"^m ■* \-hr-i ■?:. iv <( ■ ■^^^mi -'^'K,^ • ,■■■?• ■' ■ ' '¥ ■ ?•» ' ' ,t-v- ■" .4- ::,:,:-3f;, ■%■ ': -^■.:.-r , .•.''••V ,..s ,. >*!►>■'' ['.'.♦'i^.v; -:>■•.-■ ■-VV . vW^'^ >v- /v-.-;"; ■;::"( ■^:.iiir .-v^'' .*>:»'• .«'*** ^ 7 m%:il^^^^ ( A FIELD-MANUAL FOR KAILROAD ENGINEERS, BY J. C. NAGLE, M.A., M.C.E., professor nf Civil Knaineenna if the Agricultural and Medianicat Culleye of Texas. SECOXD KlUTIOX, HE VISED. SECOXI) T1JOLSAJ5-D. KEW YORK: JOHN AVILEY & SONS. London: CHAPMAN & HALL, Limited. 1903. Copyright, 1897 BY J. C. NAGLE. ROBERT DRUMMOND, ELECTROTYPER AND PRINTER, NEW YGnK. PREFACE. Ease of reference and uniformity of notation are essential in a book that is to be consulted in the field. With this in mind an effort has been made in the following pages to secure a systematic arrangement of the subject-matter and uniformity of terms and - notation. Except for a few cases Greek letters have been avoided "' and a single letter is used to designate an angle. In so far as "j)racticable each figure is intended to be self-explanatory, so that ^the explanations necessary in connection with the problems have -> been reduced to a minimum. Algebraic equations stand each in ra distinct line, thus rendering them more easily read. t A knowledge of the elements of geometry and trigonometry has ^been assumed, and only in the derivation of a few formulas in ^^connection with the theory of transition-curves will any higher mathematics be needed. But these formulas may be accepted by the reader who is unfamiliar with the calculus without in any way affecting his ability to understand their applications or to follow sabst>quent reasoning. rj One can most readily turn to what he wants in a book after hav- < ing become familiar with its contents in the classroom. Keeping "^ this in mind this book has been written so that it may be used as a text as well as for reference in the field. Wherever practicable -^.solutions to problems have been given in a rigid, general form, 5' followed by illustrative examples, so that the student need not viose sight of the principle involved while following the solution £ for a particular case. Wherever approximate solutions seemed ■ preferable they have also been given and their limitations pointed '^ut. Free use has been made of the Table of Functions of a One- degree Curve, thus reducing the labor of field computations. By defining the degree of curve with reference to short chords for I I 45577 '" IV PREFACE^ sharp curves — and, with tables of Radii, Long Chords, Mid- ordinates, etc., based on appropriate equations — the errors result- ing from assuming the radius to vary inversely with the degree of curve will generally be found to be quite small. Chapter I gives briefly the general method of making Re- connoissance; Chapter II treats of Preliminary Surveys; while Chapter III relates to Location. Chapter IV, on Transition-curves, follows the method adopted by Professor Crandall, and enables one to locate the transition- curve with rigid accuracy where such is necessary. Approximate methods are also given by means of which the curve may be as easily located as any of the more limited easement curves ordi- narily met with. Chapter V, on Frogs and Switches, contains all that is necessary for their location. The formulas have been arranged to give the desired quantities in terms of the frog number whenever the re- sulting equations would be easier of application than the trigono- metric ones usually given. The turnout tables are unusually full and give not only the theoretical lead but the stub lead as well, from which the practical lead can be at once found when the length of switch-rail is known. Chapter VI, on Construction, tells how to set slope-stakes, and gives simple methods for computing areas and volumes either directly or by the use of tables. A short table of prismoidal corrections is given for end sections level, and also a formula for three-level sections, by means of which a suitable table may be computed if desired. The tables at the end of this book have been arranged with a view to ease of reference, for, whatever the character of the text, the chief value of a field-book must depend upon the ease with which the tables may be consulted and upon their extent and accuracy. Table IX — Functions of a One-degree Curve — sepa- rates the logarithmic functions on the one side from the natural functions on the other and will be of assistance in locating these tables. Table XVI — Transition-curve Table — reading lengthwise of the page, likewise serves to separate the trigonometric tables from the miscellaneous tables that follow. Some engineers object to the use of logarithmic tables in the field, but for them the natural functions are at hand ; while for those who prefer logarithms the five-place tables of logarithmic sines, cosines, etc., will be found easy to consult and interpolate between. PREFACE, V All trigonometric tables are five-place, and others were carried to as many decimal places as their cliaracter demanded. Tables I, III, IV, and V have been computed to agree ^vith the definition of the degree of curve requiring curves sharper than 7° to be run with chords less than 100 feet in length, as described in the text. Tables XVII and XV 111 were also com- puted expressly for this book. Tables VI and XXVll are from electrotyj^es fram Ca' iiart's Field Book for Cioil FJngineers and were furnished by (linn k Co. Electrotypes of Tables II. X, Xll, XIII, XIX, XX, XXIV, XXV, XXVI, and also XVI — this last being from Crundall's book, 'I'he Transition Curve — were furnished by John Wiley & Sons. Of the others, some were arranged from standard tables and others adapted in part and extended to increase their usefulness. It will be noticed that vertical lines have been omitted wher- ever practicable, thus rendering it easier to refer to the tables. Acknowledgments are due my associate. Professor D. W. Jpence, for aid in making the tabular computations and in reading proof. J. C. Nagle. College Station, Texas, May, 1897. PREFACE TO THE SECOND EDITION. In this edition some of the typographical and other minor errors that appeared in the first edition have been eliminated. Tables XXVIII and XXIX have been added in order to increase the use- fulness of the book, and are from electrotypes of tables in Traut- v/ine's Pocket Book. A suggestion has been made by one who has had occasion to use the tables quite freely that Table XIX be extended so as to give quantities for variations of one tenth of a foot in center heights, but such extension would have increased the size of the book unduly. When closer approximations are wanted than are given by Table XIX the area for the given center height can be taken from Table XVII and by entering Table XX irith this as argument the quantity can be at once read off. For «enter heights greater than those given in Table XVII we may refer to books devoted exclusively to earthwork computations. J. C. N. College Station, Texas, Jauuary, 1899. CONTENTS. CHAPTER I. RECONNOISSANCE. Article 1. Objects of Reconnoissance — How Made. SECTION PAGE 1. Relative Importance of tlie Work of Reconnoissance and Location.. 1 2. Object of Reconnoissance 2 3. The Instruments 2 4. Use of Maps , 4 5. Making the Reconnoissance 4 CHAPTER II. PRELIMINARY SURVEYS. Article 2. Objects; The Field Corps; Duties of the Chief. 6. Objects of Preliminary Surveys 6 7. The Exploration-line .- ^ 8. Data Sought in Making Preliminary Surveys 7 9. The Field Corps , 7 10. The Chief of Party, Duties of 7 Article 3. The Transit Party, a. duties of the members. 11. Composition of the Transit Party fi 12. The Transitman 8 13-17. Other Members of the Party £ 18. Instruments S B. TRANSIT ADJUSTMENTS— THE VERNIER. 19. Kind of Transit S 20. To Adjust the Plate Levels IC 21. Parallax IC 22. To Adjust the Line of Collimation IC 23. To Adjust the Standards 11 vii Vlll CONTENTS. SECTION PAGE 24. To Adjust the Level on Telescope 12 25. Direct and Retiograde Verniers 13 26. The Least Count of a Vernier c....o 13 27. To Read a Vernier 14 C. ACCESSORIES. (1°) The Gradienter. 28. Description and Method ofUsing Gradienter 14 (2°) The Stadia, or Telemeter. 29. Principle of the Stadia 15 30. Formula for Line of Sight Horizontal 15 31 . Formulas for Line of Sight Inclined 16 32. The Instrumental Constant, To Find .^ IT 33. Reducing the Notes 17 D. FIELD-WORK. 34. station Numbers , 18 35. Hubs or Plugs 18 36. Reference-points . . 18 37. Alignment 18 38. Form of Transit Notes , 19 39. Stadia Methods for Preliminary Surveys 19 E. OBSTACLES IN TANGENT. 41. To Pass an Obstacle by Means of Parallel Lines 20 42. To Pass an Obstacle by Angular Deflections 20 43. To Measure across a River 21 Article 4. The Level Party. 44. Make-up and Instruments 23 45. Work of the Leveler 23 46. Work of the Rodman 23 adjustments of the level. 47. To Adjust the Line of Collimation ... 23 48. To Adjust the Level-bubble 24 49. To AdjntJt the Wyes 25 B. THEORY OF LEVELING. 50. True and AuDarent Level 25 51. The Error Due to Curvature 25 52. The Difference of Elevation of Two Points 26 C. FIELD-WORK. 53. The Datum 27 54 Bench-marks 27 55 Work in the Field 28 \ CONTENTS. ix SECTION , PAGE 56. Tbe Level Notes 28 57. Precautions when Using Level 29 58. TheRod /v;..^.. .,... 29 Article 5, The Topographic Party. 59. Instruments Used ; Area to be Mapped 30 80. Methods of Recording Data 30 61 . Topographers' Field-sheets o . « 31 62. Use of the Slope-level 31 63. Cross-section Rods 32 64. The Transit and Stadia in Topographical Surveying 32 Article 6. Preliminary Estimates. 66. Map of Preliminary Lines . . 32 67. The Profile 33 68. Preliminary Estimates of Quantities 33 69. Report of the Locating Engineer 34 CHAPTER III. LOCATION. Article 7. Projecting Location. 70. Problems Involved in the Paper Location 35 71. Hints Regarding Methods of Projecting the Line 35 72. Tbe Curve-protractor 36 r^'i. Work in the Field 37 Article 8. Simple Curves. A. definitions and formulas. 74. Definitions 38 75. To Find the Radius R, the Degree of Curve Being Known 40 76. To Find the Length of Curve 42 77. The Functions of a One-degree Cui've 42 79. To Find Z), -R and C Being Known 43 80. To Find the Tangent Distance T, I and B Being Known 43 81. To Find R, Given I and T 44 82. Given i and D, to Find the Long Chord i.C 44 83. Ordinates from Chord 45 84-86. To Find the External E 48 87. To Find 7?, E and 2 Given 49 88. To Find r, S and i Given 49 89. To Find the Deflection Offset from Chord Produced 49 90. To Find the Tangent Deflection Offset 50 91 . The Sub-tangential Deflection Offset 51 92. To Find the Tangent Offset z 52 ■33. Difference in Length of Arc and Long Chord 53 X CONTENTS. B LOCATING SIMPLE CURVES. SECTION PAGE 94. To Locate a Curve vvitli the Chain by Offsets from Chords ProduceU 55 95. To Locate a Curve by Offsets from Tangent 57 96. To Locate a Curve by Offsets from a Long Chord 58 97. To Locate a Curve with Transit and Chain 59 98. The Index-angle 60 99. Subdeflection-angles 60 100- lOL Transit Notes 61 C. OBSTACLES. 102. To Pass an Obstacle on a Curve 63 103. To Locate a Curve when the P. C. is Inaccessible , 64 104. To Pass to Tangent when the P. T. is Inaccessible 67 105-107. To Pass a Curve through a Given Point 69 108. To Locate a Tangent to a Curve from an Outside Point . 71 109. To Run a Tangent to Two Curves of Contrary Flexure 78 D. CHANGE OF LOCATION. 110. To Locate a Curve Parallel to a Given Curve 73 111. To Change P.O. in Order to Make P.T. Fall in a Parallel Tangent. . 74 112 To Change R and P.C to make P.T. Fall in Parallel Tangent, on Same Radial Line 75 113. To Find Change in P.C. or R for a Given Change in 7 76 114. Required the Change in P.C. and R for a Given Change in 7, the P.T' Unchanged . ... . .. 77 115. To Find Nevv Radius for a Given Change in T 77 116. To Find New 72 to Connect P.C. with a Parallel Tangent 78 Article 9. Compound Curves. A. location problems. 117. Given Both Tangents and One Radius, to Find the Other Radius ... 80 118. Given One Radius, the Long Chord and the Angles it Makes with Tangents, to Find the Other Radius and Central Angles . 82 119. Given the Radii and Central Angles, to Find the Tangents, the Long Chord, and the Angles it Makes with Tangents ,.... 82 120. Given the Long Chord and Angles Made with Tangents, to Find Both Radii when Common Tangent is Parallel to Long Chord S3 B. obstacles. 121. To Locate Second Branch when P. C. is Inaccessible 84 C. CHANGE OF LOCATION. 122. To Compound a Simple Curve so P.T. shall Fall in a Parallel Tan- gent.. . 85 123. To Find Change in P.CC. Necessary to xMake P.T. Fall in a Par- allel Tangent 86 124. To Change P.CC. and Second Radius so P.T. shall Fall in a Par- allel Tangent, on Same Radial Line , 89 CONTENTS. XI SECTION PAGE 125. To Chanjje P.C.C. and Second Radius to Cause P. T, to Fall at a New Point in Same Tangent 91 126. To Substitute a Three-centered Compound Curve for a Simple One. 94 127. To Substitute a Curve for a Tangent Uniting Two Curves 95 Article 10. Track Problems. 128. Reversed Curves, Where to Use 96 129. To Connect a Located Curve with an Intersecting Tangent 97 130. To Locate a Y 100 131. A Reversed Curve between Parallel Tangents 102 132. A Crossover between Parallel Tracks when a Fixed Length of Tan- gent is Inserted , 105 133. A Reversed Curve with Unequal Angles 106 134. A Reversed Curve between Fixed Points 106 135. To Connect Two Divergent Tangents by a Reversed Curve 107 136. To Change P.jR. a so P.T. shall Fall in a Parallel Tangent.. 108 137. To Find the Radius of a Curved Track 109 CHAPTER IV. TRANSITION-CURVES. Article 11. Theory of the Transition-ccrve. 138. Elevation of Outer Rail on Curves 110 139 Requirements of the True Transition-curve Ill 140. Notation Employed ill 141. Equation of Transition-curve 112 142. Transition-curve Angle, / 114 143. Coordinates of Points 114 144. Deflection-angles 115 145. Explanation of Transition-curve Tables 118 146. To Unite the Brandies of a Compound Curve by a Transition- curve 119 147. Length of Transition-curve to be Taken 121 Article 12. Field-work. A. field formulas. 148. When to Use the Simplified Formulas 122 149. Simplified Formulas for Transition-curves 122 1.50. Offsets 124 151. Compound^^ Curves 125 B. setting out transition-curves. 153. Location by Offsets 125 154. Location i;y Deflection-angles 126 155. Form of Transit Notes for Tran.sition-curves 128 . i Xll CONTENTS. Article 13. Transition curve Problems. SECTION PAGE 156. Tangent Distances and Exterual for Equal Offsets 129 157. Tangent Distances, Offsets Unequal 130 158. Transition-curves Inserted without Changing the Vertex of Cir- cular Curve 131 159. Transition-curves Inserted with Least Deviation from Old Track.... 133 160. Transition-curves Inserted at Ends of Long Circular Curve, Cen- tral Portion Undisturbed 133 161. Transition-curve Inserted at P.C.C. by Changing Radius of Second Branch 136 162. To Insert Transition-curves at the Ends of Two Circular Curves United by a Common Tangent 138 163. To Unite a Tangent and Circular Curve when the Offset Cannot be Directly Measured . 139 164. Inserting Transition-curves in Old Track 140 165. Remarks on Tabular Interpolations 140 CHAPTER V. FROGS AND SWITCHES. Article 14. Turnouts. A. turnouts from straight lines.^ 166. Definitions 143 167. To Find the Lead, I, and Radius, R, in Terms of the Frog Number, N, and Gauge, g 144 168. Given R and g, to Find N, I, and Frog-angle, F 146 169. To Find Theoretic Length of Switch-rail 146 170. To Find Lead and Number of Crotch-frog for a Double Turnout to Opposite Sides of Main Track 147 171. To Find Turnout Radius and Lead of Crotch-frog in Terms of Crotch-frog Number 148 172. To Find Radius of Curve from Point of Middle Frog to Point of Main Frog, Given i\r,, N, andN' 148 173. Double Turnout to Same Side of Main Track 150 174. To Find Radius of Curve between Frog-points for a Double Turn- out to Same Side of Main Track 151 175. To Unite Main Track with Siding. Reversing Point Opposite Frog . . 152 176. To Lay Out a Ladder-track .. 153 B. turnouts from curves. 177. To Find Lead and Radius for Turnout to Concave Side of Main Line 154 178. To Find Lead and Radius, Turnout to Convex Side 157 179. To Find Theoretic Length of Switch-rail <, 158 180. To Unite Main Track with a Concentric Siding 160 CONTENTS. Xlll C. THE STUB LEAD, SECTION PAGE 181. Definitions 162 182. Given N, t, and g, to Find the Stub Lead 362 183. Turnout Table and Explanation 163 184. To Stake Out a Turnout 165 185. Curving Rails 165 Article 15. Crossovers. 186. Crossover between Parallel Straight Tracks, a Tangent between Frog-points 166 187. A Crossover in the Form of a Reversed Curve 168 188. A Crossover with Fixed Length of Intermediate Tangent 168 189. A Crossover between Curved Main Tracks 168 Article 16. Crossing-frogs and Crossing-slips. A. crossing-frogs. 191. Length of Rail Intercepted between Two Intersecting Straight Tracks 170 192. Angles of a Set of Crossing frogs, One Track Curved 170 193. Angles of a Set of Crossing-frogs, Both Tracks Curved 171 B. crossing-slips. 195. Length and Radii of Slip-rails, Both Tracks Straight 172 196. Length and Radii of Slip-rails, One Track Curved 172 197. Length and Radii of Slip-rails, Both Tracks Curved! 173 CHAPTER VI. CONSTRUCTION, Article 17. Definitions ; General Considerations ; Vertical Curves ; Elevation op Outer Rail. 199. The Division Engineer 176 200. The Resident Engineer 176 201-204. Definitions 177 205. To Find the Grade-point, Longitudinal Slope Uniform 178 206. Vertical Curves , 17? 207. Elevation of Outer Rail on Curves 188 208. Easing Grade on Curves 183 Article 18. Earthwork. A. SETTING slope-stakes. 209. The Distance Out for Level Sections ISS 210. To Find Position of Slope-stakes for Surface Inclined 184 211. Cross-section Notes •. i86 212. Irregular Sections 187 Jil3. Staking Out Openings 187 XIV CONTEJSTS. SECTION PAGE 214. Manner of Marking^ Stakes , 187 215. Shrinkage— Growth 187 216. Borrow-pits, Drainage of, etc 188 B. AREAS OF SECTIONS. 218. Area of Three-level Section. 188 219. Area of Five-level Section 189 220. General Formula for Areas 190 221. Explanation of Table of Areas of Level Sections and the Three- level Correction , 191 C. VOLUME OF EARTHWORK. 222. Where Cross-sections should be Taken 192 223. Volume by Averaging End Areas 192 224. The prismoidal Formula 193 225. Form of Record 195 226. The Prismoidal Correction 195 227. Computation of Volumes when Passing from Cut to Fill 198 228. Use of Tables of Volumes in Making Preliminary Estimates 199 229. Side Ditches 199 230. Earthwork on Curves 199 231. Overhaul .-.201 Article 19. Grade and Ballast Stakes, Culverts, Bridge.^, and tunnils. 232. Grade and Center Stakes 202 233. Ballast-stakes 202 235. Openings of, for Culverts, Trestles, etc 202 236. Bridge Piers and Abutments 203 237. Tunnels , 204 Article 20. Monthly and Final Estimates. 238. Monthly Estimates 205 239. Measurements for Earthwork 206 240. Classification of Earthwork 206 211. The Progress Profile 207 242. Masonry Estimates 207 243. Bridge Estimates 207 244. Track Material ."* 207 245. Blank Estimate Sheets 208 246. Monthly Payments 208 247. Extras » 208 248. Final Estimate 208 249. Acceptance 209 TABLES. Table Showing Length of Transition -curve to be Taken 121 Table of Values of g - Vgi for Stub Lead 163 Turnout Table 164 Table of Corrections for Vertical Curves , 181 CONTENTS. XV PAGE Tabie of Elevation of Outer Rail on Curves 182 Table of rrisinoi'dal Corrections for Level Sections 196 I. Radii of Curves 212 II. Minutes in Decimals of a Degree 215 III. Tangential Offsets 216 IV. Long Chords and Actual Arcs 217 V. Mid-ordinates to Long Chords 218 VI. Logarithms of Numbers , . 220 VII. Logarithmic Sines and Cosines 238 VIII, Logarithmic Tangents and Cotangents 253 IX. Functions of a One-degree Curve 268 X. Natural Sines and Cosines 298 XI. Natural Secants and Cosecants 307 XII. Natural Tangents and Cotangents 320 XIII. Natural Versines and Exsecants 332 XIV. Coordinates for Transition-curves 355 XV Deflection-angles for Transition-curves 356 XVI. Transition-curve Table 358 XVII. Areas of Level Sections 371 XVIII. Corrections for Three-level Ground 375 XIX. Cubic Yards per 100 ft. in Terms of Center Height 376 XX. Cubic Yards per 100 ft. in Terms of Sectional Area 382 XXI. Rise per Mile of Various Grades 386 XXII. Slopes for Topography 387 XXIII. Material Required for One Mile of Track 387 KXIV. Mutual Conversion of Feet and Inches into Meters and Centi- meters 388 XXV. Mutual Conversion of Miles and Kilometers 389 XXVI. Length of V Arc of Latitude and Longitude 389 XXVIL Trigonometric and Miscellaneous Formulas 390 XXVIII. Square Roots and Cube Roots of Numbers from .1 to 28 395 XXIX. Squares. Cubes, Square Roots, and Cube Roots, of Numbers from 1 to 1000 396 A FIELD-MANUAL FOR RAILROAD ENGINEERS. CHAPTER I. RECONNOISSANCE. Article 1. Objects op Reconnoissance — How Made. 1. The question of the selection of the proper route for a Hue of railway is essentially an economic one, luvolving not only the cost of construction, but of maintenance and operation, and a consideration of the immediate and future traffic likely to pass over the completed road. The engineer upon whom devolves the duty of making the surveys for a railroad is not often called upon to determine whether it should or should not be built, though his preliminary estimate may decide those whose duty it is to do so : the problem confronting him is how to secure the best line, answering a given purpose, for the least cost. Keeping in mind the proper working of the completed road, the problem may be divided into two gen- eral parts : First. The selection of the general route between terminal points, and in some cases the selection of the terminals them- selves. Second. The fitting of the line to the ground in such a manner as will render the cost of constructing and operating the road a minimum. The first is by far the more important and difficult operation, requiring the highest grade of engineering skill — a fact too sel- dom recognized by those selecting engineers for this work. The acquirement of the necessary skill can result only from 1oT)-:iIANUAL FOR RAILROAD ENGINEERS. 4. The Map. — Before beginning the reconnoissance the engi- neer should provide himself with the best available map of the region to be traversed ; if this is a topographic one, he can at once determine from it the lines that are likely to justify an examination ; and even if it is only a sketch-map, he can get material assistance by observing the courses of the streams and remembering that their positions indicate the relative elevations of the portion of the region through which they flow. Thus the large streams follow the lines of least elevation, and the manner in which the lateral streams unite with the principal one indi- cates the general trend of the terrain. Two streams flowing nearly parallel approacli or recede from each other according as the intervening land diminishes or increases in altitude. Two streams flowing away from each other on opposite sides of a divide, and having their source therein, approach each other closest at the point of least elevation, and indicate the position of a pass or the lowest point of the dividing ridge. The study of any good contour map covering sufficient area will illustrate the laws governing the courses followed by streams. The elevations of a few correctly mapped points, when obtain- able, from the map or otherwise, serve as a guide in tentatively fixing on the maximum gradient to be employed and the amount of development needed. A skillful engineer will thus be enabled to project his lines with sufficient accuracy to enable him to select on the ground the most feasible route or routes for his preliminaries in the least possible time. He should guard against the conviction, however, that it is unnecessary for him to look elsewhere than along the projected routes ; for the inaccuracies of the map, local peculiari- ties, the nature of the excavation and embankment, the number and cost of bridges and other mechanical structures, — all these may conspire to make the most promising map-line inferior to some other whose advantages have to be sought for on the ground. 6. Having tentatively decided on the limiting grades and cur- vature to be employed, the engineer goes carefully over the ground, examining the entire area that seems likely to afford passage, in order to determine whether a suitable line may l)e secured for the grades and curves previously assumed. With his pocket-compass he takes the bearings of lines, and by means of the hand-level and aneroid determines differences of elevation. RECONNOISSANCE. O Distances are estimated by the eye, paced, and tlie count taken from the pedometer, or, if the country admits of the use of a vehicle, talceu from the odometer readings. If a well-gaited saddle-horse is used, very good results may be gotten by timing him, or by the use of the pedometer if his stride is uniform. But in all cases much dependence must be placed on the ability to estimate with the eye differences of elevation and distances. The ability to do this with even reasonable accuracy comes only from long practice and careful observation, even to the most gifted in this respect. New and unexpected conditions some- times deceive even the most practiced eye, but under ordinary conditions almost any one can train his eye to estimate horizontal distances fairly well. Vertical heights are more deceptive, pos- sibly because we have less practice in this line, and the mind seems naturally to exaggerate the vertical as compared with the horizontal ; practice, however, will enable us to make allowance for the natural tendency to overestimate heights and slopes. The ground should be gone over in both directions, for the ap- pearance may be quite different when approached from different quarters. Ruling points, such as a pass in the mountains, the crossing of a large stream, or a town or city through which the road must be built, serve to reduce the problem to a number of special ones, each having its own solution. In a mountainous region offering a limited number of possible routes, but heavy construction work, it may often happen that the location of a line is a much less difficult operation than in an open, rolling country offering a score of possible lines, between which the engineer making the reconnoissance must decide, selecting only those that in his judgment seem to justify an accurate instrumental survey. The engineer must keep constantly in mind all the factors of the general problem of economic location and maintenance, and successful operation of trains. One line may cost more for con- struction and maintenance than another, but less for operation, or may invite less traffic. In all cases, however, the question of grades, curvature, length of line, earthwork, and mechanical structures are the controlling elements to be considered. Having decided upon the route or routes over which to run preliminaries, these are marked on the map, and the engineering party organized and put in the field, with all the necessary instruments. CHAPTER II. PRELIMINARY SURVEYS. Article 2. Objects; The Field Corps ; Duties of the Chief. 6. The Objects of the preliminary surveys are to secure all the data necessary to determine wliicli one of the routes selected on reconuoissance is the most feasible, all things considered, and the approximate cost of construction. In rough country it will be economical to make two, or even three, surveys over the route se- lected for location before beginning to place the line in the yjosition it is finally to occupy. The first of these is often omitted, and is called an "exploration-line " ; it will frequently save the making of the more expensive "preliminary" over one or more of the routes. 7. The Exploration-line may be made with either transit or compass, and consists of a rapidly run line, made for the purpose of determining the maximum curvature and gradients with which to project the preliminary. It will not be necessary to make a detailed study of the region at this time, the distances and eleva- tions, with such sketch topography as may be easily taken, being all that is needed. The magnetic bearing of lines is taken by the compassman, and the chainmen align each other with the flag set by the flagman. As the progress of the level party will be slower than that of the compass party, it will be economical to add an extra rodman, and sometimes a recorder. The compassman may sketch in the features adjacent to the line while waiting for his chainmen, who may be either in front of or behind the com- pass. The stadia method of surveying — to be spoken of later — would seem to offer exceptional advantages for this work — only three or four men being needed in addition to the chief. With it, by set- ting the transit over alternate stations, very rapid progress may be made, and obstacles avoided with as much or greater ease than with the compass. The exploration-line will more than pay for itself in showing 6 PRELIMINARY SURVEYS. Y what routes it will be unnecessary to make preliminaries over, and in indicating the most feasible one. It should be run over all the routes selected on reconnoissance. 8. The Preliminary Survey follows the exploration, or, when this is omitted, comes next after the reconnoissance. It may, with advantage, be made in two parts — first and second preliminary. It is made with such instrumental accuracy as the nature of the case may demand, sufficient data being obtained to determine the best line on which to locate and the approximate cost of construc- tion. • The rapidity with which this work can be done will depend on the care with which the reconnoissance was made. The pre- liminary line should approximate, as closely as the eye can deter- mine, to the position the located line should occupy, and forms the base on which the topographic work rests. In reasonably easy country, where exploration-lines have been run, one preliminary should suffice for each route, but in difficult regions it will be best to run a second preliminary. If portions of the route are easy, fol- lowed by difficult parts, it will often be sufficient to "back up" and re-run the difficult portion until a reasonably satisfactory line has been obtained. 9. The Field Corps consists of a chief of party, transitman, leveler, rodman, two chainmen, rear rodman or "back-flag," stakeman, and two or more axemen. If a topographic party is added, as it should Ije in any but the easiest country, there will be also a topographer with two or more assistants. A cook and teamster will be needed with the camp outfit. The corps is usually divided into the following parties : (a) The Transit Party. (h) The Level Party. (c) The Topographic Party. 10. The Chief of Party receives his orders from the chief en- gineer, or such other officer as may be in charge, directs the mo- tions of the surveying corps, and is responsible for their conduct and progress. He provides accommodations and supplies, pays all expenses, taking receipts or vouchers for all outlays — in dupli- cate when required. In the less thickly populated sections he must provide tents, wagons, cook, and all necessary camping outfit and supplies. He must direct the field oj^erations in person, keep- ing in advance of the transit, establish turning-points or hubs, and direct the transitman in the proper course. He should keep 8 A FIELD-MAlS^rAL FOR RAILROAD ENGINEERS. a record— or direct the transitman and topographer to do so — of the character of earthwork likelv to be encountered, the places where drains, culverts, bridges, cattle-guards, etc., are needed; the nature of material for embankment, piling, etc., adjacent to the line ; the probable amount of clearing and grubbing, and all other features likelv to affect the cost of construction. He should see that the names of property owners and residents along the line and the positions and bearings of property lines, when possible, are noted. He should have authority to discharge assistants — except transit- man, leveler, and topographer — whose services are unsatisfactory, and in many cases it will be best for him to have entire control, engaging or discharging any member of the corps as circumstances may require. Article 3. The Transit Party. A. Duties of the Members. 11. The Transit Party should consist of a transitman, head chainman, rear chainman, rear flagman, stakeman, and as many axemen as may be required — rarely less than two even for open country. 12. The Transitman cares for his instrument, keeping it in ad- justment; directs the chainmen into line; notes the angle between successive tangents as read on plates; notes also the bearings of tangents, of highways, streams, and property lines (on location), with the plus at which the line crosses them. If there is no topographic party he must make sketches, on the right-hand page of note-book, of the surface features adjacent to the line; the red Hue down the middle of page represents the transit line, whether straight, broken, or curved, to which the sketches are adjusted. He must see that the axemen keep in line, in order that no unnecessary chopping may be done. Large trees need rarely be felled on preliminary, even when a given general course has to be followed, for small angles may be turned to avoid them, the deflections to right being made to approximately balance those to left. When the chief of party is absent the transitman is ranking man. and will take temporary charge. 13. The Head Chainman carries a range-pole or "flag," and drags the chain, which he must see is straight and horizontal PRELIMIXARY SURVEYS. 9 when setting a point for a stake. He directs the stakeman where to drive his stake, calling out the number after the rear chainman has read and called out the number on his stake; he keeps the axemen in line by setting his flag and going ahead, directing them where to cut bv keeping them in line with the flag and transit. The speed of the party is dependent on the rapidity and accuracy with which he can set his flag in position, by ranging with stakes already set between him and transit, and in seeing that the axemen make all their work count. 14. The Rear Chainman must be careful to hold his end of the chain in the proper place, and that it is kept straight and taut when the head chainman is setting a stake. He must give all pluses, note the number on each stake as he comes up to it, and see that the stakeman has marked it correctly; he must make a note of pluses for roads, fences, streams, etc., to be given to the transitman later on. 15. The Stakeman must keep himself supplied with stakes about 1^" X 2' X ~^", marking the number on them plainly, and drivino^ them as directed bv the head chainman. If sawed stakes are not provided, he must cut the stakes and face them for the numbers. He must keep on hand a number of plugs or "hubs," to be driven flush with the ground and having the point where flag rested marked with a tack. About ten or twelve inches to the left of and facing the hub a guard stake is driven, on which is marked the station number, and which enables one to find the hub at any time. 16. The Axemen do all necessary clearing and chopping in order that the transit and level parties may have a clear sightway, and yet restrict the work of clearing to a minimum. One of them may be detailed to keep the stakeman supplied with stakes. 17. The Rear Flagman holds his flag on the last turning- point for the transitman to use in back-sighting. 18. The Instruments used by the party are the transit (or compass), one-hundred-foot chain or tape, range-poles, and the necessary axes and hatchet for axemen and stakeman. B. Transit Adjustments — The Vernier. 19. For railroad work the transit is usually plain, but it is often convenient to have a clamp and tangent movement to tele- 10 A FIELD-MANUAL FOR RAILROAD ENGINEERS. scope, a vertical circle, a level on telescope, stadia wires, and a gradienter; the solar attachment will rarely be needed. 20. To Adjust the Plate Levels. — The axis of the instrument is set at right angles to the plates by the manufacturer, so that when the axis is made vertical the plates will be horizontal. In making adjustments remember that a complete reversal always doubles any existing error. Place the bubble-tube parallel to a diagonal pair of leveling- screws, and bring the bubble to the centre of its run. Revolve the instrument 180° on the vertical axis, and the level- tube will be parallel to the same pair of leveling-screws as before, but reversed. If the bubble has moved from its central position bring it /t«(/'-way back by means of the capstan-headed screws at the ends of the tube. Relevel and repeat until the bubble remains at the centre after reversal. Do the same for the- other bubble. Both bubbles should remain at the centres of their tubes during a complete reversal. 21. Parallax is an apparent movement of the cross- wires with respect to the object sighted when the eye is moved from side to side of the eyepiece, and shows that the image does not fall in the plane of the cross-wires. In precise measurements it should be removed before making an observation with the telescope. To do this, first bring the cross-wires clearl}' into view, when the object, glass is turned towards the sky, then, when sighting an object, note if there is any relative movement of cross-wires and image when the eye is moved from side to side at the eyepiece ; if there is, refocus the object-glass until this movement disappears. 22. To Adjust the Line of OoUimaiion is to make the line joining the intersection of cross-wires and optical center of objec- tive describe a plane perpendicular to the horizontal axis of instru- ment. First Method. — Level the instrument and clamp the move- ments on vertical axis. Sight some well-defined object distant about the length of an average sight, and in the same horizontal plane as telescope. Reverse the telescope on its horizontal axis, and fix a point about as far from instrument as first point, and in the same horizontal plane. Revolve the instrument on its vertical axis and sight the first point; then reverse the telescope and note if line of sight cuts the second point. If not, loosen the capstan- headed screws holding cross- wire ring and move the vertical wire PRELIMIXARY SURVEYS. 11 over one fourth tLe apparent error— since tLere were two reversals — remembering that the image of the cross- wires is inverted, while that of the object appears in its true position. Test by repetition. Second Method. — If the limb graduations can be relied on they may be used in adjusting the vertical wire. With the instru- ment level sight a well-defined point, then revolve 180° by vernier- plate, reading both verniers; reverse telescope, and note if line of sight cuts the point. If not, correct one half the apparent error by moving diaphragm ; then test by repetition. The manufacturers adjust the object-glass slide so that the ob- jective travels in the telescope axis, and this adjustment is not liable to serious derangement. It is well, however, to sometimes test by adjusting the line of collimation for both near and distant objects. If not correct for both, move the ring which guides the rear end of object-glass slide until the adjustment is correct for both positions. Next make the vertical wire vertical by noting if it coincides throughout its length with a plumb-line, or by observing if it de- viates from a point, on which the intersection has been fixed, when the telescope is elevated or depressed. Any error is corrected by turning the ring after slightly loosening the screws holding' it. The horizontal wire should also be adjusted so that the inter- section of the cross-wires will be in the axis of the telescope ; if the transit is to be used as a leveling instrument this adjustment is essential. Drive a stake close to the instrument, and with the telescope clamped as nearly horizontal as can be conveniently done read a rod held on top of the stake ; about 800 feet distant, and in line with first stake and instrument, drive a second stake and read the rod on it. Revolve 180° on vertical axis, reverse the telescope and bring the horizontal wire to the former reading when the rod is held on first stake ; if the reading on the second stake is not the same as before, correct one Jialf the apparent error by moving the cross-wire ring. Repeat as a test. The vertical wire should again be tested lest the movement of the ring may have caused it to change. 23. To Adjust the Standards is to make the plane described by the line of collimation vertical. Set up the transit about as far in front of some high building, or other tall object, as the highest ])oint that can be sighted is above the base. Level the instrument and fix the intersection of the cross-wires on the highest point that 12 A FIELD-MANUAL FOR RAILROAD ENGINEERS. can be easily sighted. Depress tlie telescope and fix a point near tlie base of the building at about the beiglit of the telescope. Un- clamp and revolve on the vertical axis until the telescope reversed cuts the lower point. Clamp the plates and raise the telescope until the cross-wires are at the height of the upper point. If they cut it the standards are in adjustment. If they do not, bring them half-way back by means of the adj ustable screws at the top of one of the standards. Repeat as a test. 24. To Adjust the Level on Telescope is to make the bubble stand at the center of its run when the line of sight is horizontal. Bring the telescope as nearly horizontal as may be convenient, and take readings on the tops of two pegs in the same vertical plane with, and equidistant from, the instrument — say 300 feet. The difference of readings will equal the difference of elevation of the pegs; this difference may be obtained with the wye-level if pre- ferred. Move the instrument to a point beyond one of the pegs and in line with both. Set up as close to nearer peg as convenient, but not so close that the rod cannot be easily read. Bring the tele- scope as nearly horizontal as possible, and read on both pegs. If the difference of readings equals their difference of elevation the line of sight is horizontal, and the bubble may be brought to the center by means of the adjustable screws attaching the level-tube to the telescope. If this is not the case, we must set the telescope so the reading on second peg equals the reading on first peg plus the difference of elevation ; then read again on first peg and pro- ceed as before until the condition is satisfied. Or we may proceed as follows : In Fig. 1 let the transit be at 0, and A and B be the pegs. AC is a horizontal through A, so that CB is the difference of elevation Fio. 1. of A and B. Suppose line of sight to cut the rods at ^and 7), we must find DO so that the target may be set at the proper read- rilELlMlNARY SURVEYS. 13 ing to make tlie line of sight horizontal. Let 0F= a, FG = b, EA = r, DB = r', CB = k. Draw DH parallel to CA and OG; then ER—T-\-k-r'. From similar triangles DG = ER-^ ={r^k-r')[~J-j Set the target at a reading GB = GB H-r', sight to G, and tl.:^ line of sight will be horizontal. Bring the bubble to the center ol its run while the telescope is in this position, and the adjust ment is complete. If desired, a correction for the curvature of the earth and re- fraction may be introduced, but for short sights this is a useless refinement. 25. The Vernier is an auxiliary scale for measuring smaller divisions than those graduated on the limb. There are two classes, the direct-reading and the retrograde, according as the fractional parts of limb readings are taken on that side of the zero of vernier scale towards which the vernier has moved with respect to the limb, or the reverse. On the direct vernier a cer- tain number of divisions on the vernier equals the same number of divisions on the limb, less one ; on the retrograde there is one more division on limb than on vernier when the same space is covered by both. 26. The Least Count of a vernier is the smallest subdivision of limb graduation that can be read by it, and equals the difference of one space on limb and one on vernier. Let I = value of one space on limb ; V = value of one space on vernier ; n = number of spaces on vernier. Then for the direct vernier nv = {n — V)l ; from which we get the least count, n For the retrograde vernier nv = {n + 1)1, 14 A FIELD-MANUAL FOR RAILROAD ENGINEERS. from which the least count is V — I = —, n the same result as found for the direct vernier. So, to find the least count : Divide the value of one limb space by the number of spaces on the vernier. For example : If the limb of a transit is divided to half-degrees and the number of spaces on the vernier is 30, the least count will be \ divided by 30, or -^^ of a degree — that is, 1 minute. 27. To Read a Vernier, take the number of the last division on limb back of the vernier zero, then look along the vernier until a line is found to coincide with a line on the limb ; add the number of this vernier line, multiplied by the least count, to the scale reading, and the result will be the required reading. C. Accessories. (V) The Gradienter. 28. The Gradienter consists of a tangent-screw having a micrometer-head, attached to one of the standards of the transit and capable of being clamped to the horizontal axis of the tele- scope. It is used — as its name indicates — in running grades, and it accurately measures a small vertical angle in terms of its tan- gent. The screw is so cut that one revolution moves the tele- scope through an angle whose tangent at one hundred feet from the instrument has a certain value, usually one foot. The grad- uated head is divided into 100 parts, so that one division corre- sponds to 0.01 ft. at 100 feet from instrument. To run a given gradient, bring the telescope level and read the micrometer-head of screw; then turn the screw as many divisions as there are hundredths of a foot rise or fall in 100 feet, and with " target set at the height of the horizontal axis, points on the surface corresponding to the given grade can be found. For example : To run a 0.75 per cent grade, move the microm- eter milled head 75 graduations from the horizontal. When used as a Telemeter, we may either measure the space on the rod moved over by the line of sight for a given number of revolutVms of the screw, or we may note the number of revolu- ticms required to move the line of sight over a certain space on rod. The second method is the more accurate, particularly for long sights. PRELIMINARY SURVEYS. J5 (3°) The Stadia, or Telemeter. 29. The Stadia is an instrument for determining the distance of a point from the observer by noting the space intercepted on a rod by a given visual angle, as determined by two auxiliary wires parallel to, and equidistant from, the horizontal wire of the transit telescope. When used with an ordinary leveling-rod the wires should be adjustable; if they are fixed (which for some reasons is preferable), the rod must be graduated to correspond. In addition to the distance of a point from the instrument, the differ- ence of elevation is determined by observing the angle made by line of sight with the horizontal when the middle horizontal wire cuts a point on the rod as high above the ground as is the centre of the telescope. The horizontal position of the point is determined from its magnetic bearing, or the azimuth of line of sight with reference to some fixed line, usually the north-south line. 30. Line of Sight Horizontal. — In Fig. 2 let a and h be the stadia wires, AB the intercept on the rod. The secondary axes A Fig. 2. aA and hB pass through the optical center 0. Let h = ah, r = AB, d — distance of cross-wires from objective, Z> — distance of rod from objective. From similar triangles. From optics. In which/ is the focal length of objective. Eliminating d from these two equations. h r d ~D d^ D i " f 16 A FIELD-MANUAL FOR RAILROAD ENGINEERS. Ijet c be tlie mean distance of objective from center of instru- mentf. Adding this to I) gives, for the distance of the rod from the center of the instrument. l = c+f+£r. f J- may be made constant, when (2) becomes ft I = a -\- kr. (2) (2') 31. Line of Sight Inclined. — When the line of sight is not level it is difficult to hold the rod perpendicular thereto ; hence the rod is held vertical, the angle of inclination measured, and a correction applied. In Fig. 3 -Vr) B let r ■■ r' : H: V n ■ Fig. 3. CD be the reading on rod held vertical ; FE, the reading perpendicular to line of sight ; AG, the horizontal distance from ^ to ^ ; BG, the difference of elevation between A and B ; BAG, the angle of inclination of line of sight. Assume angles AFB and AEB = 90^ from which they rarely differ more than 15' to 17'. Then, since FBC = n^ r z= r cos n. PRELIMINARY SURVEYS. 1 1^ By (2'). AB = a-{- kr'. Hence AB = a -{- kr cos n. From triangle ABG H = AB cos n .'. n = a cos n-\-kr cos' n (3) F= AB sin n\ . •. F = a sin n -\- kr sin n cos n. But 2 sin 71 cos w = sin 2n. Hence V ■= a sin n-\- ^kr sin 2n. (4) 32. The Instrumental Constant a [— c +/ of (2)] may be found by measuring the distance from center of instrument to mean position of objective, which equals c ; then focusing on a very distant object, preferably a star, and measuring from center of objective to plane of cross-wires, which equals/. The sum of these distances is a in formulas (3) and (4). If the stadia wires are fixed, k may be found by measuring for- ward on level ground the distance a from plumb-line, and from this point a further distance h ; then note carefully the stadia reading r when the telescope is level. Then, remembering (2)', a-\-'b = a ■\- kr. .'. k = —, a. constant ratio. r If the stadia wires are adjustable, we may so adjust k that any desired reading may be had for a given length of base. A con- venient value of k is 100, which corresponds to an intercept of 1 foot on the rod at 100 feet from a point a feet in front of the instrument, 2 feet at 200 feet in front, etc. 33. A Stadia Table based on formulas (3) and (4) 4s published by the D. Van Nostrand Company in Winslow's Stadia Surveying, and can be used more rapidly than the formulas. Johnson's Re- duction Diagram, by John Wiley & Sons, gives values of i/and V graphically. Colby's S/ide-ride, manufactured by Mahn & Co., St Louis, gives values of y for distances in feet, yards, or meters to tenths of a foot, and can be used with great rapidity. 18 A FIELD-MANUAL FOR RAILROAD ENGINEERS. D. Field-work. 34. Station Numbers should begin with zero for the initial stake, and are marked on rear side of stake, from the top down- ward, the number of the preliminary, A, B, G, etc., being marked on the forward side. The marking should be with kiel, or crayon that will withstand the action of sun and rain. Stakes may be set every hundred feet or only at even stations, as preferred. 35. Hubs, or Plugs, are transit turning-points, and are short, flat-topped stakes driven into the ground flush with the surface. The flag is held on the top and carefully aligned, the position of the point being marked by a tack. A special tack with concave bead offers a foothold for point of flag when used in backsight ing.* About 10 inches to the left of and with numbered sid' facing the hub is driven a guard-stake to mark its position. 36. Reference- points are two or more hubs, with guard -stakes in each of two lines making a good intersection angle at the point whose position they serve to locate. They should be driver, beyond reach of disturbance, and are used in replacing a dig located hub. These need rarely be used on preliminary. 37. Alignment. — It is not intended that the preliminary an(\ location lines occupy exactly tlie same position ; hence consider- able latitude is allowable in the size and number of angles turned, care being taken, however, that the maximum curvature need not be exceeded on location. Large trees and other obstruc- tions may be avoided by turning a SMiall angle until the obstacle has been passed, then making a deflection in the opposite sense. Bearings of tangents are taken with the needle, to serve as a check on the angle read on the plates. In easy country not requiring a topographic party large angles should not he turned, a succession of small ones with short inter- vening tangents being substituted in order to make the prelimi- nary profile approximate more closely to the location profile. These short tangents may conveniently be the long chords of the curve that is to follow. * ^ uch a tack is manufactured by the A. S. Aloe Co., St. Louis. PRELININARY SURVEYS. 19 38. The Transit Notes may be kept in the form below, which shows both pages of the note-book. The notes run from the bottom up, the right-hand page being reserved for sketches ; the red line up the middle of the page represents the transit line, whether straight or broken, to which the sketches must be adjusted. Sta. Angle. Calculated Course. Magnetic Course. Remarks and Sketches. 68 1 67© 20° 0' L. N. 1° 48' W. N. 1° 45' W. 66 65 64 63© 6° 2' R. N. IS" 12' E. N. 18° 15' E. 6-2 61 39. Stadia Methods for Preliminary Surveys. — Preliminary lines are usually run with the transit, but the compass will answer nearly as well in most cases, besides admitting of more rapid work. The transit and stadia method might well be em- ployed, and would effect considerable saving in the cost of pre- liminary surveys. For some reason railroad engineers have not regarded it with favor, though it is extensively employed in topographic surveying where the map is to be used for work that is often more precise than needed for railroad preliminaries. Particularly is this method applicable to exploration lines. With the transit and stadia the entire surveying corps need not exceed five or six men, the instrument man acting as transitman, leveler. and topographer all in one. The only objection would seem to be in the amount of reduction the notes would need ; however, with tables and slide-rule (see 33) this work may be very rapidly done. For vertical angles of less than one degree the horizontal reduction can be neglected, and with side readings for topography the angle may be 5 or 10 degrees without necessi- tating the correction. Vertical heights are found by the slide- rule or by charts. This method would really necessitate the making of a topo- graphic map along a narrow strip of country, from which the profile could readily be taken. With a skilled observer and two to four rodmen the progress may be more rapid, and fully as good for the purpose intended as the more expensive method usually employed. 30 A FIELD-MANUAL FOR RAILROAD EN'GIXEERS. The transit need onlv be set at alternate stations (which may be any length within the reading limits of the wires), the bearings to other stations and points off the line being taken with the needle. The horizontal angle should also be read on the plates for points on stadia line, as a check on the bearings. E. Obstacles in Tangent 40. Obstructions to vision and measurement in tangent may be avoided in a number of ways, a few of which are given in the following problems. Other methods of avoiding them will sug- gest themselves in special cases. The same devices may be used on location, but it is more im- portant to maintain a clear sight way then ; so, when possible, we should remove the obstruction. 41. To Pass an Obstacle by Means of Parallel Lines. — In Fig. 4, is the obstruction, AB the obstructed line. At B set H A B ^^^^ C Fig. 4. transit ; turn 90^ and measure BF long enough to clear obstruc- tion. Set transit at F, make BFG — 90\ and measure FG. Move to G and backsight to F, making FGC = 90". Measure GC — FB, and move to C, where the angle GCD is made equal to 90°. CD is the desired line, and BC = FG. Otherwise, at J. and B erect perpendiculars; take BF=AE; produce EF, and at G and H, beyond 0, erect perpendiculars mak- ing GG= HD — FB. CD will be the desired line, and BG = FG. 42. To Pass an Obstacle by Angular Deflections. General Case. Angle anything less than 90\ At B (Fig. 5) on the obstructed line deflect an angle a to one side and measure BC, taking C so that after deflecting 2a to the other side CZ) will clear the obstruction. Make CD = BC a-nd deflect an angle a to the same side as at B\ DE will lie in AB produced. Draw Cfi^ perpendicular to 52); then BD = BH4- HD = 2BC cos a (5) PEELIMIXARY SURVEYS. 21 F.' Fig. 5. Example.— Suppose a = 14° 10', BC = CD = 520 ft. BD = 2X 520 X 0.96959 = 1008.37 feet. Special Case. Afig/e 60 degrees. In this case the triangle BBFiFig. 6) is equilateral and BF = BD = DF. Should it be inconvenient to run to D we may stop at C, having measured BC. At C deflect 60" and measure CE; at E again de- Fig. 6. fleet 60' and make EF= BC. At i^ a final deflection of 60' in the opposite sense will put the telescope in the desired line, FG^ and BF=BC-\-CE. ...... (5a) 43. To Pass an Obstmction, such as a River, when the Pre- ceding Methods are Inapplicable. First Case. Point beyond obstruction lymble. In Fig. 7 let BC be required. Fig. 7. At B erect and measure the perpendicular BD ; set instrument at B and measure angle BBC = a ; then BC =BDt&nii (6) 52 A FIELD-MAIfUAL FOR RAILROAD ENGINEERS. Or, if a trigonometric table is not at band, make CDE ^= 90" and fix the point E where DE intersects AB ; measuring EB there results, from similar triangles, CB_BD BD ~ EB' BD'^ whence CB =-^T7r (6«) Otherwise, if a right angle at B is not convenient, measure angles CBD = b, BDC = a, and side BB. Then c = 180°- (a+t). From triangle BBC, BC = BD^^. ...... (6&) sm c Example.— « = 66\h= 70°, BB = 400 feet. By {6h), BC = 400 ^!" ^^o = 409.8 feet. •^ ' sm 54 Second Case. Point beyond obstrniction invisible. At B (Fig, 8) measure angle b and line BE ; move to E and measure angle y, and set hubs on line EG so the line BC will pass Fig. 8. between them. Angle z = ECB = 180 - (& + y). Then from tri- angle BEC BG=BE^^ (7) sm 2 Produce EB to B, where BC will be sure to clear obstruction; measure BB. From triangle BBC, tan K^ - ^) ^ BC- BB t&n {{a -\-j') BC-\-BB But a -{-x =:b, hence ^ ^. , BC-BB ^ ., ,^. tan lia - a-) = ^ ^ - . tan ^6. ... (8) PRELIMINARY SURVEYS. 23 The sum and difference of a and x are now known, so botli may be readily found. At D set off the angle a with the transit, and have the chainmen stretch a cord between the hubs set on line EC Sit C. Now signal the flagman to move his rod along this cord until the vertical wire cuts it at C. Set a hub here and place the transit over it. Sight to D or E, reverse telescope and deflect into CII. Article 4. — The Level Party. 44. The Level Party consists generally of two members, the Jeveler and a rodman ; sometimes an axeman is added to keep the lodman supplied with pegs for turning-points and in clearing the hue of sight for the level. As the party follows the transit little or no clearing will be needed. The instruments used are a level, a rod, and a hand-axe or hatchet. 45. The Leveler makes all necessary observations with his instrument, keeping a neat, accurate record of readings and ele- vations ; also the positions and elevations of benches and turning- points. He should work out elevations of stations while the rod- man is going from one station to the next ; he must see that the rodman gives him readings at points where the longitudinal slope changes suddenly, recording the plus. He must plot his profile at night, or at such times as the chief of party is likely to need it. The rodman 's readings at turning-points should be checked. 46. The Rodman holds his rod at each station, calling out the number. If stakes are set only at even stations, he must hold his rod midway between stakes, the point being found by pacing the distance. Target-readings need only be taken at turning-points and benches, and the rodman should keep a record of these in his "peg-book," checking the calculations of leveler for heights of instrument and elevations of turning-points. At any marked surface change he will hold his rod, calling out the plus to leveler. He must assist the leveler in plotting up the notes. A. Adjustments of the Level. 47. To Adjust the Line of Collimation is to bring the inter- section of the cross-wires into the optical axis of the telescope. Set up and level the instrument, then bring the vertical wire into coincidence with a plumb line or vertical edge of a building, 24 A FIELD-MAXUAL FOR RAILROAD ENGINEERS. at the mean leugth of sight, and note if the vertical wire is truly parallel thereto. If it is not, loosen the capstan -headed screws holding cross-wire ring and turn slightly so that the wire is parallel to the vertical line. Loosen the wye-clips and bring the vertical wire into coin- cidence with the line and clamp the instrument. Rotate the telescope in the wyes 180° and note if the wire coincides with the line. If not, correct 07ie half the error by loosening one and tightening the opposite, of the capstan-headed screws that hold the cross-wire ring in place, remembering that the image of the cross wires is inverted by the eyepiece. Turn the telescope until the horizontal wire is parallel to the plumb-line or edge of building, and make the same test and correction. Repeat for both wires. The horizontal wire is the one on which the accuracy of leveling depends, but it is wise to have both adjusted. Their intersection should remain on a point during a complete rotation of the telescope in the wyes. 48. To Adjust the Level-bubble is to bring the axis of IhQ level-tube into the same vertical plane with the line of collimation, and to make the bubble stand at the center when the line of sight is horizontal. Since the axis of the telescope coincides with the line joining the center of the wye-rings (which requires these to be of the same size), it is sufficient to make the axis of the bubble parallel to this line. (a) With the telescope over one diagonal pair of leveling- screws and the clips loosened, bring the bubble to the center of its run ; then turn the telescope, in the wyes, a little to either side of the vertical plane through the telescope and note if the bubble remains at the center. If not, correct the error by means of the screw at end of the level-tube case arranged for lateral movement. Repeat until the tube may be rotated half an inch or more to either side of vertical without movement of the bubble. This adjustment is made merely to prevent error from failure to set level-tube vertically beneath telescope. (b) With the wye-clips opened well out, again bring the bubble to the center of its run ; remove the telescope from wyes and turn it end for end, then carefully replace it in the wyes. Should the bubble fail to remain at the center, bring it halfway back by raising the lower or depressing the higher end of tube at the points of attachment to telescope. Relevel and repi^at as a test. PRELIMINARY SURVEYS. 35 . 49. To Adjust the Wyes is to make the axis of the telescope perpendicular to the vertical axis. Witli the wye-clips closed place the telescope over oue pair of leveling-screws aud bring the bubble to the center of its run ; then turn the telescope half-way rouud on its vertical axis, so that its ends have changed places. If there is any error, correct by bringing the bubble half-way back to center b}' means of the screws connecting wyes with level-bar. Repeat until the bubble remains in the center during a complete revolution. B. Theory of Leveling. 50. Wlien the level has been adjusted the line of collimation A'ill describe a plane parallel to the horizontal plane tangent to \ he earth's surface at the point where the instrument is placed. A level surface, such as the surface of still water, will coincide with this plane only at the point of tangency, and will depart farther and farther therefrom as the point considered recedes from the instrument. For short sights this differeuce may be neglected in railroad work, as will presently be shown, but for long sights a correction must be applied. The effect of curvature is to make objects appear lower than they really are, while the refraction of a beam of light, due to the greater density of the layers of air nearest the earth's surface, has a contrary effect. Experience shows the average error due to refraction to be about one seventh of that due to curvature. 51. The Error due to Curvature at any point is the deviation of a tangent line from true level, as the point recedes from the point of tangency. Let be the center of the earth, T the point of tangency, and iVthe point where the error due to curvature is desired. Let the notation be as shown in Fig. 9. From the right triangle DTP, we have From which Fia. 9. t^ c = 2R-\- c' Now, since c m always very small compared with 27?, the quolieut resulting from the division of t- by 27? will not differ 26 A FIELD-MANUAL FOR RAILROAD ENGINEERS. seusibly from that obtained by dividing by 2R + c. Therefore we write (9) c = 2R For t = 1 mile, M = 3963 miles, c = about 8 inches, for any other distance in miles we have, for c, c = 8 X t^ inches Hence (9a) The correction for refraction is about -c, hence we have, from (9), 7 7 IE or, closely enough, C = .85c. (10) Example. — "What is the correction for a half-mile sight? For one eighth of a mile? By (9«), c = 8 X ay = 2" for first case, and c = 8 X (i)- = 0".125 for second case. By (10) the final correction is c = 0.85 X 2 = 1".7 for first case, c = 0.85 X 0.125 = 0.106" for second case. 52, The Difference of Elevation between two points not so far apart but that a rod may be read on each from some inter- mediate point may be readily found from these rod-readings. In Fig, 10 let the instrument be at I, A and B the points whose difference of elevation is desired. Let r = AD, r' = BC. Since the line of sight, DC, is horizontal, the difference of Fig, 10. elevation will evidently be r' — r. When the distance from / to A equals that from I to B the errors due to curvature evidently balance. PRELIMINARY SURVEYS. 27 When the points are so situated that the rod cannot be read on both from one intermediate position of the instrument, an Fig. 11. auxiliary point or points must be used and readings taken on these points in pairs. Thus in Fig. 11 suppose the difference of elevation of ^ and 5 required : With the instrument at / read on A and some intermediate point E. Considering the backsights as plus and foresights as minus, the difference of elevation of A and E is AD — FE. Again, with the instrument at 1' the difference of elevation of E and B is GE — CB. The sum of these differences equals the dif- ference of elevation of A and B, and may be written (AB -\- GE) — {EF-\- CB), or, in general, the sum of the backsights less the sum of the foresights equals tJie difference of elevation. C. Field-work. 53. A Datum is a level surface so taken that it shall lie below the lowest point likely to be reached by the profile, to which the surface elevations are referred. It is often spoken of as the datum-line or datum-plane, and is the zero of elevations. 54. A Bench-mark is a permanent mark, such as a copper or other bolt let into the top of a solidly fixed stone, whose height above the datum is known; it may be simply a mark on a stone, or a tack driven into the projecting root of a tree, upon which the rod may be read. In any case it must be so situated that it cannot change its elevation nor is likely to be disturbed within the lime for which it is intended to be used as a standard of reference. The elevation should be marked on some object adjacent to the bench, with the letters B. M. indicating the nature of the point. 28 A FIELD-MANUAL FOR RAILROAD ENGINEERS. 55. The Field-work consists in finding the elevation of a number of points on the line established by transit part}' suffi- cient to give, when plotted, a faiily correct outline of the surface as seen in profile. A bench-mark is taken at the beginning of the line, and its dis- tance above mean sea-level or other datum is known or assumed. The level is set with one pair of leveling screws in the line to be run (in order that any change in the position of the bubble may be easily corrected;, and the rod is read on the bench. This read- ing plus the elevation of bench gives the height of instrument [H. I.) above the datum. Readings are taken at every hundred feet along the line, or oftener if the surface changes greatly, until a point is reached beyond which it is desired to move the level. A peg is driven firmly into the ground and the rod read on this ; the height of instrument less the rod reading will give its elevation, as it will for the. intermediate points. This point is a temporary bench and is called a turning-point. It should be marked by a guard- stake if it is desired to use it again. The instrument is now car- ried beyond the turning-point, set up, and the whole process repeated. Benches and turning points should be read to hun- dredths or thousandths of a foot, intermediate points to tenths. Turning-points are marked O or T. P. in the notes, and their positions, as also the bench-marks, noted by both leveler and rodman in their note-books. 56. The Level Notes may be kept in any convenient form that is easily understood. The following is used more exten- sively, perhaps, than any other: Sta. B. S. I H. I. F. S. Elev. Remarks. B.M. 5.613 205.613 200.0 j B. M. on root of L. O. tree 60' to ( right of line. 2.3 203.3 1 0.8 204.8 2 5.7 199.9 3 7.8 197.8 4 99 195.7 O 1.120 196.310 10.4-.i3 19.5.190 J On peg at 4 -f 30' - 20' to left of line, 1 by small P. O. tree. 5 6.;i 190.0 6 4.5 191.8 Here the elevation of tlie datum was taken 200 00 feet below the tiist bench-mark. The instrument was set up near Station 2, PRELIMIXARY SURVEYS. 29 ind a reading of 5.613 taken on the bench; this was written in the B. S. coluniu, and when added to tlie elevation of the bench gives the height of instrument, 20o.6l3. A reading of 2.3 was taken on Sta. 0, recorded in the F.iS. column, and when sub- tracted from the H.J. jields an elevation of 203.3. The eleva- tions of other points were determined in the same way. A little bej^ond Station 4 the rodmau drove a peg and held the rod on it, yielding a reading of 10.423 and an elevation of 195.190. The iuslrunieut was then moved to a point near Station 7 and a read- ing of 1.120 taken on the peg; this added to 195.190 made the new //. /. 196.310, and the process continued with this H. I. In most cases it will be sufficient to read benches and turning- points to hundredths and intermediate points to tenths. It will be seen from the notes tl»at any error in -a turning point causes the same error in all succeeding points. To guard against this the rodman is required to keep a " peg-book," in which the heights of instrument and elevations of turning-points are re- corded, and which must check with the leveler's record. 57. Wind and sunshine affect the accuracy of the work with the level, as is also the case with the transit. For very great accuracy a calm, cloud}'^ day is the best, but the railroad engineer cannot always choose the best times for his work, and must take such precautions as may be possible while he exercises the great- est care to prevent and detect errors. The adjustments should be tested at least once a week, even when the greatest care has been taken, for unequal expansion and other causes may con- spire to cause them to change. B}' making foresights and backsights to turning-points about equal the error due to curvature will be eliminated; the readings of rodman at these points should also be checked. The rodman should hold his rod vertical, which is sometimes accomplished by means of a level attached to rod; or the leveler can tell by his vertical wire when the rod is in the same vertical plane with the instrument, and by causing the rodman to wave his rod back and forth slowly, after clamping the target, he can tell if the hori- zontal wire just bisects the target at its highest position. 58. The Rod should be graduated to feet and tenths, reading by target at turning-points and benches; intermediate readings are maide by the leveler at his instrument. Siiength and dura- bility are essential qualities. The Philadelphia rod seems to 30 A FIELD-MANUAL FOR RAILROAD ENGINEERS. answer the purpose as well as any other now manufactured; the Troy rod may be used in the same manner as the Philadelphia rod, but is lighter and less able to stand rough usage. Article 5. The Topographic Party. 59. The Topographic Party follows the level and secures all the data necessary for making an accurate contour-map of a strip of country extending as far each side of the preliminary as may be needed for the intelligent projection of the location-line. This distance may vary from 50 to 300 or 400 feet, its width de- pending on the difficulties to be encountered and the degree of precision with which the preliminary approximates to the final location-line. The lateral slope of surface is obtained at the stations of prelimiuary by means of the hand-level and tape, by the slope-level or clinometer, by cross section rods, or by the transit and stadia. Strictly speaking the topography includes all the surface features, but for railroad work the surface elevations, streams, and nature of surface are the most important; it may be necessary to note the positions of roads, buildings, etc., and should always be done when practicable without imdue loss of time. A pocket-compass will be of use in observing the bear- ings of lines. 60. There are two methods of recording the data obtained; one by means of notes and sketches in a book, the other by drawing the contours directly on the field-sheet as the data are obtained. Station elevations can be taken direct from the levelerg notes, and constitute the base on which the contour elevations rest. Suppose the hand-level to be used and the notes kept in a book, to be afterwards transferred to the map. Starting with the known center elevation, the topographer notes the height of his eye above the ground and calculates the height of center above or below tlie next contour; from this the reading of the rod when held on this contour is found, being the height of station above contour {)lus the height of eye. He directs the slopeman in or out on a line at right angles to preliminary until this reading is given by the hand-level; the distance out is then measured and recorded, just as in setting slope-stakes, and the slopeman di- rected into position on the next contour, in the same manner. Thus if 5-foot contour-intervals are employed, and the station PRELIMINARY SURVEYS. 31 elevation is 321. 6 feet and the height of eye 5.3 feet, we shall have for tUe reading at the 320-foot contour 5.3 + (321.6 - 320)= 6.9. Motion the slopeman down the slope until his rod reads 6.9 and measure the distance out, suppose 21 feet. Tlie 315-foot contour will be 5 feet lower, giving a reading of 11.9, which may be found in like manner at, say, 80 feet out. As the rod reads only to about 12 feet the topographer must move out to this last point, and with the reading 5.3+ 5= 10.3 find the 310-foot contour in the same way. On the up-hill side the 325-foot contour will l^e found with a reading of 5.3 — (325 — 321.6) = 1.9 feet, and other contours in like manner. The notes may be written thus Sta. Left. Center Elev. Right. 824 305 310 315 320 193' 125' 80 ' 21 321.6 325 330 335 340 27' 56' 80' 112 The number above the line is the contour elevation, the num- ber below its flistance out from center. If preferred the elevation can be taken at regular .distances out and recorded as above; the position of the contour will then be found by interpolation when mapping the work. 61. If the topography is to be plotted in as the work progresses the topographer must have a light drawing-board with a pocket and flap on back for holding the sheets on which the transit-line has been plotted the night before ; the station elevations are marked on the line and the contour positions spotted in as ob- tained by slopemen, after which the contours are sketched in. Points where contours cross transit-line are found in the same manner as side points. The size of the sheets will depend on the taste of topographer antl size of drawing-board; 17x24 to 19x28 inches are good sizes. The topographer will soon learn to guess at the position his contours will occupy at the next station ahead, and will sketch them in lightly, to be erased and corrected when necessary. It is often sufficient to take lateral readings at every second or third station. 62, If the Slope-level is used, the inclination of the surface is obtained; then by the use of a scale constructed to show the 33 A FJ ELD-MANUAL FOR KAILROAD EXGINEERS. burizontal distance apart of contours, foi- the given conlour in- terval, for slopes varying from V to 20% the position of contours can at once be spotted on the map. Wellington recommends the use of the altazimuth as permitting the employment of either method at vs-ill — the altazimuth being merely a hand-level vfith a clinometer attached. 63. Cross-section Rods are measuring- rods 10 or 12 feet long carrying a level-bubble. By placing one end at the center, bringing the rod horizontal, and noting the height of the end of rod on the downhill side, the slope may readily be obtained and the contours Avorked in as before. For very rough, broken ground this method may be preferable to either of the others. 64. If the Transit and Stadia are employed, very elaborate topography may be taken with very little Hekl work, but the ob servations require considerable reduction. AVith a suitable topo graphic protractor and the slide-rule mentioned in 33, the large number of points that may be obtained from each setting of the transit may be readily plotted and their elevations marked on the plot, after which the contour-lines can be worked in, and otber features mapped. For small vertical angles no horizontal reduc tion is needed. While not generally favored by railroad engineers in the past, this method is probably the most lapid and economical of any so far employed in topographic work. Article 6. Preliminary Estimates. 65. After completing the field-work of the preliminary survey the party is usually disbanded, only the transitman, leveler, and topographer being retained to assist the chief of party to complete the map, profile, and estimate of cost. 66. The Map may be drawn to any suitable scale, but less than 400 feet to the inch is not to be recommended where it must be used in projecting location. The transit-line is laid down first and the topography worked in afterwards from the field-map or topogiapher's notes. If it is wanted on a continuous sheet, the tiansit-line must first be drawn on a succession of small sheets, which are added as the plotting progresses, a new sheet being slipped under the edge of the preceding and tacked tlowu when PRELIMINARY SURVEYS. 33 required. The overlapping edge is marked by a number of short lines extending over onto the sheet beneath, to enable one to re- place in the proper position. When the line has been plotted the sheets are pasted together and the whole shifted so as to bring the transit-line over the continuous sheet. Angular points are then pricked through and the line drawn on the continuous sheet. Ordinarily it will answer to have the map drawn on a succession of small sheets, to be joined together as required. The plotting had best be done by bearings, though it may be done from the delleciioii angles, provided care is used to check frequently by bearings. Otherwise an error in one angle wilL throw all the remaining portion of the line out of position. If more than one preliminar}'^ was run, they should all be shown on the same sheet whenever possible. 67. The Profile will be drawn by the leveler on profile-paper> and shows a developed vertical projection of the line. The scale will depend on the paper used. There are three scales in general use, styled respectively Plates "A," "B," and " C." There is also a metric protilo- paper. Plate "A" has the vertical exagger- ated 20 to 1 as compaied wilh the horizontal and is the best to use where much rock work is expected. The vertical exaggera- tion of Plate " B" is less than of Plate "A"; this plate is most used for ordinar}- earthwork. A strip of color laid on below the surface-line, and fading out at the lower edge, adds greatly to the appearance of the protile. The tentative grade- line and points of change should be draw-n in red. > 68. Preliminary Estimates of quantities are made by assuming a grade-line and drawing it on the protile; then the cuts and tills are taken from the protile, and the corresponding quantities ob- tained from Table XIX for the base the road is intended to have w'hen completed. The nature of the w^ork, whether ordinary earth or rock, can. of course, be only roughly estimated. Bridging is estimated from the profile where piling or framed bents may be used, but where piers and long spans are needed special surveys with soundings are required, Culverts, drains, cattle guards, cross-ties, and rails for main line and sidings, switch stands, buildings, right of way, clearing, and other factors entering into the question of cost must all be considered and allowed for in making up the estimate. 34 A FIELD-MAKUAL FOR RAILROAD ENGINEERS. Engiueeriug expenses aud unforeseen outlays that are sure to arise should have a liberal allowance. 69. The Report of the chief of party should set forth the ad- vantages and probable cost of each of the several lines run ■when there is more than one. On this report frequently depends whether or not the line is to be located, and it should be clear and exhaustive, though plainly and concisely worded. The map and profile form an integral part of the report and show from what data the estimates were derived. CHAPTER III. LOCATION. Article 7. Projecting Location. 70. After the prelimiuary lias been mapped and the topography ^•orked in, the engineer proceeds to make a paper location for his guidance in the field. The solution of the varied and complex problems that confront him are more or less interdependent. The guiding principle, applicable to all departments of engineer- ing, that the best structure is that which for the least cost test an- swers the purpose for uliich iticas intended, should control, even though the resulting structure be inferior, in point of scientific design, to some other. The best road as regards construction and grades may be a failure because of excessive first cost, while the cheapest construction will entail such heavy operating ex- penses that it may be equally unprofitable. The alignment must be as free from curves as possible, while heavy grades are at the same time excluded; these two requirements conflict and must be as well adjusted as possible. The amount of earthwork, of bridging and other structures must be kept down to the lowest limits. 71. Starting at the summit of the most diflScult portion of the route, assume a starting-point and elevation; with the dividers set at such a distance to the scale of the map as will give a fall of one contour space — or half space — for the assumed grade, step down the slope in such a way that the dividers fall each lime on the next lower contour, oi half-space, according to the fall assumed in setting dividers. If curve compensation is allowed, the dividers must be reset for each curve, for the same fall, since the giade will be slackened on curves. The points at which the dividers fall are lighily ..potted on the map and connected b}'^ a grade contour, which represents the surface-line having tlie required gradient. This line will be too broken to be used as a ]or"tiou- 35 36 A FIELD-MANUAL FOR RAILROAD ENGINEERS. Hue, so we have then to draw on the map a succession of curves and tangents that will approximate sufficiently close to it, at the same time that a proper balance is maintained between earthwork and curvature. Having lightly plotted the proposed line, the elevations are transferred to profile-paper, thus giving a profile of the line. With a fine thread stretched aloug the profile, to represent the grade-line, adjust the cuts and fills to suit the nature of the work, la general, fills are cheaper than cuts both in construction and maintenance; and especially is this true Avhere a shallow surface layer of earth is underlaid by rock. It may happen that the material from excavation must be used in embankment, when the cuts and fills must be made to balance b}' shifting the grade- line until this appears to be the case on the profile. At the stream crossings the grade-line must be kept safely above high- water mark, so that sufficient waterway is provided, and allowance made therefor. After locating the most difficult portions pass on to the easier work, returning later on .to study the effect this will have on the part first located. It may be necessary to go over the projection several limes before you can be reasonably sure that the best loca- tion has been projected; even then the study of tlie line in the field will cause many of the details to be altered, sometimes materially. Long grades are to be preferred to short ones, but questions of economy may necessitate the latter in order to lighten work; care must be taken that the grades are not so badly " chopped" that they interfere with the easy riding of the train. In projecting the line it will generally be best to strike the curves first and draw the tangents afterwards, though it some- times happens that long tangents will control the curves; when this is the case the tangents are drawn to intersection and the curves afterwards put in. When transition-curves are employed, a slight offset should be made at the beginning and end of curves to allow for their inser- tion in the field. These offsets will be so small that it is useless to attempt to show them to scale. 72. A Curve-protractor will be of material assistance in find- ing the degree of curve required to unite two tangents that have been laid down on the map. It consists of a transparent, semi- circular protractor having a series of curves from 30' up to 8° LOCATION. 37 plainly cut upon it. The curves are on both sides, those on the reverse side having their concavities turned in an opposite sense from those on the face. The scale is usually 400 feet to the inch, and in any case the map and protractor must be drawn to the same scale. Sometimes a set of cardboard or hard-rubber curves are used, but they are inferior to the curve-protractor. To use , it, simply prolong tangents to intersection and then place the protractor so thai the curve admitting of tlie best grade is tan- gent to the two straight lines. Mark the points of taugency, which will be the beginning and end of curve. When the curve is required to pass through a given point the proper curve may be immediately found by trial, whereas the calculations would require some little time. ^-«7j*^ Reversed curves should never be allowed on main lines. Suffi- cient tangent should be interposed to allow space for easing off the superelevation of outside lails, or for the insertion of tran- sition-curves when these are w be employed. 73. The Field Corps is substantially that required on the pre- liminary survey, and the methods of work pretty much the same, except that curves must now be run in, and this necessitates more clearing. If first and second location-lines are to be run (and it is real economy to run both), it will not be necessary to have the stationing continuous on the lirst, so the pluses arising from *' backing up" need only be noted and eliminated when the final location-line is run. If transition-curves are to be inserted, they need not be run the first tiixl^, the proper offset being made at vhe P. T. or P.C of the circular curves, which latter are to be run. On the final location-line the stationing must be continuous, beginning with zero. The stakes are marked as on the pre- liminary survey, and all hubs that are likely to be used again must be referenced in, the reference-hubs being set well out of the way of disturbance by the plow or scraper. The leveler should make bench-marks every 1000 or 2000 feet, to be used in running check-levels and in giving grades later on. From the paper location the notes should be made up in the office, to serve as a guide in the field; however, no attempt should be made to adhere rigidly to them, since slight errors in the mapping will affect the projected line, while in the field the line may be shifted here and there so as to fit the ground more snugly and accord more closely' with what the nature of the earthwork demands. 38 A FIELD-MANUAL FOR RAILROAD ENGINEERS. The highest skill of the engineer is required to secure the best location-line, and he should have all the time he needs. Undue haste on location — as on reconnoissance and preliminary — is almost sure to result in increased cost of construction. Article 8. Simple Curves. 4. Definitions and Formulas. 74. The Circular Curves that are usually employed to unite straight reaches of the railroad may be simple, compound, or re- versed. The use of reversed curves should, however, be limited to turnouts and cross-overs. a. A Simple Curve is the arc of a circle. b. A Compound Curve consists of two simple curves, of differ- ent radii, both on the same side of a common tangent. c. A Reversed Curve is made up of two curves of contrary flexure having the same or different radii, and a common tangent. d. The Point of Curve {P.O.) is the end of tangent and begin- ning of curve, as at A, Fig. 13. Fig, 12, e. The Point of Tangent {P.T.) is the end of curve and be- ginning of tangent, as at B of Fig. 12. /. The Point of Intersection (P.I.) is the point where the tangent at theP.C. and P.T. intersect when produced. {D of Fig. 12.) g. The Intersection Angle (7^ is the angle at the PL be- tween the tangents meeting there, and equals the angle at the center. h. The Tangent Distance (7") is the length of the produced tangent measured from the /*. 6'. ov P.T. to the P./, The term LOCATION. 39 tangent is applied to aii}'^ straight portion of the Hue, but the letter T will be used to designate the produced portion only. i. The Mid-ordinate {M) is the portion of the radius inter- cepted between the arc and chord when it cuts the chord at its middle point. j. The External {E) is the part of the radius produced to the P. I., intercepted between curve and the P.l. k. The Long Chord (L.C.) is the chord joining the P.O. and P.T. Frequently the term is applied to any chord longer than the unit chord. l. The Radius will be denoted by R. tn. The Point of Compound Curve (P. G. C. ) is the point of common tangency of the two branches of a compound curve. (See Fig. 13.) P.C. n. The Point of Reversed Curve {P. B.C.) is the point of common tangency of the two branches of a reversed curve. o. The Degree of Curve {P) is the angle at the center sub- tended by the unit chord. In the United States this chord is 100 feet, in England 66 feet, and where the metric system is em- ployed it is taken at 20 meters. Any convenient chord length may be taken, but for uniformity American engineers have adopted the chord of 100 feet, and unless otherwise stated it is always so understood when we speak of the degree of curve. Half the degree of curve is called the deflection-angle, since it is the angle to be deflected from the tangent to the chord. If there were any practical method ot measuring around the curve instead of along the chord, an accurate and convenient ratio for expressing the radius in terms of the degree would be had. Thus if D is the angle at the center subtended by the mre of unit length, we have, where a is this uuit arc, 40 A FIELD-MANUAL FOR RAILROAD ENGINEERS. Hence 27tR=z a -jj. „ a 360 • • c (11) When a equals 100 ft, this becomes 100 360 R = 27t D (11') i2 varies inversely as B, so that knowing the radius for a 1° curve, we should have only to divide this by D to get the radius for a D^ curve. Since the chord is employed instead of the arc, we determine R by means of the following problem 75. Given the Chord C, and Degree of Curve D, to Find the Radius R. In Fig. 14, AB \s the chord G, OE a perpendicular from the center upon AB From the right triangle AEO we have R sin IB = 10. Whence R = When CislOOft., . , „ = iCcosec IB. sm IB " ^ . . (12) 50 R = —. — r-^ = 50 cosec IB. sni IB . . . (12') LOCATION". 41 Comparing results givcu by formula (13') with those given by (11'), we have for a few curves: Degree of Curve. R by (12'). R by (11'). Difference. 1 5729.65 5729.58 0.07 2 2864.93 2864.79 0.14 3 1910.08 1909.86 0.22 5 1146.28 1145.92 0.36 7 819.02 818.51 0.51 10 573.69 572.96 0.73 14 410.28 409.26 1.02 20 287.94 286.48 1.46 The difference is seeu to be about one half a foot for a 7° curve, one foot for a 14° curve, and one and one-half feet for a 20° curve. Up to a 7° curve the difference is inconsiderable, and we may stake out curves with 100- foot chords. From 7 to 14 degrees 50- foot chords may be used. Therefore 25 R — - — r^ = 25 cosec \D (12'a) sin \D * For curves from 14° to 28° we should use 25-foot chords, for which 12 K R = ^-—f, = 12.5 cosec ^D {12'b) sin ^D " Above 28° shorter chords— say 10 feet— should be used, if the curve cannot be struck from the center. In this case R = - — --^ = 5 cosec 4^B (12'c) sin ■^\I) Table I of radii was computed by formulas (12'), (12'a), and (12'6). In practice it is customary to take the radius of a 1° curve as 5730 feet and to assume the radii to vary inversely as the degree ; thus for a 4° curve the radius wouhl be R = ^p^ = 1432.5 feet, while by Table I it is 1432.69 feet — a difference of only .19 foot ; for a 12° curve R = m^ = 477.5 feet, wliile by Table I it is 477.68 feet. The effect of taking 5730 instead of 5729.65 for the radius of a 1° curve is to reduce the error resulting from the assumption that U equals 5730 divided by the degree of curve. 42 A FIELD-MANUAL FOR RAILROAD ENGINEERS. 76. The Length of Curve (L) is foiiud by dividing the angle at the center (which equals the intersecliou angle) b}' the degiee of curve, the result being in chains and decimals of a chain. The number of P.O. -\- L will give the station number of P. T. Example.— The P. C. of a 4° curve having /= 26 30' is at sta. 104 + 12.5. Find L and the number of the P. T. Here Of! K X = '^ = 6.625 chains. 104.125 + 6.625 = 110.75 ; hence the number of P. T. is 110 + 75. 77. Use of the Table of Functions of a One-degree Curve. — In the location of railway curves geometrical accuracy will frequently be of less importance than rapidity of Held- work, so long as errors are kept within certain limits. On tangents slight errors of alignment may readily be detected by the unaided eye, but on curves these are not .so apparent. Moreover it is not likely that the trackmen will keep them up in the exact position of their location. To simplify and shorten the field computations engineers make use of a table of functions of a 1° curve, and assume these func- tions for other curves to vary inversely as their degree, or directly as their radii. Table IX gives values of the tangent distances, long chords, mid-ordiuates, and externals for a 1° curve, the radius of which is taken as 5730 feet. To find these functions for other curves, divide the tabular values by the degree of curve. The error resulting from this assumption will, in any practical case, amount to no more than a few tenths or hundredths of a foot. Table IX may also be used as a metric curve table, the tabular values being taken as meters instead of feet. If the unit metric chord is 20 meters long, this may be taken as one fifth of the tabular unit chord; so to use the table multiply the metric degree by 5 and enter the table with the result as a value of D. For instance, a 2° metric curve having / = 40° w^ould have a mid-ordinate equal to ,r— — .: = 34.56 meters. ^ 2X0 For the approximate radius of a metric curve divide 5730 by 5 times the degree. Thus a 4° metric curve would have R= 4X5 LOCATION". 43 — 286.5 meters. For the exact radius make use of formula (12). Thus for a 4° curve haviuff 20-mcter chords K = — — ^ = 286.54 ° sm 2 meters, a difference of only .04 meters. If a metric curve is to be retraced with a 100-ft. chain, we convert the metric degree to the degree referred to 100-ft. chords by the relation that a 100-ft. chain = 1.524 chains of 20 meters each; a 20-meter chain = 65.618 ft.; one foot = 0.8048 meters; one meter = 3.2809 ft. It will sometimes be a sufficiently close approximation to take the 20 meter chain as two thirds of a 100-ft. chain; this will make the metric curve nearly two thirds of the degree the same curve would have when laid out with a 100-ft. chain, and the curve with 100-ft. chords nearly three halves of the degree as laid out with the 20-meter chain. Thus a 4° metric curve would be equivalent ^o a 6° curve laid out with a 100-ft. chain. In the problems that follow two methods of solution will be given when practicable — the first being rigid, while the second .'s based ou the use of Table IX. To shorten the formulas the subscript 1 will be written after the letters T, L.G., M, and E when these are the functious of a 1° curve. Thus Ti '4- 28" means Mie tangent distance for a 1° curve when /=28°, L.G.i '4- 16° Mie long chord for a 1° curve when /= 16°, etc. 78. Tables of Natural and Logarithmic Circular Functions. — Many engineers prefer to work altogether by tables of natural sines, cosines, etc., and time may often be saved by their use. Nevertheless logarithmic tables are of frequent advantage, even in the field, and the more important ones, such as the logarithmic sines, cosines, tangents, and cotangents, together with the loga- rithms of numbers, are given in the back of the book along with the tables of natural functions. 