m 2 - < u tn * J- CM o 2 L LJ H. TF 34- PBEFACE. ALTHOUGH the modern railway system is but about fifty years old, yet its growth has been so rapid, and the progress in the science of railway construction so great, as to render the earlier technical books on this subject inadequate to the needs of the engineer of to-day. In the course of his practical experience as a railway engi- neer, the author was strongly impressed with the want of a more complete hand-book for field use, and finally concluded, at the solicitation of his friends, to undertake the preparation of the present volume. The aim in this work has been: First To present the general subject of railway field work in a progressive and logical order, for the benefit of beginners. Second To classify the various problems presented, so that they may be readily referred to. Third To embrace discussions of all the more important practical questions while avoiding matters non-essential. Fourth To employ throughout the work a uniform and systematic notation, easily understood and remembered, so that after one perusal the formula may be intelligible at a glance wherever referred to. Fifth To express the resulting formula of every problem in the shape best adapted to convenient numerical compu- tation. Sixth To furnish a large variety of useful tables, more com- plete and extended than any heretofore published, especially adapted to the wants of the field engineer. An elementary knowledge of algebra, geometry and trigono- metry on the part of the reader has been taken for granted, as a command of these instrumentalities is deemed essential to the education of the civil engineer. The few references to mechanics, analytical geometry, optics and the calculus may be assumed correct by those not conversant with these branches. 1823 ** IV PREFACE. Many of the problems in curves are new, yet there is hardly one that has not presented itself to the author in the course of his practice. The investigation of the valvoid curve is original, and though the mathematical discussion is somewhat difficult, yet the resulting formulae, taken in connection with Table X, are exceedingly simple and convenient for the solution of a certain class of problems. The treatment of compound curves is novel and exhaustive. A few general equations are established, which, by slight modifications, solve all the problems that can occur. No discussion of reversed curves is given, because these .are inconsistent with good practice, except in turnouts, under which head they are noticed. The chapter on levelling includes a discussion of stadia measurements, with practical formula. The chapter on earth- work contains a review of several methods for calculating quantities, and states the conditions under which these suc- ceed or fail in giving correct results. Among the tables, numbers 3, 5, 6, 10, 18, 19, 26 and 29 are original. The adoption of versed sines and external secants throughout the work, wherever these would simplify the formulae, rendered necessary the preparation of tables of these functions. The table of logarithmic versed sines and external secants has been computed from ten-place logarithmic tables of sines and tangents, so that the last decimal is to be relied on, and no pains have been spared to make the table thoroughly accurate. Tables numbers 4, 7, 8, 9, 11, 12, 13, 14 and 30 have been recalculated, enlarged, and some of them carried to more deci- mal places than similar tables heretofore published. The intention has been to give one more decimal than usual, so that in any combination of figures the result of calculation might be reliable to the last figure usually required. The tables which have been compiled and rearranged are numbers 1, 2, 15, 16, 17, 24, 25 and 31. The tables of log. sines and tangents here given are the only six-place tables which give the differences correctly for seconds. The table of logarithms of numbers is accompanied by a complete table of proportional parts, which greatly facilitates interpolation for the fifth and sixth figures. In all the tables, whether new or old, scrupulous care has PREFACE. V been taken to make the last figure correct, and the greatest diligence lias been exercised by various checks and compari- sons to eliminate every error. It is, therefore, hoped and believed that a very high degree of accuracy has been ob- tained, and that these tables will be found to stand second to none in this respect. The preparation of this work has extended over several years, as time could be spared to it from other engagements. It is, therefore, the expression of deliberate thought, based on experience, and as such is submitted to the judgment of brother engineers. If it shall prove to have even partially met the aim herein announced, and so shall serve to smooth the way of the ambitious student, or to assist the expert in his responsible duties, the labors of the author will not have been in vain. WM. IL SEARLES, C.E. NEW YORK, March 1st, 1880. TABLES. I. Geometrical Propositions 271 II. Trigonometrical Formulae 273 HI. Curve Formulae 277 IV. Radii, Offsets, and Ordinates 280 V. Corrections for Tangents and Externals 288 VI. Tangents and Externals to a One-Degree Curve 289 VTI. Long Chords and Actual Arcs 293 VIII. Middle Ordinates to Long Chords 298 IX. Linear Deflections 301 X. Curved Deflections; Valvoid Arcs 302 XI. Frog Angles and Switches 393 XII. Middle Ordinates for Rails 304 XIII. Difference in Elevation of Rails 304 XIV. Grades and Grade Angles .". 305 XV. Barometric Heights, in feet 307 XVI. Coefficient of Correction for Atmospheric Temperature.... 309 XVII. Correction for Earth's Curvature and Refraction 309 XVIII. Coefficient for Reducing Stadia Measurements 310 XIX. Logarithmic Coefficients, Stadia Measurements 311 XX. Lengths of Circular Arcs 312 XXI. Minutes in Decimals of a Degree 313 XXII. Inches in Decimals of a Foot 314 XXIII. Squares, Cubes, Roots and Reciprocals 315 XXIV. Logarithms of Numbers 333 XXV. Logarithmic Sines, Cosines, Tangents, and Cotangents. 359 XXVI. Logarithmic Versed Sines, and External Secants, 404 XXVII. Natural Sines and Cosines 449 XXVHI. Natural Tangents and Cotangents 458 XXIX. Natural Versed Sines and External Secants 470 XXX. Cubic Yards per 100 Feet in Level Prismoids 493 Useful Numbers and Formulae 498 CONTENTS. CHAPTER I. RECONNOISSANCE. SECTION PAGE 2. Topographical considerations 1 6. Use of maps 3 7. Pocket instruments 3 9. Aneroids 4 10. Formulae for aneroid 4 16. Locke level 7 17. Prismatic compass 7 CEJATTER II. PRELIMINARY SURVEY. 18. Definitions 8 19. Engineer corps 8 20. Chief engineer 8 21. Assistant engineer 9 22. Chainman 10 24. Axeman 11 26. Topographer 11 28. Leveller 13 29. Rodmen 13 32. Thecompass 14 33. The chain 14 35. The level 15 36. Therods 15 38. The clinometer 16 39. The plane-table 16 40. The transit 16 42. Transit points 17 43. Transit flags 17 44. Obstacles to alignment and measurement : 18 45. Parallel lines 18 46. Lines at a small angle - . . . . 18 47. General problem 19 48. Lines at a large angle 20 49. Selection of angles 21 53. Rocky shores; Tie-lines 22 54. System of plotting map 23 Vlll CONTENTS. CHAPTER III. THEORY OF MAXIMUM ECONOMY IN GRADES AND CURVES. SECTION PAGE 55. Choice of routes 24 56. Statement of the problem 24 57. Engine traction 25 58. Engine expense 26 60. Resistance to motion 27 61. Resistance due to grade 27 62. Resistance due to curve 28 63. Formulae for resistance 28 64. Formulae for maximum trains 21? 65. Engine-stage 30 66. Graphical solution 32 67. Train load reciprocals 33 68. Reduction of grades on curves 33 69. Example 33 70. Assisting engines 36 71. Maximum return grades 36 72. Undulating grades 37 75. Comparison of routes :~ 38 76. Value of distance saved 39 77. Conclusion 39 CHAPTER IV. LOCATION. 78. General remarks 39 79. Long tangents 39 80. Leveller's duties; Profiling 40 81. Establishing grade lines 41 CHAPTER V. SIMPLE CURVES. A. Elementary Relations, 82. Limits to curves and tangents 42 83. Definition of terms 42 84. Radius and degree of curve 43 65. Measurement of curves 44 8G. Approximate value of R 44 87. Central angle and length of curve 45 88. Definition of other elements 4* 90. Formula for tangent distance T 46 91. Formula for long chord C .- 47 92. Formula for middle ordinate M 48 93. Formula for external distance E 48 95. Formula for radius in terms of Tand A 49 96. Formula for external distance in terms of T and A 50 97. Formula for radius in terms of E and A 50 CONTENTS. SECTION PAGE 98. Formula for tangent distance in terms of E and A ....... ...... 51 99. To define the curve of an old track ............................... 51 100. Other curve formula; Table III .................................. 52 B. Location of Curves by Deflection Angles. 101. Deflection angles ................................................. 52 102. Rule for deflections ..................... , ........................ 53 103. Rule for finding direction of tangent at any point .............. 53 104. Subchords ......................................................... 54 105. Deflections for subchords ......................................... 54 106. Correction for subchords ......................................... 55 107. Ratio of correction to excess of arc .............................. 55 108. Transit work on curves ........................................... 57 109. Field notes ........................................................ 58 110. Central angle in terms of deflections ............................. 58 111. Method by deflections only ...................................... 58 C. Location of Curves by Offsets. 112. Fourmethods .................................................... 59 113. By offsets from the chords produced ............................. 59 114. Do. beginning with a subchord ................................... 60 115. Formula for subchord offsets, approximate ..................... 61 117. By middle ordinates .............................................. 61 1 18. Do. beginning with a subchord ................................... G2 119. By tangent offsets ........................................ . ....... 62 120. Do. beginning with a subchord ................................... 64 121. By ordinates from a long chord .................................. 64 122. Do. for an even number of stations ............................... 65 123. Do. for an odd number of stations ................................ 66 124. Do. for an even number of half stations ......................... 67 125. Do. beginning with any subchord ................................ 67 126. Erecting perpendiculars without instrument ..................... 69 D. Obstacles to the Location of Curves. 127. The vertex inaccessible ........................................... 69 128. The point of curve inaccessible ................................... 70 129. The vertex and point of curve inaccessible ...................... 70 130. The point of tangent inaccessible ......................... ....... 71 131. To pass an obstacle on a curve ................................... 72 E. Special Problems in Simple Curves. 132. To find the change in R and E for a given change in 'T. .......... 73 133. To find the change in R and T for a given change in E. .......... 74 134. To find the change in T and E for a given change in R ........... 75 135. General expression for elementary ratios ........................ 75 136. To find a new point of curve for a parallel tangent .............. 76 137. To find a new radius for a parallel tangent ...................... 76 138. To find new P. C. and new radius for a parallel tangent .......... 77 X COKTEKTS. SECTION PAGE 139. To find new tangent points for two parallel tangents 78 140. To find new R and P. C, for new tangent at same P. T. 80 141. To find new P. C. for a new tangent from same vertex 81 142. To find new radius for a new tangent from same vertex 81 143. To find new R and P.O. for same external distance, but new A . 82 144. To find a curve to pass through a given point 83 145. To find new radius for a given radial offset 84 146. Equation of the valvoid- 86 147. To find direction of a tangent to the valvoid at any point 87 148. To find the radius of curvature of the valvoid at any point 88 149. To find the length of arc of the valvoid 88 150. To find new position of any stake for a new radius from same P.C 89 151. To find new radius from same P. 0. for new position of any station 92 152. To find distance on any line between tangent and curve 93 153. To find a tangent to pass through a distant point 94 154. To find a line tangent to two curves 96 155. To find a line tangent to two curves reversed 98 156. Study of location on preliminary map ; Templets ; Table of con- venient curves ". 100 CHAPTER VI. COMPOUND CURVES. A. Theory of Compound Curves. 157. Definition 102 158. The circumscribing circle 102 159. The locus of the point of compound curve : 103 160. The inscribed circle of the principal radii 104 Cor. 2. Maxima and minima of the radii 104 B. General Equations. 161. Formula for radii, central angles, and sides 105 162. Given: Si S a A and R^ to find A x A 2 and R z 106 163. Given: AB, VAB, VBA and R 3 , to find A 2 A l a.ndR l 107 164. Given: 7? t A ^ R 2 A 2 , to find the triangle VAB 108 165. Given: A , the radii, and one side to find the other. . . 108 166. Given:' one side, radius and central angle to find the others 110 167. Remarks on special cases Ill 168. Obstacles; theP.C.C. inaccessible 112 C. Special Problems in Compound Curves. 169. To find a new P.C.C. for a parallel tangent 113 170. To find a new P.C.C. and last radius for a parallel tangent 115 171. To find a new P.C.C. and last radius for the same tangent 118 172. To find a new P.C.C. and last radius R a ' for new direction of tangent through same P.T 121 COKTEKTS. XI SECTION PAGE 173. To find a new P.O. C. and Jast radius -R/ for'new direction of tangent through same P.T 124 174. To replace a simple curve by a three-centred compound curve between the same tangent points 127 175. To find the distance between the middle points of a simple curve and three-centred compound curve 129 176. To replace a simple curve by a three-centred compound curve passing through the same middle point 129 I. The curve flattened at the tangents 129 II. The curve sharpened at the tangents 132 177. To replace a tangent by a curve compounded with the adjacent curves 134 I. When the perpendicular offset p is assumed 136 II. When the angle ct or ft is assumed 137 III. When the radius R z is assumed 137 IV. Locus of the centre O 2 138 178. To replace the middle arc of a three-centred compound by an arc of different radius 140 I. When the radius of the middle arc is the greatest 140 II. When the radius of the middle arc is the least 141 III. When the radius of the middle arc is intermediate 142 CHAPTER VII. TURNOUTS. 179. Definitions; Frogs and switches 147 180. Single tuwiout from straight track in terms of frog angle 148 181. Single turnout from straight track in terms of frog number 149 182. Double turnout, middle track straight, to calculate F" 151 183. Double turnout, middle track straight, and three given frogs. . . 152 184. Double turnout on same side of straight track to calculate the middle frog, F" 153 185. Double turnout on same side of straight track with three given frogs 155 a. When the middle track is a simple curve 155 6. When the middle track is straight beyond F 158 c. When the middle track is reversed at F. 159 186. Turnout on the inside of a curved track 101 187. Turnout on the outside of a curved track 1G3 188. Tongue switches 164 189. Tongue switch turnout from a straight track '. 164 190. Tongue switch double turnout to find F" 165 191. Tongue switch double turnout with three given frogs 166 192. Tongue switch double turnout on same side of straight track with three given frogs 167 a. The middle track reversed at F 167 6. The middle track compounded at F 168 c. The middle track straight beyond F 168 Xll CONTENTS. SECTION PAGE 193. To find the reversed curve for parallel siding in terms of .Fand perpendicular distance p 169 194. To find the connecting curve from frog to parallel siding on a curve in terms of F and perpendicular distance p 170 a. The siding outside of main track 171 b. The siding inside of main track 171 195. To locate a crossing between parallel tracks 172 196. To locate a reversed curve crossing between straight tracks 173 197. To locate a reversed curve crossing between curved tracks 174 198. To find the middle ordinate m, for one station in terms of D 175 199. To find the middle ordinate mi for rails, in terms of rail and R.. 175 200. Curving rails ; To find mi in terms of rail and m 176 201. To find elevation of outer rail on curves 177 202. To find a chord whose middle ordinate equals the proper eleva- tion 179 203. General remarks on elevation of rail 179 204. General remarks oa coned wheels 180 CHAPTER VIIL LEVELLING. 205. Use of the engineers' level 181 206. The datum, how assumed 181 207. Benches, how used; B.M 181 208. Height of instrument; H.I 182 09. Reading of the rod 182 210. Elevation of intermediate points 182 211. Turning points ; T.P. 182 212. Rule for backsights and foresights .* 183 213. Form of field-book; proof of extensions 183 214. Profiles 184 215. Simple levelling; test levels 185 216. Errors in reading, 'due to the level ; how avoided 185 217. Errors in reading, due to the rod ; how avoided 185 218. Errors due to curvature of the earth 1 86 219. Errors due to refraction 187 220. Radius of curvature of the earth 187 221. Levelling by transit or theodolite 188 222. To find the H. I. by observation of the horizon 189 223. Stadia measurements ; horizontal sights 191 224. Stadia measurements ; inclined sights, vertical rod 193 225. Stadia measurements; inclined sights, inclined rod 1&5 CHAPTER IX. CONSTRUCTION. 228. Organization of engineer department 196 227. Clearing and grubbing 197 228. Test levels and guard plugs , 197 CONTEXTS. xiii SECTION PAGE 2^9. Cross sections; Slopes 197 230. Cross sections, formulae for 198 231. Cross sections, staking out 200 232. Cross sections on irregular ground 201 233. Cross sections on side-hill work 201 234. Compound cross sections , 202 235. Selection of points for cross sections 203 236. Vertical curves 203 237. Form of cross-section book 204 238. Extended cross profiles 205 239. Inaccessible sections 205 240. Isolated masses 206 211. Borrow-pits 206 242. Shrinkage ; Increase 206 243. Office-work 207 244." Alteration of line 207 245. Drains and culverts 208 246. Arch culverts 209 247. Foundation pits; Bridge chords on curves 210 248. Cattle-guards 214 249. Trestle-work 214 250. Tunnels: Location; Alignment; Shafts; Curves ; Levels ; Grades; Sections; Rate of progress; Ventilation; Drain- age 216 251. Retracing the line 222 252. Side ditches and drains 223 253. Ballasting 223 254. Track-laying ; Expansion of rails ; Sidings 223 CHAPTER X. CALCULATION OF EARTHWORK. 254. Prismoids ; Choice of cross sections 225 255. Formulae for sectional areas 227 256. Prismoidal formulae for solid contents 229 257. Tables of quantities in cubic yards 229 258. Tables of equivalent depths 231 259. Formula for equivalent depth in terms of slope angle '. 232 260. Conditions necessary for correct results in use of tables 233 261. Method of mean areas ; correction required 233 262. Exact calculation of content; examples 234 2C3. Wedges and pyramids 236 264. Side-hill sections, uniform slope '. 236 265. Side-hill sections, irregular ground 237 206. Side-hill sections in terms of slope angle 237 207. Systems of diagrams 238 268. Correction for curvature in earthwork 239 269. Haul ; Centre of gravity of prismoid 243 270. Final estimate , 245 271. Monthly estimates. . , 246 XIV COHTEKTS. CHAPTER XI. TOPOGRAPHICAL SKETCHING. SECTION PAGE 272. General remarks 247 273. Artificial features 248 274. Natural features ; Contours ; Hatchings 248 275. Method of sketching 249 CHAPTER XH. ADJUSTMENT OF INSTRUMENTS. 276. The transit 250 277. The level 252 278. The theodolite 253 CHAPTER XIII. EXPLANATION OF THE TABLES 253 FIELD CHAPTER I. RECONNOISSANCE. 1. The engineering operations requisite to and preceding the construction of a railroad are in general : THE RECONNOISSANCE, THE PRELIMINARY SURVEY, and THE LOCATION. 2. The Reconnaissance is a general and somewhat hasty examination of the country through which the proposed road is to pass, for the purpose of noting its more prominent features, and acquiring a general knowledge of its topography with reference to the selection of a suitable route. The judicious selection of a route may be a very simple or com- plex problem, depending on the character of the topography, and more especially on the direction of the streams and ridges as compared with the general direction of the proposed road. 3. A road running along a water-course is most easily located. In this case the choice is to be made merely between the two banks of the stream, or between keeping one bank continuously and making occasional crossings. "When the stream is small it will usually be found best to cross it at intervals, the advantage of direct alignement outweighing the cost of bridging; but when the stream is of considerable size the solution of the problem is not so obvious, requiring patient comparison of results in the two cases to determine whether to cross or not, while in the case of the larger rivers crossing may be out of the question. When there is a choice of sides, both banks, should be traversed by the engineer on reconnoissance, and while exam- ining in detail the one side he should take a general and com- prehensive view of the other. Only thus can he gain a complete knowledge of either side. The points to be considered are the relative value of the property on either side, the number and FIELD ENGINEERING. size of tributary streams, and probable cost of crossing them, the cost of graduation as affected by the amount and character of the material to be removed, and the liability to land slides, the amount and degree of curvature required, and the proba- ble revenues which the road can command If, in respect to these points, one bank of the stream gives the more favorable result all the "way, the question, is decided at once; but in case the greater inducements are found on either bank alter- nately, as usually happens, the propriety of bridging the stream, with the costs and advantages, must be considered as an additional element in the problem. 4. When no water-course offers along which the road may be located, the difficulties of selecting a route are increased, and these usually become greatest when the streams are found to run about at right angles to the direction of the road. Val- leys and ridges are to be crossed alternately, involving the necessity of ascending and descending gradejs, diverting the road from a straight line, and increasing the distance and cur- vature. The engineer must now seek the lowest points on the ridges, and the highest banks at the stream crossings, in order to reduce as much as possible the total rise and fall, but these points must be so chosen relatively to each other as to admit of their being connected by a grade not exceeding the maxi- mum which may be allowable. The intervening country between summit and stream must usually be carefully exam- ined, even on reconnoissance, to determine where the assumed grade will find sustaining ground at a reasonable expense for graduation and right of way. In selecting stream crossings, regard should be had not only to the height of the bank, but also to the character of the bot- tom, its suitability for foundations, and its liability to be washed by the current. The direction and force of the cur- rent should be observed, and its behavior during freshets, and the extremes of high and low w r ater ascertained, if possible. An approximate estimate of the cost of bridging may be made. 5. The engineer should not only seek the best ground on the route first assumed, but should have an eye to all other possi- ble routes, holding them in consideration pending his accu- mulation of evidence, and being ready, finally, to adopt that one which promises the greatest ultimate economy. He should be able to read the face of the country like a map, and to RECCWNOISSANCE. 3 . carry in his mind a continuous idea or image of any line he is ex- amining, so as to judge with tolerable accuracy of the influence any one portion of the line may have on another as to align- ment and grade, even though many miles apart. In the success- ful prosecution of a reconnoissance he must depend mainly on his own natural tact and a judgment matured by experience. 6. The engineer will bring to his aid in the first place the most reliable maps, and those drawn on the largest scale. The sectional maps of United States surveys will be found very useful when they exist. In addition to these it is often desira- ble to prepare a map on a scale of one or two inches to a mile, on which will be drawn the principal features of the country to be traversed, such as streams, roads, towns, and the princi- pal ridges, if known, but leaving the further details to be filled in by the engineer as he progresses. Such a map furnishes a cor- rect scale for his sketches, and saves much valuable time, as he has only to sketch what the map does not contain, and occa- sionally to make corrections when he finds the map to be in error. He also notes on the map the governing points of the route, such as the best crossings of streams, ridges, or other roads, and any point where the line will evidently be com- pelled to pq,ss He may then indicate the route by a dotted line on the map drawn through the governing points. Having traversed the route in one direction he should retrace his steps, verifying or correcting his observations, and making such further notes as seem important. When in a densely wooded country, with but few openings, it may be impossible for him to get a commanding view from any point that will afford him the necessary information as to the general topography. He must then depend largely upon instrumental observations, taking these more frequently, and noting carefully all details likely to prove useful in future surveys. 7. The instruments required on an extended recon- noissance are the barometer and thermometer, the hand or Locke level, a pocket or prismatic compass, and fl telescope or strong field-glass. To these may be added a telemeter for measuring distances at sight, but when good maps are to be had this instrument is seldom needed. So also some portable astronomical instruments are necessary in a new country, for determining latitude and longitude, but would only be a use- less iucumbrance in a settled district. 4 FIELD ENGINEERING. 8. The mercurial barometer has generally been relied upon for the determination of heights, but owing to its inconvenient dimensions and the danger of breaking, it is now discarded by railroad engineers in favor of the more portable aneroid barometer, except in the case of trans-continental surveys, and when astronomical instruments are to be used also. 9. The best aneroids are designed to be self compen- sating for temperature, so that with a constant atmospheric pressure the reading shall be the same at all temperatures of the instrument. This, however, being a very delicate adjustment, is not always successfully made, so that each instrument is lia- ble to have a small error due to temperature peculiar to itself. This error will be found rarely to exceed one hundredth of an inch, plus or minus, per change of ten degrees Fah., and is frequently much less than this. Just wliat the error is in a particular instrument may be determined by careful compari- son with a standard mercurial barometer at the extremes of temperature, assuming the error found as proportional to the diiference of temperature for all intermediate degrees of heat. The error having been determined for any aneroid, it should be applied, with its proper sign, to every reading to obtain the true reading. The sizes generally used are If and 2| inches in diameter, respectively, and experience seems to prove that there is no advantage in using larger sizes, but rather the contrary. 1C. The ordinary barometric formula3 and tables have been prepared with reference to the mercurial barometer. In order that they may apply to the aneroid, it is. necessary that the latter should be adjusted to read inches of mercury identically with the mercurial column at the sea level at a temperature of 32 Fah. But as the aneroid, unlike the mercurial column, requires no correction for latitude, nor for the variation in the force of gravity due to elevation, that portion of the formula which provides for such corrections, as well as that which provides for a correction due to the temperature of the instrument itself, may be omitted when using an aneroid. Thus the general formula is very much simplified, and be- comes , = log 608818 '< .<< + <'- 64 h' \ 900 KECOKKOISSA^CE. 5 in which 7i, and h' are the readings of the aneroid in inches, and t t and t' the readings of a Fahrenheit thermometer at the lower and upper of any two stations respectively, and z is the difference in elevation in English feet of those stations. To facilitate the calculation of heights by this formula, we may write Log |j 60384.3 = [log h, - log h 1 ] 60384.3 and since only the difference of the logs, is required, this will not be affected, if we subtract unity from each. The quan- tities in Table XV. are prepared, therefore, by the formula (log 7i 1)60384. 3 for every ^ths of an mcn from 19 inches to 31 inches. Table XVI. contains values of -i^t - for every de- yUU gree of (t, + t') from 20 to 200 Fah. 11. To find the difference in elevation of any two stations by the tables : Take the difference of the quantities corresponding to Ti, and 7i' in Table XV. as an approximation, and for a correction multiply this difference by the coefficient corresponding to (,-{- 0, in Table XVI., adding or subtracting the product according to the sign of the coefficient. Example. Lower Sta. Upper Sta. in. in. Aneroid h, = 29.92 h' = 23.57 Thermometer t, = 77.6 t = 70.4 By Table XV. for 29.92 we have 28741 for 23.57 22485 Difference 6256 By Table XVI. for 77.6 + 70.4 = 148 we have -f .0933 Then 6256 X .0933 = 583.6848 and 6256 + 584 = 6840 ft. = z.Ans. 12. Certain precautions are to be observed in the use of the aneroid. When the index has been adjusted to a correct reading by means of the screw at its back, it should not be meddled with until it can again be compared with a standard mercurial barometer, and even then some engineers prefer to take note of its error, if any, rather than disturb the aneroid. 6 FIELD ENGINEERING. Again, since the principle of compensation supposes the aneroid to have a uniform temperature throughout its parts, it must be guarded against sudden changes, as otherwise the metallic case will be considerably heated or cooled before the change can affect the inner chamber, thus inducing very erro- neous results. The aneroid, therefore, should seldom be taken from its leather case, nor exposed to any radiant heat of sun or fire, nor worn so near the person as to increase its tempera- ture above that of the surrounding atmosphere. If removed to an atmosphere of decidedly different temperature, time must be allowed for the aneroid to be thoroughly permeated by the new degree of heat. The aneroid should be held with the face horizontal while being read ; it should be handled care- fully, and all concussions avoided, and it should be compared with a standard as often as practicable to make sure that it has suffered no derangement. Observing these precautions, and having a really good aneroid, the engineer should obtain excellent results in the estimation of heights. It has been found that the slight error in compensation, previously alluded to, is subject to a change during the first year or two after the instrument is made, but subsequently it becomes quite per- manent. 13. For the purpose of obtaining approximate elevations by a simple inspection of the dial, the modern aneroid is provided with a secondary scale reading hundreds of feet, which is placed outside the scale of inches. It is divided according to the following formula prepared by Prof. Airy : s = 650 3 3 ^( 1 + ^--) in which it is evident that no correction for temperature is required when the average temperature of the two stations is 50. When the two scales are engraved on the same plate the zero of the scale of feet is coincident with 31 on the scale of inches; but in some aneroids the scales are on two concentric plates, so that the zero of one may be made to coincide with any division of the other, which is in some respects an advan- tage. 14. The theory of the barometer, as expressed in the above formula, assumes the atmosphere to be at rest, and its pres- ure affected only by temperature, whereas, in fact, the pres- RECOK^OISSAKCE. 7 sure at any point is liable to sudden changes due to variations in the force of the wind, the amount of humidity, etc. The best way to eliminate errors due to these causes is to take read- ings simultaneously at the points the elevations of which are to be compared. For this purpose an assistant should be stationed at some point of known elevation contiguous to the route to be surveyed, and provided with an aneroid similar to that carried by the engineer. The aneroids, time-pieces, and thermometers having been compared at this point, the assist- ant should record the readings every ten minutes, with the time, temperature, and state of the weather. The engineer will thus have a standard with which to compare his own observations. If the survey is so extended that the same con- ditions of atmosphere are not likely to be experienced by the. two observers, the assistant should be instructed to move for- ward to a new station at a designated time; or two assistants may be employed, one at each of two stations between which the engineer intends to make a reconnoissance. Even with these precautions no attempt should be made to obtain the ele- vation of important points during, or just before, or after a storm of wind or rain. 15. When but one aneroid is used the observations at the several stations should be taken as nearly together as possible in point of time, and then repeated in inverse order, taking the mean of the observations at each station, and repeating the whole operation if necessary. Only approximate results can be hoped for, however, with a single instrument, unless the atmospheric conditions are very favorable. 10. The Locke Level is an instrument in which the bubble and the observed object may be seen at the same instant, enabling the operator to keep the instrument horizontal, while holding it in the hand, like an ordinary spy-glass. While very portable, it enables the observer to define rapidly all visi- ble points of the same elevation as his own, and to estimate from these the relative heights of other points. It may be made useful in a variety of ways which easily suggest them- selves to the engineer in cases where no great precision is required, and where a more elaborate level is not at hand. 17. The Prismatic Compass is a portable instrument with folding sights, in using which the bearing to an object may be read at the same instant that the object is observed. 8 FIELD ENGINEERING. ^ The bearings are read upon a floating card, graduated and numbered from zero to 360, so that no error can be made in substituting one quadrant for anotlier. The instrument may be held freely in the hand during an observation, though better results are obtained by giving it a firm rest. CHAPTER II. PRELIMINARY SURVEY. 18. A preliminary survey consists in an instrumental exam- ination of the country along the proposed route, for the purpose of obtaining such details of distances, elevations, topography, etc. , as may be necessary to prepare a map and profile of the route, make an approximate estimate of the cost of constructing the road, and furnish the data from which to definitely locate the line should the route be adopted. The survey is more or less elaborate, according to circumstances. In case the country is new, or the reconnoissance has been incomplete, or if several routes seem to offer almost equal inducements, the survey w r ill partake somewhat of the nature of a reconnoissance, and will be made more hastily than if but one route is to be examined, and that, perhaps, presenting serious engineering difficulties. The survey is made as expe- ditiously as possible, consistent with general accuracy, but should not usually be delayed for the sake of precision in matters of minor detail. 19. For preliminary survey the Corps of engineers is organized as follows: A chief engineer, an assistant engineer, two chainmen, one or two axemen, a stakeman, and a topographer, these forming the compass (or transit) party, to which a flagman is some- times added; a leveller and one or two rodmen, forming the level party; and to these is sometimes added a cross level party of two or three assistant rodmen. 20. The chief engineer takes command of the corps, and directs the survey. He ascertains or estimates the value of the lands passed over, the owners' names, and the boundary lines crossed by the line of survey. He examines all streams, PRELIMINARY SURVEY. 9 and estimates the size and character of the culverts and bridges which they will require; he notices existing bridges, and inquires concerning their liability to be carried away by freshet; he selects suitable sites for bridges, examines the character of the foundations, the direction of the current rela- tively to that of the line, and considers any probable change in the direction of the current during freshets; he inspects the various soils, rocks, and kinds of timber as they are met with, and takes full notes of all these and kindred items in his field book. He not unfrequently assumes in addition the duties of topographer. He should run his line as nearly as may be over the ground likely to be chosen for location, so that the infor- mation obtained may be pertinent, and so that the length of the line, the shape of the profile, and the estimate based on the survey may approximate to those of the proposed location. To this end he has due regard to the levels taken, and when they show that the line as run fails to be consistent with allowable grades, he either orders the corps back to some proper point to begin a new line, or makes an offset at once to a better position, or continues the same line with some deflection, simply noting the position and probable elevation of better ground, as in his judgment he thinks "best. He should at all times maintain a friendly attitude toward pro- prietors, and by his polite bearing endeavor to secure their cordial support of the new enterprise. If he is tolerably cer- tain that the location will follow nearly the line of the prelim- inary survey, he should have with him some blank deeds of right of way, and let these be signed by land-owners while they are favorably disposed. When this cannot be done, a blank form of agreement to allow the surveys and construc- tion of the road to proceed until such time as the terms of right of way may be agreed upon may be made very useful. The chief also selects quarters for his men, and in case of camping out he directs the movements of the camp equipage. 21. The assistant engineer takes the bearings of the courses run, and makes a minute of them, with their lengths, or the numbers of the stations where they terminate. He sees that the axemen keep in line while clearing, and the chainmen while measuring; he takes the bearings of the principal roads and streams, and of property lines when met with. In an open country he may save time by selecting some prominent 10 FIELD distant object toward which the chainmen measure without his assistance, while he goes forward and prepares to take the bearing of the course beyond. In traversing a forest with not too dense undergrowth, when the line is being run to suit the ground according to a given grade, it i^ a good plan for the assistant to go ahead of the chainmen as far as he can be seen, select his ground, take his bearing by backsight on the last station, and then have the chainmen measure toward him. In this case both he and the head chainman should be provided with a good sized red and white flag, mounted on a straight pole, to be waved at first to call attention, and afterward held vertically for alignement. Otherwise a flagman must be added to the party, who will select the ground ahead, under the in- structions of the chief, and toward whom the survey will pro- ceed in the usual manner. 22. The head chainmaii drags the chain, and carries a flag which is put into line at the end of each chain length by the assistant engineer or the rear chainman. It is his duty to know that his flag is in line and that his chain is straight and horizontal before making any measurement, and to show the stakeman where each stake is to be driven. A stake is usually driven at the end of each measured chain length, called a station, though in an open and level country the stakes at the odd stations may be omitted, in which case marking pins are used to indicate the odd stations temporarily. In case there is much clearing to be done the head chainman plants his flag in line, and ranging past it, indicates to the axemen what is to be cut, going a little in advance through the bushes so that they may work toward him. The head chainman should be a quick, active and strong man, with a good eye and a taste for his work, as very much of the real progress of the survey depends upon him. 23. The rear chainman holds his end of the chain firm- ly at the last stake or pin by his own strength, not by means of the stake. He keeps the tally by the pins when they are used, and watches the numbers on the stakes to see that they are cor- rect. The end of a course should always be chosen at the end of a chain, if possible, and if not, then at a brass tag indicating tens of feet, as thus the labor of plotting the map will be much lessened. The numbering of stations is not recommenced with each new course, but is continued from the beginning to PRELIMINARY SURVEY. 11 the end of the survey, through all its courses, and if one course ends with a portion of a chain the next course begins with the remainder of it. It is the rear chainman's duty to attend to this, holding the proper link at the compass station. Any fraction of a chain measured on the line is called a plus, and is counted in feet from the previous station. The length of an offset in the line is never included in the length of the line, but if the line should change its course by a right angle, or more, or less, the numbering would go on as usual. 24. The axemen should be accustomed to chopping and clearing, and are, therefore, to be selected in the country rather than the city. They will cut out so much of the underbrush and overhanging branches as may interfere with the sight of the assistant or leveller; but care must be taken not to cut unnecessarily wide, and no tree of considerable size should be felled, except in rare instances. When running by compass, if the assistant goes ahead of the chain, he can always select a position so that no large tree will interfere ; or, if the line must be produced and strikes a tree, the compass may be brought up and set close to the tree on the forward side as nearly in line as can b3 estimated, the slight error in offset being neglected, since the lino will 1)3 produc3d parallel to itself by the needle. 25. The stakemau prepares and marks the stakes, and drives them at the p.pints indicated by the head chainman. When no clearing is needed, the axemen keep him supplied with stakes, as the rapid progress of the chain will only give him time to drive them. The stakes should be two feet long and pointed evenly so as to drive straight, and are blazed or faced on two opposite sides, one of which is marked in red chalk with the number of the station. The stake must be driven vertically, and with the marked face to the rear, so that it may be read by the <*odman as he follows the line. 26. The topographer makes accurate sketches of all features of the country immediately on the line, and extends the sketches as far each side of the line as he can, in a book prepared for the purpose. He must never sketch in advance of the chain, nor in advance of his own position. His work should be done to scale as nearly as possible, using the same scale for distances on the line and at right angles to it. The scale adopted should never be less than that of the map to be made from the sketches. The ruled lines of a field book are 12 FIELD ENGINEERING. usually one quarter of an inch apart, so that a scale of one line to a station equals about four hundred feet to an inch. This is the smallest scale ever used. The scale of two lines to a station is most convenient for general use. Four lines to a station are needed in special cases to show details, as in pass- ing through* villages. The scale may be changed from time to time as found necessary, but no two scales should ever be used on the same page. The numbering of the stations up the page indicates the scale of the sketch. 27. When the contours of the surface are required, the topographer may join the level party in order that his esti- mates of heights and slopes may be corrected by the instru- ment. He should never draw a mass of contours indiscrimi- nately, but should sketch them as they exist at a uniform ver- tical interval. This interval may be assumed at five feet in a gently rolling country, and at twenty feet in a mountainous one, but an interval of ten feet will be found most convenient generally. If the topographer accompanies the level he can assume the contours at the even tens of feet in elevation, and mark them so, noting where a contour crosses the surveyed line, and sketching its direction and shape both ways from that point. He will estimate the rate of slope of the ground at right angles to the line as so many feet per hundred, and record it from time to time, noting ascent from the line on either side by "A," and descent by " D." If the slope changes within the limit of the page, the line of change may be sketched and the next slope recorded. When little banks or terraces occur, or bluffs and rocks, which cannot be suf- ficiently indicated by contours, they should be shown by hatchings, and the height noted. Special care should be taken to sketch roads and houses in their correct positions and dimensions, the latter to bo either measured, paced or estimated. The dimensions should also be recorded in num- bers. The outline of forests may be shown by a scalloped line, and the kind of timber, and whether dense or scattered, written within the inclosed space. Correct outlines are essen- tial, but no time should be given to shading up a sketch with conventional signs. A single sign, or the name of the thing intended, is all sufficient. Land-owners' and residents' names should be recorded whenever they can be obtained, as well as the names of roads, streams and public buildings. PRELIMINARY SURVEY. 13 28. The leveller takes charge of the level party and keeps the notes of his work. He reads the rod on benches and at turning points to hundredths of a foot and to tenths at other points. He should direct a bench to be made at least once every half mile, and in a very rough country every quarter of a mile. The benches need not be far from the line, and, if well chosen, may be used as turning points, thus saving time. The elevation of turning points must be computed when taken, so that the elevation of any one of them may bo instantly given when called for, and the other elevations will be filled in as far as may be without delaying the survey. As the levels are usually the most essential part of the survey, much care should be taken to have the instrument well ad- justed and truly level, and the rod held vertically and correctly read on turning points, but the intermediate work should not be so done as to delay the party unnecessarily. The leveller should use every endeavor to follow closely after the survey- ing party, so that the chief and topographer may have the advantage of his notes. 29. The rodman's first duty is to hold the rod vertically, and he must learn to do this in calm or windy weather, in level field or on side hill. He may carry a small disk-level, which applied to^he edge of the rod will show when it is vertical. 'The turning points are to be selected for firmness and definite- ness, and so that they will afford a clear view from beyond for a backsight. The rod is held for a reading on the ground at every stake, the number of which is called out to the level- ler as soon as the rodman arrives at it ; the rod is also to be held at every prominent change of slope on the line, as the crest and foot of every bank, the rodman calling out its dis- tance from the last stake as plus so many feet, but all gentle undulations and minor irregularities are to be neglected. The rod will always be read at the surface of a stream or pond, and also at its deepest part on the line, when possible; other- wise the depth of the water may be found by sounding, and so recorded. Should the line run along a stream the surface will be taken occasionally, opposite certain stations, and in case of a canal, the elevation of surface above and below each lock must be noted. The rodman makes inquiry for high- water marks or seeks traces of them himself in an uninhabited district, and holds the rod upon them that their elevation may 14 FIELD ENGINEERING. be determined. The rodman carries a small axe or hatchet with which to make benches and to trim out any stray branch that may intercept the leveller's view. 30. The assistant rodmen take the slope and elevations of the ground at right angles to the line, using vertical and hori- zontal rods and a pocket level, or a tape line and clinometer. The cross levels are not taken throughout the whole survey, if at all, but only where the roughness of the country seems to demand it. They may be extended to any distance from the centre line required by the chief not less, however, than fifty feet as a rule. They may be taken at the stations only, or oftener, if necessary, depending upon the roughness of the surface, the object being to define accurately the contours, and so the shape of the ground. The assistant rodmen will also take soundings when they are needed, either on the line or at right angles to it. 31. In defining the duties of the members of the corps, the instruments used have been incidentally noticed. 32. The compass is preferable to the transit on prelimi- nary surveys, because it can be operated more rapidly, is lighter, and usually has a better needle. It may have either plain sights or telescope, and be mounted on tripod or Jacob staff. The simpler forms are preferred for forest work. Not unfre- quently the engineer's transit is employed, but usinfthe needle. A preliminary line should not be run by backsights and deflec- tions, unless local attraction is found to exist to such an extent as to destroy confidence in the needle; or, in special cases, where the natural obstacles to a survey are very great. In the latter case the survey partakes of the nature of a location, and should be conducted with similar care and fidelity. 33. The chain is 100 feet long, and composed of 100 links. It should be of steel for lightness, durability, and greater accuracy. Those having rings of hard steel, unbrazed, are least apt to wear. Five marking pins are needed, each having a piece of red flannel attached, for temporal y stations, or for keeping points temporarily while measuring by parts of a chain up or down a slope. A. pointed plumb bob, with sev- eral yards of small cord wound on a carpenter's spool, is use- ful in chaining over steep declivities or bluffs. 34. The axes should be of best quality, with hand-made handles, and not too heavy. The axe of the stakeman should PRELIMINARY SURVEY. 15 have a fine edge for dressing and a broad head for driving the stakes. When the stakes are not required to be over two feet long, a stout basket, having a square, flat bottom, 26x14 inches, should be furnished the stakeman. He will then pre- pare a basketful of stakes, ready marked, and place them in the basket regularly, in the reverse order of their numbers, so that they will couie to hand as wanted. A small hand-saw no larger than the basket, with rather coarse teeth, wide set, will be found extremely useful in cutting stakes with square heads and of uniform length, and much more rapidly than can be done with an axe. When not in use, it is to be strapped to the inside of the basket, to prevent its being lost by the way. When the basket is about empty, the stakeman, with the assistance of the axemen, can soon replenish it, and the stakes being all numbered at once, there is less danger of a mistake being made in the tally than when they are marked only as wanted. 35. The level should be the regular engineer's level, the same as used on location. 36. The rod should be self -reading, i.e., to be read by the leveller, as too much time would be consumed in the constant adjustment of a target by the rodman. It should be as long as can be conveniently handled in order to reduce the number of turning points on hill sides. A very convenient rod may be made of thoroughly seasoned clear white pine, sixteen feet long and two inches wide, with a thickness of one inch at the bottom, increasing to one and a quarter inches at six feet from the bottom, and then gradually diminishing to three eighths of an inch at the top. The rod is shod with a stout strap of steel, extending five inches up the edges, and secured by screws. The top is protected for a few inches by a plate of sheet brass on the back. The face of the rod is a plain surface through- out, and is graduated from the lower edge of the steel shoe as zero. The divisions are fine cuts made with the point of a knife. At the foot and half-foot points the cuts extend across the face. For the tenths and half tenths they extend three quarters of an inch from the right hand edge, terminating in a line scribed parallel to the edge of the rod, thus forming rec- tangular blocks half a tenth wide, every other one of which is painted black, the body of the rod being white. The feet are indicated by numerals painted red on the blank part of the 1G FIELD face, each figure standing exactly on its foot mark, and being exactly one tenth high. For the figure 5 the Roman V. is sub- stituted and for 9 the Roman IX., so that in case a dumpy level is used the 5 may not be mistaken for a 3, nor the 9 for a 6. At the half-foot points a red diamond is painted, so that the graduated line bisects it. No other figures nor gradua- tions are required. With this rod the leveller can read quite accurately to hundred ths of a foot, and after some practice can estimate the half hundredths. 37. The horizontal rod for cross-levels maybe made of white pine, ten feet long and one inch thick by three wide, tipped with brass, painted white, and graduated to feet and tenths. It must be a straight edge, and is levelled by a pocket level placed upon it when needed, or by a small level embedded permanently in one edge. The vertical rod to be used with it is made of pine eight feet long and one and a quarter inches square, and graduated to feet, tenths, and half tenths. All rods when not in use should be laid on a flat surface to pre- vent their being sprung. Leaning them in a corner soon ruins them for use. 38. The clinometer is any small instrument which will measure the slope angle of the surface. The angle is always estimated from the horizon, a vertical being 90. The rise per 100 feet is found by multiplying the nat. tangent of the slope angle by 100. It may often be found more easily by the leveller reading the rod at a station and then 100 feet left or right of the line. If surface measures are taken in connec- tion with a slope angle they are reduced to horizontal meas- ures by multiplying them by the cosine of the slope angle. 39. The plane-table is rarely if ever used on prelimi- nary surveys in the United States. Occasional bearings taken to prominent objects by the assistant engineer, or the use of a prismatic compass by the topographer in connection with his sketches, is found to answer every purpose. 40. In case a survey is to be made with a tran- sit, it is necessary to add a back flagman to the party, who will hold his flng or rod on the point last occupied by the transit, so that the assistant may take a backsight upon it. The direction of a new course in each case is determined by the deflection angle to the right or left of the preceding course produced. The bearing of one long course near the beginning of the sur- PRELIMINARY SURVEY. 17 rey having been carefully ascertained, the bearing of each suc- ceeding course is calculated from the deflections, and entered in a column of the field book headed Calculated Bearings, from which the line is afterwards plotted. The magnetic bearing of each course should also be taken from the needle, and re- corded as such, but is used only as a check on the transit work. The deflections should be made in degrees, halves, or quarters, if possible, to facilitate the calculation of bearings, and to admit of using a traverse table. 41. The attached level and vertical arc of the transit are useful in determining approximately the grade of the line run in advance of the level party, or in seeking for one assumed grade to which it is desired that the line shall conform. For this purpose it is only necessary to set the vertical arc to the angle corresponding to the grade as given in Table XIV., and let the head chainman move about until a point on his rod at the same height from the ground as the telescope is covered by the horizontal cross-hair. 42. The point on the ground where a transit is set up is marked by a good-sized plug, flat headed, and driven down flush with the ground, with a tack set in the head to show the exact point or centre. This is called a transit point. When a transit point occurs at a regula? station, the stake bearing the number of that station is set three feet to the left cf the line opposite the plug and facing it. When a transit point occurs between stations the stake is driven three feet to the left of it, marked with the number of the preceding station -j- the distance from that station in feet. 43. As a transit is capable of giving a line with great pre- cision, it is important that the flags used in connection with it should be equally precise in giving"points. An excellent flag for this purpose is made of well-seasoned clear white pine ten feet long, two and a half inches wide, and one inch thick. It is tapered for the last four inches to an ^ge atone end, the edge being formed at the middle of the width. The tapered end is shod with a band of steel covering the edge of the rod, and secured by screws, and the steel is brought to a sharp edge at the point of the rod. The rod is then painted white and tipped with brass at the square or upper end. A centre line on the face is then struck from the point of the steel to the 18 FIELD ENGIKEEKING. middle of the brass tip by means of a piece of sewing silk, and a fine cut made with a knife and steel straight edge. The centre line must not be scribed parallel to one edge of the rod, as this is rarely ever straight. The face of the rod is then divided into one-foot spaces, measured from the head of the rod, and these are painted red on either side of the centre line in alternate blocks. On the back of the rod at three and a half feet from the point is placed a small ground-glass bubble-tube, mounted very simply, and attached to the rod by a brass plate and screws, and guarded by two blocks of wood for protection. The centre line of the rod is made vertical by a plumb-line while the level tube is being attached, which ever after secures a vertical rod. If only two feet of this rod can be seen over any obstruction, a point can be set with great precision, provided the level tube is in adjustment. This flag can also be used as a plumb in chaining w T ith much more satisfaction than a cord and weight, especially in windy weather. 44. A transit survey usually requires more clearing than one made by compass. When a given course is to be produced in a forest, some large trees will inevitably be encountered, but the labor and delay of felling them may be avoided by the use of auxiliary lines. These may be classified as running parallel to the main line, at a small angle with it, or at a large angle with it. 45. The parallel line is established by means of two short perpendicular offsets measured with care before reach- ing the obstacle, and the main line is established beyond the obstacle by means of two more equal offsets. But since short back-sights are to be avoided, these offsets should be at least 100 feet apart, so that it may be difficult to find a parallel line of sufficient length which does not strike some other obstacle, or at Jleast require considerable extra clearing. 46. The auxiliary lines making- a small angle with the main line are more convenient, not only on this account, but because they require a less number of transit points. By them an isosceles triangle is formed on the ground, having the intercepted portion of the main line as base, and the vertex near the obstacle. The deflections at the points where the lines leave and join the main line are similar and equal, and PRELIMINARY SURVEY. 19 the deflection at the vertex is double in amount and opposite in direction. No calculation is necessary, for the error in measurement due to the deviation is too small to be noticed, and since the main line is immediately resumed, the calculated bearings of the auxiliary lines are unnecessary. Should the point where the second line joins the main line prove irnsuit- able for a transit point, the second line may be produced to any convenient point beyond, and so go to form an isosceles triangle on the opposite side of the main line, the triangle being completed by running a third line parallel to the first, and equal to the difference of the first and second. Again, the second line may encounter a serious obstacle before reach ing the main line. To avoid this a parallel to the main line may be run from the end of the first line for a con- venient distance, and there the second line be put in parallel and equal to its first position, as before de- scribed, thus forming a trapezoid. 47. The following general solution of this problem allows the engineer to make use of any number of auxiliary lines, provided that none of them make an angle of much more than one degree with the main line, with a certainty of resuming the main line in position and direction at the extremity of any course desired, and without necessitating any trigonometrical calculation. It is based on the assumption, practically true for small angles, that the sines are proportional to their angles, and is ex- pressed by the following rule : Call all deflections to the right plus, and all to the left minus; multiply the length of each course in feet by the algebraic sum in minutes of all the auxiliary deflections made to reach that course; take the algebraic sum of these products, and when the sum equals zero the extremity of the last course will be on the main line. The deflection required at that point to give the direction of the main line is equal to the algebraic sum of all the preceding deflections, but taken with the contrary sign. Thus, if we have left the mam line at A, and run by these notes: (Fig. l.> FIG. 1. 20 Sta- Defl. Dist. Factors. Products. A 16' R 190 g> + 16 X 190 = -f 3040 B 31' L 120 | | - 15 X 120 = - 1800 C 18' R 175 1 8 + 3 X 175 = -f 525 I) 13' L 265 f | - 10 X 265 = - 2650 E 15' R * | -f 5 X (?) 3565 - 4450 and their algebraic sum is 885 Therefore to render the sum zero we must add 885 as the pro- duct of the last course. But 5' is already given as one factor, 885 so that the other factor must be - - = 177, which is the length o of the last course, giving some point F on the main line. The deflection at F from the last course to give the direction of the main line is 16 31 + 18 - 13 + 15 =-5' and changing the sign we have 5' ; that is, the deflection is to the left. The distance on the main line from A to F equals the sum of the courses, or 927 feet, but this we have by the stations, which have been kept by stakes in the ordinary way. All the stakes on the auxiliary lines will be more or less off the main line, but as these offsets are usually very small, they are con- sidered of no consequence on a preliminary survey through a forest. In Fig. 1 the offsets are very much magnified. The field notes of such auxiliary courses should be kept, not as regular notes, but on the margin or opposite page, and in such a way that, while the line may be retraced by them on the ground, the draughtsman may see that it is not necessary to plot them, when a straight line ruled and measured through is suf- ficient. It is obvious that in selecting a closing course either the deflection may be assumed and the length calculated, or vice versa ; but care should be taken to assume such values as do not involve a fraction in either factor, if possible. 48. The method of passing an obstacle on the line by auxiliary lines at a large angle with the main line will only be resorted to when circumstances are such that the other methods mentioned cannot be employed, as in passing a build- ing, pond, or densely wooded swamp. In such a case we may PRELIMINARY SURVEY. 21 turn a right angle with the transit, and measure accurately one offset, putting a transit point at its extremity, where another right angle will give a parallel line. If the offset prove too short for an accurate backsight, a temporary point at a sufficient distance may be established for that purpose on the offset line produced before the instrument is removed from the main line. If any other angle than 90 is used it should be selected, when circumstances permit, so that the distances on the inter- cepted part of the main line may be in some simple ratio to the distances measured on the auxiliary line. Thus a deflection of 60 gives a distance on the main line equal to half the length of the auxiliary course, that is, . 60 gives a ratio of i = 0.5 53 08' " " " 0.6 nearly 45 34V " " " 0.7 " 36 52" " " " 0.8 " 25 50V " " " 0.9 " the angles being taken to the nearest half minute. 49. If it be desired that the stakes on the auxiliary line should stand on perpendiculars through the true stations on the main line, a certain correction must be added to each chain length depending on the angle which the auxiliary makes with the main line. If there is a fraction of the chain at either end of the course, a proportional addition must be made for this. Thus, by referring to the table of external secants, we find that we must add a correction as follows: 2 33V- . .0.1 ft. per chain. 6 45V. ..0.7ft. per chain. 3 37' i ..0.2" 7 13V- ..0.8" 4 26' . ..0.3" 7 39V- ..0.9" 5 07' . ..0.4" 8 04' . ..1.0" 5 43' . ..0.5" 9 52' . ..1.5" " . 6 15V. .0.6" 11 22' . ..2.0" These methods of suiting the angle to an even measure are much superior to assuming an even number of degrees deflec- tion, and then calculating the distance by trigonometry. The last table, which may be extended indefinitely by reference to the table of Ex. secants, is perfectly adapted to chaining* by surface measure on regular slopes when the slope angle is 22 FIELD known, the chain being lengthened by the correction corre- sponding to the slope angle. 50. If the chain is lengthened as per above table on auxil- iary lines, the numbering of the stakes goes on as usual, but they should have an additional mark as X to show that they are -off the main line; and they may stand facing the true stations which they represent, and the length of offset, if known, may also be recorded on them. The leveller will then understand that he is to read the rod not only at the stakes as they stand, but also at the true stations, as nearly as may be. The assistant engineer will always make a diagram in his field book, showing exactly the method pursued in reference to auxiliary lines. Having passed the obstacle, it is advisable to return to the main line by a course equal in length to the first auxiliary, and making an equal angle with the main line. If this cannot be done from the end of the first course, a parallel to the main line may be run any convenient distance, and the return line then put in, forming a trapezoid. 51. When there is no obstruction to sight on the main line, but only to measurement, a transit point should be set in line beyond the obstacle before the transit leaves the main line, as St check on the other operations, and the main line should be afterward produced from this point by back- sight on the main line, rather than by deflection from an auxiliary line. 52. The main line should always be resumed as soon as practicable, making the auxiliary lines the mere exception. When a number of courses at a large angle are likely to be required before the main line can again be reached, it may be better to consider these as regular courses of the survey, and to note them as such. The simplest method is always the best, because least likely to involve mistakes. 53. When the natural obstacles are so numerous and of such magnitude as to render any continuous line of sur- vey or location extremely difficult, if not impossible, as in the case of a bold rocky shore, all the data necessary to a location should be gathered with precision on the preliminary survey, the measurements and angles being taken with the greatest care, and as many checks as possible should be introduced to verify the work. In meandering such a shore it is probable that a large number of short courses will be used, which may be measured PRELIMINARY SURVEY. 23 correctly, but there is liability to error in the angles. To verify the latter the more conspicuous transit stations on prominent points of the shore are selected, and these being named by the letters of the alphabet, the deflections between them are taken by careful observations repeated .a number of times, as for a triaugulation. These points, joined by tie- lines, then form a survey of themselves, much simpler than the full traverse. To obtain the length of these tie-lines', the angles between them and the courses meeting at the same station are measured. Then since each tie-line forms the closing side of a field, in which all the bearings are known, and all the distances, save one, that one may be calculated by latitude and departures. But the angles should first be tested for error in each complete field, and if the error be large the angles must all Be remeasured until the error is found and cor- rected, but if very small it may be distributed among the angles, or among those most probably inaccurate. Before cal- culating the traverse of any of these fields, it will be advanta- geous to assume, for an artificial meridian, a line parallel to the average direction of the shore for several miles, and to refer all courses to this meridian for their bearing. This meridian is called the axis of the survey, and all bearings referred to it are called axial bearings, as distinguished from magnetic bearings. The magnetic bearing of the axis should be some exact number of degrees, so as to facilitate the reduc- tion from one system to the other. 54. In plotting the map, the axis is first laid down, and then the lettered stations in their respective positions, after which the meandering surveys can be filled in. The map being drawn on a scale of one hundred feet to an inch, and the con- tours constructed from the notes of the level and cross-level parties, the engineer may project the location upon it with great certainty and economy of result. But he should calcu- late the traverse of the location as projected, and compare it with the traverse of the preliminary, to eliminate, all errors in drafting, before taking his notes to the field to reproduce the location on the ground. Any point where the location crosses the preliminary should have the same latitude and longitude by the traverse of either line. This system, though laborious, is the only one that will ensure a successful location under the circumstances supposed. Advantage may sometimes be taken 24 FIELD ENGINEERING. of cold weather to cross bays and inlets on the ice, but there is great liability to error in angles taken upon the ice, due both to its motion and to the sinking of the feet of the tripod into the ice as soon as exposed to the rays of the sun. CHAPTER III. THEORY OF MAXIMUM ECONOMY IN GRADES AND CURVES. 55. Before commencing the field work of location it de- volves upon the engineer to decide as to which of the surveyed routes shall be adopted as being most advantageous in all respects, and also to establish the maximum grade in each direction and the minimum radius of curve on that route. The general considerations which guide the engineer in the selection of one of several routes for location are such as were hinted at in the chapter on reconnoissance, but upon the com- pletion of the preliminary surveys he has at hand a large amount of information which enables him to consider this important question much more in detail. Unless his instruc- tions are explicitly to the contrary, he may assume it to be his duty to find the best line, or that one which, for a series of years following the completion of the road, will require the least annual expense, including interest on first cost. The finances of the company may be so limited as not to permit the construction of the best line at once, and it may then be the duty of the engineer to select the cheapest line, or that of least first cost, as a temporary expedient, with the expectation of building the road at its best when the improved credit of the company will permit. But generally he will be able to build the cheaper portions of the best line at once, only making deviations and introducing heavier grades at the expensive points to avoid a cost beyond the present means at his com- mand. The selection of the best line may be a question as between different routes or as between different grades and curves on the same route. We will consider the latter case first. 56. To solve the problem of true economy we must determine the actual expense both of building and operating MAXIMUM ECONOMY IK GBADES, ETC. 25 the line at a given maximum grade, and also what changes will be made in these expenses by a change in that maximum. We have then, on one hand, the annual interest upon the original cost, and, on the Other, the annual expense of operating the road. The best grade is that which will render the sum of these two a minimum. Both forms of expense consist of two parts: one that is affected by a change in grade, and the other that is not. Clearly the former is the only one we have to consider in either, since when the sum of the variable portions is a minimum, the sum total will be a minimum also. The varying portions then are functions of the grade, though independent of each other. If, therefore, we let z 1 represent the maximum grade in feet per mile, and let x represent the corresponding value of that portion of the annual expense which varies with the grade, and establish the relation existing between the two, we shall have x =f(z'). Similarly if we let y represent the interest on so much of the first cost as is affected by grade, we shall have y=f (z'}. The problem then is to find that value of z' which shall render x -j- y = a minimum. Let us now seek the complete expression represented by x=f(z').' The elements that enter into this expression are numerous, and will be considered in succession. 5 7 . The traction of an engine is the force with which it pulls a train, and is limited by the reaction of the drivers against the rails. It depends on the weight upon each driver, the number of drivers, and the coefficient of friction. The weight on one driver should not exceed 12,000 Ibs., and is usually less than this. If the exact proportions of engine that will be used on the road are not known, the weight per driver may be assumed at 10,000 Ibs., with 4 drivers for ordinary grades and traffic, or at 11,000 Ibs. with 6 drivers, if the grades are steep and the tonnage large. For extraordinary grades special engines are required, having 8 or 10 drivers. Tlie coefficient of friction, called also the adhesion, varies from .09 to .37, these being the extremes on record. The lowest is due to extremely unfavorable circumstances, as sleet and frost; the highest doubtless to the use of sand, though not so stated in the record. The more common range of values is from .15 to FIELD .25. For our present purpose it will be assumed at .20, so that if a 4-driver engine has 10,000 Ibs. on each driver, its traction is 40,000 X .20 = 8000 Ibs. when hauling its maximum train. 58. The expense of running an engine one mile, hauling a train, on the proposed road, can only be estimated from the experience on other roads similarly situated. The expense is composed of the items of fuel, water, oil and waste, repairs (including renewals), wages of conductor, engineer, and fire- roan, engine-house expenses, and interest on first cost of engine and engine-stall. The range and approximate average of these items is here given : ITEMS. 4-DuivER ENGINE. 4-DuiVER 6-DRiVER 8-DRIVER Lowest. Highest. Average. Average. Average. Fuel $0.050 .001 .004 .050 050 025 .0^5 $0.210 .010 .030 .150 .100 060 .038 $0.100 .004 .006 .080 .075 .035 ,030 $0.165 008 .008 .104 .075 .050 .038 $0.213 .008 .010 .133 .075 .060 .047 Water Oil and waste Repairs and renewals Wages Engine-house . Interest Totals .205 .598 .330 .446 .546 In a given case the probable value of each item should be estimated separately, and the sum taken afterwards. In the above averages each engine is supposed to haul its maximum train. The relative expense of the several classes of engines has not been established conclusively. 59. The resistance offered to the motion of a railway train is occasioned by a variety of causes, concerning which a great deal of -uncertainty exists as to their relative effect. An investigation which should seek to determine the exact amount of each partial resistance, and then by a summation derive the total, would be tedious, and, in the present state of our knowledge, unsatisfactory. We shall therefore simply group the resistances under three general heads, namely: Resistance due to uniform motion on a straight, level track ; Resistance due to grade ; Resistance due to curvature. MAXIMUM ECONOMY IN GRADES, ETC. '27 6O. The first of these, considered as an aggregate of the various items of friction in engine and train, of oscillations and impacts, and of resistance of the atmosphere, is found to vary nearly or quite as the square of the velocity. The fric- tion of an engine is greater in proportion to its weight than that of a car, owing to its many moving parts, so that the resistance of a short train is greater in proportion to its total weight than that of a long train. The resistance of the atmos- phere is greater also in proportion to the weight of a short train than of a long one. An empty train will offer more resistance in proportion to its weight than a loaded one. A formula which shall express the resistance of a train to uni- form motion must include at least the velocity and the weight of the train and engine. The following empirical formula is based upon a careful investigation of all such records of experiments on the subject, several hundred in number, as have come to the author's notice, and is believed to give results agreeing closely with the average experience and practice of tlie present day. It is designed to give the resistance per ton for all trams, whether freight or passenger, and at any velocity, under ordinary circumstances. Accidental circumstances, such as the state of the weather, and the condition of the road-bed, rails, and rolling stock, may largely modify the resistance, but these, of course, are not taken account of in the formula. Let V = velocity of train in miles per hour, " E = weight of engine and tender in tons, " W = weight of cars in tons, " T weight of freight in tons, " q resistance to uniform motion in Ibs. per ton. We then have the formula = 5.4 .006 61. The second resistance considered is that due to gravity in grades. It varies in the exact ratio of the rise to the length of the grade. Let (? 8 = rise of grade in feet per station. " G m = rise of grade in feet per mile. " q = resistance in pounds per ton due to grade. 28 FIELD Then, tf = 2240 = 23.4.0. (2) 62. The third resistance considered is that due to curvature of the track. This resistance is due to the friction of the wheels upon the top of the rail, and of their flanges upon the side of the rail. The top friction is lateral, due to the oblique position of the wheel on the rail, and longitudinal, due to the greater length of the outer rail, since both wheels are rigidly attached to the axle. The flange friction is due to the reaction of the top friction, which, combined with the parallel- ism of the axles, throws the truck into an oblique position on the track. A forward flange presses the outer rail, while a rear flange is usually in contact with the inner mil. The centri- fugal force of the car will increase the pressure on the outer rail, unless the ties are inclined at an angle sufficient to coun- terbalance this force. But if the ties are inclined too much, or the velocity is less, the pressure on the inner rail will be increased. An uneven track will cause the truck to pursue a zigzag course, increasing the resistance considerably. Experiments for determining the amount of curve resistance have been neither numerous nor very satisfactory, but the generally accepted conclusion is that the resistance is a little less than half a pound per ton on a one-degree curve, and that it varies as the degree of curve. On European roads, how- ever, it is estimated at about one pound per ton per degree of curve, owing largely to the form of rolling stock used. 63. Let g" = curve resistance in pounds per ton on any curve, and D = degree of curve. Then, assuming the resistance per ton on a one-degree curve at 0.448, we have for any other curve q" = 0.448Z> (3) To ascertain what grade upon a straight line will offer the same resistance as a given curve; substitute the value of q" for q' in eq. (2) and solve for G; whence &, = .ow i 4 #, = 1.056,0 \ MAXIMUM ECONOMY IN GRADES, ETC. 29 For definition of degree of curve, see Art. 84. O4. It is evident that grades and curves, by their resistances, fix a limit to the weight of a train which a given engine can haul over them. A locomotive is usually built with such a surplus of boiler and cylinder capacity that its power, at ordinary velocities, is limited by the adhesion of the drivers, so that the adhesion is the proper measure of the tractive force. To find an expression for the maximum train which a given engine can haul over a given grade and curve: Let P = tractive force of engine in pounds, " T' = weight of paying load in tons per maximum train, " W = weight in tons of cars carrying the load T'. Then for uniform motion, at a given velocity. Let t = average load of one car in tons " w = average weight of one car and load in tons. Then W + T' = y T', substituting which in eq. (5) we derive r^f-r^-p^-tf) (6) In this equation q = the resistance per ton due to uniform motion, q' = the resistance per ton due to the maximum grade opposed to the direction of the train, and q" = the resistance per ton due to the sharpest curve on that grade. For accelerated motion the reaction of inertia of the train must be added to the above resistances. This is estimated at ^q, in order that a train starting from rest may acquire the requisite maximum velocity, even on a maximum grade, in a reasonable time, say from 3 to 6 minutes. Therefore, for accelerated motion, Now, the values of T and q involve each other, but if we accent W and T in eq. (1) the value of q becomes that used in 30 FIELD ENGINEERING. eq. (7), and we may eliminate q between these equations, and derive the value of T' ; whence -(p-.ooo9^ 2 F*) T- Lv ~ 4 + one direc- tion is estimated at 375 000 tons per annum, and in the other direction at 125 000 tons, that it is decided to use 6-driver engines, and that the expense per engine-mile is estimated at 40 cents ; hence (2 A a)Le 20 000 000. We are now required to find the most economical maximum grade. We first select at least two other maximum grades, and having FIELD E^GLNEEKING. TABLE OF RECIPROCALS OF T'. t - 10, W = 18. y diff. y. diff. x. X. *+y. z'. 1.4 1.3 1.2 1.1 1.0 0.9 0.8 0.7 0.6 145 596 149886 155 050 1(51088 168000 175786 184446 193 980 204388 4290 5164 6038 6 912 7786 8(160 9534 10408 9298 9102 8914 8730 8548 8374 8208 8044 142934 133636 124534 115620 106890 73716 274890 274128 274414 73.92 52.80 47.52 42.24 31.68 The numbers in the fourth and fifth columns are obtained as follows : the values assumed above give us (2A a) Le = $20 000 000, and this multiplied by the tabular differences in the preceding table for a 6-driver engine, gives the numbers in the fourth column. We now observe that the differences of x and of y increase in opposite directions, therefore at some point they will be equal ; and a simple inspection shows us that this point is at or near the grade of 0.9, which is therefore the grade required. We now multiply the tabular number for 0.9, and a 6-driver engine by $20 000 000, for thu number in the fifth column, and this added to the value of y on the same line gives the sum of (x -\- y} for the most economical grade. This of course is not the total annual outlay of the road, or engine-stage, because many items of expense which are independent of a maximum grade have not been con- sidered. 36 FIELD E^GI^EEKISTG. If an 8-driver engine were to be used, and the expense per engine-mile estimated at 50 cts., then (2 A a)Lc $25 000 000 ; hence < y. diff . y. diff. x. x. x-\-y. z'. 1.1 1.0 0.9 161088 168000 175786 6912 7786 7673 7538 95810 263 810 5/* . 80 indicating a saving of $10 318 per annum in the case supposed by using 8-driver engines, although on a steeper ruling grade. On the other hand, should we adopt 4-driver engines, and esti- mate the expense per engine-mile at 30 cents, we should find the most economical grade to be 0.7 per station and (x -f- y) = $293 280, showing a loss in this case of $19 152 per annum, as compared with the results of 6-driver engines. It should be remembered that the table 67 is prepared on the assumption that the ratio = If cars are to be used giving for full loads ,ftny other ratio, , , a new table may be prepared by multiplying each tabular number by - X -3 . lo t The velocity adopted of 12 miles per hour is sufficient for ordinary grades. When the maximum grade is very low, it would be better to use 15 or 18 miles an hour in calculating the value of x. 7O. Since #, eq. (11), varies directly as L, it is important that an engine-stage having heavy grades should be short. Its length, however, must be consistent with the economical length of the adjoining engine-stages, and with the amount of work which an engine ought to perform dally. The most favorable condition for a road would be that in which all the engine-stages were operated at equal expense. But if, to secure this result, the engine-stage of heavy grades must be unreasonably reduced in length, it will be better to adapt the grades to the use of two engines per train. 7 1. The maximum grade z', opposed to the heavier tonnage A, having been determined, we have now to consider what is the limit to grades in the opposite direction. The engines are MAXIMUM ECOKOMY IN GRADES, ETC. 37 supposed to haul their maximum loads in moving the ton- nage A, and since the return tonnage, a, is less than A, the engines, in returning, will not be worked to their full capacity if they encounter no grades steeper than z' '. We therefore have a margin of power in the returning engines which may be taken advantage of to cheapen the cost of construction, or to shorten the line, by introducing grades, steeper than s 1 ', against the lighter traffic. The weight of a maximum train moving up the grade z' is, eq. (9), W -\- T' ; the weight of the train returning will be Substituting this in place of (W -f- T'), eq. (9), and solving for q', we find the resistance due to a maximum grade opposed to the returning train. Whence, by eq. (2), if we let Z the maximum return grade, and make q" = 0, . _ 33 flo9 Inasmuch as the value of Z varies with every change made in 2', the engineer, when estimating the cost of construction upon the basis of any maximum grade z', should take care. that the return grade Z nowhere exceeds its limit as .given by the last equation (14). In the example, 69, z' 47.52; hence T 203.37, eq. (8). Substituting these values, in eq. (14), we find Z= 81.25, which is therefore the limit for return grades in this case. With regard to curves on the maximum grade, see 68. qq 72. If ineq. (1) we lets = -TJ q be the grade per mile which offers a resistance equal to the resistance to uniform motion on a level, we have When F= 20 this becomes (15i) 38 FIELD which is the grade down which a train, whose weight is (E-\- W-\- T), if started at 20 miles an hour, will continue tp move at that speed without steam or brakes. As that speed is not objectionable, so the grade z, which induces it is not, pro- vided it does not exceed the values of z' or Z respectively, determined with reference to economy. For the extra work done by the engine in ascending one grade z is utilized in descending the next; and the net result is the same as though the two were replaced by a uniform grade. The engineer therefore is not warranted by economic considerations in reducing undulating grades which do not exceed z to a uni- form grade, when to do this would cause any increase in the cost of construction, unless z exceeds the grades z' or Z of maximum economy. 73. But when grades exceed z, eq. (15J), the resulting speeds of the maximum train become too great, and the neces- sary application of the brakes absorbs a portion of the power previously expended in gaining the summit, which is thus worse than wasted, since it increases the wear and tear of machinery and track. Therefore the engineer is justified in spending a certain sum of money in reducing grades which exceed z to that limit. A calculation of the loss of power due to the use of brakes on a grade, and of the cost of that lost power, together with the resulting wear and tear per annum, will give the interest on the sum that may be justifiably spent in reducing the grade from its position of cheapest construc- tion. 74. The limit z is not constant, but depends on the weight of the maximum train, which in turn depends on z' . It will not be the same in both directions unless A = a, giving z' = Z. In the example 69, E = 49.5 and W -f- T' = 366.07; hence, eq. (15|), z = 21.72 descending in the direction of the traffic A. Also W + -. T' = 230.49, whence z 23.34 descending in A the opposite direction. These are the limits in this case at which undulating grades cease to be profitable. 75. We have finally to consider the method for selecting the best line from several proposed routes. For this purpose we determine the most economical grade on each route thought worthy of consideration, and calculate the interest on the entire cost of constructing the line with that ruling grade, and LOCATION. 39 also the annual expense of operating the line, and take the sum of the two. That route is best in respect to which this sum is the least. 76. The value of saving one mile in distance on any route is found by dividing the sum of the annual operating expense and the interest on the cost of construction by the rate of interest, and the quotient by the length of the line in miles. 77. We have now fully discussed the theory and developed the formulae necessary to the determination of the most economical grades; but the value of the results in a given case depend upon the correctness of the engineer's estimates which enter into the formulae. These may seldom prove pre- cisely accurate, yet, if he can bring them within definite limits, he may determine the grades of maximum economy within corresponding limits. In the case of a finished road and in full operation, however, the elements of first cost, of traffic, and of operating expenses being known, an investiga- tion by means of the foregoing formulae becomes a critical test as to the economy of the location and grades; and should the road fail to pay dividends, or be forced to charge high rates of toll, we can determine, though perhaps too late, to what extent the location is chargeable with these results. CHAPTER IV. LOCATION. 78. A railroad is said to be located when its centre line is established on the ground in the position which it is intended finally to occupy. The location is made by an engineer corps similar in its organization to that employed on preliminary surveys. The instruments used are also the same, except that the transit is substituted for the compass, and usually the target rod for the self-reading rod. The magnetic needle is never used upon the centre line, except as a rough check on the transit work. It is used, however, to obtain the direction of property lines, roads, and other topographical data. 79. The remarks upon transit work in the preceding chapter apply to the running of straight lines on location. All 40 FIELD ENGINEERING. field-work on location should be done with accuracy and fidelity. No guesswork, nor rude approximations, are to bo tolerated. All transit, points are made as secure and permanent as possible, and the more important ones are guarded by other transit points set in safe positions near by, their distances and directions from the main point being recorded. The stakes for the stations are made neatly, and somewhat uniform in size, and they are firmly driven. Sometimes a small plug is driven down flush with the surface of the ground to indicate the station point, and the stake is then set near by as a witness. In locating a very long tangent the greatest care is re- quired to make it straight. If the tangent is produced from point to point by backsights and foresights, the observation should be repeated in every instance with reversed instrument, to eliminate any possible lack of adjustment, and to check any accidental error. (Indeed it is proper to observe this rule on curves, as well as on tangents.) When some object in the horizon can be used as a foresight, it is preferable to set the instrument by this rather than by a backsight. For final loca- tion, the line should be cleared to give as continuous a line of sight as possible, but in case of an obstacle which cannot be removed at the time, at least two independent methods of passing it should be employed, so that there may be a check upon the alignment beyond. 8O. The leveller selects his benches far enough from the line to prevent their being disturbed during the construction of the road. They should be nearly at grade, as a rule, though it is well to leave a bench near a water-course for reference in lay- ing out masonry or trestle-work. The rodman holds the rod at every station, and at every point on the centre line where the slope changes direction, so that these points may be accu- rately defined on the profile. When he uses a target rod, he sets the target as directed by the leveller, and after clamping it, takes the reading. He reads to thousandths upon turning points and benches, but only to tenths of a foot elsewhere, and announces the readings to the leveller for record. He also records the readings upon turning points and benches in his own book as a check. At the close of each day the leveller and rodman compare notes, and draw a profile of the line sur- veyed. (See also 28, 29, 30.) LOCATION. 41 81. The fixing of the grade-lines upon the profile is one of the most important operations connected with the loca- tion. It is usually performed by the engineer in charge of the locating party, as being most conversant with the general character and detailed requirements of the line. The maxi- mum gradients will have generally been determined in advance from the preliminary data by the principles laid down in the preceding chapter, but the position of each grade-line, relative to the profile of the surface, must be left to the judgment and skill of the engineer. In general, the grade-line is so placed :is to equalize the amounts of excavation and embankment, but there are various exceptions to" this rule. Thus, the exca- vation may be in excess: first, when it is necessary to pass under some other road or highway, the grade of which cannot be changed; second, when valuable property is to be avoided, the appropriation of which would cost more than the excava- tion; third, when the grade is at the maximum near a sum- mit, and cannot be raised parallel to itself without incurring too great an expense for masonry, etc., at some other part of the line. The embankment may be in excess, first, when the country is flat and wet, in order to keep the road-bed well drained; (the grade-line should be at least two feet above the average level of the surface, or above high-water mark, if the district is subject to overflow;) second, in approaching a stream, where it is necessary to raise the grade above the requirements of navigation; third, when the cuttings on the line are largely in solid rock, and a cheaper material for embankments may be conveniently had at other points; fourth, in a district subject to heavy drifts of snow, by which deep cuts would be liable to be obstructed; fifth, in side-hill work, where there is danger of land-slips; sixth, when it is determined to supply the place of a portion of an embankment by a timber trestle-work or other viaduct. The apparent equality of cut and fill on the profile does not represent an equality in fact, owing to the different bases and slopes of the sections adopted, and to the various inclinations of the natural surface transversely to the line. This is espe- cially true in side-hill work, where there are both cut and fill at every point, while the profile shows very little of either. In the latter case it is an excellent plan to combine with the pro- file of the centre line the profiles of parallel lines ten 42 FIELD ENGINEERING. or twenty feet either side of the centre, and drawn with differ- ent colored inks, as these will indicate tolerably well the relative amount of cut and fill required. But after the grade has been thus chosen, the only safe method in side-hill work is to actually compute the amounts of excavation and embankment from cross-sections, mark the amount for each cut and fill on the profile, and compare the results. Any changes required in the grade or alignment may then be discovered arid effected before the work of construction has begun. CHAPTER V. SIMPLE CURVES. A. Elementary Relations. 82. The centre line of a located road is composed alternately of straight lines and curves. The straight lines are called tangents because they are laid exactly tangent to the curves. A tangent may be indefinitely long, but should never, as a rule, be shorter than 200 feet between two curves which deflect in opposite directions, nor shorter than 500 feet between curves which deflect in the same direction. A curve should not be less than 200 feet long. When a tangent is said lo be straight, the meaning simply is that it has no deflections to the right or left; for since it fol- lows the surface of the ground, it evidently has as many undulations as the ground. But if we conceive a vertical plane to be passed through the line, a horizontal trace of this plane will accurately represent the line; and so, if we con- ceive a vertical cylinder to be passed through a curve on the surface of the ground, a horizontal trace of that cylinder will accurately represent the curve, since all distances and angles are measured horizontally, whatever be the irregularities of the surface. In all problems, therefore, relating to this sub- ject, we may consider the ground to be an absolutely level plain. 83. A Simple curve is a circular arc joining two tan- gents. It is always considered as limited by the two tangent SIMPLE CURVES. 43 points, and any part of it beyond these points is called the curve produced. The first tangent point, or the point where the curve begins, is called the Point of Curve, and is indicated by the initials P. G. The point where the curve ends, and the next tangent begins, is called the Point of Tangent, and is indi- cated by the initials P. T. When accessible, these points are always occupied by the transit in the course of the survey, and the plug driven to fix the point is guarded, not only by the usual stake bearing the number of the station, but also by another bearing the proper initials, the " degree" of the curve, and an " R" or "L" to indicate whether the deflection is to the Eight or Left. 84. A simple curve is designated either by the radius, JR, or the degree of curve, D. The Degree of Curve, D, is an angle at the centre, sub- tended by a chord of 100 feet. It is expressed by the number of degrees and minutes in that angle, or in the arc of the x curve limited by the chord of ICO feet. Therefore D equals the num- ber of degrees of arc per station. The radius R and degree of curve D can be expressed in terms of each other. Let ab, Fig. 3, be a chord of Fl - 3> 100 feet subtending an arc de- scribed with a radius ao R from the centre o. Then, by definition the angle boa D. Bisect the angle boa by a line orj, and this line will also bisect the chord ab and be perpen- dicular to it ; and in the right-angled triangle bgo we have bg = ob X sin bog Hence, to find Radius in terms of Degree of Curve:. R= (16) sin \D and to find Degree of Curve in terms of Radius: sin iD = A < 17 ) JK 44 FIELD ENGIHEERLim It is the practice of English engineers to assume the radius at some round number of feet and calculate the degree of curve, which is therefore fractional. In America, on the contrary, the degree of curve is assumed at some integral number of degrees or minutes, and the radius deduced from this. Example. What is the radius of a 3 20' curve? 50 log 1.698970 iD = l40' log sin 8. 463665 Ans. ^ = 1719.12 log 3.235305 Thus the second and third columns of Table IV. have been calculated. Example. "What is the degree of curve when the radius is 600 feet? 50 log 1.698970 .8 = 600 log 2.7781^1 \D 4 46' 48". 73 log sin 8.920819 Ans. D = 9 33' 37". 46 Measurement of Curves. 85. A railroad curve is always assumed to be measured with a 100-foot chain, and as the chain is stretched straight between stations it cannot coincide with the arc of the curve, but forms a chord to the arc, as in Fig. 3. Consequently the curve as measured from one tangent point to the other is an inscribed polygon of equal sides, each side being 100 feet. The sum of these sides (with any fraction of a side at either end of the curve) is called the Length of curve, L. This length L is evidently a little less than the length of the actual arc between the same points, but the latter we very seldom have occasion to consider. 86. If the chain lengths were taken on the arc instead of as chords of the curve, the degree of curve would be inversely proportional to the radius, and since the arc whose length is equal to radius contains 57.3 degrees nearly, we should have D : 57. 3 :: 100 : It. or 573 SIMPLE CURVES. 45 a convenient formula, but only approximately true when D is small, and seriously at fault when D is large; the error in- volved being proportional to the difference in length of a 100-foot chord, and the arc which it subtends. 87. The Central Angle of a simple curve is the angle at the centre included between the radii which pass through the tangent points (P. C.) and (P.T.). It is therefore equal to the number of degrees contained in the entire arc of the curve between those points. The central angle will be designated by the Greek letter A (delta). From the definitions of the length and degree of curve we have the proportion, D : A :: 100 : L. Hence, to find the Length of curve in terms of the central angle: X = 100 I (18) Example. What is the length of a 4 curve when the cen- tral angle is 29 ? D = 4 and A = 29 j 4)2900 Ans. L = 7 stations -f- 25 feet I 725 feet. To find the Central angle in terms of the length and degree of curve: * = m Example. What is the central angle of a 5 curve 730 feet long? = 730, = 36. 5 100 Ans. A = 36 30' To find the Degree of curve in terms of the length and ventral angle: Example. What is the degree of a curve 8 stations long, and having a central angle of 26 40' ? L = 800, A = 26.666, 100 -^f 6 = 3.333 oUU Ans. D = 3 20' 46 FIELD ENGINEERING. FIG. 4. 88. If two tangents, joined by a simple curve, are produced (one forward and the other backward) until they intersect, the point of intersection, V (Fig. 4), is called the vertex, and the exterior or deflection angle which they make with each other is equal to the central angle, A The Tangent-distance, T, is the distance from the vertex to either tangent point; thus in Fig. 4, T=AV=VB. The Long Chord, C, is the line AB joining the two tangent points. The Middle-ordinate, M, is the line QII, joining the middle point of the long chord with the middle point of the curve. The External distance, E, is the line HV, joining the middle point of the curve with the vertex. We observe that both the middle-ordinate, M, and the external distance, E, are on the radial line joining the centre, 0, with the vertex, V, and that this line is perpendicular to the long chord, C; also, that it bisects the central angle AOB= A, and its supplement A VB. (Tab. 1. 14.) We also observe that the angle VAB = VBA =-}A (Tab. I. 20); and if in the figure we draw the two chords AH and BH, the angle BAH equals one half the angle BOH, or BAH ABU i-A (Tab. I. 18); also the angle VAH= VBH=A. 89. If we have laid out two tangents on the ground, inter- secting at V, and have measured the angle, A , between them, we may then assume any other one of the elements of a simple curve before mentioned, and calculate the rest. If we assume D, for instance, we then find It by eq. (16) or by Table IV. Then, having A and E, we may proceed to calculate the other elements as they are needed, 90. To find the Tangent-distance in terms of the Radius and Central Angle : SIMPLE CUKVES. 47 In the right-angled triangle VGA, Fig. 4, we have VA=OAX tan VOA .: T = .Rtan -JA (21) Otherwise, approximately: In Table VI., opposite the central angle, take the value of T for a 1 curve and divide it by the degree of curve D. If desirable, add the correction taken from Table V., corresponding to D. Example. What is the tangent distance of a 4 curve with a central angle of 30 ? D = 4 R (Table IV.) log 3 . 156151 A = 30, i A = 15 log tan 9.428052 Ana. T = 383. 89 feet log 2.584203 Otherwise : By Table VI. 4)1535.3 Approximate ans. 383.82 Correction from Table V. .08 Ans. T 383.90 feet. 91. To find the Long Chord C, in terms of Radius and Central Angle : In the right-angled triangle BOG> Fig. 4, we have BG = BOX sin BOG ' . C = 2EsmiA (22) But in case A can be divided by D without a remainder, that is, if the curve contains an exact number of stations (not exceeding 12), we may take the long chord at once from Table VII. Example. What is the long chord of a 3 20' curve with a central angle of 36 40' ? 2 log 0.301030 D - 3 20', R (Tab. IV.) log 3.235305 A = 36 40', i A = 18 20' log sin 9.497682 Ans. C = 1081.48 feet log 3.034017 48 FIELD Otherwise : O/<> = ?!- = It stations 2} 3J- And by Table VII. C= 1081.48. 92. To find the Middle-ordinate M, in terms of Radius and Central Angle: It is evident from the figure that if the radius OH wore unity, the line OH would be the nat. versed sine of the arc BH. But the arc EH measures the angle JBOH=i&, and OH= R; A (23) But in case A can be divided by D without a remainder, that is, if the curve contains an exact number of stations (not exceeding 12), we may take the middle-ordinate at once from Table VIII. Example. What is the middle-ordinate of- a 4 30' curve with a central angle of 40 30' ? D = 4 30', J2(Tab. IV.) log 3.105022 A = 40 30', |A = 20 15' log vers 8.791049 Ans. M = 78. 717 1 . 896071 Otherwise : A 40.5 . 4 ' 5 = -475 = 9 stations and by Tab. VIII. M = 78.717 93. To find the External Distance E in terms of Radius and Central Angle. It is evident from the figure that if the radius OA were unity, the portion HV of the secant line 0V would be the external secant of the arc AH. But the arc AH measures the angle AOH A, and OA = R; .'. E = Rex sec^A (24) Otherwise, approximately: In Table VI., opposite the central angle, take the value of E for a 1 curve, and divide it by the degree of curve D. If desirable, add the proper correction corresponding to D, taken from Table V. SIMPLE CURVES. 49* Example. What is the external distance JEJ of a 7 30' curve when the central angle is 60 ? D = 7 30', R (Tab. IV.) log 2 . 883371 A = 60, i A = 30 log ex sec 9 . 189492 Am. E = 118.27 feet log 2.072863 Otherwise: By Tab. VI. 7.5)886.38 Approximate ans. 118.184 Correction for D = 7 30' (Tab. V.) .084 Ans. E = 118.268 94. But, instead of assuming D or R, we may prefer, or may find it necessary to assume, some other element of the curve, the central angle being given. If we assume the tangent distance, then: 95. To find the Radius and Degree of Curve in terms of the Tangent-distance and Central Angle. From eq. (21), and by Table II. 40, we have E=Tcot^A (25) Otherwise, approximately: Divide the tangent of a 1 curve found opposite the value of A in Table VI., by the assumed tangent distance; the quotient will be the degree of curve in degrees and decimals. Example. The exterior angle at the vertex is 54, and the tangent distance must be about 700 feet. What shall be the degree of curve? A = 54, $ A = 27 log cot . 292834 T -700 2.845098 log= 3.137932 Am. By Table IV. D = 4 10' + Otherwise: By Table VI. 700)2919.4 Ans. D = 4 10' 15" 4.1706 But as it is difficult to lay out a curve when D is fractional, we discard the fraction and assume 4 10' as the value of Z>. 50 FIELD This may require us to recalculate the value of T, which we do by eq. (21) and find T = 700.8 feet log 2.845596. If the other elements are required, they may be calculated by eqs. (22), (23), (24), or directly from Tand A, as follows: 96. To find the External distance E, in terms of the Tangent-distance and Central Angle. In Fig. 5 we have given AOB= A and AV T, to find HV= E. In the diagram draw the chord AH, and through // draw a tangent line to intersect OA pro- duced in /, and join VI. Then HI is parallel to BA, and since HI= AV= T, and AI= HV Fio g = E, VI is parallel to HA, and VIH = HAB = i A. (Tab. I. 18.) In the right-angled triangle VHIwc have tan VIH or E (26) Example. The angle at the vertex being 54 and the tan- gent-distance" 700.80 feet, how far will the curve pass from the vertex ? T= 700.80 (from last example) 2.845596 A 54 5 i A =13 30' log tan 9 . 380354 . Am. E = 168.25 feet 2.225950 (For the formulae by which to find the long chord and mid- dle-ordinate in terms of the tangent-distance and central angle, see Table III. 12 and 13.) 97. Again, it may be necessary to assume the external dis- tance in order to determine the proper degree of curve. To find the Radius and Degree of Curve in terms of the External distance and Central Angle: k ' By eq. (24) E ex sec (27) SIMPLE CURVES. 51 Otherwise: In Table VI. divide the external distance of a 1 curve, opposite the given value of A, by the assumed external dis- tance ; the quotient is the degree of curve required. Example. The angle at the vertex being 24 30', the curve is desired to pass at about 65 feet from the vertex. What is the proper degree of curve ? E = Q5 log 1.812913 A = 24 30', iA = 12 15' log ex sec 8.367345 logJ2= 3.445568 Am. By Table IV. D = 2 03' + Otherwise: By Table VI. 65)133.50 Ans. D = 203' 14" 2. 0538 We may therefore assume a 2 curve, unless required by the circumstances to be more exact, when we might use a 2 03' curve. Assuming a 2 curve, we have by eq. (24) .0=68.75 log 1.824460 Having decided on the degree of curve, we may calculate the remaining elements by eqs. (21), (22), (23), which is always the better way, but we may calculate them directly from E and A. 98. To find the Tangent-distance in terms of the External distance and Central Angle: From eq. (26), and by Table II. 40, T=Eco\,& (28) Example. The angle at the vertex is 24 30', and the curve passes 66.75 feet from the vertex. How far are the tangent points from the vertex ? E - 68 . 75 (from last cxampls) log 1 . 824460 A = 24 30', i A = 6 07' 30" log cot 0.969358 Ans. T = 622.04 feet 2.793818 99. Remark. Eqs. (27) and (28) are particularly useful in denning the curve of a railroad track where all original FIELD ENGINEERING. points are lost. Produce the centre lines of the tangents of the curve to an intersection V, and there measure the angle A . Bisect its supplement A VB, and measure the distance on the bisecting line from Fto the centre line of the track. This will give VII E. Then R and T may be calculated, and the distance T laid off from Fon the tangents, giving the tangent points A and B. (For the formulae by which to find the long chord and mid- dle-ordinate in terms of E and A, see Table III. 16 and 17.) 100. Again, having only the central angle given, we may assume the long chord, or the middle-ordinate, and from either of these and the central angle calculate the remaining ele- ments. Or, finally, the central angle being unknown, we may suppose any two of the linear elements given, and from these calculate the rest. As such problems have little practical value, their discussion is omitted. The requisite formulae for their solution are given in Table III., and the verification of them is suggested as a profitable exercise to the student. B, Location of Curves by Deflection Angles. 101. In order that the stakes at the extremities of the 100-foot chords, by which the curve is measured, shall be set exactly on the arc of the curve by transit observation, it is neces- sary at the point of curve, A, to deflect certain definite angles from the tangent AV. Let us suppose that in the curve AB, Fig. 6, the points A, a, b, c, d, etc., indicate the proper posi- tions of the stakes 100 feet apart, and that OA is the radius of the curve. In the diagram join Oa, Ob, etc., and also Aa, ab, be, etc. Then, by definition, the angle AOa = D, and by Geom. (Tab. I. 20 and 11) the angle VAa = D. Therefore if we set the transit at A, and deflect from AV the angle \D, we shall get the x direction of the cltord Aa, on which by measuring 100 feet from A we fix the stake, a, in its true position on the curve. So again, since the angle a Ob, at the centre, = D, the angle aAb, at the circumference, = %D. FIG. 6. SIMPLE CURVES. 53 If therefore, with the transit at A, we deflect the angle \I> from the chord Aa, we shall get the direction of the chord Ab; and when the stake b is on this chord it will also be on the curve, if b is 100 feet distant from a. Thus, in general, we may fix the position of any stake on the curve, by deflect- ing an angle 4Z> from the preceding stake, and at the same time measuring a chain's length from it, the chain giving the distance, while the instrument at A gives the direction of the point. \D is called the Deflection-angle of the curve ; so that in any curve, the deflection-angle is equal to one half the degree of curve. 102. Since each additional station on the curve requires an additional deflection-angle, the proper deflection to be made at the tangent point from the tangent to any stake on. the curve is equal to the deflection-angle of the curve multiplied by the number of stations in the curve up to that stake ; or it is equal to one half the angle at the centre subtended by the included arc of the curve. 103. It may happen that all the stations of a curve are not visible from the tangent point, A. When this is the case a new transit-point must be prepared at some point on the curve, by driving a plug and centre in the usual manner, and the transit moved up to it. Let us suppose that the point d, Fig. 6, has been selected for a transit-point, and that the transit has been set up over it. Before the curve can be run any farther, it is necessary to find the direction of a tangent to the curve at the point d. For this purpose we deflect from chord dA an angle Adz equal to the angle VAd previously deflected to fix the point d. (Tab. 1. 16.) Or we may adopt the following Rule : To find the direction of the tangent to a curve at the extremity of a given chord, deflect from the chord an angle equal to one half the angle at the centre subtended by the chord. (Tab. I. 20.) Having thus found the direction of the auxiliary tangent zdx, we proceed to deflect from-tfoj, (^D) for the next station e, 2 (|D) for station/, 3(|D) for station g, etc., as before. When the end of the curve is reached, a transit-point is set at the Point of Tangent, after which it only remains to find the direction of the tangent, by the above rule. Thus if g is to be 54 FIELD the point of tangent, we obtain the direction of the tangent by deflecting from the chord gd an angle equal to xdy, or to \ dOg. If this tangent VB was already established, the line gx thus obtained should coincide with it; and if it does so, the correctness of our work is proved. 104. The centre line is measured, and the stations num- bered regularly and continuously through tangents and curves from the starting point fo the end of the work. It therefore frequently happens that a curve will neither begin nor end at an even station, but at some intermediate point, or plus distance. If the Point of Curve occurs a certain number of feet beyond a station, the first chord on the curve is composed of the remaining number of feet required to make 100. Any chord less than 100 feet is called a subchord. If a curve ends with a subchord, the remainder of the 100 feet must be laid oft' on the tangent from the Point of Tangent to give the position of the next station, so that the stations may everywhere be 100 feet apart. 105. The deflection to be made for a subchord is equal to one half the arc it subtends. Let c length of any subchord in feet. " d = angle at centre subtended by subchord. Then, from eq. (22), by analogy c = 2.72 sin $d (29) 100 But by eq. (16) Z = ' (30) .-. smirf=~-sinp> (31) When D does not exceed 8 or 10, we may assume without serious error that the angles are to each other as their sines, and the last two equations become (approx.) c = 100 - (32) SIMPLE CURVES. 55 and irf=~(P>) (33) In curves sharper than 10 per station, the error involved in this assumption becomes apparent and must be corrected. 1OO. If curves were measured on the actual arc, then eqs. (32) and (33) would be true in all cases; but since a curve is measured by 100-ft. chords, it is evident that if a 100-ft. chord between any two stations were replaced by two or more subchords, these taken together would be longer than 100 feet, since they are not in the same straight line. Let us conceive the actual arc of one station to be divided into 100 equal parts; since the arc is longer than the chord, each part will be slightly longer than one foot. Now if we take an arc contain- ing any number of these parts (less than 100), the nominal length of the corresponding subchord in feet will equal the number of parts, and the deflection for the subchord will be proportional to the number of parts which the arc contains. The deflection therefore will be exactly given by eq. (33) if in that equation we let c equal the number of parts in the arc, or the nominal length of the subchord in feet. Having thus obtained the correct value of (%tf), we may introduce it into eq. (29) or (30), and obtain the true value of the subchord, which will always be a little greater than its nominal value. Suppose, for instance, that the arc of one station is to be divided into four equal portions; then each subchord will be nominally 25 feet long; and by cq. (33) which is the correct value of the deflection, whatever be the degree of curve. Substituting this value in eq. (29) .or (30) we obtain the true value of the subchord, c, a little greater than 25; the excess is called the correction of the nominal length. 1O7. This correction for any given subchord bears an almost constant ratio to the excess of arc per station, what- ever be the degree of curve. These ratios are shown in the following table for a series of subchords, and Table VII. gives the length of actual arc per -station for various degrees of curve. Subtracting 100 we have the excess of arc per station, and multiplying this excess by the ratio corresponding to the 56 FIELD ENGINEERING. nominal length of subchord we obtain as a product the proper correction for the subchord. TABLE OP THE RATIOS OF CORRECTIONS OF SUBCHORDS TO THE EXCESS OF ARC PER STATION. Nominal Length of Subchord. Ratio. Nominal Length of Subchord. Ratio. Nominal Length of Subchord. Ratio. 5 - 10 15 20 25 30 .COD .059 .099 .147 .192 .234 .273 35 40 45 50 55 60 65 .507 .33'J .358 .374 .383 .383 .374 70 75 80 85 90 95 100 .356 .327 .287 .235 .169 .OJ2 .003 We observe that the largest correction is required by a sub- chord between 55 and 60 feet in length. Example. It is proposed to run a 14 curve with a 50-ft. chain. What correction must be added to the chain? By eq. (30) c = 100 Ans. Correction = .093 Or, by Table VII., and by above table, Ans. Correction = product = ~ X 7 = 3.5 = 3 30' = 50.093 sm 7 length of arc = 100.249 excess of arc = . 249 ratio for 50 feet = . 374 = .093 Example. The P. C. of an 18 curve is fixed at -}- 55 feet beyond a station. What are the nominal and true values of the first subchord, and what the proper deflection? Nominal value = 100 55 = 45 feet Deflection = %d = - X 9 = 4. 05 = 4 03' 1UU and by eq (30) True value = c = 100 sm . ft sm9 - = 45.148 SIMPLE CURVES. 57 Or, by Table VII., excess of arc .412 by above table, ratio for 45 feet = . 358 Correction = product = . 147 Ans. True value of subchord = 45 . 147 Example. The last deflection at the end of a 40 curve is found to be 6 30'. What are the nominal and true values of the last subchord? Here $d = 6 30', and by eq. (32) / p* Nominal value, c = 100 -^- = 32.5 feet By eq. (30) True value, c = 100 ~-~ ~ = 33.098 feet sin 20 Or by Table VII., excess of arc 40 = 2.060 by above table, ratio for 32 . 5 feet = .290 Correction product = .597 Nominal value of subchord = 32 . 5 True value =33.097 1O8. For convenience in making deflections, the zeros of the instrument should always be together when the line of collimation coincides with a tangent to the curve. Thus, in beginning a curve, the transit being set at the P. C. zeros together, and line of collimation on the tangent, the read- ing of the limb for any station on the curve has simply to be made equal to the proper deflection from the tangent for that station. After the transit is moved forward from the P. C. and set at another point of the curve, the vernier is set to a reading equal to the reading used to establish that point, but on the opposite side of the zero of the limb, and the line of collimation is set on the P. C. just left. Then by simply turn- ing the zeros together again, the line of collimation will be made to coincide with a tangent to the curve through the new point, and the deflections for the succeeding stations can be read off directly, as before. Thus any number of transit points may be used in locating a curve by finding the direc- tion of the tangent through each by a deflection from the pre- ceding point, until finally the P. T. is reached, where another deflection gives the direction of the located tangent. 58 FIELD ENGINEERING. 1O9. The assistant engineer keeps neat and systematic field-notes of all his operations with the transit in running curves. The numbers of the stations are written in regular order up the first column of the left-hand page of the field- book, using every line, or every other line, as may be pre- ferred. The second column contains the initials of each transit point on the same line as the number of its station, or between lines, if the point occurs between two stations. In the third column, and opposite the initials in the second, is recorded the station and plus distance, if any, of each transit point. The fourth column contains, opposite the "P. (7.," the degree of curve used, and an R or L, showing whether the curve deflects to the right or left ; the fifth column contains the readings or deflections made from a tangent to set each station or point, written on the same line as the number of that station or point; and the sixth column contains the cen- tral angle of the whole curve, A, written opposite the "P. T" The plus distances recorded in the third column are always the nominal lengths of subchords, but if the true lengths have been calcu- lated and laid off on the ground, these should also be recorded in parenthesis. On the right-hand page are recorded the calculated bearings of the tangents and their A magnetic bearings; and on the centre line of the page, opposite the record of each transit point, a dot is made with a small circle around it, to show the relative position of the several points on the ground. Some slight topographical sketches may be made, indicating the more prominent objects, but the full sketches should be taken by the topographer in a separate book. HO. Since the deflections start from zero at each new transit point, the sum of the deflections by which the transit points are located will be equal to one half the central angle of the curve. 111. The stations on a curve may be located by deflec- tions only, Avithout linear measurements. For this purpose two transits are set at two transit points on the curve, as A FIG. 7. SIMPLE CURVES. 59 and B, Fig. 7, and the proper deflections for any station are made with both instruments, the station being located by find- ing the intersection of the two lines of collimation. This method requires that the two transit points shall have been previously established, that their distance from each other shall be known, that they shall be visible from each other, and that they shall both command a view of the stations to be located. It is not therefore generally useful, but may be resorted to to set stations which fall where chaining cannot be accurately done, as in water or swamps. The chord join- ing the two transit points becomes, in fact, a base-line, and the deflections form a series of triangulations. C. Location of Curves by Offsets. 112. A curve may be located by linear measurement only, without angular deflections. There are four general methods, By offsets from the chords produced, By middle-ordinates, By offsets from the tangents, and By ordinates from a long chord. To locate a curve by offsets from the chords produced. When the curve begins and ends at a station. 113. Let A, Fig. 8, be the P. C. of a curve taken at a station, to locate the other stations, a, b, c, etc. The chords Aa, ab, be, etc., each equal 100 feet, and since the angle AOa = D, the angle VAa \D. (Tab. I. 20.) Taking an off- set ax t, perpendicular to the tangent, we have in the right- angled triangle Axa. ax = Aa X sin \T> or t = 100 sin ^D (34) The offset t is called the tangent offset, and its value is givenfor all degrees of curve in Tab. IV. col. 4. FIG. 8. If the curve were produced backward from A, 100 feet to station z, the offset zy would 60 FIELD equal t; and if the chord zA were produced 100 feet from A to a', the offset a'x would also equal t. Therefore the distance aa' = 2t, and the angle aAa' = D. So if we produce the chord Aa 100 feet to b', the distance bb' = 2t. To lay out the curve, stretch the chain from A, keeping the forward end at a perpendicular distance, t, from the line of the tangent to locate station a. Then find the point b' by stretch- ing the chain from a in line with a and A, and then stretching the chain again from a, fix its forward end at a distance from b' equal to 2t. This gives station b. In the same way find other stations. When the last station, as d, of the curve is reached, produce the curve one station farther to c" '. Then the tangent through d is parallel to the chord ce", and laying off t from c and e" per- pendicular to this chord, the tangent c"e is found. If the work has been correctly done the tangent c"e will coincide with the given tangent VB. When the curve begins or ends with a subchord. 114. Let A, Fig. 9, be the P.O. and Aa the first sub- chord = c, and the angle VAa = ^d, and let the offset ax = ^. Then ti = c sin ^d (35) Producing the curve backward to the nearest station z, we have another subchord Az = (100 c), and the angle yAz = i (D d) t and putting the offset yz = t, t. = (100 - c) sin i(D-d) (36) Laying off the two subchords on the ground, and making the proper offsets, t t and t u , at the same time, we fix the position of the two stations a and z on the curve ; after which we may pro- duce the chord za 100 feet to b', and proceed as before until the curve is finished. If the curve ends with a sub- chord, as dB, produce the curve to the first station beyond B, as e", then calculate the two offsets for the two subchords Bd and Be", and lay them off from d and e" SIMPLE CUBVES. 61 perpendicular to the supposed direction of the tangent. If the line d"e so obtained coincides with the given tangent, VB, the work is correct. 115. We may find the values of t t and t u otherwise than by the formulae above, for in Fig. 8 we have shown that the angle aAoJ aOA, and since these triangles are isosceles, they are similar; therefore Fig. 8, OA : Aa :: Aa : aa' or R: 100:: 100 : 2t and similarly, Fig. 9, t - Hence t, : t :: c 2 : (100) 2 c 2 t Thus t t may be found by multiplying the square of the sub- chord by the value of t given in Tab. IV., and dividing the product by 10000. As c is always less than 100, so t, is always less than t. 116. In eqs. (35), (38), and (39) it is customary to use the nominal values of c, and this can produce no error in t or t, exceeding -005, when the degree of curve does not exceed ten degrees. In the case of a very sharp curve, the formulae eqs. (40) and (41) are preferable. To locate a curve by middle-ordinates. "When the curve begins and ends at a station. 117. In Fig. 10, let A be the P. C. at a station, and let a and z be the next stations on the curve either way from A. Then, since zy = ax = t, the chord za is parallel to the' tangent A V, and Ag = t. Hence, having any two consecutive stations on the curve, as z and A, we may lay off the tangent offset t from A to g on the radius, and find the next station, a, 100 feet from A on the line zg produced. Then laying off ah = t on the radius aO, a point on the line Ah produced and 100 feet from a will be the next station b. 62 FIELD On reaching the end of the curve, the tangent is found precisely as described in the method by chords produced, 113. In Fig. 10, we observe that if the radius OA were unity, gA would be the versed sine of the angle aOA = D. But gA = t, . t = R vers D (40) When the curve begins or ends with a subchord. 118. Let A, Fig. 11, be the P.O., and a and z the nearest FIG. 10. FIG. It stations. Then Aa = c, the first subchord, and aOA = d, and by analogy, we have from the last equation, if ax = t, and t, = R vers d t = (41) or eq. (39) may be used if preferred. Having found the two stations, a and z, on the curve, lay off from the forward station a, ah = t on the radius, and so continue the curve as described above. When the end of the curve is reached, produce the curve to the next station beyond, and find the tangent by offsets as described in the previous method, 114. To locate a curve by offsets from the tangents. When the curve begins at a station. 