REESE LIBRARY - OF THE UNIVERSITY OF CALIFORNIA. Received Accessions No. Shelf No. JOHN WILEY & SONS, Astor Place, New York, PUBLISH: THE RAILROAD SPIBAL. The Theory of the Compound Transition Curve reduced to Practical Formulae and Rules for Application in Field Work, with Complete Tables of Deflections and Ordinates for five hundred Spirals. By Wm. H. Seaiies, C.E., author of "Field Engineering," Member of Am. Soc. of C. E. Pocket-book form $1 50 " It should have a place in the library of every Civil Engineer in the world." Rail way ~4ye. FIELD ENGINEERING. A HAND-BOOK of the Theory and Practice of RAIL- WAY SURVEYING, LOCATION and CONSTRUCTION, de- signed for CLASS-ROOM, FIELD, and OFFICE USE, and containing a large number of Useful Tables, Original and Selected. By Wm. H. Searles, C.E., late Prof, of Geodesy at Rensselaer Polytechnic Inst., Troy. This vol- ume contains many short and unique methods of Laying Out, Locating, and Constructing Compound Curves, Side Tracks, and Railroad Lines generally. It is also intended as a text-book for Scientific Schools. Pocket-book form. Eighth edition, 1887 12mo, morocco, 3 00 " The book is admirable. The internal arrangements and ap- pearance are excellent. It is an easy work to refer to, and is plain and clear. There is no useless lumber in it. Every sen- tence belongs there." PROF. DAVIS, University of Michigan. THE CIVIL ENGINEER'S FIELD-BOOK. Designed for the use of the LOCATING ENGINEER. Containing Tables of Actual Tangents and Arqs, ex- pressed in chords of 600 feet for every minute of inter- section, from to 90, from a 1 curve to a 10 curve in- clusive. Also, Tables of Formulas applicable to Railroad Curves and the location of Frogs, together with Radii, Long Chords, Grades, Tangents, Natural Sines, Natural Versed Sines, Natural External Secants, etc. With Ex- planatory Problems. By Edward Butts, C.E. 12mo, morocco- flaps, 3 00 "The work is a monument of patience on the part of the au- thor, and should prove a labor saving investment to the pur- chaser. It is a ' Henck ' elaborated, and this is quite recommend- ation enough to the practising engineer." Engineering News. TREATISE ON LEVELLING BY VERTICAL AN- GLES, And the Method of Measuring Distances by Telescope and Rod. With Tables of Heights for all angles, from zero to 22V degrees (in minutes), for any distance re- quired. By August Faul, C.E. . . .' 8vo, limp cloth, 1 00 TABLES FOR CALCULATING THE CUBIC CON- TENTS OF EXCAVATIONS AND EMBANK- MENTS BY AN IMPROVED METHOD OF DIAGONALS AND SIDE TRIANGLES. By J. R. Hudson. New edition, with additional tables, 8vo, cloth, 1 00 "Hudson's Tables for Computing Earthwork are well known *.o Engineers, and by many are considered indispensable." Raihvay Age. CIVIL ENGINEEB'S POCKET-BOOK Of Mensuration, Trigonometry, Surveying, Hydraulics, Hydrostatics, Instruments and their adjustments, Strength of Materials, Masonry, Principles of Wooden and Iron Roof and Bridge Trusses, Stone Bridges and Culverts, Trestles, Pillars, Suspension Bridges, Dams, Railroads, Turnouts, Turning Platfornis ; Water Stations, Cost of Earthwork, Foundations, Retaining Walls, etc. In addition to which the elucidation of certain important Principles of Construction is made in a more simple man- ner than heretofore. By J. C. Trautwine, C.E. 12mo, morocco flaps, gilt edges. 28th thousand, revised and enlarged, with new illustrations, bv J. C. Trautwine, Jr., C. E. 1887 .* $5 00 "It is the best Civil Engineer's Pocket-book in existence." American Engineer. A METHOD OF CALCULATING THE CUBIC CON- TENTS OF EXCAVATIONS AND EMBANK- MENTS BY THE AID OF DIAGBAMS, Together with Directions for Estimating the Cost of Earthwork. By John C. Trautwine, C.E. Ninth edition, revised and enlarged by J. C. Trautwine, Jr. 8vo, cloth, 2 00 THE FIELD PBACTICE OF LAYING OUT CIB- CULAB CUBVES FOB BAILBOADS. By J. C. Trautwine, Civil Engineer. 12th edition, re- vised by J. C. Trautwine, Jr. . . 12mo, limp morocco, 2 50 " Probably the most complete and perfect treatise on the single subject of Railroad Curves that is published in the English language." Engineering Neu-s. THE ECONOMIC THEOBY OF THE LOCATION OF BAILWAYS. An Analysis of the Conditions controlling the laying out of Railways to effect the most judicious expenditure of capital. By Arthur M. Wellington, Chief Engineer of the Vera Cruz and Mexico Railway, etc. New and improved edition 8vo, 5 00 " Mr. Wellington has done great service to the Railroad pro- fession; more particularly to Engineers, Managers, and Superin- tendents, by bringing together in a single volume such a mass of valuable matter. It should be in every Railway Library." Railway Age. A TBEATISE UPON CABLE OB BOPE TBACTION. As applied to the working of STREET and other RAIL- WAYS. (Re vised and enlarged from Engineering.) By J. Bucknall Smith, C.E. With illustrations and folding plates 4to, cloth, 2 50 "The publication of this book seems to us to be most timely. The subject is ably handled by an experienced Engineer." American Machinist. A TBEATISE ON CIVIL ENGINEEBING. By O. H. Mahan. Revised and edited, with additions and new plates, by Prof. De Volson Wood. With an Appendix and complete Index. New edition, with chap- ter on River Improvements byF. A. Mahan. 8vo, cloth, 5 00 "This is the standard text-book on this subject." *** Will be Mailed and Prepaid on the receipt of the price. Our New Catalogue of 77 Pages for 1890 will be mailed gratis to order. AND EXPLORERS' GUIDE. ESPECIALLY ADAPTED TO THE USE OF RAILROAD ENGINEERS ON LOCATION AND CONSTRUCTION, AND TO THE NEEDS OP THE EXPLORER IN MAKING EXPLORATORY SURVEYS. BY H. C. GODWIN. NEW YORK: JOHN WILEY & SONS, 15 ASTOR PLACE. 1890. Copyright, 1890, BY JOHN WILEY & SONS. DRUMMOND & NEF, FERRIS BROS., Electrotype, Printers, 1 to 7 Hague Street, s ^ Pearl Street, New York. New York. PEEFACE. I AM publishing the following notes because I think they may possibly supply the want of a Field-book, a want which I have often felt myself and have often heard expressed which, while avoiding as much as possible the intricacies of mathematics, would be of more general application than any of the books of this class which I have as yet come across. The Railroad engineer is rarely an expert mathematician : in fact it has always seemed to me that the time which must necessarily be spent by him in attaining mathematical pro- ficiency might be very much better employed in reading up some of the more practical subjects of his profession. Bear- ing this in mind, I have endeavored to strip the following pages of all unnecessary mathematical deductions, making it mainly my object to give the results deduced, and yet at the same time giving sufficient explanation to enable any one pos- sessed of the ordinary smattering of mathematics and me- chanics to deduce the same results for himself. I have avoided the insertion of Logarithmic Tables. I am well aware that to some this will appear a serious omission; but considering that this is merely a Field-book, and not a work to be consulted in cases where accuracy in the 6th figure is usually essential, I have deemed that the exclusion of the hundred pages or so which this omission permits, amply com- pensates for the few seconds of additional labor which the lack of them may occasionally involve, ^peaking for myself, as regards Railroad work, I must say that for one time that I work by logarithms I work a hundred times by " naturals ;" and I know that most engineers would bear similar testimony. In the Astronomical problems in the latter part of the book, considerable labor may, of course, be saved by the use of iii CONTENTS. BAILBOAD LOCATION. GENERAL CONSIDERATIONS. SEC. PAGE 1. Conditions of Economical Location 1 2. Train Resistances 2 3. Rolling Resistance 2 4. Resistance due to Oscillation and Concussion 3 5. Atmospheric Resistance 3 6. Resistance due to Curvature 4 7. Resistance due to Gravity 4 8. Diagram of Resistance . 5 Limiting Velocity on any Grade , 8 9. Propelling Force of Locomotive 8 Coefficient of Adhesion 8 Sliding Friction : 8 Limiting I.H.P 9 Weight on Driving-wheels 9 Grate-surface 9 10. Diagram of Propelling Force 10 Limiting Speed for any given I.H.P 10 Internal Frictional Resistances. 11 Back-pressure, Wire-drawing, etc 11 11. Diameters of Driving-wheels 11 12. I.H.P. required at any given speed 12 Most Economical Speed 12 Limiting Grade 13 13. Weight of Locomotives and Rolling-stock 13 14. Resistance due to Inertia 13 Rotative Energy of the Wheels. . <, 14 15. Resistance caused by Application of Brakes 14 Automatic Brakes 15 Hand Brakes 15 16. Initial Velocity 15 17. Height corresponding to Velocity 16 Table of Heights corresponding to Velocity 17 18. Assumption of Mean Resistance and Mean Propelling Force.. . 17 19. Graphic Method of solving Dynamical Problems 17 20. Examples 18 V VI CONTENTS. SEC. PAGE 21. Rise and Fall 19 Profile of Velocities 20 22. Effects of Rise and Fall . 24. Maximum Grade 21 25. Economy of Locomotive 22 26. Compensation for Curvature 22 27. Compensation for Brakes 23 28. Broken Grades 23 Momentum Grades - 23 29. Danger of breaking Train and Derailment 24 30. Work done on Grades 24 31. Pusher-grades 26 Table of Pusher-grades 26 32. Maximum Curvature 26 Safe Speed on Curves 26 33. Short Tangents 27 Location of Curves 27 34. Table of Work done against Resistances 28 COST OF OPERATING. 35. Cost of Work done against Resistance 28 36. Cost per Train-mile 28 37. Economy of Construction 29 38. Cost of Operating Pusher-grades 30 39. To test Relative Cost of Various Routes 30 40. Effect of Alterations in Alignment 30 41. To estimate Effect of Ditto 31 RECEIPTS. 42. Deviating to catch Way-business 31 COST OF CONSTRUCTION. 43. Average Cost of Track 30 Average Cost of General Construction 32 Average Cost of One Mile of Track 33 Cost of Trestle-work, Trusses, Tunnels, etc 33 INSTRUMENTS. 44. Transit 34 Adjustments 34 45. Remarks 36 46. Stadia 39 47. Compass 42 Adjustments 42 Remarks 42 48. Magnetic Variation 43 Chart of Magnetic Variation 44 49. Dumpy Level 45 50. Y Level 45 CONTEXTS. Vll SEC. PAGE 51. Correction for Curvature and Refraction 46 52. Hand Level 47 THE SURVEY. 53. Reconnoissance and Preliminary Surveys 48 Running the Line to Grade 49 Table of Grades and Grade Angles 50 54. Transit Work 51 55. Latitudes and Departures 52 56. Azimuth Observations 54 57. A. Maximum Elongation of Polaris 55 B. Observation of y Cassiopeia and Polaris 56 C. Observation of Alioth and Polaris 57 58. Convergence of Meridians 58 59. Simple Triangulations 60 Offsetting the Transit-line 61 60. Levelling 61 Precision of a Line of Levels ... 62 61. Taking Topography 62 62. Contour Lines 64 Locating by means of Contour Lines 64 63. Levels and Curvature 66 64. Equations 66 65. Value of Topography 67 66. Tangents and Curves 68 67. Selection of Curves by Eye 1 68 68. Balance of Cuts and Fills 69 69. Establishing the Grades 69 Rough Estimation of Grading 69 70. Estimating by Centre Heights 71 CURVES. 71. Radius and Degree of Curves 71 72. Corrections for 50- foot Chords 72 73. Length of Curves , 74 74. Nomenclature and Symbols 74 75. Fundamental Formulae 75 PROBLEMS IN SIMPLE CURVES. 76. To lay out a curve by deflection angles: 78 To find corrected length of any sub-chord 78 Example 78 77. To locate a curve when the apex is inaccessible 81 78. To locate a curve by offsets from a tangent 82 Ditto if the apex, P.C., etc., are inaccessible 83 79. To locate a curve by offsets from the chords produced 85 80. To locate a curve by ordinates from a long chord 87 Example 87 Ditto by mid-ordinates 8 Vlll COKTEKTS. SEC. PAGK 81. To pass a curve through a fixed point, / being given 89 82. To run a tangent from a curve to any fixed point so 83. To connect two curves by a tangent 90 84. Given a curve joining two tangents, to change the P.O. so that the curve may end in a parallel tangent 91 85. To transfer a curve both at its P.O. and P.T. to parallel tangents. 92 86. Given a curve joining two tangents, to change R and the P.O. so that the new curve may end in a parallel tangent at a point opposite to the original P.T 93 87. Given a curve, to find R of another curve, which, from the same P.O., will end in a parallel tangent 93 88. Given a curve joining two tangents, to change R and the P.O. so that the curve may end in the same P.T., but with a change iu direction 93 COMPOUND CURVES. 89. Locating compound curves 94 90. To locate a C.C. when the P.C.C. is inaccessible 94 91. Given a simple curve ending in a tangent, to connect it with a parallel tangent by means of another curve 95 92. To connect a curve with a tangent by means of another curve of given radius 95 93. Given a C.C. ending in a tangent, to change the P.C.C. so that the terminal curve may end in a given parallel tangent, without changing jts radius 97 94. To connect two curves already located by means of another curve of given radius 98 95. To locate any portion of a C.C. from any station on the curve. . 99 TRANSITION CURVES. 96. Advantages of Transition Curves 100 97. Method 1 100 98. Method II 104 99. Method III 105 100. Vertical Curves . . 107 CONSTEUCTION. 101. Division of the Subject 109 A. SETTING OUT WORK. 102. Clearing Right of Way, etc 109 103. Location of Culverts, etc 109 104. System of Drainage. Ditches 110 105. Checking Benchmarks and Alignment Ill 106. Cross-sectioning Ill Setting Slope-stakes 112 Points at which Cross-sections should be taken 114 CONTENTS, ix SEC. PAGE 107. Reference Points 115 108. Staking out Borrow-pits ; 115 109. Staking out Foundation-pits for Culverts 115 110. Setting out Bridge Foundations 116 111. Setting out Trestlework 117 112. Setting out Tunnels 118 113. Giving "Grade" and centres 120 Shrinkage and Increase 121 114. Difference of Elevation on Curves 121 Effect on the Dump and on Trestles 123 Increase in Gauge on Curves 124 115. Inspecting the Grading 124 116. Running Track-centres and setting Ballast-stakes 125 117. Permanent Reference-points 125 118. Turnouts and Crossings 125 119. Locating by Offsets 127 120. Example 129 121. Turnouts and Crossings on Curves 129 122. Curving Rails 132 123. Expansion of Rails 132 B. THE ESTIMATING OF LABOR AND MATERIAL. 124. The Cost of Earthwork and Rockwork removed by Carts 133 125. Ditto, by other means. 136 126. Overhaul 136 127. The Calculation of Earthwork , 137 Areas of Cross-sections 139 128. The Pyramid, Wedge, and Prismoid 139 129. The Prismoidal Formula 140 130. The Method of Average End-areas 143 Prismoidal Corrections ... 143 131. The Method of Equivalent Level Sections 146 132. The Method of Centre-heights - 147 133. Earth-work Tables 147 134. Correction for Curvature 148 135. Contents of the Toe of a Dump 149 136. General 152 137. Timber- work 152 Table of Board Measure 151 Fractions of an Inch in Decimals of a Foot 153 138 Iron-work * 153 Weight of Bolts, Nuts, and Bars 153, 154 Railroad Spikes 154 Angle-bars and Bolts per mile 155 Weight of Rails per mile 155 139. Ballast and Ties per mile 156 CONTENTS. EXPLOEATOKY SURVEYING. SEC. PAGE 140. Introduction 157 INSTRUMENTS. 141. The Sextant. Adjustments, etc 157 142. Use of the Sextant 159 Parallax 159 143. Supplementary Arc 161 144. Observing Horizontal Angles 161 145. Eliminating Instrumental Errors 162 146. The Artificial Horizon 162 147. The Chronometer. 164 148. Barometers 165 149. Barometric Formulae 166 150. Reduction of Errors of Gradient 168 151. Taking Readings 169 152. Diurnal and Annual Gradient 169 153. The Cistern Barometer 170 154. To fill a barometer 170 155. Reading the barometer 171 156. Cleaning the barometer 171 157. The Aneroid Barometer 172 158. Elevation Scales 173 EXPLORATORY SURVEYS. 159. Division of the Subject 174 160. To find the distance apart, etc., of two inaccessible points 175 161. The " Three-point Problem" 176 162. Positions fixed by bearings 178 163. Positions fixed by intersection 178 164. Obtaining Heights of Mountains trigonometrically 178 165. Refraction of the Air 180 166. Reciprocal Angles 180 167. By Depression of the Sea Horizon 181 168. Observing Altitudes and Depressions 181 With a Sextant and Artificial Horizon 181 With a Transit * 182 169. Measurement of a Base 182 Correction for Temperature 182 Reduction to Sea-level 183 170. Example of Triangulating on Exploratory Surveys 183 171. To measure a horizontal angle without an instrument 184 172. To measure a vertical angle without an instrument 185 173. Measurement of Distance by Sound 185 174. Measurement of Time by Vibrations 185 175. Direct Measurement and Compass Courses 186 Odometers and Pedometers 186 Estimating the Rate of Progress 186 CONTEXTS. XI SEC. PAGE 176. Astronomical Observations 187 177. Solar Time.... .187 178. Equation of Time -. 188 179. Sidereal Time 189 180. Right Ascension and Declination 189 181. Correcting for Longitude, etc 190 182. Hour-angle 191 183. Examples 192 184. Refraction 194 185. Parallax 195 186. Correcting for Semi-diameter 197 Augmentation 197 187. Dip 198 188. Summary of Corrections 198 189. Latitude. By Meridian Altitudes 199 190. Remarks 201 191 . By Transits across the Prime Vertical 202 192. By an Altitude out of the Meridian 203 193. By Double Altitudes 205 194. By an Altitude of Polaris at any time 206 195. Longitude. Local Time, by an Altitude of a Star '-06 196. Local Time, by Equal Altitudes of a Star 208 197. Local Time, with a Transit 208 198. By Lunar Culminations 209 199. By Lunar Distances 210 200. By Jupiter's Satellites 214 201. To test the chronometer rate \ 214 202. To set the transit in the meridian , 214 203. Interpolation by Successive Differences 215 204. " Accidental Error" 216 205. Influence of Spheroidal Form of the Earth 218 206. Figure of the Earth 218 207. Conversion of Angular Measure into Distance and vice versd 219 208. Given the lat. and long, of two places, to find their distance apart, etc 220 209. To find the radius of a circle of latitude 221 210. Offsets to a Parallel of Latitude 221 211. Development of a Spherical Surface 221 212. Example 222 213. Star Map 225 Star Tables .-. 226, 227 MISCELLANEOUS. 214. The Horse-power of Falling Water 215. To gauge a stream roughly , 216. Sustaining Power of Wooden Piles 217. Supporting Power of Various Materials. . Xll CONTENTS. SEC. PAGE 218. Transverse Strength of Rectangular Beams 229 219. Natural Slopes of Earth 230 220. Weight of Earths, Rocks, etc., per cubic yard , 230 221. Weight of Timber and Metals per cubic foot 231 222. Mortar, Cement, and Concrete 231 223. Notes on Timber. Selection of Trees 231 224. Defects of Timber , 232 225. Felling Timber 233 226. Seasoning and preserving Timber 233 227. Decay of Timber 231 228. Tests for Steel and Iron 234 229. Strength of Rope. Manilla, Iron and Cast Steel 235 230. Properties of the Circle 236 231. Trigonometry. Plane 237 232. General Equations 240 233. Spherical 241 234. Measures of Length and Surface 243 235. Measures of Weight and Capacity 244 APPENDIX. TABLES. Table I. Radii of Curves 252 " II. Tangents and Externals to a 1 Curve 255 44 III. Tangential Offsets at 100 feet 259 IV. Mid-ordinates to 100-foot Chords 259 44 V. Long Chords 260 41 VI. Mid-ordinates to Long Chords 263 44 VH. Minutes in Decimals of a Degree 264 44 VIII. Squares, Cubes, Square and Cube Roots 265 IX. Logarithms of Numbers. 1 to 1000 282 44 X. Natural Sines and Cosines 285 41 XI. ' 4 Secants and Cosecants 294 " XII. 4 ' Tangents and Cotangents 309 44 XIII. ' 4 Versines and Exsecants 321 44 XIV. Cubic Yards per 100 feet, in terms of Centre-height. . . . 345 XV. Cubic Yards per 100 feet, in terms of Sect-:.>iial Area. . . 350 44 XVI. Mutual Conversion of Feet and Inches into Meters and Centimeters 354 44 XVII. Mutual Conversion of Miles and Kilometers 355 " XVIII. Length of 1' arcs of Latitude and Longitude 355 44 XIX. Mutual Conversion of Mean and Sidereal Time , 356 * 4 XX. Mutual Conversion of Time and Degrees 358 ERRATA. Page 4, line 10. For .27 read 3.7 " 24, " 11. " 12 " 18 ' 36, " 16. After them wi*er< say A " 36, " 24. For 1.59 read 1.84 " 55, fo0m fone. -For ZP read WP " 72, 2tn 3. For radius read degree, and for degree read radius " 78, line 17. For abc read acb " 117, ^Equation in line 8 should be Be = R f sin POM - ^- ct 128, fone 3. Far AB read AD PART I. BAILROAD LOCATION. GENERAL CONSIDERATIONS, 1. IN the early days of Railroad Building, the Locating En- gineer was forced to rely mainly on his individual ability, trusting principally to the correctness of his eye to detect the most suitable route, guided only by the very limited experience of others and his own common-sense. The man who worked his party the hardest, and covered most ground in the day, was in those days, unless any very obvious defects were visi- ble in his work, too often looked upon as the best locator. But the years of experience which have followed have been years of experiment also ; and the practice of Railroad Loca- tion has by degrees developed into a science, which, though yet far from perfect, forms a most important part of a Modern Engineering Education. In a Field book of this sort, it is impossible to do more than treat rapidly a few of the leading questions which the subject involves, and formulate, where possible, rules for guidance in the field. A knowledge of the principles of Railroad Location must be backed up by experience in Railroad Construction. For, in order to locate well, a man must have fairly accurate ideas of the suitability and cost of the various works which his lo- ^ation involves. The best location for a certain road is not that which enables the traffic to be carried on with the least amount of work, or which gives the lowest Operating Expenses, but that which, in a given time, renders the ReceiDtS - /P eratin ^ /Interest on Capital spent on\ _ p_ nfit _ VExpensesJ ( Construct. Equipment, etc .) ~ 2 KAILROAD LOCATION. a maximum. Thus we see that more or less accurate esti- mates of the probable Receipts and Operating Expenses are of the utmost importance before starting the location ; and it is only when these are arrived at that the amount which we are entitled to expend on construction can be fixed. 2. Before considering the Financial side of the question, however, we will glance hurriedly over some of the principal Mechanical Problems which occur in dealing with the motion of trains, for, without some slight knowledge of Railroad Dynamics, an intelligent application of the Laws of Location is impossible. TEAIN EESISTANCES. The Resistance due to the motion of a train on a straight level track excluding for the present the Inertia of the train may be regarded as being the sum of the three following com- ponents : 3. ROLLING RESISTANCE, which is composed of the frictional resistance at the journals and that at the wheels at the points of contact with the rails : these two may for ordinary purposes be classed together under the head of Rolling Resist- ance. Its magnitude depends largely upon the surface bear- ing at the journals ; the coefficient of friction decreasing as the load per unit-surface on the journals increases, so that the resistance is relatively higher in the case of Empty Cars than with Loaded ones ; being at ordinary speeds about 6 Ibs. per ton (2000 Ibs.) of weight of train in the former case, while with Passenger Coaches or Loaded Cars it only amounts to about 4 Ibs. By referring to the Diagram of Resistances, p. 6, we see that at the point of starting the Rolling Resistance is very high, being then about 20 Ibs. per ton, but that at a velocity of about ten miles per hour it reaches its minimum value, and from that point increases constantly by a trifling amount through the successive higher velocities. The Initial Resist- ance depends largely on the length of time the train has been standing, a stop of only a few seconds causing a resistance of about one half that given in the Diagram. Since, however, there is always more or less " give " about the couplings, no two cars at the same instant offer their maximum resistance, the front end of a long train being well under way before any motion at all is transmitted to the rear. Thus the pull on tke KAILROAD LOCATION. 3 draw-bar is not in reality so excessive as it at first appears ; for if we take the whole train into consideration, the resist- ance at the start may be set down as about 12 Ibs. instead of 20 Ibs. per ton, as in the case of a single car. The Line of Rolling Resistance starts in the Diagram from the line of the 1 p. c. grade ; thus indicating that a train left standing with the brakes off on this grade, is just on the point of starting on its own account. On any grade lighter than this, a train will usually require considerable force to set it in motion. By increasing the diameters of the wheels we slightly decrease the resistance to rolling. 4. RESISTANCE DUE TO OSCILLATION AND CON- CUSSION. The amount of this we obtain approximately by assuming that it equals .005 Ib. per ton at 1 mile per hour, and increases as the square of the velocity. Thus, e.g., at 40 m. p. h. it equals 8 Ibs. per ton. The longer the train, however, the less this resistance amounts to per ton, for each car is more or less steadied by the force which is transmitted through it to the ad- joining one; thus it is usually much more considerable in the rear than in the centre or forward end of the train. It is pro- duced in a great measure by the inequality in elevation of the two rails on an imperfect track, and thus is often found to dimin- ish on curves where the difference in elevation of the rails is not exactly suited to the speed at which the train is travelling, since it is then subjected to a lateral thrust which prevents the oscillations being so great as they otherwise would be. 5. ATMOSPHERIC RESISTANCE. This is due to two causes : (a) The opposition offered by the particles of air in the direct path of the engine, while being thrust forwards and sideways by the advancing train, together with the " suction" caused by the rear car ; and (b) The frictional resistance of the air against the surface of the train, corresponding to the " skin resistance " in the case of ships. The former (a) amounts to about 0.3 Ib. per train running through still air at a velocity of 1 mile per hour, and increases as the square of the speed : thus, e.g., at 40 m. p. h. it amounts to about 480 Ibs. Probably in ordinary trains not more than one third of this resistance causes addi- tional strain on the draw-bar, because the greater part of it is taken and overcome by the engine itself. As regards the latter 4 EAILEOAD LOCATION. resistance, (b) it may be ascertained with tolerable accuracy by allowing 0.03 Ib. per car at a speed of 1 mile per hour, and considering it to increase as the square of the velocity. Thus, if we have a train composed of 10 loaded box-cars (see Sec. 13) hauled by an engine which, together with its tender, weighs 60 tons, .the total atmospheric resistance in Ibs. at 40 m. p. h. 480 -f- 480 = 960 Ibs. (assuming that the allowance already given for the engine includes the surface resistance as well) ; and since the weight of the train inclusive of engine and tender equals about 260 tons, this is equivalent to about 3R 31 Ibs. per ton of entire train. Suppose, in the above example, we have a Head-wind blowing at the rate of 20 m. p. h., we may then consider the atmospheric resistance as being that due to a train velocity of 60 m. p. h. But if this wind were blowing in the same direction in which the train is going, then the resistance caused by it would be equal to that caused by a train velocity of 20 m. p. h. in still air. A Side-wind adds very considerably to the ordiuar}^ atmos- pheric resistances by increasing the frictional resistance at the rails, owing to the flanges of the wheels being pressed against the inner side of the leeward rail. The above resistances are peculiar to all trains at all times ; the two following, however, are accidental, and dependent on circumstances. 6. RESISTANCE TO CURVATURE. The many causes which combine to make up this resistance, and the share which each has in forming the result as a whole, have been but vaguely determined by experiment: it is known, however, that at speeds not exceeding about 5 miles per hour, it amounts to about 2 Ibs. per ton per degree of curvature, and that it decreases as the speed increases, as shown in Diagram I, till at 70 miles per hour it does not probably amount to more than i Ib. per ton. Thus, e.g., on a 5 curve it amounts at a velocity of 35 m. p. h. to about 2 Ibs. per ton. The use of Transition curves (page 100) is found to decrease it materially. 7. RESISTANCE DUE TO GRAVITY. This resistance may be termed a " mathematical " one, whereas the previous ones have been based entirely on experiment ; for though the coefficient of gravity is itself a quantity derived from experi- ment, it is merely the ratio of the inclined component AB RAILROAD LOCATION". 5 (Fig. 1) to the force of gravity A C, which enters into the question ; or, what is the same thing, the ratio of ab to ac. But since, in dealing with ordinary inclines, we may con- sider ac = cb, we may say that AB_ab AC~cb' so that the resistance caused by gravity per ton (2000 Ibs.) equals in Ibs. 20 X fate per cent of tlie grade. Thus on a 2.5 p. c. up- grade the gravity resistance equals 50 Ibs. per ton. DIAGRAM OF RESISTANCES. 8. We are now in a position to draw the Line of Resist- ance for any given train under any ordinary conditions. This line, for a train on a straight level track, is found by setting-off at the successive velocities the sum of the ordinates for the Resistances given in Sections 3, 4, and 5 ; and the line representing each of these component resistances can be read- ily plotted with the aid of the information already given. Suppose, however, that the train is running on a curve of, say, 10, we must then measure the respective ordiuates to the resistance line for the 10 curve, and #dd these to the ordi- nates already obtained. We then get the Line of Total Resist- ance on a 10 curve. If in addition to the 10 curve we have a J r 0.25 per cent grade, we have simply to add the height given on the diagram for this grade to each of the ordinates already found, in order to obtain the Line of Resistance for the train on a 10 curve and a -f-0.25 p. c. grade. If the train were descending the grade, it would be necessary to sub- tract the last ordinate instead of adding it. DIAGRAM I. TRAIN RESISTANCES IN LBS. PER TON. Engine and Tender weigh 60 tons. 10 Loaded Box-Cars, each weighing 20 tons. SCALE, 1 inch vert. ^= 10 Ibs. (6) DIAGRAM II. PROPELLING FORCE OF LOCOMOTIVE IN LBS. PER TON. Locomotive 500 I. H. P. Engine and Tender = 60 tons. f = 0.2 10 Cars, 20 tons each. SCALE, 1 inch vert. = 10 Ibs. 8 RAILROAD LOCATION. In order to find the Limiting Telocity of any train on a certain grade, moving solely under the influence of gravity, we have only to find the point of intersection of the line of total resistance, for a level track, with the horizontal line cor- responding to the grade in question, and notice the velocity corresponding to this point. Thus in Diagram I, for the train there given, running round a 10 curve down a 2 p. c. grade, the limiting velocity will be about 63 in. p. h. 9. Next comes the consideration of the counteracting force, namely : THE PEOPELLING FOKCE OF THE LOCOMOTIVE. The Coefficient of Adhesion, i.e., Static friction, between the rails and the driving-wheels of a locomotive, is found to be much the same at all speeds, but to increase rapidly as the load per unit-surface increases. It varies in ordinary Rail- road practice from about 0.33 when sand is used to about 0.18 when the rails are slippery. Under ordinary circumstances the maximum Propelling Force of a Locomotive may be con- sidered equal to one fifth the weight on its drivers, assuming 0.2 as the usual working coefficient of adhesion ; thus varying from about a ton to a ton and a half per driving-wheel, ac- cording to the type of locomotive. If on starting a train the driving-wheels are allowed to slip on the rails, the friction is no longer Static but Sliding 1 , the coefficient of which equals about 0.1, decreasing rapidly as the velocity increases ; which shows the fallacy of allowing the wheels to slip. The part of the rail, however, on which the slipping, if any, has taken place is found, if the engine is reversed, to give a coefficient of adhesion higher than else- where. Where Two or more pairs of wheels are coupled together, the adhesive force is, of course, due to the load on all the wheels coupled to the driving-wheels. Now, however great steam-producing capacity the locomo- tive may possess, its Propelling Force is limited by the coeffi- cient of adhesion ; and though it can expend its full power in spinning the wheels around, the portion of this power which RAILROAD LOCATION. can be utilized for propelling the train is limited by the amount expressed in Indicated Horse-Power : I. H. P. = 5.9 WfV, where W= total weight in tons (2000 Ibs.) on the drivers, / = coefficient of adhesion, V = velocity in miles per hour. This formula allows 10 p. c. for overcoming the Internal Resistances in the engine itself (see page 11). The friction at the journals of the driving-wheels, however, is not included among these, but is allowed for in the ordinary Rolling Re- sistance already dealt with. Thus if we take the weight on each driving-wheel as 6 tons, and/= 0.2, the above formula becomes I. H. P. = 1NV (nearly), where N= the number of driving-wheels. Thus, e.g., if, in an ordinary locomotive with four driving- wheels, we have the production of steam equivalent to 400 I. H. P., we see that it is unable to utilize its full power for propelling purposes until it attains the speed of about 14 miles per hour, at which point an} r slight increase in pressure would cause the wheels to slip. Thus up to a certain speed the propelling power of an engine is limited by the weight on its drivers, but remains more or less constant until that speed is attained, after which, instead of being limited by the adhe- sion of the wheels, it is mainly a question of the steam-pro- ducing power of the boiler, In ordinary practice, 1 square foot of Grate-surface is able, at ordinary speeds, to maintain the production of steam equivalent to 24 I. H. P. : so that if we know the total grate- surface of an engine and the load on its drivers, assuming it to be tolerably well-proportioned in its various parts, we can form a fair idea of its tractive power. The usual allowance of grate-surface varies from about 15 square feet in Passenger Engines to double this amount in some of the Heavy Freight Engines : thus the power of an ordinary Passenger Engine, when working under ordinary conditions, equals about 360 I. H. P., and in. the case of a heavy Freight Engine about 720 I. H. P. Both these classes of engines can, and often do, maintain very much higher powers than these, but to work very considerably above them over a long run is a severe tax on the economy of the engine. 10 KAILROAD LOCATION. DIAGRAM OF PROPELLING FORCE. 10. In order to ascertain the probable effect of a given lo- comotive on a certain train on various grades and curves, it is best to draw the Line of Propelling Force of the Engine i.e., the Line of Tractive Power exerted at the point of con- tact of the driving-wheels with the rails in Ibs. per ton (2000 Ibs.), of the weight of the engine and train. Suppose, as in Diagram II, we wish to find the effect of a locomotive capable of maintaining a working power of 500 I. H. P. having four drivers with 6 tons on each; and let the engine with its tender weigh 60 tons, and the train be the same as that for which the Lines of Resistance are given in Diagram I, namely, 10 loaded box cars, each weighing 20 tons/ being taken as 0.2. We then have a fair example of the working of a Light Freight Engine. Draw the Line of Propelling Force as follows : Make OA = J~~^ r- = 36.9 Ibs. per ton. lot. Weight of Train I H P Then draw Aa = -'^^^ = 17.6 miles per hour, o.y w j which (according to Sec. 9) gives the velocity above which slipping cannot occur. Now the theoretic curve of Propel- ling Force will be a hyperbola, drawn through a (AO and OH being its asymptotes). This curve may be drawn by off- sets from OA thus : At a distance along OA from equal to \OA, the offset equals lAa ; at a distance equal to \OA, the offset equals 2 Aa, and so on ; the offset varying inversely as its perpendicular distance from 0. Then C, the point of in- tersection of the Line of Propelling Force with the Line of Resistance, gives the Limiting Speed at which the engine can haul the train, under the conditions for which the line of resistance is drawn, in this case, on a straight level track. Then, taking any ordinate such as NMPQ, the part NM in- cluded between the Line of Propelling Force and the Line of Resistance gives that portion of the propelling force of the engine in Ibs. per ton (2000 Ibs.) which goes to overcome the Inertia of the train at the speed indicated. But this Line of Propelling Force assumes as we men- tioned before that 10 per cent of the I. H. P. is absorbed in BAILED AD LOCATION. 11 overcoming the Internal Frictional Resistances of the en- gine itself exclusive of the resistance at the journals inde- pendent of the velocity. At low speeds this allowance is con- siderably too much, but at high velocities it is insufficient ; for ordinary speeds, however, it will not be far from correct. The journal-friction forms probably about one third of the whole : the friction of the piston, slide-valve, valve-gear, and cross-heads also contribute considerably to the total. Very little is known as to what allowance ought to be made to cover these resistances, in fact it is so much a matter of lu- brication and mechanical detail that no general formula could be applied, but undoubtedly they increase with the velocity, and are higher in an engine hauling a heavy train than in an engine running light. Also we have Back-pressure of the steam in the cylinders, Wire-drawing, and various other causes entering into the question at high speeds which also tend to lessen the effective Horse-power. See Note A, Appendix. 11. Now since the loss of power due to these causes de- pends largely on the rotary velocity of the Driving-wheels, in the case of two engines both developing the same I. H. P. at the same speed, the cylinders being suitably-proportioned, the engine with the larger wheels will have a great advan- tage over the other at high speeds, although at low speeds the engine with the smaller wheels will have the best of it. At low speeds since the initial pressure in the cylinders then differs but little from the boiler-pressure and the back-pressure is practically nothing an engine with several small drivers will of course have an enormous advantage over an engine of the same I. H. P. with only a single pair of large drivers on ac- count of its being able to utilize so much more of its power, by reason of its higher adhesive qualities. For instance, it would probably tax the engine with Jarge drivers severely to start a train which the other engine could handle with ease; but when the speed reached, say, thirty miles per hour, the engine with the large drivers could work it much more easily and economically than the engine with the small ones. Thus where high velocities are required, whether on heavy grades or not, provided the weight on the drivers is sufficient, if the cylinders, etc., are suitably proportioned, the wheels of large diameter are decidedly the best. 12 RAILROAD LOCATION. Mr. Wellington states that in the case of ordinary Passenger Engines and trains of medium length, 50 per cent of the I. H. P. is consumed in the locomotive itself, overcoming its various resistances atmospheric, rolling, internal, etc., so that only one half of the Horse-power produced is trans- mitted through the draw-bar. From the foregoing it appears that a closer approximation to the true line of propelling force at high velocities may be found by drawing it as shown by the dotted -line in Diagram II, somewhat below the theoretic line already drawn. The intersection of this line with OH (produced) gives the maxi- mum speed of the engine if unopposed by any external resist- ances, i.e., if running free as a stationary engine, 10 per cent only of the power developed being absorbed in overcom- ing internal resistances. It must be remembered that the Line of Propelling Force shown in the Diagram is at all points the maximum which can be obtained without exceeding the I. H. P. stated ; but by taking a comparatively low value of /, and a high allowance for the internal frictional resistances of the engine at low speeds, we obtain by the method given probably as correct results as can be obtained by any mathematical process. 12. If we require to know what I. H. P. an Engine must develop to haul a certain train at a given velocity V, we can find it at once theoretically by multiplying the total weight of the engine and train in tons (2000 Ibs.) by the resist- ance in Ibs. per ton (taken from Diagram I) and multiplying the product by .003F(Fbeing in miles per hour). Thus with the train given in Diagram II, we should need an en- gine capable of developing about 950 I. H. P. in order to haul it at a speed of 50 miles per hour. The I. H. P. exerted in- creases nearly as F 3 , and the tractive force nearly as F 2 . The total amount of steam used theoretically, on a run, is nearly proportional to F' 2 . The most economical speed, as regards fuel, at which a train can be run provided the en- gine is of a power suitable to the weight of the train is found by experiment to be about 18 miles per hour, and not, as might be expected from Diagram I, at about 8 miles per hour. This is due mainly to the saving in heat owing to the engine being a shorter time on the trip, and also on account of the smaller effect produced by variations in grade at the higher BAILBQAD LOCATION". 13 velocity. To ascertain the Limiting Grade which it is possi- ble to work, we find from the diagrams that an engine and tender weighing together 60 tons, with 24 tons on the drivers, can under ordinary conditions just make head-way up a 12- per-cent grade ; and that it is just all two engines of the above description can do to haul a passenger coach up a 10- per-cent grade. 13. The following may be taken as fair examples of the WEIGHT OF AMERICAN ROLLING-STOCK: Type. No. of Drivers. Weight in tons on each Driver. Weight in tons, engine and tender, with fuel and water. Heavy Passenger Engine. . . Consolidation Engine Decapod Engine. 4 8 10 ! 7 55 75 95 feet - (1 ton ?= 2000 Ibs.) veight 10 tons. ) " 20 " f Flat " empty, " 8 " " 34 " Passenger car, empty, weight 20 tons ) t( ^ n (( loaded, " 25 " \ Drawing room car, " 35 " 3 " 50 to 60 feet. Sleeping-car, weight, 30 to 45 " " 50 to 70 " RESISTANCE DUE TO INERTIA. 14. We are now able to calculate with a fair amount of pre- cision the Propelling Force of an engine and the Total Resist- ance opposed to it at any given speed. The Difference between these two, such as is represented by NM, in Diagram II, gives the force in Ibs. per ton which goes to overcome the inertia of the train: if the Propelling Force be the greater, increasing the velocity; but if the Resistance be the greater, decreasing it. We will first consider the subject on the assumption that the accelerating force remains constant at all speeds, and that there are nofrictional resistances. It is found by experiment that a force of 1 Ib. acting on a weight of 32.2 Ibs. (which is perfectly free to move in the di- rection in which the force is acting) will, after acting on it for 1 second, give it a velocity of 1 foot per second; and that the velocity at all points increases in proportion to the interval of 14 KAILROAD LOCATION. time during which the force acts: also, that for a given force, the velocity of a body (after it has been acted on by the force for a certain interval of time) is inversely proportional to the weight of the body. Thus the value of the Accelerating Force in Ibs. per ton of train equals 1.518V t ' where t = time in minutes during which force acts, and V = velocity in miles per hour acquired in time t. But this formula takes no account of the force necessary to cause the wheels to rotate; it only allows for motion in the di- rection in which the force acts. In order to obtain the ad- ditional force required to overcome the Rotative Energy of the Wheels, we may imagine the whole weight of each wheel concentrated at a point distant from its axis by an amount equal to the Radius of Gyration of the wheel. For ordinary rolling-stock we may say that this distance equals 0.75 of the radius of the wheel; and the velocity with which a point so situated rotates round the axis equals 0.75 the velocity of the train. Now the ratio of the weight of the wheels to the total weight of a train of medium length varies from about 0.1 to 0.25, according to whether the cars are loaded or empty, the proportion in the case of Passenger Cars being about the same as with Loaded Freight Cars. Therefore the Total Force neces- sary to overcome the entire Inertia of the train varies from about to t where F = constant accelerating force in Ibs. per ton (2000 Ibs.) of train. The former value is applicable to Loaded and the latter to Empty cars. As regards the distance covered by the train from the start- ing-point to the point at which it attains the velocity V, it can be found by the formula where 8 = distance in feet. 15. Now the force required to stop a train travelling with a certain velocity, in a given time, equals the force which is necessary to give it that velocity in the same time; so that the RAILROAD LOCATION". 15 formula given above for F applies to the resistance caused by the Application of Brakes, as well as to the Propelling power of the engine. Now, since, as in the case of the driving- - wheels of a locomotive, as soon as slipping begins, the ad- hesion at the rails decreases rapidly, therefore, in applying the brakes, the pressure should be such that the wheels will just roll on the rails; i.e., the resistance on the brakes must not be allowed to exceed the resistance at the rails, but should be as near to this limit as possible. If the pressure on the brakes could be adjusted so as to effect this in practice, we should have an efficiency for the brakes equal to the coefficient of adhesion, which we have already considered under ordinary circumstances to equal 0.2. But it is found that with Automatic Brakes we cannot generally rely on a greater efficiency than 0.12, which is equal to a value of F (if the brakes are applied to the whole train) of 240 Ibs. Thus the brakes may be said to offer a resistance equivalent to a 12 p. c. grade. In the case of Hand Brakes it usually takes about four times as great a distance in which to stop a train when they are used, as with Automatic ones applied to the whole train. Suppose under the above assumption we have' a passenger- train running at a speed of 60 miles per hour. Ii steam is shut off at the same instant that the brakes are applied auto- matically with an efficiency of 0.12 to three quarters of the weight of the train, the retarding value of F would equal .75 X 240 = 180 Ibs. per ton, and thus by our previous formula gives a value for t 0.53 minutes, from which we can obtain S = 1400 feet. Had the train being going at only 30 m. p. h. instead of 60, it could have been pulled up in one half the time and one quarter the distance it required to stop it when running at 60 m. p. h. Thus in order to stop a train going at 60 m. p. h., we must apply four, times the amount of brake-resistance wliich would be required to stop it if going at 30 m. p. h. in the same time. 16. So far w r e have dealt only with a change of velocity from Rest to F, or from F to Rest. Suppose, however, in the former case that the train, instead of being at rest, before the accelerating force Fis applied, has an Initial Velocity (v). The formulae given in section 14 then become changed, F in 16 RAILROAD LOCATION. this case varying from about 1.6(F-0) 1.7 (F-t>) "- " -- ' and And just as the previous formulae applied to either an accel- erating or retarding force, so these apply equally well to the Propelling Force of the Locomotive or the Resistance of the Brakes. As an Example, suppose we take a Passenger-train run- ning at 50 miles per hour. The value of F necessary to re- duce this speed to 30 m. p. h. in one minute = 1.6 X 20 = 32 Ibs. per ton, which gives a resistance equivalent to a -f- 1.6 p. c. grade. Problems such as the above, where the value of F is assumed constant, where no account is taken of the fric- tional resistances, and in which the question of the time t is not directly involved, may often be solved more simply still by means of the Table of Equivalent Heights given below. HEIGHT CORRESPONDING TO VELOCITY. 17. In the above example of the train running at 60 m. p. h. being brought to a stand-still, if the brakes had been ap- plied to the whole train with an efficiency of 240 Ibs. per ton, it would have been stopped in a distance of about 1056 ft. ; or, putting it in another way, the train could have run up a 12 p. c. grade fora distance of 1056 feet before stopping, showing that it had stored up in it the Energy necessary to raise itself vertically through a height of about 127 feet. In a similar way without going into the subjects of Kinetic and Potential Energy every velocity may be shown to have a corresponding vertical height. Now about 5.6 p. c. of this rise, in the case of trains, is due to the Rotative Energy of the wheels (when dealing with loaded cars) and the remainder is simply the height from which a body must fall under the influence of a force equal to its own weight, i.e. .gravity, in order to obtain the velocity in ques- tion. But since this Rotative Energy is taken account of in the previous formulae, we can, by finding the value of 5 when F 2000, obtain for any given velocity the corresponding vertical height. KAILROAD LOCATION. 17 In this way the following table has been calculated for Pas- senger or Loaded Freight Cars. For a train of Empty Freight or Flat Cars, 6 p. c. should be added to the heights given. TABLE OF HEIGHTS IN FEET CORRESPONDING TO VELOCITY IN MILES PER HOUR. Vel. 1 2 3 4 5 6 7 8 9 10 3.5 4.3 5.1 5.9 6.9 7.9 9.0 10.2 11.4 12.7 20 14.1 15.5 17.0 18.6 20.2 22.0 23.8 25.7 27.6 29.6 30 31.7 33.8 36.0 38.3 40.7 43.1 45.6 48.2 50.8 53.5 40 56.3 59.2 62.1 65.1 68.2 71.3 74.5 77.8 81.2 84.6 50 88.0 91.5 95.1 98.9 102.7 106.5 110.4 114.4 118,4 122.5 60 126.7 131.0 135.3 139.7 144.2 148.7 153.3 '158.0 162.8 167.6 70 1172.5 177.4 182.5 187.6 192.8 198.0 ,203.3 |208.7 214.2 219.7 Now if we have a Passenger train running at a speed of 20 m. p. h., and we wish to know what its velocity will be after descending 1000 feet of a 3 p. c. grade ignoring as before frictional resistances we can find it at once from the Table, thus : Its velocity at the foot of the grade will be that due to the height corresponding to a velocity of 20 m. p. h. -}- 30 feet = 44.1 feet, which corresponds with the velocity required, name- ly, 35.4 miles per hour. Or, suppose we wish to know what rate of grade would be required to decrease the speed of the above train from 40 rn. p. h. to 25 m. p. h. in a distance of 1000 ft.: we have Height corresponding to 40 m. p. h. = S6.-3 feet "25 " =22.0' " Difference = 34.3 feet. Thus it is a 3.43 p. c. grade that would be required. 18. So far we have dealt only with the Inertia of the train on the supposition that the propelling force of the engine is con- stant at all speeds, and that there are no frictional resistances. A method much in use in practice which partially corrects for both these fallacies is that of allowing for the mean frictional resistance and the mean propelling force of the engine, and then, by the aid of formulae similar in effect to those given above, obtaining approximate values of 8. 19. But this method of averaging gives very unreliable re- sults when dealing with any but comparatively low velocities, so that the following Graphic Method,, which is extremely 18 KAILROAD LOCATION. simple, is in most cases preferable, since the correctness of the results obtained by it depends almost solely on the care employed in working it. Let the Lines of Resistance and Propelling Force be drawn as in Diagram II. 00 Take anyordiuate NQ, and make PQ = * Similarly take other ordinates, and thus fix other positions of the point P. Draw the curve OPD through these points. Then, if (as in Diag. II) 1 inch vertical =10 Ibs., and 1 inch horizontal = 20 miles per hour, the area (shown shaded in Diag. II) enclosed by the curve OPD, the line OS, and the ordinate corre- sponding to any given velocity gives the distance covered while attaining that velocity, using as a scale 1 square inch = 1 linear mile. (See Note B, Appendix.) And as a conse- quence of this, assuming, e.g., the train has an Initial velocity of 20 miles per hour, and a final velocity of 34 miles per hour, the area between the ordiuates of 20 and 34 in. p. h. gives the distance traversed while the speed is being raised from the lower velocity to the higher, By the ordinary method of averaging, at a speed of 34 m. p. h. the distance would be represented by the area Opg, in- stead of the shaded portion. This shows the little dependence to be placed on the averaging process, when dealing with speeds which approach the limit. But there is a correction to apply to this if we wish to allow for the Rotative Energy of the wheels ; and this, as we have already seen, varies from about 6 to 12 p. c. of the total energy of the train ; so that in the case of Passenger or Loaded Cars 6 p. c. should be added to the distance as ob- tained above, and in the case of Empty Cars 12 p. c. 20. This method may be applied to a variety of problems i-u Railroad Dynamics : thus, for example, suppose we have a train travelling at 60 m. p. h., and we wish to know how far it will run if the brakes are suddenly applied, causing an ad- ditional resistance of 20 Ibs. per ton of entire train. Then the line of total resistance will be given by the dotted line EG (Diag. II), and the value of MN at any given speed will equal the entire ordiuate from OH to the curve EG, for the * AH measured in inches on the diagram, RAILROAD LOCATION. 19 line of propelling force then coincides with OH i.e., equals zero. Or, conversely, if the train be pulled up in any known distance, we can by two or three trials ascertain the efficiency of the brakes. If in dealing with such problems as these we have in the course of the distance travelled various rates of grade and curves of different " degree," we can, without serious error, draw our line of resistance for the mean grade and the mean degree of curvature. 21. We are now able to ascertain the effects of various amounts of Rise and Fall on the velocity of a train. In the first place, we will go back to our former assumption that the engine exerts the same tractive force at all speeds, and that there are practically no frictional resistances. Of course this is a thoroughly erroneous supposition, but by adopting it we simplify matters very considerably, and yet at the same time are able to obtain results which, for practical purposes, are sufficiently correct when we limit their application to compar- atively short distances. FIG. 8. In Fig. 2 let ABGDEF represent the grades on a lim- ited portion of a certain road, then under the assumption al- ready made if we have a train running along the level tow- ards A at a uniform speed of 40 miles per hour, we obtain from the Table of Equivalent Heights in Sec. 17 Vel. Head in ft. at A = 56, because V = 40 m. p. h. " # = 56-40 = 16; .'. Fat 5 = 21m.p.h. =16 + 10 = 26; .-. " (7 = 27 " " = 264-30 = 56; .-. " Z> = 40 j= 56 + 30 = 86; .'. " #=50 " " " ^=60-30 = 56; .'. " F=Q " 20 RAILROAD LOCATION. By determining the speed at a few such points as these, and drawing through them the dotted lines as in Fig. 2, we have practically a Profile of Velocities, from which we can read approximately the speeds at different points on the grade. 22. In such a case as the above the strain on the draw-bar of the engine would at all points be constant, and the amount of work done in transporting the train from A to F would ignoring the difference in distance, which of course in prac- tice amounts to nothing be the same whether the train went along the grade ABEF, or along a level grade ADF. Now the effect of running over such a ridge as ABD is to lower the average speed: thus if running from A to D on the level, the train would arrive at D much sooner than by way of ABD. Again, in running over the grade DEF, its average velocity would be much higher than along the level DE. Thus the ridge ABD is detrimental to high speeds, but the depres- sion DEF tends to raise the average velocity. In dealing with cases where the distance AD or DF does not exceed a few hundred yards, the results obtained as above are sufficiently accurate to enable the engineer to find the effect of adopting certain grades over such a ridge as D or depression E. 23. But this theory utterly fails when applied to grades of considerable length, for the reason that the possible tractive power of the engine at any but the lower speeds decreases as the velocity increases, and the resistances increase rapidly as the speed is raised. We will now consider the result of taking these considerations into account in the case shown in Fig. 2. Now if the train comes on to the grade AB at a certain speed assuming that the Effective Horse-power remains constant it will have a ve- locitv at B appreciably greater than that which we should ob- tain for it at that point by means of the Table of Equivalent Heights. So also at D it will have a velocity greater than it had at A, although by the Table the velocity at A and D should be the same. The reason of this is, that the increase in the accelerating force is more than in proportion to the increase in the total propelling force, being due to a decrease in the re- sistances as well as to the reduction in speed. Similar reasoning applies to the down-grades BC and CD, so that by the time the .train has got to D the total amount of work done on the higher grade is relatively less than what it would have been along RAILROAD LOCATION. 21 the level AD, owing to the reduced frictional resistances. Thus the train is travelling faster at D than it was at A, although it has lost time on the way. Similarly, in the case of crossing a depression such as E, the amount of work done will be greater by the lower route than along the level, and the train will thus have at F a velocity less than it had at D, although it will have made better time between D and F by way of E, than along the level DF. But although the train arrives at D with a higher velocity than if it had proceeded along the level, yet this increase in velocity only partially makes up for the time lost between A and D. So also the decrease in speed at F does not entirely counteract the gain in time made along DEF. The amounts by which the velocities at D and F actually differ from those obtained by the Table, depends mainly in practice on the distance between A and D, or D and F. The greater these distances are, the less reliance is to be placed on the Table ; so much so in fact in dealing with long grades, as to render the energy of the train itself considered as a store of available tractive power practically worthless. 24. It is usual for Railroad Companies to adopt a cer.tain rate of grade which is not except where Pusher-grades are used to be exceeded. This is usually termed the Maxi- mum or Ruling Grade, and is selected with due considera- tion to the tractive power of the locomotives to be employed, the probable amount of traffic, the weight of trains to be hauled, and the speed required to be maintained. It is also selected in most cases so as to admit of a train starting on the grade, if by any chance it should have had to pull up. Also, it should be such that the locomotive employed can haul the train over it, altogether independent of the Momentum or more correctly Energy of the train. By means of Diagram II we can readily select the most suitable Maximum Grade by drawing the line of resistance for a level track and the line of propelling force suitable for the locomotives to be em- ployed ; the length of the ordinate NM, when scaled off, gives the equivalent resistance in Ibs. per ton of the maxi- mum grade. Thus, in the case of the example given in Diagram II, if the speed required to be maintained on the grade equals 24 miles per hour, since NM represents to scale about 17 Ibs. per ton, the maximum grade will equal 22 RAILROAD LOCATION", 0.85 p.-c. Had the required speed been only 10 miles per hour, we might then have used a 1.6 p. c. grade. But proba- bly in neither of these cases could the train start on the grade, and in order to allow for this, we must assume that the line of resistance at no point dips below 15 Ibs. per ton, i.e., 12 Ibs., in accordance with Sec. 3, and a small margin of 3 Ibs. to overcome the Inertia of the train. Thus, allowing for stop- pages, if a speed of 24 m. p. h. is to be maintained in the case shown in Diagram II, the maximum grade must not exceed 0.55 p. c. ; but if 10 m. p. h. only is required, then includ- ing allowance for stoppage the maximum grade may be 1.1 p. c. But we must remember that where the velocity required to be maintained on the maximum grade exceeds that given by Aa, in Sec. 10, some allowance should be made for the probable increase in boiler-pressure after the train has come to a stand-still ; which means that on starting, the I. H. P. of the engine may be placed considerably above its normal working power. 25. Without going into the question of the Economy of the Steam engine, we may say that a Locomotive works with its greatest efficiency when the boiler-pressure remains constant and the engine is running at a uniform velocity. Thus fluc- tuations in speed or variations in the opposing resistances are more or less detrimental to the working of the locomotive. As a consequence of this, if a certain elevation has to be at- tained, in order to make the work as easy on the engine as possible, the grade should be such as to render the sum of the resistances opposed at all points as nearly constant as possible. Thus, if the alignment be straight, the rate of grade should be uniform ; but if curves or other irregularities occur, they should be compensated for, so that a constant resistance may be maintained. 2C. Compensation for Curvature. From Diagram I we see that at 10 miles per hour the resistance for each degree of curvature is about 1 Ib. per ton, i.e., equivalent to a-|- 0.5 p. c. grade, and that at about 30 in. p. h. it is about half this. The rate, however, usually adopted is .03 p. c., which is suitable to a speed of about 25 m. p. h. Thus, if the equivalent grade on a tangent is 1.5 p. c., we must reduce it on a 3 curve to 1.41 p. c. in order that the resistance may remain constant. RAILROAD LOCATIOK. 23 27. Compensation for Brakes, etc. A point to be re- membered in running a long uniform grade which does not approach the maximum is to consider at what points the train will be required to slacken or increase its speed. For exam- ple, suppose on such a grade we have a sharp curve around which the speed is not to exceed 20 miles per hour, but that on the tangent at either end of it a speed of 40 m. p. h. can be maintained. By means of the Table of Equivalent Heights we can adapt the Energy of the train so that the velocity will be reduced without the application of the brakes, and that when the curve is passed the speed of the train can be more readily increased from 20 to 40 in. p. h. But in doing this we have to be careful that at the lower end of the curve we do not in- crease the grade so as to tax the engine too severely. At all such points as crossings, where short stoppages are required, attention should be paid to this, for by so doing we can at times save something even in the cost of construction, besides saving considerably in fuel and in wear and tear to the Roll- ing-stock. 28. But though the operating-expenses may be reduced to a minimum by the use of Long uniform (equivalent) grades, the amount necessarily expended on their construction may be too great to warrant adopting them. In such cases Broken Grades have then to be used. Now we have already seen how to obtain the effect of un- dulations on the velocity and the work done, so that we can in any particular case determine for ourselves what will be the result of selecting a certain arrangement of grades. The following "pointers," however, deduced from what has al- ready been said, may come in handy. 1. A Rise from the uniform grade is detrimental to fast traffic, and though there is a saving in actual work done on it, there is probably no saving in the consumption of fuel. 2. A Depression from the uniform grade tends to increase the mean velocity, but at the cost of a considerable amount of extra fuel. 3. Breaks in the grade which from the point where the broken grade leaves the uniform one to the point where they next intersect do not exceed, say, 1000 to 2000 feet, may be regarded as "Momentum Grades," and accordingly are not so injurious as longer breaks where the Initial Energy of the 24 EAILROAD LOCATION". train is small compared with the Total Energy to be expended on them. 4. The nearer the uniform grade approaches the "Maxi- mum grade," the more injurious do any breaks become; and the only point in connection with the "Maximum grade," where an increase in the rate is allowable, is the insertion of a "Momentum grade" at its lower end. 5. Breaks in a grade are more injurious to slow than to fast traffic as may be seen from the Table of Equivalent Heights e.g., an increase in elevation of 20 feet reduces the velocity from 30 to Ifrmiles per hour, while a velocity of 60 m. p. h. is only reduced to about 55 miles per hour. 6. Be careful in inserting Momentum grades that they will not be such as to cause the velocity at any point to exceed the safe limit. A difference in elevation of about 30 feet be- tween the Broken and the Uniform grade should generally be taken as a limit. 29. Another point to be considered, which we have not yet referred to, is the increase in Liability to Danger of Break- ing-train and Derailment to which an undulating grade gives rise. For, suppose in Fig. 2 we have a train running up the grade from A to B: as soon as the engine is over the summit the pull on the draw-bar becomes enormously in- creased, and similarly with the car-couplings throughout the entire train; so that, unless the greatest care is taken in ap- plying the brakes, the train runs a very great risk of being broken in two. Similarly, in such a hollow as E, the cars near the centre of the train are liable to get terribly jammed together, thereby greatly increasing the chances of Derail- ment. Vertical curves reduce these dangers considerably, but not entirely. It must be remembered that it is not in the least necessary that one of the grades should be an up-grade and the other a down-grade: it is the difference in flie rate of grade that has to be looked out for. (See Sec. 100.) 30. In Fig. 3, let ACB and ADB represent two different routes between A and B, the total Rise and Fall between the two points in each case being the same. The amount of work done in hauling the train from A to B by way of G will, supposing we are dealing with grades so long that the ques- RAILROAD LOCATION. 25 tion of "Momentum Grades" may be ignored, be then prac- tically the same as by way of D. Similarly, if such a point as H in Fig. 4 has to be reached, the work done in hauling the train along the uniform grade EH will be practically the same as by way of FG. It is not the amount of work done on the grades themselves that has to be considered, but the amount of extra work which is uselessly done by a heavy engine haul- ing a large surplus of dead-weight (due to its own size) over FIG. 3. grades where a lighter engine could have hauled the train equally well. If each of the divisions EF, FG, and GRwere a suitable length for one engine to work, the lower route would then be as economical probably as regards Operating Expenses as the higher. Besides this, we have the increased Fia. 4. consumption of fuel, before referred to, which always accom- panies variations in grade. If we make each of the divisions along the lower route from E to //of such a length as to keep the engine employed on each fairly busy, using a different engine on each division, the lower route is then as economical as can be wished for, but otherwise the upper route has the advantage. 26 RAILROAD LOCATION. 31. Now the average length of an Engine-stage may be considered to be about 100 miles, which is of course too long to enable us to work the lower route in the manner described above. We may often, however, by adopting a Pusher-grade, even at a point where at first it appears unnecessary, make a decided improvement in the economy of our grades. The length of this grade, if the Pusher is to be kept steadily em- ployed, depends of course on the number of trains to be taken up it each day: if there are four trains a day the engine will be kept sufficiently at work if the length of the grade is only 12 miles. As to the rate of grade which may be adopted in such cases as this, Mr. Wellington gives the following Table, which is suitable for average Consolidation Engines, the coefficient of adhesion being taken at 0.25 : TABLE OF PUSHER-GRADES. Grade worked by Net Load of GRADE POSSIBLE WITH 1 Pusher. 2 Pushers. Level. 2675 0.38 0.74 0.2 1758 0.75 1.26 05 1147 1.30 2.01 1.0 711 2.16 3.13 1.5 504 2.96 4.13 2.0 383 3.72 5.03 32. Maximum Curvature. In countries where construc- tion is comparatively easy, it is often the custom to select a cer- tain degree of curvature which is not to be exceeded. The ques- tion of the speed required to be maintained is the main one which arises in this case. Wear and tear of rails and rolling- stock is also an important factor. The question of resistance at ordinary speeds is comparatively unimportant, since at a speed of 25 miles per hour a 10 curve only offers the resist- ance of about a 0.3 p. c. grade. In rough country it is im- possible to fix a "maximum," for the additional cost of con- struction which the adoption of a limiting-grade might involve would perhaps be an inconceivably greater consideration than the loss of a few seconds or possibly minutes in time. As regards the question of the Safe Speed on Curves, it is diffi- RAILROAD LOCATION". 2 Cult to lay down any law, but it is supposed to vary inversely as the square root of the radius. Thus if we assume that 40 miles per hour is a safe speed on a 2 curve, the speed should be limited to 20 m. p. h. on an 8 curve and to 14 m. p. h. on a 16 curve. The chances of derailment and the wear and tear of rolling-stock and rails are decreased materially by the use of Transition curves. (See Sec. 96 ) 33. It is almost unnecessary to refer to the subject of Re- verse Curves. In Station-yards, where the speeds are insignifi- cant, their use is sometimes advisable; but on the Main Track an intervening tangent of at least 200 feet in length should be regarded as an absolute necessity. A fault much more fre- quently found is the insertion of a short tangent between two curves of the same direction. Getting on to a tangent from a curve is as hard work as getting on to a curve from a tangent; and since it is at the P. C. and P. T. that the curve gives its maximum resistance, the curves should at least be compounded so as to make the radius of curvature at all points as uniform as possible, for in each case the total amount of curvature will be the same. Another point to be remem- bered though it is not often that it can be applied is, that a road which has its curves at points where the speed is com- paratively low has a decided advantage over one in which the curves are located at places where a high speed is required to be maintained. Thus, if a certain amount of curvature has to be got in, in such a place as DEF in Fig. 2, it should be arranged if possible so that the curvature at-Z) and F will be sharper than at E. Curvature should also be avoided as much as possible at all points where a stoppage is required, for on starting, the resistance due to the curvature is a great con- sideration, and, as we saw in Sec. 6 and Diagram I, will probably make it as difficult for the train to start as a decided up-grade. 