79. Given R and G to Find D. From equation (12), smlD = i^ (13) JX 80. Given / and II (or D) to Find T. If I) is given, find li by (12); then in Fig. 15 from triangle OAB we get T=2iiiinU. (14) 44 A FIELD-MANUAL FOR RAILROAD ENGINEERS. By Table IX. — Fiud the tabular value of T for the given angle I; then T = D (14a) Example.— /= 35° 40', Z) = 4"; required T. By (14), T= 1432.69 tan 17° 50' = 460.91 feet. By (14a), T = — -^- = 460.85 feet, a result differing from the value found by tbe rigid method by ouly 0.06 foot. 81. Given I and Tto Find R or D From (14), T R = tan \I = TcotU. Then by Table I the degree may be found. By Table IX. D = (15) (15a) 82. Given /and D to Find the Long Chord L.G. First find R by (12) or (12'), or by Table I ; then from the triangle O^Fof Fig. 15, AF=R?>m II. AG = 2AF= L.a =2R sin ^I. . (16) LOCATION. 45 By Table IX. — Fiud the tabular L.C. for the giveu augle /; tbeu L.G.= Xy. C.J (16a) 83. Given the Radius R and any Chord G to Find the Ordinate to the Curve at any Point. First Method.— In Fig. 16 let HE be the chord C\ HK = a and KE = b, the segments into which it is divided by the ordi- nate y. Draw the radius through K; call the portion between chord aud curve y'. By geometry, from which (2R - y')y' = ah, , ah y ^n-y" But y' is small compared with 27?, and hence we write ab ^=2li (a) Now y does not differ sensibl}^ from y' in the cases met with in practice, so we write y = ah 2M (6) 46 A FIELD-MANUAL FOR RAILROAD ENGINEERS. If we write E = -, formula {b) becomes _ abD ^ ~ a X 5730 ^^^ a b Let — - = m J— = n, and substitute in (c), giving y = TTTgQ mnD = O.SldmnD, or very nearly y = |wnZ> (17) y is given in feet when m and n are in chains and decimals of a chain. At the mid-point F, m = n, and y = M. .-.31= pi^D (18) Caution. — Formulas (17) and {\S), while ver}' convenient for field use in passing obstructions, are liable to error when very long chords or large values of I) are used, since Ihey give results that are too small. If we write the arcs HN, NE for a and b, we shall get results that are too large, 3'et about Jis near the true values as by taking m and n to be the segments of the chord. To illustrate we will find a few values of J/ and compare with the true values taken from Table V. Decree Length Mid-ord. Mid-ord. Mid-ord. of of by by by Curve. Arc. M^i{HF)-^D. M=l{HGyW. Table V. 2 2 stations. 1.75 1.75 1.75 2 G " 15.69 15.75 15.69 5 2 " 4.37 4.38 4.36 5 6 " 38.51 39.38 39.06 8 2 " 6.96 7.00 6.97 8 4 " 27.29 28.00 27.75 8 5 " 42.02 43.75 43.20 8 6 " 59.43 63.00 61.93 From this it appears we may use formula (18) — and (17) as well — taking eitlici- llie segments of the arc or chord for curve.'! not exceeding 4° with arcs up to 600 ft.; for curves from 4° to 6" LOCATION. 47 tbej may be used up to 500-ft- arcs, while for curve§ Ijetwedu 6° !ind 8' uot more Ibrm 400 feet of arc may be takeu. SF.co^'D Method— First determine the mid-ordinate. In triangle OEF, 0F= \^W - \(T-\ then M=FG = R- VR' - \G' (19) To find ordinate J.C distant d from the mid-point ot EH, draw OB=d parallel to HE; draw AB at right angles to HE. Then BA = x^R' Therefore CA = y= VR' - d^ -> \/R' - \0. . . . (20) Third Method. — If the chord C is short, we mav resrard the arc as an arc of a parabola, for which it is kuo-A-n that ordi- nal es vary as the product of the segments into which they divide the chord. The mid ordinate being known, we have .•.y=4«-^. (21) From formula (b) we have for y = M, a = b = ^C, The mid-ordinate for any other chord C" is Hence M~ C .-. M, = m{^J (23) If C = ^C, this gives M,=iJtr. (23) 48 A FIELD-MANUAL FOR RAILROAD ENGINEERS. This last relation affords an easy method of staking out a curve when the mid-ordiuate of a given chord has been determined. First erect the ordinate iLT at the midpoint of the chord; then join the ends of chord with the extremity of the ordinate just measured; the lengths of these chords do not differ much from IC; at their mid-points erect ordinates equal to ^M, giving points on the curve. Proceed in like manner for other points until a sufficient number have been located. 84. Given i? and / to Find the External E. In Fig. n E=GB=OB- OG. But OB=R sec \I and OG = R. . •. E=.R{sec \l-\) = R ex sec i/. . . . (24) By Table IX. — Find E for a 1° curve for an intersection angle I; then E = D (24a) 85. Given T and / to Find E. In Fig. 17 draw BG perpendicular to AB, and produce AG \,o b/ intersect 5Cat C. BG\% parallel to AG, and the triangles AGO and GBCfiva similar; hence BC = BG = E. In the right triangle ABC, angle BAG=\BAF= \I. Therefore ^ = r tan \L (25) Exercise. — Derive equation (25) from (24). LOCATION. 49 86. Given 3/ and /to Find E. FroDi trigonometry, IT 1 sec 4/ = r>. ' COS \I Insert this in (24) and we get .1 — cos 47 E=R cos |/ (a) But from Fig. 17, M = R{\ - cos \I), Substitute in {a) : M E = cos \I = M sec 4/. (26) 87. Given £^and /to Find i?. From (24), E E R = sec |/ — 1 ex sec ^/ 88. Given /and ^ to Find /. From (25), _, cos 4/ = E 5_ vers |/ * s (27) (28 89. Given the Chord C and Degree of Curve D to Find the Chord Deflection Offset d. In Fig. 18 extend EA to H, making AH = EA = AB; join "O Fig. 18. U and 5 and draw AK to the mid-point of HB. Then /Zir = KB= C sin i/). .-. d= IJB = 20 sin W (29) 50 A FIELD-MANUAL FOR RAILROAD ENGINEEHS. When C = 100', d = 200 sin ID (29') If we write sin 1^ = ^ from (12) in formula (29), there results ^ = -^ (30) For curves up to 7^ C = 100'; hence ^ 10000 ^=-E- (3^') For curves from 7" to U\ C = 50'; therefore <^ = f» (30") 5730 For R write — , and (30'), for C = 100, becomes -i:?-----^ <-, and for C = 50. (30") becomes 2^00 7) ^-5-:^^ = .43632) = . 873.-. . . . (31') Example.— Find d for a 6° curve, C = 100 feet. By (29'), cf = 200 X 0.05234 = 10.47 feet. By (30'), d = ]^^ =10.47 feet. '' ' 955.4 By (31), d = 1.745 X 6 = 10.47 feet. 90. Given the Chord G and Degree of Curve D to Find the Tangential Deflection Offset t. lu Fig. 18 make EF (tangent at E) equal to EA, and join F with A. Draw EG to the mid-point of FA. Angle AEG = GEF = \D; hence, from the figure, AG = GF= Csm^D. : t = 2C sin ID (32) LOCATION. 51 When C = iOO feet, t = 200 siu \D. (33) Since \D is small, we may write, without material error, siu ID =: ^ sin |Z); then, writing siu W = -n, as in 89, we get t = 91 211 5730 Making C = 100 ft. and writing 11 — — ^ gives i = 10000 1) D = 0.873i>. 2 X 5730 When C = 50 feet, (33) yields t = .218Z> = .436 X D (33) (33') (33") Example.— Fiud t for a 6° curve, G = 100 ft. By r32') t = 200 sin 1° 30' = 5.24 ft. By (33'), = .873 X 6 = 5.24 ft. 91. To Find the Subtangential Deflection Offset i' for a Subchord C" First Method. — By formula (13) fiud the angle at the center subtended by the subchord C; call this angle D'. From (32), i' = 2C' sin^D'. . (34) Second Method.— In Fig. 19, with ^as center strike the arcs FG and AH, taking EF = C and EA= C; prolong EG to B. Now assuming that the chords C and G are proportional to their central angles we have From the similar sectors EFO and EAB, since EB = G, Fia, 19. AB C' t' ' (*) ^o >'Z A FIELD-MANUAL FOR RAILROAD ENGINEERS. MiiltiplyiDg (a) and (b) together, term by term, Whence C_l C^ C'~ t' ' C '-{?)■ (35) Example. — Find t' for a 7° curve when (7 = 60 ft. Here ly By (34), t' By (32'), t By (35), t' = ^ X 7° (very nearly) = 4' 12'. = 2 X 60 X 0.01832 = 2.20 ft. = 6.11 ft. 92. To Find the Tangent Offset z. In Fig. 20, EB—z is the required offset. Let AE= n chains = lOOn feet. AE=FB, the half-chord having the mid-ordinate AF = EB , hence we have, by formula (18), 2 = In-I). . . . (36) In this formula we may take n to he either the length of AE ov the arc AB, in chains. If taken equal to AE the offsets will be slightly too small, while if taken equal to AB they will be a little too large. The use of the formula is limited to small values of n and B, as was pointed out in 83. (See Caution.) Formula (36) is easy of application and of frequent use in locating curves by offsets from the tangents. For curves up to 4° 71 may be as great as 3, but for sharper curves it should be less. Example. — Find six offsets to a 4° curve at points 50 ft. apart, measured around the curve. Fig. 20. LOCATION. 53 By successive applicatious of (36) we have for 71 n n n n n 1 = 1, 2 = — 3 — 2' — 9 5 = |X i X4 = = 3, 2 = I X 1 X 4 i X I X 4 i X 4x4 I X -f - X 4 i X 9X4 0.88 feet 3 50 7.88 14.00 21.88 31.50 The last vahie of z is iu error by about 0.2 ft , but for seltiug stakes on coii.structiou this difference is not material so long as the alignment beyond this point does not depend on it. In setting irack-cenlers the completed road-bed is available and the stakes may be set with the transit, in the usual way. 93. Diflference in Length of a Circular Arc and its Long Chord. First Method. — Let the central angle be a degrees. By (13), c sin \a° ■= ^R Changing degrees to circular measure, a (in n meas.) = : — — . The length of arc is Ra = R^ir^- Then 57. o 57. o It a 180 a Arc — chord = Rzrrr^ — ^■ 5<.o (37) Second Method. — An easy approximation may be found as follows : Referring to Fig. 17, ^^=c, QF=M. luQi AG = b = ^-\-x. From the right triangle AFG 'i+-y=i+^'- From which X c -{- x (a) 54 A FIELD-MANUAL FOR RAILROAD ENGINEERS. Neglectiug the x in deuomiuator as small compared with c gives ^=T (*> Then will 26 - c = 2^ = — (38) From Huygens' approximation to the length of a circular arc (see Williamsou's Differential Calculus, p. 66), arc = — ^ — . o Therefore Arc — chord = — c = |(25 — c). . . (c) o Inserting the value of 2o> — c from (38) gives 8i/'- Arc — chord = —^ — {d) When the arc is not very great we may write c = 100/h , where 7ii is the number of chains contained in the arc AE. From (18), remembering that rii = 2n, M = 0.21Snr'J). Inserting these values of c and M\n (d), Arc - chord = | ^^^^""-^ = ^^n,WK nearly. . (39) Example. — Find the difference in length of arc and chord of a 4' curve when ni = 6 stations. The central angle is 4 X 6 = 24°; then, from Table IV, c = 595.74. By (37), Arc - chord = 1432.7 X ^ - 595.74 = 4.34 ft. By (39). . , - 6X6X6X4X4 .._. Arc - chord = ^;r = 4.32 ft Remark.— Formula (38) is interesting as showing what a com- LOCATION. 55 paratively small iucrease iu length of liuo is caused by a consid- erable lateral deflection in alignment. For instance, a lateral deflection of 2000 feet is made at the mid-point of a line 40,000 feet long ; what will be the increase in length? By (38) the increase is )^^ ^^^J^ = 200 feet, giving for the 40,000 increased length 40,200 feet. B. Locating Simple Curves. 94. To Locate a Curve with the Chain by Oft*ets from Chords Produced. In Fig. 21 let the P. C. fall at B. If BC is a full chain, prolong Fig. 21. the tangent AB to if, making 55"= BC , HO will equal t, which may be calculated by (32) or (33'). With 5 as center, strike an arc with radius BH, and with H as center and t as radius strike an arc , at G, where these arcs intersect, set a stake. Produce BC to K, making CK = BC = CD \ strike the arc ED from C as center ; make the chord KD = d, calculated from (29'), (30'), or (31). Set a stake at D and proceed in like manner for the other points until the P.T. is reached, where FP is made equal to t. Usually the P.C. does not fall at a full station ; then HC = t', which may be found by (34) or (35). Using this value of t\ we locate as above. At B make BB = t', and prolong RG to L ; make LB = t and set a stake at Z>. EM will equal d, and may be located as before. We may regard KD as equal to KL -{- t, and, finding, KL, 56 A FIFLD-MANUAL FOR RAILROAD ENGIXEERS. measure KD and set D without locatiug R. To do this we have the similar triangles BHC and CKL, from which KL t' CK BC and therefore, since KG = CD, In like manner at ^ we have PN= t^, and FP = W hence Make EQ^ = ti , prolong QF, and wc have the tangent at F. Example. — Given the P.C. of a 5" curve at 106 + 20 and the angle of intersection 22°, to locate the curve. 22 Here L = -^ = 4.4 stations. 5 Therefore the number of the P. T. is 106.20 + 4.4 = sta. 110 -f 60. BCin this case is 80 ft., and by (33') t = 0.873 X 5 = 4.37 ft. By (35), ^=4.37X (1^^=2.80 ft. Set off HC - 2.80 ft., and at B make KB = 2.80 X ^ + 4.37 = 7.87 ft. At E make ME = d = 8.72 by (31). This will be at sta. 109 ; at 110 set a stake by offsetting 8.72 ft. The last chord is 60 long, and hence the offset NF= 4.37 X Y^ + 4.37 X ' ^^^)'= 2.62 + 1.57 = 4.19 ft. Make .fi;^ = 1.57 ft , and prolong QF, the terminal tnngcnt. LOCATION". 57 H- 95. To Locate a D Degree Curve by Oflfsets from Tangent. Let AM, Fig. 22, be tangent at A, and E, F, G, etc., points on tbe curve. Tbe offsets BE, CF, ^ B etc., may be found from formula l (36), eitber by taking equal intervals, ^ AB, BC, CM along tbe tangent or by taking E, F, O, etc., at regular stations around tbe curve and using tbe arc lengtb instead of tbe tangent. Wben tbe arc AO is large, or strict accuracy is required, we proceed to find tbe offsets at regular stations and tbe lengtbs of AB, AG, etc. First find R from (12) or (12'); tben from triangle OEL, Fig. 23. BE= AL = R{1 - cos D) = R vers D, AB = LE= R sin D. lu like manner OF = AH = R{\ - cos 22)) = R vers 2D, AC= HF = R sin 2D, and so on for any number of stations. Sbould A fall at a plus station, we first find tbe angle Bi at tbe center, tben BE = R vers Bi , AB= R sin P, , CF = R vers {Br + D), AC= i?sin(A + B), etc. = etc. The ordinates BE, CF, etc., are evidently equal to tbe mid- ordinates for long chords 2LE, 2IIF, etc.; hence we can, if A, E, F, and 0, fall at full stations, take them direct from Table V; then take the long chords from Table IV and dividing these by 2, get the required coordinates. 58 A FIELD-MAJsUAL FOR RAILROAD ENGINEERS. Example.— Locate three stations of a 4' curve by offsets every 50 ft. on curve. Referring to Table V, the required offsets are 0.87, 3.49, 7.85, 13.94, 21.77, and 31.31. By Table lY the distances measured along tangent are 50.0, 99.94, 149.76, 199.39, 248.78, and 297.87. With these values we can set out the curve either way from A. Had we used formula (36) we should have had for the values of the offsets 0.87, 3.50, 7.88, 14.00, 21.87, and 31.50. 96. To Locate a Curve by Offsets from a given Long Chord. Let FK, Fig. 23, be the given chord. We may compute the offsets yi,y^. . .Mhy the methods of 83— of which formula (17), y = ImnD, is the most convenient, within the limits of its applicability— and setting off these ordinates, locate the curve. Or we may set off the mid-ordinate J/ = i^^ersi^O^ at A, and at C set off ^2 = iW — 7? vers D, making AC = HL = R sin D. GE will be yi - M - B vers 2 A find AE = 7? sin 22). Anotheii Method is to find the angle A'Oi^at the center, and by Table IX delennine BA - M ; then by Tables V and IV LOCATION. 59 delermino BL, BN, Lll, and J^^G. Then HC = M - BL, which scL oil" lit C, aud other points in like manner. Example.— Given the P.C. of a 4° curve at station 160 + 75, the angle between tangent and chord = 9°, required the offsets necessai-y to locate the curve. Here 7=2x9 = 18°. 18 ,'. L — -r = 4.50 stations. 4 Hence the P.T. falls at 160.75 + 4.50 r= sta. 165 + 25. The mid-point on curve B falls at sta. 163. By Table IX, ri M='-^ = 17.64 ft. 4 By Table V the mid -ordinate for two stations of a 4° curve is BL = 3.49. Hence HC = 17.64 - 3.49 = 14.15. By Table IV, IlL = AC = 99.94 ft. Measure AG = 99.94 ft., and set off CII= 14.15 ft., and drive a stake at //. In like manner lind GE=d.lO and ^1^= 199.39 ft. The points P and Q are also located by means of the coordi- hates just determined. If B had fallen at an odd station, the curve could have been located in the same manner, 7/ and P being 100 ft. from B, G and Q 200, etc. 97. To Locate a Curve with Transit and Chain when the Degree D or Radius E is Known. If R is given, determine 7> by (13); then, since the angle in the circumference of a circle is half the angle at the center sub- tended by the same chord, we may locate points on the curve by successive deflections from the tangent. In Fig. 24 let the P C. be at A, at which point set the transit, and with the vernier plates clamped at zero place the telescope in tangent either by sighting the P.I. or by backsighting to some point in the tangent Deflect from the tangent half the angle at the center for the sub-chord or chord, and direct the head chain man into line while the real chainnian holds his end of the chain 60 A FIELD-MANTAL FOR RAILROAD ENCtINEERS. at Ihe transit, the cbaiu beiug kepi taut. The stakeuiau drives a stake at the point where the head chainman's Q.i\g rested, aud the rear chainman advances to this point. Deflect iD from the chord AB just run, aud while the rear chainman holds his end of the chain at B direct the head chainman into line at C. Other points are located by deflecting an additional ID for each chord length measured, until a point E is reached to which it is desirable to Fig. '-M. move the transit. The angle FAE shouM not exceed about 15°. Move the transit to E, backsight to A, and deflect FEA = EAF, when the telescope will be in tangent, and the curve can be con- tinued until it is again necessary to move the transit. At the P.T. put the telescope in tangent by backsighting to the point last occupied by transit and deflecting the tangential angle as at E. The line may now be continued. 98. The Index-angle is read on the vernier-plate, and is the angle between the tangent to the curve at the P.C. and any other line passing through a point on the curve when the telescope is directed along this line. It is most frequently taken as the angle between the initial and any subsequent tangent to the curve. Thus at E the index-angle equals EFP = 2FAE. At any point on the curve the index-reading in tangent may be found by the following rule, which may be easily deduced from a figure: From double ihe index-angle that fixed the point subtract the index- angle in tangent at the last point; the remainder is the index-angle required. 99. Subdeflection-angles may be found by (13) ligidly, or approximately (and with sufficient accuracy except when D is very large) by assuming the central angles to be proportional to their chords. Thus on a 4° curve the central angle for a sub-chord of 25 ft. would be V, and the subdefleclioiianirle 30'. LOCATIOK. 61 Example. —Locaie a V curve to left when the P.C. is at sla. 81 -f- 25 and I=H2' 86'. 32.6 Here L - — ,"— = 8.15 chains. 4 Hence the P. T. will fa.l ac 81.25 -j- 8.15 = sta. 89 + 40. The rst sub-chord found by (13). first sub-chord is 75 ft. lon^, and the tirst deflection-angle will be ^'°^* = im7-»»2«" i(5 = V 30'. By the approximate rule, since ^D = 2", 45 _ 75 2 100' whence i5 = 2 X I = T 30' as before. With transit at P.C. deflect 1° 30' from tangent, measure 75 feet, and set stu. 82. Then a deflection of 3' 30' will determine 83, 5" 30' sta. 84, 7° 30 sta. 85. Now remove transit to 85, and with vernier at 7° 30' backsight to 81 + 25. Reverse telescope and set vernier at 15° 00', when the telescope will be in tangent. An index angle of 17' will fix 86, and so on. The last chord will be only 40 feet long, for which the sub- deflection-angle is {§^ of 2^ that is, 48'. The index-angle fixing the P.T. is therefore 23° 48'. To get in tangent at 89 -f 40 backsight to sta. 85, with vernier at 23° 48' ; then by the rule of 98 the index-reading is (23° 48') X 2 — 15' = 32' 36' = /. Set the vernier at this reading and run tangent. Caution —It is not good practice to set more than 4 or 5 sta- tions on curve from any one point. Mr. Siiunk gives the limit- ing angle to be deflected from tangent as 20% and sa^s 15° should rarel}' be exceeded. {Field Engineer, p. 82.) 100. The Transit Notes may be conveniently kept in the form below, which shows the notes for the last example. When possible the tangents should be run to intersection, the angle 1 measured, and the t:ingent distance calculated. Then 62 A FIELD-MANUAL FOR RAILROAD ENGINEERS a O - , ii) ■a 'Z a> •z ® X c y. ^ cs j2 Oj 03 Station. C TO i-iH II Remarks. 90 +40 QP.T. 0°48' 23° 48' 32° 36' N 37°36' E N 27°30' E 89 23° 0' 88 21° 0' 87 19° 0' 86 17° 0' 85 O 7° 30' 15° C 84 .5° 30' 83 2° 0' 3° 30' 82 1°30' 1°30' 4° C.L.; P.I. set. -f25 OP.C.4°C.L. 0° 0' 0° 0' 0° 0' I = 32° 36'; T ~ 418.9 ft. 81 N 60°12' E N60°10'E measure along tangents and set P.C. and P.T. from the P. I. When the curve is run in, the position of the P.T. thus found should agree with the one set from the P.I. If the error is greater than the circumstances of the case permit, the curve must be rerun and tanircnts remeasured. 101. Another Form of Notes, and in some respects a better one than the above, is given below. Tl^e index-readings are com- puted as though the entire curve v^-ere run from the P.C. The notes for ihe last example would appear as below : Station. otal ngle. 3 5 Remarks. M p E- R, F E I />-- . ,H C \ \ 1 ' ^"h. / • '^^.---^'r'""^ " 3, 1 Fig. 55. In Fig. 55 let arc 5(7,with center 0, and radins 7?, , be the join- ing curve. Draw O^E parallel to CF, aud OiCand OF perpen- dicuhir thereto. LOCATION. 99 From the figure, (R + -Ki) cos h = R cos a — jRi; - i?cos a — i?i ,^__. .'. cos6 = — -— -•^— - (103) Then d = 180 — h, and J^05 = b — a. The curve may now be traced on the ground. If ^C is wanted, we have AC = (i? + ^0 sin & — i? sin a. If the point -B is fixed and Ei required, there results, from (103), It (CO, a- cos b) _ 1 + cos Example. — Take the example given for the first and second cases By (103), , 5730x0.43 -1432.5 ^ . . . _,„.,, ""^ ^ = 5730 + 1432.5 = ^'^^^ = ^^ ^^ ^^ 6 - a = 81° 44' - 64° 32' = 17° 12', equivalent to 17.2 stations on located curve from A to B. Angle d — 180" — 81" 44' = 98° 16', equivalent to 24,567 stations from B to C on the 4° curve. Fourth Case, — Joining curve tangent externally to located curve, with center on opposite side of cutting line. Let O2, Fig 55, be center of joining curve, R2 its radius. From the figure, {R -\- R2) cos c = R cos a + i?2. ■ R cos a -{- R2 ,^^^, . •• cos c = — -g-^-^-- (105) If Jf is fixed and R2 required, (105) yields „ _ i?(C0S C — cos a) R(C0S C — cos a) /ir\a\ Hi — :- = — ; . . (10b) 1 — cos c versm c Example. — Take same example as in preceding cases. By (105), cos c = 0.54403 = cos 57° 02'. Then a - c = 64° 32' - 57° 2' = 7° 30', 100 A FIELD-MANUAL FOR RAILROAD ENGINEERS. calliug for a distance of 7.50 stations from A to M around 1° curve. From MtoHon 4° curve is 14.258 stations. 130. To Locate a Y A Y is made up of a system of tracks so arranged as to admit of turning an entire train. Tliree of the most used arrangements are given belovs^. First Case. — One branch of T a straight line. This is only the special case of the last problem in which the cutting line becomes tangent to both curves. In Fig. 56, if any A F C J ^ "^"v.^ ''\ ^0^^"^ ' '' ^■^\ /' \ ^ V ^^ E R / N x\l. ; 2,'^ \? "^1 '^'''"'^ Fig. 56. one of the points A, B, or G is given, the others may be located by landing the angles c and b. Draw OiE parallel to CA ; then in triangle OOiE {B 4- El) cosb = R- Ri. cos b = R — Ri R-\- Ri (107) This follows at once from (103) by making angle a = 0. Then angle c = 180 — b. If AB were a located curve and the point ^glvens formula (107) would furnish us a value for i?i. Another solution is to produce the tangent at B to cut AC at F; then AF = FG = BF. Join i^with and 0, ; it can easily be seen that angle OFOt = 90°, and, by geometry. Therefore tan BF= \/RX Ri. . BF_ /Rr and ^ ^ BF /R (108) (109) (110) LOCATION. 101 Example.— Let AB be a 3° curve, BG& 6° curve, the poiut A at station 180. By (107), cos b = !o!n ! 7 l li^A = 0-^317 = cos 70° 32'. 1910.1 4- 955.4 The nutQber of B is 180 + 23.511 = 203 + 51.1. Angle c = 109° 28', equivalent to 18.244 stations on the 6° curve. Second Case. — 77ie tliree branches curved and convex towards each other. Given the three radii and any one of the points A, B, or G, Fig. 57, we have only to find the angles at the center, then divide these angles by the degrees of the respective curves to get their lengths and locate the three branches. In the triangle OOiOi, letting ^ OOi = I, OiOi = m, OOi = n, we shall have, by trigonometry, Fia. 57. cos f m . n r I (i? H- -Ri + Bi)B^ {R-\- B^){Rx -^ B,) (111) Angles b and c may be found in like manner. The angles may be found otherwise by letting fall a perpen- dicular from one vertex upon the opposite side, as OE perpen- dicular to Oi Oi. Then from the relation OiOa : 0x0 -f 00a = OOi - 00^ : OiE - O^E determine O^E and OiE; then the right triangles 0^0 E and OiOE yield values of cosine a and cosine c, after which b may readily be obtained. Third Case. — One branch concave to the otJier two. In Fig. 58 the triangle 00, O2 may be solved for the angles at 0, 0) . and O-i ; for if the radii are given, the sides OOj = B — Bi, OOi = .K — Ri, and OiOt = Bi -\~ Bi are known and the solution 102 A FIELD-MANUAL FOR RAILROAD ENGIKEERS. is the same as for second ease. Then b is the central angle for curve AB, a' = 180 — a, the central angle for AG, and c' = 180 — c, the central angle for curve BG. Example. — If A is at sta. 820 on the 1° curve AB, AG an 8° curve, connect with a 6° curve GB. Here we have OaO = 5730 - 717 = 5013, 0x0 = 5730 - 955 = 4775, and OaOi = 955 + 717 = 1672. Solving this triangle, we get c = 88° 20', b - 19° 28', and a = 72° 12'. The number of B is therefore 820 + 19.467 = 839 + 46.7 ; the length of GBis — -— = 15.278 stations, and 107 8 of J. (7 is ' = 13.475 stations. 8 131. To Locate a Reversed Curve between Parallel Tangents. First Case. — Radii equal. (a) The equal radii B and distance p between tangents known. In Fig. 59 draw 0^ parallel to AG to meet OiB produced. From triangle OEOi, and OE = 2Ii&\na (113) LOCATION. 103 From triangle ABG, P AB - —^-~ - p cosec iitt = \/0E^ -\- p\ . (114) Sill "^Qf , Fig. 59, (b) AQ and p known, R required. Here AB = \^'AG- + p^ = k. Draw OH to the mid-point of AC. Triangles J^O^ and ABO are similar and AH = \k. Therefore K -A. \k-p* whence R = Ap (115) Example. — Connect two parallel tracks, 30 ft. c. to c. by a T reversed curve. From Table 1, B = 819 feet, and, by (113), cos a = \ 30 1638 = 0.98167 = cos 10' 59'. By (113), OE = 1638 X .19053 = 312.1 feet. By (114), AB = i/(313.1)'^ + (30)^ = 313.5 feet. If jo = 30, OE = 312.1, or AB = 313.5 had been given, we should have had, by (115) R = 1??M)! = 819 feet. XfwU 104 A FIELDoiANUAL FOR RAILROAD ENGINEERS. Second Case. — Radii unequal. (a) Suppose the radii R = OA and Ri — 0,B (Fig. 59) to be known. "We must find central angle a and AB = k. From the trjangle 00i£^ Then AB will be given by (114). {b) Suppose AB = k, p and R known, to find Ri and angle a. Triangle ABG yields sin ia = -|- (117) OiLB is similar to A OB. Hence Ri _ k LB~ p' But AO = 2R sin ^a, and LB = i{k - AC) = iCi. Inserting this value of LB and solving for Ri, «■ = ^- • • ("8) From similar triangles, Ri_ Cr ■ B k- Gi' Inserting me value of (7i = -^^ from (118) and solving for Ri , we get R, = ~~- R. . . . . . . (119) 2p Example.— ^5 = 300', p = 30', R = 819 ft., to find angle a and Rj. on By (117), sin ^a = ^ = 0.10000 = sin 5° 44'. Tliorefore angle a = 11° 28'. Bv (119\ Rx = ^^^ - 819 = 681 ft., an 8° 25' curve. LOCATION". 105 132. To Connect Two Parallel Tracks by a Crossover com- posed of two D° Curves with a Given Length of Tangent between Points of Contrary Flexure. In Fig. 60 let AFOB be the re- quired crossover, FO = l, EB=p, and OA = OB = R known ; angle a and AE = x are re- quired. Draw OM parallel to AE to meet O'B produced ; draw also „ 00 parallel and equal to FG ; join and 0'. From triangle 00' C, tan y = — , .... (120) 2B •~-'c Fig. 60. 00' = = 2R sec y = \^^R'' + ^'. . cos y (121) Then in triangle 00' M, O'M 2R - p cos 2 = y \ ^Rj cosy. (122) 00' 2R sec Now knowing y and z, a = z-y (123) Next, ic = oif= 00' sing = 2i? secy sin z. . (124) Example.— Given D=T 30', ^ = 62 ft., I = 100 ft., to locate crossover when A is at sta. 86 + 20. By (120), log tan 2^ = 2 - 3.18441 = 8.81559 = log tan 3° 44'. By (121), log 00' = 3.18441 - 9.99908 = 3.18533 = log 1532. By (122), log cos z = 3.16643 - 3.18533 = 9.98110 = log cos 16° 47'. By (123), a = 16° 47' - 3° 44' = 13° 3'. By (124), log X = 3.18533 + 9.46053 = 2.64586 = log 442.4. 106 A FIELD-MANUAL FOR RAILROAD ENGINEERS. 133. To Find the Radius of the Reversed Curve AFE, Fig ^O' 61, Given Angles / and 1' , and BG=k. From the figure, E tan ^I = BF, i?tany= OF. Adding, • i2(tan ^1 + tan i/') = BG=k. Fig. 61. Whence R = k (125) tan U + tan \I' Example.— Given / = 10°, 7'= 20% BC = 700 feet, to find B. 700 By (125), B = 0.08749 + 0.17633 = 2653 ft., a 2° 9f curve. 134. To Locate a Reversed Curve between Fixed Points. In Fig. 62 let AB = k, and angles I and / be known. We Lave to find B and the angles a and b. Draw O'G parallel to, and OG and O'F perpendicular to, AB. Angle AOG = I and BaF = I'. Then OF = B cos / and OF = B cos /'. Hence OG = i?(cos I + cos 7'). In triangle 00' G, 00 = 2B. Therefore i?(cos 1 -r cos 1') cos I -f- cos I' cos X = 2B ~ 2 an expression from which B has disappeared. . (12ff) LOCATION. 107 We uow have a = I -\- x aud b = I' -\- x. To tiud i^ we Lave AE -\- EF -\- FB = k, or i^ siu / + 2R sin x -\- R sin I' = k. Whence R — k . ^ ^ , (127) sin i -f- sin 7' -f 2 sin 05 ' * Another expression for R can be found by drawing ^iVand BL perpendicular to 00', and BN parallel thereto. Then, since 4 BAN=x, RsXn a -\- R sin h =i k cos x. k cos X sin a -\- sui Example. — Take the example of the last problem, A = 700, 1=10°, J' = 20\ By (126), cos X = 1(0.98481 4- 0.93969) = 0.96225 = cos 15° 48'. We now have a = 25° 48' aud b = 35° 48'. 700 X 0.96225 (128) By (128), R = 660.2 ft., an 8° 41' curve. 0.4^523 + 0.58496 135. To Connect Two Divergent Tangents by a Reversed Ourve. First Case. — Advancing towards the P. I. Given the radii R and A*i , the angle / and AG = k,Xo find the angles a and b (Fig. 63). ^--^ A- ^ ***** Oy / / ^^A^ ^ z < R/i"^. >^ \ M"^ H ^*' / y "^ \,.^ mT^--^ Ey ^^ okiF.. ..t. ' Q Fia. 63. Draw 00 parallel to the tangent BC to meet O^B produced. Then EF = BO = AF - AE. Therefore BG = R cos I — k sin /. 108 A FIELD-MAIsrUAL FOR RAILROAD ENGINEERS. From triangle 00 iG, Ri + BG Hi 4- RcosI- ksml ,,^^, '''' = -E+Mr'= BTB. ■• • ^''^^ Then a = M0i2^ = b - I, O.Jf being parallel to OA. Second Case. — Receding from the P.I. In Fig. 63 we have BC = ki , angle /, R, and 2?, given, to find angles a and b. Pioduce OA to meet OiL drawn parallel to CA. AL equals OiM= O^H cos I. 0,H = R, - EB = Ri - k, tan I. .*. AL = OiM — {Ri — ki tan I) cos /. Hence OL = R -{- (Ri — kx tan 7) cos / = i? + Rx cos I — k^ sin 7. From triangle OOiL, cos a OL Z? + 7?i cos 7 — kx sin 7 00, ~ i? + 7?i Evidently, 5 = a + 7. . . . (130) 136. To Change the P.R.C. so that Second Branch of Curve shall End in a Tangent Parallel to Terminal Tangent and Distant p therefrom. In Fig. 64 let MAB be the located curve, EN = p. We must M H E F .-a- / 1 ^ V B !n JQ — > ._. NO lL ^- determine the angle CO A, after which the desired curve AGE may be located. Draw HOx and LOx parallel to ^Fand NG. HL = 0,K = p. LOCATION. 109 From triangles 00, '77 and OO^L, {R + Ri) cos b = (R-\- Rx) cos a - p. P . '. cos 5 = cos a R-\-Rx' (131) Angle AOG = b — a. 137. To Find the Radius of a Curved Track. Measure any chord AB = 21, and mid-ordinate CE = M. Fig. 65. Theu in the right triangle OAE (Fig. 65), 72^- (7? - Mf = ?. . R 2M (132) \ CHAPTER IV TRANSITION-CUR VES. Article 11. — Theory op the Transition-curve. 138. Elevation of Outer Rail on Curves. — To counteract the effect of centrifugal force on curves the outer rail must be elevated above the inner one. It is shown in mechanics that the centrifugal force is F = 32.16i?' where W is the weight, v the velocity in feet per second, 32. 1(^ an average value of the acceleration of gravity in feet per second per second, and R the radius in feet. In Fig. 66 let the vertical BL represent W, the horizontal KH the centrifugal force, AB the plane of the rails, and GB = e the superelevation of outer rail From similar triangles, Equate this value of F to that given above and solve for e, giving ^^^' . (133) K,*F*iH e = 32.1622' The gauge AB should be greater on curves than on tangents to allow for flange clearance and the effect of a rigid wheel-base. AG = 4.9 feet is about the right value for the horizontal distance between centers of rail-heads for standard gauge. In formula (133) ■« is in feet per second, but the train velocity is usually given in miles per hour. Let V = velocity in miles per hour, then the 110 TRANSITION-CURVES. Ill 22 velocity in feet per second will be « = — F. Inserting these values in (133) gives 4.9X484F* V . ,,„.. This elevation will be required from the P.C. to the P.T., but obviously it cannot be introduced suddenly, so that for easy riding the rate of increase of e should be uniform. From (134) it is seen that e varies inversely with R, which requires that when e = 0, R = infinity. Hence II must decrease from infinity to the radius of the circular curve, while e increases fron\ to its maximum value. 139. The True Transition- curve should satisfy formula (134), but so far no such curve has been found that will at the same time admit of the same ease of location as the simple circular curve. According to Rankine the first use of any other than the circular curve was made by Gravatt about 1828 or 1829, the curve employed being the curve of sines. Another method described by Rankine is attributed to William Froude about 1842 ; this curve was worked up in the Engineering News by A. M. Wellington in 1890. Other approximations are the Rail- road Spiral, developed by W. H. Seailes in 1882, and the cubic parabola, described by C. D. Jameson and E. W, Crellin in the Railroad and Engineering Journal, 1889. In 1880 Ellis Holbrook described in the Railroad Gazette the true transition-curve applicable to small angles and short lengths of the curve. In 1893 C. L. Crandall published formulae and tables applicable to large central angles for both the offset and deflection methods. 140. The Notation here employed will be explained with reference to Fig. 67. The curve CBB'C is the circular curve offset at Cand C" from the tangents by the amounts (7// and C'll'. AGB and B'G'A' are the transition-curves. A is the P. 7'. C, or point of transition-curve, C the P.O., B the P.O.,, B' the P.TC.i, G the P.T., aud A' the P. 7'.,. The co-ordinates of G are All = x', HG = y'; of C, x' and TIG = F ; of B, AM = Xi and MB = y^. The length of curve from P. T. C. to any point P is I, aud the whole length from P.T.C. to P d is ^i. 112 A FIELD-MANUAL FOR KAILKOAD EXGINEERS. 141. Equation of Transition- curve. — Since the rate of change of e must be uniform, (134) may be written • e = kl — 3p' (135) Fig. 67. in which k is the rate of rise of outer rail along curve, and p the varying radius of curvature. From the calculus pd(p = dl, whence dl (136) Insert this in (135) and solve for d(p. 3yfc d(p = -y^/dl = 2mldl, (137) 2m is dependent upon V and k, and is constant for any one curve. Integrating (137),

. . . . (c) o Both S and 7i are in circular measure ; to reduce to degrees TT multiply by -^. This gives, neglecting terms involving higher powers ot 7i than the third, 5" = :^ 71-^- .00000086 7i 3,16 (151) The second term is quite small, and in most cases may be en- tirely neglected in practice. 116 A FIELD-MANUAL FOR RAILROAD ENGINEERS. With the instrumeut at auy intermediate point x"y" the deflec- tion-angle for any point xy^ measured from initial tangent, will be tan 8 = ^-^, = i(mP + ml"' -f mil") +jh?{mH' + mH"') -f ^%{mHH" 4- mHl"^)-{- ^^^^{mHH''^ + mHn"^+mHH"^)+ ..., (152) in which powers of mP higher than the third have been neglected. Substitute the value of tan d from (152) in (a), write ml^ = = 7,w^ ml'"^ = ) For transit at mid-point of transition-curve n" = |, and, from (153), 7 ° (^i°) = -5-(^' 4- i + i^) - correction, or (5^°) = ^-^A^ -B^ (156) TRANSITION-CURVES. 117 For transit at three-quarter poiut n" = | and or (^D = -t(^*' + T5 + f^) - correction. (5j°) = —^A^ — B^ ••• (157) For transit at P.C.i 7i" = 1 and // or (5x°) = -^(w' + 1 + w) - correction, o /i' (5.-) = :^^, -5,. (158) With the transit at the P.T.d it will frequently be most con- venient to measure the deflections from the tangent to the circular curve at that point. Sometimes this will also be the case for the transit at the P.C.i. Bv reference to Fig. 68 it will be seen that for the transit at B the deflection from the tangent BC which serves to fix any point on the curve, as .6, is given by the equation or, in general, (<5c°) = —Ac + B,. (159) 118 A FIELD-MANUAL FOR RAILROAD ENGINEERS. Table XV gives the values of A and B for the five positions of instrument for which equations (154) to (159), inclusive, were de. duced. The value of A must be multiplied by -^ , but B is taken o direct from the table in thousandths of a degree. If deflection-angles are wanted for other positions of the instru- ment, or for other points on the curve, they may be computed from equation (153). 145. Tables. — Three tables are given for use with transition- curves. Table XIV was computed for use with formulas (140) and (142) in determining C and E ; being assumed and G and E com- puted. . ^ Table XV gives A and B for computing the deflection-angles by (154), (155), (156), (157), (158), and (159) for 20 equidistant stations on the transition-curve. For points not given in the table A and B must be interpolated. Linear interpolation will suffice in most cases, though when Ii° is quite large second differ- ences may be preferable for A. B is given in the table in thou- sandths of a degree. Table XVI was calculated by assuming h in lengths varying by increments of 20 feet, then computing 7/ by (146), yi by (139), X, by (141), F by (147), and x' by (148). y, and x, will also be given more directly by (140) and (142) with the aid of Table XIV. The excess in length of transition curve, measured from P.T.C. to the point on offset at B.C., over x' is tabulated as e; I' is found by trial such that when inserted in (141) or (142) the same value of x' will be obtained as in (148). This may be done by as- suming l' a little less than - , then computing .i'. More than two trials will rarely be needed to find a sufficiently close value of l'; then e = I' — x'. y' is found by (139) after finding l', or 0' may be found from (b) of 144, and used in (140) in connection with Table XIV. h - I' is the length from G (Fig. 67) to the P.C.i; the difference in length between this and the length of circular curve from P. C. to P. G. i is tabulated as e' ; that is, e' = {li — V) — arc. Then e -\- e = U — {x' -\- circular arc). For values of li intermediate between those given in the table linear interpolation will suffice, 1 hough second differences may be used for ii^and yi if ])ref erred. TRANSITION -CU RYES. 119 146. To Unite the Two Branches of a Compound Curve by i\ Transition-curve. The same objections bold to compound curves as to simple curves uniting with a tangent; i.e., wbere there is a sudden change of curvature there should be a sudden cbange of super- elevation of outer rail, which of course is not allowable. Instead of compounding the curves, we may offset them at theP. C. C. and unite them by means of a portion of a transition-curve tangent to each of the simple curves. In Fig. 69 AB am] CELMare the simple curves that are to be united by the transition-curve ANE, Extend the transition-curve Fig. 69. to O, where its radius of curvature becomes infinite, and let G8 be its tangent. Call the length of transition-curve from G to A li , from O to E h , and from E to A h. E and A are points of tangency of simple and transition curves. Then l^ = It — U. The coordinates of A arc 08= Xi , SA = y^ ; and of V{WV perpendicular to GS), GW=x,', WV = F,; of E, GP = x, , EP = y^', of L {LH perpendicular to GS), GH = Xs, EL = Fz. Let BC=F^. The radius of curvature of transition-curve is inversely pro- portional to its length from G ; hence the curvature is propor- tional to the length of curve; therefore ^3 : h = B3 : Di , whence ^3 = I, B,' (160) 120 A FIELD-MAXUAL FOR RAILROAD ENGINEERS. Then h -. U - u = IM - -~\ = I, ' ~^ \ . . (161) By (138) or (143), A = U ) * Equaling the value of /a from this equation to that resulting from (146) gives ^' = ^^U) = -200 (162) WV = Fi and RL = F^ may be taken from Table XVI with h and ^3 as arguments. Then 0, W = R, + F, O^H— R^-^-Fz. Draw 0,r parallel to QS, then 0,T = WH; henc6 0,T = (R, + F,) - (R,-{.F,), and OlT=Xl'-a^'. Therefore '^^- = iR, + F.)-iR.+F.) ' . . . (163) 0.0, = {x,' - a-s') cosec a = VoJ'' + T07\ . (164) Then CB = CO, - BO,, or i^2 = i?s - (i?i + OiO,) (165) The lengths of AB and CE are ^^ = ^'°^^''° 100 (166) [3" CE= ^-^^m (167) The excess of transition-curve length over AB + C^is e, = h-~i—^^ 1 B, ) • • • (16^) TRANSITION-CURVES. 121 If AB and CE are quite sharp, we must take account of the arc excess, so that we have then Cj — frj — f ^'°^ ^ + ^^-^ J 100 + arc excess | . (168') The arc excess may be taken from the second column of Tuble IV, which gives the arc length for one station; this multi- plied by the number of stations gives the curve length, which may replace the values within the brackets in (168'). 147. Length of Transition-curve to be Taken. — In practice the rate of change of superelevation of outer rail may vary from VHhV ^^ 400* ^^^^ ^^^ ^^^^ ^ ' ^^^^ evidently kl, must equal the superelevation of outer rail for circular curve ; or, by (135), 72 Writing i? = 5730 I) and solving for li U For k = For k = 1200' 1 600' 17190A; (169) (169') ^^' ^ = 400' li = 0.035 F*i> (169") h = 0.02s V^D (169'") The following table gives values of Z, in feet per degree of circular curve for a few values of Fand k. k 30 Miles per Hour. 35 Miles per Hour 40 Miles per Hour. 45 Miles per Hour. 50 Miles per Hour. 55 Miles per Hour. 1 1200 C3 86 112 142 175 212 1 600 32 43 56 71 83 106 1 400 21 29 37 47 58 71 122 A FIELD-MANUAL FOR BAILROAD ENCiLNEERS. "\Ybeu only a short tangent intervenes between two curves shorter transition-curves* must be takeu, requiring larger values of A*, so that overlapping may be prevented. For illustration suppose a 5" curve to be eased off with a tran- sition-curve, the highest train-speed being 45 miles per hour and k = --. By the table the value of h will be 71 X 5 = 355 feet, dOO so that we should probably take a 360-ft. transition -curve, re- quiring an offset of 4.7 feet by Table XVI. Article 12. — Field-work. A Field Formulas. 148. For the cases most frequently presenting themselves in practice the foregoing formulas may be simplified so as to admit of the rapid location of points on the transition-curve with all the accuracy needed on location, though it is best to use the exact formulas and tables in setting track-centers on the finished road- bed. "When the transition-curve aogle is quite large it will be better to use the accurate methods on location also, but for the more common cases the followiuc: formulas will answer. ■o 149. Simplified Formulas.— In (189) and (140) neglect, as small, all the terms following the first, giving y = '^ = ^l= Mh^m^" (170) In (141) and (142) retain only the first two terms ^=^^"llf) = ^(l - ^) = ^ -- -OOOOSZ^^S . (171) in which the last term is small for short transition-curves and may often be neglected, x being taken equal to I. The values of m and /i remain as before : 2Rli 11460^1 ^■"=2««4=l^ ("«^ TRANSITION-CURVES. 123 In (147) expand cos 7, , giving F-y.-Ii{^-^ " 24 + 720"'--/' But 7? = -^, by (145). Substitute this for E and neglect all but the first two terms : But i,/i = 3y, , by (170), since = /i and j^ = yi when Z = ^i ; hence F=yx-^y, = iyt (173) Likewise expanding sin 7i in (148), nnd writing R = -~ as above, « But x^=l,- -^- , by (171). By. (170), ^ - 3V2J 3 "^ 8 8 2 In (154), (155), (156), (157), (158), and (159) neglect the correc tion; then (5o°) = ^^o (175) (.V)-^"^i (176) ]24 A FIELD-MANUAL FOR RAILROAD ENGINEERS. (V) = ^^i. ...... (177) (V) = ^°^i (178) {^,^) = ^A^ (179) iSc'') = ^Ac (180) 160. Offsets. — Formula (170) shows that offsets from transition- curve to tangent vary as the cube of the distance from the P. T.C., and it can be shown that offsets from the circular curve to tran- sition-curve follow the same law, reckoning from the P.G.i. Formula (36) may be written z=f{l'D), (a> in which D is the degree of curve if offset is from tangent, and the difference of degrees if offset is ])etween two curves having a com- mon point of tangeucy, I being reckoned from the tangent-point. From (136) and (137), ^ - _ J_ and the degree of transition-curve at any point is j)^ = "il^ = lUQOml = d (181) P Formula (181) shows that the degree of curvature of transition- curve at any point is a function of its length. If the D in (a) is the difference between degrees of circular and transition curves, it will equal Di — Dt , which is also a function of the length; so in {a) write D =f{l), giving z=f'{l% (182) which shows that the offset betw^een circular and transition curves varies as the cube of the distance from P.d. The offset at the P. 0. is known, being half of F, and may therefore be found for TRANSITION-CURVES. 125 Other points ; thus midway between P.C. and P.C.i it will be one eighth of its value at P.C, or ^^F 151. Compound Curves. — By trial it has been found that ^ = 0.033x23= 0.26 " three-eighths points, stas. i 40 fii' < = 0.033 x 3' = 0. 89 Stakes at the one-eighth and three-eighths points were not needed, but were worked out for illustration. 154. Location by Deflections. — The number of chord-lengths being taken as an aliquot part of 20, the deflection angles for the TRANSITION-CURVES. 127 transit at any one of five positions may be taken from Table XV by 1 ° inultiplyiug the tabular values of A by -^, Ji" being found from o Table XVI or formula (146). If the number of chords is not an aliquot part of 20, or if the transit is at some point other than one of the five for which Table XV was calculated, then the deflec- tion-angles must be computed by (153). The curve is then run out in the usual way. When 7i is not more than 15 or 20 degrees the curve may be run from the P.T.G. or P.T.G.i by neglecting the correction B as small. Even when 7, is greater than 20° the correction may be neglected, provided half the transition-curve is run from the P.T.G. and the remainder with the transit at the mid-point, the telescope being first placed parallel to original tangent. Example. — Take the example of the last section: li = 340 ft., 340 X 5 F = 4.2 ft., D = 5°. By formula (146), L = ^^^ ' = 8.5% the L° same as given by Table XVI. Then -- = 2.833°. Divide h into o 5 parts of 68 ft. each, which will be the chord-length to be used. From Table XV for transit at P. T. G. the deflections will be : For sta. 410 -f 30.1, P.T.G., {d,°)o = 0. " " 410 + 98.1, (5/).2 = 2.833 X .04 = 0.1133 = 0° 6.8'. " " 411 + 66.1, (5/).4 = 2.833 X .16 = 0.4533 = 0° 27.2'. " " 412 -\- 34.1, (<5o°).6 = 2.833 X .36 = 1° 1.2'. " " 413 + 02.1, (5o°). 8 = 2.833 X .64 = 1° 48.8' " " 413 + 70.1, ((5„°). = 2.833 X 1 =2° 50'. Having set out the transition-curve, move to P. G.i at sta. 413 + 70.1, backsight to P.T.G.i , and deflect 7,° - (V)i = 8° 30' - 2° 50 = 5° 40', and run out the circular curve to the P. 7.(7. i, which suppose to fall at sta. 420. Set the transit at this point, and cause the vernier to read zero when the telescope is in tangent to circular curve. The deflections taken from Table XV will now be : For sta. 420 + 68, (5o°).8 = 2.833 X .56 = 1° 35.2'. " " 421 -f 36. (V).6 = 2.833 X 1.04 = 2° 56.8'. " " 422 + 04, (5e°).4 = 2.833 X 1.44 = 4° 4.8'. ** " 422 -f 72, (5c°).2 = 2.833 X 1.76 = 4° 59.2'. «♦ " 423 4_ 40, (V)o = 2.833 X 2 =5° 40'. 128 A FIELD-MANUAL FOR RAILROAD ENGINEERS. Set transit at 423 + 40, the P.T.i, backsight to 420 and deflect 8° 30' — 5° 40' = 2° 50', when the telescope will be in tangent. 155. Form of Transit Notes.— The following will illustrate a form of notes that will be found to answer. Let the P.C. of a 4° aurve be at sta. 160 -f 50, and a 200-ft. transition-curve be employed. Let the intersection-angle I be 20°. By Table XVI, F = 1.16 ft., 7,° = 4°, x = 100 ft., so that P.T.G. is at 159 + 50. Take four 50-ft. stations on transition- curve and determine the deflection-angles as in the last section. Sta. Deflec- tion- angle. Central angle. Calcu- lated Course. Mag- netic Course. Remarks. 167 -1-50 166 -\-m 165 F.r.i 2° 40' 2° 15' 1°40' 0°55' 4''0' -hso© 164 16.3 16> F. T.C.I "6°'0' ' 5°0' 3°0' 1°0' 12° 0' +50© 161 -fSO 160 -+-50© 159 158 P.C, ,4 C.L. P.T.C. "]'°*20' OMS' 0° 20' 0»5' 0°0' 4°0' Set ver. at 2° 40'. B.S. to 1.59 + 50, and deflect to 0°. Run circular curve. li = 200. F= 1.16. 20 — 2 X 4 The length of circular curve was j = 3 stations, since the central angle was 12°. With transit at 161 -f 50 set the vernier to 2° 40 , backsight to 159 -h 50, and deflect into tangent with the vernier reading zero. With the transit at 164 -\- 50 cause the vernier to read zero when the transit is in the tangent to circular curve, and run the last transition-curve by deflections from this tangent. With the transit at 166 + 50 backsight to 164 + 50 and deflect 4' - 2° 40' = V 20', when the telescope will be in tangent and the line may be continued. TRANSITION-CURVES. 129 Article 13. Transition-curve Problems. 166. To Find the Tangent Distance and External when Transition-curves are Employed, Offsets Equal. In Fig. 70 let AB be the circular curve, EF and OH the tran- sition-curves. Let EK — IIK = Ti be tlie tangent distances, and J^K= El the external required. Let LK = T' . Draw PF per- V^—^-^x: pendicular to LK \ then in triangle PVK, FJ? = JP' tan \I\ LK = AP -\- VK is now known, or T' = T+FianiL (184) Hence Ti=x' + T' = X' + T+ Ftan^L . . . . (185) In triangle PVK, PK = Fsec |/, so that, letting PiV^ =z E, Ei = E+FseciI (18G) Example.— Two tangents intersect at sta. 91 + 37 8; required the tangent and external when F — 2.62 ft., / = 26° 30', D = 4". By Table XVI. I, = 300, x' = 149.9. From Table IX, T= ^^^ 4 = 337.3. Then, bj^ (185), r. = 149.9 + 337.3 + 2 62 tan 13° 15' = 487.8 ft. The station number of P.T.G. will now be 91.378 - 4.878 = 86 -f 50. 130 A FIELD-MANUAL FOR RAILROAD ENGINEERS. By (186), E, = 156.7 + 2.62 sec 13° 15' = 41.87 ft. Table XVI gives 7,° = 6°; hence the circular curve -will cover 26° 30' - 2 X 6° = 14° 30', or 3.625 stations, so that the number of the FT., will be 86.50 + (2 X 3.00 + 3.625) = 96 + 12.5. 157. Tangent Distance, Offsets Unequal. In Fig. 71, 0, 2^, and K do not lie in the same straight line. '<- —X-t.-^ <„ _T-i J c L K 9 1 '"— -"....^ -- l^ n\ 1 A -> V ''I \ 1 • 1 ^""^1 1 1 1 \t,y^* ■ ^\ 1 1 / / \\N ■ ^ / / / y"^ / ^ / ^ 1 ^ / ^/R / ^^ •■ / ^^ ^ ^ / j^ ^ -" ■^ M ^H Fig. 71. Draw PS perpendicular to NB, PQ perpendicular to LK. Let LA - F, MB = F'. or T = LK= AN-{-NP - KQ, T' = T-\- F' cosec I - Fcoil; (187) T, = X -{- T' = x' -\- T-{- F' cosec /- i^'cot /; (188) T" = MK =T - F' cot 7+ 2^ cosec i ; . . . (189) T^ = x" -\- T - F' cot I + 7^ cosec /..... (190) Example — Two tangents intersect at sta. 820 and are to b(; united by a 6° curve having F - 1.75, i^' = 2.95, and /= 3r 48'. 1632.3 By Table IX, T = 6 = 272.05 ft. By Table XVI. d = 200, I, = 260, x' = 100, xf' = 129.9. TRANSITION-CURVES. 131 By (188). T, = 100 4- 273.05 + 2.95 x cosec 31° 48'- 1.75 cot 31° 48' = 374.8. By (190). T2 = 129.9 + 272.05 -2.95 cot 31° 48' + 1.75 cosec 31° 48'=400.5. 158. To Insert Transition-curves without Changing the Position of the Vertex, B. In Fig. 72, ABC is the located curve, FGHK the curve after Fig. 72. inserting transition-curve. The radius of the circular portion has been changed from R \o R' m order to mal^e room for the offset PS= F. BM = E is the external to located curve, BL = E' the external to circular curve having radius R' and central angle I. lu the triangle LNM, LM — LN sec ^I = Fsec^I; hence E' = E - Fsecll. (191) E may be found by (24) or by means of Table IX ; then E" becomes known, and from the same table Z)' is found by dividing the tabular E by E'. D' will be larger than D, It is .sometimes more convenient to assume D' and calculate E' in the same manner as E; then, from (191), F=iE- E') cos iZ. (192) If this value of i^is too large or too small for the conditions of the problem, a new D' can be assumed and 7''' recomputed. 132 A FIELD-MAXUAL FOR RAILROAD ENGINEERS. Example. — The P.C. of a 5^ curve is at sta. 182, aud angle / = 40". Compute the data for a new curve to allow for a transition-curve with 1.5 ft. offset. From Table IX, E, = 367.7 for 7 = 40° ; therefore E = ??^ = 73.54, i^sec 20° = 1.5 X 1.0642 = 1.6 ; then, by (191), E' = 73.54 - 1.60 = 71.94, and B' = -^f^ = 5.1113° = 5° 6.678', say 5" 7'. By Table XVI, for h = 200, Z> = 5° 7' F= 1.45 4-^(1.75 - 1.45) = 1.485. For li = 220, F= 1.76 + g'j)(2.11 - 1.76) = 1.842. Then for F= 1.5, D= 5^7', I, = 200 + 20 , ^i~ ; ,Q, = 200.8 and a-' = 100.4. l.o4-& — 1.400 T5 n4R^ r° ^'^ 200.8 X 5.117 _ ^ ,^0 ., ^, By (146), L =200^ 200 -^-^^ =^ ^- The central angle for circular portion of curve is 40 — 2 X 5.13 = 29.74", equivalent to 581.2 feet around curve. In Fig. 72, B is at sta. 186 on the 5° curve, and arc BG = 290.6 ft. on the 5° 7' curve. The P.C.i is at 186 - 2.906 = sta. 183 + 09.4, the P.T.a at 183.094-2.008 = sta. 181 + 08.6, the P.T.C.i at 188 + 90.6. and the P. T., at 190 + 91.4. Had D' been assumed equal to 5° 6' or 5.1° to begin with, we should have had E' = ?|^ = 72.10 ; then, by (192), 0. 1 2<'= 1.44 X .93969 = 1.35 ft. ^1 may be found by interpolation from Table XVI as above. TRANSITIOK-CURVES. 133 159. To Insert Transition-curves on an Existing Road-bed with the Least Deviation from Old Track. To satisfy this conditiou the new track should pass about as far outside the old at the vertex as it does inside at the original W P.O.; that is, about — . "We shall now have B' = E Fsec ^- ^. (193) The remainder of the problem may be solved by 158. Transition-curves may be inserted in old track by shifting to suit the existing road-bed, thus adding materially to the safety and easy riding of cars. 160. To Insert Transition-curves at the Ends of a Long Circular Curve without Moving the Central Portion. In Fig. 73, ^Cis the circular curve. In order to make room for the offset F the ends must be sharpened by compounding. Let C be the point of compounding, li' the radius of the branch ON, HN = KB = F. Let BEG be the transition-curve ; the B A H • 1 1 K R -^-; r' X L ^'> 'Z V [? Fio. 73. closer G comes to C the better, provided the change in radius at G is kept within certain limits. The difference in degrees between the original and the sharpened curve should never exceed 3° and may usually be kept in the neighborhood of 1°. First Method. — Having decided upon the value of F, assume R so that i)' — i) is not greater than 2°. Draw O'L parallel OL to BH; 00' = R- B', and cos /' = 00 /> or 134 A FIELD-MANUAL FOR RAILROAD ENGINEERS. cos/' - ^-(^' + ^) - 1 _ _^L (194) This is the same as (69) in 122. /' being known, set the transit at G, run out the curve CN, and insert transition-curve in the usual way. If /' had been assumed in the beginning, R' could be found from (194). Second Method. — "When the circular curve is flat, and short transition-curves are employed, we ma}" compound the transition- curve with the circular at the P.d, taking care that the differ- euce of curvatures is not greater than 1° or 2". Assume the position of the P.Ci from 100 to 200 feet from the P.C.; measure the perpendicular let fall from the P.d upon the tangent at the P.C. produced; this will be yi. The central angle /i can be calculated, knowing the length of circular curve from the P.C. to the assumed P.d, or the angle between tangents may be measured with the transit. The coefficients C and E of (140) and (142) may be found from Table XIV with li =

. Therefore I. = ^^^^^^. = 150 j^, nearly. TRANSITION-CURVES 135 But nD = L°; lieuce U = 1507i ; (197] and as 100/i is the length of circular curve from P.C. to P.C.i, Ix is once and a Juilf as great. From (146), 200V ^ 3W/ ^ 2^ ^ 4^_ li ISO^i 150;i 3 From this equation it is seen that if the break in curvatures is limited to 2°, this method is admissible up to Z) = 6°, independent of the length of transition-curve. Example. — A 4° curve is to have transition-curves inserted at each end; compute the necessary data. By First Method. — Assume a 1.45-ft. offset, and the curva- ture to be changed from 4° to 5° by compounding. In Table I find R = 1433.7, E' = 1146.3; then, by (194), 1 45 cos 1=1- -1^ = .99494 = cos 5° 46'. 286.4 5 767 The length of 5° curve is -^-- — = 1.153 stations, and, by Table o XVI, /i° = 5°, so that the P.d will fall 15.3 ft. back of the 5 767 P.aC, while the P.C. will be moved forward -— 1.153 = 4 .289 stations or 28.9 ft. ; the P.T.C. being, by Table XVI, 100 ft. back of the new P.C. will fall 100 - 28.9 = 71.1 ft. back of old P.C. The transition-curve may now be located in the usual manner. By Second Method, — Assume the P.d to fall 150 ft. from the P.C, making I,° = 1.5 X 4 = 6°. From Table XV, C = .03488, and, by (36), y, = 1(1.5)^ X 4 = 7.875 ft. By (195), Now, by (146), i)' X 225.8 " " 200 ' 136 A FIELD-MANUAL FOR RAILROAD ENGINEERS. from which D' = 5.314° = 5° 18.8', which differs less than 2" from D. By (196), a-, = 225.8(1 - .0011) = 225.6 ft. To find the position of P.T.C. with reference to the old P.O. consider that the distance from P.O. to foot of perpendicular from the P.C 1 is half the chord for augle 2/i , and can be taken from 1 1197 9 Table IX. being equal to - X —r- = 149.7. Then 225.6 - 149.7 = 75.9 feet is the distance from old P.C. back to P.T.C. By Third Method.— Assume the P. C.i to be 150 ft. from the old P. C. ; then, by (197), U — 225 ft., and, by (198), the curvature of transition-curve at the P.d is | X 4° = 5° 20', giving almost the same results as by the second method. Had we taken the P.C, 160 ft. from P.C. we should have had U = 240, D' = 5° 20'; iCi = 239.7, by interpolation from Table XVI; the length along tangent from P.C. to foot of perpendicular from P.d 159.9 ft., and therefore 239.7 - 159.9 = 79.8 ft. as the distance from P.C. to P.T.C. 161. To Insert Transition-curves at the P.C. and P.C.C. of a Compound Curve by Changing the Curvatures of the First Branch. In Fig. 74 let ABV be the located curve compounding at ^. Two cases occur. First Case. — Second branch having shorter radius. The offset at P.C.C. must be to outside of located curves; let it be EB = F^ in the figure. Let CP = Fbe known or assumed. Draw the tangent BG, and draw EH parallel thereto. Let CE be the changed curve, and CQ parallel to tangent AH. Angle / may be computed from the known station numbers of A and B, or may be measured on the ground. The new tangent distance is EQ = BG - OK- HQ (or LS). From the right triangle GHK, GK = HK tan GHK = F^ cot /. Similarly, LS = LW cosec I = F cosec /. Therefore T' = EQ= T - F^cot I - Fcosec I. . . (199) T can be found from Table IX or formula (14); then T' is known from (199). The degree of new curve, Z>', may now be found by means of Table IX, or from Table I by first finding TRANSITION-CURVES. 137 R' by (15) The transilion curve at the P. C.C. may be located by 146 aud 151, while that at the P.C. may be located either by offsets or deflectious. Second Case. — Second branch having longer radius. Fia.- 74. In this case the offset must be to inside of curve, and IfS is the tangent required. From the figure, letting NB = F^, NS — T', T - r+ F^ cot I - i^cosec /. (200) The remainder of the solution is the same as for first case. Example. — A 5° curve compounds at sta. 280 with a 9° curve; the P.C. is at sta. 272. Required the change in curvature of first branch for an offset of 1.50 ft. at F.C. C. and 2.00 ft. at P.O. Here 7=8x5°^ 40°, and, by (199), T' = 417.1 - 1.5 X 1.19175 - 2 X 1.5557 = 412.2 2085.5 D' = 412.2 = 5.06° = 5° 3.6'. Da — ZX = 9 — 5.06 = 3.94° is the difference in curvatures of the two branches of the altered curve. Entering Table XVI with this value for D and F = 1.5 ft., we find : 138 A FIELD-MANUAL FOR RAILROAD ENGINEERS. for li = 220, *' lx= 240, F= 1.06+ .94(1.41 - 1.06) = 1.39; F= 1.26+ .94(1.67 - 1.26) = 1.65. 1 K 1 QQ .-. for F= 1.5, D = 3.94% h = 220 + 20 ^ _, "I- = 228.5 1 .bo — 1. 0«7 Bisect the offset at P.C.C, measure 114.25 ft. along each curve, aud set the ends of transition-curve. Midway between these points and P. CO. offset -^^ X 1.5 = 0.1 ft.; these are all the points needed. The length of transition-curve at P.O. may be found in like manner, taking Z) = 5.06° and j^=2.0 ft. as arguments in interpolating in Table XVI. 162. To Insert Transition-curves at the Ends of Two Circu- lar Curves of Contrary Flexure united by a Common Tangent. In Fig. 75 let the located line be ABCE; the tangent BC must be shifted outward at B and C to the position UG , the relative size of offsets being determined by the nature of the ground. The points B' and C at which the tangents to circular curves r ii ; I Fig. 75. will be parallel to HQ will each move towards S a distance due to the increase of central angle, which increase equals B8H= a, for which we have tan a — (201) Let the offset at B' be F, and at C", F'. Then F= (R-\- BH) cos a - B. . . . . (202) F' = (A" + CG) cosa — E' (203] TRANSITION-CURVES. 139 F aud F', being now known, the transition-curves may be located. Example. — A G° curve and a 4° curve are united by a tangent 540 ft. long; 57/ for G^ curve = 4.5 ft.; CO for 4° curve = 3 ft.; B is at sta. 180, G at 185 + 40. Find F and F'. By (201), tan a = ^^ = .0139 = tan 0° 48'. o B will be moved forward ' = .133 stas. = 13.3 ft. to sta. 6 a 180 + 13,3, and G will be moved backwards '— = .2 stas. or 20 ft. to 185 + 20. By (202), F= (955.4 + 4.5)0.99990 - 955.4 = 4.4 ft. By (203), F' = (1432.7 + 3)0.99990 - 1432.7 = 2.86 ft. These values call for ly = 317.8 ft. for 6" curve, and h - 313.3 for 4° curve. Remark. — It will frequently be found that this problem allows the line to be thrown on better ground. Should the ground require tangent to be shifted inward, the curves must be sharpened by compounding to admit of the necessary offsets. 163. Having Run a Tangent which Falls Outside a Located Curve, to Find the Offset F for a Transition-curve Uniting them. lu Fig. 76 let the tangent be AB ; CF the located curve. Set transit at some point C, and bring telescope into tangent to curve. Measure CB and move to , B, where angle ABC must be measured ; or measure CH per- pendicular to AB ; then CII sm a = — . Now FG = R vers a ; or it is the mid-ordinate for twice a, and may be found from Table IX ; then F=CR-EO= CH-nveraa. . . . (204) The point E is found from C by the relation EC = jz- The transition-curve may now be located. 140 A FIELD-MANUAL FOR RAILROAD ENGINEERS. 164. Inserting Transition-curves in Old Track. — Sections 159 aud 160 afford the means of inserting transition-curves, of which 159 is theoretically the best, though from the amount of track disturbed it may be better to employ 160. Sometimes the method of 162 may be employed to advantage when the connect- ing tangent is short. For easing the curves at point of com- pounding, the method of 161 may be made use of. The offsets must necessarily be small if the new track is re- quired to occupy the old road-bed. It may be profitable to add to the road-bed when sufficient offset cannot be secured for sharp curves, though ordinarily much good can be accomplished even when the new track is restricted to the old road-bed. Unless the theoretical P.C., P.C.C., and P.T. have been marked by monuments it may be diflicult to retrace the old lines. If there is plenty of room, the terminal tangents may be prolonged to intersection and /measured, after which the degree of curve may be found by measuring around curve aud by ap- proximate measurements of the tangent distances ; then one or two assumptions and computations will generally suffice. In cuts and rough country the curve may be run out by setting transit in center of road-bed and measuring the deflection-anglea for a few points around the curve. After the transition-curves have been inserted permanent monu- ments should be placed at each end of transition-curve to guide the trackman in keeping up the proper superelevation of oute' rail. 165. Remarks on Tabular Interpolations. — The general inter polation formula given in algebra is in which t is any term, a the first term taken, p the number of; terms from a to t, fZi the first from a of the first order of differ ences, do the first of the second order of differences, etc. In ordinary linear interpolation all terms after the second are neglected ; in interpolating by second differences all after the third, etc. In Table XIV linear interpolation will answer for C and ordi- TRANSITION-CURVES. 141 narily for ^.though second dilBt'erences may sometimes be needed for ihe latter. In Table XV, A is a quadratic function of n, as shown by for- mula (15;i), while 5 is a cubic function of that portion that lias been retained. Hence A should be interpolated by second differ- ences, while theoretically B should be interpolated by third differences ; but as B is always quite small, its second and third differences will be too small to affect results, and linear interpola- tions may be made when any are needed. In Table XVI linear interpolations will generally suffice, though when ii^and y are large it may be necessary to use second differences. The examples of 158 and 161 illustrate the method of inter- polating in Table XVI for intermediate values of F and D Values of 7^ were first found for the given degree of curve and assumed values of ^i , so taken that the true ^i should be between them. From these assumed values of Ij and F, taken with the required F, the true h was found by linear interpolation. As an extreme case suppose F and 2/1 wanted for an 18^ curve when li = 408 feet. First write a few values of ^i and F so as to obtain the first and second differences. Z, y, rfi d^ F rf, ^2 400 81.43 20.63 8.08 2.08 420 89.51 0.35 22.71 0.10 8.43 2.18 440 97.94 0.35 24.89 0.10 8.78 2.28 460 106.72 27.17 By the interpolation formula, when ?i = 408, y, = 81.43 + is X 8.08 + 4^_zi.) X 0.35 = 84.62, F = 20.63 + ^% X 2.08 + '^^^, — - X 0.10 = 21, By linear interpolation, .y, = 84.60, F= 21.40. 45. 142 A FIELD-MAXUAL FOR RAILROAD EXGINEERS. Again, suppose ^i to be wanted when h — 430. By the formula, yy = 81.43 + M X 8.08 + IMLzii-^ X 0.35 = 93.68, or y, = 89.51 4- i^ X 8.43 + ^^^ — ^- X 0.35 = 93.68. CHAPTER V. *^ FROGS AND SWITCHES. Article ^4, Turnouts. A, Turnouts from Straight Lines. 166. A Turnout is a track used in leaving the main line. A Frog is placed at the intersection of main and turnout rails. a. The Gauge-line is taken as coinciding with inside face of rail. In makiug measurements between tracks the distance be- tween corresponding gauge-lines is what is wanted. b. The Gauge of track is the distance between gauge-lines of the rails of that track. c. The Point of Switch is the point at which the turnout curve begins ; for a point switch (split switch) this is at the head-block, while with a stub switch it is the length of the switch-rail back of the head-block, which is at the toe of switch. d. The Frog-point is at the intersection of the gauge-lines of intersecting rails, and lies a few inches in front of the blunt point of frog as manufactured. The angle formed by the intersecting gauge-lines is the Frog- angle. e. The Frog-number, J^, is the ratio of the axial length to the width of base of frog. In Fig. 77, Fig. 77. AE k CB~ w 143 144 A FIELD-MANUAL FOR RAILROAD ENGINEERS. Letting the frog-angle BAG ha F, the figure yields (205) (206) ^ ^ ^w 1 cot \F= ^ = 2N. f. The Lead, I, is the distance from point of switch to point of frog, measured along that main rail in which the frog is placed. In Fig. 78, CB = I. g. The Stub-lead, s.l., is the distance along main rail from frog- point back to a point where the turnout rail diverges from main rail an amount equal to the throw. In Fig. 78, KB = s.l. = I — length of switch-rail. 7i. The Throw, i, of switch-rail is the distance the point of a split switch, or toe of stub switch, is moved in opening or closing the switch. A distance of from 5 to 5f inches is needed to give necessary clearance for flanges. k. The Frog-distance, f.d., is the length of the chord of outev rail of turnout from the point of a split switch, or toe of stub switch, to the point of frog. 167. Given the Frog-number, iV", and the Gauge, g, of a Turnout from a Straight Line, to Find the Lead, I, and Radius, M, of Center Line of Turnout. A G E Fig. 78. In Fig. IS, AC = g, CB = I, angle ABC From the figure, I = g cot iF. But, (206^ cot iF = 2N. .-. I = 2glf. . = \F. . (207) FROGS AND SWITCHES. 145 From triangle OBG, (H + kf - (^ - iff)" = ^ = ^9'N\ whence 2gR = 4g^N\ .-. R = 2gN'' = IN. (208) 5730 Taking 22 = — yy-, inserting in (208), and solving for i), -=S <-> For g = 4iii. 8^ in. these formulas become ; = 9.42iV feet, (207') 72 = 9.42iV^'' feet, (208') 2> = ^ degrees; (209') and for^ = 4 ft. 9 in., I = 9.5iV, (207") R = 9.5N\ (208") i>=|| (209") If the frog-distance AB is wanted, we have AB = |/Z* + ^', or f.d. = g ViiV^TT = 9 cosec IF. ... (210) Example. — Find I, R, D, and f.d. for a No. 8 frog and 4.75 feet gauge. By (207"). Z = 9.5 X 8 = 76.0 ft.; " (208"), i? = 9.5 X 64 = 608 ft.; " (209"), D=^ = r 25'; 64 " (210). f.d. = 4.75 t/257 = 76.14 ft. 146 A FIELI>-MA?^UAL FOR RAILROAD ENGINEERS. 168. Given 7? (or D) and g, to Find N, I, and F. From (208) and (209), , /R , /5730 53.52 2g y 2gD ^^- ' From (207) aud (211), l=2gN = 2g\/^ = |/2^ = 107 Y^. From (206) and (211), cot|^.2i.= 2/|=|/f = (211) 107 2ff ^ g \/gB F m&y also be found from triangle OBC. Fig. 78 : cos7^=4=^ (212) (213) (214) 169. To Find the Length of Switch-rail, S, when the Frog- number, 3", the Throw of Switch, t, and the Gauge, g, are Given. In Fig. 78, by geometry, HG ^' 2(i? 4- Iff) + HG' Neglecting the HG in denominator as small, JIG= -^» In like manner, KL = 2(i? + yy 2{R - W Writing AG = Aff= CK — S, and taking the mean of de- nominators, ^- 2R' whence S = y^Rt = 2JV Vgi: ... * (215) wf 7? 5730 Writmg R = -^, , 8= a/ 2t^^= 107 |/- (216) FROGS AND SWITCHES. 14' 170. Given the Main Frog-number, N, to Find the Num- ber, Ni . and Lead, ^, , of Crotch-frog for a Turnout from Both Sides of Straight Main Track. Ill triangle OCII, Fig. 79, re- memberiug that E = 2gN^, cos ^Fi = — R AN^ (217) A' + 1(7 AN'+l Tlfeea, by (206), iv^, =icotii^,.. . . (218) |_.4^_7^;:r:^^c^'i From the figure and (205), -^ — ^ I, ^ Rtau IF, =^.=g^^; (219) also. = ^r |/2iV"M^~i. .... (220) Equating these values of li and solving for Ni gives N, V2iV"^+i (221) If the \ in denominator be neglected as small compared "with 2N\ (2C1) becomes N, = -^= 0.707iV: (222) |/2 If in (220) we neglect the | under radical, tbere^results i, = g]^ i/2 = lAUgN = O.IOIl. . . . (223) The distance between main and (Totch frogs measured along main rail is I- I, =2gN- g V2JV'+i (224) or, approximately, I -Uz= 2gN - \AUgN = O.^SGgJST = 0.293i. . (225) . 148 A FIELD-MANUAL FOR RAILROAD ENGINEERS. 171. To Find the Radius, R, of Turnout and Lead, ^i , of Orotch-frog in Terms of the Crotch-frog Number, Ni From (222). iV^^ = 2N^\ Insert this in (208) and (219), giving E=2g. 2N^-' = 4gNx\ (226) h = ^-^^ = 2gN, (227) Remark.— In general the frogs kept in stock by manufacturers do not afford suitable combiuatious of numbers for double turn- outs. For instance, the theoretical number of crotch-frog for a number 8 main frog is, by (221) or (222), iVi = 5.66, and we should be compelled to use a number 5^ or 6 for the crotch-frog; this would necessitate a different rate of curvature from crotch to main frog than from head-block to crotch. 172. Given the Numbers of Middle Frog, iV, , and of Main Frogs, iVand N', to Find the Radii li^ from Point of Switch to Crotch-frog, and R and M', from Crotch to Main Frogs. In Fig. 80 we have, by (226), 0,N=B,=4gNx\ and, by (227), NO =U= 2gN,. Now if Fx , F, and F' are the an- gles of the frogs Ni , JV, and N', the angle COH = F- ^F, , and % CHQ=F-l{F-\F,) = \{F+\F,). Since CG = \g, the triangle GEO yields OH = \g cot \{F + ^F,). . (228) But, by trigonometry, ..t (xw-x- XW^ - l -taniF . tanjJ^. cot (ii^+ IF.) - -_-^^^-— p^ FROGS AND SWITCHES. 149 Assume tan ^Fi = ^ tau |F, , aud write and after simplifying and reducing, The last term is (juite small, rarely amounting to as much as one inch, aud may be neglected ; then 2gNN, _ hlf _ m, From the triangles LCO and KHO, (B + Ig) cos IF,-{H -f \g) cos ii^ = \g, whence In like manner for the curve CE, ME = y cot K^' + i^O = ^/^ _^ ^, . . . (232) i2' •+ 1^^ = -~-^- — . . . (363) -^ 2(cos ^Fi — cos F ) Example. — Given iV", = 6, iV^= 8, and N' = 9, to find the lead li, the distances GH and ME, and radii R, Ri , and R', g being 4.75 ft. By (226), i?i = 19 X 6^ = 684 ft., an 8° 23' curve. By (227), ^1 = 9.5 X 6 = 57 ft. By (230). GH = — ^f^^^ = 32.8 ft. By (232), ME = 24.4 feet. 150 A FIELD-MANUAL FOR RAILROAD ENGINEERS. By (231), R = a 10° 28' curve. By (233), li' = a 6° 32f curve. 4.75 2(cos 4° 46' - cos 7" 9) 2.38 = 547.4 ft., 4.75 2(cos 4° 46' - cos 6° 22 ) - 2.38 = 876.4 ft. 173. Given the Number, iV^, of the Two Main Frogs and the Gauge, g, to find the Crotch-frog Number, iV, , its Lead, li , and the Radius, lii , of Curve through Crotch when the Double Turnout is to Same Side of Straight Main Track. In Fig. 81 the frogs at B and G are of the same number, and may be taken as falling on the same straight line through the center 0. Angle 0,G0 = 90° - OGL = F, and the triangle 00i(r is therefore isosceles; hence OiG = 0,0 = OA - 0,0 = lOA, or whence (234) Now, by the same reasoning as in 167, 2gNi'^ ~ R, , whence ^. = /f:= 'R 4g (235) FROGS AND SWITCHES. 151 Neglecting the J under radical and willing B = 2gN' gives N, = ^_ = .707iV, (236) 1/3 which is identical with (222) for turnouts to opposite sides. For ^Cand EB, as in 167, U = 'igN, and I = 2gN. Hence GB= I- I, = 2g{]Y- J^J (287) Example. — Find Wi, i?i , and l — li where iV^ = 9 and g = 4.75 ft. By (208"), ^ = 9.5 X 81 = 769.5 ft., a 7° 27' curve. " (234), Bi = 384.75 - 1.19 = 383.56 ft., a 14" 56' curve. " (236), N, = .707 X 9 = 6.36. " (237), GB = 1 - h =9.5x 3.64 = 25.08 ft. Remark. — It may now be seen that the proper combination of frogs for a double turnout to opposite sides applies also where the turnouts are to same side of straight main line. Also they apply to turnouts from opposite sides of curved main line when its radius is not less than that required by main frog for straight track. 174. Given the Number of Main Frogs, iV", and of Crotch- frog, Ni , to Find the Radius of Curve between Frog-points of a Double Turnout to Same Side of Straight Track. Fig. 82. In Fig. 83, O-iG - B^-\- \g, and the chord CO must be deter- mined. The frogs at B and G being of the same number, 0,00 = GOOi = F and CO,E = F,. 152 A FIELD-MANUAL FOR RAILROAD ENGINEERS. Draw (?^ perpendicular to EB; then in triangle BGH QH = g cos F. Draw O^L perpendicular and (?Z parallel to EB\ from tri- angles OaG^-K'and O^CL, (ij, ^ ^^)(cos Fi - cos 'iF) = KL - GH = g cos F, whence It2 + hg = g cos F (238) (239) cosi^'i -cos2i^" • * From triangle O^CQ, since GO^G = 2F - Fi, CG = 2{R, + ^g) sin U^F - F,). . . Example.— Given N = S, JSTi =6, and g = 4.75, to locate the turnout. By (208), R = 608 ft. ; Ri = 342 ft. By (238), Bi + y = 274.5 ft. By (239), CG = 22.8 ft. 175. Given the Frog-number, N, the Gauge, g, and Distance, p, between Centers, to Unite Main Line with a Parallel Siding when the Reversing-point is at Frog-point. In Fig. 83, BOx = Ri - \g and BE are required. In triangle 50,^, BO, = R, - \9, EO, - R, + ^g ~ p, and angle BO,E = F. By trigonometry, FROGS AND SWITCHES. 153 (Br - y) + (Jg. + y-p) ^ tan i(180° - F) (i?i - ig) - (Ri + y -P) tau ^F 2R^p^cotiF p - g tan ^F whence Ei .= 2(p - g)N'' -\- \p, (240) BE = (i?x -ig)smF. (241) T>7f 7 From the similar triangles ABG and BCE, = - , from p-g g t • which BE = ^P^^ = (? - l]l. .... (242) g-l). . . . . Example.— Find Bi and BE when iV = 8, p = 12.35 ft., g = 4.75 ft. By (240), El = 15.2 X 64 + 6.2 = 979 ft., a 5° 51' curve. By (207"). i = 9.5 X 8 = 76 ft. By (242), BE = (2.6 - l),x 76 = 121.6 ft. Remark. — If space requires that the turnout get away from main line more rapidly than by the above method, we can assume the second radius equal to or less than the radius of turnout and find the reversing-point by 131, and then compute BE. 176. To Lay Out a " Ladder" Track In yardwork a number of parallel sidings may be conveniently connected with the main line by means of a ladder-track. In Fig. 84, if the frog-number iV and the distance p between center lines of track are given, it is only necessary to determine the distances BC, CE, etc., between frog-points, and BK, CL, etc., between point of switch and point of frog. From triangle BCQ, BC = -r^ = p cosec F, (243) sm ^ BK= BO - KG = p cosec F - 2gN ; . . . (244) 154 A FIELD-MAKL'AL FOR RAILROAD ENGINEERS. or, since cosec F — iV-|-— r^, (see 186,) BG =pN-\- (243') BK= (p-2g)N + V (244') Example.— For a No. 8 frog find 5Cand BKvi\ien p = 12.8 ft. and^ = 4.75 ft. By (243), BC = 12.8 X 8 + -^ = 102.8 ft. By (244), BK= 3.3 X 8 + 0.4 = 26.8 ft. B. Turnouts from Curves. 177. Given the Radius of Main Curve, the Frog-number, and the Gauge, to Find the Radius and Lead of Tvurnout from Concave Side of Main Line. In Fig. 85, AB is tlie outer rail of turnout, CB the inner rail of main track. In triangle OAB, since O2BA = GAB, OBA - OAB = F, OB A + OAB = 180'^ - 0, and OA ^ R+ ^_g. OB = R - Ig. FROGS AND SWITCHES. 155 Then, by trigonometry, (i? + ha) -\-iIi - jg) _ tan ^ (180 - 6) {R + \g) - (H- W ~ tan ^F cot^e tan hF' 9 7? 7? Reducing, cot |9 = y ^an ^2^ = ^. Then I = BC = 2{R - y) sin 49. . (245) (246) If the length of AB is wanted, we can sliow that the angle ABC = \F; and by solving the triangle ABC, since ACB = 90° + i/, g cos ^0 AB sinii^ (247) To find Ri , from triangle dAB, 2(.ff2 + Ig) sin liF + 6) = ^5. (248) Or, in triangle BO^C, {R. +_i£)_+ {R-^-_hu) ^ tan i[180 - (/^+ «)] (/?:;+ is') - (^^2 - \g) tan ^i^ coti(F+e) tan \F ' 156 A FIELD-MANUAL FOR RAILROAD ENGINP^ERS. Reducing aud solving for i?,, ^'2' tan ^F = ^ • ^«^ i^'- cot K^+ fi). . (249) But, from trigonometry cot i(^+ 6, = cot ,iF+ m = ' :jpf,::]t - Substitute this in (249) and write cot ^F = 2]Sr, tan ^F = ^r^, tan ^0 = — tan iQ - ^^ and reduce ; then For 2^iV* write i?i , the radius of turnout from straight track, and neglect the ^g in numerator as small compared with B; then «' = 4^ <2«» Now write 5730 „ 5730 „ 5730 and reduce, yielding Da = D + D, (253) Formula (252) affords an easy method of finding the degree of turnout curve, or, if preferred, the radius may be first found by (251). Draw Oi?to the mid-point of CB ; OE does not differ greatly from OB or OC ; so, if we write OE = R — ^g, there results aJSf a'^N 1 = 2{B- Ig) tan \0 = 2{R - lgf~ = 2gN - V- (253) The last term is quite small, even in the most extreme case likely to arise in practice ; for a turnout from a 6° curve with FROGS AND SWITCHES. 157 number 8 frog it nmoimts to only 2| inches ; neglecting it, we may write, as for straight main track, I = 2gN. (254) Example. — Turnout from inside of a 4" curve, ]V=S, 2 = 4° + 9^ 26' = 13" 26', for which E^ = 426.8. By (254), I = 76 ft. 178. Given the Frog-number, the Gauge and Radius of Main Curve, to Find the Lead and Radius (or Degree) of Turnout from Convex Side of Main Line. In triangle AOB of Fig. 86, ^ + ^ = 180° - 6, A -B = (180° - O^AB) -{ISO" -O^B A- F) = F. Bj'^ trigonometry, ifi^W^-j- {R - Ig) tan |(180 -0) 2R 9 tan IF or whence cot IB tan AF 27? cot IB= ~ tan IF= R (255) 9 - 9N From triangle OCB, l=CB = 2{R + Ig) sin |(3. (256) Assuming OE — R ^^ \9, i = 2iR + Ig) tan 10 Neglecting the last term as small, as in 177, l = 2gN,, . . . which is the same as (207). In triangle CO,B, 0^ = F - 0. (258) •. tauiOa = tan {IF - |0). 158 A FIELD-MANUAL FOR RAILROAD ENGINEERS. We may now follow the same Hue of reasouing by which (231) was derived, or more simply by assuming the tangent of the difference of two small angles equal to the difference of their tangents; that is, tan ^02 = tan IF — tan ^0. fjN Now it can be easily shown that tan \0-i. = ir- ; therefore Hi gN _ l__gN Ri~ 2N E* whence 7?T? from which R^ = — '- (259) it — R\ __^ . „ 5730 „ 5730 „ 5730 ,' , ^ Write ^2 = -fr-, Ri = -yr-. R = —f^ ' ^^^^ ^"^^^ ^or B-x. Z>2 = D, - A (260) in which D^. is the degree of turnout from straight track. Example. — Turnout from outside of a 4^ curve, iV = 8, g = 4.75. By (208"). i?i = 9.5 X 64 = 608 ft., a 9° 2G' curve. By (260), 7), = 9° 26' - 4° = 5" 26', for which R = 1054.9. By (258), Z = 9.5 X 8 = 76 ft. From (255) we have, by inverting, tan 19 = jj|-^ = tan 1° 31' auQ, ?jy (256), I = 2870 X sin 1° 31' = 75.97 ft.. a difference of only 0.03 ft. from the value given by (258). V V 179. To Find Theoretical Length of Switch-rail when the Turnout is from a Curved Track. A common tangent being drawn at the switch-point, we shall have, as in 169, for offset from tangent to main curve, _ S^ ^~ 2R'' FROGS AND SWITCHES. ^ 150 the offset from taugent to turnout is . yi When the turnout is from concave side of main line, therefore whence S = j/ ^J^^^ (261) r It — lit WritiDg R=^,R, = '^, and reducing, When the turnout is from convex aide of main line, t = y^-\'y = -^[j^ + -^y whence ^ = ^/-^iZ^ ; (263) ^ R-\-M^ from which «="V3i^'«VJ- • • • '^""^ In (262) and (264) Z>i is the degree of turnout from straight track, and, as these formuhis are identical with (216), it is seen that the theoretical lengtli of switch -rail on turnouts from curves is the same as on turnouts from straight line. Example.— Find S when t = 0.42, ^ = 8, ^ = 4.75. By (208"), R, = 608 feet, for which j9, = 9° 36'. 160 A FIELD-MANUAL FOR RAILROAD ENGINEERS. By (216), (362), or (264), S= 107i/-:i? = r 9.43 22.6 feet. 180. Given the Distance p between Center Lines of Curved Main Line and Side Track, the Frog-angle, F {or Number, iV^), and Gauge, g, to Find the Radius and Central Angle of Curve beyond Frog-point, First Case. — Turnout from outside of main line. In Fig. 87, is the center of main curve, Oi the center of curve "Whose radius is required. In triangle BOG, C0 = R + p-y, BO:=^R+y, By the same reasoning as in 177, , ^ 2R + p^ , „ 2R + P cot ^6=-:: — ^tan^ii^^ (265; p - g - 2N{p - g)' In triangle OOiB, OiB — R^ — \g; then, by the lavs^ of sines, sin Q Ri-k9 = (R + i9). Also, sin(i<'+ 6) BE=2{R-^^g)&m^B (266) (267) FROGS AND SWITCHES. IGl Second Case. — Turnout from inside of main track. 0,J •o Fig. 88. In Fig. 88, we have from triangle BOE, reasoning as in 178, cot ^9 = ?^-^li'tan \F = _^-f ~^. ; . . . (268) p - g 2Mp - g) and from triangle OBOi, Also, EC = 2(i? - ig) sin ^G, (270) and 5^ = 2(i?, - ^gr) sin 4(i^ - 0). . . . . (271) When 9 is greater than F, sin (F— 6) is negative, and center Oi falls on same side as 0, and 162 A FIELD-MANUAL FOR RAILROAD ENGINEERS. ii^ + i9 = ~^^-XR - m . . . (372) sin {e-Fy BE = 2(i?i + y) sin i(e - F). . . . (273) C. The Stub Lead. 181. When the frog-numher exceeds seven, the length of switch-rail required to give the necessary clearance at heel be- comes greater than is allowed in practice. To overcome this ditficulty slightly more curvature is given the switch-rail ; more- over the physical point of switch is necessarily some distance in advance of the theoretical point. The distance from heel of switch to point of main frog will then be the same as from head- block of stub switch to main -frog point, and is termed the Stub Lead. If to this distance the length of switch-rail be added, we get the distance from the head-block of a point switch to the point of main frog, which is the Short Lead required in practice. 182. Given the Throw, i, the Gauge, g, and the Frog-number, N, to Find the Stub Lead, s.l. In Fig. 89, KB is the stub lead required; GIf= KL, the throw. Fig. 89. From (207), 1= CB = 2gN, and from (215), s= CK=2N Vgi- From the figure, KB= GB - GK, or 8.1. = 2N{g - Vgt) (274^ FROGS Al^I) SWITCHES. 163 Formula (274) may be employed for turnouts from curves as well as straight lines, since it was shown that the formulas from which it was derived may be employed even when the curvature of main track is considerable. Below is a table of values of (g — ^y'gt) for some of the more common values of g and t. TABLE OP VALUES OF g — |/JT. 3 Feet Gauge. 4 Feet 8i Inch Gauge. 4 Feet 9 Inch Gauge. Throw. 9 - Vgt. Throw. 9- Vgt. 1 Throw. g - Vgt. Inches. 3 4 Feet. 2.13 2.06 2.00 Inches. 5 5i Feet. 3.308 3.239 3.206 Inches. 5 5| Feet. 3.. 343 3.275 3.242 Example. — Find the stub lead for N =^d>, g = 4.75 ft., i = 5 inches. From the table, g - \/gl = 8.343 ft.. and, by (274). s.l. = 16 X 3.343 = 53.49 ft 183. The Turnout Table on the next page gives the frog-angles, the radius of center line of turnout from a straight track and its degree, the theoretical lead, the theoretical length of switch-rail for t = 5 inches, and the stub lead for certain values of t. The frog-numbers given cover all the usual cases. Suppose it required to find the short lead for a No. 9 frog and 5-inch throw when the gauge is 4 ft. 9 inches and the length of switch-rail 18 feet. From the table the stub lead is 60.17 feet; hence the short lead is 60.17 + 18 = 78.17 feet, as against 85.50 ft. for the theoretical lead. Inspection of the table will show that it makes no very great dilierence in the tabular quantities whether the gauge be taken ^s 4 feet 8^^ inches or 4 feet 9 inches. However, the numeiical coefficients in the formulas involving g are somewhat simpler for the latter value. 164 A FIELD-MANUAL FOR RAILROAD ENGINEERS. TURNOUT TABLE FOR STRAIGHT TRACK. 4 FEET 8}4 INCH GAUGE. Degree of Turn- out. Theoret- Stub-lead for a Throw Frog No. Frog Angle. Theo- retical Lead. Turn- out Radius. ical Switch- rail for of 1 t = 5In. 5 In. 5]4 In. 5% In. o / feet feet o / feet feet feet feet 4 14 15 37.67 150.7 38 2 11.20 26.46 25.91 25.65 5 11 25 47.08 235.4 24 21 14.01 33.08 32.39 32.06 5]^ 10 23 51.79 284.9 20 7 15.41 36.39 35.63 35.27 6" 9 32 56.50 339.0 16 54 16.81 39.70 38.87 38.47 ei4 8 48 61.21 397.9 14 24 18.21 43.00 42.11 41.68 1 8 10 65.92 461.4 12 25 19.61 46.31 45.35 44.88 ^14 38 70.63 529.7 10 49 21.01 49.62 48.59 48.09 H" 1 9 75.33 602.7 9 301^ 22.41 52.93 51.82 51.30 81^ 6 44 80.04 680.4 8 25 23.81 56.24 55.06 54.50 9 6 O-J 84.75 762.7 7 31 25.21 59.54 58.30 57.71 9^ 6 J) 89.46 849.8 6 45 26.61 62.85 61.54 60.90 10 5 44 94.17 941.7 6 5 28.01 66.16 64.78 64.12 11 5 12 103.58 1139.4 5 2 30.81 72.78 71.26 70.53 12 4 46 113.00 1356.0 4 131^ 33.61 79.39 77.74 76.94 13 4 24 122.42 1591.4 3 36 36.42 86.01 84.21 83.36 14 4 5 131.83 1845.7 3 6 39.22 92.62 90.69 89.77 15 3 49 141.25 2118.7 2 42 42.02 99.24 97.17 96.18 4 FEET 9 INCH GAUGE. ( Degree of Turn- out. Theoret- Stub-lead for a Throw Frog No. Frog Angle. Theo- retical Lead. Turn- out Radius. ical Switch- rail for of feet < = 5Iu. 5 In. 51^ In. feet b% In. feet o / feet o / feet feet 4 14 15 38.00 152.0 37 42 11.26 26.74 26.20 25.94 5 11 25 47.50 237.5 24 8 14.07 33.43 32.75 32.42 5J^ 10 23 52.25 287.4 19 .56 15.48 36.77 36.03 35.66 6 9 32 57.00 342 16 46 16.88 40.12 39.30 38.90 6>^ 8 48 61.75 401.4 14 16 18.29 43.46 48.58 42.15 7 8 10 66.50 465.5 12 19 19.70 46.80 45.85 45.39 7J^ 1 38 71.25 .534 4 10 44 21.10 50.15 49.13 48.63 8 7 9 76.00 608.0 9 25^ 22.51 53.49 52.40 51. 8T 8^ 6 44 80 75 686.4 8 21 23.92 .56.83 55.68 55.11 9 6 22 >35.50 769.5 7 27 25.32 60.17 58.95 58 36 9J^ 6 2 90.25 857.4 6 41 26.73 63.52 62.23 61.60 10 5 44 95.00 950.0 6 2 28.14 66.86 65.. 50 64.84 11 5 12 104 50 1149.5 4 59 30.95 73 . 55 72.05 71.32 12 4 46 114.00 1368.0 4 11 33.77 80.23 78.60 77.81 13 4 24 123.50 1605.5 3 34 36.58 86.92 85.15 84 29 14 4 5 133.00 1862.0 3 4V-0 39.39 93.60 91.70 90.78 1 15 3 49 142.50 2137.5 2 41 42.21 100.29 98.25 97.26 L. 1 FROGS AND SWITCHES. 165 184. To Stake Out a Turnout. — If the positiou of bead- block is given, fix tlie frog-point by the foregoiug table, remembericg that it may be used for turnouts from curves, as well as from straight lines, without material error. To locate the rail between head-block and point of switch it is sufficient to do so by offsets from main rail. Consider the equa- tion (36) for tangent offsets. z = \ii^B for straight main line. z — \n\I)x ± Z>) for curved main line. At frog-point z = f/, and 7i = /?, ; hence g = ph'A or ph^Di ± D). At mid-point of curve (practically mid-point of lead), n = \ni , and 2 = I . In.'D, or I . iWi^(i>a ± D) = y. . (275j When n — \ni , z = l. j\n,W, or I . j\7H\D, ± D) = ^\g. . (276) When n = f ?ii , z = l.j%7H^B, or i . j\n\D, ± JD) =. j^,g. . (277) These formulas are for the theoretical lead, and afford an easy method of locating the outer rail of turnout with all the accuracy needed in practice. 185. Curving Rails. — In bending rails for curves the proper curvature is determined by measuring the mid-ordinate from a cord held against the inside face of rail-head. This ordinate may be determined by (18), in which 7i is the half-length of rail divided by 100. For a 30-ft. rail, M - 1(0.15)^7) = 0.0196Z> = 0.02i) (nearly). . (278) From (209'). i> = ^; and from (209"), ^ = ^- Inserting either of these values in (278) gives J/ =^,, nearly (279) 1G6 A FIELD-MANUAL FOR RAILROAD ENGINEERS. When the turnout is from a curve compute M from (279), and the mid-ordinate for a rail 30 ft. long on main curve by (278); then the mid ordinate for turnout lail will be the sum or difler- ence of these values according as the turnout is from concave or convex side of main curve \ Article 15. Crossovers. 186. To Locate a Crossover betw^een Parallel Straight Tracks w^hen the Frog-number, the Distance, p, between Cen- ters, and the Gauge are given, inserting a Tangent between Frog-points. A / i •nB / ' f G /f- F / ' i H> /^ I N ^v/ / ^>--~^ ci / M K ^\^^ P 1 / Fig. 90. In Fig. 90 it is required to find OB = KG = I, MK and NO, also HK=k. In the triangle BPM, BM=p — g\ then BE=k = BP-EP 1 or k = {p — g) cosec F — g co\. Fy . . (280) and MK=MP- KP, or MK = (p — g) cot F— g cosec F. (281) Jfrom triangle OBC of Fig. 78, ^ 00 R cos F = ^^^ = y _2R- g OB B+ig 2B + g (a) PROGS AND SWITCHES. 167 In (a) write R = 2gN^ by (208), giving 4gN^ + g 4JV* + 1 From Fig 78, triangle OBG, . ^ GB I 21 Writing I = 2gN and B = 2gN^ gives From trigonometry, taking the above values of sin F and cos^, Inserting these values in (280) and (281), k = {p-2g)N+-^ (282) MK=ip-2gW-^ (283) By (207), 0B = KG=l = 2gN; therefore IiC= 21 + MK=4gN+(p- 2g)N - ^, or NG={p + 2g)N-^=l-{-p(N ^ }i\ , . (284) Example.— Find k and MK for a No. 8 frog when ;> = 13 ft. and^ = 4.75 ft. By (282), A; = 3.5 X 8 -r 0.4 = 28.4 feet. By (283). ifii: z= 3.5 X 8 - 0.4 = 27.6 feet. 168 A field-mani;al for railroad engineers. 187. To Lay Out a Crossover in the Form of a Reversed Curve. When j9 is large, or for otlier reasons it is desirable to get away from main track more rapidly than b}' the foregoing method, we may lay out the crossover in the form of a reversed curve. A A 0, Q ■0\B X "">^^'"'^ 1 1 M / ' "^ I 1 1/ K Fig. 91. In Fig. 91 it is required to find QB = HE and LH. Find OB = HE =1 by (307), and the radius 00 = 0,C by (208). Then, from (113), we have ME = 2R sin a. The angle a is given by (112). Then LH = 2R sin a - 21. (285) 188. To Lay Out a Crossover when a Fixed Length of Tan- gent must be Interposed between Points of Reversal of Curvature. From the given frog-number determine the radius by (208) then the problem may be solved by 132. 189, To Lay Out a Crossover in the Form of a Reversed Curve -when the Tracks to be Joined are Ciarved. In Fig. 92 let the notation be as shown. Let OM = B, OiM = Bi, O-iP = O^C = B^. 0,0^ = B, + B^ = a, 00^ = B-^ p - B^ = b, FROGS AND SWITCHES. 109 00, = R-\- Bx= C, ^{a + b -\- c) = 5. Fig. 92. Then, from trigonometry, cos hA = Y jc cos ^B ac (286) (287) Ml is determined by 178, and R^ by 177, while R and p are given to begin with. The angle {A + B) determines the length of arc CP, and angle B the length of arc MP. The angle 6 is given by (255), and 0. by (245). Hence angle GOH, which determines the arc G'// measured along main track between frog-points, is GOH= A - (9 + 0i). The frog-numbers at G and B need not be equal, only providing that P falls between G and B. 170 A FIELD-MANUAL FOR RAILROAD ENGINEP:RS. Article 16. Crossing-frogs and Crossing-slips, k. Crossing-frogs. 190. When two tracks intersect each other four crossing-frogs are required at the intersection of the two sets of rails. The four frogs are sometimes called a set of crossing-frogs. 191. To Find the Length of Rails Intercepted between two Intersecting Straight Tracks w^hen the Angle of Intersection and the Two Gauges are given. In Fig. 93, from triangle ABH, AB= EG = g cosec F ; . (288) and from triangle AEG, AE = EG = g, cosec F. (289) Fig. 93. 192. Given the Angle of Intersection, a, made by the Center Lines of a Straight and Curved Track, the Gauges g\ and g, to Find the Angles of the Set of Crossing-frogs. In Fig. 94, from the triangles OS^Tand OAH, '■ . ' (R + y) cos F=R cos a + ^gi. .'. cos 2^= „ ' ^^ . . (290) ii + y In like manner, R cos a — ^gi cos Fi = i?+i^ cos Fa = B-hg R cos a -f- ^gi (291) _ R cos a — \gi ,nr.n^ cos F^ = — ^-^^. . (292) (293) N M HK 11 Fig. 94. From triangle BOC to find the chord BG. BC = 2(i? + Ig) sin 1{F, - F). ... (294) FROGS AND SWITCHES. "I f^"? t L Similarly, OE = 2(7? - Ig) sin ^{F, - F^) (295) From triangles EOM aud COL, we have EG= ML = (R+ y) sinFr - {R- ^g) sin F^. (296) In like manner, GB = JYK = {R+ hg) sin F - {R - y) sin Fs. (297) 193. Given the Angle of Intersection, a, made by the Center Lines of Two Curved Tracks, their Gauges, g and 91 , to Find the Angles of the Crossing-frogs. In Fig. 95, OA = R, 0,A = R, , and angle OAOi = a of the triangle OAOi are given; whence OOi may be determined. In triangle OBOi the side OB = R + y, 0,B= R, + ^gi, and OOi — k are known, from which we can determine the angle OBOi = Fig. 95. F- In like manner from the triangle OCOi determine Fi, and from triangle OEOi find F^. F3 may be found from triangle OQOi. To find the chord GB first find angle 0. BOxO from triangle BOiO, and angle GOiOfrom triangle GOyO; then GB = 2(i?i + ^^0 sin ^GOiB. (298) In like manner, EG = 2(i?i - y,) sin |JS'0, G, (299) BG = 2{R + Ig) sin ^BOG, . (300) GE - 2(i? - Ig) sin ^GOE. . (301) When the tracks intersect, as in Fig. 90, tlie solution is evidently similar to Fio. 96. the foregoing. 172 A field-maxual for railroad engineers. B. Crossing-slips. 194. A Crossing-slip is fin arrangement of switch-rails, in connection with a set of crossing-frogs, to connect two tracks intersecting at a small angle. 195. Given the Angle of Intersection of Two Straight Tracks, to Find the Length and Radii of Curvature of Slip-rails. In Fig. 97 determine EA and AB by 191 ; then assume GE or BH (according as EA is less or greater tlian AB) as small as the crossing- frogs will permit. Draw the radii HO and GO; AH = AG = k h the known tangent for the central angle F. Hence OG = B+y = AH cot IF =2kN, (302) 0L = R-y = 2kN - g. (303) For the theoretical length of rails, OH={R+\g) X ^1^ X i^° = {R-V y) X ^3. . . (304) (305) 196. Given the Angle of Intersection made by the Center Lines of a Straight and a Curved Track, to Find the Radii and Length of Slip-rails. FiKST Case. — Slip-rails inside main curve. In Fig. 98 determine the angles F and Fi at B and Chj 192. Then assume KC as small as constructive reasons will permit. Now sin^^-OC =-^ EG (306) b = BOK = (F, -F)- KOC, . . . (307) c = KO,H= F+h = F,- KOC. . . (308) FROGS AND SWITCHES. 173 From 129, formula (100), B. + ^g^O,H= <-^ + M "=°^ ^ - ""' '\ . (3091 ^ 1 — COS c Fig. 98. For the leugths of slip-rails, HK={R,+ ig) X EL = (i?i - ig) X 57.3 57.3 • • (310) (311) Second Case. — Slip-rails outside main curve. Find the angle F^ at S, Fz at O, and GS by the methods of 192. Assiinie ^.riV^ as small as constructive reasons permit, and calculate angle GON, as in (30G) Then NOS = F^ - F^ - GON, d= F^-i- GON =^ F^ - NOS. By 129, formula (106), R^^ \g = O2M = {R — ig)(cos d — cos F^) 1 — cos d (312) The remainder of the solution is similar to the first case. 197. Given the Angle of Intersection betw^een the Center Lines of Two Curves, to Find Radii and Length of Slip-rails. First Case — Slip-rails on concave side of curves. In Fig. 99 take JJJ as small as constructive reasons permit. Join L with . .hen 174 A fip:li)-manual for kailroad engineers. Determine angles COO,, 00,0, and side 00, by 193. Make LM = KO, ; then and MOO, = LOC+ COOu Fig. 99. In triangle MOO, two sides and the included angle are now known, and tbe triangle ma}^ be solved. On is tbe center of slip-rail curves. and O^MO, = 0^0, M = MOOi + M0,0, MO^O, = 180 - W^MOx. From the isosceles triangle OiO-^M, in which Oi iff and the three angles arc known, (314) MO, = 2 sin IMO^Oi Then B2 + l9= O^L = i?i + i = 5730 -^ R, and n = a:, in formula (86). 90.1 91.0 90.4 89.8 89 2 0.50 1.125 0.50 0.125 89.60 89 875 89.90 89.675 89 2C CONSTRUCTION. 181 Example 2. — A + 0.3^ meets a + 1.1^ grade at sta. 312, whose elevation is 155 Fiiul correctious. Take a = 0.2 in this case; Iheu I = + 0-3-(+l.l )_o 2 X .2 lions. The corrected grade heights, etc., will be as follows = 2 sta- sia. 310 311 312 313 314 154.7 155.0 156.1 157.2 0.1 0.4 0.1 0.0 154.8 155.4 156.2 157.2 Original elevation ... 154.4 Corrections 0.0 Corrected elevation 154.4 The reason for adding the corrections in this case will be evi- dent from a figure. The table below gives the corrections in feet, for certain alge- braic differences of gradients and lengths of curve, at intervals of 50 ft. each way from the vertex. When the difference of gradients is plus, tlie correction must be suhtracted from the original grade elevation ; when llie difference is minus, the cor- TABLE OP CORRECTIONS FOR VERTICAL CURVES. 