119. Let A, Fig. 12, be the P.O. at a station. Then the next station a is located by the tangeot offset t, taken from SIMPLE CURVES. 63 Tab. IV., or calculated by eq. (40). To calculate the distances and offsets for the following stations, b, c, etc. , in the diagram draw lines through the points b, c, etc., parallel to the tangent AV, intersecting the radius AO in g' , g", etc., and draw the lines bx' , ex", etc., perpendicular to the tangent. Then Ax' = g'b = Ob sin bOA and Also, or and Ax' = R sin 2Z> Ax" = It sin 3D etc. etc. bOA f = t" =R vers 3Z) etc. etc. (43) (43) But these calculations may be avoided, for as twice ag equals the chord of two stations, so twice bg' equals the chord of four stations, and twice eg" the chord of six stations, etc. So also as Ag is the middle-ordinate of two sta- tion, Ag' is the middle-ordinate of four, and Ag" the middle-ordinate of six stations, etc. Hence the rule : The distance on the tangent from the tangent point to the perpendicu- lar offset for the extremity of any arc is equal to one half the long chord for twice that arc; and the offset from the tangent to the ex- tremity of any arc is equal to the middle-ordinate of twice that arc. The long chords and middle-ordinates may be taken from Tables VII. and VIII. for 2, 4, 6, 8, etc., stations, when the P. (7. is at a station, or for 1, 3, 5, 7, etc., stations, when the P. C. is at + 50, or half a station. If the offsets from the first tangent AV prove inconveniently long, the second half of the curve may be located from the other tangent BY, beginning at the point of tangent B, and closing on a station located from the first tangent. FIG. 12. 64 FIELD When the curve begins with a subchord. 12O. If d=the angle at centre, subtended by the first subchord, we have for the distances on the tangent (Fig. 13) Ax = .R sin d Ax' = R sin (d + Z>) Ax" = E sin (d + 2i>) etc. etc. and for the offsets (Fig. 11) t, = R vers d t =R vers (d + D) t" = R vers (d -f- 2D) etc. etc. (44) (45) If the first subchord equals 50 feet (nominal), then d =^D, and the Tables VII. and VIII. may be used as explained FIG. 13. FIG. 14. above. These tables may be used in any case, by adopting a temporary tangent through any station, and laying off the dis- tances on this, and making the offsets from it. When a curve is located by offsets the chain should be car- ried around the curve, if possible, to prove that the stations are 100 feet apart. To locate a curve by ordinates from a long chord. WJien the curve begins and ends at a station. 121. In Fig. 14 draw the long chord AB, joining the tan- gent points, and from this draw ordinates to all the stations on SIMPLE CURVES. 65 the curve. We then require to know the several distances on the long chord Aa', a'b', b'c', etc., and the length of ordinate at each point. Let G = the long chord AB, then eq. (22) C=2E sin |A If a is the second station and * next to the last on the curve, join ai, and let the chord ai = C'. Then since the arc Aa ik = D, the angle at the centre subtended by ' is ( A 2D). .. (7' = 212 sin i (A -21?) Again, if we join b and h the next stations and let bh= C' G" =2R sin (A - 4D) and so on for other chords. Since Aa' = ki, C= C' + 2Aa' , , C-G 1 '' Aa =-2- Similarly, , c'-cr a V Thus we continue to find the distances up to the middle of the curve, after which they repeat themselves in inverse order. 122. When the long chord G, subtends an even number of stations (as 10 in Fig. 14), the middle ordinate of the chord is the ordinate of the middle station, as e. Since the chords AB and ai are parallel, the ordinate a' a or Hi is evidently equal to the difference of the middle ordinates of these chords. Let M, M', M", etc., be the middle-ordinates of the chords C, G', G", etc. Then eq. (23) M =^vers|A M' = 12vers|(A - 2J9) M" = levers (A - 4D) etc. , etc. And a' a = i'l M M ' b'b =7i'h = M-M" etc. etc. etc. The values of the chords and middle-ordinates may be taken at once from Tables VII. and VIII. 66 FIELD Example. It is required to locate a 4 degree curve of ten stations by offsets from the long chord. By Table VII. : ^Diff. 97. 030 = '&' = i'h' 98.481 =b'c' =h'g' 99.452 = c') d"c = 100 sin i (A - 2d - 5Z>) etc., "etc., (46) (47) When the middle point of the curve is passed the minus quantities in the parentheses become greater than A, making the parentheses negative, and, therefore, the sines negative, and indicating that such values as are determined by them must be laid off toward the long chord AB. By a proper summation of the quantities determined by eqs. (46) and (47) we obtain the distances Aa', Ab', Ac', etc., and the ordinates a' a, b'b, c'c, etc., and the curve may be located accordingly. It is well to make all the necessary calculations before beginning to lay down the lines on the ground, thus avoiding confusion and mistakes. Example. The P.O. of a 3 20' curve is fixed at -|- 25 feet beyond a station, and the central angle is 16 24' = A. It is required to locate the curve by ordinates from the long chord. We have c = 100 - 25 = 75 and d = 2 30' and D = 3 20'. Hence, eqs. (46) Aa'= 75 cos 6 57' =74.449 ab" = 100 cos 4 02' = 99.752 be" = 100 cos 42' = 99.993 d"d = 100 cos (- 2 38') = 99.894 e"e = 100 cos (- 5 58') = 99.458 e'B = 17 cos (- 7 55') = 16.838 By eqs. (47) a'a= 75 sin 6 57' = 9.075 &"&=100sin 4 02'= 7.034 c"c = 100 sin 42' = 1 .222 cd" = 100 sin (- 2 38') = - 4.594 de" = 100 sin (- 5 58') = - 10.395 ee' = 17 sin(- 7 55') = - 2.341 74.449 = Aa' 174.201 = Ab' 274.194 = ^' 374.088 = Ad' 473.546 = J.' 490.384 = AB 9.075 = a' a 16.109 = b'b 17.331 = c'c 12.737 = d'd 2.342 = e'e 0.000 . SIMPLE CURVES. 69 The same formula? can be used when the curve begins at a station by making c = 100 and d D. 126. The methods of locating curves by linear measure- ments do not require the use of a transit, although one may be used to advantage for giving true lines, turning right angles, etc. When a transit is not used the alignments should be made across plumb-lines suspended over the exact points previously marked on top of the stakes. A right ang'le may easily be obtained, without an instrument, by laying off on the ground the three sides of either of the right-angled triangles represented in the following table (or any multiples of them), always making the base coincide with the given line. TABLE OF RIGHT-ANGLED TRIANGLES. Hypothenuse. Perpendicular. 5 3 13 5 4 12 24 40 60 84 96 25 41 61 85 100 7 9 11 13 28 D. Obstacles to the Location of Curves. 127. To locate a curve joining two tangents when the in- tersection V is inaccessible. Fig. 16. From any transit.point p on one tangent run a line pq to intersect the other tangent; measure pq and the angles it makes with the tangents. Then the sum of the de- flections at p and q equals the central angle A. Solve the triangle pqV and find Vp. Having decided on the radius R of the curve, calculate the tangent distance VA by eq. (21), and lay off from p the distance pA = VA Vp to locate the point of curve. The point p being as- sumed at random, Vp may exceed VA, in which case the differ- ence pA is to be laid off toward V. In case obstacles prevent the direct alignment of any line pq, a line of several courses may be substituted for it (as FIG. 16. 70 FIELD ENGINEERING. explained in 46, 47, 48,) from which the length of pq will be deduced. The algebraic sum of the several deflections will equal A. 128. To locate a curve wlien the point of curve is inaccessible. Fig. 17. Assume any distance Ap on the curve which will reach to an accessible point p. Then by eq. (19) the angle Ap' R sin pOA p'p = R ve Vp' = VA - Ap' Measure Vp' and p'p to locate a transit point at p; and meas- ure an equal offset from some transit point on the tangent, as qq'. This gives a line pq', parallel to the tangent, from which deflect at p an angle equal to pOA for the direction of a tangent through the point p. Instead of measuring the second offset qq' w r e may deflect from pq an angle found by tan qpq' = -^-~ and so obtain the line pq' parallel to the FIG. 17. tangent. Or we may deflect from p V the angle found by tan p Vp' = -^, to obtain the line q'p pro- duced, from which the tangent to the curve at p is found as above. Again, we may lay off from V, the external distance Vh found by eq. (24) or Tab. VI on a line bisecting the angle A VB. This gives us li, the middle point of the curve, and a line at right angles to li V is tangent to the curve at U, from which the curve may be located in either direction. 129. To locate a curve wJien both the Vertex and Point of curve are inaccessible. Fig. 18. From any point p on the tangent run a line pq' to the other SIMPLE CURVES. 71 tangent, and so determine pA as in 127. Suppose the curve produced backward to p' on the perpendicular offset pp'. Then sin p' OA = ^- and pp' = R vers p' OA Having located the point p', a parallel chord p'q may be laid off, giving a point q on the curve, since p'q = 2 X pA. At q deflect from qp' an angle equal to p' OA for a tangent to the curve at q. If any obstacle prevents using the chord p'q, any other FIG. 18. Fia. 19. chord as p's may be used, by deflecting from p'q the angle qp's = $ (qOs) and laying off its length, p's = 2R sin (p'OA + qp's). At s a deflection from the chord sp' of (p'OA + qp's) -will give the tangent at s. If obstacles prevent the use of any chord, the methods de- scribed in 131 may be resorted to. 13O. To pass from a curve to the forward tangent when the Point of Tangent is inaccessible. Fig. 19. From any transit point p on the curve, near the end of the curve, run a chord parallel to the tangent. The middle point g of the chord will be on the radius through the point of tan- gent B. At any convenient point beyond this an offset equal lo pp = R vers pOB may be made to the tangent, and at some other point an equal offset will fix the direction of the tangent. FIELD ENGINEERING. Otherwise, if an unobstructed line pq can be found inter- secting the tangent at a reasonable distance from B, measure the angle q'pq = pqp', and lay off the distance pg' = pp' sin qpq to fix the point q. Then Bq=p'q p'B = pp' cot q'pq R sin pOB. Otherwise ; assume an arc of any number of stations from p to q" on the curve produced, and take the length of chord from Tab. VII. Lay off pq", and from q" lay off q"q = H vers q"OB, perpendicular to the tangent, to locate g. The angle pq"q = 90 D q'pq", and the distance qB E sin q"OB. 131. To pass an obstacle on a curve. Fig. 20. From any transit point A' on the curve take the direction of a long chord which will miss the obstacle, as A'B' . The length of this chord is 2JR sin V'A'B', V'A being tangent to the curve at A' (see eq. 22), and by measuring this distance, the point B' on the curve is obtained. If the angle V'A 'B' is made equal to the deflection for an exact number of stations, the chord may be taken from Tab. VII. If the chord which will clear the obstacles would be too long for con- venience, as A'q', we may measure a part of it as A'p', and then, by an ordinate to some station, regain the curve at p. The distance on the curve from A' to p being assumed, the distances A'p' and p'p are calculated by the methods given in 121 to 125. If p'p can be made a middle ordinate the work will be much simplified. If more convenient the middle ordinate may first be laid off from A ' to p", and the half chord afterwards measured from p" to locate p. Again, we may calculate the auxiliary tangent A ' V for any assumed length of curve A 'B', and lay off the distance A'V and V'B 1 , deflecting at V an angle equal to twice FIG. 20. SIMPLE CURVES. 73 V'A'B'. But if the point V should prore inaccessible, we may conceive the auxilliary tangents to be revolved about the chord A B' as an axis, so that V will fall at V, and the lines A'V and V"B' may be laid out accordingly. If these in turn meet obstructions, we may run a curve from A ' to B' of same radius as the given curve, but tangent to A'V" and V"B'. Again, the entire curve or any portion of it may be laid out by offsets from the tangents, or by ordinates from a long chord, as already explained, 119 to 126. In case any distance on a curve must be measured by a tri- angulation, as in crossing a stream, a long chord may be chosen, either end of which is accessible, and the triangula- tion is then performed with respect to this chord or a part of it, as upon any other straight line. SPECIAL PROBLEMS IN SIMPLE CURVES. 132. Given: a curve joining two tangents, to find the change required in the radius B, and external distance E, for an assumed change in the value of the tangent distance T. Fig. 21. FIG. 21. Let T = AV= VB = A'V= VB' Then T T' AA = the given change. Byeq. (25) = T cot^A -K'= T cot^A OG - R R ' r= (T - T 1 } cot \ A (48) 74 FIELD EHGIKEERLtfG. By eq. (26), similarly, HH' = E - E' = (T - T') tan A (49) Eqs. (48) (49) give the changes in R and E for any change in T. When Tis increased R and E will be increased also, and vice versa. Example. A 4 curve joins two tangents, making an angle of 38 = A , and it is necessary to shorten the last tangent dis- tance 80 feet. What will be the change in the radius and in the external distance? Eq. (48) TT' = 80 log 1.903090 iA 19 log cot 0.463028 Ans. R -R' 232.34 log. 2.366118 R 1432.69 R ' = 1200. 35 or about 4 46' = D '. If the tangent distance had been increased 80 feet we should add the above to R. R' = 1665.03 or about 3 26' = I) ' Eq. (49) T- T' = 80 log 1.903090 iA 9 30' log tan 9.223607 Ans. EE' 13.387 log 1.126697 133. Given: a curve joining two tangents, to find the change required in the radius K, and tangent distance T, for any assumed change in the value of the external distance E. Fig. 21. We suppose HH ' given to find OG and A A '. By eq. (24) E =R ex sec i A E' = R' ex sec-A OG = R-R'=:~ (50) ex sec i A By eq. (49) AA = T- T' = (E- E') cot A (51) SIMPLE CURVES. 75 Example. A 4 curve joins two tangents, making an angle of 38 = A , and it is necessary to bring the middle point of the curve 25 feet nearer the vertex F. What changes are re- quired in the radius and point of curve? Eq. (50) E-E'= 25 log 1.397940 | A 19 log ex sec 8.760578 Am. R-R' 433.87 log 2.637362 R 1432.69 R ' 998. 82 or about 5 44' = D' Eq. (51) E-E' 25 log 1.397940 iA 9 30 log cot 0.776393 T- T' 149.39 2.174333 or the P. C. will be moved toward the vertex 149.39 feet. But if the point H, Fig. 21, were to be moved 25 feet further from the vertex V, then R' = 1866.56 or about 3 04' = D' and the P.O. will be moved 149.39 feet further from the vertex. It is preferable to assume some radius from Table IV. near the value of R ' found as above, and from this calculate the value of T 7 ' by eq. (21). 134. Given: a curve joining two tangents, to find the change made in the tangent distance T, and external distance E, by any assumed change in the value of the radius R. Fig. 21. By eq. (48) AA' = T- T' = (R -R') tan A (52) By eq. (50) HE' = E-E' =(R- R')exseclA (53) The changes calculated by eqs. (52) (53) will be added to or subtracted from T and E respectively, according as the radius is increased or diminished. 135. Since for a constant value of the central angle A, 76 FIELD ENGINEERING. the homologous parts of any two curves are proportional to each other, we may write at once 0' (54) etc. etc. etc. 136. Given: a curve joining two tangents, to change the position of the Point of curve so that the curve may end in a parallel tangent. Fig. 22. Let AB be the given curve, AV, VB the tangents, and V'B ' the parallel tangent. Then W is the distance from one vertex to the other; and since there is no change in the form or dimensions of the t;urve, we may conceive it to be moved bodily, parallel to the line AV, until it touches the line V'B', when every point of the curve will have moved a distance equal to VV. Hence AA ' = 00 ' = BB ' = VV. There- fore, run a line from B parallel to AV, intersecting the new tangent in B', measure BB', and lay off the dis- tance from A to find A. In the figure the new tangent is taken outside the curve, and so A ' falls beyond A, but if the new tangent were taken inside the curve at V"B", the new P. C. would fall back of A at some point A". If the parallel tangent is defined by a perpendicular offset from B, as Bp; since the angle BB'p = A FIG. 22. sin A (55) 137. Given: a curve joining two tangents, to find the radius of a curve that, from the same Point of curve, will end in a parallel tangent. Fig. 23. Let AB be the given curve, AV, VB the tangents, and V'B' the parallel tangent; and let AO = R and AO' = R'. SIMPLE CURVES. 77 Since the central angle A remains unchanged, the angle JA between the tangent and long chord remains unchanged; therefore V A B ' = VAB, and the new point of tangent is on the long chord AB produced. Find on the ground the inter- section of V'B' with AB produced and measure BB'. In the diagram draw Be parallel to AO, then BeB ' = A, and by eq. (22) but Be= 00' = R' - R ... R > = R + J^ (56) '2 sin -I- A . The -f- sign is used when B ' is be- FlG - 23 - yond B, as in the figure; but if the parallel tangent is within the given curve it will cut the chord in some point B ", and then the sign must be used, since R ' will evidently be less than R. If the parallel tangent is defined by a perpendicular offset, as Bp = B 'f; since BeB ' = A Bp Be vers A = (R ' M) vers A R' = RA ?2 (57) ' vers A Add or subtract as explained above. If the long chord C = AB is known, then the new long chord C' = AB ' or AB" = C BB', and by eq. (54) 138. Given: a curve joining two tangents, to change the radius, and also the Point of curve, so that the new curve may end in a parallel tangent directly opposite the given Point of tangent. Fig. 24. Let AB be the given curve, A V, VB the tangents, V'B' the parallel tangent, and B ' the given tangent point on the radius OB produced. 78 FIELD ENGINEERING. In the diagram, produce the tangent .AFand the radius OB to intersect at K. Then BK = R exsec A B ' K = R ' exsec A Subtracting we have BB' = (R-R') exsec A exsec A from which R' is easily determined, as in 132 and 133. (59) FIG. 24. FIG. 25. To find the change A A' of the P.O., in the diagram draw O'G parallel to A' A; then or 0'G = OG tan A AA = (R R') tan A (60) By substituting the value of (R R') from eq. (59) and ob- serving Table II. 42 we have AA =BB' X cot-l-A (61) Observe that eqs. (59), (60), and (61) may be derived directly from eqs. (50), (52), and (51) respectively by writing A for i A. 139. Given: a curve joining two tangents; to find the new tangent points after each tangent Ms been moved parallel to itself any distance in either direction. Fig. 25. SIMPLE CURVES. 70 Let A and B be the given tangent points, and A' and B ' the new tangent points required. Let the known perpendicu- lar distances Aq = a, and Bp = b. We then require the unknown parallel distances q A = x and pB ' = y. Since the form and dimensions of the curve remain un- changed we may conceive the curve to be moved bodily into its new position on lines parallel and equal to the line VV joining the vertices. Then A A = 00' = BB 1 = VV. In the diagram draw VK parallel and equal to Bp = b and VH parallel and equal to Aq = a. Then VH= qA = x, and V'K= B'p = y. Since VG V = A, we have VG = -T- and GH= ^ sin A tan A and since VH= VG-GH=x b sin A tan A Similarly (62) b a nj ~ tan A sin A When the new tangents are outside of the given curve, the offsets a and b are considered positive; if either new tangent were inside of the given curve its offset would be considered negative. In solving eqs. (62) if a- and y are found to be positive they are to be laid off forwards from q and p, as in Fig. 25; if either is found to be negative it is to be laid off in the opposite direction. Example. A certain curve has a central angle of 50 A , and it is proposed to move the first tangent in 20 feet and the second tangent out 12 feet. Required, the distances on the tangents from the old tangent points to the new. Fig. 26. 80 FIELD EHGIHEEKI^G. Here a = - 20 and b = -f 12 + b 12 1.079181 -a 20 1.301030 A 50" log sin 9.884254 A 50 log tan 0.076186 15.665 1.194927 - 16.782 ,1 1.224844 x = 15.665 -(- 16.782) = + 32 450 + & 12 1.079181 -a 20 1.301030 A 50 log tan 0.076186 A 50 log sin 9.884254 10.069 1.002995 'ii''i 26.108 1.416776 y = 10.069 - (- 26.108) = +36.177 i x = - 32.450 For -J- a and -b \y=- 36.177 x- - 1.120 =- 15.989 < = + 1.120 For - a and - b \y = + 15.939 If we have a and x given to find b and y: Solving eqs. (62) for b and y we obtain = x sin A -*' cos A | //ox y = cos A sn A In which the algebraic signs of the quantities must be ob- served as above. 14O. Given: a curve joining two tangents, to find a new Radius and new position of the Point of curve, such that the curve may end at the same point as before, but with a given change in the direction of the forward tangent. Fig. 27. Let AB be the given curve, AV, VB the given tangents, V'B the new tangent, and VBV the given change in direc- tion. Let A' = A + VBV. SIMPLE CURVES. 81 In the diagram draw.Btr perpendicular to AV produced; then BG = R vers A = R' vers A' Hence vers A (64) and AA = AG A'G = Rsm A R' sin A' (65) In the figure the change in direction of tangent makes A' greater than A ; therefore V falls beyond F, and A beyond . 27. FIG. 28. A ; but if the change made A ' less than A , then V and A would fall behind V and A respectively, and R ' would be greater than R. The same formulae apply to the converse problem in which B is taken as the point of curve, and A and A as points of tangent. 141. Given a curve joining two tangents, to find the change in the Point of curve when the forward tangent takes a new direction from the vertex V. Fig. 28. By eq. (21) AA' = R (tan i A tan A^) (66) 142. Given: a curve joining two tangents, to find the new OZ FIELD EKGIKEERING. radius, R', wlien the forward tangent takes a new direc- tion from the vertex, V. Fig. 29. By eqs. (21) (25) cotiA' (67) 143. Given: a curve joining two tangents, and a given change in the direction of the forward tangent from the vertex, to find the radius and point of curve of a curve that shall pass at the same distance, VH, from th^ vertex. Fig. 30. Let AS be the given curve, BVB' the given change in FIG. 29. FIG. 30. direction of tangent, and VH' = VH. Let A' = A + BVB'-, then eq. (24) VH= -Rex sec 4 A VH' = R' ex sec iA' By eq. (28) exsec (68) FA=F5"cotiA, VA = FIT cot i A' AA = FH"(cot iA cot A') (69) But in. case A' = A BVB', A A becomes negative and must be laid off backward from A. SIMPLE CURVES. 83 Example. Given a 2 curve, A = 80 and BVB' = - 10' .-. A' = 70 1A VH IA' JR' 40 874.97 35 1 27' nearly 20 17 30' log 3.457114 log exsec 9.484879 2.941993 log exsec 9.343949 3.598044 cot 2.74748 cot 3.17159 - 0.42411 AA = 874.97 X (- .42411) = - 371.08 and must be laid off backward from A. 144. Given: two indefinite tangents, a point situated be- tween them, and tJie angle A, to find the radius R, and tan- gent distance T of a curve joining the tangents which shall pass through the given point. Fig. 31. If the given point is on the bisecting line VO, as H, meas- ure VH= E, and find R and Tas in 97, 98. When the given point, as P is not on the bisecting line VO; if a line GK is passed through P per- pendicular to VO, it will be parallel to any long chord, as AB, and the angle VGK%&. The curve pass- ing through P will intersect GK in some other point P' ; the line GK is bisected by the line VO at /, and PI= P'L If the given point P is located by a perpendicular off set from the tangent, asPZ; in the triangle ' PLG, LG PL cot | A. Lay off LG, and at G deflect VGK= |A, and measure GP and PK. Since by Geom. (Tab. I. 24) GA* = GP' X OP, and OP' - PK; FIG. 81. OA= V (70) 84 FIELD ENGINEERING. Lay off GA; and A is the Point of curve, AV= T, and R = AVcotiA. If the given point were located by an offset from BV, find B first, and make VA = BV. If the given point Pis located by a perpendicular offset IP from the bisecting line VO; produce IP to intersect the tangent at G and measure PG. Since P'G = GP -\- 2PI GA= VGP(GP+2PI) (71) whence we have the point of curve A, as before. 145. Given: a curve, AP, and the radial offset PP' to Jind a curve which shall pass through the point P ', start- ing from the same point of curve A. Fig. 32. Let b PP', and in the diagram draw P'G ' parallel to the common tangent AX, and join AP'. Then P'G' = (R ft) sin A G'A = R - (R b) cos A tan 4- A ' _ P'G'~ (R b) sin A cot A R' = (JS b) sin A sn A sin A (72) (73) When the offset is outward use R + & when it is inward use R - b. Example. Given: a 3 curve of 16 stations and a radial offset of 205 feet inward from the P. T. to find the radius of the curve passing through the extremity of the offset. SIMPLE CURVES. 85 Here A = 3 X 16 = 48; and b = 205. H 3 = 1910.08 R-b 1705.08 log 3.231745 A 48 log sin 9.871073 P'G' 3.102818 RZ log 3.281051 1.50742 0.178233 A 48 cot .90040 iA' tan .60702 = 31 15J' 2 A' 62 31' log sin 9.947995 P'G' log 3.102818 R' (about 4 01'). Ans. 3.154823 If the same offset were made outside of the curve we should find R 1 log 3.438350, or about a 2 05' curve. This solution is inconveniently long for ordinary field prac- tice. When the offset is small compared with the length of curve, we may use the following Approximate Rule : Divide twice the offset 6 by the length of curve, look for the quotient in the table of nat. sines, and take out the corresponding angle, which multiply by 100, and divide by the length of curve. The quotient is the correction for the given degree of curve ; to be subtracted when the offset is made outward, and added when the offset is made inward. This rule is expressed by the formula J- = .D T *-'. (74) Ju Li Taking the same example, we have ~ = sin 14 51' Ju and correction = 14 51' X ; ^r- = T 56' 1600 Hence D' 3 56' or D' = 2 04' 86 FIELD ENGINEERING. THE VALVOID. 14G. Given: any number of circular curves of equal length L, all starting from a common point of curve A, in a common tangent AX, to find the equation of the curve joining their extremities. Fig. 33. Let AP be any one of the given curves, " R = its radius AO, " D = its degree of curve, " A = its central angle AOP, " C = its Ion? chord AP. FIG. 33. By substituting the value of ft from eq. (16) in eq. (23) we have C=l, and letting i = OPK = VPG. i=Q -q> = lA-q> (79) Therefore, to obtain the direction of a tangent to the val- void at any point P, deflect from the radius PO an angle equal to z ' = ( A (p), on the side of PO farthest from the point ot curve A. The value of i may be found by eqs. (78) (79), but we are saved this somewhat tedious calculation by the use of Table X. 1, which contains values of the ratio = u Fl - 34< A for various values of A, and length of curve L. Multiplying A by the proper tabulated number gives the value of i OPK at once ; or = (iA -' 1 |-x 95" ) = 6 I \ A I JrL. _ A! _i_ ' _ * By eq. (80) 2u') and A " - Fia .33. and since q>* = - - (1 2u) and p'op* = (A'- A')(l-tO ; But the condition of the problem requires A' A ff = l, hence p'op" = (1 u). Therefore the length of arc p'p" for a change of 1 in the value of A is l t = r(l u) X arc 1 or (Tab. XVII.) I, = r (I u) .0174533 and since r = vL (Tab. X. 2), l, = i)(\. u)L .0174533 (83) By this formula Table X. 3 has been prepared, for various values of A and L. 15O. Given: two curves of the same length L but of different radii, starting from the same point of curve in a 90 FIELD EKGINEEKING. common tangent, to determine the direction and length of a line joining their extremities. Fig. 36. Let AX be the common tangent, and AP ', AP" the two curves, to determine the direction and length of PP". If we take the point P on the arc P'P" determined by the A' + A angle A = and draw A a tangent PK to the valvoid at P, we may assume without ma- terial error that the chord P'P" will be parallel to PK for any value of P'P" not exceeding \L, a limit not likely to be ex ceeded in practice. Let be the centre of the curve AP fixing the point P ; then AOP = , and Since P'P" is assumed parallel to PK, PP'O" = KGO" = A"- K= A"- P'P'O" = i"= A'fl + ^l A '" 2 Similarly producing P"P to any point IT, HFO' = t = whence also f = i* + A' - A" (1-u) (84) (85) (85)' The slight error involved in the above assumption is cor- rected by taking out the value of u (Table X. 1) correspond- ing t.o A", the less of the two given central angles; we have therefore written u with the double accent in equations (84) and (85). SIMPLE CUKVES. 01 When i' and i* are positive, they will be deflected as in Fig. 36, on the side of the radius farthest from A ; should i" be negative it will of course be deflected from P"0" toward A. The arc P'P" corresponds to a change of the central angle from A' to A" ; hence 1 : A'- A" :: I, : P'P" or P'P" = .(A'- A"K (86) in which I, is taken from Table X. 3 for L = AP, and As in practice, the distance P'P" is usually small compared with L, the arc and chord will be almost identical and no further calculation is necessary. If P'P" is large, it will be found that equation (86) gives the length of arc very correctly when -- 59' PP' = 54.60 which proves this method practically correct. 152. Given: a tangent and curve, and a straight line intersecting them, making a given angle with tlie tangent at a given point, to determine the distance on the line from the tangent to the curve. Fig. 37. FIG. 37. We have OA, AG, and the angle AGP to find OP. tan AGO = ^- POO = AGO - -AGP sin POO 84 FIELD ENGINEERING. When AOP = AGO, cq. (24), GP = R exsec (90 - AGO) When AGP= 90, (92), (119), R When AGP' > AGO, we have P'GO = AGP' -AGO but the other formulae remain unchanged. Example. Let R = 955.37, AG = 350, AGP= 40 R 955.37 log 2. 9801 70 AG 350. log 2.544068 PGO AGO 69 52' 47" OPI POG R PG 72.40 Ana. 69 52' 47" 40 log tan 0.436102 log sin 9. 697387 log sin 9. 972653 29 52' 47" 32 02' 36" log sin 9. 724734 log sin 8.576953 8.879566 log 2. 9801 70 2 09' 49" log 1.859736 This problem may be used in passing from a tangent to a curve when the tangent point is obstructed. The distance ^4Pon the curve is defined by the angle ^4 OP, which is readily found. If AGP' > 2 AGO the line will not cut the curve. 153. Given : a curve and a distant point to find a tangent that shall pass through the point. Fig. 38. We have the curve adg and the point P visible, but distance unknown, to find the point of tangent P. SIMPLE CUIiVES. 95 Any chord, as bf, parallel to the required tangent, if pro- duced will pass the point P at a perpendicular distance equal to the middle ordinate of that chord. Ranging across every two consecutive stakes on the curve we at first find the range falling outside of the required tangent, as bcG, cdll, etc. ; but finally the range falls inside, as deK. "We then know that the required point is between c and e. \ If the range ce falls inside the point P, a perpendicular distance equal to the middle ordinate of ce, the tangent point is at d. If the perpendicular distance is greater than this, the point B is between c and d. If less, or if the range ce falls outside of P, the point B is between d and e. The middle ordinate for ce (200 feet) equals the tangent offset for 100 feet, given in Tab. IV., and it is generally so small that it can be estimated at ^without going to lay it off. To find the exact point B, when it falls between d and e, find by trial a point x on the arc cd in range with e and a point inside of P a perpendicular distance equal to the middle ordinate of ex. The point B is at the middle point of the arc ex. If the point B is between c and d, stand at c and find a point x on the arc de in the same way. B is at one half the arc ex. The middle ordinate of any chord ex is less than M for 200 feet, and greater than m for 100 feet, necessary, its exact value m' can be found by Fro. 38. If , _ m x ex* "lOOOCT (87) and this equation is' nearly true when ex is as great at 300 or 400 feet. That is, middle ordinates on the same curve are to each other as the squares of their chords very nearly. By this method the point B is found without the use of the transit, so that the plug can be driven at B before the transit 96 FIELD ENGINEERING. is brought up from the rear. It is therefore preferable to the following solution. Fig. 39. From any two points a and c of the curve measure the angles to the point P, so that with the chord ac as a base, and the measured angles, we may find cP by the formula sin caP cP = ac sin cPa Knowing the angle c that cP makes with a tangent at c, we find the length of the chord cd by cd = 2E sin c. By Geom. Tab. I. 24, PB=Pe = VcP X dP whence we know ce. Opposite with the radius Pe, we find B. or on the arc eB described FIG. 39. Fi. 40. 154. Given: two curves exterior to each other, to find the tangent points of a line tangent to both and its length between tangent points. Fig. 40. Let B and A be the required tangent points. Let OB = R, &ndO'A = E'. On the curve of greater radius R select a point H supposed to be near the unknown tangent point B, and knowing the SIMPLE CURVES. 97 direction of the radius Oil, find on the other curve a point K having a radius 'K parallel to OH, and measure HK. In the diagram draw Ob and O'a perpendicular to HK. Then the angle KO'a = 90 - HKO' = KO'A nearly, which is the angle required. We have therefore to find the correction aO'A = x, and apply it to KO'a. Aa = R' vers KO 'a; Bb R vers KO 'a nearly. Ka = R' sin KO 'a ; Hb = R sin KO 'a Bb Aa = (RR') vers KO'a ab =HK -\- (R - R') sin KO'a (R -^ R') vers KO'a (88) KO'A = (KO'a - a-) = HOB Observe that KO 'a the angle between the tangent at K or H and the line IIK ; and KO 'A = the angle between the tangent at K or H and the required tangent BA. If, instead of H and K, the points H' and K' had been. selected, then (R-R')versH'Ob . /oov Sm X = WK^(R - R ') aiOTOft nearly> (88) and H'OB = K'O'A = II' Ob -f x. The length of BA should be obtained by measurement, but it may be calculated by AB = ab - (R -R') sinx (89) When R = R', x 0, and HKis parallel to BA. In case the curves are reverse to each other, as in Fig. 41, 'a (90) KO'A = HOB = KO'a - x If the points H' and K' are selected, Fig. 41, ' Ob H'OB = K'O'A = H'Ob + x. 98 FIELD ENGIXEERIKG. The lines IIK, AB, and 00' all intersect in a common point J, Fig. 41. " IB = VHI(HI+ 21t sin HOb) (92) (93) (94) These last three equations furnish another method of solving the same problem. They may be applied to Fig. 40 by changing the sign of R'. In Fig. 41, if R = R', then HI \HK and AE = 2IB. FIG. 41. Fio. 42. 155. Given: two curves, O and O, reverse to each other, joined by a tangent BA', and terminating in another tangent, B'F ; to change the position of the Point of Tangent B of the first curve, so that the second curve may terminate m a given parallel tangent, B"F'. Fig. 42. Let X be the required new position of B. " 0" be the corresponding position of 0'. " A' = A'O'B' and A" = A"0"B\ Since^the radii and the connecting tangent are unchanged in length, and all rotate together about as a centre, " will be on a circle passing through 0', described with a radius 00', and the required angle BOX= O'OO". SIMPLE CURVES. 99 In the diagram, produce 'A' and draw the perpendicular OO, and let a = the angle 00' G. Also, draw OK parallel and 0"K and 'H perpendicular to B'O'. In the triangle 00' G we have cot 00 ' = -, or cota= ^~ ( 95 ) and cos a The angle KOO 1 = 00 'B' =.+ A'. The angle KOO" = 00 "B" ='a -j- A". ^0 = 00". cos (a -f A"), HO = 00'. cos (a -f A'). . \ 1TJ5T = 00' [cos (a + A") - cos (a -f A')] = B 'F' B'F' cos (a + A") = cos (a + A') + -^7 (97) BOX= O'OO" = (a + A') -(a+ A") (98) If we conceive a line to be drawn through bisecting the arc O'O", the angle it makes with B"0" is a mean between B'0'0 and B"0"0 ; hence the chord O'O", perpendicular to this line, makes an angle with O'P perpendicular to B'O' of PO'O" =H(+ A ') + (<*+ A")] and since 0'Pr=PO" cotPO'O" 5'^'coti[(a+ A ') + (+ A")] (99) which gives the distance, measured on the parallel tangent, between the old tangent point and the new. This problem occurs in practice when both the connecting tangent and the radius of the last curve are at their minimum limit, and the parallel tangent is inside of the old one, as in the figure. Should the new tangent be outside, the same for- mulEe apply, only changing the sign of B'F' in eq. (97). But in this last case it is usually preferable to employ problem 136 or 137. Example. A 1 40' curve is followed by a tangent of 200 ft., and that by a 4 curve of 10 stations ending in a tangent ; 100 . FIELD ENGINEERING. and the offset to the given parallel tangent is 80 ft. on the inside. Kequired, the position of the new tangent points X and B". Here B = 3437.87, E' = 1432.69, BA = 200, B'F' = 80. Eq. (95) R + R' 4870.56 log 3.687579 BA' 200. log 2. 301030 .-. a 2 21' log cot 1.386549 Eq. (96) a 2 21' log cos 9.999635 00' : V : 3.687944 Eq. (97) B'F 80 1.903090 .01641 8.115146 a+ A' 42 21' cos .73904 a -f- A" 40 56' cos .75545 Eq. (98) BOX 1 25' .-. BX=85 ft. Ana. Eq. (99) PO'O" 41 38' 30" cot 1.12468 X 80 = 89.97 = F'B * 156. When the tangents of a proposed road are to be in general much longer than the curves, it is desirable to estab- lish the tangents first in making the location, and afterwards determine suitable curves. On the other hand, if the curves necessarily predominate, they should be first selected and adjusted to the- ground with reference to grade and easy alignment, and afterwards joined by tangents. In the latter case the field work cannot be successfully accomplished unless the location has been previously worked out upon a correct map constructed from the preliminary surveys. The map sliould show contours of the surface, and also the grade contour, or intersection of the surface and plane of the grade. In side-hill work the grade contour indicates approximately the degree and position of the necessary curves. In the work of selecting proper curves upon the map, templets or pattern curves are almost indispensable. The templets are cut to form a series of curves, the radii being taken from Table IV. to a scale corresponding to the scale of the map, which ranges from 400 to 100 feet per inch, according to the difficulty of the location. The templets should represent convenient curves, or those in which the number of minutes SIMPLE CURVES. KM per station bear a simple ratio to 100. Curves of 50' and multiples of 50' are most convenient; 40' curves and multi- ples standing next in order, and 30' curves and multiples next. TABLE OF CONVENIENT CURVES. D. Ratio of Min. to Feet. D. Ratio of Min. to Feet. D. Ratio of Min. to Feet. 50' 1 2 40' 2:5 30' 3:10 1 40' 1 1 1 20' 4:5 1 00' 3:5 2 30' 3 2 2 00' 6:5 1 30' 9:10 3 20' 2 1 2 40' 8:5 2 00' 6:5 4 10' 5 2 3 20' 2:1 2 30' 3:2 5 00' 3 1 4 00' 12:5 3 00' 9:5 5 50' 7 2 40 40 / 14:5 3 30' 21 : 10 G 40' 4 1 5 20' 16:5 4 00' 12 : 5 7 30' 9 2 6 00' 18:5 4 30' 27: 10 8 20' 5 1 6 40' 4: 1 5 00' 3: 1 9 10' 11 2 7 20 7 22:5 5 30' 33:10 10 00' 6 1 8 00' 24:5 6 00' 18:5 After drawing the curves and tangents upon the map, the tangent points and central angles are carefully determined, the latter being compared with the lengths of the curves ob- tained by a pair of stepping dividers set precisely by scale to the length of one station. Field notes are then prepared from the map,. and if the work has been well done these notes may be followed in the field with scarcely any alterations. No ordinary protractor will measure the angles closely enough for this purpose ; it is better to use a radius as large as convenient, of 50 parts. The chord of any arc drawn with this radius equals 100 times the sine of one half the angle subtended. The importance of having absolutely straight-edged rulers in such work is obvious. In case a very long line is to be projected upon the map, it is well to use a pieqe of fine sewing silk for the purpose. See 53, 54. FIELD ENGINEERING. CHAPTER VI. COMPOUND CURVES. '- A. Theory. 157. A compound curve consists of two or more consecu- tive circular arcs of different radii, having their centres on the same side of the curve ; but any two consecutive arcs must have a common tangent at their meeting point, or their radii at this point must coincide in direction. The meeting point is called the point of compound curve, or P.C.C. Compound curves are employed to bring the line of the road upon more favorable ground than could be done by the use of any simple curve. When a compound curve of two arcs connects two tangent lines, the tangent points are at unequal distances from the intersection or vertex, the shorter distance being on the line which is tangent to the arc of shorter radius. 158. Let VA, VJ3 (Fig. 43) be any two right lines inter- secting at V, and let A be the deflection angle between them. Let A and B be the tangent points of a compound curve ( VA less than VB), and let AP, PB be the two arcs of the curve. The centre Oi of the arc AP will be found on AS, drawn per- pendicular to VA ; the centre 2 of the arc PB will be found on BS produced perpendicular to VB ; and the angle ASB will evidently equal A. Join VS, and on VS as a diameter describe a circle; it will pass through the points A and B, since the angles VAS, VBS are right angles in a semicircle. Draw the chord VQ, bisecting the angle AVB, and join AQ, BQ. Then AQ, BQ are equal, since they are chords subtend- ing the equal angles AVQ, BVQ. From Q as a centre, and with radius QA, describe a circle ; it will cut the tangent lines at A and B, and also at two other points G and Y, such that VQ = VA, and VY= VB. Hence BG = AY, and the parallel chords AO, BY are perpendicular to VQ. Join AB; then AQB = ASB A , since both angles are subtended by the same chord AB. In the triangle VAB, the sum of the angles at A and B is equal to the exterior angle A between the tangents ; while their difference (A B) is equal to the angle at the centre Q COMPOUND CURVES. 103 subtended by the chord BG, which is the difference of the sides (VB - VA). For the angle VAB = VAG + GAB. and the angle VBA = VBT - ABY. But VAG = VBY and GAB = ABY, and by subtraction VAB - VBA = 2GAB = GQB, since A is* on the circumference and Q at the centre. 159. THEOREM. The circle YAGB, whose centre is Q, ie the locus of the point of compound curve P, whatever be the relative lengths of the arcs AP, PB composing the curve. FIG. 43. On the circle YAGB, and between A and G, take any point P, and on vl/Sfind a centre M from which a circular arc may be drawn cutting the circle at A and P ; also on B8 produced find a centre 2 , from which a circular arc may be drawn cutting the circle at B and P. Join PQ, POi and PO S . Since when two circles intersect, the angles are equal be- tween radii drawn to the points of intersection, QPOi= QAOi 104 FIELD ENGIHEEKIKG. \ and QP0 2 = QBO*. Draw the chord QS and it subtends the equal angles QAO l = QBO*. Hence QP0 1 = QPO* and the radius P0 l coincides in direction with the radius P0 2 , which is the condition essential to a compound curve. Now, if we imagine another point P ' to be taken on QP or on QP produced, and the arcs AP' BP', drawn from centres found on A8 and B8, it is evident that the equality of angles found in respect to P could not exist in respect to P. Hence the arcs would intersect in P' at some angle 0iP0 2 and would not form a compound curve. Therefore, Q. E. D. 16O. THEOREM. In any compound curve the radial lines passing through the three tangent points A, P, and B are all tangent to a circle having the point Q for its centre, and for its diameter the difference of the sides VB and VA. Draw the three lines QM, QN, QL perpendicular to the radial lines BO^, AS, and P0 2 respectively. Then the three right-angled triangles BQN, PQL, and AQM^are equal, since BQ = PQ AQ = radius of the circle AGB, and the angles at B, P, and A are equal by the last theorem. Hence QM = QL = QN, and if a circle be described with this radius about Q, the three lines BO^, P0 2 , and AOi produced will be tan- gent to it. Draw Ql perpendicular to VB; it will bisect the chord GB in 7; and QN = BI $BG. Hence the diameter 2QN= BG = VB VA; which was to be proved. Corollary 1. The compound curve intersects the circle AGB in the point P, at an angle equal to half the difference of the angles VAB, VBA. For QPL = QBN= BQI = iBQG. The arc AP is exterior, and the arc PB interior to the circle AGB. Cor. 2. Since both centres are on the line PL, the position of the point P fixes the lengths of the radii of a compound curve. As P is moved toward G both radii are increased, until when P reaches G-, AOi becomes AK, a maximum, while B0_/ cot / Q V = i(8 9 + /Si) cot I A , and BI = K& - Si). cot \y = -g~^ cot i A (100) By Cor. 3, Ai + A 2 = A (101) In the triangle AQM, AOi = AM MOi. But AM = MQ cot \y, and MOi = MQ cot AI. Si = i(Si Si) (cot \y cot | A i) > (102) Similarly, jR a = 4(5 a - Si) (cot \y + cot | A 2 ) ) Subtracting, J2 a - JRi = K/S'a - Si) (cot i A 2 -f COt i A (103) 106 FIELD E cot i A i = cot \y - From (102), \ (104) In the triangle ABG, ^^ AB sin BAG sinAGV or by which we find ^(S* &), when, instead of the sides and A , we have given AB, and the angles VAB and VBA. From (103), -| A T-> ox + COt ^ A ft) ! From (102), } (107) From (100), J< A + ft) = = (108) ^2 and & are found by adding and subtracting the values found by eqs. (106), (108). From (105), t ^ = L^lM ' , (109 ) - which may be used instead of (108) when the sides are not re- quired. VAB = i-( A + r) and VBA = |( A - r). 162. Given : the sides VA = Si and VB = 8 2 and the angle A; assuming the shorter radius jR lt to find Ai, A a, and jR 2 . Use equations (100), (104), (101), (102), and (18). Example. Let VA - 1899.90, VB = 1091.12, A = 74, and assume -Bi = 955.37. COMPOUND CURVES. 107 (100) K& 4- S> ) 1495. 51 log 3. 174789 (&-&) 404.39 " .2.606800 " 0.567989 A 37 cot " 0.122886 .-. \y 11 31' 01".5 cot 4.90769 " " 0.690875 (104) R, (D = 6) " 2.980170 &-&) " 2.606800 2.36249 " 0.373370 .-. A A! 21 27' cot 2.54520 (101) i A 37 A 2 15 33' " 3.59370 y " 4.90769 jt-to | (102) \ 8.50139 " 0.929490 " 2.606800 . . R* (D = 1 40') 3.536290 (18) .'. Ax = 42 54', Zi = 715; A e = 31 06', Z 8 = 1866. 163. Given: the line AB, and tJie angles VAB, VBA; assuming tJie longer radius R^, to lind A 2 , Aj, and Ri. Example. Let AB = 2437.82, VAB = 48 31', VBA = 25 29', and assume J? 2 = 3437.87. (105) \AE 1218.91 log 3.085972 \Y 11 31' sin " 9.300276 " 2.886248 37 '" " 9.779463 " 2.606785 (104) & 3.536289 8.50166 " 0.929504 11 31' cot 4.90785 .-. iAt 15 33* cot 3. 59381 (101) i A 37 .*. iAt 2127 f cot 2,54516 (102) \y " 4.90785 2.36269 log 0.373407 2.606785 (D = 6 6 ) 2.980192 108 FIELD 164. Usually a compound curve is fitted by trial to the shape of the ground, after which it may be desirable to calculate the sides VA, VB, or the line AB, and the angles VAB, VBA. Example. From the point of curve A, a 6 curve is run 715 feet to the P. C.C.; thence a 1 40' curve is run 1866 feet to the P.T. Required, the sides VA, VB, and the line AB, and angles VAB, VBA. Here fit = 935.37, AI = 42 54', J2 a = 3437.87, A 2 = 31 06'. (106) #,-!?, 2482.50 log 3.394889 iA, 21 27' cot 2. 5451 6 * A, 15 33' " 3.59370 6.13886 " 0.788088 -&) 404.39 " 2.606801 " 2.980170 2.36248 0.373369 *A, 21 27' cot 2, 54516 % ir ll31'0r.7 "4.90764 0.690873 (108) (& - AS 2.606801 3.297674 |A 37 cot " 0.122886 ... ^-[-/S 1495.51 3.174788 $a 1899.90 A 1091.12 FA# 48 31' F#A 25 29' (109) |(& - ) " 2.606801 IA 37 sin " 9.779463 2.386264 Ir 11 31' Ol'.T sin " 9.300294 .-. |^4^ 1218.91 3.085970 ^li? 2437.82 165. Given : the radii J?,, ^ 2 , ^ w//?e A, K4, or VB, to find ffo other side and the central angles A a ." Fig. 43, COMPOUND CURVES. 109 In the triangle AMQ, A0 t = AM - M0 l = IQ - MQ cot MOiQ; or J2 1 = $(S<> + S,) cot 4 A - i(& &) cot iAi whence i(& + ,) = -K& ,)~cot i A, tan 4 A + -Bi tan |A By eq. (106) Substituting this above, subtracting and reducing Si = (l? a -Ri) sin i Aa - -+- Ri tan iA But(A A]) =-jA a and 2 sin 2 A a = vers A 2 , whence c _ (1?. - l?i) vers A a + IZi vers A 1() 1 sin A Transposing, #, sin A 12i vers A vers A , = j5 s (HI) ft* Mi Similarly, from the triangle BQO* Ri = i(5 a + -S,) cot i A -f- -K^a ^i) cot i from which and eq. (106) we derive _ R* vers A QRa #0 vers At sin A and Ri vers A & sin A vers AI = ^c- n ' 110 FIELD Example. Given : VA = Si = 1091.12, A 74, and the radii R t = 955.37, R* = 3437.87, to find AI, A 2 , and a . (Ill) 8, 1091.12 log 3.037873 A 74 sin " 9.982842 1048.85 " 3.020715 Si " 2.980170 A 74 vers " 9.859956 692.03 " 2.840126 356.82 " 2.552449 " 3.394889 A 2 31 06' vers " 9.157560 A! 42 54' .... " 9.427254 3.394889 663.96 vers " 2.822143 3.536289 9.859956 2490.26 3.396245 3.261572 9.982842 1826.30 sin " A .-. 8, 1899.90 " 3.278730 166. Given : one side, and the radius and central angle of the adjacent arc, to find, the other radius and side. From eqs. (Ill), (113) we have 81 sin A Ri vers A vers A 2 (114) R? vers A 81 sin A by one of which the required radius may be found ; the required side is then found by eq. (110) or (112), as in the last problem. Example. Given : VA=8i = 1091.12 A = 74, RI = 955.37 and A , = 4254' ; to find R* A a = 74 - 43 54' = bl 06'. COMPOUND CURVES. Ill (114) & A 2 1091.12 955.37 2482.52 3437.89 74 74 3106 log 3.037873 " sin 9.982842 1048.85 692.03 356.82 3.020715 2.980170 vers 9.859956 2.840126 2.552449 vers 9. 157556 3.394893 FIG. 44. Otherwise : Fig. 44. If convenient in the field, a tan- gent PF 3 may be run from the point P to intersect the farther tangent. The distance PF 3 multiplied by cot iA 2 will equal the radius JK a by eq. (25). 167. Remarks. It the first arc AP be produced to G, Fig. 44, so that AO^G = A, then G is the tangent point of a tangent parallel to VB, and by 137, the tangent point B must be on the line P& produced. Conversely, if the point B is assumed, and the arc AG given, the point P must be on the line BG produced. The radius R* may be found by 112 FIELD 737) .R 3 = ; , BP being measured on the ground ; or by & sin -A 2 similar triangles J? 2 : 72i :: BP : GP. The distance YD, Fig. 43, from the vertex to the circle AGB is expressed by the formula If the point P falls at D, then YD is also the distance of the curve from the vertex measured on the line VQ. But when P falls at D, the radius P0 2 is perpendicular to the line AB, and AI = VAB, and A 2 = VBA. When AI is greater than VAB, the arc AP, being exterior to the circle, cuts the line YD; but when At is less than VAB, the arc PB cuts the line DQ. If the line 2 P produced passes through V, we have sin Q VL = 8 >-~ sin i A (116) giving AI = iA + QVL and A 2 4-A When AI is greater than this, we have for the external distance of the vertex E l = & ex sec AOiV in which the angle AO^. Fis found by the formula tan AO t Y= 73 , and Ei is measured on a line V0 lt making the angle Si AVO, = 90 - AO^V. When A i is less than, (-J- A -j- Q VL), we have similar expres- sions with respect to the arc BP and centre O a . 168. To locate a compound curve when tJie point of com- pound curve it inaccessible. Fig. 45. Each arc being in itself a simple curve is located as such. When the P.C.C. is accessible, the transit is placed over it, and the direction of the common tangent found, from which the second arc is then located. When the P.C.C. is not accessible, the" common tangent V\ Fa may be found by locating the points Y\ and Fa, which may be easily done, since Y\A Y\P R\ tan AI, and COMPOUND CURVES. 113 V'tP == It* tan 1 A 2 , from which each arc may then be located by offsets or otherwise, as in the case of simple curves. Should the points V\ F 2 be obstructed, the common tangent may be found by an offset IIG = LP from any convenient point 11, for knowing the angle HO^P, we have HG = J2i vers HOiP, and GP = A', sm HO, P. If the entire tangent Pi F 2 is too much obstructed for use, the parallel line HK may be employed, observing that the LP angle PO t K is found by vers P0 2 K = -=-, and the distance LK by LK = R sin PO^K, by which a point K on the second arc is found having a tangent offset KI = HG. FIG. 45. FIG. 46. Should the line HK be also obstructed, we may run the in- verted curve HP' HP and P'K = PKto find the point K from which so much of the second arc as is accessible may be located. C. Special Problems in Compound Curses. 169. Given: a compound curve ending in a tangent; to change the P.C.C. so that the curve may end in a given parallel tangent. Fig. 46. Let APE be the given curve ending in VB, " VI? be the given parallel tangent, " p = perpendicular distance between tangents. It is required to change the point P, and with it the values of AI and A 2 , so that with the same radii KI and J2 2 the new curve APB may end in the parallel tangent VB. 114 FIELD a. WJien the tangent VB' is inside of VB : and in the diagram draw 0j(? perpendicular to BO*-, then GO* = 0i02 cos A, KO-i = 0i02 cos A 2'. Subtracting, since 0;0 2 = 0i0 2 ' = (^ - -#1), and KOi - GO* = GB - KB' = p> p = (Ri Rt) (cos A a r cos A 2 ) whence COS A 2' = 7> -4- cos A a (117) Hz K, POiP' = ( A 2 A 2 ') and the point P is advanced. h. FAm tAe teusrc< F'P' w outside of VB: p = (J? a R t ) (COS A 2 COS A 3 ') whence COS A a' = COS A 2 (118) P0jP' = (A 2 ' A 2 ) and the point P is moved back and the arc AP diminished. FIG. 47. 7^ case the curve terminates with the arc of shorter radius, or R l follows R*. Fig. 47. c. When VB is inside of VB: p = (Rz R^) (ccs AI cos AO whence cos Ai' = cos AI =: =r- PO*P' (AI' Ai) and the point P is moved back. COMPOUND CURVES. 115 d. When V'B' is outside of VB: p = (Mi RJ (cos A / cos A i) whence cos A,' =cos Ai -f -^ sr P0 3 P' (Ai Ai') and the point P is advanced. Example. Let R = 2292.01, Ri = 1432.69, A 2 = 28, and p = 20.07 inside of VB ; case a. p 20.07 log 1.302547 (117) #2 - Ri 859.32 " 2.934155 .023356 " 8.368392 A 3 ' 28 cos .88295 .'. A'a 2JT " 7906306 .-. PO.P' 3 17O. Given: a compound curve terminating in a tangent, to change the P.C.C. and also the last radius, so that the curve shall end in a parallel tangent at a point on the same radial line as before. Fig. 48. FIG. 48. Let APB be the given curve ending in the tangent VB\ let V'B' be the given parallel tangent; and let p BE' = ///= the perpendicular distance between tangents. It is required to change the point P to P', and also the value of R* to RJ, so that the new curve may end in V'B' at B' inside of VB on the same radial line #0 2 . In the diagram produce the arc AP to Q to meet Oi G drawn parallel to 2 5; then POiG = A 2 . Draw the chord PB, and it will pass through G. Lay off the distance p from 116 FIELD ENGINEERING. B on BO* to find B 1 ; draw B'G and produce it to intersect the arc APG in P'. Then P' is the P,C.C. required. Join P ' Oi and produce it to meet BO- 2 produced in 0?. Then P'0 a ' = l?'0 a ' = R* the new radius, with which describe the arc P'B'. By Geom. Tab. I. 18 : PBV = i P0 3 Jff = iA a , and #' F = P'0*B = |A 2 '. PGP'=BGB' = -K A 2 - A 2 ') Draw OiJST perpendicular to J50 2 . Then 0^= B'ff=BI= Oi0 2 sin A 2 = (5 a - -Bi) sin A 2 GI- P .:,, . tan iA 2 ' = tan |A 2 - ^-__ I --- (121) In the triangle Oj0 2 2 ' sin A 3 ' : sin A 2 :: 0^ : OM :: (R* - E,) : (H* - & B 9 '- Sl= *J. (Ba - Sl ) sm A 2 ^ 2 '-(^ 2 - J R 1 )^| 2 7 + ^ (122) Sin Aa IfB' V were outside of VB; W hen the smaller radius Ri follows R z : If the given tangent B'V is inside of BV. Fig. 49. tan iA/ = tan IA, + T -^--5- (124) COMPOUND CURVES. 11' IfB'V is outside ofBV: tan $Ai f = tan |AI Ri) Bin sin Ai sin AI' (126) (135) FIG. 49. .m. 48. Let R* = 2292.01 p = 20.07 inside. " ft = 1432.69 A a = 28 (121) (122) ( ( ft Hi A 2 p tan iAa tan |Aa' AV R 2 - ft) ft' - ft) 20.07 .04975 .24933 log 2.934155 28 log sin 9.671609 2.605764 1.302547 8.696783 - \ 11 17' 22 34' sin 9.584058 2.934155 .19958 1051.25 1432.69 3.350097 28 sin 9.671609 3.021706 Ans. R z ' 2483.94 . . D = 2* 18' 25" PO,P = 28- 22 34' = 5. 26' .'. PP' = 135.83 ft. FIELD Example 2. Fig. 49. Let R* = 2292.01 p = 20.07 inside. (124) JB, - tan ^ = 1432,69 859.32 20.07 .03247 .42447 .45694 Ai JZ, 46 C 46 23 24 33^ 49 07' 46 log 2.934155 log sin 9. 856934 2.791089 1.302547 8.511458 log sin 9.878547 2.934155 3.055608 log sin 9.856934 817.60 2292.01 Ans. S = 1474.41 . . D = 3 53' 12" P0 2 P' = A / - A : = 3 07' . ' . arc PP' = il 1 ^ = 124.67 ft. Observe that in either figure both tangents must be on the same side of the point G, in order to a solution. FIG. 50. 171. Given: a compound curve ending in a tangent, to change the last radius and also the position of the P.C.C., so that the curve may end in the same tangent. Fig. 50. COMPOUND CURVES. 119 I. When the curve ends with the greater radius 7? 2 . Let APB be the compound curve in which MI MI A i and A a are known. In the diagram draw the chord PB and produce the first arc AP to meet it in 6r; draw OiG, and produce it to meet the tangent in K. Then by 137 OiK is parallel to Z B, and by eq. (57) OK = (J2 - #0 vers A 2 (127) If we assume P' as the new P. C, G., we have A a '= P'O-t'B', and the chord P'G produced will intersect the tangent at the new point of tangent B\ and J?0 3 ' =--R a '. Similar to eq. (127) we have OK (Ri Hi) vers A 2 ' and equating the two expressions, we obtain vers A a vers A a If we assume R a J we have vers A 2 ' = -|^~- vers A 2 = =* (129) ft* Mi Mi Mi In the two right-angled triangles BKG and B'EG, we have BK= B'K = and by subtraction, BB' = GK (cot \ A a ' - cot I A 3 ) (130) in which GK is obtained from eq. (127). When BB' as given by eq. (130) is negative, the p'oint B' falls between 5 and F. #" we assume ;fc distance BB' 7> COt|A 2 '=COtiA 2 ~ (131) FIELD G^being obtained from eq. (127) and K from e* (US) . ^ eq. (131) use the + sign when B' is beyond B as in the Fig.^5( II. When the given curve ends with the smaller radius B,. Fig. 51. /V FIG. 51. We have by a similar reasoning GK - (B* - Bi) vers A (133) vers (134) = GK (cot 1 A , - cot \ A i') = COt^Ai - using the - sign when B' is beyond #. Example. Fig- 51. Let ^ 2 - 2292.01, A - 1482.69, A, = 46, and le ; the P C C be moved back 200 feet from P to P ; hence P0 2 P 5' and A/ = 51; to find the new radius B,' and the d BB'. COMPOUND CURVES. Eq. (132) ft* - Ai .-. GK eq. (133) AI' R* - Ei' K* .'. Ei' eq. (135) GK COt - A i COtiAi' .-. BH Ei 859.32 707.85 2292.01 46 51 and D = 3 23 25 30' log 2.934155 " vers 9.484786 log 2.418941 " vers 9. 568999 2.849942 37' log 2.418941 log 9.413819 1.832760 1584.16 2.35585 2.09654 0.25931 68.04 172. Given: a compound curve ending in a tangent, the last radius being the greater, to change the last radius and also the position of the P.C.C. so that the curve may end at the same tangent point, but with a given difference in the direction of the tangent. Fig. 52. Fia. 52. Let APS be the given compound curve, POi Ei and P0 2 = 2 > Ei. LetF'-Sbe the new tangent, and the angle V~BV = *, the given difference in direction : to find BO* = E*', BO*P' A a' and the angle POiP\ 122 FIELD ENGINEERING. We have BO, - 0,0, = It, - CR 2 - R,} = R! BO* - 0i0 2 ' = R 2 ' - (.R a ' Ri) = R t From which we see that whatever may be the value of the new radius, the difference of the distances from B and 0i to the new centre is constant, and equal to Ri. We therefore conclude that the centres 2 and 0,' are on an hyperbola of which B and Oi are the foci, and Ri the major axis. This suggests an easy graphical method of solving the problem. Through B draw a line perpendicular to the new tangent V'B which will give the direction of the required centre 2 '. On this line lay off BK equal to Ri, and since (# a ' -Bi) = 0i a ' = KO*, if we join KOi, the triangle K0- 2 '0 l is isosceles; therefore bisect K0 t and erect a perpendicular from the mid- dle point to intersect the line BK produced in a '. Draw 2 '0, and produce it to intersect the arc AP (produced if necessary) in P'. Then P' is the new P.O.C. required, and 2?0 a ' = P'Oa = Ri.', the new radius. The analytical solution is as follows : Adopting the usual notation of the hyperbola Let 2a = Ri = the major axis, " 2c = BO , = the distance between foci. Produce the arc AP and through B draw the tangent BH, and join HOi = Ri. Then in the right-angled triangle BHOi Now by Anal. Geom., c 2 a 9 = 5 2 . Therefore 2b = BH = the minor axis. Draw the chord PB and produce the arc AP to cut it in 0, Then by Geom. (Table I. 24) %H* - PB X OB = 2R* sin i A 2 X 2( 2 - J?i) sin i A 2 . . J5// = 2 sin i A a ^-^fBa - #1) (137) COMPOUND CURVES. 123 Let a = the angle HO^, then tan a -Sj^- and BOj. = ^^ (138) In the triangle BOM let O^BO* = /3 then sin ft = a p<0 sin A 2 (139) The polar equation of the hyperbola for the branch 10M, taking the pole at B and estimating the variable angle v from the line EOi, is c . cos v a When v = fi i, r R*, and substituting the values of a, b, and c found above, we have ~DTT$ * 2 (BO, cos (/? t) - ^0 using (yS -|- *') when F' falls between F and J., as in the figure, and (fi i) when V falls beyond F. In the triangle BOM, the angle BO^'Oi = A 2 ' and 50 sin A a ' = -p-r- -^ sin (/? *) (141) iz i\ Finally P^P' = A, - (A,' ) (142) Remark, When F' falls between Fand ^4, as in Fig. 52, if the angle i be greater than the angle VBE, the curve ceases to be a compound, and becomes reversed. Therefore VBH = a ft is the maximum value of / possible in this case. When F' falls beyond F, the point P' will fall between Pand A; and the largest possible value of I will then be that which renders POiP'= AI, and makes the point P' coincide with A 124 FIELD ENGINEERING. Example. Fig. 52. Let R l = 1432.69 i =6 jR 2 = 2292.01 (137) R* - R! 859.32 Rt 2292.01 (138) 2 BH a BO, 1432.69 (139) 2 - A 2 (140) 28 42 36' 23". 7 42 36' 23".7 56 21 28' 06".3 27 28' 06".3 1727.09 1432.69 BH* J? 2 ' (141) .-. (142) .*. A Q1 Ai = dl A 2 = 56 log 2.934155 3.360217 2 ) 6.294372 3T47186 log sin 9.671609 0.301030 3Tl9825 3.156151 log tan 9T963674 log cos 9.866889 3.289262 2.934155 9.644893 log sin 9.918574 log sin 9^563467 log cos 9.948053 3.289262 294.40 X 2 = 588.80 2949.05 A a ' = 36 18' 26" POiP' = 13 41' 34" = 342.3 feet. 3.237315 2.769968 6.239650 3.469682 Remark This problem may also be solved by first finding the new sides VA, V'B, from which and the new central angle (A i), and the radius Ri, may be found AI', A a ', and Rz, as in 162. The new sides are readily found from the old ones by solving the triangle VBV. If the original sides are not given, they must be calculated as in 164. 173. Given: a compound curve ending in a tangent, the last radius being the less, to change the last radius and the position of tlie P.C.C. so that the curve may end at the same tangent point, but with a given difference in the direction of tangent. Fig. 53. COMPOUND CURVES. 125 Let APB be the given curve, and P0 2 = #2, and P0j = Ei < E-i. Let F'J5be the new tangent, and VBV = i, the given angle; to find BOi = Hi, BO t 'P' = A/, and P0 2 P'. We have BO, + O t 2 = & -f CR a - JBO = -R a P(V + Oi'Oa = / -f (# - J2/) = -Ra from which we infer that the locus of the centre Oi is an ellipse, of which B and 2 are the foci, and R* the major axis, FIG. 53. since the sum of the distances SOi and 2 0i' is always equal to P 2 . This suggests an easy graphical solution of the prob- blem, as follows : Perpendicular to V'B draw the indefinite line BK, which will contain the required centre O/, and layoff BK = i? 2 . Join KOi, bisect it, and from the middle point erect a perpen- dicular to intersect BK in 0,'. Join 2 O,', and produce the line to intersect the arc AP (produced if necessary) in P', which is the new P. C.C. required. P'O/ = SOi = J?i', the required radius, and P'O^B = AI'. The analytical solution is as follows : Adopting the usual notation of the ellipse, let 2a Hi ,= the major axis, " 2c = P0 2 = the distance between foci. At B erect 5J3" perpendicular to BO? to intersect the arc AP 126 FIELD (produced if necessary) in E, and join HO* = B*. Then BE' 2 = RJ - BO Z - = a? - 4c 2 But by Anal. Gebm., 2 - c 2 = 6 2 . Hence 2b BE = the minor axis. In the triangle B0 t 2 we know BO, R t , and 0,0* = R-i Hi, and the included angle jB0j0 2 = 180 A, ; hence by Trig. (Tab. II. 25) 27? 7? tan 1(0! 0,B - O^BOJ = - * ** tan | A , (143) Ji 2 The angles at B and 2 are then found by (Tab. II. 26). Let ft = the angle 0,BO^ ; then m = (ft-lZi)^V (144) The value of BE* above may be written BE* = (R* + B0. 2 ) (7? 2 - BOJ (145) The polar equation of the ellipse, taking the pole at B, and estimating the variable angle v from the axis BO Z , is a c . cos v When v = ? i, then r = J2/ f and substituting the values of a, b, and c, given above, we have O) using (ft i} when F' falls between Fand ^4, as in Fig. 53, and (/?+*) when F' falls beyond F. In the triangle BOi'Oi, the angle Oi'50 3 = (/?T 0, and the exterior angle BOi'P' = A/; hence J50 sin Ai' = - -- 5-7 sin (/? T ) (147) Ji 2 *6 Finally PO,P' =-(A, T a) - A/ (148) When F' is on AV, then PO^P' is negative, showing that it must be laid off from P toward A; but when V is beyond COMPOUND CURVES. 12? F, then P0 2 P' is positive, and P' will tie on AP produced. The only limits imposed on the angle i are that the resulting value of PP' shall not exceed PA, and that Ri shall not be less than a practical minimum. Example. Fig. 53. Let D 9 = 3 20' R* = 1719.12 A 2 = 23 20' D. = 6 R,= 955.37 A, = 48 * = 7 45' The resulting values are as follows: ft BO, J3H* A/ O,P' PP' 1572.42 1273.65 440.5 21 09' 32".6 54 56' 14 41' 3.196567 5.683829 3.105052 (See also remark at end of 172.) 174. Given a simple curve joining two tangents, to re- place it by a three-centred compound curve between the same tangent points. Fig. 54. J50 a P' Fid. 54. Let R = AO radius of simple curve. R v = P0 l = P'0 l R A 3 = AOi A = AOB . Since AO* is made equal to BO* and VAVB, AOiPmust equal BO$P', and the compound curve will be symmetrical about the bisecting line V0\ and the centre Oi will be on the line VO. We have at once from the figure, 2A 2 + A, = A (149) 128 FIELD ENGINEERING. In the triangle 00i0 2 we have Oi0 2 : 00-i :: sin AOV: sin whence (150) which expresses the general relation between the quantities, R and A being given. We may now assume values for Ri and J? 2 subject to the above conditions, viz., -Bi < R and Ri > R; whence sin i A i = t R) sin (151) In selecting values for R^ and Ri, the degree of curve DI should be but little greater than D of the simple curve, say from 30 to 60 minutes, while i> 2 may be taken at i-D to D. '.-Given: R = 1719.12 D - 3 20' A = 40 = 1432.69 D l = 4 , = 5729.65 Z> 2 = l a - R 4010.53 4296.96 4 A Ai AP = P'B 138.4 ft. 20' 18 36' 57" 37 13' 54" 1 23' 03" log 3.603202 " 3.633161 " 9^970041 log sin 9.534052 " " 9.504093 Again we may assume A 2 and Ri, whence Ai = A 2A 3 and _ gsinjA - Ri sin A sin (152) Example. Given: R = 1719.12 A = 40 Let R l = 1432.69 A 2 = 1 .'. Ai = 38 Am. R* = 7387.24 . . D a = 46f AP = 129. Finally we may assume A 3 aw-^ .#2, and deduce A i and Ri from eqs. (149) (150); but this is the least desirable because COMPOUND CURVES. 129 the valug> of Ei so found will not usually give a convenient value to the degree of curve Di. 175. To determine the distance HH' between the middle points of a simple curve and a three-centred compound curve joining the same tangent points AB. Fig. 54. In the triangle OOi 2 , we have > = a - . - sin i A HH' = 00 1 -f 0,H' - OH ... HH' = (Z? 2 - *i)* - (5 - *0 (153) In the first example given above HH' 14.55, and in the second HH' - 17.05 ft. In many instances the distance HH' is so great as to render this problem practically useless, unless the distance HHi is discounted beforehand by putting the simple curve AHB a sufficient distance inside of the proper location through the point H'. But the problem given below is usually preferable. 176. Given, a simple curve joining two tangents to re- place it by a three-centred Compound curve which shall pass through the same middle point H. I. The curve flattened at the tangents. Fig. 55. = AO, the radius, and A = the central angle of the simple curve AHB, and let H be the middle point. Let ^ = PO, = HO, A , = POiP' " A' and B' be the new tangent points required. We have at once, as in the last problem, 2A 2 +A 1 = A. (154) 130 FIELD Since the curve is to be symmetrical about VO, HM = HP'. PA = P'B, and A A' = SB'. In the diagram produce the arc HP to G, and draw Oi parallel to OA, and produce it to K. Then a tangent line al G will be parallel to VA\ and by 137 the point G will be OD the long chord HA, and on the long chord PA'. GK is the perpendicular distance between parallel tangents, and the problem is similar to that given in 171 ; whence by eq. (57) we have, in this case, OK = (B a - BO vers A 2 = (B .BO vers i- A . (155) for the general equation in which B and A are given. Analagous to eq. (130) we have AA' = KA - KA = GK cot GA'K - GKcot GAK. . ' . AA' = GK (cot | A 3 - cot i A) (156) in which GK is obtained from (155). We may now assume values for Bi and B 2 , making Bi < B and B 2 > B, and deduce the values of A 2> Ai, and AA. Solving eq. (155) (B - BQ vers JA _ vers A a = - 5 - D - = Eq. (154) gives AI, and eq. (156) gives AA'. COMPOUND CURVES. 131 Example. Fig. 55. Given: R = 764.489 D = 7 30' A = 40 Let^ = 716.779 D, = 8 " R 2 = 3437.870 2? 3 = 1 40' (155) R- R, 47.71 log 1.678609 | A 20 log vers 8.780370 GK log 0.458979 J2 2 - R l 2721.091 " 3.434743 A 2 (say) 2 38' log vers 7.024236 A'P 158.00 Ax =34 44' (156) iA 2 43.5081 = cot 1 19' A 5.6713 cot 10 37.8368 log 1.577914 OK " 0.458979 A A 108.87 " 2.036893 Again, we may assume A 2 and Ri < R\ whence Ai = A 2A 2 and eq. (155) GK = (R - RJ vers i A and Eq. (156) gives AJ.'. Again, we may assume A 2 and the distance A A ; whence, from eq. (156) AA> eq. (155) R, = R - -- COtiA nir (159) vers eq. (158) gives R*. Again, we may assume & < R and AA ; then, eq. (155) GK= (R- iJJOversiA and eq. (156) A A' cot i A a = cot i A + ~ (160) and eq. (158) gives R*. 132 FIELD Examjde. Fig 55. Given : R = 764.489 Let jKi = 716.779 " A A = 110. Hence by last example, D = 7 30' A = 40 GK eq. (160) AA 110. -*v log 0.458979 2.041393 (158) COtiA COt |A 2 A 2 GK xl AP' 38.2309 5.6713 1.582414 10 1 18' 18" log 1.642486 43.9022 2759.5 3476.3 157. (say) 2 37' log vers 7.018147 0.458979 3.440832 D a = 1 39' Ai = 34 46' II. The curve sharpened at the tangents. Fig. 56. This case will only occur when, with a given external dis- tance VH, a simple curve would absorb too much of the tan- gents. FIG. 56. Let AHB be the simple curve, and " A'PHP'B'the required compound curve We have from the figure, 2A 4 A 2 = O^ A, = A a = A. (161) COMPOUND CUEVES. 133 In the diagram draw 0?G parallel to OA cutting the tan- gent at K, and produce the arc HP to O. Draw the chords QH and GP, passing through A and A' respectively. We have then a discussion similar to the preceding case, and to the problem 171, Fig. 51, whence we derive the general f ormula? : GK = (R* - R,) vers A i ' = (It* - R) vers i A (162) and AA = GK(cot iAi cot iA) (163) 1. Assuming Ri < R and R* > R vers Aj = R ^ _^ = ^~^ vers |A (164) 2. Assuming Ai < iA and R! < R R vers j A - R* vers AI vers i A -vers A: 3. Assuming AJ < iA and A A ' e *= (166) (167) 1 vers | A tfx = R, -- ^ (168) vers AI 4. Assuming Ry > R and -4.4' (?.?= (,R 2 - R) vers i A j AI cot | A , = cot i A + - (169) The third assumption will usually secure most readily the desired curve. AA should be assumed as small as the nature of the case will allow, and AI should not be much smaller than A . It is evidently not necessary that the new curve should be symmetrical; for having laid out the curve A'PH, the simple curve HB may then be used, or, if desirable, some compound curve HP'B' determined by an assumed value of BB' not equal to AA. 134 FIELD ENGINEERING. These formulae (154) to (169) are readily adapted to the case of substituting a compound for a simple curve when it is necessary to keep one tangent point fixed, but to move the other a certain distance in either direction on the tangent. For if in Figs. 55, 56, we draw a tangent at H, and make // the fixed point of tangent, it is evident that the central angle of the curve will then be AOH. The only change necessary, therefore, to adopt the formulas to this case is to write A in place of i A, and to observe, instead of eqs. (154) (161), that Aj -j- A 2 = A. Example. Fig. 55. Let 11 = 1910.08 A = 84 Assume AA = 260. AI = 38 .'. A 2 = 8 Eq. (166) AA' = 260. log 2.414973 cot i AH 2.90421 19* cotiA 2.60509 21 .29912 log 9.475846 GK " 2.939127 Eq. (167) i A 42 " vers 9.409688 3384.07 3.529439 E 1910.08 R* 5294.15 D = say 1 05' Eq. (168) GK log 2.939127 Ai 38 " vers 9.326314 4100.27 3.612813 R l 1193.88 D = 448' A'P 791.67 PS =369. 23 177. Given, two curves joined by a common tangent to replace the tangent by a curve compounded with the given curves. Fig. 57. Let EI = BOi the radius of one curve, " Rs = A0 3 the radius of the other curve, > jf?i, I BA the common tangent, " J? 2 = POi = P'0 2 the radius of connecting curve. " A 2 = P# 2 P' the central angle of " " a= AO S P' and /S = BO^P. COMPOUND CURYES. 135 In the diagram join Oi0 3 and draw OiG parallel to BA. Then in the right-angled triangle OiGO a we have, 0,0, = R 3 RJ cos i (170) sin* which gives the distance between the centres of the given curves. FIG. 57. We shall now assume the following geometrical truths, which may be easily demonstrated. If two circles intersect in one point, they intersect in two points; and the line joining the two points is the common chord. The common chord is perpendicular to the line joining the centres, and when produced it bisects the common tangents. If a third circle is drawn touching the two circles, a tangent to the third circle, parallel to the common tangent, will have its tangent point on the common chord produced. Conversely, therefore, if the tangent BA be bisected at K, and a line, KI, drawn perpendicular to 0i# 8 , KI will coincide with the common chord produced, and the angle IKA = A0 3 0i = i. If on KI we assume a point / through which it is desirable that the connecting curve should pass, then / is the tangent point of a tangent parallel to BA ; consequently a line through / perpendicular to BA contains the required centre Ot. 136 FIELD ENGINEERING. I. Let p HI = the perpendicular distance between the tangents. If in the diagram we join IA and IB, and produce the chords to intersect the given curves in Pand P', then Pand P' are the points of compound curvature; and the lines P0 t and P 3 produced will intersect /0 2 in the same point 2 ; and the angles P'0 2 / = a and PO 2 7 = /?. In the triangle AIB the line KI bisects the base AB, and we have by Geom. Tab. I. 25. AI n - + P/ 2 = 2AK* -f 2KI* By eq. (56) AI = 2(# 2 - R 9 ) sin ia BI = 2(I?a -BO sin $/3 AK = # and KI = -^-^ Bint Dividing by 2 and putting vers a = 2 sin 2 a and vers ft = 2 sin 2 i/3 (Tab. II. 46) (5, - P 3 ) 2 vers a + (P 2 - J20 vers /? = *P + ^ But by eq (57) (i? 3 - JB,) vers a = (JSa J?,) vers /J = ^ (172) (173) tJJ Dill I/ From (172) vers a = ^_ ; vers fi = ^_ (174) and from the figure As = a -j- ft (175) These forrmilae solve the problem when p is assumed. If desirable we may find a and ft independently of R*, for in COMPOUND CUEVES. 137 the triangle AIB, IAB = $a and ISA = \fi\ and since HK = p cot i, A TT 17 TTTr J (176) (177) II. In case a or ft is assumed, we have from the last equa- tion P = 2(cot |a + cot = 2(cot */J - cot t) III. Jtt ease ^e radius It* is assumed, then in the triangle 1 2 6> 3 we know all three sides; for dOt = (R* Mi), O a 3 = (S a - RJ, and Oi0 3 = ^ ^ COS 2 By Trig. (Table II. 31.) _ in which * = \ sum of the three sides. Substituting values, and reducing, observing that, and that (# 3 Mi) tan i = I, we have vers A 2 = H7p - p^T-p - p-v (179) 4 In the same triangle. sin 0,0,0, = sin A 2 But from the figure 3 0,0^ = i - ft, and taking the value of 0j0 3 from eq. (171). 138 FIELD ENGINEERING. sin (i-(S) = .-. (180) We then find a from eq. (175) and p from (172). The angles a and ft may be found otherwise, for by Trig= (Tab. II. 27) we have in the triangle 0i0 a 3 sin K0i0,0 - 0,0! 0.) = l0 * ~ * 3 cos iA 3 or sin (90" - (i + 5L=A = (R-AXiOBicoBtA. \ a I Mz M\ . . cos li -\ --- ~- 1 = cos i . cos AJ (181) which is a convenient formula when i and A 2 are not too small. Having obtained ^-, we have For a constant value of I the less the difference of E 3 Ei the greater will be the value of the angle i. When J? 3 = 'R lt cot t = and i = 90 and the tangent point / will be on a per- pendicular to BA drawn through the middle point K; and a = ft. On the contrary, as (R 3 R^ increases, i becomes less, and the foot, H, of the perpendicular ///moves toward B, the tangent point of the curve of smaller radius JRi. The distance HK = p cot i. The connecting curve is farthest from the tangent BA at /. To find the ordinate from BA to the curve at any other point, subtract from p the tangent offset for the length of curve from / to the ordinate in ques- tion. 115, eq. (39) may be used on flat curves with tolera- ble accuracy, even when the distance equals several hundred feet. IY. It is evident that in this problem R* must be greater than either Hi or /? 3 . As the centre 2 is taken nearer the COMPOUND CURVES. 139 line 0i 3 , M-2 grows less, and is a minimum when 2 falls on the line Oi0 3 . In this case we have A 2 = 180, and -f R t +0 1 3 ); a minimum. (183) This limit must be regarded in assuming the value of M-,. Since 0!0 2 - 2 3 = CR 3 - M,) - (Mi - R 3 ) = (R 3 - R,) a constant value, independent of R*, we infer that the centre 2 is always on a hyperbola of which Oi and 3 are the foci; (R 3 M\) equals the diameter on the axis joining the foci; and I equals the diameter at right angles to it, for in the tri- angle OiGOs, I* = 0^ 2 - (R 3 -7?i) 2 (184) Example. Fig. 57. Given : Assume Eq. (170) R 3 Eq. (173) R> -I I i i i P * P = 1432.69 = 11.4 477.39 400. 11.4 27.64 3508.77 3342.77 M 3 = 1910.08 and I- 400. to find R 3 ,a and/?, log 2.678873 " 2.602060 39 57' 34" log cot 0.076813 39 57' 34" " sin 39 57' 34" " sin 2 log < < if 9.807701 9.615402 1.056905 1.441503 4.602060 1.056905 3.545155 Eq. (174) p - M, a P 2) 6879.18 3439.59 (say) 3437.87 11.4 1527.79 1.056905 3.184064 11.4 2005.18 ft (nearly) 7 00' log vers 7.872841 log 1.056905 " 3.302153 6 07' log vers 7. 754752 13 07' 140 FIELD Example. Fig 57. Given: ^ r^ 1432.69, R 3 = 1910.08, and I = 400. Assume R* = 3437.87, to find A 2 , ft, a and p. Eq. (179) A- A A - 3 Eq. (170) A- Z 2 A 3 \ Eq. (180) A 2 - R* Eq. (175) a Eq. (172) a -A 2. 2005.18 1527.79 477.39 400. 1527.79 400. 13 07' 22" 39 57' 34" 39 57' 34" 13 07' 22" 33 50' 39" 6 06' 55" 7 00' 27" 7 00' 27" log 0.301030 " 3.302153 " 3.184064 " 6.787247 5.204120 log vers 8.416873 log 2.678873 " 2.602060 11.41 log cot 0.076813 log sin 9.807701 " " 9.356099 log 3. 184064 log sin 2.347864 log 2. 602060 log sin 9.745804 log 3.184064 log vers 7.873309 1.057373 178. Given: a three-centred compound curve to replace the middle arc by an arc of different radius. I. When the radius of the middle arc is the greatest. Fig. 57. First find the length and direction of the common tangent AB. Let A 2 = central angle of the middle arc, R* = its radius, and Hi and E 3 the radii of the other arcs. From eq. (179). I = V2( 2 - J?0 (ft* - R s ) vers A, (1 85) Then find i by eq. (170), a and /5 by eqs. (181) (182), and p by eq. (172). For the new arc we may now assume a new value for p, or for Ri, or for a. Indicating the new values by an accent, if we assume p' we proceed as in the last problem, using eqs. (173), etc. If we assume R*, we use eq. (179), etc. If we assume a', we use eq. (178). COMPOUND CURVES. 141 II. When the radius of the middle arc is the least of the three. Fig. 58. In this case the middle arc is within the other two pro- duced; and for- the same values of RiR 3 and Oi0 3 , the locus FIG. 58. of the centre 2 is the opposite branch of the hyperbola found in 177. When the centre 2 falls on the line Oi# 3 , A 2 = 180, and Rz = K-R, + Mi Oj 3 ), a maximum. (186) Analogous to eq. (185), we have I = V2(R 1 - R*} (R 3 - J2 a ) vers A 2 (187) which gives the length of the common tangent FZ. We then have the values of i and of O t 3 by eqs. (170) (171), and of a and ft by eqs. (181) (182), and analogous to eq. (172), p = (^ _ Rs} vers a = (R*-R*) vers ft (188) in which p is the perpendicular distance HI bet ween parallel tangents. For the new arc we may now assume a new value for p, for .R 2 , or for a. Indicating the new values by an accent, if we assume p', we have, analogous to eq. (173) FIELD ENGIJSTEEKIKG. (189) and from eq. (188) vers a' = ; vers ft' = j^-~jr 11 3 Jt 2 (190) If we assume R 2 ', we have, analogous to eq. (179), M vers A 2 = (191) 2(5! - 5 2 ') (5 3 - 5 8 ') and we find a and /3 by eqs. (181) (182), and p' by eq. (188). III. WJien the radius of the middle arc has an intermedi ate value, compared with the other radii. Fig. 59. FIG. 59. \ In this case, whatever be the value of J? 2 , we have 3 2 + 0^0, = (E 3 - R*} -f CR 2 - 50 = (B 9 - 50 a constant value independent of 5 2 ; hence we infer that the locus of 2 is an ellipse, of which Oj and 3 are the foci, and (5 3 50 equal to the transverse axis. Let I = QQ' = the conjugate axis, and let i = Q0 3 0i = Q0 1 3 . Produce 3 Q to O, making QG OsQ, and join GO,. COMPOUND CURVES. 143 Then by similar triangles 0i is perpendicular to Oi0 3 , and O0 t = I; and in the right-angled triangle (r0 3 0i O0 1 sin ^ = -- O l 3 = (Rz - Ri) cos i - I cot t (193) Analogous to eqs. (185) and (187), we have I = V 2(R* - R*) (R* - Ri) vers 'A , (194) which may also be derived from the triangles 0i0 a 3 and O&Q. Let a = 2 3 0i, and ft = 2 0x0 3 Then sin a = %%- sin A 2 = ^=^ tan . sin A 2 (195) O\U From the figure ^ = A 2 a (196) In the diagram produce the line 3 0i and it will intersect all the arcs. At the points Z and T, where it cuts the inner and outer arcs, draw tangent lines perpendicular to 3 0i. Draw the radius O a J parallel to 00i, and the tangent line IL at I. Let g = ZYauAp = ZL = HI Then by the theory of parallel tangents, 137, the point J is on the chord PZ produced, and it is also on the chord P' Y-, and we have p = ZL = (R* - R 1 ) vers ft . (197) q - p = LT = (R 3 - jR 2 ) vers a (198) and q equals the sum of these. But q = ZFis.the shortest distance between the inner and outer arcs, and has a constant value independent of 1? 2 . If we assume R* = i(R 3 + Ri) the centre 2 will be at Q, and a = ft = i,and p = q. Making these substitutions above, g = (R 3 - RJ vers f. (199) Also, from the figure, 144 FIELD ENGINEERING. ZY = 3 Y- 0,Z- 0,0,, or, q = E 3 - R, - 0,03. (200) In the triangle ZIT we have by Geom. Tab. I. 26, ZP = IT* + ZY 2 - 2ZY(ZY - ZL) or ZY 2 - 2ZY.ZL = Now, ZI* 4(^2 -fti) 2 sin 2 1/3 = 2(E* E^ vers /3 1Y* = 4(E 3 -R2) 2 sin 2 la = 2(Jft s E*) vers a Hence ZP = 2(^2 Ej.) p and JF 2 = 2(E 3 E*) (q p) Substituting these values, and solving for p, we have p _ q(E 3 - E^-lg) = q(E 3 - J? 2 - lq) ^^ Also E* = (E 3 - lq) - p . ^ (202) For any other value of E^, we have Hence E*' - jR 2 = -~-? (p - p') (203) which gives the change in R 2 for a given change in the value of p Observe that as p diminishes 7? 2 increases and vice versa. Having determined the value of E*, we find p' by substitut- ing E 2 = 4 10' A 2 = 48 " R 3 = 1910.08 Z> 3 = 3 00' Let pp'= 11.30 Eq. (194) 2 log 0.301030 R 3 - R 2 534.68 " 2.728094 R 3 - Ri 593.56 " 2.773465 A 8 48 log vers 9.519657 2) 5^322246 I 458.27 log 2^661123 (192) R 3 - R l 1128.24 " 3.052402 i 23 57' 55" log sin 9J608721 (193) i 23 57' 55" log cos 9. 960847 R 3 - Ri log 3.052402 0,0, 1030.98 log *3. 013249 (195) ^ 2 - R, log 2.773465 A 2 48 log sin 9.871073 log * 2^644538 25 19' 52" log sin 9. 631289 (196) ft 22 40' 08" (203) Oj0 3 log 3.01 3249 (200) q 97.26 1.987934 q p-p' 11.30 log 1.053078 2' - R* 119.78 " 2.078393 R* 1495^18 (say) 1494.95 for 3 50' curve. 146 FIELD ENGINEERING. (201) 7? 3 - RS - \q 366.50 log 2.564074 " 1.025315 q p' 34.57 " 1.538759 (197) .K 2 ' - .Si 713.11 " 2.853157 /?' 17 55' log vers8. 685602 (198) q-p' 62.69 log 1.797198 E 3 - Ri 415.13 " 2.618184 31 54' log vers 9. 179014 A a ' 49 49' a' _ a = 6 34' .-. P'P" = 218.89 ft - ft' 45' .-. PP" = 64.77 The practical difficulty in changing the middle arc of three centred curves lies in the difference of measurement that ensues. Thus, in the last problem, although the total central angle is the same, the new curve is 6.56 feet shorter than the original, making a fractional station at P ". If the change is made during the location, it is well to re-run the last arc from P" to the tangent following, so as to eliminate the fractional station from the curve. TURNOUTS. 147 CHAPTER VII. TURNOUTS. 179. A turnout is a curved track by which a car may leave the main track for another. At the point where the outer rail of the turnout crosses the rail of the main track a frog is introduced which allows the flanges of the wheels to pass the rails. A frog consists essentially of a solid block of iron or steel having two straight channels crossing each other on the upper surface, in which the flanges of the wheels pass. The triangular portion of the upper surface formed by the channels is called the tongue of the frog, and the angle which the channels make with each other is called the frog- angle. Every railroad is provided with a set of frogs of different angles, from which may be selected one best adapted to any particular case. The frogs may be designated by their angles, but it is customary to designate them by numbers expressing the ratio of the bisecting line FC of the tongue to the base line ab, Fig. 60. Observe that F is at the intersection of the edges produced, and not at the blunt point of the tongue. In the triangle aFC, ryf ' and if we let n = the number of the frog, and F = the frog angle, then On some roads, however, the frogs are numbered arbitrarily, or according to their length in feet, while on others they are designated by letters of the alphabet. In any case the true number (n) of a frog may be determined by the above for- mula. 148 FIELD ENGINEERING. The first rail of the turnout is common to both tracks, and is called the switch-rail. It has one end free, so as to be shift- ed from one track to the other as required ; the free end, D (Fig. 61), is called the point of switch. The tangent point of the turnout, at A, is called the heel of switch, and the distance, AD, is the length of switch. The switch-rail should be several feet longer than AD, and the excess be spiked down in the line of the main track back of the point A. Then if the point D is thrown over to meet the rail of the turnout at K, the switch rail is sprung into an arc, which coincides with the arc of the turnout, provided that the length of switch AD has been prop- erly taken. The distance DK through which the point moves is called the throw of the switch. It varies on different roads from 4| to 6 inches, but is usually made about 5 inches, or 0.42 feet. A turnout should be a simple curve from the heel of the switch to the point of the frog. 18O. Owen: a main track, straight, and a frog angle F, to determine the distance BF, on the main track from the heel of switch to point of frog, the radius, r, of the centre line of the turn- out, the length of chord af, and the proper length of switch AD. Fig. 61. FIG. 61. Let C be the centre of the turnout. " F = the frog angle, HFI = FCB. " g the gauge of track AB. " r = radius, aC = fC. " DK = the throw of switch. Then the radius of the gauge side of the outer rail is (r -f- ^g), and we have TURNOUTS. 149 AB = FC . vers FOB or, 9 = (r + iff) vers F whence ;;";...; '"'-" '.''*+*> = ** ; (207) The angle and jB^ 1 AB cot J.JPB = # . cot \F (208) Again, in the triangle FCB BF=FC . sin FCB = (r + &) sin ^ (209) The chord of is, evidently of = 2r sin \F (210) Similar to eq. (207), we have DK DK vers ACD = -j^r = - -S^C7 r + 4# But since the inside rail has the same throw, while its radius is (r \g), we may, if convenient, drop the \g> and hence the length of switch is AD = r. sin ACD (211) The degree of curve corresponding to r is found from Table IV., or by eq. (17), and the centre line of the turnout may be located by transit deflections from the tangent point a, using chords of 20 or 25 feet -|- the correction found in 106, 107; or the deflection for a 20-foot chord may be calculated at once by sin (tf ) = -^ (212) 181. Simple as these formulae are, they may be rendered still more convenient by introducing the number of the frog:, n. By eq. (206) we have cot \F 2n, which substi- tuted in eq. (208) gives BF- 2gn (213) Drawing the chord AF to the outer rail, AF = V AB* -f BF* = gVl +4n* (214) 150 FIELD Make BA' = AB and join FA' ; then by similar triangles, AA'F and AFC, AA : AF :: AF : FO whence AF* FG = 4 ~ AA or (r + ft) = \g (1 -f 4^) (215) whence r = 2gn* = BF . n (216) The chord af to the arc of the centre line is to AF as r is to (f + iff} ; hence af = - eqs. (214) (215) we have ; hence / = ~I~> an( * substituting values from (217) 4/1 -f 4* Assuming that, for small angles, the tangent offsets vary as the squares of their distances from the tangent point, which will lead to no material error in this case; whence AD= AB:DK:: BF' 2 : AD* DK AB V (218) or AD = Vn*.DK-- V2r . DK It is not necessary to determine the degree of curve in order to locate the turnout, for having fixed the position of BF, the position of af is found by laying off Ba, and F/, each equal to iff. Whatever be the length of the chord af, found by eq. (217) or (210), its middle ordinate is always g, and the ordin- nates at the quarter points, f . \g ^g. Thus for the stan- dard gauge of 4.708 the middle ordinate is 1.177, and the side ordinates 0.883. By the preceding formulae Table XI. has been calculated, which gives the required parts of a turnout for various frogs when the gauge is 4 feet 8|- inches and the throw 5 inches; also for a gauge of 3 feet and throw of 4 inches. For any other throw, only AD must be calculated. For a different gauge the engineer will do well to construct a similar table, adapted to the frogs used on the road. TURNOUTS. 151 In the table the frog angle is given to seconds, in order that the results may agree, whether found by equations in 180 or 181; but in practice the nearest minute is sufficiently exact. The frogs most used for single turnouts are those from No. 7 to No. 9, inclusive. 182. In case of a double turnout from the same switch, three frogs are required, as at F,F' and F", Fig. 62., and the switch is called a three-throw switch, because its point takes three positions. The frogs F and F 1 are usually alike, and placed exactly opposite each other in the main track. . The other frog F" is placed on the centre line of the main track. Its angle F" and its distance from a are now to be determined in terms of F. In the figure we have vers F"Ca = -^7-,-^ or The distance also aF" ~(r + i?) sin aF" = r . tan \~F" (219) (220) (221) All the parts of the turnout required to locate the frogs F and F" are calculated by the formulae in the preceding sec- tions, or are taken from Table XI. If we let n" = the number of the frog F", then by eq.(206) tan \F" = -j-pr, which substituted in eq. (221) gives aF" = 2n" (222) 152 FIELD EKGItfEEKING. Also, in the triangle aF"C, Equating these and replacing r by Zgn*, we obtain If we neglect the i, we have (approx.) n" = -11. = . 707171 (233) (224) (225) Example. It F F' = 6 44', or ra = n' = 8.5, then n" = 6.0 + or F" = 9 32'. 183. In case no frog is at hand of the angle or number given by eq. (219) or (225), we may select one as nearly like it as pos- sible, and locate the turnout as a compound curve, pro- vided that F" is less than 2F. Fig. 63. Fio. 63. Let r" = C'a", and r - r' = QT = Cf Then analogous to the equations of 180, (r4-tt)= -*- vers ^" i .-. r" = (226) (227) exsec }!?" aF" = (r" + $g) sin ^^" = r" tan 117"' (228) TURNOUTS. The length of the switch, by eq. (218), is 153 AD= 1 DK The curvature of the rail between the frogs F" and F is F"CF = (F - iF"). Draw the chord F"Fand. the perpendicular F"L; then the angle LFF" = F- $(F - iF") = i(F + i^"); and since LF-\g. cot i (329) (230) (281) Example. Let F = 6 44' . Eq. (226) iff 2.354 iF" r" 569.616 Eq. (228) iF" aF" 51.839 Eq. (229) iff 2.354 i(F+iF") F"F 22.645 Eq. (231) i(F- iF") 2(r + iff) 1692.432 r "' = 10 24' log 0.371806 5 12' log exs 7.616224 2.755582 5 12' log tan 8.-959075 1.714657 When ft" > . 707ft, r will be equal F, (F" being given), then also, by substituting F' for ^i log 0.371806 5 58' log sin 9.016824 O54982 46' log sin 8.126471 3.228511 less than r". Should F ' not r' and J7.F 7 ' must be calculated eqs. (230) and (231). 184. From the same switch in a straight track it is required to lay two turnouts on the same side. Fig. 64. If we assume F' = F, and that these two frogs shall be opposite each other, we calculate all the distances of the first turnout for the angle F (or number ft) by 180, 181, whence we have the radius r = Ca. 154 MELD ENGINEERING. Let r' = C'a, the radius of the centre line of the second turnout. The angle AGF '= P., and since F' = F, the angle CF'C'= F, and the triangle CF'C 1 is isosceles, and C'F' = C'C. But C'F 1 = C'A = or (232) (233) C' FIG. 64. To calculate the remaining frog at F", we have from eq. (207) vers F" = or from eq. (216) BF' = (?' of" = 2r' sin iJ and since AO'F' = 2F, sin F = 2r" of' = 2r' sin F (234) (235) (236) (237) (238) The length of switch may be calculated by either r or r', since for r', which is about ^r, the throw of switch is double, thus giving practically identical results. If we compare the values of F" as obtained by eqs. (234) and (219), we shall find them almost identical for given values TURNOUTS. 155 of jPand g\ and since this may also be proved analytically by assuming that vers $F" = i vers F\ which is very nearly true for ordinary values of F", we conclude that a set of frogs (F = F' , and F") which is adapted to a double turnout in opposite directions from a straight line (as in Fig. 62) is also adapted to a double turnout on one side (as in Fig. 64), the curves being simple curves in every case. But this being true, the set is also adapted to a double turnout in opposite directions from any curved track the radius of which is not less than r as given for F, since any such case is intermediate between the two cases named. When, therefore, a certain frog, F, is adopted for general use on any road, another frog should also be adopted, whose angle, F ", is determined by eq. (219), or whose number n is determined by eq. (225). Thus, if F = 6 44', or n = 8|, then F" should be 9 32', or n" = 6. 185. In case no frog is at hand of the angle or number given by eqs. (234) (235), we may select one as near the same angle as possible, and, calling this F", calculate the distance BF" and the radius C"F" (Fig. 65) as for a single turnout; 180. Then assuming any other frog F'. whether equal to .For not, it is required to find the chord F"F' , and the radius C'F' of the arc F"F'. The point F' may fall either side 'of the radius CF, according to the values given to F" and F'. a. In case F' falls beyond the radius CF, we will assume first, that the entire rail from B to F' is laid with the same radius BC, and centre C. (This investigation also applies to the case when F' falls between B and the line CF.) In the diagram (Fig. 65) draw CF\ We then have 156 FIELD ENGINEERING. and QF" = (r ig) exsec BCF" (240) In the triangle F"CF', F"C - F'C : F"C + F'C ::. tan $(F"F'C- F'F"C) : cot F"CF' Now, since C'F'C = F', and BC"F" = F", .'. F"F'C= F"F'C' + F' and F'F"C= F'F"O" - C"F"C=F"F'C' - (F" -BCF") Letting U- C"F"C = (F" - BCF") and subtracting, we have F"F'G- F'F"O='F' -\- U Hence the above proportion may be written GF" : 2BC + GF" :: tan i(F' + U) : cot %F"CF' whence cot IF"CF' = 2B +p? F " tan l(F' +U) (241) (Since BCF" -f F"CF' = BCF', and we know the radius BC, the chord or arc BF' is easily obtained, which fixes the position of the frog F'; and the problem may end here, frequently, in practice.) Now in the same triangle "'F'-'OF', flic half sum of F"F'C and F'F"C is 90 - $F"CF'; while, as we have just seen, the half difference is %(F' -f- 0; and by subtracting we have the less, or F'F"C = 90 - i(F' + 17+ F"CF') (242) F' C sin F"CF' ,_ Jg(7, sin ^GF^ r ^ F ~ ' r '' r TUKKOUTS. 157; To find the angle F"C'F'; produce the line F-'O' in the dia- gram to intersect the line EG at K. Then the two triangles KC"C' and KCF' have the angle K common, and the sum of the other angles will be equal ; that is, KC"C + 'C"C K = KCF' -f CF'K or F" + F"C'F' = BCF' + F' and since BCF' BCF" + F"CF' .-. F"C'F' = F"CF' + F' - V" (244) If we denote the radius F'C' by r' + |^ < 245 > Example. Given: the three frogs ^ 6 43' 59", ^" = 6 01' 32", and ^" = 8 47' 51" to lay a double turnout on one side of a straight track. Fig. 65. By Tab. XI. BF = 80 036 r = 680.306 AD = 23.82 BF" = 61.204 r" = 397.' 826 Eq. (239) BF" 61.204 log 1.786779 (r-ig) 677.952 " 2.831199 BCF" 5 09' 38" log tan 8^55580 Eq. (240) BCF" < 5 09' 38" log exsec 7.609587 (r - ig) 677.952 log 2.831199 GF" 2.760 " 0.440786 Eq. (241) (2BC +GF") 1358.664 ' 3.133112 2.692326 tan 8.926968 i(F"CF') 122'35" " cot 1.619294 (r=338'13 ff ) 2.692326 i(F' + U) 4 49' 52".5 log Eq. (243) F'CF- 2 45' 10" " sin 8.681481 r-tff 677.952 2.831199 1.512680 l(F'+ U+F"CF') 612'27".5 " cos 9.997446 F"F' 32.752 1.515234 Eq. (245) $F"C'F' 2 34' 14". 5 " sin 8.651781 730.219 2.863453; 362.755 158 FIELD EKGINEERIKG. b. We assume, secondly, that the middle track is straight beyond F, and tangent to the curve at F. Fig. 66. Then whenever the value of F" is less than that given by eq. (234), the arc AF", produced with the same radius AC", will intersect the straight rail HF' at some point F', and the frog angles F and F' will be equal. - ' FIG. 66. For the straight rail HF' produced backwards, passes through the point A, making an angle F with the main track, since the triangles CBF and CHA are equal, and AH = BF. Now any circle, tangent to the main rail at A, will intersect the line AH in some point F', and since AF' is the chord of the arc, the angle at F' equals the angle at A, which is F. Hence F F' and the angle AC"F' = 2F, The length of the chord AF is AF' = 2AC" sin F The chord F"F' = 2F"C" sin Hence, = 2AC" sin F"F' - 2(r" -f- (246) (247) Example. Let F' - F - 6 43' 59" and j^" = 8 47' 51* By Table XI. r" - 397.826 Eq.(247)2(r" + ^) = 800.360 ' 2 20' 03".5 32.60 log 2.903285 log sin 8.609915 1.513200 If the frog F' is required to be different from F, then the inside curve must be compounded at F", giving other values to the length and radius of the arc F"F'. TURNOUTS. 159 c. "We assume, thirdly, that the curve of the middle track is reversed at F. Fig. 67. In the diagram, let Q be the centre of the reversed portion, and F' the proper position of the frog F', and C' the centre of the required arc F F'. Then Q is on the radial line CF, produced, and 6" is on the radial line F"C" produced. Join FQ and F'Q, and produce C"F" to intersect these lines in L and M respectively. Also join F"Q, and denote the angle LF'Q by (/and the angle F'QF" by Q. FIG. 67. In the triangle FF"Q we know F"F= BF- BF", and the side FQ is given; and the included angle F"FQ = 90 -f- F. Hence we may calculate (Tab. II. 25) the angle F"QFa.nd the side F"Q. The triangle CC"L gives the angle at L = F" F; and the triangle F"LQ gives LF 'Q = L F"QF .-. U= F" - F- F"QF (248) In the triangle F'QF" we have F'Q - F"Q : F'Q -f F"Q :: tan $(F'F"Q - F"F'Q) But F'F"Q = F'F"L + 7 and F"F'Q - F"F'N- F', and since F"F'N = F'F"L, we have by subtraction, F'F'-Q- F"F'Q = U-\-F' Hence cot 4 Q = *j&Jr*& ten I (U+F') (249) 160 FIELD ENGINEERING. (Now the angle FQF' = Q - F"QF, and is subtended by the chord JIF', which is therefore easily found, and serves to locate the frog F', and frequently this is all that will be required.) In the triangle F"QF', the half sum of QF"F' and QF'F" is 90 . iQ, while, as we have just seen, the half difference is F'); hence by adding, we have the greater, or ' - Q) (250) The triangle C'F'M gives F"G'.F' = P' M, while the triangle F"MQ gives M= U -\- Q; hence F".C'F' = F' - Q); and denoting the radius G 'F' by r' -j- $g, Example. Lei F = F' = 6 43' 59", F" = 8 47' 51", and FQ = 953.012. Then by Tab. XL, BF '= 80.036 and BF " = 61.204; hence ^"^ = 18.832; and the included angle is 96 43' 59". Solving the triangle FF"Q we find F"QF = 1 07 f 18", FF"Q = 82 08' 43", and F"Q = 955.402. Now FQ = ^4-^ = 957.720. (249) F'Q+F"Q 1913.122 log 3.281743 F'Q-F"Q 2.318 " 0.365113 (CTO56'34") " ^916630 F') 3 50' 16". 5 log tan 8.826231 1 02' 08". 4 " cot 1.742861 (250) 2 04' 16".8 " sin 8.558033 l(U+ F'- Q) 2 48' 08".l " cos 9.999480 8.558553 F'Q 957.720 log 2.981239 F"F' 34.633 1.539892 (251) #F' -U-Q) 1 51' 34".l log sin 8.511191 2(r' + 4#) 1068.32 3~J028701 r' 53181 TURNOUTS. 161 186. Given: a main track, curved, and a frog-angle F, to locate a turnout on tJie inside of the curve. Fig. 68. Let R = Oa = radius of main track. " r = Ca = radius of turnout. " F = CFO = the frog angle. In the diagram draw the chord JLFand produce it to inter- sect the outer rail at G; and draw FO and GO. Since the chords AF and AG coincide, and the radii AC and AO coincide, the chords subtend equal angles at C and respec- tively, and GO is parallel to FG. .(See 137.) Hence, FOG = CFO = F. Let 6 = the angle FOA. In the triangle FOA, = GFO - FAO = GFO - FGO; and in the triangle GFO, GO+ FO : GO FO :: tan l^GFO + FGO) : tan i(GFO - FGO), or ZR : g :: cot $F : tan $0 .-. tan |9 = -- cot^=.- (252) In the triangle CFO, ( y+ig ) = ( B -^) sin s j,% - < 253 > In the triangle EOF, BF = 2(R - ig) sin |0 (254) In the triangle aCf, af = 2r sin #F+ 6) (255) 162 FIELD ENGINEEKII^G. The length of switch AD, for a given throw DK, may be found thus: from Table IV. take the tangent offsets, t and t', corresponding to R and r respectively, and assuming that the offsets may vary as the squares of their distances from the tangent point, we have t-t' : DK:: (1/)0) 2 : AD* AD = V *.- a (256) This result is practically the same as that found for length of switch in a turnout from a straight line with the same frog, when R is large. Example. Let R = 1432.69 and F = 6 43' 59". Eq. (252) 2.354 R (Tab. IV.) 3 21' 59". 5 log 0.371806 log cot 1.230440 log 1.602246 " 3.156151 1 35' 59". 8 log tan 8.446095 Eq. (254) F+e 3 11' 59". 6 9 55' 58".6 (254) (255) R-ig 1430.336 r + iff 462.856 r 460.502 2 R 4<7) 1430.336 sin 8. 746786 " 9.236778 9^510008 3.155438 BF 2r 79.872 921.004 1 35' 59".8 4 57' 59".3 of 79.734 2.665446 log 0.301030 " 3.155438 log sin 8.445924 log:L902392 " 2.964262 log sin 8.937381 log 1.901643 The values of BF and af are found to be so nearly identical in this case with those determined in case of a turnout from a straight line, that the values given in Table XI may be used at once for ordinary values of R; and the degree of curve of the turnout in this problem is approximately the sum of the degree of curve of the main track and the degree of curve given in Table XI. opposite F. Thus, in the example 4 -f- 8 26' = 12 26' .-. r = 461. 7 nearly. TURNOUTS. 163 187. Given: a main track, curved, and a frog-angle F, to locate a turnout on the outside of tJie curve. Fig. 69. In the diagram draw the chord AF, and produce it to meet the inner rail at G; and draw FO and GO. The triangle? CAF and OA G are both isosceles, and have the angles at A equal; hence they are similar, and FCA = AOG. Hence FOG = HFO = F. Let R = Oa, r = Ca, and = FOA, FIG. 69. In the triangle FOA, = OAG - AFO = FGO - GFO; and in the triangle FOG; FO -f- GO : FO GO :: tan i(FOO + GFO) : tan $(FGO - GFO), or 2R : g :: cot $F : taniQ which is identical with (252). In the triangle CFO In the triangle BOF, J5^=2CK + ^)sin40 In the triangle a Cf, af=2r. sm^(F-G) For a given throw, the length of switch will be AD = j/10000 (257) (258) (259) (260) (261) 164 FIELD ENGINEERING. in which t and t' are the tangent offsets (Tab. IV.) corre- sponding to R and r. In this problem, as in the preceding, we may, for ordinary values of R, assume the values for BF and a/given in Tab. XI. The degree of curve of this turnout is, approximately, d D, taking d from Tab. XI. and D from Tab. IV. corresponding to R. Should D = d, this turnout becomes a straight line; FIG. 70. and when D > d, or when R is less than r given in Tab. XI., the centre falls on the same side as 0. Fig. 70. In this case, using the same notation, is given by eq. (257). Eq. (259) BF = 2(R + $g) sin of = 2r sin i(B - F) . (263) 188. A tongue-switch is a short, stiff switch which, when moved, revolves at the heel as on a pivot. When it is thrown over to the turnout track, it makes an abrupt angle with the main track, called the switch angle; but in this posi- tion it should be tangent to the turnout curve. The use of this switch is generally confined to yards and warehouses, where but little space can be afforded, and where the motion of the cars is always slow. 189. Given: a straight track, a frog-angle F, and the length and throw of a tongue-switch, to locate the turnout. Fig. 71. TURNOUTS. 165 Eet AD be the length, and DK the throw of switch, and let S denote the switch-angle DAK. T) Jf Then sin 8 = ~~ or 8 = 57.3 ~- (264) (Compare 86.) Let C be the centre of the required turnout, and in the dia- gram draw CK and GF\ also draw DG perpendicular to the .straight track. Then DGF = F; and in the triangle KGC, KCF = KGF -GKC, and since CKA is a right-angle, GKG - 8 .-. KCF= F- 8. Draw the chord KF, and since the triangle KCF is isosceles, the angle CFK = 90 - \(F 8). Now, CFI = 90 - F\ hence by subtraction, KFI = i(F + 8). FIG. 71. If g denote the gauge, we know KI = g DK- and in the right-angled triangle KIF, we have IF= KI . cot $(F+ 8) (265) (367) These equations are analogous to eqs. (229) (230) (231). 19O. Given: a double turnout with tongue- switch, from a straight track; to find the angle, F", of the middle frog. Assuming F' = F calculate (r -f |#) by the last equations. Since the rails of the turnouts intersect on the centre line .of 166 FIELD the straight track, as in Fig 63; if we substitute the value of F" F', eq. (229) in eq. (231), we have __lff F")sini(F- iF") iff 2 sin and by Trig. Table II. cos^F" cosF whence r + iff) (268) If the angle of the middle frog to be used does not agree with F" found by the last equation, the turnout will be com- pounded at F". 191. Given : a straight track, tfie frog-angles F f F' and F", and the switch angle S, to locate a double turnout. Fig. 72. FIG. 72. Assuming that F" shall be placed on the centre line of the straight track, let h be a point on the centre line at the point of switch. Then JiK = ig DK; and since the angle F" is bisected by the centre line the necessary formulae in this case are obtained from 189 by simply replacing ^byl^" and KI by hK; and in the first members ZZ^by hF" and r by r\ This is obvious by the similarity of the figures. TURNOUTS. Hence hF" = hK. ' "" KF" = hK sin - S) 167 (269) (270) (271) The location of the remaining frogs is a problem already discussed, 183, eq. (229), etc. 192. Given: a straight track, the frog angles F, F', F", and the switch angle S, to locate a double turnout on one side. Fig. 73. FIG. 73. The frog Fis located by 189; but for the frog F" we have evidently a double throw; hence eqs. (265) (266) (267) become IF" = (g - 2DK) cot 28) sn sin i(F" - 28) (272) (273) (274) To locate the remaining frog F' : when F' falls beyond the line CF, there are three cases. a. The middle track reversed beyond F. We find the distance F"Fby aubtracting IF", eq. (272) from IF, eq. (265) : after which the solution is identical with that given 185, C., Fig. 67. 168 FIELD b. The middle track compounded at F. Let Q be the centre of the curve beyond F, and also let Q = the angle F'QF"; and let U = the angle C"F"Q. Then by a course of reasoning analogous to that of case a, we derive ... U=F"-F+F"QF (275) cot IQ = y'.Q + y.Q tan i(U+ F') (276) Now since the radius F'Q is given, and the angle FQF 1 = Q FQF", we readily determine the distance HF', and so locate the frog F'. In the triangle F"QF', the half sum of QF"F' and QF'F" is 90 - $Q, while the half difference is $(U + F'); hence by subtraction we have the less, or F'F"Q = 90 - i( U+ F' + Q) Hence P ' F ' = FQ -^^^ (277) Join C'Q, and the quadrilateral C'QF'F" gives F' + Q = U+F"C'F' hence F"O'F' = F' - U+ Q', and denoting the radius C'F by r' -j- $g, we have Cor. Since the centre Q is assumed at pleasure, it may be made to coincide with the centre C, and then the compound curve becomes a simple curve. Then also, the above formulae will apply when F' is such that the frog will come on the arc Iff. But as FQF" will be greater than Q, the difference FQF' will be negative, indicating that the distance HF' is to be laid off backwards from H. c. The middle track straight beyond F, and tan- gent to the curve at F. Fig. 74. Let F' be the required position of the frog F'. A tangent to the curve at F' makes an angle (F' -f- F) with the main track, and a tangent at F" makes an angle of F" with the same; hence the angle they make with each other is TURNOUTS. 169 ' F"), and this is the curvature of the arc F"F' t and equals the angle F"C'F'. Produce the straight line F'H backwards to G, and draw F"G perpendicular to it. Then F"G = FH F"F. sin F, or (279) FIG. 74. In the right-angled triangle F'GF", the angle F"F'G = F' - %(F' + F- F") = i(F' + F" - F). F"F' = -? F"G sin i(F' + F" - F) and GF' = F"F' . cos \(F' +F'.-F) (281) Observe that GF' cannot be less than GH= F"F. cos F. 193. Given: a turnout with a frog angle F, and the perpen* dicular distance \> between the centre lines of the main and side FIG. 75. tracks ; to find the radius r of the curve connecting the turnout with the side track. Fig. 75. 170 FIELD ENGINEERING. Let the reversing point be taken at F, and let Q on CF pro- duced be the centre of the required curve, and draw Q,M per- pendicular to the main track. Then QM= QF= r ^g; the point M is the point of tangent, and the angle FQM = F. Now N being the intersection of the rail .B^with the radius QM, we have MN= QFvers F, but MN = p g; hence The distance FN is evidently FN=(r-ig)smF (283) and the chord to the centre line is /ra = 2r sin \F (284) Should the distance FN consume too much of the track, it may be lessened by introducing a short tangent at F, denoted by k; then by eq. (48) the radius will be shortened by an amount equal to k . cot $F, and the distance FN will be shortened by k. Since the tangent k reduces the length of the tangent offset of the entire curve by k . sin F, we have for the new radius r' When r' is fixed by a limit, we obtain k by resolving eq. (285) p-. g -(r' -^ VGT3 f> TO*" In case the main track is but slightly curved, we may at first assume it to be straight, and find r as above, eq. (282), and the degree of curve corresponding to r; but this degree of curve must then be increased or diminished by the degree of curve of the main track, according as the track is concave or convex toward Q. 194. Given : the perpendicular distance p between the centre lines of a curved main track and a parallel side track, and the frog angle Fofa turnout; to find the radius r of the connecting curve, and the length FIST, or fin, of the curve. Fig. 76. TURNOUTS. 171 Let FN be the rail of the main track, and GM the rail of the siding, adjacent to each other; let be the centre of the main track, and Q the centre of the connecting curve. Then the connecting curve will terminate at m, on the line OQ pro- duced. In the diagram draw MF, and produce it to intersect the rail MG at G, and join GO, FO, and FQ. Let R = radius of centre line of the main track; r = radius of centre line of the connecting curve; and = the angle FOM. Case a. The siding outside the main track. Fig. 76. FIG. 76. By similarity of the triangles GOM and FQM, GO is paral- lel to FQ, and the angle GOF = F; and by a process similar to that of 186, we have (287) (288) sinB sin (F-\- 0) fm = 2r . sin #F+ 0) (290) Case b. The siding inside the main track. Fig. 77. By a process entirely similar to 187, we have cot & (291) 172 FIELD ENGINEERING. r - to = ( R - i FN = 2(R - sin sin (FG) ) sin *0 fm - 2r sin \(F 0) (292) (293) (294) When = .Fin the last equations, sin (F 6) = 0, and r is infinite, and the curve FM becomes a straight line. FIG. 77. When > F, sin (JF 6) is negative, and the centre Q falls on the same side of the track as 0, and we* have fm = 2r . sin |(6 F) Equations (291) and (293) remain unchanged. (295) (296) 195. To locate a crossing between parallel tracks. Fig. 78. When a turnout from one track enters a parallel track by means of another frog and switch, the whole is called a cross- ing. The frogs are alike, and the calculation for one end of the crossing answers for the other. 180, 181. We have only to find the length of track between the two frogs. In the diagram let AF be one turnout, and A'F' the other, connected by the straight track F'G. It is required to deter- mine the length F'G, or the distance FN measured on the mam track from F to a perpendicular through F'. Produc- ing the line F'G to intersect the rail NFat H, we have two TURNOUTS. 173 right-angled triangles GFH and F'NII, having the common angle at H F. Let p = the perpendicular distance between centre lines of main tracks, and g gauge. Then GF g, = (p-ff.) F'G = F'H - OH = - - GFcot F (297) So FN = Nil - FH = (p-g) cot F - sinF (298) W7ien the main tracks are curved the distance F'G may be calculated by the same formula (297) which gives a value only a fraction too small, but in laying the track the rail F G must be curved to a radius which is to R of the main track as F'G : NF. When p is large, or the tracks are very wide apart, it will effect some saving of room to lay Hie crossing in the form of a reversed curve ; and the frogs being alike, the two arcs will be equal, and the point of reversed curve P will be midway between ^aud F'. Fig. 79. FIG. 79. In the diagram we have aPa' the centre line 'of the cross- ing, and PL the centre line between .tracks; aL = $p, and aC a'C' = r. The radius r having been found by 180 or 181, we have and vers aCP - PL r sin aCP (299) (300) 174 FIELD ENGINEERING. The distance between frogs, FN, measured on the main track is evidently FN ^(PL - BF) (301) in which BFis determined by eqs. (209), (213), or by Tab. XL 197. To lay a crossing in the form of a reversed curve, when the parallel tracks are on a curve. Fig. 80. Let be the centre of the main curve, G and C' the centres of the reversed curve. Then in the triangle GOG 1 we know all three sides; for CO R + ?; GG' = r + r' t and C'O = R -j- p r' ; and the half sum of the three sides is s R -\- r -J- %p. Denoting the angle COG' by - vers - P ( r ~ r> ~ (302) The angle

=S&^ (303) TURNOUTS. 175 The angle C'CO determines the length of the arc aP described with the radius r; the angle ((p -f- C'CO) = CC'A determines the length of the arc Pa', and P is the point of reversed curve. In this problem R is known, r is found by 187, and r' is found by 186, only observing that in this case the value of E must be increased by "p. The frog angles 7^ and F' may be equal or otherwise, only taking care that the point P shall be included between the radii G'F' and GF. The angle FOG = is given by eq. (257), and the angle F'OC' 6' is given by eq. (252) (in which the value of R is to be increased by p); hence the angle FOF* =