34. We have now dealt in a more or less superficial way with most of the mechanical problems which arise in connec- tion with railroad trains; but it is convenient, for the sake of more readily comparing the value of the various resistances to passenger and freight trains at average speeds, to tabulate their mean values (as given by Prof. Jameson) as follows: 28 BAILROAD LOCATION. TABLE SHOWING COMPARATIVE VALUES OF RESISTANCES AS REGARDS WORK DONE. Items. Distance. Curvature. Rise and Fall. 1 mile 5280 feet 600 25 feet 1 Curvature 88" 041 " 1 foot Rise and Fall- 211.2 " 24 1.0 " Rise and Fall " of course means in one direction only, and is so stated in order to take account of the Rise when run- ning in the opposite direction. Thus in Fig. 3 the total Rise and Fall between A and B by either route equals 710 feet. COST OF OPEEATING. 35. The expense involved in overcoming the resistances re- ferred to in Sec. 34 is not proportional to the amount of work which is performed on account of them. For instance, it is found by experience that hauling a train over one mile of level track costs on an average about the same as 150 feet of rise and fall, not of 25 feet, as given in the last table. Similarly, with curvature, the operating of one mile of level track is found to cost the same as about 900 of curvature (not 600); so that as regards operating -expenses the table given in Sec. 34 becomes Items. Distance. Curvature. Rise and Fall. 1 mile 5280 feet 900 150 feet 1 Curvature 1 foot Rise and Fall. 5.86 " 35.2 " 1 6 0.166 " 1.0 . " As soon, then, as we know the expense of operating one mile of level track, we can by means of this table find the probable cost of working any certain grade or any given amount of curvature. 36. Taking $1.00 it is probably nearer 90 cts. as the aver- age cost of operating one mile of level track on American Railroads for each train that runs over it (and returns) each day, we can make this our unit of operating-expenses and RAILROAD LOCATION. 29 term it the cost of one Train-mile. The items which go to make up the expense of the train-mile are as follow: ( Oil, Fuel, Waste. Motive Power ..... -j Driver, Fireman. ( Repairs. .. j ^ir^ndBenewals to Cars. Train Expenses. Road Repairs Track, Road-bed, Structures. roT1 , ( Stations, Terminal, Taxes. ' ' ] Repairs and Renewals. Taking, then, $1.00 as the cost per train-mile, and assuming the interest on the amount capitalized at 6 p. c., we obtain the following table: Unit. Value per annum per daily train. Amount Capitalized. 1 mile $350 $5 833 33 1 foot 066 1 10 1 Curvature 1 foot Rise and Fall.. . . 0.39 2.33 6,50 38.88 This assumes that each " daily" train only runs 350 days in the year, which makes a sort of allowance for Sundays, 1 ' specials, " etc. 37, From the above we see that if we have ten trains mak- ing the round-trip every day, we are entitled to spend $58,333 extra on the construction of a certain route, if by so doing we can save a mile of level track; so also we should be entitled to spend $388 in the reduction of a foot of rise and fall. Thus with 10 daily trains we might safely expend 2 X $388 = $776 in lowering (only one foot) such a summit as C in Fig. 3; but if C had been the terminus of the line AC we ought only to spend $388 in lowering it one foot. Suppose again we have two routes to select from, one of which would probably cost $40,000 more than the other, but would shorten the distance by one mile and would save a rise and fall of 100 feet. Then if there are only likely to be three trains running including returning each day, we are not entitled to spend more than ($5833+ $3888) X 3 = $29,163 to save the above distance and rise and fall ; therefore it would probably be injudicious to adopt the more expensive route. 30 RAILROAD LOCATION. 38. As regards the cost of operating Pusher grades, we find that a Pusher kept pretty busy costs on aii average about $280 per mile of incline per annum i.e., $140 per mile run " all that the engine fails to do below 100 miles per day may be assumed to cost f roin to as much as if it had been rim, and is so much added to the cost of what is run." Thus on a 5-mile incline, with only 4 trains to be taken up it each day, the probable annual expense of the Pusher will be found thus: Work done, 4 X 5 X $280 = $5,600 Work not done, 30 X ^ = 2,100 Total $7,700 Had we been able to reach the summit without adopting a Pusher-grade supposing the total rise and fall to be 1000 feet the cost of "Rise and Fall " would have been for the 4 daily trains 4 X 1000 X $2.33 = $9320, representing a differ- ence in the operating-expenses of $1620 per annum, which at 6 p. c. would have warranted our expending $27,000 more on the route which involved the Pusher-grade, assuming curvature and distance to be the same in both cases. 39. To test the merits of different routes as regards operat- ing-expenses, we may express them in terms of their Equiv- alent Lengths (L) in miles thus: L = l + m + m' where I = actual length in miles, H = total rise and fall in feet, C = total curvature in degrees. 40. As regards the increase in operating-expenses caused by any slight increase in distance, such as is the result of changes in the alignment, it is not usually the case that the cost per train-mile for any small additional distance is as high as the rate already given; for many of the items, such as station and terminal expenses, which go to make up the average cost per train-mile, are not affected by an addition in distance which does not exceed 2 or 3 p. c. of the total length of the road. Thus, in selecting the choice of two routes, the engineer RAILROAD LOCATION. 31 should not necessarily take the average cost per train-mile as his standard by which to find the probable difference in the operating-expenses, but in most cases may consider about 50 cents per train-mile an amply sufficient allowance for that portion of the longer route which is in excess of the other, when that excess does not exceed the above amount. 41. In order to approximate as closely as possible to the probable cost per train-mile on any projected road, the en- gineer must judge by the results on other roads where the conditions are more or less similar. Where changes are to be made in the alignment of a road already in operation, the value of the proposed improvements can then be found with considerable accuracy, since the cost per train-mile is then known. BECEIPTS. 42. The Receipts usually vary from about 1.5 to 2.0 the cost of operating; and it is not often that the locatiug-engineer has it in his power to affect them in any way. He may, however, by carrying the location by a slightly more circuitous route than he would otherwise have adopted, catch the traffic of some outlying village. Mr. Wellington on this subject says: " When the question comes up of lengthening the line to secure way-business, we may almost say that where there seems any room for doubt, it will almost always be policy to do so. Extra business to a railroad the engineer will rarely err in thinking is almost always clear profit. Of Passenger business this is literally true until the increase becomes con- siderable; of Freight business it is so nearly true that 80 or 90 per cent at least of the way-rate is clear profit over the usual cost of any particular shipment." Thus, suppose we are projecting a line between two points 100 miles apart, and that half-way between them lies a small town 10 miles off the direct route. The additional distance involved in running through it is about 2 miles. Suppose, as is a reasonable estimate, the average payment per head of pop- ulation is $13 per annum. Then, if there are likely to be 5 daily trains, we may put the extra cost of the two miles, in- cluding the interest on the capital spent on their construction, at about f 2000 per unuuua. Therefore, looking at the matter 32 RAILROAD LOCATION". only from this point of view, if the place contains, or is likely to contain before long, only about 150 people, it would prob- ably be wise to locale the road through it. COST OF CONSTRUCTION. 43. This is a subject which had almost better be omitted, for the range of prices is so great in different parts of the country, that values given to suit one place may be entirely mis- leading when applied to another place a few hundred miles off. I have, however, endeavored to strike the average prices as nearly as possible, and with these remarks they must be taken for what they are worth. They show more or less the relative cost of various works, and in this way may sometimes be of service. First we have the following lot common to all track : Steel rails per ton (2000 Ibs.)... $25 00 to $45 00 Angle-bars, per Ib , 02 " 03 Bolts, " 03" 05 Spikes, " 02" 04 Ties (in place), each 20" 50 Ballast Gravel, p. cu. yd 25 " 75 " Broken Stone, p. cu. yd 75" 150 Track-laying per mile 250 00 " 500 00 Then we have the following, according to circumstances : Solid Rock, per cu.yd ; $0 75 to $ 2 00 Loose Rock or Hard Pan. per cu. yd 35" 75 Earth, per cu. yd 10 " 50 1st Class Masonry, per cu. yd 10 00 " 30 00 2d " " " ... 700" 1000 3d ' " " , 500" 700 Dry rubble " " 200" 500 Riprap, per cu. yd 100" 200 Iron erected in bridge-work, per Ib .. 04" OS Timber in Trestles, per M , 25 00 " 45 00 " "Culverts, " 1500" 2500 " " Log Culverts, per M 1000" 2000 Piling driven, per lin. ft Grubbing, per Station 1200 Clearing, per acre 20 00 Overhaul, p. cu. yd. per Sta 01 Fencing per mile of track 300 00 Telegraph line Single wire 175 00 75 20 00 30 00 02 800 00 250 00 KA1LKOAD LOCATION. 33 By taking the mean prices of the first set, we obtain for an average mile of standard-gauge track (10 p. c. short rails) the following cost: 103 tons Steel rails (65 Ibs. p . yd.) $3,862 00 710 Angle-bars, 20 Ibs. each 355 00 1420 Bolts, 7 kegs, 200 Ibs. each 56 00 5670 Ibs. Spikes, 38 kegs, 150 Ibs. each 171 00 2640 Ties 924 00 Ballast, 3667 cu. yds. Gravel 1,834 00 Track-laying 375 00 Total $7,577 00 Besides these we have, of course, Right of Way, Engineer- ing, Law, and a variety of Incidental expenses. As regards the COST OF TRESTLEWORK, we find that for Low Pile Trestles say 20 ft. high assuming piling to cost 50 cents per lin. ft. driven, and the superstructure $20 per M., the cost will usually be about $6 per foot run. For a Wooden Trestle 50 feet high at $25 per M., the cost, if resting on piles or sills, will usually be about $10 per foot run ; but if 100 feet high, $20 to $25 per foot run. The cost of Iron Trestlework varies so enormously accord- ing to the design, that it is impossible to lay down any figures which might be generally applicable. Assuming, however, that the total weight of iron in the trestle equals the total weight of wood in an equally strong wooden trestle, the cost, at 5 cents per lb., would be about double that of a wooden one. These figures are of course exclusive of Masonry foun- dations, and are for single-track. As regards the COST OF TRUSSES, a Wooden Howe Truss single-track, of 100 ft. span, Lumber at $15 per M. costs, framed, somewhere about $2000; and an Iron Truss of the same span, at 5 cents per lb., costs about $5000. The cost in both cases varies pretty much as the square of the span. Erecting usually costs from $5 to $10 per liu. foot. As regards the COST OF TUNNELLING, we may say it varies from $2.50 to $7. 50 per cu. yd.; so that for a single-track tun- nel we may consider the price per foot run to vary from, about $30 to $80, including masonry. The cost of sinking a shaft or driving a heading is considerably higher in proportion than this. For more on the subject of the Cost of Grading, see Sec. 124, Part II. 34 RAILROAD LOCATION. INSTRUMENTS. 44. The principal Instruments ordinarily used on Railroad Location are: The Transit, Compass, Level, and Hand Level; and we will consider them in the order here given . (For Instru- ments used on exploratory-work, see Sees. 141 to 158.) THE TRANSIT. Before proceeding with the adjustments of the Transit, it should be seen that the object-glass is screwed firmly home, and a short scratch made on the ring of the glass and contin- ued on to the slide, so that, should the glass be taken out or work loose, it may be screwed up to exactly the same position it was in before. If this is not done, and the glass happens to be badly centred, i.e., its optical axis does not lie in the cen- tre of the telescope-tube, if by any chance the glass is moved, the Line of Collimation will also be thrown out of adjustment. The following are the usual adjustments for a Transit : A. To make the vertical axis truly vertical by means of the small bubble- tubes. Turn the vernier- plate until each of the tubes is parallel to a pair of opposite plate-screws. Bring both bubbles to the centres of the tubes. Then turn the instrument through about 180. If the bubbles are still in the centre, the adjustment of the small tubes is correct ; but if not, correct for half the error in each case by means of the adjusting screws at the ends of the tubes. This adjustment should then be correct ; if not, repeat the process until it is. B. To set the cross-hairs truly vertical and horizon- tal. After levelling up, test the vertical hair along its whole length on some fixed point, and if not correct, loosen the cap- stan-headed screws and move the diaphragm around. The horizontal hair may be tested in a similar way. C. To make the horizontal axis of the telescope truly horizontal. Level up the instrument and point the tele- scope to some object C, as in Fig. 5, at an altitude, if possi- ble, of not less than 45. Mark the point A where this verti- cal plane strikes the ground. " Reverse" the instrument, and KAILROAD LOCATION. 35 if on pointing to C and then reducing to the ground we again strike A, this adjustment is correct. c But suppose the first time the "verti- cal" plane had struck the ground at / B, and then on reversing, instead of / striking B again, it cuts through some ; ' point D. Mark a point E between D I and B, distant from D by one quarter of / DB. Then by means of the screws / under one of the pivots of the horizon- / tal axis bring the intersection of the / cross-hairs to strike the point E. This #WW*!!!< (nearly), #p' -{- (R R) sin dpp' pp' being obtained by direct measurement; and aa' = pp' -}- (R 1 R) sin dpp' (R R) sin acb, from which we can find the position of a' . But suppose, as in Fig. 35, the two curves are of opposite direction. FIG. 35. Then select p on the side of a towards the other curve. Then, as before, pea = dpp' acb; but in this case (R 4- R) vers dpp' sin acb = pp -j- (R -j- R) sin dpp - (nearly), and aa' = pp' -\- (R -\- R) sin dpp' (R' -J- R) sin acb. The distance ap should never exceed 100 feet when the curves are of the same direction, or 75 feet when of opposite direction, and should always be taken as small as possible. 84. Given a curve joining 1 two tangents, to change the P.C. so that the curve may end in a parallel tangent. Let it be required to move the P.C. at a (in Fig. 36) so that the J> x" curve ab, instead of ending at b, will end in a parallel tangent, distant from the tangent at b by the amount Then, since it is simply a. case of shifting the curve bodily in the di- rection of the tangent aa' , we have aa' = e cosec 1. FIG. 36. Had a'b' been the given curve, and it were required to shift EAILROAD LOCATION". it outwards to the parallel tangent at b, the same equation of course applies. 85. Suppose we have such a case as that shown in Fig. 87, where ab is the given curve, and it is required to shift it to parallel tangents at each end, as at a' and b'. FIG. 37. Then starting from the tangent at a, we can, as above de- scribed, shift the curve from the tangent at b to the tangent at b', and from_the tangent at a we can in the same way shift it on to the tangent at a', which gives us the required positions of a' and b'. 86. Given a curve joining two tangents, to change the radius and the P.C. so that the new curve may end in a parallel tangent at a point opposite to the original P.T. In Fig. 38 let it be required to change the radius of the curve ab and also the position of a, so that the curve, instead of ending in b, will end in a parallel tangent at b' (V being directly opposite to b). Then if is the centre of the curve ab and R its radius, and 0' the cen- tre of the curve a'b' and R' its radius, by Equation 11, Ab = R exsec /, and Ab' = R exsec 7; O'i. FIG. 88. therefore and R 1 -R = bb' exsec aa' = Had a'b' been the given curve, and it were required to shift RAILROAD LOCATION. 93 it outwards to the parallel tangent at b, the same equations of course apply. 87. Given a curve joining two tangents, to find the radius of another curve which, from the same P.C., will end in a parallel tangent. Let it be required to change the radius of the curve ab, so that it will end in a parallel tan- gent at V. "\ / "V. .-** Let be the centre of the curve ab and R its radius, and 0' be the centre of the curve ab' and R its radius. Then R - R = 00; therefore R - R = - vers /' FIG. 39. Had ab' been the given curve, and it were required to shift it outwards to the parallel tangent at b, the same equation of course applies. 88. Given a curve joining two tangents, to change the radius and position of, the P.C. so that the curve may end in the same P.T., but with a given change in direc- tion. In Fig. 40 let it be required to change the radius and P.C. of the curve ab, so that at b it will have a difference in direction equal to /' I. Then if is the centre of the curve ab and R its radius, and 0' and R' are the centre and radius of the curve a'b, Myers I = R' vers/'; therefore D , R vers 7 = - FIG - 40 - and aa' = R sin / R' sin /', 94 RAILROAD LOCATION. COMPOUND CUKVES. 89. A compound curve, being merely a series of two or more simple curves, the manner in which it is located is by setting out its components separately, each P.C.C. (Point of Com- pound Curvature) being treated as a P. C. or P.T., the direction of the.tangent at each P.C.C. being given by its Index-reading. As regards the notes, instead of keeping them for each curve independently, it is better to carry the Index-reading through continuously from the P.C. to the P.T., so that the reading for the P.T. equals half the total intersection-angle. The length and intersection-angle of eacJi component curve should be entered in the notes, and also the total length and total intersection-angle. 90. To locate a compound curve when the P.C.C. is inaccessible. FIG. 41. Suppose, as in Fig. 41, p (the P.C.C.) is inaccessible. The points e and d, if accessible, may then be found by inserting the value of the intersection-angle, in the case of each curve separately, in Equation 9, and thus obtaining for T the dis- tances ad and be. Then from the tangent de the curve can be located by offsets, as already shown. If the points d and e are also inaccessible, select in the curve some convenient point /, and from it set oft' the offset/ft = RAILROAD LOCATION". 95 of vers fop (by Equation 20). Similarly, from a point in the other branch of the curve lay off an offset ik qi vers iqp. We can then rind the position of p by Equation 21 ; thus : hp = of sin fop. 91. Given a simple curve ending in a tangent, to con- nect it with a parallel tangent by means of another curve. 1. Let ac in Fig. 42 be the given curve, and be the required curve: then we have cos C = 1 R-r' p.c.c. FIG. 42. from which we can at once find the P.C.C. 2. Let be be the given curve, and ac the required curve: then since C, the central angle, is the same for both curves, the above equation holds good also in this case. 92. To connect a curve with a tangent by means of another curve of given radius, -JL p.c.c FIG. 43. 1. Let ac in Fig. 43 be the given curve which it is required to connect with a given tangent at b. Find the point a on the given curve which has its tangent parallel to the given tan- gent, and measured: then, since cos C = 1 R- r' we can thus find the position of the P.C.C. 96 EAILROAD LOCATION. 2. But if the radius of the required curve is less than that of the other curve, then, as in Fig. 44, find the point dj&t the intersection of the tangent at b with the given curve ac, and ob- serve the angle of intersection at d = aod; then R cos (aod) r cos aop = R-r Thusj9, the P.C.C., will be sit- uated at a distance along the FlG - 44 - curve from d represented by the curvature aop aod. 3. An analogous case is that shown in Fig. 45, where^it is required to connect the curve ac with a tangent on the convex side by means of the curve pb. Then, as before, find d and observe the angle of intersection at d = aod ; then cos (aop) = R cos (aod) r from which we can find p as above. FIG. 45. Suppose in case 3 the point d were found to coincide with a; then we merely have the case of a Y located on the tangent db, in which case the above formula becomes cos (aop) R-r RAILROAD LOCATION. 97 93. Given a compound curve ending in a tangent, to change the P.C.C. so that the terminal curve may end in a given parallel tangent without changing its ra- dius. 1. In Fig. 46 let the radius of the terminal curve pb be greater than the radius of the other curve-pa ; then, A.. If we want to shift tJie curve inwards to b' , then to find p 1 , the new position of the P.C.C., \ve have FIG. 46. cos o = cos o -\- R-r' but, B. If apb' were the given curve, and it were required to shift it outwards to b, then cos o = cos o R-r' and since in both cases pqp' =0-0', we can thus find the position of p or p', as the case may be. 2. Suppose, however, the radius of the terminal curve bp is less than the radius of the other curve pa as in Fig. 46, and that it is required to shift the tangent (A) inwards to b: then cos o = cos o - R-r But (B) if ap'b' were the given FIG. 47. compound curve, and it were required to shift it outwards, then cos o = cos o' -f- -, R-r 98 RAILROAD LOCATION. Then since in both cases (A) and (B) pqp' = o' o, we can find the position of p or p' as the case may be. 94. To connect two curves, already located, by means of another curve of given radius. As in Fig. 48, let E be the radius of the easier curve, and r the radius of the sharper curve. Find the tangent ab as shown in Sec. 83, and also the distance ab by direct measure- ment or calculation ; then and tan (aqs) = qs = ab cosec (aqs). Then, since oq = op R and os = op' r, where op and op' are each equal to the radius of the required curve, we have the three sides of the triangle oqs, from which we can find the angle oqs (see Sec. 231); and aqp 180 (oqs + aqs). Thus we can find the position of p. Similarly, we can find the position of p'-, or we can calculate the angle at o, which does equally well. The radius of the required curve must exceed If R = r, then RAILROAD LOCATION. 99 95. To locate any portion of a compound curye from any station on the curve. FIG. 49. Let abce in Fig. 49 be a compound curve, and a any station on the curve, and.let it be required to establish the point e ; the P.C.C.'s at b and c being inaccessible. Assume, -for the sake of simplicity, that the chords ab, be, and ce are equal, and let the curvature of be equal twice the curvature of ab, and that of ce three times the curvature of ab. Now if d the deflection from the tangent at a for Sta. b, then, if ab be produced to /, the angle fbc = d -\- 2d = 3d. Again, if the chord be be produced to g, the angle ecg = 2d -f 3d = 5d. Then in the triangle abe, the angle at b 180 M ; and since the length of the chords can be found by Equation 16 (Sec. 74), we can find the side ae and the angles at a and c. Again, in the triangle ace, the angle at c 180 (bca -f- 5d); thus we can find the angle at a. Similarly we can find the angle subtended at a by the chord be, and thus we have the total deflections to b, c, and e. When the chords are of different lengths, as is of course usually the case in prac- tice, and the curvature varies irregularly, we can by plotting the curves and drawing the tangent at each P.C.C. see at once in each case what the deflection-angle at any P.C.C. will be from the chord produced. The principle will be just the same as in the case above described. Sec. 96 is an application of this problem. 100 RAILROAD LOCATION. TRANSITION CURVES. 96. Since the elevation and depression of the outer and inner rails, respectively, at the entrance to a curve must be made gradually, and for any given speed the difference in elevation varies inversely as the radius of curvature, it fol- lows that the curvature should also decrease gradually, having a radius equal to infinity at the P.O. and a minimum at the centre of the curve. If we assume, as is usual, that the dif- ference in elevation of the two rails increase at a uniform rate until the maximum curvature is attained, then the theoretic curve which should be adopted is a form of the elastic curve, which, on account of the trouble involved in locating it, has been supplanted by various approximations, such as the curve of sines, parabolae, etc.; these being easier to locate in the field. The use of Transition Curves is found not only to cause less resistance to the passage of trains than a similar curve whose ends are not eased off, but also generally to enable the curves to be fitted better to the ground than in the case of plain circular ones. That Transition Curves are of advantage in actual practice is shown by the fact that all Simple Curves at their P.C.'s and P.T. 's have a decided tendency to assume the form of the Elastic Curve ; and since this lateral creeping is caused by the pressure of the flanges of the wheels, increased wear and tear to rails and rolling-stock is the result. It is to be noticed that the easing of curves in many cases involves an increase in curvature at the centre of the curve, but this is usually so slight as to be practically inappreciable, and is much more than compensated for by the reduction of curva- ture at the ends of the curve. Thus, for example, where a 9 simple curve defines the limit of curvature in the case of uneased curves on any road, by inserting transition curves a 10 curve would be perfectly allowable. The three following methods of inserting transition curves are simple and easily applied: 97. Method I. Suppose, as in Fig. 50, that we have a 5 30' curve db, which it is required to ease off by means of a transition curve. RAILROAD LOCATION. 101 Now if we do not wish to shift the main curve inwards from the tangent at a, it becomes necessary to shift the tangent at a 0.50 1 "^-^^ FIG. 50. itself outwards by the amount ac, and also to throw the P.O. at a backwards by the amount oc, so that the point o becomes the new P.O. Now ac = T sin d R vers 0, and oc = Tcos d R sin C, . where Y the long chord to the end of the transition curve; d = the total deflection-angle from Sta. o to the end of the tran- sition curve (given in top line of Tables A and B in this sec- tion); C =the total curvature of the transition curve, as rep- resented by the angle esa (values of which are given in Tables A and B); and R = Radius of the main curve. The values of the first term in each of these equations are also given (i.e., T"sin d and Fcos d) in Tables A and B. Suppose we consider that a transition curve which increases its curvature by 1 in every 50 feet (as in Table A) will suit the case in question, then we want 250 feet of such a curve in order that the increase in curvature at no point may exceed 1, and in that case we find from the above formula that oc = 113.40 feet and ac = 3.06 feet; so that the tangent must be offsetted to the 102 KAILROAD LOCATION. left a distance of 3.06 feet, and the new P.O. will be situated 113.40 feet back from the original one. Set the transit up at the point o and locate the curve in the usual manner, the zero of the instrument coinciding with the direction of the tangent, the index readings being taken from the top line in Table A. The point e at Sta. 2.50 from o will then be the P.C.C. of the 5 branch of the transition curve and the 5 30' main curve. Should the point e not be visible from o, the transit may be moved up to any of the intermediate sta- tions, and the total deflection for the other stations from the tangent at any station are given in the tables; so that, suppose we had found it necessary to move up to Sta. 1.50, then we can get the zero of the instrument to coincide with the direction of the tangent at that station, by setting the vernier to the deflec- tion for Sta. 1.50 (taken from the top line in the table) when the telescope is clamped on to the back-sight at Sta. o. We then proceed as before; e.g., our index-reading for e will be 3 25', and so on. Had a change of 1 in every 50 feet extended the transition curve too much, we might have adopted the curve given in Table B. TABLE A. CHANGING 1 IN EVERY 50 FEET. TOTAL DEFLECTIONS PROM THE TANGENT AT ANY STATION, AND THE VALUES OF C, T sin d, AND Y cos d. .50 1.00 1.50 2.00 2.50 3.00 Transit. 15' 52*' \W 3 07*' 4 45' 6 42*' 15' Transit. 30' 1 22*' 2 35' 4 07*' 6 00' 37*' 030' Transit. 045' 1 52*' 3 20' 5 07*' 1 10' 107* 45' Transit. 100' 2 22*' 4 05*' 1 52*' 155' 1 37*' 100' Transit. 1 15' 2 52*' 2 45' 2 52* 2 40' ?gr Transit. 1 30' 3 47*' 4 00' 3 52*' 3 25' 2 37*' 130' Transit. C 030' 1 30' 3 00' 5 00' 7 30' 10 30' Fsind in feet. 0.22 1.09 3.05 6.54 11.98 19.80 Ycosd in feet. 50.00 99.99 149.95 199.81 249.41 298.74 RAILROAD LOCATION. 103 TABLE B. CHANGING 2 IN EVERY 50 FEET. .50 1.00 1.50 2.00 2.50 3.00 Transit. 030' 145' 3 40' 6 15' 9 30' 13 25' 030' Transit. 100' 2 45' 5 10' 8 15' 12 00' 1 15' 1 00' Transit. 1 30' 3 45' 6 40' 10 15' 2 20' 2 15' 130' Transit. 2 00' 4 45' 8 10' 3 45 3 50 3 15' 2 00' Transit. 2 30' ' 5 45' 5 30' 5 45' 5 20' 4 15' 2 30' Transit. 300 / 7 35' 8 00' 7 45' 6 50' 5 15' 3 00' Transit. C 100' 3 00' 6 00' 10 00' 15 00' 21 00' Fsind in feet. 0.44 2.18 6.10 13.06 23.89 39.37 Fcos d in feet. 50.00 99.98 149.80 199.32 248.12 295.70 The stations located as above need only be considered as ternpwary ones, by means of which the true stations may be located. These may be best obtained as follows: Suppose Sta. o falls really at Sta. 304 + 34, then Sta, 304 + 50 can be located by stretching a tape between temporary Stations o and 0.50 and setting oft' the ordinate if (Equation 24, Sec. 80) 16 feet along it from o, and so on between the different stations. Values of M are given in the following table for a 1 curve. The value of M for any other curve may be considered to vary as the curvature, so that, for example, for a 9 curve the ordi- nate at any point will be 9 times that given in the table for the corresponding distance. VALUES OF M FOR 1 CURVE, 50-FT. CHORDS. Dist. from Temp. Sta. M in feet. Dist. fiom Temp. Sta. M in feet. Dist. from Temp. Sta. M in feet. 2 ft. 4 " 6 " 8 " .011 .016 .022 .030 10ft. 12 " 14 " 16 " .035 .040 .044 .048 18ft. 20 " 22 " 24 " .050 .052 .054 .054 The principal objection which can be urged against this curve is its rigidity; this is in a great measure overcome by having the option of the two sets of curves given above, one changing by 1 every 50 feet, and the other by 2. Generally speaking, the former is adapted to curves not exceeding 7, and 104 BAILROAD LOCATION". the latter to curves of from 6* to 14 curvature; while for curves of from 5 to 8 either set may be employed. Another objection which may be brought against it, and one which is often brought against transition curves generally, is that it is not worth the trouble taken in locating it. As regards this, the use of transition curves, not only theoretically but practically, is found to reduce the resistance of the curve very materially, to lessen the cost of maintenance of way, to reduce the chances of derailment, and considerably to ease the motion of the cars. There is no need to set out the transition curves during the location, but the tangent in any instance should be run to c (Fig. 50) and the transit then offsetted to a, from which point the main curve can be located. The amount of the offset ac, and the distance oc, should be added to the notes of the curve, and also the distance ae, which represents G. The general plan of the location then shows the curves as in Fig. 16. Then when the engineer takes charge of the work for construction he has simply to "reference" the points o and e, and run in the curve by means of the above table, as easily as he would any simple curve. 98. Method II. Another form of transition curve is that shown in Fig. 51. It is especially suitable in cases where it is more convenient to offset the curve than the tangent itself. It practically converts the original simple curve into a 3-centre one, but where the curvature of the main curve is light, it answers the purpose of easing off the curvature at its ends suf- ficiently in ordinary cases. In Fig. 51, let r = radius of the FIG. 51. original main curve ab. Offset ab inwards by an amount af = e; then if R radius of the terminal curve cd, we have cosfod = 1 from which we can find the position of d ; and ca = R (r e) siufod, RAILROAD LOCATIOH. 105 from which we can find the position of c. The curve cd can then be best located with a transit from the point c. A convenient method of applying this principle in practice is to make e = 0.2 foot for every degree of curvature of ab, and to make R 3(r e); then if we make/<# = 33.9 feet, d is the P.C.C., and ca = 2(r e) sin/otf, fod being found from the formula e cos fod = 1 2(r - e) For ordinary curves ca then varies from 75 to 100 feet. 99. Method III. Another method of substituting a 3-centre curve for a simple one, when we do not wish to change the original tangent-points, is as follows: In Fig. 52 let o be the centre of the original simple curve afb, the radius of which = R ; and let Oi be the centre of the new main curve ced, whose radius = R l . And let 2 , #2 be the centre of the terminal curves ac and db, whose radii = J? 2 . 2 FIG. 52. 1. Given Ri and R* . Then (*,-*) sin and aob 106 BAILROAD LOCATIOK. Thus we obtain the position of the points c and d. 2. Given Ri and ao-tC = bo^d. Then adb coid R sin -- R l sm -~ R, = -- * -- _?__. . aob , sin - -- sin The curvature of the arc ced should never exceed that of ab by more than 1 (about 50' excess is usually a suitable amount), and Ri should equal about 3R. The distance fe = (R^-R l ) sin ao*c cosec ^~ - (R - BJ. Suppose, however, in substituting the 3-centre curve for the simple one, it is advisable for the points e and/ to coincide as in Fig. 53. 1. Given Hi and R z , we then have (R -Rivers ~ vers uo& = 5 Then a must be put back on the tangent to u, and aob I uorf aob\ the * fore H So that, substituting for R the value given in Sec. 71, and substituting V, velocity in miles per hour, for v, we have H= .00067 #F 2 sin D ', or, as an approximate formula, easy to remember, we have If we take G = 4' 8", we then have #=.0032 F 2 sinD. The following table, abbreviated from that given by Mr. Searles, calculated for the value of F parallel to the centrifu. RAILROAD CONSTRUCTION. 123 gal force, and for a distance from centre to centre of rail = 4' 10|" (suitable to the 4' 8V gauge), gives the difference in elevation of the two rails in feet, at various speeds for different degrees of curvature. Vel. in m. p. h. DEGREE OF CURVE. 1 2 3 4 5 6 7 9 12 16 10 .006 .011 .017 .023 .029 .034 .040 .051 .069 .091 20 .023 .046 .069 .091 .114 .137 .160 .20(3 .274 .365 30 .051 .103 .154 .206 .257 .308 .359 .460 .611 .809 40 .091 .183 .274 .365 .455 .545 .634 811 1.069 50 .143 .285 .427 .568 .707 .844 .979 60 .206 .410 .612 .811 1.006 1.196 A convenient rule, much used in practice for a gauge of 4' 8|", is, that the difference in elevation equals one half inch for every degree of curvature. In order to allow for the difference in elevation on the dump, the road-bed should have its outer edge higher, and its inner edge lower, than grade. To allow for it on trestles, whether in pile-bents or framed bents, the posts must be cut so as to give the required inclination to the cap on which the stringers rest : the batter of the batter-posts and the verticality of the upright posts remain unchanged. It is usual to adopt a difference in elevation in the rails suitable to the mean speed of the trains which pass over them : the consequence of which is, that the rails on both sides get worn, but in different ways the outer ones by the fast trains and the inner ones by the slow trains. The coning of wheels, which was at one time largely resorted to, is rarely used now on account of the increased oscillation and concussion (see Sec. 4) to which it gave rise, so that the flanges of the wheels, by means of their pressure against the inner sides of the rails, have themselves to keep the balance between the centrifugal force and the component of gravity which is set to counteract it, more or less. In curves uneased by transition curves, the difference in elevation at the P.O. and P.T. must be at least equal to what it is at any other Dart of the curve, so that it must begin some little distance back on the tangent and in- crease gradually until it reaches its maximum at the P.O. or 124 RAILROAD CONSTRUCTION. P.T. , as the case may be. For a 3 curve it isNisually sufficient to begin the difference in elevation about 100 feet back, aud for a 10 curve about 200 feet back on the tangent. When transition curves are used, they must be treated with a differ- ence in elevation at all points more or less suitable to their curvature ; but where the transition curve is merely a simple curve inserted to ease the approach to a sharper one, the difference in elevation for the terminal curve must begin back on the tangent as above, and for the main curve some little distance back on the terminal curve, so as to admit of its reaching its maximum at the P.C.C. It is usual to slightly increase the gauge on curves, generally by about i" for every degree of curvature up to 5. 115. Inspecting the Grading. The engineer should, if pos- sible, pass over every portion of his subdivision at least twice a week, and the oftener the better. In open country there is comparatively little chance of having the dump badly put up owing to lack of supervision, except perhaps through the use of a superabundance of " sods ;" but in timber country where there is plenty of grubbing to be done, and the work is largely let as "station-work," the engineer must be constantly on the lookout for the presence of roots and stumps in the dump. In winter too, snow, frozen moss, etc., at the bottom of a fill serve admirably as a temporary means of bringing it up to grade. He should see that there is a fair line of stumps at the side of the track after the completion of the work in places where grubbing has occurred, or that they have really been burnt ; and when there is snow on the ground he must have it swept well to the side before the filling is begun. He must see that the ditches on either side of the embankments, etc., as well as those in the cuts themselves, are taken out properly, aud thoroughly cleared of all obstructions, that the slopes are neatly dressed off and well out to the slope-stakes. For the final inspection of the road-bed, grades and centres must be carefully run, and the width tested wherever it ap- pears lacking. All litter along the side of the track must be cleared away or burnt, and anything in danger of falling on to the road-bed removed. About this latter injunction the er gineer cannot be too careful, and when in doubt as to the stabil- ity of a piece of rock or an overhanging tree, he should have it removed at any cost. He must also remember that a rock or RAILROAD CONSTRUCTION. 125 tree which at the time of inspection looks tolerably firm, may be a considerable source of danger after the disintegrat- ing effects of a hard winter, or a season of heavy rains, and that it costs very much less to have it removed during con- struction than at a later period. 110. Running Track-centres and setting Ballast-stakes. Where the ballasting is done before the track is laid, ballast- stakes must be driven every 50 feet, so that their tops indicate the elevation of the top of the ballast. They should be placed on either side of the centre-line at the foot of the ballast-slopes. Centre-stakes should also be set every 100 feet apart on tangents and every 50 feet apart on curves, to guide the track-layers ; tacks should be inserted in them. When the track is laid without first ballasting, a line of cen- tres must be given before the track is laid, and usually after- wards as well, to guide the surfacing gang, lor the centres previously put in are almost sure to have been knocked out in laying the track. It sometimes happens in hasty work that the engineer who has the track-centres to run cannot get his centres to coincide with the centre of the dump or with the centres of the bridges. As regards the centres on the dump, he must use his own judgment as to what is best to do : if it is clear that the dump is out of line, he must stand by his own centres ; but if otherwise, it is usually better for him to increase or ease his curvature a little, so as to make it conform with the centre of the road-bed. On bridges or open culverts lie must make his own centres fit the centres of the structures, and if this can- not be done without seriously affecting the adjacent track, the case must be reported at once. 117. Permanent Reference-points. After the track is laid, large hardwood stakes or better still, stone monuments should be set to mark the P.C.'s, P.C..'s, and P.T.'s. They should be placed on the outer side of the curves, at right angles to the track, usually about 5 or 6 feet from the centre. TURNOUTS AND CROSSINGS. 118. In dealing with the subject of turnouts and crossings, we will assume that the Common Stub Switch is used, since it 126 RAILROAD CONSTRUCTION. is the simplest, and the formulae for it are readily applied to any other form of switch. Let Fig. 60 represent a turnout from a straight track, A and a forming the " heel " and B and b the " toe" of the switch. FIG. 60. Then if O = gauge, 2f= number of the frog, F "Frog angle," = Angle of Intersection at F, we have cot H radius of turnout curve, AF = frog distance, AB length of switch-rail, D = degree of curve, G AF = ZGN, R = AF=(ll+} sin F, '}-- I 8 AB = X Throw. The throw according to Sec. 78 = -^=- att The number of a frog may of course always be found by measuring the tongue: thus if at a certain point we find its width to be 5 inches, this divided into the distance of that point from the theoretic point of the tongue gives the number of the frog; thus if that distance were 4' 2", it would be a No. 10 frog. RAILROAD CONSTRUCTION. 127 The following table gives these values for a gauge of 4' and a throw of 5". N F AF in feet. R in feet. D AB in ft. 4 14 15' 37.66 150.66 38 46' 11.2 5 11 25' 47.08 235.40 24 32' 14.0 6 9 32' 56.50 338.98 16 58' 16.8 7 8 10' 65.91 461.38 12 27' 19.6 8 7 09' 75.33 602.62 9 31' 22.4 9 6 22' 84.74 762.70 7 31' 25.2 10 5 43' 94.16 941.60 6 05' 28.0 11 5 12' 103.58 1139.34 5 02' 30.8 12 4 46' 112.99 1355.90 4 14' 33.6 This table may be applied to other gauges; F of course re- maining unchanged, AF and R will vary directly as the gauge ; D will, of course, vary inversely as R. Thus for a 3-foot gauge and a No. 9 Frog we must multiply the above values of AF and R by ' = .637 ; and the above value of D must be multiplied by ' =1.57. AB is of course de- o pendent on the value of the throw adopted. 119, Suppose, however, that the turnout instead of starting from a straight track, as in Fig. 60, starts from a curve as in Figs. 61 and 62 ; then we may assume that when the main curve and the turnout curve are both in the same direction, that the case, as regards the position of the frog, etc., is equiva- lent to a turnout from a straight track, the curvature of the turnout curve being equal to the difference of the curvature of the main and of the turnout curve; and if in. opposite directions, then the curvature of the turnout curve may be taken as being equal to the sum of the curvatures. FIG. 61. FIG. 62. Suppose we have two parallel tracks AD and CB, as in Fig. 63, which we wish to join by a crossing; or, having the track AD only, we wish to insert a turnout AB which shall connect the side track B with the main track AD. Since the former case differs only from the latter in the fact that the dotted 128 RAILROAD CONSTRUCTION. portion C, with the accompanying frog, is omitted, the two cases may be treated together as follows: A-J FIG. 63. Starting from the centre-line .^.j^with a given frog number, we select a certain length n, expressing the length of the branch AM in terms of 100-foot stations. The length of the offset t at M is then given, according to Sec. 78, by the formula t = R vers nD, and the distance along the track AD to this offset equals Thus by setting off the offset t at a distance T along the tangent from A, we locate the point M. The position of the frog at Fis found by taking from the above table the value of AF, and measuring it off along AD, offsetting F by an amount equal to half the gauge. Another offset y = gauge may also be set off at a tangential distance = $AF. These points, together with the toe of the switch, are usually all that are wanted in the curve AM. The length of any other offset, if required, may be found from Sec. 78. The offset t is then produced across to the centre of the other track (or the other track produced) and assuming both branches to have the same radius the offset JVe = tis set off from the point e, which point is found from the formula ce (d 2t) cot nD. We thus have the point N. The curve NB is then located by using the same value of T, and the same offsets as before, only of course in reverse order. By obtaining n from the formula vers nD -=r=^ &H which gives its limiting value, we have a simple reverse curve RAILROAD CONSTRUCTION. 129 without the intervening tangent MN : but this is bad practice when it can be avoided. Should the radius of NB be required different from that of AM, the tangential distance for NB must then be calculated afresh. The advantages of this method are, that any length of inter- vening tangent can be used, provided that the curves are carried up to the frogs, so that the engineer can select any value of n for himself; and with simply a tape, he can locate the crossing in a manner a good deal simpler than the ways ordinarily in use. 120. As an example, let d = 40 feet and let No. 8 frogs be used; and suppose we select 1.3 as a value for n. Then from the table, AF = 75.33, R = 602.62, and D = 9 31', the gauge being 4' 8i". Then from the above formulae we have t = 602.6 X vers 12 22' = 14 feet, T 602.6 X sin 12 22 = 129 feet, ce = 12 x cot 12 22' = 54.7 feet, and# = 1.2 feet. The notes for the setting out of the crossing may then be arranged as follows: 8 3 3 6 FIG. 64. When the distance between the two tracks is great, the cross- ing should be run in with a transit. 121. If the turnout or crossing falls on a curve, it is best to locate it with a transit according to one of the two following methods: 1. If the curvature of the main track is tolerably sharp and the distance d between the centres of the two parallel tracks comparatively small, we can avoid the insertion of a reverse curve without materially lengthening the crossing as follows: 130 RAILROAD CONSTRUCTION-. In Fig. 65 let D = the degree of the turnout curve AC, H = radius of the outer track A, and r = radius of the turnout curve AC The length of AC may then be found in terms of nD, thus: and the length of the tangent equals CB=(R- r) sin nD. For example, let the outer track A be on a 4 curve; then R = 1433, and let d = 40 feet, and the given frog number for the main curve = 11. Then, according to Sec. 119, D for the turnout curve must be that value which is required to make the difference in curvature of the track A and the curve AC equal about 5, both curves being in the same direction; and since this value FIQ. 65. is 9, therefore r = 637 feet. Set the instrument up at A and locate the 9 curve AC; and since by the above formula nD = 18 15', therefore the length of A C =202.7 feet, and similarly the length of CB = 249.2 feet. Thus we find the point B. To run from B to A would be simply a reversal of the above. The frog for the track B will of course be that suitable to a turnout radius equal to the radius of the track B. But suppose this method would in any particular case cover too much ground, or be unsuitable in some other respect, we can then use the following one, which, though involving the use of a reverse curve, is well enough for station-yards, etc., where no high speeds are attained. RAILROAD CONSTRUCTION. 131 2. In Fig. 66 let R = radius of the inner track B, r = radius of branch CB, TI = radius of branch AG. Then vers BEG = from which we can find the length of the branch BG\ and vers BOA = and since the angle AEG = BO A -\-BHG, we can thus find the length of the arc AG, and locate the crossing with the transit, starting from either end A or B. FIG. 66. In order to use frogs of the same number for tracks A and B, we must have r = 7*1 -j- d (nearly). The positions of the frogs maybe found according to Sec. 119. In the case of a Double Turnout the engineer can, by ap- plying the formulae given above, always locate it with ac- curacy sufficient for ordinary purposes, without the aid of special formulae. The length of switch-rails given in Table in Sec. 118 are the proper lengths for a 5" throw, but in practice a difference of 5 feet or so in the length of the rail will be of very little importance. In the same way there is no necessity for the frog to have exactly the number which it should have according to the table. The laxity which is allowable in these matters depends on the speeds at which the trains are likely to pass over the switch. 132 RAILROAD CONSTRUCTION". 122. Curving Rails. The following table gives the mid- ordinates in inches for curves of various lengths. Rails should also be tested for Uniformity of Curvature by testing one half of their length for of the inid-ordiuate. (See Sec. 80.) LENGTH OP RAILS IN FEET. DKG. OF 30 28 26 20 18 14 10 CURVE In. In. In. In. In. In. In. 1 240 .192 .156 .096 .072 .048 .024 2 .456 .408 .348 .204 .168 .096 .048 3 696 .612 .528 .312 .264 .144 .072 4 .948 .828 .720 .420 .348 .216 .108 5 1 19 1.03 .888 .528 .420 .264 .132 6 1 40 1.22 1.06 .624 .504 .312 .156 1 64 1.44 1.25 .732 .588 .360 .180 QO 1 90 1.64 1.43 .840 .672 .408 .204 10 2 35 2.05 1.78 1.04 .852 .540 .264 12 2.83 2.47 2.15 1.26 1.02 .636 .312 14 3 30 2 87 2.48 1.46 1.19 .732 .360 16 3.76 3.28 2.83 1.67 1.36 .840 .420 123. Expansion of Kails. Steel expands about 1 part in 150,000 for each degree Fah. through which its temperature is raised; so that for 30-ft. rails the spaces between their ends should vary from about T y at a temperature of 120 F. to about T V ' at a temperature of - 40 F. This must be carefully attended to. B. THE ESTIMATING OF LABOR AND MATERIAL. 124. The Expense of Grading is of course almost entirely dependent on the cost of the labor expended on it, the value of the material not entering into the question ; so that esti- mating the cost of it is simply a matter of ascertaining the time and wages which are absorbed in its execution. The following notes on the subject of handling earth and rock, which are taken from Trautwine on Excavations and Em- bankments, than whom possibly no better authority could be quoted, serve to show the relative cost of the different processes through which the material has to pass before being finally disposed of in the embankment; and, consequently, from them the aggregate cost may be obtained with a greater or less amount of precision. These processes we will consider in the order in which they occur, taking as the standard of RAILROAD CONSTRUCTION. 133 wages $1.00 per working day of 10 hours, and the expense of a horse as $0.75 (including Sundays). A. THE COST OF EARTHWORK REMOVED BY CARTS. 1. Loosening the Earth ready for the Shovellers. A two-horse plough, with two men to manage it, will loosen about 250 yards per day of strong heavy soil, about 500 yards of common loam, or about 1000 yards of light sandy soil ; thus the cost of loosening these materials per cubic yard will respectively be about 1.5 cents, 0.8 cent, and 0.4 cent i.e., assuming the total cost of the plough and men and horses con- nected with it to be about $3.87 per day. When a four-horse plough is needed, as in dealing with stiff clays or cemented gravel, the cost runs up to about 2.5 cents per cubic yard. Loosening by picks costs about three times as much as by ploughs, where the latter can work to advantage. The amount which a man can loosen with a pick in a day varies from about 14 to 60 yards, according to the material. 2. Shovelling the loosened earth into carts. The shovellers are usually actually at work from 5 to 7 hours out of the day. If we assume that each cart carries, as a working load, cu. yd., a shoveller can load it in from 5 to 7 minutes, according to the nature of the material ; and suppose he is actually shovelling for 6 hours out of the day, then in the course of the 10 hours he handles about 24 yards of light sandy soil, 20 yards of loam, and 17 of heavy soil at the cost of 4.2 cents, 5 cents, and 5.8 cents, respectively. 3. Hauling away the earth, dumping and returning. The average speed of horses when hauling is about 200 feet per minute, so that every 100 feet of lead occupies about one minute ; dumping and turning occupies about another 4 minutes; so that the number of trips per cart per day equals where M = number of minutes in the working day (here 600) and L = length of the lead in terms of 100 feet. Then %N equals the number of cubic yards moved by each cart per day ; and %N, divided into the total expense of the cart per day, gives the cost of hauling per cubic yard. Assuming that one driver attends to four carts (doing nothing else), the total cost per cart may be set at $1 .25 per day. 134 KAILROAD CONSTRUCTION. 4. Spreading on the embankment. The cost of this varies considerably, but may be said to average about 1| cents per cu. yd. When the earth is dumped over the end of the embankment, or is "wasted," cent per cu. yd. should be allowed for keeping the dumping-places clear. Keeping the hauling road in good order. This is an item highly expensive if neglected, but if well looked after, y 1 ^ cent per cu. yd. per 100 feet of lead is usually sufficient to cover it. Wear and tear of tools. "Experience shows that of a cent per cubic yard will cover this item." This also includes the interest on the cost of the tools. Besides the above, 1| cents per cubic yard should be added to cover the cost of superintendence and water-carriers, and about cent for extra trouble in ditching and trimming up. As regards the profit to the contractor, it may be set down as from about 6 to 15 per cent, according to the magnitude of the work and the risks incurred ; out of this he usually has to pay the clerks, store-keepers, cost of shanties, etc., but these as a rule cover their own expenses. The following table gives the cost, exclusive of profit to the Length of Lead in feet. Cu. yds. hauled per day per cart. TOTAL COST, PLOUGHED AND SPREAD, IN CENTS. Light sandy soil. Common loam. Qf Stiff clay or "ATon. <~ 50 44.4 10.4 12.2 13.7 14.7 100 40.0 10.8 12.5 14.0 15.0 200 33.3 11.5 13.2 14.8 15.8 300 28.6 12.2 14.0 15.5 16.5 400 25.0 12.5 14.7 16.2 17.2 600 20.0 14.4 16.1 17.7 18.7 800 16.7 15.8 17.6 19.1 20.1 1000 14.3 17.3 19.0 20.6 21.6 1200 12.5 18.8 20.5 22.0 23.0 1400 11.1 20.2 21.9 23.4 24.4 1600 10.0 21.7 23.4 24.9 259 1800 9.1 23.1 24.8 26.3 27.3 2000 8.3 246 26.3 27.8 28.8 2500 6.9 28.2 29.9 31.4 32.4 3000 5.9 31.8 33.5 35.0 36.0 4000 4.5 39.0 40.8 42.3 43.3 5000 3.7 46.4 48.1 49.6 50.6 RAILROAD CONSTRUCTION. 135 contractor, of earth when ploughed and spread in the embank. ment. When loosened with picks, from 1.3 to 4.5 cents per cu. yd. should be added to the values given, according as to whether the material is of a light sandy nature or a stiff clay. If merely dumped over the embankment, then the values given may be reduced by about 1 cent per cubic yard. B. THE COST OF ROCK REMOVED BY CARTS. The total cost of loosening hard rock with wages at $1.00 per day is usually covered by 45 cents per yard in place ; in dealing with soft shales which can be loosened by pick, being sometimes as low as 20 cents, while in shallow cuttings of tough rock, in which the strata lie unfavorably, $1.00 may be insufficient. A good churn-driller will drill from 8 to 12 feet of 2-inch holes, about 2| feet deep, per day, at a cost of about 12 to 18 cents per foot. A cart suitable for cu. yd. of earth as a working load will take about \ cu. yd. of rock. Rock takes longer to shovel into the carts than earth, so that we may say the equa- tion given above for earth becomes in the case of rock and the number of yards hauled per day is given by ^N. Loading costs about 8 cents per cu. yd., and the repair of the hauling-road about ^ cent per cu. yd. per 100 feet of lead. Thus we have, exclusive of the profit to the contractor Length of Lead in feet. No. of cu. yds. per cart per day. Cost per cu. yd. for hauling and emptying. Total cost per cu. yd. 50 18.5 6.8 60.0 100 17.1 73 60.5 200 15.0 8.3 61.7 300 13.3 9.4 63.0 500 10.9 11.5 65.5 700 9.2 13.6 68.0 1000 7.5 16.7 71.7 1500 5.7 21.9 77.9 2000 4.6 27.1 84.1 2500 3.9 32.3 90.3 3000 3.3 37.5 96.5 4000 2.6 47.9 108.9 136 RAILROAD CONSTBUCTIOtf. " Loose Rock" usually costs about 30 cents per yard less than the above cost for hard rock. 125. Both rock and earth can generally be moved at about the same cost by wheelbarrows as by carts when the lead is equal to about 200 feet ; for shorter hauls the wheelbarrows have the advantage, but for longer, the carts. As regards the cost of removal by scrapers or any other form of vehicle, it may be approximated to in the same man- ner as the removal by carts in Sec. 124. A scraper generally moves from 30 to 60 cubic yards per day with a short haul. A medium-size steam-shovel, if kept tolerably busy, should, un- der ordinary conditions, load the cars at a cost of from 2 to 3 cents per cu. yd.' Grading-machines, 8 or 12 horse, in light soil and with low fills, can generally turn over from 500 to 1000 cu. yds. per day. 126. Estimating Overhaul. It is common to allow an extra price, usually from 1 to 2 cents for every cubic yard of material, either earth or rock, for each 100 feet that it is hauled beyond what is termed FlG - 67 - the limit of free haul, represented by I in Fig. 67. Let us suppose that the material in the cut AC is just suffi- cient to make the fill CB, then the material on which overhaul must be charged is that lying between A and D (or B and E), and the distance which that material is hauled is represented by L, the distance between the centres of gravity of the two solids AD and EB; consequently the length of overhaul = L I, and if 8 represents the contents of AD (or EB), then the amount of overhaul 8(L I). Thus, for example, if L = 1000 ft., I = 600 ft. , and S = 4000 cu. yds., the cost of overhaul at 1 cent per cu. yd. per 100 ft. will be |160. But though the distance I is always given, in order to locate it on the profile we must find the points D and E, such that the material in DC = the material in EG. This may usually be done by inspection of the profile ; and in the same way the points A and B may be fixed. In cases where the centre- heights are not fair indications of volume, these points may RAILROAD CONSTRUCTION. 137 be quickly found to within a few feet, by means of the cross- section note-book. The positions of the centres of gravity of the two solids AD and EB may also usually be fixed by inspec- tion. On this subject the Engineering News says: " As quick a way as any is to plot the volumes of each solid as ordinates, as one would plot a profile, on stiff card-board, cut out the area thus drawn, and balance it on a knife-edge ; but a way which we can recommend as much the best and fairest of any, in competent hands, is to guess at it, throwing the benefit of a doubt for or against the contractor according to the character of the haul, and to some extent of the material excavated. The actual haul cannot fairly be taken at times as the crow flies, nor is it exactly fair that haul over good solid gravel should have the same allowance as haul from a shallow cut through muck. As a contract is a contract, and must be gen- eral, no considerable deviations on account of such contin- gencies as these are admissible, but no considerable ones are necessar}', the limits of error in guessing at the ' centre of mass ' being very small, and having reference to a small item of price, whereas the limits of error in one unavoidable kind of guessing which is usually going on at the same time, that of classification, are very large, and have reference to a very large item. This consideration alone ought to show the folly of any great hair-splitting in mathematical computations of the precise overhaul ; but there is a certain class of minds who are never happy unless they can find -some hair to split, and who will split it with just as much care although there may be a log of wood alongside which they can't split, to which the right half of the hair is to be added." THE CALCULATION OF EARTHWORK. 127. The three solids with which engineers have mainly to deal in the calculation of earthwork are the pyramid, the wedge, and the " prism oid ;" for though, owing to the irregu- larities of surface, these figures, mathematically speaking, are never actually met with in practice where the surface of the ground forms one or more sides of the figure, yet the contents as given by them are sufficiently accurate under ordinary cir- cumstances, when the work has been properly cross-sectioned. But before dealing with the calculation of the contents of 133 RAILROAD CONSTRUCTION. these solids, it will be well to consider the methods of obtain- ing the areas of the cross-sections themselves, on which the computations are based. 1. When the cross-section is of triangular form, as in Fig. 69, its area of course taking for instance the triangle ABO equals AB X 1 the perpendicular distance from C to AB, or AB produced. 2. When the cross-section is an ordinary 3-level one, as in Figs. 71 and 72, then if B = width of road-bed and //, h, 7i', I, and I ' are as shown in Fig. 55, which is the formula most generally in use. 3. If the surface is horizontal, then this becomes Area = 1 4. Or, if regularly inclined, where h is the greater side-height, and I its corresponding distance out from the centre, h' being the smaller side-height. 5. But it frequently happens that we have such a section as that shown in Fig. 68. Such an area may be best calculated iL._\ FIG. 68. by first finding the contents of the figure IDHL, and then deducting from it the areas DIA and HLB; thus the area of this cross section equals TD-\-EJ 2 EJ+FC 2 GK+HL FC+GK 2 fjrT ID.IA BL.HL (AL) 2 2 2 The above forms of cross sections are really all that are re- quired in practice, 1, 2, and 5 being those most generally in EA1LEOAD CONSTRUCTION. 139 use. Neither of these forms requires plotting, but it is usually advisable to plot cross-sections of large area which are very irregular even though calculated as above, for by so doing mistakes are much more readily apparent. Where the work consists largely of irregular cross sections, a good and rapid method of obtaining the areas is to plot the cross-sections and use a planimeter. The error in ordinary cross-sections, plotted on cross section paper to a scale of 10 feet to an inch, should never where the planimeter is carefully adjusted so as to allow for the shrinkage of the paper, etc. exceed 1 p. c. ; and considering that these errors to a large extent cancel each other and are free from errors of calculation, which are usually much more probable than errors in reading the planim- eter scale, the result in the long run is at least equally likely to be as near the truth as that obtained by the more laborious process of calculation. 128. The areas of the cross-sections having been obtained, the calculation of the contents of the solids which they bound is the next point to deal with, and we will consider them in the order given above. A. The Pyramid. The usual cases in which pyramids occur are those shown in Fig. 69, which need no explanation. Fro. 69. The contents of such a pyramid as ABCD are found by the formula S=ABCX~, and this rule applies to any form of base. B. The Wedge. The various forms of wedge which pre- sent themselves in calculating the contents of earthwork, of which that represented in Fig. 70 is the usual type, can only be estimated correctly by the application of the Prismoidal 140 RAILROAD CONSTRUCTION. Formula. But since at the points where the wedge form of solid occurs the cut or .fill is always small, the error involved by using the formula for the rectangular wedge is immaterial; thus we may say that the contents A fl- 8 = area ABODE X ~ : - C. The Prismoid. Though the term " prismoid " strictly applies only to such solids as are contained by 6 plane surfaces, the two end -faces being parallel, and two of the other faces being not, parallel, the extended application of the " pris- moidal formula" has corrupted its true meaning, so that it is now applied very generally in Railroad work to all solids hav- ing two parallel faces, whether plane or curved, upon which, and through every point of which, a straight line may be drawn from one of the parallel faces to the other. The contents of such a solid according to the PRISMOID AL FORMULA equal where L = the length of the solid, A and a = the areas of its two parallel faces, and M= the cross-section parallel to A and a, and half-way between them. This formula at first looks simple enough, but the calculation of M is the difficulty. 129. To explain the application of this formula, suppose we have two end-areas A and a as in Fig. 71. Now in order to obtain the mid section, we must know the points in A and a from which the straight lines joining them start, and at which they end; thus in Fig. 71, if the cross- RAILROAD CONSTRUCTION". 141 section notes simply give the elevations for the 3-level sections A and a, we assume that the upper surface between them is B Fio. 71. composed of two warped surfaces, BCcb and GDdc, which is what follows from supposing that the centre and side heights of M are the averages of the corresponding heights of A and a. So that if the surface were actually as shown in Fig. 72, FIG. 72. we should obtain entirely erroneous results by taking the value of M given by Fig, 71. Thus when the surface is such that points in A and a, other than those directly cor- responding, are to be considered as being joined by straight 142 RAILROAD CONSTRUCTION-. lines, it becomes necessary to indicate in the notes between what points in A and a the straight lines are assumed to be drawn; and then the surface, instead of being made up of two or more warped surfaces, will be composed entirely of a series of plane surfaces as in Fig. 72. This is best done, where re- quired, by drawing, in the cross-section note-book, lines con- necting the notes of the points to be joined. This would also have to be done between two cross sections A and a which did not happen to have the same number of points taken in each. At times cases occur in which it is advisable to fill in slope- lines in this way, but they are very few and very far between; for the labor involved in the calculation of M in such cases would usually have been very much better expended in actually taking a cross-section between A and a. Therefore, as a rule, where the prismoidal formula is to be used in the calculation of the contents, it is very much better to cross-section a little more closely, where necessary, and to omit the filling-in of the slope-lines, than to take cross-sections a little farther apart and fill in the slope-lines by inspection. The value of the prismoidal formula, as applied in the case of Fig. 71, is not so much to rectify irregularities in surface as to make suitable allowance for the difference in the heights of A and a, which the method of average end-areas does not do. In practice, however, where the work is properly cross-sectioned, the application of the prismoidal formula is a mathematical refinement which is entirely unnecessary, for the method of average end-areas that usually employed then gives results sufficiently satisfactory, both to the Railway Company and the Contractor. It is an interesting fact in connection with Figs. 71 and 72, that if the contents be calculated for each possible arrangement of slope-lines, the mean of the results so obtained will be equal to the result as derived by merely the joining of corre- sponding points, as in Fig. 71. The calculation of the mid-area is merely a matter of simple proportion. In dealing with such a case as Fig. 72, by plotting A and a on a sheet of cross-section paper, the drawing of the mid-sections may be done by simply drawing parallel lines ; so that this should be done as a check to the calculations and also as a means of facilitating them. RAILROAD CONSTRUCTION. 143 130. The method used nowadays almost entirely for the calculation of grading, is that of Average End-areas, which assumes that , A + a Now this method, which is the simplest of any to work, unfortunately has a considerable tendency to excess; the re- sults obtained by it are, however, the same as those given by the prismcidal formula applied as in Fig. 71, therefore presumably correct, under the following circumstances : 1. Whenever the centre-heights of A and a are the same, whatever the difference in side heights may be. 2. Whenever the entire widths between the slope stakes at A and a are the same, whatever the difference in centre- heights may be. When, however, the smaller centre-height is at the same end of the solid as the greater width between the slope-stakes, the volume as given by average end-areas will be actually de- ficient. But since these cases are the exceptions, the results as given by this method are in the long run considerably too high, unless care is taken in cross-sectioning to limit the excess. To correct for this tendency a Prismoidal Correction maybe used, found by deducting the prismoidal formula from the formula for average end-areas ; and this correction, when the surface of each end-section is horizontal, equals in cubic yards where ZTand H' are the end centre-heights in feet, s the slope- rates, and L the lengths of the solid in feet. Taking s = U and L 100, we obtain the following values for C, which serve in making up preliminary estimates to show the errors involved by a rough system of cross-sectioning when the contents are calculated by average end-areas. 144 RAILROAD CONSTRUCTION. TABLE OF PRISMOIDAL CORRECTION FOR 100 FEET IN CU. YDS. FOR HORIZONTAL SURFACES WHERE s = U. // - //' 1 2 3 4 5 6 7 8 9 834 779 10 20 93 370 1 llii 408 4 133 448 8 156 490 15 181 533 23 208 578 33 237 626 45 268 675 59 300 720 This value of C is altogether independent of the width of the road-bed ; so that, for example, suppose on ground sloping in the direction of the length of the solid we have, between two sections 100 feet apart, a dill'ereuce in centre-heights of 23 feel, if s = \\ and there is no slo{>e transversely, the contents as given by average end-areas will be 490 cubic yards too much, even with a 14-foot road-bed ; or. if the fill at one end is 2 feet and at the other end 25 feet, the prismoidal formula gives 1937 cubic yards as the volume, while the method of average end-areas gives 2447 cubic yards, or 25 p. c. too much. But the above values of the prismoidal correction only apply when the surfaces of the sections are horizontal. If, however, in dealing with 3-level sections we call Wand W the entire width between the slope-stakes at each end, then the prismoi- dal correction equals, in cubic yards, which is independent of the side-slopes and width of the road- bed. So that, having calculated the contents according to the formula for average end-areas, we have simply to find for each cross-section the value of (H ' H'} and (W - W'}, and take out from the following table, which gives the values of C, the amount in cubic yards which is to be added to the contents already obtained in order to obtain the result which would be given by the prismoidal formula. Should, however, the smaller centre-height be at the same end of the solid as the greater width between the slope slakes, then C must be sub- tracted. BAILROAD CONSTRUCTION. 