1 ... -i^ Algebraic Diflferenc of Grad eiits. Kate of Change o Grade pe Station. Distance from Vertex in Feet. 50 100 150 0.01 200 250 300 350 400 0.3 0.075 0.15 0.08 0.04 4 .10 .20 .11 .05 .01 0.5 .125 .25 .14 .06 .02 0.6 .15 .30 .17 .08 .02 0.7 .175 .35 .20 .09 .0;! 0.8 .20 .40 .23 .10 .03 9 .225 .45 .25 .11 .03 1.0 .25 .50 .28 .13 .03 11 .1833 .83 .57 .37 .21 .09 .02 1.2 .20 .90 .63 .40 .23 .10 .03 1.3 .2167 .98 .68 .44 .24 .11 .03 1.4 .2333 1.05 .73 .47 .26 .12 .03 1.5 .25 1.13 .78 .50 .28 .13 .03 1.6 .2667 1 .20 .83 .53 .30 .13 .03 1.7 .2833 1.28 .89 .57 .32 .14 .04 1.8 .30 1.35 .94 .60 .34 .15 .04 1.9 .2375 1.90 1.46 1.07 .74 .48 .27 .12 .03 2 .25 2.00 1..53 1.13 .78 .50 .28 .13 .03 2.1 .2625 2.10 1.61 1.18 .82 .53 .30 .13 .03 2.2 .275 2.20 1.68 1.24 .86 ..55 .31 .14 .03 2.3 .2875 2,. 30 1.76 1 29 .90 .58 .32 .14 .04 2.4 .30 2.40 1.S4 1.35 .94 .60 .34 .15 .04 2.5 .3125 2.50 1.91 1.41 .97 .63 .35 .16 .04 2.G .325 2.60 1 1.99 1.46 1.02 .65 .37 .16 .04 182 A FIELD-MANUAL FOR RAILROAD ENGINEERS. lection must be added. Similar tables for other lengths of curve or differences of gradients maybe computed by the eugineer, aud time in the field saved by their use. In setting grade-stakes, it will be well to set them 50 ft. apart on vertical curves, though to allow for the vertical curve at each regular station will suffice when cross-sectioning. 207. Elevation of Outer Rail on Curves. — In 138 it was sliown that the superelevation of outer over inner rail might, for standard gauge track, be given, nearly enough, by the formula e = dR' (134) in which e is the elevation in feet, and Fthe velocity in miles per 5730 * hour. Writing E = ——- in this formula gives e = 0. 000058 F'i> C319) The following table has been computed by formula (319). TABLE OF SUPERELEVATIONS OF OUTER RAIL. r V'm Degree of Curve. Miles per Hour. 1" 2° 3° 4° 5° 6° 7° 8° 9° 10° 12° 20 .02 .05 .07 .09 .12 .14 .16 .19 .21 .23 .28 30 .05 .10 .16 .21 .26 .31 .37 .42 .47 .52 .63 40 .09 .19 .28 .37 .46 .56 .65 .74 .84 .93 50 .15 .29 .44 .58 .73 .87 1.02 1.16 CO .'21 .42 .63 .84 1.04 1.25 Since grade-stakes are set at the edge of base it will be neces- sary to determine the difference between these elevations and the elevations of center line. Calling this difference h, the half-base h, and the distance between centers of rail-heads for standard gauge 4.9 feet, we shall have, from similar triangles. \ = -T-r e = 0.2be (nearly). 4.9 \ J/ (320) If the inner rail is required to remain at grade (320) will become 7i = 0.2be ± 0.5e, (320') according as the grade-stake is to be set at outer or inner edge of base. CONSTRUCTION". 183 Example, — What will be the value of 7i when e = .46, the base being 14 feet? By (320), A = 0.2 X 7 X 0.46 = 0.64 feet. The outside is this much higher than the center, the inside edge this much lower. The superelevation of outer rail should be computed for the highest speed at which trains are to be run over the curve; the maximum allowed in practice rarely exceeds 8 inches, since a greater elevation would endanger the slow-running freight trains. Even when the theoretical superelevation is given tiie outer rail, it is more worn than the inner one, either because there are other forces acting, or because of the sliding action of the outer wheel due to imperfect adjustment where the original coning has been destroyed by wear. Engineers sometimes elevate the outer rail 1 inch per degree up to 3°, and make 6 = 3i inches for a 4° curve, 4 inches for a 5° curve, and 4| inches for a 6° curve. Still other rules are in use. If transition-curves are not employed, the difference of eleva- tion is the same from P.O. to P.T., fading out to nothing on tangent. The elevation begins on tangent from 50 to 200 feet back of P.O., depending on the amount the outer rail is to be raised. 208. Easing Grades on Curves. — To compensate for the increased resistance due to curvature, it is customary to reduce the grade on curves. This resistance is taken to vary directly as the curvature; a rule often used is to reduce the gradient 0.05 foot per degree of curve \ Article 18. Earthwork. A. Setting Slope-stakes 209. Slope-stakes are set at the points where the side slopes meet the ground-surface, to mark the limits of the excavation or embankment, and to show the constructor what the cut or fill must be. In Fig. 102, if^i^ represents the ground-surface, IIBG the grade-surface. Let AB = h be the center height. Let HL = s be the side slope, which varies with the nature of the KL material; for earth-excavation the side slope will average about 1 to 1, so that s = 1, while for ordinary earth-embankment it will 184 A FIELD-MANUAL FOR RAILROAD ENGINEERS. average about 1^ to 1, so that s = 1^. The side height for level sections is the same as the center and may be found for any section, so that the distance LH is required. From the equation j*"^-*^ 6---*|<---6^--->t<--As--»4 i ' i ' i I Fig. 102. defining s, HL = KLs=Tis. Let the base EC = 2b; the "dis- tance out " from center is d.o = BL = h -\- hs. 210. Surface Inclined. — Where the ground slopes transversely the position ot the slope-stake cannot be found from the center height unless the slope of the ground-surface, us well as the side slope, is known. The slope-stakes can be most easily and rapidly set by trial. Fig. 103. In Fig. 103, FAE is the ground-surface, AB = 7i the fill at the center. We have to find the distances out of ^and F from A, and the side heights ME= hi , and NF = li^. Let OP repre- sent the plane of the instrument at a height U.I. above the datum, obtained from the known elevation of a bench or turn- ing-point. PB is the height of the i)]ane of the instrument above the grade. Call PB the Station Constant (s.c). COXSTRUCTION". 185 The fill at A will evideut^y equal the rod rcadiui; less the slatiou constant. Mark this ou the center stake. Since the ground slopes downward from J. to E, the distance out will be greater than for a level section, while for F, on the higher side, it will be less. Suppose we take a reading ^^at a distance out — h-\-lis\ the fill at that point is LK — QK — QL = r — s.c, and the corre- sponding distance out is d.o. = b -{- KL X s, which is greater than All, since LK is greater than h. If now a reading is taken at the distance out 6 + * X LK, we shall have a fill greater than LK, luiless the ground is level from ^to E, and therefore b -\- s X LK, the distance out actually used, will be less than that called for by the reading. However we shall have obtained a closer approximation to the position of E, and by repetitions of this process may come as close to its true position as the con- ditions require. The same thing can be accomplished more rapidly by estimat- ing the fall, by the eye, from ^ to a point b -f hiS out, then mul- tiply this estimated fall by the slope and add to b -\- hs. Take a reading r at this distance out; then compute the d.o. for the fill ?• — s.c. and note if this agrees with the actual d.o. If it does not, make a new trial with this reading as a guide. For ordinary work the actual and computed distance out should be such that if the rod were held at the computed distance the new distance would not differ more than a tenth from that just computed. The stake is then set at the computed distance out. After a little practice it will be found that the second setting of the rod may usually be made to fall as close to the true position as the limit requires. When the stake is marked and driven the cut or fill at that point and the distance out are recorded in the notes. As an example suppose 26 ■= 14 feet, * = H (i.e., slope 1^- to 1), H.I. = 187.3, grade elevation = 184.0. The station constant is s.c. = 187.3 — 184.0 = 3.3. Suppose the rod at center to read 8.5 ; the fill will be 8;5 — 3.3 = 5.2, which mark on stake as " F. 5.2. " The distance out, if section were level, would be d.o. — 7 -f 5.2 X 1.5= 14.8; but suppose the ground rises and we esti- mate the rise as 1 foot, which multiplied by s gives 1.5 feet to be subtracted from 14.8, since this is on the higher side of center for a section in embankment. Let the reading at 13 3 out be 7.7, which gives a fill of 7.7 — 3.3 = 4.4 feet, calling for a d.o. = 7-1-4.4 X 1.5 = 13.6. This shows we are too far in. but as a 18G A FIELD-MAX UAL FOR RAILROAD EXGINEERS. reading further out will be less, giving a correspoudingly smaller do., we try a reading at 13.5 feet out. Suppose the read- ingto be 7.6; the fill will be 7.6 — 3.3 = 4.3, calling for a distance out of 13.45 feet, which agrees almost exactly with the trial dis- tance. The stake is marked " F. 4,3," and the result recorded in the cross-section book. On the other side of the section suppose we estimate the fall to be 1.5 feet in 15; we should try a reading at 13.8 + 1.5 X 1.5 = 16.1, say 16.0 feet. Let this reading be 9.0 ; the fill will be 9.0 - 3.3 = 5.7 feet, calling for a do. = 7 + 5.7 X 1.5 = 15.6, which shows our readinir was taken too far out. Trv a reading at 15.4, which suppose 8.9 ; the fill is 8.9 — 3.3 = 5.6, and the d.o. =7+5.6 X 1.5 = 15. 4, which agrees exactly with the trial distance. In excavation the method of proceeding is the same as in em- bankment, except that s has generally a different value. For solid rock s is usually I, that is, the slope is taken as ^ to 1; for loose rock, gravel, and ordinary earth the slope may be taken as 1 to 1. The station constant in cuts is always positive, and the rod reading has to be subtracted from it to obtain the out. In fills, when the II. I. is greater than the grade height, the fill equals the difference of the rod reading and the station constant. When the///, {'^less than the grade height the rod reading plus the s.c. gives the fill. 211. The Notes may be kept in the form below, which repre- sents one page of the cross-section book. The cut or fill is written above the line, the distance out below, A plus sign indicates a cut, a minus sign a fill Sta. IGl 162 -f 20 -f 48 + 66 163 Ground. Grade. 178.8 184.0 181.6 183.0 182.2 « O »^ 182.5 ' 183.5 185.0 182.0 Left. - 4.4 13.6 - 0.8 8.2 0.0 9.0 -f 0.9 9.9 _L 2.4 11.4 4-4.4 -f 2.8 13.4 10.0 Center. - 5.2 - 1.4 - 0.6 0.0 -f 1.2 + 3.0 Right. 5.6 15.4 10.0 - 1.0 8.5 - 0.4 7.6 0.0 9.0 + 4.3 +2.6 6.1 11.6 CONSTRUCTIO]Sr. 187 212. Irregular Sections. — Wbeu readings are taken only at the center and sides it is termed a "three-level section." Very- irregular ground may require several more readings in order to determine its area ; in this case a reading is taken at each change of surface in the section, and the cut or fill, together with the dis- tance out, recorded— the distance being measured from the center to the point where the rod was held in taking the reading. When the base cuts the ground-surface the section is partly in excavation and partly in embankment, but each side will be staked out in the manner described above. The distance of grade-point from center must be found and recorded. 213. Staking Out Openings. — Where openings are to be left for trestles, culverts, and other structures, stakes must be set to mark the limits of the embankment. Stakes marked T. B. are set at the center and sides to fix the place where the top of bank is to end ; other stakes, marked F. S., are set at the foot of slope, the plus at which they fall— together with the distance out from center— being recorded in the note-book. The slope of the toe of dump should be the same as the side slope. 214. Marking Stakes. — All slope and toe stakes that limit excavation or embankment should be driven with tops inclined outward from the center. The cut or fill is marked on inside in plain figures preceded by the letter C. or F. as being more easily understood b}^ the contractor than the plus and minus signs used in the notes. The reverse side should bear the station number. 215. Shrinkage — Gi owth. — It must be remembered that earth- work in embankment will settle, or shrink in volume, even after having been compacted by the feet of the teams during construc- tion. Where the fill is not great, allowance may be made for shrinkage when setting grade-stakes, but in heavy fills allowance should be made when the stakes are set for construction. The proper allowance will vary with the nature of the material, but about 10 per cent will be a fair average. The contract should always specif}'^ the amount of shrinkage to be allowed on par- ticulai works. If the earth is measured in the borrow-pits, an equivalent allowance shoukl be made, since earth is more com- pact in embankment than l)efore excavating. With rock, however, it is found that the volume increases 188 A FIELD-MANUAL FOR RAILROAD EXGIXEERS. after excavation, and this increase is termed growth. The size of the fragments will determine the growth, wbicli will vary from one half to five eighths of the original volume — the larger the fragments the greater the increase. Little or no allowance need be made for settlement when placed in embankment. 216. Borrow-pits should be regular in form, particularl}' if the volume of earth moved is to be measured in the borrow-pit. They should be properly drained to prevent water standing in them and should have an ample berm between edge of pit and foot of slope, the width of berm increasing with the height of embankment. B. Areas of Sections. 217. Before the volume of earth in excavation or embank- ment can be computed the area of each cross-section must be found. To do this divide the section into triangles and trape- zoids, find the area of each separately, and take the sum. To shorten the calculations a few simple rules will be deduced. When the center and side heights are equal we have a one- level section; when the center and side heights differ it is a three-level section; where the height is found at five places in the section it is a five-level section ; and so on. 218. To Find the Area of a Three-level Section. Fig. 104. In Fig. 104 the area ABCDF is required. Draw EA and EG, dividing the area into four triangles. "With the notation of the figure it is seen that triangles (1) and (2) have equal bases h and CONSTKCCTION. ISO altitudes //i and hi , while (3) and (4) have the common base Zio and altitudes Bi = b -\- IIi s and di = b -\-yiiS. By geometry, Area {l)+(2) = b III + hr 2 '' Area (3) + (4) = /^o ^~ = ho g — -- The area of the whole section is therefore A = ,?i+h + nD^^ ^ ^11^ ^ ;JM:(^+^. (32,) -» * 2 2 This formula affords an easy method of obtaining the area of a three-level section, aud when written in words becomes the follow- ing llvM,Y.. — Multiply the half -sum of the side heights by the half -base and to this add the product of the center height by the half -sum of the distances out; the result icill be the area. If the linear measurements are in feet, the area will be in square feet. When Hy = hi = ho , then Di = d^ = d = b+ hos and (321) re- iuces to A = bho + ^lod = ho{b + d) = ho{2b + 7ioS). . . (322) The section is now a trapezoid or one level section for which '322) may be deduced by the usual rule for the area of a trapezoid. 219. Area of Five-level Section. Fig. 105 represents this case, where we may evidently divide N^ M D- — ^-rf-i-^^ •• Fig. 105. he area into triangles and trapezoids, computing the area of ?ach separately and taking their sum for the whole area. The 'oliowing simpler method may be preferred: 190 A FIELD-MANUAL FOR RAILROAD ENGINEERS. Write the notes as in field-book, except that center height is placed over zero and an additional - is written at each end as below. L C R Beginning at the center, multiply heights by distances out in pairs as indicated by the sloping lines, the products of members connected by full lines being plus and of those connected by dotted lines minus. Half the sum of the products will be the area, thus: ^ = 2 (323) The grouping is symmetrical for areas each side of center, as (323) exhibits; so it will be sufficient to show that the rule is cor- rect for either side. Divide the figure into trapezoids and tri- angles as shown; then Area trapezoid BCKM = |(7io + Hi)Di , ABM]^ = MR, + E,){D^ - Di\ " triangle ALN = ^H^{D2 - b). The two trapezoids include the triangle ; hence the latter must be subtracted from their sum. Doing this and simplifying, we have Al = W^oD, + E^D^ - DxH2 + S^b), which is the same as results in (323). 220. General Formula for Areas. — The method of 219 may be applied to any section no matter how irregular. Suppose there have been n levels taken on one side exclusive of the center height; the notes would appear as below: dy hi 57 h„. dn-l Tin dn 0_ h Expanding in the same manner as in 219, '^R=l\!Jlo^i+TLxdi+ . . hn-idn-^hnb) — {dJl'i-\- . dn-xhn)l (324) To show that (324) gives the true area, consider that we have n CONSTRUCTION. 191 trapezoids whose area is pc^sitive, aud oue triangle whose area is negative and equal to J lin{dn — b). Writing out the area, we have Ar = 4[(7/o -f Ai) etc. For the prism ai = a-i = am ; hence the volume is V = aj = (ui -f 4a,H + a-i)~x (a) For the wedge ay' — 2am' and a-i' = 0; tlierefore v' = a,' J ={((,'-{- 4:am' -i- a^')-^. ... (6) For the pyramids a/' = 4a»i" and a^" = 0; hence ^' = a/'l == («/' + 4a„r + ci,")^. . . (c) 194 A FIELD-MANUAL FOR RAILROAD ENGINEERS. Adding {a), (b), and (c), the total volume is V =v-i-v' + v" = [(«i+«i'+«i")+4(rt,„-ha,«'+«,„")+(«2+a2'+a2")]-| . («f But «i + tti' + «i" = -4i flm -|- C// draw BK to the midpoint of GH. The center of gravity- of this triangle is at M, two thirds of the distance BK from B. Now, by Guldin's rule (theorem of Pappus) the volume generated equals the area multiplied by the path of the center of gravity, the center of rotation being in the plane of the area. Draw BL horizontal and take N on a vertical through M; let the angle in degrees at the center be 5. The volume generated by the triangle BCH is V=BCHf^iR+BN). But the calculated volume is Fo = BCE X I = BCh'^R. Hence the curvature correction will be 180 C.C. = V- r, = BCII^r::,BN. % 2 d,-\-d^ d, + d. But BN= ^BL = ^^. -^ = —^ • •• «•<'• = ~BCH{d, + d.iYf = .OOQBCB{d^ + d.,)B°. (332) o40 When the sections are 100 ft. apart 6 = I) and the correction becomes c.c. = .OOQBCH{di + d.)D (332') The area of the triangle BCH is easily seen to be • A = l[b{?i.2 - h,) + 7>.o{d-2 - (h)]- . . (333) If the triangle BCH is on the convex side of curve the correc- tion must be added, if on the concave side it must be subtracted. For light work the correction is small, but for heavy work with steep transverse slope on sharp curves it ma}' be considerable. In practice we may use the middle for the mean area without material error Example. —Find the correction per station on an 8" curve, 28 il CONSTRUCTION 201 ft. base, side slopes 1|- to 1, iuside height 10 ft,, outside height 30 ft., end sections equal. 231. Overhaul. — Coutract prices are usually based on a certain maximum length of haul, and all material carried farther than this is termed overhaul, for which the contractor receives extra compensation. Fig. 108 In Fig. 108 let ABhe the length of free haul, the points A and B being fixed on the profile so that the volume ACE equals the volume CBK\ this may be done by trial computations, or closely enough iu some cases from the profile alone. Let the mass EFIIA be removed to BKLM, and let the centers of mass in the two positions be at g and g^ respectively; the length of overhaul to be paid for will be GG, - AB = GA + BG^. To find g and gx accurately requires that the sum of the moments of the ele- mentary masses equal the moment of the whole mass with respect to any chosen point. It will answer in practice to multiply the volume per station by the distance of its center of mass (found by dividing the station length in the inverse ratio of its end areas) from some selected point, as G, and equate this to the product of the whole mass by the unknown distance of its center of gravity from the same point, then solve for this distance. Indeed, it will answer iu most cases to find a point that divides the mass into two equal parts and treat this as the center of gravity; such a point may be readily found by trial. Sometimes it is specified that the overhaul must be found by finding the distance of the center of gravity of the wJiole mass moved from the center of gravity of the same mass after deposit- ing in embankment and deducting from this the length of free haul, the remainder being called the overhaul. This is the easier method, but requires every yard moved to be carried the entire lengtli of free haul before any overhaul whatever is counted. KxAMiM.E - Let AEVn =. 5000 cu. yds., GA = 200 ft., G,B 202 A FIELD-MANUAL FOR RAILROAD ENGINEERS. = 300 ft., and the price paid for overhaul 1 cent per cubic yard per 100 ft. The additional compensation above the contract price will be 200 + 300 100 X 5000 X .01 = $250.00. Article 19. Grade and Ballast Stakes, Culverts, Bridges, and Tunnels. 232. Grade and Center Stakes. — After the excavations and embankments have been brought approximately to the level called for on the cross-section stakes, the engineer must go carefully over the road, setting center stakes every hundred feet on tangents and flat curves, and every oC or 25 feet on sharp curves — the distance between center stakes depending on the sharpness of the curves. On tangents it will be sufficient to drive a grade-stake beside each center stake, so that its top will be at tbe height to which the finished surface must come, due allow- ance being made for shrinkage. On curves grade-stakes must be set at each side a distance equal to the half-base from the center; the proper elevation or depression of these stakes must be found by 207, formula (320). The P.C.'s and P.T.'s are recovered by means of the reference- points set during location. 233. Ballast-stakes are set on the completed sub-grade at the proper width of ballast-base — just as in slope-staking — with their lops at the level of the final grade. They should be set at inter- vals of 50 ft. ou tangents and flat curves, and at 25 ft. on sharp curves. 234. Track Centers are set for the guidance of trackmen as soon as the road-bed is ready to receive the cross-ties and rails. 235. The Opening left for a culvert, drain, or trestle bridge is measured from top of bank to top of bank; the manner in which it should be staked out is described in 213. A note of the size of drain and the material of which it is to be built, whether glazed earthenware pipe, box drain, stone culvert, etc., should be made in the note-book opposite the notes for the opening. After the culvert or drain has been built the earth is filled in CONSTRUCTION. 203 over and around it, and face or wing walls built to protect the bank at the points where culvert or drain meets its face. For trestle bridges it must be remembered that the bank-sills set back from the top of bank a distance sufficient to give firm bearing, usually about 6 ft, for ordinary earth, and allowance made therefor in staking out the opening. The length of open- ing is designated by the number of bents between bank-sills; thus a 12-bent opening, where the distance between bents is 14 ft., would be 13 X 14 — 12 = 170 ft. The bent spacing depends upon the size of timbers available and upon the weight of loco- motives to be run over the road. Whatever the nature of the structure, ample waterway should dvvays be provided for the heaviest storms; failure to do this is ;,he cause of many a costly wreck. , Center stakes are set for each trestle-bent, and if piles are to be driven a stake should mark the position of each pile. If the bridge is not at right angles to the stream it will often be best to Bet the bents askew, but this should be avoided whenever possible. After the piles have been driven cut-off levels are given by the engineer, for which a tack is set in the pile at a definite distance below the point of cut-off, allowance being made for cap, stringer, etc. If the bridge is on a grade, the rate of rise per bent must be figured out and allowed for. On curves the proper superelevation of outer rail must be computed by the method of 207. For details of trestles see Foster's Trestle Bridges. 236. The Piers and Abutments for truss bridges must be very accurately located, the spacing being done with a steel tape whose constants are known, and the center and limits being marked by stakes. On tangents the centers are easily located and referenced, but on curves this is not so easy, as the center of track cannot be taken as the center of pier on account of the clearance necessary for trains. Bridges on curves should be avoided whenever possible, but when they cannot be avoided the centers of piers are to be placed at the intersection of pier-axis and "bridge-chord." In Fig. 109 ABC is the center line of track, AE and CF the pier-axes. At the mid-point of the arc AG the tangent EF, parallel to AC, is drawn ; make AN = NE = CL = LF, and draw NL, which is the bridge-choid. The points N and L are the centers of the piers. 204 A FIELD-MANUAL FOR RAILROAD EXGIXEERS. Should L or N be inaccessible, the}'" may be locp.ted from a point P on some accessible portion of the curve. To do this lake PQ perpendicular to LN, such that PQ = RB- KB = E{veTsb-iYeTsa); . . (334) then will QL = QK + KL = Rlsin h -\- sin a). . . . (335) The manner of building the piers, determining the nature of the foundation, and erecting the bridge come properly within \ s \ ,y' \if^^' I I / y / •4/ Fio. 109. the province of the bridge engineer and require too much space to be outlined here. For preliminary estimates it will often be suflScieut for the locating engineer to make soundings with gas- pipe in order to determine the depth to a suitable foundation and the nature of the overlying deposits, the core forced up within the pipe serving for the latter purpose. 237. Tunnels, like bridges, require great nicety in the meas- urements by which they are constructed. The angular measure- ments should be made with the best available transit in the best possible adjustment, and repetitions and reversals made to elimi- nate errors as much as possible. Linear measurements should be made with a steel tape the constants of which are known, so that correction may be made for temperature, etc. If headinETS are to be driven from the ends and an unobstructed view of the summit is obtainable, a point may be fixed in the same vertical plane as the axis of the road, and will serve for giving the alignments of the headings. Sometimes several points must be located on the mountain in the plane of the axis, and triaugulation resorted to to secure the desired end. CONSTRUCTION. 205 The most accurate work can often be doue at night, sightings being made to a pliimmct-laivip, or in the early morning before the sun's heat has produced great changes in the density of the air. Within the tunnel, alignment is made by sighting to a plummet- lamp suspended from a plug V't into the roof. Work is usually carried on from both ends, so that it is necessary to secure accurate alignment. If the entire tunnel is on a tangent, this is not ditlicult when working from the ends: but when the tunnel is on a curve (the curve falling most often at the ends), or when align- ment must be transferred down a shaft, the operation is much more difficult. The Mont Cenis Tunnel, over seven miles long, was constructed from the ends — one end being on a curve — yet there was no trouble in making a fit where the headings met. When headings are driven from the foot of a shaft it is necessary to secure a point in the surface on each side of shaft in the plane of tunnel-axis and to transfer these points by plumb- lines to the bottom. By connecting these transferred points the direction of the axis may be secured. In the Hoosac Tunnel a line was transferred down a shaft 1000 ft. deep and carried 2050 ft. with a final error of only nine sixteenths of an inch. Levels are run over the surface with great care, and may be transferred down a shaft by measuring its depth with a rod or steel tape. The grade of the bottom must be sufficient for drainage. The dimensions of the tunnel will depend on the height of engines and the purposes for which it was intended. For detailed information regarding tunnels the student is referred to Drinker's and Sims's books on the subject and to the current engineering journals. Article 20. ^Fontiily and Final Estimates. 238. Monthly Estimates are made by the engineers in charge of construction about the end of each monlh, and upon these the division engineer bases his estimate, which he forwards to the chief engineer for approval. The contractor receives his compen- sation some days later, usually about the loth or 20th of the month following. Monthly estimates should always be based on actual measurements and never guessed at, jiarticularly if several classifications are to be made. The total quantity of work done 206 A FIELD-MA NCAL FOR RAILROAD ENGINEERS. or material delivered is to be estimated, then the difference between any estimate and the last preceding one will be the estimate upon which the contractor receives his installments. 239. For Earthwork, measurements (when needed) are only approximate, but it is best to make them with level and tape even for monthly estimates. It will be sufficient to compute volumes by averaging end areas, no attention being paid to the prismoidal correction. Care must be taken, however, that such estimates are not in excess — in fact, it is well to keep slightly within the actual quantities on account of the greater cost and labor required to finish the work, which would make the latter part appear so much less profitable to the contractor as sometimes to induce a disposition to abandon the work before completion. 240. The Classification of Earthwork.— It is customary to group earthwork in excavation, according to the difficulty of removal, into three classes— earth-excavation, loose rock, and solid rock — though other cla.ssifications are frequently made. Earth-excavation includes all earth, sand, loam, and loose stones that can be moved with the plow and scraper. Loose rock includes all stones and detached boulders less than from 1 to 3 cubic yards in size, and all slate, shale, or cemented gravel requiring the use of the pick and bar, but which may be removed without blasting. Solid rock includes all boulders above a certain size (usually from 1 to 3 cubic yards, as specified) and all rock masses that cannot be removed without blasting. The relative prices vary, but a ratio of 1:3:7 will not be far from an average for the more common conditions arising in rail- road work. The neces.sity for a correct classification is evident, and the engineer should keep full notes, and make careful measurements whenever a given volume involves more than one class of earth- work. It is customary to specify that his decision is final, and therefore his measurements should be cjxrefully made during the progress of the work, and notes on the nature of material taken at the same time. A notebook should be kept showing, the measurements and amounts of each class of material for each station, together with the date of completion and acceptance. CONSTRUCTION". 207 241. A Progress Profile shoultl accompauy the monthly esti- mate to exhibit graphically the amount of work done during the month, different colors being used for the different mouths. The final profile should show approximately the progress of the work. The colors may be laid on witii a brush, or hatchings made with a pen; in neither case should the color obscure the lines of the pro- file-paper. A duplicate progress profile shoultl be retained in the division engineer's office; if transparent profile-paper is employed, one may be simply traced through from the other. A further advantage of the transparent paper is that blue-prints of any por- tion of the profile may be readily made wneu duplicates are desired, provided the drawings are in black or any color admit- ting blue printing. 242. Masonry is to be measured in cubic yards, and any material on hand, but not in place, is to be measured and esti- mated. The classification of masonry must be according to specifications. Foundation-pits for piers or culverts must be measured as soon as completed, and before the masonry has been put in place. 243. Bridges must be estimated by measurement, or by checking up material in place and that on hand but not in place. For trestle bridges, or foundations requiring piling, the actual number of linear feet below cap must be measured; this neces- sitates the constant supervision of the engineer or an assistant, sometimes known as a "pile-recorder," whose duty it is to see that all piles come up to specifications and are driven in accord- ance therewith. All framing-timber in place, or delivered but not in place, is to be included in the estimate, the amount being obtained by measurement. Steel spans or trestles are to be estimated, in the same manner as wooden trestles, by checking up or measuring the material on hand and in place. 244. Track Material must be checked up either by the" mate- rial clerk " or the engineer in charge of track. Ballasting prop- erly belongs with the graduation, but may be put in place after the rails have been laid; in either case it is estimated in accord- ance with the specifications. For preliminary and monthly estimates it will be suflficient to 208 A FIELD-MAXUAL FOR RAILROAD EXGIXEERS. estimate track material by means of tables showing the numl)er of cross ties for a given spacing and the weight of steel fur a given rail section, but before the final estimate is made all mate- rial must be measured or counted. 245. Blank Estimate-sheets are sent out from the chief euiri- neer's office to be filled out by the engineers makiug estimate, who should retain a copy of each estimate rendered. On these sheets should appear the total quantit}' estimated, the amount of the last preceding estimate, and the estimate for the month, which will be the difference of the other two. The division engineer's estimate must show not onl}- the cj[uan- tity of material, but its value in dollars and cents computed from the contract price. The footings of the several columns then serve as a check upon each other. 246. The Monthly Payments are not made for the full amount estimated, but about 15 or 20 per cent is retained until after the final estimate has been made, in order to insure the completion of the work by the contractor, and to be used as a fund from which to withhold the amount of damages provided in the contract for failure to comply with all its provisions. 247. Extras incident to minor changes, or to the protection or drainage of the work, are usually shown on the final estimate, but a better way would be to require the contractor to present his bill for extras at the end of each mouth, and to incorporate them in the monthly estimate when they are just. The engineer should take measurements upon any extra work at the time of its com- pletion, and should keep a record thereof. If the extras are of a nature not admitting of measurement, he should note the com- pensation to be allowed at the time tbe extra work is done. 248. The Final Estimate must include all earthwork moved, all material in bridges, all masonry in foundations, culverts, piers, and tunnels, and all other material supplied or work done in compliance with the contract. The engineer should keep his notes full and complete during the construction of the work, in order to be able to meet the contractor's claims for extras or com- plaints as to classification. Any items that may have been over- looked in making up the monthly estimates must be includec here. i CONSTRUCTION. 209 249. Acceptance. — Until the en,i;inccr lias ijioiiounccd tlic work satisfactory and formally accepted it the contractor is liable for its couditiou, aud must make good all damage caused by accident or storm. The road-bed and track may be accepted without special test; but all spans should be subjected to a speci- fied test -load, under which they must show not more thari a cer- tain maximum deflection, so their acceptance will come last. Sometimes the contract requires a particular structure or class of structures to be maintained in good order for a certain length of time after completion, aud a percentage is retained to cover the case. After tinal acceptance the work is paid for in accordance with the final estimate. TABLES. ii 010 TABLE I. RADII. Beg. Radius. Deg. Radius. Deg. 1 Radius. \ Deg. Radius. Deg. Radius. 0° 0' j Infinite 1° 0' 5729.65 2° 0' 2864.93 3° 0' 1910.08 4° 0' 14.32.69 1 343775. 1 56:35.72 1 2841.26 1 1899.53 1 1426.74 2 171887. 2 5544.83 2 2817.97, 2 1889.09 o 1420.85 3 114592.1 S 5456.82 3 2795.06 3 1878.77 3 1415.01 4 85943.7! 4 5371.56 4 2772.53 4 1868.56 4 1409.21 5 68754.9 5 5288.92 5 2750.35 5 1858.47 5 1403.46 6 57295.8 6 5208.79 6 2728.52 6 1848.48 6 1397.76 t 49110.71 4 5131.05 7 2707.04 7 18:38.59 1:392.10 8 42971.8! 8 5055.59 8 2685.89 8 1828.82 8 1:386.49 9 33197. 2i 9 4982.33 9 2665.08 9 1819.14 9 1380.92 10 34377.5 10 4911.15 10 2644.58 10 1809.57 10 1375.40 11 31252.3 11 4841.98 11 2624.39 11 1800.10 11 1:369.92 12 28647.8 12 4774.74 12 2604.51 12 1790.73 12 1364.49 13 26444.2 13 4709. as 13 2584.93 13 1781.45 13 1359.10 14 24555.4 14 4645.69 14 2565 . 65 14 1772.27 14 1:353.75 15 22918.3 15 4583.75 15 2546.64 15 1763.18 15 1348.45 16 21485.9 16 4523 44 16 2.527.92 16 1754.19 16 1:34:3.15 17 20222.1 17 4464.70 17 2509.47, 17 1745.26 17 1.337.65 18 19098 6 18 4407.46; 18 2491.29 18 1736.48 18 1.3.32.77 19 18093.4 19 4351.67] 19 2473. :37 19 1727.75 19 1:327.63 20 17188.8 20 4297.28 20 2455.70 20 1719.12 20 1322.53 21 16370.2 21 4244.23 21 24.38.29 21 1710 56 2! 1317.46 22 15626.1 22 419-^.47 22 2421.12 22 1702.10 22 1312.43 23 14946.7 23 4141.96! 23 2404.19 1 23 1693.72 23 1307.45 24 14323.6; 24 4092.66 24 2:387.. 50 24 16,S5.42 24 1:302. 50 25 13751 25 4044.51 25 2:371.04 25 1677.20 25 1297.58 26 13222. ll 26 3997.49 26 2354.80 26 1669.06 26 1292.71 27 127.32.4 27 3951.54 27 2:3:38.78 27 1661.00 27 1287.87 28 12277.7 28 3906. 54( 28 2:322.98 28 1653.01 28 1283.07 29 11854.31 29 3862.74 29 2307. :39 29 1645.11 29 1278.30 30 11459.2 30 3819.83 30 2292.01 1 30 1637.28 30 1273.57 31 11089 6, 31 3777 . 85 31 2276.841 31 1629.52 31 1268.87 32 1074:i 32 3736.79 32 2261.86 32 1621.84 32 1264.21 33 10417.5; 33 3696 61 33 2247.08 33 1614.22 33 1259.58 34 10111.1 34 3657.29 ai 22:32. 49 i 34 1606.68 34 1254.98 35 9822.18; 3o 3618.80, a5 2218.09 1 35 1599.21 35 1250.42 36 9549. 34 1 36 .3581.10 36 2203.87 ! 36 1591.81 36 1245.89 37 9291.29' ' 37 3544.19! 37 2189.84, 1 37 1584.48 37 1241.40 38 9046.75 38 :3508.02 38 2175.98' ■ 38 1.577.21 38 1236.94 39 &S14.78: 39 3472.. 59 39 2162.30 39 1.570.01 39 12.92.51 40 8594.42 40 3437.87 40 2148.79, 1 40 1562.88' 40 1228.11 41 a384.80 41 3403.83 41 2135.44 41 1.555.81 41 1223.74 42 8185.16, 42 3370.46 42 2122.26 1 42 1.548.80 42 1219.40 1 43 7994.81! 43 33.37.74 43 2109.24 43 1541.86 43 1215.30 44 7813. ii; 44 :3305.65 44 2096. :39 ! 44 1.5:M.98 44 1210.82 45 7639.49! 45 3274.17 45 2083.68 45 1528.16 45 1206.. 57 46 7473. 42 1 46 3243.29 46 2071.13 46 1.521.40 46 1202.36 47 7314.41! 47 3212.98 47 2058.73 47 1.514.70 47 1198.17 48 7162.03 48 3183.23 48 2046.48 48 1508.06 48 1194.01 49 7015.87 49 3154.03 49 20:34. :37 1 "^9 1.501.48 49 1189.88 50 6875 . 55 50 3125.36 i 50 2022.41 50 1494.95 50 1185.78 51 6740.74 51 3097.20! 51 2010.59 51 1488.48 51 1181.71 52 6611.121 52 3069.55 52 ' 199S.90 52 1482.07 52 1177.66 53 6486.38 ': 53 3042.39 53 1987.35 53 1475.71 53 1173.65 54 6366.26, 54 , 3015.71 54 1 1975.93 54 1469.41 .54 1169.66 55 6250.51' 55 ; 2989.48 55 1964.64 55 1463.16 55 1165.70 56 6138.90, 56 2963.71 56 1953.48 56 1456.96 56 1161.76 57 6031.20 57 29:38.39 57 1942.44 57 1450.81 57 11.57. a5 58 5927.22 58 2913.49 58 19:31.. 53 ; 58 1444.72 58 11.53.97 59 .5826.76 59 2889.01 59 1920.75 59 1438.68 59 1150.11 60 1 5729.65 60 , 2864-93 60 1910.08 ! 60 1432.69 60 1146.28 TABLES. 213 TABLE I. RADII. Deg. 6° 0' Radius. 1146.28 [Deg. Radius. Deg. Radius. 818.64 Deg. Radius. Deg. Radius. 6°0' 955.37 70 0' 8°0' 716.34 9°0' 636.78 I 1142.47 1 952.72 1 816.70 1 714.85 1 635.61 2 1138.69' 2 950.09 2 814.76 2 713.37 2 634.44 3 1134.94 3 947.48 3 812.83 3 711.90 3 633.27 4 1131.21 4 944.83 4 810.92 4 710.43 4 632.10 5 1127.50 5 942.29 5 809.01 5 708.96 5 630.94 6 1123.82 6 939.72 6 807.11 6 707.51 6 629.79 7 1120.16 1 937.16 7 805.22 7 706.05 7 628.04 8 1116.52 8 934.62 8 803.34 8 704.60 8 627.49 9 1112.91 9 932.09 9 801.47 9 703.16 9 626.35 10 1109.33, 10 929.57 10 799.61 10 701.73 10 625.21 11 1105.76 11 927.07 11 797.75 11 700.30 11 624.08 12 1102.22 12 924.58 12 795.91 12 698.88 12 622.95 13 1098.70 13 922.10 13 794.07 13 697.46 13 621.82 14 1095.20: 14 919.64 14 792.24 14 696.05 14 620.70 15 1091.73 15 917.19 15 790.42 15 694.65 15 619.58 16 1088.28 16 914.75 16 788.61 16 693.24 16 618.47 17 1084.85 17 912.33 17 786.80 17 691.85 17 617.36 18 1081.44 18 909.92 18 785.01 18 690.46 18 616.25 19 1078.05 19 907.52 19 783.22 19 689.08 19 615.15 20 1074.68 20 905.13 20 781.44 20 687.70 20 614.05 21 1071.34 21 902.76 21 779.67 21 686.33 21 612.96 22 1068.01 22 900 40 22 777.91 22 684,96 22 611.87 23 1064.71 23 898.05 23 776.15 23 683.60 • 23 610.78 24 1061.43 24 895.71 24 774.40 24 682.25 24 609.70 25 1058.16 25 893.39 25 772.66 25 680.89 25 608.62 26 1054.92 26 891.08 26 770.93 26 679.55 26 607.55 27 1051.70 27 888.78 27 769.21 27 678.21 27 606.48 28 1048. 48i 28 886.49 28 767.49 2& 676.88 28 605.41 29 1045.31 29 884.21 29 765.78 29 675.54 29 604.35 30 1042.14 30 881.95 30 764.08 30 674.22 30 603.29 31 1039.00' 31 879.69 31 762.39 31 672.90 31 602.23 32 1035.87 32 877.45 32 760.70 32 671.59 32 601.18 33 1032.76 33 875.22 33 759 02 33 670.28 33 600.13 34 1029.671 34 873.00 34 757.35 34 668.98 34 599.09 35 1026.60 35 870.80 35 755.69 35 667.68 35 598.04 36 1023.55; 36 868.60 36 754.03 36 666.39 30 597.01 37 1020.51 37 866.41 37 752.38 37 665.10 37 595.97 38 1017.49 38 864.24 38 750.74 38 663.82 38 594.94 39 1014.50 39 862.08 39 749.10 39 662.54 39 593 91 40 1011.51 40 859.92 40 747.48 40 661.26 40 592.89 41 1008.55 41 857.78 41 745.86 41 659.99 41 591.87 42 1005.60 42 855.65 42 744.24 42 658.73 42 590.85 43 1002.67 43 853.53 43 742.63 43 657.47 43 .589.84 44 999.76 44 851.42 44 741.03 44 656.22 44 588.83 45 996.87 45 849.32 45 739.44 45 654.97 45 587.83 46 993.99 46 847.23 46 737.86 46 653.72 46 586.82 47 991.13 47 845.15 47 736.28 47 652.48 47 585.83 48 988.28 48 843.08 48 734.70 48 651.25 48 584.83 49 985.45 49 841.02 49 733.14 49 650.02 49 583.84 50 982.64 50 838.97 50 731.58 50 648.79 50 582.85 51 979.84 51 836.93 51 730.03 51 647.57 51 581.86 52 977.06 52 834.90 52 728.48 52 646.35 52 580.88 53 974.29 53 832.89 53 726.94 53 645.14 53 579.90 54 971.54 54 830.88 54 725.41 54 643.94 54 578.92 55 968.81 55 828.88 55 723.88 55 642.73 55 577.95 56 966.09 56 826.89 56 722.36 56 641.53 56 576.98 57 963.39 57 824.91 57 720.85 57 640.34 57 576.02 58 9G0.70 58 822.93 58 719.34 58 639.15 58 575.06 59 958.03 59 820.97 59 717.84 59 637.06 59 574.10 60 955.37 60 819.02 GO 716.34 60 636.78 60 573.14 214 A FIELD-MANUAL FOR RAILROAD ENGINEERS. TABLE I.— RADII. Deg. Radius. Deg. Radius. Deg. Radius. Deg. Radius. Deg. Radius. 10° ' 573.14 12° 477.68 14° 0' 409.32 16° 0' 358.17 18° 0' 318.39 2 571.24 2 476.36 o 'Si' 408.35 2 357.43 2 317.80 4 569.35 4 475.05 4 407.38 4 356.69 4 317.22 6 567.47 6 473.74 6 406.42 6 355.95 6 316.63 8 565.60 8 472.44 8 405.46 8 355.21 8 316.05 10 563.75 10 471.15 10 404.51 10 354.48 10 315.47 12 561.91 12 469.86 12 403.56 12 353.75 12 314.89 14 560.08 14 468.58 14 402.61 14 353.03 14 314.32 16 558.26 16 467.31 16 401.67 16 352.. 30 16 313.75 18 556.45 18 466.04 18 400.74 18 351.58 18 313.18 20 554.66 20 464.78 20 399.80 20 350.86 20 312.61 22 552.88 22 463.53 22 398.88 22 350.15 22 312.04 24 551.11 24 462.29 24 397.95 24 349.44 24 311.47 26 549.35 26 461.05 26 397.03 26 348.72 26 310.91 28 547.60 28 459.82 28 396.13 28 348.02 28 310.35 30 545.87 30 458.59 30 395.21 30 347.. 32 30 309.79 32 544.14 32 457.38 32 394.30 32 346.62 32 309.23 34 542.42 34 456.16 34 393.40 34 345.93 34 308.68 36 540.72 36 454.96 36 392.50 36 345.23 36 308.13 38 539.03 38 453.76 38 391.61 38 344.54 38 307.58 40 537.34 40 452.57 40 390.72 40 343.85 40 307.03 42 5S5 . 67 42 451.38 42 389.83 42 343.16 42 306.48 44 534.01 44 450.20 44 388.95 44 342.48 44 305.93 46 532.36 46 449.02 46 388.07 46 341.80 46 305.39 48 .•530.71 48 447.86 48 387.20 48 341.12 48 304.85 50 529.08 50 446,69 50 .386.33 50 340.45 50 304.31 52 527.46 52 445.54 52 385.47 52 339.78 52 303.77 54 525.85 54 444.39 54 384.60 54 339.11 54 303.24 56 .524.25 56 443.24 56 383.75 56 338.44 56 302.70 58 522.65 58 442.11 58 382.89 58 337.77 58 302.17 11° 0' 521.07 13° 0' 440.97 16° 0' 382.04 17° 0' 337.11 19° 0' 301.64 2 519.50 2 439.85 2 381.19 2 336.45 2 301.12 4 517.93 4 438.73 4 380.35 4 335.80 4 300.59 6 516.38 6 ■ 437.61 6 379.51 6 335.14 6 300.07 8 514.84 8 436.50 8 378.68 8 3.34.49 8 299.54 10 513.30 10 4.35.40 10 377.84 10 333.84 10 299.02 12 511.77 12 434.30 12 377.02 12 333.19 12 298.50 14 510.26 14 433.21 14 376.19 14 332.55 14 297.99 16 .508.75 16 432.12 16 375.37 16 331.91 16 297.47 18 507.25 18 431.04 18 374.55 18 331.27 18 296.96 20 505.76 20 429.96 20 373.74 20 330.63 20 296.45 22 504.28 22 428.98 22 372.93 22 330.00 22 295.94 24 .502.80 24 427.82 24 372.12 24 329.37 24 295.43 26 501.34 26 426.76 26 371.32 26 328.74 26 294.92 28 499.88 28 425.71 28 370.. 52 28 328.11 28 294.42 30 498.43 30 424.66 30 369.72 30 327.48 30 293.91 32 496.99 32 423.61 32 368.93 32 326.86 32 293.41 34 495.56 34 422.57 34 368.14 .34 V36 326.24 34 292.91 36 494.14 36 421.54 36 367.35 325.62 36 292.41 38 492.73 38 420.51 38 366.57 38 325.01 38 291.92 40 491.32 40 419.49 40 365.79 40 324.40 40 291.42 42 489.92 42 418.47 42 365.01 42 323.79 42 290.93 44 488.53 44 417.45 44 364.24 44 323.18 44 290.44 46 487.15 46 416.44 46 363.47 46 322.57 46 289.95 48 485.77 48 415.44 48 362.70 48 321.97 48 289.46 50 484.40 50 414.44 50 361.94 50 321.37 50 288.98 52 483.05 52 413.44 52 361.18 52 320.77 52 288.49 54 481.69 54 412.45 54 360.42 54 320.17 54 288.01 56 480.35 56 411.47 56 .3.59.67 56 319.57 56 287.53 58 479.01 58 410.49 58 .3.58.92 58 318.98 58 287.05 60 477.68 60 409.51 60 358.17 60 318.39 60 286.57 TABLE II.— MINUTES IN DECIMALS OF A DEGREE. 215 # 0* 10" 15' 20" SO" 40" 45' 50" / .00000 00278 .00417 ,005.56 .00833 .01111 .01250 .01389 1 .01667 .01944 .020S3 .02222 .02.500 .02778 .02917 .03055 1 2 .0333;3 .03611 .03750 .0:3889 .04167 .04444 .0458:3 .04722 2 3 .05000 .05278 .05417 05556 .0583:3 .00111 .06250 .06:389 3 4 .06667 .06944 .07083 .07222 .07500 .07778 .07917 .08056 4 5 .08333 .08611 .08750 .08889 .09167 .09444 .09.583 .09722 5 6 .10000 .10278 .10417 .10556 .10833 .11111 .11250 .11:389 6 7 .11667 .119-W . 1208:3 ■ i./V'^'W .12500 .1277'8 .12917 .13056 7 8 .13:333 .13611 .13750 .13889 .14167 .14444 . 14583 .14722 8 9 .15000 . 15278 .1.5417 . 15556 .158:33 .16111 .16250 .16:389 9 10 .16667 .16944 .17083 .17222 .17500 .17778 .17917 .18056 10 11 .18333 .18611 . 187.50 .18889 .19167 19444 .19.583 .19722 11 12 .20000 .20278 .20417 .20556 .20833 .21111 .21250 .21.389 12 13 .21607 .21944 .22083 .22222 .22500 .22778 .22917 .23056 13 14 .23333 .23611 .2:3750 .2:3889 .24167 .24444 .24583 .24722 14 15 .25000 .252;8 .25417 .25556 .25833 .26111 .26250 .26389 15 16 .26007 .26944 .27083 .27222 .27500 .2777'8 .27917 .28056 16 17 .28333 .28011 .28750 .28889 .29167 .29444 .29583 .29722 17 18 .30000 .30278 .30417 .30556 .:30833 .31111 .312.50 .31:389 18 19 .31667 .31944 .:32083 ..32500 .32778 .32917 .33056 19 20 .;33333 .3:3611 .33750 .33889 .34167 .34444 .34583 .34722 20 21 .35000 .3,5278 .35417 .355,56 ..35833 .36111 .36250 .36389 21 .36067 .36944 .37083 .37222 .37500 .3777'8 .37917 .38056 22 23 .383:3:3 .38611 .:38r.50 .38889 .39167 .39444 .39583 .39722 23 24 .40000 .40278 .40417 .40556 .408:33 .41111 .41250 .41389 24 25 .41667 .41944 .42083 .42222 .42500 .4277'8 .42917 .4:3056 25 26 .43:333 .4:3611 .43750 .43889 .44107 .44444 .44583 .44722 26 27 .4.5000 .45278 .45417 .45556 .45833 .46111 .46250 .46389 27 28 .46667 .46944 .47083 .47222 .47500 . 47'77'8 .47917 .48056 28 29 .48333 .48611 .48750 .48889 .49167 .49444 .49583 .49722 29 30 ..50000 .50278 .50417 .50556 .50833 .51111 .51250 .51389 30 31 ..5166? .5i941 .,52083 .52222 .52500 .52778 .52917 .53056 31 32 ..5:3333 .5:3011 ..537,50 ..53889 .54167 .54444 .54583 .54722 32 33 ..5.5000 .5.5278 .5.5417 .55556 .55833 .56111 .56250 .56389 33 34 ..56667 .56944 .57083 .57222 .57500 .57778 .57917 .58056 ;34 35 ..58:3:33 ..58011 ..5S7.50 ..58889 .59167 .59444 .59583 .59722 35 36 .600eX) .00278 .60417 .605.56 .fi0833 .61111 .61250 .61389 36 37 .01667 .()1944 .62083 .62222 .02500 .62778 .62917 .6:3056 37 38 .0:3:3:3:3 .0:3011 .6:37.50 .6:3889 .64167 .64444 .64583 .64722 .38 39 .6.5000 .6.5278 .6.5417 .65556 .65833 .66111 .662.50 .66:389 39 40 .06667 .66944 .67083 .67222 .67500 .67778 .67917 .68056 40 41 .68.3:33 .68611 .68750 .68889 .69167 .69444 .69583 .69722 41 42 .70000 .70278 .70417 .70556 .70833 .71111 .71250 .71.389 42 4:3 .71607 .71944 .7208:3 .72222 .72500 .72778 .72917 .73056 43 44 .73:3:33 .7:3611 .7.3750 .73889 .74167 .74444 .74583 .74722 44 45 .7.5000 .7.5278 .75417 .75556 .758.33 .76111 .76250 .76389 45 46 .76607 .76944 1 .77083 .77222 .77500 .7777'8 .77917 .780.56 46 47 .78:3:33 .78611 .78750 .78889 .79167 .79444 .79583 .79722 47 4S .80000 .80278 ; .804K»< .80556 .808:33 .81111 .81250 .81.389 48 49 .81667 .81944 .82083 .82222 .82.500 .82778 .82917 .83056 49 50 . 8:3:3:33 .83611 .8:3750 .83889 .84167 .84444 .84583 .84722 50 51 .8.5000 .85278 .8,5417 .a5556 .85833 86111 .86250 .86389 51 52 .80007 .86944 .87083 .87222 .87500 .87 < 78 .87917 .880.56 .52 53 .88:3:33 1 .88611 .88750 .88889 .89167 .89444 .89583 .89722 53 54 .90000 i .90278 .90417 .90.556 .90ft:33 .91111 .91250 .91:389 54 55 .91667 .91944 .92083 92222 .92.500 .92778 .92917 .93056 55 56 .9.3.3.3:3 .9:3611 .9.37.50 ! 93889 .94167 .94444 .94583 .94722 56 57 .9.5000 .9.5278 .9.5417 .9.5.5.56 .95^33 .96111 .962.50 .96:389 57 58 .96007 .96944 .970a3 .97222 .97.500 .97778 .97917 .98056 58 59 / .98333 .98611 .96750 .98889 .99167 .99444 .99583 .99722 59 0" 10" 1 15- 1 SO- 30" 40° 45" 50- / 21G A FIELD-MANUAL FOR RAILROAD EKGIXEERS. o -r -* L-^ cc o i - 3C c; C-. o — ' gZZ?2i3::;i2!5iri^2£ o o o — Si Si CO -^ in in Tj ?> cj -rj -ii •:» ?? -:? c« -r? ?* OJ CJ C? 7J 5< Si T? SJ 7J W SiCOJO^OCOCOCOCOCOCO , o ?* ?? eo -r o lO o t- ac 00 Ci oOT-ioicocoTi<»n«o«D I- L- I- i- I- i- i- I- i.^ i- 1-0005050'>-I--S»CO-^ t- i- l- I- X 00 X X X X e4 M 5i 7* ?J Ci -TJ 01 ■?> CJ ?J 5i S» 9» S* Oi 5* (N W OJ Oi TJ Si Si Si Si Si Si Si Si Si s^ > t3 o o o ■— ^ 'N « -r ^ o » t- i'- 9* -jj (?j CJ CJ -jj ?J si 7J W W OC C5 Ci O ^ 1? S» CO -^ O -r T TT o ir: 1.0 in »r; j,-; o cj 5> cj w sj ni si si si si m o t- 1- X ci o o — Si inoinintninoooo si si si si si si si si si si o GO 3i O O " TJ :C «> -f »C O ^ 1- Tj ::> ■:i oi tj i» tj ■:? ~« CCI-OCXCIO — — s?co SJ SJ S* S* S? CC CO CO CO CO CO -r m to o I- X 05 Ci o COCOCOCOCOCOCOCOCO"^ < •?? C» « ?» CJ 1* ?i 5> ■?? IJ OJ Si SJ S» Si Si SJ Si Si Si Si Si Si Si Si S* Si Si ©i Si i- l- i- I- I- i- i- GC 00 ClU 3D CO CO -f X,S L~ O l~ 00 X Ci Oi— — Siwcorrintso c; o; c; C-. c; c; C-. C-- C-. 05 UJ „^^„„„^^^^^ .____ "__ : : : i fa O o iO iT: iC O O iO iC O O O O 1— -^ Si CO -r -f »r; o t- 1>. X Ci c; o ■— Si s> CO -t m OTJ-^lSXOIJ-^tOCCO Si-^OQCOSi-rtoxo Si Si Si Si CO CO CO eo CO -^ Si--r»xOSi-"-^XO Tj. -^ T- -r m in in in m o o IJ ■?? ■?» CO ?3 JO T ^ -^ O O O X — -P i- O CO O 05 ^ o i.o o CD ■j:> i - l~ I- i- X -»" I- o CO o C5 Si in X -r-i CO X ci c". ci o i- i- I- X X X X c; Ci i- o CO -^ C5 Si o X o 00 CSOOOO'— ^^SiSi » -T-*-*-»-^T-!rT-^'3"-# T-»J'-3<-^'3'-<3" 23 U" D 05 '^> o 00 — 'T o o ?i m 00 -T iO »0 O » :0 13 O i- I- I- '-'-ri-oco-ooo — 'i"t>- XXXCSC-. csciooo ocooc5Siinx!Ocoo '—'-•-'— Si Si Si CO CO CO ■-K COCOCOCOCO?5COCCCO0OCO COCOCOCOCOCOCOT'S"-}' -^■^T-T'^-^-^'S'TJ'TJ' Z O o T'J O OD — < CO O Ol 7> ».- OD « o O :s I- 1- 1- i- 00 oo 00 Ci -" I- o Si in X ^ ■'t I- o OSClOOOO^i-i-Si CO tS C5 Si m i~ O OS ?S C5 Ci Si Si CO CO CO -^ '^^ -f -T < c^) 3* oi c* c« c» e* c» w c» (?j s*c»cocoeoccc<3cococo COCOCOCOCOCOPOCOCOCO a «1 t-i.-x;ooococcioc;oo i-C5S?inx— i-ri-oco O O -r-i ^ -r- Si Si Si CO CO «o o i-H -r t- o CO ;c c: Si 00 CO TT -r -^ m in in in «n pr^ ^„„rt^^^,-i^-jC» Si Si Si Si Si Si S> Si S» Si Si Si Si Si Si Si Si Si Si Si ;/5 fa fa O o <- o CO o era •?» o oc ^ CO «3 XC50S5S-. OOO — " — ^-^— ^f— s-— ^— *, _^,,,- C5 Si in X — Tji £- o CO o ^SiSiSicococOrr-T'^ oo TH -t< t- o CO o c; Si in T Si in X « T I- c; Si in X CO CO CO -5" -r 'T' — »n in in -— ■^t-OcooCi'— ■^i- ':0 ;^ O I- I- I- I- cc X X COOwOOOOOOO oooooooooo oooooooooo O J o?»Tr50ooo'c»-^coxo ^ -- r- -^ ri Ti Si-rOOOOSi-r:OXO Si Si Si Si OO CO CO CO CO -^ s>-^oxosi'*":oxo "^''^'^^Tininininino QO TABLES. 17 TABLE IV.- ■LONG CHORDS. T-\ . ___. ^ _ _ .e Actual Arc. Long Chord s. Depfree oi Curve. One Station. o 3 4 5 6 stations Stations . Stations. Stations. Stations. 0° 10' 100.000 200.00 300.00 400.00 500.00 599.99 20 .000 200.00 300.00 399.99 499.98 .599.97 30 .000 200.00 299.99 399.98 499.96 .599.93 40 .001 200.00 299.99 399.97 499.93 599.88 50 .001 200.00 299.98 399.95 499.89 .599.82 1 100.001 199.99 299.97 399.92 499.85 599.73 10 .002 199.99 299.96 399.90 499.79 .599.64 20 002 199.99 299.95 399.87 499 73 599.. 53 30 .003 199.98 299. 93 399.83 499.66 599.40 40 .003 199.98 299.92 399.79 499.. 58 .599.26 50 .004 199.97 299.90 399.74 499.49 599.11 2 100.005 199.97 299.88 399.70 499 39 .598.93 10 .006 199.96 299.86 399.64 499.29 .598.75 20 .007 199.96 299.83 399.. 59 499.17 598.. 55 80 .003 199.95 299.81 399.52 499.05 .598.34 40 .009 199.95 299.78 399.46 498.92 598.11 50 .010 199.94 299.76 399.39 498.78 597.86 3 100,011 199.93 299.73 399.32 498.63 597.60 10 .013 199.92 299.70 399.24 498.47 597.33 20 .014 199.92 299.66 399.15 498.31 597.04 30 .015 199.91 299.63 399.07 498.14 596.74 40 .017 199.90 299.. 59 398.98 497.96 .596.42 50 .019 199.89 299.55 398.88 497.77 596.09 4 100.020 199.88 299.51 398.78 497 57 .595.74 10 .022 199.87 299.47 398.68 497.36 595.38 20 .024 199.86 299.43 398.. 57 497.15 595.01 30 .0-'6 ] 99 . 85 299.38 398.40 496.92 594.62 40 .028 199.83 299.34 398.34 496.69 •594.21 50 .030 199.82 299.29 398 22 496.45 593.79 5 100.032 199 81 299.24 .398.10 496.20 593.36 10 .034 199.80 299.19 397.97 495.94 .592.91 20 .036 199 78 299.13 397.84 495.68 592.45 30 .038 199.77 299.08 397.70 495.41 591.97 40 .041 199.76 299.02 397.56 495.12 591.48 50 .043 199.74 298.96 397.41 494.83 590.97 G 100.046 199.73 298.90 397. 2G 494.53 590.45 10 .04S 199.71 298.84 397.11 494.23 .589.91 20 .051 199.70 298.78 396.95 493.91 589.36 •30 .054 199.68 298.71 396.79 493.59 588.80 40 .056 1!)9.G6 298.65 396.62 493.26 588.22 50 .059 199.64 298.. 58 .396.45 492.92 587.63 7 100.062 199.63 298.51 396.28 492.57 587.02 10 .016 199.51 298.. 30 395 91 491.97 .586.12 20 .017 199.49 298.21 395.71 491.60 585.46 30 .018 199.47 298.13 395.. 52 491.21 .584.80 40 .018 199.44 298.05 .395.32 490.82 .584.13 50 .019 199.42 297.96 395.11 490.42 583.44 S 100.020 199.39 297.87 394.90 490.01 582.72 10 .021 199.37 297.78 394.69 489.59 582.01 20 .022 199.31 297.69 394.47 489.16 .581.27 30 .023 199.31 297.60 .394.25 488.73 .580.52 40 .024 199.28 297.50 394.02 488.28 .579.76 50 .025 199.26 297.41 .393.79 487.83 578.98 9 100.026 199.23 297.31 393.. 55 487.37 .578.18 10 .027 199 20 297 21 393.31 486.90 .577.38 20 .028 199.17 297.10 393.07 486.43 576.. 56 •30 .029 199. U 297.00 .392.82 485.95 .575.73 40 .030 199 11 296.90 392.. 57 485.45 574.88 50 .031 1 99 . 08 296.79 392.31 484.95 .574.02 10 100.032 199.05 296.68 392.05 484.44 573.14 218 A FIELD-MANUAL FOR RAILROAD ENGINEERS. TABLE IV.— LONG CHORDS. Degree of Curve. 10° 11 12 13 14 0' 10 ■20 30 40 50 10 20 30 40 50 10 20 30 40 50 10 20 30 40 50 Actual Arc, One Station. 100.0.32 .033 .034 .035 .036 .037 100.038 .040 .041 .042 .043 .044 100.046 .047 .048 .050 .051 .052 100.0.54 .055 .056 .058 .059 .061 100.062 Long Chords. 12 3 4 5 Station. Stations. Stations. Stations. Stations. 99.91 99.90 99.90 99.90 99.89 99.89 99.88 99.88 99.88 99.87 99.87 99.87 99.86 99.86 99.85 99 85 99.85 99.84 99.84 99.84 99.83 99.83 99.82 99.82 99.81 199 05 199.02 198.98 198.95 198.92 198.88 198.85 198.81 198 78 198.74 198.71 198.67 198.63 198.. 59 198.55 198.51 198.48 198.43 198.39 198.35 198.31 198.27 198.23 198.18 198.14 296.68 296.57 296.45 296.33 296.22 296.10 295,98 295.86 295.74 295.61 295.48 295.35 295.22 295.09 294.95 294.82 294.68 294.54 294.40 294.26 294.11 293.96 293.81 293.66 293.51 392.05 .391.79 391.51 391.24 390.96 390.68 390.39 390.10 389.81 389.. 50 389.20 388.89 388.58 388.27 387.95 387.62 387.29 386.95 386.62 386.28 385.93 385.58 385.23 384.87 384.. 51 484.44 483.92 483.39 482.86 482.32 481.77 481.21 480.64 480.07 479.48 478.89 478.29 477.68 477.07 476.44 475.81 475.18 474.52 473.87 473.20 472.53 471.86 471.17 470.48 469.77 TABLE v.— MID-ORDINATES TO LONG CHORDS. Degree of 1 o 3 4 5 ... ., 6 Curve. Station. Stations. Stations. Stations. Stations. Stations. 0° 10' .04 .15 .33 .58 .91 1.31 20 .07 .29 .65 1.16 1.82 2.62 30 .11 .44 .98 1.75 2.73 8.93 40 .15 .58 1.31 2 33 3.64 5.24 50 .18 .73 l.tj4 2.91 4.55 6.54 1 .22 .87 1.96 3.49 5.45 7.85 10 .26 1.02 2.29 4.07 6.36 9.16 20 .29 1.16 2.62 4.65 7.27 10.47 30 .33 1.31 2.95 5.24 8.18 11.78 40 .36 1.45 3.27 5.83 9.09 13.08 50 .40 1.60 3.60 6.40 9.99 14.39 9 .44 1.75 3.93 6.98 10.90 15 69 10 .47 1.89 4.25 7.56 11.81 17.00 20 .51 2.04 4.58 8.14 12.72 18.30 30 .55 2.18 4.91 8.72 13.62 19.61 40 .58 2.33 5.23 9.30 14.53 20.91 50 .62 2.47 5.56 9.88 15.44 22.21 3 .65 2.62 5.89 10.46 16.34 23.52 10 .69 2.76 6.22 11.04 17.25 24.82 20 .73 2.91 6.54 11.62 18.15 26.12 30 .76 3.05 6.87 12.20 19.06 27.42 40 .80 3.20 7.20 12.78 19.96 28.71 50 .84 3.35 7.52 13.. 36 20.86 30.01 4 .87 3.49 7.85 13.94 21.77 31.31 TABLES. 219 TABLE v.— MID-ORUINATES TO LONG CHORDS. Degree of 1 2 3 4 5 6 Curve. Station. Stations. Stations. Stations. Stations. Stations. 4» 0' .87 3.49 7.85 13.94 21.77 31.31 10 .91 3.64 8.18 14.52 22.67 32 60 20 .95 3.78 8.50 15.10 23.57 33.90 30 .98 3.93 8.83 15.68 24.47 35.19 40 1.02 4.07 9.15 16.26 25.37 36.48 50 1.05 4 22 9.48 16.84 26.27 37.77 5 1.09 4.36 9.81 17.42 27.17 39.06 10 1.13 4.51 10.13 17.99 28.07 40.35 20 I.IG 4.65 10.46 18.57 28.97 41.63 30 1.20 4.80 10.79 19.15 29.87 42.92 40 1 24 4.94 11.11 19.72 30.76 44.20 50 1.27 5.09 11.44 20.30 31.66 45.48 6 1.31 5 23 11.76 20.88 32.55 46.76 10 1.35 5.38 12.09 21.45 .33.45 48.04 20 1.38 5.52 12.41 22 03 34.34 49.31 30 1.42 5.67 12.74 22.60 35.23 50.59 40 1.16 5.81 13.06 23.18 36.13 51.86 50 1.49 5.96 13.39 23.75 37.02 53.13 7 1 53 6.11 13.72 24.33 37.91 .54.40 10 1.56 6.25 14.03 24.89 38.79 55.64 20 1.60 6.39 14 36 25.46 39.66 56.90 30 1 64 6.54 14.68 26.04 40.55 58.16 40 1.67 6.68 15.01 26.61 41.43 59.41 .50 1.71 6.83 15.33 27.18 42.32 60.68 8 1.74 6.97 15.65 27.75 43.20 61.93 10 1.78 7.12 15.98 28.32 44.08 63.18 20 1.82 7.26 16.30 28.89 44.96 64.43 30 1.85 7.41 16.62 29.46 45.84 65.68 40 1.89 7.55 16.95 30.03 46 72 66.92 50 1.93 7.70 17.27 30.60 47.60 68.17 9 1.96 7.84 17.59 31.17 48.47 69.40 10 2.00 7.98 17.92 31.73 49.35 70.64 20 2.04 8.13 18.24 32.30 .50.22 71.88 30 2.07 8.27 18.56 32.87 51.09 73.11 40 2.11 8.42 18 88 33.43 51.96 74.34 50 2.14 8.. 56 19.21 34.00 52.83 75.56 10 2.18 8.71 19.53 .34.56 53.70 76.79 10 2 22 8.85 19.85 35.13 .54.57 20 2 25 9.00 20.17 35.69 55.43 30 2.29 9.14 20.49 36.26 56.29 40 2.. 33 9.28 20.82 36.82 57.16 50 2.. 36 9.43 21.14 37.38 58.02 11 2.40 9.57 21.46 37.94 58.88 10 2.44 9.72 21.78 38.50 59.73 20 2.47 9.86 22.10 39 00 60.58 30 2.51 10.01 22.42 39.62 61.44 40 2.54 10.15 22.74 40.18 62.30 50 2.58 10.29 23.06 40.74 63.15 12 2 62 10.44 23.38 41.30 64.00 10 2.65 10.58 23.70 41.86 64.85 20 2 69 10.73 24.02 42.41 65.69 30 2.73 10.87 24.34 42.97 66.54 40 2.76 11.01 24.66 43.52 67.38 50 2.80 11.16 24.97 44.08 68.22 13 2.83 11.30 25.29 44.63 69.06 « 10 2.87 11.45 25.61 45.18 69.90 20 2.91 11. .59 25.93 45.73 70.73 30 2.94 11.73 26 25 46.29 71.57 40 2 98 11.88 26.. 57 46.84 72.40 50 3.02 12.02 26.88 47.39 73.23 14 3 05 12.16 27.20 47 93 74.06 1 N O 12 3 4 5 6 7 8 9 100 00000 00043 00087 00130 00173 00217 00260 00303 00346 00389 1 0432 0475 0518 0561 0604 0647 0689 0732 0775 0817 2 0860 0903 0945 0988 1030 1072 1115 1157 1199 1242 3 1284 1326 1368 1410 1452 1494 1536 1578 1620 1662 4 1703 1745 1787 1828 1870 1912 1^53 1995 2036 2078 5 2119 2160 2202 2243 2284 2325 2366 2407 2449 2490 6 2531 2572 2612 2653 2694 2735 2776 2816 2857 2898 7 2938 2979 3019 3060 3100 3141 3181 3222 3262 330^ 8 3342 3383 3423 3463 3503 3543 3583 3623 3663 3703 9 3743 3782 3822 3862 3902 3941 3981 4021 4060 4100 110 04139 04179 04218 04258 04297 04336 04376 04415 04454 04493 1 4532 4571 4610 4650 4689 4727 4766 4805 4844 4883 2 4922 4961 4999 5038 5077 5115 5154 5192 5231 5269 3 5308 5346 5385 5423 5461 5500 5538 5576 5614 5652 4 5690 5729 5767 5805 5843 5881 5918 5956 5994 6032 5 6070 6108 6145 6183 6221 6258 6296 6333 6371 6408 6 6446 6483 6521 6558 6595 6633 6670 6707 6744 6781 7 6819 6856 6893 6930 6967 7004 7041 7078 7115 7151 8 7188 7225 7262 7298 7335 7372 7408 7445 7482 7518 9 7555 7591 7628 7664 7700 7737 7773 7809 7846 7882 120 07918 07954 07990 08027 08063 08099 08135 08171 08207 08243 1 8279 8314 8350 8386 8422 8458 8493 8529 8565 8600 2 8636 8672 8707 8743 8778 8814 8849 8884 8920 8955 3 8991 9026 9061 9096 9132 9167 9202 9237 9272 9307 4 9342 9377 9412 9447 9482 9517 9552 9587 9621 9656 5 9691 9726 9760 9795 9830 9864 9899 9934 9968 10003 6* 10037 10072 10106 10140 10175 10209 10243 10278 10312 0346 | 7 0380 0415 0449 0483 0517 0551 0585 0619 0653 0687 8 0721 0755 0789 0823 0857 0890 0924 0958 0992 1025 9 1059 1093 1126 1160 1193 1227 1261 1294 1327 1361 130 11394 11428 11461 11494 11528 11561 11594 11628 11661 11694 1 1727 1760 1793 1826 1860 1893 1926 1959 1992 2024 2 2057 2090 2123 2156 2189 2222 2254 2287 2320 2352 3 2385 2418 2450 2483 2516 2548 2581 2613 2646 2678 4 2710 2743 2775 2808 2840 2872 2905 2937 2969 3001 5 3033 3066 3098 3130 3162 3194 3226 3258 3290 3322 6 3354 3386 3418 3450 3481 3513 3545 3577 3609 3640 7 3672 3704 3735 3767 3799 3830 3862 3893 3925 3956 8 3988 4019 4051 4082 4114 4145 4176 4208 4239 4270 9 4301 4333 4364 4395 4426 4457 4489 4520 4551 4582 140 14613 14644 14675 14706 14737 14768 14799 14829 14860 14891 1 4922 4953 4983 5014 5045 5076 5106 5137 5168 5198 2 5229 5259 5290 5320 5351 5381 5412 5442 5473 5503 3 5534 5564 5594 5625 5655 5685 5715 5746 5776 5806 4 5836 5866 5897 5927 5957 5987 6017 6047 6077 6107 5 6137 6167 6197 6227 6256 6286 6316 6346 6376 6406 6 6435 6465 6495 6524 6554 6584 6613 6643 6673 6702 7 6732 6761 6791 6820 6850 6879 6909 6938 6967 6997 8 7026 7056 7085 7114 7143 7173 7202 7231 7260 7289 9 7319 7348 7377 7406 7435 7464 7493 7522 7551 7580 150 17609 17638 17667 17696 17725 17754 17782 17811 17840 17869 TABLE VI.— LOGARITHMS OF NUMP.EIJS. 22\ N 150 0123456789 17609 17638 17667 17696 17725 17754 17782 17811 17840 17869 1 7898 7926 7955 7984 8013 8041 8070 8099 8127 8156 2 8184 8213 8241 8270 8298 8327 8355 8384 8412 8441 3 8469 8498 8526 8554 8583 8611 8639 8667 8696 8724 4 8752 8780 8808 8837 8805 8893 8921 8949 8977 9005 5 9033 9061 9089 9117 9145 9173 9201 9229 9257 9285 6 9312 9340 9368 9396 9424 9451 9479 9507 9535 9562 7 9590 9618 9645 9673 9700 9728 9756 9783 9811 9838 8 9866 9893 9921 9948 9976 20003 20030 20058 20085 20112 9 20140 20167 20194 20222 20249 0276 0303 0330 0358 0385 160 20412 20439 20466 20493 20520 20548 20575 20602 20629 20656 1 0683 0710 0737 0763 0790 0817 0844 0871 0898 0925 2 0952 0978 1005 1032 1059 1085 1112 1139 1165 1192 3 1219 1245 1272 1299 1325 1352 1378 1405 1431 1458 4 1484 1511 1537 1564 1590 1617 1643 1669 1696 1722 5 1748 1775 1801 1827 1854 1880 1906 1932 1958 1985 6 2011 2037 2063 2089 2115 2141 2167 2194 2220 2246 7 2272 2298 2324 2350 2376 2401 2427 2453 2479 2505 8 2531 2557 2583 2608 2634 2660 2686 2712 2737 2763 9 2789 2814 2840 2866 2891 2917 2943 2968 2994 3019 170 23045 23070 23096 23121 23147 23172 23198 23223 23249 23274 1 3300 3325 3350 3376 3401 3426 3452 3477 3502 3528 2 3553 3578 3603 3629 3654 3679 3704 3729 3754 3779 3 3805 3830 3855 3880 3905 3930 3955 3980 4005 4030 4 4055 4080 4105 4130 4155 4180 4204 4229 4254 4279 5 4304 4329 4353 4378 4403 4428 4452 4477 4502 4527 6 4551 4576 4601 4625 4650 4674 4699 4724 4748 4773 7 4797 4822 4846 4871 4895 4920 4944 4969 4993 5018 8. 5042 5066 5091 5115 5139 5164 5188 5212 5237 5261 9 5285 5310 5334 5358 5382 5406 5431 5455 5479 5503 180 25527 25551 25575 25600 25624 25648 25672 25696 25720 25744 1 5768 5792 5816 5840 5864 5888 5912 5935 5959 5983 2 6007 6031 6055 6079 6102 6126 6150 6174 6198 6221 3 6245 6269 6293 6316 6340 6364 6387 6411 6435 6458 4 6482 6505 6529 6553 6576 6600 6623 6647 6670 6(594 5 6717 6741 6764 6788 6811 6834 6858 6881 6905 6928 6 6951 6975 6998 7021 7045 7068 7091 7114 7138 7T(U 7 7184 7207 7231 7254 7277 7300 7323 7346 7370 7393 8 7416 7439 7462 7485 7508 7531 7554 7577 7600 7623 9 7646 7069 7692 7715 7738 7761 7784 7807 7830' 7852 190 27875 27898 27921 27944 27967 27989 28012 28035 28058 28081 1 8103 8126 8149 8171 8194 8217 8240 8262 8285 8307 2 8330 8353 8375 8398 8421 8443 8466 8488 8511 8533 3 8556 8578 8601 8623 8646 8668 8691 8713 8735 8758 4 8780 8803 8825 8847 8870 '8892 8914 8937 895p 8981 5 0003 9026 9048 9070 9092 9115 9137 9159 9181 9203 6 9226 9248 9270 9292 9314 9336 9358 9380 9403 9425 7 9447 9469 9491 9513 9535 9557 9579 9601 9623 9645 8 9667 9688 9710 t)732 9754 9776 9798 9820 9842 9863 9 9885 9907 9929 9951 9973 9994 30016 30038 30060 30081 200 30103 30125 30146 30168 30190 30211 30233 30255 30276 30298 00 o V /^ fV TABLE VI.— LOGARITHMS OF NUMBERS. N 0123456789 200 30103 30125 30146 30168 30190 30211 30233 30255 30276 30298 1 032» 0341 0363 0384 0406 0428 0449 0471 0492 0514 2 0535 0557 0578 0600 0621 0643 0664 0685 0707 0728 3 0750 0771 0792 0814 0835 0856 0878 0899 0920 0942 4 0963 0984 1006 1027 1048 1069 1091 1112 1133 1154 5 1175 1197 1218 1239 1260 1281 1302 1323 1345 1366 6 1387 1408 1429 1450 1471 1492 1513 1534 1555 1576 7 1597 1618 1639 1660 1681 1702 1723 1744 1765 1785 8 1806 1827 1848 1869 1890 1911 1931 1952 1973 1994 9 2015 2035 2056 2077 2098 2118 2139 2160 2181 2201 210 32222 32243 32263 32284 82305 32325 32346 32366 32387 32408 1 2428 2449 2469 2490 2510 2531 2552 2572 2593 2613 2 2634 2654 2675 2695 2715 2736 2756 2777 2797 2818 3 2838 2858 2879 2899 2919 2940 2960 2980 3001 3021 4 3041 3062 3082 3102 3122 3143 3163 3183 3203 3224 5 3244 3264 3284 3304 3325 3345 3365 3385 8405 3425 6 8445 3465 3486 3506 3526 8546 3566 8586 8606 3626 7 3646 3666 8686 3706 3726 8746 8766 8786 3806 8826 8 3846 3866 3885 3905 3925 3945 3965 3985 4005 4025 9 4044 4064 4084 4104 4124 4143 4163 4183 4203 4223 220 34242 34262 34282 34301 34321 84341 34361 34380 34400 84420 1 4439 4459 4479 4498 4518 4537 4557 4577 4596 4616 2 4635 4655 4674 4694 4713 4733 4753 4772 4792 4811 3 4830 4850 4869 4889 4908 4928 4947 4967 4986 5005 4 5025 5044 5064 5083 5102 5122 5141 5160 5180 5199 5 5218 5238 5257 5276 5295 5315 5334 5353 5372 5392 6 5411 5430 5449 5468 5488 5507 5526 5545 5564 5583 7 5603 5622 5641 5660 5679 5698 5717 5736 6755 5774 8 5793 5813 5832 5851 5870 5889 5908 5927 5946 5965 9 5984 6003 6021 6040 6059 6078 6097 6116 6135 6154 230 86173 36192 36211 86229 36248 36267 36286 36305 36324 36342 1 6361 6380 6399 6418 6436 6455 6474 6493 6511 6530 2 6549 6568 6586 6605 6624 6642 6661 6680 6698 6717 3 6736 6754 6773 6791 6810 6829 6847 6866 6884 6903 4 6922 6940 6959 6977 6996 7014 7033 7051 7070 7088 5 7107 7125 7144 7162 7181 7199 7218 7236 7254 7273 6 7291 7310 7328 7346 7365 7383 7401 7420 7438 7457 7 7475 7493 7511 7530 7548 7566 7585 7603 7621 7639 8 7658 7676 7694 7712 7731 7749 7767 7785 7803 7822 9 7840 7858 7876 7894 7912 7931 7949 7967 7985 8003 240 38021 38039 38057 38075 38093 38112 38130 38148 38166 38184 1 8202 8220 8238 8256 8274 8292 8310 8328 8346 8364 2 8382 8399 8417 8435 8453 8471 8489 8507 8525 8543 3 8561 8578 8596 8614 8632 8650 8668 8686 8703 8721 4 8739 8757 8775 8792 8810 8828 8846 8863 8881 8899 5 8917 8934 8952 8970 8987 9005 9023 9041 9058 9076 6 9094 9111 9129 9146 9164 9182 9199 9217 9235 9252 7 9270 9287 9305 9322 9340 9358 9375 9393 9410 9428 8 9445 9463 9480 9498 9515 9533 9550 9568 9585 9602 9 9620 9637 9655 9672 9690 9707 9724 9742 9759 9777 250 39794 39811 39829 39846 39863 39881 39898 39915 89933 39950 TABLE VI.— LOGARITHMS OF NUMBERS. 223 N 0123456789 250 39794 39811 39829 39846 39863 39881 39898 39915 39933 39950 1 9967 9985 40002 40019 40037 40054 40071 40088 40106 40123 2 40140 40157 0175 0192 0209 0226 0243 0261 0278 0295 3 0312 0329 0346 0364 0381 0398 0415 0432 0449 0466 4 0483 0500 0518; 0535 0552 0569 0586 0603 0620 0637 5 0654 0671 0688 0705 0722 0739 0756 0773 0790 0807 6 0824 0841 0858 0875 0892 0909 0926 0943 0960 097(5 7 0993 1010 1027 1044 1061 1078 1095 1111 1128 1145 8 1162 1179 1196 1212 1229 1246 1263 1280 1296 1313 9 1330 1347 1363 1380 1397 1414 1430 1447 1464 1481 260 41497 41514 41531 41547 41564 41581 41597 41614 41631 41647 1 1664 1681 1697 1714 1731 1747 1764 1780 1797 1814 2 1830 1847 1863 1880 1896 1913 1929 1946 1963 1979 3 1996 2012 2029 2045 2062 2078 2095 2111 2127 2144 4 2160 2177 2193 2210 2226 2243 2259 2275 2292 2308 5 2325 2341 2357 2374 2390 2406 2423 2439 2455 2472 6 2488 2504 2521 2537 2553 2570 2586 2602 2619 2635 7 2651 2667 2684 2700 2716 2732 2749 2765 2781 2797 8 2813 2830 2846 2862 2878 2894 2911 2927 2943 2959 9 2975 2991 3008 3024 3040 3056 3072 3088 3104 3120 270 43136 43152 43169 43185 43201 43217 43233 43249 43265 43281 1 3297 3313 3329 3345 3361 3377 3393 3409 3425 3441 2 3457 3473 3489 3505 3521 3537 3553 3569 3584 3600 3 3616 3632 3648 3664 3680 3696 3712 3727 3743 3759 4 3775 3791 3807 3823 3838 3854 3870 3886 3902 3917 5 3933 3949 3965 3981 3996 4012 4028 4044 4059 4075 6 4091 4107 4122 4138 4154 4170 4185 4201 4217 4232 7 4248 4264 4279 4295 4311 4326 4342 4358 4373 4389 8 4404 4420 4436 4451 4467 4483 4498 4514 4529 4545 9 4560 4576 4592 4607 4623 4638 4654 4669 4685 4700 280 44716 44731 44747 44762 44778 44793 44809 44824 44840 44855 1 4871 4886 4902 4917 4932 4948 4963 4979 4994 5010 2 5025 5040 5056 5071 5086 5102 5117 5133 5148 5163 3 5179 5194 5209 5225 5240 5255 5271 5286 5301 6317 4 5332 5347 5362 5378 5393 5408 5423 5439 5454 5469 6 5484 5500 5515 5530 5545 5561 5576 5591 5606 5621 6 5637 5652 5667 5682 5697 5712 5728 5743 5758 5773 7 5788 5803 5818 5834 5849 5864 5879 5894 5909 5924 8 5939 5954 5969 5984 6000 6015 6030 6045 6060 6075 9 6090 6105 6120 6135 6150 6165 6180 6195 6210 6225 290 46240 46255 46270 46285 46300 46315 46330 46345 46359 46374 1 6389 6404 6419 6434 6449 6464 6479 6494 6509 6523 2 6538 6553 6568 6583 6598 6613 6627 6642 6657 6672 3 6687 6702 6716 6731 6746 6761 6776 6790 6805 6820 4 6835 6850 6864 6879 6894 6909 6923 6938 6953 6967 5 6982 6997 7012 7026 7041 7056 7070 7085 7100 7114 6 7129 7144 7159 7173 7188 7202 7217 7232 7246 7261 7 7276 7290 7305 7319 7334 7349 7363 7378 7392 7407 8 7422 7436 7451 7465 7480 7494 7509 7524 7538 7553 9 7567 7582 7596 7611 7625 7640 7654 7669 7683 7698 300 47712 47727 47741 47756 47770 47784 47799 47813 47828 47842 i>0 1 TABLE VI.— liOGARITHMS OF NUMBEKS. N 300 Ol 2 3456789 47712 47727 47741 47756 47770 47784 47799 47813 47828 47842 1 7857 7871 7885 7900 7914 7929 7943 7958 7972 7986 2 8001 8015 8029 8044 8058 8073 8087 8101 8116 8130 3 8144 8159 8173 8187 8202 8216 8230 8244 8259 8273 4 8287 8302 8316 8330 8344 8359 8373 8387 8401 8416 5 8430 8444 8458 8473 8487 8501 8515 8530 8544 8558 6 8572 8586 8601 8615 8629 8643 8657 8671 8686 8700 7 8714 8728 8742 8756 8770 8785 8799 8813 8827 8841 8 8855 8869 8883 8897 8911 8926 8940 8954 8968 8982 9 8996 9010 9024 9038 9052 9066 9080 9094 9108 9122 310 49136 49150 49164 49178 49192 49206 49220 49234 49248 49262 1 9276 9290 9304 9318 9332 9346 9360 9374 9388 9402 2 9415 9429 9443 9457 9471 9485 9499 9513 9527 9541 3 9554 9568 9582 9596 9610 9624 9638 9651 9665 9679 4 9693 9707 9721 9734 9748 9762" 9776 9790 9803 9817 5 9831 9845 9859 9872 9886 9900 9914 9927 9941 9955 6 9969 9982 9996 50010 50024 50037 50051 60065 50079 50092 7 50106 50120 50133 0147 0161 0174 0188 0202 0215 0229 8 0243 0256 0270 0284 0297 0311 0325 0338 0362 0365 9 0379 0393 0406 O420 0433 0447 046l 0474 0488 0501 320 50515 60629 60542 50556 60660 60683 50596 50610 60623 50637 1 0661 0664 0678 '0691 0705 0718 0732 0745 0769 0772 2 0786 0799 0813 0826 0840 0853 0866 0880 0893 0907 3 0920 0934 0947 0961 0974 0987 1001 1014 1028 1041 4 1065 1068 1081 1095 1108 1121 1135 1148 1162 1175 5 1188 1202 1215 1228 1242 1255 1268 1282 1295 1308 6 1322 1335 1348 1362 1375 1388 1402 1415 1428 1441 7 1455 1468 1481 1495 1508 1621 1634 1648 1661 1574 8 1687 1601 1614 1627 1640 1654 1667 1680 1693 1706 9 1720 1733 1746 1759 1772 1786 1799 1812 1825 1838 330 61851 61865 51878 61891 61904 61917 61930 51943 51957 51970 1 1983 1996 2009 2022 2035 2048 2061 207i 2088 2101 2 2114 2127 2140 2153 2166 2179 2192 ,2206 2218 2231 3 2244 2257 2270 2284 2297 2310 2323 233^ 2349 2362 4 2375 2388 2401 2414 2427 2440 2453 2466 2479 2492 5 2604 2517 2630 2643 2556 2669 2582 2595- 2608 2621 6 2634 2647 2660 2673 2686 2699 2711 2724 2737 2750 7 2763 2776 2789 2802 2815 2827 2840 2853 2866 2879 8 2892 2905 2917 2930 2943 2956 2969 2982 2994 3007 9 3020 3033 3046 3058 3071 3084 3097 3110 3122 3136 340 63148 53161 63173 53186 63199 63212 63224 53237 63250 53263 1 3275 3288 3301 3314 3326 3339 3352 3364 3377 3390 2 3403 3415 3428 3441 3453 3466 3479 3491 3504 3617 3 3529 3542 3555 3567 3580 3593 3605 3618 3631 ^.643 4 3656 3668 3681 3694 3706 3719 3732 3744 3757 21769 5 3782 3794 3807 3820 3832 3845 3857 3870 3882 ;-1895 6 3908 3920 3933 3945 3958 3970 3983 3995 4008 4 020 7 4033 4045 4058 4070 4083 4095 410^ 4120 4133 4145 8 4158 4170 4183 4195 4208 4220 4233 4245 4258 4270 9 4283 4295 4307 4320 4332 4345 4357 4370. 4382 4394 350 54407 54419 64432 54444 54466 54469 64481 64494 54606 64( ')18 TABLE VI.— LOGARITHMS OF NUMBERS. 22.J N 350 1 2 3 4 5 6 7 8 9 360 1 2 3 4 5 6 7 8 9 370 1 2 3 4 5 6 7 8 9 380 1 2 3 4 5 6 7 8 9 390 1 2 3 4 5 6 7 8 12 34 567 8 9 54407 54419 54432 54444 54456 54469 54481 54494 54506 54518 4531 4543 4555 4568 4580 4593 4605 4617 4630 4642 4654 4667 4679 4691 4704 4716 4728 4741 4753 4765 4777 4790 4802 4814 4827 4839 4851 4864 4876 4888 4900 4913 4925 4937 4949 4962 4974 4986 4998 5011 5023 5035 5047 5060 5072 5084 5096 5108 5121 5133 5145 5157 5169 5182 5194 5206 5218 5230 5242 5255 5267 5279 5291 5303 5315 5328 5340 5352 5364 5376 5388 5400 5413 5425 5437 5449 5461 5473 5485 5497 5509 5522 5534 5546 5558 5570 5582 5594 5606 5618 55630 55642 55654 55666 55678 55691 55703 55715 55727 55739 5751 5763 5775 .5787 5799 5811 5823 5835 5847 5859 5871 5883 5895 5907 5919 5931 5943 5955 5967 5979 5991 6003 6015 6027 6038 6050 6062 6074 6086 6098 6110 6122 6134 6146 6158 6170 6182 6194 6205 6217 6229 6241 6253 6265 6277 6289 6301 6812 6324 6336 6348 6360 6372 6384 6396 6407 6419 6431 6443 6455 6467 6478 6490 6502 6514 6526 6538 6549 6561 6573 6585 6597 6608 6620 6632 6644 6656 6667 6679 6691 6703 6714 6726 6738 6750 6761 6773 6785 6797 6808 56820 56832 56844 56855 56867 56879 56891 56902 56914 56926 6937 6949 6961 6972 6984 6996 7008 7019 7031 7043 7054 7066 7078 7089 7101 7113 7124 7136 7148 7159 7171 7183 7194 7206 7217 7229 7241 7252 7264 7276 7287 7299 7310 7322 7334 7345 7357 7368 7380 7392 7403 7415 7426 7438 7449 7461 7473 7484 7496 7507 7519 7530 7542 7553 7565 7576 7588 7600 7611 7623 7634 7646 7657 7669 7680 7692 7703 7715 7726 7738 7749 7761 7772 7784 7795 7807 7818 7830 7841 7852 7864 7875 7887 7898 7910 7921 7933 7944 7955 7967 57978 57990 58001 58013 58024 58035 58047 58058 58070 58081 8092 8104 8115 8127 8138 8149 8161 8172 8184 8195 8206 8218 8229 8240 8252 8263 8274 8286 8297 8309 8320 8331 8343 8354 8365 8377 8388 8399 8410 8422 8433 8444 8456 8467 8478 8490 8501 8512 8524 8535 8546 8557 8569 8580 8591 8602 8614 8625 8636 8647 8659 8670 8681 8692 8704 8715 8726 8737 8749 8760 8771 8782 8794 8805 8816 8827 8838 8850 8861 8872 8863 8894 8906 8917 8928 8939 8950 8961 8973 8984 8995 9006 9017 9028 9040 9051 9062 9073 9084 9095 59106 59118 59129 59140 59151 59162 59173 59184 59195 59207 9218 9229 9240 9251 9262 9273 9284 9295 9306 9318 9329 9340 9351 9362 9373 9384 9395 9406 9417 9428 9439 9450 9461 9472 9483 9494 9506 9517 9528 9539 9550 9561 9572 9583 9594 9605 9616 9627 9638 9649 9660 9671 9682 9693 9704 9715 9726 9737 9748 9759 9770 9780 9791 9802 9813 9824 9835 9846 9857 9868 9879 9890 9901 9912 9923 9934 9945 9956 9966 9977 9988 9999 60010 60021 60032 60043 60054 60065 60076 60086 9 60097 60108 0119 0130 0141 0152 0103 0173 0184 0195 400 60206 60217 60228 60239 60249 60260 60271 60282 60293 60304 226 TABLE VI.— LOGARITHMS OF NUMBERS. N 400 O 1 3 3 4 5 6 7 8 9 60206 60217 60228 60239 60249 60260 60271 60282 60293 60304 1 0314 0325 0336 0347 0358 0369 0379 0390 0401 0412 2 0423 0433 0444 0455 0466 0477 0487 0498 0509 0520 3 0531 0541 0552 0563 0574 0584 0595 0606 0617 0627 4 0638 0649 0660 0670 0681 0692 0703 0713 0724 0735 5 0746 0756 0767 0778 0788 0799 0810 0821 0831 0842 6, 0853 0863 0874 0885 0895 0906 0917 0927 0938 0949 7 0959 0970 0981 0991 1002 1013 1023 1034 1045 1055 8 1066 1077 1087 1098 1109 1119 1130 1140 1151 1162 9 1172 1183 1194 1204 1215 1225 1236 1247 1257 1268 410 61278 61289 61300 61310 61321 61331 61342 61352 61363 61374 1 1384 1395 1405 1416 1426 1437 1448 1458 1469 1479 2 1490 1500 1511 1521 1532 1542 1553 1563 1574 1584 3 1595 1606 1616 1627 1637 1648 1658 1669 1679 1690 4 1700 1711 1721 1731 1742 1752 1763 1773 1784 1794 5 1805 1815 1826 1836 1847 1857 1868 1878 1888 1899 6 1909 1920 1930 1941 1951 1962 1972 1982 1993 2003 7 2014 2024 2034 2045 2055 2066 2076 2086 2097 2107 8 2118 2128 2138 2149 2159 2170 2180 2190 2201 2211 9 2221 2232 2242 2252 2263 2273 2284 2294 2304 2315 420 62325 62335 62346 62356 62366 62377 62387 62397 62408 62418 1 2428 2439 2449 2459 2469 2480 2490 2500 2511 2521 2 2531 2542 2552 2562 2572 2583 2593 2603 2613 2624 3 2634 2644 2655 2665 2675 2685 2696 2706 2716 2726 4 2737 2747 2757 2767 2778 2788 2798 2808 2818 2829 5 2839 2849 2859 2870 2880 2890 2900 2910 2921 2931 6 2941 2951 2961 2972 2982 2992 3002 3012 3022 3033 7 3043 3053 3063 3073 3083 3094 3104 3114 3124 3134 8 3144 3155 3165 3175 3185 3195 3205 3215 3225 3236 9 3246 3256 3266 3276 3286 3296 3306 3317 3327 3337 430 63347 63357 63367 63377 63387 63397 63407 63417 63428 63438 1 3448 3458 3468 3478 3488 3498 3508 3518 3528 3538 2 3548 3558 3568 3579 3589 3599 3609 3619 3629 3639 3 3649 3659 3669 ' 3679 3689 3699 3709 3719 3729 3739 4 3749 3759 3769 3779 3789 3799 3809 3819 3829 3839 5 3849 3859 3869 3879 3889 3899 3909 3919 3929 3939 6 3949 3959 3969 3979 3988 3998 4008 4018 4028 4038 7 4048 4058 4068 4078 4088 4098 4108 4118 4128 4137 8 4147 4157 4167 4177 4187 4197 4207 4217 4227 4237 9 4246 4256 4266 4276 4286 4296 4306 4316 4326 4835 440 64345 64355 64365 64375 64385 64395 64404 64414 64424 64434 1 4444 4454 4464 4473 4483 4493 4503 4513 4523 4532 2 4542 4552 4562 4572 4582 4591 4601 4611 4621 4631 3 4640 4650 4660 4670 4680 4689 4699 4709 4719 4729 4 4738 4748 4758 4768 4777 4787 4797 4807 4816 4826 5 4836 4846 4856 4865 4875 4885 4895 4904 4914 4924 6 4933 4943 4953 4963 4972 4982 4992 5002 5011 5021 7 5031 5040 5050 5060 5070 5079 5089 5099 5108 5118 8 5128 5137 5147 5157 5167 5176 5186 5196 5205 5215 9 5225 5234 5244 5254 5263 5273 5283 5292 5302 5312 450 65321 65331 65341 65350 65360 65369 65379 65389 65398 65408 r^ TABLE VI.— J.O(JAHITIIMS OF NTMBEHS. 227 N O 1 2 3 4 5 6 7 8 9 450 65321 05831 65341 65350 65360 65360 65370 65380 65308 65408 1 5418 5427 5437 5447 5456 5466 5475 5485 5405 5504 2 5514 5523 5533 5543 5552 5562 5571 5581 5501 5600 3 5610 5610 5620 5630 5648 6658 5667 5677 5686 5606 4 5706 5715 5725 5734 5744 5753 5763 5772 5782 5702 5 5801 5811 5820 5830 5830 5840 5858 5868 5877 5887 6 5806 5006 5016 5025 5035 5044 5054 5063 5973 5082 7 5002 6001 6011 6020 6030 6030 6040 6058 6068 6077 8 6087 6006 6106 6115 6124 6134 6143 6153 6162 6172 9 6181 6101 6200 6210 6210 6220 6238 6247 6257 6266 460 66276 66285 66205 66304 66314 66323 66332 66342 66351 66361 1 6370 6380 6380 6308 6408 6417 6427 6436 6445 6455 2 6464 6474 6483 6402 6502 6511 6521 6580 6530 6540 3 6558 6567 6577 6586 6506 6605 6614 6624 6633 6642 4 6652 6661 6671 6680 6680 6600 6708 6717 6727 6736 5 6745 6755 6764 6773 6783 6702 6801 6811 6820 6820 -§ 6830 6848 6857 6867 6876 6885 6804 6004 6013 6022 7 6032 6041 6050 6060 6969 697 S 6087 6007 7006 7015 S 7025 7034 7043 7052 7062 7071 7080 7080 7000 7108 9 7il7 7127 7136 7145 7154 7164 7173 7182 7101 7201 470 67210 67210 67228 67237 67247 67256 67265 67274 67284 67203 1 7302 7311 7321 7330 7380 7848 7857 7867 7376 7885 2 7304 7403 7413 7422 7481 7440 7440 7450 7468 7477 3 7486 7405 7504 7514 7523 7532 7541 7550 7560 7560 4 7578 7587 7506 7605 7614 7624 7638 7642 7651 7660 5 7660 7670 7688 7607 7706 7715 7724 7733 7742 7752 6 7761 7770 7770 7788 7707 7806 7815 7825 7834 7843 7 7852 7861 7870 7870 7888 7807 7006 7016 7025 7034 8 7043 7052 7061 7070 7070 7088 7007 8006 8015 8024 9 8034 8043 8052 8061 8070 8070 8088 8007 8106 8115 480 68124 68133 68142 68151 68160 68160 68178 68187 68196 68205 1 8215 8224 -8233 8242 8251 8260 8260 8278 8287 8206 2 8305 8314 8323 «332 8341 8350 8350 8368 8377 8386 3 8305 8404 8413 8422 8431 8440 8440 8458 8467 8476 4 8485 8404 8502 8511 8520 8520 8588 8547 8556 8565 5 8574 8583 8502 8601 8610 8610 8628 8637 8646 8655 6 8664 8673 8681 8600 8600 8708 8717 8726 8785 8744 7 8753 8762 8771 8780 8780 8707 8806 8815 8824 8833 8 8842 8851 8860 8860 8878 8886 8805 8004 8018 8022 9 8031 8040 8040 8058 8066 8075 8084 8003 0002 0011 490 60020 60028 60037 60046 60055 60064 60073 60082 60000 60000 1 0108 0117 0126 0135 0144 0152 0161 0170 0170 0188 2 0107 0205 0214 0223 0232 0241 0240 0258 0267 0276 3 9285 0294 0302 9311 0320 0320 0338 0346 0355 9364 4 0373 0381 0300 0300 9408 9417 9425 9434 9443 9452 5 0461 0460 0478 0487 9406 0504 0513 0522 0531 0530 6 0548 0557 0566 0574 0583 0502 0601 9600 0618 0627 7 0636 0644 0653 9662 0671 0670 0688 0607 0705 0714 8 0723 0732 0740 0740 0758 0767 0775 0784 0703 0801 9 0810 0810 0827 0836 0845 0854 0862 0871 0880 0888 500 60807 60006 60014 60023 60032 60040 60040 60058 00066 60075 228 TABLE VI —LOGARITHMS OF XTMBERS. N O 1 2 3 4 5 6 7 8 9 500 69897 69906 69914 69923 69932 69940 69949 69958 69966 69975 | 1 9984 9992 70001 70010 70018 70027 70036 70044 70053 70062 2 70070 70079 0088 0096 0105 0114 0122 0131 0140 0148 3 0157 0165 0174 0183 0191 0200 0209 0217 0226 0234 4 0243 0252 0260 0269 0278 0286 0295 0303 0312 0321 5 0329 0338 0346 0355 0364 0372 0381 0389 0398 0406 6 0415 0424 0432 0441 0449 0458 0467 0475 0484 0492 7 0501 0509 0518 0526 0535 0544 0552 0561 0569 0578 8 0586 0595 0603 0612 0621 0629 0638 0646 0655 0663 9 0672 0680 0689 0697 0706 0714 0723 0731 0740 0749 510 70757 70766 70774 70783 70791 70800 70808 70817 70825 70834 1 0842 0851 0859 0868 0876 0885 0893 0902 0910 0919 2 0927 0935 0944 0952 0961 0969 0978 0986 0995 1003 3 1012 1020 1029 1037 1046 1054 1063 1071 1079 1088 4 1096 1105 1113 1122 1130 1139 1147 1155 1164 1172 5 1181 1189 1198 1206 1214 ■ 1223 1231 1240 1248 1257 6 1265 1273 1282 1290 1299 1307 1315 1324 1332 1341 7 1349 1357 1366 1374 1383 1391 1399 1408 1416 1425 8 1433 1441 1450 1458 1466 1475 1483 1492 1500 1508 9 1517 1525 1533 1542 1550 1559 1567 1575 1584 1592 520 71600 71609 71617 71625 71634 71642 71650 71659 71667 71675 1 1684 1692 1700 1709 1717 1725 1734 1742 1750 1759 2 1767 1775 1784 1792 1800 1809 1817 1825 1834 1842 3 1850 1858 1867 1875 1883 1892 1900 1908 1917 1925 4 1933 1941 1950 1958 1966 1975 1983 1991 1999 2008 5 2016 2024 2032 2041 2049 2057 2066 2074 2082 2090 6 2099 2107 2115 2123 2132 2140 2148 2156 2165 2173 7 2181 2189 2198 2206 2214 2222 2230 2239 2247 2255 8 2263 2272 2280 2288 2296 2304 2313 2321 2329 2337 9 2346 2354 2362 2370 2378 2387 2395 2403 2411 2419 530 72428 72436 72444 72452 72460 72469 72477 72485 72493 72501 1 2509 2518 2526 2534 2542 2550 2558 2567 2575 2583 2 2591 2599 2607 2616 2624 26.^ 2640 2648 2656 2665 3 2673 2681 2689 2697 2705 2713 2722 2730 2738 2746 4 2754 2762 2770 2779 2787 2795 2803 2811 2819 2827 5 2835 2843 2852 2860 2868 2876 2884 2892 2900 2908 6 2916 2925 2933 2941 2949 2957 2965 2973 2981 2989 7 2997 3006 3014 3022 3030 3038 3046 3054 3062 3070 8 3078 3086 3094 3102 3111 3119 3127 3135 3143 3151 9 3159 3167 3175 3183 3191 3199 3207 3215 3223 3231 540 73239 73247 73255 73263 73272 73280 73288 73296 73304 73312 1 3320 3328 3336 3344 3352 3360 3368 3376 3384 3392 2 3400 3408 3416 3424 3432 3440 3448 3456 3464 3472 3 3480 3488 3496 3504 3512 3520 3528 3536 3544 3552 4 3560 3568 3576 3584 3592 3600 3608 3616 3624 3632 5 3640 3648 3656 3664 3672 3679 3687 3695 3703 3711 6 3719 3727 3735 3743 3751 3759 3767 3775 3783 3791 7 3799 3807 3815 3823 3830 3838 3846 3854 3862 3870 8 3878 3886 3894 3902 3910 3918 3926 3933 3941 3949 9 3957 3965 3973 3981 3989 3997 4005 4013 4020 4028 550 74036 74044 74052 74060 74068 74076 74084 74092 74099 74107 1 TABLE VI.— LOGARITHMS OF NUMBERS. 229 N O 1 2 3 4 5 6 7 8 9 650 74036 74044 74052 74060 74068 74076 74084 74002 74099 74107 1 4115 4128 4131 4139 4147 4155 4162 4170 4178 4186 2 4194 4202 4210 4218 4225 4233 4241 4249 4257 4265 3 4273 4280 4288 4296 4304 4312 4320 4327 4335 4343 4 4351 4359 4367 4374 4382 4390 4398 4406 4414 4421 5 4429 4437 4445 4453 4461 4468 4476 4484 4492 4500 6 4507 4515 4523 4531 4539 4547 4554 4562 4570 4578 7 4586 4593 4601 4609 4617 4624 4632 4640 4648 4656 8 4663 4671 4679 4687 4695 4702 4710 4718 4726 4733 9 4741 4749 4757 4764 4772 4780 4788 4796 4803 4811 560 74819 74827 74834 74842 74850 74858 74865 74873 74881 74889 1 4896 4904 4912 4920 4927 4935 4943 4950 4958 4966 2 4974 4981 4989 4997 5005 5012 5020 5028 5035 5043 3 5051 5059 5066 5074 5082 5089 5097 5105 5113 5120 4 5128 5136 5143 5151 5159 5166 5174 5182 5189 5197 5 5205 5213 5220 5228 5236 5243 5251 5259 5266 5274 6 5282 5289 5297 5305 5312 5320 5328 5335 5343 5351 7 5358 5366 5374 5381 5389 5397 5404 5412 5420 5427 8 5435 5442 5450 5458 5465 5473 5481 5488 5496 5504 9 5511 5519 5526 5534 5542 5549 5557 5565 5572 5580 570 75587 75595 75603 75610 75618 75626 75633 75641 75648 75656 1 5664 5671 5679 5686 5694 5702 5709 5717 5724 5732 2 5740 5747 5755 5762 5770 5778 5785 5793 5800 5808 3 5815 5823 5831 5838 5846 5853 5861 5868 5876 5884 4 5891 5899 5906 5914 5921 5929 5937 5944 5952 5959 5 5967 5974 5982 5989 5997 6005 6012 6020 6027 6035 6 6042 6050 6057 6065 6072 6080 6087 6095 6103 6110 7 6118 6125 6133 6140 6148 6155 6163 6170 6178 6185 8 6193 6200 0208 6215 6223 6230 6238 6245 6253 6260 9 6268 6275 6283 6290 6298 6305 6313 6320 6328 6335 580 76343 76350 76358 76365 76373 76380 76388 76395 76403 76410 1 6418 6425 6433 6440 6448 6455 6462 6470 6477 6485 2 6492 6500 6507 6515 6522 6530 6537 6545 6552 6559 3 6567 6574 6582 6589 6597 6604 6612 6619 6626 6634 4 6641 6649 6656 6664 6671 6678 6686 6693 6701 6708 5 6716 6723 6730 6738 6745 6753 6760 6768 6775 6782 6 6790 6797 6805 6812 6819 6827 6834 6842 6849 6856 7 6864 6871 6879 6886 6893 6901 6908 6916 6923 6930 8 0938 694^ 6953 6960 6967 6975 6982 6989 6997 7004 9 7012 7019 7026 7034 7041 7048 7056 7063 7070 7078 590 77085 77093 77100 77107 77115 77122 77129 77137 77144 77151 1 7159 7166 7173 7181 7188 7195 7203 7210 7217 7225 2 7232 7240 7247 7254 7262 7269 7276 7283' 7291 7298 3 7305 7313 7320 7327 7335 7342 7349 7357 7364 7371 4 7379 7386 7393 7401 7408 7415 7422 7430 7437 7444 5 7452 7459 7466 7474 7481 7488 7495 7503 7510 7517 6 7525 7532 7539 7546 7554 7561 7568 7576 7583 7590 7 7597 7605 7612 7619 7627 7634 7641 7648 7656 7663 8 7670 7677 7685 7692 7699 7706 7714 7721 7728 7735 9 7743 7750 7757 7764 7772 7779 7786 7793 7801 7808. 600 77815 77822 77830 77837 77844 77851 77859 77866 77873 77880 TABLE VI.— LOGARITHMS OF NUMBERS. N O 1 2 3 4 5 6 7 8 9 600 77815 77822 77830 77837 77844 77851 77859 77866 77873 77880 1 7887 7895 7902 7909 7916 7924 7931 7938 7945 7952 2 7960 7967 7974 7981 7988 7996 8003 8010 8017 8025 3 8032 8039 8046 8053 8061 8068 8075 8082 8089 8097 4 8104 8111 8118 8125 8132 8140 8147 8154 8161 8168 5 8176 8183 8190 8197 8204 8211 8219 8226 8233 8240 6 8247 8254 8262 8269 8276 8283 8290 8297 8305 8312 7 8319 8326 8333 8340 8347 8355 8362 8369 8376 8383 8 8390 8398 8405 8412 8419 8426 8433 8440 8447 8455 9 8462 8469 8476 8483 8490 8497 8504 8512 8519 8526 610 78533 78540 78547 78554 78561 78569 78576 78583 78590 78597 1 8604 8611 8618 8625 8633 8640 8647 8654 8661 8668 2 8675 8682 8689 8696 8704 8711 8718 8725 8732 8739 3 8746 8753 8760 8767 8774 8781 8789 8796 8803 8810 4 8817 8824 8831 8838 8845 8852 8859 8866 8873 8880 5 8888 8895 8902 8909 8916 8923 8930 8937 8944 8951 6 8958 8965 8972 8979 8986 8993 9000 9007 9014 9021 7 9029 9036 9043 9050 9057 9064 9071 9078 9085 9092 8 9099 9106 9113 9120 9127 9134 9141 9148 9155 9162 9 9169 9176 9183 9190 9197 9204 9211 9218 9225 9232 620 79239 79246 79253 79260 79267 79274 79281 79288 79295 79302 1 9309 9316 9323 9330 9337 9344 9351 9358 9365 9372 2 9379 9386 9393 9400 9407 9414 9421 9428 9435 9442 3 9449 9456 9463 9470 9477 9484 9491 9498 9505 9511 4 9518 9525 9532 9539 9546 9553 9560 9567 9574 9581 5 9588 9595 9602 9609 9616 9623 9630 9637 9644 9650 6 9657 9664 9671 9678 9685 9692 9699 9706 9713 9720 7 9727 9734 9741 9748 9754 9761 9768 9775 9782 9789 8 9796 9803 9810 9817 9824 9831 9837 9844 9851 9858 9 9865 9872 9879 9886 9893 9900 9906 9913 9920 9927 630 79934 79941 79948 79955 79962 79969 79975 79982 79989 79996 1 80003 80010 80017 80024 80030 80037 80044 80051 80058 80065 | 2 0072 0079 0085 0092 0099 0106 0113 0120 0127 0134 3 0140 0147 0154 0161 0168 0175 0182 0188 0195 0202 4 0209 0216 0223 0229 0236 0243 0250 0257 0264 0271 5 0277 0284 0291 0298 0305 0312 0318 0325 0332 0339 6 0346 0353 0359 0366 0373 0380 0387 0393 0400 0407 7 0414 0421 0428 0434 0441 0448 0455 0462 0468 0475 8 0482 0489 0496 0502 0509 0516 0523 0530 0536 0543 9 0550 0557 0564 0570 0577 0584 0591 0598 0604 0611 640 80618 80625 80632 80638 80645 80652 80659 80665 80672 80679 | 1 0686 0693 0699 0706 0713 0720 0726 0733 0740 0747 2 0754 0760 0767 0774 0781 0787 0794 0801 0808 0814 3 0821 0828 0835 0841 0848 0855 0862 0868 0875 0882 4 0889 0895 0902 0909 0916 0922 0929 0936 0943 0949 5 0956 0963 0969 0976 0983 0990 0996 1003 1010 1017 3 1023 1030 1037 1043 1050 1057 1064 1070 1077 1084 7 1090 1097 1104 1111 1117 1124 1131 1137 1144 1151 8 1158 1164 1171 1178 1184 1191 1198 1204 1211 1218 9 1224 1231 1238 1245 1251 1258 1265 1271 1278 1285 650 1 81291 81298 81305 81311 81318 81325 81331 81338 81345 81351 TABLE VI.— LOGARITHMS OF NUMBEKB. 331 N 01234567 89 650 81291 81298 81305 81311 81318 81325 81331 81338 81345 81351 1 1358 1365 1371 1378 1385 1391 1398 1405 1411 1418 2 1425 1431 1438 1445 1451 1458 1465 1471 1478 1485 3 1491 1498 1505 1511 1518 1525 1531 1538 1544 1551 4 1558 1564 1571 1578 1584 1591 1598 1604 1611 1617 5 1624 1631 1637 1644 1651 1657 1664 1671 1677 1684 6 1690 1697 1704 1710 1717 1723 1730 1737 1743 1750 7 1757 1763 1770 1776 1783 1790 1796 1803 1809 1816 8 1823 1829 1836 1842 1849 1856 1862 1869 1875 1882 9 1889 1895 1902 1908 1915 1921 1928 1935 1941 1948 660 81954 81961 81968 81974 81981 81987 81994 82000 82007 82014 1 2020 2027 2033 2040 2046 2053 2060 2066 2073 2079 2 2086 2092 2099 2105 2112 2119 2125 2132 2138 2145 3 2151 2158 2164 2171 2178 2184 2191 2197 2204 2210 4 2217 2223 2230 2236 2243 2249 2256 2263 2269 2276 5 2282 2289 2295 2302 2308 2315 2321 2328 2334 2341 6 2347 2354 2360 2367 2373 2380 2387 2393 2400 2406 7 2413 2419 2426 2432 2439 2445 2452 2458 2465 2471 8 2478 2484 2491 2497 2504 2510 2517 2523 2530 2536 9 2543 2549 2556 2562 2569 2575 2582 2588 2595 2601 670 82607 82614 82620 82627 82633 82640 82646 82653 82659 82666 1 2672 2679 2685 2692 2698 2705 2711 2718 2724 2730 2 2737 2743 2750 2756 2763 2769 2776 2782 2789 2795 3 2802 2808 2814 2821 2827 2834 2840 2847 2853 2860 4 2866 2872 2879 2885 2892 2898 2905 2911 2918 2924 5 2930 2937 2943 2950 2956 2963 2969 2975 2982 2988 6 2995 3001 3008 3014 3020 3027 3033 3040 3046 3052 7 3059 3065 3072 3078 3085 3091 3097 3104 3110 3117 8 3123 3129 3136 3142 3149 315i 3161 3168 3174 3181 9 3187 3193 3200 3206 3213 3219 3225 3232 3238 3245 680 83251 83257 83264 83270 83276 83283 83289 83296 83302 83308 1 3315 3321 3327 3334 3340 3347 3353 3359 3366 3372 2 3378 3385 3391 3398 3404 3410 3417 3423 3429 3436 3 3442 3448 3455 3461 3467 3474 3480 3487 3493 3499 4 3506 3512 3518 3525 3531 3537 3544 3550 3556 3563 5 3569 3575 3582 3588 3594 3601 3607 3613 3620 3626 6 3632 3639 3645 3651 3658 3664 3670 3677 3683 3689 7 3696 3702 3708 3715 3721 3727 3734 3740 3746 3753 8 3759 3765 3771 3778 3784 3790 3797 3803 3809 3816 9 3822 3828 3835 3841 3847 3853 3860 3866 3872 3879 690 83885 83891 83897 83904 83910 83916 83923 83929 83935 83942 1 3948 3954 3960 3967 3973 3979 3985 3992 3998 4004 2 4011 4017 4023 4029 4036 4042 4048 4055 4061 4067 3 4073 4080 4086 4092 4098 4105 4111 4117 4123 4130 4 4136 4142 4148 4155 4161 4167 4173 4180 4186 4192 5 4198 4205 4211 4217 4223 4230 4236 4242 4248 4255 6 4261 4267 4273 4280 4286 4292 4298 4305 4311 4317 7 4323 4330 4336 4342 4348 4354 4361 4367 4373 4379 8 4386 4392 4398 4404 4410 4417 4423 4429 4435 4442 9 4448 4454 4460 4466 4473 4479 4485 4491 4497 4504 700 84510 84516 84522 84528 84535 84541 84547 84553 84559 84566 /^ >J ryj TABLE VI.— LOGARITHMS OF NUMBERS. N 0123456789 700 ' 84510 84516 84522 84528 84535 84541 84547 84553 84559 84566 1 ! 4572 4578 4584 4590 4597 4603 4609 4615 4621 4628 2 j 4634 4640 4646 4652 4658 4665 4671 4677 4683 4689 3 I 4696 4702 4708 4714 4720 4726 4733 4739 4745 4751 4 i 4757 4763 4770 4776 4782 4788 4794 4800 4807 4813 5 4819 4825 4831 4837 4844 4850 4856 4862 4868 4874 6 4880 4887 4893 4899 4905 4911 4917 4924 4930 4936 7 4942 4948 4954 4960 4967 4973 4979 4985 4991 4997 8 5003 5009 5016 5022 5028 5034 5040 5046 5052 5058 9 5065 5071 5077 5083 5089 5095 5101 5107 5114 5120 710 1 2 3 4 5 6 7 8 9 85126 85132 85138 85144 85150 85156 85163 85169 85175 85181 5187 5193 5199 5205 5211 5217 5224 5230 5236 5242 5248 5254 5260 5266 5272 5278 5285 5291 5297 5303 5309 5315 5321 5327 5333 5339 5345 5352 5358 5364 5370 5376 5382 5388 5394 5400 5406 5412 5418 5425 5431 5437 5443 5449 5455 5461 5467 5473 5479 5485 5491 5497 5503 5509 5516 5522 5528 5534 5540 5546 5552 5558 5564 5570 5576 5582 5588 5594 5600 5606 5612 5618 5625 5631 5637 5643 5649 5655 5661 5667 5673 5679 5685 5691 5697 5703 5709 5715 5721 5727 720 85733 85739 85745 85751 85757 85763 85769 85775 85781 85788 5794 5800 5806 5812 5818 5824 5830 5836 5842 5848 5854 5860 5866 5872 5878 5884 5890 5896 5902 5908 5914 5920 5926 5932 5938 5944 5950 5956 5962 5968 4 5974 5980 5986 5992 5998 6004 6010 6016 6022 6028 5 6034 6040 6046 6052 6058 0064 6070 6076 6082 6088 6 6094 6100 6106 6112 6118 6124 6130 6136 6141 6147 7 6153 6159 6165 6171 6177 6183 6189 6195 6201 6207 8 6213 6219 6225 6231 6237 6243 6249 6255 6261 6267 9 6273 6279 6285 6291 6297 6303 6308 6314 6320 6326 730 86332 86338 86344 86350 86356 86362 86368 86374 86380 86386 1 6392 6398 6404 6410 6415 6421 6427 6433 6439 6445 2 6451 6457 6463 6469 6475 6481 6487 6493 6499 6504 3 6510 6516 6522 6528 6534 6540 6546 6552 6558 6564 4 6570 6576 6581 6587 6593 6599 6605 6611 6617 6623 5 6629 6635 6641 6646 6652 6658 6664 6670 6676 6682 6 6688 6694 6700 6705 6711 6717 6723 6729 6735 6741 7 6747 6753 6759 6764 6770 6776 6782 6788 6794 6800 8 6806 6812 6817 6823 6829 6835 6841 6847 6853 6859 9 6864 6870 6876 6882 6888 6894 6900 6906 6911 6917 740 86923 86929 86935 86941 86947 86953 86958 86964 86970 86976 1 6982 6988 6994 6999 7005 7011 7017 7023 7029 7035 2 7040 7046 7052 7058 7064 7070 7075 7081 7087 7093 3 7099 7105 7111 7116 7122 7128 7134 7140 7146 7151 4 7157 7163 7169 7175 7181 7186 7192 7198 7204 7210 5 7216 7221 7227 7233 7239 7245 7251 7256 7262 7268 6 7274 7280 7286 7291 7297 7303 7309 7315 7320 7326 7 7332 7338 7344 7349 7355 7361 7367 7373 7379 7384 8 7390 7396 7402 7408 7413 7419 7425 7431 7437 7442 9 7448 7454 7460 7466 7471 7477 7483 7489 7495 7500 750 87506 87512 87518 87523 87529 87535 87541 87547 87552 87558 TABLE VI.-LO(JAIUTIIMS OF NUMBERS 'iSr] N O 1 2 3 4 5 6 7 8 9 750 87506 87512 87518 87523 87529 87535 87541 87547 87552 87558 1 7564 7570 7570 7581 7587 7593 7599 7604 7610 7616 2 7622 7628 7633 7639 7645 7651 7656 7662 7668 7(574 3 7679 7685 7691 7697 7703 7708 7714 7720 7726 7731 4 7737 7743 7749 7754 7760 7766 7772 7777 7783 7789 5 7795 7800 7806 7812 7818 7823 7829 7835 7841 7846 6 7852 7858 7864 7869 7875 7881 7887 7892 7898 7904 7 7910 7915 7921 7927 7933 7938 7944 7950 7955 79(51 8 7967 7973 7978 7984 7990 7996 8001 8007 8013 8018 9 8024 8030 8030 8041 8047 8053 8058 8064 8070 8076 760 88081 88087 88093 88098 88104 88110 88116 88121 88127 88133 1 8138 8144 8150 8156 8161 8167 8173 8178 8184 8190 2 8195 8201 8207 8213 8218 8224 8230 8235 8241 8247 3 8252 8258 8264 8270 8275 8281 8287 8292 8298 8304 4 8309 8315 8321 8326 8332 8338 8343 8349 8355 83(50 5 8366 8372 8377 8383 8389 8395 8400 8406 8412 8417 6 8423 8429 8434 8440 8446 8451 8457 8463 8468 8474 7 8480 8485 8491 8497 8502 8508 8513 8519 8525 8530 8 8536 8542 8547 8553 8559 8564 8570 8576 8581 8587 9 8593 8598 8604 8610 8615 8621 8627 8632 8638 8643 770 88649 88655 88660 88666 88672 88677 88683 88689 88694 88700 1 8705 8711 8717 8722 8728 8734 8739 8745 8750 8756 2 8762 8767 8773 8779 8784 8790 8795 8801 8807 8812 3 8818 8824 8829 8835 8840 8846 8852 8857 8863 8868 4 8874 8880 8885 8891 8897 8902 8908 8913 8919 8925 5 8930 893() 8941 8947 8953 8958 8964 8969 8975 8981 6 8986 8992 8997 9003 9009 9014 9020 9025 9031 9037 7 9042 9048 9053 9059 9064 9070 9076 9081 9087 9092 8 ^098 9104 9109 9115 9120 9126 9131 9137 9143 9148 9 9154 9159 9165 9170 9176 9182 9187 9193 9198 9204 780 89209 89215 89221 89220 89232 89237 89243 89248 89254 89260 1 9265 9271 9276 9282 9287 9293 9298 9304 9310 9315 2 9321 932<> 9332 9337 9343 9348 9354 9360 9365 9371 3 9376 9382 9387 9393 9398 9404 9409 9415 9421 9426 4 9432 9437 9443 9448 9454 9459 9465 9470 9476 9481 5 9487 9492 9498 9504 9509 9515 9520 9526 9531 9537 6 9542 9548 9553 9559 9564 9570 9575 9581 9586 9592 7 9597 9603 9609 9614 9620 9625 9631 9636 9642 9647 8 9653 9658 9664 9669 9675 9680 9686 9691 9697 9702 9 9708 9713 9719 9724 9730 9735 9741 9746 9752 9757 790 89763 89768 89774 89779 89785 89790 89796 89801 89807 89812 1 9818 9823 9829 9834 9840 9845 9851 9856 9862 9867 2 9873 9878 9883 9889 9894 9900 9905 9911 9916 9922 3 9927 9933 9938 9944 9949 9955 9960 9966 9971 9977 4 9982 9988 9993 9998 90004 90009 90015 90020 90026 90031 5 90037 90042 90048 90053 0059 0064 0069 0075 0080 008(5 6 0091 0097 0102 0108 0113 0119 0124 0129 0135 0140 7 0146 0151 0157 0162 0168 0173 0179 0184 0189 0195 8 0200 0206 0211 0217 0222 0227 0233 0238 0244 0249 9 0255 0260 02(56 0271 0276 0282 0287 0293 0298 0304 800 90309 90314 90320 90325 90331 90336 90342 90347 90352 90358 •4 TACLE VI.