | 7.23 
 25 | 1192645 | 7.95 
 26 | .193080 | 7.28 
 27 | 1193514 | 7.98 
 28 | .193948 | 7.98 
 29 | 1194382 | 7.92 
 30 | .194815 | 7.93 
 31 | 9.195249 | 7.22 
 32 | 195682 | 7.22 
 33] 196115 | 7.20 
 34] 196547 | 7.22 
 35 | .196980 | 7.20 
 36 | .197412 | 7.20 
 87 | 1197844 | 7.18 
 38 | .198275 | 7.20 
 39 | .198707 | 7.18 
 40 | .199138 | 7.18 
 41 | 9.199569 | 7.18 
 42 |  .200000 7 
 43 | .200430 | 7.18 
 
 44 . 200861 
 45 .201291 
 46 201720 
 47 . 202150 
 48 202579 
 49 203008 
 50 203437 
 
 51 | 9.203866 
 52 204294 
 53 204723 
 54 -205151 
 55 205578 
 56 . 206006 
 57 206433 
 58 . 206860 
 59 207287 
 60 | 9.207714 | 7.10 
 
 bas as doe Se Son ne See es ee Miles Sa tee Des Se ROD LOLS LN 
 ms es 
 wa or 
 
 Ex. sec. 
 
 9.253286 
 253805 
 254324 
 254843 
 255362 
 255881 
 256399 
 256918 
 257436 
 257954 
 258471 
 258989 
 259506 
 260023 
 260540 
 261057 
 26157 
 262090 
 262606 
 263122 
 263638 
 9.264154 
 264669 
 265184 
 265700 
 266214 
 266729 
 267244 
 267758 
 268272 
 268786 
 269300 
 269814 
 210327 
 2710840 
 271354 
 271866 
 272379 
 272802 
 273404 
 213916 
 9.274428 
 274940 
 RUBADR 
 275963 
 216474 
 276986 
 277496 
 278007 
 278518 
 279028 
 
 oO 
 
 =) 
 
 | 9.279538 
 
 - 280048 
 . 280558 
 .281068. 
 281577 
 282087 
 282596 
 283105 
 .283614 
 9.284122 
 
 D. 1’. 
 
 § 
 
 ot ON x HH DORRRARAAMAAR RAARARRAADAAHD 
 SSSISSSSBSE SSSSSSSSSS SSESSRKBSRRARARR 
 
 OO OVO OV oer oor or 
 SwWNOwNnwwwe GD G2. G9 MOH GS NON 
 
 OP 20 BCP GD BGO GE.GP 0 G0 GE. Go. G0 G6 G0.GO.|G0 0 60 G0 00 60 G0 40 GD Go GO A 60 GO-GO 66 40.00 60.60.60 G0 00.00 GO GD Go Gd GD G0 00 GD 
 NOT OT OTT OT Or ¢ ¢ 
 
 SSmaaonewns| ® 
 
 33° 
 
 TABLE XXVI.—LOGARITHMIC VERSED SINES 
 
 Vers. 
 
 9.207714 
 -208140 
 208566 
 . 208992 
 . 209418 
 209843 
 
 .210698 
 -211118 
 .211543 
 211967 
 
 9.212391 
 212815 
 218239 
 218662 
 214085 
 214508 
 214931 
 215854 
 
 215776 
 
 9.216620 
 217042 
 217463 
 217884 
 218305 
 -218726 
 
 219567 
 219987 
 220407 
 9.220826 
 221246 
 221665 
 222084 
 222503 
 222921 
 223340 
 223758 
 224176 
 
 9.225011 
 225428 
 225845 
 226262 
 226678 
 227095 
 227511 
 224927 
 228342 
 228758 
 229178 
 229588 
 230008 
 230418 
 230832 
 231246 
 231660 
 232074 
 232487 
 
 We} 
 
 210268 | 
 
 216198 | 
 
 -219146 | 
 
 224593 
 
 9.232901 
 
 D> D> D> SD D> D2. AE AAT AEAEAPABATAT AE HVAT GPF aP VP gg ge 
 
 Dit 
 
 Qooooeok Hee 
 SSRSRRRSSSS 
 
 Noe ye) sO < SO2aooooe eocec|ecocooo 
 PODS SESRSERSEEES CO CO C9 OL OT OT OF Ol $3 =F 
 
 2 D2 DISD DAIAHAPAUD ARDAIRMWAAwAOND 
 = ° 
 
 Ex. sec. 
 
 9.284122 
 
 -284631 
 
 -285139 
 
 -285647 
 - 286155 
 286663 
 287170 
 287678 
 . 288185 
 . 288692 
 289199 
 
 9.289705 
 290212 
 290718 
 291224 
 291730 
 290236 
 292742 
 293247 
 293753 
 294258 
 294763 
 295268 
 295772 
 296277 
 296781 
 297285 
 297789 
 298293 
 298797 
 299300 
 9.299803 
 
 300807 
 
 800809 
 
 801312 
 
 301815 
 
 802317 
 
 - 802820 
 
 - 808322 
 
 - 803824 
 
 304325 
 
 9.804827 
 . 805828 
 . 805830 
 .806331 
 806832 
 . 807333 
 807833 
 . 808334 
 808834 
 . 809334 
 
 9.309834 
 . 3103834 
 . 310884 
 .311333 
 .3811882 
 .312331 
 .312830 
 . 313329 
 . 318828 
 
 9.314326 
 
 =) 
 
 1 
 
 e 
 
 APP RL LR RR 
 
 9 OF OF OF 2 OT 3 FIV 
 
 PR oe 
 wD Go Go So Go GOH 
 
 WHWUEIWO Wh Wee 
 RVIAGIRIGROSIS BHSS 
 
 co 
 be 
 
 ‘) 
 
 ED ODD OD Oo Go GH Ow Ow Cw ow 
 SESSSE SBESRSere 
 
 oo a O0 oy O9 | 
 SSEERE 
 
AND EXTERNAL SECANTS. 
 
 257314 
 
 —————————— aan 
 
 a ee eae eC ae OE a et ee Se hee ets 
 | 
 34° | 35° 
 ( 
 ‘| Vers. | D.1".| Ex.sec.| D.1".|) ’ | Vers. D. 1”.| Ex. sec. |D. 1”. 
 0 | 9.232901 6.88 | 9.314326 8.32 0 | 9.257314 | 6.67 | 9.348949 8.15 
 1 . 233314 6.88 .814825 CPSU ike .257714 | 6.68 .344438 8.15 
 2 . 233727 6.87 -Sloaee. 8.30 9 .258115 | 6.67 344927 8.15 
 3} .234139 6.88 .315821 8.380 |} 3 | .258515 | 6.67 845416 8.13 
 4 . 234552 6.87 .3816319 8.30 || 4 .258915 | 6.65 .845904 S15 
 5 . 234964 6.87 .316817 8.28 5 .259314 | 6.67 .846393 8.13 
 6 . 235376 6.87 317314 8.28 6 259714 | 6.65 .346881 S13 
 7 .235788 6.85 .317811 8.30 || 7 .260113 | 6.65 .8473869 8.13 
 8 .236199 | 6.87 .318309 8.28 | 8 .260512 | 6.65 847857 8.13 
 9 . 236611 6.85 .318806 8.28 || 9 .260911 | 6.65 .3848345 8.83. 
 10 . 237022 6.85 .319303 8.27 || 10 | .261810 | 6.65 .348833 8.13 
 11 | 9.237483 | 6.85 | 9.319799 8.28 || 11 | 9.261709 | 6.63 | 9.349321 8.12 
 2 _237844 6.83 . 320296 8.27 || 12 .262107 | 6.63 .3849808 8.12 
 13 . 238254 6.85 .820792 §.28 13 .262505 | 6.638 .000295 8.12 
 14 | .238665 6.83 .821289 8.27 14 . 262903 | 6.63 . 850782 8.12 
 15 . 239075 6.83 .821785 8.27 | 15 .263301 | 6.62 .351269 8.12 
 16 .239485 § .82 . 322281 8.25 || 16 .263698 | 6.63 .3851756 8.12 
 ve .239894 6.83 .3822776 8.27 17 ‘264096 | 6.62 002243 8.12 
 18 240304 6.82 .3823272 8.27 || 18 .2644938 | 6.62 . 852730 8.10 
 19 . 240713 6.82 .3823768 8.25 || 19 | .264890 | 6 62 .858216 8.10 
 20 | . 241122 6.82 .324263 8.25 2 .265287 | 6.60 .300102 8.10 
 21 | 9.241531 6.82 | 9.324758 8.25 || 21 | 9.265683 | 6.62 9.354188 8.10 
 92 .241940 6.82 Soe Dee 8.25 || 22 .266080 | 6.60 .354674 8.10 
 93 . 212348 6.80 .320748 8.25 || 23 .266476 | 6.60 .3805160 8.10 
 24 242756 6.80 .326243 8.23 || 24 . 266872 | 6.58 .305646 8.08 
 25 .243164 6.80 .3826737 8.25 || 25 . 267267 | 6.60 .30601381 8.10 
 26 .243572 | 6.80 .827232 8.23 26 .267663 | 6.58 .856617 8.08 
 27 . 243980 6.7 .3826726 8.23 || 27 .268058 | 6.58 .807102 8.08 
 28 244387 6.78 .3828220 8.23 || 28 .268453 | 6.58 007587 8.08 
 29 244794 6.78 823714 8.22 || 29 .268848 | 6.58 .3858072 8.08 
 30 , 245201 6.7 .3829207 8.23 30 .269243 | 6.58 .808557 8.08 
 31 | 9.245608 6.77 | 9.829701 8.23 31 | 9.269638 | 6.57 9.359042 8.07 
 32 .246014 | 6.78 .330195 8.22 || 32 | .270032 | 6.57 .859526 8.08 
 33 246421 | 6.7 .330688 Sree O38 .270426 | 6.57 .3860011 8.07 
 34 . 246827 6.77 .331181 8.22 || 34 .270820 | 6.57 .860495 8.07 
 35 . 247233 Gree .33167 8.22 || 85 | .271214 | 6.57 .860979 8.07 
 36 .247639 | 6.75 .392167 8.20 || 36 .271608 | 6.55 ,861463 8.07 
 37 .248044 | 6.75 .332659 SEO Eas . 272001 | 6.55 .361947 8.07 
 38 .248449 6.75 Bs 13 1359 65) 8.20 38 , 272394 | 6.55 .362431 8.05 
 39 | .248854 6.75 .333644 8.22 39 272787 | 6.55 .3862914 8.07 
 40 | .249259 6.73 .334137 8.2 40 .273180 | 6.538 . 3863398 8.05 
 41 | 9.249664 6.73 | 9.324629 8.20 41°| 9.273572 | 6.55 | 9.863881 8.05 
 | 42 .250068 Gea fe saDl2t 8.18 || 4: . 273965 | 6.53 .864364 8.05 
 43 .250473 6.73 .335612 8.20 || 43 274357 | 6.53 .364847 8.05 
 44 .250877 | 6.73 .336104 8.18 || 44 274749 | 6.58 .865330 8.05 
 45 .251281 | 6.72 .335595 8.20 || 45 275141 | 6.52 .365813 8.03 
 46 .251684 6.7 831087 8.18 || 46 2755382 | 6.53 .866295 8.05 
 AT .252088 6.72 .837578 8.18 47 .275924 | 6.52 86677 8.03 
 48 .252491 | 6.72 | .338069 8.18 || 48 .276315 | 6.52 .3867260 8.03 
 49 . 252894 6.72 . 3838560 8.17 || 4 276706 | 6.52 367742 8.038 
 50 | .2538297 | 6.70 . 339050 8.18 || 50 277097 | 6.52 . 868224 8.03 
 51 | 9.263699 | 6.72 | 9.839541 8.17 51 | 9.277488 | 6.50 | 9.368706 8.03 
 2 .254102 | 6.7 .3840031 8.18 52 277878 | 6.50 .369188 8.03 
 53 254504 | 6.70 , 340522 8.17 53 | .278268 | 6.50 .3869670 8.02 
 54 . 254905 6.70 .841012 8.17 54 278658 | 6.50 .3870151 8.0% 
 55 . 255303 6.68 . 341502 8.15.1) 55 279048 | 6.50 .370632 8 03 
 56 .255709 | 6.70 .841991 8.17 | 56 279438 | 6.48 .871114 8.02 
 57 .256111 6.68 342481 8.17 BY 279827 | 6.50 011595 8.02 
 58 .256512 | 6.68 ,3842971 8.15 58 .280217 | 6.48 . 872076 8.00 
 59 .256913 | 6.68 . 843460 8.15 || 59 280606 | 6.48 7 .372556 8.02 
 60 : 9. ) 6.67 | 9.348949 §.15 60 | 9.280995 | 6.47 | 9.373037 8.02 
 
| 
 
 ~ 
 
 Vers. 
 
 Ex. sec. 
 
 TABLE XXVI.—LOGARITHMIC VERSED SINES 
 
 Vers. 
 
 o 
 
 . | Ex. sec, | 
 
 | 
 | 
 
 ODIDUIP WMHS 
 
 Vo) 
 
 © 
 
 co 
 
 ie) 
 
 9.280995 
 
 . 281883 
 251772 
 .282160 
 . 282548 
 . 282936 
 . 283324 
 .283712 
 . 284099 
 .284486 
 . 284873 
 . 285260 
 285647 
 . 286033 
 .286419 
 .286805 
 .287191 
 228757 
 
 5 28) 7962 
 . 2883848 
 . 288733 
 .289118 
 . 289502 
 289887 
 .290271 
 . 290655 
 . 291039 
 . 291423 
 . 291807 
 . 292190 
 .292573 
 . 292956 
 . 298339 
 . 2938722 
 . 294104 
 . 294486 
 , 9948) 68 
 2952. 50 
 . 295632 
 
 "996014 
 . 296395 
 296776 
 297 157 
 
 297% 538 
 
 7 297 918 
 998299 
 .298679 
 
 299059 
 
 (299439 
 
 299819 
 . 800198 
 "300577 
 800957 
 
 "301335 
 .801714 
 . 802093 
 302471 
 
 302849 | 
 
 303227 
 
 a 80: 3605 | 
 9.303983 
 
 9 OD asi corei mp oie ate a 50D GD ub f 
 
 © 
 
 Jo) 
 
 A AS AES AERTS ERE OO TEE peter SS na 
 
 9.373037 
 .0(8018 
 .343998 | 
 .3814478 
 874958 
 
 375438 
 
 375918 
 "376398 
 376877 
 
 17357 
 
 "377836 
 878315 
 318094 
 
 379273 
 
 819752 
 .880231 
 . 380709 
 .3881188 
 .881666 
 302144 
 .882622 
 .883100 | 
 888077 
 . 884055 
 "384532 
 . 885010 
 . 885487 
 885964 
 . 3886441 
 886918 
 
 388824 
 . 889300 
 889776 
 .890252 
 .3890727 
 .3891203 
 .3891678 
 392154 
 .892629 
 .3893104 
 .893579 
 . 394054 
 .3894529 
 .3895003 
 .895478 
 .395952 
 .896426 
 .3896900 | 
 8973874 
 .3897848 
 . 898322 
 .898795 
 . 399269 
 .899742 
 400215 
 400688 
 .401161 
 9.401634 
 
 © OF OD OTR WI WHO 
 
 e 
 a) 
 
 SSE Shes ee eee ee ee Se See Det he eS Be he Le Ee Eee Eee ere rerers! IAID-FWOMOMOOG 
 
 co 
 
 9.303983 
 
 304360 
 
 304738 
 
 305115 
 
 305492 
 305868 
 306245 
 306621 
 306998 
 307374 
 307749 
 308125 
 308501 
 303876 | 
 309251 
 309626 
 310001 
 31037 
 
 310750 
 311124 
 311498 
 
 .3811872 
 .812245 
 
 "312619 
 . 312992 
 
 313365 | 
 .313738 
 .ol4414 
 .314484 
 .314856 
 .315228 
 
 | 9.315600 
 815972 | 
 816344. 
 316716 
 
 317087 | 
 817458 | 
 817829 
 318200 
 31857 
 
 318941 
 
 .319311 
 .319682 
 .820051 
 820421 
 .820791 
 .821160 
 . 821530 
 .3821899 
 2822267 
 822636 
 . 823005 
 
 823373 
 
 323741 
 
 . 2374 
 
 351409 
 
 ood 77 
 
 O24845 
 
 825212 
 
 as 82! DE 580 
 
 | .825947 
 | 9.826314 
 
 So Od G2 GS. S32 S2 Gd G3 Gd Od OentesLae eee 
 WWW WWW HWY WW waa 2d 2 
 
 ee ee ee ee a eee 
 
 oO 
 
 © 
 
 Wwwwwwiwe 
 
 9.401634 
 .402107 
 .402580 
 .403052 
 .403524 
 .403997 
 .404469 
 .404941 
 .405412 
 .405884 
 .406356 
 .406827 
 .407298 
 
 407770 
 .408241 
 .408712 
 .409183 
 .409653 
 .410124 | 
 .410594 
 .411065 
 .4115385 
 .412005 
 .412475 
 .412945 
 .413415 
 .413854 
 .414854 
 ae 
 
 15293 
 
 “Ter63 
 416281 
 .416700 
 417168 
 417637 
 .418106 
 418574 
 419042 
 419511 
 .419979 
 420447 
 
 | 9. 420915 
 .421382 
 .421850 
 ,422317 
 .422785 
 4 93952 
 .423719 
 .424186 
 .424653 
 .425120 
 .425587 
 .426053 
 .426520 
 .426985 
 .427452 
 
 427918 
 
 .428384 
 .428850 
 .429316 
 9.429782 
 
 Oat IQVANINQ I 
 
 J PEER MEMENTO I a gg gg ng nz 2 
 Le ESE : ip PRT Aa hey epee NEG OA sin Dae en len Ch WE ways lI Nee gE ak 
 
AND EXTERNAL SECANTS. 
 
 89° 
 
 Vers. 
 
 el Eee SOC? 
 
 Vers. 
 
 S) 
 
 ix, sec. D. ale 
 ) 
 
 i/o) 
 
 We) 
 
 is) 
 
 9.826314 
 326681 
 827047 
 827414 
 3827780 
 328146 
 328512 
 3828878 
 
 3829243 | 
 
 .3829609 
 829974 
 
 330339 
 330704 
 
 .3831069 
 331433 
 
 831798 |: 
 
 .332162 
 832526 
 .332890 
 333254 
 333617 
 
 .38808981 | 
 
 334344 
 8384707 
 335070 
 335432 
 .8380795 
 
 836157 
 
 .836519 
 
 336881 
 
 337248 
 837605 
 3837966 
 838328 
 
 838689 | 
 
 .3839050 
 .339411 
 83977 
 .840132 
 340492 
 340852 
 841212 
 841572 
 
 341932 | 
 342291 
 
 842651 
 843010 
 3848369 
 84372 
 
 .844086 
 844445 
 2.44803 
 .845161 
 845519 
 845877 
 
 346235 
 346592 
 .846950 
 
 .347307 
 
 347664 
 
 9.348021 
 
 9.429782 
 430247 
 430713 
 .431178 
 .431643 
 432108 
 432573 
 .433038 
 433503 
 -433967 
 .4384482 
 
 9.434896 
 435361 
 435825 | 
 436289 | 
 436753 
 437217 
 437680 
 438144 
 -438608 
 439071 
 
 9.439534 
 .439997 
 .440460 
 -440923 
 .4413886 
 .441849 
 442312 
 442774 
 443237 
 443699 
 
 | 9.444161 
 444623 
 445085 
 445547 
 446009 
 446470 
 446932 
 447393 
 447855 
 448316 
 448777 
 449238 
 449699 
 450160 
 450620 
 451081 
 451541 
 452002 
 452462 
 452922 
 | 9.453382 
 453842 
 454302 | 
 454762 
 455221 
 455681 
 456140 
 456600 
 457059 
 9.457518 | 
 
 a3 <3 -3 
 | 
 
 OTOTOT OT I 
 
 a3 AF FF I HII 
 OMDMIRWUIP WW OS 
 
 C9 OT OD OTOH 
 
 co G2 Ol 
 
 a 2 FoF 
 
 : 3 Fe aPad alas = -5 
 WwWwWWNND WWHWwwwe 
 
 a2 J AJrI VWI 
 
 Oe cee Ce ew Lh oe oe he ae te he JINN VIIA Sods Sas Ha aes gs Te ad pt eS 
 
 Jes) 
 
 (Jes) 
 
 | Bap PAB AFA P AP AEAE AE PAPE EEA II 
 
 | 9.348021 
 | .348377 
 848734 
 349090 4 
 849446 
 849802 
 350158 
 
 350514 
 350869 
 351225 
 851580 
 
 9.351935 
 352290 
 352644 
 . 3802999 
 . 008000 
 .BD38707 | 
 .854062 
 .804415 
 .854769 
 .o0123 
 
 9.355476 
 .855829 
 .856182 
 . 856585 
 . 856888 
 857241 
 .8575938 
 .857945 
 358297 
 .858649 
 
 9.359001 
 . 8093853 
 359704 
 . 860056 
 .860407 
 .860758 
 .661108 
 .861459 
 .861810 
 362160 | 
 
 362510 
 362860 
 . 868219 
 .863560 
 . 363909 
 . 864259 
 .864608 
 .3864957 
 . 865806 
 . 865655 
 366003 
 .866352 
 .366700 
 .867048 
 .3867396 
 367744 
 .868091 
 368439 
 .368786 
 9 369183 
 
 CRON OT OT OT OT OT ONOTON CLOTOTOTOCOUT OUT OTOH CLOT ON OTOL OT ON OT OU N CLOTOLOT OV OT OT OT OV OT ON 
 io 2) 
 
 TOUOT OT 
 
 rovovorercy arororororere 
 QD 
 i=) 
 
 OU 
 
 e 
 
 orororor 
 ~ 
 
 Oo 
 oon 
 wo 
 
 ve) 
 
 ), 457518 
 457977 
 458436 
 458895. | 
 459355 
 459812 
 460270 
 460729 
 461187 
 461645 
 462108 
 
 9.462561 
 463019 
 463477 
 463934 
 464392 
 464849 
 465307 
 465764 
 466221 
 .466678 
 
 9.467135 
 467592 | 
 468049 
 468506 
 468962 
 469418 
 469875 
 470831 
 ATN787 
 471248 
 
 | 9.471699 
 
 AG 2155 
 
 472611 
 
 473067 
 
 473522 
 
 473978 
 
 474435 
 
 474888 
 
 475343 
 
 475798 
 
 476253 
 
 476708 
 
 477163 
 
 ATT6I8 
 
 48072 
 
 478527 
 
 478981 
 
 479435 
 
 479890 
 
 480344 
 
 .480798 
 
 481252 
 481705 
 .482159 
 482618 
 .483066 
 483520 
 .483973 
 484426 
 9.484879 
 
 | 
 
 3-3-5 
 
 AE AQ AATF P BF EET AI 
 
 BRAVA 
 
 morgorororororgrcdr ¢ 
 
 OO ort 
 
 UAT AF AQ AR ATAPAT AQP PAP PAPP PIA NIN NINN NINN NNNNNNNNNN 
 
 | 
 
TABLE XXVI.—LOGARITHMIC VERSED SINES 
 
 40° 
 
 41° 
 
 ‘|, Vers. | D. 1". | Ex. see;|.D. 1": 4] 4 | Vers. | D.1".| Ex. sec. |D. 1", 
 
 | eae} a | 
 
 0 | 9.369133 | 5.78 | 9.484879 | 7.55 || 0 | 9.389681 | 5.62 | 9.511901 | 7.45 
 
 1 869480 | 5.78 -4853832 | 7.55 1 .890018 | 5.63 012348 | 7.47 
 
 2 869827 | 5.7 485785 | 7.55 2 890356 | 5.63 | -.512796 | 7.45 
 
 3 BVO174 | 5.77 486238 | 7.55 3 390694 | 5.62 518243 | 7.47 
 
 4 870520 | 5.7 -486691 7.55 4 391031 | 5.62 -513691 | 7.45 
 
 5, .8(0867 | 5.77! .487144 | 7.53 5 3913868 | 5.62 -514138 | 7.45 
 
 6 3871213 | 5.77 |. .487596 | 7.55 6 .891705 | 5.62 -514585 | 7.47 
 
 de .oelpogy BiG -488049 | 7.53 7 -392042 | 5.62 .515033 | 7.45 
 
 8! .371905 | 5.7 .488501 | 7.53 || 8 392379 | 5.62 .515480 | 7.45 
 
 9 872251 |° 5.75 -488953 | 7.55 9} .892716 | 5.60 515927 | 7.45 
 
 10 | .372596 | 5.7 489406 | 7.53 || 10 | .393052 | 5.60 | .516374.| 7.43 
 
 11 | 9.372942 | 5.75 | 9.489858 | 7.53 || 11 | 9.393388 | 5.60 | 9.516820 | 7.45 
 
 12 38738287 | 5.7 -490310 | 7.53 || 12 893724 | 5.62 010267 | 7.45 
 
 13 373632 | 5.75: .490762 | 7.58 || 18 -894061 | 5.58 517714 | 7.48 
 
 14 .873977 | 5.75 | .491214 | 7.52 || 14 -394396 | 5.60 -518160 | 7.45 
 
 15 874822 | 5.75 .491665 | 7.53 || 15 394732 | 5.60 .518607 | 7.43 
 
 oi 16 .374667 | 5.7% 492117 | 7.53 || 16 395068 | 5.58) .519053 | 7.45 
 i 17 | .875011 |° 5.75 492569 | 7.52 |) 17 .895403 | 5.58 -519500 | 7.43 
 ie 18 | 3875856 | ° 5.7% 493020 | 7.52 |} 18 | .8957388 | 5.60 | .519946 | 7.43 
 i 19 -3875700 | 5.7% 493471 | 7.53 || 19 .896074 | 5.58 -5203892 | 7.43 
 
 i 20 876044 | 5.73 493923 | 7.52 || 20] .396409 | 5.57 520838 | 7.43 
 21 | 9.376388 | 5.73 | 9.49437 7.52 |] 21 | 9.396743 | 5.58 | 9.521284 | -7.43 
 22 8767382 | 5.7% -494825 | 7.52 || 2 897078 | 5.58 .0217380 | 7.43 
 Bil 23 377075 | 5.73; .495276 | 7.52 || 23 897413 | 5.57 022176 | 7.42 
 | 24 3((419 | 5.72 495727 | 7.52 || 24 897147 | 5.57 022021 | 7.43 
 25 307762 | 5.7% -496178 | 7.50 || 25 -898081 | 5.57 | .523067 | 7.43 
 
 fi 26 878105 | 5.'7% 496628 | 7.52 || 26) .3898415 | 5.57] .523513 | 7.42 
 
 i 27 378448 | 5.72 | .497079 | 4.52 || 27 .898749 | 5.57 523958 | 7.43 
 28 878791 | 5.7 497530 | 7.50 || 28 899083. | 5.57 .524404 | 7.42 
 
 i 29 | 3791383 | 5.7% 497980 | 7.52 || 29 899417 | 5.55 524849 | 7242 
 ' 30 | .379476 | 5.7 .4984380 | 7.5 30 | .3899750 | 5.57 | .525294 | 7.42 
 
 | 31 | 9.379818 |. 5.72 | 9.498881 | 7.48 || 81 | 9.400084 | 5.55 | 9.525739 | 7.42 
 
 : 32} .3880161 { 5.7 499331 | 7.52 || 82 | .400417 | 5.55 -526184 | 7.42 
 33 880503 | 5.7 .499781 | 7.50 || 33 | .400750 | 5.55 .026629 | 7.42 
 
 34] .380845 | 5.68! .500231 | 7.50 || 34 -401083 | 5.55 -O27074 | 7.42 
 
 35 .881186 | 5.7 .500681 |} 7.50 || 35 .401416 | 5.53 527519 | 7.42 
 
 36 .881528 | 5.68 .501131 7.50 || 36 -401748 | 5.55 .527964 | 7.42 
 ca 37 | .381869 | 5.70 | .501581 | 7.48 || 37} .402081 | 5.53] .528409 | 7.40 
 38 882211 3.68 50203 7.50 || 388 402413 | 5.53 -528853.| 7.42 
 | 3 .882552 | 5.68 502480 | 7.48 || 89] .402745 | 5.53} .529298 | 7.40 
 1 40} .882893} 5.68; .502929| 7.50 || 40] 1403077 | 5.5 -529742 | 7.42 
 41 | 9.383234 | 5.67 | 9.503379 | 7.48 || 41 | 9.403409 | 5.53 | 9.530187 | ¥.40 
 
 42 | .383574 | 5.68] .503828} 7.48 || 42] 1403741 | 5.53 .530631 | 7.40 
 
 | 43 883915 | 5.67 | .504277 | 7.48 || 43 404073 | 5.52 531075 | 7.40 
 tt 384255 | 5.67 504726 | 7.48 || 44 404404 | 5.53 -531519 | 7.40 
 
 45 6884595 | 5.67 .505175 | 7.48 |) 45 | .404736 | 5.52 .931963 | 7.40 
 
 46 .384935 [ 5.67 .505624 | 7.48 || 46 405067 | 5.52 5382407 | 7.40 
 
 7 885275 | 5.67 506073 | 7.48 || 47 | .405398 | 5.52 532851 | 7.40 
 
 48 .885615 | 5.67] .506522 | 7.48 || 48 405729 | 5.50 -533295 | 7.40 
 49 | 885955 | 5.65} .506971 | 7.47 || 49 | 406059 | 5.52 | ‘538739 | 28 
 
 50 | .386294 | 5.67 | .507419 | 7.48 || 50 -4063890 | 5.52 534162 | 7.40 
 
 o1 | 9.886634 | 5.65 | 9.507868 | 4.47 || 51 | 9.406721 | 5.50 | 9.534626 | 7.40 
 
 52 .886973 | 5.65 .508316 | 7.48 || 52 407051 | 5.50 535070 | 7.88 
 
 53; 1387312 | 5.65} .508765 | 7.47 || 538 -407381 | 5.50 85513 | 7.33 
 
 54 887651 5.63 509213 | 7.47 || 54 407711 | 5.50 535956 | 7.40 
 
 55 887989 | 5.65 -509661 | 7.47 || 55 | .408041 | 5.50 .536400 | 7.88 
 
 56 888328 | 5.63 -51)109 | 7.47 || 56 | 408371 | 5.4! 536843 7.53 
 
 57 . 3888666 5.65 .510557 Ge YN yay .408700 | 5.50 .537286 7.388 
 
 58 .889005 | 5.68 -511005 | 7.47 || 58 -409030 | 5.48 -537729 | 7.38 
 
 5D | 88938438 | 5.63 51145: 7.47 || 59 .409359 | 5.48 038172 | 7.88 
 
 60 | 9.889681 5.62 | 9.511901 7.45 || 60 | 9.409688 | 5.48 | 9.538615 7.58 
 
AND EXTERNAL SECANTS. 
 
 412972 
 
 413300 
 .413627 
 -418955 
 .414282 
 .414609 
 .414936 
 415263 
 .415589 
 415916 
 416242 
 
 .416568 
 -416894 
 .417220 
 6417546 
 .417871 
 .418197 
 .418522 
 .418848 
 419173 
 .419498 
 
 9.419822 
 
 -420147 
 420471 
 420796 
 -421120 
 .421444 
 -421768 
 422092 
 422416 
 422789 
 
 9.423063 
 
 -423586 
 -423709 
 -424082 
 ~424355 
 424677 
 
 425000 
 
 2425045 
 -425967 
 
 | 9.426289 
 
 426611 
 426953 
 427254 
 427576 
 427897 
 .428218 
 428529 
 
 428060 
 
 9.429101 
 
 Ov Or Or orez OW 
 
 oP PR 
 
 aD if 
 909 G9 G9 ves 
 
 NGS 
 C9 W GI OD 
 
 =o 
 
 OR 
 OWW WW 
 
 OV? OVO OT OT ET OT OT OVOTOU OVLOTOV OT OCOT OT OT OU OU OCrorvor ew Ov or ororor or or 
 nS 
 
 ~) 
 
 ~ 
 
 oe 
 COwCce 
 
 ree 
 
 oa 
 
 20D wwwowwomwe 
 
 NaI NNW 
 
 OP OT OCOTOUOTOUOT OT GTOTOTOTOLOUOITOTOTOU OLOTOTOTOV OT OVS Or 
 
 oo Ww Go we wows: 
 Ce SUE STOUT OT OURS Ot 
 
 Or 
 
 Vers. . | Ex. sec. 
 | 9.409688 48 | 9.588615 
 | 410017 48 } .539058 
 .410346 48 .5389500 
 .410675 48 .589943 
 .411004 47 .540386 
 .411332 47 .540828 
 .411660 48 541271 
 -411989 47 .541713 
 .412317 45 .542155 
 .412644 47 542597 
 47 
 
 .543040 
 
 9.543482 
 548924 
 .544366 
 .544807 
 545249 
 .545691 
 .546132 
 .546574 
 .5ATOI5 
 .547457 
 
 9.547898 
 
 .548339 
 
 .548781 
 
 549222 
 
 .549663 
 
 .550104 
 
 .550544 
 
 .550985 
 
 551426 
 
 .551867 
 
 .552307 
 
 .552748 
 
 .553188 
 
 .5d58029 
 
 .554069 
 
 .554509 
 
 .554949 
 
 .500889 
 
 .555829 
 
 .556269 
 
 9.556709 
 
 .557149 
 
 .557589 
 
 .558028 
 
 .558468 
 
 .558907 
 
 559847 
 
 .559786 
 
 .560226 
 
 .560665 
 
 .561104 
 
 561548 
 
 .561982 
 
 562421 
 
 .562860 
 
 568299 
 
 .5637388 
 
 256-4176 
 
 .564615 
 
 9.505053 
 
 Jo} 
 
 ve) 
 
 Vers, 
 
 Ex. sec. 
 
 COMOAIHUS WOH © 
 
 — 
 
 aS aQay WEY ATF Y AIPA IF FI IFN VAIN NRA INNS NVRNAANNNNN ANRQRARRRRRER 
 
 WMWWMWWW WWOWe 
 
 QO 
 
 wow 
 
 peo wwwWwIwN w CWO wWtD Ww 
 
 a3 3 3 3-3-7 
 
 —~ 
 29 
 nH 
 
 9.429181 
 .429502 
 429822 
 -480142 
 .430463 
 430783 
 .431103 
 431422 
 .431742 
 -432062 
 482381 
 
 9.432700 
 433020 
 .433339 
 433657 
 .433976 
 4384295 
 .434613 
 434982 
 485250 
 435568 
 
 9.435886 
 .436204 
 .436521 
 .436839 
 -437156 
 .437473 
 437791 
 438107 
 -438424 
 438741 
 
 9.439058 
 43937 
 -439690 
 -440007 
 .440328 
 -440639 
 440954 
 -441270 
 -441585 
 441901 
 
 9.442216 
 442531 
 442846 
 .443161 
 .443476 
 .443790 
 .444105 
 444419 
 .444733 
 -445047 
 .445361 
 445675 
 .445989 
 .446302 
 446616 
 -446929 
 447242 
 447555 
 447868 
 9.443181 
 
 ie) 
 
 OOO OTT OC OT OT OT UOT OTT OT, HOT OW OT OWOT UOT UT, OVWOT OWT OE 
 WNVWNVYW VWNVVVKVVVAN wz : ¢ 
 
 =) 
 
 CTO OTTO 
 
 | 9.565053 
 
 565492 
 .565930 
 .566369 
 .566807 
 567245 
 .567683 
 .568121 
 .568559 
 .568997 
 .569435 
 
 9.569873 
 .570311 
 570748 
 571186 
 .571624 
 .572061 
 572498 
 572936 
 573373 
 573810 
 
 9.574247 
 574685 
 575122 
 575558 
 .575995 
 .576482 
 .576869 
 .577306 
 
 577742 
 
 (((4t0 
 
 578179 
 .578615 
 ~5 19052 
 .59 9488 
 79924 
 .580361 
 580797 
 .581233 
 .581669 
 .582105 
 .582041 
 582977 
 .583413 
 583848 
 .584284 
 284720 
 .585155 
 .585591 
 ‘586026 
 .586462 
 
 .586897 
 
 587332 
 
 58V767 
 .588203 
 .5886538 
 589073 
 589508 
 589942 
 590377 
 .590812 
 991247 
 
 . . fs 
 
 — 
 
 OOo re rero reir iss, 
 
 a} AT AT AY AF PA PAP AHF EAE PPI I II 
 
 OF AE AF nF nF ag Ba PTF I III NNN NNN NNN 
 
TABLE XXVI.—LOGARITHMIC VERSED SINES 
 
 44° 45° 
 : i 
 
 ’ | Vers. | D..1". | Ex. sec. | D. 1’. 4 Vers. 22D, |MEx. see! Dal: 
 O | 9.448181 | 5.20 | 9.591247 | 7.23 || 0 | 9.466709 | 5.08 | 9.617224 | 7.20 
 1 .448498 | 5.22] .591681-| 7.25 1 .467014.| 5.08! 617656 | 7.18 
 2} .448806 | 5.20] .592116-) 7.25 || 2] .467319-| 5.08 |- .61808% | 7.18 
 3 | .449118 | 5.22) .592551-| 7.23 3 | .467624./ 5.07 | .618518 | 7.18 
 41 .449431 | 5.2 .592985.| 7.23 || 4] .467928 | 5.08} .618949 | 7.18 
 5 | .449748 | 5.2 .593419.| 7.25 5 | .468233.| 5.07 | .619380 | 7.18 
 6 .450055 | 5.18 .5938854.|  '7.28 6 | .468587./ 5.07 | .619811 | 7.18 
 7 | .450366 | 5.20} .594288.' 7,93 7 | 1468841] 5.07 | .620242 | 7.18 
 8 | .450678 | 5.20 | .594722.) 7.23 8 | .469145 | 5.07 | .620673 | 7.18 
 9 .450990 | 5.18 | .595156.| 7.25 9| .469449 | 5.07 | .621104 | 7.18 
 10 | .451201 } 5.18 | .595591 | 7.23 || 10] .469753 | 5.07 | .621535 | 7.18 
 11 | 9.451612 | 5.20 | 9.596025 | 7.23 || 11 | 9.470057 | 5.05. 9.621966 | 7.17 
 12} .451924 } 5.18] .596459.} 7.23 || 12 | .470360.) 5.07 | .622396 | 7.18 
 138} .4522385 | 5.18] .596893.| 7,22 13 | .470664 | 5.05 | .622827 | 7.18 
 14. | .452546 |} 5.17] .597826 | 7.23 || 14 | .470967.| 5.05 | .623258 | 7.17 
 15 | . .452856 | 5.17 | .597760.| 7.23 | 15 | .471270 | 5.05 | .628688 | 7.18 
 16 | .453167 | 5.18 | .598194.| 7.23 || 16 | .471573 | 5.05 | .624119 | 7.17 
 7 | ~ 458478) 5.1% .598628. | 7.22 || 17 | .471876 | 5.05 | .624549 | 7.18 
 18 | .458788 | 5.17 | .599061.| '7.23° | 18 | .472179 | 5.05 | .624980 |- 7.17 
 19} .454098 | 5.17] .599495.| 7.22 || 19 | .472482 | 5.08 | .625410 + “748 
 20} .454408 | 5.1% | .599928.| 7.23 || 9 472784. | 5.05 | .625841 | 7.17 
 21 | 9.454718 | 5.17 | 9.600862 | 7.22 |} 21 | 9.478087 | 5.03 | 9.626271 | 7.17 
 22 | .455028 | 5.17 | 600795 | 7.23 || 22 | - .473889 | 5.08 | .626701 | 7.17 
 23 | .455388 | 5.17 | .601229.; 7.22 || 28.) 478691 | 5.03 | .627131 | 7.17 
 24} .455648 | 5.15 .601662. | 7.22 |) 24] .473998 | 5.08 | .627561 | 7.17 
 25 | .455957 | 5.17 602095 | 7.22 || 25 .474295 | 5.03 | .627991 | 7.17 
 26 | .456267 | 5.15 .602528 | 7.23 || 26 474597 | 5.03 | .628421 | 7.17 
 27 | 1456576 | 5.15 602962 | 7.22 || 27 | .474899 | 5.02 | 628851 | 7.17 
 28 | .456885 | 5.15 603595. | 7.22 || 28) .475200 | 5.03 | .6292981 | 7.17 
 29 | .457194 | 5.15 | .603828 | 7.22 || 29 | .4'75502.| 5.02 | .629711 | 7.17 
 30.| .457503.| 5.18 604261 | 7.22 || 80 | .475803 | 5.02 | .630141 | 7.17 
 31 | 9.457811 | 5.15 | 9.604694 | 7.20 |! 31 | 9.476104 | 5.02 | 9.680571 | 7.17 
 32 |- .458120 | 5.15 | 605126 | 7.22 | 82 | .476405 | 5.02 | .631001 | 7.15 
 83} .458429-| 5.13 605559 | 7.22 || 83 | .476706 | 5.02 | .631430 | 7.17 
 34} .4587387 | 5.13 .605992.| 7.22 || 34 477007 | 5.02.| .631860 | 7.17 
 85 | .459045 | 5.13 .606425 | 7.20 || 385 | .477308 | 5.00 | .682290 | 7.15 
 36 .459353 | 5.13! .606857 | 7.22 || 36 | .477608 | 5.02 | .682719 | 7.17 
 87 | .459661 | 5.18 .607290 | 7.20 || 87 .477909 | 5.00 | .683149 | 7.15 
 88 | .459969 | 5.13 607722 | 7.22 || 88 | .478209.| 5.00 | .638578 | 7.17 
 3 .460277 | 5.12 .608155 | 7.20 |! 89 | .478509.) 5.00 | .634008 | %.15 
 40.| .460584.| 5.13 .608587 | 7.22 || 40 | .478809.| 5.00 | .684437 | 7.15 
 41 | 9.460892 | 5.12 | 9.609020 | ‘7.20 || 41 | 9.479109 | 5.00 | 9.684866 | 7.17 
 42 | .461199 } 5.12 -609452 | 7.20 |) 42°} .479409.| 5.00] .635296 | 7.15 
 43 | .461506 | 5.12] .609884 | 7.20 || 48 | .479709.| 5.00 | .635725 | 7.15 
 44 | .461813 | 5.12 .610816 | 7.22 || 44 .480009 | 4.98 | .686154 | 7.15 
 45 | .462120 | 5.12 610749 | 7.2 45 | .480308 | 5.00 | .636583 | 7.15 
 46 | .462427 | 5.12 -611181.| 7.20 || 46 | .480608 | 4.98 | .687012 | 7.15 
 47 | 462734 | 5.10 611613; 7.20+| 47 | .480907 | 4.98 | .637441 | 7.15 
 48 | .463040 | 5.12} .612045.|° 7.20 || 48 | .481206 | 4.98 | .637870 | 7.15 
 49 | .463347 | 5.10] .612477-) 7.18 || 49 | .481505.| 4.98 | .688299 | 7.15 
 50 | .463653 ) 5.10 | .612908 | 7.20 || 50) .481804 , 4.98] .688798 | 7.15 
 51 | 9.463059 | 5.10 | 9.613340 | 7.20 || 51 | 9.482103 | 4.97 | 9.639157 | 7.15 
 O% | .464265 | 5.10 | .613772.| 7.20 || 52 | 482401 | 4.98 639586 | 7.15 
 53 | .464571 | 5.10 | .614204-| 7.18 || 58] .482700 | 4.97 | .640015 | 7.13 
 54 | .464877 | 5.10] .614635.| 7.20 || 54 .482998 | 4.97 | .640443 | 7.15 
 55 | .465183 | 5.08 | .615067.| 7.20 || 55 | .483296 | 4.98 640872 | 7.15 
 56 | .465488 | 5.10-| .615499-| 7.18 |! 56 483595 | 4.97 .641301 | 7.18 
 57 | .465794 | 5.08.| .615930.| 7.20 7 | .4888938:] 4.97 | .641729 | 7. 
 
 58 | .466099 | 5.08} .616362.| 7.18 || 58 | .484191 | 4.95 | .642158 | 7. 
 
 59 | .466404 | 5.08 | .616793.} 7.18 |' 59 | .484488 | 4.97] .642586 1 7. 
 
 60 | 9.466709 | 5.08 | 9.617224 1 7.20 | 60]! 9.484786 | 4.97 | 9.643015 | 7. 
 
S BABAR WWMS | 
 
 Vers. 
 
 AND EXTERNAL SECANTS. 
 
 o 
 
 |/Ex. sec. 
 
 pt 
 
 Vers. 
 
 o 
 
 r ot EX, SCC 
 
 | 9.484786 
 .485084 
 .485381 
 
 -485678 | 
 
 .485976 
 .486273 
 .486570 
 486866 
 487163 
 
 487460 
 487756 
 
 9.488053 
 .488349 
 .483645 
 .4889 41 
 .489237 
 .489533 
 . 489828 
 
 .490124 
 
 490419 
 
 .490714 | 
 
 9.491010 
 491305 
 .491600 
 .491894 
 
 -492189 
 
 492484 
 492778 
 
 493072 | 
 
 .492367 
 .493661 
 9.493955 
 -494249 
 
 494542 | 
 .494836 | 
 
 .495130 
 .495423 
 .495716 
 .496009 
 . 496302 
 .496595 
 
 | 9.496888 
 | 1497181 
 497473 
 497766 
 .498058 
 
 -498350 
 493643 
 
 .493935 
 .499226 
 .499515 
 .499810 
 pOO101 
 .500393 
 .500684 
 .500975 
 .501266 
 
 =) 
 
 501557 
 
 .501848 
 
 502139 
 
 9.502429 
 
 le) 
 
 cers 
 
 | 9.648015 
 643443 
 643872 
 .644300 
 
 .644728 
 645156 
 
 645585. | 
 .646013 
 .§46441 
 . 646869 
 .647297 
 647725 
 .648153 
 .648581 
 .649009 
 649436 
 .649864 
 . 650292 
 .650720 
 .651147 
 .65157 
 
 9.652002 
 . 652430 
 .652857 
 . 6538285 
 .6538712 | 
 . 654140 
 . 654567 
 .654994 
 .655421 | 
 .655849 
 .656276 
 .656703 
 .657130 
 .657557 
 .657984 
 .658411 
 .658833 
 .659265 
 .659691 
 .660118 
 
 .660545 
 .660972 
 .661898 
 
 .661825 
 .662252 
 .662578 
 .663105 
 .663531 
 .663958 
 .664884 
 .664810 
 665237 
 ~665863 
 6660389 
 666515 
 666942 
 .667368 
 667794 
 668229 
 9.668646 
 
 OOP WmOr © 
 
 1?) 
 
 “I 
 
 fee peek peek peek ek et pt et 
 wowuqnwqwoowl! |= 
 
 a as 
 WWE OOM 
 
 JIIINIINNT AWAVIANANAN NNVAINABANN NAANINAAANN ANAARAASAAS 
 
 nF AFaJY-I-F-IW-I-I-F 
 
 | 9.502429 
 
 .502720 
 .503010. 
 . 503300 
 
 .503591 | 
 
 .508881 
 504171 
 504460 
 504750 
 505040 
 .505329 
 
 9.505618 
 
 .505908 
 ~506197 
 .506486 
 506775 
 .507063 
 .507852 
 .507640 
 .507929 
 .508217 
 
 9.508505 
 
 508793 
 .509081 
 .509369 
 509657 
 509945 
 . 510232 
 .510520 
 .510807 
 .511094 
 
 9.511381 
 
 .511668 
 .511955 
 512241 
 512528 
 512815 
 .513101 
 .513387 
 .513673 
 .513959 
 
 9.514245 
 
 .514531 
 .514817 
 .515102 
 .515388 
 
 .515673 | 
 
 .515959 
 516244 
 .516529 
 516814 
 
 | 5 9.517098 
 B hoe (BTTS85 
 
 .517668 
 
 .517952 | 
 
 .518236 
 518521 
 .518805 
 .519089 
 519378 
 
 9.519657 | « 
 
 PA. A ROR RR RR 
 a 
 ed 
 
 AtALA AL AAA ALAA AA AAA RADAR AAAS AR RRR ARR 
 Mp as eet BBB RIRRINAY : 
 
 W 09 09 0 OH OT OD C9 OT OF 
 
 82 
 
 4.82 
 
 C-~30000 
 
 Roots 
 
 aI FI IIIVIAVINA 
 Son! 3 
 
 QIN 
 
 09 OTOTOT SS OF 2 OT 3 
 
 | 9.668646 
 669072 
 | 669498 
 669924 
 . 670350 
 670776 
 671201 
 671627 
 672053 
 672479 
 .672904 
 
 9.673330 
 673756 
 .674181 
 674607 
 675082 
 675458 
 675883 
 .676809 
 676734 
 677159 
 
 9.677584 
 .678010 
 678435 
 678860 
 679285 
 .679710 
 .680136 
 680561 
 . 680986 
 .681411 
 
 9.681836 
 . 682260 
 682685 
 .683110 
 683535 
 .683960 
 .684385 
 . 684809 
 685234 
 685659 
 
 9.686083 
 686508 
 686933 
 687357 
 687782 
 .688206 
 688631 
 689055 
 689479 | 
 . 689904. 
 
 | 9.690328 
 
 690752 | 
 
 .691177 
 
 .691691 
 
 692025 
 
 .69 2449 
 
 -692873. | 
 
 693298 
 693722 
 
 9.694146 
 
 =z 
 
 Bap ag ay aPaT EE PT I TE TIE SE SS I I I 
 
 AF AB aS aS AS APA TBP 
 
48° 
 
 TABLE XXVI.—LOGARITHMIC VERSED SINES 
 
 Ex. sec. 
 
 * Vers, »4iD,.1" 1a 
 
 0 | 9.519657 | 4.72 | 9.694146 | 7.07 
 
 519940 | 4.73 | .694570 | 7.07 
 
 2 |» 520224 | 4.72) .694994 | 7.07 
 
 3 | .520507 | 4.73 | .695418 | 7.07 
 
 4/ > 520791 | 4.721 .605842 | 7.07 
 
 D | .521074 | 4.72 -696206 | 7.05 
 
 6 5213857 | 4.72 .696689 | 7.07 
 
 < |. .521640) 4:72 6971138 | 7.07 
 
 8 521923 | 4.72 697537 | 7.07 
 
 9 922206 | 4.70 697961 | 7.07 
 
 10 522488 , 4.72 -698385 | 7.07 
 11 | 9.52277 4.72 | 9.698809 | 7.05 
 12 | .523054 | 4.7 .699232 | 7.07 
 13 | - 523336 | 4.7 .699656 | %.07 
 14 523618 | 4.7 .700080 | 7.05 
 15 | .523900 | 4.4 .700503 | 7.07 
 16 024182 | 4.7 ~TOC92T | 7.05 
 17 024464 | 4.7 701850 | 7.07 
 18 -024746 | 4.7 TOLTT 7.07 
 19 | .525028 | 4.68) .702198 | 7.05 
 20 525309 | 4.70 | .702621 | 7.07 
 21 | 9.525591 | 4.68 | 9.703045 | 7.05 
 22; .525872 | 4.68 103468 | 7.05 
 23] 526153 | 4.7 -703891 | 7.07 
 24) 526485 | 4.68) .704315 | 7.05 
 25} .526716 | 4.68] .704788 | 7.07 
 26 526997 | 4.67; .705162 | 7.05 
 27 527277 | 4.68 705585 | 7.05 
 28 | .52755: 4.68 (06008 | 7.05 
 29 | .527839 | 4.67 706431 | 7.07 
 30 -528119 | 4.68 706855 | 7.05 
 31 | 9.528400 | 4.67 | 9.707278 | 7.05 
 32 | .528680 | 4.67 -WO7T7T01 | 7.05 
 33 | .528960 | 4.67 708124 | 7.05 
 34] .529240 | 4.67 108547 | 7.07 
 35 | .529520 | 4.67 -C08971 | 7.05 
 36 | .529800 | 4.67 709394 | 7.05 
 37 | 5380080 | 4.65 | .709817 | 7.05 
 38 -580359 | 4.67 410240 | 7.05 
 39 -530639 | 4.65 -710663 | 7.05 
 40 -080918 | 4.67 -711086 | 7.05 
 41 | 9.531198 | 4.65 | 9.711509 | 7.05 
 42 | .5381477 | 4.65 | 711982 | 7.05 
 43 | .531756 | 4.65 | .712355 | 7.05 
 44} .5820385 | 4.65 712778 | 7.03 
 45 | .532314 | 4.63 | .718200 | 7.05 
 46 | .5382592 | 4.65) .713623 | 7.05 
 47 | 582871 | 4.65} .714046 | 7.05 
 48 | .533150 | 4.63 | .714469 | 7.05 
 49 | .533428 ) 4.63] .714892 | 7.05 
 50, .533706 | 4.65 715815 | 7.08 
 51 | 9.533985 | 4.63 | 9.715787 | 7.05 
 52 | .5384263 | 4.63 | .716160 | 7.05 
 53} .534541 | 4.63] .716583 | 7.03 
 54] .534819 | 4.63] .717005 | 7.05 
 55 | .585097 | 4.62 | .717428 | 7.05 
 56] .585374 | 4.63 | .717851 | 7.08 
 57 | .585652 | 4.62 | .718273 | 7.05 
 58 | .535929 | 4.63 | .718696 | 7.03 
 59 | .5386207 | 4.62 | .719118 | 7.05 
 60 | 9.586484 | 4.62 | 9.719541 | 7.05 
 
 CwroomewHe | a 
 
 re 
 (or) 
 
 Wt 
 
 -551299 
 551571 
 -551842 
 .5d2118 
 
 552384 
 
 9. 
 
 552656 
 552927 
 
 52 
 
 _ 
 gr 
 oO 
 
 . 742330 
 742751 
 . 748173 
 743595 
 744016 
 . 744438 
 . 744859 
 
 Vers. |D.1”.| Ex. see. 
 9.586484 | 4.62 | 9.719541 | 
 .586761 | 4.62 ~ 719964 
 .587038 | 4.62 . 720886 
 .587315 | 4:62 . 726809 
 -5387592 | 4.62 . 721231 
 .5387869 | 4.60 - 721653 
 -588145 | 4.62 . 722076 
 .588422 | 4.60 . 122498 
 .5388698 | 4.60 722921 
 .538974 | 4.62 723343 
 .539251 | 4.60 1238765 
 9.539527 | 4.60 | 9.724188 
 .539803 | 4.60 . 724610 
 .540079 | 4.58 725082 
 .540854 | 4.60 725454 
 .540630 | 4.60 425877 
 -540906 | 4.58 - 726299 
 .541181 | 4.58 (26721 
 .541456 | 4.60 (27148 
 -541732 | 4.58 127565 
 .542007 | 4.58 727988 
 9.542282 | 4.58 | 9.728410 
 .542557 | 4.58 . 7288382 
 -542882 | 4157 729254 
 -543106 | 4.58 . 729676 
 .543381 | 4.57 730098 
 .543655 | 4.58 . 7380520 
 .543930 | 4.57 . 730942 
 .544204 | 4.57 . 731364 
 .544478 | 4.57 . 731786 
 544752 | 4.57 7382208 
 9.545026 | 4.57 | 9.732630 
 .5453800 | 4.57 . 738052 
 1545574) 4257 (88474 
 .545848 | 4.55 . 733896 
 .546121 | 4.57 . 734817 
 .546395 | 4.55 734739 
 .546668 | 4.55 . 7385161 
 .546941 | 4.55 1385583 
 .547214 | 4.55 . 736005 
 547487 | 4.55 . 186427 
 9.547760 | 4.55 | 9.736848 
 .548038 | 4.55 1B 270 
 .5483806 | 4.55 . 137692 
 548579 | 4.58 .7388114 
 .548851 | 4.55 . 738535 
 .649124 | 4.53 . 788957 
 .549396 | 4.53 ~ 189879 
 .549668 | 4.53 . 739800 
 .549940 | 4.538 . 740222 
 550212 | 4.53 | .740644 | 
 9.550484 | 4.53 | 9.741065 
 .550756 | 4.538 741487 
 .551028 | 4.52 . 741908 
 4 he 
 4. 
 4. 
 4. 
 4. 
 4. 
 
 . 
 CN 
 
OUR IHS | 
 
 Jo) 
 
 AND EXTERNAL SECANTS. 
 
 .5d4009 
 .554280 
 .554550 
 .554820 
 .555091 
 .555361 
 .555631 
 .555900 
 .556170 
 . 556440 
 .556709 
 .556979 
 .557248 
 Beta fecn ye 
 .557786 
 .5D8055 
 . 558324 
 558593 
 . 558862 
 .559131 
 .559899 
 .559667 
 .559936 
 .560204 
 .560472 
 .560740 
 .561008 
 .56127 
 .561544 
 .561811 
 .562079 
 . 562346 
 .562613 
 .562881 
 .563148 
 .563415 
 . 563682 
 
 9.563948 
 
 -564215 
 -564482 
 -564748 
 .565015 
 .565281 
 .565547 
 .565813 
 .566079 
 .966345 
 
 .566611 
 566877 
 .567142 
 .567408 
 567673 
 567938 
 568204 
 568469 
 .568734 
 
 9.568999 
 
 D&S 
 
 = 
 
 ee OF 
 Co 
 
 So 
 
 LAA LAA RAR A RRR RARER ABRE BRERA BRROROE 
 DOAKAIARKIARI’ QBIRVRQRDRWIRHGH BHDDHDOBDSS 
 
 ALR APRA RAR PROD 
 9028209 COW Www WwoIW OO 
 
 > 
 C29 CO Oe 
 
 S88 
 
 LA a OTC eT’ 
 
 ee 
 0 0 
 
 Ex. sec. 
 
 9.744859 
 ~ 745280 
 745702 
 -746123 
 ~ 746545 
 . 746966 
 . 747388 
 747809 
 74823 
 
 . 748652 
 . 749073 
 749494 
 .749916 
 - 750887 
 750758 
 751180 
 . 751601 
 - 752022 
 . (52443 
 . 752865 
 . 1538286 
 - 753707 
 - 754128 
 154549 
 154971 
 . 155892 
 155813 
 7156234 
 756655 
 - 757076 
 757498 
 757919 
 . 758340 
 ~ 158761 
 759182 
 759603 
 . 760024 
 . 7160445 
 . 760866 
 . 761287 
 - 761708 
 762129 
 762550 
 762971 
 763392 
 - 763813 
 - 764234 
 . 764655 
 . 765076 
 . 765497 
 . 165918 
 . 766339 
 . 766760 
 167181 
 . 767602 
 768022 
 - 768443 
 . 768864 
 . 769285 
 . 769706 
 9.770127 
 
 _ 
 S er tenernionas ia | = 
 
 md Ag ag a) oF od og a 
 
 > Ee a J 
 
 wr 
 
 ches be Se oe Se oe MS Ec rs 
 
 z 
 
 ~ 
 
 WE AB AE AT AF AF TFA 
 
 | 
 
 | 
 
 Vers. 
 
 9.568999 
 569264 
 .569528 
 569793 
 570057 
 570322 
 -570586 
 570850 
 571114 
 -571878 
 
 571642 
 
 571906 
 572170 
 972434 
 972697 
 .972960 
 -573224 
 573487 
 573750 
 -974013 
 074276 
 9.574539 
 974802 
 575064 
 575327 
 575589 
 575852 
 -976114 
 .576376 
 .5766388 
 -576900 
 
 577162 
 
 Vo) 
 
 is) 
 
 O7TT424 | 
 
 577685 
 BU947 
 578208 
 578470 
 578731 
 578992 
 T9253 
 519514 
 9.579775 
 580036 
 580297 
 580557 
 580818 
 581078 
 581339 
 581599 
 581859 
 582119 
 
 9.582879 
 
 582639" 
 
 582898 
 .583158 
 .583418 
 583677 
 583936 
 584196 
 584455 
 9.584714 
 
 Ex. sec. | 
 
 | 9.770127 
 
 770548 
 710969 
 771389 | 
 771810 
 
 72652 
 773073 
 7 
 73014 
 714335 
 
 “782330 
 82751 
 9.78317 
 “783592 
 784013 
 “784433 
 “784854 
 785215 
 “785696 
 786116 
 "786537 
 786958 
 9.787378 
 787799 
 “788220 
 “788641 
 789061 
 89482 
 “789908 
 “79032: 
 790744 
 791165 
 9.791586 
 “792006 
 “792427 
 “792848 
 “793268 
 “793689 
 “794110 
 “794581 
 “794951 
 9.795372 
 
 SEE EE TEE TE Ty a Ba EE a a aa a ng ng ng ng ng a og 
 
TABLE XXVI.—LOGARITHMIC VERSED SINES 
 
 ~ 
 
 pet 
 SODIARTR OTIS | 
 
 © 
 
 © 
 
 Vers. 
 
 9.584714 
 
 .584973 
 585232 
 .685491 
 .585749 
 .586008 
 586266 
 .5E6525 
 .586783 
 587041 
 .587299 
 
 587557 
 587815 
 .588073 
 .588331 
 588588 
 588846 
 .589103 
 589861 
 .589618 
 589875 
 590132 
 590389 
 .590646 
 .590903 
 .591160 
 .591416 
 .591673 
 591929 
 .592185 
 592442 
 
 592698 
 592954 
 593210 
 593466 
 593721 
 593977 
 594233 
 .594488 
 594748 
 .594999 
 
 .595204 
 .595509 
 .595764 
 .596019 
 .596274 
 .596528 
 .596783 
 .597038 
 .597292 
 .597546 
 
 597801 | 
 
 598055 
 .598309 
 598563 
 .598817 
 599071 
 .599324 
 .599578 
 .599831 
 9.600085 
 
 Aw. AR RRO 
 39 Go O9 G9 G9 bo G9 by by 
 SESH SUND 
 
 Vo) 
 
 Ex. sec. 
 
 ~ 
 
 Vers. 
 
 "| Ex. see 
 
 1953872 
 2795798 
 (96213 
 7966384. 
 2197055 
 T9746 
 197896 
 
 .798317 
 
 798788 
 799158 
 199579 
 .800000 
 800421 
 .800841 
 -801262 
 .801683 
 .802104 
 .802524 
 802945 
 .803366 
 
 803787 
 
 9.804207 
 , -804628 
 
 -805049 
 .805470 
 »805891 
 .806311 
 806732 
 807158 
 807574 
 «807995 
 808415 
 .808836 
 
 , +809257 
 
 .809678 
 .810099 
 .810520 
 .810940 
 .811361 
 
  ,811782 
 
 812208 
 
 812624 > 
 
 .818045 
 
 1» .818466 
 ' (813886 
 » 1814307 
 
 814728 
 .815149 
 .815570 
 .815991 
 
 ; 816412 
 
 9.816833 
 ; 1817254 
 i .817675 
 , .818096 
 
 .818517 
 
 ; 1818938 
 
 1819359, 
 .819780 
 820201 
 
 9 -820622 
 
 =) 
 vo 
 { 
 
 i=) 
 
 SOWWW 
 
 We 
 
 Oo 
 
 COOIAIOUP WMH © 
 
 Se Scoocessooe 
 _ 
 
 WD WW 
 
 S 
 OWS 
 
 AE AE AT AF AE ATVB ATA AE I NINN NNN NNNNNNNNNN NNNNNNNNNS NNN ANARANN 
 
 9.600085. | 
 
 .600338 
 .600591 
 . 600845 
 .601098 
 .6013851 
 - 601603 
 . 601856 
 . 602109 
 . 602362 
 . 602614 
 
 | 9.602866 
 
 .603119 
 .60337 
 . 603623 
 603875 
 -604127 
 .604879 
 -604631 
 .604883 
 .605134 
 
 9.605386 
 
 - 605637 
 .605888 
 -606140 
 .606391 
 -606642 
 . 606893 
 .607144 
 607394 
 - 607645 
 
 9.607896 
 
 .608146 
 .608397 
 .608647 
 608897 
 .609147 
 . 609397 
 .609647 
 .609897 
 .610147 
 
 9.610397 
 
 .610646 
 .610896 
 .611145 
 .611394 
 .611644 
 .611893 
 .612142 
 .612891 
 .612640 
 
 9.612&88 
 
 .613187 
 . 613386 
 .613634 
 .613883 
 .614131 
 
 .614379 | 
 
 . 614627 
 614876 
 
 | 9.615124 
 
 AAA AA LAA OA RAR AAR ALAA AAR ALR ALAA PAAA ARASH A PAPERS SEAA PREPRESS 
 Re d+ 2 2D Ww dD? > wwMNHNNNNH NW? 
 
 — 
 — 
 eo GO 
 
 | 9.820622 
 . 821043 
 .821464 
 . 821885 
 . 822306 
 822727 
 .823148 
 . 823569 
 . 823990 
 824411 
 . 824833 
 
 9.825254 
 
 .825675 
 
 . 826096 
 
 .826517 
 
 . 826938 
 
 .827360 
 
 827781 
 
 . 828202 - 
 
 . 828623 
 
 .829044 
 
 . 829466 
 
 829887 
 
 . 830308 
 
 . 830729 
 
 .831151 
 
 .831572 
 
 .831993 
 
 .8382415 
 
 832836 
 
 .888257 
 
 . 883679 
 
 . 834100 
 
 . 884522 
 
 . 8384943 
 
 835364 
 
 . 835786 
 
 .836207 
 
 . 836629 
 
 .887050 
 
 8387472 
 
 .837893 
 
 .8388315 
 
 . 838736 
 
 .8389158 
 
 .8389579 
 
 .840001 
 
 , 840423 
 
 . 840844 
 
 .841266 
 
 .841687 
 
 9.842109 
 . 842531 
 .842953 
 . 843374 
 . 843796 
 .844218 
 .844639 
 . 845061 
 . 845483 
 
 9.845905 
 
 AUNT AE APAYAT AT VATA AIA NII NINA ANAND NAN NN NNNNIN NNNNNNNANN RANNRANARAERR 
 
 Jat =3 
 
~ 
 
 wre | 
 
 oo, 
 
 Ct 
 
 ocosto 
 
 10 
 
 c 
 
 iJon) 
 
 © 
 
 ie) 
 
 .615619 
 .615867 
 .616115 
 .616362 
 .616610 
 .616857 
 .617104 
 .617551 
 .617599 
 617845 
 .618092 
 .618339 
 .618586 
 . 618833 
 .619079 
 .619326 
 .619572 
 .619818 
 .620065 
 
 620311 
 .620557 
 620803 
 621048 
 .621294 
 621540 
 621786 
 622031 
 
 622276 
 
 622522 
 
 ~URRVIKRA 
 . 622767 
 .623012 
 . 623257 
 . 623502 
 623747 
 . 6238992 
 . 624237 
 . 624481 
 . 624726 
 .624970 
 .625215 
 .625459 
 .625703 
 .625947 
 . 626191 
 .626435 
 . 626679 
 . 626923 
 .627166 
 .627410 
 . 627654 
 .627897 
 .628140 
 . 628384 
 . 628627 
 .628870 
 .629113 
 . 629356 
 629598 
 
 9.629841 
 
 AND EXTERNAL SECANTS. 
 