145 TABLE OF THE VALUES OF G, WHEN L = 100 FEET, W-W in feet. H H' in feet. 1 2 3 4 5 6 7 8 9 10 1 .3 .6 .9 1.2 1.5 1.8 2.1 2.4 2.7 3.1 2 .6 1.2 1.8 2.4 3.0 3.6 4.3 4.9 5.5 6.2 3 .9 1.8 2 7 3.6 4.6 5.5 6.5 7.4 8.3 9.3 4 1.2 2.4 3^6 4.9 6.2 7.4 8.6 9.8 11.1 12.3 5 1.5 3.1 4.6 6.2 7.7 9.2 10.8 12.3 13.8 15.4 6 1.8 3.6 5.5 7.4 9.2 11.1 12.9 14.8 16.6 18.5 7 2.1 4.3 65 8.6 10.8 12.9 15.1 17.3 19.4 21.5 8 2.4 4.9 7.4 9.8 12.3 14.8 17.3 19.7 22.2 24.6 9 2.7 5.5 8.3 11.1 13.8 16.6 19.4 22.2 25.0 27.7 10 3.1 6.2 9.3 12.3 15 4 18.5 21.5 24.6 27.8 30.8 11 3.4 6.8 10.2 13.6 17.0 20.3 23.7 27.1 30.6 33.9 12 3.7 7.4 11.1 14.8 18.5 22.2 25 8 29.5 33.3 37.0 1| 4.0 8.0 12.0 16.0 20.0 24.0 28.0 32.0 36.0 40.1 14 4.3 8.6 12.9 17.3 21.5 25.8 30.1 34.5 38.8 43.2 15 4.6 9.2 13.8 18.5 23.1 27.7 32.3 37.0 41.6 46.3 16 4.9 9.8 14.8 19.7 24.6 29.5 34.5 39.4 44.3 49.3 17 5.2 10.4 15.7 20.9 26.2 31.4 36 6 41.9 47.1 52.4 18 5.5 11.1 16.7 22.2 27.8 33.3 38.8 44.4 49.9 55.5 ?9 5.8 11.7 17.6 23.4 ao.s 35.1 41.0 46.9 52.7 58.6 20 6.2 12.3 18.5 24.6 30.8 37.0 43.2 49 4 55.6 61.8 21 6.5 12.9 19.4 25.8 32.3 38.8 45.3 51.8 58.3 64.8 22 6.8 13.5 20.3 27.1 33.9 40.6 47.4 54.3 61.1 67.9 23 7.1 14.2 21.3 28.4 35.4 42.5 49 6 56.8 63.9 71.0 24 7.4 14.8 22.2 29.6 37.0 44.4 51.8 59.2 66.7 74.1 25 7.7 15.4 23.1 30.8 38.5 46.2 54.0 61.7 69.4 77.1 26 8.0 16.0 24.0 32.0 40 48.1 56.1 64.1 72.1 80.2 27 8.3 16.6 24.9 33.2 41.5 49.9 58.3 66.6 74.9 83.3 28 8.6 17.2 25.8 34.5 43.1 51.8 60.5 69.1 77.7 86.4 29 8.9 17.8 26.8 35.7 44.7 53.7 62.7 71.6 80.5 89.5 30 j 9.3 18.5 27.7 37.0 46.3 55.6 64.9 74.1 83.3 92.6 There is no need to apply these corrections at the time when the quantities are worked out by average end-areas, as generally the engineer is then too much occupied in obtaining rougJi es- timates of the work ; but they can subsequently be applied, with very little trouble, to such solids as in his opinion need correcting. The application of this method undoubtedly reduces the final estimate of the grading very considerably, rarely by less than 1 p. c., and in some cases, where the cross-sectioning has been carelessly done, by as much as 4 or 5 p. c. But it must be remembered that in this way the true volume is obtained more nearly than by any other of the approximate processes, and that the results are slightly higher than those obtained by the use of such tables as " Trautwine," " Rice," etc., founded on the principle of Equivalent Level Sections. Without the 146 RAILROAD CONSTRUCTION. application of the prismoidal correction the contractor is en- tirely at the mercy of the engineer who does the cross-section- ing (if the method of average end-areas is used), who has it, often unconsciously, in his power to make a difference in the final estimate of 3 or 4 per cent, by not paying attention to the differences in centre-heights and widths of the cross-sec- tions he is taking. And though the errors in any given piece of work are in favor of the contractor, still the uncertainty to which they give rise, in the long run do him considerably more harm than good. If a correction is not used, some limiting value for (HH) X (W - W) should be estab- lished. Some standard system of measuring grading is much wanted. As it is now, a contractor on one piece of work gets the benefit, possibly of 3 p. c. due to the use of average end -areas, uii- corrected; while on the next contract he takes very likely he has the quantities actually cut down, owing to the use of tables of equivalent level sections. It is true that if the work is properly cross-sectioned the excess as given by the method of average end-areas should not exceed 1 or 2 p. c., but in the ordinary way in which cross sectioning is done, a considerable amount of trouble is taken in order to correct for small sur- face irregularities, while the great errors which are involved by the difference in centre-heights are barely considered so long as the slopes between the sections are tolerably uniform. When the cross-sections are irregular, the prismoidal correc- tion can usually be applied with sufficient accuracy by treating them as 3-level sections, and thus applying the value of C as given above. 131. The Method of Equivalent Level Sections is an incorrect means of applying the prismoidal formula by reduc- ing the end-sections to sections equivalent in area but with their surfaces horizontal, and then taking as the area of the mid-section that which is given by the mean of the corrected centre-heights. But unfortunately the results so obtained are only correct 1. When the two end -areas are " similar" i.e., the corre- sponding surface-slopes from the centre to the slope-stakes are the same at both ends, provided the road-bed is not intersected between them; 2. When the surface is regularly warped from one end to RAILROAD CONSTRUCTION. 147 the other, provided that no two of the straight lines connecting corresponding points, such as A, a, etc., in Fig. 71 are inclined to grade in opposite direction (as they are in Fig. 71). In cases where these conditions do not hold, then, assuming that the true result is given by the prismoidal formula if merely the corresponding points A, a, etc., are joined by straight lines, the method of equivalent level sections gives results too small. But if the surface is intersected by undulations, running obliquely, necessitating the use of "slope-lines" as in Fig. 72, then the results may either be too small or too great, according to circumstances. But since this latter method of applying the prismoidal formula is the exception, and the results as obtained by applying it in the manner shown in Fig. 71 more generally correct, the general tendency of the method of equivalent level sections is to deficiency, but not by an amount usually sufficient to warrant the use of a correction. The real objection to this method is the labor involved in applying it when dealing with cross-sections in the slightest degree "irregular," and even in dealing with 3-level sections the work involved is greater than that by the method of average end-areas, corrected; while the result in the former case is an approximation, in the latter it is presumably correct. 132. The method of centre-heights, which is very useful in making preliminary estimates, simply assumes that the con- tents between any two cross-sections are given according to the method of average end-areas, the area at each end being taken as the area of a horizontal section with a height equal to the actual centre-height. The results so obtained naturally err, sometimes in excess and sometimes in deficiency the tendency in the former direction being, however, the more common. But since there is no decided tendency to cumula- tive error, the result obtained as a whole- for several stations where the direction of the surface slope is varied, agrees toler- ably well with the true volume, though for anyone station the error may be very considerable. In the long run more ac- curate results are usually given by this method than by that of average end-areas. (See Sees. 69 and 70.) 133. By the use of Table XIV the labor of applying the method of Centre-heights is greatly reduced. Table XV saves considerable labor in reducing areas to cubic yards, by avoiding the necessity of multiplying by 100 148 KAILROAD CONSTRUCTION. and dividing by 27. There is no need to take the quantities out closer than to the nearest yard. In using the table for lengths other than 100 feet a good deal of trouble may be saved in the way of multiplication and division by reducing each time the simpler of the two values with which the table is entered; thus if we have an average area of 634 square feet for 50 feet, the amount opposite 317 gives the quantity required, instead of dividing 2348.2 by 2. 134. Correction for Curvature. We have hitherto as- sumed that the cross-sections are parallel to each other i.e., that the track is straight. Suppose, however, that in Fig. 73, exaggerated for the sake of clearness, o represents the centre of a certain curve whose radius R, the cross-section ACaB representing any cross-section on the curve. Now it is clear that if we have two cross-sections whose centres are 100 feet apart (along the curve) and take in each a point b, situated outside the centre by a distance y, the distance between these two selected points, measured along a line parallel to the centre-line, is to 100 feet as R -j- y is to R, arcs FIG. 73. subtended by equal angles at the centre being proportional to their radii. But instead of calculating the contents for the varying distance, it is simpler to assume that the track is straight, and to correct the sections themselves so as to allow for it : so that, instead of using the above proportion, we may consider that the area of a section at any distance y from the centre must be increased or decreased in the proportion x' = x(R where x' represents the corrected area and x the original area; y being positive if falling, as in Fig. 73, on the outside of the curve, and negative if falling inside. So that if at any point as a we measure the ordinate x and its distance from the RAILROAD CONSTRUCTION 149 centre y, the above equation gives us x' , the corrected length of x, which, being measured upwards from the point b, gives us a', the new position of a. Similarly by finding other positions of a', the curved line ACa'B being drawn through them, gives the equivalent section on a straight track. In curves of 8 and upwards, where the slope is compara- tively steep in one direction, this correction should be applied. It is best to assume an average section for two or three stations together, and to divide the radius by 10, so as to make R a distance easily scaled, and then to divide the correction so ob- tained by 10. Thus, if the section is taken as an average one for 300 feet on a 10 curve, we plot R = 57 feet, and the cor- rection so obtained which is of course equal to the difference between the contents given by the actual section and the equivalent section must itself be divided by 10; or, what is the same thing, be considered to apply only to a length of 30 feet. Two or three ordiuates are usually sufficient to locate with sufficient accuracy the surface of the equivalent section. Where the surface is level there will of course be no correction necessary, for then the excess on one side of the centre-line balances the deficiency on the other. This method is equally easy to apply to any form of cross- section, however irregular it may be. 135. The contents of the toe of a dump are commonly calculated according to the formula given in Sec. 128 for a wedge, but the result so obtained is always considerably too small; neither can the prismoidal formula be directly applied. FIG. 74. First, let us assume the surf ace of the ground to be level; then the simplest way to obtain correctly the contents of the toe is to consider each corner as a quarter of a cone; then if H equals the height of the fill in feet, and s the slope ratio, the contents 150 RAILROAD CONSTRUCTION. of the two corners together equal .523/W; so that the entire contents of the toe are given by the formula 8 = . B being the width of the road-bed in feet. This formula is easily worked out by means of Table VIII. 8 must then be divided by 27 to reduce it to cubic yards. If = 1|, then the above equation becomes But when the ground slopes downward in the direction of the toe, as is the more common case, then we may consider the toe to be divided into two portions, as shown in Fig. 74; the upper one, which we have just dealt with, having a vertical height equal H, and the lower one with a vertical height = h. Then, omitting for a moment the consideration of the circular corners, the contents of the upper portion are to the contents of the lower portion as His to h. Now, though this does not quite hold good when taking the corners into account, the error involved by assuming it to do so is immaterial; so that we may say, that when the ground slopes forward as in Fig. 74, the total contents equal r h the value of S being obtained as above. The value of h may be obtained quite well enough by plot- ting //and the slopes of the ground and the dump. If the ground slopes transversely as well, the case becomes decidedly complicated, and the engineer must then assume such values, as will when inserted in the above formulae, give what he considers fair results. In dealing with the toe of a dump less than 10 feet in height the wedge formula is sufficiently accurate, but where the fill RAILROAD CONSTRUCTION". 151 t^c^ot-^oot-^oc^ as :o - 55 i -^ i ? - .-f i - o co - o co i~ o co o r- O CO O CO O to 10 O 00 CC i O CO 50 5 CO CO CO CO co*?555*-oQOcD2scS'"*s5 toccot>cooo-coot - -oco i^t-coOiooo^ac^p^coSoGop^coXopco THrHC^C7SO^J 1 '^ l AOO?Dt;'-OOOiO'r-t'??Tt*iCCDt--CO'*- e>} co **< in co oo o 00 i-c^ciWNcoco'^'Ttininot-oooJOO^-'TJcointo i-i T-I si M TO co eo' TI< TJ< in co co' t- oo oa os o -< o so o coocotoocooococoo^cdocBTOococoStooco i-< i-< i-i w ^ w co' co' co * -^ ic CD' co i- c oo oi o o o* eo' *" ^^^^^^^c.Vco^^ir'in'cot-^ OOO'OO^OOOOOOOOO': M ^5 cct-tccoi^-tc^^n'^>^--t ' - ' o o is ~> QC ^ a in i- r- co in o .- W CO 1C O C 05 TT GO CO co gi i O CO W S^CO^SrtCOl-^' o tt ^-r-n-iT-n-i | MO??>o S f- 55 e o f- i~o "^ o J: o co in i- os r- s>j ic to x S 01 to c co i- -i ic GO oj co o i- o *T-.'T-ii-<'^^i^iyi?eo'go';crii-^Tfirix;>mt^ i^ CC T? O? O eo c\i c 218 1.46 5 X 390 2.60 6 1 310 2.07 5 9 295 1 97 6 JL 262 1.75 5 1 257 1.71 6 B 196 1.30 The following table gives the angle-bars and bolts neces- sary lor 1 mile of track : Length of Rails in feet. No. of Angle- bars. No. of Bolts. Length of Rails in feet. No. of Angle- bars. No. of Bolts. 24 880 1760 27 782 1564 25 844 1688 28 754 1508 26 812 1624 30 704 1408 The following table gives the weight of Rails required for 1 mile of track : Weight of Rail per yard. Weight per mile. Weight of Rail per yard. Weight per mile. Weight of Rail per yard. Weight per mile. Ibs. tons. Ibs. Ibs. tons. Ibs. Ibs. tons. Ibs. 40 62 1920 56 88 65 102 320 . 45 70 1600 57 89 1280 68 106 1920 48 75 960 60 94 640 70 110 50 78 1280 62 97 960 72 113 820 52 81 1600 64 100 1280 76 119 960 The weight of iron required per mile is very nearly given by the rule: Multiply the weight in Ibs. per yard by If; the product is the weight required in tons of 2000 Ibs. (the tons in the table = 2240 Ibs.) The weight of iron in Ibs. per yard is given by multiplying its sectional area in inches by 10, assuming the iron to weigh 480 Ibs. per cubic foot. Steel rails usually weigh about 490 Ibs. per cubic foot. 156 RAILROAD CONSTRUCTION. 130. BALLAST AND TIES. The following table gives the amount of ballast required per mile of road: Depth Top Width, Single Track. Top Width, Double Track. inches. 10 Ft. 11 Ft. 12 Ft. 21 Ft. 22 Ft. 23 Ft. cu. yds. cu. yds. cu. yds. cu. yds. cu. yds. cu. yds. 12 2152 2347 2543 4303 4499 4695 18 3374 3667 3960 6600 6894 7188 24 4694 5085 5474 8996 9388 9780 30 6111 6600 7087 11490 11980 1:3470 This table assumes that the side-slopes of the ballast are at the rate of 1 to 1 , and that there is a space of 6 feet clear be- tween the tracks. The following table gives the number of Ties required per mile of track : Centre to Centre in inches. No. of Ties. Centre to Centre in inches. No. of Ties. 18 3520 27 2347 20 3168 30 2112 22 2889 33 19-20 24 2640 36 1760 For useful information in connection with Construction, see Part IV. PART III. EXPLORATORY SURVEYING. 140. IN Part I we have already considered the subject of " Preliminary Surveys," made principally with the object of obtaining topography by means of which the final location for a railroad may be selected. We will here deal with the sub- ject of rough Reconnoissance and Exploratory Surveys, in which accuracy such as it is generally understood is not essential, and in which the general bearings of rivers and streams, and the elevations of mountain passes, etc., plotted to a scale of a mile or so to an inch, are the main points to be established. But before dealing with the problems which arise in explora- tory surveying it will be well to consider the Instruments usually employed in this class of work. INSTEUMENTS. 141. The Instruments generally used in Reconnoissance and Exploratory Surveys are the following: The Sextant, Chro- nometer, Artificial Horizon, and the Cistern and Aneroid Ba- rometers. To these may be added with advantage, a light portable Transit. We will treat each separately in the order here given. The Sextant. There are in common use two forms of sextant the Nautical and the 'Box sextant; but since the latter is nothing more than the former reduced into a small portable shape, we can con- sider them both under one head. For astronomical work the 157 158 EXPLORATORY SURVEYING. box-sextant may be considered almost worthless, but for taking ordinary topography it is an extremely handy instrument, and in more extensive work it is a very useful support to a nautical sextant in many ways. The ADJUSTMENTS of the sextant are as follows: A. To place the index-glass perpendicular to the plane of the instrument. Set the index to about 60, and then, looking at the image of the limb of the instrument as reflected in the index-glass, the real limb and the image should. appear to form one continuous arc. If they do not do so, the index- glass must be moved by means of the screws at its back (see Fig, 75) until it does. B. To place the horizon-glass perpendicular to the plane of the instrument. Clamp the index near to zero, and then, looking at some well-defined object, turn the tangent- screw of the index until the object, as seen directly, and its re- flected image are brought, if possible, to coincide. If they cannot be made to coincide the horizon-glass is out of adjust- ment and must be corrected by means of the adjusting screws with which it is fitted. C. To obtain the index-error. For the purpose of meas- uring the index-error when it is negative, i.e., when the cor- rection for it is to be added, the graduations of the limb are carried a short distance back from zero through what is termed the ABC OP EXCESS. The index-error is obtained by noticing the reading when the coincidence mentioned in Adjust. B is obtained. But in this case the object must be a far distant one, so that the reading may not be affected by instrumental paral- lax. Had the index been set exactly at zero when the above- mentioned coincidence was made, there would of course be no index-error, but it is usually better to apply an index-error than to attempt to obtain an exact coincidence at zero. A very accurate method of obtaining the index-error is to measure the diameter of the sun several times " on and off the arc " i.e., on the positive and negative side of zero: the mean of the readings will then be the correction, positive if on the main arc, and negative if on the arc of excess. Thus, for ex- ample, if the diameter of the sun measured on the main arc = 32' 20", and on the arc of excess 30' 40", the mean being 0' 50" on the main arc, shows that 50" has to be subtracted from all angles as read from zero on the main arc, i.e., that the coinci- EXPLORATORY SURVEYING. 159 dence mentioned in Adjust. B occurs when the reading is 50" on the main arc. D. To correct for eccentricity. A common error to which all sextants are liable is eccentricity of the centre of mo- tion of the index-arm and the centre of the graduated arc. It unfortunately admits of no adjustment, but corrections for it maybe obtained as follows: " As it has no appreciable effect on small angles, it is advisable using the artificial horizon- to take a set of altitudes, say 10, which will form a mean of about 100 on the arc, noting the time of each accurately by a trustworthy chronometer; should the time so found coincide with the known rate of the chronometer there is no error. Should the results differ by several seconds of time, it may be assumed that the error of the instrument, combined with per- sonal error, has caused it. By the rate at which the sun was rising or going down during the observations, the amount of angle due to those seconds is easily found (see Sec. 195). Half that amount will be .the error of the sextant upon that angle. As an EXAMPLE, suppose by a morning observation the true reflected altitude = 100, while the instrument made it 100 01', the calculation would make it about 3 seconds later than the truth. In the afternoon a similar error would make it 3 seconds earlier. Thus a disagreement of about 6 seconds arises for about 1' of altitude. By 4 or 5 such sets of altitudes at different parts of the arc sufficient data will be procured from which to form a table of corrections for all altitudes." 142. The sextant, unlike the transit, has the apex of the angle which it measures not coincident with any particular part of the instrument, but varying its position according to the magnitude of the angle observed. This is due to what is usually called Instrumental Parallax, and arises from the fact that the index -glass is not situated .in the direct line of sight. This may be best shown by means of Fig. 75. Suppose 8 and R are two objects, the angle between which we wish to measure. When the index-arm has been so placed that the image of S -is reflected from the index-glass 7, so as to coincide with R as seen directly through the horizon-glass H, the angle which is given by the sextant is the angle 8AE, where A is a point in the line of sight, found by producing 81 to its intersection. But suppose 8' and E were the two objects between which the angle is to be observed, then a will be the 160 EXPLORATORY SURVEYING. apex of the angle measured. Finally, if 8 is situated at s, so that si is parallel to RA, then the angle given by the sextant between s and R=Q (i-e., if there were no index-error the reading should be zero), and if the reflection of H were brought to coincide with R as directly seen, then the angle observed would be negative, and would thus be read on the " arc of ex- cess/ and be equivalent to IRA. If R is at a distance from the instrument so great that HI and RA are sensibly parallel, as was assumed in Adjustment C, the question of instrumental FIG. 75. parallax may be ignored; but in measuring angles between two objects when the object directly looked at is near at/hand, the instrument must be either so placed that the apex will coincide with the position at which the angle is to be observed, or else a correction applied, the angle as given by the sextant taking, say, the index-glass as the constant apex of the angle being always too small. In using an artificial horizon there is another form of paral- lax which sometimes needs consideration due to the apex A of the angle observed not coinciding with the artificial horizon. Let R be the image of a star 8 reflected in the artificial horizon. Then if 8 A is parallel to SR t as is sensibly the case when deal- ing with objects at a considerable distance from the instrument, the angle 8AR may be considered equal to twice the angle 8RB; i.e., the altitude read on the sextant is the " double-altitude" of the star, which needs dividing by two in order to obtain the altitude; but where Sis comparatively close at hand, then we cannot consider 8AR = 2SRB, and consequently by dividing EXPLORATORY SURVEYING. 161 the reading on the arc by two, it is not the altitude as reflected from the horizon which is observed, but from a point r so situated tbat the angle ASr is equal to the angle RSr. Suppose we select this point r in the line of sight, as in Fig. 75, then it may be easily proved that if rb is parallel to RB (the surface of the artifiofltl horizon) Srb %SAR. And since the sines of small angles may be assumed to be proportional to the angles themselves, we may consider the point r to be situated half- way between A and R. Thus in observing an altitude with the artificial horizon, where the distance RA is appreciable compared with the distance 8 A, it becomes necessary either to apply a correction, or to arrange the positions of the horizon and the instrument so that the point r may coincide with the apex of the angle which it is wished to observe. 143. A sextant is usually only graduated up to about 140. For nautical work this is amply sufficient, but where an arti- ficial horizon is used since the angle read is double the real altitude the altitude will be limited to about 70. To obviate this difficulty, sextants are often supplied with a contrivance which consists of a small mirror below the index glass, fixed in such a position that when the index is at the mark numbered 180 upon what is called the SUPPLEMENTARY ABC, those two mirrors are at right angles to each other, and the objects whose images appear to coincide in direction really lie in diametri- cally opposite directions. 144. In observing angles with the sextant, when the two objects and the observer's eye are not in the same horizontal plane, in order that the angle measured may be a horizontal one, it becomes necessary either to arrange matters in such a way that the angle observed between the objects may be the horizontal angle, or to apply a correction to the angle ob- served. In the former case two vertical rods may be ranged in line with the objects and the observer's eye, and the angle between them then measured with the plane of the sextant horizontal. But the most accurate method is to observe the angle between the objects themselves, and then to observe the angle of altitude, or depression of each. Thus, in Fig. 76, let A and B be the two objects, Othe position of the observer. Then if Z be the zenith and a and b points where the vertical planes through A and B respectively inter- 162 EXPLORATORY SURVEYING. sect the horizontal plane abO, then Aa and Bb represent respectively the altitudes of A and B, and the complement of &''- the altitude of each equals its " zenith distance," AZor BZ. Then in the spherical triangle ABZ, since we know all three sides, therefore (since ab = Z) where 8 sinAZsinBZ 145. Every possible means should be taken in observing angles with a sextant to eliminate instrumental errors. In order to do this all careful observations should be in " doubles:" thus if the observation is for latitude, a star north and a star south should be observed; the errors of the instrument will then affect the result in opposite directions, and taking the mean of the results will eliminate the errors. So also an ob- servation for time should be taken in "doubles:" namely, a star east and a star west. Also in taking Lunar Distances the sets should be taken in "doubles," one set of distances to a star east of the moon and one to a star west. The Artificial Horizon. 146. The best substance to use for an artificial horizon is mercury, mainly on account of its bright reflecting surface. In a wind, however, syrup is better than mercury, being more EXPLOKATORY SURVEYING. 163 viscous and consequently less liable to be affected by currents of air, but its reflecting surface is decidedly inferior, Oil, too, is frequently made use of. A sheet of water on a still night makes a fairly good horizon. Black glass horizons, which can be levelled up by means of adjusting screws, are sometimes used, but though at times more convenient than a liquid surface they are considerably less reliable. The best way to carry mercury is in an iron bottle, which can be made by any blacksmith out of a piece of iron pipe, fitted with a screw stopper in the cap. Mercury must be kept carefully away from all greasy substances, and also from lead, gold, or silver, with which it amalgamates. A glass cover in the form of a triangular prism is often of use in shielding the horizon from the wind; but owing to the in- creased probability of error, due to refraction in the cover it- self, it is to be avoided when possible. The mercury can usually be protected from the wind by placing it in a hole slightly below the general surface of the ground, or by build- ing up a sort of protection around it. A wooden trough makes the best form of saucer to hold it in; copper also does well. It should have an outlet at one corner to facilitate the pouring back of the mercury into the bottle. About 5 inches by 3 inches is a good size for the trough. It should also be of about uniform depth, which need not exceed half an inch. To PREPARE THE HORIZON, pour the mercury into a small chamois-leather bag, leaving, however, a little behind in the bottle as "scum," and then squeeze it out gently into the trough. The surface so obtained is usually as clear as could be wished for, but if the trough or the leather happens to have been a little dirty, a film of dust will sometimes be found on the surface. This can easily be cleared away by sweeping it lightly with a feather. The horizon is then ready for use. If a class cover is used over it, the observation should be taken twice, the cover being turned around for the second ob- servation, and the mean of the results taken; in this way the error arising from the refraction of the glass is more or less eliminated. The mercury should always be carried as steadily as possible, the bottle being kept "end up." Altitudes less than about 6 cannot be read with the artificial horizon on account of the obliquity of the rays. 164 EXPLORATORY SURVEYING. An artificial horizon is almost always to be preferred to a natural horizon, such as is given at sea, on account of the refraction of the air, as regards the horizon itself, not entering appreciably into the question. The Chronometer. 147. Chronometers have been found by experience, when subjected to the shakings and joltings which necessarily more or less accompany their transportation on laud, to be very un- reliable instruments. A small pocket-chronometer is usually almost as reliable for land work as one of larger and finer make, being less liable to derangement. As regards the care of chronometers, they should always be kept as much as possible in the same position, and be always wound at the same time of day, and wound to the butt. Also, they must be kept away from all magnetic influence, such as is often caused by their proximity to iron. They should, of course, be rated before starting out, but if they are new chro- nometers they will probably gain on their " rate. " The " shop- rate" is almost always different from the field-rate, so that really very little dependence is to be placed on them compared with that placed on chronometers at sea. But though the rate when out on the work may be entirely different from what it was before starting, yet the rate in the field will be more or less constant ; and though no great dependence can be placed on the actual position as given by a chronometer after consid- erable jogging and jolting, yet it serves to connect the various stations observed, relatively to each other, with a fair amount of accuracy when the intervals of time between the observa- tions are not great. These positions can then be finally cor- rected after the general field-rate of the chronometer has been ascertained. As regards allowing for temperature, that can only be done by an actual testing at different temperatures. Every chro- nometer goes fastest in some certain temperature which has to be calculated from the rates that it makes at three fixed tem- peratures; then as the temperature varies from that at which the chronometer goes fastest, so its rates vary in the ratio of the square of the distance in degrees of temperature from its maximum gaining temperature. A fair test for a pocket- EXPLORATORY SURVEYING. 165 chronometer is to place it in four extreme positions and let it stay in each for 24 hours ; if the rate for any position does not vary by more than rive seconds from the rate in any other position, the watch is as good as can generally be found. BAROMETERS. 148. There are two kinds of barometers used in exploratory surveying the "CISTERN" form of the mercurial barometer, and the "ANEROID." The Cistern barometer, owing to its size, is mainly suitable for use in camp as a standard with which the Aneroids may be compared. The nature of the difficulties involved in observing the dif- ference of elevation between any two points may be best shown as follows : FIG. 77. In Fig. 77, suppose we have two stations, A and B, whose difference in elevation we wish to determine. If the atmos- phere were in a state of rest there would be no difficulty in devising formulas which should give correct results, supposing the instruments themselves recorded correctly, for then the barometric reading along the horizontal line CB would at all points be the same, and we should simply have to obtain a formula founded on Boyle and Mariotte'sjaw for the pressure of gases, to obtain the difference in the heights of A and which should correspond with the observed difference in pressure. But since the atmosphere is always more or less subject to disturbing influences, such as temperature, humid- ity, etc., which cause the barometric gradient at B to assume such forms as BD or BE, no formula founded on statical principles can possibly be expected to give correct results ; yet any formula which attempts to take account of the fluctuations in gradient necessitates a knowledge of the temperature, 166 EXPLORATOEY SURVEYING. humidity, and general state of the atmosphere between A and B, which it is impossible to obtain. By taking observations at points immediately between A and B some allowance may be made for these various disturbances, but as a rule very little is gained by so doing compared with the time and labor which it involves. Since the variations in gradient are generally too rapid to allow of the state of the atmosphere at one hour being of much service in indicating its probable condition a few hours or even minutes later, it follows that labor spent in reducing barometric readings between two such stations as A and B, by applying corrections for latitude and various other require- ments which are often employed, simply results in a mathe- matical illusion which is possibly erroneous to the extent of 50 or 100 p. c. The best way to proceed in ordinary practice is to make use of formulae which assume the air to be in a state of equi- librium applying corrections for temperature which expe- rience has shown to be necessary and then to eliminate the errors due to variations in gradient as much as possible by taking the mean result of the readings on several occasions, or by observing simultaneously at the two stations, as described in Sec. 150. 149. The first information necessary in devising a formula for the reduction of barometric readings is the relative weight of mercury and air. This ratio amounts to about 1050, de- pending upon which values of the densities are employed. The barometer at the time is supposed to be at sea-level in latitude 45 at a temperature of 32 F. This ratio, if multiplied by 5.74 which is a factor obtained from Boyle and Mariotte's law that the density of a gas varies directly as the pressure to which it is subjected gives a product known &$i\ie barometric coefficient. Various values are given for this coefficient, but probably that given by Reguault is the most accurate, namely, 60,384 ; from this, taking no account of the effects of tempera- ture or latitude, we find that the difference in elevation in feet equals X= 60384 log ^, where H is the barometric reading at the lower station and 7* EXPLORATORY SURVEYING. 167 is the barometric reading at the upper station. The correction for temperature, as usually applied, assumes that the mean temperature of the air between A and B is the mean tempera- ture of the air at the two stations. If we then take .004 as the coefficient of expansion of air for 1 Centigrade, the above formula needs multiplying by 1 -\- .002(7 7 -)- t), where T and t are the temperatures on the Centigrade scale at the lower and the upper station, respectively ; and if we take Tand t as the temperatures on the Fahrenheit scale, then this factor becomes T+t-M 900 and this is usually called the " temperature term." Another factor is often employed to correct for the different effects of gravity, due to difference of latitude. According to Laplace, this " latitude term" equals 1 + . 0026 cos 2L, where L = the latitude. He also applied a correction for the effect of altitude above sea-level on the force of gravity ; but this may be altogether neglected. A correction is also some- times applied to allow for the effect of temperature on the barometers themselves which is ascertained by having ther- mometers attached to them. And since changes of tempera- ture affect both the mercury and the scales in opposite direc- tions, if we take .0001 as the relative expansion of mercury for 1 F. to the expansion of the scales, in order to correct the barometers themselves for temperature, the above value of X should be multiplied by where T' and t' are the temperatures as recorded by the " at- tached " thermometer at the lower and the upper station, respectively. Thus the complete formula becomes . 0020 cos 168 EXPLORATORY SURVEYING. A correction for humidity is sometimes applied, but it necessitates observations of the state of the air being taken with a hygrometer ; and since it is doubtful, even then, whether any material advantage is derived by so doing, we may ignore this correction entirely. We may simplify the above equa- tion considerably by dispensing with the latitude term, which in ordinary practice is never required. In aneroid barome- ters the last term of course does not enter into the question at all ; so that the formula generally applicable to aneroid barome- ters is X= 60384 log 900 If ff&nd Ti do not differ by more than about 3000 feet we may do away with the logarithms in the above equation, which thus becomes, approximately, H+ h \ 900 / The error involved by this formula is inappreciable within the limits stated. By assuming (T -}- 1) to equal 108 this formula becomes X= 55000 . which is generally known as Belville's Formula and is con- venient for rough work. ( 900 ' 150 The results which are obtained by using only one barometer, carrying it from station to station, are of course subject to all the errors of gradient ; and these errors usually increase with the distance between the two stations ; but by taking the mean of several results, the probable error becomes greatly reduced. (See Sec. 204.) Errors of gradient may be more or less eliminated by using TWO BAROMETERS, and observing simultaneously at each station, the barometers being EXPLORATORY SURVEYING. 169 T+t T+i-64 r-f* T+f-64 T+t T+J-64 T+t T-\- 1 - 64 900 900 900 900 20= -.0489 66 + .0022 112 + .0533 158 +.1044 2-2 .0-167 68 .0044 114 .0556 ' 160 .1067 24 .0444 70 .0067 116 .0578 , 162 .1089 26 .0422 72 .0089 118 .0600 164 .1111 28 .0400 i 74 .0111 120 .0622 166 .1133 30 -.0378 ! 76 +.0133 122 + .0644 168 +.1156 & .0356 78 .0156 124 .0667 170 .1178 34 .0533 80 .0178 126 .0689 172 .1200 36 .0311 82 .0200 128 .0711 174 .1222 38 .0289 84 .0222 130 .0733 176 .1244 40 .0267 86 + .0244 132 +.0756 178 +.1267 42 .0244 88 .0267 134 .0778 180 .1289 44 .0222 90 .0289 136 .0800 182 .1311 46 .0200 92 .U311 138 .0822 184 .1333 48 .0178 94 .0333 140 .0844 186 .1356 50 -.0156 96 + .0356 142 +.0867 188 +.1378 52 .0133 98 .0378 144 .0878 190 .MOO 54 .0111 100 .0400 146 .0911 192 .1422 56 .0089 102 .0422 148 .0933 194 .1444 58 .0067 104 .0444 150 .0956 196 .1467 60 -.0044 106 + .0467 152 +.0978 198 + .1489 62 .0022 108 .0489 154 .1000 200 .1511 64 .0000 110 .0511 156 .1022 202 .1533 compared before and after the observations : and these errors may of course be still further reduced by taking the mean of several simultaneous observations ; and in this way the best results can probably be obtained. But between two stations there is usually a permanent gradient dependent on local causes, such as the topography and nature of the ground, which no number of observations would -tend to eliminate, and for which allowance can rarely be made. It is largely due to this cause that the heights of mountains, calculated from the mean of a large number of observations which differ but little from each other, are often found, when obtained by more accurate means, to be very largely in error. 151. There are two or three points in connection with the READING OF BAROMETERS that are worth remembering. For instance, readings should never be taken in the im- mediate vicinity of any body which obstructs the wind. " If the barometer is observed on the windward side of a moun- tain the reading will be too higti ; if on the leeward side, too low." Neither should readings ever be taken directly before or after a storm of wind or shower of rain, as the atmosphere is then usually in an unsettled state. 152. "The pressure of the air everywhere undergoes a 170 EXPLORATORY SURVEYING. daily oscillation. The gradient introduced by this daily change is called the DIURNAL GRADIENT. The pressure has two maxima and two minima which are easily distin- guishable. Near the sea-level the barometer attains its maxi- mum about 9 or 10 A.M. In the afternoon there is a minimum about 3 to 5 P.M. ; it then rises until 10 to midnight, when it falls again until about 4 A.M., and again rises to attain its forenoon maximum. The day fluctuations are the larger." " The annual progress of the sun from tropic to tropic throws a preponderance of heat first on one side of the equator and then on the other, which produces an annual cycle of changes in the pressure, and gives rise to what has been called the ANNUAL GRADIENT. The amount of this variation is quite small, but increases rapidly toward the poles ; at the equator it rarely exceeds one quarter of an inch per year, while hi the polar regions it is often as much as two or three inches in a few days." We will now consider the barometers themselves. A, The Cistern Barometer. 153. This is an awkward instrument to carry about, but its usefulness on exploratory work usually fully makes up for the inconvenience which it causes. It is found by experience to be absolutely necessary in carrying forward an extended system of barometric observations to have at hand a standard barometer with which the aneroids may be from time to time compared. A supply of tubes and mercury should accompany the barometer in case of accident, and it should be provided with a wooden and leather case. When moved from one place to another, even across the room, it should be screwed up so that the tube and cistern will be perfectly full, and gently turned over, end for end, so that the cistern will be upper- most. In wheeled vehicles it should be carried by hand, and on horseback strapped across the rider's shoulder. By car- rying it with the cistern uppermost any particles of air which may be contained in the mercury become disengaged by the jolting, and escape at the end where they do no harm. 154. TO FILL A BAROMETER, should it become neces- sary to do so in the field, proceed as follows : Warm both the EXPLORATORY SURVEYING. 171 mercury and tube aiid filter in through a paper funnel the hole of which does not exceed ^ of an inch to about i of an inch from the top. Close the end and turn the tube on its side ; the mercury will then form a bubble which can be made to travel from end to end and gather all the small air-bubbles visible that adhered to the inside of the tube while filling. Let the bubble pass to the open end, fill up with mercury and close the tube. Reverse the tube over a basin, when, by slightly relieving the pressure against the end, some of the mercury will be forced out, forming a vacuum above, which ought not to exceed half an inch. Close up again tightly and let this vacuum-bubble traverse the length of the tube as be- fore, on the several sides, absorbing the minute portions of air still left, now greatly expanded by the reduction in pressure. Perfect freedom from air can be detected by the sharp con- cussion with which the mercury beats against the sealed end, when, with a large vacuum-bubble, the horizontally held tube is slightly moved. Any air which may still be left which will probably not affect the reading by more than a few thousandths of an inch will soon escape if the barometer is carried about cistern uppermost. Filling by boiling is a slightly more efficient method, but it is a much more difficult proceeding. 155. IN READING THE BAROMETER, first of all note the temperature on Jhe attached thermometer, then screw up the mercury in the cistern so that its surface just touches the ivory point, being careful that the barometer hangs vertically. Give a gentle tap near the top of the mercurial column to destroy the adhesion of the mercury. Set the vernier by bringing its front and back edges into the same horizontal plane with the top of the mercury ; then read. 156. Should the mercury in the cistern become so dirty that neither the ivory point nor its reflection in the mercury can be seen, the instrument must be taken apart and cleaned. To do this " screw up the adjusting screw at the bottom until the mercury entirely fills the tube, carefully invert, place the instrument firmly in an upright position, unscrew and take off the brass casing which encloses the wooden and leather parts of the cistern. Remove the screws and lift off the upper wooden piece to which the bag is attached ; the mercury will then be exposed. By then inclining the instrument a little, a 172 EXPLORATORY SURVEYING. portion of the mercury in the cistern may be poured out into a clean vessel at hand to receive it, when the end of the tube will be exposed. This is to be closed by the gloved hand, when the instrument can be inverted, the cistern emptied, and the tube brought again to the upright position. Great care must be taken not to permit any mercury to pass out of the tube. The long screws which fasten the glass portion of the cistern to the other parts can then be taken off, the various parts wiped with a clean cloth and restored to their former position." Everything used in the operation must be clean and dry, and all breathing on the parts avoided as much as possible. If the mercury is dusty or dimmed by oxide it may be cleaned by filtering through chamois leather, but if chemically impure it must be rejected and fresh mercury substituted. The cistern should then be filled as nearly as possible and the wooden portion put together and fastened. The screw at the bottom of the instrument should then be screwed up. " The instrument can then be inverted, hung up and readjusted. The tube and its contents having been undisturbed, the instrument should read the same as before." B. The Aneroid Barometer, 157. The "Aneroid" is a valuable instrument for engin- eering and exploratory purposes on account of its portability, and though not to be compared in accuracy with the mercu- rial barometer, the results given by it will often not differ from those given by the latter sufficiently to be of importance. It is in such cases as these that the aneroid is eminently useful. But it is too liable to derangement, and subject to too many defects, to warrant its being used in any other way than to supplement some more accurate form of obtaining elevations. In dealing with the mercurial barometer, after the correction for temperature has been applied, the instrumental errors which need correcting are very small ; but with an aneroid the same cannot be said. Most of the better class of aneroids are supposed to compensate automatically for changes in temperature. This compensation should be tested by com- parison at different temperatures with a standard barometer, and the errors tabulated and kept for future reference. EXPLORATORY SURVEYING. 173 While reading, Hie aneroid should always be held horizon- tally, for the weight of the parts themselves has a very considerable influence on the readings: a difference correspoud- ing'to fifty feet being not uncommon when held in different positions. The aneroid may be adjusted by means of the small screw at its back, so as to agree with the reading of a standard barometer, but when the difference is only slight it is better to regard it as an ''index error," and correct in that way, than to alter the reading. 158. Cheap aneroids commonly have the SCALE of inches subdivided so as to read the elevations above sea-level. This would be very convenient if only the corresponding pressure at the sea-level were always the same as given on the index and the atmosphere always in a state of equilibrium. The pressure at the sea-level is generally assumed as being equiv- alent to 30 inches. Another method which is convenient, though " unscientific and inaccurate," is that of having a movable scale of elevations which can be set to agree with the barometer reading at any known elevation. But the best way to obtain a reading is to observe the reading in inches, and then to reduce it by one of the formulae already given. BAROMETRIC AND ATMOSPHERIC HEIGHTS. Bar. in. Alt'de feet. Bar. in. Alt'de. feet. Bar. in. Alt'de. feet. Bar. in. Alt'de. feet. Bar. in. AltMe. feet. 21. 9900.1 23. 7375 1 25. 5060.6 27. 2924.4 29. 940.9 .1 9768.3 .1 7254.7 .1 4949.8 .1 2821.8 .1 845.4 .2 9637.1 .2 7134.7 .2 4839.5 .2 2719.6 .2 750.2 .3 9506.5 .3! 7015.3 .3 4729.6 .3 2617.8 .3 655.3 .4' 9376.4 .4' 6896.5 .4 4620.1 .4 2516.3 .4 56071 .5 9-247.0 .5 6778.1 .5 4511.0 .5 2415.2 .5 466.5 .6 9118.3 .6 6660.2 .6 44023 .6 2314.4 .6 372.6 .7 8990.0 .7 6542.8 .7 4294.0 .7 2214.0 | 19 279.0 .8; 8862.4 .8 6426.0 .8 4186.3 .8 2114.0 ; '.8 185.7 .9! 8735.3 .9 6309.6 Cj 40789 g 2014.3 .9 92.7 22. 8608.9 24. 6193.8 26. 3971.9 2S. 1915.0 3O. 0.0000 .1 8483.0 .1| 6078.3 .1 3865.4 .1 1816.0 i .1 92.5 .-> 8357.7 .2 5963.4 .2 3759.3 a 1717.4 I .2 - 184 7 .3 8233.0 .3 5848.9 .3 3653.6 * 1619.2 i .?. - 276.6 .4 8108.7 .4 5734.9 ^ &54S.3 ^ 1521.3 'A - 368.2 .5j 7985.1 .5 5621.4 '.5 3443.4 ft 1423.7 - 459.5 .6 7862.0 .6 5508.3 .6 33388 '.6 1326.5 '.'c - 550.6 .7 7739.4 .7 5395.7 t "j 3234.6 f t 1229.6 i "7 - 6414 .8 7617.5 .8 52S3.6 '.8 3130.8 '.8 1133.0 '.8 - 731.9 .9 7495.9 .9 5171.9 ( 3027.4 .9 1036.8 .9 - 822.2 174 EXPLOKATOHY SURVEYING. No advantage seems to be gained by the use of large ane- roids; in fact experience shows that when the barometer is subjected to much shaking, the best work is usually done by instruments not exceeding 3 inches in diameter. The eleva- tions according to which the elevation-scales on aneroids are usually divided are as given on the preceding page, and are ob- tained by a formula similar to those already given, assuming the temperature to be 60 Fahr. Many scales, however, adopt a temperature of 32 F., in which case the corresponding elevations will be reduced in the proportion of 1.058 to 1. The uncertainty which is connected with barometric obser- vations is greatly dependent on the latitude ; the barometric pressure being very much more regular in the tropics than in the polar regions. EXPLORATORY SURVEYS. 159. There are three distinct ways in which exploratory surveys may be carried on : A. By a series of triangulations. B. By direct measurement and compass courses. C. By astronomical observations. And though usually an explorer makes use more or less of all three methods, it will be better for the sake of clearness to consider each separately. A. By a Series of Triangulations. The method of triangulating is mainly suitable to moun- tainous country, or at any rate to country where a view of distant mountain-peaks is to be had. Before, however, considering the practical working of this system, it will be well to deal with a few of the principal trigonometrical problems which arise in work of this sort. In Sec. 59 we have already dealt with some of the simpler forms of triangulation, suitable in cases where a straight line has to be continued over an inaccessible surface ; but we will here consider the cases of obtaining distances and directions of points relatively to each other. EXPLORATORY SURVEYING. 175 160. Given two inaccessible points A and B, to find their distance apart and bearing: relatively to each other. In Fig. 78 let CD be a line the length and bearing of which are known. Observe the angles ACD, BCD, ADC, and BDC. Then in the triangle CDA we have the angles at C and D and the length CD, and can thus find CA. Similarly in the triangle CBD we can find CB. Then in the triangle CAB we have the side CA and CB and the angle at C, from which we can obtain the distance AB and its bearing relatively to CD. The following equations, however, reduce the work which the direct solution given above involves. Find an angle K such that sin ADC sin CBD t*in 7t^ -- * ~ sin CAD sin BDC ' then f CAB - ABCf\ ACB tan f 3 J = tan (45 - K) cot ; CAB + ABC . CAB -ABC 1 . sin BDC sin ACB - h 2 \ 3600 ) where T the hour required; t = the hour for which R.A. is given in the Almanac, previous to T; A = R.A. corresponding with T; a = R.A. corresponding with tf; D Increase in R.A. in 1 mean minute at time t\ d Increase in D in 1 mean hour at time t. If D is decreasing, d is of course negative. In the term in- volving the unknown value (T t), the probable value must be used, which is correct enough. We thus have the value of the Greenwich time corresponding with the observed local time of the transit of the moon's centre, the difference of which, divided by 15, gives the difference of longitude. 199. TO OBTAIN THE LONGITUDE BY LUNAR DIS- TANCES. This method is similar in principle to the preced- EXPLORATORY SURVEYING. 211 ing one, the difference being that here it is the distance from the moon to some star which is observed instead of its R. A. The present case, since it does not involve the use of a transit and admits of several observations being taken on one night, is more suitable for exploratory work, and is the method alto- gether used for checking the chronometers at sea. The dis- tances between the moon's centre and certain stars of the first and second magnitude are given in the Nautical Almanac for every three hours at Greenwich, so that it is simply a case of measuring the distance from the moon's limb to a star, and correcting for refraction, semi-diameter, etc., noting the local time of the observation, and then finding from the Almanac what hour at Greenwich corresponds with the corrected dis- tance. In Fig. 92 let M' and 8' be the positions of the moon and star at the moment of observa- tion, and Z the zenith; then M'S' , corrected for semi-diameter, equals the apparent Lunar distance, and M 'Z and 8'Z the co-altitudes. The true positions will differ from these by the differences in altitude MM' and 88': the moon, on account of the correction for parallax exceeding that for refraction, will be elevated above its apparent position ; whilst the star, on account- of re- fraction only, will be depressed below its observed position. Now, if the apparent altitudes are observed at the time of observing the lunar distance 8'M', we have the three sides of the triangle 8'ZM' , so that the angle at Zm&y be found trigo- nometrically. Then the two sides 8'Z and M' Z, being cor- rected for refraction and parallax, give the sides of the cor- rected triangle 8ZM; and since we thus have two sides and the included angle Z, we can calculate the true lunar distance SM. This operation is termed " Clearing: the lunar dis- tance." The following formula, by Borda, is probably the most con- venient to use for effecting this: D H+H' sin = cos cos 0. A % EXPLORATORY SURVEYING. where 2 ri _ cos s cos ( 8 ~ ^ cos ^7cos H' cos A cos h' cos 5 ^t - where s = and A = app. alt. of moon's centre, h' = app. alt. of star; H= true alt. of moon's centre, H' = true alt, of star; d = app. distance S'M', D true distance 8M. An error of a minute or two in the altitude makes no appreci- able difference in the distance. The vernier should be set to a division easily read off, and at the moment when the distance agrees with this reading the ob- server should call " stop," at which signal the assistant should note the time by the watch, and at the same instant, if possible, the altitudes may be observed by two assistants. But usually one observer has to do the whole work with the sextant, in which case he will have to observe the altitudes of the moon and star, both before and after the observation, and note the times, and then deduce the altitudes at the time of measuring the distance, by proportion. But a better way is to spend the time otherwise occupied in observing altitudes, in obtaining a large number of lunar dis- tances and then to compute th altitudes as follows: Since we know the time of each observation, we can obtain the hour-angle at that moment, which, in either the case of the moon or a star, is merely the difference in R.A. of the body and the sidereal time at the moment -1- 24 hours if necessary, the R.A. in the case of the moon being corrected for the time of observation by assuming a probable value for the longitude. Then if L latitude and d = co-declination, sin L sin (E+d) sin (alt.) = -^ ^ , where cot E = cot L cos h, and h = the hour-angle. If h exceeds 90 cos h is negative, which will make cot 2also negative; so that to avoid the use EXPLORATORY SURVEYING. 213 of supplements, it is simpler to say sin L sin (E - d) sin (alt.) = sin E These are of course the true altitudes. In selecting stars from which to measure the distance, it should be remembered that the mean of two distances, one measured to a star on the right and the other on the left, will be practically free from instrumental errors; so that this plan of observing should always be adopted when possible. It is well, too, to select stars the distances between which and the moon are varying most rapidly, for there is a considerable difference sometimes between the rates, and yet at the same time the altitudes should not be less than, say, 10. A complete lunar observation should consist of 6 " sets," each set including 3 simple distances; 3 of these sets should be taken to the left of the moon and 3 to the right; also two ob- servations for latitude, one in the north and one in the south, to eliminate instrumental errors; and two sets of observations for time, one to a star in the east and another in the west, one before and the other after the measuring of the distances. Having thus obtained the mean lunar distance for the mean local time, the corresponding Greenwich time may best be deduced according to the instructions and data given in the Nautical Almanac with sufficient clearness to render any further explanation superfluous, as that work must of necessity be an accompaniment to the observations. Since, however, the Nautical Almanac assumes that the computer has at hand a table of Ternary Proportional Logarithms, such as is given in Chambers' Mathematical Tables or Bowditch's Navigator, it will be well to see how these may be calculated, in the event of such not being the case. A Proportional Logarithm for any portion of a certain period is merely the difference of the logarithms of the period and of the portion. Thus, taking the period as 3 hours, since lunar distances are given in the Almanac at intervals of every 3 hours, or 10,800 seconds, the logarithm for it = 4.0334; then since the logarithm for 1 hour (= 3600 seconds) = 3.5563, the proportional logarithm for 1 hour = 0.4771. The explorer, however, should provide himself with some portable form of 214 EXPLORATORY SURVEYING. logarithmic tables if likely to have much of this sort of work to do. 200. Another method of obtaining Greenwich time is by observing with a powerful telescope the local time of the Eclipses of Jupiter's Satellites. But this method, for a variety of reasons, is considerably less reliable than those given above. The Nautical Almanac gives instructions and data as to the manner of obtaining Greenwich time by this method. TO TEST THE CHRONOMETER RATE. 201. Whenever a halt is made for over 24 hours, it is a very simple matter to check the rate of the chronometer. With a transit this can best be done by setting it in a vertical plane lying fairly north and south, and noting the moments of the passages of 3 or 4 stars. The interval of time before the respective passage of each on the following evening = 23 h 56 m 04 s . 9. With a sextant this may best be done by observing the altitudes of 3 or 4 stars lying fairly east or west their motion being greater in altitude when near the prime vertical and noting the chronometer times; after the lapse of the above in- terval, each will again be at the same altitude on the following night. TO SET THE TRANSIT IN THE MERIDIAN. 202. Three methods of obtaining a north and south line have already been given in Sec. 57; the method by Maximum Elongations of Polaris is the best, for it admits of plenty of time to reverse the instrument and establish a true north and south line. When Polaris is not convenient for this purpose, any other star (which has an elongation) may be used as shown in Note D, Appendix. In the same way, if neither Alioth nor y Cassiopeia is convenient for observation, other stars may be used as shown in Note E, Appendix. When, however, neither of these methods is exactly suitable, the azimuth of Polaris out of the meridian may be found at any moment by solving the astronomical triangle PZ8 in Fig. 87, and thus obtaining the angle at Z, which is the azimuth. To do this we have given the declination, and we must also have two of the following three : latitude, altitude, and hour- angle. Since the latitude is most easily obtained, and the EXPLORATORY SURVEYING. 215 altitude gives the best result if near the elongations, these two should then be used. If, however, the star is near the meridian, the latitude and the hour-angle should be employed. In the former case we have Z / sin s sin (s d) DS 2 ~ V ~ sin a~smT~ a, d, and I being the complement of the altitude, declination and latitude respectively, and s the half sum of a, d, and I. In the latter case we have cos a = cos d cos I -f- sin d sin I cos #, from which we obtain sin Z = sin 7i sin d cosec a. h = hour-angle. (See Sec. 182.) When the latitude and declination are of opposite signs, d = dec. -f 90. 203. In observing the altitude of the moon for time or latitude, as is often practicable in thick weather when the stars are invisible, and more accurate interpolation of its declination is necessary than is obtained by simple proportion, the method usually adopted for this purpose is that known as INTER- POLATION BY SUCCESSIVE DIFFERENCES. The interpolation formula is nd l n(n - 1) n(n - 1) (n - 2) = F+ -T + 1x2 <** + 1x2x^3 d3 +' etc " For example, suppose we wish to find the moon's declina- tion at Greenwich at 2 h 15 on Nov. 15, 1889. From the Nautical Almanac we find Uie declination given for every hour. We select the declination at the hour before the one for which we wish to interpolate (= F), and put it in the first column as below ; beneath it we put in order the decli- nations for, say, 3 or 4 following hours, as given in the Almanac. In the second column we put down the first differences of these (di) obtained by subtracting downwards and prefixing the proper algebraic sign. In the third column we place the second difference (d^) (i.e., the differences of the first differ- ences), and so on. 216 EXPLORATORY SURVEYING. Now n is the ratio of the fractional period for which we wish to interpolate, to the interval between which the values are given ; in this case 15 minutes to 1 hour, therefore n = i : so that now we have merely to insert the upper values in the columns for di , di , etc., and the above value of n, in order to find the declination at 2 h 15 m . Dec. at 2 h " 4" = 18 17' 4" - 7' 59" '-8' 05"" - 8' 10" $2 = 18 09' 5" - 18 or o" = 17 52' 50" - 6" -5" 1 Thus, F - 18 17' 4" - 1' 59".8 -j- .56" - .07" ; therefore, Dec. at 2 h 15'" = 18 15' 04".75. In such a case as the above, as it happens, the simple method of interpolation would have given F a = 18 15' 04". 2, which of course would have been amply near enough for any- thing in the way of ordinary work. But where the explorer is desirous of obtaining a really accurate observation this method is often of high value. 204. Adjustment of Observations. It is a well-recog- nized fact in practice, when making a series of measurements of any quantity, that after every possible means of eliminating and correcting for instrumental errors have been employed, there still remain certain accidental errors which no experience or skill on the part of the observer can rectify, since the causes to which they are due are themselves unknown. Thus it hap- pens that each measurement in the set may be different, al- though, judging from the care taken in observing each and the apparent similarity of the conditions under which they were taken, no such differences should exist. The question then arises as to what is to be taken as the most probable result. Now according to the Theory of Least Squares, the method usually adopted for the solution of these problems, the most probable value of any number of measurements of the same quantity, each measurement being considered to be equally reliable, is that which makes the sum of the squares of the EXPLORATORY SURVEYING. 217 " errors " a minimum ; and the value which does so is the arithmetical mean of all the measurements. The " error " in the case of each measurement being its difference from the mean. But it often happens that the circumstances under which the several measurements are made are such as to warrant greater " weight " being given to some of them than to others. These weights are often deduced from the observations themselves, or from them in connection with a special series of observa- tions ; but in ordinary field practice, weights assigned arbitrari- ly after a thoughtful perusal of all the attendant circumstan- ces are more likely to be of value than those found by a strict application of the formulas of Least Squares. Weights being thus assigned, the most probable value of the results will be found by multiplying each observed value by its weight, and dividing the sum of the products by the sum of the weights, the result being that value which renders the sum of the prod- ucts of the squares of the errors and the respective weights a minimum. And this value is termed the Weighted Mean. This may be best illustrated by an example. Suppose that we have, as several corrected measurements of a base, the following numerators, and that, .considering all the attendant circumstances, we have assigned to each the weight shown as its denominator, assuming, for the sake of simplicity, that the weight of the least reliable is expressed by unity: 2056. 32 feet , 2056.20 feet 2056 16 feet -4- -3- Then the most probable value of the result is given by 2056. 32 -+- (2056. 20 X 4) -f (2056. 16x3) 1 + 4 + 3 A fair test of precision in dealing with a set of measure- ments is afforded by means of the " probable error " of a sin- gle determination, which is found by taking the difference between each individual result and the mean, squaring these quantities, and dividing their sum by (n 1) where n repre- sents the number of individual results ; then, on extracting the square root of this quotient and multiplying by 0.674, we 218 EXPLORATORY SURVEYING. obtain the so-called Probable Error. But this terra does not mean that that error is more probable than any other, but merely that in a future observation the probability of com- mitting an error greater than the probable error is equal to the probability of committing an error less than the probable error. The probable error of the arithmetical mean may be simi- larly found, the value n(n 1) being substituted for (n 1) in the rule given above for a single determination. Errors in excess are considered positive ; those in defect, negative. 205. Having now examined the various methods of obtain- ing positions on exploratory surveys, we next come to the sub- ject of ascertaining the bearings and distances of these posi- tions relatively to each other or to other points, when taking into consideration the curvature of the earth's surface. From what has already been said in Sec. 58 on the subject of the Convergence of the Meridians, we can see what form the corrections will have to take in order to allow for the spherical or more correctly spheroidal form of the earth ; and now, by means of 3 or 4 simple problems, we can obtain all the formulae necessary for the construction of the groundwork of a map, or the calculation of courses, which are ever likely to be needed in connection with exploratory surveys. In Engineering Geodesy it is usually sufficiently accurate to assume the earth to be a sphere, the radius of which equals the mean radius of curvature of the spheroid ; but it may be as well here to examine the subject roughly, in order that the engineer may have an idea of the extent of the errors which this assumption involves. 206. THE FIGURE OF THE EARTH. According to Col. Clarke, the mean Equatorial semi-axis = 20926202 feet, and the Polar Semi-axis = 20854895 feet. Also the radius of curvature in the direction of the meridian in any latitude L equals in feet R = 20890564 - 106960 cos 2Z+ 228 cos 4; EXPLORATORY SURVEYING. and the radius of curvature in a direction perpendicular to the meridian equals in feet r = 20961932 - 35775 cos 2L + 46 cos 4. Thus at the Equator R = 20783832 feet, r = 20926203 feet ; and at the poles M = 20890564 feet, r = 20961932 feet. So that for engineering purposes we may take 20,890,000 feet as the mean radius of curvature. Again, according to the same authority, the length of a degree of latitude equals in feet D = 364609.1 - 1866.7 cos 2L -f 4 cos 4L, and the length of a degree of longitude equals in feet d = 365542.5 cos L - 311.8 cos 3 + 0.4 cos 5. The value of the foot taken above is the English standard, which is less than the American standard in the ratio of 1 mile to 1 mile and 3.677 inches. For rough work we may consider D = 364000 feet and d = D cos Lat. Table XVIII gives the true values of 1 minute of arc, to the nearest foot. 207. Now from the formula for the length of a circular arc given in Sec. 73, if we take the above value pf the mean radius of curvature, we find the length of an arc on the earth's surface in feet equals I = 6076 n (nearly), where n = the number of minutes in the arc ; and the con- verse of this, 220 EXPLORATORY SURVEYING. enables us to convert any given distance into its equivalent in augular measure. If it is desirable to obtain the value of I more accurately than by this means, we can do so by obtaining first the value of I in the direction of the meridian, either from Table XVIII, or more correctly by dividing the value of D, given in Sec. 206, by 60. Also the length of a 1' arc perpendicular to the merid- ian is needed, which may be obtained by means of the value of r, given in Sec. 206. Then if we call this latter value I', the length of an arc subtending 1' at the earth's centre, which makes an angle A with the meridian, equals Zcos 2 A + l 1 sin 2 A 208. Given the latitude and longitude of two places to obtain their distance apart, and the bearing of the course joining them. Suppose A and D in Fig. 12 are the two given places, then the arc AF&nd the arc ED represent their latitudes. Then in the spherical triangle AND, since N difference of longitude, and AN and ND are equal to the co-latitudes of A and D, we can find AD thus: cos AD = sin a sin d -j- cos a cos d cos AND, where a and d are the latitudes of A and D. And the bearing of the arc AD, which at A is represented by the angle NAD, is then given by the equation sin A = cos d cosec AD sin AND. Or, if A and D are in the same latitude, we have tan A = cot %AND cosec lat. The arc so obtained can be converted into feet as shown in Sec. 207; and this is the distance along the arc of the great circle passing through A and D, i.e., the shortest distance be- tween them on the earth's surface. Conversely, given the latitude and longitude of A, and the bearing and distance of another place J), to find the latitude and longitude of J). First convert AD into angular measure according to Sec. 207; then we have the sides EXPLORATORY SURVEYING. 221 AD, AN, and the included angle A. Then to find d we have sin d = cos AD sin a -j- sin AD cos a cos A Then AND, the difference of longitude, is given by sin AND = sin A sin AD sec d. The bearing of AD at D may be obtained from the equation sin D = sin J..ZV.D cos a cosec AZ). The formulae given in this section are simply those ordinarily used for the solution of spherical triangles. (See Sec. 233.) 