— LOGARITHMS OF XUMBERB. X O* 1 2 3 4 5 6 7 8 9 800 90309 90314 90320 90325 90331 90336 90342 90347 90352 90358 1 0363 0369 0374 0380 0385 0390 0396 0401 0407 0412 2 0417 0423 0428 0434 0439 0445 0450 0455 0461 0466 3 0472 0477 0482 0488 0493 0499 0504 0509 0515 0520 4 0526 0531 0530 0542 0547 0553 0558 0563 0569 0574 5 0580 0585 0590 0596 0601 0607 0612 0617 0623 0628 6 0634 0639 0644 0650 0655 0660 0666 0671 0677 0682 7 0687 0693 0698 0703 0709 0714 0720 0725 0730 0736 8 0741 0747 0752 0757 0763 0768 0773 0779 0784 0789 9 0795 0800 0806 0811 0816 0822 0827 0832 0838 0843 810 90849 90854 90859 90865 90870 90875 90881 90886 90891 90897 1 0902 0907 0913 0918 0924 0929 0934 0940 0945 0950 2 0956 0961 0966 0972 0977 0982 0988 0993 0998 1004 3 1009 1014 1020 1025 1030 1036 1041 1046 1052 1057 4 1062 1068 1073 1078 1084 1089 1094 1100 1105 1110 5 1116 1121 1126 1132 1137 1142 1148 1153 1158 1164 6 1169 1174 1180 1185 1190 1196 1201 1206 1212 1217 . 7 1222 1228 1233 1238 1243 1249 1254 1259 1265 1270 8 1275 1281 1286 1291 1297 1302 1307 1312 1318 1323 9 1328 1334 1339 1344 1350 1355 1360 1365 1371 1376 820 91381 91387 91392 91397 91403 91408 91413 91418 91424 91429 1 1434 1440 1445 1450 1455 1461 1466 1471 1477 1482 2 1487 1492 1498 1503 1508 1514 1519 1524 1529 1535 3 1540 1545 1551 1556 1561 1566 1572 1577 1582 1587 4 1593 1598 1603 1609 1614 1619 1624 1630 1635 1640 5 1645 1651 1656 1661 1666 1672 1677 1682 1687 1693 6 1698 1703 1709' 1714 1719 1724 1730 1735 1740 1745 7 1751 1756 1761 1766 1772 1777 1782 1787 1793 1798 8 1803 1808 1814 1819 1824 J829 1834 1840 1845 1850 9 1855 1861 1866 1871 1876 1882 1887 1892 1897 1903 830 91908 91913 91918 91924 91929 91934 91939 91944 91950 91955 1 1960 1965 1971 1976 1981 1986 1991 1997 2002 2007 2 2012 2018 2023 2028 2033 2038 2044 2049 2054 2059 3 2065 2070 2075 2080 2085 2091 2096 2101 2106 2111 4 2117 2122 2127 2132 2137 2143 2148 2153 2158 2163 5 2169 2174 2179 2184 2189 2195 2200 2205 2210 2215 6 2221 2226 2231 2236 2241 2247 2252 2257 2262 2267 7 2273 2278 2283 2288 2293 2298 2304 2309 2314 2319 8 2324 2330 2335 2340 2345 2350 2355 2361 2366 2371 9 2376 2381 2387 2392 2397 2402 2407 2412 2418 2423 840 92428 92433 92438 92443 92449 92454 92459 92464 92469 92474 1 2480 2485 2490 2495 2500 2505 2511 2516 2521 2526 2 2531 2536 2542 2547 2552 2557 2562 2567 2572 2578 3 2583 2588 2593 2598 2603 2609 2614 2619 2624 2629 4 2634 2639 2645 2650 2655 2660 2665 2670 2675 2681 5 2686 2691 2696 2701 2706 2711 2716 2722 2727 2732 6 2737 2742 2747 2752 2758 2763 2768 2773 2778 2783 7 2788 2793 2799 2804 2809 2814 2819 2824 2829 2834 2840 2845 2850 2855 2860 2865 2870 2875 2881 2886 if 2891 2896 2901 2906 2911 2916 2921 2927 2932 2937 850 92942 92947 92952 92957 92962 92967 92973 92978 92983 92988 TABLE Vr. LOGARITHMS OF NUMBERS. S35 N 01234567 89 850 92942 92947 92952 92957 92962 92967 92973 92978 92983 92988 1 2993 2998 3003 3008 3013 3018 3024 3029 3034 3039 2 3044 3049 3054 3059 3064 3069 3075 3080 3085 3090 3 3095 3100 3105 3110 3115 3120 3125 3131 3136 3141 4 3146 3151 3156 3161 3166 3171 3176 3181 3186 3192 5 3197 3202 3207 3212 3217 3222 3227 3232 3237 3242 6 3247 3252 3258 3263 3268 3273 3278 3283 3288 3293 7 3298 3303 3308 3313 3318 3323 3328 3334 3339 3344 8 3349 3354 3359 3304 3369 3374 3379 3384 3389 3394 9 3399 3404 3409 3414 3420 3425 3430 3435 3440 3445 860 93450 93455 93460 93465 93470 93475 93480 93485 93490 93495 1 3500 3505 3510 3515 3520 3526 3531 3536 3541 3546 2 3551 3556 3561 3566 3571 3576 3581 3586 3591 3596 3 3601 3606 3611 3616 3621 3626 3631 3636 3641 3646 4 3651 3656 3661 3666 3671 3676 3682 3687 3692 3697 5 3702 3707 3712 3717 3722 3727 3732 3737 3742 3747 6 3752 3757 3762 3767 3772 3777 3782 3787 3792 3797 7 3802 3807 3812 3817 3822 3827 3832 3837 3842 3847 8 3852 3857 3862 3867 3872 3877 3882 3887 3892 3897 9 3902 3907 3912 3917 3922 3927 3932 3937 3942 3947 870 93952 93957 93962 93967 93972 93977 93982 93987 93992 93997 1 4002 4007 4012 4017 4022 4027 4032 4037 4042 4047 2 4052 4057 4062 4067 4072 4077 4082 4086 4091 4096 3 4101 4106 4111 4116 4121 4126 4131 4136 4141 4146 4 4151 4156 4161 4166 4171 4176 4181 4186 4191 4196 5 4201 4206 4211 4216 4221 4226 4231 4236 4240 4245 6 4250 4255 4260 4265 4270 4275 4280 4285 4290 4295 7 4300 4305 4310 4315 4320 4325 4330 4335 4340 4345 8 4349 4354 4359 4364 4369 4374 4379 4384 4389 4394 9 4399 4404 4409 4414 4419 4424 4429 4433 4438 4443 880 94448 94453 94458 94463 94468 94473 94478 94483 94488 94493 1 4498 4503 4507 4512 4517 4522 4527 4532 4537 4542 2 4547 4552 4557 4562 4567 4571 4576 4581 4586 4591 3 4596 4601 4606 4611 4616 4621 4626 4630 4635 4640 4 4645 4650 4655 4660 4665 4670 4675 4680 4685 4689 5 4694 4699 4704 4709 4714 4719 4724 4729 4734 4738 6 4743 4748 4753 4758 4763 4768 4773 4778 4783 4787 7 4792 4797 4802 4807 4812 4817 4822 4827 4832 4836 8 4841 4846 4851 4856 4861 4866 4871 4876 4880 4885 9 4890 4895 4900 4905 4910 4915 4919 4924 4929 4934 890 94939 94944 94949 94954 94959 94963 94968 94973 94978 94983 1 4988 4993 4998 5002 5007 5012 5017 5022 5027 5032 2 5036 5041 5046 5051 5056 5061 5066 5071 5075 5080 3 5085 5090 5095 6100 5105 5109 5114 5119 5124 5129 4 5134 5139 5143 5148 5153 5158 5163 5168 5173 5177 5 5182 5187 5192 5197 5202 5207 5211 6216 6221 5226 6 5231 5236 5240 5245 5250 5255 6260 5265 5270 5274 7 5279 5284 6289 6294 5299 5303 5308 5313 6318 6323 8 6328 5332 6337 6342 5347 6362 5357 5361 6366 5371 9 5376 5381 6386 5390 6395 5400 6405 6410 6415 5419 900 95424 96429 96434 95439 95444 95448 96453 95458 96463 95468 — 236 TABLE VI.— LOGARITHMS OF NUMBERS. N O 1 2 3 4 5 6 7 8 9 900 95424 95429 95434 95439 95444 95448 95453 95458 95463 95468 1 5472 5477 5482 5487 5492 5497 5501 5506 5511 5516 2 5521 5525 5530 5535 5540 5545 5550 5554 5559 6664 3 5569 5574 5578 5583 5588 5593 5598 5602 5607 6612 4 5617 5622 5626 5631 5636 5641 5646 5650 5655 6660 5 5665 6670 5674 5679 5684 5689 5694 5698 6703 6708 6 5713 5718 5722 5727 5732 5737 5742 5746 5751 6756 7 5761 5766 5770 5775 5780 5785 5789 5794 5799 5804 8 5809 5813 5818 5823 5828 5832 5837 5842 5847 6852 9 5856 5861 5866 5871 5875 5880 5885 5890 5895 6899 910 95904 95909 95914 95918 95923 95928 95933 95938 95942 95947 1 5952 5957 5961 5966 5971 5976 5980 5985 5990 5995 2 5999 6004 6009 6014 6019 6023 6028 6033 6038 6042 3 6047 6052 6057 6061 6066 6071 6076 6080 6085 6090 4 6095 6099 6104 6109 6114 6118 6123 6128 6133 6137 5 6142 6147 6152 6156 6161 6166 6171 6175 6180 6185 6 6190 6194 6199 6204 6209 6213 6218 6223 6227 6232 7 6237 6242 6246 6251 6256 6261 6265 6270 6275 6280 8 6284 6289 6294 6298 6303 6308 6313 6317 6322 6327 9 6332 6336 6341 6346 6350 6355 6360 6365 6369 6374 920 96379 96384 96388 96393 96398 96402 96407 96412 96417 96421 1 6426 6431 6435 6440 6445 6450 6454 6459 6464 6468 2 6473 6478 6483 6487 6492 6497 6501 6506 6511 6515 3 6520 6525 6530 6534 6539 6544 6548 6553 6558 6562 4 6567 6572 6577 6581 6586 6591 6595 6600 6605 6609 5 6614 6619 6624 6628 6633 6638 6642 6647 6652 6656 6 6661 6666 6670 6675 6680 6685 6689 6694 6699 6703 7 6708 6713 6717 6722 6727 6731 6736 6741 6745 6750 8 6755 6759 6764 6769 6774 6778 6783 6788 6792 6797 9 6802 6806 6811 6816 6820 6825 6830 6834 6839 6844 930 96848 96853 96858 96862 96867 96872 96876 96881 96886 96890 1 6895 6900 6904 6909 6914 6918 6923 6928 6932 6937 2 6942 6946 6951 6956 6960 6965 6970 6974 6979 6984 3 ■6988 6993 6997 7002 7007 7011 7016 7021 7025 7030 4 7035 7039 7044 7049 7053 7058 7063 7067 7072 7077 5 7081 7086 7090 7095 7100 7104 7109 7114 7118 7123 6 7128 7132 7137 7142 7146 7151 7155 7160 7165 7169 7 7174 7179 7183 7188 7192 7197 7202 7206 7211 7216 8 7220 7225 7230 7234 7239 7243 7248 7253 7257 7262 9 7267 7271 7276 7280 7285 7290 7294 7299 7304 7308 940 97313 97317 97322 97327 97331 97336 97340 97345 97350 97354 1 7359 7364 7368 7373 7377 7382 7387 7391 7396 7400 2 7405 7410 7414 7419 7424 7428 7433 7437 7442 7447 3 7451 7456 7460 7465 7470 7474 7479 7483 7488 7493 4 7497 7502 7506 7511 7516 7520 7525 7529 7534 7539 5 7543 7548 7552 7557 7562 7566 7571 7575 7580 7585 6 7589 7594 7598 7603 7607 7612 7617 7621 7626 7630 7 7635 7640 7644 7649 7653 7658 7663 7667 7672 7676 8 7681 7685 7690 7695 7699 7704 7708 7713 7717 7722 9 7727 7731 7736 7740 7745 7749 7754 7759 7763 7768 950 97772 97777 97782 97786 97791 97795 97800 97804 97809 97813 TABLE VT. LdGA'RITILAfB OF NTAniKUS.' 337 N 0123456789 050 97772 97777 97782 97786 97791 9779;-) 97800 97804 97809 97813 1 7818 7823 7827 7832 7836 7841 7845 7850 7855 7859 2 7864 7808 7873 7877 7882 7886 7891 7896 7900 7905 3 7909 7914 7918 7923 7928 7932 7937 7941 7946 7950 4 7955 7959 7964 7968 7973 7978 7982 7987 7991 7996 5 8000 8005 8009 8014 8019 8023 8028 8032 8037 8041 6 8046 8050 8055 8059 8064 8068 8073 8078 8082 8087 7 8091 8096 8100 8105 8109 8114 8118 8123 8127 8132 8 8137 8141 8146 8150 8155 8159 8164 8168 8173 8177 9 8182 8186 8191 8195 8200 8204 8209 8214 8218 8223 960 98227 98232 98236 98241 98245 98250 98254 98259 98263 98268 1 8272 8277 8281 8286 8290 8295 8299 8304 8308 8313 2 8318 8322 8327 8331 8336 8340 8345 8349 8354 8358 3 8363 8367 8372 8376 8381 8385 8390 8394 8399 8403 4 8408 8412 8417 8421 8426 8430 8435 8439 8444 8448 5 8453 8457 8462 8466 8471 8475 8480 8484 8489 8493 6 8498 8502 8507 8511 8516 8520 8525 8529 8534 8538 7 8543 8547 8552 8556 8561 8565 8570 8574 8579 8583 8 8588 8592 8597 8601 8605 8610 8614 8619 8623 8628 9 8632 8637 8641 8646 8650 8655 8659 8664 8668 8673 D70 98677 98682 98686 98691 98695 98700 98704 98709 98713 98717 1 , 8722 8726 8731 8735 8740 8744 8749 8753 8758 8762 2 8767 8771 8776 8780 8784 8789 8793 8798 8802 8807 3 8811 8816 8820 8825 8829 8834 8838 8843 8847 8851 4 8856 8860 8865 8869 8874 8878 ' 8883 8887 8892 8896 5 8900 8905 8909 8914 8918 8923 8927 8932 8936 8941 6 8945 8949 8954 8958 8963 8967 8972 8976 8981 8985 7 8989 8994 8998 9003 9007 9012 9016 9021 9025 9029 8 9034 9038 9043 9047 9052 9056 9061 9065 9069 9074 9 9078 9083 9087 9092 9096 9100 9105 9109 9114 9118 980 99123 99127 99131 99136 99140 99145 99149 99154 99158 99162 1 9167 9171 9176 9180 9185 9189 9193 9198 9202 9207 2 9211 9216 9220 9224 9229 9233 9238 9242 9247 9251 3 9255 9260 9264 9269 9273 9277 9282 9286 9291 9295 4 9300 9304 9308 9313 9317 9322 9326 9330 9335 9339 5 9344 9348 9352 9357 9361 9366 9370 9374 9379 9383 6 9388 9392 9396 9401 9405 9410 9414 9419 9423 9427 7 9432 9436 9441 9445 9449 9454 9458 9463 9467 9471 8 9476 9480 9484 9489 9493 9498 9502 9506 9511 9515 9 9520 9524 9528 9533 9537 9542 9546 9550 9555 9559 990 99564 99568 99572 99577 99581 99585 99590 99594 99599 99603 1 9607 9612 9616 9621 9625 9629 9634 9638 9642 9647 2 9()51 9656 9660 9664 9669 9673 9677 9682 9686 9691 3 9695 9699 9704 9708 9712 9717 9721 9726 9730 9734 4 9739 9743 9747 9752 9756 9760 9765 9769 9774 9778 5 9782 9787 9791 9795 9800 9804 9808 9813 9817 9822 6 9826 9830 9835 9839 9843 9848 9852 9856 9861 98(55 7 9870 9874 9878 9883 9887 9891 9896 9900 9904 9909 8 9913 9917 9922 9926 9930 9935 9939 9944 9948 9952 9 9957 9961 9965 9970 9974 9978 9983 9987 9991 9996 1000 1 00000 00004 00009 00013 00017 00022 00026 00030 00035 00039 t 3 1- • t Vv^nm.Cm. Sin, Cos. iit^ ni 92 238 TABLE VII.— LOGARITHMIC SINES AND COSINES. / Sine 0° 1« 2° 1 / Cosine Sine - Cosine Sine Cosine — ao 10.00000 8.24186 9.99993 8.54^82 54642 9.99974 60 . 1 6.46373 00000 24903 99993 99973 59 2 76476 00000 25609 99993 54999 , 99973 58 3 94085 00000 26304 99993 55354 99972 57 4 7.06579 00000 26988 99992 55705 99972 56 5 16270 00000 27661 99992 56054 69971 55 6 24188 00000 28324 99992 56400 99971 54 7 30882 00000 28977 99992 • 56743 99970 53 8 36682 00000 29621 99992 57084 99970 52 9 41797 00000 30255 9999] 57421 99969 51 10 7.46373 10.00000 8.30879 9.99991 8.57757 9.99969 50 11 50512 00000 31495 99991 58089 99968 49 12 54291 00000 32103 99990 58419 99968 48 13 57767 ooooo 32702 99990 58747 99967 47 14 60985 00000 33292 99990 59072 99967 46 15 63982 ooooo 33875 99990 59395 99967 45 16 66784 ■ ooooo 34450 99989 59715 99960 44 17 69417 9.99999 35018 99989 60033 99966 43 18 71900 99999 35578 99989 60349 99965 42 19 74248 99999 36131 99989 60662 99964 41 20 7.76475 9.99999 8.36678 9.99988 8.60973 9.99964 40 21 78594 99999 37217 99988 • 61282 99963 39 22 80615 99999 37750 99988 61589 99963 38 23 82545 99999 38276 99987 61894 99962 37 24 84393 99999 .38796 99987 62196 99962 36 25 86166 99999 39310 99987 62497 99961 35 26 87870 99999 39818 99986 62795 99961 34 27 89509 99999 40320 99986 63091 99960 33 28 91088 99999 40816 99986 63385 99960 32 29 92612 99998 41307 99985 63678 99959 31 30 7.94084 9.99998 8.41792 9.99985 8.63968 9.99959 30 31 95508 99998 42272 9998E 64256 99958 29 32 96887 99998 42746 99984 64543 99958 28 33 98223 99998 43216 99984 64827 99957 27 34 99520 99998 43680 99984 65110 99956 26 35 8.00779 99998 44139 99983 65391 99956 25 36 02002 99998 44594 99983 65670 99955 24 37 03192 99997 45044 99983 65947 99955 23 38 04350 99997 45489 99982 66223 99954 22 39 05478 99997 45930 99982 66497 99954 21 40 8.06578 9.99997 8.46366 9.99982 8.66769 9.99953 20 41 07650 99997 46799 99981 67039 99952 19 42 08696 99997 47226 99981 67308 99952 1* 43 09718 99997 47650 99981 67575 99951 17 44 10717 99996 48069 99980 67841 99951 16 45 11693 99996 48485 99980 68104 99950 15 46 12647 99996 48896 99979 68367 99949 14 47 13581 99996 49304 99979 68627 99949 13 48 14495 99996 49708 99979 68886 99948 12 49 15391 99996 50108 99978 69144 99948 11 50 8.16268 9.99995 8.50504 9.99978 8.69400 9.99947 10 51 17128 99995 50897 99977 69654 99946 9 52 17971 99995 51287 99977 69907 99946 8 53 18798 99995 51673 99977 70159 99945 7 54 19610 99995 52055 99976 70409 99944 6 55 20407 99994 52434 99976 70658 99944 5 56 21189 99994 52810 99975 70905 99943 4 57 21958 99994 53183 99975 71151 99942 3 58 22713 99994 53552 99974 71395 99942 2 59 23456 99994 53919 99974 71638 99941 1 60 24186 99993 54282 99974 71880 99940 / Cosine Sine Cosine Sine Cosine Sine 89° 8.« o 87 »— .^/-r^ S.^-iv 1 - T — . Sin : Cn& 93 94 95 TABLE VII.— LOGARITHMIC SINES AND COSINES. 239 / 3° 4° 6° / Sine Cosine Sine Cosine Sine Cosine 8.71880 9.99940 8.843.58 9.99894 8.94030 9.99834 60 1 72120 99940 84539 99893 94174 99833 59 2 72359 99939 84718 99892 94317 99832 58 3 72597 99938 84897 . 99891 94461 99831 .57 4 72834 9993S 85075 99891 94603 99830 56 5 73069 99937 85252 99890 94746 99829 55 6 73303 99936 85429 59889 94887 99828 54 7 73535 99936 85605 99888 95029 99827 53 8 73767 99935 85780 99887 95170 99825 52 9 73997 99934 85955 99886 95310 99824 51 10 8.74226 9.99934 8.86128 9.99885 8.954.50 9.99823- 50 11 74454 99933 86301 99884 95589 99822 49 12 74680 99932 86474 99883 95728 99821 48 13 74906 99932 86645 99882 95867 99820 47 14 75130 99931 86816 99881 96005 99819 46 15 75353 99930 86987 99880 96143 99817 45 16 75575 99929 87156 99879 96280 99816 44 17 75795 99929 87325 99879 96417 99815 43 18 76015 99928 87494 99878 96553 99814 42 19 76234 99927 87661 99877 96689 99813 41 20 8.76451 9.99926 8.87829 9.99876 8.96825 9.99812 40 21 76667 99926 87995 99875 96960 99810 39 22 76883 99925 88161 9987'4 97095 99809 38 23 77097 99924 88326 99873 97229 99808 37 24 77310 99923 88490 99872 97363 99807 36 25 77522 99923 88654 99871 97496 99806 35 26 77733 99922 88817 99870 97629 99804 34 27 77943 999^1 88980 99869 97762 99803 33 28 78152 99920 89142 99868 97894 99802 32 29 78360 99920 89304 99867 98026 90801 31 30 8.78568 9.99919 8.89464 9.99866 8.981.57 9.99800 30 31 78774 99918 89625 99865 98288 99798 29 32 78979 99917 89784 99864 98419 99797 28 33 79183 99917 89943 99863 98549 99796 27 34 79386 99916 90102 99862 98679 99795 26 35 79588 99915 90260 99861 98808 99793 25 36 79789 99914 90417 99860 98937 99792 24 37 79990 99913 90574 99859 99066 99791 23 38 80189 99913 90730 99858 99194 99790 22 39 80388 99912 90885 99857 99322 99788 21 40 8.80585 9.99911 8.91040 9.99856 8.99450 -9.99787 20 41 80782 99910 91195 99855 99577 99786 19 42 80978 99909 91349 99854 99704 99785 18 43 81173 99909 91502 99853 99830 99783 17 44 81367 99908 91655 99852 99956 99782 16 45 81560 99907 91807 99851 9.00082 99781 15 46 81752 99906 91959 99850 00207 99780 14 47 81944 99905 92110 99848 00332 99778 13 48 82134 99904 92261 99847 00456 99777 12 49 82324 99904 92411 99846 00581 99776 11 50 8.82513 9.99903 8.92561 9.99845 9.00704 9.99775 10 51 82701 99902 92710 99844 00828 99773 9 52 82888 99901 92a59 99843 00951 99772 8 53 83075 99900 93007 99842 01074 99771 7 54 83261 99899 93154 99841 01196 99769. 6 55 83446 99898 93301 99840 01318 99768 5 56 83630 99898 93448 99839 01440 99767 4 57 83813 99897 93594 99838 01.561 997()5 3 58 83996 99896 93740 99a37 01682 99764 o 59 84177 99895 9.3885 99836 01803 99763 1 60 84358 99894 94030 99834 01923 99761 Cosine Sine Cosine Sine Cosine Sine / ^ 4 . 86° 86° 84° ^^nL'^)i^. -j^r^WA,. ^'^rJR!. • r 96 •>- jr*-: yb 9 7 QQ 340 TABLE VII.— LOGARITHMIC SINES AND COSINES. / 6° 7° 8» / Sine 9.01923 Cosine Sine Cosine Sine 9.14356 Cosine 9.99575 9.99761 9.08589 9.99675 fiO 1 02043 99760 08692 99674 14445 99574 Kf\J 59 58 o 02163 99759 08795 99672 14535 99572 3 02283 99757 08897 99670 14624 99570 57 56 4 02402 99756 08999 99669 14714 99568 5 02520 99755 09101 99667 14803 99566 55 54 - 6 02639 99753 09202 99666 14891 99565 7 02757 99752 09304 99664 14980 99563 53 8 02874 99751 09405 99663 15069 99561 5" 9 02992 99749 09506 99661 15157 99559 51 10 9.03109 9.99748 9.09606 9.99659 9.15245 9.99557 50 11 03226 99747 09707 99658 15333 99556 49 12 03342 99745 09807 99656 15421 99554 48 13 03458 99744 09907 99655 15508 99552 47 14 03574 99742 10006 99653 15596 99550 46 15 03690 99741 10106 99651 15683 99548 45 16 03805 99740 10205 99650 15770 99546 44 17 03920 99738 10304 99648 15857 995i5 43 18 040:34 99737 10402 99647 15944 99543 42 19 04149 99736 10501 99645 16030 99541 41 20 9.04262 9.99734 9.10599 9.99643 9.16116 9.995:39 40 21 04376 99733 10697 99642 16203 995:37 39 22 04490 99731 10795 99640 16289 99535 38 23 04603 99730 10893 99638 16374 99533 37 24 04715 99728 10990 99637 16460 99532 36 2o 04828 99727 11087 99635 16545 99530 35 26 04940 99726 11184 99633 16631 99528 34 27 05052 99724 11281 99632 16716 99526 33 28 05164 99723 11377 99630 16801 99524 32 29 05275 99721 11474 99629 16886 99522 31 30 9.05386 9.99720 9.11570 9.99627 9.16970 9.99520 30 31 05497 99718 11C66 99625 17055' 99518 29 32 05607 99717 11761 99624 17139 99517 28 33 05717 99716 11857 99622 17223 995J5 27 34 05827 99714 11952 99620 17307 99513 26 :ir, 05937 90713 12047 9961 S 17:391 99511 25 36 06046 99711 12142 99617 17474 99509 24 37 06155 99710 12236 99615 17558 99507 23 38 06264 99708 12331 99613 17641 99505 22 39 06372 99707 12425 99612 17724 99503 21 40 9.06481 9.99705 9.12519 9.99610 9.17807 9.99501 20 41 06589 99704 12612 99608 17890 99499 19 42 06696 99702 12706 99607 17973 99497 18 43 06804 99701 12799 99605 1805.-) 99495 17 44 06911" 99699 12892 99603 18137 99494 16 45 07018 99698 12985 99601 18220 99492 15 46 07124 99696 13078 99600 18:302 99490 14 47 072;:J1 99695 13171 99598 ia383 99488 13 48 07337 99693 13263 99596 18465 994^6 12 49 07442 99692 13:355 99595 18547 99484 11 50 9.07548 9.99690 9.13447 9.99.593 9.18628 9.99482 10 51 07653 99689 13539 99591 18709 99480 9 52 07758 99687 136:30 995S9 18790 99478 8 53 07863 99686 13722 99588 18871 99476 i 54 07968 99684 13813 995S6 18952 99474 6 55 08072 99683 13904 99584 190:33 99472 5 56 08176 99681 13994 99582 19113 99470 4 57 08280 99680 14085 99581 19193 99468 3 58 08383 99678 14175 99579 19273 99466 2 59 08486 99677 14266 99577 19:353 99464 1 60 08589 99675 14:356 99575 19433 99462 / Cosine Sine Cosine Sine Cosine Sinp / 83° 82° 81" ' Si^tJ^a. ^n^^^M siti, r:. £.% TABLE VII.— LOGARITIIMIC SINES AND COSINES. 241 / 9 o 10 o 11 o / Sine Cosine Sine Cosine Sine Cosine 9.19433 9.99462 9.23967 9.993.35 9.28060 9.99195 60 1 19513 99460 24039 99333 28125 99192 59 2 19592 99458 24110 99331 28190 99190 58 3 19672 99456 24181 99328 28254 99187 57 4 19751 99454 24253 99326 28319 99185 56 5 19830 99452 24324 99324 28384 99182 55 6 19909 99450 24395 99322 28448 99180 54 i 19988 99448 24466 99319 28512 99177 53 8 20067 99446 24536 99317 28577 99175 52 9 20145 99444 24607 99315 28641 99172 51 10 9.20223 9.99442 9.24677 9.99.313 9.28705 9.99170 50 11 20302 99440 24748 99310 28769 99167 49 12 20380 99438 24818 99308 28833 99165 48 13 20458 99436 24888 99306 28896 99162 47 14 20535 99434 24958 99.304 28960 99160 46 15 20613 99432 25028 99301 29024 991.57 45 16 20691 99429 25098 99299 29087 99155 44 17 207G8 99427 25168 99297 29150 99152 43 18 20845 99425 25237 99294 29214 991.50 42 19 20922 99423 25307 99292 29277 99147 41 20 9.20999 9.99421 9.25376 9.99290 9.29340 9.99145 40 21 21076 99419 25445 99288 29403 99142 39 22 21153 99417 25514 99285 29466 99140 38 23 21229 99415 25583 99283 29529 991.37 37 24 21306 99413 25652 99281 29591 99135 36 25 21382 99411 25721 99278 29654 99132 35 26 21458 99409 25790 99276 29716 99130 34 27 21534 99407 25858 99274 29779 99127 33 28 21610 99404 25927 99271 29841 99124 32 29 21685 99402 25995 99269 29903 99122 31 30 9.21761 9.99400 9.26063 9.99267 9.29966 9.99119 30 31 21836 99398 26131 99264 30028 99117 29 32 21912 99396 26199 99262 30090 99114 28 33 21987 99394 26267 99260 30151 99112 27 34 22062 99892 26335 99257 30213 99109 26 35 22137 99390 26403 99255 30275 99106 25 36 22211 99.388 26470 99252 30336 99104 24 37 22^86 99385 26538 99250 30398 99001 23 38 22361 99383 26605 99248 30459 99099 22 39 22435 99381 26672 99245 30521 99096 21 40 9.22509 9.99379 9.267.39 9.99243 9.30582 9.99093 20 41 22583 99377 26806 99241 30643 99091 19 42 22657 99375 26873 99238 30704 99088 18 43 22731 99372 26940 99236 30765 99086 17 44 22805 99370 27007 99233 30826 99083 16 45 22878 99368 27073 99231 30887 99080 15 46 22952 99.366 27140 99229 30947 99078 14 47 23025 99364 27206 99226 31008 99075 13 48 23098 99362 27273 99224 31068 99072 12 49 23171 99359 27339 99221 31129 99070 11 50 9.23244 9.99357 9.27405 9.99219 9.31189 9.99067 10 51 23317 99355 27471 99217 31250 99064 9 52 23390 99353 27537 99214 31310 99062 8 53 23462 99351 27602 99212 31370 99059 7 54 23535 99348 27668 99209 31430 99056 6 55 23607 99346 27734 99207 31490 99054 5 56 23679 99344 27799 99204 31549 99051 4 57 23752 99342 27864 99202 31609 99048 3 58 23823 99340 27930 99200 31669 99046 2 59 23895 99337 27995 99197 .31728 99043 1 60 23967 99335 28060 99195 31788 99040 / Cosine Sine Cosine Sine Cosine Sine / 80O 79° 78° 1 i 70 Jl G9 JL GS ^0^^. 1. , V— V.^ t-- 4 v.n t .nr mmiC SINES AND 242 TABLE VII.— LOG AK COSINES. / 12° 13° 14° / Sine Cosine Sine Cosine Sine Cosine 9.31788 9.99040 9.35209 9.98872 9.38368 9.98690 60 1 31847 99038 35263 98869 38418 98687 59 2 31907 99035 35318 98867 38469 98684 58 3 31966 99032 35373 98864 38519 98681 57 4 32025 99030 35427 98861 38570 98678 56 5 32084 99027 35481 98858 38620 98675 55 6 32143 99024 35536 98855 38670 98671 54 7 32202 99022 35590 98852 38721 98668 53 8 32261 99019 35644 98849 38771 98665 52 9 32319 99016 35698 98846 38821 98652 51 10 9.32378 9.99013 9.35752 9.98843 9.38871 9.98659 50 11 32437 99011 35806 98840 38921 98656 49 12 32495 99008 35860 98837 38971 98652 48 13 32553 99005 35914 98834 39021 98649 47 14 32612 99002 35968 98»^1 39071 98646 46 15 32670 99000 36022 98828 39121 98643 45 16 32728 98997 36075 98825 39170 98640 44 17 32786 98994 36129 98822 39220 98636 43 18 32844 98991 36182 98819 39270 98633 42 19 32902 98989 36336 98816 39319 98';30 41 20 9.32960 9.98986 9.36289 9.98813 9.39369 9.98627 40 21 33018 98983 36342 98810 39418 98623 39 22 33075 9S980 36395 98807 39467 98620 38 28 33133 98978 36449 98804 39517 98617 37 24 33190 98975 36502 98801 39566 98614 36 25 33248 98972 36555 98798 39815 98610 35 26 33305 98969 36608 98795 39664 98607 34 27 33362 98967 36660 98792 39713 98604 33 28 33420 98964 36713 98789 39762 98601 32 29 33477 98961 36766 98786 39811 98597 31 30 9.33534 9.98958 9.36819 9.98783 9.39860 9.98594 30 31 3359] 98955 36871 98780 39909 98591 29 32 33647 98953 36924 98777 39958 98588 88 33 33704 98950 36976 98774 40006 98584 27 34 33761 98947 37028 98771 40055 98581 26 35 33818 98944 37081 98768 40103 98578 25 36 33874 98941 37133 98765 40152 98574 24 37 38931 98938 37185 98762 40200 98571 23 38 33987 98936 37237 98759 40249 98568 22 39 34043 98933 37289 98756 40297 98565 21 40 9.34100 9.98930 9.37341 9.98753 9.40346 9.98561 20 41 34156 9S927 37393 98750 40394 98558 19 42 34212 98924 > 37445 98746 40442 98^55 18 43 34268 98921 37497 98743 40490 98551 17 44 34324 98919 37549 98740 40538 98548 16 45 34380 98916 37600 98737 40586 98545 15 46 34436 98913 87652 98734 40634 98541 14 47 34491 98910 37703 98731 40682 98538 13 48 34547 98907 37755 98728 40730 98535 12 49 34602 98904 37806 98725 40778 98531 11 50 9.34658 9.98901 9.37858 9.98722 9.40825 9.98528 10 51 34713 98898 37909 98719 40873 98525 9 52 34769 98896 37960 98715 40921 98521 8 53 34824 98893 38011 98712 40968 98518 1 54 34879 98890 38062 98709 41016 98515 6 55 34934 98887 38113 98706 41063 98.M1 5 56 34989 98884 38164 98703 41111 98508 4 57 35044 988*^1 38215 98700 41158 98505 3 58 35099 98878 38266 98697 41205 98501 2 59 35154 98875 38317 98694 41252 98498 1 60 35209 98872 38368 98690 41300 98494 / Cosine Sine Cosine Sine Cosine Sine / 77° 76° 75° A3^ 6i: O**. A^/i^ 6 PnQ i-f) ;) . •:• TABLE ^hr — LOGARlTlhnC SINES AN^) TOSINES. 243 , 1 5° 16 17° / Sine Cosine Sine Cosine Sine Cosine 9.41300 9.98494 9.44034 9.98284 9.46594 9.98060 60 1 41347 98491 44078 98281 46635 98050 59 2 41394 98488 44122 98277 46676 98052 58 3 41441 98484 44166 98273 46717 98048 57 4 41488 98481 44210 98270 467.58 98044 56 5 41535 98477 44253 98266 46800 98040 55 6 41582 98474 44297 98262 46841 98030 54 7 41028 98471 44341 98259 46882 98032 53 8 41675 98467 44385 98255 46923 98029 52 9 41722 98464 44428 98251 46964 98025 51 10 9.41768 9.98460 9.44472 9.98248 9.47005 9.98021 50 11 41815 98457 44516 98244 47045 98017 49 12 41861 98453 44559 98240 47086 98013 48 13 41908 98450 44602 98237 47127 98009 47 14 41954 98447 44646 98233 47168 98005 46 15 42001 98443 44689 98229 47209 98001 45 16 42047 98440 44733 98226 47249 97997 44 17 42093 98436 44776 98222 47290 97993 43 18 42140 98433 44819 98218 47330 97989 42 19 42186 98429 44862 98215 47371 97986 41 20 9.42232 9.98426 9.44905 9.98211 9.47411 9.97982 40 21 42278 98422 44948 98207 47452 97978 39 22 42324 98419 44992 98204 47492 97974 38 23 42370 98415 45035 98200 47533 97970 37 24 42416 98412 45077 98196 47573 97966 36 25 42461 98409 45120 98192 47613 97962 35 26 42507 98405 45163 98189 47654 97958 34 27 42553 98402 45206 98185 47694 97954 33 28 42599 9839S 45249 98181 47734 97950 32 29 42644 98395 45292 98177 4) t li 97946 31 30 9.42690 9.98391 9.453.34 9.98174 9.47814 9.97942 30 31 42735 98388 45377 98170 47854 97938 29 32 42781 98384 45419 98166 47894 97934 28 33 42826 98381 45462 98162 47934 97930 27 34 42872 98377 45504 98159 47974 97920 26 35 42917 98373 45547 98155 48014 97922 25 36 42962 98370 45589 98151 48054 97918 24 37 43008 98366 45632 98147 48094 97914 23 38 43053 98363 45674 98144 48133 97910 22 39 43098 98359 45716 98140 48173 97906 21 40 9.43143 9.98356 9.45758 9.98136 9.48213 9.97902 20 41 43188 98352 45801 98132 48252 97898 ^^ 42 43233 98349 45843 98129 48292 97894 18 43 43278 98345 45885 98125 48332 97890 17 44 43323 98342 45927 98121 4^371 97886 16 45 43367 98338 45969 98117 48411 97882 15 46 43412 98334 46011 98113 48450 97878 14 47 434.57 98331 46053 98110 48490 97874 13 48 43502 98327 46095 98106 48529 97870 12 49 43546 98324 46136 98102 48568 97806 11 50 9.43591 9.98320 9.46178 9.98098 9.48607 9.97861 10 51 43635 98317 46220 98094 48047 97857 9 52 43680 98313 46262 98090 48686 97853 8 53 43724 98309 46303 98087 48725 97849 1 54 43769 98306 46345 98083 48764 97845 6 55 43813 98302 46386 98079 48803 97841 5 56 43857 98299 46428 98075 48S42 97a37 4 57 43901 98295 46469 98071 48881 97833 3 58 43946 98291 46511 98067 48920 97 829 2 59 43990 98288 46552 98063 48959 97825 i 60 44034 98284 46594 98060 48998 97821 Cosine Sine Cosine Sine Cosine Sine 72° / 74° 73° s ^I^ZMgi^ ■7 iT ' 244 TABLE VIL— LOGARITHMIC SIXES AND COSIXES. / 18° 19° 20° / Sine Cosine Sine Cosine Sine Cosine 9.48998 9.97821 9.51264 9.97567 9.53405 9.97299 60 1 49037 97817 51301 97563 53440 97294 59 2 49076 97812 51338 97558 53475 97289 58 3 49115 97808 51374 97554 5:Bo09 97285 57 4 49153 97804 51411 97550 .5.3544 97280 50 5 49192 97800 51447 97545 53578 97276 55 6 49231 97796 51484 97.541 53613 97271 .54 7 49269 97792 51520 97536 53647 97266 53 8 49:i08 97788 51557 97532 53682 97262 52 9 49347 97784 51593 97528 53716 97257 51 10 9 49385 9.97779 9.51629 9.97523 9.. 53751 9.97252 50 11 49424 97775 51666 97519 53785 97248 49 1-2 49462 97771 51702 97515 5.3819 97243 48 13 49500 97767 51738 97510 53854 97238 47 14 49539 97763 51774 97.506 53888 97234 46 15 49577 97759 51811 97501 53922 97229 45 16 49615 97754 51847 97497 53957 97224 44 17 49654 977.50 518*3 97492 53991 97220 43 18 49692 97746 51919 97488 54025 97215 42 19 49730 97742 51955 97484 54059 97210 41 20 9.49768 9.97738 9.51091 9.97479 9.54093 9.97206 40 21 49806 97734 52027 97475 54127 97201 39 22 49844 97729 52063 97470 54161 97196 38 23 49882 97725 52099 97466 54195 97192 37 24 49920 97721 52135 97461 54229 97187 36 25 49958 97717 52171 97457 542^3 97182 :35 26 49996 97713 52207 97453 54297 97178 34 27 50034 97708 52242 97448 54331 97173 33 28 50072 97704 52278 97444 .54:365 97168 32 29 501 IQ 97700 52314 97439 54399 97163 31 30 9.50148 9.97696 9.52350 9.97435 9.54433 9.97159 30 31 50185 97691 52385 97430 54466 971.54 29 32 50223 97687 52421 97426 54500 97149 28 33 50261 9768:3 52456 97421 51.5.34 97145 27 34 50298 97679 52492 97417 54.507 97140 26 35 50336 97674 52527 97412 54601 97135 25 36 50374 97670 52563 97408 54635 971.30 24 37 50411 97666 52598 97403 54668 97126 23 38 50449 97662 52634 97399 54702 97121 22 39 50186 97657 52669 97394 547:35 97116 21 40 9.50523 9.976.53 9.52705 9.97390 9.54769 9.97111 20 41 .50501 976i9 52740 97385 54802 97107 19 42 5()59S 97645 52775 97381 548:36 97102 18 43 50635 97640 52811 97376 54869 97097 17 44 50(;73 97636 52846 97372 54903 97092 16 45 50710 97632 528S1 97367 54936 97087 15 46 50747 97628 52916 97363 54969 970S3 14 47 50784 97623 52951 973.58 55003 97078 13 48 50821 97619 52986 973.53 55036 97073 12 49 50858 97615 53021 97349 55069 97068 11 50 9.50S96 9.97610 9.53056 9.97344 9.55102 9.97063 10 51 5()9:'>3 97606 53092 97340 5.5136 97059 9 5-i 50970 97602 53126 97335 55169 97054 8 53 51007 97597 .53161 97331 55202 97049 r* i 54 51013 97593 53196 97326 55235 97044 6 55 51080 97589 53231 97322 55268 97039 5 56 i 51117 97584 53266 97317 55301 97035 4 57 51154 97580 53301 97312 55334 97030 3 58 1 51191 97576 53336 97308 55367 97025 59 51227 97571 53370 97303 55400 97020 1 60 51364 97567 53405 97299 55433 97015 / Cosine Sine Cosine Sine Cosine Sine 71° 70° — ,^ — ^i^^ 69° Q;ji 1 /-^ nrtr 111 11*> 11Q TABLE YIT. — LOGA RITHMK r SINES AND^C OSINES. Z^'O / 21° 22 o 23 o / Sine Cosine Sine Cosine Sine Cosine 9.55433 9.97015 9.573.58 9.96717 9.59188 9.96403 60 1 55466 97010 .57389 96711 59218 96397 59 o 55499 97005 57420 96706 59247 96392 58 3 55532 97001 57451 96701 59277 963-S7 57 4 55564 96996 57482 96696 59307 96381 56 5 55597 96991 57514 96691 59336 96376 55 6 55630 96986 57545 96686 59.366 96:^70 54 7 55663 96981 57576 96681 59396 9(365 53 8 55695 96976 57607 96676 59425 96360 52 9 55728 96971 57638 96670 59455 9C354 51 10 9.55761 9.96966 9.57669 9.96665 9.59484 9.96349 50 11 55793 90962 57700 96660 59514 96343 49 12 55826 96957 57731 96655 59543 96388 48 13 55858 96952 57762 96650 59573 96333 47 14 55891 96947 57793 96645 59602 96327 46 15 55923 96942 57824 96640 59632 96322 45 16 55956 96937 57855 96634 59661 96316 44 17 55988 96932 57885 96629 59690 96311 1 43 18 56021 96927 57916 96624 59720 96305 i 42 19 56053 96922 .57947 96619 59749 96300 41 20 9.56085 9.96917 9.57978 9.96614 9.. 59778 9.96294 40 21 56118 96912 58008 96608 59808 96289 39 22 56150 9()907 58039 96603 59837 96284 1 38 23 56182 96903 58070 96598 59866 96278 37 24 56215 96898 58101 96593 59895 96273 36 25 56247 96893 58131 96588 59924 96267 35 26 56279 96888 58162 96582 599.54 96262 34 27 56311 96883 58192 96577 59983 96256 33 28 56343 96878 58223 96572 60012 96-J51 32 29 56375 96873 58253 96567 60041 96245 31 30 9.56408 9.96868 9.58284 9.96562 9.60070 9.96240 30 31 56440 96863 58314 96556 60099 96234 29 32 56472 96858 58345 96,551 60128 96229 28 33 56504 96853 58375 96546 60157 962V3 27 34 56536 96848 58406 96541 60186 9*218 26 35 56568 96843 58436 96535 60215 96212 25 36 56599 96838 58467 96530 60244 96207 24 37 56631 96833 58497 96525 60273 96201 23 38 56663 96828 58527 96520 60302 96196 22 39 56695 96823 58557 96514 60331 96190 21 40 9.56727 9.96818 9.-58588 9.96509 9.60359 9.96185 20 41 56759 96813 .58618 96504 60:^88 96179 19 42 56790 96808 58648 96498 60417 96174 18 43 56822 96803 58678 96493 60446 96168 17 44 568.54 96798 58709 96488 60474 96162 16 45 56886 96793 58739 96483 60503 96157 15 46 .56917 96788 58769 96477 60532 96151 14 47 56949 96783 58799 96472 60561 96146 13 48 56980 96778 58829 96467 60589 96140 12 49 57012 96772 58859 96461 60618 96135 11 50 9.57044 9.96767 9.58889 9.96456 9.60646 9.96129 10 51 .57075 96762 58919 96451 60<;75 96123 9 52 57107 967.57 58949 96445 60704 96118 8 53 57138 96752 58979 96440 60732 96112 7 54 57169 96747 59009 96435 60761 96107 6 55 57201 96742 59039 96429 60789 96101 5 56 57232 96737 59069 96424 60818 96095 4 57 57264 96732 59098 96419 60846 96090 3 58 57295 96727 .59128 96413 60875 96084 2 59 57326 96722 59158 96408 60903 96079 1 60 / 57.358 96717 59188 96403 60931 96073 Cosine Sine Cosine Sine Cosine Sine / 68° 67° 66° ;W?5e I.* 246 TABLE W.— LOG ARli'ltflC SINES In'd'' COSINES. / 24° 25° 26° / Sine Cosine Sine Cosine Sine Co.sine 9.60931 9.96073 9.62595 9.95728 9.64184 9.95366 60 1 G0960 96067 62622 95722 64:il0 95360 59 2 60988 96062 62649 95716 64236 95354 58 3 61016 96056 62676 95710 64262 95348 57 4 61045 96050 62703 95704 64288 95341 56 5 61073 96045 62730 95698 64313 95335 55 6 61101 96039 62757 95692 64339 95329 54 61129 96034 62784 95686 64365 95323 53 8 61158 96028 62811 95680 64391 95317 52 9 61186 96022 62838 95674 64417 95310 51 10 9.61214 9.96017 9.62865 9.95668 9.64442 9.95304 50 11 61242 96011 62892 95663 64468 95298 49 12 61270 96005 62918 95657 64494 95292 48 13 61298 96000 62945 95651 64519 95286 47 14 61326 95994 62972 95645 64545 95279 46 15 61354 95988 62999 95639 64571 95273 45 16 61382 95982 63026 95633 64596 95267 44 17 61411 95977 63052 95627 64622 95201 43 18 61438 95971 63079 95621 64647 95254 42 19 61466 95965 63106 95615 64673 95248 41 20 9.61494 9.95960 9.63i:33 9.95609 9.64698 9.95242 40 21 6152i 95954 63159 95603 64724 95236 39 22 61550 95918 63186 95597 64749 95229 38 23 61578 95942 63213 95.591 64775 95223 37 24 61606 93937 63239 95585 64800 95217 36 25 61631 95931 63266 95579 64826 95211 35 26 61662 95925 63292 95573 64851 95204 34 27 61689 95920 63319 95.567 64877 95198 33 28 61717 95914 63345 95561 64902 95192 32 29 61745 95908 63372 95555 64927 95185 31 30 9.61773 9.95902 9.63398 9.95549 9.64953 9.95179 30 31 61800 95897 63425 93.543 64978 95173 29 32 61828 95891 63451 95537 65003 95167 28 33 61856 958S5 63478 95531 65029 95160 27 34 61883 95879 C3504 95525 65054 95154 26 35 61911 95873 63531 95519 65079 95148 25 36 61939 95S68 63557 95513 65104 93141 24 37 61966 95862 63583 95507 65130 95135 23 38 61994 95836 63610 93500 63155 95129 22 39 62021 95850 63636 95494 65180 95122 21 40 9.62049 9.95844 9.63662 9.95488 9.65205 9.95116 20 41 62076 958C9 63689 95482 65230 95110 19 42 62104 95833 63715 95476 65255 9.3103 18 43 62131 95827 63741 95470 65281 95097 17 44 62159 9.5821 63767 95464 65306 95090 16 45 62186 95815 63794 95458 63331 95084 15 46 62214 95810 63820 95452 65356 95078 14 47 62241 95804 63846 95446 65381 95071 13 48 62268 95798 63S72 95440 65406 93065 12 49 62296 95792 63898 95434 65431 95059 11 ■ 50 9.62323 9.95786 9.63924 9.95427 9.6.5456 9.95052 10 51 62350 95780 63950 95421 65481 95046 9 52 62377 95775 63976 95415 65508 95039 8 53 62405 95769 64002 95409 65531 93033 7 54 62432 95763 64028 95403 65565 95027 6 55 62459 90(0. 04054 95397 65580 95020 5 56 62486 95751 640S0 9.3391 65605 95014 4 57 62513 95745 64106 95384 65630 95007 3 58 62541 95739 64132 95378 65655 95001 2 59 62568 93733 64158 95372 65680 94995 1 60 62595 95728 64184 95366 65705 949R8 /. Cosine Sine Cosine Sine Cosine Sine ' 65° 64° 63° 1 — r— ^ '" /< 1 si^. <-^^^. » AA i.^'i*. ^ c^ '. Cw hm ttfVtti 9iSi fi,€c s. CQtf.Stii.Cos. Sin. Cos. Sin. n-iH- ""IR 1iO TABLE VII'.--LOGARITH^iI(; 'SINES AND COSINES. 24/ / 2 «" 28' * 29° / Sine Cosine 9.94988 Sine Cosine Sine Cosine 9.65705 9.67161 9.94.593 9.68557 9.94182 60 1 65729 94982 67186 94587 68580 94175 59 2 65754 94975 67208 94580 68603 94168 58 3 65779 94909 67232 94573 68025 94161 57 4 65804 94962 67256 94567 68648 94154 56 5 65828 94956 67280 94560 68071 94147 55 6 65853 94949 67303 94553 68694 94140 54 i 65878 94943 67327 94546 68716 94133 53 8 65902 94936 67350 94540 68739 94126 52 9 65927 94930 67374 94533 68762 94119 51 10 9.65952 9.94923 9.67398 9.94526 9.68784 9.94112 50 11 65976 94917 67421 94519 68807 94105 49 12 66001 94911 67445 94513 68829 94098 48 i3 66025 94904 67468 94506 68852 94090 47 14 66050 94898 67492 94499 6S875 94083 46 15 66075 94891 67515 94492 68897 94076 45 16 66099 94885 67539 94485 68920 94009 44 17 66124 94878 67.562 94479 68942 94062 43 18 66148 94871 67586 94472 68965 94055 42 19 66173 94865 67609 94465 68987 94048 41 20 9.66197 9.94858 9.67633 9.94458 9.69010 9.94041 40 21 66221 948.52 676.56 94451 69032 94034 39 22 66246 94845 67680 94445 69055 94027 38 23 662';0 94839 67703 94438 69077 94020 37 24 66295 9483-2 67726 94431 69100 94012 36 25 66319 94826 67750 94424 69122 94005 35 26 66343 94819 67773 94417 69144 93998 34 27 28 60368 66392 94813 94806 67796 67820 94410 94404 69167 69189 93991 93984 33 32 29 66416 94799 67843 94397 69212 93977 31 30 9.66441 9.94793 9.67866 9.94390 9.69234 9.93970 30 31 66165 94786 67890 94383 69256 93963 29 32 66489 94780 67913 94376 69279 93955 28 33 66513 94773 67936 94369 69301 93948 2i 34 66537 94767 67959 94362 69323 93941 26 35 66502 94700 67982 94355 69345 93934 25 36 665^6 94753 68006 94349 69368 93927 24 37 60610 94747 08029 94342 69390 93920 23 38 66634 94740 08052 94335 69412 93912 22 39 66658 94734 68075 94328 69434 93905 21 40 9.66682 9.94727 9.68098 9.94321 9.69456 9.93898 20 41 66706 94720 68121 94314 69479 93891 19 1 o 42 66731 94714 68144 94307 69.501 93884 18 4 r* 43 60755 94707 68167 94300 69523 93876 17 1 iS 44 66779 94'; 00 68190 94293 69545 93869 lo 45 00803 94694 68213 94286 69567 93862 15 40 60827 94687 68237 94279 69589 93855 14 -1 O 47 66851 94080 68260 94273 69611 93847 13 1 O 48 60875 94674 08283 94266 69638 93840 12 49 60899 94067 68305 94259 69655 93833 11 50 9.66922 9.94660 9.68328 9.94252 9.69677 9.93826 10 ' 9 8 51 6C946 94654 68351 94245 69699 93819 52 00970 94647 68374 94238 69721 93811 53 60994 94640 68397 94231 69743 93804 4 6 5 54 67018 94634 68420 94224 69765 93797 55 67042 94627 68443 94217 69787 93789 56 67066 94620 68466 94210 69809 93782 4 3 2 57 67090 94614 68489 94203 69831 93775 58 67113 94607 68512 94196 69853 93768 59 67137 94000 68534 94189 69875 93760 1 60 67101 94593 68557 94182 69897 93753 / Cosine Sine 62° Cosine Sine Cosine Sine / 61° 60° "7 %• 'i -1- ■). .!< W" o%f-. > ;• . ; 1 « n ^O.^^Xi ■ -J on t 2] -» o o 248 table' V!t.— logarithmic sines ano rosiNEs. t 30° 31° 32° / Sine Cosine Sine Cosine Sine Cosine 9.69897 9.93753 9.71184 9.93307 9.72421 9.92842 60 1 69919 93746 71205 93299 72441 92834 59 2 69941 93738 71226 93291 72461 92826 58 3 69963 93731 71247 93284 72482 92818 57 4 69984 93724 71268 93276 72502 92810 56 5 70006 93717 71289 93269 72522 92803 55 6 70028 93709 71.310 93261 72542 92795 54 7 70050 93702 71331 93253 72562 92787 53 8 70072 93595 71352 93246 72582 92779 52 9 70093 93687 71373 93238 72602 92771 51 10 9.70115 9.93680 9.71393 9.93230 9.72622 9.92763 50 11 70137 93673 71414 932-,'3 72043 92755 49 12 70159 93665 71435 93215 72663 92747 48 13 70180 93658 71456 93207 72683 92739 47 14 70202 93650 71477 93200 72703 92731 46 15 70224 93643 71498 93192 72723 92723 45 16 70245 93636 71519 93184 72743 92715 44 17 70267 93628 71539 93177 72763 92707 43 18 70288 93621 71560 93169 72783 92C99 42 19 70310 93614 71581 93161 72803 92691 41 20 9.70332 9.93606 9.71602 9.93154 9.72823 9.92683 40 21 70353 93599 71622 93146 72843 92675 39 22 70375 93591 71643 93138 72863 92667 38 23 70396 93584 71664 93131 72883 92659 37 24 70418 93577 71685 93123 72902 92651 o 1 36 25 70439 93569 71705 93115 72922 92643 35 26 70461 93562 71726 93108 72942 92635 34 27 70482 93554 71747 93100 72962 92627 33 28 70504 93547 71767 93092 72982 92619 32 29 70525 93539 71788 93084 73002 92611 31 30 9.70547 9.93532 9.71809 9.93077 9.73022 9.92603 30 31 70568 93525 71829 93069 73041 92595 29 32 70590 93517 71850 93061 73061 92587 28 33 70611 93510 71870 9.3053 73081 92579 27 34 70633 93502 71891 93046 73101 92571 26 35 70654 93495 71911 93038 73121 92563 25 36 70675 93487 71932 93030 73140 92555 24 37 70697 93480 71952 93022 73160 92546 23 38 70718 93472 71973 93014 73180 92538 22 39 70739 93465 71994 93007 73200 92530 21 40 9.70761 9.93457 9.72014 9.92999 9.73219 9.92522 20 41 70782 93450 72034 92991 732.39 92514 19 42 70803 93442 72055 92983 73259 92506 18 43 70824 934:35 72075 92976 73278 92498 17 44 70846- 93427 72096 92968 73298 92490 16 45 70867 93420 72116 92960 73318 92482 15 46 70888 93412 72137 92952 73337 92473 14 47 70909 93405 72157 92944 73357 92465 13 48 70931 93397 72177 92936 73377 92457 12 49 70952 93390 72198 92929 73396 92449 11 50 9.70978 9.93382 9.72218 9.92921 9.73416 9.92441 10 51 70994 93375 72238 92913 73435 92433 9 52 71015 93367 72259 92905 73455 92425 8 53 71036 93360 72279 92897 73474 92416 54 71058 93.352 72299 92889 73494 92408 6 55 71079 93344 72320 92881 73513 92400 5 56 71100 93337 72340 92874 735.33 92392 4 57 71121 93329 72360 92866 73552 92384 3 58 71142 93322 72881 92858 73572 92376 o 59 71163 93314 72401 92850 73591 92367 1 60 71184 93307 72421 92842 73611 92359 / Cosine Sine Cosine Sine Cosine Sine ^ / 69° «8° 57° ♦ A Slnt Sih 1^ .■jSIj^:Pf>y. Uos.tSiu.ios. Sm,Vofi. »m. -IQO ^*^/l "If)!- TABT.F^rIr— LOGARITkMie SINES Al^lTcfcsINES. 24D 1 33° 34° 85° / Sine Cosine Sine Cosine Sine Cosine 9.73611 9.92359 9.74756 9.918.57 9.758.59 9.91336 60 1 73630 92351 74775 91849 75877 91328 59 2 73650 92343 74794 91840 75895 91319 58 3 73669 92335 74812 91832 75913 91310 57 4 73689 92326 74831 91823 75931 91301 56 5 73708 92318 74850 91815 7.5949 91292 55 6 73727 92310 74868 91806 75967 91283 54 r* f 73747 92302 74887 91798 75985 91274 53 8 73766 92293 74906 91789 76003 91266 52 9 73785 92285 74924 91781 76021 91257 51 10 9.73805 9.92277 9.74943 9.91772 9.76039 9.91248 50 11 73824 92269 74961 91763 76057 91239 49 12 73813 922C0 74980 91755 76075 91230 48 13 73863 92252 74999 91746 76093 91221 47 14 73882 92244 75017 91738 76111 91212 46 15 73901 92235 75036 91729 76129 91203 45 16 73921 92227 75054 91720 76146 91194 44 17 73940 92219 75073 91712 76164 91185 43 18 73959 92211 75091 91703 76182 91176 42 19 73978 92202 75110 91695 76200 91167 41 20 9.73997 9.92194 9.75128 9. 91 (-86 9.76218 9.91158 40 21 74017 92186 75147 91677 76236 91149 39 22 74036 92177 75165 91669 76253 91141 38 28 74055 92169 75184 91660 76271 91132 37 24 74074 92161 75202 91651 76289 91123 36 25 74093 92152 75221 91643 76307 91114 35 26 74113 92144 75239 91634 76324 91105 34 27 74132 92136 75258 91625 76342 91096 33 28 74151 92127 75276 91617 76360 91087 32 29 74170 92119 75294 91608 76378 91078 31 30 9.74189 9.92111 9.75313 9.91599 9.76395 9.91069 30 31 74208 92102 75331 91591 76413 91060 29 32 74227 92094 75350 91582 76431 91051 28 33 74246 92086 75368 91573 76448 91042 27 34 '<4265 92077 75386 91565 76466 91033 26 35 74284 92069 75405 91556 76484 91023 25 36 74303 92060 75423 91547 76501 91014 24 37 74322 92052 75441 91538 76519 91005 23 38 74341 92044 75459 91530 76537 90996 22 39 74360 92035 75478 91521 76554 90987 21 40 9.74379 9.92027 9.75496 9.91512 9.76572 9.90978 20 41 74398 92018 75514 91504 76590 90969 19 42 74417 92010 75533 91495 76607 90960 18 43 74436 92002 75551 91486 76625 90951 17 44 74455 91993 75569 91477 76642 90942 16 45 74474 91985 7.5587 91469 76660 90933 15 46 74493 91976 75605 91460 76677 90924 14 47 74512 91968 75624 91451 76695 90915 13 48 74531 91959 75642 91442 76712 90906 12 49 74549 91951 75660 91433 76730 90896 11 50 9.74568 9.91942 9.75678 9.91425 9.76747 9.90887 10 51 74587 91934 75696 91416 76765 90878 9 52 74606 91925 7.5714 91407 76782 90869 8 53 74625 91917 75733 91398 76800 90860 1 54 74644 91908 7.5751 91.389 76817 90851 6 55 74662 91900 75769 91381 76835 90842 5 56 74681 91891 75787 91372 76852 90832 4 57 74700 91883 75805 91363 76870 90823 3 58 74719 91874 75823 91354 76887 90814 2 59 74737 91866 75841 91345 76904 90805 1 60 74756 91857 75859 91336 76922 90796 / Cosine Sine Cosine Sine Cosine Sine / 66» 65° 54» •1 -jC^ f 250 TABLE^ti.— LOGARITHMIC SINES 'aS0 COSINES. 36° 37° 1 4 J8° / Sine Cosine Sine Cosine Sine Cosine 9.76922 9.90796 9.77946 9.90235 ■ 9.78934 9.89653 60 1 76939 90787 77963 90225 78950 89643 59 2 76957 90777 77980 90216 78967 89633 58 3 76974 90768 77997 90206 78983 89624 57 4 76991 90759 78013 90197 78999 89614 56 5 77009 90750 78030 90187 79015 89604 55 6 77026 90741 78047 90178 79031 89594 54 7 77043 90731 78063 90168 79047 89584 53 8 77061 90722 78080 90159 79063 89574 52 9 77078 90713 78097 90149 79079 89564 51 10 9.77095 9.90704 9.78113 9.90139 9.79095 9.89554 50 11 77112 90694 78130 90130 79111 89544 49 12 77130 90685 78147 90120 79128 89534 48 13 77147 90676 78163 90111 79144 89524 47 14 77164 90667 78180 90101 79160 89514 46 15 77181 90657 78197 90091 79176 89504 45 16 77199 90648 78213 90082 79192 89495 44 17 77216 90639 78230 90072 79208 89485 43 18 77233 90630 78246 90063 79224 89475 42 19 77250 90620 78263 90053 79240 89465 41 20 9.77268 9.90611 9.78280 9.90043 9.79256 9.89455 40 21 77285 90602 78296 90034 79272 89445 39 22 77302 90592 78313 90024 79288 89435 38 23 77319 90583 78329 90014 79304 89425 37 24 77336 90574 7834G 90005 79319 89415 36 25 77353 90565 78362 89995 79335 89405 35 26 77370 90555 78379 89985 79351 89395 34 27 77387 90546 78395 89976 79367 89385 33 28 77405 90537 78412 89966 79383 89375 32 29 77422 90527 78428 89956 79399 89364 31 30 9.77439 9.90518 9.78445 9.89947 9.79415 9.89354 30 31 77456 90509 78461 89937 79431 89344 29 32 77473 90499 78478 89927 79447 H9334 28 33 77490 90490 78494 89918 79463 89324 27 34 77507 90480 78510 89908 79478 89314 26 35 77524 90471 78527 89898 79494 89304 25 36 77541 90462 78543 89888 79510 89294 24 37 77558 90452 78560 89879 79.526 89284 23 38 77575 90443 78576 89869 79542 89274 22 39 77592 90434 78592 89859 79558 89264 21 40 9.77609 9.90424 9.78609 9.89849 9.79573 9.89254 20 41 77626 90415 78625 89840 79589 89244 19 42 77643 90405 78642 89830 79605 89233 18 43 77660 90396 7S658 89820 78621 89223 17 44 77677 90386 78674 80810 7963(; 89213 16 45 77694 90377 78691 89801 79652 89203 15 46 77711 9036S 78707 89791 79668 89193 14 47 77728 90358 78723 89781 79684 89183 13 48 77744 90349 78739 89771 79699 89173 12 49 77761 90339 78756 89761 79715 89162 11 50 9.77778 9.90330 9.78772 9.89752 9.79731 9.89152 10 51 77795 90320 78788 89742 79746 89142 9 52 77812 90311 78805 89732 79762 89132 8 53 77829 90301 78821 89722 79778 89122 7 54 77846 90292 78837 89712 79793 89112 6 55 77862 90282 78853 89702 79809 89101 5 56 77879 90273 78869 89693 79825 89091 4 57 77896 90263 78886 89683 79840 89081 3 58 77913 90254 78902 89673 79856 89071 2 59 77930 90244 78918 89663 79872 89060 1 60 77946 90235 78934 89653 79887 89050 / Cosine Sine Cosine Sine Cosine Sine / i 53° 52° 51° ^;:k%5. sM^s. sin}cas. C^^^.Sm. C(^^^ Sin. Cqh,, Sih, TABLE Vtt.—LOGARITlilirtC SINES ANli COSINES. 251 / 89° 40° 41° / Sine Cosine Sine Cosine Sine Cosine 9.79887 9.89050 9.80807 9.88425 9.81694 9.87778 60 1 79903 89040 80822 88415 81709 87767 59 2 79918 89030 80837 88404 81723 87756 58 3 79934 89020 80852 88394 81738 87745 57 4 79950 89009 80867 88383 81752 87734 56 5 79965 88999 80882 88372 81767 87723 55 6 79981 88989 80897 88362 81781 87712 54 < 79996 88978 80912 88351 81796 87701 53 8 80012 88968 80927 88340 81810 87690 52 9 80027 88958 80942 88330 81825 87679 51 1 10 9.80043 9.88948 9.80957 9.88319 9.81839 9.87668 50 11 80058 88937 80972 88308 81854 87657 ' 49 1~' 80074 88927 80987 88298 81868 87646 48 13 80089 88917 81002 88287 81882 87635 47 14 80105 88906 81017 88276 81897 87624 46 15 80120 88896 81032 88266 81911 87613 45 16 80136 88886 81047 88255 81926 87601 44 17 80151 88875 81061 88244 81940 87590 43 18 80166 88865 81076 88234 819.55 87579 42 19 80182 88855 81091 88223 81969 87568 41 20 9.80197 9.88844 9.81106 9.88212 9.81983 9.87557 40 ^1 80213 88834 81121 88201 81998 87546 39 8'> 80228 8S824 81136 88191 82012 87535 38 2:i 80244 88813 81151 88180 82026 87524 37 24 80259 8S803 81166 88169 82041 87513 36 25 80274 88793 81180 88158 820.')5 87501 35 26 80290 88782 81195 88148 82069 87490 34 27 80305 88772 81210 88137 82084 87479 33 28 80320 88761 81225 88d26 82098 87468 32 29 80336 88751 81240 88115 82112 87457 31 30 9.80351 9.88741 9.81254 9.88105 9.82126 9.87446 30 31 80866 88730 81269 88094 82141 87434 29 32 80382 88720 81284 88083 82155 87423 28 33 80397 88709 81299 88072 82169 87412 27 34 80412 88699 81314 88061 82184 87401 26 35 80428 88688 81328 88051 82198 87390 25 36 80443 88678 81343 88040 82212 87378 24 37 80458 88668 81358 88029 82226 87367 23 38 80473 88657 81372 88018 82240 87356 22 39 80489 88647 81387 88007 82255 87345 21 40 9.80504 9.88636 9.81402 9.87996 9.82269 9.87334 20 41 80519 88626 81417 87985 82283 87322 19 42 80534 88615 81431 87975 82297 87311 18 43 80550 88605 81446 87964 82311 87300 17 44 80565 88594 81461 87953 82326 87288 16 45 80580 88584 81475 87942 82340 87277 15 46 80595 8^^573 81490 87931 82354 87266 14 47 80610 88563 81505 87920 82368 87255 13 48 80625 88552 81.519 87909 82382 87243 12 49 80641 88542 81534 87898 82396 87232 11 50 9.80656 9.88531 9.81549 9.87887 9.82410 9.87221 10 51 80671 88521 81563 87877 82424 87209 9 52 80686 88510 81578 87866 82439 87198 8 53 80701 88499 81592 87855 824.^3 87187 7 54 80716 88489 81607 87844 82467 87175 6 55 80731 88478 81622 87833 82481 87164 5 56 80746 88468 81036 87822 82495 87153 4 57 80762 88457 81651 87811 82509 87141 3 58 80777 88147 81665 87800 82523 87130 2 59 80792 88436 81680 87789 82537 87119 1 60 80807 88425 81694 87778 82551 87107 / Cosine Sine Cosine Sine Cosine Sine / 60° 49° 48° 3>^; Sii'ifCos. sin -Cos. SmlCat. m TABLE Xm. LC )GARIT HMrC SI NES A! ^D'COSIi STES. 1 42° 43° 44° / 1 Sine Cosine Sine Cosine Sine Cosine 9.82551 9.8710? 9.83378 9.86413 9.84177 9.85693 60 1 ! 8:2565 87096 8:3392 86401 84190 85681 59 o 82579 87085 83405 86389 84203 85669 58 3 82593 87073 83419 86377 84216 85657 57 4 82607 87062 83432 86366 84229 85645 56 5 82621 87050 8:3446 86354 84242 85632 55 6 82635 87039 83459 86:342 84255 85020 54 7 82649 87028 83473 86:330 84269 85608 53 8 82663 87016 83486 86318 84282 85596 52 9 82677 87005 83500 86306 84295 85583 51 10 9.82691 9.86993 9.83513 9.86295 9.84308 9.85571 50 11 82T05 86982 83527 86283 84321 85559 49 12 82719 86970 83540 86271 84334 85547 48 13 82733 86959 83554 86259 84347 85534 47 14 82747 86947 8:3567 86247 84360 85522 46 15 82761 86936 83581 86235 84373 85510 45 16 82775 86924 835^ 86223 84385 85497 44 17 82788 86913 83608 86211 84398 8.5485 43 18 82802 86902 83621 86200 84411 85473 42 19 82816 86890 83634 86188 84424 85460 41 20 - 9.82830 9.86879 9.a3648 9.86176 9.84437 9.8.5448 40 21 82844 86867 83661 86164 84450 85436 39 22 82858 86855 83674 86152 84463 85423 38 23 82872 86844 83688 86140 84476 8.5411 37 24 82885 86a32 83701 86128 84489 85399 36 25 82899 86821 8:3715 86116 84502 85386 35 26 82913 86809 83728 86104 84515 85374 34 27 82927 86798 83741 86092 84528 85361 33 28 82941 86786 83755 86080 84540 85349 32 29 82955 86775 83768 86068 84553 85337 31 30 9.82968 9.86763 9.83781 9.86056 9.84566 9.85324 ;30 31 82982 86752 83795 86044 84579 85312 29 32 82996 86740 8:3808 86032 84592 85299 28 33 83010 86728 83821 86020 84605 85287 27 34 83023 86717 83834 86008 84618 85274 26 a-j 83037 86705 83848 85996 84630 85262 25 36 83051 86694 83861 85984 84643 85250 24 37 83065 86682 &3874 85972 84656 85237 23 38 83078 86670 83887 85960 84669 85225 22 39 83092 86659 83901 85948 84682 85212 21 40 9.83106 9.86647 9.83914 9.85936 9.84694 9.85200 20 41 t 83120 866:35 83927 85924 84707 85187 19 42 83133 86624 83940 8.5912 84720 85175 18 43 83147 86612 83954 85900 84733 85162 17 44 83161 86600 83967 85888 84745 85150 16 45 83174 86589 83980 85876 84758 85137 15 46 83i8S 86577 83993 85864 84771 85125 14 47 83202 86565 84006 85851 84784 85112 13 48 83215 86554 84020 85839 84796 85100 12 49 83229 86542 84033 85827 84809 85087 11 50 9.83242 9.86530 9.84046 9.a5815 9.84822 9.85074 10 51 83256 86518 84059 85803 84835 85062 9 52 83270 86507 84072 85791 84847 85049 8 53 83283 86495 84085 85779 84860 85037 < 54 83297 86483 84098 85766 84873 85024 6 55 83310 86472 84112 85754 84885 85012 5 56 8:3:324 86460 84125 a5742 84898 84999 4 57 8a3.38 86448 84138 857:30 84911 84986 3 58 8:33ol 86436 84151 85718 84923 84974 2 59 83365 86425 84164 85706 849:36 84961 1 60 83.378 86413 84177 85693 84949 84949 / Cosine Sine Cosine Sine Cosine Sine / 470 46° 45° Oe* •- • >L OS* Sb ;. Sh t - ^, TABLE VIII.— LOG. TANGENTS AND COTANGENTS. 253 / 0" 1° 2° 1 / 1 1- 60 Tan Cotan Tan Cotan Tan Cotan — QO 00 8.24192 11.75808 8.54308 11.45692 1 6.46373 13.53627 24910 75090 54669 45331 59 2 76476 23524 25616 74384 55027 44973 58 3 94085 05915 26312 73688 55382 44618 57 4 7.06579 12.93421 26996 73004 55734 44266 56 5 16270 83730 27669 72331 56083 43917 55 6 24188 75812 28332 71668 56429 43571 54 7 30882 69118 28986 71014 66773 43227 53 8 36682 63318 29629 70371 57114 42886 52 9 41797 58203 30263 69737 57452 42548 51 10 7.46373 12.53627 8.30888 11.69112 8.57788 11.42212 50 11 50512 49488 31505 68495 58121 41879 49 12 54291 45709 32112 67888 58451 41549 48 Vi 57767 42233 32711 67289 58779 41221 47 14 60986 39014 33;W2 66698 59105 40895 46 15 63982 36018 33886 66114 59428 40572 45 16 66785 33215 34461 65539 59749 40251 44 17 69418 30582 35029 64971 C0068 39932 43 18 71900 28100 35590 64410 60384 39616 42 19 74248 25752 36143 63857 60698 39302 41 20 7 76476 12.23524 8.36689 11.63311 8.61009 11.38991 40 21 78595 21405 37-J29 62771 61319 38681 39 22 80615 19385 37762 62238 61626 38374 38 23 82546 17454 38289 61711 61931 38069 37 24 84394 15606 38809 61191 62234 37766 36 25 86167 13833 39323 60677 62535 37465 35 26 87871 12129 39832 60168 6-2834 37166 34 27 89510 10490 40334 59666 63131 36869 33 28 91089 08911 40830 59170 63426 36574 32 29 92C13 07387 41321 58679 63718 36282 31 30 7.94086 12.05914 8.41807 11.58193 8.64009 11.35991 30 31 95510 04490 42287 57713 64298 35702 29 32 96889 03111 42762 57238 64585 35415 28 33 98225 01775 43232 56768 64870 35130 27 34 99522 00478 43696 56304 65154 34846 26 35 8.00781 11.99219 44156 55844 65435 34565 25 36 02004 97996 44611 55389 65715 34285 24 37 03194 96806 45061 54939 65993 34007 23 38 04:353 95647 45507 54493 66269 33731 22 39 05481 94519 45948 54052 66543 33457 21 40 8.06581 11.93419 8.46385 11.53615 8.66816 11.33184 20 41 07653 92347 46817 53183 67087 32913 19 42 08700 91300 47245 52755 67356 32644 18 43 09722 90278 47669 52331 67624 32376 ; 17 44 10720 89280 48089 51911 67890 3-2110 ' 16 45 11696 88304 48505 51495 68154 31846 1 15 46 12651 87349 48917 51083 68417 31583 i 14 47 13585 86415 49325 50675 68678 31322 13 48 14500 85500 49729 50271 68938 31062 12 49 15395 84605 50130 49870 69196 30804 11 50 8.16273 ll.a3727 8.50527 11 .49473 8.69453 11.30547 10 51 17133 82867 50920 49080 69708 20292 9 52 179:6 82024 51310 4S690 69962 30038 8 53 18804 81196 51696 48304 70214 29786 7 54 19616 80384 52079 47921 70465 29535 6 55 20413 79587 52459 47541 70714 29286 1 5 56 21195 78805 52835 47165 70962 29038 i 4 57 21964 78036 53208 46792 71208 28792 ' 8 58 227:i0 77280 53578 464-22 714.53 28547 2 59 23462 76538 53945 46055 71697 28303 1 60 24192 75808 54308 45692 71940 28060 / Cotau Tan Cotan Tan Cotan Tan / 89° 88° 87° ' !o4 TABLE vni.—LOW. TANGENTS AND COTANGENTS / 1 3° 4» 5° 1 Tan Cotan Tan Cotan Tan Cotan 8.71940 11.28060 8.84464 11.15.536 8.94195 11.05805 60 1 72181 27819 84646 15354 94340 05660 59 2 72420 27580 84826 1.5174 94485 05515 58 3 72659 27341 85006 14994 94630 05370 57 4 72896 27104 85185 14815 94773 05227 56 5 73182 26868 85:363 14637 94917 0.508:? 55 6 73366 26634 85540 14460 95060 04940 54 7 73600 26400 85717 14283 95202 04798 53 8 73832 26168 85893 14107 95344 04656 52 9 74063 25937 86069 13931 95486 04514 51 10 8.74292 11.25708 8.86243 11.137.57 8.9.5627 11.04373 50 i 11 74521 25479 86417 1.3583 95767 04233 49 1 12 74748 25252 86591 13409 95908 04092 48 13 74974 25026 86763 132.37 96047 03953 47 14 75199 24801 86935 13065 96187 03813 46 15 75423 24577 87106 12894 96325 03675 45 16 75645 24355 87277 12723 96464 03536 44 17 75867 24133 87447 12553 96C02 03398 43 18 76087 23913 87616 12.384 96739 03261 42 19 76306 23694 8r;'85 8.87953 12215 11.12047 96877 8.97013 41 40 20 3.76525 11.23475 11.02987 21 76742 23258 88120 11880 97150 02850 39 22 76958 23042 88287 11713 97285 02715 38 23 77173 22827 88453 11547 97421 02579 37 24 77387 22613 88618 11382 97556 02444 36 25 77600 22400 88783 11217 97691 02309 35 26 77811 22189 88948 11052 97825 02175 34 27 78022 21978 89111 10SS9 979.59 02041 33 28 78232 21768 89274 10726 98092 01908 32 29 78441 21559 89437 10563 98225 01775 31 30 8.78649 11.21351 8.89598 11.10402 8.98.3.58 11.01642 30 31 78855 21145 89760 10240 98490 01510 29 32 79061 20939 89920 10080 98622 01378 28 33 79266 20734 90080 09920 987.53 01247 27 34 79470 20530 90240 09760 98884 01116 26 35 79673 20827 90399 09601 99015 00985 25 36 79S75 201S5 90557 09443 99145 00855 24 37 80076 19924 90715 09285 99275 00725 23 38 80277 19723 90872 09128 99405 00595 22 39 80476 • 19524 91029 08971 99534 00466 21 40 8.80674 11.19326 8.91185 11.08815 8.99662 11.00.338 20 41 80872 19128 91340 08660 99791 00209 19 42 81068 18932 91495 08505 99919 00081 18 43 81264 18736 916.50 0a350 9.00046 10.99954 17 44 81459 18541 91803 08197 00174 99826 16 45 816.53 18317 91957 08043 00301 99699 15 46 81H46 18154 92110 07890 00427 99573 14 1 47 82038 17962 92262 0773S 00553 99447 IS 48 82230 17770 92414 07586 00679 99321 12 49 82420 17580 92565 07435 00805 99195 11 50 8.82610 11.17390 8.92716 11.07284 9.00930 10.99070 10 51 82799 17201 92866 07134 010.i5 98945 9 52 82987 17013 93016 06984 01179 98821 8 53 83175 16825 93165 06835 01303 98697 51 83361 166.39 93313 06587 01427 98573 6 5.-. 83547 164.53 93462 06538 015.50 98450 5 56 83732 16268 93609 06391 01673 98327 4 57 83916 16084 93756 06244 01796 98204 3 58 84100 1.590O 93903 06097 01918 9^^082 2 59 84282 15718 94049 05951 02040 97960 ^ 1 60 84464 15536 94195 05805 02162 978.38 / Co tan Tan Cotan Tan Cotan Tan / S6» 8o° 84° TABLE VIII.— LOG. TANGENTS AND COTANGENTS. 255 6 B 7 o 8° 1 Tan Cotan Tan Cotan Tan Cotan 10.85220 9.02162 10.97838 9.08914 10.91086 9.14780 60 1 02283 97717 09019 90981 14872 85128 59 o 02404 97596 09123 90877 14963 85037 58 3 02525 97475 09227 90773 15054 84946 57 4 02645 97355 09330 90670 15145 84855 56 5 02766 97234 09434 90566 15236 84764 55 6 02885 97115 09537 90463 15327 84673 54 i 03005 96995 09640 90360 15417 84583 53 8 03124 96876 09742 90258 15508 84492 52 9 03242 96758 09845 90155 15598 84402 51 10 9.03361 10.96639 9.09947 10.90053 9.15688 10.84312 50- 11 03479 96521 10049 89951 15777 84228 49 12 03597 96403 10150 89850 15867 84133 48 13 03714 96286 10252 89748 15956 84044 47 14 03832 96168 10353 89647 16046 83954 46 15 03948 96052 10454 89546 16135 83865 45 16 04065 95935 10555 89445 16224 83776 44 17 04181 95819 10656 89344 16312 83688 43 18 04297 95703 10756 89244 16401 83599 42 19 04413 95587 10856 89144 16489 83511 41 20 9.04528 10.95472 9.10956 10.89044 9.16577 10.83423 40 21 04643 95357 11056 88944 16665 83335 39 22 04758 95242 11155 88845 16753 83247 38 23 04873 95127 11254 88746 16841 83159 37 24 04987 95013 11353 88647 16928 83072 36 25 05101 94899 11452 88548 17016 82984 35 26 05214 94786 11551 88449 17103 82897 34 27 05328 94672 11649 88351 17190 82810 33 28 05441 94559 11747 88253 17277 82723 32 29 05553 94447 11845 88155 17363 82637 31 30 9.05666 10.94334 9.11943 10.88057 9.17450 10.82550 30 31 05778 94222 12040 87960 17536 82464 29 32 05890 94110 12138 87862 17622 82378 28 33 06002 93998 12235 87765 17708 82292 27 34 06113 93887 12332 87668 17794 82206 26 35 06224 98776 12428 87572 17880 82120 25 36 06335 93665 12525 87475 17965 82035 24 37 06445 93555 12621 87379 18051 81949 23 38 06556 93444 12717 87283 18136 81864 22 39 06666 93334 12813 87187 18221 • 81779 21 40 9.06775 10.93225 9.12909 10.87091 9.18306 10.81694 20 41 06885 93115 13004 86996 18391 81609 19 42 06994 93006 13099 86901 18475 81525 18 43 07103 92897 13194 86806 18560 81440 17 44 07211 92789 13289 86711 18644 81356 16 45 07320 92680 13384 806 16 18728 81272 15 46 07428 92572 13478 86522 18812 81188 14 47 07536 92464 13573 86427 18896 81104 13 48 07643 92357 13667 86338 18979 81021 12 49 07751 92249 13761 86239 19063 80937 11 50 9.07858 10.92142 9.13854 10.86146 9.19146 10.80854 10 51 07964 92036 13948 86052 19229 80771 9 52 08071 91920 14041 85959 19312 80688 8 53 08177 91823 14134 85866 19395 80605 ^ 54 08283 91717 14227 85773 19478 80522 6 55 08389 91(n) 14320 85680 19561 80439 5 56 08495 91505 14412 85538 19643 80357 4 57 08600 91400 14504 85496 19725 80275 3 58 08705 91295 1 4597 85403 19807 80193 2 59 08810 91190 14688 85312 19889 80111 1 60 08914 91086 14780 85220 19971 80029 1 Cotan Tan Cotan Tan Cotan Tan / 83" 82" 81° 256 TABLE VIII.— LOG. TANGENTS AND COTANGENTS. / 9° 10" 11° t Tan Cotan Tan Cotan Tan Cotan 9.19971 10.80029 9.24632 10.75368 9.28865 10.71135 60 1 20053 79947 24706 75294 289:33 71067 59 2 20134 79866 24779 75221 29000 71000 58 3 20216 79784 24853 75147 29067 70933 57 1 20297 79703 24926 75074 29134 70866 56 5 20378 79622 25000 75000 29201 70799 55 6 20459 79541 25073 74927 29268 70732 54 7 20540 79460 25146 74854 29335 70665 53 8 20621 79379 25219 74781 29402 70598 52 9 20701 79299 25292 74708 29468 70532 51 10 9.20782 10.79218 9.25365 10.74685 9.29535 10.70465 50 11 20862 79138 25437 74563 29601 70399 49 12 20942 79058 25510 74490 29668 70332 48 13 21022 78978 25582 74418 29734 70266 47 14 21102 78898 25655 74345 29800 70200 46 15 21182 78818 25727 74273 29866 70134 45 16 21261 78739 25799 74201 29932 70068 44 17 21341 78659 25871 74129 29998 70002 43 18 21420 78580 25943 74057 30064 69936 42 19 21499 78501 26015 73985 30130 69870 41 30 9.21578 10.78422 9.26086 10.73914 9.30195 10.69805 40 21 21657 78343 26158 73842 30261 69739 39 22 21736 78264 26229 73771 30326 696T4 38 23 21814 78186 26301 73699 30391 69609 37 24 21893 78107 26372 73628 30457 69543 36 25 21971 78029 26443 7:3657 30522 69478 35 26 22049 77951 26514 73486 :30587 69413 34 27 22127 77873 26585 73415 30652 69348 33 28 22205 77795 266.55 7:3345 30717 69283 32 29 82283 77717 26726 73274 30782 69218 31 30 9.22361 10.77639 9.26797 10.73203 9.30S46 10.691.54 30 31 22438 77562 26^67 731:33 30911 69089 29 32 22516 77484 26937 73063 30975 69025 28 33 22593 77407 27008 72992 31040 68960 27 34 22670 773;J0 27078 72922 31104 68896 26 35 22747 77253 27148 72852 31168 68832 25 36 22824 77176 27218 72782 31233 68767 24 37 22901 77099 27288 72712 31297 68703 23 38 22977 77023 27357 72643 31361 68639 22 39 2.i034 76946 27427 72573 31425 68575 21 40 9.23130 10.76870 9.27496 10.72.504 9.31489 10.68511 20 41 23206 76794 27566 72434 315.52 68448 19 42 23283 76717 27635 72365 31616 68384 18 43 23359 76641 27704 72296 31679 68321 17 44 ^3435 76565 27773 72227 31743 68257 16 45 23510 76490 27842 72158 31806 68194 15 46 23586 76414 2791 1 72089 31870 681:30 14 47 2:3661 76;i39 27980 72020 319:33 68067 13 48 23737 76263 28049 71951 31996 68004 12 49 23812 76188 28117 71883 32059 67941 11 50 9.2:J887 10.76113 9.2S1S6 10.71S14 9.32122 10.67878 10 51 23962 76038 28254 71746 :32185 67815 9 52 24037 75963 28323 71677 :32248 67752 8 m 24112 75888 28391 71609 3-'3l 1 676.e9 ^ i 54 24186 7.5814 28459 71541 :32373 67627 6 55 24261 75739 28527 71473 324:30 67.564 5 .56 243:i-> 7.5665 2859.5 71405 32498 67502 4 57 24410 75590 28662 71338 3^561 K7439 3 58 ! 24484 7.5516 28730 71270 32623 67377 o 59 21.5.58 7.5442 28708 71202 32685 67315 1 60 1 \ 24632 75368 28865 71135 32747 672.53 Cotan Tan Cotan Tan Cotan Tan so- JJP 78° TABLE VIII.— LOG. TANGENTS AND COTANGENTS. 257 / 12° 13° 14° 1 / Tan Cotan Tan Cotan Tan Cotan 9.32747 10.67253 9.36336 10.63664 9.39677 10.60323 60 1 32810 67190 36394 63606 39731 60269 59 2 32872 67128 36452 63548 39785 60215 58 3 32933 670G7 36509 63491 39838 60162 .57 4 32995 67005 36566 63434 89892 60108 56 5 33057 66943 36624 63376 39945 60055 55 6 33119 6G881 36681 63319 39999 60001 54 r^ ^ 83180 66820 36738 63262 40052 59948 53 8 33-242 66758 36795 63205 40106 59894 52 9 33303 66697 36852 63148 40159 59841 51 10 9.33365 10.66C35 9.36909 10.63091 9.40212 10.59788 50 11 33426 66574 3G9G6 63034 40266 59734 49 12 33487 66513 37023 62977 40319 59681 48 13 33548 66452 37080 62920 40372 59628 47 14 33G09 66391 37137 62863 40425 59575 46 15 33670 66330 37193 62807 40478 59522 45 16 33731 66269 37250 62750 40531 59469 44 17 33792 66208 37306 62694 40584 59416 43 18 33853 6G147 37363 62637 40636 59364 42 19 33913 66087 37419 62581 40689 59311 41 20 9.33974 10.66026 9.37476 10.62524 9,40742 10.59258 40 21 34034 65966 37532 62468. 40795 59205 39 22 34095 65905 37588 62412 40847 591,53 38 23 34155 65845 37644 62356 40900 59100 37 24 34215 65785 37700 62300 40952 59048 36 25 34276 65724 37756 62244 41005 58995 35 26 34336 65664 37812 62188 41057 58943 34 27 34396 65604 37868 621.32 41109 58891 33 28 34456 65544 37924 62076 41161 58839 32 29 34516 65484 37980 62020 41214 58786 31 30 9.34576 10.65424 9.38035 10.61965 9.41266 10.. 587.34 30 31 34635 65365 38091 61909 41318 58682 29 ;32 34695 65305 38147 61853 41370 58630 28 33 34755 6.5245 38202 61798 41422 58578 27 34 34814 65186 38257 61743 41474 58526 26 35 34874 65126 38313 61687 41526 58474 25 36 34933 65067 38368 61632 41578 58422 24 37 34992 65008 3&423 61577 41629 58371 23 38 35051 64949 38479 61521 41681 58319 22 39 35111 64889 38534 61466 41733 58267 21 40 9.35170 10.6-4830 9.38589 10.61411 9.41784 10.58216 20 41 35229 64771 38644 61356 41836 .58164 19 42 35288 64712 .38699 61301 41887 58113 18 43 35347 64G53 3-^754 61246 41939 58061 17 44 35405 64595 38808 61192 41990 58010 16 45 35464 64536 38863 61137 42041 57959 15 46 35523 64477 38918 61082 42093 57907 14 47 35581 64419 3S972 61028 42144 57856 13 48 35640 64360 39027 60973 42195 57805 12 49 35G98 64302 39082 60918 42246 57754 11 50 9.35757 10.64243 9.39136 10.60864 9.42297 10.. 57703 10 51 35815 64185 39190 60810 42348 57652 9 52 35873 64127 .39245 60755 4239!» 57601 8 53 35931 64069 39299 60701 424.50 57550 7 54 35989 64011 39.^53 60647 42501 57499 6 55 36047 63953 .30407 60593 425,52 57448 5 56 36105 63895 39461 60539 42603 57397 4 57 36163 63837 39515 60485 42653 57.347 3 58 36221 63779 39569 60431 42704 57296 2 59 36279 63721 39623 60377 427.55 57245 1 60 36336 63664 39677 60323 42805 57195 1 • / Cotau Tan Cotan Tan Cotan Tan 77° 76° 75° 358 TABLE VIII.— LOG. TANGENTS AND COTANGENTS. ^ 15» 16° 17° 1 Tan Cotan Tan Cotan Tan Cotan 9.42805 10.57195 9.45750 10.54250 9.48534 10.51466 60 1 42856 57144 45797 54203 48579 51421 59 2 42906 57094 45845 54155 48624 51376 58 3 42957 57043 45892 54108 48669 51331 57 4 43007 56993 45940 54060 48714 51286 56 5 43057 56943 45987 54013 48759 51241 55 6 43108 56892 46035 53965 48804 51196 54 7 43158 56842 46082 53918 48849 51151 53 8 43208 56792 46130 53870 48894 51106 52 9 43258 56742 46177 53823 48939 51061 51 10 9.43308 10.56692 9.46224 10.53776 9.48984 10.51016 50 11 43358 56642 46271 53729 49029 50971 49 12 43408 56592 46319 53681 49073 50927 48 13 43458 56542 46366 53634 49118 50882 47 14 43508 56492 46413 53.587 49163 50837 46 15 48558 56442 46460 53540 49207 50793 45 16 43607 56393 46507 53493 49252 50748 44 17 43657 56343 46554 53446 49296 50704 43 18 43707 56293 46601 53399 49341 50659 42 19 43756 56244 46648 53352 49385 50615 41 20 9.43806 10.56194 9.46694 10 53306 9.49430 10.50570 40 21 43855 56145 46741 53259 49474 50526 ■ 39 22 43905 56095 46788 53212 49519 50481 38 23 43954 56046 46835 53165 4956:3 50437 37 24 44004 55996 46881 53119 49607 50393 36 25 44053 55947 46928 53072 49652 50348 35 26 44102 55898 46975 53025 49(i96 50304 34 27 44151 55849 47021 52979 49740 50260 33 28 44201 55799 47068 52932 49784 50216 32 29 44250 55750 47114 52886 49828 50172 31 30 9.44299 10.55701 9.47160 10.52840 9.49872 10.50128 30 31 44348 55652 47207 527J3 49916 50084 29 32 443!)7 55603 47253 52747 49960 50040 28 83 44446 55554 47299 52701 50004 49996 27 34 44495 55505 47346 526*4 50048 49952 26 35 44544 55456 47392 52608 50092 49908 25 36 44592 55408 47438 52562 50136 49864 24 37 44641 55359 474S4 52516 .50180 49820 23 38 44690 55310 47530 52470 .5(IJ23 49777 22 39 44738 55262 47576 52424 50267 49733 21 40 9.447H7 10.55213 9.47622 10.52378 9.50311 10.49689 20 41 44836 55164 47668 52332 50355 49645 19 42 44884 55116 47714 52286 50398 49602 18 43 44933 .55067 47760 52240 50442 49558 17 44 44981 55019 47806 52194 50485 49515 16 45 4502y 54971 47852 52148 .50529 49471 15 46 45078 54922 47897 52103 50572 49428 14 47 451 2(; 54874 47943 52057 .50616 49384 13 48 45174 54826 479S9 52011 50659 49341 12 49 45222 54778 48035 51965 50703 49297 11 50 9.45271 10.54729 9.48080 10.51920 9.50746 - 10.49254 10 51 45319 546S1 4Sl-,'6 51874 50789 49211 9 52 45367 54633 48171 51829 50833 491 07 8 53 45415 54585 48217 51783 50876 49124 7 51 45463 54537 48262 51738 50919 49081 6 55 45511 54489 48307 5 '693 50962 49038 5 56 45559 54441 483.53 51647 51005 48995 4 57 45()06 54394 48-598 51602 51048 4S952 3 58 45654 54346 48443 51557 51092 48908 2 59 - 45702 54298 48489 51511 51135 48*^65 1 60 45750 54250 48534 51466 51178 48822 / Co I an Tail Cotan Tan Cotan Tan / 740 73° 72° TABLE VIII.— LOG. TANGENTS AND COTANGENTS. 259 / 18» 19° 20° / Tan Cotaii Tan Cotan Tan Cotan 9.51178 10.48822 9.53697 10.46303 9.56107 10.4:3893 60 1 51221 48779 53738 46263 56146 43854 59 o 51264 48736 53779 46221 56185 43815 58 i 51306 48694 53820 46180 56224 43776 57 4 51349 48651 53861 46139 56264 43736 56 5 51392 48608 53902 46098 56303 43697 55 6 51435 48565 53943 46057 56342 43658 54 7 51478 48522 53984 46016 56381 43619 53 8 51520 48480 54025 45975 56420 43580 52 9 51563 48437 54065 45935 56459 43541 51 10 9.51606 10.48394 9.54106 10.4.5894 9.56498 10.43502 50 11 51648 48352 54147 45853 56,537 43463 49 12 51691 48309 54187 45813 56576 43424 48 13 51734 48266 54228 45772 56815 43385 47 14 51776 48224 54269 45731 56654 43346 46 15 51819 48181 54309 45691 56693 43307 45 16 51861 48139 54350 45650 56738 43268 44 17 51903 48097 54390 45610 56771 43229 43 18 51946 48054 54431 45569 56810 43190 42 19 51988 48012 54471 45529 56849 43151 41 20 9.52031 10.47969 9.54512 10.45488 9.56887 10.43113 40 21 52073 47927 545.52 45448 56926 43074 39 22 52115 47885 54593 45407 56965 43035 38 28 52157 47843 54033 45367 57004 42996 37 24 52200 47800 54673 45327 57042 42958 86 25 52242 47758 54714 45286 57081 42919 35 26 52284 47716 54754 45246 57120 42880 34 27 52326 47674 54794 45206 57158 42842 33 28 52368 47032 54835 45165 57197 42803 32 29 52410 4<590 54875 45125 57235 42765 31 30 9.52452 10.47548 9.54915 10.45085 9.57274 10.42726 30 31 52494 47506 54955 45045 5^312 42688 29 32 52536 47464 54995 45005 57351 42649 28 33 52578 47422 55035 44965 57389 42611 27 34 52620 4^380 55075 44925 57428 42572 26 35 52661 47339 5.5115 44885 57466 42534 25 36 52703 47J97 551.55 44845 57.504 42496 24 37 52745 47255 55195 44805 57543 42457 23 38 52787 47213 55235 44765 57581 42419 22 39 52829 47171 5.5275 44725 57619 42381 21 40 9.. 52870 10.47130 9.5.5315 10.44685 9.57658 10.42342 20 41 52912 47088 5.^3:)5 44645 57696 42304 19 42 52953 47047 5.5395 44605 57734 42266 18 43 52905 47005 55434 44566 57772 42228 17 44 53037 46963 55474 445-.'6 57810 42190 16 45 53078 4<;922 5.5514 44486 57849 421.51 15 46 53120 46880 5.5.554 44446 57887 42113 14 47 53161 46839 55593 44407 57925 42075 13 48 53202 46798 55633 44367 57963 42037 12 49 53244 467.56 55673 44327 58001 41999 11 50 9.53285 10.46715 9.5.5712 10.44288 9.580.39 10.41961 10 51 53327 46673 55752 44248 58077 41923 9 52 533()8 46632 55791 44209 .58115 41885 8 53 53409 46.591 55S31 44169 581.53 41847 54 53450 46.550 55870 441.30 .58191 41809 6 55 53492 46.508 5.5910 44090 58229 41771 5 56 53533 46467 .55949 44051 582(17 41:33 4 57 53574 46426 55989 44011 58304 41696 3 58 53615 46385 56028 43972 .58342 41658 2 59 53656 46.344 56067 43933 58380 41620 1 60 63697 46303 56107 43893 58418 41582 / Cotan Tan Cotan Tail Cotan Tan 69° / 71° 70° "60 TABLE VIII.— LOG. TANGENTS AND COTANGENTS. / i 21° 22° 23° / 7 i Tan Cotan Tan Cotan Tan Cotan 9.58418 10.41582 9.60641 10.39359 9.62785 10.37215 60 1 58155 41545 60677 39323 62820 37180 59 2 5S493 41507 60714 39286 62855 37145 58 3 58531 41469 60750 39250 62890 37110 57 4 5S569 41431 60786 39214 62926 37074 56 5 58606 41394 60823 39177 62961 37039 55 6 58644 41356 60859 39141 62996 37004 54 7 58681 41319 60895 .39105 63031 36909 53 8 58719 41281 60931 39069 6.3066 36934 52 9 58757 41243 60967 39033 63101 36899 51 10 9.58794 10.41206 9.61004 10.38996 9.631.35 10.36865 50 11 58832 41168 61040 38960 63170 368,30 49 12 58869 41131 61076 38924 63205 .36795 48 13 58907 41093 61112 38888 63240 36760 47 14 58944 41056 61148 38852 63275 36725 46 15 58981 41019 61184 38816 6.3310 36690 45 16 59019 40981 61220 .38780 63345 36655 44 17 59056 40i)44 61256 38744 63379 36621 43 18 59094 40906 61292 38708 63414 36586 42 19 59131 40869 61328 38672 63449 36551 41 20 9.59168 10.40832 9.61364 10.38636 9.63484 10. .36516 40 21 59205 40795 61400 38600 63519 .36481 39 22 59243 40757 614.36 38564 63553 36447 38 23 59280 40720 61472 38528 63588 36412 37 24 59317 40683 61508 3f'492 63623 .36377 36 25 59354 40646 61544 384.56 63657 36313 35 26 59391 40609 61.579 .38421 63692 36.308 34 27 59429 40571 61615 38385 63726 36274 .33 28 59466 40534 616.51 .38.549 63761 .36239 32 29 59503 40497 61687 .38313 63796 36204 31 30 9.59540 10.40460 9.61722 10.38278 9.6.38.30 10.36170 30 31 59577 40423 61758 3S242 6.3865 36135 29 32 59614 40386 61794 38206 6.3899 36101 28 33 59651 40349 61830 38170 63934 36066 27 34 59688 40312 61S65 3S1.35 63968 36032 26 35 59725 40275 61901 38099 64003 35997 25 36 59702 40238 61936 38064 64037 35963 24 37 59799 40201 01972 38028 64072 35928 23 38 59S35 40165 62008 37992 64106 35894 22 39 598';2 40128 02043 379.57 64140 35860 21 40 9.. 59909 10.40091 9.62079 10.37921 9.64175 10.35825 20 41 .59946 40054 62114 37886 64209 35791 19 42 59983 40017 62150 37850 64243 35757 18 43 60019 39981 62185 37815 64278 35722 17 44 60056 39944 62221 37779 64312 35688 16 45 60093 39907 6225(5 37744 64346 35654 15 1 46 60130 39S70 62292 37708 64381 .3.5619 14 47 60166 39834 62327 37673 64415 35585 13 48 60203 39797 62302 .37638 64449 35551 12 49 60:i40 39760 6239H 37602 64483 35517 11 50 9.00276 10.39724 9.62433 10.. 37.567 9.64.517 10.. 35483 10 51 60313 39687 62468 37.5.32 64.552 .3.5448 9 52 6034'.» 39651 62504 37496 64586 3.5414 8 53 60386 39614 62539 37461 64620 3.5380 i 54 00 4. '2 39.=.78 62574 37426 64654 35346 6 55 60459 39.541 62609 37391 64688 .3.5312 5 56 60405 39505 62645 37.3.55 64722 35278 4 57 60532 39468 62680 37320 64756 35244 3 58 60568 394.32 62715 37285 64790 35210 ') 59 60605 39395 62750 372.50 ■ 64824 35176 1 1 60 60611 39359 62785 37215 64858 35142 1 / Cotan Tan Cotan Tan Cotan Tan 68° 67° 66° TABLE VIII. LOG. TANGENTS AND COTANGENTS . 261 / 24° 1 26° 26° / Tan Cotan Tan Cotan Tan Cotan 9.64858 10.35142 9.66867 10.33133 9.68818 10.31182 60 1 64892 35108 66900 33100 68850 31150 59 64926 35074 66933 33067 68882 31118 58 3 649()0 35040 66966 33034 68914 31086 57 4 64994 35006 66999 33001 68946 31054 56 5 65028 34972 67032 32968 68978 31022 55 6 65062 34938 67065 32935 69010 30990 54 7 65096 34904 67098 32902 69042 30958 53 8 65130 31870 67131 32869 69074 30926 52 9 65164 34836 67163 32837 69106 30894 61 10 9.65197 10.34803 9.67196 10.32804 9.69138 10.30862 50 11 65'i31 34769 67229 32771 69170 30830 49 12 65265 34735 67262 32738 69202 80798 48 13 65299 34701 67295 32705 69234 30766 47 14 65333 34667 67327 32673 69266 30734 46 15 65366 34634 67360 3^640 69298 30702 45 16 65400 34600 67393 32607 69329 30671 44 17 65434 34566 67426 32574 69361 30639 43 18 65467 34533 67458 32542 69393 30607 42 . 19 65501 34499 67491 32509 69425 30575 41 20 9.65535 10.34465 9.67524 10.32476 9.69457 10.30543 40 21 65568 34432 67556 32444 69488 30512 39 22 65602 34398 67589 32411 69520 30480 38 23 65636 34364 67622 32378 69552 30448 37 24 65669 34331 67654 32346 69584 30416 36 25 65703 34297 67687 32313 69615 30385 35 26 65736 34264 67719 32281 69647 30353 34 27 65770 34230 67752 32248 •69679 30321 33 28 65803 34197 67785 32215 69710 30290 32 29 65837 34163 67817 32183 69742 30258 31 30 9.65870 10.34130 9.67850 10.32150 9.69774 10.30226 30 31 65904 34096 67882 32118 69805 30195 29 32 65937 34063 67915 32085 69837 30163 28 33 65971 34029 67947 32053 69868 30132 27 34 66004 33996 67980 32020 69900 30100 26 35 66038 33962 68012 31988 69932 30068 25 36 66071 33929 08044 31956 69963 30037 24 37 66104 33896 68077 31923 69995 30005 23 38 66138 33862 68109 31891 70026 29974 22 39 66171 33829 68142 31858 70058 29942 21 40 9.66204 10.33796 9.68174 10.31826 9.70089 10.29911 20 41 66238 33762 68206 31794 70121 29879 19 42 66271 33729 68239 31761 70152 29848 18 43 66304 33696 68271 31729 70184 29816 17 44 66337 33663 68303 31697 70215 29785 16 45 66371 33629 68336 31664 70247 29753 15 46 66104 33596 68368 31632 70278 29722 14 47 66437 33563 68400 31600 70309 29691 13 48 66470 33530 68432 31568 70341 29659 12 49 66503 33497 68465 31535 70372 29628 11 50 9.66537 10.33463 9.68497 10.31503 9.70404 10.29596 10 51 66570 33430 68529 31471 70435 29565 9 52 66603 33397 68561 31439 70466 29534 8 53 66636 33364 68593 31407 70498 29502 7 54 60669 33331 68626 31374 70529 29471 6 55 66702 33298 68658 31342 70560 29440 5 56 66735 33265 68690 31310 70592 29408 4 57 66768 33232 68722 31278 70623 29377 8 58 66801 33199 68754 31246 70654 29346 2 59 66834 33166 68786 31214 70685 29315 1 60 / 66867 33133 68818 31182 70717 29283 Cntan Tan Cotan Tan Cotan Tan / 66» 64° 68° 262 TABLE VIII.— LOG. TANGENTS AND COTANGENTS. t 2 7° * * >8° 2 9° / Tan Cotan Tan Cotan Tan Cotan 9.70717 10.29283 9.72567 10.27433 9.74375 10.25625 60 1 70748 29252 72598 27402 74405 25595 59 2 70779 29221 72628 27372 74435 25565 58 3 70810 29190 72659 27341 74405 25535 57 4 70841 29159 72689 27311 74494 25.506 56 5 70873 29127 72720 27280 74.-24 25476 55 6 70904 29096 72750 27250 74554 25446 54 1 70935 29065 72780 27220 74583 25417 53 - 8 70966 29034 72811 27189 74613 25387 52 9 70997 29003 72841 27159 74643 25357 51 10 9.71028 10.28972 9.72872 10.27128 9.74673 10.25327 50 11 71059 28941 72902 27098 74702 25298 49 Vl 71090 28910 72932 27068 74732 25268 48 13 71121 28879 72963 27037 74762 2.5238 47 14 71153 28847 72993 27007 74791 25209 46 15 71184 28816 73023 26977 74821 25179 45 16 71215 28785 73054 26946 74851 25149 44 17 71246 28754 73084 26916 74880 25120 43 18- 71277 28723 73114 26886 74910 25090 42 19 71308 28692 73144 26856 74939 25061 41 . 20 9.71339 10.28661 9.73175 10.26825 9.74969 10.25031 40 21 71370 28630 73-,'05 26795 74998 25002 39 22 71401 28.599 73235 26765 75028 24972 38 23 71431 28569 73265 26735 75058 24942 37 24 71462 28538 73295 26705 751)87 24913 36 25 71493 28507 73326 26074 75117 24883 35 26 71524 28476 7aS56 26644 75146 24854 34 27 71555 28445 733S6 26614 75176 24824 33 28 71586 28414 73416 26584 75205 24795 32 29 . 71617 28383 73446 26554 75235 24765 31 30 9.71648 10.28352 9.73476 10.26524 9.75264 10.24736 30 31 71679 28321 73507 2G493 75294 24706 29 32 71709 28291 73537 26463 75323 24677 28 33 717-10 28260 73567 26433 75353 24647 27 34 71771 28229 73597 26403 75882 24618 26 35 71802 28198 ';3627 26373 75411 24589 25 36 71833 28107 73657 26:343 75441 24559 24 37 71863 28137 73687 26313 75470 24530 23 38 71894 28105 73717 26283 75500 24500 22 39 71925 28075 73747 26253 75529 24471 21 40 9.71955 10.2S045 9.73777 10.26223 9.75558 10.24442 20 41 71986 28014 73807 26193 75588 24412 19 42 72017 27983 73837 26163 7.5617 1^383 18 43 72048 27952 73867 26133 75647 24353 17 44 72078 27922 73897 2610:^ 75676 24324 16 45 72109 27891 73927 26073 75705 24295 15 46 72140 27860 73957 26043 75735 24265 14 47 72170 27830 73987 26013 75764 24236 13 48 72201 27799 74017 25983 75703 24207 12 49 72231 27769 74047 25953 75822 24178 11 50 9.72262 10.27738 9.74077 10.2.5923 9.758.52 10.24148 10 51 72293 27707 74107 2.5893 75881 24119 9 52 72323 27677 74137 25863 75910 24090 8 53 72354 27646 74166 25834 75939 24061 i 54 72384 27616 74196 25804 . 75969 24031 6 55 72415 27585 74226 25774 75098 24002 5 56 72445 27555 74256 25744 76027 23973 4 57 72476 27524 74286 25714 76056 23944 3 58 72506 27494 74316 25684 76086 23914 2 59 72537 27463 74345 25655 76115 23885 1 60 72567 27433 74375 25625 76144 23856 / Cotan Tan Cotan Tan Cotan "an 62° 61° 60° TABLE VIII.— LOG. TANGENTS AND COTANGENTS. 263 / »0° 31° S2° / Tan Cotan Tan Cotau Tan Cotan 9.76144 10 23856 9.77877 10.22123 ' 9.79579 10.20421 60 1 70173 23827 77906 22094 79G07 20393 59 o 76-J02 23798 77935 22065 79035 20365 58 3 76231 23709 77963 22037 79663 20337 57 4 76J61 23739 77992 22008 79691 20309 56 5 76290 23710 78020 21980 79719 20281 55 6 76319 23681 78049 21951 79747 20253 54 76348 23652 78077 21923 79776 20224 53 8 IVOI 1 23623 78106 21894 79804 20196 52 9 76406 23591 78135 21865 79882 20168 51 10 9.76435 10.23505 9.78163 10.218.37 9.79800 10.20140 50 11 76464 23536 78192 21808 79888 20112 49 12 76493 23507 78220 21780 79916 20084 48 13 76522 23478 78219 21751 79944 20056 47 14 765.->l 23449 78277 21723 79972 20028 46 15 765S0 23420 78306 21694 80000 20000 45 16 76609 23391 78334 21666 80028 19972 44 17 76639 23361 78363 21637 80056 19944 43 18 76668 23332 78391 21609 80084 19916 42 19 76697 23303 78419 21581 80112 19888 41 20 9.76725 10.2.3275 9.78448 10.21552 9.80140 10.19860 40 21 76754 23246 78476 21524 80168 19832 39 22 76783 23217 78505 21495 80195 19805 38 28 76812 23188 78533 21467 80223 19777 37 24 76841 23159 78562 21438 80251 19749 36 25 768'; 23130 78590 21410 80279 19721 35 26 ';6899 23101 78618 21382 80307 19693 34 27 16928 2o072 78647 213.53 80335 19665 33 28 76957 23043 78675 21325 80363 19637 32 29 76986 23014 78704 21296 80391 19609 31 30 9.77015 10.22985 9.787.32 10.212G8 9.80419 10.19.581 30 31 77044 22956 78760 21240 80447 19.553 29 32 77073 22927 78789 21211 80474 19.526 28 33 77101 22899 78817 21183 80.502 19498 27 34 77130 22870 78845 21155 80530 19470 26 35 771.59 22841 78874 21126 805.58 19442 25 36 77)88 22812 78902 21098 80586 19414 24 37 77217 22783 78930 21070 80614 19386 23 38 7^246 22754 78959 21041 80642 19.358 22 39 77274 22726 78987 21013 80669 19331 21 40 9.77303 10.22697 9.79015 10.20985 9.80897 10.19.303 20 41 77332 22668 7904:^ 20957 80725 19275 19 42 77361 226;!9 79072 20928 80753 19247 18 43 77390 22610 79100 20900 80781 19219 17 44 77'4I8 22582 79128 20872 80808 19192 16 45 77447 22553 79156 20844 80836 19164 15 46 77476 22.524 79185 20815 80864 19136 14 47 77505 22495 79213 20787 80892 19108 13 48 77-533 22467 79241 20759 80919 19081 12 49 77562 22438 79269 20731 80947 19053 11 50 9.77.591 10.22409 9.79297 10.20703 ?. 80975 10.19025 10 51 77<;i9 22381 79326 20674 81003 18997 9 52 77648 22352 79354 20646 810.30 18970 8 53 77677 22323 79382 20618 81058 18942 < 54 77706 22294 79410 20590 81086 18914 6 55 77734 22266 79438 20562 81113 18887 5 56 77763 22237 79466 20534 81141 18859 4 57 77791 22209 79495 20505 81169 18831 3 58 77820 22180 79523 20477 81196 18804 2 59 77849 22151 79.551 20449 81224 18776 1 60 t 22123 79579 20421 81252 18748 / Tan Cotan Tan Cotnn Tan / 69° 58° 57° 2G4 TABLE VIII.— LOG. TANGENTS AND COTANGENTS. / 33<» 34° 35° / Tan Cotan Tan Cotan Tan Cotan 9.81252 10.18748 9.82899 10.17101 9.84523 10.15477 60 1 81279 18721 82926 17074 84550 15450 59 1 2 81307 18693 82953 17047 84576 15424 58 3 81335 18665 829S0 17020 84603 15397 57 4 81362 18638 83008 16992 84630 15370 56 5 81390 18610 83035 16965 84657 15343 55 6 81418 18582 83062 16938 84684 15316 54 7 81445 18555 83089 16911 84711 15289 53 1 g 81473 18527 83117 16883 84738 15262 52 9 81500 18500 83144 16856 84764 15236 51 10 9.81528 10.18472 9.83171 10.16829 9.84791 10.15209 50 11 81556 18444 83198 16802 84818 15182 49 81583 18417 83225 16775 84845 15155 48 18 81611 18389 83252 16748 84872 15128 47 14 81638 18362 83280 16720 84899 15101 46 81666 18334 83307 16693 84925 15075 45 16 81693 18307 83334 16666 84952 15048 44 17 81721 18279 83361 10639 84979 15021 43 t 1 18 81748 18252 S^i'iH 16612 85006 14994 42 19 81776 18224 83415 16585 85033 14967 41 20 9.81803 10.18197 9.83142 10.16558 9.85059 10.14941 40 21 81831 18169 83470 16530 85086 14914 39 22 81858 18142 83497 16503 85113 14887 38 23 81886 18114 83524 1&476 85140 14860 37 24 81913 18087 83551 16449 85166 14834 36 25 81941 18059 83578 16422 85193 14807 35 26 81968 18032 83605 16395 85220 14780 34 27 81996 18 = 373.1 X 5 2°0'. 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54 56 58 60 0° L. C. 0.00 3.. 33 6.67 10.00 13.33 16.67 20.00 23.33 26.67 30.00 33.33 36.67 40.00 43.33 46.67 50.00 53.33 56.67 60.00 63.33 66.67 70.00 73.33 76.67 80.00 83.33 86.67 90.00 93 33 96.67 100.00 M. E. 0.000 0.000 0.001 0.002 0.004 0.006 0.009 0.012 0.015 0.019 0.024 0.029 0.035 0.041 0.048 0.054 002 070 0.079 0.088 0.097 0.107 0.117 0.128 0.140 0.151 0.164 176 0.190 0.204 0.218 0.000 0.000 0.001 0.002 0.004 0.006 0.009 0.012 0.015 0.019 0.024 0.029 0.035 0.041 0.048 0.054 0.062 O.OfO 0.079 0.088 0.097 0.107 0.117 0.128 0.140 0.151 0.164 0.170 0.190 0.204 0.218 0.00 1.67 3.33 5.00 6.67 8.33 10.00 11.67 13.33 15.00 16.67 18.33 20.00 21.67 23.33 25.00 26.67 28.33 30.00 31.67 33 33 35.00 36. nr 38.. 33 40.00 41. or 43.. 33 45 0(» ■1(1. t;: 4,s 3.-! •5(1 (»0 1° L. C. M. E. T. 100.00 0.218 0.218 50.00 103.33 0.233 0.233 51.67 o 100.66 0.248 0.248 53.33 4 110.00 0.264 0.264 55.00 6 113.33 0.280 0.280 56.67 8 116.66 0.297 0.297 58.33 10 120 00 0.314 0.314 60.00 12 123.33 0.332 0.332 61.67 14 12G.66 0.3.50 0..350 63.33 16 130.00 0.368 0.368 65.00 18 1.33.33 0.388 0.388 66.67 20 130.66 0.407 0.407 68.33 22 140.00 0.427 0.427 70.00 24 143.33 0.448 0.448 71.67 26 146.66 0.4G9 0.4G9 73.33 28 150 00 0.491 0.491 75.00 30 153.33 0..513 0.513 76.67 32 156.66 0.536 0.536 78.33 34 160.00 0.5.59 0,559 80.00 36 163.33 0.582 0.582 81.67 38 166.66 0.606 0.606 83.. 33 40 170.00 0.630 0.630 85.00 42 173,33 0.655 0.655 86.67 44 17G.66 0.681 0.681 88.33 46 180.00 0.706 0.706 90.00 48 183.33 733 0.733 91.67 50 186.66 0.760 0.760 93.33 52 190. (K) 0.7SS 788 95.00 54 193.33 0.815 0.815 96.67 56 196 66 0.844 0.844 98.33 58 199.98 873 873 100.00 60 ( IX.— FUNCTIONS OF A ONE-DEGREE CURVE. 2fi9 t k>0 3 o / L. 0. M. E. T. L. C. M. E. T. 199.98 0.873 0.S73 100.00 299.96 1.964 1.964 1,50.07 2 203.31 0.902 0.902 101.67 303.29 2.008 2.009 151.74 2 4 206.64 0.9.32 0.932 103.34 306.62 2.053 2.054 1,53.41 4 6 209.97 0.962 0.9G2 105.01 309 95 2.098 2.099 1.55.08 6 8 213.31 0.993 0.993 106.68 313.29 2. 143 2.144 156.75 8 10 210.64 1.024 1.024 108.35 316.62 2.188 2.189 158.42 10 Vi 219.97 1.0.56 1.0.56 110.02 319.95 2.235 2.2.36 160.09 12 14 223.30 1.088 1.088 111.69 323.28 2.282 2.283 161.76 14 16 226.64 1.121 1.121 113.36 326.62 2. 329 2.3,30 163.43 16 18 229.97 1.154 1.154 115.02 329.95 2.-376 2.. 377 165.09 18 20 233.30 1.188 1.188 116.69 333.28 2.424 2.425 166.76 20 22 236.63 1.222 1.222 118.36 336 61 2.473 2.474 168.43 22 24 239.97 1.256 1.256 120.03 339.95 2.523 2.523 170.10 24 26 243.30 1.292 1.292 121.70 343.28 2. 572 2.573 171.77 26 28 246.63 1.328 1.328 123.. 37 346.61 2.622 2.623 173 44 28 30 249.96 1.364 1.364 125.03 349.94 2.672 2.673 175.10 30 ;i2 253.29 1.399 1.399 126.70 3.53 27 2.724 2.725 176.72 32 34 2.56.62 1.437 1.4.37 12S 37 3.")6 . 60 2.776 2.777 178.39 34 36 259.96 1.475 1.475 130.04 359.94 2.828 2.829 180.06 36 38 263.29 1.513 1.513 131.71 363.27 2.880 2.881 181.73 38 40 266.62 1.552 1..552 133 38 366 60 2 933 2.9.34 183.40 40 42 269.96 1.592 1..592 135.05 369.94 2.987 2.988 185.07 42 44 273.29 1.632 1.632 136.72 .373 27 3.042 3 043 186.74 44 46 276.62 1.672 1.672 138.. 38 376.60 3.096 3 097 188.40 46 48' 279.96 1.712 1.712 140.05 379.94 3.151 3.1.52 190.07 48 50 283.29 1.752 1.7.52 141.72 383.27 3.206 3.207 191.74 50 52 286.62 1.794 1.794 143 39 .386 60 3 263 3.264 193.41 52 54 289.96 1.836 1.836 145.06 389 94 3.. 320 3.321 195.08 54 56 293 29 1.878 1.878 146.73 393.27 3.. 377 3 378 196.75 56 58 296.02 1.921 1.921 148.40- 3'.»t; . 60 3.434 3.435 198.42 58 60 299.96 1.964 1.964 1.50 07 399.94 3.491 3.492 2u0 09 60 4< > 5 3 / t L. C. M. E. T. L. C M. E. T. .399.94 3.491 3.492 200.09 499.88 5.4,54 5.4.59 250.17 o 403.27 8.5.50 3.. 551 201.76 503.21 5.. ^27 5.033 251.84 2 4 406.60 3.609 3.610 203.43 506 54 5.601 5.607 253.51 4 6 409.93 3.668 3.670 205.10 509.87 5.675 5.681 255.18 6 8 413.26 3.727 3.730 206.77 513.20 5.749 5.7.55 256.85 8 10 416.59 3.787 3.790 208.44 516 .53 5.823 5.829 258.. 52 10 12 419 92 3.848 3.851 210.11 519.86 5.899 5.905 260.20 12 14 423.26 3.910 3 913 211.77 523.19 5.975 5.981 261.86 14 16 426.59 3.972 3.975 213.45 526.. 52 6.0.52 6.058 263.54 16 18 429.92 4.034 4.037 215.11 529.85 6.129 6.1-35 265.20 18 20 433.25 4.096 4.099 216.78 533.18 6.206 6.212 266.87 20 22 436.58 4.160 4.163 218.45 536 51 6.284 6.290 268.54 22 24 439.91 4.224 4.227 220.12 539.84 6.362 6.369 270.21 24 26 443.24 4.288 4.291 221.79 543.17 6.441 6.448 271.88 26 28 446.. 58 4.353 4.3.56 223.46 546.. 50 6.520 6.527 273.. 54 28 30 449.91 4.418 4.421 225.13 549.83 6.599 6.606 275.21 30 32 4.53.24 4.484 4.467 226 80 553 17 6.680 6.687 276.88 32 34 456.57 4.550 4.554 228.47 556.50 6.761 6.768 278.55 34 36 459.90 4.617 4.621 2.30,14 559.83 6.842 6.849 280.23 36 38 463.23 4.684 4.688 231.81 563,16 6.923 6.931 281.90 38 40 466.56 4.751 4.755 233 48 566.49 7.005 7 013 283.57 40 42 469.89 4.820 4.824 235.15 569.82 7.088 7.096 285.24 42 44 473.23 4.889 ^.893 2.36.82 573.15 7 171 7.180 286.91 44 46 476 .56 4.958 4 962 2.38.48 576.48 7 255 7.264 288.59 46 48 479.89 5.027 5.031 240.15 579.81 7.339 7.348 290.26 48 50 483.22 5.096 5.100 241.82 583.14 7 423 7.432 291.93 50 52 486.55 5.167 5.171 243.49 586.47 7.508 7.517 293.60 52 54 489 88 5.238 5.243 245 16 .589.80 7. 593 7.603 295.27 54 56 493.21 5.310 5.315 216 83 593.13 7 678 7 689 296 95 56 58 496 .54 5,. 382 5 3S7 248 ,50 ,596.46 7.764 7 775 298 62 58 60 499.88 5 454 5.4.59 2."0 17 599. HO 7.8.50 7 861 300.30 60 2?0 IX.— FUNCTIONS OF A ONE-DEGREE CURVE. t t° JO / L C. M. E T. L. C. M. E. T. 1 599.80 7.850 7.861 300.30 699.60 10.69 10.71 350.44 2 603.13 7.940 7.951 301 .97 702.93 10.79 10.81 3.52.11 2 4 606.46 8.030 8.041 303.64 706.26 10,90 10.92 353.79 4 6 609.78 8.120 8.131 305.81 709,. 58 11.00 11.02 355.46 6 8 613.11 8.210 8.2J1 306.98 712.91 11.11 11.13 357.13 8 10 616.44 8.300 8.311 308.65 716.24 11.21 11.23 358.81 10 12 619.76 8.390 8.401 310.32 719 .56 11.31 11.33 360.48 12 14 623.09 8.480 8.491 311.99 722.89 11.42 11.44 362.15 14 16 626.42 8.. 570 8.581 ;il3.66 726 21 11,52 11.54 363 83 16 IS 629.74 8.660 8.671 315.33 729.. 53 11.63 11.65 365.50 18 •20 633.07 8.750 8.761 317.00 732.86 1 1 . 73 11.75 367,17 20 22 636.40 8.844 8.856 318.67 736.19 11.84 11.86 368.85 22 24 639.72 8.939 8.951 320.34 739 51 11.95 11.97 370,52 24 26 643.05 9.033 9.046 322 01 742.84 12.06 12.08 372.19 26 28 646.38 9.128 9.141 323.68 746.17 12.17 12.19 373.86 28 30 649.70 9.222 9.236 325.35 749.49 12.27 12.. 30 375.54 30 32 653.03 9.317 9. 331 327.02 7.52.82 12.38 12.41 377.22 32 34 656.36 9.411 9.4-.'6 328.69 756.15 12.49 12 52 378.89 34 36 659.69 9.506 9.521 33U.37 7.59.47 12.60 12.63 380 57 36 38 663.02 9.600 9.616 332 04 762.80 12.71 12.74 382.24 38 40 666.34 9.695 9.712 3.3-3 71 766.13 12.82 12.85 383.92 40 42 669.67 9.794 9.812 335.38 769.45 12,93 12.96 385.60 42 44 673.00 9.894 9.912 337.05 < /2. (0 13,04 13.08 387.27 44 46 676.32 9.993 10.01 338. '13 776.11 13.15 13.19 388.95 46 48 679.65 10.09 10.11 340.40 779.43 13 26 13.31 390.62 48 50 682.98 10.19 10.21 342.07 782.76 13,37 13.42 392.30 50 52 686.30 10.29 10.31 343.74 786.09 13.48 13.53 393.98 52 54 689.63 10. .39 10.41 345.41 789.41 13 .59 13.65 395.65 54 56 692.96 10 49 10.51 347.08 792.74 13.70 13.76 397.33 56 58 696.28 10.59 10,61 348.76 796.07 13.81 13.88 399.01 58 60 699.60 10.69 10.71 3.50.44 799 40 13 96 13.99 400.70 60 / 8 a 8 1° / L. C. 799.40 M. 13.96 E. 13.99 T. 400.70 L. C. M. E. T. 899.10 17.66 17.71 450.95 2 802.72 14.07 14.10 402 37 902.42 17.79 17.84 452.63 2 4 806.04 14.19 14,22 404.05 905,74 17,92 17.98 454.31 4 6 809.37 14.31 14 34 405.72 909.07 18.06 18.11 455.98 6 8 812.69 14 43 14.46 407.39 91 2. 39 18.19 18.25 457.66 8 10 816.01 14.. 55 14.58 409.06 915.71 18.32 18.38 4.59,34 10 12 819.34 14.66 14.70 410.74 919.04 18.46 18.. 52 461.02 12 14 822.66 14.78 14.82 412.41 922.. 36 18.59 18.65 462.70 14 16 825.98 14.90 14.94 414 OS 925.68 18.72 18.79 464,37 16 18 829.31 15.02 15.06 415.75 929.01 18,86 18.92 466.05 18 20 832.63 15.14 15.18 417.43 932 .33 18.99 19.06 467.73 20 22 835.95 15.26 15.30 419.10 935.65 19.12 19.19 469,41 22 24 839.28 15.38 15.43 4-:o 77 9.38.98 19.26 19.33 471.08 24 26 842.60 15.51 15.. 55 422 45 942.. 30 19.40 19.47 472.76 26 28 845.92 15.63 15.68 424 12 945.62 19.54 19.61 474.43 28 30 849.25 15.75 15.80 425.79 948.95 19.68 19.75 476.10 30 32 852.57 15.88 15 93 427.47 9.52.27 19.82 19.89 477.78 32 34 8.-)5.89 16.00 16.05 429.15 955.59 19.96 20.03 479.46 .34 36 859.22 16.12 16.18 4.30.82 9.58.92 20.10 20.17 481.14 36 38 862.54 16.25 16.30 432.. 50 962.24 20.24 20.31 482.83 38 40 865.86 16.38 16.43 434.13 965.56 20.38 20.45 484.51 40 42 869.19 16.50 16.55 435.86 968.89 20 .52 20.59 486.19 42 44 872.51 16.63 16.68 437.54 972 21 20.66 20.74 487.87 44 46 875.83 16.76 16.81 439.2! 975.. 53 20.80 20.88 489.55 46 48 879.16 16.89 16.94 440.89 978 86 20.94 21.03 491.24 48 50 882.48 17.02 17.07 4 42.. 57 9S2.18 21.09 21.17 492,92 50 52 885.80 17.14 17.19 444 25 985.50 21.23 21,31 494.60 52 54 8S9.13 17.27 17 32 445.93 988.83 21.. 37 21.40 496,28 54 56 892.45 17.40 17.45 447.60 992.15 21.51 21,60 497.96 56 58 895.77 17.. 53 17 .58 449.28 995 47 21.6.T 21.75 499,65 58 60 899.10 17.66 17.71 4.50.95 998.80 21.80 21,89 501.32 60 IX. FUJSCTIONS OF A ONE-DEGREE CURVE. 27i i 10° 1 11° / / L C. M. E. T. L. C. M. E. T. 1 -» -^ 998.8 21.80 21.89 501.32 1098.4 26.38 26.50 551.74 o 1002.1 21.94 22.03 .503.00 1101.7 26.. 54 26.66 553.42 o 4 1005.4 22.09 22.18 .504.68 1105.0 26.70 26.83 555.10 4 6 1008.8 22.24 22.33 .506.. 36 1108 3 26.86 26.99 556.78 6 8 1012.1 22.39 22.48 508.04 1111.7 27.02 27.16 558.46 8 10 1015.4 22.54 22.63 509.72 1115.0 27.19 27.. 32 560.14 10 U' 1018.7 22.68 22.78 511.40 1118.3 27.35 27.48 561.82 12 14 1022.0 32.83 22.93 513.08 1121.6 27.51 ,27.65 563.50 14 16 1025.4 22.98 23.08 514.76 1124.9 27.67 27.81 565.18 16 18 1028.7 23.13 23 23 516.44 1128.2 27.83 27.98 566.86 18 20 1032.0 23.28 23.38 518 12 1131.6 28.00 28.14 568.54 20 22 1035.3 23.43 23.. 53 519.80 1134.9 2g.l7 28.30 570.22 22 24 1038.6 23.58 23 68 521.48 1138.2 28.34 28.47 .571.90 24 26 1042.0 23.73 23.84 .523 16 1141.5 28.. 50 28.64 573.58 26 28 1045 3 23.88 23.99 .524.85 1144.8 28.67 28.81 575 27 28 30 1048.6 24.04 24.14 526., 53 1148.1 28.84 28.98 .576,95 30 32 1051.9 24.19 24.30 .528.21 1151.5 29.00 29.14 578.63 32 34 1055.2 24.34 24.45 529.89 1154.8 29.17 29.31 .580.32 34 36 1058.6 24.49 24.60 531.57 11.58.1 29.34 29.48 582.00 36 38 1061.9 24.64 24.76 533.25 1161.4 29.. 50 29.65 583.69 38 40 1065.2 24.80 24.91 534.93 1164.7 29.67 29.82 585.37 40 42 1068.5 24.95 25.06 536.61 1168.0 29.84 29.99 587.05 42 44 1071.8 25.11 25.22 538.29 1171.4 30.01 30.17 588.74 44 46 1075.2 25.27 25.38 .539.97 1174.7 .30.18 30.34 590.42 46 48 1078.5 25.43 25.54 541.65 1178.0 30.35 30.52 592.11 48 50 1081.8 25.59 25.70 543., 33 1181.3 30. 53 30.69 593.79 50 52 1085.1 25.74 25.86 545.01 1184.6 .30 70 30.86 595.47 52 54 1088.4 25.90 26.02 546.69 1187.9 .30.87 31.04 597.16 54 56 1091.8 26.06 26.18 54S..S7 1191.3 31.04 31.21 598.84 56 58 1095.1 26.22 26.34 .5.50 06 1H)4.6 31.21 31.39 600.53 58 60 1098.4 26.38 26.. 50 551.74 1107.9 31.39 31.56 602.22 60 3 12° 1 13° / L. C. 31. E. T. L. C. M. E T. 1197.9 31.. 39 31.. 