 9 
 
 Ex. sec. 
 
 ~ 
 
 Vers. 
 
 Ex. sec. | 
 
 ASA AAAS AAS PRA AAA ALAA ALAR AAA ADA PARA RAR RR 
 
 4 | 
 ( 
 
 9.845905 
 
 840527 
 .846749 
 847170 
 847592 
 848014 
 848436 
 .848858 
 849280 
 .849702 
 850124 
 
 9.850546 
 850968 
 851390 
 851812 
 852234 
 852656 
 853078 
 853500 
 853923 
 854345 
 
 9.854767 
 855189 
 855612 
 856034 
 856456 
 856878 
 857301 
 857723 
 858145 
 858568 
 
 9.858990 
 859413 
 859835 
 860258 
 . 860689 
 861103 
 861525 
 .861948 
 862370 
 862793 
 
 9 863215 
 .863638 
 .864061 
 .864483 
 864906 
 865329 
 865752 
 866174 
 866597 
 867020 
 
 9.867443 
 867866 
 868289 
 868712 
 .869135 
 869558 
 869981 
 870404 
 870827 
 
 9.871250 
 
 COOVIAOoR WMH © 
 
 =) 
 
 ba en Sens Sous Sone Sen Dent Deas ae Se Silas Ses es Se De hh Le Ss hh Lt oR ae en Pa ce ce ee ee eee: 
 
 9.629841 
 .639084 
 . 630326 
 . 680569 
 . 630811 
 .631054 
 .631296 
 .63153 
 .631780 
 .682022 
 . 682264 
 
 9.632505 
 
 682747 
 
 . 632989 
 
 .633230 
 
 .683472 
 
 .633713 
 
 . 6388954 
 
 . 634196 
 
 .6384437 
 
 .634678 
 
 .684919 
 
 .685159 
 
 .6385400 
 
 .685641 
 
 .685881 
 
 .636122 
 
 . 636862 
 
 .686603 
 
 686843 
 
 .637083 
 
 9.637323 
 .687563 
 .687803 
 .638043 
 . 638283 
 .688522 
 . 638762 
 .639001 
 .689241 
 . 639480 
 
 6389719 
 . 689958 
 .640197 
 . 640486 
 .640675 
 .640914 
 .641153 
 .641891 
 .641630 
 .641868 
 642107 
 642345 , 
 . 642583 
 642522 
 .643060 
 648298 
 .648585 
 64877 
 .644011 
 9.644249 
 
 SoS ooocoodoeo°ococo 
 
 SCwwo SSSSRBBSESES CO 09 CO 09 CO 09 OT 09 OLN Or 
 
 Seooeoocose soooSesso 
 
 Sime 
 
 =) 
 oS 
 
 Sooo > 
 CoOooo CO: 
 
 le) 
 @ 
 
 MCPD COCCI CDCI COCICIDWNWNWW WOK AOD AAA AAD ARAL AAA PAD ADALE AA ARAAAREA ABE 
 
 9.87125 
 .871673 
 .872096 
 872519 
 -872942 
 .873266 
 .873789 
 874212 
 .874636 
 .875059 
 875482 
 9.875906 
 .876829 
 .876752 
 877176 
 .877599 
 .878023 
 .878446 
 878870 
 879294. 
 879717 
 9.880141 
 .880565 
 -880958 
 .881412 
 .881885 
 .882260 
 - 882583 
 -883107 
 .888531 
 .883955 
 9.884379 
 88480 
 -885227 
 .885651 
 .886075 
 .886499 
 . 886923 
 887847 
 887772 
 .888196 
 9.888620 
 889044 
 .889469 
 .889293 
 .890317 
 .890742 
 .891166 
 .891591 
 .892015 
 892440 
 
 9.892864 
 893289 
 893714 
 .894188 
 894563 
 894958 
 .895412 
 895087 
 .896262 
 
 9.896687 
 
TABLE XXVI.—LOGARITHMIC VERSED SINES 
 
 _ 
 owome- 
 
 OOP WMO 
 
 Vers Hix, secy| 1 1".71) <{ Vers... | D. 1".|, Ex. sec..|D. 1° 
 | 
 
 | 9.644249 | 3.95 | 9.896687 | 7.08 0 | 9.658356 | 3.87 | 9.922217 | 7.18 
 644456 | 3.9% 897112 | 7.08 1 658588 | 3.88 922074 | 7.13 
 644724 | 38.95 897537 | 7.08 2 .658821 | 3.87 .923102 | 7.12 
 644961 3.95 897962 | 7.08 3 .659053 | 3.88 928529 7.12 
 "645198 | 3.95 | .898387 | 7.08 || 4 | .659286 | 38.8% | .920056 | 7.135 
 "645435 | 3.97 | .898812 | 7.08 || 5 | .659518 | 8.87 | .924e84 ) 7.18 
 645673 | 3.95 | .899237 | 7.08 || 6 | .G59750 | 8.88} .924511 | 7.19 
 645910 | 3.95 | .899662 | 7.08 || 7% | .659083 | 8.87 | 925209 | 7.12 
 "46147 | 3.95 | .900087 | 7.08 || 8] .660215 | 8.87 | .9256C6 | 7.13 
 646884 | 3.93 .900512 | 7.10 9 660447 | 3.87 926094 | 7.12 
 "646620 | 3.95 | .900988 | 7.08 || 10 | .660679 | 38.85 | .926521 | 7.13 
 9.646857 | 3.95 | 9.901863 | 7.08 || 11 | 9.660910 | 3.87 | 9.926049 | 7.13 
 "47004 | 3.93 | .901788 | 7.08 || 12 | .661142 | 3.87 | .920877 | 7.12 
 .647330 | 3.95 .902213 | 7.10 |} 13 .661874 | 3.85 927804 | 7.13 
 647567 | 3.98 902689 | 7.08 || 14 .661605 | 3.87 928232 | 7.13 
 "647803 | 8.93 | 902064 | 7.10 [15 | .661837 | 3.85 | .928660 | 7.138 
 "648039 | 3.95 | .903490 | 7.08 || 16 | .662068 | 8.87 | .920088 | 7.13 
 "648276 | 3.93 | .903915 | 7.10 || 17] .662500 | 8.85 | .929516 | 7.138 
 .648512 | 3.93 904341 7.08 || 18 .662581 | 3.65 .929944 | 7.18 
 .648748 | 3.93 904766 | 7.10 || 19 .662762 | 3.85 930372 | 7.18 
 "648984 | 3.93 | .905192 | 7.08 || 20 | .662993 | 3.85 | .930800.) 7.18 
 9.64922 3.93 | 9.905617 | 7.10 |] 21 | 9.663224 | 3.85 | 9.931228 | 7.13 
 "49456 | 3.92 | .906043 | 7.10 || 22 | .663455 | 8.85 | .981056 | 7.15 
 649691 3.93 .906469 | 7.08 || 2 663666 | 8.85 982085 | 7.13 
 .649927 | 8.93 | .906804 | 7.10 || 2 663917 | 8.85 | .982513 | 7.13 
 .650163 | 38.92 907820 | 7.10 || 25 664148 | 3.83 982941 7.13 
 "650398 | 8.92 | .907746 | 7.10 || 26] .664378 | 8.85 | .988869 7) 7.15 
 650633 | 3.93 | .908172 | 7.10 || 27 | .664609 | 8.83 | .983708 | 7.13 
 "50869 | 3.92 | .908598 | 7.10 || 28] .664889 | 3.85 | .984226 | 7.15 
 .651104 | 3.92 .909024 | 7.10 || 2 .665070 | 38.83 . 984655 7.13 
 651339 | 3.92) .909450 | 7.10 || 80 | .665300 | 3.83 | .980033 | 7.15 
 9.651574 | 3.92 | 9.909876 | 7.10 || 81 | 9.665530 | 8.83 | 9.935512 | 7.15 
 .651809 | 3.92 .910802 | 7.10 || 382 .665760 | 8.83 935941 7.13 
 .652044 | 3.92 910728 | 7.10 || 33 .665990 | 3.88 9803869] 4.15 
 652279 | 3.92 .911154 | 7.10 || 34 666220 | 8.88 9386798 | 7.15 
 652514 | 3.90 911580 | 7.10 || 35 .666450 | 3.83 9387227") 7.15 
 652748 | 3.92 .912006 | 7.10 || 36 .666680 | 3.83 .987656°| 7.15 
 .652983 | 3.90 912482 | 7.12 || 37 .666910 | 8.82 9388085 | 7.138 
 .650217 | 3.92 912859 | 7.10 || 88 | .667139 | 3.83 .988513°| 7.15 
 "653452 | 3.90 | .918285 | 7.10 || 89 | .667369 | 8.83 | .988042°| 7.15 
 "653686 | 3.90 | .913711 | 7.12 || 40 | .667599 | 8.82 | .989871°) 7.1% 
 9.653920 | 3.92 | 9.914188 | 7.10 || 41 | 9.667828 | 3.82 | 9.939801 | 7.15 
 .654155 | 3.90 914564 | 7.12 || 42 .668057 | 3.83 940250") 7.15 
 .654889 | 3.90 914901 7.10 || 43 66287 | 3.82 .940659°} 7.15 
 654623 | 8.90 | .915417 | 7.12 || 44 | 668516 | 3.82 | .941088 7.15 
 654857 | 3.88 915644 | 7.10 || 45 668745 | 3.82 OATON Ts le Ele Le 
 ‘655090 | 3.90 | .91G270 | 7.12 |] 46 | .668074 | 3.82 | .941947 | 7.15 
 655824 | 3.90 | .916G07 | 7.12 || 47 | .660203 | 3.82 | .9423876 | 7.15 
 .655558 | 3.90 917124 | 7.10 || 48 | .669482 | 3.82 .942806.| 7.15 
 .655792 | 3.88 917550 | 7.12 || 49 | .669661 | 3.80 .948285°| 7.17 
 (656025 | 3.88] .917977 | 7.12 || 50 | .669889 | 3.82 | 948665.) 7.16 
 9.656258 | 3. 7.12 || 51 | 9.670118 | 3.82 | 9.944094 | 7.17 
 656492 | 3.8 7.12 || 52 | .670847 | 8.80 | .944524| 7.15 
 . 65672 3. 7.12 || 53 | .G70575 | 8.82 | .944953 | 7.17 
 .656958 | 3.8 7.12 || 54 | .670804 | 8.80 | .945383'| 7.17 
 657191 | 38. 7.12 || 55 | .671082 | 8.80 | .945813°) 7.17 
 657424 | 3.88 920589 | 7.12 || 56 .671260 | 3.80 .946243°| 7.17 
 :657657 | 8.88] .920966 | 7.12 || 57 | .671488 | 8.80 | .946673 7.17 
 657890 | 3.88 | .921303 | 7.12 || 58} .671716 | 8.82 | .947103 7.17 
 .658123 -€8 | .921620 | 7.12 |) 59 | 671945 | 3.78 | .947583°| 7.17 
 9.658356 | 3.87 | 9.922947 1 7.12 || 60 | 9.672172 | 8.80 | 9.947063! 7.1% 
 
»AND EXTERNAL SECANTS. 
 
 o re 
 
 Ne) 
 
 Vers. 
 
 | 9.672172 
 
 . 672400 
 .672623 
 . 672856 
 .673083 
 .673311 
 .673533 
 .673766 
 .673993 
 67422 
 
 674448 
 .674675 
 .674902 
 .675129 
 .675356 
 675582 
 . 675809 
 . 676036 
 676262 
 .676489 
 .676715 
 676941 
 .677168 
 .677394 
 . 677620 
 .677846 
 .678072 
 .678298 
 .678523 
 .678749 
 .678975 
 .679200 
 .679426 
 .679651 
 .679876 
 . 680102 
 . 680327 
 . 680552 
 .680777 
 .681002 
 .681227 
 
 9.681451 
 
 .681676 
 .681901 
 682125 
 . 682350 
 682574 
 682798 
 . 683023 
 683247 
 683471 
 683695 
 683919 
 
 .684143 | 
 
 . 684367 
 . 654590 
 684814 
 .685037 
 68561 
 .685484 
 
 | 9.685708 
 
 Ex, sec. 
 
 Vers. 
 
 | Jox. sec. 
 
 CERN TIN NIS 
 
 a ate FI 
 
 32 3 2 III 
 
 © OTOT OT OT OT =2 OF OT 
 
 ~3 
 OU 
 
 wWecew$c oc cotcec wow tec cote Coco 
 
 IWAN 
 
 tS We OO COW OO 
 
 9.947963 
 . 948393 
 . 948823 
 . 949253 
 . 949683 
 .950114 
 . 950544 
 . 950975 
 . 951405 
 . 951836 
 952266 
 . 952697 
 . 953128 
 .953558 
 .9538989 
 . 954420 
 .954851 
 . 955282 
 .9355713 
 956144 
 . 956575 
 9.957006 
 .957438 
 . 957869 
 . 958300 
 . 958732 
 .959163 
 .959595 
 . 960026 
 . 960458 
 . 960890 
 9.961321 
 .961753 
 .962185 
 . 962617 
 . 963049 
 . 963481 
 .963913 
 . 964345 
 96477 
 . 965210 
 9.965642 
 . 966075 
 . 966507 
 . 966940 
 . 967372 
 . 967805 
 . 968238 
 968670 
 . 969103 
 . 969536 
 
 . 969969 
 . 970402 
 . 9708385 
 . 971268 
 .971701 
 . 9721385 
 972568 
 973001 
 . 973435 
 9.973868 
 
 o 
 
 eo) 
 
 WOONROUR WOH OS 
 
 AP AE AT AT AQ APPT IVI AI III - 
 
 II RQ WII 
 
 | 9.685705 | 
 
 . 685931 
 
 .686154 | 
 
 .686377 
 . 686800 
 . 686823 
 .687046 
 
 687269 
 
 . 687492 
 687714 
 687937 
 9.688159 
 . 688382 
 688604 
 -683826 
 . 689048 
 689271 
 689493 
 .689715 
 -689937 
 .690158 
 
 9.690380 
 . 690602 
 . 690823 
 -691045 
 .691266 
 .691488 
 .691709 
 .691930 
 .692151 
 692372 
 
 9.692593 
 .692814 
 .693035 
 .693256 
 693477 
 .693697 
 -695918 
 .694138 
 .694359 
 .694579 
 
 694799 
 .695019 
 .695240 
 .695460 
 .695680 
 . 695899 
 .696119 
 
 =) 
 
 . 696339 | 
 
 . 696559 
 .696778 
 
 .696998 
 .697217 
 . 697437 
 .697656 
 697875 
 698094 
 698313 
 698532 
 698751 
 
 Je) 
 
 9.698970 
 
 co oo G9 G9 G9 G9 CD C9 OD OD OD 
 
 JIA VII MIM MMII 
 SWOwWWNwwnwwey 
 
 TOO OWooOew 
 
 Sor eS Bo 0) Bel lie en a 
 a 
 
 co 9 9 Go C8 G9 CO OF G9 CO CO 
 
 w) 
 “ 
 
 509 © 
 
 Wwiwewwtcoct cwtccocmc NOI ce co cu oN G9 Co cu to GY OO CO 
 3 Fe SS SO SAE 2 FAAARPADMH >HI AAIDP OOS 
 
 | 9.973868 
 
 974802 
 974736 
 . 975169 
 975603 
 . 976037 
 976471 
 976905 
 97733 
 
 ITT 
 
 978207 
 
 9.978641 
 
 1979075 
 .979510 
 . 979944 
 . 980879 
 . 980813 
 . 981248 
 - 981682 
 982117 
 982552 
 
 9.982987 
 
 . 983422 
 . 983857 
 - 984292 
 . 984727 
 . 985162 
 . 985597 
 - 986033 
 . 986468 
 . 986904 
 
 9.987339 
 
 lpdedou-4 
 : 987 (40 
 
 . 988210 
 . 988646 
 . 989082 
 . 989518 
 . 989954 
 . 990390 
 . 990826 
 991262 
 
 9.991698 
 
 . 992134 
 99257 
 
 . 993007 
 
 . 9938444 
 
 . 9983880 
 .994317 
 . 994754 
 . 995191 
 995627 
 
 9.996064. 
 996501 
 . 996938 
 . 997376 
 997813 
 . 998250 
 . 998687 
 999125 
 
 9.999562 
 
 10.000000 
 
 WP AE ABAD PATE A aaa INI INNIS NHN NNNINNNN NNNNNNAANEN 
 
TABLE XXVI.—LOGARITHMIC VERSED SINES 
 
 | 
 
 oumcswee | 
 
 mWMwre COONS 
 
 Vers. 
 9.698970 
 . 699189 
 .699407 | 
 .699626 
 . 699845 
 100063 
 700282 
 -700500 
 . 700718 
 . 700936 
 -701154 
 701372 
 701590 
 .701808 
 . 702026 
 . 702244 
 702462 
 102679 
 ~T02897 
 . 703114 
 . 703832 
 . 703549 
 . 703766 
 . 703983 
 704200 
 - 704417 
 . 704634 
 704851 
 . 705068 
 05285 
 . 705501 
 . 705718 
 - 105935 
 . 706151 
 106367 
 . 106584 
 . 706800 
 . 707016 
 - (OF282, 
 107448 
 . 707664 
 . 707880 
 . 708096 
 . 708811 
 708527 
 . 7087438 
 08958 
 WO91LT4A 
 .709389 
 .709604 
 . 709819 
 . 710085 
 .710250 
 . 710465 
 .710680 
 .710895 
 ~711109 
 . (113824 
 ~711589 
 111758 
 
 9.711968 
 
 NNO MMM AMARe SRRADIPOSOS CONS SSSS6O SONS SOSSSS Sane Sees: 
 
 Doo Go 
 Co OU 
 
 Ex. sec. 
 
 | 10.000000 
 
 .0004388 
 000875 
 .001313 
 001751 
 .002189 
 .002627 
 . 008065 
 . 003503 
 .003942 
 .004380 
 
 10.004818 
 005257 
 .005695 
 .006184 
 .006573 
 .007012 
 . 007450 
 007889 
 .0083828 
 008767 
 
 10 .009207 
 .009646 
 .010085 
 .010525 
 .010964 
 .011404 
 .011843 
 .012283 
 012723 
 .01381638 
 
 10.013603 
 .014043 
 .014483 
 .014923 
 .015363 
 .015804 
 .016244 
 .016684 
 017125 
 -017566 
 
 10.018007 
 .018447 
 .018888 
 .0193829 
 .019770 
 -020212 
 .020653 
 .021094 
 .021535 
 .021977 
 022419 
 . 022860 
 023302 
 023744. 
 024186 
 . 024628 
 . 025070 
 :025512 
 025954 
 
 10. 026297 
 
 aomewne | 
 
 — 
 THe WWM OOO 
 
 TEE ENE NEE a a a gt ER HEE 
 
 9 
 
 Vers. 
 
 - 711968 
 712182 
 712397 
 712611 
 712825 
 .718039 
 718253 
 - 713467 
 718681 
 
 718895 
 
 .718870 
 718582 
 . 18794 
 .719007 
 . 719219 
 .719431 
 719643 
 719855 
 . 720066 
 
 720278 , 
 
 720490 
 - 720701 
 
 Bit 
 ; 
 9.0 
 
 S 
 
 .| Ex. see, 
 
 ew 
 
 SP LOSE CORO CO COO COED CY CD CIDDEDCD OI ODCIOD CUCUCCDEDEDEDEDEODED CIDCCUUDOIODG WeDeDCcDEceDecD cE EOI ECD OD CD ON OH OD 
 : mare OS at 5 EO Ee aca Ce ese CO COLON ET OC EY Oh Oh Oe a ae a Sr ait Gr SSF SS ae Dv ee ee 
 
 1 
 
 10.026397 
 026839 
 .027281 
 
 ORT724 
 
 028167 
 
 028609 
 .029052 
 .029495 
 029938 
 .030381 
 030825 
 .031268 
 
 031711 
 
 .032155 
 . 032598 
 .033042 
 .033486 
 
 .033929 | 
 
 034373 
 .034817 
 .035261 
 10.035705 
 -036150 
 -936594 
 037038 
 .037483 
 037928 
 038372 
 . 038817 
 039262 
 089707 
 
 10.040152 
 040597 
 .041042 
 -041488 
 .041933 
 042379 
 042824 
 . 043270 
 043716 
 .044162 
 
 10.044608 
 045054 
 .045500 
 045946 
 - 046393 
 046839 
 047286 
 047732 
 .048179 
 . 048626 
 
 ».049073 
 049520 
 049967 
 050414 
 050861 
 .051309 
 051756 
 052204 
 
 052652 
 
 10.053099 
 
 pas oma, as Dent ake Sr ras Si as De dons Sons Ss So Sons Ss Se ln Son ous Sow Sows Bs Ss Sow De Ses ies Se De es be Se Se Se Rohe Me Cote te te hot te a fo toto to roe tk Per: 
 
ro | 
 
 a 
 
 a pe 
 
 AND EXTERNAL SECANTS. 
 
 ~ 
 
 Ex. sec. 
 
 SD -yo ok wrmwoRic | 
 
 59 
 
 10 | 
 
 Vers. | D.1. 
 
 9 724709 | 50 | 10.058099 
 ~ 724919 | 50 | .053547 
 . 725129 50 053995 
 "725339 | 8.50 |  .054443 
 "725549 | 3.50 |  .054892 
 
 725759 
 . .725969 
 726179 
 
 . 726388 
 
 va 
 2-2 
 5.29 
 5 
 U — 
 co) 
 
 Q 
 
 9.731193 | 
 
 731401 
 . 731609 
 731817 
 . 732025 
 . 182233 
 . 732441 
 732648 
 732356 
 ‘ 
 
 . 733064 
 
 | 9.733271 
 
 T8047 
 . 738686 
 783893 
 . 734100 
 .7384307 
 784515 
 .1384721 
 . 73492 
 785185 
 9.735342 
 730549 
 735755 
 735962 
 .736169 
 736875 
 .736581 
 
 + 
 
 7386788 
 736994 
 
 > : 
 GO | 9.737200 
 
 He ee ee 
 JOUN AEN 
 
 mH ee 
 Ot Olas 
 
 me eo 
 coor O1rordr 
 
 Cor 
 
 PP Pop wp 
 O39 OVOD C9 Oe 
 
 SepeeseeweneneweD co 09 0D CN CH C9 CO CN 09 09 co Co 09 OD 9 OD CO 
 a 
 2 
 
 iS) 
 me me 
 Co oe 
 
 .0553840 
 .055788 
 .056237 
 .056685 
 
 .057134 
 
 057583 | 
 
 10 .058032 
 058481 
 
 .058930 
 -059379 
 .059828 
 .060278 
 .060727 
 061177 
 .061626 
 .062076 
 
 ; 10:062526 
 
 .065678 
 .066129 
 
 .066580 | 
 
 10.067030 
 067482 
 .067933 
 068384 
 .068835 
 
 069287 
 
 .069738 
 .070190 
 070642 
 .071093 
 10 .071545 
 .071998 
 .072450 
 .072902 
 .073354 
 073807 
 .074260 
 074712 
 .075165 
 .075618 
 10.076671 
 076524 
 076977 
 0774381 
 LOTT 88 4 
 .078338 
 .078792 
 079245 
 079699 
 10.080153 
 
 my AP Ry ATAT AT AE Aa ag ss SS NNN NNN HSIN SN MINNA NNN NNNNNRANNN 
 
 Or or or Or cr 
 
 (oe) 
 
 Jets 
 
 47 
 
 and 
 
 Orr d 
 
 ren orargr Ororg 
 
 IA OTH -2 OU OT OT 
 
 = ~ 
 CSODMDRAUP WMHS 
 
 Vers. 
 
 Ex. sec, 
 
 9.737200 
 737406 
 737612 
 737818 
 . 738024 
 . 138230 
 738436 
 . 738642 
 738847 
 739053 
 
 139258 
 9.739464 
 
 . 739669 
 39875 
 40080 
 740285 
 740490 
 40695 
 740900 
 . 741105 
 741310 
 9.741515 
 ~TALT19 
 TAL924 
  T42129 
 142333 
 (42588 
 142742 
 742946 
 .743150 
 743355 
 9.743559 
 (48762 
 . 743967 
 44171 
 744875 
 T4457 
 744782 
 744986 
 745189 
 745893 
 9.745596 
 . 745800 
 746003 
 746206 
 . 746409 
 7466138 
 146816 
 747019 
 
 AIR | 
 
 TATARA 
 9.747627 
 (47830 
 (48033 
 48235 
 “748438 
 748640 
 . 748843 
 749045 
 749247 
 
 9.749449 
 
 oo 
 
 h ARR ARRR RAD Bw 
 
 co) 
 
 wewewwwowwi cwcocwmccwci;to¢ wtwocwietwcto 
 
 He 
 
 BPR RP a Be 
 
 IOSOWOCNWS WKHWWWWNWKWWWNH WW 
 
 Oe 
 
 — 
 oO 
 
 10080153 
 080608 
 .081062 
 
 082425 
 082880 
 083835 
 083790 
 084245 
 084700 
 
 10.085155 
 
 086066 
 .086522 
 086977 
 087433 
 .087889 
 .088845 
 088801 
 
 10. 089714 
 090171 
 080627 
 .091084 
 .091541 
 .091998 
 .092455 
 . 092912 
 .098387' 
 .098827 
 
 | 10.094285 
 
 .094743 
 
 . 095200 
 
 .095658 
 
 .096116 
 
 .096575 
 
 097491 
 097950 
 .098408 
 
 10.098867 
 099326 
 099785 
 100244 
 100704 
 101163 
 101623 
 102082 
 102542 
 103002 
 
 10. 103462 
 103922 
 104382 
 
 105803 
 . 105764 
 106224 
 . 106685 
 .107146 
 
 10.107607 
 
 | 081516 | 
 081971 
 
 .085611 
 
 089258 
 
 097033. | 
 
 104843 
 
 oe LES rn ents Sie et 8 DSI NNSA aistst Bg ya a aa aa te a es ietet SIV sIV VWs INI 
 
“ 
 
 Vers. 
 
 9.749449 
 749652 
 - 749854 
 . 750056 
 - 750258 
 . 750459 
 - 750661 
 . 750863 
 751065 
 751266 
 751468 
 
 11 | 9.751669 
 
 12 | .751871 
 
 13 152072 
 
 14 T52278 
 
 15 T2475 
 
 16 152676 
 
 17 | 752877 
 
 18 753078 
 
 19 753279 
 
 20 | .7538480 
 
 21 | 9.753681 
 22) .753881 
 
 23 | .754082 
 Ha 24 | 754283 
 
 25 | .754483 
 26 | .754684 
 27 | .754884 
 28 |} .755085 
 
 ed 
 SOHIRUR IHS | ~ 
 
 29 155285 
 30 . 155485 
 ; 31 | 9.755685 
 
 32 | .755886 
 33 | .756086 
 Baie 34 | .756286 
 
 ie 35 | .'756486 
 Wi 36 | .756685 
 wT 37 | .756885 
 ah 38 |  . 757085 
 39 | .757285 
 40 | . 757484 
 41 | 9.757684 
 42 | . 757883 
 43 | .758083 
 44 | .758282 
 45 | . 758481 
 46 | .758681 
 47 | .'758880 
 48 | .759079 
 49 | .759278 
 
 50 59477 
 51 | 9.759676 
 52 159875 
 53 160073 
 5 760272 
 
 55 | .760471 
 56 | .760669 
 57 | .760868 
 58 | .761066 
 59 | .761265 
 60 | 9.761463 
 
 TABLE XXVI.—LOGARITHMIC VERSED SINES 
 
 64° 65° 
 D. 1’.| Ex. see. | D 1’. z, Vers. | D.1".| Ex. see. 
 3.38 | 10.107607 Tel 0 | 9.761463 |.3.30 | 10.135515 
 3.07 . 108069 7.68 ] .761661 | 3.32 .135984 - 
 3/37 . 108530 “we 2 . 761860 | 3.30 . 186454 
 3.0% .108992 7.68 3 . 762058 | 3.30 . 1869238 
 3.35 . 109453 sf 4 . 762256 | 3.30 . 137393 
 Ook .109915 | 7.7 5 762454 | 3.30 .187863 | 
 Sy aye 180377 | 785 6 . 762652 | 3.30 . 138333 
 3.00 .110889 | 7.7 if . 762850 | 3.28 . 1388803 
 3.35 111301 eek 8 . 763047 | 3.30 . 139273 
 3.00 .1117638 Visite 9 - 763245. | 3.30 .1389744 
 3.35 . 112226 ne 10 . 763443 |-3.30 . 140214 
 3.37 | 10.112688 | 7.7 11 | 9.163641 | 3.28 | 10.140685 
 3.35 .113151 “ote 12 .768838 | 3.30 .141156 
 3.35 .113614 Late 13 . 7640386 | 3.28 . 141627 
 Bot .114077 | 7.72 || 14 . 764233 | 3.28 . 142098 
 3.35 .114540 112 15 . 764430 | 3.30 . 142569 
 3.59 .115003 G92 16 . 764628 | 3.28 . 143041 
 3.35 . 115466 4.42 17 . 764825 | 3.28 . 143512 
 3.35 .115929 Tate 18 . 765022 | 3.28 . 148984 - 
 3.35 . 116393 Ub Me 19 . 765219 | 3.28 . 144456 
 3.35 . 116857 Cite 2 . 165416 | 3.28 . 144928 
 8.33 | 10.117321 Vite 21 | 9.765613 | 8.28 | 10.145400 
 3.35 .117785 see 2% . 765810 | 3.28 . 145872 
 3.35 .118249 Tesi 23 . 766007 | 3.28 . 146845 
 Shas! lIS713: | 7 738 || Ba 766204 | 3.2 . 146818 
 3.35 119177 var, 25 . 766401 | 3.27 .147290 
 aya: .119642 | 72% 26 766597 | 3.28 .147763 
 3.35 . 120106 at 27 . 766794 | 3.28 . 148236 
 Side 120571 7 28 .766991 | 3.27 . 148710 
 3.33 . 121036 CM 29 . 767187 | 3.28 . 149188 
 3.33 .121501 v@ares) 30 . 767384 | 3.27 . 149657 
 8.85 | 10.121966 | 7.75 || 831 | 9.767580 | 3.27 | 10.150130 
 3.33 .1£2431 WAG || 82 S76(776- | 8227 .150604 
 3.33 122897 | 795 || 338 167972. | 3.28 .151078 
 3.33 . 123362 CHE 34 .768169. | 3.27 .151552 
 3.02 . 123828 Ceres 35 . 768365. | 3.27 . 152027 
 3.33 . 124294 TOU ve 36 . (68561. | 3.27 . 152501 
 3.33 . 124760 CAG 37 68757 | 3.27 . 152976 
 3.53 . 125226 7.70 || 38 . 768953. | 3.27 . 153450 
 3.02 . 125692 Wh 39 . 769149 | 3.25 . 158925 
 3.33 .126158 7.78 40 . 169344 | 3.27 . 154400 
 3.82 | 10.126625 |. 7.78 || 41 | 9.769540 | 8.27 | 10.154876 
 3.33 . 127092 Vsti |) 42 . 769736 | 3.25 . 1553851 
 3.382 .127558 7.78 || 43 . 769931. | 3.27 . 155826 
 8.382 . 128025 7.98 44 Stiletto. et . 156302 
 3.00 . 128492 | 7.80 || 45 .770323 | 3.25 .156778 
 3.32 . 128960 7.78 46 ~770518 | 3.25 . 157254 
 3Jo8 . 129427 @fiom Wie Ve MOM8 || 8.2% . 157730 
 3.32 . 129894 7.80 48 .770909. | 38.25 . 158206 
 3.02 . 180862 7.80 49 .771104 | 3.25 . 158683 
 3.32 . 1308380 7.80 50 ~ 701299 | 3.25 .159159 
 3.32 | 10.131298 | 7.80 |} 51 | 9.771494 | 3.25 | 10.159636 
 Susy .131766 7.80 52 .771689 | 3.25 .160113 
 3.32 132234 7.80 53 . 771884 | 3.25 . 160590 
 $.32 . 182702 7.80 54 -772079. | 8.25 . 161067 
 3.30 . 133170 7.82 55 CR2274. | 8.25 .161545 
 3.32 . 138639 7.82 56 . 772469 | 3.25 . 162022 
 3.30 . 134108 / ey 57 . 772664 | 3.23 . 162500 
 3.32 . 184577 7.82 58 772858 | 3.25 . 162978 
 3.30 . 185046 ‘eises 59 . 7730538 | 3.25 . 163456 
 3.80 | 10.1385515 7:82 (| 60 | 9.778248 | 3.23 | 10.163934 
 
AND EXTERNAL SECANTS. 
 
 / Vers Dp, 1° \dixewsec Dy I’ J Vers. |D. 1’ 
 
 0 | 9.773248 | 3.23.| 10.168934 | 7.98 0 | 9.784809 | 3.18 
 
 1 V73442 | 3.23 .164413 |. 7.97 1 .785000 | 3.18 
 
 2 773686 | 3.25 .164891 7.98 2 .785191 | 3.17 
 
 3 773831 |. 3:23 .165370 | 7.98 3 .7853881 | 3.18 
 
 4 .774025 | 3.23 .165849 | 7.98 4 7185572 | 3.18 
 
 5 . 774219 | 3.25 .166828 | 7.98 5 .(85763 | 3.17 
 
 6 .174414 | 3.23 .166807 | 7.98 6 785953 | 8.18 
 
 7 .774608 | 3.23 . 167286 8.00 fi .786144 |- 3.17 
 
 8 .774802 | 3.23 . 167766 7.98 8 .786334 | 3.17 
 
 :.9 .774996 | 3.23 . 168245 8.00 9 . 786524 | 3.18 
 
 10 .775190 | 3.23 .168725 | 8.00 || 10 ~786715 | 3.17 
 
 | 11 | 9.775384 | 3.22 | 10.169205 | 8.00 || 11 | 9.786905 | 3.17 
 | 2 TUT? | 3.23 -169685 8.00 || 12 .787095 | 3.17 
 13 i dae 8 ips) oa Re .170165 | 8.02 || 18 ~ 787285 | 3.17 
 | 14 V75965 | 3.2% .170646 8.02 || 14 (87475 | 38.17 
 15 776159 | 3.22 ih abe 8.00 || 15 .787665 | 3.17 
 : 16 7763852 | 3.23 .171607 8.02 || 16 . 787855 | 38.17 
 17 776546 | 3.22 £172088 | 8.02 || 17 .788045 | 3.17 
 
 18 776739 | 3.23 .172569 | 8.03 || 18 788235 | 3.17 
 
 19 776933 | 8.22 .173051 8.02 |} 19 .788425 | 3.15 
 
 20 UT77126 | 3.22 .1735382 | 8.03 || 20 788614} 3.17 
 
 21 | 9.777319 | 3.22 | 10.1%4014 | 8.03 || 21 | 9.788804 | 3.15 
 
 29 777512 | 3:22 .174496 | 8.03 || 22 .788993 | 3.17 
 
 23 77105 -| 3.238 .174978 | 8.038 || 28 .789188 | 38.15 
 
 24 .777899 | 3.22 .175460 | 8.03 || 24 .789372 | 3.17 
 
 25} .778092 | 3.22 .175942 | 8.05 || 25 . 789562 | 8.15 
 | 26 .778285 | 38.20 .176425 8.03 || 26 .789751 | 8.15 
 27 TBAT 3.22 . 176907 8.05 || 27 .789940 | 3.17 
 
 | 28 778670 | 3.22 .177390 | 8.05 || 28 .790180 | 3.15 
 29 778863 | 3.22 177873 8.05 || 29 .790319 | 3.15 
 
 30 779056 | 3.20 .178356 8.05 || 30 .790508 | 8.15 
 
 31 | 9.779248 |-3.22.| 10.178839 | 8.07 || 31 | 9.790697 | 3.15 
 
 82 .779441 | 3.22 .179823 8.07 || 32 .790886 | 38.15 
 
 33 .779634 | 3.2 179807 8.05 || 33 .791075 | 3.15 
 
 34 |, .779826 | 3.20 .180290 | 8.07 || 34 .791264 | 8.15 
 
 35 .780018 | 3.22 18077 8.08 ||} 35 791458 | 3.13 
 
 36 .780211 | 3.20 . 181259 8.07 || 36 .791641 | 3.15 
 
 3v .780403 | 38.20 .181743°} 8.07 || 37 .791830 | 3.15 
 
 38 .780595 | 3.20 .182227 | 8.08 || 38 .792019 | 3.13 
 
 39 .780787 | 3.22 182712 8.08 || 39 792207 | 3.15 
 
 40 .78098C | 3.20 .183197 | 8.08 || 40 . 792396 | 3.138 
 
 41 | 9.781172 | 3.20 | 10.183682 | 8 08 || 41 | 9.792584 | 3.18 
 
 42 .781364 | 8.20 .184167 | 8.10 2 192772 | 3.15 
 
 43 .781556 | 3:18 .184653 | 8.08 || 48 .792961 | 3.13 
 
 44 -781747 | 3.20 .1851388 | 8.10 || 44 .793149 | 3.13 
 
 45 .781939 | 3.20 .185624'| 8.10 || 45 2938887 4.3.13 
 46 .782131 | 3.20 .186110 | 8.10 || 46 ~793525 | 3.15 
 
 47 782823 | 3.18 .186596 | 8.10 || 47 793714 | 3.138 
 
 48 782514 | 3.20 .187082 | 8.10 || 48 .798902 | 3.13 
 
 49 .782706 | 3.18 .187568 | 8.12 || 49 .794090 | 3.12 
 
 50 . 782897 | 3.20 .188055 | 8.12 || 50 -794277 | 38.13 
 
 51 | 9.783089 | 3.18 | 10.188542 | 8.12 || 51 | 9.794465 | 3.13 
 
 52 .783280 | 3.18 . 189029 8.12 || 52 .794653 | 3.13 
 
 53 ~783471 | 3.20 . 189516 8.12 || 53 .794841 | 3.12 
 
 54 .783663 | 3.18 .190003 | 8.13 || 54 .795028 | 3.13 
 
 | 55 .783854 | 3.18 .190491 8.12 Bd .795216 | 3.13 
 
 56 .784045 | 3.18 .190978 8.13 || 56 .795404 | 3.12 
 
 5Y .784236 | 3.18 .191466 | 8.13 || 57 .795591 | 3.13 
 
 58 784427 | 3.18 .191954 |} 8.15 || 58 795779 | 3.12 
 
 | 59 784618 | 3.18 .192448 | 8.13 || 59 .795966 | 3.12 
 
 | 60 | 9.784809 | 8.18 | 10.192931 8.15 || 60 | 9.796158 | 3.13 
 
 Ex. sec, 
 
 192931 
 
 193420 
 193908 
 . 194397 
 194886 
 . 195376 
 195865 
 196355 
 196845 
 197335 
 197825 
 
 10. 
 
 198315 
 
 198806 
 199297 
 199788 
 20027 
 
 20077 
 
 201262 
 .201753 
 202245 
 202787 
 
 10. 
 
 203229 
 
 203722 
 204215 
 204707 
 - 205200 
 . 205694 
 . 206187 
 . 206681 
 207174 
 . 207668 
 
 10. 
 
 208162 
 
 208657 
 .209151 
 .209646 
 210141 
 210636 
 2111381 
 .211627 
 212123 
 212618 
 
 10. 
 
 213115 
 
 .213611 
 214107 
 214604 
 .215101 
 215598 
 .216095 
 216598 
 217090 
 217588 
 
 10. 
 
 218086 
 
 218585 
 219083 
 219582 
 220081 
 
 .220580 | 
 
 221079 
 22157 
 
 10. 
 
 222078 
 222578 
 
 Sqeqeqeg=gegeg= aaa. 
 
 AF ad a 8 HE OT 2 OL OT CO Ot 
 
68° 
 
 TABLE XXVI.—LOGARITHMIC VERSED SINES 
 
 69° 
 
 D: 1s 
 
 / Mersin. pbs 1 )' Mxeseeas Dal / Vers. Ex. sec. |D.1” 
 0 | 9.796153 | 3.18 | 10.222578 | 8.33 0 | 9.807286 | 3.07 | 10.252957 | 8.55 
 | .796341 | 3.12 .220078 8.33 i .807470 | 38.07 .2538470 -| 8.55 
 21° 796528 | 3.12 . 223578 8.3b° 1) 32 .807654 | 3.05 .2539838 | 8.57 
 S5) o  796ta)| 3:12 . 224079 8.33 | 3 .807837 | 3.07 254497 | 8.55 
 4 .796902 | 3.12 224579 8.35 || 4 .808021 | 8.05 .255010 | 8.57 
 5 ~797089 | 3.12 . 225080 8.35 || 5 .808204 | 3.07 .255524 | 8.58 
 6 797276 | 3.12 . 225581 8.3%. IG .808388 | 3.05 .256039. | 8.57 
 fi -797463' | 3.12 . 226083 Srao lt a7 .808571 | 3.07 .256553 | 8.58 
 8 .797650 | 8.12 . 226584 8.37 8 .808755 | 3.05 .251068 | 8.57 
 9 197837 | 3.10 . 227086 8.37 i 2 . 808988 | 3.05 .257582 | 8.60 
 10 .798023 | 3.12 .227588 8.37 || 10 .809121 | 3.07 .258098 | 8.58 
 11 | 9.798210 | 3.12 | 10.228090 8.37 || 11 | 9.809305 | 3.05 | 10.258613 | 8.60 
 12 .798897 | 8.10 . 228592 8.38 || 12 .809488 | 3.05 .259129 | 8.58 
 13 . 798583 | 8.12 «229095 8.38 || 18 .809671 | 3.05 .259644 | 8.60 
 14 .798770. | 3.10 . 229598 8.38 || 14 .809854 | 3.05 .260160 | 8.62 
 15 |. .798956 | 3.10 . 230101 8.38 |} 15 .8100387 | 3.05, .260677 | 8.60 
 16 .799142 | 3.12 .230604 8.38 || 16 810220 | 3.05 .261193 | 8.62 
 17 .799829 | 3.10 .231107 8.40 || 17 .810403 | 3.038 261710 | 8.62 
 18 .799515 | 3.10 .231611 8.40 || 18 .810585 | 3.05 - 262227 | 8.62 
 19 .799701 | 3.10 .232115 8.40 | 19 .810768 | 3.05 .262744 | 8.63 
 20 .799887 | 3.12 .232619 8.40 |) 2 .810951 | 3.05 .268262 | 8.62 
 21 | 9.800074 | 8.10 | 10.233128 8 40 || 21 | 9.811134 | 3.03 | 10.263877 8.63 
 22 .800260 | 3.10 . 283627 8.42 || 22 .811816 | 3.05 .264297 | 8.638 
 93 .800446 | 3.08 . 2841382 8.42 23 .811499 | 3.03 .264815 | 8.65 
 24 .800631 | 3.10 234637 8.42 24 .811681 | 3.05 .265384 | 8.65 
 25 800817 | 3.10 .205142 8.42 || 25 .811864 | 3.03 .205858 | 8.63 
 26 .801003 | 3.10 . 235647 8.48 || 26 .812046 | 3.03 .266371 | 8,67 
 27 .801189 | 3.10 . 236153 8.42 || 27 .812228 | 3.038 .266891 | 8.65 
 28 |. 801375 | 3.08 . 236658 8.43 28 .812410 | 3.05 267410 | 8.67 
 29 .801560 | 3.10 . 237164 8.43 |; 29 .8125938 | 3.03 . 267930 | 8.65 
 3 ,801746 | 3.08 2237670 8.45 || 30 812775 | 3.08 .268449 | 8.68 
 31 | 9.801931 | 3.10 | 10.238177 8.43 || 31 | 9.812957 | 8.03 | 10.268970 | 8.67 
 82 .802117 | 3.08 . 238683 8.45 82 .813139 | 8.03 .269490 | 8.68 
 33 . 802302 | 3.08 .239190 8.45 33 .8138321. | 3.03 .270011 | 8.67" 
 34 ,802487 | 3.10 . 239697 8.45 34 .813508 | 3.03 .270531 | 8.68 
 35 . 802673 | 3.08 . 240204 8.47 35 .813685 | 3.02 271052 | 8.7% 
 36 .802858 | 3.08 240712 8.45 || 86 . 813866 | 3.03 271574 | 8.68 
 37 .803043 | 3.08 .241219 8.47 || 387 .814048 | 3.03 2262095 | 8.7 
 38 .803228 | 3.08 6241727 8.47 || 38 .814230 | 8.02 212017 | Sag 
 39 , 8034138 | 3.05 , 242235 8.48 || 39 .814411 | 3.03 .2738139 | 8.7% 
 40 .803598 | 3.08 PART 8.47 || 40 .814593 | 3.08 .2738662 | 8.7 
 41 | 9.808788 | 3.08 | 10.248252 8.48 41 | 9.81477 8.02 | 10.274184 | 8.72 
 42 | .808968 | 3.08 . 243761 8.48 42 .814956 | 3.02 204707 | 8.72 
 43 .804158 | 3.08 . 244270 8.48 || 48 .8151387 | 3.03 -2c0e200 | 8% 
 44 . 804338 |. 3.07 244779 8.50 || 44 .815319 | 3.02 2051538 | 8.7 
 45 .804522 | 3.08 . 245289 8.48 || 45 .815500 | 3.02 4O2Cb| 8: te 
 46 .804707 | 8.08 245798 8.50 46 .815681 | 3.02 .276801 | 8.73 
 47 .804892 | 3.07 . 246308 | 8.50.)| 47 .815862 | 3.08 atlaee | 8. re 
 48 .805076 | 3.08 . 246818 8.52 48 .816044 | 3.02 207849 | 8.75 
 49 .805261 | 3.07 2473829 8.50 || 49 .816225 | 3.02 . 2783874 | 8.75 
 50; .805445 | 3.07 247889 8.52 || 50 .816406 | 3.02 248899 | 8.75 
 51 | 9.805629 | 3.08 | 10.248850 | 8.52 || 51 | 9.816587 | 3.00 | 10.279424 | 8.75 
 52 | .805814 | 3.07 . 248561 8.52 2 ,816767 | 8.02 . 279949 | 8.77 
 53 | .805998 | 3.07 249872 | §.52 53 .816948 | 3.02 280475 | 8.75 
 54 |) .806182 | 3.07 . 249883 8.53 || 54 .817129. | 3.02 .281000 | 8.78 
 55 | .806366 | 3.07 2503895 8.53 || 55 .817310 3.00 281527 | 8.77 
 56 | .806550 | 3.07 . 250907 8.53 | 56 .817490 | 3.02 . 282053 | 8.78 
 57 .806734 | 3.07 251419 | . 8.55 || 57 .817671 | 3.02 .282580 | 8.77 
 58 .806918 | 8.07 . 2519382 8.53 58 .§17852 | 3.00 .283106 | 8.80 
 59 .807102 | 8.07 .202444 8.55 59 .818032 | 3.02 . 283634 | 8.78 
 60 | 9.807286 | 8.07 | 10.252957 8.55 60 1 9.818278 | 3.00 | 10.284161 | 8.80 
 
AND EXTERNAL SECANTS 
 
 CHIR OMH WWOHS | 
 
 Vers. Delt iss sec. Ds 1" 4 
 | 9.818213 | 3.00 | 10.284161 | 8.80 0 
 .818393 | 3.00 284689 | 8.7 1 
 818573 | 3.02 285216 | 8.82 2 
 .818754 | 3.00 285745 | 8.80 3 
 818934 | 3.00 286273 | 8.82 4 
 819114 | 3.00 286802 | 8.82 5 
 819294 | 3.00 .287331 | 8.82 || 6 
 .819474 | 3.00 287860 | 8.82 ve 
 819654 | 3.00 288389 | 8.83 8 
 819834 | 3.00 288919 | 8.83 9 
 820014 | 3.00 289449 | 8.83 || 10 
 9.820194 | 3.00 | 10.289979 | 8.85 || 11 
 820374 | 2.98 .290510 | 8.85 || 12 
 820553 | 3.00 .291041 | 8.85 || 13 
 820733 | 3.00 ,291572 | 8.85 || 14 
 820913 | 2.98 .292103 | 8.87 || 15 
 .821092 | 3.00 292635 | 8.85 || 16 
 821272 | 2.98 .293166 | 8.87 || 17 
 821451 | 3.00 293698 | 8.88 || 18 
 821631 | 2.98 294231 | 8.88 || 19 
 .821810 | 2.98 294764 | 8.87 || 20 
 9.821989 | 2.98 | 10.295296 | 8.90 || 2 
 822168 | 3.00 295830 | 8.88 || 22 
 22348 | 2.98 | .296363 | 8.90 || 23 
 822527 | 2.98 .296897 | 8.90 || 24 
 822706 | 2.98 297431 | 8.90 || 25 
 822885 | 2.98 297965 | 8.92 || 26 
 823064 | 2.98 .298500 | 8.90 || 27 
 823243 | 2.97 299034 | 8.93 || 2 
 823421 | 2.98 299570 | 8.92 || 29 
 823600 | 2.98 .300105 | 8.93 || 30 
 9.823779 | 2.98 | 10.300641 | 8.92 || 31 
 823958 | 2.97 301176 | 8.95 || 32 
 824136 | 2.98 .301718 | 8.93 || 33 
 824315 | 2.97 302249 | 8.95 || 3 
 824493 | 2.98 302786 | 8.95 || 35 
 824672 | 2.97 303323 | 8.95 || 36 
 824850 | 2.97 303860 | 8.97 || 37 
 825028 | 2.98 804398 | 8 97 || 3% 
 825207 | 2.97 804936 | 8.97 || 39 
 825385 | 2.97 .805474 | 8.97 | 40 
 9.825563 | 2.97 | 10.306012 | 8.98 || 41 
 825741 | 2.97 .306551 | 8.98 || 42 
 825919 | 2.97 .807090 | 8.98 || 43 
 826097 | 2.97 .307629 | 9.00 || 44 
 826275 | 2.97 .308169 | 8.98 || 45 
 826453 | 2.97 .308708 | 9.02 || 46 
 826631 | 2.97 .809249 | 9.00 || 47 
 ,826809 | 2.97 | .3809789 | 9.02 || 48 
 .826987 | 2.95 810330 9.02 || 49 
 827164 | 2.97 810871 | 9.02 || 50 
 9 827342 | 2.95 | 10.311412 | 9.02 || 51 
 827519 | 2.97 811958 | 9.03 || 52 
 827697 | 2.95 812495 | 9.03 || 53 
 827874 | 2.97 313037 | 9.05 || 54 
 828052 | 2.95 .813580 | 9.03 || 55 
 828229 | 2.95 814122 | 9.05 |! 56 
 828406 | 2.97 .814665 | 9.07 || 57 
 828584 | 2.95 815209 | 9.05 || 58 
 823761 | 2.95 .815752 | 9.07 || 59 
 9.828938 | 2.95 | 10.316296 | 9.07 || 60 
 
 Vers. 
 
 9.828938 
 829115 
 829292 
 829469 
 829646 
 
 829823. | 
 .830000 | 
 
 .830177 
 830353 
 .830530 
 .830706 
 .830883 
 .831059 
 . 831236 
 .831412 
 .831589 
 .831765 
 .831941 
 .832117 
 832293 
 . 832469 
 832645 
 832821 
 832997 
 83317 
 . 8383849 
 8388525 
 .833700 
 .833876 
 834051 
 . 834227 
 
 .834402 
 .834578 
 .834753 
 . 834928 
 . 835104 
 . 835279 
 .835454 
 835629 
 . 8385804 
 .830979 
 .836154 
 . 836329 
 .836504 
 .836678 
 . 836853 
 .837028 
 .8387202 
 .837377 
 .837551 
 .837726 
 .837900 
 . 838075 
 
 ie) 
 
 © 
 
 WD WWW WWW 
 
 838249 | 
 
 838423 | 
 
 838597 
 838771 
 838945 
 .839119 
 
 839293 
 
 9.839467 
 
 2. 
 
 WN WWNWNWWWND WNWWNWNNWWWwWwW WNWWWWWNWWWYW 
 
 SO © 
 Ww 
 
 WNWWNWNWWNWNWW 
 OOOO OO OS 
 
 5 
 cS) 
 
 95 
 
 SMOnwnwoww 
 
 Ex. sec. 
 
 10.316296 
 .316844) 
 817385 
 .311929 
 .318475 
 .3819020 
 .019565 
 .820111 
 .820658 
 .3821204 
 .O21751 
 
 10.322298 
 .3822845 
 6820093 
 .823941 
 .324489 
 .825038 
 825587 
 .826136 
 .826686 
 .82(235 
 
 10.327786 
 .828336 
 828887 
 .829438 
 .829989 
 .3830541 
 .831093 
 .381645 
 .832198 
 .882750 
 
 10.333304 
 .o308807 
 .084411 
 .034965 
 .380520 
 .336074 
 .336629 
 .387185 
 .307¢41 
 .8388297 
 
 10 .3838853 
 .3839410 
 .839967 
 .840524 
 .841082 
 .3841640 
 .842198 
 .842756 
 .843315 
 .843875 
 
 10.344434 
 844994 
 345554 
 .846115 
 .3846676 
 3347237 
 .847798 
 .848360 
 .348922 
 
 10.349485 
 
 WoOooOooowwocse 
 
 lol 
 Jen) 
 
 eomowm cs 
 
 Cc 0 60 6 
 
 Ss 
 ie) 
 
 9 
 
 WOMMMOMMOMOO WHMOMmUwomwUownowowowo ovr 
 
 ASAIN 
 
TABLE XXVI.—LOGARITHMIC VERSED SINES 
 
 ~ 
 
 Ex. sec, 
 
 ~ 
 
 Vers. 
 
 Ex. sec. 
 
 ¢ DrammwwHo | 
 
 9.839467 
 
 .839641 
 .839815 
 .839989 
 .840162 
 
 .840336 | 
 
 .840510 
 .840683 
 .840857 
 .841030 
 .841204 
 
 9.841377 
 
 .841550 
 841723 
 .841896 
 .842070 
 , 842243 
 . 842416 
 .842589 
 842762 
 842934 
 
 9.843107 
 
 848280 
 _ 843458 
 843625 
 84 8798 
 848970 
 .844143 
 844315 
 844488 
 .844660 
 
 9.844832 
 
 845004 
 845177 
 845349 
 845521 
 845693 
 845865 
 .846037 
 846208 
 846380 
 
 9.846552 
 
 .846724 
 . 846895 
 .847067 
 . 847238 
 .847410 
 .847581 
 847753 
 . 847924 
 . 848095 
 848267 
 .848438 
 . 848609 
 848780 
 .848951 
 .849122 
 .849293 
 .849464 
 . 849634 
 
 9.849805 
 
 IWWWNWWWWW 
 
 09 0 
 
 9 00 
 
 DW 0 0 
 
 G90 0000 2009 G DO OD< 
 THT QO GS Tt 
 
 OO 
 
 OLOUd 
 
 | 10.849485 
 
 . 850048 
 850611 
 -do1175 
 .851738 
 802303 
 . 852867 
 303432 
 .808997 
 804563 
 80012 
 
 10. 355695 
 
 .3896261 
 .3856828 
 .857395 
 
 = -357963 
 
 -008531 
 . 809099 
 .859668 
 . 8602387 
 . 3860806 
 10361376 
 .3861946 
 .862516 
 . 863087 
 .863658 
 .864229 
 .864801 
 .865373 
 .805945 
 .3866518 
 
 10.367091 
 
 . 867665 
 . 3868239 
 . 868813 
 . 3869387 
 .869962 
 .8705388 
 .871118 
 .3871689 
 .8(2266 
 
 10.372842 
 
 8738419 
 .313997 
 87457 
 .875153 
 .810731 
 76310 
 376890 
 
 877469 
 
 .8¢8049 
 
 10.378630 
 
 .879210 
 .3 (9792 
 880373 
 . 3880955 
 .881537 
 382120 
 882703 
 . 883286 
 
 10.383870 
 
 OMNIAWIR WW © 
 
 COOOowoeo 
 TOV on en or etc = 
 
 Vela ilelleelleilelisiielie) 
 
 FITS ~ 
 ww 
 
 ~ 
 
 Ea 
 
 RSE 
 
 9.849805 
 .849976 
 .850147 
 .850317 
 . 850488 
 . 850658 
 . 850829 
 .850999 
 . 851169 
 .851340 
 .951510 
 
 9.851680 
 .851850 
 .852020 
 .852190 
 .952360 
 Sth 
 - 852700 
 .852870 
 .853040 
 . 8538209 
 
 9.853379 
 .853549 
 .853718 
 .853888 
 . 854057 
 . 854227 
 . 854396 
 . 854565 
 . 854735 
 . 854904 
 
 9 855073 
 .855242 
 . 855411 
 . 855580 
 . 855749 
 .855918 
 856087 
 . 856255 
 856424 
 . 856593 
 
 | 9.856762 
 
 . 856930 
 . 857099 
 .857267 
 . 857436 
 . 857604 
 .857772 
 .857941 
 858109 
 .858277 
 9.858445 
 . 858613 
 . 858781 
 .858949 
 .859117 
 859285 
 ~859453 
 .859621 
 859788 
 
 80 
 9.859956 | 2.80 
 
 10383870 
 
 884454 
 3850388 
 885623 
 . 886209 
 386794 
 887380 
 .387967 
 . 888554 
 .3889141 
 389728 
 
 10.390316 
 
 -890905 
 391493 
 392082 
 392672 
 393262 
 . 893852 
 394443 
 395034 
 395625 
 
 10.396217 
 
 3896809 
 .397402 
 3897995 
 898589 
 .899182 
 899777 
 -400371 
 . 400966 
 .401562 
 
 10.402158 
 
 402754 
 .403351 
 . 403948 
 404545 
 .405143 
 -405742 
 . 406340 
 406939 
 407539 
 
 10.408139 
 
 .408739 
 .409340 
 -409941 
 -410543 
 411145 
 411747 
 
 -412350 
 412954 
 413557 
 
 10.414161 
 
 .414766 
 415371 
 .415976 
 .416582 
 417189 
 417795 
 .418402 
 .419010 
 
 10.419618 
 
a so Man Ai ae i iat ae 
 
 CIRO mwwee | 
 
 | 
 | 
 | 
 
 1 
 
 AND EXTERNAL SECANTS. 
 
 ee 
 
 Vers. |D. 1".| Ex. sec. | D.1".|| ’ | Vers. 1".| Ex. sec. |D. 1’ 
 | 
 9.859956 | 2.80 | 10.419618 | 10.13 | 0.) 9.869924 | 2.75 |10.456928 | 10.60 
 "960124 | 2.78 | .420226 | 10.15 || 1] .870089 | 2.73 | .457564 | 10.62 
 860291 | 2.80 "420885 | 10.17 || 2 | .870253 | 2.75 | .458201 | 10.638 
 860459 | 2.78 "421445 | 10.15 || 3 | .870418 | 2.73 | .458839 | 10.62 
 “860626 | 2.80 422054 | 10.17 || 4] .870582 | 2.75 | .459476 | 10.65 
 860794 | 2.7% 422664 | 10.18 || 5 "0747 | 2.73 | .460115 | 10.65 
 860961 | 2.7 423275 | 10.18 || 6 | .870911 | 2.75 | .460754 | 10.65 
 861128 | 2.80 "493886 | 10.20 || 7% | .871076 | 2.73 | .461593 | 10.67 
 ,861296 | 2.78 "424498 | 10.20 || 8 | .871240 | 2.73 | .462033 | 10.67 
 861463 | 2.78 -425110 | 10.20 || 9 | .871404 | 2.73 | .462673 | 10.68 
 861630 | 2.78 425722 | 10.22 || 10 | .871568 | 2.73 | .463814 | 10 7 
 | 9.861797 | 2.78 | 10.426385 | 10.22 || 11 | 9.871782 | 2.78 '10.463956 |. 10.70 
 (861964 | 2.78 "426948 | 10.23 |) 12 | .871896 | 2.73 | .464598 | 10.70 
 862131 | 2.78 "427562 | 10.28 || 13 | .872060 | 2.73 | .465240 | 10.72 
 862298 | 2.78 428176 | 10.23 || 14 | .872224 | 2.73 465883 | 10.75 
 (862465 | 2.78 "428790 | 10.27 || 15 | .872388 | 2.73 | .466527 | 10.73 
 862632 | 2.78 "429406 | 10.25 || 16 | .872552 | 2.73 | .467171 | 10.73 
 862799 | 2.77 "430021 | 10.27 || 17 | .872716 | 2.73 | .467815 | 10.75 
 862965 | 2.78 "420637 | 10.27 || 18 | .872880 | 2.72 | .468460 | 10.77 
 863132 | 2.78 431253 | 10.28 | 19 | .873043 | 2.73 | .469106 | 10.77 
 863299 | 2.77 431870 | 10.3 | 20 | .873207 | 2.73 | .469752.| 10.77 
 9.863465 | 2.78 | 10.432488 | 10.28 || 21 | 9.878871 | 2.72 |10.470398 | 10.78 
 863632 | 2.78 | 433105 | 10.32 || 22 | .873534 | 2.7% 471045 | 10.80 
 863799 | 2.77 433724 | 10.80 |: 93 878698 | 2.72 471693 | 10.80 
 863965 | 2.77 "434342 | 10.32 || 24 | .873861 | 2.73 | .472341 | 10.82 
 864131 | 2.78 "434961 | 10.33 || 25 | .874025 | 2.72 | .472990 | 10.82 
 864298 | 2.77 "435581 | 10.33 || 26| .874188 | 2.72 | 473639] 10.88 
 864464 | 2.77 436201 | 10.33 || 27 “4351 | 2.73 , .474289 | 10.83 
 834630 | 2.78 436821 | 10.35 |; 28 4515 | 2.72 | .474939 | 10.85 
 864797 | 2.77 437442 | 10.37 || 29 | .874678 | 2.72 | .475590 | 10.87 
 864963 | 2.77 438064 | 10.37 || 80 | .874841 | 2.72 | _ 476242 | 10.85 
 9.865129 | 2.77 | 10.438686 | 10.37 || 31 | 9.875004 | 2.72 |10.476893 | 10.88 
 "965205 | 2.77 |  .439808 | 10.38 || 82 | .875167 | .2.72-| .477546 | 10.88 
 "865461 | 2.77 | 489981 | 10.38 || 83 | .875320 | 2.72 78199 | 10.88 
 865627 | 2.77 440554 | 10.40 || 34 | .875493 | 2.72 | .478852 | 10.90 
 865793 | 2.77 "441178 | 10.40 || 35 | .875656 | 2.72 | .479506 | 10.92 
 865959 | 2.75 "441802 | 10.42 || 36] .875819 | 2.72 | .480161 | 10.92 
 866124 | 2.77 442427 | 10.42 || 87 | .875982 | 2.72 | .480816 | 10.93 
 866290 | 2.77 "443052 | 10.43 || 88 | .876145 | 2.72 | .481472 | 10.93 
 866456 | 2.77 "443678 | 10.43 || 39 | .876308 | 2.70 | .482128 | 10.95 
 866622 | 2.75 "444304 | 10.45 || 40 | .876470 | 2.72 | .482785 | 10.95 
 | 9.966787 | 2.77 | 10.444931 | 10.45 || 41 | 9.876633 | 2.72 |10.483442 | 10.97 
 866953 | 2.75 445568 | 10.45 || 42 | .876796 | 2.70 | 484100 | 10.98 
 867118 | 2.77 "446185 | 10.47 || 43 | .876958 | 2.72 | .484759 | 10.98 
 867284 | 2.75 "446813 | 10.48 || 44] .877121 | 2.70 | .485418 | 10.98 
 867449 | 2.75 447442 | 10.48 || 45 | .877283 | 2.7 486077 | 10.98 
 867614 | 2.77 448071 | 10.48 |, 46 | 877445 | 2.72 | .486738 11.00 
 867780 | 2.75 448700 | 10.50 || 47 | .877608 | 2.7 487398 | 11 02 
 867945 | 2.75 "449330 | 10.52 || 48 | .877770 | 2.70 | .488059 | 11.03 
 868110 | 2.75 449961 | 10.52 || 49 | 877982 | 2.7% 488721 | 11.05 
 868275 | 2.77 450592 | 10.52 || 50 | .878095 | 2.7 489384 | 11.05 
 | 9.969441 | 2.75 | 10.451228 | 10.53 } 51 | 9.878257 | 2.70 |10.490047 | 11.05 
 "968606 | 2.75 L 451855 | 10.53 || 52} .878419 | 2.70 | .490710 | 11.07 
 868771 | 2.75 452487 | 10.55 || 53 | .878581 | 2.7 491374 | 11.08 
 868936 | 2.73 "453120 | 10.57 || 54 | .878743 | 2.7 492039 | 11.08 
 869100 | 2.75 453754 | 10.57 || 55 | .878905 | 2.7 492704 | 11.10 
 869265 | 2.75 "454383 | 10.57 || 56 | .879067 | 2.7 493370 | 11.10 
 869430 | 2.75 "455/22 | 10.58 || 57 | .879229 | 2.68 | .494036 | 11.12 
 869595 | 2.95 455657 | 10.58 | 58 | -.879390 | 2.70 | .494703 11.13 
 869760 | 2.7 456292 | 10.60 || 59 | .879552 ) 2% 495371 | 11.13 
 2 
 
 9.869924 | 2.7 
 
 10.456928 
 
 | 9.879714 
 
 4 
 70 
 
 110. 496039 
 
Vers. 
 