209. To find the radius of a Circle of Latitude. In Fig. 93 let C be the centre of the earth, N the P pole, and L any given latitude; then, consider- ing the earth to be a sphere, the angle LPG = the latitude of L, so that PL = LC cot latitude, where PL = radius of the circle of latitude. LC may be taken as equal to 20,890,000 feet. 210. To calculate the offset at aiiy point C to a parallel of latitude AC from a straight Hue AB, tangent to AC at A. We can do this by treating the parallel of latitude A C in Fig. 94 as a curve FIG. 93. to which the arc of a great circle AB is tangent at A, and thus obtain the offset CB according to Sec. 78; or, we can solve the N right-angled spherical triangle ANB, and so find the latitude of B, if we know the differ- ence of longitude N, thus: tan (lat. B) tan (lat. A) cos N. CB then equals the difference of latitude of A and B. 211. We are now in a position to consider the influence of the spherical form of the earth, assuming for the moment the earth to be a sphere, on a map the linear measurements of which have been computed on the supposition that the sur- face of the earth is a plane, 222 EXPLORATORY SURVEYING. Now a spherical surface cannot be developed on a plane surface, but can only be developed on a sphere of equal radius. Thus no map can, theoretically even, be correct to the same scale in all its parts. In nautical charts, which are gen- erally made on Mercator's Projection, this difficulty is over- come by the use of a scale of meridional parts, the scale at all points being proportional to the secant of the latitude. And this is a very convenient method, where all positions are obtained astronomically and where the error involved by calculating the courses according to " Middle Latitude Sail- ing" is of no importance. But in constructing a map this method is inconvenient; for if the same scale is used through- out, it assumes that parallels of latitude are right lines, and that there is no convergence of the meridians. In plotting exploratory surveys, simplicity is an important factor; also, the map must be adapted to the same scale throughout, and be so arranged as to be suitable to the plotting of topography as on a plane surface. To approximate as near as possible to correctness in the more important portions, and to throw the excess of error into the less important parts, is the best that can be done under any circumstances. 212. In Sec. 58 we referred to the corrections which it was necessary to make on account of the convergence of the merid- ians. By extending this method we are able, with the aid of the preceding problems, to construct the groundwork of our map without any other principles than those already explained. The best way is to take an example and work it out as if in actual practice. Suppose from A in Latitude 60 N. and Longitude 120 W. we intend starting off straight across country for B, a place which, from the maps, we find to be situated in about Lat. 59 N. and Long. 110 W., and wish before starting to lay out the groundwork of a map to be constructed from the knowledge of the topography which we intend to obtain on the way that we may have some reliable means of plotting our results as soon as obtained, and also of determining positions relatively to each other by means of bearings and distances. At A we draw, as in Fig. 95, the base-lines AS and AD. Then find the length of AC from Table XVIII, calculating as if it were in the mean latitude of A and B, i.e., 59 30' N. ; thus AC = about 10 X 60 X 3095 = say 1,857,000 feet. If EXPLORATORY SURVEYING. 223 great accuracy were required, we could find the value of d in latitude 59 30' according to Sec. 206, then AC = IQd. -t N-'^-^"^ g FIG. 95. Next we make AD = AC, and through!) draw the meridian CB, the bearing of which on the map, relatively to A, = the convergence between A and B = 8" 36'. Therefore the angle CD A = 81 24'. The length of the offset CD may be found according to Sec. 78, and is equal to about 140,000 feet; and since B lies 1 to the south of C, and on the meridian passing through D, we have DB = about 225,400 feet. Then by solving the plane triangle ADB, we obtain AB = 1,903,800 feet, and -the angle BAD = 6 44'. Thus the direct course from A to B is S. 83 16' E., and Ad = " Total departure" = AB cos 6 44' 1,890,700 feet, and Bd " Total latitude" = DB cos 8 36' = 222,800 feet. We have thus the groundwork of our map ready for the plotting of the courses, and if we use sheets of cross-section paper, with 10 divisions to the inch, and plot to a scale of 10,000 feet to an inch, we then have a map of tolerably convenient size, plotted to a scale sufficiently large to show the main features of the country, since any important parts which may have been made the subjects of special survey can be best shown separately. In order to connect the Astronomical work with that which is plotted by Latitudes and Departures, or by protractor, and which we may call our " dead-reckoning," we must draw meridians and curves of latitude at about every 30'. To fill in these meridians, divide A C equally into 20 parts, and draw the meridians perpendicular to the curve at each of these points, i.e., dividing up the convergence equally among them. The curve of latitude AC, since we know the dis- tance CD, can be drawn by assuming that the offset half-way between A and D = %CD, and so on, according to Sec. 78. 224 EXPLORATORY SURVEYING. The advantages of this method of plotting are, that we can readily connect positions taken by astronomical observations with those calculated from dead-reckoning, the former being plotted by the guidance of the parallels of latitude and the meridians, and the latter by means of the base Ad. Also, that the same scale is used throughout, and the bearings of all points may be taken off with a protractor. If the topographical positions are obtained solely by direct astronomical observations, tJien the method of Mercator's Pro- jection is more convenient than that given above. To plot our route we proceed as follows: Suppose we take rough compass courses; these we plot lightly on the map, having worked them out, say, by Latitudes and Departures, correcting the " latitudes" absolutely according to any latitude observations we may take, the " departures" being guided to a reasonable extent by the observations for longitude. Thus our course is constantly being broken, involving a new "total latitude" for each fresh start. This we can best find by scal- ing from Ad, after having plotted the position astronomically. At the end of our. journey, whatever error in longitude we may have, may usually be divided up proportionally along the whole route, if the trip has been made at a tolerably uniform pace. The error in latitude should be inappreciable. The above example shows what must be considered in plot- ting an extensive survey; and though a more rough and ready method is usually correct enough, yet where the field-work is run in such a way as to warrant a tolerably accurate plot of it being made, the little extra time involved in making a good map is time well spent. As regards the mode of procedure in keeping a course astro- nomically, Col. Frome says: "It is probably inconvenient always to obtain latitude at noon, but we can generally do so, and more correctly, at night by the meridian altitude of one or more of the stars. The local time can immediately before or after be ascertained by a single altitude of any other star out of the meridian the nearer the prime vertical the better; and if a pocket-chronometer is carried, upon which any de- pendence can be placed, the explorer has thus the means, by comparison with his local time, of obtaining his approximate longitude, and laying down his position on paper. The lon- gitude should also be obtained occasionally by Lunar Dis- EXPLORATORY SURVEYING. 225 tances, or some other method. The latitude he should always get correct to half a mile, and the longitude to 8 or 10 miles." 213. The Star Map given below will be found convenient in selecting suitable stars for observations. The stars are plotted from their R.A.'s and Decs, in the same way that a map of the earth is plotted by longitudes and latitudes, i.e., looking down on it. STAR MAP FOR *" NORTHERN HEMISPHERE. The centre is the celestial pole, and the 24 radiating lines divide the 24 hours of R.A. Now the initial point for R.A. being on the meridian at 10 P.M. about Oct. 21, we can divide the circle into 12 divisions, and arrange them so that the radi- ating line marked Hours will cut the 10 o'clock division about two thirds along it. Thus we read off that about Oct. 21 the star marked 1 will be on the meridian, i.e., due south, at 226 EXPLORATORY SURVEYING. 10 P.M. Similarly the star marked 23 will be on the meridian at 10 P.M. about Aug. 17. But suppose we want to know what star will be near the meridian about 8 P.M. on Jan. 10. Imagine the margin of the map, with the months marked on it, to be stationary, and the interior portion to rotate in the same direction as the hands of a watch, once in 23 h 56 m ; then, since the map shows the posi- tion at 10 p.m., at 8 P.M. (two hours earlier) the star marked 5 will have been near the meridian on Jan. 10, In this way we can tell at about what time any meridian ob- servation will occur without referring to the Nautical Almanac, Thus with this map and the following key and table no Nauti- cal Almanac is needed for latitude observations, by the merid- ian altitudes of stars. The Decs, and R.A.'s given are for Jan. 1, 1889. TABLE OF MAGNITUDE, DEC., AND R.A. OF THE PRINCIPAL STARS. No. in Map. NAME. i Dec. An. Var. R.A. An. Var. 1 II h. m. s. s. 1 Alpherat, a Andromedse 2.0 - - 28 28 39 + 19.88 2 39 + 3.09 2 Polaris, a Ursae Minoris.. 2.0; - - 88 42 59 + 18.90 1 18 08 + 23.15 3 y Cassiopeia? 2.0 - - 60 06 55 4- 19.56 50 01 r 3.58 4 Algol /3 Persei 2.7 - -4031 38 + 14.1'2 3 57 -3.88 5 a Persei 2.0 - -49 27 55; 4- 13.10 3 16 24 -4 26 6 Aldebaran, a Tauri 1.0 - - 16 17 07 + 7.52 4 29 33 -3.44 7 Capella, a Aurigae l.O 1 - - 45 53 03 4- 4.03 5 08 29 -442 8 a Arietis 2.0: --2256 14 + 17.7 2 55 - 3.37 9 Rigel, Orionis 1.0 -81950 + 4.40 5 09 12 - 2.88 10 Betelgeuze, a Orionis... 1.2 4 7 23 8 + 0.95 5 49 10 - 3.25 11 Sirius, a Canis Majoris.. 1.0 - 16 33 5-2 - 4.71 6 40 15 -2.64 12 Castor, a Geminorum. . . 1.7 + 32 07 53 - 7.55 7 27 31 -3.84 13 Pollux, /3 Geminorum... 1.3 4-281737 -8.41 7 38 31 -368 14 Procyon, a Canis Minoris 1.0 4- 5 30 3-2 - 8.99 7 33 29 -3.14 15 Regulus, a Leonis. 1.3 4-' 12 30 34 - 17.47110 02 28 -320 16 a Ursae Majoris 2.04-6221 - 19.36 10 56 52 - 3.75 17 y Ursae Majoris 2.34-54 1842 - 20.03 11 47 59 - 3.18 18 TJ Ursae Majoris 2.0,4-495203 - 18.08' 13 43 10 - 2.37 19 Arcturus, a Bootis 1.01+ 19 45 38 - 18.88J14 10 36 - 2.73 20 Spica, a Virginis 1.0 - 103454 18.90 13 19 21 -3.15 21 AntT/res a Scorpii 1.3 - 26 11 06 8.30 16 22 36 - 3. (57 22 23 24 25 Vega, a Lyrae 1.0 1.3 1 7 4-384050 4-3.1? 4- 8 34 32l +9.27 4 44 53 02 j + 12.72 - 301237 4-18.99 18 33 11 19 45 22 20 37 39 22 51 31 - 2.03 - 2.93 -204 - 3.8-2 Altair. a Aquilee a Cvgni Fomalhaut, a P. Aust. . . 1.3 26 Mar kiib a Pegasi. 2.0 + 14 36 29 + 19.30 22 59 14 -2.98 EXPLORATORY SURVEYING. 227 IK THE SOUTHERN HEMISPHERE WE ALSO HAVE- NAME. Mag. Dec. An. Var. R.A. An. Var. /3 Hydri 3 77 52 46 4-2028 h. m. s. 19 54 s. 4-323 Achernar, a Eridani .. Canopus, a Argus ft Argus a Crueis 1.0 1.0 1.5 1 - 57 48 03 - 52 38 07 - 69 15 36 - 62 29 02 4- 18.36 - 1.87 - 14.80 - 2001 1 33 M 6 21 29 9 11 59 12 20 26 -1-2.23 4- 1.83 + 0.68 - 3 29 /3 Centauri 1.0 - 59 50 14 17 59 13 55 59 --4-18 a Centauri a Trianguli Aust 1.0 2.0 2 - 60 22 47 - 68 49 21 _j_ 12 33 29 - 15.38 - 7.16 2 87 14 32 05 16 36 55 17 29 47 - - 4.05 - - 6.30 - - 2 78 a Gruis 2.0 47 29 53 4- 17.25 22 01 14 --3 81 In order better to recognize the positions of the stars at night, they may be pricked through on a sheet of paper, which, when turned backwards and held up towards the south, with the month at the lowest part, will correspond with the face of the sky at 10 P.M. PART IV. MISCELLANEOUS. THE following miscellaneous information may at times be found of service in the field to both the engineer and the ex- plorer: 2U. To find the Horse-power of Falling Water. H.P. = 0.00189 QH, where Q = the number of cubic feet of water passing over the fall per minute, and H= height of fall in feet. Turbines can utilize about 75 p. c. of this H.P. Thus the Effective horse-power, i.e., available for useful work, = about .0014 QH. 215. To gauge a stream, roughly. Take some body, which, when floating, will be almost entirely immersed, and throw it into the middle of the stream, in a part, if possible, unobstructed by reeds, etc., and free from slack- water, eddies, or counter-currents; and where the cross-section of the stream is fairly uniform. Observe the time T in seconds which the body takes to float a distance of 100 feet. Then if A = the cross-section of the stream in square feet, and Q = cubic feet of water that pass per minute, 5000J. This assumes that the middle surface velocity is to the mean velocity as 6 to 5, which is a fairly average ratio. 228 MISCELLANEOUS. 229 216. The Sustaining power of ordinary wooden piles in Ibs. equals FW -IT where F fall of hammer in inches, W= weight of hammer in Ibs., S = space driven by last blow in inches. This formula is generally found to give results about as re- liable as any general formula can give. 217. Supporting power of various materials. Clay 1.0 to 2.0 tons per sq. foot. Sandy clay 2.0to4.0 " Sand 3.0 to 5.0 " " Gravel 4.0 to 5.0 " Sandstone 2.0 to 4,0 " Firm Rock 10.0 ' ' These are the pressures to which the above may usually be safely loaded. 218. Transverse strength of rectangular beams. Let L = length of beam in feet between points of support, b = breadth of beam in inches, d = depth of beam in inches, W= Load at centre of beam in Ibs., /= coefficient of modulus of rupture. Then 1SWL 1 A '' and 6 = For the values of /see-following table. For example, if b = 6", d = W", and L 20 feet, if we take/= 10,000 Ibs., by the above formula W= 16,666 Ibs. ; so that with a Factor of Safety of 6 we may safely load it at its centre, and consequently at any part of it, with a weight of 2778 Ibs. A beam will carry as a centre load only half the weight that it will bear distributed uniformly over it. So that, for instance, if we wish to know what total breadth we must give to a set of stringers, where d = 16" , in order safely to carry an ordinary train over a span of 15 feet, if we take/= 10,000 Ibs. and the 230 MISCELLANEOUS. load per foot run as equivalent to 4000 Ibs., we have as the equivalent value of W, 30,000 Ibs. So that by the above formula b = about 3 inches. Therefore, taking a factor of safety of 8, b = about 24 inches; so that four 6" X 16" stringers may safely be used. The factor of safety usually adopted for wood varies from 5 to 10, according to the condition of the timber, the amount of impact caused by the load, and the possible amount of decay to which it will be subjected. For spans, in railroad bridges, less than 10 feet, 5000 Ibs. per foot run should usually be taken as the uniformly distributed load. In spans exceeding 15 feet 3500 Ibs. is usually sufficient. These values take no account of the weight of the beams them- selves. VALUES OF /. Material. Lbs. per sq. in. Material. Lbs. per sq. in. Ash 12,000 to 14,000 Red Pine . . . 7100 to 9500 Birch 11,700 Spruce . . . 9900 to 12,300 Blue Gum Elm 18,000 6000 to 9700 Brit. Oak Am. Red Oak 12,000 10600 219. Natural Slopes of Earths. Material. 1 53 Material. ! 53 Material. i o 02 Gravel 40 Vegetable Earth 28 Ruble 45 Dry Sand Sand 38 goo Compact Earth . . . Shingle 50 39 Clay (drained) Clay (wet) 45 16 220. Weight of Earths, Kocks, etc., per cubic yard, Material. Weight in Ibs. per cu. yd. Material. Weight in Ibs. per cu. yd. Material. Weight in Ibs. per cu. yd. Sand Gravel Mud Marl 3360 3360 2800 2900 Clay Chalk Sandstone. . Shale 3470 4030 4370 4480 1 ; Quarts Granite . . . Trap ; Slate. 4590 4700 4700 4810 A cubic yard of water weighs about 1680 Ibs. MISCELLANEOUS. 231 221. Weight of Timber and Metals per cubic foot. Material. Weight in Ibs. per cu. ft. Material. Weight in Ibs. per cu. ft. Material. Weight in Ibs. per cu. ft. Elni, English . Canadian Elm Maple 35 45 42 Pine, red. ... " white . . Teak 36 30 50 Iron, cast " wrought Steel 450 482 490 English Oak . . American Oak 48 50 Spruce Larch 30 34 Copper Lead 550 710 222. Mortar, Cement, etc. (common mixtures). Mortar. \ of lime to 2 or 3 of sharp river sand. Coarse Mortar. 1 of lime to 4 of coarse gravelly sand. Concrete. 1 of lime to 4 of gravel and 2 of sand. Hydraulic Mortar. 1 of blue lias lime to 2 of burnt clay, ground together. Beton. 1 of hydraulic mortar to H of angular stones. Cement. 1 of sand to 1 of cement; or if great tenacity is required the sand may be omitted. Portland Cement is composed of clayey mud and chalk ground together and afterwards calcined at a high temperature, and then ground to a fine powder. NOTES. For ordinary engineering work the following proportions make a good mortar : 1 measure of Lime; 3 to 5 measures of sand, according to the " hunger " of the sand, 1 measure of ashes, brick dust, or burnt clay. For engineering work, if exposed to dampness, of the lime in the above should be replaced by hydraulic cement ; whilst for work under water, 1 measure hydraulic cement to 2 measures of sand make a good mixture. NOTES ON TIMBER. 223. Selection of standing trees. " Scribner's Log Book." "The principal circumstances which affect the quality of growing trees are soil, climate, and aspect. "In a moist soil the wood is less firm, and decays sooner than in a dry, sandy soil ; but in the latter the timber is seldom fine : the best is that which grows in a dark soil, mixed with 232 MISCELLANEOUS. stones and gravel. This remark does not apply to the poplar, willow, cypress, and other light woods which grow best in wet situations. "Trees growing in the centre of a forest or on a plain are generally straighter and more free from limbs than those growing on the edge of the forest, in open ground, or on the sides of hills ; but the former are at the same time less hard. The toughest part of a tree will always be found on the side next the north. The aspect most sheltered from prevalent winds is generally most favorable to the growth of timber. The vicinity of salt water is favorable to the strength and hardness of white oak. "The selection of timber trees should be made before the fall of the leaf. A healthy tree is indicated by the top branches being vigorous, and well covered with leaves**, the bark is clear, smooth, and of a uniform color. If the top has a reg- ular, rounded form ; if the bark is dull, scabby, and covered with white and red spots, caused by running water or sap, the tree is unsound. The decay of the uppermost branches and the separation of the bark from the wood are infallible signs of the decline of the tree." 224. Defects of Timber Trees (especially of oak). "Sap, the white wood next to the bark, which very soon rots, should never be used, except that of hickory. There are sometimes found rings of light-colored wood surrounded by good hard wood; this may be called the second sap : it should cause the rejection of the tree. "Brash-wood is a defect generally consequent on the decline of the tree from age ; the pores of the wood are open, the wood is reddish-colored, it breaks short without splinters, and the chips crumble to pieces. " Wood which has died before being felled should in general be rejected ; so should knotty trees, and those which are covered with tubercles, etc. "Twisted wood, the grain of which ascends in a spiral form, is unfit for use in large scantling ; but if the defect is not very decided, the wood may be used for naves, and for some light pieces " Splits, checks, and cracks, extending towards the centre, if deep and strongly marked, make the wood unfit for use, un- less it is intended to be split. MISCELLANEOUS. 233 " Wind-shakes are cracks separating the concentric layers of wood from each other; if the shake extends through the entire circle, it is a ruinous defect." 225. Felling Timber. " The most suitable season for felling timber is that in which vegetation is at rest, which is the case in midwinter and in midsummer; recent opinions derived from facts incline to give preference to the latter sea- son. The tree should be allowed to attain its full maturity before being felled; this period in oak timber is generally at the age of from 75 to 100 years, or upwards, according to cir- cumstances. The age of hardwood is determined by the num- ber of rings which may be counted in a section of the tree. "The tree should be cut as near the ground as possible, the lower part being the best timber. The quality of the wood is in some degree indicated by the color, which should be nearly uniform in the heart wood, a little deeper toward the centre, and without transitions. " Felled timber should be immediately stripped of its bark, and raised from the ground. " As soon as practicable after the tree is felled the sap-wood should be taken off and the timber reduced, either by sawing or splitting, nearly to the dimensions required for use. " The best method of preventing decay is the immediate re- moval of it to a dry situation, where it should be piled in such a manner as to secure a free circulation of air around it, but without exposure to the sun and wind. .When thoroughly seasoned before cutting it up into small pieces, it is less liable to warp and twist in drying. When green, timber is not so strong as when thoroughly dry. " Lumber containing much sap is not only weaker, but de- cays much sooner than that free from sap." 226. Seasoning and Preserving Timber. "For the pur- pose of seasoning, timber should be piled under shelter, where it maybe kept dry, but not exposed to a strong current of air; at the same time there should be a free circulation of air about the timber, with which view slats or blocks of wood should be placed between the pieces that lie over each other, near enough to prevent the timber from bending. The seasoning of timber requires from two to four years, accord- ing to its size. 234 MISCELLANEOUS. " Gradual drying and seasoning in this manner is considered the most favorable to the durability and strength of timber. " Timber of large dimensions is improved by immersion in water for some weeks. Oak timber loses about one fifth of its weight in seasoning, and about one third of its weight in becoming dry." 227. Decay of Timber. There are three principal causes of decay of timber dry-rot, wet-rot, and the " teredo navalis" and other worms. Dry-rot does not usually occur where there is a free circu- lation of air, and if the timber is properly dried an occasional immersion in water should do no harm. Timber kept dry and well ventilated has been known to last for several hun- dred years without apparent deterioration. Dry-rot is caused by a species of wood fungus Merulius lacJirymans which destroys the tensile and cohesive strength, gradually convert- ing the timber into a fine powder. Wet-rot. This is the destructive agent at work more or less on all timber freely exposed to air and moisture. It is of two kinds : A. CJiemical. In this case a slow combustion takes place, and by a gradual process of oxidation the wood slowly rots away. B. Mechanical. This is the more common form, and gener- ally occurs near the water-line in timber subject to frequent immersion. It is the frequent alternate conditions of moisture and dryness that are most trying to timber, as is the case with metals. AVhen timber is constantly under water, the action of the water dissolves a portion of its substance, which is made apparent by its becoming covered with a coating of slime, and this protects the interior. If, however, it is exposed to al- ternations of moisture and dryness, as is the case with piles in tidal waters, the dissolved parts being continually removed by evaporation and the action of the water, new surfaces are be- ing frequently exposed for decomposition. Piles driven in sea-water are frequently destroyed by the " teredo navalis," and also by another species of worm called the " limnoria." They both work from about the high-water mark to the surface of the mud. 228. To test Steel and Iron. Scientific American. Nitric acid will produce a black spot on steel; the darker the MISCELLANEOUS. 235 spot the harder the steel. Iron, on the contrary, remains bright if touched with nitric acid. Good steel in its soft state has a curved fracture and a uni- form gray lustre; in its hard state, a dull, silvery, uniform white. Cracks, threads, or sparkling particles denote bad quality. Good steel will not bear a white heat without falling to pieces, and will crumble under the hammer at a brig7it-rcd heat, while at a middling heat it may be drawn out under the hammer to a fine point. Care should be taken that before at- tempting to draw it out to a point the fracture is not concave; and should it be so, the end should be filed to an obtuse point before operating. Steel should be drawn out to a fine point and plunged into cold water; the fractured point should scratch glass. To test its toughness, place a fragment on a block of cast-iron: if good, it may be driven by a blow of a hammer into the cast-iron; if poor, it will crush under the blow. Tests of Iron. A soft tough iron, if broken gradually, gives long silky fibres of leaden-gray hue, which twist to- gether and cohere before breaking. A medium even grain with fibres denotes good iron. Badly refined iron gives a short blackish fibre on fracture. A very fine grain denotes hard steely iron, likely to be cold- short and hard. Coarse grain with bright crystallized fracture or discolored spots denotes cold-short, brittle iron, which works easily when heated and welds well. Cracks on the edge of a bar are indi- cations of hot-short iron. Good iron is readily heated, is soft under the hammer, and throws out few sparks. 220. Strength of Rope. The table on following page gives some idea of the strength of ordinary Manilla Hope. It must be remembered that these values are for new ropes and that a few months' exposure to the weather will probably cause a decrease in the strength of 40 or 50 p. c. A factor of safety of 4 or 5 is generally employed to obtain their safe working strength. Ropes made of good Italian hemp are considerably stronger than these. 236 MISCELLANEOUS. TABLE OF MANILLA ROPE-3 STRANDS. SIZE OF ROPE. Breaking- strength in Ibs. SIZE OF ROPE. Breaking- strength in Ibs. Diam. in inches. Circum. in inches. Diam. in inches. Circum. in inches. 1 j 1 0.71 1.43 2.14 2.86 3.57 4.28 5.70 375 1,500 3,380 6,000 9,380 13,500 24,000 & 3 * f 6 7.14 8.57 10.0 "11.4 12.1 14.2 17.1 37,500 54.000 73.600 96.000 121.000 150,000 216,000 Wire Ropes. The following table gives the strength of iron and cast-steel wire rope : TABLE OF IRON AND CAST- STEEL WIRE ROPE. SIZE OF ROPE. BREAKING- STRENGTH IN LBS. SIZE OF ROPE. BREAKING- STRENGTH IN LBS. Diam. in In. Circum. in In. Iron. C. Steel. Diam. in In. Circum. in In. Iron. C. Steel. 1 H 1 6.960 17,280 32,000 54,000 15,000 36,000 66,000 104,000 H * 2J I 6| 78.000 108,000 130,000 148,000 154.000 212,000 250,000 310,000 These ropes have 19 wires to the strand and hemp centres. One fifth of the above breaking-strength may be taken as the safe working strength. For the strength of Iron Rods see Sec. 138. 230. Properties of the Circle. Diameter X 3.14159 Diameter X .88622( Diameter X .7071 Diameter* X .7854 Radius X 6.28318 Circumference X .31831 Circumference = 3.5449 Diameter = 1.1283 Length of arc = circumference. = side of an equal square. = side of an inscribed square. = area of circle. = circumference. = diameter. area of circle. area of circle. = number of degrees X 0.017453 radius. Arc of 1 to rad. 1 -- 0.01745329. Arc of 1' to rad. 1 = 0.000290888. Arc of 1' to rad. 1 - 0.000004848. Degrees in arc whose length = radius = 57. 2957795. TT = 3.1415926536; Log n = 0.4971499. MISCELLANEOUS. 237 231. PLANE TRIGONOMETRY.-In Fig. 96, angle OAE = 90; then in the right- Q H angled triangle ABC, if AB = Radius = unity, AF= cosec A; CE = versin A; BH = co-versin A; BD = exsec A; if the BC=sin A; AC = cos A; DE-ismA; AD = sec A-, GF=cotA; Therefore BF = co-exsec A. BC A cosec A = - Thus, sin A = cosec A ' AC COsA =AB> AB cos A = AC sec tan A = 1 cot A' An angle and its Supplement have the same Sine and Cose- cant; but the Tangents, Secants, Cosines and Cotangents, though of equal length, are of contrary signs: so that in applying to obtuse angles trigonometrical formulae which were originally intended for acute angles, the algebraic signs of the tangents, secants, cosines, and cotangents must be reversed. The sine, secant, and tangent of an angle A are respectively equal to the cosine, cosecant, and cotangent of its comple- ment (i.e., of 90 A). B = 90 - A. AC.&C AB 2 = AC* -f Area of triangle = 2 Examples of Right-angled Triangles: 1. Given A = 30 Q , and AC - 100, find BC. T>r< We see above that tan A = r^, ; therefore BC = AC tan A = 57.73. 238 MISCELLANEOUS. 2. Find the sine of 128. Since sin (180 - A) = sin A, sin 128 = sin (180 - 52) = sin 52, which from the tables we find = 0.788. Solution of Oblique-angled Triangles. FIG. 98. sm A _ sin B _ sin C a b tan A-B 2 a-+-b tan c A-\-B A *"* I " | "-* -A- 3 + g- 5 = (3) Let c = ( + &)_ A-B = <; then vers ud = cos - ft) (a c) (5) (6) A a) (7) MISCELLANEOUS. 239 Area of triangle = 4/ s(s - a) (s-b) (s -c). . , (8) = ^ sin C. ....... (9) a a? sin B sin C ~ ...... (11) The above formulae are all that are required for the ordi- nary solution of plane triangles. Remarks. Though such a formula as No. 2 simpty men- tions A and B and their opposite sides, it holds equally well whether we substitute C for A, or G for B, provided that the sides are changed to correspond also. In Equations 2, 3, 4, and 5, A is intended to represent the greater angle of the two angles A and B. Examples, 1. Given A, B, and b, find A. By Equation 1, _ b sin A ~ sin B ' 2. Given B, c, and b, find C. By Equation 1, ~ c sin B sin C = j . b 3. Given A, B, and c, find a. By Equation 11, C= 180 -(A +J5); and by Eq. 1, c sin A 4. Given B, a, and c, find A and b. By Eq. 2, A- C a-c 240 MISCELLANEOUS. from which we obtain the value of A-0 2 ' and by Eq. 11, therefore we can find A from Eq. 3. Then by Eq. 5, __ - . cos - 5. Given a, b, and c, find B. By Eq. 6, 2(8 a) (s c) or, we might equally well have used Eq. 7. 232. The following general equations are worth noting: sin A = tan A cos A = 4/ 1 cos 2 A = 2 sin cos ; a 4i cos A = cot A sin A 4/1 sin 2 A = 2 cos 2 1; vers 2^1 , .4 tan J. = sin J. sec J. = -j ; ( = exsec A cot ; sm 2J. ^ ^ cot A cos J. cosec A = - r- vers A = 1 cos A = 2 sin 2 = cos J. exsec A\ A vers J. exsec A = sec .4 - 1 = tan A tan - = 233. MISCELLANEOUS. Spherical Trigonometry. 241 FIG. 99. RIGHT-ANGLED TRIANGLES. In Fig. 99 let A = 90; then sin b = sin a sin B ; tan c = tan a cos B; cot G = cos a tan tan c = sin 5 tan (7; cos a = cos b cose; cos 5 = cos b sin (7; tan b tan 6 tan # = --- -7=; sin c = ; cos C tan B sin b sin a = sin 5 cos J9 cos a sin & sin C7 = r ; cos c = ; sin B = cos b cos 6 sin a ~ tan b tan c _ tan b cos (7 = ; tan C = = ; tan B = . ; tan a sin b sin c cos c = cos C - ~ ; sin B cos b = B . , sin C cos a = cot (7 - =. tan B b and c are of the same species respectively as B and C. Any side is greater than 90 if the other sides are of differ- ent species, and less than 90 if of the same species. B or (7 is less than 90 if the containing sides are of the same species, and less than 90 if of different species. 242 MISCELLANEOUS. Oblique-angled triangles. FIG. 100. Let ABC in Fig. 100 represent any oblique-angled spherical triangle; then sin A _ sin B _ sin G sin a ~ sin b ~ sin c ' a ~ o __ __ sm - c 2 tan tan f (7 ~2~ = cot -- cos . a = cot -^ 2' . a-f$' sm ~ cos c := cos a cos 6 -f- sin a sin 5 cos (7 ; sin A /sin (a- 6) si: 2 ~ r sin 6 si ft) sin (a c) sin 6 sin c F cos 8 cos (8 A\ 'B sin C ' (1) (8fl) (35) (4) (5) (6) a ____c where* = - and 8 = MISCELLANEOUS. 243 The greater angle is always opposite the greater side. No angle or side is greater than 180. The sum of any two sides is greater than the third side. The sum of the three sides is less than 360. Given a, b, and C, to find A and B : use Eqs. 2a and 2b. " A, B, and c, " a and b " a, b, and G, " ' c; or, given a, b, and C, " c ; " A, B, and a, " B or 6 ; " A,B, and a, " (7; " A, B, and , " c; " a, 6, and A, " C; " , b, and ^4, " c ; " -4, 5, and_c, " (7; 3a and 36. 2a, 25, and 36. 4 1 1 and 2a. 1 and 3a. 1 and 2a. 1 and 3a. 8a, 36, and 26. 234. Measures of length and surface. MEASURE OF LENGTH. Miles. Furlongs. Chains. Rods. Yards. Feet. Inches- 1 8 80 320 1760 5280 63360 0.125 1 10 40 220 ' 660 7920 0.0125 0.1 1 4 22 66 792 0.003125 0.025 0.25 1 55 16.5 198 0.00056818 0.0045454 0.045454 0.181818 1 3 36 0.0(1018939 0.00151515 0.01515151 0.0600060 0.33333 1 12 0.000015783 0.000126262 0.001262626 0.00505050 0.0277777 0.083333 1 MEASURE OF SURFACE. Sq. Miles. Acres. S. Chains. Sq. Rods. Sq. Yards. Sq. Feet. 1 640 6400 102400 3097600 27878400 0.001562 1 10 160 4840 43560 0.0001562 0.1 1 16 484 4356 0.000009764 0.00625 0625 1 30.25 272.25 0.000000323 0.0002066 0.002066 0.0330 1 9 0.0000000358 0.00002296 0.0002296 0.00367 0.1111111 1 244 MISCELLANEOUS. 235. Measures of weight and capacity. MEASURES OF WEIGHTS. AYOIRDUPOIS. Ton. Cvt. Pounds. Ounces. Drams. 1 20 2240 35840 573440 0.05 1 112 1792 28672 0.00044642 0.0089285 1 16 256 0.00002790 0.000558 0.0625 1 16 0.00000174 0.0000348 0.0016 0.0625 1 TROY. Pounds. Ounces. Dwt. Grains. Pound Avoir. 1 12 240 5760 0.822861 0.08:3333 1 20 480 0.068571 0.004166 0.05000 1 24 0.0034285 0.0001736 0.002083333 0.0416666 1 0.00014285 1.215275 14.58333 291.6666 7000 1 MEASURE OF CAPACITY, Cub. Yard. Bushel. Cub. Feet. Pecks. Gallons. Cub. inch. 1 0.03961 0.037037 0.009259 21.6962 1 0.803564 0.25 0.107421 27 1.24445 1 0.31114 0.133681 0.000547 100.987 4 3.21425 1 0.429684 0.001860 201.974 9.30918 7.4805 2.32729 1 0.004329 46656 2150.42 1728 537.605 231 1 APPENDIX. NOTE A. (See Sec. 10.) IF we knew the average pressure in the cylinders we could find the propelling force of an engine at any speed, if not limited by adhesion, by the following rule : Multiply together the square of the diameter of one piston in inches, the length of stroke in inches, and the mean pressure (above atmosphere) in Ibs. per sq. in. The product divided by the diameter of a driver in inches gives the pro- pelling force in Ibs., ignoring " internal frictional resistances." Theoretically, the mean effective cylinder- pressure in Ibs. per sq. in. equals P + 2.3P(Log) 8 -15, where P = absolute boiler-pressure in Ibs. per sq. in. and 8 = Stroke -*- part of stroke before cut-off. But owing to the contraction of the steam-ports, the initial cylinder-pressure always falls below the boiler- pressure. Similarly owing to the contraction of the exhaust-port, back- pressure always exists ; and these are matters so purely of mechanical detail that no general rule can be given which would take them into consideration. At 20 miles per hour, however, the effective initial cylinder pressure often equals only about 90 p. c. of the boiler-pres- sure, and at 50 m. p. h. about 60 p. c. Thus if P = 125 Ibs. per sq. in. and the'stroke = 24 inches ; if steam is cut off at 6 inches, the theoretical mean cylinder- pressure = 59 Ibs. per square inch, which at 50 m. p. h. will probably be reduced to about 36 Ibs. : so that if the diameter of the piston =16 inches, and of the driving-wheels 60 inches, the propelling force will equal 3680 Ibs.; and if we deduct 10 p. c. from this for internal frictioual resistances, the propelling force = 3200 Ibs. 245 246 APPENDIX. NOTE B. (See Sec. 19.) In order to reduce the quantities used in Diagram II into the same units, say ton, mile, and hour, the ordiuates of the curves must be multiplied by (3600) 2 aooo-xio x88 - 8=40(Iie8rly) to reduce them to tons weight (2000 Ibs.), in miles per hour units. Then, with the units selected, the equation of motion is But if x is the space passed over, so that and therefore OQ.d(OQ) ~ = the graphic process giving the integral. But with the scales used in Diagram II, instead of multiplying the ordinatea as above, we can simply use as a scale 1 square inch = 1 mile, which practically comes to the same thing. If the horizontal scale were ten miles per hour to one inch, the scale then to be used would be 4 square inches = 1 mile; and this is often a more convenient scale to adopt. NOTE C. (See Sec. 44.) Messrs. W. and L. E. Gurley in their Manual give the fol- lowing methods of adjusting the object-slide : To Adjust the Object-slide of a Transit." Hav- ing set up and levelled the instrument, the line of collimation being also adjusted for objects from three hundred to five APPENDIX. 247 hundred feet distant, clamp the plates securely, and fix the vertical cross-wire upon an object as distant as may be dis- tinctly seen ; then, without disturbing the instrument, throw out the object-glass, so as to bring the vertical wire upon an object as near as the range of the telescope will allow. Hav- ing this clearly in mind, unclamp the limb, turn the instru- ment half-way around, reverse the eye-end of the telescope, clamp the limb, and with the tangent-screw bring the vertical wire again upon the near object ; then draw in the object-glass slide until the distant object first sighted upon is brought into distinct vision. If the vertical wire strikes the same line as at first, the slide is correct for both near and remote objects ; and, being itself straight, for all distances. " But if there be an error, proceed as follows: First, with the thumb and forefinger twist off the thin brass tube that covers the screws. Next, with the screw-driver, turn the two screws on the opposite sides of the telescope, loosening one and tightening the other, so as apparently to increase the error, making, by estimation, one-half the correction required. "Then go over the usual adjustment of the line of collima- tion, and having it completed, repeat the operation above de- scribed ; first sighting upon the distant object, then finding a near one in line, and then reversing, making correction, etc., until the adjustment is complete." To Adjust the Object-slide of a Y-Level." The maker selects an object as distant as may be distinctly ob- served, and upon it adjusts the line of collimation, making the centre of the wires to revolve without passing either above or below the point or line assumed. "In this position, the slide will be drawn in nearly as far as the telescope-tube will allow. " He then, with the pinion-head, moves out the slide until an object, distant about ten or fifteen feet, is, brought clearly into view ; again revolving the telescope in the Y's, he observes whether the wires will reverse upon this second object. " Should this happen to be the case, he will assume that, as the line of collimation is in adjustment for these two dis- tances, it will be so for all intermediate ones, since the bear- ings of the slide are supposed to be true, and their planes parallel with each other. "If, however, as is most probable, either or both wires fail to 248 APPENDIX. reverse upon the second point, he must then, by estimation, remove half the error by the screws at right angles to the hair sought to be corrected, remembering, at the same time, that on account of the inverting property of the eye -piece he must move the slide in the direction which apparently in- creases the error. When both wires have thus been treated in succession, the line of collimatiou is adjusted on the near object, and the telescope again brought upon the most distant point ; here the tube is again revolved, the reversion of the wires upon the object once more tested, and the correction, if necessary, made in precisely the same manner. " He proceeds thus, until the wires will reverse upon both objects in succession ; the line of collimation will then be in adjustment at these and all intermediate points, and by bring- ing the screw-heads, in the course of the operation, to a firm bearing upon the washers beneath them, the adjustable ring will be fastened so as for many years to need no further ad- justment." " The centring of the eye-tube is performed after the wires have been adjusted, and is effected by moving the ring, by means of the screws shown on the outside of the tube, until the intersection of the wires is brought into the centre of the field of view." NOTE D. (See Sec. 57.) The time at which any elongation will occur may be found by the formula cos Ji cot (dec.) X tan (lat.), where Ti the hour-angle (see Sec. 182), h really being the supplement of the angle at P in the right-angled spherical triangle WZP (or EZP] in Fig. 10, the right angle beinir :l t WorE. The angle li may be reduced to mean time as shown in Part III. NOTE E. (See Sec. 57.) To find the azimuth of two stars when in the same vertical plane (Polaris being one of them) proceed as follows : Let A = the difference in R.A. of the stars, d = the declination of Polaris, and D the declination of the other star. APPENDIX. 249 Find p and m from the formulae cos A sin D tan m --, p = --- ; tan D cos m then find a from the formula cos a = p sin (d + w). Then Z, the azimuth, is given by sin A cos D cos d sin Z = - - . - , cos L sin a where L the latitude of the place. To find the interval of time which must elapse after the two stars are observed to be in the same vertical plane, before Polaris will be due north, find S from the equation . cos D sin 8= sin A - . sm a Then L4-d where h is the hour-angle in sidereal time. . To find the interval in mean time, see Sec. 179. The above steps may be easily traced by drawing the posi- tions of the star, the pole, and the zenith. It is not necessary to use Polaris ; but if any other star is selected, d refers to the star whose declination is the greater. NOTE F. (See Sec. 58.) The true value of the convergence is given by the equation . convergence . diff. of long. sin = sm 2- x sin (lat.). a . 4/ If the places are in different latitudes, as A and D in Fig. 12, we have the convergence = the difference in azimuth at 250 APPENDIX. A and D, which we can find by solving the spherical triangle AND. NOTE Gr. (See Sec. 189.) The difference in altitude in seconds of arc, between the meridian altitude and the maximum altitude of a body, is equal to d 2 where _ cos lat. cos dec. X 1.964 sin (lat. dec.) ' and d = the hourly change of declination in minutes of arc. When the declination differs in sign from the latitude, it will be negative. If the body has its declination changing towards the north in the northern hemisphere or towards the south in the southern hemisphere, the meridian altitude pre- cedes the maximum altitude, which will be the case between mid-winter and mid-summer ; but if changing towards the south in the northern hemisphere, or towards the north in the southern, the maximum altitude occurs to the east of the meridian. TABLES. TABLE I.-RADII. Deg. Radius. Deg. Radius. Deg. Radius. Deg. Radius. Deg. Radius. o o' Infinite 1 0' 5729.65 2 0' 2864.93 3 0' 1910.08 4 0' 1432.69 i 343775. 1 5635.72 1 2841.26 " 1 1899.53 1 1426^74 2 171887. 2 5544.83 2 2817.97 2 1889 09 2 1420 85 3 114592. 3 5456.82 3 2795.06 3 187S.77 3 1415.01 4 85943.7 4 5371.56 4 2772.53 4 1868.56 4 1409.21 5 68?54.9 5 5288.92 5 2750.35 5 1858.47 5 1403 46 6 57295.8 6 5208.79 6 2728 52 6 1848.48 6 1397^76 7 49110.7 7 5131.05 7 2707.04 1838 59 1392 10 8 42971.8 8 5055.59 8 2685.89 8 1828.82 8 1386.49 9 38197.2 9 4982.33 9 2665.08 9 1819.14 9 1380.92 10 34377.5 10 4911.15 10 2644.58 10 1809.57 10 1375.40 11 31252.3 11 4841.98 11 2624.39 11 1800.10 11 1369.92 12 28647.8 12 4774.74 12 2604.51 12 1790.73 12 1364.49 13 26444.2 13 4709.33 18 2584.93 13 1781.45 13 1359.10 14 24555.4 14 4645.69 14 2565.65 14 1772 27 14 1353.75 15 22918.3 15 4583.75 15 2546.64 15 1763! 18 15 1348.45 16 21485.9 16 4523.44 16 2527.92 16 1754.19 16 1343.15 17 20222.1 17 4464. 70 17 2509.47 17 1745 26 17 1337 65 18 19098.6 18 4407.46 18 2491.29 18 1736.48 18 1332.77 19 18093.4 19 4351.67 19 2473.37 19 1727.75 19 1327 63 20 17188.8 20 4297.28 20 2455.70 20 1719.12 20 1322^53 21 16370 2 21 4244.23 21 2438.29 21 1710 56 21 1317.46 22 15626.1 22 4192.47 22 2421.12 22 1702.10 22 1312.43 23 14946.7 23 4141.96 23 2404.19 23 1093.72 23 1307.45 24 14323.6 24 4092.66' 24 2387.50 24 1685.42 24 1302.50 25 13751.0; 25 4044 51 1 25 2371.04 25 1677.20 25 1297.58 26 13222.1 26 3997.49 26 2354.80 26 1669.06 26 1292.71 27 12732.41 27 3951.54 27 2338.78 1661.00 27 1287.87 28 12277.7 28 3906.54 28 2322.98 28 1653.01 28 1283 07 29 11851.3 29 3862.74 29 2307.39 29 1645.11 29 1278 30 30 11459.2 30 3819.83 30 2292.01 30 1637.28 30 1273.57 31 11089 6 31 3777.85 31 2276.84 31 1629.52 31 1268.87 32 10743 Oi 32 3736.79 32 2261.86 32 1621.84 32 1204.21 33 10417.5 88 3696 61 33 2247.08 33 1614.22 33 1 259 . 5S 34 10111.1 34 3657.29 34 2232 49 34 1606.68 34 1254^98 35 9822.18 35 3618.80! 35 2218^09 35 1599.21 35 1250.42 36 9549.34 36 3581.10; 36 2203.87 36 1591.81 36 1245.89 37 9291.29 37 3544.19 37 2189.84 37 1584.48 37 1241.40 38 9046.75 38 3508. 02 ! 38 2175. 98 | 38 1577.21 38 1236.94 39 40 8814.78 8594.42 39 40 3472.59 3437.87 39 40 2162.30! 2148.79 39 40 1570.01 1562.88 39 40 1232.51 1228.11 41 8384.80 41 3403.83 41 2135.44 41 1555.81 41 1223 74 44 8185.16 42 3370.46! 42 2122.26 42 1548.80 42 1219.40 43 7994.81 43 3337. 74 i 43 2109.24 43 1541. SO 43 1215.30 44 7813.11 44 3305.65 44 2096.39 44 15:34.98 44 1210.82 45 7639.49 - 45 3274.17 45 2083.68 45 1528.16 45 1206.57 46 7473.42 46 3243.29 40 2071.13 46 1521.40 46 1202.36 47 7314.41 47 3212.98 47 2058.73! 47 1514.70 47 1198.17 48 7162.03 48 3183.23 48 2046.48] 48 1508.06 48 1194.01 49 7015.87 49 3154.03 49 2034.37 49 1501.48 49 1189.88 50 6875.55 50 3125.36 50 2022.41 50 1494.95 50 1185.78 51 6740.74 51 3097.20 51 2010.59; 51 1488.48 51 1181.71 52 6611.12 52 3069.551 52 1998.90 52 1482.07 52 1177.66 53 6486.38 53 3042.39 58 1987. 35 1 53 1475.71- 53. 1173.65 54 6366.26 54 3015.71 54 1975.93 54 1101). 11 54 1169.66 55 6250.51 55 2989.48 55 1964.64 55 1463.16 55 1105.70 56 6138.90 56 2963. 71 ! 56 1953.48 56 1456.96 56 1161.76 57 6031.20 57 2938. 39 i 57 1942.44! 57 1450.81 57 1157.85 58 5927.22 58 2913.49 58 1931.53 58 1444.72 58 1153.97 59 5826.76! 59 2889.01 59 1920.75 59 1438.68 59 1150.11 60 5729.65 60 2864 ..93 60 1910.08 60 1432.69 60 1146.28 252 TABLE I. -RADII. Deg. adius. eg. [ Radius. eg. 1 Radius. eg. adius. | eg. adius. 5 0' 1-16.28 0' 955.366 0' 819.020) 0' 16.779 1 . 0' 37.275 1 142.47 1 952.722 1 817. 0^7 1 15. 391 ! 1 36.099 2 138.09 2 950.093 2 815.144 2 13.8101 2 34.928 3 134.94 3 947.478 3 813.238; 3 12.335 3 33.761 4 131.21 4 944.877 4 i 811.303 4 10.8651 4 32.599 5 127.50 5 i 942.291 5 809.397 5 09.402 5 31.440 6 123.82 6 939.719 6 807.499 6 07.945 6 30.286 120.16 7 937.161 7 805.611 7 06.493! 7 29.136 8 116.52 g 934. GIG 8 803.731 8 05.048 8 27.991 9 112 91 9 932.086 9 801.860 9 "03.609 9 626.849 10 109.33 10 929.569 10 799.997 10 "02.175 10 625.712 11 105.76 11 927.066 11 798.144 11 -00.748 11 624.579 12 102.2-.' 12 924 576 12 796.299 12 699.326 12 623.450 ; 13 098.70 13 922.100 13 794.462 13 697.910 13 622.325 14 095.20 14 919.637 14 792.634 14 696.499 14 621.203 15 091.73 15 917.187 15 790.814 15 195.095 15 620 087 16 088.28 16 914.750 16 789.003 16 693. 696 > 16 618.974 17 1084.85 912 326 17 787.210 17 692.302 17 617.865 18 1081.44 18 909.915 18 785.405 18 600.914 18 616.760 19 1078.05 19 907.517 19 783.618 19 689.532 19 615.660 20 1074.68 20 905.131 20 781.840 20 688.156 20 614.563 21 1071.34 21 902.758 21 780.069 21 686.785 21 613.470 22 1068.01 22 900.397 22 778.307 22 685.419 22 612.380 23 1064.71 23 898.048 23 776.552 23 684.059 23 611.295 24 1061.43 24 895.712 24 774.806 24 682.704! 24 610.214 25 1058.16 25 893.3HS 25 773.067 25 681.354 25 609.136 26 1054.92 26 891.076 26 771.336 26 680.010 26 608.062 27 1051.70 27 888.776 27 769.613 27 678.671 27 606.992 28 1048.48 28 886.488 28 767.897 28 677.338 28 605.926 29 1045.31 29 884.211 29 766.190 29 676.008 29 604.864 30 1042.14 30 881.946 30 764.489 30 674.686 30 603.805 31 1039.00 31 879.693 31 762.797 31 673.369 31 602.750 32 1035.87 32 877.451 32 761.112 32 672.056 32 601.698 33 1032.76 33 875.221 33 759.434 33 670.748 33 600.651 34 1029.67 34 873.002 34 757.764 34 669.446 34 599.607 35 1026.60 35 870.795 35 756.101 35 668.148 85 598.567 36 1023.55 36 868.598 36 754.445 36 666.856 36 597.530 37 1020.51 37 866.412 37 752.796 37 665.568 37 596.497 38 1017.49 38 864.238 38 751.155 38 -664.286 38 595.467 39 1014.50 39 862.075 39 749.521 39 663.008 39 594.441 40 1011.51 40 859.922 30 747.894 40 661.736 40 593.419 41 1008.5. 41 857.780 41 746.274 41 660.468 41 592.400 43 1005 ft 42 855.648 42 744.661 42 659.205 42 591.384 43 1002.6 43 853.527 43 743.055 4: 657.947 43 590.372 44 999.76 44 851.417 44 741.45 44 656.694 44 589.364 45 996.86 45 849.31 45 739.86 45 655.446 45 588.359 46 993.98 46 847.22 46 738.27 46 654.202 46 587.357 47 991.12 47 845.14 47 736.70 4" 652.963 47 586.359 48 988.28 48 843.08 48 735.12 48 651.729 48 585.364 49 985.45 49 841.02 49 733.56 49 650.499 49 584.373 50 982.638 50 838.97 50 732.00 50 649.274 50 583.385 5 979.84 51 836.93 51 730.45 5 648.05 5 582.400 5 977.06 52 834.90 52 728.90 5 646.83 5 581.419 53 974.29 53 832.88 53 7^7 .3 i 5 645.62 5 580 441 5 971.54 54 830.8" 54 725.83 5 644.42 5 579.466 5 968.81 55 828.8~ 55 724.31 5 643.21 578.494 5 966.00 56 826.88 56 722.79 5 642.02 577.526 5' 963.38 57 824.90 57 721.28 5' 640.82 576.561 5 960.69 58 822.9 58 719.77 5 639.63 575.599 5 958.02 59 820.9 59 718.27 5 638.45 574.641 6 955.36 60 819.0 60 716.77 60 637.27 ( 573.686 TABLE I.-RADII. Deg. Radius. Deg. Radius. Deg. Radius. Deg. Radius. Deg. Radius. 10 0' 573. C86 12 0' 478. a39 14 0' 410.275 16 0' 359.265 18 0' 319.623 2 4 571.784: 569.896 2 4 477.018 475.705 2 4 409.306 40H.341 4 358.523 357.784 2 4 319 037 318.463 6 568. (KO 6 474.400 6 407.380 6 357.048 6 317.871 8 566.156 8 473.102 8 406.424 8 356.315 8 317.292 10 564.305 10 471.810 10 405.473 10 355.585 10 316.715 12 562.466 12 470.526 12 404.526 12 354.859 12 316.139 14 560.638 14 469 249 14 403.583 14 354.135 14 315.566 16 558.823 16 467.978 16 402.645 16 353.414 16 314.993 18 557.019 18 466.715 18 401 . 712 18 352.696 18 314.426 20 555.227 20 465.459 20 400.782 20 351.981 20 313.860 22 553.447 22 464.209 22 399.857 22 351.269 22 313.295 24 551.678 24 462.966 24 398.937 24 350 560 24 312.732 26 549.920 26 461 . 729 26 398.020 26 349.854 26 312.172 28 548.174 28 460.500 28 397.108 28 349.150 28 311 613 30 546.438 30 459.276 30 396.200 30 348.450 30 311.056 32 544.714 32 458.060 32 395.296 32 347.752 32 310.502 34 543.001 34 456.850 34 394.396 34 347.057 34 309.949 36 541.298 36 455.646 36 393.501 36 346.365 36 309.399 38 539.606 38 454.449 38 392.609; 38 345.676 38 308.850 40 537.924 40 453.259 40 391.722 40 344.990 40 308.303 42 536.253 42 452.073 42 390. 838 i 42 344.306 42 307.759 44 534.593 44 450. 894 j 44 389.959 44 343.625 44 807.216 46 532.943 46 449.722! 46 389.084: 46 342.947 46 306.675 48 531.303 48 448.556; 48 388.212 48 342.271 48 306.136 50 529.673 50 447.395 50 387.345 50 341.598 50 305.599 52 528.053 52 446.241 52 386.481 52 340.928 52 305.064 54 526 443 54 445.093 54 385.621 54 340.2(50 54 304.531 56 524.843 56 443.951 56 384.765 56 339.595 56 304.000 58 523.252 58 442.814 58 383.913 58 338.933 58 303.470 11 0' 521.671 13 0' 441.684 15 0' 383.065 17 0' 338.273 19 0' 302.943 2 520.100 2 440.559: 2 382.220: 2 337.616 2 302.417 4 518.539 4 439 440: 4 381.380 4 336.962 4 301.893 6 516.986 6 488. 326 | 6 380.543 6 336.310 6 301.371 8 515.443 8 437.219 8 379.709 8 335.660 8 300.851 10 513.909 10 436.117 10 378.880i 10 335.013 10 300.3:33 12 512.385 12 435.020 12 378.054! 12 334.369 12 299.816 14 510.869 14 433.929 14 377.231! 14 333.727 14 299.302 16 509.363 16 432.844 16 376.412! 16 333.088 16 298.789 18 507.865 18 431.764 18 375.597 18 332.451 18 298.278 20 506.376 20 430.690 20 374.786 20 331.816 20 297.768 22 504.896 22 429.620 22 373. 977 j 22 331.184 22 297.260 24 503.425 24 428.557 24 373.173 24 330.555 24 296.755 26 501.96'J 26 427.498 26 372.372 26 329.928 26 296.250 28 500.507 28 426.145 28 371.574 28 329 303 28 295.748 30 499.061 30 425.396 30 370.780 30 328.689! 30 295.247 32 497.624 32 424.354 32 369.989 32 328.061 32 294.748 34 496.195 34 423.316 34 369.2021 34 327.443 34 294.251 36 494.774 36 422.283 36 368.418 36 326.828 36 293.756 38 493.361 38 421.256 38 367.637 38 326.215 38 293.262 40 491.956 40 420.233 40 366.859 40 325.604 40 292.770 42 490.559 42 419.215 42 56. 085 42 324.996 42 292.279 44 489.171 44 418.203 44 365.315 44 324 390 44 291.790 46 487.790 46 417.195 46 3(51.547 46 323.786 46 291.303 48 486.417 48 416.192 48 3f,3.783 48 323.184 48 290.818 50 485 051 50 415.194 50 363.022 50 3-22. 585 50 290.331 52 483.694 52 414.201 52 362.264 52 3-21.989 52 289.851 54 482.344 54 413.212 54 361.510 54 321.394 54 289.371 56 481 001 56 412.229 56 3(50.758 56 320.801 56 288.892 58 479.666 58 411.250 58 360.010 58 320.211 58 288.414 60 478.339 60 410.275 60 359.265 60 319.623 60 287.939 254 TABLE II. TANGENTS AND EXTERNALS TO A 1 CURVE. Angle. I. Tan- gent. T. Exter- nal. E. Angle. I. Tan- gent. T. Exter- nal. E. Angle. I. Tan- gent. T. Exter- nal. E. 1 50.00 .218 11 551.70 26.500 21 1061.9 97.577 10' 58.34 .297 10' 560.11 27.313 10' 1070.6 99.155 20 60.67 .388 20 568.53 28.137 20 1079.2 100.75 30 75.01 .491 30 576.95 28.974 30 1087.8 102.35 40 83.34 .606 40 585.36 29.824 40 1096.4 103.97 50 91.68 .733 50 593.79 30.686 50 1105.1 105.00 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 20 110.08 1.188 20 619.07 33.347 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 40 635.93 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 37.070 23 1165.7 117.38 10 158.38 2.188 10 661.25 38.031 10 1174.4 119.12 20 166.72 2.425 20 669.70 39.006 20 1183.1 120.87 30 175.06 2.674 30 678.15 39.993 30 1191.8 122.63 40 183.40 2.934 40 686.6,) 40.992 40 1200.5 124.41 50 191.74 3.207 50 695.06 42.004 50 1209.2 126.20 4 200.08 3.492 14 703.51 43.029 24 1217.9 128.00 10 208.43 3.790 10 711.97 44.066 10 1226.6 129.82 20 216.77 4.099 20 720.44 45.116 20 1235.3 131.65 30 225.12 4.421 30 728.90 46.178 30 1244.0 133.50 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 25 127'0.2 139.11 10 258.51 5.829 10 762.80 50.554 10 1279.0 141.01 20 266.86 6.211 20 771.9 51.679 20 1287.7 142.93 30 275.21 6.606 30 779.77 52.818 30 1290.5 144.85 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 1340.4 154.09 30 325.35 9.230 30 830.76 59.914 30 1349.2 156.70 40 333.71 9.710 40 &S9.27 61.141 40 1358.0 158.72 50 342.08 10.202 50 847.78 62.381 50 1366.8 160.76 7 350.44 10.707 17 856.30 63.634 27 1375.6 162.81 10 358.81 11.224 10 864.82 64.900 10 1384.4 164.86 20 367.17 11.753 20 873.35 66.178 20 1393.2 166.95 30 375.54 12.294 30 881.88 67.470 30 1402.0 169.04 40 383.91 12.847 40 890.41 68.774 40 1410.9 171.15 50 392.28 13.413 50 898.95 70.091 50 1419 7 173.27 8 400.66 13.991 18 907.49 71.421 28 1428.6 175.41 10 409.03 14.582 10 916.03 72.764 10 1437.4 177.55 20 417.41 15.184 20 924.58 74.119 20 1440.3 179.72 30 425.79 15.799 30 933.13 75.483 30 1455.1 181.89 40 434.17 16.426 40 941.69 76.869 40 1464 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 10 967.38 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 83.972 30 1508.5 195 25 40 484.49 20.447 40 993.12 85.431 40 1517 4 197.53 50 492.88 21.161 50 1001.7 86.904 50 1526.3 199 82 10 501.28 21.887 20 1010.3 88.389 30 1535 3 202.12 10 509.68 22.624 10 1018.9 89.888 10 1544.2 204.44 20 518.08 23.375 20 1027.5 91.399 20 1553.1 206.77 30 526.48 21.138 30 1036.1 92.924 30 1562.1 209 12 40 534.89 24.913 40 1044.7 94.462 40 1571 211 48 50 543.29 25.700 50 1053.3 96.013 50 1580.0 213.86 255 TABLE II. TANGENTS AND EXTERNALS TO A 1 CURVE. j Angle. Tan- | gent. B ST 1** Tan- gent. Exter- nal. Angle. Tan- Exter- gent. nal. I. T. E. I. T. E. I. T. E. 31 15H9.0 216.25 41 2142.2 387.38 51 2732.9 618.39 10 1598.0 818*66 ! 10' 2151.7 390.71 10' 2743.1 622.81 20 1606.9 221.08 1 20 2161.2 394.06 20 2753 4 627.24 30 1615.9 223.51 1 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 fU<) (5(5 32 1643.0 230.90 42 2199.4 407.64 52 2794.5 615 17 10 1652.0 233.39 10 2209.0 411.07 . 10 2804.9 649.70 20 1661.0 235.90 20 2218.6 414.52 20 2815.2 654.25 30 1670.0 238.43 i 30 2228.1 417.99 30 2825 6 658 83 40 1679.1 240.96 40 ' 2237.7 421.48 I! 40 2835.9 663.42 50 1688.1 243.52 50 ! 2247.3 424.98 50 2846.3 668.03 33 1697.2 246.08 ' 43 2257.0 428.50 53 2856.7 672. 66 10 1706.3 248.66 10 2266.6 432.04 10 2867.1 677.32 20 1715.3 251.26 20 8276.2 435.59 i 20 2877.5 681.99 30 1724.4 253.87 30 2285. !) 439.16 i! 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 2805! 2 446.35 50 i 2908 9 696.13 34 1751.7 261.80 44 2314. 9 449.98 54 i 2919.4 700.89 10 1760.8 264.47 ' 10 2324.6 453.62 10 ! 29..".) 9 705 6(5 20 1770.0 fli7.ll> 20 2334.3 457.27 i 20 i 2940.4 710.46 30 1779.1 26!). 8(5 30 2344.1 460.95 30 i 2951.0 715.28 40 1788.2 272.58 40 2353.8 464.64 40 i 2961.5 720.11 50 1797.4 275.31 ; 50 2363.5 468.35 50 2972.1 724 . 97' 35 1806.6 27'8.05 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 1834.1 286.39 30 2402.6 483.37 30 1 3014.5 744.62 40 1843.3 289.20 40 2412.4 i 487.17 , ! 40 3025.2 749.59 50 1852.5 292.02 50 2422.3 490.98 i 50 3035.8 754.57 36 1861.7 294.86 46 2432.1 494.82 56 3046.5 759.58 10 1870.9 297.72 10 2441.9 498.67 10 3057.2 764.61 ^20 1880.1 300.59 20 2451.8 502.54 20 3067.9 769 66 30 1889.4 303.47 30 2461.7 1 506.42 30 3078.7 I 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 j 795.24 20 1935.7 318.13 ! 20 2511.2 526.13 ' 20 i 3132.6 800.42 30 1945.0 321.11 30 2521.1 5:30.13 ! 30 3143.4 1 805.62 40 1954.3 324.11 40 2531 . 1 534.15 i 40 j 3154.2 '> 810.85 50 1963.6 327.12 50 2541.0 538.18 50 3165.1 816 10 38 1972.9 330.15 48 2551.0 542.23 58 i 3176.0 821.37 10 1982.2 333.19 ! 10 2561.0 546.30 ! 10 1 3186.9 826. G6 20 1991.5 886.25 20 2571.0 550.39 20 3197.8 831.98 30 2000.9 339.32 30 2581.0 554.50 30 3208.8 j 837.31 40 2010.2 342.41 40 2591.1 i 558.63 40 3219.7 842.67' 60 2019.6 345.52 50 2601.1 ! 562.77 i 50 3230.7 848.06 39 2029.0 348.64 i 49 2611.2 566.94 '. 59 3241.7 ' 853.46 10 2038.4 351.78 ! 10 2621.2 571.12 10 | 3252.7 858.89 20 2047.8 354.94 1 20 2631.3 575.32 | 20 ! 3263.7 864.34 30 2057.2 358. .11 30 2641.4 579.54 30 ' 3274.8 '. 869.82 40 2066.6 361.29 40 2651.5 583.78 1 40 3285.8 875.32 50 2076.0 364.50 50 2661.6 588.04 50 3296.9 880. HI 40 2085.4 367.72 50 2671.8 592.32 60 SJ08.0 886.38 10 2094.9 370.! IT) 10 2681.9 , 596.62 10 ! 3319.1 891.95 20 2104.3 374.20 . 20 2692.1 ! 600.93 20 8330.3 897.54 30 2113.8 377.47 30 2702.3 | 605.27 30 i 3341.4 903.15 40 2123.3 380.76 40 2712.5 609.62 40 3352.6 908.79 50 2132.7 384.06 50 2722.7 614.00 i 50 i 3363.8 914.45 256 TABLE II. TANGENTS AND EXTERNALS TO A 1 CURVE. Angle. Tan- gent. Exter- nal. Angle. Tan- gent. Exter- nal. Angle. Tan- gent. Exter- nal. I. T. E. I. T. E. T. T. 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 4937.0 1833.6 40 3420.1 ! 943.12 40 4137.4 1337.7 40 4951.5 1843.1 30 3431.4 i 948.92 50 4150.1 1345.1 50 i 4966.1 1852.6 62 3442.7 954.75 72 4162.8 1352.6 82 ; 4980.7 1862.2 10 3454.1 960.60 10 4175.6 '' 1360.1 10 i 4995.4 1871.8 20 3405.4 966.48 20 4188.5 ! 1367.6 20 5010.0 1881.5 30 3476.8 972.38 : 30 4201.2 1375.2 30 5024.8 1891.2 40 3488.3 978.31 40 4214.0 1382.8 40 5039.5 1900.9 50 3499.7 984.27 : 50 4226.8 : 1390.4 50 5054.3 1910.7 63 3511.1 990.24 73 4239.7 1398.0 83 5069.2 1920.5 10 | 3522. (5 996.24 10 4252.6 1405.7 10 5084.0 1930.4 20 ! 3534.1 1002.3 20 4265.6 1413.5 20 5099.0 1940.3 30 a545.6 1008.3 30 4278.5 1421.2 30 5113.9 1950.3 40 3557.2 1014.4 40 4291.5 ! 1429.0 40 i 5128.9 1960.2 50 3568.7 1020.5 ' 50 4304.6 1436.8 50 5143.9 1970.3 64 3580.3 1026.6 i 74 4317.6 1444.6 84 i 5159.0 1980.4 10 3691.9 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 I 2000.6 30 3615.1 1045.2 30 4356.9 1468.4 30 ' 5204.4 2010.8 40 I 3626.8 1051.4 4jO 4370.1 1476.4 40 5219.7 2021.1 50 3638.5 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 1300.5 10 5265.6 2052.1 20 ! 3673.7 1076.6 20 4423.1 1508.6 20 5281.0 2062.5 30 3685.4 1082.9 30 1 4436.4 1516.7 30 i 5296.4 i 2073.0 40 3697.2 1089.3 40 i 4449.7 1524.9 40 5311.9 2083.5 50 3709.0 1095.7 50 4463.1 1533.1 50 5327.4 2094.1 66 3720.9 1102.2 76 4476.5 1541.4 86 5343.0 ! 2104.7 10 3732.7 1108.6 i 10 4489.9 1549.7 ~ 10 5358.6 ; 2115.3 20 3744.6 1115.1 20 4503.4 1558.0 20 5374.2 2126.0 30 3756.5 1121.7 ! 30 1 4516.9 1566.3 30 5389.9 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 - 50 5421.4 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 1 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 i 3889.0 1195.2 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 2308.5 10 3950.2 1229.7 10 4737.2 1704.7 10 5646.9 2315.0 20 3962.5 1236.7 20 4751.2 1713.7 20 5663.4 2326.6 30 3974.8 1243.7 30 4765.3 1722.7 30 5679.9 2338.2 40 3987.2 1250 8 40 4779.4 1731.7 40 5696 4 2349.8 50 3999.5 1257.9 50 4793.6 1740.8 50 5713.0 2361.5 70 4011.9 1265.0 80 4807.7 1749.9 90 5729.7 2373.3 10 ; 4024.4 | 1272.1 ! 10 4822.0 1759.0 10 5746.3 2385.1 20 ' 4036.8 i 1279.3 20 4836.2 1768 2 20 5763.1 : 2397.0 30 i 4049.3 1286.5 i 80 4850.5 I 1777.4 i 30 j 5779 9 ' 2408.9 40 4061.8 1293.6 40 i 4864.8 1786.7 40 5796.7 2420.9 50 1 4074.4 1300.9 50 4879.2 1796.0 50 5813.6 2432.9 257 TABLE II. TANGENTS AND EXTERNALS TO A 1 CURVE. Angle. Tan- gent. Ex- ternal. Angle. Tan- gent. teSfal. i| Angle. Tan- gent. Ex- ternal I. T. E. I. T. E. *' T. E. 91 5830.5 2444.9 97 6476.2 ' 2917.3 103 7203 .2 3474.4 10' 5847.5 245" r.i 1 64 95.2 2!)31 (5 10 7224 _7 34 1)1.3 20 5864.6 2469.3 20 6514.3 2945.9 20 7246.3 3508.2 ao 5881.7 i 2481 .5 c 10 o.: 33.4 2960 3 30 7268 .0 33 25.2 40 5898.8 ! 249: 5.8 < K) 65 52.6 2974 40 728? .8 8! 42.4 50 5916.0 i 250t 5.1 ti Gi 71.9 2989 2 50 7811 _ / 3T 59.6 92 5933.2 2518.5 98 6591.2 3(X)3 8 104 7838 .6 3576.8 10 5950.5 2531.0 10 6610.6 3018.4 10 7855 .15 3594.2 20 5967.9 i 254: 5.5 s !0 ! 01 .80.1 3033 1 20 7377 .8 31 11.7 80 5985.3 255( 5.0 I 50 61 49.6 i 3047 i) 30 7399 9 :',( 29.2 40 6002.7 256* 4 6b 69.2 i 3062 8 40 7422 .2 .'if >46.8 50 6020.2 2581.3 50 6688.8 3077 7 50 7444 .6 3664.5 93 6037.8 2594.0 99 6708.6 3092.7 105 746! .0 a >82.3 10 6055.4 260( 5.8 1 67 28.4 3107 7 i 10 7489 .0 V 100.2 20 6073.1 261< ).7 $ 67 48.2 ! 3122 9 20 7512 .2 3' "18.2 30 6090.8 2632.6 30 6768.1 ! 3138 1 30 7534.9 3736.2 40 6108.6 264; 5.5 - 07 88.1 i 3153 3 40 7557 .7 Of '54.4 50 6126.4 265* 5.5 6 ,08.2 ! 3168 50 7580 .5 3' 72.0 94 6144.3 267 1.6 100 6828.3 3184 1 106 7603.5 3791.0 10 6162.2 2(58- L7 ] 68 48.5 i 3199 6 10 7620 .0 3* 909.4 20 6180.2 269 ".9 0> 168.8 l 3215 1 ; 20 7649 .7 3! 127.9 30 6198.3 I 2711.2 30 68 Sit. 2 32:30.8 30 7672.9 3846.5 40 6216.4 j 2724.5 40 6909.6 3240.5 40 70%. 3 3865.2 50 6234.6 | 2737.9 50 6930.1 3262.3 50 7719.7 3884.0 95 6252.8 2751.3 101 6 50.6 3278.1 107 7743.2 3902.9 10 6271.1 276- 1.8 ] 0' 6t 71.3 3294 1 10 7700 .8 3 J21.9 20 6289.4 2778.3 J 50 6992.0 3310.1 20 7790.5 3940.9 30 6307.9 279S J.O ( n 7C 12.7 3326 1 30 7814 .3 3 >60.1 40 6326.3 2805.6 40 7033.6 3342 3 40 7838.1 3979.4 50 6:344.8 281 ( ).4 j >o 7C 54.5 3358 5 50 7'80^ .1 3 398.7 96 6363.4 283, J.2 102 ^ 7075.5 3374 9 108 7886.2 4018.2 10 6382.1 ! 284 r.o 1 7( 96.6 3391 2 10 791C .4 4) J37.8 20 6400.8 286 l.O 71 17.8 i 3407 7 20 79:34 .(i 4 357.4 30 6419.5 2875.0 80 7139.0 ! 3424 8 30 7839.0 4077.2 40 6438.4 288 ).0 i 7] 60.3 3440 9 40 7983 .5 4 D97.1 50 6457.3 2903.1 50 7181.7 3457.6 50 8008,0 4117.0 CORRECTIONS FOR TANGENTS AND EXTERNALS. FOR TANGENTS, ADD FOR EXTERNALS, ADD Ang 5 10 15 20 25 30 Ang 5 10 15 20 25 30 I- Cur. Cur. Cur. 1 Cui . Cur. Cur. I ! Cur. Cur. Cur. Cur. Cur. Cur. 10 .03 .06 .09 .] J .16 i .19 10 001 .003 .004 ~T006~ .007 .008 20 .06 .13 .19 5 .32 .39 20 .006 .011 .017 .022 .O 1 JS .034 30 .10 .19 .29 '.3 ) .49 .59 30 .013 .025 .0381 .051 ;.-> .078 40 .13 .26 .40 .5, i .67 80 40 .023 .046 .0701 .093 '.117 .141 50 .17 .34 .51 .6i J .85 1 02 ! 50 .037 .075 .116 .151 .1 s<) .227 60 .21 .42 .63 .84 1.05 1.27 60 .056 .112 .168 .225 .2 S3 .340 70 .25 .51 .76 1.02 1.28 1.54 70 .oso .159 .240 .321 .4 ).", .485 80 .30 .61 .91 1.2211.53 1.84 80 .110 .220 .332! .445 .558 .071 90 .36 .72 1.09 1.45 1.83 2.20 90 .149 .299 .450 .603 .7 56 .910 100 .43 .86 1.30 1.74 2 is 2.62 100 .200 .401 .604 .809 1.0 15 1.221 110 .51 1.03 1.56 2 08 2.61 3.14 110 .268 .536 .806 1.082 1.3 55 1.683 120 .62 1.25 1.93 2.5 2 3 16 3.81 120 ,300 .72111.086 1.456 1.825 2.197 258 TABLE III.-TANGENTIAL OFFSETS 100 FT. ALONG THE CURVE. Deg. of 0' 10' 20' 30' 40' 50' Curve. 0.000 0.145 0.291 * 136 0.582 0.727 1 0.873 1.018 1.164 lii $09 1.454 - .600 2 1.745 1.89 1 2.036 2. 81 2.327 S .472 3 2.618 2.763 2.908 3.054 3.199 3.345 4 3.490 3.03 3.781 3. )26 4.071 4 .217 5 4.362 4.507 4.653 4 f -98 4.943 5.088 6 5.234 5.37 ) 5.524 5. 369 5.814 .960 6.105 6.250 6.395 6.540 6.685 ( .831 8 6.976 7.12 I 7.266 7. 111 7.556 r .701 9 7.846 7.99 I 8.136 8 ' >81 8.426 e t.57i 10 8.716 8.86 } 9.005 9J50 9.295 c .440 11 9.585 9.729 9.874 10.019 10.164 10.308 12 10.453 10.59 ~ 10.742 10. 387 11.031 1] .176 13 11.320 11.465 11.609 11.754 11.89S 12.043 14 12.187 12.33 1 12.476 12. 520 12.764 IS .908 15 13.053 13.19 13.341 13. 185 13.62E U 1.773 16 13.917 14.061 14.205 14.349 14.49S 14.637 17 14.781 14.92 5 15.069 15. >12 15.356 If ..500 18 15.643 15.787 15.931 16.074 16.218 16.361 19 16.505 16.648 16.792 16. ?35 17.078 17.222 20 17.365 17.50 3 17.651 17. 17.937 1 5.081 21 18.224 18.36 r 18.509 18. 552 18.795 11 i.938 22 19.081 19.224 19.366 19.509 19.652 >.794 23 19.937 20.07 i 20.222 20. 364 20.507 2( ).649 24 20.791 20.933 21.076' 21.218 21.36C 21.502 TABLE IV. MID-ORDINATES TO A 100-FT. CHORD. Deg. of 1 2 3 4 5 6 7 ' 8 9 Curve. 0.000 0.21* 5 0.436 0.655 0.873 1.091 1.309 1.528 1.746 1.965 10 2.183 2.405 > 2.620 2.839 3.058 3.277 3.496 3.716 3.935 4.155 20 4.374 4.59' I 4.814 5.035 5.255 5.476 5.697 5.918 6.139 6.360 259 TABLE V.-LONG CHORDS. Degree Actual Arc, LONG CHORDS. of Curve. One Station. Stations. 3 Stations. 4 | 5 Stations. | Stations. 6 Stations. 010' 100.000 200.000 299.999 399,998 499.996 599.993 20 .000 199.999 1 299.997 399.992 499.983 599.970 30 .000 199.998 : 299.992 399.981 499.962 599.9:33 40 .001 199.997 299.986 399.966 ; 499.932 599.882 i 50 .001 199. 095 299.979 399.947 i 499.894 599.815 1 100.001 199.992 299.970 399.924 499.848 599.733 10 .002 199.990 ! 299.959 399.896 499.793 j 599.637 20 .002 199.986 299.946 1 399 .'865 499.729 i 599.526 30 .003 199.983 299.932 399.829 499.657 599.401 40 .003 199.979 299.915 399.789 499.577 599.260 50 .004 199.974 299.898 399.744 499.488 599.105 2 100.005 199.970 299.878 399.695 499.391 598.934 10 .006 199.964 299.857 399 . 643 499.285 59H.750 20 .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 .010 199.939 299.756 399.389 498.778 597.862 3 100.011 199.931 299.726 399.315 ! 498.630 597.604 10 .013 199.924 299.695 399.237 498.474 I 597.331 20 .014 199.915 299.662 399.154 498.309 597.043 30 .015 199.907 1" 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 M0.528 398.782 497.566 595.744 10 .022 199.868 299.471 398.679 497.360 595.383 20 .024 199.857 299.428 398.571 497.145 595.007 30 .026 199.846 299.383 398.459 496.921 594.617 40 .028 199.834 299.337 398.343 496.689 594.212 50 .030 199.822 299.289 398.223 496.449 593.792 5 100.032 199.810 299.239 398.099 496.201 593.358 10 .034 199.797 299.187 397.970 495.944 592.909 20 .036 199.783 299.134 397.837 495.678 592.446 30 .038 199.770 299.079 397.700 495.405 591.968 40 .041 199.756 299.023 397.559 495.123 591.476 50 .043 199.741 298.964 397.413 494.832 590.970 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 7 100.062 199.627 298.509 396.273 492.568 587.021 10 .065 199.609 298.438 396.099 492.212 586.400 20 .068 199.591 298.364 395.916 491.847 685.766 30 .071 199.572 298.289 395.729 491.474 585.115 40 .075 199.553 298.212 395.538 491.093 584.451 50 .078 199.533 298.134 395.342 490.704 583.773 8 100.081 199.513 298.054 395.142 490.306 sas.ofii 10 .085 199.492 297.972 394.938 489.900 582.375 20 .088 199.471 297.888 394.731 489.486 581.654 30 .092 199.450 297.803 394.518 489.064 580.920 40 .095 199.428 297.716 394.302 488.634 580.172 50 .099 199.406 297.628 394.082 488.196 579.409 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 .111 199.887 297.352 393.396 486.832 577.039 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 29G.962 392.424 484.900 573.686 TABLE V. LONG CHORDS. Degree of Curve. Actual Arc, One Station. LONG CHORDS. 2 Stations. 3 Stations. 4 Stations. 5 Stations. 6 Stations. 1010'i 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 i 199.161 296.651 391.652 483.367 571.027 40 .145 199.1:34 296.544 391.387 482.840 570.113 50 j .149 199.107 j 296.436 391.117 482.305 569.186 11 ! 100.154 199.079 i 296.325 390.843 481.762 568.245 10 .158 i 199.051 296.214 390.565 481.211 567.292 20 .163 j 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 i .193 198.843 295.384 388.508 477.135 560.240 30 1 .199 198.811 295.259 388.197 476.521 559.180 40 ! .204 ! 198.779 295.132 387.883 475.899 558.107 50 ! .209 198.747 295.004 387.565 475.270 557.020 13 100.215 198.714 294.874 387.243 474.633 555.921 10 .220 198.681 294.742 386.916 473.988 554.809 20 .226 198.648 294.609 386.586 473.336 553.684 30 .232 198.614 294.474 386.252 472.675 552.546 40 ! .237 198.579 294.337 385.914 472.007 551.395 50 .243 198.544 294.199 385.572 471.332 550.232 14 100.249 i 198.509 294.059 385.225 470.649 549.056 10 .255 198.474 293.918 384.875 469.958 547.867 20 .261 198.437 293.774 384.521 469.260 546.666 30 267 ! 198.401 293.629 384.163 468.554 545.452 40 274 198.364 293.483 383.801 467.840 544.226 50 280 198.327 293.335 383.435 467.119 542.987 15 100.286 1 198.289 293.185 383.065 466.390 541.736 10 .292 198.251 293.034 382.691 465.654 540.472 20 .299 198.212 292.881 382.313 464.911 ! 539.196 30 .306 198.173 293.786 381.931 464.160 j 537.908 40 1 .312 198.134 292.570 381.546 463.401 536.608 50 .319 198.094 292.412 381.156 462.635 535.296 16; 100.326 i 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 17 100.368 197.803 291.261 378.320 457.069 525.776 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 37'6.179 452.875 518.625 18 100.412 197.538 290.211 375.739 452.015 517.160 10 .420 197.492 290.031 375.295 451.147 515.685 20 .428 197.446 289.849 374.848 450.373 514.198 30 .436 197.399 289.605 374.397 449.392 512.699 40 .444 197.352 289.479 373.942 448.504 511.190 , 50 .452 197.305 289.292 873.483 447.608 509.670 19 100.460 197.256 289.104 373.021 446.7'06 508.139 10 .468 197.209 288.913 372.554 445.797 506.597 20 .476 197.160 288.722 372.084 444.881 505.043 30 .484 lOr.lll 288.528 371.610 443.957 503.479 40 .493 197.082 288.333 371.133 I 443.028 501.905 ... 50 .501 197.012 i 288.137 370.652 442.091 500.320 20 100.510 196.962 287.939 370.167 441.147 498.724 261 TABLE VI.-MID-ORDINATES TO LONG CHORDS. Degree of Curve. 1 Station. g Stations. 3 Stations. 4 Stations. 5 Stations. 6 Stations. o i9 262 TABLE VI.-MID-ORDINATES TO LONG CHORDS. Degree of Curve. 1 Station. 2 Stations. 3 Stations. 4 Stations. 6 Stations. 6 Stations. 10 10' 2.219 8.860 19.870 35.164 54.619 78.083 20 2.256 9.005 20.193 35.729 55.486 79.305 30 2.293 9.150 20.516 36.294 i 56.353 80.523 40 2.329 9.295 20.838 36.859 57.218 81.739 50 2.365 9.440 21 . 160 37.423 58.081 82.951 11 2.402 i 9.585 21.483 37.986 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 60.663 86.571 30 2.511 10.019 22.448 39.673 61.521 87.772 40 2.547 10.164 22.769 40.234 62.377 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 23.7-33 41.914 64.937 92.542 20 2.693 10.742 24.053 42.473 ' 65.787 93.727 30 2.730 10.887 24.37'4 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 11.465 25.654 45.256 70.013 99.598 20 2.912 11.609 25.974 45.811 70.854 100.762 30 2.949 11.754 26.293 46.365 71.692 101.922 40 2.985 11.898 26.612 46.919 72.529 103.079 50 3.022 12.043 26.931 47.472 73.364 104.232 14 3.058 12.187 27.250 48.024 74.197 105.381 10 3.095 ! 12.331 27.569 48.575 75.029 106.527 20 3.131 12.476 27.887 49.126 75.859 107.669 30 3.168 i 12.620 28.206 49.676 76.687 108.807 40 3.204 ! 12.764 28.524 50.225 77.513 109.941 50 3.241 12.908 28.841 50.77.3 78.337 111.071 15 3.277 13.053 29.159 51.321 79.159 112.197 10 3.314 13.197 29.476 51.868 79.979 113.319 20 3.350 13.341 29.794 52.414 80.798 114.438 30 3.387 13.485 30.111 52.959 81.614 115.552 40 3.423 13.629 30.427 53.504 : 82.429 116.662 50 3.460 13.773 30.744 54.048 | 83.241 117.768 16 3.496 13.917 31.060 54.591 84.052 118.870 10 3.533 14.061 31.376 55.133 84.861 119.967 20 3.569 14.205 31.692 55.675 i 85.667 121.061 30 3.606 14.319 32.008 56.215 86.471 122.150 .40 3.643 14.493 32.323 56.755 87.274 123.235 50 3.679 14.637 32.638 57.294 88.074 124.315 17 3. 