56 602.22 1207.3 36.83 37.07 652.87 2 1201.2 31.57 31. IS 603.91 1300.6 37.02 37.26 6.54.56 o 4 1204.5 31.74 31.01 605.00 1303.9 37.21 37.46 6,56.25 4 6 1207.8 31.92 32.09 607.28 1307.2 37 40 37.65 657.93 6 8 1211 1 32 09 32.27 608.97 1310.5 37.59 37.85 650.62 8 10 1214.5 32.27 32.45 610.66 1.313.8 37 79 38.04 661.31 10 12 1217.8 32.45 32.63 612.35 1317.2 37.98 38.:i3 663.00 12 14 1221.1 32.62 32.81 614.04 1320.5 38.17 38.43 664.69 14 16 1224.4 32.80 32.99 615.72 1323.8 38.36 38.02 666.37 16 18 1227.7 32.97 33.17 617.41 1327.1 38.55 38.82 668.06 18 20 1231.0 33.15 33.35 619.10 1330.4 38.75 39.01 669.75 20 22 1231.3 33.33 33.53 6-.I0.79 1333.7 38.95 39.20 671.44 22 24 1237.7 .33.51 33.72 622.48 1337.0 39.15 39.40 673.13 24 26 1241.0 33.69 .33.90 624.16 1340.3 .39.35 39.60 674.81 26 28 1244.3 33.87 34.09 6-'5.85 1343.6 39.54 39.80 676.51 28 30 1247 6 34.06 34.27 627.. 55 1346.9 39.74 40.00 678.20 30 32 12.50.9 34.24 34.45 629.24 13.50.3 .39.94 40.19 679.89 32 34 1254.2 34.42 .34 64 630 93 13.53.6 40.13 40.. 39 681.58 34 36 1257.5 .34.60 34.82 632.61 13.56.9 40.33 40 .59 683.26 36 38 1260.8 34.78 35.01 634.30 1360.2 40.52 40.79 684.95 38 40 1264.2 34 97 35.19 6.35.99 1363.5 40.71 40.99 686.64 40 42 1267.5 35.16 35.37 637.08 1366.8 40.91 41.19 688.33 42 44 1270.8 35.34 35.56 639.37 1370 1 41.11 41.40 690.02 44 46 1274.1 35.. 53 35.75 641.05 1373.4 41.31 41.60 691.70 46 48 1277.4 35.71 35.94 642.74 1376.7 41.51 41.81 693.. 39 48 50 1280.7 35.90 36.13 644.43 1380.0 41.71 42.01 695.08 50 52 1284.0 36.09 36 31 646.12 1383.4 41.91 42.21 690.77 52 54 1287.4 36.27 3(i..50 647.81 1380.7 42 11 42.42 698.46 54 56 1290.7 36.46 30 . 69 649.49 1300.0 42.31 42.02 700.14 56 58 1294.0 36.64 .36.88 651.18 1393.3 42.51 42.83 701.83 58 60 1297.3 36 . K3 3r.07 6.52.87 1 1300.6 42 71 43.03 703.. ^:3 60 1 272 IX.— FUNCTIONS OF A ONE-DEGREE CURVE. 14" 15° f L. C. 1306.6 M. 42.71 E. 43.03 T. r03.53 L. C. M. E. T. / 1495.9 49.02 49.44 754.35 2 1399.9 42.92 43.23 43.44 705.23 1409.2 49.24 49.66 756.05 o 4 1403.2 43.12 706.92 '1502.5 49.40 49.89 757.74 4 6 1406.5 43.33 43.65 708.62 1505.8 49.08 50.11 759.44 6 8 1409.8 43.53 43.86 710.31 1500.1 49 90 50.34 761.13 8 10 1413.1 43.74 44.07 712.01 1512.4 50.12 50.56 762.83 10 12 1416.5 43.94 44.28 713.71 1515.7 50.34 50.78 764.53 12 14 1419.8 44.15 44.49 715.40 1519.0 50.56 51.01 766.22 14 16 1423.1 44.35 44.70 717.10 1522.3 50 78 51.23 767.92 16 18 1426.4 44.56 44.91 718.79 1525.6 51.00 51.46 769.61 18 20 1429.7 44.77 45.12 720.49 1528.9 51.22 51.68 771.31 20 1433.0 44.98 45.33 722.20 1532.2 51.44 51.90 773.01 22 24 1436.3 45.19 45.54 723.89 1535.5 51.67 52.13 774.70 24 26 1439.6 45.40 45.76 725.59 1538.8 51.89 52.36 776.40 26 28 1442.9 45.61 45.97 727.28 1542.1 52.12 .52.59 778.09 28 30 1446.2 45.82 46.18 72S.i»7 1545.4 52.34 52.82 779.79 30 32 1449.6 46.03 46.40 730.06 1548.7 52. 57 53.05 781 .49 32 34 1452.9 46.24 46.61 732 35 15.52.0 52.79 .53.28 783.19 34 36 1456.2 46.45 46.82 734.05 1555.3 53.02 53.51 784.89 36 38 1459.5 46.66 47.04 735.74 1558.6 53.24 53.74 786.59 38 40 1462.8 46.87 47.25 737.43 1561.9 53.47 53.97 788.29 40 42 1460.1 47. as 47.46 739.12 1565.2 53.69 54.20 789.99 42 44 1469.4 47.30 47.68 740. SI 1508.5 53.92 54.44 791.69 44 46 1472.7 47.51 47.90 742.51 1571.8 54.15 54.67 793.39 46 48 147C.0 47.73 48.12 744.20 1575.1 54.38 54.91 795.09 48 50 1479.3 47.94 48.34 745.89 15T8.4 54.61 55.14 796.79 50 52 1482.7 48.16 48.56 747.58 1581.7 54.84 55.37 798.49 52 54 1486.0 48.37 48.78 749.27 1585.0 ■55.07 55.61 800.19 54 56 1489.3 48.59 49.00 750.97 15S8.3 55.30 55.84 801.89 56 58 1492.6 48.80 49.22 752.66 1591.6 55 53 56.08 803.59 58 60 1495.9 49.02 49.44 754.35 1594.9 55.76 56.31 805.29 60 / 16° 1 17" , / L. C. M. E. T. L. C. M. E. T. 1594.9 55.76 56.31 805.29 1693.9 62 94 63.64 856.35 o 1598.2 55.99 56.. 54 806.99 1697.2 63.18 63.89 858.05 2 4 1601.5 56.23 56.78 808.64 1700.5 63.43 64.15 859.76 4 C 1604.8 56.46 57.02 810.39 1703.8 63.08 64.40 801.46 6 8 1608.1 56.70 57.26 812.09 1707.1 63 93 64.66 863.16 8 10 1611.4 56-93 57.50 813.79 1710.4 64.18 64.91 864.87 10 12 1614.7 57.17 57.74 815.49 1713.7 64.42 65 16 866.57 12 14 1618.0 57.40 57.98 817.19 1710.9 64.67 65.42 868.27 14 16 1621.3 57.64 58.22 818.89 1720.2 64.92 65 67 869.98 16 18 1624.6 57.87 58.46 820.59 1723.5 65.17 65.93 871.68 18 20 1627.9 58.11 58 TO 822.29 1726.8 65.42 66.18 873.38 20 22 1631.2 58.34 58.94 823.09 1730.1 65.67 66.43 875.09 22 24 1634.5 58.58 59 19 825.09 17:^:3.4 65.93 66.69 876.79 24 26 1637.8 58.82 59.43 827 39 17:^6.7 66. Ig 66.95 878.49 26 28 1641.1 59.06 59.68 829.09 1740.0 66.44 67.21 880.20 28 30 1644.4 59.30 59 92 830 79 1743.3 66.69 67.47 881.90 30 32 1647.7 59.54 60.16 832.49 1746.6 66.94 67.72 883.61 32 34 1651.0 59.78 60.41 834.20 1749.9 67.20 67.98 885.32 34 36 1654.3 60.02 60.65 835.90 1753.2 67.45 68.24 887.02 36 38 1657.6 60.26 60.90 837.61 1756.5 67.71 68.50 888.73 38 40 1660 9 60.50 61.14 839.31 1759.8 67.96 68.76 890.44 40 42 1664.2 60.74 61.39 841.01 1703.1 68.21 69.03 892.15 42 44 1667.5 60.99 61.64 842.72 1766 3 68.47 69.29 893.86 44 46 1670.8 61.23 61.89 &44.42 1769.6 68.73 69.56 895.56 46 48 1674.1 61.48 62.14 846.13 1772.9 68.99 69.82 897.27 48 50 1677.4 61 72 62.39 847.8:3 1776 2 69.25 70.09 898.98 50 52 1680.7 61.96 62.64 849.53 1779.5 69.50 70.36 900.69 52 54 1684.0 62.21 62.89 851 24 1782.8 69.76 70.62 902.40 54 56 1687.3 62.45 63.14 852.94 r786.1 70.02 70.89 904.10 56 58 1690.6 62.70 63 39 854.65 1789.4 70.28 51.15 905.81 58 60 1693.9 62. 9 J 03 64 8.56 35 1792.7 70.54 71.42 907.52 60 IX.— FUNCTIONS OF A ONE-DE&IiEe" CURVE. 273 / 18° 1 19° 1 / L. C. M. E. T. L. C. M. E. T. 1792.7 70.54 71.42 907.52 1S91.5 78.58 79.65 958.86 1796.0 70.80 71.69 909.23 1894.8 78.86 79.94 900.57 2 4 1799.3 71.06 71.96 910.94. 1898.1 79.13- 80.22 962.30 4 6 1802.6 71.33 72.23 912.65 1901.3 79.41 80.51 964.00 6 8 1805.9 71.59 72.50 914.36 1904.6 79.68 80.79 965.72 8 10 1809.2 71.85 72.77 916.07 1907.9 79.96 81.08 967.43 10 1-^ 1812.5 72.12 73.04 917.78 1911.2 80.24 81.37 . 969.15 12 14 1815.7 72.38 73.31 919.49 1914.5 80.51 81.65 970.86 14 16 1819.0 72.64 73.58 921.20 1917.8 80.79 81.94 972.58 16 18 1822.3 72 91 73.85 922.91 1921.0 81.07 82.22 974.29 18 20 1825.6 73.17 74.12 924. ers 1924.3 81.35 82.51 976.01 20 22 1828.9 73.43 74.39 926.34 1927.6 81.63 82.80 977.72 22 24 1832.2 73.70 74.67 928.05 1930.9 81.91 83.09 979.44 24 26 1835.5 73.97 74.94 929.76 1934.2 82.20 83.38 981.15 26 28 1838.8 74.24 75.22 931.47 1937.5 82.48 83.67 982.86 28 30 1842.1 74.51 75.49 933.18 1940.7 82.76 83.97 984.58 30 32 1845.4 74.77 75.77 934.89 1944.0 83.05 84.26 986.30 32 34 1848.7 75.04 76.04 936.60 1947.3 83.33 84.55 988.02 34 36 1852.0 75.31 76.32 938.32 1950.6 83.61 84.84 989.74 36 38 1855.3 75.58 76.59 940.03 1953.9 83.90 85.13 991.46 38 40 1858.6 75.85 76.87 941.74 1957.2 84.18 85.43 993.18 40 42 1861.9 76.12 77.14 943.45 1960.4 84.47 85.73 994.90 42 44 1865.1 76.39 77.42 945.16 1963.7 84.75 86.02 996.62 44 46 1868.4 76.67 77.70 946.88 1967.0 85.04 86.32 998.34 46 48 1871.7 76.94 77.98 948.59 1970.3 85.32 86.61 1000.0 48 50 1875.0 77.21 78.26 950.30 1973.6 85.61 86.91 1001.8 50 52 1878.3 77.49 78.53 952.01 1976.9 85.90 87.21 1003.5 52 54 1881.6 77.76 78.81 953.72 1980.1 86.19 87.50 1005.2 54 56 1884.9 78.03 79.09 955.44 1983.4 86.47 87.80 1006.9 56 58 1888.2 78.31 79.37 957.15 1986.7 86.76 88.09 1008.6 58 60 1891.5 78.58 79.65 958 86 1990.0 87.05 88.39 1010.4 60 / 20° 1 21° 1 , L. C. M. E. T. L. C. M. E. T. 1990.0 87.05 88.39 1010.4 2088.5 95.95 97.58 1062,0 o 1993.3 87.34 88.69 1012.1 2091.8 96.26 97.90 1063.7 2 4 1996.6 87.63 88.99 1013.8 2095.0 96.56 98.21 1065.4 4 6 1999.8 87.92 89.29 1015.5 2098.3 96.87 98.53 1067.2 6 8 2003.1 88.21 89.59 1017.2 2101.6 97.17 98.84 1068.9 8 10 2006.4 88.50 89.89 1019.0 2104.9 97.48 99.16 1070.6 10 12 2009.7 88.79 90.19 1020.7 2108.1 97.79 99.48 1072.4 12 14 2013.0 89.08 90.49 1022.4 2111.4 98.09 99.79 1074.1 14 16 2016 3 89.37 90.79 1024.1 2114.7 98.40 100.1 1075.8 16 18 2019.5 89.66 91.09 1025.8 2118.0 98.70 J00.4 1077.5 18 20 2022.8 89.96 91.40 1027.6 2121.2 99.00 100.7 1079.3 20 22 2026.1 90.25 91.71 1029.3 2124.5 99.30 101.1 1081.0 22 24 2029-4 90.55 92.01 1031.0 2127.8 99.60 101.4 1082.7 24 26 2032.7 90.85 92.32 1032.7 2131.0 99.90 101.7 1084.4 26 28 2036.0 91.15 9?". 62 1034.4 2134.3 100.2 102.0 1086.2 28 30 2039.2 91.45 92 .«3 1036,1 2137.6 100.5 102.3 1087.9 30 32 2042.5 91.74 93.24 1037.9 2140.9 100.8 102.7 1089.6 32 34 2045.8 92.04 93.54 1039.6 2144.1 101.1 103.0 1091.3 34 36 2049.1 92.34 93.85 1041.3 2147.4 101.4 103.3 1093.1 36 38 2052.4 92.64 94.15 10-13.0 2150.7 101.7 103.6 1094.8 38 40 2055.7 92.94 94.46 1044.8 2154.0 102.1 104.0 1096.5 40 42 2058.9 93.24 94.78 1046.5 2157.2 102.4 104.3 1098.3 42 44 2062.2 93.54 95.09 1048.2 2160.5 102.7 104.6 1100.0 44 46 2065.5 93.84 95.40 1049.9 2163.8 103.0 104.9 1101.7 46 48 2068.8 94.14 95.71 1051.7 2167.1 103.3 105.3 1103.4 48 50 2072.1 94.44 96.03 10.53.4 2170.3 103.6 105.6 1105.2 50 52 2075.4 94.74 96.34 1055.1 2173.6 103.9 105.9 1106.9 52 54 2078.6 95.04 96.65 1056.8 2176.9 104.2 106.3 1108.6 54 56 2081.9 95.34 96.96 1058.6 2180.1 104.5 106.6 1110.3 56 58 2085.2 95.64 97.27 1060. S 2183.4 104.8 106.9 1112.1 58 60 2088.5 95.95 97.58 1062.0 2188 7 105.2 107.2 1118.8 60 274 IX.— FUNCTIONS OF A ONE-DEGREE CURVE. / 22° 1 2 3° / L. C. M. E. T. L. C. 2284.8 M. 115.0 E. 117.4 T. 1165. S 2186.7 105.2 107.2 1113.8 2 2190.0 105.6 107.6 1115.5 22SS 1 115.3 117.7 1167.5 2 4 :^193.2 105.9 107.9 1117.3 2291.3 115.7 118.1 1169.2 4 6 2196.5 106.2 108.2 1119.0 2294.6 116.0 118.4 1171.0 6 8 2i99.8 106.5 108.6 1120.7 2297.8 116.4 118.8 1172.7 8 10 2203.0 106.8 10S.9 1122.4 2.301 . 1 116.7 119.1 1174.4 10 VZ 2206.3 107,1 109.2 1124.2 2.304.4 117.0 119.5 1176.2 12 14 2209.6 107.4 109.6 1125.9 2307.6 117.4 119.8 1177.9 14 16 2212 9 107.7 109.9 1127.6 2310.9 117.7 120.2 1179.7 16 18 2216.1 108.0 110.2 1129.4 2314 1 118.1 120.5 1181.4 18 20 2219.4 108.4 110.6 1131.1 2317.4 118.4 120.9 1183.1 20 22 2222.7 108.7 110.9 1132 8 2320.7 118 7 121.2 1184.9 22 24 2225.9 109 111.2 1134.6 2.323.9 119.1 121 6 1186.6 24 26 2229.2 109.4 111 6 1136.3 2327.^ 119.4 121.9 1188.4 26 28 2232.5 109.7 111.9 11.38.0 2330.4 119.8 122.3 1190.1 28 30 2235.7 110.0 112.3 1139.7 2333.7 120.1 122.6 1191.8 30 32 2239.0 110.4 112 6 1141.5 2337.0 120.4 123.0 1193.6 S2 34 2242 3 110.7 112.9 1143 2 2340.2 120.8 123.3 1195.3 34 36 2245.6 111.0 113.3 1144.9 2843.5 121.1 123.7 1197.1 36 38 2248.8 111.4 113.6 1140.7 2346.7 121.5 124.1 1198.8 38 40 2252.1 111.7 113.9 1148.4 2.3.50 121.8 124.4 1200.5 40 42 2255.4 112 114.3 11.50.1 2353 3 122. 1 124.8 1202.3 42 44 2258.6 112.3 114.6 1151.9 2356.5 122 5 125.1 1204.0 44 46 2261.9 112.7 115.0 11.53.6 2359.8 122.8 125.5 1205.8 46 48 2265.2 113.0 115.3 11.55.4 2.363.0 123.2 125.8 1207.5 48 50 2268.4 113.3 115.7 11.57.1 23(J6.3 123.5 126.2 1209.2 50 52 2271.7 113.7 iie.o 1I.5S.8 2369.6 123.8 126.6 1211.0 52 54 2275.0 114.0 116.3 1100.6 2372 8 124.2 126.9 1212.7 54 56 2278.3 114.3 116.7 1162.3 2376.1 124.5 127 3 1214.5 56 58 2281.5 114.7 117.0 1104.0 2379.3 124.9 127.6 1216.2 58 60 2284.8 115.0 117.4 1165.8 2382.6 125.2 128 1218.0 60 / 24° 1 2 5° / L. r. M. E. T. L C. M. E. T. 23.^2.6 125.2 128.0 1218.0 2480.4 135.8 139.1 1270.3 2 2.385.9 125.5 128.4 1219.7 2483.6 136.2 139.5 1272.0 2 4 23S9.1 125.9 128.7 1221.4 2486.9 136.5 139.9 1273.8 4 6 2.392.4 126.2 120.1 1223.2 24!0 1 136.9 140.3 1275.5 6 8 2395.6 126.6 129.5 1224 9 2493.4 137.2 140.6 1277.3 8 10 2398.9 126 9 129.8 1226.7 2496.6 137.6 141.0 1279.0 10 12 2402.2 127.3 130.2 1228.4 2499.9 138 141.4 1280.8 12 14 2405.4 127 6 130 6 1230.2 2.503.1 1.38.3 141 8 1282.5 14 16 24aS.7 128.0 130.9 1231.9 2506 4 138.7 142.2 1284.3 16 18 2411.9 128.3 131.3 1233.6 2509.6 139.0 142.5 1286.1 18 20 2415.2 128.7 131.7 1235 4 2512.9 139.4 142.9 1287.8 20 22 2-118 5 129.0 132.0 1237.1 2511).! 139.8 143.3 1289.6 22 24 2121.7 129.4 132.4 123^.9 2519.4 140.1 143.7 1291.3 24 26 2425.0 129.7 132 8 1240.6 2522.6 140.5 144.1 1293.1 26 28 242S.2 130.1 133 1 1242.4 2.525.9 140.8 144.5 1294.8 28 30 2431.5 130.4 133.5 1244.1 2529 1 141.2 144 9 1290.6 30 32 2434.8 130 8 133.9 1215.8 2.532.4 14; 6 145 3 1298.3 32 34 243S.0 131.1 134.2 1247.6 2535.6 142 145.6 1300.1 34 36 2441.3 131.5 134.6 1J4!).3 2.538.9 142.3 146.0 1301.8 36 38 2414.5 131.8 135.0 1251.1 2.542.1 142 7 146.4 1303.6 38 40 2447.8 132.2 135.4 12.52.8 2545.4 143.1 146.8 1305.3 40 42 2451.1 132 6 135.7 1254.6 2548.6 14 5 5 147.2 1307.1 42 44 2454.3 132.9 136.1 12.56.3 2.551.9 143.8 147.6 1308.8 44 46 2457.6 133.3 136.5 12.58.1 25.55.1 144.2 148.0 1310.6 46 48 2460.8 133.6 136.9 12.59.8 2558.4 144.5 148 4 1312.4 48 50 2464.1 134.0 137.2 1261.5 2561.6 144.9 148.8 1314.1 50 52 2467.4 134.4 137.6 1263 3 2564.9 145.3 149.2 1315.9 52 54 2170.6 134.7 138.0 1265.0 2508.1 145.7 149.5 1317.6 54 56 2473.9 135.1 138.4 1266 8 2571.4 146.0 149.9 1319.4 56 58 2477.1 135.4 138.7 126S.5 2.574.6 146.4 1.50.3 1321.1 58 60 2480.4 135.8 139.1 1270.3 2577.9 146 8 1.50.7 1322.9 60 IX.— FUNCTIONS OF A ONE-DEGKEE CURVE. 275 / 26° 27» 1 L. C. M. E. T. L. C. M. E. T. 2577.9 146.8 150.7 1322.9 2675.3 158.3 162.8 1375.6 2 2581 . 1 147.1 151.1 1324.6 2678.5 1.58.6 163.2 1377.4 2 4 2584.4 147.5 151.5 1326.4 2681.8 159.0 163.7 1379.2 4 6 2587.6 147.9 151.9 1328.1 2685.0 1.59.4 164.1 1380.9 6 8 2590.9 148.3 152.3 1329.9 2688.2 159.8 164.5 1882.7 8 10 2594.1 148.7 152.7 1331.6 2691.5 160.2 164.9 1384.5 10 12 2597.4 149.1 l.=)3.l 1333.4 2694.7 160.6 165.3 1386.2 12 14 2600.6 149.4 153.5 1335.2 2698.0 181 165.7 i:J88.0 14 10 2603.9 149.8 153.9 13.36.9 2701.2 161.4 166 1 1389.8 16 18 2607.1 150 2 154.3 1338.7 2704.4 161 8 166.5 1391.5 18 20 2610.4 150 6 154.7 1340.4 2707.7 162.2 167.0 1393.3 20 >_)•> 2613.6 151 155.1 1342.2 2710.9 162.6 167.4 1395.0 22 24 2616.9 151.4 155.5 1343.9 2714.1 163 167.8 1396.8 24 26 2620.1 151.7 155.9 1945.7 2717.4 163.4 168.2 1398.6 26 28 2623.4 152.1 156.3 1347.4 2720.6 163.8 168.6 1400.3 28 30 • 2626.6 152.5 156.7 1349.2 2723.8 164.2 169.1 1402.1 30 32 2629.8 152.9 157.1 1351.0 2727.1 164.6 169.5 1403.9 32 34 2633.1 153.3 157.5 1352.7 2730.3 165.0 169.9 1405.6 34 36 2636 3 153.7 157.9 1354 5 2733.0 165.4 170.3 1407.4 36 38 2639.6 154.0 158.3 1356.2 2736.8 165.8 170.8 1409.2 38 40 2642.8 154 4 158.7 1358.0 2740.0 166.2 171.2 1410.9 40 42 2610.1 154.8 159.1 1359.8 2743 3 166.6 171.6 1412.7 42 44 2649.3 155.2 159.5 1301.5 2746.5 167.0 172.0 1414.5 44 46 2652.6 155.6 160.0 1.363.3 2749 7 167 4 172.5 1416.3 46 48 2655.8 156.0 160.4 1365.1 2753.0 167.8 172.9 1418.0 48 50 2659.1 150.3 160.8 1366.8 2756.2 168.2 173 3 1419.8 50 52 2662.3 156.7 161.2 1368.6 13-0.4 2759.5 168.6 173.7 1421.6 52 54 2665. G 157.1 161.6 2762.7 169.0 174.1 1423.3 54 56 2668.8 1.57 5 162.0 13~2.1 2765.9 169.4 174.6 1425.1 56 58 2672.1 157.9 162.4 1373.9 2769.2 169.8 175.0 1426.9 58 60 2675.3 1.58.3 162.8 1375.6 2772.4 170.2 175.4 1428.6 60 2S i° '2S 1" / L. C. M. E. T. L.C. M. E. T. 2772.4 170.2 175 4 1428.6 2869.4 182.5 188.5 1481.9 2 2775.6 170.6 175.8 1430 4 2872.6 182.9 189.0 1483.7 2 4 2778.9 171.0 176.3 1432.2 2875.8 183 3 189.4 1485.4 4 6 2782.1 171.4 176 7 1434.0 2879.1 183.7 189.9 1487.2 6 8 278.J.3 171.8 177 1 1435.7 2882.3 184.2 190.3 1489.0 8 10 2788.6 172.2 177.6 1437.5 2885.5 184.6 190.8 1490.8 10 12 2791 .8 172.6 178.0 1439.3 2888.7 185.0 191.2 1492.6 12 14 2795.0 173 178.4 1441.1 2892.0 185.4 191.7 1494.3 14 16 2798.3 173.4 17S.9 1442 8 2895.2 185.8 192.1 1496.1 16 18 2801.5 173.8 179.3 1444.6 2898.4 186.3 192.5 1497.9 18 20 2804.7 174.3 179.7 1446.4 2901.6 186.7 193.0 1499.7 20 22 2808.0 174.7 180.2 1448.2 2904.8 187.1 193.5 1501.5 22 24 2811.2 175.1 180.6 1449.9 2908.1 187.5 193.9 1503.2 24 26 2814.4 175.5 181.0 1451.7 2911.3 188.0 194.4 1505.0 26 28 2817.7 175.9 181 5 1453.5 2914 5 188.4 194.8 1506.8 28 30 2820.9 176 3 181.9 1455.2 2917.7 188.8 195.3 1508-6 30 32 2824.1 176.7 182.3 14.57.0 2921.0 189.2 195.7 1510.4 32 34 2827 4 177.1 182.8 14.58.8 2924.2 189.7 196.2 1.512.1 34 36 28.30.6 177.5 183.2 1460.6 2927.4 190.1 196.7 1513.9 36 38 2833.8 177.9 183.6 1462 3 2930.6 190.5 197.1 1515.7 38 40 2837.1 178.4 184.1 1464.1 2933.9 190.9 197.6 1517.5 40 42 2840 3 178.8 184.5 1465.9 2937.1 191.4 198.0 1519.3 42 44 2843 5 179 2 185.0 1467.7 2940.3 191.9 198.5 1521.0 44 46 2846.8 179.6 185.4 1469,5 2943.5 192.4 198.9 1522.8 46 48 2850.0 180 185.9 1471.2 2946.8 192.8 199.4 1524.6 48 50 2853.2 180.4 186.3 1473.0 2950 193.2 199.8 1526.4 50 52 2856 5 180.8 186.8 1474.8 2953.2 193.6 200.3 1528.2 52 54 2859.7 181.2 187.2 1476.6 2956.4 194.0 200.8 1.530.0 54 56 2862.9 181.6 187.6 1478.3 29.59.6 194 4 201.2 1531.7 56 58 2866.2 182 188.1 1480.1 2962.9 194.8 201.7 1.5.33.5 58 60 2809.4 182 5 188.5 1481 9 2966.1 195.2 202.1 1.535 3 60 276 IX. FUNCTIONS OF A ONE-DEGREE CURVE. 1 30° L. C. 3 M. 1° 1 L. C. M. E. T. E. T. 2966.1 195.2 202.1 1535.3 3062.6 208.4 216 3 1589.0 2 2969.3 195.6 202.6 1537.1 3065.8 208.8 216.8 1590.8 2 4 2972.5 196.1 203.1 1538.9 3069.0 209.3 217.2 1592.6 4 6 2975.7 196.5 203.5 1540.7 3072.2 209.7 217.7 1594.4 6 8 2979.0 197.0 204.0 1542.5 3075.4 210.2 218 2 1596.2 8 10 2982.2 197 4 204.5 1544 3 3078.6 210.6 218.7 1598.0 10 12 2985.4 197.8 204.9 1546.0 3081.8 211.1 219.2 1599.8 12 14 2988.6 198.2 205.4 1547.8 3085.0 211.5 219.6 1601.6 14 16 2991.8 198.6 205 9 1549.6 3088.3 212.0 220 1 1603.4 16 18 2995 199.1 206 3 1.551.4 3091.5 212.4 220.6 1605.2 18 20 2998.3 199.5 206.8 1553.2 3094.7 212.9 221.1 1607.0 20 22 3001.5 199.9 207.3 1555.0 3097.9 213.3 221.6 1608.8 22 24 3004.7 200.4 207.7 1.556.8 3101.1 213.8 222.1 1610.6 24 26 3007.9 200.8 208.2 1558.6 3104.3 214.2 222.6 1612.4 26 28 3011.1 201.3 208.7 1560.4 3107.5 214.7 223.0 1614.2 28 30 3014.3 201.7 209.1 1562.2 3110.7 215.1 223.5 1616.0 30 32 3017.6 202 1 209.6 1564.0 3113.9 215.6 224.0 1617.8 32 34 3020.8 202 6 210.1 1 565 . 7 3117.1 216.0 224.5 1619.6 34 36 3024.0 203.0 210.5 1567.5 3120.3 216.5 225.0 1621.4 36 38 3027.2 203.5 211.0 1569.3 3123.5 216.9 225.5 1623.2 38 40 3030 4 203.9 211 5 1571.1 3126.7 217.4 226.0 1625.0 40 42 3033.6 204.3 212 1.572.9 3129.9 217. S 226.5 1626.8 42 44 \ 44 3036.9 204.8 212 4 1.574.7 3133.1 218.3 227.0 1628.6 46 3040.1 205.2 212.9 1576.5 3136.4 218.7 227.5 16-30.5 46 ' 48 3043.3 205.7 213.4 1578.3 3139.6 219.2 228.0 16:32.3 48 50 3046.5 206.1 213.9 1580.1 3142.8 219.6 228.4 16:34.1 50 52 3049.7 206.5 214.4 1581.9 3146.0 220.1 228.9 1635.9 52 54 3052.9 207 214 8 1583.7 3149.2 220.5 229.4 1637.7 54 56 3056.2 207.4 215.3 1585.5 31.52.4 221.0 2:.'9.9 16:39.5 56 58 3059.4 207.9 215 8 1587.2 315.5.6 221.5 230.4 1641.3 58 60 3062.6 208.4 216 3 1.589.0 31.58.8 222.0 230.9 1643.1 60 / 32° 1 33" '» / L. C. 3158.8 M. 222.0 E. 2.30.9 T. 1643.1 L. C M. E. T. 3254 9 2.36.0 246.1 1697 3 >■> 3162.0 222.5 231.4 1644 9 3258 1 236.4 246.6 1699.1 2 4 3165.2 222.9 231.9 1646 7 3261.3 2.36 9 247.1 1700 9 4 6 3168.4 223.4 232.4 1648.5 .3264.5 2.37 4 247.7 1702.7 6 8 3171.6 223.8 2.32.9 1650 3 3267 7 237 9 248.2 1704.5 8 10 3174.8 224.3 233.4 1652 1 .3270.8 2:38.4 248.7 1706.4 10 12 3178.0 224 8 233.9 1653.9 .3274 2:38 9 249.2 1708.2 12 14 3181.2 225.2 234.4 16.55 7 3277 2 2:39 3 249.7 1710.0 14 16 3184.4 225 7 234.9 1657 5 3280 4 2:39.8 250.2 1711.8 16 18 3187.6 226.1 2:35.4 1659.3 328:3.6 240.3 250.8 1713.6 18 20 3190.8 226.6 235.9 1661 1 .3286.8 240.8 251.3 1715.5 20 22 3194.0 227.1 236.4 1662.9 3290 241.2 251 8 1717.3 22 24 3197 2 227.5 236 9 1664.7 3203.2 241 7 252 3 1719.1 24 26 3200.4 228.0 237.4 1666 5 .3296.4 242.2 252.9 1720.9 26 28 3203.6 228.4 237 9 1668 3 .3299.6 242.7 253 4 1722.7 28 30 3206.8 228 9 238.4 1670.1 3302.7 243 2 253.9 1724.6 30 32 3210.0 229 4 239.0 1671.9 3-305 9 243.6 254.4 1726.4 32 34 3213.2 229 8 239.5 1673 7 3.309 1 244.1 255 1728.2 34 36 3216 5 230.3 240.0 1675 5 .3312 3 244.6 255.5 1730.0 36 38 3219.7 230.7 240.5 1677.4 3315 5 245.1 256.0 1731.8 38 40 3222 9 231 2 241.0 1679.2 3:318.7 245.6 256 5 1733.6 40 42 3226 1 231 7 241 5 1681.0 3.321.9 246 257 1 1735.5 42 44 3629.3 232.2 242 1682 8 3325.1 246.5 257.6 1737.3 44 46 3232.5 232.6 242.5 1684.6 3:328.3 247.0 258.1 1739.1 46 48 3235.7 233.1 243.0 1686 4 .3.3:31.5 247 5 258 6 1740.9 48 50 3238.9 233.5 243.5 1688 2 :3:3:34.6 248.0 259.2 1742.7 50 52 3242.1 234.0 244 1 1690.0 33:37.8 248.4 259.7 1744.6 52 54 3245 3 234.5 244 6 1691.8 3341 .0 248 9 260 2 1746.4 54 56 3248.5 235 245.1 1693 7 :3.344.2 249.4 260.8 1748.2 56 58 3251 . 7 235.5 245.6 1695 5 :3:347.4 249.9 261.3 1750.0 58 60 3254 9 236 246 1 1697 3 :3:3.j0.6 250.4 261 8 1751.8 60 IX.— FUNCTIONS OF A ONE-DEGREE CURVE. ^^1 / S4^ 3 0° / L. 0. M. E. T. L.C. 3146.1 M. 265.2 E. 278.1 T. 1806.7 3350.6 250.4 261.8 175I.S 2 3353.8 2.50.8 262.3 1753.7 3119.3 265.7 278.6 180S.5 4 3357.0 251.2 262.9 V>'M.n 3452.5 266.2 279.2 1810.3 4 6 3360.1 251.7 263.4 1757.3 3455.6 266.7 279.7 1812.2 6 8 3363.3 252.2 2o4.0 1759.1 3458.8 267.2 280.3 1814.0 8 10 3366.5 252.7 264.5 1761.0 3162.0 267.7 280.8 1815.8 10 \2 3369.7 253.2 265.0 1762.8 3165.2 268.2 281.4 1817.7 12 14 3372.9 253.7 265.0 1764.6 3468.3 268 7 2S1.9 1819.5 14 16 3376.1 2.54.2 ^'66.1 1 766 . 4 3171.5 269.2 2S2 . 5 1821.3 16 18 3379.2- 254.7 266.7 1768.3 3474.7 269.7 2S3.0 1823.2 18 20 33S2.4 255.2 267.2 1770.1 34. r. 9 270.2 283.6 1825.0 20 22 3385.6 255.7 267.7 1771 9 3481.0 270.7 284.2 1826.8 22 24 3388.8 256.2 268.3 1773.7 3484.2 271.2 284.7 1828.7 24 26 3392.0 256.7 268.8 1775.6 3487.4 271.7 285.3 1830.5 20 28 3395.2 257.2 269.3 1777.4 3490.6 272.2 285.9 1832.3 28 30 3398.3 257.7 269.9 1779.2 3493.7 272.7 286.4 1834.2 30 3i 3401.5 258.2 270.4 1781.0 3496.9 273.2 287.0 1836.0 32 34 3404.7 258.7 271.0 1782.9 3500.1 273. V 287.5 1837.8 34 36 3407.9 2.59.2 271.5 1784.7 3503.3 274.2 288.1 1839.7 30 38 3411.1 259.7 272 1786.5 3506.5 274.7 288.7 1841.5 38 40 3414.3 260.2 272.6 1788.4 3509.6 275.2 289.2 1843.4 40 42 3417.4 260.7 273.1 1790.2 3512.8 275.7 289.8 1845.2 42 44 3420.6 261.2 273.7 1792.0 3516.0 276 2 290.4 1847.1 44 46 3423.8 261.7 274.2 1793 9 3519.2 ^<0. i 290.9 1848.9 46 48 3427.0 262.2 274.8 1795.7 3522.3 291.5 1850.7 48 50 3430.2 262.7 275.3 1797.5 3525.5 277.7 292.0 1852.6 50 52 3433.4 263.2 275.9 1799.3 3528.7 278.2 292.6 1854.4 52 54 3436.5 263.7 276.4 1801.2 3531.9 278.7 293.2 1850.3 54 56 3439.7 264.2 277.0 1803.0 3.5.35.0 279.2 293.7 1858.1 56 58 3442.9 264.7 277.5 1804.8 3538.2 279.8 294.3 18.59.9 58 GO 3446.1 265.2 278.1 1806.7 3541.4 280.4 294.9 1861.8 60 / 3«° 1 3- ?" / L. C. M. E. T. L.C. M. E. T. 3541.4 280.4 294.9 1861.8 3636.3 296.1 312.3 1917.3 o 3544.6 280 9 295.4 1863.6 3639.5 296.6 312.8 1919.1 4 3547.7 281.4 296.0 1865.5 3642 6 297 1 313.4 1921.0 4 6 3550.9 281.9 296.6 1867.3 3645.8 297.7 314 1922.8 6 8 3554.0 282.5 297.2 1869.2 3648.9 298.2 314.6 1924.7 8 10 3557.2 283.0 297.7 1871.0 3652.1 298.7 315.2 1926.5 10 12 3560.4 283.5 298.3 1872.9 3655.2 299.3 315.8 1928.4 12 14 3563.5 284.0 298.9 1874.7 36.58.4 299.8 316.4 1930.2 14 16 3566.7 284.6 299.5 1876.5 3661.6 300.3 317.0 1932.1 16 18 3569.9 285.1 300.0 1878.4 3664.7 300.9 317.5 1933.9 18 20 3573.0 285.6 300.6 1880.2 3667.9 301.4 318.1 1935.8 20 22 3576.2 286.1 301.2 1882.1 3671.0 301.9 318.7 1937.6 22 24 3579.4 286.7 301 .8 1883.9 3674.2 302.5 319.3 1939.5 24 26 3582.5 287.2 302:3 18S5.8 3677.3 303.0 319.9 1941.3 26 28 a585.7 287.7 302.9 1887.6 3680.5 303.5 320.5 1943.2 28 30 3.588.8 288.2 303.5 1889.5 3683.6 304.1 321.1 1945.0 30 32 3.592.0 288.8 304.1 1891.3 3686.8 304.6 321.7 1946.9 32 34 3595.2 289. S 304.6 1893.2 3690.0 305.1 322.3 1948.8 34 36 3598.3 289.8 305.2 1895.0 3693.1 305.7 322.9 1950.6 36 38 3601.5 290.3 305.8 1896.9 3696.3 306.2 323.5 1952.5 38 40 3604.7 290.9 306.4 1898.7 3699.4 306.7 324.2 1954.4 40 42 3607.8 291.4 307.0 1900.6 3702.6 307.3 324.8 19.56.2 42 44 3611.0 291.9 307.5 1902.4 3705.7 307.8 325.4 1958.1 44 46 3614.1 292.4 308.1 1904.3 3708.9 308.3 326.0 1960.0 46 48 3617.3 893.0 308.7 1906.1 3712.1 308.9 326 6 1961.8 48 50 3620.5 293.5 309.3 1908.0 3715.2 309.4 327 2 1963.V 50 52 3623 6 294.0 309.9 1909.8 3718.4 309 9 327.8 1965.5 52 54 3626.8 294.5 310.5 1911.7 3721 5 310.5 328.4 1967.4 54 56 3630.0 295.1 311.1 1913.5 3724.7 311.0 329.0 1969.3 56 58 3633.1 295.6 311.7 1915.4 3727.8 311.6 329 6 1971 1 58 60 3636.3 296.1 312.3 1917.3 3731.0 312.2 330 2 1973.0 60 ^ 278 IX.— FUNCTIONS OF A ONE-DEGREE CURVE. / 38° 1 39° t L. C. M. E. T. L. C 3825 5 M 328-7 E. 348.7 T. 2029.1 3731.0 312.2 330.2 1973.0 2 3734.1 312 7 330.8 1974.9 3828.6 329.2 349 3 2031.0 2 4 3737.3 313 3 :i31.4 1976-7 3831.8 329.8 .349.9 2032 9 4 6 3740 4 313.8 332.0 1978.6 3834 9 3^0-3 .350.8 20.34.7 6 8 3743.6 314.4 332.6 1980.5 3838.0 330.9 351.2 2036.6 8 10 3746.7 314.9 333.2 1982.3 3841.2 331 5 351.8 2038.5 10 12 3749 9 315.5 ::!33.8 1984-2 3844 3 332.0 352.4 2040 4 12 14 3753 316 .334.5 1986.1 3847.4 332.6 353.1 2042 3 14 16 3756 2 316.6 335.1 1987.9 3850.6 333.2 353.7 2044.1 16 18 3759.3 317.1 335-7 1989.8 38.53 7 333 7 354-3' 2046.0 18 20 3762 5 317 7 336 3 1991.7 38.56.8 334.3 354.9 2047.9 20 22 3765.6 .318 2 336 9 1993-6 3860 334.9 355.6 2049.8 22 24 3768.8 318.8 337.5 1995.4 3863.1 ;3;35.4 3.56.2 2051 7 24 26 3771.9 319.3 338.1 1997.3 :^66.2 336. 356.9 2053.5 26 28 3775 1 319.9 338.7 1999.2 3869.4 a36.6 357 5 2055.4 28 30 3778.2 320.4 339.4 2001.0 :S872.5 337.1 358.1 2057-3 30 32 3781.4 321.0 310.0 2002.9 3875 6 337 7 3.58.8 2059.2 32 34 3784 5 321.5 310-6 2004.8 3878.8 338 3 359 4 2061.1 34 36 3787.7 322.1 341.2 2006.6 3881.9 3.38 8 .360.1 2063.0 36 38 3790 8 322.6 341.8 2008.5 3885.0 339 4 360.7 2064.8 38 40 3794.0 323 2 342.4 2010 4 3888.2 340.0 361.3 2066.7 40 42 3797 1 323.7 343.1 2012 3 3891.3 :i40.5 362.0 2068.6 42 44 3800.3 324.3 :^3 7 2014.1 3894 4 341 1 362.6 2070.5 44 46 3803.4 324.8 344.3 2016 3897.6 341.7 363.3 2072.4 46 48 3806.6 325.4 344.9 2017.9 3900 7 342 2 363.9 2074 2 48 50 3809.7 325 9 345.6 2019.7 3903.8 342.8 364.5 2076 1 50 52 3812.9 326 5 M6 2 2021.6 3907 343 4 365.2 2078.0 52 54 3816 327 346.8 2023.5 3910.1 343.9 365.8 2079.9 M 56 3819.2 327.6 347 4 2025 4 3913 2 .344.5 366.5 2081.8 56 68 3822.3 328.1 348 1 2027.2 3916.4 345 1 367 1 2083.7 58 60 3825.5 328.7 348.7 2029.1 3919 5 U5.6 367 7 2085.5 60 / 40'^ 1 41° ' L. C. M. E. T. L. C. M. E. T. 3919.5 345.6 367.7 2085.5 4013.4 362.9 387.4 2142.3 2 392J.6 346.1 368.4 2087 4 4016.5 363.4 388.1 2144.2 2 4 3925.8 346 7 369 2U89.3 4019.6 364.0 388.8 2146.1 4 6 3928 9 347 2 369.7 2091 2 4022.7 364 5 389.4 2148.0 6 8 3932 347 8 370.3 2093. 1 4025.9 365.1 390 I 2149.9 8 10 3935.1 348.4 371 2095 4029.0 365.6 390.7 21.51.9 10 12 3938.3 a48.9 371 2096 9 4032.1 366.2 391.4 2153.8 12 14 3941.4 349.5 372.3 2098.8 40-35.2 366.8 392.1 2155.7 14 16 3944.5 350.1 372.9 2100.7 4038.3 367.4 392.7 2157.6 16 18 3947.7 350.7 373.6 2102 6 4041.4 368.0 393.4 2159.5 18 20 3950.8 351.3 374.3 2104.5 4044.6 368.6 394.1 2161.4 20 22 39.53.9 351 8 374.9 2106.3 4047.7 369.2 394.7 2163.3 22 24 3957.1 352.4 375.6 2108.2 4050.8 .369.8 395.4 2165.2 24 26 3960.2 353.0 376.2 2110.1 4053 9 370.4 .396.1 2167.1 26 28 3963.3 3.53 6 376.9 2112 40.">7.0 371.0 396.8 2169.0 28 30 3966.4 354.2 377.5 2113 9 4060.1 371.6 .397.5 2170 9 30 32 3969.6 354.7 378.2 2115 8 4063.3 372.2 398.1 2172.8 32 34 3972.7 355.3 378.8 2117 7 4066.4 372.8 .398.8 2174.7 34 36 3975.8 355.9 379.5 2119 6 4069.5 373.4 399.5 2176.6 36 38 3979.0 356.5 380 1 2121.5 4072.6 374.0 400.2 2178.5 38 40 3982.1 357.1 380.8 2123.4 4075.7 .374.6 400.9 2180.4 40 42 3985.2 357 6 381 4 2125 3 4078. 8 375.2 401.5 2182.4 42 44 3988.4 358.2 382.1 2127.2 4082.0 375.8 402.2 2184.3 44 46 3991 5 358 8 382 8 2129.1 4085.1 376.4 402.9 2186.2 46 48 3994.6 359.4 383.4 2131.0 408«.2 377.0 403.6 2188.1 48 50 3997 7 360 384.1 2132.9 4091.3 377.6 404.3 2190.0 50 52 4000 9 :^60 5 384 8 2134.7 4094.4 378.2 404.9 2191.9 52 54 4004 361 1 385.4 2136.6 4097.5 .378.8 405.6 2193.8 ! 54 56 4007 1 .361.7 386.1 2138.5 4100.7 379.4 400.3 2195.7 56 58 4010 3 3ti2.3 380 8 2140 4 4103.8 380 407.0 2197.6 58 60 CO 4013 4 3'i2.9 3!S7 4 2142 3 4106.9 380.6 407.7 2199.5 ■ IX.-FUNCTIONS OF A ONE-DEGREE CURVE. 279 r 4 2" 43° / L. C. M. E. T. L. C. M. E. T. / 4106.9 380.6 407.7 2199.5 4200.1 398.7 428.6 2257.1 2 4110.0 381.2 408.3 2201.4 4 -,'03. 2 399.3 429.3 2259.0 2 4 4113.1 381.8 409.0 2203.3 4206.3 399.9 430.0 2261 .0 4 6 4116.2 382.4 409.7 2205.3 4209.4 400.5 430.7 2262.9 6 8 4119.3 383.0 410.4 2207.2 4212.5 401.1 431.4 2264.8 8 10 4122.4 383.6 411.1 2209.1 4215.6 401.7 432.1 2260.7 10 13 4125.5 3S4 . 2 411.8 2211.0 4218.7 402.4 432.8 8268.7 12 14 4128.6 384.8 412.5 2212.9 4221.8 403.0 433.5 2270.6 14 16 4131.8 38. 4 413.2 2214.9 4224.9 403.6 434.2 2272.5 16 18 4134.9 386 413.9 2216.8 4228.0 404.2 434,9 2274.5 18 20 4138.0 380 i 414.6 2218.7 4231.1 404.8 435.6 2276.4 20 22 4141.1 3S7 2 415.3 2220.6 4234.2 405 . 4 436.3 2278.3 22 24 4144.2 387.8 416.0 2222.5 4237.3 406.1 437.0 2280.2 24 26 4147 3 388.4 416.6 2224.4 4240.4 406 7 437 8 2282.2 26 2S 4150.4 389 417.3 2.226.4 4243.5 407.3 438.5 2281.1 28 30 4153.5 389.6 418.0 2228.3 4246.5 407.9 439 2 2286.0 30 32 4156.6 390,2 418.7 2230.2 4249.6 408 5 439.9 2288.0 32 34 4159.7 390.8 419.4 2232.1 4252.7 409.1 440.6 2289.9 34 36 4162 8 391.4 420.1 2234.0 4255.8 409 8 441.4 2291.8 36 38 4165.9 392.0 420.8 2-J36.0 4258.9 410.4 442.1 2293.8 38 40 4169.0 392.6 421.5 2237.9 4262.0 411.0 442.8 2295.7 40 42 4172.1 393.2 422 2 2239.8 4205.1 411.6 443.5 2297.7 42 44 4175.2 393.8 422.9 2241.7 4208.2 412 2 444.2 2299.6 44 46 4178.4 394.4 423.6 2243.6 4271.3 412.8 445.0 2.301.5 46 48 4181.5 395.0 424.3 2245.6 4274.4 413.5 445.7 2303.5 48 50 4184.6 395.6 425.0 2247.5 4277.5 414.1 446.4 2305.4 50 52 4187.7 396.2 425.7 2i49 4 4280.6 414.7 447.1 2307.3 52 54 4190.8 396.8 426.4 2251.3 4283.7 415 3 447.8 2309.3 54 56 4193.9 397 4 427.1 2--"53.3 42,S6.8 415.9 448,6 2311.2 56 58 4197.0 398 427.8 2255 2 4289.9 416.5 449.3 2313.1 58 60 4-'00.1 398.7 428.6 22.-.7.1 4293.0 417.2 450.0 2315.1 60 / 44° 1 4 5° » / L. C. M. E. T. L. C. 4385.5 M. 436.2 E. 472.1 T. 2373.4 4293.0 417.2 450.0 2315.1 2 4296 1 417.8 450.7 2317.0 4388.6 436.8 472.9 2375.4 2 4 4299.2 418.4 451.5 2319.0 4391.7 437.5 473.6 2377.3 4 6 4302.2 419.1 452.2 2320.9 4394.7 438.1 474.4 2379.3 6 8 4305.3 419.7 452 9 2322.8 4397.8 438 8 475.1 2381.2 8 10 4308.4 420.3 453.7 2324.8 4400.9 439.4 475.9 2383.2 10 12 4311.5 421.0 454.4 2326.7 4404.0 440.0 476 6 2385.2 12 14 4314.6 421 . 6 455.1 2328.7 4407.0 440.7 477.4 2387.1 14 16 4317.7 422.2 455.9 2330.6 4410.1 441.3 478.1 2389.1 16 18 4320.7 422.9 456.6 2332.6 4413.2 442.0 478 9 2391 .0 18 20 4323.8 423 5 457.3 2334.5 4416.3 442,6 479.6 2393.0 20 22 4326.9 424.1 458.1 2336.4 4419.3 443.2 480.4 2394.9 22 24 4330.0 424.8 458.8 2338.4 4422.4 443.9 481.1 2396.9 26 4333.1 425 4 459.5 2340 3 4425 . 5 444.5 481 9 2398.8 26 28 4336.2 426.0 460.3 2342.3 4428.6 445.2 482.6 2400.8 28 30 4339.2 426.7 461 2344.2 4431.6 445.8 483.4 2402.8 30 32 4342.3 427 3 461.7 2346.1 4431.7 446 4 484.2 2404.7 32 34 4345.4 427 9 462.5 2348.1 4437 . 8 447.1 484.9 2406.7 34 36 4348.5 428 6 463.2 2350 4440.9 447.7 485.7 2408.6 36 38 4351.6 429.2 463.9 2352.0 4444.0 448 3 486.5 2410 6 38 40 4354.7 429.8 464.1 2353.9 4447.0 448.9 487.2 2412.6 40 42 4357.7 430.5 465.4 2355,9 4450.1 449 5 488.0 2414.5 42 44 4360.8 431.1 466.2 23.57.8 4453 2 450.2 488.7 2416.5 44 46 4363.9 431.7 466.9 23.59.8 4456.3 450.8 489.5 2418.5 46 48 4367.0 432.4 467.7 2361 7 4459.3 451.5 490.3 2420.4 48 50 4370.1 433.0 46» 4 2363.7 4462 4 452.1 491.0 2422.4 50 52 4373.2 433.6 469.1 2365 6 4465.5 452.7 491 8 2424.4 52 54 4376 2 434.3 469.9 2367.6 4468.6 453 4 492.5 2426.3 54 56 43'^9.3 434.9 470.6 2369.5 4471.6 454 . 1 493.3 2428 . 3 56 5« 4382.4 435 b 471.4 23-^1.5 4474.7 454.8 494.1 2430 2 58 60 4385 5 436 2 472.1 2373.4 4477.8 4.55.5 494.8 2432.2 60 280 IX. FUNCTIONS OF A ONE DEGREE CURVE. / 46° 1 4 7° / L. C. M. E. T. L. C M. E. T. 4477.8 4.55 . 5 494.8 2432.2 4569.7 475.2 518.3 2491.5 2 4480 9 456.1 495.6 24.34.2 4.572.7 475.9 519 2493.4 2 4 4-183.9 456.8 496.5 2136.1 4575.8 476.5 519.8 2495.4 4 6 4487.0 457.1 407.2 2438.1 4578.8 477.2 520.6 2497.4 6 8 4490.0 458.1 497.9 2440.1 4581.9 477.8 521.4 2499.4 8 10 4493.1 458.7 498.7 2442.1 4584.9 478.5 522.2 2501.4 10 1-2 4496.2 459.4 499.5 2444.0 4588.0 479.2 .523.0 2503.4 12 14 4499.2 460.0 500.3 2446.0 4591.0 479.8 523.8 2505.4 14 16 4502.3 460.7 501.0 ^448.0 4594.1 480.5 524.6 2507.3 16 18 4505.4 461.3 501.8 2449.9 4597.1 481.1 525.4 2509.3 18 20 4508.4 462.0 502.6 2451.9 4600.2 481.7 .526.2 2511.3 20 oo 4511.5 462.7 503.4 2453.9 4603.2 482.3 527.0 2513.3 22 34 4514.6 463.3 504.1 2155 9 460G.3 483.0 527.8 2515.3 24 26 4517.6 464.0 504.9 2457.8 4609.3 483.7 528.6 2517.3 26 28 4520.7 404.6 505.7 2459.8 4612.4 484.3 529.4 2519.3 28 30 4.523.7 405.3 506.5 2461.8 4615.4 485.0 530.2 2521.2 30 32 4526.8 406.0 507.3 2463.8 4618.5 485.7 .5.31.0 2523.2 32 34 4529.9 466.6 508.0 2465.7 4621.5 486.3 531.8 2525.2 34 36 4532.9 467.3 508.8 2467.7 4624.6 487.0 532.6 2527.2 36 88 4536.0 467.9 509.6 2469.7 4627.6 487.7 533.4 2529.2 38 40 45.39.1 468.6 510.4 2471.7 4630.7 4S8.4 5.34.2 2531.2 40 42 4542.1 469.3 511.1 2473.6 403:3.7 489.1 535.0 2533.2 42 44 4545.2 469.9 511.9 2475.6 4636.8 489.8 5.35.8 2535.2 44 46 4.548.2 470.6 512.7 2477.6 4639.8 490.5 536.6 2537.2 46 48 4551.3 471.2 513.5 2479.6 4642.9 491.2 537.4 2539.2 48 50 4554.4 471.9 514.3 2481.6 4645.9 491.9 538.2 2541.2 50 52 4557.4 472.6 515.1 2483.5 4649.0 492.6 5.39.0 2543.1 52 54 4560.5 473.2 515.9 2485.5 4652.0 493.3 539.8 2.545.1 54 56 4563.6 473.9 516.7 2487.5 46.55.1 494.0 540.6 2547.1 56 58 4.566.6 474.5 517.5 2489.5 46.58.1 494.7 541.4 2549.1 58 60 4569.7 475.2 518.3 2491.5 4661.2 495.4 542.3 2551 . 1 60 / 48° 1 49° 1 / L. C. 4661.2 M. 495.4 E. 542.3 T. 2.551.1 L. C. M. E. T. 47.52.3 515.9 567.0 2611.3 2 46C4 2 496.0 543.1 2.5.53.1 4755 . 3 516.5 567.8 2613.3 o 4 4667.3 496.7 543.9 2555.1 4758.4 517.2 568.7 2615.3 4 6 4670.3 497.4 544.7 2,5.57.1 4761.4 517.9 569.5 2617.3 6 8 4673.3 498.1 545.5 2559.1 4764.4 518.6 570.3 2619.3 8 10 4676.4 498.8 546.4 2.5G1.1 4767.4 519.3 .571.2 2621.4 10 12 4679.4 499.4 547.2 2563.1 4770.5 520.0 572.0 2623.4 12 14 4682.5 500.1 .548.0 2.565.1 4773.5 .520.7 572.8 2625.4 14 16 4685.5 500.8 548.8 2.507.1 4776.5 .521.4 573.7 2627.4 16 18 4688.5 501.5 549.6 2569.1 4779.6 522.1 574.5 2629.4 18 20 4691.6 502.2 .5.50.5 2571.1 4782.6 522.8 575.3 2631 .4 20 22 4694.6 502.8 551.3 2573.1 4785.6 523.5 576.2 2633.5 22 24 , 4697.6 503.5 552.1 2575.1 4788.7 524.2 577.0 2635.5 24 26 4700.7 504.2 5.52.9 2577.1 4791.7 524.9 577.9 2637.5 26 28 4703.7 504.9 5.53.7 2.579.1 4794.7 .525.6 578.7 2639.5 28 30 4706.7 505.6 .554.6 2581.1 4797.7 526.3 579.6 2641.5 30 32 4709.8 506.2 5.55.4 2583.1 4800.8 527.0 580.4 2643.5 32 34 4712.8 506.9 .556.2 2585.1 4803.8 527.7 581.3 2645.6 34 S6 4715.9 507.6 557.0 2587.2 4806.8 528.4 582.1 2647.6 36 38 4718.9 508.3 557.8 2589.2 4809.9 529.1 583.0 2649.6 38 40 4721.9 509.0 5.58.7 2.591.2 4812.9 .529.8 583.8 2651.6 40 42 4725.0 509.6 559.5 2593.2 4815.9 530.5 584.7 26.53.7 42 44 4728.0 510.3 560.3 2595.2 4819.0 531.2 585.5 26.55.7 44 46 4731.0 511.0 561.2 2597.2 4822.0 531.9 586.4 2657.7 46 48 4734.1 511.7 562.0 2.599.2 4825.0 532.6 .587.2 2659.7 48 50 4737.1 512.4 562.8 2601.2 4828.0 .533.3 588.1 2661.8 50 52 4740.2 513.1 563.7 2603.2 4831.1 534.0 588.9 2663.8 52 54 4743.2 513.8 564.5 2605.2 4834.1 534.7 589.8 2665.8 54 56 4746.2 514.5 565.3 2607.2 4837.1 535.4 590.6 2667.8 56 58 4749.3 515.2 566.2 2609.3 4S40.2 536.1 591.5 2669.9 58 ' 60 4752.3 515.9 567.0 2611.3 4843.2 .536.8 592.4 2671.9 60 1 IX. -FUNCTIONS OF A ONE-DEGREE CURVE. 281 / 50° 1 51° / L. C. M. E. T. L. C. M. E. T. 4843.2 536.8 592.4 2671. 9 4933 6 558.2 618.5 2733.0 o 4846.2 537.5 593.2 2673 9 4936.6 558.9 619.3 2735.1 2 4 4849.2 538.2 594.1 2676.0 4939.6 559.7 620.2 2737.1 4 6 48.02.2 538.9 594.9 2678.0 4942.6 560.4 621.1 2739.2 6 8 4855.2 539.6 595.8 2680.0 4945.0 561.1 622.0 2741.2 8 10 4858.3 540.3 596 7 2682.1 4948.6 561.8 622.9 2743.3 10 12 4861.3 541.0 597.5 2684.1 4951.6 562.5 623.7 2745.3 12 14 4864.3 541.7 598.4 2686.1 49.54.6 563.3 624.6 2747.4 14 16 4867.3 542.4 599.3 2688.2 4957.6 564.0 625.5 2749.4 16 18 4870.3 543.1 600 1 2690.2 4960.6 564.7 626.4 2751.5 18 20 4873.3 543.9 601.0 2692.3 4963.6 565.4 627.3 2753.5 20 22 4876.3 544.6 601.9 2694.3 4966.6 566.2 628.2 2755.6 22 24 4879.4 545.3 602.7 2696.3 4969.6 566.9 629.9 2757.7 24 26 4882.4 546.0 603.6 2698.4 4972.6 567.6 630.0 2759.7 26 28 4885.4 546.7 604.5 2700.4 4975.6 568.3 630.9 2761.8 28 30 4888.4 547.4 605.3 2702.4 4978.6 .569.1 631.8 2763.8 30 32 4891.4 548.1 606.2 2704.5 4981.6 569.8 632.7 2765.9 32 34 4894.4 548.8 607.0 2706.5 4984.6 570.5 633.6 2767.9 34 36 4897.4 549.5 607.9 2708.6 4987.7 571.2 634.5 2770.0 36 38 4900.4 550.2 608.8 2710.6 4990.7 572.0 035.3 2772.0 38 40 4903.5 551.0 609.7 2712.6 4993.7 572.7 636.2 2774.1 40 42 4906.5 551.7 610.5 2714.7 4996.7 573.4 637.1 2776.2 42 44 4909.5 552.4 611.4 2716.7 4999.7 574.1 638.0 2778.2 44 46 4912.5 553.1 612.3 2718.8 .5002.7 574.9 638.9 2780.3 46 48 4915.5 553.8 613.2 2720.8 5005.7 575.6 639.8 2782.3 48 50 4918.5 .554.5 614.1 2722.8 5008.7 576.3 640.7 2784.4 50 52 4921.5 555.2 014.9 2724.9 .5011.7 577.0 641.6 2786.4 52 54 4924.6 555.9 615.8 2726.9 .5014.7 577.8 642.5 2788.5 54 56 4927.0 556.6 616.7 2729.0 5017.7 578.5 643.4 2790.6 56 58 4930.6 557.4 617.6 2731.0 5020.7 579.2 644.3 2792.6 58 60 4933.6 558.2 618.5 2733.0 5083.7 579.9 645.2 2794.7 60 / 52° 1 5 S° / L. C. M. E. T. L. C. M. E. T. 5023.7 579.9 645.2 2794.7 5113.5 602.0 672.7 2856.9 »> .5026.7 580.6 6)6.1 2796.8 5116.5 602.8 673.7 2858.9 2 4 5029.7 .581.3 647.0 2798.8 5119.4 603.5 674.6 2861.0 4 6 .5032.7 582.1 647.9 2800.9 5122.4 604.3 675.5 2863.1 6 8 5035.7 .582.8 648.9 2803.0 5125.4 605.0 676.4 2865.2 8 10 5038.7 583.5 649.8 2805.0 5128.4 605.8 677.4 2867.3 10 12 5041.7 584.3 6.50.7 2807.1 5131.3 606.5 678.3 2869.4 12 14 5044.7 585.0 651.6 2809.2 5134.3 607.3 679.2 2871.5 14 16 5047.7 585.7 652.5 2811.2 5137.3 608.0 680.2 2873.5 16 18 5050.7 .586.5 653.4 2813.3 5140.3 608.8 681.1 2875.6 18 20 5053.6 587.2 654.3 2815.4 5143.2 609.5 682.0 2877.7 20 22 5056.6 587.9 6.55.2 2817.4 5146.2 610.3 683.0 2879.8 22 24 5059.6 .588.7 656.2 2819.5 5149.2 611.0 683.9 2881.9 . 24 26 5062.6 .589.4 6.57.1 2821.6 5152.1 611.8 684.9 2884.0 26 28 5065.6 590.1 6.58.0 2823.6 5155.1 612.5 685.8 2886.1 28 30 5068.6 590.9 6.58.9 2825.7 51.58.1 613.3 686.7 2888.1 30 32 5071.6 591.6 659.8 2827.8 5161.1 614.0 687.7 2890.2 32 34 5074.6 592.3 660.7 2829.8 5164.0 614.8 688.6 2892.3 34 36 5077.6 593.1 661.6 2831.9 5167.0 615.5 689.6 2894.4 36 38 5080.6 593.8 662.5 2834.0 5170.0 616.3 690.5 2896.5 38 40 5083.6 594.5 663.5 2836.1 5173.0 617.0 691.5 2898.6 40 42 5086.6 595.3 664.4 2838.2 5175.9 617.8 692.4 2900.7 42 44 5089.6 596.0 065.3 2840.2 5178.9 61S.5 693.4 2902.8 44 46 5092.6 596.7 666.2 2842.3 5181.9 619.3 694.3 2904.9 46 48 5095.6 597.5 667.2 284.4.4 5184.9 620.1 695.3 2907.0 48 50 5098.6 598.2 668.1 2846.5 5187.8 620.8 696.2 2909.1 50 52 5101.6 598.9 669.0 2848.5 5190.8 621.5 697.1 2911.2 52 54 5104.6 .599.7 669.9 28.50.6 5193.8 622.3 698.1 2913.3 54 50 5107.0 600.4 670.9 2852.7 5196.7 623.0 099.0 2915.4 56 5S 5110.6 601.2 671 .8 2.S.^.4.S 5199.7 623.8 700.0 2917.5 58 60 5113.5 002.0 672.7 /- 28.)6.9 .520J.7 624.6 700.9 2919.5 60 282 IX. FUNCTIONS OF A ONE-DEGREE CURVE. 1 54 OO"* / L. C. M. E. T. L. C, M. E. T. 5202.7 624.6 700.9 2919.5 5291.7 647.4 7.9.9 2982.8 2 5205.7 625 4 701.9 2921.6 5294.6 648.1 730.9 2984.9 2 4 5208.6 626.1 70^.8 2923.8 5297.6 648.9 731.9 2987.1 4 6 5211.6 6-,'6.9 703.8 2925.9 5300.5 649.6 732.9 2989.2 6 8 5214.6 627.6 704.8 2928.0 5303.5 650.4 733.8 2991.3 8 10 5217.5 628.4 705.7 2930.1 5306.4 651.2 734.8 2993.4 10 VI 5220.5 629.2 706.7 2932.2 5309.4 652.0 735.8 2995.5 12 14 5223.5 629.9 707.7 2934.3 5312.3 652.7 736.8 2997.7 14 16 5226.4 630.7 708-6 2936.4 5315.3 653.5 737.8 2999.8 16 18 5229.4 631.4 709.6 2938.5 5318.2 654.3 738.7 3001.9 18 20 5232.4 632.2 710.5 2940.6 5321.2 655.1 739.7 3004.0 20 22 5235.3 633.0 711.5 2942.7 5.324.1 655.8 740.7 3006.2 22 24 5238.3 633.7 712.5 2944.8 5327.1 656.6 741.7 3008.3 24 26 5241.3 634.5 713.4 2946.9 5330.0 657.4 742.7 3010.4 26 28 5244.2 635.2 714.4 2949.0 5.3.33.0 658.2 743.7 3012.5 28 30 5247.2 636.0 715.3 2951 . 1 5.335.9 658.9 744.7 3014.7 30 32 5250.2 636.8 716.3 2953.2 5338.8 659.7 745 7 3016.8 a2 34 5253.1 6.37.5 717.3 2955.3 5341.8 660.5 746.7 3018.9 34 36 5256.1 638.3 718.2 2957.5 5344.7 661.3 747.7 3021.1 36 38 5259.1 639.0 719.2 2959.6 5347.7 662.0 748.7 3023.2 38 40 5262.0 639.8 720.2 2961.7 5350.6 662.8 749.7 3025.3 40 42 5265.0 640.6 721.1 2963.8 5353.6 663 6 7.50.7 3027.5 42 44 5268.0 641.3 722.1 2965.9 5356.5 664.4 751.7 3029.6 44 46 5270.9 642.1 723.1 2968.0 5359.5 665.1 752.6 3031.7 46 48 5273 9 642.8 724.1 2970.1 5362.4 665.9 753.6 3033.8 48 50 5276.9 643.6 725.0 2972.2 5365.4 666.7 754.6 3036.0 50 52 5279.8 644.4 726.0 2974 4 5368.3 667.5 755 . 6 3038.1 52 54 5282.8 645.1 727.0 2976.5 5371.3 668.3 756.6 3040.2 54 56 5285.8 645.9 728.0 2978.6 5374.2 669.1 7.57.6 3042.4 56 58 5288.7 646.6 729.0 2980.7 5377.2 669.9 758.6 .3044.5 58 60 5291.7 647.4 729 9 2982.8 5380.1 670.7 759.6 3046.6 60 / oG° 1 5 ;° / L. C. M. E. T. L. C. M. E. T. 5380.1 670.7 759.6 30 16. 6 5468.2 694.4 790.2 3111.1 2 5383.0 671.4 760.6 3048.8 5471.1 695.2 791.2 3113.3 2 4 5386.0 672.2 761.6 3050.9 5474.0 696.0 792.2 3115.4 4 6 5388.9 672.9 762.7 30-3.1 5477.0 696.8 793.3 3117.6 6 8 5391.8 673.7 763.7 30.55.2 5479.9 697.6 794.3 3119.7 8 10 5394.8 674.4 764.7 3057.4 548:i.8 698.4 795.3 3121.9 10 12 5397.7 675.2 765.7 3059.5 5485.7 699.2 796.3 3124.1 12 14 5400.7 676.0 766.7 3061.6 5488.7 700.0 797.4 3126.2 14 16 5403.6 676.8 767.7 3063.8 5491.6 700.8 798. 4 3128.4 16 18 5406.5 677.6 768.7 3065.9 .5494.5 701.6 799.4 3130.6 18 20 5409.5 678.4 769.7 3068.1 .5497.4 702.4 800.5 3132.7 20 22 5412.4 679.2 770.8 3070.2 .5500.3 703.2 801.5 3134.9 22 24 5415.3 680.0 771.8 3072.4 .5503.3 704.0 802.6 3137.0 24 26 5418.3 680.8 772.8 3U74.5 .5506.2 704.8 803.6 3139.2 26 28 5421.2 681.6 773.8 3076.6 5509.1 705.6 804.7 3141.4 28 30 5424.1 682.4 774.8 3078.8 5512.0 706.4 805.7 3143.5 30 32 5427.1 683.2 775.8 3080.9 5515.0 707.2 806.8 3145.7 32 34 5430.0 684.0 776.8 3083.1 5517.9 708.0 807.8 3147.9 34 36 54^3.0 684.8 777.8 3085.2 5520.8 708.8 808.8 3150.0 36 38 5435.9 6S5.6 778.9 3087.4 5523.7 709.6 809.9 3152.2 38 40 5438.8 686.4 779.9 3089.8 5.5-26.7 710.4 810.9 3154.4 40 42 5441.8 687.2 780. 9 3091.7 5.529.6 711.2 812.0 3)56.6 42 44 5444.7 688.0 781.9 3093.9 55.32.5 712.0 813.0 31.58.7 44 46 5447.6 688.8 783.0 3096.0 5535.4 712.8 814.1 3160.9 46 48 5450.6 689.6 784.0 3098.2 5538.4 713.6 815.1 3163.1 48 50 5453.5 690.4 785.0 3100.3 5541.3 714.4 816.2 3165.3 50 52 5456.5 691.2 786.0 3102.5 5544.2 715.2 817.2 3167.4 52 54 5459.4 692.0 787.1 3104.6 .5547.1 716.0 818.3 3169.6 54 56 54G2.3 692.8 788.1 3106.8 55.50.0 716.8 819.3 3171.8 56 58 54G.T.3 093.6 789.1 3108.9 .5553.0 717.6 820.4 3174.0 58 60 54()S.2 m\.\ 70:1.2 3111.1 5.5.55.9 71S.4 821.4 3176.1 60 IX.— FUNCTIONS OF A ONE DEGREE CURVE. 283 / o 8° 69° / L. C. M. E. T. L. C. M. E. T. 5555.9 718.4 821.4 3176.1 5643.1 742.8 8.53 5 3241.9 o 5558.8 719 2 822.5 3178.3 5646.0 743.6 8-A.G 3244 I 2 4 5561.7 720.0 823.5 3180.5 5648.9 744.4 855.7 3246.3 4 6 5564.6 720.8 824.6 3182.7 5651.8 745.3 856.8 3248.5 6 8 5567 . 5 721.6 825.7 3184.9 5054.7 746.1 8.57.9 3250.7 8 10 5570.4 722.4 826.7 3187.1 5657.6 746.9 859.0 3252.9 10 12 5573.3 723.2 827.8 3189.2 •5060.5 747.7 860.0 3255.1 12 14 .5576.2 724.0 828.9 3191.4 5663.4 748.6 861.1 3257.3 14 16 5579.2 724.8 829.9 3193.6 5666.3 749.4 862.2 3259.5 16 18 5582.1 725.0 831.0 3195.8 5669.2 750.2 863.3 3261.7 18 20 5585 ;0 726.5 832.1 3198.0 5672.1 751.1 864.4 3263.9 20 22 5587.9 727.3 833.1 3200,2 5675.0 751.9 865.5 3266.1 22 24 5590.8 728.1 834.2 3202.4 5677.9 752.7 866.6 3268.3 24 26 5593.7 728.9 835.3 3204.5 5680.8 753.5 867.7 3270.5 26 28 5596.6 729.7 836.3 3206.7 5683.7 7.54.4 868.8 3272.7 28 30 5599.5 730 5 837.4 3208.9 5086.5 755 2 869.9 3274.9 30 32 5602.4 731.3 838.4 3211.1 5689.4 756.0 871.0 3277.1 32 34 5605.3 732.1 839.5 3213.3 5692.3 756 9 872.1 3279.4 34 36 5608.2 732.9 840.6 3215.5 5695.2 757.7 873.2 3281 6 36 38 5611.1 733.7 841.0 3217.7 5698.1 758.5 874.3 3283.8 38 40 5614 734.6 842.7 3219.9 5701.0 759.4 875.4 3286.0 40 42 5616.9 735.4 843.8 3222.1 5703.9 760.2 876.5 3288.2 42 44 5619.8 736.2 844.9 3224.3 5706.8 761.0 877.6 3290.5 44 46 5622.8 737.0 846.0 3226.5 5709 7 761.9 878.7 3292.7 46 48 5625.7 737.8 847.0 3228.7 5712.6 762.7 879.8 3294.9 48 50 5628.6 738.6 848.1 3230.9 5715.5 763.5 880.9 3297.1 50 52 5631.5 739.4 849.2 3233.1 5718.4 764.4 882.0 3299.3 52 54 5634.4 740.2 850.3 3235.3 5721.3 765.2 883.1 3301 5 54 56 5637.3 741 851.4 3237.5 5724.2 766.0 884.2 3303.8 56 58 5640.2 741.9 852.5 3239.7 5727.1 766.8 885.3 3306.0 58 60 *■ 5643.1 742.8 853.5 3241.9 5730.0 767.7 886.4 3308.2 60 / 60° 1 61° / L. C. M, E. T. L. C. M. E. T. 5730.0 767.7 886.4 3308.2 5816.4 792.9 920.2 3375.2 2 5732.9 768.5 887.5 3310.4 5819.3 793.7 921.4 3377.4 2 4 5735.8 769.4 888.7 3312.7 5822.1 794.6 922.5 3379.7 4 6 5738.6 770.2 889.8 3314.9 5825.0 795.4 923.6 3381.9 6 8 5741.5 771.1 890.9 3317.1 5827.9 796.3 924.8 3384.2 8 10 5744.4 771.9 892.0 3319.3 5830.7 797.1 925.9 3386.4 10 12 5747.3 772.7 893.1 3321.6 5833.6 798.0 927.1 3388.7 12 14 5750.2 773.6 894.3 3323.8 5836.5 798.8 928.2 3390.9 14 16 5753.0 774.4 895.4 3326.0 5839.3 799.7 929.3 3393. S 16 18 5755.9 775.3 896.5 3328.3 5842.2 800.5 930.5 3395.4 18 20 5758.8 776.1 897.6 3330.5 5845.1 801.4 931.6 3397.7 20 22 5761.7 776.9 898.8 3332.7 5847.9 802.2 932.8 3399.9 22 24 5764.6 777.8 899.9 3334.9 5850.8 803.1 933.9 3402.2 24 26 5767.4 778.6 901.0 3337.2 5853.7 803.9 935.1 3404.4 26 28 5770.3 779.5 902.1 3339.4 5856.5 804.8 936.3 3406.7 28 30 5773.2 780.3 903.2 3341.6 5859.4 805.6 937.4 3408.9 30 32 5776.1 781.1 904.4 3343.9 5862.3 806.5 938.6 3411.2 32 34 5779.0 782.0 905.5 3346.1 5865.1 807.3 939.7 3413.5 34 36 5781.8 782.8 906.6 3348.3 5868.0 808.2 940.9 3415.7 36 38 5784.7 783.7 907.7 3350.6 5870.9 809.0 942.1 3418.0 38 40 5787.6 784.5 908.8 3352.8 5873.7 809.9 943.2 3420.3 40 42 5790.5 785.3 910.0 3355.0 5876.6 810.7 944.4 3422.5 42 44 5793.4 786.2 911.1 3357.3 5879.5 811.6 945.5 3424.8 44 46 5796.2 787.0 912.3 33.59.5 5882. 3 812.4 946.7 3427.1 46 48 5799.1 787.9 913.4 3361.8 5885.2 813.3 947.8 3429.3 48 50 5802.0 788.7 914.5 3364.0 5S8S.1 814.1 949.0 3431.6 50 52 5804.9 789.5 915.7 3366.2 .5890,9 815.0 9.50.2 3433.9 52 54 5807.8 790.4 916.8 3368.5 .5893.8 815.8 951.3 3436.1 54 56 .5810.6 791.2 918.0 .3370.7 5S!)6 . 7 816.7 9.52.5 3438.4 56 58 5813.5 792.1 919.1 3373.0 .5899.5 817.5 9.53.6 3440.7 58 60 5816.4 792.9 920.2 3375.2 5902.4 818.4 954.8 3442.9 60 N 284 IX.— FUNCTIONS OF A ONE-DEGREE CURVE. 1 6 2° 63» 1 / L. C. M. E. T. L. C. M. E. T. f .-902.4 818.4 954.8 3442.9 5987. 8 844.4 990.3 3511.3 2 5905.2 819.3 9.56.0 :3445.2 5990.6 845.3 991.5 3513.6 2 4 5908.1 820.1 957.2 3447.5 5993.5 846.2 992.7 3515.9 4 6 5910.9 821.0 958 3 :3449.7 .5998.3 847.1 993.9 3518.2 6 8 5913.8 821.8 9.59.5 :34.52.0 5999.1 847.9 995.1 3520.5 8 10 5916.6 822.7 960.7 .34.54.3 6002.0 848.8 996.3 3522.8 10 12 5919.5 823.6 961.9 .34.56 .« 6004.8 849.7 997.5 3.525.1 12 14 5922.3 824.4 963.0 3458.8 C007.7 850.6 998.7 3.527.4 14 16 5925.2 825.3 9G4.2 :3461.1 0010.5 851.4 999.9 3.529.7 16 18 5928.0 826.1 965.4 :3463.4 6013.3 852.3 1001.1 3532.0 18 20 5930.9 827.0 966.6 ^465. 7 0016.2 853.2 1002.3 35:34.3 20 22 .5933.7 827.9 967.8 31G7.9 0019.0 854.1 1003.5 35.36.6 22 24 5936.6 828.7 968.9 3470.2 6021.8 854.9 1004.7 3.538.9 24 26 .5939.4 829. G 970.1 3472.5 6024.7 855.8 1005.9 3541.2 26 28 59-12.3 830.4 971.3 3474.7 G027.5 8.56.7 1007.1 .3543.5 ' 28 30 5945.1 831.3 972.5 3477.0 0030. 3 857.6 1008.4 .3545.8 , 30 32 5947.9 832.2 973.6 3479.3 GO.33.2 858.4 1009.6 3548.1 : 32 34 .5950.8 833.0 974.8 3481.6 00.36.0 859.3 1010.8 3550.4 1 34 36 5953.6 833.9 976.0 ;3483.9 60:38.9 860.2 1012.0 a552.7 36 38 5956.5 834.7 977.2 3486.2 0041.7 861.1 1013.2 3555.0 38 40 .5959.3 8:35.6 978.4 3488.5 G044.5 861.9 1014.5 3557.3 40 42 5962.2 83G.5 979.6 3490.7 0047.4 862-8 1015.7 35.59.6 42 44 5965.0 837.4 980.8 ;3493.0 6050.2 863.7 1016.9 3562.0 44 46 5967.9 838.3 982.0 :3495.3 0053.0 864.6 1018.1 ;3564.3 46 48 5970.7 8:39.1 983.2 3497.6 00.55.9 865.4 1019.3 3.566.6 48 50 5973.6 840.0 984.4 3499.9 60.5^^.7 866.3 1020.6 3568.9 50 52 5976.4 840.9 985.5 :3502.2 00G1.6 867.2 1021.8 3571.2 52 54 .5979.3 841.7 9S6.7 3.504.5 6004.4 868.1 1023.0 .3573.5 54 56 5982.1 842.6 9s7.9 ;3506.S 0067.2 868.9 1024.2 :3575.8 56 58 5985.0 843.5 989.1 3.509.0 0070.1 869.8 1025.4 3578.1 58 60 5987.8 844.4 990.3 3511.3 0072.9 870.7 1026.7 3580.4 60 / 64°, 1 t .5° / L. C. M. E. T. L. C. M. E. T. 6072.9 870.7 1026.7 3.580.4 6157.5 897.3 1064.0 36.50.4 2 6075.7 871.5 1027.9 ;35S2.S 0160.3 898.2 1065.2 36.52.8 2 4 6078.5 872.4 1029.2 3585.1 6103.1 899.1 1066.5 .3655.1 4 6 6081.4 873.3 10:30.4 3587.4 6165.9 900.0 1067.7 36.57.5 6 8 6084.2 874.2 1031.7 3589.7 0168.7 900.9 1069.0 3659.8 8 10 6087.0 875.1 10:32.9 3.592.1 6171.5 901.8 1070.2 3662.2 10 12 6039.8 875.9 1034.1 :3594.4 6174.3 902.7 1071.5 3664.5 12 14 6092.6 876.8 10:35.4 3596.7 6177.1 903.6 1072.7 3666.9 14 16 6095.5 Oi 1 . t 10:36.6 :3599.1 6179.9 904.5 1074.0 3669.2 16 18 6098.3 878.6 1037.9 3601.4 6182.7 905.4 1075.2 3671. G 18 20 6101.1 879.5 1039.1 3603.7 6185.5 906.3 1076.6 3673.9 20 22 6103.9 880.3 1040.3 3606.0 6188.3 907.2 1077.8 3676.2 22 24 6106.7 881.2 1041.6 3608. 4 6191.1 908.1 1079.1 :3678.6 24 26 6109.6 882.1 1042.8 3610.7 6193.9 909.0 1080.4 3680.9 26 28 6112.4 883.0 1044.1 3613.0 6196.7 909.9 1081.7 3683.3 28 30 6115.2 88:3.9 1045.3 3615.3 6199.5 910.8 1083.0 3685.6 30 32 6118.0 884.7 1046.5 :3617.7 6202.3 911.7 1084.2 3688.0 32 34 6120.8 885.6 1047.8 3620.0 6205.1 912.6 1085.5 3690.4 U 86 6123.7 886.5 1049.0 3622.3 6208.0 913.5 1086.8 3692.7 36 38 6126.5 887.4 10.50.3 3624.7 6210.8 914.4 1088.1 :3695.1 38 40 6129.3 888.3 1051.5 :3627.0 6213.6 915.3 1089.4 .3697.4 40 42 6132.1 889.2 10.52.7 .3G29.4 6216.4 916.2 1090.0 36&9.8 42 44 6i:i4.9 890.1 10,54.0 :3631.7 6219.2 917.1 1091.9 .3702.2 44 46 6137.8 891.0 1055.2 :36:34.0 6222.0 918.0 1093.2 .3704.5 46 48 6140.6 891.9 10.56.5 36:3G.4 6224.8 918.9 1094.5 3706.9 48 50 6143.4 892.8 10.57.7 .3638.7 6-227.6 919.8 1095.8 3709.3 50 52 6146.2 893.7 10.59.0 3641.1 6230.4 920.7 1097.0 •3711.6 52 54 6149.0 894.6 1060.2 364:^.4 62:33.2 921.6 109P.3 3714.0 54 56 6151.9 895.5 1061.5 3645.7 62.36.0 922.5 1099.6 3716.3 56 58 6154.7 896.4 10G2.7 3648.1 02.38.8 923.4 1100.9 :37]S.7 58 60 6157.5 897.. S 10G4.0 36.50.4 6241.6 924.3 1102.2 .3721.1 1 60 IX.— FUNCTIONS OF A ONE-DEGREE CURVE. 285 / 66 o 67 o / L. C. M. E. T. L. C, M. E. T. 6241.6 924.3 1102.2 3721.1 6325.2 951.8 1141.5 3792.6 2 6244.4 925.2 1103.5 3723.4 6328.0 952.7 1142.8 3795.0 2 4 6247.2 9-26.1 1104.8 37-^5.8 6330.7 953.6 1144.1 3797.4 4 6 6250.0 927.0 1106.1 3728.2 6333.5 954.5 1145.4 3799.8 6 8 6252.7 927.9 1107.4 3730.6 6330.3 955.5 1146.7 3802.2 8 10 6255.5 928.8 1108.7 3732.9 6339.0 950.4 1148.1 3804.6 10 12 6258.3 929.8 1110. 3735.3 0341.8 957.3 1149.4 3807.0 12 14 6261.1 930.7 1111.3 3737.7 6344.6 958.2 1150.7 3809.4 14 16 6263.9 931 6 1112.6 3740.1 6347.4 059.2 1152.0 3811.8 16 18 6260. 7 932.5 1113.9 3742.4 6350.1 900.1 1153.3 3814.2 18 20 6269.5 '933.4 1115.2 3744.8 6352.9 901.0 1154.7 3816.6 20 22 6272.3 934.3 1116.5 3747.2 6355.7 961.9 1156.0 3819.0 22 24 6275 935.3 1117.8 3749.0 0358.4 962.9 1157.4 3821.4 24 26 6277.8 936.2 1119.1 3751.9 6361.2 963.8 1158.7 3823.8 26 28 62S0.6 937.1 1120.4 3754.3 6364.0 904.7 1160.1 3826.2 28 30 6283.4 938.0 1121.7 3756.7 6306.7 905.6 1161.4 3828.6 30 32 6286.2 938.9 1123.0 3759.1 6309.5 960.6 1162.8 3831 .0 32 34 6289.0 939.8 1124.3 3701.5 0372.3 907.5 1164.1 3833.4 34 36 6291.8 940.8 1125.6 3703.9 6375.1 968.4 1165.5 3835.9 36 38 6294.5 941.7 1126.9 3766.3 6377.8 909.3 1166.8 3838.3 38 40 6297.3 942 6 1128.3 3708.7 6380.6 970.3 1168.2 3840.7 40 42 6300.1 943.5 1129.6 3771.0 6383.4 971.2 1169.5 3843.1 42 44 6302.9 944.4 1130.9 3773.4 6380.1 972.1 1170.9 3845.5 44 46 6305.7 915.3 1132.2 3775.8 6388.9 973.0 1172.2 3847.9 46 48 6308.5 946.3 1133.5 3778.2 6391.7 974.0 1173.6 3850.4 48 50 6311.3 947.2 1134.9 3780.6 6394.4 974.9 1174.9 3852.8 50 52 6314.1 948.1 1136.2 3783.0 6397.2 975.8 1176.3 3855.2 52 54 6316.8 949.0 1137.5 3785.4 6400.0 976.8 1177.6 3857.6 54 56 6319.6 949.9 1138.8 3787.8 6402.8 977.7 1179.0 3860.0 56 58 6322.4 950.8 1140.1 3790.2 6405.5 978.6 1180.3 3862.5 58 60 6325.2 951.8 1141.5 3792.6 6408.3 979.6 1181.6 3864.9 60 / 1 6S »° 6S •° / L. C. M. E. T. L. C. M. E. T. 6108.3 979.6 1181.6 3864.9 6491.1 1007.7 1222.9 3938.1 2 6411.1 980.5 1183.0 3867.3 6493.8 1008.7 1224.3 3940.6 2 4 6413.8 981.4 1184.4 3869.7 6496.6 1009.6 1225.7 3943.0 4 6 G416.6 982.4 1185.7 3872.2 6499.3 1010.6 1227.1 3945.5 6 8 6419.3 983.3 1187.1 3874.6 6502.1 1011.5 1228.5 3947.9 8 10 6422.1 984.2 1188.5 3877.0 6504.8 1012.5 1229.9 3950.4 10 12 642 1.9 985.2 1189.8 3879.5 6.507.5 1013.4 1231.3 3952.9 12 14 6427. 6 986.1 1191.2 3881.9 6510.3 1014.4 1232.7 3955.3 14 16 6430.4 987.0 1192.6 3884.3 6513.0 1015.3 1234.1 3957.8 16 18 6433.1 988.0 1193.9 3886.8 6515.8 1016.3 1235.5 3960.2 18 20 64a5.9 988.9 1195.3 3889.2 6518.5 1017.2 1236.9 3962.7 20 22 6438.7 989.8 1196.7 3891.6 6521.2 1018.2 1238.3 3965.2 22 24 6441.4 990.8 1198.0 3894.1 6524.0 1019.1 1239.7 3967.6 24 26 6444.2 991.7 1199.4 3896.5 6520.7 1020.1 1241.1 3970.1 26 28 6446.9 992.6 1200.8 3898.9 0529.5 1021.0 1242.5 3972.5 28 30 6449.7 993.6 1202.1 3901.4 6532.2 1022.0 1243.9 3975.0 30 32 6452.5 994.5 1203.5 3903.8 6534.9 10J2.9 1245.3 3977.5 32 34 6455.2 995.4 1204.9 3900.3 6537.7 1023.9 1246.7 3980.0 34 36 6458.0 996.4 1200. 2 3908.7 6.540.4 1024.8 1248.1 3982.4 36 38 6460.7 997.3 1207.6 3911.2 6543.2 1025.8 1249.5 3984.9 38 40 6403.5 998.2 1209.0 3913.6 6545.9 1020.7 12.50.9 3987.4 40 42 6466.3 999.2 1210.3 3916.1 6548.6 1027.7 1252.3 3989.9 42 44 6469.0 1000.1 1211.7 3918.5 0551.4 1028.6 12.53.7 3992.3 44 46 6471.8 1001.0 1213.1 3921.0 6554.1 1029.6 1255.1 3994.8 46 48 6474.5 1002.0 1214.5 3923.4 6556.9 1030.5 1256.5 3997.3 48 50 6477.3 1002.9 1215.9 3925.9 6559.6 1031.5 12.57.9 3999.8 50 52 6480.1 1003.8 1217.3 3928.3 6562.3 1032.4 1259.3 4002.2 52 54 6482.8 1004.8 1218.7 39.30.8 0.505.1 1033.4 1260.7 4004.7 54 56 6485.6 1005.7 1220.1 39.33.2 6507.8 1034.3 1262.1 4007.2 56 58 6488.3 1006.7 1221.5 3935.7 0.570.6 1035 3 1263.5 4009.7 58 1 60 6491.1 1007.7 1222.9 3938.1 6573.3 1036.3 1265.0 4012.1 60 286 IX.— FUNCTIONS OF A ONE-DEGREE CURVE. 70° 71 [° L. C. 6573.3 M. 1036.3 E. 1265.0 T. 4012.1 L. C. M. E. T. / 6654.9 1065.1 1308.4 4087.1 2 6576.0 1037.3 1266.4 4014.6 6657.6 1066.1 1309.9 4089.7 2 4 6578.7 1038.2 1267.9 4017.1 6660.3 1067.0 1311.3 4092.2 4 6 6581.5 1039.2 1269.3 4019.6 6663.0 1068.0 1312.8 4094.7 6 8 6584.2 1040.1 1270.8 4022.1 6665.7 1068.9 1314.2 4097.2 8 10 6586.9 1041.1 1272.2 4024.6 6668.4 1069.9 1315.7 4099.8 10 \2 6589.6 1042.1 1273.6 4027.1 6671.1 1070.9 1317.2 4102.3 12 14 6.592.3 1043.0 1275.1 4029.6 6673.8 1071.9 1318.6 4104.8 14 16 6595.1 1044.0 1276.5 4032.1 6676.6 1072.9 1320.1 4107.3 16 18 6597.8 1044.9 1278.0 4034.6 6679.3 1073.8 1321.5 4109.8 18 20 6600.5 1045.9 1279.4 4037.1 6682.0 1074.8 1323.0 4112.4 20 22 6603.2 1046.9 1280.8 4039.6 6684.7 1075.8 1324.4 4114.9 22 24 6605.9 1047.8 1282.3 4042.1 6687.4 1076.8 1325.9 4117.4 24 26 6608.7 1048.8 1283.7 4044.6 6690.1 1077.7 1327.4 4119.9 26 28 6611.4 1049.7 1285.2 4047.1 6692.8 1078.7 1.328.9 4122.4 28 30 6614.1 1050.7 1286.6 4049.6 6695.5 1079.7 1330.4 4125.0 30 32 6616.8 1051.7 1288.0 40.52.1 6698.2 1080.7 1331.8 4127.5 32 34 6619.5 1052.6 1289.5 4054.6 6700.9 1081.6 1333.3 4130.4 34 36 6622.3 1053.6 1290.9 4057.1 6703.6 1082.6 1.334.8 4132.6 36 38 6625.0 1054.5 1292.4 4059.6 6706.3 1083.6 1336.3 4135.1 38 40 6627.7 1055.5 1293.8 4062.1 6709.0 1084.5 1337.8 4137.7 40 42 6630.4 1056.5 1295.3 4064.6 6711.7 1085.5 1339.2 4140.2 42 44 6633.1 1057.4 1296.7 4067.1 6714.4 1086.5 1340.7 4142.7 44 46 6635.9 1058 4 1298.2 4069.6 6717.2 1087.5 1342.2 4145.3 46 48 6638.6 1059.3 1299.6 4072.1 6719.9 1088.4 1343.7 4147.8 48 50 6641.3 1060.3 1301.1 4074.6 6722.6 1089.4 1345.2 4150.4 50 52 6644. C 1061.3 1302.6 4077.1 6725.3 1090.4 1.346.7 4152.9 52 54 6646.7 1062.2 1304.0 4079.6 6728.0 1091.3 1348.2 41.55.4 54 56 6649.5 1063.2 1305.5 4082.1 6730.7 1092.3 1349.7 4158.0 56 58 66.52.2 1064.1 1306.9 4084.6 6733.4 1093.3 1351.2 4160.5 58 60 L . 6654.9 1065.1 1308.4 4087.1 6736.1 1094.3 1352.7 4163.1 60 / 72 o 73 ," L. C. M. E. T. L. C. M. E. T. 6736.1 1094 3 13.52.7 4163.1 6816.6 1123.9 1398.1 4240.0 2 6738.8 1095 2 1354.2 4165.6 6819.3 1124.8 1399.6 4242.6 2 4 6741.5 1096.2 1355.7 4168.2 6821.9 1125 8 1401.2 4245.1 4 6 6744.1 1097.2 1357.2 4170.7 6824.6 1126.8 1402.7 4247.7 6 8 6746.8 1098.2 1358.7 4173.3 6827.3 1127.8 1404.2 4250.3 8 10 6749.5 1099.2 1360.2 4175.8 6830.0 1128.8 1405.8 4252.9 10 12 6752.2 1100.1 1361.7 4178.4 6832.6 1129.8 1407.3 4255.5 12 14 6754.9 1101.1 1363.2 4181.0 68.35.3 1130.8 1408.8 4258.1 14 16 6757 6 1102.1 1.364.7 4183.5 6838.0 1131.8 1410.4 4260.7 16 18 6760.2 1103.1 1366.2 4186.1 6840.7 1132.8 1411.9 4263.2 18 20 6762.9 1104.1 1367.7 4188.6 6843.3 1133.8 1413.5 4265.8 20 22 6765.6 1105.1 1369.2 4191.2 6846.0 1134 8 1415.1 4268.4 22 24 6768.3 1106.0 1370.7 4193.7 6848.7 11.35.8 1416.6 4271.0 24 26 6771.0 1107.0 1372.2 4196.3 6851.3 1136.8 1418.2 4273.6 26 28 6773.7 1108.0 1373.7 4198.8 6854. 1137.8 1419.7 4276.2 28 30 6776.3 11U9.0 1.375.2 4201.4 68.56.7 1138.8 1421.3 4278.8 30 32 6779.0 1109.9 1376.7 4204.0 6859.4 1139.8 1422.9 4281.4 32 34 6781.7 1110.9 1378.2 4206.5 6862.0 1140.8 1424.4 4284.0 34 36 6784.4 1111.9 1379.7 4209.1 6864.7 1141.8 1426.0 4286.6 36 38 6787.1 1112.9 1381.2 4211.7 6867.4 1142.8 1427.5 4289.2 38 40 6789.8 1113.9 1382.8 4214.3 6870.1 1143.8 1429.1 4291.8 40 42 6792.4 1114.9 1384.3 4216.8 6872.7 1144.8 1430.7 4294.4 42 44 6795.1 1115.9 13S5.8 4219.4 6875.4 1145.8 1432.2 4297.0 44 46 6797.8 1116.9 1387.4 4222.0 6878.1 1146.8 1433.8 4299.6 46 48 6800.5 1117.9 1388.9 4224.5 6880.8 1147.8 1435.3 4302.2 48 50 6803.2 1118.9 1390.4 4227.1 6883.4 1148 8 1436.9 4304.8 50 52 6805.9 1119.9 1.392.0 4229.7 6886.1 1149.8 1438.5 4307.4 52 54 6808.5 1120.9 1393.5 42.32.3 6888.8 1150.8 1440.0 4310.0 54 56 6811.2 1121.9 1395.0 4234.8 6891.4 1151.8 1441.6 4312.6 56 58 6813.9 1122.9 1396.6 42.37.4 6894.1 1152.8 1443.1 4315.2 58 60 6816.6 1123.9 1398.1 4240.0 6896.8 1153.8 1444.7 4317.8 60 IX.— fu:nctions op a one-degree curve. 287 2 4 6 8 10 1^> 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54 56 58 60 '4° L. C. M. E. T. 6896.8 6899.4 6902.1 6904.8 6907.4 6910.1 6912.7 6915.4 6918.0 6920.7 6923.3 6926.0 6928.6 6931.3 6933.9 6936.6 6939.2 6941.9 6944.6 6947.2 6949.9 6952.5 6955.2 6957.8 6960.5 6963.1 6965.8 6968.4 6971.1. 6973.7 6976.4 1153.8 1154.8 1155.8 1156.8 1157.8 1158.8 1159.8 1160.8 1161.8 1162.8 1163.9 1164.-9 1165.9 1166.9 1167.9 1168.9 1169.9 1170.9 1171.9 1172.9 1174.0 1175.0 1176.0 1177.0 1178.0 1179.0 1180.0 1181.0 1182.0 1183.0 1184.1 1444.7 1446.2 1447.8 1449.4 1451.0 1452.6 1454.1 1455.7 1457.3 1458.9 1460.5 1462.0 1463.6 1465.2 1466.8 1468.4 1469.9 1471.5 1473.1 1474.7 1476.4 1478.0 1479.6 1481.2 1482.8 1484.4 1486.0 1487.7 1489.3 1490.9 1492.5 4317.8 43-'0.5 4323.1 4325.7 4328.3 4330.9 4333.6 4336.2 4338.8 4341.4 4344.0 4346.7 4349.3 4351.9 4354.5 4357.1 4359.8 4362.4 4365.1 4367.7 4370.3 4373.0 4375.6 4378.3 4380.9 4383.5 4386.2 4388.8 4391.5 4394.1 4396.7 to" L. C. M. E. T. 6976.4 6979.0 6981.7 6984.3 6986.9 6989, 6992. 6994, 6997, 7000.1 7002.8 7005.4 7008.0 7010.7 7013.3 7015.9 7018.6 7021.2 7023.9 7026.5 7029.1 7031.8 7034.4 7037.0 7039.7 7042.3 7045.0 7047.6 7050.2 7052.9 7055.5 1184.1 1185.1 1186.1 1187.1 1188.1 1189.2 1190.2 1191.2 1192.2 1193.2 1194.3 1195.3 1196.3 1197.3 1198.3 1199.4 1200.4 1201.4 1202.4 1203.4 1204.5 1205.5 1206.5 1207.5 1208.5 1209.6 1210.6 1211.6 1212.6 1213.6 1214.7 1492.5 1494.1 1495.7 1497.3 1499.0 1500.6 1.502.2 1503.8 1505.4 1507.0 1508.7 1510.3 1512.0 1513.6 1515.3 1516.9 1518.5 1520.2 1521.8 1523.5 1525.1 1.526.7 1528.4 1530.0 1531.7 1533.3 1534.9 1.536.6 1.538.2 1539.9 1541.5 4396.7 4399.4 4402.1 4404.7 4407.4 4410.0 4412.7 4415.3 4418.0 4420.7 4423.3 4426.0 4428.6 4431.3 4434.0 4436.6 4439.3 4442.0 4444.6 4447.3 4450.0 4452.7 4455.3 4458.0 4460.7 4463.4 4466.0 4468.7 4471.4 4474.1 4476.7 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54 56 58 60 7 6" 7 70 L. C. M. E. T. L. C. M E. T. 7055.5 1214.7 1541.5 4476.7 7134.0 1245.6 1.591.7 4557.8 2 7058.1 1215.7 1543.2 4479.4 7136.6 1246.6 1593.4 4560.5 2 4 7060.7 1216.7 1544.9 4482.1 7139.2 1247.7 1595.1 4563.3 4 6 7063.3 1217.8 1546 5 4484.8 7141.8 1248.7 1596.8 4566.0 6 8 7066.0 1218.8 1548.2 4487.5 7144.4 1249.8 1598.5 4568.7 8 10 7068.6 1219.8 1549.9 4490.2 7147.0 12.50.8 1600.2 4571.5 10 12 7071.2 1220.9 1551.5 4492.9 7149.6 1251.8 1601.9 4574.2 12 14 7073.8 1221.9 1553.2 4495.6 7152.2 1252.9 1603.6 4576.9 14 16 7076.4 1222.9 1554.9 4498.3 7154.8 1253 9 1605.3 4579.7 16 18 7079.0 1224.0 1556.5 4.501.0 7157.4 1255 1607.0 4582.4 18 20 7081.7 1225.0 1.558.2 4.503.7 7160.0 12.56.0 1608.7 4585.1 20 22 7084.3 1226.0 1.5.59.9 4.506. S 7162.6 12.57.0 1610.4 4587.9 22 24 7086.9 1227.1 1561.5 4509.0 7165.2 12.58.1 1612.1 4^0.6 24 26 7089.5 1228.1 1.563.2 4511.7 7167.8 1259.1 1613.8 4593.3 26 28 7092.1 1229.1 1564 9 4514.4 7170.4 1260.2 1615.5 4596.0 28 30 7094.7 1230.2 1566.5 4517.1 7173.0 1261.2 1617.3 4598 8 30 32 7097.4 1231.2 1568.2 4519.8 7175.6 1262.2 1619.0 4601.5 32 34 7100.0 1232.2 1.569.9 4522.5 7178 2 1263.3 1620.7 4604.3 34 36 7102.6 1233.3 1571.5 4525.3 7180.8 1264.3 1622.4 4607.0 36 38 7105.2 1234.8 1573.2 4528.0 7183.4 1265.4 1624.1 4609^ 38 40 7107.8 1235.3 1.574.8 4530.7 7186.0 1266.4 1625.9 4612.5 40 42 7110.4 1236.4 1576.4 4.533.4 7188.6 1267.4 1627.6 4615.3 42 44 7113.1 1237.4 1578.1 45,36,1 7191.2 1268.5 1629.3 4618.0 44 46 7115.7 1238 4 1579.8 4.538.8 7193.8 1269.5 1631.0 4620.8 46 48 7118.3 1239 5 1.581 .5 4.541.5 7196.4 1270.6 1632.7 4623.5 48 50 7120.9 1240 5 1.583.2 4544.2 7199.0 1271.6 1634 5 4626.3 50 52 7123.5 1241.5 1584.9 4.547.0 7201.6 1272.7 1636.2 4629.0 52 54 7126 1 1242.6 1586.6 4.549.7 7204 2 1273.7 1637.9 4631.8 54 56 7128.8 1243.6 1588.3 4,5.52.4 7206.8 1274.8 1639.6 4634.5 56 58 7131.4 1244.6 1590 4.5.55.1 7209.4 1275.8 1641.3 4637.3 58 60 7134 1215.6 1591.7 4.557.8 7212.0 1276.9 1643.1 4640.0 60 , 288 IX.— FUNCTIONS OF A ONE-DEGREE CURVE. r 7! ^o 7! [)° / L. C. M. E. T. L. C. M. E. T. 7212.0 1276.9 1643.1 4C40.0 7289.5 1.30S.5 ;696.0 4723.4 7214.6 1278.0 1644.8 4642.8 7292.1 1309.5 1097.7 47 26.2 4 7217.2 1279.0 1646.6 4645.6 7294.6 1310.6 1699.5 4729.0 4 6 7219.7 1280.1 1648.3 4648.3 7297.2 1311.7 1701.3 4731.8 6 8 7222.3 1281.1 1650.1 4651.1 7299.7 1312.7 1703.1 47:34.7 8 10 7224.9 1282.2 1651.8 46.53.9 7302.3 1313.8 1704.9 47.37.5 10 1-2 7227.5 1283.2 1653.6 4656.7 7304.9 1314.9 1706.6 4740.3 12 14 7230.1 1284.3 1655.3 4659.4 7307.4 1315.9 1708.4 4713.1 14 16 7232.7 1285.3 1657.1 4662.2 7310.0 1317.0 1710.2 4745.9 16 18 7235.2 1286.4 1658.8 4665.0 7312.6 1318.1 1712.0 4748.7 18 20 7237.8 1287.4 1660.6 4667.7 7315.1 1.319.1 1713.8 4751.5 20 22 7240.4 1288.5 1662.3 4670.5 7317.7 1320.2 1715.6 4751.3 22 24 7243.0 1289.5 1664.1 4673.3 7320.3 1.321.3 1717.4 4757.1 24 26 7245.6 1290.6 1665.8 4676.0 7322.8 1322.3 1719.2 4760.0 26 28 7248.2 1291.6 1667.6 4678.8 7325.4 1323.4 1721.0 4762.8 28 30 7250.7 1292.7 1669.3 4681.6 7327.9 1324.5 1722.8 4765.6 30 32 7253.3 1293.7 1671.1 4684.4 7330.5 1325.5 1724.6 4768.4 32 34 7255.9 1294.8 1672.8 4687.2 7.3.33.1 1.3-26.6 1726. 4771.2 34 36 "258.5 1295.8 1674.6 4689.9 7.335.6 1327.7 1728.2 4774.1 36 38 7261.1 1296.9 1676.3 4692.7 7338.2 1328.7 17:30.0 4776.9 38 40 7263.7 1297.9 1678.2 4695.5 7.340.8 1.329.8 17:31.9 4779.7 40 42 7266.2 1299.0 1679.9 4698.3 7343.3 1330.8 1733.7 4782.6 42 44 7268.8 1300.0 1681.7 4701.1 7345.9 1331.9 17:35.5 4785.4 44 46 7271.4 1301.1 1683.5 4703.9 7348.4 13:3:3.0 1737.3 4788.2 46 48 7274.0 1302.1 1685.3 4706.7 7351.0 1334.1 1739.1 4791.0 48 50 7276.6 1303.2 1087.1 4709.5 7353.6 1:3:35.2 1740.9 4793.9 50 52 7279.2 1304.2 1688.8 4712.2 7356.1 13:36.2 1742.7 4796.7 52 54 7281.7 1305.3 1690.6 4715.0 7358.7 1337.3 1744.5 4799.5 54 56 7284.3 1306.3 1692.4 4717.8 7361.3 13:38.4 1746.3 .4802.4 56 58 7286.9 1307.4 1694.2 4720.6 7363.8 1339.5 1748.1 4805.2 58 60 7289.5 1308.5 1696.0 4723.4 7366.4 1310.6 17.50.0 4808.0 60 / 80° 1 81° / L. C. M. E. T. L. C. M. E. T. 7366.4 1340.6 1750 4808.0 7442.7 1372.8 1805.5 4893.9 2 7368 9 1341.7 1751.8 4810.9 7445 2 1373.9 1807.3 4896.8 2 4 7371.5 1342.7 1753.7 4813.7 7447.7 1375.0 1809.2 4899.7 4 6 7374.0 1343.8 1755.5 4816.6 7450.3 1:376.1 1811.1 4902.6 6 8 7376.6 1344.9 1757.4 4819.4 7452.8 1.377.1 1813.0 4905.4 8 10 7379.1 1346.0 1759.2 4822.3 7455.3 1:378.2 1814.9 4908.3 10 12 7381.7 1347.0 1761.0 4825.1 74.57.8 1:379.3 1816.8 4911.2 12 14 7:384.2 1348.1 1762.9 4828.0 7460.4 1:380.4 1818.6 4914.1 14 16 7386.7 1349.2 1764.7 4830.8 7462.9 1381.4 1820.5 4917.0 16 18 7389.3 1350.3 1766.6 4833.7 7465.4 1:382.5 1822.4 4919.9 18 20 7391.8 1351.3 1768.4 4836.5 7467.9 1:383.6 1824.2 4922.8 20 22 7394.4 1352.4 1770.2 4839.4 7470.4 1:384.7 1826.1 4925.7 24 7396.9 1353.5 1772.1 4842.2 7473.0 1.385.7 1828.0 4928.6 24 26 7399.5 13.54.6 1773.9 4845.1 7475.5 1:386.8 1829.9 4931.5 26 28 7402.0 1.355.6 1775.8 4847.9 7478.0 i;387.9 1831.8 49:34.4 28 30 7404.5 1356.7 1777.6 4850.8 7480.5 1:389.0 18:33.7 49:37.2 30 32 7407.1 1357.8 1779.4 48.53.7 7483.1 1390.1 18:35.6 4940.2 32 34 7409.6 1358.9 1781.3 48.56.5 7485.6 1391.2 18:37.5 4943.1 :34 36 7412.2 1359.9 1783.1 4859.4 7488.1 1:392.3 18:39.4 4946.0 36 38 7414.7 1361.0 1785.0 4862.3 7490.6 1393.4 1841.3 4948.9 38 40 7417.3 1362.1 1786.8 4865.1 7493.2 1394.5 1843.2 4951.8 40 42 7419 8 1363.2 1788.6 4868.0 7495.7 1:395.6 1845.1 4954.7 42 44 7422.3 1364.2 1790.5 4870.9 7498.2 1396.7 1847.0 49.57.6 44 46 7424.9 1365.3 1792.4 4873.8 7500.7 1397.8 1848.9 4960.6 46 48 7427.4 1366.4 1794 3 4876.6 7503.3 1:398.9 18.50.8 4963.5 48 50 7430.0 1367.5 1796.2 4879.5 7505.8 1400.0 1852 7 4966.4 50 52 7432.5 1368.5 1798.0 4882.4 7.508.3 1401.1 1854.6 4969.3 52 54 7435.1 1369.6 1799.9 4885 3 7510.8 1402.2 1856.5 4972.2 54 56 7437.6 1370 7 1801.8 4888. 1 7513.3 1403.3 1858.4 *4975.1 56 58 7440.1 1371. P 1803.7 4891.0 7515.9 1404 4 1860.3 4978.0 58 60 1 7442.7 1372.8 1805.5 4893.9 7518.4 1405.5 1862 3 4981.0 60 IX.— FUNCTIONS OF A ONE-DEGREE CURVE. 289 / 82° 83° / L. C. 7518.4 1405.5- E. 1862.3 T 4981.0 L. C. M. E. T. 7593.6 1438.5 1920.6 5069.4 2 7520.9 1406.6 1864.2 4983.9 7596.1 1439.6 1922.6 5072.4 2 4 T523.4 1407.7 1866.1 4986 . 8 7598.6 1440.7 1924.6 5075.4 4 6 7525.9 1408 8 1^8.1 49S9.8 7601.1 1441.8 1926.5 5078.4 6 8 7528.4 1409.9 1870.0 4992.7 7603.6 1442.9 1928.5 5081.4 8 10 7530.9 1411.0 1871.9 4995.7 7606.0 1444.0 1930,5 5084.4 10 12 7533.4 1412.1 1873 9 4998.6 7608.5 1445.1 1932.4 5087.3 12 14 7535.9 1413.2 1875.8 5001.5 7611.0 1446.2 1934.4 5090.3 14 16 7538.5 1414.3 1877.7 5004.5 7613.5 1447.3 1936.4 5093,3 16 18 7541.0 1415.4 1879.7 5007.4 7616.0 1448.4 1938.4 5096.3 18 20 7543.5 1416.5 1881.6 5010.3 7618.5 1449.6 1940.4 5099.3 20 22 7546.0 1417.6 1883.5 5013.3 7621.0 1450.7 1942.4 5102.3 22 24 7548.5 1418.7 1885.5 5016.2 7623.5 1451.8 1944.4 5105.2 24 26 7551.0 1419.8 1887.4 5019.2 7626.0 1452.9 1946.4 5108.2 26 28 7553.5 1420.9 1889.3 5022.1 7628.5 1454.0 1948.4 5111.2 28 30 7556.0 1422.0 1891. 3 5025.0 7030.9 1455.1 1950.4 5114 2 30 32 7558.5 1423.1 1893.2 5028.0 7633.4 1456.2 1952.4 5117.2 32 34 7561.0 1424.2 1895.1 5031 .0 7G35.9 1457.3 1954.4 5120.2 34 36 7563.5 1425.3 1897.1 5033.9 7038.4 1458.4 1956.4 5123.2 36 38 7566.0 1426.4 1899.0 5036.9 7640.9 1459.5 1958.4 5126.2 38 40 7568.5 1427.5 1901.0 5039.8 7643.4 1460.7 1960.4 5129.2 40 42 7571.0 1428.6 1902.9 5042.8 7645.9 1461.8 1962.4 5132.2 42 44 7573.5 1429.7 1904.9 5045.8 7648.4 1462.9 1964.4 5135.2 44 46 7576.1 1430.8 1906.9 5048.7 7650.9 1464.0 1966.4 5138.2 46 48 7578.6 1431.9 1908.8 5051.7 7653.4 1465.1 1968.4 5141.2 48 50 7581.1 1433.0 1910.8 .50.54.6 7655.8 1466.2 1970.4 5144.3 50 52 7583.6 1434.1 1912.8 5057.6 7658.3 1467.3 1972.4 5147.3 52 51 7586.1 1435.2 1914.7 .5060.6 7660.8 1468.4 1974.4 5150.3 54 56 7588.6 1436.3 1916.7 .5063.5 7663.3 1469.5 1976.4 5153.3 56 58 7591 . 1 1437.4 1918.7 5066.5 7665.8 1470.6 1978.4 5156.3 58 60 7593.6 1438.5 19:>0.6 .5069.4 7068.3 1471.8 1980.5 5159.3 60 / 84° 1 85° / L. C. M. E. T. L. C. M. E. 1 7668.3 1471.8 1980.5 5159.3 7742.4 1505.4 2041.8 5250.6 2 7670.8 1472.9 1982.5 5162.3 7744.8 1506.5 2043.9 5253.6 '2 4 7673.2 1474.0 1984.5 5165.3 7747.3 1507.6 2046.0 5256.7 4 6 7675.7 1475.1 1986.6 5168.4 7749.7 1508.8 2048.0 5259.8 6 8 7678.2 1476.2 1988.6 5171.4 7752.2 1509.9 2050.1 5262.9 8 10 7680.6 1477.4 1990.6 5174.4 7754.6 1511.0 2052.2 5266 10 12 7683.1 1478.5 1992.7 5177.5 77.57.1 1512.2 2054.2 5269.0 12 14 7685 6 1479.6 1994.7 5180.5 7759 5 1513.3 20.56.3 5272.1 14 16 7688.1 1480.7 1990.7 5183.5 7762.0 1514.4 2058.4 5275.2 16 18 7690.5 1481.8 1998.8 5186.6 7764 4 1515.6 2060.5 5278.3 18 20 7693.0 1483.0 2000.8 5189.6 7766.9 1.516.7 2062.6 5281.4 20 22 7695.5 1484 1 2002.8 5192.6 7769.3 1517.8 2064.7 5284.4 22 24 7697.9 148.T.2 2004.9 5195.6 7771.8 1519.0 2066.8 5287.5 24 26 7700.4 1486.3 2006.9 5198.7 7774.2 1520.1 2068.9 5290.6 26 28 7702.9 1487 4 2008.9 5201.7 4 1 iV. 1 1521.2 2071.0 5293.7 28 30 7705.3 1488 6 2011.0 5204.7 7779.1 1522.4 2073.1 5296.7 30 32 7707.8 1489.7 2013.0 5207.8 7781.5 1523.5 2075.2 5299.8 32 34 7710.3 1490.8 2015.0 5210 8 7784.0 1524.6 2077 3 5302.9 34 36 7712.8 1491.9 2017.0 5213.9 7786.4 1525.8 2079.4 5306.1 36 38 7715.2 1493.0 2019.1 5216.9 7788.9 1526.9 2081.5 5309.2 38 40 7717.7 1494.2 2021.2 5220.0 7791.3 1528.0 2083.7 .5312.3 40 42 7720.2 1495.3 2023.2 .5223.1 7793.8 1529.2 2085.8 5315.4 42 44 7722 6 1496.4 2025.3 5226.1 7796.2 1.530.3 2087.9 5318.5 44 46 7725.1 1497.5 2027.4 .5229.2 7798 7 1531 .4 2090.0 .5321.6 46 48 7727.6 1108.6 2029.4 .5232.2 7801 . 1 1532.6 2092.1 .5324.7 48 50 7730.0 1499 8 2031.5 5235.3 7803.6 1.533.7 2094.2 .5327.8 50 52 7732 5 1.500.9 2033.0 5238.3 7806.0 15.34 8 2096.3 5330 9 52 54 7735.0 1502.0 2035.6 5241 4 7808.5 1536.0 2098.4 5334.0 54 56 7737 5 1503.1 2037 7 .5244 5 7810.9 1.5,37.1 2100.6 .5337.1 56 58 7739 9 1504.2 2039.8 5247.5 7813 4 1538.2 2102.7 5340.2 58 60 7742.4 1505 4 2041.8 .5250.6 7815 8 1539.3 2104.8 .5343.3 1 60 i 290 IX.— FUNCTIONS OF A ONE-DEGREE CURVE. r / 86° 1 8 4° / L. C. M. E. .T. L. C. 31. 1.573.6 E. 2169.5 T. 54.37.5 r815.8 15.39.3 2104.8 5.343.3 7888.5 2 7818.2 1540.4 2106.9 5.346.4 7890.9 1574.8 2171.6 5440.7 2 4 7820.6 1541.6 2109.1 5349.5 7893.3 1575.9 2173.8 5443.9 4 6 7823.1 1542.7 2111.2 .5.3.52.7 7895.7 1577.1* 2176.0 5447.1 6 8 7825.5 1543.9 2113.4 5355.8 7898.1 1578.2 2178.2 5450.3 8 10 7827.9 1545.0 2115,5 5358.9 7900.5 1579.4 2180.4 54.53.4 10 12 7830.3 1546.1 2117.6 5362.0 7903.0 1580.5 2182.5 5456.6 12 14 78.32.8 1547.3 2119.8 5365.2 7905.4 1581.7 2184.7 5459.8 14 16 78.35.2 1548.4 2121.9 5368.3 7907.8 1.582.9 2186.9 5463.0 16 18 7837.6 1549.6 2124.1 5371.4 7910.2 1584.0 2189.1 5466.2 18 20 7840.0 15.50.7 2126.2 5374.6 7912.6 1.585.1 2191.3 5469.4 20 22 7812.4 1551.8 2128.3 5377.7 7915.0 1.586.3 2193.5 5472.5 22 24 7844.9 1.553.0 21.30.5 5?80.8 7917.4 1587.4 2195.7 5475.7 24 26 7847.3 1554.1 2132.6 •5383.9 7919.8 1588.6 2197.9 5478.9 26 28 7849.7 1555.3 2134.8 5.387.1 7922.2 1589.7 2200.1 5482.1 28 30 7852.1 15.56.4 2136.9 .5390.2 7924.6 1590.9 2202.3 5485.3 30 32 7854.6 1557.5 2139.0 5393.4 7927.1 1592.0 2204.5 5488.5 32 34 7857.0 1.5.58.7 2141.2 5.396.5 7929.5 1593.2 2206.8 5491.7 34 36 7859.4 1559.8 2143.3 5399.7 79.31.9 1594.3 2209.0 5494.9 36 38 7861.8 1561.0 2145.5 5402.8 7934.3 1595.5 2211.2 5498.1 38 40 7864.3 1.562.1 2147.7 5406.0 7936.7 1596.6 2213.4 5501.3 40 42 7866.7 1563.2 2149.8 5409.1 7939.1 1597.8 2215.6 5504.5 42 44 7869.1 1.564.4 21.52.0 5412.3 7941.5 1.598.9 2217.8 5507.7 44 46 7871.5 1.565.5 21.54.2 .5415.4 7913.9 1600.1 2220.0 5510.9 46 48 7874.0 1566.7 21.56.4 .5418.6 7946.3 1601.2 22-22.3 5514.1 48 50 7876.4 1.567.8 2158.6 5121.8 7948.7 1602.4 2224.5 5517.3 50 52 7878.8 1568.9 2160.7 5424.9 7951.2 1603.5 2226.7 5520.5 52 54 7881.2 1570.1 2162.9 5428.1 79.53.6 1604.7 2228.9 5523.7 54 56 7883.6 1.5:1.2 21C5.1 5431.2 7956.0 1605.8 2231.1 5526.9 56 58 7886.1 1.572.4 2167.3 54.34.4 7958.4 1607.0 2233.3 .5530.1 58 60 7888.5 1573.6 2169.5 5437.5 7960.8 1608.2 2235.6 55.33.3 60 > / 88° 1 89° f L. C. 7960.8 M. 1608.2 E. 2235.6 T. .5,533.3 L. C. M. E. T. 8032.4 1643.0 2303.6 5630.8 o 7963.2 1609.4 2237.8 5536.6 8034.8 1644.1 2305.9 56.34.1 2 4 7965.6 1610.5 2240.1 .5539.8 80.37.1 1645.3 2308.2 56.37.4 4 6 7968.0 1611.7 2-242.3 .5543.1 8039.5 1646.5 2310.5 .5640.7 6 8 7970.3 1612.8 2244.6 5546.3 8041.9 1647.7 2312.8 5644.0 8 10 7972.7 1614.0 2246.8 5549.5 8044.2 1648.9 2315.1 5647.3 10 1? 7975.1 1615.2 2249.1 55.52.8 8046.6 1650.0 2317.4 5650.6 12 14 7977.5 1616.3 2251.3 5556.0 8049.0 1651.2 2.319.7 56.53.9 14 16 7979.9 1617.5 2253.6 5.559.2 8051.4 1652.4 2.322.0 .56.57.1 16 Ih 7982.3 1618.6 2255.8 5562 . 5 8053.7 1653.6 2.324.3 5660.4 18 20 7984.7 1619.8 22.58.1 .5.565.7 8056.1 16.54.8 2.326.7 5663.7 20 22 7987.1 1621.0 2260.4 5568.9 8058.5 1655.9 2.329.0 5667.0 22 24 7989.4 1622.1 2262.7 .5.572.2 8060.8 16.57.1 2.331.3 5670.3 24 26 7991.8 1623.3 2264.9 5.575.4 8063.2 1658.3 2333.7 5673.6 26 28 7994.2 1624.4 2267.2 .5.578.6 8065.6 1659.5 2.3.36.0 5676.9 28 30 7996.6 1625.6 2269.5 .5581.9 8067. 9 1660.7 2338.3 .5680.2 30 32 7999.0 1626.8 2271.7 5585.1 8070.3 1661.8 2.340.7 5683.5 32 34 8001.4 1627.9 2273.9 .5588.4 8073.7 1663.0 2343.0 .5686.8 34 36 8003.8 1629.1 2276.2 5.591.7 8075.1 1664.2 2.345.3 5690.2 36 38 8006.1 16.30.2 2278.5 5594.9 8077.4 1665.4 2347.7 5693.5 38 40 8008.5 1631.4 2280.8 .5598.2 8079.8 1606.6 2.350.0 5696.8 40 42 8010.9 1632.6 2283.0 .5601.4 80S2.2 1667.7 2352.3 5700.1 42 44 8013.3 1633.7 2285.3 .5604.7 8084.5 1668.8 2.354.7 5703.4 44 46 8015.7 10:J4.9 2J87.6 5608.0 8086.9 1670.0 2:357.0 5706.8 46 48 8018.1 1636.0 2289 9 .5611.2 80S9.3 1671.2 2359.3 .5710.1 48 50 8020.5 1637.2 2292 2 .5614 5 8091.6 1672.4 2.361.7 5713.4 50 52 8022.9 16:38.4 2294! 4 5617.8 8094.0 1673.5 2.364.0 5716.7 52 54 8025.2 1639.5 2296.7 .5621.0 8096.4 1674 7 2366.3 5720.0 54 56 8027.6 1640.7 2290.0 .562 ». 3 8098.8 1675.9 2368.7 5723.4 56 58 8030.0 1641.8 2301 .3 .5627.5 8101.1 1677.1 2371.0 5726.7 58 60 80.32.4 1643.0 2303.6 5630.8 8103.5 1678.3 2373.4 .5730.0 60 IX.— FUNCTIONS OF A ONE-DEGREE CURVE. 291 / yo^ 1 91° 1 L. C. M. E. T. L. C. M. E. T. 8103.5 1678.3 2373.4 5730.0 8173.9 1713.8 2445.1 5830.9 2 8105.8 1679.5 2375.8 5733.3 8176.2 1715.0 2447.5 5834.3 2 4 8108.2 1680.6 2378.2 5736.7 8178.5 1716.2 2450.0 5837.7 4 6 8110.5 1681.8 2380.5 .5740.0 8180.9 1717.4 2452.4 5841.1 6 8 8112.9 1683.0 2382.9 5743.4 8183.2 1718.6 2454.8 5844.5 8 10 8115.2 1684.2 2385.3 .5746.7 81S5.5 1719.7 2457.2 5847.9 10 12 8117 G 1685.4 2387.6 .5750.0 8187.9 1720.9 2459.7 5851.3 12 14 8119.9 1686.5 2390.0 5753.4 8190.2 1722.1 2462.1 5854.7 14 16 8122.3 1687.7 2392.4 5756.7 8192.5 1723.3 2464.5 5858.1 16 18 8124.6 1688.9 2394.7 5760.1 8194.8 1724.5 2467.0 5861.5 18 20 8127.0 1690.1 2397.1 5763.4 8197.2 1725.7 2469.4 5864.9 20 22 8129.3 1691.3 2399.5 .5766.8 8199.5 1726.9 2471.9 5868.3 22 24 8131.7 1692.5 2401.9 .5770.1 8201.8 1728.1 2474.3 5871.8 24 26 8134.0 1693.6 2404.3 5773.5 8204.2- 1729.3 2476.7 5875.2 26 28 8136.4 1694.8 2106.6 .5776.9 8206.5 1730.5 2479.2 5878.6 28 30 8138.7 1696.0 2409.0 5780.2 8208.8 1731.7 2481.6 5882.0 30 32 8141.1 1697.2 2411.4 .5783.6 8211.1 1732.9 2484.1 5885.4 32 34 8143.4 169S.4 2413.8 5787.0 8213.5 1734.1 2486.5 5888.9 34 36 8145.8 1699.6 2416.2 5790.3 8215.8 1735.3 2489.0 .5892.3 36 38 8148.1 1700.7 2418.6 5793.7 8218.1 1736.4 2491.5 5895.7 38 40 8150.4 1701.9 2421.0 5797.1 8220.4 1737.6 2493.9 5899.2 40 42 8152.8 1703.1 2423.4 5800.4 8222.8 1738.8 2496.4 5902.6 42 44 8155.1 1704.3 2425.8 .5803.8 8225.1 1740.0 2498.9 5906.0 44 46 8157.5 1705.5 2428.2 5807.2 8227.4 1741.2 2501.3 5909.4 46 48 8159.8 1706.7 2430.6 5810.6 8229.7 1742.4 2.503.8 5912.9 48 50 8162.2 1707.9 2433.0 .5814.0 8232.0 1743.6 2506.3 5916.3 50 52 8164.5 1709.0 2435.4 5817.3 8234.3 1744.8 2.508.7 5919.8 52 54 8166.8 1710.2 2437.9 .5820.7 8236.7 1746.0 2511.2 5923.2 54 56 8169.2 1711.4 2440.3 5824 . 1 8239.0 1747.2 2513.7 5926.7 56 58 8171.5 1712.6 2442.7 5827.5 8241.3 1748.4 2516.2 5930.1 58 60 8173.9 1713.8 2445.1 5830.9 8243.6 1749.6 2518.7 5933.6 60 92° 1 93° / L. C 8243.6 M. 1749.6 E. 2518.7 T. .5933.6 L. C. M. E. T. 8312.8 1785.7 2594.2 6038.2 2 8245.9 1750.8 2521.2 5937.0 8315.1 1786.9 2596.8 6041.7 2 4 8248.2 1752.0 2.523.6 5940.5 8317.4 1<88.2 2599.3 6045.2 4 6 8250.6 1753.2 2.526.1 5944.0 8319.7 1789.4 2601.9 6048.7 6 \ 8252.9 1754.4 2528.6 5947.4 8322.0 1790,6 2604.4 6052.2 8 10 8255.2 1755.6 2531.1 59.50.9 8324.3 1791.8 2607.0 6055.8 10 12 8257.5 1756.8 2.533.6 5954.4 8326.6 1793.0 2009.6 6059.3 12 14 8259.8 1758.0 2536.1 59.57.8 8328.8 1794.2 2612.1 6062.8 14 16 8262.2 1759.2 2538.6 5961.3 8331.1 1795.4 2614.7 6066.4 16 18 8264.5 1760.4 2541.1 5964.8 8:«3.3 1796.6 2617.3 6069.9 18 20 8266.8 1761.6 2543.6 5968.2 8335.6 1797.8 2619.8 6073.4 20 22 8269.1 1762.8 2546.1 5971.7 8337.9 1799.1 2622.4 6077.0 22 24 8271.4 1764.0 2.548.6 5975.2 8340.2 1800.3 2625.0 6080.5 24 26 8273.7 1765.2 2551.2 5978.7 8342.5 1801.5 2627.6 6084.1 26 28 8276.0 1766.4 2553.7 5982.2 8344.8 1802.7 2630.2 6087.6 28 30 8278.3 1767.6 2.556.2 5985.6 8347.1 1803.9 2632.7 6091.2 30 32 8280.6 1768.8 2558.7 5989.1 8349.4 1805.1 2635.3 6094 7 32 34 8282.9 17^0.0 2561.2 5992.6 8351.7 1806.3 2637.9 6098.3 34 36 8285.2 1771.2 2563.8 5996.1 83.54.0 1807.6 2640.5 6101.8 36 38 8287.5 1772.5 2566.3 5999.6 8356.3 1808.8 2643.1 6105.4 38 40 8289.8 1773.7 2.568.8 6003.1 8358.5 1810.0 2645.7 6109.0 40 42 8292.1 1774.9 2.571.3 6006.6 8360.8 1811.2 2648.3 6112.5 42 44 8294.4 1776.1 2573.9 6010.1 8363.1 1812.4 2650.9 6116.1 44 46 8296.7 li 1 1 .6 2576.4 6013.0 8365.4 1813.6 2653.5 6119.7 46 48 8299.0 1778.5 2578.9 6017.1 8367.7 1814.9 2656.1 6123.2 48 50 8301.3 1779.7 2581.5 6020.6 8369.9 1816.1 2658.7 6126.8 50 52 8303.6 1780.9 2584.0 6024.1 8372.2 1817.3 2661.3 6130.4 52 54 8305.9 1782.1 2.586.6 6027.6 8374.5 1818.5 2663.9 6133.9 54 56 8308.2 1783.3 2589.1 6031.1 8376 8 1819.7 2666.6 6137.5 5(i 58 8310.5 1784.5 2591.7 6034.6 8379.1 1820.9 2669.2 6141.1 .5.S 60 8312.8 1785.7 2.591.2 ()n.3,S.2 8381 3 1822.2 2671.8 6114 7 60 '4\)'4 IX.— FUNCTIONS OF A ONE-DEGREE CURVE. 1 94° 1 95° / L. C. M. E. T. L. C. M. E. T. 8381.3 1822.2 2671.8 6144.7 8449.2 1858.9 2751.5 6253.2 2 8383.6 1823.4 2674.4 6148.3 8451.5 1860.1 2754.2 62.56.9 2 4 8385.9 1824.6 2677.0 6151.9 8453.7 1861.3 2756.9 6260.5 4 6 8388.1 1825.8 2679.7 6155.4 8456.0 1862.6 2759.6 6264.2 6 8 8;B90.4 1827.0 2682.3 6159.0 8458.2 1863.8 2762.3 6267.8 8 10 8392.7 1828.3 2684.9 6162.6 8460.4 1865.0 2765.0 6271.5 10 12 8395.0 1829.5 2687.6 6166.2 8462.7 1866.3 2767.7 6275.2 12 14 8397.2 1830.7 2690.2 6169.8 8464.9 1867.5 2770.4 6278.8 14 16 8399.5 1831.9 2692.8 6173.4 8467.2 1868.7 2773.1 6282.5 16 18 8401.7 1833.1 2695.6 6177.0 8469.4 1869.9 2775.8 6286.2 18 20 8404.0 1834.4 2698.1 6180.6 8471.7 1871.2 2778.5 6289.8 20 22 8406.3 1835.6 2700.8 6184.2 8473.9 1872.4 2781.2 6293.5 22 24 8408.5 1836.8 2703.4 6187.8 8476.2 1873.6 2784.0 6297.2 24 26 8410.8 1838.0 2706.1 6191.5 8478.4 1874.9 2786.7 6300.9 26 28 8413.1 18::i9.3 2708.7 6195.1 8480.7 1876.1 2789.4 6304.6 28 30 8415.3 1840.5 2711.4 6198.7 8482.9 1877.3 2792.1 6308.2 30 32 8417.6 1841.7 2714.0 6202.3 84S5.1 1878.6 2794.9 6311.9 32 34 8419.9 1842.9 2716.7 6205.9 8487.4 1879.8 2797.6 6315.6 34 36 8422.1 1844.2 2719.3 6209.5 8489.6 1881.0 2800.3 6319.3 36 38 8424.4 1845.4 2722.0 6213.2 8491.9 1882.3 2803.1 6323.0 38 40 8426.6 1846.6 2724.7 6216.8 8494.1 1883.5 2805.8 6326.7 40 42 8428.9 1847.8 2727.3 6220.4 8496.3 1884.8 2808.6 6330.4 42 44 8431.2 1849.1 2730.0 6224.1 8498.6 18S6.0 2811.3 6-334.1 44 46 8433.4 1850.3 2732.7 6227.7 8500.8 1887.2 2814.1 6.337.8 46 48 8435.7 1851.5 2735.4 6231.3 8503.0 1888.5 2816.8 6341.5 48 50 8437.9 1852.7 2738.0 6235.0 8505.3 1889.7 2819.6 6345.2 50 52 8440.2 1854.0 2740.7 6238.6 8507.5 1890 9 2822 3 6349 52 54 8442.4 18.15.2 2743.4 6242.3 8509.8 1892.2 2825.1 6352.7 54 56 8444.7 1856.4 2746.1 6245.0 8512.0 1893.4 2827.8 6356.4 56 58 8447.0 1857.6 2748.8 6249.6 8514.2 1894.6 2830.6 6360.1 58 60 8449.2 ia58.9 2751.5 62.53.2 8516.4 1895.9 2833.4 6363.8 60 / 96° 1 97" / L. C. M. E. T. L. C. M. E. T. 8516.4 1895. S 2833.4 6363.8 8583.0 1933.2 2917.5 6476.6 2 8518.7 1897.1 2836.1 6367.5 8.585.2 1934.4 2920.3 6480.4 2 4 8520.9 1898.4 2838.9 6371.3 8587.5 1935.7 2923.2 6484.2 4 6 8.523.1 1899.6 2841.7 6375.0 8589 7 1936.9 2926.0 6488.0 6 8 8.525.4 1900.8 2844.5 6378.7 8591.9 1938.2 2928.9 6491.8 8 10 8527.6 1902.1 2847.2 6382.5 8594 1 1939.4 2931.7 6495.6 10 12 a529.8 1903 3 28.50.0 6386.2 8.596.3 1940.7 2934.6 6499.4 12 14 8532.0 1904.6 2852 8 6389.9 8.59^.5 1941.9 2937.5 6503.2 14 16 8534.3 1905.8 2855 6 6393.7 8600.7 1943.2 2940 3 6.507.1 16 18 8536.5 1907.0 2858.4 6397.4 8602.9 1944.4 2943.2 6510.9 18 80 8538.7 1908.3 2861.2 6401.2 8605.1 1945.7 2946.1 6514.7 20 22 8.540 9 1909.5 2864.0 6404.9 8607.3 1946.9 2948.9 6518.5 22 24 8543.2 1910.8 2866.7 6408.7 8609.5 1948.2 2951.8 6522.3 24 26 8545.4 1912.0 2869.5 6412.4 8611.7 1949.4 2954.7 6526.2 26 28 8547. e 1913 3 2872.3 6416.2 8613.9 1950.7 2957.6 6530.0 28 30 8549 8 1914 5 2875.1 6419.9 8616 1 1952.0 2960.4 6533.8 30 32 85.52.0 1915.7 2877.9 642;3.7 8618.3 1953.2 2963.3 6537.7 32 34 8554.3 1917.0 2S80.8 6427 5 8620.5 1954.5 2966.2 6.541.5 34 36 8556.5 1918.2 2883.6 6431 2 8622.7 19.55.7 2969.1 6545.3 36 38 8558.7 1919.5 2886 4 6435 8624.9 1957.0 2972.0 6549.2 38 40 8560.9 1920.7 2889.2 6438.8 8627.1 19.58. 2 2974.9 65.53.0 40 42 8563.1 1922.0 2892.0 6442 5 8629.3 1959 5 2977.8 6.5.56.9 42 44 8565 3 1923.2 2894 8 6446.3 8631.5 1960.7 2980.7 6.560.7 44 46 8567 6 1924 5 2897.7 6450.1 8633 7 1962.0 2983.0 6564.6 46 48 8569 8 1925.7 2900.5 6453.9 8635.8 1963.2 2986.5 6568.4 48 50 8.572.0 1927.0 2903.3 64.57.6 8638.0 1961.5 2989.4 6572.3 50 52 8.574.2 1928.2 2906.1 6461.4 8640.2 1965.8 2992.3 6.576.2 52 54 8576.4 1929 4 2909.0 6465.2 8642.4 1967.0 2995 2 6.580.0 54 56 8.578.6 1930.7 2911.8 6469.0 8644.6 1968.3 2998.1 6.583.9 56 58 8.5S0.8 1931.9 2914.7 6472.8 8646 !< 1969.5 3001.1 6.587.7 58 , 60 8583.0 1933.2 2917.5 6476.6 8649 1970.8 3004.0 6591.6 60 IX.— FUNCTIONS OF A. ONE-DEGREE CURVE. 293 98° 99° .' L. C. M. £. T. L. C. M. E. T. 8649.0 1970.8 3004.0 6591.6 8714.3 2008.7 3092.9 6709.0 2 8651.2 1972.0 3006.9 6595.5 8716.4 2009.9 3095.9 6712.9 2 4 8653.3 1973.3 3009.8 6.599.4 8718.6 2011.2 3098.9 6716.9 4 6 8655.5 1974.6 3012.8 6603.2 8720.7 2012.5 3101.9 6720.8 6 8 8657.7 1975.8 3015.7 6607.1 8722.9 2013.7 3104.9 6724.8 8 10 8659.9 1977.1 3018.6 6611.0 8725.1 2015.0 3107.9 6728.8 10 12 8662.1 1978.3 3021.6 6614.9 8727.2 2016.3 3111.0 6732.7 12 14 8664.3 1979.6 3024.5 6618.8 8729.4 2017.5 3114.0 6736.7 14 16 8666.4 1980.9 3027.5 6622.7 8731.5 2018.8 3117.0 6740.7 16 18 8668.6 1982.1 3030.4 6026.6 8733.7 2020.1 3120.0 6744.6 18 20 8670.8 1983.4 3033.3 6630.5 8735.9 2021.4 3123.1 6748.6 20 22 8673.0 1984.6 3036.3 6634.4 8738.0 2022.6 3126.1 6752.6 22 24 8675.2 1985.9 3039.3 6638.3 8710.2 2023.9 3129.1 6750.6 24 26 8677.3 1987.2 3042.2 6642.2 8742.3 2025.2 3132.2 6760.6 26 28 8679.5 1988.4 8045.2 6646.1 8744.5 2026.4 3135.2 6764.6 28 30 8681.7 1989.7 3048.1 6650.0 8746.6 2027.7 3138.3 6768.6 30 32 8683.9 1991.0 3051.1 6653.9 8748.8 2029.0 3141.3 6772.6 32 34 8686.0 1992.2 3054 . 1 6657.8 8750.9 2030.3 3144.4 6776.6 34 36 8688.2 1993.5 30.57.0 6661.7 8753.1 2031.5 3147.4 6780.6 36 38 8690.4 1994.7 3060.0 6665.7 8755.3 2032.8 3150.5 6784.6 38 40 8692.6 1996.0 3063.0 6669.6 8757.4 2034.1 3153.5 6788.6 40 42 8694.7 1997.3 30U6.0 6673.5 8759.5 2035.4 3156.6 6792.6 42 44 8696.9 1998.5 3068.7 6677.4 8761.7 2036.6 3159.7 6796.6 44 46 8699.1 199;). 8 3071.9 6681.4 8763.8 2037.9 3162.7 6800.6 46 48 8701.2 2001.1 3074.9 6685.3 8766.0 2039.2 3165.8 6804.6 48 50 8703.4 2002.3 3077.9 6689.2 8768.1 2040.5 3168.9 6808.6 50 52 8705.6 2003.6 3080.9 6693.2 8770.3 2041.7 3172.0 6812.6 52 54 8707.8 2004.9 3083.9 6697.1 8772.4 2043.0 3175.1 6816.7 54 56 8709.9 2006.1 3086.9 6701.1 8774.6 2044.3 3178.1 6820.7 56 58 8712.1 2007.4 3089.9 6705.2 8776.7 2045.6 3181.2 6824.7 58 60 ^ , 8714.3 2008.7 3092.9 6709.0 8778.9 2046.8 3184.3 6828.8 60 / 100° 1 101° / L C. 8778.9 M. E. T. L. C. M. E. T. 2046.8 3184.3 6828.8 8842.8 2085.3 3278.3 6951.0 o 8781.0 2048.1 3187.4 6832.8 8844.9 2086.6 3281.5 6955.2 2 4 8783.1 2049.4 3190.5 6836.8 8847.0 2087.8 3284.7 6959.3 4 6 8785.3 20.50.7 3193.6 6840.9 8849.2 2089.1 3287.9 6963.4 6 8 8787.4 2051.9 3196.7 6S44.9 8851.3 2090.4 3291.1 6967.6 8 10 8789.6 2053.2 3199.8 6849.0 8853.4 2091.7 3294.3 6971.7 10 12 8791.7 2054.5 3202.9 6853.0 8855.5 2093.0 3297.5 6975.8 12 14 8793.9 20.55.8 3206.0 6857.1 88.57.6 2094.3 3300.7 6980.0 14 16 8796.0 20.57.1 3209.1 6861.1 8859.8 2095.6 3303.9 6984.1 16 18 8798.9 2058.3 3212.2 6865.2 8861.9 2096.9 3307.1 6988.2 18 20 8800.3 2059.6 3215.4 6869.2 8864.0 2098.2 3310.3 6992.4 20 22 8802.4 2060.9 3218.5 6873.3 8866.1 2099.4 3313.5 6996.6 22 24 8804.5 2062.2 3221.6 6877.4 8868.2 2100.7 3316.7 7000.7 24 26 8806.7 2063.5 3224.7 6881.4 8870.3 2102.0 3319.9 7004.9 26 28 8808.8 2064.7 3227.9 6885.5 8872.4 2103.3 3323.1 7009 28 30 8810.9 2066.0 3231 6889.6 8874.5 2104.6 3326.4 7013.2 30 32 8813.1 2067.3 3234.1 6893.7 8876.7 2105.9 3329.6 7017.3 32 34 8815.2 2068.6 3237.3 6897.8 8878.8 2107.2 3332.8 7021.5 34 36 8817.3 2069.9 3240.4 6901.8 8880.9 2108.5 3336.0 7025.7 36 38 8819.5 2071 . 1 3243.5 6905.9 8883.0 2109.8 3339.3 7029.9 38 40 8821.6 2072.4 3246.7 6910.0 8885.1 2111.1 3342.5 7034.0 40 42 ' 8823.7 2073.7 3249.8 6914.1 8887.2 2112.4 3345.8 7038.2 42 44 8825.8 2075.0 3253.0 6918.2 8889.3 2113.6 3349.0 7042.4 44 46 8828.0 2076.3 3256.2 6922.3 8891,4 2114.9 3352.3 7046.6 46 48 8830.1 2077.6 3259.3 6926.4 8893.5 2116.2 3355.5 7050.8 48 50 8832.2 2078.9 3262.5 6930.5 8895.6 2117.5 3358.8 7055 50 52 8834 . 3 2080.1 3265.7 6934.6 8S97.7 2118.8 3362.0 70.59.2 52 54 8^36.4 2081.4 3268.8 6938.7 8899.8 2120.1 3365.5 7063 4 54 56 8K38 6 2082.7 3272.0 6942.8 8901.9 2121.4 3368.7 7067.6 56 58 S340 7 20S4.0 3275.2 6946.9 8904.0 2122.7 3372.0 7071.8 58 60 8842 8 2085.3 3278.3 6951.0 8906.1 2124.0 3375.1 7076.0 60 294 IX.— FUNCTIONS OF A ONE-DEGREE CURVE. / 102° 103° / L. C. M. E. T. L. C. M. E. T. 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54 56 58 60 8906.1 2124.0 3375.1 7076.0 8908.2 2125.3 3378.3 7080.2 8910.3 2126.6 ;3381.6 70.84. 4 8912.4 2127.9 3384.9 7088.6 8914.5 2129.2 3388.2 7092.8 8916.6 2130.5 3391.5 7097.1 8918.7 2131.8 3394 7 7101.3 8920.8 2133.1 3398.0 7105.5 8922.9 2134.4 3401.3 7109.7 8925.0 2135.7 3404.6 7114.0 8927.0 2137.0 3407.9 7118.2 8929.1 2138.3 3411.2 7122.4 8931.2 2139.6 3414.5 7126.7 8933.3 2140.9 3417.9 7130.9 8935.4 2142.2 3421.2 7135.2 8937.5 2143.5 3424.5 71-39.4 8939.6 2144 8 3427.8 7143.7 8941.6 2146.1 3431.1 7148.0 8943.7 2147.4 3434.5 7152.2 8945.8 2148.7 3437.8 7156.5 8947.9 2150 3441.1 7160.7 8950.0 2151.3 3444.4 7165.0 8952.1 2152.6 3447.8 7169.3 8954.1 2153.9 3451.1 7173.6 8956.2 2155.2 3454.5 7177.9 89.58.3 21.56.5 34.57.8 7182.1 8960.4 2157.8 3461.2 7186.4 8962.5 2159.1 3464.5 7190.7 8964.5 2160.4 3467.9 7195.0 8966.6 2161.7 .3471.2 7199.3 8968.7 2163.0 3474.6 7203.6 8968.7 2163.0 3474.6 7203.6 8970.8 2164.3 3478.0 7207.9 8972.9 2165.6 3481.4 7212.2 8974.9 2166.9 3484.7 7216.5 8977.0 2168.2 3488.1 7220.8 8979.1 2169.5 3491.5 7225.1 8981.1 2170.8 3494.9 7229.5 8983.2 2172.1 3498.3 7233.8 8985.3 2173.4 3501.6 7238.1 8987.3 2174.7 3505.3 7242.4 8989 4 2176.1 3508.4 7246.8 8991.5 2177.4 3511.8 7251.1 8993.5 2178.7 3515.2 7255.4 8995.6 2180.0 -3518.7 7259.8 8997.7 2181.3 3522.1 7264.1 8999.7 2182.6 3525.5 7268.5 9001.8 2183.9 3528.9 7272.8 9003.9 2185.2 35-32.3 7277.2 9005.9 2186.5 3535.7 7281.5 9008.0 2187.8 3539.2 7285.9 9010.0 2189.1 3542.6 7290.3 9012.1 2190.5 3,546.0 7294.6 9014.2 2191.8 3549.5 7299,0 9016.2 2193.1 3552.9 7303.4 9018.3 2194.4 3556.3 7307.7 9020.3 2195 7 3559.8 7312.1 9022.4 2197.0 3.563.2 7316.5 9024.5 2198.3 3566.7 7320.9 9026.5 2199.6 3-570.2 7-325.3 9028.6 2200.9 3573.6 7329.7 90.30.6 2202.3 .3.577.1 7334.1 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54 56 58 60 ^ / 104° 105° / L. C. M. E. T. L. C. M. E. T. 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 . 42 44 46 48 50 52 54 56 58 60 9030.6 2202.3 3577.1 7334.1 90.32.7 2203.6 3580.5 7388.5 9034.7 2204.9 3584.0 7342.9 9036.8 2206.2 3587.5 7347.3 9038.8 2207.5 3591.0 7351.7 9040.9 2208 8 3594.4 7356.1 9042.9 2210 2 3597.9 7360.5 9045.0 2211.5 3601.4 7.364.9 9047.0 2212.8 3604.9 7369.4 9049.1 2214.1 .3608.4 7373.8 9051.1 2215.4 3611.9 7378.2 90.53.1 2216.7 3615.4 7-382.6 9055.2 2218.0 3618.9 7-387.1 90.57.2 2219.4 3622.4 7391.5 9059.3 2220.7 -3625.9 7396.0 9001.3 2222.0 3629.4 7400.4 9063.3 2223.3 3633.0 7404.8 9065.4 2224.6 3636.5 7409.3 9067.4 2226.0 3640.0 7413 8 9069.5 2227.3 3643.5 7418.2 9071.5 2228.6 .3647.1 7422.7 9073.5 2229.9 3650.6 7427.1 9075.6 2231.2 -3654.1 7431.6 9077 6 2232.6 36.57.7 7436.1 9079.6 2233.9 3661.2 7440 6 9081.7 2235.2 3664.8 7445.0 9083.7 2236.5 3668.3 7449.5 9085.7 22-37.8 3671.9 74.54.0 9087.8 2239.2 3675.4 74.58 5 90S9.8 2240.5 3679.0 7463.0 9091 8 2241.8 3682.6 7467 5 9091.8 2241.8 3682.6 7467.5 9093.9 2243.1 3686.1 7472.0 9095.9 2244.4 3689.7 V476.5 9097.9 2245.8 3693.3 7481.0 9099.9 2247.1 3696.9 7485.5 9102.0 2248.4 3700.4 7490.0 9104.0 2249.7 3704.0 7494.5 9106.0 2251.1 3707.6 7499.1 9108.0 2252.4 3711.2 7503.6 9110.1 2253.7 3714.8 7508.1 9112.1 2255.0 3718.4 7512.6 9114.1 22.56.4 3722.0 7517.2 9116.1 2257.7 3725.6 7521.7 9118.1 22.-9 3729.3 7.526.3 9120.2 2260.3 3732.9 7530.8 9122.2 2261.7 3736.5 75.35.3 9124.2 2263.0 3740.1 7539 9 9126.2 2264.3 3743.7 7544.4 9128 2 2265.7 3747.4 7.549.0 9130.2 2267.0 3751.0 7553.6 91-32.3 2268.3 .3754.6 7558.1 9134.3 2269.6 3758.3 7562.7 9136.3 2271.0 3761.9 7.567.3 9138.3 2272.3 3765.6 7571.8 9140.3 2273 6 3769.2 7.576.4 9142.3 2275.0 3772.9 7581.0 9144.3 2276.3 3776.5 7.585.6 9140 3 2277.6 3780.2 7.590.2 9148.3 2278.9 3783.9 7594.8 9150.4 22S0.3 3787.5 7.599.4 9152.4 2281.6 .37912 7604 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54 56 58 (iO IX.— FUNCTIONS OF A ONE-DEGREE CURVE. 295 / 106° 1 107» L C. M. E. T. L. C. M. E. T. 9152.4 2281.6 3791.2 7604.0 9212.2 2321.7 3903.1 7743.7 2 9154.4 2282.9 3794.9 7608.6 9214.2 2323.0 3906.9 7748.4 til 4 9156.4 2284.3 3798.6 7613.2 9216.2 2324.4 3910.7 7753.1 4 C 9158.4 2285.6 3802.3 7617.8 9218.1 2325.7 3914.5 7757 . 8 6 8 9160.4 2286.9 3805.9 7622.4 9220.1 2327.0 3918.3 7762.5 8 10 9162.4 2288.3 3809.6 7627.0 9222.1 2328.4 3922.1 7767.3 10 12 9164.4 2289.0 3813 3 7631.7 9224.1 2329.7 3925.9 7772.0 12 14 9166.4 2290.9 3817.0 7636.3 9226.1 2331.1 3929.7 7776.7 14 16 9168.4 2292.3 3820.7 7610.9 9228.1 2832.4 3933.6 7781.5 16 18 9170.4 2293.6 3824.4 7645.5 9230.0 2333.7 3937.4 7786.2 18 20 9172.4 2294.9 3828.1 7650.2 9232.0 2335.1 3941 .2 7791.0 20 22 9174.4 2296.3 3831.9 7654.8 9234.0 2336.4 3945.0 7795.7 22 24 9176.4 2297.6 3835.6 7659.5 9235.9 2337.8 3948.9 7800.5 24 26 9178.4 2298.9 3839.3 7664.1 9237.9 2339.1 3952.7 7805.2 26 28 9180.4 2300.3 3843.0 7668.8 9239.9 2340.5 3956.5 7810.0 28 30 9182.4 2301.6 3846.7 7673.4 9241.9 2341.8 3960.4 7814.7 30 32 9184.4 2302.9 3850.5 7678.1 9243.8 2343.1 3964.2 7819.5 32 34 9186.4 2304.3 3854.2 7682.7 9245.8 2344.5 3968.1 7824.3 34 36 9188.4 2305.6 3858.0 7687.4 9247.8 2345.8 3971.9 7829.1 36 38 9190.4 2306.9 3861.7 7692.1 9249.7 2347.2 3975.8 7833.8 38 40 9192.4 2308.3 3865.4 7696.7 9251.7 2348.5 3979.6 7838.6 40 42 9194.4 2309.6 3869.2 7701.4 9253.7 2349.9 3983.5 7843.4 42 44 9196.3 2311.0 3873.0 7706.1 9255.6 2351 2 3987.4 7848.2 44 46 9198.3 2312.3 3876.7 7710.8 9257.6 2352.6 3991.3 7853.0 46 48 9200.3 2313.6 3880.5 7715.5 9259.6 2353.9 3995.1 7857.8 48 50 9202.3 2315.0 3884.2 7720.1 9261.5 2355.3 3999.0 7862.6 50 52 9204 3 2316.3 3888.0 7724.8 9263.5 2356.6 4002.9 7867.4 52 54 9206.3 2317.7 3891.8 7729.5 9265.4 2358.0 4006.8 7872.2 54 56 9208.2 2319.0 3895.6 7734.2 9267.4 2359.3 4010.7 7877,0 56 58 9210.2 2320.3 3899.3 7739.0 9269.4 2360.7 4014.6 7881.9 58 60 9212.2 2321.7 3903.1 7743.7 9271 .3 2362.0 4018.5 7886.7 60 / 108° 109° - ■ - L. C. M. E. T. L. C. M. E. T. 9271.3 2362.0 4018.5 7886.7 9329.8 2402.6 4137.4 8033.2 2 9273.3 2363.3 4022.4 7891 .5 9331.7 2403.9 4141.4 8038.1 o 4 9275.3 2364.7 4026.3 7896.3 9333.6 2405.3 4145.4 8043.1 4 6 9277.2 2366.0 4030.2 7901.2 9335.6 2406.6 4149.5 8048.0 6 8 9279.2 2367.4 4034.1 7906.0 9337.5 2408.0 4153.5 8053.0 8 10 9281.1 2368.7 4038.0 7910.8 9339.4 2409.4 4157.5 8057.9 10 12 9283.1 2370.1 4042.0 7915.7 9341.4 2410.7 4161.6 8062.9 12 14 9285.0 2371.4 4045.9 7920.5 9343.3 2412.1 4165.6 8067.9 14 16 9287.0 2372.8 4049.8 7925.4 9345.2 2413.4 4169.7 8072.8 16 18 9288.9 2374.1 4053.8 7930.3 9347.2 2414.8 4173.8 8077.8 18 20 9290.9 2375.5 4057.7 7935.1 9349.1 2416.2 4177.8 8082.8 20 22 9292.8 2376.8 4061.6 7940.0 9351.0 2417.5 4181.9 8087.8 22 24 9294.8 2378.2 4065.6 7944.8 9352.9 2418.9 4186.0 8092.8 24 26 9296.7 2379.5 4069.5 7949.7 9354.9 2420.2 4190.0 8097.8 26 28 9298.7 2380.9 4073.5 7954.6 9356.8 2421.6 4193.1 8102.8 28 30 9300.6 2382.3 4077.5 7959.5 9358.7 2423.0 4198.2 8107.8 30 32 9302.6 2383.6 4081.4 7964.4 9360.6 2424.3 4202.3 8112.8 32 34 9304.5 2385.0 4085.4 7969.3 9362.6 2425.7 4206.4 8117.8 34 36 9306.5 2386.3 4089.4 7974.1 9364.5 2427.0 4210.5 8122.8 36 38 9308.4 2387.7 4093.4 7979.0 9366.4 2428.4 4214.6 8127.8 38 40 9310.4 2389.0 4097.3 7983.9 9368.3 2430.0 4218.7 8132.8 40 42 9312.3 2390.4 4101.3 7988.8 9370.2 2431.1 4222.8 8137.9 42 44 9314.2 2391.7 4105.3 7993.8 9372.2 2432.5 4226.9 8142.9 44 46 9316.2 2393.1 4109 3 7998.7 9374.1 2433.9 4231.0 8147.9 46 48 9318.1 2394.4 4113.3 8n03.6 9376.0 2435.2 4235.1 8153.0 48 50 9320.1 2395.8 4117.3 80U8.5 9377.9 2436.0 4239 3 8158.0 50 52 9322.0 2397.2 4121.3 8013.4 9379.8 2438.0 4243.4 8163.1 52 54 9323 9 2398.5 4125.3 8018.4 9381.7 24.39.3 4247.5 8168.1 54 56 9325.9 2399.9 4129.3 8023 3 9383. 7 2440.7 4251.7 8173.2 56 58 9327,8 2401.2 4 1 33 4 8028.2 93S5.6 2442.1 42.55.8 81 78 2 58 L 60 9329.8 2402 . 6 4137.4 8033 . 2 93H7.5 2443 4 4260 8183.3 60 y'^m!^m,''mm 296 IX. FUNCTIONS OF A ONE -DEGREE CURVE. / 110° 1 111 1° L. C. M. E. T. L. C. M. E. T. 9387.5 2443.4 4260.0 8183.3 9444.5 2484.5 4386.4 8337.2 2 9389.4 2444.8 4264.1 8188.4 9446.4 2485.9 4390.7 8342.4 2 4 9391.3 2446.1 4268.3 8193.4 9448.3 2487.2 4395.0 8347.6 4 6 9393 2 2447.5 4272.4 8198.5 9450.1 2488.6 4399.3 8352.8 6 8 9395.1 2448.9 4276.6 8203.6 94.52.0 2490.0 4403.6 8:358.0 8 10 9397.0 2450.2 4280.8 8208.7 9453.9 2491 ,4 4407.9 8363.2 10 12 9398.9 2451.6 4284.9 8213.8 9455.8 2492.7 4412.2 8368.5 12 14 9400.8 2453.0 4289.1 8218.9 94.57.7 2494.1 4416.5 8373.7 14 16 9402.7 2454.3 4293.3 8224.0 9459.6 2495.5 4420.8 8378 9 16 18 9404.7 2455.7 4297.5 8229.1 9461.4 2496.9 4425.1 8384.1 18 20 9406.6 2457.1 4301.7 8234.2 9463.3 2498.2 4429.5 8389.4 20 22 9408.5 2458.4 4305.9 8239.3 9465.2 2499.6 4433.8 8394 6 22 24 9410.4 24.59 8 4310.1 8244.4 9467.1 2501.0 4438.1 8399.9 24 26 9412.3 2461.2 4314.3 8249.5 9469.0 2502.4 4442.5 8405.1 26 28 9414.2 2462.6 4318.5 8254.6 9470.8 2503.8 4446.8 8410.4 28 30 9416.1 2463.9 4322.7 8259.8 9472.7 2505.1 4451.2 8415.6 30 32 9418.0 2465.3 4326.9 8264.9 9474.6 2506.5 4455.5 8420.9 32 34 9419.9 2466.7 4331.1 8270.0 9476.5 2507.9 4459.9 8426.2 34 36 9421.8 2468.0 4335,4 8275.2 9478.3 2509.3 4464.2 8431.4 36 38 9423.7 2469.4 4339.6 8280.3 9480.2 2510.6 4468.6 8436.7 38 40 9425.6 2470.8 4343.8 8285.5 9482.1 2512.0 4473.0 8442.0 40 42 9427.5 2472.1 4348.1 8290.6 9484.0 2513.4 4477.3 8447.3 42 44 9429.3 2473.5 4352.3 8295.8 9485.8 2514.8 4481.7 84.52.6 44 46 9431.2 2474.9 4356.6 8300.9 9487.7 2516.2 4486.1 8457.9 46 48 9433.1 2476.3 4360.8 •8306.1 9489.6 2.517.5 4490.5 8463.2 48 50 9435.0 2477.6 4365.1 8311.3 9491.4 2518.9 4494.9 8468.5 50 52 9436.9 2479.0 4369,3 8316.5 9493.3 2520.3 4499.3 8473.8 52 54 9438.8 2480,4 4373.6 8.321.6 9495.2 2521.7 4503.7 8479.1 54 56 9440.7 2481 .7 4377.9 8326.8 9497.0 2523.1 4508.1 8484.4 56 58 9442.6 2483.1 4382 2 8332.0 9498.9 2524.5 4512.5 8489.7 58 60 9444.5 2484.5 4386.4 8337.2 9500.8 2525.8 4516.9 8495.1 60 / 11 2» 113° / L. C. M. E. T. L. C, M. E. T. 9500.8 2525.8 4.516.9 8495.1 9.5.56.3 2.567.4 4651.6 8657.1 2 9502.6 2527.2 4521.4 8.500.4 95.58.2 2568.8 4656.2 8662.6 2 4 9504.5 2528.6 4525.8 8505.8 9.560.0 2570.2 4660.8 8668.0 4 6 9506.4 2530.0 4530.2 8511.1 9561.8 2571.6 4665.4 8673.5 6 8 9508.2 2531.4 4534.6 8516.4 9.563.7 2573.0 4669.9 8679.0 8 10 9510.1 2532.7 4.539.1 8.521.8 9.565.5 2574.4 4674.5 8684.5 10 12 9511.9 2534.1 4543.5 8527.1 9567.4 2.575.8 4679.1 8690.0 12 14 9513.8 2.535.5 4548.0 85.32.5 9569.2 2577.1 4683.7 8695.5 14 16 9515.7 2536.9 4552.4 8537.9 9571.0 2578.5 4688.3 8701.0 16 18 9517.5 2538.3 4556.9 8543.2 9.572.9 2579.9 4692.9 8706.5 18 20 9519.4 2539.7 4.561.3 8548.6 9.574.7 2581.3 4697.5 8712.0 20 22 9521.2 2541.0 4.565.8 8554.0 9576.5 2582.7 4702.1 8717.6 22 24 9523.1 2542.4 4570.3 8.5.59.4 9.578.4 2.584.1 4706.8 8723.1 24 26 9524.9 2543.8 4574.8 8.564.8 9.580 2 2585.5 4711.4 8728.6 26 28 9526.8 2545.2 4.579.3 8570.2 9.582.0 2586.9 4716.0 8734.2 28 30 9528.6 2546.6 4583.7 8575.6 9.583.8 2588.3 4720.6 8739.7 30 32 9530.5 2548.0 4588.2 8581.0 9.585.7 2589.7 4725.3 8745.3 32 34 95.32.3 2549.4 4592.7 8586.4 9587.5 2591.1 4729.9 8750.8 34 36 9534.2 2550.7 4.597.2 8591.8 9.589.3 2.592.5 4734.6 8756.4 36 38 9536 2552.1 4601.7 8597.2 9591.1 2593.9 4739.2 8761.9 38 40 9537.9 2553.5 4606 2 8602.6 9.593.0 2.595.3 4743.9 8767.5 40 42 9539.7 2554.9 4610 8 8608.0 9594.8 2596.7 4748.5 8773.1 42 44 9541.6 2556.3 4615.3 ^6l3.5 9596.6 2.598.1 4753.2 8778.6 44 46 9543.4 2557.7 4619.8 8618.9 9598,4 2599.4 4757.9 8784.2 46 48 9.545.3 2559.1 4624.3 8624.3 9600.3 2600.8 4762.6 8789.8 48 50 9.547.1 2560.5 4628.9 8629.8 9602.1 2602.2 4767.2 8795.4 50 52 9549.0 2561.8 4633 4 8635.2 9603.9 2603.6 4771.9 8801,0 52 54 9.550.8 2563 2 4638.0 8640.7 9605.7 2605 4776.6 8806.6 54 56 9552.6 2564.6 4642.5 8646.2 9607 5 2606.4 4781.3 8812.2 56 58 '.)554 5 2566.0 4647 1 8651.6 9609.4 2607.8 4786 . 8817.8 58 60 9.556 3 2567.4 4651.6 86.57.1 9611.2 2609.2 4790.7 8823.4 60 IX. FUNCTIONS OF A ONE-DEGREE CURVE. 297 / 114° 1 115° / L. C. M. E. T. L. C. M. E. T. 9611 2 2609.2 4790.7 8823,4 9665.3 2651.3 4934.4 8994.3 2 9613.0 2610.6 4795.5 8829.1 9667.1 2652.7 4939.3 9000.1 2 4 9614.8 2612.0 4800.2 8834.7 9668.8 2654 . 