 TABLE XXVI.—LOGARITHMIC VERSED SINES 
 
 76° 
 
 D. 1".| Ex, sec. 
 
 ~ 
 
 cay 
 CO 
 
 COHIOwUkK wore 
 
 ie) 
 
 =) 
 
 N=) 
 
 ay 
 
 | 
 ThFy 
 
 hae 
 Panty 
 Wy 
 eT | 
 Hien | 
 My 
 
 =) 
 
 | 9.879714 
 
 879876 
 .880037 
 
 .880199 | 
 . 880360 
 
 .880522. | 
 - 880683 | 
 .880845 | 
 
 .881006 
 
 8381167 
 
 881329 
 881490 
 881651 
 
 .881812 | 
 
 831973 
 882134 
 882295 
 882456 
 
 882617 
 
 882777 
 .882938 
 .8383099 
 
 883260 
 
 883 120 
 .8335381 
 883741 
 833902 
 
 884062 | 
 884223 
 
 884383 
 884543 
 884703 
 884864 
 885024 
 885184 
 . 885344 
 885504 
 885664 
 885824 
 885983 
 886148 
 886303 
 886463 
 886622 
 886782 
 .886941 
 -887101 
 887260 
 887420 
 887579 
 887739 
 
 9.887898 
 .888057 
 .888216 
 .888375 
 888534 
 .888693 
 .888852 
 
 .889011 
 
 889170 
 
 9.889329 
 
 VWWWNWWw®W WWW WWWWWWWW 
 
 Wwe 
 
 9 0 WW WW WWW 
 
 Se Se G2 G2 Gd Ao So 
 on NW NN 
 
 WWNWWWWNWWHDW WWWWKwwwwww we 
 . Da O 
 oe = 
 
 POA 2A2R20RRR0 
 
 WW WD 
 
 10.496039 
 
 68 496707 
 
 4973877 
 
 .68 |  .498047 
 00 498717 
 .68 .499388 
 70 .500060 
 .68 -900732 
 .68 .501405 
 
 502078 
 
 68 502752 
 
 .68 | 10.503426 
 .68 .904102 
 .68 504777 
 63 505454 
 68 .506131 
 68 506808 
 .68 507486 
 67 .508165 
 .63 503844 
 8 009524 
 68 | 10.510205 
 67 .910386 
 68 .911568 
 
 67 512250 
 68 912933 
 67 .513617 
 
 68 .914301 
 67 .514936 
 
 515572 
 516358 
 10.517045 
 517732 
 7 | .518420 
 7 | 1519109 
 7 | 1519798 
 7} 520188 
 521179 
 521870 
 7 | .522562 
 523254 
 10.523947 
 524641 
 7 | 525335 
 5 | 526030 
 7 | — 526726 
 5 | 527423 
 7 | 528120 
 5 | 528817 
 7 | .529516 
 5 | 530215 
 5 | 10.530914 
 5 | 531614 
 5 | 582315 
 5 |  .583017 
 65 |  .533719 
 165 | 534422 
 2.65 | 535126 
 2.65 | 535830 
 2.65 .586535 
 
 2.65 | 10.537241 
 
 AIRHOR WOH S 
 
 59 
 
 Vers. ol Exec, 
 9.889329 | 2.65 10.537241 
 .889488 i 537947 
 889647 588654 
 889805 .539362 
 889964 540071 
 89012: 540780 
 890281 541490 
 890440 .542200 
 .890598 542911 
 890757 543623 
 890915 544336 
 9.891073 '10.545049 
 891232 545763 
 .891390 546477 
 891548 547193 
 891706 547909 
 891864 .548626 
 892022 549343 
 .892180 “550061 
 .892338 .550780 
 892496 .551500 
 9.892654 | 552220 
 892312 | 552941 
 892969 .553663 
 893127 554385 
 893235 .555109 
 893142 555833 
 . 893600 556557 
 893753 | 557283 
 893915 .558009 
 894072 | .558736 
 9.894230 10.559463 
 894387 560192 
 894544 560921 
 894702 561651 
 894859 562381 
 895016 563113 
 895173 563845 
 .895330 564577 
 895487 5653811 
 895644 566045 
 9.895801 '10.566781 
 .895958 567516 
 896115 568253 
 896272 .568990 
 .896428 569729 
 .896585 .570468 
 896742 571207 
 .896898 .571948 
 897055 .572689 
 897211 573431 
 9.897368 10.5741'74 
 897524 574917 
 .897680 515662 
 897837 576407 
 897993 BV7153 
 .898149 .5Y7900 
 .898305 578647 
 898461 579396 
 
 9 | 898618 580145 
 60 | 9.898774 110580895 
 
 =o 
 IWW WWW 
 
 ~ 
 
 2S 
 
 ~ 
 
 Wd 
 
 BW 09 WWW 
 
 eo 
 
 Pah ek ah pak pk fk fal fe ek fk Pe Be pe 
 
 ti) 
 
 et et 
 WW WNWNNNWNWWWe 
 B09 COC wwe 
 
 © 
 ~ 
 
 fmt peek peek ped Pe ek pe 
 
 9 WWW WW 
 
AND EXTERNAL SECANT*S. 
 
 DORDIOUPFWWH OS 
 
 10 
 
 59 
 60 
 
 "| Ex. sec. 
 
 Vers. 
 
 | 9.898774 
 
 . 898930 
 . 899086 
 
 .899709 
 
 | .900176 
 | 900334 
 | 9.900487 
 | ,900642 
 | .900798 
 | .900953 
 
 . 91108 
 
 | 901264 
 | .901419 
 | 90157 
 
 . 901729 
 
 .961884 
 9.902040 
 
 | 902350 
 | 902504 
 . 902659 
 . 902814 
 . 802969 
 .908124 
 .908278 
 .903433 
 
 .903742 
 . 903897 
 .904051 
 . 904206 
 ~ 904360 
 904514 
 . 904668 
 , 904823 
 .90:977 
 905131 
 . 905285 
 .905439 
 .905593 
 
 ile) 
 
 5 | 905747 
 
 .905901 
 -906055 
 .906209 
 . 906363 
 .906516 
 9.906670 
 . 906824 
 
 907284 
 .907438 
 
 9.908051 
 
 @ 0 %~ 
 
 899241 | 
 .899397 | 
 899553 | 
 
 9% 
 
 .899865 
 F 900020 
 
 WWWWW We 
 
 SWWWNWNWWNwWNww 
 
 .902195 
 
 KQQ | 
 9.903588 
 
 .906977 
 907131 
 
 WWWNWWNNWWNOW WWW WWNWNWNWYD WBUDWWWNWWNWNWW NWNWNWWWNWWOWWW 
 
 Oorergrg 
 
 .907591 
 907744 | 
 907898 
 
 Or Oorsz Oud 
 
 | 10.580895 
 
 .581645 
 582397 
 .583149 
 588903 
 584657 
 
 .585411 
 
 .586167 
 
 .586928 
 
 .587681 
 .588439 
 10.589198 
 .589957 
 .590718 
 .591479 
 592242 
 .598005 
 .593769 
 594533 
 -595299 
 .596066 
 
 | 10.596833 
 
 597601 
 598370 
 .599140 
 599911 
 “600682 
 .601455 
 602228 
 603003 
 603778 
 10. 604554 
 “60533 
 .606108 
 606887 
 
 607667 | 
 
 608447 
 609228 
 -610010 
 610794 
 611578 
 
 10.612363 
 613148 
 .613935 
 614723 
 .615511 
 .616301 
 617091 
 617883 
 .618675 
 .619468 
 
 57 | 10.620262 
 
 .621057 
 
 .621853 
 
 .622650 
 
 .623448 
 .624247 
 
 625047 
 625848 
 
 . 626650 
 10.627452 
 
 | 9.908051 
 
 DW IAD>rkwwore © 
 
 | 9.909734 
 
 | 9.911259 
 
 fed ped ek Pek ed ped Ped Be pe RR Pt 
 wecwwwowuwwwuU wwuwwioe 
 p09 09.09 09 G9 IW DWN 
 
 908204 
 .908357 
 .908511 
 . 908664 
 . 908817 
 .908970 
 . 909123 
 . 909276 
 . 909428 
 . 909581 
 
 . 909887 
 . 910039 
 .910192 
 .910345 
 .910497 
 . 910650 
 .910802 
 .910955 
 911107 
 
 .911412 
 911564 
 911716 
 .911868 
 912020 
 912172 | 
 912324 
 912476 
 912628 
 9.912780 
 912982 
 .918084 
 913235 
 .918387 
 .913539 
 .9138690 
 . 913842 
 .918993 
 914145 
 
 9.914296 
 914448 
 .914599 
 -914750 | 
 914902 
 .915053 
 915204 
 
 .915855 
 .915506 
 .915657 
 9.915808 
 . 915959 
 .916110 
 .916261 
 .916412 
 .916562 
 .916713 
 .9168C4 
 .917014 
 
 WWWNWWWWWWW WWWww 
 
 9.917165 
 
 9 57 
 
 10. 627452 
 628256 
 . 629060 
 .629866 
 .630673 
 .681480 
 . 6382289 
 .633098 
 . 6383909 
 .634720 
 . 635533 
 
 10.686346 
 .637161 
 .6387976 
 . 688792 | 
 .639610 
 .640428 
 .641248 
 . 642068 
 . 642890 
 .643713 
 
 10.644536 
 .§45361 
 .646186 
 .647013 
 .647541 
 648670 
 .649499 
 .6503830 
 .651162 
 651995 
 
 10.652829 
 658664 
 .654501 
 .655388 
 .656176 
 .657016 
 . 657856 
 .658698 
 .659540 
 .660884 
 
 10.661229 
 .662075 
 . 662922 
 .663770 
 . 664619 
 -665470 | 
 .666321 
 607174 
 . 668028 
 . 668883 
 
 10.669739 
 .670596 
 .671454 
 672314 
 673174 
 .674036 
 .674899 
 .675768 
 .676628 
 
 10.677495 
 
 wwe wo co Go te 
 r 
 
 Heh ek peek feed Pet Red Ped Fd Pt 
 209 05 CO 
 WwOrOtror 
 
 oo 
 
 wwIwwWwKNWwWWwWww wWiwwwwwwwe 
 Oo ~ 2 Qi 20 N 
 
 — 
 
 DODO D MII 
 OA W SANS 
 
 mel ek fee pk pk pk fee fk et Pd 
 
 HS 09 09.09 9 
 Sea amieke 
 mS toto te 
 
80° 
 
 TABLE XXVI.—LOGARITHMIC VERSED SINES 
 
 Som WINS | om 
 
 rary 
 (ani lele oh) 
 
 Vers. 
 
 9.917165 
 .917316 
 .917466 
 917616 
 917767 
 917917 
 918068 
 918218 
 918368 
 -918518 
 . 918668 
 
 9.918818 
 .918968 
 .919118 
 919268 
 - 919418 
 919568 
 919718 
 919868 
 .920018 
 -920167 
 
 9.920317 
 920466 
 .920616 
 920766 
 920915 
 - 921064 
 921214 
 921363 
 .921512 
 . 921662 
 
 .921811 
 . 921960 
 . 922109 
 . 922258 
 . 922407 
 . 922556 
 . 922705 
 . 922854 
 - 923003 
 - 923152 
 
 9.923301 
 - 923449 
 923598 
 923747 
 923895 
 - 924044 
 -924192 
 924341 
 - 924489 
 924637 
 
 9.924786 
 - 924934 
 - 925082 
 925231 
 925379 
 925527 
 92567 
 925823 
 925971 
 
 9.926119 
 
 Jes) 
 
 Dal" at Ex sec! 
 2.52 | 10.677495 
 2.50 678362 
 2.50 679231 
 2.52 .680101 
 2.50 -680972 
 2.52 681845 
 2.50 .682718 
 2.50 . 683593 
 2.50 684469 
 2.50 . 685346 
 2.50 -686224 
 2.50 | 10.687104 
 2.50 687985 
 2.50 688867 
 2.50 689750 
 2.50 690634. 
 2.50 691520 
 2.50 692407 
 2.50 693295 
 2.48 -694185 
 2.50 -695075 
 2.48 | 10.695967 
 2.50 .696861 
 2.50 697755 
 2.48 - 698651 
 2.48 .699548 
 2.50 . 400446 
 2.48 .701346 
 2.48 . 002247 
 2.50 . 703149 
 2.48 «704052 
 2.48 | 10.704957 
 2.48 . 705863 
 2.48 10677 
 
 2.48 - 07680 
 2.48 . 708590 
 2.48 . 709501 
 2.48 - 710414 
 2.48 .711828 
 2.48 . 7122438 
 2.48 713160 
 2.47 | 10.714078 
 2.48 ~714998 
 2.48 715919 
 2.47 - 716841 
 2.48 717764 
 2.47 . 718689 
 2.48 - 719616 
 2.47 . 720543 
 2.47 ~ 721472 
 2.48 722408 
 2.47 | 10.723335 
 2.47 . 424268 
 2.48 725203 
 2 AW 726139 
 2.47 127077 
 2.47 . 728016 
 2.47 . 728956 
 2.47 . 729898 
 2.47 . 780842 
 
 2.47 | 10.7381786 
 
 81° 
 D. 1’. f Vers. | D.1".| Ex. see. | D. 1" 
 14.45 0 | 9.926119 | 2.47 |10.731786 | 15.78 
 14.48 1 926267 | 2.47 | .7827383 | 15.78 
 14.50 2| .926415 | 2.45] .733680 | 15.83 
 14.52 3 | .926562 | 2.47 | .7346380 | 15.83 
 14.55 4 | .926710 | 2.47 | .735580 | 15.87 
 14.55 5 | .926858 | 2.47 | .736582 | 15.90 
 14.58 6 | .927006 | 2.45 | .9787486 | 15.92 
 14.60 7 | (927153 | 2.47 | 1738441 | 15.95 
 14.62 8 | .927801 | 2.45 | .739398 | 15.97 
 14.63 9 | .927448 | 2.47 | .740356 | 16.00 
 14.67 || 10 | .927596 | 2.45 | .741316 | 16.02 | 
 14.68 || 11 | 9.927743 | 2.47 |10.'742277 | 16.03 
 14.7 12 | .927891 | 2.45 | .743239 | 16.08 
 14.72 || 18 | .928088 | 2.45 | .%44204 | 16.08 
 14.73 || 14 | .928185 | 2.47 | 745169 | 16.13 
 14.77 || 15 | .9283833 | 2.45 | 746137 | 16.13 
 14.7 16 | .928480 | 2.45 | .747105 | 16.18 
 14.80 || 17 | .928627 | 2.45 | .'748076 | 16.20 
 14.83 || 18 | .928774 | 2.45 | .749048 | 16.22 
 14.83 || 19 | .928921 | 2.45 | .750021 | 16.25 
 14.87 || 20} .929068 | 2.45 | .%50996 | 16.28 
 14.90 |} 21 | 9.929215 | 2.45 |10.751973 | 16.30 
 14.90 || 22] .929862 | 2.45 | .752951 | 16.33 
 14.93 || 23 | .929509 | 2.45 | .753931 | 16.35 
 14.95 |} 24 | .929656 | 2.45 | .754912 | 16.38 
 14.97 || 25.| .929803 | 2.45 |] 755895 | 16.42 
 15.00 || 26 | .929950 | 2.45 | .756880 | 16.43 
 15.02 || 27 | .930097 | 2.43 | .%57866 | 16.47 
 15.03 || 28 | .930248 | 2.45 | 758854 | 16.50 
 15.05 || 29 | .980890 | 2.45 | - .759844 | 16.52 
 15.08 |} 80 | .980587 | 2.43 | .760885 | 16.53 
 15.10 || 81 | 9.930688 | 2.45 |10.'761827 | 16.58 
 15.13 || 82] .930830 | 2.48 | .'7628292 | 16.60 
 15.15 || 33| .980976 | 2.45 | -.768818 | 16.62 
 15.17 || 3 .9381123 | 2.438 | .%64815 | 16.67 
 15.18 || 385 | .931269 | 2.45 | .765815 | 16:68 
 15.22 || 86 | .981416 | 2.481 .766816 | 16.72 
 15.23 || 87 | .981562 | 2.43°| .767819 | 16.7% 
 15.25 || 88 | .931708 | 2.45 | .%68828 | 16.77 
 15.28 ||. 389 | .931855 | 2.43 | .769829 | 16.80 
 15.30 || 40 | .932001 | 2.48 | 770887 | 16.82 
 15.33 || 41 | 9.982147 | 2.43 |10.771846 | 16.87 
 15.85 || 42 | .982293 | 2.43} 772858 | 16.87 
 15.37 || 43 | .982439 | 2.48! 773870 | 16.92 
 15 88 || 44°} .9382585 | 2.43 | .774885 | 16.95 
 15.42 || 45 | .9827381 | 2.48 | .775902 | 16.97 
 15.45 || 46 | .982877 | 2.48] .'776920 | 17.00 
 15.45 || 47 | .983023°| 2.48 |° .777940 | 17.02 
 15.48 || 48 | .983169 | 2:48.| . .778961 | 17.07 
 15.52 || 49 | .988315:| 2.42 | 779985 | 17.08 
 15.53 || 50 | 933460 | 2.43 781010 | 17.12 
 15.55 || 51 | 9.983606 | 2.43 |10.782087 | 17.13 
 15.58 || 52 | .933752 | 2.42 | .788065 | 17.18 
 15.60 || 58 | .933897 | 2.48] .784096 | 17.20 
 15.63 || 54 | .934043 | 2.43] .785128 | 17.93 
 15.65 || 55 | .934189°| 2.42 | .786162 | 17.97 
 15.67 || 56 | .934834 | 2 48] .787198 | 17.30 
 15.70 || 57 | .984480°| 2.42 | .788236 | 17.33 
 15.73 || 58 | .984625°| 2.42 | .789976°| 17.35 
 15.73 || 59 | .984770 | 2.438 | .7903817 | 17.40 
 15.7 60 | 9.934916 | 2.42 110.791361 | 17.42 
 
oa 
 
 82° 
 
 ~ 
 
 Vers. 
 
 9.934916 
 . 935061 
 . 935206 
 . 935352 
 .935497 
 .935642 | 
 935787 
 . 9359382 
 . 936077 
 . 936222 
 . 936367 
 9.936512 
 . 936657 
 .936801 
 .936946 
 . 937091 
 .937236 
 .937380 
 .937525 
 . 937669 
 .937814 
 . 937958 
 .938103 
 . 9388247 
 -938391 
 2 - 938536 
 26 . 938680 
 27 | .938824 
 28 . 938968 
 29 .939112 
 380 - 939257 
 
 . 939401 
 32 . 939545 
 3: . 939688 
 34 - 939832 
 35 . 939976 
 36 .940120 
 37 . 940264 
 38 . 940408 
 39 . 940551 
 40 .940695 
 
 41 | 9.940839 
 
 940982 
 43 .941126 
 44 941269 
 45 .941413 
 46 | .941556 
 7 | .941699 
 
 Ho | 
 
 SO WH 2 C2 OU HP CO 0D 
 
 rt 
 
 oo 
 hoe 
 © 
 
 > 
 4) 
 
 DWMWNWNW Wee eRe ee ee 
 OP WO COON WOH 
 eo) 
 
 WW WWW WWNWNWW WWWNWNWNWNWNWNWD WWW NWNHNWNWWND WW 
 
 48 | .941843 
 49°) .941986 | 
 50 | .942120 
 
 B1 | 9.942272 | 
 52} .942415 
 53. | .942559 
 54| .942702 
 55 | 942845 
 | 942988 | 
 57 | .943131 
 58 | .943273 
 59 | 943416 | 
 | 60 | 9.943559 | 
 
 or 
 (=r) 
 
 D. 1".| Ex. sec. 
 
 *y 
 ~) 
 
 2 
 
 F] 
 
 29 2 %~ 
 
 WWWNWWWE 
 
 WBWNWWNWWNWWWW 
 
 TWWWNWWWYW 
 
 . 792406 
 193453 
 
 . 797660 
 - 798716 
 799774 
 .800835 
 .801897 
 10.802961 
 .804027 
 805095 
 .806165 
 807237 
 .808311 
 .809387 
 .810465 
 811545 
 .812627 
 
 10.813711 
 814797 
 815385 
 816975 
 818067 
 819161 
 820257 
 821356 
 822456 
 823559 
 
 10.824664 
 82577 
 826879 
 .827990 
 829104 
 830219 
 831337 
 832456 
 “83357 
 834708 
 
 10835829 
 836957 
 838088 
 839221 
 84035? 
 841494 
 842634 
 843776 
 844921 
 846068 
 
 | 10.847217 
 
 818368 
 
 851836 
 852997 
 854161 
 8553826 
 856494 
 10.85,7665 
 
 | 10.791261 
 
 194502 © 
 
 . 849522 
 
 .850678 | 
 
 ee ee es 
 
 FEA SI RPS SES SS ST ST St SS 2 
 
 rae 
 Seen eto} eS 
 
 AND EXTERNAL SECANTS. 
 
 Vers. 
 
 9.943559 
 943702 
 . 943845 
 943987 
 . 944130 
 
 944273 | 
 
 944415 
 
 944558 
 
 944700 
 944843 
 944985 
 
 9.945127 
 945270 
 945412 
 945554 
 945696 
 945838 
 945981 
 946123 
 946265 
 - 946407 
 
 9.946549 
 946690 
 946832 
 946974 
 947116 
 947258 
 947399 
 940541 
 947683 
 947824 
 
 9.947966 
 .948107 
 948249 
 948390 
 948531 
 .948673 
 .948814 
 948955 
 .949096 
 949237 
 
 9.949379 
 949520 
 949661 
 949802 
 . 949943 
 . 950083 
 . 950224 
 . 9503865 
 . 950506 
 950647 
 
 9.950787 
 
 950928 
 
 “951069 | 
 951209 | 
 
 951350 
 . 951490 
 951631 
 951771 
 . 951911 
 9.952052 
 
 Pils. 
 
 0 WWM 
 eo ao a0 
 J 
 
 2.37 
 
 Co Ce G2 09 C9 C2 C2 
 a AININNONN 
 
 we) 
 
 37 
 
 WWNWWNWNWWWWW WWWWWWwWWwWwW WWNVNWNWNWNWWWD WNW WNNWWWWD WWNWwwnwnwnw 
 5 Rimaroe 3 oo 
 ea 
 
 Ex. sec. | D. 1". 
 
 )10.857665 | 19.55 
 .858838 | 19.58 
 .860013 | 19.68 
 .861191 | 19.67 
 862371 | 19.72 
 .863554 | 19.7 
 
 .864739 | 19.80 
 .865927 | 19.83 
 .867117 | 19.88 
 .868310 | 19.92 
 .869505 | 19.97 
 
 10.870703 | 20.00 
 .871903 | 20.05 
 .873106 | 20.10 
 .874312 | 20.18 
 .875520 | 20.18 
 876731 | 20.28 
 -877945 | 20.27 
 .879161 | 20.30 
 .880379 {| 20.37 
 .881601 | 20.40 
 
 10882825 | 20.45 
 "884052 | 20.48 
 “885281 | 20.55 
 -886514 | 20.58 
 887749 | 20.62 
 "888986 | 20.68 
 890227 } 20.72 
 891470 | 20.77 
 "892716 | 20.82 
 893965 | 20.87 
 
 10.895217 | 20.92 
 896472 | 20.95 
 897729 | 21.00 
 .898989 | 21.07 
 . 900253 | 21.10 
 .901519 | 21.15 
 .902788 | 21.20 
 .904060 | 21.25 
 .905335 | 21.30 
 .906613 | 21.33 
 
 10.907893 | 21.40 
 .909177 | 21.45 
 .910464 | 21.50 
 .911754 | 21.55 
 .918047 | 21.60 
 .914843 | 21.65 
 915642 | 21.7 
 .916944 | 21.75 
 .918249 | 21.52 
 .919558 | 21.85 
 
 10.920869 | 21.92 
 922184 | 21.97 
 .923502 | 22.02 
 924823 | 22.07 
 926147 | 22.13 
 927475 | 22.17 
 .928805 | 22.23 
 .9380139 | 22.30 
 931477 | 22.33 
 
 10.932817 22.40 
 
a eer a TS = oa —— 
 
 TABLE XXVI.—LOGARITHMIC VERSED SINES 
 
 | | 
 d : Vers. lO ass ella eal seas <5) 4 BA , Vers; | D. 1°.:! Ex. sees Dil" 
 0 | 9.952052 | 2.33 | 10.932817 | 22.40 0 | 9.960397 | 2.30 |11.020101 | 26.40 
 1 .952192 | 2.83 .984161 | 22.45 1 .960535 | 2.28 .021685 | 26.48 
 2 952382 | 2.35 .935508 | 22.52 2 .960672 | 2.30 | .028274 | 26.57 
 3 . 952473 | 2.33 -9386859 | 22.57 || 38 .960810 | 2.380 .024868 | 26.65 
 4 .952613 | 2.33 .9388213 | 22.62 4 .960948 | 2.30 .026467 | 26.78 
 5 .952753 | 2.83 .939570 | 22.68 5 .961086 | 2.28 .028071 | 26.80 
 6 .952893 | 2.33 .940931 | 22.75 6 . 961223 | 2.30 .029679 | 26.90 
 fi .9530383 | 2.83 .942296 | 22.78 || 7 .9613861 | 2.2 .031293 | 26.98 
 8 .953173 | 2.33 .9438663 | 22.85 || 8 .961498 | 2.80 .032912 | 27.07 
 9 .953313. | 2.33 .945034 | 22.92 9 .961636 | 2.28 .034536 | 27.13 
 10 .9534538 | 2.33 . 946409 | 22.97 || 10 .961773 | 2.30 .036164 |. 27.28 
 11 | 9.953593 | 2.32 | 10.947787 | 23.03 || 11 | 9.961911 | 2.28 |11.037798 | 27.38 
 2 .9537382 | 2.33 .949169 | 23.08 || 12 .962048 | 2.30 .039438 | 27.40 
 lies -953872 | 2.3: .950554 | 28.15 || 13 .962186 | 2.28 .041082 | 27.50 
 ile’ .954012 | 2.33 .951943 | 28.22 || 14 . 9623823 | 2.2 .042732 | 27.58 
 15 .954152 | 2.32 .953336 | 23.27 || 15 . 962460 | 2.28 .044387 | 27.67 
 16 .954291 | 2.33 .954732 | 23.33 || 16 . 962597 | 2.80 .046047 | 27.77 
 We .954431 | 2.33 .956132 | 23.38 || 17 .962735 | 2.28 .047713 | 27.85 
 18 .954571 | 2.32 .957585 | 28.45 || 18 . 962872 | 2.28 .049384 | 27.93 
 19 .954710 | 2.38 . 958942 | 23.52 || 19 .963009 | 2.2 .051060 | 28.08 
 20 .954850 | 2.32 .960353 | 28.57 || 2 .663146 | 2.2 .052742 |. 28.13 
 21 | 9.954989 | 2.33 | 10.961767 | 23.65 || 21 | 9.963283 | 2.28 |11.054430 | 28.22 
 Q2 .955129 | 2.32 .963186 | 23.7 22 .963420 | 2.28 .056123 | 28.30 
 93 .955268 | 2.32 .964608 | 23.77 93 . 963557 | 2.28 .057821 | 28.40 
 24 .955407 | 2.33 .966034 | 23.82 || 24 . 963694 | 2.28 .059525 | 28.50 
 25 955547 | 2.32 .967463 | 23.90 || 25 .968831 -| 2.2% .061235 | 28.60 
 26 .955686 | 2.32 .968897 | 23.95 || 26 .963968 -| 2.27 .062951 | 28.68 
 Q7 .955825 | 2.32 .970384 | 24.02 || 27 .964104 | 2.28 .064672- | 28.78 
 28 . 955964 | 2.32 OTL 75> |-24.10-1)-2 .964241 | 2.28 .066399 | 28.88 
 29 .956103 | 2.338 .9738221°| 24.15 29 .964378 -| 2.28 .068132 | 28.98 
 30 .956243 | 2.32 .974670°| 24.22 || 3 .964515 | 2.27 .069871 |. 29.08 
 31 | 9.956382 | 2.32 | 10.976123 | 24.28 || 31 | 9.964651 | 2.28 |11.071616 | 29.18 
 ae .956521 | 2.32 .977580 | 24.85 || €2 .964788 | 2.27 .078367 |. 29.28 
 Be .956660 | 2.82 .979041-| 24.42 || 33 .964924 | 2.28 .075124 | 29.38 
 34 .956799 | 2.80 .980506 | 24.48 || 34 .965061 | 2.27 076887 | 29.48 
 35 .956987 | 2.32 .981975° | 24.55 || 85,| -965197 | 2.2% .078656 | 29.58 
 36 957076 | 2.32 .983448° | 24.63 || 36 2965834.| 2.27 .080431 | 29.68 
 Bi .957215. | 2.32 .984926 | 24.68 || 37 .965460 | 2.2 .082212 | 29.80 
 38 195 7ad4 1° 2.382 .986407 | 24.77 || 88 .965607-| 2.27 .084000 | 29.90 
 39 .9574938 | 2.30 .987893 | 24.83 || 39 965743 | 2.27 .085794 | 80.00 
 40 .957631 | 2.32 .989883-| 24.90 || 40 .965879 | 2.28 .087594 | 380.12 
 41 | 9.957770 | 2.82 | 10.990877 | 24.97 || 41 | 9.966016 | 2.27 |11.089401 | 30.22 
 42 .957909 | 2.30 .992375 | 25.03 2 .966152 | 2.27 .091214 | 30.82 
 43 .958047. | 2.82 .9938877: | 25.12 || 43 . 966288 | 2.27 .093033 | 30.43 
 44 .958186 | 2.30 .995384 | 25.18 44 .966424 | 2.27 .094859 | 30.55 
 45 .958324 | 2.32 .996895- | 25.27 || 45 .996560 | 2.27 .096692 | 80.67 
 46 .958463 | 2.80 .998411 | 25.33 || 46 .966696 | 2.27 .098532 | 80.77 
 47 .958601 | 2.30 .999931 | 25.40 47 . 966882 | 2.27 .100378 | 80.87 
 48 .958739 | 2.82 11.001455 | 25.48 || 48 .966968 | 2.27 .102230 | 31.00 
 49 .958878 | 2.30 .002984: | 25.52 49 .967104 | 2.27 .104090 | 31.12 
 50 .959016 | 2.30 .004517 | 25.63 |; 50 967240 | 2.27 .105957 | 31.22 
 51 | 9.959154 | 2.30 | 11.006055 | 25.70 || 51 | 9.967376 | 2.27 |11.107830 | 31.35 
 52 .959292 | 2.32 007597 | 25.7 52 .967512-| 2.25 .109711 | 31.45 
 53 | .959431 | 2.30 | 009144 | 25.85 || 53 | .967647 | 2.27] .111598 | 31.58 
 54 . 959569 | 2.30 .010695 | 25.93 54 967783 | 2.2 113498 | 31.68 
 5D 959707 | 2.380 .012251 | 26.00 5D 967919 | 2.25 .115394 | 31.82 
 56 .959845 | 2.3 .013811 | 26.10 56 . 968054 | 2.27 .117808 | 31.98 
 57 .959983 | 2.30 .015377 | 26.17 || 57 .968190 | 2.27 .119219 | 32.07 
 58 .960121 | 2.30 .016947 | 26.23 || 58 .968326 | 2.25 .1211438 | 32.18 
 59 .960259 | 2.380 018521 | 26.33 || 59 | .968461 | 2.27 | - .128074 | 32.80 
 60 | 9.960397 | 2.30 | 11.020101 | 26.40 || GO 9.968597 | 2.25 |11.125012 | 32.43 
 
. 
 
 —s 
 SODAS wwe S | 
 
 86° 
 
 AND EXTERNAL SECANTS, 
 
 87° 
 
 a 
 
 Vers. Dp. 1°.; Ex, see, 
 
 9.968597 | 2.25 
 .968782 | 2.27 
 .968868 | 2.25 
 969003 | 2.25 
 969138 | 2.2 
 . 969274 2. 25 
 .969409 | 2.25 
 .969544 | 2.25 
 .969679. | 2.25 
 .969814 | 2.25 
 . 969949 | 2.25 
 
 9.970084 | 2.27 
 . 970220 | 2.2 
 . 970354 | 2.25 
 .970489 | 2.25 
 . 970624 :| 2.25 
 970759 | 2.25 
 . 970894 | 2.25 
 .971029 | 2.25 
 .971164 | 2.23 
 971298 | 2.25 
 
 9.971433 | 2.25 
 .971568 | 2.23 
 971702 | 2.25 
 971837 | 2.23 
 2 DELO TE 2.25 
 .972106.| 2.23 
 . 972240 | 2.23 
 . 972874 | 2.25 
 . 972509 | 2.2% 
 . 972643 | 2.23 
 
 9.972777 | 2.25 
 972912 | 2.23 
 -973046 | 2.23 
 .973180 | 2.23 
 .973314 | 2.23 
 978448 | 2.2% 
 973582 | 2.28 
 .973716 | 2.23 
 .973850 | 2.2% 
 .973984 | 2.23 
 
 9.974118 
 - 974252 
 . 974386 22 
 
 SR OE CS) 
 ~) 
 (ou) 
 
 974519 | 2.23 
 974653 | 2.23 
 974787 | 2.22 
 974920 | 2.23 
 975054 | 2.23 
 975188 | 2.22 
 975321 | 2.23 
 9.975455 | 2.22 
 975588 | 2.23 
 975722 | 2.22 
 D75855. | 2,22 
 975988 | 2.23 
 976122 | 2.22 
 976255 | 2.22 
 .976388 | 2.22 
 976521 | 2.22 
 9.976654 | 2.23 
 
 | 
 1”.| Ex. sec. 
 
 Dibts: - Vers. |D. | DOT 
 11.125012 | 32.43 || 0 | 9.976654 | 2.23 |11.257854 | 42.52 
 126958 | 32.55 1 | .976788 | 2.22 | .260405 | 42.73 
 .128911 | 32.7 2 | .976921 | 2.22 | 262969 | 42.95 
 130873 | 32.80 3 | .977054 | 2.22 | .265546 | 43.20 
 132841 | 32.95 4 | .977187 | 2.22 | .268138 | 43.42 
 .184818 | 33.07 5 | .977320 | 2.20 | .270743 , 43.67 
 136802 | 33.22 || 6 | .977452 | 2.22 | .273363 | 43.88 
 138795 | 33.33 7 | 977585 | 2.22] .275996 | 44.15 
 .140795 | 33.47 8 | .977718 | 2.22 | .278645 | 44.38 
 142803 | 33.62 9 | 977851 | 2.22 | .281308 | 44.68 
 .144820 | 33.73 || 10 | .977984 | 2.20 | . .283986 | 44.88 
 11.146844 | 33.88 || 11 | 9.978116~-| 2.22 |11.286679 | 45.13 
 .148877 | 34.02 || 12 | .978249 | 2.2: 289387 | 45.38 
 .150918 | 34.17 || 13 | .978382 | 2.20 | .292110 | 45.65 
 .152968 | 84.80 |} 14} .978514 | 2.22 | .294849 | 45.92 
 155026 | 34.43°|| 15 68617 | 2.20 | .297604 | 46.17 
 157092 | 34.60 || 16 | .978779 | 2.22 | .8003874 | 46 45 
 159168 | 34.73 || 17 | .978912 | 2.20 | -.808161 | 46.72 
 .161252 | 34.87 || 18 | .979044 | 2.22 | .305964 | 47.00 
 163344 | 35.03 || 19 | .979177.| 2.2 .808784 | 47.27 
 165446 | 35.17 || 20 | .979309 | 2.22 | - .811620 | 47.55 
 11.167556 | 35.33 || 21 | 9.979442 | 2.20 |11.314473 | 47.83 
 .169676 | 35.48 |} 22 | .979574 | 2.20 | .817348 | 48.13 
 .171805 | 35.63 || 23} .979706 |. 2.20 | .320231 | 48.43 
 173943 | 35.78 || 2. 9798388 | 2.20 | 323137 | 48.72 
 .176090 | 85.93 || 25 | .979970 | 2.22 | .826060 | 49.02 
 .178246 | 86.10 || 26 | .980103 | 2.20 | .829001 | 49.33 
 .180412 | 36.27 || 27 | .980235 | 2.20 | .331961 | 49.63 
 182588 | 86.42 || 28 | .980867 | 2.20 | .334939.| 49.93 
 .184773 | 36.58 || 29 | .980499 | 2.20 | .387985 | 50.27 
 .186968 | 36.7 30 | .980631 | 2.20 | .340951 | 50.58 
 11.189173 | 36.90 || 31 | 9.980763 | 2.20 |11.843986 | 50.92 
 .191387 | 87.08 || 82} .980895 | 2.18 | .347041 | 51.23 
 .193612 | 37.25 || 83 | .881026 | 2.20 | .850115 | 51.58 
 .195847 | 37.42 || 84 | .981158 | 2.20 | 858210 | 51.92 
 .198092 | 37.58 || 35 | .981290 | 2.20 | . 856325 | 52.25 
 .200347 | 87.77 || 86 | .981422 | 2.20 | .359460 | 52.62 
 -202613 | 87.93 || 87 | 981554 | 2.18} .862617 | 52.95 
 204889 | 38.12 || 88 | .981685 | 2.20 | .865794 | 53.32 
 207176 | 38.28 || 89 | .981817 | 2.2 868993 | 53.68 
 -209473 | 38.47 || 40 .981949 | 2.18 | .872214 | 54.07 
 11.211781 | 88.67 || 41 | 9.982080 | 2.20 |11.375458 | 54.42 
 .214101 | 88.83 | 42 | .982212 | 2.18 | .378723 | 54.80 
 .216431 | 39.03 || 43 | .982343 | 2.20 | .382011 | 55.20 
 218773 | 39.20 || 44 | .982475.| 2.18 | .885828 | 55.58 
 221125 :| 89.42 | 45 | .982606 | 2.18 | .888658 | 55.97 
 .223490 | 39.58 || 46 | .982787 | 2.20 | .892016 | 56.38 
 .225865 | 39.80 || 47 | .982869 | 2.18 | .895899 | 56.80 
 .229253 | 39.98 || 48 | .983000 | 2.18 | .398807 | 57.20 
 230652 | 40.18 || 49 | .983131 | 2.18 | .402239 | 57.62 
 .233063 | 40.38 || 50 | .988262 | 2.20 |. .405696 | 58.07 
 11.235486 | 40.58 || 51 | 9.983394 | 2.18 |11.409180 | 58.48 
 237921 | 40.78 || 52 | .983825 | 2.18 | .412689 | 58.93 
 240368 | 41.00 || 53 | .983656 | 2.18 | .416225 | 59.38 
 242828 | 41.20 | 54 | .988787 | 2.18 | .419788 | 59.83 
 245300 | 41.42 || 55 | .983918 | 2.18 | .423878 | 60.28 
 247785 | 41.63 || 56 | .984049 | 2.18 | .426995 | 60.77 
 . 250283 | 41.83 || 57 | .984180 | 2.18 | .480641 | 61.25 
 252793 | 42.07 |} 58 | .984311 | 2.18 | .434316 | 61.73 | 
 255317 | 42.2 | 59 | .984442 | 2.18 | .438020 | 62.22 
 11. 257854 | 42.52 || 60 | 9.984573 | 2.17 |11.441753 | 62.7 
 
, 
 
 CORIO WWH OS 
 
 10 
 
 Vers. 
 
 ( 9.984573 | 
 . 984703 
 .984834 | 
 . 984965 
 985096 
 - 985226 
 985357 
 985487 
 985618 
 985748 
 985879 
 
 986009 
 -986140 
 986270 
 986400 
 986531 
 
 986661 
 986791 
 986921 | 
 987051 
 
 987181 
 
 987311 
 987441 
 987571 
 987701 
 987831 
 987961 
 988091 
 988221 
 
 “988350 
 988480 | 
 
 . 988610 
 . 988739 
 . 988869 
 988998 
 . 989128 
 989257 
 . 989387 
 .989516 | 
 . 989646 
 9897715 
 . 989904 
 . 990034 
 .990163 
 . 990292 
 .990421 
 990550 
 .990679 
 990803 
 :990937 
 .991066 
 
 .991195 
 .991324 
 
 991453 
 
 . 991582 
 .991710 
 .991839 
 .991968 
 . 992096 
 . 992225 
 
 9.992354 
 
 ILWWWNWNWND NYOWNWNWNWNWWNWWW WNW WWWWNWWWWYW 
 
 WWW WWWNWWW WNWNWNWNHWWWW WWW 
 
 re 
 
 2) 
 
 =e 
 
 @O W Wes 
 
 5 
 ~ 
 
 BD WW WW 
 
 —_— 
 
 Ex. sec. 
 
 | 11.441753 
 445517 
 449311 
 -453137 
 -456994 
 -460883 
 -464805 
 -468761 
 472751 
 470775 
 480834 
 .484929 
 .489061 
 493230 
 -497437 
 .501683 
 .505968 
 .510293 
 .514659 
 -519066 
 .523516 
 .528010 
 532548 
 537131 
 .541760 
 546437 
 .551161 
 555935 
 560759 
 565634 
 570561 
 
 5735542 
 58057 
 
 585670 
 .590819 
 .596027 
 601295 
 606625 
 .612018 
 617475 
 622998 
 
 628589 
 634250 
 639982 
 645788 
 651668 
 657626 
 663663 
 669781 
 675984 
 682272 
 
 688649 
 695117 
 701679 
 
 708338 
 - 715097 
 721958 
 728925 
 736002 
 743192 
 750498 
 
 q+ 
 
 115. 29% 
 ; 9086 
 
 9215 
 9345 
 9474 
 9603 
 9732 
 9862 
 9991 
 
 120 
 
 0249 
 0378 
 0507 
 0636 
 0765 
 0894 
 1023 
 1152 
 1281 
 1410 
 1539 
 1668 
 1797 
 1925 
 2054 
 2183 
 2312 
 2440 
 2569 
 2698 
 2826 
 2955 
 3083 
 3212 
 3340 
 3469 
 3597 
 
 726 
 3854 
 3983 
 4111 
 4239 
 4368 
 4496 
 4624 
 4752 
 4881 
 
 5009 
 5137” 
 
 5265 
 
 | 5393 
 
 | 
 | 
 | 
 | 
 
 5521 
 
 5649 || 
 
 5777 
 5905 
 6033 
 6161 
 6289 
 6417 
 6545 
 6673 
 6801 
 
 15. 30*! 
 
 TABLE 20 LOS ee SIGNS AND EXTERNAL 
 
 OOIHD OUP OF TH © 
 
 Je) 
 
 °) 
 
 =) 
 
 © 
 
 Vers. 
 
 9.992354 
 . 992482 
 .992611 
 992739 
 . 992868 
 . 992996 
 .993124 
 993253 
 .993381 
 .993509 
 -993637 
 
 .993765 
 993894 
 -994022 
 994150 
 994278 
 994406 
 994534 
 . 994662 
 . 994789 
 994917 
 
 995045 
 995173 
 995301 
 995428 
 995556 
 995683 
 995811 
 995939 
 996066 
 996193 
 
 .996321 
 . 996448 
 . 996576 
 . 996703 
 . 996830 
 - 996957 
 . 997085 
 997212 
 997839 
 . 997466 
 997593 
 997720 
 997847 
 :997974 
 .998101 
 998228 
 
 998355 
 
 .998481 
 
 998608 
 
 . 998735 
 998862 
 998988 
 . 999115 
 . 999241 
 .999368 
 . 999494 
 , 999621 
 999747 
 | .999874 
 10.000000 
 
 .| Ex. sec. 
 
 TWWWWNWWWWW ; 
 
 © . 
 
 eh beh bed peek pre peek pee ek rm et ek peek beat rk re eek frm pr 
 [WOW wWWwWwWwWwo WWW WwoWww 
 
 WWWWNNWWNWWW 
 
 WWWWNWNWWD VWWWNHNWNNWWWOYWD WNW wWwnwnnwnww 
 
 % 0 
 
 WW WW Ww wwww 
 
 {1.750498 
 
 157925 
 ~ (65477 
 . 773158 
 .780973 
 (88926 
 - 797022 
 805268 
 -813668 
 822229 
 
 .880956 
 
 11.839858 
 .848940 
 858211 
 86767 
 877351 
 887239 
 897350 
 
 907697 | 
 
 - 918290 
 929141 
 
 . 940264 
 .951672 
 . 963381 
 -975408 
 11.987'769 
 12.000485 
 .013578 
 .027069 
 .040984 
 .0553852 
 12.070202 
 .085569 
 .101490 
 .118008 
 .135168 
 153024 
 .171634 
 .191066 
 .211896 
 .232712 
 12.255116 
 20872 
 
 803674 
 
 1 
 
 are 
 
 .8380129 
 
 .808285 
 .888375 
 420686 
 .455575 
 -493490 
 535009 
 
 12.580893 
 
 632172 
 
 .690291 
 . 757364 
 
 836672 
 
 12..933708 
 
 13.0587 
 234991 
 536148 
 
 Inf. pos. 
 
 0878 
 1005 
 1132 
 1259 
 1386 
 1518 
 1640 
 1767 
 1894 
 2020 
 2147 
 2274 
 2401 
 2527 
 2654 
 2781 
 2907 
 3034 
 3161 
 8287 
 3414 
 3540 
 3667 
 3793 
 8920 
 4046 
 4172 
 4299 
 4426 
 
 '15,81* 
 
TADLE XXVII.—NATURAL SINES AND COSINES, 
 
 ~ 
 
 0° 
 
 ~ 
 
 OH CD VD 
 
 2 
 
 449 
 
 “i 1° 9° 8° | 4° . 
 
 Sine |Cosin | Sine. aan Sine Cosin| Sine |Cosin 
 One. || .01745 30985 | (99939 | | 05234 99863 | 0 6976 | .99756| 60 
 One. |} 01774! .9 | 99938 | “05263 .99861 || .07005 | .99754| 59 
 One. |) .01803} . 99937 | 05292 | .99860} 07034 | .997 D2) 58 
 7; One, || .01832} . 80 . 99936 | | .05321 | .99858 || .07063| .99750) 57 
 | One. || .01862 |. }| .99935 || 05350 | 99857 || .07092| .99748 | 56 
 5| One. || 01891] .99982 9|.99934)| 05379 | 99855 || .07121} .99746) 55 
 75| One. |; .01920] .99982 || 99933 | 05408 | .99854|| .07150| 99744 54 
 One. |) .01949) .99981 || | .99932/| 05437} .99852!| .07179| .99742) 53 
 One. |/..01978] .99980 | .99931 || .05466 .99851 || 07208] 99740. 52 
 One. || .02007, .99 | 2| .99930 || .05495 | .99849|| .07237| .99733 | 51 
 | One. || .02036 | . 9! || . 99929 | | .05524 | .99847 || .07266| .99736) 50 
 20! .99999 || .02065 | . |): é . 99927 || .05553 | .99846 || .07295| .99734! 49 
 349 | .99999 || .02094! . nee 05582 | .99844|| .07324| .99731) 48 
 78! 99999 | “02123 (99977 || 3 - 99925 || .05611 | .99842|| .07853] .99729| 47 
 7| .99999 || .02152} .99977 || 03897 "99924 | .05640 | .99841 || .07382] 99727) 46 
 51. 99999 || 02181}. ||. 5| .99923|| .05669 | .99839'| .07411| .99725| 45 
 00465 | 99999 || .02211 99976 ||. 55 | .99922|| .05698 | .99838 || .07440] .99723| 44 
 5! .99999 | 02240! .99975 || .03984| .99921 || .05727! 99836 || .07469| .99721| 43 
 ' 99999 |! 02269 | .¢ 7 99919 || .05756 | 99834 || .07498] .99719| 42 
 . 99998 || .02298 | 96 2|.99918}| .05785 | .99833 | “O7527 .99716| 41 
 $2) .99998 || .02327 99917 || .05814 | .99831 || .07556| .99714| 40 
 |99998 || .02356 | . 9 {| - | 99916) .05844 99829 | .07585 . 99712) 39 
 99998 || 02385) 99972 || 04129) .99915!| .05873) .99827 | .07614) .99710) 38 
 j bese epi ced || . 99913) | .05902] .99826 | .0'7643 | 99708 | 37 
 3 | .99998 || .02443] . ie .99912/| .05931 | .99824'| .07672| 99705) 36 
 99997} | “p42 | 9 9 99911 || .05960).99822 | . ae .99703| 35 
 699997 | 02501 | . ¢ | -99910|| 05989} .99821 | .07730|.99701 | 34 
 7 85 | 99997 || 02530) . 99 | 99909 || .06018 | .99819 | . i309 .99699 | 33 
 .99997 || 02560! .99967 || . .99907 || .06047| .99817 || .07788 | .99696| 32 
 | .99996 || .02589) . 999 99906 | || 06076 .99815 || 07817} .99694| 31 
 |. 99996 || .02618| . . 99905 | | .06105 |. 99813 || .07846| .99692| 30 
 2.99996 | -02647 | . 99904 || .06134) .99812'| .07875] .99689)} 29 
 . 99996 || .02676 | . 96 -99902 || .06163 | .99810 | .07904| .99687| 28 
 -99995 || .02705 |! . | . 99901 || .06192| 99808 | .07933) .99685 | 27 
 99995 || .02734 | .999 . 99900 || .06221 | .99806 || .07962| 99683] 26 
 | .99995 || .02763}. || . 99898 | | .06250 | 99804 || .07991| 99680) 25 
 | 99995 | .02792|..9 || 04596 | .99897|| .0627'9 | .99803 || .08020} .99678 | 24 
 5} .99994 || .02821 | .99960 | 565 |.99896 || .06308|.99801 || .08049/ 99676) 23 
 9; .99994}| .02850}. 594 | 99894) | .06337'| 99799 | .08078 | 99673 | 22 
 . 99994} .0287' .99893 || .06366 | 99797 | .08107| 99671) 21 
 . 99993 || .02908 | « 99892 | | .06395 | .99795 | .08136 | .99668) 20 
 
 | | || | 
 .99993}| .02938 | .99957 82} .99890|| .06424| .997'93 | .08165] .99666| 19 
 222° .99993 || .02967 | .9 |.99889 |! .06453| 99792 | .08194 . 99664) 18 
 251 | + 99992 || .02996 | . 99955 | |. 99888 |; .06482].99790 | .08223) .99661 | 17 
 D' 99992 || .03025 | .99954 || . 99886 |; .06511) 99788 | .08252| .99659 16 
 9.99991 || .03054 | .99953 || 8} 99885 || .06540).99786 | .08281| 99657) 15 
 . 99991 || .03083 | , 99952 || .04827 - 99883 ||. 06569 | 99784 | .08810|.99654| 14 
 367 | .99991 || 03112] .99952 ||. 3} 99882 || .06598 | 99782 | .08339] 99652) 13 
 3! 99990 || ,03141 | .99951 || .04885).99881 || 06627] 99780 | 08368] .99649) 12 
 25 | .99990)| .03170} .999 |.99879'| .06656].99778 | .08397| .99647| 11 
 . 99989 || 03199}. . 99878 | 06685 a 08426 | .99644| 10 
 8399989 | .03228] .99¢ 2 | gog7e | 06714 99774 || .08455! 99642! 9 
 99989 || .03257 | .99947 || 99875 | | 06743 90772 08484 | .99639|} 8 
 542) .99988 || .03286 | 99946 || .99873 |) .06773| 99770 | .08513) 99637] 7 
 .99988 || .03316} .99945 || .05059} .99872 | | "06802 . 99768 || .08542|.99635| 6 
 .99987 || .033845 | . 99° . 99870 | | -06831 | . 99766 '| .08571 | .99632) 5 
 99987 || .03374 | .99 : 99869) .06860..99764 | 08600] .99630) 4 
 -99986 || .03403 | .99942 || .05 .99867 || .06889 99762 | .08629/.99627| 3 
 37 | 99986 || .03432] .999- 75 | .99866 || .06918 .99760!| .08658 | 99625} 2 
 716 .99985 || .03461 | .999¢ 5}.99864|| .06947 .99758 | .08687'|.99622} 1 
 5! .99985 || .03490 99863 | .06976 .99756 | .08716!.99619| 0 
 | Sine i| Cosin |. 8 (Cosin| Sine | Cosin | Sine || Cosin | Sine . 
 
 a = 
 89° || 88 87° see | 85° 
 
TABLE 
 5° | 6° 7° 
 / = = — ae Ns > sates || 
 Sine |Cosin |} Sine Cosin | Sine |Cosin | 
 0 | .08716|.99619 |) .10453) .99452 | .12187) .99255 || 
 1 | .08745/ .99617)| . 10482! .99449 || .12216) .99251 
 2 08774! .99614|| .10511| .99446 | . 12245! .99248 
 3.08803 .99612|| 10540) .99443 |} 12274] .99244 | 
 4 | 08831} .99609|! . 10569) .99440 || .12302' .99240 
 5 | .08860] .99607 || .10597| .99437 || 12331] .99237 
 6 | .08889] .99604|| 10626} .99434)| .12360 99233 
 7 | 08918] 99602) | 10655) .99431 |} .12389 99230 
 8 | 08947] .99599|| .10684| .99428 | .12418 .99226 
 9 | .08976; .99596|| .10713 | .99424 || .12447 99222) 
 10 | .09005 | .99594 || .10742 99421 12476 99219 | 
 11 | .09034) .99591 || .10771| .99418 || .12504; .99215 | 
 12 | .09063| .99588] | .10800) 99415 || .12533) .99211 | 
 13 | .09092) .99586|| .10829 | .99412|| .12562| .99208 
 14 | .09121| .99583}| .10858| .99409 || .12591 | .99204 | 
 15 | .09150! .99580}| .10887 | .99406 || .12620 | .99200 
 16 | .09179| .99578]| .10916 | .99402 || .12649| .99197 | 
 17 | .09208) .99575 || .10945 | .99399 || .12678 | .99193 
 18 | .09237) .99572|| .10973 | 99396 || .12706 | 99189 | 
 19 | .09266) .99570) | .11002) .99393 || .12735 | .99186 
 20 | .09295) .99567 || .11031 | .99390 || .12764] .99182 | 
 21 | .09324} .99564|! .11060] .99386 || .12793).99178 
 22 | 09353] 99562 |) .11089|.99383)| .12822|.99175' 
 23 | .09382| .99559]| .11118) .99380'} .12851 | 99171 | 
 24 | 09411} .99556]| .11147! 99377 | .12880) 99167 
 25 | .09440] .99553|| .11176| .99374'| .12908] .99163 
 96 | .09469| .99551}| . 11205! .99370 | .12937 | .99160 | 
 27 | 09498] .99548}| .11234!.99367 || .12966 | .99156 | 
 28 | .09527 | .99545|| .11263| .99364 | .12995| .99152 | 
 29 | 09556! .99542|| .11291| .99360 | .13024} .99148 | 
 30 | .09585 | 99540) | .11320) 99357 || .13053) .99144 | 
 31 | .09614! .99537)| .11349] .99354'| .13081 | .99141 
 32 | .09642| .995341| 11378) .99351'| .13110} .99137 
 33 | .09671] .99531]|| .11407| .99347)| .13139| .99133 | 
 34 | 09700] .99528)| .11436|.99344'| .13168) .99129 
 35 | .09729| .99526]) .11465| 99341 || .138197]| .99125 | 
 36 | .09758| .99523]| .11494| .99337 || .18226 | .99122 | 
 37 | .09787|-.99520) | 11523] .99334 || .13254] .99118 | 
 38 | .09816] .99517|| .11552) .99331 |] .13283) .99114 | 
 39 | 09845! 995141! .11580)| .99327 || .13312| .99110| 
 40 | .09874| .995114 11609) 99324, 13341 | 99106 | 
 41 | .09903] .99508}| . 11638] .99320 | .13370 .99102 | 
 42 | 09932} .99506]| .11667 | .99317 || .13399| .99098 | 
 43 | .09961| .99503|| .11696| .99314/| .13427 | .99094 
 44 | 09990] .99500}| .11725)| .99310 || .13456 | .99091 | 
 45 | 10019} .99497|| .11754|°99307 || .13485 | .99087 
 46 | .10048| 99494) |..11783|.99303)| .13514 | .99083 | 
 7 | .10077| .99491|| .11812}.99300)| .13543 | .99079 | 
 48 | .10106| .99488)| .11840| .99297 || .13572 | .99075 
 49 | .10135] .99485]| .11869| .99293 || .13600 | .99071 
 50 | .10164| .99482)| .11898 99290, 13629 | .99067 
 51 | .10192|.99479|| .11927 | 99286)! . 13658 | .99063 
 52 | 10221 |.99476|| 11956 | 99283)! 13687 | .99059 
 53 | 10250) 99473} | .11985|.99279 || .13716 | .99055 | 
 54 | .10279| 99470] | .12014).99276 || .13744|.99051 
 55 | .10808).99467|| .12043] 99272 || .13773 | .99047 | 
 56 | .10337|.994641) . 12071 | .99269 || .13802| .99043 | 
 57 | 10366! .99461]) .12100' .99265 | 13831 | .99039 | 
 58 | .10395) 99458] .12129!.99262'| .13860 | .99035 | 
 59 | 1042499455] .12158!.99258 || .13889|.99031 
 60 | .10453 | .99452| .12187| 99255 || .13917| 99027) 
 : Cosin | Sine | Cosin| Sine || Cosin | Sine | 
 | 84° 83° 82° 
 
 XXVII.—NATURAL SINES AND COSINES. 
 
 ge 
 
 g° 
 
 Cosin | 
 
 glo || 
 
 80° 
 
 450 
 
 Sine |Cosin |} Sine |Cosin 
 .13917| .99027 | .15643 | 98769) 60 
 13946) .99023 || 15672) 98764) 59 
 .13975| 99019} .15701 98760 58 
 .14004| .99015 | .15730).98755 | 57 
 | .14033} .99011|! .15758;.98751 5 
 .14061 | .99006 | .15787) 98746. 55 
 .14090 | .99002 || .15816|.98741 54 
 .14119| .98998 || 15845! .98737 53 
 . 14148} .98994 || .15873 | .98732. 52 
 .14177 | .98990 || .15902).98728' 51 
 -14205 | .98986 recat meen 50 
 .14234 | .98982 |) .15959|.98718' 49 
 .14263 | .98978 || .15988|.98714) 48 
 .14292) .98973 || .16017|.98709' 47 
 .14320 | 98969 || .16046|.98704 46 
 .14349 | .98965 || .16074| 98700! 45 
 .14378 | .98961 |} .16103|.98695! 44 
 .14407 | .98957 || .16132) .98690| 43 
 .14436 | 98953}; .16160) .98686) 42 
 14464} .98948|) . 16189) .98681) 41 
 14493 | 98944 || .16218) .98676 40 
 14522| .98940/| .16246| 98671, 39 
 .14551 | .98936 || .16275 | .98667| 38 
 .14580 | .98931 || .16304 | .98662) 37 
 .14608 | 98927 || .16333).98657) 36 
 “14637 | .98923|| .16361|.98652) 35 
 .14666| .98919}) .16390) 98648) 34 
 .14695 | .98914|| 16419 .98643! 33 
 .14723) .98910]| .16447 | 98638! 32 
 .14752| .98906 |) .16476 | .98633) 31 
 14781 | 98902), . 16505 | .98629) 30 
 .14810! .98897 || .16533|.98624) 29 
 .14838| .98893 || . 16562 | 98619} 28 
 .14867! .98889}| .16591|.98614! 27 
 .14896 | .98884|' . 16620) .98609) 26 
 .14925 | .98880|) .16648| .98604) 25 
 .14954 | .98876 || .16677'| 98600) 24 
 14982} .98871 || .16706 | .98595) 23 
 .15011 | .98867|| .16734, .98590 | 22 
 .15040 | 98863 || . 16763 | .98585) 21 
 .15069 | .98858 |) . 16792) .98580) 2 
 .15097' | .98854 |) .16820).98575| 19 
 .15126 | .98849 || .16849 .98570; 18 
 .15155 | .98845|) .16878 | .98565) 17 
 15184 | .98841 |! .16906:.98561, 16 
 .15212 | 98836) .16935 98556 | 15 
 .15241 | .98832|| .16964' 98551) 14 
 .15270 | 98827! .16992| .98546| 13 
 15299 | .98823 || 17021) .98541) 12 
 .15327 | .98818 || .17050) 98536! 11 
 -15356  .98814)| .17078 98531 10 
 15385 | .98809)) .17107|.98526, 9 
 15414! .98805 || .17186|.98521! 8 
 15442) 98800 || .17164) 98516) 7 
 “15471 | 98796 |! 17193) .98511| 6 
 15500; 98791 |}..17222: .98506| 5 
 15529. 98787 || .17250}.98501| 4 
 15557 |. 98782}; 17279 .98496' 3 
 15586. .987'78|; .17308 .98491| 2 
 17615 .98773|| .17336| .98486} 1 
 15643 .98769| | .17365!.98481} 0 
 Sine Cosin ; Sine 
 
A i ia ile ti i i i a ett ltl ee le art er Oe 
 
 TABLE XXVII.—NATURAL SINES AND COSINES. 
 
 / 
 
 el | 
 
 10° {| 11° a 2° 18° Il 14° 
 / ragits ee | ge = , > oe si / 
 Sine Cosin|| Sine ‘Cosin | Sine Cosin | Sine |Cosin || Sine |Cosin 
 0 | 17865 . 98481 i “19, 81}. 98163 | 203 ‘91 .97815 | .22495| 97437 || .24192|.97030] 60 
 1 |.17393 .98476!| .19109|.98157 | .20820 .97809 | .22523|.97430|| .24220| 97023) 59 
 2 | 17422 .98471 || 191388) .98152 20848 | 97803 | 122552 | 97424 24245 | 97015 | 58 
 8 | 17451 .98466)) .19167) .98146 | 20877 | 97797 || .22580| 97417 || .24277 | 97008) 57 
 4 }:17479 .98461)| .19195}.98140 | .20905! 97791 || .22608) .97411 |) 24 305 | 97001 | 56 
 5 | .17508 .98455 | .19224|.98135 | .20933) .97784 || .22637 | .97404 || .24383 | 96994 | 55 
 6 | 17537 .98450 | 19252] 93129 | .20962| 97778 | .22665 | 07398 || .24362|.96987) 54 
 7 | 17565 .98445|| .19281| 98124.) .20990| 97772 | .22693 | .97391 || .24390|.96980| 53 
 8 | 17594 .98440)| 19309) 93118 | .21019| 97766!) 22722] .97384)| .24418] 96973) 52 
 9 | 17623 .98435 |} .19333|.98112 | .21047|.97760 | .22750|.97378|| .24446|.96966! 51 
 10 | .17651 .98430)| .19306|.938107,) .21076).97754 | .22778| 97871 || £24474) .96959 | 50 
 11 | .17680{.98425)| .19395;.98101 || .21104 ee | -22807 | .97365 || .24503 96952 49 
 12 | .17708|.98420)| .19423| 98096 || .21182|.97742 | .22835 | .97358 || .24531|.96945) 48 
 13 | 17737 |.98414!| .19452).98C90 | .21161 ‘oir 22863 | 97351 || .24559 | 96937 | 47 
 14 | .17766!.98409 | .19181).98084 || .21189] .977 22892 | 97345 || .24587 | .96930| 46 
 15 | 17794 |.98404)| .19503; .938079 || .21218 1193 . 22020 | .97338 || .24615 | .96923| 45 
 16 | .17823 | .98399| |: 19533 | .98073 | .21246).97717 | .22948 | 97331 |) .24644} 96916) 44 
 17 | 17852) .98394|) .19565) 98067 || .21275) 97711 | 22977 | 97825 || .24672). 96909) 43 
 18 | .17880 |.98389!); 19595) .98961 || .21303|.97705 | .23005 | .97318|| .24700 .96902) 42 
 19 | .17909 | 98383 | | 119823! 98956 | .21331|.97698 | .23033 | .97311 || .24728|.96894| 41 
 20 | 17937 |.98378 | -19552) 98050 || .21360) .97692 | 23002 | .97304|| .24756 | .96887) 40 
 31 | .17966 |.98373]| .19580| 98044 | .21388} .97686 23090 | .97298 || 24784 |.96880| 39 
 22 | 17995 |.98368/! .19709| .98039 || .21417| .97680 | .23118|.97291 || .24813) 96873) 38 
 23 | . 18023 | .98362!! .19737| 93033 :) .21445|.97673 | .23146 | 97284) .24841) .96866 | 37 
 24 | 18052 |.98357 || .19765| .98927'| .21474| .97667)| .23175 |. 972 78 | | 24869 | .96858 | 36 
 25 | .18081/.98352) 19794) .98021 || .21502) .97661 || .23203 | .97271 |) .24897| .96851 | 35 
 QE | .18109 | .98347)| .19823] 938916) ) .21530) 97655 || "23231 | 97264 | 24925 | .96844! 34 
 27 | 18138) .98341 || .19851|.98010 | .21559|.97648!| .23260|.97257)| .24954| 96837) 33 
 28 | .18166|.98336!| .19880|.98004 | .21587| 97642 || .23238|.97251 ||, .24982| .96829) 32 
 29 | .18195 | .93331)| .19908} .97998 | 21616} .97635 | .23316 |. 97244 || .25010 | . 96822 | 31 
 30 | .18224 | 98325) 19937) 97992 || .21644| 97630 || .23345 | .972387 || .25088 | .96815 | 30 
 81 | .18252|.98320)| 19965) .97987 || .21672 97623 || .23373| .97230|| .25066| .96807| 29 
 32 | .18281|.98315 | .19994|.97981!| .21701| .97617 || .23401 | 97223) | .25094) .96800) 28 
 33 | .18309|.98310) 20022) .97975 | .21729) 97611 )| .23429) 97217 || .25122) .96793 | 27 
 34 | .18338/.98304) .20051|.97969') .21758) 97604 || .23458 | .97210 | (25151 |. 96786 | 26 
 35 | .18367 | .98299'| .20079).97963 || .21786| 97598 | .23486 | .97203)) .25179) 96778) 25 
 36 | 18395 | 93294)! .20108| 97958 | .21814| .97592'| .23514|.97196 || .25207| .96771| 24 
 37 | 18424! 98288 || .20136) .97952 | .21843] .97585)| .23542} 97189 || .25235 | 96764! 23 
 88 |..18452 98283!) .20165).97946 || .21871) 97579 || .28571|.97182 || .25263 | 96756) 22 
 39 | .18481 | .98277 |! .20193! 97940; | .21899! 97573); .23599|.97176 || .25291| .96749| 21 
 40 | .18509| 98272 | .20222).97934 | . 21928) .97565 || .23627 97169 | . 25820 | .96742 | 20 
 41 | .18538 |;98267 |! .20250| .97928 || .21956) 97560 || .23656 |.97162'| .25348} .96734| 19 
 2 | .18567 | .98261)) 20279) .97922 .21985 | 97553 || .23684 | .97155) | .25376) .96727| 18 
 43 | 18595! .98256)) .20307| 97916 | 22013) 97547 || .238712).97148 || .25404| 96719) 17 
 44 | 18624 | 93250) 20336] .97910 | .22041| 97541 || .23740| .97141|| .25432).96712| 16 
 45 | .18652 | .98245!' .20364) 97905 | | 22070 | 97534 || .23769 | .97134|) .25460) 96705 | 15 
 46 | .18681 | .98240;! 20393) .97390 || 22093 | 97528 || 23797 | 97127 || .25488 | .96697| 14 
 47 | .18710| 98234); .20421 | .97893 )| 2212 26| 97521 || .23825|.97120)| .25516) .96690) 13 
 48 | .18738) .98229 || .20450 97887 || .22155':.97515 || 23853 |.97113 || .25545| 96682] 12 
 49 | .18767 | 98223); .20478| .97831 || .22183|).97508 || .23882 | .97106 | 25573) .96675 | 11. 
 50 | .18795 | .98213)) . ined 91875 || .22212| 97502 -23910 | .97100 | .25601 .96667| 10 
 5 | 19824| 93212 20535) .97369 | 22240 | .97496 || .23938 |.97093 || .25629:.96660; 9 
 52 | .18852 98207); .20563 97863 | | 22268 |.97489 | .23966 | 97086 |) .25657- 96653) 5 
 53 | 18881 | .98201 || .20592) .97857 || .22297 |,.97483 || .23995 | .97079 |; 25685 .96645) 7 
 54 | 18910) .98196!! .20620) .97851 || 22325! .97476 || .24023] 97072 | .25713 .96638 6 
 55 | .18988 |.98190 |) .20649| .97845 || .22353 |.97470 || .24051 | .97065 || .2 5741 ,.96630! 5 
 56 | .18967 |.98185|| .20677) .97839 || .22382 |.97463 24079 | 97058 | |'.25769 296623 |. 4 
 57 | 18995 .98179|| .20706! 97833 |) .22410°.97457 ||.24108 | .97051 ||. .25798 .96615) 3 
 58 |.19024 | .98174 || .20734| .97827 |) 22488 .97450 ||. 24136 | 97044 || .25826 .96608 | 2 
 59} .19052 .98168/| .20763) .97821 || .22467 | .97444 | |-.24164| .97037)| .25854 .96600; 1 
 60 | 19081 .98163)) .20791 | 97815 || .22495 97437 oeeaeee fiend | .25882 .96593) 0 
 | Cosin Sine | | Cosin | Sine | | (Cosin Sine || Cosin | Sine » | Cosin Sine : 
 
 | 79° 
 
 NOES ! 
 
 | 
 
 76° ll 
 
 ‘15° 
 
16 
 
 ie) 
 
 an 
 
 18° 
 
 TABLE XXVII.—NATURAL SINES AND COSINES. 
 