716 14.781 32.953 57.832 i 88.872 125.391 10 3.752 14.925 as. 267 58.369 89.G68 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.E23 40 3.862 15.356 34.210 59.1)6 92.043 129.651 50 0.899 15.500 34.523 60.510 92.830 130.704 18 3.935 15.643 34.837 61.042 93.616 131.753 10 3.972 15.787 35.150 61.574 94.398 132.797 20 4.008 ! 15.931 35.463 62.106 95.179 133.837 30 4.045 16.074 35.775 62.636 95.957 134.872 40 4.081 16.218 36.088 63.165 96.733 135.902 50 4.118 16.361 36.400 63.693 97.506 136.928 19 4.155 16.505 36.712 64.221 98.278 137.948 10 4.191 16.648 37.023 64.747 99.047 138.964 20 4.228 16.792 37.334 65.273 99.813 139.975 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 4l 886 17.222 38.266 66.843 102.098 142.978 20 4.374 17.365 38.576 67.365 102.855 14,3.969 263 TABLE VII. -MINUTES IN DECIMALS OF A DEGREE. / 0' 10" 15' 20" 30" 40" 45" 50" / .00000 00278 .00417 .00556 .00833 .01111 .01860 .01389 1 .01(567 .01944 .03088 .02222 .02500 .02778 .02917 [08055 1 2 .03333 .03611 .03750 .03889 .04167 .04444 .04583 .04722 2 3 .05000 .05278 .05417 05556 .05833 .U0111 .00250 00389 3 4 .06(567 .06944 .07083 .07222 .07500 .07778 .07917 .08050 4 5 .08333 .08011 .08750 .08889 .09107 .09444 .09588 .09722 5 6 .10000 . 10278 .10417 .10556 .10833 .11111 .11860 .11889 (5 7 .11667 .11944 .12083 .12222 .12500 .12778 .12917 .13056 i 7 8 .13333 .13611 .18750 .13889 .14167 .14444 .14583 .14122 8 9 .15000 .15278 .15417 .15550 .15833 .10111 .10250 .10389 !) 10 .16667 .10944 .17083 .17222 .17500 .17778 .17917 . 18050 10 11 .18333 .18611 .18750 .18889 .19167 19444 .19.583 .19723 11 12 .20000 .20278 .20417 .30556 .20833 .21111 .21260 .2KJS!) !;> 13 .21667 .21944 .99081 [22228 .2251)0 .22778 .22917 .2:l( C><> 13 14 .23333 .23611 .88750 .23889 .24167 .94444 [34588 .24722 14 15 .25000 .25278 .25417 .25556 .25833 .26111 .26250 .2038!* 15 16 .26607 .20914 .27083 .27332 .37600 .27778 .27917 .28050 10 17 .283:33 .28611 .28750 .28889 .21)167 .29444 .29588 .29722 17 18 .30000 .30278 .30417 .30556 .30833 .31111 .31250 .3K38!) 18 19 .31067 .31944 .32083 .32232 .82500 .32778 .32917 ..V.050 19 20 .33333 .33011 .33750 .33889 .34167 .34444 .34583 .34722 20 21 .35000 .35278 .35417 .35556 .35833 .36111 .86860 .36389 21 22 .86667 .30944 .37083 .37222 .37500 .3777'8 .37917 3805(5 22 23 .38333 .38011 .38750 .38889 .89167 .39444 .39583 .397 22 2:5 24 .40000 .40278 .40417 .10556 .40833 .41111 .41250 .41389 21 25 .41667 .41944 .42083 .42232 .42500 .48778 .42917 .43056 25 26 .433:33 .43611 .43750 .438S9 .44167 .44444 .44588 .44722 20 27 .45000 .45278 .45417 .45550 .45833 .46111 .46250 ! .40389 27 28 .46667 .46944 .47083 .47328 .47500 .4777'8 .471)17 .-1S056 28 29 .48*33 .48611 .48750 .48889 .49167 .49441 .49588 .ID: 22 29 30 .50000 .50278 .50417 .5 36 37 .61667 .61944 .62083 .63233 .62500 .62778 .62917 .03050 37 38 .03333 .63611 .63750 .63889 .64167 .61111 .64583 .04722 38 39 .65000 .65278 .66417 .65556 .058:33 .60111 .66250 .00:3X5) 39 40 .66667 .66944 .67083 .67222 .67500 .07778 .67917 .68056 40 41 .68333 .68611 .68750 .68889 .69167 .69441 .69583 .69722 41 42 .70000 .70378 .70417 .70556 .71KU .71111 .71250 .7138!) 12 43 .71667 .71944 .720S-! .72222 .73300 178778 .7:2017 .73050 43 44 .73333 .73611 .73750 .73889 .74167 .74444 .7458-') .71722 44 45 .75000 .75278 .75417 .75556 .75833 .70111 .70250 .7'f : '3S!> 45 46 .7r,!>(;7 .7'09i4 .77083 [77223 .77500 .77778 .771117 .78066 46 47 .78333 .78611 .78750 .78889 .79167 .79444 .79588 [79728 47 48 .80000 .80378 .80417 180556 180888 .81111 .S 1250 .81:38!) is 49 .81667 .S19U .82083 .83333 .83500 .88778 [82917 .S3056 -4!) 50 .83333 .83611 .83760 .83889 .84167 .SI 111 [84688 .81722 50 51 .85000 .85278 .85417 .85556 .85833 86111 .80250 .86869 51 52 .80607 .86044 .87083 .87222 [87500 .87778 .87917 .88056 68 53 .88333 .88611 .887'50 .88889 .89167 .89444 89588 .89728 68 54 .90000 ! .90278 .90417 .90556 .90888 .91111 91250 .91889 54 55 .91667 .91944 .92083 [92222 .92500 .!)2778 92917 .93050 55 56 .93333 .(Mill .93750 .93889 94167 .94444 94588 .1)4722 j 56 57 .95000 .86878 .95417 .95556 95888 .96111 96290 .9I53M) 57 58 .%i::r,o:3 7.1855162 .002695418 372 138384 51478848 19.2873015 7.1919663 .002688172 270 TABLE VIII. Continued. No. Squares. Cubes. Square Hoots. Cube Roots. Reciprocals. 373 139129 51895117 19.3132079 7.1984050 .002680965 374 139876 52313624 | 19.3390796 ! 7.2048322 .002673797 375 140625 52734375 19.3649167 7.2112479 .002666667 376 141376 53157376 19.3907194 7.2176522 .002659574 377 142129 53582033 19.4164878 7.2240450 .002652520 378 1428*4 54010152 ! 19.4422221 7.2304268 .002645503 379 143641 54439939 19.4079^23 7.2367972 .002638522 380 144100 54872000 19.4935887 : 7.2431565 .002631579 381 145161 55306341 19.5192213 7.2495045 .002024672 38-3 145924 55742908 19.5448203 7.2558415 .002617801 38.5 14(5089 50181887 19.5703858 7.2621675 .002610966 384 147456 50023104 19.5959179 7.2684824 .002004107 385 148225 57000025 19.6214169 ! 7.2747804 .002597403 386 148996 57512456 19.6468827 7.2810794 .002590074 387 149709 57900003 19.0723156 7.2873617 .002583979 388 150541 58411072 19.6977156 7.2936330 .002577320 389 151321 5886:3809 19.7230829 7.2998936 .002570694 390 152100 59319000 19.7484177 7.3061436 .002564103 391 152881 59776471 lit. 7 737199 7.3123828 .002557545 392 153004 60236288 19.7989899 7.3180114 .002551020 393 154449 60098457 19.8242276 7.3248295 .002544529 394 155236 61162984 19.8494332 7.3310369 .002538071 395 156025 61629875 19.8746009 7.337'2339 .002531646 390 156816 62099136 19.8997487 7.3434205 .00x525253 397 157609 62570773 19.9248588 7.3495900 .002518892 398 158404 63044792 19.9499373 7.3557024 .002512563 399 159201 63521199 19.9749844 7.3619178 .0025U0206 400 160000 64000000 20.0000000 7.3680630 .002500000 401 160801 64481201 20.0249844 i 7.3741979 .002493766 402 161604 64964808 20.0499377 7.3803227 .002487562 403 162409 65450827 20.0748599 7.3864373 .002481390 404 163216 05939264 20.0997512 7.3925418 .002475248 405 164025 60430125 20.1246118 7.3986363 .002409136 400 164836 60923416 20.1494417 7.4047206 .002403054 407 165649 67419143 20.1742410 7.4107950 .002457002 408 160404 67917312 20.1990099 ! 7.4168595 .002450980 409 167281 68417929 20.2237484 7.4229142 .002444988 410 168100 68921000 20.2484567 7.4289589 .002439024 411 168921 69426531 20.2731349 7.4849938 .002433090 412 169744 69934528 20.2977831 7.4410189 .002427184 413 170509 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 420 176400 74088000 20.4939015 7.4888724 .002380952 421 177241 74618461 20.5182845 7.4948113 .002375297 422 178084 75151448 20.5426386 7.5007406 .002369668 423 178929 75686967 20.5669638 7.5066607 .002364066 424 179776 76225024 20 5912003 7.5125715 .002358491 425 180625 70765625 20.6155281 7.5184730 .002352941 426 181476 77308776 20.6397674 7.5243652 .002347418 427 182329 77854483 20.6639783 7.5302482 .002341920 428 183184 78402752 20.6881609 7.5361221 002336449 429 184041 7S953589 20.7123152 7.5419867 .002331002 430 184900 79507000 20.7364414 7.5479423 .002325581 431 185761 80062991 20.7605395 7.5536888 .002320186 432 186624 80621508 20.7846097 7.5595263 .00*314815 433 187489 1 811827R7 20.8086520 7 5053548 .002309469 434 188:356 ! 81740504 20.a326667 7.5711743 .002304147 271 TABLE VIII. Continued. No. Squares. Cubes. Square Hoots. Cube Roots. Reciprocals. 435 189225 82312875 20.8566536 7.5769849 .002298851 436 190096 82881856 20.88061:30 7.5827865 .00229^578 437 190969 83453453 20.9045450 7.5885793 .0022->8330 438 191844 84027672 20.9284495 7.5943033 .002283105 439 192721 84604519 20.9523208 7.6001385 .002277904 440 193600 85184000 20.9761770 7.6059049 .00227'27'27 441 194481 85766121 21.0000000 7.6116026 .002267574 443 195364 86350888 21.0237900 7.6174116 .002262413 443 196249 86938307 21.0475053 7.6231519 .002257336 444 197136 87528384 21.0713075 7.6288837 .002252252 445 198025 88121125 21.0950231 7. 6346067 .002247191 446 198916 88716536 21.1187121 7.6403213 .00:2242152 447 399809 89314023 21.1423745 7.6460272 .00223713(5 448 200704 89915392 21.1660105 7.6517247 .002232143 449 201601 90518849 21.1896201 7.657'4133 .002227171 450 202500 91125000 21.2132034 7.6630943 .002222222 451 203401 9173:3851 21.2367006 7'.G(587'665 .002217295 452 204304 92345408 21.2602916 7.6744303 .002212389 453 205209 92959677 21.2837967 7.6800857 .002207506 454 206116 i 93576664 21.3072758 7.6857328 .0022021543 455 207025 94196375 21.8307290 7.6913717 .002197802 456 207938 94818816 21.a541505 7.6970023 .002192982 457 208849 95443993 21.3775583 7.7026246 .002188184 458 209764 96071912 21.4009346 7.7082388 .002183406 459 210681 96702579 21.4242853 7.7138448 .002178649 460 211600 97336000 21.4476106 7.7194426 .002173913 461 212521 97972181 21.4709106 7.7'250325 .002169197 462 213444 98611128 21.4941853 7.7306141 .002164502 463 214369 99252847 21.5174348 7.7361877 .002159827 464 215296 99897344 21.5406592 7.7417'532 .002155172 465 216225 100544625 21.5638587 7.7473109 .002150538 466 217156 101194696 21.5870331 7.7528606 .002145923 467 218089 101847563 21.6101828 7.7584023 .002141328 468 219024 102503232 21.6333077 7.7639361 .002136752 469 219961 103161709 21.6564078 7.7694620 .002132196 470 220900 103823000 21.6794834 7.7749801 .002127660 471 221841 104487111 21.7025344 7.7'804904 .00212:3140 472 222784 105154048 21.7255610 7.7859928 .0021 ISO U 473 223729 105823817 21.7485632 7.7914875 .002114165 474 224676 106496424 21.7715411 7.7969745 .002109705 475 225625 107171875 21.7944947 7.8024538 .002105263 476 226576 107850176 21.8174242 7.8079254 .002100840 477 227529 108531333 21.8403297 7.8133892 .002096436 478 228484 109215352 21 8632111 7.8188450 .002092050 479 229441 109902239 21.8860686 7.8242942 .002087683 480 230400 110592000 21.9089023 7.8297353 .002088333 481 231361 111284641 21.9317122 7.8351688 .002079002 482 232324 111980168 21.9544984 7.8405949 .0020745 7.9104599 .002020202 496 246016 12202:3!W6 22.2710575 7.9157832 .002016129 TABLE VIII. Continued. Xo. Squares. Cubes. Square Koots. Cube Boots. Reciprocals. 497 347009 133763473 22.2934968 -. 9210994 .002012072 498 21S004 133505'.)! 13 22.3159136 -.9264085 .002008032 499 249001 134351499 22.3383079 7.9317104 .003004008 500 350000 125000000 22.3606798 7.9370053 .003000000 501 351001 125751501 22.3830293 -.9422931 .001996008 503 253. i04 130506008 22.4053565 7.9475739 .001993033 503 353009 13:303527 22.4376615 -.9528477 .001988073 504 354016 128024064 22.4499443 7.9581144 .001984127 505 255035 128787625 22.4722051 7.96a3743 .001980198 506 356036 129554216 22.4944438 7.9686271 .001976285 sor 357049 130323843 22.5166605 7.9738731 .001972:387 508 258064 131096512 22.5!388553 7.9791122 .001968504 509 259081 131872239 22.5610283 7.9843444 .001964637 510 260100 132651000 22.5831796 7.9895697 .001960784 511 261121 133432831 22.6053091 7.9947883 .001956947 513 263144 134217728 22. 6274170 8.0000000 .001953125 513 263169 135005697 22.6495033 8.0052049 .001949318 514 204 196 135796744 22.6715681 8.0104032 .001945525 515 205335 13659087'5 22.6936114 8.0155946 .001941748 516 260356 137388096 22.7156334 8.0207794 .001937984 517 267289 138188413 22.7376340 8.0259574 .001934236 518 268324 138991832 22.7'596134 8.0311287 .001930502 519 269361 139798359 22.7815715 8.0362935 .001936782 530 270400 140608000 22.8035085 8.0414515 .001933077 521 271441 141420761 22.8254244 8.0466030 .001919386 533 273484 142236648 22.8473193 8.0517479 .001915709 533 273539 143055667 22.8691933 8.0568862 .001913046 534 274576 143877824 22.8910463 8.0620180 .001908397 535 275625 144703125 22.9128785 8.0671432 .001904763 536 276676 145531576 22.9346899 8.07'22620 .001901141 537 277729 146363183 22.9564806 8.0773743 .001897533 538 278784 147197952 22.9783506 8.0R24800 .001893939 529 279841 148035889 23.0000000 8.0875794 .001890359 530 280900 148877000 23.0217289 8.0936733 .001886792 531 281961 1497'21291 23.043437'2 8.0977589 .001883339 533 283024 150568768 23.0651252 8.1038390 .001879699 533 284089 151419437 23.0867928 8.1079138 .001876173 534 285156 152273304 23.1084400 8.1139803 .001872659 535 286225 153130375 23.1300670 8.1180414 .001869159 536 287.396 153990656 23.1516738 8.1330963 .001865672 537 288369 ' 154a54153 23.1732605 8.1381447 .001862197 538 ' 3894-44 155720873 23.1948370 8. 1331870 .001858736 539 21)0531 156590819 23.2163735 8.1383330 .001855288 540 291600 157464000 23.2379001 8.1433539 .001851852 541 392681 15&340421 23.3594067 8.1483765 .001848429 543 293764 159330088 23.2808935 8.1532939 .001845018 543 294849 10; 103007 23.3023604 8.1583051 .001841631 544 295936 160989184 23.3238076 8.1633102 .001838335 515 297025 161878635 23.3452351 8.1683092 .001834862 546 298116 163771336 23.3666429 8.1733020 .001831502 54?' 299209 163667323 23.3880311 8.1782888 .001828154 548 300304 104566592 23.4093998 8.1&33695 .001824818 549 301401 165469149 23.4307490 8.1882441 .001821494 550 302500 106375000 23.4520788 8.1932127 .001818182 551 303601 167384151 23.4733892 8.1981753 .001814882 553 304704 168196608 23.4946802 8.2031319 .001811594 553 305S09 169113377 23.5159520 8.2080825 .001808318 854 306916 170031464 23.5372046 8.2130271 .001805054 i 555 308035 170953875 23.5584380 8.2179657 .001801802 556 309136 171879616 23.5796522 8.2228985 .001798561 557 310349 173808693 23.6008474 8.22782.54 .001795332 558 311364 173741112 23.6220236 8.2337463 .001792115 273 TABLE VIII. Continued. No. Squares. Cubes. Square Boots. Cube Roots. Reciprocals. 559 312481 174676879 23.6431808 8.2376514 .001788909 560 313600 175616000 23.6643191 8.2425706 .001785714 561 314721 176558-i81 23.6854386 8.2474740 .001782531 562 315844 177504328 23.7065392 8.2523715 .001779859 563 316969 178453547 23.7270210 8.2572633 .01)177 0199 564 318096 179406144 23.7486842 8.2621492 .001773050 565 319225 ! 180362125 23.7697286 8.2670294 .001769912 566 320350 181321496 23.7907545 8.2719039 .001766784 567 321489 182284263 88.8117818 8.2767726 .001763668 568 322624 183250432 28.8327506 8.2816355 .001760563 569 323701 184220000 H&.S687309 8.X&64928 .001757469 570 324900 185193000 23.8746728 8.2913444 .001754386 571 326041 180169411 23.8956063 8.2961903 .001751313 572 327184 187149248 23.9165215 8. 010304 .001748252 573 328329 188132517 23.9374184 8.3058651 .001745201 574 329476 189119224 2x3.9582971 8.3106941 .001742160 575 330625 190109375 23.97'91576 8.3155175 .001739130 576 331776 191102976 24.0000000 8.3203353 .001736111 577 332929 192100033 24.0208243 8.3251475 .001733102 578 334084 193100552 24.0416306 8.32995-12 .001730104 579 335241 194104539 24.0024188 8.3347553 .601727116 580 336400 195112000 24.Ca31891 8.3395509 .001724138 581 337561 196122941 24.1039416 8.3443410 .0017-21170 582 338724 197137368 24.1246762 8.3491256 .001718213 583 339889 198155287 24.1453929 8.0539047 .001715266 584 341056 199176704 24.1660919 8.3586784 .001712329 585 342225 200201625 24.1867732 8.: 634466 .001709402 586 343:396 201230056 24.2074369 8.8682065 00170(1485 587 344569 202262003 24.2280829 8.o729Gt>8 .001703578 588 345744 203297472 24.2487113 8.8777188 .001700680 589 346921 204336469 24.2693222 8.8824653 .OU1G97793 590 348100 205379000 24.2899156 8.3872065 .001694915 591 349281 206425071 24.3104916 8.3919423 .001692047 592 350464 207474688 24.3310501 8. 966729 .001089189 593 351649 208527857 24.3515913 8.4013981 .001686341 594 52836 209584584 24.3721152 8.4061180 .001683502 595 354025 210644875 24.3926218 8.4108326 .001680672 596 355216 211708736 24.4131112 8.4155419 .001677852 597 356409 2127761 73 24.4335834 8.4202460 .001675048 598 357604 213847192 24.4540385 8.4249448 .001672241 599 358801 214921799 24.47'44765 8.4296383 .001669449 600 360000 216000000 24.4948974 8.4348267 .001666G67 601 361201 217081801 24.5153013 8.4390098 .001668884 602 362404 218167208 24.5356883 8.4436877 .0016611.30 603 363609 ; 219256227 24.5560583 8.4483605 .001058375 604 364816 220:348864 24.5764115 8.4530281 .001C55629 605 366025 221445125 24.5967'478 8.4576906 .001652893 606 367236 22254501(3 24. 6170673 8.4623479 .001650165 607 368449 223648543 24.6373,00 8.4670001 .001(147446 608 369664 224755712 24.6576560 8.4716471 .001(544737 609 370881 225866529 24.6779254 8.47'02892 .001042036 610 372100 226981000 24.6981781 8.- 1809261 .001639344 611 373321 228099131 24.7184112 8.485557$) .0(11036661 612 374544 229220928 34.7886338 8.4901848 .001633987 613 375769 230346397 24.7588368 8.4948065 .001631321 614 376996 831475544 24.7790234 8.4994233 .001028664 615 378225 232608375 24.7991935 8.5040350 .001626016 616 379456 233744896 24.8193473 8.5086417 .001(123377 617 380689 234885113 i 24.83048-17 8.5132435 .001620746 618 381924 236029032 24.KV.MH5rt ! 8.517K403 .001618123 619 383161 237J76659 24.H797IO'! 8..V,>24321 .001615509 620 384400 238328000 24.8997992 8.5270189 i . 001 (51 2903 374 TABLE VIII. Continued. No. Squares Cubes. Square Roots. Cube Roots. Reciprocals. 621 385641 239483061 24.9198716 8.5316009 .001610306 622 386884 240641848 24.9399278 8.5361780 .001607717 653 388129 241804367 24.9599679 8. 540750 1 .001605136 624 389376 242SJ70624 24.9799920 8.5453173 .001602564 625 390625 244140625 25.0000000 8.5498797 .001600000 626 i 391876 245314376 25.0199920 8.-J544372 .001597444 627 ! 393129 246491883 25.0399681 8.5589899 .001594896 628 394384 247673152 SB. 0599283 8.5635377 .001592357 629 395641 248858189 25.079S7'24 8.5680807 .001589825 630 396900 250047000 25.0998008 8.5726189 .001587302 631 398161 251239591 25.1197134 8.5771523 .001584786 632 399424 252435968 25.1396102 8.5816809 .001582278 633 4U0689 253636137 25.1594913 8.5862047 .001579779 634 401956 254840104 25.1793566 8.5907238 .001577287 635 403225 256047875 25.1992063 8.5952380 .001574803 636 404496 257259456 25.2190404 8.5997476 .001572327 637 405769 258474853 25.2388589 8.6042525 .001569859 638 407044 259694072 25.2586619 8.6087'526 .001567398 639 ! 408321 260917119 25.2784493 8.6132480 .001564945 640 409600 262144000 25.2982213 8.6177388 .001562500 641 410881 263374721 25.3179778 8.6222248 .001560002 642 412164 264609288 25.3377189 8.6267063 .001557632 643 413449 265847707 25.3574447 8.6311830 .001555210 644 414736 267089984 25.3771551 8.6356551 .001552795 645 416025 268336125 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 .001536098 652 425104 277167808 25.5342907 8.6712665 .001533742 653 426409 278445077 25.5538647 8.6756974 .001531394 654 427716 279726264 25.5734237 8.6801237 .001529052 655 429025 281011375 25.5929678 8.6845456 .001526718 656 430*36 282800416 25.6124969 8.6889630 .001521390 657 431649 283593393 25.6320112 8.6933759 .001522070 658 432964 284890312 25.6515107 8.6977843 .001519757 659 434281 286191179 25.6709963 8.7021882 .001517451 660 435600 287496000 25.6904652 8.7065877 .001515152 661 436921 288804781 25.7099203 8.7109827 .001512859 662 438244 290117528 25.7293607 8.7153734 .001510574 663 439569 291434247 25.7487864 8.7197596 .001508296 664 440896 292754944 25.7681975 8.7241414 .001506024 665 442225 294079625 25.7875939 8.7285187 .0015037'59 666 443556 295408296 25.8069758 8.7328918 .001501502 667 444889 2967'40963 25.8263431 8.7372604 .001499250 668 446224 298077632 25.8456960 8.7416246 .001497'006 669 447561 299418309 25.8650343 8". 7459846 .001494708 670 448900 300763000 25.8843582 8.750:3401 .001492537 671 450241 302111711 25. 9036677 8.7546913 .001490313 672 451584 303464448 25.9229628 8.75903*3 .001488095 673 452929 304821217 25.9422435 8.76,33809 .001485884 674 454276 306182024 25.9615100 j 8.7677192 .00148:3680 675 455625 307546875 25.9807621 8.7720532 .001481481 676 450976 308915776 26.0000000 8.7763830 .001479290 677 458329 310288733 26.0192237 8.7807084 .001477105 678 459684 311665752 26.0384831 8.7850296 .001474926 679 461041 313046839 6.0576284 8.7893466 .004472754 680 462400 314432000 26.0768096 8.7936593 .001470588 681 463761 315821241 26.0959767 8.7979679 .001468429 682 465124 317214568 26.1151297 8.8022721 .001466276 375 TABLE VIII.-Confmf. No. Squares. Cubes. Square Roots. Cube Roots. Reciprocals. 683 466489 318611987 26.1342667 8.8065722 .001464129 681 467-856 320013504 26.1533937 8.8108081 .001401988 685 469225 321419125 26.1725047 8.8151598 .001459854 686 470596 322828856 26.1916017 8.8194474 .001457726 687 471969 324242703 20.2100848 8.8237307 .001455004 088 47-3;344 325600672 26.2297541 8.8280099 .001453488 689 474721 327082769 26.2488095 8.8322850 .001451379 690 476100 328509000 26.2078511 8.8365559 .001449275 691 477481 329939371 20.2808789 8.8408227 .001447178 692 478864 331373888 20.3058929 8.8450854 .001445087 693 480249 332812557 26.3248932 8:849:3440 .001443001 694 481636 334255384 26.3438797 8.8535985 .001440922 695 483025 35702375 26.3628527 8.8578489 .001438849 696 484416 337153536 26.3818119 8.8630953 .001430782 697 485809 33860887-3 26.4007576 8.8003375 .0014:347-20 698 487204 340068392 26.4190890 8.87-05757 .001432005 699 488601 341532099 26.4386081 8.8748099 .001430015 700 490000 343000000 26.4575131 8.8790400 .001428571 701 491401 344472101 26.4764046 8.8832001 .001420534 702 492804 345948408 20.4952820 8.8874883 .001424501 703 494209 347428927 20.5141472 8.8917063 .001422475 704 495616 348913604 20.5329983 8.8959204 .001420455 705 497025 350402625 20.5518361 8.9001304 .001418440 708 498436 351895816 26.5706605 8.9043300 .00141(5431 707 499849 353393243 26.5894716 8.9085387 .001414427 708 501264 &54894912 20.0082094 8.9127369 .001412429 709 502681 350400829 20 6270539 8.9169311 .001410437 710 504100 38791^000 26.6458252 8.9211214 .001408451 711 505521 359425431 26.6045833 8.9358078 .001400470 712 506944 360944128 20.0833281 8.9294902 .001404494 713 508369 362467097 20.7020598 8.9330087 .001402525 714 509796 303994344 20.7207784 8.9378433 .001400500 715 511225 305525875 2(5.7394839 8.9420140 .001398601 716 512656 367061696 20.7581703 8.9401809 .001396648 717 514089 368601813 20.77-08557 8.9503438 001394700 718 515524 370146232 20.7955220 8.9545029 .001:392758 719 516961 371694959 26.8141754 8.9580581 .001390821 720 518400 373248000 26.8328157 8.9628095 .001388889 721 519841 374805361 20.8514432 8.9669570 .(;01380963 722 521284 37'6367048 26.8700577 8.9711007 001385043 723 5227'29 377933067 20.8880593 8.9752406 .001:383120 724 524176 379503124 20.9073481 8.97937-60. .OOK381215 725 525625 381078125 86.9258840 8.9835089 .001379:310 726 527076 382657176 26.944:3872 8.9870373 .001377410 727 528529 384240583 26.9629375 8.9917020 . 0013755 10 728 529984 385828352 26.9814751 8.9958889 0013730:$ 729 531441 387420489 27.0000000 9.0000000 .001371742 730 532900 389017000 27.0185122 9.0041134 .001369863 731 534:361 890817891 27.0370117 9.0082229 .001307'!W9 732 535824 392223168 27.0554985 9.0123288 .001306120 733 537289 393832837 27.0739727 9.010430!) .001304256 734 538756 395440904 27.0924344 9.0205293 .001362398 735 540225 897065375 27.1108834 9.0246339 .001300544 736 541696 398688258 27.1293199 9.0287149 .001358(590 737 543169 400816663 27.1477439 9.0328021 .001356853 738 44644 40194727'2 27.10<>ir>54 9.0368857 .001:355014 739 546121 403583419 27.1845514 9.0409655 .001353180 740 547600 405224000 27.2029410 9.0450419 .001 351 351 741 549081 4068(59021 27.221.-J1.V3 !).( 1491 142 .OOK349528 742 550564 408518488 87.2896769 9.0531881 .001:347709 743 552049 410172407 27.2580363 9.0572482 .001345895 744 553536 411830784 27.2703034 9.0613098 .001344086 276 TABLE Vlll. Continued. No. Squares. Cubes. Square Roots. Cube Roots. Reciprocals. 745 555025 413493625 27.2946881 9.0653677 .001342288 746 556516 415160936 27.3130006 9.0694220 .001340483 747 558009 4168:32723 27.3313007 9.0734726 .00133868S 748 559504 418508992 27.3495887 9.0775197 .001336898 749 561001 420189749 27.3678644 9.0815631 .001335113 750 562500 421875000 27.3861279 9.0856030 .001333333 751 564001 423564751 27.4043792 9.0896392 .Or] 331558 752 565504 425259008 27.4226184 9.0936719 001329787 753 567009 426957777 27.4408455 9.0977010 .001328021 754 568516 428661064 27.4590604 9.1017265 001326260 755 570025 '430368875 27.4772633 9.1057485 .001324503 756 571536 432081216 27.4954542 9.1097669 .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 9.1298061 .001314060 762 580644 442450728 27.6043475 9.1338034 .001312336 763 582169 444194947 27.62-,4546 9.1377971 .001310616 764 583696 445943744 27.6405499 9.1417874 .001308901 765 585225 447697125 27.6586*34 9.1457742 .001307190 7'66 586756 449455096 27.6767050 9.1497576 .001305483 767 588289 451217663 27.6947648 9.1587375 .001303781 768 589824 452984832 27.7128129 9.1577139 .001302083 769 591361 454756609 27.7308492 9.161686'J .001300390 770 592900 456533000 27.7488739 9.1656565 .001298701 771 594441 458314011 27.7068868 9.1696225 .001297017 772 595984 460099648 27.7848880 9.1735852 .001295337 773 597529 461889917 27.802S775 9.1775445 .001293661 774 599076 463684824 27.8208555 9.1815003 .001291990 775 600625 465484375 27.8388218 9.1854527 .001290323 776 C02176 467288576 27.8567766 9.1894018 .001288660 777 (503729 469097433 27.8747197 9.1933474 .001287001 778 605284 470910952 27.8926514 9.1972897 .001285347 779 606841 472729139 27.9105715 9.2012286 .001283697 780 608400 474552000 27.9284801 9.2051641 .001282051 781 609961 476379.541 27.9463772 9.2090962 .001280410 782 611524 478211768 27.9642U29 9.2130250 .001278772 783 613089 480048687 27.9821372 9.2169505 .001277139 784 614656 481890304 28.0000000 9.2208726 .001275510 785 616225 483736625 28.0178515 9.2247914 .001273885 786 617796 485587656 28.0356915 9.2287068 .001272265 787 619369 48744:3403 28.0535203 9.2326189 .001270648 788 620944 489303872 28.07ia377 9.2365277 .001269036 789 622521 491169069 28.0891438 9.2404333 .001267427 790 624100 493039000 28.1069386 9.2443355 .001265823 791 625681 494913671 28.1247222 9.2482344 .001264223 792 627264 496793088 28.1424946 ,9.2521300 .001262626 793 628849 498677257 28.1602557 9.2560224 .001261034 794 630436 500566184 28.1780056 9.2599114 .001259446 795 632025 502459875 28.1957444 9.2637973 .001257862 796 633616 504358336 28.2134720 9.2676798 .001256281 797 635209 506261573 28.2311884 9.2715592 .001254705 798 636804 508169592 28.2488938 9.2754352 .001253133 799 638401 510082399 28.2665881 9.2793081 .001251564 800 640000 512000000 28.2842712 9.2831777 .001250000 801 641601 513922401 28.3019434 9.2870440 .001248439 802 643204 515849608 28.3196045 9.2909072 .001246883 803 644809 517781627 28.3372546 9.2947671 .001245330 804 646416 519718464 28.a548938 9.2986239 .001243781 805 648025 521660125 28.3725219 9.3024775 .001242236 806 649636 523606616 28.3901391 9.3063278 .001240695 277 TABLE Vm. Continued. No. Squares. Cubes. Square Roots. Cube Roots. Reciprocals. 807 651249 525557943 28.4077454 9.3101750 .001230157 808 652864 527514112 28.4253408 !). 3140190 .0012:37024 809 654481 529475129 28.4429253 9.3178599 .001230094 810 656100 531441000 28.4604989 9.3216975 .001 23451 M 811 657721 533411731 28.4780617 9.3255320 .00123:3040 812 659344 535387328 28.4956137 9.3293034 .001231527 813 660969 537307797 28.5131549 9.3331910 .001230012 814 662596 539353144 28.5300852 9.3370167 .001228501 815 664225 541343375 28.5482048 9.3408386 .001228094 816 665856 543338496 28.5657137 9.3446575 .0012-25190 817 667489 545338513 28.5832119 9.3484731 .00122J399U 818 669124 547343432 28.6000993 J. 3522857 .001222494 819 670761 549353259 28.6181760 9.3560U52 .001221001 820 672400 551368000 28.6356421 9.3599016 .001219512 821 674041 553:387661 28.6530976 9.3037049 .001218027 822 675684 555412248 28.6705424 9.3675051 .001210545 823 677329 557441767 28. 6871)? tit) 9.3713022 .001215007 824 678976 559476224 28.7054002 9.3750963 .001213592 825 680625 561515625 28.7228132 9.3788873 .001212121 826 682276 563559976 28.7402157 9.3820752 .00121065 i 827 683929 565609283 28.7570077 9.3804600 .0012091M) 828 685584 567003553 28.7749891 9.3902419 .001207729 829 687241 569722789 28.7923601 9.3940206 .001200273 830 688900 571787000 28.8097206 9.3977964 .001204819 831 690561 573836191 28.8270706 9.4015691 .001203309 832 692224 575930368 28.8444102 9.4053387 .001201923 833 693889 578009537 28.8617394 9.4091054 .001200480 834 695556 580093704 28.8790582 9.4128090 .001199041 835 697225 582182875 28.8963666 9.416G297 .001 197005 836 698896 584277056 28.9130046 9.4203873 .001190172 837 700569 586376253 28,9809598 9.4241420 .001194743 838 702244 588480472 28.9482297 9.4278936 .001193317 839 703921 590589719 28.9654967 9.4316423 .001191895 840 705600 592704000 28.9827535 9.4353880 .00119047u 841 707281 59482:3321 29.0000000 9.4391307 .001189001 842 708964 596947688 29.0172363 9.4428704 .001187'64'3 843 710049 599077107 29.0344023 9.4460072 .001186240 844 712336 601211584 29.0516781 9.4503410 .001184831 845 714025 60a351125 29.0688837 9.4540719 .0011 8:3 132 846 715716 605495736 29.0860791 9.4577999 .001182033 847 717409 607645423 29.1032044 9.4615249 .0011 NOT.* 5 848 719104 609800192 29.1204396 9.4652470 .001179245 849 720801 611960049 29.1370046 9.4689061 .001177850 850 722,500 614125000 29.1547595 9.4726824 .001176471 851 724201 616295051 29.1719043 0.4763957 .0011750SS 852 725904 618470208 29.1890390 9.4801061 .001173709 853 727609 620650477 29.2061037 9.4838136 .001172333 854 729316 622835864 29.2232784 9.4875182 .00117'0900 855 731025 625026375 29.2408830 9.4912200 .001 1(59591 856 732736 627222016 29.2574777 9.4949188 .00110S-221 857 734449 629422793 29.2745623 9.4980147 .0011008(11 858 736164 631628712 29.2916370 9.5023078 .001165501 859 737881 633839719 29.3087018 9.5059980 .001104144 860 739600 636056000 29.3257566 9.5096854 .001162791 861 741321 638277^381 29.3428015 9.5133699 .001161440 862 743044 640503928 29.3598365 9.5170515 .001160093 863 744769 642735647 29.3768616 9.5207303 .001158749 864 746496 644972544 29.3938769 9.5244063 .001157407 865 748225 647214625 29.4108823 9.5280794 .001156069 866 749956 649461896 29.4278779 9.5317407 .001154734 867 751689 651714363 29.4448637 9.5354172 .001153403 868 753424 653972032 29.4618397 9.5390818 .001152074 278 TABLE Vlll. Continued. No. Squares. Cubes. Square Roots. Cube Roots. Reciprocals. 8G9 755161 656234909 29.4788059 9.5427437 .001150748 870 756900 658503000 29.4957624 9.5464027 .001149425 871 758041 000770311 29.5127091 9.5500589 .001148106 87'2 700384 003U54&48 29.5290401 9.5537123 .001140789 873 762129 00f>338017 21). 5405734 9.5573630 .00114547'5 874 763876 667027024 29.50:34910 9.5010108 .001144165 875 765025 00!)92J87'5 29.5803989 9.5646559 .001142857 876 767376 67'2221376 29.ryj72i)7'2 9.5082982 .001141553 877 769129 074c26133 29.0141858 9.5719377 .001140251 878 770884 670830152 29.0310048 9.57557'45 .001138952 879 773641 679151439 29.0479342 9.5792085 .001137656 880 774400 681472000 29.6047939 9.5828397 .001136364 881 776161 6837'97841 29.6816442 9.5804082 .001135074 883 777924 686128908 29.6984848 9.5900939 .0011337'87 883 779689 688405387 29.7153159 9.5937169 .001132503 884 781456 690807104 29.7321375 9.5973373 .001131222 885 783225 69315-1125 29.7'489496 9.0009548 .001129944 886 7'84996 695500456 29.7057'521 9.0045096 .001128668 887 786769 697804103 29.7'825452 9.6081817 .001127396 888 7B8544 700227072 29.7993289 9.6117911 .001126126 889 790321 702595309 29.8101030 9.6153977 .001124859 890 792100 704969000 29.8328678 9.6190017 .00112,3596 891 793881 7W347971 29.8496231 9.6220030 .001122334 892 795664 7'097'32288 29.8003090 9.6202016 .001121076 893 797449 712121957 M. 8831056 9.6297975 .001119821 894 799236 714510984 29.8998328 9.0333907 .001118568 895 801025 710917375 29.9165506 9.6369812 .001117318 896 802816 719323136 29.9332591 9.6405690 .001116071 897 804609 7217'34273 29.9499583 9.6441542 .001114827 898 800404 724150792 29.9000481 9.6477367 .001113586 899 808201 726572699 29.9833287 9.6513100 .001112347 900 810000 729000000 30.0000000 -9.6548938 .001111111 901 811801 731432701 30.0100620 9.0584684 .001.109878 902 813604 733870808 30.0333148 9.6020403 .001108&47 903 815409 736314327 30.0499584 9.6650096 .001107420 904 817216 738763264 30.0065928 9.6691762 .0011(6195 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 30.1:330383 9.6834166 .001101322 909 826281 751089429 30.1496269 9.6869701 .001100110 910 828100 753571000 30.1002063 9.6905211 .001098901 911 829921 756058031 30.1827765 9.6940694 .001097695 912 831744 758550528 30.1993377 9.6976151 .00109&491 913 833569 761048497 30.2158899 9.7011583 .001095290 914 835396 763551944 30.2324329 9.7040989 .001094092 915 837225 766060875 30.2489069 9.7082309 .001092896 916 839056 768575296 30.2654919 9.7117723 .001091703 917 840889 771095213 30.2820079 9.7153051 .001090513 918 842724 773620632 30.2985148 9.7188354 .001089335 919 844561 776151559 30.3150128 9.7223631 .001088139 920 846400 778C88000 30.a315018 9.7258883 .001086957 921 848241 781229961 30.3479818 9.7294109 .001085776 922 850084 . 783777448 30.3644529 9.7329309 .001084599 923 851929 786330467 30.3809151 9.7364484 .001083423 924 853776 788889024 30.39736*3 9.7399634 .001082251 925 a55625 791453125 30.4138127 9.7434758 .001081081 926 857476 794022776 30.4302481 9.7469857 .001079914 927 859329 796597983 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 279 TABLE VIII.- r,,,, /,,/,.,/. No. Cubes. Square Boot* Cube Roots. I ;<- ., r V . ;.,',, j 939 n J , , . 940 883600 ::u ';:.: ii'.M S8611 .i*}]i.> 941 87621 .001062699 942 :jo.:: 9.8028036 .001061571 941 944 81807 B2711 .001060445 .00101 '.i l.-j 85&025 8438 ::u .; 9.8i; .001(1 946 894916 571180 ,i,:/j] .OOK 949 896809 80.778 "I I Oil 948 Kr.l'.i. 9.82; .00101 949 900601 KM' i .OOK - 990 902500 30.8220700 9.880476? .Odli .' 951 904401 906804 WBHI) no* tf4972 (i -:, ;:. i:,:, MB 908909 149818 964 910116 !i M i-j:,:;(; .001041 961 912085 !i ,- , I I'li'l^ l:.'d 956 918906 80.9192497 1 irj.M) 957 915849 876* 854166 15617 .001044982 958 917764 !1 7919 80.0515751 OOK 959 919681 881!)', !<)?. ::d ft 9!S614818 960 921600 8HJ^: 80 96i 141667 961 9C.2 (i-:, 1 1 1 S77128 ;;| (i|i;|:j |K Mill |1 .001039501 981 898056847 8llo MI:;:, .001088422 964 11844 81.0488494 I) s , 966 ,^1-J.') 81.0844491 19461 .(.'UP 9M 988156 lid! : 81.08( .(-dp 907 904831068 81.0966286 })> ii-.'i; 9M 90-30; :;i i 9.89 969 81.]:.' 9.8055801 .0011 970 940900 U1-J673000 81.14 (i 89! .00101 971 942841 915498611 ::i ! '.i '. .OOK 972 944784 918880048 81.1769146 j 9057817 .00102W07 971 946799 98111 ::i li Ii W.MIVfj .001027749 974 994010494 81 .01 '.I 'ji: .Cdll.Vi '.'.II 979 950696 59876 II '.:! .001026641 9M 92971 J 1 id ::i 941 9.911 OOK 917 81.251 I) I-' gn 956484 985441859 ooi< 979 958441 <.'.',:','.i 81.S8I ii 981 I lud 980 960400 {H1920nO 81,8040517 I) 93 ,0010 981 969861 94407614] .Hiir, .ooion 982 !M')!W;('|P'-S :;! ::: .00101 Ii 94 .00101 B8904 ,',|.", !i 941 (Ki!d|r,;.'(',d 9M 17097 9.9497479 ; ooioi 9M 979198 :;i 10 .001014199 '(K7 96150 I) 'i. .001018171 US* 976144 '.I '.i.' !lK.'JHII .001019146 9M 97H121 51669 81981 .OOlOlllM 990 980100 00900 ;, ;,(,i,; i; -, |(| .(H)ldididl 10096 ddidd'MiH^ 992 9B40M :i;i; 11*1488 ! ::i IIMMI815 iooioi TABLE IX. LOGARITHM OF NUMBERS FROM TO 1000. No. 1 2 3 4 5 6 7 8 9 00000 30103 47712 60206 69897 77815 84510 90309 95424 10 00000 00432 00860 01284 01703 02119 02530 02938 03342 03743 11 04139 04532 04922 05307 05690 06070 06446 06819 07188 07555 12 07918 08279 08637 08990 09342 09691 10037 10380 10721 11059 13 11394 11727 12057 12385 12710 13033 13354 13672 13988 14301 14 14613 14922 15229 15533 15836 16137 16435 16732 17026 17319 15 17609 17898 18184 18469 18752 19033 19312 19590 19866 20140 16 20412 20683 20952 21219 21484 21748 22011 22272 22531 22789 17 23045 23300 23553 23805 24055 24304 24551 24797 25042 25285 18 25527 25768 26007 26245 26482 26717 26951 27184 27416 27646 19 27875 28103 28330 28556 28780 29003 29226 29447 29667 298S5 20 30103 30320 30535 30749 30963 31175 .31386 31597 31806 32015 21 32222 32428 32633 32838 33041 33244 33445 33646 33846 34044 22 34242 34439 34635 34830 35025 35218 35411 35603 35793 35984 23 36173 36361 36549 36736 36922 37107 37291 37475 37658 37840 24 38021 38202 38382 38561 38739 38916 39094 39270 39445 39619 25 39794 39967 40140 40312 40483 40654 40824 40993 41162 41330 26 41497 41664 41830 41996 42160 42325 42488 42651 42813 42975 27 43136 43297 43457 43616 43775 43933 44091 44248 44404 44560 28 44716 44871 45025 45179 45332 45484 45637 45788 45939 46090 29 46240 46389 46538 46687 46835 46982 47129 47276 47422 47567 30 47712 47857 48001 48144 48287 48430 48572 48714 48855 48996 31 49136 49276 49415 49554 49693 49831 49969 50106 50243 50379 32 50515 50651 50783 50920 51055 51189 513-22 51455 51587 51720 33 51851 51983 52114 52244 52375 52504 5-2634 52763 52892 53020 34 53148 53275 53403 53529 53656 53782 53908 54033 54158 54283 35 54407 54531 54654 54777 54900 55022 55145 55267 55388 55509 36 55630 55751 55871 55991 56110 56229 56348 56467 56585 56703 37 56820 56937 57054 57171 57287 57403 57519 57634 577-19 57863 38 5797'8 58093 58206 58320 58433 58546 58659 58771 58883 58995 39 59106 59218 59328 59439 59550 59660 59770 59879 59989 60097 40 60206 60314 60423 60531 60638 60745 60853 60959 61066 61172 41 61278 61384 61490 61595 61700 61805 61909 62014 62118 62221 42 62325 62428 62531 62634 62737 62839 62941 63043 63144 63246 43 63347 63448 63548 63649 63749 63849 63949 64048 64147 64246 44 64345 64444 64542 64640 64738 64836 64933 65031 65128 65225 45 65321 65418 65514 65609 65706 65801 65896 65992 66087 66181 46 66276 66370 66464 66558 66652 66745 66839 6693'2 67025 67117 47 67210 67302 67394 67486 67578 67669 67761 67852 67943 68034 48 68124 68215 68305 68395 68485 68574 68664 68753 68842 68931 49 69020 69108 69197 69285 69373 69461 69548 69636 69723 69810 50 69897 69984 70070 70157 70243 70329 70415 70501 70586 70672 282 TABLE ix continued. LOGARITHM OF NUMBERS FROM o TO 1000. No. 1 2 3 4 5 6 7 8 9 51 70757 70842 70927 71012 71096 71181 71265 71349 71433 71517 52 71600 71684 71767 71850 71933 72016 72099 72181 72263 72346 53 72428 72509 72591 72673 72754 72835 72916 72997 73078 73159 54 73239 73320 73399 73480 73560 73(53!) 73719 73799 73878 73957 55 74036 74115 74194 74273 74351 74429 74507 745S6 74663 74741 56 74819 74896 74974 75051 75128 75205 75282 75358 75435 75511 57 75587 75664 75740 75815 75891 75967 76042 76118 76193 76268 58 76343 76418 76492 76567 76641 76716 76790 7G864 76938 77012 59 77085 77159 77232 77305 77379 77452 77525 77597 77610 77743 60 77815 77887 77960 78032 78104 78176 78247 78319 78390 78462 61 78533 78604 78675 78746 78817 78888 78958 79029 79099 79169 62 79239 79309 79379 79449 79518 79588 79657 79727 79796 79865 63 79934 80003 80072 80140 80209 80277 80346 80414 80482 80550 64 80618 80686 80754 80821 80889 80956 81023 81090 81158 81224 65 81291 81358 81425 81491 81558 81624 81690 81757 81823 81889 66 81954 82020 82086 82151 82217 82282 82347 82413 82478 82543 67 82607 82672 82737 82802 82866 82930 82995 83059 83123 83187 68 83251 83315 83378 83442 83506 83569 83632 836% 83759 83822 69 83885 83948 84011 84073 84136 84198 842(51 84323 84386 84448 70 84510 84572 84634 84696 84757 84819 84880 84942 85003 85065 71 85126 85187 85248 85309 85370 85431 85491 85552 85612 85673 72 85733 85794 85854 85914 85974 86034 86094 86153 86213 86273 73 86332 86392 86451 86510! 86570 86629 86688 86747 86806 86864 74 86923 86982 87040 87099 87157 87216 87274 87332 87390 87448 75 87506 87564 87622 87680 87737 87795 87852 87910 87967 88024 76 88081 88138 88196 88252 88309 88366 88423 88480 88536 88593 77 88649 88705 88762 88818 88874 88030 88986 89042 8909H 89154 78 79 89209 89763 89265 89818 89321 89873 89376 89927 89432 89982 89487 90037 89542 90091 89597 90146 89653 j 89708 90200 90255 80 90309 90363 90417 90472 90526 90580 90634 90687 90741 90795 81 90848 90902 90956 91009 91062 91116 91169 91222 91275 91328 82 91381 91434 91487 91540 91593 91645 .91698 91751 91803. 91855 83 91908 91960 92012 92065 92117 92169 92221 92273 92324 92376 84 92428 92480 92531 92583 92034 92686 92737 92789 92840 02891 85 92942 92993 93044 93095 93146 93197 93247 93298 93349, 93399 86 93450 93500 4 93551 93601 93651 93702 93752 93802 93852! 93902 87 93952 94002 94052 94101 94151 94201 94250 94300 94349; 94398 88 94448 94498 94547 94596 94645 94694 94743 94792 94841) 94890 89 94939 94988 95036 95085 95134 95182 95231 95279 95328; 95376 90 95424 95472 95521 95569 95617 95665 95713 95761 95809 95856 91 95904 95952 95999 96047 96095 96142 96190 96237 96284 96332 92 96379 96426 96473 965201 96567 96614 96661 96708 96755 96802 93 96848 96895 96942 96988 97035 97081 97128 97174 97220,' 97267 94 97313 97359 97405 97451 97497 97543 97589 97635 976811 97727 95 97772 97818 97864 979091 97955 98000 98046 98091 98137 98182 96 98227 98272 9S318 98363 98408 98453 98498 98543 98588 98632 97 98677 96722 98767 98811! 98856 98900 98945 98989 99034 99078 98 99123 99167 99211 99255 99300 99344 99388 99432 99476 99520 99 99564 99607 99651 99695 99739 99782 99826 99870 99913 99957 283 NOTE TO TABLES OF TRIGONOMETRIC FUNCTIONS. In the following Tables the values of Sines, Cosines, Tangents, Cotangents, Versines, and Exsecants are carried only to 5 places of decimals ; the Table of Secants and Cosecants, however, is given to 7 places of decimals, and from it more accurate deter- minations of the Sines, etc., may be obtained, if for any special purpose they be required. For, by Sees. 231 and 232, 1 1 sec A sin A cos A = -. ; tan A = -. , y^uo .0. -. , ICI.LI -a. j, cosec A sec A cosec A 1 cosec A vers A = 1 ; : exsec A sec A 1 ; cot A = .-. sec A sec A 284 TABLE X. SINES AND COSINES. . 1 2 30 40 / Sine ! Cosin Sine Cosin Sine Josin Sine Cosin Sine Cosin ~o TOOOOO One7 .01745 .99085 .03490 .99939 .05234 99863 .06976 .99756 60 1 . 0029 One. .01774 .99984 .03519 .99938 .05203 99861 .07005 .99754 59 2 .00058 One. \ .01803 .99984 .03548 .99937 .05292 99860 .070134 .99752! 58 3 1.00087: One. ! .01832 .99983 .03577 .99936 .05321 99858 .07063 .99750! 57 4 i. 00116 One. .01862 .99983 .03606 .99935 .05:350 99857 .07092 .99748 50 5 .00145 One. .01891 .99982 .03635 .99934 .05379 99855 .07121 .99746 55 6 .00175 One. .01920 .99982 .03664 .99933 .05408 99854 .07150 .99744 54 7 .00204 One. .01949 .99981 .03693 .99932 .05437 .99852 .07179 .99742 53 8 .00233 One. .01978 .99980 1.03723 .99931 .05466 .99851 .07208 .99740 52 9 .00262 One. .02007 .99980 .03752 .99930 .05495 .99849 .07237 .99738! 51 iO .00291 One. .02036 .99979 ;. 03781 .99929 .05524 .99847 .07266 .99736 50 11 . 00320 ! . 99999 ! .02065 .99979 .03810 .99927 .05553 .99846 .07295 .99734 49 12 .00349 .99999 . 02094 i .99978 .03839 .99920 .05582 .99844 .07324 .99731 48 13 .00378 .99999: 1.02123 ;. 99977 1 .03868 .99925 ! .05611 .99842 .07353 .997'29 47 14 .00407 .99999 .02152 .99977 ! .03897 .99924 .05640 .99841 .07382 .99727 46 15 .00436 .99999 ! :. 021 81 .99976 .03926 .99923 .05069 .99839 .07411 .99725 45 16 . 00465 . 99999 . 0221 1 i . 99976 .03955 .99922 .05698 .99838 .07440 .99723 44 17 . 00495 . 99999 . 02240 . 99975 .03984 .99921 .05727 .99836 .07469 .997'21 43 18 .00524 .99999 .02209 .99974 ! .04013 .99919 .05756 .99834 .07498 .99719 42 19 .00553 .99998 .02298 .99974 .04042 .99918 .05785 .99833 .07527 .99716 41 20 .00582 .99998; .02327 .99973 .04071 .99917 .05814 .99831 .07556 .99714 40 21 .00611 .99998 1.02356 .99972 .04100 .99916 .05H44 .99829 .07585 .99712 39 22 .00640 .9<)'.I98 .02385 .99972; .04129 .99915 .05873 .99827 .07614 .99710 38 23 .00669 .99998 i .02414 .99971 1.04159 .99913 .05902 .99826 .( 7643 .99708 37 24 . 00698 . 99998 i . 02443 i . 99970 .04188 .99912 .05931 .99824 .07672 .99705 36 25 .00727 .99997 .02472 .99969 ; .04217 .99911 .05960 .99822 .07701 .99703 35 26 .00756.99997 .02501 .99969 i .04246 .99910 ; .05989 .99821 .07730 .99701 34 27 .00785 .99997 .02530 .99968 ! .04275 .99909! .06018 .99819 .07759 .99699 33 28 .00814:. 99997 .02560 [.99967 .04304 .99907 i .06047 .99817 .07788 .99696! 32 29 .00844 .99996 .025891.99966 .04333 . 99906 : ! .06076 .99815 .07817 .99094 31 30 .00873:. 99996 .02618 .99966 .04362 .99905 | .06105 .99813 .07846 .99692 30 31 .00902 '.99996 .02647 .99965 .04391 .99904 .06134 .99812 .07875 .99689 29 32 .00931:. 99996 .02676 .99964 .04420 .99902 .06163 .99810 .07904 .99087 28 33 .00960 .99995 .02705 .99963 .04449 .99901 .06192 .99808 .07933 .99685 27 34 .00989 .99995 .02734 .99963 1.04478 .99900 .00:J21 .99800 .07962 . 99683 j 26 35 .01018 .99995 .02763 .99962 .045071.99898 .06250 .99804 .07991 .99080 25 36 .01047 .99995; .02792 .99961 .0-1536!. 99897; .00279 .99803 .08020 .99678; 24 37 .01076 .99994 .02821 .99960 .04565 .99890 .00308 .99801 .08049 .99670 23 38 .01105 .99994 .02850 .99959 .04594 .99894 .06387 .99799 .08078 .99673 22 39 .01134 .99994 .02879 .99959 .04023 .99893!:. 06306 .99797 .08107 .99671 21 40 .01164.99993 .02908 .99958 .04653 .99892 .06395 .99795 .08136 .99668 20 41 .01 193 '.99993 .02938 .99957 .04682 .99890 .06424 .99793 .08165 .99666 19 42 .01222 .99993 .02967 .99956 1.04711 .99889 .00453 .99792 .08194 .99001 1 < 43 .01251 .99992 ,02996 .99955 .04740 .99888 .00482 .99790 .08223 .99001 IT 44 .01280 .99992 .03025 .99954 .04769 .99886 .06511 .99788 .08252 .99059 10 45 .01309 .99991 .03054 .99953 .04796 .99885 .06540 .99786 .08281 .99057 15 46 .01338 .99991 .03083 .99952! .04827 .99883 .06569 .99784 .08310 .99054 14 47 .01367 .99991 .03112 .99952 '. 04856 .99882 .06598 .99782 .08339 .99052 13 48 .01396 .99990 .03141 . 99951 li. 04885 . 99881 !!. 06627 .99780 .08368 .99019 12 49 .01425 .99990 .03170 .99950! .04914 .99879 .00056 .99778 .08397 .99647 11 50 .01454 .99989 .03199 .99949 .04943 ,.99878 .00685 .99776 .08426 .99644 10 51 .01 483 '.99989 .03228 .99948 ''.04972 .99876 .06714 .99774 .08455 .99642 9 52 .01513 .99989 .03257 .99947! .05001 .99875 1.067431.99772 .08484 .99039 8 53 .01542 .99988 .03286 .99946 .05030 .99873 .06773 .99770 .08513 .99037 7 54 01571 .99988 .03316 .99945;!. 05059 .99872 .06802 .99708 .08542 .99035 6 55 .01600 .99987 .03345 .99944 1|. 05088 .99870 : .06831 .99706 .08571 .99032 5 56 .01629 .99987 .03374 .99943 1.05117 .998(59 .Of 5860 .99764 .08600 .99630 4 57 .01658 .99986 .03403 .99942 .05146 .99867 .06889.99762 .08029 .99627 3 58 | .01687 .999SO .03432 .99911 .05175 . 99866 ' .00918 .99700 .08658 .99625 2 59 .01716 . .03461 .99940 .05205 .99804 .00947 .99758 .osr,S7 .99022 1 60 .01745 .99985 .03490 .99939 .0;V>;54 .!)23 .16361 .JW552 35 26 .09469 .99551 .11205 .99370 .12937 .99160 .14666 .9891 9 .16390 .98648 34 27 .09498 .99548 .11234 .99367 .12966 .99156 .14695 .989141 .16419 .98(543 33 28 .09527 .99545 .11203 .99364 .12995 .99152 .14723 .98910 .16447 .<(SO:!S 32 29 .09556 .99542 .11291 .13024 .99148 .14752 .98906 . 16476 31 30 .09585 .99540 .11320 ! 99357 .13053 .99144 .14781 .98902; .16505 .'98029 30 31 .09614 .99537 .11349 .99354 .13081 .99141 .14810 .98897 .16533 .98624 29 32 .09642 .99534 .11378 .99351 .13110 .99137' .14838 .16502 .98019 28 33 .09671 .99531 .11407 .99347 .13139 .99133 .14867 ! 98889 .10591 .98014 27 34 .09700 .99528 .11436 .99344 .18168 .99129 .14896 .98884 .1(56,20 26 35 .09729 .99526 .11465 .99341 .13197 .99125 .14925 .98880 .16648 ! 98604 25 36 .09758 .99523 .11494 .99337 .132261.99122 .14954 .98876 .16677 .98(500 24 37 .09787 .99520 .11523 .99334 .132541.99118 .14982 .98871 .16706 .98595 23 38 .09816 .99517 .11552 .99331 .13283 .99114 .15011 .98867 .16734 .9S590 22 39 .09845 .99514 1.11580 .90327 .13312 .99110 .15040 .98863 .16763 .9S5S5 21 40 .09874 .99511 1.11609 .99324 .13341 .99106 .15069 .98858 .16792 .98580 20 41 .09903 .99508 .11638 .99320 .13370 .99102^ .15097 .98854 .16820 .98575 19 42 .09932 .oo .11667 .99317 .133!)9 .99098 .15126 .98849 .1(5849 .9857'0 18 43 .09961 .99503 .11696 .99314 .13427 .99094 .15155 .98845 .16878 .98565 17 44 .09990 .99500 .11725 .99310 .13456 .99091 .151841.98841 .1690(5 .98561' 16 45 .10019 .99497 .11754 99307 .13485 .99087 .15212 .98836 .16935 .98556 15 46 .10048 .99494 .11783 .99:303 . 13514 .99083 .15241 .JISKJ2 .16964 .98551 14 47 .10077 .99491 .11812 .99:300 .13543 .99079 .15270 .1)8827 .10992 .98546 13 48 .10106 .99488 .11840 .99297 .13572 .99075; .15299 .98823 .17021 .98541 12 49 .10135 .99485 .11869 .99293 .13600 .99071 .15327 .98818 .17050 .98536 11 50 .10164 .99482 1 .11898 .99290 .1:3029 .<)<)007 .15356 .98814 ;. 17078 .98531 10 51 .10192 .99479 ' .11927 .99286 .13658 '.99063 .15385 .98809 .17107 .98526 9 52 .10221 .99476 .11950 .99283 .13687 .99059 .15414 .<)8S05 .17130 .98521 8 53 .10250 .99473 i .11985 .9!)279 .13716 .'0055 .15442 .98800 .17164 .98516 7 54 .10279 .99470 ! . 12014 .99276 .13744 .99051 .15471 .98796 .17193 .98511 6 55 .10308 .99467 .12043 .99272 .13773 .99047 .15500 .98791 .17222 .98506 5 56 .10337 .994(54 .12071 .99269 .13802 .99043 .15529 .98787 .17250 .98,501 4 57 10366 .99461 .12100 .1:3831 .99039 .15557 .98782 ! .17279 .98496 3 58 .10395 99458 .121291.99262 .13860 .99035 .15586 .98778 .17308 .98491 2 59 .104241.99455 .12158 .99258 .13889 .99031 .15615 .98773 .17336 .98486 1 60 .10153 .99452 .12187 .99255 .13917 .99027 .15643 .98769 . 17365: .98481 J) "7 Cosiu j Sine Cosin Sine Cosiu Sine Cosin Sine Cosin Sine . 84 83 82 81 80 28G TABLE X. SINES AND COSINES. 10 || 11 12 13 || 14 Sine Cosin Sine Cosin Sine Cosin Sine Cosin Sine Cosin .17365 .98481 .19081 .98163 .20791 .!>7S15 .22495 .97437 .24192 .97030 60 .17393 .98476 .19109 .98157 .20820 .97809 .22523 .974:30 ! .24220 .97023 59 2 .17'422 .98471 .19188 .98152 .20848 .97803 . 22552 ; . 97424 .2424!) .97015 58 3 .17'451 .98466 i. 191 67 .98146 1.20877 .97797 .22580 .97417 1 .24277 .97008 57 4 .17479 .98461 .19195 .98140 .20905 .97791 .22608 .97411 .24305 .97001 56 5 .17'508 .98455 .19224 .98135 .20'j:$3 .97784 .22637 .97404 .24333 .96994 55 6 .17'537 .98450 .1925.2 .98129 .20962 .9777'8 .22665 ,07398 .24362 .96987 54 7 .17565 .98445 .19281 .98124 .20990 .97772 .22693 . 97391 ii .24390 .96980 53 8 .17591 .98440 .19309 .98118 .21019 .97766 22722 .97384! .24418 .96973 52 9 .17623 .98435 . 19338 .98112 .21047 .97760 .22750 .97378! .24446 .96966 51 10 . 17651 ,98480 .19366 .98107 .21076 .97754 .2277 .97371 .24474 .96959 50 11 .17680 .98425 .19395 .98101 .21104 .97748 .22807 .97365 .24503 .96952 49 .17708 .98420 .19423 .98096 .21132 .97742 .22835 .97:358 .24531 .96945 48 13 .17737 .98414 .19452 .98090 .21161 .97735 .22863 .97351 .24559 .96937 47 14 .17766 .98109 .19481 .98084 .21189 .97729 .22892 .97345 .24587 .96930 46 15 .17794 . 98404 .19509 .98079 .21218 .97723 .22920 .97338 .24615 .96923 45 16 .17823 .98399 .195J38 .98073 .21246 .97717 .22948 i. 97331 i! .24644 .96916 44 17 .17852 .98394! .19566 .98067 .21275 .97711 .22977 .97323 .24672 .96909 43 18 .17880 .98389 .19595 .98061 .21303 .97705 .23005 . 97318 ii .247'(K) .96902 42 19 .17909 .98:38311. 19623 .98056 .21331 .97698 .23033 .97311 .24728 .96894 41 20 .17937 .98378 |.19052 .93050 .21360 .97692 .23062 . 97304 |j .24756 .96887 40 21 .17966 .98373 .19680 .98044 .21388 .97686 .23090 .97298 .24784 .96880 39 22 23 .17995 .18023 .98368 .98362 .19709 .19737 .98039 .93033 .21417 .21445 .97680 .97673 .23118 .23146 .972911 .24813 .97284 .24841 .96873 38 .96866! 37 24 .18052 .98357! .19766 98027 .21474 .97667 .23175 .97278 .248691.96858 36 25 . 18081 .98352: .19794 93021 .21502 .97661 .23203 .97271 .24897 .96851 -35 26 .18109 .98347 .19823 98016 .21530 .97655 .232311.97264 .24925 .96844 34 27 .18138 .98341 .19851 98010 .21559 .97648 .23200 .97257 .24954 .96837 33 28 .18166 .98336 .19880 98004 ; .21587 .97642 .23288 .97251 ! .24982 .96829 32 29 . 18195 .98331 .19908 97993 ! .21616 .97636 .23316 . 97244 .25010 .96822 31 30 .18224 .98325 .19937 97992 .21644 .97630 .23345 .97237 .25038 .96815J 3 31 .18252 .98320 .19965 97987' .21672 .97623 .23373 .97230 .25066 .96807 29 32 .18281 .98315 .19994 97981 .21701 .97617 .23401 .97228 .25094 .96800 28 33 .18309 .98310 ; .20022 9797'5 .21729 .97611 .23429 .97217 .25122 .96793 27 34 .18338 .93304 .20051 97969 .21738 .97604 .23458 .97210 ! .25151 .96786 26 35 .18367 .98299 .20079 97963 .21786 .97598 .23486 .97208 .25179 .96778! 25 36 . 18395 .93294 .23108 97958 ! .21814 .97592 .23514 .97196 .25207 .96771 24 37 .18424 .98288 i .20136 97952 .21843 .97585 .23542 .97189 .25235 . 96764 i 23 38 .18452 .98283 .20165 97946 ! .21871 .97579 .23571 .97182 i .25263 .96756 22 39 .18481 .98277 .20193 97940 .21899 .!)7573 .23599 .97176: .25291 '.96749 21 40 .18509 .98272 .20222 97934 .21928 .97566 .23627 .9716U . 25320 j. 96742 20 41 .18538 .98267 .20250 97923 .21956 .97560 .23656 .97162 .2,5348 '.96734 19 42 .18567 .98261 .20279;. 97922 i .21985 .97553 .23684 .97155 .25376 .96727 18 43 .18595 .98256 .20307 .97916 .22013 .97547 .23712 .97148 .254041.96719 17 44 .18624 .98250 .20336 .97910 .22041!. 97541 ' .23740 .97141 .254321.96712 16 45 .18652 .98245 .20364 .97905 .22070 .UIWH .23769 .97134 . 25460 J.9G705 15 46 .18681 .98240, .20393 .97899 .22093:. 97528 .23797 .97127 .254881.96697 14 47 . 18710 .98234 .20421 .97893 .22126 .97521 .23825 .97120 .255161.96690 13 48 .18738 .93-2!) .20450 .97887 . 22155 1 . 97515 .23853 .97113: . 25545 i. 96682 12 49 . 18767 .'13223 .2047.3 .97881 : .22183 .97508 .2:3882 .97106" .25573 .96675 11 50 .18795 .98218 i .20507 1 .97875 j i .22212 .97502 .23810 .97100 .25601 .96667 10 51 .18824 . 98212 . 20535 . 97869 . 22240 . 97496 . 23938 .97093! .25629 .96660 9 52 .18852 .98207 .20563 .97863 .22268 .97489 .2396") .97086 .25657 .96653 8 53 .18881 .98201 .20592 .97857 .22297 .97483 .23995 .97079 .25685 .96645 7 54 .18910 .98196 .20620 .97851 .22325 .97476 .24023 .97072 .25713 .96638 6 55 .18938 .98190 .20649 .97845 .22353 .97470 .24051 .97065 .25741 .966:30 5 56 . 18967 .98185! -.20677 .97839 .22382 .97463 .24079.97058 .25769.96623 4 57 . 18995 .!S17!) .20706 .97833 .22410 .97457 .24108 .97051 .25798 .96615 3 58 .19024 .98174' .207:34 .97827 .22438 .97450 .24136 .97044 .25826 .96608 2 59 .19052 . '. M i ( iS . 20763 . 97821 ' . 22467 . 97444 . 24 1 64 . 97037 . 25854 . 96600 1 60 .19081 .98163J .20791 .97815 .22495 .97437 .24192 . 97030 ; . 25882 .96593 j Cosin Sine j Cosin i Sine Cosin Sine Cosin Sine Cosin Sine J 79 78 77 76 75 387 TABLE X. SINES AND COSINES. 15 16 17 ! 18 19 Sine Cosin Sine Cosin Sine j Cosin Sine Cosin Sine Cosin .25882 . 96593 'j. 27564 .96126 .29237 .95630 .30902 .95106 .82567 .0155-> 60 1 .25910 .96585 ; .27592 .96118 .21W!i.j .95622 .30929 .95CJ7 .82584 .945421 59 2 .35938 .96578 .27620 .96110 .29293 .95613 .30957 .95088 .32612 .915:5:! 58 8 .95966 .9(5570 .27(548 .96102 .29321 .95605 .30985 .95079 .32639 .9452:3 57 4 !. 25994 .96562 .27676 .96094 .29348 .95596 .31012 .95070 .32667 .94514 56 5 .26022 .96555 .27704 .96086 .29376 .95588 .31040 .95061 .32694 .94504 55 6 .26050 .96547 .27731 .96078 .29404 .95579 .31068 .95052 .32722 .94495 54 7 1. 26079 ! . 96540 I .27759 96070 .29432 .95571 .31095 .95043 .32749 .94485 53 8 .26107 .96532 | .27787 .96062 .294601.95562 .31123 .95033 .32777 .94476 52 9 .26135 .96524 ! .27815 .96054 .29487 .9555 i .31151 .95024 .32804 .94466 51 10 .26163:. 96517 .27843 .96046 .29515 .95545 .31178 .95015 .32832 .94457 50 11 .26191 .96509 .27871 .96037 .29543 .95536 .31206 .95006 .32859 .94447 49 12 .26219:. 96502 .27899 .96029 .29571 .95528 .81238 .94997 ! 32887 .944:38 48 13 .26247 .96494 ! .27927 .96021 .29599 .95519 .31261 .94988 .32914 .94428 47 14 .26275 .96486 .27955 .96013 .29626 .95511 .31289 .94979 .32942 .94418 46 15 .26303 .96479 .27983 .96005 .29654 .95502 .31316 .94970 .32969 .94409 45 16 .26331 .96471 .28011 . 95997 : .29682 .95493 .31344 .94961 .32997 .94399 44 17 .26359 .96463 .28039 .95989 .29710 .95485 .31372 .91953 .33024 .94390 43 18 .2(3387 .96456 .28067 .95981 .29737 .9.5476 .81899 .94948 .33051 .948801 42 19 .26415 .96448 .28095 .95972 .29765 .95467 .31427 .94933 : .33079 .94370 41 20 .26443 .96440 .28123 .95964 .29793 .95459 .31454 .94924 .33106 .94361 40 21 .26471 .96433 .28150 .951)56 .29821 .95450 .31482 .91915 .33134 .94351 39 22 .26500 .96425 .28178 .95948: .29849 .95441 .31510 .94906 .33161 .94.'3W 38 23 .26528 .96417 .28206 .95940 .29876 .95433 .31537 .94897 .33189 .94332 37 24 .26556 .96410 .28234 .95931 .299041.95424 .31565 .94888 .33216 .94822 86 25 .26584 .96402 .28262 .95923 .29932 .95415 .31593 .94878 .33244 .94313; 35 26 .26612 .96394 .28290 .959151 .29960 .O.Vldi' .31620 .94869 .33271 .9 1303 34 27 .26640 .96:386 .28318 .95907 .29987!. 95398 .31648 .94860 .33298 .91-20:] 3:3 28 .26668 .96379 .28346 .95898 .30015 .95389 .31675 .94851 .33326 .912S1 33 29 .26696!. 96371 .28374 .95890 .30043 .95:380 .31703 .94842 .33353 .91271 31 30 .26724 .96363, .28402 .95882 .30071 .95372 | .31730 .94832 .88881 .94264 30 31 .26752 .96355 .28429 .95874 .30098 .95363 1 .31758 .94823 .33408 .94251 29 32 .26780 .96347 .28457 .95865 i .30120 .95351 .31786 .94814 .33436 .942451 28 33 .26808 .96340 .28485 .95857 .30154 .95345 .31813 .94805 33463 .94235 27 34 .26836 .96:332 .28513 .95849 .30182 .95337 .31841 .94795 .33490 .94255 26 35 .26864 .96324 .285411.95841 .30209 .95328 i .31868 .94786 .33518 .94215 25 36 .26892 .96316 .28569 .95832 .30237 .95319 i .31896 .94777 .33545 .94206 24 37 .26920 .96308 .28597 .95824 .30265 .95310 i .31923 .94768 .33573 .91196 J.':} 38 39 .26948 .96301 .26976 .90293 .28625 .28652 .95816 .95807 .30292 .30320 .95301 .95293 | .31951 1 .31979 .94758 .94749 . 33600 j. 941 86 j 23 .33627 .94176] 21 40 .27004 .96285 .28680 .95799 j .30348 .95284 !.320QG .94740 .3:3655 .94167 20 41 .27032 .96277 .28708 .95791 .30376 .95275 .32034 .94730 .83682 .94157 19 42 .27060 .96269 .28?'36 .95782 .30403 .952(5(5 ' .32001 .94721 .88710 .94147 H 43 .27088 .96261 .28764 .95774 .30431 .95257 .32089 .94712 .38737 .94137 17 44 .27116 .96253 .28792 .95766 .30459 .95248 : .32116 .94702 .33764 .941271 16 45 .27144 .96246 .28820 .95757! .30486 .95240 .32144 .94693 .33792 .9111* 15 46 .27172 .96238 .28847 .95749! .30514 .95231 ! .32171 .94684 .33819 .94108 14 47 .27200 .96230 .28875 .95740 .30542 .95222 .32199 .94674 .33846 .91098 13 48 .27228 .96222 .28903 .95732 .30570 .95213 .32227 .94665 .83874 .9 tOSS 12 49 .27256 .96214; .28931 .95724 .30597 .95201 .82254 .91656 33901 .94078 11 50 .27284 .96206 .28959 .95715 .30625 .95195 .32282 .9464o' .33929 .94068 10 51 .27312 .96198 .28987 .95707 .306531.95186 i .32309 .94637 .33956 .940.58 9 52 .27340 .96190 .29015 .95698 .30680 .95177 .32337 .94627 .3:398:3 .91049 8 53 .27368 .96182 .29012 .95690 .30708 .95168 .32364 .94618 .34011 .94039 7 54 .27396 .96174 .29070 .95681 .30736 .95159 .32392 .94609 .34038 .94029 6 55 .27424 .96166 .29098 .95673 .30763 .95150 .32419 .91599 .34065 .94019 5 56 .27452 .{Mil 58 .29126 .95661 .30791 .95142 .82447 .94590 .34093 .94009 4 ! 57 .27480 .96150 .39154 .0565(i .30819 .95133 .32474 .o:$5so .:!(! 20 .93999 3 i 58 .27508 .96142 .20182 .95647 .30846 .95124 .32502 .91571 .34147 .93089 2 59 .27536:. 961 34 .29909 .9563!) .30874 .95115 .:!2529 .94561 .34175! : 9:3970 1 60 . 27564 i. 96126 .29237 .95630 .::0902 .95106 .94552 .84302 .93969 - Cosin Sine Cosin 8tne Cosin Sine Cosin Sine Cosin Sine" / 1 / 74 73 72 71 70 388 TABLE X.-SINES AND COSINES. - 20 11 21 22 23 24 / Sine Cosin Sine Cosin Sine Cosin Sine i Cosin Sine Cosin ~o .34202 .939(55) .35837 .9-3358 .37461 .92718 .39073 .92050 .40674 .91355 60 1 .34229 .93959 .35864 .93348 .37488 .98707 .39100 .92039 .40700 .91343 59 2 .34257 .93949 .35891 .93337 .37515 .92697 .39127 .92028 .40727 .91331 58 3 .34284 .93939 .35918 .93327 .37542 .92686; .39153 .92016 .40753 .91319 5? 4 .34311 .93929 .35945 .93316 .37569 .92675 .39180 .92005 .40780 .91307 56 5 .34339 .93919 .35973 .93306 .37595 .92664 .39207 .91994 .40806 .91295 55 6 .34366 .93909 .36000 .93295 .37622 .92653 .39234 .91982 .40833 .91283 54 7 .34393 .93899 .36027 .93285 .37649 .92642 .39260 .91971 .40860 .91272 53 8 .34421 .93889 .36054 .93274 .37676 .92631 .39287 .91959 .40886 .91260 52 9 .34448 .93879 .36081 .93264 .37703 .92620 .39314 .91948 .40913;. 91248 51 10 .34475 .93869 .36108 .93253 .37730 .92609 .39341 .91936 .40939;. 91236 50 11 .34503 .93859 .36135 .93243 .37757 .92598 .39367 .91925 .40966 .91224 49 12 .34530 .93849 .36162 .93232 .377841.92587 .39394 .91914 .40992 .91212 48 13 .34557 .9:3839 .36190 .93222 .37811 .92576 .39421 .91902 .41019 .91200 47 14 .34584 .93829 .36217 .93211 .37838 .92565 .39448 .91891 .41045 .91188! 46 15 .34612 .93819 .36244 .93201 .37865 .92554 .39474 .91879 .41072 .91176 46 16 .34639 .93809 .362711.93190 .37892 .92543 .39501 .91868 .41098 .91164 44 17 .34666 .93799 .36298 .93180 .37919 .92532 .39528 .91856 .41125 .9115-2 43 18 .34694 .93789 .36325 .93169 .37946 .92521 .39555 .91845 .41151 . 91140 j 42 19 .34721 .93779 .36352 .93159 .37973 j.92510 .39581 .91833 .41178 .91128 41 20 .34748 .93769 .36379 .93148 .37999 .92499 .39608 .91822 .41204 .91116 40 21 .34775 .93750 .36406 .93137 .38026 .92488 .39635 .91810 .41231 .91104 39 22 .34803 .93748 .36434 .93127 .38053 .92477 .39661 .91799 .41257 .91092: 38 23 .34830 .93738 .36461 .93116 .38080 .92466 .39688 .91787 .41284 .91080; 37 24 .34857 .93728 .36488 .93106 .38107 .92455 .39715 .91775 .41310 .91068 36 25 .34884 .93718 . 36515 !. 93095 .38134 .92444 .39741 .91764 .41337 .91056 35 26 .34912 . 93708 . 36542 . 93084 . 38161 .92432 .39768 .91752 .41363 .91044: 34 27 .34939 .93698 .36569 .93074 .38188 .92421 .39795 .91741 .41390 .91032 33 28 .34966 .93688 .36596 .93063 i .38215 .92410 .39822 .91729 .41416 .91020 32 29 .34993 .93677 .36623 .93052 1 .8241 .92399 .39846 .91718 .41443 .91008 31 30 .35021 .93667 .36650 .93042 ; .38268 .92388 .39875 .01706 .41469 .90996 30 31 .35048 .9365? .36677 .93031 i .38295 .92377 .39902 .91694 .41496 .90984 29 32 .35075 .93647 l .36704 .93020 .38322 .92366 .39928 .91668 .41522 .90972 28 33 .35102 .93637 .36731 .93010 .38349 .92355 .399551.91671 .41549 .90960 27 34 .35130 .93626 .36758 .92999 .38376 .92343 .39982 .91660 .41575 .90948 26 35 .35157 .93616 .36785 .92988 ! .38403 .92332 ! .40008 .91648 .41602 .90936 25 36 .35184 .93606 .36812 .92978 .384:30 .92321 .40035 .91636 .41628 .909241 24 37 .35211 93596 .368391.92967 .38456 .92310 .40062 .91625 .41655 .90911 23 38 39 .35239 .35266 .93585 .93575 .36867 .36894 .92956 .92945 .38483 .38510 .92299 i .40088 .92287 1 .40115 .91613 .91601 .41681 .41707 .908991 22 .908871 21 40 .35293 .93565 .36921 .92935 .38537 .92276 ! .40141 .91590 .41734 .90875: 20 41 .35320 .93555 . 36948 ! . 92924 .38564 .92265 .40168 .91578 .41760 .90863 19 42 .35347 .93544 . 36975 !. 92913 .38591 .92254 .40195 .91566 .417871.90851 18 43 .35375 [.93534 .37002 .92902 .38617 .92243 .40221 .91555 .41813 .90839 17 44 .&5402 .9a524 .37029 .92892, .38644 .92231 .40248 .91543 .41840 .90826 16 45 .35429 .93514 .37'056'. 92881 .38671 .92220 .40275 .91531 .41866 i. 90814' 15 46 .35456 .93503 . 37083 ! . 92870 .38698 .92209 .40301 .91519 .41892 .90802 14 47 .35484 .93493 .37110 .92859 .38725 .92198 .40328 .91508 .41919 .90790! 13 48 .35511 .93483 .37137). 92849 .38752 .92186 .40355 .91496 .41945 .90778 12 49 i 35538 ! 93472 .37164 .92838 .38778 .92175 .40381 .91484 .41972 .90766 11 50 .35565 .93462 .37191 .92827 .38805 .92164 .40408 .91472 .41998 .90753 10 51 .35592 .93452 .37218 .92816 .38832 .92152 .40434 .91461 .42024 .90741 9 52 .35619 .93441 .37245 .92805 .38859 .92141 .40461 .91449 .42051 .90729 8 53 .356471.93431 .37272 .92794 .38886 .92130 .40488 .91437 .42077 .90717 7 54 .35674 .93420 .37299 .92784 ! .38912 .92119 .40514 .91425 .42104 .90704 6 55 . 35701 ; . 93410 .37326 .92773 i. 38939 1.92107 .40541 .91414 .42130 .90692 5 56 .35728 .93400 .37353 .92762 ; .38966 .92096 .40567 .91402 .42156 .90680 4 57 .35755 .93389 .37:380 .92751 .38993 .92085 .40594 .91390 .42183 .90668 3 58 .35782^.93379 .37407 .92740 .39020 .92073 .40621 .91378 .42209L90655 2 59 .35810 .93368 .37434 .92729 .39046 .92062 .40647 .91366 .422351.90643 1 60 .35837 .93358; .37461 .92718 .89078 .92050 .40674 .91355 .42262 .90631 / Cosiu 'Sine Cosin j Sine Cosin Sine Cosin Sine Cosin Sine / 69 || 68 67 II 66 65 TABLE X. SINES AND COSINES. 25 I 26 27 OOo 29 Sine Cosin Sine Cosin Sine Cosin Sine | Cosin Sine Cosin / .422621.90631 .43837 .89879 .45399 .89101 .46947 .88295 .48481 .87462 60 1 ,42288|.90618 .43863 .89867 .45425 .89087 .46973 .88281 .48506 .87448 59 2 .423151.90606 .43889 .89854 .45451 .89074! .469991.88267 .48532 .87434 58 3 .42341 .90594 .43916 .89841 .45477 .89061 1.470241.88254 .48557 .87420 57 4 .42367 .90582 .43942 .89828 .45503 .89048 .47050 .88240 .48583 .87406 56 5 .42394 .90569 .43968 .89816 .45529 .89035 .47076 .88226 .48608 .87391 55 6 .42420 .90557 .43994 .89803 .45554 .89021 .47101 .88213 .48634 .87377 54 7 .42446 .90545 .44020 .89790 .45580 .89008 .47127 .88199 .48659 .87363 53 8 .42473 .90532 .44046 .89777 .45606 .88995 .47153 .88185 .48684!. 87349 52 9 .42499 .90520 .44072 .89764 .45632 .88981 .47178 .88172 .48710 .87335 51 10 .42525 .90507 .44098 .89752 .45658 .88968 .47204 .88158 .48735 .87321 50 11 .42552 .90495 .44124 .89739 .45684 .88955 .47229 .88144 .48761 .87306 49 12 .42578 .90483 .44151 .89726 .45710 .88942 .47255 .88130 .48786 .87292 48 13 .42604 .90470 .44177 .89713 .45736 .88928 .47281!. 88117 .48811 .8727-8 47 14 .42631 .90458 .44203 .89700 .45762 .88915 .47306 .88103 .48837 .87264 46 15 .42657 .90446 .44229 .89687 .45787 .88902 .47-332 .88089 .48862 .87250 45 16 .42683 .90433 .44255 .89674 .45813 .88888 .47358 .88075 .48888 .87235 44 17 .42709 .90421 .44281 .89662 .45839 .88875 .47383 .88062 .48913 .87221 43 18 .42736 .90408 .44307 .89649 .45865 .88862 .47409 .88048 .48938 .87207 42 19 .42762 .90396 .44333 .89636 .45891 .88848 .474341.88034 .48964 .87193 41 20 .42788 .90383 .44359 .89623 .45917 .88835 i .47460 .88020 .48989 .87178 40 21 .42815 .90371 .44385 .89610 .45942 .88822 .47486 .88006 .49014 .87164 39 22 .42841 .90358 .44411 .89597 .45968 .88808 .475111.87993 .49040 .87150 38 23 24 .42867 .42894 .90346 .90334 .44437 .44464 .89584 .89571 .45994 .46020 .88795 .88782 .47537 .87979 .47562 .87965 .49065 .49090 .87136 .87121 37 36 25 .42920 .90321 .44490 .89558 .46046 .88768 .47588 .87951 .49116 .87107 35 26 .42946 .90309 .44516 .89545 .46072 .88755 .47614 .87937 .49141 .87093 34 27 .42972 .90296 . 44542 l . 89532 .46097 .88741 .47639 .87923 .49166 .87079 33 28 .42999 .90284 .44568 .89519 .46123 .88728 .47665 .87909 .49192 .87064 32 29 .43025 .9)271 .44594 .89506 .46149 .88715 .47690 .87896 .49217 .87050 31 30 .43051 .90259 .44620 .89493 .46175 .88701 .47716 .87882 .49242 .87036 30 31 .43077 .90246 .44646 .89480 .46201 .88688 .47741 .87868 .49268 .87081 29 32 .43104 .90233 .44672 .89467 .46226 .88674 .47767 .87854 .49293 .87007 28 33 .43130 .90221 .44698 .89454 .46252 .88661 .47793 .87840 .49318 .86993 27 34 .43156 .90208 .44724 .89441 .46278 .88647 .47818 .87826 .49344 .86978 26 35 .43182 .90196 .44750 .89428 .4630-1 .88634 .47844 .87812 .49369 .86964 25 36 .43209 .90183 .44776 .89415 .46330 .88620 .47869 .87798 .49394 .86949 24 37 .43235 .90171 .44802 .89402 .46355 .88607 .47895 .87784 .49419 .86935 23 38 .43261 .90158 .44828 .89389 .46381 .88593 . 47920 !. 87770 .49445 .86921 22 39 .43287 .90146 .44854 .89376 .46407 1.88580 .47946 .87756 .49470 .86906 21 40 .43313 .90133 .44880 .89363 .46433 .88566 .47971;. 87743 .49495 .86892 20 41 .43340 .90120 .44906 .89350 .46458 .88553 . 47997 !. 87729 .49521 .86878 19 42 .43366 .90108 .44932 .89337 . 46484 ! . 88539 .48022 .87715 .49546 .86863 18 43 .43392 .90095 .449581.89324 .46510 .88526 .48048 .87701 .49571 .86849 17 44 .43418 .90082 . 44984 !. -89311 .46536 .88512 .48073 .87687 .49596 .86834 16 45 .43445 .90070 .45010 .89298 .46561 .88499 . 48099 ; . 87673 .49622 .86820 15 46 .43471 .90057 .450361.89285 .46587 .88485 .48124 .87659 .49647 .86805 14 47 .43497 .90045 .45062 .89272 .46613 .88472 .48160 .87645 .4967'2 .86791 13 48 .43523 .90032 .45088 .89259 .46639 .88458 .481751.87631 .49697 .86777 12 49 .43549 .90019 .45114 .89245 .46664 .88445 .48201 .87817 .49723 .86762 11 50 .43575 .90007 .45140 .89232 .46690 .88431 .48226 .87603 .49748 .86748 10 51 .43602 .89994 .45166 .89219 .46716 .88417 .48252 .87589 .49773 .86733 9 52 .43628 .89981 .45192 .89206 .46742 .88404 .48277 .87575 .49798 .86719 8 53 .43654 .89968 .45218 .89193 .46767 .88390! .48303 .87561 .49824 .86704 7 54 .43680 .89956 .45243 .89180 .46793 .88377 .48328 .87546 .49849 .86690 6 55 .43706 .89943 .45269 .89167 .46819 .88363 .48354 .87532 .49874 .86675 5 56 .43733 .89930 .45295 .89153 .46844 .88349 .48379 .87518 .49899 .86661 4 57 .43759 .89918 .45321 .89140 .46870 .88336 .48405 .87504 .49924 .86646 3 58 .43785 .89905 .45347 .89127 .46896 .88322 .48430 .87490 .49950 .86632 2 59 .43811 .89892 .45373 .89114 .46921 .88308 .48456 .87476 .49975 .86617 1 60 .43837 .89879 .45399 .89101 .46947 .88295 .48481 .87462 .50000 .86603 _0 / Cosin | Sine Cosin Sine Cosin Sine Cosin j Sine Cosin Sine / 64 63 62 61 60 TABLE X. -SINES AND COSINES. 30 31 32 33 34 / Sine Cosin Sine Cosin Sine 1 Cosin Sine ! Cosin Sine Cosin ~0 750000 .86603 ".51504 .85717 752992 : . 84805 .54464!. 83867 755919 ."82904 60 1 .50025 .86588 .51529 .85702 . 53017 ! . 84789 .54488 .83851 .55943 .82887 59 2 .50050 .86573 .51554 .85687 .53041 .84774 .54513 .83835 .55968 .82871 58 3 .500761.80559 .51579 .85672 .53066 .84759 .54537 .83819 .55992 .82855 57 4 .50101 .86544 .51604 .85657 .53091 .847'43 .54561 .83804 .56016 .82839; 56 5 .50126 .86530 .51628 .85642 .53115 .84728 .54586 '.83788 .56040 .82822! 55 6 .50151 .86515 .51653 .85627 .53140 .84712 .54610 .83772 .56064 .82806 54 7 .50176 .86501 .51678 .85612 .53164 .84697 .54635 .83756 .56088 .82790 53 8 .50201 .86486 .51703 .85597 .53189!. 84681^ .54659 .83740 .56112 .82773! 52 9 .50227 .86471 .51728 .85582 .532141.84666 .54683 .83724 .56136 . 82757 i 51 10 .50252 .86457 .51753 .85567 .53238 .84650 .54708 .83708 .56160 .82741 50 11 .50277 .86442 .51778 .85551 .53263 .84635 .54732 .83692 .56184 .82724 49 12 .50302 .86427 .51803 .85536 .53288 .84619 .54756 .83676 .56208 . 827'08' 48 13 .50327 .8(5413 .51828 .85521 .53312 . 84604 : .54781 .83660 .56232 .82692 47 14 .50352 .86398 .51852 .85506 .53337 .84588. .54805 .83645 .56256 .82675 46 15 .50377 .86384 .51877 .85491 .53361 .84573] .54829 .83629 .56280 .82659 45 16 .50403 .86369 .51902 .85476 .53386 .84557 .54854 .83613 .56305 .82643! 44 17 .50428 .86354 .51927 .85461 .53411 .84542 .54878 .83597 .56329 .82626 43 18 .50453 .86340 .51952 .85446 .53435 .84526 .54902 .83581 .56353 .82610! 42 19 .50478 .86325 .51977 .85431 .53460 .84511 f>4927 .83565 .56377 .82593 41 20 .50503 .86310 .52002 .85416 .53484 .84495 .54951 .83549 .56401 .82577 40 21 .50528 .86295 .52026 .85401 .53509 . 84480 .54975 . 83533 ! .56425 .82561 39 22 .50553 .86281 .52051 .85385 .53534 .84464 .54999 .83517 .56449 . 82544 ! 38 23 .50578 .86266 .52076 .85370 .53558 .84448 .55024 .83501 .56473 .82528; 37 24 .50603 .86251 .52101 .85355 .535a3 .84433 .55048 .83485 .56497 .82511136 25 .50628 .86237 .52126 .85340 .53607 .84417 .55072 .83469 .56521 .82495! 35 26 .50654 .86222 .52151 .85325 .53632 .84402 .55097 .83453 .56545 . 82478 34 27 .50679 .86207 .52175 .85310 .53656 .84386 .55121 .83437 .56569 .82462! 33 28 .50704 .86192 .52200 .85294 .53681 .84370 .55145 .83421 .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 .84308 .55242 .83356 .56689 .82380! 28 33 .50829 .86119 .52324 .85218 .53804 .84292 .55266 .83340 .56713 .82363 27 34 .50854 .86104 .52349 .85203 .53828 .84277 .55291 .83324 .56736 .82347 26 35 36 .50879 .50904 .86089 .86074 .52374 .52399 .85188 .851731 .53853 .53877 .84261 .84245 .55315 1.83308 .55339 .83292 .56760 .56784 .82330 .82314 25 24 37 .50929 .86059 .52423 .85157! .53902 .84230 .55363 .83276 .56808 .82297123 38 .50954 .86045 .52448 .85142 .53926 .84214 .55388 .83260 .56832 .82281 22 39 .50979 .86030 .52473 .85127: .53951 .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 .510541.85985 .52547 .85081 .54024 .81151 .55484 .83195 .56928 .82214 18 43 .51079 .85970 .52572 .85066 .54049 .84135 . 55509 ! . 831 79 'I .56952 .82198 17 44 .51104 .85956 .52597 .85051 .54073 .84120 .55533 .83163! .56976 .82181 16 45 .51129 .85941 .52621 .85035 .54097 .84104 .55557 .83147: .57000 .821651 15 46 .51154 .85926 .52646 .85020 .54122 .84088 .55581 .83131 .57024 .82148 14 47 .51179 .85911 .53671 .85005 .54146 .84072 . 55605 . 831 15 i . 57047 i . 82132 ! 13 48 .51204 .85896 .52696 .84989 .54171 .84057 .55630 .83098 .57071 .82115 12 49 .51229 .85881 .52720 .84974 .54195 .84041 .59654 .83082 .57095 .82098 11 50 .51254 .85866 .52745 .84959 .542201.84025 .55678 .83066 .57119 .82082 10 51 .51279 .85851 .52770 .84943 .54244 .84009 .55702 .83050 .57143 .82065 9 52 .51304 .85836 .52794 .84928 .54269 .83994 .55726 .83034 .57167 .82048 8 53 .51329 .85821 .52819 .84913 .542931.83978 .55750 .83017! .57191 .82032 7 54 .51354 .85806 .52844 .84897 . 54317 j. 83962 .55775 .83001! .57215 .82015 6 68 .51379 .85792 .52869 .84882 .543421.83946 .55799 .82985; .57238 .81999 5 56 .51404 .85777 .52893 .84866 .54366 .83930 .55823 .82969' .57262 .81982! 4 57 .514291.85762 .52918 .84851 .54391 .83915 .55847 .82953 .57286 .81965! 3 58 .51454 .86747 .52943 .84836 .544151.83899 .55871 .82936 .57310 .81949! 2 59 .51479 .85732 .52967 .84820 .54440 .83883 .55895 .82920 .57&S4 . 81932 ! 1 60 .51504 .85717 .52992 .84805 .54464 .83867 .55919 .82904! .57358 .81915| / Cosin Sine Cosin Sine Cosin Sine Cosin "Sine" Cosin Sine i 59 58 57 11 56 55 TABLE X.-SINES AND COSINES. / ~o 35 36 37 38 39 i\n Sine ! Cosin .57358-. 81915 : Sine Cosin 758779 .80902 Sine .60182 Cosin 79864 : Sine | Cosin Sine 01500 78801 : ~(i-> Cosin ~777'1 5 1 .57381:. 