1 4944.2 9005.9 4 6 9616.6 2613.4 4804.9 8840.3 9670.6 2655.5 4949.1 9011.6 6 8 9618.4 2614.8 4809.6 8846.0 9672.4 2656.9 4954.0 9017.4 8 10 9620.2 2616.2 4814.4 8851.6 9674.2 2658.3 4958.9 9023.2 10 12 9622.0 2617.6 4819.1 8857.2 9676.0 2659.7 4968.8 9029.0 12 14 9623.8 2619.0 4823.9 8862.9 9677.8 2661.1 4968.7 9034.8 14 16 9625.7 2620.4 4828.6 8868.5 9679.6 2662.5 4973.6 9040.7 16 18 962T.5 2621. e 4833.4 8874.2 9681.4 2663.9 4978.5 9046.5 18 20 9629.3 2623.2 4838.1 8879.9 9683.1 2665.4 4983.4 9052.3 20 22 9631 . 1 2624.6 4812.9 8885.5 9684.9 2666.8 4988.3 9058.1 22 24 9632.9 2626.0 4847.7 8891.2 9086.7 266H.2 4993.8 9064.0 24 26 9634.? 2627.4 4852.4 8896.9 9688.5 2669.6 4998.2 9069.8 26 28 9636.5 2628.8 4857.2 8902.6 9690.3 2671.0 5003.2 9075.7 28 30 9688.3 2630.2 4862.0 8908.3 9692.0 2672.4 5008.1 9081.5 30 32 9640.1 2631.6 4866.8 8914.0 9693.8 2673.8 5013.1 9087.4 32 34 9641.9 2633.0 4871.6 8919.7 9695.0 2675.2 .5018.0 9093.2 34 36 96-J3.7 2634.4 4876. 4 8925.4 9697.4 2676.6 5023.0 9099.1 36 38 9615.5 2635.8 4881.2 8931.1 9699.1 2678.0 5028.0 9105 38 40 9617.3 2637.2 4885.0 8936.8 9700.9 2679.5 5032.9 9110.8 40 42 96J9.1 2638.6 4890.9 8942.6 9702.7 2680.9 5037.9 9116.7 42 44 9650.9 ^640.0 4895.7 8948.3 9704.5 2682.3 5042.9 9122.6 44 46 9652.7 2641.4 4900.5 8954.0 9706.2 2683.7 5047.9 9128.5 46 48 9654.5 2642 9 4905.3 8959 8 9708.0 2685.1 5052.9 9134 4 48 50 9656.3 2644 3 4910.2 8965.5 9709.8 2686.5 5057.9 9140.3 50 52 9658.1 2645.7 4915.0 8971.3 9711.6 2687.9 5062.9 9146.2 52 51 9659.9 2647.1 4919 9 8977.0 9713.3 2689.3 5067.9 9152.1 54 56 9661.7 2648.5 4924.7 8982.8 9715.1 2690.7 5072.9 9158.1 56 58 9663.5 2649.9 4929.6 8988. 5 9716.9 2692.2 5078.0 9164.0 58 60 9665.3 2651 3 4934.4 8994.3 9718.6 2693.6 5083.0 9169.9 60 llfi^ 117° / L. C. M. E. T. L. C. M. E. T. 9718.6 2693.6 5083.0 9169.9 9771.3 2736.1 5236.6 9350.5 •> 9720.4 2695.0 5088 9175.9 9773.0 2737.5 5241.8 9356.6 2 4 9722.2 2696.4 5093.1 9181.8 9774.7 2738.9 5247.0 9362.7 4 6 9723.9 2697.8 5098.1 9188.8 9776.5 2740.4 .5252.2 9368.9 6 8 9725.7 2699.2 5103 2 9193.7 9778.2 2741.8 5257.4 9375.0 8 10 9727.4 2700.6 5108.2 9199.7 9779.9 2743.2 5262.6 9381.1 10 12 9729.2 2702.1 5113.3 9205.6 9781.7 2744.6 5267.9 9387.3 12 14 9731.0 2703.5 5118.4 9211.6 9783.4 2746.0 5273.1 9393.4 14 16 9732.7 2704.9 5123.4 9217.6 9785.2 2747.5 5278.4 9399.5 16 18 9734.5 2706.3 5128.5 92:.'3.6 9786.9 2748.9 5283.6 9405.7 18 20 9736.3 2707.7 5133.6 9229 9 9788.6 2750.3 5288.9 9411.9 20 22 9738.0 2709.1 5138.7 9235 5 9790.4 2751.7 5294.2 9418.0 22 24 9739.8 2710.6 5143.8 9211.5 9792.1 2753.2 5299.5 9424.2 24 20 9741.5 2712.0 5148.9 9247.6 9793.8 2754.6 5304.7 9430.4 26 28 9743. 3 2713.4 5154 9253.6 9795.6 2756.0 5310.0 9436.6 28 30 9745.0 2714.8 5159.1 9259.6 9797.3 2757.4 5315.3 9442.8 30 32 9746.8 2716.2 .5164.2 9265 6 9790.0 27.58.9 5320.6 9449.0 32 34 9748.5 2717.6 5169.4 9271.6 9800.7 2760.3 5325.9 9455.2 34 36 9750.3 2719.1 5174.5 9277 7 9802.5 2761.7 5331.2 9461.4 36 38 9752.0 2720.5 5179.7 9283.7 9804.2 2763.1 5336.5 9467.6 38 40 9753.8 2721.9 5184.8 9289.8 9805.9 2764.6 5341.8 9473.8 40 42 9755.6 2723.3 5190.0 9295.8 9807.7 2766.0 5847.2 9480.0 42 44 9757.3 2724.7 5195.1 9301.9 9809,4 •767.4 5352.5 9486.3 44 46 9759.0 2726.2 5200.3 9307.9 9811.1 2768.8 5357.9 9492.5 46 48 9760.8 2727.6 5205.4 9314.0 9812.8 2770.3 5363.2 9498.7 48 50 9762.5 2729.0 5210.6 9320.1 9814.5 2771.7 5368.5 9505.0 .50 52 9764.3 2730.4 5215.8 9326.1 9816.3 2773.1 5373.9 9511.2 52 54 9760.0 2731.8 5221.0 9332.2 9818.0 2774.6 5379.3 9517.5 54 56 9767.8 2733.3 .5226.2 9338.3 9819.7 2776.0 53S4.7 9.523.8 56 58 9769.5 2734.7 .5231.4 9344.4 9821.4 2777.4 5390.0 9530.0 58 60 9771.3 2736.1 .5236.6 93.50.5 9823.1 2778.8 5395.4 9.536 3 60 298 TABLE X.— SINES AND COSINES. ~0 0° 1 1° 1 2° 1 3° 1 40 60 sine .ooooo; Cosin One. Sine .01745 Cosin T99985 Sine 703490 Cosin Sine .05234 Cosin T99863 Sine Cosin .06976 .99756' .99939 1 .00029 One. .01774 .99984 .03519 .999:38 .05263 .99861 .07005 .99754 59 2 .OOO08 One. .01803 .99984 .03548 .999:37 .0.3292 .99860 .07034 .997521 58 3 .00087 One. .01832 .99983 .03577 .999)36 .05:321 .99858 .070631 .99750 57 4 .00116 One. .01862 .99983 .0:3606 .999:35 .05:350 .99857 .07092 .99748' 56 5 .00145 One. ; .01891 .99982; ; .036:35 .99934 .05:379' .99855 .07121 .99746 55 6 .00175 One. .01920 .99982 .0:3664 .99933 .05408 .99854 .07150 .99744 54 7 .00204 One. .01949 .999811 !. 0:3693 .999:32 .05437 .998521 .07179 .99742 53 8 .00233 One. .01978 .999801 .0:3723 .999.31 .05466 .998.51 .07208 .99740 52 9 .00262 One. .02007 .99980 .0:3752 .99930 .05495 .99849^ .072.37 .99738 51 lO .00291 One. .02036 .99979 .03781 .99929 j .05524 .99847: .07266 .99736 50 11 .00320 .99999 .02065 .99979 '.03810 .99927 i .05553 .99846 .07295 .99734 49 12 .00349 .99999 .02094 .99978 .0:3839 .99926 1 .05582 .99844 .07324 .99731 48 13 .00378 .999991 .02123 .99977 .0:3868 .99925 ! .05611 .99842 .07353 .99729 47 14 .00407 .99999 .021.52 .99977 .03897 .99924 .05640 .99841 .07:382 .99727 46 15 .00436 .99999 .02181 .99976 ; .03926 .99923 .05069 .99839 .07411 .99725 45 16 .00465 .99999 .02211 .99976 .0:3955 .99922 .05698 .99838 .07440 .99723 44 17 .00495 .99999 .02240 .99975 1.03984 .99921 .057'27 .99836, .07469 .99721 43 18 .00524 .99999: .02269 .99974 .04013 .99919 .05756 .99834 .07498 .99719 42 19 .00.553 .99998 .02298 .99974 .04042 .99918 .057M5 .998:33 .07527 .99716 41 20 .90582 .99998 .02:327 .99973 .04071 .999171 ! .05814 .99831 .07556 .99714 40 21 .00611 .99998 .02.356 .99972 .04100 .99910 ' .05844 .99829 .07585 .99712 39 22 .00640 .99998 .02385 .99972 .04129 .99915 .0.5873 .99827 .07614 .99710 38 23 .00669 .99998 .02414 .99971 1.04159 .99913 .05902 .99826 .('7643 .99708 37 24 .00698 .99998 .02443 .99970 .04188 .99912 .05931 .99824 .07672 .99705 36 25 .00727 .999971 .02472 .99969 .04217 .99911 .05960 .99822 .07701 .99703 35 26 .00756 .99997. ! .02.501 .99969 .04246 .99910 .0.5989 .99821 .07730 .99701 34 27 .00785 .99997 .025.30 .99968 : .04275 .99909 .00018 .99819 .07759 .99699 33 28 .00814 .99997 .02560 .99967 i .04.304 .99907 .06047 .99817 .07788 .99696 32 29 .00844 .99996 .02589 .99966 .04.333 .99906 .00076 .99815 .07817 .99694 31 30 .00873 .99996, .02618 .99966 .04362 .99905 '•• .06105 .99813 .07846 .99692 30 31 .00902 .99996 .02647 .99965 ! .04391 .99904 .00134 .99812 .07875 .99689 29 32 .009-31 .99996 .02676 .99964 1 .04420 .99902 .06163 .99810 .07904 .99687 28 33 .00960 .99995 .02705 .99963 .04449 .99901 .06192 .99808 .079:33 .99685 27 34 .00989 .99995 .027^4 .99963 .04478 .999(W .06221 .99806 .07962 .99683 26 35 .01018 .99995 .02763 .99962 ! .04507 .99898 i .06250 .99804 .07991 .99080 25 36 .01047 .99995! .02792 .99961 * .04.5:36 .99897 1 .06279 .99803 .08020 .99678 24 37 .01076 .99994 .02821 .99960 .04.565 1.99890 .06308 .99801 .08049 .99676 23 38 .01105 .999941 .028.50 .99959 i .04.594 '.99894 j .06337 .99799 1 .08078 .99673 22 39 .01134 .99994! .02879 .99959 .04023 .99893 .06:360 .99797 .08107 .99671 21 40 .01164 .99993 .02908 .99958 1.04653 '.99892 I .06395 .99795 .08136 .99668 20 41 .01193 .99993! .029.38 .99957 ''.04682 .99890 .06424 .99793 .08165 .99666 19 42 .01222 .99993 .02967 .99956 .04711 .99889 .06453 .99792 .08194 .99664 18 43 .01251 .99992! .02996 .99955 .04740 .99888 .00482 .99790 .08223 .99661 1 17 44 .01280 .99992; \ .03025 .999.54 .04769 .99880 .06511 .99788 .08252 .99659 16 45 .01309 .99991 .03054 .999.53 .04796 .99885 ! .06540 .997-86 .08281 .99657 15 46 .01338 .99991 .03083 .99952 .04827 .998a3 .06.569 .99784 .08310 .99654 14 47 .01367 .99991 .03112 .99953 .04856 .99882 .00598 .99782 .08.339 .99652 13 48 .01396 .99990 .03141 .99951 .04885 .99881 .06627 .99780 .08368 .99649 12 49 .01425 .999901 .03170 .99950 .04914 '.99879 ' .066.56 .99778 .08.397 .99647 11 50 .01454 .99989 .03199 .99949 .04943 ■ .99878 .06685 .99776 .08426 .99644 10 51 .01483 . 99989 ' .03228 .99948 .04972 .99876 .06714 .99774 .08455 .99642 9 52 .01513 .99989: .03257 .99947 .0.5001 .99875 .06743 .99772 .08484 .996:39 8 53 .01542 .99988 .03286 .90946 .0.5030 .99873 i .06773 .99770 .08513 .996.37 7 54 01.571 .99988! .03316 .99945 .050.59 .99872 .06802 .99768 .08542 .996:35 6 55 .01600 .99987 .0.3345 .99944 .05088 .99870 .06831 .99766 .08571 .996.32 5 56 .01629 .99987' .03374 .99943 1.0.5117 .99S69 .06860 .99764 .08600 .996:30 4 57 .01658 .99986 .0:3403 .99942 1 .0.5146 .99867 .06889 .99762 .08629 .99627 3 58 .01687 .99986 .0:34:32 .99941 .0.517-5 .99S60 1 .06918 .99760 .086.58 .99625 2 59 .01716 .99985 .03461 .99940 .0.5205 .99864 1 .0=> TABLE X— SINES AND COSINES. 299 , 5° 1 6» 70 «• 1 90 f Sine Cosin Sine Co.sin Sine Cosin ■ Sine 1 Cosin ! Sine Cosin "o .08716 799619 .10453 T99452 T12187 ^9925,5 .13917 .99027 .15643 798769 60 1 .08745 .99617 .10482 .99449 .12216 .99251 .13946 .99023 .15672 .98764 59 2 .08774 .99614 .10511 .994-16 .12245 .99248 .13975 .99019 .15701 .98760 58 3 .08803 .99612 .10540 .99443 .12274 .99244 .14004 .99015 .15730 .987.55 57 4 .08831 .99609 .10569 .99440 .12302 .'99240 .14033 .99011 ! .1575? .98751 56 5 .08800 .99607 .10597 .99437 .12331 .99237 .14061 .99006 ! .15787 .98746 55 6 .08889 .99604 [.10626 .99434 .12360 .99233 .14090 .99002 .15818 .98741 .54 7 .08918 .99602 .106.-)5 .99431 .12389 .992.30 .14119 .98998 .1.5845 .98737 53 8 .08947 .99.VJ9 . 10(]S4 .99428 .12418 .9i)226 .14148 .98994 M5873 .987:32 52 9 .08976 .99596 .10713 .99424 .12417 99222 .14177 .98990 .15902 .98728 51 10 .09005 .99594 .10742 .99421 .1247T} ! 99219 .14205 .98986 ' .15931 .98723 50 11 .09034' .99591 .10771 .99418 .12504 .99215 .14234 .98982 1.15959 .98718 49 12 . 09063 : .99588 . 10800 .994151 .12533 .99211 ' .14263 .98978 ' .15988 .98714 48 13 .09092 .99586 .10829 .99412 .12562 .99208' .14292 .98973 1.16017 .98709 47 14 09121 .99583 1 . 10858 .99409' .12591 .99204 .14320 .98969 .16046 .98704 46 15 09150 .99580 1.10887 .99406 .12620 .99200 .14340 .98962 .16074 .98700 45 16 .09179 .99578 :. 10916 .99402 .12649 .99197 .14.378 .98961 .16103 .98695 44 17 .09208 .99575 .10945 .99399; .12678 .99193 .14407 .98957 .16132 .98690 43 18 1 .09237 .99572 .10973 .99396' .12706 .99189 .14436 .98953 .16160 .98686 42 19 .09206 .99570 .11002 .993931 .12735 .99186 . 14464 .98948 .16189 .98681 41 20 .09295 .99507 .11031 .993901 .12764 .99182 .14493 .98944 .16218 .98676 40 21 .09324 .99564 L 11060 .993861 .12793 .99178' .14522 .98940 .16246 .98671 39 22 .09353 .99562 .11089 .99383 .12822 .99175 .14551 .98936 .16275 .98667 38 | 23 .09382 .99559 .11118 .99380 .128,51 .99171; .14580 .98931 1 .16304 .98662 37 24 .09411 .99556 .11147 .993771 .12880 .99167 .14608 .98927 .16333 .98657 36 25 .09440 .99.553 .11176 .993741 .12908 .99163 .14637 .98923 .16361 .986.52 35 26 .09469 .99.551 .11205 .99370' .12937 .99160 .14666 .98919 .16390 .98648 34 27 .09498 .99548 .11231 .99367! .12966 .99156 .14695 .98914 .16419 .98643 33 28 .09527 .99545 AViijii .99304 .12995 .99152 .14723 .98910 .16447 .986:38 32 29 .09556 .99.542 .11291 .99360 .13024 .99148 .14752 .98906 .16476 .986.33 31 30 .09585 .99540 .11320 .99357 .13053 .99144 .14781 .98902 .16505 .98629 30 31 .09614 .99537 .11349 .99351 .13081 .99141 .14810 .98897 .16533 .98624 29 32 .09642 .99534 .11378 .99351 .13110 .99137 .14838 .98893 .16.562 .98619 28 33 .09671 .99531 .11407 .99347 .13139 .99133 .14867 .98889 .16591 .98614 27 34 .09700 .99528 .11436 .99344, .13168 .991291 .14896 .98884 .16620 .98609 26 35 .09729 .99526 .11465 .99341! .13197 .991251 .14925 .98880 1.16648 .98604 25 36 .09758 .99523 .11494 .99337 .13226 .99122 i .14954 .98876 1.16677 .98600 24 37 .09787 .99520 .11.523 .99334 .132.54 .99118! .14982 .98871 :. 16706 .98595 23 38 .09816 .99.517 .11552 .99331, .13283 .991141 .15011 .98867 .167.34 .98.590 22 39 .09845 .99514 .11.580 .99327 .13.312 .99110 .1.5040 .98863 .16763 .98585 21 40 .09874 .99511 .11609 .99324 .13341 .99106: .15069 .98858 1.16792 .98580 20 41 ; 09903 .99.508 .11638 .99320 .13370 .99102' .1.5097 .98854 .16820 .98575 19 42 .09932 .99.506 1.11667 .99317 .13399 .99098! .1.5126 .98849 !. 168-19 .98570 18 43 .09961 .99503 .11690 .99314 .13427 .99094' .1.5155 .98845 .16878 .98565 17 44 .09990 .99.500 .11725 .99310 .134,58 .99091 .15184 .98841 .16906 .98561 16 45 .10019 .99497 .117.54 •99307 .i;i485 .990871 .15212 .98836 .169:35 .98.5.56 15 46 .10048 .99494 .11783 .99303 .13514 .99083 .1.5241 .98832 .16964 .98551 14 47 . 10077 .99491 .11812 .99300 .13.543 .99079 .15270 .98827 1.16992 .98546 13 48 .10106 .99488 .11840 .99297 .13.572 .99075! .15299 .98823 1 .17021 .98.541 12 49 .10135 .99485 .11869 .99293 .13600 .99071 .1.5.327 .98818 I .17050 .98.536 11 50 .10164 .99482 .11898 .99290 .13629 .99067 .15356 ] .98814 .17078 .98531 10 61 .10192 ! 99479 .11927 .99286 .136.58 .99063 .1.5.385 '.98809 .17107 .98526 9 52 .10221 .99470 ;. 11 9.50 .99283 .13687 .99059 .1.5414 .98805 .17136 .98.521 8 53 .10250 .99473 : .11985 .99279 .13716 .990.55 .1.5442 .98800 .17164 .98516 7 54 .10279 .99470 .12014 .99276 .13744 t. 990.51 .1.5471 .98796 .17193 .98.511 6 55 .10308 .99467 .12043 .99272 .13773 1.99047 .15.500 .98791 .17222 .98.506 5 56 .10337 .99464 .12071 .99269 .13802 i. 99043 .1.5.529 .98787 .172.50 .98.501 4 57 10366 .99461 .12100 .99265 .13831 .99039 .1.5.5.57 .98782 .17279 .98496 3 58 .10395 .994.58 .12129 .99262 .13860 .99035 .1.5.586 .98778 .17308 .98491 2 59 .10424 .994.55 .121.58 .992.58 .13889 .99031; .1.5615 .98773 .17:3:36 .98486 1 6< .10453 .99452 .12187 .99255 1.13917 .99027 .1.5643 .98769 :. 17365 .98481 ^ 1 Cosin Sine Cosin Sine ' Cosin , Sine" Cosin Sine 1 Cosin Sine f 84» 83» 82>» 81° 80» 300 Table x.— sines and cosines. 1, 10° 1 11° 12° 13° 14° ! 1 t Sine Cosin Sine Cosin Sine Cosin Sine Cosin Sine ! Cosin ~o .17365 .98481 .19081 .98163 .20791 .97815 .2-2495 .974:37 .24192 .97030 60 1 .17393 .98476 ' .19109 .98157 .20820 .97809 .2-252:3 .974:30 .24220 .970-23 59 2 .17422 .98171 .19138 .98152 .20848 .9780:3 .2-2552 .97424 .24-249 .97015 58 3 .17451 .98466 .19167 .98146 .20877 .97797 .2-2.5«0 .97417 .24277 .97008 57 4 .17479 .98461 .19195 .98140 .20905 .97791 .2-2608 .97411 24305 .97001 56 5 .17508 .98455 .19224 .98135 .209:33 .97784 .2-2637 .97404 .24:333 .96994 55 6 .17537 .98450 .19252 .98129 .20962 .97778 .2-2665 .07398 .24:362 .96987 54 7 .17565 .98445 .19281 .98124 .20990 .97772 .2-2693 .97:391 .24:390 .96980 53 8 .17594 .98440 .19:309 .98118 .21019 .97766 .2-2722 .97384 .24418 .96973 52 9 .1762:3 .984:35 .193:38 .98112 .21047 .97760 .22750 .97.378 .24446 .96966 51 10 .17651 .984:30 .19:366 .98107 .21076 .97754 .22778 .97:371 .24474 .96959 50 11 .17680 .98425 .19:395 .98101 .21104 .97748 .2-2807 .97365 .24503 .96952 49 12 .17708 .98420 .19423 .98096 .21132 .97742 .228:35 .97:358 .245:31 .96945 48 13 .17737 .98414 .19452 .98C90 .21161 .97735 .2-286:3 .97351 .24559 .96937 47 . 14 .17766 .98409 .19481 .98084 .21189 .97729 .2-2892 .97:345 .24587 .969:30 46 15 .17794 .98404 .19509 .98079 .21218 .97723 .2-2920 .97:338 .24615 .969-23 45 16 .17823 .98399 .19538 .98073 .21246 .97717 .2-2948 .973:31 .24644 .96916 44 17 .17852 .98:394 .19566 .98067 .21275 .97711 .2-2977 .97325 .24672 .96909 43 18 .17880 .98:389 .19595 .98061 .21:303 .97705 .2.3005 .97318 .24700 .96902 42 19 .17909 . 98:38:3 .1962:3 .98056 .21:3:31 .97098 .2:30:33 .97:311 .247-28 .96894 41 20 .179:37 .98:378 .19652 .98050 .21:360 .97692 .23062 .97304 .247.56 .96887 40 21 .17966 .98373 .196.80 .98044 .21:388 .97686 .23090 .97298 .247^ .96880 39 22 . 17995 .98:368 .19709 .980:39 .21417 .97680 .2:3118 .97-291 .24813 .96873 38 23 .18023 .98:362 .197:37 .980:3:3 .21445 .97673 .23146 .97284 .24841 .96866 37 24 .18052 .98:357 .19766 .98027 .21474 .97667 .23175 .97278 .24869 .968.58 36 25 .18081 .98352 .19794 .98021 .21502 .97661 .-23-203 .97271 .24897 .96851 35 26 .18109 .98;i47 .19823 .98016 .21.5:30 . 97655 .23231 .97264 .24925 .96.844 34 27 .18138 .98^1 .19851 .98010 .21559 .97648 .23260 .97257 i .24954 .968.37 .33 28 .18166 .98:336 .19880 .98004 .21.587 .97642 .23-288 .97251 ; .24982 .968-29 32 29 . 18195 .98331 .19908 .97998 .21616 .976:36 .23:316 .97244 ; .25U10 .96822 31 30 .18224 .98325 .19937 .97992 .21644 .97630 .23345 .97237 ' .25038 .96815 30 31 .18252 .98320 .19965 .97987 .21672 .97623 .23:373 .97230 .25066 .96807 29 32 .18281 .98315 .19994 .97981 .21701 .97017 .2^01 .972-23 .25094 .96800 28 33 .18309 .98310 .20022 .97975 .21729 .97611 .2i429 .97217 .25122 .96793 27 34 .18338 .98304 .20051 .97969 .21758 .97604 .2^458 .97210 .25151 .96786 26 35 .18367 .98299 .20079 .97963 .21786 .97598 .2^486 .97203 .25179 .96778 25 36 .18:395 .98294 .20108 .97958 .21814 .97592 .-2-3.514 .97196 .25-207 .96771 24 37 .18424 .98288 .201:30 .97952 .21843 .97585 .23.542 .97189 .25-235 .96764 23 38 .18452 .982*3 .20165 .97946 .21871 .97.579 .-2:3571 .97182 .25-263 .967.56 22 39 .18481 .98277 .20193 .97940 .21899 .97573 .23599 .97176 .25291 .96749 21 40 .18509 .98272 .20222 .979:34 .21928 .97560 .23627 .97169 .25:320 .96742 20 41 .18538 .98267 .202.50 .97928 .21956 .97560 .23656 .97162 .2.5348 .96734 19 42 .18567 .98261 .20279 .97922 .2198.5 .97553 .23684 .97155 .25:376 .96727 18 43 .18595 .98250 .20:307 .97916 .22013 .97547 .2:3712 .97148 .25404 .96719 17 44 .18624 .98250 .20:3:36 .97910 .22041 .97541 .23740 .97141 .254:32 .96712 16 45 .18652 .98245 .2a3t>4 .97905 .22070 .97.534 .23769 .971:^ .25460 .96705 15 46 .18681 .98240 .20:393 .97899 .22098 .97.528 .23797 .97127 .25488 .96697 14 47 .18710 .98234 .20421 .97893 .22126 .97521 .2:3825 .971-20 .2.5516 .96690 13 48 .18738 .98229 .204.50 .97887 .22155 .97515 .-2:3^53 .97113 .25545 .96682 12 49 .18767 .9822:3 .20478 .97HS1 .22183 .97508 oo^s.;-? .97106 .25.573 .96675 11 50 .18795 .98218 .20507 .97875 .22212 .97502 .'23910 .97100 .25601 .%667 10 51 .18824 .98212 .20535 .97869 .22240 .97496 .239:38 .97093 .256-29 .96660 9 52 .18852 .98207 .20563 .97863 .22268 .97489 .23966 .97086 .2.5657 .96653 8 53 .18881 .98201 .20592 .978.57 .22297 .97483 : .23995 .97079 .25685 .96645 7 54 .18910 .98196 .20620 .97851 .22:325 .97476 .240-23 .97072 .2.5713 .966:38 6 55 .18938 .98190 .20649 .97.845 .22:3.5:3 .97470 .-24051 .97065 ; .25741 .966:30 5 56 .18967 .98185 .20677 .978:39 .22:382 .97463 .-24079 .97058 ' .25769 .966-23 4 57 .18995 .98179 .20706 .9783:3 .22410 .97457 .24108 .97051 : .25798 .96615 3 58 . 190*24 .98174 .207:34 .97827 .224:38 .97450 .241:36 .97044 , .258-26 .96608 2 59 .19052 .98168 .2076:3 .97821 .22467 .97444 .241W .97(^37 ' .'25854 .96600 1 60 .19081 '.9816:3 .20791 .97815 .2-2495 .97437. 1 .24192 .97030 .25882 .96593 < Cosin Sine Cosin Sine Cosin Sine i Cosin i Sine ' Cosin Sine / 79° 1 78° 1 77° 76° 75° TABLE X.-SINES AND COSINES. 301 15° 1 1 16° 1 1 17° 1 1 18° 19° / Sine Cosin Sine Cosin Sine Cosin Sine Cosin Sine Cosin "o .25882 796593 .27564 .96126 .29237 795630 .30902 .9.5106 .32.557 794T552 60 1 .25910 .96585 .27592 .96118 .29265 .95622 .30929 .95C:)7 .32584 .94542 .59 2 .25938 .96578 .27620 .96110 .29293 .95613 .30957 .9.5088 .32612 .94533 58 3 .25966 .96570 .27648 .96102 .29321 .95605 .30985 .95079 .32639 .94523 57 4 .25994 .96562 .27676 .96094 , .29348 .95596 .31012 .95070 .32667 .94514 56 5 .26022 .96555 .27704 .96086 1 .29376 .95588 .31040 .95061 .32694 .94504 55 6 .26050 .96547; .27731 .96078 .29404 .95579 .31068 .95052 .32722 .94495 54 7 .26079 .96540' .27759 ■96070 .29432 .95571 .31095 .95043 .32749 .94485 53 8 .26107 .9()532 .27787 .96062 .29460 .95562 .31123 .95033 .32777 .94476 52 9 .26135 .96524 .27815 .9()().54 .29487 .95554 .31151 .95024 .32804 .9446(i 51 10 .26163 .96517 .27843 . 96046 .29515 .95545} .31178 .95015 .32832 .94457 50 11 .26191 .96509' .27871 .960371 1 .29543 .95536 .31206 .95006 .32859 .94447 49 12 .26219 .96502 .27899 .96029 .29571 .9.5528 .31233 .94997 .32887 .?>um 48 13 .26247 .96494 .27927 .96021' .29599 .95519 .31261 .94988 .32914 .94428 47 14 .26275 .96486 .27955 .96013 .29626 .95511 .31289 .94979 .32942 .94418 46 15 .26303 .96479 .27983 .96005 .29654 .95502 .31316 .94970 .32969 .94409 45 16 .26331 .96471 .28011 .95997 1 .29682 .95493 .31344 .94961 .32997 .94399 44 17 .26359 .96463 .28039 .95989 ' .29710 .95485 .31372 .94952 .33024 .94390 43 18 .2638? . 96456 .28067 .959811 .29737 .9:5476 .31399 .94943 .33051 .94:380 42 19 .26415 .96448 .28095 .95972 .29765 .95467 .31427 .94933 1 ,33079 .94370 41 20 .2644:3 .96440 .28123 .95964 ' .29793 .95459 .31454 .94924 .33106 • .94:361 40 21 .26471 .96433 .28150 .959.56 .29821 .95450' : .31482 .94915 .33134 .94351 39 22 .26500 .96425 .28178 .95948 .29849 .9.5441 1 .31510 .94906 .33161 .94342 38 23 .26528 .96417 .28206 .95940 .29876 .95433 .31537 .94897 .33189 .94332 37 24 .26556 .96410 .28234 .95931 .29904 .954241 ' .31565 .94888 .33216 .94:322 36 25 .26584 .96402 .28262 .95923 : .29932 .95415; ; .31593 .94878 .33:^4 .94313 35 26 .26612 .96394 .28290 .95915 .29960 .95407 .31620 .94869 .33271 .94303 34 27 .26640 .96386 .28318 .95907 : .29987 .95398 .31048 '.94860 .33298 .94293 33 28 .26668 .96379 .28346 .95898 .30015 .95389 i .31675 .94851 .33326 .94281 32 29 .26696 .96371 .28374 .95890 .30043 .95380 j .31703 .94842 .333.53 .94274 31 30 .26724 .96363 .28402 .95882 .30071 .95372 .31730 .94832 ■ .33:381 .94204 30 31 .26752 .96355 .28429 .95874 .30098 .95363 .31758 .94823 ' .33408 .94254 29 32 .26780 .96347 .28457 .95865 ! .30126 .95354' .31786 .94814 .33436 .94245 28 33 .26808 .96340, .28485 .95857 i .30154 .95345 .31813 .94805 .33463 .94235 27 34 .26836 .96332 .28513 .95849 .30182 .95337 .31841 .94795 .33490 .94225. 26 35 .26864 . 96324 i .28541 .95841 .30209 .95328 .31868 .94786 .33518 .94215 25 36 .26892 .96316 .28569 .95832 .30237 .95319 .31896 .94777 .33545 .94206 24 37 .26920 .96308 .28597 .95824 .30265 .95310 .31923 .94768 .3:3573 .94196 23 38 .26948 .96301 .28625 .95816 .30292 .95301 .31951 .94758 ; .;33600 .94186 22 3!) .26976 .96293 .28652 .9.5807 .30320 .95293 .31979 .94749 .33627 .94176 21 40 .27004 .96285 .28680 .95799 .30348 .95284 1 .32000 .94740 .33655 .94167 20 41 .27032 .9627?' .28708 .95791 .30376 .95275 ! .32034 .947'30 .33082 .941.57 19 42 .27060 .96269: .28736 .95782 .30403 .95266! 1 .32061 .94721 .33710 .94147 18 43 .27088 .96261 .28764 .95774 ' .30431 .95257 ; .32089 .94712 1 .a3737 .941:37 17 4-1 .27116 .96253 .28792 .95766 ! .30459 .95248 i .32110 .94702 .33764 .94127 16 45 .27144 .96246 .28820 .95757 .30480 . 95240 i .32144 .94693 .33792 .94118 15 46 .27172 .96238 28847 .95749 .3051> .95231 .32171 .94681 ! .33819 .94108 14 47 .27200 .96230; ."28875 .95740 .30542 .952221 .32199 .94674 ' .33846 .94098 13 48 .27228 .96222 .28903 .95732 ; .30570 .952131 .32227 .94665 i .33874 .94088 12 49 .27256 .96214 .28931 .95724 1 .30597 .95204 .32254 .946.56 3:3901 .94078 11 50 .27284 .96206 .28959 .95715 .30625 .95195 .32282 .94646 .33929 .94068 10 51 .27312 .96198 .28987 .95707 .30653 .95186 .32309 .94637 .33956 .94058 9 52 .27340 .96190 .29015 .95698] .30680 .95177 .32337 .94627 .33983 .94049 8 53 .27368 .96182 .29042 .95690 .30708 .95168 .32364 .94618 .34011 .94039 7 54 .27396 .96174 .29070 .95681 .30736 .95159 .32392 .94(109 ..^038 .94029 6 55 .27424 .96166 .29098 .95673 .307'63 .95150 .32419 .94.599 .34065 .94019 5 56 .27452 .96158 .29126 .95664 .30791 .95142 .32447 .94590 .340931 .94009 4 57 .27480 .96150 .291,54 .95656 .30819 .95133 .32474 .9.3.5801 .;34120 .93999 3 58 .27508 .96142 .2!)! 82 .95647 .;30846 .951^ .32.502 .94571 .:34147 .9:3989 2 59 .27536 .961.^ .29209 .9.5639, .30874 .95115 .32529 .94:561 .34175, .9:39791 1 60 .27564 .96126 .29237 .9.5630 .:30902i .95106 .32557 .94.552 .34202! .939691 / Cosin Sine 1 C3osin ' Sine Cosin Sine Cosin Sine Cosin 1 Sine 74» 1 73° 72° i 71° 1 to° 303 TABLE X.-SINES AND COSINES. r 20° 2] ■" 1 22 ! 23 ° il 24° 1 Sine Cosin Sine Cosin ! Sine Cosin ! Sine Cosin Sine Cosin / "o ^34202 .93969 .3.58:37 . 9:3.358 i .;37461 .92718 .3907:3 .92050; .4067'4 .913.55' 60 1 .34229 .939.59 1 .35864 .9:31*48 ; .37488 .92707 .39100' .92039! A0700 .91:343 59 2 .34257 .93949: .35891 .93:3:371 .37515. .92697; .39127' .92028: .407'27 .91:331 58 3 .34284 .93939! .35918 .9:3:327 .37542! .92686; .391.53! .92016; .40753 .91319 57 4 .34311 .93929; .35945 .93:316 .37569 .92675 .39180' .92005' .40780 .91307 56 & .34339 .939191 .35973 .93:306 .37595 .926641 .39207! .919941 .40806 .91295 55 6 .34366 .939091 .36000 .9:3295 .37622; .92653 .392.34 .91982; .40833 .91283 54 7 .34393 .93899 .:36027 .9:3285 ..37649 .92642 .39260 .919711 .40860 .91272 53 8 '.34421 .93889; .:36054 .9.3274 .37676' .92631 .39287 .919591 .40886 .91260 52 9 .34448 .93879 .,36081 .9:3264 .37703 .92620 .39:314 .91948' .40913 .91248 51 10 .34475 .93869 .36108 .9:3253 .37730 .92609 .39341 .91936; .409.39 .91236 50 11 ..34.503 .93859 .36135 .9.3243 .01 lOl .92598 .39367 .91925 .40966' .91224 49 12 .34.5.30 .93849 .36162 .9:32.32 .37784 .92587 ..39.394 .91914 .40992 .91212 48 13 .34.557 .93839; .36190 .9:3222 .37811 .92576 • .:3942I .91902; .41019 .91200; 47 14 .34.584 .938291 .36217 .0:3211 . 3 ( 8:38 .92565 .;39448 .91891 1 .41045 .91188 46 15 .34612 .93819 .36244 .9:3201; . 37865 .92554 .39474 .918791 .41072 .91176 46 IG ..34639 .93800! .36271 *.93190 .37892 .92543 .39501 .91868' .41098 .91164 44 17 ..34666 .9.3799; .36298 .93180 .37919 .92532; .39528 .91856! .41125 .91152 43 18 .34694 .93789 .36325 .93169 .37946 .92521; 1 ..S9.555 .91845; .41151 .91140 42 19 .34721 .93779 ' .36:3.52 .93159 .37973 .92510: ' .39.581 .9ia33' .41178 91128 41 iiiO .34748 .93769, j .36.379 .93148: .37999 .92499 1 .39608 .91822^ .41204 .91116 40 21 .34775 .9.375©! 1 .36406 .93137; ..38026 .92488 ' .396.35 .91810 .412.31 .91104 39 22 ..34803 .93748 ..364:34 .931271 .38053 .92-i77 .39661 .91799 ' .41257 .91092 .38 23 ..348.30 .937.38 .3(>461 .931101 .38080 .92466, .39688 .91787 ; .41284 .91080 37 24 .34857 .93728 .36488 . 93106 1 .38107 .92455' .3971.5, .91775 1 .41310 .91068 :36 25 .34884 .93718 1 .36515 .93095, .aSi34 .92444 .39741 .917(>4 .41.337 .91056 35 26 .34912 .9.3708 .36542 .9:3081 .:38101 .924:32 .39768 .91752 : .41.363 .91044 34 27 .349.39 .9.3698 ..36569 .93074 .38188 .9242. t .39795 .91741 .41390 .910:32 33 28 .34966 .9.3688 .36.596 .9.3063 .38215 .92410 .:39822 .91729 .41416 .91020 .32 29 .34993 .93677 .36623 .930.521 .:-:8241 .92:399 ..3984!^ .91718 .41443 .91008 31 30 .35021 . 93667 1 .36650 .93042 .38268 .92.388 ,.39875 .91706 : .41469 .90996 30 1 31 .35048 .9.3657 ' ..36677 .9:3031 ..38295 .92377 ! .;399<}2 .91694 .41496 .90984 29 32 .35075 .93647 ..36704 .9:3020 .:38322 .92:366 .39928 .91688 .41522 .90972 28 33 .35102 .93637 .36731 .9:3010 .38:349 .92:355 .39955 .91671 ' .41549 .90960 27 34 ..35130 .93626 .:367.58 .92999 .38376 .92:343 .39982 .9I6(;0 .41575 .90948 26 35 .36157 .93616 ..36785 .92988' .38403 .92:332 .40008 .91648 .41602 .909:36 25 36 .35184 .93606 ; ..36812 .92978 .384:30 .92:321 .400:35 .91636 .41628 .90924 24 37 .3.5211 9.3596 : ..368:39 .92967 .:3a456 .92.310 .40062 .91625 .416.55 .90911 23 38 .. 3.52:39 .93.585 ..36867 .929.56 .38483 .92299 ' .40088 .91613 .41681 ! .90899 22 39 .3.5266 .93.575 ..36891 .92945 ..38510 .92287 ! .40115 .91601 .41707 .90887 21 40 .35293 .9.3565 ..36921 .92935 .38537 .92276 ! .40141 .91590 .41734 .90875 1 20 41 .a5.320 .9.3.555 ..36948 .92924 ' .3856-1 '.92265 ' .40168 .91.578 ' .417'60 .90863 19 42 ..3.5347 .93544 , .:36975 .92913 .38.591 .922.54 .40195 .91566 .47787 .90851 18 43 . .35375 .93534 .:37002 .92902 . .38617 .92243 .40221 L 91555 .41813 .908:39 17 44 ..3.5402 .93.5.'^! .37029 .92892 ..38f)}4 .92231 : .40.248 .91.543 .41840 .90826 16 45 ..35429 .9.3514 .370.56 .92881 ' ..38671 .92220 .40275 ;. 91.5:31 .41866 .90814 15 46 .354.56 .93503 .37083 .92870 ..38698 .9^209 .#198 .40:301 !. 91.519 .41892 ;. 90802 14 47 .3^K4 .9:^^493 .37110 .92a59 ..•i8725 .40328 .91.508 .41919 .90790 13 48 . 3551 1 .93483 .3; 1:37 .9:^849 , 00 t 04 .92180 .40:355 .91496 .41945 .90778 12 49 ..3.5.538 .93472 ..37164 .928:38 .38778 .92175 .40381 1.91484 .41972 .90766 11 50 .35565 .93462 .37191 .92827 .38805 .92164 .40408 .91472 .41998 _. 90753 10 51 .35592 .93452 ..37218 .92816 .388.32 .921.52 i .40434 .91461 .42024 ' .90741 9 52 ..3.5619 .93441 ..37245 .92805! ..388.59 .92141 • .40461 .91449 .42051 .90729 I 8 53 .3.5647 .9:3431 .:37272 .92794 ..38886 .921.30; ' .40488 .91437 .42077 !. 90717 : 7 54 ..35574 .93420 .37299 .92784; ; .38912 .92119! 1 .40.514 .91425 .42101 '.907'n4' 6 55 ..3.5701 .9.3410 .37326 .92773 ..389:39 .92107; i .40.541 .91414 .421.30 .90692 5 56 .35728 .9.3400 ..37:353 .92762 ..38966 .92096' ! .40.567 .91402 .421.56 .90680 1 4 57 ..3.57.55 .93.389 .:37.380 .92751 i .38993 .92085 .40.594 .91390 .42183 .90068 . 3 58 .35782 .9.3:379 .37407 .92740 .:39O20 .9207-31 .40621 .91.378 42209 .906,55 59 ..35810 .9.3:368 ..37434 .92729' ..39046 .92062 .40647 .91366 .42235 .90643 1 60 .358-37 .93358 .37461 .92718 ..39073 .920.50 .40674 .913.55 .42262 .90631 li Cosiu Sine 1 Cosin Sine Cosin Sine | Cosin Sine Cosiu 1 Sine / 69° ! 68' ! 67° 66° 65° .'ABLE X.-SINES AND COSINES. 303 / 25° 26° 1 27° 28° : 29° / Sine Cosin Sine Cosin Sine Cosin Sine Cosin ' Sine Cosin "o .422(52 .90631 .4:3837 789879 .4.5399 .89101 1 .4(5947 788295 .48481 T87462 60 1 .42288 .90618 .43863 .89807 .45425 .89087 .46973 .88281 .48506 .87448 59 2 .42315 .90606 .43889 .89854 .45451 .89074 .46999 .88267 .48532 .87434 58 3 .42341 .90.594 ; .4:3916 .89841 .45477 .89061 .47024 .882541 .48557 .87420 57 4 .42367 .90582 .43942 .89828 .45.503 .89048 .47050 .88240; .48583 .87406 56 5 .42394 .90569 .43968 .898161 .45.529 .890:35 .47076 .88226 .48608 .87391 55 6 .42120 .90557 .4:39i)4 .898031 .455.54 .89021 .47101 .88213 .48634 . 87377 54 7 .42446 .90545 .44020 .89790 .4,5.580 .89008 .47127 .88199 .48659 .87.363 53 8 .42473 .90.532 .44046 .89777 .45606 .88995 .471.53 .88185 .48684 .87.349 52 9 .42199 .90520 .44072 .89764 .45(5:32 .88981 .47178 .88172 .48710 .873,35 51 10 .42525 .90507 .44098 .89752 .45658 .88968 .47204 .88158 .48735 .87.321 50 11 .425.52 .90495 .44124 .89739 .45684 .88955 .47229 .88144 .48761 .87.306 49 12 .42578 .90183 .44ir.i .89726 .45710 .83942 .472.55 .88130 .48786 .87292 4i^ 1-3 .42604 .90470 .44177 .80713 i .457:36 .88928 .47281 .88117 .48811 .87278 47 14 .42631 .90458 .44203 .89700 .45;'62 .88915 .47306 .88103 .488:37 .87264 46 15 .42657 .90446 .44229 .89687 .45787 .88902 .47332 .88089 .48862 .87250 45 16 .42683 .904:33 .44255 .89674 .45813 .88888 .473.58 .88075 .48888 .87235 44 17 .42709 .90421 .44281 .896631 .4.58:39 .88875 1 .47.383 .88062 1 .48913 .87221 43 18 .42736 .90408 .44:307 .89649' .4.5865 .88862 ' .47409 .88048 1 .48938 .87207 42 19 .42762 .90:396 : .44:3:53 .896:36! .45891 .88848 .474:34 .88034 .48964 .87193 41 20 .42788 .90383 .44359 .89623: .45917 .88835 .47460 .88020 ' .48989 .87178 40 21 .42815 .90:371 .44.385 .89610; .45942 .88822 1.47486 .88006 i .49014 .87164 39 22 .42841 . 90358 .44411 .89597 .4.5968 .88808 .47.511 .87993 .49040 .871.50 38 23 .42867 .90346 .444:37 .89.584' .4.5994 .88795 .47537 .87979 .49065 .87136 37 24 .42894 .903:34 .44464 .89.571- .46020 .88782 .47562 .87965 .49090 .87121 36 25 .42920 .90:321 .44490 .89.558; .46046 .88768 1 .47588 .87951 .49116 .87107 35 2H .42946 .90309 .44516 .89.5451 .46072 .88755 .47614 .87937 .49141 .87093 34 27 .42972 .90296 .44.542 .89.532 .46097 .88741 .476:39 .87923 .49166 .87079 33 2S .42999 .90284 .44508 .89519 ' .46123 .88728 .47665 .87909 .49192 .87064 32 29 .43025 .91271 .44.594 .89506 .46149 .88715 .47690 .87896 .49217 .870.50 31 30 .43051 .902.59 .44620 .89493 .46175 .88701 .47716 .87882 .49242 .87030 30 31 .4.3077 .90246 .44046 .89480 .46201 .88688 .47741 .87868 .49268 .87021 29 32 .43104 .902:33 .44672 .89407 .46226 .88674 .47767 .87854 .49293 .87007 28 33 .43130 .90221 .44698 .89454 .46252 .88661 .47793 .87840 .49.318 .8(5993 27 34 .43156 .90208 .44724 .89441 .46278 .886471 .47818 .87826 .49.344 .86978 26 35 .43182 .901961 .44750 .89428 .46.3(34 .88(5.34' .47844 .87'812 .49369 .86964 25 3(5 .4.3209 .90ia3 .44776 .89415 .46:3.30 .88620 .47869 .87798 .49394 .86949 24 37 .43235 .90171 .44802 .89402 ' .4(5:355 .88607 .47895 .877-84 .49419 .86935 23 1 38 .43261 .90158 .44828 .89:389 .46.381 .88593 .47920 .87770 .49445 .8(5921 22 ! 39 .43287 .90146 .448.54 .89:376 .46407 .88.580 .47946 .877.56 .49470 .8(5906 21 1 40 .43313 .901:33, .44880 .89:363 .46433 .88566 .47971 .87743 .49495 1 .86392 20 1 41 .43.340 .90120 .44906 .89350 .464.58 .88553 .47997 .877-29 1 .49521 .86878 19 ' 42 .4:!366 .90108 .449:32 .89.337 .4648-1 .885.39 .48022 .87'715 .49546 .86863 18 ' 43 .4.i392 .90095 .449.58 .89:324 .4(5510 .88526 .48048 .87701 .49571 .86849 17 44 .4?>418 .90082 .44984 .89311 .46.5:36 .88512 .4807'3 .87-G87 .49596 .80831 16 45 4;M45 .90070 .4.5010 .89298 .46.561 .88499 .48099 .87-673 .49622 .86820 15 46 .4:3471 .900.57 .45036 .89285 .46587 .88485 .48124 .87659 .49647 .86805 14 47 .4:3497 .900151 .4.5062 .89272 .40613 .88472 .48150 .87'645 .49672 .80791 13 48 .4:3523 .900:32 .4.5088 .89259 .466:39 .88458 .48175 .87631 .49697 .8677V 12 49 .4.3.549 .90019 .45114 .89245 .4 00000 00016 00063 00140 00248 00387 00557 58 3 00000 01K)17 00064 00142 00250 00390 00560 57 4 00000 00017 00065 00143 00252 00392 00563 56 5 t)0000 00018 00066 00145 00254 00395 00566 55 f) 00000 00018 00067 00147 00257 00397 00569 54 t 00000 00019 , 00068 00148 00259 00400 00573 53 8 00000 00020 00069 00150 0026 1 00403 00576 52 9 00000 00020 00070 00151 00263 00405 00579 51 10 00000 00021 00072 00153 00265 00408 00582 50 11 1.00001 1.00021 1.00073 1.00155 1.00267 1.00411 1.00585 49 12 00001 00022 00074 00156 00269 00413 00588 48 13 00001 00023 00075 00158 00271 00416 00592 47 14 00001 00023 00076 00159 00274 00419 00595 46 15 00001 00024 00077 00161 00276 00421 00598 45 16 00001 00021 00078 00163 00278 00424 00601 44 17 00001 00025 00079 00164 00280 00427 00604 43 18 00001 00026 00081 00166 00282 00429 00608 42 19 00002 00026 00082 00168 00284 00432 00611 41 20 00002 00027 00083 00169 00287 00435 00014 40 21 1.00002 1.00028 1.00084 1.00171 1.00289 1.00438 1.00617 39 22 00002 00028 00085 00173 00291 00440 00621 38 23 00002 00029 00087 00175 00293 00443 00624 37 24 00002 00030 00088 00176 00296 00446 00627 36 25 00003 00031 00089 00178 00298 00449 00630 35 26 00003 00031 00090 00180 00300 00451 00034 34 27 00003 00032 00091 00182 00302 00454 00037 33 28 00003 00033 00093 00183 00305 00457 00640 32 29 00004 00034 00094 00185 00307 00460 00644 31 30 00004 00034 00095 00187 00309 00463 00647 30 31 1.00004 1.00035 1.00097 1.00189 1.00312 1.00465 1.00650 29 SZ 00004 00036 00098 001 !)0 00314 00468 00654 28 33 00005 00037 00099 00192 00316 00471 00657 27 34 00005 00037 00100 00194 00318 00474 00660 26 35 00005 00038 00102 00196 00321 00477 00064 25 36 00005 00039 00103 00198 00323 00480 00667 24 37 00U06 00040 00104 00200 00326 00482 00671 23 38 00006 00041 00106 00201 00328 00485 00674 22 39 00006 00041 00107 00203 00330 00488 00677 21 40 00007 00042 00108 00205 00333 00491 00681 20 41 1.00007 1.00043 1.00110 1.00207 1.00335 1.00494 1.00684 19 42 00007 00044 00111 00209 00337 00497 00688 18 43 00008 00045 00113 00211 00340 00500 00691 17 44 00008 00046 00114 00213 00342 00503 00695 16 45 00009 00047 00115 00215 00345 00506 00698 1.5 46 00009 00048 00117 00216 00347 00509 00701 14 47 00009 00048 00118 00218 00350 00512 00705 13 48 00010 00049 00120 00220 00352 00515 00708 12 49 00010 00050 00121 00222 00354 00518 00712 11 50 00011 00051 00122 00224 00357 00521 00715 10 51 1.00011 1.00052 1.00124 1.00226 1.00359 1.00524 1.00719 9 52 00011 00053 00125 00228 00362 00527 00722 8 53 00012 00054 00127 00230 00364 00530 00726 54 00012 00055 00128 00232 00367 00533 00730 6 55 00013 00056 00130 00234 00369 00536 00733 5 56 00013 00057 00131 00236 00372 00539 00737 .4 57 00014 00058 001.33 00238 00374 00542 00740 1 58 00014 00059 00134 00240 00377 00545 00744 59 00015 00060 00136 00242 00379 00548 00747 T 60 00015 00061 00137 00244 00382 00551 00751 f 89° 88° 87° 86° 85° 84° 83° / COSECANTS. 308 XI. -NATURAL SECANTS AND COSECANTS. / SECANTS. / 7° 8° 9° 10° 11° 12° 13° 1.00751 1.00983 1.01247 1.01543 1.01872 1.02234 1.02630 60 1 00755 00987 01251 01.548 01877 02240 02687 59 2 00758 00991 012.56 01553 01883 02247 02644 58 3 00762 00995 01261 01.5.58 01889 02253 02051 57 4 00765 00999 01205 01564 01895 02259 02658 56 5 00769 01004 01270 01569 01901 02266 02665 55 6 00773 01008 01275 01.574 01906 02272 02672 54 7 00776 01012 01279 01.579 01912 02279 02679 53 8 00780 01016 01281 01585 01918 02285 02686 52 ' 9 00784 01020 01289 01590 01924 02291 02693 51 10 00787 01024 01294 01595 01030 02298 02700 50 11 1.00791 1.01029 1.01298 1.01601 1.01936 1.02304 1.02707 49 12 00"; 95 01033 01303 01606 01941 02311 02714 48 13 00799 01037 01308 01011 01947 02317 02721 47 14 00802 01011 01313 01616 01953 02323 02728 46 15 00806 01046 01318 01622 019.59 02330 02735 45 16 00810 01050 01322 01627 01965 02336 02742 44 17 00813 01054 01327 01633 01971 02313 02749 43 18 00817 01059 01332 01038 01977 02349 02756 42 19 00821 01063 01337 01643 01983 02356 02763 41 'JO 00825 01067 01342 01649 01989 02362 02770 40 21 1.00828 1.01071 1.01316 1.01654 1.01995 1.02369 1.02777 39 22 00832 01076 01351 016.59 02001 02375 02784 38 23 00836 01080 013.56 01665 02007 023S2 02791 37 24 00840 01084 01361 01670 02013 02388 02799 36 25 00844 01089 01366 01676 02019 02395 02806 35 26 00848 01093 01371 01681 02025 02402 02813 34 27 00851 01097 01376 01687 02031 02408 02820 33 28 00855 01102 01381 01692 02037 02415 02827 32 29 00859 01106 01386 01698 02043 02421 02834 31 30 00863 01111 01391 01703 02049 02428 02842 30 31 1.00867 1.01115 1.01395 1.01709 1.02055 1.02435 1.02849 29 32 00871 01119 01-100 01714 02061 02441 02856 28 33 00875 01124 01405 01720 02067 02418 028C3 27 34 00878 01128 01410 01725 02073 024.54 02870 26 35 00882 01133 01415 01731 02079 02461 02878 25 36 00S86 01137 01420 01736 (.2085 02468 028S5 24 37 0i890 01142 01425 01742 02091 02474 02892 23 38 00894 01146 01430 01747 02097 02481 02899 22 39 00898 01151 01135 01753 02103 024S8 02907 21 40 00902 01155 01410 017.58 02110 02494 02914 20 41 1.00906 1 01160 1.01445 1.01764 1.02116 1.02501 1.02921 19 42 00910 01164 OI4.0O 01769 02122 02508 02928 18 43 00914 01169 01455 01775 02128 02515 02936 17 44 00918 01173 01461 01T81 02134 02521 02913 16 45 00922 01178 01466 01786 02140 02528 029.50 15 46 00926 01182 01471 01792 02146 02535 02958 14 47 00930 01187 01476 01798 02153 02542 02965 13 48 009:^4 01191 01481 01803 02159 02548 02972 12 49 0093S 01196 014-6 01809 02165 02555 02980 11 50 00942 01200 01491 01815 02171 02562 02987 10 51 1.00946 1.01205 1.01496 1.01820 1.02178 1.02.569 1.02994 9 52 00950 01209 01501 01826 02184 02576 03002 8 53 00954 0121 01506 01832 02190 02582 03009 7 54 00958 01219 01512 01837 02196 02589 03017 6 55 00962 012:3 01517 01843 02203 02596 03024 5 56 00966 01228 01522 01849 02209 02603 03032 4 57 00970 01233 01527 018.14 02215 02610 03039 3 58 00975 01237 01532 01860 02221 02617 03046 59 00979 01242 01.537 01866 02228 02624 0:^054 I 60 00983 01247 01543 01872 02234 02630 03061 / 82° 81° 80° 79° 78° 77° 70° / COSECANTS. XI.— NATURAL SECANTS AND COSECANTS. 309 / SECANTS. / 14° 15° 16° 17° 18° 19° 20° 1.03061 1.03528 1.04030 1.04.569 1.05146 1.0.5762 1.06418 60 1 03069 03536 04039 04578 051,56 05773 06429 59 o 03076 03544 04047 04588 051G6 05783 06440 58 3 030S4 03552 04056 04.597 05176 05794 06452 57 4 03091 03560 04065 04606 05186 05805 06463 56 5 03099 03568 04073 04616 05196 0.5815 06474 55 6 03106 03576 040S2 04025 05206 05826 06486 54 i 03114 03584 04091 04635 0.5216 05836 06497 13 8 03121 03592 04100 04641 05226 05847 08.508 52 9 03129 03601 01108 04653 05236 05858 06520 51 10 03137 03609 04117 01663 05246 05869 06531 50 11 1.03144 1.03617 1.04126 1.01672 1.052.56 1.0.5879 1.06542 49 1'^ 03152 03625 04135 04682 05266 05890 06554 48 Vi 031.59 03633 04144 04691 05276 05901 06565 47 11 03167 03642 04152 04700 05286 0.5911 06577 46 15 03175 03650 04161 04710 05297 05922 065S8 45 10 03182 03658 04170 04719 05307 05933 06600 44 IT 03190 03666 04179 04729 05317 05944 06611 43 18 03198 03674 04188 04738 05.327 05955 06622 42 1!) 03-J05 03683 04197 04748 0.5337 0.5965 06634 41 20 03il3 03691 04206 04757 0.5347 05976 06645 40 21 1.03221 1.03699 1.04214 1.01767 1.0.53.^7 1.05987 1.066.57 39 2i 03228 03708 .04223 04776 05367 0.5998 06668 38 23 03236 03716 04232 04786 05378 06009 06680 37 21 03244 03724 04241 04795 05388 06020 06691 36 25 03251 03732 042.50 04805 05398 06030 06703 35 2G 032.59 03741 04259 04815 05408 06041 06715 34 27 03:67 03749 04268 01824 0.5418 06052 06726 33 28 03-^75 03:5s 04277 04834 05429 06063 00738 32 29 03282 03766 042S6 04843 05439 06074 06749 31 30 03290 03774 04295 04853 05449 06085 06761 30 31 1 .03298 1.0.3783 1.04.304 1.04863 1.0.5400 1.06096 1.06773 29 32 03306 03791 04313 04872 05470 06107 06784 28 33 03313 03799 04322 04882 05180 06118 06796 27 34 03321 03808 04331 04891 05490 06129 06807' 26 35 03329 03816 01340 04901 05501 06140 06819 25 3(5 03337 03825 04349 04911 05511 06151 06831 24 37 03345 03833 04358 04920 0.5521 06162 06843 23 38 03353 0.3842 04367 04930 05532 06173 06854 22 39 03360 03850 04376 04940 05542 06184 06866 21 40 03368 03858 04385 049.50 05552 06195 06878 20 41 1.0.3376 1.0.3867 1.04394 1.04959 1.0.5.503 1.06206 1.06889 19 42 03384 03875 04403 04969 05573 06217 06901 18 43 03392 03884 04413 04979 05584 06228 06913 17 44 03400 03892 04422 04989 0.5594 06239 06925 16 45 03408 03901 01431 04998 05604 06250 069.36 15 4G 03416 03909 04140 05008 0.5615 06261 06948 14 47 03424 0.3918 04449 05018 05625 06272 06960 13 48 03432 03927 04458 0502S 05636 06283 06972 12 49 0.3439 03935 04468 05038 0.5646 06295 06984 11 50 03447 03944 04477 05047 05657 06306 06995 10 51 1.0.34.55 1 03952 1.04486 1 .050.57 1.05667 1.06317 1.07007 9 52 03463 03961 04495 05067 05678 06328 07019 8 53 03471 03969 04504 05077 05fi88 06339 07031 54 03179 03978 04514 05087 05699 06350 07043 6 55 03487 03987 01523 05097 05709 06362 07055 5 5(5 03495 03995 04.532 05107 05720 06373 07067 4 57 03503 04004 04541 05116 05730 063S4 07079 3 58 03512 04013 04551 05126 05741 06395 07091 2 59 03520 04021 04560 05136 0.5751 06407 07103 1 60 03528 04030 04569 05146 05762 06418 07115 / 75° 74° 7S° 72° 71° 70° 69° f COSECANTS. 5iO XI.— XATUHAL SECANTS AND COSECANTS. 1 / SECANTS. "1 / 21° 22° 23° 24° 25° 20° 27° 1.07115 1.07853 1.08636 1.09464 1.10338 1.11260 1.12233 eo 1 07126 07866 08649 09478 10353 11276 12249 59 >) 07138 07879 08663 09492 10368 11292 12266 58 3 07150 07892 08070 09506 1038:! 11308 12283 4 07162 07904 08690 09520 10398 11323 12299 56 5 07174 07917 08703 09535 10413 11339 12316 55 6 07 186 07930 08^17 09.549 10428 11355 12333 54 f ( 07199 07943 08730 09563 10443 11371 12349 53 8 07211 07955 08744 09577 10458 11.387 12366 52 9 07223 07968 08757 09592 10473 11403 12383 51 10 07235 07981 08771 09606 10488 11419 12400 50 H 1.07247 1 07994 1.08784 1.09620 1.10503 1.11435 1.12416 49 12 07259 08006 08798 09035 10518 11451 12433 48 13 07271 08019 08811 09649 10.533 11467 12450 47 14 07283 08032 08825 09663 10549 11483 13467 46 15 07295 08045 08839 09678 10564 11499 124S4 45 16 07307 08058 08852 09692 10579 11515 12501 44 17 07320 08071 08866 09707 10594 11.531 12518 43 18 07332 08084 08880 09721 10609 11547 12534 42 19 07344 08097 08893 09735 10025 11563 12551 41 20 07356 08109 08907 09750 10640 11579 12568 40 21 1.07368 1.08122 1.08921 1.00764 1.106.55 1.11595 1.12585 39 22 07380 08135 08934 09779 10070 11611 12602 38 23 07393 08148 08948 09793 10686 11627 12619 37 24 07405 08101 08962 09808 10701 11643 12636 36 25 07417 08174 08975 09822 10716 11659 12653 35 26 07429 08187 08989 09837 10731 11675 12670 34 27 07442 08200 09003 09851 10747 11691 12687 33 28 07454 08213 09017 09866 10762 11708 12704 32 29 07466 08226 09030 098S0 10777 11724 12721 31 30 07479 08239 09044 09895 10793 117-10 12738 30 31 1.07491 1.08252 1.090.58 1 .09909 1.10808 1.11756 1.12755 29 32 07503 08205 09072 09924 10824 11772 12772 28 33 07516 08278 09086 09939 10839 11789 12789 27 34 07528 08291 09099 09953 10854 11805 12807 26 35 07540 08305 09113 09908 10870 11821 12824 25 36 07553 08318 09127 09982 10885 11838 12841 24 37 07505 08331 09141 09997 10901 118.54 12858 23 .38 07578 08344 09155 10012 10916 11870 12875 22 39 07590 08357 09169 10026 10932 11886 12892 21 40 07602 083^0 09183 10041 10947 11903 12910 20 41 1.07615 1.08383 1.09197 1.10055 1.10963 1.11919 1.12927 19 42 07627 0S397 09211 10071 10978 11936 12944 18 43 07640 08410 09224 10085 10994 11952 12961 17 44 07652 08423 09238 10100 11009 11968 12979 16 45 07665 08436 09252 10115 11025 11985 12996 15 46 07677 08449 09266 10130 11041 12001 13013 14 47 07690 08403 09280 10144 11056 12018 13031 13 48 07702 08476 09294 101.59 11072 12034 13048 12 49 07715 08489 09308 10174 11087 12051 13065 11 50 07727 08503 09323 10189 11103 12067 13083 10 51 1.07740 1.08516 1.09337 1.10204 1.11119 1.12084 1.13100 9 52 07752 08529 09351 10218 11134 12100 13117 8 53 07765 08542 09365 10233 11150 12117 13135 7 54 07778 08556 09379 10248 11166 12133 13152 6 55 07790 08569 09393 10263 11181 12150 13170 5 56 07803 08582 09407 10278 11197 12166 13187 i 57 07816 08596 09421 10293 11213 12183 1.3205 3 58 07828 08609 09435 10308 11229 12199 13222 2 59 07841 08623 0!»449 10323 11244 12216 13240 1 60 07853 08636 09464 10338 11260 12233 13257 / 68° 67° 66° 65° 64° 63° 62° / 1 CO SECANTS. XI.-NATURAL SECANTS AND COSECANTS. ' ;ni / SECANTS / 28° 29" 30° 31° 32° 33° 34° 1.13257 1.14335 1.15470 1 . 1 6663 1.17918 1.192.36 1.206V" U) 1 13275 143.54 15 ISO lt;US4 17939 1 92.-9 201 4 r9 o 13292 H372 15509 16704 1.7961 19281 20669 5H 3 13310 14391 1.5528 16725 17982 19304 2(1693 .^7 4 133i7 14109 15.548 16745 18004 193J7 2o;ir ."6 5 13345 14428 15567 16766 18025 19349 20740 "<.'i G 13362 14446 1.5587 16786 18047 19372 20764 51 13380 1 4465 15(;06 16806 18068 19394 2(17 ^ 8 5i 8 13398 14483 15626 16S27 18090 19417 2(:8]2 .-,•) 9 13415 1 1502 1.5645 16848 18111 19440 20Si6 r.i' 10 13433 14521 1.5605 16S68 18133 19463 20859 50 11 1.13451 1.14539 1.1 5684 1.16889 1.18155 1.19485 1.20883 49 1-J 13468 14558 15704 16909 18176 19508 20907 48 13 13486 14.576 15724 16930 18198 19.531 20931 47 14 13504 14595 15743 16950 18220 19.554 20955 46 15 13521 14614 1.5763 16971 18241 19576 20979 45 16 13539 14632 15782 16992 18263 19.599 21003 44 17 13557 14651 15802 17012 18285 19622 21027 43 18 13575 14670 15822 17033 18307 " 19645 21051 42 19 13593 1 4689 15841 17054 18328 19668 21075 41 20 13610 14707 15861 17075 18350 19691 21099 40 21 1.13628 1.14726 1.15881 1.17095 1.18372 1.19713 1.21123 39 22 13646 14745 15901 17116 18394 19736 21147 38 23 13G64 14764 15920 17137 18416 19759 21171 37 24 13682 14782 15940 171.58 18437 19782 21195 36 2r) 13700 14801 15960 17178 18459 19805 21220 35 26 13718 14820 1.5980 17199 18481 19828 21244 34 27 13735 14839 16000 17220 18503 19851 21268 33 28 137.53 14858 16019 17241 18525 19874 21292 32 21) 13771 14877 16039 17262 18547 19897 21316 31 30 13789 14896 16059 17283 18569 19920 21341 30 31 1.13807 1.14914 1.16070 1.17304 1.18591 \. 19944 1.21365 29 32 13825 1 4933 16099 17325 18613 19967 21389 28 33 13843 14952 16119 17346 18635 19990 21414 27 34 13861 14971 16139 17367 18657 20013 21438 26 35 13879 14990 161.59 17388 18679 20036 21462 25 36 13897 15009 16179 17409 18701 20059 21487 24 37 13916 15028 16199 17430 18723 20083 21511 23 38 13934 1.5047 16219 17451 18745 20106 21535 22 39 139.52 15066 16239 17472 18767 20129 21560 21 40 13970 15085 16259 17493 18790 20152 21584 20 41 1.13988 1.15105 1.16279 1.17514 1.18812 1.20176 1.21609 19 42 14006 15124 16299 17535 18834 20199 21633 18 43 14024 15143 16319 17556 18856 20222 21658 17 44 14042 15162 16339 17577 18878 20246 21682 16 45 14061 15181 163.59 17598 18901 20269 21707 15 . 46 14079 15200 16380 17620 18923 20292 21731 14 47 14097 15219 16400 17641 18945 20316 21756 13 48 14115 15239 16420 17662 18967 20339 21781 12 49 14134 15258 16440 17683 18990 20363 21805 11 50 141.52 15277 16460 17704 19012 20386 21830 10 51 1.14170 1.1.5296 1.16481 1.17726 1.19034 1.20410 1.21855 9 52 14188 1.5315 16.501 17747 19057 20433 21879 8 53 14207 15335 16521 17768 19079 20457 21904 1 54 1 4225 15354 16541 17790 19102 20480 21929 6 55 14243 15373 16562 17811 19124 20504 21953 5 56 14262 1.5393 16582 17832 19146 20527 21978 4 57 14280 1.5412 16602 17854 19169 20.551 22003 3 58 14209 15431 16623 17875 19191 20575 22028 2 59 14317 1.5451 16643 17896 19214 20598 22053 1 60 14335 15470 16663 17918 19236 20622 22077 / / L.- - - «1° 00° 59° 58° 57° 5«° 65° CHJSECANTS. 312 XI.— NATURAL SECANTS AND COSECANTS. / SECANTS . 1 60 35° 36° 37° 38° 39° 40° 41° 1.22077 1.23607 1.25214 1.26902 1.28G76 1.;B0541 1.32501 1 22102 23638 25241 26931 28706 30573 325:35 59 2 22127 23659 25269 26960 28737 30605 32568 58 3 22152 23685 25296 20988 28767 30636 32602 57 4 22177 23711 25324 27017 28797 :30668 32636 56 5 22202 23738 25351 27046 28828 30700 32669 55 6 22227 23764 25379 27075 28858 30732 3270:3 54 < 22252 23790 25406 27104 28889 30764 327:37 53 8 22277 23816 25434 27133 2S919 30796 32770 52 9 22302 23843 25462 27162 28950 30829 32804 51 10 22327 23869 25489 27191 28980 30861 32838 50 11 1.22352 1.23895 1.25517 1.27221 1.29011 1.30893 1.32872 49 12 22377 23922 25545 27250 29042 30925 32905 48 13 22402 23948 25572 27279 29072 30957 32939 47 14 ^22428 23975 25600 27308 29103 30989 32973 46 15 22153 24001 25628 27337 291:33 31022 33007 45 16 22478 24028 25656 27366 29164 31054 33041 44 IT 22503 24054 25683 27396 29195 31086 3:3075 43 18 22528 24081 2.5711 27425 29226 31119 33109 42 19 22554 24107 25739 27454 29256 31151 33143 41 20 22579 24134 25707 27483 29287 31183 33177 40 21 1.22604 1.24160 1.25795 1.27513 1.29318 1.31216 l.;33211 39 22 22629 24187 25S23 27542 29349 31248 33245 38 2:3 22655 24213 2.5851 27572 29380 31281 :33279 37 24 22680 24240 25879 27601 29411 31313 33314 36 25 22706 24267 25;J07 27630 29442 31346 33348 35 26 22731 24293 25935 27660 29473 31378 33:382 34 27 22756 24320 25963 27689 29504 31411 3:3416 33 28 22782 24347 25991 27719 29535 31443 33451 32 29 22807 24373 26019 27748 29566 31476 :334S5 31 30 22833 244 OO 26047 27778 29597 31509 33519 30 31 1.22858 1.24427 1.26075 1.27807 1.29628 1.31541 l.:3:3554 29 32 22884 24454 26104 27837 29659 31574 33588 28 33 22909 24181 26132 27867 29690 31607 a3622 27 34 22935 24508 26160 27896 29721 31640 ;33657 26 35 22960 24534 26188 27926 29752 31672 3:3691 25 36 22986 24561 26216 27956 29784 31705 33726 24 37 2;i012 24588 26245 27985 29815 317;B8 33760 23 38 23037 24615 26273 28015 29846 31771 33795 22 39 23063 24642 26301 28045 29877 31804 3:3830 21 40 23089 24669 20330 28075 29909 31837 33864 20 41 1.23114 1.24696 1.26358 1.28105 1.29940 1.31870 l.:33899 19 42 23140 24723 26387 28134 29971 31903 33934 18 43 23166 24750 26415 28164 30003 31936 33968 17 44 23192 24777 26443 28194 30034 31969 34003 16 45 23217 24804 26472 28224 30066 32002 34038 15 46 23243 24832 26500 28254 30097 32035 3407o 14 47 23269 24859 26529 28284 30129 32068 :34108 13 48 23295 24886 26557 28314 30160 32101 34142 12 49 23321 24913 26586 28344 30192 32134 34177 11 50 23347 24910 26615 28374 :30223 ;32168 34212 10 51 1.23373 1.24967 1.26643 1.28404 l.:30255 1.32201 1.34247 9 52 23399 24995 26672 28434 302^7 32234 34282 8 53 23424 25022 26701 28464 30318 32267 :34317 i 54 23450 25049 26729 28495 30350 32:301 34:352 6 55 23476 25077 26758 28525 30382 32334 34:387 5 56 23502 25104 26787 28555 30413 32368 ;34423 4 57 28529 2.^131 2081 5 28585 30445 ;32401 34458 3 58 23555 25159 26844 28015 30477 32434 ;34493 2 59 23581 25186 26873 28046 30509 32468 3452S 1 60 23607 25214 26902 28676 :30541 32.501 34563 / 54° 53° 52° 51° 50° 49° 48° 1 1 COSECANTS. f ,. 1 XL— NATURAL SECANTS AND COSECANTS. '61'6 1 SECANTS. • / 42° 43° 44° 45° 46° 47° 48° 1.34503 1.36733 1.39016 1.41421 1.43956 1.46628 1.49448 60 1 34599 36770 :i9055 41463 43999 46674 49496 59 "i. 34634 36807 39095 41.504 44042 46719 49544 58 3 34669 36841 391.34 41.545 44086 40765 49593 57' 4 34704 36881 39173 41586 44129 46811 49641 56 5 34740 36919 39212 41627 44173 46857 49690 55 6 34775 36956 39251 41669 44217 46903 497.38 54 7 34811 36993 39291 41710 44260 46949 49787 53 8 34846 37030 39330 41752 44304 46995 49835 52 9 34882 37068 39369 41793 44347 47041 49884 51 10 34917 37105 39409 41835 44391 47087 49933 50 11 1.34953 1.37143 1.39448 1.41876 1.44435 1.47134 1.49981 49 12 34988 37180 39487 41918 44479 47180 50030 48 13 35024 37218 39527 419.59 44523 47226 50079 47 14 35060 37255 39566 42001 44567 47272 50128 46 15 35095 37293 39606 42042 44610 47319 50177 45 16 35131 37330 39646 42084 44654 446^8 47365 50226 44 17 35167 37368 39685 42126 47411 50275 43 18 35203 37406 39725 42168 44742 47458 50324 42 19 :i5238 37443 39764 42210 44787 47504 50373 41 20 35274 37481 39804 42251 44831 47551 50422 40 21 1.35310 1.37519 1.39844 1.42293 1.44875 1.47598 1.50471 39 22 35346 37556 39884 42335 44919 47644 50521 38 23 35382 37594 39924 42377 44963 47691 50570 37 24 35418 37632 39963 42419 45007 47738 50619 36 25 35454 37670 40003 42461 45052 47784 50669 35 26 35490 37708 40043 42503 45096 47831 50718 84 27 35526 37746 40083 42545 45141 47878 50767 33 28 35562 37784 40123 42587 45185 47925 50817 32 29 35598 37822 40163 42630 45229 47972 50866 31 30 35634 37860 40203 42672 45274 48013 50916 30 31 1.35670 1.37898 1.40243 1.42714 1.45319 1.48066 1.50966 29 32 35707 37936 40283 42756 45363 48113 51015 28 33 35743 37974 40324 42799 45408 48160 51065 27 34 35779 38012 40.364 42841 45452 48207 51115 26 35 35815 38051 40404 42883 45497 48254 51165 25 36 35852 38089 40444 42926 45542 48301 51215 24 37 35888 38127 40485 42968 45587 48349 51265 23 38 35924 38165 40525 43011 4.5631 48396 51314 22 39 35961 38204 40565 43053 45676 48443 51.364 21 40 35997 38242 40606 43096 45721 48491 51415 20 41 1.36034 1.3S280 1.40646 1.431.39 1.45766 '.48538 1.51465 19 42 36070 38319 40687 43181 4.5811 48586 51515 18 43 36107 38357 40727 43224 45856 48633 51565 17 44 36143 38396 40768 43267 45901 48681 51615 16 45 36180 38431 40808 43310 45946 48728 51665 15 46 36217 38473 40849 43352 45992 48776 51716 14 47 36253 38512 40890 43395 46037 48824 51766 13 48 36290 38550 40930 4.3438 46082 48871 51817 12 49 36327 38589 40971 43481 46127 48919 51867 11 50 36363 38628 41012 43524 46178 48967 51918 10 51 1.36400 1.38666 1.41053 1.43567 1.46218 1.49015 1.51968 9 52 36437 38705 41093 43610 46263 49063 52019 8 53 36474 38744 41134 43653 46309 49111 52069 7 54 36511 38783 41175 43696 46354 49159 52120 6 55 36548 38822 41-.'16 437.39 46400 49207 52171 5 56 36585 38860 41257 43783 46445 49255 52222 4 57 36622 38899 41298 43826 46491 49303 52273 3 58 36659 38938 41339 43869 46537 49351 52323 2 59 36<)96 38977 41380 43912 46582 49399 52374 1 60 36733 39016 41421 43956 46628 49448 52425 / 47° 46° 45° 44" 43° 42° 41° / COSECANTS. h514 XL NATURAL SECANTS AND COSECANTS. ■ SECANTS. / 49° 60° 51° 52° 53° 54° 55° 1.52425 1.55572 1.58902 1.62427 1.66164 1.70130 1.74345 60 1 52476 55626 58959 62487 66228 70198 74417 59 o 52527 55680 59016 62548 66292 70267 74490 58 3 52579 55734 59073 62609 66357 70335 74562 57 4 52630 55789 59130 62669 66421 70403 74635 56 5 52681 55843 59188 62730 66486 70472 74708 55 6 52^32 55897 59245 62791 66550 70540 74781 54 7 52784 55951 59302 62852 66615 70609 74854 53 8 52835 56005 59360 62913 66679 70677 74927 52 9 52886 56060 59418 62974 66744 70746 75000 51 10 52938 56114 59475 63035 66809 70815 75073 50 11 1.52989 1.56169 1.59533 1.63096 1.66873 1.70884 1.75146 49 12 53041 56223 59590 63157 66938 70953 75219 48 13 53092 56278 59048 63218 67003 71022 75293 47 14 53144 50332 59706 63279 67068 71091 75366 40 15 53196 56387 59764 63341 67133 71160 75440 45 16 53247 5G442 59S22 63402 67199 71229 75513 44 17 53299 56497 598S0 63464 67264 71298 75587 43 18 53351 56551 59938 63525 67329 71368 75661 42 19 53403 56606 59996 63587 67394 71437 75734 41 20 53455 5606 1 60054 63648 67460 71506 75808 40 21 1.53507 1.56716 1.60112 1.63710 1.67525 1.71576 1.75882 39 22 53559 56771 601T1 63772 67591 71646 75956 38 23 53611 56826 60229 63834 67656 71715 76031 37 24 53663 56S81 60287 63895 67722 71785 70105 36 25 53715 56937 60346 63957 67788 71855 70179 35 26 53768 50992 60404 61019 67853 71925 76253 34 27 53820 57047 60463 64081 67919 71995 76328 S3 28 53872 57103 60521 64144 67985 72065 76402 32 29 53924 57158 605S0 64206 68051 72135 76477 31 30 53977 57213 60639 04268 68117 72205 76552 30 31 1.54029 1.57269 1.6069S 1.61330 1.68183 1.72275 1.76626 29 32 54082 57324 60756 (i4393 68250 72346 76701 28 33 54134 57380 60815 64455 68316 72416 76776 27 34 54187 57436 60874 64518 68382 72487 76851 26 35 51240 57491 6093S 64580 68449 72557 76926 25 36 54292 57547 60992 61643 68515 72628 77001 24 37 54345 57603 61051 64705 68582 72698 77077 23 38 54398 57659 61111 64768 68648 72769 77152 22 39 54451 57715 61170 64831 68715 72840 77227 21 40 54504 57771 61229 64894 68782 72911 77303 20 41 1.54557 1.57827 1.61288 1.64957 1.68848 1.72982 1.77378 19 42 54610 57883 61348 65020 68915 73053 77454 18 43 54663 57939 61407 65083 68982 73124 77530 17 44 54716 57995 61^67 65146 69049 73195 77606 16 45 54769 58051 61526 65209 69116 73207 77681 15 46 54822 58108 61586 65272 69183 73338 77757 14 47 54876 58164 61646 65336 69250 73409 77833 13 48 54929 58221 61705 65399 69318 73481 77910 12 49 54982 58277 61765 65462 69385 73552 77986 11 50 55036 58333 61825 65526 69452 73624 78062 10 51 1.55089 1.58390 1.61885 1.65589 1.69520 1.73696 1.78138 9 52 55143 58447 61945 65653 69587 73768 78215 8 53 55196 58503 62005 65717 69655 73840 78291 7 54 55250 58560 62065 657F0 69723 73911 78368 6 55 55303 58617 62125 65844 69790 73983 78445 5 56 55357 58674 62185 65908 6985S 74056 78521 4 57 55411 58731 62246 65972 69926 74128 78598 3 58 55465 58788 (i2306 66036 69994 74200 78675 2 59 55518 58845 62366 66100 70062 74272 7875:? 1 60 55572 58902 62427 66164 70130 74345 78829 / / 40° 39° 38° 37° 36° 35° 34° i - COSECANTS. XI.— NATURAL SECANTS AND COSECANTS. 315 I SECANTS. / 56° 57° 58° 69° 60° 61° 62° 1.78859 1.83608 1.88708 1.94160 2.00000 2.06267 2 13005 60 1 789C6 83690 88796 94254 00101 00375 13122 59 2 78984 83773 88884 94349 00202 00483 1.32-39 58 3 79061 83855 88972 94443 00303 06592 i;i356 57 4 79138 83938 89060 94537 00404 06701 i;W73 56 5 79216 84020 89148 94632 00505 00809 13590 55 6 79293 84103 89237 94726 00007 06918 13707 54 t 79371 84186 89325 948-21 00708 07027 13825 53 8 79449 84269 89414 94916 00810 07137 13942 52 9 79527 84352 89.503 95011 00912 07246 14060 51 10 79604 84435 89591 95106 01014 07356 14178 50 n 1.79682 1.84518 1.89680 1.95201 2.01116 2.07465 2.14296 49 i-j 79701 84601 89709 95296 01218 07575 14414 48 13 798:39 84685 89858 95392 01320 07685 14533 47 14 79917 84768 89948 9.5487 01422 07795 14651 46 15 79995 84852 90037 95583 01525 07905 14770 45 IC. 80074 8-1935 90126 95678 01628 08015 14889 44 ir 80152 85019 90216 95774 01730 08126 15008 43 18 80231 85103 90305 95870 01833 OS-230 15127 42 I'J 80309 85187 90395 95966 01930 08347 15246 41 L'O 80388 85271 90485 96062 02039 08458 15366 40 'Jl 1.80467 1.8.5355 1.90575 1.96158 2.0-2143 2 08569 2.15485 39 »•> 80546 85439 90665 96255 02246 08680 15(i05 •38 •J3 80025 85523 90755 90351 02349 08791 15725 37 •z\ 80704 85008 90845 90448 02453 08903 15845 36 X!5 80783 85692 90935 90544 02557 09014 15905 35 ^'G 80802 85777 91026 90641 02661 09126 16085 34 ^ 1 80942 85861 91110 96738 02765 09238 16206 33 •JS 81021 85946 91207 96835 02809 09350 16326 32 2".) 81101 80031 91297 90932 02973 0'.»462 . 16447 31 30 81180 86116 91388 97029 03077 09574 10568 30 1 31 1.81260 1 .86201 1.91479 1.97127 2.03182 2.09686 2.16689 29 1 32 81340 80286 91570 972-24 03286 09799 16810 28 I 38 81419 80371 91661 97322 03391 09911 16932 27 34 81499 80457 917.52 974-20 03496 10024 17053 26 35 81.579 80542 91844 97517 03601 10137 17175 25 31J 81059 86627 91935 97015 . 03706 10250 17-397 24 37 81740 86713 920-.27 97713 03811 10363 17419 23 38 81820 80799 92118 97811 03916 10477 17541 22 3'.) 81900 80885 92210 97910 04022 10590 17663 21 40 81981 80990 9230-2 98008 04128 10704 17786 20 41 1.82061 1.87056 1.92394 1.98107 2.04233 2.10817 2.17909 19 42 8-J142 87142 9-,'486 98-205 04339 10931 18031 18 43 822-,>2 87-229 92578 98304 04445 11045 18154 17 44 8-2303 87315 92670 98403 04551 111.59 18-277 16 45 82384 87401 92762 98502 04658 11274 18401 15 40 82465 87488 92855 98601 04764 11388 18524 14 47 82546 87574 92947 98700 04870 11503 18648 13 48 82627 87661 93040 98799 04977 11617 18772 12 49 82709 87748 93133 98899 05084 11732 18895 11 50 82790 87834 93:226 98998 05191 11847 19019 10 51 1.8-2871 1.87921 1.93319 1.99098 2.05298 2.11903 2.19144 9 5',> 809.53 88008 93412 99198 0.5405 12078 19268 8 53 83034 88095 93505 99298 05512 12193 19393 7 54 83116 8S183 93598 99398 0.5619 12309 19517 6 55 83198 88270 9309-2 99498 05727 124-25 19642 5 50 8;;2S0 88357 93785 99598 05835 12.540 19767 4 '•> i 83362 88445 93879 9!I09S 05942 1-2657 1 9892 3 58 83444 8S532 93973 99799 00050 12773 20018 o 59 83520 88020 94000 99899 00158 12889 20143 1 60 1 / 83008 88708 94100 2.00000 06207 13005 20269 27° 33° 32° 81° 30° 29° 28° / COSECANTS. ue XI. NATURAL SECANTS AND COSEC. \NTS. / SECANTS. 1 60 63° 64° 65° 66° 67° 68° 69° 2.20269 2.28117 2.36620 2.45859 2.559.30 2.66947 2.79043 1 20395 28^:53 36768 46020 56106 67139 792.54 59 2 20521 28390 36916 46181 56282 67332 79466 58 3 20647 28526 37064 46342 56458 67525 79079 57 4 20773 28663 37212 46504 56634 67718 79891 56 5 20900 28800 37361 46665 56811 67911 80104 55 6 21026 28937 37509 46S27 56988 68105 80318 51 7 21153 29074 376.58 46989 57165 68299 80531 53 8 21280 29211 37808 47152 57342 68494 80746 52 9 21407 29349 37957 47314 57520 68689 80960 51 10 21535 29487 38107 47477 57698 68884 81175 50 11 2.21662 2.29625 2.38256 2.47640 2.57876 2.69079 2.81390 49 12 21790 29763 38406 47804 58054 69275 81605 48 13 21918 29901 38556 47967 58233 69471 81821 47 14 22045 30040 38707 48131 58412 69667 82037 46 15 22174 30179 38857 48295 58591 69864 82254 45 16 22302 30318 39008 48459 58771 70061 82471 44 17 22430 30457 391.59 48624 58950 70258 82688 43 18 22559 30596 39311 48789 59130 70455 82906 42 19 22688 .•i0735 39402 489,54 59311 70653 83124 41 20 22817 30875 39614 49119 59491 70851 83342 40 21 2.22946 2 31015 2.39766 2.49284 2.. 5967 2 2.710.50 2.83561 39 ' 22 23075 31155 39918 49450 59853 71249 83780 38 23 23205 31295 40070 49616 60035 71448 83999 37 24 23334 31436 40222 49782 60217 71647 84219 36 25 23464 31576 40375 49948 60399 71847 84439 35 26 23594 31717 40528 .50115 60581 72047 846.59 34 27 23724 31858 40681 502S2 60763 72247 84880 33 28 23855 31999 40835 .50449 60946 72448 85102 32 29 23985 32140 409.^8 .50617 61129 72649 85323 31 30 24116 3228 i 41112 50784 61313 72850 85.545 30 31 2.24247 2.32424 2 41296 2.509.52 2.61496 2.73052 2.8,5767 29 32 24378 32566 41450 51120 61680 73254 85990 28 33 24509 32708 41(;05 51289 61864 734.56 80213 27 34 24640 32850 41760 51457 62049 73659 86437 26 35 24772 32993 41914 51626 62234 73S62 86661 25 36 24903 33135 42070 51795 62419 74065 86885 24 37 25035 332^8 42225 51965 62604 74269 87109 23 38 25167 33422 423.S0 52134 62790 74473 87334 22 39 25300 33565 42536 52304 62976 74677 87560 21 40 25432 33708 42692 52474 63162 74881 87785 20 41 2.25565 2.33852 2.42848 2.52645 2.63348 2.750S6 2.88011 19 42 25697 33996 43005 52815 63.^>35 75202 88238 18 -1 ^ 43 25830 34140 43162 52986 63722 75497 88465 1 1 t I* 44 25963 34284 43318 53157 63909 75703 88692 16 45 26097 34429 43476 53329 64097 75909 8S9-J0 15 14 13 1 k) 46 26230 34.573 43633 53500 64285 76116 89148 47 26364 34718 43790 5367-2 64473 76323 8937'6 48 26498 34863 43948 53845 64662 76530 89605 12 11 10 49 26632 3.5009 44106 54017 64851 7C)737 898:J4 50 26766 35154 44264 54190 65040 70945 90063 51 2.26900 2.35.300 2.44423 2 54363 2.6.5229 2.771.54 2.90293 9 8 52 27035 3.5446 44.582 54536 6.5419 ( ( .502 90524 53 27169 35592 44741 54709 65609 77571 90754 1 6 5 4 3 v> 54 27304 35738 44900 54883 65799 77780 90986 55 27439 35885 45059 5.5057 65989 77990 91217 56 27574 36031 4.5219 .55231 66180 78200 91449 57 27710 36178 45378 55405 6(1371 78410 91681 58 27845 36325 45539 5.5580 66563 7S021 91914 1 59 27981 36473 45699 55755 667.55 78832 92147 60 28117 36620 458.59 55930 66947 79043 92380 J / 26° 25° 24° 23° 22° OSECANTS. 21° 20° ' C< XI.— NATURAL SECANTS AND COSECANTS. 317 / SECANTS. / 70° 71° 7-2° 73° 74° 75° 76° 2.92380 3.07155 3.23607 3.42030 3.62796 3.86370 4.13357 60 1 92614 07415 23897 42356 63164 86790 13839 59 a 92849 07675 24187 42683 63533 8721 1 14323 58 3 93083 07936 24478 43010 6i90:] 87633 14809 57 4 93318 08197 24770 43337 64274 88056 15295 56 5 93554 08459 25062 43666 64645 88179 15782 55 6 93790 08721 25355 439!»5 65018 88904 16271 54 7 94026 08983 25648 44324 65391 89330 16761 53 8 94263 09246 25942 44655 65765 89756 172.52 52 9 94500 09510 26237 44986 66]40 90184 17744 51 10 94737 09774 26531 45317 66515 90613 18238 50 11 2.94975 3.10038 3.26827 3.45650 3.66892 3.91042 4.18733 49 12 95213 10303 27123 45983 67269 91473 19228 48 18 95452 10568 27420 46316 67647 91904 19725 47 14 95691 10834 27717 46651 68025 92337 20224 46 15 95931 11101 28015 46986 68405 92770 20723 45 16 96171 11367 28313 47321 68785 93204 21224 44 17 96411 11635 28612 47658 69107 93640 21726 43 18 96652 11903 28912 47995 69549 91076 22229 42 19 96^93 12171 29212 48333 69931 94514 22734 41 20 97135 12440 29512 48671 70315 94952 23239 40 21 2.97377 3.12709 3.29814 3.49010 3.70700 3.95392 4.23746 39 22 97619 12979 30115 49350 71085 95832 242.55 38 28 97862 13249 30418 49691 71471 96274 24764 37 24 98106 13520 30721 50032 71858 96716 2.5275 36 25 98349 13791 31024 50374 72246 97160 25787 35 26 9S594 14063 31328 50716 72635 97604 26300 34 27 98838 14335 31633 51060 73024 98050 26814 33 28 99083 14608 31939 51404 73414 98497 27330 32 29 993-.'9 14881 32244 51748 73806 98944 27847 31 30 99574 15155 32551 52094 74198 993f)3 28366 30 31 2.99821 3.15429 3.328.58 3.52440 3,74591 3.99843 4.288S5 29 32 3.00067 15704 33166 52787 74984 4.00293 29406 28 33 00315 15979 33474 53134 75379 00745 29929 27 34 00562 16255 33783 53482 75775 01198 30452 26 35 00810 16531 34092 53831 76171 01052 30977 25 36 01059 16808 34403 54181 76.568 02107 31503 24 3V 01308 17085 34713 54531 76966 02563 32031 23 38 01557 17363 35025 54883 77365 03020 32560 22 39 01807 17641 35336 55235 77765 03479 33090 21 40 02057 17920 35649 .55587 78166 03938 33622 20 41 3.02308 3.18199 3.35962 3.55940 3.78568 4.04398 4.34154 19 42 02559 18479 36276 56294 78970 04800 34689 18 43 02810 18759 36590 56649 79374 05322 35224 17 44 03062 19040 30905 57005 79778 05786 35761 16 45 03315 19322 37221 57361 80183 06251 36299 15 46 03568 19604 37537 57718 80589 06717 36839 14 47 03821 19886 37854 58076 80996 07184 37380 13 48 04075 20169 38171 58434 81404 07652 37923 12 49 04329 20453 38489 58794 81813 08121 38466 11 50 04584 20737 38808 59154 82223 08591 39012 10 51 3.04839 3.21021 3.39128 3.59514 3.82633 4.09063 4.395.58 9 52 05094 21306 39448 59876 83045 09535 40106 8 53 05350 21592 39768 60238 83457 10009 40656 7 54 05607 21878 40089 60601 83871 10484 41206 6 55 05864 22165 40411 60965 84285 10960 41759 5 56 00121 22452 40734 61330 84700 11437 42312 4 57 06379 22740 41057 61695 8.5116 11915 42867 3 58 06637 23028 41381 62061 85533 12394 43424 2 59 06896 23317 41705 62428 85951 12875 43982 1 60 07155 23607 42030 62796 86370 13357 44541 / 19" 18° 17° 16" 16° 14° 13° / COSECANTS. 318 XL— NATURAL SECANTS AND COSECANTS. / SECANTS. / 77° 78° 79° 80° 81° 82° 83° 4.44541 4.80973 5.24084 5. (Obi i 6.39245 7.18530 8.205.51 60 1 45102 81633 24870 76829 40422 20020 22500 59 2 45664 82294 25658 77784 41602 21517 244.57 58 3 46228 82956 26448 78742 42787 23019 26425 57 4 46793 83621 27241 79703 43977 24529 28402 56 5 47360 84288 28036 80667 45171 26044 .30388 55 6 479r^8 84956 28833 81635 46369 27566 321384 54 1 48498 85627 29634 82606 47572 29095 34390 53 8 49069 86299 30436 83581 48779 30630 36405 52 9 49642 86973 31241 84558 49991 32171 38431 51 10 50216 87649 32049 85.539 51208 33719 40466 50 11 4.50791 4.88327 5.32859 5.86524 6.52429 7.35274 8.42511 49 12 51368 89(X>7 33t571 87511 53655 36835 44566 48 13 51947 89689 34486 88502 54886 38403 46632 47 14 52527 90373 35304 89497 56121 39978 48707 46 15 53109 91058 36124 90495 57361 41560 50793 45 16 53692 91746 36947 91496 58606 43148 52889 44 17 54277 92436 37772 92501 598.55 44743 54996 43 18 54863 93128 38000 93.509 61110 46346 .57113 42 19 55451 9.3821 39430 94.521 62369 479.55 59241 41 20 56041 94517 40263 95536 63'J33 49571 61379 40 21 4.56632 4.95215 5.41099 5.965.55 6.64902 7.51194 8.6.3.528 39 22 57224 95914 41937 97.577 66176 52825 65688 38 23 57819 96616 42778 98603 6^4.54 54462 67859 37 24 58414 97320 43622 99633 68738 56107 70041 36 25 59012 98025 444r,8 6.00666 70027 57759 72234 35 26 59611 98733 45317 01703 71321 59418 74438 34 27 60211 99443 46169 02743 72620 61085 76653 33 28 60813 5.00155 47023 03787 73924 62759 78880 32 29 61417 00869 47881 04834 75233 64441 81118 31 30 62023 01.585 48740 05886 76547 66130 83367 30 31 4 62630 5.02303 5.49603 6.06941 6.77866 7.67826 8.8.5628 29 32 63238 03024 50468 08000 79191 69530 87901 28 33 63849 03746 51337 09062 80521 71242 90186 27 34 64461 04471 52208 10129 818.56 72962 92482 26 35 65074 05197 53081 11199 83196 74689 94791 25 36 65690 05926 53958 12273 84542 76424 97111 24 37 66307 06657 54837 1.3350 8.5893 78167 99444 23 38 06925 0:.39C 55720 14432 87250 79918 9.01788 22 39 67545 08125 56605 1.5517 88612 81677 04146 21 40 68167 0S863 57493 16607 89979 83443 06515 20 41 4.68791 5.09602 5.5S383 6.17700 6.913.52 7.85218 9.08897 19 42 69417 10344 59277 18797 92731 87001 11292 18 43 70044 11088 60174 19898 94115 88792 13699 17 44 70673 11835 01073 21004 95505 90592 16120 16 45 71303 12.58? 61976 22113 96900 92400 18553 15 46 71935 13334 62881 23226 98301 94216 20999 14 47 72569 14087 63790 24343 99708 96040 2.3459 13 48 73205 14842 64701 25464 7.01120 97873 25931 12 49 73843 15.599 6.5616 26590 025.38 99714 28417 11 50 74482 16359 66533 27719 03962 8.01565 .30917 10 51 4.7.5123 5.1:121 5.67454 6.288.53 7.0.5392 8.03423 9.33430 9 52 75766 17880 68377 29991 06828 05291 35957 8 53 76411 18652 69304 31133 08269 07167 .38497 54 77057 19421 70234 32279 09717 09052 410.52 '6 55 77705 20193 71166 33429 11171 10946 43620 5 56 783.55 20966 72102 34.584 12630 12849 46203 4 57 79007 21742 73041 35743 14096 14760 48800 3 58 79661 22521 73983 36906 1.5508 16681 51411 2 59 80316 23301 74929 380^3 17046 18612 54037 1 60 80973 24084 75877 39245 18530 20551 56677 / 12° 11° 10° 9° 8° 7° 6° J COSECANTS. XI.— NATURAL SF.CANTS AND COSECANTS. 319 / SECANTS. / 84^ 8.5° 86° 87° 88° 89° 9.56677 11.47.371 14.3.3.5.59 19.10732 28.6.5371 57.29869 60 1 59332 51199 39.547 21397 89440 58.26976 59 o 62002 55052 455S6 32182 29.13917 59.27431 58 3 64687 58932 51676 43088 38812 60.31411 57 4 67387 62837 .57817 54119 64137 61.39105 56 5 70103 06769 61011 65275 89903 62.. 5071 5 55 6 72833 70728 70258 76.560 30.16120 63.66460 54 7 75579 74714 76558 87976 42802 04.86.572 53 8 78341 78727 82913 99524 69960 66.11304 52 9 81119 82768 89323 20.11208 97607 67.40927 51 10 83912 86837 95788 23028 31 .25758 68.75730 TjO 11 9.80722 11.90934 15.02310 20.34989 31.. 54425 70.16047 49 12 89547 95060 08890 47093 83023 71.62285 48 13 92389 99214 1.5.527 .59341 32.13.366 73.14.583 47 14 95248 12.03.397 22223 71737 43671 74.73586 46 15 98123 07610 28979 84283 74554 76.396.55 45 16 10.01015 11852 35795 96982 33.06030 78.1.3274 44 17 03923 16125 42072 21.09838 38118 79.94968 43 18 00849 20427 49611 22852 70835 81.85315 42 19 09792 24761 ■ 56614 36027 34.04199 83.84947 41 20 12752 29125 63679 49368 38232 85.94561 40 21 10.1.5730 12.33.521 15.70810 21.62876 34.72952 88.14924 39 22 18725 37948 78005 76555 35.08380 90.46S86 38 23 21739 42408 85268 90409 44.539 92.91387 37 24 24770 46900 92597 22.04440 81452 95.49471 36 2?^ 27819 51424 99995 18653 36.19141 98.22303 35 26 30887 55982 16.07462 33050 57633 101.11185 34 27 33973 60572 14999 47635 96953 104.17574 33 28 37077 65197 22607 62413 37.37127 107.43114 32 29 40201 69856 30287 77386 78185 110.89656 31 30 43343 74550 38041 92.559 38.20155 114.59301 30 31 10.46505 12.79278 16.4.5869 23.07935 38.63068 11 8,. 54440 29 3i 49685 84042 53772 23520 39.06957 122.77803 28 33 52886 88841 61751 39316 51855 127.32526 27 34 56106 93677 69808 55329 97797 1.32.22229 26 35 59346 98549 77944 71503 40.44820 137.51108 25 3(; 62605 13.034.58 86159 88022 92963 143.24061 24 37 65S85 08040 944.56 24.04712 41.42266 149.46837 23 38 69186 13388 17.02835 21637 92772 156.26228 22 3'J 72507 18411 11297 38802 42.44525 163.70325 21 40 75S49 23472 19843 56212 97571 171.88831 20 41 10.79212 13.28572 17.28476 24.7.3873 43.51961 180.93496 19 42 82.596 3.3712 .37196 91790 44.07746 190.98680 18 43 80001 38891 40005 25.09969 64980 202.22122 17 44 89428 44112 51903 28414 45.23720 214.8.5995 16 4'> 92877 49373 63893 47134 84026 229.18385 15 4C> 96:348 54076 72975 66132 46.4.5963 245.55402 14 47 99841 60021 82152 85417 47.09.596 264.44269 13 48 11.033.^6 65408 91424 26.04994 74997 286.47948 12 49 06894 70838 18.00794 24869 48.42241 312.52297 11 50 10455 76312 10262 45051 49.11406 343.77516 10 51 11.14039 13.«1829 18.19830 26.65546 49.82.576 381.972.30 9 52 17646 87391 29501 86360 50.. 5.5840 429.71873 8 r)3 21277 92999 39274 27.07.503 51 .31290 491.10702 7 54 24932 98651 49153 28981 52.09027 572.9.5809 6 55 28610 14 04350 .59139 .50804 891.56 687.54960 5 50 32313 10096 69233 72978 .53.71790 8.59.43689 4 57 36040 1.5889 79438 9.5513 54.57046 1145.9157 3 58 397!»2 21730 897.55 28.18417 .55.4.^^053 1718.8735 2 59 43569 27620 19.00185 41700 56.35946 3437.7468 1 60 47371 33559 10732 65371 57.29869 00 1 5° 4° 3° 2° 1° 0° / COSECANTS. 320 TABLE XU.— TANdENTS AND COTANGENTS. "o 0» 1 1 1° ! 2° 1 3 ° / 60 . Tang .00000 Cotang Tang .01746 Cotang Tang .0.3492 Cotang Tang ! .05241 Cotang Infinite. 57.2900 28.6363 19.0811 1 .00029 3437.75 .01775 56.3506 .0.3.521 28.3994 .05270 18.9755 59 2 .00058 1718.87 .01804 55.4415 .03550 28.1664 .05299 18.8711 1 58 3 .00087 1145.92 .01833 54.5613 .03579 27.9.372 .05328 18.7678 57 ' 4 .00116 859.436 .01862 53.7036 .03609 27.7117 .053.57 18.6656 56 5 .00145 6S7.549 .01891 52.0821 .03638 27.4899 .05387 18.5645 55 6 .00175 572.957 .01920 52.0807 .03667 27.2715 .0.5416 18.4645 54 7 .00204 491.106 .01949 51.. 30.32 : .0.3696 27.0566 .05445 18.3655 .53 8 .00233 429.713 .01978 50.5485 : .03725 26.8450 .05474 18.2677 52 9 .00262 381.971 .02007 49.81.^7 1 .03754 26.6.367 .05503 18.1708 51 10 .00291 a43.774 .02036 49.1039 .03783 26.4316 .05533 18.0750 50 11 .00320 312.521 .02066 48.4121 .0.3312 26.2296 .05562 17.9802 49 12 .00349 286.478 .02095 47.7395 .03842 26.0307 .05501 17.8863 48 13 .00378 264.441 .02124 47.0853 .03871 25.8;M8 .05620 17.7934 47 14 .00407 245.552 .02153 46.4489 .03900 25.6418 .05649 17.7015 46 15 .00436 229.182 .02182 45.8294 .0.3929 25.4517 .05678 17.6106 45 IG .004G5 214.858 .022U 45.2261 .03958 25.2644 .05708 17.5205 44 17 .00495 2U2.219 .02240 44.6386 .0.3987 25.0798 .05737 17.4314 43 18 .00524 190.984 .02269 44.0661 .04016 24.8978 .05766 17.3432 42 19 .00553 180.932 .02298 43.. 5081 .04046 24.7185 .05795 17.2558 41 20 .00582 171.885 .02328 42.9641 .04075 34.5418 .05824 17.1693 40 21 .00611 163.700 .02357 42.4.335 .04104 24.3675 .05854 17.08.37 39 22 .00040 156.259 .02386 41.9158 .04133 24.19.57 .05883 16.9990 38 23 .00669 149.465 .02415 41.4106 .04162 24.0263 .05912 16.9150 37 24 .00698 143.237 .02444 40.9174 .04191 23.8593 .05941 16.8319 36 25 .00727 137.507 .02473 40.4.358 .04220 23.6945 .05970 16.7496 35 26 .00756 132.219 ,02502 39.9655 .04250 23.5.321 .05999 16.6681 M 27 .00785 127.321 .02531 39.50.59 .04279 23.3718 .06029 16.5874 33 28 .00815 122.774 .02560 39.0.568 .04308 23.21,37 .06058 16.5075 32 29 .00844 118.540 .02589 .38.6177 .04337 23.0.577 .06087 16.4283 31 30 .00873 114.589 .02619 38.1885 .04366 22.9038 .06116 16.^99 30 31 .00902 110.892 .02648 .37.7688 .04.395 22.7519 .06145 16.2722 29 32 .00931 107.426 .02677 37.. 3579 .04424 22.6020 .06175 16.1952 28 33 .00960 104.171 .02706 36.9.560 .04454 22.4541 .06204 16.1190 27 34 .00989 101.107 .02735 36.. 5627 .04483 22.3081 .06233 16.0435 26 35 .01018 98.2179 ,02764 36.1776 .04512 22.1640 .06262 15.9687 25 36 .01047 95.4895 .02793 .35.8006 .04541 22.0217 .06291 15.8945 24 37 .01076 92.9085 .02822 ;35. 4.313 .04570 21.8813 .06321 15.8211 23 38 .01105 90.4633 .02851 35.0695 .04599 21.7426 .06350 15.7483 22 39 .01135 88.1436 .02881 34.7151 .04628 21.6056 .06379 15.6762 21 40 .01164 85.9398 .02910 S4.3678 .04658 21.4704 .OfrlOS 15.6048 20 41 .01193 as. 84.35 .029.39 34.0273 .04687 21.3.369 .064.37 15.5340 19 42 .01222 81.8470 .02968 33.6935 .04716 21.2049 .06467 15.4638 18 43 .01251 79.9434 .02997 33.. 3662 ,04745 21.0747 .06496 15.3943 17 44 .01280 78.1263 .03026 33.04.52 .04774 20.9460 .06525 15.3254 16 45 .01309 76.3900 .03055 32.7.3<;3 .04803 20.8188 .06.554 15.2.571 15 46 .C1338 74.7292 .030^1 32.4..,13 .048.33 20.69.32 .06584 15.1893 14 47 .01367 73.1.390 .03114 32.1181 .04862 20.5691 .06613 15.1222 13 48 .01396 71.6151 .03143 31.8205 .04891 20.4465 .06642 15.0.557 12 49 .01425 70.1.5.33 .03172 31.. 5284 .04920 20.3253 .06671 14.9898 11 50 .01455 68.7501 .03201 31.2416 .04949 20.2056 .06700 14.9244 10 51 .01484 67.4019 .03230 30.9599 .04978 20.0872 .067.30 14.8,596 9 52 .01513 66.1055 .a3259 30.6833 .05007 19.9702 .06759 14.79.54 8 53 .01542 64.8580 .0.3288 30.4116 .0.5037 19.8546 .06788 14.7.317 i 54 .01571 63.6567 .03317 30.1446 .05066 19.7403 .06817 14.6685 C 55 .01608 62.4992 .0.3.346 29.8823 .05095 19.6273 .06847 14.6059 5 56 .01629 61.3829 .03.376 29.6245 .0.5124 19.5156 .06876 14.54.38 4 57 .01658 60.3058 .0.3405 29.3711 .0.51.53 19.4051 .06905 14.4823 3 58 .01687 59.2659 .0^434 29.1220 .05182 19.2959 .06934 14.4212 o 59 .01716 58.2612 .03463 28.8771 .05212 19.1879 .06963 14.. 3607 1 60 / .-■■1 ■! .01746 Cotang 57.2900 .03492 28.6363 .05241 Cotang 19.0811 .06993 Cotang 14.3007 Tang f Tang ' Cotang Tang Tang 8 9° 88° i 87» 8 6° TABLE XII.-TAiSraENTS AND COTANGENTS. 321 40 1 5° ■ 1 6^ •J '0 60 / Tang .0(5993 Cotang Tang .08749 Cotang 11.4:301 Tang .10510 Cotang Tang .12278 Cotang 8.14435 T) 14.3007 9.51436 1 .07022 14.2411 .08778 11.3919 .10540 9.48781 .12308 8.12481 59 2 .07051 14.1821 .08807 11.3^0 .10569 9.46141 .12338 8.10536 58 3 .07080 14.1235 .08837 11.3163 .10599 9.43515 .12367 8.08600 57 4 .07110 14.0655 .08866 11.2789 .10628 9.40904 .12397 8.06674 56 5 .07139 14.0079 .08895 11.5W17 .10657 9.38307 .12426 8.04756 55 6 .07168 13.9507 I .08925 11.2048 .10687 9.35724 .12456 8.02848 54 7 .07197 13.8940 .08954 11.1681 .10716 9.33155 .12485 8.00948 53 8 .07227 13.8378 .08983 11.1316 .10746 9.30599 .12515 7.99058 52 9 .07256 13.7821 .09013 11.0954 .10775 9.28058 .12544 7.97176 51 10 .07285 13.7267 .09042 11.0594 .10805 9.25530 .12574 7.95302 50 11 .07314 13.6719 .09071 11.0237 .10834 9.23016 .12603 7.93438 49 12 .07344 13.6174 .09101 10.9882 .10863 9.20516 .12633 7.91582 48 13 .07373 13.56:^4 .09130 10.9529 .10893 9.18028 .12662 7.89734 47 14 .07402 13.5098 .09159 10.9178 .10922 9.15554 .12692 7.87895 46 15 .07431 13.4566 .09189 10.8829 .10952 9.13093 .12722 7.86064 •45 IG .07461 13.4039 .09218 10.8483 .10981 9.10646 .12751 7.84242 44 17 .07490 13.3515 .09247 10.8139 .11011 9.08211 .12781 7.82428 43 18 .07ol9 13.2996 1 .09277 10.7797 .11040 9.05789 .12810 7.80622 42 19 .07548 13.2480 .09306 10.7157 .11070 9.03379 .12840 7.78825 41 20 .07578 13.1969 .09335 10.7119 .11099 9.00983 .12869 7.77035 40 21 .07607 13.1461 .09365 10.6783 .11128 8.98598 .12899 7.75254 39 22 .07636 13.0958 .09394 10.6450 .11158 8.96227 .12929 7.73480 38 23 .07665 13.0458 .09423 10.6118 .11187 8.93867 .12958 7.71715 37 24 .07695 12.9962 .09453 10.5789 .11217 8.91520 .12988 7.69957 36 25 .07724 12.9469 .09482 10.5402 i .11246 8.89185 .13017 7.68208 35 26 .07753 12.8981 .09511 10.5136 .11276 8.86862 .13047 7.66466 34 27 .07782 12.8496 .09541 10.4813 .11305 8.84551 .13076 7.64732 33 28 .07812 12.8014 .09570 10.4491 .11335 8.82252 .13106 7.63005 32 29 .07841 12.7536 .09600 10. 4172 .11364 8.79964 .13136 7.61287 31 30 .07870 12.7062 .09629 10.3854 .11394 8.77689 .13165 7.59575 30 31 .07899 12.6591 .09658 10.3538 .11423 8.75425 .13195 7.57872 29 32 .07929 12.61;24 .09088 10.3224 ■ .11452 8.73172 .13224 7.56176 28 33 .07958 12.5660 .09717 10.2913 .11482 8.70931 .13254 7.54487 27 34 .07987 12.5199 .09746 10.2602 .11511 8.68701 .13284 7.52806 26 35 .08017 12.4742 .09776 10.2294 .11541 8.66482 .13313 7.51132 25 36 .08046 12.4288 .09805 10.1988 .11570 8.64275 .13343 7.49465 24 37 .08075 12.3838 .09834 10.1683 .11600 8.62078 .13372 7.47806 23 38 .08104 12.3390 .09804 10.1381 .11629 8.59893 .13402 7.46154 22 39 .08134 12.2946 .09893 10.1080 .IK ' 8.57718 .13432 7.44509 21 40 .08163 12.2505 .09923 10.0780 .11688 8.55555 .13461 7.42871 20 41 .08192 12.2067 .0995?^ 10.0483 .11718 P. 53402 .13491 7.41240 19 42 .08221 12.1632 .09981 10.0187 .11747 8.51259 .13521 7.39616 18 43 .08251 12.1201 .10011 9.98931 .11777 8.49128 .13550 7.37999 17 44 .08280 12.0772 .10040 9.96007 .11800 8.47007 .13580 7.36389 16 45. .08309 12.0346 .10069 9.93101 .11836 8.44896 .13609 7.34786 15 46 .08389 11.9923 .10099 9.90211 .11805 8.42795 .13639 7.33190 14 ^•; .08368 11.9504 .10128 9.87338 .11895 8.40705 .13669 7.31600 13 iS .08397 11.9087 .10158 9.ai482 .11924 8.38625 .13698 7.30018. 12 49 .08427 11.8673 .10187 9.81641 .11954 8.36555 .13728 7.28442- 11 50 .08456 11.8262 .10216 9.78817 .11983 8.34496 .13758 7.26873 10 51 .08485 11.7853 .10246 8.76009 .12013 8.32446 .13787 7.25310 9 52 .08514 11.7448 .10275 9.73217 .12042 8.30406 .13817 7.23754 8 53 .0a544 11.7045 .10305 9.70441 .12072 8.28376 .13846 7.22204 7 54 .08573 11.6645 .10334 9.67680 .12101 8.26355 .13876 7.20661 6 55 .08602 11.6248 .10363 9.64935 .12131 8.24345 .13906 7.19125 5 56 .08632 11.5853 .10.393 9.62205 .12160 8.22344 .13935 7.17594 4 57 .08661 11.5461 .10422 9.59490 .12190 8.20352 .13965 7.16071 3 58 .08690 11.5072 .10452 9.. 56791 1 .12219 8.18370 .13995 7.14553 2 59 .08720 11.4685 .10481 9.54106 .12249 8.16398 .14024 7.13042 1 60 .08749 Cotang 11.4301 Taug j .10510 Cotang 8 9.51436 I Tang 1 40 1 .12278 Cotang 8.14435 ! .14054 Cotang 7.11537 1 / Tang Tang 8 5° 8 3° 8 2° 622 TA&uE XII.-iANUENTS AND COTANGENTS. / 8° ! 9° 10° 11° / Tang j Cotang Tang Cotang Tang Cotang Tang 1 Cotang ~0 .14054 i 7.11537 .158^38 1 6.31375 .176:33 5.67128 .194:38 5.14455 60 1 .14084 7.10038 .15S68 6.30189 .17663 5.66165 .19468 5.13658 59 2 .14113 7.08.546 . 15898 6.29007 .17693 5.6.5205 .19498 i 5.12862 58 3 .14143 7.07059 .15928 6.27829 .17723 5.64248 .19.529 i 5.12069 57 4 .14173 7.05579 .15958 6.26655 .17753 5.63295 .195.59 5.11279 56 5 .14202 7.04105 .15988 6.2.>i86 .17783 5.62344 .19589 5.10490 55 6 .14232 7.02637 .16017 6.24.321 .17813 5.61397 .19619 5.09704 54 7 .14262 6.91174 i .16047 6.23160 .17843 5.60452 .19649 5.08921 53 8 .14291 6.99718 .16077 6.22003 .17873 5.59511 .19680 5.08139 52 9 .14321 6.98268 .16107 0.20851 .17903 5.58573 .19710- 5.07360 51 10 .14351 6.96823 .16137 6.19703 .17933 5.57638 .19740 5.06584 50 11 .14381 6.9.5385 .16167 6.18559 ' .17963 5.56706 .19770 5.05809 49 12 '.14410 6.93952 ' .16196 6.17419 .17993 5.55777 .19801 5.05037 48 13 .14440 6.92525 .16226 0.16283 .18023 5.54851 .19831 5.042G7 47 14 .14470 6.91104 .162.56 6.1.5151 ( .18053 5.53927 .19861 5.03499 46 15 .14499 6.89688 .16286 0.14023 .18083 5.53007 .19891 5.02734 45 16 .14529 6.88278 .16316 6.12899 .18113 5.52090 .19921 5.01971 44 17 .14559 6.86874 .16346 6.11779 , .18143 5.51176 .19952 5.01210 43 18 .14588 6.85475 1 .16376 6.10064 i .18173 5.50264 .1^982 5.00451 42 19 .14618 6.84082 i .16405 0.09552 , .18203 5.49356 .20012 4.99695 41 20 .14648 6.82694 .16435 1 0.08444 1 .18233 5.48451 .20012 4.98940 40 21 .14678 6.81312 .16465 6.07340 .18263 5.47548 .20073 4.98188 39 22 .14707 6.79936 .16495 6.06240 .18293 5.46648 .20103 4.97438 38 23 .14737 6.7W564 .16525 6.05143 .18:323 5.4.5751 .20133 4.96690 37 24 .14767 6.77199 . 16555 6.04051 j .18:3.53 5.44857 .20164 4.95945 36 25 .14796 6.75838 ; .16585 6.02902 .18:384 5.4:3966 .20194 4.95201 35 26 .14826 6.74483 .16615 6.01878 .18414 5.43077 .20224 4.94460 34 27 .1^856 6.73133 .16645 6.0079'? .18444 5.42192 .20254 4.93721 33 28 .14886 6.71789 .166;-4 5.99720 .18474 5.41309 .20285 4.92984 02 29 .14915 6.70450 .16704 5. 9% 46 ' .18504 5.40429 .20315 4.92249 31 30 .14945 6.69116 .16734 0.97576 .18534 5.39552 .20345 4.91516 30 31 .14975 6.67787 .16764 5.96.510 .18.564 5.38677 .20376 4.90785 29 32 . 15005 6.66463 .16794 5.95448 .18.594 5.37805 .20406 4.90056 28 33 .15034 6.65144 .16824 5.94390 .18624 5.36936 .20436 4.89330 27 34 .15064 6.63831 .16854 5.93335 .18654 5.36070 .20466 4.88605 26 35 .1.J094 6.62523 .163S4 5.92283 : .18684 5.35206 .20497 4.87882 25 36 .15124 6.61219 .16914 5.91236 .18714 5.34S45 .20527 4.87162 24 37 .15153 6.59921 .16944 5.90191 .18745 5.33487 .20557 4.86444 23 38 .15183 6.5S627 .16974 5.89151 . 18775 5.326:31 .20.588 4.85727 22 39 .15213 6.. 57339 .17004 5. 881 14 .18805 5.31778 .20618 4.85013 21 40 .15243 6.56055 .17033 5.87080 .18835 5.30928 .20648 4.84300 20 41 .1.5272 6.54777 .17063 5.86a51 .18865 5.. 30080 .20679 4.83590 19 42 .15302 6.53.503 .17093 5.85024 ' .18895 5.29235 .20709 4.82882 18 43 .15:332 6.52234 .17123 5.84001 .18925 5.28393 .20739 4.82175 17 44 .15362 6.. 50970 .17153 5.82982 .189.55 5.275.53 .20770 4.81471 16 45 . 15391 '6.49710 .17183 5.81966 .18986 5.26715 , .20800 4.80769 15 46 .15421 6.48456 1 .17213 5.80953 .19016 5.25.880 . .20830 4.80068 14 47 .15451 6.47206 ; .17243 5.79944 .19046 5.25048 .20861 4.79370 13 48 .15481 6.4.5961 1 ; .17273 5.78938 .19076 5.24218 .20891 4.78673 12 49 .15511 6.44720 1 ! .17303 5.77936 .19106 5.23:391 .20921 4.77978 11 50 .15540 6.43484 i 1 .17333 5.76937 .19136 5.22566 .20952 4.77286 10 51 .15570 6.42253 i .17363 5.7.5941 .19166 5.21744 .20982 4.76595 9 52 .15600 6.41026 ! .17393 5.74949 .19197 5.20925 .21013 4.75906 8 53 .15630 6.39804 ! .17423 5.73960 .19227 5.2<1107 .21043 4.75iil9 7 54 .15660 6.38587 .174.53 5.72974 .192.57 5.19293 .21073 4.74.534 6 55 .15689 6.37374 .17483 5.71992 .19287 5.18480 .21104 4.73851 5 56 .15719 6.. 361 65 .17513 5.71013 .19:317 5.17671 .21134 4.73170 4 57 .15749 6.. 34961 .17.54:3 5.700:37 j .19:347 5.16863 .21164 4.72490 3 TjS .15779 6.33761 .17573 5.69064 .19:378 5.16058 .21195 4.71813 2 :>9 .1.5809 6.32.566 .17603 5.68094 .19408 5.152.56 .21225 4.711:37 1 CO .15838 6.31375 1 .17633 Cotang 8 5.67128 , .194:38 Cotang 5.1+4.55 .21256 Cotang 4,70463 Cotang Tang Tang 1 0° ' Tang Tang 81° ' 79° 7 8° TABLE XIL— TANGENTS AND (JUTaNUENTS. :333 / "o 12° 1 13° 1 14° 1 15° 1 / 60 -Tang .21256 Cotang Tang .23087 Cotang Tang .249;^ Cotang 1 Tang .26795 Cotang 4.70463 4.33148 4.01078 3.73205 1 .21286 4.69791 .23117 4.32573 .24964 4.tK)582 .26826 3.72771 59 2 .21316 4.69121 .23148 4.32001 .24995 4.00086 .26857 3.72X38 58 3 .21347 4.68452 .23179 4.31430 i .25026 3.99592 .26888 3.71907 57 4 .21377 4.67786 .23209 4.30860 .25056 3.99099 .26920 3.71476 56 5 .21408 4.67121 .23240 4.30291 .25087 3.98607 .26951 3.71046 55 6 .21438 4.66458 .23271 4.29724 .25118 3.98117 .26982 3.70616 54 7 .21469 4.65797 .23301 4.29159 .25149 3.97627 .27013 3.70188 53 8 .21499 4.65138 .23332 4.28595 .25180 3.97139 .27044 3.69761 52 9 .215^9 4.64480 .23363 4.28032 .25211 3.96651 .27076 3.69335 51 10 .21560 4.63825 .23393 4.27471 .25242 3.96165 .27107 3.68909 50 11 .21590 4.63171 .23424 4.26911 .25273 3.95680 .27138 3.68485 49 12 .21621 4.62518 .23455 4.26352 .25304 3.95196 .27169 3.68061 48 13 .21651 4.61868 .23485 4.25795 .2.5335 3.94713 .27201 3.67638 47 14 .21682 4.61219 1 .23516 4.25239 .25366 3.94232 .27232 3.67217 46 15 .21712 4.60572 .23547 4.24685 .25397 3.93751 .27263 3.66796 45 1 16 .21743 4.59927 .23578 4.24132 .25428 3.93271 .27294 3.66376 44 17 .21773 4.59283 .23608 4.23580 .254.59 3.92793 .27326 3.65957 43 ■i8 .21804 4.58641 .23639 4.23030 .25490 3.92316 .27357 3.65538 42 19 .21834 4.5800f .23670 4.22481 .25.521 3.91839 .27388 3.6.5121 41 20 .21864 4.57363 .23700 4.21933 .25552 3.91364 .27419 3.64705 40 21 .21895 4.56726 .23731 4.21387 .25583 3.90890 .27451 3.64289 39 22 .219C5 4.56091 .23762 4.20842 .25614 3.90417 .27482 3.63874 38 23 .21956 4.5.5458 .23793 4.20298 .25645 3.89945 .27513 3.63461 37 24 .21986 4.54826 .23823 4.19756 : .25676 3.89474 .27545 3.63048 36 25 .52017 4.54196 .23^54 4.19215 .25707 3.89004 .27576 3.62636 35 26 .22047 4.53.568 .23885 4.18675 .25738 3.88536 .27607 3.62224 34 27 .22078 4.52941 .23916 4.18137 .25769 3.88068 .27638 3.61814 33 28 .22108 4.52316 .23946 4.17600 .25800 3.87601 .27670 3.61405 32 29 .22139 4.51693 .23977 4.17064 .25831 3.87136 .27701 3.60996 31 30 .22109 4.51071 .24008 4.16530 .25862 3.86671 .27732 3 60588 30 31 .22200 4.50451 .24039 4.15997 .25893 3.86208 .27764 3.60181 29 32 .22231 4.49832 .24069 4.15465 .2.5924 3.85745 .27795 3.59775 28^ 33 .22201 4.49215 .24100 4.14934 .25955 3.85284 .27826 3.59370 27 34 .22292 4.48600 .24131 4.14405 .25986 3.84824 .278.58 3.58966 26 35 .22322 4.47986 .24162 4.13877 .26017 3.84364 .27889 3.58562 25 36 .22353 4.47374 .24193 4.13350 .26048 3.83906 .27921 3.58160 24 37 .22383 4.46764 .24223 4.12825 20079 3.83449 .27952 3.57758 38 .224U 4.46155 .24254 4.12301 .26110 3.82992 .27983 3.57357 39 .22444 4.45548 .24285 4.11778 .26141 3.82537 .28015 3.56957 v.\ 40 .22475 4.44942 .24316 4.11256 .26172 3.82083 .28046 3.b6557 20 41 .22505 4.44338 ,24347 4.10736 .26203 3.81630 .28077 3.56159 19 t2 .22536 4.43735 .24377 4.10216 .26235 3.81177 .28109 3.. 55761 18 43 .22567 4.43134 .24408 4.09699 .26266 3.80726 .28140 3.553G4 17 44 .22597 4.42534 .24439 4.09182 .26297 3.80276 .28172 3.54968 16 45 .22628 4.41936 .24470 4.08666 .26328 3.79827 .28203 3.54573 15 4G .22658 4.41340 .24501 4.08152 .26359 3.79378 .28234 3.. 54179 14 47 .22GS9 4.40745 .24532 4.07639 .26390 3.78931 .28266 3.53785 13 48 .22719 4.401.52 .24562 4.07127 .26421 3.78485 .28297 3.. 53393 12 49 .22750 4.39560 .24593 4.06616 .264.52 3.78040 .28329 3.53001 11 50 .22781 4.38969 .24624 4.06107 .26483 3.77595 .28360 3.52609 10 51 .22811 4.38381 .24655 4.05599 .26515 3.77152 .28391 3.52219 9 52 .22842 4.37793 .24686 4.05092 .26546 3.76709 .28423 3.51829 8 53 .32872 4.37207 .24717 4.04.586 .26577 3.76268 .28454 3.51441 7 54 .22903 4.36623 .24747 4.04081 .26608 3.75828 .28486 3.51053 6 55 .22934 4.36040 .24778 4.03578 .26639 S . 75388 .28517 3.50666 5 56 .22964 4.354.59 : .24809 4.03076 .26670 3.74950 .28549 3.50279 4 57 .22995 4.a4879 ! .24840 4.02.574 .26701 3.74512 .28580 3.49894 3 58 .23026 4.. 3^1300 ' .24871 4.02074 .26733 3.74075 .28612 3.49.509 2 5!) .23056 4.33723 .21902 4.01.576 .2()764 3.73640 .28643 3.49125 1 60 / .2;i087 Cotang 4.33148 .249:i3 Cotang 4.01078 .26795 Cotang 3.73205 .28675 , Cotang 3.48741 Tang Tang Tang Tang 77° ! 76° 75° 7 4° 324 TABLE XII.— TANGENTS AND COTANGENTS. "o 16° 1 17° 1 i 18° 1 19° / 60 Tang .28675 Cotang Tang ..30573 Cotang Tang .32492 Cotang Tang .34433 Cotang 3.48741 3.27085 3.07768 2.90421 1 .28706 3.48359 .30605 3.26745 .32524 3.07464 .34465 2.90147 59 2 .28738 3.47977 .30637 3.26406 1 .32556 3.07160 .34498 2.89873 58 3 .28769 3.47596 .30669 3.26067 i .32588 3.06857 .34530 2.89600 57 4 .28800 3.47216 .30700 3.25729 i .32621 3.06554 .34563 2.89327 56 5 .28832 3.46837 .30732 3.25392 .32653 3.06252 .34596 2.89055 55 6 .28864 3.46458 .30764 3.25055 .32685 3.05950 .34628 2.88783 54 7 .28895 3.46080 1 .30796 3.24719 .32717 3.05649 .34661 2.88511 .53 8 .28927 3.45703 .30828 3.24383 .32749 3.05349 .34693 2.88240 52 • 9 .28958 3.45.327 .30860 3.24049 .32782 3.05049 .34726 2.87970 51 10 .28990 3.44951 .30891 3.23714 .32814 3.04749 .34758 2 87700 50 11 .29021 3.44576 .30923 3.23381 .32846 3.04450 .34791 2.874.30 h9 12 .29053 3.44202 .30955 3.23048 .32878 3.041.52 .34824 2.87161 48 13 .29084 3.43829 .30987 3.22715 .32911 3.0.3854 .34856 2.86892 47 14 .29116 3.43456 .31019 3.22384 .32943 3.03556 .34889 2.86624 46 1.5 .29147 3.4.3084 .31051 3.22053 .32975 3.03260 .34922 2.86356 45 16 .29179 3.42713 .31083 3.21722 .33007 3.02963 .34954 2.86089 44 17 .29210 3.42343 .31115 3.21392 .33040 3.02667 .34987 2.8.5822 43 18 .29242 3.41973 1 .31147 3.21063 .33072 3.02372 .35020 2.85555 42 19 .29274 3.41604 i .31178 3.20734 .33104 3.02077 .35052 2.85289 41 20 .29305 3.41236 .31210 3.20406 .33136 3.01783 .35085 2.85023 40 21 .29337 3.40809 .31242 3.20079 .33169 3.01489 .35118 2.84758 39 22 .29368 3.40502 ..31274 3.19752 .3.3201 3.01196 .35150 2.84494 38 23 .29400 3.40136 .31306 3.19426 .3.3233 3.00903 .35183 2.84229 37 24 .29432 3.39771 .31338 3.19100 .33266 3.00611 .35216 2.8.3965 36 25 .29463 3.39406 1 .31370 3.18775 .33298 3.00319 .35248 2.83702 35 26 .29495 3.. 39042 .31402 3.18451 .33330 3.00028 .35281 2.83439 34 27 .29526 3.38079 .31434 3.18127 .33363 2.99738 .35314 2.8,3176 33 28 .29.558 3.. 3831 7 .31466 3.17804 .33395 2.99447 .35.346 2.82914 32 29 .29590 3.379.55 .31498 3.17481 .33427 2.99158 .35379 2.82653 31 30 .29621 3.37594 .31530 3.17159 .33460 2.98868 .35412 2.82391 30 31 .296.53 3.372.34 .31.562 3.16838 .33492 2.98.580 .35445 2.82130 29 32 .29685 3.36875 .31594 3.16517 .o3524 2.98292 .35477 2.81870 28 33 .29716 3.36516 .31626 3.16197 .3.3557 2.98004 .35510 2.81610 27 34 .29748 3. 361.58 .31658 3.15877 .3.3589 2.97717 .35543 2.81350 26 35 .29780 3.. 3.5800 .31690 3.1.55.58 1 .33621 2.97430 .35576 2.81091 25 36 .29811 3.35443 .31722 3.15240 .33654 2.97144 .35608 2.80833 24 37 .29843 3.35087 .317.54 • 3.14922 .33686 2.96858 .35641 2.80574 23 38 .29875 3.34732 .31786 3.14605 .3.3718 2.90573 .35674 2.80316 22 39 .29906 3.34377 .31818 3.14288 .33751 2.96288 1 .35707 2.80059 21 40 .29938 3.34023 .31850 3.13972 .33783 2.96004 .35740 2.79802 20 41 .29970 3.33670 .31883 3.13656 ..33816 2.95721 .35772 2.79545 19 42 .30001 3.33317 .31914 3.13341 .33848 2.95437 .3.5805 2.79289 18 43 .30033 3.32965 .31946 3.13027 .3.3881 2.9.5155 i .35838 2.79033 17 44 .30065 3.32614 .31978 3.12713 .3.3913 2.94872 .35871 2.78778 16 45 .30097 3.. 32264 .32010 3.12400 .a3945 2.94591 .35904 2.78523 15 46 ..30128 3.. 31914 .32042 3.12087 .33978 2.94309 .35937 2.78269 14 47 .30160 3.31.565 .32074 3.11775 .34010 2.94028 .35969 2.78014 13 48 .30193 3.31216 .32106 3.11464 .34043 2.93748 .36002 2.77761 12 49 .30224 3,. 30868 .32139 3.11153 .34075 2.9.3468 .36035 2.^7507 11 50 .30255 3.30521 .32171 3.10843 .34108 2.93189 .36068 2.77254 10 51 .30287 3.30174 .32203 3.10532 .34140 2.92910 .36101 2.77002 9 52 .30319 3.29829 .32235 3.10223 .34173 2.92632 .36134 2.767.50 8 53 .30351 3.29483 .32267 3.09914 .34205 2.92354 .36167 2.76498 7 54 .30382 3.29139 .32299 3.09606 .34238 2.92076 .36199 2.76247 6 55 .30414 3.28795 .32.331 3.09298 ! .34270 2.91799 .36232 2.75996 5 56 .30446 3.28452 .32:363 3.08991 .34303 2.91.523 .36265 2.75746 4 57 .30478 3.28109 .32.396 3.08685 .343.35 2.91246 .36298 2.7.5496 3 58 .30509 3.27767 .32428 3.08379 .34.368 2.90971 .363;31 2.75246 2 59 .30541 3.27426 ..32460 3.08073 .34400 2.90696 .36364 2.74997 1 60 / .30573 Cotang 3.270R5 .32492 3.07768 .34433 Cotang 2.90421 .36397 Cotang 2.74748 Tang Cotang Tang Tang Tang 73° ■1 72° 1 7 1° 70° TABLE XII.— ^ rANGENI rS AND COTANGENTS. "o 20° 21" 22° 1 23'^ 60 Tang .36397 Cotang 2.74748 Tang .38386 Cotang 2.00509 Tang .40403 Cotang 2.47509 1 Tang Cotang .42447 2.35585 1 .36430 2.74499 .38420 2.60283 -.40436 2.47302 .42482 2.35395 59 2 ] .36463 2.74251 .384.53 2.60057 .40470 2.47095 .42516 2.35205 58 3 .36496 2.74004 .38487 2..59S31 .40504 2.46888 .42551 2.35015 57 4 .36529 2.73756 .38520 2.59006 .40538 2.46682 .42585 2.^4825 56 5 .36562 2.73509 .38553 2.59381 .40572 2.46476 .42619 2.^4636 55 6 .36595 2.73263 .38587 2.59156 .40606 2.46270 .42654 2.34447 54 7 .36628 2.73017 .38620 2.589.32 .40640 2.46065 .42688 2.34258 53 8 ' .36661 2.72771 .38654 2.58708 .40674 2.45860 .42722 2.34069 52 9 .36691 2.72526 .38687 2.58484 .40707 2.4.5655 .42757 2.33881 51 10 .36727 2.72281 .38721 2.58261 .40741 2.4&451 .42791 2.33693 50 11 .36760 2.72036 .38754 2.58038 .40775 2.45246 .42826 2.33505 49 12 .36793 2.71792 .38787 2.57815 .40809 2.45043 .42860 2.33317 48 13 .36826 2.71548 .38821 2.57593 .40843 2.44839 .42894 2.33130 47 14 .36859 2.71305 .38854 2.57371 .40877 2.44636 .42929 2.32943 40 15 .36892 2.71062 .38888 2.57150 .40911 2.44433 .42963 2.32756 45 16 .36925 2.70819 .38921 2.56928 .40945 2.44230 .42998 2.32570 44 17 .36958 2.70577 .38955 2.56707 .40979 2.44027 .43032 2.32383 43 18 .36991 2.70335 .38988 2.56487 .41018 2.43825 .43067 2.. 321 97 42 19 .37024 2.70094 .39022 2.56266 .41047 2.43023 .43101 2.32012 41 20 .37057 2.69853 .39055 2.56046 .41081 2.4*422 .43136 2.31826 40 21 .37090 2.69612 .39089 2.55827 .41115 2.43220 .43170 2.31641 39 22 .37123 2.69371 .39122 2.55608 .41149 2.43019 .43205 2.31456 38 23 .37157 2.69131 .39156 2.55389 .41183 2.42819 .43239 2.31271 37 24 .37190 2.68892 .39190 2.55170 .41217 2.42618 .43274 2.31086 30 25 .37223 2.68653 .39223 2.54952 .41251 2.42418 .43308 2.30902 35 26 .37256 2.68414 .39257 2.54734 .41285 2.42218 .43343 2.30718 34 27 .37289 2.68175 .39290 2.54516 .41319 2.42019 .43378 2.30534 33 28 .37322 2 07937 .39324 2.54299 .41353 2.41819 .43412 2.30351 32 29 .37355 2.67700 .39357 2.54082 .41387 2.41020 .43447 2.30167 31 30 .37388 2.67462 .39391 2.53865 .41421 2.41421 .43481 2.29984 30 31 .37422 2.67225 .39425 2.53648 .41455 2.41223 .43510 2.29801 29 32 .37455 2.66989 .39458 2.53433 .41490 2.41025 .43550 2.29619 28 33 .37488 2.66752 .39492 2.53217 .41524 2.40827 .43585 2.29437 27 34 .37521 2.66516 .39526 2.53001 .41558 2.40629 .43620 2.29254 26 35 .37554 2.66281 .395.59 2.52786 .41592 2.40432 .43054 2.29073 25 36 .37588 2.66046 .39593 2.52571 .41626 2.40235 .43689 2.28891 24 37 .37621 2.65811 .39626 2.52357 .41660 2.40038 .43724 2.28710 23 38 .37654 2.65576 .39660 2.52142 .41694 2.39841 .437o8 2.28528 22 39 .37687 2.65342 .39694 2.51929 .41728 2.39645 .43793 2.28348 21 40 .37720 2.65109 .39727 2.51715 .41763 2.39449 .43828 2.28167 20 41 .37754 2.64875 .39761 2.51502 .41797 2.39253 .43862 2.27987 19 42 .37787 2.64642 .39795 2.51289 .41831. 2.39058 .43897 2.27806 18 43 .37820 2.64410 .39829 2.. 