Table XVII. is calculated by this formula, assuming a mean value of R = 20,913,650 feet. 220. The form of the earth is approximately an el- lipsoid of revolution. Its meridian section at the mean level of the sea is an ellipse, the semi-axes of which are, according to Clarke, at the equator A = 6378206 metres [6.8046985] at the poles =6356584 " [6.8032238] According to the same authority 1 metre = 3.280869 feet [0.5159889] Therefore the semi-axes expressed in feet are A = 20 926 058 feet [7.3206874] B = 20 855 119 " [7.3192J.27] Then the radius of curvature of the meridian at the equator, ~ = E o = 20 784 422 ft. [7.3177379] at the poles, ~ = J? = 20 997 240 " [7.3221622] 188 FIELD ENGINEERING. In latitude 40 the radius of curvature of the meridian is 20 871 900, and of a section at right angles to the meridian, 20 955 400; the mean value, or R = 20 913 650 [7.320430], be- ing adopted for general use. The error in the correction H Q eq. (316) due to this assumption will usually be much less than that due to the assumed value of the radius of refraction. 221. Levelling by Transit or Theodolite. When a transit has a level-tube attached to the telescope, it may be used as a Theodolite for levelling, and for taking vertical angles. If the instrument be in perfect adjustment, the line of sight will be horizontal when the bubble stands at the middle point of the tube, and the reading of the vertical circle will be zero. Should there be a small reading when the line of sight is horizontal it is called the index error. When the line of sight is not horizontal, the angle which it makes with the plane of the horizon is called an angle of elevation, or of de- pression, according as the object upon which the line of sight is directed is above or below the telescope. This angle is measured on the vertical circle, being the difference of the reading and the index error, when both are on the same side of the zero mark, and their sum, when they are on opposite sides. When the distance to an observed object is known, and its angle of elevation or depression is measured, we may calculate its vertical height above or below the telescope. . ( elevation Let a angle of - _ f depression " L = the horizontal distance " L' = the distance parallel to line of sight " h = difference in elevation of object and instrument. Then for short distances, \ h = L tan a = L' sin a (317) FIG. 83. For long distances the curvature of the earth and refraction must be considered. Fig. 83. Let / be the place of the instrument, and F the object observed. LEVELLING. 189 Let L = the distance, measured on the chord of the level arc ID, passing through the instrument; and let ty = the number of seconds in the arc ID hence, since for ordinary distances the chord and arc are sensibly equal, ^ = L R 306264 "- 8 [5.314425] or giving to R its mean value, 220, if^ = L X .0098627 [7.993995] or a fraction less than 1" per 100 feet. Let IF be the arc of the refracted ray, and assuming that its radius is 7R , the arc will contain }th the number of seconds of the arc IF IF', tangent to IF, is the direction of the telescope; IF is the chord of the arc IF, and IE is the horizontal. Let a = EIF' = observed angle of elevation. Then EIF ' true angle of elevation = EIF' F'IF= a . \ip a .071^. The angle EID = $i/> .-'. DIP = &f> + a - .071^; and IDF = 90 + W . .'. IFD = 90 - (# -f a - .071^). "We now solve the triangle IFD for the side DF = h, and find sin (ifl + g -. 0710 ^ cos -- For an observed angle of depression make a negative in the formula. The coefficient .071 is called the coefficient of refraction, this being a fair average value, while its extreme range is from .067 to .100 under varying conditions of the atmosphere, and values of the angle a. When the difference in elevation of two or more distant objects is required, we obtain the elevation of each separately, and subtract one elevation from another. The elevation of the observed object is given by (//. /.) h. 222. To find the Height of Instrument of a transit or t/ieodoUte by an observation of the horizon. Fig. 84. 190 FIELD ENGINEERING. Let I be the place of the instrument, and let a = observed angle of depression of the horizon. Let F be the point where the refracted ray meets the level surface, and draw the chords IF and AF. Let iff = the angle ACF, let h = AI, and let k = the coeffi- cient of refraction. In the triangle IAF, IAF = 90 + i#, AFI = iif> - faf}, AIF = 90 - (iff - kip] Hence FIE = if> k$. But FIE = a -f faj> ^=__^_ (319) Let F" be the tangent point of a right line drawn through /; E FIG. 84 then AI = CF' exsec ACF", but CF" = R , and, since $ is 1 & always very small, ACF' = #$ + a) very nearly = . = ft exsec = XT a (320) I -2k Giving to E its mean value, 220, and assuming k = T V log 7* = 7.320430 -f log exsec 1.0801 a (321) LEVELLING. 191 Otherwise, we may solve the triangle AIF since AF = 2B sin & = 2R sin ^ " ^ sin(i0-*0) and fc_.__ cos Y When k = T V h = 2R n sin A a . - (323) cos if a Example. The observed dip of the sea horizon is 24' = a- What is the height of the instrument above thejsea? By eq. (321) 1.0801 X a X 60 = 1555". 34 3.191825 2 6.383650 Table XXVI. (q - 2 9. 070130 R 7.320430 h = 594.58 2.774210 Methods of determining heights by distant observations can- not be relied on for more than approximate results, since they necessarily involve the uncertain element of refraction, and usually a lack of precision in the vertical angle, the arc reading only to minutes in ordinary instruments. These methods, how- ever, are useful where no great accuracy is required, as for a temporary purpose until levels can be taken in the regular way, or for interpolating between points of established elevation. 223. Stadia Measurements. It is sometLnes convenient to determine distances by instru- mental observation For this purpose two additional cross- hairs may be placed in the telescope parallel to each other and equidistant from the central cross-hair. These are called stadia hairs, and distances determined by them are called stadia measurements. The stadia hairs are adjusted so as to inter- cept a certain space on a rod held at a certain distance from the instrument and perpendicular to the line of sight. For any 192 FIELD ENGINEERING. other place of the rod, the distances and intercepted spaces are nearly proportional. The exact relation is given below. Fig. 85. Let I AB, the distance of the rod from the vertical axis of the instrument; c = the distance from the axis to the ob- ject glass of the telescope; a = the distance from the object- FIG. glass to the rod ; i = the space between the stadia hairs ; s = CD the space intercepted by them on the rod; and/ = the focal distance of the object-glass. We then have by optics, '- = ~- f whence a f=-.s; and since a = I c . \ I (/-|- c) = -s. Now in any given instrument the focal distance /, and the space between the stadia hairs i are constant, while s and c vary with I. For any other distance I', we then have I' (/+ c') = -s', and combining the two equations (324) s ' is usually assumed at 1 foot and I' (/ + c ') at 100 feet, and the stadia hairs are then adjusted accordingly. The focal distance /may be found by removing the object glass and ex- posing it to the rays of the sun and noting at what distance from the surface of the lens the rays form a perfect and min ute image of the sun on a smooth surface; the distance c' is measured on the telescope when the rod is clearly in focus, at the assumed distance. To measure any other distance, the rod is again observed at the desired point, and the space s noted, which, placed in eq. (324), gives I (f-\- c) at once. We then measure c on the telescope, and adding (/ -f- c), obtain I, the distance re- quired. LEVELLING. 193 But inasmuch as c has but a small range of values, it will usually be sufficient to assume for it a mean value, as a con- stant. In this case we may find the value of (/ -f- c) = IF for the instrument used. Making c = c in eq. (324), and solv- ing for (/-f- c), we have =? (325) and by laying off on level ground any two distances from the instrument for I' and I, as 100 and 500, and observing the corresponding spaces s' and s intercepted on a rod, we insert them in eq. (325) and find (/+ c). Having found (/+ c), lay off (100 +/+ c) from the instru- ment and adjust the stadia hairs to inclose just one foot on the rod at that distance. Any other distance is then found by the formal a, (326) Example. At I' = 100 we finds' = 1.00, and at I = 500 we find s = 5.061. Hence, eq. (325) /+ c = ^ = 1-502 and eq. (326) I = 100 s + 1.5; provided the stadia hairs be ad- justed so as to intercept 1 foot at 101.5 feet distance from the centre of the instrument. 224. The foregoing formulae are all that are necessary for horizontal sights, but since the line of collimation is generally inclined more or less to the horizon, it follows that the stadia hairs will intercept a larger space on the vertical rod than that due to the true horizontal distance. We therefore require a formula for reducing* inclined measurements to the horizontal. Fig. 86. Let a = EFG = the angle of inclination of the -line of colli- mation IG ; " = CFD -- the visual angle defined by the stadia hairs; " s = CD = space intercepted on a vertical rod. Then (Fig. 85), tan *= = !. (327) 194 FIELD ENGINEERING. Ill Fig. 86 = CE DE = EF [tan (a -f $6) - tan (a - while the true value (for the same distance) would be Dividing one by the other we derive C'D' 2 tan s tan (a -f- 0) tan (a $0) By giving to ' and I' (/-j- c) in eq. (327) their customary -ID' FIG. 86. values, viz., 1 and 100, we have tan $0 = .005 .-. = 34' 22". 63 and by Trig. Table II. 70, tan (a 4--J-0) tan (a J) = - - cos (a -j- |0) cos (a &) Since is small, we have sensibly sin = 2 tan 0, and cos (a -|- 0) cos (a 6) = cos 2 a and the last equation reduces sensibly to SlS-^&^a (328) which is the coefficient of reduction required by which to multiply the observed space s in case of inclined sights. Hence the formula for distance (eq. 326) becomes in this case without sensible error I = 100 s cos 2 a + (/+ c) (329) Tables XVIII. and XIX. Lave been calculated by the exact formula for the coefficient. LEYELLlKG. 195 Example. Given : a = 8 20' and s = 9.221; what is the horizontal distance to the rod? Eq. (329) 100 log. 2. s 9.221 " 0.964778 a 8 20' Tab. XIX. " 9.990780 902.7 2.955558 f-\-c 1.5 .'. Aw. 904.2ft. The rod man should have a disk level to insure keeping the rod vertical. 225. Another method of procedure is that in which the rod is always held perpendicular to the line of collimation, however much inclined the latter may be. To secure this posi- tion of the rod, a small brass bar is attached, having sights upon it through which the rodman watches the instrument during an observation, the line of sight being at right angles to the rod. The distance thus obtained is of course parallel to the line of collimation, and requires to be reduced to the hori- zontal. For this purpose, we have (Fig. 87). FIG. 87. IE = IG cos a -f- BG sin a or IE = (100 s + /+ c) cos a -f- r sin a (330) in which r is the reading of the rod by the line of collimatioa For the elevation of the point B above /, EB = HG - GB cos a or EB = (100 s +/+ c) sin a - r cos a (331) FIELD EJ^GISTEERI^G. When the distances are sufficiently great, correction must be made for curvature of the earth and refraction, as already ex- plained. This method is employed by the topographical parties of the U. S. Coast Survey in connection with the plane table. Their instruments, however, arc so constructed as to give distances in metres, and heights in feet, requiring a modification of the above formulae. CHAPTER IX. CONSTRUCTION. 226. The engineer department of a railway com- pany is usually reorganized for the construction of the road, as follows : Chief engineer, Division engineers, Resident engineers, -Assistant engineers. On some roads the division engineers are styled "Principal Assistants;" the resident engineers, "Assistants;" and the assistant engineers are de- signated according to their duties, as "leveller," " roclman," etc. A resident engineer has charge of a few miles of line, limited to so much as he can personally superintend and direct. He has one or more assistants and an axman in his party. All instrumental work is done and all measurements taken by the resident engineer and his assistants. A division engineer has charge of several residencies, and inspects the progress of the work on his division once or twice a week. In his office, which should be centrally located, all maps, profiles, plans, and most of the working drawings required on his division are prepared. To him the resilient engineers make detailed reports once a month, or oftener if necessary, which he passes upon as to their cor- rectness, and from which he makes up a monthly report, or estimate, of the amount and value of the work done and ma- terials provided by each contractor on his division. The esti- mates are forwarded about the first of each month to the chief engineer, who examines and approves them, returning for modification any that seem to require it. CONSTRUCTION. 197 The chief engineer lias charge of the entire work, and directs the general business of the engineer department, lie occasionally inspects the work along the line. 227. Clearing and Grubbing'. The first step in the work of construction is to clear off all growth of timber within the limits of the right of way. The resident engineer with his party passes over the line, making offsets to the right and left, and blazing the trees which stand on, or just within, the limits of the company's property. The blazed spot is marked with a letter 0, as a guide to the contractor. After felling, the valuable timber should be piled near the boun- dary lines, to be saved as the property of the company. The brushwood is burned. Where a deep cut is to be made, the stumps are left to be removed as the earth is excavated. In very shallow cuts and fills the contractor will generally prefer to tear up the trees by their roots at once, rather than to grub out the stumps after clearing. Where the embankments will be over three feet high, grubbing is not necessary; but the trees require to be low-chopped, leaving no stump above the roots. The engi- neer should indicate to the contractor the localities where each process is suitable. 228. While the clearing is in progress, the engineer should run a line of test levels touching on all the benches to verify their elevations ; he may also rerun the centre line, replacing any stakes that may have disappeared, and setting guard plugs to any important transit points which may not have been previously guarded. If any changes in the alignment have been ordered, these may be made at the same time. 220. Cross Sections. The resident engineer is fur- nished with a profile of the portion of the line in his charge, upon which is plainly indicated by line and figures the estab- lished grade. From this he calculates the elevation of grade at each station, and by subtracting this from the elevation of the surface, he derives the depth of cut or fill (-f- or ) to be made at each point. The grade given on the profile is that which is subsequently called the subgrade, being the surface of the road-bed. The final or true grade is the upper surface of the ties after the track is laid. 198 FIELD ENGINEERING. The base of a cross section is identical with the width of the road-bed. It is made wider in cuts than in fills to allow for the side ditches. Six feet should be allowed in earth, and four feet in rock cuts. The ratio of the side slopes depends upon the material. The usual slope ratio for earth is 1| horizontal to 1 vertical for both excavation and embank- ment. Damp clay and solid gravel beds will stand for a time in cuts at 1 to 1, or an angle of 45, but this cannot be perma- nently depended on. On the other hand, fine sand and very wet clay may require slopes of If to 1 or 2 to 1. Exceptional cases require slopes of 3 or 4 to 1. In rock work the slopes are usually made at to 1 for solid, | to 1 for loose, and 1 to 1 for very loose rock, liable to disintegrate. Rock embankments stand at 1 to 1. 23O. All cross sections are taken in Vertical planes at right angles to the direction of the centre line. Figs. 88, 89. Formulae. Let b = AB, the base of section, or road-bed. " s = = the slope ratio " d = CO = the cut (or fill) at the centre stake. " h = DHor EN= the cut (or fill) at the side stake. " x = CD = the "distance out" from centre to side stake. " y = h-d = KD. We have at once from the figures the general formula x=$b-\-8h (332) When the ground is level transversely; h d, and x = $b -f- sd. For embankment use the same formula, considering d or h as positive in this case also, the figure being simply inverted. When the ground is inclined transversely; h = CO -j- DK d -f- y on the upper side in cuts; x=\b + sd-\-sy (333) and h = EN =d y on the lower side in cuts *. * = \b -f sd - sy (334) CONSTRUCTION. 199 "For embankments use the same formulae, but apply eq. (333) to the lower side and eq. (334) to the upper side, the figure being inverted. The points D and E on the ground are usually found by trial, such that the corresponding values of x and y will verify the formulae. When the natural slope FD or LE is uniform its ratio s' may be found by measuring along the section the horizontal dis- tance necessary to change the reading of the rod 1 foot (or halt the distance necessary to change it 2 feet, etc.). Then, having found the depths of cut (or fill) at J^and L, distant %b from the centre C, we have BE = sh = s'(h - BF) and AN = s?i = s'(AL - h) From these we have, for the upper side in cuts, and lower side in fills. h = y-*^ BF . . x = \b + --- BF (335) also, for the lower side in cuts, and upper side in fills, h = yVj AL ' x = ^ b + 7^77 AL We also have and (337) whence the points D and E may be found by the level. But points D and E thus calculated should have their posi- tions verified by the general formula, eq. (332), lest the slope s' may not have been perfectly uniform. When the natural surface intersects the base between the points A and B, the section is said to be in side hill work, Fig. 90. Both portions of the section are then determined by eq. (333), or where the slope s' is regular, by eq. (335) measuring in every case from the centre stake C; but observing that when the centre is in cut and one side in fill, or vice versa, that d must be considered negative for that side, whence eq. (333) becomes for this case x = ib - sd -f *y (333)' 200 FIELD EKGINEEKLtfG. 231. Staking- out Earthwork. Beginning at a point on the centre line where the grade cuts the natural sur- face, the engineer drives a grade stake (marked 0.0) and notes the point in the cross-section book. If the line of intersection of the road-bed and surface would make an acute angle with the centre line, he also finds the points where the edges of the proposed road-bed will intersect the surface, drives grade stakes, and also stakes out a cross section through each of those points, if necessary. Then advancing to the next point on the centre line where a section is required, he finds its elevation with the level (veri- fying or correcting the elevation taken on the location), calcu- lates the depth of cut or fill CG, which is then marked upon the back of a stake there driven; a cut being designated by G and a fill by F. If the ground is level transversely (Fig. 88), ha calculates x by eq. (382) and lays off this distance at right angles to the centre line, driving slope stakes at the points D and E, marked with the depth of cut or fill. The marked side of slope stakes should face the centre line. If the ground is inclined transversely (Fig. 89), he first measures FIG. the distance, |&, to F, and finds the depth BFior record. He then proceeds to find the point D. If the natural slope be uni- form, D may be found by eq. (335) or (337), verifying the result by eq. (332). The point E of the other slope may be found similarly, using eq. (336) or eq. (337); verifying by eq. (332). CONSTRUCTION. 201 232. If the ground be irregular, the depth of cut or fill is found not only at the centre and edges of the road-bed, but also at every other point along the cross section where the sur- face slope changes, all of which depths are recorded, together witk their respective distances from the centre. To find the point D : assume a point supposed to be near D, and there take a reading of the rod. The difference of the readings at that point and at C equals y' for that point, which inserted in eq. (333) gives a value x. If x agrees with the horizontal dis- tance of the assumed point from C, the true position of D has been found. If x' be greater than this, by subtracting the eq. x = ib -\- sd -\- sy' from eq. (333) we derive X = X>+ S (y-y") (338) the last term of which shows the correction to be added to x '. Now in advancing from the assumed point to the extremity of x', the rise of the surface is approximately (y y'}, and if, in going the additional distance, s(y y'}, a further rise is en- countered, this last, multiplied by s, must also be added to x' , and so on until the additional advance makes no change in the value of y. The point thus found, verified by eq. (332), is the point D required. But if x' be less than the distance of the assumed point from C, we have x = x'-s(y' -y} (338)' the corrections being subtractive. The point E on the other slope is found in a similar manner, using eq. (334) for the value of x ; if x' be greater than the as- sumed distance, we have x = x -s(y- y'} (339) the corrections being subtractive ; but if x'. be less than the as- sumed distance, *=*' + (y'-y) (339)' the corrections being additive. 233. In side-hill work (Fig. 90) proceed in the same manner, using eqs. (333) or (333)' and (338) in all cases of un- even ground. When the surface slope s' is uniform, eq. (335) may be used, if preferred, on either side. In addition to the 202 FIELD centre and side stakes, a grade stake is driven at the point 0, where the surface intersects the grade, the stake facing down hill. To find a grade point, set the target to a reading equal to the height of instrument less the elevation of grade, and stand the rod at various points along the given line until the target coin- cides with the line of collimation. FIG. 90. 234. When two materials are^foundin the same section, as rock overlaid with earth, each material "requires its own slope, and a compound section is the result. To stake out work of this description, the depth of earth to the rock must be known, and may be nearly ascertained by reference to an adjacent section already excavated. Fig. 91. Then h Let ai be the depth of earth at C " a* " " " POT " Si be the ratio of rock slope " s 3 " " earth slope s,(d a i y,) -j- 2 (a 2 (340) in which y\ difference of rod readings on the rock at C and Di, or C and EI ; and y. z difference of rod readings on the surface at P and D 2 , or at Q and E*. The upper sign applies to the upper side, the lower sign to the lower, CONSTRUCTION. 203 It is better, however, to make an indefinite cross profile at first, driving two reference stakes quite beyond the section limits; and when the contractor has removed the earth from between DI and Ei t indicate to him those exact points by marks on the rock, and also set the slope stakes at D* and E*. 235. The frequency with which cross sections should be taken depends entirely upon the form of the surface; where this is regular, a section at each station is sufficient. A cross section should be taken, not only at every point on the centre line where there is an angle in the profile, but also wherever an angle would be found in the profile of a line joining a series of slope stakes on either side, even though the profile of the centre line maybe quite regular at the corresponding point: the object being, not only to indicate the proper outlines of the earthwork, but to furnish the data necessary to calculate correctly the quantities of material removed. Rock work will generally require more frequent sections than earthwork. 236. Vertical Curves. The grades as given on the profile are right lines, which intersect each other with angles more or less abrupt. These angles require to be replaced by vertical curves, slightly changing the grade at and near the point of intersection. A vertical curve rarely need extend more than 200 feet each way from that point. Fig. 92. Let AB, SO, be two grades in profile, intersecting at station 1?, and let A and C be the adjacent stations. It is required to join the grades by a vertical curve extending from A to C. Suppose a chord drawn from A to C; the elevation of the middle point of the chord will be a mean of the elevations of grade at A and C; and one half of the difference between this 204 FIELD ENGINEERING. and the elevation of grade at B will be the middle ordinatc of the curve. Hence we have in which M the correction in grade for the point B. The correction for any other point is proportional to the square of its distance from A or C. Thus the correction at A -\- 25 is ^M ; at A -{- 50 it is \M ; at ^l -f- 75 it is ^M; and the same for corresponding points on the other side of B. The correc- tions in the case shown are subtractine, since M is negative. They are additive when M is positive, and the curve concave upward. These corrections are made at the time the cross sections are taken, and the corrected grades are entered in the field- book opposite the numbers of the respective stations. 237. Form of Field-book. A complete record of all cross-section work is kept in the cross-section book. On the left-hand page is recorded, in the first column, the numbers of the stations and other points where sections are taken ; in the second, the elevations of those points, copied in part from the location level-book, but verified or corrected at the time the section is taken ; in the third, the elevation of the grade for the same points; in the fourth, the width of base b ; in the fifth, the slope ratios, s ; and in the sixth, the surface ratio *' when uniform. The right-hand page has a central column, in which, and opposite the number of the station, is recorded the centre depth of -the section, marked -f- or , to indicate cut or fill, as the case may require. To the right of this are recorded the notes of that portion of the section which lies on the right of the centre line, as the line was run, and to the left, the notes of the left side. The distance from the centre to each point noted is recorded as the numerator of a fraction, and the cut or fill at the point as the denominator, prefixed by a -f- or as the case may require. The denominator for a grade point is zero. The numbers of the stations should increase up the page, as in a transit book, so that there may be no confusion as to the right and left side of the line. The several points being noted in order as they occur from the centre outwards, the notes far- CONSTRUCTION. 205 thest from the centre of the page usually appertain to the slope stakes ; but in case the cross profile is extended beyond the slope stake, the note of the latter should be surrounded by a circle to distinguish it. The following form is a specimen of a right-hand page, with the first column only of the left- hand page : Sta. Cross Sections. 83 + 60 82 + 38 + 27 + 19 81 80 22.9 16.5 10 5 4 o 10 20 +25.6 24 32 55.6 + 8.6 +14 +17.7 17.5 10 + 21.5 o + 9^4 o +20.8 10 +28.3 '42.6 +30.4 + 5.0 +10 14.2 +13.2 10 +14.7 6 +20.1 10 ' +21.7 31.6 + 2.8 21.7 + 5.4 10 + 8.5 10 +11.6 19.3 15 - 5.3 18 +14.4 25.6 7 + 2.8 - 4.7 + 3.8 10 - 9.8 25.9 -12.6 33.4 - 5.6 7 7 -11.2 7 -12 -17.6 -16.4 -17.6 -19.6 -19.1 -12.4 238. In case there is a liability to land-slips, the profiles of cross sections should be carried beyond the slope stakes, on the upper side of the cut, to any distance thought neces- sary to reach firm ground, and stakes driven for future refer- ence. When a number of consecutive cross profiles are to be considerably extended, it is well to first run, instrumentally, a line parallel to the centre line, and set stakes opposite the stations, taking their elevations. The intermediate surface of the sections may then be taken with cross-section rods if more convenient. See 37. . 239. In case of inaccessible ground, preventing a regular staking out, an indefinite profile of the section may generally be obtained, referred to the datum for elevation and to the centre line for position, which being plotted on cross- section paper, and the grade Hue and side slopes added, shows to scale where the slope stakes should be. 206 FIELD 240. Any isolated mass of rock or earth which oc- curs within the limits of the slope stakes, but not included in the regular notes, is separately .measured and noted, so that its contents may be computed and added to the sum of the same material found in the cross sections. 241. Borrow-pits. When the excavations will not suffice to complete the embankments, material may be taken from other localities, termed borrow-pits. These should be staked out by the engineer and their contents calculated, unless the contractor is to be paid for work by embankment measurements. A number of cross profiles are taken of the original surface, and (on the same lines) of the bottom of the pit after it is excavated, which furnish the depth of cutting at each required point. Borrow-pits should be regularly ex- cavated, so that they may not present an unsightly appear- ance when abandoned. Borrow-pits may be avoided by widening the cut uniformly at the time it is staked out, so that it may furnish sufficient material; provided the material is suitable, the embankment accessible, and the distance not too great. When the excavation is in excess, the surplus ma- terial should be uniformly distributed by widening the adja- cent embankments, if possible; otherwise it is deposited at' convenient places indicated by the engineer and is said to be wasted, 242. Shrinkage. In estimating the relative amounts of excavation and embankment required, allowance must be made for difference in the spaces occupied by the material before ex- cavation and after it is settled in embankment. The various earths will be more compact in embankment, rock less so. The difference in volume is called shrinkage in the one case, and increase in the other. Shrinkage in 1000 cu. yds. Material. Of excavation. Of settled erubkt. Sand and gravel .............. 80 C. Yds. 87 C. Yds. Clay ......................... 100 " 111 " Loam ........................ 120 " 136 " Wet soil ......... . ........... 150 " 200 " Increase in 1000 cu. yds. Rock, large fragments ......... 600 C. Yds. 375 C. Yds. " medium fragments ...... 700 4< 413 " " small " . 800 " 444 " CONSTRUCTION. 207 Thus, an excavation of sand and gravel measuring 1000 cubic yards will form only about 920 cubic yards of embankment; or an embankment of 1000 cubic yards will require 1087 cubic yards of sand or gravel measured in excavation to fill it ; but will 'require only 587 cubic yards of rock excavation, the rock being broken into medium-sized fragments; while 1000 cubic yards of the latter, measured in excavation, will form 1700 cubic yards of embankment. The lineal settlement of an earth embankment will be about in the ratio given above, therefore the contractor should be instructed in setting his poles to guide him as to the height of grade on an earth embankment, to add 10 per cent (average) to the fill marked on the stakes. In rock embankments this is not necessary. The engineer should see that all embank- ments are made full width at first, out to the slope stakes, and by measure at or above grade, so that the whole may settle in a compact mass. Additions to the width made subsequently are likely to slide off. 243. The cross-section notes should be traced in ink at the first opportunity to secure their permanence. An office copy should also be made to serve in case of loss or damage to the original. 244. Alteration of Line. Inasmuch as the centre line at grade is the base of reference for all measurements and cal- culations in earthwork, any change made in it after the work of grading has begun should be most carefully recorded and explained. The centre stakes of the old line should be left standing until after the new line is established, so that the per- pendicular offset from the old line to the new, at each station, may be measured, as also the distance that the new station may be in advance of, or behind the old one. The date of the change should be recorded. The original cross sections are extended any amount requisite, the distance out being still reckoned from the old centre, while a marginal note states the amount by which the centre has been shifted. The difference in length of the lines will make a long or short station at the point of closing. The exact length of such a station should be recorded, so that it may be observed in re- tracing the line at any time, and in calculating the quantity of 208 FIELD ENGINEERING. earthwork. The original transit notes of the altered line should be preserved, but marked as " abandoned," with a reference to the notes of the new line on another page. 245. Drains and Culverts. The engineer should ex-' amine the nature and extent of each depression in the profile with reference to the kind of opening required for the passage of water. For small springs, and for a limited surface of rain- fall, cement pipes, in sizes varying from 12 to 24 inches diame- ter, serve an excellent purpose as drains. These are easily laid down, and if properly bedded, with the earth tamped about them, are very permanent ; but their upper surface should be at least 2$ feet below grade. The embankment is protected at the upper end of the drain by a bit of vertical wall, enclosing the end of the pipe. If necessary, a paved gutter may lead to it. Where stone abounds, the bed of a dry ravine may be partly filled with loose stone, extending beyond the slopes a few feet, which will prevent the accumulation of water. When the flow of water is estimated to be too great for two lines of the largest cement pipe, or when the embankment is too shallow to admit them safely, a culvert is required. A pavement is laid one foot thick, protected by a curb of stone or wood 3 feet deep at each end, and wide enough to allow the Avails to be built upon it. It should have a uniform slope, usu- ally between the limits of 50 to 1 and 100 to 1 to ensure the ready flow of water. In firm soils the foundation pit is exca- vated one foot below the bed of the stream, but if mud is found this must be removed and the space filled with riprap, the up- per course of which is arranged to form the pavement at the proper level. In a V-shaped ravine, requiring too much ex- cavation at the sides, and where the fall is considerable, riprap may be used to advantage, the bed of the stream above the culvert being graded up by the same material to meet the pave- ment. In some cases a curtain, or cross wall, is necessary on the lower end to retain the riprap. Culverts should be laid out at right angles to the centre line whenever practicable, the bed of the stream being altered if necessary. The length of an open culvert is the entire, distance between slope stakes, the walls being parallel throughout, or the length may be taken somewhat less than this, and the walls CONSTRUCTION. 209 turned at right angles on the upper end, forming a facing to the foot of the slope. The walls are carried up to grade for the width of the road-bed, and are stepped down to suit the slopes. A course is afterwards added to retain the ballast. In box culverts the span varies from 2 to 5 feet, the height in the clear from 2 to 6 feet; the thickness of walls from 3 to 4 feet; the thickness of cover from 12 to 18 inches, and its length at least 2 feet greater than the span. The walls terminate in short head-walls built parallel to the centre line, the top course being a continuation of the cover. The length of a head- wall, measured on the outer face, is equal to the height of the culvert in the clear multiplied by the slope ratio of the embankment. The perpendicular distance from the centre line to the face of a head-wall is equal to one half the road-bed, plus the depth of the top of the wall below grade multiplied by the slope ratio, or b -{- sk. A coping is sometimes added. 24(>. Arch culverts are used when the span required is more than 5 feet, and the embankment too high to warrant carrying the walls up to grade as an open culvert. The span varies from 6 to 20 feet; the arch is a semicircle, the thickness varying from 10 or 12 inches to 18 or 20 inches. The height of abutments to the springing line varies from 2 to 10 feet, the thickness at the springing line from 3 to 5 feet, and at the base from 3 to 6 feet, the back of the abutment receiving the batter. The foundations are laid broader and deeper than in box cul- verts, each abutment having its own pit, carried to any depth found necessary. The half length of the culvert is \b -j- sk, in which k is the depth of the crown of the arch below grade. The abutments are carried up half way from the spring to the level of the crown of the arch, and thence sloped off toward the crown. The face walls are carried up to the crown, and coped. The wing walls stand at an angle of 30 C with the axis of the culvert, they receive a batter on the face, and are stepped (or sloped) down to suit the embankment. Their thickness, at the base, is the same as that of the abutment; at the outer end 3 feet. They stop about 3 feet short of the foot ot the slope. They need not be curved in plan. Any stone structure of dimensions greater than those given above, scarcely comes under the head of culverts, and should be made the subject of a special design by the engineer. 210 FIELD 247. Staking- out Foundation Pits. For box culverts. The engineer having decided upon the size of cul- vert required, makes a diagram of it in plan, on a page of his masonry book, recording all the dimensions, stating the sta tion and plus at which its centre is taken, the span and height of the opening, etc. He then sets the transit at the centre A, Fig. 93, measures the angle between the centre line and axis, Fio. 93. (making it 90 if practicable) ; on the axis he lays off the dis- tances to the ends of the culvert and drives stakes at B and G. Perpendicular to BC he lays off the half widths of the pit, set- ting stakes at D and E, and laying off DFaud EH = AB; and DG and El = AC. On IG produced he lays off CJ = OK, and perpendicular to this JM and KL, and finds the intersections and N. A stake is driven at each angle, and upon it is marked the cut required to reach the assumed level for the foundation. These cuts are recorded on the corresponding angles of the diagram. The pit is thus no larger than the plan of the proposed masonry, and the sides are vertical, which answers the purpose for shallow pits. For arch culverts. The pit for each abutment when shallow may be of the same dimensions as the lower founda- tion course . if more than five feet deep, it should be enlarged by an extra space of one foot all around. In Fig. 94 the inside CONSTRUCTION. 211 lines show the plan of the abutments at the neat-lines ; the outside lines represent the pits. Having prepared a plan of the structure suited to the locality, and made a diagram of the same in the masonry book, set the transit at A, and drive stakes at D, E, N and on the centre line. Then turning to the axis BC, lay off AC, and set stakes at O and /. With Or as a centre, and a radius tqual to 2DJS, describe on the ground >>c Fl B! I/i ^'' j" H: s<. y2 '' FIG. 94. an arc cutting El in X or (IX = DE . cot 30) may be calcu- lated; and on XG produced lay off Q K, and perpendicular to this, KL. From N lay off NP, parallel to AC, and measure PL as a check. Drive a stake at each angle, marked with the proper cutting, and record the same on the diagram. The locality may require the wings to be of different lengths and angles, of which the engineer will judge. Guard-plugs should be driven in line with the intended face of one or both abut- ments, so that the neat-lines can be readily given when re- quired. In case the material is not likely to stand vertically, the pit must be staked out with sloping sides, as described below. - : '-* ': ; For bridge abutments. A design for every impor- tant structure is usually prepared in the office after a survey of the site. The foundation pit is then laid out from dimen- sions furnished on a tracing, but a diagram of the pit should be made in the masonry book as usual. When the bridge is on a tan- gent, Fig. 95, set the transit at A on the centre line at its inter- section with the axis BC oi the abutment , at the level of the seat. 212 FIELD ENGINEERING. Deflect from the tangent the angle giving the direction of BC, and lay off AC, AB, setting plugs at B and C, and reference plugs (two on each side) on BC produced. After staking out the sides of the pit parallel to BC, set the transit at C, and deflect the angle for the wing, laying off CD, and driving stakes at the corners E and F. Two reference points are then set on the line CD produced. The other wing being FIG. 95. staked out in the same manner, the cut is found at each stake and marked and recorded. Cross sections are then taken near each corner, perpendicular to each side, and slope stakes (marked "slope") are driven where the slope runs out. Inter- mediate sections are taken when the unevenness of the ground makes- it necessary, and the lines joining the slope stakes are produced to intersect, and other stakes are driven at the inter- sections. The position of each stake is shown on the diagram, and the cut recorded. A slope of 1 to 1 is usually sufficient for pits. If the material will not stand at 1^ to 1, or if space cannot be spared for the slope, the sides may be carried down vertically, supported by sheet piling braced from within. The reference points should be so chosen that the points A, B and (7 may be found by intersection, on any course of the masonry, during the progress of construction. When the bridge is on a curve, the bridge-chord should be found and the abutments laid out from this. Fig. 96. The bridge-chord is a line AB, midway between the chord of the curve CD, joining the centres of the abutments, and a tan- gent to the curve at the middle point of the span. Hence CONSTRUCTION. 213 CA = DB = $MN t which may be laid off, and A and B are the true centres of the abutments, from which the foundations are staked out as before. The distance CE = DF to the points where the bridge-chord cuts the curve is 0.147(71). Should an abutment site on a curve be inaccessible, as when FIG. 96. under water, from any transit point P on the curve lay off. PX perpendicular to the tangent at M, observing that PX MQ A C = R (vers PM \ vers CM) and A X = PQ-%AB = R (sin PM - The point A may then be found by intersection, or by direct measurement with a steel tape or wire, driving a long stout stake to show the point above the water. Other points may then be approximately found, sufficient to begin operations. In case of a bridge of several spans, the piers are laid out in the same manner, from a centre point and axis. If on a curve, each span has its own bridge chord, but for convenience, the centre of a pier may be taken on the centre line during its con- struction, and the bridge-chord only found for the purpose of placing the bridge; the piers being long enough to allow of the shift. 214: FIELD To locate the centres of piers, a base line is re quired on one or both shores, and two transits are used to give the intersections by calculated angles. When practicable the spans should also be measured with a steel tape or wire. The bed of a pit for any sort of structure should receive the closest scrutiny of the engineer, it being his duty to judge whether the material will resist the load to be im- posed upon it. A pit may require to be excavated to a greater depth than first ordered, while sometimes a less depth will answer, as when solid rock is found. When a good material is reached, if any doubt exist as to its thickness, or as to the character of the underlying stratum, borings should be made or sounding rods driven down. Piles may be driven to gain the requisite firmness, and a layer of riprap, of beton, or of timber may be used to afford a uniform bearing. When satis- fied of the stability of the bed, the engineer finds the original centres, and gives points for the courses of masonry. A com- plete record is kept of the amount and kind of excavation, the materials used in foundation under the masonry, and of the size and thickness of each foundation course of masomy; the notes should be taken at the time the work is done, it being generally impossible to take measurements thereafter. 248. Cattle-guards are shallow pita placed at right angles across the road at the fence lines to prevent the passage of cattle. They are either entirely open, in which case they should be at least 4 feet deep, or they are covered in part with wooden rails laid a few inches apart. The open guard is preferred. It is built like an open culvert except that no pavement is required. The stringers carrying the rails over any opening should be no longer than the span plus the thick- ness of the walls. 249. Trestle Work. No wooden culverts should ever be used. If stone cannot be had at first, two trestle bents may be erected, leaving between them a space sufficient to contain the stone structure to be built when the material for it can be brought by rail. The bents may be backed by plank to retain the embankment, and the stringers are then notched down an inch on the caps to receive the pressure of the earth, and render the bents mutually sustaining. The sills are prevented from yielding to the pressure cf the earth by being sunk in COKSTBUCTIOK. 215 a trench, or by sheet piling. Should the span be too long, a central bent may be used, so as not to interfere with building the wall. Sometimes pile-bents may be used with greater ad- vantage, the piles being driven in rows of four each, and cap- ped to receive the stringers. In districts where suitable stone is entirely wanting, pile or trestle abutments and piers are used for the support of bridges, the piles or posts being arranged in groups and capped to receive the direct weight of the trusses. They should not sustain the embankment, but should be connected with it by a short trestle work. Trestle work is frequently used as a substitute for embank- ment, either to lessen the first cost, or to hasten the completion of the line, or for lack of suitable material with which to form an embankment. The cost of trestle work, however, is not less than that of an earth embankment formed from borrow pits, unless its height exceeds about 15 feet, depending on the relative prices of materials and labor. When not exceeding 30 feet in height, the bents, for single track, are usually composed of two posts, a cap and sill, each 12 X 12, and two batter posts, 10 X 12, inclined at th to 1, all framed together. Two lengths of 3-inch plank are spiked on diagonally on opposite sides of the bent as braces. The length of the caps should equal the width of the embankment; the posts should be 5 feet from centre to centre, ai^d the batter posts 2 feet from the posts at the cap. The sill should extend about two feet beyond the foot of the batter post. A masonry foundation for the bent is preferable, though pile foundations are not uncommon, and some temporary structures are placed directly on a firm soil, supported only by mudsills laid crosswise under the sill. The spans, or distance between bents, may vary from 12 to 16 feet. The stringers should consist of 4 pieces, 2 under each rail, bolted together, with packing blocks to separate them 2 or 3 inches. Over each bent and at the centre of each span a piece of thick plank about 4 feet long should be placed on edge between the two pair of beams to preserve the proper distance between them, while rods pass through the beams and strain them up to the ends of the plank, to increase the stability of the beams and prevent their buckling under a load. The string- ers should be able to carry safely the heaviest load without bracing against the posts. The bents, however, if high, must be braced against each other. The stringers should be con- 216 FIELD linuous, the two pieces breaking joints with each other at the bents, to which they are firmly bolted. They may rest directly on the caps, or corbels may intervene. The spans on a curve should be shorter than on a tangent. The ties should be notched down to fit the stringers closely, and guard rails, cither wood or iron, secured to them firmly. Unless the spans are very short, horizontal bracing should be employed consisting of 3-inch plank, extending from the centre ol each span to the ends of the caps, which are notched down to receive the plank. For trestles much higher than 30 feet the cluster l>ent is preferable, so termed because each vertical post is composed of a cluster of four pieces, 8x8, standing a little apart to allow the horizontal members to pass between them. The verticals are continuous, breaking joints, two and two, while the hori- zontals pass the posts and are bolted to them at the joints; the framing is accomplished entirely by packing blocks and bolts. The batter posts consist each of two pieces 8x8; the horizon- tals may be 4 X 10, and extend not only across the bent, but from one bent to another. Proper bracing is also used in every direction. When very high, a secondary pair of batter posts may be introduced in the lower part of the structure. The batter need not exceed tli to 1. In some instances two adjoin- ing bents are strongly braced together, forming a tower or pier, and the piers placed from 50 to 100 feet apart, the roadway being carried on trussed bridges. The cluster bent admits of any piece being removed and a new one inserted when neces- sary. Iron trestles are now adopted where a permanent struc- ture is desired. Owing to the expansion of the metal by heat, the bents cannot be continuously connected with each other as in a wooden trestle; hence the pier form is resorted to, having spans varying from 30 to 150 feet, covered by trussed bridges, and the whole structure is more properly styled a viaduct. 2oO. Tunnels. Tunnels are adopted in certain cases to avoid excessive excavations, steep grades, high summits, and circuitous routes. Their disadvantages are the increased time and cost of their construction compared with an open line, and their lack of light and fresh air when in use. It is desirable that they should be on a tangent throughout, both for the ad- mission of light and for convenience of alignment. Many , CONSTRUCTION". 217 tunnels, however, liave been built with a curve at one or both ends.* The location of a tunnel, other things being equal, should be such as to make not only the tunnel proper, but also its im- mediate approaches by open cut as short as possible ; and the latter should be selected so as not to be subject to overflow, nor liable to land slides. The material to be encountered may frequently be determined with tolerable accuracy by a study of the geological formation in the vicinity, or by actual borings. The most favorable material for tunnelling is a homogeneous self-supporting rock, devoid of springs, which does not disin- tegrate on exposure to the atmosphere. The worst materials are saturated earth and quicksands. The presence of water in any material increases the cost considerably. The alignment of a tunnel is made the subject of special survey, after the general location is decided, and this is more or less elaborate according to the length of tunnel. A perma- nent station is established at the highest point crossed by the tunnel tangent, from which, if possible, monuments are set in each direction at points beyond the ends of the tunnel. If there are two principal summits, stations on these will define the tangent, which may then be produced. The monuments established beyond the tunnel should be sufficiently distant to afford. a perfect backsight from the ends of the tunnel, where other monuments arc also established. The first quality of in- struments only should be used, and these perfectly adjusted, and the observations should be repeated many times until it is certain that all perceptible errors are eliminated. Since the line of collimation will be frequently inclined to the horizon at a considerable angle, it is important that it should revolve in a vertical plane; and to secure this, a sensitive bubble tube should be attached to the horizontal axis, at right angles to the telescope of the transit. The distance may be obtained by tri- angulation, though direct measurement is to be preferred. A steel tape is convenient and accurate, providing that allowance be made for variations due to temperature, from an assumed standard. The rods described in 43 may be used instead of * The Mont Cenis tunnel, requiring a curve at each end, was first opened on the tangent produced, giving a straight line through, and the curves were excavated subsequently. 218 FIELD ENGINEERING. plumb lines, the tape being held at right angles to them, and therefore horizontal. A plug should be driven for each rod to stand on, and a centre set to indicate the line and measure- ment. As the excavation of the tunnel proceeds, the centre line is given at short intervals by points either on the floor or roof. Overhead points are generally preferred, from which short plumb lines may be hung, constantly indicating the line, with little danger of being disturbed. When a new transit point is required in the tunnel, it should be established directty under an overhead point, which serves as a check upon its perma- nence, and as a backsight when needed. Shafts are sometimes opened to give access to several points of the tunnel at the same time, thus facilitating the work, though at an increased cost. They also serve for ventilation during the progress of the work, though they are worse than useless for this purpose afterward, except possibly in the case of a single shaft near the centre of the tunnel. Some of the longest tun- nels have been formed without shafts, while many shorter ones have had several, which have generally been closed after the tunnel was completed. Shafts are either vertical, inclined, or nearly horizontal ; in the latter case they are called adits. In- clined shafts should make an angle of at least 60 with the ver- tical. Vertical shafts may be either rectangular, round, or oval. Their dimensions vary, depending on their depth and the material encountered, between 8 and 25 feet. They are usually sunk on the centre line of the tunnel, though some- times at one side. When over the tunnel the alignment below is obtained directly from two plumb lines of fine wire suspended on opposite sides of the shaft from points very carefully deter- mined at the surface. The plummets are suspended in water to lessen their vibrations, and as soon as the transit can be set up at a sufficient distance to bring the lines into focus, it is shifted by trial into exact line with the mean of their oscilla- tions, the latter being very limited. Permanent points may then be set, but should be repeatedly verified. As soon as the workings from a shaft communicate with those from either end, or from another shaft, the alignment thus found is tested, and revised if necessary. These operations require the greatest nicety of observation and delicacy of manipulation to obtain satisfactory results. CONSTRUCTION. 219 From plumb lines in the central shaft of the Hoosac tunnel, the line was produced three tenths of a mile, and met the line produced 2.1 miles from the west end with an error in offset of five sixteenths of an inch. In the Mont Cenis tunnel the lines met from opposite ends with " no appreciable" error in alignment, while the error in measurement was about 45 feet in a total length of 7.6 miles. When a curve occurs in a tunnel it is usually near one end. The tunnel tangent is produced and established as before described, and a second tangent from some point on the curve outside the tunnel is produced to intersect it, the inter- section being precisely determined and the angle measured with many repetitions. The tangent distances are then calcu- lated, and the position of the tangent points corrected by precise measurements, and permanent monuments are estab- lished. As the tunnel advances, points may be set at short intervals on the curve in the usual manner; but at intervals of 100 feet the regular stations should be defined with finely centred monuments, using a 100-foot steel tape carefully sup- ported in a horizontal position. When it is necessary to use a subchord, its exact length should be calculated as shown in 107. When the curve has advanced so far as to render a new transit point necessary, this should be established at a full station. The subtangents from the two transit points should then be produced to intersect, and measured for equality with each other and with their calculated length. The distance from their intersection to the middle of the long chord should also be measured as a check on the deflections. When no perceptible errors remain, the curve may be produced as before until the P. T. is reached. It is evident that correct measure is indispensable to correct alignment on curves. Should obstacles on the surface necessitate triangiilation, more than ordinary care must be exercised, and as many checks introduced as possible. The triangles should be so arranged that all of the angles and most of the sides may' be measured. Test levels are carried over the surface with great care, each turning point being made a permanent bench, and its elevation determined with a probable error not exceeding 0.005 foot. Levels may be carried down a shaft on a series of bolts or spikes about 12 feet apart in the same v.ertical line, the distances being measured by the same level-rod as that 220 FIELD ENGINEERING. with which the benches are determined. The measures should be taken between two graduations of the rod, not using the end of the rod, w T hich may be slightly worn. Fine horizontal lines on the heads of the bolts may be used to mark the exact distances. After the shaft reaches the level of the tunnel, the depth may be measured more directly with a steel tape, the entire length of which has been corrected at the given tem- perature, by comparison with the same rod. If the grade of a tunnel is to be continuous, it should be assumed at something less than the maximum of the road, but not less than 0.10 per station, which is required for drainage. If a summit is to be made in the tunnel, the grade from the upper end should not exceed 0.10 per station. Grades are given in the tunnel from day to day, or as often as required by the progress of the work, the marks being made on the sides at some arbitrary distance above grade. Turning points should be taken on permanent benches? The least width of a tunnel in the clear should be, for single track about 15 feet, and for double track 26 feet. The least height in the clear above the tie should be 18.5 feet for single track, and 16.5 feet at the outside rails for double track, allowing for tie and ballast; the roof at the centre of the section should be at least 20 feet above subgrade, and with a full centred arch 22 or 23 feet for double track. The form of section depends somewhat on the material traversed. In perfectly solid rock a nearly rectangular section may be used, the roof being slightly rounded. In dry clay, and stratified rock, a flat arch may be used, and in other cases a full-centred arch. The latter form is rather to be preferred on account of the better ventilation afforded. The sides are made vertical, battered or curved, as necessity or taste may dictate. In wet and infirm soil an invert floor may be required, otherwise it is made level transversely. When a lining is required the original section must of course be made large enough to allow for the masonry, and the temporary timber supports behind it. Hard burned brick is usually adopted for arching, being durable and easily handled. In loose rock the arching may be from 13 to 26 inches thick, in wet and yielding soil a thickness of from 2G to 39 inches may be necessary. The walls may be from 2J to 6 feet thick. In forming a tunnel, a heading or gallery of smaller CONSTRUCTION. 221 cross section is first driven and afterwards enlarged to the full size required. In firm clay or loose rock which will tem- porarily support itself until the masonry can be put in, it is better to drive the heading along the floor (at subgrade) of the tunnel, the remaining material being then easily thrown down in sections as the arching is advanced. In solid rock, or wet earth, a top-heading (along the roof) is generally preferred. The dimensions of a heading driven by hand are usually 8 feet high by 8 or 10 feet wide, but in solid rock where drilling machinery is introduced, it is advantageous to make the head- ing as wide as the tunnel at once. By drilling holes into the face at points about five feet each side of the centre, and con- verging on the centre line at a depth of about ten feet, a tri- angular mass of rock may be blown out, and the space thus gained facilitates the blasting of the adjacent rock on either side. An advance of about 10 feet in each day of 24 working hours may thus be made, using nitroglycerine in some form as the explosive agent. Owing, however, to unavoidable delays from various causes, this rate of progress cannot always be maintained. At the Hoosac tunnel the greatest advance in one week was 50 feet; in one month 184 feet at one heading. At the Musconetcong tunnel a heading 8 X 22 feet in syenitic gneiss was advanced at the average rate of 137 feet per month for 6 months, the maximum being 144 feet the enlargement of the tunnel to full size going on at the same time, a few hundred feet behind. At the St. Gothard tunnel the north heading 2. 5 X 3 metres was advanced in mica gneiss, during the year 1875 at the average daily rate of 3.71 metres, with a maximum of about 4 metres, but the en- largement was not made. The south heading advanced at the rate of 2 metres a day, timbering being at times necessary. In ordinary clay a heading may be driven at from 75 to 180 ft. per month, according to circumstances, where timbering is put in. The enlargement, including timbering and masonry, may be advanced at from 20 to 60 ft. per month.' Small tun- nels for water conduits are driven through dry clay at the rate of 10 ft. per day, the masonry following at once without tim- bering. The compressed air used to drive the drilling machinery serves to supply ventilation also. When this is wanting or proves insufficient, exhaust fans are used. At Mont Cenis a FIELD ENGINEERING. horizontal bi'attice or partition was built in the tunnel, dividing it so as to secure a circulation of air. When foul gases are en- countered, ventilation becomes a serious question, and in one instance an important work was abandoned for this cause. Cross sections of the heading, and also of the tunnel en- largement, should be measured at intervals of about 20 feet, as soon as opened, to see that the sides, roof, and floor are taken out to the prescribed lines, at the same time that the latter are exceeded as little as possible. In solid rock, since some ma- terial outside of the true section will necessarily be thrown down, leaving an irregular outline, it is well to take two cross sections at the same point, one following the projections and the other the recesses of the rock, from which an average sec- tion may be estimated. A daily, or at least a weekly, record of operations should be kept in tabular form, and the progress indicated by a profile and cross sections drawn on a sufficiently large scale to show details. The drainage of a tunnel is best secured by a line of stoneware or cement pipe laid in a trench along each side, and covered with ballast or other loose material. The entire floor is thus made available for the use of the trackmen. When an invert is used, the drain is placed in the centre between tracks. If the amount of water is large, drain pipe may be laid behind the walls, and the back of the arch may be covered with as- phaltum, or coal tar, to prevent a constant dripping on the track. Retracing the Line. As the grading pro- gresses, in either excavation or embankment, the principal transit points are established on the road-bed from the points of reference, and the centre line is retraced, setting stakes at every 50 feet. Transit points on grade should be fixed upon stout, durable posts firmly set in the ground, and standing high enough to be easily reached after the ballast is laid. To recover the old line, any discrepancies in measurement must be left between the transit points where they occur, and not carried forward. In retracing a curve, if the transit is placed at the forward point, allowing the chain to ad- vance toward it, slight differences in measurement will not affect the position of the curve. If any short or long sta- CONSTRUCTION. 223 * tions have been introduced on the location, their position on the line must not be changed in retracing. The chain may be adjusted so that its measures will agree with the recorded distances between transit points. Offsets are made right and left from the new stakes to see that the road-bed is of the full width at all points. The levels are also .carried over the grade, and any remaining cut or fill found necessary is marked on the back of the stakes, due allowance being made for the probable settlement of embankments. 252. As the work approaches completion the contractor goes over the line dressing it to grade and opening .the side ditches if this has not been previously done. Drain-tile should be laid at the bottom of these ditches and lightly covered with earth, particularly if the cut be wet. These not only prevent the water from reaching the ballast, but by keeping the foot of the slope comparatively dry pre- vent the earth from sliding down and filling up the cut. There is also a marked economy in their use, as the cost is trifling, and all further excavation of mud and water from the cut is generally obviated. Should any springs appear in the slope a branch line of smaller tile may be laid to meet it. If the slope is liable to be overflowed from the surface above, an open ditch should be dug a few feet beyond the slope stakes, leading the surface water to discharge elsewhere. 253. The road-bed being prepared, ballast stakes are driven at every half station, giving the width of the ballast at its base, while the tops of the stakes indicate the proper level of its upper surface, which is the under side of the tie. These stakes should be set so as to give the proper elevation to the outer rails on curves when the ballast is graded to them. The ballast should be about one foot deep before the ties are laid. Broken stone or a mixture of coarse and fine gravel is the best material, affording elasticity and good drainage. The side slopes of the ballast are made 1 to 1 ; its width at the under side of the tie should be one foot greater than the length of the tie. 254. Track-laying. After the ballast has been laid and graded, the centre line is retraced upon it ; short stakes 224 FIELD ENGINEERING. i are used, each of which is centred. On long tangents, one stake in every 200 feet is sufficient, on ordinary curves one in every 50 feet, and on very sharp curves one In every 25 feet. The ties are then spaced evenly according to the number prescribed per mile, or per rail length ; but a tie should not be allowed to cover a transit point. Ties for the standard gauge are 8 or 9 feet long; they should be sawed off square at the ends and in uniform lengths for appearance sake when laid. Specifications usually call for ties having a thickness of 6 inches and a width of from 7 to 10 inches. The ends of the ties are aligned on one side of the road, though if cut into uniform lengths both ends will be equally well aligned. The rails are then laid on, and spiked to gauge. The first spikes are driven in the ties near a centre stake, the centre mark of the gauge bar being kept over the centre on the stake. Upon curves the rails must be sprung to the proper arc before they are laid ( 199). All the ties required in a given distance should be laid before the rails are brought upon them. The practice of laying only joint and middle ties at first subjects the rails to the danger of bending from passing loads. Owing to the expansion of the rails by heat, a space must be left at the rail-joints. The highest temperature of a rail in the summer sun is about 130 Fah. The expansion of iron or steel per 100 is .0007 per foot; or for a 30-foot rail .021 foot or .252 inch. Therefore when 30-foot rails are laid at a temperature near the freezing point, or 100 below the maximum, the space allowed must be at least a quarter of an inch. At 80 Fah. or 50 below the maximum, it need be only half as much. The space required is also proportional to the length of rail used. The exact space should be given, as less would result in the rails being forced up by expansion, while more than necessary space gives a rough road, and hastens the destruction of the rail. Wherever siding's are required, the necessary frogs and long switch- ties should be provided in advance, so that they may be put in place at the time of laying the main track. For every road crossing at grade, heavy oak plank should be pro- vided, and laid upon the ties as soon as the rails are spiked, so that the highway travel may not be impeded. CALCULATION OF EARTHWORK. 225 CHAPTER X. CALCULATION OF EARTHWORK. 254. The first step toward finding the cubical content of an excavation is to divide it into a number of prismoids by several cross sections. A prismoid is a solid having plane parallel bases or ends, and bounded on the sides either by planes, or by such surfaces as may be generated by a right line moving continuously along the edges of the bases as directrices. The positions of the cross sections must be so selected that the solid included between any two consecutive sections may be a prismoid as nearly as possible. Upon a tangent the roadbed and side slopes are planes, so that the prismoidal character of a given solid depends upon the shape of the natu- ral surface. When the natural surface is a plane, the sections are taken only at the regular stations, 100 feet apart; when it is curved, warped, irregular, or broken, the sections must be more numerous, so that the surface limited by any two shall be composed substantially of right-lined elements extending from one section to the other. If two end sections of a prismoid are somewhat similar, we infer that the corresponding points are connected by right- lined elements, forming in each case the axis of a ridge or of a hollow. If one section has less breaks than the next, some of these ridges or hollows must vanish; and in order that the solid may be a prismoid, they must vanish in the section of least breaks ; therefore a cross section must be taken on the ground through the point where each ridge or hollow vanishes, and the distance of that point from the centre line noted, so that it may be coupled with the proper point in the next section for exact calculation of content. When ridges or hollows run diagonally across the line of roa'd, cross sections must be taken where they are intersected not only by the centre line but also by the side slopes; that is, sections must be taken so that a side stake may stand on top of 226 FIELD ENGINEERING. each ridge and at bottom of each hollow. In case the centre line intersects at right angles a retaining wall or other vertical surface, two cross sections are required at the same point, one at top and the other at base of wall, in order to furnish the data necessary to calculate the content each way from the ver- tical surface. (See Art. 235.) Every thorough cut terminates in either side -hill cutting, a pyramid, or a wedge; the latter happens only when the con- tour of the natural surface is at right angles to the line of road. Sections should always be taken through the points where the edges of the road bed meet the surface, as these are the points of separation between thorough and side hill work. Such sec- tions also serve to define terminal pyramids when they occur as is illustrated by Fig. 97. In side-hill work the foregoing Fio. 97. rules apply as well, but sections will generally be more numer- ous than in thorough cuts, The same rules apply also to em- bankment, but as grading is preferably paid for in excavation, the same precision in determining the quantities in embank- ment is not usually necessary. CALCULATION OF EARTHWORK. 227 255. Formulae for Sectional Areas. Let b = base of section or width of road-bed, horizontal s = slope ratio = : i vertical " d = depth at centre stake. " h, k = depths at side stakes. " m, n horizontal distances from centre to side stakes. For ground level transversely, the section is a parallelogram, and the area is evidently (342) or directly from the field notes, A = $(b + m + ri)d (343) For ground of uniform transverse slope between slope stakes, Fig. 98, the section consists of the parallelogram ABOE and the triangle EOD. Hence A = &AB+ EO)EN+ EO(DH-EN) A - \(AB . EN+ EO . DH) or A = i[ also (344) A = $[bk+7i(b-{-28k-)]} From which also A tyh 4- mk ) and [ (345) These formulae are independent of the centre depth. They are convenient for calculating the area of a plotted section 228 FIELD ENGINEERING. having an irregular surface after the surface line has been averaged by stretching a silk thread over it. The points where the thread intersects the slope lines determine the values of h, k, m, and n respectively. When the ground has uniform slopes transversely from the centre to the side stakes: Fig. 99 : If in the diagram we draw D !r FIG. 99. EG and DG, the section will be divided into four triangles, two having the common base CG d and respective heights GN= m and GH= n, and two having the equal bases AG GB = $ and the respective heights EN = h and DH = A;. Hence we have for the area of section A = (346) Othencise, if the slope lines are produced to meet below grade at P, then GP = --. The area of CEPD is 8 ,i/j NII= %(d + |J (m + 7i). The area of ABP is J. G GP -7- Hence we have for the area of the section X (347) Both these formulas are convenient, and as the values of the several letters can be substituted directly from the field notes, it is unnecessary to plot such sections. . When the. surface of the ground is irregular, verticals are con- ceived to be drawn to the grade line through the slope stakes, CALCULATION OF EARTHWORK. 229 and through each break in the surface line, giving a number of trapezoids, the areas of which are severally calculated, and from their sum is subtracted the area of the two triangles EN A and DUE. The remainder is the area of section required. This calculation maybe made directly from the data furnished by the field notes without plotting; but if the ground has a number of small breaks, it is generally better to plot the sec- tions and stretch an averaging line over them, finding the areas by eq. (345). Or two averaging lines may be employed extend- ing from the centre stake, each way, when the area may be cal- culated by cq. (34G) or (347). 256. Prismoidal Formulae for Solid Contents. The content of a prismoid may be exactly calculated by means of the Prismoidal Formula, which is 8 = [A + **+*} (348) 8 = cubic yards, I = length in feet, A, A' the areas at the two parallel ends, and M the area of a section midway be- tween the ends. This area is not a mean of the other two, but the linear dimensions of the mid-section are means of the cor- responding dimensions severally of the end sections; from which therefore the area of the mid -section may be computed. The labor of calculating the middle area may be avoided in many instances by substituting in the prismoidal formula, eq. (348), for A, A, and M, their values as given in eq. (342) for ground level transversely. .. t d 4- d! . d*-\-2dd'-{-d'* A=bd-}-sd* A' = bd' -f- sd'* M=b~^ -- h*-- 1 - 7 8 = in which S is expressed in terms of the end dimensions. 257. Tables of cubic yards may be constructed upon this formula which are very convenient in practice. The constant values in any one table are I which is taken at 100, and b and s which are given values corresponding to the road-bed and slope ratio. The variables are d and d'. The columns in the table 230 FIELD ENGINEERING. will be headed by the successive values of d' , while each hori- zontal line will be headed by a value of d. For any one column therefore d' is constant, and the only variable is d. Assuming any value for d', the values of 8 in that column may be computed, letting d take a series of values differing by unity from zero upwards, and the corresponding values of S will be placed in the column d' opposite the several values of d. But instead of solving the eq. (349) for each value of S re- quired, the process of rilling the table may be much abbre- viated by observing that since the equation is of the second degree with respect to the variable d, the second difference of the values of S will be a constant and equal to twice the co- efficient of $ t or d" = 8 Also the first term in the series D X <* 7 of first differences of 8 in the column d' (i.e. between d = and d = 1) is expressed by the sum of the coefficients of d* and d' t or The first value of S in any column d ' is found by solving eq. (349) after making d = 0; or, Starting with these values we may fill any column d ' simply by successive additions. The values of d' for the several columns should also differ by unity. The final value of S in each column should be calculated by formula as a check; or since all the final quantities in the same line d of the table form a series of which the second difference is d", if on taking their differences this result is obtained, the quantities are proved to be correct. Example. Given a base of 18 feet and slopes 1-J to 1, to fill the column of d' = 6 in a table of cubic yards for level cross sections. Here I = 100, b = 18, s = f, d' 6. Hence d' = 3.7037+, - |w)] or e = #b -\- n w) Ae b -4- n w wk and C = -R~- ^ XTT . ( 3o9 > The correction c will be plus or minus as before explained. This formula applies to all side-hill triangular sections, whether there be cut or fill at the centre stake. Example 1. Thorough cut; base 20; slopes H 1- I = 100; 8 curve, left; R = 716.78 4' +-- + - _ 2 " " h 20 Then K = i X 58 X 12 -f i X 20 X 32 = 508 J5T=i X 16 X 12 + i X 20 X 4 = 116.\ 4 = 624 K- H 392 (4 + c) 637.49 JT ; = | x 40 X 8 + i X 20 X 20 = 260 H' = i X 13 X 8 + i X 20 X 2 = 62 . . A' = 322 K-H= 198 J3 + 40_ _ ~ 3 X 716.78 l = 326.87 From which we obtain $ = 1758 cub. yds. Ans. Without correction we have 1726 " " Showing a difference of 32 " " Had the curve been to the right with same notes, c would have been minus, and 8 would = 1694. CALCULATION OF EARTHWORK. 243 Example 2. Side-hill cut; base 20; slopes li : 1 I = 60; 10 curve, right; E = 573.69 6 40 _0__L_2__, 37 0.8^0.0"^ "18 A = i X 16 X20 = 160 (J. - e) = 156.42 = $ x 8 X 18 = 72 (A! -c)= 69.95 Hence 8 = 248 cub. yds. Without correction S would = 255 " " Difference 7 269. Haul. The cost of removing excavated material, when the distance does not exceed a certain specified limit, is included in the price per cubic yard of the material as meas- ured in the cutting. But when the material must be carried beyond this limit, the extra distance is paid for at a stipulated price per cubic yard, per 100 feet. The extra distance is known by the name of haul, and is to be computed by the engineer with respect to so much of the material as is affected by it. The contractor is entitled to the benefit of all short hauls (less than the specified limit), and material so moved should not be averaged against that which is carried beyond the limit. Therefore, in all cuts, the material of which is all deposited within the limiting distance, no calculation of liaul is to be made. On the other hand, the company is entitled, in cases of long haul, to free transportation for that portion of the cutting, no one yard of which is carried beyond the specified limit. There- fore, this portion is first to be determined in respect to its ex- tent; and the number of cubic yards contained in it is to be de- 244 FIELD ENGINEERING. ducted from the total content of the cutting, before estimating the haul upon the remainder. Find on the profile of the line two points, one in excavation, and the other in embankment, such, that while the distance between them equals the specified limit, the included quantities of excavation and embankment shall just balance. These points are easily found by trial, with the .aid of the cross sections and calculated quantities, and be- come the starting points from which the haul of the remainder of the material is to be estimated. FIG. 106. Pig. 106 represents a cut and fill in profile. The distance AB is the limit of free haul. The materials taken from AO just make the fill OB and without charge for haul; but the haul of every cubic yard taken from AC, and carried to the fill BD, is subject to charge for the distance it is carried, less AB. It would be impossible to find the distance that each separate yard is carried, but we know from mechanics that the average dis- tance for the entire number of yards is the distance between the centres of gravity of the cut AC, and of the fill BD which is made from it. If, therefore, X and T represent the centres of gravity, the actual average haul is the sum of the distances (AX-\-BY), and this (expressed in stations) multiplied by the number of cubic yards in the cut AC, gives the product to which the price for haul applies. But the product of AX by the number of cubic yards in AC is equal to the sum of the products obtained by multiplying the contents of each prismoid in AC by the distance of its own centre of gravity from A. The distance of the centre of gravity of a prismoid from its mid-section is expressed by the formula _ I* (A -A) 12X27S If we replace 8 by its approximate value, J , which (861 > will produce no important error in this case, we have A -A' CALCULATION OF EARTHWORK. 245 in which A should always represent the more remote end area from the starting point A, fig. 106. Hence, x may be -j- or , and it must be applied, with its proper sign, to the distance of the mid-section from the starting point A, before multiplying by the contents 8. Each partial product is thus obtained. By a similar process with respect to the prismoids composing the mass BD, and using the point B as the starting point, we obtain finally a sum of the products representing this portion of the haul. If a cut is divided, and parts are carried in opposite direc- tions, the calculation of each part terminates at the dividing line. If a portion of the material in AC is wasted, it must be deducted, and the haul calculated only on the remainder. The specified limit is sometimes made as low as 100 feet, sometimes as high as 1000 feet. A limit of about 300 feet, how- ever is usually most convenient, as it includes the wheelbarrow work, and a large part of the carting, while it protects the con- tractor on such long hauls as may occur. 27O. The Final Estimate is a complete statement in detail, of the amount of work done and materials provided, in the construction of the road, and is the basis of final settlement between the company and contractor. Its preparation should be begun as soon as possible after the work is in progress, and should be continued, as fast as the necessary data are accumu- lated, while the circumstances are still fresh in mind, and when any omissions in the field notes may be readily supplied. The content of each prismoid, the classification of its material, and the length of haul to which it is subject, should be matters of special record in a book provided for that purpose. These re- sults having been carefully computed by exact methods form a standard of comparison for those approximate results which must be had from time to time during the progress of the work, and furnish a limit to the amounts of the monthly estimates. The same remark applies to all other items of labor and mate- rial. The notes and record of the final estimate should be par- ticularly full and exact in respect to all such items as will be inaccessible to measurement at the completion of the work, such as foundation pits, foundation courses of masonry, cul- verts, and works under water. 246 FIELD 271. Monthly Estimates. On or before the last day of every month during the progress of construction, measure- ments are taken to determine the total amount of work done and material provided up to that date. The estimates based on these measurements are called Monthly Estimates. It is fre- quently necessary to take measurements for both monthly and final estimates at other times than the end of the month, as in the case of foundations which are not long accessible. With respect to each piece of work satisfactorily completed, the monthly estimate should be exact, and identical in amount with the final estimate. With respect, however, to items of work in progress at the time of measurement, the monthly estimate is only approximate, yet should be as precise as the nature of the case will allow; and the quantities stated should not be in excess of fair proportion of the total quantities given on the final estimate for the same piece of work. A special field book is devoted to monthly estimate notes. Each page should be dated with the day on which the notes upon it were taken. The notes consist of measurements of all sorts, principally of cross sections partially excavated. These sections should be at the same points on the line as the original sections, so that comparisons may be made. Where- ever the excavation is finished to grade, it is only necessary to write " completed " opposite such stations, and the quantities may be taken from the final estimate or computed from the original notes. It is frequently necessary to retrace portions of the centre line in taking estimate notes, so that all the field instruments are required, but a party of three or four men is usually sufficient. If the contractor has provided materials, such as stone, lum- ber, etc., which are not as yet put into any structure when the estimate is taken, these should be measured and entered under the head of temporary allowance, an arbitrary price be- ing used somewhat below the actual value of the material as delivered. Such allowances should never be copied from one month's estimate to the next, but made anew on such material as may be found that seems to require it. But all completed items of contract work, and of extra work when ordered by the engineer, are necessarily copied from one monthly esti- mate to the next during the continuance of the contract. A blank form is used by the resident engineer in report TOPOGRAPHICAL SKETCHING. 247 ing monthly estimates, on which a column is provided for each class of material and work required by the contract, while the several lines, headed by the numbers of the proper stations, are devoted to the different cuttings, structures, etc., in consecu- tive order as they occur on the line of road. The estimates are made out and reported separately for the several sections into which the line of road is divided for letting. These reports are reviewed by the division engineer, and the footings copied upon another blank, which is the monthly estimate proper; the prices are attached to the items, and the amounts extended and summed up. This sum indicates ap- proximately the total amount earned by the contractor up to date, from which is deducted a certain percentage (usually 15 per cent.), which is retained by the company until the comple- tion of the contract. From the remainder is deducted the amount of previous payments, which leaves the amount due the contractor on the present estimate. A blank form of re- ceipt is appended, to be signed by the contractor. CHAPTER XI. TOPOGRAPHICAL SKETCHING. 272. Topographical sketches taken on preliminary surveys are usually of great value in projecting a line for location; they should be made therefore as accurate and complete as possible. In too many instances sketches are presented having a picturesque appearance, but conveying little information, if not tending to mislead the map-maker. The aim of the topog- rapher should be to record the topographical features either side of the line with as much precision as those directly upon the line, without taking actual measurements, except in rare instances. The eye and the judgment must be usually depended on for distances and dimensions. The sketch of a tract ex- tending to 400 feet each side of the line ought to be accurate enough to warrant its being copied literally upon the map. If a much wider range is required it may be advisable to use the plane-table; but an approximation to plane-table methods may be employed in ordinary sketching. 248 FIELD ENGINEERING. 273. As artificial features are the most readily de- fined and located these should first receive attention in making a sketch. When recorded they form a skeleton upon which the natural features can be drawn with more precision than if the order were reversed. The point where each fence crosses the line and the angle between the two may be sketched exact- ly. The distance along the fence to any object may be esti- mated, and checked (in case of an oblique angle) by observing where a line from the object perpendicular to the centre line would intersect the latter. The book may be rested on any support, the centre-line of the page coinciding with the line of survey, and the direction of objects defined by a small ruler laid on the page. This operation being repeated from another point gives intersections which locate the several objects on the sketch. If the bearings are taken they may be plotted on the page as well as recorded, giving the same results. The eye may be trained to estimate distances correctly by observ- ing the appearance of objects along the measured line, the dis- tances to which are therefore known. 274. After the artificial objectslhe more distinct natural features are to be sketched, as streams, shores, margins of swamps, forests, etc., ravines, ridges, and bluffs, taking care that all these outlines intersect the features of the sketch already delineated at the proper points. The correct repre- sentation of contours is the most difficult part of sketching, since these lines are quite imaginary, yet for railroad maps they are usually as important as any others. It is desirable to know not only the locality of a hill or slope, but also its shape, steepness, and height. This information is best given by con- tour lines. A contour is the intersection of the surface of the ground by an imaginary level surface. When the surface is real, like that of a lake, the intersection is called a shore. If the water should rise a certain height a new shore would be defined, and rising double that height still another shore would result, each of which, on the subsidence of the water, would be a contour. A practiced eye is able to follow on the ground the course of a contour with all its windings; but in sketching them due allowance must be made for the fore- shortening effect of distance. All contours on the same sketch should have the same vertical interval, so that by counting TOPOGRAPHICAL SKETCHING. 249 them the height of the hill may be known. The spaces on the sketch between contours vary as the cotangent of the slope angle, so that the width of the spaces indicates the degree of steepness. The contours nearest the topographer should gene- rally be sketched first, although if there be a shore that is apt to be the best guide to the shape of the slopes. If the height of the hill is known and the upper contour located, the other contours can be spaced between with less difficulty, the proper number being ascertained by dividing the height by the assumed verti- cal interval. A special line of levels up an inclined ravine or sloping ridge to fix the contour points is often of the greatest service in obtaining correct results. Other random lines are sometimes run to locate the contours more definitely. These should be made to cross several contours rather than to trace a single one. Old preliminary lines which have proved useless in themselves often furnish by their profiles valuable informa- tion in respect to contours. The use of hatchings should be avoided in the sketch-book, except to represent precipitous banks, or slight terraces, which would not be sufficiently defined by the contour system. Hatchings freely used consume too much time, and fail to give an accurate idea of either slope or height, while they obscure the page for the representation of other objects. 275. The centre line on the page is straight, and for sketching purposes the surveyed line on the ground is assumed to be so also. Slight deflections in the course of a preliminary line may be ignored in the sketch ; but if a large angle occurs it is better to terminate the sketch with the course, and begin again, leaving a few blank lines between the two .sketches. On a located line with curves, the sketch is continuous. The curved line in the field is represented by the straight line on the page, and the radial lines through the stations are repre- sented by the parallel lines ruled across the page. All objects are sketched at the proper offset distance by scale, from the centre line; but longitudinally the sketch is necessarily dimin- ished outside of the curve, and magnified inside of the curve, Consequently topographical lines which are straight in fact ap- pear curved in the sketch, concave to the centre line if inside the curve, and convex if outside of it. Such features are cor- rectly sketched by means of offsets estimated or measured 250 FIELD from each station of the curve on the radial lines. This kind of distortion creates no confusion if properly done, for in mak- ing the map, after drawing the curve and the radial lines, the same offsets will give the correct positions of the objects delin- eated. This method is preferable to drawing a curved line on the page to represent the centre line, as it is difficult to draw it correctly; it will cross the ruled lines obliquely, rendering them of no service for offsets or scale, and moreover is likely to run off the page altogether. CHAPTER XII. ADJUSTMENT OF INSTRUMENTS. Every adjustment consists of two processes: first the test, and second the correction. Inasmuch as the amount of correction is made by estimation, the test must always be repeated until no further lack of adjustment is observable. 276. THE TRANSIT. Tlie level tubes should be parallel to the vernier plate. Test : Place the tubes in position over the levelling screws, and turn the latter till the hubbies are centred; revolve the plate 180. The bubbles should remain centred; if they have retreated Correction : Bring them half way back to the centre by turning the adjusting screws which attach the tubes to the plate. The line of collimation should be perpendi- cular to the horizontal axis. Test: Clamp the limb, and by the tangent screws bring the intersection of the cross-hairs to cover a well-defined point about on a level with the telescope ; plunge the telescope to look in the opposite direction, and note any point about on a level with the telescope and about equidistant with the first point, which the intersection of the cross-hairs now happens to cover. Now unclamp the limb and turn through 180, and repeat the above operation, using the same first point as before. ADJUSTMENT OF INSTRUMENTS. 251 The third point obtained should be identical' with the second; if not Correction : Move the vertical cross-hair over one fourth of the apparent distance from the third to the second point, by turning the adjusting screws at the side of the telescope. The horizontal axis should be parallel to the vernier plate. Test : After completing the above adjustments level the limb, clamp it, and bring the intersection of the cross-hairs to cover some high point so that the telescope may be elevated to a large angle; depress the telescope and note some point on the ground now covered by the intersection of the cross-hairs. Now unclamp the limb, turn it through 180, and repeat the above operation, using the same high point as before. The third point found should be identical with the second; if not Correction : Kaise the end of the axis opposite the second point (or lower the other end) by a small amount, by turning the adjusting screws in the standard. The amount of motion required is only determined by repeated trials until the test is satisfied. The intersection of the cross-hairs should appear in the centre of the field of view. Test : Bring the cross-hairs into focus and direct the tele- scope toward the sky, or hold a sheet of blank paper in front of it. If the intersection appear eccentric Correction : Turn the screws (by pairs) which support the end of the eyepiece until the desired result is obtained. If there be a level on the telescope it should be parallel to the line of collimatioii. Drive two stakes equidistant from the instrument in exactly opposite directions, and having perfected the previous adjust- ments, level the plate carefully, clamp the telescope in about a horizontal position, and observe a rod placed on each stake. Have the stakes driven by trial until the rod reads alike on both. The heads of the stakes are then on a level. Re- move the instrument beyond one stake, and set it up in line with the two, level the plate, and elevate or depress the telescope to a position which will again give equal readings on the stakes. The line of collimation is now level 252 FIELD ENGINEERING. Test : "While in this position the bubble of the attached level should stand centred; if not Correction : Bring the bubble to the centre by turning the nuts at one end of the tube, while the cross-hair continues to give equal readings. 277. THE Y LEVEL. The line of collimation should coincide with the axis of the telescope. Test : Clamp the spindle, and bring the intersection of the cross-hairs to cover a well-defined point by the tangent and levelling screws; revolve the telescope half over in the Ys, so that the level tube is on top. The intersection of the cross- hairs should still cover the point. If either hair has departed Correction : bring it half way back by means of the pair of adjusting screws at the extremities of the other hair. The attached .level should be parallel to the axis of the telescope. Test : Bring the telescope over one pair of levelling screws, clamp the spindle, open the clips, and bring the bubble to the centre. Then gently remove the telescope from the Ys, and replace it end for end. If the Ys have not been disturbed, the bubble should return to the centre. If it does not Correction : bring the bubble half way back by turning the nuts at one end of the tube. But as now the level tube and telescope may only lie in parallel planes, and yet not be parallel to each other Test: bring the bubble to the centre as before, and turn the telescope on its axis so as to bring the level tube out to one side. The bubble should remain centred. If it has departed Correction : bring it back to the centre by the adjusting screws at one end. The axis of the telescope should be at rig-ht angles to the spindle. Test : Having completed the above adjustments (and not before), fasten down the clips, unclamp the spindle, and bring the bubble to the centre over each pair of levelling screws in succession, then swing the telescope end for end on the spin- dle. The bubble should settle at the centre. If it do not- Correction : bring it half way back by the large nuts at one end of the bar. EXPLANATION OF TABLES. 253 278. THE THEODOLITE, This instrument being a combination of Transit and Level, its several adjustments are to be made according to the rules already given for those instruments. CHAPTER XIII. EXPLANATION OF TABLES. TABLE I. Contains concise statements of such geometrical truths as are applicable to the various discussions in this volume. References are given to Dayies' Geometry, in which the demon- strations of the propositions majr be found. TABLE II. Contains all the formulae necessary to the solu- tion of any plane triangle ; also, a select list of miscellaneous formulas. A few formulae with respect to the versed sine and external secant are new. TABLE III. Contains a complete list of formulae expressing the relations between the radius, tangent, chord, versed sine, external secant, and central angle of a railway curve; also, the relations between the radius, degree of curve, length of curve, and central angle. The notation is identical with that used elsewhere in the book. TABLE IV. Contains the radius, and its logarithm, for every degree of curve to single minutes up to 10 degrees, and thence by larger intervals up to 50 degrees. With the radius is given also the perpendicular off-set, t, from the tangent to a point on the curve at the end of the first 100-foot chord from the tan- gent-point, and the middle ordinate, m, of a 100-foot chord. See eqs. (16, 34, 37, 40, and 305). TABLE V. Contains the corrections to be added to the tan- gents and externals of any railroad curve, as obtained by refe- rence to Table VI., according to the degree of the given curve (found at head of columns), and its central angle, (found in the 254: FIELD ENGINEERING. first column.) If the given degree of curve, or central angle, does not appear in the table, the exact value of the correction may be easily obtained by interpolation. TABLE VI. Contains the exact values of the tangents, T, and externals, E, to a 1 degree curve, for every 10 minutes of central angle, from 1 to 120 50'. Approximate values of the tangent and external to any other degree of curve may be had by simply dividing the tabular values opposite the given cen- tral angle by the given degree of curve, expressed in degrees. These approximations may be made exact by adding the proper corrections taken from Table V. See eqs. (21) and (24). TABLE VII. Contains the value of Long Chords of from 2 to 12 stations, for every 10 mmutes of degree of curve from to 15, and of a less number of stations for degrees of curve be- tween 15 and 30. As the chord of one station is always 100 feet, the column of the first station gives instead the length of arc subtended by the chord of 100 feet. See 121, 122, 123, 124, 125. TABLE VIII. Contains the values of; Middle Ordinates to long chords of from 2 to 12 stations, for every 10 minutes of degree of curve from to 10, and of from 2 to 6 stations for every curve from 10 to 20, at 10-minute intervals. The table may be used, not only to fix the middle point of an arc, but also, in conjunction with the table of long chords, to locate in- termediate stations. See 121, 122, 123, 124, 125. TABLE IX. Contains the chords of a series of angles vary- ing by half degrees up to 30 for radii varying by 100 feet up to 1000 feet. It shows, therefore, the linear opening between the extremities of two equal lines at any given number of hun- dred feet from their intersection, when the angle does not ex- ceed 30. For any distance exceeding 1000 we have only to add to the value found in that column, the value found in the column headed by the excess of distance over 1000 feet. Con- versely, the table gives the angular deflection required between two equal lines, in order that at a given distance from the point of intersection they may be separated a given amount. EXPLANATION OF TABLES. 255 - TAULE X. 1. Contains values of the ratio u , accord - A ing to the notation of 147 for finding the angle i (Fig. 34) between the radius PO of the curve at any point P, and the tangent PK to the valvoid arc PX by the simple formula eq. (80)=?A. The table embraces lengths of curve from 300 to 2000 feet, and central angles from 10 to 120. When = 60 u = $, and for hasty approximation this 1000 value of u may be assumed in any case without consulting the table. T 2. Contains values of the ratio = - for finding the radius of the valvoid arc at the point P (Fig. 35) in terms of the length of curve L = AP by the simple formula, eq. (82), r = vL. 3. Contains values of the length I, of a valvoid arc limited by two curves of equal length laid out from the same tangent and same P. C. , but whose central angles djft'er by 1. The length L of each curve is given in the first column, and the half sum of their central angles I ~ I is given at the head of the other columns. When the central angles of two curves of equal length differ by x degrees the length I of the valvoid arc Joining their extremities is expressed by the simple formula, Fig. 36, cq. (86) l=P'P" =(A' - A"K in which l t is taken from the column headed by - ~ and opposite the given value of Z;or l t is found by inter- polation if necessary. See 150 and example. TABLE XI. Contains the measurements necessary to lay down a turnout with frogs of given numbers or angles for both a standard and a three-foot gauge. The distance BF is measured on the rail of the given track from the heel of the switch to the point of the frog, while of is the chord of the centre line of the turnout between the same points. The radius r applies to the centre line of the turnout. The dis- tance aF" is measured on the centre line of the straight track 256 FIELD from the lied of the switch to the point of the middle frog. The length of switch AD should conform to the tabular values unless the throw is to he different from that assumed in the table. See 180, 181, 182. TABLE XII. Contains the middle ordinates of chords vary- ing in length from 10 to 32 feet, and for degrees of curve vary- ing from 1 to 50. The use of the table is obvious. See 199. TABLE XIII. Gives the proper difference in elevation of rails on curves of various degrees from 1 to 50 for veloci- ties varying from 10 to 60 miles per hour. See 201. TABLE XIV. Gives the rise of grades in feet per mile and their angle of inclination corresponding to a rise per station (100 feet) varying from 0.01 foot to 10 feet. TABLE XV. Contains values of the formula (log h 1) 60384.3 in which li = reading of the barometer in inches. The inches and tenths of the readings are in the left-hand column, while the hundredths are found at the top of the other columns. The difference of any two values corresponding to two read- ings taken simultaneously at any two stations is the differ- ence in elevation in feet of those stations. But the differ- ence in height so found is subject to a correction for tempera- ture given in the next table. See 10. TABLE XVI. Contains coefficients of correction for atmos- pheric temperature, by which the approximate heights ob- tained by Table XV. are to be multiplied for a correction of these heights, which correction is to be added or subtracted according as the coefficient given in the table is marked -f or -. See 11. TABLE XVII. Contains corrections in feet, required by the curvature of the earth and the refraction of the atmosphere, to be applied to the elevation of a distant object as obtained by a level or theodolite observation for distances ranging from 300 feet to 10 miles. See 119. TABLE XVIII. Contains the coefficients for reducing the space on a vertical rod intercepted by the stadia hairs when EXPLANATION OF TABLES. 257 the line of collimation is inclined to the horizon, to the space that would be intercepted were the line of collimation horizon- tal; provided, that the visual angle denned by the stadia hairs is such that tan ^0 = .005 or fl = 34 22". 63, which is its customary value in surveying instruments. The angle of in clination a is taken at every 10 minutes through half a quad- rant. TABLE XIX. Contains the logarithms of the coefficients given in Table XVIII. TABLE XX. Gives the lengths of circular arcs to a radius = 1. To find the length of any arc expressed in degrees, minutes, and seconds, take from the table the lengths of the given num- ber of degrees, minutes, and seconds respectively, and multi- ply their sum by the length of the radius. The product is the length of arc required. TABLE XXI. Contains the values of minutes and seconds expressed in decimals of a degree, for every 10 seconds of arc, and also for quarter minutes up to one degree. TABLE XXII. Contains the values of inches and fractions expressed in decimals of a foot for every 32d of an inch up to one foot. TABLE XXIII. Contains the squares, cubes, square roots, cube roots, and reciprocals of numbers from 1 to 1054. Its use may be greatly extended by observing that if any number is multiplied by n its square is multiplied by 2 , its cube by n 3 , and its reciprocal by . , . . TABLE XXIV. The logarithm of a number consists of two parts, a whole number called the characteristic, and a deci- mal called the mantissa. All numbers which consist of the same figures standing in the same order have the same man- tissa, regardless of the position of the decimal point in the number, or of the number of ciphers which precede or follow the significant figures of the number. The value of the char- acteristic depends entirely on the position of the decimal point in the number. It is always one less than the number of 258 FIELD ENGINEERING. figures in the number to the left of the decimal point. The value is therefore diminished by one every time the decimal point of the number is removed one place to the left, and vice versa. Thus Number. Logarithm. 13840. 4.141136 1384.0 3.141136 138.40 2.141136 13.84 1.141138 1.384 0.141136 .1384 - 1.141136 .01384 -2.141136 .001384 -3.141136 etc. etc. The mantissa is always positive even when the characteristic is negative. We may avoid the use of a negative characteristic by arbitrarily adding 10, which may be neglected at the close of the calculation. By this rule we have Number. Logarithm. 1.384 0.141136 .1384 9.141136 .01384 8.141136 .001384 7.141136 etc, etc. No confusion need arise from this method in finding a number from its logarithm; for although the logarithm 6.141136 repre- sents either the number 1,384,000, or the decimal .0001384, yet these are so diverse in their values that we can never be uncer- tain in a given problem which to adopt. The table XXIV. contains the mantissas of logarithms, car- ried to six places of decimals, for numbers between 1 and 9999, inclusive. The first three figures of a number are given in the first column, the fourth at the top of the other columns. The first two figures of the mantissa are given only in the second column, but these are understood to apply to the remaining four figures in either column following, which are comprised between the same horizontal lines with the two. If a number (after cutting off the ciphers at either end) con- sists of not more than four figures, the mantissa may be taken direct from the table ; but by interpolation the logarithm of a number having six figures may be obtained. The last column contains the average difference of consecutive logarithms on EXPLANATION OF TABLES. 259 the same line, but for a given case the difference needs to be verified by actual subtraction, at least so far as the last figure is concerned. The lower part of the page contains a complete list of differences, with their multiples divided by 10. To find the logarithm of a number having- six figures : Take out the mantissa for the four superior places directly from the table, and find the difference between this mantissa and the next greater in the table. Add to the man- tissa taken out the quantity found in the table of proportional parts, opposite the difference, and in the column headed by the fifth figure of the number; also add ^ the quantity in the col- umn headed by the sixth figure. The sum is the mantissa required, to which must be prefixed a decimal point and the proper characteristic. Example. Find the log of 23.4275. For 2342 mantissa is 369587 " diff. 185 col. 7 129.5 " " " " 5 9.2 Ans. For 23.4275 log is 1.369726 The decimals of the corrections are added together to deter- mine the nearest value of the sixth figure of the mantissa. To find the number corresponding to a given logarithm. If the given mantissa is not in the table find the one next less, and take out the four figures corresponding to it; divide the difference between the two mantissas t>y the tabu- lar difference in that part of the table, and annex the figures of the quotient to the four figures already taken out. Finally, place the decimal point according to the rule for characteristics, prefixing or annexing ciphers if necessary. The division re- quired is facilitated by the table of proportional parts, which furnishes by inspection the figures of the quotient. Example. Find the number of which the logarithm is 8.263927 8.2(53927 First 4 figures 1836 from 263873 Diff. 5To Tabular diff. =236 .-. 5th fig. =2 47.2 6.80 6th fig. = 3 7.08 Ans. No. = .0188623 or 183,623,000. 2GO FIELD ENGINEERING. The number derived from a six-place logarithm is not reliable beyond the sixth figure. At the end of table XXIV. is a small table of logarithms of numbers from 1 to 100, with the characteristic prefixed, for easy reference when the given number does not exceed two digits. But the same mantissas may be found in the larger table. TABLE XXV. The logarithmic sine, tangent, etc. of an arc is the logarithm of the natural sine, tangent, etc. of the same arc, but with 10 added to the characteristic to avoid negatives. This table gives log sines, tangents, cosines, and cotangents for every minute of the quadrant. With the number of degrees at the left side of the page are to be read the minutes in the left-hand column ; with the degrees on the right-hand side are to be read the minutes in the right-hand column. When the degrees appear at the top Qf the page the top headings must be observed, when at the bottom those at the bottom. Since the values found for arcs in the first quad- rant arc duplicated in the second, the degrees are given from to 180. The differences in the logarithms due to a change of one second in the arc are given in adjoining columns. To find the log. sin, cos, tan, or cot of a given arc. : Take out from the proper -column of the table the log- arithm corresponding to the given number of degrees and minutes. If there be any seconds multiply them by the ad- joining tabular difference, and apply their product as a cor- rection to the logarithm already taken out. The correction is to be added if the logarithms of the table are increasing with the angle, or subtracted if they are decreasing as the angle in- creases. In the first quadrant the log sines and tangents in- crease, and the log. cosines and cotangents decrease as the angle increases. Example. Find the log sin of 9 28' 20". Log sin of 9 28' is 9.216097 Add correction 20 X 12.62 252 Ans. 9.216349 Exampk.. Find the log cot of 9 28' 20". Log cotan of 9 28' is 10.777948 Subtract correction 20 X 12.97 259 EXPLANATION" OP TABLES. 261 To find the angle or arc corresponding to a given logarithmic sine, tangent, cosine, or co- tangent. If the given logarithm is found in the proper column take out the degrees and minutes directly; if not, find the two consecutive logarithms between which the given logarithm would fall, and adopt that one which corresponds to the least number of minutes; which minutes take out with the degrees, and divide the difference between this logarithm and the given one by. the adjoining tabular difference for a quo- tient, which will be the required number of seconds. With logarithms to six places of decimals the quotient is not reliable beyond the tenth of a second. Example. 9.383731 is the log tan of what angle? Next less 9.383682 gives 13 36' Diff. ~~49.00 -^ 9.20 = 05".3 Ans. 13 36' 05". 3 Example. 9.249348 is the log cos of what angle? Next greater 583 gives 79 45' Diff. 235 -*- 11.65 = 20".2 Ans. 79 45 20". 2 The above rules do not apply to the first two pages of this table (except for the column headed cosine at top) because here the differences vary so rapidly that interpolation made by them in the usual way will not give exact results. On the first two pages, the first column contains the number of seconds for every minute from I'to2; the minutes are given in the second, the log. sin. in the third, and in the fourth arc the last three figures of a logarithm which is the difference between the log sin and the logarithm of the number of sec- onds in the first column. The first three figures and the char- acteristic of this logarithm are placed, once for all, at the head of the column. To find the log sin of an arc less than 2 given to seconds. Reduce the given arc to seconds, and take the logarithm of the number of seconds from the table of loga- rithms, and add to this the logarithm from the fourth column opposite the same number of seconds. The sum is the log sin required. The logarithm in the fourth column may need a slight inter- 262 FIELD ENGINEERING. polation of the last figure, to make it correspond closely to the given number of seconds. Example. Find the log sm of 1 39' 14". 4. 1 39' 14".4 = 5954".4 log 3.774838 add (q - I) 4.685515 Ans. log sin 8.4G0353 Log tangents of small arcs are found in the same way, only taking the last four figures of (q 1) from thejift7i column. Example. Find the log tan of 52' 35". add (q - I) 4.685609 52' 35" = (3120" -f 35") = 3155" log 3.498999 I) Ans. log tan 8.184608 To find the log- cotangent of an angle less than 2 given to seconds. Take from the column headed ( q-\- 1) the logarithm corresponding to the given angle, interpolating for the last figure if necessary, and from this subtract the loga- rithm of the number of seconds in the given angle. Example. Find the log cotan of 1 44' 22". 5. q + I 15.314292 6240" + 22",5 = 6262,5 log 3.796748 Ans. 11.517544 These two pages may be used in the same way when the given angle lies between 88 and 92 , or between 178 and 180 ; but if the number of degrees be found at the bottom of the page, the title of each column will be found there also; and if the number of degrees be found on the right hand side of the page, the number of minutes must be found in the right hand col- umn, and since here the minutes increase upward, the number of seconds on the same line in the first column must be dimin- ished by the odd seconds in the given angle to obtain the num- ber whose logarithm Is to be used with (q I) taken from the table. Example. Find the log cos of 88 41' 12". 5 (q -I) 4,685537 4740" - 12".5 = 4727.5 log 3,674631 Ans. 8.360168 EXPLANATION OE TABLES. 263 Example. Find the log tan of 90 30' 50". q 4- 1 15.314413' 1800" + 50' = 1850' log -3.267172 Ans. 127047241 To find the arc corresponding to a given log sin, cos, tan, or cotan which falls within the limits of the first two pages of Table XXV. Find in the proper column two consecutive logarithms be- tween which the given logarithm falls. If the title of the given function is found at the top of that column read the degrees from the top of the page; if at the bottom read from the bottom. :M- . Find the value of (q I) or (q -fr I), as the case may require, corresponding to the given log (interpolating for the last figure if necessary). Then if q given log and I = log of number of seconds, n, in the required arc, we have at once I = q (q I) or I = (q -j- ~~ Q> whence n is easily found. Find in the first column two consecutive quantities between which the number n falls, and if the degrees are read from the left hand side of the page, adopt the less, take out the minutes from the second column, and take for the seconds the difference between the quantity adopted and the number n. But if the degrees are read from the right hand side of the page, adopt the greater quantity, take out the minutes on the same line from the right-hand column, and for the seconds take the difference between the number adopted and the num- ber n. Example. 11.734268 is the log cot of what arc? q + I 15.314376 q 11.734268 .'. n= 3802.8 ~~&580l08 For 1 adopt 3780. giving 03' Difference 22". 8 An*. 1 03' 22".8 or 178 56' 37".2. Example. 8.201795 is the log cos of what arc? 9 - ^ 4.685556 9 ^8.201795 .' n= 3282". 8 3.516239 For 89 adopt 3300. giving 05' Difference 17". 2 Am. 89 05' 17".2 or 90 54' 42".8. 264 FIELD EffGLffEEBLNTG. TABLE XXYI. Contains logarithmic versed sines and ex- ternal secants for every minute of the quadrant, with the differences of the same corresponding to a change of 1 second in the arc or angle. Interpolation for seconds is made in the same manner as with log sines of the preceding table, except on the first two pages. For angles less than 4 the differences vary so rapidly that interpolation by direct proportion will not give exact values. . On the first two pages the column headed q 21 contains the difference between the log versed sine (or log ex secant) of an arc and twice the logarithm of the number of seconds in the same arc. The characteristic, and first three decimals (9.070) are common to all the logarithms in these columns up to 3 19' for log vers sines, where it changes to (9.069), as shown at the foot of the column; and up to 4 for log ex secants, where it changes to (9.071). At the point of change a cipher is replaced by the mark + to call attention. To find the log vers sin, or log ex sec of an angle given to seconds. Reduce the angle to seconds, take the logarithm of this number, multiply it by 2, and add the product to the logarithm in the column (q 21) found op- posite the given angle. The log (q 21) should be corrected by interpolation for the fractional part of a minute in the given angle. Example. What is the log ex secant of 2 14' 43". 7? 2 14' 43". 7 - 8040" -f 43.7 = 8083". 7 log 3.907610 2 21 7.815220 (q - 20 9.070064 Ans. .'. q 6.885284 To find the arc corresponding 1 to a given log" vers, or log ex sec. Find in the column of log vers, or log ex sec the two values between which the given log falls,, and take out from the column (q 21) the logarithm corres- ponding to the given log (interpolating for the value of the last figure if necessary). Subtract this from the given logarithm and divide by 2. The quotient is the logarithm of the num- ber of seconds in the required arc. EXPLANATION OF TABLES. 265 Example. 7.344728 is the log vers of what arc? q 7.344728 3 48' -f- (q - 21) 9.069960 2)8.274768 13720". 9 .-. I 4.137384 13680. Ans. 3 48' 40".9 To find the log ex sec of an arc greater than 88 given to seconds. Take from the column (q-\-l) the logarithm corresponding to the given arc, interpolating for the fraction of a minute. From this subtract the logarithm of the number of seconds in the complement of the given arc. Example. What is the log ex sec of 88 24' 20". 5? For 88 24' q -4- 1 15.302183 on fi Correction 129 X ~-= 44 q + l 15.302227 Comp. 88 24' 20".5 _= 5739".5 log 3.758874 Ans. log ex sec 11.543353 To find the angle corresponding to a given log ex sec when the angle is greater than 88. Find in the table the two consecutive log ex secants between which the given one falls, and then find by interpolation the value of the log (q-\-l) corresponding to the given log ex sec and subtract the latter from it. The difference will be the logarithm of the number of seconds in the complement of the required angle, which is then easily found. Example. 11.924368 is the log ex sec of what arc? Given log ex sec 11.924368 Next less 11.918290 ? + * 15.309225 Diff. 6078 q + I 15.309296 Given log ex sec Jl. 924368 40' 26".2 = 2426".2 .'. log 3.384928 An*. 89 19 33".8. 266 FIELD EKGIKEEBIKG. TABLE XXVII. Contains natural sines and cosines, to five places of decimals for every minute of the quadrant. Correc- tions for fractions of a minute are made directly proportional to the* difference of consecutive values in the table; positive for sines, negative for cosines. TABLE XXVIII. Contains natural tangents and cotangents to five places of decimals for every minute of the quadrant. Corrections for fractions of a minute are made directly propor- tional to the difference of consecutive values in the table ; positive for tangents, negative for cotangents. TABLE XXIX. Contains natural versed sines and external secants to five places of decimals for every minute of the quadrant. Corrections for fractions of a minute are made directly proportional to the difference of consecutive values. They are positive in every case. TABLE XXX. Contains the number of cubic yards con- tained in prismoids of various side slopes, bases, and depths, as indicated by the title and the numbers in the first column. Each prismoid is supposed to have a uniform level cross sec- tion throughout. These tables are chiefly useful in making up preliminary estimates from the profile, or in other cases where only approximate results are required. For monthly and final estimates more elaborate tables are required, such as are des- cribed in 257. To make an approximate estimate of quanti- ties from a profile by use of Table XXX. Select the proper column for base and slopes, and if the outline of a cut on the profile is roughly a four-sided figure, stretch a fine silk thread over the surface line to average it, note the depth from thread to grade line midway of the cutting, and multiply the tabular number opposite this depth by the average length of the cutting in stations of 100 feet. (By average length is meant the half sum of the length of the grade line in the cutting and of so much of the surface line as is covered by the thread.) If the area of a cutting as seen on the profile is approximately triangular, stretch an averaging line over each incline, and note the depth from the intersection of these lines to grade, and multiply the tabular number opposite this depth by one- EXPLANATION OF TABLES. 267 half the length of the cut measured on the grade line in sta- tions. The resulting quantities will be slightly in excess if the ground is level transversely, but may be too small if the trans- verse slope is steep, and cutting on the centre line is small. In general they furnish a good approximation. Quantities in embankments may, of course, be found similarly. A cut or fill may be divided on the profile into several portions, and the contents of each portion found separately if preferred. The content of a prismoid, level transversely, but having different end depths, maybe found correctly by this table thus: add together the quantities opposite each end-depth and 4 times the quantity opposite the half sum of the depths; multiply the sum by the length in feet, and divide by 600. TABLE XXXI. Contains a variety of useful numbers and formulae. The logarithms are here given to seven places of decimals. TABLES. 269 TABLE I. GEOMETRICAL PROPOSITIONS. The References are to Davies* Legendre, Revised Edition. No. REFERENCE. HYPOTHESES. i CONSEQUENCES. 5 i i IV., XI If a triangle is right angled, The square on the hypothe- nuse is equal to the sum of the squares on the other two sides. 2 3 L, XL, Cor. 1.... I XI If a triangle is equilateral, If a triangle is isosceles, It is equiangular. The angles at the base are equal. 4 L, XL, Cor. 2.... If a straight line from the vertex of an isosceles triangle bisects the base, It bisects the vertical angle. And is perpendicular to the base. 5 I., XXV,, Cor. 6.. If one side of a tri- angle is pro- duced, The exterior angle is equal to the sum of the two interior and opposite angles. 6 IV., XX If two triangles are mutually equian- gular, They are similar. And their corresponding sides are proportional. 7 L, XXVII If the sides of a polygon are" pro- duced in the same order, The sum of the exterior angles equals four right angles. 8 9 L, XXVL, Cor. 1. L, XXVIII If a figure is a quadrilateral, If a figure is a parallelogram, The sum of the interior angles equals four right angles. The opposite sides are equal. The opposite angles are equal. It is bisected by its diagonal. And its diagonals bisect each other. L, XXXI. 10 III., VII If three points are not in the same straight line, A circle may be passed through them. 11 in., xvn If two arcs are in- tercepted on the same circle, They are proportional to the corresponding angles at the centre. 12 13 14 15 V M XIII., Cor. 2.. If two arcs are similar, If two areas are circles, If a radius is per- pendicular to a chord, If a straight line is tangent to a circle, They are proportional to their radii. They are proportional to the squares on their radii. It bisects the chord. And it bisects the arc subtended by the chord. It meets it in only one point. And it is perpendicular to the radius drawn to that point. III., VI Ill IX 16 1 in., xrv., cor. . . If from a point without a circle tangents are drawn to touch the circle, There are but two. They are equal. And they make equal angles with the chord joining the tangent points. 271 TABLE I. GEOMETRICAL PROPOSITIONS. The References are to Davies' Legendre, Revised Edition. No. REFERENCE. HYPOTHESES. CONSEQUENCES. 17 m.,x If two lines are parallel chords or a tangent and parallel chord, They intercept equal arcs of a circle. 18 III., XVIH If an angle at the circumference of a circle is sub- tended by the same arc as an angle at the cen- tre, The angle at the circumfer- ence is equal to half the angle at the centre. 19 in.,xvm.,cor.3 If an angle is in- scribed in a semi- circle, It is a right angle. 20 in., xxi If an angle is formed by a tan- gent and chord, It is measured by one half of the intercepted arc. 21 IV.,XXVin.,Cor. If two chords, in- tersect each oth- er in a circle, The rectangle of the seg- ments of the one, equals the rectangle of the segments of the other. 22 IV.,XXin.,Cor.2 And if one chord is a diameter, and the other per- pendicular to it, The rectangle of the seg- ments of the diameter is equal to the square on half the other chord. And the half chord is a mean pro- portional between the seg- ments of the diameter. 23 IV., XXIX., Cor.. If two secants meet without the circle, The rectangles of each secant and its external segment are equal. 24 25 IV., XXX. IV., XIV If a secant and tangent meet, If a straight line from the vertex of a triangle bi- sects its base, The rectangle of the secant and its external segment is equal to the square on the tangent. And the tangent is a mean proportional be- tween the secant and its external segment. The sum of the squares on the two sides is equal to twice the square of half the \ base increased by twice the square of the bisecting line. 26 IV., XII If a perpendicular be drawn from the vertex of a triangle to the base, The square of a side opposite an acute angle is equal to the sum of the squares of the other side and the base, diminished by twice the rectangle of the base and the distance from the ver- tex of the acute angle to the foot of the perpendicu- lar. 272 TABLE II.-TRIGONOMETRIC FORMULA. TRIGONOMETRIC FUNCTIONS. Let A (Fig. 107) = angle BAG = arc BF, and let the radius AF ~ AB = H=\. We then have sin .4 cos A tan A cot A sec A cosec = AG = DF = HG -AD = AG versin .4 = CF = covers .4 = j?A' = exsec A = > coexsec -4 = .Bo? chord ^4 = BF chord 2 ^ = 51 = In the right-angled ti-iangle ABC (Fig. 107) Let AB = c,AC = b, and BC = a. We then have : FIG. 107. 1. sin A 2. cos A 3. tan A 4. cot A 5. sec A = cos B c = sin 5 c = tan B a c -7- = cosec B cosec A = a sec.S c - 6 7. vers J. = = covers B 8. exsec A = , = coexsec # o 9. covers A = = versin B 11. a = c sin yl = 6 tan A 12. & = c cos .A = a cot A sin ^4 cos A 14. a = c cos B = 6 cot B 15. 6 c sin B = tan 5 ir __ B cos B ~ sin B 17. rt = 18. 6 = 19. c = (c- a) v a 2 + 6 20. C = 90 = A 21. area = - 273 TABLE II. -TRIGONOMETRIC FORMULAE. SOLUTION OF OBLIQUE TRIANGLES. Fia. 108. GIVEN. SOUGHT. FORMULA. 22 A, B,a C, 6, c sin ad, f ^ cin ( A 1 R" - sin^ SmU 23 A,a,b B,C,c sin 5 = S1 ^ .6, C = 180 - (A -f B), 24 C,a,b <**> y* (A -f B) = 90 - % c 25 MU- ran ^ (A &) tan ^ (A -f- ^; 26 A, B B = %(A-}- B) }&(A B) 27 c ., cos l(A-{-B) , , sin ^(^4 + -P) 28 area ^- = 1^ a 6 sin C. 29 a, 6, c A ^.^(a + .+O^in^y^^c, 30 co^^/^^.tan^^y^^ . ^ 2Vs(s - a) (s 6)(s c). 32 area JT = Vs (s - a) (s - 6) (s - c) 33 A, B, C, a area g. _ a 2 sin S . sin C 274 TABLE II.-TRIGOXOMETRIC FORMULAE. GENERAL FORMULAE. sin A - . = V 1 cos a A = tan A cos A cosecA J4 vers 2 ^4 = = 2 sin J4 A cos J^ A = 36 87 cos .4 = - - = 4/ f sin 2 A = cot ^4 sin A sec ^ cos A = 1 - vers .4 = 2 cos 2 y^A 1 = 12 sin 2 J^ JL cos ^4 = cos 3 J4 ^ sin2 ^ A = 1 sin A tan A - T , SBB - , = */ sec 2 ^L 1 cot A cos ^1 45 46 48 50 tan J. = i-/ 5-3 1 = y cos 2 X " cos^4 = 1 4- cos 2 .4 1 cos 2 A vers 2 ^4 tan A = - . = ; - - = exsec A cot 3^ -4 sin 2 A sin 2 ^4 1 cos .4 sjn2^4 sin 2 ^4 1 + cos 2 -^ : 1 ~cos2^ : vers 2 J. sin 2 A vers J. = 1 cos A = sin A tan J= -g- 3 A, D i^lOO-^ 4 D, L A Di 5 A, L D J> = 100^- 6 R, A r T = R tan ^ A T * c C = 2 J? sin ^ A 8 M M M= R vers ^ A a M E E = R exsec J^ A 10 T r A R .R= Tcot^ A ii " E E = T tan J4 A 12 M C C = 2 T cos ^ A IS 14 E, A M R Jf = T cot 1^ A . vers J^ A K ~ exsec J^ A 15 " T T = E cot J4 A IS " C exsec Jx A 17 M Jf=#COS^A 18 C, A R jB = !Tsin^~A" 19 20 M T Jf = U C tan M A T ^ 2 COS ^ A 21 r E F u r exsec ^ A 22 Jf, A R *. ^ * vers J^ A 23 C C = 2 M cot M A 24 25 K T E r _ Jf tan ^ A ~ COS J^ A 277 TABLE III.-CURVE FORMULAE. 26 GIVEN. SOUGHT. FORMULAE. R, T A tan 14 A = 27 28 29 R, C M A T sin J^ A = COS ^A=, 1|/( B + -|)(S-|-) 30 R, M Jf vers J^ A = 31 " " COS J^ A = ^ 32 R, E A exsec J^ A = p- 33 " cosK 3 A =ir ^ 34 35 T, c u A COS * A = -^ tan y A = |/l|^g 36 T,E A tan M A = ^~ 37 if COS J^ A = y- a T ^;a 38 39 C 1 Hf -.: A ; J$ 23f cos ^ A = -Qf-f+tfi 40 41 42 43 44 Jf , R,'T A 3/ . COS ^ A = -=- jE/ _ 6 J. \f M R E = V 2 1 2 -f- -K 2 -B 45 46 47 R, C tt r .if .E r _ CK y(*+i)(-l) J?2 = v(HF*c?H~X C) ~ R 278 TABLE III. -CURVE FORMULAE. GIVEN. SOUGHT. FORMULAE. 48 49 A M T C * _ ' T - - C = 2VM(2R - M) 50 51 R, E E T E = ^~^~M T = VE&R + E) 52 53 54 T, C C M R f^, _ 2 R /j/ E (2 R 4 E) B + E CT ~ \'(2T+C) (2T-C) 55 " M M =y*c\/l T T ~ + c c - 56 " E E=T \/'2T^- - 57 T, E r> R=I ^Jp 58 c 2 T (T 2 jE 12 ) ~~ T^'+E* ' 59 " M M =~T*+E*- 60 C, M R E <=- M "'\-M~ 61 62 63 M, E T E R C (C 2 4- 4 M 2 ) 2(C^-T~.M2) L^: 4 " 2 ,; 5 64 T r = i/!: 65 C c =^V : ^-^ 66 T, M R 2 J/ 67 68 . E C E 3 + 2 M ET 2 4- MT ~ = C' 3 4 2 TC 2 4 4 M 2 C - 8 Jf 2 T _ o 69 C, E R v + v^ljjLCL -B ~ - C 1 E = 70 " T g r s _ r a c 2 T^ 2 - C7* = 71 M 4 4 TABLE IV. RADII, LOGARITHMS, OFFSETS, ETC. f- ' Dog. Radius. Loga- rithm. Tan. Off. Mid. Ord. [ Deg. Radius. Loga- rithm. Tang. Off Mid. Ord. I>. 11. log. K. t. in. D. R. log. K. t. in. o' Infinite Infinite .000 .000 1 0' 5729.65 3.758128 .873 .218 i 343775. 5 530274 .015 .001 1 5(535.72 .750950 .887 .222 2 171887. 5 235244 .029 .007 2 5544.83 .743888 .902 .225 3 114592. 5.059153 .044 .011 3 5456.82 .736939 .916 .229 4 85943.7 4.934214 .058 .015 4 5371.56 .730100 .931 .233 5 68754.9 .837304 .073 .018 5 5288.92 .723367 .945 .236 6 57295.8 .758123 .087 .022 6 5208.79 .716737 .960 .240 7 49110.7 .691176 .102 .025 7 5131.05 .710206 .974 .244 8 42971.8 .633184 .116 .023 8 5055.59 .703772 .939 .247 9 38197.2 .532031 .131 ..033 9 4982.33 .697432 1.004 .251 10 34377.5 4.536274 .145 .036 10 4911.15 3.691183 1.018 .255 11 3125-3.3 4.494831 .160 .040 11 4841.98 3.685023 1.033 .258 12 23347.8 .457o:H .175 .044 12 4774.74 .678949 1.047 .262 13 23441.2 .4323*1 .189 .047, 13 4709.33 .67'2959 .062 .265 14 24.yr>.4 .330146 .204 .051 14 4645.69 .667051 .076 .269 15 2291H.3 .360183 .218 .055 15 4583.75 .661221 .091 .273 16 21H5.9 .332154 .2-33 .658 16 4523.44 .655469 .105 .276 17 2)233.1 .305825 .247 .032 17 4464.70 .649792 .120 .230 18 19033.0 .231003 .232 .065 18 4407.46 .644189 .134 .284 19 18333.4 .257521 .276 .069 19 4:351.67 .638656 .149 .287 20 17188.8 4.235244 .291 .073 23 4297.28 3.6=33194 .164 .291 21 16370.2 4.214055 .305 .076 21 4244.23 3.627799 .178 .295 22 15626.1 .193852 .320 .0:30 22 4192.47 .622470 .193 .298 23 14:>46.7 .174547 .335 .034 23 4141.96 .6172J6 .207 .302 24 143-33.0 .156034 .349 .087 24 4092.66 .612005 222 .305 25 13751.0 .138335 .364 .091 25 4044.51 .606866 iaae .309 26 13-333.1 .121302 .378 .095 26 3997.49 .601787 .251 .313 27 13732.4 .104911 .393 .093 1 27 3951.54 .596766 .235 .316 28 12377.7 .089117 .407 .102 1 23 3906.54 .591803 .230 .320 29 11854.3 .073877 .422 .105 29 3862.74 .586396 1.294 .324 30 11459.2 4.059154 .436 .109 30 3819.83 3.582044 1.309 .327 31 11039.6 4.044914 .451 .113 31 3777.85 3.577245 1.324 .331 32 10743.0 .031125 .465 .116 32 3736.79 .572499 1.338 .335 33 10417.5 .017762 .480 .120 33 3396.61 .567804 1.353 .338 34 10U1.1 4.004797 .495 .124 34 3657.29 .563160 1.367 .342 35 9i33.l3 3.998203 .509 .127 &5 3318.80 .5585(54 1.382 .345 36 9.549.34 .979973 .524 .131 36 ,3581.10 .554017 1.396 .349 37 9331.23 .938074 .538 .135 37 3544.19 .549517 1.411 .353 38 904(5.75 .956493 .553 .138 38 3508.02 .545063 1.425 .356 39 8S14.7-J .945212 .567 .142 39 3472-59 .540354 1.440 .360 40 8.594.42 3.934216 .58* .145 40 3437.87 3.53G283 1.454 .3(34 41 8384.80 3.923493 .596 .149 41 3403.83 3.531963 1.469 .367 42 8185.16 .913027 .(511 .153 42 3370.46 .527690 1.483 .371 43 7934.81 .932803 .635 .156 43 8-337.74 .523453 .493 .375 44 7813.11 .892824 .640 .160 44 3305.65 .519257 .513 .378 10 7iB-).4'.) .8-53065 .654 .164 45 3-374.17 .515101 .527 .382 41) 7473.4-3 .873519 .669 167 46 3243.29 .510985 .542 .385 47 7314.41 .864179 .684 .171 47 3212.93 .503908 .556 .389 48 710-3.05 .855036 .(593 .174 48 3183.23 .5Q3868 .571 .393 49 ri)ir,.s7 .846382 .713 .178 49 3154.03 .498866 .585 .396 50 6875.55 3.837303 .727 .182 50 3125.33 3.494900 .600 .400 51 6740.74 3.823703 .742 .185 51 3097.20 3.490970 .614 .404 52 0611.12 .820275 .756 .189 52 3069.55 .487075 .629 .407 53 6483.33 .812002 .771 .193 53 3042.39 .483215 .643 .411 - 54 6366.2(5 .803885 .785 .196 54 3315.71 .479389 .658 .414 55 (J-3.50.5l .795916 .800 .200 55 2939.43 . 475596 .673 .418 56 6138.93 .783091 .814 .204 56 2933.71 .471836 .687 .422 57 6031.23 .780404 .829 .207 57 2338.39 .468109 .702 .425 58 5927.22 .772851 .844 .211 58 2913.49 .464413 .716 .429 59 5826.76 .765427 .858 .215 59 2389.01 .460749 1.731 .433 i 60 5729.65 3.758128 .873 .218 60 2364.93 3.457115 1.745 .436 280 TABLE IV. RADII, LOGARITHMS, OFFSETS, ETC. Deg. D. Radius. K. Loga- riuim. log. K. T S g ' t. Mid. ! Orel. m. | Deg. . Radius. B. Loga- rithm. log. R. 'S3?- t. M;!. Ord. in. 2 0' 2864 93 3 457115 1.745 .436 3 0' 1910.08 3.281051 2.618 .054 i 2841 : 26 .453511 1.760 .440 1 1899.53 .278646 2.632 ioES 2 2817.97 .44UU87 1 . 774 .444 -i 2 1889.09 .276253 2.647 .002 3 2795.06 .440392 .789 .447 3 1878.77 .273874 2.601 .665 4 2i 72. 53 .442876 .803 .451 4 1868.56 .2710)8 2.076 .009 5 2750.85 .43U&8 .818 .454 5 1858.47 .269155 2.600 .073 6 2728.52 .4359x8 .832 .458 6 1848.48 .266814 2.705 .676 2707.04 .432495 .847 .462 7 1838.59 .264486 2.719 .080 8 2685.89 .429089 ,862 .465 ! 8 1828.82 .262170 2.734 .084 9 2665.08 .425710 .876 .469 9 1819.14 .259867 2.749 .687 10 2644.58 3 422356 .891 .473 10 1809.57 3.257576 2.763 .691 11 2624.39 3.419029 .05 .476 11 1800.10 3.255296 2.778 .694 1:3 2604.51 .415727 .920 .480 12 1790.73 .253029 2.792 .698 13 2584.93 .412449 .934 .484 13 1781.45 .250774 2.807 .702 14 2565.65 .409197 .949 .487 14 1772.27 .248530 2 821 .705 15 2546. 64 .405968 .963 .491 15 1763.18 .246297 2.836 .7'09 10 2527.92 .402763 .978 .494 16 1754.19 .244077 2.850 .713 17 2509.47 .399582 .992 .498 17 1745.26 .241867 2.805 .716 18 2491.29 .396424 2.007 .502 18 1736.48 .239669 2.87'9 .720 19 2473.37 .393289 2.022 .505 19 1727.75 .237481 2.894 .723 20 2455.70 3 390176 2.036 .509 20 1719.12 8.235305 2.908 .727 21 2438.29 3.887085 2.051 .513 21 1710.56 3.233140 2.923 .731 22 2421.12 .384016 2.065 .516 22 1702.10 .230985 2.988 .734 23 2404.19 .380969 2.080 .520 23 1693.72 .228841 2.952 .738 24 2387.50 .377943 2.094 .524 24 16a5.42 .226707 2.967 .742 26 2371.04 .374938 2.109 .527 25 1677.20 .224584 2.981 .745 26 2354. KJ .371954 2.123 .531 26 1669.06 .222472 2.996 7'49 27 2&S8.78 .368990 2.138 .534 27 1661.00 .220369 3.010 .753 28 2322.98 .366046 2.152 .538 28 1653.01 .218277 8.025 .756 29 2307.39 .363122 2.167 .542 29 1645.11 .216195 3.039 .700 30 2292.01 3.360217 2.181 .545 30 1637.28 3.214122 3.054 .763 31 2276.84 3.357332 2.196 .549 31 1629.52 3.212060 3. 008 .767 32 2261.86 .&54466 2.211 .553 32 1621.84 .210007 3.083 . 771 33 2247.08 .351618 2.225 .556 33 "1614.22 .207964 3.097 .774 34 2232.49 .348789 2.240 .560 34 1606.68 .205930 3.112 .778 35 2218.09 .345979 2.254 .564 35 1599.21 .203906 3.127 .782 36 2203.87 .343187 2.269 .567 36 1591.81 .201892 3.141 .785 37 2189.84 .340412 2.283 .571 37 1584.48 .199886 3.1.-0 .789 38 2175.98 .337655 2.298 .574 38 1577.21 .197890 3.170 .793 39 2162.30 .334916 2.312 .578 , 39 1570.01 .195903 a! 185 . 7'IHJ 40 2148.79 3.332193 2.327 .582' 40 1562.88 3.1m 3.19!) .800 41 2135.44 3.329488 2.341 .585 41 1555.81 3.191950 3.214 .803 42 2122.26 .326799 2.356 .589 42 1548.80 .189:)% 3.228 .807 43 2109.24 .324127 2 371 .593 ; 43 1541.86 .188045 3.243 .811 44 2096. 39 .321471 2.385 .596 44 1584.98 .18(3103 3.157 .814 45 2083.68 .318832 2.400 .600 ! 45 1528.16 .1K41C9 3.272 .818 46 2071.13 .316208 2.414 .604 46 1521.40 .182244 3.2S6 .822 47 2058.73 .313600 2.429 .607 ! 47 1514.70 .1808S7 3.301 .825 48 2046.48 .311008 2.443 .611 i 48 1508.06 .178419 3.316 .829 49 2034 37 .308431 2.458 .614 49 1501.48 .176519 8.830 .832 50 2022.41 3.305869 2.47'2 .618 50 1494.95 3.174027 3.345 .830 51 2010.59 3.30,3323 2 487 .622 51 1488.48 3.17S744 3. 39 .840 52 1998.90 .300791 2.501 .625 52 1482.07 .170868 3.374 .843 53 1987.35 .298274 2.516 .629 53 1475.71 .169001 3.388 .847 54 1975.93 .295771 2-530 .633 54 1469.41 .167142 3.403 ,651 55 1964.64 .293283 2.545 .636 55 1463.16 .165291 3.417 .854 56 19:53.48 .290809 2.560 .640 56 1456.96 .163447 3 . m .858 57 1942.44 .288349 2.574 .644 57 1450.81 .161612 3.446 .802 58 1931.53 .285902 2 589 .647 58 1444.72 .159784 3.461 .865 59 1920 75 .283470 2.603 .651 59 1438.68 .157963 3.475 .809 60 1910.08 3.281051 2.618 .654 60 1432.69 3.156151 3.490 .872 281 TABLE IV. RADII, LOGARITHMS, OFFSETS, ETC. Deg. D. "Hiss: R. log. K. Tang. oc t. Mid. Ord. 111. Deg. D. ** HtS K, ! log.R. "*& t. Mid. Ord. in. 4 0' 1433.09 3.156151 3.490 .872 5 0' 1146.28 3.059290 4.362 1.091 1 1426.74 .154346 3.505 .876 1 I 1142.47 .057846 4.376 1.094 2 ! 1420.85 .152548 3.519 .880 2 i 1138.69 .056407 4.391 1.098 3 i 1415.01 i .150758 3.534 .883 3 i 1134.94 .054972 4.405 1.102 1 4 1409.21 I .148975 3.548 .887 4 1131.21 .053542 4.420 1.105 ! 5 1403.46 .147200 3.503 .891 5 1127.50 .052116 4.435 1.109 6 1397.76 .145431 3.577 .894 6 1123.82 .050696 4.449 1.112 7 1392.10 j .143670 3.592 .898 7 1120.16 .049280 4.464 1.116 8 138(5.49 .141916 j 3.606 .902 8 1116.52 .047868 4.478 1.120 9 1380.92 i .140170 3.621 .905 9 1112.91 .046462 4.493 1.123 10 i 1375.40 i3.138430 3.635 .909 10 1109.33 |3.045059 4.507 1.127 11 1369.92 3.136697 3.650 .912 11 1105.76 3.043662 4.522 1.181 12 1364.49 .134971 3.664 .916 12 1102.22 .042268 4.536 1.134 13 1359.10 .133251 3.679 .920 13 1098.70 .040880 4.551 1.138 14 1353.75 .131539 1 3.693 .923 14 1095.20 .039495 4.565 1.142 15 1348.45 .1298:33 3.708 .927 15 1091.73 .038115 4.580 1.146 16 1343.15 .128134 3.723 .931 16 1088.28 .036740 4.594 1.149 17 1337.65 .120442 3.736 .934 17 1084. 85 .035808 4.609 1.153 18 | 1332.77 .124756 3.752 .938 18 1081.44 .034002 4.623 1.157 19 1327.63 .123077 3.766 .942 19 1078.05 .032639 4.638 1.160 20 1322.53 |3.121404 3.781 .945 20 1074.68 3.031281 4.653 1.164 21 1317.46 3.119738 3.795 .949 21 1071.34 3.029927 4.667 1.168 22 1312.43 .118078 3.810 .952 22 1068.01 .028577 4.682 1.171 23 1307.45 .116424 3.824 .956 23 1064.71 .127231 4.696 1.175 24 1302.50 .114777 3.839 .960 24 1061.43 .025890 4.711 1.179 25 1297.58 .113136 3.853 .963 25 1058.16 .024552 4.725 1.182 26 1292.71 .111501 3.868 .967 26 1054.92 .023219 4.740 1.186 27 1287.87 .109872 3.882 .971 27 1051.70 .021890 4.754 1.190 28 1283.07 .108249 3.897 .974 28 1048.48 .020505 4.709 1.193 29 | 12; 8. 30 .100032 3.911 .978 29 1045.31 .019244 4.783 1.197 30 1273.57 3.105022 3.926 .982 30 1042.14 3.017927 4.798 1.200 31 1268.87 3.103417 3.941 .985 31 1039.00 3.016314 4.812 1.204 32 1264.21 .101818 3.955 .989 32 1035.87 .015305 4.827 1.208 33 1259.58 .100225 3.970 .993 33 1032.76 .013999 4.841 1.211 34 1264.08 .098638 3.984 .996 34 1029.67 .012098 4.856 1.215 35 1250.42 .097057 3.999 1.000 35 1026.60 .011401 4.870 1.218 36 1245.80 .095481 4.013 1.003 36 1023.55 .010107 4.885 1.222 37 1241.40 .093912 4.028 1.007 37 1020.51 .008818 4.900 1.226 38 1236.94 .092347 4.042 1.011 38 1017.49 .00.032 4.914 1.229 39 1232.51 .090789 4.057 1.014 39 1014.50 .006250 4.929 1.233 40 1228.11 3.089236 4.071 1.018 40 1011.51 3.004972 4.943 1.237 41 1223.74 3.087689 4.086 1.022 41 1008.55 3.003698 4.958 1.240 42 1219.40 .086147 4.100 1.025 42 1005.60 .002427 4.972 1.244 43 1215.30 .084610 4.115 1.029 43 1002.67 3.001160 4.987 1.247 44 1210.82 .083079 4.129 1.032 44 999.762 2.999897 5.001 1.251 45 1206.57 .081553 4.144 1.036 45 996.867 .998637 5.016 1.255 46 1202.36 .080033 4.159 1.040 46 993.988 .997381 5.030 1.258 47 1198.17 .078518 4.173 1.043 47 991.126 .996129 5.045 1.262 48 1194.01 .077008 1 4.188 1.047 48 988.280 . 994H80 5.059 1.266 49 1189.88 .075504 4.202 1.051 49 985.451 .993635 6.074 1.269 50 1185.78 3.074005 4.217 1.054 50 982.638 2.992393 5.088 1.273 51 1181.71 3.072511 4.231 1.058 51 979.840 2.991155 5.103 1.277 52 1177.06 .071022 4.246 1.062 52 977.060 .989921 5.117 1.280 53 1173.65 .069538 4.260 1.065 53 974.294 .988690 5.132 1.284 ^ 54 1169.06 .068059 4.275 1.069 54 971.544 .987463 5.146 1.288 55 1165.70 .066585 4.289 1.073 55 968.810 .986238 5.161 1.291 56 i 1161.76 .065116 4.304 1.076 56 966.091 .985018 5,175 1.295 57 1157.85 .063653 4.318 1.080 57 963.387 i .983801 5.190 1.298 58 1153.97 | .062194 4.333 1.083 58 960.698 .982587 5.205 1.302 59 1150.11 j .060740 4.347 1.088 59 958.025 .981377 5.219 1.306 60 1146.28 3.059290 4.362 1.091 60 955.366 2.980170 5.234 1.309 TABLE IV. RADII, LOGARITHMS, OFFSETS, ETC. Deg. Radius. Loga- rithm. Tang. Off. Mid. Ord. Deg. ! Radius. Loga- Tang, rithin. Off. Mid. Ord. I>. R. log. R. t. in. D. R. log. R. t. m. 6 O'j 955. 3G6 2.980170 5.234 I 1.300 7 0' 819.020 12.913235 6.105 1.528 1 952.7'22 .978966 5.248 1.313 1 817.077 .912:363 6.119 1.531 2 950.093 .977766 5.263 1.317 2 815.144 .911234 1 6.134 1.535 3 947.478 .976569 5.277 1.320 3 813.238 .910208 6.148 1.539 4 944.877 .975375 5.292 1.324 4 811.303 .909183 6.163 1.543 5 942.291 974185 5.306 1.327 5 809.397 .908162 6.177 1.546 6 939.719 .972998 5.321 1.331 6 807.499 .907142 1 6.192 1.550 7 937.161 .971814 5.335 1.335 7 805.611 .906125 I 6.206 1.553 8 934.616 .970633 5.350 1.338 8 803.731 . 905111 6.221 1.557 9 932.086 .909456 5.364 1.342 P 801.860 .904098 6.236 1.501 10 920.569 2.968282 5.379 1.346 10 799.997 2.903089 6.250 1.564 11 927.066 2.967111 5.393 1.349 11 798.144 2.902081 6.265 1.508 12 924.576 .965943 5.408 1.353 12 796.299 .901076 6.27-9 1.57-2 13 922.100 .964778 5.422 1.356 13 794.462 .900073 6.294 1.575 14 919.637 .963616 1 5.437 1.360 14 792.634 .899073 6.803 1.579 15 917.187 l .96245815.451 1.364 15 790.814 .898074 6.323 1.582 16 914.750 .961303 5.466 1.368 16 789.003 ^97078 6.337 1.586 17 912.326 .900150 5.480 1.371 17 787.210 .896085 6.352 1.590 18 909.915 .959001 5.495 1.375 18 785.405 .895094 6.366 1.593 19 907.517 .957855 5.510 1.378 19 783.618 .894105 6.381 1.597 20 905.131 2.956711 5.524 1.382 20 781.840 2.89^118 6.395 1.600 21 902.758 2.955571 5.539 1.386 21 780.089 : 2. 892133 6.410 1.604 22 900.397 .954434 5.553 1.389 22 778.307 .891151 6.424 1.608 23 898 048 .953300 5.568 1.393 23 776.552 .890171 6.439 1.011 24 895.712 .952168 5.582 1.397 24 774.806 .889193 6.453 1.615 25 893,388 .951040 5.597 1.400 25 773.067 .888217 0.408 1.019 26 891.076 .949915 5.611 1.404 26 771.386 .887244 6.482 1.023 27 888.776 .948792 5.626 1.407 27 769.613 .88627-2 6.497 1.026 28 886.488 ,94707-3 5.640 1.411 28 767.897 .885303 6.511 1.6EO 29 884.211 .946556 5.655 1.415 2!) 706.190 .884336 6.526 1.083 30 881.946 2.945442 5.669 1.418 30 764.489 2.883371 6.540 1.037 31 879.693 2.944331 5.684 1.422 31 762.797 2.882409 6.555 1.041 32 877.451 .943223 5.698 1.426 32 761.112 .881448 6.569 1.044 33 875.221 .942118 5.713 1.429 33 759.434 .880490 6.584 1.04' 5 34 873.002 .941015 ! 