81899 .58802 .80885 .60205 .79846 .61589: .78783! 62955 .77090 59 2 .57405 .81882 .58826 .80867; . 60228 . 79829 .6161* . 78705 ! | . 62977 1 . 77678 58 3 .57429 .81865; .58849 .80850: . 00251 . 7981 1 ; . 01 035 . 78747 i . 63000 .77660! 57 4 .57453 .81848, .58873 .80833 .60274!. 79793 .61058 .78729 .63022 .77641 50 5 .57477 .81832! .58896 .80816 .60298 .79776 .61681 .787'lli .030451.77623! 55 6 .57501 .81815 .58920 .80799 .60321 .79758 .61704 .78694 .63068 .77005 54 7 .57524 .81798 .58943 .80782 .60344 .79741 .61726 .78676 .63090 .77586 53 8 .57548 .81782 .58967 .80765 .60307 .79723 .61749 .78658 .63113 .77568 52 9 .57572 .81765 .58990 .80748 .60390 .79706 .61772 .78640; .63135 .77550 51 10 .57'596 .81748 .59014 .80730 .60414 .79688 .61795 .78022; .63158 .77531 50 11 .57619 .81731 .59037 .80713 .60437 .79071 .61818 .78004' .63180 .77513 40 12 .57643 .81714 .59001 .80096 .60460 .79053 .618411.78586; .03203 .77494 48 13 .57667 .81698 .59084 .80079 .60483 .79635 .C1864 .78568 .63225 .77476 47 14 .57691 .81681 .59108 .80002 .60506 .79618 .61887 .78550 .63248 .77458 46 15 .57715 .81664 .59131 .800441 .60529 .79000 .61909 .78532 .03271 .77439 45 16 .57738 .81647 .59154 .80027 .60553 .79583 .61932 .78514 .63293 .77421 44 . 17 j .57762 .81631 .59178 .80610 .60576 .79505 .61955 .78496 .63316 .77402 43 18 .57786 .81614 .59201 .80593 .60599 .79547 .61978 .78478 .63338 .77384 42 19 l .57810 .81597 .59225 .80576 .60622 .79530 .62001 .78460 .63361 .77366 41 20 .57833 .81580 .59248 .80558 .60645 .79512 .62024 .78442 .63383 .77347 40 21 .57857 .81563 .59272 .80541 .60668 .79494 .62046 .78424 .63406 .77329 39 22 .57881 .81546 .59295 .80524 .00091 .79477 .62069 .78405 .63428 .77310! 38 23 .57904 .81530 .59318 .80507, .60714 .79459 .62092 .78387 .63451 .77292 37 24 .57928 .81513 .59342 .80489 .60738 . 79441 1 .62115 .78369 .63473 .77273 36 25 .57952 .81496 .59365 .80472 .60761 .79424! .62138 .78351! .63496 .77255 35 26 .57976 .81479 .59389 .80455 .60784 .79406i .62160 .78333 .63518 .77236 34 27 .57999 .81462 .59412 .80438 .60807 .79388; .62183 .78315! .03540 .77218 33 28 .58023 .81445 .59436 .80420 .60830 .79371 i .62206!. 78297 i .63563 .77199 32 29 .58047 .81428 .594591.80403 60853 .79353 .62229 .78279 .63585 .77181 31 30 .58070 .81412 .59482 .80386; .60876 .79335 .62251 .78261 .63608 .77162 30 31 .58094 .81395 .59506 .80368 .60899 .79318 .62274 .78243 .63630 .77144 29 32 .58118 .81378 .59529 .80351 .60922 .79300 .62297 . 78225 ; .63653 .77125 28 33 .581411.81361 .59552 .80334 .60945 .79282 .62320 .78206 .63675 .77107 27 34 . 58165 j. 81344, .595761.80316 .60968 .79264 .62342 .78188 .63698 .77088 26 35 .581891.81327 .59599J.80299 .60991 .79247 .62365 .78170 .63720 .77070 25 36 .58212 .81310 .59622 .80282 .61015 .79229i! .62388 .78152 i .63742 .77051 24 37 .58236f.81293 .59646 .80264! .61038 . 79211 ji. 62411 .78134 ; .03705 .77033 23 38 . 58260 |. 81276 .59669 .80247; .61061 .79193 .62433 .781 16 i .03787 .77014 22 39 .58283L 81259 .59693 .80230 .61084 .79176 .62456 .780981 .03810 .76996 21 40 .58307!. 81242 .59716 .80212 .61107 .79158; .62479 .78079; .63832 .76977 20 41 .58330 .81225 .59739 .80195 .61130 .79140 .62502 . 78061 ' .63854 .76959 19 42 !. 58354 i. 81208 .59763 . 80178 ; .61153 .79122 .625241.78043 .63877 .70940! 18 43 .58378|.81191 .59786 .80160 .61176 .79105 . 62547 i. 78025 .63899 .70921 17 44 .584011.81174 .59809 .80143 .61199!. 7 9C87 .62570 .78007 .63922 .76903! 16 45 .584251.81157 .59832 .80125 .61222;. 79069 .62592 .77988 .63944 .76884! 15 46 .58449;. 81 140 .59856 .80108 .61245). 79051 .62015 .77970 .03966 .76866 14 47 .584721.81123 .59879 .80091 .61208 .79033 .62038 .77952 .63989 .76847 13 48 .58496 .81106 .59902 .80073 .61291 .79016 .62000!. 77934 .640111.76828 12 49 .58519 .81089 .59926 . 80056 i .61314!. 78998 .62683!. 77916 .64033 .76810 11 50 .58543 .81072 .59949 .80038 .61337 .78980 .62706 .77897 .64056 .76791 j 10 51 .58567 .81055 .59972 .80021 .61360 .78962 ! .62728 .77879 .64078 .76772 9 52 .68590 .81038 .59995 80003 .61383 .78944 .62751 .77861 .64100 .70754: -8 53 .58614 .81021 .60019 .79986 .61406 .78926 .62774 .77843' .64123 .70735! 7 54 .58637 .81004 .60042 .79968 .61429 .78908 .62796 .77824 .64145 .79717 6 55 .58001 .80987 .00005 .79951 .61451 .78891 .62819 .77806 .64167 . 76698 5 56 .58684 .80970 .60089 .79934 .61474 .78873 .62842 .77788 .64190 .76679! 4 57 .587'08 .80953 .60112 .79916 .61497 .78855 j .62864 .77709 .64212 .76001 3 58 ! .58731 .80936 .60135 .79899 .61520 .78837 ; .62887 .77751 .64234 .76642! 2 59 ! .58755 .80919 .60158 .79881 .61543 .78819 .02909 .777-33 .64256 .76623 1 00 .58779 .80902 .60182 .79804 .61566 .78801 .02932 .77715 .64279 .70001 Cosin Sine Cosin Sine Cosin Sine Cosin "Sine Cosin Sine i 1 j . . 54 53 52 51 50 TABLE X. SINES A.ND 40 i! 41 || 42 43 i 44 i \ Sine Cosin ' Sine Cosin ' i Sine Cosin Sine ; Cosin Sine Cosin .64279 .76004 .65600 .75471 .00913 .74314 .68200 .73135 .69466 .71934 60 1 64301 . 70586 .65628 .75452 .66935 .74295 .68221 .73116 .69487 .71914 59 2 .64323 .76567 .65650 .75433 .66956 .74276 .68242 .73096 .69508 .71894 58 3 ' 043 16 .76548 .65672 .75414 .66978 .74256 .68264 .73076 .69529 .71873 57 4 .64308 70530 .65694 .75395 .66999 .74237 .68285 .73056 .69549 .71853 56 5 . 04390 .76511 65716 .75375 .67021 .74217 .08306 .73036 .6957'0 .71833 55 6 '64412 ! 76492 .65738 . 75356 i .67043 .74198 .68327 .73016 .695911.71813 54 7 .044351.76473 .65759 .753371 .67064 .74178 .68349 .72996 .69612 .71792 53 8 .04457 76455 .65781 .75318 .67086 .74159 .68370 .72976 .69633 .71772 52 g .04179 7iU3R .65803 .75299 .67107 .74139 .68391 .72957 .69654 .71752 51 10 .04501;. 76417 .65825 .75280! .67129 .74120 .68412 .72937 .69675 .71732 50 11 .04524 .76398 .65847 .75261 .67151 .74100 .68434 .72917 .69696 .71711 49 12 .04546 70330 .65309 .75241 .67172 .74080 .68455 '.72897 .69717 .71691 48 13 .64508 .70301 .65891 .75232 .67194 .74061 .68476 .72877 .69737 .71671 47 It .64590 70342 .65913 .75203 .67215 .74041 .68497 .72857 .69758 .71650 46 15 .64612 .76323 .65935 .75184 .67237 .74022 .68518 .72837 .69779 : 71630 45 16 .64635 . 76304 .65956 .75165 .67258 .74002 .68539 .72817 .69800 .71610 44 17 .64057 .70236 .65978 .75146 .67280 .73983 .68561 .72797 .69821 .71590 43 18 .04679 .76267 .66000 .75126 .67301 .73963 .68582 72777 .69842 .71569 42 19 .64701 .76248 .60022 .75107 .67323 .73944 .68603 ! 72757 .69862 .71549 41 20 .64723 .76229, .66044 .75088 .67344 .73924 .68624 .72737 .69883 .71529 40 21 .64746 .76210 ' L66066 .75069 .67366 .73904 .68645 .72717 .69904 .71508 39 22 .64768 .76192 .66033 . 75050 j .67387 .73885 .68606 .72697 .69925 .71488 38 .64790 .761731 .68109 .75030! .67409 .73865 .68688 .72677 .69940 .71468! 37 24 .64812 . 76154 ' .66131 .75011! .67430 .73846 .687-09 .72657 .69966 .71447 36 25 .61834 .76135 .66155 .74932 .67452 .78826 .68730 .72037 .69987 .71427 35 26 .64856 .76116 .66175 .74973 .67473 .73806 .68751 .72617 .70008 .71407 34 27 .04S78 .76097 .66197 . 74953 i .67495 .73787 .68772 .72597 .70029 .71386 33 28 .64901 .76078 .6(3218 ,74034 -.67516 .73767 .68793 .72577| .70049 .71306 32 29 .64923 .76059 .66240 .74915 .67538 .73747 .68814 .72557 .70070 .71345 31 30 .64945 .76041 .66262 .74896; .67559 .73728 .68835 . 72537 j .70091 .71325 30 31 .64967 .76022! .66284 74876 .67580 .73708 .68857 .72517; .70112 .71305 29 32 .64989 .76003 i .66:306 74857 .67602 .73688 .68878 .72497, 70132 .71284 28 33 .65011 .75984 .66327 74333 .67623 .73609 .68899 .72477, .70153 .71264 27 34 .65033 .75965 .66349 74818 .07645 .73649 .68920 .72457 .70174 .71243 26 35 .65055 .75946 .66371 74799 .67666 .73629 .68941 .72437 .70195 .71223 35 36 .65077 .75927 .66393 74780 .67688 .73610 .68962 .7241? .7C215 .71203 24 37 .65100 .75903 .66414 .74760 .67709 .73590 .68983 .72397 .70236 .71182 23 38 .65122 .75889 .66436 74741 .67730 73570 .69004 .72377 .7U257 .71162 22 39 .65144 .75870 .68458 74722 .67752 .73551 .69025 .72357 .70277 .71141 21 40 .65166 .75851 .66480 .74703; .67773 .73531 .69046 .72337 .70298 .71121 20 41 .65188 .75832 .66501 .74683 .67795 .73511 .69067 .72317 .70319 .71100 19 42 .65210 .75813 .66523 .74004 .67816 .73491 .69088 .72297 .70339 .71080 18 43 .65232 .75794 .68545 .74644 .67837 .73472 .69109 .72277 .70360 .710.M) 17 44 .65254 .75775 .66566 .74625 .67859 .73452 .69130 .72257 .70381 .71039! 16 45 .65276 .75756 .665881.74606 .67880 .73432 .69151 .72236 .70401 .710191 15 46 .65298 .75738 .60610 .74586 .r,79itl .73413 .69172 .72216 .70422 .70998! 14 47 .65320 .75719 .66632 '.74567 ! .67923 .73393 .69193 .72196 i .70443 .70978 13 48 .65342 .75700 .66653 .74548 .67944 .73373 .69214 .72176 .70465 .709571 12 49 .65364 .75680 .66675 .74523 .67965 .73353 .69235 .72156 .70484 .70937; 11 50 .65386 .75661; .666971.74509 .079S7 .73333 .69256 .72136 .70505 .70916 10 51 .65408 .75642 .66718 .74489 .68008 .73314 T69277 .72116 .70525 .70896 9 52 .65430 .75623; .66740 .74470 .68029 .73294 ! .69298 .72095 .70546 .70875' 8 53 54 .65452 .75604 .65474!. 75585 .66762 .74451 ; .68051 .66783 .74431 .68072 .73274 .73254 . 69319 j. 72075 .69340 .72055 .70567 .70587 .70855: 7 .70834 6 55 .65496 .75566 .66805 .74412 ! .68093 .73234 .69361 .72035 .70608 . 70813 i 5 56 .65518 .75547: .66827 .74392 .68115 .73215 .69382 .72015 .70628 .70793 4 57 .65540 .75528 .66848 .74373 .68136 .73195 .69403 .71995 .70649 .70772 3 58 .65562 .75509 .66870 .74353 .68157 .73175 j.69424 .71974 .70670 .70752 2 59 .65584 .75490 .66891 .74834 .68179 .73155 i .69445 .71954 .70690 .70731 1 60 .65606 .75471 .66913 .74314: .68200 .73135 .69466 . 71934 .70711 .70711 / Cosin Sine Cosin Sine Cosin Sine Cosin Sine : Cosin Sine f 49 48 47 46 45 293 TABLE XL SECANTS AND COSECANTS. SECANTS. ' 1 2 3 4 5 t 1-0000000 1-0001523 1-0006095 1-0013723 1-0024419 1-0038198 60 1 1-0000000 1-0001574 1-0006198 1-0013877 1-0024623 1-0038454 1 59 2 1-0000002 1 -0001627 1-0006300 1-0014030 1-0024829 1-0038711 58 3 1-0000004 1-0001679 1-0006404 1-0014185 1-0025035 1-0038969 57 4 1-0000007 1-0001733 1-0006509 1-0014341 1-0025241 1-0039227 56 5 1-0000011 1-0001788 1-0006614 1-0014497 1-0025449 1-0039486 55 6 1-0000015 1-0001843 1-0006721 1-0014655 1-0025658 1-0039747 54 7 l-00000i>l 1-0001900 1-0006828 1-0014813 1-0025867 1-0040008 53 8 1-0000027 1-0001957 1-0006936 1-0014972 1-0026078 1-0040270 62 9 1-0000034 1-0002015 1-0007045 1-0015132 1-0026289 1-0040533 51 10 1-0000042 1-0002073 1-0007154 1-0015293 1-0026501 1-0040796 50 11 1-0000051 1-0002133 1-0007265 1-0015454 1-0026714 1-0041061 49 12 1-0000061 1-0002194 1-0007376 1-0015617 1-0026928 1-0041326 48 13 1-0000072 1-0002255 1-0007489 1-0015780 1-0027142 1-0041592 47 14 1-0000083 1-0002317 1-0007602 1-0015944 1-0027358 1-0041859 46 15 1-0000095 1-0002380 1-0007716 1-0016109 1-0027574 1-0042127 45 16 1-0000108 1-0002444 I 1 0007830 1-0016275 1-0027791 1-0042396 44 17 1-0000122 1-0002509 1-0007946 1-0016442 1-0028009 1-0042666 43 18 1-0000137 1-0002575 1-0008063 1-0010609 1-0028228 1-0042937 42 19 1-0000153 1-0002641 1-0008180 1-0016778 1-0028448 1-0043208 41 20 1-0000169 1-0002708 1-0008298 1-0016947 1-0028669 1-0043480 40 21 1-0000187 1-0002776 1-0008417 1-0017117 1-0028890 1-0043753 39 22 1-0000205 1-0002845 1-0008537 1-0017288 1-0029112 1-0044028 38 23 1-0000224 1-0002915 1-0008658 1-0017460 1-0029336 1-0044302 37 24 1-0000244 1-0002986 1-0008779 1-0017633 1-0029560 1-0044578 36 25 1-0000264 1-0003058 1-0008902 1-0017806 1-0029785 1-0044855 35 26 1-0000286 1-0003130 1-0009025 1-0017981 1-0030010 1-0045132 34 27 1-0000308 1-0003203 1-0009149 1-0018156 1-0030237 1-0045411 33 28 1-0000332 1-0003277 1-0009274 1-0018332 1-0030464 1-0045690 32 29 1-0000356 1-0003352 1-0009400 1-0018509 1-0030693 1-0045970 31 30 1-0000381 1-0003428 1-0009527 1-0018687 1-0030922 1-0040251 30 31 1-0000407 1-0003505 1-0009654 1-0018866 1-0031152 1-0046533 29 32 1-0000433 1-0003582 1-0009783 1-0019045 1-0031383 1-0046815 28 33 1-0000461 1-0003660 1-0009912 1-0019225 1-0031615 1-0047099 27 34 1-0000489 1-0003739 1-0010042 1-0019407 1-0031847 1-0047383 26 35 1-0000518 1-0003820 1-0010173 1-0019589 1-0032081 1-0047669 25 36 1-0000548 1-0003900 1-0010305 1-0019772 1-0032315 1-0047955 24 37 1-0000579 1-0003982 1-0010438 1-0019956 1-0032551 1-0048242 23 38 1-0000611 1-0004065 1-0010571 1-0020140 1-0032787 1-0048530 22 39 1-0000644 1-0004148 1-0010705 1-0020326 1-0033024 1-0048819 21 40 1-0000677 1-0004232 1-0010841 1-0020512 1-0033261 1-0049108 41 1-0000711 1-0004317 1-0010977 1-0020699 1-0033500 1-0049399 19 42 1-0000746 1-0004403 1-0011114 1-0020887 1-0033740 1-0049690 18 43 1-0000782 1-00044SK) 1-0011251 1-0021076 1-00339SO 1-0049982 17 44 1-0000819 1-0004578 1-0011390 1-0021266 1-0034221 1-0050275 IS 45 1-0000857 1-0004066 1-0011529 1-0021457 1-0034463 1-0050569 15 46 1-0000895 1-0004756 1-0011670 1-0021648 1-0034706 1-0050864 14 47 1-0000935 1-0004846 1-0011811 1-0021841 1-0034950 1-0051160 13 48 1-0000975 1-0004937 1-0011953 1-0022034 1-0035195 1-0051456 12 49 1-0001016 1-0005029 1-0012096 1-0022228 1-0035440 1-0051754 11 50 1-0001058 1-0005121 1-0012239 1-0022423 1-0035687 1-0052052 10 51 1-0001101 1-0005215 1-0012384 1-0022619 1-0035934 1-0052351 9 52 1-0001144 1-0005309 1-0012529 1-0022815 1-0036182 1-0052651 8 53 1-0001189 1-0005405 1-0012670 1-00-23013 1-0036431 1-0052952 7 54 1-0001234 1-0005501 1-0012323 1-00-23211 1-0036681 1-0053254 6 55 1-0001280 1-0005598 1-0012971 1-00-23410 1-0036932 1-0053557 5 56 1-0001327 1-0005696 1-0013120 1-0023610 1-0037183 1-0053860 4 57 1-0001375 1-0005794 1-0013269 1-0023811 1-0037436 1-0054164 3 58 1-000142;} 1-0005894 1-0013420 1-0024013 1-0037689 1-0054470 2 69 1-0001473 1-0005994 1-0013571 1-0024216 1-0037943 1-0054776 1 60 1-0001523 1-OOOG095 1-0013723 1-0024419 1-0038198 1-0055083 i 89 88 87 86 85 84 ' COSECANTS. TABLE XI. SECANTS AND COSECANTS. SECANTS. / 6 7 8 9 10 11 t o 1-0075098 1-6098276 1-0124651 1-0154266 1-0187167 60 i 1-0055083 1-0075159 1-0098689 1-0125113 1-0154787 1-0187743 59 2 1-0055391 1-0075820 1-0099103 1-0125586 1-0155310 1-0188321 58 3 1-0055099 1-0070182 1-0099518 1-0126055 1-0155833 1-0188899 57 4 1-0056009 1-0076545 1-0099934 1-0126524 1-0156357 1-0189478 56 5 1-005C319 1-005G631 1 -0078908 1-0100351 1-0120993 1-015G882 1-0190059 55 6 1-0077273 1-0100769 1-0127466 1-0157408 1-0190640 54 7 1-0056943 1-007/639 1-0101187 1-0127939 1-0157934 1-0191222 53 8 1-0057256 1-0078005 1-0101607 1-0128412 1-0158462 1-0191805 52 9 1-0057570 1-0078372 1-0102027 1-01*8880 1-0158991 1-0192389 51 10 1-0057885 1-0058200 1-0078741 1-0102449 1-0129361 1-0159520 1-0192973 50 11 1 'fin^Qfn? 1-0079110 1-0102871 1-0129837 1-0160050 1-0193559 49 12 J. UUOoOl/ 1 '0058834 1-0079480 1-0103294 1-0130314 1-0160582 1-0194146 48 13 1 '0059153 1-0079851 1-0103718 1-0130791 1-0161114 1-0194734 47 14 l'00. r )9472 1-OOS0222 1-0104143 1-0131270 1-0161647 1-0195322 46 15 1-0059792 1-0080595 1-0104568 1-0131750 1-0162181 1-0195912 45 16 1 '0000113 1-0080968 1-0104995 1-0132230 1-0162716 1-0196502 44 17 1 -0000435 1-0081343 1-0105422 1-0132711 1-0163252 1-0197093 43 18 1-0060757 1-OU81718 1-0105851 1-0133194 1-0163789 1-01976S6 42 19 1-0061081 1-0082094 1-0106-280 1-0133677 1-0164327 1-0198279 41 20 1-0061405 1-0082471 1-0106710 1-0134161 r-0161865 1-0198873 40 21 1-0061731 1-0082S49 1-0107141 1-0134646 1-0165405 1-0199468 39 22 1-0062057 1-0083228 1-0107573 1-0135132 1-0165946 1-0200064 38 23 1-006-2384 1-0083607 1-0108006 1-0135618 1-0166487 1-0200661 37 24 1-0002712 1-0083988 1-0108440 1-0136106 1-0167029 1-0201259 30 25 1-0063040 1-0084369 1-0108875 1-0136595 1-0167573 1-0201858 35 26 1-0063370 1-0084752 1-0109310 1-0137084 1-0168117 1-0202457 34 27 1-0063701 1-0085135 1-0109747 1-0137574 1-0168662 1-0203058 33 28 1-0064032 1-0085519 1-0110184 1-0138006 1-0169208 1-020.3660 32 29 1-0064364 1-0085904 1-0110622 1-0138558 1-0169755 1-0204262 31 30 1-0064697 1-0086290 1-0111061 1-0139051 1-0170303 1-0204866 30 31 1-0005031 1-0086676 1-0111501 1-0139545 1-0170851 1-0205470 29 32 1-0065366 1-0087064 1-0111942 1-0140040 1-0171401 1-0206075 28 33 1-0065702 1-0087452 1-0112384 1-0140536 1-0171952 1 '0200082 27 34 1-0066039 1-0087842 1-0112827 1-0141032 1-0172503 1-0207289 26 35 1-0066376 1-0088232 1-0113270 1-0141530 1-0173056 1 -0-207897 25 36 1-0066714 1-0088623 1-0113755 1-0142029 1-0173609 1-0208506 24 37 1-0067054 1'009015 1-0114100 1-0142528 1-0174163 1-0209116 23 38 1-0007394 1-0089408 1-01H60G 1-01430-28 1-0174719 1-0209727 23 39 1-0067735 1-0089802 1-0115054 1-0143530 1-0175275 1-0210339 21 40 1-006S077 1-0090196 1-0115502 1-0144032 1-0175832 1-0210952 20' 41 1-0068419 1-0090592 1-0115951 1-0144535 1-0176390 1-0211566 19 42 1-0063763 1-0090088 1-0116(00 1-1145039 1-0176949 1-0212180 18 43 1 "0069108 1-0001386 1-0116851 1-014554 4 110177509 1-0212796 17 44 1 -0069453 l .fwo7aa 1-0091784 1-0117303 1-0146050 1-0178069 1-0213413 16 45 i uuowyy 1-0092183 1-0117755 1-0146556 1-0178081 1-0214030 15 46 1*0070146 1 '0070494 1-0092583 1-0118209 1-0147064 1-0179194 1-0214649 14 47 1 "0070843 l-0092l84 1-0118663 1-0147572 1-0179757 1-0215208 13 43 1 "0071193 1-0093386 1-0119118 1-0148082 1-0180321 1-0215888 12 49 1 "007 15 44 1-0093788 1-0119575 1-0 14859* 1-0180887 1-0216510 11 50 1-0094192 1-0120032 1-0149103 1-0181453 1-0217132 10 51 1-0071895 1-0072248 1-0094596 1-0120489 1-0149616 1-0182020 1-0217755 9 02 1-0072601 1-00951,01 1-0120948 1-0150129 1-0182588 1-0218379 8 03 1-0072955 1-0095408 1-0121408 1-01506J3 1-0183158 1-0219004 7 04 1-0073310 1-0095815 1-0121869 1-0151158 1-0183728 1-0219630 6 55 1-0096223 1-0122330 1-0151673 1-0184298 l-0220-'57 5 1-0073666 50 1-0074023 1-0096631 1-0122793 1-0152190 1-0184870 1 -0220865 4 07 1-00743SO 1-0097041 1-0123206 1-0152708 1-0185443 1-0221514 3 08 1-0074739 1-0097452 1-0123720 1-0153226 10180017 1-0222144 2 09 1-0075098 1-0097863 1-0124185 1-0153746 1-0186591 1-0222774 1 CO 1-0098276 1-0124651 1-0154266 1-0187167 1-0223 406 / 83 82 81 80 79 78 t COSECANTS. TABLE XI. SECANTS AND COSECANTS. SECANTS. ' 12 13 14 15 16 17 i > 1-0223406 1-0263041 1-0306136 1-0352762 1-0402994 1-045R918 60 1-0224039 1-0263731 1-0306884 1-0353569 1-0403863 1-0457848 59 1 1-0224672 1-0264421 1-0307633 1-0354373 1-0404732 1-0458780 58 2 1-0225307 1-0265113 1-0308383 1-0305187 1-0405602 1-0459712 57 3 1-0225942 1-0265806 1-0309134 1-0355998 1-0406473 1-0460646 56 4 5 1-0226573 1-0266499 1-0309886 1-0356809 1-0407346 1-0461581 55 1 g 1-0227216 1-0267194 1-0310639 1-0357621 , 1-0408219 1-0462516 54 1-0227854 1-0267889 1-0311393 1-0358435 1-0409094 1-0463453 53 g 1-0228493 1-0268586 1-0312147 1-0359249 1-0409969 1-0464391 52 9 1-0229133 1-0269283 1-0312903 1-0360065 1-0410845 1-0465330 51 10 1-0229774 1-0269902 1-0313660 1-0360881 1-0411723 1-0466270 50 1 _ 1-0230416 1-02706S1 1-0314418 1-0361699 1-0412601 1-0467211 49 12 1-0231059 1-0271381 1-0315177 1-0362517 1-0413481 1-046^153 48 13 1-0231703 1-0272082 1-0315936 1-0363337 1-0414362 l-046i'096 47 14 1-0232348 1-0272785 1-03 16U97 1-0364157 1-0415243 1-0470040 46 15 1-0232994 1-0273488 1-0317459 1-0364979 1-0416126 1-0470986 45 16 1-0233641 1-0274192 1-0318222 1-0365SOI 1-0417009 1-0471932 44 17 1-0234288 1-0274897 1-0318US5 1-0366625 1-0417894 1-0472879 43 18 1-0234937 1-0275603 1-0319750 1-0367449 1-0418780 1-0473823 42 19 1-0235587 1-0276310 1-0320516 1-0368275. 1-0419667 1-0474777 41 j 20 1-0236237 1-0277018 1-0321282 1-0369101 1-0420554 1-0475728 40 i 21 10236889 1-0277727 1-0322059 1-0369929 1-0421443 1-0476679 39 22 1-0237541 1-0278437 1-03-22818 1-0370757 1-0422333 1-0477632 38 23 1-0238195 1-0279148 1-0323588 1-0371587 1-0423224 1-0478586 37 24 1-0238849 1-0279860 1-0324359 1-0372417 1-0424116 1-0479540 36 25 1-0239304 1-0280573 1-0325130 1-0373249 1-0425009 1-0480496 35 26 1-0240161 1-0281287 1-0325903 1-0374082 1-0425903 1-0481453 34 27 1-0240818 1-02S20U2 1-0326676 1-0374915 1-0426798 1-0482411 33 28 1-0241476 1-0282717 1-0327451 1-0375750 1-0427694 1-0483370 32 29 1-0242135 1-0283434 l-032t>227 1-0376585 1-0428591 1-0484330 31 30 1-0242795 1-0284152 1-0329003 1-0377422 1-0429489 1-0485291 30 31 1-0243456 1-0284871 1-0329781 1-0378260 1-0430388 1-0486253 29 32 1-0244118 1-0285590 1-0330559 1-0379098 1-0431289 1-0487217 28 33 1-0244781 1-0286311 1-0331339 1-0379938 1-0432190 1-0488181 27 34 1-0245445 1-0287033 1-0332119 1-0380779 1-0433092 1-0489146 26 35 1-0246110 1-0287755 1-0332901 1-0381621 1-0433995 1-0490113 25 36 1-0246776 1-0288479 1-0333683 1-0382463 1-0434900 1-0491080 24 37 1-0247442 1-0289203 1-0334467 1-0383307 1-0435805 1-0492049 23 38 1-0248110 1-0289929 1-0335251 1-0384152 1-0436712 1-0493019 22 39 1-0248779 1 -0290655 1-0336037 1-0384998 1-0437619 1-0493989 21 40 1-0249448 1-0291383 1-0336823 1-0385844 1-0438528 1-0494961 20 41 1-0250119 1-0292111 1-0337611 1-0386692 1-0439437 1-0495934 19 42 1-0250790 1-0292840 1-0338399 1-0387541 1-Q440348 1-0496908 18 43 1-0251463 1-0293571 1-03391S8 1-0388391 1-0441259 1-0497883 17 44 1-0252136 1-0-29430-2 1-0339979 1-0389242 1-0442172 1-0498859 16 45 1-0252811 1-0295034 1-0340770 1-0390094 1-0443086 1-0499836 15 48 1-0253486 1-02957G3 1-0341563 1-0390947 1-0444001 1-0500815 14 47 1-0254162 1-0296502 1-0342356 1-0391800 1-0444917 1-0501794 13 48 49 1-0254839 1-0297237 1-0343151 1-0392655 1-0445833 1-0302774 12 1-0255518 1-0297973 1-0343946 1-0393511 1-0446751 1-0503756 11 50 1-0256197 1-0298711 1-0344743 1-0394368 1-0447670 1-0504738 10 51 52 1-0256877 1-0299449 1-0345540 1-0395226 1-0448590 1-0505722 9 53 1-0257558 1-0300188 1-0346338 1-0396085 1-0449511 1-050706 8 54 1-0258240 i-0300928 1-0347138 1-0396945 1-0450433 1-0507692 7 55 1-025S923 1-0301669 1-0347938 1-0397806 1-0451357 1-0508679 6 1-0259607 1-0302411 1-0348740 1-0398669 1-0452281 1-0509667 5 56 57 1-0260292 1-0303154 1-0349542 1-039953-2 1-0453206 1-0510656 4 58 1-0260978 1-0303898 1-0350346 1-0400396 1-045413-2 1-0511646 3 59 1-0261665 1-0304C43 1-0351150 1-0401261 1-0455060 1-0512637 2 60 1-026235-2 1-0305389 1:0351955 1-0402127 1-0455988 1-0513629 1 1-0263011 1-0306138 1-0302762 1-0402994 1-0456918 1-0514622 ' 77 76 75 74 73 72 ' COSECANTS. SECANTS. 18 19 20 21 22 23 i 1-0514622 1-0576207 1-0641778 1-0711450 1-0785317 1-OB63604 GO 1 1-0515617 1-0577267 1-064-2905 1-0712647 1-0786616 1-0864946 59 2 1-0.")16612 1-0978323 1-0644033 1-0713814 1-0787885 1-0866289 58 3 1-0517608 1-0579390 1-0645163 1-0715043 1-0789156 1-0867631 57 | 1-0518606 1-0530453 1-0646294 1-0716244 1-0790427 1-0868979 5G 5 1-0519605 1-0581517 1-0647425 1-0717445 1-0791700 1-0870326 55 6 1-0520604 1-0582583 1-0648558 1-0718647 1-0792975 1 -087 1675 54 7 1-05-21605 1-0583649 1-0649693 1-0719851 1-0794250 1-0873021 53 g 1 -0522607 1-0584717 1-0(550828 1-0721056 1-0795527 1-0874375 52 9 1 '0523610 1-0585786 1-0651964 1-0722262 1-0796805 1-0875727 51 10 1-0524614 1-0586855 1-0653102 1-0723469 1-0798084 1-0877080 50 H 1-0525619 1-0587925 1-0654240 1-0724678 1-0799364 1-0878435 49 l-052(i625 1-0588999 1-0655380 1-0725887 1-0800646 1-0879791 4S 3 1-0527633 1-0590072 1-0656521 1-0727098 1-OS01928 1-0881148 47 1 1-05-28641 1-0591146 1-0657663 1-0728310 1-0803212 1-0882506- 46 15 1-0529651 1-0592221 1-0638S07 1-0729523 1-0804497 1-0883866 45 16 1-0530(561 1-0593298 1-0659951 1-0730737 1 C805784 1-088522(5 44 17 1-0531673 1-0594376 1-0661097 1-0731953 1-OS07071 1 0886589 43 18 1-0532685 1-0595451 1-0662243 1-0733170 -10808360 1 0887952 4-2 19 1-0533699 1-0596531 1-0663391 1-0734388 1-0809650 1-08S9317 41 20 1-0534714 1-0597615 1-0664540 1-0735607 1-0810942 10890682 40 21 1-0535730 1-0598697 1-0665690 1-073(5827 1-0812234 1-0892050 39 22 1-0536747 1-0599781 1-0(566843 1-0738048 1-0813528 1-0893418 38 23 1-0537765 l-06008f)5 1-0667994 1-0739271 1-0814823 1-0894788 37 21 1-0538785 1-0601951 1-0669148 1-0740495 1-0816119 1-08S6159 3 25 1-0539805 1 -0603037 1-0670302 1-0741720 1-0817417 1-0897531 35 26 1-0540826 1-0604125 1-0671458 1-0742946 1-0818715 1-0898904 34 27 1-0541849 1-0605211 1-0672615 1-0744173 1-0820015 1-0900279 33 28 1-0542873 1-0606301 1-0673774 1-0745402 1-08-21316 1-0901655 32 29 1-0543897 1-0607395 1-0674933 1-0746631 1-0822618 1-0903032 31 30 1-0544923 1-0608487 1-0676094 1-0747862 1-0823922 1-0904411 30 31 1-0545950 1-0609580 1-0677255 1-0749095 1-0825227 1-0905791 29 32 1-054G978 1-0610675 1-0678418 1-0750328 1-0826533 1-0907172 28 33 1-0548007 1-0611770 1-0679582 1-0751562 1-0827840 1-0908554 27 34 1-0549037 1-0612867 1-0680747 1-0752798 '1-0829149 1-0909938 96 35 1-0550068 1-0613965 1-OU81914 1-0754035 1-0830458 1-0911323 25 36 1-0551101 1-0615064 1-0683081 1-0755273 1-0831769 1-0912709 21 37 1-0552134 1-0616161 1-0684250 1-0756512 1-0833081 1-0914097 23 38 1-0553169 1-0617265 1-0685420 1-0757753 1-0834395 1-0315485 22 39 1-0554204 1-0618367 1-0686591 1-0758995 1-0835709 1 -0916876 21 10 1-0555211 1-0619171 1-0687763 1-0760237 1-0837025 1-0918267 20 41 1-0556279 1-0620575 1-0688936 1-0761481 1-0838342 1-0919659 19 42 1-0557318 1-0621681 1-0690110 1-0762727 1-0839661 1-0921053 18 43 1-0558358 1-06-22788 1-0691286 1-0763973 1-0840980 1-0922448 17 41 1-0559399 1-0623896 1-0692463 1-0765221 1-0842301 1-0923845 16 15 1-0560441 1-0625005 1-0693641 1-0766470 1-0843623 1-0925243 15 46 1-05(51485 1-0626115 1-0694820 1-0767720 1-0844947 1-0926642 14 47 1-0562529. 1-06-27227 1-0096000 1-0768971 1-0846271 1-0928042 13 48 1-0563575 1-0628339 1-0697182 1-0770224 1-0847597 1-0929444 12 49 1-0564621 1-0629453 1-0698364 1-0771477 , 1-0848924 1-0930846 11 50 1-0565669 1-0630568 1-0699548 1-0772732 1-0850252 1-0932251 10 51 1-0566718 1-0631681 1-0700733 1-0773988 1-0851582 1-0933656 9 52 1-0567768 1-063-2801 1-0701919 1-0775246 1-0852913 1-0935003 8 53 1-0568819 1-0633919 1-0703103 1-0776504 1-0854245 1-0936471 7 51 1-0569871 1-0635038 1-0704295 1-0777764 1-0855578 1-0937880 6 55 1-0570924 1-06.36158 1-0705484 1-0779025 1-0856912 1-0939291 5 56 1-0571978 1-0637280 1-0706675 1-0780287 1-0858248 1-0940702 4 57 1-0573034 1-06384J3 1-0707867 1-0781550 1-0859585 1-0942116 3 01 1-0574090 1-0639527 1-0709060 1-0782815 1-0800924 1-0943530 2 59 1-0575148 1-0640652 1-0710254 1-0784080 1-0862263 1-0944946 60 1-0576207 1-0641778 1-0711450 1-0785347 1-0863604 1-0946363 71 70 69 68 67 66 / COSECANTS. 29; TABLE XI. SECANTS AND COSECANTS. SECANTS. / 24 25 26 27 28 29 t 1-0946363 1-1033779 1-1126019 1-1223262 1-1325701 1-1433541 60 1 1-0947781 1-1035277 1-1127599 1-1224927 1-13-27453 1-1435385 59 1-0949201 1-1036775 1-1129179 1-1226592 1-1 329207 1-1437231 58 3 1 -09506-2-2 1-1038275 1-1130701 1-1228259 1-1330962 1-1139078 57 4 1 -0952044 1-1039777 1-1132345 1-1229928 1-1332719 1-1440927 56 5 1-0953107 1-1041279 1-1133929 1-1231598 1-1334478 1-1442778 55 6 1-0954892 1-1042783 1-1135516 1-1233269 1-1336238 1-1444630 54 7 1-0956318 1-1044289 1*1137103 1-1231942 1-1337999 1-1446484 53 8 1-0957746 1-1045795 l-1138(i92 1-1236016 1-1339762 1-1448339 52 9 1-0959174 1-1047303 1-1140282 M23S292 1-1341527 1-1450196 51 10 1-0960604 1-1048813 1-1141874 1-1239969 1-1313293 1-1452055 50 11 1-0962036 1-1050324 1-1143467 1-1211618 1-1345060 1-1453915 49 12 1-0963468 1-1051836 1-11450G-2 1-1243328 1-1340829 1-1455776 48 13 1-0964902 1-1053319 MH6053 1-1245010 1-1348500 1-1457639 47 14 1-0906337 1-1054864 1-1148255 1-1246693 1 1350372 1-1459504 46 15 1-0967774 1-1056380 1-11*9854 1-1248377 1-135-2146 1-1461371 45 16 1-0969212 1-1057898 1-1151454 M250063 1-1353921 1-1463238 44 17 1-0970651 1-1059417 1-1153056 1-1251750 1-1355697 1-1465108 43 18 1-0972091 1-1060937 1-1154659 1-1253439 1-1357476 1-1466979 42 19 1-0973533 1-1062458 l-1156l'G3 1-1255130 1-1359255 1-1468852 41 20 1-0974976 1-1063981 1-1157869 1-1256821 1-1361036 1-1470726 40 21 1-0976420 1-106550G 1-1159476 1-1258514 1-1362819 1-1472602 39 22 1-0977866 1-1067031 1-1161084 M 960209 1-1364603 1-1474479 38 23 1-0979313 M068558 1-1162691 l-126iy05 1 -1300389 1-1476358 37 21 1-0980761 1-1070087 1-1164300 1-1203003 M368176 1-1478-239 36 25 1-0982211 M071616 1-1165919 1-1265302 1-1369965 1-1480121 35 26 1-0983662 1-1073147 1-1167533 1-1267003 1-1371755 1-1482005 34 27 1-0985114 1-1074C80 1-1169118 1-1208705 1-1373547 1-1483890 33 28 1-0986568 1-1076214 1-1 170706 1-1270408 1-1375341 1-1185777 32 29 1-0988023 1-1077749 1-1172384 1-1272113 1-1377135 1-1487065 31 80 1-0989479 1-1079285 1-1174004 1-1273819 1-1378932 1-1489555 30 31 1-0990336 1-1080823 1-1175625 1-1275527 J -1380730 1-1491447 29 32 1-0992395 1-1082363 1-1177248 1-1277237 1-1382529 1-1493340 23 33 1-0993855 1-1083903 1-1178872 1-1278948 1-1381330 1-1495235 27 31 1-0995317 1-1085445 1-1180498 l-1280CfiO 1-1380133 1-1497132 26 35 1-0996779 1-1086989 1-1182124 1-1282374 1-1387937 1-1499030 25 36 1-0998243 1-1088533 1-1183753 1-1284089 1-1389742 1-1500930 24 37 1-0999709 1-1090079 M1853o3 1-1285806 1-1391550 1-1502831 23 38 1-1001175 1-1091627 1-1187014 1-1287524 1-1393358 1-1501734 23 39 1-1002644 1-1093176 1-1188647 1-1289244 1-1390169 1-1500638 21 iu 1-1004113 1-10S4726 M190281 1-1290965 1-13U6980 1-1508544 20 41 M005584 1-1096277 1-1191916 1-1292687 1-1398794 1-1510452 19 42 1-1007056 1-1097830 1-1193553 1-1294412 1-1400608 1-1512361 18 43 1-1008529 1-1099385 1-1195191 1-1206137 1-1102425 1-1514272 17 44 1-1010004 1-1100940 1-119G831 1-1297864 1-UU42J3 1-1510185 16 45 1-1011480 1-1102498 1-1198472 1-1299593 1-1400062 1-1518099 15 46 1-1012957 1-1104056 1-1200115 1-1301323 1-1407883 1-1520015 14 47 1-1014436 1-1105616 i-1201739 1-1303035 1-1409706 1-1521932 13 48 1-1015916 1-1107177 1-1203405 1-1304788 1-1411530 1-1523801 12 49 1-1017397 1-1108740 1-1:4)5051 M30G522 1-1413356 1-1525772 11 50 1-1018879 M1103J4 1-1200700 1-1308258 1-1415183 1-1527094 10 51 1-1020363 1-1111869 1-1208350 M309996 1-1417012 1-1529618 9 52 1-1021849 1-1113436 1-1210001 1-1311735 1-1418842 1-1531513 8 53 1-1023335 1-1115004 1-1211S53 1-1313475 1-1,20674 1-1533170 7 54 1-1024823 1-1116573 1-1213308 M315217 1-1422507 1-1535399 6 55 1-1026313 1-1118144 1-12149C3 1-1 3 16961 1-1424342 1-1537329 56 1-1027803 1-1119716 1-121GC20 1-1318706 1-1426179 1-1539261 4 67 1-1029295 1-1121290 1-1218278 1-1320452 1-1428017 1-1541195 3 58 1-1030789 1-1122865 1 -1219938 1-1322200 1-1429857 1-1543130 2 59 1-1032283 1-1124442 1-1221600 1-1323950 1-1431698 M545007 1 60 M033779 1-1120019 M223282 1-1325701 1-1433541 1-1547005 t 65 64 63 62 61 60 / COSECANTS. TABLE XL-SECANTS AND COSECANTS. . SECANTS. / 30 31 32 33 34 35 i 11547005 1-1666334 1-1791784 1-1923633 1-2062179 1-2207746 60 1 1-1518945 1-1668374 1-1793928 1-1925886 1-2064547 1-2210233 59 2 1-1550887 1-1670416 1-1796074 1-1928142 1-2006917 1-2212723 58 3 1-1552830 1-1672459 1-1798222 1-1930399 1-2069288 1-2215215 57 4 1-1554775 1-1674504 1-1800372 1-1932658 1-2071662 1-2217708 56 5 1-1556722 1-1676551 1-1802523 1-1934918 1-2074037 1-2220204 55 6 1-1558670 1-1678599 1-1804676 1-1937181 1-2076415 1-2222702 54 7 1-1560620 1-1680649 1-1806831 1-1939446 1-2078794 1-2225202 53 8 M56257S 1-1682701 1-1808988 1-1941712 1-2081175 1-2227703 52 9 1-1564525 1-1684755 1-1811146 1-1943980 1-2083559 1-2230207 51 10 1-1560480 1-1686810 1-1813307 1-1946251 1-20^5944 1-2232713 50 11 1-1568436 1-1688867 1-1815469 1-1948523 1-2088331 1-2235222 49 12 1-1570394 1-1690926 1-1817633 1-1950796 1-2090720 1-2237732 48 13 1-1572354 1-1692986 1-1819798 1-1953072 1-2093112 1-2240244 47 11 1-1574315 1-1695048 1-1821966 1-1955350 1-2095505 1-2242758 46 15 1-1576278 1-1697112 1-1824135 1-1957629 1-2097900 1-2245274 45 16 1-1578243 1-1699178 1-1826306 1-1959911 1-2100297 1-2247793 44 17 1-1580209 1-1701245 1-1828479 1-1962194 1-2102696 1-2250313 43 18. 1-1582177 1-1703314 1-1830654 1-1964479 1-2105097 1-2252836 42 19 1-1584146 1 '1705385 T1832830 1-1966767 1-2107500 1-2255361 41 20 1 -1586118 1-1707457 1-1835008 1-1969056 1-2109905 1-2257887 40 21 1-1588091 1-1709531 1-1837188 1-1971346 1-2112312 1-2260416 39 22 1-1590065 1-1711607 1-1839370 1-1973639 1-2114721 1-2262947 38 23 1-1592041 1-1713685 1-1841554 1-1975934 1-2117132 1-2265180 37 24 1-1591019 1-1715764 1-1843739 1-1978230 1-2119545 1-2268015 3d 25 1-1595999 1-1717845 1-1845927 1-1980529 1-2121960 1-2270552 35 26 1-1597980 1-1719928 1-1848116 1-1982829 1-2124377 1-2273091 34 27 1-1599963 1-1722013 1-1850307 1-1985131 1-2126795 1-2275633 33 23 1-1601947 1-1724099 1-1852500 1-1987435 1-2129216 1-2278176 32 29 1-1603U33 1-1726187 1 -1854694 1-1989741 1-2131639 1-2280722 31 30 1-1605921 1-1728277 1-1856890 1-1992049 1-21340G4 1-2283269 30 31 1-1607911 1730368 1-1859089 1-1994359 1-2136191 1-2285819 29 32 1-1609902 1732462 1-1861289 1-199W71 1-2138920 1-2288371 28 33 1-1611894 1734557 1-1863490 1-19981)85 1-2141351 1-2290924 27 31 M6138S9 1736-053 1-1863694 1-2001300 '1-2143784 1-2293480 26 35 1-1615885 1738752 1-1867900 1-2003618 1-2146218 1-2296039 25 36 1-1617883 1740852 1-1870107 1-2005937 1-2148655 1-2298599 24 37 M619882 1742954 1-1872310 1-2008258 1-2151094 1-2301161 23 38 1-1621883 1745058 1-1874527 1-2010582 1-2153535 1-2303725 22 39 1-1623886 1747163 1-1876740 1-2012907 1-2155978 1 -2306292 21 40 1-1625891 1749270 1-1878954 1-2015234 T2158423 1-2308861 20 41 1-1627897 1751379 1-1881171 1-2017563 1-2160870 1-2311432 19 42 1-1G29905 1703490 1-1883389 1-2019894 1-2163319 1-2314004 18 43 1-1631914 1705603 1-1685609 1-2022226 1-2165770 1-2316579 17 41 1-1633925 1757717 1-1867831 1-2024561 1-2168223 1-2319156 16 45 1-1635938 1759833 1-1890055 1 -2026898 1-2170673 1-2321736 15 46 1-1637953 17C1951 1-1892280 1-2029236 1-2173135 1-2324317 14 47 1-1639969 1764070 1-1894508 1-2031577 1-2175594 1 2320!X)0 13 48 1-1641987 1766191 1-1896737 1-2033919 1-2178055 1 '2329486 12 49 50 1-1644007 1-1646028 1768314 1770439 1-1898963 1-1901201 1 -2036261 , 1-2038610 1-2180518 1-2182983 1-2332(374 1-2334G61 1) 10 51 1-1648051 1-1772566 1-190343G 1-2040958 1-2185450 1-2337256 9 52 1-1650076 1-1774694 1-1D05K73 1-2043308 1-2187919 1*2339850 8 53 1-1652102 1-1776824 1-11)07911 1-2045660 1-2190390 1-2342446 7 51 1-11554130 1-1778956 1-1910152 1-2048014 1-2192864 T2345044 6 55 1-1656160 1 -178. 1069 1-1912394 1-2050370 1-2195339 1-2347645 5 56 1-1658191 1-1783225 1-1914638 1-2052728 1-2197816 1-2350248 4 57 1-1660224 1-17&5362 M91K8S4 1-2055088 1-2200296 1 '2352852 3 58 1-1C&-2239 M787001 1-191SH32 1-2057450 1-2202777 1 -2355453 2 59 1-1664296 1-17S9U42 1-19213-J1 1-2059814 1-2200260 1 -235S069 1 CO 1-1666331 1-1791784 1-19236J3 1-2062179 1-2207746 1-2360680 69 68 67 66 65 64 / COSECANTS. 299 TABLE XL SECANTS AND COSECANTS. SECANTS. 36 37 38 39 40 41 9 1-2360680 1-2521357 1-2690182 1-2S67596 1-3051073 1-3250130 60 1-2363293 1-2521102 1-2693067 1-2370628 1-3057261 1-3253182 59 1-2365909 1-2526850 1-2695955 1-2873663 1-3060451 1-3256837 53 1-2368526 1-2529G01 1-2698845 1-2876700 1-3063614 l-3iU0191 57 1-2371146 1-2532353 1-2701737 1-2879740 1-3066839 1-3203554 56 1-2373768 1-2535108 1-2704632 1-2882782 1-3070038 1-326CJ918 55 1-2376393 1-2537865 1-2707529 1-2885827 1-3073239 1 -3270281 04 1-2379019 1-2510625 1-2710429 1-2888875 1-3076142 1-3273653 53 1-2381647 1-2513387 1-2713331 1-2891925 1-3079619 1-3277024 52 1-2384278 1-2516151 1-2716235 1-2894977 1-3082853 1-3280399 51 1-2386911 1-2518917 1-2719142 1-2898032 1-3086069 . 1-3283776 CO 10 1-2389546 1-2551685 1-2722052 1-2901090 1-3089284 1-3287156 49 11 1-2392183 1-2554456 1-2721963 1-2904150 1-3092501 1-3290539 43 12 1-2394823 l-25572-:3 1-2727877 1-2907213 1-3095720 1-3293925 47 13 1-2397464 1-2060005 1-2730794 1-2910278 1-3098943 r32'J7314 46 14 1-2400108 1-2562782 1-2733712 1-2913316 1-3102168 1-3300706 45 15 1 2102754 1-2565502 1-2736634 1-2916116 1-3105396 1-3301100 44 16 1-2405402 1-25G8315 1-2739557 1-2919189 1-3108676 1-3307197 43 17 1-2108052 1 '257 11 29 1-2712184 1-2922564 1-3111860 1-3310897 42 18 1-2410704 1-2573916 1-2745412 1-2925642 1-3115095 1-3314301 41 19 1-2413339 1-2576705 1-2748313 1-2928723 1-3118334 1-3317707 40 20 1-211G016 1-2079497 1-2751276 1-2931806 1-3121575 1-3321115 39 21 1-2118675 1 -'2582291 1-2751212 1-2934892 1-3121820 1-3321527 38 22 1-2121336 1-2585087 1-2757151 1-2937980 1-3128066 1-33-27912 37 23 1-2423999 1-2587885 1-2700091 1-2911071 1-3131316 1-3331359 33 1-2661460 1-2837411 1-302-2313 1-3216705 1-3121232 10 00 1-2196746 1-2664322 1-2840418 1-3025004 1-3220089 1-3121728 9 01 1-2499471 1-26G7186 1-2843423 1-3028667 1-3223116 1-3128227 8 02 1-2502199 1-2070052 1-2346140 1-3031831 1-3220745 1 3131729 7 53 1-2501929 1-2672S21 1-2849455 1-3035003 1-3230078 1-3435234 6 04 1-2507661 1-2675792 1-2852472 1-3038175 1-3233413 1-3133712 05 1-2510396 1-2678665 1-2855492 1-3041349 1-3236750 1-341225$ 4 06 1-2513133 1-2681041 1-2858514 1-3044526 1-3210091 1-3440767 3 07 1-2516872 1-2681419 1-2861539 1-3047706 1 3243135 1-3449284 2 53 1-2518613 1-2687299 1-2864566 1-3050888 1-324G781 1-3102S04 1 09 1-2521357 1-2690182 1-2867596 1-3054073 1-8250120 1-3106327 60 63 52 61 60 49 48 t COSECANTS. AINU UUSJ1.UAJN 1 R>. DECANTS. / 42 43 44 45 46 47- / 1 -345(5327 1-3673275 1-3901636 1-4142136 1-4395565 1-4662792 60 1 1-3459853 1-3676985 1-390J543 1-4146251 1-4399904 1-4667368 59 2 1-3463382 1-3680699 1-3909453 1-4150370 1-4404246 1-4671943 58 3 1-34W5914 1-3684416 1-3913366 1-4154493 1-4408592 1-4676532 57 4 1-3470449 1-3688136 1 -3917283 1-4158619 1-4412941 1-4681120 56 5 1-3473987 1-3691859 1-3921203 1-4162749 1-4417295 1-4G85713 55 6 1-3477528 1-3695586 1-3925127 1-4166883 1-4421652 1-4690309 54 7 1-3481072 1-3699315 1-3929054 1-4171020 1-4426013 1-4694910 53 8 1-3484G19 1-3703048 1-3932985 1-4175161 1-4430379 1-46995U 52 9 1-3188168 1-3706784 1-3936918 1-4179306 1-4434748 1-4704123 51 10 1-3491721 1-3710523 1-3940856 1-4183454 1-4439120 1-4708736 50 11 1-3495277 1-3714266 1-3914796 1-4187605 1-4443497 1-47M354 49 12 1-3498836 1-3718011 1-3948740 1-4191761 1-4447878 1-4717975 48 13 1-3502398 1-3721760 1-3952688 1-4195920 1-4452262 1-4722600 47 14 1-3505963 1-3725512 1-3956639 l-4->00082 1-4*56651 1-4727230 46 15 1-3509531 1-3729868 1-3960593 1-4204248 1-4461043 1-4731864 45 16 1-3513102 1-3733025 1-3964551 1-4208418 1-4465439 1-4736502 44 17 1-3516677 1-3736788 1-3968512 1-4212592 1-4469839 1-4741144 43 18 1-3520254 1-3740553 K-3972477 1-4216769 1-4474243 1-4745790 42 19 1-3523834 1-3744321 1-3976445 1-4220950 1-4478651 1-4750440 41 20 1-3527417 1-3748092 1-3980416 1-4225134. 1-4483063 1-4755095 40 21 1-3531003 1-3751867 1-3984391 1-4229323 1-4487478 1-4759751 39 22 1-3531593 1-3755645 1-3988369 1-4233514 1-4491898 1-4764417 33 23 1-3538185 1-3759426 1-3992351 1-4237710 1-4496322 1-4769084 37 24 1-3541780 1-3763210 1-3996336 1-4241909 1-4500749 1-4773755 36 25 1-3545379 1-3766998 1-4000325 1-4246112 1-4505181 1-4778431 35 26 1-3548980 1-3770789 1-4004317 1-4250319 1-4509616 1-4788111 34 27 1-3552585 1-3774583 1-4008313 1-4254529 1-4514055 1-4787795 33 28 1-3556193 1-3778380 1-4012312 1-425874S 1-4518498 1-4792483 32 -29 1-3559803 1-3782181 1-4016315 1-4262961 1-4522946 1-4797176 31 30 1-3563417 1-3785985 1-4020321 1-4267182 1-4527397 1-4801872 30 31 1-3567034 1-3789792 1-4024330 1-4271407 1-4531852 1-4806573 29 32 1-3570654 1-3793(502 1-4028343 1-4275636 1-4536311 1-4811278 23 33 1-3574277 1-3797416 1-4032360 1-4279868 1-4540774 1-4815988 27 34 1-3577903 1-3801233 1-4036380 1-4284105 1-4545241 1-4820702 26 35 1-3581532 1-3805053 1-4040403 1-4288345 1-4549712 1-4825420 25 36 1-3585164 1-3808877 1-4044430 1-4292588 1-4554187 1-4830142 24 37 1-3588800 1-3812704 1-4048461 1-4296836 1-4558666 1-4834868 23 38 1-3592438 1-3816534. 1-4052494 1-4301087 1-4563149 1-4839599 22 39 1-3596080 1-3820367 1-4050532 1-4305342 1-4567636 1-4844334 21 40 1-3599725 1-3824204 1-4060573 1-4309600 1-4572127 1-4849073 20 41 1-3603372 1-3828044 1-4064(517 1-4313863 1-4576621 1-4853817 19 42 1- 3607023 1-3831887 1-4068665 1-4318129 1-4581120 1-4858565 18 43 1-3610677 1-3835734 1-4072717 1-4322399 1-4585623 1-4863317 17 44 1-3614334 1-3839584 1-4076772 1-4326672 1-4590130 1-4868078 16 45 1-3617995 1-3843437 1-4080831 1-4330950 1-4594641 1-4872834 15 46 1-3621658 1-3847294 1-4084893 1-4335231 1-4599156 1-4877599 14 47 1-3625324 1-3851153 1-4088958 1-4339516 1-4603675 1-4882369 13 48 1-3628994 1-3855017 1-4093028 1-4343805 1-4608198 1-4887142 12 49 1-3632667 1-3858883 1-4097100 1-4348097 - 1-4612726 1-4891920 11 50 1-3636343 1-3862753 1-4101177 1-4352393 1-4617257 1-4896703 10 51 1-3640022 1-3866626 1-4105257 1-4356693 1-4621792 1-4901489 9 52 1-3643704 1-3870503 1-4109340 1-4360997 1-4626331 1-4906280 8 53 1-3647389 1-3874383 1-4113427 1-4365305 1-4630875 1-4911076 7 54 1-3651078 1-3878286 1-4117517 1-4 3696 16 1-4635422 1-4915876 6 55 1-3654770 1-3882153 1-4121612 1-4373932 1-4639973 1-4920680 5 56 1-3658464 . 1-3886043 1-4125709 1-4378251 1-4644529 1-4925483 4 57 1-3662162 1-3889936 1-4129810 1-4382574 1-4649089 1-4930301 3 58 1-3665863 1 -3893832 1-4133915 1-4386900 1-4653652 1-4935118 2 59 1-3669567 1-3897733 1-4138024 1-4391231 1-4658220 1-4939940 1 60 1-3673275 1-3901636 1-4142136 1-4395565 1-4662792 1-4944765 / 47 46 45 44 43 42 / COSECANTS. J SOI TABLE XI. SECANTS AND COSECANTS. SECANTS. / 48 49 50 51 52 53 1-4911765 1-5242531 1-5557238 1-5890157 1-6242692 1-6616101 60 1 1-4949596 1-5247634 1-5562634 1-5895868 1-6248743 1-6622819 2 1-4954431 1 -5252741 1-5568035 1-5901584 1-6254799 1-6629213 j 3 1-4959270 T5257854 1-5573441 1-5907306 1-6260861 1-6635673 57 4 1-19(54113 1-5262971 1-5578852 1-5913033 1-6266929 1-6642110 56 5 1-4908961 1-5268093 1-5584268 1-5918766 1-6273003 1-6648553 55 6 1-4973813 1-5273219 1-5589689 1-5924504 1-6279083 1-6655002 54 7 1-4978670 1-5278351 1-5595115 1-5930247 1-6285169 1-6661458 53 8 1 -1983531 1-5283487 1-5600546 1-5935996 1-6291261 1-6667920 52 9 1-4988897 1-5288627 1-5605982 1-5941751 1-6297359 1-6674389 51 10 1-4993267 1-5293773 1-5611424 1-5947511 1-6303462 1-6680864 50 11 1-4998141 1-5298923 1-5616871 1-5953276 1-6309572 16687345 49 12 1-0003020 1-5304078 1-5622322 1-5959043 1-6315688 1-6693833 48 13 1-5007903 1-530923S 1-5627779 1-5964821 l'-6321809 1-6700328 47 u 1-5012791 1-5314403 1-5633241 1-5970606 1-6327937 1-6706828 46 15 1-5017C83 1-5319572 1-5638708 1-5976394 1-6334070 1-6713336 45 16 1-50 -22580 1-5324746 1-5644181 1-5982187 1-6340210 1-6719850 44 17 1 '5027481 1-5329925 1-5649658 1-5987986 1-6346355 1-6726370 43 18 1-5032387 1-5335109 1-5655141 1-5993790 1-6352507 1-6732897 a 19 1-5037297 1-5340297 1-5660628 1-5999600 1-6358661 1-6739430 41 20 1-5012211 1-5345191 1-5666121 1-6005416 1-6364828 1-6745970 40 21 1-5017131 1-5350689 1-5671619 1-6011237 1-6370997 1-6752517 39 22 1-5052054 1-5355892 1-5677123 1-6017064 1-6377173 1-6759070 38 23 1-5056982 1-5361100 1-5682631 1-6022896 1-6383355 1-6765629 37 24 1-5061915 1-5366313 1-5688145 1-6028734 1-6389512 1-6772195 36 25 1-5066S52 1-5371530 1-5693664 1-6034577 1-6395736 1-6778768 35 26 1T.071793 1-5376752 1-5699188 1-6040126 1-6401936 1-67853 17 31 27 1-5070739 1-5381980 1-5704717 1-6046281 1-6408142 1-6791933 33 28 1-50S1G90 1-5387212 1-5710252 1-6052142 1-6414354 1-6798525 32 29 1-5086S45 1-5392449 1-5715792 1-6058008 1-6420572 1-6805124 31 30 1-5091505 1-5392690 1-5721337 1-6063879 1-6126796 1-6811730 30 31 I-50965G9 1-5402937 1-5726887 1-6069757 1-6133027 1-6818342 29 32 1-510153S 1-5408189 1-5732443 1-6075640 1-6139263 1-6824961 28 33 1-5106511 1-5413445 1-5738004 1-6081528 1-6445506 1-6831586 27 31 1-5111489 1-5418706 1-5743570 1-6087423 1-6151754 1-6838219 26 35 1-5116472 1-5423973 1-5749141 1-6093323 1-6458009 1-6844857 25 36 1-5121459 1-5429244 1-5754718 1-6099228 1-6461270 1-6851503 24 37 1-5126150 1-5134520 1-5760300 1-6105140 1-6170537 1-6858155 23 38 1-5131446 1-5439801 1-5765887 1-6111057 1-6476811 1-6864814 22 39 1-5136417 1-5445087 1-5771479 1-6116980 1-6483090 1-6S71479 21 40 1-5141452 1-5150378 1-5777077 1-6122908 1-6489376 1-6878151 20 41 1-514(5462 1-5155673 1-5782680 1-6128843 1*6495668 1-6884830 19 42 1-5151477 1-5160974 1-5783289 16134788 1-6501966 1-6891516 18 43 1-5106496 1-5466280 1-5793902 1-6140728 1-6508270 1-6898208 17 44 1-5161520 1-5471590 1-5799521 1-6146680 1-6514581 1-6904907 16 45 1-5166548 1-5176906 1-5305146 1-6152637 1-6520898 1-6911613 15 46 1-5171581 1-5482226 1-5810776 1-6158600 1-6527221 1-6918325 14 47 1-5176619 1-5487552 1-5816411 1-6164569 1-6533550 1-6925045 13 .48 1-5181661 1-5492882 1-58-22051 1-6170514 1-6539885 1-6931771 12 49 1-5186708 1-5498-218 1-5827097 1-6 176524 1-6546227 1-6938504 11 50 1-5191759 1-5503558 1-5333348 1-6182510 1-6552575 1-6945244 10 01 1-5196815 1-5508904 1-5839005 1-6188502 1-6558929 1-6951990 9 52 1-5201876 1-5514251 1-5844667 1-6194500 1-6565290 1-6958714 8 53 l-5i'06942 1-5519610 1-5850334 1-6200504 1-6571657 1-6965504 7 51 1-5212012 1-5524970 1-5356007 1-6206513 1-6578030 1-6972271 6 55 1-5217087 1-5530335 1-5861.685 1-6212523 1-6584109 1-6979044 5 5G 1-5222166 1-5535706 1 -5S67369 1-C218549 1-6590795 1-6985825 4 67 1-5227-250 1-5541081 1-5S73058 1-6224576 1-6597187 1-6992612 3 53 1-52:12339 1-5546462 1-5878752 1-6230609 1-6603586 16999407 2 59 1-5237433 1-5551848 1-5884452 1-0236648 1-6609990 1-7006208 1 1-5242D31 1-5557238 1-5890157 1-6212692 1-6616101 1-7013016 / 41 40 39 38 37 36 ' COSECANTS. 802 SECANTS. * 54 55 56 57 58 59 t 1-7013016 1-7134468 1-7882916 1-8360785 1-8870799 1-9116040 60 1 1-7019831 1-7441715 1-7890633 1-8369013 1-8879589 1-9425445 59 2 1-7026653 1-7448969 1-7898357 l-8377-'5l 1-8888388 1-9134861 58 3 1-7033482 1-7456230 1-790G090 1-838J498 1 -8897197 1-9114288 57 4 1-7010318 1-7463499 1-7913831 1-8393753 1-8906016 1-9453725 56 5 1-70471(50 1-7470776 1-7921580 1-8402018 1-8914845 1-9463173 55 6 1-7054010 1-7478060 1-7929337 1-8410292 1-8923684 1-9472632 54 7 1-7060867 1-7485352 1-7937102 1-8418574 1-8932532 1-9482102 53 8 1-7067730 1-7492651 1-7944876 1-8426866 1-8941391 1 9491583 52 9 1-7074601 1-7499958 1-7952658 1-8435166 1-8950259 1-9501075 51 10 1-7081478 1-7507273 1-7960449 1-8143476 1-8059138 1-9510577 50 11 1-7088362 1-7514595 1-7968247 1-8451795 1-8968026 1-9520091 49 12 1-7095254 1-7521924 1-7976054 1-8460123 1-8976924 1-95-29615 48 13 1-7102152 1-7529262 1-7983S69 1-8168460 1-8985832 1-9539150 47 14 1-7109058 1-7536607 1-7991693 1-8476806 1-8994750 1-9548697 . 16 15 1-7115970 1-7513959 1-7999524 1-8485161 1-9003678 1-9558254 45 16 1-7122890 1-7551320 1-8007365 1-8493525 1-9012616 1-9567822 44 17 1-7129817 l-75, r i8687 1-8015213 1-8501898 1-9021564 1-9577102 13 13 1-7136750 1-7566063 1-8023070 1-8510281 1-9030522 1-9586992 4-2 19 J -71 43691 1-7573146 1-8030935 1-8518672 1-9030491 1-9596593 41 20 1-7150639 1-7580837 1-8038809 1 -8527073 1-9018469 1-9606206 40 21 1-7157594 1-7588236 1-804G691 1-8535483 1-9057457 1-9615329 39 22 1-7164556 1-7595642 1-8051582 1-8543903 1-9066456 1-9625464 38 23 1-7171525 1-7603057 1-8062481 1-8552331 -1-9075464 1-9635110 37 21 1-7178501 1-7610478 1-8070388 1-8560769 1-9084483 1-9644767 36 25 1-7185484 1-7617903 1-8078304 1-85C9216 1-9093512 1-9654435 35 26 1-7192475 1-7625345 1 -8086 223 1-8577672 1-9102551 1-9664114 34 27 1-7199472 1-7632791 1-8094161 1-8586138 1-9111600 1-9673805 33 28 1-7206477 1-7610244 1-8102102 1-8594612 1-9120659 1-9683507 32 29 1-7213489 1-7647704 1-8110052 1-8603097 1-9129729 1-9693220 31 30 1-7220508 1-7655173 1-8118010 1-8611590 1-9138809 1-9702944 30 31 1-7227534 1-7662619 1-8125977 1-8620093 1-9147899 1-9712680 29 32 1-7231568 1-7670133 1-8133953 1-86-28605 1-9156999 1-9722427 28 33 1-7241609 1-7677625 1-8141937 1-8637126 1-9106110 1-973-2185 27 31 1-7248657 1-7685125 1-8149929 1-8645657 1-9175230 1-9741954 26 35 1-7255712 1-7692633 1-8157930 1-8054197 1-9184362 1-9751735 25 36 1-7262774 1-7700149 1-8165940 1-8662747 1-9193503 1-9761527 24 37 1-7269844 1-7707072 1-8173958 1-8671306 1-9202655 1-9771331 23 38 1-7276921 1-7715204 1-8181985 1-8679875 1-9211817 1-9731146 22 39 1-7284005 1-7722743 1-8190021 1-8688453 1-9220990 1-9790972 21 10 1-7-291096 1-7730230 1-3198065 1-8699040 1-9230173 1-9800810 20 41 1-7298195 1-77378 15 1-8206118 1-8705637 1-9239366 1-9810659 19 12 1-7305301 1-7745409 1-8214179 1-8714244 1-9248570 1-98205-20 18 13 1-7312414 1-7752980 1-8-22-J-249 1-87-22859 1-9257784 1-9830393 17 11 1-7319535 1-7760559 1-8-230328 1-8731185 1-9267009 1-9840276 16 15 1 '7526683 1-7768146 1-8238416 1-8740120 1-9276244 1-9850172 15 16 1-7333798 1-7775741 1-8246512 1-8748764 1-92854CO 1-9860080 14 17 1-7310941 1-7783344 1-8254017 1-8757419 1-9294746 1-9869997 13 18 1-7348091 1-7790955 1-8^-2731 1-8766082 1-9304013 1-9879927 12 19 1-7355248 1-7798574 1-8270854 1-8774755 1-9313290 1-9889869 11 50 1-7362413 1-7806201 1-8278985 1-8783438 '1-9322578 1-9899822 10 51 1-7369585 1-7813836 1-8287125 1-8792131 1-9331876 1-9909787 9 52 1-7376764 1-7821479 1-8295274 1-8800833 1-9341185 1-9919764 8 53 1-7383951 1-7829131 1-8303432 1-8809545 1-9350505 1-9929752 7 51 1-7391145 1-7836790 1-8311599 1-8818266 1-9359835 1-9939753 6 55 1-7398347 1-7844457 1-8319774: 1-8826998 1-9369176 1-9949764 5 56 1-7405556 1-7852133 1-83-27959 1-8835738 1-9378527 1-9959783 4 57 1-7412773 1-7859817 1-8336152 1-8844489 1-9387889 1-C969823 3 58 1-7119997 1-7867508 1-8344354 1-8858-249 1-9397-262 1-9979870 2 59 1-7427229 1-7875208 1-8352565 1-8862019 1-9406646 1-9989929 1 1-7434468 1-7882916 1-8360785 1-8870799 1-9416040 2-0000000 / 35 34 > 33 32 31 30 t COSECANTS. 303 SECANTS. , 60 61 62' 63 64 65 ' "2-0000000 2-0626G33 2-1300545 2-2026893 2-2811720 2-3662016 60 2-0010083 2-0637484 2-1312205 2-2039476 2-2825335 2-3676787 19 2 2-0020177 2-06483-28 2-1323830 2-2052075 2-2838967 2-3691578 >8 3 2-0030283 2-0659186 2-1335570 2-2064691 2-2852618 2-3706390 i7 2-0040402 2-0670056 2-1347274 2-2077323 2-2866286 2-3721222 >6 5 2-0050532 2-0680940 2-1358993 2-2089972 2-2879974 2-3736075 55 2-0000674 2-0691836 2-1370726 2-2102637 2-2893679 2-3750949 54 7 2-0070828 2-0702746 2-1382475 2-2115318 2-2907403 2-3765843 >3 g 2-0080994 2-0713670 2-1394238 2-2128016 2-2921145 2-3780758 - 2-0091172 2-0724606 2-1406015 2-2140730 2-2934906 2-3795694 H 10 2-0101362 2-0735556 2-1417808 2-2153460 2-2948685 2-3810650 50 2-0111564 2-0746519 2-1429615 2-2166208 2-2962483 2-3S25627 49 12 2-0121779 2-0757496 2-1441438 2-2178971 2-2976299 2-3S40625 48 A.1 13 2-0132005 2*0768486 2-1453275 2-2191752 2-2990134 2-3855645 4* Ad 14 2-0142243 2-0779489 2-1465127 2-2204548 2-3003988 2-3870685 40 15 2-0152494 2-0790506 2-1476993 2-2217362 2-3017860 2-3885746 45 16 2-0162756 2-0801536 2-1488875 2-2230192 2-3031751 2-3900828 44 17 2-0173031 2-0812580 -1500772 2-2243039 2-3045660 2-3915931 43 18 2-0183318 2-0823637 2-1512684 2-2255903 2-3059588 2-3931055 42 19 2-0193618 2-0834708 2-1524611 2-2268783 2-3073536 2-3946-201 41 20 2-0203929 2-0845792 2-1536553 2-2281681 2-3087501 2-3961367 40 21 2-0214253 2-0856890 2-1548510 2-2294.595 2-3101486 2-3976555 39 22 2-0224589 2-0868002 2-1560482 2-2307526 2-3115490 2-3991764 38 07 23 2-0234937 2-0879127 2-1572469 2-2320474 2-3129513 2-4006995 Of 24 2-0245297 2-0890265 2-1584471 2-2333438 2-3143554 2-4022247 36 25 2-0255670 2-0901418 2-1596489 2-2346420 2-3157615 2-4037520 35 26 2-0266056 2-0912584 2-1608522 2-2359419 2-3171695 2-4052815 34 00 27 2-0276453 2-0923764 2-1620570 2-2372435 2-3185794 2-4068132 66 28 2-0286863 2-0934957 2-1632633 2-2385168 2-3199912 2-4083469 33 29 2-0297286 2-0946164 2-1644712 2-2398517 2-3214049 2-4098829 31 30 2-0307720 2-0957385 2-1656806 2-2411585 2-3228205 2-4114210 30' 31 2-0318168 2-0968620 2-1668915 2-2424669 2-3242381 2-4129613 29 32 2-0328628 2-0979869 2-1681040 2-2437770 2-3256575 2-4145038 7 M 2-0339100 2-0991131 2-1693180 2-2450889 2-3270790 2-4160484 ** H 34 2-0349585 2-1002408 2-1705335 2-2464025 2-3285023 2-4175952 26 25 35 2-0360082 2-1013698 2-1717506 2-2477178 2-3299276 2-4191442 36 2-0370592 2-1025002 2-1729693 2-2490348 2-3313548 2-4206954 24 00 37 2-0381114 2-1036320 2-1741895 2-2503536 2-3327840 2-4222488 26 38 2-0391649 2-1047652 2-1754113 2-2516741 2-3342152 2-4238044 9 22 39 2-0402197 2-1058998 2-1766346 2-2529964 2-3356482 2-4253622 21 40 2-0412757 2-1070359 2-1778595 2-2543204 2-3370833 2-4269222 20 2-0423330 2-1081733 2-1790859 2-2556461 2-3385203 2-4284844 19 42 2-0433916 2-1093121 2-1803139 2-2569736 2-3399593 2-4300489 17 43 2-0444515 2-1104523 2-1815435 2-2583029 2-3414002 2-4316155 ifi 44 2-0455126 2-1115940 2-1827746 2-2596339 2-3428432 2-4331844 10 1 r 45 2-0465750 2-1127371 2-1840074 2-2609667 2-3442881 2-4347555 Id 46 2-0476386 2-1138815 2-1852417 2-2623012 2-3457349 2-4363289 14 1 o 47 2-0487036 2-1150274 2-1864775 2-2636376 2-3471838 2-4379045 lo 48 2-0497698 2-1161748 2-1877150 2-2649756 2-3486347 2- J 394823 12 II 49 2-0508373 2-1173235 2-1889541 2-2663155 2-3500875 2-4410624 in 50 2-0519061 2-1184737 2-1901947 2-2676571 2-3515424 2-4426448 IV 51 2-0529762 2-1196253 2-1914370 2-2690005 2-3529992 2-4442294 9 <; 52 2-0540476 2-1207783 2-1926808 2'270.-i457 2-35445S1 -4458163 53 2-0551203 2-1219328 2-1939262 2-2716927 2-3559189 2-4474054 KJ 2-0561942 2-1230887 2-1951733 2-2730415 2-3573818 2-4189968 04 55 2-0572695 2-1242460 2-1964219 22743921 2-3588467 2-4505905 56 2-0583460 2-1254048 2-1976721 2-2757445 2-3603136 2-4521865 j 57 2-0594239 2-1265651 2-1989240 2-2770987 2-3617826 2-4537848 * 58 2-0605031 2-1277267 2-2001775 2-2784546 2-3632535 2-4553853 J 59 2-0615836 2-1288899 2-2014326 2-2798124 2-3647i65 2-4569882 j . 60 2-0626653 2-1300545 2-2026893 2-2311720 2Q662016 2-4585933 / 29 28 27 26 25 24 COSECANTS. 304 TABLE JLL SECANTS AND COSECANTS. SECANTS. ' 66 67 68 69 70 71 2-4585933 2-5593047 2-6694672 2-7904281 2-923S044 S-0715335 c 2-4602008 2-5610599 26713906 2-7925444 2-3201431 3-0741507 5 24618106 2'56i8176 2-6733171 2-7946641 2-923-1858 3-07670ij 5 2 -4 S 34 227 2-5645781 2-6752465 2-7967873 2-9308326 3-0793590 5 2-4650371 2-5663412 2-6771790 2-7989140 2-9331833 3-081 U702 2-4000538 2-5681069 2-6791145 2-8010441 2-9305380 3-Obl5b60 5 6 2-4682729 2-5698752 2-6810530 2-8031777 2-9378968 3-0872066 5 7 2-4693943 2-5716162 2-6829945 2-8053148 2-9402597 3-0898319 5 8 2-4715181 2-5734199 2-6849391 2-8074554- 2'9l2026j 3-0924020 9 2-4731442 2-5751963 2-6868867 2-8095995 2-9449975 3-09JJP67 5 10 2-1747726 2-5769753 2-6888374 2-8117471 2-9473725 3-0977363 C 11 2-4764034 2-5787570 2-6907912 2-8138982 2-9497516 3-1003805 4 12 2-4780366 2-5805114 2-6927480 2-8160029 2-9521348 3-1030298 4 13 2-4796721 2-5823284 2-6947079 2-8182111 2-9545221 3-1050835 4 14 2-4813100 2-5841182 2-6966709 2-8203729 2-9509135 3'10d3i22 4 15 2-4829503 2-5859107 2-6986370 2-8225382 2 9593090 3-1110057 4 16 2 1845929 2-5877058 2-7006061 2-8247071 2-9617087 3-1136740 4 17 2 4862380 2-5895037 2-7025784 2-8268796 2-9611125 3-1163472 4 18 2-4878854 2-5913043 2-7045038 2-8290556 2-9065205 3-1190252 4 19 2-4S95352 2-5931077 2-7065323 2-8312353 2-9089327 3-1217081 4 20 2-4911874 2-5949137 2-7085139 2-8334185 2-9713490 3-1243959 1 21 2-4528421 2-5967225 2-7104987 2-8356054 2-9737695 3-1270S86 3 22 2-4944991 2-5985341 2-7124866 2-8377958 2-9761942 3-1297862 3 23 2-4961586 2-6003484 2-7144777 2-8399899 2-9786231 3-1321887 3 24 2-4978204 2-C021654 2-7164719 2-8421877 2-9810563 3-1351962 3 25 2-4994848 2-6039852 2-7184693 2-8443891 2-983493S 3-1379086 3 26 2-5011515 2-C05S078 2-7204698 2-8465941 2-9859352 3-1406259 3 27 2-5028207 2-6076332 2-7224735 2-8488028 2-9883811 3-1433483 3 28 2-5044923 2-6094613 2-7244804 2-8510152 2-9908312 3-1460706 3 29 2-5061663 2-6112922 2-7261905 2-8532312 2-9932856 3'148b079 3 30 2-0078428 2-6131259 2-7285038 2-8554510 2-9957443 3-1515103 . 31 31 2-5095218 2-6149624 2-7305203 2-8576744 2-9982073 3-1542877 2C 32 2-5112032 2-6168018 2-7325400 2-8599015 3-0006746 3-1570351 2* 33 2-5128871 2-6186439 2-7345630 2-8621324 3-0031462 3-1597876 2" 34 2-5145735 2-6204888 2-7363892 2-8643670 - 3-0056221 3-1625452 it 35 2-5162624 2-6223366 2-7386186 2-8666053 3-0081024 3-1653078 23 36 2-5179537 2-6241872 2-7406512 2-8688474 3-0105870 3-1680756 24 37 2-5196475 2-6260406 2-7420871 2-8710932 3-0130760 3-1708484 23 38 2-5213438 2-62789G9 2-7447263 2-8733428 3-0155694 3-1736264 22 39 Atl 2-5230426 2-6297560 2-7467687 2-8755961 3-0180672 3-17C4095 21 ftv 2-5247440 2-5316180 2-7488144 2-8778J32 8-0205693 3-1791978 20 41 2-5264478 2-6334828 2-7508634 2-8801142 3-0230759 3-1819913 19 42 43 2-5281541 2-6353506 2-7529157 2-8S23789 3-0255868 3-1847899 18 44 2-5298630 2-6372211 2-7549712 2-8846474 3-C281023 3-1875937 17 JK 2-5315744 2-6390946 2-7570301 2-8869198 3-0306i21 3-1904028 16 40 2-5332883 2-6409710 2-7590923 2-8891960 3-0331464 3-1932170 15 46 2-5350048 2-6428502 2-7611578 2-8914760 3-0356752 3-1960365 14 47 48 2-5367238 2-6447323 2-7632267 2-8937598 3-0382084 3-ly8b613 13 49 2-5384453 2-6466174 2-7652988 2-8960475 3-0407462 3-2016913 12 r,r\ 2-5401694 2-6485054 2-7673744 2-8983391 , 3-0432S84 3-2045266 U ou 2-5418961 2-6503962 2-7694532 2-9006346 3-0458352 3-2073673 10 51 2-5436253 2-6522901 2-7715355 2-9029339 3-0483864 3-2102132 9 2 53 2-5453571 2-6541868 2-7736211 2-9052372 3-0509423 3-2130644 8 2-5470915 2-6560865 2-7757100 2-9075443 3-0535026 3-2159210 7 KK 2-5488284 2-6579891 2-7778024 2-9098553 3-0560075 3-2187830 6 00 2-5505680 2-6598947 2-7798982 2-9121703 3-0086370 3-2216503 5 C6 57 2-5523101 2-6618033 2-7819973 2-9144892 3-0012111 3-2245230 4 68 2-5540518 2-6637148 2-7840P99 2-9168121 3-01.37898 3- 2274011 3 59 2-5558022 2-6656292 2-7862059 2-9191389 3-0603731 3-2302346 2 60 2-5575521 2-6675167 2-7883153 2-9214697 3 Oi>b9610 3 2331736 1 2-5593047 2-6694672 2-7901281 2-9238044 3 0715535 3-2360680 23 22 21 20 19 18' / COSEC ANTS. 305 TABLE XI.-SECANTS AND COSECANTS. SECANTS. r 72 73 74- 75 76 77 3-2360680 3-4203036 3-6279553 3-8637033 4-1335655 4-4454115 60 1 3-2389I.78 3-4235611 3-6316395 3-8679025 4-13*3939 4-4510198 59 2 3-2418732 3-4263-251 3-63533J6 3-8721112 4-143-2339 4-4/J664-28 58 3 3-2447840 3-430095(5 3-6390315 3-8763293 4-1180856 4-4622803 57 4 3-2477003 3-4333727 3-6427392 3-8805570 4-1529191- 4-4679324 56 5 3-2506222 3-4366563 3-6164548 3-8847913 4-1578243 4-4735993 55 6 3-2535496 3-4399465 3-6501783 3-8890411 4-16-27114 4-4792S10 04 7 3-2564825 3-4432433 3-6539097 3-8932976 4-1676102 4-1819775 53 8 3-2594211 3-4465167 3-6576491 3-8975637 4-17-20210 4-4906889 52 9 3-2623652 3-4498368 3-6613964 3-9018395 4-1774138 4-4964152 51 10 3-2653149 3-4531735 3-6651513 3-9061250 4-1823785 4-5021065 59 11 3-2682702 3-4564969 3-6689151 3-9104203 4-1873252 4-5079129 49 12 3-2712311 3-459S269 3-6726865 3-9147-254 4-1922840 4-5136814 48 13 3-2741977 3-4631637 3-6764660 3-9190403 4-1972519 4-5194711 47 14 3-2771700 3-4665073 3-680-2036 3-9233651 4-2022380 4-5252730 46 15 3-2801479 3-4698576 3:6840493 3-9276997 4-2072333 4-5310903 45 16 3-2831316 3-4732146 3-6878532 3-9320443 4-2122408 4-5369229 44 17 3-2S61209 3-4765785 3-6916652 3-93U3988 4-2172606 4-54-27709 43 18 3-2891160 3-4799492 3-6951854 3-9107633 4-2222928 4-5186314 42 19 3-2921168 3-4833267 3- 6993139 3-9151379 4-2273373 4-5510134' 41 20 3-2951234 3-4867110 3-7031506 3-9495224 4-2323943 4-0601030 40 21 3-2981357 3-4901023 3-7069956 3-9539171 4-2374637 4-5663183 39 22 3-3011539 3-4935004 3-7108489 3-9583219 4-2425457 4-5722444 33 23 3-3041778 3-4969055 3-7147105 3-9627369 4-2476402 4-5781862 37 21 3-3072076 3-5003175 3-7185805 3-9671621 4-2527474 4-5811139 36 25 3-3102432 3-5037365 3-7224589 3-9715975 4-2573671 4-5901174 35 26 3-3132347 3-5071625 3-7263157 3-9760431 4-262f>996 4-5961070 31 27 3-3163320 3-5105954 3-7302409 3-9804991 4-2681 419 4-6021126 33 28 3-3193853 3-5140354 3-7341446 3-9849654 4-2733029 4-6081343 32 29 3-3224444 3-5174824 3-73S0563 3-9894,21 4-2784733 4-6141722 31 30 3-3255095 3-5209365 3-7419775 3-9939292 4-2836576 4-6-202263 30 31 3-3285805 3-5243977 3-7459063 3-9984267 4-2888513 4-6262967 29 32 3-3316575 3-5278660 3-7498447 4-0029347 4-2940640 4-6323835 28 33 3-3347105 3-5313414 3-7537911 4-0074532 4-299!.'867 4-6381867 27 31 3-3378-294 3-5348210 3-7577162 4-0119823 4-30452-25 4-6146061 20 35 3-3409244 3-5383138 3-7617100 4-0165219 4-3097715 4-6507427 25 36 3-3440254 3-5418107 3-7636824 4-0210722 4-3150336 4-6568956 24 37 3-3471324 3-5453149 3-7696636 4-0256332 4-3203090 4-6630652 23 .J8 3-3502455 3-5488263 37736535 4-0302048 4-3255977 4-6692516 22 39 3-3533647 3-5523450 3-7776522 4-0347872 4-3308996 4-6754518 21 40 3-3564900 3-5558710 3-7816596 4-0393804 4-3362150 4-6816718 20 41 3-3596214 3-5591042 3-7856760 4-0439844 4-3415438 4-6879119 19 42 3-3627589 3-5629448 3-7897011 4-0485992 4-3468861 4 -69 4 1600 18 43 3-3659026 3-5661928 3-7937352 4-0032219 4-3522419 4-7004372 17 44 3-3690524 3-5700481 3-7977782 4-0578615 4-3576113 4-7067256 16 45 3-3722084 3-5736108 3-8018301 4-0625091 4-3629943 4-7130313 15 46 3-3753707 3-5771810 3-8058911 4-0671677 4-3683910 4-7193542 14 47 3-3785391 3-5807586 3-8099610 4-0718374 4-3738015 4-7250945 13 48 3-3817138 3-5843137 3-8140399 4-0765181 4-3792257 4-7320521 12 49 3-3848948 3'58793(J2 3-8181280 4-0812100 4-3816638 4-7384277 11 50 3-38S0820 S'5915363 3-8222251 4-08591)60 4-3901153 4-7448206 10 51 3-3912755 3-5951439 3-8263313 4-090C272 4-3955817 4-7512312 9 52 3-3944754 3-5987590 3-8304167 4-0953526 4-4010616 4-7576596 8 53 3-3976816 3-6023818 3-8345713 4-1000893 4-4065556 4-7641028 7 54 3-4008941 3-6060121 3-8387052 4-1048374 4-4120637 4-7705699 55 3-4041130 3-6096501 3-8428482 4-1095967 4-4175859 4-7770519 56 3-4073382 3-6132957 3-8470006 4-1143675 4-4231224 4-7835520 57 3-41056'J9 3-6169490 3-8511622 4-1191498 4-4286731 4-7900702 58 3-4138080 3-6206101 38563332 4-1-239435 4-4312382 4-7966066 59 3-4170526 3-624-27s8 3-8595135 4-1287487 4-4398176 4-8031613 60 3-4203036 3-6279553 3-8637033 4-1335655 4-4454115 4-fcU!>?3i3 * 17 16 15 14 13 12 ' COSECANTS. TABLE XL SECANTS AND COSECANTS. SECANTS. 78 79 80 81 82 83 4-8097343 6-2408431 8-7587705 6-3924532 7-1852965 8-2055090 4-8163258 5-2486979 5-7682867 6-4042154 7-2001996 8-2249952 4-8229357 5-2065768 5-7778350 6-4160216 7-2151653 8-2445748 4-8295643 5-2644798 5-7874153 -4278719 7-2301940 8-2642485 4-8362114 5-2724070 5-7970280 6-4397666 7-2452859 8-2840171 4-8428774 5-2803587 5-8066732 6-4517059 7-2604417 8-3038812 4-8495621 5-2883347 6-8163510 6-4636901 7-2756616 8-3238415 4-8562657 5-2963354 5-8260617 6-4757195 7-2909460 8-34389S6 4-8629883 5-3043608 5-8358053 6-4877944 7-3062954 8-3640534 4-8697299 8-3124109 5-8455820 6-4999148 7-3217102 8-38430(i5 4'8764907 5-3204860 5-8553921 6-5120812 7-3371909 8-4046586 4-8832707 5-3285861 5-8652356 6-5242938 7-3527377 8-4251105 4-8900700 5-3367114 5-8751123 6-5365528 7-3683512 8-4456629 4-8968886 5-3448620 5-8850238 6-5488586 7-3840318 8-4663165 4-9037267 5-3530379 5-8949688 6-5612113 7-3997798 8-4S70721 4-9105844 5-3612393 5-9049479 6-5736112 7-4155959 8-5079304 4-9174616 5-3694664 5-9149614 6-5860587 7-4314S03 8-5288923 4-9243586 6-3777192 6-9250095 6-5985540 7-4474335 8-549D584 4-9312754 5-3859979 5-9350922 6-6110973 7-4(534560 8-5711295 4-9382120 5-3943026 5-9452098 6-6236S90 7-4795482 8-5921065 4-9451687 5-4026333 5-9553625 6-6363293 7-4957106 8-6137901 4-9521453 5-4109903 5-9655504 6-6490184 7-5119437 8-6352812 4-9591421 5-4193737 5-9757737 6-6617568 7-5282478 8-6568805 4-9661591 5-4277835 5-9860326 6-6740446 7-5446236 8-6785889 4-9731964 5-4362199 5-9963274 6-6873822 7-5610713 8-7004071 4-9S02541 5-444C831 6-OOC6581 6-7002699 7-5775916 8-7223361 4-9873323 5-4531731 6-0170250 6-7132079 7-5941849 8-7443766 4-99*4311 5-4616901 6-0274282 6-7261965 7-6108516 8-7665295 6-0015505 5-4702342 6-0378680 6-7392360 7-6275923 8-7887957 5-0086907 5-4788056 6-0483445 6.-7S23268 7-6444075 8-8111761 6-0158517 5-4874043 6-0588580 6-7654691 7-6612976 8-8336715 6-0230337 5-4960305 6-0694085 6-778C632 7-6782631 8-8562828 6-0302367 5-5046843 6-0799964 6-7919095 7-G953047 8-8790109 5-0374607 5-5133659 6-0906219 6-8052082 7-7124227 8-6018567 5-0447060 5-5220754 6-1012850 6-8185597 7-7296176 8-9248211 5-0519726 5-5308129 6-1119861 6-8319642 7-7468901 8-9479051 5-0592606 5-5395786 6-1227253 6-8454222 7-7642406 8-9711095 5-OG65701 5-5483726 6-1335028 6-8589338 7-7816697 8-9944354 5-0739012 5-5571951 6-1443189 6-8724995 7-7991778 9-0178837 5-0812539 5-5660460 6-1551736 6-8861195 7-816765S 9-0414553 5-0886284 5-5749258 6-1600674 6-8997942 - 7-8344335 9-0651512 6-0960248 5-5838343 6-1770003 6-9135239 7-8521821 9-0889725 5-1034431 5-5927719 6-1879725 6-9273C89 7-8700120 9-1129200 5-1108835 5 -CO 17386 6-1989843 6-9111496 7-8879238 9-1369949 8-1183461 5-6107345 6-2100359 6-9550464 7-9059179 9-1611980 5-1258309 5-6197599 6-2211275 6-9689994 7-9239950 9-1855305 5-1333381 5-6288148 6-2322.194 6-9830092 7-9421556 9-2699934 5-1408677 5-6378995 6-2431316 6-9970760 7-9604003 9-2345877 5-1484199 5-6470140 6-25JG446 7-0112001 7-9787298 9-2593145 6-1559948 5-6561584 6-2658984 7-0253820 7-9971445 9-2841749 5-1636924 5-6653331 6-2771933 7-0396220 8-0156150 9-3091699 8-1712123 6-6745380 6-2885295 7-0539205 8-0342321 9-3343006 5-1788563 5-6837734 6-29y9073 7-0682777 8-0529062 9-3595B82 5-1865228 5-6930393 6-5113269 7-0826941 8-0716681 9-3849738 5-19i2i25 5-7023360 6-3227884 7-0971700 8-0905182 9-4105184 6-2019254 6-7116636 6-3342923 7-1117059 8-1094573 9-4362033 5-2096618 5-7210223 8-3458386 7-1263019 8-1284860 9-4620296 5-2174216 5-7304121 6-3574276 7-1409587 8-1476048 9-4879984 5-2252050 5-7398333 6-3690595 7-1556764 8-1668145 9-5141110 6-2330121 6-7492861 6-3807347 7-1704556 8-1861157 9-5403686 5 -2408431 5-7087705 6-3924532 7-1852965 8-20SOOyQ 9-5667722 11 10 9 8 7 6 COSECANTS. TABLE XI.-SECANTS AND COSECANTS. SECJ LNTS. ' 84 85 86 r 87 88 89 f 9-5667722 11-473713 14-335587 19-107323 28-653708 67-2986S8 60 9-5933233 11-511990 14-395471 19-213970 28-894398 58-263755 69 9-6200229 11-550523 14-455859 19-321816 29-139169 59-274308 9-6468724 11-589316 14-516757 19-430882 29-388121 60-314110 67 9-6738730 11-628372 14-578172 19-541187 29-641373 61-391050 5ff 8-7010260 11-667693 11-640109 19-652751 29-899026 62-507153 65 9-7283327 11-707282 11-702576 19-765604 30-161201 63-664595 54 9-7557944 11-747141 14-765580 19-879758 30-128017 64-865716 53 9-7834124 11-787274 11-829128 19-995241 30-699598 66-113036 52 9-8111880 11-827683 11-893226 20-112075 30-976071 67-409272 51 10 9-8391227 11-868370 11-957882 20-230284 31-257577 68-757360 60 11 9-8672176 11-909340 15-023103 20-349893 31-544246 70-160474 49 12 9-8954744 11-950595 15-088896 20-470926 31-8362-25 71-622052 48 13 9-9238943 11-992137 15-155270 20-593109 32-133663 73-145827 47 14 9-9524787* 12-033970 15-222231 20-717368 . 32-436713 74-735856 48 15 9-9812291 12-076098 15-289788 20-842830 32-745537 76-39U551 45 16 10-010117 12-118522 15-357949 20-969824 33-060300 78-132742 44 17 10-039234 12-161246 15-426721 21-098376 33-381176 79-949684 43 18 10-068491 12-204274 15-496114 21-228515 33-708345 81-853150 42 19 10-097920 12-247608 15-566135 21-360272 34-041U94 &3-849470 41 20 10-127522 12-291252 15-636793 21-493670 34-382316 85-945609 40 21 10-157300 12-335210 15-708096 21-628759 34-729515 88-149244 39 22 10-187254 12-37^484 15-780051 21-765553 35-083800 90-408863 38 23 10-217386 12-424078 15-852676 21-904090 35-445391 92-913869 37 21 10-247697 12-468995 15-925971 22-044403 35-814517 95-494711 33 25 10-278190 12-514240 15-999918 22-1865-J8 36-191414 98-223033 35 26 10-308866 12-559815 16-074017 22-330499 36-576332 101-11185 34 27 10-339726 12-605724: 16-149987 22-476353 36-969528 104-17574 33 28 10-370772 12-651971 16-226069 22-624126 37-371273 107-43114 32 9 10-402007 12-698560 16-302873 22-773857 37-781849 110-89656 31 30 10-433431 12-745495 16-380408 22-9255S6 38-201550 114-59301 30 81 10-465046 12-792779 16-458686 23-079351 38-630683 118-54449 29 32 10-496854 12-810416 16-537717 23-235196 39-069571 122-77803 28 83 10-528857 12-888410 16-617512 23-393161 39-518549 127-32526 27 34 10-561057 12-936765 16-098082 23-553291 39-977969 132-22229 26 83 10-593455 12-985486 16-779439 23-715630 40-448201 137-51108 25 36 10-626054 13-034576 16-861591 23-880224 40-929630 143-24061 24 27 10-658851 13-084040 16-944559 24-047121 41-422660 149-46837 23 88 10-691859 13-133882 17-028346 24-216370 41-927717 150- 202 i8 22 89 10-725070 13-184106 17-112966 24-388020 42 445245 163-70325 21 40 10-758488 13-234717 17-198431 24-562123 42-975713 171-88831 20 41 10-792117 13-285719 17-284761 24-738731 43-519612 18033496 19 43 10-825957 13-337116 17-371960 24-917900 44-077458 190-98680 18 43 10-860011 13-388911 17-460046 25-099685 44-649795 202-22122 17 44 10-894281 13-141118 17-549030 25-234144 45-237195 214-85995 16 45 10-928768 H-493731 17-638928 25-471337 45-840:160 229-18385 15 46 10-963476 13-540758 17-729753 25-661324 46-459625 245-55402 14 47 10-998406 13-G00205 17-821520 25-854169 47-095961 264-44269 13 48 11-033560 13-654077 17-914243 26-049937 47-749974 2S6-47948 12 49 11-068940 13-708379 18-007937 26-248094 48-422411 312-52297 11 50 11-104519 13-763115 18-102619 26-450510 49-114002 343-77516 10 61 11-140389 13-818291 18-198303 26-655455 49-825762 381-97230 9 62 11-176462 13-873913 18-295005 26-863603 50-558396 429-71873 8 63 11-212770 13-929985 18-392742 27-075030 61-312902 491 10702 7 51 11-249316 13-986511 18-491530 27-289814 62-090272 672-95809 6 55 11-286101 11-013501 18-591387 27-508035 52-891564 687-54963 5 66 11-323129 11-100963 18-692330 27-729777 63-717895 859-43689 4 67 11-360102 11-158894 18-794377 27-955125 54-570464 1145-9157 3 68 11-397922 14-217304 18-897545 28-184168 55-450531 1718-8735 2 69 11-435692 14-276200 19-001854 28-416997 56-359462 3437-7468 1 60 11-473713 14-335587 19-107323 28-653708 67-298688 Infinite. ' 6" 4 3 2 1 7 COSE CANTS. TABLE XII. TANGENTS AND COTANGENTS. 1 li 2 3 S Tang Cotang Tang Cotang Tang Cotang Tang Cotang .00000 Infinite. .01746 57.2900 .03492 28.6363 .05241 19.0811 60 1 .00029 3437.75 .01775 56.3506 .03521 28.3994 .05270 18.9755 59 2 .00058 1718.87 .01804 55.4415 .03550 28.1664 .05299 18.8711 |58 3 .00087 1145.92 .01833 54.5613 .03579 27.9372 .05328 18.7678 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 .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 S .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 13 .00378 264.441 .02124 47.0853 .03871 25.8348 .05620 17.7934 47 14 .00407 245.552 .02153 46.4489 .03900 25.6418 .05649 17.7015 46 15 .00436 229.182 .02182 45.8294 .03929 25.4517 .05678 17.6106 45 16 .00465 214.858 .02211 45.2261 .03958 25.2644 .05708 17.5205 44 17 .00495 202.219 .02240 44.6386 .03987 25.0798 .05737 17.4314 43 18 | .00534 190.984 .02269 44.0661 .04016 24.8978 .05766 17.3432 42 1!) .00553 180.932 .02298 43.5081 .04046 24.7185 .05795 17.2558 41 SO .00582 171.885 .02328 42.9641 .04075 24.5418 .05824 17.1693 40 21 .00611 163.700 .02357 42.4335 .04104 24.3675 .05854 17.0837 39 22 .00640 156.259 .02386 41.9158 .04133 24.1957 .05883 16.9990 38 2:5 .00609 149.465 .02415 41.4106 .04162 24.0263 .05912 16.9150 37 2! .00698 143.237 .02444 40.9174 .04191 23.8593 .05941 16.8319 36 25 .00727 137.507 .02473 40.4358 .04220 23.6945 .05970 16.7496 35 96 .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 33 28 .00815 122.774 .02560 39.0568 .04308 23.2137 .06058 16.5075 32 ;.,' .00844 118.540 .02589 38.6177 .04337 23.0577 .06087 16.4283 31 yu .00873 114.589 .02619 38.1885 .04366 22.9038 .06116 16.3499 30 31 .00902 110.892 .02648 37.7680 .04395 22.7519 .06145 16.2722 29 82 .00931 107.426 .02677 37.3579 .04424 22.6020 .06175 16.1952 28 83 .00960 104.171 .02706 36.9560 .04454 22.4541 .06204 16.1190 27 34 .00989 101.107 .02735 36.5627 .04483 22.3081 .06233 16.0435 26 88 .01018 98.2179 .02764 38.1776 .04512 22.1640 .06262 15.9687 25 86 .01047 95.4895 .02793 35.8006 .04541 22.0217 .06291 15.8945 24 a; .01076 92.9085 .02822 35.4313 .04570 21.8813 .06321 15.8211 23 as .01105 90.4633 .02851 35.0695 .04599 21.7426 .06350 15.7483 22 89 .01135 88.1436 .02881 34.7151 .04628 21.6056 .06379 15.6762 21 40 .01164 85.931)8 .02910 34.3678 .04658 21.4704 .06408 15.6048 20 41 .01193 83.8435 .02939 34.0273 .04687 21.3369 .06437 15.5340 19 12 .01222 81.8470 .02968 33.6935 .04716 21.2049 .06467 15.4638 18 13 .01251 79.9434 .02997 33.3662 .04745 21.0747 .06496 15.3943 17 41 .01280 78.1263 .03026 33.0452 .04774 20.9460 .06525 15.3254 10 45 .01309 76.3900 .03055 32.7303 .04803 20.8188 .06554 15.2571 15 10 .1338 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 48 .01396 71.6151 .03143 31.8205 .04891 20.4465 .06642 15.0557 12 40 .01425 70.1533 .03172 31.5284 .04920 20.3253 .06671 14.9898 11 50 .01455 68.7501 .03201 31.2416 .04949 20.2056 .06700 14.9244 10 51 .01484 67.4019 .03230 30.9590 .04978 20.0872 .06730 14.8596 9 52 .01513 66.1055 .03259 30.6833 .05007 19.9702 .06759 14.7954 8 68 .01542 64.8580 .03288 30.4116 .05037 19.8546 .06788 14.7317 T 51 .01571 63.6567 .03317 30.1446 1 .05086 19.7403 .06817 14.6685 6 55 .01600 62.4992 .03346 29.8823 .05095 19.6273 .06847 14.6059 5 66 .01629 61.3829 .0.3376 29.6245 .05124 19.5156 .06876 14.5438 4 57 .01658 60.3058 .03405 29.3711 .05153 19.4051 .06905 14.4823 3 58 .01687 59.2659 .03434 29.1220 .05182 19.2959 .06934 14.4212 2 60 .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 f Cotangj Tang Cotang Tang Cotang Tang Cotang j Tang f 89 88 I 87 86 309 TABLE XII. TANGENTS AND COTANGENTS. 