51076 .41865 2.38863 .43932 2.27626 17 44 .37853 2.64177 .39862 2.50864 41899 2.38668 .43066 2.27447 16 45 .37887 2.63945 .39896 2.50652 .41933 2.38473 .44001 2.27267 15 46 .37920 2.63714 .39930 2.50440 .41968 2.38279 .44036 2.27088 14 47 .37953 2.63483 .39963 2.50229 .42002 2.38084 .44071 2.26909 13 48 .37986 2.63252 .39997 2.50018 .42036 2.37891 ,44105 2.26730 12 49 .38030 2.G3021 .40031 2.49807 .42070 2,37697 .44140 2.20.552 11 50 .38053 2.62791 .40065 2.49597 .42105 2.37504 .44175 2.26374 10 51 .38086 2.62561 .40098 2.49386 .42139 2.37311 .44210 2.26196 9 52 ..38120 2.62332 .40132 -2.40177 .42173 2.37118 .44244 2.26018 8 53 .38153 2.62103 .40166 2.48967 .42-307 2.36925 .44279 2.25840 7 54 .38186 2.61874 .40200 2.48758 .42242 2.36?3;3 .44314 2.25663 6 55 .38220 2.61646 .40234 2.48549 .422^6 2.36541 .44349 2.25486 5 56 .38253 2.61418 .40267 2 48340 .42310 2.36349 .44:384 2.25309 4 57 .38286 2.61190 .40301 2.48132 .42345 2.36158 ..44418 2.2.5132 3 58 .38320 2.60963 .40335 2.47924 .42379 2.3.5967 .44453 «. 24956 2 59 .38353 2.60736 .40369 , 2.47716 .42413 2.35776 .44488 2.24780 1 60 / .;38386 Cotang 2.60509 .40403 1 Cotang ! 2.47509 Tang 1 .42447 Cotang 2.3.5.585 .^523 2.24604 Tang Tang Cotang : Tang 69» 6 8<^ 67» 66° 326 TABLE XII.— TANGENTS AND COTANGENTS. ~0 24° 1 25° 26° 27° 60 Tang .44523 Cotang Tang .46631 Cotang ' Tang .48773 Cotang ' 2.05U30 Tang .50953 Cotang 2.24604 2.14451 1.96261 1 .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 .46737 2.13963 .48881 2.04577 .51063 1.95838 57 4 .44062 2.23902 .46772 2.1:3801 .48917 2.04426 .51099 1.95698 '56 1 5 .44097 2.23727 .46808 2.1:3039 .48953 2.04276 .51136 1.95557 55 6 .44733 2.23553 .46.S43 2.13477 .48989 2.04125 .51173 1.95417 54 7 .44767 2.2:3378 .46879 2.1.3:316 .49026 2.03975 .51209 1.95277 53 8 .44802 2.23204 .46914 2.1:3154 .49063 2.0:3825 .51246 1.95137 52 9 .44837 2.2.30.30 .46950 2.12993 I .49098 2.0.3(575 .51283 1.94997 51 10 .44872 2.22857 ,46985 2.12833 .49134 2.03526 .51319 1.94858 50 11 .44907 2.22683 .47021 2.12671 ! .49170 2.03.376 .51356 1.94718 49 12 .44942 2.22510 .47056 2.12511 .49206 2.03227 .51393 1.94579 48 13 .44977 2.223.37 .47092 2.12350 .49242 2.03078 .51430 1.94440 47 14 .43012 2.22164 .47128 2.12190 .49278 2.02929 .51467 1.94.301 46 15 .45047 2.21992 .47163 2.12030 .49315 2.02780 .51503 1.94162 45 16 .45082 2.21819 .47199 2.11871 .49351 2.02631 ! .51540 1.94023 44 17 .45117 2.21647 .472:34 2.11711 .49387 2.0248:3 .51o< i 1.93885 43 18 .45153 2.21475 .47270 2.11.552 .49423 2.02:335 .51614 1.93746 42 19 .45187 2.21304 .47305 2.11.392 .49459 2.02187 .51651 1.93608 41 20 .45222 2.21132 .47^41 2.11233 .49495 2.02039 .51688 1.93470 40 21 .45257 2.20961 .47377 2.11075 .495.32 2.01891 .51724 1.9:3.332 39 22 .45292 2.20790 ' .47412 2.10916 .49.568 2.01743 .51761 1.93195 38 23 .45327 2.20019 i .47448 2.10758 .49604 2.01596 .51798 1.93057 37 24 .45362 2.20449 .47483 2.10000 .49640 2.01449 .51&35 1.92920 36 25 .45397 2.20278 .47519 2.10442 .49077 2.01:302 .51872 1.92782 35 26 .45-132 2.20108 .47555 2.10284 i .49713 2.01155 I .51909 1.92645 34 27 .45467 2.199:38 .47590 2.10126 .49749 2.01008 .51946 1.92508 33 28 .45502 2.19769 .47026 2.09969 .49786 2.00862 .51983 1.92:371 32 29 .4.5538 2.19.599 .47002 2.09811 .49822 2.00715 ..52020 1.92235 31 30 .45573 2.1^430 .47698 2.09654 .49858 2.00569 .52057 1.92098 30 31 .45608 2.19261 .477a3 2.09498 .49894 2.00423 .52094 1.91962 29 32 .45643 2.19093 .47769 2.09341 .49931 2.00277 .521:31 .1.91826 28 33 .45678 2.18923 .47805 2.09184 .49967 2.00131 .52108 1.91690 27 3i .45713 2.18755 .47840 2.09028- .50004 1.99986 .52205 1.91554 26 35 .45748 2.18587 .47876 2.08872 .50040 1.99841 .52242 1.91418 25 36 .45784 2.18419 .47913 2.08716 .5007-6 1.99695 .52279 1.91282 24 37 .45819 2.18251 .47948 .2.08560 .50113 1.99550 .52316 1.91147 23 38 .45854 2.18084 .47984 2.08405 .50149 1.99406 .52:353 1.91012 22 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 20 41 .45960 2.17.582 .48091 2.07939 .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 2.07'630 .50331 1.98684 ..525.38 1.903:37 17 44 .46065 2.17083 .48198 2.07476 ' .50.368 1.98.540 . 52575 1.90203 16 45 .46101 2.16917 ' .482:34 2.07321 .50404 1.98396 .52613 1.90069 15 46 .46136 2.16751 1 .48270 2.07167 .50441 1.98253 .52650 1.899:35 14 47 .46171 2.16585 ' .48.306 2.07014 .50477 1.98110 .52687 1.8G801 13 48 .46206 2.16420 .48342 2.06860 .50514 1.97966 .52724 1.89667 12 49 .46242 2.162.55 .48378 2.06706 .50550 1.97823 .52761 1.89.533 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 1.8926b 9 52 .46:348 2.15760 .48486 2.06247 .50660 1.97:395. .52873 1.89133 81 53 .46:383 2.15596 .48521 2.06094 1 .50696 1.97253 .52910 1. 89000 V 7' 54 .4^118 2.154:32 .48557 2.05942 .50733 1.97111 .52947 1.88867 6 55 .46454 2.15268 .48593 2.05790 .50769 1.96969 .52985 1.887.34 5I 56 .46489 2.15104 .48629 2.05637 .50806 1.96827 .53022 1.88602 4 57 .46525 2.14940 .48665 2.05485 ..50843 1.96685 .53059 1.88469 3 58 .46560 2.14777 .48701 2.05:3:33 .50879 1.90544 .53096 1.88337 2 59 .46595 2.14614 1 .487:37 2.05182 .50916 1.96402 .53ia4 1.88205 1 60 „ .46631 Cotaii^ 2.144.51 j Taug 1 .48773 2.0.50.30 .509.53 Cotang 1.96201 .53171 Cotang 1.88073 / Cotang Taug Tang } Tang 65° 1 1 64° ' i 6 3° 1 6 2» TABLE XII.-TANGENTS AND COTANGENTS. 327 28° 1 . 29° 30° ' 31° 60 Tang .53171 Cotang Tang .55431 Cotang Tang .57735 Cotang ' 1.73205 1 Tang .60086 Cotang 1.66428 1.88073 ' 1.80405 i 1 .53208 1.87941 ! .55469 1.80281 .57774 1.73089 ' .60126 1.66.318 59 2 .53246 1.8'; 809 ; .55507 1.80158 .57813 1.72973 .60165 1.60209 58 3 .53283 1.87677 .55545 1.80034 .57851 1.72857 .60205 1 . 06099 57 4 .53320 1.87546 1 .55583 1.79911 .57890 1.72741 .60245 1.0.5990 56 5 .53358 1.87415 i .55621 1.79788 , .57929 1.72625 i .60284 1.65881 55 6 .5:3395 1.87283 1 .55659 1.79665 ! .57968 1.72509 1 .60324 1.65772 54 7 .53432 1.87152 t .55697 1.79542 ] .58007 1.72393 .60304 1.65668 53 8 .53470 1.87021 ' ,55736 1.79419 I .58046 1.72278 .60403 1.6.5554 .52 9 .53507 1.86891 .55774 1.79296 .58085 1.72163 .60443 1.65445 51 10 .53545 1.86760 .55813 1.79174 : .58124 1.72047 .60483 1.65337 50 11 .53582 1.86630 .55850 1.79051 .58162 1.71933 .60522 1.65228 49 12 .53620 1.86499 .55888 1.78929 1 .58201 1.71817 ! .60562 1.65120 48 13 .53657 1.86369 .55926 1.78807 .58240 1.71702 .60602 1.65011 47 14 .53694 1.86239 .55904 1.7B685 .58279 1.71588 .00642 1.64903 46 15 .53732 1.86109 .56003 1.78563 .58318 1.71473 .60681 1.64795 45 16 .53769 1.85979 .56041 1.78441 , .58:357 1.71358 .60721 1.04687 44 17 .53807 1.85850 .56079 1.78:319 .58396 1.71244 .60761 1.64579 43 18 .53844 1.85720 i .56117 1.78198 , .58435 1.71129 .60801 1.04471 42 19 .53882 1.85591 .56156 1.78077 .58474 1.71015 .60841 1.64363 41 20 .53920 1.85462 .56194 1.77955 .58513 1.70901 .60881 1.04256 40 21 .53957 1.853.33 .56232 1.77834 .58552 1.70787 .60921 1.04148 39 22 .53995 1.85204 '■ .56270 1.77713 ..58591 1.70673 .60960 1.04041 38 23 .54032 1.85075 .56309 1.77592 .58631 1.70560 .61000 1.0.39:34 37 24 .54070 1.84946 .56:^47 1.77471 .58670 1.70416 .61040 1.63826 36 25 .54107 1.84818 .56335 1.77351 .58709 1.70332 .61080 1.63719 35 26 .54145 1.84689 .56424 1.77230 .58748 1.70219 .61120 1.6.3612 34 27 .54183 1.84561 ! .56462 1.77110 .58787 1.70106 .61160 1.63505 33 28 .54220 1.84433 .56501 1.76990 .58826 1.69992 .61200 1.63.398 32 29 .54258 1.84305 .56539 1.76869 i .58865 1.69879 .61240 1.6.3292 31 30 .54296 1,84177 .56577 1.76749 1 .58905 1.69766 .61280 1.63185 30 31 .54333 1.84049 i .56616 1.76629 .58944 1.69653' ! .61320 1.63079 29 32 .54371 1.83922 .56654 1.76510 1 .58983 1.09541 1 .61360 1.62972 28 33 .54409 1.8;i794 .56693 1.76390 ' .59022 1.69428 .61400 1.62866 27 34 .54446 1.83667 .56731 1.76271 .59001 1.69316 .61440 1.62760 26 35 .54484 1.83540 .56769 1.76151 .59101 1.69203 .61480 1.62654 25 36 . O'iO/w'V 1.83413 .56808 1.76032 .59140 1.69091 .61520 1.62548 24 37 .54560 1.83280 .50846 1.75913 .59179 1.68979 .61501 1.62442 23 38 .54597 1.83159 .50885 1.75794 .59218 1.68866 ..61601 1.62336 22 39 .54635 1.83033 j .56923 1.75675 ' .59258 1.68754 .61641 1.62230 21 40 .54673 1.82906 .56962 1.75556 .59297 1.68043 .61681 1.02125 20 41 .54711 1.82780 .57000 1.75437 .59336 1.68531 .61721 1.62019 19 42 .54748 1.82654 .570:30 1.75:319 .59:376 1.68419 .61761 1.61914 18 43 .54786 1.82528 .57078 1.75200 .59415 1.68308 .61801 1 61808 17 44 .54824 1.82402 .57116 1.75082 .59454 1.68190 .61842 1.61703 16 45 .54862 1.82276 .57155 1.74964 : .59494 1.68085 .61882 1.61598 15 46 .54900 1.82150 .57193 1.74846 i .59533 1.67974 .61922 1.61493 14 47 .54938 1.82025 .572:32 1.74728 1 .59573 1.67863 61962 1.61388 13 48 .54975 1.81899 .57271 1.74610 .59612 1.07752 .62003 1.61283 12 49 .5.5013 1.81774 ' .57:309 1.74402 .59651 1. 67641 .62043 1.61179 11 50 .55051 1.81649 .57:348 1.7'4375 .59691 1.67530 .62083 1.61074 10 51 ; 55089 1.81524 .57:386 1.74257 ' .59730 1.67419 .62124 1.60970 9 52 .55127 1.81399 .57425 1.74140 .59770 1.67:309 .62164 1.60865 8 53 .55165 1.81274 .57464 1.74022 i .59809 1.67198 .62204 1.60761 t 54 .55203 1.81150 .57503 1.73905 .59849 1.67088 .62245 1.60657 6 55 .55241 1.81025 .57541 1.7:3788 .59888 1.66978 .622a5 1.00553 5 56 .55279 1.80901 : .57-580 1.7:3671 .59928 1.66867 .62:325 1.60449 4 57 .55317 1.S0777 1 .57619 1.73555 ' .59967 1.66757 .62366 1.60.345 3 58 . 55:i55 1.S0653 I .57657 1.73438 .60007 1.66647 .62406 1.60241 2 50 .55393 1.80529 ' .57696 1.7:3:321 .60046 1.66538 .62446 1.60137 1 60 .554:J1 Cotaijg ! 6 1.M0405 i Tang ; i° 1 .57735 Cotant^ G 1.73205 Tang I 0^ ! .60086 Cotang 1.66428 .62487 Cotang 1.600:33 Tang Tang 6 9° 5 8° rA.TA"A* TABLE XII.— TANGENTS AND COTANGENTS. / ~0 32° i 33° 11 34° 1 35° 1 / 60 Tang 1 .6^487 Cotangr 1.60033 Tang .64941 Cotang Tang Cotang i Tang .70021 Cotang 1.53986 .67451 1.48256 1.42815 1 .62527 1.. 59930 .64982 1.53888 .67493 1.48163 .70064 1.42726 59 2 .62568 1.59826 .65024 1.53791 .67536 1.48070 .70107 1.42638 58 3 .62608 1.59723 .65065 1.53693 .67578 1.47977 .70151 1.42550 57 4 .62649 1.59620 .65106 1.53595 .67620 1.47885 .70194 1.42462 56 5 .62689 1.59517 .65148 1.53497 .67663 1.47792 .70238 1.42,374 55 6 .62730 1.59414 .65189 1.53400 .67705 1.47699 .70281 1.42286 54 7 .62770 1.59311 .65231 1.53302 .67748 1.47607 .70325 1.42198 53 8 .62811 1.59208 .65272 1.53205 .67790 1.47514 .70388 1.42110 52 9 .628.52 1.59105 .6.5314 1.53107 .67832 1.47422 .70412 1.42022 51 10 .62892 1.59002 .65355 1.53010 .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 .63055 1.58593 .65521 1.52622 .68045 1.46962 .70629 1.41584 46 15 .63095 1.58490 .65563 1.52525 .08088 1.46870 .70673 1.41497 45 16 .63136 1.58388 .65604 1.52429 .68130 1.46778 .70717 1.41409 44 ir .63177 1.58286 .65646 1.52332 ' .68173 1.46686 .70760 1.41.322 43 18 .63217 1.58184 .65688 1.52235 .68215 1.46595 .70804 1.41235 42 19 .63258 1.58083 .65729 1.52139 .68258 1.46503 .70848 1.41148 41 20 .63299 1.57981 .65771 1.52043 .68301 1.46411 .70891 1.41061 40 21 .63340 1.57879 .65813 1.51940 .68343 1.46320 .70935 1.40974 39 22 .63380 1.57778 .65854 1.51850 .68386 1.46229 .70979 1.40887 .38 23 .63421 1.57676 .65896 1.51754 .68429 1.46137 .71023 1.40800 37 24 .63462 1.5(Oio .65938 1.51658 .68471 1.46046 .71066 1.40714 36 25 .63503 1.57474 .65980 1.51562 .68514 1.45955 .71110 1.40627 35 26 .63544 1.57372 .66021 1.51466 .68557 1.45864 .71154 1.40540 34 27 .63584 1.572^1 .66063 1.51370 .68600 1.45773 .71198 1.40454 33 28 .63625 1.57170 .66105 1.51275 .68642 1.45682 .71242 1.40.367 32 29 .63666 1.57069 .66147 1.51179 .68685 '1.4.5592 .71285 1.40281 31 30 .63707 1.56969 .66189 1.51084 .68728 1.45501 .71329 1.40195 30 31 .63748 1.56868 .66230 1.50988 .68771 1.4.5410 .71373 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 1.4.5229 .71461 1.39936 27 M .63871 1.56566 .66356 1.50702 .68900 1.45139 .71505 1.39850 26 35 .63912 1.56460 .66398 1.50607 .68942 1.45049 .71549 1.39764 25 36 .63953 1.. 56366 .66440 1.50512 .08985 1.44958 .71593 1.. 39679 24 37 .63994 1.56265 .66482 1.50417 1 .69028 1.44868 .71637 1.39593 23 38 .64035 1.56165 .66524 1.50322 1 .09071 ■ 1.44778 .71681 1.39507 22 39 .64076 1.560G5 .66566 1.50228 .69114 1.44688 .71725 1.39421 21 40 .64117 1.55966 .66608 1.50133 .69157 1.44598 .71769 1.39336 20 41 .64158 1.55866 .66650 1.50038 .69200 1.44508 .71813 1.. 39250 19 42 .64199 1.55766 .66092 1.49944 .69243 1.44418 .71857 1.39165 18 43 .64240 1.55666 .66734 1.49849 .69286 1.44329 .71901 1.39079 17 44 .64281 1.55567 .66776 1.49755 .69329 1.44239 .71946 1.38994 16 45 .64322 1.55467 .66818 1.49661 .69372 1.44149 .71990 1.. 38909 15 46 .64;3G3 1.55308 .66860 1.49566 .69416 1.44060 .72034 1.38824 14 47 .64404 1.55269 .66902 1.49472 .69459 1.43970 .72078 1.38738 13 48 .64446 1.55170 .66944 1.49378 .69502 1.4.3881 .72122 1.38653 12 49 .64487 1.55071 .60986 1.49284 .69545 1.43792 .72167 1. 38.508 11 50 .64528 1.54972 .67028 1.49190 .69588 1.43703 .72211 1.384&4 10 51 .64569 1.54873 .67071 1.49097 .69631 1.43614 .72255 1.38399 9 52 .64610 1.54774 .67113 1.49003 .69675 1.4.3525 .72299 1.38314 8 53 .64652 1.54675 .67155 1.48909 .69718 1.43436 .72344 1.38229 7 54 .64693 1.54576 .67197 1.48816 .69761 1.43347 .72388 1.38145 6 55 .64734 1.54478 .67239 1.48722 .69804 1.43258 .72432 1.38060 5 56 .64775 1.54379 I .67282 1.48629 .69847 1.4.3169 .7^77 1.37976 4 57 .64817 1.54281 .67324 1.48536 .69891 1.43080 .72521 1.37891 3 5S .64858 1.54183 .67366 1.48442 .09934 1.42992 .72505 1.37807 2 5S .64899 1.54085 .67409 1.48.349 .69977 1 .42903 .72610 1.37722 1 6C 1 .64941 Cotang 1.. 53986 i .67451 1.48256 .70021 1.42815 .72654 Cotang 1.376.38 > J 1 Tang Cotang Tang Cotang Tang Tang 57° i 56° 55° 54° TABLE XII. -TANGENTS AND COTANGENTS. 329 "o 36° 1 37° 1 38° 39° 1 / 60 Tang 1 .72654 ■ Cotang Tang i .75355 Cotang 1.32704 Tang 1 .78129 Cotang Tang 1 .80978 Cotang 1.2:3490 1.37638 1.27994 1 .72699 1.37554 .75401 1.32024 .78175 1.27917 .81027 1.23416 59 2 .72743 1.37470 .75447 1.32544 .78222 1.27841 .81075 1.23343 58 3 .72788 1.37386 .75492 1.32404 .78269 1.27764 .81123 1.23270 57 4 72832 1.37302 .75538 1.32384 .78:316 1.27688 .81171 1.23196 66 5 .72877 1.37218 .75584 1.32304 .78363 1.27611 .81220 1.23123 55 6 .72921 1.37134 .75629 1.32224 .78410 1.27535 .81268 1.23050 54 7 .72966 1.37050 .75675 1.32144 .78457 1.27458 .81316 1.22977 53 8 .73010 1.36907 .75721 1.32004 .7.S504 1.27382 .81364 1.22904 52 9 .73055 1.3()883 .75707 1.31984 .78551 1.27306 .81413 1.228:51 51 10 .73100 1.36800 .75812 1.31904 .78598 1.27230 .81461 1.22758 50 11 .73144 1.36716 .75858 1.31825 .78645 1.27153 .81510 1.22685 49 12 .73189 1.36633 .75904 1.31745 .78692 1.27077 .81558 1.22612 48 13 .73234 1.36549 .75950 1.31666 .78739 1.27001 .81606 1.225:59 47 14 .73278 1.36466 .75996 1.31586 .78786 1.26925 .81655 1.22467 46 15 .73323 1.30S83 .76042 1.31507 .78834 1.20849 .81703 1.22394 45 16 .73308 1.36300 .76088 1.31427 .78881 1.20774 .81752 1.22321 44 17 .73413 1.36217 .761M 1.31348 .78928 1.26698 .81800 1.22249 43 18 .73457 1.36134 .76180 1.31269 .78975 1.26622 .81849 1.22176 42 19 .73502 1.360.51 .76226 1.31190 .79022 1.26546 ! .81898 1.22104 41 20 .73547 1.35968 .76272 1.31110 .79070 1.26471 .81946 1.22031 40 21 .73592 1.35885 .76318 1.31031 .79117 1.26395 .81995 1.219.59 39 32 .73637 1.35802 .76364 1.30952 .79104 1.26319 ' .82044 1.21886 38 23 .73681 1.35719 .76410 1.30873 .79212 1.26244 i .82092 1.21814 37 21 .73726 1.35037 .76456 1.30795 .79259 1.26169 .82141 1.21742 36 25 .tOIll 1.35554 .76502 1.30716 .79306 1.26093 .82190 1.21670 35 26 .73816 1.35472 .76548 1.30637 .79354 1.26018 .82238 1.21598 34 27 .73801 1.35389 i .76594 1.305.58 .79401 1.25943 .82287 1.21526 33 28 .73906 1.35307 .70040 1.30480 .79449 1.25867 .82336 1.21454 32 29 .73951 1.35224 .70086 1.30401 .79496 1.25702 .82385 1.21382 31 30 .73996 1.35142 .76733 i. 30323 .79544 1.25717 .824:34 1.21310 30 51 .74041 1.3.5060 .76779 1.30244 .79591 1.25642 .82483 1.21238 29 32 .74086 1.34978 .70825 1.30166 .79639 1.25567 .82531 1.21100 28 3:5 .74131 1.34896 .70871 1.30087 .79086 1.25492 .82580 1.21094 27 34 .74176 1.34814 .76918 1.30009 .797:34 1.25417 .82629 1.21023 26 35 .74221 l.;W732 .70964 1.29931 .79781 1.25343 .82678 1.20951 25 36 .74267 1.34650 .77010 1.29853 .79829 1.25268 1 .82727 1.20879 24 37 .74312 1.34568 .77057 1.29775 .79877 1.25193 i .82776 1.20808 23 38 .74357 1.34487 .77103 1.29696 .79924 1.25118 .82825 1.20736 22 39 .74402 1.34405 .77149 1.29618 .79972 1.25044 .82874 1.20605 21 40 .74447 1.34323 .77196 1.29541 .80020 1.24969 .82923 1.20593 20 41 .74492 1.34242 .77242 1.29463 .80067 1.24895 .82972 1.20522 19 42 .74538 1.34160 .77289 1.29385 .80115 1.24820 .83022 1.204.51 18 43 .74583 1.34079 .77335 1.29307 .80163 1.24746 .83071 1.20379 17 44 .74628 1.33998 .77382 1.29229 .80211 1.24672 .83120 1.20308 16 45 .74074 1.33916 .77428 1.29152 .80258 1.24597 .83169 1.20237 15 4G .74719 1.33835 .77475 1.29074 .80306 1.24523 .8:5218 1.20106 14 47 .74764 1.33754 .77521 1.28997 .80354 1.24449 .83268 1.20095 13 48 .74810 1.33673 .77568 1.28919 .80402 1.24375 .83317 1.20024 12 49 .74855 1.3.3.592 .77615 1.28842 .804.50 1.24301 .83366 1.19953 11 50 .74900 1.33511 .77661 1.28764 .80498 1.24227 .83415 1.19882 10 51 .74946 1.33430 .77708 1.28687 .80.546 1.24153 .83465 1.19811 9 52 .74991 1.33349 .77754 1.28610 .80594 1.24079 .83514 1.19740 8 53 75037 1.33268 .77'801 1.28533 .80042 1.24005 .83564 1.19069 7 54 .75082 1.3;5187 .77848 1.28456 .80690 1.23931 .83613 1.19599 6 55 .75128 1.33107 .77895 1.28:579 .807:38 1.23858 .83662 1.19528 5 56 .75173 1.3^3026 .77941 1.28302 .80786 1.23784 .83712 1.19457 4 57 .75219 1.32946 .77988 1.28225 .808:34 1.2:3710 .83761 1.19387 3 58 .75264 1.32865 .78035 1.28148 .80882 1.23637 .83811 1.19316 2 59 .75310 1.32785 .78082 1.28071 .80930 1.2:5563 .83860 1.19iM6 1 00 .75:555 jCotang 1.32704 .78129 Cotang 1.27994 Tang .80978 Cotang 1.2:3490 Tang .83910 Cotang 1.19175 _0 Tang Tang 63° 52° 51° 1 60° 330 TABLE XIl.— TANGENTS AND COTANGENTS. 40° 41° 42° 43° 1 60 Tang Cotang Tang .86929 Cotang 1.15037 Jang .90040 Cotang 1.11061 Tang .93252 Cotang .83910 1.19175 1.07237 1 1 .83960 1.19105 .86980 1.14969 .90093 1.10996 .93306 1.07174 59 2 .8W09 1.1903.5 .87031 1.14902 .90146 1.109:31 ^ .93360 1.07112 1 58 3 .^059 1.18964 1 .87082 1.148*4 .90199 1.10^67 i .93415 1.07049 57 4 .81108 1.18894 1 .87ia3 1.14767 .90251 1.10802 1 .9*469 1.06987 56 5 .84158 1.18824 ' .87184 1.14699 .90:304 1.107:37 .9:3524 ■ 1.06925 55 6 .84208 1.18754 .87236 1.14632 .90:357 1.10672 .93578 1.06862 54 7 .84:',t8 1.18684 .87287 1 . 14565 .90410 1 . 10607 .93633 1.06800 53 8 .8*307 i 1.18614 .87338 1.14498 .90463 1.10543 .93688 1.06738 52 9 .84;357 1 1.1S544 i .87389 1.144:30 .90516 1.10478 .93742 1.06676 51 10 .81407 1.18474 : .87441 1.14363 .90569 1.10414 .93797 1.06613 50 11 .84457 j 1.18404 i .87492 , 1.14296 .90621 1.10349 .93852 1.06551 49 12 .84507 1 1.18334 .87543 1.14229 .90674 1.10285 .93906 1.06489 ■fe 13 .84556 1 . 182G4 .87595 i 1.14162 .90727 1.10220 .93961 1.06427 47 14 .&4606 1.18194 .87646 1.14095 .90781 1.10156 .94016 1.06365 46 15 1 .84656 1.18125 .87698 1.14028 .90834 1 . 10091 .94071 1.06303 45 16: .84706 ; 1.18055 .87749 1.13961 .90887 1.10027 .94125 1.06241 44 17 .84756 i 1.17986 .87801 1.1:3894 .90940 1.09963 .94180 1.06179 43 18 .84806 1.17916 .87852 1.13828 .90993 1.09899 .94235 1.06117 42 19 .84856 1.17^46 .87904 1.13761 .91046 1.09^34 .94290 1.06056 41 20 .84906 1 . 17777 . 87955 1.13694 .91099 1.09770 .94345 1.05994 40 21 .84956 1.17708 .88007 1.13627 .91153 1.09706 .94400 1.05932 39 22 .85006 1.17638 .88059 1.1:3561 .91206 1.09(>42 .94455 1.05870 38 23 .85057 1.17569 .88110 1.1:3494 .91259 1.09578 .94510 1.05809 37 ^ .85107 1.17500 .88162 1.1:3428 .91:313 1.09514 .94565 1.05747 36 25 .85157 1.174:30 .88214 1.13:361 .91366 1.09450 .94620 1.05685 35 26 .85207 1.17361 .88265 1.1:3295 ' .91419 1.09386 .94676 1.05624 34 27 .a5-:'o7 1.17292 .88^317 1.1:3228 .91473 1.09.322 .94731 1.05562 33 28 .85308 1.17223 .aS369 1.131G2 .91526 1.09258 .94786 1.05501 32 29 .85358 1.171.54 .88421 1.1:3096 ' .91580 1.09195 .94&41 1.054.39 31 30 .85408 1.17085 ! .8t^73 1.13029 .9163:3 1.09131 .94896 1.05378 30 31 .85458 1.17016 .88524 1.12963 .91687 1.09067 .94952 1.05317 29 32. .85509 1.1G947 .88576 1.12897 ! .91740 1.09003 .95007 1.052.55 28 33 .85559 1.16878 .88628 1.128:31 .91794 1.08940 .95062 1.05194 27 34 .85609 1.16809 ; .88680 1.12765 .91847 1.08876 .95118 1.05133 26 35 .85660 1.16741 ; .88732 1.12699 .91901 1.08813 .95173 1.05072 25 36 .85710 1.16672 1 .88784 1 . 126:33 91955 1.08749 .95229 1.05010 24 37 .85761 i.ieeas .888:36 1.12567 .92008 1.08686 .95284 1.04949 23 38 .85811 1.1G535 .88888 1.12501 .92062 1.08622 .95.^40 1.04888 22 39 .85862 1.1G4G6 i .88940 1.124:35 .92116 1.08559 .95.395 1.04827 21 40 .85912 1.16398 1 .88992 1.12369 .92170 1.08496 .95451 1.04766 20 41 .85963 1.16.329 .89045 1.12.303 .92224 1.084.32 .95506 1.04705 19 42 .86014 1.1G261 ! .89097 1.122:3s .92277 1.08:369 .95562 1.04644 18 43 .86064 1.16192 .89149 1.12172 .92:331 1.08306 .95618 1.04.5a3 17 44 .86115 1.16124 .89201 1.12106 .92:385 1.08243 .95673 1.04.522 16 45 .86166 1.16056 i .892.53 1.12041 92439 1.08179 .95729 1.04461 15 46 .86216 1.159S7 .89:306 1.11975 .92493 1.08116 .95785 1.04401 14 47 .86207 1.15919 .89:358 1.11909 .92^7 1.08053 .95841 1.04.340 13 48 .86318 1.1.5851 ; .89410 1.11.844 .92601 1 .07990 .95897 1.04279 12 49 .86.368 1.15783 1 .89463 1 11778 .92655 1.07927 .95952 1.04218 11 50 .86419 1.15715 ; .89515 1.11713 .92709 1.07864 ; .96008 1.04158 10 51 ' .8&470 1.15647 ' .89567 1.11648 ' .92763 1.07801 .96064 1.04097 9 52 .86521 1.15579 .89620 1.11582 .92817 1.07738 .96120 1.040:36 8 53 , .86572 1.15511 ' .89672 1.11517 .92872 1.07676 .96176 1.03976 7 54 1 .86623 ' 1.1544:3 : .89725 1.11452 .92926 1.07613 .96232 l.a3915 6 55 '.86G74 1 . 15375 :| .89777 1.11387 .92980 1.07550 .96288 1.0:3855 5 56 .86725 i 1.15:308 .89830 1.11.321 .9:30:34 1.07487 .96*44 1.03794 4 57 .86776 1.15240 : .89883 1.112.56 .9308.8 1.07425 .96400 1.0.37:34 3 58 .86827 1.15172 !| .899.35 1.11191 .9:3143 1.07.362 1 .964.57 1.0:3674 2 59 .86878 1.1.5104 .89988 1.11126 .9:3197 1.07299 ' .96513 1.0:3613 1 60 1 .86929 Cotang 1 1.tO:37 Tang .90040 1.11061 .93252 Cotang 4 1.07237 Tang 70 1 .96.569 Cotang 1 03553 ' Cotang Tang Tang 49° 4 8° 46° TABLE XII. -TANGENTS AND COTANGENTS. 331 44° 440 1 44° 1 / t 60 / 1 ao / / / Tang Cotang 1.0:3553 Tang 1 Cotang Tang .98843 Cotang .96569 .97700 1.02:355 40 40 1.01170 20 1 .96625 1.0:i493 59 21 .97756 1.02295 39 41 .98901 1.01112 19 2 .96681 1.0:34:3:3 58 i 22 .97813 1.02236 .38 42 .98958 1.01053 18 3 .967:58 1.0:3:372 57 1 23 .97870 1.02176 37 4:3 .99016 1.00994 17 4 .9679^1 1.0:3:312 56 1 24 .97927 1.02117 :36 44 .99073 1.009:35 16 5 .968.50 1.0:3252 00 25 .97984 1.02057 :3.5 45 .99131 l.(X).S76 15 6 .96907 1.0:3192 54 1 26 .98041 1.01998 34 46 .99189 l."0818 14 7 .96963 1.031:32 53 j 27 .98098 1.019:39 :3:3 47 .99247 1.00759 13 8 .97020 1.0:3072 52 28 .981.55 1.01879 :32 48 .99304 1.00701 12 9 .97076 1.03012 51 29 .98213 1.01820 31 49 .99362 1.00642 11 10 .9713:3 1.02952 50 30 .98270 1.01761 30 |50 .99420 1.00583 10 11 .97189 1.02892 49 31 .98:327 1.01702 29 51 .99478 1.00.525 9 12 .97246 1.028^ 48 32 .98384 1.01642 28 52 .99536 1.00467 8 13 .97:302 1.02772 47 3:3 .98441 1.01.583 27 53 .99594 1.00408 1^ i 14 .97:359 1.02713 46 .34 .98499 1.01.524 26 54 .99652 1.0a3.54 .00127 ; .00127 .00230 .002:30 53 54 .00012 .00012 .000.55 .00055 ; .00128 .00128 .002:32 .002:32 54 55 .00013 .00013 .00056 .000.56 .001:30 .001:30 .002:34 .002:34 55 56 .00013 .00013 .00057 .000.57 .00131 i .001:31 .002:36 .002:36 56 57 .00014 .00014 .000.58 .000.58 .00133 I .001:3:3 .00238 .00238 57 58 .00014 .00014 .00059 .00059 .001:34 .001:34 .00240 .00240 58 59 .00015 .00015 . .0<¥)60 .00060 .00136 .001:36 .00242 .00242 59 60 .00015 .00015 1 .00061 .00061 .00137 .00137 : .00244 .00244 60 1 TABLE XIII.-VERSINES AND EXSECANTS. 333 ~o" 40 »° 6° 7 Vers. Exsec. 1 Vers. .00381 Exsec. .00382 Vers. 1 Exsec. Vers. Exsec. .00244 .00244 ' .00548 .00551 1 .00745 .00751 1 .00246 .00246 i .00383 .00385 1 .00551 .00554 i .00749 .00755 1 o .00248 .00248 .00386 .00387 ! .00554 .00557 .00752 .00758 2 3 .00250 .00250 .00388 .00390 .00557 .00560 1 .00756 .00762 3 4 .00252 .00252 ' .00391 .00392 .00560 .00563 .00760 .00765 4 5 .00254 .00254 .00393 .00395 i .00563 .00566 .00763 .00769 5 6 .00256 .00257 .00396 .00397 1 .00566 .00569 .00767 .00773 6 7 .00258 .00259 .00398 .00400 i .00569 .00573 .00770 .00776 7 8 .002(50 .00261 \ .00401 .00403 i .00572 .00576 ■ .00774 .00780 8 9 .00202 .002(53 .00404 .00405 1 .00576 .00579 1 .00778 .00784 9 10 .00264 .00265 .00406 .00408 .00579 .00582 ! .00781 .00787 10 11 .00206 .00207 .00409 .00411 1 00582 .00585 ! .00785 .00791 11 12 .00269 .002(59 1 .00412 .00413 .00585 .00588 i .00789 .00795 12 13 .00271 .00271 .00414 .00416 ; .00588 .00592 ! .00792 .00799 13 14 .00273 .00274 .00417 .00419 .00591 .00595 i 00796 .00802 14 15 .00275 .00276 .00420 .00421 ! 00594 ..00598 .00800 .00806 15 16 .00277 .00278 .00422 .00424 1 .00598 .00601 .00803 .00810 16 17 .00279 .00280 .00425 .00427 ' .00601 .00604 .00807 .00813 17 18 .00281 .00282 .00428 .00429 ; 00604 .00608 ; .00811 .00817 18 19 .00284 .00284 .00430 .00432 .00607 .00611 .00814 .00821 19 20 .00286 .00287 .00433 .00435 .00610 .00614 .00818 .00825 20 21 .00288 .00289 .0043(5 .00438 i .00614 .00617 .00822 .00828 21 22 .00290 .00291 ' .(X)438 .00440 ; .00617 .00621 1 .00825 .00832 23 23 .00293 .00293 .00441 .00443 i .00620 .00624 .00829 .00836 23 24 .00295 .00296 .00444 .00446 .00623 .00627 ! .00833 .00840 24 25 .00297 .00298 .00447 .00449 .0062(5 .00630 ; .00837 .00844 25 2(5 .00299 .00300 1 .00449 .00451 ! .00630 .00634 ! .00840 .00848 26 27 .00301 .00302 1 .00452 .00454 i .00633 .00637 .00844 .00851 27 28 .00304 .00305 .00455 .00457 .00636 .00640 .00848 .00855 28 29 .00306 .00307 ; .00458 .00460 .00640 .00644 .00852 .00859 29 30 .00308 .00309 .00460 .00463 , .00643 .00647 .00856 .00863 30 31 .00311 .00312 , .00463 .00465 , .00646 .00650 .00859 .00867 31 32 .00313 .00314 I .00466 .00408 .00649 .00654 .00863 .00871 32 33 .00315 .00316 i .00469 .00471 .00653 .00657 .00867 .00875 33 34 .00317 .00318 .00472 .00474 .00656 .00660 \ .00871 .00878 34 35 .00320 .00321 .00474 .00477 .00659 .00664 : .00875 .00882 35 36 .00323 .00323 i .00477 .00480 1 .006(53 .00667 ; .00878 .00886 36 37 .00324 .00326 ! .00480 .00482 .00666 ,00671 .0(J882 .00890 37 38 .00327 .00328 i .00483 .00485 1 .00669 .00674 ! .00886 .00894 38 39 .00329 .00330 i .00480 .00488 ; .00673 .00677 I .00890 .00898 39 40 .00333 .00333 ! .00489 .00491 .00676 .00681 1 .00894 .00902 40 41 .00334 .00335 j .00492 .00494 .00680 .00684 .00898 .00906 41 42 .00336 .00337 .00494 .00197 .00683 .00(588 .00902 .00910 42 43 .00339 .00340 .00497 .00500 .00686 .00691 .00906 .00914 43 44 .00341 .00342 ' .00500 .00503 i 00690 .00695 .00909 .00918 44 45 .00343 .00345 ' .00503 .00506 ; .00693 .00698 00913 .00922 45 46 .00346 .00347 .00506 .00509 ■ .00697 .00701 00917 .00926 46 47 .00348 .00350 1 .00509 .00512 1 .00700 .00705 .00921 .00930 47 48 .00351 .00352 ' .00512 .00515 ; .00703 .00708 .00925 .00934 48 49 .00353 .00354 .00515 .00518 .00707 .00712 00929 .00938 49 50 .00350 .00357 .00518 .00521 1 .00710 .00715 ; .00933 .00942 50 51 .00358 .00359 .00521 .00524 j .00714 .00719 .00937 .00946 51 52 .00361 .00362 .00524 .00527 .00717 .00722 .00941 .00950 52 53 .003(53 .00364 .00527 .00530 .00721 .00726 .00945 .00954 53 54 .00365 .003(57 .00530 .00533 .00724 .00730 .00949 .00958 54 55 .00368 .00369 .00533 .00536 .00728 .007;ij .00953 .00962 55 56 .00370 .0037'2 .00.536 .00539 .00731 .00737 .00957 .00966 56 57 .00373 .00374 .00539 .00.^)42 .00735 .00740 .00961 .00970 57 58 .00375 .00377 .00542 .00545 ! .00738 .00744 .00965 .00975 58 59 .0(J378 .00379 .00.545 .00548 1 .00742 .00747 .009(59 .00979 59 60 .00381 .00382 .00548 .00551 1 .00745 .00751 .00973 .00983 60 '#ArA~Av r TABLE XIII.-VERSINES AND EXSECANTS. 1 8 o 9 o lo- 11 ~0 Vers. Exsec. Vers. Exsec. vers. Exsec. Vers. Exsec. .00973 .00983 : .01231 .01247 .01.519 .01543 .01837 .01872 1 .00977 .00987 .01236 .01251 1 .01.524 .01548 1 .01843 j .01877 1 2 .00981 .00991 : .01240 .01256 \ .01529 .01553 .01848 .01883 2 3 .00985 .00995 ! .01245 .01261 .01534 .01558 .01854 .01889 3 4 .00989 .00999 , .01249 .01265 .01540 .01564 .01860 .01895 4 5 .00994 .01004 .01254 .01270 .01545 .01569 .01865 .01901 5 6 .00998 .01008 .01259 .01275 I .01.550 .01574 .01871 .01906 6 1 .01002 .01012 .01263 .01279 .01555 .01579 .01876 .01912 7 8 .01006 .01016 .01268 .01284 .01.560 .01585 .01882 .01918 8 9 .01010 .01020 ! .01272 .01289 .01.565 .01590 .01888 .01924 9 10 .01014 .01024 .01277 .01294 .01570 .01595 .01893 .01930 10 11 .01018 .01029 .01282 .01298 .01575 .01601 .01899 .01936 11 12 .01022 .01033 1 .01286 .01303 .01580 .01606 ; .01904 .01941 12 13 .01027 .01037 1 .01291 .01:308 .01586 .01611 .01910 .01947 13 14 .01031 .01041 ' .01296 .01313 .01591 .01616 .01916 .01953 14 15 .010:35 .01046 .01300 .01:318 .01596 .01622 .01921 .01959 15 16 .01039 .01050 .01:305 .01323 .01601 .01627 .01927 .01965 16 ir .01043 .01054 .01310 .01327 .01606 .01633 .019.33 .01971 17 18 .01047 .01059 .01:314 .01332 .01612 .01638 .01939 .01977 18 19 .01052 .0106:3 .01319 .01:337 i .01617 .01643 .01944 .019a3 19 20 .01056 .01067 .01324 .01342 i .01622 .01649 .01950 .01989 20 21 .01060 .01071 ' .01329 .01346 i .01627 .01654 ' .01956 .01995 21 22 .01064 .01076 .013:33 .01351 .01632 .01659 ! .01961 .02001 22 23 .01069 .01080 .01338 .01356 1 .01638 .01665 ; .01967 .02007 23 21 .01073 .01084 .01343 .01:361 ' .01643 .01670 ; .01973 .02013 24 25 .01077 .01089 1 .01348 .01366 .01648 .01676 ! .01979 .02019 25 26 .01081 .01093 .01352 .01371 .01653 .01681 , .01984 .02025 26 27 .oias6 .01097 .01357 .01376 .01659 .01687 1 .01990 .02031 27 28 .01090 .01102 .01362 .01:381 ' .01664 .01692 .01996 .020:37 28 29 .01094 .01106 .01367 .01:386 ; .01669 .01698 j .02002 .02043 29 30 .01098 .01111 .01371 .01391 1 .01675 .01703 .02008 .02049 30 31 .01103 .01115 .01376 .01395 ' .01680 .01709 ; .02013 .02055 31 32 .01107 .01119 .01:381 .01400 .01685 .01714 i .02019 .02061 32 :« .01111 .01124 .01.386 .01405 .01690 .01720 ' .02025 .02067 33 34 .01116 .01128 .01:391 .01410 i .01696 .01725 .02031 .02073 34 35 .01120 .01133 .01396 .01415 i .01701 .01731 .02037 .02079 35 36 .01124 .011:37 .01400 .01420 ! .01706 .01736 .02042 .02085 36 37 .01129 .01142 .01405 .01425 .01712 .01742 .02048 .02091 37 38 .01133 .01146 .01410 .014:30 i .01717 .01747 , .02054 .02097 38 39 .01137 .01151 .01415 .014:35 1 .01723 .017.53 .02000 .02103 39 40 .01142 .01155 .01420 .01440 .01728 .01758 .02066 .02110 40 41 .01146 .01160 .01425 .01445 .01733 .01764 .02072 .02116 41 42 .01151 .01164 .01430 .014.50 .017:39 .01769 .02078 .02122 42 43 .01155 .01169 .01435 .01455 .01744 .01775 .02084 .02128 43 44 .01159 .01173 .01439 .01461 .01750 .01781 .02090 .02134 44 45 .01164 .01178 .01444 .01466 ; .017.55 .01786 .02095 .02140 45 46 .01168 .01182 .01449 .01471 .01760 .01792 ' .02101 .02146 46 47 .01173 .01187 .01454 ■01476 .01766 .01798 .02107 .02153 47 48 .01177 .01191 i .01459 .01481 .01771 .01803 .02113 .02159 48 49 .01182 .01196 .01464 .01486 .01777 .01809 .02119 .02165 49 50 .01186 .01200 1 .01469 .01491 .01782 .01815 .02125 .02171 50 51 .01191 .01205 .01474 .01496 .01788 .01820 .02131 .02178 51 52 .01195 .01209 .01479 .01.501 .01703 .01826 .021:37 .02184 52 53 .01200 .01214 .01484 .01506 .01799 .018:32 .02143 .02190 53 54 .01204 .01219 ; .01489 .01.512 .01804 .018.37 .02149 .02196 54 55 .01209 .01223 .01494 .01.517 .01810 .01843 i .02155 .02203 55 56 .01213 .01228 .01499 .01.522 .01815 .01849 .02161 .02209 56 57 .01218 .0123:3 .01504 .01.527 ! .01821 .01854 .02167 .02215 57 58 .01222 .012:37 .01.509 .01.5:32 .01826 .01860 .02173 .02221 58 59 .01227 .01242 .01514 .015:37 .01^32 .01866 .02179 .02228 59 60 .01231 .01247 .01.519 .01.543 .01837 .01872 .02185 .022:34 60 TAPLE XIir.-VERSINES AND EXSECANTS. 335 1 12° 13° 14° 16° / Vers. 1 Exsec. , Vers. Exsec. Vers. Exsec. Vers. Exsec. .02185 .022.34 1 .02.563 .02630 .02970 .03061 .03407 .03528 1 .02191 .02240 .02570 .02637 .02977 .03069 .03415 .03536 1 2 .02197 .02247 .02576 .02644 .02985 .03076 : .03422 .03544 2 3 .02203 .02253 .02583 .02651 .02992 .03084 .03430 .03,5.52 3 4 .02210 .02259 .02.589 .02658 .02999 .03091 { .034.38 .03560 4 5 .02216 .02266 .02596 .02665 .03006 .03099 .03445 .03568 5 6 .02222 .02272 .02602 .02672 .03013 .03106 .0;3453 .03,576 6 7 .02228 .02-279 .02609 .02679 .03020 .03114 .03460 .03.584 7 8 .02234 .02285 .02616 .02686 .03027 .03121 .03468 .03592 8 9 .02240 .02291 .02622 .02693 .03034 .03129 .03476 .03601 9 10 .02246 .02298 .02629 .02700 .03041 .03137 .03483 .03609 10 11 .022.52 .02304 .02635 .02707 .03048 .03144 .03491 .03617 11 12 .02258 .02311 .02642 .02714 .03055 .031.52 .0.3498 .03625 12 13 .02205 .02317 .026-49 .02721 .03063 .03159 ! .03506 03633 13 14 .02271 .02323 .02655 .02728 .0.3070 .03167 .03514 .0,3642 14 15 .02277 .02330 .02662 .02735 .03077 .03175 .03521 .03650 15 16 .02283 .02336 .02669 .02742 .03084 .03182 .03529 .0.36.58 16 17 .02289 .02343 .02075 .02749 .03091 .03190 .035,37 .03666 17 18 .02295 .02349 .02682 .02756 .03098 .03198 .03544 .0,3674 18 19 .02302 .02356 .026S9 .02763 .03106 .03205 .03552 .03683 19 20 .02308 .02362 ,02096 .02770 .03113 .03213 .03560 .03691 20 21 .02314 .02369 .02702 .02777 .03120 .03221 .03567 .03699 21 22 .02320 .02375 .02709 .02784 .03127 .03228 .03575 .03708 22 23 .02327 .02382 .02716 .02791 .03134 .032.36 .03583 .03716 23 24 .02333 .02388 .02722 .02799 .03142 .03244 .03590 .03724 24 25 .02339 .02395 1 .02729 .02806 .03149 .03251 .03598 .03732 25 26 .02345 .02402 • .02736 .02813 .03156 .032.59 .0,3606 .03741 26 27 .023.52 .02408 .02743 .02820 .03163 .03267 .0.3614 .03749 27 28 .02358 .02415 .02749 .02827 .03171 .03275 .03621 .037.58 28 20 .02364 .02421 1 .027.56 .02834 .03178 .03282 .0.3629 .03766 29 30 .02370 .02428 ! .02763 .02842 .03185 .03290 .03637 .03774 30 31 .02377 .02435 .02770 .02849 .03193 .03298 .0,3645 .0,3783 31 32 .02383 .02441 .02777 .02856 .03200 .03306 .03653 .03791 32 33 .02389 .02448 .02783 .02863 .0.3207 .03313 .03660 .03799 33 34 .02396 .02454 .02790 .02870 .03214 .0.3321 .03668 .0.3808 34 35 .02402 .02461 .02797 .02878 .03222 .0.3329 .03676 .03816 35 36 .02408 .02468 .02804 .02885 .03229 .03337 .03684 .03825 36 37 .02415 .02474 .02811 .02892 .03236 .03345 .0.3692 .0,38.33 37 38 .02421 .02481 .02818 .02899 .03244 .0.3353 .0,3699 .03842 38 39 .02427 .024SS .02824 .02907 .0.3251 .03360 .03707 .038,50 39 40 .02434 .02494 .02831 .02914 .03258 .03368 .03715 .03858 40 41 .02440 .02.501 .02838 .02921 .03266 .03376 .03723 .03867 41 42 .02447 .02508 .02845 .02928 .03273 .03384 .0,3731 .0,3875 42 43 .024.53 .02515 .02852 .02936 .03281 .0,3392 .03739 .03884 43 44 .02459 .02521 .028.59 .02943 .03288 .03400 .03747 .03892 44 45 .02466 .02528 1 .02866 .029.50 .03295 .03408 .0,3754 .0,3901 45 46 .02472 .02.535 1 .0287'3 .02958 .03303 .03416 .0,3762 .0,3909 46 47 .02479 .02542 1 .02880 .02965 .0.3310 .03424 .0.3770 .0,3918 47 48 .0:M85 .02548 1 .02887 .02972 .03318 .03432 .03778 .03927 48 49 .02492 .02555 .02894 .02980 .03325 .03439 .03786 .0,3935 49 50 .02498 .02.562 .02900 .02987 .03333 .03447 .03794 .03944 50 51 .02504 .02509 .02907 .02994 .0.3.340 .03455 .0.3802 .0,39.52 51 52 .02511 .02576 .02914 .03002 .03347 .0.3463 .03810 .03961 52 53 .02517 .02582 .02921 .03009 .03355 .0.3471 .0.3818 .0.3969 53 51 .02524 .02589 .02928 .03017 .03362 .0.3479 .03826 .0.3978 .54 55 .02.530 .02.596 .02935 .0.3024 .03370 .03487 .0.3834 .03987 55 56 .02537 .02603 .02942 .03032 .03377 .03495 j .03842 .03995 56 57 .02.543 .02610 .02949 .03039 .03385 03503 .0,38.50 .04004 57 58 .025.50 .02617 .029.56 .03046 .03392 .03512 .0,38.58 .04013 58 50 .02556 .02624 .02963 .030.54 .03400 .03520 .03866 .(M021 59 60 .02563 .02630 .02970 .03061 1 .03407 .03528 .03874 .04030 60 TABLE XIII.— VERSINES AND EXSECANTS. r 16° 17° 18 ;° 19° / Vers. Exsec. Vers. 1 Exsec. j Vers. Exsec. Vers. Exsec. .(m874 .04080 .04370 .04569 .04894 .05146 .05448 .05762 1 .(^82 .04039 .04378 .04578 .04903 .05156 .05458 .05773 1 o .03S90 .04047 1 .04387 .04588 .04912 .05166 .05467 .05783 2 3 .03898 .04056 j .04395 .04597 ! .04921 .05176 .05477 .05794 3 4 .03906 .04065 .04404 .04606 .04930 .05186 .05486 .05805 4 5 .03914 .04073 .04412 .046^6 .04939 .05196 ! .05496 .05815 5 6 .03922 .04082 .04421 .04625 ! .04948 .05206 i .0.5505 .05826 6 7 .03930 .04091 .04429 .04635 i j .04957 .05216 1 .05545 .05836 7 8 .03938 .04100 .04438 .04644 .04907 .05226 1 .05524 .05847 8 9 .03946 .04108 .04446 .04653 .04976 .05236 .05534 .05858 9 10 .03954 .04117 .04455 .04663 .04985 .05246 .05543 .05869 10 11 .03903 .04126 .04464 .04672 .04994 .05256 ' .05553 .05879 11 V2 .03971 .04135 .04472 .04683 .05003 .05266 .05562 .05890 12 13 .03979 .04144 .04481 .04691 .05012 .05276 .05572 .05901 13 14 .03987 .04152 .04489 .04700 .05021 .05286 .05582 .05911 14 15 .03995 .04161 .04498 .04710 .05030 .05297 .05591 .05922 15 16 .01003 .04170 .04507 .04719 .05039 .05307 .05601 .05933 16 17 .04011 .04179 i .04515 .04729 1 .05048 .05317 .05610 .05944 17 18 .04019 .04188 ' .04524 .04738 .05057 .05327 .05620 .059,55 18 19 .04028 .04197 .04533 .04748 1 .050G7 .05337 .05630 .05965 19 20 .04036 .04206 .04541 .04757 .05076 .05347 .05639 .05976 20 21 .04044 .04214 .04550 .04767 ' .05085 .05357 .05649 .05987 21 22 .04052 .04223 .04559 .04776 , .05094 .05367 .05658 .0,5998 22 23 .04060 .04232 .04567 .04786 .05103 .05378 .05668 .06009 23 24 .04069 .04241 .04576 .04795 .05112 .05388 .05678 .06020 24 25 .04077 .04250 : .04585 .04805 i .05122 .05398 .05687 .06030 25 26 .04085 .04259 .04593 .04815 i .05131 .05408 .05697 .06041 26 27 .04093 .04268 .04602 .04824 . .05140 .05418 .05707 .060,52 27 28 .04102 .04^7 .04611 .04834 .05149 .05429 .05716 .06063 28 29 .04110 .04286 .04620 .04843 : .05158 .05439 .05726 .06074 29 30 .04118 .04295 .04628 .04853 .05168 .05449 .05736 .06085 30 31 .04126 .04304 .04637 .04863 .05177 .05460 ' .05746 .06096 31 32 .04135 .04313 .04046 .04872 .05186 .05470 ! .05755 .06107 32 33 .04143 .04322 .04055 .04882 .05195 .05480 ' .05765 .06118 33 34 .04151 .04331 .04663 .04891 .05205 .0.5490 .05775 .06129 34 35 .04159 .04:340 .04673 .04901 ' .05214 .05501 ' .05785 .06140 35 36 .04168 .04349 .04681 .04911 .05223 .05511 ' .05794 .06151 36 37 .04176 .04358 .04690 .04920 .05232 .05521 .05804 .06162 37 38 .04184 .04367 .04699 .04930 .05242 .05532 .05814 .06173 38 39 .04193 .04376 .04707 .04940 .05251 .05542 .05824 .06184 39 40 .04201 .04385 j .04716 .04950 .05260 .05552 .05833 .06195 40 41 .04209 .04394 .04725 .04959 .05270 .05563 .05843 .06206 41 42 .04218 .04403 .04734 .04969 .05279 .05573 .05853 .06217 42 43 .04226 .04413 .04743 .04979 .05288 .05584 .0.5863 .06228 43 41 .04234 .04422 .04752 .04989 .05298 .05594 .05873 .06239 44 45 .04243 .04431 .04760 .04998 .05307 .05604 .05882 .062.50 45 46 04251 .04440 .04769 .05008 .05316 .05015 .05892 .06261 46 47 .04260 .04449 .04778 .05018 .05.326 .05625 .05902 .06272 47 48 .04268 .04458 .04787 .05028 .05335 .0.5636 .05912 .06283 48 49 .04276 .04468 .04796 .05038 .05344 .05646 .05922 .06295 49 50 .04285 .04477 .04805 .05047 .05354 .05657 .05932 .06306 50 51 .04293 .04486 1 .04814 .05057 .0.5363 .05667 .05942 .06317 51 52 .04302 .04495 ' .04823 .0.5067 .05373 .05078 .05951 .06328 52 53 .04310 .04504 .04832 .05077 ; .05382 .05688 .0.5961 .06339 53 54 .04319 .04514 .04841 .05087 .05391 .0,5699 .05971 .06,350 54 55 .04327 .04523 .04850 .05097 .05401 .0.5709 .0,5981 .06362 55 56 .04336 .04532 .04858 .05107 .05410 .05720 .0.5991 .06373 56 57 .04344 .04541 i .04867 .05116 .05420 .05730 .06001 .06384 57 58 .04353 .04551 .04876 .05126 .05429 .0.5741 .06011 .06,^95 58 59 .04361 .04.560 I .04885 .0.5136 .0.5439 .05751 .06021 .06407 59 60 .04370 .04569 ! .04894 .05146 ; .05448 .05762 .06031 .06418 60 TABLE XIII.— VERSINES AND EXSECANTS. 337 / 20" 21° 22° 23° / Vers. Exsec. Vers. Exsec. Vers. Exsec. Vers. Exsec. .06031 .06418 .06642 .07115 .07282 .07853 .07950 .08636 1 .06041 .06429 .01)652 .07126 .07293 .07866 .07961 .08649 1 2 .06051 .06440 .06663 .07i:i8 .07303 .07879 .07972 .08663 2 3 .06061 .06452 .06673 .07150 .07314 .07892 .07984 .08676 3 4 .06071 .00463 .06684 .07162 .07325 .07904 .07995 .08690 4 5 .06081 .06474 .06691 .07174 .07336 .07917 .08006 .08703 5 6 .06091 .06486 .06705 .07186 .07347 .07930 .08018 .08717 6 7 .06101 .06497 .06715 .07199 .07358 07043 .08029 .08730 7 8 .06111 .06508 .06726 .07211 .07369 .07955 .08041 .08714 8 9 .06121 .06520 .00736 .07223 .07380 .07968 .08052 .08757 9 10 .06131 .06531 .06747 .07235 .07391 .07981 .08064 .08771 10 11 .06141 .06542 .06757 .07247 .07402 .07994 .08075 .08784 11 l:i .06151 .00554 .06768 .07259 .07413 .08006 .1)8086 .08798 12 13 .06161 .06565 .06778 .07271 .07424 .08019 .08098 .08811 13 14 .06171 .06577 .06789 .07283 .07435 .08032 .08109 .08825 14 15 .06181 .06588 .06799 .07295 .07446 .08045 .08121 .08839 15 16 .06191 .06000 .06810 .07307 .07457 .08058 .08132 .08852 16 17 .06201 .06611 .06820 .07320 ! .07468 .08071 .08144 .08866 17 18 .06211 .06622 .06831 .07332 .07479 .08084 .08155 .08880 18 19 .06221 .06634 .06841 .07344 .07490 .08097 .08167 .08893 19 20 .06231 .06645 .06852 .07356 .07501 .08109 .08178 .08907 20 21 .06241 .06657 .06863 .07368 .07512 .08122 .08190 .08921 21 22 .06252 .06668 .06873 .07380 .07523 .08135 .08201 .08934 22 23 .06263 .06680 .06884 .07393 i .07534 .08148 .08213 .08948 23 24 .06272 .06691 .06894 .07405 1 .07545 .08161 .08225 .08962 24 25 .06282 .06703' .06905 .07417 1 .07556 .08174 .08236 .08975 25 26 .06292 .06715 .06916 .07429 ; .07568 .08187 .08248 .08989 26 27 .06302 .06726 .06926 .07142 i .07579 .08200 .08259 .09003 27 28 .06312 .06738 .06937 .07454 .07590 .08213 .08271 .09017 28 29 .06323 .06749 .06948 .07466 .07601 .08226 .08282 .09030 29 30 .06333 .06761 .06958 .07479 .07612 .08239 .08294 .09044 30 31 .06343 .06773 .06969 .07491 .07623 .08252 .08306 .09058 31 32 .06353 .06784 .06980 .07503 .07634 .08265 .08317 .09072 32 33 .06363 .06796 .06990 .07516 .07645 .08278 .08329 .09086 33 34 .06374 .06807 ,07001 .07528 .07657 .08291 .08340 .09099 34 35 .06384 .06819 .07012 .07540 .07668 .08305 .08352 .09113 35 36 .06394 .06831 .07022 .07553 .07679 .08318 .08364 .09127 36 37 .06404 .06843 .07033 .07565 .07690 .08331 .08375 .09141 37 38 .06415 .06854 i .07044 .07578 ' .07701 .08344 .08387 .09155 38 39 .06425 .06866 1 .070.55 .07590 , .07713 .08357 .08399 .09169 39 40 .06435 .06878 j .07065 .07602 .07724 .08370 .08410 .09183 40 41 .06445 .06889 .07076 .07615 .07735 ..08383 .08422 .09197 41 42 .06456 .06901 1 .07087 .07627 ; .07746 .08397 .08434 .09211 42 43 .06466 .06913 ! .07098 .07640 .07757 .08410 .08445 .09224 43 44 .06476 .06925 .07108 .07652 i .07769 .08423 .08457 .09238 44 45 .06486 .06936 .07119 .07665 ! .07780 .08436 .08469 .09252 45 46 .06497 .06948 .07130 .07677 1 .07791 .08449 I .08481 .09266 46 47 .06507 .06960 .07141 .07690 .07802 .08463 .08492 .09280 47 48 .06517 .06972 .07151 .07702 .07814 .08476 .08504 .09294 48 40 .06528 .06984 .07162 .07715 .07825 .08489 .08516 .09308 49 50 .06538 .06995 j .07173 .07727 .07836 .08503 .08528 .09323 50 51 .06548 .07007 .07184 .07740 .07848 .08516 .08539 .09337 51 52 .065.59 ,07019 .07195 .07752 .07859 .08529 .08551 .09351 52 53 .06569 .07031 .07206 .07765 , .07870 .08.542 .08563 .09365 53 54 .06580 .07043 .07216 .07778 .07881 .08556 .08575 .09379 54 ^ 55 .06590 .07055 .07227 .07790 .07893 .08569 .08.586 .09393 55 56 .06600 .070()7 .07238 .07803 .07904 .08582 .08598 .09407 56 57 .06611 .07079 .07^9 .07816 ! .07915 .08596 .08610 .09421 57 58 .06621 .07091 ' .07260 .07828 1 .07927 .081)09 .08622 .09435 58 59 .06632 .07103 1 .07271 .07841 .07938 .08623 i .086:i4 .09449 59 60 .06642 .07115 ! .07282 .07853 .079.50 .08636 1 .08(i45 .094W 60 338 TABLE Xm.— VERSINES AND EXSECANTS. § 24- 25° 26° 27° w Vers. 1 Exsec. .09404 Vers. .09300 Exsec. Vers. Exsec. j Vers. Exsec, 1 .08645 .10:538 .10121 .11260 .10899 .122.33 1 1 .08657 .09478 .09382 .10:353 .10133 .11276 ; .10913 .12249 1 .08669 .09492 .09394 .10368 .10146 .11292 ! .10926 .12266 2 3 .a8681 .09506 .09406 .10383 ; .10159 .11:308 .10939 .12283 3 4 .08693 i .09520 .09418 .10:398 .10172 .11:323 .10952 .12299 4 5 .08705 .09535 .09431 .10413 .10184 .113:39 .10965 .12:316 5 6 .08717 .09549 .09443 .10428 .10197 .11:355 ! .10979 .12:3a3 6 7 ! .08728 .09563 .09455 .10443 , .10210 .11:371 .10992 .12349 7 8 1 .08740 .09577 .09468 .10458 1 .10223 .11387 ; .11005 .12366 8 9 .08752 .09592 .09480 .10473 ; .102:36 .11403 .11019 .12383 9 10 .08764 .09606 .09493 .10488 ' .10248 .11419 .11032 .12400 10 11 .08776 .09020 .09505 .10503 .10261 .11435 .11045 .12416 11 12 .08788 .09635 .09517 .10518 .10274 .11451 .11058 .1243:3 12 13 .08800 .09649 .09530 . 105:33 .10287 .11467 .11072 .12450 13 14 .08812 .09663 .09542 .10549 1 .10300 .11483 .11085 .12467 14 15 .08824 .09678 \ .09554 .10564 .10:313 .11499 .11098 .12484 15 ' 16 .08836 .09692 .09567 .10579 .10326 .11515 1 .11112 .12501 16 17 .08848 .09707 .09579 .10594 .10338 .11531 ' .11125 .12518 17 18 .08860 .09721 .09592 .10609 .10351 .11547 .11138 .12534 18 19 .08872 .097.35 .09604 .10625 .10364 .11.563 i .11152 .12.5.51 19 20 .08884 .09750 .09617 .10640 1 . 10377 .11579 .11165 .12568 20 21 .08896 .09764 .09629 .10655 .10.390 .11595 .11178 .12585 21 22 .08908 .09779 .09042 .10070 .10403 .11611 .11192 .12602 22 23 .08920 .09793 .09654 .10686 .10416 .11627 .11205 .12619 23 24 .08932 .09808 .09666 .10701 ! .10429 .11643 .11218 .126.36 24 25 .08944 .09822 .09679 .10716 ' .10442 .11659 .112:32 .12653 25 26 .08956 .09837 .09691 .10731 .10455 .11675 .11245 .12670 26 27 .08968 .09851 .09704 .10747 .10468 .11691 .11259 .12687 27 28 .03980 .09866 .09716 .10762 .10481 .11708 .11272 .12704 28 29 .08992 .09880 .09729 .10777 i .10494 .11724 .11285 .12721 29 30 .09004 .09895 ; .09741 .10793 .10507 .11740 .11299 .12738 30 31 .09016 .09909 .09754 .10808 .10520 .11756 .11312 .12755 31 32 .09028 .09924 .09767 .10824 .10533 .11772 .11326 .12772 32 33 .09040 .09939 .09779 . 10839 .10546 .11789 .11339 .12789 33 34 .09052 .099.53 .09792 .10854 .10559 .11805 .11353 .12807 34 35 .09064 .09908 .09804 .10870 .10572 .11821 .11:366 .12824 35 36 .09076 .09982 .09817 .10885 .10585 .11838 .11380 .12841 36 37 .09089 .09997 .09829 .10901 .10598 .11854 .11393 .12858 37 38 .09101 .10012 .09842 .10916 .10611 .11870 .11407 .12875 38 39 .09113 .10026 .09854 .109:32 .10624 .11880 .11420 .12892 39 40 .09125 .10041 .09867 .10947 .10637 .11903 .11434 .12910 10 41 .09137 .10055 .09880 .10963 .10650 .11919 .11447 .12927 41 42 .09149 .10071 1 .09892 .10978 .10663 .119:36 .11461 .12944 42 43 .09101 .10085 1 .09905 .10994 ' .10676 .11952 .11474 .12961 43 44 .09174 .10100 ' .09918 .11009 .10689 .11968 .11488 .12979 44 45 .09186 .10115 .09930 .11025 .10702 .11985 .11501 .12996 4,5 46 .09198 .10130 .09943 .11041 .10715 .12(X)1 .11515 .1:3013 46 47 .09210 .10144 .09955 .11056 .10728 .12018 .11528 .13031 47 48 .09222 .10159 .09968 .11072 .10741 .12034 .11542 .13048 48 49 .09234 .10174 .09981 .11087 .10755 .12051 .11555 .13065 49 50 .09247 .10189 .09993 .11103 .10768 .12067 .11569 .13083 50 51 .09259 .10204 .10000 .11119 .10781 .12084 .11583 .13100 51 52 .09271 .10218 .10019 .111:34 .10794 .12100 ! .11596 .13117 52 53 .09283 .10233 .10032 .11150 ; .10807 .12117 .11610 .13135 53 54 .09296 .10248 .10044 .11166 .10820 . 121:33 .11623 .13152 .54 55 .09308 .10263 .10057 .11181 .10833 .12150 .11037 .13170 55 56 .09320 .1027'8 .10070 .11197 .10847 .12166 .11651 .13187 56 57 .093.32 .10293 . 10082 .11213 .10860 .12183 .11664 .1:3205 57 58 .09345 .10308 .10095 .11229 .10*^73 .12199 .11678 .13222 58 59 .003.i7 .10323 .10108 .11244 .10886 .12216 .11(592 .13240 59 60 .09369 1 .10338 .10121 .11260 .10899 .12233 .11705 .13257 60 TABLE XIII.-VERSINES AND EXSECANTS. :J39 / 28° 29° 30° 31» / Vers. Exsec. Vers. Exsec. Vers. Exsec. Vers. Exsec. .11705 .13257 .12538 .14335 .13397 .15470 .14283 .10003 1 .11719 .13275 .12552 .14354 .13412 .15489 .14298 .10084 1 2 .11733 .13292 .12506 .14372 .r5427 .15509 .14313 .107'04 2 3 .11746 .13310 .12580 .14391 .J.3441 ,15528 .14328 .10725 3 4 .11700 .13327 .12595 .14409 .13456 . 15548 .14343 .16745 4 5 .11774 .13345 .12609 .14428 .1^70 . 15567 .14358 .16766 5 6 .11787 .13362 .12023 .14446 .13485 .15587 .14373 .16786 6 7 .11801 .13380 .12037 .14405 .13499 .15606 .14388 .10800 7 8 .11815 .13398 .12651 .14483 .13514 .15026 .14403 : 10827 8 9 .11828 .13415 .12065 .14502 .13529 .15045 .14418 .10848 9 10 .11342 .13133 .12679 .14521 .13543 .15065 .14433 .10808 10 11 .11856 .13451 .12694 .14539 .13.558 .15684 .14449 .10889 11 12 .11870 .i;«68 .12708 .14558 .13573 .15704 .14404 .10909 12 13 .11883 .13486 .12722 .14576 .13.587 .15724 .14479 .16930 13 14 .11897 .13504 .12736 .14595 .13602 .15743 .14494 .16950 14 15 .11911 .13521 .12750 .14014 .13010 .15763 .14509 .16971 15 16 .11925 .13539 .12705 .14632 .13031 .15782 .14524 .10992 16 17 .11938 .13557 .12779 .14651 .13646 .15802 .14539 .17012 17 18 .11952 .13.575 .12793 .14070 .13600 .15822 .14554 .17033 18 19 .]1%6 .13593 .12807 .14689 .13075 .15841 .14569 .17054 19 20 .11980 .13610 .12822 .14707 .13090 .15801 .14584 .17075 20 21 .11994 .13628 .12836 .14726 .13705 .15881 .14599 .17095 21 23 .12007 .13646 .12850 .14745 .13719 .15901 .14615 .17116 22 23 .12021 .13664 .12804 .14704 .13734 .15920 .14630 .17137 23 24 .12035 .13682 .1287'9 .14782 .13749 .15940 .14645 .17158 24 25 .12049 .13700 .12893 .14801 .13703 .15960 .14660 .17178 25 20 .12063 .13718 .12907 .14820 .13778 .15980 .14675 .17199 26 27 .12077 .13735 .12921 .14839 .13793 .16000 .14690 .17220 27 28 .12091 .13753 .12936 .14858 .13808 .16019 .14706 .17241 28 29 .12104 .13771 .12950 .14877 .13822 .10039 .14721 .17262 29 30 .12118 .13789 .12904 .14896 .13837 .16059 .14736 .17283 30 31 .12132 .13807 .12079 .14914 .13852 .16079 .14751 .17304 31 32 .12146 .13825 .12993 .14933 .13807 .16099 .14766 .17325 32 33 .12160 .13843 .13007 .1495'> .13881 .16119 .14782 .17346 33 34 .12174 .13861 .13022 .14971 .13896 .16139 .14797 .1.7307 34 35 .12188 .13879 .13036 .14990 .13911 .16159 .14812 .17388 35 36 .12202 .13897 .13051 .15009 .13926 .16179 .14827 .17409 36 37 .12216 .13916 .13005 .15028 .13941 .16199 .14843 .17430 37 38 .12230 .139:34 .13079 .15047 .13955 .16219 .14858 .17451 38 39 .12244 .13952 .13094 .15006 .13970 .10239 .14873 .17472 39 40 .12257 .13970 .13108 .15085 .13985 .16259 .14888 .17493 40 41 .12271 .13988 .13122 .15105 .14000 .16279 .14904 .17514 41 42 .12285 .14006 .13137 .15124 .14015 .16299 .14919 .17535 42 43 .12299 .14024 .131.51 .15143 .14030 .16319 .14934 .17556 43 44 .12313 .14042 .13106 .15102 .14044 .16339 .14949 .17577 44 45 .12327 .14061 .13180 .15181 .14059 .16359 .14965 .17598 45 46 .12:^41 .14079 .13195 .15200 .14074 .16380 .14980 .17020 46 47 .12355 .14097 .13209 .15219 .14089 .16400 .14995 .17641 47 48 .12309 .14115 .13223 .15239 .14104 .16420 .15011 . 17062 48 49 .12383 .14134 .13238 .15258 .14119 .16440 .15026 .17083 49 50 .12397 .14152 .132.52 .15277 .14134 .16460 .15041 .17704 50 51 .12411 .14170 .13207 .15296 .14149 .16481 .15057 .17726 51 52 .12425 .14188 .13281 .15315 .14104 .16501 .15072 .17747 52 53 .12439 .14207 .13296 .15335 .14179 .16521 .15087 .17768 53 54 .12454 .14225 .13310 .15354 .14194 .16.541 .15103 .17790 54 55 .12408 .14243 .13325 .15373 .14208 .16562 .15118 .17811 55 56 .12482 .14262 .13339 .15393 .14223 .16582 1 .15134 .17832 56 57 .12490 .14280 .13354 .15412 .14238 .16602 1 .15149 .17854 57 58 .12.510 .14299 .13368 .15431 .14253 .16623 1 .15164 .17875 58 59 12524 .14317 .13383 .154.51 .14268 .16043 .15180 .17890 59 60 .12538 .14335 A^m .15470 .14283 .16663 .15195 .17918 60 uo TABLE XIII.-VERSINES AND EXSECANTS. 32" 33° 84° 35° J Vers. Exsec. Vers. Exsec. Vers. Exsec. Vers. Exsec. / .15195 .17918 .16133 .19236 .17096 .20622 .18085 .22077 1 .15211 .17939 .16149 .1.9259 .17113 .20645 .18101 .22102 1 2 .15226 .17961 .16165 .19281 .17129 .20669 .18118 .22127 2 3 .15241 .17982 .16181 .19304 .17145 .20693 .18135 .22152 3 4 .15257 .18004 .16196 .19327 .17161 .20717 .18152 .22177 4 5 .15272 .18025 .16212 .19349 .17178 .20740 .18168 .22202 5 6 .15288 .18047 .16228 .19372 .17194 .20764 .18185 .22227 6 7 .15303. .18068 .16244 .19394 .17210 .20788 .18202 .22252 7 8 .15319 .18090 .16260 .19417 .17227 .20812 .18218 .22277 8 9 .15334 .18111 .16276 .19440 .17243 .20836 .18235 .22302 9 10 .15350 .18133 .16292 .19463 .17259 .20859 .18253 .22327 10 11 .15365 .18155 .16308 .19485 .17276 .20883 .18269 .22352 11 12 .15381 .18176 .16324 .19508 .17292 .20907 .18286 .22377 12 13 .15396 .18198 .16340 .19531 .17308 .20931 .18302 .22402 13 14 .15412 .18220 .16355 .19554 .17325 .20955 .183^9 .22428 14 15 .15427 .18241 .16371 .19576 1 .17341 .20979 .18336 .22453 15 16 .15443 .18263 .16387 .19599 .17357 .21003 .18:353 .22478 16 17 .15458 .18285 .16403 .19622 .17374 .21027 .18369 .22503 17 18 .15474 .18307 .16419 .19645 .17390 .21051 .18386 .22528 18 19 .15489 .18328 .16435 .19668 .17407 .21075 .18403 .22554 19 20 .15505 .18350 .16451 .19691 .17423 .21099 .18420 .22579 20 21 .15520 .18372 .10467 .19713 .17439 .21123 .18437 .22604 21 22 .15536 .18394 .16483 .19736 .17456 .21147 1 .18454 .22629 22 23 .15552 .18416 .16499 .19759 .17472 .21171 [ .18470 .22655 23 24 .15567 .18437 .16515 .i':'782 ; .17489 .21195 1 .18187 .22680 24 25 .1558;3 .184.59 .16531 .19805 1 1 .17505 .21220 .18504 .22706 25 26 .15598 .18481 .16547 .19828 .17522 .21244 .18521 .22731 26 27 .15614 .18.503 .16503 .19851 .17538 .21208 .185.38 .22756 27 28 .15630 .18525 .16579 .19874 .17554 .21292 .18555 .22782 28 29 .15645 .18547 .16595 .19897 .17571 .21316 .18572 .22807 29 30 .15661 .18569 ; .16611 .19920 .17587 .21341 .18588 .22833 30 31 .15676 .18591 .16627 .19944 .17604 .21365 .18605 .22858 31 32 .15692 .18613 .16644 .19967 .17620 .21389 .18622 .22884 32 33 .15708 .18635 .16660 .19990 .17637 .21414 .18639 .22909 33 34 .15723 1 .15739 ! .18657 .16676 .20013 .17653 .21438 .18656 .22935 34 35 .18679 .16692 .20036 .17670 .21462 .18073 .22960 35 36 .15755 .18701 .16708 .20059 .17686 .21487 .18690 .22986 36 37 .15 .TO .18723 .16724 .200&3 1 .17703 .21511 .18707 .23012 37 38 .15780 .18745 .16740 : .20106 .17719 .21535 .18724 .23037 38 39 .15802 .18767 .16756 i .20129 .17736 .21560 i .18741 .23063 39 40 .15818 . 18790 .16773 1 .20153 .17752 .21584 .18758 .23089 40 41 .15833 .18812 .16788 i .20176 .17769 .21609 : .18775 .23114 41 42 .15849 .18834 .16805 .20199 .17786 .21633 .18792 .23140 42 43 .1.5865 .18856 .16821 .20222 .17802 .21658 .18809 .23166 43 44 .15880 .18878 .16837 1 .20246 .17819 .21682 .18826 .23192 44 45 . 15896 .18901 .16853 .20269 .17835 .21707 i .18843 .23217 45 46 .15912 .18923 .16869 .20292 .17852 .21731 s .18860 .23243 46 47 .1592S .18945 .16885 .20316 .17868 .21756 .18877 .23269 47 48 .15943 .18967 .16902 1 .20339 ! .17885 .21781 .18894 .23295 48 49 .15959 .18990 .16918 .20363 .17902 .21805 .18911 .23321 49 50 .15975 .19013 .16934 .20386 .17918 .21830 .18928 .23347 50 51 .15991 .19034 .16950 .20410 .17935 .218.55 .18945 .23373 51 52 .1G006 .19057 .16906 .20433 .17952 .21879 .18962 .23399 52 53 .16022 .19079 .16983 .20457 .17968 .21904 .18979 .23424 53 54 .10038 .19102 .16999 .20480 .17985 .21929 .18996 .23450 54 55 .16054 .19124 : .17015 .20504 .18001 .21953 .19013 .23476 55 56 .16070 .19146 .17031 .20.527 .18018 .21978 .19030 .23502 56 57 .16085 .19109 j .17047 .20551 i .18035 .23003 .19047 .23529 57 58 .16101 .19191 .17064 .20575 ' .18051 .22028 .19064 ! .23555 58 59 .16117 .19214 .17080 .20598 .18068 .220.53 .19081 .23581 59 60 .16133 i .19236 .17096 .20622 .18085 .22077 .19098 .23607 60 TABLE XIII.— VERSINES AND EXSECANTS. 341 36" Vers. Exsec. 37° Vers. Exsec. 1 2 3 4 5 6 7 8 9 10 11 12 13 U 15 16 17 18 19 20 21 23 24 25 20 27 28 29 30 3t 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 52 53 55 56 57 58 59 60 .19098 .19115 .19133 .19150 .19167 .1918! .19201 .19218 .19235 .19252 .19270 .19287 .19304 .19321 .19:338 .19356 .19373 .19390 .19407 .19424 .19442 .19459 .19476 .19493 .19511 .19528 .19545 .19502 .19580 .19597 .19614 .19632 .19649 .19666 .1'J684 .19701 .19718 .19736 .197.53 .19770 .19788 .19805 .19822 .19840 .198.57 .19875 .19892 .19909 .19927 .19944 .19962 .19979 .19997 .20014 .20032 .20049 .200(56 .20084 .20101 .20119 .20136 .23007 .2:3633 .23659 .23685 .23711 .2:37:38 .23764 .2:3790 .2:3816 .2:3843 .23869 .23895 .23922 .23948 .23975 .24001 .24028 .24054 .24081 .24107 .24134 .24160 .24187 .24213 .^i^O .24267 .24293 .24320 .24347 .24:373 .24400 .24427 .24454 .24481 .24508 .:215;34 .21561 .24588 .24615 .24&42 .24669 .24696 .24723 .24750 .24777 .24804 .248:32 .24859 .24880 .24913 .21940 .24967 .j^995 .25022 .25049 .2.5077 .25104 .25131 .25159 .25186 .25214 .201. SO .20154 .20171 .20189 .20207 .20224 .20242 .20259 .20277 .20294 .20312 .20329 .20347 .20365 .20382 .20400 .20417 .20435 .20453 .20470 .20488 .20506 .20523 .20541 .20559 .20576 .20594 .20612 .20629 .20647 .20665 .20682 .20700 .20718 .20736 .20753 .20771 .20789 .20807 .20824 .20842 .20860 .20878 .20895 .20913 .20931 .20949 .20967 .20985 .21002 .21020 .21038 .21056 .21074 .21092 .21109 .21127 .21145 .21163 .21181 .21199 .2.-^214 ! 25269 I .25296 i .25324 .25351 ; .25379 .25406 .25434 .25462 ! .25489 I .25517 .25545 .25572 ; .25600 .25628 ! .25656 i .25683 .25711 j .25739 .25767 .25795 .25823 .25851 .25879 .25907 .25935 I .25963 .25991 .26019 .26047 I .26075 .26104 .26132 .26160 .26188 .26216 .26245 .26273 r .26301 .26330 .26.3.58 .263G7 .26415 i .26443 .26472 .26500 .26529 .26557 .26586 .26615 .26643 I .26672 .26701 .26729 .26758 .26787 .26815 .26844 .26873 : .26902 I 38» 3 d° Vers. Exsec. Vers. Exsec. : .21199 .26902 .22285 .2;-676 .21217 .26931 .22:304 .28706 : .212:35 .26960 .22:322 .28737 .212.53 .26988 .22340 .28767 .21271 .27017 .22359 .28797 .21289 .27046 .22377 .28823 .21307 .27075 .22395 .28858 .21.324 .27104 .22414 .28889 .21.342 .271:33 .22432 .28919 .21 300 .27162 .22450 .289.50 .21378 .27191 t .22469 .28980 .21390 .27221 .22487 .29011 .21414 .27250 .22506 .29042 .21432 .27279 .22524 .29072 .21450 .273aS .22542 .29103 .21468 .27337 .22501 .29133 .21486 .27366 .22579 .29164 .21504 .27.396 .22598 .29195 .21522 .27425 .22616 .29226 .21540 .274.>1 .22634 .29256 .21558 .2748:3 .22653 .29287 .21576 .27513 .22671 .29318 .21595 .27^2 i .22690 .29349 .21613 .27572 .22708 .29380 .21631 .27601 .22727 .29411 .21649 .27630 .22745 .29442 .21667 .27660 1 .22764 .29473 .21685 .27689 .22782 .29504 .21703 .27719 .22801 .29.535 .21721 .27748 .22819 .29506 .21739 .27778 .22838 .29597 .21757 .27807 .22856 .29628 .21775 .278:37 .22875 .29659 .21794 .27867 .22893 .29690 .21812 .27896 .22912 .29721 .218.30 .27926 .22930 .29752 .21848 .27956 .22949 .29784 .21866 .27985 .22967 .29815 .21884 .28015 .22986 .29^46 .21902 .28045 .2.3004 .29877 .21921 .28075 ' .23023 .29909 .21939 .28105 .23041 .29940 .21957 .281:34 .23060 .29971 .21975 .28164 .23079 .30003 .21993 .28194 .23097 .300.34 .22012 .282^ .23116 .30006 .22030 .28254 .23134 .3lX)97 .22048 .28284 .23153 .30129 .22006 .28314 .23172 .30160 .22084 .28344 .2:3190 .30192 .22103 .28374 .23209 .30223 .22121 .28404 .23228 ..30255 .22139 .28434 .23246 .;30287 .22157 .28464 .23205 .30318 .22176 .28495 .2:3283 .30a50 .22194 .28525 .2.3302 .30382 .22212 .285.55 .23321 .30413 .22231 .28585 .23.3.39 .30445 .22249 .28615 .2:3.3.58 ..30477 .22267 .28646 .2:3.377 .;30509 .22285 .28076 .23396 .30.541 o 10 Table xiii.— versines and exsecakts. / 40» 41» 42° 43'» f Vers. Kxsec. Vers. Exsec. 1 Vers. Exsec. Vers. Exsec. .23:396 .30541 • .24529 .32501 .25686 .34.563 .26865 .367^3 1 .2:3414 .30573 .24548 .32535 .25705 .34599 ' .26884 .36770 1 2 .2:34:33 .30605 .24567 ..32568 1 .25724 .34634 ! .26904 .36807 2 3 .2^452 .30636 .24556 .32C02 .2.5744 .34669 .26924 .36844 3 4 .2^470 .30668 .24605 .:32G:36 .25763 .34704 .26944 ..36881 4 5 .2^489 .30700 .24625 .32669 .2.5783 .34740 .26964 .36919 5 6 .23508 .307:32 .24644 .32703 .25802 .34:75 .26984 .36956 6 7 .2:3527 .30764 .24663 .327:37 .25822 .34811 .27004 .36993 7 8 .23545 .30796 .24682 ..32770 i .25841 .34846 .27024 .37030 8 9 .23564 .308-29 .24701 .32804 i .25861 .34882 .27043 .37068 9 10 .23583 .30861 .24720 .32838 ' .25880 .34917 .27063 .37105 10 11 .23603 .30893 .24739 ..32872 .25900 .34953 .27083 .37143 11 12 .23620 .30925 .24759 .32905 .2.5920 .34988 .27103 .37180 12 13 .236:39 .3t)9o7 .24778 .32939 1 .259:39 .35024 .27123 .37218 13 14 .236.i8 .:309S9 .24797 .;32973 .25959 .35060 .27143 .37255 14 15 .23677 .31022 .24816 .3:3007 .25978 .3.5095 .27163 .37293 15 16 .2:3696 .31054 .24835 .3:3041 .2.5998 .351:31 .27183 .37330 16 17 .2:3714 .31086 .24854 .3:3075 .26017 .35167 .272113 .37368 17 18 .2:37:33 .31119 .24874 ..3:31(^ .26a37 .35203 .27223 .37406 IS 19 .23752 .31151 .24893 .3:3143 .26056 .a52:38 .27243 .37443 19 20 .23771 .31183 .24912 .33177 .26076 .35274 .27263 .37481 20 21 .23790 .31216 .24931 .3:3211 .20096 .35.310 .27283 .37519 21 22 .2:3.S06 .3:3451 ' .262:3:3 .35562 .27423 .37784 28 29 .2:3941 .31476 i .25085 .3S485 .262.53 .a5598 .27443 .37822 29 30 .23959 .31509 .25104 .33519 , .26272 .35634 .27403 .37860 30 31 .23978 .31541 .25124 .a3554 ! .26292 .35670 .2^483 .37898 31 32 .23997 .31574 .25143 .3:3588 .26312 .35707 .27503 .37936 ^2 33 .24016 .31607 .25162 .33622 . .263:31 .35743 .27523 .37974 33 S4 .240:35 .31640 .25182 .a3657 1 .26.351 .a5779 .27543 .38012 34 35 .240.S4 .31672 ; .25201 .33691 .26:371 ..3.5815 .27563 .38051 35 36 .24073 .:31705 .25220 .a3726 .26:390 .35852 .27583 ..38089 36 37 .24092 .317:38 ' .25240 ..3:3760 .26410 .35888 .27603 .38127 37 38 .24111 .31771 .25259 .3:3795 .2f>4.30 ..a5924 .27623 .38165 38 39 .241:30 .31804 .2.5278 ..3:38:30 .26449 .a5061 .27643 .38204 39 40 .24149 .318:37 .25297 .33864 .26469 .35997 .27663 ..38242 40 41 .24108 .31870 .25317 .3.3899 ' .25489 ..360.34 .27&«3 .38280 41 42 .241'<7 .31903 .253:36 .3:3934 : .26509 .36070 .27703 .38319 42 43 .24206 .319:36 .253.56 .3:3968 .2(3528 .36107 .27723 .38357 43 44 .24225 .31909 .25375 ..34003 .26548 .36143 .27743 .38396 44 45 .24244 .32002 .25394 ..340.38 .26568 .36180 .27764 .384:34 45 46 .24262 .320:35 .25414 .34073 .26588 .30217 .27784 .38473 46 47 .24281 .32008 .254:3:3 .34108 .26607 .362.5:3 .27804 .38512 47 48 .24:3«X) .32101 .2.54.52 .34142 .26627 .36290 .27824 .38550 48 49 .24320 .321:34 .25472 .ail 77 .26647 .36:327 .27844 ..38.589 49 50 .24:3:39 .32168 .25491 .34212 ! .26667 .36363 .27864 .38628 50 51 .24358 .32201 .25.511 .34247 .26686 ..36400 .27884 ..a%68 51 52 .24377 .3-22:34 j .25530 ..31282 .26706 .36437 .27905 .38705 52 53 .24396 ..32267 I .25549 .34:317 .26726 ..36474 .27925 .38744 53 54 .24415 .32301 .2.5.569 .34:3.52 .26746 ..36.511 .27945 .387^3 54 55 .244:34 .323:34 .25588 .34:387 .26766 .36.548 .27965 .38822 55 56 .244.53 .32:368 ! .25608 ..34423 .26785 ..36585 .27985 .38860 56 57 .24472 .32401 .25627 .34458 .26805 .36822 .28005 .38899 57 58 .24491 .324-^ .25647 .34493 i .26825 .36659 .28026 .3^9-38 53 59 .24510 .32468 1 .2.5666 .34.528 [ .26845 .36696 . .28046 .38977 5J 60 .24529 .32501 .25686 .34563 .26865 .36733 .28066 .30C16 i TABLE XIII.— VERSINES AND EXSECANTS. 343 1 44° 45° 46° 470 Vers. Exsec. .39016 Vers. 1 .29289 Exsec. j .41421 Vers. 1 Exsec. Vers. Exsec. .28066 .30.5:34 .43956 .31800 .46628 1 .28086 .39055 .29310 .41463 \ .30555 .4^3999 .31821 .46674 1 2 .28106 .39095 .29330 .41504 1 .30576 .44042 .31843 .46719 2 3 .28127 .39134 .29351 .41545 .30597 .44086 .31864 .46765 3 4 .28147 .39173 i .29372 .41586 j .:30618 .44129 1 .318a5 .46811 4 5 .28167 .39212 ! .29392 .41627 .306.39 .44173 \ .31907 .46857 5 6 .28187 .39251 1 .29413 .41669 .30660 .44217 t .31928 .46903 6 7 .28208 .39291 .294:3:3 .41710 .30681 .44260 1 .31949 .46949 7 8 .28228 .39330 .29454 .41752 .30702 .44304 i .;31971 .46995 8 9 .28218 .39369 .29175 .41793 .30723 .44.347 .31992 .47041 9 10 .28268 .39409 .29495 .41835 .30744 .44391 ..32013 .47087 10 11 .28289 .39448 .29516 .41876 .30765 .44435 .32035 .47134 11 VZ .28309 .39487 .29537 .41918 .30786 .44479 .32056 .47180 12 13 .28329 .39527 .29557 .41959 .30807 .44523 .32077 .47226 13 14 .28350 .39566 .29578 .42001 .30828 .44567 ..32099 .47272 14 15 .28370 .39606 .29599 .42042 .30849 .44610 ..32120 .47319 15 l(j .28390 .39646 .29619 .42084 .30870 .44654 .32141 .47365 16 17 .28410 .;i!)6S5 .29640 .42126 .30891 .44698 .;32163 .47411 17 18 .28431 .39725 .29661 .42168 ..30912 .447'42 .32184 .47458 18 19 .28451 .39764 .29681 .42210 ..30933 .44787 .32205 .47504 19 5iO .284:1 .39804 .29702 .42251 .30954 .44831 .32227 .47551 20 21 .28492 .39844 .29723 .42293 .30975 .44875 .32248 .47598 21 22 .28512 .39884 .29743 .42335 .30996 .44919 ..32270 .47644 22 23 .28532 .39924 .29761 .42377 .31017 .44903 .32291 .47691 23 24 .28553 .39963 .29785 .42419 .310.38 .45007 ..32312 .477.38 24 25 .28573 .40003 .29805 .42461 .31059 .45052 ..32334 .47784 25 26 .285'J3 .40043 .29826 .42503 .31080 .45096 ..32355 .47831 26 2?' .28614 .40083 .29847 .42.545 .31101 .4,5141 .32377 .47878 27 28 .28634 .40123 .29808 .42587 .31122 .45185 .32398 .47925 28 2J .28655 .40163 .29888 .42630 .31143 .45229 ..32420 .47'972 29 30 .28675 .40203 .29909 .42672 .31105 .45274 .32441 .48019 30 31 .28605 .40243 .29930 .42714 .31186 .45319 .32462 .48066 31 32 .28716 .40283 .29951 .42756 .31207 .45363 .32484 .48113 32 33 .28736 .40324 .29971 .42799 .31228 .45408 ..32505 .48160 33 34 .28757 .40364 .29992 .42841 .31249 .45452 ..32527 .43207 34 35 .28777 .40404 .30013 .42883 .31270 .45497 ..32548 .48254 35 3G .28797 .40444 .30034 .42926 .31291 .4.5542 .32570 .48:301 36 37 .28818 .40485 .30054 .42968 .31:312 .45587 .32591 .48349 37 38 .28838- .40525 .30075 .43011 .31.3:34 .456:31 ..32613 .48.396 38 39 .28859 .40565 .30096 .4:3053 .31.3.55 .45676 ..32634 .48443 39 40 .28879 .40608 .30117 .43096 .31376 .45721 .32656 .48491 40 41 .28900 .40646 .30138 .43139 .31397 .45766 .32677 .48.5.38 41 42 .28920 .40687 .30158 .4:3181 ..31418 .4.5811 .32699 .48.586 42 43 .28941 .40727 .30179 .4:3224 .31439 .4.5856 .;32720 .48633 43 44 .28961 .40768 ,30200 .43267 .31461 .45901 .32742 .48()81 44 45 .28981 .40808 .30221 .4.3310 .31482 .45946 .32763 .48728 45 46 .29002 .40849 .30242 .4:3352 .31503 .45992 .32785 .48776 46 47 .29022 .40890 .30263 .43:395 .31524 .46037 ..32806 .48824 47 48 .29043 .40930 .30283 .43438 .31545 .46082 .32828 .48871 48 49 .29063 .40971 .30:304 .4:3-1*^1 .31.567 .46127 ..32849 .48919 49 50 .29084 .41012 .30325 .435:24 .31588 .46173 .32871 .48967 50 51 .29104 .41053 .30^46 .43567 .31609 .46218 ! ..32893 .49015 51 52 .29125 .41093 .;30:367 .43610 .316.39 .462(i3 1 ..32914 .49063 52 53 .29145 .41134 .30388 .4:3653 .31651 .4().309 : ..329:36 .49111 53 54 .29166 .41175 .30409 .43696 i .31673 .46:354 .32957 .49159 54 55 .29187 .41216 .304:30 .43739 .31694 .46400 .;32979 .49207 55 56 .29207 .41257 .3ai51 .43783 .31715 .46445 .83001 .492.55 56 57 .29228 .41298 .30471 .4:3826 .317.36 .46491 .,33022 .49:303 57 58 .29:248 .41339 .30492 .4:3869 .31758 .46.537 .3.3044 .49:351 58 59 .29269 .41380 .;30513 .43912 .:31779 .46.582 .3:i065 .49:399 59 60 .29289 .41121 .305^4 .43956 1 .31800 .46628 .33087 .49418 60 344 TABLE XUI.-VERSIXES AND EXSECANTS. / 48» 49' 50° 61» / Vers. Exsec. Vers. Exsec. i Vers. Exsec. Vers. Exsec. .33067 .49448 .34:394 .52425 .35721 .55572 .37068 .58902 1 .33109 .49496 .:i4416 .52476 .35744 .55626 .37091 .58959 1 2 .33130 .49544 .344:38 .52527 .35766 .55680 .37113 .59016 2 3 .33152 .49593 .34460 .52579 .35788 .55734 .37136 .59073 3 4 .33173 .49641 .34482 .5263<3 .35810 .55789 i .3n58 .59130 4 5 .33195 .49690 .34504 .52681 .35833 .5584:3 .37181 .59188 5 6 .aS217 .49738 .34526 .52732 .35855 .55897 .37204 .59245 6 7 .33238 .497W 1 .34548 .52784 .35S77 .55951 .37226 .59302 1 8 .33260 .49835 i 1 .34570 .52835 .3590«) .56005 .37249 .59360 8 9 .33282 .49884 1 .34592 .52886 .:359->2 .56060 .37272 .59418 9 10 .3:W>3 .49933 .34614 .52938 1 .35944 .56114 1 .37294 .59475 10 11 .33325 .49981 .34636 .52989 .35967 .56169 .37317 .59533 11 12 .33:}47 .500:30 .34658 .5:3041 .35989 .56223 .37340 .59590 12 13 .:B3;368 .50079 : ■ .34680 .53<392 1 .36011 .56278 .37362 .5964S 13 U .33390 .50128 .34702 .53144 .36031 .56332 .37385 .59706 14 15 .3:^412 .50177 .34724 .5:3196 .36(K6 .56387 1 .37408 .59764 15 16 .334S4 .50226 .34746 .5:3247 .36078 .56442 .37430 .59822 16 IT .3*455 .50275 .:34768 .53299 .36101 .56497 .37453 .59880 17 18 .3:i4rr .50:324 ..34790 .53351 .36123 .56551 1 .37476 .59938 18 19 .3:i499 .50:373 .34812 .5340:3 .36146 .56606 .37498 .59996 19 20 .3:^520 .50422 .:i4S;34 .53455 .36168 .56661 .37521 .60054 20 21 .33.542 .50471 .34856 .53507 .36190 .56716 .37544 .60112 21 22 .3;55t>4 .50521 .34878 .53559 .36213 .56771 .37567 .60171 23 .3:3586 .50570 .34900 ,53611 .362:35 .56826 .37589 .6>229 23 24 .33607 .50619 .34923 .53663 .36258 .56881 .37612 .60287 24 25 .3:3629 .50669 1 .34945 .53715 .36280 .56937 .37635 .60:i46 25 26 .3:3651 .50718 .34967 .53768 .36302 .56992 .37658 .60404 26 27 .$3073 .50767 .34989 .5:3820 1 .36:325 .57047 .37680 .60463 27 28 1 .3:3694 .50617 .35011 .53872 1 .36:347 .57103 .37703 .60521 28 29 : .3:3716 ..5«D866 .350:3:3 .53924 , .3<3370 .571.58 .37726 .60580 29 30 .33738 .50916 .35055 .53977 I .36392 .57213 .37749 .60639 30 31 .33760 .50966 .35077 .54029 .36415 .57269 .37m .60698 31 32 .3:3782 .51015 .35c>99 .54082 .364:37 .57324 .37794 .60756 32 33 , .33803 .51065 .3.5122 .541:34 t .36460 .57380 .37817 .60815 33 U ' .33825 .51115 .35144 .54187 .36482 .57436 1 .37840 .60874 ai 35 .33847 .51165 .35166 .54^40 1 .36504 .57491 .37862 .60933 35 36 .33869 .51215 .35188 .54292 .36527 .57547 .37885 .60992 36 37 .33891 .51265 ' .35210 .54345 i .36549 .57603 .37908 .61051 37 38 .33912 -51314 . .352:32 .54393 .36572 .57659 .37931 • .61111 38 39 .33934 .5i:3(>4 .:35254 .5«51 , , .36594 .5. (15 .37954 .61170 39 40 .33^6 .51415 ..35:>77 .54504 1 .36617 .5,..l .37976 .61229 40 41 .33978 .51465 .35299 .54557 .36639 .57827 .37999 .61288 41 42 .Mm .51515 .35321 .54610 .36662 .57883 .38022 .61348 42 43 .34022 .51565 .35:^3 .54663 .36684 .57939 .38045 .61407 43 44 .34044 .51615 .35:365 .54716 .36707 .57995 .38068 .61467 44 45 .34065 .51665 .35:388 .54769 .36729 .58051 .38091 .61526 45 46 .34087 .51716 I .35410 .54822 ! .36752 .58108 .38113 .61.586 46 47 .ail09 .51766 .354:32 .54876 .36775 .58164 .38136 .61646 47 48 .341:31 .51817 .35454 .54929 .36797 .58221 .38159 .61705 48 49 .341S3 .51867 .35476 .54982 .36820 .58377 .38182 .61765 49 50 .34175 .51918 .35499 .55036 .36842 .58333 .38205 .61825 50 51 .34197 .51968 .,35521 .55089 .36865 .58390 .38228 .61885 51 52 .34219 .52019 .35543 .5.5143 .36887 .58447 .38251 .61945 52 53 .34241 .52«169 .35565 .55196 .36910 .58503 .38274 .62005 53 54 .34262 .52120 .a5588 .55250 .36932 .5a560 .38296 .62065 54 55 .34284 .52171 .35610 .553a3 .36955 .58617 .38319 .62125 55 56 .34:306 .52222 .a5632 .55357 .36978 .58674 .38342 .62185 56 57 .34:328 .52273 .35654 .55411 .37000 .58731 .38365 .62246 57 58 .34:350 .52:323 .a567r .55465 .37i>23 .58788 .38388 .62306 58 59 .34372 .52:374 .35699 .55518 .37045 .5*=!845 .38411 .62366 59 60 .34394 .52425 .35721 .55572 .37068 .58902 .38434 1 .62427 60 TABLE Xni.— VERSINES AN'D EXSECANTS. 345 : 1 ■ 52^ 63° 54° i 55° / Vers. Exsec. ' Vers. Exsec. Vers. Exsec. Vers. Exsec. .38434 .62427 ..39819 .06164 .41221 .701:30 42&42 .74345 1 .38457 .6^487 .39^2 .66228 1 .41245 .70198 .42«66 .74417 1 2 .38480 . 62548 .39865 .66292 .41269 1 .70267 .42690 .74490 2 3 .38503 .62609 .39888 .66357 .41292 1 .703a5 .42714 .74562 3 4 .38.526 .62669 .39911 .66421 ! .41316 ' .70403 .427:38 .746:35 4 5 .38549 .627:30 .399:35 .66486 ; .413:39 .70472 .42762 .74708 5 6 .38571 .62791 .39958 .66550 i .41:363 .70540 .42785 .74781 6 7 .38594 .62852 .39981 .66615 ! .41:386 .70609 .42809 .74854 7 8 .38617 .62913 .40005 .66679 1 .41410 . 7(*677 .428:3:3 .74927 8 9 .38640 .62974 .40028 .66744 .41433 .70746 .42857 .75000 9 10 .38663 .63035 .40051 .66809 .41457 .70815 .42881 .75073 10 11 .38686 .63096 .40074 ' .66873 .41481 .70884 .42905 .75146 11 12 .38709 .6.3157 .40098 .66938 .41504 .7095:3 .42929 .75219 12 13 .;38r32 .6.3218 .40121 .67003 ! .41528 .71022 .4295:3 .75293 13 14 ..38755 .6:3279 .40144 .67068 .41551 .71091 .42976 .75366 14 15 .38778 .6:3341 .40168 .67133 ' .41575 .71160 .4:3000 .75440 15 IG .38801 .6.3402 .40191 .en 99 .41599 .71229 .43024 .75513 16 17 .38824 .6:3464 .40214 .67264 .41622 .71298 .43048 .75587 17 18 .38847 .6:3525 .402:37 .67329 .41646 .71368 .43072 .75661 18 19 .38870 .63587 .40261 .67394 .41670 .71437 .43096 .75734 19 20 .38893 .63648 .40284 .67460 .41693 .71506 .43120 .75808 20 21 .38916 .6.3710 .40307 .67525 ' .41717 .71576 .43144 .75882 21 22 ..38939 .63772 .40331 .67591 .41740 .71646 .43168 .75956 22 23 -38962 .63834 .40354 .67656 , .41764 .71715 .43192 .76031 23 24 .389^5 .6.3895 .40378 .67722 .41788 .71785 .43216 .76105 24 25 ..39009 .6:3957 .40401 .67788 .41811 .71855 .43240 .76179 25 20 .390.32 .64019 .40424 .6785:3 .418:35 .71925 .43264 .76253 26 27 .39055 .64081 .4044.8 .67919 .41859 .71995 .43287 .76328 27 28 .3907^ .64144 .40471 .67985 .41882 .72065 .4:3311 .76402 28 29 .39101 .64206 .40494 .68051 .41906 .72135 .43:3:35 .76477 29 30 .39124 .61268 .40518 .68117 1 .41930 .72205 , .43359 .76552 30 31 .39147 .043.30 .40541 .68183 .4195:3 .72275 .43383 .76626 31 32 .39170 .64393 .40565 .68250 .41977 .72346 .4:3407 .76701 32 33 .39193 .64455 .405S8 .68316 .42001 .72416 .4:34:31 .76776 33 34 .39216 .64518 .40011 .68382 .42024 .72487 .43455 .76851 34 35 .39239 .64580 .406.35 .68449 .42048 .72557 .43479 .76926 35 36 .39262 .64643 .40658 .68515 .42072 .72628 .43503 .77001 36 37 .39286 .64705 .40682 .68582 i .42096 .72698 .43527 .77077 37 38 .39309 .64768 .40705 .68648 i .42119 .72769 .43551 .77152 38 39 .39332 .648:31 .40728 .68715 1 .42143 .72840 .43575 .77227 39 40 .39355 .64894 .40752 .68782 i .42167 .72911 .43599 .77^03 40 41 .39378 .64957 .40775 .68848 i .42191 .72982 .43623 .77378 41 42 .39401 .6.5020 .40799 .68915 ' .42214 .73053 .43647 .77454 42 43 .39424 .65083 .40822 .68982 .422:38 .73124 .43671 .77530 43 44 .39447 .65146 .40846 .69049 .42262 .73195 .43695 .77606 44 45 .39471 .65209 .40869 .69116 .42285 .73267 .43720 .77681 45 46 .39494 .65272 .40893 .69183 .42309 .73338 .43744 . 1 1 lOi 46 47 .39517 .65.3.36 .40916 .69250 .42:3:33 .7:3409 .43768 .77833 47 48 .39540 .65:399 .40939 .69318 .42357 .73481 .43792 .77910 48 49 .39563 .65462 .4096:3 .69385 .42381 .73552 .43816 .77986 49 50 .39586 .65526 1 .40986 .69452 , .42404 .73624 .43840 .78062 50 51 .39610 .65589 .41010 .69520 .42423 .73696 .43864 .78138 51 52 .396.33 .65653 .410.33 .69587 .42452 .73768 .43888 .78215 52 5:3 .39656 .65717 .41057 .69655 .42476 .73840 .43912 .78291 53 54 .39079 ■ .65780 .41080 .69723 .42499 .7:3911 .439:36 .78368 54 55 .39702 .65844 .41104 .69790 .42523 .73983 .43960 .78445 55 56 .39726 .65908 .41127 .69858 .42547 74056 .43984 .78521 56 57 .39749 .65972 .41151 .69926 .42571 .74128 .44008 .78598 57 58 .39772 .66036 .41174 .69994 .4'J595 .74200 .44032 .78075 58 59 .39795 .66100 .41198 .70062 .42619 .74272 .440=.7 .78752 59 60 .39819 1 .661W .41221 .70130 1 .42642 .74:345 .44081 .78829 GO 346 TABLE XIII.— VERSINES AND EXSECANTS. / 56° 1 1 57° 58° 59° / Vers, Exsec. Vers. Exsec. : Vers. Exsec. 1 Vers. Exsec. .44081 .78829 .45536 .83608 .47008 .88708 .48496 .94160 1 .44105 .78906 .45560 .83690 ! .47033 .88796 .48521 .94254 1 2 .44129 .78984 .45585 . 83773 .47057 .88884 .48546 .94349 2 3 .44153 .79061 .45609 .83855 1 .47082 .88972 .48571 .94443 3 4 .44177 .79138 .456:i4 .8:3938 .47107 .89060 .48596 .945:37 4 5 .44201 .79216 t .45658 .&4020 .47131 .89148 .48621 .946:32 5 6 .44225 .79293 .45683 .84103 .47156 .89237 .48646 .94726 6 7 .44250 .79371 .45707 .84186 1 .47181 .89325 .48671 .94821 7 8 .44274 .79449 .45731 .84269 . .47206 .89414 .48696 .94916 8 9 .44298 .79527 . 45756 .843.52 .472:30 .89503 i .48721 .95011 9 JO .44322 .79604 .45780 .&4435 i .47255 .89591 .48746 .95106 10 11 .44346 .79682 .45805 .84518 .47280 .89680 ; 1 .48771 .95201 11 12 .44370 .79761 .45829 .84601 .47304 .89769 .48796 .9.5296 12 13 .44395 .79839 .45854 .S4685 .47329 .89858 1 .48821 .9.5392 13 14 .44419 .79917 .45878 .W768 .47:354 .89948 .48846 .95487 14 15 .44443 . 79995 .4.5903 .84852 .47379 .90037 1 .48871 .95583 15 16 .44467 .80074 .4.5927 .849:35 .47403 .90126 1 .48896 .95678 16 17 .44491 .80152 .45951 .85019 .47428 .90216 .48921 .95774 17 18 .44516 .80231 .4.5976 .8.5103 ■ , .4745:3 .90305 \ .48946 .95870 18 19 .44540 .80309 .46000 .85187 .47478 .90:395 .48971 .95966 19 20 .44564 .80388 .46023 .85271 .47502 .90485 .48996 .90062 20 21 .44588 .80467 .46049 .8.5355 .47.527 .90575 .49021 .96158 21 22 .44612 .80546 .46074 .85439 ' .47562 .90665 .49046 .96255 22 23 .44637 .80625 .46098 .85523 : , .47577 .90755 .49071 .96:351 23 24 .44661 .80704 j .46123 .85608 .47601 .90845 i .49096 .96448 24 25 .44685 .80783 1 .46147 .85692 1 .47626 .90935 .49121 .96544 25 26 .44709 .80862 i .46172 . 85777 .47651 .91026 .49146 .96641 26 27 .44734 .80942 ' .46196 .85861 ! ! .47676 .91116 ' .49171 .96738 27 28 .447.58 .81021 .46221 .85946 1 1 .47701 .91207 .49196 .96835 28 29 .44782 .81101 .46246 .b6031 .47725 .91297 .49221 .96932 29 30 .44806 .81180 .46270 .86116 .47750 .91388 .49246 .97029 30 31 .44831 .81260 i .46295 .86201 ' .47775 .91479 .49271 .97127 31 32 .44855 .81340 .46319 .86286 .47800 .91570 .49296 .97224 :J2 33 .44879 .81419 .46:344 .86371 .47825 .91661 .49321 .97.322 :33 34 .44903 .81499 1 .46368 .86457 .47849 .91752 .49346 .97420 34 35 .44928 .81579 ' .46393 .86542 .47874 .91844 .49372 .97517 35 36 .44952 .81659 .46417 .86627 .47899 .91935 ! .49397 .97615 36 37 .44976 .81740 .46442 .86713 ' .47924 .92027 .49422 .97713 37 38 .45001 .81820 .46466 .86799 i .47949 .92118 .49447 .97811 38 39 .45025 .81900 .40491 .86885 .47974 .92210 I .40472 .97910 39 40 .45049 .81981 .46516 .86990 .47998 .92302 .49497 .98008 40 41 .45073 .82061 .46.540 .87056 .48023 .92394 .49522 .98107 41 42 .45098 .82142 .46565 .87142 .48048 .92486 .49547 .98205 42 43 .45122 .82222 .46589 .87229 ; .48073 .9-2578 .49572 .98304 43 44 .45146 .82303 .46614 .87315 .48098 .92670 1 .49597 .98403 44 45 .45171 .82384 1 .46639 .87401 ' .48123 .92762 ' .49623 .98502 45 46 .45195 .82465 .46663 .87488 .48148 .92855 .49648 .98601 46 47 .45219 .82546 .46688 .87574 , .48172 .92947 .49673 .98700 47 48 .45244 .82627 .46712 .87661 I .48197 .9:3040 .49698 .98799 48 49 .45268 .82709 ! .46737 .87748 .48222 .93133 1 .49723 .98899 49 50 .45292 .82790 i .48763 .87834 .48247 .93226 i i .49748 .98998 50 51 .45317 .82871 .46786 .87921 .48272 .93319 1 .49773 .99098 51 52 .45341 .82953 .46811 .88008 .48207 .93412 .49799 .99198 52 53 .45365 .a3034 .468:36 .88095 , .48322 .9.3505 1 .49824 .99298 53 54 .45390 .83116 .46860 .88183 .48347 .93598 1 .49849 .99398 54 55 .45414 .83198 .46885 .88270 .48372 .93692 1 .49874 .99498 55 56 .45439 .&3280 .46909 .88357 : .48:396 .9:3785 1 .49899 .99598 56 57 .45463 .8.3.362 1 .469:34 .88445 .48421 .9.3879 .49924 .99698 57 58 .45487 .8.3444 .46959 .88532 .48446 .93973 .499.50 .99799 58 59 .45512 .83.526 1 .46983 .88620 .48471 .04066 1 .49975 .99899 59 60 .4S5;:6 .83008 . .47008 .88708 ! .48496 .941 CO 1 .50000 i. 00000 60 TABLE XIII.-VEllSINES AND EXSECANTS. 347 / 60» Vers. Exsec. .50000 1.00000 1 .50025 1.00101 o .50050 1.00202 3 .50076 1.00303 4 .50101 1.00404 5 .50126 1.00505 6 .50151 1.00607 i .50176 1.00708 8 .50202 1.00810 9 .50227 1.00912 10 .50252 1.01014 11 .50277 1.01116 12 .50303 1.01218 13 .50328 1.01320 14 .50353 1.01422 15 .50378 1.01525 16 .50404 1.01628 17 .504.29 1.01730 18 .5(1454 1.01833 19 .50479 1.01936 20 .50505 1.02039 21 .50530 1.02143 22 .50555 1 02246 23 .50581 1.02349 ^ .50606 1.02453 25 .50631 1.02557 26 .50656 1.02661 27 .50682 1.02765 28 .50707 1.02869 29 ..50732 1.02973 30 ..50758 1.03077 31 .50783 1.03182 32 .50808 1.03286 33 .50834 1.03391 M .50859 1.03496 35 .50884 1.03601 36 .50910 1.03706 37 ..50935 1.03811 38 .50960 1.03916 39 .50986 1.04022 40 .51011 1.04128 41 .51036 1.04233 42 .51062 1.04339 43 .51087 1.04445 44 .51113 1.04551 45 .51138 1.046.58 46 .51163 1.047&4 47 .51189 1.04870 48 .51214 1.04977 49 .51239 1.05084 50 .51265 1.05191 51 .5*290 1.05298 52 ..51316 1.05405 53 .51341 1.05512 54 .51366 1.0.5619 55 .51392 1.0.5727 56 .51417 1.05835 57! .51443 1.05942 58 .51468 1.060.50 59, .51494 1.061.58 601 .51519 1.06267 61' 62" 63« Vers. Exsec Vers. Exsec .51519 .51W4 .51570 .51595 .51621 .51646 .51672 ..51697 .51723 .51748 .51774 .51799 .51825 .51850 .51876 .51901 .51927 .51952 ..51978 .52003 .52029 .52054 ..52080 .52105 .52131 .52156 .52182 .52207 .52233 .52259 .52310 .52335 ..52361 .52386 ..52412 .52438 .52463 .52489 .52514 .52540 .52566 .52591 .52617 .52642 .52668 .52694 .52719 .52745 .52771 .52796 .52822 .52848 ..52873 ..52899 .-52924 .529.50 .52976 .53001 .53027 .53053 1.06267 1 .53053 1.06375 .53079 1.06483 .53104 1.06592 .53130 1.06701 .53156 1.06809 .53181 1.06918 .53207 1.07027 .53233 1.07137 .53258 1.07246 .53284 1.07356 .53310 1.07465 .53336 1.07575 .53361 1.07685 .5.3387 1.07795 .53413 1.07905 .534;^9 1.08015 .53464 1.08126 .53490 1.08236 .53516 1.08347 .5.3542 1.08458 .53567 1.08569 ..53593 1.08680 .53619 1.08791 .53645 1.08903 .53670 1.09014 .53696 1.09126 ..53722 1.09238 ..53748 1.09350 .53774 1.09462 .53799 1.09574 .53825 1.09686 .53851 1.09799 .53877 1.09911 .53903 1.10024 .53928 1.10137 .53954 1.10250 .53980 1.10363 .54006 1.10477 .54032 1.10590 .540.58 1.10704 .54083 1.10817 ..54109 1.10931 .54135 1.11045 .54161 1.11159 .54187 1.11274 ..54213 1.11388 .54238 1.11503 ..54264 1.11617 ! .54290 1.117.32 .54316 1.11847 .54343 1.11963 .54368 1.12078 .54394 1.12193 1 ..54420 1.12309 ' ..54446 1.12425 1 .54471 1.12540 ' ..'>1497 1.126.57 .54523 1.12773 1 .54M9 1. 128^9 . .W575 1.13005 i .54601 1 .1.3005 .13122 .13239 .13.356 .13473 .13590 .13707 .13825 .13942 .14060 .14178 .14296 .14414 .145.33 .14651 .14770 .14889 .15008 .1.5127 :15346 .15366 .15485 .1.5605 .15725 .15845 .15965 .16085 .16206 .16326 .16447 .16568 .16689 .16810 .16932 .17053 .17175 .17297 .17419 .17541 .17663 .17786 .17909 .18031 .18154 .18277 .18401 .18524 .18648 .18772 .18895 .19019 .19144 .19268 .19393 .19517 .19642 .19767 .19892 .20018 .20143 .20269 Vers. .54601 .54627 .54653 .54679 .&4705 .54731 .54757 .54782 .54808 .54834 .54800 ..54886 .54912 .54938 .54964 .54990 .55016 .55042 .55068 .55094 .55120 ..55146 .55172 .55198 .55224 .55250 .55276 .55302 .55328 .55354 .55380 .55406 .55432 .55458 .55484 .55510 .555.36 .55563 .55589 .55615 .55641 .55667 .55693 .55719 .55745 .55771 .55797 .55823 .55849 .55876 .55902 .55928 .559.54 .55980 .56006 .56032 .56058 .56084 .56111 .56137 .56163 Exsec. .20269 .20395 .20521 .20647 .20773 .20900 .21026 .21153 .21280 .21407 .21535 .21662 .21790 .21918 .22045 .22174 .22.302 .22430 .22559 .22688 .22817 .22946 .2.3075 .23205 .23334 .23404 .23594 .23724 .2:5855 .23985 .24116 .24247 .31 .24378 -32 .24509 .33 .24640 .24772 .249(3 .25035 25167 .2.5300 139 .25432 40 .25565 .25697 .25830 .25963 .26097 .26230 .26364 .26498 .26632 .26766 .26900 1 51 .27035 1 52 .27169 153 .27304 54 .27439 .27574 .27710 .27845 .27981 .28117 348 TABLE Xm.— VERSINES AND EXSECANTS. / 64° \ 65° ' ! 1 ' 66' 67» Vers. Exsec. Vers. Exsec. ' Vers. 1 1 Exsec. Vers. Exsec. .56163 1.28117 .57738 l.:36620 ' i .59326 1.45859 .60927 1.55930 1 .56189 1.28253 .57765 l.;36768 .59353 1.46020 .60954 1.56106 1 2 .56215 1.28390 .57791 l.;36916 .59379 1.46181 .60980 1.56282 2 3 .56241 1.28526 .57817 1.37064 ' .59406 1.46342 .61007 1.56458 3 4 .56267 1.28663 .57844 1.37212 .59433 1.46504 .61034 1.566a4 4 5 .56294 1.28800 .57870 1.37361 .59459 1.46665 .61061 1.56811 5 6 .56320 1.28937 , .57896 1.37509 .59486 1.46827 .61088 1.56988 6 7 .56ai6 1.29074 1 .57923 1.37658 .59512 1.46989 .61114 1.57165 7 8 .56372 1.29211 , .57949 1.37808 : .59539 1.47152 .61141 1.57342 8 9 .56398 1.29349 ' .57976 1.37957 : .59566 1.47314 .61168 1.57520 9 10 .56425 1.29487 i .58002 1.3810? : 1 .59592 1.47477 .61195 1.57698 10 11 .56451 1.29625 .58028 1.382.56 .59619 1.47640 .61222 1.57876 11 12 .56477 1-29763 ' .580.55 1.38406 1 .59645 1.47804 .61248 1.580.54 12 13 .56503 1:29901 .58081 1.38556 .59672 1.47967 .61275 1.58233 13 14 .56529 1.30040 .58108 1.38707 : .59699 1.48131 .61302 1.58412 14 15 .56555 1.30179 .58134 1.388.57 1 .59725 1.48295 .61:329 1.58591 15 16 .56582 1.30318 .58160 1.39008 ' .59752 1.48459 .61:356 1.58771 16 17 .56608 1.. 30457 .58187 1.39159 .59779 1.48624 .61383 1.58950 17 18 .56634 1.30596 .58213 1.39311 .59805 1.48789 .61409 1.59130 18 19 .56660 1.307:35 .582-iO 1.39462 .59832 1.48954 .614:36 1.59:311 19 20 .56687 1.30875 .58266 1.39614 .59859 1.49119 .61463 1.59491 20 21 .56713 1.31015 .58293 1.39766 .59885 1.49284 .61490 1.59672 21 22 .56739 1.31155 '■ .58319 1.39918 .59912 1.49450 .61517 1.59853 22 23 .56765 1.31295 .5^345 1.40070 : .59938 1.49616 .61.544 1.C0035 23 24 .56791 1.31436 .5^372 1.40222 .59965 1.49782 .61570 1.60217 24 25 .56818 1.31576 .58:398 1.40375 .59992 1.49W8 1 .61597 1.60399 25 26 .56844 1.31717 .58425 1.40528 .60018 1.50115 .61624 1.60581 26 27 .56870 1.31858 . .58451 1.40681 .60045 1.50282 .61651 1.60763 27 28 .56896 1.31999 1 .58478 1.408:35 I .60072 1.50449 .61678 1.60946 28 29 .56923 1.32140 ' .58504 1.40988 ; .00098 1.50617 .61705 1.61129 29 30 .56949 1.32282 ; .58531 1.41142 1 .60125 1.50784- [ .61732 1.61313 30 31 .56975 1.32424 '■ .58557 1.41296 ' ' .60152 1.50952 .617.59 1.61496 31 32 .57001 1.32566 .58584 1.414.50 .60178 1.51120 .61785 1.61680 32 33 .57028 1.32708 .58610 1.41605 .60205 1.. 51289 .61812 1.61864 33 34 .57054 1.32850 1 .586:37 1.41760 .60232 1.51457 .618:39 1.62049 34 35 .57080 1.32993 1 .58663 1.41914 ' .60259 1.51626 .61866 1.622.34 35 36 .57106 i.asi.ss .58690 1.42070 1 .60285 1.51795 .61893 1.62419 36 37 .571.33 1.33278 .58716 1.42225 .60312 1.51965 .61920 1.62604 37 38 .57159 1.. 3.3422 .58743 1.42380 .60339 1.52134 .61947 1.62790 38 39 .57185 1.3:3565 .58769 1.42536 i .60365 1.52304 .61974 1.62976 39 40 .57212 1.33708 1 .58796 1.42692 ; .60392 1.5^474 .62001 1.63162 40 41 .57238 1.33852 .58822 1.42848 i .60419 1.52645 .62027 1.63348 41 42 .57264 1.3.3996 ' .58849 1.43005 .60445 1.52815 .62054 1.635:35 42 43 .57291 1.34140 1 .58875 1.43162 : .60472 1.52986 .62081 1.63722 43 ■^4 .57317 1.34284 .58902 1.43318 ! .60499 1.531.57 .62108 1.6:3909 44 45 .57343 1.34429 .58928 1.4.3476 .60526 1.5.3.329 .621:35 1.64097 45 4G .57369 1.34573 .58955 1.43633 .605.52 1.53500 .62162 1.64285 46 47 .57396 l.a4718 .58981 1.43790 .60579 1.53672 .62189 1.64473 47 48 .57422 1.34863 .59008 1.43948 .60606 1.53845 .62216 1.64662 48 49 .57448 1.35009 .590:i4 1.44106 .60633 1.54017 .62243 1.64851 49 50 .57475 1.35154 i .59061 1.44264 .60659 1.54190 .62270 1.65040 50 51 .57501 1.35300 ' .59087 1.44423 .60686 1.54363 ! .62297 1.65229 51 52 .57527 1.35446 .59114 1.44582 .60713 1.545:36 .62:324 1.65419 52 53 .57554 1.35592 : .59140 1.44741 1 .60740 1.54709 .62351 1.65609 53 54 .57580 1.3,5738 : .59167 1.44900 .60766 1.54883 .62378 1.65799 54 55 .57606 1.35885 .59194 1.45059 .60793 1.5505? i .62405 1.65989 55 56 .57633 1.36031 .59220 1.45219 .60820 1.55231 .62431 1.66180 56 57 .57659 1.36178 .59247 1.45378 .60847 1.55405 .62458 1.66:371 57 58 .57685 1.36325 .59273 1.45539 ; .60873 1.55580 .62485 1.66.563 58 59 .57712 1.36473 .59:300 1.45699 .60900 1.557.55 .62512 1.66755 59 60 .57738 1.36620 1 .59326 iAuS:.d .60D27 1.55930 i .62539 1.66947 60 TABLE XIII.— VERSINES AND EXSECANTS. U9 68» 69° 1 70° 71° ! > Vers. Exsec. Vers. Exsec. Vers. Exsec. Vers. Exsec. .63539 ' 1.66947 .64163 1.79043 ' .65798 1.92380 .67443 2.07155 1 .62.-)66 1 1.67139 .64190 1,79254 .65825 1.92614 .67471 2.07415 1 o .62593 1.67332 .64218 1.79466 .65853 1.92849 .67498 2.07675 2 3 .62620 i 1.67525 .64245 1.79679 .65880 1.93083 .67526 2.07930 3 4 .62647 1.67718 .64272 1.79891 .6.5907 1.93318 .67553 2.08197 4 5 .62674 1.67911 .64299 1.80104 .65935 1.93554 .67581 2.08459 5 G .62701 1.68105 1 .64326 1.80318 .65962 1.93790 .67608 2.08721 6 1 .62728 1.68299 1 .64353 1.80531 .65989 1.94026 .67636 2.08983 7 8 .62755 1.68494 .64381 1.80746 .66017 1.94263 .67663 2.09246 8 9 .62782 1.6S689 .64408 1.80960 ' .66044 1.94500 .67691 2.09510 9 10 .62809 1.68884 .04435 1.81175 .66071 1.94737 .67718 2.09774 10 11 .62836 1 69079 .64462 1.81390 .66099 1.94975 .67746 2.10038 11 12 .62863 1 1.69275 .64489 1.81605 .66126 1.95213 .67773 2.10303 12 13 .62890 i 1.69471 : .64517 1.81821 •: .66154 1.95452 .67801 2.10568 13 14 .62917 1.69667 .64544 1.820.37 .66181 1.95691 .67829 2.10834 14 15 .62944 1.69864 .64571 1.82254 .66208 1.95931 .67856 2.11101 15 16 .62971 1.70061 .64598 1.82471 .662:36 1.96171 .67884 2.11367 16 17 .62998 1.70258 .6462.5 1.82688 .66263 1.96411 .67911 2.11635 17 18 .63025 1.704.55 .646.53 1.82900 1 .66290 1.96652 .67939 2.11903 18 19 .630.32 1.70653 .64680 1.83124 .66318 1.96893 .67966 2.12171 19 20 .63079 1.70851 .64707 1.83342 j .66345 1.97135 .67994 2.12440 20 21 .63106 1.71050 .64734 1.83.561 .66373 1.97377 .68021 2.12709 21 22 .63133 1.71249 .64701 1.83780 .66400 1.97619 .68049 2.12979 22 23 .63161 1.71448 .64789 1.83999 .66427 1.97862 .68077 2.13249 23 24 .63188 1.71647 .64816 1.84219 1 .66455 1.98106 .68104 2.13520 24 25 .63215 1.71847 .64843 1.84439 [ .66482 1.98349 .68132 2.13791 25 26 .6.3342 1.72047 .64870 1.84659 .66510 1.9S594 .68159 2.14063 26 27 .63269 1.72247 .64898 1.84880 .66537 1.98838 .68187 2.14335 27 28 .63296 1.72448 .64925 1.85102 .66564 1.99083 .68214 2.14608 28 29 .63323 1.72649 .64952 1.85323 .66592 1.99329 .68242 2.14881 29 30 .63350 1.72850 .64979 1.85545 .66619 1 1.99574 .68270 2.15155 30 31 .63377 1.73052 .65007 1.85767 1 .66647 1.99821 .68297 2.15429 31 32 .6;i404 1.73254 .65034 1.85990 i .66674 2.00067 .68325 2.15704 32 33 .634^31 1.73456 .65061 1.86213 i .66702 2.00315 .68352 2.15979 33 34 .63458 1.73659 1 .05088 1.86437 .66729 2.00562 .68380 2.16255 34 35 .63485 1.73862 .6.5116 1.86661 i . 66756 2.00810 .68408 2.16531 35 36 .63512 1.74065 : .6.5143 1.86885 ! .66784 2.01059 .68435 2.16808 36 37 .63.539 1.74269 .65170 1.87109 ! .66811 2.01308 .68463 2.17085 37 38 .6.3566 1.74473 i .65197 1.87334 .66839 2.01557 .68490 2.17363 38 39 .63594 1.74677 .65225 1 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 1.75086 .65279 1.88011 .66921 2.02308 .68573 2.18199 41 42 .63675 1.75292 .65306 ; 1.88238 .66949 2.02559 .68601 2.18479 42 43 .63702 1.75497 .65334 1.88465 .66976 2.02810 .68628 2.18759 43 44 .63729 1.75703 : .65361 1.88692 .67003 2.03062 .68656 2.19040 44 45 .63756 1.75909 .65388 1 1.88920 .67031 2.03315 .686&4 2.19322 45 46 .63783 1.76116 .6.5416 ! 1.89148 .67058 2.03568 .68711 2.19604 46 47 .6.3810 1.76323 .65443 1.89376 .67086 2.03821 .68739 2.19886 47 48 .63838 1 1.76.530 .6.5470 1.89605 .67113 2.04075 .68767 2.20169 48 49 .63865 1.707.37 .6.5497 1.898.34 .67141 2.04.329 .68794 2.20453 49 50 .63892 1.76945 .65.525 1.90063 .67168 2.04584 .68822 2.20737 50 51 .63919 1.771.S4 i .65552 1.90293 .67196 2.04839 ,68849 2.21021 51 52 .63946 1.77362 . 65579 1.90524 .67223 2.0.5094 .68877 2.21306 52 53 .63973 1.77571 .65607 1.90754 .67251 2.05350 .68905 2.21592 53 54 .64000 1 77780 .6.5634 1.90986 . 67278 2.05607 .68932 2.21878 1 54 55 .64027 1.77990 .6.5661 1.91217 .67306 2.05864 ! .68960 2.22165 55 56 .64055 1.78200 .6.5689 1.91449 .67333 2.06121 .68988 2.22452 : 56 57 . 640S2 1.78410 .6.5716 1.91681 .67361 2.06379 .69015 2.22740 I 57 58 .64109 1.78621 .6.5743 1.91914 .67388 2.06637 .69043 2.23028 38 59 .641.36 1.7S832 i .6.5771 1.92147 .67416 2.06896 .69071 2.23317 59 60 .64163 1.79043 1 .65798 1.92380 .67443 1 2.07155 1 .69098 2.23607 50 350 TABLE XIII.— VERSINES AND EXSECANTS. 72° 7 3° 74° 1 75° Vers. 1 Exsec. Vers. Exsec. Vers. Exsec. Vers. Exsec. .69098 2.23607 .70763 2.42030 .72436 2.62796 .74118 2.86370 1 .69126 2.2:3897 .70791 2.42356 .72464 2.63164 .74146 2.86790 4 2 .69154 2.24187 .70818 2.426a3 .72492 2 63533 .74174 2.87211 2 3 .69181 2.24478 .70846 2.43010 .72520 2.63903 1 .74202 2.87633 3 4 .69209 2.24770 .70874 2.43337 .72548 2.64274 ! .74231 2.88056 4 5 .69237 2.2.5062 ! .70902 2.43666 . 72576 2.&4645 i .74259 2.88479 5 6 69264 2.25355 i .70930 2.43995 1 .72604 2.65018 .74287 2.88904 6 7 .69292 2.25648 i .70958 2.44324 ' .726:32 2.65391 .74315 2.89.3.30 7 8 .69320 2.25942 ! .70985 2.44655 .72660 2.65765 .74343 2.89756 8 9 .69:^47 2 26237 ' .71013 2.44986 .72688 2.66140 .74371 2.90184 9 10 .69375 2.26531 j .71041 2.45317 .72716 2.66515 .74399 2.90613 10 11 .69403 2.26827 ' .71069 2.45650 .72744 2.66892 .74427 2.01042 11 12 .69430 2.27123 .71097 2.45983 .72772 2.67269 .74455 2.01473 12 13 .69458 2.27420 .71125 2.46316 .72800 2.67047 ! .74484 2.01904 13 14 .69486 2.27717 .711.53 2.46651 .72828 2.68025 .74512 2.02337 14 15 .69514 2.28015 .71180 2.46986 .72856 2.65v405 .74540 2.92770 15 16 .69541 2.28313 .71208 2.47321 1 .72884 2.68785 .74568 2.9:3204 16 17 .69569 2.28612 .71236 2.47658 1 .72912 2.69167 .74596 2.9.3040 17 18 .69597 2.28912 .71264 2.47995 .72940 2.09549 .74624 2.94076 18 19 .69624 2.29212 .71202 2.4&333 .72968 2.^0931 .74652 2.94.514 19 20 .69652 2.29512 1 .71320 2.48671 .72996 2.70315 .74680 2.04952 20 21 .69680 2.29814 ! .71^48 2.49010 .73024 2.70700 '■ .74709 2.0.5.392 21 22 .69708 2.30115 .71375 2.49350 .7:3052 2.71085 i .74737 2.05832 22 23 .697::55 2.30418 .71403 2.4%91 .73080 2.71471 .74765 2.96274 23 24 .69763 2.30721 .71431 2.50032 .73108 2.71858 .74793 2.96716 ai 25 .69791 2.31024 ! .71459 2.50374 .731:36 2.72^6 .74821 2.97160 25 26 .69818 2.31328 .71487 2.50716 1 .73164 2.72635 ' .74S49 2.97604 26 HI .69840 2.31633 .71515 2.51060 ! .73192 2.73024 .74878 2.98050 27 28 .69874 2.31939 .71543 2.51404 .73220 2.7:3414 .74906 2.98497 28 29 .60002 2.32244 1 .71571 2.51748 .73248 2.73806 .74934 2.98044 29 30 .69929 2.32551 .71598 2.52094 .73276 2.74198 .74962 2.99393 30 31 .69957 2.32858 .71626 2.52440 ' 73304 2.74591 .74990 2.09843 31 32 .69985 2.33166 .71654 2.52787 .73:332 2.74984 1 .75018 3.00293 32 33 .70013 2.3;}474 1 .71682 2.53134 1 .73360 2.75379 .75047 3.00745 .3:3 ai .70040 2.33783 • .71710 2.5^482 .7^388 2.75775 ! .75075 3.01198 34 35 .70068 2.34092 i .71738 2.5;3831 .7^416 2.76171 .75103 3.01652 35 36 .70096 2.31403 .71766 2.54181 .73444 2.76568 , .75131 3.O2107 36 37 .70124 2.34713 .71794 2.54531 ' .73472 2.76966 1 ! .7.5159 3.02563 37 38 .70151 2.. 35025 .71822 2.54883 i .73500 2.77:365 , .75187 3.0.3O20 .38 39 .70179 2.. 3.5336 .71850 2.55235 ; .73529 2.77765 I i .75216 3.03479 39 40 .70207 2.35&49 .71877 2.55587 ! .73557 2.78166 1 .75244 3.03938 40 41 .70235 2.3.5962 .71905 2.. 55940 ! .73585 2.78568 .75272 3.04398 41 42 .70263 2.36276 .71933 2.56294 .7:3613 2.78970 1 1 .75300 3.04860 42 4-3 .70290 2.36590 ' .71961 2.56&49 .73641 2.79374 .75328 3.05.322 43 44 .70318 2.36905 ! .71989 2.57005 .73669 2.79778 .75356 3.05786 44 45 .70346 2.37221 ; .72017 2.57361 ; .73697 2.80183 ; .75385 3.062.51 45 46 .70374 2.37537 .72045 2.. 57718 .73725 2.80589 ! .75413 3.06717 46 47, .70401 2.37854 .72073 2.58076 .73753 2..S0996 .75441 3.07184 47 48 .70429 2.38171 .72101 2.. 5^34 .73781 2.81404 .75469 3.07652 48 49 .70457 2.3*489 .72129 2.-58794 i .73809 2.81813 .75497 3.08121 49 50 .70485 2.38808 .72157 2.59154 .73837 2.82223 j . 75526 3.08591 50 51 .70513 2.39128 .72ia5 2.59514 .73865 2.826a3 .75554 3.09063 51 52 .70540 2.39448 .72213 2.59876 .73893 2.8:3045 .75582 3.09535 52 53 .70568 2.39768 .72241 2.60238 .73921 2.8:3457 .75610 1 3.10009 53 54 .70596 ' 2.40089 • .72269 2.60601 ; .73950 ! 2.83871 .75639 3.10484 1 54 55 .70624 2.40411 .72296 2.60965 1 .73978 2.84285 .75667 3.10960 55 56 .70652 2.40734 .72324 i 2.61330 .74006 2.84700 .7.5695 3.114:37 5G 57; .70679 2.41057 .72352 2.61695 .740*4 ! 2.85116 .75723 3.11915 57 58 .70707 i 2.41381 .72380 2.62061 .74062 ; 2.8.5.5:33 .7.5751 1 3.12.394 58, ^9 .70735 2.41705 .72408 2.62428 .74090 2.8.5951 .7.5780 i 3.12875 59 €0 .70763 2.42030 1 .72436 2.6279G 1 . .74118 2.86370 .75808 3.13357 60, TABLE XIII.-VERSINES AND EXSECANTS. 351 76° 1 ' 77° 78° 1 1 7 9° Vers. Exsec. Vers. Exsec. Vers. Exsec. 1 Vers. Exsec. .75808 3.13357 .77505 3.44541 .79209 3.80973 1 .80919 4.24084 1 .75836 3.13839 .77533 3.45102 .79237 3.81033 .80948 4.24870 1 2 .75864 3.14323 .77562 3.45064 .79266 3.82294 .80976 4.25658 2 3 .75892 3.14809 .77590 3.46228 .79294 3.82956 .81005 4.26448 3 4 .75921 3.15295 .77618, 3.46793 .79323 3.83021 .81033 4.27241 4 5 .75949 3.15782 .77647 3.47360 .79351 3.84288 , .81062 4.28036 5 6 .75977 3.10271 . 77075 3.47928 .79:^0 3.84956 .81090 4.28833 6 7 .70005 3.10761 .77703 3.48498 .79408 3.85627 .81119 4.29034 7 8 .70034 3.172.52 .77732 3.49069 .79437 3.86299 .81148 4.30430 8 9 .76002 3.17744 .7.V00 3.49642 .79465 3.80973 .81176 4.31241 9 10 .70090 3.18238 , i i too 3.50216 .79493 3.87649 1 .81205 4.32049 10 11 .76118 3.18733 .77817 3.. 50791 .79522 3.88327 .81233 4.32859 11 12 .76147 3.19228 .77845 3.51368 .79550 3.89007 .81262 4.33671 12 13 .76175 3.19725 .77874 ■ 3.51947 .79579 3.89689 .81290 4.34486 13 14 .76203 3.20224 .77902 3.. 52527 .79607 3.90373 .81319 4.35304 14 15 .76231 3.20723 .77930 3.. 53109 .79636 3.910.58 .81348 4.30124 15 10 .76260 3.21224 .77959 3.53692 .79064 3.91746 .81376 4.30947 16 17 .76288 3.21726 .77987 3.54277 .79093 3.92436 .81405 4.37772 17 18 .76316 3.22229 .78015 3.. 54863 .79721 3.93128 .81433 4.38000 18 19 .76344 3.22734 .78044 3.55451 .79750 3.93821 1 .81462 4.3943') 19 20 .76373 3.23239 .78072 3.56041 .79r78 3.94517 ; .81491 4.40263 20 21 .76401 3.23746 .78101 3.56632 .79807 3.95215 .81519 4.41099 21 22 .76429 3.21255 .78129 3.57224 .79835 3.95914 .81548 4.41937 22 23 .76458 3.24764 .781.57 3.57819 .79864 3.90616 1 .81576 4.42778 23 24 .70486 3.25275 .78186 3.58414 .79892 3.97320 .81605 4.43622 24 25 .76514 3.2.5787 .7«214 3.59012 .79921 3.98025 ' .81633 4.44468 25 26 .;'(i542 3.26300 .7S242 3.59611 .79949 3.98733 .81602 4.45317 26 27 .76571 3.26814 .78271 3.60211 .79978 3.994J3 .81691 4.46169 27 28 .76599 3.27330 .78299 3.60813 .80006 4.00155 .81719 4.47023 28 29 .76627 3.27847 .78328 3.61417 .80035 4.00809 .81748 4.47881 29 30 .76655 3.28366 .78356 3.62023 .80063 4.01585 .81776 4.48740 30 31 .76684 3.28885 .78384 3.62630 .80092 4.02303 .81805 4.49603 31 32 .76712 3.29406 .78413 3.63238 .80120 4.03024 .81834 4.. 50408 32 33 .70740 3.29929 .78441 3.63849 .80149 4.03746 1 .81862 4.51337 33 34 .70769 3.30452 .78470 3.64461 .80177 4.04471 .81891 4.52208 34 35 .76797 3.30977 .78498 3.65074 .80206 4.05197 .81919 4.53081 35 36 .76825 3.31503 .78526 3.65090 .80234 4.05926 .81948 4.53958 36 37 .76854 3.. 