5.7'27 1.433 34 757.764 .87'9534 6.598 1.651 35 870.795 .939916 5.742 1.437 S5 756.101 .87'8580 | 6.613 1.655 36 868.598 .938319 5.756 1.440 88 754.445 .877627 6.627 1.659 37 866.412 .937725 5.771 1.444 37 752.796 .876678 6.642 1.662 38 864.238 .936033 5.785 1.447 38 7'51.155 .8757-30 6.656 1.0C6 39 862.075 .935545 5.800 1.451 39 7'49.521 .874784 0.671 1.070 40 859.922 2.034459 5.814 1.455 40 747.894 2.873840 6.085 1.073 41 857.780 2/J33376 5.829 1.458 41 746 271 2.872898 6.700 1.677 42 855.648 .932295 5.844 1.462 42 744.001 .871859 6.714 1.080 43 853.52? .931218 5.858 1.466 43 743.055 .871021 6.729 1.084 44 851.417 .930142 5873 1.469 44 741.456 .870086 6.743 1.688 45 849.317 .929070 5.887 1.473 45 739.864 ,869152 6.758 1.691 46 847.228 .928000 5.902 1.476 46 738.279 .868221 6.773 1.695 47 845.148 .926933 5.916 j 1.480 47 736.701 .867'291 6.787 1.699 48 843 080 .925809 5.931 1.484 48 735.12:) .866363 6.802 1.702 49 841.021 .924807 5.945 1.487 49 7'33. 564 ! .865438 6.816 1.706 50 838.972 2.923747 ; 5.960 1.491 50 732.005 J2. 864514 6.831 1.710 51 836.933 2.922691 5.974 1.495 51 730.454 2.863593 6.845 1.713 52 834.904 .921637 5.989 1.498 52 798.909 .862673 6.860 1.717 53 832.885 .920585 6.003 1.502 53 727.370 .8617-55 6.874 1.7SQ 51 &30.876 .919536 6.018 1.505 54 725.838 .8608-10 6.889 1.724 55 828.876 .918489 6.032 1.510 55 7-24.312 .859926 6.903 1.728 56 826.886 .917446 i 6.047 1.513 56 722.793 .859014 6.918 1.731 57 824.905 .916404 6.061 1.517 57 7'21.280 .858104 6.932 1.735 58 822.934 .915365 6.076 1.520 58 719.774 .857196 6.947 1.739' 59 820.973 .914329 6.090 1.524 59 718.273 .856290 1 6.961 1.742 60 819.020 2.913295 6.105 1.528 60 716.779 2.855385 6.976 1.746 TABLE IV. RADII, LOGARITHMS, OFFSETS, ETC. Deg. D. Radius. K. Loga- rithm. log. B. 155? t. Mid. Ord. in. Deg. . Radius. K. Loga- rithm. log. K. 185? t. Mid. Ord. in. 8 0' 716.779 2.855385 6.976 1.746 9 0' C37.275 2.804327 7. 846 1.965 1 715.291 .854483 6.990 1.749 l 636.099 .803525 7.860 1 968 2 713.810 .853583 7.005 1.753 2 634.928 .8027^4 7.875 1.972 3 712.335 .852684 .019 1.756 3 633.761 .801C26 7.889 1.975 4 710.865 .851787 .034 1.761 4 632.599 .801128 7.904 1.979 b 709.402 .t 50892 .048 1.704 5 631.440 .800332 r.i8 1 . 983 6 707.945 .849999 .063 1.768 6 630.286 .799538 7.933 1.987 7 706.493 .849108 .077 1.771 7 629.136 .798745 7.947 1.990 8 705.048 .848219 .092 1.775 8 627.991 .797953 7.962 1.994 9 703.609 .847331 .106 .778 9 626.849 .797163 7 976 1.998 10 702.175 2.846445 .121 .782 10 625.712 2.796374 7.991 2.001 11 700.748 2.845562 .135 .786 11 624.579 2.795587 8.005 2.005 12 699.326 .844679 .150 .790 12 623.450 .794801 8.020 2.008 13 697.910 .84875)9 .164 .793 13 622.325 .794017 8.034 2.012 14 696.499 : 842921 .179 .797 14 621.203 .793234 8.049 2.016 15 695.095 .842044 .193 .801 19 620.087 .^92453 8.063 2.019 16 693.696 .841169 .208 .804 16 618.974 .791673 0.078 2.023 17 692.302 .840296 .222 .807 17 617.865 .790894 8.092 2.026 18 690.914 .839424 .237 .811 18 616.760 .790117 8.107 2.030 19 689.532 [838555 .251 .815 19 615. 6CO .789341 8.121 2.034 20 688.156 2.837687 .266 .819 20 614.C63 2.788566 8.136 2.037 21 686.785 2.836821 .280 .822 21 613.470 2.787793 8.150 2.041 22 685.419 .8M5si.50 .295 .826 22 612.380 .787021 8.165 2.045 23 684.059 .835093 .09 .829 23 611.2!J5 .786251 8.179 2.C48 24 682.704 .834232 .324 .833 24 610.214 .785482 8.194 2.052 23 681.454 .S33 t $;'3 .338 .837 25 609.136 .784714 8.208 2.056 26 680.010 .832515 .353 .840 1 26 608.062 .783948 8.223 2.060 27 678.671 .831(360 .367 1.844 27 606.992 .788188 8.237 2.003 28 677.338 .8';OH05 .382 1.848 23 605.926 .782420 8.252 2. 066 29 676.008 .KtfJ!!.53 .396 1.851 29 604.864 .781657 K.266 2.070 30 674.686 2.821)102 .411 1.855 30 603.805 2.780897 8.281 2.074' 31 673.369 2.828253 .425 1.858 i 31 602.750 2.780137 8.295 2.077 32 G72.056 .827405 .440 1.862 i 32 601.698 .779379 8.310 2.081 33 WO. 748 .836500 .454 .866 i 33 600.651 .778622 8.324 2.084 34 600.440 .8*5715 .469 .869 34 599.607 .777867 8.339 2.088 35 668.148 .834878 .483 .873 35 598.567 .777112 8.853 2.092 36 666.856 .824032 .598 .877 36 597.530 .776360 8. 68 2.096 37 665.668 .823193 .512 .880 37 596.497 .775608 8.382 2.099 38 004. 281 i .822855 .527 .884 38 595.467 .774858 8.397 2.103 39 61 13. 008 .821519 .541 .887 39 594.441 .774109 8.411 2.106 40 661.736 2.S2U685 .556 .892 40 593.419 2.773361 8.426 2. 110 41 660.468 2.819852 .570 .895 41 592.400 2.772615 8.440 2.113 42 0.59. :>( .819021 .585 .899 42 591.384 .771870 8.455 2.117 43 657.947 .818191 .599 .903 43 590.37'2 .771126 8.469 2.12j. . 44 656.694 .817363 .614 .906 44 589.364 .770383 H.484 2.125 45 655.446 .816537 .628 .910 45 588.359 .769642 8.498 2.128 46 654.202 .815712 .643 .914 46 587.357 .768002 8.513 2.132 47 652.963 .814889 .657 .918 47 586.359 .768164 8.527 2.135 48 651.729 .8140(37 .<;7'2 .921 48 585.364 .767426 8.542 2.139 49 650.41)9 .813247 .686 .924 49 584.373 .766690 8.556 2.142 50 649.214 2.812428 .701 .928 50 583.385 2.765955 8.571 2.147 51 648.054 2.811611 .715 .932 51 582.400 2.765221 8.585 2.150 52 646.838 810796 .730 .95 52 581.419 .764489 8.600 2.154 53 645.627 .7'44 .99 53 580.441 .763758 8.614 2.158 54 644.420 .809169 .759 .943 54 579.466 .763028 8.629 2.161 55 643.218 .808358 .773 9-6 j 55 578.494 .762299 8.643 2.105 56 642.021 .807549 .788 .950 56 577.526 .761572 8.658 2.168 57 640.828 .806741 .802 .953 57 576.561 .7'60845 8.67'2 2 172 58 639.639 .805935 .817 .957 58 575.599 .760120 8.687 2.175 59 638.455 .805130 .831 .961 59 574.641 .759397 8.701 2.179 GO 637.275 2.804327 .846 .965 60 573.686 2.758674 8.716 2.183 284 TABLE IV. RADII, LOGARITHMS, OFFSETS, ETC. Deg. Radius. Loga- rithm. Tang. Off Mid. Ord. Deg. Radius. Loga- rithm. Tang. Off. Mid. Ord. I). R. log. R. t. in. D. K. log. K. t. in. 10 0' 573.686 2.758674 8.716 2.183 12 0' 478.339 2.679735 10.453 2.020 2 571.784 .757232 8.745 2.190 2 477.018 j .G7NW-> 10.482 2.028 4 569.896 .755796 8.774 2.198 4 475.7t)5 .077338 10.511 2.0:35 6 508.020 .754:304 8.803 2.205 6 47'4.400 ! .070145 10.540 2.042 8 566,166 .752937 8.831 2.212 8 473.102 .074954 H-.569 2.050 10 504.305 .751514 8.800 2.219 10 471.810 .073707 10.597 2.657 12 502.406 .750096 8.889 2.227 12 470.526 .072584 10.020 2.004 14 660.688 .748683 8.918 2.234 14 409.249 .671403 10.055 2.671 16 558.823 .747274 8.947 2.241 16 467.978 .070226 10.684 2.079 18 557.019 2.745870 8.976 2.234 18 466.715 12.609052 10.713 2.686 20 555.227 2.744471 9.005 2.256 20 405.459 2.607881 10.742 2.093 22 553.447 .743076 9.034 2.263 22 404.209 .600713 10.771 2.701 24 551.678 .741686 9.003 2.27'0 ?4 402.900 .605549 io.eoo 2.708 20 549.920 .740300 9.092 2.278 26 401.729 ! .004387 10.829 2.715 28 548.174 .7:38918 $.121 2.285 28 400.500 I (i(jf<22!) 10.858 2.722 30 546.438 .737541 9.150 2.293 30 459.276 ! .602074 10.887 2.730 32 544.714 .7'36169 9.179 2.300 32 458.000 .0001)22 10.916 2.737 34 543.001 .734800 9.208 2.307 34 450.850 .!77:$ 10.945 2.744 36 541.298 .733436 9.237 2.314 36 455.040 .U5W*S 10.973 2 752 38 539.006 2.732077 -9.266 2.3*1 38 454.449 i2. 057485 11.002 2.759 40 537.924 ! 2. 730721 9.295 2.329 40 453.259 2.050345 11.031 2.706 42 526.253 .729370 9.324 2.330 42 452.073 .055208 11.000 2.774 44 534.593 .728023 9.353 2.343 44 450.894 i .054075 11.089 46 532.943 .7'26681 9.382 2.351 46 449.722 .052944 11.118 2 788 48 1 531.303 .725342 9.411 2.358 48 448.550 .051810 11.147 2! 795 50 529.673 .724008 9.440 2.365 EO 447.395 .050G91 11.170 2.803 52 528.053 .722677 9.409 2.372 52 446.241 .6456 34 496.195 ,695052 i 10. 077 2.526 ! 34 423.316 .020005 11.K12 2.1JG3 36 494.774 .694407 10.106 2.5S3 36 422.288 025004 11. WO 2.971 38 493.301 2.693165 10.135 2.540 38 421.250 2.024540 11.809 2.978 40 491.950 2.691926 10.164 2.547 40 420.2-33 2.023490 11.898 2.985 42 490.559 .690692 ; 10.192 2.555 ; 42 419 215 .622437 11.927 2.992 44 489.171 .689460 J10.221 2.562 j 44 418.203 .621:387 11.856 3.000 46 487.790 .688233 10.250 2.509 46 417.195 .02033!) 11.685 3.007 48 480.417 .687008 10 279 2.577 48 416.192 .019294 12.014 3.014 50 485 051 685788 110.308 2.584 CO 415.194 .018251 12.043 3.022 52 483.094 .684570 ! 10. 337 2.591 52 414.201 .017211 12.071 3.029 54 482.344 .683357 ;10.306 2.598 64 413.212 .010173 12.100 3.036 56 481.001 .682146 110 395 2.606 56 412.229 .015138 12.129 3.044 58 479.000 .680939 j 10. 424 2.613 ! 58 1 411.250 .614106 12.158 3.051 60 i 478.339 2.679735 !l0.453 2.620 ! 60 i 410.275 i 2. 613075 12.187 3.058 285 TABLE IV. RADII, LOGARITHMS, OFFSETS, ETC. Deg. I). Radius. Loga- rithm. log. R. Tan. Oil. t. Mid. Orel. 111. Deg. B. Radius. K. Loga- rithm. log. R. Tan. Off. t. Mid. Ord. m. 14 0' 410.273 2.613075 12.187 3.058 16 ('' 359.265 2.555415 13.917 3.496 2 409. 8 Jtf .612048 12.216 3.065 2 358.528 .554517 13.946 3.504 4 408.341 .611023 12.245 3.073 4 a57.784 .553621 13.975 3.511 6 407.390 .610000 12.274 3.080 6 357.048 .552727 14.004 3.518 8 408.424 .603980 12.302 3.087 8 a56.31i; .551834 14.0a3 3.526 10 40.1.473 .607962 12.331 3.095 10 [355.585 .550944 14.061 3.533 12 404.528 .606946 12.360 3.102 12 354.859 .550055 14.090 3.540 14 403.583 .605933 12.389 3.109 14 354.1:35 .549169 14.119 3.547 10 403.645 .604923 12.418 3.117 16 a53.414 .548284 14.148 3.555 18 401.712 .603914 12.447 3.124 18 352.696 .547401 14.177 3.562 20 400.782 2.602903 12.476 3.131 20 351.981 2.546519 14.205 3.569 22 899.857 .601905 12.504 3.1:38 22 351.269 .545640 14.234 3 . 577 2i 398.937 .603904 12. 533 j 3. 146 24 350.560 .544762 14.263 3.584 26 398.020 .59J905 12.562 3.153 26 349.854 .543887 14.292 3.591 28 397.108 .59893$ 12.591 3.160 28 349.150 .543013 14.320 3.599 30 398.200 .597914 12.620 3.168 30 348.450 .542140 14.349 3.606 32 39.-). 29ti .59692:2 12.649 3.175 32 347. 75 .541270 14.378 3.613 34 391.396 .595933 12.678 3.182 34 347.057 .540401 14.407 3.621 36 393.501 .594945 12. 708 13.190 36 346.365 .539535 14.436 3.628 38 392.609 .593960 12. 735 j 3.197 38 345.676 .538670 14.464 3.635 40 391.722 2. 592:) 78 12.764 3.204 40 344.990 2.537806 14.493 3.643 42 390.83$' .5919'.)7 12.793:3.211 42 344.306 . 53(5945 14.522 3.650 44 389.959 .591019 , 12.822 3.219 44 343.625 .536085 14.551 3.657 46 389.084 .590043 12.851 3.2215 46 342.947 .5:35227 14.580 3.664 48 388.212 .5890J9 12.830 3.233 48 342.271 . 534370 14.608 3.672 50 837.845 .5330.) 7 12.908 3.241 50 341.598 .533516 14.637 3.679 52 888.481 .587128 12.937 3.243 52 340. 92 i .5:32663 14.666 3.686 54 885.621 .586161 12.986 3.255 54 340.260 .531811 14.695 3.694 56 384.765 .585196 12.99") 3.263 58 339.595 .530962 14.723 3.701 5i 383.913 .584233 13.024 3.270 58 338.933 .530114 14.752 3.708 15 a 833.035 2.583272 13.053 3.277 17 338.273 2.529268 14.781 3.716 2 83*. 2*0 .532314 13.031 3.284 2 337.616 .528424 14.810 3.723 4 881. 88 J .581358 13.110 3.21)2 4 3:36.982 .527581 14.838 3.7:30 6 380.543 .580403 13.139 3.299 6 336.310 .526740 14.867 3.738 8 37;). 70.) .579451 13.163 3.30J 8 335.6601 .525900 14.896 3.745 10 378.8*1 .578501 13.197 3314 10 335.013 .525062 14.925 3.752 12 378.051 .577553 13.22) 3.3-21 12 334.369 .524226 14.954 3.760 14 377.251 .576608 13.254 3.323 14 333.727 .523392 14.982 3.767 1(5 376.412 .575664 13.233 3.336 16 333.033 .522,559 15.011 3.774 18 375.597 .574722 13.312 3.343 18 332.451 .521728 15.040 3.781 20 374.788 2.573783 13.341 3.350 20 331.816 2.520898 15.069 3.789 22 373.977 .572845 13.370 3.353 22 331.184 .520070 15.097 3.796 24 373.173 .571910 13.399 3.365 24 330.555 .519244 15.126 3.803 26 872.372 .570977 13.427 3.372 26 329.923 .518419 15.155 3.811 28 371.574 .570045 13.455 3. >79 28 329.303! .517596 15.184 3.818 30 370.780 .569116 13.435 3.337 30 323 639 .516774 15.212 3.825 32 369. 93 ') .568189 13.514 3.394 32 323.061 .515954 15.241 3.8:33 31 36 361). 202 368.418 .567264 13.543 .566340 13.572 3.401 3.409 34 327.413 36 32li. 8-28 515138; 15.270 .514:319| 15.299 3.840 3.847 38 367. 63 7 .565419 13.60) 3.416 38 326.215 .'5ia504 15.327 3.855 40 868.859 2.564530 13.629 3.423 40 325.604 2.512690 15.a56 3.862 42 366.085 .563532 13.658 3.431 42 324.996 .511878! 15.385 3.869 44 365.315 JS63367 13.687 3.438 44 324.390 .511067 15.414 3.877 46 364.547 .561754 13.716 3.445 46 323.736 .510258 15.442 3.884 48 363.783 .560343 13.744 3.452 48 323.184 .509451 15.471 3.891 50 363.022 .559933 13.773 3.460 50 322.585 .508645 15.500 3.899 52 362.264 .559028 13.802 3.467 52 321.939 .507'840 15.528 3.906 54 361.510 .558120 13.831 3.474 54 321.394 .507037 15.557 3.013 56 360.758 .557216 13.860 3.482 56 320.801 .50(5236 15.586 3.920 58 360.010 .556315 13.889 3.489 58 320.211 .505436 15.615 3.928 60 359.265 2.555415 13.917 3.496' 60 1319.623 2.504638 15.643 3.935 280 TABLE IV. RADII, LOGARITHMS, OFFSETS, ETC. Deg. Radius. Loga- rithm. Tang. Off. Mid. Ord. Deg. *"- ias: Tang. Off. Mid. Ord. IX R. log.R. t. m. i IX R. log. R. t. m. 18 0' 319.623 2.504638 15.643 3.985 20 0' 287. 939 ! 2. 459300 17.365 4.374 2 319.037 .503841 15.672 3.942 1 10 i 285.583 .4557:33 17.508 4 411 4 318.453 .503045 15.701 3.950 20 283.267 .452195 17.651 4.448 6 317.871 .502251 15.730 3.957 30 280.988 .448688 17.794 4.484 8 317.292 .501459 15.758 3.964 40 278.746 .445209 17.937 4.521 10 316.715 .500668 15.787 3.972 50 1 276.541 .441759 18.081 4.55S 12 316.139 .499879 15.816 3 979 21 ()' 274.370 2.438337 18.224 4.594 14 315.566 .499091 15.845 3.986 N 10 272.234 .434943 18.367 4.631 16 314.993 .498:304 15.873 3.994 20 271.032 .431576 IS. 509 4.668 18 314.42ii .497519 15.902 4.001! 30 268.002 18.652 4.704 20 22 313.860 2.496736 313.295! .495953 15.931 4.008! 15.959 4 016 i 40 50 266.024 .424921 264.018 .421633 18.795 18.938 4.741 4.778 24 312.732 .495173 15.988 4.023 Q2 0' 262.042 2.418371 19.081 4.814 26 312.172 .494393 16,017 4.030 10 260.098 .4151:34 19.224 4.851 28 311.613 .493616 16.046 4.038 20 258.180 .411922 4 888 30 311.056 .492839 16.074 4.045' 30 i 256.292 408734 19! 509 4.925 32 310.502 .492064 16.103 4.052i; 40 254.431 .405571 19.052 4.961 34 309.949 .491291 16.132 4.060 j 50 252 599 ! 402431 19.7SI4 4.998 36 309.399 .490518 16.160 4. 067!! 23 0' 250. 793 ' 2. 399:315 19.937 5.035 38 308.850 .489748 16.189 4.074 10 249.013! .39Q222 20.079 5. Oil 40 308 303 ! 2 488978 16.218 4 081 247.258 .393151 20.222 5.108 42 44 307.759 307.216 .488210 .487444 16.246 4.089i| 16.275 4.09(5 1245.473; .390103 243.825 .387077 20.364 20.507 5.145 5.182 46 306.675 .486679 16.304 4.103 242.144 .384074 20.64J 5.218 48 50 52 54 56 58 30(5.136 305.599 305.064 304.531 304.000 303.470 .485915 .485152 .484391 .483632 .482873 .482116 16.333 16.361 16.390 16.419 16.447 16.476 4.111 4.118 4.125 4.133 4.140 4.147 24 0'; 240. 487 10:238.853 20 237.241 30 235.052 40 234.084 50 232.537 2.381091 .378130 .375190 .372270 .369371 .3(56492 20.791 20.933 21.076 21.218 21.360 21.502 5.255 5.292 5.329 5.306 5.402 5.439 19 0' 302.943 2.481361 16.505 4.155 25 0' 231.011 2.363633 21.644 5.476 2 302.417 .480607 16.533 4.162 10 229.50(3 .360794 21.786 5.513 4 301.893 .479854 16.562 4.169 20 228.020 .357974 21.928 5.549 -6 301.371 .479102 16.591 4.177! 30 '226.555 .355173 22.070 5.586 8 '300.851 .478:352 16. 620 ! 4. 184! 40 225.108 .352391 22.212 5.623 10 300.333 .477(503 16. 648 14. 191 j 50 223.680 .349627 22.353 5.660 12 14 16 18 i 299. 816 299.302 298.789 ! 298.278 .476855 .476109 .475364 .474621 16. 677 j 4. 199' 16. 706! 4. 206 16.734 4.213 16. 763| 4. 221 26 10 20 30 222.271 220.879 219.506 218.150 2.346882 .344155 .341446 .338755 22.495 22.1537 22.778 22.920 5.697 5.734 5.770 5.807 20 1297.768 2.473878 16.792 4.228 40 216.811 .336081 23.062 5.844 22 1297.260 .473137 16.8201 4. 235 50 '215.489 .333424 23.203 5.881 24 '296.755 .472398 16.849 4.243l;27 : 214. 183 2.330785 23.345 5.918 26 296.250 .471659 16.878 4.250 ! 10 212.893; .328162 23. 4S6 5.955 28 295.748 .470922 16.906 4.257 20 211.620 .325556 23.627 5.992 30 i 295.247 .470186 16.! 135 4.265 30 210.362 .322967 23.7(59 6. (129 32 1294.748 .469452 16.964 4.272 40 209.119 .320:393 23.910 6.065 34 294.251 .468718 1 .992 4.279 i 50 207.891 .317836 24.051 6.102 36 38 29:3.756 .467986 293.262 .467256 'oS ' 1'Sl 28 0' 206.678 12.315295 24.192 10 205 . 480 . 312769 24 . 333 6.139 6.176 40 292.770 2.466526 .078 ! 4.301 20 ! 204.296 .310259 24 .47'4 6.213 42 292.279 .4(55798 .10714.308 30 1203.125 .307764 24.615 6.250 44 291.790 .465071 .186 4.816 40 1 201. 969; .305285 24.756 6.287 46 291.303 .464345 .164 4.323 50 200.826 : .302820 24.897 6.324 48 290.818 .463621 ,198 [4.880 29 0' 199.696 2.300370 25.038 6.360 50 290. 334 .462897 1 .222 4.338 10 198.580 .297935 25.179 6.398 52 289. 851 .462175 .250 4.345 20 197.476 .295515 25.320 6.435 54 ! 289.371 .461455 .279 4.352 30 196. 3&5 .293108 25.4CO 6.472 56 : 288.892 .460735 .308 4.360 1 40 195.306 .290716 25.601 6.509 58 i 288.414 .460017 17.336 4.367;i 50 194.240 .288338 25.741 6.545 60 '< 287.939 2.459300 17.365 4.374 80 193.185! 2. 285974 25.882 6.583 287 TABLE IV. RADII, LOGARITHMS, OFFSETS, ETC. ! 1 Deg. Radius. Loga- Tang. ' Mid. rithm. Off. Ord. Deg. Radius. Loga- rithm. Tang. Off? Mid. I Ord. 1 D. B. log. R. t. in. D. K. log. K. t. in. 30 20' 191.111 2.281286 5 J6.163 6.657 38^ 30' 151.657 2.180863 32 .969 8.479 4 18 J.083 .2701 552 ; 26.443 6.731 V 49.787 .175 t::. 33 .381 8.592 31 V 187.099 .272071 26.724 6.805 30 147.965 .170160 88 .792 8.704 X t) 18 5.158 .267.' >41 15 27.004 ft. 879 40< ()' 1 46.190 .164 918 84 .202 8.816 -- A 18. 1.258 .26* )62 j 27.284 6.958 30 1 44.460 .15 747 34 .612 8.929 32 0' 181.398 .258632 27.564 .027 41 0' 142.773 .154645 35.024 9.041 2 17 J.577 .254; 250 , 27.843 .101 30 1 41.127 .14S 6i< 35 .429 9.154 40 17 7.794 .249916 { 28.123 .175 42 0' i 39.521 .144641 35.837 9.267 33 (V 7 E5.047 .245( 528 , 28.402 .250 :>'.' 1 37.97)5 i >( 36 .244 9.380 t 7 L336 .241: 58(5 28.680 .324 ; 43 5 0' 1 3(5.425 .134 syr :,(i .650 9.493 40 72.659 .237188 i 28.959 .398 : 80 134.932 .130114 37 .056 9.606 34 (V 7 1.015 2.23:1035 29.237 .473 44 ' (i 133.4732.125395 37 .461 9.719 2 f6 J.404 .228 ( )24 29.515 .547 i 30 1 32.049 .12( 1784 ;;; .865 9.8512 4 16 7.825 .224S 65 29.793 .621 45 = 1 30.6561 .IK i:;i 38 .268 9.946 35 0' 1(5 3. 275 .220 08 10.071 30 1 29.296! .111 584 88 .671 10.059 20 1(54.75(5 .216842 10.348 7.770 46 0' 127.9651 .KV7092 39.073 10.173 4 16. 1.2(50 .212 W5 10.625 7.845 20 1 26.664 .10$ !65 81 .474 10.286 36 0' 161.803 .20S988 30.902 7.919 ! 47 125.392 .098270 3! .875 10.400 2 o ie 0.3(58 .205 119 11.178 7.994 80 1 24.148 .09f !'.i:> -l< .275 10.516 4 15 8.960 .201 388 11.454 8 068 1 , 48 1 22.930! .08' ;.->; 41 .674 10.628 37 0' 15 7.577 .lit; 494 31.730 80 121.738! .085425 41 .072 10.742 20 156.220 .193736 32 006 8*218 49 0' 120.5711 . 0812-13 41<469 10.856 40 : 154.8S7 38 0'; 153.578 .190014 i 32. 282 2.186328 32.557 8.292 8 36; 50 SO 119.429 .077109 41.866 0' 118.3102.073022 42.262 10.970 11.085 I TABLE V. -CORRECTIONS FOR TANGENTS AND EXTERNALS. FOR TANGENTS, ADD FOR EXTERNALS, ADD Ang 5 10 15 20 25 30 Ang 5 10 15 2 25 30 A Cur. Cur. Cur. Cur. Cur. Cur. A Cur. Cur. Cur. Cur. Cur . Cur. 10 .03 .06 .09 .13 .16 .19 10 001 .003 .004 .006 .007 .008 20 .06 .13 .19 .26 .32 .39 20 .OC Mi .011 .017 .( 23 .028 .034 30 .10 .19 .29 .39 .49 .59 30 .0 8 .025 .038 .( 51 .065 .078 40 .13 .26 .40 .53 .67 80 40 M ;:i .046 .07'0 .C 98 .117 .141 50 .17 .34 .51 .68 .85 1 02 50 .() g .075 .116 .1 61 .189 .227 60 .21 .42 .63 .84 1.051 1.27 60 .0. ,i; .112 .168 1 .S 25 .28.' J .340 70 .25 .51 .76 1.02 il.28jl.54 70 .0 .159 .240 .321 .40; J .485 80 .30 .61 .91 1.22 1.53 1.84 i 80 .1 .220 .332 1 - 4 45 .558 ! .671 90 .36 .72 1.09 1.45 1.83 2.20 90 .149 .299 .450 M .0:1 .756 .910 100 .43 .86 1.30 1 74 2 18 I 2.62 100 .2( (0 .401 .604 1 .1 '(19 1.015 1.221 110 .51 1.Q8 1.56 2 08 2.61 3.14 110 .8 is .536 .806 i.( >S2 1.S55 1.633 120 .62 1.25 1.93 2.52 3 16 3.81 120 .3 50 .721 1.086 1.456 1.82. 5 2.197 ] i 288 TABLE VI. TANGENTS AND EXTERNALS TO A 1 CURVE. r" j II Angle. Tan- gent. Exter- nal. Angle. Tan- gent. Exter- nal. Angle. Tan- gent. Exter- nal. A T. E. A T. E. A T. E. 1 50.00 .218 11 551.70 26.500 81 1061.9 i 97.577 10' 58. iW .297 10' 1. 560.11 27.313 10' 1070.6 99.155 20 60.07 .388 20 568.53 28.137 20 1079.2 : 100.75 30 75.01 .491 30 ' 576.95 28.974 30 i 1087.8 102.35 40^ 83.34 .606 40 585.36 29.S24 40 1096.4 103.97 50 91.68 .733 50 593.79 30.686 50 1105.1 105.60 2 100.01 .873 12 602.21 31.561 22 1113.7 107.24 10 108.35 1.024 10 610.64 32.447 10 1122.4 108.90 20 116.68 1.188 20 619.07 as. 347 20 1131.0 110.57 30 ! 125.02 1.364 30 627.50 34.259 30 1139.7 : 112.25 40 133.36 1.552 40 6:35.93 35.183 40 1148.4 : 113.95 50 141.70 1.752 50 644.37 36.120 50 1157.0 ! 115.66 3 150.04 1.964 13 652.81 i 37.070 23 1165.7 117.38 10 158.38 2.188 10 661.25 ; 38.031 i 10 1174.4 i 119.12 20 166.72 2.425 ! 20 669.70 39.000 20 1183.1 1 120.87 30 175.06 2.674 ! 30 678.15 : 39.993 . 30 1191.8 ! 122.63 40 183.40 2.934 j 40 686. 6. 40.993 !! 40 1200.5 124.41 50 191.74 3.207 i 50 695.06 ! 42.004 ; ! 50 1209.2 i 126.20 4 200.08 3.492 'l4 703.51 < 43.029 24 1217.9 128.00 10 208.43 3.790 i 10 711.97 44.066 10 1226.6 1 129.82 20 216.77 4.099 20 720.44 45.116 20 1235.3 131.65 30 225.12 4.421 30 728.90 46.178 30 1244.0 133.50 40 233.47 4.755 40 737.37 47.253 40 1252.8 135. ,35 50 241.81 5.100 50 745.85 48.341 ;| 50 1261.5 ! 137.23 5 250.16 5.459 15 754.32 49.441 25 1270.2 j 139.11 10 258.51 5.829 10 762.80 50.554 10 1279.0 141.01 20 266.86 6.211 20 771.99 51.679 20 1287.7 142.93 30 275.21 6.606 30 779.77 52.818 30 1296.5 144. 85 40 283.57 7.013 40 788.26 53.969 40 1305.3 146.79 50 291.92 7.432 50 796.75 55.132 50 1314.0 148.75 6 300.28 7.863 16 805.25 56.309 26 1322.8 150.71 10 308.64 8.307 10 813.75 57.498 10 1331.6 152.69 20 316.99 8.762 20 822.25 58.699 20 1340.4 154.69 30 325.35 9.230 30 830.76 59.914 30 1349.2 156.70 40 333.71 9.710 40 839.27 61.141 40 1358.0 158.72 50 342.08 10.202 50 847.78 62.381 50 1366.8 160.76 7 &50.44 10.707 17 856.30 63.634 27 1375.6 162.81 10 &58.81 11.224 10 864.82 64.9(30 10 1384.4 104.86 20 367.17 11.753 20 873.35 66.178 : 20 131)3.3 166.95 30 375.54 12.294 30 881.88 67.470 30 1402.0 1189.04 40 383.91 12.847 40 890.41 68.774 : 40 1410.9 171.15 50 392.28 : 13.413 50 898.95 ; 70.091 i, 50 1419.7 173.27 8 400.66 i 13.991 18 907.49 i 71.421 28 1428.6 175.41 10 409.03 14.582 10 916.03 72.764 10 1437.4 177.55 20 417.41 15.184 20 924.58 74.119 !| 20 1446.3 179.72 30 425.79 15.799 30 933.13 75.488 ! 30 1455.1 181.89 40 434.17 16.426 40 941.69 76.869 40 1464.0 184.08 50 442.55 17.065 50 950.25 78.264 50 1472.9 186.29 9 450.93 17.717 19 958.81 79.671 ! 29 1481.8 188.51 10 459.32 18.381 10 967.88 81.092 10 1490.7 190.74 20 467.71 19.058 20 975.96 82.525 20 1499.6 192.99 30 476.10 19.746 30 984.53 88.973 i 30 1508.5 195.25 40 484.49 20.447 40 993.12 85.431 40 1517.4 197.53 50 492.88 21.161 50 1001.7 86.904 i 50 1526.3 199.82 10 501.28 21.887 20 1010.3 88.389 ! 30 1535.3 202.12 10 509.68 22.624 10 1018.9 89.888 10 1544.2 204.44 20 518.08 23.375 20 1027.5 91.399 20 1553.1 206.77 30 526.48 21.138 30 1036.1 92.924 ! 30 1562.1 209.12 40 534.89 24.913 40 1044.7 94.462 40 1571.0 211.48 50 k. 543.29 25.700 50 1053.3 i 96.013 50 1580.0 213.86 TABLE VI. TANGENTS AND EXTERNALS TO A 1 CURVE. Angle. A Tan- gent. T. Exter- nal. E. 1 Angle. A Tan- gent. T. Exter- nal. E. Anglo. A Tan- gent. T. Exter- nal. E. 31 1589.0 216.25 I 41 2142.2 387.38 51 2732.9 618.39 10 1598.0 218.66 10' 2151.7 390.71 10' 2743.1 622.81 20 1606.9 221.08 20 2161.2 394.06 20 2753.4 627.24 30 1615.9 223.51 30 2170.8 397.43 30 2763.7 631.69 40 1624.9 225.96 40 2180.3 400.82 40 2773.9 636.17 50 1633.9 228.42 50 2189.9 404.22 50 2784.2 640.66 32 1643.0 2:30.90 42 2199.4 407.64 52 2794.5 645.17 10 1652.0 ! 233.39 10 2209.0 411.07 10 2804.9 649.70 20 1661.0 2&5.90 * 20 2218.6 414.52 20 2815.2 654.25 30 1670.0 2:38.43 30 2228.1 417.99 30 2825.6 658.83 40 1679.1 240.96 40 2237.7 421.48 40 2835.9 663.42 50 1688.1 I 243.52 50 2247.3 424.98 50 2846.3 668.03 33 1697.2 246.08 43 2257.0 428.50 53 28S6.7 672.66- 10 1706.3 248.66 10 2266.6 432.04 10 2867.1 677.32 20 1715.3 251.26 20 2276.2 435.59 ! 20 2877.5 681.99 30 1724.4 253.87 ; 30 2285.9 439.16 ; 30 2888.0 686.68 40 1733.5 256.50 40 2295.6 422.75 ! 40 2898.4 691 .40 50 1742.6 259.14 50 2305.2 446. &5 ; 50 2908.9 696.13 34 1751.7 261.80 44 2314.9 449.98 : 54 2919.4 700.89 10 1760.8 264.47 10 2324.6 453.62 i 10 2929.9 705.66 20 1770.0 267.16 20 2^334.3 457.27 i 20 2940.4 710.46 30 1779.1 269.86 30 2:344.1 460.95 30 2951.0 715.28 40 1788.2 272.58 40 2353.8 464.64 40 2961.5 720.11 50 1797.4 275.31 50 2363.5 468.35 50 2972.1 724.97 35 1806.6 278.05 45 2373.3 ! 472.08 55 2982.7 729.85 10 1815.7 280.82 10 2383.1 475.82 10 2993.3 734.76 20 1824.9 283.60 20 2392.8 479.59 20 3003.9 739.68 30 1834.1 280.39 30 2402.6 483.37 30 3014.5 744.62 40 1843.3 289.20 40 2412.4 487.17 40 3025.2 749.59 50 1852.5 292.02 50 2422.3 490.98 50 3035.8 754.57 36 1861.7 294.86 46 2432.1 j 494.82 56 3046.5 759.58 10 1870.9 297.72 10 2441.9 498.67 i 10 3057.2 764.61 20 1880.1 300.59 20 2451.8 i 502.54 20 3067.9 769.66 30 1889.4 ! 303.47 30 2461.7 506.42 30 3078.7 774.73 40 1898.6 306.37 40 2471.5 510.33 40 3089.4 779.83" 50 1907.9 309.29 50 2481.4 514.25 50 3100.2 784.94 37 1917.1 312.22 47 2491.3 518.20 57 3110.9 790.08 10 1926.4 315.17 10 2501.2 522.16 10 3121.7 795.24 20 1935.7 318.13 20 2511.2 526.13 20 3132.6 800.42 30 i 1945.0 321.11 30 2521.1 530.13 ! 30 ! 3143.4 805.62 40 1954.3 324.11 40 2531.1 534.15 40 I 3154.2 : 810.85 50 1963.6 327.12 50 2541.0 538.18 50 1 3165.1 816.10 38 1972.9 330.15 48 2.551.0 542.23 58 i 3176.0 821.37 10 1982.2 1 333.19 10 2561.0 546.30 > 10 i Ml 86. 9 826.66 20 1991.5 336.25 20 2571.0 ] 550.39 20 3197.8 K3i UK 30 200J3.9 339.32 30 2581.0 | 554.50 30 3208.8 837.31 40 2010.2 1 342.41 40 2591.1 i .'58.63 40 3219.7 842.67 50 2019.6 345.52 50 2601.1 562.77 .50 3230.7 848.06 39 2029.0 S48.64 49 2611.2 566.94 59 3241.7 853.46 10 20:38.4 &51.78 10 2621.2 571.12 10 3252.7 858.89 20 2047.8 354.94 20 2631.3 1 575.32 i 20 3263.7 I 864.34 30 2057.2 358.11 30 2641.4 579.54 i 30 3274.8 869.82 40 2066.6 361.29 40 2651.5 583.78 ' 40 ! 3285.8 875.32 50 2076.0 364.50 50 2661.6 588.04 i 50 3296.9 880.84 40 2085.4 367.72 50 2671.8 592.32 60 3308.0 886.38 10 2094.9 I 370.95 10 2681.9 596.62 10 3319.1 891.95 20 2104.3 374.20 ; 20 2692.1 600.93 20 j 3330.3 897. B4 30 2113-.8 377.47 | 30 2702.3 605.27 30 3341.4 903.15 40 2123.3 380.76 40 2712.5 609.62 i 40 3352.6 908.79 50 2132.7 - 384.06 j 50 27,22.7 614.00 50 3363.8 914.45 290 TABLE VI. TANGENTS AND EXTERNALS TO A 1 CURVE. Angle. A Tan- gent. T. Exter- nal. E. Angle. A Tan- gent. T. Exter- nal. . i i Angle. A Tan- gent. T. Exter- nal. E. 61 3375.0 920.14 71 4086.9 1308.2 81 4893.6 1805.3 ; 10' 3386.3 925.85 ! 10' 4099.5 1315.6 10' 4908.0 1814.7 20 3397.5 931.58 20 4112.1 1322.9 20 4922.5 1824.1 30 3408.8 937.34 30 4124.8 1330.3 30 4!;c7 1633.6 40 3420.1 943.12 40 4137.4 1337.7 40 4951 ! 5 1843.1 8481.4 948.92 50 4150.1 1345.1 50 4966.1 It 52 . 6 62 3442.7 954.75 72 4162.8 1352.6 82 4980.7 1E6&8 10 3454.1 960.60 10 4175.6 1360,1 10 4995.4 1871.8 20 i 3405.4 966.48 20 4188.5 j 1367.6 20 5010.0 iesi.5 30 ! 3476.8 972.38 30 4201.2 I 1375.2 30 024.8 1F91.2 40 : 3488.3 978.31 ! 40 4214.0 1382.8 40 .5039.5 18C0.9 50 13499.7 984.27 50 4226.8 1390.4 50 5054.3 1910.7 *63 3511.1 990.24 73 4239.7 1398.0 i! 83 5069.2 1920.5 10 3522. G 996.24 10 4252.6 1405.7 10 084.0 1 1930.4 20 3534.1 1002.3 20 4265.6 1413.5 i 20 099.0 1940.3 30 3545.6 1008.3 30 4278.5 1421.2 j 30 5113.9 i 1950.3 40 3557.2 1014.4 40 4291.5 1429.0 i 40 5128.9 ! 1SC0.2 50 3568.7 1020.5 50 4304.6 1436.8 50 5143.9 ; 1970.3 64 3580.3 1026.6 74 4317.6 1444.6 j 84 5159.0 j 1980.4 10 3591.9 1032.8 10 4330.7 1452.5 10 "5174.1 1990.5 20 3603.5 1039.0 20 4343.8 1460.4 20 5189.3 2C00.6 30 3615.1 1045.2 30 4356.9 1468.4 20 5204.4 2010.8 40 3626.8 1051.4 40 4370.1 1476.4 40 5219.7 i 2021.1 50 8638.5 1057.7 50 4383.3 1484.4 50 5234.9 2031.4 65 3650.2 1063.9 75 4396.5 1492.4 85 5250.3 2041.7 10 3661.9 1070.2 10 4409.8 1500.5 10 C-265.6 i 2052.1 20 3673.7 1076.6 20 4423.1 1508.6 20 5281.0 i 2062.5 30 3685.4 1082.9 30 4436.4 1516.7 30 96. 4 2073.0 40 3697.2 1089.3 40 4449.7 1524.9 40 5311.9 083.5 CO 3709.0 1095.7 50 4463.1 1533.1 50 5327.4 ; 2C94.1 66 3720.9 1102.2 76 4476.5 1541.4 86 5343.0 i 2104.7 10 3732.7 1108.6 ' 10 4489.9 1549.7 10 5358.6 2115.3 20 3744.6 1115.1 20 4503.4 1558.0 20 5S74.2 2126.0 80 3756.5 1121.7 30 4516.9 1566.3 30 53KL9 1 2136.7 40 3768.5 1128.2 40 4530.4 1574.7 40 5405.6 2147.5 50 3780.4 1134.8 50 4544.0 1583.1 CO 5421.4 I 2158.4 67 3792.4 1141.4 77 4557.6 1591.6 87 5437.2 2169.2 10 3804.4 1148.0 ! 10 4571.2 1600.1 10 5453.1 2180.2 20 3816.4 1154.7 ; 20 4584.8 1608.6 20 5469.0 2191.1 30 3828.4 1161.3 30 4598.5 1617.1 30 5484.9 2202.2 40 3840.5 1168.1 40 4612.2 1625.7 40 5500.9 2213.2 50 3852.6 1174.8 50 4626.0 1634.4 50 5517.0 2224.3 68 3864.7 1181.6 78 4639.8 1643.0 88 5533.1 2235.5 10 3873.8 1188.4 10 4653.6 1651.7 10 5549.2 2246.7 20 3889.0 1195.2 i -20 4667.4 1660.5 20 5565.4 2258.0 30 3901.2 1202.0 || 30 4681.3 1669.2 30 5581.6 2269.3 40 3913.4 1208.9 40 4695.2 1678.1 40 5597.8 2280.6 50 3925.6 1215.8 50 4709.2 1686.9 50 5614.2 2292.0 69 3937.9 1222.7 79 4723.2 1695.8 89 5630.5 203.5 10 3950.2 1229.7 10 4737.2 1704.7 10 646.9 2315.0 20 I 3962.5 1236.7 20 4751 .2 1713.7 20 5663.4 2326.6 30 I 3974.8 1243.7 30 4765.3 1722.7 30 679.9 2338.2 40 3987.2 12508 1 40 4779.4 1731.7 40 56964 2349.8 50 3999.5 1257.9 I 50 4793.6 1740.8 50 5713.0. 2361.5 70 4011.9 1265.0 1 80 4807.7 1749.9 90 6729.7 2373 3 10 4024.4 1272.1 1 10 4822.0 1759.0 10 5746.3 i 2385.1 20 4036.8 1279.3 i 20 4836.2 1768.2 20 5763.1 . 2S97.0 30 40493 1286.5 |i 30 4850.5 1777.4 30 5779 9 ! 2408.9 40 4061.8 1293.6 ! 40 4864.8 1786.7 40 5796.7 i 2420.9 50 1 4074.4 1300.9 50 4879.2 1796.0 50 5813.6 j 2432.9 .291 :AELE VI.-TANGENTS AND EXTERNALS TO A 1 CURVE. Angle. A Tan- gent. T. Ex- ternal. E. Angle. A Tan- ' gent. T. temaY ' Angle. E. A Tan- gent. T. Ex- ternal E. 91 5830.5 2444.9 101 6950.6 i 3278.1 111 8336.7 4386.1 10' 5847.5 2457.1 10' j 0971.3 i 3294.1 n 10' 8362.7 4407.6 20 5864.6 2469.3 20 6992.0 i 3310.1 20 8388.9 4429.2 30 5881.7 2481.5 30 7012.7 ; 3326.1 30 8415.1 4450.9 40 5898.8 2493.8 40 i 7033.6 ; 3342.3 i 40 8441.5 4472.7 50 5916.0 2506.1 50 1-7054.5 3358.5 i 50 8468.0 4494.6 92 5933.2 2518.5 102 7075.5 3374.9 112 8-194.6 4516.6 10 5950.5 2531.0 I 10 7096.6 3391.2 i 10 8521.3 4538.8 20 5067.9 2543.5 t 20 7117.8 3407.7 20 8548.1 4561.1 30 5985.3 2556.0 30 ! 7139.0 3424.3 30 8575.0 4583.4 40 6002.7 2568.6 40 7160.3 3440.9 40 8602.1 4606.0 50 6020.2 2581.3 50 7181.7 3457.6 50 8629.3 | 4628.6 93 6037.8 2594.0 103 7203.2 3474.4 113 8656.6 4651.3 10 6055.4 260(5.8 10 7224.7 3491.3 10 8684.0 467'4.2 20 6073.1 2019.7 20 7246.3 3508.2 20 1 8711.5 4697.2 30 6030.8 2632.6 30 7268.0 3525.2 30 8739.2 4720.3 40 6108.6 2645.5 40 i 7289.8 i &542.4 40 8767.0 4743.6 50 6126.4 2658.5 1 50 i 7311.7 3559.6 i 50 8794.9 4766.9 94 6144.3 2671.6 104 7333.6 3576.8 114 8822.9 4790.4 10 6162.2 2684.7 10 7355.6 3594.2 10 8851.0 4814.1 20 6180.2 2697.9 1 20 7377.8 3611.7 20 8879.3 4837.8 30 6198.3 2711.2 1 30 7399.9 3629.2 30 8907.7 4861.7 40 6210.4 2724.5 40 7422.2 3646.8 40 8936.3 4885.7 50 6234.6 2737.9 50 7444.6 3664.5 50 8965 4909.9 95 6252.8 2751.3 105 7467.0 3682.3 115 8993.8 4934.1 10 6271.1 2764.8 10 7489.6 3700.2 10 9022.7 4958.6 20 6289.4 2778.3 20 7512.2 3718.2 20 9051.7 ! 4983.1 30 6307.9 2792.0 30 | 7534.9 3736.2 30 9080.9 I 5007.8 40 6326.3 2305.6 40 7557.7 3754.4 40 9110.3 5032.6 50 6344.8 2819.4 50 7580.5 3772.6 50 9139.8 5057.6 98 6353.4 2333.2 106 7603.5 3791.0 116 9169.4 5082.7 10 6382.1 2847.0 10 7626.6 3809.4 10 9199.1 5107.9 20 0400.8 2861.0 20 7649.7 3827.9 20 9229.0 5133.3 30 i 6419.5 2875.0 30 7672.9 3846.5 30 9259.0 5158.8 40 643S.4 2889.0 40 i 7096.3" 3865.2 40 9289.2 5184.5 50 6457.3 2903.1 . 50 7719.7 3884.0 CO 9319.5 5210.3 97 6476.2 2917.3 107 7743.2 3902.9 117 9349.9 5236.2 10 0495.2 2931.6 10 7766.8 3921.9 10 9380.5 5262.3 20 6514.3 2945.9 20 7790.5 3940.9 20 9411.3 5288.6 30 6533.4 2960.3 30 7814.3 3960.1 30 9442.2 5315.0 40 655:2.6 2974.7 40 7838.1 3979.4 40 9473.2 5341.5 50 6571.9 2989.2 50 7862.1 3998.7 50 9504.4 5368.2 98 6591.2 3003.8 108 7886.2 4018.2 118 9535.7 5895.1 10 6610.6 3018.4 10 7910.4 4037.8 10 9567.2 5422.1 20 6630.1 3033.1 20 7934.6 4057.4 20 9598.9 5449.2 30 6649.6 3047.9 30 7959.0 4077.2 30 9630.7 5476.5 40 6669.2 3062.8 40 7983.5 4097.1 40 9662.6 5504.0 50 6688.8 3077.7 50 8008 4117.0 50 9694.7 5531.7 99 6708.6 3092.7 109 8032.7 4137.1 119 9727.0 5559.4 10 0723.4 3107.7 10 8057.4 4157.3 10 9759.4 5587.4 20 6748.2 3122.9 20 8082.3 4177.5 20 9792.0 5615.5 30 ! 6768.1 3138.1 30 8107.3 4197.9 30 9824.8 5643.8 40 6788.1 3153.3 40 8132.3 4218.4 40 9857.7 567'2.3 50 6808.2 3168.7 50 ; 8157 5 4239.0 50 9890.8 57v0.9 100 6828.3 3184.1 110 i 8182.8 4259.7 120 9924.0 5729.7 10 6848.5 ! 3199.6 10 8208.2 4280.5 10 9957.5 5758.6 20 6868.8 3215.1 20 8233.7 4301.4 20 9991.0 ; 5787.7 30 6889.2 3230.8 30 8259.3 4322.4 30 10025.0 ! 5817.0 40 6909.6 3246.5 40 8285.0 ^1343.6 40 10059.0 5846.5 50 6930.1 3262.3 50 8310.8 4364.8 50 10093.0 5876.1 TABLE VII. LONG CHORDS. Degree 0' Curve. Actual Are, One Station. LONG CHORDS. 2 3 Stations. Stations. 4 Stations. 5 Stations. 6 Stations. 10' 100.000 200.000 299.999 399998 499.996 599.993 20 .000 199.999 1 299.997 899.992 499.983 599.970 30 .000 199.998 | 299.992 399.981 499.962 599.933 40 .001 199.997 I 299.986 399.966 499.932 599.882 50 .001 199.995 | 299.979 399.947 499.894 599.815 1 100.001 199.992 | 299.970 399.924 499.848 599.733 10 .002 199.990 299.959 399.89(5 499.793 599.637 20 .002 199.986 299.946 399.865 499.729 599.526 30 .003 199. 983 299.932 399.829 499.657 599.401 40 .003 199.979 299.915 899.789 499.577 599.260 50 .004 199.974 299.898 399.744 499.488 599.105 2 100.005 199.970 299.878 399.695 499.391 598.934 10 .000 199.964 299.857 399.643 499.285 598.750 8@ .007 199.959 299.834 399.586 499.171 598.550 30 .008 199.952 299.810 399.524 499.049 598. 336 40 .009 199.946 299.783 399.459 498.918 598.106 50 .010 199.939 299.756 399.389 498.778 597.8(52 3 100.011 199.931 299.726 399.315 498.6:30 597.604 10 .013 199.924 299.695 399.237 498.474 597.331 20 .014 199.915 299.662 399.154 -498.309 597.043 yo .015 199.907 299.627 399.068 498.136 596.740 40 017 199.898 299.591 398.977 497.955 596.423 50 .019 199.888 299.553 398.882 497.765 596.091 4 100.020 199.878 299.513 398.782 497.566 595.744 10 .022 199.868 299.471 398.679 497.360 595.383 20 .024 199.857 299.428 398.571 497.145 595.007 bO .026 199.846 299.383 398.459 496.921 594.617 40 .028 199.834 299.337 398.343 496.689 594.212 50 .030 199.822 299.289 39S.223 496.449 593.792 5 100.032 199.810 299.239 398.099 496.201 593.358 10 .034 199.797 299.187 397.970 495.944 592.909 20 .036 199.783 299.134 397.837 495.678 592.446 30 .0=38 199.770 299.079 397.700 495.405 591.968 40 .041 199.756 299.023 397.559 495.123 591.476 50 .043 199.741 298.964 397.413 494.832 590.970 6 100.046 199.726 298.904 397.264 494.534 590.449 10 .048 199.710 298.843 397.110 494.227 589.913 20 .051 199.695 298.779 396.952 493.912 589.364 30 .054 199.678 298.714 396.790 493.588 588.800 40 .056 199.662 298.648 396.623 493.257 588.221 50 .059 199.644 298.579 396.453 492.917 587.628 7 100.062 199.627 298.509 396.278 492.568 587.021 10 .065 199.609 298.438 396.099 492.212 586.400 20 .068 199.591 298.364 395.916 491.847 585.765 30 .071 199.572 298.289 393.729 491.474 585.115 40 .075 199.553 298.212 395.538 491.093 584.451 50 .078 199.533 298.134 395.342 490.704 583.773 8 100.081 199.513 298.054 395.142 490.306 583.081 10 .085 199.492 297.972 394.9:38 ! 489.900 582.375 20 .088 199.471 297.888 394.731 ! 489.486 581.654 30 .092 199.450 297.803 394.518 489.064 580.920 40 .095 199.428 297.716 394.302 488.634 580.172 5-J .099 199.406 297.628 394.082 488.196 579.409 9 100.103 199.383 297.538 393.857 487.749 578.633 10 .107 199.360 297.446 393.629 487.294 577.843 20 .111 199.337 297.352 393.396 486.832 577.039 30 .115 199.313 297.257 393.159 486.361 576.222 40 .119 199.289 297.160 392.918 485.882 575.390 50 .123 199.264 297.062 392.673 485.395 574.545 10 100.127 199.239 296.962 392.424 484.900 573.686 - 298 TABLE VII.-LONG CHORDS. Degree of Curve. LONG CHORDS. 7 Stations. 8 Stations. Stations. 10 Stations. Stations. 12 Stations. 010' 699.988 799.982 1 899.974 999.965 1099.95 1199.94 20 699.953 799.929 899.899 999.860 1099.81 1199.76 30 099.893 799.840 899.772 999.686 1099.58 1199.46 40 699.810 799 716 899.594 999.442 1099.25 1199.03 50 699.704 799.550 899.365 999.128 1098.84 1198.49 1 699.574 799.360 899.086 998.744 1098.33 1197.82 10 699.420 799.130 898.757 998.290 1097.72 1O7.04 20 699.242 798.863 898.376 997.708 1097.02 11JJ6.13 30 699.041 798.562 897.945 997.175 1096.23 1195.11 40 698.816 798.224 897.464 996.513 1095.35 1193.90 50 1 698.567 797.852 896.931 995.782 1094.38 1192.09 2 j 698.295 797.444 896.349 994.981 1093.31 1191.31 10 698.000 797 000 895.716 994.112 1092.15 11MJ.80 20 697.080 790.522 895.033 993.173 1090.90 lit: 8. 18 30 607.838 796.008 894.299 992.165 1089.56 1180.43 40 696.971 795.459 893.515 991.088 1088.12 1184.57 50 090.581 794.874 892.681 989.943 10S6.60 11^-2.59 3 690.108 794.255 891.798 988.729 1084.98 1180.49 10 095.731 793.600 890.864 987.447 1083.28 1178.28 20 095.271 792.911 889.880 986.096 1081.48 1175.94 30 094.787 792.186 888.846 984.677 1079.59 1173.49 40 094.280 791.427 887.703 983.190 1077.61 1170.93 50 093.750 790.032 886.630 981.036 1075.54 1168.25 4 693.196 789.803 885.448 980.014 1073.38 1105.45 10 608.619 788.939 884.217 978.325 1071.14 1102.54 20 692.018 788.040 882.936 970.509 10B8.81 1159.51 30 091.395 787.108 881.606 974.746 1006.38 1156.37 40 690.748 786.140 880.228 i 972.856 1003.87 1153.12 50 090.079 785.138 878.800 970.900 1061.27 1149.76 5 089.380 784.101 877.324 968. 877 1058.59 1140.28 10 688.070 783.030 875.800 966.788 1055.81 1142.09 20 687.930 781.925 874.227 964.634 1052.95 1138.99 30 687.169 780.786 872.605 962.415 1050.01 1135.18 40 080.384 779.012 870.936 960.130 1046.97 1131.26 50 Oa5.576 778.406 869.219 957.780 1043.86 1127.24 6 684.745 777.165 867.454 955.366 1040.66 1123.10 10 083.892 775.890 865.642 952.888 1037.37 1118.86 20 083.016 774.582 803.782 950.345 1034.01 1114.51 30 082.117 773.240 801.875 947.7'39 1030.55 1110.05 40 081 . 195 771.804 859.922 945.069 1027.02 1105.49 50 680.251 770.455 857.921 942.337 1023.40 1100.83 7 079.285 769.014 a55.874 939.542 1019.70 1096. CG 10 : 678.296 707.539 853.780 936.084 1015.93 1091.19 20 i 677.284 706.030 851.640 933.764 1012.07 1086.22 30 676.250 764.490 849.455 930.783 1008.13 1081.15 40 ; 675.194 762.916 847.224 927.7'41 1004.11 1075.98 50 ; 07'4.116 761.309 844.947 924.038 1000.01 1070.71 8 673.015 759.670 842.625 921.474 -095.834 | 1065.34 10 071.892 757.999 840.258 918.250 991.580 | 1059.88 20 670.748 756.295 837.845 914.906 987.250 1054.32 30 069.581 754.560 835.389 911.023 982.844 1048.00 40 668.393 752.792 832.888 908.221 978.302 1042.91 50 667.182 750.993 830.342 904.761 973.806 1037.00 9 605.950 749.161 827.75-1 901.242 969.175 1031.13 10 064.697 747.299 825.121 897.667 964.471 1025.11 20 663.421 745.404 822.445 894. 033 959.094 1018.99 30 662.124 743.479 819.726 890.343 954.844 1012.79 40 660.806 741.522 816.965 886.597 949.924 1006.49 50 659.406 739.535 814.160 882.795 944.933 1000.12 10 658.105 787.516 811.314 878.938 939.871 993.653 294 TABLE VII. LONG CHORDS. Degree of Curve. Actual Arc, One Station. LONG CHORDS. 2 Stations. 3 Stations. 4 Stations. 5 Stations. 6 Stations. \ 10 10' 100.131 199.213 296.860 392.171 484.397 572.813 20 .136 199.187 296.756 391.914 483.886 571.926 30 .140 199.161 296.651 391.652 483.367 571 .027 40 .145 199.134 296.544 , 391.387 482.840 570.113 50 .149 199.107 296.436 391.117 482.305 i 569. 1MB 11 100.154 199.079 296.325 390.843 481.762 568.245 10 i .158 199.051 296.214 390.565 481.211 567.292 20 .163 199.023 296.100 390.284 480.653 566.324 30 .168 198.994 295.985 389.998 480.086 565.343 40 .173 198.964 295.868 389.708 479.511 564.349 50 .178 198.935 295.750 389.414 478.929 563.341 12 100.183 198.904 295.629 389.116 478.338 562.321 10 .188 198.874 295.508 388.814 477.740 561.287 20 .193 198.843 295.384 388.508 477.135 560.240 30 .199 198. 8il 295.259 ; 388.197 476.521 559.180 40 .204 198.779 295.132 ' 387.883 475.899 558.107 50 .209 198.747 295.004 387.565 475.270 557.020 13 100.215 1 19S.714 294.874 387.243 474.633 555.921 10 .220 198.081 294.742 386.916 473.988 554.809 20 -226 198.648 294.609 386.586 -473.336 553.684 30 ; .232 i 198.614 294.474 38(5.252 472.675 552.546 40 .237 ! 198.579 294.337 385.914 472.007 551.395 50 .213 198.544 294.199 385.572 471.332 550.232 14 100.249 I 198.509 294.059 ! 385.225 470.649 549.056 10 .255 1 198.474 293.918 384.87'5 469.958 547.867 20 .261 i 198.437 293.774 \ 384.521 469.260 546.666 30 .267 i 198.401 293.629 ' 384.163 468.554 545.452 40 .274 i 198.364 293.483 ! 383.801 1 467.840 ' 544.226 50 .280 i 198.327 293.335 383.4;i5 467.119 ! 542.987 15 100.286 i 198.289 293.185 383.065 466.390 541.736 10 292 198.251 293.034 382.691 465.6.54 540.472 20 299 198.212 292.881 ! 382.313 464.911 539.196 30 .306 198.173 292.726 ! 381.931 464.160 537.908 40 .312 198.134 292.570 | 381.546 463.401 536.608 50 .319 198.094 292.412 381.156 462.635 535.296 is: 100.326 198.054 292.252 380.763 461.862 533.972 10 .333 198.013 292.091 380.365 461.081 532.635 20 .339 197.972 291.928 379.964 460.293 531.287 30 .346 197.930 291.764 379.559 459.498 529.927 40 .353 197.888 291.598 379.150 458.695 528.555 50 .361 197.846 291.430 378.737 457.886 527.171 17 100.368 197.803 291.261 378.320 457.069 525.778 10 .375 197.760 291.090 377.900 456.244 524.369 20 .382 197.716 290.918 377.475 455.413 522.950 30 .390 197.672 290.743 377.047 454.574 521.519 40 .397 197.628 290.568 376.615 453.728 520.078 50 .405 197.583 290.390 376.179 452.875 518.625 18 100.412 197.538 290.211 375.739 452.015 517.160 10 .420 197.492 290.031 375.205 451.147 515.685 20- .428 197.446 289.849 374.848 450.373 514.198 30 ! .436 197.399 289.665 374.397 j 449.392 512.699 40 .444 197.352 289.479 373.942 448.504 511.190 50 .452 197.305 289.292 373.483 447.608 509.H70 19 100.460 197.256 289.104 373.021 446.706 508.139 10 .468 197.209 288.913 i 372.554 445.797 506.597 20 .476 197.160 288.722 i 372.084 444.881 505.043 30 .484 j 197.111 288.528 371.610 443.957 503.479 40 .493 197.062 288.333 871.133 443.028 501.905 50 .501 197.012 288.137 370.652 i 442.091 500.320 20 100.510 196.962 287.939 i 370.167 441.147 498.724 295 TABLE VDL LONG CHORDS. Degree of Curve. LONG CHORDS. 7 Stations. 8 Stations. 9 10 Stations. Stations. 11 Stations. 12 Stations. i 10 10' 656. 723 735.467 808.426 875.025 934.741 987.105 20 655.320 733.387 805.495 871.058 929.542 980.473 30 653.895 731.277 802.524 867.038 924.276 973.760 40 652.450 729.137 799.512 862.963 918.943 966.967 50 650. 983 726.967 796.458 858.836 913.544 960.093 11 649.496 724.767 793.364 854.656 908.080 953.141 10 647.989 722.537 790.230 850.425 902.550 946.112 20 646.460 720.278 787.056 846.140 896.957 939.007 30 644.911 717.990 783.843 841.808 891.303 i 931.828 40 643.342 715.672 780.590 837.424 885.586 ! 924.575 50 641.752 713.325 777.298 832.990 879.807 917.250 12 640.142 710.950 773.968 828.507 873.968 9C9.&54 10 638.512 708.546 770.600 823.974 868.070 902.389 20 636.862 706.113 767.193 819.394 862.113 894.855 30 635.191 703.653 763.749 814.766 856.099 887.254 40 6:33.501 7'01.164 760.268 810.092 850.028 879.588 50 631.792 698.647 756.749 805.370 843.900 871.857 13 630.062 696.103 753.194 800.602 837.718 864.063 10 628.313 693.531 749.603 795.790 831.482 856.208 20 626.544 690.932 745.976 790.932 825.192 848.293 30 624.756 688.306 742.313 786.030 818.850 840.318 40 622.949 685.653 738.616 781.065 812.457 832.286 50 621.123 682.974 734.883 776.096 806.013 824.198 14 619.278 680.268 731.116 771.066 799.520 816.056 10 617.413 677.535 727.315 765.993 792.979 807.860 20 615.530 674.777 723.480 760.879 786.389 799.612 30 613.628 671.993 719.612 755.725 779.753 791.313 40 611.708 669.183 715.711 750.531 773.072 782.966 50 609.769 666.348 711.777 745.297 766.345 774.571 15 607.812 663.488 707.811 740.024 759.575 766.130 10 605.836 660.603 703.814 734.714 752.763 20 603.842 657.693 699.785 729.366 745.908 30 601,881 654.758 695.725 723.982 739.014 40 599.801 651.799 691.634 718.561 732.078 50 597.753 648.817 687.513 713.105 725.104 16 595.688 645.810 683.362 707.614 718.092 10 593.605 642.780 679.182 702.088 711.043 20 591.505 639.727 674.973 696.529 703.959 30 589.388 636.650 670.735 690.938 40 587.253 633.550 666.469 685.314 50 585.101 630.428 662.175 679.659 17 582.933 627.283 657.854 .673.972 10 580.747 624.117 653.506 668.256 20 578.545 620.928 649.131 662.510 30 576.326 617.717 644.730 656.735 40 574.091 614.485 640.304 650.933 50 571.839 611.232 635 . 852 645.103 18 569.571 607.958 631.375 639.245 10 567.287 604.664 626.874 20 564.988 601.349 622.349 30 562.673 598.013 617.801 40 560.342 594.658 613.229 50 557.996 591.283 608.635 19 555.634 587.888 604.018 10 553.257 584.475 599.379 20 550.864 581.012 594.720 30 548.457 577.591 590.039 40 546. 035 574.121 585.339 50 543.599 570.634 580.618 20 541.147 i 567.128 575.877 1 296 TABLE VII. LONG CHORDS. LONG CHORDS. Degree of Actual Arc, [!>UTV6 One 3 3 4 5 6 Station. Stations. Stations. Stations. Stations. Stations. 21 100.562 196.651 286.716 367.179 435.845 488.931 22 100.617 196.325 285. 437 364.060 429.305 478.775 23 100.675 195.985 284.101 3(50.810 423.033 468.270 24 100.735 195.630 282.709 357.433 416.535 457.433 25 100.798 195.259 281.262 353.930 489,819 446.280 20 100.863 194.874 279.759 350.303 402.891 1 434.827 27 100.931 194.474 278.201 346.555 395.758 423.092 28 101.002 194.059 276.589 342.688 388.428 411.092 29 101.075 193.630 274.924 338.704 380.908 398.846 30 101.152 193.185 273.205 334.607 373.205 386.370 i 297 TABLE VIII -MIDDLE ORDIMTES. TABLE VIII. -MIDDLE ORDIXATE3. Degree of Curve. 1 Station. 2 Stations. 3 Stations. 4 Stations. 5 Stations. 6 Stations. j Itf .036 ,145 .32? .582 .909 1.309 20 .073 .291 .654 1.164 1.818 2.618 30 .109 .436 .982 1.745 2.727 3.926 40 .145 .582 1.309 2.327 3. 036 5.235 50 .182 .727 1.636 2.909 4.545 6.544 1 .218 .873 1.963 3.490 5.453 7.852 10 .255 1.018 2.291 4.072 6.362 9.160 20 .291 1.164 2.618 4.654 7.270 10.468 30 .327 1.309 2.945 5.235 8.179 11.775 40 .364 1.454 3.272 5.816 9.087 13.082 50 .400 1.600 3.599 6.393 9.994 14.389 2 .430 1.745 3.926 6.979 10.902 15.694 10 .473 1.891 4.253 7.560 11.809 17.000 20 .509 2.036 4.580 8.141 12.716 18.304 30 .545 2.181 4.907 8.722 13.623 19.608 40 .582 2.327 5.234 9.303 14.529 20.912 50 .618 2.472 5.561 9. 83 15.485 22.214 3 .654 2.618 5.888 10.464 16.341 23.516 10 .691 2.763 6.215 11.044 17.246 24.817 20 .727 2.908 6.542 11.624 v 18.151 26.117 30 .763 3.054 6.868 12.204 19.055 27.416 40 .800 3.199 7.195 12.784 19.959 28.714 50 .836 3.345 7.522 13.303 20.863 30.012 4 .872 3.490 7.848 13.943 21.766 31.308 10 .909 3.635 8.175 14.522 22.668 32.603 20 .945 3.781 8.501 15.101 23.570 ,33.896 30 .98-2 3.926 8.828 15.680 24.471 35.189 40 1.018 4.071 9.154 16.258 25.372 36.480 50 1.054 4.217 9.480 16.837 26.272 37.770 5 1.091 4.362 9.807 17.415 27.171 39.059 10 1.127 4.507 10.133 17.992 28.070 40.346 20 1.164 4.653 10.459 18.570 28.968 41.631 30 1.200 4.798 10.785 19.147 29.666 42.916 40 1.237 4.943 11.111 19.724 30.762 44.198 50 1.273 5.038 11.436 20.301 31.658 45.479 6 1.303 5.234 11.762 20.877 32.553 46.759 10 1.346 5.379 12.088 21.453 33.448 48.037 20 1.332 5.524 12.413 22.029 34.341 49.313 30 1.418 5.669 12.7'39 22.604 35.234 50.587 40 1.45.") 5.814 13.064 23.179 36.126 51.860 50 1.491 5.960 13.389 23.754 37.017 53.130 7 1.528 6.105 13.715 24.328 37". 907 54.399 10 1.564 6.250 14.040 24.902 38.796 55.666 20 1.600 6.395 14.365 25.476 39.684 56.931 30 1.G37 6.540 14.689 26.049 40.571 58.193 40 1.673 6.685 15.014 26.622 41.458 59.45-4 50 1.710 6.831 15.339 27.195 42.343 60.712 8 1.746 6.976 15.663 27.767 43.227 61.969 10 1.71:2 7.121 15.988 28.338 44.110 63.223 20 1.819 7.266 16.312 28.910 44.992 64.475 30 1.855 7.411 16.636 29.481 45.873 65.724 40 1.892 7.556 16.960 30.051 46.753 66.97'2 50 1.923 7.701 17.284 30.621 47.632 68.216 9 1.965 7.846 17.608 31.190 48.510 69.459 10 2.001 7.991 17.932 31.759 49.386 70.699 20 2.037 8.136 18.255 32.328 50.261 71.936 30 2.074 8.281 18.578 32.896 51.135 73.171 40 2.110 8.426 18.902 33.464 52.008 74 403 50 2.147 8.571 19.225 34.031 52.880 75.632 10 2.183 8.716 19.548 34.597 53.750 7'6.859 298 TABLE VIII. MIDDLE ORDINATES. Degree of Curve. 7 Stations. 8 Stations. 9 Stations. 1O Stations, 11 Stations. Stations. 10' 1.782 2.327 2.945 3.636 4.400 5.236 20 8.668 4.654 5.890 7.272 8.799 10.471 30 5.S45 6.981 8.835 10.907 13.197 15.704 40 7.126 9.307 11.778 14.540 17.593 20.936 50 8.907 11.632 14.721 18.173 21.987 26.164 1 10.687 13.957 17.663 21,803 26.378 31.388 10 12.467 16.281 20.603 25.431 30.766 36.607 20 14.246 18.604 23.541 29.057 35.150 41.821 30 16.024 20.925 26.477 32.679 39.530 47.028 40 17.802 23.246 29.411 36.298 43.905 52.229 50 19.579 25.564 32.343 39.914 48.274 57.422 2 21.355 27.881 35.272 43.525 52.637 62.606 10 23.130 30.197 38.198 47.131 56.993 67.780 20 24.903 32.510 41.121 50.733 61.343 72.945 30 26.676 34.821 44.040 54.330 65.684 78.098 40 28.447 37.K30 46.956 57.921 70.018 83.240 50 30.21(5 39.436 49.868 61.506 74.342 88.370 3 31.984 41.740 52.776 65.084 78.657 93.486 10 33.751 44.041 65.679 68.656 82.963 98.588 20 35.516 46.. 339 58.577 72.221 87.258 108.675 30 37.279 48.634 61.471 75.778 91.542 108.747 40 39.040 50.926 64.360 79.328 95.814 113.803 50 40.800 53.215 67.243 82.869 100.075 118.841 4 42.557 55.500 70.121 86.402 104.323 123.862 10 44.312 57.781 7'2.992 89.925 108.558 128.864 20 46.065 60.059 75.858 93.440 112.779 133.847 30 47.816 62.3=33 78.717 96.945 116.986 138.810 40 49.564 64.602 81.570 100.489 121.178 143.753 50 51.310 66.868 84.416 103.924 125. a56 148.674 5 53.053 69.129 87.255 107.397 129.517 153.572 10 54.794 71.386 90.087 110.860 133.663 158.448 20 56.532 73.638 92.911 114.311 137.791 163.300 30 58.267 75.885 95.728 117.751 141.903 168.128 40 59.999 78.127 98.536 121.178 145.997 172.931 50 61.729 80.364 101.337 124.593 150.072 177.708 6 63.455 82.596 104.129 127.995 154.129 182.459 10 65.178 84.822 106.912 131.384 158. 1C6 187.182 20 66.898 87.043 109.686 134.759 162.184 191.878 30 68.615 89.258 112.452 138.120 166.182 196.545 40 70.328 91.468 115.298 141.468 170.159 201.183 50 72.037 93.671 117.954 144.800 174.114 205.792 7 73.744 95.868 120.691 148.118 178.048 210.370 10 75.446 98.059 123.417 151.421 181.960 214.916 20 77.145 100.244 126.134 154.708 185.850 219.431 30 78.840 102.422 128.840 157.979 189.716 223.914 40 80.531 104.594 131.535 161.234 193.559 228.363 50 82.218 106.758 134.219 164.473 197.377 232.779 8 as. 901 108.916 136.893 167.695 201.171 237.160 10 85.580 111.067 139.555 170.899 204.941 241.507 20 87.254 113.210 142.205 174.086 208.685 245.818 30 88.924 115.346 i 144.844 177.255 212.403 250.093 40 90.590 117.475 147.470 180.407 216.095 254.331 50 92.252 119.596 150.085 183.539 219.760 258.531 9 93.909 121.709 152.687 186.653 223.398 262.694 10 95.561 123.814 155.277 189.748 227.008 266.818 20 97.208 125.911 157.854 192.824 230.591 270.904 30 98.851 128.000 160.417 195.880 234.145 274.949 40 100.489 130.081 162.968 198.916 237.670 278.955 50 102.122 132.153 165.505 201.932 241.167 282.919 10 103.750 134.217 168.029 204.928 244.633 286.843 TABLE VIII. MIDDLE ORDINATES. Degree of Curve. Station. 2 Stations. 3 Stations. 4 Stations. 5 Stations. 6 Stations. 10 10' 2.219 8.860 19.87'0 35.164 54.619 7'8.083 20 2.256 9.005 20.15)3 85.729 55.486 79.305 30 2.293 9.150 20.516 36.294 56.358 80.523 40 2.329 9.295 20.888 36.859 57.218 81.739 50 2.3C5 9.440 21 . ICO 37.428 58.081 82.951 11 2.402 9.585 21.483 87.9HG 58.943 84.161 10 2.438 9.729 21.804 38.549 59.804 85.368 20 2.475 9.874 22.126 39.111 60.603 86.571 30 2.511 10.019 22.448 39.673 61.521 87.772 40 2.547 10.164 22.769 40.234 62.S77 88.969 50 2.584 10.308 23.090 40.795 63.232 90.164 12 2.620 10.453 23.412 41.355 64.085 91.355 10 2.657 10.597 23.732 41.914 64.937 92.542 20 2.693 10.742 24.053 42.473 05.787 93.727 30 2.730 10.887 24.374 43.031 66.636 94.908 40 2.766 11.031 24.694 43.588 67.482 96.086 50 2.803 11.176 25.014 44.145 68.328 97.260 13 2.839 11.320 25.334 44.701 69.171 98.431 10 2.876 11.405 25.654 45.256 70.013 < 19. 5118 20 2.912 11.609 25.97'4 45.811 .70.854 100.702 30 2.949 11.75-1 26.293 46.365 71.692 101.922 40 2.985 11.898 26.612 46.919 i 72.529 108.079 50 3.022 12.043 26.931 47.47'2 73.864 104.232 14 3.058 12.187 27.250 48.024 74.197 105.381 10 3.095 12.331 27.569 48.575 75.020 106.527 20 3.131 12.476 27.887 49.126 7T>.K)9 107.669 30 3.168 12.620 28.206 49.676 76.687 108.807 40 3.204 12.764 28.524 50.225 77.513 109.941 5J 3.241 12.908 28.841 50.773 7'8.337 111.071 15 3.277 13.053 29.159 51.321 7'9.159 112.197 10 3.314 13.197 29.476 51.868 79 . U79 113.319 20 3.350 13.341 29.794 52.414 80.7C8 114.438 30 3.387 13.485 30.111 52.959 81.614 115. E52 40 3.423 13.629 30.427 53.504 82.429 116.662 50 3.460 13.773 30.744 54.048 83.241 117.7fc8 1C 3.496 13.917 31.060 54.591 84.052 118.870 10 3.533 14.061 31.376 55.183 84.861 119.5:67 20 3.569 14.205 31.692 55.675 85 . 667 121. C61 30 3.606 14.349 32.008 56.215 86.471 122.150 40 3.643 14.493 32.323 56.755 87.274 123.225 50 3.679 14.637 32.6:38 57.294 88.074 124.315 17 3.716 14.781 32.953 57.832 88.872 125.891 10 3.752 14.925 a3. 267 58.369 89.668 126.463 20 3.789 15.069 33.582 58.906 90.462 127.530 30 3.825 15.212 33.896 59.441 91.254 128.593 40 3.862 15.356 34.210 59.976 92.043 129.651 50 3.899 15.500 34.523 60.510 92.830 180.704 18 3.935 15.643 34.837 61.042 93.616 131.753 10 3.972 15.787 35.150 61.574 94.38 132.797 3D 4.008 15.931 35.463 62.106 05.179 133.837 30 4.045 16.074 35.775 ea v 6S6 95.957 134.872 40 4.081 16 218 36.088 63.165 96.783 135. C02 50 4.118 16.361 36.400 63.693 117.5* Hi 136.928 19 4.155: 16.505 36.712 64.221 98 278 137.948 10 4.191 : 16.648 37.023 64.747 !!'.. 017 138. SC4 20 4.228 16.792 ar.834 65.2?3 90.818 13(t!)75 30 4.265 16 9.35 37.645 65.797 100.577 140.981 40 4.301 17.078 37.956 66.321 101. 339 141.982 50 4.338 17.222 38.266.: 66.843 102.098 142.978 20 4.374 17.365. :| 38.576 : 67.365 102. 855 143.969 300 TABLE IX. LINEAR DEFLECTION TABLE. Deflec- tion. 100. 200. 300. 400. 500. 600. 700. ceo. 900. 1000. 30' 0.87 1.7'5 2.62 3.49 4.30 5.24 6.11 6.98 7.85 8.73 1 1.75 3.49 5.21 6.98 8.73 10.47 12.22 13.96 15.71 17, 45 3D 2.03 i 5.24 7.8.3 10.47 13.09 15.71 18.33 20.94 23.56 26.18 2 3 49 ' 6.98 10.47 13.90 17.45 20.94 24.43 27.92 31.41 34.90 30 4.36 ! 8.73 13. OJ 17.45 21.81 26.18 30.54 34.90 39.27 43.63 3 5.34 10.47 15.71 20.94 20.18 31.41 36.65 41.88 47.12 52.35 30 6.11 12.22 18.32 24.4-3 30.54 36.65! 42.75 48.86 54.97 61.08 4 6.98 13.90 20.91 27.93 34.90 41.88 48.86 55.84 06.82 69.80' 30 7.85 15.70 33.5'j 31.41 39.26 47.11 54.96 62.82 70.67 78.52 5 8.73 17.45 26.17 34.89 43.62 52.134 61.07 69.79 78.51 87.24 30 9.60 19.19 28.79 38.33 47.98 57.57 67.17 76.70 86.36 95.96 6 10.47 20.93 31.40, 41.87 53.34 62.80 73.27 83.74 94.20 104.67 30 11.34 22.68 34.03 45.35 56.67 68.03 79.37 90.71 102.05 113.39 7 13.21 24.42 36.63 48.84 61.05 73.26 85.47 97.08 109.89 122.10 30 13.03 26.16 39.24; 52.32 65.40 78.48 91.56 104.64 117.73 130.81 g 13.95 27.901 41.85! 55.80 69.76 83.71 97.00 111.01 125.56 139.51 30 14.82 29.64i 44.47; 59.29 74.11 88.93 103.75 118.57 1&3.40 148.22 9 15.69 31.38: 47.03! 62.77 78.46 94.15 109.84 125.53 141.23 156.92 30 16.56 33.12 49.68j 66.25 82.81 99.37 115.93 182.49 149.05 165.62 10 17.43 34.86 52.29 69.72 87.16 104.59 122.02 139.45 150.88)174.31 30 18.30 36.60 54.90 73.20 91.50 109.80 128.10 146.40 164.701183.00 11 19.17 38.34 57.51 76.68 95.85 115.01 134.18 153. So 172.52 191.69 30 20.04 40.03 60.11 80.15 100.19 120.23 140.26 160.30 180.34 200.38 12 20.91 41.81 62.72 83.62 104.53 125.43 140.34 107.25 188.15 209.06 30 21.77 43.55 65.32 87.09 108.87 130.1 54 152.41 174.19 195.96 217.73 13 23.64 45.23 67.92 90.56 113.20 135.84 158.48 181.13 203.77 226.41 30 23.51 47.01 70.521 94.03 117.54 141.04 164.55 188.00 211.57 235.07 14 24.37 48.75 73.12 97.50 121.87 140.24 170.62 194.99 219.36 243.74 30 25.24 50.48 75,72 100.90 126.20 151.44 170.68 201.92 227.16 252.40 15 26.11 52.21 78.321 104.42 130.53 156. 63 182.74 208.84 234.95 261.05 30 2<5.97 5'3.! 4 80.91 107.83 1:34. 85 161.82 188.79 215.70 242.73 269 70 16 37.83 55.67: 83.50 111.34 139.17 167.01 194.84 222.08 250.51 278 .'35 80 2170 57.40 88.10 114.79 143.49 17'2.19 200.89 229.59 258.29 286.99 17 2;). 56 59.12; 83.69 118.23 147.81 177.37 ''00 ').') 236.50 266.06 295.62 30 30.43 00.85 91.27; 121.70 153.12 182.55 212^97 243.40 273.82 304.25 18 31.23 03.57 93.33 125.15 156.43 187.72 219.01 250.30 281.58 312.87 80 32.15 04.3) 93.45 123.59 160.74 192.89 225.04 257.19 289.34 321.49 19 33.01 00.02 90.03 132.04 105.05 198.06 231.07 204.08 297.08 330.09 30 33.87 67.7, 131.01 135.43 169.35 203.22 237.09 270.96 304.83 338.70 20 34.73 69.46 104.19 138.92 173.65 208.38 243.11 277.84 312.57 347.30 30 .35.59 71.13 108.77! 142.35 177.94 213.53 249.12 284.71 320.30 &55..S9 21 36.45 73.89 109.34 145.79 183.24 218.68 255.13 291.58 328.02 364.47 30 37.30 74.61 111.91 149.22 186.52 223.83 261.13 298.44 335.74 373.05 22 33.16 76.33 114.49 152.65 190.81 228.97 267.13 305.20 343.46 381.62 30 39.02 78.04 117.05 156.07 195.00 234.11 27'3.13 312.14 &51.16 390.18 23 39.87 79.75 119.63 159.49 199. 37 i 239. 24 279.12 318.89 S58.86 398.74 30 40.73 81.46 133.19 168.91 203.641244.37 285.10 325.83 866.50 407.28 21 41.53 83.16 134.75 1:56.33 207. 911849. 49 291.08 332.06 37'4.24 415.82 30 43.44 84.87 137.31: 109.74 212.18:254.61 297.05 330.48 381.92 424,36 25 43.29! 86.58 ! 129.8G ! 173.15 216.44:259.73 303.02 364.30 389.59 432.88 30 44. 14' 83. 33 : 133. 43 176.50 220.70 264.84 308.98 353.12 397.26 441.39 26 44.99 89.98 134.97 179.9(5 234.95 269.94 314.913 859.92 404.91 449,90 30 45.84 91.68 137.53 183.30 229.20.275.04 330.88 366.72 412.56 458.40 27 46.69 93.38 140.07 180.70 233.45 280.14 33(5.83 373.51 420.20 466.89 30 47.54 95.07 142.61 190.15 237.09 285.22 332.7(5 380.30 427.83 475.37 28 43.33 96.77 145.15 193.5! 241.93 290.31 3:38.09 387.08 4:35.46 4R3.84 30 49.23) 98.46 147.09 196.92 246.15 295.38 344.63 393.85 443.08 492.31 29 50.08 100.15 150.23 200.30 250.38 300.46 350.53 400.10 450.68 500.76 30 50.92 101.84 152.76 203.08 254.00 305.52 350.44 407.36 458.28 509.20 30 51.76 103.53 155. 291207. 06 258. 82! 310. 59 362.35 414.11 465. 87 1517. 64 301 TABLE X. -COEFFICIENTS FOR VALVOID ARCS. I. RATIO OF u = - t A L 10 20 30 40 50 60 70 80 90 100 110 120 300 .3518 J .3516 .3514 .3510 .3506 .a500 .3493 .3485 .3476 .3466 .S455 .3444 400 .3437 .3436 .3433 .3430 .3426 .3421 .3415 .3408 .3399 .3390 .3:380 .3368 500 .3400 .3398 .3396 .3393 .3389 .3383. .3379 .3373 .3364 .3356 .3345 .3835 600 .3379 .3378 .3376 .3373 .3369 .3365 .3359 .3353 .3:345 3337 .3327 .8317 700 .3367 .3366 .3364 .3361 .aS57 .3:353 .3347 .aS41 .3334 '.332(j .:3316 .3306 800 .3359 .3358 .3356 .3a53 .3349 .3345 .3340 .3333 .3326 asis .3309 .329;) 900 .3353 .3352 .3350 .3344 .3340 .3334 .3328 .!3321 '.3313 .330-1 .3294 1000 .3350 .3348 .3346 '.3344 .3340 .aase .3331 .3324 .3317 .3310 .3301 .3291 1200 .3345 .a343 .3341 .3339 .3336 . 3331 .3326 .3320 .13313 .3305 .3296 .3286 1500 .3340 .8339 .aS37 .3335 .3331 .327 .3322 .asi6 .3309 .3301 .3292 .3283 2000 .3337 .3336 .3333 .3331 .3328 .3324 .3319 .3313 .3306 .3298 .3289 .3280 n. RATIO OF r v = L, L 10 20 30 40 50 60 70 80 90 100 110 120 300 .7706 .7683 .76431.7588 .75181.7432 .7aS2 .7218 .7090 .6949 .6795 .6630 400 .7611 .7588 .7549 .7495 .7425 .73-11 . 7243 .7130 .7004 .6865 .6714 .6551 500 .7568 .7'545 .7506 .7452 .7384 .7300 .7202 .7091 .69(56 .6828 .6678 .6516 600 .7545 .7522 .7483 .7430 .73611.7278 .7181 .7070 .6946 .6808 .6659 .6498 700 .7531 .7508 .7469 .7416 .73481.7265 .7168 .7057 .69:33 .6797 .6648 .6487 800 .7522 .7499 .7461 .7407 .73391.7257 .7160 .7040 .6926 .6789 .6640 .6480 900 .7516 .7492 .7454 .7401 .7888.7251 .7151 .7041 .6920 .678-1 .6635 .6475 1000 .7512 .7489 .7450 .7397 .7329 .7247 .7150 .704< .(5917 .6780 .6682 .6472 1200 .7505 .7483 .7444 .7391 .7324 .7241 .7145 .7035 .6912 .67751.6627 .6468 1500 .7501 .7478 .7440 .7887 .73191.7237 i.7141 .7031 |.69Cte .6772 .(5624 .6464 2000 .7497 .7474 .7436 .7383 .7316 .7234 i.7137 .702* ( .69041.6769 .6621 .6461 JTT RATIO I ' A' - A" TO A CHANGE OF ONE DEGREE IN THE ANGLE A. L 10 20 i 30 40 50 60 70 80 90 100 110 120 300 2.62 2.61 1 2.60 2.59 2.57 2.55 2.52 2.49 2.46 ! 2.42 2.38 2.34 400 3.49 3.48 3.46! 8.44 3.42 3.38 3.35 3.30 i 3.25 I 3.20 3.14 3.08 500 4.36 4.35 4.33J 4.30 4.26 4.22 4.17 4.11 4.05i 3.98 3.90 3.81 600 5.23 5.22 5.19 5.10 5.11 5.06 1 4.99 4.92 4.84 4.75 4.65 i 4.55 700 6.10 6.09 6.06 6.02 5.96 5.90 i 5.83 5.74 5.651 5.54 5.43 5.31 800 6.97 6.95 6.92! 6.87 6.82 6.74 i 6.66 6.56 6.45 6. as 6.20 6.06 900 7.85 7.79 7.73 7.67 7.59 1 7.49 7.38 ! 7.26 7.13 6.98 ; 6.82 1000 8.72 8 69 8.65i 8.59 8.52 8.43 8.32 8.20 8.07 i 7.92 7.75 7.58 1100 9.59 9^56 9.52 9.45 9.37| 9.27 ! 9.16 i 9.02 8.87 i 8.71 8.53 8.34 120) 10.46 10.43 10.38 10.31 10.22 10.11 ! 9.99 ! 9.84 9.68 9.50 9.31 9.09 1300 11.33 11.30 11.25 11.17 11.07 10.96 10.8'i 10.66 10.49 S10.29 10.08 9.85 1400 12.21 12.17 12. 11 12. as 11.93 11.80 ill. 65 11.48 11.29 111. 08110. 86 10.61 1500 13.08 13.04 12.98 ! 12.89 12.78!l2.64 12.48 12.30 12 10 11.88 11.68 ! 11.37 1600 13.95 13.91 13.84 13.75.13.63 13.49 13.32 13.12 12.91 12.67 12.41 12.13 1700 14.82 14.78 14.71 14.61 14.48 14.33 14.15 13.94 13.71 13.46 13.18 12.88 1800 15.69 15.65 15.5715.47 15.33 15.17 14.98 14.76 14.52 14.25 13.96 13.64 1900 16.57 16.52 16.44116.33 16.19116. 01 15.81 15.58 is. as 15.04 14.73 i 14.40 2000 17.44 17.39 17.30il7.19 117.04 16.86 16. 65 16. 40 1613 15.83 15.51 15.16 302 TABLE XL-TURNOUTS AND SWITCHES FROM A STRAIGHT TRACK. 180, 181, 182. GAUGE, 4 FEET 8^3 INCHES = 4 .708. THRO w, 5 INCHES = . 41 7. No. Angle Dist. Chord i Switch Radius Log'thm. Degree n. F. BF. Of. AD. r. log. r. of Curve. 4 14 15' 00" 37.664 37.373 11.209 150.656 2.177986 38 45' 57" 4)^ j 12 10 49 42 .372 4 2.113 1 2.610 i 190. 674 2.2S 292 I ) 24 09 5 11 25 16 47.080 46.846 14.012 235.400 2.371806 24 31 36 10 23 20 51 .188 5 1.575 1 5.413 284 834 2Ai >4592 2( ) 13 13 6 ~ 9 31 39 56.496 56.301 16.814 338 1)76 2.530169 ]( J 57 52 OK 8 < 17 51 61 .204 6 1.024 1 8.215 397 820 2.5t mm ft I 26 25 7 8 10 16 65.912 65.744 19.616 i 461 384 2.664063 12 26 34 T^ij 7 57 41 70 .620 r* 0.464 2 1.017 529 (i.-,0 2.7$ S989 1( ) 50 02 8 " 7 ( )9 10 It .328 7 5.181 2 2.418 602 2.7S S0046 ! ) 31 07 8V 6 43 59 8 .036 79.898 23.820 680 3 2.832704 I J 25 47 9 6 5 11 35 81 .744 8 4.613 2 5.221 762 (Mi 2. 2352 r 31 04 6 01 32 H .452 89.328 26.622 849 704 2.929314 i J 44 46 10 ~ 5 < 13 29 94 .160 9 4.043 2 8.023 941 ooo 2.9 r '3866 ( 5 05 16 5 27 09 98.868 98.756 i 9.424 1038 114 3.01 6245 5 31 17 11 S 5 1 2 18 103 .576 10 3.469 . a 0.825 1139 330 3. Of .6652 > 01 50 HJ^ 4 i rfj 45 108 .284 10 S.182 3 2.227 1245. 21 M; 3.0J 5262 ^ I 36 08 12 4 4 6 19 112.992 112.894 33.628 1355. !HM 3.132229 4 13 36 GAUGE, 3 FEET. THROW, 4 INCHES = 0.333. No. Angle Dist. Chord 1 Switch Radius Log'thm. Degree n. F. BF. a/. AD. r. log. r. of Curve. 4 14 15' 00" 24 23.815 8 96.0 1.982 271 6$ 1 46' 34" 4/^ 12 40 49 27 26.835 9 121. 5 2.084576 48 36 04 5 11 25 16 30 2 9.851 10 150. 2.176 091 i ! 56 33 5^ 10 23 20 as 3 2.865 11 181. 5 2.25S #77 31 58 55 6 9 31 39 36 35.876 12 216.0 2.334454 26 46 07 6V> 8 47 51 39 3 8.885 13 253. 5 2.40S 978 2$ J 45 04 7 " 8 10 16 42 41.893 14 294. 2.46 ,347 19 35 01 7V> 7 37 41 45 4 4.900 15 v 337. 5 2.52* 274 r ' 02 21 8 ~ 7 09 10 48 4 7.906 16 384. 2.584 331 14 I 57 48 8^> G 43 59 51 50.912 17 433. 5 2.636989 13 14 47 9 ~ G 21 35 54 5 3.917 18 486. 2.68(3 636 11 48 37 i)V G 01 32 57 5 6.921 19 541. 5 2.73S 598 H ) 35 46 10 ~ 5 43 29 60 59.925 20 600. 2.778151 < > 33 38 lOVo 5 27 09 63 6 2.929 21 661. 5 2.82C 530 i i 40 12 11 ~ 5 12 18 66 65.932 22 726. 2.86C 1937 7 53 54 11*6 4 5S 45 69 6 8.935 23 793. 5 2.89S 547 r * 13 32 12 4 46 19 72 71.938 24 864. 2.936514 ( > 8 06 ANGLE AND DISTANCE OF MIDDLE FROG, F' Gauge Gauge Gauge Gauge No. No. Angle 4, 8Vjj>. 3. No. No. An^le 4, 8Va- 3. w. n". F". Dist. Dist. i n. n*. F* Dist Dist. aF'. of: I aF". aF". 4 2.817 20 07' 36" 26.736 17.037 8 5.651 10 06' 44" 53.317 33.974 4U 3.172 17 54 52 30.054 19.151 6.005 9 31 08 56.643 36.094 5 3.527 16 08 19 33 374 21.266 i 9 6.359 8 59 30 59.969 38.213 5Va 3.881 14 40 58 36.695 23.383 6.713 8 31 10 63.296 40.333 6 4.235 13 27 57 40.018 25.500 10 7.067 8 05 40 66.623 42.458 6V 4.589 12 26 07 | 43.342 27.618 7/420 7 42 35 69.950 44.573 7 4.943 11 33 04 46.666 29.736 11 7.774 7 21 36 73.277 46.693 7U 5.297 10 47 02 49.991 31.855 8.128 7 02 26 76.605 48.813 8 5.651 10 06 44 53.317 33.974 12 8.482 6 44 51 79.932 50.934 803 TABLE XII. MIDDLE DEDICATES FOR CURVING* RAILS. 103. LENGTH OF RAIL-CHORD. ~ 1 D D 32 30 28 26 24 22 s 10 18 16 14 12 10 1 .022 .020 .017 .015 013 .011 ".009 .007 .006 .004 .003 .002 1 2 .045 .039 .034 .030 025 0^1 .( m .01 4 .011 .009 .0 ti .004 2 3 .067 .059 .051 .044 !038 ' .032 .026 .021 .017 .013 .009 .007 3 4 .089 .079 .0138 .059 050 :, 04^ 5 .Oi S .022 .017 .0 u 009 4 5 .112 .098 .036 .074 Oc3 053 .'044 ;085 .028 .021 .016 .011 5 6 .134 .118 .103 .088 075 ^ U03 tt2 .OJ -.' .034 .026 .013 ft 7 .156 .137 .120 .103 .'(H8 on '.061 .049 .039 .0:30 .022 .015 7 8 .179 .157 .137 .118 100 a w .( KTO .0 7 .045 .0:34 .0 25 .017! 8 9 .201 .177 .154 .133 J13 Hto .078 .064 .050 .0:38 .028 .020 9 10 .223 .196 .171 .147 126 105 )S7 .07 1 .05(5 018 .0 81 .022 ' 10 11 .245 .216 .188 .162 138 Hi} !096 .078 .061 .047 .035 .024; 11 12 .268 .235 .205 .177 151 127 05 .0!- 15 .067 .051 .0 :;s .026 12 14 .312 .274 .2:38 .206 175 147 !l22 .099 .078 .<60 .014 .030 14 16 .356 .313 .273 .2:35 200 KiS i 39 .11 8 .089 .068 .0 50 .035 16 18 .400 .352 .307 .264 225 189 'l56 .127 .100 .077 .056 .039 18 20 .445 .391 .340 .293 250 j MO 71 .14 1 .111 .085 M .043 20 24 .531 .467 .40?' .351 299 f r.i ~> J07 .!( s .133 .102 .0 75 .052 24 28 .618 .543 .473 .408 347 '.241 .195 .154 .118 .087 .060 28 33 .705 .619 .539 .465 396 ( V i 275 .!& 8 .176 .135 .0 99 .069 32 36 .791 .696 .603 .522 445 ft';} i 309 .250 .197 .151 .111 .077 36 40 .878 .772 .672 .579 493 114 ., U2 .2" 7 .219, .168 .1 88 .086 40 45 .983 .863 .752 .648 552 !463 .383 .305 .188 .137 .096 i 45 50 jl.087 .955 .831 .716 610 .512 .423 .343 ^271 .207 .152 .106 50 TABLE XIII. -DIFFERENCE IN ELEVATION OF RAILS ON CURVES 201. VELOCITY IN MILES PER HOUR. D D 10 15 20 25 30 35 40 45 50 60 1 .006 .013 .023 .03 i .051 .070 191 .116 !. .143 206 1 2 .Qll .026 .046 .071 .103 .140 188 .231 .285 410 1 2 3 .017 .039 .069 .10 .If 4 .2 10 874 .346 .42 7 612 3 4 .023 .051 .091 .143 .20 6 .9 <0 !365 .461 .568 .811 4 5 .029 .064 .114 .17 1 .2 7 .a J', 455 .574 .7(1 7 1 006 5 6 1 .0:34 .077 .137 .214 .3C 18 .418 .'545 .687 .844 1 196 6 7 .040 .090 .160 .25 ) .3 g .41 37 ( ttl .798 .97 g 8 .046 .103 .183 .285 .410 .556 i'23 .908 1.112 9 .(151 .116 .206 .32 ) .4fi i .6, .'l >] 1 1 .017 10 .057 .129 .228 .35 1 .51 i .(i 12 ^'.)( v> . 1 .124 11 .063 .142 .251 .391 .561 7 10 !984 12 .069 .154 .274 .42 .61 t . 6 1. JO!) 14 ! .080 .180 .319 497 .711 ,! 16 .091 .206 .365 56 .80 !) 1 Ot >8 18 .102 .231 .410 .637 .906 20 .114 .256 .4oo .70 1.00 2 25 .141 .318 .563 .77 5 30 .168 .380 .672 .844 35 .195 .441 .778 40 .222 .501 .831 50 .276 1 .618 80-1 TABLE IIY.-GRADES AM) GRADE ANGLES. TABLE XIV. GRADES AND GRADE ANGLES. Feet per Sta- tion. Feet per Mile. Inclina- tion. Feet per Sta- tion. Feet per Mile. Inclina- tion. Feet S tion. Feet per Mile. Inclin- ation. / / / .01 .528 21 .51 26.928 17 32 1.01 53.328 34 43 .02 1.056 41 .52 27.456 17 53 .02 53.856 35 04 .03 1.584 1 02 .58 27.984 18 13 .03 54.384 35 24 .04 .05 2.112 2.640 1 23 1 43 .54 .55 28.512 29.040 18 34 1854 .04 .05 54.912 55.440 35 45 36 05 .06 3.168 2 04 .56 29.568 19 15 i .06 55.968 36 26 .07 3.696 2 24 .57 30.096 19 36 ! .07 56.496 36 47 .08 4.224 245 .58 30.624 19 56 .08 57.024 37 08 .09 4.752 3 06 .59 31.152 20 17 ! .09 57.552 3728 .10 5.280 3 26 .60 31.680 20 38 ; l.io 58.080 3749 .11 5.808 3 47 .61 32.208 20 58 .11 58.608 38 09 .12 6.336 4 08 .62 32.736 21 19 ! .12 59.136 38 30 .13 6.864 4 28 .63 33.264 21 39 ! .13 59.064 38 51 .14 7.392 4 49 .64 33.792 22 00 1 - 14 60.192 39 11 .15 7.920 509 ,65 34.320 22 21 .15 60.720 39 32 .16 8.448 5 30 .66 34.848 2241 ! .16 C1.248 39 53 .17 8.976 5 51 .67 35.376 23 02 .17 C1.776 40 13 .18 9.504 6 11 .68 35.904 23 23 .18 62.304 40 34 .19 10.032 6 32 .69 36.432 23 43 .19 02.832 40 54 .20 10.560 6 53 .70 36.960 2404 .20 63.360 41 15 .2T 11.088 7 13 .71 37.488 24 24 .21 63.888 41 35 .22 11.616 7 34 .72 38.016 24 45 .22 64.416 41 56 .23 12.144 7 54 .73 38.544 25 06 .23 64.944 42 17 .24 12.672 8 15 .74 39.072 25 26 .24 65.472 42 38 .25 13.200 8 36 .75 39.600 25 47 .25 66.000 42 58 .26 13.728 8 56 .76 40.128 2608 .26 C6.528 43 19 .27 14.256 9 17 .77 40.656 26 28 .27 67.056 43 39 .28 14.784 938 .78 41.184 2649 .28 07.584 44 00 .29 15.312 9 58 .79 41.712 27 09 .29 68.112 44 21 .30 15.840 10 19 .80 42.240 27 30 .30 68.1540 44 41 .31 16.368 1039 .81 42.768 27 51 .31 69.168 45 02 .32 16.896 11 00 .82 43.296 28 11 .32 69.696 45 23 .33 17.424 11 21 .83 43.824 28 32 ..33 70.224 45 43 .34 17.952 11 41 .84 44.352 28 53 .34 70.752 4604 .35 18.480 12 02 .85 44.880 29 13 .35 71.280 46 24 .36 19.008 12 23 .86 45.408 2934 .36 71.808 4645 .37 19.536 1243 .87 45.936 29 54 .37 72.336 47 06 .38 20.064 13 04 .88 46.464 30 15 .38 72.864 47 26 .39 20.592 1324 .89 46.992 30 36 .39 73.392 47 47 .40 21.120 13 45 .90 47.520 30 57 .40 73.920 4808 .41 21.648 14 06 .91 48.048 31 17 .41 74.448 4828 .42 22.176 14 26 .92 48.576 31 38 .42 74.976 48 49 .43 22.704 14 47 .93 49.104 31 58 .43 75.504 49 09 .44 23.232 15 08 .94 49.632 32 19 i .44 76.032 49 30 .45 23.760 15 28 .95 50.160 32 39 .45 76.560 49 51 .46 24.288 15 49 .96 50.688 33 00 .46 77.088 50 11 .47 24.816 1609 .97 51.216 33 21 1.47 77.616 50 32 .48 25.344 16 30 .98 51.744 33 41 1.48 78.144 5052 .49 25.872 16 51 .9!) 52.272 3402 1.49 78.672 51 13 .50 26.400 1711 1.00 52.800 3423 1.50 79.200 51 34 C05 TABLE XI\r. GRADES GRADE ANGLES. Feet per j Sta- tion. Feet per Mile. Inclina- tion. Feet per Sta- tion. Feet per Mile. || Inclina- tion. Feet per Sta- tion. Feet per Mile. Inclina- tion. at" / / 51 79.728 51 54 2.05 108.240 1 10 28 5.10 269.280 2 55 10 52 80.256 52 15 2.10 110.880 ) 1 12 11 5.20 874.660 25836 .53 80.784 52 36 2.15 113.520 1 13 54 5.30 219.840 3 02 09 54 81.312 52 56 2.20 116.160 1 15 37 5.40 285.120 3 05 27 .55 81.840 53 17 2.25 118.800 j 1 17 20 5.50 290.400 3 08 53 .56 82.368 53 37 2.30 121.440 1 19 03 5.60 295.680 3 12 19 .57 82.896 53 58 2.35 124.080 1 20 46 5.10 300. S60 3 15 44 .58 | 83.424 54 19 2.40 126.720 1 22 29 5.80 806.240 3 19 10 .59 83.952 54.39 2.45 129.360 1 24 12 5.90 311.520 3 22 36 .60 84.480 55 00 2.50 132.000 1 25 56 6.00 316.800 326 01 .61 85.008 55 21 2.55 134.640 1 27 89 6.10 822.080 3 29 27 .62 85-536 55 41 2.60 137.280 1 29 22 I 6.20 827.860 3 32 52 .63 86.064 56 02 2.65 139.920 1 31 05 | 6.10 882.640 3 36 18 .64 86.592 56 22 2.70 142. 60 1 32 48 6.40 837.920 1 3 89 43 .65 87.120 56 43 2.75 145.200 1 34 31 6. CO 343. 2CO 3 43 08 .66 ! 87-648 57 04 2.80 147.840 1 86 14 6. tO 348.480 3 46 34 .67 88.176 57 24 2.85 150.460 1 37 57 6.10 353. 7CO 349 59 .68 88.704 57 45 2.90" 153.120 1 39 40 6.80 359.040 3 53 24 .69 89.232 58 06 2.95 155.760 1 41 23 6.0 864.820 3 56 50 .70 i 89.760 58 26 3.00 158.400 1 43 06 7.CO 869.600 4 CO 15 .71 90.288 58 47 3.05 161.040 1 44 49 7.10 S74.880 4 03 40 .72 i 90.816 59 07 3.10 163.680 1 46 82 7.20 880.160 4 0706 .73 ! 91.344 59 28 3.15 166.320 1 48 15 7. SO 885.440 4 10 31 .74 91.872 59 49 3.20 168.960 1 49 58 7.40 SCO. 120 4 13 56 .75 92.400 1 00 09 , 3.25 171.600 1 51 41 7.50 SC6.0CO 4 17 21 .76 92.928 1 00 30 ' 3.30 174.240 1 53 24 7.60 401.280 4 20 46 .77 93.456 1 00 51 I 3.35 176.880 1 55 07 7.70 4C6.5CO 4 24 11 .78 93.984 1 01 11 3.40 179.520 1 6 50 7.80 411.840 4 27 86 .79 94.512 1 01 32 3.45 182.160 1 8 83 7. GO 417.120 4 81 01 .80 95.040 1 01 52 3.50 184.800 2 CO 16 8. CO 422. 4CO 4 84 26 .81 95.568 1 02 13 3.55 187.440 2 01 59 8.10 427.680 4 87 51 .82 96.096 1 02 34 i 3.60 190.080 2 03 42 8.20 482. 9CO 4 41 16 .83 96.624 1 03 64 3.65 192.720 2 05 25 8.80 488.240 4 44 41 .84 97.152 1 03 15 3.70 195.360 2 07 08 8.40 443.520 4 48 C6 .85 97.680 1 0335 3.75 198.000 2 08 51 8.EO 448. 8CO 4 51 80 .86 i 98.208 1 03 56 3.80 200.640 2 10 34 8.60 454. C80 4 54 55 .87 98.736 1 04 17 3.85 203.280 2 12 17 8.70 459.860 4 58 20 .88 99.264 1 04 37 3.90 205.920 2 14 00 8.80 1 464.640 5 01 44 .89 99.792 1 04 58 3.95 208.560 2 15 43 8.90 469.920 5 05 10 .90 100.320 1 05 19 4.00 211.200 2 17 26 9.00 415. 2CO 5 C8 34 .91 100.848 1 05 39 4.10 216.480 220 52 9.10 480.480 5 11 59 .92 101.376 1 06 00 4.20 221.760 2 24 18 9.20 485. 7CO 5 15 23 .93 101.904 1 06 20 4.30 227.040 2 27 44 9 30 491.040 5 18 48 .94 102.432 1 06 41 4.40 232.320 2 31 10 9.40 496.320 5 22 12 .95 102.960 1 07 02 4.50 237.600 2 34 36 9 50 501. 6CO 5 25 37 .96 103.488 1 07 22 4.60 242.880 23801 9.60 506.880 5 29 01 .97 104.016 1 07 43 4.70 248.160 241 27 9.70 512. ICO 5 32 25 1.98 104.544 1 08 04 4.80 253.440 2 44 53 9.80 517.440 5 35 50 1.99 105.072 1 0824 4.90 258.720 2 48 19 9.90 522.720 5 39 14 2.00 105.600 1 08 45 5.00 264.000 2 51 45 10.00 528.000 5 42 38 306 TABLE XV. FOR OBTAINING BAROMETRIC HEIGHTS IN FEET. Barom- eter. Inches 0.00 0.02 0.04 0.06 0.08 Diff. per .002 in. 19. .1 .2 16832 16970 17107 16860 16997 17134 16888 17025 17162 16915 17052 17189 16943 17080 17216 2.8 2.8 2.7 .3 17243 17270 17298 17325 17852 2.7 .4 17379 17406 17433 17460 17487 2.7 .5 17514 17540 17567 17594 17621 2.7 .6 17648 17674 17701 17728 17755 2.7 .7 17781 17803 17&34 17861 17887 2.7 .8 17914 17940 17967 17993 18020 2.7 .9 18046 18072 18099 18125 18151 2.6 20. 18178 18204 18230 18256 18282 2.6 .1 18308 18334 18360 18386 18413 2.6 .2 18438 18464 18490 18516 18542 2.6 18568 18594 18820 18645 18671 2.6 A 18697 18723 18748 18774 18799 2.6 .5 18825 18851 18S76 18902 18927 2.6 .6 18953 18978 19004 19029 19054 2.5 .7 19030 19105 19130 19156 19181 2.5 .8 19208 19231 19256 19232 19307 2.5 .9 19332 19357 19382 19407 19432 2.5 21. o 19457 19482 19507 19532 19557 2.5 .1 19582 19808 19(531 19656 19681 2.5 .2 19708 19730 19755 19780 19804 2.5 .3 19823 19854 19378 19903 19927 2.5 .4 19952 19376 20001 20025 20050 2.5 .5 20074 20098 20123 20147 20172 2.5 .6 20196 20220 20244 20269 20233 2.4 .7 20317 20311 20365 20389 20413 2.4 .8 20438 20462 20486 20510 20534 2.4 .9 20558 20581 20605 20629 20653 2.4 22 .0 20677 20701 20725 20748 20772 2.4 .1 20793 20820 20843 20867 20891 2.4 .2 20914 20938 20962 20985 21009 2.4 .3 21032 21056 21079 21103 21126 2.4 .4 21150 21173 21196 21220 21243 2.3 .5 21266 21230 21313 21336 21359 2.3 .6 21383 21408 21429 21452 21475 2.3 .7 21498 21522 21545 21568 21591 2.3 .8 21614 21637 21660 21683 21706 2.3 .9 21728 21751 21774 21797 21820 23 23. 21843 21866 21888 21911 21934 2.3 .1 21957 21979 22002 22025 22C47 2.3 .2 22070 220:2 22115 22138 22160 2.3 .3 22183 22205 22228 22250 22272 2.2 .4 22295 22317 22340 22362 22:384 2.2 .5 22407 22429 22451 23474 22496 2.2 .6 22518 22540 22562 22585 22607 2.2 .7 22629 22351 22673 22695 22717 2.2 .8 22739 22761 227a3 22805 22827 2.2 .9 22849 22871 S3893 22915 22937 2.2 24 22959 22981 23003 23024 23046 2.2 .1 23088 23090 23111 23133 23155 2.2 .2 23176 23198 23220 23241 23263 2.2 .3 23205 23306 23328 23349 23371 2.2 .4 23392 23414 234.35 23457 23478 2.2 .5 23500 23521 23542 23564 23585 2.1 .6 23608 23628 23649 23670 23()92 2.1 .7 23713 23734 23755 23776 23798 2.1 g 23819 23840 23861 23882 23903 2.1 .'9 23924 23945 23966 23987 24008 2 1 TABLE XV.-FOR OBTAINING BAROMETRIC HEIGHTS IN FEET. Barom- eter. Inches 0.00 0.02 0.04 0.06 0.08 Diff . per .002 in. 25.0 --.I .2 24029 24134 24238 24050 24155 24259 24071 24176 24280 24092 24197 24301 24113 24217 24321 2.1 2.1 2.1 .3 24342 24363 243*4 24404 24425 2.1 .4 24446 24466 24487 24508 24528 2.1 .5 24649 24569 24590 24610 24(531 2.1 .6 24651 24672 24692 24713 24733 2.0 .7 24754 24774 24794 24815 24835 2.0 .8 24855 24876' 24896 24916 24937 2.0 .9 24957 24977 24997 25018 25038 2.0 26. o 25058 25078 25098 25118 25138 2.0 .1 25159 25179 25199 25219 25239 2.0 .2 25259 25279 25299 25319 25*39 2.0 .3 25359 25379 25399 25419 25438 2.0 .4 25458 25478 25498 25518 25538 2.0 .5 25557 25577 23597 25617 25687 2.0 .6 25656 25676 25696 25715 25786 2.0 .7 2575;5 25774 25794 25818 25888 2.0 .8 25853 25872 25892 25911 25931 2.0 /J 25950 25970 25989 26009 26028 2.0 27. 26048 26067 26086 26106 26125 1.9 .1 26145 26164 26183 26203 26222 1.9 .2 26241 245260 26280 26299 26318 1.9 .3 26337 26&57 26376 26395 26414 1.9 .4 26433 26452 26472 2(5491 26510 1.9 .5 26529 26548 26567 26586 26605 1.9 .6 26624 26643 26662 26681 26700 1.9 7 26719 26738 26757 26776 26795 1.9 .'8 26813 26832 26851 2-. -870 26889 1.9 .9 26908 26926 26945 26964 26983 1.9 28.0 27001 27020 27039 27058 27076 1.9 .1 27095 27114 7132 27151 27169 > 1-9 .2 27188 27207' 27225 27244 27262 1.9 .3 27281 27299 27318 27336 27855 1.8 .4 27373 27392 27410 27429 27447 1.8 .5 27466 27484 27502 27521 27539 1.8 .6 27557 27576 27594 27612 27631 1.8 .7 27649 27(567 27685 27704 27722 1.8 .8 27740 27758 27777 27795 27'813 1.8 .9 27881 27849 27867 27885 27904 1.8 29. 27922 27940 27958 27976 27994 1.8 .1 28012 28030 28048 28066 28084 1.8 .2 28102 28120 28188 28156 28174 1.8 .3 28192 28209 28227 28245 28263 1.8 .4 23281 28299 28317 28834 28853 1.8 .5 28370 28388 28405 28423 28441 1.8 .6 28459 28476 28494 28512 28529 1.8 .7 88547 23565 28582 28600 88618 1.8 .8 28635 28653 28(570 28688 28706 1.8 .9 28723 28741 28758 28776 28793 1.8 30. 28811 28828 28846 28863 28881 1.8 .1 28898 28915 28933 28950 28968 1.8 .2 23985 2900.3 29020 29037 29054 1.7 .3 29072 29089 29106 29124 29141 1.7 .4 29158 29175 20192 29210 29227 1.7 .5 29244 29261 29278 29296 29313 1.7 .6 29330 29347 29364 29381 25)398 1.7 7 29416 29433 29450 29467 29484 1.7 [fi 29501 29518 29535 29552 29569 1.7 .9 29586 29C03 9620 29637 29654 , 1.7 COS TABLE XVI. COEFFICIENT OF CORRECTION FOR TEMPERATURE. .+* t + t'-W 900 t+t' \t+t' - 64 t + t' t + f - 64 + , t + f - 64 900 900 900 20 .0489 65 . .0011 110 _|_ .0511 155 .1011 21 .0478 66 .0022 111 .0522 156 .1022 23 .0467 67 .0033 IIS 1 . .0688 157 .1033 23 .0456 68 .0044 113 .0544 158 .1044 24 .0444 69 .0056 114 ir .0556 159 .1056 25 .0433 70 .0067 115 .0567 160 .1067 26 .0422 71 .0078 lie i .0578 161 .1078 27 .0411 72 .0089 117 .0589 .1089 28 .0400 73 .0100 11* \ .0600 163 .1100 29 .0:389 74 .0111 ni > .0611 164 .1111 30 .0378 75 .0122 120 -f .0622 165 .1122 31 .0367 76 -4- .0133 121 .06133 166 + .1133 32 .0356 77 .0144 12* i .0644 167 .1144 33 .0344 78 .0156 123 .0656 ! 168 .1156 34 .0333 79 .0167 124 l .0667 169 .1167 35 .0322 80 .0178 125 .0678 170 .1178 36 .0311 81 .0189 126 .0689 171 .1189 37 .0300 82 .0200 127 .0700 172 .1200 38 .0289 83 .0211 1& \ .0711 173 .1211 39 .0278 84 .0222 129 .0722 174 .1222 40 .0267 85 .0233 13( -f .0733 175 .1233 41 _ .0256 86 + .0244 131 .0744 176 + .1244 42 .0244 87 .0256 13$ ! .0756 177 .1256 43 .0233 88 .0267 133 .0767 J 178 .1267 44 .0222 89 .0278 134 .0778 179 .1278 45 .0211 90 .0289 13 1 .0789 180 .1289 46 .0200 91 .0300 136 .0800 181 .1300 47 .0189 92 .0311 137 .0811 182 .1311 48 .0178 93 .0322 138 .0822 183 .1322 49 .0167 94 .0333 m .0833 184 .1333 50 _ .0156 95 .0344 140 _|_ .0844 185 .1344 51 .0144 96 + .0356 141 .0856 186 -j- 1356 52 .0133 . 97 .0367 142 .0867 187 !l367 53 .0122 98 <* .0378 143 .0878 188 .1378 54 .0111 99 .0389 144 .0889 189 .1389 55 .0100 100 .0400 145 .0900 190 .1400 56 .0089 101 .0411 146 .0911 191 .1411 57 .0078 102 .0422 147 .0922 192 .1422 58 .0067 103 .0433 148 .0933 193 .1433 59 .0056 j 104 .0444 148 .0944 194 .1444 60 .0044 i 105 .0456 15C ' -f .0956 195 .1456 61 _ .0033 106 _j_ .0467 151 .0967 196 + .1467 62 .0022 107 .0478 152 .0978 i 197 .1478 63 .0011 108 .0489 153 .0'J&9 198 .1489 64 .0000 109 .0500 154 .1000 199 .1500 1 TABLE XVIL-CORRECTION FOR EARTH'S CURVATURE AND REFRACTION. 119. L H L H V H 1? H L H Miles H 300 .002 | 1300 .0.35 I 8300 .108 asoo .223 4300 .879 1 .571 400 003 1400 .040 2400 .118 3400 .237 4400 .397 2 2.285 500 .005 ! 1500! .046 2500 128 3500 .251 4500 .415 3 5.142 600 007 1600 .052 2600 .139 3600 .266 4600 .434 4 9.141 700 .010 1700; .059 : 2700 .149 3700 .281 4700 .453 5 14.282 800 .013 1800 .066 , 2800 .161 3800 .298 4800 .472 6 20.567 900 .017 191)0 .074 2900 .172 3900 .312 4900 | .492 7 27.994 1000 020. > 2000 082 ! 3000 .184 4000 .328 50001 .512 8 36.563 1100 1200 .025 .030 : 2100 2200 .090 j .099 3100 3200 .197 .210 4100 j ,345 4200 i .362 5100 i .533 5200 .554 9 i 46.275 10 57.130 009 TABLE XVIII.-COEFFICIENT FOR REDUCING INCLINED STADIA MEASUREMENTS TO THE HORIZONTAL. 224. a 0' 10' 20' 30' 40' 50' 1.000000 .999992 .999967 .999924 .999865 .999789 1 .99C696 .999586 .999459 .999315 .999154 .998977 2 .998782 .998571 .998343 .998098 .997836 .997557 3 .997261 .996949 .996619 .996273 .995910 .995531 4 .995134 .994721 .994291 .993844 .993381 .992901 5 .992404 .991891 .991360 .990814 .990250 .989670 6 .989074 .988461 .987831 .987185 .986522 .985843 7 .985148 .984436 .983708 .982963 .982202 .981424 8 .980631 .979821 .978995 .978152 .977294 .976419 9 .975528 .974621 .973698 .972759 .971804 .970833 10 .969846 .968843 .967824 .966790 .965739 .964673 11 .96:3591 .962494 .961380 .960252 .959107 .957948 12 .956772 .955581 .954375 .953153 .951916 .950664 13 .949396 .948113 .946815 .945.502 .944174 .942831 14 .941473 .940100 .938711 .937309 .935891 .934459 15 .933011 .931550 .930073 .928582 .927077 .925557 16 .924022 .922474 .920911 .919334 .917742 .916137 17 .914517 .912883 .911236 .909574 .907899 .906209 18 .904507 .902790 .901060 .899:316 .897558 .895787 19 .894003 .892206 .890395 .888571 .886733 .884883 20 .883020 .881143 .879254 .877352 .875437 .873510 21 .871569 .869617 .867652 .865674 .863684 .861681 22 .859667 .857(540 .855601 .853550 .851487 .849412 23 .847326 .845227 .843117 .840996 .838862 .836718 24 .834561 .832394 .830215 .828025 .825825 .823613 25 .821390 .819156 .816911 .814656 .812390 .810113 26 .807826 .805529 .803221 .800903 .798575 .796236 27 .793888 .791529 .789161 .786783 .784396 .781998 28 .779591 .777175 .774749 .772314 .769870 .767416 29 .764934 .762483 .760002 .757513 .755015 .752509 30 .749994 .747471 .744939 .742399 .739850 .737294 31 .734729 .732157 .729577 .726989 .724393 721790 32 .719179 .716561 .713935 .711302 .708662 .706015 33 .703361 .700700 .698033 .695358 692677 .689990 34 .687296 .684595 .681889 .679176 .676457 .673733 35 .671002 .668266 .665524 .662776 660023 .657264 36 .654500 .651731 .648957 .646177 .643393 .640604 37 .637810 .635011 .632208 .629401 .626588 623772 38 39 .620952 .603946 .618127 .601099 .615299 .598248 .612466 .595395 .609630 .592537 .606790 .589677 40 .586814 .5&3948 .581079 .578207 .575332 .572455 41 .569576 .566694 .563810 .660924 .558036 .555145 42 43 44 .552253 .534867 .517438 .549a59 .531964 .514530 .546464 .529061 .511622 .543567 .526156 .508714 .540668 .523251 .505805 .537768 .520345 .502897 45 .499988 .497079 .494170 .491261 .488353 .485445 310 TABLE XIX. - LOGARITHM OF COEFFICIENT FOR REDUCING IN- CLINED STADIA MEASUREMENTS TO THE HORIZONTAL. fc!>24. a 0' 10' 2(X 30' 40' 50' i 2 3 4 5 0.000000 9.999868 .999471 .998809 .997882 .996689 9.999996 .999820 .999379 .998673 .997701 .996464 9.999985 .999765 .999280 .998529 .997514 .990232 9.999967 .999702 .999173 .998379 .997318 .995992 9.999941 .999683 .999059 .998220 .997116 .995745 9.999908 .999555 .998938 .998055 .996906 .995491 6 7 8 9 10 9.995229 .993501 .991506 .989240 .986703 9.994959 .993187 .991147 .988836 .986253 9.994683 .992866 .990780 .988424 .985797 9.994399 .992537 .990406 .988005 .985332 9.994107 .992201 .990025 .987579 .'84860 9.993808 .991857 .989636 .987144 .984:380 11 JO 13 14 15 9.983893 .980808 .977447 .973808 .969887 9.983398 .980268 .976860 .973174 .969206 9.982895 .979719 .976265 .972532 .968517 9.982385 .979163 .975663 .971883 .967820 9.981867 .978599 .975052 .971225 .967116 9.981342 .978027' .974434 .970560 .966403 16 17 18 19 20 9.965683 .961192 .956412 .951339 .945970 9.964954 .960415 .955587 .950465 .945047 9.964218 .959631 .954753 .949583 .944114 9.963473 .958838 .953912 .948692 .943174 9.962721 .958087 .958068 .947793 .942225 9.961960 .957229 .952205 .946886 .941268 21 22 23 24 25 9.940302 .934330 .928050 .921458 .914549 9.939328 .933305 .926974 .920329 .913366 9.938345 .932271 .925888 .919191 .912175 9.937354 .931229 .924794 .918044 .910974 9:936355 .930178 .923691 .916888 .909764 9.935347 .929119 .922579 .915723 .908546 26 27 28 29 CO 9.907318 .899759 .891867 .883635 .875058 9.906081 .898467 .890519 .882230 .873594 9.904835 .897166 .889161 .880815 .872121 9.903580 .895855 .887794 .879390 .870637 9.902316 .894535 .886417 .877956 .869144 9.901042 .893.206 .885031 .876512 .867641 31 32 33 34 35 9.866127 .a56837 .847178 .837144 .826724 9.864604 .855253 .845532 .835434 .824949 9.863071 .858659 .843876 .833714 .823163 9.861528 .852054 .842209 .&31982 .821367 9.859974 .850439 .840531 .830240 .819559 9.858411 .848814 .838843 .828488 .817740 36 37 38 39 40 9.815910 .804691 .793058 .780998 .768500 9.814068 .802781 .791078 .778946 .766374 9.812216 .8008(50 .789086 .776882 .764235 9.810352 .798927 .787'082 .774805 .762083 9.808476 .796982 .785066 .772716 .759919 9.806589 .795026 .7830:38 .770614 .757742 41 42 43 44 45 9.755552 .742138 .728246 .713858 9.698959 9.753349 .739857 ,725883 .711411 9.696425 9.751133 .737561 .723506 .708950 9.693876 9.748904 .735253 .721115 .706474 9.691313 9.746662 .732931 .718710 ! 708988 9.688734 9.744407 .730595 .716291 .701479 9.686140 TABLE XX.-LENGTHS OF CIRCULAR ARCS; RADIUS = 1. Sec. 1 ,- . *:. | Length. Min. Length. 1 Deg. Length. Deg. | I Length. 1 .0000048 1 .0002909 1 .0174533 61 1.0646508 2 .0000097 2 .0005818 2 .0349066 62 1.0821041 3 .0000145 3 .0008727 3 .0523599 63 1 .0995574 4 .0000194 4 .0011636 4 .0698132 64 | 1.1170107 5 .0000242 5 .0014544 5 .0872665 65 1.1344640 6 .0000291 6 .0017453 6 .1047198 66 1.1519173 7 .0000339 7 .0020362 7 .1221730 67 1.1693706 8 .0000388 8 .0023271 8 .1396263 68 1.1868289 9 .0000436 9 .0026180 9 .1570796 69 1.2042772 10 .0000485 10 .0029089 10 .1745329 70 1.2217305 11 .0000533 11 .0031998 11 .1919862 71 1.2391838 12 .0000582 12 .0034907 12 .094395 72 1.2566371 13 .0000630 13 .0037815 13 .2268928 73 1.2740804 14 .0000679 14 .0040724 14 .2443461 74 1.2915436 15 .0000727 15 .004S633 15 .2617894 75 1.3089969 16 .0000776 16 .0046542 16 .2792527 76 1.3264502 17 .0000824 17 .0049451 17 .2967060 77 1.3439035 18 .0000873 18 .0052360 18 .3141593 78 1.8613568 19 .0000921 19 .0055269 19 .3316126 79 1.3788101 20 .0000970 20 .0058178 20 .3490659 1.3962634 21 .0001018 21 .0061087 21 .3665191 81 1.4137167 22 .0001067 22 .0063995 22 .SfcS9724 82 1.4311700 23 .0001115 23 .0066904 23 .4014257 83 1. 44 J- 6233 24 .0001164 24 .0069813 24 .4188790 84 1.46fc07C6 25 .0001212 25 .0072722 25 .4368323 85 1.4885299 26 .0001261 26 .0075631 26 .4537856 86 1.6008fc82 07 .0001309 27 .0078540 27 .4712389 i 87 1.5184864 28 .0001357 28 .0081449 28 .4886922 88 1.58Efcfc97 29 .0001406 29 .0084358 29 .5061455 89 1. 583430 30 .0001454 30 .0087266 30 .5235988 90 1.5707963 31 .0001503 31 .0090175 31 .5410521 91 1. 5882486 32 .0001551 32 .0093084 32 .5685054 92 1.C057U29 33 .0001600 33 .0095993 33 .5768E87 93 1.C2315G2 34 .0001648 34 .0098902 34 .5934119 94 1.640COG5 35 .0001(597 35 .0101811 35 .6108652 95 1. 580028 36 .0001745 36 .0104720 36 .6283185 96 1.6755161 37 .0001794 37 .0107C29 37 .6457718 97 1. 6828694 38 .0001842 38 .0110538 38 .6632251 98 1.7104227 39 .0001891 39 .0113446 39 .6806784 99 1.72787CO 40 .0001939 40 .0116355 40 .6981317 100 1.7453293 41 .0001988 41 .0119264 41 .7155850 101 1.7627825 42 .0002036 42 .0122173 42 .7380883 102 1.7802358 43 .0002085 43 .0125082 43 .7504916 103 1.797C891 44 .0062133 44 .0127991 44 .7679449 104 1.8151424 45 .0002182 45 .0130900 45 .7858982 105 1.8825857 46 .00022430 46 .0133809 46 .8028515 106 1.8*600480 47 .0002279 47 .0136717 47 .8208047 107 1.867 023 48 .0002327 48 .0139626 48 .8377580 108 1.8849556 49 .0002376 49 .0142535 49 .8552113 109 1.9024089 50 .0002424 50 .0145444 50 .8726646 110 1.9168622 51 .0002473 51 .0148353 51 .8901179 111 1.9373155 52 .0002521 52 .0151262 52 .9075712 112 1.9547688 53 .0002570 53 .0154171 53 .9250245 113 1.9722221 54 .0002618 54 .0157080 54 .9424778 114 1.8886753 55 .0002666 55 .0159989 55 .9599311 115 2.C071286 56 .0002715 56 .0162897 56 .9773844 116 2.024819 57 .0002763 57 .0165806 57 .9948377 117 2.042C852 58 .0002812 58 .0168715 58 1.0122910 118 2 0594885 69 .0002800 1 59 .0171624 59 1.0297443 119 2.0769418 60 .0002909 60 .0174533 60 1.0471976 12J 1 2.0943951 TABLE XXI. -MINUTES IN DECIMALS OF A DEGREE < or 10" 15" 1 | 20" 30" 40" ! 45" 50" ' .00000 00278 .00417 .00556 .00833 .01111 I .01250 .01389 1 .01667 .01944 .02083 .02222 .02500 .02778 .02917 .03055 | 1 2 .03333 .03611 .03750 .03889 .04167 .04444 .04583 .04722 2 3 .05000 .05278 .05417 .05556 .05833 .06111 .06250 .06:389 3 4 .06S67 .06944 .07083 .07222 .07500 .07778 .07917 .08056 4 5 .08333 .08611 .08750 i .08889 .09167 .09444 .09583 .09722 5 6 .10000 . 10278 .10417 .10556 .10833 .11111 .11250 | .11389 6 7 .11667 .11944 .12083 . 12222 .12500 .12778 .12917 .13056 7 8 13333 .13611 .13; 50 .13889 .14167 .14444 .14583 .14722 8 9 . .15000 .15278 .15417 .15556 .15833 .16111 .16250 .16389 9 10 .16667 .16944 .17083 .17222 .175JO .17778 .17917 .18056 10 11 .18333 .18611 .18750 .18889 .19167 19444 .19583 .19722 11 12 .20000 .20278 .20417 .20556 .20833 .21111 .21250 .21389 12 13 .21667 .21944 .22083 j .22222 .22500 .22778 .22917 j .23056 13 14 .23333 .23611 .23750 .23889 .24167 .24444 .24583 | .24722 14 15 .25000 .25278 .25417 .25556 .25833 .26111 .26250 .26389 15 16 .26667 .26944 .27083 .27222 .27500 | .27778 .27917 .28056 16 17 .28333 .28611 .28750 .28889 .29167 .29444 .29583 .29722 17 18 .30000 .30278 .30417 .30556 .30833 .31111 .31250 .31389 18 19 .31667 .31944 .32083 .32222 .32500 .32778 .32917 .33056 i 19 20 .33333 .33611 .33750 .33389 .34167 .34444 .34583 .34722 20 21 .3.5000 .35278 .35417 .35556 .35833 .36111 .36250 .36389 21 22 .36667 .36944 .37083 .37222 .37500 .37778 ."37917 .38056 22 23 .38333 .38611 .38750 .38889 .39167 .39444 .395a3 .39722 23 24 .40000 .40278 .40417 .40556 .40833 .41111 .41250 .41389 24 25 .41667 .41944 .42083 .42222 .42500 .42778 .42917 .43056 25 26 .43333 .43611 .43750 .43889 .44167 .44444 .44583 .44722 26 27 .45000 .45278 .45417 .45556 .45833 .46111 .46250 .46389 27 28 .46667 .46944 .47083 .47222 .47500 .47778 .47917 .48056 28 29 .48333 .48611 .48750 .48889 .49167 .49444 .49583 .49722 29 30 .50000 .50278 .50417 .5^556 .50833 .51111 .51250 .51389 30 31 .51667 .51944 .52083 .52222 .52501 .52778 .E2917 .530.56 31 32 .53333 .53611 .53750 .53889 .54167 54444 .54583 .54722 32 33 .55000 .55278 .55417 .55556 .55833 iseni .56250 .56389 33 34 .56667 .56944 .57083 .57222 .57500 .57778 .57917 .58056 34 35 .58-333 .58611 .58750 .58889 .59167 .59444 .59583 .59722 35 36 .60000 .60278 .60417 .60556 .60833 .61111 .61250 .61389 36 37 .61667 .61944 .62083 .62222 .62500 .62778 .62917 .63056 37 38 .63333 .63611 .63750 .63889 .64167 .64444 .64583 .64722 38 39 .65000 .65278 .65417 .65556 .65833 .66111 .66250 .66389 39 40 .66667 .66944 .67083 .67222 .67500 .67778 .67917 .68056 40 41 .68333 .68611 .68750 .68889 .69167 .69444 .69583 .69722 41 42 .70000 .70278 .70417 .70556 .708a3 .71111 .71250 .71389 42 43 .71667 .71944 .72083 .72222 .72500 .72778 .72917 .73056 43 44 .73333 .73611 .73750 .73889 .74167 .74444 .74583 .74722 44 45 .75000 .75278 .75417 .75556 .75833 .76111 .76250 .76389 45 46 .76667 .76944 .77083 .77222 .77500 .77778 .77917 .78056 46 47 .78333 .78611 .78750 .78889 .79167 .79444 .79583 .79722 47 48 .80000 .80278 .80417 .80556 .80833 .81111 .81250 .81389 48 49 .81667 .81944 .82083 .82222 .82500 .82778 .82917 .83056 49 50 .83*33 .83611 .83750 .83889 .84167 .84444 .84583 .847'22 50 51 .85000 .85278 .85417 .85556 .85833 86111 .86250 .86389 51 52 .86667 .86944 .87083 .87222 .87500 .87778 .87917 .88056 52 53 .88333 .88611 .88750 .88889 .89167 .89444 .89583 .89722 53 51 .90000 .90278 .90417 .90556 .90833 .91111 .91250 .91389 54 55 .91667 .91944 .92083 .92222 .92500 .92778 .92917 .93056 55 56 .93333 .93611 .93750 .93889 .94167 .94444 .945&3 .94722 56 57 .95000 .95278 .95417 .95556 .95833 .96111 .96250 .96389 57 58 .96667 .96944 .97083 .97222 .97500 .97778 .97917 .98056 58 59 .98333 .98611 .99750 .98889 .99167 .99444 .99583 .99722 59 ' 0- 10" 15" 20" 30" 40" 45" 50" 31 o TABLE XXII.- INCHES IN DECIMALS OF A FOOT. . ' In. 1 2 3 4 5 6 7 8 9 i 10 11 In. i Foot .0833 .1667 . 25001. 333 .4167 .5000 .5833 .6667 .7500! .8333 .9167 1-32 .0026 .0859 .1693 .2526 .3:359 .4193 .5026 .5859 .6693 .75261.8359 .9193 1-32 1-16 .0052 .088.") .1719 .2552 .3385 .4219 .5052 .58851.6719 .75521.8385 .9219 1-16 3-32 .0078 .0911 .1745 .2578 .3411 .4245 .5078 .5911 .6745 .75781.8411 .9245 3-32 1-8 .0104 .09:38 .1771 .2604 .3438 .4271 .5104 .5938 .0771 .7604L8438 .9271 1-8 5-32 .0130 .0964 .1797 .2630 .34641 .42971. 5130 .5964 .6797 .7630 .8464 .9297 5-32 3-16 .0156 .0990 .1823 .2650 .3490 .4323 .5156 .5990 .6823 .7656 .8490 .9323 3-16 7-32 .0182 .1016 .1849 .2682 .3516 .4349 .5182 .6016 .6849 .7682:. 8516 .9349 7-32 l-l .0208 .1042 .1875 .2708 .3542 .4375 .5208 .6042 .6875 .7708*. 8542 .9375 1-4 9-32 .0234 .1068 .1901 .2734 .3568 .4401 .5234 .6068 .6901 .7734 ! .8568 .9401 9-32 5-16 .0260 .1094 . 1927 . 2760: .3594i. 4427|.5260 .60941.6927 .7760!. 8594 .9427 5-16 11-32 .0280 .1120 .1953 .2780 .3020 .4453 .5286 .6120 .6953 .7786 .8620 .9453 11-32 3-8 .0313 .1146 . 1979 .2813 .3646 .4479 .5313 .6146 .6979 7813 ! .8646 .9479 3-8 13-32 .0339 .1172 .2005 .2839 .36721 .4505[ .5339| .6172| .7005 .7839 .867'2 .1-505 13-32 7-16 0365 .1198 .2031 .2865 .3698 .4531 .5365 .6198 .7031 .7865 .8698 .9531 7-16 15-32 .0391 .1224 .2057 .2891 .3724 .45571.5391 .6224 .7057 . 7891 :. 8724 .9557 15-32 1-2 .0417 .1250 .2083 .2917 .3750 .4583 .5417 .6250 .7083 .7917 .8750 .9583 1-2 17-32 .0443 .1276 .2109 .2943J .3776 .4609 .5443 .6276J. 7109 .7943 .8776 .9609 17-32 9-16 .0469 .1302 .2135 .2969 .3802 .46:35 .5469 .6302 .7135 .7969 s .8802 .9635 9-16 19-32 .0495 .1328 .2161 .2995 .3828 .4661 .5495 .7161 .7995 ! .8828 .9661 19-32 5-8 .0521 .1354 .2188 . 3021 1.3854 .4688 .55211.6354 .7188 .8021,. 8854 .968- 5-8 21-32 .0547 .1380 .2214 .3047 .3880 .4714 .5547 .7214 .8047; .8880 .9714 21-32 11-16 .0573 .1406 .2240 .3073 .3906 .4740 .5573 .7240 .8073 .8906 .9740 11-16 23-32 .0599 .1432 .2260 .3099 .3932 .4766 .5599 .7266 .8099,. 8932 .9766 23-32 3-4 .0625 .1458 .2292 .3125 .3958 .4792 .5625 .6458 .7292 .8125L8958 .0792 3-4 25-32 .0651 .1484 .2318 .3151 !3984 .4818 .5651 .6484 .7318 .8151 .8984 .9818 25-32 13-16 27-32 7-8 .0677 .0703 .0729 .1510 .1536 .1563 .2344 .3177 .2370 .3203 .2396 .3229 .4010 .4036 .4063 .4844 .4870 .4896 .5677 .5703 .5729 .6510 .7344 .6536 .7370 .6563.7396 .81771. 90101. 9844 13-1 6 .8203 .9036 .987027-32 .8229 .9063!. 9896) 7-8 29-32 .0755 .1589 .24221.3255 .4089 .4922 .5755 .65891.7422 .8255 .9089 .992229-32 15-16 .0781 .1615 .24481 .3281 .4115 .4948 .5781 .6615 .7448 .8281 .9115 .9948 15-16 31-32 .0807 .1641 .2474 .3307 .4141 .4974 .5807 .6641 .7474 .8307: .9141 .9974 31-32 ,0 1 2 3 4 5 6 7 8 9 i 10 11 : .'.'. 814* TABLE XXIII. SQUARES, CUBES, SQUARE ROOTS. No. Squares. Cubes. Square Roots. Cube Roots. 1 Reciprocals. 1 1 1 1.0000000 1.0000000 1.000000000 2 4 8 1.4142136 1.2599210 .500000000 3 9 27 1.7320508 1.4422496 .333333333 4 16 64 2.0000000 1.5874011 .250000000 5 25 125 2.2360680 1.7099759 .200000000 6 36 216 2 4494897 1.8171206 . 166666667 7 49 343 2.6457513 1.9129312 . 142857 143 8 64 512 2.8284271 2.0000000 .125000000 9 81 729 3.0000000 2.0800837 .111111111 10 100 1000 3.1622777 2.1544347 .100000000 11 121 1331 3.3166248 2.2239801 .090909091 12 '144 1728 3.4641016 2.2894286 .083333333 13 169 2197 3.6055513 2.3513347 .076923077 14 196 2744 3.7416574 2. -1101422 .071428571 15 225 3375 3.8729833 2.4662121 .066666667 16 256 4096 4.0000000 2.5198421 .062500000 17 289 4913 4.1231056 2.5712816 .058823529 18 324 5832 4.2426407 2.6207414 .055555556 It) 361 6859 4.3588989 2.6684016 .052631579 20 400 8000 4.4721360 2.7144177 .050000000 !tl 441 9261 4.5825757 2.7589243 .047619048 22 484 10C48 4.6904158 2. 80*0393 .045454545 23 529 . 12167 4.7958315 2.8438610 .043478261 24 576 13824 4.8989795 2.8844991 .041666667 25 625 15625 5.0000000 2.1)240177 .04COOOOOO 20 676 17576 5.0990195 2.9624060 .038461538 27 729 19683 5.1961524 3.000COOO .037037037 28 784 21952 5.2915026 3.0365889 .035714286 29 ' 841 24389 5.3851648 3.0723168 .034482759 30 900 27000 5.4772256 3.1072325 .033333333 31 961 29791 5.5677644 3.1413806 .032258065 32 1024 32768 5.6568542 3.1748021 .031250000 33 1089 35937 5.7445626 3.2075343 .030303030 34 1156 39304 5.8309519 3.2396118 .029411765 35 1225 43875 5.9160798 3.2710663 .028671429 3(3 1296 46656 6.0000000 3.3019S72 .027777778 37 1369 50653 6.0827625 3.3322218 .027027027 38 1444 54872 6.1644140 3.3619754 .026315789 39 1521 59319 6.2449980 3.3912114 .025641026 40 1600 64000 6.3245553 3.4199519 .025000000 41 1681 68921 6.4031242 3.4482172 .024390244 42 1764 74088 6.4807407 3.476C266 .023809524 43 18-49 79507 6.5574385 3.5033981 .023255814 44 1936 85184 6.6332496 3.5303483 .0227'27273 45 2025 91125 6.708.2039 3.5568933 .022222222 46 2116 97336 6.7823300 3.5830479 .021739130 47 2209 103823 6.8556546 3.6088261 .02127(5600 48 2304 110592 6.9282032 3.6342411 .020833333 49 2401 117649 7.0000000 3.6593057 .020408163 50 2500 125000 7.0710678 3.6840314 .020000000 51 2601 132651 7.1414284 3.7084298 .019607843 52 2704 140608 7.2111026 3.7325111 .019230769 53 2809 148877 7.2801099 3.7562858 .0188fJ7!i2r> 54 2916 157464 7.3484692 3.7797631 .018518519 55 3025 166375 7.4161985 3.8029525 .018181818 56 3136 175616 7.4833148 3.8258624 .017857143 57 3249 185193 7.5498344 3.8485011 .017543860 58 3364 195112 7.6157731 3.8708766 .017241379 59 3481 205379 7.6811457 3.8929965 .016949153 60 3600 216000 7.7459667 3.9148676 .016666667 61 3721 226981 7.8102497 3.93(54972 .016393443 62 3844 238828 7.8740079 3.9578915 .016129032 UL5 CUBE ROOTS, AND RECIPROCALS. r No. Squares. Cubes. Square Roots. Cube Roots. Reciprocals. 63 3969 250047 7.9372539 3.9790571 .015873016 64 4096 262144 8.0000000 4.0000000 .015625000 6,5 4225 274625 8.0622577 4.0207256 .015:384615 60 4356 287496 8.1240384 4.0412401 .015151515 67 4489 300763 8.1853528 4.0615480 .01492,5373 68 4024 314432 8.2462113 4.0816551 .014705882 69 4761 328509 8.3066239 4.1015661 .014492754 70 4900 343000 8.3666003 4.1212853 .014285714 71 5041 357911 8.4261498 4.1408178 .014084507 72 5184 373248 8.4852814 4.1601676 .013888889 73 5329 389017 8.5440037 4.1793390 .013698630 74 5476 405224 8.6023253 4.1983364 .013513514 75 5625 421875 8.6602540 4.2171633 .013*33333 76 5776 438976 8.7177979 4.2358236 .013157895 77 5929 456533 8.7749644 4.2543210 .012b870l3 78 6084 474552 8.8317609 4.2726586 .012820513 79 6241 493039 8.8881944 4.2908404 .012658228 80 6400 512000 8.9442719 4.3088695 .012500000 81 6561 531441 9.0000000 4.3267487 .012345679 82 6724 551368 9.0553851 4.3444815 .012195122 83 6889 571787 9.1104336 4.3620707 .012048193 84 7056 592704 9.1651514 4.3795191 .011904762 85 7225 614125 9.219.5445 4.3988296 .011764706 86 7396 636056 9.2736185 4.4140049 .011627907 87 7569 658508 9.3273791 4.4310476 .011494253 83 7744 681472 9.3808315 4.4479602 .01136:3636 89 7921 704969 9.4:339811 4.4647451 .011235955 90 8100 723000 9.4868330 4.4814047 .011111111 91 8231 753571 9.5393920 4.4979414 .010989011 92 8464 778683 9.5916630 4.5143574 .010369565 93 8849 804357 9.6436508 4.5306549 .010r52688 94 8836 830584 9.6953597 4.5468359 .010638298 M 9025 857375 9.7467943 4.5629026 .010526316 96 9216 884736 9.7979590 4.5788570 .010416667 87 9403 912673 9.8488578 4.5947009 .010309278 98 9504 941192 9.8994949 4.6104363 .010204082 90 9801 970299 9.9498744 4.6260350 .010101010 ioa 10000 1000000 10.0000000 4.6415888 .010000000 101 10201 1030301 10.0498756 4.6570095 .009900990 102 10404 1061208 10.0995049 4.6723287 .009303922 103 10609 1092727 10.1488916 4.6875482 .0097087:38 104 10816 1124864 10.1980390 4.7026694 .009615385 105 11025 H678S5 10.2469508 4.7176940 .009523810 106 11236 1191016 1Q. 2956301 4.7326235 .009433962 107 11449 1225043 10.3440804 4.7474594 .009345794 108 11664 1259712 10.3923048 4.7622032 .00:'259259 109 11881 1295029 10.4403065 4.7768562 .009174312 110 12100 1331000 ; 10.488C885 4.7914199 .009090909 111 12321 1367631 10.5356538 4.8058955 .009009009 112 12544 1404928 10.5830052 4.8202845 . 0089285 il 113 12769 1442897 10.G301458 4.8345881 ' .008849558 114 12996 1481544 10. (770783 4.848S076 .008771930 ' 115 13225 1520875 10.7238053 4.8629442 .008695652 116 13456 1560396 10.7703296 4.8769990 .008620690 117 13689 1601613 10. 8166538 4.8909732 .003547009 118 13924 1643032 10.8627805 4.9048681 .008474576 119 14161 1685159 10.9087121 4.9186847 .008403361 120 14400 1728000 10.9544512 4.9324249 .008333333 121 14641 1771561 11.00.0000 4.9460874 .0032(54463 188 14884 1815848 11.0453610 4.95967'57 .008196721 123 15129 1860867 11.0905365 4.9731898 .008130081 124 15376 1906624 ll.ia>5287 4.9866310 .008064516 31G TABLE XXIII. SQUARES, CUBES, SQUARE ROOTS. No. Squares. Cubes. Square Roots. Cube Roots. Reciprocals. 125 15625 1953125 11.1803399 5.0000000 .008000000 126 15876 2000376 11.22497'22 5.0132979 .007936508 127 16129 2048383 11.2694277 5.0265257 .007874016 128 16384 2097152 11.3137085 5.0396842 .007812500 129 16641 2146689 11.3578167 5.0527743 .007751938 130 16900 2197000 11.4017543 5.0657970 .007692308 131 17161 2248091 11.4455231 5.0787531 .007633588 132 17424 2299968 11.4891253 5.0916434 .007575758 133 17689 2352637 11.5325626 5.1044687 .007518797 134 17956 2406104 11.5758369 5.1172299 .007462687 135 18225 2460375 11.6189500 5.1299278 .007407407 136 18496 2515456 11.6619038 5.1425632 .007352941 137 18769 2571353 11.7046999 5.1551367 .007299270 138 19044 2628072 11.7473401 5.1676493 .007246377 139 19321 2685619 11.7898261 5.1801015 .007194245 140 19600 2744000 11.8321596 5.1924941 .007142857 141 19881 2803221 11.8743421 5.2048279 .007092199 142 20164 2863288 11.9163753 5.2171034 .007042254 143 20449 2924207 11.9582607 5.2293215 .006993007 144 20736 2985984 12.0000000 5.2414828 .006944444 145 21025 3018625 12.0415946 5.2535879 .006896552 146 21316 3112136 12.0830460 5.2656374 .006849315 147 21609 3176523 12.1243557 5.2776321 .006802721 148 21904 3241792 12.1655251 5.2895725 .006756757 149 22201 8307949 12.2065556 5.3014592 .006711409 150 22500 3375000 12.2474487 5.3132928 .006666667 lol 22801 3442951 12.2882057 5.3250740 .006622517 152 23104 3511808 12.3288280 5.33G8033 .006578947 153 23409 3581577 12.3693169 5.3484812 .006535948 154 23716 3652264 12.4096736 5.3601084 .006493506 155 24025 3723875 12.449899(5 5.3716854 .006451613 156 24336 3796416 12.48999GO 5.3832126 .006410256 157 24649 3869893 12.5299641 5 3946907 .006369427 158 24964 3944312 12.5698051 5.4061202 .006329114 159 25281 4019679 12.6095202 5.4175015 .006289308 160 25600 4096000 12.6491106 5.4288352 .006250000 101 25921 4173281 12.6885775 5.4401218 .006211180 162 26244 4251528 12.7279221 5.4513618 .006172840 163 26569 4330747 12.7671453 5.4625556 .006134969 164 26896 4410944 12.8062485 5.4737037 .006097561 165 27225 4492125 12.8452326 5.4848066 .006060606 166 27556 4574296 12.8840987 6.4958647 .006024096 167 27889 4657463 12.9228480 5.5068784 .005988024 168 28224 4741632 12.9614814 5.5178484 .005952381 169 28561 4826S09 13.0000000 5.5287748 .005917160 170 28900 4913000 13.03^4048 5.5396583 .0058821353 171 29241 5000211 13.0766968 5.5504991 .005847953 172 29584 5088448 13.1148770 5.5612978 .005813953 173 29929 5177717 13. 1529464 5.5720546 .005780347 174 30276 5268024 13.1909060 5.5827702 .005747126 175 30625 5359375 13.2287566 5.5934447 .005714286 176 30976 5451776 13.2664992 5.6040787 .005681818 177 31329 5545233 13.3041347 5.6146724 .005649718 178 31684 5639752 13.3416641 6.6252263 .005617978 179 32041 5735339 13.3790882 5.6357408 .005586592 180 32400 5832000 13.4164079 5.6462162 .005555556 181 32761 5929741 13.4536240 5.6566528 .005524862 182 33124 6028568 13.4907376 5.6670511 .005494505 183 33489 6128487 13.5277493 5 6774114 .005464481 1&4 33856 6229504 13.5646600 5.6877340 .005434783 185 84225 6331625 13.6014705 5.6980192 .005405405 186 34596 6434856 13.6381817 6.7082675 .005376344 317 CUBE ROOTS, AND RECIPROCALS. No. Squares. Cubes. Square Roots. Cube Roots. Reciprocals. 187 34969 6539203 13.6747943 5.7184791 .005347594 188 35344 6644672 13.7113092 6.7286543 .005319149 189 35721 6751269 13.7477271 5.7387936 .005291005 190 36100 6859000 13.7840488 6.7488971 .005263158 191 36481 6967871 13.8202750 6.7588652 .005235602 192 36864 7077888 13.8564065 5.7669982 .005208333 193 37249 7189057 13.8924440 5.7789966 .005181347 194 37636 7301384 13.9283883 5.7889604 .005154639 195 38025 7414875 13.9642400 5.7988900 .005128205 196 38416 7529536 14.0000000 5.8087857 .005102041 197 38809 7645373 14.0356688 5.8186479 .005076142 198 39204 7762392 14.0712473 5.8284767 .005050505 199 39601 7880599 14.1067360 5.8382725 .005025126 200 40000 8000000 14.1421356 6.8480355 .005000000 201 40401 8120601 14.1774469 5.8577660 .004975124 202 40804 8242408 14.2126704 6.8674643 .004950495 203 41209 8365427 14.2478068 5.8771307 .004926108 204 41616 8489664 14.2828569 5.8867653 .004901961 205 42025 8615125 14.3178211 5.8963685 .004878049 206 42436 8741816 14.3527001 5.9059406 .004854369 207 42849 8869743 14.3874946 5.9154817 .00480918 208 43264 8998912 14.4222051 5.9249921 .004807692 209 43681 9129329 14.4568323 5.9344721 .004784689 210 44100 9261000 14.4913767 5.9439220 .004761905 211 44521 9393931 14.5258890 5.8533418 .004739336 212 44944 9528128 14.5C02198 5.9627320 .004716981 213 45369 9663597 14.5945195 5.9720926 .004694836 214 45796 9800344 14.6287388 ! 5.8814240 .004672897 215 46225 9938375 14.6628783 5.9907264 .004651163 216 46656 101)77696 14.6969385 6.0COOCOO .004629630 217 47089 10218313 14.7309199 6.CCS2450 .004608295 218 47524 10360232 14.7648231 6.0184617 .004587156 219 47961 10503459 14.7986486 6.0276502 .C04E66210 220 48400 10048000 14.8S23970 6.0368107 .004545455 221 48841 10793861 14.8660687 6.0459435 .004524887 222 49284 10941048 14.8996644 6.0250489 . 004504 Ea5 223 49729 11089567 14.9331845 6.C641270 .004484305 224 50176 11239424 14.9666295 6.0731779 .004464286 225 50625 11390625 15.0000000 6.C8S2020 .004444444 226 51076 11543176 15.0332964 6.0911994 .004424779 227 51529 11697083 15.0665192 6.1C01702 .004405286 228 51984 Iia52352 15.0996689 6.1091147 .004385965 229 52441 12008989 15.1327460 6.1160332 .004366812 230 52900 12167000 15.1657509 6.1269257 .004347826 231 53361 12326391 15.1986842 6.1357924 .004329004 . 232 53824 12487168 15.2315462 6.1446337 .004310345 233 54289 12649337 15.2643375 6.1 34495 .004291845 234 54756 12812904 15.2970585 6.1622401 .004273504 235 55225 12977875 15.3297097 6.1710058 .004255319 236 55696 13144256 15.3622915 6.1797466 .004237288 237 56169 13312053 15.3948043 6.1884628. .004219409 238 56644 13481272 15.4272486 6.1971544 .004201681 239 57121 13651919 15.4596248 6.2058218 .004184100 240 57600 13824000 15.4919334 6.2144650 .004166667 241 58081 13997521 15.5241747 6.2230843 .C04149378 242 58564 14172488 15.5563492 6.2316797 .00412231 243 59049 14348907 15.5884573 6.2402515 ' .004115226 244 59536 14526784 15.6204994 6.2487998 .C040C8261 245 60025 14706125 15.6524758 6.2573248 .004081 63 246 60516 14886936 15.6843871 6.2658266 .004C65041 247 61009 15069223 15.7162336 6.2743054 .004048583 248 61504 15252992 15.7480157 6.2827613 .004032258 318 TABLE XXIII. SQUARES, CUBES, SQUARE ROOTS, No. Squares. Cubes. Square Boots. Cube Roots. Reciprocals. 249 62001 15438249 15.7797338 6.2911946 .004016064 250 62500 15625000 15.8113883 6.2996053 .004000000 13.31 63001 15813251 15.8429795 6.30799:35 .0039840(54 2&1 63504 16003008 15.8745079 6.3163596 .003968254 988 6400J 16194277 15.9059737 6.3247035 .003952569 254 64516 16387064 15.9373775 6.3330256 .003937008 355 65025 16581375 15.9687194 6.3413257 .003921569 255 65536 16777216 16.0000000 6.3496042 .003906250 257 66049 16974593 16.0.J12195 6.3578611 .003891051 .258 66564 17173512 16.0623784 6.3600968 .00:3875969 259 67081 17373979 16.0934769 6.3743111 .00:3861004 260 67600 17576000 16.1245155 6.3825043 .003846154 261 68121 17779581 16.1554944 6.3906765 .00:3831418 263 68644 17984723 16.1864141 6.3988279 .003816794 263 69169 18191447 16.2172747 6.4069585 .003802281 264 69696 18399744 16.2480768 6.4150687 .003787879 265 70225 18(509625 16.2788206 6.4231583 .003773585 266 70756 18821096 16.3095064 6.4312276 .003759398 267- 71289 19034163 16.3401346 6.4392767 .0037 45318 268 71821 19248832 16.37W055 6.447:3057 .003731343 269 72361 19465109 16.4012195 6.4553148 .003717472 270 72900 19683000 16.4316767 6.4633041 .003703704 271 73441 19902511 16.4620776 6.4712736 .003690037 27'2 73984 20123648 16.4924225 6.4792236 .003676471 273 7452 ) 20316417 16.5227116 6.4871541 .003663004 27'4 750 io 2057'0824 16.5529454 6.4950653 .003649635 275 75625 20796875 16.5831240 6.5029572 .003636364 276 76176 21024576 16.6132477 6.5108300 .003623188 277 76729 21258933 16.6433170 6,5186839 .003610108 278 . 77284 21484952 16.6733320 6.5265189 .003597122 279 77841 21717639 16.7032931 6.5343351 .003584229 280 78400 21952000 16.7332005 6.5421326 .003571429 281 78961 22188041 16.7630546 6.5499116 .003558719 282 79524 22425768 16.7928556 6.5576722 .003,546099 283 80089 22665187 16.8226038 6.5654144 .00:35:33569 284 80656 22906304 16.8522995 6.5731385 .003521127 285 81225 23149J25 16.8819430 6.5808443 .003508772 286 81796 2*393656 16.9115345 6.5885323 .003496503 287 82369 23639903 16.9410743 6.5962023 .003484321 288 82944 23887872 16.9705627 6.60:38545 .003472222 289 83521 24137569 17.0000000 6.6114890 .003460208 200 84100 24389000 17.0293864 6.6191060 .003448270 291 84681 24642171 7.0587221 6.6267054 .0034:36426 292 85264 24807088 7.0880075 6.6342874 .00:3424658 21)3 85849 25153757 7.1172428 6.6418522 .00:3412969 294 86433 23412184 7.1464282 6.649399S .003401361 . 295 87025 25672375 7.1755640 6.6569302 .00:3389831 296 87616 23934336 7.2046505 6.6644437 .003378378 297 88209 26198073 7.2336879 6.6719403 . 003367033 298 88804 26463592 7.2626765 6.6794200 .003:355705 299 89401 2673089;) 7.2916165 C. 6868831 .003344182 300 90000 27000000 7.3205081 G. 4943295 .003333333 301 90601 27270901 7.:>493516 6.7017593 .00:3:322259 302 91204 27543608 7.37'81472 6.7091729 .003:311258 303 91809 27818127 7.4068952 6.7165700 .0033103:30 304 92416 28094464 7.4355958 6.7239508 .003289474 305 93025 :* 28372625 7.4642492 6.7313155 .003278689 306 93636 '28052616 7.4928557 6.7386641 .003267974 307 94249 28934443 7.5214155 6.7459967 .0032.57329 308 94864 29218112 7.5499288 6.7533134 .00:3246753 . 309 95481 29503629 7.5783958 6.7606143 ' .003236246 310 96100 29791000 17.6068169 6.7678995 ' .003225806 819 CUBE ROOTS, AND RECIPROCALS. No. Squares. Cubes. Square Koots. Cube Root 8. Reciprocals. 311 96721 30080231 17.6351921 6.7751690 .003215434 312 97344 30371328 17.0035217 6.7824229 .003205128 313 97909 30604297 7.6918000 6.7'890013 .0031948b8 314 98596 30959144 7.7200451 6.71/08844 .003184713 315 99223 31255875 7.7482393 6.8U40921 .003174603 310 99850 31554496 7.7763888 6.8112847 .003164557 317 100489 31855013 7.80449:38 6.8184620 .003154574 318 101124 ' 32157432 7.8325545 6.8256242 .003144054 319 101701 32461759 17.8005711 6.8327714 .003134790 320 102400 32768000 17.8885438 6.8399037 .003125000 321 103041 33076161 17.9104729 6.8470213 .003115265 p 103084 33386248 17.9443584 6.8541240 .003105590 828 104329 33698267 17.9722008 6.8612120 .003095975 324 104976 34012224 18.0000000 6.8682855 .003086420 325 105625 34328125 18.0277564 6.8753443 .003076923 338 100270 34645976 18.0554701 6.8823888 .003067485 327 100929 349(55783 18.0831413 6.8894188 .003058104 328 107584 35287552 18.1107703 6.8964345 .003048780 329 108241 35611289 18.1383571 6.9034359 .003039514 330 108900 35937000 18.1659021 6.9104232 .003030303 331 109501 3620401)1 18.1934054 6.917'3964 .003021148 332 110224 30594368 18.2208072 6.9243556 .003012048 ' 333 110389 30J20037 18.2482870 6.931:3008 .003003003 334 111556 37259704 18.2756669 6.9382321 .002994012 335 118225 37595375 18.3030052 6.9451490 .002985075 330 118896 37933056 18.3303028 6.9520533 .00297'6190 337 113569 3827'2753 18.3575598 6.9589434 .002907359 338 114244 38614472 18.3847763 6.9658198 .002958580 339 114921 38958219 18.4119526 6.9726826 .002949853 340 115000 39304000 18.4390889 6.9795321 .002941176 * 341 1 16281 39651821 18-.4661853 6.9803081 .002932551 342 110904 40001688 18.4932420 6.9931806 .002923977 343 117049 4035:3007 18.5202592 7.0000000 .002915452 344 118330 40707584 18.5472370 .0007962 .002906977 315 119025 41001302;") 18.5741756 0135791 .002b98551 340 119710 41421736 18.6010752 .0203490 .002890173 847 12040'.) 41781923 18.027'9360 .0271058 .002881844 348 121104 42144192 18.6547581 .0338497 .002873503 349 121801 42508549 18.0815417 .040:;806 .002865330 "50 122500 42875000 18. 7'0828G9 .0472987 .002a57143 351 123201 43243551 18.7349940 .0540041 .00284<>00:3 352 123904 43614208 18.76166:30 .0606967 .002840909 853 124009 43980977 18.7882942 .0673767 .C02832h61 ' 354 125310 44361864 18.8148877 .0740440 .002824859 355 120025 44738875 18.8414437 .0806988 .002810901 350 120730 45118016 18.8679023 .0873411 .002808989 357 127449 45499293 18.8944436 .0939709 .002801120 358 128164 45882712 18.9208879 .1005885 .002793296 359 128881 46268279 18.9472953 .1071937 .002785515 300 129000 46656000 18.9736660 .1137'86f> .002777778 3ffl 130321 47045881 19.0000000 .1203674 .002770083 :Jtu 131044 47437928 19.0262976 .1269360 .002762431 383 131709 4?832147 19.0525589 .1334925 .002754821 304 132490 48228544 19.0787840 .1400370 .002747253 305 133225 48627125 19.1049732 .1465695 .0027397'26 300 133950 49027896 19.1311265 .15:30901 . 002732240 307 134689 49430863 19.1572441 .1595988 .00272479.6 308 135424 49830032 19.1833261 7.1660957 .002717391 309 130101 50243409 19.2093727 7.1725809 .002710027 370 136900 50653000 19.2353841 7.1790544 .002702703 371 137641 51004811 19.2813603 7.1855162 .002695418 372 138384 51478848 19.2873015 7.1919663 .00268S172 320 TABLE XXIII.-SQUARES, CUBES, SQUARE ROOTS, No. Squares. nnh M Square Cubes. Koot& Cube Roots. Reciprocals. 373 139129 51895117 19.3132079 7.1984050 .002680965 374 139876 62313624 19.3390796 7.2048322 .002673797 375 140625 52734375 19.3649167 7.2112479 .00266(5667 376 141376 53157376 19.3907194 7.2176522 .002659574 377 142129 53582633 19.4164878 7.2240450 .002652520 &78 ' 142884 54010152 19.4422221 ! 7.2304268 . 002645503 379 143641 54439939 19.4679^23 7.2367972 .002638522 380 144400 54872000 19.4935887 7.2431565 .002631579 381 145161 55306341 19.5192213 i 7.2495045 .002624672 38-3 145924 55742968 19.5448203 7.2558415 .002617801 383 146689 56181887 19.5703858 7.2621675 .002610966 384 147456 56623104 19.5959179 7.2684824 .002604167 385 148225 57066625 19.6214169 7. 2747864 .002597403 386 148996 57512456 19.6468827 7.2810794 .002590674 387 149769 57960603 19.6723156 7.2873617 .002583979 388 150544 58411072 19.6977156 7.2936330 .002577320 389 151321 58863869 19.7230829 7.2998936 .002570894 390 152100 59319000 19.7484177 7.3061436 .002564103 391 152881 59776471 19.7737199 i 7.3123828 .002557545 392 153664 60236288 19.7989899 j 7.3186114 .002551020 393 154449 60698457 19.8242276 7.3248295 .002544529 394 15523G 61162984 19.8494332 7.3310369 .002538071 895 156025 61629875 19.8746069 7.3372339 .002531646 396 156816 62099136 19.8997487 7.34S4205 .OOS525253 397 157609 62570773 19.9248588 7.3495966 .002518892 398 158404 63044792 19.9499373 7.3557624 .002512563 399 159201 63521199 19.9749844 7.3619178 .002506266 400 160000 64000000 20.0000000 7.3680630 .002500000 401 160801 64481201 20.0249844 7.3741979 .002493706 402 161604 64964808 20.0499377 7.3803227 .002487562- 403 162409 65450827 20.0748599 7.3864373 .002481390 404 163216 65939264 20.0997512 7.3925418 .002475248 405 164025 66430125 20.1246118 7.3986863 .002469136 406 164836 66923416 20.1494417 7.4047206 .002463054 407 165649 67419143 20.1742410 7.4107950 .002457002 408 166464 67917312 20.1990099 7.4168595 .002450980 409 167'281 68417929 0.2237484 7.4229142 .002444988 410 168100 68921000 20.2484567 7.4289589 .002439024 411 168921 69426531 20.2731349 7.4349938 . 0024:33090 412 169744 69934528 20.2977831 j 7.4410189 .002427184 413 170569 70444997 20.3224014 ! 7.4470342 .002421308 414 171396 70957944 20.3469899 7.4530399 .002415459 415 172225 71473375 20.3715488 7.4590359 .002409639 416 173056 71991296 20.3960781 7.4650223 .002403846 417 173889 72511713 20.4205779 7.4709991 .002398082 418 174724 73034632 20.4450483 7.4769664 .002392344 419 175561 73560059 20.4694895 7.4829242 .002386635 430 176400 74088000 20.4939015 7.4888724 .002380952 421 177241 74618461 20.5182845 7.4948113 .002375297 422 178084 75151448 20.5426386 7.5007406 .008369668 423 178929 75686967 20.5669638 7.5066607 .002364066 424 179776 76225024 20 5912603 7,5125715 .002358491 425 180625 76765685 20.6155281 7.5184730 .002352941 426 181476 77308776 20.6397674 7.5243652 " .002347418 427 182329 T7&54483 20.6639783 7.5302482 .002341920 428 183184 78402752 20.6881609 7.5361221 .002336449 429 184041 78953589 20.7123152 7.5419867 .002331002 430 184900 79507000 20.7364414 7.5478423 .002325581 431 185761 8C062991 20.7605395 7.5536888 .002320186 432 186624 80621568 20.7846097 7.5595263 .002314815 433 187489 81182737 20.8086520 ! 7.5653548 .002309469 434 188356 81746504 20.8826667 i 7.5711743 .002304147 321 CUBE ROOTS, AND RECIPROCALS. No. 'Squares. Cubes. Square Roots. Cube Roots. Reciprocals. 435 189225 82312875 20.8566536 7.5769849 .002298851 436 190036 82881856 20.8806130 7.5827865 .002293578 437 190969 83453453 20.9045450 7.5885793 .002258330 438 191844 8402,672 20.9284495 7.5943633 .002283105 439 192721 84604519 20.9523268 7.6001385 .OJ2277904 440 193600 85184000 20.9761770 7.6059049 .032272727 441 194481 85766121 21.0000000 7.6116626 .002267574 442 195364 86350888 21.0237960 7.6174116 .002262443 443 196249 86938307 21.0475652 7.6231519 .002257336 444 197136 87528:384 21.0713075 7.0288837 .002252252 445 198025 88121125 21.0350231 7.63460G7 .002247191 446 198916 88716536 21.1187121 7.6403213 .002242152 447 199809 89314623 21.1423745 7.6460272 .002237136 448 200704 89915392 21.1660105 7.6517247 .002232143 449 201601 90518849 21.1896201 7. 657413 J .002227171 450 202500 91125000 21.2132034 7.6630943 .002222222 451 203401 91733851 21.2367608 7.6687665 .002217295 452 204301 92345438 21.2602916 7.6744303 .002212389 453 205203 92959677 21.2837967 7.6800857 .002207506 454 206116 93576664 21.3072758 7.6857323 .002202643 455 207025 94196375 21.3307230 7.6913717 .002197802 456 207936 94818816 21.3541565 7.6970023 .002192932 457 203849 95443993 21.3775583 7.7026246 .002188184 458 203764 96071912 21.4009346 7.7082388 .002183406 459 210881 96702579 21.4242853 7.7138443 .002178649 460 211600 97336000 21.4476103 7.7194423 .002173913 461 212521 97972181 21.4703106 7.7250325 .002169197 462 213444 98611128 21.4941853 .7336141 .002164502 4(53 214369 99252847 21.5174348 .7361877 .002159827 464 215236 99897344 21.5406592 .7417532 .002155172 485 216225 100544625 21.5638587 .7473109 .002150538 466 217156 101 194636 21.5870331 .7528606 .002145923 4(57 218039 101847563 21.6101828 .7584023 .002141328 468 219024 102503232 21.6333077 .7639361 .002136752 469 219931 103161709 21.6564078 .7694620 .002132196 470 220900 103823000 21.6794834 .7749801 .002127660 471 221841 104487111 21.7025344 .7804904 .002123142 472 222784 105154048 21.7'255610 .7859923 .002118644 473 223729 105823817 21.7485632 .7914875 .002114165 474 221676 103496424 21.7715411 .7-369745 .002109705 475 225625 107171875 21.7944947 .8024538 .002105263 476 226576 107850176 21.8174242 .8079254 .002100840 477 227529 103531333 21.8403297 .8133892 .002096436 478 22S484 109215352 21 8632111 .8188456 .002092050 479 229441 109902239 21.8860686 .8242942 .002087683 480 230400 110592000 21.9089023 .8297353 .002088333 481 231361 111284641 21.9317122 .8:351688 .002079002 482 232324 111980168 21.9544984 .8405949 .002074689 483 233289 112678587 21.9772610 .8460134 .002070393 484 234256 11:3379904 22.0000000 .8514244 . .002066116 485 235225 114084125 22.0227155 .8568281 .002061850 486 236196 114791256 22.0454077 .8622242 .002057613 487 237169 115501303 22.0680765 .8676130 .002053388 488 238144 116214272 22.09072.20 .8729844 .002049180 489 239121 116930169 22.11&3444 .8783684 .002044990 490 240100 117649000 22.1359436 .8837352 .002040316 491 241081 118370771 22.1585198 .8890946 .002036660 492 242064 119095488 22.1810730 .8944463 .002032520 493 243049 119823157 22.2036033 .8997917 .00.2028398 494 244036 120553784 22.2261108 7.9051294 .002024291 495 245025 121287375 22.2485955 7.9104599 .002020202 496 246016 122023936 22.2710575 7.9157832 .002016129 TABLE XXIII. SQUARES, CUBES, SQUARE ROOTS. ,' No. Squares. Cubes. Square Roots. Cube Roots. Reciprocals. 497 217009 122763473 22.2934968 .9210994 .002012072 498 248004 123505992 22.3159136 .9264085 .002008032 409 249001 124251499 22.3383079 .9317104 .002004006 500 250000 125000000 22.3606798 .9370053 .002000000 591 251001 125751501 22.3830293 .9422*31 .001996008 5P3 252 04 126506008 22.4053565 .9475733 .001992032 503 253009 127263527 22.4276615 ,9528477 .001988072 504 254016 128024064 22.4499443 .9581144 .001984127 505 255025 128787625 22.4722051 .9633743 .001980198 50(3 256036 129554216 22.4944438 .9686271 .001976285 507 257049 130323843 22.5166605 .9738731 .001972:387 508 258064 131096512 22.5388553 .9791122 .001968504 509 259081 131872229 22.5610283 .9843444 .001964637 510 260100 132651000 22.5831796 7.9895697 .001960784 511 261121 133432831 22.6053091 7.9947883 .001956947 512 262144 134217728 22.6274170 8.0000000 .001953125 513 263169 135005697 22.6495033 8.0052049 .001949318 514 264196 135796744 22.6715681 8.0104032 .001945525 515 265225 136590875 22.6936114 8.0155946 .001941748 516 286256 137388096 22.7156334 8.0207794 .001937984 517 267289 138188413 22.7376340 8.02oC574 .001934236 518 268324 138991832 22.7596134 8.0311287 .001930502 519 269361 139798359 22. 7815715 8.0362935 .001926782 520 270400 140608000 22.8035085 8.0414515 .001923077 521 271441 141420761 22.8254244 8.0466030 .001919386 522 272484 142236648 22.8473193 8.0517479 .0019157'09 523 273529 143055667 22.8691933 8.0568862 .001912046 524 274576 143877824 22.8910463 8.0620180 .001908397 525 275625 144703125 22.9128785 8.0671432 .001904762 526 276676 145531576 22.9346899 8.0722620 .0019,11141 527 277729 146363183 22.9564806 8.0773743 .001897533 528 278784 147197952 22.9782506 8.0824800 .001893939 529 279841 148035889 23.0000000 8.0875794 .001890359 530 280900 148877000 23.0217289 8.0926723 .001886792 531 281961 149721291 23.0434372 8.0977589 .001883239 532 283024 150568768 23.0651252 8.1028390 .001879699 533 284089 151419437 23.0867'928 8.1079128 .001876173 534 285156 152273304 23.1084400 8.1129803 .001872659 535 286225 153130375 23.1300670 8.1180414 .001869159 536 287296 153990656 23.1516738 8.1230962 .001865672 537 288369 154854153 23.1732605 8.1281447 .001862197 538 289444 155720872 23.1948270 8.1331870 .001858736 539 290521 156590819 23.2163735 8.1382230 .001855288 540 291600 157464000 23.2379001 8.1432529 .001851852 541 292681 158340421 23.2594067 8.1482765 .001848429 542 293764 159220088 23.2808935 8.1532939 .001845018 543 294849 16>103007 23.3023604 8.1583051 .001841621 544 295936 160989184 23.3238070 8.16:33102 .001838.235 545 297025 161878625 23.3452-351 8.1683092 .001834862 546 208116 162771336 23.3666429 8.1733020 .001831502 547 299209 163667323 23.3880311 8.1782888 .001828154 548 300:304 164566592 23.4093998 8.ia32695 .001824818 549 301401 165469149 23.4307490 8.1882441 .001821494 550 302500 iflKKoOOO 23.4520788 8.1932127 .001818182 551 303601 167284151 23.4733892 8.1981758 .001814882 552 304704 168196608 23.4946802 8.2031319 .001811594 553 305809 169112377 23.5159520 8.2030825 .001.808318 554 306916 170031464 23.. 5372046 8.2130271 .001805054 r,55 308025 170953875 23.5584380 8.2179657 .001801802 556 309136 171879616 23.5796522 8.2228085 .001798561 557 310249 172808693 23.6008474 8.2278254 .001795332 558 311364 173741112 23.6220236 8.2327463 .001792115 323 CUBE ROOTS, AND RECIPROCALS. No. U;[uares. Cubes. Square Roots. Cube Roots. Reciprocals. 659 312481 174676879 23.6431808 1 8.2376614 .001788909" 560 313600 175616000 23.6643191 8.2425706 .001785714 561 314721 176558481 23.6854386 8.2474740 .001782531 563 315844 177504328 23.7065392 8.2523715 .001779359 563 316969 178453547 23.7'276210 8.2572633 .001776199 564 318096 179406144 23.7486842 8.2621492 .001773050 565 319225 180362125 23.7697286 8.2670294 ,001769912 566 320356 181321496 23.7907545 8.2719039 .001766784 567 321489 182284263 23.8117618 8.2767726 .001768668 568 322624 183250432 23.8327506 8.2816855 .001760563 569 323761 1842.iOUUO 2U.868&909 8.2864928 .001757469 570 1 324900 185193000 23.8746728 8.2913444 .001754386 571 ! 326041 186169411 23.8956063 8.2161903 .001751313 572 327184 187149248 83.9165215 8.3010304 .001748252 573 328329 188132517 23.9374184 8.2058651 .001745201 574 329476 189119224 23.9582971 8.3106941 .001742160 575 330025 190109375 23.9791576 8.3155175 .001739130 576 831776 191102976 24.0000000 8.3203353 .001736111 577 332929 192100033 24.0208243 8.3251475 .001733102 578 334084 193100552 24.0416306 8.3299542 .00173010-1 579 335241 194104539 24.0624188 8.3347553 .001727116 580 336400 195112000 24.0831891 8.3S95509 .001724138 581 837561 196122941 24.1039416 8.3443410 .001721170 582 338724 197137368 24.1246762 8.3491256 .001718218 583 339889 198155287 24.1453929 8.3539047 .C017 15266 584 341056 199176704 24.1660919 8.3586784 .001712329 585 342225 200201625 24.1867732 8.8634466 .C01709402 586 343396 201230056 24.2074369 8.3G82095 ,001706485 587 344569 202262003 24.2280829 8.3729668 .001703578 588 345744 203297472 24.2487113 8.3777188 .001700680 589 316921 204336469 24.2693222 8.3824653 .001697793 590 848100 205379000 24.2899156 8.387'2CC5 .001694915 591 349281 206425071 24.3104916 8.3919423 .001692047 592 350464 207474688 24.3310501 8.2966729 .001689189 593 ! 351649 208527857 24.3515913 8.4013881 .001686341 594 352836 209584584 24.3721152 8.4061180 .001683502 595 354025 210644875 24.3926218 8.4108326 .001680672 596 355216 211708736 24.4131112 8.4155419 .001677852 597 356409 212776173 24.4335834 8.4JC024CO .001675042 598 3:>7604 213847192 24.4540S85 8.4249448 .001672241 599 358801 214921799 24.4744765 8.4296383 .001669449 GOO 360000 216000000 24.4948974 8.4343267 .001666667 001 361201 217'081801 24.5153013 8.4390098 .001668894 602 362404 218167208 24.5356883 8.4486877 .001661180 603 363609 219256227 24.5560583 8.4483605 .001658375 604 364816 220348864 24.5764115 8.4530281 .001C55629 605 366025 221445125 24.5967478 8.457CG06 .001652893 606 S67236 222545016 24.6170673 8.4623479 .001650165 607 368449 223648543 24.6373,00 8. 4670001 .001647446 608 369664 224755712 24.6576560 8.4716471 .001644737 609 370881 225866529 24.6779254 8.4762892 * .001642036 610 372100 226981000 24.6981781 8.4809261 .001639344 Gil 373321 228099131 24.7184142 8.4855579 .001636(561 C12 374544 229220928 24.7386.338 8.4901848 .001633987 613 375769 230346397 24.7588368 8.4948065 .001631321 614 376996 231475544 24.7790234 8.4994233 .001628664 615 378225 232608375 24.7991935 8.5040350 .061626016 616 379456 233744896 24.8193473 8.5086417 .001623377 617 380689 234885113 24.8394847 8.5132435 .001 620746 618 3S1924 236029032 24.8596058 8.5178403 .001618123 619 383161 237176659 24.8797106 8.5224321 .001615509 620 384400 238328000 24.8997992 8.5270189 .001612903 324 TABLE XXIII.-SQUARES, CUBES, SQUARE ROOTS, No. Squares. Cubes. Square Hoots. Cube Roots. Reciprocals. 621 385641 239483061 24.9198716 8.5316009 .001610306 622 386884 240641848 24.93U9278 8.5361780 .001607717 623 388129 241804367 24.9599079 8.5407501 .001605136 624 389376 242970624 24.9799920 8.5453173 .001602564 625 390625 244140625 25.0000000 8.5498797 .001600000 626 391876 245314376 25.0199920 8. )544372 .001597444 627 393129 246491883 25.0399681 8.5589899 .001594896 628 394384 247673152 25.0599*82 8.5635377 .001592:357 6.29 395641 248858189 25.0798724 8.5680807 .001589825 630 396900 250047000 25.0998008 8.5726189 .001587302 631 398161 251239591 25.1197134 8.5771523 .0015847'86 632 399424 252435968 25.1396102 8.5816809 .001582278 633 4l"0689 253636137 25.1594913 8.5862047 .001579779 634 401956 254840104 25.1793566 8.59072:38 .001577287 635 403225 256047875 25.1992003 8.5952380 .001574803 636 404496 257259456 25.2190404 8.5997476 .001572327 637 405769 258474853 25.2388589 8.6042525 .001569859 638 407044 259694072 25.2586619 8.6087526 .001567398 639 408321 260917119 25.2784493 8.6132480 .001564945 640 409800 262144000 25.2982213 8.6177388 .001562500 641 410881 283374721 25.3179778 8.6222248 .001560062 642 412164 264603288 25.3377189 8.6267:063 .001557632 643 413449 265847707 25.3574447 8.6311830 .001555210 644 414736 267089984 25.3771551 8.6356551 .001552795 645 416025 268335125 25.3968502 8.6401226 .001550388 646 417316 269586136 25.4165301 8.6445855 .001547988 647 418609 270840023 25.4361947 8.6490437 .001545595 648 419904 272097792 25.4558441 8.6534974 .001543210 649 421201 273359449 25.4754784 8.6579465 .001540832 650 422500 274625000 25.4950976 8.6623911 .001538462 651 423801 275894451 25.5147016 8.6668310 .00153601)8 652 425104 277167808 25.5342907 8.6712665 .001533742 653 426409 278445077 25.5538647 8.6756974 .001531394 654 427716 27972o2o4 25.5734237 8.6801237 .001529052 655 429025 281011375 25.5929678 8.6845456 .001526718 656 430338 282300416 25.6124969 8.68896:30 .001524390 657 431649 283593393 25.6320112 8.69*3759 .001522070 658 " 432964 284890312 25.6515107 8.6977843 .001519757 659 434281 286191179 25.6709953 8.7021882 .001517451 660 435600 287498000 25.6904352 8.7065877 .001515152 661 436921 288804781 25.7099203 8.7109827 .001512859 662 438244 290117528 25.7293607 8.71537'34 .001510574 663 439569 291434247 25.7487864 8.7197596 .001508296 664 440893 292754944 25.7681975 8.7241414 .001506024 665 442225 294079625 25.7875939 8.7285187 .001503759 666 443556 1895408296 25.8069758 8.7328918 .001501502 667 444889 296740963 25.8263431 8.737'2604 .001499250 668 446224 298077632 25.8456960 8.7'416246 .001497'000 669 447561 299418309 25.8650343 8.7459846 .001494768 670 448900 300763000 25.8843582 8.7503401 .001492537 671 450241 302111711 25.9036677 8.7546913 .001490313 672 451584 30:5464448 25.9229628 8.7590:383 .001488095 673 452929 304821217 25.9422435 8.7683809 .001485884 674 454276 306182024 25.9615100 8.7677192 .001483680 675 4.55625 307546875 25.9807621 8.7720532 .001481481 676 456976 308915776 26.0000000 8.776:3830 .001479290 677 458329 310288733 26.0192237 8.7807084 .001477105 678 459684 311665752 26.0:384.331 8.7850296 .001474926 679 461041 313046839 6.0576284 8.7893466 .004472754 680 462400 314432000 26.0768006 8.7936593 .001470.588 681 463761 315821241 26.0959767 8.7979679 .001468429 682 | 465124 317214568 26.1151297 8.8022721 .001466276 325 CUBE ROOTS, AND RECIPROCALS. No. Squares. Cubes. Square Roots. Cube Roots. Reciprocals. 683 466489 318611987 26.13426^7 8.8065722 .001464129 68 1 I 467850 320013504 26.1533937 8.8108681 .001461988 6*5 469225 321419125 26.1725047 8.8151598 .001459854 68(5 470596 322828850 26.1916017 8.8194474 .001457720 687 471909 324242703 26.2100848 8.8237307 .001455604 688 473344 325660072 20.2297541 8.8280099 .001453488 689 474721 327082769 26.2488095 8.8=322850 .001451379 690 476100 828509000 26.2678511 8.8365559 .001449275 691 477481 329939371 20.2868789 8.8408227 .001447178 692 .478864 33137I3888 20.3058929 8.8450854 .001445087 693 480249 332812557 26.3248932 8.8493440 .001443001 694 481636 334255384 26.3438797 8.8535985 .001440922 695 483025 335702375 26.3628527 8.8578489 .001438849 696 484416 &37153536 26.3818119 8.8620952 .001436782 697 485809 338608873 26.4007576 8.8663375 .001434720 698 487204 340008392 26.4196896 8.8705757 .001432665 699 488601 341532099 26.4386081 8.8748099 .001430615 700 490000 343000000 26.4575131 8.8790400 .001428571 701 491401 344472101 26.4764046 8.8832661 .001426534 703 492804 345948408 26.4952820 8.8874882 .001424501 703 494209 347428927 26.5141472 8.8917063 001422475 704 495616 348913664 26.5329983 8.8959204 .001420455 705 497025 350402625 26.5518361 8.9001304 .001418440 706 498436 351895816 26.5706605 8.9043366 .001416431 707 499849 353393243 26.5894716 8.9085:387 .001414427 708 501204 354894912 26.6082094 8.9127369 001412429 709 502081 356400829 26 6270539 8.9109311 .001410437 710 504100 357911000 26.6458252 8.9211214 001408451 711 505521 359425431 26.6645833 8.9253078 .001406470 712 506944 360944128 26.6833281 8.9294902 .001404494 713 508369 362467097 26.7020598 8.9336687 .001402525 714 509796 363994344 26.7207784 8.9378433 .001400560 715 511225 365525875 20.7394839 8.9420140 .001398601 716 512656 367061696 26.7581763 8.9461809 .001396648 717 514089 368601813 26.7768557 8.9503438 001394700 718 515524 370146232 26.7955220 8.9545029 .001392758 719 516961 371694959 26.8141754 8.9586581 .001390821 720 518400 373248000 26.8328157 8.9628095 .001388889 721 519841 374803361 26.8514432 8.9609570 .