40 50 6 7 Tang Cotang Tang Cotang Tang Cotang Tang ; Cotang ' .06993 14.3007 .08749 11.4301 .10510 9.51436 .12278 i 8.14485 00 1 .07022 14.2411 .08778 11.3919 .10540 9.48781 I .12308 8.12481 59 2 .07051 14.1821 i .08807 11.3540 .10569 9.46141 1 .12338 8.10536 58 3 .07U80 14.1235 j .08837 11.3163 .10599 9.43515 .12367 8.08600 57 4 .07110 14.0655 ! .08866 11.2789 .10628 9.40904 .12397 8.06674 5(> 5 .07139 14.0079 i .08895 11.2417 .10657 9.38307 , .12426 8.04756 55 6 .07168 13.9507 1 .08925 11.2048 .10687 9.35724 .12456 8.02848 54 r* .07197 13.8940 .08954 11.1681 .10716 9.33155 .12485 8.00948 53 8 .07227 13.8378 ! .08983 11.1316 .10746 9.30599 >J2515 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 .12608 7.93438 49 12 .07344 13.6174 .09101 , 10.9882 .10863 9.20516 .12633 7.91582 -IS 13 .07373 13.5634 .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 16 .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 48 18 .07519 13.2996 .09277 10.7797 .11040 9.05789 .12810 7.80622 42 19 .07548 13.2480 .09306 10.7457 .11070 9.03379 .12840 7.78825 41 20 .07578 13.1969 .09335 10.7119 .11099 9.00983 .12869 7.77035 40 21 .07607 13.1461 .09365 10.6783 .11128 8.98598 .12899 7.75254 39 22 .07636 13.0958 .09394 10.6450 .11158 8.96227 .12929 7.73480 m 23 .07665 13.0458 .09423 10.6118 .11187 8.93867 .12958 7.71715 37 24 .07695 12.9962 .09453 10.5789 .11217 8.91520 .12988 7.69957 86 25 .07724 12.9469 .09482 10.5462 .11246 8.89185 .13017 7.68208 36 2(i .07753 12.8981 .09511 10.5136 .11276 8.86862 .13047 7.66466 34 27 .07782 12.8496 .09541 10.4813 .11305 8.84551 .13076 7.64732 m 28 .07812 12.8014 .09570 10.4491 .11335 8.82252 .13106 7.63005 32 29 .07841 12.7536 .09600 10.4172 .11364 8.79964 .13136 7.61287 81 30 .07870 12.7062 .09629 10.3854 .11394 8.77689 .13165 7.59575 30 31 .07899 12.6591 .09658 10.3538 .11423 8.75425 .13195 7.57872 29 32 .07929 12.6124 .09688 10.3224 i .11452 8.73172 .13224 7.56176 28 33 .07958 12.5660 .09717 10.2913 i .11482 8.70931 .13254 7.54487 27 34 .07987 12.5199 .09746 10.2602 1 .11511 8.68701 .13284 7.52806 20 35 .08017 12.4742 .09776 10.2294 i .11541 8.66482 .13313 7.51132 .T, 36 .08046 12.4288 .09805 10.1988 .11570 8.64275 .13343 7.49465 24 37 .08075 12.3838 .09834 10.1683 .11600 8.62078 .13372 7.47806 23 88 .08104 12.3390 .09864 10.1381 .11629 8.59893 .13402 7.46154 >> 39 .08134 12.2946 .09893 10.1080 ! .IK i 8.57718 .13432 7.44509 21 40 .08163 12.2505 .09923 10.0780 .11688 8.55555 .13461 7.42871 20 41 .08192 12.2067 .09952 10.0483 ! .11718 P. 53402 .13491 7.41240 19 42 .08221 12.1632 .09981 10.0187 ! .11747 8.51259 .13521 7.39616 18 43 .08251 12.1201 .10011 9.98931 8.49128 .13550 7.37999 17 44 .08280 12.0772 .10040 9.96007 .'11806 8.47007 .13580 7.36389 16 45 .08309 12.0346 .10069 9.93101 .11836 8.44896 .13609 7.34786 15 46 .08339 11.9923 .10099 9.90211 .11865 8.42795 .13639 7.33190 14 47 .08368 11.9504 .10128 9.87338 .11895 8.40705 .13669 7.31600 18 48 .08397 11.9087 .10158 9.84482 .11924 8.38625 .13698 7.30018 12 49 .08427 11.8673 .10187 9.81641 .11954 8.36555 .13728 7.28442 11 50 .08456 11.8262 .10216 9.78817 .11983 8.34496 .13758 7.26873 10 51 .08485 11.7853 .10246 8.76009 .12013 8.32446 .13787 7.25310 9 52 .08514 11.7448 .10275 9.73217 .12042 8.30406 .13817 7.23754 8 53 .08544 11.7045 .10305 9.70441 .12072 8.28376 .13846 7.22204 7 54 .08573 11.6645 .10334 9.67680 .12101 8.26355 .13876 7.20661 6 55 .08602 11.6248 .10363 9.64935 .12131 8.24345 .13906 7.19125 5 56 .08632 11.5853 .10393 9.62205 .12160 8.22344 .13935 7.17594 4 57 .08661 11.5461 .10422 9.59490 .12190 8.20352 .13965 7.16071 8 58 .08690 11 5072 .10452 9.56791 .12219 8.18370 .13995 7.14553 2 59 .08720 11.4685 .10481 9.54106 .12249 8.16398 .14024 7.13042 1 CO .08749 11.4301 .10510 9.51-436 .12278 8.14435 .14054 7.11537 / Cotang Tang Cotang Tang Cotang Tang Cotang Tang / 85 84 il 83 82 310 TABLE XII.-TANGENTS AND COTANGENTS. 8 - i 9 10 li 11 / Tang | Cotang | Tang Cotang Tang Cotang Tang Cotang o .14054 7.11537 .15838 6.31375 .17633 5.67128 .19438 5.14455 GO 1 .140,84 7.10038 .15868 6.30189 .17663 5.66165 .19468 5.13658 59 2 .14113 7.08546 .15898 6.29007 .17693 5.65205 .19498 5.12862 58 3 .14143 7.07059 .15928 6.27829 .17723 5.64248 .19529 5.12069 57 4 .14173 7.05579 .15958 6.20655 .17753 5.63295 .19559 5.11279 56 5 .14202 7.04105 .15988 6.25486 .17783 5.62344 .19589 5.10490 55 6 .14232 7.02637 .16017 6.24321 .17813 5.61397 .19619 5.09704 54 7 .14262 6.91174 .16047 6.23160 .17843 5.60452 .19649 5.08921 58 8 .14291 6.99718 .16077 6.22003 .17873 5.59511 .19680 5.08139 68 9 .14321 6.98268 .16107 6.20851 .17903 5.58573 .19710 5.07360 51 10 .14351 6.96823 .16137 6.19703 .17933 5.57638 .19740 5.06584 50 11 .14381 6.95385 .16167 6.18559 .17963 5.56706 .19770 5.05809 49 12 .14410 6.93952 .16196 ! 6.17419 .17993 5.55777 .19801 5.05037 48 13 .14440 6.92525 .16226 6.16283 .18023 5.54851 .19831 5.042G7 47 14 .14470 6.91104 i .16256 ft 15151 .18053 5.53927 .19861 5.03499 46 15 .14499 6.89688 .16286 6.14023 .18083 5.53007 .19891 5.02734 45 16 .14589 6.88278 .16316 6.12899 .18113 5.52090 .19921 5.01971 44 17 .14559 6.86874 .16346 6.11779 .18143 5.51176 .19952 5.01210 43 18 .14588 6.85475 .16376 6.10664 .18173 5.50264 .19982 5.00451 42 19 .14618 6.84082 .16405 6.09552 .18203 5.49356 .20012 4.99695 41 20 .14648 6.82694 .16435 6.08444 .18233 5.48451 .20042 4.98940 40 21 .14678 6.81312 .16465 6.07340 .18263 5.47548 .26073 4.98188 39 22 . 14707 6.79936 .16495 6.06240 .18293 5.46648 .20103 4.97438 38 23 .14737 6.7'8564 .16525 6.05143 .18323 5.45751 .20133 4.96690 37 24 .14767 6.77199 .16555 6.04051 .18353 5.44857 .20164 4.95945 36 25 .14796 6.75838 .16585 6.02962 .18384 5.43966 .20194 4.95201 35 26 .14826 6.74483 .16615 6.01878 .18414 5.43077 .20224 4.94460 34 27 .14856 6.73133 .16645 6.00797 .18444 5.42192 .20254 4.93721 33 28 .14886 6.71789 .16674 5.99720 .18474 5.41309 .20285 4.92984 32 29 .14915 6.70450 .16704 5.98646 .18504 5.40429 .20315 4.92249 31 30 .14945 6.69116 .16734 5.97576 .18534 5.39552 .20345 4.91516 30 81 .14975 6.67787 .16764 5.96510 .18564 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 .18624 5.36936 .20436 4.89330 27 34 .15064 6.63831 .16854 5.93335 .18654 5,36070 .20466 4.88605 26 35 .15094 6.62523 .16884 5.92283 .18684 5.35206 .20497 4.87882 25 36 .15124 6.61219 .16914 5.91236 .18714 5.34345 .20527 4.87162 24 37 .15153 6.59921 .16944 5.90191 .18745 5.33487 .20557 4.86444 23 38 .15183 6.58627 .16974 5.89151 .18775 5.32631 .20588 4.85727 22 39 .15213 6.57339 .17004 5.88114 .18805 5.31778 .20618 4.85013 21 40 .15243 6.56055 .17033 5.87080 .18835 5.30928 .20648 4.84300 20 41 .15272 6.54777 .17063 5.86051 .18865 5.30080 .20679 4.83590 19 42 .15302 6.53503 .17093 5.85024 .18895 5.29235 .20709 4.82882 18 43 .15332 6.52234 .17123 5.84001 .18925 5.28393 .207'39 4.82175 17 44 .15362 6.50970 .17153 5.82982 .18955 5.27553 .20770 4.81471 16 45 .15391 6.49710 .17183 5.81966 .18986 5.26715 .20800 4.80769 15 46 .15421 6.48456 .17213 5.80953 .19016 5.25880 .20830 4.80068 14 47 .15451 6.47206 .17243 5.79944 .19046 5.25048 .20861 4.79370 13 48 .15481 6.45961 .17273 5.78938 .19076 5.24218 .20891 4.78673 12 49 .15511 6.44720 .17303 5.77936 .19106 5.23391 .20921 4.77978 11 50 .15640 6.43484 .17333 5.76937 .19136 .5.22566 .20952 4.77286 10 51 .15570 6.42253 .17363 5.75941 .19166 5.21744 .20982 4.76595 9 52 .15600 6.41026 .17393 5.74949 .19197 5.20925 .21013 4.75906 8 53 .15630 6.39804 .17423 5.73960 .19227 5.20107 .21043 4.75219 7 54 .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 56 .15719 6.36165 .17513 5.71013 .19317 5.17671 .21134 4.73170 4 57 .15749 6.34961 .17543 5.70037 .19347 5.16863 .21164 4.72490 8 58 .15779 6.33761 .17573 5.69064 .19378 5.16058 .21195 4.71813 2 59 .15809 6.32566 .17603 5.68094 [I .19408 5.15256 .21225 4.71137 1 60 .isass 6.31375 .176:33 5.67128 .19438 5.14455 .21256 4.70463 f Cotang Tang Cotang Tang Cotang Tang Cotang Tang / 81 80 79 78 311 TABLE XII. TANGENTS AND COTANGENTS. 12 13 14 15 Tang Cotang Tang Cotang Tang i Cotang Tang Cotang i .21256 4.70463 .23087 4.33148 .24933 j 4.01078 .26795 i 3.73205 60 1 .21286 4.69791 .23117 | 4.32573 .24964 ; 4.00582 .26826 ! 3.72771 59 a .21316 4.69121 .23148 4.32001 .24995 i 4.00086 .26857 3.72338 58 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 i 3.71476 56 5 .21408 4.67121 .23240 4.30291 .25087 3.98607 .26951 3.71046 55 G .21438 4.66458 .23271 4.29724 .25118 3.98117 .26982 3.7'0616 54 7 .21469 4.65797 .23301 4.29159 .25149 3.97627 .27013 3.70188 53 8 .21499 4.65138 .23332 4.28595 .25180 3.97139 .27044 3.69761 52 9 .2159 4.64480 .23363 4.28032 .25211 3.96651 .27076 3.69a35 51 10 .21560 4.63825 .23393 4.27471 .25242 3.96165 .27107 3.68909 50 11 .21590 4.63171 .23424 4.26911 .25273 3.95680 .27138 3.68485 49 12 .21621 4.62518 .23455 4.26352 .25304 3.95196 .27169 3.68061 48 13 .21651 4.61868 .23485 4.25795 .25335 3.94713 .27201 3.67638 47 14 .21682 4.61219 .23516 4.25239 .25366 3.94232 .27232 3.67'217 46 15 .21712 4.60572 .23547 4.24685 .25397 3.937ol .27263 3.66796 45 16 .21743 4.59927 .23578 4.24132 .25428 ; 3.93271 .27294 3.66376 44 17 .21773 4.59283 .23608 4.23580 .25459 3.92793 .27326 3.65957 43 18 .21804 4.58641 .23639 4.23030 .25490 3.92316 .27357 3.65538 42 19 .21834 4.58001 .23670 4.22481 .25521 3.91839 .27388 3.65121 41 20 .81864 4.57363 .23700 4.21933 .25552 3.91364 .27419 3.647'05 40 21 .21895 4.56726 .23731 4.21387 .25583 ! 3.90890 .27451 3.64289 39 22 .21925 4.56091 .23762 4.20842 .25614 ] 3.90417 .27482 3.63874 38 23 .21956 4.55458 .23793 4.20298 .25645 3.89945 .27513 3.63461 37 21 .21986 4.54826 .23823 4.19756 .25676 3.89474 .27545 3.63048 36 25 .52017 4.54196 .23854 4.19215 .25707 3.89004 .27576 3.62636 35 26 .22047 4.53568 .23885 4.18675 .25738 3.88536 .27607 3.62224 34 27 .22078 4.52941 .23916 4.18137 .25769 3.88068 .27638 3.61814 33 28 .22108 4.52316 .23946 4.17600 .25800 : 3.87601 .27670 3.61405 32 29 .22139 4.51693 .23977 4.17064 .25831 I 3.87136 .27701 3.60996 !31 30 .22169 4.51071 .24008 4.16530 .25862 3.86671 .27732 | 3 60588 30 31 .22200 4.50451 .24039 4.15997 .25893 3.86208 .27764 1 3.60181 29 32 .22231 4.49832 .24069 4.15465 .25924 3.85745 .27795 i 3.59775 28 33 .22261 4.49215 .24100 4.14934 .25955 3.85284 .27826 3.59370 27 34 .22292 4.48600 .24131 4.14405 .25986 3.84824 .27858 3.58966 26 85 .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 24 87 .22383 4.46764 .24223 4.12825 26079 ; 3.83449 .27952 3.57758 23 38 .22414 4.46155 .24254 4.12301 .26110 i 3.82992 .27983 3.57357 22 39 .22444 4.45548 .24285 4.11778 .26141 3.82537 .28015 3.56957 21 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 19 42 .22536 4.43735 .24377 4.10216 .26235 ! 3.81177 .28109 3.55761 18 43 .22567 4.43134 .24408 4.09699 .26266 ! 3.80726 .28140 3.55364 17 44 .22597 4.42534 .24439 4.09182 .26297 3.80276 .28172 3.r>l%8 16 45 .22628 4.41936 .24470 4.08666 .26328 3.79827 .28203 3.54573 15 46 j .22658 4.41340 .24501 4.08152 .26359 3.79378 .28234 3.54179 14 47 .22689 4.40745 .24532 4.07639 .26390 3.78931 .28266 3.53785 13 48 1 .22719 4.40152 .24562 4.07127 .26421 3.78485 .28297 3.53393 12 49 .22750 4.39560 .24593 4.06C16 .26452 3.78040 .28329 ,3.53001 11 50 .22781 4.38969 .24624 4.06107 .20183 3.77595 .28360 3.52609 10 51 .22811 4.38381 .24655 4.05599 .26515 3.77152 .28391 3.52219 9 5-3 .22842 4.37793 .24686 4.05092 .26546 3.76709 .28423 3.51829 8 63 .22872 4.37207 .24717 4.04586 .26577 i 3.76268 .28454 3.51441 7 r>4 .22903 4.36623 .24747 4.04081 .26608 3.75828 .28486 3.51053 6 55 .22934 4.36040 .24778 4.03578 .26639 3.75388 .28517 3.50666 *5 f,<5 .22964 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 3 58 .23026 4.34300 .24871 4.02074 .26733 3.74075 .28612 3.49509 2 59 .23056 4.33723 .24902 4.01576 .26764 3.73640 .28043 3.49125 1 60 .23087 4.33148 .24933 4.01078 .26796 3.73205 .28675 3.48741 j Cotang Tang Cotang Tang Cotang Tang Cotang Tang 7 77 76 T5 74 313 TABLE xn. TANGENTS AND COTANGENTS. 16 ' 17 18 19 Tang Cotang Tang j Cotang Tang Cotang Tang Cotang o 3.48741 .30573 3.27085 .32492 3.07768 .34433" 2.90421 60 1 .28706 3.48359 .30605 3.26745 .32524 3.07464 .34465 2.90147 59 2 .287:38 3.47977 .30637 3.26406 .32556 3.07160 .34498 2.89873 58 3 .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<> f> .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 53 8i .28927 3.45703 .30828 3.24383 .32749 3.05349 .34693 2.88240 52 9' .28958 3.45327 .30860 3.24049 .32782 3.05049 .34726 2.87970 51 10 | .28990 3.44951 .30891 3.23714 .32814 3.04749 .34758 2 87700 50 11 .29021 3.44576 .30923 3.23381 .32846 3.04450 .34791 2.87430 49 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 14 .29116 3.43456 .31019 3.22384 .32943 3.03556 .34889 2.86624 46 15! .29147 3.43084 .31051 3.22053 .32975 3.03260 .34922 2.86356 45 16 .29179 3.42713 .31083 3.21722 .33007 3.02963 .34954 2.86089 44 17 .29210 3.42343 .31115 3.21392 .33040 3.02667 .34987 2.85822 43 18 .29242 3.41973 .31147 3.21063 .33072 3.02372 .35020 2.85555 42 19 .29274 3.41604 .31178 3.20734 .33104 3.02077 .35052 2.85289 41 20 .29305 3.41236 .31210 3.20406 .33136 3.01783 .35085 2.85023 40 Oj .29337 3.40869 .31242 3.20079 .33169 3.01489 .8S1HB 2.84758 39 28 .29368 3.40502 .31274 3.19752 .33201 3.01196 .35150 2.84494 38 23 .29400 3.40136 .31306 3.19426 .33233 3.00903 .35183 2.84229 87 24 .29432 3.39771 .31338 3.19100 .33266 3.00611 .35216 2.&S96:. 86 25 .29463 3.39406 .31370 3.18775 .33298 3.00319 .35248 2.83702 35 28 .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 88 28 .21)558 3.38317 .31466 3.17804 .33395 2.99447 .35346 2.82914 32 29 .29590 3.37955 .31498 3.17481 .33427 2.99158 .35379 2.82653 31 30 .29621 3.37594 .31530 3.17159 .33460 2.98868 .35412 2.82391 30 33 .29653 3.37234 .31562 3.16838 .33492 2.98580 .35445 2.82130 29 32 .29685 3.36875 .31594 3.10517 .33524 2.98292 .35477 2.81870 28 88 .29716 3.3G516 .31626 3.16197 .a3557 2.98004 .35510 2.81610 27 : J 1 .29748 3.36158 .31658 3.15877 .33589 2.97717 .35543 2.81350 M 38 .29780 3.35800 .31690 3.15558 .33621 2.97430 .35576 2.81091 25 36 .29811 3.35443 .31722 3.15240 .33654 2.97144 .35608 2.80833 24 37 .29843 3.35087 .31754 3.1492.2 .33686 2.96858 .35641 2.80574 23 88! .29875 3.34732 .31786 3.14605 .33718 S. 96573 .35674 2.80316 22 89 .29906 3.34377 .31818 3.14288 .33751 2.96288 .35707 2.80059 21 W .29938 3.34023 .31850 3.13972 .33783 2.96004 .35740 2.79802 96 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 11 .30065 3.32614 .31978 3.12713 .33913 2.94872 .35871 2.78778 1(5 45 .30097 3.32264 .32010 3.12400 ,aS945 2.94591 .35904 2.78523 15 46 .30128 3.31914 .32042 3.12087 .33978 2.94309 .35937 2.78269 14 47! .30160 3.31565 .32074 3.11775 .34010 2.94028 .35969 2.78014 13 18 .30192 3.31216 .32106 3.11464 .34043 2.93748 .36002 2.77761 12 I!) .:!0-."J4 3.30868 .32139 3.11153 .34075 2.93468 .36035 2.77507 11 50 .30255 3.30521 .32171 3.10842 .34108 2.93189 .36068 2.77254 10 51 .30287 3.30174 .32203 3.10532 .34140 2.92910 .36101 2.77002 9 58 .30319 3.29829 .32235 3.10223 .34173 2.92632 .36K34 2.76750 8 :>:>, .30351 3.29483 .32267 3.09914 .34205 2.92354 .36167 2.76498 7 54 ' .30382 3.29139 .32299 3.09606 .34238 2.92076 .36199 2.76247 6 55 .30414 3.28795 .32331 3.09298 .34270 2.91799 .36232 2.75996 5 58 .30446 3.28452 .32363 3.08991 .34303 2.91523 .36265 2.75746 4 57 .30478 3.28109 .32396 3.08685 .343:35 2.91246 .36298 2.75496 3 58 .30509 3.27767 .32428 3.08379 .34368 2.90971 .38381 2.75246 2 59 .30541 3.27426 .32460 3.08073 .34400 2.90696 .36364 2.74997 1 60 .30573 3.27085 .32492 3.07768 .34433 2.90421 .36397 2.74748 / Cotang Tang Cotang Tang Cotang Tang Cotang Tang / 73 72 71 II 70 A BIB TABLE XIL TANGENTS AND COTANGENTS. 20 ! 21 ! 22 i 23 Tang | Cotang Tang Cotang Tang Cotang ' Tang Cotang .36397 | 2.74748 ! .38386 2.60509 .40403 2.47509 .42447 | 2.35585 60 1 .33430 2.74499 .38420 2.602&3 ! .40436 2.47302 .42482 2.35395 59 2 .36463 I 2.74251 .38453 2.60057 .40470 2.47095 .42516 2.35205 58 3 .36496 2.74004 .38487 2.59831 .40504 2.46888 ! .42551 2.35015 57 4 .36529 2.73756 .38520 2.59606 .40538 2.46682 .42585 2.34825 56 5 .36562 2.73509 .38553 2.59381 .40572 2.46476 .42619 2.34636 55 6 .36595 2.73263 .38587 2.59156 .40606 2.46270 .42654 2.34447 54 7 .36628 2.73017 .38620 2.58932 .40640 2.46065 .42688 2.34258 58 8 .36661 2.72771 .38654 2.58708 .40674 2.45860 42722 2.34069 52 9 .36694 2.72526 .38687 2.58484 .40707 2.45655 '.42757 2.33881 51 10-1 .36727 2.72281 .38721 2.58261 .40741 2.45451 .42791 2.33693 50 11 .36760 8.72036 .38754 2.58038 .40775 2.45246 .42826 2.33505 49 12 .36793 2.71792 .38787 2.57815 .40809 2.45043 .42860 2.33317 18 13 .36826 2.71548 .38821 2.57593 .40843 2.44839 .42894 2.33130 47 14 .36859 2.71305 .38854 2.57371 .40877 2.44636 .42929 2.32943 4G 15 .36892 2.71062 .38888 2.57150 .40911 2.44433 .42963 2.32756 45 16 .36925 2.70819 .38921 2.56928 .40945 2.44230 .42998 2.32570 44 17 .36958 2.70577 .38955 2.56707 .40979 2.44027 .43032 2.32383 43 18 .36991 2.70335 .38988 2.56487 .41013 2.43825 .43067 2.32197 42 19 .37024 2.70094 .39022 2.56266 .41047 2.43623 .43101 2.32012 41 20 .37057 2.69853 .39055 2.56046 .41081 2.43422 .43136 2.31826 10 21 .37090 2.69612 .39089 2.55827 .41115 2.43220 .43170 2.31641 39 22 .37123 2.69371 .39122 2.55608 .41149 2.43019 .43205 2.31456 138 .37157 2.69131 .39156 2.55389 .41183 2.42819 .43239 2.31271 37 24 .37190 2.68892 .39190 2.55170 .41217 2.42618 .43274 2.31086 86 25 .37223 2.68653 .39223 2.54952 .41251 2.42418 .43308 2.30902 35 26 .37256 2.68414 .39257 2.54734 .41285 2.42218 .43343 2.30718 34 27 .37289 2.68175 .39290 2.54516 .41319 2.42019 .43378 2.30534 33 28 .37322 2 67937 .39324 2.54299 .41353 2.41819 .43412 2.30351 3-3 2'J .37355 2.67700 .39357 2.54082 .41387 2.41620 .43447 2.30167 31 30 .37388 2.67462 .39391 2.53865 .41421 2.41421 .43481 2.29984 30 31 .37422 2.67225 .39425 2.53648 .41455 2.41223 .43516 2.29801 20 32 .37455 2.66989 .39458 2.53432 .41490 2.41025 .43550 2.29619 28 33 .37488 2.66752 .39492 2.53217 .41524 2.40827 .43585 2.29437 27 34 .37521 2.66516 .39526 2.53001 .41558 2.40G29 .43620 2.29254 26 35 .37554 2.66281 .39559 2.52786 .41592 2.40432 .43654 2.29073 25 36 .37588 2.66046 .39593 2.52571 .41626 2.40235 .43689 2.28891 21 37 37621 2.65811 .39626 2.52357 .41660 2.40038 .43724 2.28710 23 38 .37654 2.65576 .39660 2.52142 .41694 2.39841 .43758 2.28528 go 39 37687 2.65342 .39694 2.51929 .41728 2.39645 .43793 2.28348 21 40 .37720 2.65109 .39727 2.51715 .41763 2.39449 .43828 2.28167 20 41 .37754 2.64875 .39761 2.51502 .41797 2.39253 .43862 2.27987 19 42 .37787 2 64642 i 39795 2.51289 .41831 2.39058 .43897 2.27806 18 43 .37820 2.64410 i .39829 2.51076 .41865 2.38863 .43933 2.27626 17 44 37853 2.64177 ! .39862 2.50864 41899 2.38668 .43966 2.27447 16 45 .37887 2.63945 i .39896 2.50652 .41933 2.38473 .44001 2.27267 15 46 47 .37920 .37953 2.63714 2.63483 .39930 .39963 2.50440 2.50229 .41968 .42002 2.38279 2.38084 .44036 .44071 2.27088 2.26909 11 13 48 .37986 2.63252 i .39997 2.50018 .42036 2.37891 .44105 2.26730 12 49 38020 2.63021 ! .40031 2.49807 .42070 2.37697 .44140 2.26552 11 50 .38053 2.62791 i .40065 2.49597 .42105 2.37504 .44175 2.26374 10 51 38086 2.62561 .40098 2.49386 .42139 2.37311 .44210 2.26196 9 52 .38120 2.62332 .40132 2.49177 .42173 2.37118 .44244 2.26018 H 53 .38153 2.62103 .40166 2.48967 .42207 2.36925 .44279 2.25840 7 54 .38186 2.61874 .40200 2.48758 .42242 2.36733 .44314 2.25663 6 55 38220 2.61646 .40234 2.48549 .42276 2.36541 .44349 2.25486 5 56 57 .38253 38286 2.61418 2.61190 .40267 .40301 2 48340 2.48132 .42310 .42345 2.36349 2.36158 .44384 .44418 2.25309 2.25132 4 3 58 .38320 2.60963 .40335 2.47924 .42379 2.35967 .44453 2.24956 2 59 90 .38353 .38386 2.60736 2.60509 .40369 2.47716 .40403 2.47509 .42413 .42447 2.35776 2 35585 .44488 2 24780 .44523 2.24604 1 Cotang Tang Cotang Tang h Cotang Tang Cotang Tang , 1 69 68 67 ii 66 314 TABLE m TANGENTS AND COTANGENTS. 24 'I! 25 26 ! 27 Tang j Cotang Tang Cotang Tang Cotang Tang Cotang T44523 2.24604 .46631" 2.14451 ; .48773 2.05030 .50953 1.96261 60 1 .44558 2.24428 .46666 2.14288 1 .48809 2.04879 .50989 1.96120 '59 8 .44593 2.24252 .46702 2.14125 i .48845 2.04728 .51026 1.95979 58 8 .44627 2.24077 .46737 2.13963 j .48881 2.04577 .51063 1.95838 157 4 .44662 2.23902 .46772 2.13801 .48917 2.04426 .51099 1.95698 '56 5 .44697 2.23727 .46808 2.13639 ! .48953 2.04276 .51136 1.95557 55 6 .44732 2.23553 .46843 2.1:3477 I .48989 2.04125 .51173 1.95417 54 7 .44767 2.23378 .46879 2.13316 .49026 2.03975 .51209 1.95277 53 8 .44802 2.23204 .46914 2.13154 ! .49062 2.03825 .51246 1.95137 52 9 .44837 2.23030 .46950 2.12993 i .49098 2.03675 .51283 1.94997 51 10 .44872 2.22857 .46985 2.12832 .49134 2.03526 .51319 1.94858 50 11 .44907 2.22683 ' .47021 2.12671 .49170 2.03376 .51356 1.94718 49 12 .44942 2.22510 .47056 2.12511 .49206 2.03227 .51393 1.94579 48 13 .44977 2.22337 .47092 2.12350 .49242 2.03078 .51430 1.94440 47 14 .45012 2.22164 .47128 2.12190 .49278 2.02929 .51467 1.94301 46 15 .45047 2.21992 .47163 2.12030 .49315 2.02780 .51503 1.94162 t5 16 .45082 2.21819 .47199 2.11871 .49351 2.02631 .51540 1.94023 44 17 .45117 2.21647 .472:34 2.11711 .49387 2.02483 .51577 1.93885 ; 43 18 .45152 2.21475 .47-270 2.11552 .49423 2.02335 .51614 1.93746 |42 11) .45187 2.21304 .47305 2.11392 .49459 2.02187 .51651 1.93608 141 20 .45222 2.21132 .47341 2.11233 .49495 2.02039 .51688 1.93470 40 21 .45257 2.20961 .47377 2.11075 .49532 2.01891 .51724 1.93332 39 22 .45292 2.20790 .r412 2.10916 .49568 2.01743 .51761 1.93195 38 23 .45327 2.20619 .47448 2.10758 .49604 2.01596 .51798 1.93057 37' 24 .45362 2.20449 .47483 2.10600 ! .49640 2.01449 .51835 1.92920 36 25 .45397 2.20278 .47519 2.10442 ! .49677 2.01302 .51872 1.92782 35 26 .45432 2.20108 .47555 2.10284 i .49713 2.01155 .51909 1.92645 34 27 .454(57 2.19938 .47590 ! 2.10126 ! .49749 2.01008 .51946 1.92508 33 28 .45502 2.19769 .47626 2.09969 ! .49786 2.00862 .51983 1.92371 3-2 29 .45538 2.19599 .47062 2.09811 .49822 : 2.00715 .52020 1.92235 31 30 .45573 2.19430 .47698 2.09654 .49858 2.00569 .52057 1.92098 30 31 .45608 2.19261 .47733 2.09498 .49894 2.00423 .52094 1.91962 29 3-2 .45643 2.19092 .47769 2.09341 ! .49931 2.00277 .52131 1.91826 28 33 .45678 2.18923 .47805 2.09184 .49967 2.00131 .52168 1.91690 27 34 .45713 2.18755 .47840 2.09028 .50004 1.99986 .52205 1.91554 26 35 .45748 2.18587 .4787'6 2.08872 .50040 1.99841 .52242 1.91418 25 36 .45784 2.18419 .47912 2.08716 .500J'6 1.99695 .52279 1.91282 24 37 .45819 2.18251 .47948 2.08560 .50113 1.99550 .52316 1.91147 23 38 .45854 2.18084 .47984 ! 2.0S405 .50149 1.99406 .52353 1.91012 22 39 .45889 2.17916 .48019 2.08250 .50185 1.99261 .52390 1.90876 21 40 .45924 2.17749 .48055 2.08094 .50222 1.99116 .52427 1.90741 20 41 .45960 2.17582 .48091 2.07939 .50258 1.98972 .52464 1.90607 19 42 .45995 2.17416 .48127 2.07785 .50295 1.98828 .52501 1.90472 18 43 .46030 2.17249 .48163 2.07630 .50331 1.98684 .52538 1.90337 17 44 .46065 2.17083 .48198 2.07476 | .50368 1.98540 .52575 1.90203 1C 45! .46101 2.16917 .48234 2.07321 : .50404 ! 1.98396 .52613 1.90069 15 46 .46136 ! 2.16751 .48270 2.07167 i .50441 1.98253 .52650 1.89935 114 47 .46171 2.16585 .48306 2.07014 i .50477 1.98110 .52687 1.8S801 Sl3 48 .46206 2.16420 .48342 2.06860 i .50514 1.97'966 .52724 1.89667 12 49! .46242 2.16255 .48378 2.067'06 ! .50550 1.97823 .52761 1.895:33 11 50 .46277 2.16090 .48414 2.06553 i .50587 1.97681 .52798 1.89400 10 51 .46312 2.15925 .48450 2.06400 .50623 1.97538 .52836 1.8926b 9 52 .46348 2.15760 .48486 2.06247 .50660 1.97395 .52873 1.89133 8 53 .46383 2.15596 .48521 2.0G094 .50696 1.97253 .52910 1.89000 7 54 .46418 2.15432 .48557 2.05942 .50733 1.97111 .52947 1.88867 6 55 .46454 2.15268 .48593 2.05790 .50769 1.96969 .52985 1.88734 5 56 .46489 2.15104 148629 2.05637 .50806 1.96827 .53022 1.88602 4 57 i .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 2.05182 .50916 1.96402 .53134 1.88205 1 60 .46631 2.14451 .48773 ' 2.05030 .50953 1.96261 .53171 1.88073 / Cotang Tang Cotang \ Tang Cotang Tang Cotang Tang r 65 64 ; 63 62 315 TABLE XII.-TANGENTS AND COTANGENTS. 2 8 2 9 3 3 1 / Tang Cotang 1 Tang Cotang Tang Cotang Tang Cotang- / .53171 .88073 .55431 1.80405 .57735 1.73205 .60086 1.66428 60 1 .53208 .87941 .55469 1.80281 .57774 1.73089 .60126 1.66318 59 2 .53246 .87809 .55507 1.80158 .57813 1.72973 .60165 1.66209 58 3 .53283 .87677 .55545 1.80034 .57851 1.72857 .60205 1.66099 57 4 .53320 .87546 .55583 1.79911 .57890 1.72741 .60245 1.65990 56 5 .53358 .87415 .55621 1.79788 .57929 1.72625 .602S4 1.65881 55 G .53395 .87283 .55659 1.79665 .57968 1.72509 .60324 1.65772 54 7 .53432 .87152 .55697 1.79542 .58007 1.72393 .60364 1.65663 58 8 .53470 .87021 .55736 1.79419 .58046 1.72278 .60403 1.65554 52 9 .53507 .86891 .55774 1.79296 .58085 1.72163 .60443 1.65445 51 10 .53545 .86760 .55812 1.79174 : .58124 1.72047 .60483 1.65337 50 11 .53582 .86630 .55850 1.79051 .58162 1.71932 .60522 1.65228 -10 13 .53620 .86499 .55888 1.78929 1 .58201 1.71817 .60562 1.65120 48 13 .53657 .86369 .55926 1.78807 .58240 1.71702 .60602 1.65011 47 14 .53694 .86239 .55964 1.78685 .58279 1.71588 .60642 1.64903 46 15 .53732 .86109 .56003 1.78563 .58318 1.71473 .60681 1.64795 45 16 .53769 .85979 .56041 1.78441 .58357 1.71358 .60721 1.64687 44 17 .53807 .85850 .56079 1.78819 i .58396 1.71244 .60761 1.64579 48 18 .53844 .85720 .56117 1.78198 .58435 1.71129 .60801 1.64471 42 10 .53882 .85591 .56156 1.78077 ; .58474 1.71015 .60841 1.64363 41 20 .53920 .85462 .56194 1.77955 .58513 1.70901 .60881 1.64256 40 21 .53957 .85333 .56232 1.77834 1 .58552 1.70787 .60921 1.64148 89 22 .53995 .85204 .56270 1.77713 .58591 1.70673 .60060 1.64041 88 23 .54032 .85075 .56309 1.77592 .58631 1.70560 .61000 1. 63934 87 24 .54070 .84946 .56347 1.77471 .58670 1.70446 ! .61040 1.63826 36 25 .54107 .84818 .56385 1.77351 .58709 1.70332 i .61080 1.63719 35 2G .54145 .84689 .56424 1.77230 .58748 1.70219 1 .61120 1.63612 31 27 .54183 .84561 .56462 1.77110 .58787 1.70106 .61160 1.63505 33 28 .54220 .84433 .56501 1.76990 .58826 1.69992 .61200 1.63398 32 2!) .54258 .84305 .5(5539 1.76869 .58865 1.69879 .61240 1.63292 31 30 .54296 ,84177 .56577 1.76749 .58905 1.69766 .61280 1.63185 80 31 .54333 .84049 .56616 1.76629 .58944 1.69653 ! .61320 1.63079 29 32 .54371 .83922 .56654 1.76510 .58988 1.61)541 j .61360 1.62972 28 33 .54409 .a3794 .56693 1.76390 ; .59022 1.69428 .61400 1.62866 27 34 .54446 .83667 .56731 1.76271 .59061 1.69316 .61440 1.62760 26 35 .54484 .83540 .56769 1.76151 .59101 1.69203 .61480 1.62654 25 36 .54522 .83413 .56808 1.76032 .59140 1.69091 .61520 1.62548 24 37 .5-1560 .83286 .56846 1.75913 .59179 1.68979 .61561 1.62442 23 38 .54597 .83159 .56885 1.75794 : .59218 1.68866 .61601 1.62336 22 3'J .54635 .83033 .56923 1.75675 .59258 1.68754 .61641 1.62230 21 40 .54673 .82906 .56962 1.75556 .59297 1.68643 .61681 1.62125 20 41 .54711 .82780 .57000 1.75437 .59386 1.68531 .61721 1.62019 1!) 42 .54748 .82654 .57039 1.75319 i .59376 1.68419 .61761 1.61914 18 43 .54786 .82528 . 57078 1.75200 .59415 1.68308 .61801 1.61808 17 44 .54824 .82402 .57116 1.75082 .59454 1.68196 .61842 1.61703 16 45 .54862 .82276 i .57155 1.74964 .59494 1.68085 .61882 1.61598 15 46 .54900 .82150 1 .57193 1.74846 ! .59533 1.67974 .61922 1.61493 14 47 .54938 .82025 .57232 1.74728 .59573 1.67863 .61962 1.61388 13 48 .54975 .81899 .57271 1.74610 .59612 1.67752 , .62003 1.61283 12 4! .55013 .81774 .57309 1.74492 .59651 1.67641 .68043 1.6117!) 11 no .55051 .81649 .57348 1.74375 .59691 1.67530 .62083 1.61074 10 51 ,55089 .81524 .57386 1.74257 .59730 1.67419 .62124 l.fiOflTO 9 52 .55127 .81399 .57425 1.74140 ; .59770 1.67309 .62164 1.60865 8 53 .55165 1.81274 .57464 1.74022 | .59809 1.67198 .62204 1.60761 7' 54 .55203 1.81150 .57503 1.73905 ! .59849 1.67088 .62245 1.60657 6 55 .55241 1.81025 .57541 1.73788 .59888 1.66978 .62285 1.60553 5 56 .55279 1.80901 l .57580 1.73671 .59928 1.66867 .62325 1.60449 4 57 .55317 1.80777 .57619 1.73555 .59967 1.66757 .62366 1.60345 8 58 .55355 L80658 .57657 1.73438 .60007 1.66647 .62406 1.60241 2 59 .55393 1.80529 .57696 1.78821 .60046 1.66538 .62446 1.60137 1 GO .55431 1.80405 | .57735 1.73205 .60086 1.66428 .62487 1.60033 / Cotaug Tang Cotang Tang Cotang Tang Cotang Tung ( 6 1 1 6 5 5 5 8 316 TABLE XII. TANGENTS AND COTANGENTS. 32 . i 33 j 34 35 Tang Co tang ' Tang Cotang | Tang Cotang Tang Cotang .t 1:2187 1.6003:-] .64941 1.53986 .07451 1.48256 .70021 1.42815 60 1 .625-27 1.59930 .64982 1.53888 .67493 1.48163 .70064 1.42726 59 2 .62568 1.59826 .65024 1 .53791 .67536 1.48070 .70107 1.42638 58 3 .62608 1.59723 .650(15 1.53693 .67578 1.47977 .70151 1.42550 57 4 .62649 1.59620 .65106 1.53595 .67620 1.47885 .70194 1.42462 56 5 .62689 1.59517 .65148 1.53497 .67663 1.47792 .70238 1.42374 55 6 .62730 1.59414 .65189 1.53400 .67705 1.47699 .70281 1.42286 54 y .62770 1.59311 .65231 1.53302 .67748 1.47607 .70325 1.42198 53 8 .62811 1.59208 .65272 1.53205 .67790 1.47514 .70368 1.42110 52 9 .62852 1.59105 .65314 1.53107 .67832 1.47422 .70412 1.42022 51 10 .62892 1.59002 .65355 1.53010 .67875 1.47330 .70455 1.41934 50 11 .62933 1.58900 .65397 1.52913 .67917 1.47238 .70499 1.41847 49 12 .62973 1.58797 .65438 1.52816 .67960 1.47146 .70542 1.41759 48 13 .63014 1.58695 .65480 1.52719 .68002 1.47053 .70586 1.41672 47 14 .63055 1.58593 .65521 1.52622 .68045 1.46962 .70629 1.41584 46 15 .63095 1.58490 .65563 1.52525 .68088 1.46870 .70673 1.41497 45 6 .63136 1.58388 .65604 1.52429 .68130 1.46778 .70717 1.41409 44 .63177 1.58286 .65646 1.52332 .68173 1.46686 .7'0760 1.41322 43 8 .63217 1.58184 .65688 1.52235 .68215 1.46595 .70804 1.41235 42 19 .63258 1 .58083 .65729 1.52139 : .68258 1.46503 .70848 1.41148 41 20 .63299 1.57981 .65771 1.52043 .68301 1.46411 .70891 1.41061 40 21 .63340 1.57879 .65813 1.51946 ' .68343 1.46320 .70935 1.40974 39 22 .63380 1.57778 .65851 1.51850 I .68386 1.46229 .70979 1.40887 38 23 .63421 1.57676 .65896 1.51754 i .68429 1.46137 .71033 1.40800 37 24 .63462 1.57575 .65938 1.51658 .68471 1.46046 .71066 1.40714 36 25 .63503 1.57474 .65980 1.51562 .68514 1.45955 .71110 1.40627 35 26 .63544 1.57372 .66021 1.51466 .68557 1.45864 .71154 1.40540 34 27 .63584 1.57271 .66063 1.51370 .68600 1.45773 .71198 1.40454 33 23 .63625 1.57170 .66105 1.51275 .68642 1.45682 .71242 1.40367 32 2'J .63666 1.57069 .66147 1.51179 .68685 1.45592 .71285 1.40281 31 30 .63707 1.56969 .66189 1.51084 .68728 1.45501 .71329 1.40195 30 31 .63748 1.56868 .66230 1.50988 .68771 1.45410 .71373 1.40109 29 32 .63789 1.56767 .66272 1.50893 .68814 1.45320 .71417 1.40022 28 33 .63830 1.56667 .66314 1.50797 .68857 1.45229 .71461 1.39936 27 34 .63871 1.56566 .66356 1.50702 .68900 1.45139 .71505 1.39850 26 35 .63912 1.56466 .66398 i; 50607 .68942 1.45049 .71549 1.39764 25 36 .63953 1.56366 .66440 1.50512 .68985 1.44958 .71593 1.39679 24 37 .63994 1.56265 .66482 1.50417 .69028 1.44868 .71637 1.39593 23 38 .64035 1.56165 .66524 1.50322 .69071 1.44778 .71681 1.39507 22 39 .64076 1.56005 .66566 1.50228 .69114 1.44688 .71725 1.39421 21 40 .64117 1.55966 .66608 1.50133 .69157 1.44598 .71769 1.39336 20 41 .64158 1.55866 .66650 1.50038 .69200 1.44508 .71813 1.39250 19 42 .64199 1.55766 .66692 1.49944 .69243 1.44418 .71857 1.39165 18 43 .64240 1.55666 .66734 1.49849 .69286 1.44329 .71901 1.39079 17 44 .64281 1.55567 .66776 1.49755 .69329 1.44239 .71946 1.38994 16 45 .64322 1.55467 .66818 1.49661 .69372 1.44149 .71990 1.38909 15 46 .64363 1.55368 .66860 1.49566 .69416 1.44060 .72034 1.38824 14 47 .64404 1.55269 ' .66902 1.49472 .69459 1.43970 .72078 1.38738 13 48 .64446 1.55170 ! .66944 1.49378 .69502 1.43881 .72122 1.38653 12 49 .64487 1.55071 i .66986 1.49284 .69545 1.43792 .72167 1.38568 11 50 .64528 1.54972 .67028 1.49190 .69588 >. 43703 72211 1.38484 10 51 .64569 1.54873 .67071 1.49097 .69631 1.43614 .72255 1.38399 9 52 .64610 1.54774 .67113 1.49003 .69675 1.43525 .72299 1.38314 8 53 .64652 1.54675 ! .67155 1.48909 .69718 1.43436 .72344 1 38229 54 .64693 1.54576 ! .67197 1.48816 .69761 1.43347 .72388 1.38145 6 55 .64734 1.54478 j .67239 1.48722 .69804 1.43258 .72432 1 38060 g 56 .64775 1.54379 .67282 1.48629 .69847 1.43169 .72477 1.37976 ^ 57 .64817 1.54281 .67324 1.48536 .69891 1.43080 .72521 1 37891 3 58 .64858 1.54183 .67366 1.48442 .69934 1.42992 .72565 1.37807 2 58 .64899 1.54085 .67409 1.48349 .69977 1.42903 .72610 1.37722 1 6C .64941 1.53986 .67451 1.48256 .70021 1.42815 .72654 1.37638 / Cotang Tang Cotang Tang ' Cotang I Tang Cotang Tang f 57 56 55 54 317 TABLE XII. TANGENTS AND COTANGENTS. 3 s 3 7 3 5 3 9 Tang Cotang Tang Cotang Tang Cotang Tang Cotang .72654 1.37638 .75355 1.32704 .78121) 1.27994 .80978 1.23490 on 1 .72699 1.37554 .75401 1.32624 i .78175 1.27917 .81027 1.23416 69 2 .72743 1.37470 .75447 1.32544 .78222 1.27841 .81075 1.23343 58 8 .72788 1.37386 .75492 1.32464 .78269 1.27764 .81123 1.23270 67 4 .72832 1.37302 .75538 1.32384 .78316 1.27688 .81171 1.23196 56 5 .72877 1.37218 .75584 1.32304 .78363 1.27611 .81220 1.23123 66 6 .72921 1.37134 .75629 1.32224 .78410 1.27535 .81268 1.23050 64 7 .72966 1.37050 .75675 1.32144 .78457 1.27458 .81316 1.22977 63 8 .73010 1.36967 .75721 .32064 .78504 1.27382 .81364 1.22904 52 9 .73055 1.36883 .75767 .31984 .78551 1.27306 .81413 1.22831 51 10 .73100 1.36800 .75812 .31904 .78598 1.27230 .81461 1.22758 60 11 .73144 1.36716 .75858 .31825 .78645 1.27153 .81510 1.22685 1! 18 .73189 1.36033 .75904 .31745 .78692 1.27077 .81558 1.22612 18 13 .73234 1.36549 .75950 .31666 .78739 1.27001 .81606 1.22539 47 14 .73278 1.36466 .75996 .31586 .78786 1.26925 .81655 1.22467 46 15 .73323 1.36383 .76042 .31507 .78834 1.26849 .81703 1.22394 45 16 .73368 1.36300 .76088 .31427 .78881 1.26774 .81752 1.22321 44 17 .73413 1.36217 .76134 .31348 .78928 1.26698 .81800 1.22249 43 18 .73457 1.36134 .76180 1.31269 .78975 1.26622 .81849 1.22176 42 19 .73502 1.36051 .76226 1.31190 .79022 1.26,546 .81898 1.22104 11 20 .73547 1.35968 .76272 1.31110 .79070 1.26471 .81946 1.22031 40 21 .73592 1.35885 .76318 1.31031 .79117 1.26395 .81995 1.21959 39 22 .73637 1.35802 .76364 1.30952 .79164 1.26319 182044 1.21886 38 23 .73681 1.35719 .76410 1.30873 .79212 1.26244 .82092 1.21814 37 24 .73726 1.35637 .76456 1.30795 .79259 1.26169 .82141 1.21742 36 25 .73771 1.35554 .76502 1.30716 .79306 1.26093 .82190 1.21670 36 26 .73816 1.35472 .76548 1.30637 .79354 1.26018 .82238 .21598 34 2V .73861 1.35389 .76594 1.30558 .79401 1.25943 .82287 .21526 33 28 .73906 1.35307 .76640 1.30480 .79449 1.25867 .82336 .21454 32 29 .73951 1.35224 .76686 1.30401 .79496 1.25792 .82385 .21382 81 30 .73996 1.35142 .76733 1.30323 .79544 1.25717 .824:34 .21310 30 81 .74041 1.35060 .76779 1.30244 .79591 1.25642 .82483 .21238 29 32 .74086 1.34978 .76825 1.30166 .79639 1.25567 .82531 .21166 28 83 .74131 1.34896 .76871 1.30087 .79686 1.25492 .82580 .21094 27 34 .74176 1.34814 .76918 1.30009 .79734 1.25417 .82629 .21023 26 35 .74221 1.34732 .76964 1.29931 .79781 1.25343 .82678 .20951 25 86 .74267 1.34650 .77010 1.29853 .79829 1.25268 8 4> 727 .20879 21 37 .74312 1.34568 .77057 .29775 .79877 1.25193 ! 82776 .20808 23 88 .74357 1.34487 .77103 .29696 .79924 1.25118 .82825 .20736 2 39 .74402 1.34405 .77149 .29618 .79972 1.25044 .82874 .20665 21 40 .74447 1.34323 .77196 .29541 .80020 1.24969 .82923 .20593 20 41 .74492 1.34242 .77242 .29463 .80067 1.24895 .82972 .20522 19 42 .74538 1.34100 .77289 1.29385 .80115 1.24820 .83022 .20451 18 43 .74583 1.34079 .77335 1.29307 .80163 1.24746 .83071 .20379 17 44 .74628 1.33998 .77382 1.29229 .80211 1.24672 .83120 .20308 16 46 .74674 1.33916 .77428 1.29152 .80258 1.24597 .8:11 69 .20237 15 46 .74719 1.33835 . 77475 1.29074 .80306 1.24523 .83218 .20166 11 47 .74764 1.33754 .77521 1.28997 .80354 1.24449 .83268 .20095 18 48 .74810 1.33673 .77568 1.28919 .80402 1.24375 .83317 .20024 12 48 .74855 1.33592 .77615 1.28842 .80450 1.24301 ,83366 .19953 11 60 .74900 1.33511 .77661 1.28764 .80498 1.24227 .83415 .19882 10 51 .74946 1.33430 .77708 1.28687 .80546 1.24153 .83465 .19811 9 52 .74991 1.33349 .77754 1.28610 .80594 1.24079 .83514 .197'40 8 53 .75037 1.33268 .77801 1.28533 .80642 1.24005 .83564 .19669 7 54 .75082 1.33187 .77848 .28-156 .80690 1.23931 .83613 .19599 6 55 .75128 1.33107 .77895 .28379 .80738 1.23858 .83662 .19528 5 66 .75173 1.33026 .77941 .28302 .80786 1.23784 .83712 1.19457 4 67 .75219 1.32946 .77988 .28225 .80834 1.23710 .83761 1.19387 3 58 .75264 1.3:28(15 .78035 .28148 .80882 1.23637 .&38H 1.19316 2 59 .75310 1.32785 .78082 1.28071 .80930 1.23563 i .83860 1.19246 1 60 .75355 1.32704 .78129 1.27994 .80978 1.23490 .83910 1.19175 / Cotang Tang Cotang Tang Cotang Tang ; Cotang Tang . i 3 5 2 5 1 ! 5 318 TABLE XII.-TANGENTS AND COTANGENTS. 40 . 41 ! 42 43 Tang 1 Cotang | Tang Cotang Tang Cotang Tang Cotang .S3'.) 10 1.19175 .86929 1.15037 .90040 1.11061 .93252 1.07237 60 1 .83960 1.19105 .86980 1.14969 .90093 1.10996 .93306 1.07174 59 2 .84009 1.19035 .87031 .14902 .90146 1.10931 .93360 1.07112 58 3 .84059 1.18964 .87082 .14834 .90199 1.10867 .93415 1.07049 57 4 .84108 1.18894 .87133 .14767 .90251 1.10802 .93469 1.06987 56 6 .84158 1.18824 .87184 .14699 .90304 1.10737 .93524 1.06925 55 6 .84208 1.18754 .87236 .14632 .90357 1.10672 .93578 1.06862 51 7 .84258 1.18684 .87287 .14565 .90410 1.10607 .93633 1.06800 53 S .84307 1.18614 .87338 i .14498 .90463 1.10543 .93688 1.06738 5:3 <) .84357 1.18544 .87389 .14430 .90516 i 1.10478 .93742 1.06676 51 10 .84407 1.18474 .87441 .14363 .90569 1.10414 .93797 1.06613 50 11 .84457 1.18404 .87492 .14296 .90621 1.10349 .93852 .06551 41) 12 .84507 1.18334 .87543 .14229 .90674 1.10285 ! .93906 .06489 4S 13 .84556 1.18264 .87595 .14162 .90727 1.10220 .93961 .06427 47 14 .84606 1.18194 .87646 .14095 .90781 1.10156 .94016 .06365 46 15 .84656 1.18125 .87698 1.14028 .90834 1.10091 ' .94071 .06303 45 16 .84706 1.18055 .87749 1.13961 ! .90887 1.10027 .94125 .06241 44 17 .84756 1.17986 .87801 1.13894 i .90940 ! 1.09963 ! .94180 .06179 43 18 .84806 1.17916 .87852 1.13828 .90993 ' 1.09899 .94235 .06117 42 19 .84856 1.17846 .87904 1.13761 .91046 1.09834 ! .94290 .06056 41 90 .84906 1.17777 .87955 1.13694 .91099 1.09770 .94345 .05994 40 21 .84956 1.17708 .88007 1.13627 .91153 1.09706 .94400 .05932 89 22 .85006 1.17638 .88059 1.13561 .91206 1.09642 .94455 .05870 38 23 .85057 1.17569 .88110 1.13494 .91259 1.09578 .94510 .05809 37 34 .85107 1 . 17500 .88162 1.13428 .91313 1.09514 .94565 .05747 36 25 .85157 1.17430 .88214 1.13361 .91366 1.09450 i .94620 .05685 85 26 .85207 1.17361 .88265 1.13295 .91419 1.09386 .94676 .05624 84 27 .85257 1.17292 .88317 1.13223 .91473 1.09322 .94731 .05562 33 28 .85308 1.17223 .88369 1.13162 .91526 1.09258 .94786 .05501 32 291 .85358 1.17154 .88421 1.13096 .91580 1.09195 .94841 .05439 31 30 .85408 1.17085 .88473 1.13029 .91633 1.09131 .94896 .05378 30 81 .85458 1.17016 .88524 1.12963 .91687 1.09067 .94952 .05317 29 32 .85509 1.16947 .88576 1.12897 .91740 1.09003 .95007 .05255 28 83 .85559 1.16878 .88628 1.12831 .91794 1.08940 .95062 .05194 27 34 .85609 1.16809 .88680 1.12765 .91847 1.. 08876 .95118 .05133 26 85 .85660 1.16741 .88732 1.12699 .91901 1.08813 .95173 .05072 25 36 .85710 1.16672 .88784 1.12633 .91955 1.08749 .95229 .05010 21 37 .85761 1.16603 .88836 1.12567 .92008 1.08686 .95284 .04949 23 38 .85811 1.16535 .88888 1.12501 .92062 1 1.08622 .95340 1.04888 22 39 .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 41 .85963 1.16329 .89045 1.12303 .92224 1.08432 .95506 1.04705 19 42 .86014 1.16261 .89097 1.12238 .92277 1.08369 .95562 1.04644 18 43 .86064 1.16192 .89149 1.12172 .92331 1.08306 .95618 1.04583 17 44 .86115 1.16124 .89201 1.12106 .92385 1.08243 .95673 1.04522 16 45 .86166 1.16056 .89253 1.12041 .92439 i 1.08179 .95729 1.04461 15 46 .86216 1.15987 .89306 1.11975 .92493 1.08116 .95785 1.04401 14 47 .86267 1.15919 .89358 1.11909 .92547 1.08053 .95841 1.04340 13 48 .86318 1.15851 .89410 1.11844 .92601 1-07990 .95897 1.04279 12 491 .86368 1 . 15783 .89463 1 11778 .92655 1.07927 .95952 1.04218 11 50 .86419 1.15715 .89515 1.11713 .92709 ,1.07864 .96008 1.04158 10 51 .86470 1.15647 .89567 1.11648 .92763 1.07801 .96064 1.04097 9 52! .86521 1 . 15579 .89620 1.11582 .92817 1.07738 .96120 1.04036 8 53 .86572 1.15511 .8967'2 1.11517 .92872 1.07676 .96176 1.03976 7 54 .86623 1.15443 .89725 1.11452 .92926 1.07613 .96232 1.03915 6 55 .86674 1.15375 .89777 1.11387 .92980 1.07550 .96288 1.03855 5 56 .86725 1.15308 ..89&30 1.11321 .93034 1.07487 .96344 1.03794 4 57 .86776 1.15240 .89883 1.11256 .93088 1.07425 .96400 1.03734 3 58 .86827 1.15172 .89935 1.11191 .93143 1.07362 .96457 1.03674 2 59 .86878 1.15104 .89988 1.11126 i .93197 1.07299 .96513 1.03613 1 60 .86929 1.15037 .90040 1.11061 J| .93252 1.07237 .96569 1.03553 f Cotang Tang Cotang Tang Cotang Tang jCotang Tang 49 !l 48 i 47 li 46 319 TABLE XII.-TANGENTS AND COTANGENTS. f 4 t4 4 t4 4 Tang Cotang Tang Cotang Tang Cotang .96569 1.08553 60 20 .97700 1.02355 41) 40 .98843 1.01170 20 1 .96625 1.03493 59 21 .97756 1.02295 39 141 .98901 1.01112 19 2 .96681 1.03433 58 22 .97813 1.02236 as I42 .98958 1.01053 18 3 .96738 1.03372 57 23 .97870 1.02176 37 43 .99016 1.00994 17 4 .96794 1.03312 56 24 .97927 1.02117 36 44 .99073 1.00935 16 5 .96850 1.03252 55 25 .97984 1.02057 35 45 .99131 1.00876 15 6 .96907 1.03192 54 26 .98041 1.01998 34 4(5 .99189 1. d0818 14 7 .96963 1.03132 53 27 .98098 1.01939 33 !47 .99247 1.00759 13 8 .97020 1.03072 53 28 .98155 1.01879 32 48 .99304 1.00701 12 9 .97076 1.03012 51 29 .98213 1.01820 31 49 .99362 1.00642 11 10 .971&3 1.02952 50 1 30 .98270 1.01761 30 J50 .99420 1.00583 10 11 .97189 1.02892 49 81 .98327 .01702 29 51 .99478 1.00525 9 12 .97246 1.02832 48 32 .98384 .01642 28 52 .99536 1.00467 8 13 .97302 1.02772 47 38 .98441 .01583 27 53 .99594 1.00408 7 14 .97359 1.02713 46 34 .98499 .01524 26 54 .99652 1.00350 6 15 .97416 1.02653 45 35 .98556 .01465 25 55 .99710 1.00291 5 10 .97472 1.02593 44 36 .98613 .01406 24 56 .99768 1.00233 4 1? .97529 1.02533 43 37 .98671 .01347 23 57 .99826 1.00175 3 18 .97586 1.02474 42 38 .98728 .01288 22 58 .99884 1.00116 a 19 .97643 1.02414 41 39 .98786 .01229 21 59 .99942 1.00058 i 20 .97700 1.02355 40 40 .98843 1.01170 20 60 1.00000 1.00000 Cotang Tang i Cotang Tang f j / Cotang Tang / 4 5 4 5 4 5 320 TABLE XIII.-VERSINES AND EXSECANTS. 0\ ; 1 2 3 / Vers. Exsec. Vers. Exsec. Vers. Exsec. Vers. Exsec. .00000 .00000 .00015 .00015 .00061 .00061 .00137 .00137 1 .00000 .00000 .00016 .00016 .00062 .00062 .00139 00139 1 2 .00000 .00000 .00016 .00016 .00063 .00063 .00140 .00140 2 3 .00000 ! .00000 .00017 .00017 .00064 .00064 i .00142 .00142 3 4 .00000 .00000 .00017 .00017 .00065 .00065 .00143 .00143 .4 5 .00000 .00000 [ .00018 .00018 .00066 .00066 .00145 .00145 5 6 .00000 .00000 .00018 .00018 .00067 .00067 I .00146 .00147 6 y .00000 .00000 .00019 .00019 .00068 .00068 i .00148 .00148 7 8 .00000 .00000 ! .00020 .00020 .00069 .00069 i .00150 .00150 8 9 .00000 .00000 i .00020 .00020 .00070 .00070 .00151 .00151 9 10 .00000 .00000 j .00021 .00021 .00071 .00072 .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 .00159 .00159 14 15 .00001 .00001 .00024 .00024 .00077 .00077 .00161 .00161 15 16 .00001 .00001 : .00024 .00024 .00078 .00078 .00162 .00163 16 17 .00001 .00001 .00025 .00025 .00079 .00079 .00164 .00164 17 18 .00001 .00001 .00026 .00026 .00081 .00081 .00166 .00166 18 19 .00002 .00002 i .00026 .00026 .00082 .00082 .00168 .00168 19 20 .00002 .00002 .00027 .00027 .00083 .00083 .00169 .00169 20 21 .00002 .00002 .00028 .00028 .00084 .09084 .00171 .00171 21 22 .00002 .00002 .00028 .00028 .00085 .00085 .00173 .00173 22 23 .00002 .00002 .00029 .00029 .00087 .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 .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 .000:38 .00038 .00102 .00102 .00196 .00196 35 30 : .00005 .00005 .00039 .00039 .00103 .00103 .00197 .00198 36 37 .00006 .00006 1 .00040 .00040 .00104 .00104 .00199 .00200 37 38 .00006 .00006 i .00041 .00041 .00106 .00106 .00201 .00201 38 39 .00006 .00006 .00041 .00041 .00107 .00107 .00203 .00203 39 40 .00007 .00007 i .00042 .00042 .00108 .00108 .00205 .00205 40 41 .00007 .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 .00216 .00216 46 47 .00009 .00009 .00048 .00048 .00118 r00118 .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 -.00054 .00054 .00127 .00127 .00230 .00230 53 54 .00012 .00012 1 .00055 .00055 .00128 .00128 .00232 .00232 54 55 .00013 .00013 .00056 .00056 .00130 .00130 .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 j .00059 .00059 .00134 .00134 ! .00240 .00240 58 59 .00015 .00015 .00060 .00060 i .00136 .00136 .00242 .00242 59 60 .00015 .00015 .00061 .00061 i .00137 .00137 .00244 .00244 60 321 TABLE XIII. VERSINES AND EXSECANTS. 4 5 C 6 7 ' Vers. Exsec. Vers.- Exsec. Vers. Exsec. Vers. Exsec. .00244 .00244 .00:381 .00382 .00548 .00551 .00745 .00751 1 .00246 .00246 .00383 .00385 .00551 .00554 .00749 .00755 ! 1 2 .00248 .00248 .00386 .00387 .00554 .00557 .00752 .(.0758 2 3 .00250 .00250 .00388 .00390 .00557 .00560 .00756 .00762 3 4 .00252 .00252 .00391 .00392 .00560 .00563 .00760 .00765 4 5 .00254 .00254 .00393 .00395 .00563 .00566 i .00763 .00769 5 6 .00256 .00257 .00396 .00397 .00566 .00569 .00767 .00773 ! 6 7 .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 .00791 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 .00594 ..00598 ! .00800 .00806 15 16 .00277 .00278 .00422 ,00424 .00598 .00601 .00803 .00810 16 17 .00279 .00280 .00425 .00427 .00601 .00604 .00807 .00813 17 18 .00281 .00282 .00428 .00429 .00604 .00608 j .00811 .00817 18 19 .00284 .00284 .00430 .00432 .00607 .00611 .00814 .00821 19 20 .00286 .00287 .00433 .00435 .00610 .00614 .00818 .00825 20 21 .00288 .00289 .00436 .00438 .00614 .00617 .00822 .00828 21 22 .00290 .00291 .00438 .00440 .00617 .00621 .00825 .00832 22 23 .00293 .00293 .00441 .00443 .00620 .00624 .00829 .00836 23 24 .00295 .00296 .00444 .00446 .00623 .00627 i .00833 .00840 24 25 .00297 .00298 .00447 .00449 .00626 .00630 .00837 .00844 25 26 .00299 .00300 .00449 .00451 .00630 .00634 .00840 .00848 26 27 .00301 .00302 .00452 .00454 ! 00633 .00637 .00844 ; .00851 27 28 .00304 .00305 .00455 .00457 .00636 .00640 .00848 .00855 | 28 29 .00306 .00307 .00458 .00460 .00640 .00644 .00852 .00859 29 30 .00308 .00309 .00460 .00463 .00643 .00647 .00856 .00863 30 31 .00311 .00312 .00463 .00465 .00646 .00650 .00859 .00867 31 32 .00353 .00314 .00466 .00468 .00649 .00654 i .00863 .00871 32 33 .00315 .00316 .00469 .00471 .00653 .00657 ..00867 ! .00875 33 34 .00317 .00318 .00472 .00474 .00656 .00660 .00871 ! .00878 34 35 .00320 .00321 .00474 .00477 .00659 .00664 .00875 i .00882 35 36 .00322 .00323 .00477 .00480 .00663 .00667 0087'8 ; .00886 36 37 .00324 .00326 .00480 .00482 .00666 .00671 j .00882 : .00890 37 38 .00327 .00328 .00483 .00485 .00669 .00674 .00886 .00894 38 39 .00329 .00330 .00486 .00488 .00673 .00677 00890 .00898 39 40 .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 .00010 42 43 .00339 .00340 .00497 .00500 .00686 .00691 .00906 ! .00914 43 44 .00341 .00342 .00500 .00503 .00690 .00695 .00901) .00918 44 45 .00:343 .00345 .00503 .00506 .00693 .00698 .00913 .00922 45 46 .00346 .00347 .00506 .00509 .00697 .00701 .00917 ' .00926 46 47 .00348 .00350 .00509 .00512 .00700 .00705 .00921 : .00930 47 48 .00351 .00352 .00512 .00515 .00703 .00708 .000% .001134 48 49 .00353 .00354 .00515 .00518 .00707 .00712 .00929 .00938 49 50 .00356 .00357 .00518 . 00081 .00710 .00715 .00933 .00942 50 51 00358 .00359 .00521 .008*4 .00714 .00719 .00937 00946 51 52 ! 00361 .00362 .00524 .00527 .00717 .00722 .00941 .00960 52 53 .00363 .00364 .00527 .00530 .00721 .00726 .00945 .00954 53 54 .00365 .00367 .00530 .00533 .00724 .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 .00542 .00735 .00740 .009(11 .00970 57 58 .00375 .00377 .00542 .00545 .00738 .00744 .00965 .00975 58 59 .00378 .00379 .00545 .00548 .00742 .00747 .00969 .00979 59 60 .00381 .00382 .00548 .00551 .00745 .00751 .00973 .00983 60 823 TABLE XIII.-VERSINES AND EXSECANTS. / 8 9 10 11 / Vers. Exsec. Vers. Exsec. Vers. Exsec. Vers. Exsec. .00973 .00983 .01231 .01247 .01519 .01543 .01837 .01872 1 .00977 .00987 .01236 .01251 I .01524 .01548 I .01843 1 .01877 1 2 .00981 .00991 .01240 .01256 .01529 .01553 .01848 .01883 2 3 .00985 .00995 .01245 .01261 .01534 .01558 .01854 .01889 3 4 .00989 .00999 .01249 .01265 .01540 .01564 .01860 .01895 4 5 .00994 .01004 .01254 .01270 .01545 .01569 .01865 .01901 5 6 .00998 .01008 ! .01259 .01275 .01550 .01574 .01871 .01906 6 7 .01002 .01012 .01263 .01279 .01555 .01579 .01876 .01912 7 8 .01006 .01016 ! .01268 .01284 .01560 .01585 .01882 .01918 8 9 .01010 .01020 .01272 .01289 .01565 .01590 .01888 .01924 9 10 .01014 .01024 .01277 .01294 .01570 .01595 .01893 .01930 10 11 .01018 .01029 .01282 .01298 .01575 .01601 1 .01899 .01936 11 13 .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 .01322 .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 i .01319 .01.337 .01617 .01643 .01944 .01983 19 20 .01056 .01067 .01324 .01342 .01622 .01649 .01950 .01989 20 21 .01060 .01071 .01329 .01346 .01627 .01654 .01956 .01995 21 22 .01064 .01076 .01333 .01351 .01032 .01659 .01961 .02001 22 28 .01069 .01080 .01338 .01356 .01638 .01665 .01967 .02007 23 24 .01073 .01084 .01343 .01361 .01643 .01670 .01973 .02013 24 23 .01077 .01089 .01348 .01366 .01648 .01676 .01979 .02019 25 23 .01081 .01093 .01352 .01371 .01653 .01681 .01984 .02025 26 27 .01086 .01097 : .01357 .01376 .01659 .01687 .01990 .02031 27 28 .01090 .01102 i .01362 .01381 .01664 .01692 .01996 .02037 28 29 .01094 .01106 ! .01367 .01386 .01669 .01698 .02002 .02043 29 30 .01098 .01111 .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 32 33 .01111 .01124 : .01386 .01405 .01690 .01720 .02025 .02067 33 34 .01116 .01128 j .01891 .01410 .01696 .01725 .02031 .02073 34 35 .01120 .01133 .01396 .01415 .01701 .01731 .02037 .02079 35 36 .01124 .01137 i .01400 .01420 .01706 .0-1736 .02042 .02085 36 37 .01129 .01142 ! .01405 .01425 .01712 .01742 .02048 .02091 37 38 .01133 .01146 j .01410 .01-130 .01717 .01747 .02054 .0.2097 38 33 .01137 .01151 ! .01415 .01435 .01723 .01753 .02000 .02103 39 40 .01142 .01155 .01420 .01440 .01728 .01758 .02066 .02110 40 41 .01146 .01160 .01425 .01445 .01733 .01764 .02072 .02116 41 42 .01151 .01164 : .01430 .01450 .01739 .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 01476 .01766 .01798 .02107 .02153 47 48 .01177 .01191 .01459 .01481 .01771 .01803 .02113 .02159 '48 49 .01182 .01196 .01464 .01486 .01777 .01809 .02119 .02165 49 50 .01186 .01200 .01469 .01491 .01782 .01815 .02125 .02171 50 51 .01191 .01205 .01474 .01496 .01788 .01820 .02131 .02178 51 52 01195 .01209 .01479 .01501 .01793 .01826 .02137 .02184 52 53 .01200 .01214 .01484 .01506 .01799 .01832 .02143 .02190 53 54 .01204 .01219 .01489 .01512 .01804 .01837 , .02149 .02196 54 55 .01209 .01223 .01494 .01517 .01810 .01843 | .02155 .02203 55 56 .01213 .01228 .01499 .01522 .01815 .01849 : .02161 .02209 56 57 .01218 .01233 .01504 .01527 .01821 .01854 .02167 .02215 ' 57 58 .01222 .01237 .01509 .01532 .01826 .01860 .02173 .02221 58 59 .01227 .01242 ! .01514 .01537 .01832 .01866 .02179 .02228 59 60 .01231 .01247 .01519 .01543 .01837 .01872 1 .02ia5 .02234 60 323 TABLE XIII.-VERSINES AND EXSECANTS. 1 12 13 14 15 1 I Vers. Exsec. Vers. Exsec. Vers. Exsec. Vers. Exsec. .02185 .02234 .02563 .02630 .02970 .03061 .03407 .03528 1 .02191 .02240 .02570 .02637 .02977 .03069 .03-115 .03536 1 2 .02197 .02247 .02576 .02644 .02985 .03076 .03422 .03544 2 3 .02203 .02253 .02583 .02651 .02992 .03084 .03430 .03552 3 4 .02210 .02259 .02589 .02658 .02999 .03091 .03438 .03560 4 5 .02216 .02206 .02596 .02665 .03006 j .03099 .03445 .03568 5 6 .02222 .02272 .02602 .02672 .03013 .03106 .03453 .03576 6 7 .02228 .02279 .02609 .0267'9 .03020 .03114 .03460 .03584 7* 8 .02234 .02285 .02616 .C2686 .03027 .03121 .03468 .0:3592 8 9 .02240 .02291 .02622 .02093 .03034 .03129 .03176 .03601 9 10 .02246 .02298 .02629 .02700 .03041 .03137 .03483 .03609 10 11 .02252 .02304 .02635 .02707 .08048 .03144 .03491 .03617 11 12 .02258 .02311 .02642 .02714 .03055 .03152 .03498 .03625 12 13 .02265 .02317 .02649 .02721 .03063 .03159 .03506 .03633 13 14 .02271 .02323 .02655 .02728 .03070 .03167 .03514 .03642 14 15 .02277 .02330 .02602 .02735 .03077 .03175 .03521 .03650 15 16 .02283 .02336 .02669 .02742 .03084 .03182 .03529 .03658 16 17 .02289 .02343 .02675 .02749 .03091 .03190 .03537 .03666 17 18 .02295 .02349 .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 .037'08 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 26 .02345 .02402 .02736 .02813 .03156 .03259 .03606 .03741 26 27 .02352 .02408 .02743 .02820 .03163 .03267 .03614 .03719 27 28 .02358 .02415 .02749 .02827 .03171 .03275 .03621 .03758 23 29 .02364 .02421 .02756 .02834 .03178 .03282 .03629 .03766 29 30 .02370 .02428 .02763 .02842 .03185 .03290 .03637 .03774 30 31 .02377 .02435 .02770 .028-19 .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 .03816 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 .02535 .02873 .02958 .03303 .03416 .03762 .03909 46 47 .02479 .02542 .02880 .02965 .03310 .03421 .03770 .03918 47 48 .02485 .02548 .02887 .02972 .03318 .03432 .0377'8 .03927 48 49 .02492 .02555 .02894 .02980 .03325 .03439 .03786 .08835 49 50 .02498 .02562 .02900 .02987 .03333 .03447 .03794 .03944 50 51 .02504 .02569 .02907 .02994 .03340 .03455 .03802 .03952 51 52 .02511 .02576 .02914 .03002 .03347 .03463 .03810 .03961 52 53 .02517 .02582 .02921 .03009 .03355 .03471 .03818 .03969 53 54 .02524 .02589 .02928 .03017 .03362 .03479 .03826 .03978 54 55 .02530 .02596 .02935 .03024 .03370 .03487 ; .03834 .o:',nsr ; 55 56 .02537 .02603 .02942 .03032 .03377 .0341)5 .03842 .03995 56 57 .02543 .02C.10 .02949 .03039 .03385 .03503 .03850 .04004 57 ' 58 .02550 .02617 i .02456 .03046 .03392 .03512 .03858 .04013 58 59 .02556 .02624 I .02JM53 .03054 .03400 .03520 .038(515 .04021 59 60 .02563 .02630 II .02970 .03061 .03407 .03528 .03874 .04030 60 324 TABLE XIII.-VERSINES AND EXSECANTS. t 16 17 18 19 / Vers. Exsec. Vers. Exsec. Vers. Exsec. Vers. Exsec. .03874 .04030 .04370 .04569 i .04894 .05146 .05448 i .05762 1 .03882 .04039 .04378 .04578 .04903 .05156 .05458 ; .05773 1 2 .03890 .04047 .04387 .04588 '! .04912 .05166 .05467 i .05783 2 3 .03898 .04056 .04395 .04597 i .04921 .05176 .05477 .05794 3 4 .03906 .04065 i .04404 .04606 \ .04930 .05186 .05486 i .05805 4 5 .03914 .04073 .04412 .04616 .04939 .05196 .05496 : .05815 5 6 .03922 .04082 .04421 .04625 ' .04948 .05206 .05505 : .05826 6 7 .03930 .04091 .04429 .04635 .04957 .05216 .05515 i .05836 7 8 .03938 .04100 ! .04438 .04644 .04967 .05226 .05524 i .05847 8 9 .03946 .04108 I .04446 .04653 | .04976 .05236 .05534 ! .05858 9 10 .03954 .04117 .04455 .04663 ij .04985 .05246 .05543 .05869 10 11 .03963 .04120 .04464 .04672 .04994 .05256 .05553 .05879 11 12 .03971 .04135 .04472 .04682 .05003 .05266 .05562 .05890 12 13 .03979 .04144 .04481 .04691 .05012 .05276 .05572 .05901 13 14 .03987 .04152 .04489 .04700 .05021 .05286 .05582 .05911 14 15 .03995 .04161 .04493 .04710 .05030 .05297 .05591 .05922 15 10 .04003 .04170 .04507 .04719 .05039 .05307 .05601 .05933 16 17 .04011 .04179 : .04515 .04729 .05048 .05317 .05610 ! .05944 17 18 .04019 .04188 ! .04524 .04738 , .05057 .05327 .05620 .05955 18 19 .04028 .04197 .04533 .04748 .050G7 .05a37 , .05630 ( .05965 19 20 .04036 .04206 .04541 .04757 .05076 .05347 .05639 .05976 20 21 .04044 .04214 .04550 .04767 ; .05085 .05357 .05649 .05987 21 22 .04052 .04223 ! .04559 .04776 : .05094 .05367 .05658 .05998 22 23 .04060 .04232 .04507 .04786 i .05103 .05378 .05668 .06009 23 24 .04069 .04241 .04576 .04795 .05112 .05388 .05678 .06020 24 25 .04077 .04250 .04585 .04805 : .05122 .05398 .05687 .060:30 25 26 .04085 .04259 i .04593 .04815 i .05131 .05408 .05697 ! .06041 26 27 .04093 .04268 .04602 .04824 1 .05140 .05418 .05707 .06052 27 28 .04102 .04277 .04611 .04834 i .05149 i .05429 .05716 .06063 28 29 .04110 .04286 .04620 .04843 1 .05158 .05439 .05726 .06074 29 30 .04118 .04295 .04628 .04853 ; .05168 .05449 .05736 .06085 30 31 .04126 .04304 .04637 .04863 ' .05177 .05460 .05746 .06096 31 32 .04135 .04313 .04646 .04872 .05186 .05470 .05755 .06107 32 33 .04143 .04322 .04655 .04882 .05195 .05480 .05765 .06118 33 34 .04151 .04:331 .04663 .04891 .05205 .05490 .05775 .06129 34 35 .04159 .04340 .04672 .04901 .05214 .05501 .05785 .06140 35 36 .04168 .04349 .04681 .04911 .05223 .05511 .05794 .06151 36 37 .04176 .04358 .04690 .04920 .05232 .05521 .05804 .06162 37 38 .04184 .04367 .04699 .04930 .05242 .05532 .05814 .06173 38 30 .04193 .04376 .04707 .04940 i .05251 .C5542 .05824 .06184 39 40 .04201 .04385 ; .04716 .04950 .05260 .05552 .05833 .06195 40 41 .04209 .04394 .04725 .04959 .05270 .05563 .05843 .06206 41 42 .04218 .04403 .04734 .04969 .05279 .05573 .05853 .06217 42 43 .04226 .04413 .04743 .04979 ! .05288 .05584 .05863 .06228 43 44 .04234 .04422 i .04752 .04989 i .05298 .05594 .05873 .06239 44 45 .04243 .04431 .04760 .04998 .05307 .05604 .05882 .06250 45 46 .04251 .04440 ; .04769 .05008 .05316 ,.05615 .05892 .06261 46 47 .04260 .04449 .04778 .05018 | .05326 .05625 .05902 .06272 47 48 .04268 .04458 .04787 .05028 ! .05335 .05636 .05912 .06283 48 49 .04276 .04468 .04796 .05088 .05344 .05646 .05922 .00295 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 .0.5067 .05373 .0567S .05951 .06328 52 53 .04310 .04504 .04832 .06077 .03383 .05688 .05961 .06339 53 54 .04319 .04514 .04841 .05087 ll .05391 .05699 1 .05971 .06350 54 55 .04327 .04523 .04850 .05097 .05401 .05709 1 .05981 .06362 55 56 .04,336 .04532 .04858 .05107 .05410 .05720 1 .05991 .06373 56 57 .04344 .04541 .04867 .05116 .05420 .05730 .06001 .06384 57 58 .04353 .04551 i .0-1876 .05126 .05429 .05741 .06011 .06895 58 59 . 04361 .045(50 .04885 .05136 .05439 .05751 .06021 .0640? 59 60 .04370 .04569 .04894 .05146 || .05448 .05762 .06031 .06418 i 60 325 TABLE XIII. VERSINES AND EXSECANTS. 2( ) 2] P * 5 2( J / Vers. Exsec. Vers. Exsec. Vers. Exsec. Vers. Exsec. 1 .0(5031 .06418 .06642 .07115 .07282 .07853 .07950 .06686 1 .06011 .06429 I .06652 .07126 .07293 .07866 .07961 .OS649 1 2 .06051 .06440 .06663 .07138 .07303 .07879 .07972 .08603 2 3 .06061 .06452 .06673 .07150 .07314 .07892 .07984 .08676 3 4 .06071 .06463 .06684 .07102 .07325 .07904 .07995 .08690 4 5 .06081 .06474 .06694 .07174 .07336 .07917 .08006 .08703 5 6 .06091 .06486 .06705 .07186 .07347 .07930 .08018 .08717 6 7 .06101 .06497 .06715 .07199 .07358 .07943 .08029 .08730 7 8 .06111 .06508 .06726 .07211 I .07369 .07955 .08041 .08744 8 9 .06121 .06520 .06736 .07*23 .07380 .07968 .08052 .08757 9 10 .06131 .06531 .06747 .07235 i .07391 .07981 .08004 .08771 10 11 .06141 .06.542 .06757 .07247 .07402 .07994 .08075 .08784 11 12 .06151 .06554 .06768 .07259 .07413 .08006 .08086 .08798 12 13 .06161 .06565 .06778 .07271 .07424 .08019 .08098 .08811 13 14 .06171 .06577 .06789 .07283 .07435 .08032 .08109 .08825 14 15 .06181 .06588 .06799 .07295 .07446 .08045 .08121 .08839 15 16 .06191 .06600 .06810 .07307 ! .07457 .08058 .08132 .08852 1(5 IT .06201 .06611 .06820 .07320 .07468 .08071 .08144 .08806 17 18 .06211 .06622 .06831 .07332 .0747'9 .08084 .08155 .08880 18 19 .06221 .06634 .06841 .07344 .07490 .08097 .08167 .08893 19 20 .06231 .06645 .06852 .07356 .07501 .08109 : .08178 .08907 20 21 .06241 .06657 .06863 .07368 .07512 .08122 .08190 .08921 21 22 .06252 .06668 .06873 .07380 .07523 .08135 .08201 .08934 22 23 .06262 .06680 .06884 .07393 .07534 .08148 .08213 .08948 > 23 24 .06272 .06691 .06894 .07405 .07545 .08161 .08225 .08962 24 25 .06282 .06703 .06905 .07417 .07556 .08174 .08236 .08975 25 26 .06292 .06715 .06916 .07429 .07568 .08187 .08248 .08989 26 27 .06302 .06726 .06926 .07442 .07579 .08200 .08259 .09003 27 28 .06312 .06738 .06937 .07454 .07590 .08213 .08271 .09017 23 29 .06323 .06749 ! .06948 .07466 .07601 .08226 .0828-2 .09030 29 30 .06333 .06761 .06958 .07479 .07612 .08239 .08294 .09044 30 31 .06343 .06773 .06969 .07491 .07623 .08252 .08306 .09058 31 32 .06353 .06784 .06980 .07503 .07634 .08265 .08317 .01)072 32 33 .06363 .06796 .06990 .07516 .07645 .08278 .08329 .09080 33 34 .06374 .06807 .07001 .07528 .07657 .08291 .08340 .09099 34 35 .06384 .06819 i .07012 .07540 .07668 .08305 .08352 .09113 35 36 .06394 .06831 i .07022 .07553 .07'079 .08318 .08364 .09127 36 37 .06404 .06843 , ! .07033 .07565 ; .07090 .08331 .08375 .09141 37 38 .06415 .06854 .07044 .07578 .07701 .08344 .08387 .09155 38 39 .06425 .06866 .07055 .07590 i .07713 .08357 .08399 .09169 39 40 .06435 .06878 .07065 .07602 .07724 .08370 .08410 .09183 40 41 .06445 .06889 .07076 .07615 .07735 .08383 .08422 .09197 41 42 .06456 .06901 .07087 .07627 .07746 .08397 .08434 .09211 42 43 .06466 .06913 .07098 .07640 .07757 .08410 .08445 .09224 43 44 .06476 .06925 .07108 .07652 .07769 .08423 .08457 .09238 44 45 .06486 .06936 I .07119 .07665 1 .07780 .08436 .08469 .09252 45 46 .06497 .06948 .07130 .07677 .07791 .08449 ! .08481 .09266 4(5 47 .06507 .06960 .07141 .07690 .07802 .08463 .08492 .09280 47 48 .06517 .06972 .07151 .07702 .07814 .08476 .08.504 .09294 4S 49 .06528 .06984 .07162 .07715 .07835 .08489 .08516 .09308 49 50 .06538 .06995 .07173 .07727 .07836 .08503 .08528 .09323 60 51 .06548 .07007 .07184 .07740 .07848 .08516 .08539 .09&37 51 52 .06559 .07019 .07195 .07752 .07859 .08529 .08551 .09351 52 53 .06569 .07031 1 .07206 .07765 . .07870 .08548 .08563 .09305 53 54 .06580 .07043 ; .07216 .07778 .07881 .08556 .08575 .09379 54 55 .06590 .07055 : .07227 .07790 . 07893 .0856!) : .08586 .09393 55 56 .06600 .07067 .07238 .07803 .07904 .08582 1 .08598 .09407 50 57 .06611 .07079 .07249 .07811) .07915 .08596 .08610 .09421 57 58 .06621 .0701)1 .07260 .07828 ! .07927 .08(509 .08622 .09435 58 59 .06632 .07103 .07271 .07841 .079:38 .0862;? .08634 .09449 59 60 .06642 .07115 .0722 .07853 .07950 .08036 ! .06645 .09404 60 326 TABLE XIII.-VERSINES AND EXSECANTS. / 24 25 26 27 / Vers. Exsec. Vers. Exsec. Yers. Exsec. Vers. Exsec. .08645 .094(54 .09369 .10338 | .10121 .11260 ! .10899 .12233 1 .08057 .09478 .09382 .10353 1 .10133 .11276 .10913 .1.2249 1 2 .08069 .09492 .09394 .10368 .10146 .11292 .10926 .12266 2 .08681 .09506 .09106 .10383 i .10159 .11308 .10939 .12283 3 4 .08693 .09520 .09418 .10398 .10172 .11323 ! .10952 .12299 4 5 .08705 .09535 .09431 .10413 .10184 .11389 .10965 .12316 5 .08717 .09549 .09443 .10428 i .10197 .11355 .10979 .12333 G .08728 .09563 .09455 .10443 j .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 .087G4 .09606 .09493 .10488 .10248 .11419 i .11032 .12400 10 11 .08776 .09620 .09505 .10503 .10261 .11435 .11045 .12-116 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 14 15 .08824 .09678 .09554 .10564 .10313 .11499 .11098 .12484 15 1(5 .08836 .09692 .09567 .10579 .10326 .11515 .11112 .12501 16 17 .08848 ! .09707 .09579 .10594 .10338 .11531 .11125 .12518 17 18 .08860 i .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 .11105 .12568 20 21 .08896 .09764 .09629 .10655 .10390 .11595 .11178 .12585 21 .089G3 .0977!) .09042 .10670 .10403 .11611 .11198 .12602 22 23 .08920 .097'93 .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 26 27 .08968 .09851 .097'04 .10747 .10468 .11691 .11259 .12687 27 2S .08980 .09866 .09716 .10762 .10481 .11708 .11273 .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 .11772 .11326 .12772 32 33 .09040 .09939 .09779 .10839 .10546 .11789 .11339 .12789 33 34 .09052 .09953 .09792 .10854 .10559 .11805 .11353 .12807 34 35 .09064 .09968 .09804 .10870 .1057-2 .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 39 40 .09125 .10041 .09867 .10947 .10(537 .11903 .11434 .12910 40 41 .09137 .10055 .09880 .10963 .10650 .11919 .11447 .12927 41 42 .09149 .10071 .09892 .10978 .10663 .11936 .11461 .12944 42 4:} .09161 .10085 .09905 .10994 .10676 .11952 .11474 .12961 43 44 .09174 .10100 .09918 .11009 .10689 .11968 .11488 .12979 44 45 .09186 .10115 .09930 .11025 .10702 .11985 .11501 .12996 4") 46 .09198 .10130 .09943 .11041 .10715 -12001 .11515 .13013 46 47 .09210 .10144 .09955 .11056 .10728 .12018 .11523 .13031 47 48 .09222 .10159 .09968 .11072 .10741 .12034 .11542 .13048 48 49 .09234 .10174 .09981 .11087 .10755 .12051 .11555 .13065 49 50 .09247 .10189 .09993 .11103 .10768 .12067 .11569 .13083 50 51 .09259 .10204 .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 .13222 58 59 .09357 .10323 .10108 .11244 .10886 .12216 ! .11692 .13240 59 60 .09369 .10338 .10121 .11260 .10899 .12233 i .11705 .13257 60 TABLE XIII.-VERSINES AND EXSECANTS. ' 28 29 30 31 / Vers. Exsec. Vers. Exsec. Vers. | Exsec. Vers. Exsec. .11705 .13257 .12538 .14335 .13397 .15470 .14283 .16603 ~T 1 .11719 .13275 .12552 .14354 .13412 .15489 .14298 .10084 i 2 .11733 .13292 .12566 .14372 .13427 .15509 .14313 .16704 2 ft .11746 .13310 .12580 .14391 .13441 .15528 .14328 .16725 3 4 ,11760 .13327 .12595 .14409 .13456 .15548 .14343 .16745 4 5 .11774 .13345 .12609 .14428 .13470 .15567 .14358 .16766 5 6 .11787 .13362 .12623 .14446 .13485 .15587 .14373 .16786 6 7 .11801 .13380 .12637 .14465 .13499 .15606 .14388 .16806 7 8 .11815 .13398 .12651 .14483 .13514 .15026 .14)03 .16827 8 9 .11828 .13415 .12665 .14502 .13529 .15645 .14418 .10848 9 10 .11842 .13433 .12679 .14521 .13543 .15665 .14488 .16868 10 11 .11856 .13451 .12694 .14539 .13558 .15684 .14119 .16889 11 12 .1187'0 .13468 .12708 .14558 .13573 .15704 .14464 .10909 12 13 .11883 .13486 .12722 .14570 .13587 .15724 .14479 .16930 13 14 .11897 .1&504 1 .12736 .14595 .13602 .15743 .14194 .10950 14 15 .11911 .13521 1 .12750 .14614 .13016 .15763 .14509 .16971 15 16 .11925 .13539 .12705 .14632 .13631 .15782 .115:24 .10092 16 17 .11938 .13557 .12779 .14651 .13646 .15802 .14539 .17012 17 18 .11952 .13575 .12793 .14070 .13660 .15823 .14554 .17033 18 19 .11966 .13593 .12807 .14GS9 .13675 .15841 .11509 .17 .11 19 20 .11980 .13610 .12822 .14707 .13690 .15861 .14584 .17075 20 21 .11994 .13628 .12836 .14726 .13705 .15881 .14599 .17095 21 22 .12007 .13646 .12850 .14745 .13719 .15901 .14615 .17116 22 23 .12021 .13664 .12864 .14764 .13734 .15920 .14630 .17137 23 24 .12035 .13682 .12879 .14782 .13749 .15940 .14645 .17158 24 25 .12049 .13700 .12893 .14801 .13763 .15960 .14600 .17178 25 26 .12063 .13718 .12907 .14820 .13778 .15980 .14675 .17199 26 27 .12077 .13735 .12921 .14839 .13793 .16000 .14690 .17320 27 28 .12091 .13753 .12936 .14858 .136C8 .10019 .14706 .17.041 28 29 .12104 .13771 .12950 .14877 .13822 .16039 .14721 .17262 29 30 .12118 .13789 .12964 .14896 .13837 .16059 .14736 .17283 30 31 .12132 .13807 .12979 .14914 .13852 .16079 .14751 .17304 31 32 .12146 .13825 .12993 .14933 .13867 .16099 .14766 .17325 32 33 .12160 .13843 .13007 .1495? .13881 .16119 .14782 .17346 33 34 .12174 .13861 .13022 .14971 .13896 .16139 .14797 .17307 34 35 .12188 .13879 .13036 .14990 .13911 .16159 .1481:3 .17388 35 36 .12202 .13897 .13051 .15009 .13926 .16179 .14827 .17409 36 37 .12216 .13916 .13065 .15028 .13941 .16199 .14843 .17-430 37 38 .12230 .13934 .13079 .15047 .13955 .16219 .14858 .17451 38 39 .12244 .13952 .13094 .15066 .13970 .16239 .14873 .17472 39 40 .12257 .13970 .13108 .15085 .13985 .16259 .14888 .17493 40 41 .12271 .13988 .13122 .15105 .14000 .16279 .14004 .17514 41 42 .12285 .14006 .13137 .15124 .14015 .1 0:21)9 .14919 .17535 42 43 .12299 .14024 .13151 .15143 .14030 .16319 .14934 .17556 43 44 .12313 .14042 .13166 .15102 .14044 .16339 .14949 .17577 44 45 .12327 .14061 .13180 .15181 .14059 .16359 .14905 .17598 45 46 .12341 .14079 .13195 .15200 .14074 .16380 .14980 .17620 46 47 .12355 .14097 .13209 .15219 .11089 .16400 .14995 .17641 47 48 .12369 .14115 .13228 .15239 .14104 .16420 .15011 .17662 48 49 .12383 .14134 .13238 .15258 .14119 .16440 .150-26 .17688 49 50 .12397 .14152 .13252 .15277 .141:54 .10460 .15041 .17704 50 51 .12411 .14170 .13267 .15296 .14149 .16481 .15057 .17726 51 52 .12425 .14188 .18281 .15315 .14164 .16501 .15072 .17747 52 53 .12439 .14207 .13296 .15335 .14179 .16521 .15087 .17768 53 54 .12454 .14225 .13310 .15354 .14194 .16541 .15103 .17790 54 55 .12468 .14243 .18326 .15373 .14208 .16562 .15118 .17811 55 56 .12482 .14262 .13339 .15393 .14223 .16582 .15184 .17833 56 - 57 .12496 .14280 .13864 .15412 .14888 .16602 .15149 .17854 57 58 .12510 .14299 .18868 .16431 .14253 .100:23 .15164 .17875 58 59 .12524 .14317 .13383 .15451 .14268 .16643 .15180 .17896 59 60 .12538 .14335 .13397 .15470 .14283 .16663 .15195 .17918 60 TABLE XIII. VERSINES AND EXSECANTS. 32 a 33 34 35 / Vers. Exsec. Vers. Exsec. Vers. Exsec. Vers. Exsec. .15195 .17918 .16133 .19236 .17096 .20622 .18085 .22077 1 .15211 .17939 .16149 .192.59 .17113 .20645 .18101 .22102 1 2 .15226 .17961 .16165 .19281 .17129 .20669 .18118 .22127 2 3 .15241 .17982 .16181 .19304 .17145 .20693 .18135 .22152 3 4 .15257 .18004 .16196 .19327 .17161 .20717 .18152 .22177 4 5 .15272 .18025 .16212 .19349 .17178 .20740 .18168 .22202 5 c . 15288 .18047 .16228 .19372 .17194 .20764 .18185 .22227 6 7 .15303 .18068 .16244 .19394 .17210 .20788 .18202 .22252 7 8 .15319 .18090 .16260 .19417 .17227 .20812 .18218 .22277 8 9 ! 15334 .18111 .16276 .19440 .17243 .20836 .18235 .22302 9 10 .15350 .18133 .16292 .19463 .17259 .20859 .18252 .22327 10 11 .15885 .18155 .16308 .19485 .17276 .20883 .18269 .22352 11 12 .15381 .18176 .16324 .19508 .17292 .20907 .18286 .22377 12 13 .15396 .18198 .16340 .19531 .17308 .20931 .18302 .22402 13 14 .15412 ! .18220 .16355 .19554 .17325 .20955 .183-9 .22428 14 15 .15427 .18241 .16371 .1957'6 .17341 .20979 .18336 .22453 15 16 .15443 i .18263 .16387 .19599 .17357 .21003 .18353 .22478 16 17 .15458 .18285 .16403 .19622 .17374 .21027 .18369 .22503 17 18 .15474 .18307 .16419 .19645 .17390 .21051 .18386 .22528 18 19 .15489 .18328 .16435 .19668 .17407 .21075 .18403 .22554 19 20 .15505 .18350 .16451 .19691 .17423 .21099 .18420 .22579 20 21 .15520 .18372 .16467 .19713 .17439 .21123 .18437 .22604 21 22 .15536 .18394 .16483 .19736 .17456 .21147 .18454 .22629 22 23 .15552 .18416 .16499 .19759 i .17472 .21171 .18470 .22655 23 24 .15567 .18437 .16515 .19782 .17489 .21195 .18487 .22680 24 25 .15583 .18459 .16531 .19805 .17505 .21220 .18504 .227'06 25 26 .15598 .18481 .16547 .19828 .17522 .21244 .18521 .22731 26 27 .15614 ; .18503 .16563 .19851 .17538 .21268 .18538 .227'56 27 28 .15630 .18525 .16579 .19874 .17554 .21292 ! .18555 : .22782 28 29 .15645 .18547 .16595 .19897 .17571 .21316 I .18572 .22807 29 30 .15661 .18569 .16611 .19920 .17587 ..21341 .18588 .22833 30 31 .15676 .18591 .16627 .19944 ! .17604 .21365 .18605 .22858 31 32 .15692 .18613 .16644 .19967 i .17620 .21389 .18622 .22884 32 33 .15708 .18635 .16660 .19990 .17037 .21414 .18639 .22909 33 34 .15723 i .18657 .16676 .20013 .17653 .21438 .18656 .22935 34 35 .15739 ! .18679 .16692 .20036 .17670 .21462 .18673 .22960 35 36 .15755 .18701 .16708 .20059 .17686 .21487 .18690 .22986 36 37 .15770 .18723 .16724 .20083 .17703 .21511 .18707 .23012 37 38 .15786 .18745 .16740 .20106 .17719 .21535 .18724 .23037 38 39 .15802 I .18767 .16756 .20129 .17738 .21560 .18741 .23063 39 40 .15818 .18790 .16773 .20152 .17752 .21584 .18758 .23089 40 41 .15833 .18812 .16788 .20176 .17769 .21609 .18775 .23114 41 42 .15849 i .18834 .16805 .20199 .17786 .21633 .18792 .23140 42 43 .15865 .18856 .16821 .2< )-_> .17802 .21058 .18809 .23166 43 44 .15880 .18878 .16837 .2021(3 .17819 .21682 .18826 .23192 44 45 .15896 .18901 .16853 .20269 .17835 .21707 .18843 .23217 45 46 .1.7.) 12 .18923 .16869 .20292 .17852- .21731 .18860 .23243 46 47 .159-3-} .18945 .16885 .2C316 i .17868 .21756 .18877 .23269 47 48 .15943 ! .18967 .16902 .20:339 .17885 .21781 .18894 .23295 48 49 .15950 .18990 .16918 .20363 .17902 .21805 .18911 .233.21 49 50 .15975 .19012 .16934 -.20386 .17918 .21830 48928 .23347 50 51 .15991 .19034 .16950 .20410 .17935 .21855 .18945 .23373 51 52 .16006 .19057 .16966 .20433 .17952 .21879 .18962 .23399 52 53 .16022 .19079 .16983 .20157 .17968 .21904 .18979 .23424 53 54 .160:38 .19102 i .16999 .20480 j .17985 .21929 .18996 .23450 54 55 .16054 .19124 .17015 .20501 : .18001 .21953 .19013 .23476 55 56 .16070 .19146 .17031 .20527 1 .18018 .21978 .19030 .23502 56 57 .16085 .19169 .17047 .20551 .18035 .22003 .19047 .23529 57 58 .16101 .19191 .17064 .30575 .18051 .22028 .19064 .23555 58 59 .16117 .19214 . 17080 .20598 il .18068 .22053 .19081 .23581 59 60 .16133 1 .19236 .17096 .20622 1 .18085 1 .22077 .19098 .23607 60 TABLE XIII. VEKSINES AND EXSECANTS. / 36 37 38 39 Vers. Exsec. Vers. Exsec. Vers. Exsec. Vers. Exsec. ~ .19098 .23007 .201&6 .25214 .21199 .26902 .22285 .28676 1 .19115 .23633 .20154 .25241 .21217 .26931 .32304 .287'06 1 2 .19133 .23659 .20171 .25269 .21235 .26960 .22322 .28737 2 3 .19150 .23685 .20189 .25296 .21253 .26988 .22340 .28767 3 4 .19167 .23711 .20207 .25324 .21271 .27017 .22359 .28797 4 5 .1918! .23738 .20224 .25351 .21289 .27046 .22377 .28828 5 6 .19201 .23764 .20242 .25379 .21307 .27075 .22395 .28853 6 7 .19218 .23790 .20259 .25406 .21324 .27104 .22414 .28889 7 8 .19235 .23816 .20277 .25434 .21342 .27133 .22432 .28919 8 9 .19252 .23843 .20294 .25462 .213(50 .27162 .22450 .28950 9 10 .19270 .23869 .20312 .25489 .21378 .27191 .22409 .28980 10 n .19287 .23895 .20329 .25517 .21396 .27221 .22487 .29011 11 13 .19304 .23922 .20347 .25545 .21414 .27250 .22506 .29042 12 13 .19321 .23948 .20365 .25572 .21432 .27'27'D .22521 .29072 13 14 .19338 .23975 .20382 .25600 .21450 .27308 .22543 .29103 14 15 .19356 .24001 .20400 .25628 .21468 .27'337 .225(51 .29133 15 16 .19373 .24028 ! .20417 .25656 .21486 .373(56 .2257'!) .29164 16 17 .19390 .24054 .20135 .25683 .21504 .37396 .32598 .29195 17 18 .19407 .24081 .20453 .25711 .21522 .27425 .22616 .29226 18 19 .19424 .24107 .20470 .25739 .21540 .27454 .22634 .29256 19 20 .19442 .24134 II .20488 .25767 .21558 .27-483 .22653 .29287 20 21 .19459 .24160 .20506 .25705 .21576 .27513 .22671 .29318 21 22 .19476 .24187 ! .20523 .35823 .21595 .37543 .22690 .29349 22 23 .19493 .24213 .20541 .25851 .21613 .2;5;2 .22708 .29380 23 24 .19511 .24240 .20559 .25879 ; .21631 .27(501 , .227'27 .29411 24 25 .19528 .24267 .20576 .25907 , .21649 .27630 1 .22745 .29442 25 26 .19545 .24293 .20594 .25935 t .21667 .27660 .22764 .29473 i 26 27 .195(52 .24320 .20612 .25963 .21685 .27689 .22782 .29504 27 28 .19580 .24347 . 