 __19° 
 
 Sine |Cosin 
 27564 | .96126 | 
 .96118 | 
 96110! 
 96102 | 
 96094 | 
 . 96086 | 
 96078 | 
 96070 || 
 96062 | 
 . 96054 | 
 96046 | 
 
 . 96037 
 . 96029 
 ; 96021 
 27955! . 
 . 96005 
 95997 
 . 95989 
 . 95981 
 95972 | 
 . 95964 
 . 95956 
 95948 
 . 95940 
 95931 
 . 96923 
 .95915 
 28318) . 
 95898 
 .95890 
 . 95882 
 
 27592 | 
 . 27620 
 27648 | 
 27676 | 
 27704 | 
 27781 
 27759 
 20787 | 
 .27815' 
 27843 | 
 Ri 71! 
 27899 | 
 27927 | 
 
 27983 | 
 .28011 
 28039 | 
 .28067 
 28095 | 
 . 28123 | 
 .28150 
 28178! 
 . 28206 | 
 28234 | 
 28262 | 
 28290 | 
 
 28346 | 
 28374 | 
 28402 | 
 
 28429 | 
 28457 | 
 28485 | 
 28513 | 
 28541 
 28569 | 
 28597 
 28625 
 28652 | 
 28680. 
 28708 | 
 28736 
 
 28792 | 
 28520 
 28847 | 
 28875 | 
 
 28908 | 
 28931 | 
 
 . 28987 | 
 29015: 
 29042 
 
 29070 
 
 29098 | 
 29126 | . 
 29154 
 29182 
 . 29209: 
 
 29287 | 
 
 Cosin | 
 
 95782 
 28764 |. 
 95766 
 95757 
 95749 
 "951740 
 95732 
 95724 
 28959. 
 95707 
 95698 
 95690 
 95681 
 95673 | | 
 
 95656 
 . 95647 
 
 . 95630 
 
 96013 
 
 95907 
 
 95874 
 . 95865 
 95857 
 . 95849 
 . 95841 
 . 95832 
 . 95824 
 . 95816 
 . 95807 
 95799 
 
 95791 
 
 9577 
 
 95715 
 
 95664 
 
 95639 
 
 Sine 
 
 | 
 | 
 | 
 | 
 
 | .82997 | .94399 
 
 | 33983 | 94049 
 | 84011 | .94039 
 | 84038 | .94029 
 
 Sine |Cosin 
 » BR557 | 94552 | 
 
 82584 | 94542 | 
 
 . 32612} .94533) ¢ 
 
 82639 | 94523 | 
 32667 | 94514 | 
 82694 | 94504 | 
 182722 | 94495 | 
 382749 | 94485 
 82777 |. 94476 | 
 82804 | 94466 | 
 - 82832 | 94457 
 82859 | 94447 
 . B2887 | 94438 
 .82914) 94428 | 
 .82942| 94418 
 32969 | 94409 
 
 .33024| 94390 
 33051 } 94380 
 33079 | .94370 
 33106 | .94361 
 
 33134 | 94351 
 33161 | 94342 
 33189 | 94332 | 
 33216 | 94322 
 | 88244! 94313 
 | .33271 | 94303 
 | 83298 94293 | 
 
 . 383326 | .94284 
 | .83353 | .94274 
 . 83381 | 94264 
 
 33408 | .94254 | 
 33436 | 94245 
 83463 | 94235 
 33490 | 94225 
 33518 | .94215 
 33545 | 94206 
 33573 | .94196 | 
 33600 | 94186 
 33627 | 94176 
 33655 | 94167 
 
 83682} 94157 | 
 .33710| 94147 
 | 88737 | 94137 
 | 83764 | .94127 
 | 83792 | 94118 
 33819 94108 
 33846 | .94098 | 
 33874 | .94088 
 33901 | 94078 
 .33929 | .94068 
 
 33956 | .94058 
 
 84065 | .94019 
 
 . 84093 | 94009 
 .94120 | .93999 
 
 34147 | 93989 | 
 || 84175 | 98979 
 | 34202) 98969 
 
 Cosin | Sine |: 
 ' 
 
 x, 15° 
 Sine | Cosin 
 0 | .25882] 96593 | 
 1 | .25910) .96585 | 
 2 | .25938) 96578 | 
 8 | .25966! .96570| 
 4 | 25994) .96562| 
 5 | .26022) .96555 
 6 | .26050) 96547 | 
 7 | .26079) .96540| 
 8 | .26107) 96532} 
 9 | 26135) 96524 
 10 | .26163! 96517’ 
 11 | .26191| 96509) 
 12 | .26219) 96502 | 
 HL 13 | .26247) 96494 | 
 14 | .26275| .96486 | 
 | | 15 | .26303] .96479 | 
 16 | 26331) .96471 
 17 | .26359| .96463 | 
 . 18 |. 26387 | 96456 | 
 We 19 | .26415} .96448| 
 Ha 20 | .26443 06140) 
 vi 21 | .26471| 96433; 
 Vill 22 | 26500) .96425 | 
 Bat 23 | 26528) 96417; 
 . 24 | 26556) 96410, 
 HE it 25 | .26584) .96402 
 i 26 | .26612) .96394 | 
 hil 27 | .26640) .96386 | 
 28 | .26668 | 96379! 
 29 | .26696) .96371 | 
 80 | .26724| 96363) 
 31 | .26752| .96355 | 
 2 | 26780) .96347 | 
 | 33 | .26808] 96340] 
 fi 34 | .26836| .96332 | 
 WW 35 | 26864] .96324 | 
 i 36 | .26892) .96316 | 
 Hh, 37 | £26920} .96308 | 
 { 38 | .26948| 96301 | 
 Tt 39 | .26976| .96293 | 
 ii 40 | .27004! . 96285 | 
 41 | 27032) . 96277 | 
 Bee: 2 | .27060) .96269 | 
 aN 43 | .27088} .96261 | 
 Bae il 44 | .27116| .96253 
 45 | .27144| .96246} 
 HEL 46 | .27172) . 96238 | 
 \ 47 | .27200| .96230! 
 WM) 48 | 27228} 96222) 
 . 49 | 27256] .96214 | 
 50 | .27284) .96206 
 51 | .27312| .96198 
 52 | 27340] 96190 
 53 | 27368} .96182 
 54 | 27396) .96174 
 55 | .27424! .96166 
 56 | 27452! .96158' 
 57 | .27480) .96150 
 58 | .27508) .96142 
 59 | 27536) .96134 
 60 | .27564! 96126; 
 Cosin| Sine 
 / 
 74° | 
 
 73° 
 
 Sine Cosin |, Sine |Cosin || 
 . 29237 .95630 || .30902 | .95106 
 29265 | .95622 |! .30929 | .95097 
 29293 .95613 .30957 | .95088 
 .29321 | 95605 .30985| 95079 
 .29348 | .95596 |. .31012| .95070 
 .29376| 95588 .31040| 95061 
 29404) 95579 .31068| .95052 
 29432] .95571) | 81095 | .95043 
 29460! 95562) .31123)| 95033 
 29487 | .95554|) 81151 | .95024 
 .29515} .95545 | .31178]| .95015 
 29543|-.95536 || .81206/.95006 
 29571 | 95528! 31233) .94997 
 . 29599 | -95519 |) .81261| .94988 
 | .29626| .95511 |! .31289) .94979 
 | 29654} .95502 || .31316' .94970 
 .29682) 95493 |; 31344) 94961 
 .29710| .95485 || .31372) .94952 
 .29737|".95476 || 813899 | .94943 
 .29765 | .95467 || .381427| 94933 
 29793) .95459 | 81454 94924 
 | .29821 | .95450!| .31482) .94915 
 | 29849 | 95441 |) 81510! .94906 
 .29876 | 95433 || .81537| .94897 
 .29904) .95424|) 81565 | .94888 || 
 .29982) 95415 |) .81593| .94878 
 . 29960} .95407 || .31620/ .94869 
 .29987| 95398 || .31648| .94860 
 80015} .95389 || 31675! 94851 
 80043 | 95380 || .31703! 94842 
 80071 95372, .81730| .94832 
 80098] .95363 | .31758| .94823 
 .80126 | 95354)! .81786| .94814 
 80154} .95345 |) .31813| .94805 
 .80182| .95387 || .81841 | .94795 
 .80209| 95328 || .81868] .94786 
 . 80237 | .95319|| .31896| .94777 
 .80265 | .95310|| .31923] .94768 
 . 80292). 95301 || .81951| .94758 
 . 80320) . 95293 || .81979| .94749 
 . 80348} .95284 || .82006| .94'740 
 .80376| .95275 || .82034] .94730 
 .80403 | .95266 || .32061 | .94721 
 . 80431 | .95257 || 32089! .94712 
 80459 | 95248!) .32116| 94702 
 80486 | .95240 || 32144] 94693 
 .80514| 95281 || .32171| .94684 
 80542! 95222 }| .82199| .94674 
 .80570! .95213 || .82227) .94665 || 
 .80597 | 95204 || .32254] .94656 
 . 80625 | 95195 || .82282} 94646 
 .80653 | .95186 || .82309| .94637 
 | 80680] .95177 || .82337| .94627 
 | .30708|.95168 | 82364] .94618 
 .80736 |.95159 || .82892] .94609 
 . 80763 | .95150 || .82419| .94599 
 | .80791 | .95142)| .82447] 94590 
 .80819 | .95133 || .382474| .93580 | | 
 30846 |. 95124 || .82502! 94571 | 
 30874 |.95115 || .32529) .94561 
 .80902 |.95106 || 82557) 94552 
 Cosin | Sine || Cosin} Sine | 
 ee 71° i 
 
 70° 
 
 452 
 
 , 
 
 60 
 59 
 
 hes ig ia cen cameos 
 
 » 
 
TABLE XXVII.—NATURAL SINES AND COSINES. 
 
 gz 20° |_21°_ji_2ae 23° 94° 
 Sine Cosin | Sine Cosin | Sine Cosin | Sine \Cosin |! Sine Cosin | 
 0 | 34202). 93969 || 35837 | 93358)! .387461 92718 | _39073 | 92050 || .40674 .91355' 60 
 1 | .34229/ 93959 |) 85864 93348) 37488 | .92707 || 39100! .92089 || .40700 .91343 59 
 9 | 3425793949 || 35891 | .93837|| 37515 | .92697) 39127 | .92028 || .40727 .91331! 58 
 3 | .34284| .93939|| .35918 93327 || .37542| .92686 ||'.39153| 92016 || 40753 .91319 57 
 4 | 34311) .93929]|| 35945) .93316|| 87569) .92675 || .39180| .92005 || .40780 .91807 56 
 5 | .34339)| .93919|| 85973) .93306'| 37595! .92664|| .89207| .91994 || .40806 .91295 55 
 || 6 | .34366| .93909 || .36000) 93295 '| .37622| .92653|| .39234 | .91982 || .40883|.91283 54 
 7 | 34393] .93899 || .36027 |. 93285 37649 |. 92642|| .89260| .91971 || .40860 .91272 53 
 . 8 | 34421 | 93889 || .36054! 93274) | .37676 | .92631|| .89287| .91959 || .40886 | .91260 52 
 9 | .34448| .93879|| .36081 | 93264 | .37°703 | .92620|| .89314|.91948}| .40913! 91248 51 
 10 | 34475] .93869|| .36108 | .93253)| .37730] .92609|| .89341 | 91936 || .40939|.91286 50 
 | 11 | 34503 | .93859 86135) .93243]| 37757) .92598}| .89367| .91925]| 40966] 91224! 49 
 | 12 | 34530| .93849|| .36162/ .93232|| .387784| .92587|| .89394| .91914|| .40992) 91212 48 
 ie | 13 | .34557| 93839 || .36190] .93222 ‘37811 || 92576 || .89421 | .91902|| .41019| .91200) 47 
 . 14 | 34584! .93829|| 86217) .93211 "37838 | .92565 || 89448] 91891 || .41045|.91188) 46 
 | 45 | 84612) .93819|| .36244| .93201 | 37865 | .92554 .89474| 91879 || 41072] .91176| 45 
 16 | .34639| .93809|| .36271 | .93190|| .87892/ .92543}| 89501 | .91868 || .41098| .91164 | 44 
 | 17 | 34666} .93799|| .86298| .93180|} 37919! .92532|| .89528 | .91856 || .41125|.91152 43 
 18 | 34694) .93789 | 86825] .93169 37946 | 92521 || 89555 91845 || .41151|.91140; 42 
 . 19 | .34721|.93779|| .36352/ .93159)| .87973] .92510|| .89581 | 91833 || 41178) 91128) 41 
 . 20 | 34748 93769 ‘36379 03118) .87999 | .92499|| .39608 | .91822/| .41204|.91116) 40 
 | 21 | 34775 | .93759|| .86406| .93187!| .88026 | .92488 || .89635| .91810)| .41231}.91104) 39 
 2 | 34803 | .93748 || 86434) .93127|| .88053 | .92477|| .39661| .91799|| .41257|.91092) 38 
 23 | 34830] .93738 || .86461 | .93116|| .88080) .92466 || 89688 | 91787 || 41284) .91080 37 
 24 | 84857! .93728 || .36488] .93106]| .88107 | .92455 || .89715| .91775 || .41310| 91068) 36 
 | 25 | 84884! 93718 || 86515 | .93095)| .88134! .92444|| 39741 | 91764 | 41387) .91056 35 
 | 26 | .84912| 93708 || .86542 .93084|| .88161 | .92432)| 89768] 91752 || .41363|.91044) 34 
 27 | 34939 | 93698]! .86569| 93074 |; 88188] .92421 || .39795| .91741 || .41390) 91032! 33 
 | 28 | 34966 | 93688 || .36596 | .93063)| .88215  .92410)| 39822} 91729 |] .41416| .91020 32 
 29 | 34993 | 93677 || .86623/ 93052]; 28241! .92899|| 89848} .91718|| .41443] 91008 | 31 
 | 30 | 35021 | .93667 -36650| .93042) eng: .89875 | 91706 || .41469) .90996 30 
 31 | .35048| .9365? || .86677 | 93031 || .38295| 92377]! 39902) .91694 || .41496 90984! 2 
 82 | .35075| .93647 || "36704 -93020|! .88322 ees .89928| .91683 || .41522| 90972) 2 
 33 | .35102! .93637 || .86731! .93010|| .88349 | .92355|| 89955) .91671 || 41549} 90960) 27 
 . | 94 | 351301. 93626)! .86758| 929991! 88376! .92343 || .39982) .91660}| .41575| .90948) 26 
 35 | .35157 | .93616 || 86785 | .92988'| .88403| .92332]) .40008|.91648|| .41602/ .90936 | 25 
 36 | 35184! :93606 | 36812 .92978 | .38430! .92321 || .40035| 91636 || .41628].90924' 24 
 37 | .35211| 93596 | 86839 .92967 | .38456) .92310|| .40062) .91625|| .41655).90911| 23 
 38.| 85239 .93585 | .86867 | 92956 | .88483 | .92299|| .40088| 91618 || .41681 | .90899| 22 
 39 | .35266|.93575 | .36894 | .92945 | 88510) .92287|| .40115| .91601 || .41707| 90887! 21 
 40 | .85293| .93565_ 36921 |. 2935 || .88537'| 92276 || .40141 | .91590|| .41784 | .90875 | 20 
 ‘41 | 85320) .93555 | .86948 | .92924 || .88564| 92265 || .40168] .91578 || .41760) .90863) 19 
 42 | 35347| 93544 | 36975 | .92913|| _88591| _92254/| .40195) .91566 |) .41787| 90851! 18 
 | 43 | .35375|.93534 | .87002/ .92902!| .88617) 92243] .40221 | .91555 || .41813) 90839 17 
 | 44 | 35402! _93524'| .3702 | 92892 || 38644) 92231 || 40248] 91543 | .41840/ .90826| 16 
 45 | 35429! .93514 | .87056| .92881 || .88671 | .92220|| .40275| .91531 || .41866| .90814| 15 
 . \ 46 | 35456! .93503.| .87083! .92870 || .88698 | .922 309 | "40301 .91519|| .41892/.90802) 14 
 7 | 35484 93493 | 87110 .92859|| .38725' 92 2198 | .40828 | .91508 || .41919| .90790| 18 
 48 | .35511|.93483 | .87137| .92849|| .88752 | .92186 || .40855 | .91496|| .41945) 90778) 12 
 49 | 35538) .93472 | .37164!.92838 |} .38778| .92175|| .40881 | .91484 || .41972|.90766| 11 
 50 | .35565 | .98462 | .87191}.92827 |! .88805).92164|| .40408 .91472|| .41998] .90753) 10 
 51 | 35592! .93452 | 87218] 92816]! .38882! .92152/| .40434 .91461 || .42024].90741} 9 
 ! 52 | 35619! .93441 | .87245} 92805 || .88859 | 92141 || .40461 .91449 || .42051| .90729| 8 
 53 | .35647| .93431 || .87272| .92794 | 38886 | 02150 40488 .91437|| .42077|.90717| 7 
 | 54 | 35674) 93420 | .37299| .92784|| .88912' 92119) .40514 .91425]| .42104|.90704| 6 
 55 | 35701! 93410 | .37326| .92773|' .38939!.92107!| .40541 .91414)/ .42130|.90692] 5 
 | 56 | 35728} .93400 | .87353| .92762!) 88966 .92096 || .40567 .91402 || .42156|.90680| 4 
 7 | .85755| .93389 | .87380| 92751 |) .88993'.92085 |) 40594 .91390 || .42183] 90668] 3 
 58 | 357821 .93379 | .87407 "92740 39020 .92073 || .40621 .91378|| .42209!.90655; 2 
 59 | 35810! .93368 | .87434).92729|) 39046’ .92062||.40647 .91366 || .42235/.90643| 1 
 60 | 85837 93358 | .37461|.92718) 39073 .92050 | 40674 91355 || 42262) -90631| 0 
 | : Cosin | Sine || Cosin | Sine | Cosin | Sine | Cosin Sine Cosin | Sine 
 . 69° | 682 67> ~—s«d||=s«G 65° 
 
 458 
 
TABLE XXVII.—NATURAL 
 
 SINES AND COSINES. 
 
 iz | 
 
 62° 
 
 Co OT = COC 
 
 ||| a a ee Cy aaa | ue ae” 
 Sine |Cosin || Sine |Cosin || Sine |Cosin | Sine |Cosin | Sine |Cosin 
 0 | .42262 | .90631 || .43837|.89879|| .45399 | 89101 |) .46947 | .88295 | .48481 | 87462] 60 
 1 | .42288/ .90618 || .48863) .89867 || .45425 | .89037 || .46973 | .88281 || .48506 | .87448) 59 
 2 | .42315 | .90606 || .48889| .89854 || .45451| 89074 || .46999 | .88267 || .48582 | .87434) 58 
 3 | .42341 | 90594 || 48916} 89841 |) .45477) .89061 || .47024| 88254 || .48557 | .87420|. 57 
 4 | .42367 | .90582 || .48942] .89828 |} .45503| 89048 || .47050| .88240 || .48583 | 87406) 5 
 5 | .42394 | .90569 || 48968 | .89816 || .45529 | .89035 || .4'7076 | .88226 || .48608 | .87391} 55 
 6 | .42420) .90557 || .48994| .89803 || .45554| 89021 }| .47101 | .88213 || .48634) .87377| 54 
 7 | .42446 ; 90545 |) .44020) .89790 || .45580 | .89008 || .47127| 88199 || .48659 | .87363! 53 
 8 | .42473 | 90532 |) .44046 | 89777 || 45606 | £88995 |) .47153 | .88185 || .48684 | .87349 | 52 
 9 | 42499) .90520 |) .4407'2| 89764 || .45632] .88981 || .47178 | .88172 || .48710| .87335| 51 
 10 | .42525/.90507 || .44098| 89752), .45658| .88968 || .47204 | .88158 || .48735 | .87321 | 40 
 11 | .42552! .90495 |) .44124) .89739 |) .45684| .88955 || .47229| .88144 || .48761 | .87306) 49 
 | 12 | .42578| 90483 |] .44151/ .89726]|| 45710] .88942 || .47255 | .88130 || .48786| .87292| 48 
 ) 13 | .42604) 90470 || .44177' .89713 || .45736 | .88928 || .47281] .88117 || .48811 | 87278} 47 
 14 | .42631 | 90458 || 44203; .89700 || .45762] .88915 || .47806 | .88103 || .48837 | .87264| 46 
 15 | .42657 | .90446 || .44229 | .89687'|| .45787 | .88902 || .47332] 88089 || .488621.87250) 45 
 16 | .42683| 90433 |) 44255) .89674 || .45813].88888 |) .47358] .88075 || .48888|.87235} 44 | 
 ‘ 17 | .42709) .90421 || .44281| .89662}| .45839 | .68875 || .47383 | 88062 || .48913! 87221 | 43 
 18 | .42736 ;.90403 || .44307} .89649 || .45865 | .88862 || .47409|.88048 || .48938|.87207| 42 
 19 | .42762| .90396 || .44333! .89636 || .45891 | 88848 || .47434 | .88034 |) .48964 -87193 | 41 
 20 | .42788 | 90383 | 44359 | 89623 || .45917 | .88835 || .47460 | 88020 || .48989 | .87178| 40 
 21 | 42815] .90371 || .44885).89610|| .45942| .88822 || 47486 |. 88006 || 49014 .87164| 39 
 2 | 42841) 90358 || .44411| .89597 || .45968| 88808 || .47511| .87993 || .49040|.87150] 38 
 23 | .42867 | 90346 |) 44437! 89584 || .45994) .88795 || .47537 | 87979 || .49065 | .87136| 37 
 24 | 42894) 90334 || 44464) 89571 || .46020|.88782!) .47562|.87965 || .49090] .87121| 36 
 i 25 | .42920) .90321 || .44490] .89558 || .46046| .88768 || .47588 | 87951 || .49116]| .87107 | 35 
 i 26 | .42946| .90309 || .44516) .89545 || .46072|.88755 || .47614| .87937 || .49141|.87093| 34 
 27 | 42972} 90296 |) 44542) 89532 || .46097 | .88741 || .47639 | 87923 || .49166| .87079| 3: 
 Hii 28 | .42999 | 90284 || 44563! .89519 || .46123| .88728 || .47665 | .87909 || .49192] .87064 | 32 
 | 29 | .43025 | 9271 |) .44594) 89506 || .46149| .887'15 |} .47690 | 87896 || .49217) 87050} 31 
 4 30 | .48051| .90259 |! .44620) .89493 || .46175| .88701 || .47716 |. 87882 || .49242) .87036} 30 
 a) 31 | .48077) .90246 | .44646] .89480|| .46201 | .88688 || .47741 | .87868 || .49268}.87021] 29 
 Hy 2 |.43104/ 90233 || .44672) .89467|| .46226 | 88674 |! .47767 | .87854 || .49293] .87007| 28 
 Pit 33 | 43130) .90221 || .44698} .89454]] .46252| .88661 || .47793 | .87840 || .49318} .86993} 27 
 i 34 | 43156} .90203 || .44724) .89441|| .46278| 88647 || .47818 | 87826 || .49344| .86978| 2 
 iM 35 | .48182) .90195 || 44750} .89428 || .46304| .88634 || .47844| .87812|| .49369] 86964) 25 
 i 36 | 43209) .90183 || .447'76) .89415]| .46330| .88620|| .47869 |.87798 || .49394| .86949|-2 
 Wh 37 | 48235) 90171 || 44802) .89402]) .46355| .88607 || .47895 | 87784 || .49419| .86935| 23 
 HI 88 | .43261 | .90153 || .44828) .89389]| .46381 | .88593 || .47920 | .877'70 || .49445| .86921 | 22 
 39 | .43287) 90146 | 44854 .89376 || .46407 | .88580)| .47946 | .87756 || .49470} 86906} 21 
 40 | .43313 | .90133 | .44880) .89363 || .46433 .88566 || .47971 | 87743 || .49495| .86892| 2 
 41.| .48340] 90120 | .44906| .89350 |! .46458 | .88553}| .4'7997’| .87729 || .49521].86878] 19 
 42 | .43366 90103 | 44932) .89337|| .46484! .88539 || .48022) .87715 || .49546| .86863| 18 
 43 | .43392 | 90095 || .44958) .80324|| .46510) .88526 || .48048) .87701 || 49571} 86849} 17 
 44 | .48418) 90052 || 44984) .89311]| .46536 | .88512]| .48073 | .87687 || .49596| .86834| 16 
 45 | .43445} 9000 || .45010; .89298 || .46561 | .88499|| .48099! .87673 || .49622] .86820| 15 
 46 | .48471) .90057 || 45036) .89285 || .46587 | .88485 | -48124 | .87659 || 49647) .86805| 14 
 | 47 | 48497) 90045 |) 45062) .89272 |) .46613 | .88472|| .48150|.87645 || .49672| 86791] 13 
 ' 48 | 43523] .90032 || .45088| .89259]| .46639 | .88458]| .48175 |.87631 || .49697 | .8677'7| 12 
 49 | .43549 90019}; 45114) .89245 |) .46664' .88445 |! .48201 | .87617|| .49723] 86762} 11 
 50 | .48575/ 90007 || .45140] .89232) .46690|.88431 || .48226 |.87603 || .49748] .86748| 10 
 51 | 43602 89994 | 45166 | .89219|' .46716 | .88417| | .48252].87589 || .49773} .86733 
 52 | .43628] .89981 |) .45192| .89205| .46742| 8840! .48277] .87575 || .49798| .86719 
 53 | .43654/ .89968 |) .45218] .89193|! .46757 | .68399|| .48303|.87561 || .49824| .86704 
 54 | 43680} .89956 || .45243|.89180), 46793 | .88377'|| .48328 |.87546 || .49849 | .86690 
 55 | .43706 | .89943 || .45269|.89167|' .46819| .88363|| .48354|.87582|| .49874] .86675 
 56 | .48733] .89930 || 45295} .89153|) .46844 | .88349| | .48379|.87518|| .49899| .86661 
 57 | .43759| .89918 || .45321 | 89140} .46870|.88336 | |’.48405 |.87504/| .49924] 86646 | 
 58 | .43785) .89905 || .45347! .89127 || .46896 | .88322 || .48430 |.87490 || .49950| .86632| 2 
 | 59 | .438811) 89892 || .45373}.89114|| .46921 | 88308 || .48456 | 87476 || .49975|.86617) 1 
 | 60 | 43837 | .89879 || .45399| 89101 || -46947 | 88295 |! .48481 | 87462) | .50000) .86603| 0 
 ; Reco Sine || Cosin| Sine | Cosin | Sine |, Cosin } Sine p eae Sine f 
 ee ee ~ \ | = 
 | 64° | 63° 61° 60° 
 
TABLiC 
 
 XAVi 
 
 © | .50000 | :86603 | 
 | 50025 | .86588 | 
 | 50050} .86573 | 
 3 | .50076 | .86559 | 
 50101 | .86544 | 
 .50126 | .86530 | 
 | .50151|.86515 
 .50176 | .86501 | 
 | 50201 | .86486 
 
 | 50227 | 86471 
 | 50252] .86457 
 
 2 | 50302 | .86427 
 50327 | .86413 
 | .50352} .86398 
 
 | 50377 | .86384 
 50403 | .86369 
 -50428| .86354 
 50453 | 86340 | 
 
 ne | 
 fek e at ae 
 NQOUBWWRH COOVIMWOILWWeH 
 
 (@ 2) 
 
 rw 
 
 .50503 | .86310 
 21 | .50528 | .86295 
 22 | .50553 | .86281 | 
 23 | 50578 | .86266 
 
 Lav) 
 Bag 
 
 1D 2 
 v 
 
 50628 |. 86237 
 26 | .50654 | .86222 
 97 | 50679 .86207 
 28 | .50704| .86192| 
 29 | 50729) .86178 | 
 30 | .50754|.86163 
 31 | .50779| .86148 | 
 32 | .50804|.86133 | 
 | .50829|.86119 
 34 | 50854 86104 
 35 | .50879|.86089 | 
 
 | 
 3G | 50004| 86074 
 | 
 | 
 i 
 | 
 | 
 
 ri 
 oO 
 
 oe 
 co 
 
 ~) 
 
 ) 
 2 
 
 37 | .50929 | .86059 
 38 | 509541 86045 
 | 00979 | .86030 
 40 | 51004! .86015. 
 41 | .51029) .86000 | 
 42 | 51054|.85983 | 
 43 | 51079 |.85970 
 44 | 511041" 85956 
 45 | 51129) .85941 | 
 | 46 | 51154) .85926 
 ; 47 | .51179|.85911 | 
 48 | 51204! .85896 | 
 49 | 51229). 85881 | 
 50 | .51254 | .85866 
 51 | .51279|.85851. 
 52 | 51304 | .85836 | 
 53 | .51329) 85821 | 
 | 54 | .51354|. 85806 
 55 | 51379) 85792 
 56 | 51404] .85777 
 BY | 51429). 85762 
 58 | 51454! 85747 | 
 59 | 51479) 85732 
 60 | 51504! 85717 
 
 oo 
 oc 
 
 - 50277 | 86442 
 
 | 50478 | 86325 
 
 | 50603 | .86251 
 
 | .51823 
 .51852 
 
 | .52026 
 
 opel | ae ae 
 Sine |Cosin | 
 .51504 
 51529 | 
 51554 | 
 .51579 
 | 51604 
 .51628 
 | 51653 
 .51678 
 .51703 
 51728 
 51753 | 
 
 51778 | 
 51803 | 
 
 51877 
 .51902 
 .01927 
 .51952 
 51977 
 52002 | 
 
 52051 
 52076 
 .52101 
 .52126 
 02151 
 .O2175 
 .52200 | 
 52225 
 
 92250 
 
 52275 
 02299 
 52324 
 52349 
 52374 
 .52399 
 
 52428 
 52448 
 02473 
 52498 
 02522 
 S247 
 52072 
 
 le 
 
 -02597 | 
 .52621 
 .52646 
 .52671 
 . 52696 
 .52720 
 .52745 
 20% 
 
 .52794 
 .52819 
 52844 
 .52869 
 .52893 
 .52918 
 .52943 
 .52967 | 
 . 52992 
 
 Cosin 
 
 85717 || .52992' 84805 
 
 .85702 || .53017) .84789 
 
 85687 || .53041} 84774 
 .85672 || 58066) .64759 
 85657 || .53091 | .84743 
 
 85642 | .53115) .84728 
 .85627 |! .58140) .84712 
 85612 || .58164| .84697 
 
 85597 || .58189| .84681 
 
 85582 || .53214| .84666 
 85567 || .532388 | .84650 
 85551 || 58263} .84635 | 
 85536 || .538288!.84619 
 85521 || .53312 | .84604 
 85506 || .53337 | .84588 
 85491 || .53361 | 
 85476 || .53386 | 84557 | 
 .85461 ||-.53411) .84542 | 
 85446 || 53435 | 84526 
 85431 || .58460) 84511 | 
 85416 || .53484) .84495 | 
 
 .85401 || .53509 | .84480, 
 85385 || .53534) 84464 
 .85370 | .53558 | 84448 
 85355 || .53583 | .84433 | 
 .85340 || .53607 | 
 .85325 || .53632 | 84402 
 .85310 || .53656 | 84386 
 .85294'| .53681 | 84370 
 
 .85279 | 53705 | 84355 
 
 see .53730 | 84339 
 
 85249 || 53754 | 84324 
 .85234 || .537'79 | .84308 
 85218 || .53804 | 84292 
 .85203 || 53828 | 84277 
 85188 || .538853 | .84261 
 .85173 || .53877 | 84245 
 .85157 || .53902} .84230 
 .85142 || .53926 | 84214 
 85127 || .53951 | 84198 
 .85112 | 53975! 84182 
 .85096 || .54000 
 85081 || .54024| 84151 | 
 .85066 || .54049 
 .85051 || .54073| 84120 
 .85035 || .54097|.84104 | 
 .85020 || .54122| 84088 
 .85005 | .54146}| .84072: 
 .84989 || 54171} .84057 | 
 84974 | 54195) .84041 || 
 84959 || 54220) .84025 | 
 
 84943 '| 54244! 84009 | 
 84928 || 54269) .83994 | 
 84913 || 54293 | 
 84897 || 54317! .83962 | 
 84882 | 54342 |.83946| 
 (84866 || .54366 | .83930) 
 84851 || 54391! 83915 | 
 84836 | if 
 84820 | .54440' 83883 | 
 84805 | 54464) .83867 
 
 Sine |Cosin 
 
 84575 
 
 84417 
 
 ,84167 | 
 
 84135 | 
 
 .83978 
 
 .54415! 83899 
 
 Sine |Cosin| Sine || Cosin | 
 
 | Cosin | Sine | 
 | 
 
 i Soe. | 
 
 33° 
 
 IL—NATURAL SINES AND COSINES. 
 
 34° 
 
 54854 | 
 '54878 | 
 54902 
 4927 
 54951 | 
 
 AQT | 
 54999 | 
 55024 
 55048 | 
 (55072 | 
 55097 | 
 "55121 | 
 155145 | 
 55169 
 55194 | 
 
 .55218 | 
 55242 | 
 
 £5266 | 
 
 55291 | 
 55315 | 
 55339 | 
 55363 | 
 55388 | 
 55412 
 .55436 | 
 55460 | 
 55484 | 
 .55509 
 55533 | 
 LBBBDT | 
 55581 | 
 55605 | 
 55630 
 B5GD4 | 
 55678 | 
 55702 
 55726 | 
 .BDT5O 
 B5G75 | 
 5S799 
 .5b823 
 55847 | 
 55871 
 55895 
 55919 | 
 
 Sine |Cosin 
 .54464 
 .54488 
 .54513 
 .54537 | 
 .54561 
 | 54586 
 .54610 
 54635 | 
 | 54659 
 .54683 
 .54708 
 54732 | 
 .54756 
 54781 
 54805 | 
 .54829 | 
 
 . 83645 || .56256 
 83629 || .56280 
 83618 || .56805 
 83597 || .56329 
 83581 || .56353 
 83565 || 563877 
 .83549 || .56401 | . 
 
 83533 || 56425 
 (83517 ||-.56449 
 | 83501 | 56473 
 83485 | .56497 
 (83469 || .56521 
 (83453 || .56545 
 83437 || .56569 
 (83421 || .56593 
 -83405 || .56617 
 .88389 || .56641 | 
 83373 || .56665 
 (88356 ||. .56689 
 .83340 || 56713 
 88324 || .56736 
 .83308 || .56760 
 83292 || .56784 | 
 88276 || .56808 | 
 “$3260 | .56882 
 83244 || 56856 
 88228 | .56880 
 88212 || 56904 
 83195 | 
 83179 
 83163 || .56976! 
 .83147 | 
 83131 || ..57024| 
 (88115 || 57047 | 
 88098 .| 57071 
 -83082 | 57095 
 -§3066 |; .57119). 
 83050 
 | 83084 ||..57167 
 88017 | .57191 
 83001 || 57215 
 -82985 ||57238 
 82969 
 ~ 82953 | 1-. 57286 
 “82936 |. 
 82920 | 
 82904 
 Sine || C 
 
 .56088 
 
 .56160 | 
 
 | 
 
 56928 | 
 .56952 
 
 57000 | 
 
 57143 
 
 pt ard 
 .57 262 
 
 Sine |Cosin 
 
 .83867 || 55919 | 
 83851 || 55943 | 
 83835 || .55968 
 83819 || .55992 | 
 .83804|! .56016). 
 .83788 || .56040 | 
 83772 || 56064 
 83756 | 
 83740 || .561121. 
 .83724 || 56136 | 
 .837u8 
 
 .83692|| .56184 
 83676 | | .56208 
 83660] | .56232 
 
 82904) 60 
 
 82887 | 
 82871 
 82855 
 82839 
 
 RVQOW 
 
 -OROKG + 
 
 82806 
 
 .82790 5: 
 
 eorry 
 e 
 
 82757 
 82741 
 82724 
 82708 
 
 82692 | ¢ 
 
 82675 | 
 82659 
 82643 
 82626 
 82610 
 82593 
 
 82577 
 
 | 
 82561 
 82544 
 
 .82028 * 
 
 82511 
 82495 
 82478 
 82462 
 82446 
 82429 
 
 82413 § 
 
 82396 
 82380 
 82363 
 82347 
 82330) 
 
 82314! 2 
 
 82297 | 
 
 82281 
 
 82264 | 2 
 
 82248 
 
 5008 | 
 82214 | 
 82198 | 
 82181 | 
 82165 | 
 82148 
 82132 
 82115 | 
 “82098 
 82082 
 82065 | 
 82048 | 
 82082 | 
 82015 | 
 81999 | 
 81982 | 
 “81965 
 
 81949 
 | 81932 
 58) 81915 | 
 
 sae | mee 
 
 56° 
 
 455 
 
 Sine | ; 
 
TABLE XXVII.—NATURAL SINES AND COSINES., 
 
 35° 
 
 ~ 
 
 Sine Cosin 
 
 | 
 | 
 
 ee 
 
 38° 
 
 | 
 | 
 | 
 
 | 
 
 | 36° 
 Sine |Cosin 
 58779. 80902 | 
 | 58802} .80885 
 | .58826 |. 80867 
 | .58849 |. 80850 | 
 | 58873 | .80833 
 58896 | .80816 
 | 58920 | .80799 | 
 | 58943 | 80782 | 
 .58967 | 80765 
 58990 | .80748 
 .59014 | .80730 
 .59037 | 80713 
 59061 | .80696 
 .59084 |. 80679 | 
 59108 | 80662 
 59131 | 80644 
 59154! 80627 | 
 .59178 | .80610 
 .59201 | .80593 
 59225 | 80576 
 59248} .80558 
 .59272 | 80541 
 59295 |. 80524 
 .59318} .80507 
 -59342) .80489 
 59365 | .80472 
 59389 | 80455 
 59412) 80438 
 59436} .80420 
 .59459 | .80403 
 . 59482! 80386 
 59506 | .80368 
 .59529 | .80351 
 .59552 | 80334 
 .59576 | .80316 
 59599} .80299 
 | .59622/ 80282 
 . 59646 | . 80264 
 .59669 | 80247 ; 
 | .59693 | 80230 | 
 | 59716 | 80212, 
 .59739 | .80195 
 .59763 | .80178 | 
 .59786 | .80160 | 
 | .59809/ .80143 
 .59832 | .80125 || 
 59856 | .80108 | 
 .59879 | .80091 
 .59902 | .80073 | 
 59926 | 80056 | 
 .59949 -80038 | 
 59972} 80021 | 
 .59995! 80003 | 
 60019} .79986 
 | .60042 | .7'9968 
 | 60065 | .79951 
 60089 | ..79934. 
 | .60112).79916 
 | .60135 | .79899 
 .60158 | .79881 
 60182) .79864 
 |; Cosin | Sine | 
 
 Sine \Cosin | 
 .60182 | .79864 | 
 60205 | .79846 | 
 . 60228 | .79829 
 60251 | .79811 | 
 60274 | .79793 
 
 60298 |. 79776 
 
 . 60321 | 79758 | 
 60344 | 79741 | 
 .60367 | .79723 | 
 . 60390! .'79706 | 
 60414 | 79688 | 
 .60437 | 79671 | 
 60460 .79653 
 .60483 | .79635 
 .60506 | .79618 
 .60529 | .79600 | 
 . 60553 | .'79583 | 
 60576 | . 79565 | 
 60599 | 79547 
 
 61566 | 
 61589 | 
 
 60622 | 79580 
 .60645 | .79512 
 60668 | 79494 
 60691 | 79477 
 60714 | .79459 
 60738 | .79441 
 60761 | 79424 
 60784 | 79406 | 
 .60807 | .79388 
 60830 | 79871 | 
 60853 | .79353 | 
 .60876 | .79335 
 .60899 | 79318 
 .60922 | 79300 
 60945 | 79282 | 
 60968 | 79264 | 
 60991 | 79247 
 61015 | 79229 | 
 61038 | 79211 | 
 61061 | 79193 | 
 .61084 | 79176 | 
 61107 | 79158, 
 .61130| 79140! 
 61153 | 79122 | 
 61176 |. 
 61199 | 
 61222 | 
 61245 | 
 61268 | 
 61291 
 61314 
 61337 
 61360) 
 61383 | 
 .61406 | 
 61429 
 61451 | 
 61474 
 61497 | 
 61520 
 
 79105 || 
 79087 | 
 79069 | 
 79051 
 . (9033 
 79016 || 
 78998 | | 
 78980 | | 
 | 
 
 . 78962 | 
 78944 || 
 . 78926 | | 
 . 78908 | | 
 78891 | | 
 78873 | | 
 . 78855 | | 
 78837 | | 
 61543 | 78819 | 
 .61566 | . 78801 
 Cosin | Sine 
 
 || 
 
 . 78801 | | 
 T8783 | | 
 
 Wee ae 
 
 Sine |Cosin || Sine 
 
 62932 
 62955 | 
 
 Cosin 
 
 fd 
 . T7696) 5$ 
 
 oO 
 
 15 
 
 pipded 
 
 oe 
 
 0 | 57358! 81915! 
 .57B81 : .81899 
 2 57405. .81882 
 3 | .57429' 81865 | 
 4 | .57453' 81848 
 5 | 57477} 81832) 
 6  .57501! .81815/ 
 7 | 57524) 81798 
 8 .57548) .81782 
 9 | .57572| 81765 
 10 | .57596) .81748 
 11 | 57619! .81731 
 12 | .57643|.81714| 
 13 | .57667| .81698 
 14 | .57691|.81681 
 15 | 57715! .81664 
 16 | 57738) 81647 | 
 17 | 57762) 81631 
 18 | 57786) .81614| 
 . 19 | 57810! .81597| 
 nh 20 | .57833! .81580 
 a 21 | 57857! 81563 
 Hirata 22 | 57881) .81546 
 Be 23 | 57904) 81530) 
 i 24 | 57928) 81513 | 
 25 | 57952) 81496 :. 
 26 | .57976| .81479| 
 27 | .57999| .81462' 
 28 | 58023! 81445 | 
 29 | 58047) 81428 | 
 i 80 | .58070.81412) 
 31 | 58094) 81395 | 
 : 82 | 58118) 81378) 
 33 | 58141) .81361 | 
 i 34 | 58165) .81344, 
 it 85 | 58189) .81327 | 
 Ny 36 | 58212] 81310. 
 87 | .58236) .81293 | 
 | 38 | .58260/ .81276 | 
 at 39 | 58283 | 81259. 
 i 40 | .58307/.81242 
 pH 41 " 58380) 812951 
 42 | 58354! .81208 | 
 | 43 | 58378) 81191 | 
 44 | 58401) .81174' 
 45 | 58425) .81157| 
 46 | 58449, 81140 / 
 4 | 58472; 81193} 
 48 | 58496) .81106 | 
 49 | 58519} .81089 | 
 50 | .58543}.81072 | 
 51 | .58567!.81055 
 52 | .58590) .81038 | 
 53 58614) 81021, 
 54 | 58637) 81004 | 
 55 | 58661 | 80987 
 56 | 58684 | .80970 || 
 57 | 58708] 80953 
 58 | .58731/ .80936 
 59 .58755| .80919 
 60 | .58779! .80902 | 
 : 'Cosin | Sine 
 | = 
 
 53°C 
 
 52° 
 
 .61612 | .78765 || .629771 .7%678| 58 
 .61635 | .78747 || .63000' .77660') 57 
 .61658 | .78729 | .63022 .77641 | 56 
 .61681 | 78711 || 68045 | .77623| 55 
 .61704 | .78694 || .63068' 77605) 54 
 .61726 | .78676 || .63090' .'77586! 53 
 .61749| . 78658 || 63113! .77568! 52 
 .61772| .78640 || .63135 °.7'7550! 51 
 61795 | .78622 | 63158) .77581 50 
 61818] 78604 | .63180! 77513} 40 
 .61841 | 78586 || .63203!.77494| 48 
 61864 | 78568 || .63225 .7'7476| 47 
 .61887 | 78550 || .63248 77458] 46 
 61909 | 78582 || 68271 77439) 45 
 .61932 | 78514 || .63293'.7'7421| 44 
 .61955 | .78496 || .63316 .'77402) 43 
 .61978 | 78478 || 63338 .7'7384| 42 
 .62001 | .78460|| .63361 .77366) 41 
 62024) .78442 || 63383 .77347) 40 
 
 "62046| 78424 | .63406 | .7'7329 39 
 
 | 62069 |. 78405 || .63428 77310) 88 
 
 | -62092| .78387 |) .63451 °.77292 37 
 62115 | .78369 | 63473! _77273| 36 
 .62188) .78351 || .63496 | .77255' 35 
 
 | 62160-78333 || |63518 | 77236) B4 
 
 | 62183 | .’78315 |) .63540'.77218| 33 
 .62206 | 78297 || .63563 | 77199) 32 
 .62229] .78279 || .63585/ .77181] 31 
 
 | -62251 78261 .63608 | .7'7162| 30 
 
 | 62274 | 78243 |] 63630! .7'7144| 29 
 62297 | .78225 || .63653 | .777125 | 28 
 . 62320 | .78206 || .63675 | .77107 | 27 
 . 62842) 78188 || .63698 | .7’7088 | 26 
 . 62365 | .78170 || .63720! .77070| 25 
 
 | .62888 | .78152|| .63742| .7'7051| 24 
 62411 | 78184 | .63765| 77033 | 23 
 .62433 | 78116 || .63787|.77014/ 22 
 .62456| .'78098 || .63810! .76996| 21 
 .62479 | .78079 | .63832) .76977) 20 
 .62502 | .78061 || . 63854 76959 | 19 
 . 62524 | .78043 || .6387'7| 76940) 18 
 .62547 | .78025 | .63899 | 76921 | 17 
 .62570 | 78007 || .63922! 76903 16 
 .62592 | .77988 | .63944| .76884/ 15 
 .62615| .77970|| .63966 |. 76866) 14 
 62638 | .77952 | .63989 | 76847 13 
 .62660 | .77934 | .64011|.76828) 12 
 .62683 | .7'7916 || .64033| 76810) 11 | 
 62706 | .77897,| .64056 | .76791 | 10 | 
 62728 | .77879|| 64078! "6772! 9 
 62751 | 77861 || 64100) .76754) 8 
 62774 | 77843 || 64123) .76735| 7 
 62796 | .77824)| .64145|. 76717) 6 
 .62819|.7'7806. | .64167|.76698| 5 
 .62842) .'77788 || .64190'.76679! 4 
 62864 | .'7769/| 64212 .76661! 3 
 62887 | .77'751|| 64234-76642; 2 
 .62909 | .77733)| .64256 16628 | 1 
 .62932|.7'7715 || .64279' 76604; 0 
 
 Cosin | Sine | Cosin | Sine 
 
 51° 50° 
 
TABLE XXVIL—NATURAL SINES AND COSINES. 
 
 : 
 . 
 a 
 ie]. 40e.|j__ 41° 42° ate aa4 
 7 || ae j a hs Gere re LS Soa Pe SS / 
 | Sine |Cosin |; Sine |Cosin | Sine |Cosin|| Sine .Cosin || Sine Cosin|} 
 | $70 | 64279 76604 |) 65600 |. 75471, | .66913} 74314 |) .68200 |.73150 | C9466 .71934' 60 
 | 1 | 64301 |. 76586 ,| .65628|.75452 | .66935] 74295 || 68221 | .78116 | .COIS7 71914) 59 
 9 | 6432376567 || .65650|.75433 || .66956 | .174276 || .68242 | 73096 || 69303. T1594! 58 
 3 | 64346] .76548 || .65672).75414 | 66978 74256 || .68204| .738076 || 69529 71873, 57 
 | 4 | 164368! .76530 || .65694 | .'75395 || 66999] .'74237 || .68285 | 78056 || .69549 .71853) 56 
 5 | .64390|. 76511 || .65716| 75375 || .67021) 74217 || .68306 | 730364) 69570 71833) 55 
 G | 64412! .76492)| .65738 | 75356 | 67043) 74198 .68327 | .73016 || .69591 | .71813) 54 
 . 7 | 64435 |. 76473 || .65759|.75337 || .67064) .74178)| .68349 | 72996 69612 .71792) 53 
 8 | 64457-76455 | .65781|.75318 || .67086 | .74159 || .68370 | 72976 || 69683 | 712, 52 
 . 9 | 64179 | 76436 || .65803}.75299 |) .67107  .74139}] 68391 |.72957 || 69654! 71752, 51 
 10 | “ake Nonna 65825 | .75280 || .67129 | .74120]| .68412 | .72937 69675 1732, 50 
 11 | .64524) .76398 || .65847 75251 || .67151| .74100]| .68434 | .'72917 || .69696' .7 ml 49 
 . 2 | 64546 | .76380 || .65869 |. 75241 || .67172) .74080}] 68455 | .72897 69717 | .71691| 48 
 13 | 61568 .76361 || 65891) .75222 || 67194) .74061 || 68476) 72877 69737 | .71671| 47 
 || 1£ | 64590) 76342)/ 65913 |.75203 | “67215! .74041]} .68497 | .72857 || .69758 | 71650, 46 
 || .15 | 64612) .76323 | -65935 | .75184 | "67237 | .74022|| .68518 | .72837' || .6977'9 |.71630) 45 
 16 | .64635' .76304/| .65956 "75165 || .67258| .74002|| .68539| .72817 || .69800|.71610) 44 
 17 64057 76236 | .65978|.75146 || .67280| .73983|| .68561|.72797 || .69821 | 71590) 43 
 : 18 },.64679 | .76267 | .63000 | .75126 | .67301| ..73963 || .68582 | 72777 || .69842 71569! 42 
 19 | 64701) .76243 | .66022}.75107 || .67323) .73944)| .68608 T2757 || 69862 | .71549| 41 
 : 20 pai eae | .66044 | .75088 | 67344 73924 || .68624| .72737 || .69883 | .71529| 40 
 : 21 | .61746' .76210'| .66066 | .75059 | 67366) .739041| .68645| .72717 || :69904.| 71508) 39 
 22 | 64768 .76192 | .63038|.75050 || .67387| . 78885 || .68666 | . 72697 | .69925 | .71488) 38 
 i 23 | 64790, .76173 | 65109]. 75030 | .67409) . 78805 68688 | 72677 || .69946) 71468) 37 
 . 54 | 64812! 76154 | .63131|.75011 || .67430] .73846 || .68709] .72657 || .69966 | .71447) 36 
 25.6183) 76135 | .63153).74092 || .67452) .73826|| 68730 72637 | .69987 | .71427| 35 
 25 | 64856 .76116 | 65175) .74973 | .67473} .78806)| 68751 72617 || .70008 | .71407| 34 
 97 | 64873-76097 ,| 66197 |. 7.1953 | .67495| .73787|| .68772 | .72597 || .70029 | 71386) 33 
 98 | .64901|.76078 | .65218|.74934 || 67516] .73767]| .68793 72577 || .70049| 71366) 32 
 | 29 | 64923) 76059 | .66240|.74915 | .67538| 73747 |) 63814] .72557 | 70070) 71345) 31 
 | 30 | 64945). 76041 | . 66262 | .74896 || .67559 23728 || .68835 | .'72537 || .70091 | :713825) 30 
 31 | 64967) .76022 | .66284 74876 || £67580! .737038 || .63857 72517|| 70112! .'71805| 29 
 39 | 64939 .76003 | 63306 |.74857 | .67602| 73633 || .68878|.72497 || 70132) .71284) 28 
 | 33 | 6501173934 | .66327|.74838 | .67523] .73669|| .63899].72477 | .70153 | 71264) 27 
 34 | 65033-75935 | .66349). 74313 | 67645} 73649 || 68920) 72457 -70174 | .'71243| 26 
 35 | 65055! .75946 | .63371 |.74799 || .67686| .73629]| .68941 | .72437 |) .70195 (1228) 35 
 : 36 | 65077) 75927 || 65393 |.74780 | .67638) .73610|| .68962 72417 | .70215| 71203) 24 
 37 | 65100, 73903. | .68414|.74760 || 67709) .73590|| .68983] .72397 || 70236) 71182) 23 
 98 | 65122. 75889 || .66436|. 74741 || .67730! 73570 || .69004| 72377 || . 70257 71162) 22 
 39 | 65144) .75870 | .60453 |. 74722 | 67752] 78551 |) .69025 | 72357 |) 70277 .71141| 21 
 40 preety be .66480 74003 || 67773 | «73531 || .69046 | .72387 || .'70298 | .71121) 20 
 41 | 65188! 75832 66501 | .74633 | .67795) . 73511 69067 | 72317 || :70319| .71100| 19 
 2 | 65210! .75813|| .66523|.74654 | 67816) 73491 || 69088 72297 |; 40339 71080) 18 
 43 | .65232|.75794 | .66545|. 74644 | 67837) .73472|| .69109].72277 || 70860 71059] 17 
 { 44 | 65254! .75775 | .66566 | .74625 | 67859) .73452)| .69130) .7% 7\1,.70881 | 71039} 16 
 45 | .65276, .75756 | .66588| 74605 | .67880) .73432)| .69151).72 | 704011 .71019) 15 
 46 | 65298-75738 | .66610| 74535 | .67901! .73413|| .69172|.72216 || .70422 70998) 14 
 | 47 | .65320; .75719 | .66632) .74567 || .67923| .73393 || .69193| 72196 | 70443! 70978, 13 
 | 48 | 65342, .75700 | .66653 | 74548 || .67944) .73373|| .69214|.72176|| .70463 70957) 12 
 | 49 | 65364! .75680 | .66675 | .74528 | .67965 ) 73353 || .69235 | 72156 | .70484|.70937| 11 
 | 50 | .65336) . 75661 | 66697 |.74509 | .67987 73333 || .69256 | .72136 || .70505|.70916; 10 
 51 | 65408! .75642 || .66718|.74489 | .68008) .73314 || 69277} .72116|| .70525|.70896) 9 
 52 | 65430| 75623 | 66740 |.74470 | .68029] .73294 || .69298) .72095 |) .70546 | . 70875) 8 
 53 | 65452! .75604 | .66762 74451 | .68051 | .73274 |) .69319 72075 || 70567} .70855| 7 
 B41 | 65474! 75585 | 66783 . 74431 | .68072|.73254|| .69340|.72055 |) .70587'. 70884) 6 
 55 | 65496| .75566|| .66805 .74412 | .68093 | .73234]| .69361 | .72035 || .70608'.70813) 5 
 56 | 65518! 75547 || 60827 .74392 | 68115) .73215|| .69382| .72015 || .70628) 70793) 4 
 57 | 65540. .75528 | .66848 .74373 | .68136|.73195|| .69403 | .71995 || . 70649 “70772| 8 
 58 | 65562) .75509 || .66870 .74352 || .68157).73175 || .69424|.71974 || .70670 | 70752) 2 
 59 | 65584) .75490 | .66891 .74334 | 68179 .73155 || .69445 | .71954 70690 .70731| 1 
 60 | .65606 | .75471 |) .66913 74314 | 68200). 73135 || .69466 | .71934 70711|.70711| 0 
 , (Cosin| Sine ||Cosin Sine | Cosin| Sine | Cosin | Sine | Cosin | Sine 
 4g°.- || 47 46° {|= 45° 
 
 49° | 
 
 457 
 
TABLE XXVII.—NATURAL TANGENTS AND COTANGENTS. 
 
 | el 0° Pe OO A ae 3° by 
 Tang | Cotang Tang | Cotang || Tang | Cotang || Tang | Cotang | 
 0; .00000 | Infinite.||} .0174 57.2500 || .03492 | 28.6363 |; .05241 | 19.0811 ‘60 
 i} .00029 | 8437.75 || .01775 | 56.3506 |} .03521 | 28.3994 || .05270 | 18.9755 59 
 2} .00058 | 1718.87 || .01804 | 55.4415 || .03550 | 28.1664 || .05299 | 18.8711 158 
 8} .00087 | 1145.92 .01833 | 54.5613 || .03579 937. .05328 | 18.7678 157 
 4} .00116 | 859.436 || .01862 | 53.7086 || .03609 | 27.7117 || .05857 | 18.6656 (56 
 5} .00145 87.549 || .01891 | 52.8821 -03638 | 27.4899 .05387 | 18.5645 | 55 | 
 6} .00175 | 572.957 || .01920 | 52.0807 || .08667 | 27.2715 || .05416 | 18.4645 154 | 
 Z| 00204 | 491.106 || .01949 | 51.3082 || .03696 | 27.0566 |! .05445 | 18.3655 | 53 
 8} .00233 | 429.718 || .01978 | 50.5485 || .03725 | 26.8450 |} .05474 | 18.2677 | 52 | 
 9} .00262 | 381.97 .02007 | 49.8157 03754 | 26.6267 || .05503 | 18.1708 |5 
 10} .00291 | 343.77 .02036 | 49.1039 || .03783 | 26.4316 || .05533 | 18.0750 50 
 11; .CO0320 ) 312.521 || .02066 | 48.4121 || .03812 | 26.2296 || .05562 |. 17.9802 | 49 | 
 12; .00349 | 286.478 || .02095 | 47.7395 || .03842 | 26.0807 || .05591 | 17.8863 | 48 | 
 13} .00378 | 264.441 || .02124 | 47.0853 | .08871 | 25.8348 || .05620 | 17.7934 | 47 | 
 14} .00407 | 245.552 || .02153 | 46.4489 |} .08900 | 25.6418 || .05649 | 17.7015 4G | 
 15; .00486 | 229.182 02182 | 45.8294 |; .03929 | 25.4517 .05678 | 17.6106 | 45 | 
 16| .00465 | 214.858 || .02211 | 45.2261 || .03958 | 25.2644 || .05708 | 17.5205 | 44 
 Z| -00495 | 202.219 || .02240 | 44.6386 || .03987 | 25.0798 || .05737 | 17.4314 (43 
 18} .00524 | 190.984 || .02269 | 44.0661 || .04016 | 24.897 .05766 | 17.8432 | 42 
 19/ ,00553 | 180.982 || .02298 | 43.5081 || .04046 | 24.7185 || .05795 | 17.2558 |4 
 20} .00582 | 171.885 || .02328 | 42.9641 || .04075 | 24.5418 || .05824 | 17.1693 | 40 
 21} .00611 | 163.700 || .02357 | 42.4885 || .04104 | 24.3675 || .05854 | 17.0887 |39 
 22} .00640 | 156.259 || .02386 | 41.9158 || .04138 | 24.1957 || .05883 | 16.9990 | 3: 
 23} .00669 | 149.465 || .02415 | 41.4106 || .04162 | 24.0268 || .05912 | 16.9150 |37 
 24] .00698 | 143.237 |! .02444 | 40.9174 || .04191 | 23.8593 || .05941 | 16.8319 136 
 25| .00727 | 187.507 || .02473 | 40.4358 || .04220 | 23.6945 || .05970 | 16.7496 | 35 
 26| .00756 | 182.219 || .02502 | 39.9655 || .04250 | 23.5321 |} .05999 , 16.6681 | 34 
 27| .00785 | 127.32 02531 | 39.5059 || .04279 | 23.3718 || .06029 | 16.5874 |33 
 28; .00815 | 122.77 -02560 | 39.0568 || .04308 | 23.2137 || .06058 | 16.5075 | 32 
 29} .00844 | 118.540 || .02589 | 88.6177 || .04337 | 23.0577 || .06087 | 16.4283 | 31 
 30; .00873 | 114.589 || .02619 | 38.1885 |) .04366 | 22.9038 || .06116 | 16.3499 |30 
 31} .00902 | 110.892 || .02648 | 37.7686 || .04395 | 22.7519 I! .06145 | 16.2792 |2 
 32] .00931 | 107.426 |} .02677 | 37.3579 N .04424 | 22.6020 || .06175 | 16.1952 | 9s 
 33] .00960 | 104.171 |} .02706 | 86.9560 || .04454 | 22.4541 || .06204 ; 16.1190 | 27 
 34). .00989 | 101.107 |; .02735 | 36.5627 || .04483 | 22.3081 || .06283 | 16.0435 |26 
 85} .01018 | 98.217 02764 | 36.177 04512 | 22.1640 || .06262 | 15.9687 |25 | 
 36; .01047 | 95.4895 || .02793 | 35.8006 || .04541 | 22.0217 || .06291 | 15.8945 |9 
 37| .01076 | 92.9085 || .02822 | 35.4313 || .04570 | 21.8818 || .06321 | 15.8911 |93 
 88! .01105 | 90.4633 .02851 | 35.0695 -04599 | 21.7426 -06350 | 15.7483 |22 
 89) .01185 | 88.1436 -O2881 | 34.7151 .04628 | 21.6056 .06379 | 15.6762 | 21 
 40} .01164 | 85.9398 || .02910 | 34.3678 || .04658 | 21.4704 || .06408 | 15.6048 | 20 
 41} .01193 | 83.8435 || .02939 | 34.0273 || .04687 | 21.3369 || .06437 | 15.5340 | 19 
 42} .01222,| 81.8470 || .02968 | 33.6935 || .04716 | 21.2049 || .06467 | 15.4638 118 
 3| .01251 | 79.9434 .02597 | 33.3662 .04745 | 21.0747 .06496 | 15.3943 |17 
 44} .01280 | 78.1265 .03026 | 33.0452 .04774 | 20 9460 .06525 | 15.3254 | 16 
 45} .01309 | 76.3900 || .03055 | 32.7308 || .04803 | 20.8188 || .06554 | 15.2571 | 15 
 46) .C1338 | 74.7292 || .03084 | 82.4213 || .04833 | 20.6932 || .06584 | 15.1893 114 
 47} .01867 | 738.1390 .03114 | 32.1181 .04862 | 20.5691 .06613 | 15.1222 |13 
 48) .01396 | 71.6151 || .03143 | 31.8205 || .04891 | 20.4465 || .06642 | 15.0557 112 
 49; .01425 | 70.1533 .08172 | 31.5284 .04920 | 20.3253 .06671 | 14.9898 | 11 
 50; .01455 | 68.7501 .08201 | 381.2416 .04949 | 20.2056 .06700 | 14.9244 |10 
 ‘1; .01484 | 67.4019 || .03230 | 30.9599 || .04978 | 20.0872 || .06730 | 14.8596:| 9 
 2) .01513 | 66.1055 |} .03259 | 30.6833 || .059°7 | 19.9702 || .06759 | 14.7954 | 8 
 3| .01542 | 64.8580 || .03288 | 30.4116 .05037 | 19.8546 .06788 | 14.7317! 7 
 d4; .01571 | 68.6567 .03317 | 30.1446 || .05066 | 19.7403 .06817 | 14.6685 | 6 
 55 | .01600 | 62.4992 .03846 | 29.8823 || .05095 | 19.6273 .06847 | 14.6059 | 5 
 00} 01629 |’ 61.3829 .03376 | 29.6245 || .05124 | 19.5156 .06876 | 14.5438 | 4 
 57) .01658 | 60.3058 -03405 | 29.3711 || .05153 | 19.4051 .06905 | 14.4823 | 3 
 58) .01687 | 59.2659 .03434 | 29.1220 || .05182 | 19.2959 .06934 | 14.4212 | 2 
 59| .01716 | 58.2612 -03463 | 28.8771 || .05212 | 19.1879 .06963 | 14.3607 | 1 
 60} .01746 | 57.2900 .03492 | 28.6363 |} .05241 | 19.0811 | .06993 : 14.3007 | 0 
 , Cotang) Tang ||Cotang| Tang ||Cotan g| Tang ||Cotang| Tang ; 
 89° 88° 87° 86° 
 
 458 
 
 $$! 
 
PABLE XXVIUIL—NATURAL TANGENTS AND COTANGENTS. 
 