(,01386963 722 521284 376367048 20.8700577 8. 971 1007 .001385042 723 522729 377933067 26.8886593 8.9752406 .001383126 724 524176 379503424 26.9072481 8.9793766 .001381215 725 525625 381078125 26.9258240 8.9835089 .001379310 726 527076 382G57176 26.9443872 8.9876373 .001377410 727 528529 384240583 26.9629375 8.9917620 .001375516 723 529984 385828352 26.9814751 8.9958829 .001373626 729 531441 387420489 27.0000000 9.0000000 .001371742 730 532900 389017000 27.01R5122 9.0041134 .001369863 731 534361 390617891 27.0370117 9.0082229 .001367989 732 535824 392223168 27.0554985 9.0123288 ' .001366120 733 537289 398833837 27.0739727 9.0164309 .001364256 734 538756 395446904 27.0924344 9.0205293 .001362398 735 540*25 397065375 27.1108834 9.0246239 .001360544 736 541696 398688256 27.1293199 9.0287149 .00ia58696 737 543169 400:315553 27.1477439 9.0328021 .001356852 738 544644 401947272 27.1661554 9.0368857 .001355014 739 546121 403583419 27.1845544 9.0409655 .001353180 740 547600 405224000 27.2029410 9.0450419 .001351351 741 549081 406869021 27.2213152 9.0491142 .001349528 742 550564 408518488 27.2396769 9.0531831 .001347709 743 552049 410172407 27.2580263 9.0572482 .001345895 744 553536 411830784 27.2763634 9.0613098 .001344086 TABLE XXIII. SQUARES, CUBES, SQUARE ROOTS. No. Squares. Cubes. Square Roots. Cube Roots. Reciprocals. 745 555025 413493625 27.2946881 9.0653677 .001342282 746 556.116 415160936 27.3130006 9.0694220 .001340483 747 558009 416832723 27.3313007 9.0734726 .0013386^8 748 559504 418508992 27.3495887 9.0775197 .001330898 749 561001 420189749 27.3678644 9.0815031 .001335113 750 562500 421875000 27.3861279 9.0a56030 .001333333 751 564001 423564751 27.4043792 9. 0890392 .OM 331558 752 565504 425259008 27.4226184 9.0930719 .001329787 753 567009 426957777 27.4408455 9.0977010 .0013*8021 754 568516 428661064 27.4590604 9.1017265 .0013*6200 755 570025 430368875 27'. 4772633 9.1057485 .001324503 756 571536 432081216 27.4954542 9.1097C69 .001322751 757 573049 433798093 27.5136330 9.1137818 .001321004 758 574564 435519512 27.5317998 9.1177931 .001319261 759 576081 437245479 27'. 5499546 9.1218010 .001317523 760 577600 438976000 27.5680975 9.1258053 .001315789 761 579121 440711081 27.5862284 9. 1*98)61 .001314000 762 580844 442450728 27.0043475 9.1338ft34 .C01312336 763 5821 69 444194947 27. 62; 4546 9.1377971 .C0131U016 764 58-3696 445943744 27.6405499 9.1417874 .001308901 765 585225 447697125 27.6580334 9.14577'42 .001307190 766 586756 449455096 27.6767050 9.1497576 .001305483 767 588289 451217663 27.6947648 9.1537375 .001303781 768 589824 452984832 27.7128129 9.1577189 .G01S02083 769 591361 454756009 27.7308492 9.1610869 .001300390 770 592900 456533000 27.7488739 9.1656565 .001298701 771 594441 458314011 27.7068808 9.1(JOG**5 .001297017 772 595984 460099648 27.7848880 9.1735852 .001295337 773 597529 461889917 27.8028775 9.1775445 .001293001 774 599076 4(53684824 27.8208555 9.1815003 .001*919iiO 775 600625 465484375 27.8388218 9.1854527 .0012^0323 776 602176 467288576 27.8507766 9.1894018 .0012H8060 777 603729 469097433 27.8747197 9.1933474 .001287001 778 605284 470910952 27.8920514 9.1972897 .C01*h5247 779 606841 472729139 27.9105715 9.2012286 .01283097 780 608400 474552000 27.9284801 9.2051641 .C01282051 781 609961 . 476379541 27.9463772 9.20 f JC902 .0012h'0410 782 611524 478211768 27.9642029 9.2120250 .(i01*7'8772 783 613089 480048687 27.982137'2 9.2109505 .001277139 784 614656 481890304 28.0000000 9.2208726 .001275510 785 616225 483736625 28.0178515 9.2247914 .C01*7'3fc8o 786 617796 485587656 28.0356915 9.2287008 .001272*65 787 619369 4874434C3 28.0535203 9.2326189 .C01 270648 788 620944 489303872 28.0713377 9. 2265277 .001209036 789 622521 491169069 28.0891438 8.24C4333 .C01207427 790 624100 493039000 28.1069386 9.2443355 .001265823 791 625681 494913671 28.1247222 9.2482344 .01264*23 792 627264 496793088 28.1424946 9.2E21300 .001202026 793 628849 498677257 28.1602557 9.2EG0224 .001201034 794 630436 500566184 28.1780056 9.2599114 .001*59446 795 632025 502459875 28.1957444 9.21S7973 .101257802 796 633616 504358336 28.2134720 9.2676798 .C0125C281 797 635209 506261573 28.2311884 9.2715592 .001*54705 798 636804 508169592 28.2488938 9.27'54352 .001253133 799 638401 510082399 28.2665881 9.2793081 .C01251504 800 640000 512000000 28.2842712 9.2831777 .001250000 801 641601 513922401 28.3019434 9.2870440 .001248439 802 G43204 515849608 28.3196045 9.2909072 .001246883 803 644809 517781627 28.8372546 9.2947671 .001245330 804 646416 519718464 28.. 3548938 9.2986239 .0012437'81 805 648025 521660125 28.3725219 9.3024775 .001242230 806 649636 523606616 28.3901391 9.3063278 .001240695 327 CUBE ROOTS, AND RECIPROCALS. No. Squares. Cubes. Square Roots. Cube Roots. Reciprocals. 807 651249 525557943 28.4077454 9.3101750 .001239157 803 6528(54 527514112 28.4253408 9.3140190 .001237624 809 654481 529475129 28.4429253 9.317a599 .0012:36094 810 656100 531441000 28.4604989 9.3216975 .001234568 811 657721 533411731 28.4780617 9.3255320 .001233046 812 659344 535387328 28.4956137 9.3293634 .001231527 813 660969 537367797 28.5131549 9. 3331916 .001230012 814 662596 539353144 28.5306852 9.3370167 .001228501 815 684225 541343375 23.5482048 9.3408386 .001226994 si a 665856 543338490 28.5657137 9.3446575 .001225490 817 667489 545338513 28.5832119 9.3484731 .001223990 818 669124 547343432 28.6006993 9.3522857 .001222494 819 670761 549353259 28.6181760 9.3:63352 .001221001 820 672400 551368000 28.6356421 9.3599016 .001219512 821 674041 553:387661 28.6530976 9.3637049 .001218027 822 675684 555412248 28.6705424 9.3675051 .001216545 823 677329 557441767 28.6879766 9.3713022 .001215067 824 678976 559476224 28:7054098 9.3750903 .001213592 825 680625 56151562.> 28.7228132 9.3788873 .001212121 826 682276 56:3559976 28.7402157 9.3828752 .001210654 827 683929 565609283 28.7576077 9.3864600 . 0012091 W) 828 685584 567863558 28.77'49891 9.3902419 .001207729 829 687241 569722789 23.7923601 9.3940206 .001206273 830 688900 571787000 23.8097206 9.3977964 .001204819 831 690561 573856191 28.8270706 9.4015691 .001203369 832 692224 575930368 28.8444102 9.4053387 .001201923 833 693889 578009537 28.8617394 9.4091054 .001200480 834 695556 580093704 28.8790582 9.4128690 .001199041 835 697225 582182875 23.8963666 9.4166297 .001197605 836 698898 584277056 28.9136646 9.4203873 .001196172 837 700569 586376253 28.9309523 9.4241420 .001194743 833 702244 588480472 28.9482297 9.4278936 .001193817 839 703921 590589719 23.9654967 9.4316423 .001191895 840 705600 592704000 23.9827535 9.4353880 .001190476 841 707281 594823321 29.0000000 9.4391307 .001189061 812 708964 596947688 29.0172363 9.4428704 .001187648 843 710649 599077107 29.0344623 9.4466072 .001186240 844 712336 601211584 29.0516781 9.4503410 .001184834 845 714025 603351125 29.0688837 9.4540719 .001183432 846 715716 605495736 29.0860791 9.4577999 .001182033 847 717409 607645423 29.1032644 9.4615249 .001180638 848 719101 609800192 20.1204396 9.4652470 .001179245 849 720301 611960049 29.1376046 9.4689661 .001177856 850 722500 614125000 29.1547595 9.4726824 .001176471 851 724201 616295051 29.1719043 9.4763957 .001175088 852 725904 618470208 29.1890390 9.4801061 .001173709 853 727609 620650477 29.2061637 9.4838136 .001172333 854 729316 622835864 29.2232784 9.4875182 .001170900 855 731025 625026375 29.2403830 9.4912200 .001169591 856 732736 627222016 29.2574777 9.4949188 .001168224 857 734449 629432793 29.2745623 9.4986147 .001166861 858 73G1G4 631628712 29.2916370 9.5023078 .001165501 859 737831 6338397,'9 29.3087018 9.5059980 .001164144 860 739000 636056000 29.3257566 9.5098a54 .001162791 831 741321 638277381 29.3428015 9.51:33699 .001161-140 8b2 743344 640503928 29.3598365 9.5170515 .001160093 883 744769 6427-35647 29.3768616 9.5207303 .001158749 864 746196 644972544 29.3938769 9.5244063 .001157407 8G5 748225 647214625 29.4108823 9.5280794 .001156069 806 749956 649461896 29.4278779 9.5317497 .001154734 807 751689 651714363 29.4448637 9.5354172 .001153403 868 753424 653972032 29.4618397 9.5390818 001152074 328 TABLE XXIII.-SQUARES, CUBES, SQUARE ROOTS. No. Squares. Cubes. Square Roots. Cube Roots. Reciprocals. i : ' ' 869 . 755161 656234909 29.4788059 9.5427437 .001150748 870 756900 658503000 29.4957624 9.5464027 .001149425 871 758641 660776311 29.5127091 9.5500589 .001148106 872 760:384 G63054848 29.5296461 9.5537123 .0011407*!) 873 762129 665338617 29.5465734 9.5573630 .001145475 874 763876 667627624 29.5634910 9.5610108 .001144105 875 765625 66992187'5 29.5803989 9.5640559 .001142857 876 767376 672221376 29.5972972 9.5682982 .001141553 877 769129 674526133 29.6141858 9.5719377 .001140251 878 770884 67(5836152 29*6310648 9.5755745 .001138952 879 772641 679151439 29.6479342 9.5792085 .001137050 880 774400 681472000 29.6647939 9.5828397 .001136364 881 776161 683797841 29.6816442 9.5864082 .00113.5074 882 777924 686128968 29.6984848 9.5900939 .001133787 883 779689 688465387 29. 7153159 9.5937169 .001132503 884 781456 690807104 29.7321375 9.5973878 .001131222 885 783225 693154125 29.7489496 9.0009548 .001129944 886 78-1996 695506456 29.7657521 9.6045096 .001128668 887 786769 69:864103 29.7825452 9.6081817 .001127396 888 788544 700227072 29.7993289 9.6117911 .001126126 889 790321 702595369 29.8161030 9.6153977 .001124859 890 792100 704969000 29.8328678 9.6f90017 .001123596 891 793881 707347971 29.8496231 9.0220030 .001122334 892 795664 709732288 29.8663(590 9.0202016 .001121076 893 797449 712121957 29.8831056 9.6297975 .001119821 894 799236 714516984 29.8998328 9.6333907 .0*31118568 895 801025 716917375 29.9105506 9. 6309812 .001117318 896 802816 719323136 29.9332591 9.0405690 .001116071 897 804609 721734273 29.9499583 9.6441542 .001114827 898 806404 724150792 29.9666481 9.6477367 .001113586 899 808201 726572699 29.9833287 9.05131C6 .001112347 900 810000 729000000 30.0000000 9.6548938 .001111111 901 811801 731432701 30.0166620 9.0584684 .001109878 902 813604 733870808 30.0333148 9.6620403 .001108047 903 815409 736314327 30.0499584 9.6650096 .001107420 904 817216 738763264 30.0665928 9.6691762 .0011115195 905 819025 741217625 30.0832179 9.6727403 .001104972 906 820836 743677416 30.0998339 9.6763017 .001103753 907 822649 746142643 30.1164407 9.6798604 .001102536 908 824464 748613312 30.1330383 9.6834166 .001101322 909 826281 751089429 30.1496269 9.6869701 .001100110 910 828100 753571000 30.1662063 9.6905211 .001098901 911 829921 756058031 30.1827765 9.6940694 .001097695 912 831744 758550528 30.1998377 9.6976151 j .001096491 913 833569 761048497 30.2158899 9.7011583 j .001095290 914 835396 763551944 30. 2324329 9.7040989 .001094092 915 837225 766060875 30.2489669 9.7082369 .001092896 916 839056 768575296 30.2654919 9.7117723 .001091703 917 840889 771095213 30.2820079 9.7153051 .001090r>13 918 842724 - 773620632 30.2985148 9.7188354 .001089&S 919 844561 776151559 30.3150128 9.7223631 .001088139 020 846400 778688000 30.3315018 9.7258883 .001086957 921 848241 781229961 30.3479818 9.7294109 .001085770 922 850084 783777448 30.3644529 9.7329309 .001084599 923 &51929 786330467 30.3809151 9.73(54484 .001083423 924 853776 788889024 30.3973683 9.7399634 .001082251 925 855625 791453125 30.4138127 9.7434758 .001081081 926 857476 794022776 30.4302481 9.7469857 .001079914 927 859329 790597983 30.4466747 9.7504930 .001078749 928 861184 799178752 30.4630924 9.7539979 .001077586 929 863041 801765089 30.4795013 9.7575002 .001076426 930 864900 804357000 30.4959014 9.7610001 .001075269 329 . GUBE ROOTS, AND RECIPROCALS. No. Squares. Cubes. Square Hoots. 1 Cube Roots. Reciprocals. 931 866761 | 806954491 30.5122926 9.7644974 .001074114 932 868624 809557568 30.52b67oO 9.7679922 .001072961 933 870489 812100237 30.5450487 9.7714845 .001071811 934 872356 814780504 30.5614136 9.77'49743 .001070(564 935 874*25 817400375 30.5777697 9.7784616 .001069519 936 876096 820025856 30.5941171 9.7819466 .001008376 937 877969 822656953 30.6104557 9.7854288 .001067236 988 879844 825293672 30.6267857 8.788908f .001066098 939 881721 827936019 30.6431069 9.792861 .001064963 940 883600 830584000 30.6594194 9.7958611 .001068830 941 885481 833237621 0.6757233 9.7998386 .001062699 942 887364 835896888 30.6920185 9.8028036 .001061571 943 889249 8385(51807 30.7083051 9.8062711 .001060445 944 891136 841232384 :;0. 7245830 9.8097362 .001059322 945 893025 843908625 30.7408523 9.8131989 .001058201 94(5 894916 846590536 30.7571130 9.8166591 .001057082 947 896809 849278123 30.7733651 9.8201169 .001055966 948 898704 851971392 30.7896086 9.8235723 .001054852 949 900601 854670349 30.80^8436 9.8270252 .001053741 950 902200 857375000 SO. 8220700 9.8304757 .001052632 951 904401 860085351 0.8382879 9.8339238 .001051525 952 906304 862801408 30.8544972 9.8373695 .001050420 953 908209 865523177 30.8706981 9.8408127 .001049318 954 910116 868250664 80.8868904 9.8442536 .001048218 955 912025 87098S875 0.i;C30743 9.847CC20 .001047120 9(56 913936 873722816 0.9192497 9.6511280 .001046025 957 915849 87(5467493 30.9354166 9. i 545617 .001044932 958 917764 879217912 0.9515751 9.8579929 .001043841 959 919681 881974079 80.9677251 9.6614218 .001042753 960 921COO 884736000 80.9838C68 9.8648483 .00104.1667 961 923521 687508681 31.0COOCOO 9.8682724 .001040583 962 925444 8S0277128 31.0161248 9.8716941 .001089501 963 987868 693056847 31.0822413 9.8751135 .C01088422 964 929296 895841344 31.0483494 9.8785305 .001087344 965 931225 8986S2125 31.0644491 9.8819451 .C010S6269 966 933156 901428696 81.0805405 9.8868574 .C01C85197 967 935089 9042310&3 31.0966236 9.8887673 .001(34126 968 937'024 907039232 31.1126984 9.8921749 .001058058 969 938961 909853209 31.1287648 9.6955801 .001031192 970 940900 912673000 31.1448230 9.888830 .001030928 971 942841 915498611 31.16C8729 9. IX 8635 .001029866 972 944784 918330048 31.1769145 9. 057 817 .C01G26607 973 946729 921167317 31.19S9479 9.9091776 .001027749 974 948676 924010424 31. 089731 9.9125712 .C01026C94 975 950625- 926859375 31.2249900 9.915624 .001025641 976 952576 929714176 31.24C9987 9.9198513 .001024590 977 954529 932574&S3 31.2569992 9.9227379 .C01023541 978 956484 9a5441352 31.2729915 9.S261222 .0010S2495 979 958441 938313739 31.2889757 9.9295042 .001021450 980 960400 941192000 31.3049517 9.9328889* .C01020408 981 988861 944076141 31.82C9195 9. 86261 3 .00101t'868 982 964324 946966168 31.83C8792 9.9896863 .001016380 983 966289 949862087 31.3528808 9.9430092 .001017294 984 968256 952763904 31.3687743 9.9463797 . 00101 C260 985 970225 955671625 31.3847097 9.9497479 ,001015228 986 972196 958585256 31.4006369 9.9531188 .001014199 987 974169 961504803 31.4165561 9.95(54775 .001013171 988 976144 964430272 31.4824673 9.9598389 .001012146 989 978121 967361669 31.4483704 9.9631981 .001011122 9PO 980100 970299000 31.4642654 9.S665549 .001010101 S91 982081 973242271 31.4801525 9.9699095 .001009082 992 984064 976191488 31.4960315 9.9732619 .001008065 330 TABLE XXIII. SQUARES, CUBES, ETC. No. Squares. Cubes. Square Roots. Cube Roots. Reciprocals. 903 98G049 979146657 31.5119025 9.9766120 .001007049 994 988038 982107784 31.5277655 9.9799599 .001006036 995 990025 985074875 31.5436206 9.9833055 .001005025 996 992016 988047936 31.5594677 9.9866488 .001004016 997 9*1009 991026973 31.5753068 9.9899900 .001003009 998 936004 994011093 31.5911380 9.99=33289 I .001002004 999 998001 997002999 1 31.6069613 9.9966656 .001001001 1000 1000000 1000000000 31.6237766 10.0000000 .001000000 1001 1002001 1003003001 31.6:385840 10.00:33322 .0009990010 100-2 1034004 10060120J3 31.6543836 10.0006623 .0009980040 1003 1005009 1009027027 31.6701752 10.0099899 . 0009970090 1004 1008016 1012 43064 31.6859590 10.0133155 .0009960159 1005 1010035 1015075125 1 31.7017349 10.0166389 .0009950249 1006 1012036 1018103216 I 31.7175030 10.0199601 .0009940358 1007 1014049 1021147343 ! 31.7332633 10.0332791 .0009930487 1003 1016084 1024192512 ! 31.7490157 10.0265958 .00099206:35 1009 1018031 1027243739 i 31.7647603 10.0299104 .0009910803 1010 1030100 1033301000 i 31.7804972 10.0332228 .0009900990 1011 1032121 1033364331 j 31.7963362 10.0365330 .0^,09891197 1013 1034144 1038433723 31.8119474 10.0398410 .0009881423 1013 1026169 1039509197 | 31.8276609 10.0431469 .0009871668 1011 1038195 1042590744 31.8433666 10.0464506 .0009861933 1015 1030235 1045678375 31.8590646 10.0497521 .0009852217 1016 1032256 1048772096 31.8747549 10.0530514 .0009842520 1017 1034389 1051871913 31.8904374 10.0563485 .0009832842 1018 1035324 1054977833 31.9081123 10.0596435 .0009823183 1019 1038361 1053039859 31.9217794 10.0629364 .0009818543 10.30 1040400 1031208000 31.9374388 10.0662271 .0009803922 1021 1042441 1064332261 31.9530906 10.0695156 .0009794319 1022 . 1044484 1087462648 31.9637347 10.0723020 .0009784736 1023 1046529 10r0599167 31.9843712 10.0760863 .0009775171 102 i 1048576 1073741824 3.3.t>000030 10.0793684 .0009765625 1033 1050325 1076890335 33.0153212 10.0823484 .0009756098 1023 1052376 1030045576 32.0312348 10.0359203 .0009746589 1027 1054729 10333)8833 32.0463407 10.0392019 .0009737098 1028 1056784 1036373953 32.0834391 10.0924755 .0009727626 1029 1058841 1039547339 33.0780293 10.0957469 . .0009718173 1030 1030900 1092 727000 32.0933131 10.0990163 .0009708738 1031 1052961 1095912791 32.1091887 10.1023835 .0009699321 1032 1035024 1099104763 32.1347563 10.1055487 .00 9689922 1033 1067039 1102302937 32.1403173 10.1088117 .0009680542 1031 1069156 1105507304 32.1558704 10.1120726 .0009671180 1035 1071225 1103717875 32.1714159 10.1153314 .000960 IHSli 1036 1073296 1111934656 32.18(59539 10.1185882 .0009(552510 1037 1075389 1115157653 32.2024844 10.1218428 .0009643202 1038 1077444 1118338372 32.2180074 10.1250953 .0009633911 1039 1079521 1121622319 32.2335229 10.1283457 .0009634639 1010 1031600 1124884000 32.2490310 10.1315941 .0009615385 1041 1033631 1128111931 32:2645316 10.1348403 .0009606148 1012 1035734 1131333038 32.2800248 10.1380845 .0009596929 1043 1037849 1134626507 32.2955105 10.1413266 .0009587738 1044 1039936 1137893184 32.3109883 10.1445667 .0009578544 1045 1093035 1141166125 32.3264598 10.1478047 .0039569378 1046 1094116 1144445336 32.3419233 10.1510406 .Ot309560229 1047 1093209 1147730823 33.3573794 10.1542744 .0009551098 1048 1093304 1151022592 83. 3728-381 10.1575062 .0009541985 1049 1100401 1154330848 32.3882695 10.1607-359 .0009532888 1050 1102500 1157625000 32.4037035 10.1639636 .0009523810 1051 1104601 1160935651 32.4191301 10.1671893 .0009514748 1053 1108704 1164252608 32.434.5495 10.1704129 .0009505703 1053 1108809 1167575877 32.449D615 10.1736344 .0009496676 1054 1110916 1170905464 32.4653662 10.1768539 .0009487666 331 TABLE XXIV.- LOGARITHMS OF NUMBERS. No. 100 L. 000.] [No. 109 L. 040. N. 1 2 8 4 5 6 7 8 9 Diff. 100 000000 0434 0868 1301 1734 2166 2598 3029 3461 3891 432 1 4321 1 4751 5181 5609 6038 6466 6894 7321 7748 8174 428 * 8600 QftOfi 9451 9876 0300 0724 1147 1570 1993 2415- . 3 012837 3259 3680 4100 4521 4940 5360 JOlU 5779 6197 6616 420 4" 7033 VAX.! 7868 8284 8700 i 9116 9532 0361 0775 416 5 021189 1603 2016 2428 2841 j 3252 3664 4075 4486 4896 412 6 5306 5715 6125 6533 6942 | 7350 7757 8164 8571 8978 408 9384 9789 0195 0600 1004 1408 1812 2216 2619 3021 404 8 033424 3826 4227 4628 5029 5430 5*30 6230 6629 7028 400 7426 7825 8223 8620 9017 9414 9311 04 0207 0602 0998 397 PROPORTIONAL PARTS. Diff. 1 2 3 4 5 6 7 8 9 434 43.4 86.8 130.2 173.6 217.0 260.4 3( >3.8 347.2 390.6 433 43.3 86.6 12 9.9 173.2 216.5 259 8 $ B.I 346.4 389.7 432 43.2 86.4 129.6 172.8 216.0 259.2 302.4 345.6 388.8 431 43.1 86.2 12 9.3 172.4 215.5 258 6 3( H.7 344.8 387.9 430 | 43.0 86.0 12 9.0 172.0 215.0 258 ft W.O 344.0 387.0 429 42.9 85.8 128.7 171.6 214.5 257.4 300.3 343.2 386.1 428 42.8 85.6 12 8.4 171.2 214.0 256 8 21 )!).6 342.4 385.2 427 42.7 85.4 128.1 170.8 213.5 256 2 298.9 341.6 384.3 426 42.6 85.2 12 7.8 170.4 213.0 255 6 2< 340.8 383.4 425 42.5 85.0 127.5 170.0 212.5 255 297.5 340.0 382.5 424 42.4 84.8 127.2 169.6 212.0 254 4 2< )6.8 339.2 381.6 423 42.3 84.6 12 6.9 169.2 211.5 253 8 2< )6.1 338.4 380.7 422 42.2 84.4 126.6 168.8 211.0 253 2 295.4 337.6 379.8 421 42.1 84.2 12 6.3 168.4 210.5 252 6 2< M.7 336.8 378.9 420 42.0 84.0 126.0 168.0 210.0 252 294.0 336.0 378.0 419 41.9 83.8 IS 5.7 167.6 209.5 251 4 2< )3.3 335.2 377 1 418 1 41.8 83.6 IS 5.4 167.2 209.0 250 a 9 )2.6 334.4 376! 2 417 41.7 83.4 125.1 166.8 208.5 250.2 a 333.6 375.3 416 ! 41.6 83.2 IS 4.8 166.4 208.0 249 i; 2 )ll2 332.8 374.4 415 41.5 83.0 124.5 166.0 207.5 249.0 290.5 332.0 373.5 414 41.4 82.8 124.2 165.6 207.0 i 248 4 2! 39.8 331.2 372.6 413 41.3 82.6 123.9 165.2 206.5 1 247 .8 39.1 330.4 371.7 412 41.2 82.4 IS 3.6 164.8 206.0 247 .2 a 38.4 329.6 370.8 411 41.1 82. S 123.3 164.4 205.5 246 .(i 287.7 328.8 369.9 410 41.0 82. C IS 3.0 164.0 205.0 246 .0 21 37.0 328.0 369.0 40C 40.9 81.8 IS 2.7 163.6 204.5 245 .4 21 36.3 327.2 368.1 408 40 8 81. 122.4 163.2 204.0 244 .8 285.6 326.4 367.2 40? 40.7 81.4 li !2.1 162.8 203.5 244 .2 21 34.9 325.6 366.3 406 40.6 81. S 121.8 162.4 203.0 243 6 284.2 324.8 365.4 405 40.5 81.0 121.5 162.0 202.5 243.0 21 33.5 324.0 364.5 404 40.4 80.* ! 121.2 161.6 202.0 242.4 a 32.8 323.2 363.6 40c , 40.3 80. ( > V JO. 9 161.2 201.5 241 .8 8 32.1 322.4 362.7 40$ 40.2 80.' I V JO. 6 160.8 201.0 241 2 S 31.4 321.6 361.8 401 40.1 80. $ I V JO. 3 160.4 200.5 240 .0 2 BO. 7 320.8 360.9 400' 40.0 80-0 120.0 160.0 200.0 240.0 280.0 320.0 360.0 39< ) 39.9 79. i J 1 19.7 159.6 199.5 239 .4 2 79.3 319.2 359.1 3 } 39.8 79.1 i 119.4 159.2 199.0 238.8 278.6 318.4 358.2 39' r 39.7 79.' 1 1 19.1 158.8 198.5 238 .3 2 77.9 317.6 357.3 396 39.6 79.2 118.8 158.4 198.0 237 .6 277.2 316.8 356.4 39 5 39.5 79.0 118.5 158.0 197.5 237 .0 276.5 316 355.5 332 TABLE XXIV.-LOGARITHMS OF NUMBERS. No. 110 L. 041.] [No. 119 L. 078. N. 1 2 8 4 5 6 7 8 9 Diff. 110 041393 1787 2182 2576 2969 3362 3755 4148 4540 4982 C3 1 5o^i 5714 0105 6495 6bfc5 7275 76C4 8053 8442 8880 IW) 2 92 Id DOUG 1/1)1)3 (Y^J-irt (Y?(\(\ 1 1 JV-l 1 J\^>i f ro 8 153078 3403 3S40 UooU l/fOO 4230 4613 1 1 OO i 1 .)oo 4WU 53,8 5700 0142 IC24 8b3 4 1)905 r*.K'(t r 'ti(i(i 8046 8420 bh05 9185 D;")(>'~] !) ( )4 C20 9r " 5 060698 1075 1452 1829 2206 2582 2958 3333 3709 4083 376 6 4458 4832 5206 5580 5953 6326 6699 7071 7443 7815 373 7 8186 8557 8928 9298 9668 AAQQ nr-i*"f -i-i AK -tK-tA Q7 A 8 071882 2250 2617 2985 3352 UUoo 3718 4085 Ul i D 1140 4451 4810 1O14 5182 otv 366 9 5547 5912 6276 6640 7004 7368 7731 8094 8457 8819 363 1 PROPORTIONAL PARTS. Diff. 1 2 3 4 5 6 -7 8 9 395 394 39.5 39.4 79.0 78.8 118.5 118.2 158.0 157.6 197.5 197.0 237 236 .0 .4 276.5 275.8 316.0 315.2 355.5 354.6 393 39.3 78.6 11 7.9 157.2 196.5 235 .S 275.1 314.4 353.7 392 39.2 78.4 117.6 156.8 196.0 235.2 274.4 HJ.-J.6 352.8 391 39.1 78.2 11 7.3 156.4 195.5 234 .6 273.7 312.8 351.9 390 39.0 78.0 117.0 156.0 195.0 234.0 273.0 312.0 351.0 389 38.9 77.8 11 6.7 155.6 194.5 283 .4 272.3 311.2 850.1 388 38.8 77.6 116.4 155.2 194.0 232.8 271.6 310.4 349.2 387 38 .-7 77.4 11 6.1 154.8 193.5 232 .2 270.9 309.6 3-48.8 386 38.6 77.2 115.8 154.4 193.0 231 270.2 3C8.8 847.4 385 38.5 77.0 115.5 154.0 192.5 231 '.Q 209.5 308.. 346.5 384 38.4 76.8 ' 115.2 153.6 192.0 2S0.4 268.8 307.2 ''845.6 383 38.3 76.6 11 4.9 153.2 191.5 228 .8 208.1 306. 4 344.7 38.2 76.4 114.6 152.8 191.0 229.2 267.4 305.6 343.8 381 38.1 76.2 11 4.3 152.4 ISO. 5 228 .6 266.7 804.8 342.9 380 38.0 76.0 114.0 152.0 190.0 228.0 206.0 04.0 3J2.0 379 37.9 75.8 11 3.7 151.6 189.5 227 .4 205.3 E03.2 841.1 378 37.8 75.6 113.4 151.2 189.0 226.8 264.6 02.4 340.2 377 37.7 75.4 11 3.1 150.8 188.5 221 .18 203.9 301.6 839.3 376 37.6 75.2 112.8 150.4 188.0 5.6 203.2 300.8 838.4 375 37.5 75.0 112.5 150.0 187.5 225.0 202.5 300.0 337.5 374 37.4 74.8 112.2 149.6 187.0 224.4 201.8 299.2 386.6 373 37.3 74.6 11 1.9 149.2 180.5 Kl .s 201 . 1 288.4 835.7 372 37.2 74.4 111.6 148.8 186.0 223.2 260.4 297.6 884.8 371 37.1 74.2 11 1.3 148.4 185.5 * .0 259.7 296.8 383.9 37'0 37.0 74.0 111.0 148.0 185.0 22.0 259.0 2 1 36.4 170.5 204.6 238.7 272.8 300.9 840 34.0 68.0 102.0 136.0 170.0 204.0 238.0 27'2.0 306.0 339 33.9 67.8 101 .7 1 35.6 169.5 203.4 237.3 271.2 305.1 338 33.8 67.6 101 .4 135.2 169.0 202.8 236.6 270.4 304.2 837 33.7 67.4 101 .1 1 34.8 168.5 202.2 235.9 269.6 303.3 '.', ,ij 33. (3 67.2 100.8 134.4 168.0 201.6 235.2 2C8.8 302.4 836 83.5 67.0 100.5 134.0 167.5 201.0 234 5 268.0 301.5 384 OQ. t 66.8 100 .2 1 33.6 167.0 200.4 233.8 267.2 300.6 333 88.8 (3(3 (5 99.9 133.2 166.5 199.8 233.1 266.4 299.7 332 33.2 6(5! 4 99 .r, 1 32.8 166.0 199.2 232.4 265.6 298.8 331 33.1 66.2 9G .3 132.4 165.5 198.6 231.7 264.8 297.9 330 33.0 66.0 9fl .0 : 32.0 165.0 198.0 231.0 264.0 297.0 829 32.9 65.8 98 .7 31.6 164.5 197.4 230.3 263.2 296.1 328 32.8 65.6 98.4 131.2 164.0 196.8 229.6 262.4 295.2 327 32.7 65.4 98 1 30.8 163.5 196.2 228.9 261.6 294.3 326 32.6 65.2 97.8 130.4 163.0 195.6 228.2 260.8 293.4 325 32.5 65.0 97.5 130.0 162.5 195.0 227.5 SfiO.O 292.5 321 32.4 64.8 97.2 129.6 162.0 194.4 226.8 259.2 291.6 323 32.3 64.6 96 .!) 1 29.2 161.5 193.8 226.1 258.4 290.7 322 32.2 64.4 ! 96.6 I 28.8 161.0 193.2 225.4 257.6 289.8 334 TABLE XXIV. LOGARITHMS OF NUMBERS. No. ia5 L. 130.] [No. 149 L. 175. N. 1 2 * 45 C 7 8 9 Diff. 135 130334 0655 0977 1298 1619 : 1939 2260 2580 2900 ! 3219 321 6 3539 3858 4177 4496 4814 I 5133 \ 5451 5769 6086 64US 318 7 6721 7037 7354 7671 7987 ; 8303 ! 8618 8934 9249 9564 316 g 9879 0194 0508 0822 1136 ! 1450 1763 2076 2389 2702 314 q 143015 3327 | 3639 3951 4263 4574 4885 5196 5507 5818 311 140 6128 6438 6748 7058 7367 7676 7985 8294 8603 8911 309 9219 9527 i 9835 ^ - . I QT'Sfi i/v*o 1*5*70 1 llTfi 1 0QO OA 1 ? 2 152288 2594 2900 3205 3510 3815 JLV/UU 4120 loiu 4424 1 U i 1 Jo w 4728 5032 OUrf 305 8 5336 5640 5943 6246 6549 i 6852 7154 7457 7759 8061 303 A 8362 8664 89C5 9567 1 9868 0469 0769 1068 301 5 161368 1667 1967 2266 2564 2863 3161 3460 3758 4055 299 6 4358 4650 4947 5244 5541 5838 6134 6430 6726 7022 297 "< 7317 7613 7908 8203 8497 8792 9086 9380 9674 9968 295 8 170262 0555 0848 1141 1434 1726 2019 2311 2603 2895 293 9 3186 3478 3769 4060 4351 4641 4932 5222 5512 5802 291 PROPORTIONAL PARTS. Diff. 1 2 3 4 5 6 7 8 9 a-ai 32.1 64.2 96.3 128.4 160.5 192.6 224.7 256.8 288.9 320 32.0 64.0 96.0 128.0 160.0 192.0 224.0 256.0 288.0 319 81.9 63.8 95 7 127.6 159.5 191 4 3.3 255.2 i 287.1 318 31.8 63.6 95 4 127.2 159.0 190 8 2S 2.6 251.4 286.2 3 17 31.7 63.4 95 1 126.8 158.5 190 a 221.9 253.6 285.3 310 31.6 63.2 94 8 126.4 158.0 189.6 221.2 252.8 284.4 815 31.5 63.0 94* 5 126.0 157.5 189 2X 0.5 252.0 283.5 314 31.4 62.8 94 2 125.6 157.0 188 4 21 9.8 251.2 282.6 313 31.3 62.6 93.9 125.2 156.5 187.8 219.1 250.4 ' 281.7 312 31.2 62.4 93 G 124.8 156.0 187 2 218.4 249.6 280.8 311 31.1 62.2 93.3 124.4 155.5 186 6 217.7 248.8 279.9 310 31.0 62.0 93.0 124.0 155.0 186.0 217.0 248.0 279.0 809 30.9 61.8 9i 7 123.6 154.5 185 4 21 6.3 247.2 278.1 308 30.8 61.6 92.4 123.2 154.0 184.8 215.6 246.4 217.2 307 30.7 61.4 92 1 122.8 153.5 184 8 21 4.9 245 6 276.3 30G 30.6 61.2 91 8 122.4 153.0 183.6 214.2 244.8 275.4 305 30.5 61.0 91 5 122.0 152.5 183 21 3.5 244.0 274 .5 304 30.4 60.8 91 2 121.6 152.0 182 4 212.8 243.2 273.6 303 30.3 60.6 90 9 121.2 151.5 181 S 21 2.1 242.4 272.7 302 30.2 60.4 90.6 120.8 151.0 181.2 211.4 241.6 271.8 301 30.1 60.2 90.3 120.4 150.5 180.6 210.7 240.8 270.9 300 30.0 60.0 90 120.0 150.0 180 21 0.0 240.0 270.0 299 29.9 59.8 89 f 119.6 149.5 179 4 2C 9.3 239.2 269.1 293 29.8 59.6 89.4 119.2 149.0 178.8 208.6 .238.4 268.2 297 29.7 59.4 89 .1 118.8 148.5 178 a 207.9 237.6 267.3 296 29.6 59.2 88 .8 118.4 148.0 177 (i * >7.2 236.8 266.4 295 29.5 59.0 88 5 118.0 147.5 177 at >6.5 236.0 265.5 294 29.4 53.8 88.2 117.6 147.0 176 4 2C 5.8 235.2 264.6 293 29.3 58.6 87 g 117.2 146.5 175 8 205.1 234.4 263.7 292 29.2 58.4 87.6 116.8 146.0 175 $ 24.4 233.6 262.8 291 29.1 58.2 87.3 116.4 145^5 174 (I 203.7 232.8 261.9 290 29.0 58.0 87 116.0 145.0 174 2t 3.0 232.0 261.0 289 28.9 57.8 86 7 115.6 144.5 173 4 2( 2.3 231.2 260.1 288 28.8 57.6 86 4 115.2 , 144.0 172 8 2C 1.6 230.4 259.2 287 28.7 57.4 86.1 114.8 143.5 172 2 200.9 229.6 258.3 286 28.6 57.2 85 .8 114.4 143.0 171 6 200.2 228.8 ! 257.4 335 TABLE XXIV. LOGARITHMS OF NUMBERS. No. 150 L. 176.1 [No. 169 L. 230. N. 1 2 3 !! 7 8 9 Diff. 150 176091 6881 61,70 6959 7248 i 7536 7825 8113 8401 8689 289 9839 89(7 J264 Jo52 0126 !' 0413 0699 0986 1272 1558 287 2 181844 2129 2415 2700 2985 3270 3555 3839 4123 4407 285 3 4691 4975 5259 5542 5825 6108 6391 6674 6956 : 7239 283 4" 7521 CSilW Kl K 1 8306 8647 i j 8928 Q*>f> ( > Qzion 9771 5 2846 279 190aS2 0612 i 0892 1171 1451 : 1730 2010 2289 2567 6 3125 3403 ! 3681 8860 4237 4514 4792 5069 5346 5623 278 7 5 617(5 8057 8932 6453 9206 6729 9481 7005 9755 7281 7556 7832 8107 8382 276 0029 1 0303 0577 , 1 1 9 A MM 9 201397 167'0 1943 2216 2488 2761 3033 3305 3577 3848 272 160 4120 4391 4663 4934 5204 5475 i 5746 6016 6286 6556 271 1 6826 7096 7365 7634 7904' 8173 8441 8710 8979 9247 269 2 9515 97'83 0051 0319 0586 0853 1121 1388 1654 1921 267 3 "212188 2454 2720 2986 3252 3518 3783 4049 4314 4579 266 4 4844 5109 5373 5638 5902 6166 6430 6694 6957 7221 264 5 7484 7747 8010 8273 8536 8798 9060 9323 9585 9846 262 6 220108 0370 0631 0892 1153 1414 1675 1936 2196 2456 261 7 2716 2976 3236 3496 3755 4015 4274 4533 4792 5051 259 8 5309 5568 5826 6084 6342 6600 6858 7115 7372 7C30 258 9 7'887 8144 8400 8657 8913 I 9170 9426 9682 9938 23 0193 256 PROPORTIONAL PARTS. Diff. 1 2 3 4 5 6 7 8 9 285 28.5 57.0 85 .5 114.0 142.5 171.0 199.5 228.0 256.5 284 ! 28.4 56.8 85 .2 113.6 142.0 170.4 198.8 227.2 255.6 283 28.3 56.6 84 .9 113.2 141.5 169.8 198.1 226.4 254.7 282 28.2 56.4 84 .6 112.8 141.0 169.2 197.4 225.6 253.8 281 i 28.1 56.2 84 .3 112 4 140.5 I 168.6 196.7 224.8 252.9 280 28.0 56.0 84 .0 112.0 140.0 1 168.0 196.0 i 224.0 252.0 279 27.9 55.8 83 .7 111.6 139.5 167.4 195.3 223.2 251.1 278 27.8 55.6 83 .4 111.2 139.0 166.8 194.6 22.4 250.2 277 27.7 55.4 83 .1 110.8 138.5 166.2 193.9 221.6 249.3 27'6 27.6 55.2 82.8 110.4 138.0' 165.6 193.2 220.8 248.4 275 27.5 55.0 82.5 110.0 137.5 165.0 192.5 220.0 2-17.5 274 27.4 54.8 82 .2 109.6 137.0 164.4 191.8 219.2 246.6 273 27.3 54.6 81 .9 109.2 136.5 163.8 191.1 218.4 245.7 272 27.2 54.4 81.6 108.8 136.0 163.2 190.4 i 217.6 244.8 271 27.1 54.2 81 .3 108.4 " 135.5 162.6 189.7 216.8 243.9 270 27.0 54.0 81.0 108.0 135.0 162.0 189.0 216.0 243.0 269 26.9 53.8 80 .7 107.6 134.5 161.4 188.3 ; 215.2 242.1 268 26.8 53.6 80 .4 107.2 134.0 160.8 187.6 214.4 241.2 207 26.7 53.4 80.1 106.8 133.5 160.2 ! 186.9 1 213.6 240.3 266 26.6 53.2 79.8 106.4 133.0 159.6 186.2 ! 212.8 239.4 385 26.5 53.0 79.5 106.0 132.5 159.0 185.5 ' 212.0 238.5 204 26.4 52.8 79 .2 105.6 132.0 158.4 184.8 \ 211.2 237.6 203 26.3 52.6 78 .9 105.2 131.5 157.8 184.1 210.4 236.7 202 26.2 52.4 78.6 104.8 131.0 157.2 183.4 i 209.6 235.8 201 26.1 52.2 78 .3 104.4 130.5 156.6 182.7 208.8 234.9 260 26.0 52.0 78.0 104.0 130.0 156.0 182.0 208.0 234.0 259 25.9 51.8 77 .7 103.6 129.5 155.4 181.3 ! 207.2 233.1 258 25.8 51.6 77 .4 103.2 129.0 154.8 180.6 206.4 232.2 257 25.7 51.4 77 .1 102.8 128.5 154.2 179.9 205.6 231.3 256 25.6 51.2 76 .8 102.4 128.0 153.6 179.2 204.8 230.4 255 25.5 ! 51.0 76.5 102.0 1^7.5 153.0 178.5 ; 204.0 229.5 336 TABLE XXIV. -LOGARITHMS OF NUMBERS. No. 170 L. 230.] [No. 139 L. 270. N 1 8 A i-vr*i . V UllL. 170 230449 0704 09JO 1215 1470 ! 1724 1979 223-1 2488 2742 255 1 2996 8250 8.304 3757 4011 4^64 4517 4770 5023 5276 253 2 6528 5781 (5033 62H5 6537 6789 7041 7292 7'544 7795 252 ft04fi 8297 8548 8799 049 9299 9550 9800 oU^tO AAFCA AOAA 1 n,^n 4 240549 | 0799 1048 1297 1546 1795 2044 2233 UUtXJ 2541 UoUU 2790 f**J\) 243 5 3038 3286 3534 37t2 4030 4277 4525 477'2 5019 5266 243 G 5513 5759 6006 6252 0499 6745 6991 7237 7482 7728 246 7 7973 8219 8464 8709 89o4 9198 9443 9687 9932 0176 24ft 8 250420 0664 0908 1151 1395 1638 1881 2125 2368 2610 iC-k\) 243 9 2853 3096 3338 3580 3822 4064 4306 4548 4790 5031 242 180 5273 5514 5755 5996 6237 6477 6718 6958 7198 7439 241 1 7679 7918 8158 8398 8637 8877 9116 9355 9594 9833 239 2 260071 0310 0548 0787 1025 1263 1501 1739 1976 2214 238 3 2451 2688 2925 3162 3399 3636 3873 4109 4346 4582 237 4 4818 5054 5290 5525 5761 5996 6232 6467 6702 6937 235 5 7172 7406 7641 7875 8110 8344 8578 8812 9046 9279 234 g 9513 9746 9980 0213 0446 0679 0312 1144 1377 1609 233 7 271842 2074 2303 2538 2770 3001 3233 * MS 3464^ 3696 3927 232 8 4158 4389- 4620 4850 5081 5311 5542 5772 6002 6232 230 6462 6692 6921 7151 7380 7609 7S:-iS 8067 8296 8525 229 PROPORTIONAL PARTS. Diff. 1 2 3 4 5 6 7 8 9 255 25.5 51.0 76.5 102.0 127.5 153 178.5 204.0 229.5 254 25.4 50.8 76.2 101.6 127.0 152 4 17 7.8 203.2 228.6 253 25.3 50.6 75.9 101.2 126.5 151 8 17 7.1 202.4 227.7 252 25.2 50.4 75.6 100.8 126.0 151 2 176.4 201.6 226.8 251 25.1 50.2 75.3 100.4 125.5 150 6 17 5.7 200.8 225.9 250 25 50.0 75.0 100.0 125.0 150 175.0 200.0 225.0 249 24.9 49.8 74.7 99.6 124,5 149 4 17 4.3 199.2 224.1 248 24.8 49.6 74.4 99.2 124.0 148 8 17 3.6- 198.4 223.2 247 24.7 49.4 74.1 98.8 123.5 148.2 17 2.9 197.6 222.3 246 24.6 49.2 73.8 98.4 123.0 147 6 17 2.2 196.8 221.4 245 24.5 49.0 73.5 98.0 122.5 147.0 171.5 196.0 220.5 244 24.4 48.8 73.2 97.6 122.0 146 4 170.8 195.2 219.6 243 24.3 48.6 72.9 97.2 121.5 145. S 17 0.1 194.4 218.7 242 24.2 48.4 72.6 96.8 121.0 145.2 169.4 193.6 217.8 241 24.1 48.2 72.3 96.4 120.5 144. (5 16 8.7 192.8 216.9 240 24.0 48.0 72.0 96.0 120.0 144. 168.0 192.0 216.0 239 23.9 47.8 71 .7 95.6 119.5 143. 4 16 7.3 191.2 215.1 238 23.8 47.6 71.4 95.2 119.0 142. s 10 6.6 190.4 214.2 237 23.7 47.4 71.1 94.8 118.5 142. 3 165.9 189.6 213.3 236 23.6 47.2 70.8 94.4 118.0 141. (i 16 5.2 188.8 212.4 235 23.5 47.0 70.5 94.0 117.5 141. 164.5 188.0 211.5 234 23.4 46.8 70.2 93.6 117.0 140. 4 163.8 187.2 210.6 233 23.3 46.6 69.9 93.2 116.5 139. 8 16 3.1 186.4 209.7 232 23.2 46.4 69.6 92.8 116.0 139. X! 162.4 185.6 208.8 231 23.1 46.2 69. 3 92.4 115.5 138. 6 16 1.7 184.8 207.9 230 23.0 46.0 69.0 92.0 115.0 138. 16 1.0 184.0 207.0 229 22.9 45.8 68.7 91.6 114.5 137.4 160.3 183.2 206.1 228 22.8 45.6 68.4 91.2 114.0 136. 6 15 9.6 182.4 205.2 227 22.7 45.4 68.1 90.8 113.5 136. 2 158.9 181.6 204.3 226 22.6 45.2 67.8 90.4 113.0 135.6 158 2 180.8 203.4 337 TABLE XXIV. -LOGARITHMS OF NUMBERS. No. 190 L. 278.] [No. 214 L. 332. N. 1 2 3 4 5 6 7 8 9 Diff. 190 278754 8982 9211 9439 9667 9895 0123 0351 0578 0806 228 1 281033 1261 1488 1715 1942 2169 2396 2G22 2849 3075 227 2 3301 3527 3753 3979 4205 4431 4656 4882 5107 5332 226 3 5557 5i'82 6007 6232 6456 : 6681 6905 7130 7354 7578 225 4 7802 8026 8249 8473 8(596 ! 8920 9143 9366 9589 9812 223 5 290035 0257 0480 0702 0925 1147 1369 1591 1813 2034 223 6 2256 2478 2699 2920 3141 88JJ3 3584 3804 4025 4246 221 | 4466 4687 4907 5127 5347 5567 5787 6007 6226 6446 220 8 6665 6884 7104 7323 7542 7761 7979 8198 8416 8635 219 oor:Q ( XV~l gown 9507 9725 on i oooo y\ji i if&OJ jy-to 0161 0378 0595 0813 218 200 801030 1247 1464 1681 1898 2114 2331 2547 2764 2980 217 1 3196 3412 362S 3844 4059 i 4275 4491 4706 4921 5136 216 2 5351 5566 : 5781 5996 6211 6425 6639 6854 7068 7282 215 3 4. 7496 nijoA 7710 i 7924 0040 i 8137 8351 8564 8778 8991 9204 9417 213 : WBU Jtyo 0056 0268 0481 0693 0906 1118 1330 1542 212 5 311751 1966 2177 2889 2600 2812 30243 3234 3445 3656 211 G 3867 4078 4289 4499 4710 4920 5130 5340 5551 5760 210 7 5970 6180 6390 6599 6809 7018 7227 7436 7646 7854 209 8 8063 8272 8481 8689 8898 9106 9314 9522 9730 9938 208 9 320146 0354 0562 07G9 0977 1184 1391 1598 1805 2012 207 210 2219 2426 2633 2839 3046 3252 3458 3665 3871 4077 206 1 4282 4488 4694 4899 5105 5310 5516 5721 5926 6131 205 2 6336 6541 6745 6950 7155 7359 7563 7767 7972 8176 204 3 8380 8^5l 8787 8991 9194 9398 9601 9805 0008 0211 203 4 330414 C617 0819 1022 1225 1427 1630 1832 2034 2236 202 PROPORTIONAL PARTS. Diff. i 2 3 4 5 6 7 8 9 225 22.5 45.0 67.5 90.0 112.5 135.0 157.5 180.0 202.5 224 22.4 44.8 67.2 89.6 112.0 134.4 156.8 179.2 201.6 223 22.3 44.6 66.9 89.2 111.5 133.8 156.1 178.4 200.7 222 22 2 44.4 66.6 88.8 111.0 133.2 155.4 177.6 199.8 221 22 .'l 44.2 68.3 88.4 110.5 132.6 154.7 176.8 i 198.9 220 22.0 44.0 66.0 H8.0 110.0 132.0 154.0 176.0 198.0 219 21.9 43.8 65.7 87.6 109.5 131.4 153.3 175.2 197.1 218 21.8 43.6 65.4 87.2 109.0 130.8 152.6 174.4 196.2 217 21.7 43.4 65.1 86.8 108.5 130.2 151.9 173.6 195.3 21G 21.6 43.2 64.8 86.4 108.0 129.6 151.2 172.8 194.4 215 21.5 43.0 64.5 86.0 107.5 129.0 150.fi 172.0 193.5 211 21.4 42.8 64.2 85.6 107.0 128.4 149.8 171.2 192.6 213 21.3 42.6 63.9 85.2 106.5 127.8 149.1 170.4 191.7 212 21.2 42.4 63.6 F4.8 106.0 127.2 148.4 169.6 190.8 211 21.1 42.2 63.3 84.4 105.5 126.6 147.7 168.8 189.9 210 21.0 42.0 63.0 84.0 105.0 126.0 147.0 168.0 189.0 209 20.9 41.8 62.7 83.6 104.5 125.4 146.3 167.2 188.1 208 20.8 41.6 62.4 83.2 104.0 124.8 145.6 166 4 187.2 207 20.7 41.4 62.1 82.8 103.5 124.2 144.9 165.6 186.3 20G 20.6 41.2 61.8 82.4 103.0 123.6 144.2 164.8' 185.4 205 20.5 41.0 61.5 82.0 102.5 123.0 143.5 164.0 184.5 204 20.4 40.8 61.2 81.6 102.0 122.4 142.8 163.2 183.6 203 20.3 40.6 60.9 81.2 101.5 121.8 142.1 162.4 182.7. 202 20.2 40.4 60.6 80.8 101.0 121.2 141.4 161.6 181.8 TABLE XXIV. LOGARITHMS OF NUMBERS. No. 215 L. 332.] [No. 239 L. 380. N' T^lff . w Dill. 215 332438 2640 2842 8044 3246 3417 3649 3850 4051 4253 2C2 6 4434 4055 4856 50c7 5257 5458 5658 5859 6059 6260 201 7 6460 6060 6860 7060 7260 7459 7659 7858 8058 8257 200 3 8456 8656 8855 9054 9253 9451 9650 9849 0047 OMft 1 OO 9 340444 0042 0841 1039 1237 1435 1632 1830 UvHi 2028 UA.40 2225 1UJ 158 220 2423 2620 2817 3014 3212 3409 3606 3802 3999 4196 197 1 4392 4589 4785 4981 5178 5374 5570 5766 5 ( J62 6157 196 2 6353 6549 6744 6939 7135 7330 7525 7720 7915 8110 195 3 8305 8500 8694 8889 9083 9278 9472 9666 9860 {\t\KA 4 350248 0442 0636 0829 1023 1216 1410 1603 1796 UUt> 1989 193 5 2183 2375 2568 2761 2954 3147 3339 3532 3724 3916 193 C 4108 4301 4493 4685 4876 5068 5260 5452 5643 5834 192 7 6026 6217 6408 6599 6790 6981 717'2 7363 7554 7744 191 8 7935 8125 8316 8506 8696 8886 9076 9266 9456 9646 190 g 9835 0025 0215 0404 0593 0783 0972 1161 1350 1539 JQQ 230 361728 1917 2105 2294 2482 2671 2859 3048 3236 3424 icy 188 1 3612 3800 3988 4176 4363 4551 4739 4926 5113 5301 188 2 5488 5675 5862 6049 6236 ; 6423 6610 6796 6983 7169 187 3 7356 7542 7729 7915 8101 i 8287 8473 8659 8845 9030 186 4 9216 9401 9587 9772 9958 0143 0328 0513 0698 0883 185 5 ~371068 1253~ 1437~ 1622 1806 1991 2175 2360 2544 2728 184 G 2912 3096 3280 3464 3647 8831 4015 4198 4382 4565 184 7 4748 4932 5115 5298 5481 5664 5846 6029 6212 6394 183 8 6577 6759 6942 7124 7306 7488 7670 7852 8034 8216 182 9 8398 8580 8761 8943 9124 9306 9487 9668 9849 38 0030 181 PROPORTIONAL PARTS. Diflf. 1 2 3 4 5 6 7 8 9 202 201 20.2 20.1 40.4 40.2 60.6 60.3 80.8 80.4 101.0 100.5 121.2 120.6 141.4 140.7 161.6 160.8 181.8 180.9 200 20.0 40.0 60.0 80.0 100.0 120.0 140.0 160.0 180.0 199 19.9 39.8 59.7 79.6 99.5 119.4 139.3 159.2 ! 179.1 198 19.8 39.6 59.4 79.2 99.0 118.8' 138.6 158.4 178.2 197 19.7 39.4 59.1 78.8 98.5 118.2 137.9 157.6 177.3 196 19.6 39.2 58.8 78.4 98.0 117.6 137.2 156.8 176.4 195 19.5 39.0 58.5 78.0 97.5 117 136.5 156.0 175.5 194 19.4 38.8 58.2 77.6 97.0 116.4 135.8 155.2 174.6 193 19.3 38. G 57.9 77.2 9G.5 115.8 135.1 154.4 173 7 192 19.2 38.4 57.6 76.8 96.0 115.2 134.4 153.6 172.8 1 1 19.1 38.2 57 3 76.4 95.5 114.6 133.7 152.8 171.9 190 19.0 38.0 57.0 76.0 95.0 114.0 ias.0 152.0 171.0 189 18.9 37.8 56.7 75. G 94.5 113.4 132.3 151.2 170.1 188 18.8 37.6 56.4 75.2 94.0 112.8 131.6 150.4 169.2 187 18.7 37 4 56.1 74.8 93 5 112.2 130.9 149.6 168.3 186 18.6 37.2 55.8 74.4 93^0 111.6 130.2 148.8 167.4 185 18.5 37.0 55.5 74.0 92.5 111.0 129.5 148.0 166.5 184 18.4 36.8 55.2 73.6 92.0 110.4 128.8 147.2 165.6 183 18.3 36.6 54.9 73.2 91.5 109.8 128.1 146.4 164.7 182 18.2 36.4 54.6 72.8 61.0 109.2 127.4 145.6 163.8 181 18.1 36.2 54.3 72.4 90.5 108.6 126.7 144.8 162.9 180 18.0 36.0 54.0 72.0 90.0 108.0 126.0 144.0 162.0 179 17.9 35.8 53.7 71.6 89.5 107.4 125.3 143.2 161.1 _* TABLE XXIV. -LOGARITHMS OF NUMBERS. No. 240 L. 380.] [No. 269 L. 431. N. 1 2 3 4 6 6 7 8 9 Difl. 240 380211 0392 0573 0754 0934 i 1115 1296 1476 1656 1837 181 1 2017 2197 2377 2557 2737 j 2917 3097 | 3 ^77 345 5 3636 180 2 3815 3995 4174 4353 4533 4712 4891 5070 5249 5423 179 3 5636 5785 5964 6142 6321 6499 6677 6 356 703 1 7212 178 4 7390 7568 7746 924 8101 8279 8456 8 634 881 1 8989 178 9166 9343 9520 9698 9875 1 0051 O998 A ins 058 j 0759 177 6 390935 1112 1288 1464 1641 1817 1993 U-*vu 2169 2345 2521 176 y 2697 2873 3048 3224 3400 a575 3751 3 926 410 1 4277 176 8 4152 4627 4802' 4977 5152 5326 5501 5 676 585 ) 6025 175 9 6199 6374 6548 6722 6896 7071 7245 7419 7592 7766 174 250 7940 8114 9674 i 9847 8287 8461 8634 8808 8981 9154 9328 9501 173 * 0020 0192 0365 0538 0711 A QQ-t 10?: a iooQ 173 2 401401 1573 1745 1917 2089 2261 2433 ULX*J 2605 lUOu j.x^u 2777 2949 172 3 3121 3292 3464 3635 .3807 3978 4149 4 320 449 .> 4663 171 4 4834 5005 5176 5346 5517 5688 5858 6 029 619 9 6370 171 5 6540 6710 6881 7051 7221 7491 7561 731 7901 8070 170 6 8240 8410 8579 8749 8918 ' 9087 9257 426 i 959 5 9764 169 *7 9933 0102 0271 0440 0609 i 0777 0946 1114 1283 1451 169 8 411620 1788 1956 2124 2293 ! 2461 2629 2 796 296 i 3132 168 9 3300 3467 3635 3803 3970 i 4137 4305 4472 463 y 4806 167 260 4973 5140 5307 5474 5641 5808 5974 6141 6308 6474 167 1 6641 6807 6973 7139 7306 7472 7(538 7 804 797 . 8135 166 2 8301 8467 8633 8798 8964 9129 9295 9460 9625 9791 165 3 9956 0121 0286 0451 0616 0781 0945 1110 1275 1439 165 4 421604 1768 1933 2097 2261 2426 2590 2 754 291 s 3082 164 5 3246 3410 3574 3737 3901 4065 4228 4392 4555 4718 164 6 4882 5045 5208 saw 5534 5697 5860 6 023 618 ; 6349 163 7 6511 6074 GH36 6999 7161 7324 7486 1 648 781 i 7973 162 8 81% 8297 8459 8621 8783 i 8944 9106 ! 268 9429 9591 162 g 9752 ' flQl * 43 0075 0236 0398 0559 0720 0881 1042 ! 1203 161 PROPORTIONAL PARTS. Diflf. 1 2 3 4 5 6 106.8 7 8 9 160.2 178 7.8 35.6 53 4 71.2 89.0 124.6 142.4 177 7.7 35.4 53 1 70.8 88.5 106.2 123.9 141.6 159.3 176 7.6 35. 2 52 8 70.4 88.0 105.6 123.2 140.8 158.4 175 7.5 a5.0 52 5 70.0 7.5 105.0 122.5 140.0 157.5 174 7.4 34.8 52 2 69.6 87.0 104.4 12U3 139.2 156.6 173 17.3 34.6 51 9 69.2 86.5 103.8 121.1 133.4 155.7 172 17.2 34.4 51 68.8 86.0 103.2 120.4 137.6 154.8 171 17.1 34.2 51 3 68.4 85.5 102.6 119.7 136.8 153.9 170 17.0 34.0 51.0 68.0 85.0 102.0 119.0 136.0 153.0 169 16.9 33.8 50.7 67.6 84.5 101.4 118.3 135.2 152.1 168 16.8 33.6 50 4 67.2 84.0 100.8 117.6 134 4 151 2 167 16.7 33.4 50 1 66.8 83.5 100.2 116.9 133.6 150.3 166 16.6 33.2 49 8 66.4 83.0 99.6 116.2 132.8 149.4 165 16.5 33.0 49 5 66.0 82.5 99.0 115.5 132.0 148 5 164 16.4 32.8 49.2 65.6 82.0 98.4 114.8 131.2 147.6 163 16.3 32.6 48 65.2 81.5 97.8 114.1 130.4 146.7 162 16.2 32.4 48 5 64.8 81.0 97.2 113.4 129.6 145 8 161 16.1 32.2 48 3 64.4 80.5 96.6 112.7 128.8 144.9 340 TABLE XXIV. LOGARITHMS OF NUMBERS. No. 270 L. 431.] [No. 299 L. 476. N. 1 2 3 4 5 6 7 8 9 Diff. 270 431364 1525 1685 1846 2007 2167 2328 2488 2649 2809 161 1 29G9 8130 1 3290 3450 3610 3770 3930 4090 4249 4409 160 2 4.o69 4729 4838 5048 5207 5367 , 5526 5685 5844 6004 159 3 6163 6322 6481 6640 6799 6957 7116 7275 74:33 755W 159 4 7751 7909 8067 8226 8384 8542 8701 8859 9017 9175 158 5 9333 9491 9(>48 9806 9964 1T"~ " 1 K.Q 6 440909 1066 1224 1381 1538 1695 1852 2009 2166 2323 loo 157 7 2480 2637 2793 2950 3106 3~>G3 3419 3576 37:32 i 3889 157 8 4045 4201 4357 4513 4669 4825 ! 4981 5137 5293 1 5449 156 9 5604 5760 5915 6071 6226 6382 6537 6692 6848 7003 155 280 7158 7313 7468 7623 7778 7933 8088 824'i 8397 8552 155 1 8706 8861 9015 9170 9324 9478 9633 9787 9941 2 450249 0403 1 0557 0711 0865 1018 1172 1326 1479 1633 154 3 1786 1940 2093 2247 2400 2553 2706 2859 3012 3165 153 4 3318 3471 3624 3777 3930 4082 4235 4387 4540 4692 153 5 4845 4997 5150 5302 5454 5606 5758 5910 6062 6214 152 6 6366 6518 6670 6821 6973 7125 7276 7428 757'9 7731 152 7 7882 80:33 8184 8336 8487 8638 8789 8940. 9091 9242 151 g 9392 9543 9694 9845 9995 0146 0296 f\AA" CfXfJ a 1M 9 460898 1048 1198 1348 1499 1649 1799 1948 UOU i 2098 2248 lol 150 290 2398 2548 2697 2847 2997 3146 3296 3445 3594 3744 ISO 1 3893 4042 4191 4340 4490 4639 4788 4936 5085 5234 149 2 5383 5532 5680 5829 5977 6126 6274 6423 6571 6719 149 3 6868 7016 7164 7312 7460 7608 7756 7904 8052 8200 148 4 8347 8495 8643 8790 8938 9085 9233 9380 9527 9675 148 g 9822 9969 0116 0263 0410 0557 0704 0851 0998 1145 147 6 471292 1438" 1585 1732 1878 2025 2171 2318 2464 2610 146 7 2756 2903 3049 3195 3341 3487 3633 3779 3925 4071 146 8 4216 4362 4508 4G53 4799 4944 I 5090 5235 5381 5526 146 9 5671 5816 5962 6107 6252 6397 ; 6542 6687 6832 6976 145 PROPORTIONAL PARTS. Diff. 1 2 3 4 5 6 7 8 9 161 16.1 32.2 48.3 64.4 80.5 96.6 112.7 128.8 144.9 160 1 16.0 32.0 48.0 64 80.0 96.0 112.0 128.0 144.0 159 15.9 31.8 47.7 63.6 79.5 95.4 111.3 127.2 143.1 158 15.8 31.6 . 47.4 63.2 79.0 94.8 110.6 126.4 142.3 157 15.7 31.4 47.1 62.8 78.5 94.2 109.9 125.6 141.3 156 15.6 31.2 46.8 62.4 78.0 93.6 109.2 124.8 140.4 155 15.5 31.0 46.5 62.0 77.5 93.0 108.5 124.0 139.5 154 15.4 30.8 46.2 61.6 77.0 92.4 107.8 123.2 138. G 153 15.3 30.6 45.9 61.2 76.5 91.8 107.1 122.4 137.7 152 15.2 30.4 45.6 60.8 76.0 91.2 106.4 121.6 136.8 151 15.1 30.2 45.3 60.4 75.5 90.6 105.7 120.8 135.9 150 15.0 30.0 45.0 60.0 75.0 90.0 105.0 120.0 135.0 149 14.9 29.8 44.7 59.6 74.5 89.4 104.3 119.2 134.1 148 14.8 29.6 44.4 59.2 74.0 88.8 103.6 118.4 138.8 147 14.7 29.4 44.1 58.8 73.5 88.2 102.9 117.6 132.3 146 14.6 29.2 43.8 58.4 73.0 87.6 102.2 116.8 131.4 145 14.5 29.0 43.5 58.0 72.5 87.0 101.5 116.0 130.5 144 14.4 28.8 43.2 57.6 72.0 86.4 100.8 115.2 129.6 143 14.3 28.6 42.9 57.2 71.5 85.8 100.1 114.4 128.7 142 14.2 28.4 42.6 56.8 71.0 85 2 99.4 113.6 127.8 141 14.1 28.2 42.3 56.4 70.5 84.6 98.7 112.8 126.9 140 14.0 28.0 42.0 56.0 70.0 84.0 98.0 112.0 126.0 TABLE XXIV.-LOQARITHMS OF NUMBERS. No. 300 L. 477.] [No. 339 L. 531. N. 1 2 i 4 5 6 7 8 9 Diff. 145 144 144 143 143 142 142 141 141 140 140 139 139 139 138 188 1S7 137 186 136 136 135 135 134 134 133 183 183 182 182 131 131 131 130 180 29 29 29 28 28 300 1 2 3 4 5 6 8 9 310 1 2 3 4 5 6 7 8 9 320 1 3 4 5 6 8 9 aso 1 a 4 5 6 7 8 9 477121 850(3 7266 8711 7411 8855 7555 8999 7700 9143 7844 9287 7989 9431 8133 ! 957o 8278 8422 9719 9663 480007 1443 2874 4300 5721 7138 8551 9958 0151 1586 3016 4442 5863 7280 8692 0099 1502 2900 4294 5683 7068 8448 9824 0294 1 72*) 3159 4585 6005 7421 8833 0438 1872 3302 4727 6147 7563 8974 0582 2016 3445 4869 6289 7704 9114 0725 2159 3587 5011 6430 7845 9255 0869 2302' j 3730 ! 5153 j 6572 ' 7986 9396 1012 2445 S872 5295 0714 6127 6537 1156 2568 4015 5437 6855 6269 677 1C81 2481 8876 6267 CC53 6C35 S412 0765 2154 8518 4678 284' 7E66 6984 277 1616 2C51 482 ECC9 6C82 8251 9EC6 0676 2183 8486 4785 C81 7372 8t(0 SC43 1299 27,31 4157 E579 6897 8410 9818 1222 2621 4015 5406 6791 8173 C5EO 0922 2291 655 014 CS70 7721 018 0239 1642 3040 4433 5822 7206 8586 9962 0380 1782 3179 4572 5960 7344 8724 "0099" 1470 2837 4199 5557 6911 8260 9606 0520 1922 3319 4711 6099 7483 8862 0236 1607 2973 4335 5693 7046 8395 9740 0661 2062 3458 4850 6238 7621 8999 ~0374 1744 , 3109 4471 5828 ! 7181 ! 8530 9874 1215 2E51 3883 | 5211 6535 7855 9171 0801 2201 3597 4989 6376 7759 9137 0511 1880 3246 4607 5964 7316 8664 0941 2341 3737 5128 6515 78S7 9275 0648 2017 8882 4743 CCG9 7451 6799 0143 1482 2818 4149 5476 68CO 8119 9434 491302 2760 4135 5544 6930 8311 9687 501059 2427 3791 5150 6505 7856 9203 510645 1883 3218 4548 5874 7196 8514 9828 521138 2444 ..3746 5045 8388 7630 8917 1196 2564 3927 5286 6640 7991 9337 13:33 2700 40G3 5421 6776 8126 9471 OOC9 1349 2684 4016 344 6(568 7987 S303 0411 17EO 8C84 4415 5741 7CC4 682 CC97 1CC7 2314 616 4915 (210 7t01 8788 0679 2017 a35i 4681 6006 7328 8646 9959 1269 2575 3876 5174 64(59 7759 9045 0813 2151 3484 4813 6139 7460 8777 0947 2284 3617 4946 6271 7592 8909 1081 2418 3750 5079 6403 7724 9040 0090 1400 2705 4006 5304 6598 7888 9174 0221 1530 2835 4136 5434 6727 8016 9:;02 0333 1661 2966 4266 6EG3 6F56 8145 9430 0464 1792 3096 4396 5693 6985 8274 9559 0615 1922 3226 4526 6822 7114 8402 9687 0745 2053 3356 4626 951 7243 8531 815 C072 1351 530200 0328 1 0456 0584 0712 0840 0968 | IC96 1223 PROPORTIONAL PARTS. Diff. 1 2 J J 4 55.6 55.2 54.8 54.4 54.0 53.6 53.2 52.8 52.4 52.0 51.6 51.2 50.8 5 6 83.4,. 82.8 82.2 81.6 81.0 80.4 79.8 79.2 78.6 70.0 77.4 76.8 76.2 r. 97.3 96.6 95.9 95.2 94.5 93.8 93.1 92.4 91.7 91.0 90.3 89.6 88.9 8 111.2 110.4 109.6 108.8 1C8.0 107.2 TC6.4 105.6 104.8 104.0 103.2 102.4 101.6 9 125.1 124.2 123.3 122.4 121.5 120.6 119.7 118.8 117.9 117.0 116.1 115.2 114.3 139 13.9 138 13.8 137 13.7 136 13.6 135 13.5 134 13.4 133 13.3 132 13.2 131 13.1 130 13.0 129 12.9 128 12.8 127 12 7 27.8 27.6 27.4 27.2 27.0 26.8 26.6 26.4 26.2 26.0 25.8 25.6 25.4 41.7 41.4 41.1 40.8 40.5 40.2 39.9 39.6 39.3 39.0 38.7 38.4 38.1 69.5 69.0 68.5 68.0 67.5 67.0 66.5 66.0 65.5 65.0 64.5 64.0 63.5 !U9. TABLE XXIV. LOGARITHMS OF NUMBERS. No. 340 L. 531.] [No. 379 L. 679. N. 1 2 8 H! 6 6 7 8 9 Diff. 340 1 2 3 4 5 6 7 8 9 350 1 2 3 4 5 6 8 9 360 1 2 3 4 5 6 7 8 9 370 1 2 3 4 5 6 7 8 9 531479 2754 4026 5294 6558 7819 gore 1607 288-i 4153 5421 0085 7945 9202 1734 1802 3009 3136 4280 4407 5547 i 5074 0811 6937 8071 ! 8197 9327 9452 1990 ii 2117 2245 3204 3391 i 3518 4534 4001 4787 5800 5927 0053 7003 7189 7315 8322 ; 8448 8574 9578 9703 9829 2372 3045 4914 61PO 7441 8699 9954 2500 3772 5041 0300 7507 8825 0079 1330 25 Jo 3820 5060 0290 7529 8758 9984 2627 3899 5107 6432 7093 8951 0204 1454 2701 3944 5183 6419 7652 8881 128 127 127 126 120 126 125 125 125 124 124 124 123 123 123 122 122 121 121 121 120 120 120 119 119 119 119 118 118 118 117 117 117 116 116 116 115 115 115 114 540329 1579 2825 4068 5307 6543 7775 9003 0455 1704 2950 4192 5431 6066 7898 9126 0580 1829 3074 4316 5555 6789 8021 9249 0705 1953 3199 4440 5078 0913 8144 9371 0830 0955 2078 2203 3323 3447 4564 1 4688 5802 5925 7036 7159 8267 8389 9494 9016 1080 2327 3571 4812 0049 7282 8512 9739 1205 2452 3096 4936 6172 7405 8035 9801 0106 1328 2547 3762 4973 6182 7387 8589 9787 550228 1450 2668 3883 5094 6303 7507 8709 9907 561101 2293 3481 4066 5848 7026 8202 9374 0351 1572 2790 4004 5215 6423 7627 8829 0473 1694 2911 4126 5336 6544 7748 8948 0595 1816 3033 4247 5457 6664 7868 9068 0717 1938 3155 4308 5578 6785 7988 9188 0385 1578 2769 3955 5139 6320 7497 8671 C842 1010 2174 3336 4494 5050 6802 7951 9097 0840 2000 3276 4489 5699 6905 8108 9308 0962 2181 3398 4610 5820 7026 8228 9428 1084 2303 3519 4731 594U 7146 8349 9548 1206 2425 3640 4852 6001 7267 8469 9607 0026 12^1 2412 3000 4784 5966 7144 8319 9491 0146 1340 2531 3718 4903 0084 7262 8436 9608 0205 1459 2050 3837 5021 6202 7379 8554 9725 0504 1698 2887 4074 5257 6437 7614 8788 9959 0024 1817 3006 4192 5376 6555 7732 8905 0743 1936 3125 4311 5494 0073 7849 9023 0863 2055 3244 4429 5612 6791 7967 9140 0982 2174 3302 4548 5730 6909 80&4 9257 0076 1243 2407 3508 4726 5880 7032 8181 9326 0193 1359 2523 3084 4841 5996 7147 8295 9441 0309 1476 .2639 3800 4957 6111 7262 8410 9555 0420 1592 2755 3915 5072. 6226 7377 8525 9009 570543 1709 2872 4031 5188 6341 7492 8639 0660 1825 2988 4147 5303 6457 7607 8754 0776 1942 3104 4263 5419 6572 7722 8868 0893 2058 3220 4379 5534 6687 7836 8983 1126 2291 ; 3452 | 4610 5765 6917 ! 8006 9212 PROPORTIONAL, PARTS. Diff. 1 234 5 078 9 123 12.8 127 12.7 126 12.6 125 12.5 124 12.4 123 12.3 122 12.2 121 12.1 120 12.0 119 11.9 25.6 38.4 51.2 25.4 38.1 50.8 25.2 37.8 50.4 25.0 37.5 50.0 24.8 37.2 49.6 24.6 36.9 49.2 24.4 36.6 48.8 24.2 36.3 48.4 24.0 36.0 48.0 23.8 35.7 47.6 64.0 63.5 63.0 62.5 62.0 61.5 61.0 60.5 60.0 59.5 76.0 89.6 76.2 88.9 75.6 88.2 75.0 87.5 74.4 86.8 73.8 86.1 73.2 85.4 72.6 84.7 72.0 84.0 71.4 83.3 102.4 101.6 100.8 100.0 99.2 98.4 97.0 96.8 96.0 95.2 115.2 114.3 113.4 112.5 111.6 110.7 109.8 108.9 108.0 107.1 343 TABLE XXIV.-LOGARITHMS OF NUMBERS. No. 380. L. 579.] [No. 414 L. 617. N. 1 2 3 4 5 7 8 9 Diff. 380 579784 9898 I 0128 0355 "0469~ 0583 0697 | 0811 114 0012 0241 1 580925 1039 1153 j 1267 1381 1495 1608 ' 1722 1836 1950 2063 2177 2291 2404 2518 2631 ! 2745 2S58 2972 3085 3 3199 3312 3 426 853 ) 3652 3765 i 3879 i 39 92 4105 4218 4 4331 4444 4 557 467 9 4783 4896 i 5009 51 22 5235 5348 113 5 5461 5574 5686 5799 5912 6024 i 6137 6250 63G2 6475 6 6587 6700 6 312 692 & 7037 7149 7262 73 74 7486 7599 7 7711 7823 7 (35 804 r 8160 1 8272 | 8384 84 96 8608 8720 112 8 8832 8944 9056 9167 9279 i 9391 j 9503 9615 9726 9838 9 99j 0061 0173 0284 0396 0507 0619 0730 0842 0953 30 591065 1176 1287 1399 1510 1621 1732 1843 1955 2066 1 2177 2288 2399 2510 2621 2732 2843 2954 3064 3175 111 2 3286 3397 3 508 361 8 3729 3840 3950 4( 161 4171 4282 3 4393 4503 4614 4724 4834 4945 5055 5165 527'6 5386 4 5495 5606 5 717 582 7 5937 i 6047 6157 65 G7 6S77 6487 5 6597 6707 t 817 692 7 7037. 7146 7256 7 JOG 747'6 7586 110 6 7G'J5 7805 914 8024 8134 8243 8353 8462 8572 8681- , 7 8791 8900 9009 9119 | 9228 9337 9446 ! 9556 9665 9774 8 9883 9992 _ ri . ' ' 109 0101 0210 (319 i 0428 0537 0646 0755 0864 9 ^00973 1082 1191 1299 1408 j 1517 1625 1734 1843 1951 400 2060 2169 2277 238 ft 2494 2603 2711 2819 2928 3036 1 3144 3253 g 361 34(3 9 3577 ! 3686 3794 I Sf to-,' 4010 4118 108 2 4326 4334 4442 4550 4658 4766 4874 4< )S2 5089 5197 3 5305 5413 5 521 562 18 5736 5844 5951 6( r,',) 6166 6274 4 6381 G4H9 6596 6704 1 6811 6919 7026 7133 7241 7348 5 7455 7562 7 669 777 7 i 7884 7991 8098 8; .'05 8312 8419 107 6 8526 8633 8740 8847 i 8954 9061 9167 9274 9381 9488 7 QTI1 r S(~N 991 A 1 f y i ui OUo 0021 0128 0234 0341 0447 0554 8 610660 0767 0873 097 1086 1192 1298 1405 1511 1617 9 1723 1829 1 936 2042 2148 2254 2360 2466 2572 2678 106 410 27'84 2890 2996 3102 3207 3313 3419 3525 3630 3736 1 3842 3947 4 053 41 9 4264 4370 4475 4 Wl 4686 4792 2 4897 5003 I >108 52 3 5319 5424 5529 | 5 134 5740 5845 3 5950 6055 6160 6265 6370 6476 6581 6686 6790 6895 105 4 7000 7105 7210 7315 7420 I! 7525 7629 7734 7839 7943 PROPORTIONAL PARTS. Diff. j 1 2 3 4 5 6 7 8 9 118 11.8 23.6 35.4 47.2 59.0 70.8 82.6 94.4 106.2 117 11.7 23.4 35.1 46.8 58.5 70.2 81.9 93.6 105.3 116 11.6 23.2 34.8 46.4 58.0 69.6 81.2 92.8 104.4 115 11.5 23.0 34.5 46.0 57.5 69.0 80.5- 92.0 103.5 114 11.1 22.8 34.2 45.6 57.0 68.4 79.8 91.2 102.6 113 11.3 22^6 33.9 45.2 56.5 67.8 79.1 90.4 101.7 112 11.2 22.4 33.6 44.8 56.0 67.2 78.4 89.6 100.8 111 11.1 22.2 33.3 44.4 55.5 66.6 77.7 88.8 99.9 110 11.0 22.0 33.0 44.0 55.0 66.0 77.0 88.0 99.0 109 10.9 21.8 32.7 43.6 54.5 65.4 76.3 87.2 98.1 108 10.8 21.6 32.4 43.2 54. 64.8 75.6 86.4 97.2 107 10.7 21.4 32.1 42.8 53.5 64.2 74.9 85.6 96.3 106 10.6 21.2 31.8 42.4 53.0 63.6 74.2 84.8 95.4 105 10.5 21.0 31.5 42.0 52.5 63.0 73.5 84.0 94.5 105 10.5 21.0 31.5 42.0 52.5 63.0 73.5 84.0 94.5 104 10.4 20.8 31.2 41.6 52.0 62.4 72.8 83.2 93.6 344. TABLE XXIV. LOGARITHMS OF NUMBERS. No. 415 L. 618.] [No. 459 L. 662 i 9 Diff. 415 618048 8153 ! 8257 8362 8466 8571 8676 8780 8884 8989 105 6 9093 9198 i 9302 9406 9511 9615 9719 9824 9928 7 C20136 0240 : 0344 0448 0552 0656 0760 0864 ! 0968 1072 104 8 1176 1280 i 1384 1488 1592 1695 1799 1903 2C07 2110 9 2214 2318 2421 2525 2628 2732 2835 2939 3042 3146 420 3249 3353 3456 3559 3663 3766 3869 3973 4076 4179 1 4282 4385 4488 4591 4695 4798 4901 5004 5107 5210 103 2 5312 5415 5518 5621 5724 5827 5929 -6032 6135 6238 3 6340 6443 6546 6648 6751 6853 6956 70E8 7161 72(i3 4 7366 7468 7571 7673 7775 7878 7980 8082 8185 8287 5 8389 8491 8593 8695 8797 8900 9002 9104 C206 9308 102 0410 9512 9613 9715 9817 9919 *j*i\j sinoi n-lOO noo/4 ! /v-joA 7 630428 0530 0631 0733 0835 1 0926 VUV.1 10b8 Ul^tJ 1139 \j&vt Ud^O 1241 i 1342 8 1444 1545 1647 1748 1849 1951 2052 2153 : 2255 2356 9 2457 2559 2660 2761 2862 2963 8064 3165 3266 3367 430 3468 3569 3670 3771 3872 3973 4074 4175 4276 4376 101 1 4477 4578 4679 4779 4880 i 4981 i 5081 5182 5^83 5S83 2 5484 5584 5685 5785 5886 5986 i 6087 6187 6287 6388 i r 3 6488 6588 6688 6789 6889 6989 7089 718Q 7290 7390 4 7490 7590 7690 7790 7890 7990 8090 8190 8290 8389 inn 5 8489 8589 8689 8789 8888 8988 9088 9188 9287 9387 -~ 6 9486 9586 9686 9785 9885 9984 0084 0183 0283 0382 7 640481 0581 0680 0779 0879 0978 1077 1177 1276 1375 8 1474 1573 1672 1771 1871 1970 2069 2168 2267 2366 9 2465 2563 2662 2761 2b60 2959 8058 3156 3255 3354 99 440 3453 3551 3650 3749 3847 3946 4044 4143 4242 4340 1 4439 4537 4636 4734 4832 4931 5029 5127 5226 5324 2 5422 5521 5619 5717 5815 5913 6011 6110 6208 6306 3 6404 6502 6600 6698 6796 6894 6992 7089 7187 7285 98 4 7383 7481 7579 7676 7774 7872 7969 8067 8165 8262 5 8360 8458 8555 8653 8750 8848 8945 9043 9140 9237 6 9335 9432 9530 9627 9724 9821 9919 0016 0113 0210 rr 650308 0405 0502 0599 0696 0793 0890 0987 1084 1181 8 1278 1375 1472 1569 1666 1762 1859 1956 2053 2150 87 9 2246 2343 2440 2536 2633 2730 2826 2923 8019 3116 450 3213 3309 3405 3502 3598 | 3695 3791 3888 3984 ! 4080 1 4177 4273 4369 4465 4562 4658 4754 4850 : 4946 j 5042 2 5138 5235 5331 5427 5523 ; 5619 j 5715 5810 ! 5906 6002 96 3 6098 6194 6290 6482 6577 6673 6769 6864 6960 4 7056 7152 7247 7343 7438 7534 7629 7725 i 7820 1 7916 5 8011 8107 8202 8488 8584 8679 8774 J 8870 6 8965 9060 9155 9250 9346 9441 9536 9631 j 9726 9821 . " 16 0011 0106 0201 0296 1 0391 0486 0581 0676 0771 95 8 660865 0960 1055 1150 1245 1339 1434 1529 1623 1718 9 1813 1907 2002 2096 2191 2286 2380 2475 2569 2663 , PROPORTIONAL PARTS. Diff . 1 234 5 678 9 105 10.5 21.0 31.5 42.0 52.5 63.0 73.5 84.0 94.5 104 10.4 20.8 31.2 41.6 52.0 ! 62.4 72 8 83.2 93.6 103 10.3 20.6 30.9 41.2 51.5 61.8 721 82.4 92.7 102 10.2 20.4 30.6 40.8 51.0 61.2 714 81.6 91.8 101 10.1 20.2 30.3 40.4 50.5 60.6 70 7 80.8 90.9 100 10.0 20.0 30.0 40.0 50.0 60.0 70 80.0 90.0 99 9.9 19.8 29.7 39.6 49.5 59.4 69.3 79.2 89.1 i 34.0 TABLE XXIV. LOGARITHMS OF NUMBERS. No. 460 L. 662.] [No. 499 L. 698. T)ifF . _ , Ulll. 460 662758 2852 2947 3041 31-35 3230 3324 3418 3512 3607 1 3701 3795 3889 3983 4078 4172 4266 4360 4454 4548 2 4642 4736 4830 4924 5018 5112 5206 5299 5393 5487 94 3 5581 5675 5769 5862 5956 6050 6143 6237 6331 6424 4 i 6518 6612 6705 8799 6892 6986 7079 7173 7266 7360 5 7453 7546 7640 7733 7826 7920 8013 8106 8199 8293 6 8386 \ 8479 8572 8665 8759 8852 8945 9038 9131 9224 9317 9410 9503 9596 9689 9782 9875 9967 ' 0060 0153 93 8 670246 0339 0431 0524 i 0617 0710 0802 0895 0988 1080 9 1173 1265 1358 1451 1543 1636 1728 1821 1913 2005 470 2098 2190 2283 2375 2467 2560 2652 2744 2836 2929 1 3021 3113 3205 3297 3390 3482 3574 3666 3758 3850 2 3942 4034 4126 4218 4310 4402 4494 4586 4677 4769 92 3 4861 4953 5045 5137 5228 5320 5412 5503 5595 5687 4 5778 5870 5962 6053 6145 , 6236 6328 6419 6511 6602 5 6694 6785 6876 6968 7059 i 7151 7242 7333 7424 7516 6 7607 7698 7789 7881 7972 8063 8154 8245 8336 8427 7 8518 8609 8700 8791 8882 ! 8973 9064 9155 9246 9337 91 8 9428 9519 9610 9700 9791 9882 9973 OOfi'3 0154 024^ 9 i 680336 0426 0517 0607 0698 0789 0879 lAJuo 0970 1060 U-*-O 1151 480 1241 1332 1422 1513 1603 1693 1784 1874 1964 2055 1 i 2145 2235 2326 2416 2506 2596 2686 2777 2867 2957 2 3047 3137 3227 3317 3407 3497 3587 3677 3767 3857 90 3 3947 4037 4127 4217 4307 4396 4486 4576 4666 4756 4 4845 4935 5025 5114 5204 5294 5383 5473 5563 5652 5 5742 5831 5921 6010 6100 6189 6279 6368 6458 6547 6 ! 6636 6726 6815 6904 6994 7083 7172 7261 7351 7440 7 I 7529 7G18 7707 7796 7'886 7975 8064 8153 8242 8331 cq 8 8420 8oO'J 8598 8687 8776 ; 8865 8953 9042 9131 9220 OJ 9 9309 9398 9486 i 9575 9664 QTftS 9841 9930 0010 0107 490 690196 0285 0373 0462 0550 0639 0728 0816 UUI J 0905 UlUrf 0993 1 1081 1170 1258 1347 14&5 I 1524 1612 1700 1789 1817 2 1905 2053 2142 2230 2318 j 2406 2494 2583 2671 2759 3 2847 2935 3023 3111 3199 ! 3287 &375 3463 8551 3639 88 4 3727 3815 3903 3991 4078 4166 4254 4342 4430 4517 5 4605 4693 4781 4868 4956 5044 5131 5219 5307 5394 6 5482 5569 5657 5744 5832 5919 6007 6094 6182 6269 7 6356 6444 6531 6618 6706 6793 6880 6968 7055 7142 8 7229 7317 7404 7491 7578 7665 7752 7839 7926 8014 9 8100 8188 8275 8362 8449 1 8535 8622 8709 8796 8883 87 II TROPORTIOXAL PARTS. Diff. 1 2 3 4 5 6 7 . 8 9 98 9.8 19.6 29.4 39.2 49.0 58.8 68.6 78.4 88.2 97 9.7 19.4 29.1 38.8 48.5 58.2 67.9 77.6 87 3 96 9.6 19.2 28.8 38.4 48.0 57.6 . 67.2 76.8 86.4 95 9.5 19.0 28.5 38.0 47.5 57.0 66.5 76.0 85 5 94 9.4 18.8 23.2 37.6 47.0 56.4 65.8 75.2 84.6 ,93 9.3 18.6 27.9 37.2 46.5 55.8 65.1 74.4 83.7 92 9.2 18.4 27.6 36.8 46.0 55.2 64.4 73.6 82.8 91 9.1 18.2 27.3 36.4 45.5 54.6 63.7 72.8 81 9 90 9.0 18.0 27.0 36.0 45.0 54.0 63.0 72.0 81.0 89 8.9 17.8 26.7 a<5.6 44.5 53.4 62.3 71.2 80 1 88 8.8 17.6 26.4 35.2 44.0 52.8 61.6 70.4 79 2 87 8.7 17.4 26.1 34.'8 43.5 52.2 -60.9 69.0 7:3 86 8.6 17.2 25.8 34.4 .43.0 51.6 60.2 68.8 77.4 TABLE XXIV.-LOGARITHMS OF NUMBERS. No. 500 L. 698.1 [No. 544 L. 736. N. 1 2 8 4 5 6 7 8 9 Diff. 500 698970 9057 9144 9231 9317 9404 9491 9578 9664 9751 -j 9838 9924 0011 0098 0184 0271 0358 0444 0531 i 0617 2 "TOOToT 0790 0877 0963 1050 1136 1222 1309 1395 1 1482 3 1568 1654 1741 1827 1913 1999 2086 2172 2258 i 2344 4 2431 2517 2603 2689 2775 2861 2947 3033 3119 3205 5 3291 3377 3463 3549 3635 3721 3807 3893 3979 4065 | 86 C 4151 4236 i 4322 4408 4494 4579 4665 4751 4837 4922 7 5008 5094 i 5179 5265 5350 5436 5522 5607 5693 5778 8 5864 5949 6035 6120 6206 i 6291 6376 6462 6547 6632 9 6718 6803 6888 6974 7059 i 7144 7'229 7315 7400 7485 510 7570 7655 7740 7826 7911 7996 8081 8166 8251 8336 QS; 1 8421 8506 8591 8676 8761 8846 8931 9015 9100 9185 OO 2 9270 9355 9440 9524 9609 9694 9779 9863 9948 nnQQ 3 710117 0202 0287 0371 0456 0540 0625 0710 0794 0879 4 0963 1048 1432 1217 1301 1385 1470 1554 1639 1723 5 1807 1892 1976 2060 2144 2229 2313 2397 2481 2566 6 2650 2734 2818 2902 2986 3070 3154 3238 3323 3407 Rl 7 3491 3575 3659 3742 3826 3910 3994 4078 4162 4246 04 8 4330 4414 4497 4581 4665 4749 4833 4916- 5000 508-1 9 5167 5251 5335 5418 5502 5586 5669 5753 5836 5920 520 6003 6087 6170 6254 6337 6421 6504 6588 6671 6754 1 6838 6921 7004 7088 7171 7254 7338 7421 7504 7587 2 7671 7754 7837 7920 8003 8086 8169 8253 8336 8419 oo 3 8502 8585 8668 8751 8834 . 8917 9000 9083 9165 9248 oo 4 9331 9414 9497 9580 9663 i 9745 9828 9911 9994 5 720159 0242 0325 0407 0490 i 0573 0655 0738 0821 0903 6 0986 1068 1151 1233 1316 ! 1398 1481 1563 1646 1728 7 1811 1893 1975 2058 2140 2222 2305 2387 2469 2552 8 2634 2716 2798 2881 2963 3045 3127 3209 3291 3374 9 3456 8538 3620 3702 3784 3866 3948 4030 4112 4194 82 530 4276 4358 4440 4522 4604 ' 4685 4767 4849 4931 5013 1 5095 5176 5258 5340 5422 5503 5585 5667 5748 5830 2 5912 5993 6075 6156 6238 6320 | 6401 6483 (55(54 6646 3 6727 6809 ! 6890 6972 7053 7134 i 7216 7297 7379 7460 4 7541 7623 7704 7785 7866 7948 8029 8110 8191 8273 5 8:354 8435 8516 8597 8678 8759 8841 8922 9003 9084 6 9165 J9<4 9246 9327 9408 1489 9570 9651 9732 9813 9893 81 0055 m cm 0217 0298 0378 0459 0540 0621 0702 8 730782 0863 0944 1024 1105 I 1186 1266 1347 1428 1508 9 1589 1669 1750 1830 1911 i 1991 2072 2152 2233 2313 540 2394 2474 2555 2635 2715 ' 2796 2876 2956 3037 3117 1 3197 3278 3358 3438 3518 i 3598 3679 3759 3839 3919 2 3999 4079 4160 4240 4320 ; 4400 4480 4560 4640 4720 Of) 3 4800 4880 4960 5040 5120 5279 5359 5439 5519 OU 4 5599 5679 5759 5838 5918 5998 6078 6157 6237 6317 PROPORTIONAL PARTS. Diff. 1 234 5 678 9 87 8.7 17.4 26.1 34.8 43.5 52.2 60.9 69.6 78.3 86 8.6 17.2 25.8 34.4 43.0 51.6 60.2 68.8 77.4 85 8.5 17.0 25.5 34.0 42.5 51.0 59.5 68.0 76.5 84 8.4 16.8 25.2 33.6 42.0 50.4 58.8 67.2 75.6 347 TABLE XXIV. LOGARITHMS OF NUMBERS. No. 545 L. 736.] [No. 584 L. 767. 1C1 . A v uiu. i 545 736397 6476 6556 6635 6715 6795 6874 6954 7034 7113 6 7193 7272 7352 743 1 I 7511 i! 7'590 7070 7' 40 7829 1 7908 7 7987 8067 j 8146 8225 i 8305 || 8384 8463 8543 8022 ' 8701 8 8781 8869 8939 901 8 9097 : 9177 9256 i )35 9414 | 9493 9 9572 9651 9731 961 9889 i GOKS 0047 o >r. 0205 >&/i 7Q 550 740363 0442 0521 0600 0678- 0757 0836 0915 0994 1073 19 1 1152 1230 1 1309 1388 1467 1546 1624 1703 1782 1860 2 1939 2018 2096 217 5 2254 2332 2411 & I8!t 2568 2647 3 2725 2804 2882 290 1 3039 3118 3196 S& 375 3353 3431 4 3510 3588 ; 3667 3745 i 3F23 3902 3980 4058 4136 4215 5 4293 4371 4449 452 8 4606 4684 4762 4 vll) 4919 4997 6 5075 5153 5231 5309 5387 5465 5543 5621 5699 5777 78 7 5855 5933 i 6011 608 6167 6245 6323 * 101 6479 6556 8 6034 0712 6790 68C 8 6945 7'023 7101 7!) 7256 7334 9 7412 7489 7567 7645 7722 7800 | 7878 7955 8033 8110 560 8188 8266 8343 8421 8498 8576 8653 8731 8808 8885 1 8963 9040 9118 9195 9272 9350 9427 9504 9582 9659 2 9730 9814 9891 996 a 0045 ' 0123 0200 0277 ! 0354 0431 3 i 750508 0586 0663 07'. 0817 C894 0971 1 )48 1125 1202 4 I 1279 1356 1433 151 1587 1604 1741 1 318 1895 1972 5 2048 2125 2202 2279 2356 2433 2509 2 586 2063 2740 77 6 2816 2893 2970 3& 7 3123 3200 | 3277 ft 353 3430 3506 7 3583 3660 3736 381 3 3889 3966 i 4042 4 119 i 4195 427'2 8 4348 4425 4501 4578 4654 4730 i 4807 4883 1 4960 50:^6 9 5112 5189 5265 5341 5417 5494 5570 5646 5722 5799 570 5875 5951 6027 6103 6180 6256 6332 6408 6484 6560 1 6636 6712 6788 ese 4 6940 7016 7092 7 HIM 7244 73SO 76 2 7396 7472 7548 7624 7700 7775 7851 7927 8003 8079 3 8155 8230 8306 mi 8 8458 8533 8609 8 186 8761 8836 4 8912 8988 9063 9U 9 9214 9290 9366 !) 441 9517 9592 5 9008 9743 9819 '.IS', 4 9970 0045 m 01 Q Kir. no TO 0347 6 760422 0498 0573 0649 0724 0799 0875 0950 1025 1101 7 1176 1251 1326 1402 1477 i 1552 1"627 1 102 1778 1853 8 1928 2003 2078 8M :>> 2228 2303 2378 9 453 2529 2604 9 2679 2754 2829 2904 2978 3053 3128 3 m 3278 3353 75 580 3428 3503 3578 3653 3727 | 3802 3877 3 m 4027 4101 1 4176 4251 4326 4400 4475 4550 4624 4699 4774 4848 2 4923 4998 i 5072 51^ 17 5221 | 5296 5370 5 445 520 5594 3 5069 5743 5818 58! 5966 i 6041 6115 6190 6264 C338 4 6413 6487 6562 GO: (i 6710 ; 6785 6859 1 6 m 7007 7082 PROPORTIONAL PARTS. Diff. 1 2 3 4 5 6 7 8 9 &3 8.3 16.6 24.9 33.2 41.5 49.8 58.1 66.4 74.7 82 8.2 16.4 24.6 32.8 41.0 49.2 57.4 65.6 73.8 81 8.1 16.2 24.3 32.4 40.5 48.6 56.7 64.8 72.9 80 8.0 16.0 24.0 32.0 40.0 48.0 56.0 64.0 72.0 79 7.9 15.8 23.7 31.6 39.5 47.4 55.3 63.2 71.1 78 ".8 15.6 23.4 31.2 39.0 46.8 54.6 62.4 70.2 77 ".1 15.4 23.1 30.8 38.5 46.2 53.9 61.6 69.3 76 ".6 15.2 22.8 30.4 38.0 45.6 53.2 GO. 8 68.4 75 7.5 15.0 22.5 30.0 37.5 45.0 52.5 60.0 67.5 74 | ".4 14.8 22.2 29.6 37.0 44.4 51.8 59.2 66.6 348 TABLE XXIV. -LOGARITHMS OF NUMBERS. No. 585 L. 767.] [No. 629 L. 799. N. 1 2 3 4 | 5 i . 7 89 Diff. 685 ~76715G 7230 7304 i 7379 7453 7527 7601 7675 7749 7823 i 6 7898 7972 8046 i 81 <50 8194 8268 8342 8416 8490 ! 8564 74 7 8638 8712 1 8786 1 8860 j 8934 9008 9082 9156 9230 9303 8 9377 9451 9525 95 99 9673 1 974B 9820 9894 0068 j Jt/UO - 9 770115 0189 0263 0336 0410 j 0484 0557 0631 ; ! 0042 0705 0778 590 0852 0926 0999 1073 1146 1520 1293 1367 1440 ! 1514 1 1587 1661 1734 18 )8 1881 1955 2028 2102 2175 2248 2 2322 2395 2468 25- 12 2615 2688 2762 2835 : 2908 2981 i 3 3055 3128 3201 3274 3348 i 3421 3494 3567 3640 3713 4 3786 3860 3933 4006 4079 ! 4152 4225 4298 4371 | 4444 73 5 4517 4590 4663 47 36 i 4809 4882 4955 5028 5100 5173 6 5246 5319 5392 5465 j 5538 5610 5683 5756 5829 ! 5902 7 5974 6047 6120 bl )3 i 6265 6338 6411 6483 6556 0629 8 6701 6774 6846 69 19 6992 ! 7'064 7137 7209 7282 i 7354 9 7427 7499 7572 7644 7717 | 7789 7862 793-1 8006 I 8079 600 8151 8224 8296 83 58 8441 1 8513 8585 8658 8730 8802 1 8874 8947 9019 90 )1 9163 9236 9308 9380 9452 9524 2 9596 9669 9741 U8 & 9885 i 9957 fiAOO 3 780317 0389 0461 0533 0605 ; 0677 UU/vU 0749 0821 01*3 0893 0965 72 4 1037 1109 1181 12, S8 1324 1396 1468 1540 1612 1684 5 1755 1827 1899 1971 2042 2114 2186 2258 2329 2401 6 2473 2544 2616 26* >s 8759 2831 2902 2974 3046 3117 7 3189 3260 3332 3403 3475 3546 3618 3689 3761 3832 8 3904 3975 4046 41 8 4189 4261 4332 4403 4475 4546 9 4617 4689 4760 4831 4902 4974 5045 5116 5187 5259 610 5330 5401 5472 5543 5615 5686 5757 5828 5899 5970 1 6041 6112 6183 6 )4 ! 6325 t 6396 6467 6538 6609 6680 ! 71 2 6751 6822 6893 m J4 7035 I 7106 7177 7248 7319 : 7390 3 7460 7531 7602 7673 7744 i 7815 7885 7956 8027 80! )8 4 8168 8239 8310 Jl 8451 8522 8593 8663 8734 8804 5 8875 8946 9016 9087 9157 9228 9299 9369 9440 9510 6 9581 9651 9722 971 \2 9863 9933 AAA i 1 /vv* t [ t 7 790285 0356 0426 0496 0567 0637 UUt>4 0707 0778 0144 0848 11O 0918 8 0988 1059 1129 1199 1269 i 1340 1410 1480 l.")0 1620 ! 9 1691 1761 1831 1901 1971 1 2041 2111 2181 2252 2322 ; 620 2392 2462 2532 2602 2672 2742 2812 2882 2952 3022 : 70 1 3092 3162 3231 33C )1 3371 ! 3441 3511 3581 3651 3721 i 2 3790 3860 3930 4000 i 4070 |] 4139 4209 4279 4349 4418 \ 3 4488 4558 I 4627 461 7 4767 \\ 4836 4906 4976 5045 i 5115 4 5185 5254 5324 5393 5463 || 5532 5602 5672 5741 5811 i 5 5880 5949 6019 60 8 6158 6227 6297 (5366 6436 6505 6 6574 6644 6713 67> 2 6852 !i 6921 6990 7060 7129 7198 7 7268 7337 7406 747 5 7545 i: 7614 7683 7752 7821 7890 8 7960 8029 8098 8167 8236 8305 8374 8443 8513 8582 9 8651 8720 8789 885 6 8927 ! 8996 i 9065 9134 9203 9272 69 PROPORTIONAL PARTS. Diff 1 2 3 4 5 678 9 75 7.5 15.0 22 5 30.0 37.5 45.0 52.5 60.0 67.5 74 7.4 14.8 22 2 29.6 37.0 44.4 51 .8 59.2 66.6 73 7.3 14.6 21 9 29.2 36.5 43.8 51 .1 58.4 65.7 72 7.2 14.4 21 6 28.8 36.0 43.2 50.4 57.6 64.8 71 7.1 14.2 21 3 28.4 &5.5 42.6 49 .7 56.8 63.9 70 7.0 14.0 21 28.0 35.0 42.0 49 .0 56.0 63.0 69 6.9 13.8 20 7 27.6 34.5 41.4 48 .3 55.2 62.1 349 TABLE XXIV. LOGARITHMS OF NUMBERS. No. 630 L. 799.] [No. 674 L. 829. j 8 a T\Z-CP . J. SB \ \ i/m. 630 799341 9409 9478 9547 9616 9685 9754 9823 9892 9961 1 800029 0098 0167 0236 0305 0373 0442 0511 0580 0648 ! 2 j 0717 0786 0854 0923 0992 1061 1129 1198 | 1266 1335 3 i 1404 1472 1541 1609 1678 1747 1815 1884 1952 2021 ! 4 1 2089 2158 2226 2295 i 2363 2432 2500 2568 ! 2637 2705 5 2774 2842 2910 2979 3047 3116 3184 3252 I 3321 3389 6 i 3457 3525 3594 3662 ! 3730 3798 3867 3935 4003 4071 7 4139 4208 4276 4344 4412 4480 4548 4616* 4685 4753 8 4821 4889 4957 5025 5093 5161 5229 5297 5365 5433 68 9 5501 5569 5637 5705 5773 5841 5908 5976 6044 6112 640 j 806180 6248 6316 6384 6451 6519 6587 6655 6723 6790 1 6858 6926 6994 7061 7129 |i 7197 7264 7332 7400 7467 2 7535 7603 7670 i 7738 7806 i| 7873 7941 8008 8076 8143 3 ! 8211 8279 8346 I 8414 8481 8549 8616 8684 8751 8818 4 8886 8953 9021 9088 9156 9223 9290 9358 9425 9492 9560 9627 9694 ("*'(!> 9829 9896 9964 d 0031 0098 n-jcK 6 810938 0300 0367 0434 0501 0569 0636 0703 0770 0837 7 0904 0971 1039 1106 1173 1240 1307 1374 1441 1508 67 8 1575 1642 1709 1776 1843 1910 1977 2044 2111 2178 9 2245 2312 2379 2445 2512 2579 2646 2713 2780 2847 650 2913 2980 3047 3114 3181 3247 3314 3381 3448 3514 1 3581 3648 3714 3781 3848 3914 3981 4048 4114 4181 2 4248 4314 4381 4447 4514 4581 4647 4714 4780 4847 3 4913 4980 5046 5113 5179 5246 5312 5378 5445 5511 4 *5578 5644 5711 5777 5843 5910 5976 6042 i 6109 6175 5 6241 6308 6374 6440 6506 6573 6639 6705 6771 6838 6904 6970 7036 7102 7169 7235 7:301 7367 7433 7499 7 7565 7631 7698 7764 7830 7896 7962 8028 8094 8160 D 8226 8292 8358 8424 8490 8556 8622 8688 8754 8820 9 8885 8951 9017 9083 9149 9215 9281 9346 9412 9478 66 660 9544 9610 9676 9741 9807 9873 9939 fl070 niqc 1 820201 0267 0333 I 0399 0464 0530 0595 0661 UUiU 0727 UloO 0792 2 0858 0924 0989 1055 1120 1186 1251 1317 1382 1448 3 1514 1579 1645 1710 1775 1841 1906 1972 2037 2103 4 2168 2233 2299 2364 2430 2495 2560 2626 2691 2756 5 2822 2887 2952 3018 3083 3148 3213 3279 3344 3409 6 3474 3539 3605 3670 3735 3800 3865 3930 3996 4061 7 4126 4191 4256 4321 4386 4451 4516 4581 4(546 4711 >K 8 4776 4841 4906 4971 5036 5101 5166 5231 5296 5361 DO 9 5426 5491 5556 5621 5686 5751 5815 5880 5945 6010 670 6075 6140 6204 6269 6334 6399 6464* 6528 6593 6658 1 6723 6787 6852 6917 6981 7046 7111 7175 7240 7305 2 7369 7434 7499 7563 7628 7692 7757 7821 7886 7951 3 8015 8080 8144 8209 8273 8338 8402 8467 8531 8595 4 8660 8724 8789 8853 8918 8982 9046 9111 9175 9239 PROPORTIONAL PARTS. Diff. 1 2-34 5 678 9 68 6.8 13.6 1 20.4 27.2 34.0 40.8 47.6 54.4 61.2 67 6.7 13.4 1 20.1 26.8 33.5 40.2 46.9 53.6 60.3 66 6.6 13.2 19.8 26.4 33.0 39.6 46.2 52.8 59.4 65 6.5 13.0 19.5 26.0 32.5 39.0 45.5 52.0 58.5 64 6.4 IS. 8 19.2 25.6 32.0 38.4 44.8 51.2 57.6 350 TABLE XXIV. LOGARITHMS OF NUMBERS. No. 675 L. 829.] [No. 719 L. 857. N. 1 2 8 9 ! Diff. I 675 829304 9368 9432 9497 9561 ! ! 9625 9690 9754 9818 9882 6 9947 0011 i 0075 Olu *o i ncvM I noco nQQo i rwo 7 830589 0053 0717 0781 0845 I 0909 0973 i 1037 1102 1106 8 1230 1294 1358 ! 14S J2 1480 1550 1014 1078 1742 IfcOG i 64 9 1870 1934 1998 99 >2 2120 ' 2189 2253 2317 2381 2445 680 2509 2573 2637 2700 2764 2828 2892 2956 3020 S083 i 1 3147 3211 3275 3338 3402 || 3400 3530 3593 3657 3721 i 2 3784 3848 3912 39" "5 4039 4103 41C6 4230 4294 4357 i 3 4421 4484 4548 4011 4075 I 4739 4802 4HGO 4929 4993 j 4 5056 5120 5183 FrtjX, 17 5310 5373 5437 ! 5500 5504 5027 5 5691 5754 5817 5881 5944 6007 0071 0134 0197 0201 6 6324 6387 6451 65] 4 6577 6641 6704 ! 6767 0830 0894 7 6957 7020 7'083 71^ 10 7210 i 7273 7336 | 7399 7402 7525 8 7588 7652 7715 7778 7841 7904 7'967 8030 8093 8156 9 8219 8282 8345 8408 8471 8534 8597 8000 8723 8786 63 690 8849 8912 8975 90; B 9101 9164 9227 9289 9352 9415 1 9478 9541 9004 9007 9729 j 9792 9855 9918 9981 0043 2 3 840106 0733 0169 0790 0232 0859 0294 0921 0357 0420 0984 1046 0482 0545 1109 1172 0008 1234 0671 1297 4 1359 1422 1485 1547 1010 1672 1735 1797 1800 1922 5 1985 2047 2110 21' a 2235 2297 2200 2422 2484 2547 6 2609 2072 2734 2796 2859 2921 2983 3046 3108 8170 7 3233 3295 3357 34i .'(.) 3482 3544 3006 3009 3731 3793 8 3855 3918 3980 4042 4104 4100 4229 4291 4353 4415* 9 4477 4539 4001 4604 4726 4788 4850 4912 4974 5036 700 5098 51GO 5222 5284 5346 5408 5470 5532 5594 5656 62 1 5718 5780 5842 59( 14 5966 6028 0090 0151 0213 0275 2 6337 0399 0401 6523 6585 6046 6708 0770 0832 6894 3 6955 7017 7079 71, 1 7202 ' 7264 7326 7388 7449 7511 4 7573 7634 7096 7758 -7819 7881 7943 8004 80GG 8128 5 8189 8251 8312 83" 4 8435 , 8497 8559 8020 8082 8743 6 8805 8800 8928 Kfr B 9051 9112 9174 9235 9297 9358 r 9419 9481 9542 9004 9665 9726 9788 9849 9911 9972 8 850033 0095 0156 0217 0279 i 0340 0401 0402 0524 0585 9 0046 0707 0709 0830 0891 | 0952 1014 1075 1130 1197 710 1258 132.0 1381 1442 1503 j 1564 1625 1686 1747 1809 1 1870 1931 1992 SOt a 2114 2175 2236 2297 2358 2419 2 2480 2541 2602 2( .3 27'24 i 2785 2846 2907 25)08 3029 61 3 3090 3150 3211 27 2 3333 3394 3455 3516 3577 2037 4 3698 3759 3820 38 1 3941 4002 4063 4124 4185 4245 5 4306 4307 44-28 4488 4549 i 4610 4670 4731 4792 4858 6 4913 4974 5034 50S 5 5150 5216 5277 5337 5398 5459 7 5519 5580 5640 5701 5761 5822 5882 5943 0003 6064 8 6124 6185 6245 63C G 6366 : 6427 6487 6548 OG08 CG68 9 6729 6789 6850 6910 6970 i 7031 7091 7152 7212 7272 PROPORTIONAL PARTS. Diff. 1 2 3 4 5 6 . r 8 9 65 6.5 13.0 19.5 2G.O 32.5 .39.0 45.5 52.0 58.5 64 6.4 12.8 19.2 25. G 32.0 38.4 44 .8 51.2 57.6 63 6.3 12.6 18.9 25.2 31.5 i 37.8 44 .1 50.4 56.7 62 6.2 12.4 18.6 24.8 31.0 37.2 43.4 . 49.6 55.8 61 6.1 12.2 18.3 24.4 30.5 36.6 42 .7 48.8 54.9 60 6.0 12.0 18.0 24.0 30.0 36.0 42.0 48.0 54.0 351 TABLE XXIV. LOGARITHMS OF NUMBERS. No. 720 L. 857.] [No. 764 L. 883. N. 1 2 3 4 6 6 7 8 9 Diff. 720 857332 7393 7453 7513 7574 7634 7694 7755 7815 7875 1 7935 7995 8056 8116 8170 8236 8297 8357 8417 8477 8537 8597 8s557 8718 8778 8838 8898 8958 9018 9078 3 9138 9198 I 9258 9318 9379 \ 9439 9499 9559 9619 9679 60 A 9739 97'99 9859 9918 9978 0038 0098 01^8 0218 0278 5 860338 0398 0458 0518 057d 0637 0697 0757 0817 0877 6 0937 0996 1056 1116 1170 123*5 1295 1355 1415 1475 7 1534 1594 1654 1714 1773 1833 1893 1952 2012 2072 8 2131 2191 2251 2310 2370 2430 2489 2549 2608 2668 9 2728 2787 2847 2906 2966 3025 3085 3114 3204 3263 730 3323 3382 3442 3501 3561 3620 3680 3739 3799 3858 1 3917 3977 4036 4096 4155 4214 4274 4333 4392 4452 2 4511 4570 4630 4689 4748 4808 4867 4926 4985 5045 3 5104 5163 5222 5282 5341 5400 5459 5519 5578 5637 4 5690 5755 5814 5874 5933 5992 6051 6110 6169 6228 5 6287 6346 6405 6465 6524 6583 6642 67'01 6760 6819 6 6878 6937 6996 7055 7114 7173 7232 7291 7350 7409 59 7 7467 7526 7585 7644 7703 7762 7821 7880 7939 7998 8 ; 8056 8115 8174 8233 8292 8350 8409 8468 8527 8586 9 i 8044 8703 8762 8821 8879 8938 8997 9056 9114 91,3 740 9232 9290 9349 9408 9466 9525 9584 9642 9701 9760 1 9818 9877 9935 9994 0053 0111 0170 ftoos noc"? 0345 2 870404 0462 0521 0579 06:38 0696 0755 0813 0872 0930 3 0989 1047 1106 1164 1223 1281 1339 1398 1456 1515 4 i 1573 1631 1690 1748 1806 1865 1923 1981 2040 2098 5 2150 2215 2273 2331 2389 2448 250(5 2564 2622 2681 6 2739 2797 2855 2913 2972 3030 3088 3146 3204 3262 7 3321 3379 3437 3495 3553 3611 3669 3727 3785 3844 8 1 3902 3969 4018 4076 4134 4192 4250 4308 4366 442-4 58 9 4482 4540 4598 4656 4714 4772 4830 4888 4945 5003 750 5061 5119 5177 5235 5293 5351 5409 5466 5524 5582 1 5640 5698 57'56 5813 5871 5929 5987 6045 6102 6160 2 6218 0270 6333 6391 6449 6507 6564 6622 6680 6737 3 6795 6853 6910 6968 7026 7083 7141 7199 7256 7314 4 7371 7429 7487 7544 7602 7659 7717 7774 7832 7889 5 7947 8004 8062 8119 8177 82:34 8292 8349 8407 8464 6 8522 8579 8037 8094 8752 8809 8866 8924 8981 9039 7 9098 9153 9211 9208 9325 9383 9440 9497 9555 9612 g 9669 9726 9784 9841 9898 9956 0013 0070 , . 9 8802-42 0299 0356 0413 0471 0528 0585 0642 0699 0756 760 0814 0871 0928 0985 1042 1099 1156 1213 1271 1328 1 1385 1442 1499 1556 1613 1670 1727 1784 1841 1898 2 1955 2012 2069 2126 2183 2240 2297 2354 2411 2468 57 3 2525 2581 2638 2695 2752 2809 2866 2923 2980 3037 4 3093 3150 32C7 3264 3321 3377 34:34 3491 3548 3605 PROPORTIONAL PARTS. Diff 1 2 3 4 5 678 9 59 5.9 11.8 17.7 23.6 29.5 35.4 41.3 47.2 53.1 58 5.8 11.6 17.4 23.2 29.0 34.8 40.6 46.4 52.2 57 5.7 11.4 17.1 22.8 28.5 34.2 39.9 45.6 51 3 56 5.6 11.2 16.8 22.4 28.0 33.6 39.2 44.8 50.4 353 TABLE XXIV. LOGARITHMS OF NUMBERS. No. 765 L. 883.] [No. 809 L. 908. N. 1 2 Lfi 1 ! _ ! 6 6 7 8 9 Diff. 765 883661 3718 3775 3832 3888 3945 4002 4059 4115 4172 6 4229 4285 4342 4399 4455 4512 4569 4625 4682 4739 7 4795 4852 4909 4965 5022 5078 5135 5192 5248 5305 8 5361 5418 5474 5531 5587 1 5644 5700 5757 5813 5870 9 5926 5983 6039 6096 6152 6209 6265 6321 6378 6434 770 6491 6547 6604 6660 6716 6773 6829 6885 6942 6998 1 7054 7111 7167 7223 7280 7336 7392 7449 7505 7561 2 7617 7674' 7730 7786 7842 7898 7955 8011 8067 8123 3 8179 8236 8292 8348 8404 8460 8516 8573 8629 8685 4 8741 8797 8853 8909 8965 9021 9077 9134 9190 9246 5 9302 9358 9414 9470 9526 9582 %38 9694 9750 9806 j 56 6 9862 9918 9974 0030 C086 i _ 1 ... 01 Q7 AOAO f.npr I 7 890421 0477 0533 0589 0645 0700 uiy < 0756 0812 \JO\J\J 0868 0924 8 0980 1035 1091 1147 1203 1259 1314 1370 1426 1482 9 1537 1593 1649 1705 1760 1816 1872 1928 1983 2039 780 2095 2150 2206 2262 2317 2373 2429 2484 2540 2595 1 2651 2707 27'62 2818 2873 2929 2985 3040 3096 3151 2 3207 3262 3318 3373 3429 3484 3540 3595 3651 3706 3 3762 3817 3873 3928 3984 4039 4094 4150 4205 4261 4 4316 4371 4427 4482 4538 4593 4648 4704 47'59 4814 5 4870 4925 4980 5036 5091 5146 5201 5257 5312 5367 6 5423 5478 5533 5588 5644 5699 5754 5809 5864 5920 7 5975 6030 6085 6140 6195 6251 6306 6361 6416 6471 8 6526 6581 6636 6692 6747 6802 6857 6912 6907 7022 9 7077 7132 7187 7242 7297 7352 7407 7462 7517 7572 790 7627 7682 7737 7792 7847 7902 7957 8012 8067 8122 55 1 8176 8231 8286 8341 8396 8451 8506 8561 8615 8670 2 8725 8780 8835 8890 8944 8999 9054 9109 9164 9218 3 4 9273 9821 9328 9875 9383 9930 9437 9985 9492 9547 9602 9656 9711 9766 5 0586 0640 0695 0749 0804 0859 900367 0422 0476 0531 6 0913 0968 1022 1077 1131 1186 1240 1295 1349 1404 7 1458 1513 1567 1622 1676 1731 1785 1840 1894 1948 8 2003 2057 2112 2166 2221 2275 2329 2384 2438 2492 9 2547 2601 2655 2710 2764 2818 287'3 2927 2981 3036 800 3090 3144 3199 3253 3307 asei 3416 3470 3524 3578 1 3633 3687 3741 3795 3849 3904 3958 4012 4066 4120 2 4174 4229 4283 4337 4391 4445 4499 4553 4607 4661 3 4716 4770 4824 4878 4932 4986 5040 5094 5148 5202 . 4 5256 5310 5364 5418 5472 5526 5580 5634 5688 5742 54 5 5796 5850 5904 5958 6012 6066 6119 6173 6227 6281 6 6335 6389 6443 6497 6551 6604 6658 6712 6766 6820 7 6874 6927 6981 7035 7089 7143 7196 7250 7'304 8 7411 7465 7519 7573 7626 7680 7734 7787 7841 7895 9 7949 8002 8056 8110 8163 II 8217 8270 832-1 8378 8431 II ! PROPORTIONAL, PARTS. Diff. 1 234 5 678 9 57 5.7 11.4 17.1 22.8 28.5 34.2 39.9 45.6 51.3 56 5.6 11.2 16.8 22.4 28.0 33.6 39.2 44.8 50.4 55 5.5 11.0 16.5 22.0 27.5 33.0 38.5 44.0 49.5 54 5.4 10.8 16.2 21.6 27.0 32.4 37.8 43.2 48.6 353 TABLE XXIV.-LOGARITHMS OF NUMBERS. No. 810 L. 908.] [No. 854 L. 931. N. 1 2 I 4 5 6 7 8 9 Diff. 810 908485 8539 8592 8646 8699 8753 8807 8860 8914 8967 1 9021 9074 9128 918 1 9235 i 9289 9342 9S 96 9449 9503 2 9556 9610 9663 971 J 9770 i 9823 9877 9 'iit 9081 nrwr 3 910091 0144 0197 0251 0304 0358 0411 0464 0518 0571 4 0624 0678 0731 0784 0838 0891 0944 (!< '.IS 1051 1104 5 1158 1211 1264 131 7 1371 1424 1477 i; 30 1584 1637 6 1690 1743 1797 185 ) 1903 1956 2009 i c,: 2116 2169 2222 2275 2328 2381 24:35 2488 2541 2594 2647 2700 8 2753 2806 2859 291 3 2966 8D19 3072 31 25 3178 3231 9 3284 3337 3390 3443 3496 3549 3602 3655 3708 3761 53 820 3814 3867 3920 397 3 4026 4079 4132 4184 4237 4290 1 4343 4396 4449 4502 4555 4608 4C60 4713 4766 4819 2 4872 4925 4977 503 q 5083 ; 5136 5189 341 i 5294 5347 3 ! 5400 5453 5505 555 s 5611 1! 5664 5716 ,-, "69 5822 5875 4 5927 5980 6033 60&5 6138 j! 6191 6243 6296 6349 6401 5 6454 6507 6559 661 2 6664 ! 6717 6770 ( <22 6875 6927 6 6980 7033 7085 713 8 7190 7243 7295 7 348 7400 7453 7 7506 7558 7611 7663 7716 7768 7820 7873 7925 7978 8 8030 8083 81 a5 818 8 8240 8293 8345 8 397 8450 8502 9 8555 8607 8659 8712 8764 8816 8869 8921 8973 9026 830 9078 9130 9183 9235 9287 9340 9392 9444 94G6 9549 9601 9653 9706 r -' K j^ 9810 S862 9914 g 367 1 ^ I nmo 0071 2 920123 0178 0228 0280 0332 1 0384 0436 0489 0541 UUrf 1 0593 3 0645 0697 0749 08C 1 0853 0906 0958 1 HO 1062 1114 xo 4 1166 1218 1270 132 2 1374 1426 1478 1 530 1582 1634 066 5 1686 1738 1790 1842 1894 j 1946 1998 2050 2102 2154 6 2206 2258 2310 23G 2 2414 |l 24C6 2518 2 ~' 2622 2674 7 2725 2777 2829 28F 1 2933 ! 2985 3037 S 3140 3192 8 3244 3296 3348 3399 3451 II 3503 3555 8607 3C58 3710 9 S7'62 3814 3865 ! 3917 3969 4021 4072 4124 4176 4228 840 4279 4331 4383 4434 4486 4538 4589 4641 4693 4744 1 4796 4848 i 4899 4951 5003 5054 5106 5157 209 5261 2 5312 5364 5415 54 5518 i 5570 5621 m 57'25 5776 3 5828 5879 5931 59F 1 6034 C085 6137 (i 188 6240 6291 4 6342 6394 6445 6497 6548 6600 6651 (i 702 6754 ceos 5 6857 6908 6959 701 1 7062 7114 7165 7 <>16 7268 7319 6 7370 7422 7473 75$ 4 7576 7627 7'678 7 7KO 7781 7'822 7 7883 7935 7986 8037 8088 8140 8191 8242 W93 8345 8 8396 8447 8498 854 9 8601 8652 8703 8 754 8805 8857 9 8908 8959 9010 9061 9112 9163 9215 9266 j 9317 93C8 850 9419 9470 9521 9572 9623 9674 9725 9776 9827 9879 9930 9981 51 fW}O /V)L n-ioe P9<*fi o tea 0338 0389 2 930440 0491 0542 UUr osr i 0643 UloO 0694 0745 0796 0847 0898 3 0949 1000 1051 1102 1153 1204 1254 1305 1356 1407 4 1458 1509 1560 1610 1661 1712 1763 1814 J865 1915 PROPORTIONAL PARTS. Diff. 1 2 3 4 5 6 7 8 9 53 5.3 10.6 15.9 21.2 26.5 31.8 37.1 42.4 47.7 52 5.2 10.4 15.6 20.8 26.0 31.2 36.4 41.6 46.8 51 5.1 10.2 15.3 20.4 25.5 30.6 35.7 40.8 45.9 50 5.0 10.0 15.0 20.0 25.0 30.0 35.0 40.0 45.0 TABLE XXIV. -LOGARITHMS OF NUMBERS No. 855 L. 931.1 [No. 899 L. 954. 1 1 N. 1 2 3 4 5 6 7 8 9 Diff. 855 931966 2017 2068 2118 2169 2220 2271 2322 2372 2423 6 2474 2524 2575 26 26 2677 2727 2778 2829 28' 2930 7 2981 3031 3082 3133 3183 ! 3234 3285 3335 13386 3437 8 3487 3538 3589 3C 39 3090 3740 3791 3841 i 38! )2 3943 9 3993 4044 4094 4145 4195 4246 4296 4347 4397 4448 860 4498 4549 4599 4650 4700 4751 4801 4852 4902 4953 1 5003 5054 ! 5104 5154 5205 || 5255 5306 5356 5406 : 5457 2 5507 5558 5608 n 58 5709 5759 5809 5860 59 5960 3 6011 6061 6111 61 (52 6212 6262 6313 6363 64 3 6463 4 6514 6564 6614 6665 6715 ! 6765 6815 ' 6865 69] (.; 6966 5 7016 7066 7116 7167 7217 I 7267 7317 7'367 7418 7468 6 7518 7568 7618 76 68 7718 i 7769 7819 7869 79] e 7969 7 8019 8069 8119 81 69 8219 8269 8390 8370 84i 8470 50 8 8520 8570 8620 8670 8720 8770 8820 i 8870 8920 8970 9 9020 9070 9120 91 70 9220 9270 9320 9369 9419 9469 870 9519 9569 9619 9669 9719 9769 9819 9S69 9918 9968 1 940018 0068 1 0118 0168 | 0218 ; 0267 0317 0367 0417 0467 2 0516 0566 i 0616 06 66 ! 0716 0765 0815 0865 09] 5 0964 3 1014 1064 1114 1163 ! 1213 1263 1313 ; 1362" 1412 1462 4 1511 1561 1611 16 60 1710 1760 1809 : 1859 19( )<) 1958 5 2008 2058 2107 2157 2207 2256 2306 2355 2405 2455 6 2504 2&54 2603 26 58 2702 2752 2801 2851 29( H 2950 7 3000 3049 3099 3148 3198 3247 3297 3346 3396 3445 8 3495 3544 3593 36 43 3692 3742 3791 3841 88! 3939 9 3989 4038 4088 4137 4186 | 4236 4285 4335 4384 4433 880 4483 4532 4581 4631 4680 4729 4779 48.28 4877 4927 4976 5025 5074 51 24 5173 5222 5272 5321 537 5419 2 5469 5518 5567 5616 5665 || 5715 5764 | 5813 5862 5912 3 5961 6010 6059 61 )8 6157 6207 6256 6305 63c 4 6403 4 6452 6501 6551 6600 6649 6698 6747 i 6796 6845 6894 5 6943 6992 i 7041 70 X) 7139 7189 7238 7287 733 8 7385 6 7434 7483 7532 7581 7630 7679 7728 7826 7875 49 7 7924 7973 8022 80 70 8119 8168 8217 8266 831 5 8364 8 8413 8462 8511 8560 8608 8657 8706 i 8755 8804 8853 9 8902 8951 8999 9048 9097 9146 9195 9244 9292 9341 890 1" 9390 QS7R 9439 9926 9488 9975 9536 9585 9634 9683 9731 j 9780 9829 y~< o 0024 0073 |l 0121 0170 i 0219 0267 0316 2 950365 0414 0462 0511 0560 ! 0608 0657 0706 0754 0803 3 0851 0900 0949 0997 1046 1095 1143 1192 1240 1289 4 1338 1386 1435 141 tf 1532 1580 1629 1677 172 6 1775 5 1823 1872 1920 1969 2017 ! 2066 2114 2163 2211 2260 6 2308 2356 2405 24 $8 2502 ! 2550 2599 2647 269 6 2744 7 2792 2841 2889 29 W 2986 i 3034 3083 3131 318 322H 8 3276 3325 3373 3421 3470 i 3518 3566 3615 366 3 3711 9 3760 3808 3856 9 )5 3953 4001 4049 4098 414 S 4194 1 PROPORTIONAL PARTS. Diff 1 2 3 4 5 6 7 8 9 51 5.1 10.2 15.3 20.4 25.5 30.6 35 7 40.8 45 Q 50 5.0 10.0 15.0 20.0 25.0 30.0 &5.0 40 450 49 4.9 9.8 14.7 19.6 24.5 29.4 34.3 39.2 44.1 48 4.8 9.6 14.4 19.2 24.0 28.8 33.6 38.4 43.2 355 TABLE XXIV. LOGARITHMS OF NUMBERS. No 900 L. 954.1 [No. 944 L. 975. I i 7 8 N. 1 2 I t 4 5 6 9 Diff. 900 954243 4291 .4339 4387 4435 4484 4532 4580 4628 4677 1 4725 4773 4821 4869 4918 4966 5014 5062 5110 f 5158 2 5207 5255 5303 53, 51 5399 5447 5495 5 543 559 2 5640 3 5688 5736 5784 5832 5880 5928 5976 6024 6072 \ 6120 4 6168 6216 6265 63 8 6361 6409 6457 6 505 655 3 6601 5 6649 6697 6745 6793 6840 l 6888 6936 6984 7032 7080 48 6 7128 7176 7224 72 8 7320 i 7368 7416 7 164 751 z 7559 7 7607 7655 7703 77 H 7799 7847 7894 7 M2 799 ) 8038 8 8086 8134 8181 8229 8277 8325 8373 8421 8468 8516 9 8564 8612 8659 8707 8755 8803 8850 8898 8946 8994 910 9041 9089 9137 9185 9232 9280 9328 9375 9423 9471 1 2 9518 9995 9566 9614 9661 9709 9757 9804 9852 9900 9947 0042 0090 0138 0185 0233 0280 L'S 0376 0423 3 960471 0518 0566 0613 0661 0709 0756 0804 0851 0899 4 0946 0994 1041 10 u 1136 1184 1231 1 .279 132 B 1374 5 1421 1469 1516 15 53 1611 1658 1706 1 1W 180 l 1848 6 1895 1943 1990 2038 2085 2132 2180 2227 2275 2322 7 2369 2417 2464 25 11 2559 2606 2653 2 roi 274 s 2795 8 2843 2890 2937 29 <5 3032 3079 3126 3 174 322 1 3268 9 3316 3363 3410 3457 3504 3552 3599 3646 369 3 3741 920 3788 3835 3882 3929 3977 4024 4071 4118 4165 4212 1 4260 4307 4354 44 )1 4448 4495 4542 4 590 463 7 4684 2 4731 4778 4825 48 a 4919 4966 5013 5 061 : 510 8 5155 3 5202 5249 5296 5343 5390 5437 5484 5531 5578 5625 4 5672 5719 5766 58 3 5860 5907 5954 & 101 604 B 6095 47 5 6142 6189 6236 6283 6329 6376 6423 6470 6517 6564 6 6611 6658 6705 67 >2 6799 6845 6892 6 l.-,9 698 ; 7033 7 7080 7127 7173 7220 7267 7314 7361 7408 7454 7501 8 7548 7595 7642 76 7735 7782 7829 7 ^75 792 2 7969 9 8016 8062 8109 8156 8203 8249 8296 8343 839 [) 8436 930 8483 8530 8576 86, 8670 8716 8763 8810 8856 8903 1 8950 8996 9043 9090 9136 9183 9229 9276 9323 9369 2 9416 9463 9509 95, >(j 9602 9649 9695 9 978 1 9835 3 9882 9928 OQ7* OOL >1 nru:e 0114 0161 0* )H7 0251 0997 1044 1090 1 137 118, 5 1229 6 1276 1322 1369 1415 1461 1508 1554 1601 1647 1693 7 1740 1786 1832 18 r '9 1925 1971 2018 2( )<>4 211 ) 2157 8 2203 2249 2295 2342 2388 2434 2481 2527 2573 2619 9 2666 2712 2758 2804 2851 2897 2943 2989 3035 3082 940 3128 3174 3220 3266 asis 8869 3405 3451 3497 3543 1 3590 3636 3682 37$ 3774 1 3820 3866 )13 395< ) 4005 2 4051 4097 4143 4*1 1 4235 ! 4281 4327 4. 04 442( ) 4466 3 4512 4558 4604 4& 4696 4742 4788 488( ) 4926 4 4972 5018 5064 5110 5156 5202 5248 5294 5340 5386 46 PROPORTIONAL PARTS. Diflf. 1 2 3 4 5 6 7 8 9 47 4.7 9.4 14.1 18.8 23.5 28.2 32.9 37.6 42.3 46 .4.6 9.2 13.8 18.4 23.0 27.6 32.2 36.8 41.4 35G TABLE XXIV. LOGARITHMS OF NUMBERS. No. 945 L. 975.] [Xo. 989 L. 995. N. 1 2 3 4 5 C 7 6 9 Diff. 945 975432 5478 5524 5570 5616 5662 5707 5753 5799 5845 6 5891' .937 5983 6029 6075 6121 6167 6212 6258 ; 6304 7 6350 6396 6442 6488 6533 6579 6625 6671 6717 6763 8 6808 j 6854 6900 6946 6992 7037 7083 7129 7175 7'220 9 7266 7312 7358 7403 7449 7495 7541 7586 7632 7678 950 7724 7769 7815 7861 7906 7952 7998 8043 8089 8135 1 8181 i 8226 8272 8317 8363 8409 8454 8500 8546 8591 2 8637 8683 j 8728 8774 8819 8865 8911 8956 8002 9047 3 9003 ! 9138 9184 9230 9275 9321 9366 9412 9457 9503 4 95-18 9594 9639 9685 9730 9776 9821 9867 9912 9958 5 980003 C049 I 0094 0140 0185 0231 0276 0322 0367 0412 G (458 0503 ; 0549 0594 0640 0685 0730 0776 Ob21 ! 0867 7 0912 : 0957 1003 1048 1093 1139 1184 1229 1275 1320 8 1366 ; 1411 1456 1501 1547 1592 1637 1683 1728 1773 9 1819 1864 1909 1954 2000 2045 2090 2135 2181 2226 960 2271 2316 2362 2407 2452 2497 2543 2588 2633 2678 1 2723 ; 2769 2814 2859 2904 2949 2994 3040 8085 3130 2 3175 3220 3265 3310 3356 3401 3446 3491- 3536 3581 3 3626 3671 3716 3762 3807 3852 3897 3942 3987 4032 4 4077 ! 4122 4167 4212 4257 4302 4347 4392 4437 4482 5 4527 4572 4617 4662 4707 4752 4797 4842 4887 4932 45 6 4977 5022 5067 5112 5157 5202 5247 5292 5337 5382 7 5426 5471 5516 5561 5606 5651 5696 5741 5786 5830 8 5875 5920 5965 C010 C055 6100 6144 6189 6234 6279 9 6324 6369 6413 6458 6503 6548 6593 6637 6682 6727 970 6772 6817 6861 6906 6951 6996 7040 7085 7130 7175 1 7219 7264 7--:09 7353 7398 7443 7488 7'532 7577 7622 2 7666 7711 7756 7800 7846 7890 7934 7979 8024 8068 3 8113 8157 8202 8247 8291 8336 8381 8425 8470 8514 4 8559 8604 8648 8693 8737 8782 8826 8871 8916 8960 5 9005 9049 9094 9138 9183 9227 9272 9316 9361 9405 6 0450 9895 9494 9939 9539 onOQ 9583 9628 9672 9717 9761 9806 9850 t7jOO C028 0072 0117 0161 0206 0250 0294 8 990389 0383 0428 0472 0516 0561 0605 0650 0694 0738 9 0783 0827 0871 0916 0960 1004 1049 1093 1137 1182 980 1226 1270 1315 1359 1403 1448 1492 1536 1580 1625 1 16G9 1713 1758 1802 1846 1890 1935 1979 2023 2067 2 2111 2156 2200 2244 2288 2333 2377 2421 2465 2509 3 2554 2598 2642 2686 2730 2774 2819 2863 2907 2951 4 2995 3039 3083 3127 3172 3216 3260 3304 3348 3392 5 3436 3480 &524 3568 3613 3657 3701 3745 3789 3833 G 3877 3921 3965 4009 4053 4097 4141 4185 4229 4273 7 4317 4361 4405 4449 4493 4537 4581 4625 4669 4713 44 8 4757 4801 4845 4889 4933 4977 5021 5065 5108 5152 9 5196 5240 5284 5328 5372 5416 5460 55U4 5547 5591 PROPORTIONAL PARTS. Diff 1 234 5 G 7 8 9 46 4.6 9.2 13.8 18.4 23.0 27.6 32.2 36.8 41.4 45 4.5 9.0 13.5 18.0 22.5 27.0 31.5 36.0 40.5 44 4.4 8.8 13.2 17.6 22.0 26.4 30.8 35.2 39.6 43 4.3 8.6 12.9 17.2 21.5 25.8 30.1 34.4 38.7 357 TABLE XXIV.- LOGARITHMS OF NUMBERS. No. 990 L. 995.] [No. 999 L. 999. N. 1 2 3 4 5 6 7 8 9 Diff. 990 995G35 5679 5723 5767 5811 5854 5398 5942 5986 6030 1 I 6074 6117 6161 6205 I 6249 6 293 6:337 b3b .) 6424 6468 44 2 6512 6555 6599 6643 6687 (i 731 6774 681 8 6862 6906 3 6949 6993 7037 7080 7124 3 168 7212 7255 7299 7343 4 7386 7430 7474 7517 7561 3 605 7648 76 ( J 2 7736 7779 7823 7867 7910 7954 7998 8 041 8085 812 '. 8172 8216 6 8259 8303 8347 8390 8434 8477 8521 8564 8608 8652 rr 8695 8739 8782 8826 8869 8 913 8956 90C 6 904S 9087 8 9131 9174 9218 9261 9305 9348 9392 9435 947 9522 9 9565 9609 9C52 9696 9739 9783 9826 9870 991c t 9957 43 LOGARITHMS OF NUMBERS FROM 1 TO 100. N. Log. 1 N. Log. N. Log. N. Log. N. Log. 1 0.000000 21 1.522219 41 1 .612784 61 1.785330 81 1.908485 2 0.301 m 22 1.342423 42 1 .6 282 1 ) 62 j_ '.):. 392 82 1.913814 3 0.477121 23 1.361728 43 1 .633468 63 1^799341 83 1.919078 4 0.602 )60 24 1.380211 44 1 .6 1845 J 64 1. 77 1. : 090 100 2.000000 ;. Value at 0. Sign in 1st Quad. ValiK at 90 > Sign 5 in2d Quad. Value at 180. Sign in3d Quad. Value at 270 - Sign in 4th Quad. Value nt 360. Sin _j_ R a. o R o Tan Sec 8 f 00 CO 1 R + co CO 4- R Versin.... Cos R R + 2R R + R 1 O R Cot GO _i_ co i O CO Cosec CO + R + CO ' R 00 R signifies equal to rad ; co signifies infinite ; O signifies evanescent. 353 TABLE XXV. -LOGARITHMIC SINES 179' " > Sine. q-l Tang. Cotang. q + l Dl" Cosine. / 4.685 15.314 Inf. neg. 575 |575 Inf. neg. Inf. pos. 425 ten 60 60 1 6.463726 575 575 6.463726 13.536274 425 ten 59 120 2 .764756 575 08 .764756 .235244 425 ten 58 180 3 6.940847 575 575 6.940847 13.059153 425 ten 57 240 4 7.065786 575 575 7.065786 12.934214 425 ten 56 800 5 .162696 575 575 .162696 .837304 425 ten 55 360 6 .241877 575 575 .241878 .758122 425 .02 9.999999 54 420 7 .308824 575 575 .308825 .691175 425 .00 .999999 53 480 .366816 574 I 576 .366817 .633183 424 .00 .999999 52 540 9 .417968 574 576 .417970 .582030 424 .00 .999999 51 GOO 10 .463726 574 576 .463727 .536273 424 .02 .999998 50 660 11 7.505118 574 ^576 7.505120 12.494880 434 .00 9.999998 49 720 12 .542906 574 577 .542C09 .457091 423 .02 .999997 48 780 13 .577668 574 577 .577672 .422328 423 .00 .999997 47 840 14 .609853 574 577 .609857 .390143 423 .02 .999996 46 900 15 .639816 573 :578 .639820 .360180 422 .00 .9999% 45 960 16 .667845 573 578 .667849 .332151 422 .02 .999995 44 1020 17 .694173 573 578 .694179 .305821 422 .00 .999995 43 1080 18 .718997 573 ; 579 .719003 .280997 421 .02 .999994 42 1140 19 .742478 573 579 .742484 .257516 421 .02 .999993 41 1200 20 .764754 572 580 .764761 .235239 420 .00 .999993 40 1260 21 7.785943 572 J580 7.785951 12.214049 420 .02 9.999992 39 1320 22 .806146 572 : 581 .806155 .193845 419 .02 .999991 38 1380 23 .825451 572 j 581 .825460 .174540 419 .02 .999990 37 1440 24 .843934 571 582 .843944 .156056 418 .02 .999989 36 1500 25 .861662 571 583 .861674 .138326 417 .00 .999989 35 1560 26 .878695 571 283 .878708 .121292 417 .02 .999988 34 1620 27 .895085 570 584 .895099 .104901 416 .02 .999987 33 1680 28 .910879 570 584 .910894 .089106 416 .02 .999986 32 1740 29 .926119 570 585 .926134 .073866 415 .02 .999985 31 1800 30 .940842 569 ;586 .940858 .059142 414 .03 .999983 30 I860 31 7.955082 569 ' 587 7.955100 12.044900 413 .02 9.999982 29 1920 32 .968870 569 587 .968889 .031111 413 .02 .999981 28 1980 33 .982233 568 588 .982253 .017747 412 .02 .999980 27 2040 34 7.995198 568 589 7.995219 12.004781 411 .02 .999979 26 2100 35 8.007787 567 590 8.007809 11.5)92191 410 .03 .9*9977 25 2160 36 .020021 567 : 591 .020044 .979956 409 .02 .999976 24 2220 37 .031919 566 592 .031945 .968055 408 .02 .999975 23 2280 38 .043501 566 593 .043527 .956473 407 .03 .999973 22 2340 39 .054781 566 593 .054809 .945191 407 .02 .999972 21 2400 40 .065776 565 [594 .065806 .934194 406 .02 .999971 20 2460 41 8.076500 565 595 8.076531 11.923469 405 .03 9.999969 19 2520 42 .086965 564 596 .086997 .913003 404 .02 .999968 18 2580 1 43 .097183 564 598 .097217 .902783 402 .03 .999966 17 2640 44 .107167 563 599 .107203 .892797 401 .03 .999964 16 2700 45 .116926 562 600 .116963 .883037 400 .02 .999963 15 2760 46 .126471 562 601 .126510 .873490 399 ,03 .999961 14 2820 ! 47 .135810 561 602 .135851 .864149 398 .03 .999959 13 2880 48 .144953 561 603 .144996 .855004 397 .02 .999958 12 2940 49 .153907 560 604 .153952 .846048 396 .03 .999956 11 3000 50 .162681 560 605 .162727 .837273 895 '.03 .999954 10 8060 51 8.171280 559 : 607 8.171328 ! 11.828672 393 .03 9 999952 9 3120 52 .X79713 558 608 .179763 .820237 392 .03 .999950 8 3180 53 .187985 558 609 .188036 .811964 391 .03 .999948 7 3240 54 .196102 557 .611 .196156 .803844 389 .03 .999946 6 3300 55 .204070 556 612 .204126 .795874 388 .03 .999944 5 8)60 56 .211895 556 613 .211953 .788047 387 .03 .999942 4 3120 57 .219581 555 615 .219641 .780-359 385 .03 .999940 3 3 180 58 .227134 554 1616 .227195 .772805 384 .03 .999938 2 3)40 59 .234557 554 618 .234621 .765379 382 .03 .999936 1 3300 60 8.241855 553 619 8.241921 11.758079 381 9.999934 4.685 15.314 // / Cosine. q I Cotang. Tang. q + l Dl" Sine. ' 90 89 C TABLE XXV. -LOGARITHMIC SINES, 178' // / Sine. q-l Tang. Cotang. q + l Dl" Cosine. , i ! 4.685 15.314 3600 8.241855 553 1619 8.241921 11.758079 381 i nn ' 9.999934 60 3660 1 .249033 552 620 ! .249102 .750898 380 " .999932 59 3720 2 .258094 551 622 .256165 .743835 378 ' 'Xo .999929 58 3780 3 .263042 551 623 .263115 .736885 377 'SI j .999927 57 3840 4 .269881 550 625 .269956 .730044 375 ryS; .999925 6 3900 5 .276614 549 627 .276691 .723:309 373 'Xo .999922 55 3960 6 .283243 548 (W8 .283323 .716677 372 .Vfj .999920 54 4020 7 .289773 547 .289856 .710144 370 .03 .998918 53 4080 8 .296207 546 632 .296292 .7037'08 268 -JS .999915 52 4140 j 9 ! .302546 546 688 .302634 .697366 887 HJ .999913 51 4200 10 .308794 545 635 .308884 .691116 365 1 - 05 .999910 4260 11 i 8.314954 544 637 8.315046 11.684954 363 JJ5 9.999907 49 4320 12 .321027 543 638 .321122 .67'8878 362 .Uo .896805 48 4380 13 .327016 542 640 .327114 .672886 860 } .989902 47 4440 14 .332924 541 642J .333025 .660875 38 -XX .998899 46 4500 15 .338753 540 644 .338856 .661144 QR .03 do n - .999897 45 4560 16 .344504 539 646 .344610 .65380 354 " .998894 44 4620 17 .350181 539 648 .350289 .649711 352 -S .999891 43 4680 18 .355783 538 649 .355895 .644105 351 -J* .996888 42 4740 19 .361315 537 651 .361430 .638570 349 - .998885 41 4800 20 .366777 536 653 .366895 .633105 347 ' uo ' .998882 40 4860 21 8.372171 535 ! 655 8.372292 11.627708 345 9.999873 39 4920 22 .377499 534 657 .377622 .622378 343 ' 'ne .999876 38 4980 23 .382762 538 659 .382889 .617111 341 .uu .998873 37 5040 24 .387962 532 661 .388092 .611808 339 '! .966870 36 5100 25 .393101 531 663 .393234 .606766 337 .05 .966867 35 5160 26 .398179 530 666 .398315 .601685 334 } .986864 34 5220 27' .403199 529 668 .403338 .86662 332 .uu .966861 33 5280 28 .408161 527 670 .408304 .91686 330 .05 fif? .998858 22 5340 29 .413068 526 672 .413213 .586787 328 .Ui r\K .99684 31 5400 30 .417919 525 674 .418068 .581932 326 i ' uu .988851 5460 31 8.422717 524 i 676 8.422869 11.577131 324 .05 rv? 9.999848 29 5520 32 .427462 523' 679 .427618 .572382 321 .Ui .688844 28 5580 33 .432156 522 681 .432315 .67685 319 f\e .898841 27 5* w*> .050832 55 G .948874 55 -22 !. .998277 .^U ; 1 tt .950597 /SO.O/& oo '"o .049403 54 7 .950287 |-5a .998266 . lo .952021 m, to .047979 53 8 .951696 SS'S -998255 .18 on i .953441 28.67 OQ KQ .040559 52 9 .953100 Sj-! .998243 .^U 1 Q .954856 30. uQ .045144 51 10 .954499 i:i ! * .lo .20 .956267 23.52 23.45 .043733 50 11 8.955894 9 o 17 9.998220 18 8.957674 90 0*: 11.042326 49 12 .957284 oo iA l .998209 . lo .959075 V * . OO .040925 48 13 .958670 SS'Jx .998197 *1S i .960473 90*99 .039527 47 14 .960052 22 95 '' - 9 9 818G . lo .961866 OQ - .038134 46 15 .961429 OO ft 1 "' .998174 *18 .963255 OQ AI** .036745 45 16 17 18 19 20 .962801 .904170 .965534 .966893 .968249 & .o< 22.82 22.73 22.65 22.60 22.52 .998163 .998151 .998139 .998128 .998116 . lo .20 .20 .18 .20 i .20 .964639 .966019 .967394 .908766 .970133 i&j . U i 23.00 22.92 22.87 22.78 22. 72 .035361 .033981 .032606 .081584 .029867 44 43 42 41 40 21 22 8.969600 .970947 22.45 9.998104 .998092 .20 8.G71496 .972855 22.65 11.028504 .027145 39 38 23 24 25 26 27 .972289 .973628 .974962 .976393 .977619 22^32 22.23 22.18 22.10 oo no .998080 .998068 .998050 .998044 .998032 !20 .20 .20 .20 .974209 .975560 .976906 .978248 .979586 22.57 22.52 22.43 22.37 22.30 oo OK .025791 .024440 .023094 .021752 .020414 37 86 35 34 33 28 29 30 .978941 .980259 .981573 at . Uo 21.97 21.90 21.83 .998020 .998008 .997996 !20 .20 .20 .980921 .982251 .983577 3DP*KD 22.17 22.10 22.03 .019079 .017749 .016423 32 31 30 31 32 33 8.982883 .984189 .985491 21.77 21.72 9.997984 .997972 .997959 .20 .22 8.984899 .986217 .987532 21.97 21.92 11.015101 .013783 .012408 29 28 27 34 35 36 37 38 39 40 .986789 .988083 .989374 .990660 .991943 .993222 .994497 21^57 21.52 21.43 21.38 21.32 21.25 21.18 .997947 .997935 .997922 .997910 .997897 .997885 .997872 '.20 .22 .20 .22 .20 22 ^20 .988842 .990149 .991451 .992750 .994045 .995337 .996624 21 '.78 21.70 21.65 21.58 21.53 21.45 21.40 .011158 .009851 .008549 .007250 .005955 .004003 .003376 26 25 24 23 22 21 20 41 8.995768 9.997860 oo 8.997908 11.C02092 19 42 43 .997036 .998299 21 .13 21.05 .997847 .997835 ,aS .20 oo 8.999188 9.000465 21 .33 21.28 11.000812 10.999535 18 17 44 45 46 47 48 8.999560 9.000816 .002039 .003318 .004503 21 .02 20.93 20.88 20.82 20.75 on ^n .997822 .997809 .997797 .997784 .997771 ,309 .22 .20 .22 .22 OO .001738 .003007 .004272 .005534 .006792 21 .22 21.15 21.08 21.03 20.97 .998202 .996993 .995728 .944406 .993208 16 15 14 18 12 49 50 .005805 .007044 40 . lO 20.65 20.57 .997758 .997745 122 .22 .008047 .009298 20^85 20.80 .991953 .990702 11 10 51 9.008278 9.99773-2 9.010546 20 7'3 10.989454 9 52 53 .009510 .010737 90.45 .997719 , .997700 !22 oo .011790 .013031 20 .'68 .988210 .986969 8 7 51 55 56 57 .011902 .013182 .014400 .015613 20.42 20.33 20.30 20.22 .997693 .997680 .997667 .997654 .let .22 .22 .22 .014208 .015502 .016732 .017959 20^57 20.50 20.45 OA AC\ .985732 .984498 .83208 .982041 6 5 4 3 58 59 60 .016824 .018031 9.019235 20.18 20.12 20.07 .997641 .997628 9.997014 '.22 .23 .019183 .020403 9.021620 IcU .4U 20.33 20.28 .980817 .979597 10.978380 2 1 ' Cosine. D.I*. Sine. D.I". Cotang. D. r. Tang. ' 95' 6 COSINES, TANGENTS, AND COTANGENTS. 173 ' Sine. D. 1". ! Cosine. 1 D. 1". Tang. D. r. Cotang. ' 1 2 9.019235 .020435 .021632 an no : 9.997614 W. UU | qn-fsni 10 Q-^ . yyn 1 b ? 10.978380 .977166 .975956 60 59 58 3 4 5 6 7 .022825 .024016 .025203 .026386 .027567 ly.oo 19.85 19.78 19.72 19.68 .997-574 .997561 .997547 .997534 .997520 , .22 .23 .22 .23 .025251 .026455 .027655 .028852 .030046 4\) . 4 20.07 20.00 19.95 19.90 in Q?C .974749 .973545 .972345 .971148 .969954 57 56 55 54 53 8 9 .028744 .029918 19.62 19.57 .997507 .997'493 .22 .23 oo .031237 .032425 19. oo 19.80 1 (1 '"Q .968763 .967575 52 51 10 .031089 19 52 19.47 .997480 rMB .23 .033609 ly. .938870 .937760 .936652 .935547 .934444 26 25 24 23 22 39 40 .063724 .064806 lo. Uo 18.03 17.98 .997068 .997053 '.25 .23 .066655 .067752 lo.o-i 18.28 18.25 .933345 .932248 21 20 41 42 43 44 45 46 47 48 49 50 9.065885 .066962 .068036 .069107 .070176 .071242 .072306 .073366 .074424 .075480 17.95 17.90 17.85 17.82 17.77 17.73 17.67 17. 63 17.60 17.55 9.997039 .997024 .997009 .996994 .996979 .996964 .996949 .996934 .996919 .996904 .25 .25 .25 .25 .25 .25 .25 .25 .25 .25 9.068846 .069938 .071027 .072113 .073197 .074278 .075356 .076432 .077505 .078576 18.20 18.15 18.10 18.07 18.02 17.97 17.93 17.88 17.85 17.80 10.931154 .930062 .928973 .927887 .926803 .925722 .924644 .923568 .922495 .921424 19 18 17 16 15 14 13 12 11 10 51 9.076533 9.996889 9.079644 10.920356 9 52 53 54 55 56 57 58 .077583 .078631 .079676 .080719 .081759 .082797 .083832 17^47 17.42 17.38 17.33 17.30 17.25 If* on .996874 .996a58 1 .996843 .996828 .996812 .996797 .996782 '.27 .25 .27 .27 .25 .25 .080710 .081773 .082833 .083891 .084947 .086000 .087050 17.77 17.72 17.67 17.63 17.60 17.55 17.50 .919290 .918227 .917167 .916109 .915053 .914000 .912950 8 7 6 5 4 3 2 59 60 .084864 9.085894 i .M 17.17 .996766 9.996751 .27 .25 .088098 9.089144 17.47 17.43 .911902 10.910856 1 ' Cosine. ix r. Sine. D. 1". Cotang. D. 1". Tang. ~ 96 305 83= TABLE XXV. LOGARITHMIC SINES, 172= / Sine. D. 1". Cosine. D. 1". Tang. D. r. Cotang. ' 1 2 3 4 5 6 7 8 9 10 9.085894 .080922 .087947 .088970 .089990 .05)1008 .092024 .093037 .094047 .095056 .096062 17.13 17.08 17.05 ; 17.00 i 16.97 16.93 16.88 16.83 16.82 16.77 16.72 9.996751 .996735 .996720 .996704 .996688 .996673 .996657 .996641 .996625 .996610 .996594 .27 .25 .27 .27 .25 .27 .27 .27 .25 .27 .27 9.089144 .090187 .091228 .092206 .093302 .094336 .095367 .090395 .097422 .098446 .099468 17.38 17.35 17.30 17.27 17.23 17.18 17.13 17.12 17.07 17.03 16.98 10.910856 .909813 .908712 .907134 .900698 .90C664 .904033 .90SC05 .902518 .801554 .900532 00 59 58 57 56 55 54 53 52 51 50 11 12 13 14 9.097065 .098066 .099065 .100002 16.68 16.65 16.62 it* Kiy 9.996578 .996562 .996546 .996530 .27 .27 'I? 9.100487 .101504 .1C2519 .103532 16.95 16.92 16.88 -1 P CO 10.899513 .898496 .897481 .896408 49 48 47 46 15 16 17 18 19 20 .101056 .102048 .103037 .104025 .105010 .105992 lO.Of 16.53 16.48 16.47 16.42 16.37 16.35 .996514 .996498 .996482 .996465 .996449 .996433 .%{ .27 .27 .28 .27 .27 .27 .104542 .105550 .106556 .107559 .108560 .109559 4O.OO 16.80 16.77 16.72 16.68 16.65 16.62 .895458 .894450 .893444 .892441 .891440 .890441 45 44 43 42 41 40 21 9.106973 1ft 9n 9.996417 OQ 9.110556 1 A Q 10.889444 39 22 23 24 25 26 27 28 .107951 .108927 .109901 .110873 .111842 .112809 .113774 10. *^U 16.27 16.23 16.20 16.15 16.12 16.08 16 OT .990400 .996384 .996368 .990351 .996335 .996318 .996302 .48 .27 .27 .28 .27 .28 .27 OQ .111551 .112543 .113533 .114521 .115507 .116491 .117472 lo.oo 16.53 16.50 16.47 16.43 16.40 16.35 1 t\ *-?*3 .888449 .887457 .886467 .885479 .884493 .888509 .882528 38 37 36 35 34 33 32 29 30 .114737 .115698 1O.UO 16.02 15.97 .996285 .996269 ,**O .27 .28 .118452 .119429 10. oo 16.28 16.25 .881548 .880571 31 SO 31 9.116656 1 CC flK 9.996252 OQ 9.120404 1 (* OO 10.879596 29 32 33 34 35 36 37 38 .117613 .118567 .119519 .120469 .121417 .122362 .123306 10. 'JO 15.90 15.87 15.83 15.80 15.75 15.73 1 H '""fi .996235 .996219 .996202 .996185 .996168 .996151 .996134 ./iO .27 .28 .28 .28 .28 .28 OQ .121377 .122348 .123317 .124284 .125249 .126211 .127172 ib.ZZ 16.18 16.15 16.12 16.08 16.03 16.02 1 K 0*7 .81-8023 .877052 .870083 .815716 .874751 .873189 .812828 28 27 26 25 24 23 22 39 .124248 lo . .857131 6 55 .139037 15.15 .995841 .oU .143196 u . 45 .856804 5 56 .139944 15.12 .995823 .30 .144121 15.42 .855879 4 57 58 .140850 .141754 15.10 15.07 i P; no .995806 .995788 .28 .30 OQ .145044 .145966 15.38 15.37 -* pr QO .854956 .854034 3 2 59 .142655 10.U/6 -1 K AA .995771 ,<;o QA .146885 1O.O,4 .853115 1 60 9.143555 10. UU 9.995753 .60 9.147803 15.30 10.852197 / Cosine. D. 1". Sine. D. 1". ! Cotang. D. 1". Tang. ' 97 8 COSINES, TANGENTS, AND COTANGENTS. m c ' Sine. D. r. Cosine. D. 1'. Tang. D. 1". Cotang. ' 9.143555 9.995753 on 9.147803 1 X. OX 10.852197 60 . 1 2 3 4 5 6 7 8 9 10 .144453 .145349 .146243 .147136 .148026 .148U15 .1498J2 .150686 .151569 .152451 14.97 14.93 14.90 14.88 14.83 14.82 14.78 14.73 14.72 14.70 14.65 .995735 .995717 .995699 .995681 .995664 .995646 .995628 .995610 .995591 .995573 .oU .30 .30 .30 .28 .30 .30 .30 .32 .30 .30 .148718 .149632 .150544 .151454 i .152363 ! .153269 .154174 .155077 .155978 .156877 lo.^o 15.23 15.20 15.17 15.15 15.10 15.08 15.05 15.02 14.98 14.97 .851282 .850368 .849456 .848546 .847637 .846731 .845826 .844923 .844022 .843123 59 58 57 56 55 54 53 52 51 50 11 9.153330 1 A O 9.995555 HI 9.157775 ~\A QQ 10.842225 49 12 .154208 14. DO MKQ .995537 .ou QA .158671 14. oo MQA .841329 48 13 14 .155083 . 155957 .58 14.57 j .995519 .995501 .oU .50 oo .159565 .160457 . yu 14.87 1/1 QQ .840435 .839543 47 46 15 16 17 18 19 20 .156830 .157700 .158569 . 159435 .160301 .161164 14.55 14.50 i 14.48 1 14.43 14.43 14.38 14.35 .995482 .995464 .995446 .995427 .995409 .995390 .oJs .CO .30 .32 .30 .32 .30 .161347 .162236 .163123 .164008 .164892 .16577-4 14. oo 14.82 14.78 14.75 14.73 14.70 14.67 .838653 .837764 .836877 .835992 .835108 .834226 45 44 43 42 41 40 21 9.162025 -t 4 00 9.995372 oo 9.166654 1 A O 10.833346 89 22 23 24 .1G28S5 .163743 .164600 14. oo 14.30 14.28 1 A OQ i .995353 .995:334 .995316 '.32 .30 .167532 .168409 .169284 14. DO 14.62 14.58 {A KK .832468 .831591 .830716 38 37 36 25 .165454 14. ^O .995297 'oo .170157 14 .OO ~\A f;Q .829843 35 26 27 28 29 . 166307 .167159 .168008 .168856 14. 23 14.20 j 14.15 14.13 j .995278 .995260 .995241 .995222 '.30 .32 .32 QO .171029 .171899 .172767 .173634 14. oo 14.50 14.47 14.45 .828971 .828101 .827233 .826366 34 33 32 31 30 .169702 14.10 l 14.08 .995203 .0x5 .32 .174499 14.42 14.38 .825501 30 31 9.170547 1 \ no. 9.995184 M 9.175362 14 VI 10.824638 29 32 .171389 14 . U-J 1 -i n*> .995165 QO .176224 14. ot HQQ .823776 28 33 .172230 1 1 . 04 . 995146 *9O .177084 .OO -t A OA .822916 27 34 ; 35 36 37 J .173070 .173903 .174744 .175578 14.00 13.97 i 13.93 13.90 1 O GG .995127 .995108 .995089 .995070 .o4 .32 .32 .32 : QO i .177942 .178799 .179655 .180508 14. 60 14.28 1427 14.22 M on .822058 .821201 .820345 .819492 26 25 24 23 38 ! 39 40 .176411 . 177242 .178072 19. OO : 13.85 ; 13.83 13.80 .995051 .99.5032 .995013 .O/C .32 .32 .33 .181360 .182211 .183059 14. *u 14.18 14.13 14.13 .818640 .817789 .816941 22 21 20^ 41 42 43 41 45 | 9.178900 .179726 .180551 .181374 .182196 13.77 13.75 : 13.72 13.70 ' 1 Q n~f \ 9.994993 .994974 .994955 .994935 .994916 .32 32 .33 .32 QO 9.183907 .184752 .185597 .186439 .187280 14.08 14.08 14.03 14.02 10.816093 .815248 .814403 .813561 .812720 19 18 17 16 15 46 .183016 lo.6< i 1 O Q .9948U6 .OO QO .188120 14.00 .811880 14 47 .183834 1 . OO ; .994877 .0*5 QO .188958 13.97 .811042 13 48 i .184651 13.62 ! -1C ^O .994857 .OO QO .189794 13.93 10 oo .810206 12 49 .185466 1 o . Oo 10 " ^ .994838 ,4K QQ .190629 lo.yy HO QCf .809371 11 50 .180280 lO.O< ] 13.53 .994818 .00 .33 .191462 lo.oo 13.87 .808538 10 51 53 54 9.187'092 .187903 . 188712 .189519 13.52 13.48 13.45 -t O AQ 9.994798 .994779 .994759 .994739 .32 .33 .33 oo 9.192294 .193124 .193953 , .194780 13.83 13.82 13.78 10.807706 .806876 .806047 .805220 9 8 7 6 55 56 57 58 59 60 .190325 .191130 .191933 .1927734 .193534 9.194332 lo.4o ' 13.42 ! 13.38 ! 13.35 13. as 13.30 .994720 .9947-00 .994680 .994660 .994640 9.994620 .34 M ! .33 .33 .33 .33 .195606 .196430 .197253 .198074 .198894 9.199713 13 . 77 13.73 13.72 13.68 13.67 13.65 .804394 .803570 .802747 .801926 .801106 10.800287 5 4 3 2 1 ' Cosine. D. 1". j Sine, i D.I". Cotang. D.I'. Tang. ' 367 81 C TABLE XXV. LOGARITHMIC SINES, 1.70 C ' Sine. D. 1'. Cosine. D. 1". Tang. D. r. Cotang. ' 1 2 3 4 5 G 7 8 9 10 9.194332 .195129 .195925 .196719 .197511 .198302 .199091 .199879 .200666 .201451 .202234 13.28 13.27 13.23 13.20 13.18 13.15 13.13 13.12 13.08 13.05 13.05 9.994620 .994600 .994580 .994560 .994540 .994519 .994499 .994479 .994459 .994438 .994418 .33 .33 .33 .33 .35 .33 .33 .33 .35 .33 .33 9.199713 .200529 .201345 .202159 .202971 .203782 .204592 .205400 .206207 ; .207013 .207817 13.60 13.60 13.57 13.53 13.52 13.50 13.47 13.45 13.43 13.40 10.800287 .799471 .7'98655 .797841 .797029 .796218 .795408 .794600 .793793 .792987 .792183 60 59 58 57 56 55 54 53 52 51 50 11 12 13 14 9.203017 .203797 .204577 .205354 13.00 13.00 12.95 9.994398 .994377 .994357 .994336 .35 .33 .35 9.208619 .209420 .210220 .211018 13.35 13.33 13.30 10.791381 .790580 .789780 .788982 49 48 47 46 15 16 17 18 19 20 .206131 .206906 .207679 .208452 .209222 .209992 12.92 12.88 12.88 12.83 12.83 12.80 .994316 .994295 .994274 .994254 .994233 .994212 .35 .35 .33 .35 .35 .35 .211815 .212611 .213405 .214198 .214989 .215780 13.28 13.27 13.23 13.22 13.18 13.18 13.13 .788185 .787389 .786595 .785802 .785011 .784220 45 44 43 42 41 40 21 22 24 25 26 27 28 29 9.210760 .211526 .212291 .213055 .213818 .214579 .215338 .216097 .216854 12.77 12.75 12.73 12.72 12.68 12.65 12.65 12.62 9.994191 .994171 .994150 .994129 .991108 .994087 .994066 .994045 .994024 .33 .35 .35 .35 .35 .35 .35 .35 9.216568 .217356 .218142 .218926 .219710 .220492 .221272 .222052 .222830 13.13 13.10 13.07 13.07 13.03 13.00 13.00 12.97 10.783432 .782644 .781858 .781074 .780290 .779508 .778728 ,777948 .777170 39 38 37 3 r 34 33 32 31 30 .217609 12.57 .994003 .35 .223607 12.92 .776393 30 31 32 33 34 35 36 9.218363 .219116 .219868 .220618 .221367 .222115 12.55 12.53 12.50 12.48 12.47 12 43 9.993982 .993960 .993939 .993918 .993897 .993875 .37 .35 .35 .35 .37 OK 9.224382 .225156 .225929 .226700 .227471 .228239 12.90 12.88 12.85 12.85 12.80 10.775618 .774844 .774071 .773300 .772529 .771761 29 28 27 26 25 24 37 38 39 .222861 .223606 .224349 .225092 12.42 12.38 12.38 12.35 .993854 .993832 .993811 .893789 .37 .35 .37 .35 .229007 .229773 .230539 .231302 12.77 12.77 12.72 12.72 .770993 .770227 .769461 .768698 23 22 21 20 41 42 43 9.225&S3 .226573 .227311 12.33 12.30 9.993768 .993746 .993725 .37 .35 9.232065 .232826 .233586 12.68 12.67 10.767935 .767174 .760414 19 18 17 44 45 .228048 .228784 12.27 .993703 .993681 .37 .234345 .235103 12.63 .765655 .764897 16 15 46 47 .229518 .230252 12.23 .993660 .993638 .37 .235859 .236614 12.58 .764141 .763386 34 13 48 49 50 .230984 .231715 .232444 12.18 12.15 12.13 .993616 .993594 .993572 .37 .37 .37 .237368 .238120 .238872 12.57 12.53 12.53 12.50 .762632 .761880 .761128 12 11 10 51 52 53 54 55 56 57 58 59 9.233172 .233899 .234625 .235349 .236073 .236795 .237515 .238235 .238953 12.12 12.10 12.07 12.07 12.03 12.00 12.00 11.97 9.993550 .993528 .993506 .993484 .993462 .993440 .993418 .993396 .993374 .37 .37 .37 .37 .37 .37 9.239622 .240371 .241118 .241865 .242610 .243354 .244097 .244839 .245579 12.48 12.45 12.45 12:42 12.40 12.38 12.37 12.33 10.760378 .759629 .788882 .758135 .757390 .756646 .755903 .755161 .754421 9 8 7 6 5 4 3 2 1 60 9.239670 9.993351 9.246319 10.753681 ' Cosine. D. 1". Sine. D. 1". 1 Cotang. D. 1'. Tang. ' 3fiS. 80 10 COSINES, TANGENTS, AND COTANGENTS. 169' Sine, i D. 1". | Cosine. D. 1". \ Tang. D. 1". Cotang, 100 C 9.239670 .2401386 .241101 .241814 .242526 .248947 .244656 .245363 .246069 .246775 9.247478 .248181 .849583 .250980 .251677 .252:373 .253067 .253761 9.254453 .255144 .255834 .256523 .257211 .257898 .258583 .259268 .2.V.W31 9.261314 .261994 .263351 .264027 .264703 .205377 .266051 .2(57395 9.268065 .2687:34 .269402 .270069 .270735 .271400 .272064 .272726 .273388 9.274708 .275367 .276025 .276681 .277337 .277991 .278645 .279297 .279948 9.280599 11.93 11.92 11.88 11.87 11.85 11.83 11.82 11.78 11.77 11.77 11.72 11.72 11.70 11.67 11.65 11.63 11.62 11.60 11.57 11.57 11.53 11.52 11.50 11.48 11.47 11.45 11.42 11.42 11.38 11.37 11.35 11.33 11.32 11.30 11.27 11.27 11.23 11.23 11.20 11.20 11.17 11.15 11.13 11.12 11.10 11.08 11.07 11.03 11.03 11.02 10.98 10.98 10.97 10.93 10.93 10.90 10.90 10.87 10.85 10.85 .'.193329 .998284 .993240 .993217 .993195 .993172 .993149 .993127 .993036 .992759 .992736 .992713 .992666 .992478 .992190 9.992166 .992118 .991996 .991971 9.991947 .37 .246319 .247057 .248580 .249264 .250730 | .251461 .252191 .252920 .253648 .254374 .255100 .255824 .256547 .257269 .257990 .258710 .259429 .260146 9.261578 .262292 .263005 .263717 .265138 .265847 .266555 .267261 .267967 9.268671 .269375 .270077 .270779 .271479 .272178 .272876 .273573 .274269 .274964 9.275658 .276351 .277043 .277734 .278424 .279113 .279801 .280488 .281174 .231858 9.282542 .283225 .283907 .285947 .287301 .287977 9.288652 12.30 12.28 12.27 12.23 12.23 12.20 12.18 12.17 12.15 12.13 12.10 12.10 12.07 12.05 12.03 12.02 12.00 11.98 11.95 11.95 11.92 11.90 11.88 11.87 11.85 11.83 11.82 11.80 11.77 11.77 11.73 11.73 11.70 11.70 11.67 11.65 11.63 11.62 11.60 11.58 11.57 11.55 11.53 11.52 11.50 11.48 11.47 11.45 11.^3 11.40 11.40 11.38 11.37 11.35 11.33 11.32 11.28 11.28 11.27 11.25 10.753681 60 .752943 59 .752206 58 .751470 i 57 .750736 ! 56 .750002 ! 55 .749270 54 .748539 I 53 .747809 52 .747080 51 .746352 I 60 10.745626 i 49 .744900 48 .744176 47 .743453 46 742731 i 45 .742010 .741290 .740571 .739854 I 41 .739137 40 10.738422 .737708 j 38 .736995 I 37 .736283 36 .735572 .734862 .734153 I as .733445 j 32 .732739 i 31 .732033 30 10.731329 29 .730625 28 .729923 27 .729221 I 26 .728521 .727822 .727124 .726427 .725731 .725036 10.724342 .723649 .722957 .722266 .721576 .720887 i 14 .720199 .719512 .718826 .718142 10.717458 .716775 .716093 .715412 .714732 .714053 .713376 .712699 10.711348 Cosine. I D. 1". Sine. I D.I". 1 1 Cotang. I D. 1". ! Tang. 79 TABLE XXV. -LOGARITHMIC SINES, 168 1 Sine. D. 1". Cosine. D. 1". Tang. D. 1". Cotang. ' 9.280599 9.991947 4O 9.288652 10.711348 CO 1 2 3 4 5 .281248 .281897 .282544 .283190 .283836 10.82 10.78 10.77 10.77 .99192^2 .991897 .991W73 .991848 .991823 .42 .40 .42 .42 .289999 .290671 .291312 .292013 11.22 11.20 11.18 11.18 .710674 .710001 .709:329 .708658 .707987 59 58 57 56 55 6 7 8 9 10 .284480 .285124 .285T66 .286408 .237048 10.73 10.70 10.70 10.67 10.67 .991799 .991774 .991749 .991724 .991699 .42 .42 .42 * .42 .42 .292682 .293350 .294017 .294684 .295349 11.13 11.12 11.12 11.08 11.07 .707318 .706650 .705983 .705316 .704651 54 53 52 51 50 11 12 13 14 15 16 17 18 19 20 9.287688 .288326 .288964 .289600 .290236 .290870 .291504 .292137 .292768 .293399 10.63 10.63 10.60 10.60 10.57 10.57 10.55 10.52 10.52 10.50 9.991674 .991649 .991624 .991599 .991574 .991549 .991524 i .991498 I .991473 .991448 .42 .42 .42 .42 .42 .42 .43 .42 .42 .43 9.296013 .236677 .297'3S9 I .298001 .298662 .299322 .299980 .300688 .301295 .301951 11.07 11.03 11.03 11.02 11.00 10.97 10.97 10.95 10.93 10:93 10.703987 .703323 .702661 .701999 .701338 .700678 .700020 .699362 .698705 .698049 49 48 47 46 45 44 43 42 41 40 21 22 23 24 25 23 27 28 29 30 9.294029 .294658 .295286 .295913 .296539 .297164 .297788 .298412 .299034 .299655 10.48 10.47 10.45 10.43 10.42 10.40 10.40 10.37 10.35 10.35 9.991422 .991397 .991372 ! .991346 .991321 1 .991295 .991270 .991244 .991218 \ .991193 .42 .42 .43 .42 .43 .42 .43 .43 .42 .43 9.302607 .303261 .303914 .304567 .305218 .305869 .306519 .307168 .307816 .308463 10.90 10.88 10.88 10.85 10.85 10.83 10.82 10.80 10.78 10.77 10.697393 .096789 .696086 .695433 .6947'82 .694131 .693481 .692832 .692184 .691537 39 38 37 36 35 34 33 32 31 80 31 32 33 34 9.300276 .300895 .301514 .302132 10.32 10.32 10.30 9.991167 .991141 .991115 i .991090 .43 .43 .42 9.309109 .309754 .310399 .311042 10.75 10.75 10.72 10.690891 .690246 .689601 .688958 29 28 27 26 35 36 37 .302748 .303364 .303979 10.27 10.25 .991064 ! .991038 1 .991012 .43 .43 .311685 .312327 .312968 10 . 72 10.70 10.68 .688315 .687673 .687082 25 24 23 38 .304593 .99C986 .313608 10.67 .686392 22 39 40 .305.307 .3C5819 10.20 10.18 .990960 ! .990934 .43 .43 .314247 .314885 10.63 10.63 .6867&S .685115 21 20 41 9.303430 9.990908 9.315523 10.684477 19 42 43 .307041 .307650 10.15 .990882 .990855 .45 .316159 .316795 10.60 .683841 .683205 18 17 44 45 46 47 48 49 50 .308259 .308867 .3C9474 .310080 .810685 .311289 .311893 10.13 10.12 10.10 10.08 10.07 10.07 i 10.03 i .990829 .990803 .990777 .990750 .990724 .990697 .990671 .43 .43 .45 .43 .45 .43 .43 .317430 .318064 .318697 .319330 .319961 .320592 .321222 10.57 10.55 10.55 10.52 10.52 10.50 10.48 .682570 .681986 .(581303 .680670 .680039 .679408 .678778 16 15 14 13 12 11 10 51 52 53 i 9.312495 .313097 .313698 10.08 ! 10.62 | n OQ ' 9.990C45 .990(518 .990591 .45 .45 9.321851 .322479 .323106 10.47 10.45 10.678140 .677521 .676894 9 8 7 54 55' 56 ,314297 .314897 .315495 10.00 9.97 ! .990505 .9905:38 .990511 .45 .45 .3237X3 .324358 i .324983 10.42 10.42 .676367 ! (57501 7 : 6 5 4 57 58 .316092 .316689 9.95 .990485 .990458 .45 i .325607 .326231 10.40 .674393 i .673769 i 3 2 59 60 .317284 9.317879 9.92 .990431 9.990404 .45 .326853 9.327475 10.37 .678147 10.672525 1 ' Cosine. D. 1*. Sine. D. 1". Cotang. i D. 1". 1 Tang. I ' ior 370 73 COSINES, TANGENTS, AND COTANGENTS. 167 ' Sine. D. r. ' Cosine. D. 1". Tang. D. 1". Cotang. ' 1 2 3 4 9.317879 .318.173 .319066 .319658 .320249 9.90 9.88 9.87 i 9.85 9.990404 i .990378 .990:351 .990324 .990297 .43 .45 .45 .45 9.327475 .328095 .328715 . 321)334 .829953 10.33 10.33 10.32 10.32 10.673525 .671905 .671285 .670666 .670047 60 59 58 57 56 s ; G 7 8 9 10 .320840 .321430 .322019 .322607 .323194 .323780 9.85 9.83 i 9.82 9.80 9.78 9.77 9.77 .990270 .990243 .990215 .990188 .990161 .9901:34 !45 .47 .45 .45 .45 .45 .330570 .331187 .331803 .332418 .333033 .333646 I0i28 10.27 10.25 10.25 10.22 10.22 .669430 .668813 .668197 .667582 .666967 .666354 55 54 53 52 51 50 11 12 13 14 15 If) 9.324366 .324950 .325534 .326117 .326700 .327281 9.73 9.73 9.72 9.72 9.68 9.990107 .990079 .990052 .990025 .989997 .989970 .47 .45 .45 .47 .45 9.334259 .334871 .3135482 .336093 .336702 .337311 10.20 10.18 10.18 10.15 10.15 m IQ 10.665741 .665129 .664518 .663907 .663298 .662689 49 48 47 46 45 44 17 18 19 20 .327862 .328442 .329021 .329599 9.68 9.67 9.65 9.63 9.62 .989942 .989915 .989887 : .989860 .47 .45 .47 .45 .47 .337919 .338527 .339ia3 .339739 1U. 10 10.13 10.10 10.10 10.08 '.661473 .660867 .660261 43 42 41 40 21 22 23 24 25 26 27 28 0.330176 .:30753 .331329 .331903 .3324; 8 .838051 .383624 .334195 9.62 9.60 9.57 9.58 9.55 9.55 9.52 9.989832 .989804 .989777 .989749 .9897'21 .989693 .989665 .989637 .47 .45 .47 .47 .47 .47 .47 9.340344 .340948 .341552 .342155 .342757 .343358 .343958 .344558 10.07 10.07 10.05 10.03 10.02 10.00 10.00 Q (J.J 10.659656 .659052 .658448 .657845 .657243 .656642 .656042 .655442 39 38 37 36 35 34 33 32 29 30 .384767 .335337 9^50 9.48 .989610 .989582 .45 .47 .48 .345157 .345755 9! 97 9.97 .654843 .654245 31 SO 31 9. 35906 9AQ 9.989553 9.346353 9qq 10.653647 29 32 .3.36475 .4o .989525 'jfo .346949 .yo .653051 28 33 34 35 36 .337043 .3-37610 .338176 .338742 9.47 9.45 9.43 9.43 .989497 .989469 .989441 .989413 '.47 .47 .47 .347545 .348141 .3-18735 .349329 9^93 9.90 9.90 90Q .652455 .651859 .651265 .650671 27 26 25 24 37 38 39 40 .339307 .339871 .340434 .340996 9.42 9.40 9.38 9.37 9.37 .989385 .989356 .989328 .989300 .47 .48 .47 .47 .48 .349922 .350514 .351106 .351697 .CO 9.87 9.87 9.85 9.83 .650078 .649486 . .648894 .648303 23 22 21 20 41 9.341558 9QK 9.989271 9.352287 9ft*} 10.647713 19 42 .342119 .00 .989243 .47 .352876 .O* .647124 18 43 .342679 9.33 9QQ .989214 .48 .353465 9. 8/2 90A .6465a5 17 44 45 46 .343239 .348797 .344355 .00 9.30 9.30 .989186 .989157 .989128 !48 .48 .354053 . 54(540 .355227 .oU 9.78 9.78 9tjf!t .645947 .645360 .644773 16 15 14 47 .344912 9.28 .989100 .47 .355813 . ( i 9r"r .644187 13 48 49 .845469 .346024 9.28 9.25 .989071 .989042 .48 .48 .356398 .856983 . PV .641269 .640687 8 7 54 55 56 .348792 .349343 .349893 JU8 9.17 91 "7 .988898 .988869 .988840 .48 .48 .48 .859898 .360474 .361053 .Ol 9.68 9. 65 9 AX .640107 .639526 .6:38947 6 5 4 57 .850443 . 1 ( 9 15 .988811 .48 .361632 .uo .638368 3 58 .850993 .988782 .4o .362210 J.uo .637790 2 59 GO .351540 9., 352088 9.13 9.13 .988753 9.988724 .48 .48 i .3627'87 ! 9.368364 9.62 9.62 .637213 10.636636 1 ' Cosine. 1 D. 1 . Sine, D. r. 1 Cotang. D. r. . Tang. ' 102 371 77 13 TABLE XXV. LOGARITHMIC SINEP5, 166 ' Sine. D. 1". Cosine. D. 1". Tang. t ' Cotang. ' 1 2 3 4 5 6 7 8 9.352088 .352635 .353181 .353726 .354271 .354815 .355358 .355901 .356443 9.12 9.10 9.08 9.08 9.07 9.05 9.05 9.03 902 9.988724 >J88G06 .988636 .988007 .988578 .988548 .988519 .988489 .48 ! .48 .50 ! .48 .48 .50 .48 .50 9.363364 .363940 .364515 .365090 .365664 .366237 .366810 .367882 .367953 9.60 9.58 9.58 9.57 9.55 9.55 9.53 9.52 10.C86GS6 .636060 .085485 .634910 .634886 .633763 .688190 .632618 .632047 CO 59 57 56 55 54 53 52 9 10 .356984 .357524 9.00 9.00 .988460 .988430 .50 .48 1 .368524 .369094 9.50 9.48 .631476 .630906 51 CO 11 12 13 9.358064 .358603 .359141 8.98 8.97 9.988401 .988371 .988342 .50 .48 9.369663 .370232 .370799 9.48 9.45 10.630337 .629768 .629201 49 48 47 14 15 16 17 18 19 .359678 .360215 .360752 .361287 .361822 .362356 8.95 8.95 8.92 8.92 8.90 .988312 .988282 .988252 .988223 .988193 .988163 .50 .50 .48 .50 .50 .371367 .371933 .372499 .373064 .373629 .374193 9.47 9.43 9.43 9.42 9.42 9.40 .628683 .628067 .627301 .626936 .626371 .625807 46 45 44 43 42 41 20 .362889 8.88 .988133 .0 .374756 9.38 9.38 .625244 40 21 22 23 24 25 9.363422 .363954 .364485 .365016 .365546 8.87 8.85 8.85 8.83 809 9.988103 .988073 .988043 .988013 .987983 ' .50 .50 .50 .50 9.375319 .375881 .376442 .377003 .377563 9.37 9.35 9.35 9.33 10.624681 .624119 .628558 .622997 .622437 9 88 37 6 CO 2(5 .366075 .987953 .378122 .621878 34 27 .366604 870 1 .987922 .378681 9.32 .621319 28 29 .367131 .367659 8.80 .987892 .987862 .50 .879289 .379797 9.30 .620761 .620203 2 31 30 .368185 8.77 .987832 .50 .52 .380354 9.28 9.27 .619646 oO 31 32 9.368711 .369236 8.75 87* 9.987801 .987771 .50 i 9.380910 I .381466 9.27 10.619090 .618534 9 8 33 34 35 .369761 '. 370808 8.72 8.72 .987740 .987710 .987679 .50 .52 .382020 .382575 t .383129 9.25 9.23 .617980 .617425 1616871 26 25 36 37 .371330 .371852 8.70 .987649 .987618 .52 1 .383682 .884234 9.20 .616318 .615766 24 23 38 .372373 .987588 i .384786 .615214 22 39 40 .372894 .373414' 8.67 8.65 .987557 .987526 .52 .50 , .385337 .385888 9.18 9.18 9.17 .614663 .614112 21 20 41 42 43 44 9.373933 .374452 .374970 .375487 8.65 8.63 8.62 9.987496 .987465 .987434 .987403 .52 .52 .52 9.386438 .386987 .387536 .388084 9.15 9.15 9.13 10.618562 .613013 .612464 .611916 19 18 17 16 45 46 .376003 .376519 8.60 .987372 .987341 .52 .888631 .889178 9.12 .611869 - .61C822 15 14 47 .377035 .987310 .52 .389724 .610276 13 48 49 50 .377549 .378063 .378577 8.57 8.57 8.53 .987279 1 .987248 ! .987217 .52 .52 .52 .390270 .390815 .391360 9.08 9.08 9.05 .609730 .C09185 .608640 12 11 10 51 52 9-379089 .379601 8.53 9.987186 .987155 .52 1 9.391903 .392447 9.07 10.608097 .607553 9 8 53 54 .380113 .380624 8.52 .987124 .987092 .53 .392989 .393531 9.03 .607011 .606469 6 55 56 57 58 .381134 .381643 .382152 .382661 8.48 8.48 8.48 8.45 ! .9S7'061 .987080 .986998 .986967 OCfJCWA .52 .53 .52 .52 | .394073 .394614 . .395154 .395694 qonoqo 9.02 9.00 9.00 8.98 .605927 .C05S86 .604846 .604306 6037'67 5 4 o 2 60 9.383675 8.45 9.986904 .53 9.396771 8.97 10.603229 ' Cosine. D. 1". Sine. D. 1". , Cotang. D. 1". Tang. ' 103 872 76' COSINES, TANGENTS, AND COTANGENTS. 165 |i I ' Sine. D.I". Cosine. ! D. 1". | Tang, D. 1". Cotang, 1 9.383675 8AK. 9.986904 trt 9.396771 8 97 10.603229 60 1 .384182 .45 .986873 O to .397309 8 95 .602691 59 2 .384687 8^42 .986841 QQAflAn .Do .53 .397846 QQUOQO 8^95 4602154 fllll (11 ^ 58 3 4 .385192 .385697 8.42 .yoooUy .986778 .52 . oyoooo .398919 8.93 8AM .OU1U1 i .601081 57 56 5 ! '. 386201 8.40 8f)Q .986746 .53 to .399455 -Uo 800 .600545 55 6 1 .386704 7 .387207 8 .387709 .00 8.38 8.37 8 OK .986714 .986683 .986651 .00 .52 .53 .399990 .400524 .401058 9/9 8.90 8.90 8QQ .600010 .599476 .598942 54 53 52 9 .388210 .o5 Q Q'- .986619 .53 K*> .401591 .00 800 .598409 51 10 .388711 o.oo 8.33 .986587 .Oo .53 .402124 .oO 8.87 .597876 50 11 9.389211 800 9.986555 to 9.402656 80* 10.597344 49 12 j .389711 13 .390210 .00 8.32 o on .986523 .986491 ,OO .53 to .403187 .403718 .oO 8.85 8QK .596813 .596282 48 47 14 .390708 o.oU .986459 .00 KQ .404249 .oD .595751 46 15 16 17 18 19 20 .391206 .391703 .392199 .392695 .393191 .393685 8.30 8.23 8.27 8.27 8.27 8.23 8.23 .986427 .986395 .986363 .986331 .986299 .986266 .OO .53 .53 .53 .53 .55 .53 .404778 .405308 .405836 .406364 .406892 .407419 8^83 8.80 8.80 8.80 8.78 8.77 .595222 .594692 .5941(54 .593636 .593108 .592581 45 44 43 42 41 40 21 I 9-394179 22 .394(373 8.23 9.986234 .986202 .53 9.407945 .408471 8.77 8i~- 10.592055 .591529 39 38 23 24 25 .395166 .395658 .396150 8. ',20 8.20 8-IQ .986169 .986137 .986104 !53 .55 KQ .408996 .409521 .410045 . (3 8.75 8.73 8i~q .591004 .590479 .589955 37 36 35 26 .396641 . lo 8-IQ .986072 .DO .410569 . t'J 8rv) .589431 34 27 .397132 .Ip .986039 .55 .411092 . (it 8r*o .588908 33 28 29 .397621 .398111 8.15 8.17 P 1 " .986007 .985974 .53 .55 .411615 .412137 . Cii, 8.70 8/0 .588385 .587863 32 31 30 .398UIX) 8.18 .985942 .53 .55 .412658 .00 8.68 .587342 30 31 9.3990S8 9.985909 9.413179 10.580821 29 32 .399575 33 .400062 154 .400549 35 1 .401035 8J2 8.12 8.10 8 no .985876 .985843 .985811 .985778 !ss .53 .55 KK .413699 .414219 .414738 .415257 o. 6* 8.67 8.65 8.65 8fn .586301 .585781 .585262 .584743 28 27 26 25 36 .401520 37 .402005 38 .402489 39 .402972 40 | .403455 .Oo 8.08 8.07 8.05 8.05 8.05 .985745 .985712 .985679 .985646 .985613 .55 .5') .55 .55 .55 .55 .415775 .416293 .416810 .417326 .417842 .DO 8.63 8.62 8.60 8.60 8.60 .584225 .583707 .583190 .582674 .582158 24 22 21 20 41 9.403938 8 no 9.985580 KK 9.418358 8 to 10.581642 19 42 .404420 .Oo .985547 .00 .418873 .Oo Stfft .581127 18 43 .404901 8.02 .985514 .55 .419387 J)i 8 toy .580613 17 44 45 46 47 .405382 .405862 .406341 .406820 8^00 7.98 7.98 .985480 .985447 -.985414 .985381 !55 .55 .55 .419901 .420415 .420927 .421440 ,o< 8.57 8.55 8.55 8KO .580099 .579585 .579073 .578560 16 15 14 13 48 49 .407299 .407777 7.98 7.97 7nt .985347 .985314 .57 .55 .421952 .422463 .5o 8.52 8Kt> .578048 .577537 12 11 CO .408254 . yo 7.95 .985280 .Of .55 .422974 .O/w 8.50 .577026 10 51 9.408731 9.985247 p-r. 9.423484 10.576516 9 52 53 .409207 .409682 7.93 7.93 .985213 .985180 .55 .423993 .424503 8.48 8.50 8A<~ .576007 .575-197 8 54 .410157 7.92 .985146 .57 .425011 .4< 8 A** .574989 6 55 56 .410632 .411106 7.92 7.90 7 DQ .985113 .985079 .55 .57 .425519 .426027 .4* 8.47 O A~ .574481 .573973 5 4 57 .411579 . OO r* QQ .985045 .57 .426534 2'iE .573466 3 58 .412052 59 .412524 4 .00 7.87 r- on* .985011 .9849?'8 ,5t .55 .427041 .427547 O.1O 8.43 .572959 .572453 2 1 60 9.412996 < .87 9.984944 .57 9.428052 8.42 10.571948 1 1 Cosine. D. 1'. Sine. D. 1". Cotang. D. 1". Tang. ' 101 373 75' 15 TABLE XXV. LOGARITHMIC SINES, / Sine. D. 1". Cosine. D. r. ! Tang. D. 1'. Cotang. / 1 2 9.412996 .413467 .413*8 7.85 7.85 700 9.984944 .984910 .984876 .57 .57 t'V 9-428052 .42H55H .429062 8.43 8.40 10.571948 .571442 .570938 60 59 58 3 4 5 6 7 8 .414408 .414878 .415347 .415815 .416283 .416751 .00 7.83 7.82 7.80 7.80 7.80 .984842 .984808 .984774 .984740 .984706 .984672 .Di .57 .57 .57 .57 .57 .429566 .430070 .430573 .431075 .431577 .432079 8^40 8.38 8.37 8.37 8.37 .570434 .5091)80 .569427 .568925 .568423 .567921 57 56 55 54 53 52 9 10 .417217 .417684 7i 78 7.77 .984638 .984603 .57 .58 .57 .432580 .433080 8.35 8.33 8.33 .567420 .566920 51 50 11 9.418150 9.984569 5>7 9.433580 8QO 10.566420 49 12 13 .418615 .419079 7.75 7.73 .984535 .984500 i .58 .434080 .434579 .OO 8.32 8QO .565920 .565421 48 47 14 15 .419544 .420007 7.75 7.72 t .984466 .984432 .'57 .435078 .435576 .04 8.30 .564922 .564424 46 45 16 .420470 7.72 .984397 .58 .436073 8.28 .563927 44 17 .420933 7.72 .984363 .57 .436570 8.28 .563430 43 IS 19 20 .421395 .421857 .422318 7.70 7.70 7.68 7.67 .984328 .984294 .984259 .58 .57 .58 .58 .437067 .437503 .438059 8.28 8.27 8.27 8-25 .562333 .5(12487 .561941 42 41 40 21 22 9.422778 .423238 7.67 9.984224 .984190 .57 9.438554 .489048 8.23 10.561446 .560952 39 38 23 .423697 7.65 .984155 .58 RQ .439543 8.25 .560457 37 24 .424156 7.65 .984120 .OO to .440036 8.22 800 .559964 36 25 26 27 28 29 30 .424615 .425073 .425530 425987 .426443 .426899 7.65 7.63 7.62 7.62 7.60 7.60 .984085 .984050 .984015 .983981 .983946 .983911 .OO .58 .58 .57 .58 .58 .440529 .441022 .441514 i .442006 ! .442497 .442988 39 8.22 8.20 8.20 8.18 8.18 .559471 .558978 .558486 .557994 .557503 .557012 85 34 33 32 31 30 7.58 .60 8.18 31 32 9.427354 .427809 7.58 9.983875 .983840 .58 KQ 1 9.443479 .443968 8.15 Ci 1 *7 10.55G521 .556032 29 28 &3 .428263 7.57 .983805 .OO to i .444458 0. 1 t .555542 27 34 .428717 7.57 .983770 .OO to .444947 !Q 1 Q .555053 26 35 36 .429170 .429623 7.55 7.55 .983735 .983700 .Do .-58 .445435 .445923 o.lo 8.13 U 1 Q .554565 .554077 25 24 37 38 .430075 .430527 7.53 7.53 .983664 .983629 !58 KQ .446411 .446898 0. 1*3 8.12 81 n .553589 .553102 23 22 39 40 .430978 .431429 7.52 7.52 7.50 .983594 .983558 .Of) .60 .58 .447384 .447870 . lu 8.10 8.10 .552616 .552130 21 20 41 9.431879 ff KA 9.983523 9.448S56 AQ 10.551P.44 19 42 43 .432329 .438778 * .5U 7.48 .983487 .983452 '.58 .448841 .449326 o. Uo 8.08 .551159 .550674 IS 17 44 45 .433226 .433675 7.47 7.48 .983416 .988381 .60 .58 .449810 .450294 8.07 8.07 8A* .550190 .549706 16 15 46 47 .434122 .434569 7.45 7.45 .983345 .983309 .60 .60 .450777 .451260 .05 8.05 8/\K .549223 .548740 14 13 48 .435016 7.45 .983273 .60 to .451743 .05 8 no .54K57 12 49 .435462 7.43 .983238 .00 .452225 .Uo . 5-177 7 5 11 50 .435908 7.43 7.42 .983202 .60 .60 .452706 8.02 .547294 10 51 52 53 54 9.436353 .436798 .437242 .437683 7.42 7.40 7.40 9.983166 .983130 .983094 .983058 .60 .60 .60 9.453187 .453668 .454148 .454628 8.02 8.00 8.00 10.546813 .540332 .545152 .545S7'2 9 8 7 6 55 .438129 7.38 .983022 .60 .455107 'Q '.544893 5 BO . 