32031 .7'8555 3.00307 .80263 4.066.57 .81977 4.54837 37 38 .7'6882 3.32500 .78583 3.00925 .80291 4.07390 .82005 4.55720 38 39 .70910 3.33090 .78612 3.67545 .80320 4.08125 .82034 4.56605 39 40 .70938 3.33622 .7'8640 3.68167 .80348 4.08863 .82063 4.57493 40 41 .70967 3.34154 .78669 3.68791 .80377' 4.09602 .82091 4.58383 41 42 .76995 3.34689 .78697 3.69417 .80405 4.10344 ; .82120 4.59277 42 43 .77023 3.35224' .78725 3.70044 .80434 4.11088 .82148 4.60174 43 44 .77052 3.35761 .78754 3.70673 .80462 4.11835 .82177 4.61073 44 45 .77080 3.36299 .78782 3.71303 .80491 4.12583 .82206 4.61976 45 46 .77108 3.36839 .7'8811 3.71935 .80520 4.13334 ! .82234 4.62881 46 47 .77137 3.37380 .78839 3.72569 .80548 4.14087 .82203 4.63790 47 48 .77165 3.37923 .78868 3.73205 .80577 4.14842 .82292 4.64701 48 49 .77193 3.38466 .78896 3.73843 .80605 4.15.7J9 .82320 4.65616 49 50 .77222 3.39012 .78924 3.74482 .80634 4.16359 .82349 4.66533 50 51 .77250 3.39558 .78953 3.75123 .80662 4.17121 .82377 4.67454 51 52 .77278 3.40106 .78981 3.75766 .80691 4.17886 .82406 4.68377 52 53 .77307 3.40656 .79010 3.76411 .80719 4.18652 .82435 4.69304 53 54 .77335 3.41206 .79038 3.77057 .80748 4.19421 .82463 4.70234 54 55 .77303 3.41759 .79067 3.77705 .80770 4.20193 .82492 4.71166 55 56 .7739S S.4231i4 .79095 3.783.55 .80805 4.20900 .825i.'l 4.72102 56 1 57 .77420 3.42867 .79123 3.79007 .80833 4.21742 82549 4.73041 57 58 .77448 3.43424 .791.52 3.79661 .80862 4.22.521 .8257'8 4.73983 58 59 .77477 3.4^^982 .79180 3.80.316 .80891 4.23301 .82007 4.74929 59 60 .77505 3.44541 .79209 3.80973 .80919 4.24084 .82635 4.75877 60 352 TABLE XIII.— VERSINES AND EXSECANTS. ~0 80° 81° 82° 83° / Vers. Exsec. Vers. Exsec. Vers. Exsec. Vers. Exsec. .82635 4.75877 .84357 5.39245 .86083 6.18.5:30 .87813 7.20551 1 .82664 4.76829 .84:385 5.40422 .86112 6.20020 .87842 7.22500 1 2 .82692 4.77784 .84414 5.41602 .86140 6.21.517 .87871 7.244.57 2 3 .82721 4.78742 .84443 5.42787 .86169 6.23019 .87900 7.26425 3 4 .82750 4.79703 .84471 5.4:3977 .86198 6.24529 .87929 7.28402 4 5 .82778 4.80667 .84.500 5.45171 .86227 6.20044 I .879.57 7.:3e.388 5 6 .82807 4.81635 .84529 5.46:369 .86256 6.27566 .87986 7.. 32.384 6 7 .82a36 4.82606 .84558 5.47572 .86284 6.29095 .88015 7.. 34390 7 8 .82864 4.a3581 1 .84586 5.48779 .86313 6.300.30 .88044 7.:36405 8 9 .82893 4.84.558 .84615 5.49991 .86.342 6.32171 .88073 7.. 384:31 9 10 .82922 4.85539 .84644 5.51208 .86371 6.. 3:3719 .88102 7.40466 10 11 .82950 4.86524 .84673 5.52429 .86400 6.. 3.5274 .88131 7.42511 11 12 .82979 4.87511 .84701 5 . 53655 .86428 6.;3GS:35 .88160 7.44566 12 13 .83003 4.88502 .847:30 5.54886 .864.57 6.. 38403 .88188 7.466:32 13 14 .83036 4.89497 .84759 5.56121 .86486 6.. 39978 .88217 7.48707 14 15 .83065 4.90495 .84788 5.57361 .86515 6. 41. 560 .88246 7.. 50793 15 16 .83094 4.91496 .84816 5.58606 .86544 6.4.3148 .88275 7.52889 16 17 .83122 4.92501 .84845 5.59855 .86.573 6.44743 .88304 7.. 54996 17 18 .83151 4.93509 .84874 5.61110 .86601 6.46:346 .8a333 7.57113 18 19 .83180 4.94521 .84903 5.62:369 .86630 6 . 4 r955 .88.362 7.. 59241 19 20 .83208 4.95536 .84931 5.63633 .86659 6.49571 .88391 7.61379 20 21 .83237 4.96555 .84960 5.04902 .866aS 6.51194 .88420 7.6.3.528 21 22 .83266 4.97577 .84989 5.66176 .86717 0.. 52825 .88448 7.65688 22 23 .8:3294 4.98603 .85018 5.67454 ] .86746 6.. 54462 .88477 7.67859 23 24 .8a323 4.99633 .85046 5.687:38 i .80774 6.. 56107 .88506 7.70^)41 24 25 .83352 5.00666 .85075 5.70027 ' .86803 6.577.59 ,88535 7.722.34 25 26 .a3380 5.01703 .85104 5.71321 .86a32 6.. 59418 .88504 7.74438 26 27 .8^409 5.0-3743 1 .&5i:33 5.72620 .86861 6.61085 1 .a8593 7.76653 27 28 .83438 5.0:3787 .85162 5.73924 .86890 0.627.59 .88622 7.78880 28 29 .a3467 5.048:34 .85190 5.75233 1 .8()919 0.64441 .88651 7.81118 29 30 .83495 5.05886 .85219 5.76547 .86947 0.66130 .88680 7.83367 30 31 .a3524 5.06941 .85-^ 5.77866 ' .86976 6.67826 .aS709 7.85628 31 32 .83553 5.08000 .85277 5.79191 .87005 6.69.5:30 .887:37 7.87901 32 33 .83581 5.09062 .85305 5.80521 .870.34 6.71242 .88766 7.90186 33 34 .83610 5.10129 .8.5:3:34 5.81856 .87063 6.72962 .88795 7.92482 .34 35 .83639 5.11199 1 .85:363 5.83196 .87092 6.74689 .88824 7.94791 35 36 .83667 5.12273 1 .85392 5.84542 .87120 6.76424 .88853 7.97111 36 37 .83696 5.13350 ' .85420 5.85893 .87149 6.78167 .88882 7.99444 37 38 .83725 5.144:32 ': .8.5449 5.87250 i .87178 6.79918 .88911 8.01788 38 39 .83754 5.15517 1 .85478 5.88612 .87207 6.81677 .88940 8.04146 39 40 .83782 5.16607 .85507 5.89979 .872.36 6.8.3443 .88969 8.06515 40 41 .83811 5.17700 ' .85536 5.91.3.52 ' .87265 6.85218 .88998 8.08897 41 42 .83840 5.18797 .85564 5.92731 1 .87294 6.87001 .89027 8.11292 42 43 .83868 5.19898 i .85593 5.94115 ! .87322 6.88792 .89055 8.1.3699 43 44 .83897 5.21004 ! .85622 5.95505 .87.351 6.90592 .89084 8.16120 44 45 .8:3926 5.22113 .85651 5.96900 .87:380 6.92400 .89113 8.18553 45 46 .a39.>4 5.23226 1 .85680 5.98301 ' .87409 6.94216 .89142 8.20999 46 47 .83983 5.24:343 .85708 5.99708 .874a8 6.96040 .89171 8. 2^459 47 48 .84012 5.25464 .85737 6.01120 .87467 6.97873 .89200 8.25931 48 49 .84041 5.26590 .&5766 6.025:38 .87496 6.99714 .89229 8.28417 49 50 .84069 5.27719 .85795 6.03962 .87524 7.01565 .89258 8.30917 50 51 .84098 5.28853 .85823 6.05392 .87.5.53 7.0.3423 .89287 8.. 3.3430 51 52 .84127 5.29991 .85852 6.06828 , .87582 7.05291 .89316 8.-35957 52 53 .84155 5.31133 .85881 6.08269 1 .87611 7.07167 .89.345 8.38497 53 54 .84184 5.32279 .85910 6.09717 ' .87640 7.09052 .89374 8.41052 54 55 .84213 5.33429 .a59:39 C. 11171 .87669 7.10946 .89403 8.43620 55 56 .84242 5.34584 .85967 6.126:30 .87698 7.12849 .89431 8.46203 ^6 57 .84270 5.35743 .85996 6.14096 .87726 7.14760 .89460 8.48800 57 58 .84299 5.36906 .86025 6.1.5568 .877.55 7.16681 .89489 8.51411 58 50 .84328 5.:38073 .86054 6.17046 .87784 7.18612 .89518 8.540:37 59 60 .84357 5.39245 .86083 6.18530 .87813 7.20551 i .89547 8.5G077 60 TABLE XIII.— VERSINE8 AND EXSECANTS. 3 k: "^ 84" Vers. Exsec. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 I .89547 .89576 .89605 .896:54 .89663 .89692 .89721 .89750 .89779 .89808 .89836 .89865 .89894 .89923 .89952 .89981 .90010 .90039 .90068 .90007 .90126 .90155 .90184 .90213 .90242 .90271 .90300 .90329 .90358 .90386 .90415 .90444 .90473 .90502 .90531 .90560 .90.589 .90618 .90647 .90676 .90705 .90734 .90763 .90792 .90821 .90850 .90879 .90908 .90937 .90966 .00995 .91024 .91053 .91082 .91111 .91140 .91169 .91197 .91226 .91255 .91284 8.56677 8.59332 8.62002 8.64687 8.67387 8.70103 8.72833 8.7.5579 8.78a41 8.81119 8.83912 8.86722 8.89547 8.92389 8.95248 8.98123 9.01015 9.0392;i 9.06849 9.09792 9.12752 9.15730 9.18725 9.21739 9.24770 9.27819 9.30887 9.33973 9.37077 9.40201 9.4:3:343 9.46505 9.49685 9.52886 9.56106 9.59346 9.62605 9.65885 9.G9186 9.72507 9.75849 9.79212 9.82596 9.80001 9.89428 9 92877 9.96348 9.99841 10.0:3356 10.06894 10.10455 10.14039 10.17646 10.21277 10.^932 10.28610 10.32313 10.36040 10.39792 10.43569 10.47371 85= Vers, .91284 .91:313 .91342 .91371 .91400 .91429 .91458 .91487 .91516 .91545 .91574 .91603 .91632 .91661 .91690 .91719 .91748 .91777 .91806 .91835 .91864 .91893 .91922 .91951 .91980 .92009 .920:38 .92067 .92096 .92154 .92183 .92212 .922-11 .92270 .92299 .92328 .92357 .92386 .92415 .92444 .92473 .92502 .92531 .92560 .92589 .92618 .92647 .92676 .92705 .92734 .92763 .92821 .92850 .92879 .92908 .92937 .92966 .92995 .93024 Exsec. 10.47371 10.51199 10.55052 10.58932 10.62837 10.66769 10.70728 10.74714 10.78727 10.82768 10.86837 10.90934 10.95060 10.99214 11.03397 11.07610 11.11852 11.16125 11.20427 11.24761 11.29125 11.33521 11.37948 11.42408 11.46900 11.51424 11.55982 11.60572 11.65197 11.69856 11.74550 11.79278 11.84042 11.88841 11.93677 11.98549 12.03458 12.08040 12.13388 12.18411 12.23472 12.28572 12.33712 12.38891 12.44112 12.49373 12.54676 12.60021 12.65408 12.708:38 12.76312 12.81829 12.87391 12.92999 12.98651 13 04350 13.10096 13.15889 13.21730 13.27620 13.33559 w / Vers. Exsec. .93024 13.33559 .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 13.765.58 7 .93257 13.82913 8 .93286 13.89323 9 .93315 13.95788 10 .93344 14.02310 11 .9:3:373 14.08890 12 .93402 14.15527 13 .93431 14.22223 14 .93400 14.28979 15 .93489 14.35795 16 .93518 14.42672 17 .93547 14.49611 18 .93576 14.56614 19 .93605 14.6367'9 20 .93634 14.70810 21 .93663 14.78005 22 .93692 14.85268 23 .9:3721 14.92597 24 .93750 14.99995 25 .93779 15.07462 26 .93808 15.14999 27 .93837 15.22607 28 .93866 15.30287 29 .93895 15.38041 30 .93924 15.45869 31 .93953 15.53772 32 .93982 15.61751 33 .94011 15.69808 34 .94040 15.77944 35 .94069 15.86159 36 .94098 15.94456 37 .94127 16.02835 38 .94156 16.11297 39 .94186 16.19843 40 .94215 16.28476 41 .94244 16.37196 42 .94273 16.46005 43 .94302 16.54903 44 .94331 16.63893 45 .94360 16.72975 46 .94:389 16.82152 47 .9«18 16.91424 48 .94447 17.00794 49 .94476 17.10262 50 .94505 17.19830 51 .94534 17.29501 52 .94563 17.39274 53 .94592 17.49153 54 .94621 17.59139 55 .94650 17.69233 56 .94679 17.79438 57 .94708 17.89755 58 .947:37 18.00185 59 .94766 18.107:32 GO 354 TABLE XIII.-VERSINES AND EXSECANTS. / 87° 88° 89° / Vers. Exsec. Vers. Exsec. Vers. Exsec. .94766 18.10732 .96510 27.65371 .98255 56.29869 1 .94795 18.21397 .96539 27.89440 .982^4 57.26976 1 2 .94825 18 32182 .96568 28.13917 .98313 58.27431 2 3 .94854 18.43088 .96597 28.38812 .98:342 59.31411 3 4 .94883 18.54119 .96626 28.64137 .98:371 60.. 391 05 4 5 .94912 18.65275 .96655 28.89903 1 .9^00 61.50715 5 6 .94941 18.76560 .96684 29.16120 1 .9^429 62.66460 6 7 .94970 18.87976 .96714 29.42802 1 .98458 63.86572 7 8 .94999 18.99524 .96743 29.69960 .9&487 65.11304 8 9 .95028 19.11208 .96772 29.97607 i .98517 66.40927 9 10 .95057 19.23028 .96801 30.25758 .98546 67 . 75736 10 11 .95086 19.34989 .96830 30.54425 .98575 69.16047 11 12 .95115 19.47093 .96859 30.8:3623 .98604 70.62285 12 13 .95144 19.59341 .96888 31.1.3366 : .98633 72.14583 13 14 .95173 19.71737 .96917 31.436jl i .98662 73.7.3586 14 15 .95202 19.&4283 .96946 31.74554 ! .98691 75.:39655 15 16 .95231 19.96982 .96975 32.060:30 i .98720 77.1:3274 16 17 .95260 20.098:38 .97004 32.38118 , .98749 78.94968 17 18 .95289 20.22852 .97033 32.708:35 .98778 80.85315 18 19 .95318 20.36027 .97062 3:3.04199 .98807 82.84947 19 20 .95347 20.49368 .97092 a3. 382:32 .98836 84.94561 20 21 .9.5377 2C. 62876 .97121 a3. 72952 .98866 87.14924 21 22 .95406 20.76555 .97150 34.08380 .98895 89.46886 22 23 .9.5435 20.90403 .97179 .34.44539 .98924 91.91.387 23 24 .95464 21.04440 .97208 :34. 814.52 ! .98953 94.49471 24 25 .95493 21.18653 .97237 35.19141 .98982 97.22.303 25 2(5 .95522 21.33050 .97266 .35.. 576:33 ' .99011 10<').1119 26 27 .95551 21.47635 .97295 ,35.96953 , .99040 1 .99069 103.17.57 27 28 .95580 21.62413 .97324 36.37127 106.4:311 28 2'J .95609 21.77386 .97353 36.78185 .99098 109.8966 29 30 .956;i8 21.92559 .97382 37.20155 .99127 113.5930 30 31 ,95667 22.07935 .97411 .37.6.3068 .r.91.56 117.5444 31 32 .95696 22.2:3520 .97440 38.06957 .99186 121.7780 .32 33 .95725 22.39316 .97470 38.518.55 .99215 126.3253 .33 34 .957.54 22.55:329 .97499 38.97797 .99244 131.2223 .3-1 35 .95783 22.71563 .97528 39.44820 ' .99273 1.36.5111 .35 36 .9.5812 22.88022 . 97557 39.92963 i .99302 142.2406 36 37 .95842 23.04712 .97586 40.42266 .99.3:31 148.4684 .37 38 .95871 23.2163; .97615 40.92772 .99360 155.2623 38 39 .95900 23.. 38802 .97644 41.44.52c i .99.389 162.7033 39 40 .95929 23.56212 .97673 41.97571 .99418 170.8883 40 41 .9.5958 23.73873 .97702 42.51961 .99447 179.9.350 41 42 .95987 23.91790 .97731 43.07746 .90476 189.9868 42 43 .96016 24.09969 .97760 43.04980 .99505 201.2212 43 44 .96045 24.28414 .97789 44.23720 .99535 213.8600 44 45 .96074 24.47134 .97819 44.84026 .99564 228.1839 45 4(; .96103 24.66132 .97^48 4.5.45963 .99593 244.5540 46 47 .96132 24.85417 .97877 46.09596 .99622 263.4427 47 48 .96161 25.04994 .97906 46.74997 .99651 285.4795 48 49 .96190 25.24869 .97935 47.42241 .99680 311.5230 49 50 .96219 25.45051 .97964 48.11406 .99709 a42.7752 50 51 .96248 25.6.5546 .97993 48.82.576 .997.38 380.9723 51 52 .96277 25.86360 .98022 49.. 55840 .99767 428.7187 52 53 .96307 26.07.503 .98051 50.. 31290 .99796 490.1070 53 54 .96336 26.28981 .98080 51.09027 i .99825 571.9581 54 55 .96365 26.. 50804 .98109 51.891.56 i .99855 686.5496 55 56 .96394 26.72978 .98138 52.71790 .99884 858.4369 56 57 .9&423 26.95.513 .98168 5:3. 57046 .99913 1144.916 57 58 .96452 27.18417 .98197 54.4.50.53 .99942 1717.874 58 59 .96481 27.41700 .98^226 55.-35946 .99971 34:36.747 59 60 .90510 27.65371 1 .98255 56.29869 : 1.00000 Infinite 1 60 XIV.— TRANSITION-CURVE COORDINATES. 355 x = l{l- E) 0° y = W 0° 10 10' 20 30 40 50 10 20 30 40 50 10 20 30 40 50 10 20 30 40 50 10 20 30 40 50 10 20 30 40 50 10 20 30 40 50 10 20 30 40 50 10 20 30 40 50 10 20 30 40 50 Dif. 0.00097 194 291 388 485 0.00582 679 776 873 970 10G7 0.01164 1260 1357 1454 1.551 1648 0.01745 1842 1939 2036 2133 oooo 0.02326 2423 2520 2617 2714 2810 0.02907 3004 3101 3198 3294 3391 0.03488 3585 3681 3778 3875 3971 0.04068 4165 4261 4358 4455 4551 0. 04648 4744 4841 4937 5034 5130 0.05227 5323 5420 5516 5612 5709 5805 97 97 97 97 97 97 97 97 97 97 97 96 97 97 97 97 97 97 97 97 97 96 97 97 97 97 97 96 97 97 97 97 96 97 97 97 96 97 97 96 97 97 96 97 97 96 97 96 97 96 97 96 97 96 97 96 96 97 96 90 x= h\ -E) E Dif. 0.00000 1 1 2 0.00003 4 5 7 8 10 0.00012 14 17 19 2'' 24 0.00027 31 34 37 41 45 0.00049 53 57 62 66 71 0.00076 81 87 92 98 104 O.OOIIO 116 122 129 135 142 0.00149 156 164 171 179 187 0.00195 203 211 220 229 237 0.00246 256 265 275 284 294 304 1 1 1 1 1 o 1 Q 4 3 3 4 4 4 4 4 5 4 5 5 5 6 5 6 6 6 6 6 i 6 7 I 8 8 8 8 8 9 9 8 9 10 9 10 9 10 10 10 <^' y = lC C Dif. 10° 10' 20 30 40 50 11 13 14 lo 16 17 18 19 20 10 20 30 40 50 10 20 30 40 50 10 20 30 40 50 10 20 30 40 50 10 20 30 40 50 10 20 30 40 50 10 20 30 40 50 10 20 30 40 50 10 20 30 40 50 0.05901 5998 6094 0190 6286 0.06383 6479 6575 6671 6767 6863 0.06959 7055 7151 7247 7343 7439 0.07535 7631 7727 7823 7919 8015 0.08110 8206 8302 8397 8493 8588 ,08684 8780 8875 8970 9066 9101 0.09257 9352 9447 9543 9638 9733 0.09828 9923 10018 10113 1U208 10303 0. 10398 10493 10.588 10683 10778 10873 .10967 11062 11157 11251 11346 11440 11535 97 96 96 96 97 96 96 96 96 96 96 96 96 96 96 96 96 96 96 96 96 96 95 96 96 95 9o 95 96 96 95 95 96 95 96 95 95 96 95 95 95 95 95 95 95 95 95 95 95 95 95 95 94 95 95 94 95 94 95 18S E Dif. 0.00314 325 335 346 357 0.00368 379 391 402 414 426 0.00438 450 462 475 488 502 0.00514 527 540 554 567 581 0.00595 610 624 639 653 668 0.00683 698 714 729 745 761 0.00777 793 810 826 843 860 0.00877 894 911 929 947 964 0.00982 1001 1019 1038 1056 1075 0.01094 1113 1133 1152 1172 1192 1212 11 10 11 11 11 11 12 11 12 12 12 12 12 13 13 14 12 13 13 14 13 14 14 15 14 15 14 15 15 15 16 15 16 16 16 16 17 16 17 17 17 17 17 18 18 17 18 19 18 19 18 19 19 19 20 19 20 20 '.'0 40 356 XIV. —TRANSITION-CURVE COORDINATES. ' 20° 20' 40 21 20 40 22 20 40 23 20 40 24 20 40 26 28 29 .30 20 40 20 40 20 40 20 40 20 40 y=iC C Dif. X = l{\ - E) E Dif. 11912 12101 12289 12477 0.12665 12853 13040 13228 13415 13602 0.13789 13975 14162 14348 14534 14720 0.14905 15091 15276 15461 15645 15830 0.16014 16198 16382 16565 16749 16932 17114 189 189 188 188 188 188 187 188 187 187 187 186 187 186 186 186 185 186 185 185 184 185 184 184 184 183 184 183 182 274 01252 1293 1335 1377 1421 01464 1509 1554 1599 1646 1693 01740 1789 1838 1887 1937 1988 0.02040 2092 2144 2198 2252 2307 0.02362 2118 2474 2531 2589 2G48 2707 41 42 42 44 43 45 45 45 47 47 47 49 49 49 50 51 52 52 52 54 54 55 55 56 56 57 58 59 59 90 «/>' 30° 30' 31 30 32 I 30 30 30 30 30 30 30 30 33 \m 85 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 y = lC C Dif. 0.17388 17661 17934 18206 18478 0.18749 19019 19288 19557 19826 20094 0.20361 20627 20893 21158 21423 21686 0.21949 22212 22474 22995 23513 24028 24540 0.25049 25554 26057 26556 27052 273 273 272 272 271 210 269 269 269 268 267 266 266 265 265 263 263 263 262 521 518 515 512 509 505 503 499 496 492 X = 1(\ - E) E Dif. 0.02797 2888 2981 3075 3170 0.03267 3365 3464 3565 3667 8771 0.03876 3983 4090 4199 4310 4422 0.04535 4649 4765 5001 5241 5487 5739 0.05995 6256 6523 6794 7070 7352 91 93 94 95 97 98 99 101 102 104 105 107 107 109 111 112 113 114 116 236 240 246 252 256 261 267 271 276 282 TABLE XV.- DEFLECTION-AXGLES FOR TRANSITION-CURVES. Transit at P.T.C., n" = 0. Tr. at quarter-point, n" = h /i° I,° (5o° )-i-Ao- -Bo (6.°) - • A^ Br (^ for Bo for^ = ^1 for ^ = n ^-1 A^ 3 ^\ 4 3 n .0 .00 40 go go 10° 12° 14° 16° 4° 8° 12° 14° 16° .0 .00 .0625 .05 .0075 .0025 .0775 .05 .1 .03 .01 .0975 .1 .15 .0675 .0225 .1225 .15 .2 .12 .04 .1525 .2 .25 .1875 .0625 .1875 .25 .3 .27 .09 .2275 .3 .3.-) .3675 .1225 .2725 .35 .4 .48 .16 .3225 .4 .4.5 .6075 .2025 1 1 .3775 .45 .5 .75 .25 1 1 1 .4375 .5 ..55 9075 .3025 1 1 2 3 .5025 .55 .6 1.08 .36 1 1 2 3 4 .5725 1 .6 .65 1 2675 .4225 1 2 3 5 7 .6475 1 1 2 .65 . 1 1.47 .49 1 1 3 5 F- i 11 .7275 12 3 4 .75 1.6875 .5625 1 2 4 7 11 17 .8125 13 4 6 .75 .8 1.92 .64 1 3 6 11 17 25 .9025 14 6 9 .8 .85 2.1675 .7225 1 2 4 9 15 24 36 .9975 2 6 10 15 .85 .9 2 43 .81 1 3 6 12 21 34 51 1.0975 3 9 15 22 .9 .95 2 7075 .9025 1 4 9 17 .30 47 71 1 1.2025 1 4 14 22 33 .95 1. 3. 1. 1 5 12 23 41 64 97 1.3125 1 6 20 31 47 1. DEFLECTION-ANGLES FOR TRANSITION CURVES. 357 TABLE XV.-DEFLECTION ANGLES FOR TRANSITION CURVES. Transit at mid-point, n" = \. Tr. at three-quarter point .71" = 1 n <^ for Ai 2 I ° fiifor4-= 5 3 ^i Bltor'^ - 71 6° 10° 14° 16° 8» 12° 16° .0 .05 .1 .15 .2 .00 .0075 .03 .0675 .12 .25 .2775 .31 .3475 .39 1 1 1 1 1 .5625 .6025 .6475 .6975 .7525 1 4 11 17 1 4 10 15 1 3 8 12 1 2 7 10 2 6 8 .0 .05 .1 .15 .2 25 .3 .35 .4 .45 .1875 .27 .3675 .48 .6075 .4375 .49 .5475 .61 .6775 .8125 .8775 .9475 1 0225 1.1025 1 4 6 1 3 4 12 3 1 2 1 1 .25 .3 .35 .4 .45 .5 .55 .6 .65 • i .75 .9075 1.08 1.2675 1.47 .75 .8275 .91 .9975 1.09 1.1875 1.2775 1.3725 1.4725 1.5775 1 .5 .55 .6 .65 .7 .75 .8 .85 .9 .95 1. 1.6875 1.92 2.1675 2.43 2.7075 3. 1.1875 1.29 1.3975 1.51 1.6275 1.75 1 1 1 1 3 1 2 5 1 3 8 2 6 14 1.6875 1.8025 1.9225 2.0475 2.1775 2.3125 1 1 .75 .8 .85 .9 .95 1. Transit at (S|°) = P.C.i, n" = 1. T o Transit at P.T.C.u or P. flections from tangent cular curve. (5c°) = Y-^e + B C.,, de- to cir- i n .0 .05 .1 ,15 .2 <}> for 1=' ^1 B. for I = Ac B.tor\ - n .0 .05 .1 .15 .2 4° 6° 8° 10° 12° 14° 16° 4° 6° 8° 10° 12° 14° 16° .00 .0075 .03 .0675 .12 1. 1 .0525 1.11 1.1725 1.24 1 5 12 23 41 64 97 1 4 11 20 36 58 86 1 4 9 18 32 51 76 1 3 8 16 28 44 66 13 7 13 23 37 56 2. 1 .9475 1.89 1.8275 1.76 1 5 12 23 41 64 1 4 11 20 36 58 14 9 18 32 51 13 8 16 28 44 13 7 13 23 37 97 86 76 66 56 .25 .3 .33 .4 .45 .1875 .•J7 .3675 .48 .6075 1.3125 1.39 1.4725 1.56 1.6525 12 6 11 20 31 47 12 5 9 16 26 39 2 4 7 13 21 31 1 3 6 10 16 24 12 4 8 12 18 1.6875 1.61 1.5275 1.44 1.3475 12 6 11 20 31 12 5 9 16 26 2 4 7 13 21 1 3 6 10 16 1 2 4 8 12 47 39 31 24 18 .25 .3 .35 .4 .45 .5 .55 .6 .65 .7 .75 .9075 1.08 1.2675 1.47 1.75 1.8525 1.96 2.0725 2.19 1 2 3 6 9 14 12 4 6 9 113 4 6 12 3 4 1 1 2 1.25 1.1475 1.04 .9275 .81 12 3 6 9 12 4 6 113 4 1 2 3 1 1 14 9 6 4 2 .5 ..55 .6 .65 . 1 .75 .8 .85 .9 .95 ll. 1.6875 1.92 2.1675 2.43 2.7075 l3. 2.3125 2.44 2.5725 2.71 2.8.525 3. 1 1 .6875 .56 .4275 .29 .1475 .00 1 1 .75 .8 .85 .9 .95 1 358 TABLE XVI.— TRANSITION CURVE TABLE. oooo oo ooo •JJ T3» «0 GO O T— ) 1— t T-l f— I ^ ooooo ooooo CJ T «0 QO o TO TOCO CO TT ooooo ooooo ooooo c-j-^-ooco ai -mo OD a s^rrtoooo Tt''^'n*Tr*0 iQlOlOiOCO OCO^sOi.^ "s- (?» m- 1-1 .-1 CJ « TO TT coco -rr ooo lO «5 I- 00 03 C* CD ^ 1-- Tf ^ <;< ■* lO i- (?»»-■ — ^ ■?* 0-. 1-1 TO <0 t~- iJ" 00 C< CO C? 05 ^ TTCO Oi 0; CO -^ CO TO 1-1 Tf i.- TO T-i s* <;j CI oi CJTO TOM TO •^ rji rji in in H s CO «0 1-1 1~ CO ■«*< iO t- 00 «0?0 0Di-cO O (?? Tf I- Oi CO c^? TO in C5 o» inoo ^ Tj< in cj 1-1 *» in OOOJCOO^ 0> in TO TO TT QDTOOOTOOO t^ C» 00 !- I- TO 05 -V OCO Ol ffi CJ TO TO TO ■>* -^in o in CO CD t'-O 00 00 0500 1—1 1-^ • .-( 1-1 CI OJ w s (NTOTO-a'O 1- 00 05 O 1-1 »— 1 1—1 cj -^ CO 00 1— 1 1— 1 1— 1 1— « Ci CJ T)« t- TO 0«CiC>TOTO OiClCJOlTO CO TO ^ ^ 10 CDCOCOJ> t- S5 TO -^O QO CD ^ CO '^» 05 Oft- c-i xi rr coco O '^ iO CJ O C: 00 00 CO QC O CV O C50 TO 05 00 00 in 00 00 CO 00 TJ" O l- 1-1 00 CO in TT TOTO TO 00 {- TO CD CO IT lO 05 CJCO y-l 1-1 QtiKn •^m «« CO t- 00 O ^ CJ TO 1— 1 1— t 1— « 1-H in CO ooo 1-1 l-( T-l 1-1 CI C* TO m t- oj — CJ CJ Ci C« CO TO in t- cj COTO TO -^ ■^ ^ H oooo TOXjO O O 00 00 I- I- i-- CO o m tJ> Tj- CO C» 01 00 1- in Tj-cjoooco O Ci Cs Ci Ci 0< TO IC I- 05 CJ Ol !?< C* 5* ClCSCiClO — CO Ol- C5 TO TO CO TO CO cs c: c:; c: o i—TOin I- o ^1 ^1 ^* ^* ^31 Ct GC 00 GO 00 1-1 TO in I- 05 in m m in 00 00 00 1- 1-- ■^co in t-C5 COCOOCO«3 £. OOOO » 05 ->» o ooooo CBi-iTTl-O OOOOO TO CO Ci C/ O ooooo 00 1— < "T i.— O OOOOO CO to c; c* in ooooo 00 -^ 's- 1- ooooo TO C-. oj in T-l »— 1 r-i Ci 0* C* CO TO TO TO 1^ Tl^ Tjiin omo OOCO J-i- i- 00 00 00 05 05 05 0500 r^ 1—1 > o o o TO »! "TOint- 0-*05-i<0» T-* 1— 1 1— 1 c^ Ci in c> CR t> in TO'^ iS'incD rr Tf Ti< in «D t- 00 05 1-1 c< Tf inooD CO 00 -^ 1-1 OS 1-1 C* Tf CO OS t- CO CO CO i- OS 1-1 TO in Tl 1—1 1-1 C* CI CJ CJ CJCJTOTOM H oooo ct-Q0 OTf QOrpr-i t-00O5^TO O5Q0 00OC0 TfCOOO r-TO «o«ao»nTO in 000 TOCO eoi^cooseo OS CI in CO CJ osinTOCJC? m OS TO I- 1-1 1—1 T— » l— ^ri C? C* CJCJTOTOTO TO Tti Tji ■^ in in in coco i> D c» 1— ( 1-^ 1-H r Ci CJ Ci Oi TO Tt ■<*<■* in CO t- J> 00 OS OS i-iCJTO -^ *— 1 1— 1 1— ' 1— 1 1— 1 CiWCCCC'*' OCJTOO Tl< ij<05 05CO 00 i-c ■<»' 00 CO ■^ in TO CO -T ooeo05in cj eoTO is< 00 05 1- in •'S' TO inco •^Oii< 0( in TO CO CO i^ OS CI in OS in 1-1 1^ t- TOODTOOOiS" »-» 1— 1 1— 1 Oi o» TO TO -^ in in CO i- 00 05 -^ CJ TO -^ in CD CO 05 1— 1 1— 1 T-1 1— 1 CJ CI TO in CO 00 CI CJ CJ CJ CJ H oooo i3< CO Xi 1— t ooooo ci -^ CO Qo 1—1 1-1 1—1 1—1 c* C5 05 C5 05 05 OOOO 00 l~ t- t- CO coinmrj-TO TOCJ^OO OOC5C5C5 c*Troi-<35 C< Cfi Cl ~i Ci CS 05 C-. Oi 05 1-1 TO ifi £- 05 TOTOTOTOTO OS OS OS 05 OS T- TO 4 - C3 Tj< -"J" TJ> -^ -rr OS C5 OS 05 OS 1— TO m I- C3 in in in in in OS CSC5 OS 1— CO in i - cs CO CO £. oooo ■^ CO GO .... ooooo CJ rr CO 30 OOOOO C J rr CO 00 OOOOO Ci'S'COOOO ooooo CI ^ CO CO o_ ooooo CJ TT CD 00 ooooo CI -r CO 00 1-1 y-, -,-, ■,-,-,-, ct cJCjcjcjTO coTOTOTO-^ 'jiTjiTT'Tin iniOinino cocococoi^ oooo ooooo ■^OGOO CI-^icOQ0 »— I ^-H rH f— t 1— ( C< OOOOO (N -rj" o OO O (?» T* C< 0« CO ooooo ooooo ooooo ooooo 0JT5000O CJ TT^O OC'O »»T13000 W-rOOCO ^cocooo-a" rKTfTrrio iCOiCiCO ;o;i — "-si- Ss CO«O0»Q0 O •-■ rl «0 CO <- OS 00 C* CO Tj< lO I- oo in CO ojco QDO!N •^O cooinc*o 00 ,-r c>3 CO 05 o — ' w I- w ci lO 00 t— ' *n OOO l-QO »-i OKWcoOin iC Ol CO -» rj- Ol -O 00 CO 00 ^—i irt T-< »-H «o(^»(^J(^» CO CO CO V ^ »ilO 05 «.- 00 Ci O 01010 05 CJl OlOOOOCO I- i- I- CO CO ^ in rr CO CO C^J — O O 05 001010105 •-1 »— O* CO ■^ 05 Ol 03 Ol 05 m CO *.- 00 <» T— 1 »-l *^ »-^ r-1 Ol CI Ol CI 05 O '- 0> CO TJ" Ol C: a CI Ol incc t - 00 Ol <31 Ol Ol O: 3D O 1- OJ CO •* coco CO M CO fe. ocoeoo o T-iweo 5J »- CO 00 lO »f5 t- Oi T-l TJI 5D oi o m t- t- O Tf 00 (N ^SfS^S o CO 00 cc t- TfO CO CO o ^ooi-oin CO m CO s>» o ?o m o: cc in ciofc t-i-t- »-^ T-l ^Oiotweo CO ■^ "a" iftin <0 l> I- 00 C5 C5 o !-> <; J CO y—i 1—1 1— « 1—1 co -^ in cc i - »— « th C^ CO o -f- (^ T-l 1-1 t-1 OJ o «ncooooo» cj (NCI CO CO ■^ Tj" Tj< in in o l' o CD CO t- t- QD CJ CI CO CO CO Tt o CO 1- 1^ 00 05 f— ( 1— 1 1-» 1- 0* 0» 0} OJ S>5 coo* com 05 lO 0» '-< T-l OOOi-4-QO CO l- N 05 I- OCOCCTOO J^OO'-iCD o» 00 i- CO Ol C-i O5 0D 050 CO «n O O -rr C< 2^^f?2 ^ -*CC COOl to CJ Ol 00 l- T-l cvj 1 J CO -^ in l^ 00 05 i-H CO TTCDOOOCO •n GO o COCO Oi CJ CO CO CO Ol cj in CO Of CO TT TT tt in in o: o> CO o iTi in CO CO i- H Ti<«)ooo Cicsoo 00 1^ i> CO in ■^CO i-H OQO O 'S- i-i 05 t- CO O I- CO <3> TT o in o in O O Ci O Oi O* TJ* lO {— Oi ^—i T^ t— I »— 1 ¥—1 Ol 05 Ol Ol 05 ^co ini-Oi C5 Ol C5 05 00 — • CO lO {- 05 COCO COCO CO 00 00 CO c- i- ^ CO in t^ 05 TT TT TT ■ CO rf CO CO CO CO CO fc. ■^oco Ol Oi-iT-iO» cj I- in •>* CO Tj« in i-* Ol 1— 1 1-1 1- CO 00 o» 13' CO 05 CJ CO Qocot-oin C5 CO t~ C* CO cj CJ incico 1-1 CO " CD CJ in I- 1- 1- in CO TJ< 1-1 I- "^ m CO 1* C) CJ 1-1 00 CD ^ CJ i-« T-1 1-1 1-1 OJ O* ococo ij" -^ o in CO CO t- t- CO Ol Ol o 1—1 1-1 ^ CJ CO ij" > « « 1-1 l-H o • TiOifNeoeo ^Mn w I- 05 T-i •rr ini-o »— 1 1— 1 1— 1 1— ■ d CO in CO Ol c> CJ CJ CJ CJ CO CO CO -^ in in CO *-> O o czmco-^io O CD i> t- CO 05 1—11—11—11^ o ■ 00 0»-*CO i-i •v t- 1-1 t^ CO 00 1- in CO OJ 05 I- CO c> 05 in 1-1 CO CO CO 00 1-1 rj< OC0C0C0 05 05 rr O l~ in oicicoinct iji i^int-o 00 tciro in CO 00 CO o t- in Tr TT in i- »— 1 TlQiOinr)! in o I- c» o 1-1 1-1 CO »n CO 00 1-11—11—11—11—1 o c» f CO Ol oj CJ ci CJ cf 1— coco CJl rt CO CO CO CO -^ •^ t-O CO CD rr M" in m in H ■^coooo 1-1 OiOi 05 00 00 t- t- COinrfCOC? T-i O Ol I- lO CO 1-IC5CO Tfl 1-1 00 m CJ CO 0000105 o O o "sj o o lO o» c-« cc o -o x; o CO o ■;? o X C< -^ i.- C5 Si O CO X O LO 50 CS 1! ^ O o OS ir; ?J — 1 ■^X M X CO cj lO o I- o XCOCSiitO t- O TC O X « 1J c: o St 1—1 t-l 1-11-1 .-1 !» OJ W CO CO TJi ttoin 50 CO -" CO CO CO CO CO fc. GC 30 0O ^ o — • lo o CO O O il f t- o ?o o o O CO t Ci l- •>5" C5 1* c; o O t- X CO OJ C* X O so t-1 lO ?J CO X r- CS X I- o o ox OS t CO t- 1- X o oj O CO -f C. X m-o col- 1-— I^OJ iN3«soeOTj< io«n«>t-x X OS O 1- 0» 1— 1 1— 1 1-1 CO ■s- lO I- X 1-^ 1—1 1-^ 1-^ I— 1 es o e? CO -r ^ Si OJ SI St — c ff» CI C» o l' C* W ffj Ci 1-1 o l' 1-1 1-1 o i-iO*Oi€Oeo 00 r-r t J> C? 1^ 1—1 1—1 0* eo-fli-^Tjiio in«ot^QOos o r- oj CO lO • o xmsooo CJ X Tf C: ■* X Ci iO X o CJ CO -^ Tfi CO OS35 OCiCl 1- so O t- 05 05 C5 O CI 05 1— CO JO I- CI CJ 7* ?» ^J Oi X X X X i- — ?? u- I- Cj CO CO CO CO CO r- « o lO m 1— CO O l- OS TT Tf T -VT ■t 1* 0? OJ 5J ■^ CO lO £- o lO ift uO lO iTi a. OOOO TJ> 1-1 00 o C< S5 O CO O ooooo I- m-1 X o 88S^S g§2gS ?,g§gs ooooo I- -T ^ X lO --cj^too •V 1* o o i- t- 00 CI OS o 1—1 ■^ -r- C> CO — T L-; o o t- X X o o ^ ■— 5> CO so t (N St Si SI SI 1 si > a; o Si woo to OOi-'CJ CO TT O I- 00 lo lO t-- — 1 ?o O W» I- OS CO St St -^ X OJ O X 1-1 r)< CO O OS o ■?> oD 11 «r; o -s* O OJ OS X OS X CO I- ■:» t- •^ lO o t- o CO X t OS irt w :>* c» so CO CO T •* o o «n «c o I- t- 00 00 OS OS o 1-^ oooo C5C5C5 0SCS ci OS c; o C5 O ^ IJ CO rr X X X £.- I- C5 OS CV C: OS «-0 O i- X OS 1> O lO o t o o c; o OS c» oj 6* ci o» CO IJ T* ^ O c: OS cs O cs c* c* cJ c* c» CS t- O O T X X X X X O — SJ CO Tt CO CO CO CO so i-i 1-11-1 fc. Oi-i?*-^ CO o c? ■;? o O X ^ ^ i- 1- ^ m TJ ?> 1-1 O C5 ■T CS o CO -t>xo TJ- O O W CS X CO « CO X » -T TJ O 00 OXCO ?jTti I- o o o o ocs — ^-^- J;- 1- OS o -> ■r-< t— 1 1-^ (?» c» ©J CO cc Tt> O 500 l-XOSOO — oj CO -t o O i- X O 1-1 » 1 1 1— « ^J ^^ "" '"""' '"'"' 1-1 j-trsrH O 1 .01 -.01 --0100 ■^ o + 1-1 1-1 1-1 CJ ffj "CX-HTtOO 1-^1—11—1 (N C« ©* (NO* CO so fe 1-1 r» t- o 00 1.'; rf IT o C5 -r -* X ooo t O t- 50 o CO r? O CO so GC r- O ^ X §8g§^ ■^ O OS "TJ X OS « CO CO so X — r- 1- o lO OS so c; i- 1-1 r-i (NM-^oea X o — CO o 1— 1 -^ 1— 1 1— 1 I- O OJ ifil- ^ c» c» ;» c^j O CO OO CO CO CO CO T T o o •«• X ■?> TT lO lO lO O O O lO OS -1- O t- t- J-OO fT OOOO TTOOOO ciox 00 I- o »o ■* CO 1-1 CS I- lO St O I- CO O {- C? t- 5? {^ 1-1 m X ^ ■* o ©< 55 o T- ci c. o C5 c; C5 — CO lO i- OS C» Si S> St St C5 X X X X 1-^0 lO I- C5 COTO CO CO CO X t- t- I- o — CO 1.0 t- cs IT ■o 1* T 1— CO lO {-OS lO lO lO lO o CO •?> 1? — o ^ so m t- cs 0=00 «o o oooo ?iao-<3'0 ooooo o (J* X -^ o ooooo -o ij X -r o 8?,g?S ooooo » 5> X TT O ooooo «a 1? X -* o ooooo O 5J X T O i-i !-"?» CO eo- CO ooooo ff J Tf O X o CO CO 0-: CO Ti> ooooo 'T? -r o X o TT TT -r t lO lO O lO lO sO 82SOO O O O O i- TABLE XVI.— TRANSITION CURVE TABLE. 361 OOO O O O O O OOOOQ ooooo coooo ooooo ooooo OQOO C'-^OOOO T^-rOaC/O 7>-^wGt/0 CJTCOCCO C«-T-ii-ii-ii-iC< c coo — C5C5 CO t.- Ci O 1-1 C> t- -r -f J- t- C* X rj> O c» C5 X o o I- CO o X ".o 1- o o» o o» CO — cs i-«o T-l t^ 1-1 T-i ©» 5* C* 00 CO -T ■'I' o O O O t- X xooo^ C* CO CO -T o H oooo ?* ro •^ o CiCi Ci Ci 00 X t- i-»o-^co cj 1-1 ox I- »o C0 1-1 cs i- Tt o» o X o ooocsc; «5 l'. cc ao c» C5 c; C-. c: C5 O r- 1> CO -T C5 c; c; c; Ci lO ec i- X C5 1—1 1— « 1—1 1— t f— 1 C-. C-. n X X o — ■;? CO -r CJ 5* ;» 5* It X X Xi-l- JO ^ I- X OS C-» O* O* Ci O-f 1- l~l-S'.0 O 1— ■:• CO -^ 00 CO CO CO CO fe, Tf X ao -:> — 05 1.< V5 ^ ^ O O 7> -? X r-i-iS> — x C5 LO »0 — CO O lO -T T T C5 — XC5» Tj-oi-cricj X O t- -f o LO O CO X CO CO lO 5» ■* r- OS jn C> O I- T-. 1-1 CI CJ CO CO -a-jOiO «0 l^ X C5 O rH T-i cj CO Tj- o i^XOr-SO »-i 1- c« c* ^ ■<»• O X OS r- ?» c* c^ e^ CO 1-1 r-l TJ « ■* o 1-1 1-1 ri CJ o oo^J^p ■^ O O CO 1— • C* CO CO T O OO t"050 f-iec »o«ox oweoiot- C» CO CO '*' o «0 X C5 1-1 W 1—1 1— « 1-1 1-1 CJ 0» OJ ot CO CO CO 't' ■<*> mm cc o t* JS; ■>.r05«><0 t- eo o i~ tfs SSS^iJ C5--COOCS «0 1-1 i- «.~ o OJ LO X o 0? X T* X i- i- Tji Lo CD ;c in 05 CO O i- i- CO o 55 o CO cs CO X o o t-iO» ec o « a; o S^iSizS'ol «» O ;0 i~ 1-1 C*CO CO CO ■■?' -T iO u-; u-MS OS o o — ?? o {- X X a» X i-- 1- X o cs c ^ — r» •— T.^ 1— 1 Tl H o -T 55 00 OS 00 t- O lO CO c* Oi t~co OO — Oi-i ■t 1- O 0> eo I* -^ w o t- C8 C L- C5 1-1 cscicinci 1— CO ir; i- C5 cr. c; cc X X 1— CO O I- C5 Ct IJ SJ I* C^J X t- 1- o o 1- CO O i- o coco CO CO CO «ri -T -^ CO ^j — CO L- l- o -T -T T -«T •«" -^ o c; X o »n o o »r; o /C -^ O O OS O OJ — « I- £. oooo 3C I- o o ooooo TT CC' ?« — O ooooo CS X i- :s O ooooo -T CO 7< i-i o OOOOO 05 X i.- o o ooooo T CO ■:> ■— o OOOOO c; X i.- CO in 1-1 J* CO ■^ »n o I- 00 o O O — 1? CO 1-^ T—l 1-^ rH TJ< Lt -O I- X T—1 1—1 1—1 1—1 1—1 X OOi-iOJ ^ — ?) s*o* 00 -r ir; --i i- Ci i7t ;* w a t- X cv o 1-" 0» J>(r» e>3 CO > Vi S^^^ CJ t- ■* -!" o •V «0 l>- OS 1-1 O £~ O t- — -T CO O ffJ » t^ io if: I- o» o CO I- — o osxo-i^os omr-co — t- i- O iC -< t- CO O CO CO O — -r OCD C I- -T '— o 1— t 1-1 1-1 T- 0* 0» C> CO CO ■>!)• ■'9' m o CO CO t- t- X o o o 1—1 1-1- 5? CO CO T— 1-^1—1—1 1—1 H oooo 5> CO T O OS OS CSOSXXX i> t- CO m o ■^00 0»r-iO 00 J> «D -<* 0» 1- O O IT ©I §gS5i? ooooo o — 7* CO -rr o oooo I.- CO t- X o ooooo O ^- T' CO •" CJ C* 0> 7? ct X xxxx m CO {- X o a St Si St St X r- 1- 1^ t- CO CO C? CO CO r C5 — t-X o s^eoo T Tt CS X OJ X — T X CO T- LO CO CO CO X CO o m o» m 1-1 ?» t- i- O i~ i-O CO T» c< — m T 1- CJ SJ 0* CO -^T ^g^^S •r- CO in X o oj CO — CO 0* ^- i-ir- 0« CJ CiJ CO ■* o mcoi-xo c — o» CO -r Ift CO X o o ^—1 1—1 1—1 1—1 c* e» CO m CO X « St ct ct ct > i-iffjco-* o o CO I- O CO CO 0» OJ CJ CO T JO CD l> t- X C5 o o o» -^ Tf CO o o 0» o c>» Si CO eo -^ »0«Dl>XO 1—1 TH 1-1 1-1 C> C* ■f^ T— 1— 1 r- c» w eo CO eo 1-1 1-1 1-1 o» ?» Tj'-ii'-^min X 11 t--»j -r CO X COCO 12 CO CO £, S??.8 XCOTTOJO OOOOO X CO T Of o ooooo X CO -^ OJ o S8?BS X 2 ^?j o gS?S58 ^c»co- CO -f «n CO OJ-X oo ^ 1-1 1- — c» ?,^?l?55! ^^fi^iig OOOO OOOOQ TTCOXO o»-~coxo 1—1 1-H — ^ 1-^ 1— » St ooooo ooooo ooooo ooooo ooooo ooooo c-»-t;oxo ci-r-oxo cfTcoxo ej->i-coxo o»»rcoxo St St St ct ^ cocococo-T ■TiT'T-rin minminco cococccot- 362 TABLE XVI.— TRANSITIOX CURVE TABLE. ooo ooooo o o c: o o ooooo ooooo ooooo o o o c o coooo cj-*i»coo Cf-poooo ot^-^iXKZ) o-roQOo -r/^^^scQcS c?-tS-7o ■r-f TH^— 11— 11— iC^ C?f7iC.t'7.fCO COCOCCCO"^ ■TrTj''^TjJt-iO CO i- ■<*• TT { - 05 00 00 00 00 ;* T— I T-t T-« »-l CJ (7* CO CO TT ■^ »o in CO ca I- 00 OS o o 1— e* CO Tf 1—1 r-i 1— 1 T-« 1— ( ■i— oo Oi-^t^ccoi oooo 7iCO-^ O O O 05 CS 05 CO i.- i'- 00 0> Oi0505Ci05 050500SOS OOOOOOQOOO l-t-i^i^CO O— 'QiSO-^ lOCOt-0005 Oi-ilJCO-^ lOcOt-OOOl 1-1 i-i i-c 1-1 T-i r-i 1-1 1-1 1-1 1-1 a '71 ^ ->{ ot c* f^i n in -^ O -^ -7? CO T CO CO CO CO CO O o f^ CO C5 — o 1- Win 00 in CO -r x> ci i-iino in 1-1 CO o o m i- 00 CO -^ "J 1-1 ino^QOO* 1- 0* CO 191 t- 1— 1 i.^ Oi 00 CO CO i- W 00 ■^ -r^COO Tfl •^ 1-1 00 CO ■v ■>* 1-1 rr cj coco CO TT in i-ii-tOi W CO CO -* in CO t~ QOCSOi- w 1—1 1-^ 1-^ -^ m CO CD 05 1—1 1—1 1—1 1—1 1—1 1-1 CO -"CO 00 W^ CD 00 CO CO CO CO CO IJ) ^oo«ni>- ocoQocooD in-^'?>i?jco O r^i—r-c^a coTfincoi- QOOi-icorf oooocoin t>oeot-o I cj (?} w w CO eo CO -rji i-H TH i-i 1-1 oj cc^iocDQO Oi—eoinco o 1-1 1-1 1-1 1— T-i oj oj CO CO Tf< Ti< Tji in CO CO i>-i>oo0505 1-1 in in in 1-1 c\f in CO 00 in CO ?J ^ CO i- CO CO -t" CO CO ■^ CO in Ci CO in 1—1 1—1 CO in in i- ^ i- CO 00 in CO --Ci I- 1- i- CO OC0 05<-G0 in ^i c? -}< ^ CO 05 1- 05 05 in T m «.- ;35 i-iC CO 00 7? 1-^ 1—1 in 00 1-1 -r 00 »-i 1-1 o* o» c* o> CO 1-1 in CO CO 1* rr in in 1-1 CO oj 00 »0 CO CO £- I- m -^ 00 in oj 00 05 Oi — 1—1 1—1 05 {- in CO 1-1 1— c? CO •^ m 1—1 1—1 1—1 1—1 1—1 OS oot-coinco ot^TOin o-^t-ooi noi^Oi^^o a t>- -^ -rr •■tj inTri-it-c* 00001 OiOiOi aoi ciQOGDooi^ {-coinm-«< coiNt-icjsoo i~^in-^T?o cocO'^i— ct; "^coooos i-icoini-05 1-1 03 in i~ 03 1— ccint-os i-icomcooo o'W-*ooo osi— coinco 1-11-11-11-11-1 Of ciQi i7i ■:> cocococoeo tji-iji-"^-<3< oinininin mcoococo 0000 7J CO ■^ .n ooooo ooooo o I- 3c oi o 1— ci CO Tf in ooooo ooooo CO i~ X' oi o 1-i (?j CO Tf in ooooo ooooo cci-oscno i-iC-icOrrin 71 ys ma coi-cccii-i c? CO Tji in CO t- 00 03 o "7? i-ii-irli-ii-l T-ii-ii-i(J«W CO -f in CO t- 00 o o ^ CO c-fcj CO CO eo •^ in CO f- CO CO CO CO CO CO > u o "SV CO CO coco 1-1 I* CO cj 1-1 CO 30 in in t- OS 1-1 -J" in OS m •* in CO t- 1—1 1—1 1—1 — 1-1 H 0000 W CO -T 10 OS OS OS o:oooot-l> com Tcoct T- OS t- CO -^ c? 00 in CO 00 00 1-' I- 1- inco t- xo OJ c-» o? w c» t^ 1* £- 00 OS C3SOS CO i- I- 00 OS OS OS OS OS OS '^ 1-J CO TJ< OS OSC: OS OS in CO I- CO OS OS 00 00 00 00 -^ ':> CO Tj< (N OJ C* CJ (M t- CO CO •— ?J CO -^ COCO CO CO CO fc< T) Of- CO — ■?! -r l- in CO coo 1-1 rf CO coos 1-1 00000^ m 1-1 OS CO in c> 1- 00 in t' •Tfcoco -^ in inos OS ir in I- OS •;* CO 1-1 CO CO 1-1 in CO o-j OS Tti Tj< OS OS rr CO rr OJ 1-1 1-1 i-< r-t 1— 1 C? C-J CO -^TT 10 CO i>OOOSOi-" T-I 1-1 0» CO in CO 00 1—1 1—1 1—1 1—1 1—1 OS ^ c» -I" in »-i C< 0> 0> CI i^ OS —1 CO in siMcoeoco 1-1 1-1 C» r}< CO 00 1-1 in OS iji l-r- — 1 00 OS 1-1 OJ CO in i> DJ T}* CO OS c» •«r eocorffinco CO OS ■-H CO 1—1 1—1 i-H 1—1 1—1 c»c*(Nooeo 1-1 1-11-1 cj cj CO ■^1^ 100 CJ Ci (Ti CO CO «ocot«ii>ao ^ t~ m CO 1-1 TOOO OS CSO 'S* 1-11-1 1-1 I- -"J- I* CO Tfi OS 1-1 in I- in I- f7> ,_, CO 1— CO CD rf> -* 1-1 1-1 CJ TfiOOCO CO l- X) ^ i.- 00 J- 1- 00 in ■ CO CO H OS OSQO i-O -J< l^ CO OS in m OS cj inocot-o ■<*» CO TT CO I- 00 OS 1-H 1-1 o CO -rr m CO J-OC OS 1-1 1-1,1-^ 1-1 7i 1— CO -^ in cct-xoso C< C* C< Ol CO ^- '7J 00 -n* lO CO cc CO CO a: 0000 ooooo ooooo TjiCDCOO C*-^CDCOO OJ-^COCOO 1-1 1-1 r-1 1-1 r^ CJ Ol 01 C* OJ CO ooooo OOOOO C» T O X O "M -r CD X O cocococo"* TrTriTTin ooooo Ot -r :D X O kT^ iSl iC 1^ ^ OOOOO C > -T CD V ^ CD CD CD O i.- TABLE XVI.— TIJANSITION CURVE TABLE. 3G3 oooo o o ooo ■rj -T oooo 1-H T— 1 1— t 1— 1 ^^ ooooo C* -V JO 00 o C< W 0< (?? CO ooooo C< TT O OO O CO CO CO CO "^ ooooo Oi TT CO X'O ^r ^1 ^^ "^ in OOOOO C» T CD U5 O in in in in CD ooooo Ci TT coooo CO CO CO CO t- > o 00 1—1 1s> I- t- O {- O T-iCO Tf 00 O*!-" CO 00 o^t^T^Sot OS O S OJ CO t- rr O i^ TT 00 CO i- oj o 1-1 o: I- CO in i-iincic»in ^ CO CO CO CO 1- o Ci t- in T} Ci 1-1 1-1 1—1 coco 05 O* lO 1-1 1-1 1-1 W 0* H 1— 1 H 1— 1 O? eo 00 -rj" •<*■ JO Ci Ci Ci CO CO lOCOt-t-OO Tji Tji -^ in in 050th Ci CO 1— 1 TH 1—1 TH a^ O<»C*Q0 «5 CO Tf I- T«> O CO OJ t~ Tl in 1-t 1-1 C'* CO «n I- ci o* o 1—1 1—1 00 1-1 lO 05 CO 1-1 0» O* « CO OCCOOOCOC35 CO -^ Tf lO »o in 1-1 00 in o» CO £~ I'- 00 05 Oi I- in CO 1-1 05 o 1-1 Ci CO 1—1 1—1 1—1 TH o C5 00 1- 1- Tji Tf in CO t- T- 1— 1 1— 1 TH l-< U C5 05 00 CO «n coo CO 0» I- N iO OOOi-iOOO m c^cD o oj Ci 1-1 CO TT 00 O 1-1 OCDtH oocscn C5 C5 C5 05 O >— CO lO l^ crs 00 00 i- 1- CO T- CO lO J- 05 in »o -«• CO 1-1 1-1 CO m I- C5 CO CO CO CO CO oo: i-coij' i-OJ-tcOOO ^p ^5^ ^^ '^Jl Tj^ Ci o i- in Ci o c> CO in I- O I- -^ O I- «:3 o Ci ■* in in CO CO CO CO fi. oooo ooooo 00 i-( r»- l- O ooooo CO CO C5 OJ iO ooooo 00 »-i 'a- 1- o CO CD 05 in ooooo 00 T-l Tf i- o CO CO osciin ?> M »0 ^ t- CS O ^ CO T-1 —^ 1-1 ■^ »no 00 05 1—1 1—1 1—1 1-H 1—1 o o» CO -r CO O* 0» Ci (W Oi 1>Q0C?5 -r- OJ C^ CJ 0'} CO 00 CO m CD I- c:5 CO CO CO CO CO o T- Ci -I^ 1-1 1-1 -^ CO 00 i-i ■<*■ i- o o cooo TH lO C5 'S- 05 CO 05 05 W l-- -^05 »n ooo {> 05 in -^ in in o? o CO CO 05 I- 00 — t- ■<*< CO Ci Ci 1-1 cooo coo o TH 1- Ci CO TJ1 T-l rl T-1 c< r^} cj CO CO ■* -^ incoco i- 00 «s> cs o T-l 1-1 Ci CO -^ in CO t- 00 OS o 1—1 TH TH TH Ci H oooo C5 05 05 00 00£> J>CD IC Tj 0> CD CO i- 00 05 C:5 05 C; C3 C75 O -H o* CO r}i 1—1 rH 1—1 1—1 T-t 05 C5 05 Ci 00 in o i~ 00 05 1— 1 1— I 1— i 1— 1 T-l OO 00 00 00 I- O !-• 1^ CO ir C* o C» I- CO 05 CO -T in Ci o CD CO TH O OS 05 CO CO in o Ci O TH CO CO C7S T-iT-iotQin TTio in CO t>. OOOi-iO»CO T--1 1— 1 1— 1 1— 1 in CO 00 05 T-l 1—1 1—1 1—1 1—1 Oi CO in t^ coo Ci Ci Ci Ci CO CO in t- 05 TH CO CO CO CO 1^ S; coin 00 o y-i T-{ 1-1 a so o i-ilO o t~ ■># 1-1 T- »j ?» CO c» o» «* in 00 Tji in CO t- 00 ON-^tcooo o Ci in 00 1-1 Tj<00i-i»0O r} 1-1 CO lO I- OS 1—1 1—1 1—1 1—1 T—1 00 cr; CO I- I- 1-1 CO lO I- C5 c? Si ci o» ci CD in -h -^ CO 1-ico 1': <- 05 CO CO CO coco 1-1 o crsoo CO 1— CO -f cooo TJ1 1^ 1^ ■^ ■^ •rti CO TH CR CO o Ci f m I- in in in in in Tti thOs ooo 05 TH Oi -!■ CO in oco coco oooo ooooo C» -J< O X/ o OOOOO C* -^ CO 00 O ooooo O* TTCOQOO Oi rr CD CO O OOOOO OJ -^ CD 00 O ooooo Oi -^CD 00 O •:< CO ■<* o t- 00 C5 O 0* 1-t 1— t CO -^ lO CO CO 05 O 1-1 O? f T-c at -1- CD on o CO CO CO CO -31 ■<9> •"iji -^ I*! in ooooo C -^ O 00 o in in »n in CD 8?8S8 CO O O CD I- 364 TABLE XVI.— TRANSITION CURVE TABLE. ooooo ooooo OOOO — ^ r-" 1— t »— « *— I t— < 7> O? C? T* GVJ ooooo oi -^ 'o ao o oieo oc ecioco -^ ooooo CJ -^ CC Xi o ■^ "^ "Tjl TJ- 40 OOOOO (>j -p a: 3c o in If; iO lO 13 ooooo C^ ■«■ O CO cs to 13 ?C 50 l- "bi 05 ^ t-OO o ?»«»o 00 i-i -^00 «5 05 ■;> o o in 1- 00 00 in n in C5 X o to 00 in CO c) o to o 00 o in 05 C5 X 05 05 ■»}< to C» — C? o 1 CO in i- CO rf in 05 in OS cj in 00 cj l-C 1-1 i-c (?i ci CO CO •*■ o in to I- 00 05 CR O — C» CO 1—1 1— 1 H T— 1 in --o i - X OS T-1 H 1—1 1—1 11 825??^^ "ii ?<00 rr iO C5 C5 OlOO l- I- in Tp CO '^ c. I- in T» C5 to CO 05 Tf o ino Tf CO n Tf t^OSO n lO CO i- 00 oi 05 OS O Oi O o I -:> CO rr 1— < 1— 1 1— ( 1— f 1— t 00 CC 00 X I- in tot- 00 C5 1—1 T— 1 1— « 1— 1 1—* t- 1- to to to O " TJ CO -^ in in Tf> CO CO in toi-cc 05 cj T-. o o OS O n li CC CO CO CO CO CO CO fc, 1-1 rr t- -^ O 0» OS I- o i-i 00 CO t- o to OQO O -"S" ococc r> o 00 CO C5 O Tf to — in t- CO o* o» (?< CO in t- -^ 05 CO -^ QO CJ CO OJ 00 CO o in X OS »n CO n o o *"■ i-l(J* 0* so TT «n CO 1- 05 o 1— < n CO ■* to X 1— ( 1— ( 1—1 1— 1 1—1 O ff? TJi to CO Oi ?J o< w w ocomooo CO CO CO CO -T CO to OS 'T? in ■f IT IT m in > .01 .02 ■^ t- O ■<1< o 1-ci-iCJ i^ in c» o CO (?» CO -^ in to 00 c c» -t t- CO 0? in o CO t>. n O n to c» 00 1 oj o O e i-i CI 01 Tf o tocoocoto 1—" 1— • 1— « 1—1 1—1 1—1 »—« Cf -?» CO CO 'T n Oi W IJ CO •^ in to to j^ CO -14" ■^ in in 00 OS O n OJ CO tot- 00 OS 00 1* cot- 00 <» n n n T-l -I ,1 -rt n fe <35 O t- 1-1 •-t o i-Hooo in l^ 05 05 s> com i-iO CO 05 in -^ to ?> 05 05 ej 00 00 OO -r- Tjit- li t- to CO CO ooooo CO n CO I- CO CO m CO CO CO in in o I- 1- OS y^Qi^ ?o C5 «-' m XI t— 1 1— 1 1— » to — o ^ cj c? '^ CO ^^ to 7 J a: in 1* ■^ »n in to i'- o {- in CO o> CC' 00 05 O 1 n n n O 05 O OS cj CC CO Tt> in H 11 n n n OS O O n 5^ COQOOSOn n n n S< C^ H OGO I- lO C'j ci in C5 10 to CO 00 t- ■* o 1 1~ C» t- Tj< OS ?? CO n 1- n CJ ^QOnCJO iSSS Oi 05 05 GC 00 ■I-I CO in i- OS 11 11 11 1—1 n I- 1- o in -^ •^ CO in I- 05 CO 5? 11 05 i- ncoinoQO CO CO CO CO CO ineonQOto o TJ Tf in I- •^ -"T -"T TJI -«< CO O to CO 05 OS n c» •* in Tf in in in o in o to n to I- OS o c'» CO in in to CO to 2s ■WOOtTO ttXNOOTfO tOWODTfO tCNOOTTO to WOOrfiO CO 0* 00 rji o cowQO-^o CO •fl'ooo OS 1-1 r> Tf CD n 1—* 1— ( 1—1 t- 05 O •?? -^ y~ y-i Ot a* at in i~ CO o "Tf cic* w coco CO in to CO o coco coco -^ — CO T to Xi •^ ■* -T Tf Tjl OS 1 C» -f to T»< in in in in CO X C'J n o n CO in CO oso -^co L- en CO to o in ii-< m CO IT OS T o; in rJiCSX — X 0O(r»O5OS?J n t- -^ G-J OS t^ to -^ CO CO OS OS CO o o oi cj CO -^ in co OS i-x ■;> to I- OS ^ TT S) T-inO* Ol C> CO CO TJ1 in in to i- i- X OS o n o» ci? rrm to t-- X c:5o w CO > U ososoDX t-t-mTjioo 'Noxto-^ cjostoeoos innt^cjt- (jjtoocoto oo o o wco-<9"in OOSC5050S 050S0S0S0S C505XXX OOi-l-l-tO COCOt-XOS OiOJCOtJ- inCOt-XOS OiCJCOT nilniin nnilnil Ot '7^ Ot Oi Oi tc to in in -?< in o I- X OS CJ I* o» w w -t- JO CO CJ n O n 0» CO Tf CO coco coco &H CO t- iC Of ^cotoo t- OS O 05 CO -^ OS to 5» o S?Sgg2J tOODOSX-^ CO to o in n xooto — t— ifl "^^ 1— 1 1— 1 CO 'T' OS CO -* n (?? CO to OS CO OS O^ CO X CO I- CO OS in 11 11 nW CO •^ mo CO I- OS Onco -^ to H H H H Tl t- OS — CO in T-cnOJOl!?* t- OS — CO in cj wcoeoco X o CO m X C13 -rr IT "^ "^ cj-ittDO ino»oseDtD O n nWOJCO'^ CDt-XOffJ iS'COODii^ t-0'J CO TT too to t- o in o» ooos -r ^ o» — nCO to o to CO CO to CO ■^ OS -f OS m ojTToxo s^?;sJJ X — o m I- OS 11 in 11 o I* to r>*'» tn C} to CO OJ CO n ■?* -* in t- o CO to H n irl OS CO «- — CO ^ C< M CO CO n to 0> J> rr rfi TTin in to o t- -^ •" OS I- t- X 05 OS to in CC •;> ^ O ^ 7> CO rr O OS O 05 05 in in CO I- X OSX t-OfnX TOSift-O nCJuOStO -"int-Xl- Tj-OTftOtO -TO-^tOin 05 05 O5O5O5O500 ■*to4.-C5 ncoine-os xr-t-toto in-ttOO rfiXCltSO TQOOJCOO "^XOJCOO -"^OOCJCOO -^QOCJCOO xosnOiTfi intoxo5-H 1—1 1F1 H 11 1—1 1—1 H .0} oj CO m to 00 0*0*0*0*0* OS o o> CO in O* CO CO CO CO CO »- OS o o* CO CO CO ■'T •n" CO TTtOl^OS ^J* '^J' ^J' '^Ij' ^^ oooo ooooo TCOOOO OJ-r-OOOO 1—1 1—1 1—1 1—1 1—1 o* OOOOO Oi TtitOXO O* O* O* O* 03 OOOOO o> -r to 00 o coco coco TT ooooo 0» -T tOODO •^ T»i ■^ -T in ooooo O* -f to GO O in in in in to OOOOO o> -^ to X o to to to CO I- TABLE XVT.— TRANSITION CrRVE TABLE. 365 Ioooo ooooo ooooo C? "* O 00 o Oi c> r>t cj CO ooooo (T> -^ ^ 00 O CC CO CO CO "^ ooooo W TCOGCO ■^ -^ Tf •<«< o OOOOO oj -T o 00 o lO »0 iO »o o ooooo CJ T «0 OD O ;0 CO ;o ^ (.~ 00 CO CO OC OS ■^ T-1 00 lOCO 'oo ri ,-1 Ti ni ot n-^-^ir>io i-ooooi-' cjcorfici-i lOOi-i'-'ira Tfccsot^'?? T-IC00505— < c;J„^^^ eieoict-o eocooscooo ooosocico 1-1 t-i 0< Ci Oi to •^ «o I- c* c* i-OJ t- eo lO COQOOi >- ei Ml O o O V V oj oiocoDi-o »ocoi-icio eoocowoo eot-f-cinxi oiwco-^rji eoi-iostoco 00003 0505C500S 0050GOOO coco I- {-=5 ^eo^-* iooi-oooj o^otn^ ooi-jocs 50 lO in T)< CO O >— C J CO TT W Ot (M »(?o ^;j5Sooo co<»05CJO omoiin^ I-Tfojooo ooi-i-xo ,-i »-(»-•>-< cici oococoTtm looi-ooQO 050»-i'?jtj< O <^*''*'^*' SSSSw eOT}ocoto cypjuj ^^„ oieoeoTTO jowi-oooo cro.-';-'•<»< 1--OS* SSSS ?^coini--CT> T=.coini-o-- ^'>}-3''oco o^co^so i-osocjco »o«di-050 ■n-lCJl. Wi ^1;^^^^ OJC ?o 00 o in in m m to Ci ■* to 00 o to to to to L~- U-igJ TT to ■* OO l~ ■^ o 05 m CO T-tTiOJW cocoTfino toi-oDOio i-i«-?cortn-^ mo eo t- 05 (TJ ■* into to to in O O Ci 03 03 in to I- 00 oj O O 35 05 00 o 1-1 : -^ cc 00 00 i-J- in to I- QC 05 t~ to to i.-: lO O " 0< CO rr -^eo (MC* 1-1 in to I- 00 05 O*Qi0tQtOt Q 05 00 I- to O O " CJ CO CO CO CO CO CO _.j_,4..^ oDto^cog* i-iooooi-- <-i-toinco i—csi-'^Q SItcooco aDincoc»J00tJ 03 to CO ■« in in to I- oo i-i-^OCO-^ l-i-i-ftOtO m o; I- cC' g? 05 o CO Ci 00 CO to m o: 00 O ■» »-i O TJ 05<-toinin ■^icmicto ocosi— icom 00 o3 o 1— 1 2» CO "^ m to I— 00 05 1— 1 1?» r? 1-^1— II— I 1—11—11—11—11—1 1— '1— iC^C?o tocootoo •«i'tot-tOT>> 0'*t-t-in i-imtOTt-o 050500 tocootoo •«i'tot-tOT>> 0'*t-t-in i-imtOTt-o -rr -^ 'it t^ a> 0050505 0505050000 I— tOiT^CO SJOOOtO-^ 0J05tCC0O tOgfOOeOQO ■^-Si-os T^coin?-c:5 i-.coint-05 i-co-T to d to i~ OS o c* in ic in to to t0-<*©»0 00t0T>>(J?O OOOTtOO QOtOrfOJO QOtO'^OJO OOtO'^WO ih in ccint>03 oofi^tooo 05--coin{- 1-11-1 1—1 1-1 1-1 1—1 ©J i^ o* c* 00 o cf ■»*■ to I- 05 1— CO lO ©» CO CO CO CO CO CO "^ !*■ T tOQD< rji ij> 1 IC < - 05 ^- CO m m m t£ to OOOQ ooooo TptooDO o»irocoo 1—1 1—1 1—1 1—1 1^ g* ooooo ooooo ooooo ooooo OJTtOOOO *»'*O0DO OJ-PtOODO WTPtOODO ©lOJCfOJCo cococoeois" ■^■*i*"Ti'in ininmmto oo ooo o> " » « o to CO to to {- r^GG TABLE XVI.— TRAXSITIOX CUKVE TABLE. - - |35?| — 1- — ^ 5<< CJ 3>J ?? T> ro •?5 ^ 3 X S CO « r3 CO .n" i^SxS rr — -r r' 1.-3 1.0 lT. lt: to 00000 rj 7T to X p > o T "si Tf — --C l~ — ro LT X ■?< i^ ?< i~ ■«»• tr; s» o ■^ cs T-i o I- « in CS iQ £- W rr 1.0 -^ .0 •rl-O ^X — — TTO CO X CO X -^ C50 CO to ■-- Ci to ■7J X in ^ T- Jj ?* eo TT TJ. O » l- X c; e — CO ~r '~. i- X c» 1—1 0) T in £~ at ^7} a OJ f,^u^^ « C5 X J-iS-J- ?JOi--!- = O ?> i2 O TT I, ~ 1-1 ri « 0; to coo rr t~ 7J CO iS 2 i- X C5 c: s; X X X O — 1? :o ■* L^ £- -.O 13 1.': L- O i- X S5 1-1 OJ CO -* C>7* NO»S* X £- to TJ. in in to £^ X CI W C< 5* 5* co^o X to C3 •" 1— 7* o> CO en CO 00 &H »o -^ •v ;s Ci ?J M Tl< O -O X t-{- — X r? c: O L.- r- 00 C5 .-^ W in X — L- CO ?? m X — ^ 7j i- ci I? -T l-^ 1- -^ 7* ;j 7i .^ t- OJ I- CO ci o> m X « C? CO CO CO "^ O? X £- OJ CO CO CO d x' 71 d d •^ -3- uo m «o in CO X CO X 1-1 c; 1— d f- d d • O 30 Tj> — ^ -> ;i ■'S' o X — X t- CJ m -^ O I- CS TJ O lO X r? =; o 1-1 — 7» ?» ^?' O CO I- 5» £- 1- IJ 7J ec CO CO c; -.o -r JO ■^ -r lO — £- Ci 0> CO 0? ?J i? -- to X d d —^ 0? CO X OJ Tfi CiCOX-^^ CO 1.0 CO xd 1— « l-H t— t 1— • 7? CO X 7? 1* — .— ' 1— 1 1— 1 — — 0? o» 0? d 7? CO CO CO ;a ■« o r? CM~ O JO I- ?J i- ^ ^ — ?» w C*^ * '-^ c^ ^^ 7> Of o CO -;> OJ X X CO ?? ci e- o ^ o ;o i>" X C5 o I 0> IC — 0> to CO CO i- X d -^ — 0» CO -rr to COi^CO0l« d d x — ^ -«' to — t- I- 7?OC: CI £- ■— m X 7) CO m to £- 7J 7> 7* 7> 7» cix-o c? X ;j o o c; X X I- » •^ ^ ~r ifi -^ O -r^ •:>=:' X — TCi-l-X c: 0? -^ :o X O CO -6 -o 0) o o> -^ .o i- co CO CO CO ec CO T-i ^ t- 05 X -r d CO £-^ OC — CO T»< CO d £- — C5 — ' ^ £- — ' to £- X — -r •^ T in in CO 7? to -T X CO -f -r -r CO 7J CO -fin to »n in in in ».n 2. XTJOO '^ X C'J :C O rr u; r: r *-^ 1— 1 r^ T* T^ f 00 ?» o o ?c X — CO :o 7> T» M W M X d CO lO X eo TT .3. — Tj. -r X 7J CO oj 01- iiO uo to ."S- X ?$ to o> -r £- d 7> CO CO to CO £- t X 7> to -r to SJ ^ t I- £- £- X 06 > -J o -^» 5s i.'^ to « ?- o ■?-H O O lO ^— 1-1 TO O ~> O X L- CO .— o CO 7? LO CO <- OS Ci S-. -- 'T to OJ 0> to CO.OX-.-T -1" uO t- £- X 7> i- " CO c. -r s; s: ^ i-CC X -T »-n-H ^ ;» « CO TT L'J O t- i- X ci — 0? CO -f 1- X 0: ■— 7J -r 1- 7J 7* 7J 7J £- X — CO 7J 7J CO CO r? H o 7> ■n T T c; X i~ -.i L- ci jrj :r; ~. « «--: :o I- X Cl CO 1— * Ci O CO C5 si X X X O ^ "J CO -T O'-O 7J{-— . X i~ t- d d i~ ;s i- X C5 LO cr; — CO i-o -r -r CO oj — OJ CO -T Ot 0» 0* 0> to to un -r 7» In 3 Si-x OJ 7> 7J 7J 0» c; m — to S-. 1.0 -^ CO -^ 3: C5 — 7' 7? 7> CO CO CO CO &H ox ->o ?* ci o -o ^—1 »-< o r? X o {- rr — o — M o OiCO — IJ-TJ tS — t- -T ?» I- C5 O 1? TT 1.0 — X -^ 1.0 1- 0? CO to to x"o -:> 1— ^ T* C» 0> cox -OlftO in — cs I- CO {- 0> d X CJ CO CO CO CO fCS^XCO <- X -^ -r C3 ^ -r x' d •.+ ■» TT TTin ^0 X 3=!^ X 7? d d d m CO to to {- ■•—1 o l- C» X t~ I- Oiftcoirt O »--: ;i> X o CO « c; CO t-^ CO cj X «n i» OXt-£-t- X '7? 10 c< CO •* ;o C5 'NlCC5-1>CS ^ ^ -. CJ T? ■~"7* 7>C0 Tji TT m » I- CO TJ. -<9. CO X C5 1- 7* t- t- OD CS Tji m I- X 1- CO -f m £- 7> 1* to X 1— » 1— » 1—1 1.^ 1-^ 1— 1 1-- 7> 7? 7J 7» 71 ro ■?» o n X X X {- ;o r? •^ — -:» u- ro gi^::^ X LO i.O m lO i~ I- .- Oi '-i OJ lO CO iO I- I- I- I- Oto — -r-r 1-1 1- I- c; Tj. Oi I- {> £- to ^ t-i CO I- CO " s* ■«»• o C5 i» -o o o '-' i-i 5J CM O =C ?' Ci -o CO CO rr -T O co -H ox X CO i-XX S5 £-i- X X s; 1-1 7J CO -r i'- X ?. p 7> in X — . m 7> CO -T to i- 7>7J7>7' 7f i7 CiXO TT O 1" SI '-' ^ o o o ■:> •^X 7» CO »n T X Ci m X CO ^ 1-1 £- C: to C5 I- O ci c; ci '3'«nt-C5 C-. C-. X i-l- ■r- d l- {~ C5 CO .- CO OJ o — CO •--: i - ci 0} 7> 7) 1J T> X L- CO I- ":• -r -o i- CO CO CO CO CO CO cc uo in c: oj — .n CO -r T -^ .* o-r X 5J £- X ~. — 7? ".- -r T in in CO ^ 3 t*- X in i-o in i-o »n Ch .*oxo OJ-VOQC O o> T>< o 00 o C>TJ -roxoco CJ 7J 1J CO CO CO r; COT -r CO X OJ m ■^T 1^ in in in £- Ci — CO to in m to CO CO Sf^eV-J- - 1-I oooo §§111 cj cj •?; cj CO 00000 o>-rto X CO CO CO CO •<»• o» — 5 X ■Tf ^ ~r -r •.'. 1-0 in in to iiiii TABLE XVI.— TRANSITION CURVE TABLE. 36^ oo ooooo ;— oci~ ooooo ooooo ooooo ooooooo • C* CO -T to Oi-XSSO »— T* 00 -T O « I- X Oi O — 1» PI — v- -^ I- .»-iT-i»-iT-. .-•i-i.-ir-iS'* ©»o»5*ff*T* o»2*7*~*w oocc©?ec«orcco Si too I- oo If 1-1 woo i- CO i-O'T'—O O— -TOO -r-^OOO -^OXCii-c^O ^ w 00 o to X o ij -J" i- o ■:? o ci T* o o « x •:* to i-" to •- to c* i- « x t o i- cc o .,_( ,_,__^2j 5*7t?rjoeo ■^Tooto cdt-t^xx cjcsoi— i->?»;» H OO o -^ o ■^ o o o o o it o -r cj -T o I- X X C5 o -rt-X t^ X CO O O — ^ T» X r? I- t> i- C^ COCO-T— • o u- to to r^ i- X fe, CO — soo • Tj-oo-rcc t- o cc to o 1—1 1—1 1—1 *? osocci-to T c; -r o o Ci 5J TO JO -T X -r 1" o i- — X O 5J O O O to t- X X d d r-' ? j 13.50 14.01 15.09 10.81 17.97 d o 1-^ 2* -r O t- 1- I» ?< ct ct V « * X — TJ OJ CO T O O «C<3-^»0» t-xcsow « 5» (N CO 00 •^lOt^C5i-i 1— • 1— ( 1— " 1— < 5? CO ■^'*i9' O 00 CO 00 1— -^ t-- o oioicicocrivi •*' oo tOcDl-XCl X » 2.93 3 99 5.20 57 8.11 9.80 11.05 13 00 15.83 18.15 oj -r ~> o iJ CO -:> o o o o CO -.o X ■:> (NIJ-IJ TJCO O — 5> i- to 0? to 1- I' o o X •:? o d CO COT-* -^ C5 O O t- CO T O I- O O CO I- — to d oooot- „ H OC5 o'ci CC-V X X I- o ■<»• O O t- X ci c»oi-- O O I* 00 ©J o o rrX5*cOO 's-xoitoo ■*xc»coo~x O C- X O — '7> •«*• o to X o — . •— ii-i^i-i w CJ CO o to X CN ;j TJ 0< 5» o o ■:) CO CO -r to I- C5 T O X ij* 1-1 ^ CO O X 1-1 .^" ^ ^ .^ 5» oo wo — •^ i- O CO I- c» ;j CO CO CO c: X cs — o o T X CO I- ■T Tf IS- o O CO to I- 1- X p k"! r 2E £r *- 3c {- 0?X CO o o — a6c^.c^.~i6^ oi 1— 1 1— ' 1— 1 1— 1 H OO 3* tJ o oo o o d -r o CO CO CO ■^r "T O X X I- t- O O O O I- oooTreo d -c" d t' d t- X X o o 104.2 109.1 113.9 118.8 123.7 128.5 133.3 138.1 142.9 147.7 ^5*05 1- — 1-X o> f-^ — ' to' — to d o o to to I- i- X ^ o t- X CO — CO o CO O CJ O X 1— 1— • 1— • X — X OTt< ?* I- — to ■?> ci •:» r: CO t 0» T 05 X O X' -?■ o I- o •V o o to I- 8.26 9.00 9.90 10.77 11.07 1- o o o CO CO o to O I- C? CO -^ o o* 17.85 19.00 20.18 21.. 39 22 01 23.92 25,23 > 5 — OJ CO o O to O ?J o o T- .^ T-( C? CO ?? -* o T^ O to C^ 1-^ C* CO CO — o coxes 1— CO 1-* 1-H CO i^ X o o 1-1 Ci ?» C» C» Oi ^ CO O O X TO CO CO ^ ^ C 7> T- to O ^ -f Si 5J si si si 00 CO O O CO to t- t- I- ^ 1.21 1.89 2 72 3.70 4 83 — o 1-^ X 7> ?• X i-X X 1— « -. ^. ~. ^. ij ;* 5/ 5i 32 81 35.95 39.23 42.01 40.19 49.87 53.07 57.00 01.00 05.85 CO X 0» X O — -T T-. O 1- i~ O -f -T d — o CO X CO X i- L- t- X X O C-. 1 « oo O d O XI- o o X ci CO o i X to CO X X o 1— o o o o ;- 1- to i~ to<.-x o 1— • — ^^ »— 1 CO to X o ■— o -^ r? CO ■:» O — '' CO ■•?• »- O O i- T i-'dx'<-d o to to i- X St SI St St SI O O O CO O t- X O r^ C» O X to -rf 5j f? CO CO 00 CO CO o CIO o to' X — — t-o ^- CO CO 1—* t4 ^^ TJ O X O X 55 — •Wl-O *> CO "— to si St cj ci CO o o ;> o t- X d — -> Cl 3* Si CO CO X — -r I- o 00 CO CO CO CO CO to O C> O X — O — 7» -• 1- t; X ijj- 1J1 -VT T -^ -V OO ooooo ooooo TO to i- X O C: T- o> ?r — 1-: oo ooooo ooeoo ooooo ooooo ooooooo c. ~ T- o> ?r — o tr . - X c: o — '^.f co -r o to i- x o o — ■:» rr — i- to jt ^ 1 1- ~- •- ^ — ^ .-. c> c» S» O* 01 SJ St St SI S> T-: r? r? rr rr rr -: c-- 368 TABLE XVI.— TRANSITION (TRVE TABLE. •-« ooo M ■^ O ooooo ;^ i- ao oi o ooooo — ?> CO -fin ooooo ooooo 1— OI CO •n- in OI 01 01 01 OI ooooo CO i- X 35 O Ct Ol 7^1 01 CO OOOOOO — 01 CO -T in CO '?0 CO 9Q 00 JO CO > o o "b) i^r>u I- T*< -JO L-J O •"J- O 3D O 03 t- O GO C» Oi in 00 .— in GO QCC5OJ00 5O 0» O i-c O O CO X OIX CO in o CO 1-1 1-. CO t^ o m OI CO 35 CO 0IO5 o o o CI CO — O CO O I- 1* 01 1-H f— 1 ^ i-iO? Ol t^in ^X'j<^t- OI X CO X 01 CO O-TCi-f 05 CO CO CO "T "T T 05 ^^ 35 •I' in o CO <» I- 05 T»< 05 00 GO {- CO 00 35 35 CO X CO t- OI OO' 01 {- r-1 to 1-1 O 01 CO CO Tfi iji O Tt 35 00 X 0? in m m CO CO t- fc. "* 00 t- ^^ -< C5 ■?» O i-H to in o cs 05 -^ ^ I- 00 o X) eO GO GO o» •— 1 eo T COCO CO in ■-< X t- o CO -*" in J.-0 f- 05 CO t- OI 01 in 05 CO X i-iinci-f oo CO X ■«• o I- •* 1-1 '-TJC* CO CO -^ in o O £.- 00 35 O i-i 01 CO ■* CO 1—1 1— H i-H 1—1 1—1 t- X 05 •rt 01 1-1 1-1 1-1 01 (M ■»r in t- 05 o 01 CI CI 01 CI CO CO « V O ?o*j GO inrfi ^ C« W CO T>< msot-oso 01 Tf CO X o C0C0 05 0J10 OS CO t» 1-H CO 1— • i-i^i^oi CO o Tfinooco T-C »-< 1-1 OI GO •»*< 05 •n 35 -* OJ 1-" S-Si2g?J m i- CO 01 01 t- 35 00 05C0 00 O I- X CO IT in coo •* CI CO 1-- Tt -r in o t- m m -o X —IIJ CO irsoaoo 1^ CJ "* t- O CO 1- .-H-H »? ?{ « O CO oo 0» ?i CO CO -^ TT X CO i- OI TT ir m in CO t- It i~ -it X CO I- 1- X X ^■oini-f-co 05 05 O — 1-1 01 r^ H OCJ QO e- •« CO o t- CO X CO i- O OJ CO CO r» O I- CO I- o OI 01 1- X -T 05 CI CO CI o t- 00 05 c; C5 s; 05 C5 O O I- 00 Oi GO 30 i- I- O O -^OJCO-J* to in -r CO o> m O { - 30 35 1— 35X CO in OI OI 01 01 01 CO —■ 35 CO T)" ■T in in CO I- (?! O) 01 II OI — 05 CO CO O CO XX050 — — CI OI CI CO CO CO a •-1 ■^C-JO ooOtJi wo GO so 'T 'MO GO --0 -* o» o X CO -T 01 o X CO TT 01 o xcotjioiox irt £-OS O ?»-><«> GO ^^ .-1 7^ I-H .-1 05 '-' CO in i- — . OJ T« 7» o» 00 O 01 -^ CO oi CO 00 CO 00 t-o> rt CO in CO CO i1» TT ^ CO X o OI T •^ ■T lO in in •n i- 35 — CO IT •n in in CO o CO > as a O o C3 ?» ooocs .-1 1-c 0* OJ e- rp Tf «o r(" O <- 05 i-H 05:0Tf irtGO 00 O 05 1J o C0005000 35 CO ;o ^ in X in -rti in i~ 05 1* 05 -^05 CI X CO in CO in o CO 01 X X 01 1- t 01 01 1* — I- •«• 1- X »-i ■•-1 T^ y-l 7t Qi 0? CO CO T}< -"l" mnmcoco t- X X O 35 O 1-1 -1 01 CO CO H lO o »o 05 Ot 05 Oi 00 00 1~- to m r*< CO 0? 1- OS X CO -cf 01 o t- in 01 35 in CIXr)"0 in 1-1 O -fOl -^05 CO coco rr Tf •rC 35 •* 35 'S" in m o o I- 35 -f 35 1*00 t- GO GO 35 C5 ooxcoxco O O rH — 01 {, '>j J, ,-, o OI CO CO IT -rr — in o in 35 -»i in in CO CO CO t- fe. c» «> o rf -^ Oi irt ao—cTfooco 00 coo in ?j — t-O-HO C^ O -T CO 01 CO O 01 3S 35 — -- ^ .-■ 01 CO OI -t 1-1 OI •»■ CO X i-c -f- CO m t- rr -^ 00 t- 1-1 in o ino •-I rl T-l W o» CO CO -'t in in ;o t- 00 OS O 1-1 01 CO Tt< m CO t^ 35 o 1— 1— 1-1 ,-c 01 — CO Tf CO I- 35 CI 0? 01 CI CI 01 "b « o :^< 00 »n I- o 00 1- -H lO^H — 1 — 0» OI CO TflOCDt-CO OS^OITTCO X o CO m X — < Tji I- 1-1 in 35 y-> T-i T^ /a -it o 00 ■* in t- GO »-i 1-1 1-1 OJ 01 1-^ 1—1 t— 1 1— 1 0? CO CO CO ^ 1-1 01 OI 01 OI CO CO CO t T IT CO r- X X 35 o ~ t-i o lO CO o in CO in 35 o 1?/ GO 35QO -KGO ^ C? in O 50 05 1- 0?CO ^ in m 1—1 01 o O X X 35 1-1 o m m 35 1- CD —X CO CO X — x«-x o t- o CO X t 0) •^^» cT* in I- C5 — CO in GO o Tl — i I-H T-H 01 CO CO 35 ?> O CI OI 01 00 00 O CO {- — CO Tji 1* iji in in O lO 35 ^r 35 CO CO CO I- l"- f-1 1^ 1— ' 5? 05SS GOt-tO-T'?? (35 35 O: 35 C5 »n O I- 33 35 O J- CO 05 Tj< 35 :o 30 I- I- O — 1J CO -T 00 OI in t- 30 Co' --o' lO -*• CO •n CO t- 30 35 35 X I- -T —1 OI 1-1 O 35 30 -- OI 01 CO 01 01 01 01 01 COO CO in CO CO in CO — 35 't m CO J- t- o» 01 01 OI c^t in ■»# 1-1 CO o CO t;J in 00 o x in X o5 o ^ — 01 CI 01 CO CO CO CO GCrt-O O -O CO rr o O 0» 30 tT o CO 01 X ■* o --0 01 X -r o CO 0>XTf o CO 01 X Tf O CO Tf-OOO 05 T-i 1? Tt" :0 C>» 35 O ■?» T»< .-Hl-H 0» OI a o o 5 "Si -too t- 00 O GO CO OSOCOOX OS {- o -t -t in in CO 1^ OS I- o »-» T-^ T— « ^oj wcoco COiJi-^OO O I- t~ 00 OS OS O --H 'TJ ■?» CO Tt o o o J- H C5 0> C5 GfJ l- I- o o CO T -^ OS -r OS th in in o o I- OS CO 00 CO 00 1^ CO 00 OS C3S O t- C* O ^ oo i-H rH 0< O O -t OS CO 0> coco CO ■» J- C* O O ir OS ■t o o o tc o &H (MOOD 1-1 »n o lO '-I ■^ ?o in -- i-H oo O CO (W i-H oooc!seooj O 1-1 1-1 CO o 00 OS O I- -*< i- O -TOOCO t- O i- Tf CD oo -a- O I- -T 00 o o o -t> -r OJOCJSX i-i- tH rt 0» TJ CO eo -^ o o t- 00 OS o — o? 1— » 1— « 1— « CO O CO l- OS 1—1 1—1 1—1 1—1 1—1 o cj -t o I- OiOiOtOi ot OS ^ o> -rt O X 0» CO CO CO CO CO "^ V T— 1 o COTOOCOOO O W 1-1 (?J o o»eo-*oo Qoosi-ceoo QOi-1 TjlX 1-1 lO OS CO 00 CO OS o 1-1 00 in CO •^ .-1 CJ CO T o o 00 o CJ Tt< 1-1 1-1 1-1 0? (?J (7J 0? CO 1-1 OJ CJ C4 CO Tj< Tji ooo CO CO 1^ i^* in I- 1- 00 OS o O O I- I- X OS .-1 C4 CO Tt o o T-l 1-1 1-1 1-^ 1-^ 1-1 ^ ;a lO -* 05 T-H O T-H 05 -^ -rooto in 01 .-• c* o «» o o t^W (M r- C^ O 1-1 OS X I- t- I- ODXOrrO l-O-IJOi- oocoocooo o X o I- e* O CO CO -f I- OSGOOOXXl- o in rix o o l-H(JiCO rfOOOOW in CO -^ -TOO 1-1 1-1 0* i?J c» Ti O O "^ OS CO CO "^ *^ T}< eox-i>os-t< in o o o L- ooo* GO -r cocoes OS o 1— < 1- I- Tf" O I- -t 1—1 T-l OJ CO CO ^ H 05 Ci 00 t- io c» 05 in OOOOOi-H 1-1 OS o i-H in 1>X ?>-^os w -t cooo 00 OS 00 Tf ooo O 05 05 00 GO in CD L- 00 05 00 4- O Ol O O -^ CJro TT -r ?f — o 00 O O i-QOCO CD -f 'T' 1-1 OJ O* CJ Ci O C8 OS O 01 CO -r -f o o oo -rt" O CD ^ t- O I- X X cs OS OiO*0tOt 01 ot ^H <£coo ojTfoooo ej Tj. o 00 o OJ-tOXO oj -to ooo C? Tt O 00 o 0* It o X o o? OQO'- M in t- 05 c* T-i 1— 1 T-t f— t Oi c»Oic 1 1- fife O t- J> I- i- i- o >S 2§5^ o? -- CO t-- ■* in i- 05 1— ** Tl i- in ■<* CO 0* o 00 o Tt« i-iOOO N05 O 1— O C-f t^ i-iooeoo OS C:i f OS Tf 05 C^ CO CO tT« "^ ■^ o> -ros -r in »n o o i>- cjj Tt xeooo I- 00 00 OS OS CO t- 01 t- — O O — ^ Of (N coco rr "!r CSCO 00 0>O o -r o o o CD t- fc. ?D«oeo rfMtomos O ■^00 CO 00 o o I- in GO o ^ooo-^ oooscoco CO CO C* CO -3< 00 X CO CO 00 o t- o CO o 00 00 CO 00 t- o o o o 1-1 oooo«>aoco oo Tf Of C3S i- o 1-1 1-1 t-hW C* CO ■* rj" oo t- X OS O -H 1—1 T~i CJ CO O O I- OS O CJ CO lO T-l CJ Oi o* o* O X O — CO o Ci T> 00 CO CO CO o CO in L- ^ in 1-< T-l 0{- o Tfco Ti C» CO -rr o eooDO^co in i- o oi o 00 oj o o Tt oscox -t o o •.-n-cjeo-* o inooooeo 1-1 1-1 T^ T-» 1^ ©»©JO»©»eo TH i— 5< •?{ Q} e0rt<-<*'0 o 04 CO CO -S" ■<*• o o i^ 00 OS Tf O O O t- i- OO — OJCO -r ;a O 00 o> i- tC O «0 CO 1-1 53 ■* com CO -* i~ I- OS O CO -^ o CO OS OO o 1- l-O i- OS CO OS J - CO o CO OS i- I- 00 Tj< 00 00 ■?> OS 1-1 O ^ OS I- o Tfo; o CO —1 X OS ^ o o o «»-l^> tP in L- 05 '-' CO O OS T» O 1-1 1-1 r-r (?J W OS o> o o -# 04 CO CO ^ ^T OS CO 00 CO oo i^i o ooo -?■ OS in o CD V- ..- 00 OS OS OJX Oi-lX -* o o 1-1 oj ■^J CO 1—1 1—1 1—1 1— 1 1-1 1— 1 05CO I- O T)< l-H 00 ^T OS CO l> OS i-ii-iox -rr O -* O I- o itO'^Ot- O CO 00 1-1 o» OJ oojoj CO CO "^ 05 OCJ5 ooo in o i- CO OS 00 1^ I- o o O 1- 0> CO -# 1—1 1— 1 1— » T— 1 1— « O "* CO 1-1 o O Ot-X CTS T— » 1-1 ^-* 1— I 1— • OS i - O CO ■— 1 OSO — CJ CO osi- -r «oo CO -T O O CD 01 ot o» rj Qt O OfOO O 1-1 {- l-OOOOOS o o OJ OJ OJ OJ CO CO e- ooo ooooo COOOO ooooo OOCOO ooooo oooooo i:^ 00 o ©» -»< «0 QC o *f -:»< O X O CI Ct C< Ol CO CJ TT O GO O CO CO CO CO ■^ CJ rji O X O ■ CO -» o o CO CO CO CO 00 00 370 TABLE XVI.— TRANSITION CURVE TABLE. f^ c^ rr in OOOOO CO I- 00 C5 O OOOOO !-• 1* CO -r in 8R? > o § o o o in ooo CO T in ooooo OI-00C5O 1-H ooooo — O^CO-^iO *-H 1— • »-^ *-^ ^^ ooo ~ l~ 3J > o o in V. O) 'T? O ■•-" CO CO C5 C-J O CO -^ in o 00 C5 c? CO ^ in o CD "I- ^«.^ 00 O I- CO CO I- O CO I- ^ ■ "■ -^ « c> t--i< -^x in in o in o o 0» CO 00 TJ> -^i ociin sjx in in in CO o ^- ^H »— 1 c» c* eo eo Tp •^tDkO H ino in O5Cft00i>-CO in-^c^^oo coeoo C5 •no ■* 05 X X I- in Tjn 0»XTf o; -f Oi -r ci C» CO CO Tj< T Tf C5 TJ-Oi CO in in CD CO I- 00 "0 00 i-QOOO 05 rr 05 -r 05 OJ CO 00 "S" ■» •^05-^00 00 in o o o I-- 00 0? J- fe, to-r C0C0 05 CO m ^- -}< in 00 QO T*" O I- CO 00 '-o I- in CO CO CO CO 00«5 int-oi o; OC5 «oi-o i>eoj>oir» in ■>-■ i~ in CO i-icj5inxo (?»"CJeoo 0500 Q0?/O ^y-^C'inn T»> m CO I- 00 050'- T— " '—I '"' i-OJ WOOTT «n o £.- X o 0>CO 5/» o TT QO ** 00 CO .-• T-. C* etj-^tot-Oi T-ieoco o o t- w X in in '-i — o» o:) inoxoN laaoi-' Ti (?» CO in CO ,1 -. l-H 1J CJ T-« l-H »-H CO CO ■v C> CO Tf o » »-n-c ?> C» O CO -^ TC s> ?§^|I ?> CO C5 '^ O -p CO O ^ 05 00 T' CO O ** 05 C0 05QOC5 OCiO? OJOOl- in <-oo Tf I- CO — . *» CJ -rr O 05 »-i 5» 00 CO 0? X o -r in 05 in 05 in o •^ o o ~ Cfoo in I- 05 oj ^ i- -- -t oo ■?> »-iC>Oi ?JCO l-^CD CO '* TT "-•CJ TT ox '-cot- o -too cf t- OJ W?J CO CO (Nt-CO ■^ "Tf in o 05 05 1- tjcott in CO C5 in a> 05 05 00 00 l- inco {-00O5 CO in in -^ ©» I- o' in -t CO o — -r* CO -r 51.8 160.1 168.3 C5Q01- 05 05"c> Tfii-iOOCO 0-' C55' x' CO I- t O I- X 05 TJ0 o i.n -r o» ^ O i-i 0> CO -* T-lOt- 05 i- -r ■^ in o oof o CO o» 00 -K o O 1? 00 -* o O T>00 ooo C5 c» in r-i 1— ooooo X -H 'I" I- o' i-io ;>o CO ooooo CO o c:5 -;? in OOOOCO TT TT OOO CC-^'T TC U.' Uti {-oeo m 00 o CO CO ^ i-i 0» !N 0» 00 -^ CO -O 05 C» CO CO CO CO ■^ -r CO rr -^ TT > o 00 V. ccocec — CJ TT c> in >-i o 'Tt o 00 — -»• I- I- CD I- -M 05 O Tt 00 CO I- OCOCTS CO 00 CO «OJ« "CO in 00 05 05 OJ C5 I- 0-. CJ O Oi o in c* CO t- ■rr 00 CO X CO in in I- 05 in 1-1 f-» »— 1 l-H oj c» e» 00 00 Tj> 'ij" in T— 1 1^ f^ « O CO CO -^ rJ CO CO -3> TT •^ 35 TT C5 TT in m o CD I- oooox ^o*o» 05 rCOl-rr Oi CJ CO 00 -T -n" ■*05 -r X CO in in o o i- XCOt- l-XX fc, -NCOt- CO moo in o o» ^ t- nt I- oj CO T>< osoocoin -r* ^-<05 X J- t- 05 0t~ I- C5 i^ino* cooo O 05 C5 t~ CO ■^ 05 in cj o t-ooomx X C-1--X C5 g5^SB ^T-iSioxn ■^ -rf in CO t- 00 05^ »-i 1— 1 '— i C* CO TT Tj* in o t- X O^Ci \ V 5" o ^'-^!S^ C« CO -VT O I- c:5i-ico o in 05 in '-I o «-i ©J CO Tj< in I- 05 5? CClC GO i-i ;? CO ■v in t- C5 — 'T t- OJINCO cj CO -^ in o T-l 1-1 ©J OJ ©» CO 00 "^ in CO 00 ^ ?J c? -^ O O I- C5 C5 O 00 00 ^ I- in ooo TH CO 1— < T— 1 34.61 38.91 43.43 ooo TT coo .-1 C? TT CO ■?> '-^ -^^ ■'H 00 c; COOO in I- d CO o' ••-1 05 in X J- CO X 4- X ?J 05 oj o o in r-i C^ 5* CO CO 39.92 44.81 49.93 'n 05 O5 00 CD •* l-H t- 0» 1-0 X CO o 050050N in coo 35Ci O T^COT CJ5C5 05 00 00 lOCOl-0DO5 !•- 1- CD in T>> O »-> C-? CO -T eO'-o m oi- CV05 C5 T* CO TT O 05X00 c- in o t- X OS O O rr CO -?) O^ 1? CO •* in in o o ■NCOO Tf 00 I'J O O ■* 00 0? o o tJ-OCN •^e> o XO-^CJO 00 O ■* CJ o QCOTJ" l-OiTJ ■^ CO 05 T-( Tfi CD 00 ^ coco W CJ 00 CO CO xoco corr TT X — -t o 05 o» in X OCOOC5 0> CO CO CO CO ^ -rt-o T*4 2G-i3:» 60 22667 23111 23556 24000 24444 26222 260G7 S'.'IU 380 XIX —CUBIC YARDS PER 100 FEET. SLOPES 2 : 1. Depth 1 2 3 4 5 6 p- 4 8 9 10 11 12 13 14 15 16 17 18 19 20 31 3-2 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 S2 53 54 55 56 57 58 59 CO Base 12 Base 14 Base 16 53 119 200 296 407 533 674 830 1000 1185 1385 1600 1830 2074 2.333 2607 2896 3-200 3.319 385-2 21 4200 22 4563 23 4941 24 5333 25 5741 26 6163 27 6600 28 7052 29 7519 30 8000 8496 9007 9533 10074 iog;3j) 11200 11785 1-2385 13)>;k) 13630 14274 14' 133 15607 16-296 17000 17719 1SJ5-2 19-200 19963 20741 81.^33 22.^41 23163 24000 24852 25719 26600 27496 28407 39333 59 133 222 3-26 444 578 726 889 1067 1259 1467 1689 19-26 2178 2444 2726 3022 S;533 3659 4000 4356 4130 5111 5511 5926 6356 6800 7259 I I CO 82-22 87*26 9-244 9:78 10326 1C889 11467 1-2059 1-2667 13-289 139-26 14578 15244 159-26 16622 1733:3 18059 18800 19556 20326 20711 21911 22726 2J556 24400 •25-259 26133 27022 279-26 28844 29778 67 148 244 356 481 622 778 948 1133 1333 1548 1778 2022 2281 2556 2844 3148 3467 3800 4148 4511 4889 5281 5689 6111 6548 7000 7467 7948 8444 Base 18 Base 20 14881 155.^6 16224 16948 17667 18400 19148 19911 20689 21481 2-2289 23111 23948 24800 25667 26548 27444 28:356 29-281 30222 74 163 267 385 519 667 &30 1007 1200 1407 16:30 1867 2119 2-385 2667 2963 3274 3600 3941 4296 4667 5052 5452 5867 6296 6741 7200 7674 8163 8667 8956 9185 94,S1 9719 10022 10-267 10578 10830 11148 11407 117:33 120(X) 1-2333 1-2607 12948 13230 l:35?8 13867 14222 14519 15185 15867 1656:3 17274 mjoo 18741 19496 20267 21052 21852 22667 23496 24:341 252C0 2K01'4 26963 27867 28785 29719 30667 81 178 289 415 556 711 881 1067 1267 1481 1711 1956 2-215 2489 2778 3081 3400 3733 4081 4444 4822 5215 5622 6044 6481 60:33 7400 7881 a378 8SS9 9415 9956 10511 11081 11667 12-267 12881 1:3511 14156 14815 15189 16178 16881 17600 ia333 19081 19S44 20622 21415 22222 23044 2:3881 24733 25600 26481 27378 28-289 29215 30156 31111 Ba-e 28 111 237 378 53:3 704 889 1089 1304 1533 1778 2037 2311 2600 2904 3222 3556 3904 4267 4644 5037 5444 5S67 C304 6756 7-222 7704 8200 8711 9237 9778 10333 10904 11489 12089 12704 13:333 1:3978 146:37 15311 16000 16704 17^22 18156 18904 19667 20444 21237 2-2044 22867 23704 24556 25422 26:304 27200 28111 290:37 20978 :30933 31904 32889 Base 30 119 252 400 563 741 933 1141 i:363 1600 1852 2119 2400 2696 3007 3S33 3674 4030 4400 4785 5185 5600 60:30 6474 6933 7407 7896 8400 8919 9452 100(X> 10563 11141 117:33 1-2341 1-296:3 13600 14252 14919 156 XX.— CUBIC YARDS IN 100 FEET LENGTH. 383 Area. Cubic Area. Sq. Ft. Cubic Area. Sq. Ft. Cubic Area. &^- Ft. Cubic Area. Sq. Ft. Cubic Sq. Ft. Yards. Yards. Yards. Yards. Yards. 2.51 929.6 301 1114.8 351 1300.0 401 1485.2 451 1670.4 253 933.3 302 1118.5 352 1303.7 402 1488.9 152 1674.1 2.53 937.0 303 1122.2 353 1307.4 403 1492.6 453 1677.8 2,51 940.7 304 1125.9 354 1311.1 404 1496.3 4.54 1681.5 25.5 944.4 305 1129.6 355 1314.8 405 1500.0 4.55 1685.2 256 948 2 306 1133.3 3.56 1318.5 406 1.503.7 4.56 1688.9 257 951.9 307 1137.0 357 1322.2 407 1507.4 457 1692.6 258 955.6 308 1140.7 358 1325.9 408 1511.1 458 1696.3 250 9.59.3 309 1144.4 359 1329.6 409 1514.8 459 1700.0 260 963.0 310 1148.2 360 1333.3 410 1518.5 460 1703.7 261 966.7 311 1151.9 361 1337.0 411 1522.2 461 1707.4 262 970.4 312 1155.6 362 1340.7 412 1525.9 462 1711.1 263 974.1 313 11.59.3 363 1344.4 413 1529.6 463 1714.8 264 977.8 314 1163.0 364 1348.2 414 1533.3 464 1718.5 265 981.5 315 1166.7 365 1351.9 415 1537.0 465 1722.2 266 985.2 316 1170.4 366 1355.6 416 1540.7 466 1725.9 26? 988.9 317 1174.1 367 1359.3 417 1544.4 467 1729.6 268 992.6 318 1177.8 368 1363.0 418 1.548.2 468 1733.3 26!) 996.3 319 1181.5 369 1366.7 419 1551.9 469 1737.0 270 1000 320 1185.2 370 i:i70.4 420 1555.6 470 1740.7 271 1U03.7 321 1188.9 .371 1374.1 421 15.59.3 471 1744.4 272 1007.4 322 1192.6 372 1377.8 422 1563.0 472 1748.2 273 1011.1 323 1196.3 373 1381.5 423 1566.7 473 1751.9 271 1014 8 324 1200 374 1385.2 424 1570.4 474 1755.6 275 1018.5 325 1203.7 375 1388.9 425 1574.1 475 1759.3 276 1022.2 326 1207.4 376 1392.6 426 1577.8 476 1763.0 277 1025.9 327 1211.1 377 1396.3 427 1581.5 477 1766.7 278 1029.6 328 1214.8 378 1400.0 428 1585.2 478 1770.4 279 1033.3 .329 1218.5 379 1403. 7 429 1588.9 479 1774.1 280 10,37.0 330 1222 2 380 1407.4 430 1592.6 480 1777.8 281 1040.7 331 1225.9 381 1411.1 431 1596.3 481 1781.5 282 1044.4 332 1229 6 382 1414.8 432 1600.0 482 1785.2 283 1048.2 333 1233.3 383 1418.5 433 1603.7 483 1788.9 284 10.51.9 334 1237.0 384 1422.2 434 1607.4 484 1792.6 285 1055.6 335 1240.7 385 1425.9 ; 435 1611.1 485 1796.3 286 10.59.3 336 1244.4 386 1429.6 1 436 1614.8 486 1800.0 287 1063.0 337 1248.2 387 1433.3 1 437 1618.5 487 1803.7 288 1066.7 338 1251.9 388 1437.0 1 438 1622.2 488 1807.4 289 1070.4 339 12.55.6 389 1440.7 439 1625.9 i89 1811.1 290 1074.1 340 12.59.3 390 1444.4 440 1629.6 490 1814.8 291 1077.8 341 1263.0 391 1448.2 441 1633.3 491 1818.5 292 1081.5 342 1266.7 392 1451.9 442 1637.0 492 1822.2 293 1085.2 : 343 1270.4 393 14.55.6 443 1640.7 493 1825.9 294 1088.9 344 1274.1 394 14.59.3 444 1 1644.4 494 1829.6 295 1092.6 .345 1277.8 395 1463.0 445 1648 2 495 1833.3 296 1096.3 346 1281.5 396 1466.7 446 1651.9 496 18.37.0 297 1100.0 347 1285.2 397 1470.4 447 1 1655.6 497 1840.7 298 1103.7 348 1288.9 398 1474.1 448 16.59.3 498 1844.4 299 1107.4 349 1292.6 399 1477.8 449 1663.0 499 1818.2 300 1111.1 350 1296.3 400 1 1481.5 450 1666.7 500 1851.9 rXTW-AW • ■! »- •■n 384 TABLE XX.— CUBIC YARDS IN 100 FEET LENGTH. Area. Sq. Ft. 501 Cubic Area. Sq. Ft. Cubic 1 Area. Sq. Ft. Cubic Area. Ft. 1 Cubic Area. Ft. Cubic Yards. Yards. Yai-ds. Yards. Yard?. 1855.6 551 2040.7 601 2225.9 651 1 2411.1 701 2596.3 502 1859.3 552 2044.4 602 2229.6 652 2414.8 702 2600.0 503 1863.0 5.53 2048.2 603 2233.3 653 2418.5 703 2603.7 504 1866.7 5.54 2051.9 604 2237.0 654 2422.2 704 2607.4 505 1870.4 555 2055.6 605 2240.7 655 2425.9 705 2611.1 506 1874.1 556 20.59.3 606 2244.4 656 2429.6 706 2614.8 507 1877.8 557 2063.0 i 607 2248.2 657 2433.3 707 2618.5 508 1881.5 558 2066.7 608 2251.9 658 2437.0 708 2622.2 .509 1885.2 559 2070.4 609 2255.6 659 2440.7 1 709 2625.9 510 188S.9 560 2074.1 610 2259.3 660 2444.4 710 2629.6 .511 1892.6 .561 2077.8 611 9263 661 2448.2 711 2633.3 512 1896.3 562 2081.5 612 2266.7 662 2451.9 712 2637.0 5i:^ 1900.0 563 2085.2 613 2270.4 663 2455.6 713 2640.7 514 1903.7 564 2088.9 614 2274.1 664 2459.3 714 2644.4 .515 1907.4 5(35 2092.6 615 2277.8 665 2463.0 i 715 2648.2 516 1911.1 566 2096.3 616 2281.5 666 2466.7 1 716 2651.9 51 : 1914.8 567 2100.0 617 2285.2 667 2170.4 717 2655.6 518 1918.5 568 2103.7 618 2288.9 668 •2474.1 1 718 2659.3 519 19-22.2 569 2107.4 619 ^292.6 669 2477.8 ! 719 2663.0 .520 1925.9 570 2111.1 620 2296.3 670 2481.5 i 720 2666.7 521 1929 6 571 2114.8 621 2300.0 671 2485.2 721 2670.4 522 1933.3 572 2118.5 1 622 2.303.7 672 2488.9 722 2674.1 523 19.37.0 573 2122.2 ' 623 2307.4 673 2492 6 ; 723 2677.8 524 1940.7 574 2)25.9 624 2311.1 674 2496.3 724 2681.5 525 1944.4 575 2129.6 625 2314.8 675 ■2500.0 725 2685.2 526 1948.2 576 2133.3 626 2318.5 676 2503.7 726 2688.9 .527 1951.9 577 2137.0 627 2322.2 677 2507.4 727 2692.6 528 1955.6 578 2140.7 628 2325.9 678 2511.1 728 2696.3 529 1959.3 579 2144.4 629 2329.6 679 2514.8 729 2700.0 530 1963.0 580 2148 2 : 630 2333.3 680 2518.5 730 2703.7 531 1966.7 581 2151.9 j 631 2337.0 681 2522.2 731 2707.4 532 1970.4 582 21.55.6 632 2*40.7 682 2525.9 732 2711.1 533 1974.1 583 21.59.3 633 2344.4 683 2529.6 733 2714.8 534 1977.8 584 2163.0 i 634 2348.2 684 2533.3 734 2718.5 535 1981.5 585 2166.7 , 635 2:3.51.9 685 2537.0 735 2722.2 536 1985.2 586 2170.4 636 23.55.6 686 2540 7 736 2725.9 537 1988.9 587 2174.1 637 23.59.3 687 2544.4 737 2729.6 538 1992.6 588 2177.8 638 2363 688 2.548.2 738 2733.3 539 1996.3 589 2181.5 639 2366.7 689 2.551.9 739 2737.0 540 2000.0 590 2185.2 640 2370.4 690 2.555.6 740 2740.7 .541 2003.7 591 2188.9 641 2374.1 691 2.5.59.3 741 2744.4 542 2007.4 ■: 592 2192.(! 642 2377.8 692 2563.0 742 2748.2 543 2011.1 593 2196.3 643 2381.5 693 2566.7 743 2751.9 544 2014.8 594 2200.0 644 2.385.2 694 2.570.4 744 2755.6 545 2018.5 595 2203.7 645 2388.9 695 2.574.1 745 27.59.3 546 2022.2 596 2207.4 646 2.392.6 696 2577.8 746 2763.0 547 2025.9 597 2211.1 647 2396.3 697 2581.5 747 2766.7 548 2029.6 598 2214.8 648 2400.0 698 2.585.2 748 2770.4 549 2033.3 599 2218.5 649 2403.7 699 2588.9 749 2774.1 550 2037.0 600 2222.2 650 2407.4 700 2592.6 750 2777.8 TABLE XX.— CUBU; YARDS IN 100 FEET LENGTH. 385 Area . Cubic Area. Sq. Ft. Cubic Area. Sq. 1 Ft. Cubic Area. Sq. Fi. Cubic Area. Sq. Ft. Cubic Sq. Ft. Yards. Yards. Yards. I Yards. Yards. 751 2781.5 801 2966.7 ! 851 3151.9 901 3337.0 951 3522.2 75-^ :^785.^> 802 2970.4 852 3155.6 902 3340.7 952 3525.9 753 2788.9 80? 2974.1 ! 853 3159.3 903 3344.4 953 3529.6 754 2792.6 8K4 2977.8 854 3163.0 904 3348.2 954 3533.3 755 2796.3 805 2981.5 855 3166.7 905 3351 . 9 955 3537.0 75'3 2800.0 806 2985.2 1 856 3170.4 906 3355.6 956 3540.7 757 2803.7 807 2988.9 857 3174.1 907 3359.3 957 3544.4 758 2807.4 808 2992.6 858 3177.8 908 3363.0 958 3548.2 759 2811.1 809 2996.3 ! 859 3181.5 909 3366.7 959 3551 9 760 2811.8 810 3000.0 i 860 3185.2 910 3370.4 960 3555.6 761 2818.5 811 3003 7 861 3188.9 ; 911 3374.1 961 35.59.3 76-.> 2822 2 812 3007.4 862 3192.6 912 3377.8 962 3563.0 763 2825.9 813 3011.1 863 3196.3 913 3381.5 963 3566.7 764 2829.6 814 3014.8 864 3200.0 914 3385.2 964 .%70.4 705 2833 3 815 3018.5 865 3203.7 915 3388.9 965 3574.1 766 2837.0 816 3022.2 866 3207.4 916 3392.6 966 3577.8 767 J840.7 817 3025.9 \ 867 321 1 . 1 917 3396.3 967 3581.5 768 2844.4 818 3029.6 868 3214.8 918 3400.0 i 968 3585.2 769 2848.2 819 3033.3 i 869 3218.5 i 919 3403.7 969 3588.9 770 2851.9 820 30S7.0 ! 870 3222.2 ' 920 3407.4 970 3592.6 771 J855.6 821 3040.7 1 871 3225.9 921 3411.1 971 3596.3 77i 2859.3 822 3044.4 ! 872 3229.6 922 3414.8 972 3600.0 773 2863.0 823 3048.2 873 3233.3 923 3418.5 973 3603.7 771 2866.7 824 3051.9 - 874 3237.0 ' 924 3422.2 974 3607.4 775 2870.4 825 3055.6 1 875 3240.7 925 3425.9 975 3611.1 776 2874.1 826 3059.3 876 3244.4 926 3429.6 976 3614.8 1 1 1 2877.8 827 3063.0 877 3248.2 927 3433.3 977 3618.5 77K 2881.5 828 3066.7 j 87'8 3251.9 928 3437.0 978 3622.2 779 2885.2 829 3070.4 ! 879 3255.6 929 3440.7 979 3625.9 780 2888.9 830 3074.1 ; 880 3259.3 930 3444.4 980 3629.6 781 2892.6 831 3077.8 881 3263.0 ' 931 3448.2 981 3633.3 78:i 2896.3 832 3081.5 882 3266.7 932 3451.9 982 3637.0 783 2900.0 833 3085.2 1 883 3270.4 933 3455.6 983 3640,7 784 290^ 7 834 3088.9 884 3274.1 934 3459.3 984 3644.4 785 2907.4 835 3092.6 1 885 3277.8 935 3463.0 985 3648.2 786 4911.1 836 3096 3 : 886 3281.5 936 3466.7 986 3651.9 787 2914.8 837 3100.0 : 887 3285.2 937 3470.4 987 3655.6 788 2918.5 838 3103.7 1 888 3288.9 938 3474.1 988 3659.3 789 2922.2 839 3107.4 ! 889 3292.6 939 3477.8 989 3663.0 790 2925.9 840 3111.1 ! 890 3296.3 940 3481.5 into 3666.7 791 2929.6 841 3114.8 ' 891 3300.0 941 3485.2 991 3670.4 79-3 2933.3 842 3118.5 892 3303.7 942 3488.9 992 3674.1 793 2937.0 843 3122.2 893 3307.4 943 3492.6 993 3677.8 794 2940.7 844 3125.9 894 3311.1 944 3496.3 994 3681.5 795 2944.4 845 3129.6 895 3314.8 945 3.'-.00.0 995 3685.2 796 2948.2 846 3133.3 896 3318.5 946 3.503.7 996 3688 . 9 797 2951.9 847 3137.0 897 3322.2 947 3507.4 997 3692.6 798 29.55.6 848 3140.7 898 3325,9 948 3511.1 998 3696.3 799 29.59.3 849 3144.4 i 899 3329.6 949 3.514.8 999 3700.0 800 2963.0 850 3148.2 900 3333.3 ; 950 3518.5 1000 3703.7 rz^ ^ ^ A M * 386 XXI.— RISE PER MILE OF VARIOUS GRADES. Rise per Cent. .01 .02 .03 .04 .05 .06 .07 .08 .09 .10 .11 .12 .13 .14 .15 .16 .17 .18 .19 .20 .21 .22 .23 .24 .25 .26 .27 .28 .29 .30 .31 .32 .33 .34 .35 .36 .37 .38 .39 .40 .41 .42 .43 .44 .45 .46 .47 .48 .49 .50 .51 .52 .53 .54 .55 .56 .57 .58 .59 .60 Rise Feet per 3Iile. per Cent. .528 .61 1.056 .62 1.5S4 .63 2.112 .64 2.640 .65 3.168 .66 3.696 .67 4.224 .68 4.752 .69 5.280 .70 5.808 .71 6.336 .72 6.864 .73 7.392 ! .74 7.920 .75 8.448 .76 8.976 .77 9.504 .78 10.032 .79 10,560 .80 11.088 .81 11.616 .82 12.144 .83 12.672 .&4 13.200 .85 13.728 .86 14.256 .87 14.784 .88 15.312 .89 15.840 .CO 16.368 .91 16.896 .92 17 424 .93 17.952 .94 18.480 .95 19.008 .96 19.5.36 .97 20.064 .98 20.592 .99 21.120 1.00 21.648 1.01 22.176 1.02 22.704 • 1.03 23.232 1.04 23.760 1.05 24.288 1.06 24.816 1 1.07 25.344 1.08 25.872 I 1.09 26.400 1.10 26 928 1.11 27.456 1.12 27.984 1.13 28.512 1.14 29.040 1.15 29.568 1.16 30.096 1.17 30.624 ' 1.18 31 152 1.19 31.080 i 1.20 1 Feet per Mile. 1 1 Rise per Cent. 32.208 1.21 32.736 1 1.22 33 264 1 1.23 33.792 1.24 34.320 1.25 .34.848 1.26 35.. 376 1.27 35.904 1 1.28 36.432 1.29 36.900 1 1 1.30 37.488 1.31 38.016 1.32 38.544 1.33 39.072 1..34 39.600 1.35 40.128 1.36 40.656 1.37 41.184 1.38 41.712 1.39 42.240 1.40 42.768 1.41 43.296 1 1.42 43.824 ! 1.43 44.3.52 1.44 44.880 ; 1 45 45.408 1 1.46 45.936 1.47 46.464 1.48 46.992 1.49 47.520 1.50 48.048 1.51 48.576 1..52 49.104 1 1..53 49.632 1.54 50.160 1.55 50.688 1.56 51.216 1.57 51 744 1.58 52.272 1.59 52.800 1.60 53.328 1.61 53.8.56 1.62 5 4.. 384 1.63 54.912 1.64 55.440 1.65 55.908 1.66 .56.496 1.67 57.024 1 68 57.552 1.69 58.080 1.70 58.608 1.71 .59.i:36 1.72 .59.664 1.73 60.192 1.74 60 720 ; 1.75 61.248 1 1.76 61.776 1.77 62 304 1 78 62.^32 1.79 63.360 1 l.SO Feet per j ' Rise 1 -^^^ Mile. ' per < Cent. 63.888 1 1.81 64 416 1.82 64.944 1.83 65.472 1.84 66.000 1.85 66.528 1.86 67.056 1.87 67.584 \ 1.88 i 68.112 1.89 68.640 1.90 69.168 1 91 69.696 1.92 70.224 1.93 70.752 1.94 71.280 1.95 71.808 1.96 72.336 1.97 72.864 1.98 73.392 1.99 i 73.920 2 00 ; 74.448 2.10 74 976 2.20 75.504 2.30 ! 76.032 2.40 76.560 2.50 77.088 2.ti0 i 77.616 2.70 78.144 2 80 78.672 2.90 79.200 3.00 79.728 3.10 80.2.56 3.20 80.784 3.30 ' 81.312 3.40 81.840 3.50 82.368 3.60 . 82.896 3.70 83.424 3 80 83.952 3.90 84.480 4.00 85.008 4.10 85.5;B6 4.20 86.064 4.30 86.592 4.40 87.120 4.50 87.648 4.60 88.176 • 4.70 88.704 4.80 89.232 4.90 89.760 5.00 90.288 5.10 90 816 5.20 91.344 5.30 91.872 5.40 92.400 5 50 92.928 5.60 93.4.56 5.70 93.984 5 80 94.512 5.90 95.040 , 6.00 Feet per Mile. 95.568 96.096 96.624 97.152 97.680 98.208 98.736 99.264 99.792 100.320 100.848 101.376 101.904 102.432 102.960 103.488 104.016 104.544 105.072 105.600 110.880 116.160 121.440 126.720 132.000 137.280 142.560 147.840 153,120 158.400 163.680 168.960 174.240 179 520 184.800 190.080 195.360 200,640 205.920 211.200 216.480 221.760 227.040 232,. 320 237.600 242.880 248.160 252.440 258.720 264.000 269.280 274.560 279.840 285.120 290.400 295.680 .300.960 .306.240 311.520 316.800 TABLE XXII.— SLOPES FOR TOPOGRAPHY. 387 fl ^ od a « -; od a a> -i od o CO cS *-• y-l o 2 .5 o W OS boa _ o •^-^ o 0) •- W r2 ©«-i o5 c3 -imt o c tea 1 a a •^'- o - Cffi c 2 n O at V3 •,-.'^(5 oQ aJ u-t oj O C (5 a — , o 'SO «y o a B ^ o5.2 — ^05 oQoj < 0= 20' > w K ! < > ffi .58 1 1718.9 1 7° 30' 12.87 i t , t 16° 28.67 34.9 40 1.16 859.4 40 13.46 74.3 17 30.57 32.7 ] 1.75 572.9 8 14.05 71.2 18 32.49 30.8 20 2.33 429.6 20 14.65 68.3 19 34.43 29.0 40 2.91 343.7 40 15.24 65.6 20 36.40 27.5 2 3.49 286.4 9 15.84 63.1 21 38.39 26.1 20 4.08 245.4 20 16.44 60.8 22 40.40 24.8 40 4.66 214.7 40 17.03 58.7 23 42.45 23.6 3 5.24 190.8 10 17.63 56.7 24 44.52 22.5 20 5.83 171.7 20 18.23 54.8 25 46.63 21.4 10 6.41 156.0 40 18.84 53.1 26 48.77 20.5 4 6.99 143.0 11 19.44 51.4 27 50.95 19.6 20 7.. 58 133.0 30 20.35 49.2 28 53.17 18.8 40 8.16 122.5 12 21.26 47.0 29 55.43 18.0 5 8.75 114.3 30 23.17 45.1 30 57.74 17.3 20 9.34 107.1 13 23.09 43.3 m 70.02 14.3 40 9.92 100.8 30 24.01 41.7 40 83.91 11.9 6 10.51 95.1 14 24.93 40.1 45 100.00 10.0 20 11.10 90.1 30 25.86 38.7 50 119.18 8.4 40 11.69 85.6 15 36.79 37.3 55 142.81 7.0 7 12.28 81.4 i 1 30 27.73 36.1 60 173 21 5.8 TABLE XXIII.- -MATERIAL ] REQUIRED FOR ONE MILE OF TRACK. RAIL WEIGHTS. RAILROAD SPIKES. ] o 4-1 o «J . C CO OjO •Ck> Required for OJ u 3 S5 ^ be I) a^ 73 5* • Ties 3 ft. Apart. For Rails Weighing 'sS) a OS t-O bC-v ... '^ - 7;o O jj 5>^ .§- a oi UK 9J OJo oW Pi 02 1-3 02 < !^.S ^ 12 21.12 18.857 ^iXl% 360 5870 29.3 45 to 70 lbs. 16 28.16 35.148 5 Xt«b 400 5380 26.4 40 " 56 ' 20 35.30 31 429 5 X i 450 4690 23.5 35 " 40 " 25 44.00 39 386 4iX i 530 3980 19.9 28 " 35 ' 30 .53.80 47.143 4 X i 600 3530 17.6 24 " 35 ' 35 61.60 55.000 4iXT'B 680 3110 15.5 20 " 30 ' 40 70.40 63.857 4 X/b 720 3930 14.7 20 " 30 ' 45 79.20 70.714 3^X/5 900 3350 11.7 16 " 35 ' 56 98.. 56 88.000 4X1 1000 3110 10.6 16 " 35 ' 60 105.60 94.386 > 34 X f 1190 1770 8.9 16 " 20 ' 70 133.20 110.000 3X1 1240 1700 8.5 16 " 20 ' 80 140.80 125.714 2iX f 1342 ! 1570 7.9 12 " 16 " , 'number of splice-joints. ' NUMBER OF CROSS-TIES. Two Bars with Four Bolts and Nuts to Each Joint. ! Distanoe apa irt, c. to c, in Feet. Length of Rail in Feet. 1.5 1.75 2.0 2.25 2.50 20 24 26 28 30 3520 3017 3640 2347 3112 528 440 406 377 352 388 TABLE XXIV. CONVERSION OF ENGLISH INCHES INTO CENTIMETRES • Ins. 1 2 3 4 5 6 i 8 9 10 20 30 40 50 60 70 80 90 100 Cm. 0.000 25.40 50.80 76.20 101.60 127.00 152.40 177.80 203.20 228.60 254.00 Cm. 2.540 27.94 53.34 78.74 104.14 129.54 154.94 180.34 205.74 231.14 256.54 Cm. 5.080 30.48 55.88 81.28 106.68 132.08 157.48 182.88 208.28 233.68 2.59.08 Cm. 7.620 33.02 58.42 83.82 109.22 134.62 160.02 185.42 210.82 236.22 261.62 Cm. 10.16 35.56 60.96 86.36 111.76 137.16 162.56 187.96 213.36 238.76 264.16 Cm. 12.70 38.10 63.50 88.90 114.30 139.70 165.10 190.50 215.90 241.30 266.70 Cm. 15.24 40.64 66.04 91.44 116.84 142.^4 167.64 193.04 218.44 243.84 269.24 Cm. 17.78 43.18 68.58 93.98 119.38 144.78 170.18 195.58 220.98 246.38 271.78 Cm. 20.32 45.72 71.12 96.52 121.92 147.32 172.72 198.12 223.52 248.92 274.32 Cm. 22.86 48.26 73.66 99.06 124.46 149. f>6 175.96 200. Si6 226. (« 251.46 276. J^e CONVERSION OF CENTIMETRES INTO ENGLISH INCHES . Cm. 1 2 3 4 5 6 7 8 9 Ins. Ins. Ins. Ins. Ins. Ins. Ins. Ins. Ins. Ins. 0.000 0.394 0.787 1.181 1.575 1.969 2.362 2.756 3.150 3.543 10 3.937 4.331 4.742 5.118 5.512 5.906i 6.299 6.693 7.087 7.480 20 7.874 8.268 8.662 9.055 9.449 9.843110.236 10.630 11.024 11.41S 30 11.811 12.205 12.599 12.992 13.386 13.780:14.173 14.567 14.961 15.355 40 15.748 16.142 16.530 16.929 17.323 17.71718.111 18.504 18.898 19.292 50 19.685 20.079 20.473 20.867 21.260 21.654:22.048 22.441 22.835 23.229 60 23.622 24.016 24.410 24.804 25.197 25 591 25.985 26.378 26.772 27.166 70 27.560 27.953 28.347 28.741 29.134 29.528 29.922 30.316 30.709 31.103 80 31.497 31.890: 32.2S4 32.678 33.071 33.465 33.859 34.253 34.646 35.040 90 35.434 35.827| 36.221 36.615 37.009 37.402 37.796 38.190 38.583 38.977 100 39.370 39.764 40.158 40.552 40.945 41. 339141. 733 42.126 42.520 42.914 CONVERSION OF ENGLISH FEET INTO METRES. Feet. 1 2 3 4 5 6 7 8 9 Met. Met. Met. Met. Met. Met. Met. Met. Met. Met. 0.000 0.3048 0.'6096 0.9144 1.2192 1.5239 1.8287,2.1335 2.4383 2.7431 10 3.0479 3.3527 3.6575 3.9623 4.2671 4.5719 4.87675.1815 5.4863 5.7911 20 6.09.59 6.4006 6 . 7055 7.0102 7.3150 7.6198 7.9246 8.2294 8.5342 8.8390 30 9.1438 9.4486 9.7534 10.058 10.363 10.668 10.972 11.277 11.582 11.887 40 12.192 12.496 12.801 13.106 13.411 13.716 14.020 14.325 14.630 14.935 50 15.239 15.544 15.849 16.154 16.459 16.763 17.068 17.373 17.678 17.983 60 18.287 18.592 18.897 19 202 19.507 19.811 20.116 20.421 20.726 21.031 70 21.335 21.640 21.945 22.250 22.555 22.859 23.164 23.469 23.774 24.079 80 24.383 24.688 24.993 25.298 25.602 25.907 26.212 26.517 26.822 27.126 90 27.431 27.736 28.041 28.346 28.651 28.955 29.260 29.565 29.870 30.174 100 30.479 30.784 31.089 31.394 31.698 32.003 32.308 32.613 32.918 33.222 CONVERSION OF METRES INTO ENGLISH FEET. Met. 10 20 30 40 50 60 70 80 90 100 1 Feet. 0.000 32.809 65.618 98.4271 131.24 164.04 196.85] 229.66 262.47 295.28 328.09 Feet. 3.2809 36.090 68.899 101.71 134.52 167.33 200.13 232.94 265.75 298.56 331.37 Feet. I 6.5618 39.371 72.179 104.99 137.80 170 61 203.42 236.22 269.03 391.84 334.651 Feet. 9.8437 42.651 75.461 108.27 141.08 173.89 206.70 239.51 272.31 305.12 337.93 Feet. 13.123 45.932 78.741 111.55 144.36 177.17 209.98 242.79 275.60 308.40 341.21 Feet. 16.404 49.213 82.022 114.83 147.64 180.45 213.26 246.07 278.88 311.69 344.49, Feet. 19.685 52.494 85.303 118.11 150.92 183.73 216.54 249.35 282.16 314.97 347.78 Feet. 22.966 55.775 88.584 121.39 154.20 187.01 219.82 252.63 285.44 318.25 351.06 8 Feet. 26.247 59.0.56 91.865 124.67 157.48 190.29 223.10 255.91 321.53 ! 354. 34 9 Feet. 29.528 62.337 95.146 127.96 160.76 193. 57 226.38 259.19 292.00 324.81 357.152 TABLE XXV. 380 CONVERSION OF ENGLISH STATUTE-MILES INTO KILOMETRES. Miles. 10 20 30 40 50 60 70 80 90 100 Kilo. 0.0000 16.093 32.186 48.2T9 64.372 80.46.5 96 558 112.65 128.74 144.85 160.93 Kilo. 1.6093 17.702 33.795 49.888 65.981 82.074 98.167 114.26 130.35 146.44 16«.53 3 Kilo. 3.2186 19.312 35.405 51.498 67.591 83.684 99 777 115.87 131.96 148.05 164 14 Kilo. 4.8279 20.921 37.014 53.107 69.200 85.293 101.39 117.48 133.57 149.66 165 75 Kilo 6.437 22.530 38.623 54.716 70.809 86.902 102.99 119.08 135.17 151.26 167.35 Kilo. 8.0465 24.139 40.232 56.325 72.418 88.511 104.60 120.69 136.78 152.87 168.96 Kilo. 9.6558 25 749 41.842 57.935 74.028 90.121 106.21 122.30 138.39 154.48 170.57 8 Kilo. 11.2652 1 27.358t 43 451 59.544 75.637 91 730 107.82; 123.91 i 140.001 156.09 172.181 Kilo. 12.8745 28.967! 45.060! 61.153 i7 246! 93.339 109.43 125.52 141.61 157.70 173.79 Kilo. 14.4818 30.577 46,670 62.763 78.856 94.949 ill. 04 127 13 143.22 159 31 175.40 CONVERSION OF KILOMETRES INTO ENGLISH STATUTE-MILES. Kilom. 10 20 30 40 50 60 70 80 90 100 Miles. 0.0000 6.2138 12.427 18.641 24.8.55 31.069 37.282 43.497 49.711 55.924 62.138 Miles. 0.6214 6.8352 13.049 19.263 25.477 31.690 37.904 44.118 50.332 56.545 62.759 Miles. 1.2427 7.4565 13.670 19.884 26.098 32.311 38.525 44.739 50.953 57.166 ,63.380 Miles. 1.8641 8.0780 14.292 20.506 26.720 32.933 39.147 45.361 51 . 575 57.788 64.002 Miles. 2.4855 8.6994 14.913 21.127 27.341 33.554 39.768 45.982 52.196 58.409 64.623 Miles. 3.1069 9.3208 15.534 21.748 27.962 34.175 40.389 46.603 52.817 59.030 65.244 Miles. 3.7282 9.9421 16.156 23.370 28.584 34.797 41.011 47,225 53.439 59.652 65.866 Miles. 4.3497 10.562 16.776 22.990 29.204 35.417 41.631 47.845 54.059 60.272 66.486 8 Miles. 4.9711 11.185 17.399 23.613 29.827 36.040 42.254 48.468 54.682 60.895 67.109 Miles. 5.5924 11.805 18.019 24.233 30.447 36.660 42.874 49.088 55.302 61.515 67.729 LENGTH IN FEET OF 1' TABLE XXVI. ARCS OF LATITUDE AND LONGITUDE. Lat. 1' Lat. r Long. Lat. V Lat. y Long. 1° 6045 6085 31^ 6061 5222 2° 6045 6083 32° 6062 5166 3° 6045 6078 33° 6063 5109 4» 6045 6071 34° 6064 5051 5° 6045 6063 35» 6065 4991 6° 6045 6053 36» 6066 4930 7° 6046 6041 37° 6067 4867 8° 6046 6027 38° 6068 4802 9» 6046 6012 39° 6070 4736 10° 6047 5994 40° 6071 4669 11° 6047 5975 41° 6072 4600 12° 6048 5954 42° 6073 4530 13° 6048 5931 43° 6074 4458 14° 6049 5907 44° 6075 4385 15° 6049 5880 45° 6076 4311 16° 6050 5852 46° 6077 4235 17° 6050 5822 47° 6078 4158 18° 6051 5790 48° 6079 4080 19° 6052 5757 49° 6080 4001 20° 6052 5721 50° 6081 3920 21° 6053 5684 51° 6082 3838 22° 6054 5646 52" 6084 3755 23° 6054 5605 53° 6085 3671 24° 6055 5563 54° 6086 3586 25° 6056 5519 • 55° 6087 3499 26° 6057 5474 56° 6088 3413 27° 6058 5427 57° 0089 3323 28° 6059 5378 58° 6090 3233 29° 6060 5327 59° 6091 3142 30° 6061 5275 60° 6092 3051 390 tRIGONOMETRlC FORMULAS. TABLE XXVII. -TKIGONOMETRIC AND MISCELLANEOUS FORMULAS. TRIGONOMETRIC FORMULAS. In Fig 99, let DCE be the arc of a quadrant, ABC o. right triangle, the angle BA C subtended by the arc CE = A , and consider the radius .1 ( ' = unity. Then BC =sin^. AB =cos^. H^=-tan^. DF = QotA. AH=s,ecA. AF=co^ecA. BE ^=\eis.m.A. Zfl = coversin-4. CH=exsecA. CF =coexsec J.. Using the small letters a, h, c, to represent the sides of a right triangle in Fig. 98 or 99, v;e may Avrite sin^ cos^ a V cosec A .-. sin^ a cosec A 1 sec-4^-; .-. q.osA:= c sec A tan-4 = -; cot^=-; .-. tan^ ^ c a cotA N SOLUTION OF TRIANGLES. 301 TABLE XXVII.— TRIGONOMETRIC AND MISCELLANEOUS FORMULAS. SOLUTION OF RIGHT TRIANGLES. Required. Given. A, C, C a, b A, C, b a, c C, 6, c A, a C, a, c A, b C, a, b A, c Formulas. sin A = cos C = - ; c = ^{b + a) {b — a)^ tSinA =cotB a b = Va-^ + c2. C ^^ 90° — A ; c = a cot A ; b = a cosec vl. C = 90° — A ; a = bsmA ; c = b cosin A, C = 90° — A; a = ctan^; 6 = csec^. SOLUTION OF OBLIQUE TRIANGLES. Required. b B 1,{A + B) h{A - B) A B Area Area Area Given. A, B, a A, a, b \ a.b, C <> a, 6, c > < A^ &, c A, B, c Formulas. b = a sin B sin -4 bsinA sin B - i{A + B) = ^{m-C) /^z^^^^^'^'^ tani(^ - B) =^^^tan'iM + B) a + 6 • A--=^i(A + B) + i(A-B) B=.i(A + B)-i{A- B) If s=i{a+b-\-c), smiA=^l ^^~^l^^~'') \ be \ s (s — a) -^inl 2x/s(8-a)(s-6)(s- -c) be Area = -v s (s — a){s — b) (s — c) Area = I be sin A c^ sin ^ sin B Area = 2sin(^+ jB) •^•^J'- GEXERAL FORMULAS. TABLE XXVII.-TRIGONOMETRIC AND MISCELLANEOUS FORMULAS. GENERAL FORMULAS. sin^ = V 1 — cos^^ = tan^ cos A. sin A =2 sin ^-4 cos ^-4. sin A = ^ = V'i(l — C0S24). cosec-4 cos^ = = V^l — sin2^ = cot^ sin^- secA cosA =1—2 sin2^^ ^ 1 — vers^. cosA = ^i + icos2^ = cos2^^ — sin2^^. tan^ = ^^ = Vsec'^A - 1. cos A tan -4 cos^ 1 + cos2yl 1 1— cos2^ cot^ sin 2^ ^^+ A 1 cos -4 / T-z rr cot A = = = V cosec^^ — 1. tan A sin A cot^ = _5HlM_ = LL^^^M. 1 — cos2^ sin 2^ sec -4 = = the reciprocal of any expression for cos^. cos J. cosec A 1 = the sin A vers^ — 1 - - cosA — 2sinH^. exsec A — sec A- -1 — vers^ cos^ Bini^ = JIUSS^ = /ZH s^ G ENTERAL FORMULAS. 393 TABLE XXVII. -TRIGONOMETRIC AND MISCELLANEOUS FORMULAS. 2 tan A 1 — cos^ sin -4 coiiA = 1 + sec^ siii^ 1 + cos A 1 + cos^ sinA sin A 1 — cos J. sin 2 A =2 sin A cos ^. cos2^ == cos2 J. — sin2 J. = 2 cos^^ — L 2tan^ tan 2^ cot 2^ 1 — tan2^ cot2^ — 1 2cot^ sin {A ± 7^) ^ sin ^ cos B ± cos A sin B, cos(yl ± 7^) = cosyl cos5 ip sin J. sinJ?. tan(^ ± B) = tan^±tanJ?^ 1 q= tan A tan B &m.A + sin J? = 2 sin |( J. + i?) cos|(J. — B). sin A — sin Ji = 2 cos i{A + B) sin i (^ — 5). cos^ + cos 7? =2 cos i (J. + B) cosi{A — B). cosB — cos^ = 2sini(^ + 5)sini(^ — B). sin2^ — sin2 5 = cos2 5 — cosS^l = sin (J. + B) Gin(^ — B). cos2^ - sin2J5 = cos {A + B) cos {A — B). tan^±tan£ = ^HiiA±JZ). cos^ cos^ cotA±o.oiB^±^''^^M, 6iaA '6m.B 391 MISCELLANEOUS FORMULAS. TABLE XXVII. —TRIGONOMETRIC AND MISCELLANEOUS FORMULAS. MISCELLANEOUS FORMULAS. Required. Area of Trapezoid Regular Polygon Circle Ellipse Parabola Surface of Cone Cylinder Sphere Zone Voliune of Prism or cylinder l^yraniid or cone Frustum of Pyramid or cone Sphere Given. Parallel sides = m and n Perp. dist. bet. them =p Length of side = I Number of sides = n Radius = r Semi-axes = a and b Base = 6, height = h Radius of base = r Slant height = s Radius = r, height = h Radius = r Height = h Radius of its sphere = r Area of base Height = h Area of base Height = h h h Area of bases h r b and b' Height : Radius Formulas. P 2 (m + n) — cot • 4 n 7tr^ [;r = 3.1416] Ttab Ibh Ttrs 2 7trh 4 7cr^ 2 7trh bh bh 3 - (b +V + ^/bb' 3 |;rr3 TABLE XXVIII. — SQUARE AND CUBE ROOTS. 305 Sq uare Roots and Cube Roots of Numbers from .1 to 28. Ko errors. No. Square. Cube. Sq. Bt. C. Rt. No. 9q. at. C. Et. No. Sq. Rt. C. Rt. .1 .01 .001 .316 .464 .7 2.387 1.786 .4 3.661 2.375 .15 .0225 .0034 .387 .531 .8 2.408 1.797 .6 3.688 2.387 .2 .04 .008 .447 .585 .9 2.429 1.807 .8 3.715 2.399 .25 .0625 .0156 .500 .630 6. 2.449 1.817 14. 3.742 2.410 .3 .09 .027 .548 .669 .1 2.470 1.827 .2 3.768 2.422 .35 .1225 .0429 .592 .705 .2 2.490 1.837 .4 3.795 2.433 .4 .16 .064 .633 .737 .3 2.510 1.847 .6 3.821 2.44A .45 .2025 .0911 .671 .766 .4 2.530 1.857 .8 3.847 2,455 .5 .25 .125 .707 .794 .5 2.550 1.866 15. 3.873 2.466 .85 .3025 .1664 .742 .819 .6 2.569 1.876 .2 3.899 2.477 .6 .36 .216 .775 .843 .7 2.588 1.885 .4 3.924 2.488 .65 .4225 .2746 .806 .866 .8 2.608 1.895 .6 3.950 2.499 .7 .49 .343 837 .888 .9 2.627 1.904 .8 3.975 2.509 .75 .5625 .4219 .866 .909 7. 2.646 1.913 16. 4. 2.520 .8 .64 .512 .894 .928 .1 2.665 1.922 .2 4.025 2.530 .85 .7225 .6141 .922 .947 .2 2.683 1.931 .4 4.050 2.541 .9 .81 .729 .949 .965 .3 2.702 1.940 .6 4.074 2.551 » .9025 .8574 .975 .983 .4 2.720 1.949 .8 4.099 2.561 ]. 1.000 1.000 1.000 1.000 .5 2.739 1.957 17. 4.123 2.571 .05 1.103 1.158 1.025 1.016 .6 2.757 1.966 .2 4.147 2.581 ).l 1.210 1.331 1.049 1.032 .7 2.775 1.975 .4 4.171 2.591 .15 1.323 1.521 1.072 1.048 .8 2.793 1.983 .6 4.195 2.601 1.2 1.440 1.728 1.095 1.063 .9 2.811 1.992 .8 4.219 2.611 .25 1.563 1.963 1.118 1.077 8. 2.828 2.000 18. 4.243 2.621 5.3 1.690 2.197 1.140 1.091 .1 2.846 2.008 .2 4.266 2.630 .35 1.823 2.460 1.162 1.105 .2 2.864 2.017 .4 4.290 2.640 1.4 1.960 2.744 1.183 1.119 .3 2.881 2.025 .6 4.313 2.650 .45 2.103 3.049 1.204 1.132 .4 2.898 2.033 .8 4.336 2.659 1.5 2.250 3.375 1.225 1.145 .5 2.915 2.041 19. 4.359 2.668 .55 2.403 3.724 1.245 1.157 .6 2.93S 2.049 .2 4.382 2.67tt 1.6 2.560 4.096 1.265 1.170 .7 2.950 2.057 .4 4.405 2.687 ,65 2.723 4.492 1.285 1.182 .8 2.966 2.065 .6 4.427 2.696 ).7 2.890 4.913 1.304 1.193 .9 2.983 2.072 .8 4.450 2.705 .75 3.063 5.359 1.323 1.205 9. 3. 2.080 20. 4.472 2.714 1.8 3.240 5.832 1.342 1.216 .1 3.017 2.088 .2 4.494 2.723 .85 3.423 6.332 1.360 1.228 .2 3.033 2.095 .4 4.517 2.732 1,9 3.610 6.859 1.378 1.239 .3 3.050 2.103 .6 4.539 2.741 .95 3.803 7.415 1.396 1.249 .4 3.066 2.110 .8 4.561 2.750 til. 4.000 8.000 1.414 1.260 .5 3.082 2.118 21. 4.583 2.759 .1 4.410 9.261 1.449 1.281 .6 3.098 2.125 .2 4.604 2.768 .2 4.840 10.65 1.483 1.301 .7 3.114 2.133 .4 4.626 2.776 .3 5.290 12.17 1.517 1.320 .8 3.130 2.140 .6 4.648 2.785 .4 5.760 13.82 1.549 1.339 .9 3.146 2.147 .8 4.669 2.794 2.802 .5 6.250 15.63 1.581 1.357 10. 3.162 2.154 22. 4.690 .6 6.760 17.58 1.612 1.375 .1 3.178 2.162 .2 4.712 2.810 .7 7.290 19.68 1.643 1.392 .2 3.194 2.169 .4 4.733 2.819 .8 7.840 21.95 1.673 1.409 .3 3.209 2.176 .6 4.754 2.827 .9 8.410 24.39 1.703 1.426 .4 3.225 2.183 .8 4.775 2.836 8. 9. 27. 1.732 1.442 .5 3.240 2.190 23. 4.796 2.844 .1 9.61 29.79 1.761 1.458 .6 3.256 2.197 .2 4.817 2.852 .2 10.24 32.77 1.789 1.474 .7 3.271 2.204 .4 4.837 2.860 .3 10.89 35.94 1.817 1.489 .8 3.286 2.210 .6 4.858 2.868 .4 11.56 39.30 1.844 1.504 .9 3.302 2.217 .8 4.879 2.876 .5 12.25 42.88 1.871 1.518 11. 3.317 2.224 24. 4.899 2.884 .6 12.96 46.66 1.897 1.533 .1 3.332 2.231 .2 4.919 2.892 .7 13.69 50.65 1.924 1.547 .2 3.347 2.287 .4 4.940 2.900 .8 14.44 54.87 1.949 1.560 .3 3.362 2.244 .6 4.960 2.908 .9 15.21 59.32 1.975 1.574 .4 3.376 2.251 .8 4.980 2.916 4. 16. 64. . 2. 1.587 .5 3.391 2.257 25. 5. 2.924 .1 16.81 68.92 2.025 1.601 .6 3.406 2.264 .2 5.020 2.932 .2 17.64 74.09 2.049 1.613 .7 3.421 2.270 .4 5.040 2.940 .3 18.49 79.51 2.074 1.626 .8 3.435 2.277 .6 5.060 2.947 .4 19.36 85.18 2.098 1.639 .9 3.450 2.283 .8 5.079 2.955 .5 20.25 91.13 2.121 1.651 12. 3.464 2.289 26. 5.099 2.962 .6 21.16 97.34 2.145 1.663 .1 3.479 2.296 .2 5.119 2.970 .7 22.09 103.8 2.168 1.675 .2 3.493 2.302 .4 5.138 2.978 .8 23.04 110.6 2.191 1.687 .3 3.507 2.308 .6 5.158 2.985 .9 24.01 117.6 2.214 1.698 .4 3.521 2.315 .8 5.177 2.993 ,1 25. 125. 2.236 1.710 .5 3.536 2.321 27. 5.196 3.000 26.01 132.7 2.258 1.721 .6 3.550 2.327 .2 5.215 3.007 .2 27.04 140.6 2.280 1.732 .7 3.564 2.333 .4 5.235 3.015 .3 28.09 148.9 2.302 1.744 .8 3.578 2.339 .6 5.254 3.022 ,4 29.16 157.5 2.324 1.754 .9 3.592 2.345 .8 5.273 3.029 .6 30.25 166.4 2.345 1.765 13. 3.606 2..351 28. 5.292 8.037 .8 S1.3« 176.6 2.366 1.776 .2 3.633 2.363 .2 &.310 3.044 S9e TABLE XXIX.— SQUARES, CUBES, AKD ROOTS. TABL.E of Squares, Cubes, Square Roots, and Cube Roets« of ^''umbers from 1 to 1000. Kemabk OS THB poLLowiNs Taelk. Wherever the effect of a fifth decimal in the roots would be t* idd I to tte fourth and final decimal in the table, the addition has been made. No errors. No. Square. Cube. Sq. Kt. C. Bt. No. Square Cube. Sq. Kt. C. Kt. 1 1 1 1.0000 1.0000 61 3721 226981 7.8102 3.9365 2 4 8 1.4142 1.2599 62 3844 238328 7.8740 3.9579 3 9 27 1.7321 ■ 1.4422 63 3969 250047 7.9373 3.9791 4 16 64 2.0000 1.5874 64 4096 262144 8.0000 4. 5 25 125 2.2361 1.7100 65 4225 274625 8.0623 4.0207 e 36 216 2.4495 1.8171 66 4356 287496 8.1240 4.0412 7 49 343 2.6458 1.9129 67 4489 300763 8.1854 4.0615 8 64 512 2.8284 2.0000 68 4624 314432 6.2462 4.0817 8 81 729 3.0000 2.0801 69 4761 328509 8.3066 4.1016 M 100 1000 3.1623 2.1544 70 4900 343000 8.3666 4.1213 11 121 1331 3.3166 2.2240 71 5041 357911 8.4-261 4.1408 12 144 1728 3.4641 2.2894 72 5184 373248 8.4853 4.1602 13 169 2197 3.6056 2.3513 73 5329 389017 8.5440 4.1793 14 196 2744 3.7417 2.4101 74 5476 405224 8.6023 4.1983 15 225 3375 3.8730 2.4662 75 5625 421875 8.6603 4.2173 16 256 4096 4.0000 2.5198 76 5776 438976 8.7178 4.2358 17 2S9 4913 4.1231 2.5713 77 5929 456533 8.7750 4.-2543 18 324 5&32 4.2426 2.6207 78 6084 474552 8.8318 4.2727 19 361 6859 4.3589 2.6684 79 6241 493039 8.8882 4.-2908 •X 400 8000 4.4721 2.7144 80 6400 512000 8.9443 4.3089 21 441 9261 4.5826 2.7589 81 6561 531441 9. 4.3267 22 484 10648 4.6904 2.8020 82 6724 551368 9.0554 4.3445 23 529 12167 4.7958 2.8439 83 6889 5e Roots, of 3f uiubers irom 1 to 1000 — (^Cominukd.) No. Square. 901 90-2 903 905 906 907 908 909 910 911 312 913 914 915 910 917 918 919 920 921 922 923 921 925 926 927 928 929 930 931 932 933 931 935 936 93- 938 939 910 911 912 913 914 945 916 917 918 9*9 950 811801 813604 815109 817216 819025 820836 822619 821164 826281 828100 829921 831714 833569 835396 837225 839056 810H89 812721 811561 816100 818211 850084 851929 853776 855625 857176 859329 861181 863011 861900 866761 868621 870189 872356 871225 876096 877969 879^11 881721 883600 885181 887361 889219 891 136 893025 891916 896S09 898701 900601 902500 Cube. 731432701 733870808 736314327 738763264 741217625 743677416 746142643 748613312 751089129 753571000 756058031 758550528 761048497 763551944 766060875 768575296 771095213 773620632 776151559 778688000 781229961 783777448 786330167 788889024 791453125 794022776 796597983 799178752 801765089 804357000 806954491 809557568 812166237 -14780501 817400375 820025856 822656!)53 825293672 827936019 830584000 833237621 835S968.Si8 83S501hO7 8U23238J 843908C25 846590536 81927.H123 851971392 8546703 19 857375000 Sq. Rt. 30.0167 30.0333 30.0500 30.0666 30.0832 30.0998 30.1164 30.1330 30.1496 30.1662 30.1828 30.1993 30.2159 .30.2324 30.2490 30.2655 30.2820 30.2985 .30.3150 30.3315 .30.3180 30.3645 30.3809 30.3974 30.4138 .30.4302 30.4467 30.4631 30.4795 30.4959 30.5123 30.5287 30.5450 .30.5614 30.5778 30.5941 30.6105 30.6268 30.6131 30.6594 30.6757 30.6920 30.7083 30.7246 30.7409 30.7571 30.7734 30.7896 30.8058 30.8221 C. Rt. No. 9.6585 9.6620 9.6656 9.6092 9.6727 9.6763 9.6799 9.6834 9.6870 9.6905 9.6941 9.6976 9.7012 9.7047 9.7082 9.7118 9.7153 9.7188 9.7224 9.7259 9.7294 9.7329 9.7364 9.7400 9.7435 9.7470 9.7505 9.7510 9.7575 9.7610 9.7615 9.7680 9.7715 9.7750 9.7785 9.7819 9.7854 9.7889 9.7921 9.7959 9.7993 9.8028 9.80G3 9.8097 9.8132 9.8167 9.8201 9.8236 9.8270 9.8305 951 952 953 954 955 956 957 958 959 960 961 962 9G3 964 965 966 967 968 969 970 971 972 973 974 975 976 977 978 979 980 981 982 983 984 985 986 987 988 989 990 991 992 993 994 995 996 997 998 999 1000 Square. 901401 906304 908209 910116 912025 913936 915849 917764 919681 921600 923521 925444 927369 929296 931225 9331.56 935089 937024 938961 940900 942S41 911781 916729 918676 950625 952576 951529 956184 958141 960400 962361 961321 966289 968256 , 970225 972196 971169 976144 978121 980100 982081 981064 986049 988036 990025 992016 994009 996001 998001 1000000 Cube. 860085351 862401408 865523177 868250664 870983875 873722816 876467493 879217912 881974079 884736000 887503681 890277128 893056347 895841344 898632125 901428696 904231063 907039232 909853209 912673000 915498611 918330048 92116731 924010424 926859375 929714176 932574833 935441352 938313739 941192000 944076141 946966168 949862087 952763904 955671625 958585256 961504803 9644.30272 967361669 970299000 973242271 976191488 979146657 982107784 985074875 988047936 991026973 991011992 997002999 1000000000 Sq. Rt. 30.8383 30.8545 30.8707 30.8869 30.9031 30.9192 30.9354 80.9516 30.9677 30.9839 31. 31.0161 31.0322 31.0483 31.0644 31.0805 31.0966 31.1127 31.1288 31.1448 31.1609 31.1769 31.1929 31.2090 31.2250 31.2410 31.2570 31.2730 31.2890 31.3050 31.3209 31.3369 31.3528 31.3688 31.3847 31.4006 31.4166 31.4325 31.4484 31.4643 31.4802 31.4960 31.5119 31.5278 31.5436 ,31.5595 31.5753 31.5911 31.6070 31.6228 C. Rt. 9.8.339 9.8374 9.8408 9.8tl3 9.8477 9.8511 9.8546 9.8580 9.8614 9.8618 9.8683 9.8717 9.8751 9.8785 9.8819 9.8854 9.8888 9.8922 9.8956 9.8990 9.9024 9.9058 9.909-2 9.9126 9.9160 9.9194 9.9227 9.9261 9.9295 9.9329 9.9.363 9.9396 9.9430 9.9164 9.9497 9.9531 9.9565 9.9598 9.9632 9.9606 9.9699 9.9733 9.9766 9.9800 9.9833 9.9866 9.9900 9.99.33 9.9967 10. lo find the square or ceibe of any nliole number ending wilBi ciphers. Fir.st, omit all the fiual tiphcr.s. Take from the table the .square or cufic (as the case may be) of the rest of the number. To this square add ttvice as many ciphers as there were tiuul ci|ihers in the orignial number. To the ciihc aiid tliree times .isfinany :i3 in the original number. Thus, for 905(Kr^ ; 90o2 — ,sl9025. Add twice 2 ci|ihers. obtaining 8ii(025i;UOO. For 905003, 9053 = 711217625. Add 3 times 2 ciphers, obtaining 7412176250ii(MlOO. e,-^/eV' o/^ ^^^'^y ^.X,-^ A:- c:^, -^i^ y... „v-x- , J . * r v ■■ ■ » :r.. -r-. 4 ,4c % ■A;-