 4° 
 
 5° 
 
 9, .07256 
 10); .07285 
 
 11| .07314 
 12) .07344 
 
 3| 07387: 
 | 07402 
 15| .074381 
 16\ .07461 
 
 —_ 
 ie 
 
 18| 07519 
 
 t 
 
 : 
 
 : 
 
 20| .0757 
 | 21| .07607 
 ; 22| 07686 
 i | 28] 07665 
 07695 
 251 07724 
 26| .07753 
 o7| 07782 
 28| .07812 
 | 29| .07841 
 | 
 : 
 
 31) .07899 
 32| .07929 
 33| .07958 
 34| .07987 
 35| .08017 
 26| .08046 
 37| .08075 
 88 |» .08104 
 
 | .08192 
 42 08221 
 43| .08251 
 
 ee 
 re 
 
 7| .07490 | 
 19| .07548 | 
 
 30| 07870 | 
 
 a9| .08184 
 40| 08163 
 
 | 
 | 
 
 QQ) .06993 
 £; .O07022 | 
 2° 07051 
 3 .07080 | 
 4° ‘07110 | 
 5) .07139 
 6. .07168 
 G 07197 | 
 8! 07227 
 
 | | 44| .08280 
 
 45| .08309 
 
 | 46| .98339 
 47| .08368 | 
 
 48| .08397 
 49| .08427 
 
 50| .08456 
 
 +51! .08485 
 
 52) .08514 
 3 08544 
 
 ) 
 
 / 54} .08573 
 
 155} .08602 
 56! .08632 | 
 | 57! .08661 
 
 58} .08690 | 
 
 159) .08720. | 
 
 60) .08749 
 
 | 
 
 |Cotang 
 / 
 
 14 
 
 feed Red ee Re et Rt 
 
 ras) 
 
 vi) 
 
 feck ek fk peek tek eh pk pk fk beh et fk ed fet pt bet 
 
 11 
 
 11 
 
 ~ Tang 
 
 WMWNWWWWHW KCWWNWWWWW 
 
 Ww wwe 
 
 m0 We 
 
 3007 
 .2411 
 .1821 
 .1235 
 .0655 
 0079 
 
 29507 | 
 3.8940 | 
 
 led 
 .597 
 
 3.7821 
 
 T267 
 6719 
 
 6174 | 
 
 56384 
 .5098 
 .4566 
 .4039 
 8515 
 
 3.2996 
 
 2480 
 1969 
 1461 
 0958 
 0458 
 9962 
 469 
 8981 
 8496 
 8014 
 7536 
 7062 
 6591 
 6124 
 £660 
 5199 
 4742 
 4288 
 
 .8390 
 2946 
 2905 
 
 . 1632 
 1201 
 O72 
 .0346 
 9923 
 
 9504 | 
 
 9087 
 8673 
 8262 
 7853 
 7448 
 . 7045 
 6645 
 6248 
 .5853 
 
 461 |} 
 
 5072 | 
 
 .4685 
 .4301 
 
 8858 | 
 
 2067 || 
 
 | 
 le 
 | Tang | Cotang 
 
 Tang 
 
 | .09746 
 
 | 
 
 Cotang | 
 
 .10011 | 9 
 .10040 | 9 
 .10069 | 9 
 
 .10099 | 9 
 .10128 | 9 
 
 .10510 | 9 
 | Cotang 
 
 Tang 
 
 "08749 | 11.4301 
 08778 | 11.3919 
 
 | .08807 | 11.3540 
 .08837 | 11.8168 
 .O8866 | 11.2789 
 
 | 08895 | 11.2417 
 .08925 | 11.2048 
 .08954 | 11.1681 
 .08983 | 11.1816 
 .09013 | 11.0954 
 .09042 | 11.0594 
 
 09071 | 11.0237 
 "09101 | 10.9882 
 | “09130 | 10.9529 
 | "99159 | 10.9178 | 
 | 09189 | 10.8829 | 
 "09218 | 10.8483 
 | "09247 | 10.8139 
 | “09277 | 10.7797 
 "09806 | 10.7457 
 "09385 | 10.7119 
 
 .09365 | 10.6788 
 109394 | 10.6450 
 (09423 | 10.6118 | 
 '09453 | 10.5789 
 09482 | 10.5462 
 (09511 | 10.5186 
 (09541 | 10.4813 
 | :09570 | 10.4491 
 09600 | 10.4172 
 109629 | 10.3854 
 
 | .09658 | 10.3538 
 .09688 | 10.8224 
 .09717 | 10.2913 | 
 10.2602 
 .09776 | 10.2294 
 .09805 | 10.1988 | 
 .09834 | 10.1683 
 .09864 | 10.1881 
 | .09893 | 10.1080 || 
 .09923 | 10.0780 
 
 .09952 | 10.0483 
 .09981 | 10.0187 | 
 
 .98981 
 .96007 
 .93101 
 90211 
 87338 
 
 .51456 
 
 .10158 | 9.84482 
 | .10187 | 9.81641 
 || .10216 | 9.<8817 
 10246 | 8.76009 | 
 | 10275 | 9.73217 
 | 10805 | 9.70441 
 | .10334 | 9.67680 
  .10863 | 9.64935 || 
 | 10393 | 9.62205 
 10422 | 9.59490 
 10452 | 9.56791 
 10481 | 9.54106 
 
 7e i \ 
 
 | 85° 
 
 || 
 
 84° 
 
 459 
 
 % SESAME cs PSO: PF 
 Tang : Cotang | Tang: ! Cotang 
 10510 | 9.51436 || .12278 | 8.14435 | 60 
 10540 | 9.48781 || .12803 | 8.12451 |59 
 10569 | 9.46144 || .12358 | 8.10586 | 58 
 10599 | 9.43515 || .12867 | 8.08600 |57 
 10628 | 9.40904 || .12507 | 8.06674 |56 
 
 | .10657 | 9.88807 || .12426 | 8.04756 [55 
 
 | 10687 | 9.85724 || .12456 | 8.02848 |54 
 (10716 | 9.88155 || .12485 | 8.00948 |53 
 10746 | 9.30599 || .12515 | 7.99058 |52 
 10775 | 9.28058 || .12544 | 7.97176 |51 
 10805 | 9.25530 || .12574 | 7.95802 |5 
 
 | .10834 | 9.23016 || .12603 | 7.93488 | 49 
 
 | 10863 | 9.20516 || .12683 | 7.91582 | 48 
 
 | 10863 | 9.18028 || .12C62 | 7.89734 |47 
 10922 | 9.15554 || .12692 | 7.87895 |46 
 10952 | 9.18098 || .12722 | 7.86064 | 45 
 10981 | 9.10646 || .12751 | 7.84242 | 44 
 
 || .11011 | 9.08211 || .12781 | 7.82428 | 43 
 
 | (11040 | 9.05789 || .12810 | 7.80622 |42 
 11070 | 9.08879 || .12840. | 7.78825 |41 
 11089 | 9.00983 |} .12869 | 7.77035 | 40 
 
 | .11128 | 8.98598 || .12899 | 7.75254 |39 
 | (11158 | 8.96227 || .12929 | 7.73480 |38 
 | .11187 | 8.93867 || .12058 | 7.71715 |387 
 11217 | 8.91520 || .12988 | 7.69957 | 36 
 
 | 11246 | 8.89185 || .18017 | 7.68208 | 35 
 
 || .112%6 | 8.86862 || .13047 | 7.66466 [34 
 (11305 | 8.84551 || .12076 | 7.64782 | 33 
 
 | 11885 | 8.82252 || .18106 | 7.63005 | 32 
 
 | 112864 | 8.79964 || .18186 | 7.61287 |31 
 
 | 11394 | 8.77689 || .18165 | 7.59575 | 30 
 
 | .11423 | 8.75425 || .18195 7 57872. |29 
 
 | .11452 | 8.73172 || .18224 | 7.56176 | 28 
 
 | 11482 | 8.70981 || .18254 | 7.54487 [27 
 
 | 11511 | 8.68701 || .18284 | 7.52806 | 26 
 
 | 11541 | 8.66482 || .18813 | 7.511382 |25 
 
 | 111570 | 8.64275 || 18843 | 7.49465 |24 
 “11600 | 8.62078 || .18872° | 7.47806 | 23 
 “11629 | 8.59893 || .13402 | 7.46154 | 22 
 “11 1) | 8.57718 || .13482 | 7.44509 | 21 
 
 | 114688 | 8.55555 || .13461 | 7.42871 | 20 
 
 | 11718 | ¢ 58402 || .18491 | 7.41240 | 19 
 
 | “11747 | 8.51259 || .13521 | 7.39616 |18 
 
 | 114777 | 8.49128 |} .18550 | 7.387999 | 1% 
 "11806 | 8.47007 || .18580 | 7.86889 | 16 
 
 | -41886 | 8.44896 |} .13609 | 7.34786 | 15 
 
 | “41865 | 8.42795 || .18639 | 7.33190 | 14 
 "11895 | 8.40705 || .18669 | 7.31600 | 13 
 
 | “41924 | 8.38625 || .18698 | 7.30018 |12 
 "41954 | 8.36555 || .13728 | 7.28442 | 11 
 "11983 | 8.34496 || .13758 | 7.26873 | 10 
 12013 | 8.32446 |) .18787 | 7.25310 | 9 
 "42042 | 8.30406 || .18817 | 7.23754 | 8 
 "12072 | 8.28376 || .18846 | 7.22204 | 7 
 
 | '12101 | 8.26355 || .18876 | 7.20661 | 6 
 "42131 | 8.24345 || .18906 | 7.19125 | 5 
 
 || 12160 | 8.22844 || .18985 | 7.17594 | 4 
 "42190 | 8.20852 || .18965 | 7.16071 | 3 
 
 | 142219 | 8.18370 || .13995 | 7.14553 | 2 
 “42249 | 8.16398 |, .14024 | 7.15042 | 1 
 
 | 142978 | 8.14435 |! .14054 | 7.11537 | 0 
 
 \Cotang| Tang | Cotang| Tang | | 
 
 83° §2° 
 
~~ 
 
 CHOIR OUMwWoHS| 
 
 10 
 
 _Tang 
 
 14054 
 
 - 14084 
 14113 
 . 14143 
 -14173 
 - 14202 
 - 14232 
 - 14262 
 14291 
 14321 
 14351 
 14381 
 .14410 
 - 14440) 
 - 14470 
 - 14499 
 14529 
 - 14559 
 - 14588 
 14618 
 . 14648 
 . 14678 
 14707 
 -14737 
 14767 
 14796 
 14826 
 . 14856 
 - 14886 
 - 14915 
 - 14945 
 
 .14975 
 . 15005 
 . 15034 
 . 15064 
 . 15094 
 . 15124 
 .15153 
 . 15183 
 . 15213 
 . 15243 
 115272 
 . 15302 
 . 15332 
 . 15362 
 . 15391 
 . 15421 
 . 15451 
 . 15481 
 15511 
 . 15540 
 
 . 15570 
 . 15600 
 . 15630 
 . 15660 
 15689 
 .15719 
 15749 
 15779 
 
 | .15809 
 
 . 15838 
 
 8° 
 
 TABLE XXVIII—NATURAL TANGENTS AND COTANGENTS, 
 
 9° 10° 11° 
 Cotang |} Tang | Cotang || Tang Cotang || Tang | Cotang 
 7.11587 || .15888 | 6.31375 .17633 | 5.67128 -19438 | 5.14455 
 7.10038 .15868 | 6.30189 .17663 | 5.66165 -19468 | 5.13658 
 7.08546 -15893 | 6.29007 |} .17693 | 5.65205 .19498 | 5.12862 
 7.07059 . 15928 . 27829 17723 | 5.64248 -19529 | 5.12069 
 7.05579 -15958 | 6.26655 || .17753 | 5.63295 -19559 | 5.11279 
 7.04105 -15988 | 6.25486 17783 .62544 || .19589 | 5. 10499 
 7.02637 -16017 | 6.24321 .17813 | 5.613897 -19619 | 5.09704 
 7.01174 16047 | 6.23160 || .17843 | 5.60452 -19649 | 5.08921 
 6.99718 |; .16077 | 6.22003 || .17873 | 5.59511 -19680 | 5.08139 
 6.98268 .16107 | 6.20851 .17903 | 5.58573 :19710 | 5.07360 
 6.96823 |} .16137 | 6.19703 |] .17933 | 5.57638 |! .19740 5.06584 
 6.95385 || .16167 | 6.18559 || .17963 5.56706 || .19770 | 5.05809 
 6.93952 |} .16196 | 6.17419 || .17998 | 5.5577 -19801 | 5.05037 
 6.92525 || .16226 | 6.16283 || .18023 | 5.54851 |] .19881 5.04267 
 6.91104 || .16256 | 6.15151 |} .18053 | 5.53927 || .19861 5.03499 
 6.89688 |/ .16286 | 6.14023 || .18083 | 5.53007 || .19891 5.02784 
 6.88278 - 16316 | 6.12899 .18113 | 5.52090 -19921 | 5.01971 
 6.86874 || .16346 | 6.1177 -18143 | 5.51176 || .19952 | 5.01210 
 6.85475 -16376 | 6.10664 || .18173 | 5.50264 || .19982 5.00451 
 6.84082 || .16405 | 6.09552 || 18208 | 5.49356 -20012 | 4.99695 
 6.82694 || .16435 | 6.08444 || .18233 | 5.48451 || 20042 4.98940 
 6.81312 || .16465 | 6.07340 || .18263 | 5.47548 || .20073 4.98188 
 6.79936 || .16495 | 6.06240 ;| .18293 .46648 -20103 | 4.97438 
 6.78564 || .16525 | 6.05143 || .18323 | 5.45751 -20133 | 4.96690 
 6.77199 || .16555 | 6.04051 || .18353 | 5.44857 -20164 | 4.95945 
 6.75838 || .16585 | 6.02962 |} .18384 | 5.43966 -20194 | 4.95201 
 6.74483 || .16615 | 6.01878 |] .18414 | 5.48077 || _20224 4.94460 
 6.73133 || .16645 | 6.00797 || .18444 | 5.42192 -20254 | 4.93721 
 6.71789 || .16674 | 5.99720 || .18474 | 5.41309 || .20985 4.92984 
 6.70450 || .16704 | 5.98646 || .18504 | 5.40429 -20315 | 4.92249 
 6.69116 16734 | 5.97576 || .18534 | 5.39552 || .20345 4.91516 
 6.67787 || .16764 | 5.96510 |! .18564 | 5.38677 || 20376 4.90785 
 6.66463 || .16794 | 5.95448 || .18594 | 5.37805 || .20406 4.90056 
 6.65144 |/ .16824 | 5.94390 |} .18624 | 5.36936 || .20436 4.89330 
 6.63831 || .16854 | 5.93385 || .18654 | 5.3607 -20466 | 4.88605 
 6.62523 || .16884 | 5.92283 || .18684 | 5.35206 || .20497 4.87882 
 6.61219 .16914 | 5.91236 18714 | 5.34345 -20527 | 4.87162 
 6.59921 || .16944 | 5.90191 || .18745 | 5.33487 || .20557 | 4 86444 
 6.58627 || .16974 | 5.89151 18775 | 5.32631 .20588 | 4.85727 
 6.57339 .17004 | 5.88114 18805 | 5.31778 || .20618 | 4.85013 
 6.56055 |} .17033 | 5.87080 || .18835 | 5.30928 |! .20648 4.84300 
 6.54777 || .17063 | 5.86051 || .18865 | 5.30080 |] .20679 4.83590 
 6.53503 || .17093 | 5.85024 -18895 | 5.29235 -20709 | 4.82882 
 6.52234 || £17123 | 5.84001 18925 | 5.28393 || .20739 | 4.82175 
 6.50970 || .17153 | 5.82982 || .18955 | 5.27553 || /20770 | 4.81471 
 6.49710 17183 | 5.81966 || .18986 | 5: 26715 -20800 || 4.80769 
 6.48456 17213 | 5.80953 || .19016 | 5.25880 -20830 | 4.80068 
 6.47206 17243 | 5.79944 -19046 | 5.25048 || .20861 | 4.79370 
 6.45961 17278 | 5.78938 -19076 | 5.24218 -20891 4.78673 
 6.44720 || .17303 | 5.77936 -19106 | 5.23391 |' .20921 | 4.77978 
 6.43484 . 17333 | 5.76937 -19136 | 5.22566 || .20952 | 4. 77286 
 6.42253 || .17363 | 5.75941 || .19166 | 5.21744 || .20982 | 4.76595 
 6.41026 || .17893 | 5.74949 .19197. | 5.20925 -21013 | 4.75906 
 6.39804 |) .17423 | 5.73960 || .19227 | 5.20107 .21043 | 4.75219 
 6.38587 || .17453 | 5.72074 -19257 | 5.19293 -21073 | 4.74534 
 6.3737 .17483 | 5.71992 .19287 | 5.18480 .21104 | 4.73851 
 6.36165 7513 | 5.71013 229817): Del T6%1 21134 | 4.73171 
 6.34961 .17548 | 5.70037 .19347 | 5.16863 .21164 | 4.72490 
 6.33761 || .17573 | 5.69064 1987 5.16058 -21195 | 4.71813 
 6.32566 || .17603 | 5.68094 || .19408 | 5.15256 || .212295 | 4.71137 
 6.31375 . 17683 | 5.67128 .19438 | 5.14455 || .21256 | 4.70463 
 \Cotang| Tang , Cotang| Tang ||Cotang| Tang Cotang| Tang 
 81° | 80° 79° 78° 
 
 eeeaeniaae = 
 
TABLE XXVIIIL—NATURAL TANGENTS AND COTANGENTS. 
 
 | ry 19° 13° jae | 15° 
 : Tang Cotang || Tang | Cotang |; Tang | Cotang || Tang | Cotang 
 | Q| .21256 | 4.70463 || .23087 | 4.33148 || .24933 | 4.01078 || .26795 | 3.73205 
 | 1] .21286 | 4.69791 || .23117 | 4.82573 || .24964 | 4.00582 || .26826 | 3.7277 
 | 2) .21816 | 4.69121 .28148 | 4.32001 .24995 | 4.00086 || .26857 | 3.42338 
 3} .213847 | 4.68452 || .23179 | 4.31430 || .25026 | 3.99592 || .26888 | 3.71907 
 4} .21877 | 4.67786 || .23209 | 4.80860 || .25056 | 3.99099 || .26920 | 8.71476 | 
 5} .21408 | 4.67121 .23240 | 4.30291 .25087 | 3.98607 || .26951 | 3.71046 
 6| .21438 | 4.66458 || .28271 |. 4.29724 || .25118 | 3.98117 || .26982 | 3.70616 
 7) .21469 | 4.65797 |} .283801 | 4.29159 || .25149 | 3.97627 || .27013 | 3.70188 
 8} .21499 | 4.65188 || .2383832 | 4.28595 || .25180 | 3.97189 || .27044 | 3.69761 
 9} .215¢9 | 4.64480 || .23363 | 4.28032 || .25211 | 8.96651 .27076 | 3.69385 
 10} .21560 | 4.63825 || .28393 | 4.27471 || .25242 | 3.96165 || .27107 | 3.68909 
 11} .21590 | 4.63171 .23424 | 4.26911 .25273 | 3.95680 || .27188 | 3.68485 
 12} .21621 | 4.62518 || .23455 | 4.26852 || .25804 | 3.95196 ‘| .27169 | 3.68061 
 | 13} .21651 | 4.61868 23485 | 4.25795 || .25385 | 3.94718 || .27201 | 3.67638 
 14! .21682 | 4.61219 23516 | 4.25289 || .25366 | 3.94282 || .27282 | 3.67217 
 15} .21712 | 4.60572 23547 | 4.24685 || .25897 | 3.98751 .27263 | 3.66796 
 | 16] .21743 | 4.59927 23578 | 4.24182 || .25428 | 3.93271 .27294 | 3.66376 
 17| .21773 | 4.59283 23608 | 4.28580 || .25459 | 3.92798 || .273826 | 3.65957 
 18) .21804 | 4.58641 23639 | 4.23080 || .25490 | 3.92816 || .27357 | 8.65538 
 19} .21834 | 4.58001 23670 | 4.22481 .25521 | 3.91839 || .27888 | 8.65121 
 bal -21864 | 4.57363 23700 | 4.21983 || .25552 | 3.91364 |} .27419 | 8.64705 
 21) .21895 | 4.56726 28731 | 4.21887 || .25583 | 8.90890 || .27451 | 8.64289 
 22} .21925 | 4.56091 23762 | 4.20842 || 25614 | 3.90417 || .27482 | 3.63874 
 23) .21956 | 4.55458 23793 | 4.20298 || .25645 | 3.89945 || .27513 | 3.63461 
 | 24] .21986 | 4.54826 238823 | 4.19756 || .25676 | 8.89474 || .27545 | 3.63048 
 25} -.22017 | 4.54196 23854 | 4.19215 || .25707 | 3.89004 || .27576 | 3.62636 
 26) .22047 | 4.53568 28885 | 4.18675 || .25738 | 3.885386 || .27607 | 8.62224 
 Q7| .22078 | 4.52941 23916 | 4.18187 || .25769 | 8.88068 || .27638 | 3.61814 
 28} .22108 | 4.52316 23946 | 4.17600 ||} .25800 | 3.87601 .27670 | 3.61405 
 29| .22139 | 4.51693 23977 | 4.17064 || .258381 | 3.87136 || .27701 | 8.60996 
 | 30| .22169 | 4.51071 24008 | 4.16530 || .25862 | 3.86671 || .277382 | 3 60588 | 
 | 81} .22200 | 4.50451 24039 | 4.15997 |} .258938 | 3.86208 || .27764 | 3.60181 
 382| .22231 | 4.49832 24069 | 4.15465 |} .25924 | 3.85745 || .27795 | 8.59775 
 33] .22261 | 4.49215 24100 | 4.14984 || .25955 | 8.85284 || .27826 | 3.59370 
 | 34| .22292 | 4.48600 || .24181 |°4.14405 || .25986 | 3.84824 || .27858 | 3.58966 
 35] .22322 | 4.47986 24162 | 4.18877 || .26017 | 3.84364 |} .27889 | 3.58562 
 386} .22353 | 4.47374 || .24193 | 4.13850 || .26048, | 3.83906 || .27921 | 8.58160 
 37| .223883 | 4.46764 || .24223 | 4.12825 26079 | 3.83449 || .27952 | 3.57758 
 38] 22414 | 4.46155 24954 | 4.12301 .26110 | 3.82992 219838 | 3.57357 
 39| .22444 | 4.45548 24285 | 4.11778 || .26141 | 8.82587 || .28015 | 3.56957 
 40| .22475 | 4.44942 24316 | 4.11256 || .26172 | 3.82088 || .28046 | 3.56557 
 41} .22505 | 4.44338 || .243847 | 4.10786 || .26203 | 3.81630 || .28077 | 3.56159 
 . 42) 1.22536 | 4.48785 || .243877 | 4.10216 || .26235 | 3.81177 || .28109 | 3.55761 
 } 43] .22567 | 4.43134 |) .24408 | 4.09699 || .26266 | 3.80726 || .28140 | 3.55364 
 | 44| .22597 | 4.42534 .24489 | 4.09182 .26297 | 3.80276 28172 | 3.54968 
 45| 122628 | 4.41936 || .24470 | 4.08666 || .26328 | 8.79827 || .28203 | 8.54573 
 . 46, .22658 | 4.41340 || .24501 | 4.08152 || .26359 | 3.79378 || .28234 | 3.54179 
 | 47| .22689 | 4.40745 245382 | 4.07639 |} .26890 | 3.778931 .28266 | 3.53785 
 i 48! .22719 | 4.40152 24562 | 4.07127 || .26421 | 8.78485 || .28297 | 3.53393 
 49; .22750 | 4.39560 24593 | 4.06616 || .26452 | 3.78040 || .28329 | 3.53001 
 | 50) .22781 | 4.388969 24624 | 4.06107 || .26483 | 3.77595 || .28860 | 3.52609 
 51) .22811 | 4.38381 24655 | 4.05599 |} .26515 | 3.77152 || .283891 | 3.52219 
 52) .22842 | 4.37793 24686 | 4.05092 || .26546 | 3.76709 || .28423 | 3.51829 
 53] .22872 | 4.37207 24717 | 4.04586 |} .26577 | 3.76268 || .28454 | 3.51441 
 54! .22903 | 4.36623 || .24747 | 4.04081 || .26608 | 3.75828 || .28486 | 3.51053 
 55} .22924 | 4.36040 || .24778 | 4.038578 || .26689 | 8.753888 || .28517 | 3.50666 
 56] .22964 | 4.35459 24809 | 4.03076 |} .26670 | 3.774950 || .28549 | 3.5027’ 
 57|_.22995 | 4.34879 24840 | 4.0257. .26701 | 3.74512 || .28580 | 3.49894 
 58| .23026 | 4.34300 24871 | 4.02074 || .26783 | 3.74075 || .28612 | 3.49509 
 59} .28056 | 4.33723 24902 | 4.01576 || .26764 | 3.73640 || .28643 | 8.49125 
 60) .238087 | 4.33148 24933 | 4.01078 || .26795 | 3.738205 ||} .28675 | 3.48741 
 \Cotang| Tang | Cotang Tang ||Cotang| Tang |/Cotang| Tang 
 / ee ee eee ee ee iat ae tne bead EE hes ' 
 | Pea - | 76° | 75° 74° 
 
 lomeocwmoa20w 
 
‘ 16° 17° 18° \| 19° : 
 Tang | Cotang || Tang | Cotang |; Tang | Cotang || Tang | Cotang 
 QO} .28675 | 3.48741 || .80573 | 5.27085 .82492 | 3.07768 84433 | 2.90421 |60 | 
 1} .28706 | 3.48359 || .80605 | 3.26745 || .82524 | 3.07464 || .384465 | 2.90147 |59 
 2| .28788 | 3.47977 || .380687 | 3.26406 || .82556 | 3.07160 || .84498 | 2.89873 |58 
 8| .28769 | 3.47596 || .80669 | 3.26067 || .382588 | 3.06857 || .384530 | 2.89600 |57 
 4; .28800 | 3.47216 || .380700 | 3.25729 || .82621 | 3.06554 || .384563 | 2.89827 |56 
 5| .288382 ) 3.46837 || .80782 | 3.25892 || .82653 | 3.06252 || .84596 | 2.89055 155 
 6| .28864 | 8.46458 || .80764 | 3.25055 || .82685 | 3.05950 || .384628 | 2.88788 |54 
 7| .28895 | 3.46086 || .380796 | 3.24719 || .382717 | 3.05649 || .34661 | 2.88511 | 53 
 8} .28927 | 8.45703 || .80828 | 3.24383 || .32749 | 3.05349 || .34693 | 2.88240 | 52 
 9} .28958 | 8.45327 || .80860 | 3.24049 || .82782 | 3.05049 || .34726 | 2.87970 |51 
 10; .28990 | 3.44951 |} .80891 | 3.28714 || .82814 | 3.04749 || .84758 | 2 87700 |50 
 11) .29021.| 3.44576 || .80923 | 3.23381: || .82846 | 3.04450 || .84791 | 2.87430 | 49 
 12} .29053 | 3.44202 || .80955 | 3.23048 |, .82878 | 3.04152 |; .84824 | 2.87161 | 48 
 13| .29084 | 3.48829 || .80987 | 3.22715 || .82911 | 3.03854 || .384856 | 2.86892 | 47 
 14} .29116 | 3.43456 || .81019 | 3.22884 || .382943 | 3.038556 || .384889 | 2.86624 | 46 
 15} .29147 | 3.43084 || .81051 | 3.22053 || .82975 | 3.08260 || .34922 | 2.86356 | 45 
 16} .29179 | 3.42713 || .381083 | 3.21722 || .388007 | 3.02963 || .84954 | 2.86089 | 44 
 7| .29210 | 3.42343 || .81115 | 3.21892 || .338040 | 3.02667 || .384987 | 2.85822 | 43 
 18} .29242 ) 3.41973 || .81147 | 3.21068 || .388072 | 3.02872 || .385020 | 2.85555 | 42 
 19} .29274 | 3.41604 || .81178 | 3.20734 || .33104 | 3.02077 || .35052 | 2.85289 | 41 
 20} .29805 | 3.41286 |) .81210 | 3.20406 || .83186 | 3.01783 |; .85085 | 2.85023 | 40 
 
 21) .29337 | 3.40869 || .81242 | 3.20079 || .33169 | 8.01489 || .385118 | 2.84758 | 39 
 a 22| .29368 | 3.40502 || .81274 | 3.19752 || .388201 | 8.01196 || .85150 | 2.84494 | 38 
 2 | 23| .29400 | 3.40136 || .81806 | 3.19426 || .383233 | 3.00903 || .85183 | 2.84229 | 37 
 24| .29432 | 3.3977 .31338 | 3.19100 || .83266 | 3.00611 || .85216 | 2.83965 | 36 
 Ha a8 25} .29463 | 3.39406 || .381870 | 3.18775 || .338298 | 3.00819 || .85248 | 2.83702 | 35 
 
 26| .29495 | 8.39042 || .381402 | 3.18451 || .383330 | 3.00028 || .85281 | 2.83439 | 34 
 27| .29526 | 3.88679 || .81434 | 3.18127 || .383363 99738 || .385314 | 2.83176 | 33 
 
 97144 || .85608 | 2.80833 | 24 
 
 96858 || .85641 | 2.80574 | 23 | 
 96573 || .85674 | 2.80316 | 22 | 
 96288 || .85707 80059 | 21 | 
 .96004 || .85740 
 
 36/ .29811 
 37| .29848 
 38| .29875 
 
 .00443 || .381722 
 85087 || .81754 .14922 || .38686 
 
 3 
 
 3 
 
 3 
 
 3 
 
 3 
 
 3 
 
 3 
 
 3.15240 || .38654 
 
 3 
 84732 || .81786 | 3.14605 || .33718 
 
 3 
 
 3 
 
 3 
 
 3 
 
 3 
 
 3 
 
 3 
 
 3 
 
 3 
 
 i 28| .29558 | 3.38317 || .81466 | 3.17804 || .38395 99447 || .85346 | 2.82914 | 32 
 Hy 29; .29590 | 3.387955 || .81498 .17481 || .83427 99158 || .85379 | 2.82653 | 31 
 ap 80] .29621 | 3.387594 || .81530 .17159 .|| .33460 98868 || .385412 | 2.82391 | 30 
 Hee ia i 31} .29653 | 3.37234 || .31562 .16838 || .383492 98580 || .85445 | 2.82130 | 29 
 aie 82] .29685 | 3.36875 || .381594 .16517 || .33524 98292 || .85477 | 2.81870 | 28 
 bo 33} .29716 | 3.36516 || .31626 .16197 || .388557 98004 || .85510 | 2.81610 | 27 
 Tan eh |i 34| .29748 | 8.36158 || .31658 15877 || .3838589 9717 || .85543 | 2.81350 | 26 
 He 85] .29780 : .85800 || .81690 .15558 |} .88621 97430 || .85576 | 2.81091 | 25 
 
 3 
 
 3 
 
 ii 39} .29906 
 He | 40} .29938 
 
 WWWWNWNWWWWW WWWYW 
 
 34377 || .81818 14288 |} .33751 
 34023 || .81850 | 3.13972 || .83783 
 
 % 0 
 
 79802 120} 
 
 3. 
 3. 
 a 41| .29970 | 3.33670 || .81882 | 3.18656 || .33816 | 2.95721 || .85772 | 2.79545 119 
 aaah 2| .30001 | 3.33317 || .81914 | 3.13841 || .83848 | 2.95437 || .85805 | 2.79289 |18| | 
 AAT 43| .30038 | 3.32965 || .81946 | 3.18027 || .88881 | 2.95155 || .85838 | 2.79033 17] 
 au 44| .30065 | 8.32614 || .81978 | 3.12713 || .83913 | 2.94872 || .85871 | 2.78778 |16} _ 
 Nae 45) .30097 | 3.32264 || .82010 | 3.12400 || .33945 | 2.94591 |] .85904 | 2.78523 /15] 
 Mane | 46} .30128 | 8.31914 || .82042 | 3.12087 || .33978 | 2.94309 || .85937 | 2.78269 | 14 
 | 47| .30160 | 8.31565 || .82074 | 3.11775 || .84010 | 2.94028 || .35969 | 2.78014 |13 
 48) .30192 | 8.31216 || .82106 | 3.11464 || .34043 | 2.98748 || .36002 | 2.77761 |12 
 ij 4)| .30224 | 3.30868 || .382189 | 8.11153 || .34075 | 2.93468 || .86035 | 2.77507 | 11 
 50; .80255 | 8.30521 || .82171 | 8.10842 || .34108 | 2.93189 || .386068 | 2.77254 | 10 
 51| .30287 | 3.30174 || .82208 | 3.10582 || .84140 | 2.92910 || .36101 | 2.77002-| 9 
 52] .80319 | 8.29829 || .82235 | 3.10223 |} .34173 | 2.92632 || .36134 | 2.76750 | 8 
 53] .80351 | 3.29488 || .82267 | 3.09914 |] .34205 | 2.92354 || .86167 | 2.76498 | 7 
 54| .30382 | 3.29139 |} .82299 | 3.09606 || .34238 | 2.92076 || .36199 | 2.76247 | 6 
 55| .80414 | 3.28795 || .32831 | 3.09298 || .84270 | 2.91799 || .36232 | 2.75996 | 5 
 56| .30446 | 8.28452 || .82363 | 3.08991 || .34303 | 2.91523 || .86265 | 2.75746 | 4 
 57| .80478 | 8.28109 || .82396 | 3.08685 || .34335 | 2.91246 || .36298 | 2.75496 | 3 
 58] .30509 | 8.27767 || .32428 | 3.08379 || .34368 | 2.90971 || .36331 | 2.75246 | 2 
 59| .80541 | 8.27426 || .82460 | 8.08073 || .34400 | 2.90696 || .36364 | 2.74997 | 1] | 
 60} .80573 | 3.27085 | 32492 | 8.07768 |] .34433 | 2.90421 || 36397 | 2.74748 | 0] 
 Cotang| Tang CObeaNS Tang |/Cotang| Tang || Cotang | Tang ; | 
 de eee nn a ee eer | One Deere. Saeeee enna | PO ene Een eee ||| tene |e al ee 
 73° | 72° | "1° | 70° 
 ln re sn ee es ee 
 
 462 
 
TABLE XXVII.—NATURAL TANGINTS AND COTANGINTS. 
 
 | 
 
 20° || ie 22° | 23° 
 
 |! Tang | Cotang || Tang | Cotang!; Tang | Cotang 
 60509 || .40408 | 2.47500 ;| .42447 | 2.85585 
 60283 .40486 | 2.473802 || .42482 | 2.85395 
 60057 || .40470 .47095 || .42516 | 2.85205 
 593831 || .40504 | 2.46888 || .42551 .80015 
 .88520 | 2.59606. || .40588 | 2.46682 || .42585 | 2.34825 
 £88558 59381 .40572 | 2.46476 || .42619 | 2.34636 
 .88587 .59156 || .40606 | 2.4627 .42654 84447 
 .88620 58932 || .40640 | 2.46065 || .42688 . 84258 
 88654 58708 || .40674 .4586 42722 | 2.34069 
 88687 58484 || .40707 | 2.45655 42757 83881 
 38721 .58261 || .40741 .4545 .42791 383693 
 .88754 .58038 || .4077 452 .42826 33505 
 .88787 57815 |; .40809 .45043 |} .42860 BBol? 
 80821 57593 || .40843 .44839 || .42894 . 83180 
 88854 573871 || .40877 «4 .42929 82943 
 | .38888 57150 || .40911 444: -42963 82756 
 .88921 56928 .40945 | 2.442% .42998 .82570 
 88955 56707 || .40979 .44027 || .48082 82383 
 88988 .56487 || .41013 .43825 || .48067 82197 
 | .39022 56266 || .41047 365 .43101 .82012 
 || .89055 56046 || .41081 .43136 
 
 .31826 
 .69612 || .39089 55827 |} .41115 4B R% .43170 31641 
 .69371 .89122 55608 |; .41149 43015 43205 .81456 
 .69131 || .89156 55389 || .41183 428 43239 81271 
 .68892 ||} .389190 5517 41217 43274 381086 
 .68653 |; .38922: 54952 || .41251 .43308 80902 
 .68414 || 89257 54734 41285 43343 80718 
 .68175 || .39290 54516 || .41819 43378 80534 
 O7T93T || .89324 54299 || .41853 .43412 80351 
 .67709 || .89357 54082 || .41887 43447 80167 
 .67462 || .89091 .58865 || .41421 43481, 
 
 29984 
 67225 || .39425 41455 41223 || .48516 .29801 | 
 66989 || .389458 .41490 41025 || .48550 29619 
 .66752 || .89492 41524 40827 || .48585 29437 12 
 .66516 || .89526 41558 4062 .43620 29254 |% 
 .66281 |! .89559 .41592 .40482 || .48054 29073 | 2 
 .66046 |) .89593 .41626 .40285 || .438689 28891 
 .65811 || .89626 .41660 .400388 |) 48724 28710 | 2 
 .65576 || ~.89660 741604 89841 || .48758 28528 | 2% 
 65342 || .389694 41728 .89645 || .48793 28348 
 .65109 || .39727 41763 89449 || .43828 28167 
 .64875 |} .389761 1797 89253 || .48862 27987 
 64642 || .39795, 41831 39058 || .43897 27806 
 64410 || .89829 41865 .88863 || .45982 27626 
 64177 |; .389862 41899 .88668 || .48966 27447 
 63945 || .89896 41933 .88473 || .44001 27267 
 .63714 |! .89930 .41968 88279 || .44036 27088 
 .63483 || .89963 .42002 | 2.38084 || .44071 | 2.26909 
 63252 || .89997 .42036 87891 || .44105 26730 
 63021 || .40031 49807 || .42070 | 2.87697 || .44140 26552 | 
 62791 || .40065 49597 || .42105 .87504 || 44175 2637 
 62561 || .40098 | 2.49886 || .42139 37311 || .44210 26196 
 62332 || .40132 49177 || .421%73 87118 || .44244 | 2.26018 
 62103 || .40166 A8967 || .42207 | 2.36925 |} .44279 2584 
 .61874 || .40200 48758 || .42242 | 2.36783 || .44814 25663 
 61646 || .40234 | 2.48549 || .42276 .86541 || .44849 25486 
 61418 || .40267 | 2 48340 || .42310 36349 || .44884 25609 
 .61190 || .40301 48132 || .42345 | 2.36158 || .44418 25132 
 .60963 || .40335 | 2.47924 || .42379 85967 || .44453 24956 
 60736. || .40369 | 2.47716 || .42413 85776 .44488 | 2.24780 
 60509 || .40403 | 2.47509 || .42447 | 2.85585 || .44523 2.24604 
 Tang ||Cotang; Tang | Cotang| Tang \Cotang | Tang 
 ) 68° 67° 
 463 
 
 ~ 
 
 83336 
 || 88420 | 
 | .88453 
 38487 
 
 Co) 
 
 W 
 
 0 ®D WW WWW 
 
 oomwmnc! 
 
 36661 
 36694 
 86727 
 
 .386760 
 .86793 
 . 36826 | 
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TABLE XXVIIL—NATURAL TANGENTS AND COTANGENTS, 
 
 ; 24° | 25° 26° | 27° | P 
 Tang | Cotang || Tang | Cotang |; Tang | Cotang || Tang | Cotang 
 O| .44523 | 2.24604 .46631 | 2.14451 || .48778 | 2.05080 || .50953 | 1.96261 | 60} 
 * J} .44558 | 2.24428 .46666 | 2.14288 .48809 | 2.04879 || .50989 | 1.96120 |59 | 
 2] .44593 | 2.24252 46702 | 2.14125 || .48845 | 2.04728 || .51026 | 1.95979 [58 | 
 3| .44627 | 2.24077 || .46787 | 2.13963 || .48881 | 2.04577 || .51063 | 1.95838 [57 | 
 4| .44662 | -2.23902 .46772 | 2.138801 48917 | 2.04426 .51099 | 1.95698 | 56 | 
 5| .44697 | 2.23727 || .46808 | 2.13639 || .48953 | 2.0427 .51186 | 1.95557 | 55 | 
 6| .44782 | 2.23553 || .46843 | 2.13477 || .48989 | 2.04125 }| .51173 | 1.95417 | 54) 
 7| 44767 | 2.23378 || .46879 | 2.18316 || .49026 | 2.03975 || .51209 | 1.95277 153 | 
 8| .44802 | 2.23204 || .46914 | 2.18154 || .49062 | 2.03825 || .51246 | 1.951387 | 52 | 
 9| .44837 | 2.23030 || .46950 | 2.12993 || .49098 | 2.03675 || .51288 | 1.94997 |51 | 
 10| .44872 | 2.22857 .46985 | 2.12882 || .49184 | 2.03526 || .51319 | 1.94858 |50 | 
 11| .44907 | 2.22683 |! .47021 | 2.12671 || .49170 | 2.03376 || .513856 | 1.94718 | 49 
 2) .44942 | 2.22510 | .47056 | 2.12511 || .49206 | 2.08227 || .51393 | 1.94579 |48 
 13) .44977 | 2.22337 || .47092 | 2.12350 || .49242 | 2.08078 || .51430 | 1.94440 | 47 | 
 14| .45012 | 2.22164 || .47128 | 2.12190 || .49278 | 2.02929 || .51467 | 1.94301 |46 
 15| .45047 | 2.21992 || .47163 | 2.12030 || .49315 | 2.02780 || .51503 | 1.94162 | 45 
 16| .45082 | 2.21819 || .47199 | 2.11871 || .49351 | 2.02681 || .51540 | 1.94023 | 44 
 17| .45117 | 2.21647 || .47234 | 2.11711 || .49387 | 2.02483 || .51577 | 1.98885 |43 
 18) .45152 | 2.21475 || .47270 | 2.11552 || .49423 | 2.02385 || .51614 | 1.93746 | 42 
 19| .45187 | 2.21304 || .47805 | 2.11892 || .49459 | 2.02187. || .51651 | 1.93608 | 41 
 20| .45222 | 2.21182 || .47841 | 2.11288 || .49495 | 2.02039 || .51688 | 1.93470 | 40 
 21| .45257 | 2.20961 || .473877 | 2.11075 || .49532 | 2.01891 || .51724 | 1.98332 |39 
 92] .45292 | 2.20790 || .47412 | 2.10916 || .49568 | 2.01743 || .51761 | 1.93195 | 38 | 
 93] .45327 | 2.20619 || .47448 | 2.10758 || .49604 | 2.01596 || .51798 | 1.93057 | 37 
 24| .45362 | 2.20449 || .47483 | 2.10600 || .49640 | 2.01449 || .51835 | 1.92920 | 36 
 25] .45397 | 2.20278 || .47519 | 2.10442 || .49677 | 2.01302 || .51872 | 1.92782 |35 
 26| .45432 | 2.20108 || .47555 | 2.10284 || .49713 | 2.01155 || .51909 | 1.92645 | 34 
 27| .45467 | 2.19938 || .47590 | 2.10126 || .49749 | 2.01008 || .51946 } 1.92508 | 33 
 28} .45502 | 2.19769 .47626 | 2.09969 || .49786 | 2.00862 |} .51983 | 1.92871 |32 
 29| .45538 | 2.19599 47062 | 2.09811 || .49822 | 2.00715 .52020 | 1.922385 131 
 30| .45573 | 2.19430 || .47698 | 2.09654 || .49858 | 2.00569 |} .52057 | 1.92098 | 30 
 31| .45608 | 2.19261 || .47783 | 2.09498 || .49894 | 2.00423 || .52094.| 1.91962 |29 
 32] .45643 | 2.19092 || .47769 | 2.09841 || .49931 | 2.00277 || .52131 | 1.91826 |28 
 83| .45678 | 2.18928 47805 | 2.09154 .49967 | 2.00131 .52168 | 1.91690 |27 
 84] .45713 | 2.18755 || .47840 | 2.09028 || .50004 | 1.99986 |} .52205 | 1.91554 | 26 
 85! .45748 | 2.18587 || .47876 | 2.08872 || .50040 | 1.99841 || .52242 | 1.91418 | 25 
 36| .45784 | 2.18419 || .47912 | 2.08716 || .50076 | 1.99695 || .52279 | 1.91282 | 24 
 | .45819 | 2.18251 .47948 | 2.08560 |} .50118 | 1.99550 52316 | 1.91147 | 2¢ 
 88) .45854 | 2.18084 .47984 | 2.08405 || .50149 | 1.99406 52353 | 1.91012 | 2; 
 39] .45889 | 2.17916 .48019 | 2.08250 .50185 | 1.99261 .52390 | 1.90876 | 21 
 40| .45924 | 2.17749 || .48055 | 2.08094 |} .50222 | 1.99116 || .52427 | 1.90741 |2 
 41| .45960 | 2.17582 || .48091 | 2.07989 || .50258 | 1.98972 || .52464 | 1.90607 | 19 
 42| .45995 | 2.17416 || .48127 | 2.07785 || .50295 | 1.98828 || .52501 | 1.90472 |18 
 43| .46030 | 2.17249 || .48163 | 2.07630 || .50331 | 1.98684 || .52538 | 1.90337 |17 
 44| .46065 | 2.17083 || .48198 | 2.07476 || .50368 | 1.98540 || .52575 | 1.90203 | 16 | 
 45| .46101 | 2.16917 || .48234 | 2.07821 || .50404 | 1.98396 || .52613 | 1.90069 |15 
 46| .46186 | 2.16751 || .48270 | 2.07167 || .50441 | 1.98253 || .52650 | 1.89935 | 14 
 47| .46171 | 2.16585 || .48306 | 2.07014 |} .50477 | 1.98110 || .52687 | 1.86801 |13 
 48) .46206 | 2.16420 | .48342 | 2.06860 || .50514 | 1.97966 || .52724 | 1.89667 | 12 
 49| .46242 | 2.16255 || .48378 | 2.06706 || .50550 | 1.97823 |} .52761 | 1.89533 |11 [ 
 50! .46277 | 2.16090 || .48414 | 2.06553 || .50587 | 1.97681 |} .52798 | 1.89400 | 10 
 51! .46312 | 2.15925 || .48450 | 2.06400 || .50623 | 1.97538 || .52836 | 1.89266, | 9 
 2\ .46348 | 2.15760 || .48486 | 2.06247 |) .50660 | 1.97395 || .52873 | 1.89133 | 8 
 53| .46383 | 2.15596 || .48521 | 2.06094 || .50696 | 1.97253 || .52910 | 1.89000 | 7 
 54| .46418 | 2.15432 || .48557 | 2.05942 || .50733 | 1.97111 || .52947 | 1.88867 | 6 
 55| .46454 | 2.15268 || .48593 | 2.05790 |} .50769 | 1.96969 || .52985 | 1.88734 | 5 
 56) .46489 | 2.15104 || .48629 | 2.05637 .50806 | 1.96827 || .58022 | 1.88602 | 4 
 57| .46525 | 2.14940 || .48665 | 2.05485 || .50843 | 1.96685 |) .53059 | 1.88469 | 3 
 58| .46560 | 2.14777 || .48701 | 2.05333 || .50879 | 1.96544 || .58096 1.88837 | 2 
 59| .46595 | 2.14614 || .48737 | 2.05182 |; .50916 | 1.96402 || .53134 | 1.88205 | 1 
 60| 46631 | 2.14451 || .48773 | 2.05030 |; .50953 | 1.96261 || .53171 1.88073 | 0 
 Cotang| Tang ||Cotang; Tang |,Cotang| Tang |/Cotang Tang 
 ; | a 
 65° 64° [yw ABP 62° | 
 
COTANGENTS. 
 
 TABLE XXVIIL—NATURAL TANGENTS AND 
 |, | 28° \| 29° 30° 
 | Tang | Cotang | Tang | Cotang |! Tang | Cotang || 
 0} .58171 | 1.88073 |) .55431 | 1.80405 | 57785 | 1.73205 | 
 1} .53208 | 1.87941 || .55469 | 1.80281 || .57774 | 1.73089 | 
 2) .53246 | 1.87809 || .55507 | 1.80158 |; .57813 | 1.72973 
 3| .53283 | 1.87677 || .55545 | 1.80084 || .57851 | 1.72857 | 
 4| .538320 | 1.87546 || .55583 | 1.79911 || .57890 | 1.72741 
 5} .58858 | 1.87415 || .55621 | 1.79788 || .57929 | 1.72625 
 6| .533895 | 1.87283 || .55659 | 1.79665 |; .57968 | 1.72509 
 7| .584382 | 1.87152 |} .55697 | 1.79542 |; .58007 | 1.72893 
 8| .53470 | 1.87021 || .557386 | 1.79419 || .58046 | 1.72278 
 9} .538507 | 1.86891 || .55774 | 1.79296 || .58085 | 1.72168 
 10| .53545 | 1.86760 || .55812 | 1.79174 |, .58124 | 1.72047 
 11} .53582 | 1.86630 || .55850 | 1.79051 || .58162 | 1.71932 | 
 12} .53620 | 1.86499 || .55888 | 1.78929 || .58201 | 1.71817 
 18| .538657 | 1.86869 |; .55926 | 1.78807 |! .58240 | 1.71702 
 14} .53694 | 1.86289 || .55964 | 1.78685 || .58279 | 1.71588 
 15| .53732 | 1.86109 || .56003 | 1.78563 || .583818 | 1.71473 
 16} .538769 | 1.85979 |; .56041 | 1.78441 || .58857 | 1.71358 | 
 17| .53807 | 1.85850 || .56079 | 1.78319 || .583896 | 1.71244 
 18| .53844 | 1.85720 || .56117 | 1.78198 |! .58435 | 1.71129 
 19} .58882 | 1.85591 || .56156 | 1.78077 || .58474 | 1.71015 
 20| .53920 | 1.85462 |} .56194 | 1.77955 |; .58513 | 1.70901 
 21| .53957 | 1.85333 || .56282 | 1.77834 || .58552 | 1.70787 
 22| .538995 | 1.85204 || .56270 | 1.77713 || .58591 | 1.70673 
 23| .54032 | 1.85075 || .56809 | 1.77592 || .58631 | 1.70560 
 24| .54070 | 1.84946 || .56347 | 1.77471 || .58670 | 1.70446 
 25; .54107 | 1.84818 || .563885 | 1.77351 || .58709 | 1.70382 
 25} 54145 | 1.84689 || .56424 | 1.77280 || .58748 | 1.70219 
 27| .54183 | 1.84561 || .56462 | 1.77110 |) .58787 | 1.70106 
 28} .54220 | 1.84433 || .56501 | 1.76990 |; .58826 | 1.69992 
 29! .54258 | 1.84305 || .56639 | 1.76869 || .58865 | 1.69879 
 80} .54296 | 1.84177 |} .56577 | 1.76749 |; .58905 | 1.69766 
 31| .54333 | 1.84049 || .56616 | 1.76629 || .58944 | 1.69653 
 82| .54371 | 1.838922 || .56654 | 1.76510 || .58983 | 1.69541 
 33| .54409 | 1.88794 || .56693 | 1.76390 |! .59022 | 1.69428 
 84| .54446 | 1.83667 || .56731 | 1.76271 |: .59061 | 1.69316 
 85| .54484 | 1.83540 || .56769 | 1.76151 |; .59101 | 1.69203 
 36| .54522 | 1.838413 |; .56808 | 1.76032 |; .59140 | 1.69091 
 87| .54560 | 1.83286 || .56846 | 1.75913 || .59179 | 1.68979 
 38|' 54597 | 1.83159 || .56885 | 1.75794 || .59218 | 1.68866 
 39| .54635 | 1.83033 || .56923 | 1.75675 |’ .59258 | 1.68754 
 40} .54673 | 1.82906 || .56962 | 1.75556 |: .59297 | 1.68643 
 41; .54711 | 1.82780 || .57000 | 1.75487 |' .59336 | 1.68531 
 42| .54748 | 1.82654 || .57039 | 1.75319 || .59376 | 1.68419 
 43| .54786 | 1.82528 || .57078 | 1.75200 || .59415 | 1.68308 
 44} .54824 | 1.82402 |; .57116 | 1.75082 |; .59454 | 1.68196 
 45| .54862 | 1.82276 || .57155 | 1.74964 || .59494 | 1.68085 
 46| .54900 | 1.82150 || .57193 | 1.74846 || .59533 | 1.67974 
 47| .54938 | 1.82025 || .57232 | 1.74728 | .59573 | 1.67863 
 48; .54975 | 1.81899 || .57271 | 1.74610 |! .59612 | 1.67°752 
 49| .55013 | 1.81774 || .57309 | 1.74492 | .59651 | 1.67641 
 50) 55051 | 1.81649 || .57348 | 1.74375 |; .59691 | 1.67530 
 51! .55089 | 1.81524 || .57886 | 1.74257 || .59730 | 1.67419 
 521 .55127 | 1.81399 || .57425 | 1.74140 || .59770 | 1.67309 
 53: 255165 | 1.8127 57464 | 1.74022 || .59809 | 1.67198 
 | 54| .55203 | 1.81150 |! .57503 | 1.73905 || .59849 | 1.67088 
 55| .55241 | 1.81025 || .57541 | 1.73788 || .59888 | 1.66978 
 56| .55279 | 1.80901 || .57580 ; 1.73671 |! 159928 | 1.66867 
 57| .55317 | 1.80777 || .57619 | 1.78555 |! .59967 | 1.66757 
 58) .55855 | 1.80658 | .57657 | 1.73438 .60007 | 1.66647 
 59| 55393 | 1.80529 || .57696 | 1.733821 60046 | 1.66538 
 60) .55431 | 1.80405 || .57785 | 1.78205 | .60086 | 1.66428 
 \Cotang| Tang |,Cotang: Tang | Cotang| Tang 
 , | | 
 
 61° 
 
 EEE Ee eRe cee OR ee RR ee ss 
 465 
 
 60° 
 
 59° 
 
 58° 
 
 31° ; 
 Tang | Cotang 
 60086 | 1.66428 | 60 
 .60126 | 1.66318 | 59 
 60165 | 1.66209 |58 
 .60205 | 1.66099 | 57 
 60245 | 1.65990 | 56 
 60284 | 1.65881 |55 
 60824 | 1.65772 | 54 
 .60364 | 1.65663 |53 
 60403 | 1.65554 | 52 
 60448 | 1.65445 |51 
 .60483 | 1.65337 | 50 
 60522 | 1.65228 | 49 
 60562 | 1.65120 | 48 
 60602 | 1.65011 | 47 
 .60642 | 1.64903 | 46 
 .60681 | 1.64795 | 45 
 60721 | 1.64687 | 44 
 60761 | 1.64579 | 43 
 60801 | 1.64471 | 42 
 60841 | 1.64363 | 41 
 60881 | 1.64256 | 40 
 60921 | 1.64148 |39 
 .60960 | 1.64041 |38 
 .61000 | 1.63934 | 37 
 .61040 | 1.63826 |36 
 61080 | 1.63719 | 35 
 .61120 | 1.63612 | 34 
 .61160 | 1.63505 | 33 
 .61200 | 1.63898 | 32 
 61240 | 1.63292 | 31 
 61280 | 1.63185 | 30 
 .61820 | 1.63079 | 29 
 .61860 | 1.62972 | 28 
 .61400 | 1.62866 | 27 
 .61440 | 1.62760 | 26 
 61480 | 1.62654 |25 
 61520 | 1.62548 |24 
 .61561 | 1.62442 |23 
 61601 | 1.62336 | 22 
 .61641 | 1.62230 | 21 
 .61681 | 1.62125 | 20 
 61721 | 1.62019 |19 
 61761 | 1.61914 |18 
 61801 | 1.61808 |17 
 61842 | 1.61703 |16 
 61882 | 1.61598 | 15 
 61922 | 1.61493 |14 
 .61962 | 1.61888 |13 
 62003 | 1.61283 112 
 62043 | 1.61179 |11 
 62083 | 1.61074 | 10 
 .62124 | 1.60970 | 9 
 62164 | 1.60865 | 8 
 62204 | 1.60761 | 7 
 .62245 | 1.60657 | 6 
 62285 | 1.60553 | 5 
 62325 | 1.60449 | 4 
 62366 | 1.60845 | 3 
 62406 | 1.60241 | 2 
 62446 | 1.60137 | 1 
 .62487 | 1.60033 | 0 
 
 Cotang; Tang 
 
1 
 | 
 | 
 
 TABLE XXVIIIL_—NATURAL TANGENTS AND COTANGENTS., 
 
 39° 
 
 li 54° 
 
 “4 | $36 84° T 35° 
 Tang | Cotang || Tang | Cotang || Tang | Cotang : Tang | Cotang 
 0| .62487 | 1.60033 || .64941 | 1.53986 .67451 | 1.48256 -(0021 | 1.42815 | 60 
 1) .62527 | 1.59930 ||} .64982 | 1.53888 67493 | 1.48163 || .70064 | 1.42796 | 59 
 2| .62568 | 1.59826 || .65024 | 1.53791 .67536 | 1.48070 -(0107 | 1.42638 |58 
 3} .62608 | 1.59723 || .65065 | 1.53693 67578 | 1.47977 || 270151 | 1.42550 |57 
 4; .62649 | 1.59620 .65106 | 1.53595 -67620 | 1.47885 || .70194 | 1.42462 156 
 5] .62689 | 1.59517 || .65148 | 1.53497 || “67663 1.47792 || .70288 | 1.49874 |55 
 6} .62730 | 1.59414 || .65189 | 1.53400 || .67705 | 1.47699 || “70281 1.42286 | 54 
 “| .62770 | 1.59311 65231 | 1.53802 || .67748 | 1.47607 || .70325 | 1.42198 | 53 
 8} .62811 | 1.59208 65272 | 1.58205 67790 | 1.47514 || .70868 | 1.42110 152 
 9} .62852 | 1.59105 65814 | 1.58107 67832 | 1.47422 || .70412 | 1.42022 | 51 
 10} .62892 | 1.59002 65355 | 1.58010 || .67875 | 1.47330 || 70455 | 1.41934 |50 
 11} .62933 | 1.58900 65397 | 1.52913 | .67917 | 1.47238 || .70499 | 1.41847 | 49 
 12} .62973 | 1.58797 65438 | 1.52816 | .67960 | 1.47146 || .70542 | 1.41759 |48 
 13} .63014 | 1.58695 65480 | 1.52719 | .68002 | 1.47053 || .70586 | 1.41672 | 47 
 14| .68055 | 1.58593 65521 | 1.52622 68045 | 1.46962 || .70629 | 1.41584 | 46 
 15| .63095 | 1.58490 || .65563 | 1.52525 .68088 | 1.46870 (0673 | 1.41497 | 45 
 16| .63136 | 1.58388 | .65604 | 1.52429 .68130 | 1.46778 .VOT17 | 1.41409 | 44 
 17; .63177 | 1.58286 65646 | 1.52332 68173 | 1.46686 || .70760 | 1.41322 | 43 
 18} .68217 | 1.58184 65688 | 1.52235 | .68215 | 1.46595 || 70804 | 1.41935 | 49 
 19; 63258 | 1.58083 || .65729 | 1.52139 || .68958 | 1.46503 70848 | 1.41148 | 41 
 20; .638299 | 1.57981 65771 | 1.52043 | .68301 | 1.46411 |) .70891 | 1.41061 | 40 
 21] .63340-| 1.57879 |) .65813 | 1.51946 ‘| 68343 | 1.46320 70935 | 1.40974 |39 
 22) .63330 | 1.57778 || .65854 | 1.51850 || |68886 | 1.46299 70979 | 1.40887 | 38 
 23| .63421 | 1.57676 |) 165896 | 1.51754 || -68429 | 1.46137 -71023 | 1.40800 | 37 
 24) .63462 | 1.57575 || .65938 | 1.51658 || .68471 | 1.46046 -71066 | 1.40714 | 36 
 25| .63503 | 1.57474 || .65980 | 1.51562 || .68514 | 1.45955 || .71110 | 1.40697 135 
 26} .68544 | 1.57372 .66021 | 1.51466 || .68557 | 1.45864 71154 | 1.40540 | 34 
 27} .63584 | 1.57271 |) .66063 | 1.51370 |} -68600 | 1.45773 |! .71198 | 1.40454 |33 
 28) .63625 | 1.57170 || .66105 | 1.51275 || .68642 | 1.45689 71242 | 1.40367 | 32 
 29] .63666 | 1.57069 || .66147 | 1.51179 || .68685 | 1.45599 71285 | 1.40281 | 31 
 30} .63707 | 1.56969 || .66189 | 1.51084 || /68728 | 1.45501 71829 | 1.40195 | 80 
 31} .63748 | 1.56868 || .66230 | 1.50988 || .68771 | 1.45410 || .71873 | 1.40109 |29 
 32} .63789 | 1.56767 || .66272 | 1.50893 .68814 | 1.45320 71417 | 1.40022 | 28 
 33} .63830 | 1.56667 || .66314 | 1.50797 || |68857 | 1.45999 71461 | 1.389936 | 27 
 34| .63871 | 1.56566 |} .66356 | 1.50702 || .68900 | 1.45189 || .71505 | 1.39850 |26 
 35} .63912 | 1.56466 || .66398 | 1.50607 || 68942 | 1.45049 71549 | 1.39764 | 25 
 36} .63953 | 1.56366 | .66440 | 1.50512 || .68985 | 1.44958 71593 | 1.39679 | 24 
 37| .68994 | 1.56265 || .66482 | 1.50417 || /¢9028 | 1.44868 71637 | 1.389593 | 23 
 38| .64035 | 1.56165 || .66524 | 1.50322 || 69071 | 1 44778 71681 | 1.39507 | 22 
 39} .64076 | 1.56065 || .66566 | 1.50228 |} 69114 | 1.44688 11725 | 1.39421 | 21 
 40) .64117 | 1.55966 || .66608 | 1.50138 || 69157 | 4 44598 || .71769 | 1.39336 | 20 
 41} .64158 | 1.55866 |} .66650 | 1.50038 || .69200 | 1.44508 71813 | 1.89250 | 19 
 42) .64199 | 1.55766 || .66692 | 1.49944 .69243 | 1.44418 71857 | 1.39165 |18 
 43} .64240 | 1.55666 || .66734 | 1.49849 || |69286 | 1144329 71901 | 1.39079 |17 
 44) .64281 | 1.55567 || .66776 | 1.49755 || 69399 | 1.44939 71946 | 1.38994 |16 
 45} .64322 | 1.55467 || .66818 | 1.49661 || 69372 | 1144149 71990 | 1.38909 | 15 
 46| .64363 | 1.55368 || .66860 | 1.49566 || |69416 | 1.44060 (2084 | 1.38824 | 14 
 47| 64404 | 1.55269 |! .66902 | 1.49472 || |69459 | 1.43970 | .@2078 | 1.387388 |13 
 48| .64446 | 1.55170 || .66944 | 1.49378 || |69502 | 1.43881 || 12122 | 1.38653 | 12 
 49) .64487 | 1.55071 |) .66986 | 1.49284-|| 169545 | 1.43792 72167 | 1.38568 | 11 
 50| .64528 | 1.54972 || .67028 | 1.49190 i} .69588 | 1.437038 72211 | 1.388484 | 10 
 51} .64569 | 1.54873 | .67071 | 1.49097 | .69631 | 1.43614 || .7@2255 | 1.38399 | 9 
 o2| 64610 | 1.54774 |, .67113 | 1.49003 || /69G75 | 1.43595 || .72299 | 1.88314 | 8 
 53| .64652 | 1.54675 .67155 | 1.48909 69718 | 1.43486 || .72844 | 1.38229 | 7 
 d4| .64693 | 1.54576 .67197 | 1.48816 || .69761 | 1.43347 | .72888 | 1.38145 | 6 
 50| .64734 | 1.54478 .67239 | 1.48722 69804 | 1.43258 || 72432 | 1.38060 | 5 
 56} .64775 | 1.54879 67282 | 1.48629 69847 | 1.43169 || .72477 | 1.87976 | 4 
 O7) .G4817 | 1.54281 |) .67824 | 1.48536 || |c9s91 | 1.43080 || .@2521 | 1.37891 | 8 
 103! .64858 | 1.54183 .67366 | 1.48442 69934 | 1.42992 || .72565 | 1.37807 | 2 
 59} .64899 | 1.54085 67409 | 1.48349 || .69977 | 1.42903 || .72610 1377221 
 | 60| .64941 | 1.53986 67451 | 1.48256 || .70021 | 1.42815 72654 | 1.387688 | 0 
 : Cotang; Tang | Cotang| Tang || Cotang | Tang ||Cotang| Tang | 
 57° 56° 55° 
 
TABLE XXVIII.—_NATURAL TANGENTS AND COTANGENTS. 
 
 ~ 
 
 SOA mmwwHol 
 
 = 
 cose) 
 
 alll aarti amet ceed coeealll come 
 IRON WCOre 
 
 | Tang 
 
 36° 
 72654 | 1 
 72699 | 1 
 
 Cotang 
 
 "1.37638 | 
 87554 
 
 37° 
 
 38° 
 
 3 
 
 ge 
 
 Tang 
 
 5 | 1.32704 
 
 53° 
 
 | 
 
 | 
 { 
 
 5 1 
 .72743 | 1.37470 | 5447 | 1.382514 | 
 (2788 | 1.37386 | 5492 | 1.32464 | 
 72832 | 1.37302 || .75538 | 1.82384 | 
 £72877 | 1.87218 || .75584 | 1.382804 
 72921 | 1.37184 || .75629 | 1.32224 
 .72966 | 1.37050 || .75675 | 1.32144 
 .738010 | 1.86967 || .7572 1.32064 | 
 .73055 | 1.36883 || .75767 | 1.31984 
 .73100: | 1.36800 || .75812 | 1.31904 
 78144 | 1.36716 || .75858 | 1.31825 | 
 .73189 | 1.86633 || .75904 | 1.81745 
 78234 | 1.36549 || .75950 | 1.31666 
 73278 | 1.36466 || .75996 | 1.31586 
 .738323 | 1.36383 || .76042 | 1.31507 
 43368 | 1.36300 || .76088 | 1.31427 
 .78413 | 1.86217 || .761384 | 1.381848 
 T3457 | 1.36134 |; .76180 | 1 31269 
 .73502 | 1.36051 || .76226 | 1.31190 
 73547 | 1.35968 || .16272 | 1.31110 
 73592 | 1.385885 || .763818 | 1.31031 
 .73637 | 1.35802 || .76364 | 1.80952 
 . 13681 | 1 || .76410 | 1.30873 
 13726 | 1 ‘| .76456 | 1.30795 
 W377 1.355 . 76502 | 1.30716 
 T3816 | 1 | .76548 | 1.30637 
 73861 | 1 76594 | 1.380558 
 | .73905 | 1 .76640 | 1.380480 
 | .%5951 | 1 . 76686 | 1.80401 
 73996 | 1 76733 | 1.80323 
 - 74041 | 1 76779 | 1.80244 
 .74086 | 1 .76825 | 1.30166 
 .74131 | 1 (6871 | 1.380087 
 .T4176 | 1 .76918 | 1.380009 
 (4221 | 1 76964 | 1.29931 
 14267 | 1 .77010 | 1.29853 
 .14312 | 1 T1057 | 1.29775 
 |) .74357 | 1 .77103 | 1.29696 
 .74402 | 1 77149 | 1.29618 
 74447 | 1 || 77196 | 1.29541 
 74492 | 1 77242 | 1.29463 
 -74588 | 1 77289 | 1.29885 
 74583 | 1. | 77335 | 1.29807 
 74628 | 1.33998 || 77382 | 1.29229 
 | .74674 | 1.83916 || .77428 | 1.29152 
 74719 | 1.388835 || .77475 | 1.29074 
 74764 | 1.338754 || .77521 | 1.28997 
 .74810 | 1.33673 || .77568 | 1.28919 
 | .74855 | 1.388592 || . 77615 1.28842 
 | .74900 | 1.38511 || .77661 | 1.28764 
 74946 | 1.33430 || .77708 | 1.28687 
 “74991 | 1.383849 || . 77754 | 1.28610 
 75037 | 1.38268 .77801 | 1.28583 
 75082 | 1.388187 || .77848 | 1.28456 
 75128 | 1.383107 || .77895 | 1.28379 
 75173 | 1.33026 || .77941 | 1.28302 
 75219 | 1.382946 || .77988 | 1.28225 
 75264 | 1.32865 | 78035 | 1.28148 
 75310 | 1.382785 . 78082 | 1.28071 
 75355 | 1.32704 || .78129 | 1.27994 
 \Cotang| Tang |\Cotang| Tang 
 
 52° 
 
 | Cotang 
 
 82624 | 
 
 || .78129 
 8175 
 
 78269 
 78316 
 78363 
 .78410 
 18457 
 . 78504 
 || .78551 
 78598 
 78645 
 (8692 
 (8039 
 78786 
 . 78834 
 18881 
 78928 
 68975 
 T9022 
 79070 | 
 TOIL? 
 79164 
 (9212 
 79259 
 .79306 
 19354 
 79401 
 T9449 
 79496 
 
 79591 
 79639 
 79686 
 T9734 
 9781 
 79829 
 1987 
 79924 
 79972 
 .80020 
 
 80067 
 80115 
 .80163 
 | .80211 
 
 80258 
 .80306 
 
 | Cotang 
 
 1 
 1 
 (8222 | 1.8 
 i 
 1 
 1 
 
 | Tang | Cotang 
 ~ 78129 | 1.27994 
 
 20917 
 
 27841 
 
 27764 
 .27688 
 27611 
 27589 
 27458 | 
 27382 
 27806 
 27230 
 
 271538 
 24077 
 27001 
 26925 
 26849 
 2677 
 
 26698 
 26622 
 26546 
 
 26471 
 26395 
 26319 
 26244 
 26169 
 26093 | 
 26018 | 
 25943 
 25867 
 
 95792 
 
 we 
 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 i 
 1 
 1 
 if 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 79544 | 1, 
 a 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 i 
 1 
 1 
 
 25717 
 
 25642 
 25567 
 25492 | 
 25417 
 25343 
 
 25268 
 25193 
 
 25118 
 25044 | 
 24969 
 
 24895 
 24820 
 24746 | 
 24672 
 (24597 
 24593 
 
 80854 24449 
 80402 24375 
 80450 24301 
 80498 -2ARR7 
 80546 .24153 
 /80594 24079 
 
 * 80642 .24005 
 .80690 | 1.28931 
 .80738 | 1.23858 
 
 |] .80786 | 1.23784 
 || .80884 | 1.28710 
 .80882 | 1.238637 
 
 | ,80930 | 1.23563 
 || .80978 | 1.23490 
 Tang 
 
 | 
 
 wha 
 
 8277 
 
 82825 
 82874 
 82923 
 .82972 
 83022 
 .83071 
 .83120 
 .83169 
 83218 
 .83268 
 83317 
 83366 
 83415 
 83465 
 83514 
 83564 
 .83613 
 88662 
 88712 
 83761 
 83811 
 
 -83860 
 
 83910 
 
 iCotang | Tang 
 
 50° 
 
 1 
 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 al 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 
 1 
 18, 
 1 
 
 | Tang | Cotang | 
 80978 | 
 .81027 | 
 .81075 
 .81123 
 .81171 
 
 .81220 
 
 .81268 
 .81316 
 
 .81364 
 .814138 
 .81461 
 
 .81510 
 81558 
 81606 
 81655 
 81708 
 81752 
 | .81800 
 
 | .81849 
 
 81898 
 81946 
 
 81995 
 “82044 | 
 82092 
 2141 
 82190 
 82228 
 82287 
 82336 
 82385 
 82434 
 82483 
 82531 
 82580 
 82629 
 82678 
 82727 
 
 / 
 
 .238490 | 60 
 23416 | 59 
 23343 | 58 
 23270 | 57 
 23196 | 56 
 23123 | 5D 
 .23050 | 54 
 22977 [53 
 23904 | 52 
 22831 | 51 
 22758 | 50 
 22685 | 49 
 22612 | 48 
 22589 | 47 
 22467 | 46 
 22394 | 45 
 22321 | 44 
 22249 | 43 
 22176 | 42 
 22104 | 41 
 .23031 | 40 
 21959. |39 
 21886 | 38 
 21814 |37 
 21742 | 36 
 21670 [35 
 21598 | 34 
 .21526 | 33 
 .21454 |32 
 .21382 | 31 
 .21810 | 30 
 21238 | 29 
 21166 | 28 
 21094 | 27 
 21023 | 26 
 20951 | 25 
 20879 |2 
 
 20808 | 2 
 
 20736 | 22 
 20665 | 21 
 .20593 | 20 
 20522 | 19 
 20451 |18 
 20379 |17 
 20308 | 16 
 20237 |15 
 20166 |14 
 20095 | 13 
 20024 | 12 
 19958 | 11 
 19882 | 10 
 .19811 | 9 
 19740 | 8 
 19669 | 7 
 -19599 | 6 
 19528 | 5 
 19457 | 4 
 19387 | 38 
 19316 | 2 
 19246 | 1 
 ‘19175 | 0 
 
= 
 
 DHNIOMBwWoHO! 
 