43857-2 7.38 .982986 .CO .455586 ' ^ .544414 4 57 58 .439014 .439456 7.37 7.37 .982950 .982914 .60 .60 .456064 .456542 7.9< 7.97 7QK .548CS6 .543458 8 2 59 60 .439897 9.440338 7.35 7.35 .982878 9.982842 '.GO .457019 9.457496 .yo 7.95 .542981 10.542504 1 ' Cosine. D. 1". I Sine. D. r. Cotang. D. 1". Tang. ' 105 74 374 16" COSINES, TANGENTS, AND COTANGENTS. 163 / Sine. D. 1". Cosine. D. 1". Tang. D. 1". Cotang. / 1 2 3 9.440338 .440778 .441218 .441(558 7.33 7.33 7.33 9.982842 .982805 .982769 ! 062733 .62 .60 .60 f'O 9.457496 .457973 .458449 .458925 .35 .93 .93 no 10.542504 ,542027 541551 .541075 60 59 58 57 4 .4-12096 7.30 700 .982696 .vz f*A .459400 ,V5o no .540600 56 5 .4425:35 .o* r" *>n .982660 .OU nr\ .459875 . . ' w f\f\ .540125 55 G .442973 i .oU r* k x> .982624 .uu .460349 .Uu .539651 54 7 .443410 7. /so i*f OQ .982587 .02 OA .460823 .90 f\f\ .539177 53 8 .443847 i . /co r* oo .982551 .O(J no ; .461297 . UU 00 .538703 52 9 .444284 t .40 r? oc* ' .982514 *M RO h .461770 .00 ft^ .538230 51 10 .444720 I . ** 7.25 .982477 .O/* .60 .462242 .Ol .88 .537758 50 11 9.445155 r* OR 9.982441 no 9.462715 OK 10.587285 49 13 .4455! JO . .-) f o-r .982404 ,0/J on I .463186 .OO .536814 48 13 .446025 . . 7 9^ .982867 .U4 f*{\ .463658 .87 00 .536342 47 14 .44(5459 t .-,> r oo .982331 ,uU ro .464128 . OO Q-; .535872 46 15 .446893 < ,O .982294 .b** ,.,) .464599 .oO QO .535401 45 16 .447326 7.22 700 .982257 .vz />0 ! .465069 .00 00 .534931 44 17 .447759 ,& 7 2O .982220 .0-6 fi2 .465539 .00 QO .534461 43 18 .448191 t . ,t; 17 OA .982183 ,O/5 (-.) .466008 .O/5 CO .533992 42 10 20 .448623 .449054 / . .472532 . *-* r 'O .527468 28 33 .454(51!) I . Uo 7 Oft .981625 ,o f*O .472995 . - 1 .527005 27 31 35 .455044 .4554(5!) i . Uo 7.08 7 07 .981587 .981549 .Do .63 fO .473457 .473919 _ . .70 7'0 .526543 .526081 26 25 3(5 .455893 < . VM ^ 0% .981512 OM A*> .474381 *fift .525619 24 37 .456-316 f .UO r* Apr .981474 .OO 4*0 .474842 .Do iD .525158 23 38 .456739 4 . UO r* A-: .981436 .uo RO .475303 . Oo .524697 22 39 4571152 i . UO r* no .981399 |X0 /o .475763 . U i Off .524237 21 40 ,157584 i . (JO 7.03 .981361 .UO .63 .476223 . Di .67 .523777 20 41 9.458006 r* /v> 9.981323 /Q 9.476683 f*K 10.523317 19 42 .458427 < .UJ 7- 02 .981285 .UO ftQ .477142 . UO 65 .522858 18 43 .458848 t .U^ r- AA .981247 .UO /> .477601 f S'-l .522399 17 44 .459268 < . UU r* AA .981209 .UO 00 .478059 .uo r/ .981095 .981057 .OO .63 PQ .479432 .479889 .0^ .62 f\f\ .520568 .520111 13 12 49 .461364 . \n 6f\ry .981019 .M n .48ft345 .DV i*A .519655 11 5J .461782 .Jt 6.95 .980981 .OO .65 .480801 .IJU .60 .519199 10 51 9.462199 .462616 6,95 9.980942 .980904 .63 9.481257 .481712 .58 to 10.5187'43 .518288 9 8 53 .468032 6 . 93 6 no .980866 .63 ftp; .482167 .OO .517833 54 55 .463448 .463804 . ;;o 6.93 .980827 .980789 .05 .63 .482621 .483075 . 5 i 7.57 .517379 .516925 6 5 56 .464279 (i . 92 /> rw> .980750 .65 /0 .488529 7.67 r p;^ .516471 4 57 .464094 o.yji .980712 .DO / K .483982 I .OO r* tr .516018 3 58 .465108 6.90 6 on .980673 ,o5 AQ .484435 < .OU 7 ^^ .515565 2 59 .465522 . yu GQO .980635 ,OO at .484887 1 .OO r* t> .515113 1 60 9.465935 .00 9.980596 .UO 9.485339 4 .OO 10.514661 ' Cosine. D. 1". Sine. D. r. Cotang. D. 1". Tang. / 375 73 TABLE XXV. LOGARITHMIC SINES, 162 ' Sine. D. r. Cosine. D. 1". Tang. D. 1'. Cotang. ' 9.465935 9.980596 9.485339 10.514661 60 1 2 .466348 .466761 6.88 6.88 .980558 .980519 '.G5 StK .485791 .486242 7.53 7.52 .514209 .513758 59 58 3 4 5 .467173 .467585 .467996 6.87 6.87 6.85 6Q~ .980480 .980442 .980403 .DO .63 .65 AX .48(5693 .487143 .487593 7.52 7.50 7.50 r RTV .513307 .512857 .512407 57 56 55 6 .46S407 .OO 6QO .980364 JDO AX .488043 1 .OU ry -o .511957 54 7 8 .468817 .469227 .So 6.83 Goo .980325 .980286 .OO .65 AX .488492 .488941 ( .4o 7.48 J7 AQ .511508 .511059 53 52 9 .469637 .00 6QO .980247 .OD A" .489390 7'47 .510610 51 10 .470046 .OSS 6.82 .930208 .tw .65 .489838 7i47 '.510162 50 11 12 9.470455 .470863 6.83 6QA 9.930169 .930130 .65 ft" 9.490286 .4907*3 7.45 10.509714 .509267 8 13 14 15 .471271 .471679 .472036 .OU 6.83 6.78 .930091 .980352 .980312 .DO .65 .67 .491180 .491627 .492073 7.45 7.45 7.43 .508820 .508373 .507927 2 45 16 .472492 Grirf .979973 />- .492519 " i > .507481 44 17 .472398 . ( ( .979934 .DO .492965 ( .4o .t,07035 43 18 19 23 .473304 .473710 .474115 6.77 6.77 6.75 6.73 .979895 .979855 .979816 .65 .67 .65 .67 .493410 .498854 .494299 7.42 7.40 7.42 7.40 .506590 .506146 .505701 42 41 40 21 22 23 9.474519 .474923 .475337 6.73 6.73 9.979776 .979737 .979697 .65 .67 9.494743 .495186 .495633 7.38 7.40 r- OQ 10.505257 .504814 .504370 39 38 371 24 .475730 6.72 6*?o .979858 .60 .496073 ( .OO .503927 36: 25 .476133 . 16 .979618 .67 .496515 9& .5034a5 35 26 .476536 6r?{\ .979579 .65 .496957 r/or' .503043 34 27 .476938 . (O 6ryf\ .979539 .67 AT .497399 J-W .502601 33 23 .477340 . t U 6AQ .979499 .Of Off .497841 r* O" .502159 32 29 30 .477741 .478142 .DO 6.68 6.67 .979459 .979420 .o7 .65 .67 .498282 .498722 i .OO 7.33 7.35 .501718 .501278 31 30 31 32 33 9.478542 .478942 .479342 6.67 6.67 9.979380 .979340 .979300 .67 .67 9.499163 .499603 .500042 7.33 7.32 10.500837 .500397 .499958 29 28 27 34 .479741 6.65 6PX .979260 .67 .500481 *S .499519 26 35 .480140 .00 .979220 .67 .500920 700 .499080 25 36 .480539 M'M .979180 .0< .501359 .o/^ rf QA .498641 24 37 .480937 5'2 .979140 .67 .501797 ( .60 .498203 23 38 39 .481334 .481731 D.02 6.62 .979100 .979059 '.GS />rv .502235 .502672 7^28 n OQ .497765 .497328 22 21 40 .482128 6 '.62 .979019 .O* .67 .503109 7.'28 .496891 20 41 42 43 44 9.482525 .482921 .483316 .483712 6.60 6.58 6.60 9.978979 .978939 .978898 .978858 .67 .68 .67 9.503546 .503982 .504418 .504854 7.27 7.27 7.27 10.496454 .496018 .49,5582 .495146 18 s 45 46 47 48 .484107 .484501 .484895 .485289 6.58 6.57 6.57 6.57 6 PC.** .978817 .978777 .978737 .978696 .68 .67- .67 .68 CQ .505289 .505724 .506159 .5C6593 7.25 7.25 7.25 7.23 7OQ .494711 .494276 .493841 .493407 15 14 13 12 49 .485682 .55 6KK .978655 .OO .507027 .0 9.507393 .508325 .508759 .509191 .22 .22 .20 1 Q 10.492107 .491674 .491241 .490,809 9 8 7 6 55 .488034 6.52 6t/\ .978411 .Go CO. .509622 . lo .490378 5 56 .488424 .DU 6c-A .978370 .uo .510054 1 Q .489946 4 57 .488814 .50 Cfc/\ 973329 po .510485 .lo 1ft .489515 3 58 .439204 .DU fi 4ft .978288 />0 .51C916 . lo 1*7 .489084 2 59 60 .489593 9.489982 6^48 ! .978247 i 9.873206 .00 .63 .511346 9.511776 . 1 1 7.17 .488654 10.488224 1 ' Cosine. D 1". i Sine. D.-r. Cotang. D. r. Tang. ' 107 C 376 COSINES, TANGENTS, AND COTANGENTS. 161' ' Sine. D. r. | Cosine. D. r. Tang. .D. r. Cotang. ' 1 1 9.489982 .490371 6.48 9.978206 .978165 .68 9.511776 .512206. 7.17 10.488224 .487794 60 59 2 .45)0759 6.47 I .978124 AQ .512635 7-|K i .487365 58 3 4 .491147 I .491535 ; 6.47 i 6.47 .978083 .978042 .DO .68 /Q .513064 .513493 . !) ! 7.15 71 ^ .486936 .486507 57 56 5 6 .491922 .492308 6.45 , 6.43 .978001 .977959 .UO .70 .513921 .514349 . lo 7.13 r-' 1 Q .486079 .485651 E5 54 7 i .492695 6.45 Gjn .977918 .68 .514777 i .lO .485223 53 8 ! 9 .493081 .493400 A6 6.42 .977877 .977835 !70 AQ .515204 .515631 7:i2 7 10 .484796 .484369 52 51 10 .493S51 6.42 6.42 .977794 .DO .70 .516057 i . 1U i 7.12 .483943 11 13 9.494236 .494621 .495005 6.42 i 6.40 6QQ 9.977752 .977711 .977669 ' .68 .70 9.516484 .516910 .517335 7.10 7.08 7 in 10.483516 .483090 .482665 49 48 47 14 15 Hi .495388 .4! 15772 .496154 .00 6.40 6.37 i GQO .977628 .977586 .977544 !70 .70 Ati .517761 .518186 .518610 . IU 7.08 7.07 r ' CV7 .482239 .481814 .481390 46 45 44 17 .496537 .OO .977503 .OO .519034 i . UY .480966 43 IS .496919 6.37 ! .977461 .70 .519458 7 07 .480542 42 19 .497301 . t .977419 "r-A .519882 7ns .480118 41 20 .497682 , ' ? .977377 . IV .520305 .Uo r* nr .479695 40 6.35 i .70 i .Uo 21 22 23 9.498064 .498444 .498825 6.33 6.35 ! /> oo 1 9.977335 .977293 .977251 .70 .70 r*A 9.520728 .521151 .521573 7.05 7.03 r* rvo 10.479272 .478849 .478427 39 38 37 24 .499204 D . O-4 .977209 . it 1 .521995 t .Uo ri no .478005 36 25 .499584 6.33 f* QO ' .977167 .70 .522417 i .Uo .477583 35 26 27 .499963 .500342 D . Op 6.32 .977125 .977083 .'70 .522838 .523259 7".02 .477162 .476741 34 33 28 .500721 6.32 .977041 .70 .523680 7.02 7 ro .476320 32 29 30 .501099 .501470 6.30 6.28 6.30 .976999 .976957' !70 .72 .524100 .524520 7!oo 7.00 .475900 .475480 31 30 31 9.501854 9.976914 9.524940 10.475060 29 32 .502231 6.28 Goo .976872 .70 .525359 fi QS .474641 28 33 .502007 .^i .976830 .525778 fi QQ .474222 27 34 35 36 .502984 .503360 .503735 6.28 6.27 6.25 .976787 .976745 .976702 '.70 .72 .526197 .520015 .527033 6^97 6.97 .473803 .473385 .472967 26 25 24 37 .504110 6.25 6 ox .976660 .70 .527451 R OK .472549 23 38 39 40 .504485 .504800 .505234 ,/vO 6.25 6.23 6.23 .976617 .970574 .976532 !72 .70 72 .527868 .528285 .528702 6 '.95 6.95 6.95 .472132 .471715 .471298 22 21 20 41 42 43 44 9.505008 .505981 ! 506727 6.22 6.22 6.22 ! 9.976489 .97'644G .976404 .976361 .72 .70 .72 9.529119 .529535 .529951 ^530366 6.93 6.93 6.92 GOO 10.470881 .470405 .470049 .469634 19 18 17 16 45 46 .507099 .507471 6^20 j GOn .976318 .97627o '.72 .530781 .531196 -SB 6.92 6. 92 .469219 .468804 15 14 47 .507843 .(*\J , 61 Q .976232 i~O .531611 Con .468389 13 48 .50S214 . lo 61Q .976189 X .532025 . yu Gon .467975 12 49 .508585 .lo 61 fi .970140 -99 .532439 .yu * .467561 11 50 .508956 . lo 6.17 .976103 !72 .582853 6.' 88 .467147 10 51 9.509326 9.976060 9.533266 COO 10.466734 9 52 .509696 6.17 61 x .976017 .72 mg .533679 .00 600 .466321 8 53 54 .510065 .510434 . lo 0.15 .975974 .975930 !73 .534092 .534504 . Cf > 6.87 .465908 .465496 6 55 .510803 6.15 61 X .975887 . 7 '3 .534916 6.87 .465084 5 56 .511172 . lo 61 Q .975844 * .535328 6.87 .46467'2 4 57 58 59 .511540 .511907 .512275 .19 6.12 6.13 6 1O .975800 .975757 .975714 !72 .72 .535739 .536150 .530501 6.85 6.85 6.85 .464261 .463850 .463439 3 2 1 60 9.512642 .1,3 9.975670 .73 9.536972 6.85 10.463028 ' Cosine. 1 D. r. Sine. D. i". Cotang. D. r. Tang. ' 108 377 19 TABLE XXV. LOGARITHMIC SINES, 160 , i 1 Sine. D.1-.J Cosine. D.,-. Tang. D. r. Cotang. i o 9.512642 9.975670 9.536972 600 10.463028 60 1 2 .513009 .513375 6.12 6.10 .975627 .975583 ira .537382 .537792 -OO 6.83 6QQ .462618 .462208 59 58 3 .513741 6.10 .975539 .73 .538202 .00 .461798 57 4 .514107 6. 10 / . O ' .975496 .72 m .538611 6.82 .461389 6 5 6 7 8 9 10 .514472 .514837 .515202 .515566 .515930 .516294 o.to 6.08 6.08 6.07 6.07 i 6.07 1 6.05 ; .975452 . 975408 .975365 .975321 .975277 .975233 . (0 .73 .72 .73 .73 .73 .73 .539020 .539429 .539837 .540245 .540653 .541061 6^82 6.80 6.80 6.80 6.80 6.78 .400980 .460571 .460163 .459755 .459347 .458939 55 54 53 52 51 50 11 9.516657 _ 9.975189 9.541468 10.458532 49 12 .517020 b.Oo 6/VJ .975145 .7o ! .541875 6.78 .458125 48 13 .517382 .Uo .975101 .73 i .542281 6.77 .457719 47 14 .517745 6.05 .975057 .73 .542688 6.78 .457312 46 15 .518107 6.03 .975013 .73 .543094 6.77 6i~e .4r.O!X)0 45 16 .518468 'A? .9749C9 "fO .543499 . 10 .466501 44 17 .518829 o . 02 .974925 .10 : .543905 6.77 .456095 43 18 19 20 .519190 .519551 .519911 6.02 6.02 6.00 6.00 .974880 .974836 .974792 .75 .73 .73 .73 .544310 .544715 .545119 6.75 6.75 --6 73 6.75 .455690 .455285 .454881 42 41 40 21 22 9.520271 .520631 6.00 5 no 9. 974748 .974703 .75 9.545524 ..545928 6.7'3 10.454476 .454072 39 38 23 .520990 .Uo .974659 .73 .546331 6.72 .45800'.) 37 24 .521349 5.98 .974014 .75 .546735 6.73 .453265 36 25 .521707 5.97 .974570 .73 .547'! 38 6.72 .452862 35 26 27 .5220(56 .522424 5.98 5.97 .974525 .974481 .75 .73 .547540 .547943 6.7'0 6.7'2 .452460 .452057 34 33 28 .522781 5.95 i .974436 .75 .548345 6.70 .451055 32 29 30 .523138 .523495 5 . 95 5.95 .974391 .974347 .75 .73 .548747 .549149 6.70 6.7'0 .451253 .450851 31 30 5.95 .75 6.68 31 9.523852 9.974302 9.549550 10.4E0450 29 32 .524208 5.93 5 no .974257 .75 .549951 6.68 .450049 28 33 .524564 .9o .974212 .75 .550352 6.08 .449648 27 34 .524920 5.93 .974167 .75 .550752 6.07 .449248 26 35 .525275 5.92 .974122 .75 .551153 6.68 .448847 25 36 .525630 5.92 .974077 .75 .551552 6.65 .448448 24 37 38 39 40 .525984 .526a39 .526(593 .527046 5.90 5.92 5.90 5.88 5.90 .974032 .973987 .973942 .973897 .75 .75 .75 .75 .75 .551952 .552351 .552750 .553149 6.67 6 65 6.65 6.65 6.65 .448048 .447649 !447250 .446851 23 22 21 20 41 42 43 44 45 9.527400 .527753 .528105 .528458 .528810 5.88 5.87 5.88 5.87 9.973&52 .97:3807 .973761 .973716 .973671 .75 .77 .75 .75 9.553548 .553946 .554344 .554741 .555139 6.63 6.63 6.62 0.03 10.446452 .446054 .445656 .445259 .444861 19 18 17 16 15 46 47 48 49 50 .529161 .529513 .529864 .530215 .530565 5.85 5.87 5.85 5.a5 5.83 5.83 .973625 .973580 .973535 .973489 .973444 In .75 .77 .75 .77 .55' 536 .555933 . 55(5329 .556725 .557121 . 02 6.62 6.60 6.60 6.60 6.60 .444464 .444067 .443071 .443275 .442879 14 13 12 11 10 51 52 53 54 55 56 57 9.530915 .531265 .531614 .531963 .532312 .532601 ! .533009 5.83 5.82 5.82 5.82 5.82 5.80 9.973398 .973352 .973307 .973261 .973215 .973169 .973124 .77 .75 .77 .77 .77 .75 9.557517 .557913 .558308 .558703 .559097 .559491 .559885 6.60 6.58 6.58 6.57 6.57 6.57 10.442483 .442087 .441692 .441297 .440903 .440509 .440115 9 8 7' 6 5 . 4 3 58 .53,3357 5.80 5r-o .973078 'i~ .560279 6.57 6p-*-* .430721 2 59 .533704 . 4W:3 .5450M .545338 5.62 5.63 5AA 9.971540 .971493 .971446 .78 .78 QA 9.573123 .573507 .572892 6.40 6.42 10.426877 .426493 .426108 29 28 27 34 85 .545674 .546011 .oU 5.62 5TA .971398 .971351 .oU .78 QA. .574276 .574660 G!40 GA(\ .425724 .425340 26 25 36 37 .546347 .546683 .OU 5.60 .971303 .971256 .oU .78 QA .575044 .575427 ,4U G.38 600 .424956 .424573 24 23 38 .547019 o.60 .971208 .OU r-o .575810 . OO i\ OQ .424190 22 39 .547'354 5.58 .971161 . 10 QA .576193 0. oo 6OQ .423807 21 40 .547689 5.58 5.58 .971113 ,oU .78 .576576 .00 6.38 .423424 20 41 9.548024 9.971066 QA ! 9.576959 vt 10.423041 19 42 43 .548:359 .548603 5.58 5.57 j .971018 .970970 .80 .80 QA .577341 .577723 6^37 GOK .422659 .422277 18 17 44 45 .549027 .5=19360 5.57 5.55 i .970922 .970874 .oU .80 r*Q .578104 .578486 .uu 6.37 6 QK .421896 .421514 16 15 46 .549693 5.55 i 5tpr .970827 . to QA .578867 .OO 6 OK .421133 14 47 48 .550026 .550359 .OD i 5.55 | 5fr- .970779 .970731 .oU .80 OA .579248 .57S629 .CO 6.35 600 .420752 .420371 13 12 49 50 .550692 .551024 .OO ! 5.53 1 5.53 ' .970683 .970635 .OU .0 .82 .580009 .580389 .00 6.33 6.33 .419991 .419611 11 10 51 9.551350 5.52 ! 9.970586 on 9.580769 GOQ 10.419231 9 52 .551687 f) fc>> .970538 .oU OA .581149 .OO COO .418851 8 53 54 .552. 18 .552349 .552680 5. '53 5.52 .970490 .970^42 .970394 .oO .80 .80 .581528 .581C07 .582x86 .66 6.32 6.32 .418472 .418093 .417714 7 6 5 57 .553010 .553341 5 .50 5.52 5AQ .970345 .970297 .82 .80 .582C65 .583044 6.32 6.32 GOA .417335 .416956 4 3 58 .553670 .4o .970249 .80 .683422 .OU 6QA .416578 2 59 .554000 5AO .970200 CA .583800 .oU GOG .416200 1 60 9.554329 .4o 9.970152 .oU 9.584177 . 9.968128 CO 9.599459 6-(O 10.400541 If, 42 43 44 .567'904 .568222 .568539 o.Jto 5.30 5.28 .968078 .968027 i .967977 .00 .85 '.83 .599827 .600194 .600562 . lo 6.12 6.13 61 O .400178 .399806 .399438 1! ; 17 1C 45 46 47 .568856 .569172 .569488 5.28 5.27 5.27 .967927 .967876 .967826 .83 .85 .83 .600929 .601296 .601663 .! 6.12 6.12 U~ 1fl .399071 .398704 .398337 16 ! 14 13 48- 49 50 .569804 .570120 .570435 5.27 5.27 5.25 5.27 : .967775 .SG7725 .007074 !83 .85 .83 .602029 .C02395 .6C2761 . lu 6.10 6.10 6.10 .397'971 .397605 .397239 1 - 11 10 51 52 9.570751 .571066 5.25. 9.9G7G24 .1)67573 .85 9.603-127 .603493 6.10 p no 10.3S6873 .396507 9 8 53 54 .571380 .571695 5.23 5.25 5OQ .967522 i .967471 .85 .85 QQ .603858 .604223 ().0o 6.08 ( (\Q .396142 .395777 i 7' 6 55 56 57 58 59 60 .57'2009 .572323 .572636 .572950 .573263 9.573575 .60 5.23 5.22 5.23 5.22 5.20 .967421 ; .967370 ! .967319 .967268 ! .967217 | 9.967166 .CO .85 .85 .85 .85 .85 .604588 .6C4053 .GC5317 .605682 .606046 9.606410 o.uo 6.08 6.07 6.08 6.07 6.07 .395412 .395047 .394683 .394318 .393954 10.393590 5 4 | 1 ! o ' Cosine. D. 1". Sine. D. 1". ; Cotang. D. 1". Tang. ' iir COSINES, TANGENTS, AND COTANGENTS. 157< ' Sine. D. r. Cosine. D. r. j Tang. D. 1". Cotang. , |j 1 2 3 4 9.573575 .573888 .574200 .574512 .574824 5.22 5.20 5.20 5.20 i 9.967166 .967115 .967064 .967013 .966961 .85 .85 .85 .67 9.606410 .606773 .607137 .607500 .607863 6.05 6.07 6.C'5 6.05 6 no 10.393580 .893227 .8C2663 .892800 .392137 60 59 8 57 56 5 6 .57'5136 .575447 5;18 5-4 Q .966910 .966859 '.S> I Off .608225 .608588 .Uo 6.05 6 no .891775 .391412 55 54 8 .575758 .576069 .lo 5.18 .066808 .966756 CO .67 ! OK .608950 .609312 .Do 6.03 .891050 .390688 53 52 9 .576379 5.17 .966705 .CO ! .609674 6.03 .81)0326 51 10 .57UC89 5.17 j 5.17 i .966653 .87 i .85 : .610036 6.03 6.02 .889964 50 11 9.576999 5-ity 9.966602 9.610397 6 no 10.S89C03 49 12 13 .577809 .577618 .li 5.15 I 51 %. .966550 .966499 !85 ! or* .610759 .611120 .Uo 6.02 6 fin .889241 .8686'80 48 47 14 .577927 .10 .966447 .or .611480 .uu .888520 46 15 .578286 5.15 .966395 .87 .611841 6.02 6nn .888159 45 16 .578545 5.15 .966344 .65 j .612201 .00 .887799 44 17 .578853 5. 13 .966S92 .87 ! .612561 -}j" .387439 43 18 .579162 5.15 5-1 Q .966240 g ! .612921 O.UU .887079 42 19 579470 .lo .966188 ?i i .613281 6.1 .386719 41 20 .579777 5.12 5.13 .866136 !8o - 613641 6.00 5.98 .886859 40 21 9.580085 51 O 9.966085 en* 9.614000 , QQ 10.386000 89 22 1 .580392 . 1/& .S66033 .o7 .614359 'no .385641 38 23 i .580699 5.12 5 in .965981 jg .614718 g -Eg .885282 37 24 25 .581005 .581312 . JU 5.12 .965929 .965876 !68 .615077 .615435 U. tJO 5.97 .384923 .384565 36 85 26 27 29 .581618 .581924 .582229 .582535 5. 10 5.10 5.08 5.10 K nfi .965824 .965772 .965720 .965668 .87 ' .87 .87 .87 CO .615793 .616151 .616509 .616867 5 . 97 5.9/ 5.97 5.97 e nt .884207 .883849 .383491 .383133 34 33 32 31 30 .582840 o.Uo 5.08 .965615 .CO .87 .617224 o.yo 5.97 .382776 30 31 32 9.583145 .583149 5.07 f AQ 9.965563 .965511 .67 9.617582 .617939 5.95 5 no 10.382418 ! 28 .382061 28 33 34 36 37 88 39 40 .583754 .584058 .584361 .584665 .584968 .585272 .585574 .585877 o.Oo 5.07 5.05 5.07 5.05 i 5.07 5.03 5.05 "5.03 .965458 .965406 .965353 .965301 .965248 .965195 .965143 .965090 .88 .87 .88 .87 .88 ! .88 .87 .88 .88 .618295 .918652 .619008 .619364 .619720 .620076 .620432 .620787 .yo 5.95 5.93 5.93 5.93 5.93 5.93 5.92 5.92 .381705 .881348 .380992 .380636 .880280 .379924 .378E68 .379213 27 26 25 24 23 22 21 20 41 ! 9.580179 9.965037 oo 9.621142 10.378858 ! 19 42 .586482 *'no .964984 -}5 .621497 5.92 5 no .378503 18 43 ! .586783 K f>Q .964931 i ;g2 .621852 .MB .378148 17 44 i .587085 o.Oo .964879 i -E .622207 5.92 .377793 16 45 ' .587386 5.02 .964826 I -SS .622561 5. 90 .377439 15 46 47 48 .587688 .587989 .688289 5.03 5.02 5.00 .964773 .964720 .964666 .00 .88 .90 .622915 .623269 .623623 5.90 5.90 5.90 .377085 .376731 .376377 14 13 12 49 .58S5JK) 5.02 < .964613 -58 .623976 5.8? .376024 11 50 .588890 .964560 .624330 5.90 .375670 10 5 . UO ; .88 5.88 51 52 53 9.589190 .589489 .589789 4.98 5.00 9.964507 .964454 .964400 .88 .90 OQ 9.624683 .625036 .625388 5.88 5.87 500 10.375317 .374664 .374612 9 8 54 .590088 4 . Jo i .964347 .CO .625741 .00 .374259 6 55 .590387 4. J8 .964294 .88 | .626093 5.87 .373907 5 66 .590686 40" .964240 .90 1 oo .626445 5.87 .373555 4 57 58 .590984 .591282 4. '97 A 07 .964187 .964133 .CO .90 QQ .626797 .627149 5 . 87 5.87 5 or* .373203 .372851 3 2 59 60 .591580 9.591878 4. yV 4.97 .064080 9.964026 .CO .90 .627501 9.627852 .Ol 5.85 .372499 10.372148 1 ' \ Cosine, i D. 1". || Sine. D. 1". | Cotang. | D. 1". Tang. ' 67' 23 TABLE XXV. -LOGARITHMIC SINES, 156 ' Sine. D. 1". Cosine. D. 1". j Tang. D. r. Cotang. ' 1 9.591878 .592170 4.97 4 OX 9.964020 .963972 .90 00 9.627K52 .628203 5.85 10.372148 .371797 60 59 2 .592473 iva 4Q~ .963919 .TO on ; .628554 5.85 5 Off .371446 58 3 4 5 .592770 .593357 .593303 . 7i) 4.95 4.93 A QO .963805 .983811 .963757 .yu .90 i .90 go .628905 .629255 .029800 .00 5.83 5.85 5QQ .371095 .370745 .370394 57' 56 55 6 7 8 9 10 .593653 .593955 .534251 .594547 .594342 4. \i Sine. D.I". Cosine. D. 1". Tang. D. 1". i Cotang. ' 9.609313 473 | 9.960730 no 9.648583 fc P*? i 10.351417 CO I 3 4 5 .609597 .609880 .610104 .610447 .610729 4!72 4.73 ! 4.72 4.70 .960674 .960618 .960561 .960505 .960448 .Uo .93 .95 : .93 .95 OQ s .648923 .649263 .649602 .649942 .650281 O.U 1 5.67 1 5.65 i 5.67 5.65 K *r .351077 .350737 .350398 .350058 .349719 59 58 57 56 65 6 7 .611012 .611294 4 ""2 ' 4.70 4 70 ' .9(50392 .960385 .UtJ ! .95 | .650620 .650959 O.OO 5.65 .349380 .349041 54 53 8 .611576 4 Ma .960279 a~ .651297 5ftX .348703 52 9 10 .611858 .612140 . i\J 4.70 4.68 .960222 .960165 !95 .93 .651636 .651974 .DO 5.63 5.63 .348364 .348026 51 50 11 9.612421 PQ 9.960109 9.652312 5fn 10.347688 49 .612702 ' .OO .960052 q? .652650 .63 5AQ .347350 48 13 .612983 > AQ .959995 Q- .652988 .Do .347012 47 14 .0132(54 .DO !oo .959938 no .653326 5 . 63 .346674 46 15 .01:3545 .DO ; . a~ .959882 QX .653663 5.02 t /rt .346.337 45 16 .613825 * .Of prt ] .959825 .yo .654000 O.O4 .346000 44 17 .614105 .or .959768 "on .654337 5.62 .345663 43 18 .614385 4/r* .959711 . yo n~ .654674 5.62 .345326 42 19 .614(565 .Ol .959654 .yo 9** .655011 5.62 .344989 41 20 .614944 4. DO 4.65 .959596 i .95 .655348 5.62 5.60 .344652 40 21 22 23 24 25 26 9.615223 .615502 .615781 .616060 .616338 .616610 4.65 4.65 4.65 4.63 4.63 4 OB 9.959539 .959482 .959425 .959368 .959310 .959253 .95 .95 .95 .97 .95 9.655684 .656020 .656356 .656692 i .657028 ! .6573(54 5.60 5.60 5.60 5.60 5.60 K to 10.344316 .343980 .343644 .343308 .342972 .342636 39 38 37 3G . 34 27 28 29 30 .616894 .617172 .617450 .017727 .bo 4.63 4.63 4.62 4.02 .959195 .959138 .959080 .959023 !95 .97 .95 .97 .657699 .658034 1 .658309 .658704 O.OO 5.58 5.58 5.58 5.58 .342301 .341966 .341631 .341296 33 32 31 30 31 9.618004 9.958965 9.659039 10.340961 29 32 33 34 .618&81 .618558 .618834 4! 08 4.60 .958908 .958850 .958792 .97 .97 .659373 .659708 .660042 5^58 5.57 .340627 .340292 .39958 28 27 26 35 .619110 4.60 .9587:34 .97 .660376 5.57 .339624 25 36 37 .619386 .61960.2 4.60 4.60 .958677 .958619 .95 .97 .660710 .661043 5.57 5.55 .339290 .338957 24 23 38 39 40 .619938 .620213 .620488 4.60 4.58 4.58 4.58 .958561 .958503 , '.958445 .97 .97 .97 .97 .661377 .661710 .662043 5.57 5.55 5.55 5.55 .338623 .338290 .337957 22 20 41 42 43 44 45 46 47 48 49 50 9.620763 .621038 .621313 .621587 .621861 .622135 .622409 .622682 .622956 .623229 4.58 4.58 4.57 4.57 4.57 4.57 4.55 4.57 4.55 4.55 1 9.958387 .958329 .958271 .958213 .958154 .958096 | .958038 ! .957979 i .957921 .957863 .97 .97 .97 .98 .97 .97 .98 .97 .97 .98 9.662376 .662709 .668042 .608375 .663707 i .664039 i .664371 : .664703 i .665035 .665366 5.55 5.55 5.55 5.53 5.53 5.53 5.53 5.53 5.52 5.53 10.337624 7337291 .336958 .336625 .366293 .335961 .335629 .335297 .834965 .3346:34 19 18 17 10 15 11 13 12 11 10 51 52 53 54 55 56 57 9.623502 .623774 .624047 .624:319 .624591 .624863 .6251:35 4.53 4.55 4.53 4.53 4.53 4.53 4K9 i 9.957804 .957746 .957687 .957628 .957570 .957.. 11 .957452 .97 .98 .98 .97 .98 .98 1 9.665698 1 .666029 .666360 .666691 .667('21 .667352 .667682 5.52 5.52 5.52 5.50 5.52 5.50 10.334302 .ami .333640 .333309 .332979 .332648 .832318 9 8 7 6 5 4 3 58 .625406 ,095 .957393 *Q7 .668013 5.52 5trk .331987 59 .625677 4 CO .957335 ' a i .668843 .uU .881657 1 60 9.625948 tCSf 9.957276 , 9.668673 5.50 10.33132> ' Cosine. D. 1". : Sine. D. 1". Cotang. D. 1". Tang. ' 383 65 25 TABLE XXV. LOGARITHMIC SINES, 154 ' Sine. D. 1". Cosine. D. 1". j Tang. D. 1'. Cotang. ' 9.625948 9.957276 9.668673 , df 10.331327 60 1 .626219 i'ro .9572W .o no .669002 '* .330998 59 2 .626190 *" .957158 .yo no .669332 rS ! .330668 ! 58 3 .G2o7(iO J-92 .957099 .'Jo nfl .669681 grn .330339 57 4 5 .627030 J'J . 62730 J fJS .957040 .956981 .Uo .98 Inn .669001 .670:320 5.50 5.48 .330009 .829(580 ! 56 55 6 .627570 j J-|2 .956921 .UU MS .670649 5.48 .329351 54 7 .627840 *; .9568(52 . \to f\O .670977 5.47 .329023 53 8 .628109 2'*g .956803 -JS .671306 5.48 54Q .328694 i 52 9 10 .62&S78 .628647 4.48 4.48 .9.56741 .956684 1.00 .98 .671635 .671963 Ao 5.47 5.47 .328365 .328037 51 50 11 13 13 14 15 16 17 18 19 20 9.628916 .629185 .629453 .629721 .629989 .6:30257 .630524 .630792 .631059 .631326 4.48 4.47 4.47 4.47 4.47 4.45 4.47 4.45 4.45 4.45 9.956625 .956566 .956506 .956447 .956:387 .956327 .956268 .956208 .956148 .956089 .98 1.00 .98 1.00 1.00 .98 1.00 1.00 .98 1.00 9.672291 .672619 .672947 .673274 .673602 .673929 .674257 .674584 .674911 .675237 5.47 5.47 5.45 5.47 5.45 5.47 5.45 5.45 5.43 5.45 10.327709 .327381 .327053 .326726 .326398 .326071 .325743 .325416 .325089 .324763 49 48 47 46 45 44 43 42 41 40 21 22 9.631593 .631859 4.43 9.956029 .955969 1.00 1/Vl 9.675564 .675890 . 40 j 10.324436 g'J i .324110 39 38 23 .632125 4.4<3 .955909 .UU .676217 g'Jg .323783 37 24 .632392 4.45 .955849 1 .00 .676543 36 25 1 .632658 4 Aft .955789 1.00 1AA .676869 M2 .'323131 35 26 .632923 27 .633189 28 1 .633454 .4% 4.43 4.42 .955729 .955669 .955609 .00 1.00 1.00 no .677194 '^ .677520 5-43 .677846 MS .322806 .322480 .322154 34 88 32 29 .633719 30 i .633984 4^42 4.42 .955548 .955488 .Uo 1.00 1.00 .678171 .678496 5.42 5.42 .321829 .321504 31 30 31 32 9.634249 .634514 4.42 9.955428 .955368 1.00 Ino 9.678821 .679146 5.42 5iO 10.321179 .320854 29 28 33 34 35 36 .634778 .635042 .635306 .635570 4^40 4.40 4.40 4Af\ .955307 .955247 .955186 .955126 .Ui 1.00 1.02 1.00 1AO .679471 .679795 .680120 .680444 .4.J 5.40 5.42 5.40 5A(\ .320529 .320205 .319880 .319556 27 26 25 24 37 qb .635834 ,4U 4.38 .955065 .(j% 1.00 .680768 .4U 5.40 .319232 o-t contt 23 .).) OO 39 ! 636360 4 38 40Q .'954944 1.02 1681416 5.40 . o J oyUo .318584 AH 21 40 .636623 .00 4.38 .954883 i!oo .681740 5.40 5.38 .318260 20 41 42 9.636886 .637*48 4.37 4QQ 9.954823 .954762 1.02 9.682063 .682387 5.40 500 10.317937 .317613 19 18 43 44 45 .637411 ! 687678 .00 4.37 ' 4.37 .954701 .954640 .9.54579 1.02 1.02 .682710 .683033 .683356 .00 5.38 5.38 .317290 .316967 .316644 17 16 15 46 ! 638197 4.37 4 OK .954518 J'iS .683679 5.38 K. Q'? .316321 14 47 .638458 *-*i .954457 1 .US .684001 a.6f 500 .315999 13 48 .638720 T'S .954396 i'^S .684324 .00 .315676 12 49 50 .638981 .639242 4.OO 4.35 4.35 .954.335 .954274 1.02 1.02 .684646 .684968 5.37 5.37 5.37 .315354 .315032 11 10 51 52 9.639503 .639764 4.35 4QO 9.954213 .954152 1.02 Ino 9.685290 i .685612 5.37 10.314710 .314388 9 8 53 .640024 .OO A fin .984090 .0.3 .685934 R-'OK .314066 7 54 .640284 **' .954029 1.02 COBCIK" O.OO .Ohb2oo t nr, .313745 6 55 .640544 i 7'SJ .953968 i'Xo .686577 U.OI .313423 5 56 .640804 t.Oit .953906 i'X2 .686898 .313102 4 57 .641064 4.33 .953845 J .VX .687219 5.35 .312781 3 58 59 .641324 .641583 4. '32 4QO .953783 .953722 1 .03 1.02 * no .687'540 .687861 5.35 5.35 ** QK .312460 .312139 2 1 60 3. 641842 .OA 9.953660 llVO 9.688182 10.311818 ' Cosine. D. r. j Sine. D. 1". Cotang. D. 1". Tang. ' 28 COSINES, TANGENTS, AND COTANGENTS. 153< ' j Sine. D. 1". Cosine. D.r. Tang. D.r. Cotang. i i 9.641842 9.953660 i An 9.688182 500 10.311818 60 1 .042101 4.32 4Q>> .953599 J'XS .688502 .00 K 00 .311498 59 .642360 .O* 4QA .953537 J. . Vt> 1" fiQ .688823 >.'< . 500 .311177 58 Jj .642618 .OU .953475 .Uo Ino .6891-13 .00 500 .310857 57 4 .042877 4.32 4QA .95:3413 .Uo .689463 .00 500 .310,537 56 .043135 .oU 4 on .953352 Ino .689783 .00 500 .310217 55 6 .043393 .oU .953290 .Uo 1 03 i .690103 .00 K OO .309897 54 7 > .043650 A OA .953228 1 03 .090423 O.OO 5QO .309577 53 8 i .643908 *' .953166 .090742 .04 500 .309258 52 9 10 ! .644165 .644423 t.TOQ 4.30 4.28 .953104 .953042 1^03 1.03 j .691062 .691381 .OO 5.32 5.32 .308938 .308619 51 50 11 12 ! 9.644680 i 4 . i .644936 I'S 9.952980 .952918 1.03 Ins ! 9.691700 j .692019 5.32 r >> 10.308300 .307981 49 48 13 14 .645193 .615450 t.xo 4.28 .C52855 .952793 .UO 1.03 .6923.8 .692656 5. Ox! 5.30 5QO .307662 .307344 47 46 15 16 17 18 .015706 .645962 .646218 .046474 4.27 4.27 4.27 4.27 A OK .952731 .952669 .952000 .952544 1.'03 1.05 1.03 Ine .692975 .693293 .693612 .693930 .04 5.30 5.32 5.30 .307025 .306707 .300388 .300070 45 44 43 42 19 .640729 ? .952481 .UO Ino .694248 5.30 Son .305752 41 20 ! .046984 *:27 .952419 .Uo 1.05 .694500 .OU 5.28. .305434 40 21 22 23 24 25 ' 26 27 28 29 30 9.047240 .647494 ! .647749 ' .648004 .048258 .048512 .648760 i .049020 .049274 .649527 4.23 4.25 4.25 4.23 4.23 4.23 4.23 4.23 4.22 4.23 9.952356 .952294 .952231 .952168 .952106 .952043 .951980 .951917 .951854 .951791 1.03 1.05 1.05 1.03 1.05 1.05 1.05 1.05 1.05 1.05 1 9.094883 .695201 .095518 .695836 .696153 .696470 ,696787 .697103 .697420 .697736 5.30 5.28 5.30 5.28 5.28 5.28 5.27 5.28 5.27 5.28 10.305117 .304799 .304482 .304164 .303847 .303530 .303213 .302897 .302580 .302264 39 38 37 36 35 34 33 32 31 30 31 32 33 34 35 36 37 38 39 40 9.649781 .650034 .650287 .650589 .650792 .651044 .651297 .651549 .651800 .652052 4.22 4.22 4.20 4.22 4.20 4.22 4.20 4.18 4.20 4.20 9.951728 .951665 .951602 .951539 .951470 .951412 .951349 .951280 .951222 .951159 1.05 1.05 1.05 1.05 1.07 1.05 1.05 1.07 1 05 iLoa 9.698053 .698369 .698685 .699001 .69931(5 .699032 .699947 .700263 .700578 .700893 5.27 5.27 5.27 5.25 5.27 5.25 5.27 5.25 5.25 5.25 10.301947 .301631 .301315 .300999 .300684 .300368 .300053 .299737 .299422 .299107 29 28 27 26 25 24 23 22 21 20 41 42 43 44 45 46 9.652304 .652555 .652800 .053057 053308 .053558 4.18 4.18 4.18 4.18 4.17 9.951090 .951032 .850968 .950905 .950841 .950778 1.07 1.07 1.05 1.07 1.05 107 9.701208 .701523 .701837 .702152 .702406 .702781 5.25 5.23 5.25 5.23 5.25 f OO 10.298792 .298477 .298163 .297848 .297534 .297219 19 18 17 16 15 14 47 48 49 50 .053808 .654059 .05430!) .654558 4.17 4.18 4.17 4.15 4.17 .950714 .9.50050 .958586 .950522 .VI 1.07 1.07 1.07 1.07 .703095 .703409 .703722 .704036 O.xJo 5.23. 5.22 5.23 5.23 .296905 .296591 .296278 .295964 13 12 11 10 51 9.654808 44|M 9.950458 9 704350 10.295650 9 52 63 54 55 56 57 58 59 60 .655058 .655307 .655556 .655805 .650054 056302 .656551 .656799 9.657047 .17 4.15 4.15 4.15 4.15 4.13 4.15 4.13 4.13 .950394 .950,330 .950200 .950202 .950138 .950074 .950010 .949945 I 9.949881 l!()7 1.07 1.07 1.07 1.07 1.07 1.08 1.07 ! 704663 .704976 .705290 .705603 .705916 .706228 .706541 .706854 9.707166 5.22 5.22 5.23 5.22 5.22 5.20 5.22 5.22 5.20 .295337 ! .295024 ! .294710 .294397 .294084 .293772 .293459 .293146 10.292834 8 7 6 5 4 3 2 1 ' \ Cosine. D. 1". Sine, ; D. 1". Cotang. D. 1". Tang. i 385 63 C 27< TABLE XXV. LOGARITHMIC SINES, 152 i Sine. D. r. Cosine. D. 1'. Tang. D. 1". Cotang. ' 1 2 9.657047 .657295 .657542 .13 .12 1Q 9.949881 .94! (81 6 .949752 1.08 1.07 9.707166 .70747'8 .707790 5.20 5.20 10.292&34 .292522 .292210 60 59 58 3 4 5 6 7 8 .657790 .658037 .658284 .658531 .658778 .659025 ' .14 .12 .12 .12 .12 .12 i A .949688 .949623 .949558 .949494 .949429 .949364 1 .07 1.08 1.08 1.07 1.08 1.08 .708102 .708414 .708726 .709037 .709349 .709660 5.20 5.20 5.20 5.18 5.20 5.18 5 -to .291898 .291586 .891274 .290963 .290651 . .290340 56 55 54 53 52 9 .659271 1 .lu .949300 1 .07 .709971 .1(5 .29002!) 51 10 . 659517 1 .10 .10 .949235 1.08 1.08 .710282 5.18 5.18 .289718 50 11 12 9.659763 .660009 .10 [ 9.949170 ! .949105 1.08 9.710593 .710904 5.18 10.289407 .28901)6 49 48 13 .660255 1 . 10 1ft .949040 1.08 1AQ .711215 5.18 r i " .288785 47 14 .660501 .lu AQ .948975 .Uo .711525 0.17 .288475 46 15 .660746 < .08 Ai2 .948910 1.08 1AQ .711&36 5.18 .288164 45 16 17 18 19 .660991 .661236 .661481 .661726 < .03 4.08 4.08 4.08 4(Y? i- .948845 i .948780 ! .948715 1 .948650 .08 1.08 1.08 1.08 .712146 .712456 .712766 .713076 5.17 5.17 5.17 5.17 .287854 .287544 . 287234 .286924 44 43 42 41 20 .661970 .\J( 4.07 .948584 1 .10 1.08 .713386 5.17 5.17 .286614 40 21 9.662214 A AQ 9.948519 1AQ 9.713696 K 1 X 10.286304 39 22 23 24 25 26 27 28 .662459 .662703 .662946 .663190 .6634:33 .663677 .663920 4 . Uo 4.07 4.05 4.07 4.05 4.07 4.05 4 A" .948454 .948388 .948323 .948257 .948192 .948126 .948060 .Uo 1.10 1.08 1.10 1.08 1.10 1.10 .714005 .714314 .714624 .714933 .715242 .715551 .715860 0. lO 5,15 5.17 5.15 5.15 5.15 5.15 .285995 .285686 .285376 .285007 .2847'58 .284449 .884140 Ajy 86 35 34- 33 32 29 30 .664163 .664406 .Oo 4.05 4.03 .947995 .947929 1 .08 1.10 1.10 .716168 .716477 5.13 5.15 5.13 .283832 .283523 31 30 31 32 33 9.664648 .664891 .665133 4.05 4.03 9.947863 .947797 .947731 1.10 1.10 9.716785 .717093 .717401 5.13 5.13 10.283215 .282907 .282599 29 28 27 34 35 36 37 .665375 .665617 .665859 .666100 4.03 4.03 4.03 4.02 4 no .947665 .947600 .947533 .947467 1.10 1.08 1.12 1.10 11 A .717709 .718017 .718325 .718633 5.13 5.13 5.13 5.13 .282291 .281983 .281677, .281367 26 25 24 23 38 39 40 .666342 .666583 \666824 .Uo 4.02 4.02 4.02 .947401 .947335 .947269 . 1U 1.10 1.10 1.10 .718940 .719248 .719555 5.12 5.13 5.12 5.12 .281060 .280752 .280445 1? 20 41 42 43 44 45 46 47 48 49 9.667065 .667305 .667546 .667786 .668027 .668267 .668506 .668746 .668986 4.00 4.02 4.00 4.02 4.00 3.98 4.00 4.00 9.947203 .947136 .947070 .947004 .946937 .946871 .946804 .946738 .946671 1.12 1.10 1.10 1.12 1.10 1.12 1.10 1.12 9.719862 .720169 .72047'6 .720783 .721089 .7'21396 .721702 .722009 .762315 5.12 5.12 5.12 5.10 5.12 5.10 5.12 5.10 10.280138 .279831 .279524 .279217 .278911 .278604 .278298 .277991 .277685 19 18 17 16 15 14 13 12 11 50 .669225 3.98 3.98 .946604 1.12 1.10 .722621 5.10 5.10 .277379 10 51 52 9.669464 .669703 3.98 9.946538 .946471 1.12 9.722927 .723232 5.08 10.277073 .276768 9 8 53 .669942 3.98 3f\Q .946404 1.12 11O .728538 5.10 51 A .276462 7 54 55 56 57 58 59 CO .670181 .670419 .670658 .670896 .671134 .671372 9.671609 .\K) 3.97 3.98 3.97 3.97 3.97 3.95 .946337 .946270 .946203 .946136 .946069 .946002 9.945935 .1* 1.12 1.12 1.12 1.12 1.12 j 1.12 .7881844 .724149 .724454 .7'24760 .725065 .725370 ' 9.725674 .1U 5.08 5.08 5.10 5.08 5.08 5.07 .276156 .275351 .273546 .275240 .274935 .274630 10.274326 G 5 4 3 2 1 ' Cosine. D. r. Sine. D. r. 1 Cotang. D. 1". Tang. 1 117 C 3SG COSINES, TANGENTS, AND COTANGENTS. 151' ' Sine. D. 1". Cosine. D. r. Tang. D. 1". Cotang. ' 1 2 3 4 5 6 8 9 10 9.671609 .671847 .672084 .672321 .672558 .672795 .673032 .673268 .673505 .673741 .673977 3.97 3.95 3.95 3.95 3.95 3.95 3.93 3.95 3.93 3.93 3.93 9.9459,35 .945868 .945800 .945733 .945666 .945598 .945531 .945464 .945396 .9453S8 .945261 1.12 1.13 1.12 1.12 1.13 1.12 1.12 1.13 1.13 1.12 1.13 9.725674 .735979 .726284 .726588 .726892 .727197 .7'27501 .7'27805 .7-28109 .728412 .7'28716 5.08 5.08 5.07 5.07 5.05 5.07 5.07 5.07 5.05 5.07 5.07 10.274326 .274021 .273716 .273412 .273108 .272803 .272499 .272195 .271891 .271588 .271284 60 59 58 57 56 55 54 53 52 51 50 11 12 13 14 9.674213 .67-4448 .674684 .674919 3.92 3.93 3.92 9.945193 .945125 .945058 .944990 1.13 .12 .13 9.729020 .729323 .729626 .729929 5.05 5.05 5.05 10.270980 .270677 .270374 .27-0071 49 48 47 46 15 16 17 .G7T) 155 .675390 .675624 3.93 3.92 3.90 .944922 .944854 .944786 .18 .13 .13 .730233 .730535 .730838 5.07 5.03 5.05 .269767 .269465 .269162 45 44 43 18 19 .675859 .676094 3.92 3.92 .944718 .944650 .13 1.13 .731141 .731444 5.05 5.05 .268859 .268556 42 41 20 .676328 3.90 3.90 .944582 1.13 1.13 .731746 5.03 5.03 .268254 40 21 22 23 24 25 26 27 9.676562 .676796 .677030 .677264 .677498 .677731 .677964 3.90 3.90 3.90 3.90 3.88 3.88 9.944514 .944446 .944377 .944309 .944241 .944172 .944104 1.13 1.15 1.13 1.13 1.15 1.13 9.732048 .732351 .732653 .732955 .783257 .733558 .733860 5.05 5.03 5.03 5.03 5.02 5.03 10.267952 .267649 .267347 .267045 .266743 .266442 .266140 89 38 37 36 35 34 33 28 29 30 .678197 .678430 .678663 3.88 3.88 3.88 3.87 .944036 .943967 .942899 1.13 1.15 1.13 1.15 .734162 .734463 .734764 5 . 03 5.02 5.02 5.03 .265838 .265537 .265236 82 31 30 31 32 33 34 35 36 37 38 9.678895 .679128 .679360 .679592 .679824 .680056 .680288 .680519 3.88 3.87 1 3.87 3.87 3.87 3.87 3.85 9.943830 .943761 .943693 .943624 .943555 .943486 .943417 .943348 1.15 1.13 1.15 1.15 1.15 1.15 1.15 9.7.35066 .7-35367 .735668 .735969 .736269 .736570 .736870 .737171 5.02 5.02 5.02 5.00 5.02 5.00 5.02 10.264934 .264633 .264332 .264031 .263731 .263430 .263130 .262829 29 28 27 26 25 24 23 22 39 40 .680750 .680982 3 . 85 3.87 3.85 .943279 .943210 1.15 1.15 1.15 .737471 .737771 5.00 5.00 5.00 .262529 .262229 21 20 41 42 43 44 9.681213 .681443 .681674 .681905 3. as 3.85 3.85 9.943141 .94>:072 .943003 .942934 1.15 1.15 1.15 9.738071 .738371 .738671 .7'38971 5.00 5.00 5.00 10.261929 .261629 .261329 .261029 19 18 17 16 45 .682135 3. as .942864 1.17 .739271 5.00 .260729 15 46 47 48 .682365 .682595 .682825 3.83 3.83 3.83 .942795 .942726 .942656 1.15 1.15 1.17 .739570 .739870 .740169 4.98 5.00 4.98 A no ^ .260430 .260130 .259831 14 13 12 49 .683055 3. as .942587 1.15 .740468 4.. 98 .2E9532 11 50 .683284 3' 83 .942517 1.17 1.15 .740767 4!98 .259233 10 51 9.683514 9.942448 9.741066 4 no 10.258934 9 52 .683743 ?** .942378 1 .17 .741365 .Do .258635 8 58 54 .683972 .684201 3! 82 .942308 .942239 1 .17 1.15 .741664 .741962 4 .98 4.97 4 no .258336 .258038 7 6 55 .684430 3 on .942169 1 . 17 .742261 .WO .257739 5 56 57 58 59 60 .684658 .684887 .686115 .685343 9.685571 .oO 3.82 3.80 3.80 3.80 .942099 .942029 .941959 .941889 9.941819 1.17 ; 1.17 ! 1.17 ! 1.17 i 1.17 .742559 .742858 .743156 .743454 9.743752 4.97 4.98 4.97 4.97 4.97 .257441 .257142 .256844 .256546 10.256248 4 3 2 1 ' Cosine. D. r. Sine. D. 1". Cotang. D. 1". Tang. i TABLE XXV. LOGARITHMIC SINES, 150 ' Sine. D. r. Cosine. D. r. Tang. D. 1'. Cotang. ' 1 9.685571 .685799 3.80 9.941819 .941749 1.17 9.743752 .744050 .97 10.256248 .266950 60] 59, 2 3 4 5 .686027 .080254 .680482 .686701) 3.78 3.80 3.78 .941679 .941609 .941539 .941469 1.17 1.17 1.17 1 18 .744348 .744645 .744943 .745240 .95 .97 .95 .255652 '. 255057 .254760 58' 57 56 55 6 7 8 9 10 .686936 .687163 .687389 .687616 .687843 3.78 3.77 3.78 3.78 , 3.77 i .941398 .941328 .941258 .941187 .941117 1.17 1.17 1.18 1.17 1.18 .745538 .7^335 .74ul32 .740429 .746726 .95 4.95 4.95 4.95 4.95 .254462 .254165 .253868 .253571 .25:32;4 54 53 52 51 50 11 12 13 14 15 16 17 18 19 23 9. 638069 .688295 .688521 .688747 .688972 .689198 .689423 .689648 .68937'3 .690098 3.77 3.77 3.77 3.75 3.77 3.75 3.75 3.75 3.75 3.75 9.941046 .940975 .940905 .940834 .940763 .940693 .940622 .940551 .940480 .940409 1.18 1.17 1.18 1.18 1.17 1.18 1.18 1.18 1.18 1.18 9.747023 .747319 .747616 .747913 .748209 .748505 .748801 .749097 .749393 ! .749689 4.93 4.95 4.95 4.93 4.93 4.93 4.93 4.93 ^4.93 4.93 10.252977 .252081 .252384 .252087 .251791 .251495 .251199 .250903 .260607 .250311 49 48 47 46 45 44 43 42 41 40 21 22 23 24 25 25 27 28 29 30 9.690323 .090548 .090772 .690996 .691220 .691444 .091668 .691892 .692115 .692339 , 3.75 3.73 3.73 3.73 3.73 3.73 3.73 3.72 3.73 3.72 9.940338 .940267 .940196 .940125 .940054 .C33982 .939911 .939840 .939768 .939697 1.18 1.18 1.18 1.18 1.20 1.18 1.18 1.20 1.18 1.20 9.749985 .7'50281 .750576 .750872 .751167 .751462 .751757 .752052 .752347 .752642 4.93 4.92 4.93 4.92 4.92 4.92 4.92 4.92 4.92 4.92 10.250015 .249719 .249424 .249128 .248833 .248538 .248243 .247948 .247053 .247358 39 38 37 36 35 84 33 32 31 30 31 32 33 9.692562 .692785 .693008 3.72 3.72 9.939625 .939554 .939482 1.18 1.20 9.752937 .753231 .753526 4.90 4 92 A on 10.247063 .240709 .24647'4 29 28 27 34 ,35 .693231 .693453 3.70 .939410 .939339 1.18 .75:3820 .754115 4.92 .246180 .245885 26 25 36 37 38 39 40 .693676 .693898 .694120 .694342 .6945(54 3.70 3.70 3.70 3.70 3.70 .939267 .939195 .939123 .939052 .938980 1.20 1.20 1.18 1.20 1.20 .751409 .754703 .754997 .755291 .755585 4.90 4.90 4.90 4.90 4.88 .245591 .245297 .245003 .244709 .244415 24 23 22 21 20 41 42 43 44 9.694786 .695007 .695229 .695450 3.68 3.70 3.68 9.938908 .938836 .938703 .938091 1.20 1.22 1.20 9.755878 .756172 .756465 .7507'59 4.90 4.88 4.90 10.244122 .24=3828 .243535 .243241 19 18 17 16 45 46 .695671 .695892 3.68 .9:38619 .938547 1.20 .757052 .757345 4.88 .242948 .242655 15 14 47 .696113 .938475 .757038 A 00 .242362 13 48 49 .696834 .696554 3.67 .938402 .9-38330 1.20 ! 758224 4.88 400 .242069 .241776 12 11 50 .696775 3.67 .938258 t'M .758517 4.88 .241483 10 51 9.696995 9.9.38185 9.758810 10.241190 9 52 .697215 .938113 .759102 A QQ .240898 8 53 54 55 56 57 58 59 .697435 .097654 .097874 .698094 .698313 .698532 .698751 3.65 3.67 3.67 3.65 3.65 3.65 .938040 .937907 .937895 .937822 .937749 .937070 .937604 1.22 1.20 1.22 1.22 1.22 1.20 .75931)5 .759087 .759979 .700272 .700504 .760856 .701148 4.87 4.87 4.88 4.87 4.87 4.87 .240605 .240313 .240021 .239728 .239436 .239144 .238852 7 6 5 4 3 2 1 60 9.698970 9.937531 1.22 9.761439 4.85 j 10.238561 ' I Cosine. D. 1". Sine. D. r. : Cotang. D. r. i Tang. ' 119= QQQ COSINES, TANGENTS, AND COTANGENTS. 149 / Sine. D.r. Cosine. D. 1". Tang. D. 1". Cotang. / 9.G98970 3/~ 9.937531 OO 9.761439 4OQ 10.238561 CO 1 .099189 .UO 3ftft .937458 JG3 oo .761731 .no 4' W7 .238209 59 2 .699407 .Oo ; .937385 jEo .762023 .o< .237977 58 3 .699626 3.05 q eq .937312 .22 i Oq .762314 4.85 A QW .237686 57 4 .699844 o. Do Q |;O .937238 .^O OO .762606 4. o< A OK .237394 56 5 .700062 o.Uo 3Co ; .937165 ijPe OO .762897 4.OO 4Qff .237103 55 G .700280 . UO i o rfQ 1 .937092 .3960 OO .763188 .00 .236812 64 7 8 I .700498 .700716 .700933 6 . UO I 3.03 3.02 o /q .937019 .936946 .936872 .,& .22 .%* OO .763479 .763770 .764061 4.85 4.85 4.85 4 fix .236521 .236230 .235939 53 52 51 10 .701151 o. Do 3.62 .936799 JQ0 .23 .764352 .00 4.85 .235648 50 11 9.701368 o /?o 9.936725 oo 9.764643 400 10.235357 49 12 .7 1585 o.O/i .936652 . . - OQ .764933 .OO 4QK .235067 48 13 14 15 .701802 .702019 .702236 3.02 3.62 3.62 ti. A .93(5578 .936505 .936431 .^o .22 .23 oq .765224 .765514 .765805 .OO 4.83 4.85 400 .234776 .234486 .234195 47 46 45 16 .702452 o. OU O |*O .936357 . /*o fc .766095 . OO 4OQ .233905 44 17 .702669 Io . u* A .936284 ./ oq .766385 OO 400 .233615 43 18 .702885 o.OU q I*A .936210 .*O oq .766675 .OO 4QO 233325 42 19 .703101 o.OU (A .936136 .^o oq .766965 .OO 4QQ 1233035 41 20 .703317 fj . OU 3.60 .936062 . 4>fj .23 .767255 .00 4.83 .232745 40 21 9.703533 3fiO 9.9a5988 OQ 9.767545 4QO 10.232455 39 22 .703749 . DU 3 CO .935914 .-^O oq .767834 .0x5 4QQ .232166 38 23 .703964 .Do 3CQ .935840 . --> OQ .768124 .OO 4QQ .231876 37 24 .704179 . UO 3(*C\ .935766 . &j OQ .768414 .OO .231586 36 25 .704395 .00 3EQ .935692 . ./CO oq .768703 4.82 4QO .231297 35 26 .701610 .Oo q tO .935618 . ,'8RJ OfC .768992 .ox5 4QO .231008 34 27 .704825 O.OO KQ .935543 ,^o oq .769281 .t52 4QQ .230719 33 28 .705040 o .Do 3x.>y .935469 ,ao OQ .769571 .OO 4vO .230429 32 29 .7052,54 . Of q KQ .935395 .*<^O OK .769860 .0/& 4ttrt .230140 31 30 .705469 O. Do 3.57 .935320 .-^0 .23 .770148 .ol/ 4.82 .229852 30 31 9.705683 3F=Q 9.9-35246 on 9.770437 4QO 10.229563 29 89 .705898 . Do 3 erf .983171 . vO OQ .770726 .Osi 4QO .229274 28 33 .700112 .y t 3SfV . 935097 .itRS OS 1 .771015 .OX .228985 27 34 35 .706326 .706539 .5* 3.55 3 try .9.35022 .934948 .xiO .23 i OK .771303 .771592 4.80 4.82 4QA .228697 .228408 26 25 36 .706753 . Ol q E7 .934873 .^a OK .771880 .oO 4Qf\ .228120 24 37 38 .706967 .707180 O . Ol 3.55 S.cr: .934798 .934723 ./^O .25 00 .772168 .772457 .OU 4.82 4OTV .227832 .221543 23 22 39 40 .707393 .707606 OO 3.55 3.55 .934649 .934574 .-^0 : .25 .25 .772745 .773033 . OU 4.80 4.80 .227255 .226967 21 20 41 42 9.707819 .708032 '3.55 3e~ 9.934499 .934424 .25 9.773321 .773608* 4.78 10.22667'9 .226392 19 18 43 .708245 .OO 3K.K .934349 .25 ok .773896 4.80 4QA .226104 17 44 .708458 ,OO 3cq .934274 ./&> OFC .774184 .OU 4r*o .225816 16 45 .708670 .00 3eq .934199 .!&) O^ .774471 . *O 4QA .225529 15 46 47 48 .708882 .709094 .709306 .00 3.53 3.53 3 tea .934123 .934048 .933973 . i .25 1.25 1OK .774759 .775046 .775333 .oU 4.78 4.78 40A .225241 .224954 .224667 14 13 12 49 50 .709518 .7097:30 . OO 3.53 3.52 .933898 .933822 .to 1.27 1.25 .775621 .775908 .oU 4.78 4.78 .224379 .224092 11 10 51 52 53 54 55 56 57 58 ' 59 CO 9.709941 .710153 .710364 .710575 .710786 .710997 .711208 .711419 .711629 9.711839 3.53 3.52 3.52 3.52 3.52 3.52 3.52 3.50 3.50 9.933747 .933671 .9:33596 .933520 933445 .933369 .933293 .933217 .933141 9.932066 1.27 1.25 1.27 1.25 1.27 1.27 1.27 1.27 1.25 9.776195 .776482 .776768 .777055 .777'342 .777628 .777915 .778201 .778488 9.778774 4.78 4.77 4.78 4.78 4.77 4.78 4.77 4.78 4.77 10.223805 .223518 .223232 .222945 .222658 .222372 .222085 .221799 .221512 10.221226 9 8 7 6 5 4 3 2 1 / Cosine. D. I', i Sine. D. r. Cotang. D. r. Tang. / 389 SI- TABLE XXV.-LOGARITHMIC SINES, K8' I ' Sine. D. 1". | Cosine. D. 1". Tang. D. r. Cotang. ' 9.711839 9.933066 i 9.778774 10.221226 60 1 2 3 4 5 .712050 .712260 .712469 .712679 .712889 3^50 3.48 3.50 3.50 3AQ .932990 .932914 .932838 .932762 .932685 I '.27 1.27 ! 1.27 ! 1.28 .779060 .77984(5 .779632 .779918 .780203 4.77 4.77 4.77 4.77 4.75 .220940 .220654 .220368 .220082 .219797 59 58 57 56 55 6 7 .713098 .713308 ,4o 3.50 340 .932609 .932533 1 i27* .780489 .780775 4.77 4.77 .219511 .219225 54 53 8 9 10 .713517 .713726 .713935 .4o 3.48 3.48 3.48 .932457 .932380 .932304 i!sj 1.27 1.27 .781060 .781346 .781631 4 .75 4.77 4.75 4.75 .218940 .218654 .218369 52 51 50 11 12 13 14 9.714144 .714352 .714561 .714769 3.47 3.48 3.47 o 40 9.932228 .932151 .932075 .931998 1.28 1.27 1.28 9.7'81916 .782201 .782486 .7^2771 4.75 4.75 4.75 10.218084 .217799 .217514 .217229 49 48 47 46 15 16 .714978 .715186 o .4o 3.47 3/1*7 .931921 .931845 l!*7 .783C56 .783341 4.75 4.75 .216944 .216659 45 44 17 18 19 20 .715394 .715602 .715809 .716017 .4* 3.47 3.45 1 3.47 3.45 .931768 .931691 .931614 .931537 i.ls 1.28 1.28 1.28 .783626 .783910 .784105 .784479 4.75 4.73 ^4.75 4.73 4.75 .216374 .216080 .215805 .215521 43 42 41 40 21 9.716224 347 9.931460 1 9ft 9.784764 4r-o 10.215236 39 22 23 .716432 .716639 ,4i ; 3.45 3JJC .931383 .931306 1 . -vO 1.28 .765048 .785832 . to 4.73 4i~Q .214952 .214668 38 37 24 .716846 .40 .931229 1 OQ .1 8:61(5 . (O .214384 36 25 .717'053 3/1Q ' .931152 J .-co 1 98 .785900 4.73 4r-o .214100 35 26 .717259 .4o 33*3* .931075 1 . <^O j .786184 . to .213816 34 27 .717466 .40 o jrc .930998 1 *28 .786468 4.73 4ft"Q .213532 33 28 29 .717673 .717879 3^43 3/1Q .930921 .930843 l^SO .786752 .787036 . to 4.73 4r*o .213248 .212964 32 31 30 .718085 ,4u 3.43 .930766 l'.30 .787319 . fx 4.73 .212681 SO 31 9.718291 3,40 9.930688 9.787603 10.212397 29 32 as .718497 .718703 .4o 3.43 Q A*) .930611 .930533 l!30 . 7 87886 .7*8170 4.72 4.73 .212114 .211830 28 27 34 .718909 o . 4o ! .930456 1 -~ i .788453 4.72 .211547 26 35 .719114 3 An \ .930378 ** .788736 4.72 .211264 25 36 .719320 .4o 349 .930300 1 28 .789019 j'wn .210981 24 37 .719525 .4, 3 An .930223 'on .789202 4. t/i .210698 23 38 39 .719730 .719935 .4^ 3.42 3 1O .930145 .930067 "30 on .789585 .789868 4^72 .210415 .210132 22 21 40 .7'20140 .4/5 3.42 .929989 oU .30 .790151 4.72 4.72 .209849 20 41 9.7'20345 3Af\ 9.929911 30 9.790434 10.209566 19 42 .7'20549 .4U 3A) * .929833 .790716 A l~-> .209284 18 43 44 45 46 .720754 .720958 .721163 .721366 .4/4 3.40 3.40 3.40 .929755 .929677 .929509 .929521 !30 30 .30 .790999 .791281 .791563 .791846 4. i 4.70 4.7'0 4.72 .209001 .208719 .208437' .208154 17 16 15 14 47 .721570 3.40 .929442 'on .792128 4.70 4r-n .207872 13 48 .721774 3.40 O Af\ .929364 . oU on .792410 . l(J .07590 12 49 .721978 d.40 .929286 .oU .792692 r"\ .207308 11 50 .72:2181 3.38 3.40 .929207 '30 .792974 4. 70 .207026 10 51 9.722385 3QQ 9.929129 32 9.793256 4 70 10.206744 9 52 .722588 .00 O OQ .929050 'on ! .7935:38 4AQ .206462 8 53 54 55 56 57 58 59 .722791 .7'22994 .723197 .723400 .723603 .723805 .724007 O.OO 3.38 3.38 3.38 3.38 3.37 3.37 .928972 .928893 .928815 .928736 .928657 .928578 .928499 .OU .32 iss l'32 1.32 .71)3819 .794101 .794383 .794664 .794916 .795227 .795508 .00 4.70 4.70 4.68 4.70 4.68 4.68 4fQ .206181 .205899 .205617 .205336 .205054 .204773 .204492 6 5 4 3 2 1 60 9.724210 3.38 9.928420 9. 79578 J .OO 10.204211 ' Cosine. D. r. Sine. D. r. Cotang. D.I". Tang. ' 58= COSINES, TANGENTS, AND COTANGENTS. 147= ' Sine. D. r. Cosine. D. 1". Tang. D. 1". Cotang. ' 9.724210 o o~ 9.928420 i ofi 9.795789 4Afi 10.204211 GO 1 2 .724412 ! 2-Si .928342 .724614 i ggi ! .928283 I .oU 1.32 -1 OO .79607'0 !79C851 . Oo 4.68 4f* Q .203930 j 59 .203649 58 3 .7'2-!8l6 ! o-'o- .988183 J-2| .796632 .UO .203368 57 4 5 .7-25017 .725219 | O . 00 3.37 3 ME .928104 HO .928025 J'S .796913 .797194 4i08 4 .b< .203087 56 .202806 55 6 .725420 .OO 30*7 .9271)46 J .tj/f .797474 4AQ .202526 54 8 9 .725622 .7-25823 .726024 .01 : 3.35 3.35 O Of .927867 .927787 .927708 li33 1.32 .797755 .798036 .798316 . 1x5 4.68 4.67 4A 1 ** .202245 .201964 .201684 53 52. 51 10 .726225 o'^ . 92762 J 1.'33 .798596 . O< 4.68 .201404 50 11 9.720426 9.927549 100 9.798877 10.201123 49 12 .720626 13 1 .726827 3^35 300 .927470 .927'390 UN9 i.as 100 .799157 .799437 4.67 4.67 4/rf .200843 .200563 48 47 14 ! .7'2r027 . OO 3 OK .927'310 .00 IQO .799717 . Ol .200283 46 15 .727228 .00 300 .927231 .Si .799997 .z .200003 45 16 .727428 .00 .927151 Ion .800277 4.67 .199723 44 17 .7'27628 j - .927071 .00 .800557 4.67 .199443 43 IS .727828 2-S 19 .723027 S-2 .926991 .926911 1 .33 1.33 .800836 .801116 4.65 4.67 4r*t* .199164 .198884 42 41 20 .728227 |;5 | .926831 l!33 .801396 .u< 4.65 .198604 40 21 fl. 728437 22 .723026 23 .728825 3.32 3.32 9.926751 .926671 .926591 1.33 1.33 IQO 9.801675 .801955 .802234 4.67 4.65 A f*~ 10.198325 .198045 .197766 39 38 37 24 .729024 QO .926511 .OO IQO .802513 I ?! .197487 36 25 .729223 o . BBS 300 .926431 .OO IQO .802792 4/>r .197208 35 26 I .729422 . >>* 3 Ml .926351 .OO 1QK .803072 .u< .196928 34 27 . 729621 28 .729820 29 ! .730018 .9m 3.32 3.30 3Qo . 926270 ! .926190 .926110 .OO 1.33 i.a3 Ifur !$08851 .803630 .803909 4^65 4.65 .196649 .196370 .196091 33 32 31 30 .730217 ,O 3.30 .926029 .OO 1.33 .804187 4.63 4.65 .195813 30 31 9 730415 9.925949 1 ort 9.804466 10.195534 29 32 .730013 33 i .730811 34 ! .731009 siao 3.30 3AQ .925868 .925788 .925707 li'33 1.35 1QK .804745 .805023 .805302 4.65 4.63 4 (55 .195255 .194977 .194698 28 27 26 35 .731203 .40 3QA .925626 .OO .805580 4.63 4/- .194420 25 36 37 38 .731404 .731602 .7'31799 .oU 3.30 3.28 OQ .925.545 .925465 .92538-4 1^33 1.35 IQs: .805859 .803137 .806415 .00 4 63 4.63 4/>0 .194141 .193863 .193585 24 23 22 39 ! 7-31996 O . O .925303 .OO i IQrt .806693 .00 .19:3307 21 40 .732193 3^28 .925222 .OO 1.35 .806971 4.63 4.63 .193029 20 41 9.732390 9.925141 10- 9.807249 10.192751 19 42 .7-32587 2'Sl .925060 .00 .807'527 4.63 .19247'3 18 43 44 .7-32784 - .732980 ** .924979 .924897 1 .35 | 1-37 i 10- t .807805 .80S083 4.63 4.63 .192195 .191917 17 16 45 46 .733177 *-~2 .7-33:373 I-**, .924816 .9247:35 .A) 1.85 IQX ! .808361 1808688 4.63 4.62 .191639 .191362 15 14 47 4S 49 50 .7*3569 ! 733765 .733961 .734157 O.j*f : 3.27 3.27 i 3.27 3.27 ; .924654 .924572 .924491 .924409 .00 ] 1.37 ! 1.35 1.37 1.35 .808916 .809193 .809471 .809748 4.63 4.62 4.63 4.62 4.62 .191084 .190807 .190529 .190252 13 12 11 10 51 52 53 9.734a53 .734549 .7347'44 3 27 sias 9.924328 .924246 .924164 1.37 1.37 10- 9.810025 .810302 .810580 4.62 4.63 10.189975 .189698 . 189420 9 8 54 .734939 Q O? .924083 .00 .810857 4.62 .189143 6 55 .735135 56 j .735:330 57 i .735525 5$ .735719 59 i .735914 60 : 9.7361C9 3^25 3.25 3.23 3.25 3.25 .924001 .923919 .923837 .923755 .923673 9.923591 1 .37 1.37 1.37 1 37 1.37 1.37 .811134 .811410 .811687 .811964 .812241 9.812517 4.62 4.60 4.62 4.62 4.62 4.60 .188866 .188590 .188313 .188036 .187759 10.187483 5 4 3 2 1 Cosine. | D. 1". Sine. D. 1'. i Cotang. i D. 1". Tang. ' 391 57' 33 TABLE XXV. LOGARITHMIC SINES, ' Sine. D. r. Cosine. D. 1'. Tang. D. 1". Cotang. ' 9.736109 300 9.9-23591 1Q** 9.812517 10.187483 60 1 o 3 4 5 .736303 .736498 .736692 .736886 .737080 ."o 3.25 3.23 3.23 3.23 300 .033509 .923427 .923345 .923263 .923181 .94 1.37 1.37 1.37 1.37 1QO .812794 .81:3070 .813347 .813023 .813899 4^60 4.62 4.60 4.60 .1872J6 .180930 . 180653 .180377 .180101 59 57 56 55 6 7 .737274 .737467 .46 3.22 300 .923098 .923016 .OO 1.37 1QO .814176 .814452 4.62 4.60 .185824 .185548 54 53 -8 9 10 .737061 .737855 .738048 JRI 3.23 3.22 3.22 .922933 .922851 .922768 .OO 1.37 1.38 1.37 .814728 .815004 .815280 4.60 4.60 4.60 4.58 .185272 .184990 .184720 52 51 50 11 12 13 14 15 16 17 9.738241 .738434 .738627 .738820 .739013 .739206 .739398 3.22 3 22 3.22 3 22 3.22 3.20 9.922680 .922003 .922520 .922438 .923355 .922272 .922189 1.38 1.38 1.37 1.38 1.38 1.38 1QQ 9.815555 .815831 .810107 .816382 .816658 .816933 .817209 4.60 4.60 4.58 4.60 4.58 4.60 4MB 10.184445 .184169 .183893 .183618 .183343 .183067 .182791 49 48 47 46 45 44 43 18 .739590 300 .922106 OO 10Q .817'484 .OO 4 fro .182516 42 19 .739783 . .922023 .OO 1QQ .817759 . *>O ^ (*ft .182241 41 20 .739975 3^20 .921940 .00 1.38 .818035 4.0U 4.58 .181965 40 21 9.740107 390 9.921857 1 ''^ 9.818310 4 pro 10.181690 39 22 .740359 ,iiO .921774 1 .Oo .818585 . Oo .181415 38 23 24 .740550 .740742 3.18 3.20 3 on .921691 .921607 1.38 1.40 1QQ .818860 .819135 4.58 4.58 4 to .181140 .180865 37 30 25 26 27 .740934 .741125 .741316 cU 3.18 3.18 .921524 .921441 .921357 .OO 1.38 ' 1.40 1QO .819410 .819684 .819959 .00 4.57 4.58 4 pro . 180590 .180316 .180041 35 34 33 28 .741508 3.~0 .921274 .OO .820234 .OO .179706 32 29 30 .741699 .741889 3.18 3.17 3.18 .921190 .921107 1.40 1.38 1.40 .820508 .820783 4.57 4.58 4.57 .179492 .179217 31 30 31 32 9.742080 .742271 3.18 9.921023 .920939 1.40 9.821057 .821,332 4.58 10.178943 .178668 29 28 33 34 35 36 .742462 .742652 .742842 .743033 3 18 3.17 3.17 3.18 .920856 .920772 .920688 .920604 1.38 1.40 1.40 1.40 .821606 .821880 .822154 .822429 4.57 4.57 4.57 4.58 .178394 .178120 .177840 .177571 27 26 25 24 37 38 39 40 .743223 .743413 .743602 .743792 3.17 3.17 3.15 3.17 3.17 .920520 .920436 .920352 .920268 1.40 1.40 1.40 1.40 1.40 .822703 .822977 .823251 .823524 4.57 4.57 4.57 4.55 4.57 .177*97 .177023 .176749 .176470 23 22 21 20 41 42 43 44 45 46 47 48 9.743982 .744171 .744361 .744550 .744739 .744928 .745117 .745306 3.15 3.17 3.15 3.15 3.15 3.15 3.15 31 o 9.920184 .920099 .920015 .919931 .919840 .919762 .919677 .919593 1.42 1.40 1.40 1.42 1.40 1.42 1.40 9.823798 .824072 .824345 .824019 .824893 .825106 .825439 .825713 4.57 4.55 4.57- 4.57 4.55 4.55 4.57 4*r?r 10.176202 .175928 .175655 .175381 .175107 .174834 .174501 .174287 19 18 17 16 15 14 13 12 49 50 .745494 .745683 . lo 3.15 3.13 .919508 .919424 l!40 1.42 i .825986 .826259 .DO 4.55 4.55 .174014 .173741 11 10 51 9.745871 9. 919339 9.826532 10.17-3468 9 52 .746060 3.15 31 O .919254 * 49 .82(5805 4.55 .173195 8 53 54 55 56 .746248 .746436 .746624 .746812 .lo 3.13 3.13 3.13 39m .919169 .919085 .919000 .918915 l!40 1.42 1.42 .827078 .827351 .827'624 .827897 4.oo 4.55 4.55 4.55 4KK .172922 .172649 .172376 .172103 7 6 5 4 57 .746999 .-1* .918830 1 .42 .828170 .OO A KQ .171830 3 58 59 60 .747187 .747374 9.747562 3.13 3.12 3.13 .918745 .918659 9.918574 1 .42 1.43 1.42 ! .828442 .828715 9.828987 4.5o 4.55 4.53 .171558 .171285 10.171013 2 1 ' 1 Cosine. D. 1'. Sine. D. 1". ! Cotang. D. 1". Tang, i ' 123 3C2 COSINES, TANGENTS, AND COTANGENTS. 145= ' Sine. D. 1". Cosine. D. 1". ! Tang. D. 1". Cotang. i 1 9.747562 .747749 3.12 310 9.918574 .918489 1.42 14O 9.828987 .829260 4.55 4 5S 10.171013 .170740 60 59 2 3 4 5 .747936 .748123 .748310 .748497 .Ms 3.12 3.12 3.12 .918404 .918318 .9182:33 .918147 .4/6 1.43 1.42 1.43 .829532 .829805 .830077 .830349 4.55 4.53 4.53 4KB .170468 .170195 .169923 .169651 58 57 56 55 6 .748683 .748870 3.10 3.12 .918062 .91797'6 1 .42 1.43 .830621 .830893 .OO 4.53 4P-Q .169379 .169107 54 53 8 9 .749056 .749243 3.10 3.12 .917891 .917805 1 .42 1.43 .831165 .831437 .OO 4.53 4 pro .168835 .168563 52 51 10 .749429 3.10 3.10 .917719 !42 .831709 :.OO 4.53 .168291 50 11 12 13 14 15 16 17 18 19 20 9.749615 .749801 .749987 .750172 .750858 .750543 ! 750914 .751099 .751284 3.10 3.10 3.08 3.10 3.08 3.10 3.08 3.08 3.08 3.08 9.917634 .917548 .917462 .91737-6 .917290 .917'204 .917118 .917'032 .916946 .916859 .43 .43 .43 .43 .43 1.43 1.43 1.43 1.45 1.43 9.831981 .832253 .832525 .832796 .833068 .83.3339 .833611 .833882 .834154 .834425 4.53 4.53 4.52 4.53 4.52 4.53 4.52 4.53 4.52 4.52 10.168019 .167747 .167475 .167204 .16(5932 .166661 .166389 .166118 .165846 .165575 49 48 47 46 45 44 43 42 41 40 21 22 23 24 25 9.751469 .751654 .751839 .752023 .752208 3.08 3.08 3.07 3.08 9.916773 .916687 .916600 .916514 .916427 1.43 1.45 1.43 1.45 9.834696 .834967 .835238 .835509 .835780 4.52 4.52 4.52 4.52 10.165304 .1650:33 .164762 .164491 .164220 39 38 37 36 35 26 27 28 29 30 .752392 .752576 .752760 .752044 .753128 3.07 3.07 3.07 3.07 ! 3.07 3.07 .916341 .916254 .916167 .916081 .915994 1.43 1.45 1.45 1.43 1.45 J.45 .836051 .836322 .836593 .836864 .837134 4.52 4.52 4.52 4.52 4.50 4.52 .163949 .163678 .163407 .163136 .162866 34 33 32 31 80 31 32 33 34 35 36 37 38 9.753312 .753495 .75:3679 .753862 .754046 .754229 .754412 .754595 3.05 3.07 3.07 3.07 3.05 305 3.05 3f\~ 9.915907 .915820 .915733 .915646 .915559 .915472 .915385 .915297 1.45 1.45 1.45 1.45 1.45 1.45 1.47 9.837405 .&S7675 .837946 .838216 .833487 .838757 .839027 .839297 4.50 4.52 4.50 4.52 4.50 4. .50 4.50 4 ICO 10.162595 .162325 .162054 .161784 .161513 .161243 .160973 .160703 20 28 27 26 25 94 23 22 39 40 .754778 .754960 .Oo 3.03 3.05 .915210 .915123 1 .45 1.45 1.47 .839568 .839838 .5,6 4.50 4.50 .160432 .160162 21 20 41 42 9.755143 .755326 3.05 9.915035 .914948 1.45 9.840108 .840878 4.50 4-rA 10.159892 .159622 19 18 43 44 ,755508 .755690 3. 03 .914860 .914773 1 .47 1.45 .840648 .U0917 .oU 4.48 .159352 .159083 17 16 45 .755872 3.03 3AQ .914685 1.47 .841187 4.50 .158813 15 46 47 48 49 50 .756054 .756236 .756418 ] 756600 .756782 . Uo 3.03 i 3.03 3.03 3.03 3.02 .914598 .914510 .914422 .914334 .914246 1 .45 1.47 1.47 1.47 1.47 1.47 .841457 .841727 .841996 .842266 .842535 4^50 4.48 . 4.50 4.48 4.50 .158543 .158273 .158004 . 157734 .157465 14 13 12 11 10 51 9.756963 9.914158 9.842805 10.1VT195 9 52 .757144 3.03 3 no .914070 1.47 .843074 4.48 4AQ 156926 8 53 54 55 56 57 58 59 .757326 .757507 .757688 .757869 .758050 .758230 .758411 Uo 3.02 3.02 3.02 3.02 3.00 3.02 3 An .913982 .913894 .913806 .913718 .913630 913541 .913453 1 .47 1.47 1.47 1.47 1.47 1.48 1.47 .843343 .843612 .84:3882 .844151 .844420 .844689 .844958 .4o 4.48 4.50 4.48 4.48 4.48 4.48 .156657 .156388 .156118 .155849. .155580 .155311 .155042 6 5 4 3 2 1 60 9.758591 . W 9.91:3365 1.47 9.845227 4.48 10.154773 ' Cosine. D. r. Sine. D. r. ! Cotang. D. 1". Tang. > 393 55 35= TABLE XXV. LOGARITHMIC SINES, i | Sine. D. 1". Cosine. D. 1'. Tang. D. 1". Cotang. ' 9.758591 9.913365 1AQ 9.845227 4AQ 10.154773 60 1 2 .758772 .758952 3.02 3.00 .913270 .913187 .4o 1.48 .845496 .845764 :.4O 4.47 .154504 .154236 59 58 3 .759132 3.00 .913099 1.47 .846033 4.48 .153967 57' 4 5 6 .759312 .759492 .759672 3.00 3.00 3.00 .913010 .912922 .912833 1 .48 1.47 1.48 11V .846302 .8-16570 .846839 4.48 4.47 4.48 4AQ .153698 .153430 .153161 56 55 54 7 8 9 .759852 .760031 .760211 2^98 3.00 .912744 .912655 .912566 ,4o 1.48 1.48 1AQ .847108 .847376 .847644 .4o 4.47 4.47 4AQ .152892 .152624 .152356 53 52 51 10 .760390 2! 98 .912477 .4o 1.48 .847913 .4o 4.47 .152087 50 11 9.760569 9.912388 9.848181 4 AW 10.151819 49 12 .760748 2(\Q .912299 1AQ .848449 .4i .151551 48 13 .760927 .yo O OQ .912210 .4o .848717 4.47 4AQ. .151283 47 14 .761106 .912121 1 .48 1KA .848986 .4o .151014 46 15 .761285 2. Jo .913031 .ou .849254 4.47 .150746 45 16 17 18 19 20 .761464 .761642 .761821 .761999 .762177 2.98 2.97 2.98 2.97 2.97 2.98 .911942 .911853 .911763 .91167'4 .911584 1 .48 1.48 1.50 1.48 1.50 1.48 .849522 .849790 .850057 .850325 .850593 4.47 4.47 4.45 4.47 4.47 4.47 .150478 .150210 .149943 .149675 .149407 44 43 42 41 40 21 22 23 24 25 26 9.762356 .762534 .762712 .762889 .763067 .763245 2.97 2.97 2.95 2.97 2.97 9.911495 .911405 ,911315 .911226 .911136 .911046 1.50 1.50 1.48 1.50 1.50 9.850861 .851129 .851396 .851664 .851931 .852199 4.47 4.45 4.47 4.45 4.47 10.149139 .148871 .148004 .148336 .148069 .147801 39 38 37 36 35 34 27 .763422 2.95 2O7 .910956 1 .50 1&A .852466 4. 45 .147534 33 28 .763600 .y< .910866 . .ou Irn .8527'33 A A .147267 32 29 30 .763777 .763954 2.95 2.95 2.95 .910776 .910686 .OU 1.50 1.50 . .853001 .853268 4^45 4.45 .146999 .146732 31 30 31 32 9.764131 .764308 2.95 9.910596 .910506 1.50 9.853535 .853802 4.45 10.146465 .146198 29 28 34 35 36 37 .764485 .764662 .764838 .765015 .765191 2.95 2.95 2.93 2.95 2.93 .910415 .910325 .910235 .910144 .910054 1 . 52 1.50 1.50 1.52 1.50 .854069 .854336 .854603 .854870 .&55137 4^45 4.45 4.45 4.4 .145931 .145004 .145397 .145130 .1-14803 27 26 25 24 23 38 .765367 2. Jo .909963 1 .52 -i rn .855404 4AZ. .144596 22 39 .765544 2.95 .909873 1 ,OU .855671 . 4o .144329 21 40 .765720 2.93 2.93 .909782 1.52 1.52 .855938 4.45 4.43 .144062 20 41 42 43 9.765896 .766072 .766247 2.93 2.92 9.909691 .909601 .909510 1.50 1.52 1RO 9.&56204 .856471 .856737 4.45 4.43 4Ae 10.143796 .143529 .143263 19 18 17 44 45 46 .766423 .766598 .766774 2^92 2.93 .909-119 .909328 .909237 .'M 1.52 1.52 .857004 .857270 .857537 .4u 4.43 4.45 .142996 .142730 .142463 16 15 14 47 48 49 50 .766949 .767124 .797800 .707475 2.92 2.92 2.93 2.92 2.90 .909146 .909055 .908964 .908873 1^52 1.52 1.52 1.53 .857803 .858069 .858336 .858803 4^43 4.45 4.43 4.43 .142197 .141931 .141664 .141398 13 12 11 10 51 52 53 54 55 56 9.767649 .767824 .767999 .768173 .768348 .768522 2.92 2.92 2.90 2.92 2.90 9.908781 .908690 .908599 .908507 .908416 .90832-1 1.52 1.52 1.53 1.52 1.53 9.858868 .859134 .859400 .859666 .859982 .800198 4.43 4.43 4.43 4.43 4.43 4 4^ 10.141132 .140800 .140600 .140334 .140008 .139802 9 8 6 5 4 57 58 59 60 .768697 .768871 .769045 9.769219 2^90 2.90 2.90 .908333 .908141 .908049 9.907958 1.53 1.53 1.52 .860464. .860730 .860995 9.861261 4.'43 4.42 4.43 .139536 .139270 .139005 10.138739 3 2 1 ' Cosine. I D. 1". il Sine. D. 1". Cotang. D. i". i Tang. ' 125 394 COSINES, TANGENTS, AND COTANGENTS. 143' t Sine. D. 1". Cosine. D. 1". Tang. D. 1". Cotang. ' 1 2 9.769219 .769393 .769566 2.90 2.88 9.907'958 .907866 .907774 .53 .53 pro 9.861261 .861527 .861792 4.43 4.42 4,|O 10.138739 .138473 .138208 60 59 58 3 .769740 2.90 .907682 .Do Mk .862058 .4o .137942 57 4 5 .769913 .770087 2.88 2.90 2QO .907590 .907498 .OO .53 tQ .862323 .862589 4 '.43 .137677 .137411 56 55 6 7 .770260 .770433 .OO 2.88 O OO .907406 .907314 .OO .53 eo .862854 .863119 4'.42 4/1Q .137146 .136881 54 53 8 9 10 .770608 .770779 .770952 2.oo 2.88 2.88 2.88 .907222 .907129 .907037 .Do .55 1.53 1.53 .863385 .863650 .863915 .4o 4.42 4.42 4.43 .136615 .136350 .136085 52 51 50 11 12 13 14 15 16 17 18 19 20 9.771125 .771298 .171470 .771643 .771815 .771987 .772159 .772331 .772503 .772675 2.88 2.87 2.88 2.87 2.87 2.87 2.87 2.87 2.87 2.87 9.906945 .906&52 .9067'60 .906667 .906575 .906482 .906389 .906296 .906204 .906111 1.55 1.53 1.55 1.53 1.55 1.55 1.55 1.53 1.55 1.55 9.864180 .864445 .864710 .864975 .865240 .865505 .865770 .866035 .866300 .866564 4.42 4.42 4.42 4.42 4.42 ' 4.42 4.42 4.42 4.40 4.43 10.135820 .i:555 .135290 .135025 .1:34760 .134495 .134230 .1:33965 .133700 .133436 49 48 47 46 45 44 43 42 41 40 21 22 23 24 25 26 27 28 29 30 9.772847 .773018 .773190 .773361 .773533 ! 778704 .773875 .774046 .774217 .774388 2.85 2.87 2.85 2.87 2.85 2.85 2.85 2.85 2.85 2.83 9.906018 .905925 .905832 .905739 .905645 .905552 .905459 .905366 .905272 .905179 1.55 1.55 1.55 1.57 1.55 1.55 1.55 1.57 1.55 1.57 9.866829 .867094 .867358 .867623 .867887 .868152 .868416 .868680 .868945 .869209 4.42 4.40 4.42 4.40 4.42 4.40 4.40 4 42 4.40 4.40 10.133171 .132906 .132642 .132377 .132113 .131848 .131584 .131320 .131055 .130791 39 38 37 36 35 34 33 32 31 30 31 32 9.774558 .774729 2.05 9.905085 .904992 1.55 9.869473 .86,7737 4.40 10.130527 .130263 29 28 33 34 35 .774899 .775070 .775240 2.83 2.85 2.83 2QQ .904898 .904804 .904711 1 .57 1.57 1.55 Itr* .87'0001 .870265 .870529 4.40 4.40 4.40 4 /in .129999 .129785 .129471 27 26 25 36 37 .775410 .775580 .00 2.83 .904617 .904523 . D* 1.57 .870793 .871057 .40 4.40 .129207 . 128943 24 23 38 .775750 2.83 2 CO .904429 1 .57 .871321 4.40- 4Af\ . 128679 22 39 40 .775920 .776090 .00 2.83 2.82_ .904335 .904241 1 .57 1.57 1.57 .871585 .871849 .40 4.40 4.38 .128415 .128151 21 20 41 9.776259 2QO 9.904147 1 57 1 9.872112 44f\ 10.127888 19 42 .776429 .OO .904053 IKty .872376 :.4U 4A(\ .127624 18 43 44 45 .776598 .776768 .776937 2^83 2.82 .903959 .903864 .903770 .Df 1.58 1.57 .872640 .872903 .873167 .40 4.38 4.40 .127360 .127097 .126833 17 16 15 46 .777106 2.82 .903676 1 .57 .873430 4.38 .126570 14 47 .777275 2.82 200 .903581 1 .58 1frf*f .873694 4.40 4QQ * .126306 13 48 .777444 .0,4 .903487 .Of .873957 .00 .126043 12 49 50 .777613 .777781 2.82 2.80 2.82 .903392 .903298 1.57 1.58. .874220 .874484 4.38 4.40 4.38 .125780 .125516 11 10 51 9.777950 9.903203 Ip-Q 9.874747 400 10.125253 9 52 .778119 2. o3 .903108 .08 IfcM .875010 .00 .124990 8 53 54 55 56 57 .778287 .778455 .778624 .77'8792 .778960 2!80 2.82 2.80 2.80 O QA .903014 .902919 .902824 .902729 .902634 .Of 1.58 .58 .58 .58 . .875273 .875537 .875800 .876063 .876326 4.38 4.40 4.38 4.38 4.38 .124727 .124463 .124200 .123937 .123674 7 6 5 4 3 58 .779128 /w.OU .902539 .58 .876589 4.38 .123411 2 59 60 .779295 9.779463 2.78 2.80 .902444 9.902349 .58 1.58 .876852 9.877114 4.38 4.37 .123148 10.122886 1 * Cosine. D. 1". Sine. D. 1". Cotang. D.r. Tang. ' 395 53 37 C TABLE XXV. -LOGARITHMIC SINES, , Sine. D. 1". Cosine. D. 1'. i Tang. D. 1". Cotang. | ' '' 1 1 I 3 4 5 6 7 8 9.779463 .779631 .779798 .779966 .780133 .780300 .780467 .780334 .780801 2.80 2.78 2.80 2.78 2.78 2.78 2.78 2.78 2r-o 9.902349 .902253 .902158 .902063 .901967 .901872 .901776 .901681 .901585 1.60 1.58 1.58 1.60 1.58 1.60 1.58 1.60 ]eo 9.&77114 .877377 .877640 .8771:03 .878165 .878428 .878691 .878953 .879216 4.38 4.38 4.38 4.37 4.38 4.38 4.37 4.38 A Q7 10.122886 .122623 .122360 .122097 .121835 .121572 .121809 .121047 .120784 60 59 58 57 56 55 . 54 53 52 9 .780968 . (8 .901490 .OO 1AA .879478 4 . Gi 4OQ . 120522 51 10 .781134 2.77 2.78 .901394 .00 1.60 .879741 .GO 4.37 .120259 50 11 12 9.7'81301 .781468 2.78 9.901298 .901202 1.60 9.880003 .880265 4.37 4OQ 10.119997 .119735 49 48 13 14 .781634 .781800 2.77 2.77 .901106 .901010 140 .880528 .8H0790 .GO 4.37 4 S7* .119472 .119210 47 46 15 16 .781966 .782133 2.77 2.77 .900914 .900818 140 i .881052 .881314 4^37 A qo .118948 .118686 45 44 17 18 .782298 .782464 2.77 2.77 .900722 .900626 I'M 1AO .881577 .881839 4 . oo 4.37 4Q*7 .118423 .118161 43 42 19 .782630 2-11 .900529 .TO 1AA .882101 .04 A Q7 .117895) 41 20 .782796 2. it 2.75 .900433 . DU 1.60 .882363 4 . O* 4.37 .117637 40 21 OO 23 9.782961 .783127 .783292 2.77 2.75 9.900337 .900240 .900144 1.62 1.60 9.882625 .882887 .883148 4.37 4.35 40.7 10.117375 .117113 .116852 39 38 37 24 25 26 27 .783458 .783623 .783788 .783953 2.77 2.75 2.75 2.75 .900047 .899951 .899854 .899757 1:00 1.62 1.62 .883410 .883672 .883934 .884196 .G< 4.37 4.37 4.37 4 OK .116590 .116328 .1160(56 .115804 36 35 34 33 28 29 .784118 .784282 2.75 2.73 .899660 .899564 140 .884457 .884719 .GO 4.37 A OK .115543 .115281 32 31 30 .784447 2.75 2.75 .899467 148 .884980 4. GO 4.37 .115020 30 31 32 33 34 9.784612 .784776 .784941 .785105 2.73 2.75 2.73 9.899370 .899273 .899176 .899078 1.62 1.62 1.63 1AO 9.885242 .885504 .885765 .886026 4.37 4.35 4.35 A Q7 10.114758 .114496 .114235 .113974 29 28 27 26 35 36 37 38 39 40 .785269 .785433 .785597 .785761 .785925 .786089 2^73 2.73 2.73 2.73 2.73 2.7'2 .898981 .898884 .898787 .898689 .898592 .898494 .0x5 1.62 1.62 1.63 1.62 1.63 1.62 .886288 i .886549 .886811 1 .887072 .887883 .887594 4.o< 4.35 4.37 4.35 4.35 4.35 4.35 .113712 .113451 .113189 . .112928 .112667 .112406 25 24 23 22 ! 21 20 41 42 43 44 45 46 47 48 49 9.786252 .786416 .786579 .786742 .786906 .787069 .787232 .787395 .787557 2.73 2.72 2.72 2.73 2.72 2.72 2.72 2.70 9.898397 .898299 .898202 .898104 .898006 .897908 .807810 .897712 .897614 1.63 1.62 1.63 1.63 1.63 1.63 1.63 1.63 ! 9.887855 i .888116 .888378 .888639 .888900 .889161 .889421 .889682 .889943 4.35 4.37 4.35 4.35 4.35 4.33 4.35 4.35 4O-r 10.112145 | 19 .111H84 ; 18 .111(522 I 17 .111361 16 .111100 15 .110839 ! 14 .110579 j 13 .110318 12 .110057 11 50 .787720 2.72 2.72 .897516 1 .63 1.68 .890204 . y> 4.35 .109796 10 51 52 53 54 55 56 57 58 59 GO 9.787883 .788045 .788208 .788370 .788532 .788694 .788856 .789018 .789180 9.789342 2.70 2.72 2.70 2.70 2.70 2.70 2.70 2.70 2.70 9.897418 .897320 .897222 ,897123 .897025 .896926 .896828 .896729 .896631 9.896532 1.63 1.63 1.65 1.63 1.65 1.63 1.65 1.63 1.65 9.890465 .890725 .890986 .891247 .891507 .89176S .892028 .892289 .892549 9.892810 4.33 4.35 4.35 4.33 4.35 4.33 4.35 4.33 4.35 10.109535 .109275 .109014 .108753 .108493 .106282 .107972 .107711 .107451 10.107190 9 ! 8 6 5 4 3 2 1 ' Cosine. D 1". il Sine, i D. 1. ! Cotang. D. 1". Tang, i ' 127 C COSINES, TANGENTS, AND COTANGENTS. 141' ' Sine. D. 1". Cosine. D. 1'. Tang. D. 1". Cotang. / 9.789342 .789504 2.70 9.896532 .890433 1.65 1 r*3 9.892810 .893070 4.33 4 QC 10.107190 .106930 60 59 2 3 .789665 .789827 2.68 2.70 2AQ .896335 .896236 . Do 1.65 1/" .893331 .89:3591 .OO 4.33 4OQ .106669 .106409 58 57 4 .789988 .DO .896137 . DO .893851 .OO 400 .106149 56 5 .790149 2.08 2AQ .896038 1 .65 COL .894111 .OO 4oe .105889 55 6 .790310 .DO 2/o .895939 . DO AX ' .894372 . oO 400 .105628 54 .790171 .OO 2AQ .895840 .DO ; ftrr .894632 .OO A QQ .105368 53 8 .790632 .Do 2AQ .895741 .DO .894892 4.Oi* 4QO .105108 52 9 .790793 .DO .895641 .67 ; .895152 .00 4QQ .104848 51 10 .790954 2.68 2.68 .895542 ! .05 .65 i .895412 .00 4.33 .10-1588 50 11 12 13 14 15 16 9.791115 .791275 .791436 .791596 .791757 .791917 2.67 2.68 2.67 2.68 2.67 9.895443 .895343 .895244 .895145 .895045 .894945 .67 : .65 .65 .67- ! .67 9.895672 .895932 .896192 .898452 .896712 .898971 4.33 4.33 4.33 4.33 4.32 400 10.104328 .104068 .103808 .103548 .103288 .103029 49 48 47 46 45 44 17 18 19 20 .792077 .792237 .792397 .792557 2.67 2.67 2.67 2.67 2.65 .894846 .894746 .894646 .894546 .65 .67 ! .67 i .67 ; .67 .897231 .897491 .897751 .898010 .00 4.33 4.33 4.32 4.33 .102769 .102509 .102249 .101990 43 42 41 40 21 22 23 24 25 26 27 9.792716 .793376 .793033 .793195 .793354 .79S514 .793673 2.67 2.65 2.67 2.65 2.67 2.65 9.894446 .894346 .894246 .894146 .894046 .893946 .893846 .67 .67 .67 .67 .67 .67 />Q 9.898270 .898530 .898789 .899049 .899308 .899568 .899827 4.33 4.32 4. as 4.32 4.33 4.32 400 10.101730 .101470 .101211 .100951 .100692 .100432 .100173 39 38 37 36 35 34 33 28 29 .793832 .793991 2.65 2.65 o /~ .893745 .893645 .DO .67 fQ .900087 .900346 .00 4.32 4- QO .099913 .099654 32 31 30 .794150 . uo 2.63 .893544 .OO .67 .900605 .34 4.32 .099395 30 31 9.794303 2 n* 9.893444 AQ 9.900864 400 10.099136 29 32 .794467 .DO o /'X. .893343 .DO A*7 .901124 .00 40.) .098876 2S 33 .794G26 /&.DO 2 OS .893243 .Di AQ .901383 ,oSs 4na .098617 27 34 .794784 .00 .893142 .DO f*Q .901642 .O^ .098358 26 ,35 .794942 2.63 2A^ .893041 .DO ro .901901 4.32 400 .098099 25 36 .795101 .DO .892940 . vO fQ .902160 . - * - 4QQ .097840 24 37 .795259 2.63 2>q .892839 .DO A*7 .902420 .00 4QO .097580 23 38 .795417 . DO O AQ .892739 .D( AQ .902679 .O 4QO .097321 22 39 40 .795575 .795733 /. Do 2.63 2.63 .892638 .892536 .DO .70 .68 .902938 .903197 .0^ 4.32 4.32 .097062 .096803 21 20 41 9.795891 2fiO 9.892435 AQ 9.903456 4OA 10.096544 19 42 .796049 .DO 2 A3 .892334 .DO AQ .903714 .OV 4QO .096286 18 43 .796205 .O^ 2/o .892233 .DO 00 .903973 CEQ .096027 17 44 .796364 .DO 21'.) .892132 .DO r-rt .904232 4.32 4QO .095768 16 45 .796521 . D.4 t> no .892030 . rfU AQ .904491 .O/v 400 .095509 15 40 .796679 ^.DO 2j'.> .891929 .DO . 904750 .06 4OA .095250 14 47 .796836 . ' ) - O AO .891827 /Q .905008 .oU 4QO * .094992 13 48 .796993 < . o/ .891726 .DO .905267 .0*5 4QO .0947:33 12 49 .797150 O AO .891624 .70 AQ .905526 .64 400 .004474 11 50 .797307 A, U-J 2.02 .891523 .DO .70 .905785 blSI 4.30 .094215 10 51 9.797464 O AO 9.891421 9.906043 4QO 10.093957 9 52 53 54 55 56 57 58 .797621 .797777. .797934 .798091 .798247 .798403 .798560 f\i . \) .77 .921247 .so 4.27 .078753 10 51 52 9. -806709 .806860 2.52 9.885205 .885100 .75 9.921503 .921760 4.28 10.078497 .078240 9 8 53 .807011 2.52 .884994 ' r"~ .922017 a Is .077983 7 54 .807163 2.53 .884889 'X~ .922274 4 27 .077726 6 55 .807314 2.52 .884783 . l 1 .922530 .077470 5 56 .807465 2.52 .884677 .77 .922787 4.28 .077213 4 57 58 .807615 .807766 2.50 2.52 .884572 .884466 I 7 .923044 .923300 4^27 4OQ .076956 .076700 3 2 59 .807917 A.oA .884360 . 77 .923557 . ^5o 4OQ .076443 1 60 9.808067 2.50 9.884254 .77 9.923814 .**& 10.076186 ' Cosine. D. 1". Sine. D. r. 1 Cotang. D. r. Tang. ' 129 303 COSINES, TANGENTS, AND COTANGENTS. 139 C * Sine. D. r. Cosine. D. r. Tang. D. 1'. Cotang. '- 1 9.808067 .808218 2.52 9.884254 .884148 : .77 9.923814 .924070 4.27 10.076186 .075930 60 59 2 3 4 5 6 7 8 9 10 .808:368 .808519 .808669 .808819 .808960 .809119 .809269 .809419 .809569 2 '.52 2.50 2.50 2.50 i 2.50 1 2.50 2.50 2.50 2.48 .884042 .883936 .883829 .883723 .883617 .883510 .883404 .883297 .883191 '.77 .78 .77 .77 .78 .77 .78 .77 .78 .924327 .924583 .924840 .925096 .925352 .925609 .925865 .926122 .926378 4'. 27 4.28 4.27 4.27 4.28 4.27 4.28 4.27 4.27 .075673 .075417 .075160 .074904 .074648 .074391 .074135 .073878 .073622 58 57 56 55 54 53 52 51 58 11 12 9.809718 .809868 2.50 2/itt 9.883084 .882977 .78 9.926634 .926890 4.27 4OQ 10.07a366 .073110 49 48 13 .810017 .4o .882871 '. .77 .927147 . 4.25 4.25 .059051 .058796 .058541 53 52 51 K) .818392 2^40 .876678 .oO 1.83 .941713 4^25 .058287 50 11 9.818536 .818681 2.42 9.876568 .876457 1.85 i OO 9.941968 .942223 4.25 10.058032 .057777 49 48 13 14 .818825 .818989 2.40 2.40 .876347 i f'S2 .876236 ?-S .942478 .942733 4.25 4.25 .057522 .057267 47 46 15 16 17 .819113 .819257 .819401 2.40 2.40 2.40 2A(\ .876125 .876014 .875904 1 .Oi> 1.85 1.83 1QX .942988 | .943243 ! .943498 4.25 4.25 4.25 4OQ .057012 .056757 .056502 45 44 43 18 19 20 .819515 .819889 .819332 .4U 2.40 2.33 2.40 .875793 .87568-2 .875571 .OO 1.85 1.85 1.87 i .943752 ! .944007 I .944262 . -> 4-. 25 4.25 4.25 .056248 .055993 .055738 42 41 40 21 22 9.819976 .820120 2.40 2OQ 9.875459 .875348 1.85 9.944517 .944771 4.23 10.055483 .055229 39 38 23 .820233 .OO .875237 1 C" .945026 **'~~ .054974 37 24 25 26 .820408 .820550 .820893 2^40 2.33 2QQ .875126 .875014 .874903 1 .00 1.87 1.85 .945281 .945535 .945790 4 . 2o 4.23 4.25 .054719 .054465 .054210 36 35 34 27 .820836 . OO aOO .874791 Q- .946045 A OQ .058955 33 23 29 .820979 .821122 .OO 2.38 2QQ .874680 .874568 1.87 .946299 .946554 4*25 .053701 i 32 .053446 ! 31 30 .821265 .00 2.37 .874456 1.87 .946808 4 "25 .053192 30 31 32 33 34 9.8-21407 .821550 .8-21693 .8218'35 2.38 2.38 2.37 o 07 9.874344 .874232 .874121 .874009 1.87 1.85 1.87 100 9.947063 .947318 .947572 .947827 4.25 4.23 4.25 10.052937 .052682 .052428 .052173 29 28 27 26 35 36 37 38 39 40 .821977 .822120 .822262 .822404 .823546 .822688 2^38 2.37 2.37 2.37 2.37 2.37 .873896 .873784 .873672 .873560 .873448 .873335 .00 1.87 1.87 1.87 1.87 1.88 1.87 .948081 .948335 .948590 .948844 .949099 .949353 4/23 4.25 4.23 4.25 4.23 4.25 .051919 .051665 .051410 .051156 .050901 .050647 25 24 22 21 20 41 42 43 44 45 9.822330 .822972 .823114 .823255 .8-23397 2.37 2.37 2.35 2.37 9.873223 .873110 .872998 .872885 .872772 1.88 1.87 1.88 1.88 1QQ 9.949608 .949862 .950116 .950371 .950625 4.23 4.23 4.25 4.23 4QO 10.050392 .050K38 .049884 .049629 .049375 19 18 17 16 15 46 .823539 .4.67 .872659 .OO IQrt .950879 . 3 4OQ .049121 1 14 47 .823680 2.35 .872547 .of 100 .951133 .&) .048867 1 13 48 .823821 2.35 20? .872434 .00 100 .951388 4.25 .048612 12 49 .823963 .of O QK. .872321 .00 1 ftH .951642 I'M .048858 i 11 50 .824104 ,.OO 2.35 .872208 l. .951896 4:23 .018104 10 51 52 9.824245 .824386 2.35 i 9.872095 ! .871981 1.90 -, 00 9.952150 .952405 4.25 10.047850 .047595 9 8 53 54 .824527 .824668 2.35 2.35 200 .871868 .871755 1 .00 1.88 IfWl .952659 .952913 4^23 .047341 .047087 7 6 55 .824808 .00 2 OK .871641 . \f\J 100 .953167 AIM .046833 5 56 57 .824949 .825090 .OO 2.35 i .871528 .871414 .00 1.90 100 . 953421 .953675 4i23 .046579 .046325 3 58 .825230 **25 .871301 .00 1(\f\ .953929 AIM .046071 2 59 60 .825371 9.825511 2.35 2.33 .871187 9.871073 ,W 1.90 .954183 9.954437 4 '.28 .045817 10.045563 ' Cosine. D. r. Sine. D.I". Cotang. 1 D. 1". 1 Tang. ' 4UO 48 COSINES, TANGENTS, AND COTANGENTS. 137 C ' Sine. D. 1". Cosine. D. r. Tang. D. 1". Cotang. ' 1 2 9.825511 .825651 .825791 2. as 2.33 9.871073 .870960 .870846 1.88 1.90 9.954437 .954691 .954946 4.23 4 25 10.045563 .045309 .045054 60 59 58 3 4 5 6 8 9 10 .825931 .826071 .826211 .826351 .826491 .826631 .826770 .826910 2^33 2.33 2.33 2.33 2.33 2.32 2.33 2.32 .870732 .870618 .870504 .870390 .870276 .870161 .870047 .869933 1 .90 1.90 1.90 1.90 1.90 1.92 1.90 1.90 1.92 .955200 .955454 .955708 .955961 .956215 .956469 .956723 ,956977 4.23 4.23 4.23 4.22 4.23 4.23 4.23 4.2Z .044800 .044546 .044292 .044039 .043785 .04a531 .043277 .043023 57 56 55 54 53 52 51 50 11 9.827049 9.869818 9.957231 10.042769 49 a .827189 .827328 2.33 2.32 .869704 .869589 1 .90 1.92 .957485 .957739 4.23 .042515 .042261 48 47 14 15 16 17 .827467 .827606 .827745 .827884 2.32 2.32 2.32 2.32 .869474 .869360 .869245 .869130 1 .92 1.90 1.92 1.92 .957993 .958247 .958500 .958754 4". 23 4.22 4.23 4 CM .042007 .041753 .041500 .041246 46 45 44 43 18 .828023 2.32 .869015 1.92 .959008 .&> A OQ .040992 42 19 20 .828162 .828301 2.32 2.32 2.30 .868900 .868785 1 .92 1.92 1.92 .959262 .959516 4.>4010 .86:3892 1.95 1.97 ! 9.969656 .969909 .970162 4.22 4.22 4OQ 10.020344 .030091 .029838 60 59 58 3 .834189 &J5 .863774 1 . J7 i .970416 WO .029584 57 4 .834325 6.6^ .863656 1 . 97 j 1 Q7 .970669 4.2J 4 fc ^2 .029331 56 5 ft. 7 .834460 .834595 .834730 2^25 2.25 2 OK, .863538 .86:3419 .863301 l!98 i 1.97 S .970922 .971175 .971429 4.92 4.23 .029078 .028825 .028571 55 54 53 8 .834865 .*O .863183 1 Q .971682 1 99 .028318 52 9 10 .834999 .835134 2^25 2.25 .863064 .862946 l'.97 1 1.98 .9719&5 .972188 4^22 4.22 .028065 .027812 51 50 11 9.835209 00 9.862827 10*7 9.972441 10.027559 49 12 .&35403 .-. . ',') Q OPv .862709 .y i .972695 4., Vers. D. 1". Ex. sec. D. 1". 8.181022 24.05 8.188271 24.42 8.264176 21.85 8.272229 22.27 i .183065 24.00 .189736 24.37 1 .265487 21.82 .273565 22.22 2 .184505 23.97 .191198 24.35 2 .266796 21.78 .274898 22.20 3 .185943 23.93 .192659 24.30 3 .268103 21.75 .276230 22.17 4 .187379 23.88 .194117 24.25 4 .269408 21.72 .277560 22.13 5 .188812 23.85 .195572 24.22 5 .270711 21.68 .278888 22.08 6 .190243 23.80 . 197025 24.18 ; 6 .272012 21.65 .280213 22.07 7 .191671 23.77 .198476 24.15 1 7 .273311 21.62 .281537 22.03 8 .193097 23.73 .199925 24.10 ! 8 .274608 21.58 .282859 22.00 9 .194521 23.68 .201371 24.07 9 .275903 21.57 .284179 21.98 10 .195942 23.65 .202815 24.03 10 .277197 21.52 .285498 21.93 11 8.197361 23.62 8.204257 23.98 11 8.278488 21.48 8.286814 21.90 12 .198778 23.57 .205696 23.95 12 .279777 21.47 .288128 21.88 13 .200192 23.53 .207133 23.92 13 .281065 21.42 .289441 21.83 14 .201604 23.50 .208568 23.88 14 .282350 21.40 .290751 21.82 13 .203014 23.45 .210001 23.83 15 .283634 21.37 .292060 21.78 18 .204421 23.42 .211431 23.80 16 .284916 21.33 1 .293367 21.75 17 .205826 23.38 .212859 23.77 17 .286196 21.28 .294672 21.72 18 .207229 23.35 .2142S5 23.72 18 .287473 21.27 .29597'5 21.68 19 .208630 ! 23.30 .215708 23.70 19 .288749 21.25 .297276 21.67 20 .2100;>8 23.27 .217130 23.65 20 .290024 21.20 .298576 21.62 21 8.211424 23.23 8.218549 23.62 21 8.291296 21.17 8.299873 21.60 22 .212318 23.18 .219966 23.57 22 .292566 21.15 .301169 21.57 23 .214209 23.17 .221380 23.55 23 .293335 21.10 .302463 21.53 24 .215599 23.12 .222793 23.50 24 .295101 21.08 .303755 21.50 25 .216986 23.08 .224203 23.47 A*O .296366 21.05 .305045 21.48 26 .218371 23.03 .225611 23.43 26 .297629 21.02 .306334 21.43 27 .219753 23.00 .227017 23.40 27 .298890 20.98 .307620 21.42 28 .221133 22.98 .223421 23.35 28 .300149 20.95 .308905 21.38 29 .222512 22.93 .229822 23.32 29 .301406 20.93 .310188 21 .35 30 .223888 22.88 .231221 23.30 30 .302662 20.90 .311469 21.33 31 8.225261 22.87 8.232619 23.25 31 ,8.303916 20.85 8.312749 21.28 32 .226633 22.82 .234014 23.22 32 .305167 20.85 .314026 21.27 33 .228002 22 78 .235407 23.17 33 .306418 20.80 .315302 21.23 34 .229369 23! 77 .236797 23.15 34 .307666 20.77 .316576 21.22 35 .230735 22.70 .238186 23.10 35 .308912 20.75 .317849 21 17 36 .232097 22.68 .239572 23.08 36 .310157 20.73 .319119 21.15 37 .2-33458 22.65 .240957 23.03 37 .311400 20.68 .320388 21.12 38 .234817 22.60 .242339 23.00 38 .312641 20.65 .321655 21.08 39 .236173 22.57 .243719 22.97 39 .313880 20.62 .322920 21.05 40 .237'527 22.55 .245097 22.93 40 .315117 20.60 .324183 21.03 41 8.238880 22.50 8.246473 22.90 41 8.316353 20.57 8.325445 21.00 42 .2J0230 22.47 .247847 22.87 42 .317587 20.53 .326705 20.98 43 .241578 22.43 .249219 22.83 43 .318819 20.50 .327964 20.93 44 .242924 22.38 .250589 22.80 44 .320049 20.48 .329220 20.92 45 .244267 22.37 .251957 22.75 45 .321278 20.45* .330475 20.88 46 .245609 22.32 .253322 22.73 46 .322505 20.42 .331728 20.87 47 .246948 22.30 .254686 22.68 47 .323730 20.38 .a32980 20.82 48 .248286 22.25 .256047 22.67 48 .324953 20.37 .334229 20.80 49 .240321 22.23 .257407 22.62 49 .326175 20.33 .335477 20.78 50 .250955 22.18 .258764 22.60 50 .327395 20.30 .336724 20.73 5i 8.252286 22.15 8.260120 22.55 51 8. 3280 13 20.27 3.337968 20.72 52 .253615 22.12 .261473 22.53 52 .329829 20.25 .339211 20.70 53 .254942 22.10 .262825 22.48 53 .331044 20.22 .340453 20.65 54 ,256268 22.05 .264174 22.47 54 .332257 20.18 .341692 20.63 55 .257591 22 02 .265522 22.42 55 .3a3468 20.17 .342930 20.60 56 .258912 21.98 .206867 22.40 56 .334678 20.13 .344166 20.58 57 .260231 21.95 -208211 22.35 57 .335886 20.10 .345401 20.55 58 .261548 21.92 .269552 22.33 58 .a37092 20.07 .346634 20.52 59 .262863 21.88 .270892 22 28 59 .3:38296 20.05 .,347865 20.50 60 8.264176 21.85 8.272229 22.27 , 60 8 339499 20.02 8.349095 20.47 409 TABLE XXVI. LOGARITHMIC VERSED SINES 12 ! 13 / Vers. D.I'. Ex. sec. D. r. / Vers. D. r. Ex. sec. D. r. 8.339499 20.02 8.349095 20.47 8.408748 18.47 8.420024 18.95 1 .340700 20.00 .350323 20.43 1 .409856 18.43 .421101 18.93 2 .341900 19.95 .351549 20.42 2 .410962 18.42 .422297 18.90 3 .343097 19.95 .352774 20. as 3 .412067 18.40 .423431 18.88 4 .344294 19.90 .353997 20.35 4 .413171 18.38 .424564 18.87 5 .345488 19.88 .355218 20.33 5 .414274 18.35 .425096 18.83 G .346681 19.85 .356438 20.30 6 .415375 18.32 .426826 18.82 7 .347872 19.82 .357656 20.28 7 .416474 18.30 .427955 18.80 8 .349061 19.80 .358873 20.25 8 .417572 18.28 .429083 18.77 9 .350249 19.77 .360088 20.22 9 .418669 18.25 .430209 18.75 10 .351435 19.75 .361301 20.20 10 .419764 18.23 .431334 18.73 11 8.352G20 19.72 8.362513 20.18 11 8.420858 18.22 8.432458 18.70 12 .353803 19.68 .363724 20.13 12 .421951 18.18 .433580 18.67 13 .354984 19.67 .864932 20.12 13 .423042 18.17 .434700 18.67 14 .356164 19.63 .366139 20.10 14 .424132 18.13 .435820 18.63 15 .357342 19.60 .367345 20.07 15 .425220 18.12 .436938 18.62 16 .358518 19.58 .368549 20.03 16 .426307 18.10 .438(tt5 18.58 17 .359693 19.55 .369751 20.02 17 .427393 18.07 .439170 18.57 18 .360366 19.53 .370952 19.98 1 18 .428477 18.05 .440284 18.55 19 .362038 19.50 .372151 19.95 19 .429560 18.02 .44131)7 18.53 20 .363208 19.48 .373348 19.95 20 .430641 18.02 .442509 18.50 21 8.364377 19.43 8.374545 19.90 21 8.431722 17.97 8.443619 18.47 22 .335.543 19.43 .375739 19.88 22 .432800 17.97 .444727 18.47 23 .306709 19.38 .376932 19.85 j 23 .433878 17.93 .445835 18.43 24 .3G787'2 19.37 .378123 19.83 24 .434954 17.92 .446941 18.42 25 .369034 19.35 .379313 19.82 25 .436029 17.88 .448046 18.38 26 .370195 19.32 .380502 19.78 | 28 .437102 17.87 .449149 18.38 27 .371354 19.28 .381689 19.75 27 .438174 17.85 .450252 18.35 28. .37'2511 19.27 .382874 1S.73 28 .439245 17.82 .451353 18.32 29 .373667 19.25 .384058 19.70 29 .440314 17.80 .452452 18.32 30 .374822 19.20 .385240 19.68 30 .441382 17.78 .453551 18.28 31 8.375974 19.18 8.386421 19.65 31 8.442449 17.75 8.454648 18.25 32 .37/125 19.17 .387600 19.63 32 .443514 17.73 .455743 18.25 33 .378275 19.13 .388778 19.60 33 .444578 17.72 .4568:38 18.22 34 .379423 19.12 .389954 19.58 i 34 .445641 17.68 .457931 18.20 35 .380570 19.08 .391129 19.55 85 .446702 17.68 .459023 18.18 3G .381715 19.05 .392302 19.53 36 .447763 17.63 .460114 18.15 37 .382858 19.03 .393474 19.50 37 .448821 17.63 .461203 18.13 38 .381000 19.02 .394644 19.48 38 .449879 17.62 .462291 18.12 3;) .385141 18.98 .395813 19.45 39 .450935 17.58 .463:378 18.10 40 .386280 18.95 .396980 19.43 40 .451990 17.55 .464464 18.07 41 8.387417 18.93 8.398146 19.42 41 8.453043 17.55 8.465548 18.05 42 .388553 18.92 .399311 19.38 42 .454096 17.52 .466631 18.03 43 .389688 18.88 .400474 19.35 43 .455147 17.48 .467713 18.00 44 .390821 18.85 .401635 19.33 44 .456196 17.48 .4(58793 18.00 45 .391952 18.83 .402795 19.32 45 .45/245 17.45 .469873 r.97 46 .393082 18.82 .403954 19.28 46 .458292 17.43 .470951 r.95 47 .394211 18.78 .405111 19.27 i 47 .459338 17.40 .472028 r.92 48 .395338 18.75 .406267 19.23 48 .460382 17.40 .473103 r.9o 49 .396463 18.73 .407421 19.22 49 .461426 17.37 .474177 r.9o 50 .397587 18.72 .408574 19.18 50 .462468 17.35 .475251 r.ss 51 8.398710 18. G8 8.409725 19.17 51 8.46:5509 17.32 8.476322 17.85 5.2 .399831 18.67 .410875 19.13 62 .464548 17.30 .477393 r.83 53 .400951 18.63 .412023 19.13 53 .465586 17.28 .478463 r.so 54 .402069 18.62 .413171 19.08 54 .466623 17.27 .479531 17.78 55 .403186 18.58 .414316 19.08 i 55 .467659 17.23 .480598 r.77 56 .404301 18.57 .415461 19.03 i 56 .468693 17.23 .481664 r.73 5? .405415 18.53 .416603 19.03 i 57 .469727 17.20 .482728 1-.73 58 .406527 18.53 .417745 19.00 58 .470759 17.17 .483792 r.7o 59 .407638 18.50 .418885 18.98 59 .471789 17.17 .484854 r.68 60 8.408748 18.47 8.420024 18. 05 60 8.472819 17.13 8.485915 r.67 410 AND EXTERNAL SECANTS. 14 15 ' Vers. D.I". Ex. sec. D.I". > Vers. D.I". Ex. sec. D. r. 8.472819 17.13 8.485915 7.67 : 8.532425 15.98 8.547482 16.53 1 .473847 17.12 .486975 7.63 i .533384 15.97 .548474 16.58 2 .474874 17.10 .4880133 7.63 2 .534342 15.95 .549466 16.52 8 .475900 17.08 .489091 7.60 i 3 .535299 15.93 .550457 16.50 4 .476925 17.05 .490147 7.58 j 4 .530255 15.92 .551447 16.48 5 .477948 17.03 .491202 7.57 ! 5 .537210 15.88 .552436 16.47 G .478970 17.02 .492256 7.53 ! 6 .538163 15.88 .553424 10.43 7 .479991 17.00 .493308 7.53 i 7 .539116 15.87 .554410 10.43 8 .481011 16.97 .494360 7.50 : 8 .540068 15.83 .555396 16.42 9 .482029 16.95 .495410 7 48 9 .541018 15.83 .556381 16.38 10 .483040 16.93 .496459 7.47 ! 10 .541968 15.80 .557364 16.88 11 8.484062 1C. 92 8.497507 7.45 it 8.542916 15.78 8.558347 16.37 12 .485077 16.90 .498554 7.43 12 .543863 15.78 .559329 16.33 13 .486091 16.87 .499GOO 7.40 ; 13 .544810 15.75 .5GOE09 16.33 14 .487103 16.87 .500644 7.38 | 14 .545755 15.73 .561289 10.30 15 .488115 16.83 .501087 7.38 15 .546699 15.72 .502267 10.30 16 .489125 16.82 .502730 7.35 i 16 .547042 15.70 .563245 16.28 17 .490134 16.78 .503771 7.32 17 .548584 15.68 .564222 16.25 IB .491141 16.78 .504810 7.32 18 .549525 15.67 .565197 16.25 19 .492148 16.75 .505849 7.30 19 .550465 15.65 .566172 16.22 20 .493153 16.73 .506887 7.27 20 .551404 15.63 .567145 16.22 21 8.494157 16.72 8.507923 7.25 21 8.552342 15.62 8.5G8118 16.20 22 .495160 16 70 .508958 7.25 i 22 .553279 15.00 .509090 16.17 23 .496162 16.67 .509993 7 22 ! 23 .554215 15.58 .570060 16.17 24 .497162 16.67 .511026 ?!l8 i 24 .555150 15.57 .571030 16.15 25 .498162 16.63 .512057 7.18 I 25 .556084 15.55 .571999 16.12 26 .499160 16.62 .513088. 7.17 26 .557017 15.53 .57'29G6 16.12 27 .500157 16.60 .514118 7.13 27 .557949 15.50 .573933 16.10 28 .501153 16.53 .515146 7.13 28 .558879 15.50 .574899 16.08 29 .50:3148 16.57 .516174 7.10 29 .559809 15.48 .575864 16.05 30 .503142 16.53 .517200 7.08 30 .560738 15.47 .576827 16.05 31 8.504134 1G.52 8.518225 7.07 31 8.561666 15.43 8.577790 10.03 32 .505125 16.52 .519249 7.05 32 .562592 15.43 .578752 16 C2 33 .506116 16.48 .520272 7.03 ! 33 .563518 15 42 .579713 16.00 31 .507105 1G.47 .521294 7.02 1 34 .564443 15.40 .580673 15. 9H 35 .508092 16.47 .522315 16.98 i 35 .565367 15.37 .581632 15.97 30 .509079 16.43 .523334 16.98 ! 36 .566289 15 37 .582590 15.95 37 .510065 16.40 .524353 16.95 i 37 .567211 15. a5 .583547 15.93 38 .511049 16.40 .525370 16.95 38 .568132 15.33 .584503 15.92 39 .512033 16.37 .526387 16.92 39 .569052 15.30 .585458 15.90 40 .513015 16.35 .527402 16.90 ! 40 .569970 15.30 .586412 15.88 41 8.513996 16.33 8.528416 16.88 41 8.570888 15.28 8.587365 15.88 42 .514976 10.32 .529429 16.87 42 .571805 15.27 .588318 15.85 43 .515955 16.28 .530441 16.85 43 .572721 15.25 .589269 15.83 44 .516932 16.28 .531452 16.83 44 .573636 15.22 .590219 15.83 45 .517909 16.25 .532462 16.82 45 .574549 15.22 .591169 15.80 4(5 .518884 16.25 .533471 16.78 i 46 .575462 15.20 .592117 15.80 47 .519859 16.22 .534478 16.78 ! 47 .576374 15.18 .592065 15.78 43 .520832 16.20 .5&>485 16.75 48 .577285 15.17 .594012 15.75 49 .521804 16.18 .536490 16.75 49 .578195 15.15 .594957 15.75 50 .522775 16.17 .537495 16.73 50 .579104 15.13 .595902 15.73 51 8.523745 16.15 8.538498 16.72 51 8.580012 15.12 8.596846 15.72 52 .524714 16.13 .539501 16.68 52 .580919 15.10 .597789 15.70 53 .525682 16.10 .540502 16.67 53 .581825 15.08 .598731 15.68 54 .526648 16.10 .541502 16.65 54 .582730 15.07 .599C7'2 15.67 55 ,527'614 16.07 .542501 16.63 55 .583634 15.05 .000612 15.05 5G .528578 16.07 .543499 16.63 56 .584537 15.05 .601551 15.65 57 .529542 16.03 .544497 16.60 i 57 .585440 15.02 . 602490 15.62 58 .5305' 14 16.02 .545493 16.58 i 58 .586341 15.00 .603427 15.00 59 .531465 16.00 .546488 16.57 ! 59 .587241 15.00 .604363 15.60 60 8.532425 15.98 8.547483 16.53 60 8.588141 14.97 8.605299 15.58 411 TABLE XXVI.-LOGARITHMIC VERSED SINES 16 17 / Vers. D. 1". Ex. sec. D. r. ' Vers. D. 1". Ex. sec. D. r. 8.58S141 14.97 8.605299 15.58 8.640434 14.08 8.659838 14.72 i .589339 14.95 .606234 15.55 1 .641279 14.07 .660721 14.72 2 .589936 14.95 .607167 15.5.3 2 .642123 14.05 .661604 14.70 3 .591)333 14.93 .603100 15.53 ! 3 .642966 14.05 .662486 14.68 4 .591729 14.93 .603032 15.52 4 .643809 14.02 .663367 14.68 5 .592523 14.90 .609933 15.50 5 1 .644650 14.02 .664248 14.65 6 .593517 14.88 .610333 15.50 6 .645491 14.00 .665127 14.65 7 .594410 14.87 .611823 15.47 7 .646331 13.98 .665006 14.63 8 .595302 14.83 .612751 15.45 8 .647170 13.97 .666884 14.62 9 .595192 14.83 .613378 15.45 9 .648008 13.95 .667761 14.60 10 .597082 14.82 .614605 15.43 10 .648845 13.95 .6G8G37 14.60 11 8.597971 14.82 8.615331 15.42 11 8.649632 13.93 8.669513 14.58 12 .593350 14.78 .616456 15.38 12 .6.30518 13.92 .670388 14.57 13 .599747 14.77 .617379 15.38 13 .651353 13.90 .671262 14.55 14 .603333 14.75 .618302 15.33 14 .652187 13.88 .672135 14.55 15 .601518 14.75 .619225 15.35 15 .653020 13.87 .673003 14.52 16 .6024)3 14.72 .620146 15.33 16 .653852 13.87 .673379 14.52 17 .633233 14.72 .621053 15.33 17 .654684 13.85 .674750 14.50 18 .604163 14.70 .621933 15.30 18 . 653515 13.83 .675620 14.50 19 .633331 14.67 .622334 15.30 19 .653345 13.82 .676490 14.47 2J .635931 14.67 .62J322 15.38 20 .657174 13.84 .677358 14.47 21 8.633311 14.65 8.621739 15.27 21 8.658003 13.78 8.678226 14.45 22 .637633 14.63 .623335 15.25 22 .658330 13.78 .679093 14.45 2} .633333 14.62 .623370 15.23 23 .659557 13.77 .679960 14.42 24 _633445 14.60 .627434 15.23 24 .680483 13.75 .680825 14.42 23 .610321 14.60 .628393 15.20 25 .661338 13.73 .681690 14.40 23 .611197 14.57 .629310 15.20 23 .662132 13.73 .682354 14.38 27 .612371 14.57 .630222 15.18 27 .662356 13.72 .683417 14.38 23 .612315 14.53 .63113} 15.17 28 .663779 13.70 .684280 14.35 23 .613317 14.53 .632313 15.15 29 .664601 13.63 .685141 14.35 33 .6141533 14.52 .632332 15.13 30 .665422 13.67 .686002 14.35 31 8.615563 14.50 8.633350 15.13 31 8.666242 13.67 8.686S63 14.32 33 .616433 14.43 .634763 15.10 32 .667032 13.65 .687722 14.32 33 .617293 14.47 .635574 15.10 33 .667881 13.63 .688581 14.30 31 .61816? 14.45 .633330 15.08 34 .663699 13.62 .689439 14.28 33 .619034 14.45 .637485 15.07 35 .639516 13.60 .690296 14.28 33 .619331 14.42 .6'333 15.05 36 .670332 13.60 .691153 14.25 37 .623766 14.42 .633232 15.05 37 .671148 13.58 .692008 14.25 33 .621631 14.40 .640195 15.02 33 .671963 13.57 .692863 14.25 33 .622495 14.33 .641035 15.02 39 .672777 13.55 .693718 14.22 40 .623353 14.37 .641937 15.00 40 .673590 13.55 .694571 14.22 41 8.624220 14.35 8.642397 14.98 41 8.674403 13.53 8.695424 14.20 '42 .625031 14.33 .643793 14.97 42 .675215 13.52 .696276 14.18 43 .625941 14.33 .644694 14.95 43 .676026 13.50 ! .697127 14.18 41 .6-26801 14.30 .645591 14.95 44 .676836 13.48 j .697978 14.17 45 .627659 14.30 .64648S 14.93 45 .677(545 13.48 .698828 14.15 46 .6-28517 14.23 .647334 14.92 43 .678154 13.47 .699677 ! 14.13 47 .629374 14.27 .648279 14.90 47 .679262 13.45 .700525 14.13 48 .630233 14.25 .649173 14.83 48 .680069 13.43 .701373 14.12 49 .631035 14.23 .650333 1 14.87 49 .680875 13.43 .702220 14.10 50 .631939 14.22 .650958 14.87 50 .681681 13.42 .703066 14.10 51 8.632792 14.22 8.651850 14.85 51 8.6S2486 13.40 8.703912 14.07 52 .633645 14.18 .65-2741 14.83 52 .683290 13.38 .704756 14.07 53 .634495 14.18 .653631 14.82 53 ' .684093 13.38 .705600 14.07 54 .635347 14.17 .654520 14.80 54 .684896 13.35 .706444 14.03 55 .636197 14.15 .655408 14.80 55 .685697 13.35 .707286 14.03 56 .637046 14.13 .656296 14.77 56 .686498 13.35 .708128 14.02 57 .637894 14.13 .657182 14.77 57 .6S7'299 13.32 .708969 14.02 38 .638742 14.10 .658068 14.77 58 ! .6881)98 13.32 .709810 14.00 59 .639588 14.10 .6.38954 14.73 59 .688897 13.30 .710650 13.98 60 8.640434 14.08 8.659838 14.72 II 60 8.68)695 13.23 i8. 71 1489 13.97 412 AND EXTERNAL SECANTS. 18 19 / Vers. D. 1'. Ex. sec. D, 1". / Vers. D. 1". Ex. sec. D. 1'. 8.089695 ! 13.28 8.711489 13.97 8.730248 12.58 8.760578 13.30 1 690492 13.28 .712327 13.95 1 .737003 12.57 .761376 13.30 2 .691289 13.25 .713104 13.95 2 .737757 12.55 .762174 13.28 3 .692084 13.25 .714001 13.95 3 .738510 12.55 .762971 13.27 4 .692879 13.25 .714838 13.92 4 .739263 12.53 .763767 13.27 5 .693674 13.22 .715673 13.92 5 .740015 12.52 .764563 13 25 6 .694467 13.22 .716508 13.90 6 .740766 12 ..50 .765358 13.23 7 .695260 13.20 .717342 13.88 7 .741516 12.50 .766152 13.23 8 .696052 13.18 .718175 13.88 8 .742266 12.50 .766946 13.22 9 .696843 13.18 .719008 13.87 9 .743016 12.47 .767739 13.20 10 .697634 13.17 .719840 13.85 10 .743764 12.47 .768531 13.20 11 8.698424 13.15 8.720671 13.85 11 8.744512 12.45 8.769323 13.18 12 .699213 13.13 .721502 13.83 12 .745259 12.43 .770114 13.18 13 .700001 13.13 .722332 13.82 13 .746006 12.45 .770905 13.17 14 .700789 13.12 .723161 13.80 14 .746752 12.42 .771695 13.15 15 .701576 13.10 .723989 13.80 15 .747497 12.42 .772484 13.15 16 .702362 13.08 .724817 13.78 16 .748242 12.40 .773273 13.13 17 .703147 13.08 .725644 13.78 17 .748986 12.38 .774061 13.13 18 .703932 13.07 .726471 13.77 18 .749729 12.38 .774849 13.12 19 .704716 13.05 .727297 13.75 19 .750472 12.37 .775636 13.10 20 .705499 13.05 .728122 13.73 30 .751214 12.35 .776422 13.08 21 8.706282 13.02 8.728946 13.73 21 8.751955 12.35 8.777207 13.10 22 .707063 13. (2 .729770 13.72 22 .752696 12.33 .777993 13.07 23 .707844 13.02 .730593 13.70 23 .753436 12.32 .778777 13.07 24 .708625 12.98 .731415 13.70 24 .754175 12.32 .779561 13.05 25 .709404 12.98 .732237 13.68 25 .754914 12.30 .780344 13.05 26 .710183 12.98 .7,33058 13.67 26 .755652 12.28 .781127 13.03 27 .710961 12.95 .7,33878 13.67 27 .756389 12.28 .781909 13.02 28 .711739 12.95 .734698 13.65 28 .757126 12.27 .782690 13.03 29 .712516 12.93 .735517 13.63 29 .757862 12.27 .7&3471 13.00 30 .713292 12.92 .736886 13.63 30 .758598 12.25 .784251 13.00 31 8.714067 12.92 8.737153 13.62 31 8.759333 12.23 8.785031 12.98 32 .714842 12.90 .737970 13.60 32 .760067 12.23 .785810 12.97 33 .715016 12.88 .738786 13.00 33 .760801 12.22 .786588 12.97 34 .716889 12.87 .739602 13.58 34 .761534 12.20 .787366 12.97 35 .717161 12.87 .740417 13.57 35 .762266 12.20 .788144 12.93 36 .717988 12.85 .741231 13.57 36 .762998 12.18 .788920 12.93 37 .718704 12.85 .742045 13.55 ! 37 .763729- 12.17 .789696 12 93 88 .719475 12.82 .742858 13.53 38 .764459 12.17 .790472 12.92 39 .720214 12.82 .743070 13.53 i 39 .765189 12.15 .791247 12.90 40 .721013 12.82 .744482 13.52 | 40 .765918 12.15 .792021 12.90 41 8.721782 12.78 8.745293 13.50 41 8.766647 12.12 8.792795 12.88 42 .722549 12.78 .746103 13.50 42 .767374 12.13 .793568 12.87 43 . 728316 12.78 .746913 13.48 43 .768102 12.10 .794340 12.87 44 .724083 12.75 .747722 13.47 44 .768828 12.10. .795112 12.87 45 .724848 12.75 .748530 13.47 45 .769554 12.10 .795884 12.83 46 .725013 12.73 .749338 13.45 46 .770280 12.08 .796654 12.85 47 .726377 12.72 .750145 13.43 47 .771005 12.07 .797425 12.82 48 .727140 12.72 .750951 13.43 48 .771729 12.05 .798194 12.82 49 .727903 12.70 .751757 13.42 49 .772452 12.05 .798963 12.82 50 .728665 12.70 .752562 13.42 50 .773175 12.05 .799732 12.80 51 8.729427 12.67 8.75:3367 13.40 51 8.773898 12.02 8.800500 12.78 52 .730187 12.67 .754171 13.38 52 .774619 12.02 .801267 12.78 53 .730947 12.67 .7541)74 13.37 53 .775340 12.02 .802034 12.77 54 .731707 12.63 .755776 13.37 51 .770061 12.00 .802800 12.75 55 .732405 12.63 .756578 13.37 55 .776781 11.98 .803565 12.75 56 .733223 12. 63 .757380 13.33 50 .777500 11.97 .804330 12.75 57 .7*3981 12. 00 .758180 13.33 | 57 .778218 11.97 .805095 12.73 58 .784787 12.00 .758980 13.33 I 58 .778936 11.97 .805859 12.72 59 .735493 12.58 .759780 13.30 59 .779654 11.93 .806622 12.72 60 8.736248 12.58 8.760578 13.30 l| 60 8.780370 11.95 8.807385 12.70 413 TABLE XXVI.-LOGARITHMIC VERSED SINES 20 21 / Vers. D. r. Ex. sec. D. 1'. ' Vers. D. r. Ex. sec. D. 1". ~0~ 8.780370 11.95 8.807385 12.70 8.822296 11.35 8.852144 i 12.17 1 .781087 11.92 .808147 12.68 1 .822977 11.35 .852874 12.17 2 .781802 11.92 .808908 12.68 2 .823658 11.33 .853604 12.13 3 .782517 11.90 .809669 12.68 3 .824338 11.33 .854*32 12.15 4 .783231 11.90 .810430 12.67 4 .825018 11.32 .855061 I 12.13 5 .783945 11.88 .811190 12.65 5 .825697 11.32 .855789 12.12 G .784658 11.88 .811949 12.65 6 .826375 11.30 .856516 12.12 7 .785371 11.87 .812708 12.63 7 .827051 11.28 .857243 ! 12.10 8 .786083 11.85 .813466 12.63 8 .82', 731 11.28 .857969 12.10 9 .786794 11.85 .814224 12.62 9 .828408 11.28 .858695 12 08 10 .787505 11.83 .814981 12.60 10 .829085 11.27 .859430 12.08 11 8.788215 11.82 8.815737 12.60 11 8.829761 11.25 8.860145 12. 0V 12 .788924 11.82 .816493 12.60 12 .830436 11.25 .860869 12.07 13 .7896-33 11.82 .817249 12.58 13 .831111 11.23 .861593 12.05 14 .790342 11.78 .818004 12.57 14 .831785 11.23 .862316 1 12.05 15 .791049 11.78 .818758 12.57 ! 15 .832459 11.22 .863039 12.03 16 .791756 11.78 .819512 12.55 16 .833132 11.20 .863761 ! 12.03 17 .792463 11.77 .820265 12.55 17 .833804 11.20 .864483 j 12.02 18 .793169 11.75 .821018 12.53 18 .834476 11.20 .865204 12.02 19 .793874 11.75 .821770 12.52 19 .8.35148 11.18 .865925 12.02 20 .794579 11.73 .822521 12.52 20 .835819 11.17 .866646 | 11.98 21 8.795283 11.73 8.823272 12.52 21 8.836489 11.17 8.867365 12.00 22 .795987 11.72 .824023 12.50 i 22 .837159 11.17 .868085 11.98 23 .796690 11.70 .824773 12.48 i 23 .837829 11.15 .868804 11.97 24 .797392 11.70 .825522 12.48 24 .838498 11.13 .869522 11.97 25 .798094 11.68 .826271 12.47 25 .839166 11.13 .870240 11.95 26 .798795 11.68 .827019 12.47 26 .839834 11.12 .870957 11. 5 27 .799496 11.67 .827767 12.45 j 27 .840501 11.12 .871674 11.93 28 .800196 11.67 .828514 12.45 i 28 .841168 j 11.10 .872390 | 11.93 29 .800896 11.63 .82-261 12.43 || 29 .841834 11.10 .873106 i 11. S3 30 .801594 11.65 .830007 12.42 30 .842500 11.08 .873822 11.62 31 8.802293 11.63 8.830753 12.42 i 31 8.843165 11.07 8.874537 11.90 32 .802991 11.62 .831497 12.42 1 32 .843829 11.07 .875251 11.90 33 .803688 11.60 .832242 j 12.40 li 33 .844493 11.07 .875965 11.88 34 .804384 11.60 .832986 ! 12.38 || 34 .845157 11.05 .876678 i 11.88 35 .805080 11.60 .833729 12.38 ii 35 .845820 11.05 .877391 ; 11.88 36 .805776 11.58 .834472 12.38 || 36 .846483 11.03 .878104 11.87 37 .806471 11.57 .835815 12.37 !! 37 .847145 11.02 .878816 11.87 38 .807165 11.57 .835957 12.35 i 38 .847806 11.02 .879528 11.85 39 .807859 11.55 .836698 12.35 ji 39 .848467 11.00 .880239 11.83 40 .808552 11.53 .837439 12.33 40 .S49127 11.00 .880949 11.83 41 8.809244 11.53 8.838179 12.33 41 8.849787 11.00 8.881659 11.83 42 .809936 11.53 .838919 12.32 42 .850447 10.98 .882369 11.82 43 .810628 11.52 .639058 12.30 43 .851106 10.97 .883078 11.82 44 .811319 11.50 .840396 12.32 44 .851764 10.97 .883787 11. SO 45 .812009 11.50 .841135 12.28 45 .852422 10.95 .884495 11.80 46 .812699 11.48 .841872 12.28 ! 46 .853079 10.95 .885203 11.78 47 .813388 11.48 .842609 12.28 j 47 .853736 10.93 .885910 11.78 48 .814077 11.47 .843346 12.27 48 .854392 10.93 .886617 i 11.77 49 .814765 11.45 .844082 12.25 49 .855048 10.92 ' .887323 11.77 5'J .815452 11.45 .844817 12.25 || 50 .855703 10.92 .888029 11.75 51 8.816139 11.43 8.845552 12.25 51 8.856358 10.90 ! 8. 888734 11.75 52 .816825 11.43 .846287 12.23 52 .a57012 10.90 .889439 11.75 53 .817511 11.42 .847021 12.22 53 .857666 10.88 .890144 11.73 54 .818196 11.42 .847754 12.22 54 .858319 10.88 .890848 11.72 55 .818881 11.40 .848487 12.22 55 .858972 10.87 .891551 11.72 56 .819565 11.40 .849220 12.20 ! 56 .859634 10.87 .892254 11.72 57 .820249 11.38 .849952 12.18 57 .860276 10.85 .892957 11.70 58 .820932 11.37 .850683 12.18 |j 58 .860937 10. &5 .893659 11.70 59 .821614 11.37 .851414 12.17 59 .861578 10.83 .894361 11.68 60 8.822296 11.35 8.852144 12.171 6018.862228 10.82 8.895062 11.68 414 ANL> EXTERNAL SECANTS. 22 23 f Vers. D. r. Ex. sec. D. 1". \ > Vers. D. r. Ex. sec. ; D. 1". 8.862228 10.82 8.895062 11.68 o 8.900341 10.33 8.936315 11.23 1 .862877 10.83 .895763 11.67 1 .900961 10. 35 .936989 11.23 2 .863527 10.80 .896463 11.67 1 2 .901582 10.32 .937663 I 11.22 3 .864175 10.80 .897163 11.65 3 .902201 10. 33 .938336 ; 11.22 4 .864823 10.80 .897862 11.65 4 .902821 10.32 .939009 11.22 5 .865471 10.78 .898561 11.63 5 .903440 10.30 .939682 11.20 6 .866118 10. rs .899259 11.63 6 .904058 10.30 .940354 11.20 7 .86676') 10.77 .899957 11.63 || 7 .904676 10.28 .941026 11.20 8 .867411 10.77 .900655 11.62 8 .905293 10.28 .941698 11.18 9 .868057 10.75 .901352 11.60 9 .905910 10.28 .942369 i 11.17 10 .868702 10.73 .902048 11.62 10 .906527 10.27 .943039 ( 11. Id 11 8.869346 10.75 8.902745 11.58 11 8.907143 10.27 8.943710 11.15 12 .869991 10.72 .903440 11.60 12 .907759 10.25 .944379 11.17 13 .870634 10.72 .904136 11.57 13 .908:374 10.25 .945049 11.15 14 .871277 10.72 .904830 11.58 14 .908989 10.23 i .945718 11.18 15 .871920 10.70 .905525 11.57 15 .909603 10.23 .946386 11.13 16 .872.562 10.70 .906219 11.55 16 .910217 10.22 .947054 11.13 17 .873204 10.68 .906912 11.55 17 .910830 10.22 i .947722 11.12 18 .873845 10.68 .907605 11.53 18 .911443 10.22 ! .948389 11.12 19 .874486 10.67 .908298 11.53 19 .912056 10.20 .949056 11.12 20 .875123 10.67 .908990 11.52 20 .912668 10.18 .949723 11.10 21 8.875766 10.65 8.909681 11.52 21 8.913279 10.20 8.950389 11.10 22 .876405 10.65 .910372 11.52 22 .913891 10.17 .951055 11.08 23 .877044 10.63 .911063 11.52 1 23 .914501 10.17 .951720 11.08 24 .877682 10.63 .911754 11.48 24 .915111 10.17 | .952385 11.07 25 .878320 10.62 .912443 11.50 25 .915721 10.17 j .953049 11.07 23 .878957 10.62 .913133 11.48 ! 26 .916331 10.15 .953713 11.07 27 .879594 10.60 .913822 11.47 | 27 .916940 10.13 .954377 11.05 23 .880230 10.60 .914510 11.47 28 .917548 10.13 .955040 11.05 29 .8808 10.58 .915198 11.47 29 .918156 10.13 .955703 11.05 30 .881501 10.58 .915886 11.45 30 .918764 10.12 .956366 11.03 31 8.882136 10.58 ! 8.916573 11.45 31 8.919371 10.10 ^8. 957028 11.03 32 .882771 10.57 .917260 11.43 32 .919977 10.12 .957690 ; 11.02 33 .883405 10.55 .917946 11.43 88 .920584 10.10 .958351 11.02 34 .8840:38 10. 55 .918632 11.43 i 34 .921190 10. (?8 i .959012 11.00 35 .884671 10.53 .919318 11.42 35 .921795 10.08 .959672 11.00 36 .885:303 10.53 .920003 11.40 36 .922400 10.07 .960332 11.00 37 .885935 10.53 .920687 11.42 i 37 .923004 10.07 .960992 10.98 38 .886567 10.52 .921372 11.38 1 33 .923608 10.07 .9616.-)! 10.98 39 .887198 10.52 .922055 11.40 39 .924212 10.05 .962310 10.98 40 .887829 10.50 .922739 11.37 40 .9.24815 10.05 i .962969 10.97 41 8.888459 10.48 8.923421 11.38 41 8.925418 10.03 ^ 8. 963627 10.97 42 .889088 10.48 .924104 11.37 42 .926020 10.03 .964285 10.95 43 l .889717 10.48 .924786 11. a5 43 .926622 10.03 .964942 10.95 44 i .890346 10.47 .925467 11.37 44 .927224 10.02 .965599 10.95 45 .890974 10.47 .926149 11.33 45 .927825 10.00 .966256 10.93 46 .891602 10.45 .926829 11.35 46 .928425 10.00 .966912 10.93 47 .8922.29 10.45 .927510 11.32 47 .929025 10.00 .967568 10.92 48 .892856 10.43 .928189 11.33 48 .939625 9.98 .968223 10.92 49 .893482 10.48 .928869 11.32 49 .930224 9.98 .968878 i 10.92 50 .894108 10.42 .929548 11.30 50 .930823 9.9? .969533 j 10.90 51 8.894733 10.42 .8.930226 11.32 51 8.931421 9.97 8.970187 10.90 52 | .895358 10.42 .930905 11.28 52 .932019 9.97 .970841 i 10.88 53 .895983 10.40 .931582 11.30 53 .932617 9.95 .971494 10.88 54 .896607 10.38 .932260 11.27 ! 54 .933214 9.95 .972147 10.88 55 .897230 ; 10.38 .932936 11.28 55 .933811 9.93 .972800 10.87 56 .897853 ' 10.38 .933613 11.27 56 .934407 9.93 .973452 10.87 57 i .898476 10.37 .934289 11.27 57 .935003 9.92 .974104 10.87 58 j .899098 10.35 .934965 11.25 58 .9a5598 1 9.92 .974756 j 10.85 59 , .899719 10.37 .935640 11.25 1 59 .936193 9.92 .975407 10.85 60 ; 8.900341 i 10. :i3 ! 8.936315 11.23 i! 60 8.936788 ! 9.90 8.976058 I 10.83 415 TABLE XXVI. LOGARITHMIC VERSED SINES 24 25 / Vers. D. 1". Ex. sec. D. r. / Vers. D.I". Ex. sec. i D. r. \ 8.936788 9.90 8.976058 10.83 8.971703 9.50 9.014428 i 10.47 1 .937382 9.90 .976708 10.83 1 .972273 9.48 .015056 10.48 2 .937976 9.88 .977358 I 10.83 2 .972842 9.48 .015685 ! 10.45 3 .938569 9.88 .978008 10.82 3 .973411 9.48 .016312 < 10.47 4 .939162 9.87 .978657 10.82 4 .973980 9.47 .016940 1 10.45 5 .939754 9.87 .979306 10.80 5 .974548 9.47 .017567 10.45 6 .940346 9.87 .979954 10.80 6 .975116 9.45 .018194 10.45 7 .940938 9.85 .980602 10.80 7 .975683 9.45 .018821 10.43 8 .941529 9.85 .981250 10.80 8 .976250 9.43 .019447 i 10.43 9 .942120 9.83 .981898 10.78 9 .976816 9.43 .020073 i 10.42 10 .942710 9.83 .982545 10.77 10 .977382 9.43 .020698 10.42 11 8.943300 9.82 8.9&3191 10.77 11 8.977948 9.43 9.021323 10.42 12 .943889 9.83 .983837 10.77 12 .978514 9.42 .021948 10.40 13 .944479 9.80 .984483 10.77 13 .979079 9.40 .022573 10.42 14 .945067 9.80 .985129 10.75 14 .979643 9.40 .023197 10.38 15 .945655 9.80 .985774 10.75 15 .980207 9.40 .023820 10.40 16 .946243 9.80 .986419 10.73 16 .980771 9^40 .024444 10.38 17 .946831 9.78 .987063 10.73 17 .981386 9.38 .025067 10.38 18 .947418 9.77 .987707 10.73 18 .981898 9.37 .025690 10.37 19 .948004 9.77 .988351 10.72 19 .982460 9.38 .026313 10.37 20 .948590 9.77 .988994 10.72 20 .983023 9.37 .026934 10.37 21 8.949176* 9.75 8.989637 10.70 21 8.983585 9.35 9.027556 10. a5 22 .949761 9.75 .990279 10.72 22 .984146 9.35 .028177 10.35 23 .950346 9.75 .990922 10.68 23 .984707 9.35 .028798 10.35 24 .950931 9.73 .991563 10.70 24 .985268 9.38 .029419 10.83 25 .951515 9.73 .992205 10.68 25 .985838 9. -33 .030039 10.33 26 .952099 9.72 .992846 10.68 26 .986388 9.33 .031)659 10.33 27 .952682 9.72 .993487 10.67 27 .986948 9.32 .031279 10.33 28 .953265 9.70 .994127 10.67 28 .987507 9.32 .031899 10 32 29 .953847 9.70 .994767 10.65 29 .988066 9.32 .032518 10.30 30 .954429 9.70 .995406 10.67 30 .988625 ! 9.30 .033136 10.28 31 8.955011 9.68 8.996046 10.65 31 8.989183 9.28 9.033755 10.30 32 .955592 9.68 .996685 10.63 32 .989740 9.30 .034373 10.30 33 .956173 9.67 .997323 10.63 33 .990298 9.28 .034991 10.28 34 .956753 9.68 .997961 10. 63 ! 34 .990855 9.27 .Oa5608 10.28 35 .957334 9.65 .998599 10.62 j 35 .991411 9.28 .036225 10.28 36 .957913 9.65 .999236 10.62 36 .991968 9.25 .036842 10.27 37 .958492 9.65 8.999873 10.62 37 .992523 9.27 .037458 10.27 38 .959071 9.65 9.000510 10.60 38 .993079 9.25 .038074 10.27 39 .959650 9.63 .001146 10.62 39 .993634 9.25 .038(590 10. as 40 .960228 9.62 .001783 10.58 40 .994189 9.23 .039305 10.25 41 8.960805 9.62 9.002418 10.58 41 8.994743 9.23 9.039920 10.25 42 .961382 9.62 .003053 10.58 42 .995297 9.23 .0405:35 10.25 43 .961959 9.60 .003688 10.58 43 .995851 9.22 .041150 10.23 44 .962535 9.60 .004323 10.57 44 .996404 9.22 .011764 10.23 45 .963111 9.60 .004957 10.57 .45 .996957 9.20 .042378 10.22 46 .963687 9.58 .005591 10.55 46 .997509 9.22 .042991 10.22 47 .964262 9.58 .006224 10.57 47 .998062 9.18 .043604 10.22 48 .964837 9.57 .006858 10.53 48 .998613 9.20 .044217 10.22 49 .965411 9.57 .007490 10.55 49 .999165 9.18 .044830 10.20 50 .965985 9.57 .008123 10.53 50 8.999716 9.17 .045442 10.20 51 8.966559 9.55 9.008755 10.53 51 9.000266 9.18 9.046054 10.18 52 .967132 9.55 009387 10.52 52 .000817 9.17 .046665 10.18 53 .967705 9.53 .010018 10.52 53 .001367 9 15 .047276 10.18 54 .968277 9.53 .010649 1 10.52 54 .001916 9.17 .047887 10.18 55 .968849 9.53 .011280 10.50 55 .002466 9.13 .048498 10.17 56 .969421 9.52 .011910 10.50 56 .003014 9.15 .049108 10.17 57 .969992 9.52 .012540 10.50 57 .003563 9.13 .049718 10.17 58 .970563 9.50 .013170 10.48 ! 58 .004111 9.13 .050328 10.15 59 .971ia3 9.50 .013799 10.48 59 .004659 9.12 .030937 10.15 60 8.971703 9.50 9.014428 10.47 60 9.005206 9.12 9.051546 10.15 416 AND EXTERNAL SECANTS. 26 27 ' Vers. D. 1". Ex. sec. D. 1". ' Vers. D.I". Ex. sec. D. 1". o 9.005206 9.12 9.051546 10.15 9.037401 8.77 9.087520 9.83 1 .005753 9.12 .052155 10.13 1 .037927 8.75 .088110 9.83 2 .006300 9.10 .05276:3 10.13 2 .038452 8.77 .088700 9.83 3 .006846 9.10 .053371 10.13 3 .038978 8.75 .089290 9.83 4 .007392 9.10 .053979 10.12 4 .039503 8.73 .089880 i 9.82 5 .007938 9.08 .054586 10.12 5 .040027 8.75 .090469 9.82 6 .0084&3 9.08 .055193 10.12 6 .040552 8.73 .091058 9.82 7 .009028 9.07 .055800 10.10 7 .041076 8.72 .091647 9.80 8 .009573 9.07 .056406 10.10 8 .041599 8.73 .092235 9.80 9 .010116 9.07 .057012 10.10 9 .042123 8.72 .092823 9.80 10 .010600 9.05 .057618 10.10 10 ..042646 8.70 .093411 9.78 11 9.011203 9.05 9.058224 10.08 11 9.043168 8.72 9.093998 9.80 18 .011746 9.05 .058829 10.08 12 .043691 8.70 .094586 9.78 13 .012289 9.03 .059434 10.07 13 .044213 8.70 .095173 9.77 14 .012831 9.03 .060638 10.08 14 .044735 8.68 .095759 9.78 15 .013373 9.03 .060643 10.07 15 .045256 8.68 .096346 9.77 16 .013915 9.02 .061247 10.05 16 .045777 8.68 .096^32 9.77 17 .014456 9.02 .061850 10.07 17 .046298 8.67 .097518 9.75 18 .014997 9.02 .062454 10.05 18 .046818 8.67 .098103 9.77 19 .015538 9.00 .063057 10.03 19 .047*38 8.67 .098689 9.75 20 .016078 9.00 .063659 10.05 20 .047858 8.65 .099274 9.73 21 9.016618 8.98 9.064262 10.03 21 9.048377 8.65 9.099a58 9.75 22 .017157 9.00 .064864 10.03 22 .048896 8.65 .100443 9.73 23 .917697 8.97 .065466 10.02 23 .049415 8.63 .101027 9.73 21 .018235 8.98 .066067 10.02 24 .049933 8.63 .101611 9.72 86 .018774 8.97 .066668 10.02 25 .050451 8.63 .102194 9.73 26 .019312 8.97 .067269 10.02 26 .050969 8.63 .102778 9.72 .019850 8.95 .067870 10.00 i 27 .051487 8.62 .103361 9.70 28 .020387 8.95 .068470 10.00 28 .052004 8^60 .103943 9.72 29 .020924 8.95 .069070 10.00 29 .052520 8.62 .104526 9.70 30 .021461 8.93 .069670 9.98 30 .053037 8.60 .105108 9.70 31 9.021997 8.93 9.070269 9.98 31 0.053553 8 60 9.105690 9.68 88 .022533 8.93 .070868 9.98 32 .054069 8 58 .106271 9.70 33 .023069 8.92 .071467 9.97 33 .054584 8.58 .106853 9.68 84 .023604 8.92 .072065 9.97 34 .055099 8.58 .107434 | 9.68 85 .024139 8.90 .072663 9.97 35 .055614 8.58 .108015 9.67 85 .024673 8.92 .073261 9.97 36 .056129 8.57 .108695 9.67 37 .025208 8.90 .073859 9.95 37 .056643 8.57 .109175 9.67 38 .025742 8.88 .074456 9 95 j 38 .057157 8.55 .109755 9.67 39 .026275 8.88 .075053 9.93 39 .05767'0 8.55 .110335 9.65 40 .026808 8.88 .075649 9.95 40 .058183 8.55 .110914 9.67 41 9.027341 8.88 9.076246 9.93 41 9.058696 8.55 9.111494 9.63 42 .027874 8.87 .076842 9.92 42 .059209 8.53 .112072 9.65 43 .028406 8.87 .077437 9.93 43 .059721 8.53 .112651 9.63 41 -.028938 8.85 .078033 9.92 44 .0602:33 8.53 .113229 9.63 45 .029469 8.85 .078628 9.92 45 .060745 8.52 .113807 9.63 46 .030000 8.85 .07'9223 9.90 46 .061256 8.52 .114385 9.63 47 .0:50531 8.85 .079817 9.92 1 47 .0617'67 8.50 .114963 9.62 43 .031062 8.83 .080412 9.90 1 48 .062277 8.52 .115540 9.62 49 .031592 8.83 .081006 9.88 | 49 .062788 8.50 .116117 9.60 50 .032122 8.82 .081599 9.90 50 .0(53298 8.48 .116693 9.62 51 9.032651 8.82 9.0S2193 9.88 51 9.063807 8.50 9.117270 9.60 52 .033180 8.82 .082786 9.87 52 .064317 8.48 .117846 9.60 53 .033709 8.80 .083378 9.88 1 53 .064826 8.48 .118422 9.58 54 .034237 8.80 .083971 9.87 54 .065335 8.47 .118997 9.60 55 .084765 8.80 .084563 9.87 55 .065843 8.47 .119573 9.58 56 .035293 8.78 .085155 9.87 56 .066351 8.47 .120148 9.58 57 .Oa5820 8.78 .085747 9.85 57 .066859 8.45 .120723 9.57 58 .036347 8.78 .086338 9.85 58 .067366 8.47 .121297 9.57 59 .036874 8.78 .086929 9.85 59 .067874 8.43 .121871 9.57 60 9.037401 8.77 9.087520 9.83 II 60 9.068380 8.45 9.122445 9.57 417 TABLE XXVI. LOGARITHMIC VERSEb SINES 28 C 29 ' Vers. D. r. Ex. sec. D. 1'. ' Vers. D. r. Ex. sec. D. 1". 9.068380 8.45 9.122445 9.57 1 9.098229 8.15 9.156410 9.30 1 .068887 8.43 .123019 9.57 | 1 .098718 8.13 .156968 9.32 2 .069393 8.43 .123593 9.55 2 .099206 8.12 .157527 9.28 3 .069899 8.43 .124166 9.55 3 .099693 8.13 .158084 9.30 4 .070405 8.42 .124739 9.53 4 .100181 8.12 .158642 9.30 5 .070910 8.42 .125311 9.55 I 5 .100668 8.12 .159200 9.28 6 .071415 8.40 .125884 9.53 6 .101155 8.12 .159757 9.28 7 .071919 8.42 .126456 9.53 7 .101642 8.10 .160314 9.27 8 .072424 8.40 .127028 9.52 8 .102128 8.10 .160870 9.28 9 .072928 8.40 .127599 9.53 9 .102614 8.10 .161427 9.27 10 .073432 8.38 .128171 9.52 10 .103100 8.08 .161983 9.27 11 9.073935 8.38 9.128742 9.52 11 9.103585 8.08 9.162539 9.27 12 .074438 8.38 .129313 9.50 12 . 104070 8.08 .163095 9.25 13 .074941 8.37 .129883 9.50 13 .104555 8.08 .163650 9.25 14 .075443 8.38 . 130453 9.50 14 .105040 8.07 .164205 9.25 15 .075946 8.35 .131023 9.50 1 15 . 103324 8.07 .164760 9.25 16 .076447 8.37 .131593 9.50 16 .106008 8?05 .165315 9.25 17 .076949 8.35 .132163 9.48 17 .106491 8.07 .165870 9.23 18 .077450 8.35 .132732 9.48 18 .106975 8.05 .166424 9.23 19 .077951 8.35 .133301 9.48 19 .107458 8.05 .16697'8 9.23 20 .078452 8.33 .133870 9.47 20 .107941 8.03 .167532 9.22 21 9.078952 8.33 9.134438 9.47 21 9.108423 8.05 9.168085 9.23 22 .079452 8.33 .135008 9.47 22 .108908 8.03 .168639 9.22 23 .079952 8.32 .135574 9.47 1 23 .109388 8.02 .169192 9.22 24 .030451 8.32 .136142 9.45 24 .109869 8.03 .169745 9.20 25 .080950 8.32 .136709 9.47 25 .110351 8.02 .170297 9.22 26 .081449 8.32 .137277 9.45 26 .110832 8.02 .170850 9.20 27 .081948 8.30 .1*7844 9.43 27 .111313 8.00 .171402 9.20 28 .082446 8.30 .138410 9.45 28 .111793 8. CO .171954 9.18 29 .082944 8.28 .138977 9.43 29 .112273 8.00 .172505 9.20 30 .083441 8.30 .139543 9.43 30 .112753 8.00 .173057 9.18 31 9.083939 8.28 9.140109 9.42 31 9.1132.33 8.00 9.173608 9.18 32 .084436 8.27 .140674 9.43 32 .113713 7.98 .174159 9.18 33 .084932 8.28 .141240 9.42 33 .114192 7.98 . 74710 9.17 34 ' .085429 8.27 .141805 9.42 34 .114671 .97 . 75260 9.17 35 .085925 8.25 .142370 9.40 35 .115149 .97 .175810 9.17 36 .086420 8.27 .142934 9.42 36 .115627 .97 .176360 9.17 37 .086916 8.25 .143199 9.40 37 .116105 .97 .176910 9.17 38 .087411 8.25 .144063 9.40 38 .116583 .97 . 77460 9.15 39 .087906 8.23 .144827 9.38 39 .117061 .95 .178009 9.15 40 .088400 8.25 . 145190 9.40 40 .117538 .95 .178558 9.15 41 9.088895 8.23 9.145754 9.38 I 41 9.118015 .93 9.179107 9.15 42 .089389 8.22 .146317 9.38 42 .118491 .95 .179656 9.13 43 .089882 8.23 .146880 9.37 43 .118968 .93 .180204 9.13 44 .090376 8.22 .147442 9.38 44 .119444 .92 .180752 9.13 45 .090869 8.22 .148005 9.37 45 .119919 .90 .181300 9.13 46 .091362 8.20 .148567 9.37 46 .120395 .92 .181848 9.12 47 .091854 8.20 .149129 9.35 47 .120870 .92 .182395 9.13 48 .092346 8.20 .149690 9.35 48 .121345 .92 .182943 9.12 49 .092838 8.20 .150251 9.37 49 .121820 .90 .183490 9.10 50 .093330 8.18 .150813 9.33 | 50 .122294 .90 .184036 9.12 51 9.093821 8 18 9.151373 9.35 i 51 9.122768 .90 9.184583 9.10 52 094312 8.18 .151934 9.33 l 52 .123242 .88 .185129 9.10 53 094803 8.17 . 152494 9.35 53 . 1237 15 .90 .185675 9.10 54 .095293 8.17 .153055 9.32 54 .124189 .88 .186221 9.10 55 .095783 8.17 .153614 9.33 55 .124662 .87 .186767 9.08 56 .096273 8.17 .154174 9.32 56 .125134 .88 .187312 9.10 57 .096763 8.15 .154733 9.33 57 .125607 .87 .187'858 9.08 58 .097252 8.15 .155293 9.30 58 .126079 .87 .188403 9.07 59 097741 8.13 .155851 9.32 59 .126551 .85 .188947 9.08 60 9.098229 8.15 9.156410 9.30 60 9.127022 .87 9.189492 9.07 418 AND EXTERNAL SECANTS. 30 31 / Vers. D. r. Ex. sec. D. 1'. / Vers. D. r. Ex. sec. D. 1'. 9.127022 .87 9.189492 9.07 9.154828 7.58 9.221762 8.85 1 .127494 .85 .190036 9.07 1 .1552813 7.58 .222293 8.87 2 .127965 .85 .190580 9.07 2 .155738 7.58 .222825 8.83 3 .128436 .83 .191124 9.07 3 .156193 7.58 .223355 8.85 4 .128906 .83 .191668 9.05 4 .156648 7.57 .223886 8.R5 5 .129376 .83 .192211 9.05 5 .157102 7.57 .224417 8.83 6 .129846 .83 .192754 9.05 G .157556 7.57 .224947 8.83 .130316 .82 .193297 9.05 7 .158010 7.57 .225477 8.83 8 .130785 .83 .193840 9.03 8 .158464 7.55 .226007 8.83 9 .131255 .82 .194382 9.05 9 .158917 7.55 .226537 8.82 10 .131724 7.80 .194925 9.03 10 .159370 7.55 .227066 8.82 11 9.132192 7.80 9.195467 9.03 11 9.159823 7.55 9.227595 8.83 12 .132660 7.82 .196009 9.02 12 .160276 7.53 .228125 8.80 13 .133129 7.78 .196550 9.03 13 .160728 7.53 .228653 8.82 14 .133596 7.83 .197092 9.02 14 .161180 7.53 .229182 8.82 15 .134064 7.78 .197633 9.02 15 .161632 7.52 .229711 8.80 16 .13-1531 7.78 .198174 9.02 16 .162083 7.53 .230239 8.80 17 .134998 7.78 .198715 9.00 17 .162535 7.52 .230767 8.80 18 .135465 7.77 .199255 9.00 18 .162986 7.52 .231295 8.78 19 .135931 7.77 .199795 9.00 19 .163437 7.50 .231822 8.80 20 .136397 7.77 .200335 9.00 20 .163887 7.52 .232350 8.78 21 9.136863 7.77 9.200875 9.00 21 9.164338 7.50 9.232877 8.78 22 .137329 7.75 .201415 8.98 22 .164788 7.48 .233404 8.78 23 .137794 7.77 .201954 9.00 23 .165237 7.50 . .233931 8.78 24 .138260 7.73 .202494 8.97 24 .165687 7.48 .234458 8.77 25 .138724 7.75 .203032 8.98 25 .166186 7.48 .234984 8.77 26 .139189 7.73 .203571 8.98 26 .166585 7.48 .235510 8.77 27 .139653 '7.73 .204110 8.97 27 .167034 7.48 .236036 8.77 28 .140117 7.73 .204648 8.97 28 .167483 7.47 .236562 8.77 29 .140581 7.73 .205186 8.97 29 .167931 7.47 .237088 8.75 30 .141045 7.72 .205724 8.97 30 .168379 7.47 .237613 8.77 31 9.141508 7.72 9.206262 8.95 31 9.168827 7.47 9.238139 8.75 3-2 .141971 7.72 .206799 8.97 32 .169275 7.45 .238664 8.75 33 .142434 7.70 .207:337 8.95 33 .169722 7.45 .239189 8.73 3-1 .142896 7.70 .207874 8.93 34 .170169 7.45 .239713 8.75 35 .143358 7.70 .208410 8.95 35 .170616 7.43 .240238 8.73 36 .143820 7.70 .208947 8.93 36 .171062 7.45 .240762 8.73 O^ .1442*3 7.68 .209483 8.95 37 .171509 7.43 .241286 8.73 38 .144743 7.68 .210020 8.93 38 .171955 7.42 .241810 8.72 39 .145204 7.68 .210556 8.92 39 .172400 7.43 .242333 8.73 40 .145665 7.68 .211091 8.93 40 .172846 7.42 .242857 8.72 41 9.146126 7.67 9.211627 8.92 41 9.173291 7.42 9.243380 8.72 42 .146586 7.67 .212162 8.92 42 .173736 7.42 .243903 8.72 43 .147046 7.67 .212697 8.92 43 .174181 7.42 .244426 8.72 44 .147506 7.67 .213232 8.92 44 .174626 7.40 .244949 8.70 45 .147966 7.65 .213767 8.90 45 .175070 7.40 .245471 8.72 46 .148425 .65 .214301 8.92 46 .175514 7.40 .245994 8.70 47 .148884 .65 .214836 8.90 47 .175958 7.40 .246516 8.70 48 .149:343 .63 .215370 8.90 48 .176402 7.38 .247'038 8.68 49 .149801 .63 .215904 8.88 49 .176845 7.38 .247559 8.70 50 .150259 .63 .216437 8.90 50 .177288 7.38 .248081 8.68 51 9.150717 .63 9.216971 8.88 51 9.177731 7.38 9.248602 8.68 52 .151175 .63 .217504 8.88 52 .178174 7.37 .249123 8.68 53 .151633 M .218037 8.88 53 .178616 7.37 .249644 8.68 54 .152090 .62 .218570 8.87 54 .179058 7.37 .250165 8.68 55 .152547 .60 .219102 8.88 55 .179500 7.37 .250686 8.67 56 .153003 .62 .219635 8.87 56 .179942 7.S5 .251206 8.67 57 .153460 .60 .220167 8.87 57 .180383 7.37 .251726 8.67 58 153916 .60 .220699 8.87 58 .180825 7.33 .252246 8.67 59 .154372 .60 .221231 8.85 59 .181265 7.35 .252766 8.67 60 9.154828 7.58 9.221762 8.85 1 60 9.181706 7.35 9.253286 8.65 419 TABLE XXVI. LOGARITHMIC VERSED SINES 32 33 / Vers. D.I". Ex. sec. D.r. / Vers. D.r. Ex. sec. D.r. o 9.181706 7.35 9.25328(5 8.65 9.207714 7.10 9.284122 8.48 1 .182147 7.33 .258805 8.65 1 .208140 .10 .284631 8.47 2 .182587 7.33 .254324 8.05 2 .208566 .10 .285139 8.47 3 .183027 7.32 .254843 8.65 3 .208992 .10 .285047 8.47 4 .183466 7.33 .255302 8.05 4 .209418 .08 .280155 8.47 5 .183906 7.32 .255881 8.03 5 .209843 .08 .280063 8.45 6 .184345 7.32 .256399 8.05 6 .210208 7.08 .287170 8.47 7 .184784 7.32 .256918 8.03 7 .210093 7.08 .287078 8.45 ! 8 .185223 7.32 .257436 8.03 8 .211118 7.08 .288185 8.45 9 .185682 7.80 .257954 8.02 9 .211543 7.07 .288092 8.45 10 .186100 7.30 .258471 8.03 ! 10 .211907 7.07 .289199 843 11 9.186538 7.30 9.258989 8.62 11 9.212391 7.07 9.289705 8.45 12 .186976 7.28 .259506 8.62 12 .212815 7.07 .290212 8.43 13 .187413 7.30 .260023 8. '62 13 .213239 7.05 .290718 8.43 14 .187851 7.28 .260540 8.62 14 .213062 7.05 .291224 8.43 15 .188288 7.27 .261057 8.62 i 15 .214085 7,05 .291730 8.43 10 .188724 7.28 .261574 8.60 16 .214508 7.05 .292230 8.43 17 .189161 7.27 .262090 8.00 17 .214931 7.05 .292742 8.42 18 .189597 7.28 .202606 8.60 18 .215354 7.03 .293247 8.43 19 .190034 7.25 .263122 8.00 19 .215776 7.03 .25)3753 8.42 20 .190469 7.27 .263638 8.00 20 .216198 7.03 .294258 8.42 21 9.190905 7.27 9.264154 8.58 21 9.216620 7.03 9.294763 8.42 22 .191341 7.25 .264669 8.58 22 .217042 7.02 .295268 8.40 23 .191776 7.25 .265184 8.00 23 .217403 7.02 .295772 8.42 24 .192211 7.23 .266700 8.57 24 .217884 7.02 .290277 8.40 25 .192645 7.25 .266214 8.58 25 .218305 7.02 .290781 8.40 26 .193080 7.23 .266729 8.58 26 .218720 7.00 .297285 8.40 27 . 194514 7.23 .267244 8.57 27 .219140 7.02 .297789 8.40 28 .193948 7.23 .267758 8.57 28 .219567 7.00 .298293 8.40 29 .194382 7.22 .268272 8.57 29 .219987 7.00 .298797 8.38 30 .194815 7.23 .268786 8.57 30 .220407 0.98 .299300 8.38 31 9.195249 7.22 9.269300 8.57 31 9.220826 7.00 9.299803 8.40 32 .195682 7.22 .269814 8.55 32 .221246 0.98 .300307 8.37 33 .196115 7.20 .270327 8.55 ! 33 .221005 0.98 .00809 8.38 34 .196547 7.22 .270840 8.57 ! M .222084 0.98 .301312 8.38 35 .196980 7.20 .271354 8.53 35 .222503 0.97 .201815 8.37 36 .197412 7.20 .271866 8.55 36 .222921 0.98 .302317 8.38 37 .197844 7.18 .272379 8.55 37 .223340 0.97 .02820 8.37 38 .198275 7.20 .272892 8.53 38 .223758 0.97 .303322 8.37 39 .198707 7.18 .273404 8.53 39 .224176 6.95 . 03824 8.35 40 .199138 7.18 .273916 8.53 40 .224593 6.97 .204325 8.37 41 9.199569 7.18 9.274428 8.53 41 9.225011 6.95 9.204827 8.35 42 .200000 7.17 .274940 8.53 42 .225428 6.95 .205328 8.37 43 .200430 7.18 .275452 8.52 43 .225845 6.95 .805880 8.85 44 .200861 7.17 .275963 8.52 44 .220202 0.93 .200331 8.35 45 .201291 7.15 .270474 8.53 45 .220078 0.95 .306832 8.35 46 .201720 7.17 .270986 8.50 46 .227095 0.93 .307333 8.33 47 .202150 7.15 .277496 8.52 47 .227511 0.93 .307833 8.35 48 .202579 7.15 .278007 8.52 48 .227927 6.92 .308334 8.33 49 .203008 7.15 .278518 8.50 49 .228342 6.93 .308834 8.33 50 .203437 7.15 .279028 8.50 50 .228758 6.92 .309334 8.33 51 9.203866 7.13 9.279538 8.50 51 9.229173 6.92 9.309834 8.33 52 .204294 7.15 .280048 8.50 52 .229588 0.92 .310334 8.33 53 .204723 7.13 .280558 8.50 53 .230003 0.92 .310834 8.22 54 .205151 7.12 .281088. 8.48 ! 54 .230418 0.90 .311883 8.32 55 .205578 7.13 .281577 8.50 | 55 .230832 6.90 .311832 8.32 56 .206006 7.13 .282087 8.48 56 .281246 6.90 .312331 8.32 57 .2064*3 7.12 .282590 8.48 | 57 .231000 6.90 .312830 8.32 58 .206860 7.12 .283105 8.48 58 .232074 6.88 .313329 8.32 59 .207287 7.12 .283614 8.47 59 .232487 6.90 .313828 8.30 60 9.207714 7.10 9.284122 8.48 i 60 9.232901 6.88 9.314326 8.32 420 AND EXTERNAL SECANTS. 34 | 35 1 Vers. D. 1". Ex. sec. D. 1". I / Vers D r. Ex. sec. D. 1". 9.23-3301 6.88 9.31432(5 8.32 9.257314 6.67 9.343949 8.15 1 .233314 6.88 .314825 8.30 ! 1 .257714 6.68 .344438 8.15 2 .233727 6.87 .315323 8.30 i 2 .258115 6.67 .344927 8.15 3 .234139 6.88 .315821 8.30 3 .258515 6.67 .345416 8.13 4 .234552 6.87 .316319 8.30 4 .258915 6.65 .345904 8.15 5 .234934 6.87 .316817 8.28 i 5 .259314 6.67 .346393 8.13 6 .235376 6.87 .317314 8.28 i 6 .259714 6.65 .346881 8.13 7 .235783 6.85 .317811 8.30 ! r- .260113 6.65 .347369 8.13 8 .23(5199 6.87 .318309 8.28 i 8 .260512 C.65 .347857 8.13 9 .233311 6.85 .318300 8.28 l 9 .260911 6.65 .348345 8.13 10 .237022 6.85 .319:303 8.27 | 10 .261310 6.65 .348833 8.13 11 9.237433 6.85 9.319799 8.28 ! 11 9.261709 6.63 9.349321 8.12 12 .237844 6.83 .320290 8.27 ! 12 .262107 6.63 .349808 8.12 13 .238254 6.85 .320792 8.28 i 13 .262505 6.63 .350295 8.12 14 .2331505 6.83 .321289 8.27 1 14 .262903 6.63 .350782 8.12 15 .239375 6.83 .321785 8.27 i 15 .263301 6.62 .351269 8.12 16 .239485 6.82 .322281 8.25 16 .263698 6.63 .351756 8.12 17 .239394 6.83 .322776 8.27 17 .264096 6.62 .352243 8.12 18 .240304 6.82 .323272 8.27 18 .264493 6.62 .352730 8.10 19 .240713 6.82 .323763 8.25 19 .264890 6 62 .353216 8.10 20 .211122 6.82 .324263 8.25 20 .265287 6.60 .353702 8.10 21 9.241531 6.82 9.324758 8.25 21 9.265683 6.62 9.a54188 8.10 22 .241940 6.82 .325253 8.25 22 .266080 6.60 .354674 8.10 23 .242348 6.80 .325748 8.25 23 .266476 6.60 .355160 8.10 24 .242756 6.80 .326243 8.23 24 .266872 6.58 .355646 8.08 23 .2431(54 6.80 .326737 8.25 25 .267267 6.60 .356131 8.10 28 .243572 6.80 .327232 8.23 26 .267663 6.58 .a56617 8.08 27 .243930 6.78 .327723 8.23 27 .268058 6.58 .357102 8.08 23 .2413S7 6.78 .323220 8.23 28 .268453 6.58 .357587 8.08 29 .244794 6.78 .323714 8.22 29 .268848 6.58 .358072 8.08 30 .215.201 6.78 .329207 8.23 30 .269243 6.58 .358557 8.08 31 9.245(>03 6.7? 9.329701 8.23 31 9.269638 6.57 9.359042 8.07 32 .246014 6.78 .330195 8.22 32 .270032 6.57 .359526 8.08 33 .245421 6.77 .330383 8.22 33 .270426 6.57 .360011 8.07 34 .24(5327 6.77 .&31181 8.22 34 .270820 6.57 .360495 8.07 35 .247233 6.77 .331674 8.22 35 .271214 6.57 .360979 8.07 30 .247633 6.75 .332167 8.20 36 .271608 6.55 .361463 8.07 37 .243944 6.75 .332659 8.22 37 .272001 6.55 .361947 8.07 33 .243449 6.75 .333152 8.20 38 .272394 6.55 .362431 8.05 39 .248854 6.75 .333644 8.22 39 .272787 6.55 .362914 8.07 40 .249259 6.75 .334137 8.20 40 .273180 6.53 .363398 8.05 41 9.249064 6.73 9.334629 8.20 41 9.273572 6.55 9.363881 8.05 42 .250003 6.75 .335121 8.18 42 .273965 6.53 .364364 8.05 43 .250473 6.73 .335612 8.20 43 .274357 6.53 .364847 8.05 44 .250377 6.73 .336104 8.18 44 .274749 6.3 .365330 8.05 45 .251281 6.72 .336595 8.20 j 45 .275141 6.52 .365813 8.03 46 .251034 6.73 ; 337087 8.18 46 .275532 6.53 .366295 8.05 47 .252038 6.72 .337578 8.18 47 .275924 6.52 .366778 8.03 48 .232491 6.72 .3:38069 8.18 48 .276315 6.52 .367260 8.03 49 .252394 6.72 .338560 8.17 49 .276706 6.52 .367742 8.03 50 .253297 6.70 .339050 8.18 50 .277097 6.52 .368224 8.03 51 9.253699 6.72 9.339541 8.17 51 9.277488 6.50 9.368706 8.03 52 .254102 6.70 .340031 8.18 52 .277878 6.50 .369188 8.03 53 .2545)4 6.70 .340522 8.17 53 .278268 6.50 .369670 8.02 51 .2549;)iJ 6.70 .341012 8.17 54 .278658 6.50 .370151 8.05 55 .255303 6.63 .341502 8.15 ! 55 .279048 6.50 .370632 8 03 56 .255709 6.70 .341991 8.17 5b .2794:38 6.48 .371114 8.02 57 .230111 6.68 .342481 8.17 | 57 .279827 6.50 .371595 8.02 58 .256512 6.68 .342971 8.15 1 58 .280217 6.48 .372076 8.00 59 .256913 6.68 .343460 8.15 59 .280606 6.48 .372556 8.02 60 9.257314 6.67 9.343949 8.15 i 60 9.280995 6.47 9.373037 8.02 TABLE XXVI. LOGARITHMIC VERSED SINES 1 36 37' > / Vers. D. 1. Ex. sec. D. I'. ' Vers. D.I'. Ex. sec. D. r. 9.280995 6.47 9.373037 8.02 j 9.303983 6.28 9.401634 .88 1 .281383 6.48 .373518 8.00 1 .304360 6.30 .402107 .88 .281772 6.47 .373998 8.00 1 2 .304738 6.28 .402580 .87 3 .283160 6.47 .374478 8.00 I .305115 6.28 .403052 .87 4 .282548 6.47 .374958 8.00 | 4 .305492 6.27 j .403524 .88 5 .282936 6.47 .375438 8.00 5 .305868 6.28 : .403997 .88 6 .283324 6.47 .375918 8.00 6 .306245 6.27 .404469 .87 7 .283712 6.45 .376398 7.98 7 .306621 6.28 .404941 .87 8 .284099 6.45 .37(5877 8.00 1 8 .306998 6.27 I .405412 .87 9 .284486 6.45 .377357 7.98 Q .307374 6.25 .405884 .87 10 .284873 6.45 .377836 7.98 10 .307749 6.27 .406356 .85 11 9.285260 G.45 9.378315 7. -98 11 9.308125 6.27 9.406827 .85 12 .285647 6.43 .378794 7.98 12 .308501 6.25 .407298 .87 13 .286033 6.43 .379273 7. -98 13 .308876 6.25 ! .407770 .85 14 .286419 6.43 .379752 7.98 14 .309251 6.25 .408241 .85 15 .286805 (5.43 .380231 7.97 15 .309626 6.25 .408712 .85 16 .287191 6.43 .380709 7.98 16 .310001 6.^3 .409183 .83 17 .287577 6.42 .381188 7.97 17 .310375 6.25 .409653 .85 18 .287962 6.40 .381666 7.97 18 .310750 6.23 .410124 .83 19 .288348 6.42 .382144 7.97 19 .311124 6.23 .410594 .85 20 .288733 6.42 .382622 7.97 20 .311498 6.23 .411065 .83 21 9.289118 6.40 9.383100 7.95 21 9.311872 6.22 i 9.411535 .83 2-2 .289502 6.42 .383577 7.97 28 .312245 6.23 .412005 .83 23 .289887 6.40 .384055 7.95 23 .312619 6.22 .412475 .83 24 .290271 6.40 .384532 7.97 24 .312992 6.22 .412945 .83 25 .290655 6.40 .385010 7.95 25 .31.3365 6.22 .413415 .8 26 .291039 6.40 .385487 7.95 26 .313738 6.22 .413884 .83 27 .291423 6.40 .385964 7.95 27 .314111 0.22 .414:354 .82 28 .291807 6.38 .386441 7.95 28 .314484 6.20 .414823 .83 29 .292190 6.38 .386918 7.93 29 .314856 6.20 .415293 .82 30 .292573 6.38 .387394 7.95 30 .315228 6.20 .415762 .82 31 9.292956 6.38 9.387871 7.93 31 9.315600 6.20 9.416231 .82 32 .293339 6.38 .388347 7.95 32 .315972 6.20 .416700 .80 33 .293722 6.37 .388824 7.93 33 .316344 6.20 .417168 .82 34 .294104 6.37 .389300 7.93 34 .316716 6.18 .417637 .82 35 .294486 6.37 .389776 7.93 35 .317087 6.18 .418106 .80 3G .294868 6.37 .390252 7.92 36 .317458 6.18 .418574 .80 37 .295250 6.37 .390727 7.93 37 .317829 6.18 .419042 .82 38 .295632 6.37 .391203 7.92 38 .318200 6.18 .419511 .80 39 .296014 6.35 .391678 7.93 39 .318571 6.17 .419979 .80 40 .296395 6.35 .392154 7.92 40 .318941 6.17 .420447 .80 41 9.296776 6.35 9.392629 7.92 41 9.319311 6.18 9.420915 .78 42 .297157 6.35 .393104 7.92 42 .319682 6.15 .421382 .80 43 .297o38 6.33 .393579 7.92 43 .320051 6.17 .42ia50 .78 44 .297918 6.35 .394054 7.92 44 .320421 6.17 .422317 .80 45 .298299 6.33 .394529 7.90 45 .320791 6.15 .422785 .78 46 .298679 6.33 .395003 7.92 40 .321160 6.17 .423252 .78 47 .299059 6.33 .395478 7.90 47 .321530 6.15 .423719 .78 48 .299439 6.33 .395952 7.90 i 48 .821899 6.13 .424186 .78 49 .299819 6.32 .396426 7.90 49 .322267 6.15 .424653 .78 50 .300198 6.32 .396900 7.90 50 .322636 6.15 .425120 .78 51 9-300577 6.33 0.397374 7.90 ni 9.323005 6.13 9.425587 .77 52 .300957 6.30 .397848 7.!>0 :>:> .3*3373 6.13 .4215053 .78 53 .301835 6.32 .398322 7.88 53 .32)741 fi.13 .426520 54 .301714 6.32 .398795 7.90 54 .324109 6.13 .426986 ' '.7" 55 .302093 6.30 .399269 7.88 | 55 .324477 6.13 .427452 p '.7~ 5G .302471 6.30 .399742 7.88 66 .324845 6.12 .427918 7 7" 57 .302849 6.30 .400215 7.88 57 .325212 6.13 .428384 7> 58 .303227 6.30 .400688 7.88 58 i .325580 6.12 .428850 7.7" 59 .303605 6.30 .401161 7.88 59 ! .325047 6.12 .429316 r~ i~r 60 9.303983 6.28 9.401634 7.88 60 9.326314 6.12 9.429782 7! 75 422 AND EXTERNAL SECANTS. 38 39 / Vers. D. 1". Ex. sec. D. r. ' Vers. ID. 1". Ex. sec. D. 1". I 9.326314 6.12 9.429782 7.75 9.348021 5.93 9.457518 i 7.65 j .326(381 6.10 .430247 7.77 1 .348377 5.95 .457977 7.65 2 .327047 6.12 .430713 7.75 2 .348734 5.93 .458436 7.65 3 .327414 6.10 .431178 7.75 3 .349090 5.93 .458895 7.63 .327780 i 6.10 .431643 7.75 4 .349446 5.93 .459:353 7.65 K .328146 6.10 .432108 7.75 5 .349802 5.93 .459812 7.63 6 .328512 6.10 .432573 1 7.75 6 .850158 5.93 .460270 7.65 7 .328878 6.08 .433038 7.75 7 .350514 5.92 .460729 7.63 8 .329243 6.10 .433503 7.73 8 .350869 5.93 .461187 7.63 9 .329609 6.08 .433967 7.75 9 .351225 5.92 .461645 7.63 10 .329974 6.08 .434432 7.73 10 .351580 6.92 .462103 7.63 11 9.330339 6.08 9.434896 7.75 11 9.351935 6.92 9.462561 7.63 12 .330704 6.08 .435361 7.73 12 .352290 5.90 .463019 7.63 13 .331069 6.07 .435825 7.73 13 .352644 5.92 .463477 7.62 14 .3314.33 i 6.08 .436289 7.73 14 ..352999 5.90 .463934 7.63 15 .331798 6.07 .436753 7.73 15 .353353 5.90 .464392 7.62 16 .332162 6.07 .437217 7.72 16 .353707 5.92 .464849 7.63 17 .332526 6.07 .437680 7.73 17 .354062 5.88 .465307 7.62 18 ' .332890 6.07 .438144 7.73 18 .354415 5.90 .465764 7.62 19 .333254 6.05 .438608 7.72 19 .354769 5.90 .466221 7.62 20 .333617 6.07 .439071 7.72 20 .355123 5.88 .466678 7.62 21 9.333981 6.05 9.439534 21 9.355476 5.88 9.467135 7.62 22 .334344 ! 6.05 .439997 7 72 22 .855829 5.88 .467592 7.62 23 .334707 6.05 .440460 7^72 23 .356182 5.88 .468049 7.62 24 .335070 6.03 .440923 7.72 24 .356535 5.88 .468506 7.60 25 .335432 6.05 .441386 7.72 25 .356888 5.88 .468962 7.60 26 .335795 6.03 .441849 7.72 26 .357241 5.87 .469418 7.62 27 .336157 6.03 .442312 7.70 27 .357593 5.87 .469875 7.60 28 .336519 6.03 .442774 7.72 28 .357945 5.87 .470331 7.60 29 .336881 6.03 .443237 7.70 29 .358297 5.87 .470787 7.60 30 .337243 6.03 .443699 7.70 30 .358649 5.87 .471243 7.60 31 9.337605 6.02 9.444161 7.70 31 9.359001 5.87 9.471699 7.60 32 .337966 6.03 .444623 7.70 32 .359353 5.85 .472155 7.60 33 .338328 6.02 .445085 7.70 33 .359704 6.87 .472611 7.60 34 .338689 6.02 .445547 7.70 34 .860056 5.85 .473067 7.58 35 .339050 6.02 .446009 7.68 35 .360407 5.85 .473522. 7.eo 36 .339411 6.00 .446470 7.70 1 36 .360758 5.83 .473978 7.58 37 .339771 6.02 .446932 7.68 37 .861108 5.85 .474433 7.58 38 .340132 6.00 .447393 7.70 38 .361459 5.85 .474888 7.58 39 .340492 6.00 .447855 7.68 89 .361810 5.83 .475343 7.58 40 340852 6.00 .448316 7.68 40 .362160 5.83 .475798 7.58 41 9.341212 6.00 9.448777 7.68 41 9.S62510 5.83 9.476253 7.58 42 .341572 6.00 .449238 7.68 42 .362860 5.83 .476708 7.58 43 .341932 5.98 .449699 7.68 43 .363210 5.83 .477163 7.58 44 .342291 6.00 .450160 7.67 44 .363560 5.82 .477618 7.57 45 .342651 5.98 .450620 7 68 45 .363909 5.83 .478072 7.58 46 .343010 5.98 .451081 7.67 46 .364259 5.82 .478527 7.57 47 .343369 5.98 .451541 7.68 47 .364608 5.82 .478981 7.57 48 .343728 5.97 .452002 7.67 48 .364957 5.82 .479435 7.58 49 .344086 5.98 .452462 7.67 49 .365306 5.82 .479890 7.57 50 .344445 5.97 .452922 7.67 50 .365655 5.80 .480344 7.57 51 9.344803 5.97 9.453382 7.67 61 9.366003 5.82 9.480798 7.57 52 .345161 5.97 .453842 7.67 52 .366352 5.80 .481252 7.55 53 .345519 5.97 .454302 7.67 53 .366700 5.80 .481705 7.57 54 .345877 5.97 .454762 7.65 54 .367048 5.80 .482159 1 7.57 55 .346235 5.95 .455221 7.67 55 .367396 5.80 .482613 7.55 56 .346592 5.97 .455681 7.65 56 .367744 5.78 .483066 7.57 57 .346950 5.95 .456140 7.67 57 H .368091 5.80 .48:3520 7.55 58 .347:307 5.95 .456600 7.65 58 1 .368439 5.78 .483973 7.55 59 .347664 5.95 .457059 7.65 59 .368786 5.78 .484426 7.55 60 9.348021 5.93 9.457518 7.65 60 9 369133 5.78 9.484879 7.55 423 TABLE XXVI. LOGARITHMIC VERSED SINES 40 41 / Vers. D. r. Ex. sec. D. 1". ' Vers. D. 1". Ex. sec. D. 1". 9.369133 5.78 9.484879 7.55 i 9.389681 5.62 9.511901 7.42 1 .369480 5.78 .485332 7.55 1 .390018 i 5.63 .512348 7.47 2 .369827 5.78 .485785 7.55 2 .390a56 5.63 .512796 7 45 3 .370174 5.77 .486238 7.55 3 .390094 | 5.62 .513243 7.47 4 .370520 5.78 .486691 7.55 4 .391031 5.62 .513091 7.45 5 .370867 i 5.77 .487144 7.53 5 .391368 5.62 .514138 7.45 G .371213 5.77 i .487596 7.55 6 .391705 5.62 .514585 7.47 7 .371559 5.77 .488049 7.53 7 .392042 5.62 .515033 7.45 8 .371905 5.77 .488501 7.53 8 .392379 5.02 .515480 7.45 9 .372351 5.75 .488953 7.55 9 .392716 5.60 .515927 7 45 10 ! .372598 1 5.77 .489406 7.53 10 .393052 5.60 .516374 7.43 11 ! 9.372942 5.75 9.489858 7.53 11 9.393388 5.60 9.516820 7 45 12 .373287 5.75 .490310 7.53 12 .393724 5.62 .517267 7.45 13 .373632 5.75 .490762 7.53 13 .394061 5.58 .517714 7.43 14 .373977 5.75 .491214 7.52 14 .394396 5.00 .518100 7.45 15 .374322 5 . 75 .491665 7.53 15 .394732 5.60 .518007 7.43 10 .374667 5.73 .492117 7.53 16 .395068 5.58 .519053 7.45 17 .375011 5.75 .492569 7.52 17 .395403 5.58 .519500 7 43 18 .375356 5.73 .493020 7.52 18 .895788 5.60 .519946 7.43 19 .375700 5.73 .493471 7.53 19 .396074 5.58 .520392 7.43 20 .376044 5.73 .493923 7.52 20 .396409 5.57 .520838 7.43 21 9.376388 5.73 9.494374 7.52 21 9.396743 5.58 9.521284 7.43 22 .376732 5.72 .494825 7.52 21 .397078 5.58 .521730 7 43 23 .377075 5.73 .495276 7.52 23 .397413 5.57 .522176 7.42 24 .377419 5.72 .495727 7.52 24 .397747 5.57 .522621 7 43 25 .377762 5.72 .496178 7.50 25 .398081 5.57 .523067 7.43 23 .378105 5.72 .496028 7.52 26 .398415 5.57 .523513 7 42 27 .378448 5.72 .497079 7.52 27 .398749 5.57 .523958 7.43 23 .378791 5.70 .497530 7.50 28 .3990a3 5.57 .524404 7 42 23 .379133 5.72 .497980 7.52 29 .399417 5.55 .524849 7.42 30 .379476 5.70 .498430 7.50 30 .399750 5.57 .525294 7.42 31 9.379818 5.72 9.498881 7.48 31 9.400084 5.55 9.525?39 7.42 32 .380161 5.70 .499331 7.52 32 .400417 5.55 .520184 7.42 33 .389503 5.70 .499781 7.50 33 .400750 5.55 .526629 7.42 34 .380845 5.68 .500231 7.50 34 .401083 5.55 .527074 7.42 35 .381186 5.70 .500681 7.50 &5 .401416 5.53 .527519 7 42 3(3 .381528 5.68 .501131 7.50 36 .4017'48 5.55 .527964 7.42 37 .381869 5.70 .501581 7.48 37 .402081 6.53 .528409 7 40 38 .882211 5.68 .502030 7.50 38 .402413 5.53 .528853 7.42 39 .382552 5.68 .502480 7.48 39 .402745 5.53 .529298 7 40 40 .382893 6.68 .502929 7.50 40 .403077 5.53 .529742 7.42 41 9.383234 5.67 9.503379 7.48 41 9.403409 5.53 9.530187 7 40 42 .383574 5.68 .503828 7.48 42 .403741 5.53 .530031 7.40 43 .383915 5.67 .504277 7.48 43 .404073 5.52 .531075 7.40 44 .384255 5.67 .504726 7.48 44 .404104 5.53 .531519 7.40 45 .384595 5.67 .505175 7.48 45 .404736 5.52 .531903 7.40 46 .384935 5.67 .505024 7.48 46 .405067 5.52 .532407 7.40 47 .385275 5.67 .506073 7.48 47 .405398 5.52 .532851 7 40 48 .385615 5.67 .506522 7.48 48 .405729 5.50 .533295 7.40 49 .385955 5.65 .506971 7.47 49 .406059 5.52 .533789 7.38 50 .386294 5.67 .507419 7.48 | 50 .406390 5.52 .534182 7.40 51 9.386634 5.65 9.507868 7.47 51 9.406721 5.50 9.534628 7.40 52 .386973 5.65 .508316 7.48 52 .407051 5.50 .636070 7.38 53 .887312 5.65 .508765 7.47 53 .407381 5.50 .535513 7.38 54 .387651 5.63 .509213 7.47 54 .407711 5.50 .535956 7.40 55 .387989 5.65 .509661 7.47 55 .408041 5.50 .530400 7.38 56 .388328 5.63 .510109 7.47 56 .408371 5.48 .530843 7.38 57 .3386(56 5.65 .510557 . 7.47 57 .408700 5.50 .537280 7.38 58 .389005 5.63 .511005 7.47 58 .409030 5.48 .537729 7.38 59 .389343 5.63 .511453 7.47 59 .409359 5.48 .533172 7.38 60 9.389681 5.62 9.511901 7.45 60 i 9.409088 5.48 9.538015 7.38 424 AND EXTERNAL SECANTS. 42 ' "I 43 / Vers. D.I'. Ex. sec. D. r. / Vers. D. r. Ex. sec. D. 1". 0.409688 5.48 9.538615 7.38 9.429181 5.35 9.565053 7.32 I .410017 5.48 .539058 7.37 1 .429502 5.33 .565492 7.30 2 .410346 5.48 .539500 7.38 2 .429822 5.a3 .565930 7.32 3 .410675 5.48 .539943 7.38 3 .430142 5.35 .566369 7.30 4 .411004 5.47 .540386 7.37 4 .430463 5.33 .566807 7.30 5 .411332 5.47 .540828 7.38 5 .430783 5.a3 .567245 7.30 6 .411600 5.48 .541271 7.37 6 .431103 5.32 .567683 7.30 7 .411989 5.47 .541713 7.37 7 .431422 5.33 .568121 7.30 8 .412317 5.45 .542155 7.37 8 .431742 5.33 .568559 7.30 9 .412644 5.47 .542597 7.38 9 .432062 5.32 .568997 7.30 10 .412972 5.47 .543040 7.37 10 .432381 5.32 .569435 7.30 11 9.413300* 5.45 9.543482 7.37 11 9.432700 5.33 9.569873 7.30 13 .413627 5.47 .543924 7.37 12 .4830-20 5.32 .570311 7.28 13 .413955 5.45 .544366 7.35 13 .4a3339 5.30 .570748 7.30 14 .414282 5.45 .544807 7.37 14 .433657 5.32 .571186 7.30 15 .414609 5.45 .545249 7.37 15 .433976 5 32 .571624 7.28 16 .414936 5.45 .545691 7.a5 16 .434295 5.30 .572061 7.28 17 .415263 5.43 .546132 7.37 17 .434613 5.32 .572498 7.30 18 .413389 5.45 .546574 7.. 35 18 .434932 5.30 .572936 7.28 19 .415916 5.43 .547015 7.37 19 .4a5250 5.30 .573373 7.28 20 .416242 5.43 .547457 7.35 20 .435568 5.30 .573810 7.28 21 9.416568 5.43 9.547898 7.a5 21 9.435886 5.30 9.574247 7.30 22 .41(5894 5.43 .548339 7 37 22 .436204 5.28 .574685 7.28 23 .417220 5.43 .548781 7.35 23 .436521 5.30 .575122 7.27 24 .417546 5.42 .549222 7.35 24 .436839 5.28 .575558 7.30 25 .417871 5.43 .549663 7.35 25 .437156 5.28 .575995 7.28 26 .418197 5.42 .550104 7.33 26 .437473 5.30 .576432 7.28 27 .418522 5.43 .550544 7.35 27 .437791 5.27 .576869 7.28 28 .418848 5.42 .550085 7 '85 28 .438107 5.28 .577306 7.27 29 .419173 5.42 .551426 7.35 29 .438424 5.28 .577742 7.28- 30 .419498 5.40 .551867 7.33 30 .438741 5.28 .578179 7.27 31 9.419822 5.42 9.552307 7.35 31 9.439058 5.27 9.578615 7.28 32 .420147 5.40 .552748 7.33 32 .439374 5.27 .579052 7.27' .33 .420171 5.42 .553188 7.35 33 .439690 5.28 .579488 7.27 34 .420796 5.40 .553629 7.33 34 .440007 5.27 .579924 7.28 35 .421120 5.40 .554069 7.33 35 .440:323 5.27 .580361 7.27 36 .421414 5.40 .554509 7.33 36 .440639 5.25 .580797 7.27 37 .421768 5.40 .554949 7.33 37 .440954 6.27 .581233 7.27. as .422092 5.40 .555389 7.33 38 .441270 5.25 .581609 7 27 39 .422416 5.38 .555829 7.33 39 .441585 5.27 .582105 7^27 40 .422739 5.40 .556269 7.33 40 .441901 5.25 .582541 7.27 41 9.423063 5.38 9.556709 7.33 41 9.442216 5.25 9.582977 7.27 42 .423386 5.38 .557149 7.33 42 .442531 5.25 .583413 7.25 43 .423709 5.38 .557c89 7.32 43 .442846 5.25 .583848 7.27 41 .424032 5.38 .558028 7.33 44 .443161 5.28 .584284 7.27 45 .424355 5.37 .558468 7.32 45 .443476 5.23 .584720 7.25 46 .424677 5.38 .558907 7.as 46 .443790 5.25 .585155 7.27 47 .425000 5.37 .559347 7.32 47 .444105 5.23 .585591 7.25 48 .425322 5.38 .559786 7.33 48 .444119 5.23 .586026 7.27 49 .425645 5.37 .560226 7.32 49 .444733 5.23 .586462 7.25 50 .425967 5.37 .560665 7. S3 50 .445047 5.23 .586897 7.25 51 9.426289 5.37 9.561104 7.32 51 9.445361 5.23 9.587332 7.25 52 .426611 5.37 .561543 7.32 52 .445675 5.23 .587767 7.27 53 .426933 5.35 .561982 7.32 53 .445989 5.22 .588203 7.25 51 .427254 5.37 .562421 7.32 54 .446302 5.23 .588638 7.25 55 .427576 5.35 .562860 7.32 55 .446616 5.22 .589073 7.25 56 .427897 5.35 .563299 7.32 56 .446929 5.22 .589508 7.23 57 .428218 5.35 .563738 7.30 57 .447242 5.22 .589942 7.25 58 .428539 5.35 .564176 7.32 58 .447555 5.22 .590377 7.25 59 .428860 5.35 .564615 7.30 59 .447868 5.22 .590812 7.25 6) 9.429181 5.33 9.565053 7.32 60 9.448181 5.20 9.591247 7.23 435 TABLE XXVI. -LOGARITHMIC VERSED SINES 44 45 / Vers. D. r. Ex. sec. D. 1". ' Vers. D.I-. Ex. sec. D. r. 9.448181 5.20 9.591247 7.23 9.466709 5.08 9.617224 7.20 1 .448493 5.22 .591681 7.25 1 .467014 5.08 .617656 7.18 2 .448806 5.20 .592116 7.25 2 .467319 5.08 .618087 7.18 3 .449118 5.22 .592551 7.23 3 .467624 5.07 .618518 7.18 4 .449431 5.20 .592985 7.23 4 .467928 5.08 .618949 ".18 5 .449743 5.20 .593419 7.25 5 .468233 5.07 .619380 7.18 6 .450055 5.18 .593854 7.23 6 .468537 5.07 .619811 ".18 7 .450366 5.20 .594288 7.23 7 .468841 5.07 .620242 ".18 8 .450678 5.20 .594722 7.23 8 .469145 5.07 .620673 ".18 9 .450990 5.18 .595156 7.25 9 .469449 5.07 .621104 ".18 10 ,45101 5.18 .595591 7.23 10 .469753 5.07 .621535 ".18 11 9.451612 5.20 9.596025 7.23 11 9.470057 5.05. 9.621966 ".17 12 .451924 5.18 .596459 7.23 12 .470360 5.07 .622396 ".18 13 .452235 5.18 .596893 7.22 13 .470664 5.05 .622827 ".18 14 .452546 5.17 .597326 7.23 14 .470967 5.05 .623258 ".17 15 .452856 5.17 .597760 7.23 ! 15 .471270 5?05 .623688 ".18 16 .453167 5.18 .598194 7.23 ! 16 .471573 5.05 .624119 ".17 17 .453478 5.17 .598628 7.22 17 .471876 5.05 .624549 ".18 18 .453788 5.17 .599061 7.23' 18 .472179 5.05 .624980 ".17 19 ' .454098 5.17 .599495 7.22 19 .472482 5.03 .625410 ".18 20 .454408 5.17 .599928 7.23 1 20 .472784 5.05 .625841 ".17 21 9.454718 5.17 9.G003G2 7.22 21 9.473087 5.03 9.C26271 ".17 22 .455028 5.17 .600795 7.23 22 .473389 5.03 .626701 ".17 23 .455338 5.17 .601229 7 22 23 .473691 5.03 .627131 ".17 24 .455648 5.15 .601662 7^22 24 .473993 5.03 .627561 7.17 25 .455957 5.17 .602095 7.22 25 .474295 5.03 .627991 .17 26 .456267 5.15 .602,328 7.23 26 .474597 5.03 .628421 .17 27 .456576 5.15 .602062 7.22 27 .474899 5.02 .628851 .17 28 .456885 5.15 .603395 7.22 28 .475200 5.03 .629281 .17 29 .457194 5.15 .603828 7.22 29 .475502 5.02 .629711 .17 30 .457503 5.13 .604261 7.22 30 .475803 5.02 .630141 .17 31 9.457811 5.15 9.604694 7.20 31 9.476104 5.02 9.630571 .17 32 .458120 5.15 .605126 7.22 i i 32 .476405 5.02 .631001 .15 33 .458429 5.13 .605559 7.22 ! i 33 .476706 5.02 .631430 .17 34 .458737 5.13 .C05992 7.22 34 .477007 5.02 .631860 .17 35 .450045 5.13 .606425 7.20 35 .477308 5. CO .632280 .15 36 .459353 5.13 .606857 7.22 j 36 .477608 5.02 .632719 .17 37 .459061 5.13 .607290 7.20 37 .477909 5. CO .633149 .15 33 .450969 5.13 .607722 7.2'2 38 .478209 5. CO .633578 .17 30 .460277 5.12 .608155 7.20 ! 39 .478509 5.00 .634008 .15 40 .460584 5.13 .608587 7.22 40 . .478809 5. CO .634437 .15 41 9.460S92 5.12 9.609020 7.20 41 9.479109 5. CO 9.634866 .17 42 .461199 5.12 .609452 7.20 42 .479409 5. CO .635296 .15 43 .461506 5.12 .609884 7.20 43 .4797'09 5.00 .635725 .15 44 .461813 5.12 .61031(3 7.22 44 .480009 4 98 .636154 .15 ft .462120 5.12 .610749 7.20 45 .480308 5. CO .636583 .15 40 .462427 5.12 .611181 7.20 46 .480608 4.98 .637012 .15 47 .462734 5.10 .611613 7.20 47 .480907 4.98 .637441 .15 48 .463040 5.12 .612045 7.20 48 .481206 4.98 .637870 .15 49 .463347 5.10 .612477 7.18 49 .481505 4.98 .688299- .15 50 .463653 5.10 .612908 7.20 i 50 .481804 4.98 .028728 .15 51 9.463f5D 5.10 9.613340 7.20 51 9.482103 4.97 9.639157 .15 52 .464265 5.10 .613772 7.20 52 .482401 4.98 .639586 .15 53 .464571 5.10 .614204 7.18 53 .4P2700 4.97 .640015 .13 54 .464877 5.10 .614635 7.20 1 54 .482988 4.97 .640443 .15 55 .465183 5.08 .615067 7.20 55 .483296 4.98 .64087'2 .15 50 .465488 5.10 .615499 7.18 i 56 .483595 4.97 .641301 .13 57 .465794 5.08 .615930 7.20 57 .483893 4:97 .641729 .15 58 .4fi5099 5.08 .616362 7.18 58 .484191 4.95 ! 642158 .13 59 .466404 5.08 .616793 7.18 59 .484488 4.97 .642586 .15 60 9.466709 5.08 9.617224 7.20 60 9.484786 4.97 9.643015 .13 426 AND EXTERNAL SECANTS. 46 o 470 / Vers. D. 1. Ex. sec. D.I". / Vers. D.I*. Ex. sec. D. 1". 9.484786 4.97 9.643015 7.13 9.502429 4.85 9.668646 7.10 1 .4&5084 4.95 .643443 7.15 1 .502720 4.83 .669072 7.10 2 .485381 4.95 .643872 7.13 2 .503010 4.83 .669498 7.10 3 .485678 4.97 .644300 7.13 3 .503300 4.85 .669924 7.10 4 .485976 4.95 .644728 7.13 4 .503591 4.83 .670350 7.10 5 .486273 4.95 .645156 7.15 5 .503881 4.83 .670776 7.08 6 .486570 4.93 .645585 7.13 i 6 .504171 4.82 .671201 7.10 7 .486866 4.95 .646013 7.13 i 7 .504460 4.83 .671627 7.10 8 .487163 4.95 .646441 7.13 8 .504750 4.83 .672053 7.10 9 .487460 4.93 .646869 7.13 9 .505040 4.82 .672479 7.08 10 | .487750 4.95 .647297 7.13 10 .505329 4.82 .672904 7.10 11 9.483053 4.93 9.647725 7.13 11 9.505618 4.83 9.673330 7.10 12 .483349 4.93 .648153 7.13 12 .505908 4.82 .673756 7.08 13 .483(345 4.93 .648581 7.13 13 .506197 4.82 .674181 7.10 14 .488941 ! 4.93 .649009 7.12 i 14 .506486 4.82 .674607 7.08 15 .489237 4.93 .649436 7.13 i 15 ! .506775 4.80 .675032 7.10 16 .489533 4.92 .649864 7.13 16 .507063 4.82 .675458 7.08 17 .489323 4.93 .650292 7.13 i 17 .507352 4.80 .675883 7.10 18 .490124 4.92 .650720 7.12 1 18 .507640 4.82 1 .676309 7.08 19 .490119 4.92 .651147 7.13 i 19 .507929 4.80 .676734 7.08 23 .430714 4.93 .651575 7.12 ! 20 .508217 4.80 .677159 7.08 21 9.491010 4.92 i 9.652002 7.13 i 21 9.508505 4.80 9.677584 7.10 22 .491305 4.9:3 .652430 7.12 ! 22 .508793 4.80 .678010 7.08 23 .491600 4.93 .652857 7.13 i 23 .509081 4.80 .678435 7.08 24 .491894 4.92 .653285 7 12 24 .509369 4.80 .678860 7.08 25 .492189 4.92 .653712 7^13 1 25 .509657 4.80 .679285 7.08 28 .492484 4.90 .654140 7.12 26 .503945 4.78 .679710 7.10 27 .492778 4.90 .654567 7.12 27 .510232 4.80 .680136 7.08 23 .493072 4.92 .654994 7.12 28 .510520 4.78 .680561 7.08 29 .493367 4.90 .655421 7.13 29 .510807 4.78 .680986 7.08 30 .493661 4.90 .655849 | 7.12 30 .511094 4.78 .681411 7.08 31 9.493955 4.90 9.656276 i 7.12 31 9.511381 4.78 9.681836 7.07 32 .494249 4.83 .656703 7.12 ! 32 .511668 4.78 .682260 7.08 33 .494542 4.90 .657130 7.12 i 33 .511955 4.77 .682685 7.08 34 .494836 4.90 .657557 7.12 ! 34 .512241 4.78 .683110 7.08 35 .495130 4.88 .657984 7.12 i 35 .512528 4.78 .683535 7.08 36 .495423 4.88 .658411 7.12 i 36 .512815 4.77 .683960 7.08 37 .495716 4.88 .658833 7.12 ! 37 .513101 4.77 .684385 7.07 38 .493009 4.88 .659265 7.10 38 .513387 4.77 .684809 7.08 39 .493302 4.88 .659891 7.12 39 .513673 4.77 .685234 7.08 40 .498595 4.88 .660118 7.12 | 40 .513959 4.77 .685659 7.07 41 9.496883 4.83 9.660545 7.12 41 9.514245 4.77 9.686083 7.08 42 .497181 4.87 .660972 7.10 | 42 .514531 4.77 .686508 7.08 43 .497473 4.83 .661398 7.12 43 .514817 4.75 . .686933 7.07 44 .497766 4.87 .661825 7.12 | 44 .515102 4.77 .687a57 7.08 45 .493058 4.87 .662252 7.10 ! 45 .515388 4.75 .687782 7.07 46 .493:350 4.88 .662678 7.12 46 .515673 4.77 .688206 7.08 47 .493643 4.87 .683105 7.10 47 .515959 4.75 .688631 7.07 48 .493935 4.S5 .663531 7.12 48 .516244 4.75 .689055 7.07 49 .499896 4.87 .663958 7.10 49 .516529 4.75 .689479 7.08 50 .499518 4.87 .664384 7.10 50 .516814 4.73 .689904 7.07 51 9.499810 4.R5 9.664810 7.12 51 9.517093 4.75 9.690328 7.07 52 .500101 4.87 .665237 7.10 52 .517383 4.75 .690752 7.08 53 .500393 4.85 .665663 7.10 53 .517663 4.73 .691177 7.07 54 .500634 4.8'> .666080 7.10 54 .517952 4.73 .691601 7.07 55 .500975 4.85 .666515 7.12 55 .518236 4.75 .692025 7.07 56 .501266 4.85 .666942 7.10 SB .518521 4.73 .692449 7.07 57 .501557 4.85 .667368 7.10 57 .618805 4.73 .692873 7.08 58 .501848 4.85 .667794 7.10 58 .519089 4.73 .693298 7.07 59 .502199 4.83 .668220 7.10 59 .519373 4.73 .693722 7.07 60 9.502429 4.85 9.668646 7.10 60 9.519657 4.72 9.694146 7.07 427 TABLE XXVI. LOGARITHMIC VERSED SINES 48 43 Vers. D.I". Ex. sec. D. 1'. ' Vers. D. r. Ex. sec. D. 1". 9.519657 4.72 9.694146 7.07 9.536484 4.62 9.719541 7.05 1 .519940 4.73 .694570 7.07 1 .536761 4.62 .719964 7.03 2 .520224 4.72 .694994 7.07 2 .537038 4.62 .720386 .05 3 .520507 4.73 .695418 7.07 3 .537315 4.62 .720809 .03 4 .520791 4.72 .695842 7.07 4 .537592 4.62 .721231 .03 5 .521074 4.72 .696266 7.05 i 5 .537869 4.60 .721053 .05 6 .521357 4.72 .696689 7.07 i G .538145 4.62 .722076 .03 7 .521040 4.72 .697113 7.07 7 .538422 4.60 .722498 .05 8 .521923 4.72 .697537 7.07 8 .538698 4.60 .722921 .03 g .522206 4.70 .697961 7.07 9 .538974 4.62 .723343 .03 10 .522488 4.72 .698385 7.07 10 .539251 4.60 .723765 .05 11 9.522771 4.72 9.698809 7.05 11 9.539527 4.60 9.724188 .03 12 .523054 4.70 .699232 7.07 12 .539803 .60 .724010 .03 13 .523336 4.70 .699656 7.07 13 .540079 .58 .725032 .03 14 .523618 4.70 .700080 7.05 14 .540354 .60 .725454 .05 15 .523900 4.70 .700503 7.07 15 .540630 .60 .725877 .03 16 .524182 4.70 .700927 7.05 16 .540906 .58 .726299 .03 17 .524464 4.70 .701350 7.07 17 .541181 .58 .7267'21 .03 18 .524746 4.70 .701774 7.07 18 .541456 .60 .727143 .03 19 .525028 4.68 .702198 7.05 19 .541732 .58 .727565 .05 20 .525309 4.70 .702621 7. 07 20 .542007 .58 .727988 .03 21 9.525591 4.68 9.703045 7.05 21 9.542282 4.58 9.728410 .03 22 .525872 4.68 .703468 7.05 22 .542557 4.58 .728832 .03 23 .526153 4.70 .703891 7.07 23 .542832 4.57 .729254 .03 24 .526435 4.68 .704315 7.05 24 .543106 4.58 .729676 .03 25 .526716 4.68 .704738 7.07 25 .543881 4.57 .730098 .03 26 .526997 4.67 .705162 7.05 26 .543655 4.58 .730520 .03 27 .527277 4.68 .705585 7.05 27 .543930 4.57 .730942 .03 28 .527558 4.68 .7060; 8 7.05 28 .544204 4.57 .731364 .03 29 .527839 4.67 .706431 7.07 29 .544478 4.57 .731786 .03 30 .528119 4.68 .706855 7.05 30 .544752 4.57 .732208 .03 31 9.528400 4.67 9.707278 7.05 31 9.545026 4.57 9.732630 .03 32 .528680 4.67 .707701 7.05 32 .545300 4.57 .733052 .03 33 .528960 4.67 .708124 7.05 33 .545574 4.57 .733474 .03 34 .529240 4.67 .708547 7.07 34 .545848 4.55 .733896 .02 35 .529520 4.67 .708971 7.05 35 .546121 4.57 .734317 .03 36 .529800 4.67 .709394 7.05 i 36 .546395 4.55 .734739 .03 37 .530080 4.65 .709817 7.05 37 .540668 4.55 .735161 .03 38 .530359 4.67 .710240 7.05 38 .546941 4.55 .735583 .03 39 .530639 4.65 .710663 7.05 39 .547214 4.55 .736005 .03 40 .530918 4.67 .711086 7.05 40 .547487 4.55 .736427 .02 41 9.531198 4.65 9.711509 7.05 41 9.547760 4.55 9.736848 .03 42 .531477 4.65 .711932 7.05 42 .548033 4.55 .737270 .03 43 .531756 4.65 .712355 7.05 43 .548306 4.55 .737692 .03 44 .532035 4.65 .712778 7.03 44 .548579 4.53 .738114 .02 45 .532314 4.63 .713200 7.05 4.-) .5-18851 4.55 .738535 .08 46 .532592 4.65 .713623 7.05 46 .549124 4.53 .738957 .03 47 .532871 4.65 .714046 7.C5 47 .549396 4.53 .739379 .02 48 .533150 4.63 .714469 7.05 48 .549668 4.53 .739800 .03 49 .533428 4.63 .714892 7.05 49 .549940 4.53 .740222 .03 50 .533706 4.65 .715315 7.03 50 .550212 4.53 .740644 .02 51 9.533985 4.63 9.715737 7.05 51 9.550484 4.53 9.741065 .03 52 .5:34203 4.63 .716160 7.05 52 .550?56 4.53 .741487 .02 53 .534541 4.63 .716583 7.03 53 .551028 4.52 .741908 .03 54 .534819 .63 .717005 7.05 54 .551299 4.53 .742330 .02 55 .535097 .62 .717428 7.05 55 .551571 4.52 .742751 .03 56 .535374 .63 .717851 7.03 56 .551842 4.52 .743173 .03 57 .535652 .62 .718273 7.05 57 .552113 4.52 .74X595 .02 58 .535929 .63 .718696 7.03 58 .552384 4.53 .744016 .03 59 .536207 .62 .719118 7.05 59 .552656 4.52 .744438 7.02 60 9.536484 .62 9.719641 7.05 60 9.552927 4.50 9.744859 7.02 AND EXTERNAL SECANTS. 50 1 51 / Vers. D. 1*. Ex. sec. D. 1", / Vers. D 1" Ex. sec. D. 1'. 9.552927 4.50 9.744859 7.02 9.568999 4.42 9.770127 7.02 1 .553197 4.52 .745280 7.03 1 .569264 4.40 .770548 7.02 2 .553468 4.52 .745702 7.02 2 .569528 4.42 .770969 7.00 3 .553739 4.50 .746123 7.03 3 .569793 4.40 .771389 7.02 4 .554009 4.52 .746545 7.02 4 .570057 4.42 .771810 7.02 5 .554280 4.50 .746966 7.03 5 .570322 4.40 .772231 7.02 6 .554550 4.50 .747388 7.02 6 .570586 4.40 .772652 7.02 7 .554820 4.52 .747809 7.02 7 .570850 4.40 .773073 7.02 8 .555091 4.50 .748230 7.03 8 .571114 4.40 .773494 7.00 9 .555361 4.50 .748652 7.02 9 .571378 4.40 .773914 7.02 10 .555631 4.48 .749073 7.02 10 .571642 4.40 .774335 7.02 11 9.555900 4.50 9.749494 7.03 11 9.571906 4.40 9.774756 7.02 12 .556170 4.50 .749916 7.02 12 .572170 4.40 .775177 7.02 13 .556440 4.48 .750337 7.02 13 .572434 4.38 .775598 7.00 14 .556709 4.50 .750758 Y.03 14 .572697 4.38 .776018 7.02 15 .556979 4.48 .751180 7.02 15 .5?2960 4.40 .776439 7.02 ia .557248 4.48 .751601 7.02 16 .573224 4.38 .776860 7.02 17 .557517 4.48 .752022 7.02 17 .57:3487 4.38 .777281 7.02 10 .557786 4.48 .752443 7.03 18 .573750 4.38 .777702 7.00 19 .558055 4.48 .752865 7.02 19 .57'4013 4.38 .778122 7.02 20 .558324 4.48 .753286 7.02 20 .574276 4.38 .778543 7.02 21 9.558393 4.48 9.753707 7.02 21 9.574539 4.38 9.778964 7.02 22 .558862 4.48 .754128 7.02 22 .574802 4.37 .779385 7. CO 23 .559131 4.47 .754549 7.03 23 .575064 4.38 .779805 7.02 24 .559399 4.47 .754971 7.C2 24 .575327 4.37 .780226 7.02 25 .559667 4.48 .755392 7.02 25 .575589 4.38 .780647 7.02 20 .559936 4.47 .755813 7.02 26 .575852 4.37 .781068 7.00 27 .560204 4.47 .756234 7.02 27 .576114 4.37 .781488 7.02 28 .560472 4.47 .756655 7.02 28 .576376 4.37 .781909 7.02 29 .560740 4.47 .757076 7.03 29 .576638 4.37 .782330 7.02 30 .561008 4.47 .757498 7.02 30 .576900 4.37 .782751 7.00 31 9.561276 4.47 9.757919 7.02 31 9.577162 4.37 9.783171 7.02 32 .5C1544 4.45 .758340 7.02 32 .577424 4.35 .783592 7.02 33 .561811 4.47 .758701 7.02 33 .577685 4.37 .784013 7.00 34 .562079 4.45 .759182 7.02 34 .577947 4.35 .784433 7.02 35 .562346 4.45 .759603 7.02 35 .578208 4.37 .784854 7.02 36 .562613 4.47 .760024 7.02 36 .578470 4.35 .785275 7.02 37 .562881 4.45 .760445 7.02 37 .578731 4.35 .785696 7.00 38 .563148 4.45 .760866 7.02 38 .578992 4.35 .786116 7.02 39 .563415 4.45 .761287 7.02 39 .579253 4.35 .786537 7.02 40 .563682 4.43 .761708 7.02 40 .579514 4.35 .786958 7.00 41 9.563948 4.45 9.762129 7.02 41 9.579775 4.35 9.787378 7.02 42 .564215 4.45 .762550 7.02 42 .580036 4.35 .787799 7.02 43 .564482 4.43 .762971 7.02 43 .580297 4.33 -.788220 7.02 44 .564748 4.45 .763392 7.02 44 .580557 4.35 .788641 7.00 45 .565015 4.43 .763813 7.02 45 .580818 .33 .789061 7.02 46 .565281 4.43 .764234 7.02 46 .581078 .35 .789482 7.02 47 .565547 4.43 .764655 7.02 47 .581339 .33 .789903 7.00 48 .565813 4.43 .765076 7.02 48 .581599 .33 .790323 7.02 49 .566079 4.43 .765497 7.02 49 .581859 .33 .790744 7.02 50 .560345 4.43 .765918 7.02 50 .582119 .33 .791165 7.02 51 9.566611 4.43 9.766339 7.02 51 9.582379 4.33 9.7915S6 7.00 52 .566877 4.42 . 7*36760 7.02 52 .582639 4.32 .792006 7.02 53 .567142 4.43 .767181 7.02 i 53 .582898 4.33 .792427 7.02 54 .567408 4.42 .767602 7.00 54 .583158 4.33 .792848 7. CO 55 .567673 4.42 .768022 7.02 55 .583-118 4.32 .793268 7.02 56 .567938 4.43 .768443 7.02 56 .583677 4.32 .793G89 7.02 57 .5(iS204 4.42 .768864 7.02 57 .583936 4.33 .794110 7.02 58 .568169 4.42 .769285 7.02 58 .584196 4!32 .794531 7.00 59 .568734 4.42 .769706 7.02 59 .584455 4.32 .794951 7.02 GO 9.568999 4.42 9.77-0127 7.02 60 9.584714 4.32 9.795372 7.02 TABLE XXVI.-LOGARITHMIC VERSED SINES 52 63 / Vers. D. r. Ex. sec. P.r. / 1 Vers. D. 1". Ex. sec. D.I'. 9.584714 4.32 9.795372 7.02 9.600083 4.22 9.820622 7.02 i .584973 4.32 .795793 7.00 1 .600338 4.22 .821043 7.02 2 .585232 4.32 .796213 7.02 2 .600591 4.23 .821464 7.02 3 .585491 4.30 .796634 7.02 3 .600845 4.22 .821885 7.02 4 .585749 4.32 .797055 7.02 4 .601098 4.22 .822306 7.02 5 .586008 4.30 .797476 7.00 5 .601351 4.20 .822727 7.02 6 .586266 4.32 .797896 7.02 6 .601603 4.22 .823148 7.02 7 .5E658) 4.30 .798317 7.02 7 .601856 4.22 .823569 7.02 8 .58678) 4.30 .798738 7.00 8 .602109 4.22 .823990 7.02 9 .537041 4.30 .799158 7.02 9 .602362 4.20 .824411 7.03 10 .587299 4.30 .799579 7.02 10 .602614 4.20 .824833 7.02 11 9.587557 4.30 9.800000 7.02 11 9.602866 4.22 9.825254 7.02 12 .587815 4.30 .800421 7.00 12 .603119 4.20 .825675 7.02 13 .588073 4.30 .800841 7.02 13 .603371 4.20 .826096 7.02 14 .588331 4.28 .801262 7.02 14 .603623 4.20 .826517 7.02 15 .588588 4.28 .801683 7.02 15 .60:3875 ^4.20 .826938 7.03 16 .588846 4.28 .802104 7.00 ! 16 .604127 4.20 .827360 7.02 17 .583103 4.30 .802524 7.02 17 .604379 4.20 .827781 7.02 18 .580351 4.28 .802945 7.02 18 .604631 4.20 .828202 7.02 19 .589618 4.28 .803366 7.02 19 .604883 4.18 .828623 7.02 20 .589875 4.28 .803787 7.00 20 .605134 4.20 .829044 7.03 21 9.590132 4.28 9.804207 7.02 21 9.605386 4.18 9.829466 7.02 22 .590389 4.28 .804628 7.02 22 .605637 4.18 .829887 7.02 23 .593646 4.28 .805049 7.02 23 .605888 4.20 .830308 7.02 24 .590903 4.23 .805470 7.02 24 .606140 4.18 .830729 7.03 25 .591160 4.27 .805891 7.00 25 .606391 4.18 .831151 7.02 26 .591416 4.28 .806311 7.02 26 .606642 4.18 .831572 7.02 27 .591673 4.27 .806732 7.02 27 .606893 4.18 .831893 7.03 28 .591929 4.27 .807153 7.02 28 .607144 4.17 .832415 7.02 29 .592185 4.28 .807574 7.02 29 .607394 4.18 .832836 7.02 30 .592443 4.27 .807995 7.00 30 .607645 4.18 .833257 7.03 31 9.592698 4.27 9.808415 7.02 31 9.607896 4.17 9.833679 7.02 32 .592954 4.27 .808836 7.02 32 .608146 4.18 .834100 7.03 33 .593210 4.27 .809257 7.02 33 .608397 4.17 .834522 7.02 34 .593466 4.25 .809678 7.02 34 .608647 4.17 .834943 7.02 35 .593721 4.27 .810099 7.02 35 .608897 4.17 .835364 7.03 36 .593977 4.27 .810520 7.00 36 .609147 4.17 .835786 7.02 37 .594233 4.25 .810940 7.02 37 .609397 4.17 .836207 7.03 38 .594488 4.25 .811361 7.02 38 .609647 4.17 .836629 7.02 39 .594743 4.27 .811782 7.02 39 .609897 4.17 .837050 7.03 40 .594999 4.25 .812203 7.02 40 .610147 4.17 .837472 7.02 41 9.595254 4.25 9.812624 7.02 41 9.610397 4.15 9.837893 7.03 42 .595509 4.25 .813045 7.02 42 .610646 4.17 .838315 7.02 43 .595761 4.25 .813466 7.00 43 .610896 4.15 .838736 7.03 44 .596019 4.25 .813886 7. l>2 44 .611115 4.15 .839158 7.02 45 .596274 4.23 .814307 7.02 45 .611394 4.17 .839579 7.03 46 .596528 4.25 .814728 7.02 46 .611644 4.15 .840001 7.03 47 .596783 4.25 .815149 7.02 47 .611893 4.15 .840423 7.02 48 .597038 4.23 .815570 7.02 48 .612142 4.15 .840844 7.03 49 .597292 4.23 .815991 7.02 49 .612391 4.15 .841266 7.02 50 .597546 4.25 .816412 7.02 50 .612640 4.13 .841687 7.03 51 9.597801 4.23 9.816833 7.02 51 9.612S88 4.15 9.842109 7.03 52 .598055 4.23 .817254 7.02 52 .613137 4.15 .842531 7.03 53 .598309 4.23 .817675 7.02 53 .613386 4.13 .842953 7.02 54 .598563 4.23 .818096 7.02 54 .613634 4.15 .843374 7.03 55 .598817 4.23 .818517 7.02 55 .613883 4.13 .843796 7.03 56 .599071 4.22 .818938 7.02 56 .614131 4.13 .844218 7.02 57 .599324 4.23 .819359 7.02 57 .614379 4.13 .844639 7.03 58 .599578 4.22 .819780 7.02 58 .614627 4.15 .845061 7.03 59 .599831 4.23 .820201 7.02 59 .614876 4.13 .845483 7.03 60 9.600085 4.22 9.820622 7.02 60 9.615124 4.12 9.845905 7.03 430 AND EXTERNAL SECANTS. 54 55 ' Vers. D. 1". Ex. sec. D. 1". Vers. D. 1'. Ex. sec. D. 1'. 9.615124 4.12 9.845905 7.03 9.629841 .05 9.871250 7.05 1 .615371 4.13 .846327 7.03 1 .630084 .03 .871673 7.05 2 .615619 4.13 .846749 7.02 2 .630326 .05 .872096 7.05 3 .615867 4.13 .847170 7.03 3 .630569 .03 .872519 .05 4 .616115 4.12 .847592 7.03 4 .6:30811 .05 .872943 .07 5 .616362 4.13 .848014 7.03 5 .631054 4.03 .873366 .05 G .616610 4.12 .848436 7.03 6 .631296 4.03 .873789 .05 7 .616857 4.12 .848858 7.03 7 .631538 4.03 .874212 .07 .617104 4.12 .849280 7.03 8 .631780 4.03 .874636 .05 .017351 4.13 .849702 7.03 9 .632022 4.03 .875059 .05 10 .617599 4.10 .850124 7.03 10 .632264 4.03 .875482 .07 11 9.617845 4.12 9.850546 7.03 11 9.632505 4.03 9.875906 .05 13 .618092 4.12 .850968 7.03 12 .632747 4.03 .876329 .05 13 .618339 4.12 .a51390 7.03 13 .632989 4.02 .876752 .07 14 .618586 4.12 .851812 7.03 14 .633230 4.03 .877176 .05 15 .618833 4.10 .852234 7.03 15 .633472 4.02 .877599 .07 16 .619079 4.12 .852656 7.03 16 .633713 4.02 .878023 .05 17 .619326 4.10 .853078 7.03 17 .633954 4.03 .87B446 .07 18 .019572 4.10 .853500 7.05 18 .634196 4.02 .878870 .07 19 .619818 4.12 .853923 7.03 19 .634437 4.02 .879294 .05 20 .620065 4.10 .854345 7.03 20 .634678 4.03 .879717 .07 21 9.620311 4.10 9.854767 7.03 21 9.634919 4.00 9.880141 .07 22 .620557 4.10 .855189 7.05 22 .635159 .02 .880565 .05 23 .620803 4.08 .855612 7.03 ! 23 .635400 .02 .880988 .07 24 .621048 4 10 .856034 7.03 ! 24 .635641 .00 .881412 7.07 25 .621294 4.10 .856456 7.03 25 .635881 .02 .881836 7.07 26 .621540 4.10 .856878 7.05 26 .636122 .00 .882260 7.05 27 .621786 4.08 .857301 7.03 27 .636362 .02 .882683 7.07 23 .622031 4.08 .857723 7.03 28 .636603 .00 .883107 7.07 29 .622276 4.10 .858145 7.05 29 .636843 .00 .883531 7.07 30 .622522 4.08 .858568 7.03 30 .637083 .00 .883955 7.07 31 9.622767 4.08 9.858990 7.05 31 9.637323 4.00 9.884379 7.07 32 .623012 4.08 .859413 7.03 32 .637563 4.00 .884803 7.07 33 .623257 4.08 .859835 7.05 33 .637803 4.00 .885227 7.07 34 .623502 4.08 .860258 7.03 34 .638043 4.00 .885651 7.07 35 .623747 4.08 .86068-3 7.05 35 .638283 3.98 .886075 7.07 36 .623992 4.08 .861103 7.03 36 .638522 4.00 .886499 7.07 37 .624237 4.07 .861525 7.05 37 .638762 3.98 .886923 7.07 38 .624481 4.08 .8(51948 7.03 38 .639001 4.00 .887347 7.08 39 .624726 4.07 .862370 7.05 .639241 3.98 .887772 7.07 40 .624970 4.08 .862793 7.03 40 .639480 3.98 .888196 7.07 41 9.625215 4 07 9.863215 7.05 41 9.639719 3.98 9.888620 7.07 42 .035459 4.07 .863638 7.05 42 .639958 3.98 . .88904-1 7.08 43 .625703 4.07 .864061 7.03 43 .640197 3.98 .88Q469 7.07 41 .625947 4.07 .864483 7.05 44 .640436 3.98 .889893 7.07 45 .626191 4.07 .864906 7.05 45 .640675 3.98 .890317 7.08 46 .626435 4.07 .865329 7.05 ! 46 .640914 3.98 .890742 7.07 47 .626679 4.07 .865752 7.03 1 47 .641153 3.97 .891166 7.08 48 .626923 4.05 .866174 7.05 ! 48 .641391 8.98 .891591 7.07 49 .627166 4.07 .86(5597 7.05 | 49 .641630 3.97 .892015 7.08 50 .627410 4.07 .867020 7.05 50 .641868 3.98 .892440 7.07 51 9.627654 4.05 9.867443 7.05 51 9.642107 3.97 9.892864 7.08 52 .627897 4.05 .867866 7.05 52 .642:545 3.97 .893289 7.08 53 .628140 4.07 - .868289 7.05 53 .642583 3.98 .893714 7.07 54 .628384 4.05 .868712 7.05 54 .642822 3.97 .894138 7.08 55 .623627 4.05 .869135 7.05 55 .643060 3.97 .894563 7.08 56 .628870 4.05 .869558 7.05 56 .643298 3.95 .894988 7.07 57 .629113 4.05 .869981 7.05 57 .643535 3.97 .895412 7.08 58 .629356 4.03 .870404 7.05 ! 58 .643773 3.97 .895837 7.08 59 .629598 4.05 .870827 7.05 | 59 .644011 3.97 .896262 7.08 60 9.629841 4.05 9.871250 7.05 i 60 9.644249 3.95 9.896687 7.08 431 TABLE XXVI. -LOGARITHMIC VERSED SINES 56 I 1 57 / Vers. D. 1'. Ex. sec. D 1". / Vers. D. 1". Ex. sec. D. 1". 9.644249 3.95 9.896687 7.08 9.658356 3.87 9.922247 7.12 1 .644486 8.97 .897112 7.08 1 .658588 3.88 .922674 7.13 2 .044724 8.1)5 .897537 7.08 2 .658821 3.87 .923102 7.12 3 .644961 3.95 .897962 7.08 3 .659053 3.88 .923529 7.12 4 .645198 8.95 .898387 7.08 4 .659286 3.87 .923956 7.13 5 .645435 8.97 .898812 7.08 5 .659518 3.87 .924384 7.12 6 .645673 3.95 .899237 7.08 6 .659750 3.88 .924811 7.13 7 .645910 3.95 .899662 7.08 7 .659983 3.87 .9252S9 7.12 8 .646147 3.95 .900087 7.08 8 .660215 3.87 .925GG6 7.13 9 .646384 3.93 .900512 7.10 9 .660447 3.87 .926094 7.12 10 .646620 3.95 .900938 7.08 10 .660679 3.85 .926521 7.13 11 9.646857 3.95 9.901363 7.08 11 9.660910 3.87 9.926949 7.13 12 .647094 3.93 .901788 7.08 12 .661142 3.87 .927377 7.12 13 .647330 3.95 .902213 7.10 13 .661374 3.85 .927804 7.13 14 .647567 3.93 .902039 7.08 14 .661605 3.87 .928232 7.13 15 .647803 3.93 .908064 7.10 15 .661837 -3.85 .928060 7.13 16 .648039 3.95 .903490 7.08 16 .662068 3.87 .929088 7.13 17 .648276 3.93 .903915 7.10 17 .662300 8.85 .929516 7.13 18 .648512 3.93 .904341 7.08 18 .662531 3.85 .929944 7.13 19 .648748 3.93 . .904766 7.10 19 .662762 3.85 .930372 7.13 20 .648984 3.93 .905192 7.08 20 .662993 3.85 .930800 7.13 21 9.649220 3.93 9.905617 7.10 21 9.663224 3.85 9.931228 7.13 22 .649450 3.92 .906043 7.10 22 .003455 3.85 .931656 7.15 23 .649691 3.93 .906469 7.08 23 .G03G86 3.85 .932085 7.13 24 .649927 3.93 .906894 7.10 24 .663917 3.85 .932513 7.13 25 .650163 3.92 .907320 7.10 25 .664148 3.83 .932941 7.13 26 .650398 3.92 .907746 7.10 26 .664378 3.85 .933369 7.15 27 .650633 3.93 .908172 7.10 27 .664009 3.83 .933798 7.13 28 .650869 3.92 .908598 7.10 28 .664839 3.85 .934226 7.15 29 .651104 3.92 .909024 7.10 29 .665070 3. as .934655 7.13 30 .651339 3.92 .909450 7.10 30 .665300 3.83 .935083 7.15 31 9.651574 3.92 9.909876 7.10 31 9.665530 3.83 9.935512 7.15 32 .651809 3.92 .910302 7.10 32 .665700 3.83 .935941 7.13 33 .652044 3.92 .910728 7.10 33 .665990 3.83 .936369 7.15 34 .652279 3.92 .911154 7.10 34 .G66220 3.83 .936798 7.15 35 .652514 3.90 .911580 7.10 35 .666450 3.83 .937227 7.15 36 .652748 3.92 .912006 7.10 36 .666680 3.83 .937656 7.15 37 .052983 3.90 .912432 7 12 37 .666910 3.82 .938085 7.13 38 .653217 3.92 .912859 7!lO 38 .667139 3.83 .938513 7M 39 .653452 3.90 .913285 7.10 39 .667369 3.83 .938942 7.15 40 .653686 3.90 .913711 7.12 40 .667599 3.82 .939371 7.17 41 9.653920 3.92 9.914138 7.10 41 9.667828 3.82 9.939801 7.15 42 .654155 3.90 .914564 7.12 42 .668057 3.83 .940230 7.15 43 .654389 3.90 .914991 7.10 43 .668287 3.82 .940659 7.15 44 .654623 3.90 .915417 7.12 44 .668516 3.82 .941088 7.15 45 .654857 3.88 .915844 7.10 45 .6687'45 3.82 .941517 7.17 46 .655090 3.90 .916270 7.12 46 .668974 3.82 .941947 7.15 47 .655324 3.90 .91GG97 7.12 47 .669203 3.82 .94:2370 7.15 48 .655558 3.90 .917124 7.10 48 .6G9432 3.62 .942806. 7.15 49 .655792 3.88 .917550 7.12 49 .G6CGG1 3.80 .943235 7.17 60 .056025 3.88 .917977 7.12 50 .669889 3.82 .943665 7.15 51 9.656258 8.90 9.918404 7.12 51 9.670118 3.82 9.944094 7.17 52 .65G492 3.88 .918831 7.12 52 .670347 3.80 .944524 7.15 53 .650725 8.88 .910558 7.12 53 .G7'0575 3.82 .944953 7. 7 54 .G5GS58 8.88 .919385 7.12 54 .670804 3.80 .945383 7. 7 55 .657191 3.88 .920112 7.12 55 .671032 3.80 .945813 7. 7 56 G ?I 4 ?i 3.88 .920539 7.12 56 .671260 3.80 .946243 7. 7 57 .657G57 8.88 .920966 7.12 57 .671488 3.80 .94GG73 7. 7 58 .657890 3.88 .921393 7.12 58 .671716 3.82 .947103 7. 7 59 .658123 3.88 .921820 7.12 59 .671945 3.78 .947533 7. 7 60 9.658356 3.87 9.922247 7.12 GO 9.672172 3.80 9.947963 7. 7 AND EXTERNAL SECANTS. 58 59 ' Vers. D.I'. Ex. sec. D.I". / Vers. D.I". Ex. sec. D. r. 9.G72172 3.80 9.947963 7.17 9.685703 3.72 9.973868 7.23 1 .672400 3.80 .948:393 7.17 1 .685931 3.72 .974302 7.23 2 .07:3023 3.80 .948883 7.17 2 .080154 3.72 .974736 7.22 3 .072356 3.78 .949253 7.17 3 .686377 3.72 .975169 7.23 4 .673083 3.80 .949683 7.18 4 .686600 3.72 .975003 7.23 5 .673311 3.78 .950114 7.17 5 .686823 3.72 .976037 7.23 6 .673533 3.80 .950544 7.18 6 .687046 3.72 .970471 7.23 7 .073766 3.78 .950975 7.17 7 .687269 3.72 .976905 7.23 8 .673993 3.78 .951405 7.18 8 .687492 3.70 .977339 7.23 9 .674220 3.80 .951836 7.17 9 .687714 3.72 .977773 7.23 10 .674448 3.78 .952266 7.18 10 .687937 3.70 .978207 7.23 11 9.674675 3.78 9.952697 7.18 11 9.688159 3.72 9.978641 7.33 12 .074902 3.78 .953128 7.17 12 .688382 3.70 .979075 7.25 13 .075123 3.78 .953558 7.18 13 .688604 3.70 .979510 7.23 14 .075356 3.77 .953989 7.18 14 .688826 3.70 .979944 7.25 15 .075582 3.78 .954420 7.18 15 .689048 3.72 .980379 7.23 16 .075809 3.78 .954851 7.18 16 .089271 3.70 .980813 7.25 17 .076030 3.77 .955282 7.18 17 .689493 3.70 .981248 7.23 18 .676262 3.78 .95.-)? 13 7.18 18 .689715 3.70 .981682 7.25 19 .670489 3.77 .956144 7.18 19 .689937 3.08 .982117 7.25 20 .076715 3.77 .956575 7.18 20 .690158 3.70 .982552 7 25 21 9.676941 3.78 9.957006 7.20 21 9.G90380 3.70 9.982987 7.25 2:2 .077108 3.77 .95i'438 7.18 22 .690602 3.08 .983422 7.25 23 .077394 3.77 .957869 7.18 23 .690823 3.70 .983857 7.25 24 . 077020 3.77 .958300 7.20 ; 24 .691045 3.08 .984292 7.25 23 .677846 3.77 .953732 7.18 | 25 .691266 3.70 .984727 7.25 20 .678072 3.77 .959163 7.20 1 26 .691488 3.08 .985162 7.25 27 .678298 3.75 .939595 7.18 27 .691709 3.08 .985597 7.27 23 .078523 3.77 .960026 7.20 28 .691930 3.08 .986033 7.25 29 .078749 3.77 .980458 7.20 29 .692151 3.08 .980408 7.27 30 .078975 3.75 .9(30390 7.18 ! 30 .692372 3.68 .986904 7.25 31 9.679200 3.77 9.961321 7.20 31 9.692593 3.68 9.987339 7.27 32 .079426 3.75 .901753 7.20 32 .692814 3.68 .987775 7.25 33 .679651 3.75 .962185 7.20 ! 33 .693035 3.08 .988210 7.27 34 .079 -(76 3.77 .962617 7.20 34 .693256 3.08 .988646 7.27 35 .680103 3.75 .963049 7.20 35 .693477 3.67 .989082 7.27 36 .080327 3.75 .963481 7.20 36 .693697 3.68 .989518 7.27 37 .680552 3.75 .963913 7.20 37 .693918 3.67 .989954 7.27 38 .680777 3.75 .964345 7.22 38 .694138 3.08 .990390 7.27 39 .681002 3.75 .964778 7.20 39 .694359 3.07 .990826 7.27 40 .681227 3.73 .955210 7.20 40 .694579 3.67 .991262 7.27 41 9.681451 3.75 9.905642 7.22 41 9.694799 3.67 9.991698 7.27 42 .681 ' 71 ' Vers. D. r. Ex. sec. D. r. 'i Vers. D.I'. Ex. sec. D.I'. 9.818213 3.00 10.284161 8.80 9.828938 2.95 10.316296 9.07 1 .818393 3.00 .284689 8.78 1 .829115 2.95 .316840 9.08 2 .818573 3.02 .285216 8.82 2 .829292 2.95 .317385 9.07 3 .818754 3.00 .285745 8.80 3 .829469 2.95 .317929 9.10 4 .818934 3.00 .286273 8.82 4 .829646 2.95 .318475 9.08 5 819114 3.00 .286802 8.82 5 .829823 2.95 .319020 9.08 6 .819294 3.00 .287331 8.82 6 .830000 2.95 .319565 9.10 .819474 3.00 .287860 8.82 7 .830177 2.93 .320111 9.12 8 .819654 3.00 .288389 8.83 8 .830&53 2.95 .320658 9.10 9 .819834 3.00 .288919 8.83 9 .830530 2.93 .321204 9.12 10 .820014 3.00 .289449 8.83 10 .830706 2.95 .321751 9.12 11 9.820194 3.00 10.289979 8.85 11 9.830883 2.93 10.322298 9.12 12 .820374 2.98 .290*10 8.85 12 .831059 2.95 .322845 9.13 13 .820553 3.00 .291041 8.85 13 .831236 2.93 .323393 9.13 14 .820733 3.00 .291572 8.85 14 .831412 2.95 .323941 9.13 15 .820913 2.98 .292103 8.87 15 .831589 2.93 .324489 9.15 16 .821092 3.00 .292635 8.85 16 .831765 2.93 .325038 9.15 17 .821272 2.98 .293166 8.87 17 .831941 2.93 .325587 9.t5 18 .821451 3.00 .293698 8.88 18 .832117 2.93 .326136 9.17 19 .821631 2.98 .294231 8.88 19 .832293 2.93 .326686 9.15 20 .821810 2.98 .294764 8.87 20 .832469 2.93 .327235 9.18 21 9.821989 2.98 10.295296 8.90 21 9.832645 2.93 10.327786 9.17 21 .822168 3.00 .295830 8.88 22 .832821 2 93 .328336 9.18 23 .822348 2.98 .296363 8.90 23 .832997 2^93 .328887 9.18 24 .822527 2.98 .296897 8.00 24 .833173 2.93 .329438 9.18 23 .822706 2.98 .207431 8.90 25 .833349 2.93 .329989 9.20 2(3 .822885 2.98 .297965 8.92 26 .833525 2.92 .330541 9.20 27 .823064 2. 98 .298500 8.90 27 .833700 2.93 .331093 9.20 23 .823243 2.07 .299034 8.93 28 .8:33876 2.92 .331645 9.22 29 .823421 2.98 .299570 8.92 29 834051 2.93 .a32198 9.20 30 .823600 2.98 .300105 8.93 30 .834227 2.92 .332750 9.23 31 9.823779 2.98 10.300041 8.92 31 9.834402 2.93 10.333304 9.22 32 .823958 2.97 .301176 8.95 32 .834578 2.92 .333857 9.23 33 .824136 2.98 .301713 8.93 33 .834753 2.92 .334411 9.23 34 .824315 2.97 .302249 8.95 34 .834928 2.93 .334965 9.25 35 .824493 2.98 .302786 8.95 35 .835104 2.92 .335520 9.23 30 .824672 2.97 .303323 8.95 | 36 .835279 2.92 .336074 9.25 37 .824850 2.97 .303860 8.97 37 .835454 2.92 .336629 9.27 38 .8250.38 2.98 .304398 8 97 38 .835629 2.92 .337185 9.27 33 .825207 2.97 .304936 8.97 39 .835804 2.92 .337741 9.27 40 .825385 2.97 .305474 8.97 40 .835979 2.92 .338297 9.27 41 9.825563 2.97 10.306012 8.98 41 9.836154 2.92 10.338853 9.28 42 .825741 2.97 .306551 8.98 42 .836329 2.92 .339410 9.28 43 .825919 2.97 .307090 8.98 43 .836504 2.90 .339967 9.28 44 .826037 2.97 .307629 9.00 44 .836678 2.92 .340524 9.30 45 .826275 2.97 .308169 8.98 45 .836853 2.92 .341082 9.30 46 .826453 2.97 .308708 9.02 46 .837028 2.90 .341640 9.30 47 .826631 2.97 .309249 9.00 47 .837202 2.88 .342198 9.30 48 .826809 2.97 .309789 9.02 48 .837377 2.90 .342756 9.32 49 .826987 2.95 .310330 9.02 49 .837551 2.92 .343315 9.33 50 .827164 2.97 .310871 9.02 50 .837726 2.90 .343875 9.32 51 9.827342 2.95 10.311412 9.02 51 9.837900 2.92 10.344434 9.33 52 .827519 2.97 .311953 9.03 52 .838075 2.90 .344994 9.33 53 .827697 2.95 .312495 9.03 53 .838249 2.90 .345554 9.35 54 .827874 2.97 .313037 9.05 54 .838423 2.90 .346115 9.35 55 .828052 2.95 .313580 9.03 55 . 38597 2.90 .346676 9.35 56 .828229 2.95 .314122 9.05 56 .838771 2.90 .347237 9.35 57 .828400 2.97 .314665 9.07 57 .838945 2.90 .347798 9.37 58 .828584 2.95 .315209 9.05 58 .889119 2.90 .348360 9.37 59 .823761 2.95 .315752 9.07 | 59 .839293 2.90 .348922 9.38 60 9.828938 2.95 10.316296 9.07 60 9.839467 2.90 10.349485 9.38 489 TABLE XXVI. LOGARITHMIC VERSED SINES 72 73 / Vers. D. 1. Ex. sec. D.r. Vers. D. 1". Ex. sec. D.I". IT 9.839467 2.90 10.349485 9.38 9.849805 2.85 10.383870 9.73 i .839641 2.90 .350048 9.38 ! 1 .849976 2.85 .384454 9.73 2 .839815 2.90 .350611 9.40 2 .850147 2.83 .385038 9.75 3 .839989 2.88 .351175 9.38 3 .850317' 2.85 .385623 9.77 4 .840162 2.90 .351738 9.42 4 .850488 2.83 .386209 9.75 5 .840336 2.90 .352303 9.40 5 .850658 2. 5 .386794 9.77 .840510 2.88 .352867 9.42 C .850829 2.83 .387380 9.78 7 .840683 2.90 .353432 9.42 7 .850999 2.83 .387967 9.78 8 .840857 2.88 .353997 9.43 8 .851169 2.85 .388554 9.78 9 .841030 2.90 .354563 9.43 9 .851340 2. 3 .389141 9.78 10 .841204 2.88 .355129 9.43 10 .851510 2.83 .389728 9.80 11 9.841377 2.88 10.355695 9.43 11 9.851680 2.83 10.390316 9.82 12 .841550 2.88 .356261 9.45 12 .851*50 2.83 .390905 9.80 13 .841723 2.88 .356828 9.45 13 .852020 2.83 .391493 9.82 14 .841896 2.90 .357395 9.47 14 .852190 2.83 .392082 9.83 15 .842070 2.88 .357963 9.47 15 .852360 2.83 .392672 9.83 16 .842243 2.88 .358531 9.47 16 .852530 2.83 .393262 9.83 17 .842416 2.88 .359099 9.48 17 .852700 2.83 .393852 9.85 18 .842589 2.88 .359668 9.48 18 .852870 2.83 .394443 9.85 19 .842762 2.87 .360237 9.48 19 .853040 2.82 .3950:34 9.85 20 .842934 2.88 .360806 9.50 20 .853209 2.83 .395625 9.87 21 9.843107 2.88 10.361376 9.50 21 9.853379 2.83 10.396217 9.87 22 .843280 2.88 .361946 9.50 22 .853549 2.82 .396809 9.88 23 .843453 2.87 .362516 9.52 23 .853718 2.83 .397402 9.88 24 .843625 2.88 .363087 9.52 24 .853888 2.82 .397995 9.90 25 .843798 2.87 .363658 9.52 25 .854057 2.83 .398589 9.88 26 .843970 2.88 .364229 9.53 26 .854227 2.82 .399182 9.92 27 .844143 2.87 .364801 9.53 27 .854396 2.82 .399777 9.90 28 .844315 2.88 .365373 9.53 28 .854565 2.83 .400371 9.92 29 .844488 2.87 .365945 9.55 29 .854735 2.82 .400966 9.93 30 .844660 2.87 .366518 9.55 30 .854904 2.82 .401562 9.93 31 9.844832 2.87 10.367'091 9.57 31 9.855073 2.82 10.402158 9.93 32 .845004 2.88 .367665 9.57 32 .855242 2.82 .402754 9.95 33 .845177 2.87 .368239 9.57 33 .855411 2.82 .403351 9.95 34 .845349 2.87 .368813 9.57 34 .55580 2.82 .403948 9.95 35 .845521 2.87 .369387 9.58 35 .855749 2.82 .404545 9.97 36 .845693 2.87 .369962 9.60 36 .855918 2.82 .405143 9.98 37 .845865 2.87 .370538 9.58 37 .856087 2.80 .405742 9.97 38 .846037 2.85 .371113 9.60 38 .856255 2.82 .400340 9.98 39 .846208 2.87 .371689 9.62 39 .856424 2.82 .406939 10.00 40 .846380 2.87 .372266 9.60 40 .856593 2.82 .407539 10.00 41 9.846552 2.87 10.372842 9.62 41 9.856762 2.80 10.408139 10.00 42 .846724 2.85 .373419 9.63 42 .856930 2.82 .408739 10.02 43 .846895 2.87 .373997 9.63 43 .857099 2.80 .40U340 10.02 44 .847067 2 85 .374575 9.63 44 .857267 2.82 .409941 10.03 45 .847238 2.87 .375153 9.03 45 .857436 2.80 .410543 10.03 46 .847410 2.85 .375731 9.65 46 .857604 2.80 .411145 10.03 47 .847581 2.87 .376310 9.67 47 .857772 2.82 .411747 10.05 48 .847753 2.85 .376890 9.65 48 .857941 2.80 .412350 10.07 49 .847924 2.85 .377469 9.67 49 .858109 2.80 .412954 10.05 50 .848095 2.87 .378049 9.68 50 .858277 2.80 .413557 10.07 51 9.848267 2.85 10.378630 9.67 51 9.858445 2.80 10.414161 10.08 52 .848438 2.85 .378210 9.70 52 .858613" 2.80 .414766 10.08 53 .848609 2.85 .379792 9.68 ' 53 .858781 2.80 .415371 10.08 4 .848780 2.85 .380373 9.70 54 .858949 2.80 .415976 10.10 5 .848951 2.85 .380955 9.70 55 .859117 2.80 .416582 10.12 56 .849122 2.85 .381537 9.72 56 .859285 2.80 .417189 10.10 57 .849293 2.85 .382120 9.72 57 .859453 2.80 .417795 10.12 58 .849464 2.83 .382703 9.72 5* .59621 2.78 .-118402 10.13 59 .849634 2.85 .383286 9.73 59 .859788 2.80 .419010 10.13 60 9.849805 2.85 10.383870 9.73 68 9.859956 2.80 10.419618 10.13 440. AND EXTERNAL SECANTS. ?4 o 75 ' I Vers. D. r. Ex. sec. D. 1". ' Vers. D. 1". Ex. sec. D. 1". 9.8J9056 2.80 10.419618 10.13 j 9.869924 2.75 ; 10. 456928 10.60 1 .860124 2.78 .420226 10.15 |l 1 .870089 2.73 .457564 10.62 o .860291 2.80 .4208:35 10.17 i 2 .870253 2.75 .458201 10.63 3 .860459 2.78 .421445 10.15 3 .870418 2.73 .458839 10.62 4 .860626 2.80 .422054 10.17 4 .870582 2.75 .459476 10.65 5 .860794 2.7'8 .422064 10.18 5 .870747 2.73 .460115 10.65 6 .860961 2.78 .423275 10.18 6 .870911 2.75 .460754 10.65 7 .861128 2.80 .423880 10.20 7 .871076 1 2.73 .461&93 10.67 8 .861296 2.78 .424498 10.20 8 .871240 | 2.7-3 .462033 10.67 9 .861403 2.78 .425110 10.20 9 .871404 2.73 .462673 10.68 10 .801030 2.78 .425722 10.22 10 .871568 2.73 .463314 10 70 11 | 9.861797 2.78 10.420335 10.22 11 9.871732 2.73 10.463956 10.70 12 i .Hlil '.Hi t 2.78 .420948 10.23 12 .871896 2.73 .464598 10.70 13 1 .802131 2.78 .427502 10.23 13 .872060 2.73 .465240 10.72 14 .802298 2.78 .428176 10.23 14 .87'2224 2.73 465883 10.73 15 .802465 2.78 .428790 10.27 15 .872388 2.73 .466527 10.73 16 .862632 2.78 .420406 10.25 10 .872552 2.73 .467171 10.73 17 .862799 2.77 .430021 10.27 17 .872716 2.73 .467815 10.75 18 .862965 2.78 .430037 10.27 18 .872880 2.72 .468460 10.77 19 .863132 2.78 .431253 10.28 1!) .873043 2.73 .469106 10.77 20 .863299 2.77 .431870 10.30 20 .873207 2.73 .469752 10.77 21 9.8G3465 2.78 10.432488 10.28 21 9.873371 2.72 10.470398 10.78 22 .863632 2.78 .433105 10.32 22 .87'3534 2.73 .471045 10.80 23 .863799 2.77 .488784 10.30 23 .873698 2.72 .471693 10.80 24 .868986 2.77 .434342 10.32 24 .873861 2.73 .472341 10.82 25 .864131 2.78 .434961 10.33 25 .874025 2.72 .472990 10.82 2(3 .864298 2.77 .485681 10.33 26 .874188 2.72 .473639 10.83 27 .864464 2.77 .436201 10.33 27 .874351 2.73 .474289 10.83 28 .834630 2.78 .430821 10.35 28 .874515 2.72 .474939 10.85 29 .804797 2.77 .437442 10.37 29 .874678 2.72 .475590 10.87 30 .864363 2.77 .438004 10.37 30 .874841 2.72 .476242 10.85 31 9.8G5129 2.77 10.438686 10.37 31 9.875004 2.72 10.476893 10.88 32 .805295 2.77 .439308 10.38 32 .875167 2.72 .477546 10.88 33 .865461 2.77 .439931 10.38 33 .875330 2.72 .478199 10.88 34 .865627 2.77 .440654 10.40 34 .875493 2.72 .47B852 10.90 35 .865793 2.77 .441178 10.40 35 .875656 2.72 .479506 10.92 36 .865959 2.75 .441802 10.42 36 .875819 2.72 .480161 10.92 37 .866124 2.77 .442427 10.42 37 .875982 2.72 .480816 10.93 38 .866290 2.77 .443052 10.43 38 .876145 2.72 .481472 10.93 39 .866456 2.77 .443678 10.43 39 .87'6308 2.70 .482128 10.95 40 .806022 2.75 .444304 10.45 40 .876470 2.72 .482785 10.05 41 9.866787 2.77 10.444931 10.45 41 9.876633 2.72 10.483442 10.97 42 .8(ir><)53 2.75 .445558 10.45 42 .876796 2.70 .484100 10.1)8 43 .Si 17118 2.77 .446185 10.47 43 .876958 2.72 .484759 10.98 44 .867284 2.75 .446813 10.48 44 .877121 2.70 .485418 10.JJ8 45 .867449 2.75 .447442 10.48 45 .877283 2.70 .486077 10.1)8 46 .8i;rc,i i 2.77 .448071 10.48 46 .877445 2.72 .486738 11.00 47 .867780 2.75 .448700 10.50 47 .877608 2.70 .487398 11 02 48 .867945 2.75 .449*30 10.52 48 .877770 2.70 .488059 11.03 49 .808110 2.75 .449961 10.52 j 49 .877932 2.7*2 .488721 11.05 50 .808275 2.77 .450592 10.52 50 .878095 2.70 .489384 11.05 51 9.868441 2.75 10.451223 10.53 51 9.878257 2.70 10.490047 11.05 52 .868606 2.75 .451855 10.53 52 .878419 2.70 .490710 11.07 53 .868771 2.75 .452487 10.55 53 .878581 2.70 .491374 11.08 54 .868936 2.73 .453120 I 10.57 54 .878743 2.70 .492039 11.08 55 .809100 2.75 .453754 10.57 55 .878905 2.70 .4927'04 11.10 56 .869205 2.75 .454888 10.57 56 .879067 2.70 .493370 11.10 57 .889430 2.75 .4551)22 10.58 57 .879229 2.68 .494036 11.12 58 .869595 2.75 i'46MB7 10.58 58 .879390 2.70 .494703 11.13 59 .809760 2.73 . .450292 10.60 59 .879552 2.70 .495371 11.13 60 9.809924 2.75 10.456928 10.00 60 9.879714 2.70 10.496039 11.18 441 TABLE XXVI. LOGARITHMIC VERSED SINES 76 o- 77 f Vers. D. r. Ex. sec. D. r. , Vers. D.r. Ex. sec. D. 1". | Q 9.879714 2.70 10.496039 11.13 9.889329 2.65 10.537241 11.77 1 .879876 2.68 .496707 11.17 1 .889488 2.65 .537947 11.78 2 .880037 2.70 .497377 11.17 2 .889647 2.63 .538654 11.80 3 .880199 2.68 .498047 11.17 3 .889805 2.65 .539362 11.82 4 .880360 2.70 .498717 11.18 | 4 .889964 2.65 .540071 11.82 5 .8805-22 2.68 .499388 11.20 5 .890123 2.63 .540780 11.83 6 .880633 2.70 .500380 11.20 6 .890281 2.65 .541490 11.83 7 .880345 2.68 .500732 11.22 7 .890440 .2.63 .542200 11.85 8 .881006 2.68 .501405 11.22 8 .890598 2.65 .542911 11.87 9 .881167 2.70 .502078 11.23 9 .890757 2.63 .543623 11.88 10 .881329 2.68 .502752 11.23 10 .890915 2.63 .544336 11.88 11 9.881490 2.68 10.503423 11.27 11 9.891073 2.65 10.545049 11.90 12 .881651 2.68 .504102 11.25 12 .891232 2.63 .545763 11.90 13 .881812 2.68 .504777 11.23 13 .891390 2.63 .546477 11.93 14 .881973 2.63 .505454 11.23 | 14 .891548 2.63 .547193 11.93 15 .882134 2.63 .503131 11.23 ! 15 .891706 ST. 63 .547909 11.95 16 .882295 2.68 .505303 11.30 16 .891864 2.63 .548626 11.95 17 .882456 2.63 .507486 11.32 17 .892022 2.63 .549343 11.97 18 .882517 2.67 .508165 11.32 18 .892180 2.63 .550061 11.93 19 .882777 2.63 .503344 11.33 19 .892338 2.63 .550780 12.00 20 .882933 2.68 .509524 11.35 20 .892496 2.63 .551500 12.00 21 9.833039 2.63 10.510205 11.35 21 9.892654 2.63 10.552220 12.02 22 .833230 2.67 .510383 11.37 22 .892312 2.62 .552941 12.03 23 .833120 2.68 .511568 11.37 23 .892989 2.63 .553663 12.03 2t .833531 2.67 .512253 11.33 24 .893127 2.63 .554385 12.07 25 .883741 2.63 .512933 11.40 25 .893235 2.62 .555109 12.07 23 .83*902 2.67 .513617 11.40 28 .893142 2.63 .555833 12.07 27 .834032 2.63 .514301 11.42 27 .893600 2.63 .556557 12.10 28 .834223 2.67 .514986 11.43 23 .893758 2.62 .557283 12.10 29 .884383 2.67 .515672 11.43 29 .893915 2.62 .558009 12.12 30 .884543 2.6Z .516358 11.45 30 .894072 2.63 .558736 12.12 31 9.884703 2.63 10.517015 11.45 i 31 9.894230 2.62 10.559463 12.15 33 .884864 2.67 .51773-2 11.47 I 32 .894337 2.62 .560192 12.15 33 .885024 2.67 .518420 11.48 33 .894544 2.63 .560921 12.17 34 .885184 2.67 .519109 11.48 34 .894702 2.62 .561651 12.17 35 .885344 2.67 .519793 11.50 35 .894859 2.62 .562381 12.20 36 .885504 2.67 .520433 11.52 36 .89.5016 2.62 .563113 12.20 37 .835364 2.67 .521179 11.50 37 .895173 2.62 .563845 12.20 38 .885824 2.65 .5:21870 11.53 38 .895330 2.62 .564577 12.23 39 .885983 2.67 .522552 11.53 39 .895487 2.62 .565311 12.23 40 .886143 2.67 .523:254 11.55 40 .895644 2.62 .566045 12.27 41 9.886303 2.67 10.523947 11.57 41 9.895801 2.62 10.566781 12.25 42 .886463 2.65 .5-21641 11.57 42 .895958 2.62 .567516 12.28 43 .886622 2.67 .525335 11.58 43 .896115 2.62 .568253 12.28 41 .886782 2.65 .526030 11.60 44 .896272 2.60 .568990 12.32 45 .886941 2.67 .526726 11.62 45 .896428 2.62 .569729 12.32 46 .887101 2.65 .527423 11.62 46 .896585 2.62 .570468 12.32 47 .887260 2.67 .528120 11.62 47 .896742 2.60 .571207 12.35 48 .887420 2.65 .528817 11.65 48 .896898 2.62 .571948 12.35 49 .887579 2.67 .529316 11.65 49 .897055 2.60 .572(5.89 12.37 50 .887739 2.65 .530215 11.65 50 .897211 2.62 .573431 12.38 51 9.887898 2.65 10.530914 11.67 51 9.897368 2.60 10.574174 12.38 52 .888057 2.65 .531614 11.68 52 .897524 2.60 .574917 12.42 53 .888216 2.65 .532315 11.70 53 .897680 2.62 .575662 12.42 54 .888375 2.65 .533017 11.70 54 .897837 2.60 .576407 12.43 55 .888534 2.65 .533719 11.72 55 .897993 2. GO .577153 12.45 56 888693 2.65 .534422 11.73 50 .898149 2.60 .577900 12.45 57 .888852 2.65 .535126 11.73 57 .898305 2.60 .578647 12.48 58 .889011 2.65 .535830 11.75 58 .898461 2.62 .579396 12.48 59 889170 2.65 .5315535 11.77 59 .898618 2.60 .580145 12.50 60 9.889329 2.65 10.537241 11.77 60 9.898774 2.60 10.580895 12.50 443 AND EXTERNAL SECANTS. 78 79 / Vers. D. r. Ex. sec. D. r. / Vers. D.I". Ex. sec. D. 1'. ! 9.898774 2.60 10.580895 12.50 \\ 9.908051 2.55 10.627452 13.40 1 .898930 2.(iO .581645 12.53 1 .908204 2.55 .628256 13.40 2 .899086 2.58 .582397 12. 53 2 .908357 2 57 ' .629060 13.43 3 .899241 2.60 .583149 12.57 3 .908511 2^55 .629866 13.45 4 .899397 2.60 .583903 12.57 4 .908664 2.55 .630673 13.45 5 .899553 2.60 .584657 12.57 5 .908817 2.55 .631480 13.48 G .899709 2.60 .585411 12.60 i G .908970 2.55 .632289 13.48 7 .899865 2.58 .586167 12.60 7 .909123 2.55 .633098 13.52 8 .900020 2.60 .586923 12.63 8 .909276 2.53 .6&3909 13.52 9 .900176 2.58 .587681 12.63 9 .909428 2.55 .634720 13.55 10 .900331 2.60 .588439 12.65 10 .909581 2.55 .635533 13.55 11 9.900487 2.58 10.589198 12.65 11 9.909734 2.55 10.636346 13.58 12 .900642 2.60 .589957 12.68 12 .909887- 2.53 .637161 13.58 13 .900798 2.58 .590718 12.68 13 .910039 2.55 .637976 13.60 14 .900953 2.58 .591479 12.72 14 .910192 2.55 .638792 13.63 15 .9,)1108 2.60 .592242 12.72 15 .910345 2.53 .639610 13.63 1(5 .901264 2.58 .593005 12.73 16 .910497 2.55 .640428 13.67 17 .901419 2.58 .593769 12.73 17 .910650 2.53 .641248 13.67 18 .901574 2.58 .594533 12.77 18 .910802 2.55 .642068 13.70 19 ! .901729 2.58 .595299 12.78 19 .910955 2.53 .642890 13.72 20 .901884 2.60 .596066 12.78 20 .911107 2.53 .643713 13.72 21 9.902040 2.58 10.596833 12.80 21 9.911259 2.55 10.644536 13.75 22 .902195 2.58 .597601 12.82 22 .911412 2.53 .645361 13.75 23 .902350 2 57 .598370 12.83 23 .911564 2.53 .646186 13.78 24 .902504 2.58 .599140 12.85 24 .911716 2.53 .647013 13.80 25 .902659 2.58 .599911 12.85 25 .911868 2.53 .647341 13.82 26 .902814 2.58 .600682 12.88 26 .912020 2.53 .648670 13.82 27 .902969 2.58 .60M55 12.88 27 .912172 2.53 .649499 13.85 28 .903124 2.57 .602228 12.92 28 .912324 2.53 .650330 13.87 29 .903278 2.58 .603003 12.92 29 .912476 2.53 .651162 13.88 30 .903433 2.58 .603778 12.93 30 .912628 2.53 .651995 13.90 31 9.903588 2.57 10.604554 12.95 31 9.912780 2.53 10.652829 13.92 32 .903742 2.58 .605331 12.95 32 .912932 2.53 .653664 13.95 33 .903897 2.57 .606108 12.98 33 .913084 2.52 .654501 13.95 34 .904051 2.58 .606887 13.00 34 .913235 2.53 .655338 13.97 35 .904206 2.57 .607667 13.00 35 .913387 2.53 .656176 14. CO 36 .904360 2.57 .608447 13.02 36 .913539 2.52 .657016 14. CO 37 .904514 2.57 .609228 13.03 37 .913690 2.53 .657856 14.03 38 .904668 2.58 .610010 13.07 38 .913842 2.52 .658698 14.03 39 .904823 2.57 .610794 1307 39 .913993 2.53 .659540 14.07 40 .90J977 2.57 .611578 13.08 40 .914145 2.52 .660384 14.08 41 9.905131 2.57 10.612363 13.08 41 9.914296 2.53 10.661229 14.10 42 .905285 2.57 .613148 13.12 42 .914448 2.52 .662075 14.12 43 .905439 2.57 .613935 13.13 43 .914599 2.52 .662922 14.13 44 .905593 2.57 .614723 13.13 44 .914750 2.53 .663770 14.15 45 .905747 2.57 .615511 13.17 45 .914902 2.52 .664619 14.18 46 .905901 2.57 .616301 13.17 46 .915053 2.52 .665470 14.18 47 .906055 2.57 .617091 13.20 47 .915204 2.52 .666321 14.22 48 .906209 2.57 .617883 13.20 48 .915355 2.52 .667174 14.23 49 .906363 2.55 .618675 13.22 49 .915506 2.52 .668028 14.25 50 .906516 2.57 .619468 13.23 50 .915657 2.52 .668883 14.27 51 9.906670 2.57 10.620262 13.25 51 9.915808 2.52 10.669739 14.28 52 .906824 2.55 .621057 13.27 52 .915959 2.52 .670596 14.30 53 .906977 2.57 .621853 13.28 53 .916110 2.52 .671454 14.33 54 .907131 2.55 .622650 13.30 54 .916261 2.52 .672314 14.33 55 .907284 2.57 .623448 13.32 55 .916412 2.50 .673174 14.37 56 .907438 2.55 .624247 13.33 56 .916562 2.52 .674036 14.38 57 .907591 2.55 .625047 13.35 57 .916713 2.52 .674899 14.40 58 .907744 2.57 .625848 13.37 58 .9168C4 2.50 .675763 14.42 59 .907898 2.55 .626650 13.37 59 .917014 2.52 .676628 14.45 60 9.908051 2.55 10.627452 13.40 60 9.917165 2.52 10.677495 14.45 443 _ TABLE XXVI.-LOGARITHMIC VERSED SINES 80 3 I 81 / Vers. D. 1". Ex. sec. D. 1", / Vers. D. 1". Ex. sec. D.I'. 9.917165 2.52 10.677495 14.45 9.926119 2.47 10.731786 15.78 .917316 2.50 .678362 14.48 1 .926267 2.47 .732733 15.78 2 .917466 2.50 .679231 14.50 2 .926415 2.45 .788680 15.83 3 .917616 2.52 .680101 14.52 3 .926562 2.47 .734630 15.83 4 .917767 2.50 .680972 14.55 4 .926710 2.47 .735580 15.87 5 .917917 2.52 .681845 14.55 5 .926858 2.47 .736532 15.90 6 .918068 2.50 .682718 14.58 6 .927006 2.45 .737486 15.92 7 .918218 2.50 .683593 14.60 7 .927153 2.47 .738441 15.95 8 .918368 2.50 .684469 14.62 8 .927301 2.45 .739398 15.97 9 .918518 2.50 .685346 14.63 9 .927448 2.47 .740356 16.00 10 .918668 2.50 .686224 14.67 10 .927596 2.45 .741316 16.02 11 9.918818 2.50 10.687104 14.68 11 9.927743 2.47 10.742277 16.03 12 .918968 2.50 -.687985 14.70 12 .927891 2.45 .743239 16.08 13 .919118 2.50 .688867 14.72 13 .928038 2.45 .744204 16.08 14 .919268 2.50 .689750 14.73 14 .92S185 2.47 .745109 16.13 15 .919418 2.50 .690634 14.77 15 .928333 2-. 45 .740137 10.13 16 .919568 2.50 .691520 14.78 16 .928480 2.45 .747105 10. IK 17 .919718 2.50 .692407 14.80 17 .928627 2.45 .748076 16.20 18 .919868 2.50 .693295 14.83 18 .928774 2.45 .749048 16.22 19 .920018 2.48 .694185 14.83 19 .928921 2.45 .750021 16.25 20 .920167 2.50 .695075 14.87 20 .929068 2.45 .750996 10 .8 21 9.920317 2.48 10.695967 14.90 21 9.929215 2.45 10.751973 16. SO 22 .920466 2.50 .690861 14.90 22 .929868 2.45 .752951 10.33 23 .920616 2./>0 .697755 14.93 23 .929509 2.45 .753931 10.35 24 .920766 2.48 .698651 14.95 24 .929656 2.45 .7'54912 10.38 23 .920915 2.48 .699548 14.97 25 .929803 2.45 .755895 16.42 26 .921064 2.50 .700446 15.00 26 .929950 2.45 .756880 10.43 27 .921214 2.48 .701346 15.02 27 .930097 2.43 .757866 16.47 28 .921363 2.48 .702247 15.03 28 .930243 2.45 .758854 16.50 29 .921512 2.50 .703149 15.05 29 .930390 2.45 .759844 10.52 30 .921662 2.48 .704052 15.08 30 .930537 2.43 .760835 10.53 31 9.921811 2.48 10.704957 15.10 31 9.930683 2.45 10.761827 16.58 32 .921960 2.48 .705863 15.13 32 .930830 2.43 .762822 10. GO 33 .92-2109 2.48 .706771 15.15 33 .930976 2.45 .763818 1C.C2 34 .922258 2.48 .707680 15.17 34 .931123 2.43 .764815 10.07 35 .922407 2.48 .708590 15.18 35 .931269 2.45 .765815 10.08 3(3 .922556 2.48 .709501 15.22 36 .931416 2.43 .766816 16 .TO 37 .922705 2.48 .710414 15.23 37 .931562 2.43 .767819 16.73 38 .922854 2.48 .711328 15.25 38 .931708 2.45 .768823 16.77 89 .923003 2.48 .712243 15.28 39 .931855 2.43 .769859 10.80 40 .923152 2.48 .713160 15.30 40 .932001 2.43 .770837 10.82 41 9.923301 2.47 10.714078 15.33 41 9.932147 2.43 10.771840 10.87 42 .923449 2.48 .714998 15.35 42 .932293 2.43 .7'7'2t5S lelg? 43 .923593 2.48 .715919 15.37 43 .932439 2.43 [773870 16. 2 44 .923747 2.47 .716841 15 8 44 .932585 2.43 .774885 1C. 85 45 .923895 2.48 .717764 15.42 45 .032731 2.43 .775902 10.1-7 46 .924044 2.47 .718689 15.45 46 .932877 2.43 .770920 1 .CO 47 .924192 2.48 .719616 15.45 47 .933023 2.43 .777940 1 .02 48 .924241 2.47 .720543 15.48 48 .933169 2.43 .77'8961 1 .07 49 .904489 2.47 .121472 15.52 49 .983315 2.42 .779985 : .08 50 .924637 2.48 .722403 15.53 50 .933460 2.43 .781010 1 .12 51 9.9247B6 2.47 10.72J3335 15.55 51 9.933006 2.43 10.782037 1 .13 ca .924934 2.47 .7242(58 15.58 52 .933752 2.42 .78COC5 1 .18 53 .925082 2.48 .725203 15. GO 53 .933897 2.43 .784096 1 .20 54 .925231 2.47 .726139 15.63 54 .934043 2.43 .783128 1 .2.'} 55 .925379 2.47 .727077 15.65 55 .934189 2.42 .786162 1 .27 16 .925527 2.47 .728016 15.07 66 .9:34334 2 43 .787198 1 .30 57 .925075 2.47 .728956 15.70 57 .934480 2.42 .788236 I .33 58 .925823 2.47 .729898 15.73 58 .934625 2.42 .7'89276 1 .35 59 .925971 2.47 .730842 15.73 59 .934770 2.43 .790317 1 .40 CO 9.926119 2.47 10.731786 15.78 60 9.934916 2.42 10.791361 1 .42 444 AND EXTERNAL SECANTS. 82 83 Vers. D. 1'. Ex. sec. D. 1". , Vers. D. 1". Ex. sec. D.I*. 1 I 9.934916 2.42 10.791301 17.42 o 9.943559 2.38 10.857665 19.55 1 .9:55061 2.42 .792406 17.45 1 .943702 2.38 .858838 19.58 2 .935206 2.43 .793453 17.48 2 .943845 2.37 .860013 19.63 3 .935352 2.42 ,794502 17.50 3 .943987 2.38 .861191 19.67 4 .935497 2.42 .795552 17.55 4 .944130 2.38 .862371 19.72 5 .935642 2.42 .796605 17.58 5 .944273 2.37 .863554 19.75 6 .935787 2.42 .797680 17.60 6 .944415 2.38 .864739 19.80 7 .935932 2.42 .798716 17.63 7 .944558 i 2.37 .865927 19.83 8 .936077 2.42 .799774 17.68 8 .944700 2.38 .867117 19.88 9 .936222 2.42 .800835 17.70 1 9 .944843 1 2.37 .868310 19.92 10 .936367 2.42 .801897 17.73 ! 10 .944985 2.37 .869505 19.97 11 9.936512 2.42 10.802961 17.77 11 9.945127 2.38 10.870703 20.00 19 .936657 2.40 .804027 17.80 12 .945270 ! 2.37 .871903 20.05 18 .936801 2.42 .805095 17.83 1 13 .945412 i 2.37 .873106 20.10 11 .936946 2.42 .806165 17.87 14 .945554 j 2.37 .874312 20.13 15 .937091 2.42 .807237 17.90 1 15 .945696 1 2.37 .875520 20.18 16 .937236 2.40 .808311 17.93 16 .945838 i 2.38 .876731 ' 20.23 17 .937380 2.42 .809387 17.97 17 .945981 2.37 .877945 ! 20.27 18 .937525 2.40 .810465 18.00 1 18 .946123 2.37 .879161 i 20.30 11) .937669 2.42 .811545 18.03 19 .946265 2.37 .880379 20.37 20 .937814 2.40 .812627 18.07 20 .946407 2.37 .881601 20.40 21 9.937958 2.42 10.813711 18.10 21 9.946549 2.35 10.882825 20.45 23 .938103 2.40 .814797 18.13 22 .946690 2.37 .884052 20.48 23 .938847 2.40 .815885 18.17 .946832 2.37 .885281 20.55 24 .938391 2.42 .816975 18.20 24 .946974 2.37 .886514 20.58 25 .938536 2.40 .818067 18.23 25 .947116 2.37 .887749 20.62 20 .938680 2.40 .819161 18.27 i 26 .947258 2.35 .888986 20.68 27 .938824 2.40 .820257 18.32 ! 27 .947399 2.37 .8905227 20.72 28 .938968 2.40 .821356 18. as 1 28 .947541 2.37 .891470 20.77 20 .939112 2.38 .822456 i 18.38 ji 29 .947683 2.35 .892716 20.82 30 .939257 2.40 .823559 i 18.42 30 .947824 2.37 .893965 20.87 31 9.939101 2.40 10.824664 18.43 81 9.947966 2.35 10.895217 20.92 32 .939545 2.38 .825770 18.48 32 .948107 2.37 .896472 20.95 33 9396.88 2.40 .826879 ! 18.52 33 .948249 2.35 .897729 21.00 34 . 939832 2.40 .827990 ; 18.57 34 .948390 2.35 .898989 21.07 35 ! 939976 ! 2.40 .829104 18.58 35 .948531 2.37 .900253 21.10 36 .940120 i 2.40 .830219 18.63 36 .948673 2.35 .901519 21.15 37 .940264 ! 2.40 .831337 18.65 37 .948814 2.. 35 .902788 21.20 3S .940408 | 2.38 .83-3456 18.70 38 .948955 2.35 .904060 21.25 39 .940551 (2.40 .a33578 18.75 39 .949096 2 -.35 .9053:35 21.30 40 .940695 2.40 .834703 18.77 40 .949237 2.37 .906613 21.33 41 9.910839 2.38 10.8.35829 18.80 41 9.949379 2.35 10.907893 21.40 42 .940982 2.40 .836957 18.85 42 .949520 2.35 .909177 21.45 43 .911126 2.38 .&38088 18.88 43 .949661 2.35 .910464 21.50 44 .941369 2.40 .839221 18.93 44 .949802 2.35 .911754 21.55 45 .911413 2.38 .840357 18.95 45 .949943 2.33 .913047 21.60 46 .9115513 2.38 .841494 19.00 46 .950083 2.35 .914343 21.65 47 .941699 2.40 .842634 19.03 47 .950224 2.35 .915642 21.70 48 .941843 2.38 .843778 19.08 48 .950365 2.35 .916944 21.75 49 .941936 ! 2.38 .841921 19.12 49 .950306 2.35 .918249 ! 21.82 50 .942129 2.33 .8460G8 19.15 50 .95C647 2.33 .919558 21.85 51 9.942272 2.38 10.847217 19.18 51 9.950787 2 35 10.920869 21.92 52 .942415 2.40 .848368 19.23 52 .950928 2.35 .922184 21.97 53 .942559 2.38 .849522 19.27 53 .951069 2.33 .923502 22.02 54 .942702 i 2.38 .850678 19.30 i 54 .951209 2.35 .924823 j 22.07 55 .912845 i 2.38 .851836 19.35 55 .951350 2.33 .926147 22.13 56 .942988 \ 2.38 .852997 19.40 i 56 .951,490 2.35 .927475 22.17 57 .943131 i 2.37 .854161 19.42 ' 57 .951631 2.33 .928805 22.23 58 .943273 ! 2.38 .855326 19.47 58 .951771 2.33 .930139 22.30 59 .943416 2.38 .856494 19.52 59 .951911 2.35 .931477 22.33 60 9.943559 i 2.38 10.8:-7665 19.55 1 60 9.952052 2.83 10.932817 22.40 445 TABLE XXVI. LOGARITHMIC VERSED SINES 84 85 ' Vers. D. 1". Ex. sec. H i ; Vers. D. 1'. Ex. sec. D.I". 9.952052 2.33 10.932817 22.40 9.960397 2.30 ill. 020101 26.40 1 .952192 2.33 .934161 22.45 1 .900535 2.28 .021085 20.48 2 .852388 2.% .935508 22.52 2 .960672 2.30 .023274 26.57 3 .952473 2.33 .936859 22.57 S .960810 2.30 .024868 26 05 4 .932613 2.33 .938213 22.62 4 .960948 2.30 .020407 26.73 5 .952753 2.33 .939570 22.68 5 .961086 2.28 .028071 26.80 6 .952893 2.33 .940931 22.75 6 .961223 2.30 .029079 26.90 7 .953033 2.33 .942296 22.78 7 .961361 2.28 .031293 26.98 8 .953173 2.33 .943663 22.85 8 .961498 2.30 .032912 .27.07 9 .953313 2.33 .915034 22.92 9 .961636 2.28 .034530 27.13 10 .953453 2.33 .946409 22.97 10 .961773 2.30 .030164 27.23 11 9.953593 2.32 10.947787 23.03 11 9.961911 2.28 ill. 037798 27.33 12 .953732 2.33 .949169 23 08 12 .962048 2.30 .039438 27.40 18 .953872 2.33 .950554 23.15 13 .962186 2.28 .041082 27.50 14 .954012 2.33 .951943 23.22 14 .962323 2.28 .042732 27.58 15 .954152 2 32 .953336 23.27 15 .962460 2.28 .044387 27.67 1G .954291 2. '33 .954732 23.33 16 .962597 2. SO .046047 27.77 17 .954431 2.33 .956132 23.38 17 .962735 2.28 .047713 27. H5 18 .954571 2.32 .1157535 23.45 18 .962872 2.28 .049384 27.93 19 .954710 2.33 .958942 23.52 19 .963009 2.28 .051060 28.03 20 .954850 2.32 .960353 23.57 20 .663146 2.28 .052712 28.13 21 9.954989 2.33 10.961767 23. G5 21 9.963283 2.28 11.054430 28.22 22 .955129 2.32 .963186 23.70 22 .963420 2.28 .050123 28.30 23 .955268 2.32 .964608 23.77 23 .963557 2.28 .057821 28.40 24 .955407 2.33 .966034 23.82 24 .903694 2.28 .059525 28.50 25 .955547 2.32 .967463 23.90 25 .963831 2.28 .061235 28.60 26 .955686 2.32 .968897 23.95 26 .963968 2.27 .002951 28.68 27 .955825 2.32 .970334 24.02 27 .904104 2.28 .004672 28.78 28 .955964 2.32 .971775 24.10 || 28 .964241 2.28 .000399 28.88 29 .9561 as 2.33 .973221 24.15 1 29 .964378 2.28 .008132 28.98 30 .950243 2.32 .074070 24.22 30 .964515 2.27 .009871 29.08 31 9.95G382 2.32 10.976123 24.28 31 9.964651 2.28 11.071616 29.18 32 .956521 2.32 .977580 24.35 32 .964788 2.27 .073367 29.28 33 .956660 2.32 .9790H 24.42 33 .964924 2.28 .075124 29.38 34 .956799 2.30 .980506 24.48 34 .965061 2.27 .070387 29.48 35 .956937 2.32 .981975 24.55 1 35 .965197 2,28 .078656 29.58 36 .957076 2.32 .983448 24.63 1 36 .965334 2.27 .080431 29.68 37 .957215 2.32 .984926 24.68 37 .965460 2.28 .082212 29.80 38 .957354 2.32 .986407 24.77 38 .965607 2.27 .084000 29.90 39 .957493 2.30- .987893 24.83 1 39 .965743 2.27 .085794 30.00 40 .957631 2.32 .989383 24.90 40 .965879 2.28 .087594 30.12 41 9.957770 2.32 10.990877 24.97 41 9.966016 2.27 11.089401 30.22 42 .957909 2.30 .992375 25.03 42 .966152 2.27 .091214 30.32 43 .958047 2.32 .993877 25.12 43 .966288 2.27 .093033 30.43 44 .958186 2.30 .995384 25.18 44 .966424 2.27 .094859 30.55 45 .958324 2.32 .996895 25.27 45 .996560 2.27 .096692 30.07 46 .958463 2.30 .998411 25.33 46 .966696 2 27 .098532 30.77 47 .958601 2.30 .999931 25.40 47 .966832 2. '27 .100378 30.87 48 .958739 2.32 11.001455 25.48 48 .966968 2.27 .102230 31.00 49 .958878 2.30 .002984 25.52 '49 .967104 2'.27 .104090 31.12 50 .959016 2.30 .004517 25.03 50 .967240 2.27 .105957 31.22 51 9.959154 2.30 11.006055 25.70 51 9.967376 2.27 11.107830 31 .35 52 .959292 2.32 .007597 25.78 52 .9(57512 2.25 .109711 31.45 53 .959431 2.30 .009144 25. 85 53 .907647 2.27 .111598 31.58 54 .959569 2.30 .010095 25.93 I 54 .967783 2.27 .113493 31.68 55 .959707 2.30 .012251 26. OJ [ 55 .967919 2.25 .115394 31.82 56 .959845 2.30 .013811 26.10 ' 56 .968054 2.27 .117303 31.93 57 .959983 2.30 .015377 26.17 57 .968190 2.27 .119219 32.07 58 .960121 2.30 .016947 26.23 58 .968326 2.25 .121143 32.18 59 .960259 2.30 .018521 26.33 59 .968401 2.27 .123074 ! 32.30 60 9.960397 2.30 11.020101 26.40 ! 60 9.968597 2.25 11.125012 i 32.43 AND EXTERNAL SECANTS. 86 87 ' Vers. D. 1". Ex. sec. D 1". : i ' Vers. D. 1".| Ex. sec. D. r. 9.96859T 1 2.25 11.125012 32.43 9.976654 2.23 11.257854 42.52 1 .908732 2.27 .126958 32.55 1 .976788 2.22 .260405 42.73 2 .968808 2 25 .128911 32.70 2 .976921 2.22 .262969 42.95 3 .909003 2.25 .130873 32.80 3 .977054 ! 2.22 .265546 43.20 4 .969138 2.27 .132841 32.95 4 .977187 2 22 .268138 43.42 5 .969274 2.25 .134818 ,33.07 5 .977320 2^20 .270743 43.67 G .909409 2.25 .136802 33.22 6 .977452 2.22 .273363 43.88 7 .969544 2.25 .138795 as. as 7 .977585 2.22 .275996 44.15 .969079 2.25 .140795 33.47 8 .977718 2.22 .278645 44.38 9 .909814 2.25 .142803 33.62 9 .977851 2.22 .281308 44.63 10 .909949 2.25 .144820 33.73 10 .977984 .2.20 .283986 44.88 11 9.970084 2.27 11.146844 as. 88 11 9.978116 2.22 11.286679 45.13 12 .970220 2.23 .148877 34.02 12 .978249 2.22 .289387 45.38 13 .970a54 2.25 .150918 34.17 13 .978382 2.20 .292110 45.65 14 .97-0489 2.25 .152908 34.30 14 .978514 2.22 .294849 45.92 15 .970024 2.25 .155026 34.43- 15 .978617 2.20 .297604 46.17 16 .970759 2.25 .157092 34.00 1 16 .978779 2.22 .300374 46 45 17 .970894 2.25 .150108 34.73 17 .978912 2.20 .303161 46 72 18 .971029 2.25 .161252 34.87 18 .979044 2.22 .305964 47.00 19 .971104 2.23 .103344 35.03 19 .979177 2.20 .308784 47.27 20 .971298 2.25 .105446 35.17 20 .979309 2.22 .311620 47.55 21 9.971433 2.25 11.107-550 a5.as 21 9.979442 2.20 11.314473 47.83 22 .971568 2.23 .169676 35.48 22 .979574 2.20 .317343 48.13 23 .971702 2.25 .171805 35.63 23 .979706 2.20 .320231 48.43 24 .971837 2.23 .173943 35.78 24 .979838 2.20 .323137 48.72 25 .971971 2.25 .176090 35.93 25 .979970 2.22 .326060 49.02 26 .972106.; 2.23 .178246 36.10 26 .980103 2.20 .329001 49.33 27 .972240 2.23 .180412 30.27 ! 27 .980235 2.20 .331961 49.63 28 .972374 2.25 .182588 ; 36.42 ! 28 .980307 2.20 .334939 49.93 29 .972.509 2.23 .184773 l 36.58 ! 29 .980499 2.20 .337935 50.27 30 .972043 2.23 .186968 36.75 30 .980631 2.20 .310951 50.58 31 9.972777 2.25 11.189173 36.90 31 9.980763 2.20 11.343986 50.92 32 .972912 2.23 .191387 37.08 82 .980895 2.18 .347041 51.23 33 .97-3046 2.23 .193612 37.25 33 .881026 2.20 .350115 51.58 34 .973180 2.23 .195847 i 37.42 34 .981158 2.20 .353210 51.92 35 .973314 2.23 .198092 37.58 1 35 .981290 2.20 .356325 52.25 30 .97'3448 2.23 .200347 37.77 36 .981422 2.20 .359460 52.62 37 .973582 2.23 .202613 37.93 37 .981554 2.18 .362617 52.95 38 .973716 2.23 .204889 38.12 38 .981685 2.fiO .365794 53.32 39 .973850 2.23 .207170 38.28 i 39 .981817 2.20 .368993 53.68 40 .973984 2.23 .209473 38.47 40 .981949 2.18 .372214 54.07 41 9.974118 2.23 11.211781 38.67 41 9.982080 2.20 11.375458 54.42 42 .974252 2.23 .214101 38.83 i 42 .982212 2.18 . .378723 54.80 43 .974J386 2.22 .216431 39.03 : 43 .982343 2.20 .382011 55.20 44 .974519 2.23 .218773 39.20 i 44 .982475 2.18 .385323 55.58 45 .974653 2.23 .221125 39.42 jl 45 .982606 2.18 .388658 55.97 40 .974787 2.22 .223490 39.58 1 46 .982737 2.20 .392016 56.38 47 .974920 i 2.23 .225805 39.80 || 47 .982869 2.18 .395399 56.80 48 .975054 2.23 .228253 39.98 !i 48 .983000 2.18 .398807 57.20 49 .975188 i 2.22 .230(152 : 40.18 | 49 .983131 2.18 .402239 57.62 50 .975321 2.23 .233063 40.38 | 50 .983202 2.20 .405696 58.07 51 9.975455 2.22 11.235486 40.58 51 9.983394 2.18 11.409180 58.48 52 .975588 2.23 .237921 : 40.78 | 52 .983525 2.18 .412689 58.93 53 .975722 2.22 .240368 41.00 53 .983656 2.18 .416225 59.38 54 .975855 2 22 .242828 41.20 i 54 .983787 2.18 .419788 59.83 55 .975988 2^23 .245300 41.42 1 55 .983918 2.18 .423378 60.28 50 .970122 2.22 .247785 41.63 50 .984049 2.18 .426995 60.77 57 .976255 S 2.22 .250283 41. as 57 .984180 2.18 .430641 61.25 58 .976:388 2.22 .252793 42.07 58 .984311 2.18 .434316 ,61. 73 59 .976521 2.22 .255317 42.28 59 .984442 2.18 .438020 62.22 60 9.970654 2.23 11.257854 42.52 60 9.984573 2.17 11.441753 62 73 447 TABLE XXVI. LOGARITHMIC VERSED SIGNS AND EXTERNAL SECANTS. 88 ! 89 t Vers. D. r. Ex. sec. q+l Vers. D. 1". Ex. sec. Q+J 15. 29* 15 30* 9.984573 2.17 11.441753 9086 : 9.992334 2.13 11.750498 6801 1 .9847-03 2.18 .445517 9215 ! 1 .992482 2.15 .757925 6929 2 .984834 2.18 .449311 9345 2 .992611 2.13 .765477 7056 3 .984965 2.18 .453137 9474 ' 3 .992739 2.15 .773158 718-1 4 .985096 2.17 .456994 9603 4 .992868 2.13 .780978 7312 5 .985226 2.18 .460883 9732 i 5 .992996 2.13 .788926 7440 6 .985357 2.17 .464805 9HO-2 6 .993124 2.15 .797-022 7507 7 .985487 2.18 .468761 9991 7 .993233 2.13 .805268 7005 8 .985618 2.17 .472751 4-120 8 .993:381 2.13 .8136(58 7823 9 .985748 2.18 .476775 0249 9 .993509 2.13 .822229 7950 10 .985879 2.17 .480834 0378 10 .993637 2.13 .830956 8078 11 9.986009 2.18 11.484929 0507 11 9.993765 2.15 11.839858 8205 12 .986140 2.17 .489061 0636 12 .993894 2.13 .848940 8333 13 .986270 2.17 .493230 0765 13 .994022 2.13 .858211 8460 14 .986400 2.18 .497437 0894 14 .994150 2.13 .867079 8588 15 .986531 2.17 .501683 1023 15 .994278 2.13 .877351 8715 16 .986661 2.17 .505968 1152 16 .994406 2.13 .887239 8843 17 .986791 2.17 .510293 1281 17 .994534 2.13 .897350 8970 18 .986921 2.17 .514659 1410 18 .994662 2.12 .907697 9097 19 .937051 2.17 .519066 1539 19 .994789 2.13 .918290 9225 20 .987181 2.17 .523516 1668 20 .994917 2.13 .929141 9352 21 9.987311 2.17 11.528010 1797 21 9.995045 2.13 11.940264 9479 22 .987441 2.17 .532548 1925 22 .995173 2.13 .951672 9607 21 .987571 2.17 .537131 2054 23 .995301 2.12 .963381 9734 21 .987701 2.17 .541760 2183 24 .995428 2.13 .975408 9862 2~) .987831 2.17 .546437 2312 25 .995556 2.12 11.987769 9988 26 .987961 2.17 .551161 2440 26 .995683 2.13 12.000485 27 .988031 2.17 .555935 2369 27 .995811 2.13 .013578 0243 28 .988221 2.15 .560759 2698 28 .995939 2.12 .027069 0370 29 .98S330 2.17 .565634 2826 29 .996066 2.12 .040984 0497 30 .988480 2.17 .570561 2955 30 .996193 2.13 .055352 0024 31 9.988610 2.15 11.575542 3083 31 9.996321 2 12 12.070202 0751 32 .988739 2.17 .580578 3212 32 .996448 2! 13 .085569 087'8 33 .988869 2.15 .585670 3:340 33 .996576 2.12 .101490 1005 34 .988998 2.17 .590819 3469 34 .996703 2.12 .118008 1132 35 .989128 2.15 .596027 3597 35 .996830 2.12 .185168 1259 36 .989257 2.17 .601295 3726 36 .996957 2.13 .153024 1386 37 .989337 2.15 .600625 3854 37 .997085 2.12 .171634 1513 38 .989516 2.17 .612018 3983 38 .997212 2.12 .191066 1(540 39 .989046 2.15 .617475 4111 39 .997339 2.12 .211396 1767 40 .9897r5 2.15 .622998 4239 40 .997466 2.12 .232712 1894 41 9.989904 2.17 11.628589 4368 41 9.997593 2.12 12.255116 2020 42 .990034 2.15 .634250 4496 42 .997720 2.12 .278723 2147 43 .990163 2.15 .639982 4(524 43 .997847 2.12 .303674 2274 44 .990292 2.15 .645788 4752 44 .997974 2.12 .3:30129 2401 45 .990421 2.15 .651668 4881 45 .998101 2.12 .358285 2527 46 .990.550 2.15 .657626 5009 46 .998228 2.12 .388375 2654 47 .990679 2.15 .663663 5137 47 .998355 2.10 .420686 2781 48 .990808 2.15 .669781 5265 48 .998481 2.12 .455575 2907 49 .990937 2.15 .675984 5393 49 .908608 2.12 .493490 3034 50 .9.91066 2.15 .682272 5521 50 .9! 1ST. '35 2.12 .535009 3161 51 9.991195 2.15 11.688649 5649 51 9 99RS62 2.10 12.5S0893 3287 52 .991324 2.15 .695U7 5777 52 .998988 2.12 .0321 73 3414 53 .991453 2.15 .701679 5905 53 .999115 2.10 .690291 8540 54 .991582 2.13 .708338 008:3 54 .999241 2.12 ! 757364 3067 55 .991710 2.15 .715097 6161 :>:> .999368 2.10 .8:36072 3793 56 .991839 2.15 .721958 6289 56 .999494 2.12 32.933708 31KO 57 .991968 2.13 .728925 6417 57 .999621 2.10 13.058774 4046 58 .992096 2.15 .736002 6.545 58 .999747 2.12 .234991 4172 59 .992225 2.15 .743192 6673 no .099S74 2.10 .536148 4299 eo'* 9.992354 2.13 11.750498 6801 63 10.000000 2.10 Inf. pos. 4426 15.30*. 15.81* TABLE XXVII. NATURAL SINES AND COSINES. 1 . 2 i 3 40 Sine ' Cosin i Sine | Cosin Sine Cosin Sine | Cosin j Sine Cosin ToOOOoTOneT .01745 999Sr> 703490 ! .99939 .052341.99803 .06976 .99756 60 1 .10029 One. .01774 .99984 .03519! .99938 .05263:. 99801 .07005 .99754 59 .00058: One. .01803 .9998-1 .03548 .99937 .o:,292 .99800 .07034 .99752 58 3 .00087! One. i .01832 .99983' .03577 .99936; .05321 .99858 ! .07063 .99750 57 4 i. 00116: One. j. .01802 .99983 .03600 .99935 .05350!. 99857, .07092 .99748 50 5 1 .00145 One. ' .01891 .99982 .03635 ! 99934 .05379 .99855 .07121 .99740 55 6 .00175 One. .01920 .999S2 .03604 .99933 .05408 .99854 .07150 .99744 54 .00204! One. .01949 .99981 .03093 .99932! .05437 .99852 .07179 .99742 53 8 .00233 'One. | .01978 .99980 .03723 .99931 .05466 .99851 .07208 .99740! 52 9 .00262: One. .02007 .99980 .03752 .99930 .05495 .99849 j .07237 .99738! 51 10 .00291 One. .02036 .99979 ; . 03781 .99929; .05524 .99847 .07266 .99730, 50 11 . 00320 ! . 99999 .02065 .99979 1.03810 .99927" .05553 .99846 .07295 .99731 49 12 . 00349 . 99999 . Oil 194 . 99978 .03839 .99926 .05582 .99844 .07324 .997315 48 13 .00378 .99999 .02123 .99977 .03808 .99925 .05011 .99842 .07353 .99729! 47 14 . 00407 . 99999 . 02152 > . 99977 .03897 .99924 .05640 .99841 .07382 .99727 40 15 .00436 .99999 .021811.99970 .03926 .99923 .05069 .99839 .07411 .99725 45 16 .0040.-, .99999 .02211 .99976 .08955 .99922 .05098 .99838 .07440 .99723 44 17 .00495 .99999 .022401.99975 .03984 .99921 .05727 .99836 .07469 .99721 43 18 .00524 . '.19999 .0220'.) .999T4 .04013 .99919! .05756 .99834 .07498 .99719 42 19 . 00553 . ! >9! )! IS . Oii'.KS ! . 99974 .04042 .99918! .05785 .99833 .07527 .99716 41 20 .00582 .99998 1.02327 j. 99973 .04071 .99917; .05814 .99831 .07556 .99714 40 21 .00611 .99998 .02356!. 99972 .04100 .99916: ! .05^44 .99829 .07585 .99712 39 22 .(KM) 40 .99998 .02:385 1.99972 .04129 .99915 .05873 .99827! .07614 .99710 38 23 . 00669 . 99998 . 0241 4 j . 99971 .04159!. 99913 i ! .05902 .99826 .(7043 .99708 37 24 . 00698 t. 99998 .02443 .99970 .04188 .99912 ; .05931 .99824 .07672 .99705 36 25 . 00727 ! . 99997 . 02472 ! . 99969 .04217 .99911 i .05960 .99822 .07701 .99703 35 26 .00756 .99997 .0*501 .99909 .04246). 99913 : .05989 .99821 .07730 .99701 34 27 .00785.99997 .02530 .99908 !. 04275 .99909 .06018 .99819 .07759 .99099 33 28 .00814 .99997 .02500 .99907 .04304 .99907 .06047 .99817 .07788 .99696 32 29 . 00844 '. 99996 . 02589 : .99966 .04333 .99906 ' .06076 .99815 .07817 .99694 31 30 .00873 .99996: .02618 .99968 .04362 .99905 .06105 .99813 .07846 .99692 30 31 . 00902 ! . 99996 j .02647 .99965 .04391 .99904 .06134 .99812 .07875 .99689 29 32 .00931 .99990 : .02076 .99904 .04420 .99902 i .06163 .99810 .07904 .99687 28 33 .00960 .99995 :| .02705 1.99963 .04449 .99901 . 06192 . 99808 . 07933 . 99685 27 34 .00989 .99995 .02734 .99963 .04478 .9990(3 .06221 .99806 . 07962 :. 99683 26 35 .01018 .99995 .02703 .99962 .04507 .99898 ! .06250 .99804 .07991!. 99680 25 36 .01047 .99995 ! .02792 .99961 .04536 .99897 :. 00279 .99803 . 08020 j. 99078 24 .01076 .99991 .02821 .99900 1.045651.99896 .00508 .99801 .08049!. 99676 23 38 .01105 .99994 .02850 .99959 .04594 .99894 .06337 .99799 .080781.99073 22 39 .01134 .99994 .02879 .99959 1.04623 .99893 .06366 .99797 .08107 !. 99071 21 40 .01164 .99993 .02908 .99958 .04653 .99892 .06395 .99795 .08136 .99068 20 41 . Oil 93 ! . 99993 .02938 .99957 .04682 .99890 .06424 .99793 .081 65 '.99666 19 42 .01222 .99993 .02967 .99956 .04711 99>"H9 . 06453 . 99792 . 08194 i . 99664 ! 18 43 .01251 .99992 .02996 .99955 .04740 .'91)888 .06482 .99790 .08223 .99661 17 44 .01280 .99992 .03025 .99954 .04709 .99886 .06511 .99788 .08252 .99659 16 45 .01309 .99991 .03054 .99953 ..04798 .99885 .06540 .99786 1.08281 .99657 15 46 .01338 .99991' .03083 .99952 .04827 .99883 .06569 .99784 f*8310 .99654 14 47 . 01367 : . 99991 .03112 .99952 1.04856 .99882 .06598 .99782 !08339 .99652 13 48 .01396 .99990 .03141 .99951 .04886 .99881 .06627 .99780 .08368 .99649 12 49 .01425:. 99990 .03170 .99950 .04914 .99879 .06656 .99778 .08397 .99647 11 50 .01454!. 99989 .03199 .99949 .04943 .99878 .06685 .99776 .08426 .99644 10 51 . 01483 ! . 99989 .03228 .99948 .04078 . 99876 ' .06714 .99774 .08455 .99642 9 52 .01513 .099S9 .03857 .99947 .05001 .99875 .06743 .99772 .08484 .99639 8 53 .01542:. 99988 ! 03236 .99946 !. 05030 .99873 .06773 .99770 .08513 .99037 7 54 01571 .99988 .03316 .99945 .05059 .99872 !. 06802 .99768 .08542 .99035 6 55 01600 .99987 .03345 .99944 .05088 .99870 .00831 .99706 .08571 .99032 5 56 .01629 .99987 > .03374 .99943 .05117 .99809 .06860 .99764 .08600 .99630 4 57 .01658 .99986 .03403 .99942 .05146 .99867 .OOSS9 .99762 .08629 .99627 3 58 .01087 99986 .03432 .99941 .08175 .99800 .06918 .99760! .08658 .99025 2 59 .01716 .99985 .03401 .99940 i.05205 .99864 .06947 .99758 .08687 .99022 1 60 .01745 .99985 .03490 .99939 |. 05234 .99863 .06976 .99756 .087161.99619 f Cosin Sine Cosin Sine Cosin Sine Cosin Sine Cosin Sine ~ 89 88 87 86 85 449 TABLE XXVII. NATURAL SINES AND COSINE3. 5 6 7 . 8' 9 ' Sine Cosin Sine Cosin Sine Cosin Sine Cosin Sine Cosin f .08716 .99619 T0453 .99452 .12187 .99255 .13917 .99027 715643 .98769 60 1 .08745 .99617 .10482 .99449 .12216 .99251 .13946 .99023 .15672 .98764 59 2 .08774 .99614 .10511 .99446 .12245 .99248 .13975 .99019; .15701 .98760 58 3 .08803 .99612 .10540 .99443 .12274 .99244 .14004 .99015! .15730 .98755 57 4 .08831 .99609 .10569 .99440 .12:302 .99240 .14033 .99011 .15758 .98751 56 5 .08860 .99607 .10597 .123311.99237 .14061 .99006 : .15787 .98746 55 6 .08889 .99604 .10626 iium .12360 .992*3 1 .14090 .99002 .15816 .98741 54 7 .08918 99602 .10655 .99431 .123891.99230 i .14119! .98998 .15845 .98737 53 8 .08947 .99599 .10684 .99428 .12418 .99226 .14148 .98994 .15873 .98732 52 9 .08976 .99596 .10713 .99424 .12447 .99222 .14177 .98990 1.15902 .98728 51 10 .09005 .99594 .10742 .99421 .12476 .99219 . 14205 j. 98986 .15931 .98723 50 11 .09034 .99591 .10771 .99418 .12504 .99215 .14234 .98982 .15959 .98718 49 12 .01)063 .99588 .10800 .99415 .12533 .99211 .14263 .98978 i .15988 .98714 48 13 i .09092 .99586 .10829 .99412 .12562 .99208 .1 4292 !. 98973 i .16017 .98709 47 14 ! . 09121 i. 99583 .10858 .99403; .12591 .99204 .14320 .98969 i.16046 : 1)870 J 46 15 .09150 .99580 .10387 .12620 .99200 .143491.98965 .16074 .98700 45 16 .09179 .99578 .10916 '.in 103 .12649!. 99197 .14378 .98961 .16103 .98(595 44 17 .09208 .99575 .10945 .93393 .126781.99193' .14407 .98957 . 161 32 ! . 98690 43 1 8 1. 092371. 99572 .10973 99 59 i .12106 .99189 .14436 .98953 .16160 . 9SOS(5 42 19 i .09266 .99570 .11002 ! 99391 .12735 . 99186 ' 14464 .98948 .1(5189 .98(581 41 20 .09295 .99567 .11031 .9339,); .12764 .99182 .14493 .98944; .162181.98676 40 21 .09324 .99564 .11060 .99^(5 .12793 .99178 .14522 .98940' .16246 .98671 39 22 .09353 .99562 .11089 .993S3 .12822 .99175 .145511.98936 .16275 .98667 38 23 .03:382 .99559 .11118 .13851 .99171 .14580 .98931 .16304 .98662 37 24 .09411 .99556 1.11147 .99377 .12880 .99167 .14608 .981*27 .16333 .98657' 36 25 .09440 .99553 .1117(5 .93374 .12908 .93163 .14(537 .1)8923 .16361 .98652 35 26 .09469 .99551 .11205 .99370 ,12937 .99160 .1 1000 .98019 .16390 .9S01S 34 27 .09498 .99548 .11234 .99 lf>7 .129(56 .99156 .11695 .98914 .16419 .98(543 33 28 i .095271.99545 .11263 .99364! .12995 .9915-2 .14723 .98910 .16-147 .98638 32 29 .09556^.99542 .11231 .99330 .13024 .99148 .14752 .981100 .1(5476 .986*3 31 30 .09585 .99540 .11320 .99357, .1:3053 .99144 .11781 .98902 1.16505 .98629 30 31 .09614 .99537 .11349 .99354 .13081 .99141 ! .14810 .98897 .16533 .98624 29 32 .09642 .99534 .11378 .93351 .13110 .99137 .14838 98893 1(5502 .98619 28 33 .09671 .99531 .11407 .99347 .13139 .99133 .148(57 !! 98889 .16591 .98614 27 34 .09700 .99528 .11436 .93344; .13163 .99129 .1489(5 L 98884 .16(520 .9S009 26 35 .09729 .99526 .11465 .93341 .13197 .99125! .11925 .98880: .16648 .98(504 25 36 .09758 .99523 .11494 .99337 .99122 .14954 .98876 .16677 .98600 24 37 .09787 S. 99520 .11523 .93334 ! 13254 .991181 .14982 . 98871 .16706 .98595 23 38 1 .09816 .99517 .11552 .99331 .13283 .99114 .15011 !. 98867 .16734 .98590 22 39 .098451.99514 .11580 .93327 .13312 .99110 .15040 .988631 .16763 .98585 21 40 .09874 .99511 .11609 .99324 .13341 .99106 .15069 :.98858 .16792 .98580 20 41 .09903 .99508 .11638 .99320 .13370 .99102 .15097 1.98854 .16820 .98575 19 42 .09932 .99506 .11667 .99317 .13399 .99098 .15126 .98849 .16849 .98570 18 43 .09961 .99503 !. 11696 .99314 .134271.99094 .15155 ;. 98845. .10878 .98505 17 44 .09990 .99500 .11725 .99310 .13456 .99091 .15184 .98841' .1(590(5 .98561 16 45 .10019 .99497 .117541-99307 .1:3485 .99087 .15212 .98836 .16935 .98556 15 46 .10048 .99494 .11783 .99:303 1 13514 .99083 .15241 .<:8S;;2 .169(54 .98551 14 47 .10077 .99491 .11812 .99300 .13543 .99079 .15270 .H8827' .16992 . 98.546 13 48 .101061.99488 .11840 .99297 .13572 .95)075 .15299 .98823 .17021 .98541 12 49 .10135 .99485 .11869 .99293 .13600 .99071 .15327 .!)8818 .17050 .9853(5 11 50 .10164 .99482 . 11893 i. 99230 .13629 . 99067 .15356 .C8814 .17078 .98531 10 51 .10192 .99479 .11927 L 99286 .1 3658 '. 99063 .15385 .98809 ". 7107 .96526 9 52 .10221 .99J7< : .119561.992*3 .1368T .99059 .15414 .98805 . 7136 .98521 8 53 .10250 .9947': ! .11985 .93279 ! .13716 .99055 .16442 98800 . 71(54 .9851(5 7 54 .10279 .9917( .12014 .99376 .13744 .UlH-,1 .15471 .9879(5 . 7193 .98511 6 55 .10308 .99467 .12043 .99272 .13773 .1)1)047 . 5500 .9N791 . 7222 .9S50I5 5 56 . 10337 . 99 4(5- .12071 .992(59 .13802 .99043 i . 5529 .98787 . 726C 9S501 4 57 10366 .99461! .12100 .992(55 .138:31 .99039 . 5557 .98782 . 7271 .U8190 3 58 .10395 .99458 .12129 .99262 138(50 990:35 . 558C 98778 l . 7308 .984D1 2 59 .10424 .99455 .12158 .99258 .138S9 .99031 . .-(iir .98773 .17381) .98486 1 60 .10453 .99452 i. 12187 i.99255 .13917 .99027 .15(513 ;. 98769 .1 7:565 . 9S 181 ! Cosin Sine 1 Cosin | Sine Cosin Sine Cosin i Sine Cosin Sine i 84 83 82 il x 81 [1 80 450 TABLE XXVII. NATURAL SINES AND COSINES. 10 | 11 1 12 13 14 ' Sine Cosin Sine Cosin I Sine i Cosin Sine Cosin Sine Cosin .17305 .1)848) .19081 .98163 .20791 .97815 .22495 .97437 .24192 T97030 60 1 .17393 .98476 1. 19109 .98157 .20820 .97809 .22523 .97430 .24220 .97023 59 2 .17422 .98471 .19138 .98152 .20848:. 97803: .22552 .97424 .24249 .97015 58 3 .17451 .98466 .19107 .98146 .20877 .97797 .22580 .97417 .24277 .97008 57 4 .17479 .98461 .19195 .98140 .20905 .97791 .22608 .97411 .24305 .97001! 56 5 .17508 .98455 .19224 .98185] .209:33 .97784 .22637 .97404 .24333 .86994 55 6 .17537 .98450 .19252 .9812!) .20962;. 97778 .22065 -97398 ! .24362 .96987 54 7 .17565 .98445 .19281 .98124 .20990 .97772 .22093 .9739l! .24390 .96980 53 8 .17591 .98440 .19309 .98118 .21019 .97766 22722 .97384: .24418 .96973 52 9 .17623 .98435 . 19338 .98112 .21047 .97700 .22750 .97378 .24446 .96966 51 10 .1765JL .98430 .19366 .98107: .21076 .97754 .22778 .97371 .24474 .96959 50 11 .176801.98425 .19395 .98101 .21104 .97748 .22807 .97365 .24503 .96952 49 12 . 17708 !. 98 420 .19423 .21132 .97742 .22835 .97358 .24531 .96945 48 13 .17737 .98414 . 19452 I.98C90 .21161 .97735 .22363 .97351 .24559 .96937 47 14 .17700 .9.8409 .19481 .98084 .21189 .97729 .2289:2 .97345 .24587 .96930 46 15 .17794 .98404 .195091.98079 .21218 .97723 .22920 .97*38 .24615 .90923' 45 16 .178231.98399 .19538 .98078 .21246 .97717 .,22948 j. 97331 .24644 .96916 44 17 .17832 .93394 .19500 .98067 .21275 .97711 22977 .97325 .24672 .96909 43 18 .17X80 .9S381) .19593 .93061 .21303 .97705 ! 2:3005 .97318 .24700 .96902 42 19 .17909 .983831 .19623 .98056 .21331 .97698 .230331.97311 .24728 .96894! 41 20 .17937:. 98378 ; .19552 .98050, .21360 .97692 .23002 1.97304 .24756 . 96887 ! 40 21 .17966 .98878 .19680 .98044 ! .21388 .97686 .23090 .97298 .24784 .96880 39 .17!)'.<5 .!)S3li8 .19709 .93039 .21417 .97680 .23118!. 97291!! .24813 .96873 38 23 18()'*3 ')S J ')'* .19737 1K)33 .21445 .97673 .23146 .972.-U .24841 .96866 37 24 ,'lS052 \ 1)8357 .19766 .98027 .21474 .97667 .23175 .97278 .24869 .96858 36 25 .13081 .98352 .19794 93021 .21502 .976t51 .23203 .97271 .24897 .96851 35 26 .18109 .98347 . 19323 .93316 .21530 .97055 .23231 .97264 .24925 .96844 34 27 .18138 .98341 .19851 .98910 .21559 .97648 .23260 .97257 .241)54 .96837 33 28 .18100 .98330 .19880 .98004 .21587 .97612 .23288 .97251 .24982 .96829 32 29 .18195 .93331: .19908 .979:)-! .21616 .97633 .23316 .97244 .25010 .90822 31 30 .18224;. 98325; .19937 .97932 .21644 .97030 .23:345 .97237 .25038 .96815 30 31 .18252 '.98320 .19965 .97937; .21672 .97623 .23373 .97230 .25066 .96807 29 32 .1823 1 .93315 .19994 .97331 .21701 .97017 .23401 .97223 .25094 .96800 28 33 .1330!) .98310 .20322 .!)7i>;5 .21729 .97611 .23423 .97217 .25122 .96793] 27 34 .18338 .1K301 .20051 .97969 .21758 .97604 .2:3458 .97210 .25151 .967861 26 35 .188671.98299 .23379 .!)7;3 .21786 .97533 .23486 .1)7203 .25179 .96778 25 30 .18393 .98294 .23103 .97958 .21814 .97592 .23514 .97196l| .25207 .96771 24 37 .18124 .98238 .20136 .97952 .21843 .97535 .23542 .97189 | .25235 .96764 23 38 .18452 .98283 .20165 .97946 | .21871 .97579 .23571 .97182 l .25263 .96756! 22 39 .18481 i.98277 .20193 .97940 .21839 .97573 .23599 .'37170 .25291 . 96749 j 21 40 .18509 .98272 .20222 .97934 .21923 .97566 .23627 .97109 .25320 .96742 20 41 .1 8538 . 98267 .20250 .971)23 .21956 .97560 .23656 .97162 .25348 .96734! 19 42 .18507 .{W261 .2037'!) .<)r:)22 .21985 .97553 .23084 .97155 .25376 .96727! 18 43 .18595 .98250 ^20307 .97916 .22013 .97547; .23712 .97148 ; .25404 .96719 17 44 .18024 .1)8250 .20330 .97910 .22041 .97541 .23740 .97141 ii .25432 1.90712 16 45 .181552 .9821-) .20364 .97903 .22070 .97534 .23709 .97134'! .25460 .967051 15 46 .18081 .1IS240 . 20393 L 97899 ! .22093 .97528 .23797 .97127 II .25488 .96697! 14 47 .18710 .98234 .20121 .97833 .221261.97521 .2:3825 .9712(7, .25516!. 966901 13 48 .18738 .93229 .204,50 .97887; .22155 .97515 .23853 .97113 .25545 .96082 12 49 .187'07 .98223 .20478 .97331 .22183 .97508 .23882 .1)7100 .25573 .96673 11 50 .18795 .98218 .20507 .97875 .22212.97502 .23910 \ .97100 .25001'. 96667 10 51 .18S21 .98212 .20535 .97809 .22240 .97496 .239381.97093 .25629 .96660 9 52 .18852 .93207 .97863 .22208 .97489 .23966 .970861 .25057 . 90053 i 8 53 .18881 .1)82,11 ! 20592 .97857 .22297 .97483 .23995 .1)707'!) .25085 . 96645 1 7 54 .18910 .98196 .20820 .1)7851 .22325 .97476 .24023 .97072 .25713 .96638 6 55 .189381.98190; .20649 .97845 .22353 .97470 .24051 .97065 .25741 '.90030 5 56 .18907 .98185 .20677 .9783!) .22382 .97463 .24079 .97058 .23709 .96623 4 57 .18995 .98179 .20706 .97833 .22410 .97457 .24108 .1)7051 .257!)8 .90015 3 58 .19024 .98174 .207:34 .97827 .22438 .97450 .24130 .97044 .25820 .90(508 2 59 .19052 .98168 .20763 .97821 .22407.97444 .24104 .97037 .25854 .96600 1 60 .19081 .<)S 103 .20791 .97815 .22495 .97437 .2111)2 .97030 .25882 .96593 Cosin , Sine Cosin Sine Cosin Sine Cosin Sine Cosin Sine " L 79 ! 7tf !! 77 76 75 451 TABLE XXVII. NATURAL SINES AND COSINES. 15 16 17 18 19 ' Sine Cosin Sine i Cosin Sine Cosin Sine Cosin Sine Cosin .25882 .96593 .27564 .96126 S .29237 .95630, ~ 30902 .95106 .325.57 .94552 60 1 .25910 '.1658.-, .27592 .96118 .jJ'.tttM .95622 .30929 .<5097 .32584 .94542 59 2 .25938.9(5578 .27620 .96110 ; .29293 .95613 .30957 .95088 .;woi2 .94533 58 3 .25966 .96570 .27648 .90102 .29321 .95605 .30985 .95079 .32639 .94523 57 4 .25994 .9fi562 .27676 .96094 .29348 .95596 .31012 .95070 .32667 .94514 56 5 26022 96555 .27704.960861.29376 .95588 .31040 .95061 .32694 .94504 55 6 .260501.96547 .27731 .960781 .29404 .95579 .31068 .95052 .32722 .94495 54 7 .26079 .96540 .27759 -96070 1;. 29432 .95571 .31095 .95043 .32749 .94485 53 8 .26107 .96532 .27787 .96062 .29460 .95562 .31123 .95033 .32777 .94470 52 9 .26135 .96524 .27815 .96054 .29487 .95554 .31151 .95024 .32804 .94400 51 10 -2IH63 .W517 .27843 .9604(5 .29515 .95545 .31178 .95015 .32832 .94457 50 11 . 26191 |. 96509 .27871 -.96037 i .29543 .95536 .31206 .95006 .32859 .94447 I!) 12 .26219 .96502 .27899 .96029 i .29571 .95528 .31233 .94997! .32887 .944:38 48 13 .26247 .96494 .27927 .96021 .29599 .95519 .31261 .94988 .32914 .94428 47 1 4 . 26275 . 96488 j . 27955 . 960 1 3 . 29626 .95511, .31289 .94979 .32942 .94418 46 15 .26303.964791.27983.96005 .29654 .95502 .31316 .94970 .:-W'.i9 .94to:) 15 16 i . 26331 .96471 .28011 .9599? .29682 .95493 .31344 .94961 .32997 .94399 44 17 .26359 .96463 i. 28039 .95989 .29710 .95485 .31372 .94JI52 .33024 .94890 43 18 .26387 .96456 .28067 .95981 .29737 .95476; .31399!. 94943 .33051 .94380 42 19 !. 26415 .96448 .28095 .95972 .29765 .95467 .31427 .94933 .33079 .94370 41 20 .26443 .96440 .28123 .95964 j .29793 .95459 . 31454 j. 94924 .33100 .94361 40 21 .26471 .96433 .28150 .95956 .29821 .95450 !. 31482 .94915 .33134 .94&51 39 22 .26500 .96425 .28178 .95948 .29849 . 95441 i .31510 .94906 .83161 .94342 i 38 23 .26528 .96417 .28206 .95940 '.29876 .95433 .31537 .94897 .33189 .94832 1 37 24 .26556 .96410, .28234 .95931 i .29904 .95424 .31565 .94888 .33216 .94822 36 25 .26584 .96402 .28262 .95923 ; .29932 .95415 .31593;. 94878 .33244 .94313 35 26 1.26612 .96394 .28290 .95915 .29960 .95407 '' .31620 .94869 .33271 1.94303 34 27 .26640 .96386 .28318 .95907 .29987 .95398 i .31648 .94860 .33298 1.94293 33 28 .26668 .96379 .28346 .95898 j. 30015 .95389 1 .31675 .94851 .33326 ;. 94284 32 29 .26696 .96371 .28374 .95890 .30043 .95380 L31 703 .94842 .33353 .94274 31 30 .26724 .96363 .28402 .95882 ; .30071 .95372 .31730.94832 .33:381 .94264 ; 30 31 .26752 .96355 .28429 ,.95874 .30098 .95363 .31758 .94823 .33408 .94254 !29 32 .26780 .96347; .28457 .95865 .30126 .95354 .:!!>(> .94814 .33436 .94245 1 28 33 I .26808 .96340 .28485;. 95857 .301K4 .95:345 .31813 .94805 .33463 ! 942ft 27 34 ! .26836 .96.332 .28513;. 95849 .30182 .95337 .31841 .94795 .33490 .94223 26 35 j .26864 .9(5324 .285411.95841 .30209 .95328 I .31868 .94786! .33518 .94215 i 25 36 1 .26892 .96316 .28569i.95832 .30237 I. 95319 .31896 .94777 .33545 .94906 24 37 1 .26920 .96308 .28597:. 95824 .30265 1.95310 .31923 .94708 .33573 .94196 23 38 ! .26948 .98301 . 28625 L 9581 6 .30292 :. 95301 .31951 .94758 .33600 .94186) 22 39 1 .26976 .96293 .28652 .95807 .30320 .95293 .31970 .94749 i .33627 .94170 21 40 .27004 .96285; .28680 .95799 .30340 .95284 .32006 .94740 .33655 .94107 20 41 .27032 .96277 .28708 .95791 .30370 .95275 .32034 .94730 .33682 .94157 19 42 .270601.96269 .28736 .95782 .30403 .95266 .32061 .94721 .33710 .94147 18 43 .27088 .96261; .28764 .95774 .30431 .95257 .32089 .94712 .33737 .94187 i 17 44 .27116 .96253 .28792 .95766 i .30459 .95248 ! .32116 .94702 .33764 .94127 16 45 .27144 .96246 .28820 .95757 ; .30480 .95240 .32144 1.94693 .33792 .94118 15 46 .27172 .96238 .28847;. 95749 .30514 .95231 .32171 .94684 .3:3819 .94108 14 47 .27200 .96230 .288751.95740 .30542 .95222 .32199 .94674 .33846 .94098 13 48 .27228 .96222 .28903 .95732 .30570 .95213 .32227 .9460.5 .33874 .94088 i 12 49 .27256 .96214 .28931 .95724 .30597 .95204 .32254 .94656 33<>01 .94078 i 11 50 .27284 .96206 .28959 .95715 .30625 .95195 .32282 . 9464d p. 33929 .94068 10 51 .27312 .96198 .28987 .95707 .30653 . 95186 : .32309 .94637 .33956 .94058 9 52 .27340 .96190 .29015 .95698 .30680 . 95177 , .32337 .94027 .33983 .94049 8 53 .27368 .061P-2 .29042 .95690 .30708 .95168 .82364.94618 .34011 .94039 7 54 1.27396 .96174 .29070 .95681 .30736 .95159 .32392 .94609 .34038 .94029 6 55 .27424 .9616;! .29098 .95673 .30763 .95150 .324191.94599 .34065 .94019 5 56 .27452 .96158 .29126:. 95664 .30791 .95142 .:!,' 117 .94590 .34093 .94009 4 57 .27480 .96150 .29154 .95656 .30819 . 95133 .82474 .93580 .34120 .93999 3 58 .27508 .96142 .29182 .95647 .30846 .95124 .32502 .94571 .34147 . 93989 i 2 59 .27536 .96134 .29209 .95639 .30874 .95115 .32529 .94561 .34175 .93979 1 60 .27564 .96126 .2JW37 . 95630 i .30902 .95106 .325571.94552 .31202 .93969 Cosin; Sine ' Cosin Sine Cosin Sine" Cosin Sine Cosin Sine / 74 73 72 71 70 452 TABLE XXVII.-NATURAL SINES AND COSINES. 20 21 22 23 24 ' Sine Cosin Sine Cosin Sine Cosin Sine Cosin Sine Cosin ~o .34202 .93909 .86837 .93858 .37461 .92718 .39073 .92050 .40674 .91355 60 1 .34229 .'.mm .&5864 .93348 .37488 .92707! .39100 .920391 .40700 .91343 59 2 .34257 .513949 .35891; .93337' .37515 . 92697 i .39127 .92028 .40727 .91331 58 8 '.342S4 .93939 .35918 i 93327 .37512 . 92686 : .39153 .92016 .40753 .91319 5? 4 . 343 11 .93929 .35945 .93316 .37569 .92675 .39180 .92005! .40780 .91307 56 5 i. 34339 .93919 1 .35973 .93: 106 .37595 .92664 .39207 .91994 .40806 .91295 55 6 1.34366 .98909 .36000 .93295 .37622 .92653 .39234 .91982: .40833 .91283 54 7 .34393 .93899! .36027 .'.KkKi .37649 .92642 .39260 .91971 .40860 .91272 53 8 .34121 .93SS9 .30054 .93274 .37(57(5 .92631 .39287 .91 959 ; .40886 .91260 52 !) .344 IS .9:JS7i) .36081 .i)3264 .37703!. 92620 .39314 .91948 .40913 .91248 51 10 .344 75 .93869 .36108 .93253 .37730 .92609 .39341 .91936 .40939 .91236 50 11 .34503 .93859 .36135 .93243 .37757 '.92598 .39367 .91925 . 40966 i. 91 224 ! 49 12 .34530 .93849 .36162 .93232 .37784 .925871 .39394 .91914 .40992 .91212 48 13 .34551' .H:5S;J9 .3(il90 .1*3222 .37't!ll .92576! .39421 .91902 .41019 .91200 47 14 .34581 .93S29 .36217 .93211 !37S38 .92565 . 39448 i. 91 891 .41045 .91188 46 15 .34012 .1)381!) .36241 .93201! .37865 .92554 .39474 (.91879 .41072 .91176 45 1(5 .34039 .93S09 .36271 .93190 .37892 .92543 .39501 .91868 .41098 .91164 44 17 .34066^.93799 .36298 .93180 .37919 .92532 .39528 .91856 .41125 .91152 43 18 !. 34691 .93789 .36325 .93169 .37946 .92521 .39555 .91845 .41151 .91140 42 19 1. 34721 1.93775) i 3(5352 .93159 .37W3 .92510 .39581 .91833 .41178 .91128 41 20 1.34748 .93769 .36379 .93148 .37999 .92499 .39608 .91822 .41204 .91116 40 21 .34775 .93759 .36406 .93137 .38026!. 92488 .39635 .91810 .41231 .91104 39 22 .34803 .9374S .36434 .93127 .SK053 .92477 .39661 .91799 .41257 .91092 38 23 .34830 .93738 .36461 .93116! .86080 .98466 .39688 .91787 .41284 .91080 37 21 .34857 .93728 .36488 .93100 .381071.92455 .39715 .91775 .41310 .91068 36 25 .34884 .'.WIS .36515 . 93095 i .38134 .92444 .39741 .91764 .41337 .91056 35 26 .31912 .937'OS .36512 .!i:5i KJ .38161 .92432 .39768 .91752 .41363 .91044 34 27 .34939 .9369S .:-J(356!) .93074 .38188 .924.21 .39795 .91741 .41390 .91032 33 28 .34966 ..)36SS .36591; .9:3063 .38215 .92110 .39822 .91729 .41416 .91020 32 29 .34993 .931)77 .3(562-3 .93052 .:-.8241 .92399 .88848 .91718 .41443 .91008 31 30 .35021 .93667 .36650 .93042 .38268 .! 12388 .39875 .91706 .41469 .90996 30 31 .350 IS .9365? .36677 .93031 .38295 1.92377 .39902 .91694 .41496 .90984 29 82 .35075 .93047 .36704 .93020 .38322 .92366 .39928 .916^3 .41522 .90972 28 33 .35102 .93637 .36731 .93010 .38349 .92855 .39955 .91671 .41549 .90960 27 34 .35130 .9.J626 -3(i758 .92999 .38376 '.9234 3 .39982 .91660 .41575 .90948 26 35 .3515;- .93616 .3(5785 .92988 .38403J. 92332 .40008 .91648 .41602 .90936 95 36 .35184 .93(KI6 .36812 .9297S ! 88480 .92821 .40035 .91636 .41628 .90924 24 37 .;i-)2 11 93596 .36839 .92!)(i7 ..'58456 .92310 .40062 .91625 i .41655 .90911 23 38 .353391.93585 i 3158(57 .92956 .38488 1.92299 .40088 .91613 .41681 .90899 22 39 .352;iii .93575 .36894 .92945 .38510 .92287 .40115 .91601 .41707 .90887 21 40 .35293 .93565 .36921 .92935 .38537 .92276 .40141 .91590 .41734 .90875 20 1 41 .35320 .93555 .36948 .92924 . 38564 i.C2265 .40168 .91578 .41760 .90863 19 42 .35347 .93541 .36975 .92913 . 38591 i. 92254 .40195 .91566 .41787 .90851 18 43 .35375 .9:5534 .37002 .92902 .88617 .92243 .40221 .91555 .41813 .90839 17 44 .35402 .93524 .37029 .92892 .38644 .92231 .40248 .91543 .41840 .90826 16 45 .35429 .93514 .37056 .92881 ! .38671 .92220 .40275 .91531 .41866 .90814 15 46 .35456 .93503 .37083 .92870 .38698 .92209 .40301 .91519 | .41892 .90802 14 47 .35484 .93493 .37110- .92859 .387251.92198 .40328 .91508* .41919 .90790 13 48 .35511 .93483 .37137 .92849 .38752 .92186 j .40355 .91496 . 41945 i. 90778 12 43 .3553S .93472 .37164 .92838 .38778,.92175 .403811.91484 .41972 .90766 11 50 .35565 .93462 .37191 .92827 .38805 j. 931 04 .40408 '.91472 .41998 .90753 10 51 .35592 .93452 .37218 .92816 .38833 .92152 .40434 .91461 .42024 .90741 9 52 .35619 .93441 .37245 .92805 .38859 .92141! .40461 .91449 .42051 .90729 8 53 '. 35647 .93431 .37272 .92794 .38886 .92130 .40488 .91437 i .42077 .90717 7 54 .35I574 ..9342:) .37299 .92784 ! .38912:. 921 19 .40514.91425 .42104 .90704 6 55 .35701 .93410 .37326 .92773 .38939 .92107 .40541 .914141 .42130 .90692 5 56 .3572S .93400 .37353 . 92762 .889(515 .92096 .405(57 .91402 .42156 .90680 4 57 :. 35755 .933-!) ! 37380 .92751 : .88993 .92085 .40594 .91390 .42183 .90668 3 5S .3578-2 .9337! .37407 .92:40 .raNyo .(fc073 .40(521 .91378 .42209 .90655 2 59 .35S10 .933(58 .37434 .92729 .3;K> .53534 .84464 .54999 .83517 .56449 .82544i 38 23 .50578 .86266 .52076 .85370 .53558 .84448: .55024 .83501 .56473 .82528! 37 24 .50603 .86251 .52101 .85355 .53583 . 84433 i .550481.83485 .56497 .82511 36 25 .50628 .HI)-,':',?' .52126 .85340 .53607 .84417: .55072 .83469 .56521 .82495 35 26 .50654 .86222 .52151 .85325 .53632 .84402 .55097 .83453 .56545 . 82478 i 34 27 .50679 .86207 .52175 .85310 .53656 .84386 .55121 .83437[ .56569 .82462 33 28 .50704 .86192 .52200 .85294 .536811.84370 .55145 .834211 .56593 .82446! 32 29 .50729 .86178 .52225 .85279 .53705 .84355 .55169 .83405! .56617 .82429 31 30 .50754 .86163 .52250 .85264 .53730 .84339: .55194 .83389 .56641 .82413 30 31 .50779 .86148 .52275 .85249 .53754 .84324 .55218 .83373 .56665 .82396 29 32 .50804 .86133 .52299 .85234 .53779 .84303 .55242 .83356 .56689 .82380 28 33 .50829 .86119 .52324 .85218 .5:3804 .84292 .55266 .83340 .56713 .82363 27 34 .50854 .86104 .52349 .85203 .53828 .84277; .55291 .833241 .56736 .82347 26 35 .50879 .86089 .52374 .85188 .53853 .84261;. .55315 .83308 .56760 .82330 25 36 .50904 .86074 .52399 .85173 .53877 .84245 . 55339 i. 83292 .56784 .82314 24 37 .50929 .86059 .52423 .85157 .53902 .84230 .55363 .83276 .56808 .82297 23 38 .509541.86045 .52448 .85142 .539261.84214 .55388 .&3260 .56832 .82281 | 22 39 .50979 .860:30 .52473 .85127 .539511.84198 .55412 .83244 .56856 .82264 21 40 .51004 .86015 .52498 .85112 .53975 .84182; .55436 .83228 .56880 .82248 20 41 .51029 .86000 '.52522 .85096 .54000 .84167 .55460 .83212 .56904 .82231 19 42 .51054 .85985 .52547 .85081 .54024 .84151 .554841.831951 .56928 .822141 18 43 .51079 .85970 .52572 .85086 .54049 .84135 .55509 .83179 .56952 .82198 17 44 .511041.85956 .52597 .85051 .54073!. 84120 .5b533!.83163 .56976 .82181 16 45* .51 129:. 85941 .52621 .85035 .540971.84104 .55557 .83147 .57000 .82165 15 46 .51154 1.85926 .52646 .85020 .54122 .84088 .55581 .83131 .57024 .82148 14 47 .51179 .85911 .52671 .85005 .54146 .84072 .556051.83115 '.57047 .82132 13 48 .512041.85896 .52696 .84989 .54171 .84057 .55630 .83098: .57071 .82115 12 49 . 51229 . 85881 .52720 .84974 .54195 .84041 .55654 .83082! .57095 .82098 11 50 .51254i.85866 .52745 .84959 .54220 .84025 .55678 .83066; .57119 .82082 10 51 .51279 .85851 .52770 .84943 .54244 .84009! .55702 .83050 .57143 .82065 9 52 .513041.85836 .52794 .84928 .54269 .83994 .55726 .83034 .57167 .82048 8 53 .513291.85821 .52819 .84913 .54293J.83978 .55750 .83017; .57191 .82032 7 54 .51354 |.85806 .52844 .84897 .54317 .83902 .55775 .83001! .57215 .82015 6 55 .513791.85792 .52869 .84882 .54342 .83946 .55799 . 82985 j .57238 .81999 5 56 57 .514041.85777 .51429 .85762 .52893 .52918 .84866 .84851 .54366 .83930! .543911.83915 .55823 .5:5847 .82969! .82953 .57262 .57286 .81982 .81965 4 3 58 .51454 .85747 .52943 .84836 .54415 .83899 .55871 .82936 .57310 .81949 2 59 .51479 .85732 .52967 .84820 .544401.83883 1.55895 .82920; .57334 .81932 i 60 .51504 .85717 .52992 .84805 . 54464 | , 83867 .55919 .82904! .57358 .81915 6 Cosin Sine / Gosin| Sine Cosiu j Sine i Cosin Sine Cosin Sine / 59 58 57 1 56 55 _J 455 TABLE XXVII. NATURAL SINES AND COSINES. 35 36 | 37 38 1 39 Sine ! Cosin Sine Cosin Sine Cosin Sine Cosin Sine Cosin .57358 .81915! .58779 .80902! ,001 82 .79864"! .61566 .78801 .02932 .77715 60 1 .57381 .818!)!); .58802 .80885 .60205 .79846 .61589 .7-8783 .62955 .7701)6 59 2 .57405 .81882! .58826 .80867: .60228 .79829 .61612 .7-87(55 .62977 .77678 58 3 .57429 .81865!- .58849 .80850; .60251 .79811 .61635 .78747 .63000 .77660 57 4 .57453I.81848' .58873 .80833 .60274 .79793 .61658|.78729 .63022 .77641 56 5 .57477J.81 832; .58896 .80816 ! 60298 ! ! 79776 ! .61681 .78711 .63045 .77623 55 6 .575011.81815! .58920 .80799 .60321;. 79758; .61704 .78694 .68008 .77605 54' 7 .57524 1.81 798 : .58943 .80782 .60344!. 79741; .61726 .78676 .63090 .77586 53 8 .57548S.81782! .58967 .80765 .60367 .79723, .61749 .78658 .63113 .77568 52 9 .57572 .81765 .58990 .80748 .60390 .79706 .61772 .78640 .63135 .77550 51 10 .57596 .81748 .59014 .80730 .60414 .79688 .61795 .78622! .63158 .77531 50 11 .57619 .81731 .590371. 80713 ' .60437 .79671 .61818 .78604' .63180 .77513 40 12 .57(543 . 81714 |j. 59061 1.80696! .60460 .79653 .61841 .78586 .63203 .77494 48 13 .57667 .81698 1 .59084 .80679 .60483 .79635' .01864 .78568 .63225 .77476 47 14 .57691 .81681 .59108 .80662! .60506 .79618 .61887 .78550 .63248 .77458 46 15 .57715 .81664 .59131 .80(544 .60529 .79600 .61909 .78532 .03271 .77439 45 16 .57738 .81647 .59154 .80627 .605531.79583 .61932 .7-8514 .63293 .77421 44 17 .57762 .81631 .59178 .80610 .605761.79565 .61955 .78496 .6:3316 .77402 43 18 19 .57786 .81614 .57810!. 815971 .59201 .59225 .80593 .80576 .60599 |.79547 .60622 .79r>::0 .61978 .62001 .78478 .78460 .68838 .68861 .77-384 .77366 41 20 .57833L81580I .59248 .80558 .60645 .79512 .02024 .78442 .63383 .77347 40 21 .57857 .81563 .592721.80541 .0608 .79494 .6204(5 .78424! .63406 .77329 39 22 23 .57881J. 81 546 . 57-904 j. 81530 : .59295 .59318 .80524 .80507 .60691 .79477 ! .62069 .607141.79459! .62092 .78405 .78387' .634128 .63451 .77310 .77292 38 37 24 .579281.815131 .59342 .80489 .607381.79441! .62115 .78369 .63473 .77273 36 25 .57952!. 81496! .59365 .80472 .60761 . 79424 .62138 .78351! .63496 .77255 35 26 .679761.81479 .59:389 .80455 .60784 .79406 ; .62160 .78333 .63518 .77236 34 27 .57999 . 81462 ; .59412 .80438 .60807 .79388 .62183 .78315 .6:3540 .77218 33 28 .58023 .81445 .59436 .80420 .60830 .79371 .(52206 .78297; .63563 .77199 32 29 .58047J.81428! .59459 .80403 60853 .62229 .78279- .63585 .77181 31 30 .58070 .81412; .59482 .80-386 .60876 ; 79335 .62251 .78261 .63608 .77162 30 31 .58094 .81395 .59506 .80368 .60899 .79318 .62274 .78243 .63630 .77144 29 32 .58118 .8137'S .5952'.) .80351 .60923 .79300 .(5-J2S17 .78225 .68658 .77125 28 33 .58141 .81361 .59552 .80334 .60945 .7-1:282 .62320 .78206 .63675 .77107 27 34 .581651.81344! .595761.80316 .60968 .792641 .62342 .78188 .63698 .77088 26 35 .581 89:. 81 327 .59599 .80299 .60991 .792471 1.62365 1.7 81 70 .63720 .77070 25 36 .58212 .81310 .59622 8( )-.'82 .61015 .79229 .62388 .781 52 : .68748 .77051 24 37 . 58236 ; . 81 293 . 59646 | . 80264 .010:18 .71)211 .02411 .78134 .63765 .77033 23 38 .58260 .81276 .59669 .80247 .61061 .79193 .024:!:; .78116 .63787 .77014 22 39 .58283 .81259 .59693 .80230] .610841.79176 .62456 .78098 .03810 .70996 21 40 . 58307 j. 81242| .59716 .80212 .611071.79158 .62479 .78079; .'63832 .70977 20 41 .58330 .81225 .59739 .80195 1 .611301.79140 .62502 .78061 .63854 .76959 1!) 42 .58354:. 81308 .5!)7'(>:! .80178! .61153 .79122 .62524 .78043 .63877 .76940 18 43 .58378 .81191 .59786 .80160 .61176 .79105 .02517 .78025 .63899 .76921 17 44 .58401 .811741 .59809 .80143 .61199 .79C87 .62570 .78007 .63922 .70903 16 45 .58425 .81157! .59832 .80125 .61222 .79069 .62592 .77988 .63944 .70884 15 46 .58449 .81140- .59856!. 80108! .61245!. 79051 .62615 .77970 .63906 .76800 14 47' . 58472;. 811 23 i .59879 .80091! .61268!. 70033 .0.-JO.S8 .77952 .63989 .70847 13 48 .58496 .81106i .59902 .80073 .61291 1.79016 .02000 . 77934 .64011 .76828 12 49 .58519 .81089! .59926 .80056 .61314 .7'89!!S .02083 .77916 .64033 .76810 11 50 .68545 .81072| .59949 .80038; .61337 .78980 .62706 .77897 . 04056 .76791 10 51 .58567 .81055! .59972 , 80021 ! .613GO .7F962 .62728 .77879' .64078 .76772 9 52 .88590 .81038! .59995 80003 1 .6138:3 .7S944 .7!H5 . 77824 .64145 ! 76717 55 .58(561 .80987J .600651.79951 .61451 .7*S91 .62819 .77806 .64167 .76698 5 56 .58684 .80970 .(50089 .79934 .61474 .78873 .0:284 2 .77788 .64190 .76679 4 57 .58708 .80953 .60112 .79916 I .61497 .78855 .0:2804 .777011 .04212 .70661 3 58 .58731 .80936 .60135 .79899 .61520 .78837 .62887 .77751 .64234 .76642 2 59 .58755 .80919 .60158 .79881 .61543 .78819 .62!)09 .77733 .64256 .70623 1 60 .58779 .80902 : .60182 ! .79864 .615(56 .78801 ; .62932 .77715 .64279 .76604 f Cosin Sine Cosin Sine Cosin Sine Cosin Sine Cosin Sine f 54 53 52 51 i 50 450 TABLE XXVII. NATURAL SIXES AND COSINES. 40 1 41 42 43 ' 44 ' Sine ! Cosin Sine ! Cosin Sine i Cosin Sine Cosin Sine Cosin / ~o g l~Q .76604 .65606 .754711 .66913 .74314 768200 773135 .09466 .71934 60 i 61301 .76586 .65628 .75152 .06935 .74295 .68221 .73116 .69487L 71914 59 2 .64323 .76567 .65650 .75433 .66956 .74276 .68242 .73096 .69508 .71894 58 3 .64346 .70548 .65672 .75414 .66978 .74256 .68264 .73076 .69529 .71873 57 4 .643681.76530 .63M4 .75395 .66999 .74237 .68285 .73056 .C9549 i .71853 56 5 .61390 .76511 .65716 .75375 .67021|.74217 .68306 .73036 .69570 .71833 55 6 .01112 .76492 .65738 .75356 .67043 .74198 .68327 .73016 .69591 .71813 54 .61435 .70473 .65759 .75337 .67004!. 74178 .68349 .72996 .69012 .71792 53 8 .64457 .76455 .65781 .75318 .67086 .74159 .68370 .72976 .69033 .71772 52 9 .61479 .7043(5 .65803 .75299 .671071.74139 .68391 .72957 .69654 .71 752 51 10 .64501 j.76417 .65825 .75280 .67129 .74120 .68412 .72937 .69675 .71732 50 11 .64524 .76398 .65847 .75231 .67151 .74100 .68434 .72917 .69696 .71711 49 12 .64546 .76380 .65869 .75211 .671731.74080 .68455 .72897 .6971 7 '.71 691 48 13 .61568 .70361 .65891 .75222 .67194:. 74061 .68476 .72877 .69737U 71671 47 14 .64590 70H2 .65913 .75203 .67215 .74041 .68497 .72857 .697581.71650 46 15 .61612 .76323 .65935 .75184' .67237 .74022 .68518 .72837 . 69779 i. 71630 45 16 .64635 .76301 .65956 .75165 .67258 .74002 .68539 .72817 .69800 .71610 44 17 .64657 .70-283 .65378 .75146 .67280 .73983 .68561 .72797 .69821 .71590 43 18 .64679 . 7(5.2:57 .68330 .75128 .67301 .73963 .68582 .72777 .69842 .71569! 42 19 .61701 .76248 .63022 75107 .67323 .73944 .68603 .72757 . 69862 :. 71 549 | 41 20 .64723 .76229 .66044 75088 |. 67344 .73924 .68624 .72737 .69883 .71529 40 21 .64746 . 76210 '' .65088 75039 .67366 .73904 .68645 .72717 .69904 .71508 39 22 .64763 .76192 .63388 .75350; .67387 .73833 .68666 .72697 . 69925 !. 71488! 38 23 .61790 .76173; .63103 .75033 .67409 .7'3865 .63688 .72677 .69946k 71468 37 24 .64812 .761541 .63131 .75911 .67430 .73846 .68709 .7'2G57 .69966 .71447 36 25 .61S3t .76135 .66153 .74932 .67452 .73828 .68730 .72637 .69987 .71427 35 26 .04S50 .76116; .63175 .74973 s .67473 .73803 .68751 .72017 .70008 .71407 34 27 .64878 .76397 .66197 .74953 .67495 .73787 .68772 .72597 .70029 .71386 33 28 .64901 ,76078 .63218 .74934 .67516 .73767 .68793 .72577 .700491.71366 32 29 .64923 . 76359 ; .63240 .74915' .67538 .7'3747 .63814 .72557 .70070 .71345 31 30 .64945 .76041 .66262 . 74896 i .67559 .73728 .68835 .72537 .70091 .71325 30 31 .61967 .76022 .63284 . 74876 : .67580 .73708 .68857 .72517 .70112 .71305 29 32 .6 49 Si) .76003 .63338 .74857 .67633 .73033 .68878 .72497 70132 .71284 28 33 .65011 .75984! .63327 .74833 .67023 .73669 .68899 .724771 .70153 .71264 27 34 .05013 ! 75985 .63319 .74818 .67615 .73649 .68920 . 72457 ' .7017-4 .7124& 26 35 .05,155 .75946 .63371 .74793 .67636 .73023 .63941 .72437; .70195 .71223 35 36 .65077 .75927 .63333 .74783 .67688 .73610 .68962 .72417 .7C215 .71203 24 37 .65100 .75908 .015111 .74763 .67709 .73590 .68983 .72397 .70236 .71182 23 38 .05122 .75889 .684381.74741! .67730 73570 .69004 .72377 .70257 .711621 22 39 .65144 .75870 .63458 . 74722 : ,67752 .73551 .69025 .72357 .70277 .7114l| 21 40 .65166 .75851 .68480 .74703 j&"7773 .73531 .69046 .72337 .70298 .71121 20 41 .65188 .75832 .68501 .74683: .67795 .7X511 .69067 .72317 .70319 .71100 19 42 .65210 .75813 .63523 .7i!i:H .678161.73491 .69088 .72297 .703391.71080 18 43 .65232 .75791 .68545 .74644! .67837 .73472 .69109 .72277 .70360J.71059 17 44 .65251 .75775 .6-3563 .74625; .67859 .73452 .69130 .72257 .70381 .71039 16 45 .65276 .75756 .68588 .74603: .67880 .73433 .69151 .72236 .70401 .71019 15 4(i .65293 . 75738 .68610 .74533 .67901 .73413 .69172 .72216 .70422 .7'0998 14 47 .63330 48 .65342 .75719 .75700 .68832 .68(553 .745671 .74548' . 67923 i. 7.3393 .679441.73373 .69193 .69214 .72196 .721761 .704431.70978 13 .70463 .70957 12 4! .05301 .75680 .666751.74528 .67965 .73353 .69235 . 72156 j .70484i. 70937 11 50 .65336 .75661 .66697 .74509 .67987 .73333 .69256 .72136 .70505 .70916 10 51 .65108 .75642 .68718 .74489' .68008 .7asi4 .69277 .72116 .70525 .70R96 9 52 .65130 .75623 .66740 .74470 .68029 .73291 .69298 .720951 .70546 .70875! 8 .->:! .1)5152 .75604 .66762 .74451 .68051 .73274 . 69319 i. 72075i .70567.70355 7 54 i .65474 .75585 .66783 .74431 . 68072 !. 73254 .693401.72055 .705871.70834 6 55 .65131) .75566 .Or>Si)5 .74112 .6.8093 .7*234 .69301 .720a5 .70608 .70813 5 56 .65518 .75547 .60827 .74392 .68115 .73215 .69382 .72015 .70628 .70793 4 57 .05510 .75528 .60S 18 .74373 .68136 .73195 .694031.71995 .70649 '.70772 3 58 .05562 .75509 .66870 .74353 .68157 .73175 .694241.71^4 .70670 .70752 2 59 .65584 .75190 .66891 .74331 .68179 .73155 .69445 .71954 .70690 .70731 1 00 .6561)6 .75471 . 66913 . 74314 ', . 68200 . 73135 .09100 .71934 . 7071 1 ! . 70711 Cjsiu Sine Cosin Sine Cosin Sine Cosin Sine |j Cosin Sine / . 49 48 ii 47 1! 46 i 45 457 TABLE XHVIIL NATURAL TANGENTS AND COTANGENTS. 1 2 3 i Tang Cotang Tang Cotang Tang Cotang Tang Cotang .00000 Infinite. .01746 57.2900 .03492 28.6363 .05241 19.0811 CO I .00029 3437.75 .01775 56.3506 .03521 28.3994 | .05270 18.9755 59 2 .00058 1718.87 .01804 55.4415 .03550 28.1064 .05299 18.8711 58 3 .00087 1145.92 .01833 54.5613 .03579 27.9372 .05328 18.7078 '57 4 .00116 859.436 .01862 53.7086 .03609 27.7117 .05357 18.6656 56 5 .00145 687.549 .01891 52.8821 .03638 27.4899 .05387 18.5645 55 6 .00175 572.957 .01920 52.0807 .03667 27.2715 .05416 18.4645 54 7 .00204 491.106 .01949 51.3032 .03696 27.0566 .05445 18.3655 53 8 .00233 429.718 .01978 50.5485 ! .03725 26.8450 .05474 18.2677 52 g .00262 381.971 .02007 49.8157 .03754 26.6367 .05503 18.1708 51 10 .00201 343.774 .02036 49.1039 .03783' 26.4316 .05533 18.0750 50 11 .00320 312.521 .02066 48.4121 .03812 26.2296 .05562 17.9802 49 12 .00349 286.478 .02095 47.7395 .03842 26.0307 .05591 17.8863 48 18 .00378 2G4.441 .02124 47.0853 .03871 25.8348 .05620 17.7934 47 14 .00407 245.552 .02153 46.4489 .03900 25.6418 ! .05649 17.7015 1 46 15 1 .0043(5 229.182 .02182 45.8294 i .03929 25.4517 .05678 17.6106 i45 16; .00465 214.858 .02211 45.2261 j .03958 ! 25.2644 .05708 17.5205 44 17 .00495 202.219 .02240 44.6386 .03987 25.0798 i .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 | .040?'5 21.5418 .05824 17.1693 40 21 .00611 163.700 ! .02357 42.4335 1 .04104 24.3675 .05854 17.0837 09 22 .00640 156.259 .02386 41.9158 .04133 24.1957 .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 40.4a58 .04220 23.01)45 .05970 16.7496 35 26 .00756 132.219 ,02502 39.9655 .04250 23.5321 .05999 16.6681 34 27 .00785 127.321 .02531 39.5059 .04279 23.3718 .06029 16.5874 88 98 .00815 122.774 .02560 39.0568 .04308 23.2137 .'06058 16.5075 132 29 .00844 118.540 .02589 38.6177 .04337 23.0577 .00087 16.4283 31 30 .00873 114.589 .02619 38.1885 .04366 22.9038 .06116 16.3499 30 31 .00902 110.892 .02648 37.7686 .04395 22.7519 .06145 16.2722 29 32 .00931 107.426 .02677 37.3579 .04424 22.6020 .00 175 16.1952 28 :;:! .00960 104.171 .02706 36.9560 .04454 22.4541 .06204 16.1190 i27 34 .00989 101.107 .02735 36.5627 .04483 22.3081 .06233 16.0435 26 86 .01018 98.2179 .02764 36.1776 .04512 22.1640 .06262 15.9687 25 3l> .01047 95.4895 .02793 35.8006 .04541 2.0317 .06291 15.8945 24 87 .01076 92.9085 .02822 35.4313 .04570 21.8813 .06321 15.8211 2-} as .01105-' 90.4633 .02851 85.0695 .04599 21.7426 .06350 15.7483 2-1 89 .01135 88.1436 .02881 34.7151 .04628 21.6056 .06379 15.6762 21 40 .01164 85.9398 .02910 34.3678 .04658 21.4704 .06408 15.6048 ;20 41 .01193 83.8435 .02939 34.0273 .04687 21.3369 .06437 15.5340 !19 & .01222 81.8470 .02968 33.6935 .04716 21.2049 .06467 15.4638 IS 43 .01251 79.9434 .02997 33.3662 .04745 21.0747 .06496 15.3943 17 44 .01280 78.1263 .03026 33.0452 .04774 20 9460 .06525 15.3254 16 45 .01309 76.3900 .03055 32.7303 .04803 20.8188 .06554 15.2571 !15 M .C1338 74.7292 .03084 32.4213 .04833 20.6932 .06584 15.1893 14 47 .01367 73.1390 .03114 32.1181 .04862 20.5691 .06613 15.1222 13 IS .01396 71.6151 .03143 31.8205 .04891 20.4465 .06642 15.0557 12 49 .01425 70.1533 .03172 31.5284 .04920 20.3253 .08671 14.9898 11 50 .01455 68.7501 .03201 31.2416 .04949 20.2056 .06700 14.9244 10 51 1 .01484 67.4019 .03230 30.9599 .04978 20.0872 .06730 14.8596 9 52! .01513 66.1055 .03259 30.6833 05907 19.9702 .06759 14.7954 8 53! .01542 64.8580 .03288 30.4116 .05037 19.8546 .06788 14.7317 rv 54! .01571 63.6567 .03317 30.1446 .05066 19.7403 .06817 14.6685 6 55 1 .01600 62.4992 .03346 29.8823 i .05095 19.6273 .06847 14.6059 5 56! .01629 61.3829 .03376 29.6245 ! .05124 19.5156 .06876 14.5438 4 57; .01658 60.3058 1 .03405 29.3711 ! .05153 19.4051 .06905 14.4823 8 58! .01687 59.2659 .03434 29.1220 i .05182 19.2959 .06934 14.4212 g 59 .01716 58.2612 .03463 28.8771 .05212 19.1879 .06963 14.3607 1 60! .01746 57.2900 .03492 28.6363 .05241 19.0811 .06993 14.3007 t | Cotangj Tang Cotang Tang Cotang Tang j Cotang i Tang / ! 89 88 87 86 458 TABLE XXVIII.-NATURAL TANGENTS AND COTANGENTS. 4 i 5 ! 6 '! 7 Tang Cotang Tang i Cotang Tang Cotang Tang i Cotang .06993 14.3007 .08749 11.4301 .10510 9.51480 .12278 8.14435 60 1' .07022 14.2411 .08778 11.3919 .10540 | 9.48781 .12308 8.12481 59 2 .07051 14.1821 .08807 11.3540 .10569 9.46141 .12338 8.10536 58 8 .07080 14.1285 .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 11.2417 .10657 9.38307 .12426 8.04756 56 6 -07108 13.9507 .08925 11.2048 .10687 9.35724 .12456 8.02848 54 7| .07197 13.8940 .08954 11.1681 .10716 9.&3155 .12485 8.00948 58 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 n .07344 13,6174 .09101 10.9882 .10863 9.20516 .12033 7.91582 48 18 .07373 13.56:34 .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 10 .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 .07519 13.2996 .01)277 10.7797 .11040 9.05789 .12810 7.80622 42 19 .07'548 13.2480 .09306 10.7457 .11070 9.03379 J2840 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 2 .07636 13.0958 .09394 10.6450 .11158 8.96227 .12929 7.73480 38 23 .07666 13.0458 .09423 10.6118 .11187 8.93867 .12958 7.71715 37 21 .07695 12.9962 .09453 10.5789 .11217 8.91520 .12988 7.69957 36 25 .07724 12.9469 .09482 10.5462 |l .11246 8.89185 .13017 7.68208 36 2!) 3S .07929 12.6124 .09688 10.3224 .11452 8.73172 .18224 7.56176 2S 88 .07958 12.5660 .09717 10.2913 1 .11482 8.70931 .13254 7.54487 27 34 .07987 12.5199 .09746 10.2602 i .11511 8.C8701 .13284 7.52806 26 86 .08017 12.4742 .09776 10.2294 .11541 8.66482 .13313 7.51132 -.'5 3<5 .08046 12.4288 .09805 10.1988 : .11570 8.64275 .13343 7.49465 24 87 .08075 12.3838 .09834 10.1683 .11600 8.62078 .13372 7.47806 23 38 .08104 12.3390 .09864 10.1381 .11629 8.59893 .13402 7.46154 22 30 .08134 12.2946 .09893 10.1080 | .11659 8.57718 .13432 7.44509 21 40 ,08163 12.2505 ij .09923 10.0780 ; .11688 8.55555 .13461 7.42871 20 41 .08192 12.2067 !J .09952 10.0483 .11718 8.53402 .13491 7.41240 19 42 .08221 12.1632 .09981 10.0187 .11747 8.51259 .13521 7.39616 18 48 .08251 12.1201 .10011 9.98931 .11777 8.49128 .13550 7.37999 17 44 .08280 12.0772 .10040 9.96007 .11806 8.47007 .13580 .36389 16 45 .08309 12.0346 .10069 i 9.93101 .11836 8.44896 .13609 .34786 15 46 .08339 11.0023 .10099 9.90211 .11865 8.42795 .13639 .33190 14 47 .08368 j 11.9504 .10128 9.87338 .11895 8.40705 .13069 .31600 18 46 .08397 11.9087 ! .10158 9.84482 .11924 8.38625 .13698 .30018 12 49 .08427 11.8673 | .10187 9.81641 .11954 8.36555 .13728 .28442 11 50 .08456 11.8262 .10216 9.7'8817 .11983 8.34496 .137'58 .20873 10 51 .08485 11.7853 .10246 8.76009 .12013 8.32446 .13787 .25310 !) 52 .08514 11.7448 .10275 9.73217 .12042 8.30406 .13817 .23754 s 53 .08544 11.7045 .10305 9.70441 .12072 8.28376 .13846 .22204 7 54 .OS573 11.6645 .10334 9.67680 .12101 8.26355 .13876 .20661 6 55 .08602 11.6248 I .10363 9.64935 .12131 8.24345 .13906 .19125 5 56 .08632 11.5853 .10393 9.62205 i .12160 8.22344 .13935 .17594 4 57 .08661 11.5461 .10422 9.59490 l .12190 8.20a52 .13965 .16071 3 58! .08690 11.5072 .10452 9.56791 i .12219 8.18370 .13995 .14553 2 59 .08720 11.4685 | .10481 9.54106 i .12249 8.16398 .14024 .13042 1 60 .08749 11.4301 .10510 9.51436 | .12278 8.14435 .14054 .11537 / Cotang Tang Cotang Tang i Cotang Tang Cotang Tang f 85 84 i 83 82 459 TABLE XXVIII.-NATURAL TANGENTS AND COTANGENTS. 8 || 9 | 10 11 Tang Cotang | Tang Cotang Tang Cotang Tang Cotang .14054 7.11537 .15838 6.31376 .17633 5.07128 .19138 5.14455 60 1 .14084 7.10038 .15S08 6.30189 .17603 5.00165 .19468 5.13058 59 2 .1*113 7.08546 .15898 6.29007 .17093 5.05205 .19498 5.]2S(W 58 .14143 7.07059 .15928 6.27'829 .17723 5.64248 .19529 5.12009 57 4 .14173 7.05579 .15958 6.26655 .17753 5.63295 .19559 5.11279 56 5 .14202 7.04105 .15988 6.25486 .17783 5.62344 .19589 5.10490 55 a .14232 7.02637 .16017 6.24321 .17813 5.61397 .19019 5.09704 54 7 .14202 6.91174 .16047 6.23160 .17843 5.60452 .19049 5.08921 53 8 .14291 6.99718 .16077 6.22003 .17873 5.59511 .19080 5.08139 52 9 .14321 6.98268 .16107 6.20851 .17903 5.58573 .19710 5.07360 51 10 .14351 6.96823 .10137 6.19703 .17933 5.57638 .19740 5.06584 50 11 .14381 6.95385 .16167 6.18559 .17963 5.56706 .19770 5.05809 49 IS .14410 6.93952 f. 16196 6.17419 .17993 5.55777 .19801 5.05037 i48 18 .14440 6.92525 .16226 6.16283 .18023 5.54851 .19831 5.042ii7 47 14 .14470 6.91104 .16256 6.15151 .18038 5.53927 .19801 5.03499 40 LI .14499 6.89088 .16286 6.14023 .18083 5.53007 .19891 5.02734 45 If] .14529 6.88278 .16316 6.12899 i .18113 5.52090 .19921 5.01971 !44 17 .14559 6.86874 .16346 6.11779 .18143 5.51176 .19952 5.01210 43 IS .14588 6.85475 .16376 6.10664 .18173 5.50264 .19982 5.00451 42 19 .14618 6.84082 .16405 6.09552 .18203 5.49356" .20012 4.99695 :41 80 .14048 6.82694 .16435 6.08444 .18233 5.48451 .20042 4.98940 40 fcl .14078 6.81312 .16465 6.07340 .18203 5.47548 .20073 4.98188 39 22 .14707 6.79936 .16495 6.06240 .18293 5.46048 .20103 4.97438 38 23 .14737 6.78564 .16525 6.05143 .18323 5.45751 .20133 4.90090 37 24 .14707 6.77199 .16555 6.04051 i .18353 5.44857 .20104 4.95945 30 25 .14796 6.75838 .16585 6.02962 .18384 5.43900 .20194 4.95201 35 26 .14826 6.74483 .16615 6.01878 .18414 5.43077 .20224 4.94400 34 27 .14856 6.73133 .16645 6.00797 .18444 5.42192 .20254 4.93721 33 2H .14886 6.71789 .16674 5.99720 .18474 5.41309 .20285 4.92984 32 29 .14915 6.70450 .16704 5.98646 ; .18504 5.40429 .20315 4.92349 31 30 . 14945 6.69116 .16734 5.97576 .18534 5.39552 .20345 4.91516 30 31 .14975 6.67787 .16764 5.96510 .1S564 5.38677 .20376 4.90785 29 32 .15005 6.66463 .16794 5.95448 .18594 5.37805 .20406 4.90056 28 33 .15034 6.65144 .16824 5.94390 .18024 5.36936 .20436 4.89330 27 34 .15034 6.63831 .16854 5.93335 . 18654 5.36070 .20466 4.88605 26 85 .15094 6.62523 .16884 5.92283 .18084 5.35206 .20497 4.878S2 25 30 .15124 6.61219 .16914 5.91236 .18714 5.34345 .20527 4.87102 j24 37 .15153 6.59321 .16944 5.90191 .18745 5.33487 .20557 4.80444 (23 :s .15183 6.53627 .16974 5.89151 .18775 5.32631 .20588 4.85727 I 28 31) .15213 6,57339 .17034 5.83114 .18805 5.31778 .20618 4.85013 21 40 .15243 6.56055 .17033 5.87080 .18835 5.30928 .20048 4.84300 20 41 .15272 6.54777 .17063 5.86051 .18865 5.30080 .20079 4.83590 19 42 .15302 6.53503 .17093 5.85024 .18895 5.29235 .20709 4.82882 18 43 .15332 6.52254 .17123 5.84001 .38925 5.28393 .20739 4.82175 17 41 .15362 6.50970 .17153 5.82982 .18955 5.27553 .20770 4.81471 16 45 .15391 6.49710 .17183 5.81966 .18980 5.20715 .20800 4.80769 15 46 .15421 6.48456 .17213 5.80953 .19016 5.25880 .20830 4.80008 14 47 .15451 6.47203 .17243 5.79344 .19046 5.25048 .20861 4.79370 13 48 .15481 6.45961 .17273 5.78938 .19076 5.24218 .20891 4! 78673 12 48 .15511 6.44720 I .17303 5.77936 .19106 5.23391 .20921 4.77978 11 50 .15540 6.43484 I .17333 5.76937 .19136 5.22566 .20952 4.77280 10 51 .15570 6.42253 .17363 5.75941 .19166 5.21744 .20982 4.70595 9 58 . 15600 6.41026 .17393 5.74949 .19197 5.20925 .21013 4. 75 WO 8 53 .156:30 6.39804 .17423 5.73900 .19227 5.20107 .21043 4.75219 7 51 .15660 6.38587 .17453 5.72974 .19257 5.19293 .21073 4.74534 6 55 .15689 6.37374 .17483 5.71992 .19287 5.18480 .21104 4.73851 5 5(5 .15719 6.36165 1 .17513 5.71013 .19317 5.17671 .21134 4.73170 4 57 .15749 6.34961 .17543 5.70037 ! .19347 5.16863 .21104 4.72490 3 58 .15779 6.33701 .17573 5.69064 .19378 5.16058 .21195 4.71813 2 59 .15809 6.32500 .17603 5.68094 ! .19408 5.15250 .21225 4.71137 1 60 .15838 0.31375 || .J7G33 5.67128 i .10488 5.14465 .21256 4.70463 / Cotang Tang Cotang | Tang Cotang 1 Tang Cotang Tang / 81 i 80 i 79 i 78 460 TABLE XXVIII. -NATURAL TANGENTS AND COTANGENTS. 1 2 ! 1 30 1 40 1 5 / Tang Cotang Tang Cotang Tang Cotang Tang Cotang ~0 .21256 4.70463 .23087 4.33148 .24933 4.01078 .26795 3.73205 80 1 .21286 4.69791 .23117 4.32573 .2491 J4 4.00582 .26826 3.72771 56 2 .21316 4.69121 .23148 4.32001 .24995 4.00086 .26857 3.72338 56 3 .21347 4.68452 .23179 4.31430 .25026 3.99592 .26888 3.71907 57 4 .21377 4.67786 .23209 4.30860 .25056 3.99099 .26920 3.71476 66 5 .21408 4.67121 .23240 4.30291 .25087 3.98607 .26951 3.71046 65 (i .21438 4.66458 .23271 4.29724 .25118 3.98117 .26982 3.70616 54 7 .21469 4.65797 .23301 4.29159 .25149 3.97.627 .27013 3.70188 98 8 .21499 4.05138 .23332 4.28595 .25180 3.97139 .27044 3.69761 K g .215^9 4.64480 .23363 4.28032 .25211 3.96651 .27076 3.69335 51 H) .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 I!) 12 .21621 4.62518 .23455 4.26352 .25304 3.95196 .27169 3.68061 -IS 18 .21651 4.61868 .23485 4.25795 .25335 3.94713 .27201 3.67638 47 14 .21682 4.61219 .23516 4.25239 .25366 3.94232 .272432 3.67217 46 15 .21712 4.60572 .23547 4.24685 .25397 3.93751 .27263 3.66796 45 19 .21743 4.59927 .2357'8 4.24132 .25428 3.93271 .27294 3.66376 U 17 .21773 4.59283 .23608 4.23580 .25459 3.92793 .27320 3.65957 43 18 .21804 4.5S641 .23639 4.23030 .25490 3.92316 .27357 3.65538 43 19 .21&34 4.58001 .23670 4.22481 .25521 3.91839 .27388 3.65121 41 90 .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 !W 22 .219:25 4.56091 .23762 4.20842 .25614 3.90417 .27482 3.63874 88 2:5 .21956 4.55458 .23793 4.20298 .25645 3.89945 .27513 3.63461 87 34 .21986 4.54826 .23823 4.19756 .25676 3.89474 .27545 3.63048 80 25 .22017 4.54196 .23834 4.19215 .25707 3.89004 . 27576 3.62636 85 86 .22047 4.53568 .23885 4.18675 .25738 3.88536 27607 3.62224 81 87 .22078 4.52941 .23916 4.18137 .25769 3.88068 ! 27638 3.61814 83 28 .22108 4.52316 .23946 4.17600 .25800 3.87601 .27670 3.61405 32 2!) .22139 4.51693 .23977 4.17064 .25831 3.87136 .27701 3.60996 Ml 30 .22169 4.51071 .24008 4.16530 .25862 3.86671 .27732 3 60588 au 81 .22200 4.50451 .24039 4.15997 .25893 3.86208 .27764' 3.60181 20 32 .22231 4.49832 .24069 4.15465 .25924 3.85745 .27795 3.59775 28 83 .22261 4.49215 .24100 4.14934 .25955 3.85284 .27826 3.59370 27 84 .22292 4.48600 .24131 4.14405 .25986 3.84824 .27858 3.58966 20 35 .22322 4.47986 .24162 4.13877 .26017 3.84364 .27889 3.58562 25 86 .22353 4.47374 .24193 4.13350 .26048 3.83906 .27921 3.58160 21 87 .22383 4.46764 .24223 4.12825 26079 3.83449 .27952 3.57758 23 :w .22414 4.46155 .24254 4.12301 .26110 3.82992 .27983 3.57.357 22 39 .22444 4.45548 .24285 4.11778 .26141 3.82537 .28015 3.56957 :i 40 .22475 4.44942 .24316 4.11256 .26172 3.82083 .28046 3.56557 20 41 .22505 4.44338 .24347 4.10736 .26203 3.81630 .28077 3.56159 1!) !2 .22536 4.437'35 .24377 4.10216 .26235 3.81177 .28109 3.55761 is 43 .22567 4.43134 .24408 4.09899 .26266 3.80726 .28140 3.55364 17 It .22597 4.42534 .24439 4.09182 .26297 3.80276 .28172 3.54968 H; 43 .22G28 4.41936 .24470 4.08666 .26328 3.79827 .28203 3.54573 i:. !!i .22658 4.41340 .24501 4.08152 .26359 3.79378 .28234 3.54179 M 47 .22689 4.40745 .24532 4.07639 .26390 3.78931 .28566 8.33785 18 48 .22719 4.40152 .24562 4.07127 .26421 3.78485 .28297 3.53393 12 19 .22750 4.39560 .24593 4.06616 .26452 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 5-2 .22842 4.37793 .24686 4.05092 .26546 3.76709 .28423 3.51829 S 5:5 .22872 4.37207 .24717 4.04586 .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 6 56 .22984 4.35459 .24809 4.03076 .26670 3.74950 .28549 3.50279 4 57 .22995 4.34879 .24840 4.02574 .26701 3.74512 .28580 3.49894 8 58 .2:3026 4.34300 .24871 4.02074 .26733 3.74075 .28612 3.49509 2 59 .23056 4.33723 .24902 4.01576 .26764 3.73640 .28643 3.49125 1 80 .23087 4.33148 .24933 4.01078 .26795 3.73205 .28675 3.48741 Cotang Tang Cotang Tang Cotang Tang Cotang Tang 7 7 ' 7 8? ! 7 5 \ 7 4 461 TABLE XXVIIL NATURAL TANGENTS AND COTANGENTS. 16 17 18 ji 19 | Tang Cotang Tang Cotang Tang ! Cotang Tang Cotang | .28675 3.48741 .30573 3.27085 .32492 3.07768 .31433 2.90421 60 1 .28706 3.48359 .30605 3.26745 .32524 1 3.07464 .34465 j 2.90147 |59 8 .28738 3.47977 .30637 3.26406 .32556 i 3.07160 .34498 2.89873 J58 8 .28769 3.47596 .30669 3.26067 .32588 ! 3.06857 .34530 2.89600 57 4 .28800 3.47216 .30700 3.25729 .32621 ! 3.06554 .34563 2.89327 5(5 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 .30796 3.24719 .32717 | 3.05649 .34661 2.88511 88 8i .28927 3.45703 .30828 3.24383 .32749 i 3.05349 .34693 2.8824C 52 9! .28958 3.45327 .30860 3.24049 .32782 i 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 2.87430 4!) 12 .'29053 3.44202 .30955 3.23048 .32878 3.04152 .34824 2.87161 48 13 .29084 3.43829 .30987 3.22715 .32911 3.03854 .34856 2.86892 47 141 .29116 3.43456 .31019 3.22384 .32943 3.03556 .34889 2.86624 40 15 .29147 3.43084 .31051 3.22053 .32975 3.03260 .34922 2.86356 48 1(5 .29179 3.42713 .31083 3.21722 .33007 3.02963 .34954 2.86089 4* ir .29210 3.42343 .31115 3.21392 .33040 3.02667 .34987 2.85822 43 IS .29242 3.41973 .31147 3.21063 .33072 3.02372 .35020 1 2.85555 42 1!) .29274 3.41604 .31178 3.20734 .33104 3.02077 .35052 1 2.85289 41 20 .29305 3.41236 .31210 3.20406 .33136 3.01783 .35085 2.85023 40 21 .29337 3.40869 .31242 3.20079 .88169 3.01489 .35118 2.84758 99 2* .29368 3.40502 .31274 3.19752 .33201 3.01196 .35150 2.84494 38 M .29400 3.40136 .31306 3.19426 .33233 3.00903 .35183 2.84229 37 24 .29432 3.39771 .31338 3.19100 .33206 3.00611 .35216 2.83965 86 25 .29463 3.3^406 .31370 3.18775 .33298 3.00319 .35248 2.83702 135 86 .29495 3.39042 .31402 3.18451 .33330 3.00028 .35281 2.83439 34 27 .29526 3.38679 .31434 3.18127 .33363 2.99738 .35314 2.83176 133 88 .29558 3.38317 .31466 3.17804 .33395 2.99447 .35346 2.82914 i 32 2!) .29590 3.37955 .31498 3.17481 .33427 2.99158 .35379 2.82653 131 SO .29621 3.37594 .31530 3.17159 .33460 2.98868 .35412 2.82391 30 81 .29653 3.37234 .31562 3.16838 .33492 2.98580 .35445 2.82130 29 83 .29685 3.36875 .31594 3.16517 .33524 2.98292 -.35477 2.81870 28 33 .29716 3.36516 .31626 3.16197 .33557 2.98004 .35510 2.81610 27 34 .29748 3.36158 .31658 3.15877 ; .33589 2.97717 .35543 2.81350 2(5 85 .29780 3.35800 .31690 3.15558 ! .33621 2.97430 .35576 2.81091 2G 3(5 .29811 3.35443 .31722 3.15240 .33654 2.97144 .35608 2.80833 24 87 .29843 3.35087 .31754 3.14922 .33686 2.96858 .35641 2.80574 23 88 .29875 3.34732 .31786 3.14605 .33718 2.96573 .35674 2.80316 22 39 .29906 3.34377 .31818 3.14288 .33751 "2.96288 .35707 2.80059 21 40 .29938 3.34023 .31850 3.13972 .33783 2.96004 .35740 2.79802 20 41 .29970 3.33670 .31882 3.13656 .33816 2.95721 .35772 2.79545 19 42 .30001 3.33317 .31914 3.13341 .33848 2.95437 ! .35805 2.79289 18 43 .30033 3.32965 .31946 3.13027 .33881 2.95155 .35838 2.79033 17 44 .30065 3.32614 .31978 3.12713 .33913 2.9487'2 .35871 2.78778 16 45 .30097 3.32264 .32010 3.12400 ! .33945 2.94591 .35904 2.78523 16 46 .30128 3.31914 .32042 3.12087 .33978 2.94309 .35937 2.78269 14 47 .30160 3.31565 .32074 3.11775 .34010 2.94028 .85989 2.78014 13 48 .30192 3.31216 .32106 3.11464 .34043 2.93748 .36002 2.77761 12 49! .30224 3.30868 .32139 3.11153 ! .34075 2:98468 .36035 2.77507 11 50 .30255 3.30521 .32171 3.10843 .34108 2.93189 .36068 2.77'254 10 SI .30287 3.30174 .32203 3.10532 .34140 2.92910 .36101 2.77002 B 52 .30319 3.29829 .32235 3.10223 .34173 2.92632 .36134 2.76750 s 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 86 .30414 3.28795 .32331 3.09298 .34270 2.91799 .36232 2.75996 5 66 .30446 3.28452 .32363 3.08991 .34303 2.91523 .36265 2.75746 4 57 .30478 3.28109 .32396 3.08685 .34335 2.91246 .36298 2.75496 8 58 .30509 3.27767 .32428 3.08379 .34368 2.90971 .36331 2.75246 2 59 .30541 3.27426 .32460 3.08073 .34400 2.90696 .36364 2.74997 1 GO .30573 3.27085 .32492 3.07768 .34433 2.90421 .36397 2.74748 / Cotang Tang Cotang Tang j Cotang Tang Cotang | Tang / 73 72 71 II 70 462 TABLE XXVIIL NATURAL TANGENTS AND COTANGENTS. ! 20 ! 21 22 I! 23 | Tang 1 Cotang Tang Cotang Tang Cotang ! ! Tang Cotang 0, .30397 i 2.74748 ! .38386 2.005U9 .40403 2.47509 1 .42447 2.35585 00 1 .36430 2.74499 .38-120 2.60283 .40436 2.47302 .42482 2.35395 59 2 .36463 2.74251 .38453 2.60057 .40470 2.47095 .42516 2.35205 58 '.