 TABLE XXVITI.—NATURAL TANGENTS AND COTANGENTS. 
 
 40° 41° | 42° | 43° 
 Tang | Cotang || Tang | Cotang |} Tang | Cotang | Tang | Cotang H 
 -83910 | 1.19175 || .86929 | 1.15037 || .90040°| 1.11061 || .938252 | 1.07237 | 60 | 
 -83960 | 1.19105 |} .86980 | 1.14969 || .90093 | 1.10996 || .93806 | 1.07174 |59 
 .84009 | 1.19035 ||} .87031 | 1.14902 |} .90146 | 1.109381 | ,933860 | 1.07112 |58 
 .84059 | 1.18964 .87082 | 1.14834 .90199 | 1.10867 || .93415 | 1.07049 | 57 
 .84108 | 1.18894 .87133 | 1.14767 .90251 | 1.10802 || .93469 | 1.06987 156 
 .84158 | 1.18824 .87184 | 1.14699 .90304 | 1.10737 || .93524 | 1.06925 155 
 .84208 | 1.18754 87236 | 1.146382 .90357 | 1.10672 .93578 | 1.06862 | 54 
 -84258 | 1.18684 || .87287 | 1.14565 || .90410 | 1.10607 || .93633 | 1.06800 |53 
 84307 | 1.18614 || .87338 | 1.14498 || .90463 | 1.10543 || .93688 | 1.06738 |52 
 .84357 | 1.18544 .87389 |; 1.14430 .90516 | 1.10478 .93742 | 1.06676 | 51 
 .84407 | 1.18474 || .87441 | 1.14363 || .90569 | 1.10414 .93797 | 1.06613 |50 
 84457 | 1.18404 || .87492 | 1.14296 || .90621 | 1.10349 | .93852 | 1.06551 | 49 
 84507 | 1.18334 || .87543 | 1.14229 || .90674 | 1.10285 |; .93906 | 1.06489 | 48 
 84556 | 1.18264 || .87595 | 1.14162 || .90727 | 1.10220 || .93961 | 1.06427-| 47 
 -84606 | 1.18194 |) .87646 | 1.14095 || .90781 | 1.10156 || .94016 | 1.06365 | 46 
 -84656 | 1.18125 |; .87698 | 1.14028 || .90834 | 1.10091 |! .94071 | 1.06303 | 45 
 -84706 | 1.18055 || .87749 | 1.13961 || .90887 | 1.10027 || .94125 | 1.06241 |44 
 .84756 | 1.17986 || .87801 | 1.18894 || .90940 | 1.09963 |! .94180 | 1.06179 | 43 
 .84306 | 1.17916 || .87852 | 1.13828 || .90993 | 1.09899 || .94235 | 1.06117 | 42 
 84856 | 1.17846 || .87904 | 1.13761 || .91046 | 1.09834 || .94290 | 1.06056 | 41 
 .84906 + 1.1777’ 87955 | 1.18694 || .91099 | 1.09770 || .94345 | 1.05994 | 40 
 84956 | 1.17708 || .88007 | 1.13627 || .91153 | 1.09706 || .94400 | 1.05932 | 39 
 -85006 | 1.17638 || .88059 | 1.13561 || .91206 | 1.09642 || .94455 | 1.05870 |38 
 -85057 | 1.17569 || .88110 | 1.18494 |} .91259 | 1.09578 || .94510 | 1.05809 | 37 
 85107 | 1.17500 || .88162 | 1.13428 || .91313 | 1.09514 || .94565 | 1.05747 136 
 85157 | 1.17430 || .88214 | 1.18361 || .91366 | 1.09450 || .94620 | 1.05685 135 
 85207 | 1.17361 || .88265 | 1.18295 || .91419 | 1.09386 || .94676 | 1.05624 | 34 
 85257 | 1.17292 || .88317 | 1.13228 || .91473 | 1.09322 || .94731 | 1.05562 | 33 
 -85308 | 1.17223 || .88369 | 1.18162 |} .91526 | 1.09258 || .94786 | 1.05501 | 32 
 .85358 | 1.17154 || .88421 | 1.13096 || .91580 | 1.09195 || .94841 | 1.05439 | 31 
 85408 | 1.17085 || .88473 | 1.13029 |} .91633 | 1.09131 || .94896 | 1.05378 |30 
 -85458 | 1.17016 || .88524 | 1.12963 || .91687 | 1.09067 || .94952 | 1.65317 |29 
 85509 | 1.16947 |) .88576 | 1.12897 || .91740 | 1.09003 |! -.95007 | 1.05255 |28 
 .85559 | 1.16878 || .88628 | 1.12831 || .91794 | 1.08940 || .95062 | 1.05194 | 27 
 -85609 | 1.16809 || .88680 | 1.12765 || .91847 | 1.08876 || .95118 | 1.05133 | 26 
 85660 | 1.16741 || .88782 | 1.12699 || .91901 | 1.08813 || .95173 | 1.05072 | 25 
 85710 | 1.16672 || .88784 | 1.12633 || .91955 | 1.08749 |) .95229 | 1.05010 |24 
 .85761 | 1.16603 || .88836 | 1.12567 || .92008 | 1.08686 || .95284 | 1.04949 |93 
 -85811 | 1.16535 || .88888 | 1.12501 |/ .92062 | 1.08622 || .95340 | 1.04888 | 22 
 .85862 | 1.16466 || .88940 | 1.12435 || .92116 | 1.08559 || .95395 | 1.04897 |91 
 85912 | 1.16398 |; .88992 | 1.12369 || .92170 | 1.08496 || .95451 | 1.04766 |20 
 -85963 | 1.16329 || .89045 | 1.12303 || .92224 | 1.08432 || .95506 | 1.04705 |19 
 86014 | 1.16261 |} .89097 | 1.12238 || .92277 | 1.08369 || .95562 | 1.04644 |18 
 86064 | 1.16192 || .89149 | 1.12172 |) .92331 | 1.08306 || .95618 | 1.04583 |17 
 86115 | 1.16124 || .89201 | 1.12106 || .92385 | 1.08243 |) .95673 | 1.04522 116 
 86166 | 1.16056 || .89253 } 1.12041 || .92439 | 1.08179 || .95729 | 1.04461 |15 
 86216 | 1.15987 || .89306 | 1.11975 || .92493 | 1.08116 || .95785 | 1.04401 |14 
 86267 | 1.15919 || .89358 | 1.11909 || .92547 | 1.08053 || .95841 | 1.04340 113 
 .86318 | 1.15851 || .89410 | 1.11844 || .92601 | 1.07990 || .95897 | 1.04279 | 12 
 -86368 | 1.15783 || .89463 | 1 11778 || .92655 | 1.07927 | .95952 | 1.04918 111 
 86419 | 1.15715 || .89515 | 1.11718 || .92709 | 1.07864 || .96008 | 1.04158 |10 
 86470 | 1.15647 || .89567 | 1.11648 || .92763 | 1.07801 ‘| .96064 | 4.04097 | 9 
 86521 | 1.15579 |] .89620 | 1.11582.|| .92817 | 1.07738 || .96120 | 1.04036 | 8 
 86572 | 1.15511 || .89672 | 1.11517 |) .92°72 | 1.07676 || .96176 | 1.03976 | 7 
 .86623 | 1.15443 -89725 | 1.11452 || .92926 | 1.07613 .96232 | 1.08915 | 6 
 86674 | 1.153875 || .89777 | 1.11887 || .92980 | 1.07550 || .96288 | 1.08855 | 5 
 .86725 | 1.15308 .89830 | 1.11321 .93034 | 1.07487 || .96344 | 1.03794 | 4 
 .86776 | 1.15240 89883 | 1.11256 |} .93088 | 1.07425 .96400 | 1.03734 | 3 
 .86827 | 1.15172 || .89935 | 1.11191 .93143 | 1.07362 || .96457 | 1.03674 | 2 
 .86878 | 1.15104 .89988 | 1.11126 || .93197 | 1.07299 | .96513 | 1.03613 | 1 
 .86929 | 1.15037 || 90040 | 1.11061 || .93252 | 1.07237 .96569 | 1.03553 | 0 
 Cotang| Tang ||Cotang Tang | Cotang| Tang '|Cotang | Tang 
 
 49° 48° 46° 
 
TABLE XXVIIL—NATURAL TANGENTS AND COTANGENTS, 
 
 —— 
 44° 44° 44° 
 
 , / , ————————___—_—_—__—_—_| / / —————ES 
 Tang | Cotang | Tang | Cotang Tang | Cotang 
 
 0 | .96569 | 1.03553 | 60 || 20} .977 1.02355 | 40|} 40) .98843 | 1.01170 
 1 | 96625 | 1.03493 | 59 || 21} .97756 | 1.02295 | 39}; 41 .98901 | 1.01112 
 29 | 96681 | 1.03433 | 58 || 22| .97813 | 1.02236 | 38/|42| .98958 1.01053 
 3 | 96738 | 1.03372 | 57 || 23) .97870 | 1.02176 | 37|| 43} .99016 | 1.00994 
 4 | .96794 | 1.03312 | 56 || 24| .97927 | 1.02117 | 36|| 44} .99073 | 1.00935 
 
 | 5 | .96850 | 1.03252 | 55 |) 25 97984 | 1.02057 | 85 |} 45} .99181 | 1.00876 
 | 6 | .96907 | 1.03192 | 54 || 26 98041 | 1.01998 | 34|| 46] .99189 | 1.00818 
 | 7 | .96963 | 1.03182 | 53 || 27| .98098 1.01939 | 33]/47| .99247 | 1.00759 
 8 | .97020 | 1.03072 | 52 || 28) .98155 | 1.01879 | 32 || 48 | .99304 1.00701 
 9 | .97076 | 1.03012 | 51 || 29} .98213 | 1.01820 | 31 || 49 | .99362 | 1.00642 
 10 | .97133 | 1.02952 | 50 || 30| .98270 | 1.01761 | 30); 50] .99420 1.00583 
 11 | .97189 | 1.02892 | 49 || 81] .98327 | 1.01702 | 29))51| .99478 | 1.00525 
 12 | .97246 | 1.02832 | 48 || 32| .98384 | 1.01642 | 28||52| .99536 | 1.00467 
 13 | .97302 | 1.02772 | 47 || 33| .98441 | 1.01583 | 27||53| .99594 | 1.00408 
 14 | .97359 | 1.02713 | 46 || 34] .98499 | 1.01524 | 26|| 54) .99652 | 1.00350 
 15 | .97416 | 1.02653 | 45 || 35| .98556 | 1.01465 | 25||55) .99710 | 1.00291 
 16 | 97472 | 1.02593 | 44 || 36] .98613 | 1.01406 | 24||56| .99768 | 1.00233 
 17 | .97529 | 1.02533 | 43 || 37| .98671 | 1.01347 | 23||57 | .99826 | 1.00175 
 18 | .97586 | 1.02474 | 42 || 38) .98728 | 1.01288 22/58 | . 99884 1.00116 
 19 | 97643 | 1.02414 | 41 || 39| .98786 | 1.01229 | 21)/59| .99942 | 1.00058 
 20 | .977 1.02355 | 40 || 40} .98843 | 1.01170 | 20)) 60 | 1.00000 | 1.00000 
 
 \Cotang| Tang ; Cotang| Tang jes Cotang| Tang 
 / mets a SS eee 
 45° 45° 
 
 45° 
 
 | Se wmWROIMD-AIDOO 
 
 et OP 
 
.00061 
 
 ATO 
 
 Vers. |Ex.sec.|' Vers. |Ex.sec.|| Vers, |Ex. sec. 
 0 | .00000 | .00000 |} .00015 | .00015 .00061 | .00061 
 1 | .00000 | .00000 .00016 | .00016 .00062 | .00062 
 2 .00000 | .00000 .00016 | .00016 -Q0063. | .00063 
 38 | .00000 | .00000 -00017 | .00017 .00064 4. .00064 
 4 | .00000 | .00000 .00017 | .00017 || .00065 | .00065 
 5 | .00000 | .00000 .00018 | .00018 .00066 | .00066 
 6 | .00000 | .00000 .00018 | .00018 .00067 | .00067 
 7 | .00000 | .00000 .00019 | .00019 .00068 | .00068 
 8 | .00000 | .00000 |} .00020 | .00020 .00069 | .00069 | 
 9 | .00000 | .00000 .00020 | .00020 .00070 | .00070 | 
 10 | .00000 | .00000 .00021 | .00021 .00071 | .90072 | 
 .00001 | .00001 .00021 | .Q0021 .00073 | .00073 
 .00001 | .00001 .00022 | .00022 00074 | .00074 
 00001 | .00001 .00023 | .00023 .00075 | .00075 | 
 .00001 |. .00001 .00023 | .00023 .00076 | .00076 
 .00001 | .00001 -00024 | .00024 .00077 | .00077 
 .00001 | .00001 .00024 | .00024 .00078 | .00078 | 
 .00001-| .00001 .00025 | .00025 .00079 | .00079 
 .00001 | .00001 -00026 | .00026 .00081 | .00081 
 .00002 | .00002 .00026 | .00026 .00082 | .00082 
 .00002 | .00002 || .00027 | .00027 || .00083 | .00083 | 
 .00002 | .00002 .0002E .00028 .00084 | .00084 
 .00002 | .00002 .00028 | .00028 .00085 | .00085 
 .00002 | .00002 .00029 | .00029 .00087 | .00087 
 .00002 | .00002 . 00030 | .00080 .00088 | .00088 | 
 .00003 | .00003 00031 | .00031 .00089 | .00089 | 
 .00003 | .00003 .00031 | .00031 .00090 | .00090. | 
 .00003 | .00003 .00032 | .00032 .00091 | .00091 | 
 .00003 | .00003 |} .00033 | .00033 |} .00093 | .00093 
 -00004 | .00004 || .00034 | .00034 |) .00094 | .00094 
 .00004 | .00004 || .00034 | .00034 |} .00095 | .00095 
 .00004 | .00004 || .00035 | .00035 || .00096 | .00097 
 .00004 | .00004 .00036 | .000386 .00098 | .00098 
 .00005 | .00005 |} .00037 | .00037 .00099 | .00099 
 .00005 | .00005 .00037 | .00037 .00100 | .00100 
 .00005 | .00005 .00038 | .00038 .00102 | .00102 
 .00005 | .00005 .00039 | .00089 .00103 | .00103 
 .00006 | .00006 .00040 | .00040 .00104 | .00104 
 .00006 | .00006 .00041 | .00041 .00106 | .00106 
 .00006 | .00006 00041 | .00041 .00107 | .00107 | 
 .00007 | .00007 .00042 | .00042 .00108 | .00108 
 .00007 | .00007 || .00043 | .00043 || .00110 | .00110 
 .00007 | .00007 .00044 | .00044 00111 | .00111 
 .00008 | .00008 .00045 | .00045 .00112 | .00113 | 
 .00008 | .00008 .00046 | .00046 .00114 | .00114 | 
 .00009 | .00009 .00047 | .00047 .O0115 |. .00115 
 .00009 | .00009 .00048 | .00048 .00117 | .00117 
 .00069 | .00009 .00048 | .00048 .00118 | .00118 
 .00010 | .00010 .00049 | .00049 .00119 | .00120 
 .00010 | .00010 .00050 | .00050 .00121 | .00121 
 | .00011 | .00011 .00051 | .00051 .00122 | .00122 
 .00011 | .00011 .00052 | .00052 .00124 | .00124 
 .00011 | .00011 || .00053 | .00053 4) .00125 | .00125 
 -00012 | .00012 || .00054 | .00054 |} .00127 | .00127 
 | .00012 | .00012 || .00055 | .00055 || .00128 | .00128 | 
 .09013 | .00013 || .00056 | .00056 ||} .09130 | .00130 
 .00013 | .00013 |; .00057 | .00057 || .00131 | .00131 
 .00014 | .00014 || .00058 | .00058 || .00133 | .00133 | 
 | .00014 | .0001% |} .00059 | .00059 || .00134 | .00134 
 00015 | .00015 -00060 | .00060 .00136 | .00136 
 .00015 | .00015 .00061 .00187 | .00137 
 
 TABLE XXIX.—NATURAL VERSED SINES AND EXTERNAL SECANTS. 
 
 Be std 
 
 A 
 
 3° 
 4 
 Vers. |Ex.sec.| | 
 .00137 | .00187 | O > 
 .00139 00139 1 
 .00140 | .00140 2 
 .00142 | .00142 3 
 .00143 | .00143 4 
 -00145 | .00145 5 
 .00146 | .00147 | 6 
 .00148 | .00148 | .7 
 .00150 | .00150 | 8 
 .00151 | .00151 9 
 .00153 | .00153 | 10 
 .00154 | .00155 | 11 
 .00156 | .00156 | 12 
 .00158 | .00158 | 1: 
 .00159 | .00159 | 14 
 .00161 | .00161 | 15 
 .00162 | .00163 | 16 
 .00164 | .00164 | 17 
 .00166 » .00166 | 18 
 .00168 | .00168 | 19 
 .00169 | .00169 | 20 
 -00171.| .00171 | 21 
 .00173 | .00173 | 22 
 0017 .00175 | 23 
 .00176 | .00176 | 24 
 .00178 | .00178 | 25 
 .00179 | .00180 | 26 
 .00181 | .00182 | 27 
 .00183 | .00183 | 28 
 ~ 00185 | .00185 | 29 
 .00187 | .00187 | 30 
 .00188 | .00189 | 31 
 -00190 | .00190 | 32 
 .00192 | .00192 | 33 
 .00194 | .00194 | 34 
 .00196 | .00196 | 35 
 .00197 | .00198 | 36 
 .00199 | .00200 | 87 
 .00201 | .00201 | 38 
 .00203 | .00203 | 39 
 .00205 | .00205 | 40 
 .00207 | .00207 | 41 
 .00208 | .00209 | 42 
 .00210 | .00211 | 43 
 .00212 | .00213 | 44 
 .00214 | .00215 | 45 
 .00216 | .00216 | 46 
 .00218 | .00218 | 47 
 .00220 ; .00220 | 48 
 .00222 | .00222 | 49 
 .00224 | .00224 | 50 
 .00226 | .00226 | 51 
 .00228 | .00228 | 52 
 .00230 | .00280 | 53 
 .00232 | .00232 | 54 
 .00234 | .00234 | 55 
 .00236 | .00236 | 56 
 .00238 | .00238 | 57 
 -00240 | .00240 | 58 
 .00242 | .00242 | 59 
 .00244 | .00244 | 60 ~| 
 
TABLE XXIX.—NATURAL VERSED SINES AND EXTERNAL SECANTS 
 
 | 
 
 4° 5° 6° ve: 
 
 Yi) | 5 ee ee Fa eee aS, 
 
 | | | 1 
 
 Vers. |Ex.sec.!| Vers. 'Ex.sec.|| Vers. |Ex.sec.|| Vers. |Ex.sec.' 
 0. | .00244 | .00244 .00381 | .00882 || .00548 | .00551 | .00745 | .00751 
 1 | .00246 | .00246 .003883 | .00385 00551 00554 00749 | .00755 
 2] .00248 | .00248 || .00886 | .00387 .00554 | .00557 00752 | .00758 
 3 | .00250 | .00250 .00388 -| .00390 .00557 | .00560 .00756 | .00762 
 4} 00252 | .00252 .00891 .00392 .00560 | .00563 || .00760 | .00765 
 5 |) .00254 |-.00254 || .00393 | .00895 .00563 | .00566 .00763 | .00769 
 6 | .00256 | .00257 .00396 | .00897 || .00566 | .00569 00767 | .0077% 
 7) .00258 | .00259 .00898 | .00400 .00569 | .00573 .00770 | .00776 
 8 | .00260 | .00261 .00401 | .00403 00572 | .00576 | 00774 | .00780 
 9 | .00262 | .00263 .00404 | .00405 || .00576 | .00579 || .00778 | .00784 
 10 | .00264 | .00265 .00406 | .00408 || .00579 | .00582 00781 | .00787 
 |} 11 | .00266 | .00267 .00409 | .00411 .00582 | .00585 | 00785 | .00791 
 12 | .00269 | .00269 .00412 | .00413 .00585 | .00588 || .00789 | .00795 
 13 | .00271 | .00271 .00414 | .00416 .00588 | .00592 .00792 | .00799 
 14 | .00273 | .00274 .00417 | .00419 00591 .00595 .00796 | .00802 
 15 | .00275 | .00276 .00420 | 00421 00594 |..00598 || .00800 | .00806 
 16 | .00277 00278 .00422 | .00424 .00598 | .00601 .00803 | .00810 
 ote ee .00280 .00425 | .00427 .00601 | .00604 00807 | .00813 
 18 |. .00281 | .00282 .00428 | .00429 .00604 | .00608 .00811 | .00817 
 19 | .00284 | .00284 .00430 | .00482 .00607 | .00611 || .00814 | .00821 
 20 | .00286 | .00287 .00483 | .00485 .00610 | .00614 .00818 | .00825 
 21 | .00288 | .00289 .00436 | .00488 .00614 | .00617 .00822 | .00828 
 22 | .00290 | .00291 .00438 | .00440 .00617 | .00621 ||} .00825 . 00832 
 23 | .00293 | .00293 .00441 | .00443 00620 | .00624 || .00829 | .00836 
 24 | .00295 | .00296 00444 | .00446 || .00623 | ..00627 .00883 | .00840 
 25 | .00297 | .00298 00447 | .00449 .00626 | .00630 .60837 | .00844 
 26 | .00299 | .003800 .00449 | .00451 .00630 | .00634 | .00840 | .C0848 
 27 | .00301 | .00302 || .00452 | .00454 .00633 | .00637 | .00844 | .00851 
 28 | .003804 | .003805 .00455 | .00457 .00636 | .00640 .00848 | . 00855 
 29 | .00306 | .00307 .00458 | .00460 .00640 | .00644 .00852 | .00859 
 80 | .09308 | .00309 || .00460 "| .00463 .00643 | .00647 || .00856 | .00863 
 31 | .00811 | .00312 .00463 | .00465 00646 | .00650 |} .00859 | .00867 
 382 | .00318 | .003814 .00466 | .00468 .00649 | .00654 || .00863 | .00871 
 83 | .00315 | .00316 .00469 | .00471 || .00653 | .00657 || .00867 | 00875 
 34 | .00317 | .00318 .00472 | .00474 .00656 | .00660 || .00871 .00878 
 39 | .00320 | .00321 .00474 | .00477 .00659 | .00664 || .00875_ | 00882 
 36 | 00322 | 00323 || .00477 | :00480 || .00663_| .00667 || .00878 | .00886 
 37 | .00824 | .00326 .00480 | .60482 .00666 | .00671 .00882 | .00890 
 88 | .00327 | .00328 || .00483 | .00485 .00669 | .00674 || .00886 | .00894 
 39 | .00329 | .00380 .00486 | .00488 00673 | .00677 || .00890 | .00898 
 40 | .00332 } .00333 .00489 | .00491 .00676 | .00681 .00894 | .00902 
 41 .003834 | .00335 .00492 | .00494 .00680 | .00684 || .00898 | .00906 
 42 | .003836 | .00337 00494 | .00497 00683 | .00688 || .00902 | .00910 
 43 | .00339 | .00340 .00497 | .00500 .00686 | .00691 | 00906 | .00914 
 4) .00341 | .00342 .00500 | .00503 00690 | .00695 || .00909 | .00918 
 45 | .00343 | .00345 || .005038 | .00506 .00693 | .00698 .00913 | .00922 
 A6 | .00346 | .00347 .00506 | .00509 00697 | .00701 |} .00917 | .00926 
 47 | .00348 | .003850 .00509 | .00512 00700 | .00705 .00921 | .00930 
 48 | .00351 .00352 || .00512 | .00515 00703 | .00708 || .00925 , .00934 
 49 | .003853 | .00354 00515 | .00518 || .00707 | .00712 |} .00929 | .00938 
 50 | .00856 | .00857 .00518 | .00521 00710 | .00715 || .00933 | .00942 
 51 | .00358 | .00359 .00521 | .00524 00714 | .00719 || .00937 | .00946 
 2 | .00361 00862 || .00524 | .00527 00717 | .00722 .00941 .00950 
 53 | .00363 | .00364 | 00527 | .00530 |} .00721 00726 00945 | .00954 
 | 54 | 100365 | :00367 || .00530 | 00533 || .60724 | .00730.| .00949 | .00958 
 | 5d | .00368 | .00869 .0053¢ .00536 .00728 | .00733 || .009538 | .00962 
 | 56 | 00370 | .00372 || -00536 | |00539 || :00731 | .00737 || 00957 | .00966 
 57 | .00373 | .00374 || .00539 | .00542 || .00735 | .00740 || .00961 | .00970 
 58 | .00375 | .00377 00542) .00545 .00738 | .00744 .00965 | .00975 
 59 | .00878 | .00379 .00545 | .00548 00742 | .00747 | .00969 | .00979 
 | 60 | .00381 | .00382 .00551 00745 | .00751 |! .00983 
 
 00548 
 
 00978 
 
 Sequnceeus| 
 
{ 8° 
 
 a | 
 
 Vers. |Ex. sec 
 0 | .00973 | .00983 
 1 | .00977 | .00987 
 2} .00981 | .00991 | 
 3 | .00985 | .00995 
 4 | .00989 | .00999 | 
 5 | .00994 | .01004 
 6 | .00998 | .01008 
 7 | .01002 | .01012 
 8 | .01006 ; .01016 
 9 | .01010 | .01020 | 
 10 | .01014 | ,01024 
 11 | .01018 | .01029 | 
 12 | .01022 | .01033 | 
 18 | .01027 | .01037 
 14 | .01031 | .01041 
 15 | .01035 | .01046 | 
 16 | .01039 | .01050 
 17 | .01043 | .01054 
 18 | .01047 | .01059 
 19 | .01052 | .01063 | 
 20 | .01056 | .01067 
 21 | .01060 | .0107 
 22 | .01064 | .01076 | 
 23 | .01069 | .01080 
 24 | .01073 | .01084 | 
 25 | .01077 | .01089 
 26 | .01081 | .01093 | 
 27 | .01086 | .01097 | 
 28 | .01090 | :01102 
 29 | .01094 | .01106 | 
 30 | .01098 | .01111 | 
 31 | .01103 | .01115 | 
 82 | .01107 | .01119 | 
 33 | .01111 | .01124 | 
 34 | .01116 | .01128 
 35 | .01120 | .01133 
 36 | .01124 | .01137 | 
 37 | .01129 | .01142 | 
 38 | .01133 | .01146 
 89 | .01137 | .01151 
 40 | .01142 | .01155 
 41 | .01146 | .01160 
 42 | .01151 | .01164 
 43 | .01155. | .01169 
 44 | .01159 | .01173 
 45 | .01164 | .01178 | 
 46 | .01168 | .01182 | 
 47 | .01173 | .01187 
 48 | .01177 | .01191 
 49 | .01182 | .01196 
 50 | .01186 | .01200 
 51 | .01191 | .01205 | 
 52 | .01195 | .01209 | 
 53 | .01200 | .01214 
 54] .01204 | .01219 | 
 55 | .01209 | .01223 
 56 | .01213 | .01298 
 57 | .01218 | .01233 | 
 58 | .01222 | .01237 | 
 59 | .01227 | .01242 | 
 60 | .01231 | .01247 | 
 
 9° 10° 11° 
 Vers. |Ex.sec.|| Vers. |Ex.sec.|| Vers. |Ex. sec.| 
 || .012381 | .01247 |} .01519 | .01543 .018387 | .01872 
 .01236 | .01251 .01524 | .01548 .01843 | .01877 
 .01240 | .01256 || .01529 | .01553 .01848 | .01883 
 .01245 | .01261 .015384 | .01558 .01854 | .01889 
 .01249 | .01265 .01540 | .01564 .01860 | .01895 | 
 .01254 | .01270 -01545 | 01569 .01865 | .01901 
 .01259. | .0127 -01550 | .01574 .01871 | .01906 
 .01263 | .01279 .01555 | .01579 .01876 | .01912 
 .01268 | .01284 .01560 | .01585 .01882 | .01918 
 .01272 | .01289 .01565 | .01590 .01888 | .01924 
 01277 | .01294 .01570 | .01595 .01893 | .01980 
 -01282 | .01298 || .01575 | .01601 || .01899 | .01936 
 .01286 | .01303 .01580 | .01606 .01904 | .01941 
 .01291 | .01308 .01586 | .01611 .01910 | .01947 
 .01296 | .01313 .01591 | .01616 -01916 | .01953 
 -01300 | .01318 || .01596 | .01622 || .01921 | _01959 Ve J 
 .01305 | .01322 -01601 | .01627 .01927 | .01965 
 .01310 | .01327 -01606 | .01633 .01933 | .01971 
 .01314 | .01332 .01612 | .01638 .01939 | .01977 - 
 -01319 | .01837 .01617 | .01643 .01944 | .01983 
 -O1324 | .01342 || .01622 | .01649 || .01950 | _01989 
 -01329 | .01346 |) .01627 | .01654 || .01956 | .01995 
 .01833 | .01351 .01632 | .01659 || .01961 | .02001 
 .01338 | .01356 .01638 | .01665 .01967 | .02007 
 -01343 | .01361 || .01643 | .01670 || .01973 | (02013 
 .01348 | .01366 -01648 | .01676 || .01979 | 02019 
 .01352 | .01371 .01653 | .01681 .01984 | .02025 
 .01357 | 01376 .01659 | .01687 .01990 | .02031 
 .01362 | .01381 .01664 | .01692 .01996 | .02037 
 .01367 | .01386 .01669 | .01698 .02002 | .02043 
 01871 | .01391 || .01675 | .01703 || .02008 | |02049 
 -01376 | .01395 || .01680 | .01709 || .02013 | .o2055 
 .01381 | .01400 .01685 | .01714 .02019 | .02061 
 .01386 | .01405 .01690 | .01720 :02025 | .02067 
 .01391 | .01410 -01696 | .01725 || .02031 | .02073 
 .01396 | .01415 -O1701 | .01731 || .02037 | .o2079 
 .01400 | .01420 .01706 | .01736 .02042 | .02085 
 .01405 | .01425 .01712 | .01742 .02048 | .02091 
 -01410 | .01430 || .01717 | .01747 || .02054 | |02097 
 .01415 | .01435 01723 | .01753 .02060 | .02103 
 -01420 | .01440 |) .01728 | .01758 || .02066 | .02110 
 -01425 | .01445 || .01733 | .01764 {| .02072 | .02116 
 -01430 | .01450 || .01739 | .01'769 || .02078 | .02122 
 .01435 | .01455 .01744 | .0177 .02084 | .02128 
 .01489 | .01461 .01750 | .01781 -02090 | .02134 
 -01444 | .01466 .01755 | .01786 || .02095 | .02140 
 .01449 | .01471 -01760 | .01792 || .02101 | .02146 
 -01454 | *01476 ||. .01766 | .01798 | .02107 | .02158 
 -01459 | .01481 || .0177 .01803 .02113 | .02159 
 .01464 | .01486 || .01777 | .01809 || .02119 | .02165 
 -01469 | .01491 || .01782 | .01815 .02125 | .02171 
 .01474 | .01496 .01788 | .01820 || .02131 | .02178 | 
 .01479 | .01501 .01798 | .01826 .02137 | .021484 
 .01484 | .01506 .01799 | .01832 || .02143 | .02190 
 .01489 | .01512 01804 | .01837 || .02149 | .02196 
 -01494 | 01517 .01810 | .01843 || .02155 | .02203 
 -01499 | .01522 .01815 | .01849 || .02161 | .02209 | 
 -01504 | .01527 .01821 | .01854 .02167 | .02215 
 -01509 | .01532 -01826 | .01860 || -02173 | .02221 
 -01514 | .01537 .01832 | .01866 | -02179 | .02228 
 .01519 | .01543 .01837 | .01872 || .02185 02234 
 
 wcomewHe | 
 
 | 
 
 ee eae 
 
k 
 
 esl 
 
 THe COD 
 
 e 
 Oo 
 
 Mm-Io 
 
 12° 
 
 14° 
 
 15° 
 
 02447 
 
 .02508 
 
 | 02453 | .02515 
 02459 | .02521 
 02466 | .02528 
 02472 | .02535 
 02479 | .02542 
 102485 | .02548 
 (02492 | .02555 
 02498 | .02562 
 .02504 | .02569 
 02511 | .02576 
 02517 | .02582 
 02524 | .02589 
 .02530 | .02596 
 .02537 | .02603 | 
 02543 | 02610 | 
 02550 | .02617 
 02556 | .02624 
 02563 | .02630 
 
 02845 
 02852 
 02859 
 02866 
 
 | .02873 
 | .02880 
 | .02887 
 
 02894 
 .02900 
 
 .02907 
 .02914 
 .02921 
 .02928 
 .02935 
 02942 
 .02949 
 .02956 
 .02963 
 .02970 
 
 02928 
 .02936 
 02943 
 02950 
 02958 
 02965 
 02972 
 .02980 
 .02987 
 .02994 
 03002 
 03009 
 03017 
 03024 
 03032 
 03039 
 
 03046 
 03054 
 .03061 
 
 Vers. |Ex.sec.|| Vers. |Ex.sec.|| Vers. 
 .02185 | .02234 || .02563 | .02630 || .02970 
 -02191 | .02240 .02570 | .02637 || .02977 
 .02197:| .02247 || .02576 | .02644 02985 
 | .02203 | .02253 .02583 | .02651 .02992 
 .02210 | .02259 .02589 | .02658 .02999 
 02216 | .02266 || .02596 | .02665 .08006 
 02222 | .02272 || .02602 | .02672 .03013 
 02228 | .02279 || .02609 | .02679 . 03020 
 .02234 | .02285 || .02616 C2686 .08027 
 .02240 | .02291 || .02622 | .02693 .08084 
 02246 | .02298 |} .02629 | .02700 .08041 
 | .02252 | .023804 .026385 | .02707 .03048 
 .02258 | .02311 || .02642 | .02714 || .08055 
 .02265 | .023817 || 2649 | .02721 .08063 
 02271 | .02323 || .02655 | .02728 || .03070 
 .02277 | .02330 || .02662 | .02735 .03077 
 .022838 | .02336 .02669 | .02742 || .03084 
 .02289 | .02343 .02675 | .02749 .038091 
 .02295 | .02349 || .02682 | .02756 .03098 
 02302 | .023856 || .02689 | .02763 || .08106 
 .02808 | .02862 |} .02696 | .02770 || .03113 
 .02814 | .02369 || 02702 | .02777 |) .03120 
 0232 .02375 02709 | .0272°4 .03127 
 .02327 | .02382 02716 | 02791 .08134 
 .02333 | .02388 -02722 | .02799 .08142 
 02339 | .02395' || .02729 | .02806 .08149 
 02345 | .(2402 || .02736 | .02813 || .08156 
 .02352 | .02408 || .02743 | .02820 || .03163 
 .02358.| .02415 || .02749 | .02827 038171 
 .02364 | .02421 || .02756 | .02834 08178 
 .02370 | .02428 .02763 | .02842 |) .03185 
 .02377 | .02485 || .02770 | .02849 .03193 
 02383 | .02441 02777 .02856 || .03200 
 | .02889 | .02448 || .02783 | .02863 || .038207 
 02896 | .02454 .02790 | .02870 2038214 
 02402 | .02461 || .02797 | .02878 08222 
 .02408 | .02468 || .02804 | .02885 .03229 
 y | 02415 | .02474 .02811 | .02892 .038236 
 02421 | .02481 || .02818 | .02899 || .08244 
 02427 | 02488 || .02824 | .02907 .03251 
 02434 | .02494 || .028381 | .02914 || .08258 
 .02440 | .02501 .028388 | .02921 .03266 
 
 | .032738 
 .03281 
 03288 
 .08295 
 03303 
 03310 
 .03318 
 .03325 
 03333 
 .03340 
 03347 
 -U8500 
 | .0383862 
 .08370 
 03377 
 .03385 
 
 03355 
 
 03392 
 
 03400 
 
 Ex. sec. | Vers. |Hx. sec. 
 03061 03407 | .03528 
 .03069 03415 | .03536 
 .08076 || .03422 | .08544 
 .08084-|| .03480 | .08552 
 .03091 .03488 | .03560 
 
 .03099 
 
 .03106 
 
 .03114 
 .08121 
 
 .03129 
 
 .03137 
 .03144 
 03152 
 
 .08159 
 .03167 
 .08175 
 
 03182 
 
 .03190 
 03198. | 
 
 .08205 
 
 08213 | 
 
 03221 
 .03228 
 .032386 
 05244 
 .08251 
 .08259 
 08267 
 .08275 
 08282 
 
 .03290 || 
 .03298 
 
 03306 
 
 .033813 
 
 .03821 
 .03329 
 03337 
 .03345 
 .03353 
 .03360 
 03368 
 .03376 
 .08384 
 03392 
 
 .03400 
 
 .03408 
 .03416 
 .03424 
 
 03432 
 
 03439 
 03447 
 .03455 
 03468 
 08471 
 03479 
 .08487 
 
 .03495 
 03503 
 
 03512 
 
 03407 
 
 .03520 
 03528 
 
 .03445 | .08568 
 .03453 | .08576 
 
 .03460 | .03584 
 .03468 | .03592 
 
 03476 
 03483 
 .03491 
 03498 
 03506 
 .03514 
 03521 
 .03529 
 03537 
 03544 
 03552 
 03560 
 
 03601 
 .08609 
 
 .03617 
 
 .03633 
 .03642 
 03650 
 .03658 
 .03666 
 03674 
 03683 
 .03691 
 
 .03567 | .03699 
 03575 | .03708 
 
 .03583 
 03590 
 .03598 
 .03606 
 .03614 
 .03621 | .08758 
 .03629 | .08766 
 .03637 | .G8774 
 
 .03716 
 038724 
 03732 
 03741 
 03749 
 
 .038645 | .03783 
 .03653 | .08791 
 .03660 | .08799 
 .03668 | .03808 
 .03676 | .03816 
 03684 | .03825 
 03692 | .03833 
 03699 | .03842 
 03707 | .03850 
 .03715 | .03858 
 .03723 | .08867 
 087381 | .08875 
 .03739 | .08884 
 08747 | .038892 
 
 .08754 | .038901 
 .03762 | .03909 
 
 0877 .03918 
 0377 .08927 
 -03786 | .03935 
 .03794 | .03944 
 
 .08802 | .03952 
 
 .03810 | .08961 
 .03818 | .03969 
 03826 | .03978 
 .03834 | .08987 
 .03842 | .03995 
 .03850 | .04004 
 .03858 | .04013 
 .03866 | .04021 
 
 03874 | .04030 
 
 473 
 
 03625 | 
 
 <XIX.—NATURAL VERSED SINES AND EXTERNAL SECANTS. 
 
 Serecmowno| 
 
16° 
 
 Vers. | 
 { 
 
 Ex. sec.'!| Vers. |/Ex.sec. | Vers 
 | | 
 0 | 04370 | .04569 || .04894 | 05448 
 } | .04378 | .04578 || .04903 | 05458 
 2 | .04887 | ,04588 || .04912 | 05467 
 3 -04395 | .04597 |, ,04921 05477 |. 
 4 | .04404 | .04606 || .04930 | | .05486 
 5 .04412 | .04616 || .04939 | | .05496 
 6 .04421 | .04625 || 04948 | .05505 | 
 fi .04429 | .04635 || .04957 | .05515 | 
 8 | 04488 | .04644 || .04967 | .05524 1, 
 9 .04446 | .04653 || .04976 05584 : . 
 | .04455 | .04663 || .04985 05543 
 .04464 | .04672 || .04994 .05553 | 
 | .04472 | .04682 || .05003 05562 
 | .04481 | .04691 || .05012 | .05572 | 
 | .04489 | .04700 || .05021 | | 05582 
 | .04498 | .04710 || .05030 | | .05591 
 | | .04507 | .04719 || .05089 | .05601 | 
 04515 | .04729 || .05048 | 05610 | 
 .04524 | .04738 || .05057 | ‘| .05620 | 
 .04533 | .04748 || 05067 7 || .05680 | 
 .04541 | .04757 || .05076 1} .05639 
 .04550 | .04767 || .05085 | .05649 | 
 '| .04559 | .04776 || .05094 || 05658 | 
 .04567 | .C4786 |! 05103 | | .05668 
 .04576 | .04795 |) 05112 05678 
 .04585 | .04805 || .05122 | 05687 
 .04593 | .04815 |} .05131 | 05697 
 1} .04602 | .04824 || .05140 | | 05707 | . 
 .04611 | .04834 |} .05149 05716 | . 
 || .04620 | .04848 |] .05158 05726 
 04628 | .04853 || .05168 05736 
 .04637 | .04€63 || .05177 05746 | 
 .04646 | .04872 || .05186 05755 
 .04655 | .04882 || .05195 05765 
 .04663 | .04891 || .05205 SO57%5 |. 
 04672 | .04901 || .05214 05785 |. 
 .04681 | .04911 || .05293 05794 | 
 .04690 | .04920 || .05232 05804 
 .04699 | .04980 || ,05242 05814 
 .04707 | .04940 || .05251 05824. | 
 .04716 | .04950 || .05260 .05833 | 
 .04725 | .04959 || .0527¢ 05843 
 i} .0473 .04969 || .05279 '} 05853 
 i| .04743 | .04979 || .05288 i| .05863. | 
 || .04752 | .04989 || .05298 05873 
 || .04760 | .04998 || .05307 05882 
 .04769 | .05008 |} .05316 .05892 | 
 | .04778 | .05018 ||. .05326 05902 
 | .04787 | .05028 || .05335 .05912 
 .04796 | .05038 || .05344 || .05922 
 .04805 | .05047 |) .05354 || .05932 
 .04814 | .05057 || .05363 .05942 
 '| .04823 | .05067 |! .05373 05951 
 ‘| .04832 | .05077 || .05382 | . 05961 | 
 04841 | .05087 || .05391 05971 
 .04850 | .05097 || .05401 05981 
 || -04858 | .05107 || .05410 05991 
 || .04867 | .05116 || .05420 06001 
 || .04876 | .05126 || .05429 06011 
 .04885 | .05136 || .05439 .06021 
 .04894 | .05146 || .05448 .06031 
 
TABLE XXIX.—NATURAL VERSED SINHS AND EXTERNAL 
 
 an 
 
 10 
 
 ) 
 SCmrIaumeIne| 
 
 | | 
 06031 | .06418 || .06642 
 06041 | .06429 || .06652 
 06051 | .06440 || .06663 
 06061 | .06452 || .06673 
 06071 | .06463 || .06684 
 .06081 | 706474 || .06694 
 .06091 | .06486 || .06705 
 .06101 | .06497 || .06715 
 .06111 | .06508 06726 
 .06121 | .06520 .06736 
 .08131 | .06531 || .06747 
 .06141 | .06542 .06757 
 .06151 | .06554 || .06768 
 .06161 | .06565 || .06778 
 06171 | .06577 || .06789 
 06181 | .06588 || .06799 
 .06191 | .05600 || .0G8i0 
 | .06201 | .06611 || .06820 
 .06211 | .06622 06831 
 06221 | .06634 |} .06341 
 06231 | .06645 || .06552 
 06241 | .06057 || .06°63 
 06252 | .00668 || .06873 
 06262 | .03580 || .06884 
 06272 | .06691 || .05894 
 .06282 | .05703 .06905 
 06292 | .0S715 || .06916 
 .06302 | .05726 || .06926 
 .06312 | .06738 || .06937 
 .06323 | .06749 || .06948 
 .06333 | .05761 ||} :06958 
 .06343 | .03773 || .06969 
 .06353 | .06784 || .06980 
 06363 | .06796 || .06990 
 .06374 | .06807 O7F00L 
 .06384 | .06819 || .07012 
 .06394 | .06831 |} .07022 
 .06404 | .06843 || .07033 
 .06415 | .06854 || .07044 
 | 06425 | .06866 || .07055 
 | .06485 | .06878 .07065 
 | .06445 | .06889 07076 
 06456 | .06901 || .07087 
 .06466 | .06913 || .07098 
 .06476 | .06925 || .07108 
 .06486 | .06936 || .07119 
 06497 | .06948 || .07130 | 
 | .06507 | .06960 || .07141 
 06517 | .06972 || .07151 
 | .06528 | .06984 || .07162 
 06538 | .06995 || .07173 
 06548 ; .07007 || .07184 
 06559 | .07019 || .07195 | 
 .06589 | .07031 || .07206 
 .06580 | .07043 || .07216 | 
 
 20° 
 
 Vers. |Ex.sec.| 
 
 21° 
 
 SECANTS. 
 
 Vers. |Ex. sec. 
 
 06590 | .07055 || 
 “06600 | .07067 || 
 06611 | .07079 | 
 
 .06632 
 
 | 
 06621 | . 
 06642 | . 
 
 07091 
 7103 
 07115 | 
 
 07227 
 0723 
 
 07249 
 07260 
 07271. | 
 07282 | 
 
 .07602 
 .07615 
 07627 
 .07640 
 .07652 
 .07665 
 07677 
 .07690 
 .07 702 
 Oia ks) 
 (07725 
 07740 | 
 97752 
 07765 
 07778 
 .07790 
 .07803 
 07816 
 .07828 
 07841 
 .07853 
 
 | 
 
 O7904 | 
 07915 
 
 SORO2T 4 
 
 OTT 24 
 .07735 
 .07746 
 OTTST 
 07769 
 07780 
 07791 | 
 07802 
 07814 
 07829 
 07336 
 .07818 
 07859 
 07870 
 07881 
 
 07955 
 
 7393 
 
 079338 
 
 08370 
 08383 
 083897 
 08410 
 08423 
 08436 
 
 08449 || 
 
 . 08463 
 
 .08476 
 
 08489 || 
 .08505 || 
 
 3516 
 08529 | 
 
 (08542 || 
 
 .08555 
 08569 
 . 038582 
 
 08596 || 
 .0860.) |} 
 08623 || 
 08636 |; 
 
 .08410 | .09183 
 
 08422 | .09197 
 .08434 | .09211 
 
 .08445 | .09224 | 
 .08457 | .09238 
 .08469 | .09252 | 
 .08481 | .09266 | 
 .08492 | .09280 | 
 .08504 | .09294 | 
 .08516 | .09305 
 .08528 | .09823 
 
 .08539 | .09837 
 
 .08551 | .09351 
 .08563 | .09365 
 08575 | .09379 | 
 .08586 | .09398 
 
 | 08598 | .09407 
 .O8610 | .09421 
 .08622 | .09435 
 08634 | .09449 
 .08645 | .09464 
 
 | 22° 238 
 | = ie , 
 | Vers, |Ex sec. | Vers. |Ex. sec.| 
 07115 || .07282 | .07853 || .07950 | .08636 | 0 
 07126 || .07293 | .07866 || .07961 | .08649 | 1 
 07138 || .07303 | .07879 || .07972 | .08663 | 2 
 - 07150 || .07314 | .07892 || .07984 | .08676 | 3 
 07162 || .07325 | .07904 || .07995 | .08690 | 4 
 07174 || .07336 | .07917 || .c8006 | .08703 | 5 
 07186 || .07347 | .07930 |} .08018 | .08717 | 6G 
 07199 || .07358 | .07943 || .08029 | .08730 | 7 
 07211 || .07369 | .07955 || .08041 | .08744 | 8 
 f223 || .07380 | .07968. |) .08052 | .08757 | 9 
 n935 || .07391 | .07981 || .08064 | .08771 | 10 
 07247 || .07402 | .07994 || .08075 | .08%84 | 11 
 07259 || .07413 | .08006 || .08086 | .08798 | 12 
 0727 07424 | .08019 |} .08098 | .O8811 | 13 
 07288 || .07435 | .08032 |} .08109 | .08825 | 14 
 07295 || .07446 | .08045 || .08121 | .08839 | 15 
 07207 || .07457 | .08058 || .08182 | .08852 | 16 
 .07320 || .07468 | .08071 || .08144 | .08866 | 17 
 "073232 || .07479 | .08084 || .08155 | .08880 | 18 
 07344 || .07490 | .08097 |} .08167 | .08893 | 19 
 .07356 || .07501 08109 || .08178 | .08907 | 20 
 07368 || .07512 | .08122 || .08190 | .08921 | 21 
 07380 || 68135 || .08201 | .08934 | 22 
 07393 08148. || .08213 | .08948 | 28 
 07405 || 08161 || .08225 | .08962 | 24 
 "O7417 || .07556 | .08174 || .08236 | .08975 | 25 
 07429 || .07568 | .08187 || .08248 | .08989 | 26 
 07442 || .07579 | .08200 || .08259 | .09003 | 27 
 07454 |) .07590 | .08213 || .08271 | .09017 | 28 
 07466 || .07601 | .08226 || .08282 | .09030 | 29 
 07479 || .07612 | .08239 || .08294 | .09044 | 30 
 07491 || .07628 | .08252 || .08306 | .09058 31 
 07503 || .07634 | .08265 || .08817 | .09072 | 32 
 07516 || .07645 | .08278 || .08829 | .09086 | 33 
 07528 || .07657 | .08291 || .083840 | .09099 | 34 
 “07540 || .07668 | .08305 || .08852 | .09113 | 35 
 07553 || .07679 | .083818 || .08864 | .09127 | 36 
 07565 || .07690 | .08833 08375 | .09141 | 37 
 07578 || .07701 | .08844 || .08887 | .09155 | 38 
 07590 || .07713. | ..08357 || .08399 | .09169 | 389 
 
 ATS 
 
24° 
 
 25° 
 
 26° 
 
 27° 
 
 Vers. | Ex. sec. 
 
 | .08645 
 08657 | 
 
 .O8669 
 
 | .08681 
 | .08693 
 
 08705 
 08717 
 
 | .08728 
 
 .08740 
 08752 
 . 08764 
 0877’ 
 
 .08788 
 . 08800 
 
 | .08812 
 
 08824 
 -08836 
 . 08848 
 .O8860 
 03872 
 03884 
 .08896 
 -08908 
 .08920 
 .089382 
 .08944 
 . 08956 
 .08968 
 .08980 
 
 08992 | 
 
 .09004 
 
 .09016 
 09028 
 .09040 
 .09052 
 .09064 
 .09076 
 
 | .09089 
 
 .09101 
 99118 
 .09125 
 
 .09137 
 09149 
 -O9161 
 
 .09186 
 .09198 
 .09210 
 
 - 09222 
 
 .09234 
 
 .09247 | 
 
 . 00259 
 
 .09271 | 
 
 09283 
 
 09296 | 
 09308 | 
 
 .09320 
 
 | 09174 | 
 
 09832 | 
 
 09345 
 
 09857 } 
 
 .09369 | 
 
 = | bore? 
 | 
 Vers. |Ex.sec.|| Vers. |Ex. sec.|| Vers. |Ex. sec. | 
 | { 
 .09464 || .09369 | .10338 |} .10121 | .11260 || 210899 | 1228371 20 
 .09478 |) .09382 | .10353 || .10183 | .11276 || .10913 | .12949 1 
 . 09492 .09394 | .10368 || .10146 | .11292 |} 10926 | .12266 | 2 
 -09506 |; .09406 | .10383 || .10159 | .11308 || .10989 | .12283 | 3 
 .09520 .09418 | .10398 10172 | .11823 |} .10952 | .12299 4 
 09535 || .09431 | .10413 |} .10184 | .11339 || .10965 | .12316 | 5 
 .09549 09442 | .10428 || .10197 | .11355 || .10979 | .12333 6 
 -09563 || .09455 | .10443 || .10210 | .11371 |' .10992 | .12349 | 7 
 09577 || .09468 | .10458 || .10223 | .11387 || .11005 | .12366 8 
 -09592 || .09480 | .10473 || .10236 | .11403 |; .11019 | .12883 | 9 
 -09606 |} .09493 | .10488 |} .10248 | .11419 } 11032 | .12400 | 10 
 -09620 || .09505 | .10503 || .10261 | .11435 |) .11045 | .12416 | 11 
 .09635 || .09517 | .10518 || .10274 | .11451 || .11058 | .12433 | 12 
 -09649 || .09530 | .10533 || .10287 | .11467 || .11072 | .12450 | 13 
 09663 |} .09542 | .10549 || .10300 | .11483 || .11085 | .12467:| 14 
 -09678 || .09554 | .10564 |] .10313 | .11499 |} .11098 | .12484 | 15 
 -09692 |; .09567 | .1057 .10826 | .11515 -11112 | .12501 | 16 
 09707 || .09579 | . 10594 |} .10388 | .11531 || .11125 | .12518 | 17 
 .09721 -09592 | .10609 || .10351 | .11547 || .11138 | .12534 | 18 
 .09735 |) .09604 | .10625 -10864 | .11563 |; .11152 | .12551 | 19 
 09750 || .09617 | .10640 |} .10377 | .11579 || .11165 | .12568 | 20 
 -09764 |} .09629 | .10655 || .10390 | .11595 || .11178 | .12585 | 21 
 09779 .09642 | .10670 -10403 | .11611 || .11192'| .12602 | 22 
 09793 || .09654 | .10686 || .10416 | .11627 11205 | .12619 | 23 
 -09808 || .09666 | .10701 || .10429 | .11643 |} .11218 | .12636 | 24 
 -09822 || .09679 | .10716 || .10442 | .11659 |} .11232 | .12653 | 25 
 -09887 |} .09691 | .10781 || .10455 | .11675 |} .11245 | .12670 | 26 
 09851 || .09704 | .10747 || .10468.} .11691 || .11259 | .12687 | 97 
 -09866 || .09716 | .10762 || .10481 | .11708 || .11272 | .19704 | 98 
 .09880 || .09729 | .10777 }} .10494 | .11724 | -11285 | .12721 | 29 
 .09895 || .09741 | .10793 .10507 «| .11740 | 11299 | .12788 |}, 380 
 -09909 |; .09754 | .10808 || .10520 | .11756 || .11812 | .19755 | 31 
 09924 || .09767 | .10824 -10533 | .11772 ||- .11826 | .12772 | 82 
 -09939 || .09779 | .10839 |} .10546 | .11789 |) .11389 | .12789 | 33 
 -09953 || .09792 | <10854 |! .10559 | .11805 || 11358 | .19807 | 34 
 -09968 |} .09804 | .10870 |; .10572 | .11821 || .11366 | .19824 | 35 
 -09982 || .09817 | .10885 .10585 | .11838 .11380 | .12841 | 86 
 .09997 .09829 | .10901 -10598 | .11854 |! .11893 | .12858 | 37 
 -10012 |} .09842 | .10916 |} .10611 | .11870 |} .11407 | .19875 | 38 
 - 10026 .09854 | .10982 || .10624 | .11886 | .11420 | .12892 | 39 
 -10041 |) .09867 | .10947 |; .10637 | .11903 |} .11434 | .19910 | 40 
 -10055 || .09880 | .10963 || .10650 | .11919 |) .11447 | .12907 | 44 
 -10071 |; .09592 | .10978 || .10663 | .11936 |} .11461 12944 | 42 
 - 10085 09905 | .10994 |} .10676 | .11952 |) .11474 | .12961 | 43 
 -10100 |} .09918 | .11009 || .10689 | .11968.|| .11488 | .12979 | 44 
 -10115 || .09980 | .11025 |; .10702 | .11985 |} .11501 | .12996 | 45 
 10139 |] .09943 | .11041 |} .10715 | .12001 .11515 | .13013 | 46 
 10144 || .09955 | .11056 |, .10728 |. .12018 ;| 11528 | .138081 ; 47 
 -10159 || .09963 | .11072 |] .10741 |. .12034:|] .11542 | .13048 | 48 
 10174 || .09981 | .11087 || .10755 | .12051 .11555 | .18065 | 49 
 -10189 || .09993 | .11103 || .10768 | .12067 || .11569 | .13083 | 50 
 -10204 || .10006 | .11119 || .10781 | .12084 || .11583 | .13100 | 51 
 -10218 |} .10019 |. .11134 || .10%794 | .12100 | .11596 | .18117 | 52 
 -10233 |) .10082 | .11150 || .10807 | .12117 || .11610 | .13135 | 53 
 -10248 || .10044 | .11166 |] .10820 | .12138 |} .11623 | .13152 | 54 
 -10263 |} .10057 | .11181 || .10833 | .12150 || .11637 | .13170 | AS 
 -10278 || .10070 | .11197 || .10847 | .12166 |} .11651 | .13187 | 56 
 -10293 || .10082 | .11213 |} .10860 | .12183 |! .11664 . 18205 | 57 
 -10308 || .10095 | .11229 || .10873 | .12199 |! .11678 .18222 | 58 
 -10323 |] .10108 | .11244 || .10886 | .12216 || .11692 | 13240 | 59 
 -10338 || .10121 | .11260 || .10899 | .12233 |} .11705 | .13257 | 60 | 
 
TADLE XXIX._NATURAL VERSED SINES AND EXTERNAL SECANTS. 
 
 29° 
 
 31° 
 
 | 
 
 Vers. |Ex. sec Vers. |Ex. sec.|| 
 | | 
 0 | .11705 | .18257 |] .12588 | .14385 
 4} .11719 | .18275 || .12552 | .14354 
 2 | 141733 | 113292 || 119566 11372 
 8 | .11746 | .13310 || .12580 4391 
 4 | .11760 | .13827 || .12595 ar 
 5 | 1177 13345 || .12609 | .14428 
 6 | .11787 | .13362 2623 | 14446 
 * | .11801 | .13380 || .12687 | .14465 
 8 | .11815 | .13398 || .12651-| .14488 
 9 | .11828 | .13415 || .12665 | .14502 
 10 | 11842 | .13433, || .12679 | .14521 
 11 | .11856 | .13451 || .12694 | .14539 
 12 | .11870 | .13468 || .12708 | .14558 
 18 | 11888 | .13486 || .12722 | .14576 
 14 | .11897 | .13504 || .12736 | .14595 
 15 | .11911 18521 12750 | .14614 
 16 | .11925 3539 || .12765 | .14632 
 17 | .11938 “5667 1277 14651 
 18 | .11952 | .13575 || 12793 | .14670 
 19 | .11966 | .18593 || .12807 | .14689 
 20 | .11980 | .13610 |} .12822 | .14707 
 21 | .11994 | .13628 || .12836 | .14726 
 22 | 12007 | .13646 || .12850 | .14745 
 23 | .12021 | .13664 || .12864 | .14764 
 24 | .12035 | .13682 |} .12879 | .14782 
 25 | .12049 | .13700 || .12893 | .14801 
 26 | .12063 | .13718 || .12907 | .14820 
 27 | .12077 | .18785 || .12921 | .14889 
 28 | ,.12091 | .13753 || .12936 | .14858 
 29 | .12104 | .1377 12950 | 14877 
 30 | .12118 | .13789 || .12964 | .14896 
 31 | .12132 | .13807 || .12979 | .14914 
 82 | .12146 | .13825 || .12993 | .14933 
 33 | .12160 | .13843 || .13007 | .14952 
 34 | .12174 | .18861 || .13022 | .14971 
 35 | .12188 | .13879 || .13036 | .14990 
 36 | .12202 | .13897 || .13051 | .15009 
 37 | .12216 | .13916 || .13065 | .15028 
 38 | .12230 | .13934 || .13079 | .15047 
 39 | .12244 | .13952 || .18094 | .15066 
 40 | .12257 | .13970 || .13108 | .15085 
 41 | .12271 | .13988 || .13122 | .15105 
 42 | .12285 | .14006 |} .13137 | .15124 
 43 | .12299 | .14024 || .18151 | .15143 
 44 | .12313 | .14042 |] .13166 | .15162 
 45 | .12827 | .14061 || .13180 | .15181 
 46 | .12341 | .14079 || .13195 | .15200 
 47 | .12355 | .14097 |} .18209 | .15219 
 48 | .12869 ) .14115 || .13228 | .15939 
 49 | .12388 | .14134 |] .13288 | .15258 
 50 | .12897 | .14152 || .18252 | 15277 
 51 | .12411 | .14170 || .13267 | .15296 
 52 | 12425 | .14188 |] .13281 | .15815 
 53 | .12489 | .14207 || .13206 | .15335 
 54 | 112454 | .14225 || .18810 | .15354 
 55 | .12468 | .14243 || .13825 | .15373 
 56 | .12482 | .14262 || .13839 | .15393 
 57 | .12496 | .14280 |] .13854 | .15412 
 58 | .12510 | .14299 || .13868 | .15431 
 59 | .12524 | .14317 |] .13883 | .15451 
 60 | .12538 | .14385 || .13897 | .15470 
 
 | 
 | 
 || 
 
 47% 
 
 Vers. | Ex. sec. 
 