} .36496 2.74004 .38487 2.59831 .40504 2.46888 .42551 2.35015 57 4 .30530 2.73756 .38520 2.59606 .40538 2.46682 .42585 2.34825 56 6 .36562 2.73509 .38553 2.59381 .40572 2.46476 | .42619 2.34036 55 6 .36505 2.732G3 .38587 2.59156 .40606 2.46270 .42654 2.34447 51 7 .36628 2.73017 .38620 2.58932 .40640 2.46065 .42688 2.34258 53 8 .36601 2.72771 .38654 2.58708 .40674 2.45860 .42722 2.34069 5:2 9 .30001 2.7:2526 .38687 2.58484 .40707 2.45655 .42757 2.3J3881 51 10 | .36737 2.72281 .38721 2.58261 .40741 2.45451 .42791 2.33093 50 11 .36760 2.72036 .38754 2.58038 .40775 2.45246 .42826 2.33505 49 1-2 .86703 2.71792 .38787 2.57815 .40809 2.45043 .42860 2.33317 48 ta .36836 2.71518 .38821 2.57593 .40843 2.44839 .42894 2.33130 47 j i .36850 2.71305 .38854 2.57371 .40877 2.44036 .42929 2.32943 46 15 .38802 8.71062 .38888 2.57150 .40911 2.44433 .42963 2.32756 45 10 .36025 2.70819 .38921 2.50928 .40945 2.44230 .42998 2.32570 41 17 .36958 g. 70577 .38955 2.50707 .40979 2.44027 .43032 2.32383 43 18 .36901 2.7033r> .38988 2.56487 .41013 2.43825 .43067 2.32197 42 18 .37024 2.70094 .39022 2.50266 .41047 2.43623 .43101 2.32012 41 20 .37057 2.09853 ; .39055 2.50046 .41081 2.4:3422 .43136 2.31826 40 21 .37090 2.09612 1 .39089 2.55827 .41115 2.43220 .43170 2.31641 80 88 .37123 2.09371 i .39122 2.55608 .41149 2.43019 .43205 2.31456 88 23 .37157 2.09131 .39156 2.55389 .41183 2.42819 .43239 2.31271 37 31 .37190 2.68892 ! .39190 2.55170 .41217 2.42618 .43274 2.31086 31 i 85 .37223 S! 68653 i .39233 2.54952 .41251 2.42418 .43308 2.30902 35 20 .37256 2.08414 .39257 2.54734 .41285 2.42218 .43343 2.30718 84 37 .37-28!) 2.68175 1 .39290 2.54516 .41319 2.42019 .43378 2.30534 S3 2* .87823 2 (57! 137 ! .39334 2.54299 .41353 2.41819 .43412 2.30351 82 89 .37355 2.67700 i .39357 2.510H2 .41387 2.41620 .43447 2.30167 31 90 .37388 2.07462 .39391 2.53865 .41421 2.41421 .43481 2.29984 80 31 .37422 2.07225 .39425 2.53648 .41455 2.41223 .43516 2.29801 29 33 . 37455 2.C.09S'.) .39458 2.53432 .41490 2.41025 .43550 2.29619 2S 33 87488 2.60752 .30492 2.53217 .41521 2.40827 .43585 2.29437 27 34 .37531 2.66516 ! .39526 2.53001 .41558 2.40629 .43620 2.29254 26 35 .37554 2.66281 ! .39559 2.52780 .41592 2.40432 .43654 2.29073 85 31; .37588 2.66046 I .39593 2.52571 .41626 2.40235 .43689 2.28891 24 37 .37621 2.05811 i .39626 2.52357 .41600 2.40038 .43724 2.28710 33 38 .37654 2.65576 .39660 2.52142 .41694 2.39841 .43758 2.28528 82 39 .37687 2.65342 .39694 2.51929 .41728 2.39045 .43793 2.28348 121 40 .37730 2.65109' .39727 2.51715 .41763 2.39449 .43828 2.28167 90 41 .37754 2.64875 .39701 2.51502 .41797 2.39253 .43862 2.27987 in 42 .37787 2.64042 .39795 2.51289 .41831 2.39058 .43897 2.27806 18 18 .37820 2.04410 .39829 2.51070 .41805 2.38803 .43932 2.27626 IV 44 .37853 2.64177 .39802 2.50804 41899 2.38668 .43966 2.27447 Hi 45 j .37887 2.63945 .39896 2.50052 .41933 2.38473 .44001 2.27267 15 46 j .379.20 2.63714 .39930 2.50440 .41968 2.38279 .44030 2.27088 11 47 .37058 2.63483 .39903 2.50229 .42002 2.38084 .44071 2.20909 18 48 .37980 2.0*253 .39997 2.50018 .42036 2.37891 .44105 2.20730 12 1!) .38020 2.03021 .40031 2.49807 .4207'0 2.37697 .44140 2.26552 11 M .38053 2.62791 .40005 2.49597 .42105 2.87504 .44175 2.26374 10 51 .38086 2.62561 .40098 2.49336 .42139 2.37311 .44210 2.20196 9 52! .38130 2.633358 .40132 2.49177 1! .42173 2.37118 .44244 2.26018 8 53 .38153 2.62103 .40166 2.48907 .42207 2.30925 .44279 2.25840 7 54 .38186 2.61874 .40200 2.48758 .42242 2.36733 .44314 2.25663 6 55 .3S330 2.61646 .40234 2.48549 .42276 2.36541 .44349 2.25486 5 56 .38253 2.61418 .40267 2 48340 .42310 2.30349 .44384 2.25309 4 57 .38280 2.61190 .40301 2.48132 .42345 2.30158 .44418 2.25132 3 58 .3S3-20 2.60963 .40335 2.47924 .42379 2.35967 .44453 2.24956 3 .V.) .38353 2.60736 i .40369 2.47716 .42113 2.35776 .44488 2 24780 1 ui) .:-5S3si5 2. 0.c.o!) .40403 j 2.47509 .1-2!!;' 2,85585 . i 1523 2.24604 Cotang | Tang Cotang Tang Cotang Tang Cotang Tang f 69 ! 68 67 li 66 463 TABLE XXV1IL NATURAL TANGENTS AND COTANGENTS. 2 4 1 2 5 ! 2 6 2 r Tang Cotang Tang Cotang Tang Cotang ! Tang Cotang .44523 2.24(304 ' .46631 2.14451 .48773 2.05030 i .50953 1.96261 60 ! 1 .44558 2.24428 .46666 2.14288 .48809 2.04879 .50989 .96120 59 8 .44593 2.24252 .46702 2.14125 .48845 2.04728 .51026 .95979 58 a .44627 2.24077 .46737 2.13963 .48881 2.04577 .51063 .95838 57 4 .44662 2.23902 .46772 2.13801 .48917 2.04426 .51099 .95698 56 5 .44697 2. "23727 .46808 2.13639 .48953 2.04276 j .51136 .95557 55 6 .44732 2.23553 i .46843 2.1:3477 .48989 2.04125 .51173 .95417 54 7 .44767 2.23378 | .46879 2.13316 .49026 2.0397-5 .51209 .95277 53 ,s .44802 2.23204 i .46914 2.13154 .49062 2.03825 .51246 .95137 52 9 .44a37 2.23030 t .46950 2.12993 .49098 2.03675 .51283 .94997 51 10 .44872 2.22857 { .4(3985 2.12832 .49134 2.03526 .51319 .94858 50 11 .44907 2.22683 .47021 2.12671 .49170 2.03376 .51356 .94718 49 12 .44942 2.22510 .47056 2.12511 .49206 2.03227 .51393 .94579 48 13 .44977 2.22337 1 .47092 2.12350 .49242 2.03078 .51430 .94440 47 14 .45012 2.22164 .47128 2.12190 .49278 2.02929 .51467 .94301 46 15 .45047 2.21992 .47163 2.12030 .49315 2.02780 .51.503 .94162 45 18 .45082 2.21819 .47199 2.11871 .49a51 2.02631 .51540 .94023 44 J7 .45117 2.21647 .47234 2.11711 .49387 2.02483 .51577 .93885 43 IS .45152 2.21475 .47270 2.11552 .49423 2.02335 .51614 .93746 42 1!) .45187 2.21304 .47305 2.11392 .49459 2.02187- .51651 .93608 41 90 .45222 2.21132 .47341 2.11233 .49495 2.02039 .51688 .93470 40 81 .45257 2.20961 .47377 2.11075 .49532 2.01891 .51724 .93332 39 -2 .45292 2.20790 .47412 2.10916 .49568 2.01743 .51761 .93195 38 28 .45327 2.20619 '.47448 2.10758 .49604 2.01596 .51798 .93057 37 24 .45362 2.20449 .47483 2.10600 .49640 2.01449 .51835 .92920 36 2:, .45397 2.20278 .47519 2.10442 .49677 2.01302 .51872 .92782 35 26 .45432 2.20108 .47555 2.10284 .49713 2.01155 .51909 .92645 34 27' .45467 2.19938 .47590 2.10126 .49749 2.01008 .51946 .92508 33 2S .45502 2.19769 .47626 2.09969 .49786 2.00862 .51983 .92371 32 29 .45538 2.19599 .47662 2.09811 .49822 2.00715 .52020 1.92235 31 30 .45573 2.19430 .47698 2.09654 .49858 2.00569 .52057 1.92098 30 81 .45608 2.19261 .47733 2 09498 .49894 2.00423 .52094 1.91962 29 ;w .45643 2.19092 .47769 2. 09J341 .49931 2.00277 .52131 1.91826 28 83 .45678 2.18923 .47805 2.09184 .49967 2.00131 : 521 68 1.91690 27 84 .45713 2.187'55 .47840 2.09028 .50004 1.99986 .52205 1.91554 26 86 .45748 2.18587 .47876 2.08872 | .50040 1.99841 .52242 1.91418 25 86 .45784 2.18419 .47912 2.08716 .5007'6 1.99695 .52279 1.91282 24 87 .45819 2.18251 .47948 2.08560 .50113 .99550 .52316 1.91147 23 86 .45854 2.18084 .47984 2.03405 : .50149 .99406 .52353 1.91012 22 89 .45889 2.17916 .48019 2.08250 .50185 .99261 .52390 1.90876 21 40 .45924 2.17749 .48055 2.08094 .50222 .99116 .52427 1.90741 20 41 .45960 2.17582 .48091 2.07939 .50258 .98972 .52464 .9060? 19 48 .45995 2.17416 .48127 2.07785 .50295 .98828 .52501 .90472 18 43 .46030 2.17249 .48163 2.07630 .50331 .98684 .52538 .90337 17 44 .46065 2.17083 .48198 2.07476 | .50368 .9&540 .52575 .90203 16 45 .46101 2.16917 .482:34 2.07321 i .50404 1.98396 .52613 .90069 15 46 .46136 2.16751 .48270 2.07167 ! .50441 1.98253 .52650 .89935 li 47 .46171 2.165S5 .48306 2.07014 j .50477 1.98110 .52687 .8C801 18 4S .46206 2.16420 .48342 2.06860 i .50514 1.97966 .52724 .89667 12 4!) .46242 2.16255 .48378 2.06706 i .50550 1.97823 .52761 .89533 11 50 .46277 2.16090 .48414 2.06553 ; .50587 1.97681 .52798 1.89400 10 51 .46312 C. 15925 .48450 2.06400 1 .50623 1.97538 .52836 1.89266 9 52 .413348 2.15760 .48486 2.06247 .50660 1.97395 .52873 1.89133 8 53 .46:383 2.15596 .48521 2.06094 .50696 1.97253 .52910 1.89000 7 51 .46418 2.15432 .48557 2.05942 : .50733 1.97111 .52947 1.88867 6 55 .46454 2.15268 .48593 2.05790 i .50769 1.96969 .52985 1.88734 5 56 46489 2.15104 .48629 2.05637 i .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 2.05333 ' .50879 1.96544 .53096 1.88337 2 59 .46595 2.14614 .48737 2.05183 ; .50916 1.96402 .53134 1.88205 1 GO .46631 2.14451 .4877'3 2.05030 i .50953 1.96261 .53171 1.88073 Cotang Taiig Cotang Tang s Cotang Tang Cotang Tang / 5 i 6 4 6 ,3 i 6 2 464 TABLE XXVIII. NATURAL TANGENTS AND COTANGENTS. 28 II 29 |l 30 31 Tang- Corang Tang 1 Cotang Tang ;otangi| Tang | Cotang .53171 1.88073 .55431 1.804(15 .57735 1.73205 .60086 1.66428 60 1 .53208 1.87941 1 .55469 1.802H1 .57774 .73089 .60126 1.66318 59 2 .53-246 1.87809 ; .55507 1.80158 i .57813 .72973 .60165 1.6G209 58 3 1 .53283 1.87677 i .55545 1.80034 1! .57851 .72857 i .60205 1.66099 57 4 .53320 1.87.54(5 .55583 1.79911 .57890 .72741 .60245 1.65990 56 5 .53358 1.87415 || .55621 1.79788 : .57929 .72625 .60284 1.65881 55 6 .53395 1.87283 .55659 1.79665 .57968 .72509 .60324 1.65772 54 7 .53432 1.87152 .55697 1.79542 i .58007 .72393 ; .60364 1.65663 153 8 .53470 1.87021 .55736 1.79419 I .58046 .72278 .60403 1.65554 52 B .53507 1.86891 .55774 1.79296 .58085 .72163 .60443 1.65445 151 10 .53545 1.86760 | .55812 1.79174 .58124 .72047 .60483 1.65337 50 11 .53582 1.86630 .55850 1.79051 ! .58162 .71932 1 .60522 1.65228 49 12 .53620 1.86499 .55888 ! 78929 .58201 .71817 ! .60562 i 1.65120 IS 13 .53(557 1.86369 i .55<2G .78807 i .58240 .71702 : .60602 1.65011 47 14 .53094 1.86239 ' .55964 .78685 i 1 .58279 .71588 .60642 1.64903 46 15 .53732 1.86109 .56003 .78563 j; .58:318 .71473 .60681 1.64795 45 16 .537(59 1.85979 .56041 .78441 : .58857 .71358 i .60721 1.64687 44 17 .53807 1.85850 .56079 .78319 j .58396 .71244 i .60761 1.64579 43 18 .53844 1.85720 .56117 .78198 I! .58435 .71129 .60801 1.64471 42 1'.) .5.-JSS2 1.85591 .56156 .78077 |! .58474 .71015 .60841 1.64363 41 20 .53920 1.85462 .56194 .77955 |j .58513 .70901 .60881 1.64256 40 21 .53957 1.85383 .56232 .77834 i .58552 .707'87 .60921 1.64148 39 22 .53995 1.85204 .56270 .77713 .58591 .70673 .60960 1.64041 38 23 .54032 1.85075 .56309 .77592 .58631 .70560 .61000 1.63934 137 24 .5107(1 1.84946 .66847 .77471 i .58670 .70446 i .61040 1.63826 i36 25 .54107 1.84818 .56385 .77351 .58709 .70332 .61080 1.63719 35 26 .54145 1.K10S9 .56424 .772:30 .587'48 .70219 .61120 1.63612 34 87 .54183 ! 1.84561 .56462 .77110 .58787 .70106 .61160 1.63505 33 88 .5422.! 1.84433 .56501 .76990 .58826 .69992 .61200 1.63398 32 29 .54258 1.84805 .56539 .76869 .58865 .69879 .61240 1.63292 31 80 .54296 1,84177 .56577 .76749 ! .58905 .69766 .61280 1.63185 30 81 .54333 1.84049 ! .56616 .76629 .58944 .69653 ! .61320 1.63079 29 & .54371 1.8392-2 ' .56654 .76510 .58983 .69541 .61360 1.62972 28 83 .54409 1.83794 ! .56693 .7V>390 .59022 .69428 .61400 1.62866 27 84 .54446 1.83667 .56731 .76271 .59061 .69316 i .61440 1.62760 26 85 .54484 1.83540 ! .507(5!) .76151 l .59101 .69203 .61480 1.62654 25 36 .54522 1.83413 : .56808 i .76032 .59140 .69091 i .61520 1.62548 24 37 .515(5(1 1.83286 .50816 : .75913 .59179 .68979 |l .61561 1.62442 23 :>o .5i5!)r 1.83159 1 .56885 i .75794 .59218 .68866 .61601 1.62336 22 !5'.i .5<;;:!5 1.83033 .56923 .75075 .59258 .68754 .61641 1.62230 21 40 .54673 1.82906 .56962 .75556 .59297 .68643 .61681 1.62125 20 11 .54711 1.82780 j ! .57000 .75437 1 .50336 .68531 .61721 1.62019 19 42 .54748 1.82654 ! .57039. .75319 ; .59376 .68419 .61761 1.61914 18 43 .5J7NI5 1.82528 ! .57078 .75200 I .59415 .88308 ! .61801 1.61808 47 44 i .54K21 1.82402 .57116 .75082 .59454 .68196 .61842 1.617'03 16 45 .51802 1.82276 .57155 .74964 .59494 .68085 .61882 1.61598 15 ir, .54JHX) l.s-21.50 .57193 .74846 .59533 .67974 .61922 1.61493 14 47 .54938 1.82025 .57232 .74728 .59573 .67863 | .614:62 1.61388 13 48 .51975 1.81899 .57271 .74610 .59612 .67752 l| .62003 1.61283 12 49 .55013 1.81774 ' .57309 .74-192 .59651 .67641 .62043 1.61179 11 50 .55051 1.81649 .57318 .74375 | .59691 .67530 .62083 1.61074 10 51 .550.-'!) 1.81524 .57386 .74257 , .59730 .67419 .62124 1.60970 9 52 i .55127 1.813119 .57425 .74140 .59770 : .67309 .62164 1.60865 8 53 .55] (',5 1.81274 .57464 .74022 ' .59809 .67198 .62204 1.60761 7 51 .55203 1.81150 .57503 .73905 i .55)849 .67088 .62245 1.60657 6 55 . 5.Y;! II 1.81025 .57541 .73788 ; .59888 .66978 .62285 1.60553 5 56 .5527-9 1.80901 .57-580 .73671 .59928 .66867 .62325 1.60449 4- 57 .55317 1.80777 1 .57619 ! .73555 .59967 .66757 .62366 1.60345 3 58 i .55335 1.80653 1 .57657 j .73438 .60007 .66647 .62406 1.60241 2 59 i .55393 1.80529 .57696 .7*321 .60046 1. 665138 .62446 i 1.60137 1 GO 55431 1.80405 i| .57735 .73205 j .60086 1.66428 .62487 1.60033 t Cotang Tang Cotang , Tang ; Cotang j Tang Cotang Tang 61 60 59 !! 58 465 TABLE XXVIII. -NATURAL TANGENTS AND COTANGENTS. 32 33 34 35 Tang Cotang Tang Cotang | Tang Cotang Tang Cotang .02487 l.r,00:;:; .(141)11 1.53980 .07451 1.48250 .70021 1.42815 00 1 .025-27 1.51)930 .64983 1.53888 .67493 1.48103 .70004 1.42726 59 2 68568 1.59820 .05024 1.53791 .67530 1 48070 .70107 1.42038 58 3 .02608 1.59723 .05005 1.53693 .67678 1.47977 .70161 1.42550 ;>7 4 .02049 1.59020 .<;.-> 100 1,58695 .67620 1:47885 .70194 1.42462 HO 5 .G2G89 1.59517 .05148 1.53497 .07003 1.4771)2 .70288 1.42374 55 6 .02730 1.59114 .65189 1.53400 .07705 1.470W .70281 1.42286 54 7 .62770 1.59311 .05231 1.53302 .07748 1.47007 .70325 1.42198 63 8 .62811 1.59208 .65272 1.53205 .67790 1.47514 .70388 1.42110 52 9 .02852 1.59105 .05314 1.53107 .07832 1.47422 .70412 1.43022 51 10 .62892 1.59002 .G5&55 1.53010 .07875 1.47330 .70455 1.41934 50 11 .02933 1.53900 .05397 1.52913 .67917 1.47238 .70499 1.41847 49 12 .02973 1.58797 .05438 1.52810 .67960 1.47140 .70.542 1.41759 48 13 .03014 1.58695 .05480 1.52719 .68002 1.47053 .70586 1. 4H72 47 14! .03055 1.58593 .05521 1.52022 .68045 1.40902 .70029 1.41.-.81 40 15 1 .O30.)5 1.58490 .05503 1.52525 .08088 1 146870 .70073 1.4MD7 45 IB! .03130 1.58388 .05004 1.52429 .68130 1.407'78 .70717 1.41 409 44 17 .03177 1 .58280 .05041) 1.52332 .68173 1.40080 .70760 1.41322 43 18 .63217 1.58184 .G5G88 1.52235 .08215 1. 46595 > .70804 1.41235 [42 19 .63-258 1.58083 .05729 1.52139 .08258 1.40503 .70848 1.41148 ! 41 20 .03299 1.57981 .65771 1.52043 .08301 1.40411 .70891 1.41001 140 21 .63340 1.57879 .65813 1.51940 ' .08343 1.40320 .70935 1.40971 39 22 .63380 1.57778 .65854 1.51850 .08:380 1.46229 .70979 U40887 38 23 .03421 1.57'G76 \ .05890 1.51754 .08429 1.40137 .71023 1.40SOO 37 24 .03402 1.57575 .051)38 1.51058 ! .68471 1.40040 .71066 1.40714 30 25 .63503 1.57474 .05980 1.51502 ] .08514 1.45955 .71110 1.40627 85 26 .63544 1.57372 \ .00021 1.51400 ; .08557 1.40864 .71154 1.40540 '!4 27 .63584 1.57271 .06063 1.51370 .08600 1.45773 .71198 1.40454 33 28 .03025 1.57170 .66105 1.51275 .08042 K45682 .71242 1.40867 32 29 .63606 1.57009 .66147 1.51179 .08085 1.45592 .71285 1.40281 31 30 .63707 1.50909 .66189 1.51084 .08728 1.45501 .71329 1.40195 80 31 .63748 '1.50808 .66230 1.50988 .08771 1.45410 .71373 1.40109 29 32 .03789 1.50707 .60272 1.50893 .08814 1.45320 .71417 1.40022 28 33 .03830 1.50007 .00314 1.50797 .08857 1.45229 .71401 1.3!)!(; 27 34 .63871 1.56566 .00350 1,50702 .08900 1.45139 .71505 1.39850 2(i 35 .63912 1.56400 .00398 1.50007 .08942 1.45049 .71549 1. 397(51 2.-> 30 .G31>:>3 1.50300 .06440 1.50512 .08985 1.44958 .71593 1.39679 21 37 .03994 1.50265 .00482 1.50417 .09028 1.44808 .71637 1.39593 23 38 .04035 1.56165 .00524 1.50322 .09071 1.44778 .71681 1.39507 2.2 39 .64076 1.56065 1 .00500 1.50228 .09114 1.44688 .71725 1.39421 21 40 .64117 1.55966 1 .66008 1.50133 .69157 1.44598 .71769 1.39330 |20 41 .64158 1.55806 ! .66650 1.50038 .69200 1.44508 .71813 1.39250 19 42 .64199 1.55700 OOOH2 1.49944 .69243 1.44418 .71857 1.39105 IS 43T .04240 1.55606 .00734 1.49849 .69:286 1.44329 i .71901 1.31!0;i) 17 44 .04281 1.55567 .00776 1.49755 .69329 1.44239 .71946 1.38994 10 45 .04322 1. 55467 .66818 1.490.01 .09372 1.44149 .71990 1.389011 15 40 .0-1303 1.55368 .66800 1.49:,00 .09416 1.44060 .72034 1.388-,'t 11 47 .04404 1.55269 .00902 1.4947'2 .09459 1.43970 .72078 1.387'38 13 48 .04446 1.55170 .00944 . 1.40378 i .(;:):>( 12 1.43881 .73122 1.38053 12 1!) .ilUST 1.55071 .06980 1.49284 ' .0r;i:> 1.43792 .72107 1.38568 11 50 .04528 1.54972 .67028 1.49190 .69588 1.43703 .72211 1.38184 10 51 .04509 1.54873 .67071 1.49097 .09631 1.43614 .72255 1.38399 ( 52 .04610 1.54774 .07113 1.49003 .69675 1.43525 .72-21)11 1.38314 8 53 .04052 1.54075 .07155 1.48909 .69718 1.43430 .72344 1.38229 54 .64693 1.545rO .67197 1.48810 .69701 1.48847 .72388 1.381 !"> 6 55 .64734 1.54478 .07239 1.48722 .09804 1.43258 .72432 L38060 5 5b .64715 1.54879 ! 67282 1.48029 .09847 1.43109 .72477 1.37976 i 5? .64817 1.54281 .67324 1.48536 .09891 1.43080 .72521 j 56 .04358 1.54183 .07300 1.48442 .69934 1.42992 .72565 1.37807 2 5C ,64399 1.54085 ! .07409 1.483-19 .09977 1.42903 ! 72610 1.37722 -\ GC .04941 1.53980 .07451 1.48250 .70021 1.42815 .72654 1.87638 ' Cotang | Tang Cotang | Tang Cotang Tang C<>tang| Tang j ^ 57 ! 56 55 ii 54 TABLE XXVIII. -NATURAL TANGENTS AND COTANGENTS. 36 37 38 39 Tang Cotang 1 1 Tang Cotang Tang | Cotang Tang Cotang .72054 1.37038 .75355 1.82704 .78129 1.27994 .80978 1.2:3490 60 1 .72699 1.37554 .75401 1.32024 .78175 1.27917 .81027 1.23416 59 8 72748 1.37470 .75447 1.32514 .78222 1.27841 .81075 1.2:3343 58 8 ! 72788 1.37380 .75492 1.32464 .78209 1.27764 .81123 1.23270 57 4 .72832 1.37302 .75538 1.32384 i .7'8316 1.27688 .81171 1.23196 56 5 .72877 1.37218 .75584 1.32304 1 .78863 1.27611 .81220 1.23123 55 i; .72921 1.371:34 .75029 1.32224 .78410 1.27535 .81268 1.23050 54 1 .72966 1.37050 i .75075 1.32144 .78457 1.27458 .81316 1.22977 53 8 .73010 1.36907 .75721 1.32004 . 78504 1.27382 .81364 1.22904 52 91 .73055 1.86883 .75707 1.31984 ; .7'8551 1.27306 .81413 1.22831 51 10 .73100 1.30800 .75812 1.81904 .78588 1.27230 .81461 1.22758 50 11 .73144 1.30716 ! .75858 1.31825 .78645 1.27153 .81510 .22685 49 12 .73189 1.30(133 .75901 1.31745 .78692 1.27077 .81558 .22012 48 13: .73231 1.30549 .75950 1.31000 .787'39 1.27001 .81606 .22539 47 14 i .73-78 1 . 30400 .75990 1.31586 .78786 1.26925 .81655 .22407 46 1.-) .73:333 1.36383 .70042 1.31507 j .78834 1.26849 .81703 .22394 45 16 .73308 1.36300 .76088 1.31427 i .78881 1.26774 .81752 .22321 44 17 .73413 1.36217 .76134 1.31348 1 .78928 1.26698 .81800 .2224!) 43 18! .73457 1.36134 .70180 1.31269 1 .78975 1.26622 ' .81849 .22176 42 19 .7350-3 1.36051 .7022(1 1.31190 i .79022 1.26546 .81898 .22101 41 20 .73547 1.85968 .70272 1.31110 .79070 1.26471 .81946 .22031 40 21 .73502 1.35885 .76318 1.31031 1 .79117 1.26395 .81995 .21959 39 22 .73037 1.35802 .76304 1.30952 ' .79164 1.26319 .82044 .21880 38 23 .73081 1.35719 .76410 1.30873 .79212 1.20244 .82092 .21814 37 :.'! .73720 1.35037 .70450 1.30795 .79259 1.26169 .82141 .21742 36 25 .73771 1.35554 .70502 1.30716 .79306 1.26093 .82190 .21670 35 20 .73816 1.35472 .76548 1.30G37 .79354 1.26018 .82238 .21598 34 27 .73861 1.351389 .78594 1.30558 .79401 1.25943 .82287 .21526 33 28 .73! n 5 1.35307 .70040 1.30480 .79449 1.25867 .82336 .21454 32 2!) .78951 1.35224 .76086 1.30401 .79498 1.25792 .82385 .21382 31 80 .73990 1.35142 .76733 1.30323 .79544 1.25717 .824134 .21310 30 ::i .74041 1.35080 .76779 1.30244 .79591 1.25642 .S24&3 .21238 29 82 .74086 1.34978 .76825 1.30106 .79639 1.25507 .82531 .21166 28 88 .74131 1.34896 .70871 1.30087 .79086 1.25492 .82580 .21094 27 34 .74170 1.34814 .70918 1.30009 179784 1.25417 .82629 .21023 S3 35 .74221 1.34732 .76964 1.29931 .79781 1.25343 .82678 .20951 25 86 .74207 1.34650 .77010 1.29853 .79829 1.25268 .82727 .20879 24 31 .74312 1.34508 .77057 1.29775 .79877 1.25193 :82776 .20808 23 88 .74857 1.34487 .77103 1.29696 .79924 1.25118 .82825 .20786 22 39 .74402 1.34405 .77149 1.29618 .79972 1.25044 .82874 .20665 21 40 .74447 1.34323 .77196 1.29541 .80020 1.24969 .82923 .20593 20 n .74492 1.34242 77242 1.29463 .80067 1.24895 .82972 .20522 19 42 .74538 1.34160 ! 77289 1.29385 .80115 1.24820 .83022 .20451 18 43 .7458:] 1.34079 .77335 1.29307 .80163 1.24746 .83071 .20379 17 44 .7402S 1.33998 .77382 1.29229 .80211 1.24672 .83120 .20308 16 45 .74(174 1.83916 .77428 1.29152 .80258 1.24597 .83169 .20287 15 46 .7471!) 1. .33835 .77475 1.29074 .80306 1.24523 .83218 1.20100 1-! 17 .74764 1.33754 .77521 1.28997 .80354 1.24449 .83268 1.20095 13 48 .74N10 1.33073 .77568 1.28919 .80402 1.24375 .&3317 1.20024 12 4!) .718.V> 1.33592 .77015 1.28842 .80450 1.24301 .83366 1.19953 11 50 .74900 1.33511 .77601 1.28764 .80498 1.24227 .83415 1.19882 10 51 .74946 1.33430 .77708 1.28687 .80546 1.24153 .83465 .19811' 9 552 .74991 1.33349 . 77754 1.28610 .80594 1.24079 .88514 .19740 8 53 .75037 1.33868 .77801 1.28533 .80642 1.24005 .83564 .19009 54 .75082 1.33187 .77848 1.28456 .80690 1.23931 .83613 .19599 6 B5 .75128 1.33107 .77895 1 128379 .807:38 1.2:3858 .83602 .19528 5 6fl .75173 1.83096 .77941 1.28302 .80786 1.23784 .83712 .19457 4 r>; .7521'.) 1.32940 .77988 1.28225 .808:34 1.23710 .88781 .19387 3 58 .75864 1.328(15 .78035 1.28148 1 .80882 1.23637 .83811 .19316 2 59 .75310 1.32785 .79082 1.28071 .809:30 1.23503 .83860 .19246 1 (,. .75865 1.32704 .78129 1.27994 .80978 1.23490 .83910 .19175 ! Cbtang Tang Cotang Tang Cotang Tang Cotang j Tang 53 il 52 ii 51 1 50 1- 407 TABLE XXVIII. NATURAL TANGENTS AND COTANGENTS. 40 41 o p 42 o 43 o / Tang 1 Cotang Tang Cotang j Tang Cotang Tang Cotang .83910 1.19175 .80929 1.15037 ji .90040 1.11061 .9:5252 1.07237 60 1 .83960 1.19105 .86980 1 . 14969 .90093 i 1.10996 .93806 1.07174 59 8 .84009 1.19035 .87031 1.14902 .90146 I 1.10931 .93300 1.07112 58 8 .B4059 1.18964 .87082 1.14834 .90199 1.10867 .98415 L. 07049 57 4 .84108 1.18894 .87133 1.14767 ! 90251 1.10802 .93469 1.00987 56 :, .84158 1.18824 .87184 1.14699 .90:304 1.10737 .93524 1.00925 55 6 .84208 1.18754 .87236 1.11032 .90357 1.10672 .93578 1.06862 54 r i .84258 1.18684 .87287 1.14565 .90410 1.10607 .93033 1.06800 53 H .81307 1.18614 .87338 1.1 H'.IS .90463 1.10543 .93088 ; 1.067:58 52 9 .84357 1.18544 .87389 ! 1.11430 .90516 1.10478 .9:',743 i 1.00676 51 10 .84407 1.18474 .87441 1.14363 .90569 1.10414 i .93797 1.00613 50 11 .84457 1.18404 .87492 1.14296 .90621 1.10349 .93852 1.06551 49 (2 .84507 1.18334 .87543 1.14229 .90674 1.10285 ; .93906 1.06489 48 18 .84556 1.18264 .87595 1.14162 .90727 i 1.10220 .93961 1.06427 47 n .84606 1.18194 .87646 1.14095 .90781 ! 1.10150 .94010 1.00365 40 15 .84656 1.18125 .87698 1.1402S .90834 ! 1.10091 .94071 1.00303 45 16 .1706 1.18055 .87749 1.13961 .90887 1.10027 .94125 1.00241 44 17 .84756 1.17986 .87801 1.13894 .90940 1.0!H!03 .94180 1.06179 43 IS .84806 1.17916 .87852 i 1.13828 .90993 1.09S99- .94235 1.00117 42 1!) .84850 1.17846 .87904 1.13761 .91040 1 .09834 .94290 1.06056 j41 20 .84906 .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 1.13561 .91206 1.09042 .94455 1.05870 138 -.'.! .85057 1.17569 .88110 1.13494 .91259 1.09578 .94510 1.05809 37 23 .85107 1.17500 .88162 1.13428 .91313 1. 09514 .94565 1.05747 86 as .85157 1 . 17430 .88214 1.13361 .91366 1.09450 .94620 1.05685 85 26 .85207 1.17361 .88265 1.13295 .91419 1.09380 .94676 1.05624 31 27 .85257 1.17292 .88317 1.13223 .9147'3 1.09322 .94731 1.05562 33 28 .85308 1.17223 .88369 1.13162 .91526 1.09258 .94786 1.05501 32 26 .85358 1.17154 .88421 1.13096 .91580 1.09195 .94841 1.05489 31 30 .85408 1.17085 .88473 1.13029 .916*3 1.09131 .94896 i 1.05378 30 31 .85458 1.17016 .88684 1.12963 .91687 1.09067 .94952 1.05817 80 82 .85509 1.16947 .88576 1.12897 .91740 1.09003 .95007 i 1.05255 28 88 .85559 1.16878 .88628 1.12831 .91794 1.08940 .95002 1.05194 27 34 .85609 1.16809 .88680 1.12765 .91847 1.08876 .95118 1.05133 20 85 .85660 1 . 16741 .88732 1.12690 .91901 1.08813 .95173 1.05072 125 86 .85710 1.16672 .88784 1.12633 .91955 1.08749 .95>9 1.05010 24 37 .85761 1.16603 .88836 1.12567 .92008 1.08080 .95284 1.04949 23 88 .85811 1.165:35 .88888 1.12501 .92i;(>3 1.08622 .95840 1.04888 22 80 .85862 1.16466 .88940 1.12435 .92116 1.08559 .95395 1.04827 21 40 .85912 1.16398 .88992 1.12369 .92170 | 1.08496 .95451 1.04766 20 4\ .85968 1.16329 .89045 1.12303 .92224 1.08432 .95500 1.04705 19 43 .86014 1.16261 .89097 1.12238 .92277 1.088(51) .95562 1.04644 18 43 .86064 1.16192 .89149 1.12172 .92331 j 1.08300 .95618 1.04583 J17 44 .86115 I.llil24 .89201 1.12106 .92385 1.08243 .95673 1.04522 Il6 45 ,86166 1.16056 .89253 1.12041 .92139 1.08179 .95729 1.01401 15 4G .86216 1.15987 .89306 1.11975 .92493 1.08116 .95785 1.04401 1! 4; .86267 1.15919 .89358 1.11909 .92547 1.08053 .95841 1.04340 18 48 .86318 1.15851 .89410 1.11844 .92601 1 -07990 .95897 1.04279 12 49 ! .803C.S 1.15783 .89463 1 11778 .92055 1.07927 .95952 1.04218 11 50 .80419 1.15715 .89515 1.11713 .92709 1.07864 .96008 1.04158 10 51 .86470 1.15647 .89567 1.11648 .92703 1.07801 .96064 1.04097 53 .86521 1 . 15579 .89620 1.115B2 .92817 1.07738 .96120 1.04030 s 58 .86572 1.15511 -.89672 1.11517 .92P72 1.07070 .96176 1.03976 7 M .86623 1.15443 .81)725 1.11452 .92920 1.07613 .96232 1.03915 6 55 .86674 1.15375 .89777 1.11387 .92980 1.07550 .96288 1.03855 r, 56 .867'25 1.15308 .89830 1.11321 .93034 1.07'487 .96344 1.03791 4 57 .86776 1.15240 .89883 1.11256 .93088 1.07425 .96400 1.03734 3 :,s .86827 1.15172 .8993:. 1.11191 .93143 1.07-362 .96457 1.03674 2 50 .8687'8 1.15104 .89988 1.11126 .93197 1.07299 .96513 1.03013 1 60 .86929 1.15037 I .90040 1.11061 . 93252 1.07237 .96569 1.03553 / Cotang Tang s Cotang Tang j Cotang Tang Cotang Tang , 1 49 ! 48 47 i 46 468 TABLE XXVin. -NATURAL TANGENTS AND COTANGENTS. 44 , I * I r 44 Tang Cotang Tang Cotang Tang Cotang .96569 1.03553 60 20 .97700 1.02355 (40 40 .98843 1.01170 20 1 .96625 1.03493 59 21 .97756 .02295 39 | 41 .98901 1.01112 19 2 .966S1 1.03433 58 22 .97813 .02236 38 1 42 .98938 .01053 18 3 4 .96738 .98791 1.03372 1.03312 s 23 24 .97870 .97927 .02176 .02117 37 1 ! 43 36 II 44 .99016 .99073 .00994 .009% 17 16 5 .96850 1.C3252 55 25 .97984 .02057 35 45 .99131 : .90876 15 6 .96907 1.03192 54 26 .98041 .01998 34 i 46 .99189 .(-0818 14 '7 .96963 1.03132 63 27 .98098 .01939 33 47 .99247 .00759 13 8 .97020 1.03072 52 28 .98155 .01879 48 .99304 .00701 12 9 .97076 1.0J3012 51 29 .98213 .01820 81 49 .99362 .00642 11 10 .97133 1.02952 50 30 .98270 .01761 90 50 .99420 .00583 10 11 .97189 1.02892 49 51 .98327 .01702 01) 51 .99478 .00525 9 12 .97246 1.02832 48 82 .98381 .01642 28 52 .99536 .00467 8 18 .97302 1.02772 47 38 .98441 .01583 271 53 .99594 .00408 7 14 .97359 1.02713 46 34 .98499 .01524 26! 54 .99652 .00.350 6 15 .97416 1.02653 45 35 .98556 .01465 25 55 .99710 .00291 5 16 .97472 1.02593 44 36 .98613 .01406 24 56 .99768 : .00233 4 17 .97529 1.02533 43 37 .98671 1.01347 23 1 ! 57 .99826 .00175 3 18 .97586 1.02474 42 38 .98728 1.01288 22 58 .99884 : .00116 a 19 .97643 1.02414 41 39 .98786 1.01229 81 59 .99942 .00058 1 20 .97700 1.02355 40 40 .98843 1.01170 20 60 1.00000 .00000 o Cotang Tang Cotang Tang Cotang Tang 45 45 45 TABLE XXIX. NATURAL VERSED SINES AND EXTERNAL SECANTS. / 1 1 2 3 i Vers. Ex. sec. i Vers. Ex. sec. Vers. Ex. sec. Vers. Ex. sec. .00000 .00000 .00015 .00015 .00061 .0006T .00137" .00137 1 .00000 .00000 .00016 .00016 .000(32 .OODli2 .00139 00139 1 2 .00000 .00000 .00016 .00016 .00083 .00063 .00140 .00140 I 2 3 .00000 .00000 .00017 .00017 [00064 .00064 .00142 .00142 I 3 4 .00000 .00000 .00017 .00017 .00066 .00065 ' .00143 .00143 4 5 .00000 .o;jooo .00018 .00018 .mom; .000(><> .00145 .00145 5 6 .00000 .00000 .00018 .00018 ! .00067 .00067 .00146 .00147 6 7 .00000 .00000 .00019 .00019 .00068 .00068 ! .00148 -.00148 7 8 .00000 .00000 .00020 .a*)2o .00069 .00069 : .00150 .00150 8 9 .00000 .00000 .00020 .00020 .00070 .00070 .00151 .00151 9 10 .00000 .00000 .00021 .00021 .00071 .00072 I .00153 .00153 10 11 .00001 .00001 .00021 .00021 .00073 .00073 .00154 .00155 11 12 .00001 .00001 .00022 .00022 .00074 .00074 .00156 .00156 12 13 .00001 .00001 .00023 .00023 .00075 .00075 .00158 .00158 13 14 .00001 .00001 .00023 .00023 .00076 .00076 .00150 .00159 14 15 .00001 .00001 .00024 .00024 .00077 .00077 \00101 .00161 15 16 .00001 .00001 .00024 .00024 .00078 .00078 I .00162 .00163 16 17 .00001 .00001 .00025 .0002.) .00079 .00079 .001(54 .OOKi 4 17 18 .00001 .00001 .00026 .00026 .00081 .00081 .00106 .00166 18 19 .00002 .00002 .00026 .00026 .00082 .00082 .00168 .00168 19 20 .00002 .00002 .00027 .00027 .00083 .00083 .00109 .00169 20 21 .00002 .00002 .00028 .00028 .00084 .00084 .00171 .00171 21 22 .00002 .00002 .00023 .00028 .00085 .00085 .00173 .00173 22 23 .00002 .00002 .00029 .00029 .00037 .00087 ' .00174 .00175 23 24 .00002 .00002 .00030 .00030 .00088 .00088 .00176 .00176 24 25 .00003 .00003 .00031 .00031 .00089 .00089 ! .00178 .00178 25 26 .00003 .00003 .00031 .00031 .00090 .00090 ! .00179 .00180 26 27 .00003 .00003 .00032 .00032 .00091 .00091 .00181 .00182 27 28 .00003 .00003 .00033 .00033 .00093 .00093 | .00183 .00183 28 29 .00004 .00004 .00034 .00034 .00094 .00094 ! .00185 .00185 29 30 .00004 .00004 .00034 .00034 .00095 .00095 .00187 .00187 30 31 .00004 .00004 .00035 .00035 .00096 .00097 i .00188 .00189 31 32 .00004 .00004 .00036 .00036 .00098 .00098 | .00190 .00190 32 .33 .00005 .00005 .00037 .00037 .00099 .00099 ] .00192 .00192 33 34 .00005 .00005 .00037 .00037 .00100 .00100 .00194 .00194 34 35 .00005 .00005 .00038 .00038 .00102 .00102 .00196 .00196 35 36 .00005 .00005 .00039 .00039 .00103 .00103 ; .00197 .00198 36 37 .00006 .00006 .00040 .00040 .00104 .00104 i .00199 .00200 37 38 .00006 .00006 .00041 .00041 .00106 .00106 i .00201 .00201 38 39 .00006 .00006 .00041 .00041 .00107 .00107 : .00203 .00203 40 .00007- .00007 .00042 .00042 .00108 .00108 i .00205 .00205 40 41 .oooor .00007 .00043 .00043 .00110 .00110 .00207 .00207 41 42 .00007 .00007 .00044 .00044 .00111 .00111 .00208 .00209 42 43 .00008 .00008 .00045 .00045 .00112 .00113 .00210 .00211 43 44 .00008 .00008 .00046 .00046 .00114 .00114 .00212 .00213 44 45 .00009 .00009 .00047 .00047 .00115 .00115 .00214 .00215 45 46 .00009 .00009 .00048 .00048 .00117 .00117 .002W5 .00216 46 47 .00009 .00009 .00048 .00048 .00118 .00118 .00218 .00218 47 48 .00010 .00010 .00049 .00049 .00119 .00120 .00220 .00220 48 49 .00010 .00010 .00050 .00050 .00121 .00121 .00222 .00222 49 50 .00011 .00011 .00051 .00051 .00122 .00122 .00224 .00224 50 51 .00011 .00011 .00052 .00052 .00124 .00124 .00226 .00226 51 52 .00011 .00011 .00053 .00053 .00125 .00125 .00228 .00228 52 53 .00012 .00012 .000.54 .00054 .00127 .00127 i .00230 .00230 53 54 .00012 .00012 .00055 .00055 .00128 .00128 .00232 .00232 54 55 .00013 .00013 .00056 .00056 .00130 .001:30 .00234 .00234 55 56 .00013 .00013 .00057 .00057 .00131 .00131 .00236 .00236 56 57 .00014 .00014 .00058 .00058 .00133 .00133 .00238 .00238 57 58 .00014 .00014 .00059 .00059 .00134 .001:34 .00240 .00240 58 59 .00015 .00015 .00060 .00060 .00136 .00136 .00242 .00242 59 60 .00015 .00015 .00061 .00061 .00137 .00137 .00244 .00244 60 470 TABLE XXIX. NATURAL VERSED SINES AND EXTERNAL SECANTS. 40 5 6 7 Vers. Ex. sec. Vers. Ex. sec. Vers. Ex. sec. I Vers. Ex. sec. | .00244 .00244 .00381 .00382 .00548 .00551 .00745 .00751 1 .00246 .00846 .00383 .00385 .00551 .00554 .00749 .00755 1 2 .00248 .00248 .00386 .00387 .00554 .00557 .00752 .corns 2 3 .00250 .00250 .00388 .00390 .00557 .00560 .00756 .00762 3 4 .00253 .00252 ! 00891 .00392 .00560 .00563 .00760 .00765 4 5 .00254 .00-254 .00393 .00395 .00563 .00566 .00763 .00769 5 6 .00256 .00257 .00396 .00397 .00566 .00569 .00767 .00773 ! 6 .00258 .00259 .00398 .00400 .00569 .00573 .00770 .00776 7 8 .00260 .00261 .00401 .00403 .00572 .00576 .00774 00780 8 9 .00262 .00263 .00404 .00405 .00576 .00579 .00778 .00784 9 10 .00264 .00265 .00406 .00408 ; .00579 .00582 .00781 .00787 10 11 .00266 .00267 .00409 .00411 .00582 .00585 .00785 .007'91 11 12 .00269 .00269 .00412 .00413 .00585 .00588 .00789 .00795 12 13 .00271 .00271 .00414 .00416 .00588 .00592 .00792 .00799 13 14 .00273 .00274 .00417 .00419 .00591 .00595 .00796 .00802 14 15 .00275 .00276 .00420 .00421 .00994 . .00598 .00800 .0080(5 15 16 .00277 .00278 .00422 .00424 .00598 .00601 .00803 .00810 16 17 .00279 .00280 .00425 .00427 .00601 .00604 1 .00807 .00813 17 18 .00281 .00282 .00438 .00429 .00(504 .00(508 i .00811 .00817 18 19 .00284 .00284 .00430 .00432 .00607 .00611 I .00814 .00821 19 20 .00286 .00287 .00433 .00435 .00610 .00614 .00818 .00825 20 21 .00288 .00289 .00436 .00438 .00614 .00617 .00822 .00828 21 22 .00290 .00291 .00438 .00440 .00617 .00621 i .00825 .00832 22 23 .00293 .00293 .00441 .00443 .00620 .00624 .00829 .00836 23 24 .00295 .00296 .00114 .00446 .00623 .00627 .00833 .00840 24 25 .00297 .00298 .00447 .00449 .00626 .00630 .00837 .00844 25 26 .00299 .00300 .00449 .00451 .00630 ! 00684 \ .00840 1 .00848 26 27 .00301 .00302 .00452 .00454 .00633 .00637 .00844 i .00851 27 28 .00304 .00305 .00455 .00457 .00636 .00640 i .00848 .00855 28 2!) .00309 .00307 .00458 .00460 .00640 .00644 1 .00852 .ooa59 29 30 .00308 .00309 .00460 .00463 .00643 .00647 .00856 .00863 30 31 .00311 .00312 .00463 .00465 .00646 .00650 .00859 .00867 31 32 .00313 .00314 .00466 .00468 .00(549 .00654 .00863 .00871 32 33 .00315 .00316 .00469 .00471 .00653 .00657 .00867 .00875 33 34 .00317 .00318 .00472 .00474 .00656 .00660 .00871 .00878 34 86 .00320 .00321 .00474 .00-177 .00659 .00664 .00875 .008S2 35 88 .00322 .00323 .00477 .00480 .00668 .00667 .00878 .00886 36 37 .00324 .00326 .00480 .00482 .00666 .00671 .00882 .00890 37 38 .00327 .00328 .00483 .00485 .00669 .00674 .00886 .00894 38 39 .00329 .00330 .00486 .00488 .00673 .00677 .00890 .00898 39 40 .00X32 .00333 .00489 .00491 .00676 .00681 .00894 .00902 40 41 .00334 .00335 .00492 .00494 .00680 .00684 .00898 .00906 41 42 .00336 .00337 .00494 .00497 .00683 .00688 .00902 .00910 42 43 .00339 .00340 .00497 .00500 .00686 .00691 .00906 .00914 43 44 .00311 .00342 .00500 .00503 .00690 .00695 .00909 .00918 44 45 .00343 .00345 .00503 .00506 .00693 .00698 .00913 .00922 45 46 .00346 .00317 .00506 .00509 .00697 .00701 .00917 .00926 46 47 .00348 .00350 .00509 .00512 .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 .00356 .00357 .00518 .00521 .00710 .00715 .00933 .00942 50 51 .00358 .00359 .00521 .00524 .00714 .00719 .00937 .00946 51 52 .00361 .00362 .00524 .00527 .00717 .00722 .00941 .00950 52 53 .00363 .00364 ,00527 .00530 .00721 .00726 .00945 .00954 53 54 .00865 .00367 .00530 .00533 .00784 .00730 .00949 .00958 54 55 .00368 .00369 .00533 .00536 .00728 .00733 .00953 .00962 55 56 .00370 .00372 .00536 .00539 .00731 .00737 .00957 .00966 56 57 .00373 .00374 .00539 .00512 .00735 .00740 .00961 .00970 57 58 .00375 .00377 .00542 .00545 .00738 .00744 .00965 .00975 58 59 .00378 .00379 .00545 .00518 .00742 .00747 .00969 .00979 59 60 .00381 .00382 .00548 .00551 .00745 .00751 1 .00973 .00983 60 TABLE XXIX.-NATURAL VERSED SINES AND EXTERNAL SECANTS. II i i I / 8 9 10 11 Vers. Ex. sec. Vers. Ex. sec. : Vers. Ex. sec. | Vers. Ex. sec. ~ .00973 .oooas i .01231 .01247 .01519 .01543 j .01837 .01872 1 .00977 .00987 .01236 .01251 .01524 .01548 ' .01843 .01877 1 2 .00981 .00991 .01240 .01256 ; .01529 .03553 .01848 .01883 2 3 .00985 .00995 ! .01245 .012(51 .015:34 .01558 .01854 .01889 3 4 .00989 .00999 .01249 .01265 .01540 .01564 .01860 .01895 4 5 .00994 .01004 l .01254 .01270 .01545 .01569 .01805 .01901 5 6 .00998 .01008 .01259 .01275 !| .01550 .01574 .01871 .01906 6 7 .01002 .01012 ': .01263 .01279 .01555 .01579 1 .01876 .01912 7 8 .01006 .01016 .01268 .01284 !; .()15(iO .01585 .01882 .01918 8 9 .01010 .01020 .01272 .01289 i .Ojr.li.l .01590 .01888 .01924 9 10 .01014 .01024 ; .01277 .012-J4 .01570 .01595 .01893 .01930 10 11 .01018 .01029 ! .01282 .01298 .01575 .01601 ' .01809 .01936 11 12 .01022 .01033 .01286 .01303 .01580 .01606 .01904 .01941 12 13 .01027 .01037 .01291 .01308 .01586 .01611 .01910 .01947 13 14 .01031 .01041 .01296 .01313 .01591 .01616 .01916 .01953 14 15 .01035 .01046 .01300 .01318 .01596 .01622 .01921 .01959 15 16 .01039 .01050 .01305 .01323 .01601 .01627 .01927 .01965 16 17 .01043 .01054 .01310 .01327 .01606 .01633 .01933 .01971 17 18 .01047 .01059 .01314 .01332 .01612 .01638 .01939 .01977 18 19 .01052 .01063 .01319 .01337 .01617 .01643 .01944 .01983 19 20 .01056 .01067 .01324 .01342 .01622 .01649 .01950 .01989 20 21 .01080 .01071 .01329 .01346 .01627 .01654 .01956 .01995 21 22 .01034 .01076 .01333 .01351 .01632 .01659 .01961 .02001 22 23 .01069 .01030 .01338 .01356 .01638 .01665 .01%7 .02007 23 24 .01073 .01034 .01343 .01361 .01643 .01670 .01973 .02013 24 25 .01077 .01089 .01348 .01366 .01648 .01676 .01979 .0201'.) 25 26 .01081 .01093 : .01352 .01371 .01653 .01681 .01984 .02025 26 27 .01036 .01097 .01357 .01376 .01659 .01687 .01990 .02031 27 28 .01090 .01102 .01362 .01381 .01664 .01692 .01 !.5 .020:37 28 29 .01094 .01106 .01367 .01386 .01669 .01698 .02J02 .02043 29 30 .01098 .01111 j .01371 .01391 .01675 .01703 .02008 .02049 30 31 .01103 .01115 ! .01376 .01395 .01680 .01709 .02013 .02055 31 32 .01107 .01119 .01381 .01400 .01685 .01714 .02019 .02061 82 33 .01111 .01124 .01386 .01405 .01690 .01720 .02025 .02067 83 34 .01116 .01128 .01391 .01410 .011)96 .01725 .02031 .02073 34 35 .01120 .01133 ; .01396 .01415 .01701 .01731 .02037 .02079 35 36 .01124 .01137 i .01400 .01420 .01701) .01736 .02042 .02085 36 37 .01129 .01142 .01405 .01425 .01712 .01742 .02048 .02091 ! 37 38 .01133 .01146 .01410 .014:30 .01717 .01747 .02054 .02097 38 39 .01137 .01151 1 .01415 .01435 .01723 .01753 .02050 .02103 89 40 .01142 .01155 .01420 .01440 .01728 .01758 .0-2066 .02110 40 41 .01146 .01160 .01425 .01445 .01733 .01764 .02072 .02116 41 42 .01151 .01164 .01430 .01450 .01739 1 .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 .01755 .01786 .02095 .02140 45 46 .01168 .01182 .01449 ! .01471 .01760 .01792 .02101 .02146 46 47 .01173 .01187 .01451 i '01476 .01766 .01798 .02l<7 .02153 47 48 .01177 .01191 .01451) .01481 .01771 .01803 ! .02113 .02159 48 49 .01182 .01196 .01404 .0148(5 .01777 .OWK). .0:2119 .02165 49 50 .01186 .012CO .01461) .01491 .01782 .01815 .02125 .02171 50 i 51 .01191 .01205 .01474 .01496 .01788 .01820 .02131 .02178 51 52 .01195 .0120!) .01479 1 .01501 .01793 .01826 .02137 .02184 52 53 .01200 .01214 .01484 .01608 .'01799 .018 .02143 .02190 53 54 .01204 .01219 .01489 .01 5] -3 .01804 .01837 .02149 .02196 54 55 .01209 .01223 .01494 .1)1517 .01810 .01843 .02155 .02203 55 56 .01213 .01228 .014!)!) .01522 .01815 .01849 .02161 .02205) 56 57 .01218 .01233 .01504 .01527 .01821 .01854 ,02167 .02215 57 58 .01222 .01237 .015(11) .01532 .01826 .01860 .02173 .02221 58 59 .01227 .01242 .01514 .01537 .01882 .018(56 .02179 .02228 59 60 .01231 .01247 i .01519 .01543 .01837 .0|S7v! .U-MS5 -02234 60 472 TABLE XXIX. NATURAL VERSED SINES AND EXTERNAL SECANTS. / 12 13 14 15 / Yers. Ex. sec. 1 Vers. Ex. sec. II j Vers. Ex. sec. Vers. Ex. sec. .02185 .02234 .02563 .026:30 .02970 .03061 .03407 .03528 1 .02191 .02240 .02570 .02637 .02977 .OJ3069 .03415 .03536 1 2 .02197 .02247 .02576 .02644 .02985 .03076 .03422 .03544 2 3 .0220.3 .02253 .02583 .02651 .02992 .03084 ; .03430 .03552 3 4 .02210 .02259 .02589 .02658 .02999 .03091 .03438 .03560 4 5 .02216 .022(56 .02596 .02665 .03006 .03099 i .03445 .03568 5 6 .02222 .02272 .02602 .02672 .03013 .03106 ! .03453 .03576 6 7 .02228 .02279 .02609 .02679 .03020 .03114 .03460 .03584 7 8 .02234 .02285 .02616 .C2686 .03027 .03121 .03468 .03592 8 .02240 .02291 .02622 .02693 .03034 .03129 .03476 .03601 9 10 .02246 .02298 .02629 .02700 . .03041 .03137 .03483 .03609 10 11 .02252 .02304 .02635 .02707 .03048 .03144 .03491 .03617 11 12 .03268 .02311 .02642 .02714 .03055 .03152 , .03498 .03625 12 13 .02205 .02317 .02649 .02721 .03063 .03159 ' .03506 .03633 13 14 .02271 .02323 .02655 .02728 .03070 .03167 .03514 .03642 14 15 .02277 .023:30 .02662 .02735 .03077 .03175 .03521 .03650 15 16 .02283 .02:336 .02669 .02742 .03084 .03182 .03529 .03658 16 17 .02289 .02343 .02675 .02749 .03091 .03190 .0&537 .03666 17 18 .02295 .02:349 .02682 .02756 .03098 .03198 .03544 .03674 18 19 .02302 .02356 .02689 .02763 .03106 .03205 .03552 .03683 19 20 .02308 .02362 .02696 .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 .03236 .03583 .03716 23 24 .02333 .02388 .02722 .02799 .03142 .03244 .03590 .03724 24 25 .02339 .02395 .02729 .02806 .03149 .03251 .03598 .03732 25 2(3 .02345 .(2402 .02736 .02813 .03156 .03259 .03606 .03741 26 27 .02352 .02408 .02743 .02820 .03163 .03267 .03614 .03749 27 28 .02358 .02415 .02749 .02827 .03171 .03275 .03621 .03758 28 29 .02364 .02421 I .02756 .02834 .03178 .03282 .03629 .03766 29 30 .02370 .02428 i .02763 .02842 .03185 .03290 .03637 .03774 30 31 .02377 .02435 .02770 .02849 .03193 .03298 .03645 .03783 31 32 .02383 .02441 .02777 .02856 .03200 .03306 .03653 .03791 32 33 .02389 .02448 .02783 .02863 .03207 .03313 .03660 .03799 33 34 .02396 .02454 .02790 .02870 .03214 .03321 .03668 .03808 34 35 .02402 .02461 .02797 .02878 .03222 .03329 .03676 .03810 35 36 .02408 .02468 ! .02804 .02885 .03229 .03337 .03684 .03825 36 37 .02415 .02474 .02811 .02892 .03236 .03345 .03692 .03833 37 38 .02421 .02481 .02818 .02899 .03244 .03353 .03699 .03842 38 39 .02427 .02488 .02824 .02907 .03251 .03360 .03707 .03850 39 40 .02434 .02494 .02831 .02914 .03258 .03368 .03715 .03858 40 41 .02440 .02501 .02838 .02921 .03266 .03376 .03723 .03867 41 42 .02447 .02508 .02845 .02928 .03273 .03384 .03731 .03875 42 43 .02453 .02515 .02852 .02936 .03281 .03392 ,03739 .03884 43 44 .02459 .02521 .02859 .02943 .03288 .03400 .03747 .03892 44 45 .02466 .02528 .02866 .02950 .03295 .03408 .03754 .03901 45 46 .02472 .025:35 .02873 .02958 .03303 .03416 .03762 .03909 46 47 .02479 .02542 .02880 .02965 .03310 .03424 .03770 .03918 47 48 .02485 .02548 .02887 .02972 .03318 .0:3432 .03778 .03927 48 49 .02492 .02555 .02894 .02980 .03325 .03439 .03786 .03935 49 50 .0249S .025U2 ' .02900 .02987 .03333 .03447 .03794 .03944 50 51 .02504 .02509 .02907 .02994 .03340 .03455 .03802 .03952 51 ;V2 .02511 .02576 .02914 .03002 1 .0:3347 .03463 .03810 .03!)<;i 52 53 .02517 .02582 .02921 .03009 .0:3355 .03471 .03818 .03969 53 54 .02524 .02589 .02928 .03017 .03362 .03479 : .03826 .03978 54 55 .02530 .02596 .02935 .03024 .03370 .03487 .03834 .03987 55 56 .02537 .02003 .02942 .03032 .03377 .03495 .03842 .03995 56 57 .02543 .02610 .02949 .03039 .03385 .03503 .03a50 .04004 57 58 .02550 .02617 .02956 .03046 .0,3392 .03512 .08858 .04013 58 59 .02556 .02624 .02963 .03054 .03400 .03520 .03866 .04021 59 60 .02563 .02630 .02970 .03061 .03407 .03528 .03874 .04030 60 TABLE XXIX. NATURAL VERSED SINES AND EXTERNAL SECANTS. / K > 1-3 " i ** ; 1! )" / Vers. Ex. sec. ; Vers. Ex. sec. Vers. Ex. sec. Vers. Ex. sec. .03874 .04030 .04370 .04569 .04894 .05146 .05448 .05762 i .03882 .04039 .04378 .04578 .04903 .05156 .05458 .05773 1 2 .03890 .04047 .04387 .04588 i .04912 .05160 .05467 .05783 2 3 .03898 .04056 .04395 .04597 .04921 .05176 i .05477 .05794 3 4 .03906 .04065 i .04404 .04606 .04930 .05186 .06486 .05805 4 5 .03914 .0-1073 ; .04412 .04616 .04939 .05196 .054% .05815 5 6 .03922 .04082 j .04421 .04625 .04948 .05206 .05505 .0,1S20 6 .03930 .04091 .04429 .04635 ! .04957 .05216 .05515 .05830 7 8 .03938 .04100 .04438 .04644 i .04967 .05226 .055:24 .05847 8 9 .03946 .04108 .04446 .04653 .04976 .05230 .05534 .05868 9 10 .03954 .04117 .04455 .04663 .04985 .05246 .05543 .05809 10 11 .03963 .04126 ' .04464 .04672 .04994 .05256 .05553 .05879 11 12 .03971 .04135 .04472 .04082 .05003 .05266 .05502 .05890 12 13 .03979 .04144 i .04481 .04091 .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 .04003 .04170 ; .04507 .04719 .05039 .05307 .05(501 .05933 16 17 .04011 .04179 ! .04515 .04729 .05048 .05317 : .05010 .05944 17 18 .04019 .04188 I .04524 .04738 .05057 .05327 .05020 .05955 18 19 .04028 .04197 i .04533 .04748 .05007 .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 .04-^3 .04559 .04776 , .05094 .05367 .05058 .05998 22 23 .04060 .04232 .04567 .04786 1 .05103 .05378 1 .05008 .00009 23 24 .04069 .04241 .04576 .04795 .05112 .05388 .05078 .06020 24 25 .04077 .04250 i .04585 .04805 .05122 .05398 .05687 .06030 25 20 .04085 .04259 .04593 .04815 .05131 .05408 .05697 .06041 26 27 .04093 .04268 .04602 .04824 .05140 .05418 .05707 .00052 27 28 .04102 .04277 .04611 .04834 .05149 .05429 .05710 .00003 28 29 .04110 .04286 .04620 .04843 | .05158 .05439 i .05726 .06074 29 30 .04118 .04295 .04628 .04853 I .05168 .05449 .05736 .06085 30 31 .04126 .04304 .04637 .04863 .05177 .05460 .05746 .00096 31 32 .04135 .04313 .04646 .04872 .05186 .05470 .05755 .00107 32 33 .04143 .04322 .04655 .04882 .05195 .05480 .05705 .06118 33 34 .04151 .04:331 .04663 .04891 .05205 .05490 ' .05775 .00129 34 35 .04159 .04340 .04672 .04901 .05214 .05501 ' .05785 .00140 35 36 .04168 .04349 .04681 .04911 .05223 .05511 .05794 .00151 36 37 .04176 .04358 .04690 .04920 .05232 .05521 .05S04 .00102 37 38 .04184 .04367 .04699 .04930 .05242 .05532 .05814 .00173 38 39 .04193 .04376 | .04707 .04940 .05251 .(5542 .05824 .00184 39 40 .04201 .04385 .04716 .04951) .05260 .05552 .'05833 .00195 40 41 .04209 .04394 .04725 .04959 .05270 .05563 .05843 .06206 41 42 .04218 .04403 .04734 .04909 .05279 .05573 .05853 .06217 42 43 .04226 .04413 .04743 .04979 .05288 .05584 .05803 .08228 43 44 .04234 .04422 .04752 .04989 .05298 .05594 : .05873 .06239 44 45 .04243 .04431 .04760 .04998 .05307 .05604 .05HK2 .00250 45 46 .04251 .04440 .04769 .05008 .05316 .05615 .05892 .00061 46 47 .04260 .04449 .04778 .05018 i .05326 .05625 .05902 .00272 47 48 .04268 .04458 .04787 .05028 | .05335 .05630 .05912 .00283 48 49 .04276 .04468 ' .04796 .05038 i .05344 .05040 .05922 .OIW95 49 50 .04285 .04477 .04805 .05047 .05354 .05657 .05932 .06306 50 51 .04293 .04486 .04814 .05057 .05363 .05667 .05942 .06317 51 52 .04302 .04495 .04823 .05007 .05373 .05678 .05951 .06828 52 53 .04310 .04504 .04832 .05077 .05382 .05688 .059(51 !06389 53 54 .04319 .04514 .04841 .05087 .05391 .05699 .05971 .06350 54 55 .04327 .04523 .04850 .05097 .05401 .05709 .05981 .0(5362 55 56 .04336 .04532 .04858 .05107 i .05410 .05720 .05991 .00373 56 57 .04344 .04541 .04867 .05116 .05420 .05730 .0(5001 .06384 57 58 .04353 .04551 j .04876 .05126 .0.5429 .05741 .00011 .(Xvrfw 58 59 .04361 .04560 .04885 .05136 .05439 .05751 .00021 .00407 59 60 .04370 .04569 .04894 .05146 i .05448 .05762 .00031 .06418 60 474 TABLE XXIX. NATURAL VERSED SINES AND EXTERNAL SECANTS. 2( > 2] 1 22 22 / Vers. Ex. sec. Vers. Ex. sec. Vers. Ex. sec. Vers. Ex. sec. .06031 .06418 .06642 .07115 .07282 .07853 .07950 .08636 1 .06041 .06429 .06652 .07126 - .07293 .07866 .07961 .08649 1 2 .06051 .06440 .06663 .071:38 .07-303 .07879 .07972 .08663 2 3 .06061 .06452 .06673 .07150 .07314 .07892 .07984 .08676 3 4 .06071 .06463 .08684 .07162 | .07325 .07904 .07995 .08690 4 5 .06081 .06474 .06(394 .07174 .07336 .07917 .08006 .08703 5 6 .06091 .06486 .06705 .07186 .07347 .07930 .08018 .08717 6 7 .06101 .06497 .06715 .07199 i .07358 .07943 .08029 .08730 7 8 .06111 .06508 .06726 .07211 .07369 .07955 .08041 .08744 8 9 .06121 .065.20 .06736 .07*23 .07380 .07968 .08052 .08757 9 10 .00131 .08531 .06747 .07235 .07391 .07981 .08064 .08771 10 11 .06141 .06542 .06757 .07247 .07402 .07994 .08075 .08784 11 12 .00151 .08364 .06768 .07259 .07413 .08006 .08086 .08798 12 13 .06161 .06565 .06778 .07271 .07424 .08019 .08098 .08811 13 14 .06171 .06577 .00789 .07283 .07435 .08032 .08109 .08825 14 15 .06181 .06588 .06799 .07295 .07446 .08045 .08121 .08839 15 16 .06191 .08600 .06810 .07307 .07457 .08058 .08132 .08852 16 IT .06201 .06611 .06820 .07320 .07468 .08071 .08144 .08866 17 18 .06211 .08622 .06831 .07332 .07479 .08084 .08155 .08880 18 19 .06221 .05634 . 0(5841 .07344 .07490 .08097 .08167 .08893 19 20 .06231 .06645 .06852 .07356 .07501 .08109 .08178 .08907 20 21 .08341 .06657 .06863 .07368 .07512 .08122 .08190 .08921 21 22 .08252 .08668 .06873 .07380 .07523 .08135 .08201 .08934 22 23 .06262 .08680 .068,84 .07393 .07534 .08148 .08213 .08948 23 24 .06272 .08891 [08894 .07405 .07545 .08161 .08225 .08962 24 25 .06282 .087'03 .06005 .07417 .07556 .08174 .08236 .08975 25 26 .06292 .06715 .06916 .07429 .07568 .08187 .08248 .08989 26 27 .06302 .0372(i i .06926 .07442 .07579 .08200 .08259 .09003 27 28 .08312 .06738 i .06937 .07454 .07590 .08213 .08271 .09017 28 29 .06323 .06749 i .06948 .07466 i .07601 .08226 .08282 .09030 29 30 .06333 .03761 1 .06958 .07479 .07612 .08239 .08294 .09044 30 31 .06343 .03773 .06969 .07491 .07623 .08252 .08306 .09058 31 32 .06-553 .06784 .06980 .07503 .07'634 .08265 .08317 .09072 32 33 .06363 .08796 .06890 .07516 .07645 .08278 .08329 .09086 33 34 .015374 .0680? i ! .070J1 .07528 .07657 .08291 .08340 .09099 34 35 .06:384 .06819 i .07012 .07540 .07668 .08805 .08352 .09113 35 36 .06394 .06831 .07022 .Or553 .07679 .08318 .08364 .09127 36 37 .06104 .06843 .07U-J3 .07565 .07690 .08331 .08375 .09141 37 33 .06415 .08854 .07044 .07578 .07701 .08344 .08387 .09155 38 39 .06125 .06866 ! .07055 .07590 .07713 .08357 .08399 .09169 39 40 .06435 .06878 .07085 .07602 .07724 .08370 .08410 .09183 40 41 .06145 .06889 .07076 .07615 .077'35 .08383 .08422 .09197 41 42 .08456 .06901 i i .07087 .07627 .07746 .08397 .08484 .09211 42 43 .06466 .06913 i .07098 .07640 i .07757 .08410 .084,15 .09224 43 44 .06476 .06925 .07108 .07652 .07769 .08423 .08457 .09238 44 45 .06486 .06936 .07119 .07665 ! .07780 .08436 .08469 .09252 45 40 .08497 .08948 .07130 .07677 .07791 .03449 .08481 .09266 46 47 .05507 .08980 .07141 .07690 ! .07-802 .08463 .08492 .09280 47 48 .06517 .08972 .07151 .07702 .07814 .08476 .08504 .09294 48 49 .08528 .03984 ! .07162 .07715 .07825 .08489 .08516 .09308 49 50 .08538 .06995 : .07173 .07727 .07836 .08503 .08528 .09323 50 51 .06548 .07007 i .07184 .07740 .07848 .08516 .08539 .09&37 51 52 .08559 .o;-oi9 .07195 .07752 .07859 .08529 .08551 .09351 52 53 .06569 .07031 .07206 .07765 i .07'870 .08542 1 .08563 .093(55 53 54 .06580 .07043 .07216 .0777-8 .07881 .0855(5 .08575 .09379 54 55 .06590 .07055 .07227 .07790 .07893 ! 08569 .08586 .09393 55 56 .06600 .07087 .07238 .07803 i .07904 .08582 .08598 .09407 56 57 .06611 .07079 .07249 .07810 i .07915 .08596 .08610 .09421 57 58 .06621 .07091 .07260 .07828 .07927 .08609 .08622 .09435 58 59 .06632 .07103 .07271 .07841 .07938 .08623 i .08634 .09449 59 60 .06642 .07115 .07282 .07853 .0795.) .08636 ! .08645 .09464 60 475 TABLE XXIX. NATURAL VERSED SINES AND EXTERNAL SECANTS 24 25 26 27 Vers. Ex. sec. Vers. Ex. sec. Vers. Ex. sec. Vers. Ex. sec. .08645 .09464 I .09369 .10:338 .10121 .11260 .10899 .1SS33 1 .08657 .09478 I .09382 .10353- .10133 .11276 .10913 .12249 1 2 .08669 .09492 .09394 . 10368 .10146 .11292 .10926 .12206 2 3 .08681 .09506 .09406 .10:383 .10159 .11308 .10939 .12283 8 4 .08693 .09520 .09418 .10398 .10172 .11323 .10952 .12291) 4 5 .08705 .09535 .09431 .10413 .10184 .11339 .10965 .12316 5 6 .08717 .09549 .09443 .10428 .10197 .11355 .10979 .12333 6 7' .08728 .09563 .09455 .10443 .10210 .11371 .10992 .12349 7 8 .08740 .09577 .09468 .10458 .10223 .11387 .11005 .12366 8 9 .08752 .09592 .09480 .10473 .10236 .11403 .11019 .12383 9 10 .08764 .09606 .09493 .10488 .10248 .11419 j .11032 .12400 10 11 .08776 .09620 .09505 .10503 .10201 .11435 .11045 .12416 11 12 .08788 .09635 .09517 .10518 .10274 .11451 .11058 .12433 12 13 .08800 .09649 .09530 .10533 .10287 .11467 .11072 .12450 13 14 .08812 .09663 .09542 .10549 .10300 .11483 .11085 .12467 M 15 .08824 .09678 .09554 .10564 .10313 .11499^ .11098 .12484 15 16 .08836 .09692 .09567 .10579 .10326 .11515 .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 .09735 .09604 .10625 .10364 .11563 .11152 .12551 19 20 .08884 .09750 .09617 .10640 .10377 .11579 .11165 .12568 20 21 .08896 .09764 .09629 .10655 .10390 .11595 .11178 .12585 21 22 .08908 .09779 .09642 .10670 .10403 .11611 .11192 .12602 22 23 .08920 .09793 .09654 .10686 .10416 .11627 .11205 .12619 23 24 .08932 .09808 .09666 .10701 .10429 .11643 .11218 .12636 24 25 .08944 .09822 .09679 .10716 .10442 .11659 .11232 .12653 25 26 .08956 .09837 .09691 .10731 .10455 .11675 .11245 .12670 20 27 .08968 .09851 ij .097'04 .10747 .10468 .11691 .11259 .12687 27 28 .03980 .09866 .09716- .10762 .10481 .11708 .11272 .12704 28 29 .08992 .09880 ! .09729 .10777 .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 .1177'2 i .11326 12772 32 33 .09040 .09939 .0977'9 .10839 .10546 .11789 .11339 .12789 33 34 .09052 .09953 .09792 .10854 .10559 .11805 .11353 .12807 34 35 .09064 .09968 .09804 .10870 i .10572 .11821 .11366 .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 .10932 .10624 .11886 .11420 .12892 30 40 .C9125 .10041 .09867 .10947 .10037 .11903 .11434 .12910 40 41 .09137 .10055 .09880 .10963 .10650 .11919 .11447 .12927 41 42 .09149 .10071 .09892 .10978 .10663 . 1 1936 .11461 .12944 42 43 .09161 .10085 .09905 .10994 .10676 .11952 .11474 .12961 43 44 .09174 .10100 .09918 .11009 .10689 .11968 i .11488 .12979 44 45 .09186 .10115 .09930 .11025 .10703 .11985 .11501 .12996 45 4(J .09198 .10130 .09943 .11041 .10715 .12001 ; .11515 .13013 46 47 .09210 .10144 .09955 .11056 .10728 .12018 .11528 .13031 47 48 .09232 .10159 .09968 .11072 .10741 .12034 .11542 .13048 48 49 .09234 .10174 .09981 .11087 .1075o .12051 .11558 .13065 49 50 .09247 .10189 .09993 .11103 .10768 .12067 .11509 .13083 50 51 .09259 .10204 .10006 .11119 .10781 .12084 .11583 .13100 51 52 .09271 .10218 .10019 .11134 .10794 .12100 .11596 .13117 52 53 .09283 .10233 .10032 .11150 .10807 .12117 .11610 .13135 53 54 .09296 .10248 .10044 .11166 .10820 .12133 .11623 .13152 54 55 .09308 .10263 .10057 .11181 .10833 .12150 .11637 .13170 55 56 .09320 .10278 .10070 .11197 .10847 .12166 .11651 .13187 56 57 .09332 .10293 .10082 .11213 .10860 .12183 .11664 .13205 57 58 .09345 .10308 .10095 .11229 .10873 .12199 .11678 .132-22 58 59 .09357 .10323 .10108 .11244 .10886 .12216 .11692 .13240 59 60 .09369 .10:338 .10121 .11260 ! .10899 . 12233 .11705 .13257 60 476 TABLE XXIX. NATURAL VERSED SINES AND EXTERNAL SECANTS. 2 5 2 9= 3( > 3] L / Vers. Ex. sec. Vers. Ex. sec. Vers. Ex. sec. Vers. Ex. sec. .11705 .13257 .12538 .14335 .13397 .15470 .14283 .16663 1 .11719 .13275 .12552 .14354 .13412 .15489 .14298 .16684 1 2 .11733 .13292 .12566 .14372 .13427 .15509 .14313 .16704 2 3 .11746 .13310 .12580 .14391 .13441 .15528 .14328 .16725 3 4 .11760 .1.3327 .12595 .14409 .13450 . 15548 .14343 .16745 4 5 .11774 .13345 .12609 .14428 .13470 .15567 .14358 .16766 5 .11787 .13362 .12623 .14446 .13485 .15587 .14373 .16786 6 7 .11801 .13380 .12637 .14465 .13499 .15606 .14388 ~. 16800 7 8 .11815 .13398 .12651 .14483 .13514 .15626 .14403 .16827 8 9 .11828 .13415 .12665 .14502 .13529 .15645 .14418 .16848 9 10 .11842 .13-133 .12679 .14521 .13543 .15665 .14433 .16868 10 It .11856 .13451 .12694 .14539 .13558 .15684 .14449 .16889 11 12 .11870 .13468 .12708. .14558 .13573 .15704 .14464 .16909 12 13 .11883 .13486 .12722 .14576 .13587 .15724 .14479 .16930 13 14 .11897 .13504 .12736 .14595 .13602 .15743 .14494 .16950 14 15 .11911 .13521 .12750 .14014 .13616 .15763 .14509 .16971 15 16 .11925 .13539 .12765 .14632 .13631 .15782 .14524 .16992 16 17 .11938 .13557 .12779 .14651 .13646 .15802 .14539 .17012 17 18 .11952 .13575 .12793 .14670 .13600 .15822 .14554 .17033 18 19 .11966 .13593 .12807 .14689 .13675 .15841 .14569 .17054 19 20 .11980 .13610 i .12822 .14707 i .13690 .15861 .14584 .17075 20 2t .1*394 .13628 .12836 .14726 .13705 .15881 .14599 .17095 21 22 .12007 .13046 j .12850 .14745 .13719 .15901 .14615 .17116 22 23 .12021 .13664 .12864 .14764 .13734 .15920 .14630 .17137 23 24 .12035 .13682 j .12879 .14782 .13749 .15940 .14645 .17158 24 25 .12049 .13700 ! .12893 .14801 .13763 .15960 .14600 .17178 25 20 .12063 .13718 ! .12907 .14820 .13778 .15980 .14675 .17199 26 27 .12077 .137:35 i .12921 .14839 .13793 .16000 .14090 .17220 27 28 .12091 .13753 i .12936 .14858 .13808 .16019 .14706 .17241 28 29 .12101 .13771 1 .12950 .14877 .13822 .16039 .14721 .17262 29 30 .12118 .13789 .12904 .14896 .13837 .16059 .14726 .17283 30 31 .12132 .13807 .12979 .14914 .13852 .16079 .14751 .17304 31 32 .12146 .13825 j .12993 .14933 ! .13867 .16099 .14766 .17325 32 33 .12160 .13843 .13007 .14952 ! .13881 .16119 .14782 .17346 33 34 .12174 .13861 ) .13022 . 14971 .13896 .16139 .14797 .17367 34 35 .12188 .13879 j .13030 .14990 .13911 .16159 .14812 .17388 35 36 .12202 .13897 .13051 .15003 .13926 .10179 .14827 .17409 36 37 .12216 .13916 .13005 .15028 .13941 .16199 .14843 .17430 37 38 .12230 .13934 .13079 .15047 .13955 .16219 .14858 .17451 38 39 .12244 .13952 .13094 .15006 .13970 .16239 .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 48 .12299 .14024 .13151 .15143 .14030 .16319 .14034 .17556 43 44 .12313 .14042 .13106 .15102 .14044 .16339 .14949 .17577 44 45 .12327 .14061 .13180 .15181 .14059 .16359 .14965 .17598 45 46 .12341 .14079 .13195 .15200 .14074 .10380 .14980 .17620 40 47 .12355 .14097 .13209 .15219 .14089 .10400 .14995 .17641 47 48 .12369 .14115 .13223 .15239 .14104 .16420 .15011 .17662 48 49 .12383 .14134 .13238 .15258 .14119 .10440 .15026 .17683 40 50 .12397 .14152 .13252 .15277 .14134 .16460 .15041 .17704 50 51 .12411 .14170 .13267 .15296 .14149 .16481 .15057 .17726 51 52 .12425 .14188 .13281 .15315 .14104 .16501 .15072 .17747 52 63 .12439 .14207 .13296 .15335 .14179 .16521 .15087 .17768 53 54 .12454 .14225 .13310 .15354 .14194 .16541 .15103 .17790 54 55 .124C8 .14243 .13325 .15373 .14208 .16562 .15118 .17811 55 56 .12482 .14262 .13339 .15393 .14223 .16582 .15134 .17832 56 57 .12496 .14280 . 13354 .15412 .14238 .16602 .15149 .17854 57 58 .12510 .14299 .13368 .15431 .14253 .16623 .15164 .17875 58 59 .12524 .14317 .13383 .15451 .14208 .10043 .15180 .17896 59 60 .12538 .14335 .13397 .15470 .14283 .10663 .15195 .17918 60 477. TABLE XXIX. NATURAL VERSED SINES AND EXTERNAL SECANTS. / 3 2 3 3 ! s 4 ! * 5 Vers. Ex. sec. Vers. Ex. sec. Vers. Ex. sec. Vers. Ex. sec. .15195 .17918 .161*3 .19230 .17096 .20022 .18085 .22077 1 .15211 .17939 .10149 .19259 .17113 .20045 .18101 .22102 1 2 .15220 .17901 .10105 .19281 .17129 .20009 .18118 .2:2127 2 3 .15241 .17982 .10181 .19304 .17145 .20093 .18135 .22152 3 4 .15257 .18004 .16196 .19327 .17101 .20717 .18152 .22177 4 5 .15272 .18025 .16212 .19349 '. 17178 .20740 .18168 .22202 5 G .15288 .18047 .16228 .19372 .17194 .20764 .18185 .22227 6 7 .15303 .18008 .16244 .19394 .17210 .20788 .18202 .22252 7 8 .15319 .18090 .16200 .19417 .17227 .20812 .18218 .22277 8 9 .15334 .18111 .16276 .19440 .17243 .20836 .18235 .22302 9 10 .15350 .18133 .16292 .19463 .17259 .20859 .18252 .22327 10 11 .15365 .18155 .16308 .19485 .17276 .20883 .18269 .22352 11 12 .15381 .18176 .16324 .19508 .17292 .20907 .183S6 .28377 13 13 .15396 .18198 .16340 .19531 .17808 .20931 .18302 .22402 13 14 .15412 .18220 .16355 .19554 .17325 .20955 .18319 .22428 14 15 .15427 .18241 .16371 .19576 .17341 .20979 .18336 .22453 15 16 .15443 .18263 .16387 .19599 .17357 .21003 .18353 .22478 16 17 .15458 .18285 .16403 .19622 .17374 .21027 .ia369 .22503 17 18 .15474 .18307 .16419 .19645 : .17390 .21051 ! .18386 .225-28 18 19 .15489 .18328 .16435 .19668 .17407 .21075 j .18403 .22554 19 20 .15505 .18350 .16451 .19691 : .17423 .21099 1 .18420 .22579 20 21 .15520 .18372 .16467 .19713 .17439 .21123 .18437 .22604 21 22 . 1653(5 .18394 .16483 .19736 .17456 .21147 ! .18454 .22029 22 23 .15552 .18416 .16499 .19759 .17472 .21171 .18470 .22055 23 24 .15567 .18437 .16515 .19782 .17489 .21195 .18487 .22680 24 25 .15583 .18459 .16531 .19805 .17505 .21220 .18504 .22706 25 26 .15598 .18481 .16547 .19828 ! .17522 .21244 .18521 .237-31 26 27 .15614 .1S503 .16563 .19851 .17538 .01268 .m538 .22756 27 28 .15630 .18525 .16579 .19874 .17554 .21292 .18555 .22782 28 29 .15645 .18547 .16595 .19897 .17571 .21310 .18572 .22F07 29 30 .15661 .18569 .16611 .19920 .17587 .21341 .18588 .22833 30 31 .15676 .18591 .16627 .19944 ! .17604 .21365 .18005 .22858 31 32 .15(392 .18613 .16644 .19907 .17020 .213S9 .18022 .22S84 32 33 .15708 .18635 .16660 .19990 1 .17037 .21414 .18639 .22009 33 34 .15723 .18657 .16676 .20013 | .17053 .21438 .18056 .22935 34 35 . 15739 .18679 .16692 .20036 .17670 .21402 .18673 .22960 35 36 .1575.") .18701 .16708 .20059 .17086 .21487 .18090 .22986 36 37 .15770 .18723 .16724 .20083 .17703 .21511 .18707 .23012 37 38 .15780 .18745 ,16740 .20106 .17719 .21535 .18724 .23037 38 39 .15803 .18767 .16756 .20129 .17736 .21500 .18741 .23003 39 40 .15818 .18790 .16772 .20152 .17752 .31584 .18758 .23089 40 41 .15833 .18812 .16788 .20176 .17709 .21609 .18775 .23114 41 42 .15849 .18834 .16805 .20199 .17786 .21033 .18792 .23140 42 43 .15365 .18836 .10821 .20222 .17302 .21058 .18809 .23166 43 44 .15880 .18878 .16837 .20246 .17819 .21682 .18826 .23192 44 45 .15896 .18901 .16853 .20209 . 17835 .21707 .18843 .23217 45 46 .15912 .1892:) .16809 .20292 .17852 .21731 .18860 .23243 46 47 .15923 .18945 .16885 .2C316 .17808 .21756 .18877 .23209 47 48 .15943 .18967 .16902 .20339 .17885 .21781 .18894 .23295 48 49 .15959 .18990 i .16918 .20303 .17902 .21805 .18911 .233-21 49 50 .15975 .19012 .16934 .20388 .17018 .21830 .18928 .23347 50 51 .15991 .19034 .16950 .20110 .17035 .21855 .18945 .23373 51 52 .16008 .19057 .16986 .20433 .17052 .21879 .18962 .23399 52 53 .10022 .19079 .169*3 : 20 157 .17908 .21904 .18979 .23424 53 54 .10038 .19102 .16909 .20180 .17985 .21029 .18996 .23450 54 55 .16054 .19124 | .17015 .20504 .18001 .21953 .19013 .23476 55 56 .16070 .19146 .17031 .30527 .18018 .21978 .19030 .23502 50 57 .16J85 .19169 .17047 .20551 .18035 .22003 .19047 .23529 57 58 .16101 .19191 .17064 .20575 .18051 .230-28 .19004 .23555 58 59 .16117 .19214 .17080 .2050S .18068 .22053 .19081 .23581 59 60 .16133 .19330 .17096 .20022 .18085 .22077 .19098 .23607 60 478 TABLE XXIX. NATURAL VERSED SINES AND EXTERNAL SECANTS. 3 8 3' r 3 jo i 3 / Vers. Ex. sec. Vers. Ex. sec. Vers. Ex. sec. Vers. Ex. sec. .19098 .23607 .201S6 .25214 .21199 .26902 .22285 .28676 1 .19115 .23633 .20154 .25241 .21217 .26931 .22304 .28706 1 2 .19133 .231)59 .20171 .25269 .21235 .26960. .22322 .28737 2 8 .19150 .23685 .20189 .25296 .21253 .26988 .22:340 .28767 3 4 .19167 .23711 .20207 .25324 .21271 .27017 .22359 .28797 4 5 .19184 .23738 .20224 .25351 .21289 .27046 .22377 .28828 5 6 . 19201 .23704 .20242 .25379 .21307 .27075 .22395 .28858 6 .19218 .23790 .20259 .25406 .21324 .27104 .22414 .28889 7 8 .19235 .23816 .20277 . .25434 .21342 .27133 .22432 .28919 8 o .19252 .23843 .20294 .25462 .21360 .27162 .22450 .28950 9 10 .19270 .23869 .20312 .25489 .21378 .27191 .22469 .28980 10 11 .19287 .23895 .20339 .25517 .21396 .27221 .22487 .29011 11 12 .19304 .23922 .20347 .25545 j .21414 .27250 .22506 .29042 12 18 .19381 .23918 .20305 .25572 .21432 .27279 .22524 .29072 13 11 .19338 .23975 .20382 .25600 .21450 .27308 .22542 .29103 14 15 .11)35(3 .24)01 .20400 .25628 .21468 .27337 .22561 .29133 15 10 .19373 .24028 .20417 .25656 .21486 .27306 .22579 .29164 16 17 .19390 .24054 .20435 .25683 .21504 .27396 .22598 .29195 17 18 .19407 .24081 .20453 .25711 .21522 .27425 .22010 .29220 18 19 .19424 .24107 .20470 .25739 .21540 .27454 .22634 .29256 19 20 .19442 .24134 .20488 .25767 .21558 .27483 .22653 .29287 20 21 .19459 .24160 .20506 .25795 : .21576 .27513 .22671 .29318 21 22 .19476 .24187 .20523 .25823 ! .21595 .27542 .22090 .29319 22 23 .19493 .24213 .20541 .25851 .21613 .27572 .22708 .29380 23 24 .19511 .21210 .20559 .25879 .21631 .27001 .22727 .29411 24 25 .19528 .24267 .20576 .25907 .21649 .27630 .22745 .29442 25 26 .19545 .24293 .20594 .25935 .21667 .27660 .22764 .29473 26 7 .19562 .24320 .20612 .25963 .21685 .27689 .22782 .29504 27 28 .19580 .24347 .20029 .25991 .21703 .27719 .22801 .29535 28 29 .19597 .24373 .20647 .26019 .21721 .27748 .22819 .29506 29 30 . 19014 .24400 .20665 .26047 .21739 .27778 .22838 .29597 30 31 .19032 .24427 .20682 .26075 .21757 .27807 .22856 .29628 31 32 .19049 .21454 .20700 .26104 : .21775 .27837 .22875 .29659 32 33 .19666 .24481 .20718 .26132 ! .21794 .27867 .22893 .29090 33 34 .1J084 .24508 .20736 .26160 .21812 .27896 .22912 .29721 34 35 .19701 .24534 .20753 .26188 .21830 .27926 .22030 .29752 35 36 .19718 .24561 .20771 .26216 .21848 .27956 .22949 .29784 36 37 ! 19736 .24588 .20789 .26245 .21866 .27985 .22967 .29815 37 38 . 19753 .24615 .20807 .26273 .21884 .28015 .22986 .29846 38 39 .19770 .21642 .20824 .26301 .21902 .28045 .23004 .29877 39 40 .19788 .21009 .20842 .26330 .21921 .28075 .23023 .29909 40 41 .19805 .24696 .20860 .26.358 .21939 .28105 .23041 .29940 41 42 .19822 .24723 .20878 .26387 .21957 .28134 .23060 .29971 42 43 .19840 .24750 .20895 .26415 .21975 .28164 .2*379 .30003 43 44 .19857 .24777 .20913 .26443 .21993 .28194 .23097 .30034 44 45 .19875 .24804 .20931 .26472 .22012 .28224 .23116 .30066 45 46 .19892 .24832 .20949 .26500 .22030 .28254 i .23134 .30097 46 47 .19909 .24859 .20967 .26529 .22048 .28284 j .23153 .30129 47 48 .19927 .24886 .20985 .26557 .22066 .28314 ! .23172 .30160 48 49 .19944 .24913 .21002 .26586 .22084 .28344 i .23190 .30192 49 50 .19962 .24940 .21020 .26615 .22103 .28374 ' .23209 .30223 50 51 .19979 .24967 .21038 .26643 .22121 .28404 .23228 .30255 51 52 .19997 .24995 .21056 .20072 .22139 .28434 .23246 .30287 62 53 .20014 .25022 .21074 .26701 .22157 .28464 : .23265 .30318 53 54 .20032 .25049 .21092 .26729 .22176 .28495 .23283 .30350 54 55 .20049 .25077 .21109 .26758 : .22194 .28525 i .23302 .30382 55 56 .20066 .25104 .21127 .20787 ! 22212 .28555 .23321 .30413 56 57 .20084 .25131 .21145 .26815 1 ! 22231 .28585 i .23339 .30445 57 58 .20101 .25159 .21163 .26844 i .22249 .28615 .2:3358 .30477 58 59 .20119 .25186 i .21181 .26873 .22207 .28646 .23377 .30509 59 60 .20136 .25214 .21199 .20902 | .22285 .28076 .23396 .30541 479 TABLE XXIX. NATURAL VERSED SINES AND EXTERNAL SECANTS / 40 42 43 Vers. Ex. sec. Vers. Ex. sec. Vers. Ex. sec. Vers. Ex. sec. ~0 .23396 .30541 .2-1529 1 .32501 .25686 .34563 .26865 .36733 1 .23414 .30573 .24548 .32535 .25705 .34599 .26884 .36770 1 2 .23433 .30605 -.24567 .32568 .257'24 .34634 .26904 .36807 2 3 .23452 .30036 .24586 .32C02 .25744 .34669 .26924 .36844 3 4 .23470 .30668 .24605 .32636 .25763 .34704 .26944 .36881 4 5 .23489 .30700 .24625 .32669 .25783 .34740 .26964 .36919 5 6 .23508 .30732 .24644 .32703 j .25802 .34775 .26984 .36956 6 7 .23527 .30764 .24663 .32737 .25822 .34811 .27004 .36993 7 8 .23545 .30796 .24682 .32770 .25841 .34846 .27084 .37030 8 9 .23564 .30829 .24701 .32804 .25861 .34882 .27043 .37068 9 10 .28583 .30861 .24720 .32838 .25880 .34917 .27063 .37105 10 11 .23602 .30893 .24739 .32872 .25900 .24353 .27083 .37143 11 12 .23620 .30925 .247'59 .32905 .25920 .34988 .27103 .37180 12 13 .23639 .30957 .24778 .32939 .25989 .35024 .27123 .37218 13 14 .23658 .30989 .24797 .32973 .25959 .35060 .27143 ,37'255 14 15 .23677 .31022 .24816 .33007 .25978 .35095 .27163 .87293 15 16 .23696 .31054 .24835 .33041 .25998 .35131 .27183 .37330 16 17 .23714 .31086 .24854 .33075 .26017 .35167 .27'203 .37368 17 18 .23733 .31119 .24874 .33109 .26037 .35203 .27223 .37406 18 19 .23752 .31151 .24893 .33143 .26056 .35238 .27243 .37443 19 98 .23771 .31183 .24912 .33177 .26076 .35274 .27263 .37481 20 21 .23790 .31216 .24931 .33211 .26096 .35310 .27283 .37519 21 22 .23808 .31248 .24950 .33245 .26115 .35346 .27303 .37556 22 23 .23827 .31281 .24970 .33279 .26135 .35382 .27823 .37594 23 24 .23346 .31313 .24989 .a3314 .26154 .35418 .27343 .37632 24 25 .23865 .31346 .25008 .33348 .26174 .35454 .27363 .37670 25 26 .23884 .31378 .25027 .33382 .26194 .35490 .27383 .37708 26 27 .23903 .31411 .25047 .83416 .26213 .35526 .27403 .37746 27 28 .23922 .31443 .25066 .33451 .26233 .353(52 .27423 .87784 28 29 .23941 .31476 .25085 .33485 .26253 .35598 .27443 .37822 29 30 .23959 .31509 .25104 .33519 .26272 .35634 .27463 .37860 30 31 .23978 .31541 .25124 .33554 .26292 .35670 .27483 .37898 31 32 .23997 .31574 .25143 .33588 .26312 .35707 .27503 .37936 32 33 .24016 .31607 .25162 .33622 .26331 .35743 .27528 .37974 33 34 .24083 .31610 .25182 .33657 .26351 .35779 .27543 .38012 34 35 .24054 .31672 .25201 .33691 .26371 .35815 .27563 .38051 35 36 .24073 .31705 .25220 .33726 .26390 .35H52 .27583 .38089 36 37 .24092 .31738 .25240 .-33760 .26410 .35888 .27603 .38127 37 38 .24111 .31771 .25259 .33795 .26430 .35924 .27'623 .38165 38 39 .24130 .31804 .25278 .33830 .26449 .35961 .27043 .38204 39 40 .24149 .31837 .25297 .33864 .26469 .35997 .27063 .38242 40 41 .24168 .3187'0 .25317 .3:5899 .26489 .36034 .27683 .38280 41 42 .24187 .31903 .25336 .33934 .26509 .36070 .277-03 .38319 42 43 .24206 .31936 .25356 .33968 .26528 .36107 .27723 .38357 43 44 .24225 .31969 .25375 .34003 .26548 .36143 .27743 .38396 44 45 .24244 .32002 .25394 .34038 .26568 .36180 .27764 .38434 45 46 .24262 .32035 .25414 .34073 .26588 .36217 .27784 .3&47'3 46 47 .24281 .32068 .25433 .34108 .26607 .36253 .27804 .38512 47 48 .2-1300 .32101 .25452 .34142 .20627 .36290 .27824 .38550 48 49 .24320 .32134 .25472 .34177 .26647 .36327 .27844 .38589 49 50 .24339 .32168 .25491 .34212 .26667 .36363 .27804 .38028 50 51 .24358 .32201 .25511 .34247 .26686 .36400 .27884 .38600 51 52 .21377 .32234 .25530 .34282 .26706 .36437 .27905 .88705 52 53 .24396 .,32267 .25549 .34317 .26726 .36474 .27925 .38744 53 54 .21415 .32301 .25569 .34352 .21)746 .36511 .27945 .38783 54 55 .21134 .32334 .25588 .34387 .2(5766 .36548 .27'965 .38822 55 56 .24453 .32368 .25608 .34423 .26785 .36585 .27985 .38860 56 57 .24472 .32401 .25627 .34458 .26805 .36622 .28005 .38899 57 58 .24491 .32434 .25647 .34493 .26825 .36659 .28026 .83938 58 59 .24510 .32468 .25666 .34528 .26845 .30696 ..28046 .38977 59 60 .24529 .32501 .25686 .34563 .26805 .36733 .28006 .39016 60 480 TABLE XXIX. -NATURAL VERSED SINES AND EXTERNAL SECANTS. 4 4 4 5 4 6 4 7 Yers. Ex. sec. Vers. Ex. sec. Yers. Ex. sec. Vers. Ex. sec. .28066 .39016 1 .29289 .41421 .305:34 .43956 .31800 .46628 1 .23086 .39055 .29310 .41463 .30555 .43999 .31821 .46674 1 2 .28106 .39095 i .29330 .41504 .30576 .44042 .31843 .46719 2 3 .28127 .39134 i .29351 .'41545 .30597 .44086 .31864 .46765 3 4 .28147 .39173 i .29372 .41586 .30618 .44129 .31885 .46811 4 5 .23167 .39212 1 .29392 .41627 .30639 .44173 .31907 .46857 5 6 .28187 .39251 i .29413 .41669 .30660 .44217 .31928 .46903 6 7 .28208 .39291 .29433 .41710 .30681 .44260 .31949 .46949 7 8 .28228 .39330 .29454 .41752 .30702 .44304 .31971 .46995 8 9 .28248 .39369 .29475 .41793 .30723 .44347 .31992 .47041 9 10 .23268 .39409 .29495 .41835 .30744 .44391 .32013 .47087 10 11 .28289 .39448 .29516 .41876 .30765 .44435 .32035 .47134 11 12 .28:309 .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 .39806 .29599 .42043 .30849 .44610 .32120 .47319 15 16 .28390 .39646 .29619 .42084 .30870 .44654 .32141 .47365 16 17 .28410 .39685 .29640 .42126 . .30891 .44698 .32163 .47411 17 18 .28431 ! 39725 .29661 .42168 .30912 .44742 .32184 .47458 18 10 .23451 .39764 .29881 .42210 .30933 .44787 .32205 .47504 19 20 .28471 .39804 .29702 .42251 .30954 .44831 .32227 .47551 20 21 .28492 .39844 .29723 .42293 .30975 .44875 .32248 .47598 21 22 .28512 .89884 .29743 .42*35 .30996 .44919 .32270 .47644 22 23 .2^52 .39924 .29764 .42377 .31017 .44963 .32291 .47691 23 24 .28553 .39963 .29785 .42419 .31038 .45007 .32312 .47738 24 25 . 28573 .40003 .29805 .42461 .31059 .45052 .32334 .47784 25 26 .28593 .40043 .29826 .42503 .31080 .45096 .32355 .47831 26 27 .28614 .40083 .29847 .42545 .31101 .45141 .32377 .47878 27 28 .28634 .40123 .29868 .42587 .31122 .45185 .32398 .47925 28 29 .28655 .40163 .29888 .42630 .31143 .45229 .32420 .47972 29 30 .28675 .40203 .29909 .42672 .31165 .45274 .32441 .48019 30 31 .28695 .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 .23757 .40364 .29992 .42841 .31249 .45452 .32527 .48207 34 35 .28777 .40404 .30013 .42883 .31270 .45497 .32548 .48254 35 36 .28797 .40444 .30034 .42926 .31291 .45542 .32570 .48301 36 37 .28818 .40485 .30054 .42968 .31312 .45587 .32591 .48349 37 38 .28838 .40525 .30075 .43011 .31334 .45631 .32613 .48396 38 39 .28859 .40565 .30096 .43053 .31355 .45676 .3263-4 .48443 39 40 .28879 .40606 .30117 .43096 .31376 .45721 .32656 .48491 40 41 .28900 .40646 .30138 .43139 .31397 .45766 .32677 .48538 41 42 .28920 .40687 .30158 .43181 .31418 .45811 .32699 .48586 42 43 .28941 .40727 .30179 .43224 .31439 .45856 .32720 .48633 43 44 .28961 .40768 .30200 .43267 .31461 .45901 .32742 .48681 44 45 .23981 .40808 30221 .43310 .31482 .45946 .32763 .48728 45 46 .29002 .40849 .'30242 .43352 .31503 .45992 .32785 .48776 46 47 .29023 .40890 .30263 .43395 .31524 .46037 .32806 .48824 47 48 .29043 H .40930 .30283 .43438 .31545 .46082 .32828 .48871 48 49 .29063 .40971 .30304 .43481 .31567 .46127 .32849 .48919 49 50 .29034 .41012 .30325 .43524 .31588 .46173 .32871 .48967 50 51 .29104 .41053 .30346 .43567 .31609 .46218 .32893 .49015 51 52 .29125 .41093 .30367 .43610 .31630 .46263 .32914 .49063 52 53 .29145 .41134 .30388 .43653 i .31651 .46309 .32936 .49111 53 54 ,29166 .41175 .30409 .43696 ! .31673 .46354 .32957 .49159 54 55 .29187 .41216 .30430 .43739, ; .31694 .46400 .32979 .49207 55 56 .29207 .41257 .30451 .43783 i .31715 .46445 .33001 .49255 56 57 .29228 .41298 .30471 .43826 .31736 .46491 .33022 .49303 57 53 .21)248 .41339 .30492 .43869 .31758 .46537 .33044 .49351 58 59 .29269 .41380 .30513 .43912 .31779 .46582 .33065 .49399 59 69 .29289 .41421 ! .30534 .43956 .31809 .46628 1 .33987 .49448 60 481 TABLE XXIX. -NATURAL VERSED SINES AND EXTERNAL SECANTS. 48 49 50 51 Vers. Ex. sec. Vers. Ex. sec. Vers. Ex. sec. | Vers. Ex. sec. .33087 .49448 i .34394 .52425 .35721 .55572 .37068 .58902 ~0~ 1 .33109 .49496 .34416 .52476 .35744 .55626 .37091 .58959 1 2 .33130 .49544 .34438 .52527 .85766 .55680 .37113 .59016 2 3 .33152 .49593 .34460 .38579 .85788 .55734 .37136 .59073 3 4 .33173 .49641 .34482 .52630 .35810 .55789 .:',; ir>s .59130 4 5 .33195 .49690 .34504 .52681 .35833 .55843 .37181 .59188 5 6 ! 83217 .49738 .34526 .52732 .35855 .55897 .37204 .5<)215 6 7 .83288 .49787 .34548 .52781 .35877 .55951 .87226 .59302 7 8 .33260 .49835 .34570 .52835 .35900 .56005 .37249 .59360 8 9 .a3282 .49884 .34592 .52886 .35922 .56060 .37272 .59418 9 10 .33303 .49933 .34614 .52938 .35944 .56114 .37294 .59475 10 11 .88386 .49981 .34636 .52989 .35967 .56169 .37317 .59533 11 12 .33347 .50030 .34658 .53041 .35989 .56223 .37340 .59590 12 13 .33368 .50079 .34680 .53092 .36011 .56278 .37362 .59618 13 14 .33390 .50128 .34702 .53144 .36034 .56332 .37385 .59706 14 15 .33412 .50177 .34724 .53196 .36056 .56387- .37408 .59764 15 1(5 .33434 .50226 .34746 .53247 .36078 .58442 .37130 .59822 16 ir .33455 .50275 .34768 .53299 .36101 .56497 .37453 .59880 17 18 .33477 .50324 .34790 .53351' .36123 .56551 .37476 .59938 18 19 .33199 .50373 .34812 .53403 .86146 .50606 .37498 .59996 19 20 .33520 .50422 .34834 .53455 .3016? .56661 .37521 .60054 20 21 .33542 .50471 .34856 .53507 .36190 .56716 .37544 .00112 21 22 .33564 .50521 .34878 .53559 .30-213 .56771 .37567 .00171 22 23 .33586 .50570 .34900 .53611 .36235 .56826 .37589 .60229 23 24 .33607 .50619 .34923 .53663 .36258 .50881 .37612 .60287 24 25 .33629 .50669 .34945 .53715 .36280 .56937 .376:35 .60346 25 26 .33651 .50718 .34967 .53768 .36302 .56992 .37658 .00404 26 O 1 "* .33673 .50767 .34989 .53820 .36325 .57047 .37080 .60-463 27 28 .33694 .50317 .35011 .53872 .36347 .57103 .37703 .(50521 28 29 .33716 .50866 .35033 .53924 .36370 .57158 .37726 .60580 29 30 .33738 .50916 .35055 .53977 .36392 .57213 .37749 .60639 30 31 .33760 .50966 .35077 .54029 .36415 .57269 .37771 .60698 31 32 .33782 .51015 .35099 .54082 .36437 .57324 .37794 .00756 32 33 .33803 .51065 .35122 .54134 .36460 .57380 .37817 .60815 33 31 .33325 .51115 .35144 .54187 .36482 .57436 .37840 .60874 34 35 .33847 .51165 .35166 .54240 .36504 .57491 .37802 .60933 35 36 1 .33869 .51215 .35188 .54292 .36527 .57547 .37885 .60992 36 37 .33891 .51265 .35210 .54345 .36549 .57603 .37908 .61051 37 38 .33912 .51314 .35232 .54398 .86573 .57659 .37931 .61111 38 39 .33934 .51364 .35254 .54451 .36594 .57715 .37954 .61170 39 40 .33956 .51415 .35277 .5450i .36617 .57771 .37976 .61229 40 41 .33978 .51465 .35299 .54557 .36639 .57827 .37999 .61288 41 42 .34000 .51515 .35321 .54610 .36662 .57883 .38022 .61348 42 43 .34022 .51565 .35343 .54663 .36084 .57939 .38045 .01407 43 44 .34044 .51615 .35365 .54716 .36707 .57995 .38068 ; 61467 44 45 .34065 .51665 .35388 .54769 .36729 .5.8051 .3S091 .61526 45 46 .34087 .51716 .35410 .54822 .36752 .58108 : .38113 .61586 46 47 .34109 .51766 .35432 .54876 .36775 .58164 .38136 .61646 47 48 .34131 .51817 .35454 .54929 .36797 .58221 .38159 .61705 48 49 .31153 .51867 .35476 .5-19S2 .36820 .58277 i .38182 .61705 49 50 .34175 .51918 .35499 .55036 .36842 .58333 i .38205 .61825 50 51 .34197 .51968 .35521 .55089 ! .368C5 .58390 .88228 .61885 51 52 .34219 .52019 .35543 .55143 ; .36887 .58-147 .3S251 .61945 52 53 .34241 .52069 .35565 .55196 .36910 .58503 .3827'4 .62005 53 54 .34202 .52120 .awSS .55250 .36932 .58560 .38296 .62005 54 55 .34284 .52171 .35610 .55303 .36955 .58617 .38319 .02125 55 56 .34306 .52222 .35632 .55357 .36978 .58674 .38342 .62185 56 57 .34328 .52273 .35654 .55411 .37000 .58731 .38365 .62046 57 58 .34350 .52323 .35677 .55465 .37023 .58788 .38388 .62306 58 59 .34372 .52374 .35699 .55518 .37045 .58845 .38411 .62?,GO 59 CO .34394 .52425 .35721 .55372 .37068 .58902 .38434 .62427 CO 4S2 TABLE XXIX. NATURAL VERSED SINES AND EXTERNAL SECANTS. 5 , 5 3 5' 1 5 ) Vers. Ex. sec. Vers. Ex. sec. 1 Vers. Ex. sec. Vers. Ex. sec. i .38434 .02427 .39819 .00104 ! .41221 .70130 42642 .74345 1 .38457 .62487 .39842 166228 < .41245 .70198 .42666 .74417 1 o .38480 . 62548 .39865 .66292 .41209 .70267 .42690 .74490 2 3 138508 .62609 .39888 .66357 i .41292 .70335 .42714 .74562 3 4 .38526 .62669 .39911 .66421 j .41316 .70403 .42738- .74635 4 5 .38649 .62730 .39935 .66486 i .41339 .70472 .42702 .74708 5 6 .38571 .62791 .39958 .66550 .41363 .70540 .42785 .74781 6 7 .38594 .62852 .39981 .66615 .41386 .70009 .42809 .74854 7 8 .38017 .62913 .40005 .66679 .41410 .70677 i .42833 .74927 8 9 .38640 .62974 .40028 .66744 .41433 .70746 1 .42857 .75000 9 10 .38663 .63035 .40051 .66809 .41457 .70815 .42881 .75073 10 11 .38086 .63096 .40074 .66873 .41481 .70884 .42905 .75146 11 12 .38709 .63157 .40098 .66938 ! .41504 .70953 .42929 .75219 12 ' 13 .88732 .63218 .40121 .67003 ' .41528 .71022 .42953 .75293 13 14 .38755 .03279 .40144 .67068 .41551 .71091 .42976 .75366 14 15 .38778 .63811 .40168 .67133 .41575 .71160 .43000 .75440 15 16 .38801 .63402 .401 91 .67199 .41599 .71229 .43024 .75513 16 17 .38824 .63164 .40214 .67264 .41622 .71298 .43048 .75587 17 18 .38847 .63525 .40237 .67329 .41646 .71368 .43072 .75661 18 19 .38870 .63537 .40201 .67394 .41670 .71437 .43096 .75734 19 20 .38893 .63648 .40284 .67460 i .41693 .71506 .43120 ,75808 20 21 .38910 .63710 .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 .38985 .63895 .40378 .67722 .41788 .71785 .43216 .76105 24 25 .39009 .63957 .40401 .67788 .41811 .71855 .43240 .76179 25 26 .39032 .64019 j .40424 .67853 i .4iass .71925 .43264 .76253 26 27 .39055 .64081 .40448 .67919 .41859 .71995 .43287 .76328 27 28 .39078 .04144 .40471 .67985 .41882 .72065 .43311 .76402 28 29 .39101 .04200 .40494 .68051 .41906 .72135 .43335 .76477 29 30 .39124 .64268 I .40518 .68117 .41930 .72205 .43359 .76552 30 31 .39147 .64330 .40541 .68183 .41953 .72275 .43383 .76626 31 32 .39170 .01:593 .40565 .68250 .41977 .72346 .43407 .76701 32 33 .39193 .64455 .40588 .68316 .42001 .73416 .43431 .76776 33 34 .39216 .64518 .40011 .68382 .42024 .72487 .43455 .76851 34 85 .39239 .64580 .40635 .68449 .42048 .72557 .43479 .76926 35 36 .39202 .64643 .40658 .68515 1 .42072 .72628 .43503 .77001 36 37 .39286 .64705 .40682 .68582 .42096 .72698 .43527 .77077 37 38 .39309 .04708 .40705 .68648 .42119 .72769 .43551 .77152 38 89 .39333 .64831 .40728 .68715 .42143 .72840 1 .43575 .77227 39 40 .39355 .64894 .40752 .68782 .42167 .72911 .43599 .77303 40 41 .39378 .64957 .40775 .68848 .42191 .72982 .43623 .77378 41 42 .39401 ' .65020 .40799 .68915 .42214 .73053 j .43647 .77454 42 4.'] .39124 .65083 .40822 .68982 .42238 .73124 i .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 i .43744 .77757 46 47 .39517 .65336 .40916 .69250 .43333 .73409 .43768 .77833 47 48 .39540 .65399 .40939 .69318 .42357 .73481 .43792 .77910 48 49 .39563 .65402 .40963 .69a85 .42381 .73552 .43816 .77986 49 50 .39586 .65526 .40986 .69452 .42404 .73624 .43840 .78062 50 51 .39010 .65589 ! .41010 .69520 .42428 .73696 .43864 .78138 51 5-J .39033 .05653 .41033 .69587 .42452 .7.3768 .43888 .78215 52 58 .39656 .05717 .41057 .69655 .42476 .73840 .43912 .78291 53 54 .39079 .65780 .41080 .69723 .42499 .73911 .43936 .78368 54 55 .39702 .058-14 .41104 .69790 .42523 .73983 .43960 .78445 55 50 .39720 .65908 .41127 .69858 .42547 74056 .43984 .78521 56 57 .39749 .65972 .41151 .69926 .42571 .74128 .44008 .78598 57 58 .39772 .66036 .41174 .69994 .42595 .74200 .44032 .78675 58 59 .39795 .66100 .41198 .70062 .42019 .7427'2 .44057 .78752 59 60 .39819 .66164 .41221 .70130 .42042 .74345 .44081 .78829 60 483 TABLE XXIX. -NATURAL VERSED SINES AND EXTERNAL SECANTS. ' 56 57 58 59 ' Vers. Ex. sec. Vers. 'Ex. sec. Vers. Ex. sec. Vers. Ex. sec. ! .44081 .78829 .45536 .83608 .47008 .88708 .48496 . .94160 1 I .44105 .78906 .45560 i .83690 .47033 ! .S8796 .48521 i .942.54 1 2 ! .44129 .78984 .45585 .83773 .47057 | .88884 .48546 ; .94:349 >> 3 I .44153 .79061 .45609 .8:3855 i .47082 .88972 .48571 .94443 3 4 i .44177 .79138 .45634 .83938 i .47107 .89060 .48596 : .94537 4 5 .44201 .79216 .45658 .84020 ! .47131 ! .89148 .48621 .94632 5 6 .44225 .79293 .45683 .84103 .47156 .89237 .48640 .94726 6 7 .44250 .79371 .45707 .84186 .47181 : .89325 .48671 .94821 rt 8 .44274 .79449 .45731 .84269 .47206 .89414 .48696 .94916 8 9 .44298 .79527 .45756 .84352 .47230 .89503 1 1 .48721 i .95011 9 10 .44322 .79604 .45780 .84435 .47255 .89591 h .48746 .95106 10 11 .44346 .79682 .45805 .84518 i .47280 .89680 ! .48771 .95201 11 12 .44370 .79761 .45829 ! .84601 .47304 .89769 1 .48796 .95296 12 13 .44395 .79839 .45854 .84685 .47329 .89858 I 1 .48821 .95392 13 14 .44419 .79917 .45878 .84768 .47354 .89948 i .48846 .95487 14 15 .44443 .79995 .45903 .84852 .47379 .90037 .48871 .95583 15 16 .44467 .80074 .45927 .84935 .47403 .901261 -48896 .95678 16 17 .44491 .80152 .45951 .85019 1 .474.28 .90216 I .48921 .95774 17 18 .44516 .80231 .45976 .85103 .47453 .90305 .48946 .95870 18 19 .44540 .80309 .46000 .85187 ! .47478 .90395 .48971 .95966 19 20 .44564 .80388 .46025 .85271 i .47502 .90485 .48996 .96062 20 21 .44588 .80467 .46049 .85355 i .47527 .90575 .49021 .96158 21 23 .44612 .80546 .46074 .85439 .47552 .90665 .49046 .96255 22 23 .44637 .80625 .46098 .85533 .47577 .90755 .49071 .96351 23 24 .44661 .80704 .46123 i .85608 .47601 .90845 .49096 i .96448 24 25 .44685 .80783 .46147 i .85692 .47626 .90935 .49121 .96544 25 26 .44709 .80862 .46172 .85777 .47651 .91026 .49146 .96641 26 27 .44734 .80942 | .46196 .85861 .47676 .91116 i .49171 .96738 27 28 .44758 .81021 .46221 .85946 .47701 .91207 ! .49196 .96835 28 29 .44782 .81101 .46246 .b6031 .47725 .91297 ' .49221 i .96932 29 30 .44806 .81180 .46270 .86116 .47750 .91388 .49246 I .97029 30 31 .44831 .81260 .46295 .86201 .47775 .91479 .49271 .97127 31 32 .44855 .81340 .46319 .86286 .47800 .91570 ; .49296 .97224 32 33 .44879 .81419 .46344 .86371 .47825 .91(561 ! .49321 .97'322 33 34 .44903 .81499 .46368 .86457 ! .47849 .91752 i .49346 .97420 34 35 .44923 .81579 .46393 .86542 .47874 .91844 .49372 j .97517 35 36 .44952 .81659 .46417 .86627 .47899 .91935 .49397 .97615 36 37 .44976 .81740 .46442 .86713 ! .47924 .92027 .49422 j .97713 37* 38 .45001 .81820 .46466 .86799 1 .47949 .92118 i .49447 .97811 38 39 .45025 .81900 .46491 .86885 .47974 .92210 l ! .4947'2 i .97910 39 40 .45049 .81981 .46516 .86990 .47998 .92302 i .49497 .98008 40 41 .45073 .82061 .46540 .87056 1 .48023 .92394 1 .49522 ! .98107 41 42 .45098 .82142 .46565 .87142 i .48048 .92486 i .49547 .98205 42 43 .45122 .82222 .46589 .87229 1 .48073 .92578 .49572 .9830-1 43 44 .45146 .82303 .46614 .87315 : .48098 .92670 .49597 .98403 44 45 .45171 .82384 .46639 .87401 .48123 .92762 .49623 .98502 45 46 .45195 .82465 .46663 .87488 i .48148 .92855 i .49648 .98601 46 47 .45219 .82546 .46688 .87574 .48172 .92947 .49673 .987XX) 47 48 .45244 .82627 .46712 .87661 ! .48197 .93040 ; .49698 .98799 48 49 .45268 .82709 .46737 .87748 .48222 .93133 .497'23 .98899 49 50 .45292 .82790 .46762 .87834 .48247 .93226 .49748 .98998 50 51 .45317 .82871 .46786 .87921 .48272 .93319 .49773 .99098 51 52 .45341 .82953 .46811 .88008 .48297 .93412 .49799 .99198 52 53 ,45365 .83034 .46836 .88095 .46322 .93505 .49824 .99298 53 54 .45390 .&3116 .46860 .88183 ! .48347 .93598 |i .49849 .99398 54 55 .45444 .83198 ! .46885 .88270 .48372 .93692 ! .49874 .99498 55 56 .45439 .83280 i .46909 .88357 .48396 .93785 : .49899 .99598 56 57 .45463 .83362 .46934 .88445 j .48421 .93879 i .49924 .99698 57 58 .45487 .83444 .46959 .88532 .48446 .93973 .49950 .99799 58 59 .45512 .83526 .46983 .88620 .48471 .94066 I .49975 .99899 59 60 .45536 .83608 1 .47008 .88708 .48496 .94160 1 .50000 1.00000 60 484 TABLE XXIX.- NATURAL VERSED SINES AND EXTERNAL SECANTST 6 - ) i 6 2 6 3 Vers. Ex .se Vers. ; Ex. sec. Vers. Ex. sec. Vers. Ex. sec. i .50000 .00000 .51519 i 1.06267 .53053 1.13005 | .54601 1.20269 1 50025 00101 .51544 1.06375 .53079 1.13122 .54627 1.20395 1 2 .50050 .00202 .51570 1.06483 .53104 1.13239 .54653 .20521 2 3 50076 .001303 .51595 1.06592 .53130 1.13,356 .54679 .20647 3 4 .50101 00404 .51621 1.06701 .53156 1.13473 .54705 .20773 4 5 .50126 .00505 .51646 1.06809 .53181 1.13590 .54731 .20900 5 (1 50151 1.00607 .51672 1.06918 .53207 1.13707 .54757 .21026 6 7 .50176 1.00708 .51697 1.07027 .53233 1.13825 .54782 .21153 r* ,s 50202 1.00810 .51723 1.07137 .53258 1.13942 .54808 .21280 8 9 .50227 1.00912 .51748 1.07246 .53284 1.14060 .54834 .21407 9 10 .50252 1.01014 .51774 1.07356 .53310 1.14178 .54860 .21535 10 11 .50277 1.01116 .51799 1.07465 .53330 1.14296 .54886 .21662 11 12 .50303 1.01218 .51825 1.07575 i .53361 1.14414 .54912 .21790 12 13 .50328 1.01320 .51850 1.07685 i .53387 1.14533 .54938 .21918 13 14 .50353 1.01422 .51876 1.07795 .53413 1.14651 1 .54964 .22045 14 15 .50378 1.01525 .51901 1.07905 .53439 1.14770 .54990 .22174 15 Ki .50404 1.01628 .51927 1.08015 .53464 1.14889 .55016 .22302 16 17 .50429 1.01730 .51952 1.08126 .53490 1.15008 .55042 .22430 17 18 .50454 1.01833 .51978 1.08236 .53516 1.15127 .55068 .22559 18 19 .50479 1.01936 .52003 1.08347 .53542 1.15246 .55094 .22688 19 80 .50505 1.02039 .52029 1.08458 .53567 1.15366 .55120 .22817 20 '.31 50530 1.02143 .520,54 1.08569 .53593 1.15485 .55146 .22946 21 2-2 .50555 1.02246 .52080 1.08680 .53619 1.15605 .55172 .23075 22 93 .50581 1.02349 .52105 1.08791 .53645 1.15725 .55198 .23205 23 21 .50606 1.02453 .52131 1.08903 .53670 1.15845 .55224 .23334 24 25 .50631 1.02557 .52156 1.09014 .53096 1.15965 .55250 .23464 25 26 50656 1.02661 .52182 1.09126 .53722 1.16085 .55276 .23594 26 27 .50682 1.02765 .52207 1.09238 .53748 1.16206 .55302 .23724 27 28 .50707 1.02869 .52233 1.09350 .53774 1.16326 .55328 .23855 28 .". 50732 1.02973 .52259 1.09462 .53799 1.16447 .55354 .23985 29 30 .50758 1.03077 .52284 1.09574 .53825 1.16568 .55380 .24116 30 \\ 50783 1.03182 .52310 1.09686 .53851 1.16689 .55406 .24247 31 32 .50808 1.03286 .52335 1.09799 .53877 1.16810 .55432 .24378 32 33 .50834 1.03391 .52361 1.09911 ! .53903 1.16932 .55458 .24509 33 34 .50859 1.03496 .52386 1.10024 .53928 1.17053 .55484 .24640 34 35 .50884 1.03601 .52412 1.10137 ..53954 1.17175 .55510 .24772 35 36 50910 1.03706 .52438 1.10250 .53980 1.17297 .55536 .24903 36 37 .50935 1.03811 .52463 1.10363 .54006 1.17419 .55563 .25035 37 88 50960 1.0391(5 .52489 1.10477 .54032 1.17541 . : -!5589 25167 38 39 .50986 1.04022 .52514 1.10590 .54058 1.17663 .55615 .25300 39 40 .51011 1.04128 .52540 1.10704 .54083 1.17786 .55641 .25432 40 41 .51036 1.04233 .52566 1.10817 .54109 1.17909 .55667 .25565 41 42 .51062 1.04339 .52591 1.10931 .54135 1.18031 .55693 .25697 42 W .51087 1.04445 .52617 1.11045 .54161 1.18154 .55719 .25830 43 It .51113 1.04551 .52642 1.11159 .54187 1.18277 .5^745 .25963 44 46 .51138 1.04658 .52668 1.11274 .54213 1.18401 .55771 1.26097 4,-> 46 .51163 1.04764 .52694 1.11388 .54238 1.18524 .55797 1.26230 46 47 .51189 1.04870 .52719 1.11503 .54264 1.18648 .55823 1.26364 47 is .51214 1.04977 | .52745 1.11617 .54290 1.18772 .55849 1.26498 48 IS .51239 1.05084 .52771 1.11732 .54816 1.18895 .55876 1.26632 49 50 .51265 1.05191 .52796 1.11847 .54342 1.19019 .55902 1.26766 50 51 .54290 1.05298 .52822 1.11963 .54368 1.19144 .55928 1.26900 51 52 .51316 1.05405 .52848 1.12078 i .54394 1.19268 .55954 1.27035 ]52 53 .51341 1.05512 .52873 1.12193 ! .54420 1.19393 .55980 1.27169 J53 54 .51366 1.05619 .52899 1.12309 .;V1H<) 1.19517 .56006 1.27304 54 55 .51392 1.05727 .52924 1.12425 .5-1471 1.19642 .56032 1.27439 155 66 .51417 1.05835 .52950 1.12540 .54497 1.1976? .56058 1.27574 56 57 .51443 1.05942 .52976 1.12657 .54523 1.19892 .56084 1.27710 57 68 i .51468 1.06050 .53001 1.12773 j .54549 1.20018 .56111. 1.27845 153 ,VJ .51494 1.06158 .53027 1.12889 .54575 1.20143 .56137 1.27981 59 GO 1 .51519 1 1.06267 .53053 1.13005 .54601 1.20269 .56163 1.28117 60 485 TABLE XXIX. -NATURAL VERSED SINES AND EXTERNAL SECANTS. ' 64 65 66 C 67 ' Vers. Ex. sec. I Vers. Ex. sec. Vers. : Ex. sec. Vers. Ex. sec. .56163 i 1.28117 i .57738 1.36620 ' .59326 | .45859 .60927 1.559:30 (i 1 .56189 i 1.28253 .57765 1.36768 .59*53 ! .46020 .60954 1.56106 1 ') .56215 i 1.28390 i .57791 1.36916 ! .59379 .46181 i .60980 1.56282 2 3 .56241 1.28326 M .57817 1.37064 1 .59406 .46342 il .61007 ! 1.56458 3 4 .56267 1.28663 | .57844 1.37212 i .594:33 .46504 .61034 1.56634 4 5 .56294 1.28800 .57870 1.37361 .59459 ; .46665 .61061 1.56811 5 8 .56320 1.28937 ! .57896 1.37509 .59486 i .46827 .61088 i 1.56988 6 7 .56346 1.39074 i .57923 1.37658 i .59512 i .46989 .61114 i 1.57165 7 8 .56372 1.29211 j .57949 1.37808 .59539 .47152 .61141 1.57:342 8 9 .56398 1.29349 .57976 1.37957 .59566 .47314 .61168 1-57520 9 10 .56425 1.29487 .58002 1.38107 .59592 .47477 'i .61195 1.57698 10 11 .56451 1.29625 .58028 1.38256 .59619 .47640 .61222 1.57876 11 12 .56477 1-29763 .58055 1.38406 i .59645 .47'804 .61248 1>)M)54 12 18 .56503 1.29901 .58081 1.38556 i .59672 .47967 .61275 1.58233 18 14 .56529 1.30040 .58108 1.38707 ; .59699 .48131 .61302 1.58412 U 15 .56555 1.30179 .581134 1.38857 .59725 .48295 i .61329 1.58591 16 16 .56582 1.30318 .58160 1.39008 ! .59752 .48459 i .61356 1.58771 1C, 17 .56608 1.30457 .58187 1.39159 i .59779 .48624 .61383 1.58950 17 18 .56(534 1.30596 j .58213 1.39311 ; .59805 .48789 | .61409 1.59130 18 19! .566(50 1.30735 ; .58240 i 1.39462 .59832 .48954 .61436 1.59311 1!) 20 .56687 1 1.30875 .58266 1.39614 j .59859 .49119 .61463 1.59491 20 21 .56713 1.31015 .58293 1.39766 ' ,59885 .49284 .61490 1.59672 21 22 .56739 1.31155 .58319 1.3.1918 ; .59912 .49450 .61517 1.59853 22 28 .56765 1.31295 .58345 1.40070 .599:38 .49616 ; .61544 1.60035 28 24 .56791 1.31436 .58372 1.40222 .59965 .49782 : .61570 1.60217 24 ! .56818 ! 1.31576 .58398 1.40375 .59992 .49948 .61597 1.60399 25 -,'i; .56844 1.31717 .58425 1.40528 .60018 .50115 .61624 1.60581 26 27 .56870 1.31858 .58451 1.40681 .60045 .50282 .61651 1.60763 27 88 .56896 1.31999 .58478 1.40835 .60072 .5044!* .61678 1.60946 2S 20 .56923 1.32140 .58504 1.40988 .60098 .50617 .61705 1.61129 29 30 .56949 1.32282 .58531 1.41142 .60125 .50784 .61732 1.61313 30 81 .56975 1.32424 . 58557 1.41296 .60152 .50952 .61759 1.61496 31 33 .57001 1.32566 .58584 1.41450 .60178 .51120 .61785 1.61680 32 33 .57028 1.327'08 .58610 1.41605 .60205 .51289 .61812 1.61864 88 34 1 .57054 1.32850 .58637 1.41760 .60232 .51457 .61839 1.62049 m 35 .57080 1.32993 .58663 1.41914 .60259 .51626 .61866 1.62234 35 36 .57106 1.33135 i .58690 1.42070 .60285 .51795 ! 61893 1.62419 36 37 .57133 1.33278 .58716 1.42225 .60312 .51965 .61920 1.62604 37 38 .57159 1.33422 .58743 1.42380 .60339 .521:34 .61947 1.62790 38 39 .57185 1.33565 .58769 1.42536 , .60365 .52304 : .61974 1.62976 39 lit .57212 1.33708 .58796 1.42692 .60392 .52474 .62001 1.63162 40 11 .57238 1.33852 : .58822 1.42848 .60419 .52645 .62027 1.63348 41 4S .57264 1.33996 i .58849 1.43005 .60445 .52815 .62054 .63535 42 43 .57291 1.34140 i .58875 1.43162 .60472 .52986 , .62081 .63722 U 44 .57317 1.34284 .58902 1.43318 .60499 .53157 .62108 .63909 44 45 .57343 1.34429 ! .58928 1.43476 .60526 .5,3329 .62135 .64097 45 46 .57369 1.3-1573 .58955 1.43633 .60552 .53300 .62162 .64285 40 47 .57396 1.34718 i .58981 I 1.43790 .60579 .53672 .62189 .64473 47 48 i .57422 ! 1.34863 i .59008 1.43948 .60606 .53845 .62216 .64662 48 49 .57448 1.35009 .59034 i 1.44106 .60633 .54017 i .62243 .64851 49 50 .57475 1.35154 .59061 1.44264 .60659 .54190 . .62270 .65040 50 51 .57501 1.35300 1 .59087 1.44423 .60686 .54363 .62297 .65229 51 58 .57527 i 1.35446 , .59114 1.44582 .60713 .54536 .62324 .65419 52 58 .57554 i 1.35592 , .59140 i 1.44741 .60740 .54709 .62351 .65609 153 r,i .57580 1.35738 .59167 1.44900 .60766 .54883 .62378 .65799 54 55 .57606 1.3.-SS5 .59194 1.45059 .60793 .55057 .62405 .65989 .53 56 .57'633 1.36031 .59220 1.45219 .60820 .55231 .62431 .66180 56 57 .57659 1.36178 .59247 1.45378 .60847 .55405 .62458 .66371 57 ns .57685 I 1.36325 .59273 1.45689 .60873 .55580 .62485 .66563 58 59 .57712 1.36473 .59300 1.45699 .60900 . 55755 .62512 1.66755 !5'J (SO .57738 ! 1.36620 i .59326 1.45859 .60927 .55930 .62539 1.66947 100 TABLE XXIX. NATURAL VERSED SINES AND EXTERNAL SECANTS. / 68 69 70 71. ; / Vers. Ex. sec. Vers. Ex. sec. , Vers. I Ex. sec. Vers. Ex. sec. .62539 1.66947 .64163 1.7'9043 .65798 1.92380 .67443 ! 2.07155 ( 1 .62566 1.67139 .64190 1.79254 .65825 1.92614 .67471 ! 2.07415 2 .62593 1.67333 .64218 1.79466 i .65853 1.92849 .67498 2.07-67-5 < 3 .62620 1.67525 .64245 1.79679 .65880 1.93083 .67526 2.07936 j 4 .62647 1.67718 .64272 1.79891 ! .65907 1.93318 .67553 2.08197 i 5 .6267'4 1.67911 .64299 1.80104 .65935 1.93554 .67581 2.08459 { 6 .62701 1.68105 .64326 1.80318 .65962 1.93790 .67608 2.08721 1 7 .62723 1.68299 .64353 1.80531 .65989 1.94026 .67636 2.08983 \ 8 .62755 1.68494 .64381 1.807'46 .66017 1.94263 .67663 2.09246 j 9 .62782 1.68689 .64408 1.80960 .66044 1.94500 .67691 2.09510 c 10 .62309 1.68=84 .64435 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 1.69275 .64489 1.81605 .66126 1.95213 .67773 2.10303 12 13 .62890 1.69471 .64517 1.81821 i .66154 1.95452 .67801 2.10568 13 14 .62917 1.69667 .64544 1.82037 i .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 .66236 1.96171 .67884 2.11367 16 17 .62998 1.70258 .64625 1.82688 .66263 1.96411 .67911 2.11635 17 18 .63025 1.70455 .64653 1.82906 .66290 i 1.96652 .67939 2.11903 18 19 .63052 1.70653 .64680 1.83124 .66318 1.96893 .67966 2.12171 19 20 .63079 1.70851 .64707 1.8a342 .66345 1.97135 .67994 2.12440 20 21 .6310S 1.71050 .64734 1.83561 .66373 1.97377 .68021 2.12709 21 22 .63133 1.71249 .64761 1.83780 .66400 1.97619 .68049 2.12979 i22 23 .63161 1.71448 .(54789 1.83999 ! .66427 1.97862 .68077 2.13249 23 24 .63188 1.71647 .64816 1.84219 II .66455 1.98106 .68104 2.13520 24 25 .63215 1.71847 .64843 1.84439 ! .66482 1.98349 .68132 2.13791 25 26 .63242 1.72047 i .64870 1.84659 .66510 1.98594 .68159 2.14063 26 27 .63269 1.72247 ; .64898 1.84880 .66537 1.98838 .68187 2.14335 27 28 .63296 1.72448 i .64925 1.85102 .66564 1.99088 .68214 2.14608 28 29 .63323 1.72649 ! .64952 I 1.85323 .66592 1.99329 .68242 2.14881 2 ( 30 .63350 1.72850 .64979 1.85545 .66619 1.99574 | .68270 2.15155 30 31 .63377 1.73052 .65007 1.85767 .66647 1.99821 .68297 2.15429 31 32 .63404 1.73254 .65034 1.85990 .66674 2.00067 .68325 2.15704 32 33 .63431 1.73456 .65061 1.86213 .66702 2.00315 .68352 2.15979 33 34 .63458 1.73659 .65088 1.86437 i .66729 2.00562 .68380 2.16255 34 35 .63485 1.73862 .65116 1.86661 1 .66756 2.00810 .68408 2.16531 35 36 .63512 1.74065 .65143 1.86885 .66784 2.01059 .68435 2.16808 36 37 .63539 1.74269 i .65170 1.87109 .66811 2.01308 .68463 2.17085 37 38 .63566 1.74473 j .65197 1.87334 .66839 2.01557 .68490 2.17363 38 39 .63594 1 1.74677 .65225 1.87560 .66866 2.01807 .68518 2.17641 39 40 .63621 1.74881 .65252 1.87785 .66894 j 2.02057 .68546 2.17920 40 41 .63648 1.7.5036 .65279 1.88011 .66921 2.02308 .68573 2.18199 41 42 .68675 1.75292 j .65306 1.88238 .66949 i 2.02559 .6QP01 2.18479 42 43 .63702 1.75497 i .65334 1.88465 .6697'6 2.02810 .68628 2.18759 43 44 .63729 1.75703 ; .65361 1.88692 .67-003 2.03062 .68656 2.19040 44 S .63756 1.7'5909 .65388 1.88920 .67031 2.03315 .68684 2.19322 45 [6 .63783 1.76116 .65416 1.89148 .67-058 2.03568 .68711 2.19604 46 [7 .63810 1.76323 .65443 1.89376 ii .67086 2.03821 .68739 2.19886 47 48 .63838 1.76530 1 .6547'0 1.89605 .67113 i 2.04075 .68767 2.20169 48 49 .63865 1.767'37 .65497 1.89834 i; .67141 2.04329 .68794 2.20453 49 50 .63892 1.76945 .65525 1.90063 !| .67168 2.04584 .68822 2.20737 50 51 .63919 1.77154 ; .65552 1.90293 i .67196 2.04839 .68849 2.21021 51 52 .63946 1.77362 i .65579 1.90524 .67223 2.05094 .68877 2.21306 52 53 .63973 1.7757* .65607 1.90754 .67'251 2.05350 .68905 2.21592 53 54 .64000 1.77780 i .65634 1.90986 | .67278 2.05607 .68932 2.21878 54 >5 .64027 1.77990 i .65661 1.91217 .67306 2.05864 .68960 2.22165 55 56 .64055 1.78200 : .65689 1.91449 .67333 2.06121 .68988 2.22452 56 57 .64082 1.78410 ; .65716 1.91681 .67361 2.06379 .69015 2.22740 57 58 .64109 1.78621 ! .65743 1.91914 .67388 2.06637 .69043 2.23028 58 59 .64136 1.78832 ! .65771 1.92147 .67416 2.06896 .69071 2.23317 59 60 .64163 1.79043 .65798 1.92380 .67443 i 2.07155 1 .69098 2.23607 30 487 TABLE XXIX. NATURAL VERSED SINES AND EXTERNAL SECANTS. < 72 73 74 75 ' Vers. Ex. sec. \ Vers. Ex. sec.: Vers. Ex. sec. Vers. Ex. sec. .69098 2.23607 'i .70763 i 2.42030 .72436 2.62796 : .74118 2.86370 < 1 .69126 2.23897 1! .70791 i 2.42356 .72464 2.63164 ; .74146 2.86790 5 g .69154 2.24187 ! .70818 2.42683 .72492 1 2 63533 1 .74174 2.87211 2 3 .69181 2.24478 .70846 2.43010 .72520 2.03903 1 .74202 2.87633 3 4 .69209 2.24770 .70874 2.43337 .72548 2.64274 ; .74231 2.88056 4 5 .69237 2.25062 .70902 ; 2.43666 .72576 j 2.64045 .74259 2.88479 5 ,69264 2.25355 .70930 2.43995 .72604 2.65018 .74287 2.88S01 6 7 .69292 2.25648 .70958 2.44324 .72632 2.65391 .74315 2.89330 7 8 .69320 2.25942 .70985 2.44655 .72660 1 2.65705 .74343 2.89750 8 9 .69347 2 26237 .71013 2.44986 .72688 2.66140 i .74371 2.90184 9 10 .69375 2.26531 .71041 2.45317 .72710 2.66515 ; .74399 2.9CG13 10 11 .69403 2.26827 .71069 2.45650 .72744 2.60892 .74427 2.91042 11 19 .69430 2.27123 .71097 2.45983 .72772 2.67269 ^ .74455 2.91473 j 12 13 .69458 2.27420 .71125 2.46316 .72800 2.67647 .74484 2.91904 !13 11 .69486 2.27717 .71153 2.46651 .72828 2. 68025 J .74512 2.92337 ;14 15 .69514 2.28015 .71180 2.46986 .72850 2.68405 .74540 2.92770 i 15 10 .69541 2.28313 .71208 2.47321 .72884 2.68785 : .74568 2.93204 16 17 .69569 2.28612 .71236 2.47658 .72912 2.69167 .74596 2.93640 17 IS .69597 2.28912 .71204 2.47995 .72940 2.69549 .74624 2.94076 [18 1!) .69624 2.29212 .71292 2.48333 .72968 2.69931 .74652 2.94514 119 20 .69652 2.29512 .71320 2.48671 .72996 2.70315 i .74680 2.94952 20 21 .69680 2.29814 .71348 2.49010 .73024 2.70700 .74709 2.95892 21 28 .69708 2.30115 .71375 2.49350 .73052 2.71085 .74737 ',' . 1)5832 ' 22 23 .69735 2.30418 .71403 2.49691 .73080 2.71471 .747'65 2! 1)6274 23 24 .69763 2.30721 .71431 2.50032 .73108 2.71858 .747'93 2.90716 124 25 .69791 2.31024 .71459 2.50374 .73136 2.72246 .74821 2.97160 35 JO .69818 2.31328 .71487 2.50716 .73164 2.72635 .74849 2.97604 2C 27 .69846 2.31633 .71515 2.51060 .73192 2.73024 .74878 2.98050 27 ;j,S .69874 2.31939 .71543 2.51404 .73220 2.73414 ' .74906 2.98497 28 i;!) .69902 2,32244 .71571 2.51748 .73248 2.73806 .74934 -'.US'. Ml 29 80 .69929 2.32551 .71598 2.52094 .73276 2.74198 .74962 2.99393 30 31 .69957 2.32858 .71626 2.52440 .73304 2.74591 .74990 2.99843 31 3-J .69985 2.33166 .71654 2.52787 .73332 2.74984 .75018 3.0021)3 82 00 .70013 2.33474 .71682 2.53134 .73360 2.75379 .75047 3.00745 38 34 .70040 2.33783 .71710 2.53482 .73388 2.75775 .75075 3.01198 84 86 .70068 2.34092 .71738 2.53831 .73416 2.76171 .75103 3.01052 85 30 .70096 2.34403 .71766 2.54181 .73444 2.70508 .75131 3.02107 86 37 .70124 2.34713 .71794 2.54531 .73472 2.76960 .7'5159 3.02563 87 38 .70151 2.35025 .71822 ! 2.54883 .73500 2.77365 .75187 3.03020 88 89 .70179 2.35336 .71850 2.55235 .73521) 2.77765 .75216 3.03479 89 40 .70207 2.35649 .71877 2.55587 .73557 2.78166 .75244 3.03938 fi) 41 .70235 2.35962 .71905 2.55940 .73585 2.78568 .75272 3.04398 41 49 .70263 2.36276 .71933 2.56294 .73613 2.7897'0 .75300 3.04800 42 4:5 .70290 2.36590 .71961 2.56649 .73641 2.79374 .75328 8.05322 13 It .70318 2.36905 .71989 2.57005 .73669 2.79778 .75356 3.05786 J44 45 .70346 2.37221 .72017 2.573G1 .73697 2.80183 .75385 3.06251 45 46 .70374 2.37537 .72045 2.57718 .73725 2.80589 .75413 3.06717 40 47 .70401 2.37854 .72073 2.58076 .73753 2.80996 .75441 3.07184 47 48 .70429 2.38171 .72101 2.58434 .73781 2.81404 .75469 3.07652 48 49 .70457 2.38489 .72129 2.58794 .73809 2.81813 .75497 3.08121 49 50 .70485 2.38808 .72157 2.59154 .73837 2.82223 .75526 3.08591 50 51 .70513 2.39128 .72185 2.59514 .73865 2.826a3 .75554 3.09063 51 .70540 2.39448 .72213 2.59876 .73893 2.83045 .75582 3.09535 52 53 .70568 2.39768 .72241 2.60238 .73921 2.&S457 J5G10 3.10009 8 54 .70596 2.40089 .72269 ! 2.60601 .73950 2.83871 ^5639 3.10484 54 56 .70624 2.40411 .72296 2.60965 .73978 2.84285 .75667 3.10960 55 50 .70652 2.40734 .72324 2.61330 .74006 2.84700 i .75695 3.11437 56 57 .70679 2.41057 .72352 2.61695 .74034 2.85116 .75723 3.11915 57 58 .70707 2.41381 .72380 2.62061 .74062 2.85533 1 .75751 3.12394 58 59 .70735 2.41705 .72408 2.62428 .74090 2.85951 ! .75780 3.12875 59 60 .70763 2.42030 .72436 2.6279G .74118 2.86370 i .75808 3.13357 60 488 TABLK XXIX. NATURAL VERSED SINES AND EXTERNAL SECANTS 76 77 78 79 Vers. Ex. sec. i Vers. Ex. sec. ; Vers. j Ex. sec. Vers. Ex. sec. .75808 3.1335? i .77505 3.44541 |j .79209 3. 80973 i .80919 4.24084 li .75836 : 3.13839 .77533 3.45102 .79237 3.816&3 i .80948 4.24870 1 2 ] .75864 I 3.14323 .77562 3.45664 .79266 3.82294 ! .80976 4.25658 2 3 .75892 3.14809 i .77590 3.46228 .79294 3.82956 i .81005 4.26448 8 4 .75921 3.15295 i .77618 3.46793 .79323 3.83621 i .81033 4.27241 -! 5 .75W.) 3.15782 i .77647 3.47360 .79351 3.84288 ; .81062 4.28036 5 6 1 .75977 3.16271 .7767o 3.47928 .79380 3.84956 .81090 4.28833 6 .76005 3.16761 .77703 3.48498 .79408 3.85627 i .81119 4.29634 8 .76031 347252 .77732 3.49069 .79437 3.86299 ; .81148 4. 3043(5 8 y, .76062 3.17744 .77760 3.49642 .79465 3.86973 ! .81176 4.31241 9 10 1 .76090 3.18238 .77788 3.50210 .79493 3.87649 ' .81205 4.32049 Hi 11 .76118 3.1S733 j .77817 3.50791 .79522 3.88327 ! .81233 4.32859 1! 12 .76147 3.19228 : .77845 3.51368 .79550 3.89007 i .81262 4.33671 !12 13 .76175 3.19725 .77S74 3.51947 .79579 3.89689 " .81290 4.34486 13 14 .76-303 3.20224 .77902 3.52527 .79607 3.90373 i .81319 4.35304 11 15 .76231 3.20723 .77930 3.53109 .79636 3.91058 .81348 4.36124 15 16 .76260 3.21224 .77959 3.53692 .79664 3.91746 ' .81376 4.36947 :16 17 i .7.; .76158 8,24764 .78157 3.57819 .79864 : 3.96616 .81576 4.42778 ;.':; 24 .76486 3.25275 .78186 3.58414 .79892 3.97320 .81605 4.43622 24 ;.':> .76514 3.35787 .78214 3.59012 .79921 3.98025 .81633 4.44468 25 jr. .76543 3.26300 .78242 3.59611 ..79949 3.98733 ; .81662 4.45317 26 . .76571 3.26814 .78271 3.60211 .7997'8 3.99443 .81691 4.46169 87 ys .76599 3.27330 .78299 3.60813 .80006 4.00155 .81719 4.47023 28 29 .76627 3.27847 .78328 3.61417 .80035 4.00869 1 .81748 4.47881 -,".) 80 .76655 3.28366 .78356 3.62023 .80063 4.01585 .81776 4.48740 80 31 .76684 3.28885 .78384 3.62630 .80092 4.02303 .81805 4.49603 81 82 .76712 3.29406 .78413 3.63238 .80120 4.03024 .81834 4.50468 32 ',:', .76740 3.29929 .78441 3.63849 .80149 4.03746 .81862 4.51337 38 34 .76769 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 85 36 .76825 3.31503 .78526 3.65690 .80234 4.05926 .81948 4.53958 |36 :>; .70854 3.32031 .78555 3.66307 .80263 4.06657 .81977 4.54837 !37 38 .76882 3.32560 .78583 3.66925 .80291 4.07'390 .82005 4.55720 38 39 .76910 3.33090 .78612 3.67545 .80320 4.08125 .82034 4.56605 39 in .76938 3.33622 .78640 3.6S167 .80348 4.08863 .82063 4.57493 40 !! .76967 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 .77'0r>3 3.35224 .78725 3.70044 .804:34 4.11088 .82148 4.6017'4 43 44 .77052 3.35761 .78754 8.7Q678 .80462 4.11835 .82177 4.61073 II 45 .77080 3.36299 .78782 3.71303 .80491 4.12583 .82206 4.61976 15 46 .77108 3.36839 [78811 3.71935 .80520 4.13334 .82234 4.62881 ]<; 47 .77137 3.37380 .78839 3.72569 .80548 4.14087 .82263 4.63790 47 48 i .77165 3.37923 .78868 3.73205 .80577 4.14842 .82292 4.64701 ;48 49 .77193 3.38466 .78896 3' 78843 .80605 4.15590 .82320 4.65616 i49 50 .77222 3.39012 .78924 3.7'4482 j .80634 4.16359 .82349 4.66533 50 M .77250 3.39558 .78953 3.75123 .80662 4.17121 .82377 4.67454 1 51 52 .77'27'8 3.40106 .78981 3.75766 .80691 4.17886 .82406 4.68377 152 53 .77307 3.406*6 .79010 3.76411 ,.80719 i 4.18652 .82435 4.69304 53 54 .77335 3.41206 .79038 3.77057 .80748 1 4.19421 .82463 4.70234 54 5o! .77363 3.41759 .79067 3.77705 .80776 4.20193 .82492 4.71166 .",") 56 .77392 3.42312 .79095 3.78355 .80805 4.20966 .82521 4.72102 56 57 .77420 3.42867 .79123 3.79007 .80833 4.21742 .82549 4.73041 67. 58 .77448 3.43424 .79152 3.79661 .80862 4.22521 .82578 4.73983 68 59 .77477 3.43982 .79180 3.80316 .80891 4.23301 .82607 4.74929 59 80 .77505 3.44541 1 .79209 3.8097'3 .80919 4.24084 .82635 4.75877' 60 489 TABLE XXIX.-NATURAL VERSED SIXES AND EXTERNAL SECANTS. 80 81 82 83 ' Vers. Ex. sec. Vers. Ex. sec. Vers. Ex. sec. Vers. Ex. sec. .82635 4.75877 .84357 5.39245 .1 .86083 6.18530 .87813 1 7.20551 ! 1 .82664 4.76829 .84385 5.40422 ; i .86112 6.20020 .87842 7.22500 1 .82892 4.77784 .84414 5.41602 : | .86140 6.21517 i .87871 7.24457 2 3 .82721 4.78742 .84443 5.42787 ii .86169 6.23019 ! .87900 7.26425 3 4 .82750 4.79703 .84471 5.43977 i -86198 6.24529 .87929 7.28402 i 5 .82778 4.80667 .84500 5.45171 .86227 6.26044 .87957 7.30388 5 .82807 4.81635 .84529 5.46369 .86256 6.27566 .87986 7.32384 6 J .82836 4.82606 .84558 5.47572 .86284 6.29095 .88015 7.34390 7 8 .82864 4.83581 .84586 5.48779 i .86313 6.30630 .88044 7.36405 8 9 .82893 4.84558 .84615 5.49991 .86:342 6.32171 .88073 7.38431 g 10 .82922 4.85539 .84644 5.51208 !; .86371 6.33719 .88102 7.404U6 to 11 .82950 4.86524 .84673 5.52429 .86400 6.35274 .88131 7.42511 n 12 .82979 4.87511 .84701 5.53655 ; .86428 6.36835 .88160 7.44566 12 13 .83003 4.88502 .84730 5.54886 : .86457 6.38403 , .88188 7.46632 i 13 14 .83036 4.89497 .84759 5.56121 .86486 6.39978 \ .88217 7.48707 14 15 .83065 4.90495 ! .84788 5.57361 ! .86515 6.41560 I .88246 7.50793 15 16 | .83094 4.91496 ; .84816 5.58606 .86544 6.43148 ! .88275 7.52S89 16 17 .83122 4.92501 i .84845 5.59855 i .86573 6.44743 .88304 7.54996 17 18 .83151 4.93509 .84874 5.61110 .86601 6.46346 .88333 7.57113 18 1!) .83180 4.94521 .84903 5.62369 : ! .86630 6.47955 .88362 7.59241 ;19 20 .83208 4.95536 .84931 5.63633 ;i .86659 6.49571 .88391 7.61379 ; 20 21 .83237 4.96555 .84960 5.64902 ! .86688 6.51194 ! .88420 7.6a528 21 32 .83266 4.97577 .84989 5.66176 ! .86717 6.52825 Si .88448 7.65688 ,22 23 .83294 4.98603 .85018 5.67454 |i .86746 6.54462 : .88477 7.67859 23 24 .83323 4.99633 .85046 5.68738 1 .86774 6.56107 .88506 7.70041 24 25 .83352 5.00666 .85075 5.70027 : .86803 6.57759 .88535 7.72234 25 86 .83380 5.01703 .85104 5.71321 ! .86832 6.59418 ': .88564 7.74438 26 .83409 5.02743 .85133 5.72620 .86861 6.61085 .88593 7.76653 27 28 .83438 5.03787 ! .85162 5.73924 .86890 6.62759 .88622 7.78880 28 29 .83467 5.04834 i .85190 5.75233 .86919 6.64441 .88651 7.81118 29 30 .83495 5.05886 .85219 5.76547 .86947 6.66130 .88680 7.83367 30 31 .83524 5.06941 .85248 5.77866 .86976 6.67826 .88709 7.85628 3! 3v> .83553 5.08000 1 .85277 5.79191 j .87005 | 6.69530 .88737 7.87901 32 33 .83581 I 5.09062 il .85305 5.80521 i .87034 6.71242 .88766 7.90186 83 84 83610 5.10129 .85334 5.81856 ; .87063 6.72962 .88795 7.92482 31 35 .83639 5.11199 1 .85363 5.83196 I .87092 6.74689 .88824 7.94791 135 86 .83667 5.12273 .85392 5.84542 ! .87120 6.76424 .88853 7.97111 |36 37 .83696 5.13350 .85420 5.85893 .87149 6.78167 .88882 7.99444 87 88 83?'25 5.14432 .85449 5.87250 .87178 6.79918 .88911 8.01788 38 39 .83754 5.15517 i .85478 5.88612 .87207 6.81677 .88940 8.04146 89 40 .83782 5.16607 | .85507 5.89979 .87236 6.83443 .88969 8.06515 40 41 83811 5 17700 ! .85536 5.91352 .87265 6.85218 .88998 8.08897 11 42 i ! 83840 5.18797 'I .85564 5.92731 i .87294 6.87001 .89027 8.11292 42 43! .83368 5.19898 |! .85593 5.94115 .87322 6.88792 .89055 8.13699 13 44 .83897 5.21004 .856.22 5.95505 .87*51 6.90592 .89084 8.16120 44 45 83926 5 22113 .85651 5.96900 .87380 6.92400 .89113 8.18553 45 46 .83954 5.23226 .85680 5.98301 .87409 6.94216 .89142 8.20999 46 47! 83933 5 24343 : .85708 5.99708 .87438 6.96040 ! .89171 8.23459 47 48 .84012 5.25464 .85737 6.01120 .87467 6.97873 .89200 8.25931 48 49 84041 5.26590 .85766 6.02538 .87496 6.99714 .89229 8.28417 49 50 ' .84069 5.27719 1 .85795 6.03962 .87524 7.01565 .89258 8.30917 J50 51 .84098 5.28853 ; .85823 C. 05392 1 .87553 7.03423 .89287 8.33430 51 84127 5 29991 .85852 6.06828 .87582 7.05291 .89316 8.35957 52 53 54 .84155 .84184 5.81133 5.32279 .85881 .85910 6.08269 6.09717 .87611 .87640 7.07167 7.09052 , .89345 .89374 8.38497 53 8.41052 54 55 50 57 58 .84213 .84242 .84270 .84299 5.33429 5.34584 5.35743 5.36906 .85939 .85967 .85996 .86025 6.11171 6.12630 6.14096 6.15568 .87669 .87698 .87726 .87755 7.10946 7.12849 7.14760 7.16681 .89403 .89431 .89460 .89489 8.43620 i55 8.46203 56 8.48800 157 8.51411 58 59 00 .84328 .84357 5.38073 5.39245 .86054 .86083 6.17046 6.18530 .87784 ! .87813 7.18612 7.20551 1 .89518 1 .89547 8.54037 |59 8.56677 !60 490 TABLE XXIX. NATURAL VERSED SINES AND EXTERNAL SECANTS. J | 4 o * J5 6 / Vers. Ex. sec. Yers. Ex. sec. Yers. Ex. sec. .89547 8.56677 .91234 10.47371 .93024 13.33559 1 .89576 8.59332 .91313 10.51199 .93053 13.39547 1 2 .89605 8.62002 .91342 10.55052 .93082 13.45586 2 3 .89034 8.64687 .91871 10.58932 .93111 13.51676 3 4 .89663 8.67387 .91400 10.62837 .93140 13.57817 4 5 .89892 8.70103 .91429 10.66769 .93169 13.64011 5 6 .89721 8.72333 .91458 10.70728 .93198 13.70258 6 7 '.89750 8.75579 .91487 10.74714 .93227 13.76558 7 8 .89779 8.78341 .91516 10.78727 .93257 13.82913 8 9 .89808 8.81119 .91545 10.82768 .93286 13.89323 9 10 .89836 8.83912 .91574 10.86837 .93315 13.95788 10 11 .89865 8.86722 .91603 10.90934 .93344 14.02310 11 12 .89894 8.89547 .91032 10.95060 .93373 14.08890 12 13 .89923 8.92389 .91061 10.99214 .9:3402 14.15527 13 14 .89952 8.95248 .91690 11.03397 .93431 14.22223 14 15 .89931 8.98123 .91719 11.07610 .93460 14.28979 15 16 .90010 9.01015 .91748 11.11852 .93489 14.35795 16 17 .90J39 9.03923 .91777 11.16125 .93518 14.42672 17 18 .90038 9.08349 .91806 11.20127 .93547 14.49611 18 19 .90097 0.09792 .91835 11.24761 .93576 14.56614 19 20 .9C126 9.12752 .91864 11.29125 .93605 14.63679 20 21 .90155 9.15730 .91893 11.33521 .93634 14.70810 21 22 .90184 9.18725 .91922 11.37948 .93663 14.78005 22 23 .90213 9.21739 ! .91951 11.42408 .93692 14.85268 23 24 .90242 9.24770 i .91980 11.46900 .93721 14.92597 24 25 .90271 9.27819 .92009 11.51424 .93750 14.99995 25 20 .90300 9.30887 .92038 11.55982 .93779 15.07462 26 27 .'JO;-.'!) 9.33973 .92067 11.60572 .93803 ' 15.14999 27 28 .90338 9.37077 .92096 11.65197 .93837 15.22607 28 29 .90-^86 9.40201 .92125 11.69356 .93866 15.30237 29 30 .90415 9.43343 .92154 11.74550 .93895 15.38041 30 31 .90444 9.46505 .92183 11.79073 .93924 15.45869 31 32 .90473 9.49685 .92212 11.84042 .93953 15.53772 32 33 .90502 9.52886 .92241 11.88841 .93982 15.61751 33 34 .90531 9.56106 .92270 11.93677 .94011 15.69808 34 35 .90560 9.59346 .92299 11.98549 i .94040 15.77944 35 36 .90589 9.62605 .92328 12.03450 1 .94069 15.86159 36 37 .90618 9.65885 .92357 12.08040 ; .94098 15.94456 37 38 .90647 9.69186 .92386 12.13388 .94127 16.02835 38 39 .90676 9.72507 .92415 12.18411 .94156 16.11297 39 40 .90705 9.75849 .92444 12.23472 .94186 16.19843 40 41 .90734 9.79212 .92473 12.2857-3 .94215 16.28476 41 42 .90763 9.82596 .92502 12.33712 .94244 16.37196 42 43 .90792 9.86001 .92531 12.38891 .94273 16.46005 43 44 .90821 9.89428 .92560 12.44112 .94302 16.54903 44 45 .90350 9.92877 .92589 12.49373 .94331 16.63893 45 46 .90879 9.96348 .92618 12.54676 .94360 16.72975 46 47 .90908 9.99841 .92647 12.60021 .94389 16.82152 47 48 .90937 10.03356 .92676 12.65408 .94418 16.91424 48 49 .90966 10.06894 .92705 12.70838 .94447 17.00794 49 50 .90995 10.10455 .92734 12.76312 .94476 17.10262 50 51 .91024 10.14039 .92763 12.81829 .94505 17.19830 51 5-3 .91053 10.17646 .92792 12.87391 .94534 17.29501 52 63 .91082 10.21277 .92821 12.92999 .94563 17.39274 53 54 .91111 10.24932 .92850 12.98651 .94592 17.49153 54 55 .91140 10.28610 .92879 13.04350 .94621 17.59139 55 56 .91169 10.32313 .92908 13.10096 .94650 17.69233 56 57 .91197 10.36040 .92937 13.15889 .94679 17.79438 57 58 .91226 10.39792 .92966 13.21730 .94708 17.89755 58 59 .91255 10.43569 .92995 13.27'620 .94737 18.00185 5,0 60 .91284 10.47371 .93024 13.33559 .94766 18.10732 60 491 TABLE XXIX. NATURAL VERSED SINES AND EXTERNAL SECANTS. / { S7 8 8 B 9 Vers. Ex. sec. Vers. Ex. sec. Vers. Ex. sec. .94766 18.10732 .96510 27.65371 .98255 56.29869 1 .94795 18.21397 .96539 27.89440 .98284 57.26976 1 2 .94825 18.32182 .96568 28.13917 .98313 58.27431 2 3 .94854 18.43088 .96597 28.38812 i .98342 59.31411 3 4 .94883 18.54119 .96626 28.64137 .98371 60.39105 4 5 .94912 18.65275 .96655 28.89903 .98400 61.50715 5 6 .94941 18.76560 .96684 29.16120 ! .98429 62.66460 6 7 .94970 18.87976 .96714 29.42802 .98458 63.86572 7 .94999 18.99524 .96743 29.69960 .98487 65.11304 8 9 .95028 19.11208 .90772 29.97607 .98517 66.40927 9 10 .95057 19.23028 .96801 30.25758 .98546 67.75736 10 11 .95086 19.34989 .96830 30.54425 .98575 G9. 16047 11 12 .95115 19.47093 .96859 30.83623 .98604 70.62285 12 13 .95144 19.59341 .96888 31.13366 .98633 72.14583 13 11 .95173 19.71737 .96917 31.43671 .98662 73.7'3580 14 15 .95202 19.84283 .96946 31.74554 .98691 75.39655 15 10 .95231 19.96982 .96975 32.06030 .98720 77.13274 16 17 .95280 20.09838 .97004 32.38118 .98749 78.94968 17 18 .95289 20.22852 .97033 32.70835 .98778 80.85315 18 19 .95318 20.36027 .97062 33.04199 .98807 82.84947 19 20 .95347 20.49368 .97092 33.38232 .98836 84.94501 20, 21 .95377 20.62876 .97121 33.72952 .88866 87.14924 21 22 .95406 20.76555 .97150 34.08380 .98895 89.4G8SO 22 23 .95435 20.90403 .97179 34.44539 .98924 91.91387 23 24 .954(54 21.04440 .97208 34.81452 .98953 94.49471 24 25 .95-193 21.18653 .97237 35.19141 .98982 97.22303 25 20 .95522 21.33050 .97206 35.57633 .C9011 100.1119 26 27 .95551 21.47635 .97295 35.96953 .99040 103.1757 27 28 .95580 21.62413 .97324 36.37127 .99069 106.4311 28 29 .95609 21 . 77386 .97a53 36.78185 .99098 109.8906 29 30 .95638 21.92559 .97382 37.20155 .99127 113.5930 30 31 .95667 22.07935 .97411 37.63068 .9156 117.5444 31 32 .95696 22.23520 .97440 38.06957 .99186 121.7780 32 33 .95725 22.39316 .97470 38.51855 .99215 126.3253 33 34 .95754 22.55329 .97499 38.977 ( J7 .99244 131.2223 34 35 .95783 22.71563 .97523 39.44820 .99273 136.5111 35 36 .95812 22.88022 ' .97557 39.92903 .99302 142.2406 36 37 .95842 23.04712 .97'586 40.42206 .99331 148.4084 37 38 .95871 23.21637 .97615 40.92772 .99860 155.2623 38 39 .95900 23.38802 .97644 41.44525 .99-389 162.7033 39 40 .95929 23.56212 .97673 41.97571 .S9418 170.8883 40 41 .95958 23.73873 .97702 42.51961 .99447 17'9.9350 41 42 .95987 23.91790 .97731 43.07745 .9947'6 189.9808 42 43 .96016 24.09969 .97760 43.64980 .99505 201.2212 43 44 .96045 24.28414 .97789 44.23720 .995&5 213.8600 44 45 .96074 24.47134 .97819 44.84026 .99564 228.18:39 45 46 .96103 24.66132 .97848 45.45963 .99593 244.5540 46 47 .96132 24.85417 .97877 46.09596 .99622 263.4427 48 .96161 25.04994 .97906 46.74997 .99651 285.4795 48 49 .96190 25.24869 .97935 47.42241 .99080 311.5230 49 50 .96219 25.45051 .97964 48.11406 .99709 342.7752 50 51 .96248 25.65546 ; .97993 48.82576 .99738 380.9723 51 53 .96277 25.86300 .98022 49.55840 .99707 428.7187 52 53 .96307 26.07503 .98051 50.31290 .99796 490.1070 53 54 .96336 26.28981 98080 51.09027 ! .99825 571.9581 54 55 .96365 26.50804 !98109 51.89150 .99855 686.5490 55 56 .96394 26.72978 .98138 52.71790 .99884 858.4309 56 57 .96423 26.95513 .98168 53.57046 .99913 1144.910 57 58 .96452 27.18417 .98197 54.450.-.:} .99942 1717.874 58 59 .96481 27.41700 .98226 55.. 35946 .99971 3436.747 59 60 .96510 27.65371 .98255 56.29869 1.00000 Infinite 60 TABLE XXX.-CUBIC YARDS PER 100 FEET. SLOPES Depth Base Base | Base I Base Base Base Base Base d 12 14 16 | 18 22 24 26 28 1 45 53 60 68 82 90 97 105 2 93 107 122 137 167 181 196 211 3 142 163 186 208 253 275 297 319 4 193 222 252 281 341 370 400 430 5 245 282 319 356 431 468 505 542 6 300 344 389 433 522 567 611 656 356 408 460 612 616 668 719 771 8 415 474 533 593 711 770 830 889 9 475 542 608 675 808 875 942 1008 10 537 611 685 759 9or 981 1056 1130 11 601 682 764 845 1008 1090 1171 1253 12 667 756 844 933 1111 1200 1289 1378 13 734 831 926 1023 1216 1312 1408 1505 14 804 907 1010 1115 1322 1426 1530 1633 15 875 986 1096 1208 1431 1542 1653 1764 16 948 1067 1184 1304 1541 1659 1778 1896 1 7 1023 1149 1274 1401 1653 1779 1905 2031. 18 1100 1233 1366 1500 1767 1900 2033 2167 19 1179 1319 1460 1601 1882 2023 2164 2305 2> 1259 1407 1555 1704 2000 2148 2296 2444 21 1342 1497 1653 1808 2119 2275 2431 2586 K 1426 1589 1752 1915 2241 2404 2567 2730 23 1512 1682 ia53 2023 2364 2534 2705 2875 2i 1600 1778 1955 2133 2489 2667 2844 3022 25 1690 1875 2060 2245 2616 2801 2986 3171 26 1781 1974 2166 2359 2744 2937 3130 - 3322 1875 2075 2274 2475 2875 3075 3275 3475 23 1970 2178 2384 2593 3007 3215 3422 3630 29 2068 2282 2496 2712 3142 3356 3571 3786 33 2167 2389 2610 2833 3278 3500 3722 3944 31 2233 2497 2726 2956 3416 3645 3875 4105 32 2370 2607 2844 3081 3556 3793 4030 4267 33 2475 2719 2964 3208 3697 3942 4186 4431 34 2581 2833 3085 3337 3841 4093 4344 4596 35 2690 2949 3208 3468 3986 4245 4505 4764 36 2800 3067 3333 3600 4133 4400 4667 4933 37 2912 3186 3460 3734 4282 4556 4831 5105 38 3026 3307 3589 3870 4433 4715 4996 5278 39 3142 3431 3719 4008 4586 4875 5164 5453 40 3259 3550 3852 4148 4741 5037 5333 5630 41 3379 3682 3986 4290 4897 5201 5505 5F08 4-2 3500 3811 4122 4433 5056 5367 5678 5989 43 3623 3942 4260 4579 5216 5534 5853 6171 44 3748 4074 4400 4726 5378 5704 6030 6356 45 3875 4208 4541 4875 5542 5875. 6208 6542 40 4004 4344 4684 5026 5707 6048 6389 6730 47 4134 4482 4830 5179 5875 6223 6571 6919 48 4267 4622 4978 5333 6044 6400 6756 7111 49 4401 4764 5127 5490 6216 6579 6942 7305 50 4537 4907 5278 5648 6389 ; 6759 7130 7500 51 4675 5053 5430 5808 6564 6942 7319 7697 52 4815 5200 5584 5970 6741 7126 7511 7S96 53 4950 5349 5741 6134 6919 7312 7705 8097 54 5100 5500 5900 6300 7100 7500 7900 8300 55 5245 5653 6060 6468 7282 7690 8097 8505 56 5393 5807 6222 6637 74(57 7881 8296 8711 57 5542 5964 6386 6808 7653 8075 8497 8919 58 5693 6122 6552 6981 7841 8270 8700 9130 59 5845 6282 6719 7156 8031 8468 8905 9342 60 6000 6444 6889 7333 8222 8667 9111 9556 i 493 TABLE XXX. CUBIC YARDS PER 100 FEET. SLOPES ^ : 1. Depth d Base 12 Base ! Base 14 16 Base 18 Base 22 Base Base 24 26 Base 28 1 46 54 Gl 69 83 91 98 106 2 96 111 126 j 141 17'0 185 200 215 3 150 172 194 217 261 283 306 328 4 207 237 267 296 356 385 415 444 5 269 306 343 380 454 491 528 565 6 333 378 422 467 556 600 644 689 402 454 506 557 661 713 765 817 8 474 533 593 652 770 830 889 948 9 550 617 683 750 883 950 1017 1083 10 630 704 778 852 1000 1074 1148 1222 11 713 794 876 957 1120 1202 1283 1365 12 800 889 978 1067 1244 1333 1422 1511 13 891 987 1083 1180 1372 1469 1565 1661 14 985 1089 1193 1296 1504 1607 1711 1815 15 1083 1194 1306 1417 1639 1750 18G1 1972 16 1185 1304 1422 1541 1779 1896 2015 2133 % 17 1291 1417 1543 1669 1920 2046 2172 ! 2298 18 1400 1533 1667 1800 20G7 2200 2333 2467 19 1513 1654 1794 1935 2217 2357 2498 2639 20 1630 1778 1926 2074 2370 2519 2667 2815 21 1750 1906 2061 2217 2528 2683 2839 2994 22 1874 2037 2200 2363 2689 2852 3015 3178 23 2002 2172 2343 2513 2854 3024 3194 3365 24 2133 2311 2489 2667 3022 3200 3378 3556 25 2269 2454 2639 2824 3194 3380 3565 87oO 26- 2407 2600 2793 2085 3370 35C3 3756 3948 27 2550 2750 2950 3150 3550 3750 3950 4151 28 2696 2904 3111 3319 3733 3941 4148 4356 29 2846 3061 3276 3491 3920 4135 ! 4350 4565 30 3000 3222 3444 3667 4111 4333 4556 4778 31 3157 3387 3617 3846 4306 4535 4765 4994 32 3319 3556 3793 4030 4504 4741 4978 5215 33 3483 3728 3972 4217 4706 4950 5194 5439 34 3652 3904 4156 4407 4911 5163 5415 6667 35 3824 4083 4343 4602 5120 5380 5639 5898 36 4000 4267 4533 4800 5333 5600 5867 6133 37 4180 4454 4728 5002 5550 5824 6098 6372 38 4363 4644 4926 5207 5770 1 G052 6333 6615 39 4550 4839 5128 5417 5994 6283 6572 6861 40 4741 5037 5333 5630 6222 6519 6815 7111 41 4935 5239 5543 5846 6454 6757 7061 7365 42 5133 5444 5756 6067 6689 7000 7311 7623 43 5335 5654 5972 6291 6928 7246 7565 7883 44 5541 5867 6193 6519 7170 7496 7822 8148 45 5750 6083 6417 6750 7417 7750 8083 8417 46 5963 6304 6644 6985 7667 8007 8348 8G89 47 6180 6528 6876 7224 7920 8269 8617 89G5 48 49 6400 6624 6756 6987 7111 7350 7467 7713 8178 8439 8802 m 9244 9528 50 6852 7222 7593 7963 87C4 9074 9444 9815 51 7083 7461 7839 8217 8972 9350 9728 10106 52 7319 7704 8089 8474 9244 9630 10015 10400 53 7557 7950 8343 8735 9520 9913 10306 10G98 54 7800 8200 8600 9000 9800 10200 10GOO 11000 55 8046 8454 8861 92G9 10083 10491 10898 11806 56 8296 8711 9126 9541 10370 10785 11200 11615 57 8550 8972 9394 9817 10661 11083 11506 11928 58 8807 9237 9667 10096 10956 11385 11815 i 12244 59 9069 9506 9943 10380 11254 11691 12128 12565 60 9333 9778 ! 10222 10667 11556 12000 12444 12889 ' 494 TABLE XXX. CUBIC YARDS PER 100 FEET. SLOPES 1 : 1. Depth Base Base Base Base Base Base Base Base d 12 14 16 18 20 23 30 32 1 48 56 63 70 78 107 115 122 2 104 119 183 148' 163 222 237 252 3 167 189 211 233 ' 256 344 367 389 4 237 267 296 326 356 474 504 533 5 315 352 389 426 463 611 648 685 6 400 444 489 533 578 756 800 844 493 544 596 648 700 907 959 1011 8 593 652 711 770 830 1067 1126 1185 9 700 767 833 900 967 1233 1300 1367 10 815 889 963 1037 1111 1407 1481 1556 11 937 1019 1100 1181 1263 1589 1670 1752 12 1067 1156 1244 13*3 1422 1778 1867 1956 13 1204 1300 1396 1493 1589 1974 2070 2167 14 1348 1452 1556 1659 1763 2178 2281 2385 15 1500 1611 1722 1833 1944 2389 2500 2611 16 1659 1778 1896 2015 2133 2607 2726 2844 17 1826 1952 2078 2204 2330 2833 2959 3085 18 2000 2133 2267 2400 2533 3067 3200 3333 19 2181 2322 2*3 2604 2744 3307 3448 3589 20 2370 2519 2667 2815 2963 3556 3704 3852 21 2567 2722 2878 3033 3189 3811 3967 4122 22 2770 2933 3096 3259 3422 4074 4237 4444 23 2981 3152 3322 3493 3663 4344 4515 4685 24 3200 3378 3556 3733 3911 4622 4800 4978 25 3426 3611 3796 3981 4167 4907 5093 5278 26 3659 3852 4044 4237 4430 5200 5393 5585 2?' 3900 4100 4300 4500 4700 5500 5700 5900 28 4148 4356 4563 4770 4978 5807 6015 6222 29 4404 4619 4833 5048 5263 6122 6337 6552 30 4667 4889 5111 5333 5556 6444 6667 6889 31 4937 5167 5396 5626 5856 6774 7004 7233 32 5215 5452 5689 5926 6163 7111 7348 7585 33 5500 5744 5989 6233 6478 7456 7700 7944 84 5793 6044 6296 6548 6800 7807 8059 8311 35 6093 6352 6611 6870 7130 8167 8426 8685 36 6400 6667 6933 7200 7467 8533 8800 9067 37 6715 6989 7263 7537 7811 8907 9181 9456 33 7037 7319 7600 7881 8163 9289 9570 9852 39 7367 7656 7944 8233 8522 9678 9967 10256 40 7704 8000 8296 8593 8889 10074 10370 10667 41 8048 8352 8656 8959 9263 10478 10781 11085 42 8400 8711 9022 9333 9644 10889 11200 11511 43 8759 9078 9396 9715 10033 11307 11626 11944 44 9126 9452 9778 10104 10430 11733 12059 12385 45 9500 9833 10167 10500 10833 12167 12500 12833 46 9881 10222 10563 10904 11244 12607 12948 13289 47 10270 10619 10967 11315 11663 13056 13404 13752 48 10667 11022 11378 11733 12089 13511 13867 14222 49 11070 11433 11796 12159 12522 13974 14337 14700 50 11481 11852 12222 12593 12963 14444 14815 15185 51 11900 12278 12656 13033 13411 14922 15300 15678 52 12326 12711 13096 13481 13867 15407 15793 16178 53 12759 13152 13544 13937 14330 15900 16293 1G685 54 13200 13600 14000 14400 14800 16400 16800 17200 55 13648 14056 14463 14870 15278 16907 17315 17722 56 14104 14519 14933 15348 15763 17422 17837 18252 57 14567 14989 15411 15833 16256 17944 18367 18789 58 15037 15467 15896 16326 16756 18474 18904 19333 59 15515 15952 16389 16826 17263 19011 19448 19885 60 16000 16444 16889 17333 17778 19556 20000 20444 495 TABLE XXX. CUBIC YARDS PER 103 FEET. SLOPES \y> : 1. Depth d Base 12 Base 14 Base 16 Base 18 Base 20 Base 28 Base 30 Base 32 1 50 57 65 73 80 109 117 124 2 111 126 141- 158 170 230 244 259 3 183 206 228 250 272 361 383 406 4 267 296 326 356 385 504 533 563 5 361 398 435 47'2 509 037 694 731 6 4G7 511 556 GOO 644 822 867 911 7 583 635 687 739 791 998 J 1050 1102 8 711 770 830 889 948 1185 ! 1244 1304 9 850 917 983 1050 1116 1383 1450 1517 10 1000 1074 1148 1222 1296 1593 1667 1741 11 1161 1243 1324 1406 1487 1813 1894 1976 12 1333 1422 1511 1600 1689 2044 2133 2222 13 1517 1613 1709 1806 1902 2287 2383 2480 14 1711 1815 1919 2022 2126 2541 i i 27-48 15 31)17 2028 2139 2250 2361 2806 2917 3028 16 2133 2252 2370 2489 2607 3081 3200 3319 17 2301 2487 2613 2739 2865 3369 3494 3620 18 2600 2733 2867 3000 was 3667 3800 3933 19 2850 2991 3131 3272 3413 3976 4117 4257 . 2.0 3111 3259 3407 3556 3704 4296 4444 4592 21 3383 3539 3694 3850 4005 4628 4783 4939 22 3667 3830 3993 4156 4318 4970 5133 5296 23 3961 4131 4302 4472 4642 5324 5494 ' 5665 24 4267 4444 4622 4800 4078 5689 58G7 6044 25 4583 4769 4954 5139 5324 6065 6250 6435 26 4911 5104 5296 5489 5681 6452 66-44 6837 27 5250 5450 5650 5850 6050 6850 7050 7250 28 5600 5807 6015 6222 6430 7259 7467 7674 29 5961 6176 6391 6606 6820 7680 7894 8109 30 6333 6556 6778 7000 7222 8111 8333 8555 31 6717 6946 7176 7406 7635 8554 8783 9013 32 7111 7348 7585 7822 8059 9007 !41 9482 33 7517 7761 8006 8250 8494 1472 9717 9962 34 7933 8185 8437 8689 8941 9948 10800 10452 35 8361 8620 8880 9139 9398 10435 10694 10954 36 8800 9067 9333 9600 9867 10933 11200 11467 37 9250 9524 9798 10072 10346 11443 11717 11991 38 9711 9993 10274 10556 10837 11963 12244 1252G 39 10183 1047'2 ! 10761 11050 11339 12494 12788 13072 40 10667 10963 11259 11556 11852 13037 13333 13630 41 11161 11465 11769 12072 12376 13591 13894 14198 42 11667 11978 ' 12289 12600 12911 14156 14467 14778 43 12183 12502 i 12820 13139 13457 14731 15050 15369 44 12711 13037 ! 1.3363 13689 14015 15319 15644 15970 45 13250 13583 13917 14250 14583 15917 16250 16583 46 13800 14141 14481 14822 15163 16526 168G7 17207 47 14361 14709 15057 15406 15754 17146 17194 17843 48 14933 15289 15644 16000 16356 17778 18133 18489 49 15517 15880 16243 16606 16968 18420 1S7K3 ' 1011(5 50 16111 16481 16852 17222 17592 19074 11)111 19815 51 16717 17094 17472 17850 18228 19739 20117 2045)4 52 173.53 17719 18104 18489 18874 20415 r:nsiW 21 1S5 53 17961 18354 1S7'46 19139 19531 21102 21494 21887 54 18300 19000 19400 10800 20200 21800 22200 22600 55 19250 19657 20065 20472 20880 22509 22917 ! S332-i 56 19911 20326 20741 21156 21570 23230 23044 240.-)'.) 57 20583 21006 21428 21850 2227'2 23961 243K-J 24805 58 21267 21696 22126 22556 22985 247'04 25133 25563 59 21961 22398 2288S5 83272 237'09 85457 2.V'!ll 2G332 GO 2-20(17 23111 23556 24000 24444 26222 26G67 27111 49G TABLE XXX. CUBIC YARDS PER 100 FEET. SLOPES 2 : 1. Depth d Base Bass 12 14 Base 16 Base 18 Base 20 Base 28 Base 30 Base 32 l 56 63 70 78 85 ~15 122 130 2 133 148 163 178 193 252 267 281 3 233 256 278 300 322 411 433 456 4 356 385 415 444 47'4 593 622 652 5 500 537 574 611 648 796 833 87'0 6 667 711 756 800 844 1022 1067 1111 856 907 959 1011 1063 1270 1322 1374 8 1067 1126 1185 1244 1304 1541 1600 1659 9 1300 1367 1433 1500 1567 1833 1900 196?' 10 1556 1630 1704 1778 1852 2148 2222 2296 11 1833 1915 1996 2078 2159 2485 2567 2648 12 2133 2222 2311 2400 2489 2844 2933 3022 m 2456 2552 2648 2744 2841 3226 3322 3419 14 2800 2904 3007 3111 3215 3630 3733 3837 15 3167 3278 3389 3500 3611 4056 4167 4278 16 3556 3674 3793 3911 4030 4504 4622 4741 17 3967 4093 4219 4344 4470 4974 5100 5226 18 4400 4533 4667 4800 4933 5467 5600 6738 19 4856 4996 5137 5278 5419 5981 6122 6263 20 5333 5481 5630 5778 5926 6519 6667 6815 21 5833 5989 C144 6300 6456 707'8 7233 7389 >> 6356 0519 6681 6844 7007 7659 7822 7985 23 6900 707'0 7241 7411 7581 8263 8433 8504 24 7487 7644 7822 8000 8178 8889 9067 9144 25 8056 8241 8426 8611 8796 9537 rt7'22 9807 2C, 8667 8859 9052 9244 9437 10207 10400 10593 27 9300 9500 9700 9900 10100 10900 11100 11200 28 9956 1 10163 10370 10578 10785 11615 11822 12030 29 10633 10848 11063 11278 11493 12352 12567 12781 30 11333 11556 11778 12000 12222 13111 13333 13556 31 - 12056 12285 12515 12744 12974 13893 14122 14352 32 12800 13037 13274 13511 13748 14U96 14933 1517'0 S3 13567 13811 14056 14300 14544 15522 15767 16011 34 14356 14607 14859 15111 15363 16370 16622 16874 35' 15167 15426 15685 15944 16204 17241 17500 17759 36 16000 16267 16533 16800 17067 18133 18400 18667 37 16856 17130 17404 17678 17952 19048 19322 19596 38 17733 18015 18296 18578 18859 19985 20267 20548 39 18633 18922 19211 19500 19789 20944 21233 21522 40 19556 19852 20148 20444 20741 21926 22222 22516 41 20500 20804 21107 21411 21715 22930 23233 23537 42 21467 21778 22089 22400 22711 23956 24267 24578 43 23456 22774 23093 23411 23730 25004 25322 25641 44 23467 23793 24119 24444 24770 26074 26400 26726 45 21500 24833 25167 25500 25833 27167 27500 27833 46 25556 25896 26237 26578 26919 28281 28622 28963 47 26633 26981 27330 27678 28026 29419 29767 30115 48 27733 28089 28444 28800 29156 30578 30933 31289 49 28856 29219 29581 29944 30307 31759 32122 32485 50 30000 | 30370 30741 31111 31481 32963 33333 33704 51 31167 31544 31922 32300 32678 34189 34567 34944 52 32a56 32741 33126 33511 33396 35437 35822 3C207 53 33567 33959 34352 34744 35137 36707 37100 37493 54 34800 35200 35600 36000 36400 38000 38400 38800 55 36056 36463 36870 3?'278 37(185 39315 39722 40130 56 37333 37748 38163 38578 38993 40652 41067 41481 57 38633 39056 39478 39900 40322 42011 42433 42856 58 39956 40385 40815 41244 41674 i 43393 43822 44252 59 41300 41737 42174 42611 43048 ] 44796 45233 45670 60 42667 43111 43556 44000 44444 46222 46667 47111 497 TABLE XXXI.-USEFUL NUMBERS AND FORMULJE. Title. Symbol. Number. Loga- rithm. Ratio of circumference to diameter Tt 3.1415927 0.4971409 Reciprocal of same 1 0.3183099 9.5028501 Tt 180 Degrees in arc of length equal to radius Tt 57.295780 1.7581226 10800' Minutes " " " " 3437.7468 3.5362739 Tt 648000" Seconds " ' 206264.81 5.3144251 Tt Length of 1 arc radius unity . Tt .01745329 8.2418774 Tt Length of 1' arc, " " .00029089 6.4637261 10800 Lcnsrth of 1" arc " 4t Tt .000004848 4.6855749 618000 Radius by \vhich 1 foot of arc = 1 degree . 57.295780 1.7581226 Radius " " ^ " " = 1 minute. 343.77468 2.5362739 Radius" " TTT " " =10 seconds 206.28481 2.3144251 Factors for dividing a line into extreme j 0.6180340 9.7910124 and mean ratio ) 0.. 3819660 9.5820248 Base of hyperbolic logarithms 2.7182818 0.4342945 Modulus of common system of logs. = log M 0.4342945 9.6377843 Reciprocal of same = hyp. log. 10 i 2.3025851 0. "0221 57 Length of seconds pendulum at New York in inches 39.11256 1.51E81G2 Length of seconds pendulum at New York ill f 66 1 3.25938 0.5131350 Acceleration due to gravity at New York. . . 32.1688 1.5074347 Square root of same / ~ 5.67175 0.7537173 Yards in 1 pietro U 1.093623 0.0388676 Feet in 1 " 3.280869 0.5159889 Inches in 1 u ..... 39.37043 1.5951701 Metres in 1 foot 0.304797 9.4840111 Metres in 1 yard 0.914392 9.9611324 Metres in 1 mile . . 1609.330 3.2066450 498 TABLE XXXI. -USEFUL NUMBERS AND FORMULAE. Title. Cubic inches in 1 U. S. gallon " " " 1 Imperial gallon " " 1 U. S. bushel Cubic feet in 1 U. S. gallon " " " 1 Imperial gallon " " 1 U. S. bushel Weight of 1 cub. foot of water, barom. 30 in. ther. 39.83 Fah. ; pounds. . " 62 Weight in grains, 1 cubic inch, at 62 Fah. . No. of grains in 1 pound avoir " " " 1 ounce " . Symbol. Number. 231. 277.274 2150.42 0.133681 0.160459 1.244456 62.379 62.321 252.458 7000. 437.5 Loga- rithm. 2.363G120 3.3325233 9.1260683 9.2053655 0.0949796 1.7950384 1.7946349 2.4021892 3.8450980 2.6409781 r = radius of circular arc ; I = length of arc ; a = degrees in same arc. I 180 a = . it I 180 ~ a ' n I = ar . a' 180 Radius by which the length of chord c in feet = in minutes; 10 sin Hyp. log x com. log x X -j- f , or com. log (hyp. log x) = com. log (com. log x) -f ; 3322157 Com. log x = M X hyp. log x ; or com. log (com. log x) = 9.6377843 -f com. log (hyp. log x) Circumference of circle (radius = r) ................................ Area of circle ....................................................... Area of sector (length of arc = I) ......................... . ..... .... Area of sector (angle of arc = a) .................................. Approximate area of segment (chord = c, mid. ord. = m) ......... 499 - litr TTr 2 APPENDIX. Verification of eq. (77). sin 8 Eq. (76) p = =-. sin Q , cosec d,J Q i 00 Q- = cos 6 . cosec - , sin . cot . cosec (76J) d/j _.*> A _ _ j_ ~ C7 f 7\ Verification of eq. (81). Differentiating eq. (76$) d 2 p 2 00 G . r = - sin cosec - cos cot cosec -^. l i 2 - sin cof cosec --^ + -^- 3 - sin coseca -^ 2P p { \ = ~ P ~ ~N~ C0t 9 C0t 2\T ^ "J\T \ COt2 ]V + COSGC2 A^/ d-p / 2 i "uo" j = p \ ~ ! " ^ cot cot ]y + A-*" (a cota ,v APPENDIX. 501 No\r dp d 2 P in which substitute for -, and for /-, and let 1 ^5 cot Q cot -^ = a 4. 2/3 2 ( _ a)2 _ p2 _ i _ 1 cot cot |p + -~ (2 cot" ^ a^-f cotQ.cot .--cof . - p 2" ' i ~~i o 1 - + a "-+ cot -|p -fa (a:- I cot -) (1 4- 2 )- (81) 10872 Soi