 80° 
 Vers. |Ex.sec.| 
 8397 | .15470 
 .18412 | .15489 
 18427 | .15509 
 .18441 | .15528 
 .18456 |. .15548 
 .18470 | .15567 
 .18485 | .15587 
 .13499 | .15606 
 .13514 | .15626 
 .18529 ; .15645 
 .18548 | .15665 
 .13558_| .15684 
 slopioe| .1ov04 
 18587 | 115724 
 . 18602 ee 
 .18616 | .15763 
 .13681 | .15782 
 .13646 | .15802 
 .18660 | .15822 
 .18675 | .15841 
 .18690 | .15861 
 .18705 | .15881 
 .18719 | .15901 
 .18784 | .15920 
 .18749 | .15940 
 .18763 | .15960 
 1377 . 15980 
 .18798 | .16000 
 .18808 | .16019 
 .18822 | .16089 
 .138837 | .16059 
 .138852 | .16079 
 18 8867 .16099 
 £18881 | .16119 
 .13896 | .16139 
 .138911 | .16159 
 .138926 | .16179 
 .18941 | .16199 
 .18955 | .16219 
 .18970 | .16239 
 .18985 | .16259 
 .14000 | .1627 
 .14015 | .16299 
 .14030 | .16319 
 .14044 | .163839 
 -14059 | .16359 
 .14074 | .16380 
 14089 | .16400 
 .14104 | .16420 
 14119 | .16440 
 .14134 | .16460 
 .14149 | .16481 
 .14164 | .16501 
 AAD) 16521 
 14194 | .16541 
 .14208 | .16562 
 .14223 | .16582 
 .14288 | .16602 
 .14253 | .16623 
 .14268 | .16643 
 .14283 | .16663 
 
 .14283 | .16663 
 -14298 | .16684 
 .14813 | .16704 
 ~14828 | .16725 
 14843 | .16745 
 .14858 | .16766 
 14373 | .16786 
 .14388 | .16806 
 .14403 | .16827 
 .14418 | .16848 
 14433 | .16868 
 
 .14449 | .16889 
 14464 | .16909 
 14479 | .16930 
 .14494 | .16950 
 -14509 | .16971 
 .14524 | .16992 
 
 145389 | ..17012 
 .14554 | .17033 
 
 14569 | .17054 
 14584 | .17075 
 
 .14599 | .17095 
 
 -14615 | .17116 
 
 .146380 | .17187 
 
 .14645 | .17158 
 .14660 | .17178 
 .14675 | .17199 
 
 -14690 | .17220 
 
 14706 | .17241 
 14721 | -.17262 
 -14786 | 172838 
 14751 | .17804 
 
 14766 | .173825 
 14782 | .17346 
 
 14797 | .17367 
 .14812 | .17388 
 
 14827 | .17409 
 
 .14843 | .17430 
 .14858 | .17451 
 14873 | 17475 
 .14888 | .17493 
 .14904 | .17514 
 14919 | .17535 
 14934 | .17556 
 .14949 | .17577 
 .14965 | .17598 
 .14980 | .17620 
 14995 | .17641 
 15011 | .17662 
 15026 | .17683 
 .15041 | .17704 
 .15057 | .17726 
 15072 | .17747 
 .15087 | .17768 
 .15103 | .17790 
 15118 | .17811 
 .15134 | .17832 
 .15149 | .17854 
 15164 | .17875 
 .15180 | .17896 
 .15195 | .17918 
 
 — 
 —— 
 
338° 
 
 .16183 
 
 . 19236 
 
 COOBNAMTIR WWE © 
 
 / 
 Vers. |Ex.sec.|| Vers... |Ex. sec. Ex. see.|| Vers. |Ex. sec. 
 15195 17918 .16133 | .19236 . 20622 ~ 18085 | 22077 
 15211 17939 .16149 | .19259 | 20645 | .18101 | .22102 
 15226 17961 .16165 | .19281 .20669 LOIS ji eoalet 
 15241 17982 .16181 | .19304 | .20693 || .18185 | .22152 
 15257 18004 .16196 | .19327 | .20717 18152. |. 22177 
 5272 18025 .16212 | .19349 .20740 .18168 | .22202 
 15288 | .18047 || .16228 19372 | ,20764 .18185 | .22227 
 15303 | .18068 .16244 | .19894 -20788 || .18202 | .22252 
 15319 | .18090 .16260 | .19417 . 20812 .18218 | .22277 
 1533 .18111 .1627 .19440 . 20836 18235 | .22802 | 
 15350 | .18133 .16292 | .19463 .20859 £18252 |: .22327, | 10 
 .15365 | .18155 || .16808 | .19485 .20883 || .18269 | .22352 | tl 
 .15381 | .18176 || .16324 | .19508 | .20907 || .18286 | .22377 | 12 
 .15396 | .18198 || .16340 | .195381 .20981 |} .18302 | .22402 | 13 
 .15412 | .18220 .16355 | .19554 . 20955 .183"9 | .22428 | 14 
 .15427 | .18241 .16871 | .19576 | .20979 |; .18886 | .22453 | 15 
 15443 | .18263 .16387 | .19599 .21003 .18353 | .22478 | 16 
 .15458 | .18285 -16403 | .19622 || .21027 .18869 | .22503 | 17 
 .15474 | .18307 .16419 | .19645 || .21051 || .18886 | .2252 / 18 
 .15489 | .18328 .16485 | .19668 | .21075 || .18403 | .22554. | 19 
 .15505 | .18850 |} .16451 | .19691 | .21099 || .1842 222579 | 20 
 15520 | .18372 || .16467 | .19713 || | .21128 || .18437 | .22604 | 21 
 .15536 | .18394 || .16483 | .19736 || .21147 || .18454 | .22629 | 22 
 .15552 | .18416 || .16499 | .19759 || 21171 || .18470 | .22655 | 23 
 615567 | .18437 || .16515 | .19782 || .21195 || .18487 | .22680 | 24 
 .15583 | .18459 .16531 | .19805 || sele20 .18504 | .22706 | 25 
 .15598 | .18481 -16547 | .19828 .21244 || .18521 | 22731 | 2 
 .15614 | .18503 .16563 | .19851 .21268 || .18588 | .22756 | 27 
 .156380 | .18525 .16579 | .19874 .21292 || .18555 | .22782 | 28 
 .15645 | .18547 || .16595 | .19897 .21316 || £18572 | .22807 | 29 
 .15661 | .18569. || .16611 | .19920 21841 || .18588 | .22533 | 30 
 .15676 | .18591 .16627 | .19944 .21365 || £18605 | .22858 | 31 
 .15692 | .18613 .16644 | .19967 .21389 || .18622 | .22884 | 32 
 .15708 | .18635 .16660 | .19990 .21414 .18639 | .22909 | 33 
 »15728 | 518657 -16676 | .20013 21438 || 118656 | .22935 | 34 
 .15739 | .18679 .16692 | .20036 -21462 .18673 | .22960 | 35 
 .15755 | .18701 || .16708 | .20059 .21487 .18690 | .22986 | 36 
 .15770 | .18723 .16724 | .20083 ~21511 || 18707 | 28012 | 37 
 .15786 | .18745 .16740 | .20106 .21585 .18724 | .23037 | 38 
 .15802 | .18767 .16756 | .20129 .21560 .18741 | .23063 | 39 
 .15818 | .18790 || .16772 | .20152 .21584 || .18758 | .28089 | 40 
 .15833 | .18812-}| .16788 | .20176 .21609 || .18775 | .238114 | 41 
 .15849 | .18834 -16805 | .20199 .216383 18792 | .28140 | 42 
 .15865 | .18856 .16821 | .20222 | .21658 .18809 | .23166 | 43 
 .15880 | .18878 .16837 | .20246 | .21682 18826 | .238192 | 44 
 .15896 | .18901 .16853 | .20269 | 21707 .18848 | .28217 | 45 
 .15912 | .18923 .16869 | .20292 peliol .18860 | .28243 | 46 
 .15923 | .18945 .16885 | .2C316 | .21756 || .18877 | .238269 | 47 
 .15943 | .18967 ,16902 | .20389 .21781 .18894 | .238295 | 48 
 .15959 | .18990 .16918 | .20363 .21805 18911 | .28321 | 49 
 .15975 | .19012 || .16934 | .20386 .21830 || .18928 | .283847 | 50 
 15991 | .19034 || .16950 | .20410 .21855 || .18945 | .28373 | 51 
 .16006 | .19057 .16966 | .204133 | .21879 -18962 | .28399 | 52 
 .16022 | .19079 .16983 | .20457 .21904 .18979 | .23424 | 53 
 .16038 | .19102 || .16999 | .20480 .21929 .18996 | .238450 | 54 
 .16054 | .19124 || .17015 | .20504 .21953 .19018 | 223476 | 55 
 .16070 | .19146 |} .170381 | .20527 -21978 .19030 | .23502 | 56 
 216085 | .19169 .17047 | .20551 . 22003 -19047 | .28529 | 57 
 .16101 | .19191 .17064 | .20575 . 22028 .19064 | .28555 | 58 
 .16117 | .19214 || .17080 | .20598 . 22053 .19081 | .28581 | 59 
 .17096 | .20622 22077 .19098 | .23607 | 60 
 
| 87° | . 88° 3 
 , / 
 Vers. |Ex.sec. | Vers, |Ex.sec.|| Vers. |Ex.sec.|| Vers. |Ex.sec. 
 0 | .19098 | .23607 20186 | .25214 || .21199 | .26902 ||" .22285 | .28676 | .0 
 1 | .19115 | .23638 || .20154 | .25241 || .21217 | ..26981 || .223804 | .28706 1 
 2 | .19133 | .23659 || .20171 | .25269 || .21285 | .26960 || .22822 | .28737 | 2 
 31 .19150 | .23685 || .20189 | .25296 || .21253 | .26988 || .22340 | .28767 | 3 
 4} .19167_| .23711 20207 | .25824 |) .21271 | .27017 || .22859 | .28797 |) 4 
 5 | .19184 | .23738 || .20224 | .25351 || .21289 | .27046 |} .22377 | .28828 | 5 
 6 | .19201 | .23764 || .20242 | .25379 || .21307 | .27075 || .22895 | .28858 | 6 
 ~ | (19218 | .23790 || .20259 | .25406 || .21824 | .27104 || .22414 | .2888) | 7 
 8 | 19235 | .23816 || .20277 | .25434 |) .21842 | .27138 || .22432 | .28919 | 8 
 9 | .19252 | .23843 |! .20294 | .25462 || .213860 | .27162 || .22450 | .28950 | 9 
 10 | .19270 | .23869 |} .203812 | .2548 21378 | .27191 || .22469 | .28980 | 10 
 
 19287 
 19804 
 
 19321 
 19338 
 
 19356 
 19873 
 ,19390 
 19407 
 ,19424 
 
 19442 
 
 19459 
 
 .19476 
 .19493 
 .19511 
 .19528 
 .19545 
 . 19562 
 .19580 
 .19597 
 .19614 
 .19632 
 .19649 
 .19666 
 1684 
 19701 
 19718 
 .19736 
 19753 
 19770 
 19788 | 
 
 .19805 
 
 - 19822 
 .19840 
 
 .19857 
 19875 
 .19892 
 .19999 
 .19927 
 .19944 
 .19962 
 .19979 
 .19997 
 .20014 
 20082 
 20049 
 20066 
 . 20084 
 .20101 
 .20119 
 . 20136 
 
 .23895 
 
 WY. 
 ~ 23922 || 
 
 23948 
 . 23975 
 .24901 
 24028 
 . 24054 
 . 24081 
 .24107 
 .24134 
 
 24160 | 
 24187 
 
 24913 
 24240 
 24267 
 24293 
 24320 
 24347 
 24373 
 24400 
 
 AAQ™ 
 24427 
 
 24454 
 -24481 
 .24508 
 24534 
 .24561 
 -24588 
 .24615 
 . 21642 
 .24669 
 24696 
 s24 (25 
 24750 
 QATTT 
 24804 
 24832 
 24859 
 . 24886 
 ,21913 
 .24940 
 24967 
 24995 
 
 £25022 
 
 25049 | 
 225077 
 .25104 
 
 .20347 
 20365 
 20382 
 20400 
 
 20470 
 
 20594 
 20612 
 .20629 
 .20647 
 20665 
 . 20682 
 .20700 
 
 .20718 
 20736 
 20758 
 20071 
 20789 
 
 . 20329 
 
 20417 | .25656 
 ny~ ~Poce | 
 20435 | .25683 | 
 
 | (20453 | .25711 
 
 20488 | .25767 | 
 20506 | .25795 
 "20523 | 125823 | 
 (20541 | 125851 
 "20559 | .25879 
 (20576 | 125907 | 
 
 20807 
 20824 
 20842 
 .20860 
 
 nQ 
 .20878 
 
 20895 
 .20913 
 20931 
 .20949 
 20967 
 20985 
 21002 
 -21020 
 21038 | 
 .21056 
 21074 | 
 21092 
 21109 
 21127 
 21145 
 21163 
 .21181 
 21199 
 
 ape aty Wf 
 .25545 
 +L0018 
 .25600 
 25628 
 
 25739 
 
 25935 
 25963 
 
 .25991 
 .26019 
 26047 
 
 26075 
 26104 
 26132 
 26160 
 :26188 
 26216 
 (26245 
 26273 | 
 
 26500. || 
 
 ORO 
 
 26801 
 . 263380 
 
 26529 
 20557 
 26586 
 26615 
 26643 
 26672 
 26701 
 26729 
 
 Bem 
 20787 | 
 
 21396 
 21414 
 21482 
 .21450 
 21468 
 21486 
 21504 
 21522 
 .21540 
 21558 
 . 21576 
 21595 
 21613 
 .21631 
 .21649 
 .21667 
 .21685 
 21708 
 21621 
 21739 
 
 221057 
 
 606 
 
 21705 
 21794 
 .21812 
 .21830 
 21848 
 .21866 
 .21884 
 21902 
 21921 
 
 .26858 || .21939 
 26387 || .21957 
 26415 || .21975 
 26448 || .21993 
 26472 || .22012 
 
 22030 
 22048 
 22066 
 22084 
 .22103 
 eal 
 22139 
 222157 
 22176 
 22194 
 
 wouehe 
 
 26815 || -,2223 
 26844 || .22249 
 26873 | 22267 
 26902 || .22285 
 
 efo)779) 
 
 20221 
 120200 
 evel) 
 27308 
 
 21888 
 27866 | 
 .27296 
 27425 
 
 27454 
 27483 
 627513 
 
 .20042 
 
 OrRroD 
 
 ~wHIIA 
 .27601 
 .27630 
 .27660 
 .27689 
 Peels) 
 27748 
 226018 
 
 27807 
 
 orRar 
 
 etl O0d 
 
 27267 
 
 #0 
 
 .27896 
 24926 
 .27956 
 .27985 
 .28015 
 28045 
 28075 
 228105 
 28184 
 .28164 
 £28194 
 28224 
 ~29204 
 28284 
 .28314 | 
 28344 
 28374 
 
 28404 
 28484 
 28464 
 
 28495 
 
 » 
 
 ee he 
 
 s 20000 
 -28585 
 .28615 
 . 28646 
 £28676 
 
 22487 
 22006 
 99594 
 
 s U, 
 
 . 22542 
 . 22561 
 22579 
 , 22598 
 22616 
 22634 
 .22653 
 22671 
 . 22690 
 22708 
 OAH | 
 622745 
 22764 
 -22¢82 
 .22801 
 -22819 
 22838 
 . 22856 
 22875 
 .22893 
 .22912 
 £22330 
 . 22949 
 . 22967 
 . 22986 
 . 28004 
 123023 
 .23041 
 . 23060 
 . 23079 
 .238097 
 .23116 
 .23134 
 .23153 
 23979 
 .23190 
 $23209 
 28298 
 .23246 
 .23265 
 . 238283 
 ~28002 
 523821 
 20808 
 28358 
 
 oonr 
 . 2 wid 
 
 23396 
 
 479 
 
 OU 
 
 .80255 | 5i 
 .80287 | 52 
 .80818 | 53 
 .80350 | 54 
 
 80382 | 55 
 .80413 | 56 
 .80445 | 57 
 80477 | 58 
 .80509 | 59 
 .80541 | 60 
 
 .29011 | 11 
 29042 | 12 
 .29072 | 13 
 .291038 | 14 
 .29188 | 15 
 .29164 | 16 
 229195 4) Ty 
 29226 | 18 
 29256 | 19 
 .29287 | 20 
 
 29818 | 21 
 129349 | 22 
 29380 | 23 
 120411 | 24 
 "29442 | 25 
 129473 | 26 
 129504 | 27 
 29535 | 28 
 "29566 | 29 
 (29597 | 80 
 
 .29628 i 
 .29659 | 32 
 .29690 | 33 
 £29721 | 34 
 .29752 | 85 
 .29784 | 36 
 .29815 | 37 
 29846 | 38 
 29877 | 39 
 .29909 | 40 
 
 .29940 | 41 
 299701 2 
 .80003 | 43 
 380034 | 44 
 .80066 | 45 
 .80097 | 46 
 .80129 | 47 
 .380160 | 48 
 .30192 | 49 
 
 30228 | 50 
 
 TABLE XXIX.—NATURAL VERSED SINES AND EXTERNAL SECANTS., 
 
 | 
 
Vers. 
 
 203396 
 20414 
 23433 
 23452 
 238470 
 20489 
 28508 
 20027 
 238545 
 . 23564 
 -20083 
 23602 
 23620 
 23639 
 23658 
 28677 
 23696 
 23014 
 23733 
 .28152 
 
 oQrr 
 roe | 
 
 23790 
 23808 
 23827 
 23846 
 23865 
 23884 
 23903 
 23922 
 23941 
 23959 
 23978 
 23997 
 24016 
 24035 
 24054 
 24073 
 24092 
 24111 
 24130 
 24149 
 
 24168 
 24187 
 24206 
 24225 
 24244 
 24262 
 2d281 
 .24500 
 24320 
 24339 
 24358 
 24377 
 24396 
 24415, 
 24484 
 .24453 
 
 24472 
 
 24491 
 -24510 
 «24529 
 
 40° 
 
 Ex. sec. 
 
 .80541 
 . 80573 
 .80605 
 . 30636 
 . 380668 
 .80700 
 .80732 
 .80764 
 .80796 
 80829 
 .30861 
 
 380893 
 . 80925 
 80957 
 . 80989 
 .31022 
 -81054 
 .31086 
 .31119 
 .d1151 
 31183 
 
 81216 
 .381248 
 81281 
 -81313 
 
 81546 | 
 
 31378 
 31411 
 .31443 
 31476 
 .381509 
 31541 
 381574 
 .31607 
 .31640 
 81672 
 81705 
 31738 
 O1T71 
 31804 
 381837 
 
 .31870 
 .51908 
 . 31936 
 . 381969 
 . 82002 
 82035 
 - 82068 
 .82101 
 8215: 
 
 .82168 
 
 32201 
 
 99909 
 . 8223 
 
 82267 
 
 32801 
 82834 
 
 .82868 
 .82401 
 
 82434 
 
 . 382468 
 82501 
 
 41° 
 
 Vers. 
 
 24529 
 24548 
 24567 
 24586 
 24605 
 24625 
 24644 
 24663 
 24682 
 24701 
 24720 
 24739 
 24759 
 2407 
 
 24797 
 24816 
 24835 
 24854 
 24874 
 24893 
 24912 
 
 .24931 
 
 24950 
 24970 
 24989 
 25008 
 25027 
 25047 
 25066 
 25085 
 25104 
 25124 
 25143 
 25162 
 25182 
 25201 
 25220 
 25240 
 
 «20259 
 
 95278 
 
 «On 
 
 20297 
 
 25317 
 . 25336 
 203856 
 253875 
 20394 
 .20414 
 25433 
 20452 
 25472 
 20491 
 
 .20511 
 .25530 
 25549 
 .25569 
 -25588 
 .25608 
 25627 
 - 20047 
 25666 
 .25686 
 
 Ex. sec. 
 
 32501 
 82535 
 82568 
 382602 
 - 32636 
 82669 
 32703 
 £82737 
 82770 
 82804 
 328388 
 382872 
 82905 
 382939 
 82973 
 .83007 
 .338041 
 83075 
 .33109 
 .83143 
 83177 
 838211 
 83245 
 38327 
 
 33314 
 . 33348 
 33382 
 33416 
 -838451 
 . 338485 
 .33519 
 33004 
 33098 
 83622 
 83657 
 .33691 
 33726 
 .33760 
 .83795 
 33830 
 33864 
 
 .33899 
 33934 
 .383968 
 - 840038 
 .84088 
 84073 
 84108 
 .84142 
 84177 
 84212 
 84247 
 . 84282 
 34317 
 . 34352 
 34387 
 .84423 
 .34458 
 84493 
 34528 
 
 84563 
 
 43° 
 
 43° 
 
 Vers. Ex. sec. 
 25686 | .34563 
 25705 | .84599 
 20724 | .34634 
 20044 | .34669 | 
 .207163 | .84704 
 .20783 | .84740 
 .20802 | .84775 
 .20822 | .384811 
 .25841 | .84846 
 .20061 | .384€82 
 .20880 | .384917 
 .25900 | .24953 
 .25920 | .84988 
 .25939 | .385024 
 .25959 | .85060 
 .20978 | .85095 
 .20998 | .35131 
 -26017 | .85167 
 26037 | .35203 
 -26056 | .85238 
 26076 | .3527 
 .26096 | .35310 
 .26115 | .85346 
 .26135 | .385382 
 .26154 | .385418 
 .26174 | .385454 
 .26194 | .385490 
 .26213 | .85526 
 .26283 | .85562 
 .26253 | .85598 
 -26272 | .385634 
 -26292 | .35670 
 26312 | .385707 
 .26031 | .85743 
 -26351 | 38577 
 26871 | .85815 
 .26390 | .85852 
 .26410 | .35888 
 -26430 | .35924 
 -26449 | .385961 
 26469 | .35997 
 .26489 | .386034 
 .26509 | .386070 
 .26528 | .386107 
 .26548 | .386143 
 .26568 | .386180 
 .26588 | .36217 
 »26607 | .86253 
 .26627 | .86290 
 .26647 | .86327 
 .26667 | .36363 
 .26686 | .386400 
 .26706 | .36437 
 26726 | .86474 
 .26746 | .386511 
 .26766 | .86548 
 .26785 | .86585 
 -26805 | .386622 
 -26825 | .36659 
 -26845 | .386696 
 .26865 | .38673 
 
 Vers. 
 
 26865 
 - 26884 
 26904 
 .26924 
 26944 
 26964 
 
 | 26984 
 | .27'004 
 | 27024 
 
 .24043 
 .21063 
 -27083 
 .21103 
 -21128 
 27143 
 27163 
 21183 
 21203 
 21223 
 21243 
 21263 
 27283 
 .27803 
 27823 
 27843 
 21363 
 21883 
 21403 
 21423 
 -27443 
 21463 
 «27483 
 27503 
 221523 
 -27543 
 621563 
 -27583 
 27603 
 271623 
 271643 
 -27663 
 -27683 
 221103 
 £20023 
 2771483 
 27764 
 20784 
 27804 
 21824 
 27844 
 27864 
 27884 
 .27905 
 .21925 
 20945 
 27965 
 24985 
 28005 
 28026 
 
 .28046 
 28066 
 
 Ex. sec. 
 
 86788 
 
 86770 
 .86807 
 86844 
 -56881 
 .86919 
 .86956 
 
 -87293 
 .87880 
 .87368 
 -387406 
 .87443 
 37481 
 387519 
 
 387898 
 .37936 
 81974 
 - 38012 
 .38051 
 88089 
 38127 
 38165 
 - 88204 
 38242 
 
 388280 
 38319 
 . 88357 
 38396 
 .38434 
 .388473 
 38512 
 - 388550 
 . 88589 
 . 88628 
 . 38666 
 . 88705 
 38744 
 88783 
 . 38822 
 . 8&860 
 . 38899 
 .33938 
 88977 
 
 .39016 
 
| 
 
 45° 
 
 | 
 
 28066 
 28086 
 .28106 
 28127 
 28147 
 23167 
 28187 
 28208 
 2822! 
 28248 
 28268 
 
 11 | .28289 
 12 | .28309 
 13 | .28329 
 14 | .28350 
 15 | .28370 
 16 | .28390 
 17-| 28410 
 18 | .28431 
 19 | .28451 
 20 | .28471 
 21 | .28492 
 22 | .23512 
 23 | .28532 
 24 | 128553 
 
 25 | .28573 
 26 | .28593 
 27 | .28614 
 28 | .28634 
 29 | .28655 
 30 | .28675 
 
 —_ pt 
 _ Bhan Rees 
 
 31 | .28695 
 2 | .28716 
 
 33 | .28786 
 34 | .28757 
 85 | .28777 
 36 | .28797 
 87 | .28818 
 88 | .28838 
 89 | .28859 
 40 | .28879 
 41 | .28900 
 42 | .28920 
 43 |. .28941 
 44 | .28961 
 45 | .28981 
 46 | .29002 
 47 | .29022 
 48 | .29043 
 49 | .29063 
 
 50 | .29084 
 51! .29104 | 
 52 | .29125 | 
 53 | 29145 | 
 54 | .29166 
 55 | .29187 
 56 | .29207 
 Bz | "99228 
 58 | .29248 
 59 | 29269 
 GO | .29289 
 
 .89685 
 39725 
 
 Vers. |Ex. sec. 
 
 .39016 
 89053 
 89095 
 .89134 
 .389173 
 .89212 
 89251 
 .39291 
 89330 
 .39369 
 .389409 
 
 89448 
 89487 
 389527 
 39566 
 389606 
 .39646 
 
 39764 
 .dd804 
 .39844 
 39884 
 .39924 
 .89963 
 .40003 
 .40043 
 .40083 
 .40123 
 .40163 
 .40203 
 .40243 
 .40283 
 -40324 
 .40364 
 .40404 
 .40414 
 .40485 
 -40525 
 .40565 
 .40606 
 
 40646 
 40687 
 40727 
 40768 
 .40808 
 40849 
 40890 
 40930 
 40971 
 41012 
 41058 
 41098 
 41134 
 41175 
 41216 
 41257 
 41208 
 “41339 
 41380 
 41421 
 
 | ,29289 
 , 29310 
 
 | 120454 
 
 | .29805 
 
 Vers. |Ex.sec.|| Vers. |Ex. sec. 
 
 i 41421 
 "41463 
 
 29330 
 29351 
 29372 
 29392 
 29413 
 29433 
 
 29475 
 .29495 
 .29516 
 £29537 
 .29557 
 29578 
 .29599 
 .29619 
 .29540 
 . 29561 
 29581 
 .29702 
 . 297238 
 £29743 
 .297'64 
 .29785 
 
 29826 
 29847 
 29868 
 .29888 
 .29909 
 29930 
 .29951 
 -29971 
 29992 
 .80013 
 80034 
 380054 
 80075 
 .380096 
 .30117 
 
 .301388 
 .380158 
 .30179 
 .380200 
 ~8022 
 .80242 
 .30263 
 .380283 
 .30304 
 .80825 
 .80346 
 .80367 
 .80388 
 00409 
 .380430 
 .30451 
 .80471 
 80492 
 .30513 
 80584 
 
 .42126 
 .42168 
 . 42210. 
 .42251 
 42293 
 .42335 
 42377 
 .42419 
 .42461. 
 -42503 
 
 2545. 
 .42587 
 .42630 
 42672 
 42714 
 .42756. 
 .42799 
 .42841 
 .42883 
 . 42926 
 .42968 
 .43011 
 .43053 
 .43096 
 
 438189 
 43181 
 43224 
 43267 
 43310 
 43352 
 43395 
 43438 
 43481 
 43524 
 43567 | 
 43610 
 
 43653 | 
 43596 
 43739 
 43783 
 43826 
 
 43869 
 43912 
 
 43956 
 
 481 
 
 46° 
 .30534 | .43956 
 30555 | .43999 
 || .80576 | .44042 
 || .80597 | .44086 
 || 80618 | .44129 
 30639 | .4417% 
 || 80660 | .44217 
 | 30681 | .44260 
 .30702 | .44304 
 -30723 | .44347 
 30744 | .44301 
 .30765 | .44435 
 30786 | .4447 
 30907 | .44523 
 30828 | .44567 
 30849 | .44610 
 -80870 | .44654 
 .30891 | .44698 
 30912 | .44742 
 .80988 | .44787 
 80954 | .44831 
 .809%5 | .44875 
 .30096 | .44919 
 /31017 | .44963 
 .31088 | .45007 
 .31059.| .45052 
 .31030.| .45096 
 .31101 | .45141 
 81122 | .45185 
 181143 | .45229 
 81165 | .4527 
 .81186 | .45319 
 31207 | .45363 
 (31228 | .45408 
 (81249 | .45452 
 31270 | .45497 
 81291 | .45542 
 '81312.| .45587 
 31384. .45631 
 81355 | .45676 
 181376 | .45721 
 .81397 | .45766 
 (31418.| .45811 
 | 131489 | .45856 
 131461 | .45901 
 | 181482 | .45946 
 "31503 | .45992 
 131524 | .46037 
 31545 | .46082 
 .81567.| .46127 
 181588 | .4617 
 .81609 | .46218 
 .381630.| .46263 
 '31651 | .46309 
 | :31673 | .46354 
 || 81694 | .46400 
 || 81715 | .46445 
 "31736 | .46491 
 "81758 | .46537 
 (31779 | .46582 
 "31800 | .46628 
 
 | 
 
 .31800 
 31821 
 81843 
 .31864 
 81885 
 .31907 
 31928 
 .381949 
 31971 
 -31992 
 82018 
 
 82035 
 82056 
 82077 
 82099 
 382120 
 32141 
 32163 
 82184 
 82205 
 82227 
 32248 
 82270 
 82291 
 -32312 
 82384 
 823595 
 82377 
 82398 
 82420 
 82441 
 
 . 82462 
 . 32484 
 . 32005 
 o2027 
 .82548 
 .o200 
 
 .382091 
 | .3826138 
 82634 
 -382656 
 02677 
 -32699 
 .82120 
 02742 
 .82163 
 ~82780 
 . 02006 
 -82828 
 -82849 
 
 82871 
 
 .82893 
 82914 
 
 47° 
 
 Vers. 
 
 | .82936 
 
 82957 
 82979 
 .83001 
 80022 
 .83044 
 .38065 
 
 | 38087 
 
 Ex. sec. 
 
 .46628 
 46674 
 .46719 
 46765 
 .46811 
 46857 
 .46903 
 .46949 
 .46995 
 47041 
 47087 
 47134 
 47180 
 47226 
 P2772 
 47319 
 -47365 
 47411 
 47458 
 47504 
 47551 
 47598 
 47644 
 47691 
 
 MAIOQ 
 . 4 (80 
 
 ATT84 
 47831 
 . 4787! 
 
 47925 
 47972 
 
 48019 
 
 -48066 
 .48113 
 -48160 
 48207 
 48254 
 .48301 
 -48349 
 -48396 
 .48448 
 -48491 
 
 .48538 
 48586 
 .486383 
 48681 
 48728 
 4877 
 
 48824 
 48871 
 48919 
 -48967 
 
 .49015 
 .49063 
 49111 
 .49159 
 49207 
 -49255 
 .49303 
 .49351 
 .49399 
 49448 
 
 TABLE XXIX.—NATURAL VERSED SINES AND EXTERNAL SECANTS. 
 
 ee a a eee 
 
 44° 
 
oy 
 | 
 
 TABLE XXIX.—NATURAL VERSED SINES AND EXTERNAL SECANTS. 
 
 Se ee " | 
 SS ae 50° ! 51° | 
 
 ! | 
 Vers. |Ex. sec. Vers. |Ex.sec.!} Vers. |Ex. sec. || Vers. |Ex. sec. 
 
 Sia ae : | = AA | te 
 0 | .33087 | .49448 || .343894 | .52495 || 85721 | .55572 {| .87068 | .58902 Oo.) 
 1 | .83109 | .49496 384416 | .52476 .385744 | .55626 87091 | .58959 1 
 2 | .3381380 | .49544 || .34438 | .59597 || .85766 | .55680 i| .87113 | .59016 2 
 8 | .388152 | .49593 .84466 | 252579 .85788 | 55734 .387186 | .59073 3 
 4 .383173 | .49641 -384482 | .52630 .85810 | .55789 .387158 | .59130 4 
 5 | .83195. | .49690 .84504 | .52681 || .35833 | .55843 .87181 | .59188 5 
 6 | .388217 | .49738 .384526 | .52732 85855 | .55897 || .87204 | .59245 6 
 7 | .83238. || .49787 .34548 | .52784 385877 | 255951 .87226 | .59302 iy 
 8 | .83260 | .49835 .84570 | .52835 .85900 | .56005 .87249 | .59360 8 
 9 | .83282 | .49884 .384592 | .52886 .85922 | .56060 87272 | .59418 9 
 
 10 | .33303.| .49933 || .34614 | .52938 || .35944 | . 56114 || .387294 | .59475.| 10 
 11 | .33325.| .49981 || .84636 | .52989 |] .35967 | . 56169 || .87317 | .59583 | 11 
 12 | .383347 | .50030 .384658 | .53041 .85989 | .56223 .387340 | .59590 | 12 
 13 | .38868 | .50079 . 84680 | .53092 .86011 . 56278 .87362 | .59648 | 13 
 
 14 | .33390.| .50128 || .34702 | .53144 || .36034 | _56339 .87385 | .59706 | 14 
 15 | .33412 | .50177 || .84724 | .53196 || .36056 | . 56387 || .87408 | .59764 | 15 
 16 | .33434 | .50226 || .84746 | .53247 || .36078 -56442 || .87430 | .59822 | 16 
 17 | .383455.| .50275 || .84768 | .58299 || .36101 56497 || .387453.| .59880 | 17 
 18 | .33477.| .50324 || .84790 | .53351 || .36123 | | 56551 || .87476 | .59938 | 18 
 19 | .83499.| .5087% -34812 | .53403 || .386146 | .56606 || .87498 | .59996 19 
 20 | .33520.} .50422 || .34834 | .538455 || .36168 .56661 .87521 | .60054 | 20 
 
 21 | .83542 | .50471 .384856 | .53507 |} .86190 | .56716 || .87544 | .60112 21 
 22.| .83564.} .50521 84878 | .53559 || .86213 | .56771 .87567 | .60171 | 22 
 23 | .33586.| .50570 |} .84900 | .53611 || .36935 | | 56826 || .87589 | .60229 | 23 
 24} 33607 | .50619 || .84923 | .53663 || .36258 -56881 .87612 | .60287 | 24 
 25 | .83629 | .50669 || .84945 | .53715 || .36980 | - 56987 || .87635 | .60346 | 25 
 26 | .33651.} .50718 || .84967 | .53768 || .36802 | | 56992 || .87658 | .60404 | 2 
 
 27 | .83673.| .50767 |] .84989 | 53820 || .36305 | | 57047 || .87680 | .60463 | 27 
 28 | .383694./ .50817 || .85011 | .538872 || .36347 -57103 || .87703 | .60521 | 28 
 29 | .383716.| .50866 || .85033 |} .53924 || .36370 ~7158 || .37'726 | .60580 | 2 
 
 30 | .38738.| .50916 || .85055 | .53977 || .36392 | 7 57213 || .87749 | .60639 | 30 
 31 | .83760.] .50966 || .85077 | .54029 || .36415 | . 57269 || .8777 .60698.| 31 
 32 | .33782 | .51015 || .85099 | .54082 || .36¢437 | | 57324 || .87794 | .60756.] 32 
 33 | .33803.] .51065 || .35122 | .54134 || .364co0 | | 57380 |} .387817 | .60815 | 3° 
 
 34, .338825 | .51115 || .85144 | .54187 || .36¢482 | | 57436 || .87840.| .60874.| 34 
 35 | .83847 | .51165 || .85166 | .54240 || 36504 57491 .37862.| .60983 | 35 
 36 | .33869 | .51215 || .85188 | .54292 || /36507 57547 || .87885.| .60992 | 36 
 37 | .33891.) .51265 || .85210 | .54345 || .36549 | | 57603 || .87908 | .61051 | 37 
 38 | .83912 | .51314 || .35932 | .54398 || .36572 | | 7659 || .87931 | .61111 | 38 
 39 | .83934 | .51364.|} .35254 | .54451 || 36594 57715 || .87954.| .61170 | 39 
 40 | .33956.] .51415.|| .85277.| .54504 || .36617 | 5777 .37976.| .61229 | 40 
 
 41 | .83978 | .51465 || .85299 | .54557 || .36689 | . 7827 || .87999 | .61288 | 41 
 42 | .84000 | .51515 || .3532 -54610 || .386662 | .57883 |] .388022 | .61348 | 42 
 43 | .384022 | .51565 || .35343 | .54663 || .36684 | |b? 939 |} .88045.] .61407 | 43 
 44°| .34044 | .51615 || .85365 | .54716 || .36707 | | 57995 || .88068 | .61467 | 44 
 45 | .384065 | .51665 || .85388 | .54769 || .36799 | | 58051 || .88091 | .61526 | 45 
 46 | .34087 | .51716 || .85410 | .54822 || .3¢6759 .58108 4 .38113 | .61586 | 46 
 
 7 | .84109 | .51766 || .85432 | .54876 || 36775 .58164 || .88136 |} .61646 | 47 
 48 | .34131 ; .51817 || .85454 | .54929 || .36797 58221 || .88159 | .61705.| 48 
 49 | .34153 | .51867 || .85476 | .54982 || .36890 58277 || .38182 | .61765 | 49 
 
 0 | .34175 | .51918 || .85499 | .55086 || .36842 58333 || .38205 | .61825 | 50 
 
 1 | .34197 | .51968 || .85521 | .55089 || . 36865 58390 || .8822 .61885 | 51 
 52 | .84219 | .52019 || .85543 | 155143 |) |36887 58447 ||| 388251 | .61945 | 52 
 
 3 | .84241 | .52069 || .85565 | .55196 || .36910 | | 58503 || .88274 | .62005 | 53 
 S4 | .34262 | .52120 || .85588 | .55250 || .36932 | “585 30 || .88296 | .62065 | 54 
 5d | .34284 | .52171 || .35610 | .55303 || .36955 | | 58617 || .88819 | .62125 | 55 
 56 | .34306 | .52222 || .85632 | .55357 || .36978 .58674 || .38342 | .62185 | 56 
 57 | .34828 | .52273 || .35654 | .55411 || .37000 .58731 .88365 | .62246 | 57 
 58 | .34350 | .52323 || .35677 | .55465 || |aro9¢ -58788 || .38388 | .62306 | 58 
 
 9 | .84372 | .5237 -385699 | .55518 |) .87045 | .58845 || .388411 | _62366 | 59 
 60) .34894 | .52425 || .85721 | .55572 || 37068 | | 8902 || .38434 | .62427 | 60 
 
 482 
 
TABLE XXIX.—NATURAL VERSED SINES AND EXTERNAL SECANTS, 
 
 ~ 
 
 52° 
 
 53° 
 
 54° 
 
 55° 
 
 Vers. |Ex.sec.|| Vers. |Ex.sec.|| Vers. |Ex.sec.|| Vers. |Ex. sec. 
 | | 
 0 | .38434 | .62427 || .89819 | .66164 || .41221 | .70180 || .42642 | .74345 
 1 | .38457 | .62487 || .39842 | .66228 || .41245 | .70198 || .42666 | .74417 
 2 | 38480 | .62548 || .39865 | .66292 || .41269 | .70267 |) .42690 | .74490 
 3 | 38503 | .62609 || .39888 | .66357 || .41292 | .70335 || .42714 | . 74562 
 4 | 38526 | .62669 |} .39911 | .66421 || .41316 | .70403 || .42738 | .74635 
 5 | .38549 | .62730 || .89935 | .66486 || .41339 | .70472 || .42762 | .74708 
 6 | .38571 | .62791 || .89958 | .66550 |} .41863 | .70540 |; .42785 T4781 
 7 | 138594 | .62852 || .89981 | .66615 || .41386 | .70609 || .42809 | .74854 
 g | 38617 | .62913 || .40005 | 66679 || .41410 | .70677 || .42833 | . 74927 | 
 9 | 38640 | .62974 || .40028 | .66744 || .41433 | .70746 || .42857 | 75000 | 
 10 | .38663 | .63035 || .40051 | .66809 || .41457 | .70815 || .42881 | .750%3 | 
 3866 | .63096 || .40074 | .66873 || .41481 | .70884 || .42905 | .75146 | 
 "39709 | .63157 || .40098 | .66938 || .41504 | .70953 || .42029 | .75219 | 
 “gare | (63218 || .40121 | .67003 || .41528 | .71022 || .42953 | .75293 | 
 38755 | .68279 || .40144 | .67068 || .41551 | .71091 || .42976 | .75366 | 
 8877 63341 40168 | .67133 || .41575 | .71160 || 43000 | .75440 | 
 88801 | .63402 || .40191 | .67199 || .41599 | .71229 || .48624 | . 75513 | 
 "38824 | .63464 || .40214 | .67264 || .41622 | .71298 || .43048 15587 
 "99947 | _63525 || .40237 | .67329 || .41646 | .71368 || .48072 | .75661 
 | 38870 | .63537 || .40261 | .67394 || .41670 | .71437 || .48096 | .75734 
 | “388903 | .63648 || .40284 | .67460 || .41693 | .71506 || .43120 | .75808 | 
 38916 | .63710 || .40807 | .67525 || .41717 | .71576 || .43144 | .75882 | 
 "38939 | .63772 || .40381 | .67591 || .41740 | .71646 || .43168 | .75956 | 2 
 '38962 | .68834 || .40354 | .67656 || .41764 | 71745 || .43192 | .76081 
 | /38985 | .63895 || .40378 | .67722 || .41788 | .71785 || .48216 | .76105 | 
 "39009 | .63957 || .40401 | .67788 || .41811 | .71855 |) .48240 | .76179 
 39032 | .64019 || .40424 | .67853 || .41835 | .71925 || .43264 | .76253 | 2 
 "39055 | .64081 || .40448 | .67919 || .41859 | .71995 || .48287 | .76328 | 2 
 39078 | .64144 || .40471 | .67985 || .41882 | .72065 |) .43311 | 76402 | 
 39101 | .64206 || .40494 | .68051 41906 | .72135 || .48385 | .7647% 
 "39124 | .64268 || .40518 | .68117 || .41930 | .72205 || .48359 | .76552 | 
 39147 | .64330 || .40541 | .68183 || .41953 | .72275 || .48383 | .76626 | 
 "39170 | .64393 || .40565 | .68250 || .41977 | .72846 || .4340¥ | .7 701 
 “39193 | .64455 || .40588 | .68316 || .42001 | .72416 || .43431 76776 
 39216 | .64518 || .40611 | .68382 || .42024 | .72487 || .43455 | .76851 
 "39239 | .64580 || .40635 | .68449 || .42048 | .72557 || .43479 | .76926 | 
 39262 | .64643 || .40658! .68515 || .42072 | .72628 || .48503 | .77001 
 "39286 | .64705 || .40682 | .68582 || .42096 | .72698 || .48527 | 700% 
 "39309 | .64768 || .40705 | .68648 || .42119 | .72769 || .48551 | 77152 
 . 39332 | .64831 40728 | .68715 || .42143 | .72840 || .48575 | 77227 
 "39355 | .64894 || .40752 | .68782 || .42167 | .72911 || .48599 | .77308 
 39378 | .64957 || .40775 | .68848 || .42191 | .72982 || .48623 | .7737 
 "39401 | .65020 || .40799 | .68915 || .42214 | .73053 || .48647 | .774 54 
 "39494 | 65083 || .40822 | .68982 || .42288 | .73124 || .48671 | .775e 
 99447 | .65146 || .40846 | .69049 || .42262 | .73195 || .48695 -77606 
 "39471 | .65209 || .40869 | .69116 || .42285 | .73267 || .48720 | .77681 
 "39494 | .65272 || .40893 | .69183 || .42309 | .73338 || 48744 | 7757 
 "39517 | .65336 || .40916 ; .69250 || .42333 | .73409 || .48768 | .77883 
 39540 | .65399 || .40939 | .69318 || .42857 | .73481 | 48792 | . 77910 
 "39563 | .65462 || .40963 | .69385 || .42381 | .78552 || 43816 77986 
 -39586 | .65526 || .40986 | .69452 || .42404 | .73624 | 43840 | .78062 
 39610 | .65589 || .41010 | .69520 || .42428 | .73696 || .48864 | .7818E 
 39633 | .65653 || .41033 | .69587 42452 | .78768 || .48888 | .78215 
 | 39656 | .65717 || .41057 | .69655 || .42476 |. . 73840 | .48912 | .78291 
 39679 | .65780 || .41080 | .69723 42499 | .73911 || .48986 78268 
 "39702 | .65844 || .41104 | .69790 || .42523 | .73983 || .48960 78445 
 | 39726 | .65908 |} .41127 | .69858 || .42547 | 74056 || .48984 78521 
 39749 | .65972 || .41151 | .69926 || .4257 74128 || .44008 | .78598 
 "39772 | .66086 || .41174 | .69994 || .42595 | .74200 || .44032 78675 
 "39795 | .66100 |} .41198 | .70062 || .42619 | .74272 || .44057 18752 
 "39819 | .66164 || .41221 | .70180 Il .42642 | .74345 |! .44081 . 78829 
 
 483 
 
 WODVIATIKRWMH OS 
 
TABLE XXIX.—NATURAL VERSED SINES AND EXTERNAL SECANTS, 
 
 | I 
 
 56° 57° | 58° 59° 
 yom | ——w —o 
 
 | Vers. |Ex.sec.|| Vers, |Ex.sec.'| Vers. \Ex. sec.|; Vers. Ex. sec. 
 
 i | ae ey | r " aes near ul 
 0 | .44081 | .78829 | .45536 | .83608 -47008 | .88708 || .48496 | .94160 | 
 1 | .44105 | .78906 .45560 | .83690 || .47033 | .88796 -48521 | .94254 | 
 2 | .44129 | .78984 45585 | .83773 || .47057 | . 88884 |; .48546 | .94349 
 3 | .44153 | .79061 45609 | .83855 || .47082 | .88972 || (48571 | .94443 
 4 | .44177 | .79138 45634 | .83938 || .47107 | .89060 || .48596 | .94537 | 
 5 | .44201 | .'79216 45658 | .84020 || .471381 | .89148 .| .48621 | .94632 | 
 6 | .44225 | .79293 45683 | .84103 || .47156 | .89237 || .48646 | -94726 
 7 | .44250 | 79871 45707 | .84186 |! .47181 | .89825 || .48671 | .94821 
 8 | .44274 | .79449 45731 | .84269 || .47206 | .89414 || .48696 | .94916 
 9 | .44298 | .79527 45756 | .84352 || .47230 | .89503 || .48721 | .95011 
 10 | .44822 | .79604 45780 | .84435 || .47255 | .89591 || .48746 | .95106 
 11 | .44346 | .79682 45805 | .84518 | .47280 | .89680 || .48771 | .95201 
 12 | .44370 | .79761 45829 | .84601 || .47304 | .89769 -48796 | .95296 
 13 | .44895 ; .79839 45854 | .84685 |! .47329 | .89858 -48821 | .95392 
 14 | .44419 | .79917 || .45878 | .84768 47354 | .89948 || .48846 | .95487 | 
 15 | .44443 | .79995 45903 | .84852 || .47379 | .90037 |} .48871 | .95583 
 16 | .44467 | .80074 45927 | .84935 || .47403 | .90126 || .48896 | .95678 
 17 | .44491 | .80152 45951 | .85019 || .47428 | .90216 -48921 | .9577 
 18 | .44516 | .80231 45976 | .85103 || .47453 | .90305 -48946 | .95870 
 19 | .44540 | .80809 46000 | .85187 || .47478 | .90395 -48971 | .95966 
 20 | .44564 | .80388 || .46025 | .85271 || .47502 | .90485 || .48996 | .96062 
 21 44588 | .80467 -46049 | .85355 || .47527 | .9057: -49021 | .96158 
 22 | .44612 | .80546 46074 | .85489 || .47552 | .90665 -49046 | .96255 
 23 | .44637 | .80625 -46098 | .85523 || .47577 | .90755 || .49071 | .96351 
 24 | .44661 | .80704 -46123 | .85608 47601 | .90845 || .49096 | .96448 
 25 | .44685 | .80783 .46147 | .85692 -47626 | .90935 |} .49121 | .96544 
 26 | .44709 | .80862 46172 | .85777 -47651 | .91026 || .49146 | .96641 
 27 | .44734 | .80942 .46196 | .85861 -47676 | .91116 || .49171 | .96738 
 28 | .44758 | .81021 -46221 | .85946 -47701 | .91207 || .49196 | .96835 
 29 | .44782 | .81101 -46246 | .66031 || .47725 | .91297 || .49221 | .96932 
 30 | .44806 | .81180 -46270 | .86116 47750 | .91888 || .49246 | .97029 
 31 | .44831 | .81260 || .46295 | .86201 || .47775 | .91479 || .49971 | .97197 
 32 | .44855 | .81340 -40319 | .86286 .47800 | .9157 -49296 | .97224 
 33 44879 | .81419 .46344 | .86871 47825 | .91661 -49321 | .97322 
 34 | .44903 | .81499 -46368 | .86457 -47849 | .91752 || .49346 | .97420 
 35 | .44928 | .81579 -463893 | .86542 || .4787 . 91844 49372 | 97517 
 36 | .44952 | .81659 .46417 | .86627 .47899 | .91935 -49397 | .97615 
 37 | .44976 | .81740 -46442 | .86713 || .47924 | .92027 || .49422 | .97713 
 38 | .45001 | .81820 -46466 | .86799 || .47949 | .92118 ; .49447 | .97811 
 39 | .45025 | .81900 -46491 | .86885 || .47974 | .92210 || .49472 | .97910 
 40 | .45049 | .81981 || .46516 | .86990 || .47998 | .92302 |! .49497 | .98008 
 41 | .45073 | .82061 || .46540 | .87056 || .48023 | .92394 |! .49522 | .98107 
 42 | .45098 | .82142 -46565 |; .87142 -48048 | .92486 .49547 | .98205 
 43 | .45122 | .82299 -46589 | .87229 48073 | .92578 || .49572 | .98304 
 44 | .45146 | .82303 .46614 | .87315 || .48098 | .92670 || .49597 | .98403 
 45 | .45171 | .82384 .46639 | .87401 +48123 | .92762 || .49623 | .98502 
 46 | .45195 | .82465 || .46663 | .87488 | .48148 | .92855 .49648 | .98601 
 47 | .45219 | .82546 || .46688 | .87574 |} «48172 | .92947 |) .49673 | .98700 
 48 | .45244 | .82627 .46712 | .87661 || .48197 | .93040 || .49698 | .98799 | 
 49 | .45268 | .82709 46787 | .87748 || .48222 | .93133 .49723 | .98899 
 50 | .45292 | .82790 || .46762 | .87834 || .48247 | .93226 || .49748 | .98998 
 S1 | .45317 | .82871 || .46786 | .87921 || .48272 | .93319 |) .49773 | .99098 
 52 | .45841 | .82953 |} .46811 | .88008 || .48297 | .93412 || .49799 | .99198 
 53 | .45865 | .83034 || .46836 | .88095 || .48322 | .93505 || .49824 | .99298 
 54 | .45390 | .83116 || .46860 | .88183 || .48347 | .93598 || .49849 | .99398 
 55 | .45414 | .83198 46885 | .88270 || .48372 | .93692 || .49874 | .99498 
 56 | .45489 | .83280 .46909 | .88357 || .48396 | .93785 || .49899 | .99598 
 57 | .45463 | .83362 .46934 | .88445 || .48421 | .93879 || .49924 | .99698 
 58 | .45487 | .83444 -46959 | .88532 || .48446 | .93973 .49950 | .99799 
 59 | .45512 | .83526 46983 | .88620 |} .48471 | .94066 || .49975 | .99899 
 60 |! .45536 | .838608 47008 | .88708 || .48496 | .94160 .50000 '1.00000 
 
 COIATRWEEH OS 
 
 ' 
 
SECANTS., 
 
 TABLE XXIX.—NATURAL VERSED SINES AND EXTERNAL 
 
 i] 
 
 60° | 61° 62° 63° 
 
 (| | | 
 
 Vers. | Ex se || Vers. | Ex. sec.|| Vers,. |Ex. sec. || Vers. | Ex. sec. 
 0} .50000 | 1.00000 || .51519 | 1.06267 || .53053 | 1.13005 || .54601 .20269 | 0 
 1| .50025 .00101 51544 06375 5307 18122 || .54627 .20395 | 1 
 2|. .50050 | 1.00202 || .5157 .06488 || .58104 13239 .54653 20521 | 2 
 3| .50076 .00303 .51595 . 06592 .531380 . 13356 .54679 .20647 | 3 
 4| .50101 .00404 .51621 .06701 538156 . 18473 .54705 .20773 | 4 
 5} .50126 00505 || .51646 | 1.06809 .53181 .18590 || .54731 | 1.20900 | 5 
 6| .50151 00607 .51672 .06918 || .58207 .18707 || .54757 .21026 | 6 
 7|- .50176 .00708 || .51697 7027 || .58233 .138825 || .54782 21153. | 7 
 8} .50202 .00810 || .51723 | M137 || .538258 13942 || .54808 | 1.21280 | 8 
 9| .50227 .00912 .51748 .07246 || .538284 .14060 || .54834 .21407 | 9 
 10] .50252 01014 |) .5177 .07356 || .53310 .14178 || .54860 .21535 |10 
 11| .50277 01116 || .51799 | 1.07465 || .53336 .14296 || .54886 .21662 |11 
 12} .50303 .01218 || .51825 | 1.07575 || .53861 .14414 || .54912 .21790 |12 
 
 21918 | 13 
 22045 | 14 
 22174 | 15 
 22302 
 22430 |17 
 22559 |18 
 22688 | 19 
 22817 | 20 
 22946 
 23075. | 22 
 23205. | 28 
 123334 | 24 
 23464 | 25 
 
 14533 || .54938 
 .14651 || .54964 
 14770 || .54990 
 .14889 |!) .55016 
 .15008 || .55042 
 .15127 || .55068 
 15246 || .55094 
 15366 || .55120 
 
 15485 ||} .55146 
 15605 || .55172 
 15725 || .55198 
 15845 || .55224 
 15965 || .55250 
 
 07685 || .538387 
 07795 || .53413 
 .07905 || .53439 
 -08015 || .53464 
 -08126 |} .538490 
 .08236 || .53516 | 
 08347 || .53542 
 08458 || .53567 
 C8569 || .58593 
 08680 || .53619 
 08791 || .58645 
 08903. || .53670 
 .09014 || .53696 
 
 13} 50328 
 14| .50353 
 15| .50378 
 16] .50404 
 17| .50429 
 18| .50454 
 19| .50479 
 20| .50505 
 
 21| .50530 
 22| .50555 
 23} .50581 
 24| .50606 
 25} .50631 
 
 -01320 || .51850 
 
 .01422 || .51876 
 .01525 || .51901 
 .61628 || .51927 
 
 .01730 |) .51952 
 .01833 || .51978 
 .01936. || .52003 
 .02039 || .52029 
 
 .02143 || .52054 
 02246 || .52080 
 02849 || .52105 
 02453 || .52131 
 .02557. || .52156 
 
 EA 
 ar) 
 
 wo 
 pat 
 
 26| .50656 .02661 || .52182 09126 || .58722 16085 || .55276 .23594 | 26 
 27| .50682 .02765. || .52207 09238 || .53748 16206 || .55302 23004. | 27 
 28} .50707 .02869 || .52233 .09350 |) .5377¢ .16326 || .55828 .20905 | 28 
 29| .50732 02973. || .52259 .09462 || .53799 .16447 || .55354 .23985. | 29 
 30} .50758 .03077 || .52284 .09574 || .58825 .16568 || .55380 .24116 | 30 
 31} .50783 03182 || .52310 .09686 || .53851 .16689 |} .55406 24247 | 31 
 32| .50808 03286. || .52385 09799 || .538877 .16810 || .55432 24378 | 32 
 33} .50834 .03391. || .52861 .09911. || .53908 16932 || .55458 .24509 | 33 
 34) .50859 .03496. || .52386 .10024 || .58928 .17053 || .55484 .24640 | 34 
 | 35} .50884 | 1.03601. || .52412 .10137 || .58954 17175 || .55510 -PATI2. | 85 
 
 24903 | 
 (25085 | 37 
 25167 138 
 25300. | 39 
 
 go 
 ror 
 
 17297 || .55586 
 17419 || .55563 
 17541 || .55589 
 17663. |) .55615 
 
 1 36| 50910 
 | 371 250985 
 38| 50960 
 39| 50986 
 
 08706. || .52438 
 .03811 || .52463 
 .038916 || .52489 
 .04022 || .52514 
 
 10250 |} .53980 
 10363 || .54006 
 10477 || .54082 
 10590 || .54058 
 
 Pre meh ek ph ph ph eh fre ph fr, fmeh fomeh feel foc focal fore fem fred frm fem fh frre frcmh fom fomech fem fremch fem fm fom fmm fame fame formed fk ponh fomh fou feck fom me ame peed fe frmh frm Pr feck freak fremch fom femh femmh ek femeh fd fk feed peek fed fed 
 Pr Fre pre fre face face fre peak mek emer ph eek mh frm fcc famed pam famed — are feed fh free peek meh freak fcc feed frmh — meh fame fd fmm form fom fmm fod frrmh peek fod fom fre fool meh freed fmt fred fmm frm — frceh femd feme fod Leek free peek ped feed bed bad 
 
 Ph bed peek ph peek eek fred peek peak ch ek pk pemeh fed red rek rah feck fk pa eh fh eh fem fra fed fd fred free femme feck fe pe fom famed mh fh fd Pe fee. fom. rk peed rach feck fh feed Pek bed peed feed fel bed beak bed Ped Peed ek Bk 
 Pre beh peek frm pened fram fk pak faced fem fh ph fom fem fm fm fom freak free fol fk free peak frm fred prmh foe fk fk fem fed fem fk fom frock pak pk fk fk fh fed fom perk pooh fk fk fk pk ped ph fed feel feed peak Ped pak bod fed pad pod bad 
 
 40| .51011 04128. || .52540 10704 |} .54083 17786 || .55641 . 25482. | 40 
 41} .51036 04233 || .52566 .10817 || .54109 .17909 || .55667 . 25565. | 41 
 42; .51062 .04339 || .52591 .10981 || .54135 .18031 || .55693 25697. | 42 
 43| .51087 | 1.04445 || .52617 .11045 || .54161 18154 || .55719 25880 | 43 
 44| .51118 ! 1.04551 || .52642 11159 || .54187 18277 || .55745 .25963 | 44 
 5} .51188 | 1.04658 || .52668 11274 || .54213 .18401 || .55771 .26097 | 45 
 46| .51163 .04764 || .52694 11888 || .54238 .18524 || .55797 . 26230 | 46 
 47| .51189 .04870 || .52719 ; 1.11508 || .54264 .18648 |} .55823 26864. | 47 
 48} .51214 04977 || .52745 | 1.11617); .54290 18772. || .55849 .26498 | 48 
 49) .51239 | 1.05084 || .5277 11782 || .54816 .18895 || .55876 .26632 | 49 
 50; .51265 | 1.05191 || .52796 .11847 || .54342 .19019 || .55902 . 26766 | 50 
 151) .54290 ; 1.05298 || .52822 11963 || .54368 .19144 |) .55928 .26900 | 51 
 52} .51316 | 1.05405 || .52848 12078 || .54394 .19268 || .55954 .2¢0385 | 52 
 53| .51341 .05512 || .52878 .12193.|| .54420 19393 || .55980 27169 |53 
 54| .51566 .05619 |} .52899 12809 || .54446 .19517 || .56006 .27804 | 54 
 55| .51892 05727 || .52924 12425 || .54471 .19642 || .56082 .27439 | 55 
 56| .51417 05835 || .52950 .12540 || .54497 19767 56058 .27574 156 
 57| .51443 .05942 |} .52976 12657 || .54523 .19892.|| .56084 27710 | 57 
 58| .51468 .06050 || .53001 127% 54549 .20018 || .56111 .27845 | 58 
 59| .51494 .06158 || .53027 .12889 || .54575 .20143 || .56137 .27981 | 59 
 60! .51519 .06267 || .53053 .18005 |! .54601 .20269 || .56163 .28117 160 
 
TABLE XXIX.—NATURAL VERSED SINES AND EXTERNAL SECANTS, 
 
 / | ————_————— || — 
 
 Vers. Ex, sec.|| Vers. | Ex. sec.| Vers. | Ex. sec. | Vers. | Ex. sec. 
 | ‘ t ! | 
 56163 | 1.28117 || .57788 | 1.36620 || .59326 
 .56189 | 1.28253 || .57765 | 1.36768 || .59353 
 56215, | 1.28390 |, .57791 36916 || .59379 
 56241 | 1.28526 || .57817 
 1 
 1 
 1 
 
 64° | 65° i 66° | 67° | 
 
 55930 
 56106 | 
 
 56282 
 
 .45859 | .60927 
 46020 .| .60954 
 46181 | _60980 
 
 | | | 1.37064 || .59406 | 1.46342 | 61007 1.56458 
 56267 | 1.28663 || .57844 | 1.37212 || .59433 | 1.46504 | 61034, 1.56634 | 
 -56294 | 1.28800 || .57870 | 1.37361 | .59459 | 1.46665 | 61061 | 1.56811 | 
 
 .56320 28937 || .57896 | 
 56346 
 56372 
 .56398 
 -56425 | 
 HI Hh 11} .56451 
 12} .56477 
 13} .56503 
 
 .46827 || .61088 
 
 .46989 || .61114 
 
 .47152 || .61141 
 
 .47314 || .61168 
 
 47477 |! 61195 
 .47640 || .61222 
 47804. |} 61248 
 .47967 || 61275 
 .48131 || .61302 
 .48295 || .61329 
 .48459 || .61356 
 .48624 || .61383 
 .48789 | .61409 
 48954 || .61436 
 49119 | .61463 
 
 .49284 || .61490 
 49450 || .61517 
 49616 || .61544 
 49782 || .61570 
 .49948 .61597 
 50115 || .61624 
 .50282 || .61651 
 50449 | .61678 
 50617 | 61705 
 41142 || .60125 | 1.50784 || .61732 | 1.61313 |30 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 56988 
 1 
 1 
 ii 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 i 
 1 
 
 41296 || .60152 | 1.50952 || .61759 .61496 | 31 
 
 1 
 1 
 i 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 di 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 i! 
 1 
 1 
 1 
 
 OlG5) |" 
 
 57342 | 8 | 
 57520 
 .57698 | 10 
 57876 |11 
 58054 | 12 
 .58233 | 13 
 58412 | 14 
 58591 | 15 
 58771 | 16 
 58950 |17 
 .59130 |18 
 .59311 |19 
 59491 | 20 
 
 59672 |21 
 .59853 | 22 
 .60085 | 23 
 .60217 | 24 
 .60399 |25 
 .60581 | 26 
 60763 | 27 
 .60946 | 28 
 .61129 |2 
 
 29074 || .57928 37658 || .59512 
 -29211 || .57949 
 .29349 || .57976 
 
 37509 ‘| .59486 
 
 ee 
 
 .29487 || .58002 
 | 
 
 37808 || .59539 
 
 ol 
 eo) 
 
 “BT957 
 38107 
 38256 || 
 
 .59566 
 59592 
 
 .59619 
 
 ok 
 SDOMDVIROAR WMWHS 
 
 -29625 || .58028 
 
 29763 || .58055 38406 || .59645 
 88556 |! .59672 
 
 -38707 || .59699 
 
 -29901 ;| .58081 
 
 14| .56529 
 15| .56555 
 16] .56582 
 HE | 17| .56608 
 pea 18] 56634 
 Wie 19| 56660 
 
 ee 20} .56687 
 
 .30040 || .58108 
 .30179 ;| 58134 
 .30318 |} .58160 
 .30457 || .58187 
 .30596 || .58218 
 .30735 || .58240 
 .30875 || .58266 
 
 .81015 || .58293 
 31155 || .58319 
 31295 |] .58345 
 .31436 || .58372 
 .31576 || 58398 
 .B1717 || 58425 
 .B1858 || .58451 
 .31999 || .58478 
 82140 || 58504 
 32282 || .58531 
 
 1 
 i 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 82424 || .58557 | 1 
 1 
 1 
 1 
 u 
 il 
 1 
 1 
 1 
 1 
 13 
 i i 
 i. 
 dus 
 li 
 ET. 
 A 
 i 
 i 
 ai 
 Te 
 aM 
 is 
 ut 
 id 
 si 
 i% 
 1. 
 t. 
 1. 
 
 38857 || .59725 
 
 39008 || .59752 
 
 39159 |) .5977 
 
 .89311 |) .59805 
 89462 |} 59832 
 39614 || 59859 
 .39766 || .59885 
 89918 || .59912 
 .40070 || .59988 
 40222 || .59965 
 40375 || .59992 
 
 21| .56713 
 22| .56739 
 Hay, 23| .56765 
 i 24] .56791 
 He 25| .56818 
 
 26] .56844 
 27| .56870 
 28] .56896 
 eae 29] 56923 
 30| .56949 
 
 ai 31] .56975 
 
 .40528 |; .60018 
 .40681 || .60045 
 .40835 || .60072 
 .40988 |} .60098 
 
 SSS 
 
 1 
 4 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 il 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 32| .57001 | 1.382566 || .58584 
 i 33] .57028 | 1.32708 || .58610 
 ill 34} .57054 | 1.382850 || .58637 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 I 
 1 
 1 
 J 
 1 
 1 
 1 
 1 
 
 .51289 || .61812 
 .51457 || .61839 
 .61866 
 
 41760 || .60232 62049 | 34 
 41914 || .60259 
 42070 || .60285 
 .42295 || .60312 
 .42380 || .60339 
 .42536 || .60365 
 .42692 || .60392 
 42848 
 43005 
 43162 
 43318 
 43476 
 
 43633 
 
 We | 35| .57080 | 1.32993 || .58663 
 Wi 36| .57106 | 1.33185 || .58690 
 Wd 37| .57133 | 1.33278 || .58716 
 iit 38| .57159 | 1.33422 || 58743 
 
 a 39| .57185 | 1.33565 || .58769 
 . 40| .57212 | 1.83708 || .58796 
 
 41] .57288 .83852_ || .58822 
 42) 57264 .89996 || .58849 
 43} 57291 84140 || .58875 
 84284 || .58902 
 .84429 || .58928 
 .84573 || 58955 
 
 62284 |35 
 .62419 | 36 
 .62604 | 37 
 .62790 | 38 
 .62976 | 39 
 63162. | 40 
 
 .63348 | 41 
 .63585 | 42 
 63722 | 43 
 .63909 | 44 
 .64097 | 45 
 64285 
 
 51626 
 51795. || .61893 
 51965 || .61920 
 52134 || .61947 
 52304 || .61974 
 52474 || .62001 
 
 52645 || .62027 
 52815 || .62054 
 .52986 || .62081 
 53157 
 53329 
 53500 
 
 “41605 || .60205 61864 |33 
 
 .60419 
 .60445 
 .60472 
 .60499 
 . 60526 
 . 60552 
 
 .62108 
 .62185 
 .62162 
 
 7 
 oO 
 
 ( j 1.84718 || .58981 43790 |! .60579 .53672 || .62189 64473 | 47 | 
 48 D7422 ; 1.34863 .59008 43948 || .60606 .538845 || .62216 .64662 | 48 
 
 . 85009 .59034 
 .80154 || .59061 
 
 44106 
 44264 
 
 .60633 
 .60659 
 
 54017 || .62243 
 .54190 || .6227 
 
 64851 | 49 | 
 65040 | 50 | 
 
 1 
 | 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 J 
 1 
 1 
 1 
 I 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 I 
 41450 || .60178 .51120 || .61785 | 1.61680 | 32 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 1 
 I 
 1 
 1 
 1 
 
 51| .57501 | -59087 | 1.44428 || .60686 | 1.54363 || .62297 | 1.65229 [51 | 
 52) .5Y527 | 59114 | 1.44582 || .60713 | 1.54536 || .62324 | 1.65419 | 52 | 
 53) .57554 | | .59140 | 1.44741 || .60740 | 1.54709 || .62351 | 1.65609 |53 
 54| 57580 | | .59167 | 1.44900 || .60766 | 1.54883 || .62378 | 1.65799 |54 
 55| .57606 | | 59194 | 1.45059 || .60793 | 1.55057 || .62405 | 1.65989 | 55 
 56| .57633 | 59220 | 1.45219 || .60820 | 1.55231 |) .62431 | 1.66180 |56 
 57| .57659 | 1.36178 || .59247 | 1.45378 || .60847 | 1.55405 || 162458 | 1.66371 [57 
 58) .57685 | 1.36325 |) .59273 | 1.45530 || .60873 | 1.55580 || .62485 | 1.66563 | 58 
 59) .57712 45699 || .60900 | 1.55755 || .62512 | 1.66755 59 
 | 
 
 36473 | .59300 
 
 60) .57738 .86620 |] .59326 66947 
 
 SS 
 
 45859 |! .60927 .55930 !] .62539 30 | 
 
68° | 69° 70° 
 f | / 
 Vers. | Ex. sec.|| Vers. | Ex. sec.|| Vers. | Ex. sec.|] Vers. | Ex. sec.) 
 0 "62539 1.66947 || .64163 | 1.79043 || 65798 | 1.92380 || .67443 | 2.07155 | 0 
 1] .62566 | 1.67139 "64190 | 1.79254 || .65825 | 1.92614 || .67471 | 2.07415 1 
 2} .62593 | 1.67332 "64218 | 1.79466 || .65853 | 1.92849 || .67498 | 2.07675 2 
 3| .62620 | 1.67525 64245 | 1.79679 || .65880 ee 93083 .67526.| 2.07936 | 3 
 4| .62647 | 1.67718 || .64272 | 1.79891 || 65907 | 1.93318 || .67553 | 2.08197 | 4 
 5 | 62674 | 1.67911 || .64299 | 1.80104 || .65935 | 1. 93554 67581 | 2.08459 | 5 
 6| .62701 | 1.68105 |) .64826 1.80318 || .65962 | 1.93790 || .67608 | 2.08721 | 6 
 ¢ 62723 | 1.68299 | 64353 | 1.8053 “65989 | 1.94026 || .67636 | 2.08983 | 7 
 8} .62755 | 1.68494 || .64381 | 1.80746 || .66017 | 1.94263 || - 7663 | 2.09246 | 8 
 9 .62782 1.68689 |; .64408 | 1.80960 66044 | 1.94500 || .67691 | 2.09510 | 9 
 10! .62809 | 1.68834 |) .64435 | 1.81175 || .66071 | 1.947837 7718 | 2.09774 | 10 
 11} .62836 | 1 69079 .64462 | 1.81390 || .66099 1.94975 || .67746 | 2.10088 | 11 
 12| .62863 | 1.69275 || .64489 1.81605 || .66126 | 1.95213 || .67773 | 2.10303 |12 
 13 -62390 1.69471 || 64517 1.81821 || .66154 | 1. 95452 .67801 | 2.10568 | 13 
 1h yo rend | Syren baer | ee 95691 || .67829 | 2.10834 | 14 
 5| .628 69% 6457 82254 || . ) 959¢ 678 : 
 16} .62971 | 1.70061 || .64598 | 1.82471 | [66286 | 1. ‘goril ‘Gres 2 11307 16 
 : i yee eves ne S a a donee i 96411 || .67911 | 2.11635 |17 
 .63025 | 1.70455 |} .0400 82906 || .66290 96652 || .67989 | 2.11903 |18 
 Af ee | i | nee ree | oak a 96893 || .67966 | 2.12171 |19 
 20) .630% 7085 647 1.83842 || .66345 97135 || .67994 | 2.12440 | 20 
 21| .63106 | 1.71050 || .64784 | 1.83561 66373 | 1.97377 || .68021 | 2.12709 |2 
 95| .63133 | 1.71249 |) .64761 | 1.83780 |} 66400 | 1.97619 || .68049 | 2.12979 | 22 
 23 | -63161 1.71448 || .64789 | 1.83999 66427 | 1.97862 || .68077 | 2.18249 | 23 
 24 aie ist || ye Laie 219 || .66455 | 1.98106 .68104 Sera 24 
 25) .63215 71847 || .6484¢ 84439 || .66482 | 1.98349 || .68182 18791 | 25 
 26 63242 1.72047 64870 | 1.84659 | "66510 | 1.96594 || .68159 | 2.14063 | 26 
 ad 63269 1.72247 | . 64898 1.84880 || .66537 | 1.98838 || .68187 | 2.14385 27 
 98| 63296 | 1.72448 || .64925 | 1.85102 || .66564 1.99083 || .68214 | 2.14608 | 28 
 29| .63323 | 1.72649 || .64952 | 1.85823 || "66592 | 1.99829 |} .68242 | 2.14881 | 29 
 30| 63350 | 1.72850 || .64979 | 1.85545 || .66619 1.9957. 68270 | 2.15155 | 3 
 31] .63377 | 1.73052 |; .65007 | 1.85767 || .66647 1.99821 || .68297 | 2.15429 | 31 
 32! 63404 | 1.73254 || .65084 | 1.85990 || "66674 | 2.00067 || .68825 | 2.15704 | 382 
 93| _63431.| 1.73456 || .65061 | 1.86213 || .66702 2.00315 || .68352 | 2.15979 | 33 
 34| .63458. | 1.73659 "65088 | 1.86437 || .66729 | 2.00562 || .68 380 | 2.16255 | 34 
 35|} .63485 | 1.73862 || .65116 | 1.86661 |) .66756 2.00810 || .68408 | 2.16531 | 35 
 36 63512 1.74065 "65143 | 1.86885 || .66784 | 2.01059 || .G8435 2.16808 | 86 
 3% .63939 1.74269 || .65170 | 1.87109 || .66811 | 2.01308, .68463 | 2.17085 | 3% 
 381 .63566 | 1.74473 || .65197 | 1.873834 "66839 | 2.01557 || .68490 | 2.17863 38 
 39| .63594 | 1.74677 || .65225 | 1.87560 "66866 | 2.01807 |} .68518 | 2.17641 | 39 
 40 “63621 1.74881 65252 | 1.87785 || .66894 | 2.02057 68546 | 2.17920 | 40 
 41} .63648 | 1.75036 6527 1.88011 66921 | 2.02808 || .68573 | 2.18199 | 41 
 2| 63675 | 1.75292 || .65806 , 1.88238 "66949 | 2.02559 |} .68601 | 2.18479 | 42 
 43) .63702 | 1.75497 || .65834 1.88465 || .66976 | 2.02810 || .68628 2.18759 | 43 
 | 44} 63729 1.75703 || .65361 1188692 "87003 | 2.03062 || .68656 | 2.19040 | 44 
 45) .63755 | 1.75909 || .65388 | 1.88920 "67031 | 2.03315 || .68684 | 2.19822 | 45 
 46) 63783 | 1.76116 || .65416 | 1.89148 "67058 | 2.03568 || .68711 | 2.19604 46 
 47| .63310 | 1.76323 .65443. | 1.89376 67086 | 2.08821 68739 | 2.19886 | 47 
 48} .63338 | 1.76530 || .65470 | 1.89605 || 67113 | 2.04075 68767 | 2.20169 | 48 
 | 49) 63365 | 1.76737 || .65497 | 1.89832 | "67141 | 2.04329 || .68794 | 2.20453 | 49 
 50) 63892 | 1.76945 65525 | 1.90063 || "67168 | 2.04584 || .68822 | 2.20737 | 50 
 51} .63919 | 1.77154 || .65552 | 1.90293 || .o7196 | 2.04839 || .68849 | 2.21021 | 51 
 52! 63946 | 1.77362: | .65579 | 1.90524 || .67223 | 2.05094 |) .688V7 | 2. 21306 | 52 
 53| .63973 | 1.77571 || .65607 | 1.90754 |) .67251 | 2.05350 “68905 | 2.21592 |53 
 54| .64000 | 1 77780 65634 | 1.90986 || “67278 | 2.05607 || .68932 | 2.21878 | 54 
 55| 161027 | 1.77990 | .65661 | 1.91217 || .67306 | 2.05864 || .68960 | 2.22165 | 55 
 | 56 .64055 / 1.78200 | .65689 1.91449 || .67333 | 2.06121 || .68988 | 2.22452 | 56 
 | 57| .64032 | 1.78410 || .65716 |-1.91681 || .67361 | 2.06379 "9015 | 2.22740 |57 
 58| .64109 | 1.78621 65743 | 1.91914 || .67388 | 2.06637 69043 | 2.23028 |58 
 59| .64136 | 1.78832 | .65771 | 1.92147 "67416 | 2.06896 || .69071 | 2.23317 |59 
 64163 | 1.79043 || .65798 | 1.92380 || “@7443 | 2.07155 || .69098 | 2.28607 | 60 
 
 NATU Ré AL 
 
 TERSED SINES AND EXTERNAL SECANTS. 
 
 71° 
 
72° 
 
 TABLE XXIX.—NATURAL VERSED SINES AND EXTERNAL SECANTS. 
 
 | . 70763 | 
 
 | Vers, | Ix. sec.'| Vers. Ex. sec.'| Vers. | Ex. see 
 0; .69098 | 2.28607 10763 2.62796 || .74118 | 2.86370 | 0 
 1} .69126 | 2.28897 40791 | 2. 2.63164 . 74146 | 2.86790 | 1 
 2| .69154. | 2.24187 . 70818 | 2. 2 68533 74174 | 2.87211.) 2 
 3] .6918 2.24478 .10846 | 2. 2.638903 || .74202 | 2.87633 | 3 
 4) .69209 | 2.24770 (0874 | 2. 2.64274 || 74231 2.88056 | 4 
 5] .69237 | 2.25062 - 70902 | 2. 2.64645 || .74259 | 2.88479 | 5 
 6] 69264.| 2.25355 . 70930 | 2.4 2.65018 T4287 | 2.88904 | 6 
 7) .69292 | 2.25648 . (0958 | 2.¢ 2.65391 || .74815 | 2.893880 | 7 
 8} .69320 | 2.25942 . 70985 | 2. 2.65765 . 74843 | 2.89756 | 8 
 | -69347 | 2 26237 || .71013 | 2: 2.66140 || .74371 | 2.90184 
 | .69375 | 2.26531 . 71041 | 2.4 2.66515 || .74899 | 2.90613 
 .69403 | 2.26827 - 71069 | 2. 2.66892 || .74427 | 2.91042 
 .69430. | 2.27123 (1097 | 2.4 2.67269 | (4455 | 2.91473 
 .69458 | 2.27420 Ail fe apa le Hs: 2.67647 (4484 | 2.91904 
 .69486 | 2.27717 ANG beaks eal! Puhee! 2.68025 . 74512 | 2.92337 
 .69514 | 2.28015 Epil tet | allem 2.68405 . 74540 | 2.9277 
 .69541 | 2.28313 - 71208 | 2.2 2.68785 || .74568 | 2.93204 
 .69569 | 2.28612 41236 | 2.2 2.69167 || .74596 | 2.93640 
 .69597 | 2.28912 ~ 61264} 2. 2.69549 . 74624 | 2.94076 
 .69624 | 2.29219 . 71292 | 2. 2.69931 . 74652 | 2.94514 
 .69652 | 2.29512 -(1820 | 2. 2.70815 - 74680 | 2.94952 
 .69680 | 2.29814 +(1348~| "2: 2.70700. || .74709 | 2.95392 
 .69708 | 2.30115 BY re WP 2.71085 || . 747387 | 2.95832 
 .69785 | 2.30418 (1403, 1-252 2.71471 74765 | 2.96274 
 .69763 | 2.30721 oe abba De 2.71858 . 74793 | 2.96716 
 .69791 | 2.31024 (1459 | 2. 2.72246 . 74821 | 2.97160 
 .69818 | 2.31328 (1487172). 2.72635 . 74849 | 2.97604 
 .69846. | 2.31633 Bi gusy teyai ps 2.78024 || .74878 | 2.98050 
 .69874 | 2.31939 (1543, | 2. 2.73414 . 74906 | 2.98497 
 .69902 | 2.82244 |; .71571 | 2. 2.73806 || .74934 | 2.98944 
 -69929 | 2.32551 wh Loose net 2.74198 || .74962 2.99393 
 .69957 | 2.32858 . 71626 | 2. 2.74591 || .74990 | 2.998438 
 .69985 | 2.33166 (1654 | 2. 2.74984 || .75018 | 3.00293 
 - 70013 | 2.33474 MC LOSe les: yeh dao N . 75047 | 3.00745 
 . 70040 | 2.33783 a EOS Pay: 2.75775 75075 | 3.01198 
 . (0068 | 2.34092 Rdldon ee 2.76171 || .75103 | 8.01652 
 .70096 | 2.34403 HAUL AL OH 2.76568 || .75181 | 3.02107 
 .10124.| 2.34713 (1794 | 2.5 2.76966 75159 | 8.02563 
 . 70151 | 2.35025 SRSA ely ab) 2.773865 | 15187 | 8.038020 
 (0179 | 2.85336 ~71850 | 2.2 2.77765 75216 | 3.03479 
 - 70207 | 2.35649 PilOre ieee 2.78166 75244 | 3.03938 
 - 70235 | 2.35962 ~(1905 | 2. 2.78568 75272 | 8.04398 
 . 70263 | 2.36276 41983 | °2..2 2.78970 75800 | 8.04860 
 3} .70290 | 2.36590 71961 | 2. 2.79874 (5828 | 8.05822 | 
 . 40318 | 2.36905 .71989 | 2.2 2.7977 75856 | 8.05786 
 . 10346. | 2.37221 =i BI Wome | | 2.80183 || .75885 | 3.06251 
 10874 | 2.37587 72045. | 2.! 2.80589 75413 | 8.06717 
 ‘| .70401 | 2.37854 (2073. |e eae 2.80996 || .75441 | 8.07184 
 | .70429 | 2.88171 72101 | 2.! 2.81404 75469 | 8.07652 
 | .70457 | 2.38489 mY he be1) pa Mey | 2.81813 (5497 | 8.08121 
 | .40485 | 2.38808 Bi to eal fe Pains 2.82223 75526 | 8.08591 
 .705138 | 2.39128 72185 | 2. 2.82633 || .75554 | 3.09063 | 
 52! .70540 | 2.89448 PAA PA aSY INB 2p | 2.83045 |! .'75582 | 3.09585 | 
 3} .70568 | 2.39768 72241 | 2. | 2.88457 || .75610 | 8.10009 |! 
 | 70596 | 2.40089 72269 | 2. 2.83871 || .'75689 | 3.10484 
 | .V0624. | 2.40411 (2296. | 2. 2.84285 75667 | 8.10960 
 3; .70652 | 2.4073 12024 | 2. 2.84700 75695 | 8.11437 
 | .70679 | 2.41057 72352 | 2 2.85116 75723 | 8.11915 
 | . 70707 | 2.41381 72880 2.85533 75751 | 3.12394 
 9| .70735 | 2.41705 . 72408 2.85951 75780 | 8.12875 
 2.420380 72436 2.86370 75808 | 8.13857 
 
TABLE XXIX.—NATURAL VERSED SINES AND EXTERNAL SECANTS. 
 
 76° 
 4 
 Vers. | Ex. sec. 
 0|'.75808 | 3.138357 
 1} .75886 | 3.138839 
 2] .75864 | 3.14323 
 3| .7%75892 | 3.14809 
 4| .75921 | 8.15295 
 5| .75949 | 3.15782 
 6} .75977 | 3.16271 
 7|. .76005 | 3.16761 
 8) .76034 | 3.17252 
 9) .76062 | 3.17744 
 10) .76090 | 3.18238 
 11| ..76118 | 3.18733 
 12} .76147 | 3.19228 
 13| «76175 | 3.19725 
 14| .76208 | 3.20224 
 15) .'762381 | 3.20723 
 16) .76260 | 3.21224 
 7) .76288 | 3.21726 
 18| .76316 | 3.22229 
 1 . 763844 | 3.227384 
 20| .763873 | 3.23239 
 21| .76401 | 3.23746 
 22| .76429 | 3.24255 
 23| .76458 | 3.24764 
 24] .76486 | 3.25275 
 25| .76514 | 3.25787 
 26| .76542 | 3.26300 
 27 | .76571 | 3.26814 
 28 .76599 | 3.27380 
 29] .76627 | 3.27847 
 30| .76655 | 3.28366 
 81] .76684 | 3.28885 
 32| .76712 | 3.29406 
 | 33| .76740 | 3.29929 
 34|, .76769 | 3.30452 
 385| .76797 | 3.30977 
 36} .76825 | 3.81503 
 37} .76854 | 3.32031 
 38} .76882 | 3.32560 
 39] .76910 | 3.38090 
 40| .76938°*| 3.338622 
 41) .76967 | 3.34154 
 42| .76995 | 3.34689 
 43) .77023 | 3.385224 
 44! .77052 | 3.35761 
 45| .77080 | 3.386299 
 46| .77108 | 3.36839 
 47| .77187 | 3.37380 
 48} .77165 | 3.37923 
 49) .7719% 8.88466 
 50} .77222 | 3.39012 
 51} .7%250 | 3.89558 
 52] .77278 | 8.40106 
 53| .77307 | 3.40656 
 54] .773385 | 3.41206 
 55) .773863 | 3.41759 
 56) .77892 | 3.42312 
 57| .77420 | 3.42867 
 58 | 77448 | 8.43424 | 
 591 .77477-| 3.43982 
 77505 | 8.44541 
 
 78° 
 
 . 79209 
 
 17° | 
 | 
 Vers | Ex. sec.!| Vers. 
 | 
 77505 | 3.44541 . 79209 
 .77583 | 3.45102 .79237 
 £77562 | 8.45664 .79266 
 .77590 | 3.46228 79294 
 .77618 | 3.46793 79823 
 77647 | 3.47360 793851 
 T1675. | 3.47928 .79380 
 .777038 | 8.48498 .79408 
 .77782 | 3.49069 .79437 
 .77760 | 8.49642 . 79465 
 W7788 | 3.50216 || .79493 
 T7817 | 8.50791 79522 | 
 . 77845 | 3.513868 79550 
 V7874 | 3.51947 £1957 
 .77902 | 8.52527 || .79607 
 77930 | 8.58109 . 79636 
 T7959 | 3.538692 . 79664 | 
 77987 | 3.54277 .79693 
 78015 | 3.54863 79721 
 78044 | 8.55451 79750 
 78072 | 8.56041 T9778 
 78101 | 8.56682 || .79807 
 78129 | 3.57224 . 79885 
 .78157 | 3.57819 || .79864 
 .78186 | 3.58414 || .'79892 
 .78214 | 3.59012 || .79921 
 . 78242 | 3.59611 79949 
 | .78271 | 3.60211 .79978 
 . 78299 | 3.60813 .80006 
 .783828 | 3.61417 || .80035 
 . 78856 | 3.620238 .80063 
 . 78884 | 3.626380 .80092 
 .78413 | 3.63288 .80120 
 .78441 | 3.63849 .80149 
 . 78470 | 3.64461 80177 
 .78498 | 3.65074 .80206 
 .78526 | 3.65690 || .80234 
 . 78555 | 3.66807 .80263 
 .785838 | 3.66925 .80291 
 .78612 | 3.67545 || .803820 
 .78640 | 8.68167 || .80848 
 . 78669 | 3.68791 80377 
 (8697 | 3.69417 .80405 
 ~ 78725 | 8.70044 80434 
 78754 | 3.70673 .80462 
 | .78782 | 3.713803 80491 
 .78811 | 3.71985 || .80520 
 .78839 | 3.72569 || .80548 
 .78868 | 3.78205 .80577 
 .18896 | 3.78843 . 80605 
 . 78924 | 8.74482 .80634 
 .78953 | 3.75128 || .80662 
 78981 | 3.75766 .80691 
 .79010 | 3.76411 |) .80719 
 .79038 | 3.77057 || 80748 
 79067 | 8.77705 .80776 
 .79095 | 3.78355 . 80805 
 79123 | 3.79007 . 80833 
 .79152 | 3.79661 .80862 
 .79180 | 3.80316 80891 
 3.80973 .80919 
 
 | 
 | 
 
 EX. sec. 
 
 | 3.80973 
 
 3.81633 
 3.82294 
 3.82956 
 3.83621 
 3.84288 
 3.84956 
 3.85627 
 3.86299 
 3.86973 
 3.87649 
 3.88327 
 8.89007 
 3.89689 
 3.90373 
 8.91058 
 3.91746 
 3.92436 
 
 93128 
 
 93821 
 
 94517 
 
 wocwmc qwewt) 
 DODO 
 
 3) 
 
 ive) 
 
 _— 
 
 os 
 
 17121 
 17886 
 18652 
 19421 
 20193 
 20966 
 21742 
 22521 
 
 79° 
 
 | Vers, | Ex. sec. 
 | 80919 | 4.24084 
 .80948 | 4.24870 
 80976 | 4.25658 
 .81005 | 4.26448 
 | .81083. | 4.27241 
 .81062 | 4.280386 
 .81090 | 4.28833 
 £81119 | 4.29634 
 .81148 | 4.30436 
 81176 | 4.31241 
 .81205 | 4.82049 
 | .81233 | 4.82859 
 | ,81262 | 4.83671 
 .81290 | 4.34486 
 .81319 | 4.35804 
 .81348 | 4.36124 
 .81376 | 4.36947 
 .81405 | 4.37772 
 
 .81483 | 4.88600 
 .81462 | 4.89430 
 .81491 | 4.40263 
 
 |; .81519 | 4.41099 
 .81548 | 4.41937 | 
 
 81576 | 4.42778 
 
 | 81605 | 4.43622 |2 
 81633 | 4.44468 
 
 .81662 | 4.45317 
 
 | 81691 | 4.46169 
 81719 | 4.47023 
 
 .81748 | 4.47881 
 .81776 | 4.48740 
 .81805 | 4.49603 
 
 .81834 | 4.50468 
 .81862 | 4.51337 
 .81891 | 4.52208 
 .81919 | 4.58081 
 .81948 | 4.58958 
 81977 | 4.54837 
 ; 82005 | 4.55720 
 | .820384 | 4.56605 
 820638 | 4.57493 
 82091 | 4.58383 
 82120 | 4.59277 
 .82148 | 4.60174 
 82177 | 4.61073 
 .82206 | 4.61976 
 .822384 | 4.62881 
 82263 | 4.63790 
 .82292 | 4.64701 
 .82320 | 4.65616 
 82349 | 4.66538 
 82377 | 4.67454 
 82406 | 4.68877 
 82485 | 4.69804 
 82463 | 4.70284 
 82492 | 4.71166 
 82521 | 4.72102 
 82549 | 4.73041 
 82578 | 4.73983 
 82607 | 4.74929 
 826385 | 4.75877 
 
 — 
 Boasorpemre | 
 
 i. 
 = 
 
 489 
 
Vers. | Ex.sec. || Vers. | Ex. see. 
 82635 | 4.75877 || .84357 | 5.39245 
 .82664 | 4.76829 || .84385 | 5.40422 
 82692 | 4.77784 || .84414 | 5.41602 
 .82721 | 4.78742 '| .84443 | 5.42787 
 82750 | 4.79703 || .84471 | 5.43977 
 82778 | 4.80667 || .84500 | 5.45171 
 82807 | 4.81635 || .84529 | 5.46369 
 82836 | 4.82606 || .84558 | 5.47572 
 .82864 | 4.83581 |) .84586 | 5.48779 
 .82893 | 4.84558 || .84615 | 5.49991 
 82922 | 4.85539 || .84644 | 5.51208 
 .82950 | 4.86524 |} .84673 | 5.52429 
 82979 | 4.87511 || .84701 | 5.53655 
 .83008 | 4.88502 || .84730 | 5.54886 
 .83036 | 4.89497 || .84759 | 5.56121 
 .83065 | 4.90495 || .84788 | 5.57361 
 .83094 | 4.91496 || .84816 | 5.58606 
 83122 | 4.92501 || .84845 | 5.59855 
 .83151 | 4.93509 || .84874 | 5.61110 
 .83180 | 4.94521 || .84903 | 5.62369 
 .83208 | 4.95586 || .84931 | 5.63633 
 .83237 | 4.96555 || .84960 | 5.64902 
 83266 | 4.97577 || .84989 | 5.66176 
 .83294:| 4.98603 || .85018 | 5.67454 
 .83323 | 4.99633 || .85046 | 5.68738 
 83352 | 5.00666 || .85075 | 5.70027 
 .83380 | 5.01703 || .85104 | 5.71321 
 .83409 | 5.02743 || .85133 | 5.72620 
 83438 | 5.03787 || .85162 | 5.73924 
 .83467 | 5.04834 || .85190 | 5.75233 
 .88495 | 5.05886 || .85219 | 5.76547 
 83524 | 5.06941 || .85248 | 5.77866 
 .83553 | 5.08000 || .85277 | 5.79191 
 .83581 | 5.09062 || .85305 | 5.80521 
 .83610 | 5.10129 || .85334 | 5.81856 
 .83639: | 5.11199 || .85363 | 5.83196 
 83667 | 5.12273 || .85392 | 5.84542 
 .83696 | 5.18350 || .85420 | 5.85893 
 .83725 | 5.14432 || .85449 | 5.87250 | 
 83754 | 5.15517 || .85478 | 5.88612 
 .83782 | 5.16607 || .85507 | 5.89979 
 .83811 | 5.17700 || .85536 | 5.91352 
 83840 | 5.18797 || .85564 | 5.92731 
 .83868 | 5.19898 || .85593 | 5.94115 
 .83897 | 5.21004 || .85622 | 5.95505 
 .83926 | 5.22113 || .85651 | 5.96900 
 | .83954 | 5.23226 || .85680 | 5.98801 
 | .83983 | 5.24343 || .85708 | 5.99708 
 .84012 | 5.25464 || .85737 | 6.01120 
 .841041 | 5.26590 |} .85766 | 6.02538 
 84069 | 5.27719 || .85795 | 6.08962 
 .84093 | 5.28853 || .85823 | 6.05392 
 .84127 | 5.29991 || .85852 | 6.06828° 
 .84155 | 5.31133 || .85881 | 6.68269 
 84184 | 5.32279 || .85910 | 6.09717 
 .84213 | 5.33429 || .85939 | 6.11171 
 .84242 | 5.34584 || .85967 | 6.12630 | 
 .84270 | 5.35743 || .85996 | 6.14096 
 84299 | 5.36906 || .86025 | 6.15568 
 84328 | 5.38073 || .86054 | 6.17046 
 84357 | 5.39245 || .86083 | 6.18530 
 
 490 
 
 Vers. 
 
 82° 
 
 . 86083 
 .86112 
 .86140 
 .86169 
 .86198 
 .86227 
 . 86256 
 . 86284 
 .86313 
 . 86342 
 .86371 
 -86400 
 .86428 
 86457 
 .86486 
 ~86515 
 -86544 
 .86573 
 .86601 
 . 86630 
 .86659 
 . 86688 
 | .86717 
 | .86746 
 8677. 
 
 | .86803 
 . 86832 
 . 86861 
 . 86890 
 .86919 
 .86947 
 . 86976 
 | .87005 
 .87034 
 .87063 
 -87092 
 .87120 
 .87149 
 87178 
 87207 
 . 87236 
 
 87265 
 
 TABLE XXIX.—NATURAL VERSED SINES AND EXTERNAL SECANTS, 
 
 Ex. sec, || Vers. 
 6.18530 || .87813 
 6.20020 || .87842 
 6.21517 |} .87871 
 6.23019 |} .87900 
 6.24529 || .87929 
 6.26044 || .87957 
 6.27566 || .87986 
 6.29095 . 88015 
 6.30630 . 88044 
 6.382171 || .88073 
 6.33719 . 88102 
 6.35274 || .88131 
 6.36835 || .88160 
 6.38403 .88188 
 6.39978 || .88217 
 6.41560 . 88246 
 6.43148 88275 
 6.44743 || .88304 
 6.46346 .88333 
 6.47955 . 88362 
 6.49571 . 88391 
 6.51194 . 88420 
 6.52825 . 88448 
 6.54462 || .88477 
 6.56107 || .88506 
 6.57759 || .88535 
 6.59418 || .88564 
 6.61085 .88593 
 6.62759 . 88622 
 6.64441 .88651 
 6.661380 . 88680 
 6.67826 . 88709 
 6.69530 || .88737 
 6.71242 . 88766 
 6.72962 || .88795 
 6.74689 || .88824 
 6.76424 - 88853 
 6.78167 . 88882 
 6.79918 .88911 
 6.81677 .88940 
 6.83443 .88969 
 6.85218 .88998 
 6.87001 || .89027 
 6.88792 || .89055 
 6.90592 |} .89084 
 6.92400 || .89113 
 6.94216 || .89142 
 6.96040 || .89171 
 6.9787 .89200 
 6.99714 |} .89229 
 "01565 .89258 
 
 | 7.03423 || .89287 
 7.05291 || .89816 
 7.07167 || .89845 
 4.09052 .893874 
 7.10946 .89403 
 7.12849 || .89431 
 7.14760 || .89460 | 
 7.16681 || .89489 
 7.18612 || .89518 
 % 20551 || .89547 
 
 © WF 3 OTR CO IDE O | 
 
 PIII YWIAI IAI III 
 
 SESE AE AEE eB ge 3 + 
 
 09 00 09 00 G0. 0D 00 G0 G00 OOO -t-3-F-3 ~ 
 cS 
 
TABLE XXIX.—NATURAL VERSED SINES AND EXTERNAL SECANTS. 
 
 fat et 
 Se DODWIHUPWWHO 
 
 84° 
 
 85° 
 
 Vers. Ex. sec. 
 89547 8.56677 
 .89576 8.59882 
 .89605 8.62002 
 .89634 8.64687 
 .89663 8.67387 
 .89692 8.70103 
 .89721 8.728383 
 .89750 8.7557 
 
 89779 8.78341 
 .89808 8.81119 
 .89836 8.83912 
 . 89865 8.86722 
 ,89894 8.89547 
 . 89923 8.92389 
 .89952 8.95248 
 .89981 8.9812 
 .90010 9.01015 
 .90039 9.03923 
 .90068 9.06849 
 90097 9.09792 
 :9C126 9.12752 
 .90155 9.15780 
 .90184 9.18725 
 .90213 9.21739 
 . 90242 9.2477 
 
 . 90271 9.27819 
 . 90300 9.30387 
 . 90329 9.33973 
 . 90358 9.387077 
 . 90386 9.40201 
 90415 9.43343 
 . 90444 9.46505 
 90473 9.49685 
 . 90502 9.52886 
 . 90531 9.56106 
 .90560 9.59346 
 .90589 9.62605 
 .90618 9.65885 
 90647 9.69186 
 .90676 9.72507 
 90705 9.75849 
 90734 9.79212 
 . 90768 9.82596 
 .90792 9.86001 
 .90821 9.89428 
 90850 9.92877 
 .90879 9.96348 
 .90908 9.99841 
 . 90937 10 .03356 
 . 90966 10.06894 
 .90995 10.10455 
 .91024 10.140389 
 ~91053 10.17646 
 .91082 10212377 
 .91111 10.24932 
 .91140 10.28610 
 .91169 10.32313 
 .91197 10.86040 
 - 91226 10.39792 
 .91255 10.43569 
 . 91284 10.47371 
 
 Vers. 
 
 Ex. sec. 
 
 91284 
 .91813 
 .913842 
 91871 
 .91400 
 -91429 
 .91458 
 91487 
 91516 
 
 91545 © 
 
 91574 
 
 .91603 
 .916382 
 . 91661 
 .91690 
 .91719 
 91748 
 91777 
 .91806 
 .518385 
 . 91864 
 .91893 
 - 91922 
 .91951 
 .91980 
 . 98009 
 . 92038 
 . 920587 
 . 92096 
 .92125 
 
 92154 
 
 92183 
 92212 
 92241 
 92270 
 92299 
 92328 
 92357 
 92386 
 92415 
 92444 
 92478 
 92502 
 92581 
 92560 
 92589 
 .92618 
 92647 
 .92676 
 92705 
 927384 
 92763 
 92792 
 92821 
 9285 
 
 92879 
 92908 
 92937 
 929366 
 .92995 
 . 93024 
 
 10.4737 
 10.51199 
 
 11.07610 
 11.11852 
 11 -16125 
 11.20427 
 11.24761 
 11.29125 
 
 11.33521 
 11.37948 
 11.4408 
 11.46900 
 11.5142 
 1.55982 
 11.60572 
 11.65197 
 11. 69856 
 11.4550 
 11.%9278 
 1184042 
 11.88841 
 11.9367 
 1198549 
 12.03458 
 1208040 
 12. 13388 
 12.18411 
 12.23472 
 12.28572 
 12.33712 
 12.38891 
 12.44112 
 12.49373 
 12.54676 
 12.0021 
 12.65408 
 2.70038 
 16512 
 
 04850 
 .10096 
 .1588 
 .21730 
 13.27620 
 18:38559 
 
 94766 
 
 86° 
 } 
 Vers Ex. sec. 
 
 - 93024 13.33559 0 
 .93053 13.39547 1 
 . 93082 13.45586 2 
 .93111 13.51676 3 
 .93140 13.57817 4 
 .93169 13.64011 5 
 . 93198 13 .70258 6 
 93227 13.76558 v4 
 93257 13.82913 8 
 93286. | 13.89323 | 9 
 .93315 13.95788 10 
 .93844 14.02310 11 
 93373 14.08890 12 
 . 93402 14.15527 13 
 93431 14. 222293 14 
 .93460 14. 28979 15 
 .93489 14.35795 16 
 - 93518 14. 42672 17 
 93547 14.49611 18 
 98576 14.56614 19 
 . 93605 14.68679 | 20 
 . 93634 14.70810 | 21 
 . 93663 14.78005 2 
 93692 14.85268 23 
 93721 14.92597 24 
 . 938750 14.99995 25 
 9377 15.07462 | 2 
 . 938808 15.14999 27 
 98837 15.22607 |-28 
 . 93866 15.30287 | 29 
 93895 15.38041 380 
 . 93924 15.45869 81 
 . 93953 15.58772 | 32 
 . 938982 15.61751 33 
 94011 15.69808 | 34 
 . 94040 15.77944 85 
 . 94069 15.86159 36 
 . 94098 15.94456 v4 
 94127 16.02885 38 
 .94156 16.11297 39 
 .94186 16.19848 | 40 
 94215 16.28476 41 
 94244 16.37196 o 
 . 94273 16.46005 | 43 
 . 94302 16.549038 44 
 94331 16.63893 | 45 
 .94560 16.72975 | 46 
 94889 16.82152 ie 
 94418 16.91424 4S} 
 94447 1700794 49 
 . 94476 17.10262 | 50 
 .94505 17.19830 51 
 9458 17.29501 52 
 . 94563 17.3927 53 
 94592 17.49153 | 54 
 94621 17.59189 5D 
 94650 17.69233 | 56 
 .91679 17.79438 57 
 94708 17.89755 58 
 94-737 18.00185 59 
 18.10782 60 
 
87° 88° 89° 
 / 
 Vers. Ex. sec. Vers. Ex. sec. Vers. Ex. sec. 
 0 - 94766 18.10732 . 96510 27 65371 - 98255 56. 29869 
 1 94795 18 .21397 .96539 2789440 98284 57 .26976 
 2 . 94825 18 382182 . 96568 28 18917 98313 |. 58.27431 
 3 . 94854 18.43088 . 96597 28.388812 98342 59.31411 _ 
 4 .94883 18.54119 . 96626 28 .64137 98371 60.39105 ° 
 5 .94912 18. 65275 . 96655 28 .89903 98400 61.50715 
 6 .94941 18.76560 . 96684 29.16120 98429 52. 66460 
 ff . 94970 18.87976 96714 29 .42802 98458 63. 86572 
 8 . 94999 18.99524 .96743 29.69960 98487 65.11304 
 9 . 95028 19.11208 96772 29.97607 98517 66 .40927 
 10 . 95057 19 .23028 . 96801 80.25758 . 98546 57 . 75736 
 11 .95086 19.34989 . 96830 30.54425 .98575 69.16047 
 12 .95115 19.47093 . 96859 80.83623 . 98604. 70. 62285 
 13 .95144 19.59341 .96888 81.13366 98633 92.14583 
 14 .95173 19.71737 .96917 31.48671 98662 3.73586 
 15 95202 19.84283 . 96946 81.74554 .98691 75 .39655 
 16 . 95231 19.96982 . 96975 382.06030 . 98720 77.1827. 
 4 - 95260 20.09838 - 97004 82.38118 .98749 78.94968 
 18 . 95289 20. 22852 - 970383 82. 708385 -98778 80.85315 
 19 . 95318 20.36027 . 97062 383 .04199 . 98807 82.84947 
 20 . 95347 20.49368 .97092 83. 38232 . 98836 84.94561 
 2 .95877 2C0.62876 .97121 83. 72952 . 98866 87.14924 | 
 22 .95406 20.76555 .97150 84.08380 .98895 89.46886 
 2 . 95435 20.90409 .97179 84.44539 . 98924 91. 91387 
 2: . 95464 21.04440 . 97208 84.81452 . 98953 9449471 
 25 . 95493 21.18653 . 97237 85.19141 . 98982 97 .22303 
 1 26 . 95522 21.38050 . 97266 85 .57633 .99011 100.1119 
 4 .95551 2147635 97295 85 .96953 .99040 103.1757 
 28 . 95580 21.62413 . 97324 36.37127 . 99069 106.4311 
 29 . 95609 21 .77386 . 97353 36. 78185 . 99098 109.8966 
 i | 30 . 95638 21. 92559 . 97882 87.20155 . 99127 118.5930 
 i 95667 22 .07935 97411 387. 68068 .§9156 117.5444 
 et 32 . 95696 22 . 28520 .97440 88 .06957 .99186 121.77 
 33 . 95725 22 .39316 .97470 88 .51855 . 99215 126 . 382538 
 34 . 95754 22553829 .97499 38 .97797 . 99244 131.2223 
 35 .95783 22.71563 97528 39 .44820 992738 136.5111 
 36 . 95812 22.88022 97557 39 .92963 . $9802 142.2406 
 i 37 . 95842 23.04712 - 97586 40 . 42266 - 99331 148 .4684 
 Haye | 38 95871 23 .21637 .97615 40. 92772 .99860 | 155.2623 
 A 39 -95900 23 . 388802 . 97644 41 .44525 . 99889 162.7083 
 40 . 95929 23.56212 .97673 41.97571 .99418 170.8883 
 41 . 95958 23.78878 - 97702 42.51961 .99447 179.9350 
 42 . 95987 23 .91790 97731 43 .07746 . 99476 189.9868 
 43 . 96016 24 .09969 97760 43 .64980 . 99505 201 .2212 
 iii 44 - 96045 24.28414 97789 44 23720 . 99585 213.8600 
 45 . 96074. 24.47134 97819 44 84026 99564 228 .1839 
 : 46 . 96103 24 661382 97848 45 .45963 . 99593 244.5540 
 47 . 96132 24.85417 97877 =|. 46.09596 - 99622 263 .4427 
 48 . 96161 25 .04994 97906 46 .74997 . 99651 285 .4795 
 49 . 96190 25 .24869 . 97985 47 42241 . 99680 311.5230 
 50 . 96219 25.45051 . 97964 48 .11406 . 99709 842.7752 
 51 . 96248 25 . 65546 .97998 48 82576 . 99738 880. 9723 
 52 - 9627 25. 86360 . 98022 49 55840 . 99767 428.7187 
 53 . 96307 26 .07503 . 98051 50.31290 . 99796 490.1070 
 54 . 96336 26. 28981 .98080 51.09027 . 99825 571.9581 
 55 . 96365 26 .50804 .98109 51.89156 . 99855 686.5496 
 56 . 96394 26.2978 .98138 52.1790 . 99884 858 .4369 
 57 . 96423 26 .95513 .98168 58 .57046 . 99913 1144.916 
 58 -96452 7.18417 . 98197 54.45053 . 99942 1717 .874 
 59 . 96481 27.41700 . 98226 55.35946 . 99971 3436 .747 
 60 .96510 27 .65371 . 98255 56.29869 1.00000 Infinite 
 
 TABLE XXIX.—NATURAL VERSED SINES AND EXTERNAL SECANTS, 
 Ss 
 
 Let pemed 
 aa Somyamncwue| 
 
 12 
 
 Le as a Sete ll ee cea 
 492 
 
CUBIC YARDS PER i00 FEET. 
 
 SuOPES 4:1. 
 
 Base | 3ase Base | Base | Base Base Base Base 
 12 | 14 16 18 22 | 24 26 28 
 45 53 60. We SS) 2b Bet HAS OO. 97 1 £105 
 93h ETE 107 122 137 [°° 167 181 196 | 211 
 142 | 163 186 208 253 275 297 | 319 
 193 | 222 252 281 341 37 400 430 
 a4, | 2892 | 319 -| 356 431 468 505 | 542 
 300 | 811 | 38) | 433 | 590 | sev | 611 | 656 
 356 408 | 460 512 616 668 719 v71 
 "415 74 533 593 "11 77 830 889 
 475 542 | 608 675 808 875 942 | 1008 
 537 611 | 685 %59 907 981 1056 113 
 601 682 | 764 845 1008 | 1090 | 1171 1253 
 667 756 844 933 1111 4200 | 1289 1378 
 734 831 | 926 1023 1216 1312 | 1408 1505 
 804 907 1010 1115 1322 1426 | 1530 | 1633 
 875 986 1098 | 1208 1431 1542 1653 | 1764 
 948 1067 | 1184 | 1304 1541 1659 | 1778 | 1896 
 1023 1149 127 1401 1653 1779 | 1905 | 2031 
 1100 1233 | 1366 1500 1767 4900 | 20383 | 2167 
 1179 | 1319 1460 | 1601 1882 | 2023 | 2164 | 2305 
 1259 1407 | 1555 | 1704 | 2000 | 2148 | 2296 | 2444 
 1342 1497 1652 1808 2119 | 2275 | 2431 2586 
 1426 | 1589 1752 «| «1915 | 2241 2404 | 2567 730 
 1512 1682 1853 | 2023 | 2864 | 2534 “05 | 2875 
 1600 1778 1955 | 2133 | 2489 | 2667 | 2844 | 3022 
 1690 | 1875 2060 | 2245 | 2616 | 2801 | 2986 | 3171 
 1781 | 1974 9166 | 2359 | 2744 | 2987 | 38130 | 3822 
 1875 2075 9074 o475 | 9875 | 3075 | 8275 | 3475 
 1970 | 2178 2384 2593 3007 215 | 3422 | 3630 
 2063 | 2282 2496 912 | 3142 | 3356 | 3571 3786 
 2167 2389 2610 2833 327 3500 | 3722 | 8944 
 2258 2497 2726 2956 | 3416 3645 | 8875 | 4105 
 2370 2607 2844 3081 3556 3793 | 4030 | 4267 
 2475 | 2719 2964 $208 3697 | 8942 | 4186 | 4431 
 2581 | 2833 3085 3337 3841 4093 | 4844 | 4596 
 2690 | 2949 3208 3468 | 3986 4245 | 4505 | 4764 
 2800 3067 3333 3600 4133 | 4400 | 4667 | 4933 
 2912 | 3186 3460 WA | 4282 | 4656 | 4831 | 5105 
 3026 | 3307 3589 3870 | 4438 | 4715 | 4996 | 5278 
 3142 3431 719 4008 4586 ASTD 5164 | 5453 
 3259 | 3556 3852 4148 | 4¢41 5037 | 5333 | 5630 
 3379 3682 3986 | 4290 | 4897 201 5505 | 5808 
 3500 3811 4422 | 4433 | 5056 | 5367 | 5678 5989 
 3623 | 3942 | 4260 4579 | 5216 5534 | 5853 | 6171 
 3748 4074 4400 472 5378 | 5704 | 6030 | 6856 
 3875 4208 4541 487 B42 «| «5875 6208 6542 
 4004 4344 4684. | 5026 | 5707 | 6048 | 6389 | 6730 
 4134 4482 | 4830 5179 5875 | 6223 | 6571 6919 
 4267 4622 | 4978 | 5833 | 6044 | 6400 "56 | W111 
 4401 4764. | 5127 | 5490 | 6216 6579 | 6942 | 7305 
 4537 | 4907 | 5278 | 5648 | 6389 6759 | 7130 | '%500 
 4675 | 5053 | 5480 | 5808 | 6564 6942 | 7319 | 7697 
 4815 | 5200 | 5584 | 59% 6741 7126 | 511 7396 
 4956 | 5349 | 5741 6134 | 6919 7312 | 7705 | 8097 
 5100 | 5500 | 5900 | 63800 | 7100 | 7500 | 7900 | 8300 
 5245 | 5653 | 6060 | 6468 weg2 | 690 | 8097 | 8505 
 53903 | 5807 | 6222 | 6637 | 7467 "881 8296 S711 
 5542 | 5964 | 6386 | 6808 | %653 | 807 8497 | 8919 
 5693 | 6122 | 6552 | 6981 "841 g270 | 8700 | 9130 
 5845 | 6282 | 6719 | 7156 8031 8468 | 8905 | 9342 
 6000 | 6444 | 6889 | 7333 | 8222 | 8667 | 9111 9556 
 
 Anteater iat EE ts 
 
TABLE XXX.—CUBIC YARDS PER 100 FEET. 
 
 Depth 
 d 
 
 Base 
 
 18 
 
 SLOPES \%: 1. 
 
 Baso 
 
 22 
 
 } Base 
 
 26 
 
 od 
 OR OWFH CDOONOUR WOH 
 
 bet et be 
 
 9817 
 10096 
 * 10380 
 ; 10667 
 han dee te oe OE ee eda ee 
 494 
 
 5994 
 
 6222 
 
 6454 
 6689 
 6928 
 7170 
 "417 
 1667 
 7920 
 8178 
 8439 
 8704 
 8972 
 9244 
 9520 
 9800 
 10083 
 10370 
 10661 
 10956 
 11254 
 11556 
 
 1861 
 2015 
 2172 
 2333 
 2498 
 2667 
 2839 
 3015 
 3194 
 3378 
 8565 
 8756 
 3950 
 4148 
 4850 
 4556 
 4765 
 4978 
 5194 
 5415 
 5639 
 5867 
 6098 
 6333 
 6572 
 6815 
 7061 
 
 11506 
 11815 
 12128 
 12444 
 
Depth 
 
 d 
 
 _ 
 
 ~) 
 
 — 
 COUR WMD DOCOMO CO? 
 
 beh ed peek Red ek 
 
 a 
 yor OTe 
 Gre OO 
 
 TABLE XXX.--CUBIC YARDS PER 100 FERT. 
 
 Base | 
 
 14 
 
 14989 
 15467 
 15952 
 
 16444 
 
 SLOPES 1: 1. 
 
 Base 
 
 30 
 
 115 
 237 
 
 367 
 
 Base 
 
 32 
 
 7233 
 7585 
 7944 
 8311 
 8685 
 9067 
 9456 
 9852 
 10256 
 10667 
 
 11085 
 11511 
 11944 
 12385 
 12883 
 13289 
 13752 
 14222 
 14700 
 
 15185 
 
 | 15678 
 
 16178 
 16685 
 17200 
 17722 
 18252 
 18789 
 19333 
 19885 
 20444 
 
TABLE XXX.—CUBIC YARDS PER 
 
 100 FEET. 
 
 SLOPES 14: 1. 
 
 Depth Base Base Base Base Base Base Base Base 
 / d 12 14 16 18 20 28 30 32 
 1 49 56 64 71 7 108 116 123 
 2 107 122 137 152 167 226 241 256 
 3 175 197 219 242 264 353 375 897 
 4 252 281 311 341 370 489 519 548 
 5 338 3875 412 449 486 634 671 708 
 6 433 478 522 567 611 789 833 878 
 Fe 538 590 642 694 745 953 1005 1056 
 8 652 711 V7 830 889 1126 1185 1244 
 9 775 842 908 75 1042 1308 1375 1442 
 10 907 981 1056 1130 1204 1500 1574 1648 
 11 1049 1131 1212 1294 1375 701 1782 1864 
 12 1200 1289 1378 1467 1556 1911 2000 2089 
 13 1360 1456 1553 1649 1745 2131 2227 2323 
 14 1530 1633 1737 1841 1944 2359 2463 2567 
 15 1708 1819 1931 2042 2153 2597 2708 2819 
 16 1896 2015 2133 2252 2370 2844 2963 8081 
 vi 2094 2219 2345 2471 2597 3101 3227 3353 
 18 2300 2433 2567 2700 2833 3367 3500 8633 
 19 2516 2656 2797 2938 8079 3642 3782 3923 
 I 20 2741 2889 3037 3185 8333 3926 4074 4222 
 | 21 2975 31381 8286 3442 8597 4220 437 4531 
 22 3219 3381 3544 3707 3870 4522 4685 4848 
 93 3471 3642 3812 8982 4153 4834 5005 5175 
 24 3733 3911 4089 4267 4444 5156 5333 5511 
 il 25 4005 4190 4375 4560 4745 5486 5671 5856 
 i 26 4285 4478 4670 4863 5056 5826 6019 6211 
 HT 27 457. 477 4975 5175 5375 6175 6375 6575 
 Hi 28 4874 5081 5289 5496 5704 6523 - 741 6948 
 iy 29 5182 5397 5612 5827 6042 6901 7116 7331 
 Hf 30 5500 5722 5944 6167 6389 278 4500 7722 
 rt 31 5827 6056 6286 6516 6745 7664 7894 8123 
 al 382 6163 6400 6637 6874 vale ial 8059 8296 8533 
 if 33 6508 6753 6997 7242 7486 8464 8708 8953 
 Hat 34 6863 7118 7367 7619 787 887 9130 9381 
 i 385 (227 7486 1745 8005 8264 9301 | . 9560 9819 
 th 36 7600 7867 8133 8400 8667 9733 10000 10267 
 i 37 7982 8256 8531 8805 9079 10175 =| 10449 | 10723 
 nt 38 8374 8656 8937 9219 9500 10626 10907 | 11189 
 39 877 9064 9353 9642 9931 11086 | 11375 | 11664 
 Vall 40 9185 9481 9778 10074 10370 11556 11852 12148 
 | i 41 9605 9908 10212 10516 | 10819 | 12034 | 12338 12642 
 Hy 42 10083 10344 | 10656 10967 | 11278 | 12522 12333 13144 
 | 43 1047 10790 | 11108 | 11427 | 11745 | 13020 | 13338 | 13656 
 44 10919 | 11244 | 11570 11896 12222 | 18526 | 18852 / 14178 
 45 11875 11708 | 12042 | 12875 | 12708 14042 | 14875 14708 
 } 46 11844 12181 12522 | 12863 | 18204 | 14567 | 14907 | 15248 
 47 12316 | 12664 | 13912 | 13360 | 13708 15101 15449 15797 
 48 12800 | 13156 | 13511 13867 | 14222 | 15644 16000 16356 
 49 18294 13656 14019 14382 | 14745 16197 16569 16923 
 50 13796 14167 | 14537 14907: || 15278 | 16759 17130 | 17500 
 51 14308 14686 | 15064 15442 15819 | 17331 7708 18086 
 52 14830 | 15215 | 15600 15985 16370 | 17911 18296 18681 
 53 15360 15753 | 16145 | 16538 16931 18501 18894 | 19236 
 54 15900 | 16300 | 16700 | 17100 | 17500 | 19100 | 19500 |- 19900 
 55 16449 | 16856 | 17264 17671 18079 | 19708 | 20116 | 20523 
 56 17007 17422 | 17837 18252 | 18667 | 20326 | 20741 21156 
 57 Lier: 17997 =| 18419 18442 | 19264 | 20953 | 21375 | 21797 
 58 18152 18581 19011 19441 19870 | 21589 | 22019 | 22448 
 59 18738 19175 19612 .| 20049 | 20486 | 22284 | 22671 23108 
 60 19383 1977 23222 | 20667 | 21111 22889 | 23333 | 2377 
 
 496 
 
© OI OTR OO tO 
 
 Base 
 
 16481 
 
 17094 
 17719 
 18354 
 19000 
 19657 
 20326 
 21006 
 21696 
 22398 
 23111 
 
 Base 
 
 17222 
 17850 
 18489 
 19139 
 19800 
 20472 
 21156 
 21850 
 22556 
 Qa202 
 
 24000 
 
 TABLE XXX.—CUBIC YARDS PER 100 FEET. 
 
 Base 
 
 3133 
 8413 
 3704 
 
 4005 
 4318 
 4642 
 4978 
 53824 
 5681 
 6050 
 6430 
 6820 
 (222 
 7635 
 8059 
 8494 
 8941 
 9398 
 9867 
 10346 
 10837 
 11339 
 11852 
 
 12376 
 12911 
 13457 
 14015 
 14583 
 15163 
 15754 
 16356 
 16968 
 17592 
 18228 
 18874 
 19531 
 20200 
 20880 
 21570 
 22272 
 22985 
 23709 
 24444 
 
 Base 
 
 1050 
 1244 
 1450 
 1667 
 
 1894 
 2133 
 2383 
 2644 
 2917 
 8200 
 3494 
 3800 
 417 
 Sid 
 
 4783 
 51383 
 5494 
 5867 
 6250 
 6644 
 7050 
 7467 
 7894 
 8333 
 
 8783 
 
 9244 
 
 9717 
 10200 
 10694 
 11200 
 11717 
 12244 
 12783 
 13333 
 13894 
 14467 
 15050 
 15644 
 16250 
 16867 
 17494 
 18133 
 18783 
 19444 
 20117 
 20800 
 21494 
 22200 
 22917 
 23644 
 24383 
 25133 
 25894 
 26667 
 
 SLOPES 114: 1. 
 
 Base 
 
 8555 
 
 9013 
 
 9482 
 
 9962 
 10452 
 10954 
 11467 
 11991 
 12526 
 13072 
 13630 
 
 14198 
 1477 
 15369 
 15970 
 16583 
 (207 
 17843 
 18489 
 19146 
 19815 
 20494 
 21185 
 21887 
 22600 
 28824 
 24059 
 24805 
 25563 
 26332 
 ryeae| 
 
YABLE XXX.—CUBIC YARDS PER 100 FEET, 
 
 SLOPES 2 ; 1. 
 
 es 
 Depth | Base | Base | Base | Base | Base | Base |.Base | Base 
 d 12 14 16 18 20 28 30 32 
 a 52 59 67 74 81 111 119 126 
 2 119 133 148 163 178 230 252 267 
 3 200 222 244 267 289 378 400 422 
 4 296 326 356 385 415 533 563 593 
 5 407 444 481 519 556 704 741 78 
 6 533 578 622 657 711 889 933 978 
 ( 674 726 V7 8380 881 1089 1141 1193 
 8 830 889 948 1007 1067 1304 1363 1422 
 9 1000 1067 1133 1200 1267 1533 1600 1667 
 10 1185 1259 1333 1407 1481 1778 1852 1926 
 11 1385 1467 1548 1630 1711 2037 2119 2200 
 12 1600 1689 177 1857 1956 2311 2400 2489 
 13 1830 1926 2022 2119 2215 2600 2696 2793 
 14 2074 2178 281 2385 2489 2904 8007 3111 
 15 2333 2444 2556 2667 2778 8222 3333 3444 
 16 2607 2726 2844 2963 3081 8556 3674 3793 
 17 2896 8022 3148 327 3400 3904 4030 4156 
 . 18 3200 $323 3457 3600 37383 4267 4400 4533 
 i 19 3519 38659 8800 3941 4081 4644 4785 4926 
 ; 20 3852 4000 4148 4296 4444 5037 5185 5333 - 
 Hg 21 4200 4356 4511 4667 4822 5444 5600 5756 
 ea 22 45638 730 4889 5052 5215 5867 6030 6193 
 23 4941 5114 5281 5452 5622 6304 6474 6644 
 24 5333 5511 5689 5867 6044 6756 69338 7111 
 25 5741 5926 6111 6296 6481 7222 7407 7593 
 ih} 26 6163 6356 6548 6741 6933 7704 7896 8089 
 tal vi 6600 6800 7000 7200 7400 8200 8400 8600 
 iti 28 7052 (259 7467 VOT4 78381 711 8919 9126 
 { 29 7519 7 3¢ 7948 8163 8378 9237 9452 9667 
 Mi 30 8000 8222 8444 8667 8889 9778 10000 10222 
 | 31 8496 8726 8956 9185 9415 10333 10563 10793 
 ln 32 9007 9244 9481 9719 9956 10904 11141 11378 
 i 33 9533 97°78 10022 10267 10511 11489 11733 11978 
 i 34 10074 10326 10578 10880 11081 12089 12341 12593 
 35 10630 10889 11148 11407 11667 12704 12963 13222 
 . 36 11200 11467 11733 12000 12267 13333 13600 13867 
 37 11785 12059 12333 12607 12381 13978 14252 14526 
 f 38 12385 12667 12948 13280 13511 14637 14919 15200 
 i 39 13000 | 13289 | 13578 | 13867 | 14156 | 15311 | 15600 | 15889 
 Ha 40 18630 13926 14222 14519 14815 16000 16296 16593 
 Hit 41 14274 14578 14881 15185 15 {89 16704 7007 17311 
 42 14983 15244 15556 15867 16178 17422 17733 18044 
 43 15607 15926 16224 16563 16881 18156 18474 18793 
 44 16296 16622 16948 17274 17600 18904 19230 19556 
 45 17000 17283 17667 18000 18333 19667 | 20000 | 20383 
 1 46 17719 18059 18400 18741 19081 20444 | 20785 | 21126 
 47 18452 18800 19148 19496 19844 | 212387 | 21585 | 21933 
 48 19200 19556 19911 20267 | 20622 | 22044 | 22400 | 22756 
 49 19963 20826 | 20689 21052 | 21415 22867 23230 23593 
 50 20741 20711 21481 21852 | 22222 | 23704 24074 | 24444 
 51 21288 © | 21911 22289 22667 23044 | 24556 | 24983 | 25311 
 52 22341 22726. |.23111 23496 23881 25422 | 25807 26193 
 53 23163 | 23556 | 23948 | 24341 24733 | 26304 26696 | 27089 
 54 24000 24400 | 24800 25200 25600 27200 27600 28000 
 55 24852 | 25259 | 25667 | 26074 | 26481 28111 28519 | 28926 
 56 25719 | 261838 | 26548 | 26963 | 27878 | 29037 | 29452 | 29867 
 57 26600 | 27022 | 27444 | 27867 | 28289 | 29978 | 30400 | 30822. 
 58 27496 27926 | 283856 | 28785 | 29215 | 30933 31363 | 31793 
 | 59 28407 28844 | 29281 29719 | 80156 | 31904 | 82341 3277 
 60 29333 | 29778 | 30222 | 30667 | 31111 32889 | 338333 | 3377 
 
 498 
 
| 
 | 
 
 Depth 
 
 d 
 
 ODF Oo OTR CO DD 
 
 10 
 
 Base 
 
 12 
 
 56 
 133 
 933 
 856 
 500 
 667 
 856 
 
 1067 
 1300 
 1556 
 
 1833 
 2133 
 2456 
 2800 
 3167 
 3556 
 3967 
 4400 
 4856 
 5333 
 
 5833 
 6356 
 6900 
 7467 
 8056 
 8667 
 9300 
 9956 
 10633 
 11333 
 12056 
 12800 
 13567 
 14356 
 15167 
 16000 
 16856 
 17733 
 18633 
 19556 
 20500 
 21467 
 
 87395 
 
 88633 
 39956 
 41300 
 42667 
 
 TABLE XXX.—CUBIO YARDS PER 100 FERT. 
 
 Base Base Base Base 
 14 16 18 20 
 63 7 78 85 
 148 163 78 1938 
 256 78 800 822 
 385 415 444 474 
 537 574 611 648 
 711 756 800 844 
 907 959 1011 1063 
 1126 1185 1244 1304 
 1367 1433 1500 1567 
 1630 1704 1778 1852 
 1915 1996 207 2159 
 2222 9311 2400 2489 
 2552 2648 2744 2841 
 2904 3007 8111 8215 
 8278 3389 8500 3611 
 8674 3793 3911 4030 
 4093 4219 4344 447 
 4533 4667 4800 4933 
 4996 5137 5271 5419 
 5481 5630 16 5926 
 5989 6144 6300 6456 
 6519 6681 8 7007 
 7070 7241 411 7581 
 7644 7822 8000 8178 
 241 8426 8611 8796 
 8859 9052 9244 9437 ° 
 9500 9700 9900 | 10100 
 10163 | 1037 1057¢ 10785 
 10848 | 11063 11278 11493 
 11556 TA 12000 12222 
 12285 12515 12744 12974 
 18037 138274 13511 13748 
 18811 14056 14300 14544 
 14607 14859 | 15111 15°63 
 15426 15685 15944 16204 
 16267 16533 16800 17067 
 17130 | 17404 | 17678 17952 
 18015 18296 18578 18859 
 18922 19211 19500 19789 
 19852 | 20148 | 20444 | 20741 
 20804 | 21107 | 21411 Q1715 
 2177 22089 | 22400 | 22711 
 2277 23093 23411 23730 
 237938 | 24119 24444 | 2477 
 24833 «| 25167 25500 | 25833 
 25896 | 262387 | 26578 | 26919 
 26981 27330 |. 27678 |. 28026 
 28089 | 28444 | 28800 | 29156 
 29219 | 29581 29944 | 30307 
 80370 80741 81111 381481 
 31544 81922 | 32300 ° | 32678 
 82741 33126 838511 33396 
 83959 34352 34744 85137 
 35200 85600 36000 36400 
 36463 36870 8727 37685 
 7748 88163 | 8857 38993 
 89056 89478 89900 40322 
 40385 40815 41244 41674 
 41737 42174 42611 43048 
 43111 43556 | 44000 | 44444 
 
 30 
 
 22222 
 23283 
 24267 
 25322 
 
 45670 
 47111 
 
Title, 
 
 Number. 
 
 Ratio of circumference to diameter........ 1 4 3.1415927 
 
 Reciprocal Of Same'.........c.scccceeeccecs - 0.3183099 
 t 180° 
 
 Degrees in arc of length equal to radius... es 57.295780 
 " 
 
 Minutes ‘ hy a ‘ ne _— 8487 .'7468 
 
 Seconds 66 66 66 66 66 ey ae 206264 .81 
 
 a4 Z 
 
 Length of 1° arc, radius unity............... 180° -01745829 
 
 Lengthof 1’ are, “ = “ _7 _ | oooggoso 
 tol > @ececeeeeeeeecsese 10800 
 
 Length of 1” are, ‘ Sarah ts ORES f. Pid 000004848 
 . 648000 
 
 Radius by which 1 foot of arc = 1 degree. 57 295780 
 
 Radius “ SOT st, TUS “Ss = 1 minute. 343 .'77468 
 
 Radius ‘‘ Sor hs ‘* = 10 seconds 206. 26481 
 
 Factors for dividing a line into extreme 0.6180340 
 
 and ‘mean ratio. . G87. |... HAE in.. Poon 0.3819660 
 
 Base of hyperbolic logarithms.............. € 2. 7182818 
 
 Modulus of common system of logs. = log ¢ M 0.4342945 
 
 Reciprocal of same = hyp. log. 10.......... + 2.3025851 
 
 Length of seconds pendulum at New York 
 
 iInpinches Gcscns 2. baeo. Fees eee C. 8911256 
 
 Length of seconds pendulum at New York 
 
 Tre LeGb.|... ese <.hisee: | Aas} eee 3.25938 
 Acceleration due to gravity at New York... g 82.1688 
 Square root of same ............0.ecccese pai Vg 5.67175 
 
 Yands in) ] metre, ssc... ees ok Sale cc 
 
 Feet 
 
 1.093623 
 3.280869 
 89.37043 
 0.304797 
 0.914892 
 1609 .330 
 
 in’ Lacs 
 
 Se eeeeeeosereseeeaseessereseos 
 
 Inchesin 1 ‘* 
 
 Se eeseceee verse sees sso ee Peres 
 
 Metresiin! 12006, .). . acing. «i. < dala iete clue tele s de 
 
 Metres ine diyard 2 cesses bors cetade cieeeee de 
 
 Metres in mile... eis. esic cate. ee 
 
 TABLE XXXI.—USEFUL NUMBERS AND FORMULA, 
 
 0.4971499 | 
 
 9.5028501 | 
 1. 7581226 
 8.5362739 
 5.3144251 
 82418774 
 6. 4637261 
 4.6855749 
 
 1.7581226 
 2.5862739 
 2.3144251 
 9.7910124 
 9.5820248 
 
 0.4342945 
 9.6377843 
 0.3622157 
 
 1.59238162 
 
 0.5131850 
 1.5074347 
 0. 7537178 
 
 0.0388676 
 0.5159889 
 1.5951701 
 9.4840111 
 9.9611324 
 3.2066450 
 
 4 
 
TABLE XXXI.—USEFUL NUMBERS AND FORMULA®. 
 
 i ; Loga- 
 Title. Symbol. | Number. | jithm. 
 Cubic inches in 1 U.S. gallon.............. 231. 2.3636120 
 . «© 86 ‘1 Imperial gallon.......... 277.274 | 2.4429092 
 no Semel Ease DUSNOl anne se cores = 2150.42 | 3.33252383 
 Cubic feet in 1 U.S. gallon................. 0.133681 | 9.1260683 
 * “4 Imperial gallon? .3 4.7. .f- 5 0.160459 | 9.2053655 
 = Ks STU USM Clos cs aes cere cae 1.244456 | 0.0949796 
 Weight of 1 cub. foot of water, barom. 30 in. 
 ther. 39°.88 Fah.; pounds. . 62.379 | 1.'7950384 
 se eaOe sy re < ie 62.321 | 1.946349 
 Weight in grains, 1 cubic inch, at 62° Fah.. 252.458 | 2.4021892 
 No. of grains in 1 pound avoir...........+..- 7000. 3.8450980 
 < a LOSI e sp ESAS es apenas 437.5 2.6409781 
 Se RO ee ee 
 Ege.) 1m0e 
 ac=—, 
 y = radius of circular arc; r “4 
 ~t 180° 
 1 = length of arc; r= — 
 Os) 40 
 a° = degrees in same arc. 
 
 Radius by which the length of chord c in feet = a in minutes; 
 
 ea’ 
 
 10 sin 44a’ 
 
 Hyp. log x = com. log x x a or 
 
 com. log (hyp. log x) = com. log (com. log x) + 0.3622157 
 Com. log « = M x hyp. log x ; or 
 
 com, log (com. log x) = 9.6377843 + com. log (hyp. log x) 
 
 Circumference of circle (radius = 1)............ceeeeeee seer eeeeees 2r 
 MMPESOL CUTIO .. . rasa ss Ledeen nes eo ea a Fae aa ects aine SNe yoo s mr? 
 Area of sector (length of arc = 1)......... ee eeee cece rece cer ereece Lélr 
 Area of sector (angle Of ATC = A°).....2 ce eee e eee cece eee e eee tenes sap wr? 
 
 Approximate area of segment (chord = c, mid. ord, = HO) ween eee 
 
APPENDIX, 
 
 Verification of eq. (77). 
 
 a sin f) . 
 Hq. (76) p= = aR = sin 7}. cosec W 
 sin a 
 
 ay = cos (9. cose : in f) . cot ¥ cosec (76% 
 
 do = ° eae WV - Sin. N 5 > N ((0$) 
 
 do iG . 
 
 ag =P cot 0) — V cot W (7) 
 Verification of eq. (81). | 
 
 Differentiating eq. (763) | 
 aa } 60 9 6 6 
 "FT eee sin  cosee Wo cos 0 cot W cosec W + 
 
 jim 1 6 
 ere Det e@ Lf aie ie@3 
 We? sin @ cot W cosee WV +- Wa sin @ cosée V 
 
 20 6 p 6 6 
 oceaaty « ey aie cot 9 . cot wy a. We cot? V + cosec? Wv 
 
 BRS ( 1 : t E 1 (DP eot2 y 
 “an? S478) sn scar cot 9 cot N SP we! cot NV + 1) 
 
: APPENDIX. 303 
 
 ap” \? 
 oie i Sane 
 Now (» Z oe 
 Pires: ase. == ——— 
 ape a’ 
 s (3 en ee ee 
 pre dj? Pag 
 
 in which substitute for “f. and for ae and let 
 t : t a 
 cot @ — wv © RG ste a 
 
 (0? + p2(— ay?) 
 Oye Penk eta. a 
 p? +293 (—a)? — p?(-1- 5s cot 0 cot + + ya 2 cot? 7 +0) 
 
 3 
 2)2 
 a Ps ue 7 
 : 1 0 1 6 1 
 1+az+ W CC a ~ Ws cot® W 2? 
 La ae . 
 2 
 
 1 pe re, { ae 6 
 Sa pean 2 et — 4 a< ee sé 
 1 an2 t & “+ ; cot N 00d N cot 7) 
 
 (+a)! 
 
 Me nd ( ai Sota) 
 5 N2 i hee N 
 
 (+a?) 
 
 1 — syq — acotg 
 
 r) Pre) 
 
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