20629 , .25991 .21703 .27719 .22801 .29535 38 29 .19597 .24373 .20647 .26019 .21721 .27748 .22819 .29568 2:) 30 .19614. .24400 .20665 .26047 .21739 .27778 i .22838 .28597 SO 81 .19632 .24427 .20682 .26075 .21757 .27807 .22856 .29628 31 33 .19649 .24454 .20700 .26104 : .217:5 .27837 .228:5 .29659 33 33 .19666 .24481 .20718 .26132 | .21794 .27867 .22893 .29(590 83 34 .1J684 .24508 .20736 .26160 .21812 .27896 j .22912 .297-21 31 35 .19701 .24534 .20753 .26188 i .21830 .27926 I .22030 .39752 35 36 .19718 .24561 .20771 .26216 .21848 .27956 .22949 .29781 36 37 .19736 .24588 .20789 .26245 .21866 .37985 ! .22967 .29815 7 38 .19753 .24615 .20807 .26273 I .21884 .28015 .22986 .29846 33 39 .19770 .21642 .20824 .26301 , .21902 .28045 .23004 .39877 39 40 .19788 .24669 .20842 .26330 .21921 .28075 .23023 .29909 40 41 .19805 .24696 .20860 .26358 .21939 .28105 .23041 .29910 41 .19822 .34733 .2087'8 .26387 .211)57 .28134 .23060 .39971 42 43 .19840 .24750 .20895 .26415 .21975 .28164 .2307'9 .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 .231:34 .30097 46 47 .19909 .34859 .20967 .26529 .22048 .28284 ] .23153 .30129 47 48 .19927 .24886 .20985 .26557 .22066 .28314 1 .23172 .301(50 43 49 .19944 .21913 .21002 .26586 .22084 .28344 .23190 .30192 49 50 .19962 [34940 .21020 .26615 i .22103 .28374 .23209 .30223 50 51 .1997-9 .24967 .21038 .26643 .22121 .28404 i .23228 .30255 51 52 .19997 .24995 .21056 .26G7'2 .22139 .28434 .23246 .30287 52 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 .23302 .30382 55 56 .20066 .25104 ; .21127 .2(5787 .33313 .28555 ! .23321 .30413 56 57 .20084 .25131 .21145 .26815 .22231 .28585 .23339 .30445 57 58 .20101 .25159 .21163 .26844 22219 .38(51.-) 133358 .30477 58 59 .20119 .25186 | .21181 .26873 ! 33267 .28646 .23377 .30509 59 60 .20136 .25214 .21199 .26902 i .22285 .28676 .23396 .30541 60 TABLE XIII.-VERSINES AND EXSECANTS. ' 40 41 42 43 / Vers. Exsec. Vers. Exsec. Vers. Exsec. Vers. Exsec. .23396 .30541 .24529 .32501 .25086 .3-1563 .26865 .36733 c 1 .23414 .30573 .24548 .32535 .25705 .34599 .26884 .36770 1 2 .2:3433 .30605 .24567 .32508 .25724 .34634 .26904 .36807 2 3 .23452 .30036 .24586 .32002 .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 .25802 .34775 .26984 .36956 6 7 .23527 .30764 .24663 .32737 .25822 .34811 .27004 .36993 7 8 .23545 .30796 .24682 .32770 .25841 .34846 .27024 .37030 8 9 .23564 .30829 .24701 .32804 .25801 .34882 .27043 .37'068 9 10 .23583 .30861 .24720 .32838 .25880 .34917 .27063 .371.05 10 11 .23603 .30893 .24739 .32872 .25900 .34953 .27083 .37143 11 12 .28620 .30925 .24759 .32905 .25920 .341)88 .27103 .37180 12 13 .23G39 .30957 .24778 .32939 .259S9 .35024 .27123 .87218 13 14 .23658 .30989 .24797 .32973 .25959 .35C60 .27143 .37255 14 15 .23677 .31022 .24816 .33007 .2597'8 .35095 .27163 .37293 15 16 .23696 .31054 .24835 .33041 .25998 .35131 .27183 .37330 16 17 .23714 .31086 .24854 .33075 .26017 .35167 .27203 .37368 17 18 .23733 .31119 .24874 .33109 .26037 .35203 .27223 .37406 18 19 .28753 .31151 .24893 .33143 .26056 .35238 .27243 .37443 19 20 .23771 .31183 .24912 .33177 .26076 .35274 .27263 .37481 20 21 .23790 .31216 .24931 .33211 .26096 .35310 .27'2S3 .37519 21 22 .23808 .31248 .24950 .33245 .20115 .35346 .27303 .37556 22 23 .23827 .31281 .24970 .33279 .26135 .35382 .27323 .37594 23 24 .23846 .31313 .24989 .33314 .26154 .35418 .27343 .37632 24 25 .23865 .31346 .25008 .33348 .26174 .35454 .27363 ,3767'0 25 26 .23884 .31378 .25027 .33382 .26194 .35490 .27383 .37708 20 27 .23903 .31411 .25047 .33416 .26213 .35526 .27403 .37746 27 28 .23922 .81443 .25066 .33451 .26233 .35562 .27423 .37784 28 29 .23941 .31476 .25065 .33485 .26253 .85598 .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 .31007 .25162 .33622 .26331 .35743 .27523 .37974 33 34 .24035 .31610 .25182 .33657 .26351 .35779 .27543 .38012 34 35 .24054 .31672 .25201 .33691 .26371 .35815 .27563 .38051 85 36 .24073 .31705 .25220 .33726 .26390 .35852 .27583 .38089 36 37 .24092 .31738 .25240 .33760 .26-110 .358B8 .27603 .38127 37' 38 .24111 .31771 .25259 .33795 .26430 .35924 .27623 .38165 86 39 .24130 .31804 .25278 .33830 .20449 .35961 .27643 .38204 39 40 .24149 .31837 .25297 .33864 .26469 .35997 .27663 .38243 40 41 .24168 .31870 .25317 .33899 .26489 .36034 .27683 .38280 41 42 .24187 .31903 .25336 .33934 .26509 .36070 .277'03 .38319 42 43 .24206 .31936 .25356 .33908 .26528 .36107 .27723 .38357 43 44 .24:225 .31909 .25375 .34003 .26548 .36143 .27743 .38396 44 45 .24244 .32002 .25394 .34038 .26568 .36180 .27764 .38434 45 46 .24262 .32035 .25414 .34073 .26588 .30217 .27784 .38473 46 47 .24281 .32068 .25438 .34108 .26607 .36253 .27'804 .38512 47 48 .24300 .32101 .25452 .34142 .26627 .36290 .27824 .38550 48 49 .2-1320 .32134 .25472 .31177 .26647 .36327 .27844 .38589 49 50 .24339 .32168 .25491 .34212 .26667 .36363 .27864 .38628 50 51 .24358 .32201 .25511 .34247 .26686 .36400 .27884 .38666 51 52 .24377 .32234 ,25530 .34282 .26706 .36437 .27905 .38705 52 53 .24396 .32207 .25549 .34317 .26726 .36474 .27925 .38744 53 54 .24415 .32301 .25569 .34352 .26746 .36511 .27945 .38783 54 55 .24434 .323:34 .25588 .34387 .26766 .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 .38938 58 59 .24:310 .32468 .25066 .34528 .26845 .36696 .28046 .38977 59 60 .24529 .32501 .25686 .34563 .26865 .36733 .28066 .39016 60 331 TABLE XIII. VERSINES AND EXSECANTS. 4 4 4 5 4 6 4 7 Vers. Exsec. Vers. Exsec. , Vers. Exsec. Vers. Exsec. .28066 .39016 .29289 .41421 .30534 .43956 .31800 .46628 1 .28086 .39055 .29310 .41463 .30)55 .43999 .31821 .46674 1 2 .28106 .39095 .29330 .41504 .30576 .44042 .31843 .46719 2 3 .28127 .39134 .29351 .41545 .30597 .44086 .31864 .40765 3 4 .28147 .39173 .29372 .41586 .30618 .44129 .31885 .46811 4 5 .28167 .39212 .29392 .41627 .30639 .44173 .31907 .46857 5 6 .28187 .39251 .29413 .41669 .30660 .44217 .31928 .46903 6 7 .28208 .39291 .29433 .41710 .30081 .44260 .31949 .46949 7 8 .28228 .39330 .29454 .41752 .30702 .44304 .31971 .46995 8 9 .28218 .39369 .29475 .41793 .30723 .41347 .31992 .47041 9 10 .28263 .39409 .29495 .41835 .30744 .44391 .32013 .47087 10 11 .28289 .39448 .29516 .41876 .30765 .44435 .32035 .47134 11 1:2 .28309 .39487 .29537 .41918 .30700 .44479 .32056 .47180 12 13 .28329 .39527 .29557 .419.59 .30807 .44523 .32077 .47220 13 14 .28350 .39566 .29578 .42001 .30828 .44567 .32090 .47'272 14 15 .28370 .39606 .29599 .42042 .30849 .44610 .32120 .47310 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 19 .28451 .39764 .29681 .42210 .30933 .44787 .32205 .47501 19 23 .28471 .39804 .29702 .42251 .30954 .44831 .32227 .47551 20 21 .28492 .39844 .29723 .42293 .30975 .44875 .32248 .47598 21 22 .28512 .39884 .29743 .42335 .30906 .44919 .32270 .47644 22 23 .28533 .39924 .29764 .42377 .31017 .449(53 .32291 .47691 23 24 .28553 .39963 .29785 .42419 .31038 .45007 .32312 .47738 24 25 .28573 .40003 .29805 .42461 .31059 .45052 .32334 .47784 25 26 .28503 .40043 .298J6 .42503 .31080 .45096 .32355 .47831 26 27 .28614 .40083 .298-17 .42545 .31101 .45141 .32377 .47878 27 28 .28634 .40123 .29868 .42587 .31122 .45185 .32398 .47925 28 29 .28655 .40163 .29888 .42630 .31113 .45229 .32420 .47972 29 30 .28675 .40203 .29909 .42672 .31165 .45274 .32441 .48019 30 31 .28605 .40243 .29930 .42714 .31186 .45319 .32462 .48066 31 32 .28716 .40283 .29951 .42756 .31207 .45363 .32484 .48113 32 33 .28736 .40324 .29971 .42799 .31228 .45408 .82505 .48160 33 34 .28757 .40364 .29992 .42841 .31249 .45452 .32527 .48207 34 35 .28777 .40404 .80013 .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 .313r,5 .45678 .32634 .48443 39 40 .28879 .40606 .30117 .43096 .31376 .45721 .32656 .48491 40 41 .28900 .40646 .30138 .43139 .31397 .45766 ! .32677 .48538 41 42 .28920 .40087 .30158 .43181 .31418 .45811 .32(5:19 148586 42 43 .28941 .40727 .30179 .43224 .31439 .45856 .82720 .48633 43 44 .28961 .40768 .30200 .432(57 .31461 .45901 .30712 .48(581 44 45 .28981 .40808 .30221 .43310 .31482 .45046 .32763 .48728 45 46 .29002 .40849 .30242 .43352 .31503 .45992 .32785 .48776 46 47 .29022 .40890 .30263 .48896 .31524 .46037 .32806 .48824 47 48 .29043 .40930 .30283 .43438 .31515 .46082 .32828 .48871 48 49 .29063 .40971 .30304 .43481 .315117 .46127 .32849 .480 10 49 50 .29084 .41012 .30325 .43524 .31588 .46173 .32871 .48967 50 51 .29104 .41053 .30346 .43507 .31609 .46218 .32893 .49015 51 52 .29125 .41093 .30367 .43610 .31630 .46263 .32914 .490(53 52 53 .29145 .41134 .30388 .43653 .31651 .46309 .32936 .49111 53 54 .29166 .41175 .30409 .43696 .31673 .40354 .32957 .49159 54 55 .29187 .41216 .30430 .437':!9 .31694 .46-400 .32979 .40207 55 56 .29207 .41257 .30451 .43783 .31715 .46445 .83001 .49255 56 57 .29228 .41298 .30471 .43826 .31736 .46491 .33022 .49308 57 '-58 .2:);.' 18 .41888 .30102 .43869 .31758 .46537 .33044 .49351 58 59 .29209 .41380 .30513 .43012 .31779 .46582 .33065 .49399 59 60 .29289 ,41421 .30534 .43956 .31800 .46628 .33087 .49448 60 333 TABLE XIII.-VERSINES AND EXSECANTS. t 48 49 50 51 f Vers. Exsec. Vers. Exsec. Vers. Exsec. Vers. Exsec. .33087 .49448 ! .34394 .52425 .35721 .55572 .37068 .58902 1 .33109 .49496 ,34416 .52476 .35744 .55626 .37091 .58959 1 2 .33130 .49544 .34438 .52527 .35766 .55680 .37113 .59016 2 .33152 .49593 .34460 .52579 .35788 .55734 .37136 .59073 3 4 .&3173 .49641 .34482 .52630 .35810 .55789 .37158 .59130 4 5 .33195 .49690 .34504 .52681 .35833 .55843 .37181 .59188 5 6 .33217 .49738 .34526 .52732 .35855 .55897 .37204 .59245 6 7 .33238 .49787 .34548 .52784 .35877 .55951 .37226 .59302 7 8 .33260 .49835 .34570 .52835 .35900 .56005 .37249 .59360 8 9 .33282 .49884 .34592 .52886 .35922 .56060 .37272 .59418 9 10 .33303 .49933 .34614 .52938 .35944 .56114 .37294 .59475 10 11 .33325 .49981 .34636 .52989 .35967 .56169 .37317 .59533 11 12 .33347 .50030 .34658 .53041 .35989 .56223 .37340 .59590 12 13 .33368 .5007'9 .34680 .53092 .36011 .56278 .37362 .5964H 13 14 .33390 .50128 .34702 .53144 .36031 -56332 .37385 .59700 14 15 .33412 .50177 .34724 .53196 .36056 .56387 .37408 .59764 15 16 .33434 .50226 .34746 .53247 .36078 .56442 .37430 .59822 16 17 .a3455 .50275 .34768 .53299 .36101 .56497 .37453 .59880 17 18 .33477 .50324 .34790 .53351 .36123 .56551 .37476 .59938 18 19 .33499 .50373 .34812 .53403 .36146 .56606 .37498 .59996 19 20 .33520 .50422 .34834 .53455 .36168 .56661 .37521 .60054 20 21 .33542 .50471 .34856 .53507 .36190 .56716 .37544 .60112 21 22 .33564 .50521 .34878 .53559 .36213 .56771 .87567 .60171 22 23 .33586 .50570 .34900 .53611 .36235 .56826 .37589 .60229 23 24 .33607 .50619 .34923 .53663 .36258 .56881 .37612 .60287 24 25 .33829 .50669 .34945 .53715 .36280 .56937 .37635 .60346 25 26 .33651 .50718 .34967 .53768 .36302 .56992 .37658 .60404 26 27 .33673 .50767 .34989 .53820 .36325 .57047 .37680 .60463 27 28 .33694 .50817 .35011 .53872 .36347 .57103 .37703 .60521 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 .60756 32 33 .33803 .51065 .35122 .54134 .36460 .57380 .37817 .60815 33 34 .33825 .51115 .35144 .54187 .36482 .57436 .37840 .60874 34 35 .33847 .51165 .35166 .54240 .36504 .57491 .37862 .60933 35 36 .33869 .51215 .35188 .54292 .36527 .57547 .37885 .60992 36 37 .33891 .51265 .35210 .54345 .36549 .57603 .37908 .61051 37 38 .33912 .51314 .35232 .54398 .36572 .57659 .37931 .61111 38 39 .33934 .51364 .35254 .54451 .36594 .57715 .37954 .61170 39 40 .33956 .51415 .35277 .54504 .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 .84022 .51565 .35343 .54663 .36684 .57939 .38045 .61407 43 44 i .340-14 .51615 .35365 .54716 .36707 .57995 .38068 .61467 44 45 i .34005 .51665 .35388 .54769 .36729 .58051 .38091 .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 .34153 .51867 .35476 .54982 .36820 .58277 .38182 .61765 49 50 .34175 .51918 .35499 .55036 .36842 .58333 .38205 .61825 50 51 .34197 .51968 .35521 .55089 .36865 .58390 .38228 .61885 51 52 .34219 .52019 .35543 .55143 .36887 .58447 .38251 .61945 52 53 .34241 .52069 ' .35565 .55196 .36910 .58503 .38274 .62005 53 54 .34262 .52120 .35588 .55250 .36932 .58560 I .38296 .62065 54 55 .34284 .52171 .35610 .55303 .36955 .58617 i .38319 .62125 55 56 .34306 .52222 .35632 .55357 .3697'8 .58674 ! .38342 .62185 56 57 .34328 .52273 .35654 .55411 .S7000 .58731 i .38365 .62246 57 58 .34.350 .52323 .35677 .55465 i .&r003 .58788 .38388 .62306 58 59 .34372 .52374 .35699 .55518 .37015 .58845 | .38411 .62366 59 60 .34394 .52425 .35721 .55572 1 .37068 .58902 1 .38434 .62427 60 TABLE XIIL VERSINES AND EXSECANTS. 5 2 5 3 5 i 5 5 Vers. Exsec. Vers. Exsec. Vers. Exsec. Vers. Exsec. .38434 .62427 .39819 .66164 .41221 .70130 42042 .74345 1 .38457 .62487 .39842 .66228 .41245 .70198 | .42600 .74417 1 2 .38480 .62548 .89665 .66292 .41209 .70207 I .42090 .74490 2 3 .38503 .62609 .39888 .66357 .41292 .70,335 .42714 .74562 3 4 .38526 .62669 .39911 .66421 .41316 .70403 ; .42738 .74035 4 5 .38549 .62730 .39935 .60486 .41339 .70472 .42762 .74708 5 6 .38571 .62791 .39958 .66550 .41303 .70540 .42785 .74781 6 7 .38594 .62852 .39981 .66615 .41386 .70609 .42809 .74a54 7 8 .38617 .62913 .40005 .66679 .41410 .70677 .42833 .74927 8 9 .38640 .62974 .40028 .66744 .41433 .70746 .42857 .75000 9 10 .38663 .63035 .40051 .60809 .41457 .70815 .42881 .75073 10 11 .38686 .63096 .40074 .66873 .41481 .70884 .42905 .75146 11 12 .38709 .63157 .40098 .66938 .41504 .70953 .42929 .75219 12 13 .38732 .63218 .40121 .67003 .41528 .71022 .42953 .75293 13 14 .38755 .63279 .40144 .67'068 .41551 ,71091 .42970 .75366 14 15 .38778 . 63341 .40168 .67133 .41575 .71100 .43000 .75440 15 16 .38801 .63402 .40191 .67199 .41599 .71229 .43024 .75513 16 17 .38824 .63464 .40214 .67264 .41622 .71298 .43048 .75587 17 18 .38847 .63525 .40237 .67329 .41646 .71368 i .43072 .75661 18 19 .38870 .63587 .40261 .07394 .41670 .71437 i .43096 .75734 19 20 .38893 .63648 .40284 .67460 .41693 .71506 ; .43120 .75808 20 21 .38916 .63710 .40307 .67525 .41717 .71576 ] .43144 .75882 21 22 .38939 .63772 .40331 .67591 .41740 .71040 .43168 .75956 22 23 .38962 .63834 i .40354 .67656 .41764 .71715 .43192 .76031 23 24 .38985 .63895 I .40378 .67722 .41788 .71785 : .43216 .76105 24 25 .39009 .63957 ! .40401 .67788 .41811 .71855 i .43240 .76179 25 26 .39032 .64019 ; .40424 .67853 .41835 .71925 I .43204 .76253 26 27 .39055 .64081 .40448 .67919 .41859 .71995 .43287 .76328 27 28 .39078 .64144 .40471 .67985 .41882 .72065 .43311 .76402 28 29 .39101 .64206 .40494 .68051 .41906 .72135 .43335 .76477 29 30 .39124 .64268 .40518 .68117 .41930 .72205 .43359 .76552 30 31 .39147 .64330 .40541 .68183 .41953 .72275 .43383 .76626 31 32 .39170 .64393 .40565 .68250 .41977 .72346 i .43407 .76701 32 33 .39193 .64455 .40588 .68316 .42001 .72416 .43431 .76776 33 34 .39216 .64518 .40611 .68382 .42024 .72487 .43455 .76851 34 a5 .39239 .64580 .40635 .68449 .42048 .72557 .43479 .76926 35 36 .39262 .64643 .40658 .68515 .42072 .72628 .43503 .77001 36 37 .39286 .64705 .40682 .68582 .42096 .72698 .43527 .77077 37 38 .39309 .64768 .40705 .68648 .42119 .72769 .43551 .77152 38 89 .39332 .64831 .40728 .68715 .42143 .72840 .43575 .77227 39 40 .39355 .64894 .40752 .687'82 .42107 .72911 .43599 .77303 40 41 .39378 .64957 .40775 .68848 .42191 .72982 .43623 .77378 41 42 .39401 .65020 .40799 .68915 .42214 .7-3053 .43047 .77454 42 43 .39424 .65083 .40822 .68982 .42238 .73124 .43671 .77530 43 44 .39447 .65146 .40846 .69049 .42262 .73195 .43695 .77606 44 45 .39471 .65209 .40809 .09116 .42285 .73267 .43720 .77'681 45 46 .39494 .65272 .40893 .09183 .42309 .73338 .43744 .77757 46 47 .39517 .G5336 .40916 .09250 .42333 .73409 .43708 .77833 47 48 .39540 .65399 i .40939 .09318 .42357 .73481 , .43792 .77910 48 49 .39563 .65462 .40963 .69385 .42381 .78552 .43816 .77986 49 50 .39586 .65526 .40986 .69452 .42404 .73634 .43840 .78062 50 51 .39610 .65589 .41010 .69520 .42428 .73696 .43864 .78138 51 52 .39633 .65653 .41033 .69587 .42452 .78768 .43888 .78215 52 53 .39056 .65717 .41057 .69655 .424:0 .7'3840 , .43912 .78291 53 54 .39679 .65780 .4108Q .69723 .42199 .73911 .43936 .78308 54 55 .39702 .65844 .41104 .69790 .42523 178988 .43900 .78445 55 56 .39726 .65908 .41127 .69858 .42547 74056 .43984 .78521 56 57 .39749 .65972 .41151 .69926 .42571 .74128 .nons .78f)!)8 57 '-58 .39772 .66036 .41174 .69994 .42505 .74200 .44032 .78075 58 59 .39795 .66100 .41198 .70063 .42019 .74272 .44057 .78752 59 60 .39819 .66164 .41221 .70130 .42642 .74345 .44081 .78829 60 334 TABLE XIII. VERSINES AND EXSECANTS. 5 6 5 7 5 > 5 |? / Vers. Exsec. Vers. Exsec. i Vers. Exsec. Vers. Exsec. .44081 .78829 .45536 .83608 .47008 .88708 .48496 .94160 1 .44105 .78906 .45560 .83690 .47033 .88796 .48521 .94254 1 2 .44129 .78984 .45585 .83773 .47057 .88884 .48546 .94:349 2 3 .44153 .79061 .45609 .83855 .47082 .88972 i .48571 .94443 3 4 .44177 .79138 .45634 .83938 .47107 .89060 .48596 .94537 4 5 .44201 .79216 .45658 .84020 .47131 .89148 .48621 .94632 5 6 .44225 .79293 .45683 .84103 .47156 .89237 .48646 .94726 6 7 .44250 .79371 .45707 .84186 .47181 .89325 .48671 .94821 7 8 .44274 .79449 .45731 .84269 .47206 .89414 .48696 .94916 8 9 .44295 .79527 .45756 .84352 .472:30 .89503 .48721 .95011 9 10 .44322 .79604 .45780 .84435 .47255 .89591 .48746 .95106 10 11 .44346 .79682 .45805 .84518 .47280 .89680 .48771 .95201 11 12 .44370 .79761 .45829 .84601 .47304 .89769 .48796 .95296 12 13 .44395 .79839 .45854 .84685 .47329 .89858 .48821 .95392 13 14 .44419 , .79917 .45878 .84768 .47354 .89948 .48846 .95487 14 15 .44443 .79995 .45903 .84852 .47379 .90037 .48871 .95583 15 16 .44467 .80074 .45927 .84935 .47403 .90126 .48896 .95678 16 17 .44491 .80152 .45951 .85019 .47428 .90216 .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 .47502 .90485 .48996 .96062 20 21 .44588 .80467 .46049 .85355 .47527 .90575 .49021 .96158 21 oo .44612 .80546 .46074 .85439 .47552 .90665 .49046 .96255 22 23 .44637 .80625 .46098 .85523 .47577 .90755 .49071 .96351 23 24 .44661 .80704 .46123 .85608 .47601 .90845 .49096 .96448 24 25 .446S5 .80783 .46147 .85692 .47626 .90935 .49121 .96544 25 26 .44709 .80862 .46172 .85777 .47651 .91026 .49146 .96641 26 27 .44734 .80942 .46196 .85861 .47676 .91116 .49171 .967'38 27 28 .44758 .81021 .46221 .85946 .47701 .91207 .49196 .96835 28 29 .44782 .81101 .46246 .S6031 .47725 .91297 .49221 .96932 29 30 .44806 .81180 .46270 .86116 .47750 .91388 .49246 .97029 30 31 .44831 .81260 .46295 .86201 .47775 .91479 .49271 .97127 31 32 .44855 .81340 .46319 .86286 .47800 .91570 .49296 .97224 33 33 .44879 .81419 .46344 .86371 .47825 .91661 .49321 .97322 33 34 .44903 .81499 .46368 .86457 .47-849 .91733 .49346 .97420 34 85 .44928 .81579 .46393 .86542 .47874 .91844 .49372 .97517 35 36 .44952 .81659 .46417 .86627 .47899 .91935 .49397 .97-615 36 37 .44976 .81740 .46442 .86713 .47924 .92027 .49422 .97713 37 38 .45001 .81820 .46466 .86799 .47949 .92118 .49447 .97811 38 39 .45025 .81900 .46491 .86885 .47974 .92210 .49472 .97910 39 40 .45049 .81981 .46516 .86990 .47998 .92302 .49497 .98008 40 41 .45073 .82061 .46540 .87056 .48023 .92394 .49522 .98107 41 42 .45098 .82142 .46505 .87142 .48048 .92486 .49547 .98205 42 43 .45122 .82222 .46589 .87229 .48073 .92578 .49572 .98304 43 44 .45146 .82303 .46614 .87315 .48098 .92670 ! .49597 .98403 44 45 .45171 .82384 .46639 .87401 .48123 .92762 1 .49623 .98502 45 46 .45195 .82465 .46663 .87488 ! .48148 .92855 i .49648 .98601 46 47 .45219 .82546 .46688 .87574 .48173 - .92947 .49673 .98700 47 48 .45244 .82627 .46712 .87661 .48197 .93040 .49698 .98799 48 49 .45268 .82709 .46737 .87748 .48222 .93133 .49723 .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 .48207 .93412 .49799 .99198 52 53 .45365 .83034 .46836 .88095 .48322 . 93505 .49824 .99298 53 54 .45390 .83116 .46860 .88183 .48347 .93598 .49849 .99398 54 55 .45414 .83198 .46885 .88270 .48373 .93692 .49874 .99498 55 56 .45439 .83280 .46909 .88357 ! .48396 .93785 .49899 .99598 56 57 .45463 .83362 .46934 .88445 : .48421 .93879 .49924 .99698 57 58 .45487 .83444 .46959 .88532 .48446 .93973 .49950 .99799 58 59 .45512 .83526 .46983 .8K020 .48471 .94066 .49975 .99899 59 60 .45536 .83608 .47008 .88708 .48496 .94160 .50000 .00000 60 335 TABLE XIII.-VERSINES AND EXSECANTS. / 60 || 61 62 63 Vers. Exsec. Vers. Exsec. Vers. Exsec. Vers. Exsec. ( .50000 1.00000 1 .51519 1.06267 .53053 1.13005 .54601 1.20269 ; .50025 1.00101 .51544 1.06375 .53079 1.13122 .54627 1.20395 1 j .50050 1.00202 .51570 1.06483 .53104 1 . 13239 .54653 1.20521 2 j .50076 1.00303 .51595 1.0G5JW .53130 1.13356 .54679 1.20647 J t .50101 1.00404 .51621 1.06701 .53156 1.13473 .54705 1.20773 4 5 .50126 1.00505 .51646 1.06809 .53181 1.13590 .54731 1.20900 G .50151 1.00607 .51672 1.06918 .53207 1.13707 .54757 1.21026 t r .50176 1.00708 .51697 1.07027 .53233 1.13825 .547'82 1.21153 ij 8 .50202 1.00810 i .51723 1.07137 .53258 1.13942 ! .54808 1.21280 8 ( .50227 1.0091-2 .51748 1.07246 .53284 1.14060 [ .54834 1.21407 1C .50252 1.01014 i .51774 1.07356 .53310 1.14178 .54860 1.21535 1C 11 .50277 1.01116 .51799 1.07465 .53-336 1.14296 .54886 1.21662 11 12 .50303 1.01218 .51825 1.07575 .53361 1.14414 .54912 1.21790 12 13 .50328 1.01320 .51850 i 1.07685 .53387 1.14533 .54938 1.21918 14 .50353 1.01422 .51876 1.07795 .53413 1.14651 1 .54964 1.22045 14 15 .50378 1.01525 .51901 1.07905 .53489 1.14770 .54990 1.22174 15 16 .50404 1.01628 .51927 1.08015 .53464 1.14889 .550113 1.22302 16 17 .50429 1.01730 .51952 1.08126 .53490 1.15008 .55042 1.22430 17 18 .50454 1.01833 .51978 1.08236 .53516 1.15127 .55068 1.225:.!) 18 19 .50479 1.01936 .52003 1.08347 .53542 1.15246 .55094 1.22688 19 20 .50505 1.02039 .52029 1.08458 .53507 1.15306 .55120 1.2-2817 20 21 .50530 1.02143 .520.54 1.08569 .53503 1.15485 .55146 1.22946 21 22 .50555 1.02246 .52080 1.08680 .58619 1.15605 .55172 1.23075 22 23 .50581 1.02349 .52105 1.08791 .58645 1.15725 .55198 1.23205 O'.> 24 .50606 1.02453 .52131 1.08903 .53670 1.15845 .55224 1.23334 24 25 .50631 1.02557 .52156 1.09014 .53696 1.15965 .55250 L28464 25 26 .50656 1.02661 .52182 1.09126 .53722 1.16085 .55276 1.28594 2(5 27 .50682 1.02765 .52207 1.09238 .53748 1.16208 .55302 1.28724 27 28 .50707 1.02869 .52233 1.09350 .53774 1.16326 .55328 l.K"> !> 29 .50732 1.02973 .52259 1.09462 .53799 1.16447 .55354 1.23985 29 30 .50758 1.03077 .52284 1.09574 .53825 1.16568 .55380 1.24116 30 31 .50783 1.03182 .52310 1.09686 .53851 1.16689 .55406 1.24247 31 32 .50808 1.03286 .52335 1.09799 .53877 1.16810 .55432 1.24378 32 33 .50834 1.03391 .52361 1.09911 .53903 1.16932 .55458 1.24509 133 34 .50859 1.03496 .52386 1.10024 .53928 1.17053 .55484 1.2415-10 :-Jt 35 .50884 1.03601 .52412 1.10137 .53954 1.17175 .55510 1.2477'2 35 36 .50910 1.03706 .52438 1.10250 .53980 1.17297 .55536 1.2491 .3 36 37 .50935 1.03811 .52463 1.10363 .54006 1.17419 .55563 1.25035 37 38 .50960 1.03916 .52489 1.10477 .54032 1.17541 .55589 1 25167 !38 30 .50986 1.04022 .52514 1.10590 .54058 1.17663 .55615 l.Ji.VJOO !39 40 .51011 1.04128 .52540 1.10704 .54083 1.17786 .55641 1.S543S 40 41 .51036 1.04233 .52566 1.10817 .54109 1.17909 .55667 1.25505 41 42 .51062 1.04339 .52591 1.10931 .54135 1.18031 .55693 1.26697 42 43 .51087 i 1.04445 .52617 1.11045 .54161 1.18154 .55719 1.25830 43 44 .51113 1.04551 .52642 1.11159 .54187 1.18277 .55745 1.26968 44 45 .51138 1.04658 .52668 1.11274 .54213 1.18401 . 55771 1.26097 |45 46 .51163 1.04764 .52694 1.11388 ; .54238 1.18524 .55797 1.26230 1 46 47 .51189 1.0487'0 .52719 1.11503 .54264 1.18648 .55823 1.26364 47 48 .51214 1.04977 .52745 1.11617 i .54290 1.18772 .55849 1.26498 1 48 49 .51239 1.05084 .52771 1.11732 .54316 1.18895 .55876 1.2i(W2 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 .52&J8 1.12078 ! .54394 1.19268 .55954 1. 271)85 52 53 .51341 1.05512 .52878 1.12193 .54420 1.WJ93 .55980 1.27169 53 54 .51366 1.05619 .52899 1.12309 .54446 1.19517 .56006 1.27304 54 55 .51392 1.05727 .52924 1.12425 .54471 1.19648 .56032 i 1.274:1!) 55 56 .51417 1.05a35 .52950 1.12540 54497 1.19767 .56058 1.27574 r><; >7 .51443 1.05942 .5297'6 1.12657 .54523 1.19892 .56084 1.27710 57 58 .51468 1.06050 .58001 1.1 277-:! .54549 1.20018 .56111 1.27845 |58 59 .51494 1.06158 .53027 1.12889 .54575 1.20143 .56137 1.27981 59 60 .51519 1 1.06267 .53053 1.13005 .54001 1.20269 .56163 1.28117 60 336 TABLE XIII. VERSINES AND EXSECANTS. 6 4 6 5 6 6 6 7 Vers. Exsec. j Vers. Exsec. Vers. Exsec. Vers. Exsec. .56163 1.28117 .57738 1.36620 ' .59326 1.45859 .60927 1.55930 1 .56189 1.28253 .57765 1.36768 r .59353 1.46020 .60954 1.56106 1 '.) .56215 1.28390 .57791 1.36916 .59379 1.46181 .60980 1.56282 2 3 .56241 1.28526 .57817 1.37064 .59406 1.46342 .61007 1.56458 3 4 .56267 128663 .57844 1.37212 .59433 1.46504 .61034 1.56634 4 5 .56294 .28800 .57870 1.37361 .59459 1.46665 .61061 1.56811 5 6 .50*30 .28937 i .57896 1.37509 i .59486 1.46827 .61088 1.56988 6 ; .56346 .29074 i .57923 1.37658 ! .59512 1.46989 .61114 1.57165 7 8 .56372 1.29211 i .57949 1.37808 i .59539 1.47152 .61141 1.57342 8 9 .56398 .29349 i .57976 1.37957 ! .59566 1.47314 .61168 1.57520 9 10 .56425 1 .29487 i .58002 1.38107 i .59592 1.47477 .61195 1 .57698 10 11 .56451 1.29625 i .58028 1.38256 .59619 1.47640 .61222 1.57876 11 12 .56477 1-29763 1 .58055 1.38406 .59645 1.47804 .61248 1.58054 12 13 .56503 1.29901 i .58081 1.38556 .59672 1.47967 .61275 1.58233 13 14 .56529 1.30040 ! .58108 1.38707 .59699 1.48131 .61302 1.58412 14 15 .56555 1.30179 : .58134 1.38857 .59725 1.48295 .61329 1.58591 15 1C, .56582 1.30318 .58160 1.39008 .59752 1.48459 .61356 1.58771 16 17 .56608 1.30457 .58187 1.39159 .59779 1.48624 .61383 1.58950 17 18 .5(5(534 .30596 i .58213 1.39311 .59805 1.48789 .61409 1.59130 18 19 .566(50 .30735 ! .58240 1.39462 .59832 1.48954 .61436 1.59311 19 20 .56687 .30875 .58266 1.39614 ! .59859 1.49119 .61463 1.59491 20 21 .56713 .31015 .58293 1.39766 .59885 1.49284 .61490 1.59672 21 -.'2 .50739 .31155 .58319 1.39918 .59912 1.49450 .61517 1.59853 22 23 .56765 .31295 ! .583-15 1.40070 .59938 1.49616 .61544 1.60035 23 24 .56791 .31436 i .58372 1.40222 .59965 1.49782 .61570 1.60217 24 25 .5(5818 .31576 ! .58398 1.40375 .59992 1.49948 1 .61597 1.60399 25 -,><; 156844 .31717 .58425 1.40528 .60018 1.50115 .61624 1.60581 26 27 .515870 .31858 i .58451 1.40681 .60045 1.50282 .61651 1.60763 27 2S .56896 .31999 .58478 1.40835 .60072 1.50449 .61678 1.60946 28 29 .56923 .32140 .58504 1.40988 .60098 1.50617 .61705 1.61129 29 30 .56949 .32282 .58531 1.41142 .60125 1.50784 .61732 1.61313 30 31 .56975 .32424 .58557 1.41296 .60152 1.50952 .61759 1.61496 31 8-2 .57001 .32566 .58584 1.41450 .60178 1.51120 .61785 1.61680 32 33 .57028 .32708 .58610 1.41605 .60205 1.51289 .61812 1.61864 33 34 .57054 .32850 1 .58637 1.41760 .60232 1.51457 .61839 1.62049 34 35 .57080 .32993 ! .58663 1.41914 .60259 1.51626 .61866 1.62234 35 30 .57106 .33135 .58690 1.42070 .60285 1.51795 .61893 1.62419 3(5 37 .57133 .33278 .58716 1.42225 .60312 1.51965 .61920 1.62604 37 88 .57159 .33422 .58743 1.42380 .60339 1.52134 .61947 1.62790 88 39 .57185 .33565 I .58769 1.42536 .60365 1.52304 .61974 '1.62976 39 40 .57212 .33708 .58796 1.42692 .60392 1.52474 .62001 1.63162 40 11 .57288 .33852 .58822 1.42848 .60419 1.52645 .62027 1.63348 41 i;2 .57264 ! 33996 .58849 1.43005 .60445 1.52815 ! .62054 1.63535 4-2 48 .57201 .34140 .58875 1.43162 .60472 1.52986 .62081 1.63722 43 44 .57317 .34284 ; .58902 1.43318 .60499 1.53157 .62108 1.63909 44 15 .57343 .34429 .58928 1.43476 .60526 1.53329 .62135 1.64097 45 46 .57369 .34573 .58955 1.43633 .60552 J. 53500 .62162 1.64285 46 47 .57396 .34718 i .58981 1.43790 .60579 1.53672 .62189 1.64473 47 48 .57422 .34863 i .59008 1.43948 .60606 1.53845 .62216 1.64662 48 19 .57 IIS .35009 .59034 1.44106 .60633 1.54017 .62243 1.64851 49 :,i) .57475 .35154 .59061 1.4-i264 .60659 1.54190 .62270 1.65040 50 51 .57501 .35300 .59087 1.44423 .60686 1.54363 .62297 1.65229 51 52 .57527 .35446 .59114 1.44582 .60713 1.54536 .62324 1.65419 52 58 .5 75.") 4 .35592 .59140 1.44741 .60740 1.54709 .62351 1.65609 53 :i .57580 .35738 | .59167 1.44900 .60766 1.54883 .62378 1.65799 54 55 .57606 .35885 i .59194 1.45059 .60793 1.55057 .62405 1.65989 55 56 .57633 .36031 .59220 1.45219 .60820 1.55231 .62431 1.66180 5(5 57 .57659 .36178 .59247 1.45378 .60847 1.55405 .62458 1.66371 57 58 .57685 1.36325 .59273 1.45539 .60873 1.55580 .62485 1.66563 58 59 .57712 1.36473 .59300 1.45699 .60900 1.55755 .62512 1.66755 59 GO .57738 1.36620 .59326 1.45859 .60927 1.55930 .62539 1.66947 60 337 TABLE XIII. VERSIXES AND EXSECAXTS. ' 68* 69 70 71 | 1 Vers, Exsec. Vers. Exsec. Vers. Exsec. Vers. Exsec. .62539 ! 1.60947 .64163 1.79043 .05798 1.98880 .67443 i 2.07155 1 .62566 1.67139 .64190 1.78864. .65825 1.92014 .07471 2.07415 1 2 .62593 1.67332 .(54218 1.79466 .65853 1.92849 .67498 2.07-675 2 3 .62620 1.67525 .64245 1.79079 .65880 1.98083 .07520 2.07-936 4 .62647 1.67718 .04272 1.79891 .65907 1.93318 .67553 2.08197 4 5 .62674 1.67911 I .0429!) 1.80104 .65935 1.93554 .(>7: .61 I 2.08459 5 6 .62701 1.68105 .64326 1.80318 .65902 1.93790 .(;;v,08 i 2.08721 ( 7 .62728 1.6*299 .64353 1.80531 .05989 1.94026 .67036 2.08983 ' { 8 .62755 1.68194 .64381 1.80746 .66017 1.94203 .67063 2.09246 8 9 .62782 l.OSOS;) .64408 1.80960 .66044 1.94500 .67091 2.09510 9 10 .62809 1.68884 .64435 1.81175 .60071 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 .07773 2.10303 12 13 .62890 1.69471 .64517 1.81821 .66154 1.95452 .07801 2.10508 13 14 .62917 1.69667 .64544 1.82037 .66181 1.95691 .07829 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 1.90171 .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 .60290 1.96652 .67939 2.11903 18 19 .63052 1.70653 .64680 1.88184 .66318 1.96893 .U7966 2.12171 19 20 .63079 1.70851 .64707 1.83342 .66345 1.97135 .67994 2.12440 20 21 .63106 1.71050 .64734 1.83561 .66373 1.97377 .68021 2.12709 21 22 .631:33 1.71249 .64761 1.88780 .66400 1.97619 .08049 2.12979 22 23 .63161 1.71448 .64789 1.83999 .60427 1.97862 .68077 2.13249 23 24 .63188 1.71647 .64816 1.84219 .66455 1.98100 .OHIO! 2.13520 24 25 .63215 1.71847 .64843 1.84439 .06482 1.98349 .08132 2.13791 25 26 .63242 1.72047 .64870 1.84659 i .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 .64925 1.85102 .06504 1.99083 .68214 2.14608 28 29 .63323 1.72649 .64952 1.85323 i .66592 1.99329 .68242 2.14881 29 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 j .66074 2.00067 .68325 2.15704 32 33 .63431 1.73456 .65061 1.86213 .66702 2.00315 .08352 2.15979 33 34 .63458 1.73659 .65088 1.86437 .66729 2.00562 .68380 2.16255 34 35 .63485 1.73862 .65116 1.80C61 .66756 2.00810 .68408 2.16531 35 36 .63512 1.74065 .65143 1.86885 i .66784 2.01059 .68435 2.16808 36 37 .63539 1.74269 165170 1.87109 .66811 2.01308 .68463 2.17085 37' 38 .63566 1.74473 j .65197 1.87884 ; .66839 2.01557 .68490 2.17363 38 39 .63594 1.74677 .65225 1.87560 ! .66800 2.01807 .08518 ! 2.17641 39 40 .63621 1.74881 .65252 1.87785 .66894 2.02057 .68546 2.17920 40 41 .63648 1.75086 .65279 1.88011 .66921 2.02308 .68573 2.18199 41 42 .63675 1.75292 .65:306 1.88388 .66949 2.02559 .68601 2.18479 42 43 .63702 1.75497 i .65334 1.88465 .66976 2.02810 .68028 2.18759 43 44 .63729 1.75703 .65361 1.88692 .67003 2.03062 .68056 2.19040 44 45 .63756 1.75909 .65388 1.88920 .67031 2.03315 .68684 2.19322 45 16 .63783 1.76116 i .65416 1.89148 .67058 2.03568 .68711 2.19604 40 17 .63810 1.7IJ323 .65143 1.89376 .67-066 2.03821 .687'39 2.19886 47 48 .63838 1.76530 I .65470 1.89605 .07113 2.04075 .68707 2.20169 48 49 .63865 1.76737 .65497 1.89834 .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 .67196 2.04839 .68849 2.21021 51 52 .63946 1.77:302 .65579 1.90524 .67223 2.05094 .68877 2.21306 52 53 .63973 ! 1.77571 .65607 1.90754 .67351 2.05350 .68905 2.21592 53 54 .64000 1 77780 .65684 1.90986 67278 2.05607 .68932 2.21878 54 55 .64027 1.77990 .65661 1.91217 .67:306 2. 058(5 1 , .68900 2.22165 55 56 .64055 1.78200 .65689 1.91449 .67333 : 2.00121 .68988 2.22452 56 57 64082 1.78410 .65716 1.91681 .67301 y.(MW7!) .69015 2.22740 57 "58 .64109 1.78681 .65743 1.91914 .(i7:is8 y.own? .69043 2.2:3028 158 59 .64136 1.7H832 .65771 1.JWN7 .<57H( 2.or,s<)0 .69071 2.2:3317 59 60 .64163 1.79043 i .65798 1.92380 I| .67443 2.07155 .69098 2.23607 160 TABLE XIII. VERSINES AND EXSECANTS. 7 2 7 I 3 7 4 7 5 , Vers. Exsec. Vers. Exsec. Vers. Exsec. Vers. Exsec. .69098 2.23607 .70763 2.42030 .72436 2.62796 .74118 2.86370 1 .09 120 2.23897 .70791 2.42350 .72404 2.63164 .74146 2.86790 1 a .09154 2.24187 .70818 2.42683 .7'2492 2 63533 .74174 2.87211 g a .69181 2.2447'8 .70846 2.43010 .72520 2.63903 .74202 2.87633 3 4 .69209 2.24770 .70874 2.43337 .72548 2.64274 .74231 2.88056 4 5 .69:237 2.25062 .70902 2.43666 .72576 2.64645 .71259 2.88479 5 G 69264 2.25355 .70930 2.43995 .72604 2.65018 .7'4287 2.88904 6 7 .69292 2.25648 .70958 2.44324 .72632 2.65391 .74315 2.89330 7 8 .69320 2.25942 .70985 2.44655 .72660 2.65765 .74343 2.89756 8 ( J .69347 2 26237 .71013 2.44986 .72688 2.66140 .74371 2.90184 9 10 .69375 2.26531 .71041 2.45317 .72716 2.60515 .74399 2.90613 10 11 .69403 2.26827 .71069 2.45650 .72744 2.66892 .74427 2.91042 11 12 .09430 2.27123 .71097 2.45983 .72772 2.67209 .74455 8.91473 12 18 .69458 2.27420 .71125 2.46316 .72800 2.07'647 .74484 2.91904 13 14 .69486 2.27717 .71153 2.46651 .72828 2.08025 .74512 2.92337 14 15 .69514 2.28015 .71180 2.46986 .72856 2.68405 .74540 2.92770 15 16 .69541 2.28313 .71208 2.47321 .72884 2.68785 .74568 2.93204 1(1 17 .69569 2.28612 .71236 2.47658 .72912 2.69167 .74596 2.9364C 17 IS .69597 2.28912 .71204 2.47995 .72940 2.69549 . .74624 2.94076 18 19 .69624 2.29212 .71293 2.48333 .72968 2.09931 1 .74652 2.94514 1!) 90 .69652 2.29512 .71320 2.48671 .72996 2.70315 .74680 2.94952 80 21 .69680 2.29814 .71348 2.49010 .73024 2.70700 .74709 2.95392 81 22 .69708 2.30115 ,71375 2.49350 .73052 2.71085 .74737 2.95832 22 23 .69735 2.30418 -71403 2.49691 .73080 2.71471 j .74765 2.'.)(i274 23 534 .69763 2.30721 .71431 2.50032 .73108 2.71858 ' .74793 2.96716 84 85 .69791 2.31024 .71459 2.50374 .73136 2.72246 .74821 2.97160 25 86 .69818 2.31328 .71487 2.50716 .73164 2.72635 .74849 2.97604 20 27 .69846 2.31633 .71515 2.51060 .73192 2.73024 .74878 2.98050 27 28 .69874 2.31939 .71543 2.51404 .73220 2.73414 .74906 2.98497 2S 29 .69902 2.32244 .71571 2.51748 .73248 2.73806 .74934 2.98944 29 80 .69929 2.32551 .71598 2.52094 .73276 2.74198 .74962 2.99393 80 31 .69957 2.32858 .71026 2.52440 ,73304 2.74591 .74990 2.99843 :J1 32 .69985 2.33166 .71654 2.52787 .73332 2.74984 .75018 3.00293 82 3:3 .70013 2.33474 .71682 2.53134 .73360 2.75379 .75047 3.00745 83 84 .70040 2.33783 .71710 2.53482 .73388 2.75775 .75075 3.01198 31 85 .70068 2.34092 .71738 2.53831 .73416 2.76171 .75103 3.01652 85 30 .70096 2.34403 .71766 2.54181 .73444 2.76568 : .75131 3.02107 36 87 .70124 2.34713 .71794 2.54531 .73472 2.76966 .75159 3.02563 87 88 .70151 2.35025 .71822 2.54883 .73500 2.77305 .75187 3.03020 88 89 .70179 2.35336 .71850 2.55235 .73529 2.77765 .75216 3.03479 89 40 .70207 2.35649 .71877 2.55587 .73557 2.78166 .75244 3.03938 4 41 .70235 2.35962 .71905 2.55940 .73585 2.78568 .75272 3.0-1398 41 45 .70263 2.36270 .71933 2.56294 .73613 2.78970 .75300 3.04860 42 43 .70290 2.30590 .71961 2.56649 .73641 2.79374 .75328 3.05322 43 44 .70318 2.30905 .71989 2.57005 .73009 2.79778 .75356 8.05786 41 45 .70:546 2.37221 .72017 2.57361 .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 .7207'3 2.5807'6 .73753 2.80996 .75441 3.07184 47' 4S .70429 2.38171 .72101 2.58434 .73781 2.81404 .75469 3.07652 4S 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 CO 51 .70513 2.39128 .72185 2.59514 .73865 2. 82633 .75554 3.09063 51 52 .70540 2.39448 .72213 2.59876 .73893 2.8:3045 .75582 3.095&5 52 r,.-i .70568 2.39768 .72241 2.60238 .73921 2.83457 .75610 3.10009 53 54 .70596 2.40089 .72269 2.00601 .73950 2.83871 .75639 3.10484 51 5.") .70624 2.40411 .72296 2.60965 .73978 2.84285 .75667 8.10960 55 56 .70652 2.40734 .72324 2.61330 .74006 2.84700 .75695 3.11437 50 57 .70679 2.41057 .72352 2.61695 .74034 2.85116 ,75723 3.11915 57 58 .70707 2.41381 .72380 2 62061 .74062 2.85533 .75751 3.12394 58 59 .70735 2.41705 .72-408 2.62428 .74090 2.85951 .75780 3.12875 59 GO .70763 2.42030 .72436 2.62796 1 .74118 2.86370 .75808 3.13357 80 339 TABLE XIII.-VERSINES AND EXSECANTS. 7 6 7 7 7 8 7 Qo Vers. Exsec. Vers. Exsec. Yers. Exsec. Yers. Exsec. .75808 3.13357 . 77505 3.44541 i .79209 3.80973 .80919 4.24084 1 .75836 3.13839 .77533 3.45102 . .79237 3.81033 .80948 4.24870 1 2 .75864 3.14323 .77562 ! 3.45664 .79200 3.82294 .80976 4.2505S 2 3 .75892 3.14809 .77590 i 3.46228 .79294 3.82956 .81005 4.26448 3 4 .75921 3.15295 .77618 : 3.46793 .79323 3.83621 .81033 4.27241 4 5 .75949 3.15782 .77647 3.47360 .79351 3.84288 .81062 4.28036 5 6 .75977 3.16271 .77675 3.47938 .79380 3.84956 .81090 4.28833 6 7 .76005 3.16761 .77703 3.48498 .79408 3.85627 .81119 4.29634 7 8 .76034 3.17252 .77732 ! 3.49069 .79437 3.86299 .81148 4.30436 8 g .76062 3.17744 .77760 3.49642 .79-165 3.86973 .81176 4.31241 9 10 .76090 3.18238 .77788 3.50216 .79493 3.87649 .81205 4.32049 10 11 .76118 3.1S733 .77817 3.50791 .79522 3.88327 .81233 4.32859 11 13 .76147 3.19228 .77845 3.51368 .79550 3.89007 .81202 4.33671 12 13 .76175 3.19725 .77874 3.51947 .79579 3.89689 i .81290 4.34486 13 14 .76203 3.20224 .77902 3.52527 .79607 3.90373 .81319 4.35304 14 15 .76231 3.20723 .77930 3.53109 .79636 3.91058 .81348 4.36124 15 18 .76260 3.21224 .77959 3.53692 .79664 3.91746 : .81376 4.365)47 16 17 .76288 3.21726 .77987 3.54277 .79093 3.92436 .81405 4.37772 17 18 .76316 3.22229 .78015 3.54863 .79721 3.93128 .81433 4.38000 18 19 .76344 3.22734 .78044 3.55451 .79750 3.93821 1 .81462 4.3943') 19 20 .76373 3.23239 .78072 3.56041 .79778 3.94517 .81491 4.40263 20 21 .76401 3.23746 .78101 3.56632 .79807 3.95215 .81519 4.41099 21 2-2 .76429 3.24255 .78129 3.57224 .79835 3.95914 .81548 4.41937 22 88 .76458 3.24764 .78157 3.57819 .79864 3.96616 ' .81576 4.42778 23 24 .76486 3.25275 ! .78186 3.58414 .79892 3.97320 .81605 4.43022 24 X .76514 3.25787 .78214 3.59012 .79921 3.98025 1 .81633 4.44468 25 96 .76542 3.26300 .78242 3.59611 .79949 3.98733 .81662 4.45317 26 2; .76571 3.26814 i .78271 3.60211 .79978 8.99443 .81691 4.46169 27 88 .76599 3.27330 ! .78299 3.60813 .80006 4.00155 i .81719 4.47023 28 89 .76627 3.27847 i .78328 3.61417 .80035 4.00869 .81748 4.47881 29 80 .76655 3.28366 .7'8356 3.62023 .80063 4.01585 .81776 4.48740 SO 81 .76684 3.28885 .78384 3.62630 .80092 4.02303 .81805 4.49603 31 82 .76712 3.29406 .78413 3.63238 .80120 4.03024 .81834 4.50468 32 88 .76740 3.29929 .78441 3.63849 .80149 4.03746 < .81862 4.51337 33 84 .76769 3.30452 1 .7847'0 3.64461 .80177 4.04471 : .81891 4.52208 34 86 .76797 3.30977 .78498 3.65074 .80206 4.05197 .81919 4.53081 35 30 .76825 3.31503 | .78526 3.65690 .80234 4.05926 i .81948 4.53958 30 87 .76854 3.32031 .78555 3.66307 .80263 4.06657 .81977 4.54837 37 88 .76882 3.32560 .78583 3.66925 .80291 4.07390 .82005 4.55720 38 39 .76910 3.33090 .78612 3.67545 .80320 4.08125 .82034 4.56605 39 40 .76938 3.33622 .78640 3.68167 .80348 4.08863 .82063 4.57493 40 41 .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 .77023 3.35224 .78725 3.70044 .80434 4.11088 .82148 4.00174 43 44 .77052 3.35761 .78754 3.70673 .80462 4.11835 .82177 4.01073 44 45 .77080 3.36299 .78782 3.71303 .80491 4.12583 .82206 4.61U76 45 46 .77108 3.36839 .78811 3.71935 .80520 4.13334 .82234 4.62881 46 47 .77137 3.37380 .78839 3.72569 .80548 4.14087 .82203 4.03790 47 48 .77165 3.37923 .78868 3.73205 .80577 4.14842 .82292 4.047'01 48 49 .77193 3.38466 .78896 8.7'3843 .80605 4.15599 .82320 4.65616 49 60 .77222 3.39012 .78924 3.74482 .80634 4.16359 .82349 4.06533 50 51 .77250 3.39558 .78953 3.75123 .80662 4.17121 .82377 4.07451 51 52 .77278 3.40106 >8981 8.75766 .80691 4.17886 .82406 4.08377 52 58 .77307 3.40656 .79010 3.76411 .80719 4.18652 .82435 4.69304 53 64 .77335 3.41206 .79038 3.77057 .80748 4.19421 .82463 4.70234 54 55 .77363 3.41759 .79067 3.77705 .8077'6 4.20193 .82492 4.71166 55 66 .77392 3.42312 .79095 8.78365 .80805 4.20966 .82521 , 4.72102 56 57 .77420 3.42867 .79123 3.79007 .80833 4.21742 82549 4.73041 57 n .77448 3.43424 .79158 3.79001 .80868 4.22521 .82578 4.73983 58 69 .77477 3.43982 .79180 3.80316 .80891 4.23801 .82607 4.74929 59 GO .77505 3.44541 .79209 3.80973 1 .80919 4.24084 .82635 4.75877 60 340 TABLE XIII.-VERSINES AND EXSECANTS. ' 80 81 82 83 ' Vers. Exsec. Vers. Exsec. Vers. Exsec. Vers. Exsec. .82635 4.75877 .84357 5.39245 .86083 6.18530 .87813 7.20551 1 .82664 4.76829 .84385 5.40422 .86112 6.20020 .87842 7.22500 1 2 .82693 4.77784 .84414 5.41602 i .86140 6.21517 .87871 7.24457 2 3; .83721 4.78742 .84443 5.42787 .86169 6.23019 .87900 7.26425 3 4 .82:50 4.79703 .84471 5.43977 1 .86198 6.24529 .87929 7.28402 4 51 .82:78 4.80667 .84500 5.45171 .86227 6.26044 .87957 7.36388 5 6| .85307 4.81635 .84529 5.46369 .86256 6.27566 .87986 7.32384 6 7 .83336 4.82606 .84558 5.47572 .86284 6.29095 .88015 7.34390 7' 8 .83864 4.83581 .84536 5.48779 .86313 6.30630 .88044 7.36405 8 9 .&IS93 4.84558 .84615 5.43991 .86342 6.32171 .88073 7.38431 9 10 .82923 4.85539 .84644 5.51208 .86371 6.33719 .88102 7.40466 10 11 .82950 4.86534 .84673 5.52429 .86400 6.35274 .88131 7.42511 11 12 i .83979 4.87511 .84701 5.53655 .86428 6.36835 .88160 7.44566 12 13 .83003 4.88503 .84730 5.54886 .86457 6.38403 .88188 7.46632 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 .88246 7.50793 15 16 .83094 4.91496 .84816 5.58606 .86544 6.43148 .88275 7.52889 il6 17 .83123 4.92501 .84845 5.59855 .86573 6.44743 .88304 7.54996 !l7 18 .83151 4.93509 .84874 5.61110 .86601 6.46346 .88333 7.57113 18 11) .83180 4.91521 .84903 6. 62309 .86630 6.47933 .88362 7.59241 ; 19 90 .83208 4.95536 .84931 5.63633 .86659 6.49571 .88391 7.61379 i 20 -I .83237 4.96555 .84960 5.64902 .86688 6.51194 .88420 7.6.3528 21 .1.) .83266 4.97577 .84989 5.66176 .86717 6.53825 .88448 7.65688 : 22 as .83294 4.98603 .85018 5.67454 .867'46 6.54462 .88477 7.67859 (23 24 .83323 4.99633 .85046 5.68738 i .86774 6.56107 .88506 7.70041 ; 24 25 .83352 5.00666 .85075 5.70027 ! .86803 6.57759 .88535 7.72234 i25 26 .83380 5.01703 .85104 5.71321 .86832 6.59418 .88564 7.74438 2(5 27 .83409 5.03743 .85133 5.72620 .86861 6.61085 .88593 7.76653 27 28 .83438 5.03787 .85162 5.73924 .86890 6.62759 .88623 7.78880 's 29! .83-107 5.04834 .85190 5.75233 .86919 6.64441 .88651 7.81118 129 30 .83495 5.05886 .85219 5.76547 .86947 6.66130 .88680 7.83367 80 81 .83534 5.06941 .85248 5.77866 .86976 6.67826 .88709 7.85628 81 83 .83553 5.03000 .85277 5.79191 .87005 6.69530 .887'37 7.87901 32 3:5 .83581 5.09062 .85305 5.80521 .87-034 6.71242 .88766 7.90186 83 84 .83610 5.10129 .85334 5.81856 .87063 6.72962 .88795 7.92482 34 :J5 .83639 5.11199 .85363 5.83196 .87092 6.74689 .88824 7.94791 35 yr, .83667 5.12273 .85392 5.84542 .87120 6.76424 .88853 7.97111 36 87 .83696 5.13330 .85420 5.85893 .87149 6.78167 .88882 7.99444 87 3H .83735 5.14433 .85449 5.87250 .87178 6.79918 .88911 8.01788 38 30 .83754 5.15517 .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 41 42 ! .83840 5.18797 .85564 5.92731 .87294 6.87001 .89027 8.11292 4-2 43 .83868 5.19898 .85593 5.94115 .87322 6.88792 .89055 8.13699 ta U .83897 5.21004 .85622 5.95505 .87351 6.90592 .89084 8.16120 44 45 .83936 5.23113 .85651 5.96900 .87380 6.92400 .89113 8.18553 45 46 .83954 5.23226 .85680 5.98301 i .87409 6.94216 .89142 8.20999 46 47 1 .830S3 5.24343 .85708 5.997'08 i .87438 6.96040 .89171 8.23459 41' 48 : .84012 5.25464 .85737 6.01120 .87467 6.97873 .89200 8.25931 48 49 | .81041 5.26590 .85766 6.02538 .87496 6.99714 .89229 8.28417 : 49 50 .84069 5.27719 .85795 6.03962 .87524 7.01565 .89258 8.30917 50 51 .84098 5.28853 .85823 6.05392 .87553 7.03423 .89287 8.33430 51 52 .84127 5.29991 : 85852 6.06828 .87582 7.05291 .89316 8.35957 52 53 .84155 5.31133 .85881 6.08269 .87611 7.07167 .89345 8.38497 53 54 .84184 5.32279 .85910 6.09717 .87640 7.09052 .89374 8.41052 51 55 .84213 5.33429 .85939 6.11171 .87669 7.10946 .89403 8.43620 55 56 .84242 5.34584 .85967 6.12630 .87698 7.12849 .89431 8.46203 56 57 .84270 5.35743 .85996 6.14096 .87726 7.14760 .89460 8.48800 57 58 .84299 5.36906 .86025 6.15568 .87755 7.16681 .89489 8.51411 58 58 .84328 5.38073 .86054 6.17046 .87784 7.18612 .89518 8.54037 59 CO .84357 5.39345 .86083 6.18530 .87813 7,20551 .89547 8.56677 60 341 TABLE xin. VERSINES AND EXSECANTS. / 84 85 86 ' Vers. Exsec. Vers. Exsec. Vers. Exsec. .89547 8.5(3677 .91284 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 .89634 8.64687 .91371 10.58932 .93111 13.51676 3 4 .89663 8.67387 .91400 10.62837 .93140 13.57817 4 5 .89692 8.70103 .91429 10.66769 .93169 13.64011 5 6 .89721 8.72833 .91458 10.7'0728 .93198 13.70258 6 7 .89750 8.75579 .91487 10.74714 .93227 13.76558 7 8 .89779 8.78341 .91516 10.787'27 .93257 13.82913 8 9 .89808 8.81119 .91545 10.82768 ' .93286 13.89823 9 10 .89836 8.83912 .91574 10.86837 .93315 13.957B8 10 11 .89865 8.86722 .91603 10.90934 .93344 14.02310 11 12 .89894 8.89547 .91632 10.95060 .98373 14.08890 12 13 .89923 8.92389 .91661 10.99214 .93102 14.15527 13 14 .89952 8.95248 .91690 11.03397 .93431 14.22223 14 15 .89981 8.98123 .91719 11.07610 .93460 14.28979 15 16 .90010 9.01015 .91748 11.11852 .93489 14.35795 16 17 .90039 9.03923 .91777 11.16125 .93518 14.42672 17 18 .90068 9.06849 .91806 11.20427 .93547 14.49611 18 19 .90097 9.09792 .91835 11.24761 .93576 14.56614 19 20 .90126 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 .91980 11.40900 .93721 14.92597 24 25 .90271 9.27819 .92009 11.51424 .93750 14.99995 25 26 .90300 9.30887 .92038 11.55982 .93779 15.07462 26 27 .90329 9.33973 .92067 11.60572 .93808 15.14999 27 28 .90358 9.37077 .92096 11.65197 .93837 15.22607 28 29 .90386 9.40201 .92125 11.69856 .93866 15.30287 29 30 .90415 9.43343 .92154 11.74550 .93895 15.38041 30 31 .90444 9.46505 .92183 11.79278 .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 .94040 15.77944 35 36 .90589 9.62605 .92328 12.03458 .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 32.18411 .94156 16.11297 39 40 .90705 9.75849 .92444 12.23472 .94186 16.198-13 40 41 .90734 9.79212 .92473 12.28572 1 .94215 16.28476 41 42 .90763 9.82596 .92502 12.33712 .94244 16.37196 4:3 43 .90792 9.86001 .92531 12.38891 .94273 16.46005 43 44 .90821 9.89428 .92560 12.44112 .94302 16.54903 41 45 .90850 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 .91418 16.91424 48 49 .90966 10.06894 .92705 12.70838 .94447 17.00794 49 50 .90995 10.10455 .93734 12.76312 .94476 17.10262 50 51 .91024 10.14039 .92763 12.81829 .94505 17.19830 51 52 .91053 10.17646 .92792 12.87391 .94534 17.29501 52 53 .91082 10.21277 .92821 12.92999 .94563 17.39274 53 54 .91111 10.24932 .92850 12.98651 .94592 17.49153 64 55 .91140 10.28610 i .92879 13 04350 .94621 17.59139 55 56 .91169 10.32313 .92908 13.1001)0 .94050 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 13.27630 .94737 18.00185 59 60 .91284 10.47371 .93024 13.33559 .94766 18.10732 60 343 TABLE XHI. VERSINES AND EXSECANTS. / 87 88 89 / Vers. Exsec. Vers. Exsec. Vers. Exsec. .94766 18.10732 .96510 27.65371 .98255 56.29869 1 .94795 18.21397 .96539 27.89440 .98284 57.26976 1 2 .94825 18 32182 .96568 28.13917 .98313 58.27'431 2 3 .94854 18.43088 .96597 28.38812 .98342 59.31411 3 4 .94883 18.54119 .96626 28,64137 .98371 60.39105 4 5 .91912 18.65275 .96655 28.89903 .98400 61.50715 5 6 .91941 18.76560 .96684 29.16120 .98429 62.66460 6 7 .94970 18.87976 .96714 29.42802 .98458 63.86572 7 8 .94999 18.99524 .96743 29.69960 .98487 65.11304 8 9 .95028 19.11208 .96772 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 69.16047 11 12 .95115 19.47'093 .96859 30.83(523 .98604 70.62285 12 13 .95144 19.59341 .96888 31.13366 .98633 72.14583 13 14 .95173 19.71737 .96917 31.43671 .98662 73.7'3586 14 15 .95202 19.84283 .96946 31.74554 .98691 75.39655 15 16 .95231 19.96982 .96975 32.06030 .98720 77.13274 16 17 .95260 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 84.94561 20 21 .95377 3C. 62876 .97121 33.72952 .98866 87.14924 21 22 .95406 20.76555 .97150 34.08380 .98895 89.46886 22 23 .95485 20.90409 .97179 34.44539 .98924 91.91387 23 21 .95464 21.04440 .97208 34.81452 .98953 94.49471 24 95 .95493 21.18653 .97237 35.19141 .98982 97.22303 25 86 .95522 21.33050 .97266 35.57633 .99011 100.1119 26 27 .95551 21.47635 .97295 35.96953 .99040 103.1757 27 23 .95580 21.62413 .97324 36.37127 .99069 106.4311 28 29 .95609 21.77386 .97353 36.78185 .99098 109.8966 29 30 .95638 21.92559 .97382 37.20155 .99127 113.5930 30 31 .95667 22.07935 .97411 37.63068 .99156 117.5444 81 33 .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 38.97797 .99244 131.2223 34 35 .95783 22.71563 .97528 39.44820 .99273 136.5111 35 36 .95812 22.88022 .97557 39.92963 .99302 142.2406 36 37 .95842 23.04712 .97'586 40.42266 .99331 148.4684 37 38 .95871 23.21637 .97615 40.92772 .99360 155.2623 38 39 .95900 23.38802 .97644 41.44525 .99389 162.7033 39 40 .95929 23.56212 .97673 41.97571 .99418 170.8883 40 41 .95958 23.73873 .97702 42.51961 .99447 179.9350 41 42 .95987 23.91790 .97731 43.07746 .99476 189.9868 42 43 l .96016 24.09969 .97760 43.64980 .99505 201.2212 43 44 .96045 24.28414 .97789 44.23720 .99535 213.8600 44 45 ! .96074 24.47134 .97819 44.84026 .99564 228.1839 45 4(5 .96103 24.66132 .97848 45.459GS .99593 244.5540 46 47 .96132 24.85417 .97877 46.09596 .99622 263.4427 47 48 .96161 25.04994 .97906 46.74997 .99651 285.4795 48 49 .96190 25.24869 .97935 47.42241 .99680 311.5230 49 50 .90219 25.45051 ,07964 48.11406 .99709 342.7752 50 51 .96248 25.65546 .97993 48.82576 .99738 380.9723 51 52 .96277 25.86360 .98022 49.55840 .99767 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 51.89156 .99855 686.5496 55 56 .96394 26.72978 .98138 52.71790 .99884 858.4369 56 57 .96423 26.95513 .'.tSKiS 53.57046 .99913 1144.916 57 58 .96452 27.18417 .98197 54.45053 .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 60 343 TABLE XIV.- CUBIC YARDS PER 100 FEET. SLOPES '4 : 1. Depth Base 12 Base 14 Base 16 Base 18 Base 22 Base 24 Base 26 Base 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 7 356 408 460 612 616 668 719 771 8 415 474 533 593 711 770 830 889 9 475 542 608 675 808 875 942 1008 10 537 611 685 759 907 981 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 17 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 20 1259 1407 1555 1704 2000 2148 2296 2444 21 1342 1497 1653 1808 2119 2275 2431 2586 22 1426 1589 1752 1915 2241 2404 2567 2730 23 1512 1682 1853 2023 2364 2534 2705 2875 24 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 33:22 27 1875 2075 2274 2475 2875 3075 3275 3475 28 1970 2178 2384 2593 3007 3215 3422 30:30 29 2068 2282 2496 2712 3142 3358 3571 3786 30 2167 2389 2610 2833 3278 3500 3722 3944 31 2268 2497 2726 2956 3416 3645 3875 4105 32 2370 2607 2844 3081 3556 3793 4030 4207 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 88 3026 3307 3589 3870 4433 4715 4996 527'8 39 3142 3431 3719 4008 4586 4875 5164 5-153 40 3259 3556 3852 4148 4741 5037 5333 5630 41 3379 3682 3986 4290 4897 5201 5505 5808 42 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 46 4004 4344 4684 5026 5707 6048 6389 6730 47 4134 4482 4830 5179 5875 6223 6571 6919 48 4267 4(522 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 597'0 6741 7126 7511 7896 53 4956 5349 5741 6134 6919 7312 7705 8097 54 5100 5500 5900 6300 7100 7500 7900 8300 55 5245 5653 6060 6468 7282 7690 8097 &505 56 5393 5807 6222 6637 7467 7881 8296 8711 57 5542 5964 6386 6808 7653 8075 8497 8919 58 5693 6122 6552 6981 78-11 8270 8700 9130 59 5845 6282 6719 7156 8031 8468 8905 9342 60 6000 6444 6889 7333 8222 8667 9111 9556 344 TABLE xiv. -CUBIC YARDS PER 100 FEET. SLOPES D^pth Base 12 Base 14 Base 16 Base 18 Base 22 Base 24 Base 26 Base 28 1 46 54 61 69 83 91 98 106 2 96 111 126 141 170 185 200 215 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 7 402 454 506 557 661 713 765 817 8 474 533 593 652 770 &30 889 948 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 It 985 1089 1193 1296 1504 1607 1711 1815 15 1083 1194 1306 1417 1639 1750 1861 1972 16 1185 1304 1422 1541 1779 1896 2015 2133 17 1291 1417 1543 1669 1920 2046 2172 2298 18 1400 1533 1667 1800 2067 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 3750 26 2407 2600 2793 2985 3370 35C3 3756 3948 27 2550 2750 2950 3150 &550 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 397'2 4217 4706 ' 4950 5194 5439 34 3652 3904 4156 4407 4911 5163 5415 5G67 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 6052 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 40 5133 5444 5756 6067 6689 7000 7311 7622 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 C644 6985 7667 8007 8348 8689 47 6180 6528 6876 7224 7920 8269 8617 8965 48 49 6400 6624 6756 6987 7111 7350 m 8178 843M 8889 9165 9244 9528 50 6852 7222 7593 7963 87G4 9074 9444 9815 51 7083 7461 7839 8217 8972 9350 9728 10106 52 7319 7704 8089 847'4 9244 9630 10015 10400 53 7557 7950 8343 8735 9520 9913 10306 10698 54 7800 8200 8000 9000 9800 10200 10600 11000 55 8046 8454 8861 9269 10083 10491 10898 11306 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 i 11815 12244 59 9069 9506 9943 10380 11254 11691 12128 12565 60 9333 9778 10222 10667 11556 12000 12444 12889 345 TABLE XIV. -CUBIC YARDS PER 100 FEET. SLOPES 1 1. Depth Base 12 Base 14 Base 16 Base 1.8 Base 20 Base 28 Base- 30 Base 32 1 48 56 03 70 78 107 115 122 2 104 119 138 148 163 222 237 252 3 167 189 211 283 256 344 367 389 4 237 267 296 326 356 474 504 5133 5 315 352 389 426 463 611 648 685 6 400 444 489 533 578 756 800 844 7 493 544 596 648 700 907 959 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 13:33 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 2463 2604 2744 3307 3448 3589 20 2370 2519 2667 2815 2963 3556 3704 3852 21 2567 2722 2878 3033 31 R9 3811 3967 4122 22 2770 2933 3096 3259 34-^2 4074 4237 4444 23 2981 3152 3322 3493 3663 4344 4515 4685 24 3200 3378 3556 8733 3911 4622 4800 4978 25 3426 3611 3796 3981 4167 4907 5093 5278 26 3659 3852 4044 4237 4430 5200 5393 5585 27 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 9X83 32 5215 5452 5689 5926 6163 7111 7318 7585 33 5500 5744 5989 6233 6478 7456 1700 7944 34 5793 6044 6296 6518 6800 7807 8059 8311 35 6093 6352 6611 6870 7130 8167 8426 8685 36 6400 6GG7 69:33 7200 7467 8533 8800 90G7 37 6715 6989 7263 7537 7811 8907 9181 9456 38 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 11300 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 13C56 1:3404 13752 48 10667 11022 11378 11733 12089 13511 13867 14223 49 11070 11433 11796 12159 13523 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 16685 54 13200 13600 14000 14400 14800 16400 16800 17200 55 13648 1405(1 14463 14870 15278 16907 17315 17722 56 14104 14519 14933 15348 15763 17423 17837 1R252 57 14567 14989 15411 15833 16256 17944 18367 18789 58 15037 15467 15896 16326 16756 18474 18904 19333 59 15515 15952 16389 KiSei) 17263 19011 10118 19885 60 16000 16444 16889 17333 17778 19556 20000 20444 346 TABLE XIV. CUBIC YARDS PER 100 FEET. SLOPES : 1. Depth Base 12 Base 14 Base 16 Base 18 Base 20 Base 28 Base 30 Base 32 1 50 57 65 72 80 109 117 124 2 111 126 141 156 170 230 244 259 3 183 206 228 250 272 361 383 406 4 267 296 326 356 385 504 533 563 5 301 398 435 472 509 657 694 731 6 467 511 556 600 644 822 867 911 7 583 635 687 739 791 998 1050 1102 8 711 770 830 889 948 1185 1244 1304 9 850 917 983 1050 1116 1883 1450 1517 10 ' 1000 1074 1148 1222 1296 1593 1667 1741 11 1161 1243 1324 1406 1487 1813 1894 1976 12 1888 1422 1511 1600 1689 2044 2133 2222 13 1517 1613 1709 1806 1902 2287 2383 ' 2480 14 1711 1815 1919 2022 2126 2541 2644 2748 15 1917 2028 2139 2250 2361 2806 2917 3028 16 2133 2252 2370 2489 2607 3081 3200 3319 17 2361 2487 2613 2739 2865 3369 3494 3620 18 2600 2733 2867 3000 3133 3(567 3800 3933 19 2850 2991 3131 3272 3413 3976 4117 4257 20 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 4978 5689 5867 6044 25 4583 4769 4954 5139 5324 6065 6250 6435 26 4911 5104 5296 5489 5681 6452 6644 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 7'222 8111 8333 8555 31 6717 6946 7176 7406 7635 8554 8783 9013 32 7111 7348 7585 7822 8059 9007 9244 9482 33 7517 7761 8006 825'0 8494 9472 9717 9962 34 7933 8185 8437 86S9 8941 9948 10200 10452 35 8361 I 8620 8880 9139 9398 10435 10694 10954 36 8800 | 9067 9333 9600 9867 10933 11200 11467 37 9250 9524 9708 1007'2 10346 11443 11717 11991 38 9711 9993 10274 10556 10837 11963 12244 12526 '39 10183 10472 10761 11050 11339- 12494 12783 13072 40 10667 10963 11259 11556 11852 13037 13333 13630 41 11161 11465 11769 12072 12376 13591 13894 14198 42 11667 11978 1*289 12600 12911 14156 14467 14778 43 12383 12502 12820 13139 13457 14731 15050 15369 44 12711 13037 133G3 13689 14015 15319 15644 15970 45 13250 13583 ! 13917 14250 14583 15917 16250 1 16583 46 13800 14141 14481 11822 15163 16526 16867 17207 47 14361 14709 15057 15406 15754 17146 17494 17843 48 14933 15289 15644 16000 16356 17778 18133 18489 49 15517 15880 16243 16606 16968 18420 18783 19146 50 16111 16481 16852 17222 17592 19074 19444 19815 51 16717 17094 17472 17850 18228 19739 20117 20494 52 17333 17719 18104 18489 18874 20415 20800 21185 53 17961 18354 18746 19139 19531 21102 21494 21887 54 18600 19000 19400 19800 20200 21800 22200 22600 55 . 19250 19657 20065 20472 20880 22509 22917 23324 56 19911 20326 20741 21156 21570 23230 23644 24059 57 205&3 21006 21428 21850 22272 23961 243a3 24805 58 21267 21G96 22126 22556 22985 24704 25133 25563 ' 59 21961 22398 2:2835 23272 23709 25457 25894 26332 | ao 22667 23111 23556 24000 24444 26222 26667 27111 347 TABLE XIV. CUBIC YARDS PER 100 FEET. SLOPES 2 ; 1. Depth Base 12 Base 14 Base 16 Base 18 Base 20 Ba-e 28 Base 30 Base 32 1 52 59 67 74 81 111 119 126 2 119 133 148 163 178 237 252 267 3 200 222 244 267 289 378 400 422 4 296 326 356 385 415 5&3 563 693 5 407 444 481 519 556 704 741 778 6 533 578 622 637 711 889 933 978 7 674 726 778 830 881 1089 1141 1193 8 830 889 948 1007 1067 1304 1363 1422 9 1000 1067 1133 1200 1267 1533 1600 1667 10 1185 1259 1333 1407 1481 1778 1852 1926 11 1385 1467 1548 1630 1711 " 2037 2119 2200 12 1600 1689 1778 1867 1956 2311 2400 2489 13 1830 1926 2022 2119 2215 2600 2696 2793 14 2074 2178 2281 2385 2489 2904 3007 8111 15 2333 2444 2556 2667 2778 3222 3333 3444 16 2607 2^6 2844 2963 3081 3556 3674 3793 ir 2896 3022 3148 3274 3400 3904 4030 4156 18 3200 o333 3467 3600 3733 4267 4400 4533 19 3519 3659 3800 3941 4081 4644 4785 4926 20 3852 4000 4148 4296 4444 5037 5185 5333 21 4200 4356 4511 4667 4822 5444 5600 5756 22 4563 4rso 4889 5052 5215 5867 6030 6193 23 4941 5111 5281 545S 5622 6304 6474 6644 24 5333 5511 5689 5867 6014 6756 6933 7111 25 5741 5926 6111 6296 6481 7222 7407 7593 26 6163 6356 6548 6741 6933 7704 7896 8089 27 6600 6800 7000 7200 7400 8200 8400 8600 28 7052 7259 7467 7674 7881 8711 8919 9126 29 7519 77ii3 7948 8163 8378 9237 9452 9667 30 8000 8222 8444 8667 8389 9778 10000 10222 31 8496 8726 8956 9185 9415 10333 10563 10793 32 9007 9244 0481 9719 9956 10904 11141 11378 33 9533 9778 10022 10267 10511 11489 11733 11978 34 10074 10326 10578 10830 11081 12089 12341 12593 35 10030 10889 11148 11407 11(567 12704 12963 13222 36 11200 11467 11733 12000 12267 13333 13600 13867 37 11785 12059 12333 12607 12.81 13978 14252 14526 38 12385 12667 12948 13230 13511 14637 14919 15200 39 13000 13289 13578 13867 14156 15311 15600 15889 40 13630 13926 14222 14519 14815 16000 16296 16593 41 14274 14578 14H81 15185 15489 16704 17007 17311 42 14'.)33 15244 155L6 15867 10178 17122 17733 18044 43 15607 15926 16224 16563 16881 18156 18474 18793 44 16296 16622 16948 17274 17600 18904 19230 19556 45 17000 17333 17667 18000 18333 19667 2001 :0 20333 46 17719 18059 18400 18741 19081 20444 20785 21126 47 18452 18800 19148 19496 19844 21237 21585 21933 48 19200 19556 19911 20267 20622 22044 22400 22756 49 19963 20326 20689 21052 21415 22867 23230 23593 50 20741 20711 21481 21852 22222 23704 24074 24444 51 2U33 21911 22289 22667 23044 24556 24933 25311 52 22-341 22726 23111 23496 23881 25422 25807 2619S 53 23163 23556 23948 24341 24733 26304 26696 27089 54 24000 24400 24HOO 2,V2<;0 2">600 27200 27600 28000 55 24852 25259 25667 26074 26481 28111 28519 28926 56 25719 26133 26548 26963 27378 29037 29452 29867 57 2d600 27022 27444 27867 28289 29978 30400 30822. 58 27496 27926 28356 28785 29215 30933 31363 31793 59 28407 28844 29281 29719 30156 31904 32341 32778 60 29333 29778 30222 30667 31111 32889 33333 33778 348 TABLE XIV. CUBIC YARDS PER 100 FEET. SLOPES 8:1. Depth Base Base Base Base Base Base Base Base 12 14 16 18 20 28 30 32 1 56 63 70 78 85 115 122 130 2 1:33 148 163 178 193 252 267 281 3 233 256 278 300 322 411 433 456 4 &56 385 415 444 474 593 622 652 5 500 537 574 611 648 796 833 870 6 667 711 756 800 844 1022 1067 1111 7 856 907 959 1011 1063 1270 1322 1374 8 1067 1126 1185 1244 1304 1541 1600 1659 9 1300 1367 1433 1500 1567 1833 1900 1967 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 13 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 5733 19 4856 4996 5137 5278 5419 5981 6122 6263 20 5333 5481 5630 5778 5926 6519 6667 6815 21 5833 5989 6144 6300 6456 7078 7233 7389 22 6356 6519 6681 6844 7007 7659 7822 7985 23 6900 7070 7241 7411 7581 8263 8433 8504 24 7467 7644 7822 8000 8178 8889 9067 9144 25 8056 8241 8426 8611 8?'96 9537 9722 9807 26 8667 8859 9052 9244 9437 10207 10400 10593 27 9300 9500 9700 9900 10100 10900 11100 11300 28 9956 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 14123 14352 32 12800 13037 132?'4 13511 13748 14696 14933 15170 33 13567 13811 14056 14300 14544 15522 15767 16011 34 14356 14607 14859 15111 15-J63 16370 16622 16874 35 1516? 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 2:3537 4 21467 21778 22089 22400 22711 23956 24267 24578 43 22456 22774 23093 23411 23730 25004 25322 25641 44 23467 23793 24119 24444 24770 26074 26400 26726 45 21500 24833 25167 25.500 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 2885G I 29219 20581 2994-4 30307 31759 32122 32485 50 . 30000 30370 30741 31111 31481 32963 33333 33704 51 31167 31544 31922 32300 32678 34189 34567 34944 52 32356 32741 33126 33511 33896 35437 35822 36207 53 33567 33959 343&i 34744 35137 36707 37100 37493 54 34800 35200 35600 36000 36400 38000 38400 38800 55 36056 36463 36870 37278 37685 39315 39722 4(130 56 37333 37748 38163 38578 38993 40652 41067 41481 57 38633 39056 39478 39900 40322 42011 42433 42856 58 39956 40385 40815 41244 41674 43393 43822 44252 59 41300 41737 4-2174 42611 43048 44796 45233 45670 60 42667 43111 43556 44000 44444 46222 46667 47111 349 TABLE XV. CUBIC YARDS IN 100 FEET LENGTH. Area. & Cubic Yards. Area. f?: Cubic Yards. Area. 9t Cubic Yards. Area. 11 Cubic Yards. Area. ft Cubic Yards. 1 3.7 51 188.9 101 374.1 151 559.3 201 744.4 2 7.4 52 192.6 102 377.8 152 563.0 202 748.2 3 11.1 53 19(5.3 103 381.5 153 500.7 203 . 751.9 4 14.8 54 200.0 104 385.2 154 570.4 204 755.6 5 18.5 55 203.7 105 388.9 155 574.1 205 759.3 6 22 2 56 207.4 106 392.6 166 577.8 206 763.0 7 25 '.9 57 211.1 107 396.3 157 581.5 207 766.7 8 29.6 58 214.8 108 400.0 158 585.2 208 770.4 9 33.3 59 218.5 109 403.7 159 588.9 209 774.1 10 37.0 60 2-22.2 110 407.4 160 592.6 210 777 . 8 11 40.7 61 225.9 111 411.1 161 596.3 211 781.5 12 44.4 62 2-^9.6 112 414.8 162 600.0 212 785.2 13 48.1 63 233.3 113 418.5 163 603.7 213 788.9 14 51.9 64 237.0 114 422.2 164 607.4 214 792.6 15 55.6 65 240.7 \ 115 425.9 165 61 1 . 1 215 796.3 16 59.3 66 244.4 116 429.6 166 614 8 216 800.0 17 63.0 67 248.2 117 433.3 167 618.5 217 803.7 18 66.7 68 251.9 118 437.0 168 622.2 218 807.4 19 70.4 69 255.6 119 440.7 169 625.9 219 811.1 20 74.1 70 259.3 120 444 .4 170 629.6 220 814.8 21 77.8 71 263.0 121 448.2 171 633.3 221 818.5 22 81.5 72 266.7 122 451.9 172 637.0 222 822.2 23 85.2 73 270.4 123 455 6 178 640.7 223 825.9 24 88.9 74 274.1 124 459.3 174 644.4 224 829.6 25 92.6 75 277.8 125 403.0 175 648.2 225 833.3 26 96.3 76 281.5 126 466.7 176 651.9 226 837.0 27 100.0 77 285 2 127 470.4 177 655.6 227 840.7 28 103.7 78 288.9 128 474 1 178 659.3 228 844.4 29 107.4 79 292.6 129 477.8 179 003.0 229 848.2 30 111.1 80 296.3 130 481.5 180 666.7 230 851.9 31 114.8 81 300.0 131 485.2 181 670.4 231 855.6 32 118.5 82 303.7 132 488 9 182 674.1 232 859.3 33 122.2 83 307.4 188 492.6 183 677.8 233 863.0 34 125.9 84 311.1 134 496.3 184 681.5 234 866.7 35 129 6 85 314.8 135 500.0 185 685.2 235 870.4 36 133.3 86 318.5 136 503.7 186 688.9 236 874.1 37 137.0 87 322.2 137 507.4 187 692.6 237 877.8 38 140.7 88 325.9 138 511.1 188 696.3 238 881.5 39 144 4 89 329.6 139 514.8 189 700.0 239 885.2 40 148.2 90 333.3 \ 140 518.5 190 703.7 240 888.9 41 151.9 91 337.0 141 522.2 191 707.4 241 892.6 42 155.6 92 340.7 I 142 525.9 192 711.1 242 896.3 43 159.3 93 344.4 j 143 529.6 193 714.8 243 900.0 44 103.0 94 348.2 \ 144 533.3 194 718.5 244 903.7 45 166.7 95 351.9 145 537.0 195 722.2 245 907.4 46 170.4 96 355.6 146 540.7 196 725.9 246 911.1 47 174.1 97 359.3 147 544.4 197 729.6 247 914.8 48 177.8 98 363.0 148 548.2 198 733.3 248 918.5 49 181.5 99 366.7 149 551.9 199 737.0 249 9C2.2 50 185.2 100 370.4 150 555.6 200 740.7 250 925.9 350 TABLE XV. CUBIC YARDS IN 100 FEET LENGTH. Area Sq. Ft. Cubic Yards. Area f Cubic Yards Area & Cubic Yards. Area & Cubic Yards. Area Sq. Ft. Cubic Yards. 251 929.6 301 1114.8 351 1300.0 401 1485.2 451 1670.4 252 933.3 302 1118.5 352 1303.7 402 1488.9 452 1674.1 253 937.0 303 1122.2 353 1307.4 403 1492.6 453 1677.8 254 940.7 304 1125.9 354 1311.1 404 1496.3 454 1681.5 255 944.4 305 1129.6 355 1314.8 405 1500.0 455 1685.2 256 948.2 306 1133.3 356 1318.5 406 1503.7 456 1688.9 257 951.9 307 1137.0 357 1322.2 407 1507.4 457 1692.6 258 955.8 308 1140.7 358 1325.9 408 1511.1 458 1696.3 859 959.3 309 1144.4 359 1329.6 409 1514.8 459 1700.0 260 963.0 310 1148.2 360 1333.3 410 1518.5 460 1703.7 261 9(56.7 311 1151.9 361 1337.0 411 1522 2 461 1707.4 262 970.4 312 1155.6 362 1340.7 412 1525.9 462 1711.1 263 974.1 313 1159.3 363 1:344.4 413 1529.6 463 1714.8 264 977.8 314 1163.0 364 1348.2 414 1533.3 464 1718.5 265 981.5 315 1166.7 365 1351.9 415 1537.0 465 1722.2 26(5 985.2 316 1170.4 366 1355.6 416 1540.7 466 1725.9 267 988.9 317 1174.1 367 1359.3 417 1544.4 467 1729.6 268 992.6 318 1177.8 368 1363.0 418 1548.2 468 1733.3 269 996.3 319 1181.5 369 1360.7 419 1551.9 469 1737.0 270 1000.0 320 1185.2 370 1370.4 420 1555.6 470 1740.7 271 1003.7 321 1188.9 371 1374.1 421 1559.3 471 1744.4 272 1007.4 322 1192.6 372 1377.8 422 1563.0 472 1748.2 273 1011.1 323 1196.3 373 1381.5 423 1566.7 473 1751.9 274 1014.8 324 1200.0 374 1385.2 424 1570.4 474 1755.6 275 1018.5 325 1203.7 375 1388.9 425 1574.1 475 1759.3 276 1022.2 326 1207.4 376 1392.6 426 1577.8 476 1763.0 277 1025.9 327 1211.1 377 1396.3 427 1581.5 477 1766.7 278 10-29.6 328 1214.8 378 1400.0 428 1585.2 478 1770.4 279 1033.3 329 1218.5 379 1103.7 429 1588.9 479 1774.1 280 1037.0 330 1222 2 380 1407.4 430 1592.6 480 1777.8 281 1040.7 331 1325.9 381 1411.1 431 1596.3 481 1781.5 282 1044.4 332 1229.6 382 1414.8 432 1600.0 482 1785.2 283 1048.2 333 1233.3 383 1418.5 433 1603.7 4&3 1788.9 284 1051.9 334 1237.0 384 1422.2 434 1607.4 484 1792.6 285 1055.6 335 1240.7 385 1425.9 4-35 1611.1 485 1796.3 286 1059.3 336 1244.4 386 1429 6 436 1614.8 486 1800.0 287 1063.0 337 1248.2 387 1433.3 437 1618.5 487 1803.7 288 1066.7 338 1251.9 388 1437.0 438 1622.2 488 1807.4 289 1070.4 339 1255.6 389 1440.7 439 1625.9 489 1811.1 290 1074.1 340 1259.3 390 1444.4 440 1629.6 490 1814.8 291 1077.8 341 1263.0 391 1448.2 441 1633.3 491 1818.5 292 1081.5 342 1266.7 392 1451.9 442 1637.0 492 1822.2 293 1085.2 343 1270.4 393 1455.6 443 1640.7 493 1825.9 294 1088.9 344 1274.1 394 1459.3 444 1644.4 494 1829.6 295 1092.6 345 1277.8 395 1463.0 445 1648 2 495 1833.3 296 1096.3 346 1281.5 396 1466.7 446 1651.9 496 1837.0 297 1100.0 347 1285.2 397 1470.4 447 1655.6 497 1840.7 298 1103.7 348 1288.9 398 1474.1 448 1659.3 498 1844.4 299 1107.4 349 1292.6 399 1477.8 449 1663.0 499 1818.2 300 1111.1 350 1296.3 400 1481.5 450 1666.7 500 1851.9 351 TABLE XV. CUBIC YARDS IN 100 FEET LENGTH. Area. St Cubic Yards. Area. 1?: Cubic Yards. Area. It Cubic Yards. Area. 11 Cubic Yards. Area. 11 Cubic Yards. 501 1855.6 551 2040.7 601 2225.9 651 2411.1 701 2596.3 502 1859.3 552 2044.4 602 2229.6 652 2414.8 702 2600.0 503 1863.0 553 2048.2 603 2233.3 653 2418.5 703 2603.7 504 1866.7 554 2051.9 604 2237.0 654 2422.2 704 2607. 4 505 1870.4 555 2055.6 605 2240.7 655 2425.9 705 2611.1 506 1874.1 556 2059.3 606 2244.4 656 2429.6 706 2614.8 507 1877.8 557 2063.0 607 2248.2 657 2433.3 707 2618.5 508 1881.5 558 2066.7 608 2251.9 658 2437.0 708 2622.2 509 1885.2 559 2070.4 609 2255.6 659 2440.7 709 2(525.9 510 1888.9 560 2074.1 610 2259.3 660 2444.4 710 2629.6 511 1892.6 561 2077.8 611 2263.0 661 2448.2 711 2633.3 512 1896.3 562 2081.5 612 2266.7 662' 2451.9 712 2637.0 513 1900.0 563 2085.2 613 2270.4 663 2455.6 713 2640.7 514 1903.7 564 2088.9 614 2274.1 664 2459.3 714 2644.4 515 1907.4 565 2092.6 615 2277.8 665 2463.0 715 2648.2 516 1911.1 566 2096.3 616 2281.5 666 2466.7 716 2651.9 517 1914.8 567 2100.0 617 2285.2 667 2470.4 717 2655.6 518 1918.5 568 2103.7 618 2288.9 668 2474.1 718 2659.3 519 1922.2 569 2107.4 619 2292.6 669 2477.8 719 2663.0 520 1925.9 570 2111.1 620 2296.3 670 2481.5 720 2666.7 521 1929 6 571 2114.8 621 2300.0 671 2485.2 721 2670.4 522 1933.3 572 2118.5 622 2303.7 672 2488.9 722 2674.1 523 1937.0 573 2122.2 623 2307.4 673 2492 6 723 2677.8 524 1940.7 574 2125.9 624 2311.1 674 2496.3 724 2C81.5 525 1944.4 575 2129.6 625 \ 2314.8 675 2500.0 725 2685.2 526 1948.2 576 2133.3 626 2318.5 676 2503.7 726 2(188.9 527 1951.9 577 2137.0 627 2322.2 677 2507.4 727 2(i92.6 528 1955.6 578 2140.7 628 2325.9 678 2511.1 728 2(1%. 3 529 1959.3 579 2144.4 629 ! 2329.6 679 2514.8 729 2700.0 530 1963.0 580 2148 2 630 2333.3 680 2518.5 730 2703.7 531 1966.7 581 2151.9 631 2337.0 681 2522.2 731 2707.4 532 1970.4 582 2155.6 632 2340.7 682 2525.9 732 2711.1 533 1974.1 583 2159.3 633 2344.4 683 2529.6 733 2714.8 534 1977.8 584 2163.0 634 2348.2 684 2533.3 734 2718.5 535 1981.5 585 2166.7 635 2351.9 685 2537.0 735 2722.2 536 1985.2 586 2170.4 636 2355.6 686 2540 7 736 2725.9 537 1988.9 587 2174.1 637 2359.3 687 2544.4 737 2729.6 538 1992.6 588 2177.8 638 2363 688 2548.2 738 2733.3 539 1996.3 589 2181.5 639 2366.7 689 2551.9 739 2737.0 540 2000.0 590 2185.2 640 2370.4 690 2555.6 740 2740.7 541 2003.7 591 2188.9 641 2374.1 691 2559.3 741 2744.4 542 2007.4 592 2192.6 642 2377.8 692 2563.0 742 2748.2 543 2011.1 593 2196.3 643 2381.5 693 2566.7 743 2751.9 544 2014.8 594 2200.0 644 2385.2 694 2570.4 744 2755.6 545 2018.5 595 2203.7 645 2388.9 695 2574.1 745 2759.3 546 2022.2 596 2207.4 646 2392.6 696 2577.8 746 2763.0 547 2025.9 597 2211.1 647 2396.3 697 2581.5 747 2766.7 548 2029.6 598 2214.8 648. 2400.0 698 2585.2 748 2770.4 549 2033.3 599 2218.5 649 2403.7 699 2588.9 749 2774.1 550 2037.0 600 2222.2 650 2407.4 700 2592.6 750 2777.8 352 TABLE XV.-CUBIC YARDS IN 100 FEET LENGTH. Area. ft Cubic Yards. Area. Sq. K, Cubic Yards. Area. Cubic Yards. Area. 11: Cubic Yards. Area. ft Cubic Yards. 751 2781.5 801 2966.7 851 3151.9 901 3337.0 951 3522.2 752 2785.2 802 2970.4 852 3155.6 902 3340.7 952 3525.9 753 2788.9 803 2974.1 853 3159.3 : 903 3344.4 953 3529.6 754 2792.6 804 2977.8 854 3163.0 904 3348.2 954 3533.3 755 2796.3 805 2981.5 ! 855 3166.7 905 3351.9 955 3537.0 756 2800.0 806 2985.2 856 3170.4 906 3355.6 956 3540.7 757 2803.7 807 2988.9 857 3174.1 907 3359.3 957 3544.4 758 2807.4 808 2992.6 858 3177.8 908 3363.0 958 3548.2 759 2811.1 809 2996.3 859 3181.5 909 3366.7 959 3551.9 760 2814.8 810 3100.0 860 3185.2 : 910 3370.4 960 3555.6 761 2818.5 811 3003.7 861 3188.9 911 3374.1 961 3559.3 762 2822.2 812 3007.4 i 862 3192.6 912 3377.8 962 3563.0 763 2825.9 813 3011.1 863 3196.3 913 3381.5 963 3566.7 764 2829.6 814 3014.8 864 3200.0 914 3385.2 964 3570.4 765 2833 3 815 3018.5 865 3203.7 915 3388.9 965 3574.1 766 2837.0 816 3022.2 866 3207.4 916 3392.6 966 3577.8 767 2840.7 817 3025.9 867 3211.1 917 3396.3 967 3581.5 768 2844.4 818 3029.6 868 3214.8 918 3400.0 968 3585.2 769 2848.2 819 3033.3 869 3218.5 919 3403.7 969 3588.9 770 2851.9 820 3037.0 870 3222.2 920 3407.4 970 3592.6 771 2855.6 821 3040.7 871 3225.9 921 3411.1 971 3596.3 772 2859.3 822 3044.4 872 3229.6 922 3414.8 972 3600.0 773 2863.0 823 3048.2 873 3233.3 923 3418.5 973 3603.7 774 2866.7 824 3051.9 ; 874 3237.0 924 3422.2 974 3607.4 775 2870.4 825 3055.6 I 875 3240.7 925 3425.9 975 3611.1 776 2874.1 826 3059.3 876 3244.4 926 3429.6 976 3614.8 777 2877.8 827 3063.0 1 877 3248.2 927 3433.3 977 3618.5 778 2881.5 828 3066.7 < 878 3251.9 928 3437.0 978 3622.2 779 2885.2 829 3070.4 \ 879 3255 6 \ 929 3440.7 979 3625.9 780 2888.9 830 3074.1 880 3259.3 ! 930 3444.4 980 3629.6 781 2892.6 831 3077.8 881 3263.0 j 931 3448.2 981 3633.3 782 2896.3 832 3081.5 i 882 3266.7 932 3451.!) 982 3637.0 783 2900.0 833 3085.2 j 883 3270.4 933 3455.6 983 3640.7 784 2903.7 834 3088.9 884 3274.1 934 3459.3 984 3644.4 785 2907.4 835 3092.6 885 3277.8 935 3463.0 985 3648.2 786 2911.1 836 3096.3 i 886 3281.5 936 3466.7 986 3651.9 787 2914.8 837 3100.0 ; 887 3285.2 937 3470.4 987 3655.6 788 2918.5 838 3103.7 888 3288.9 938 3474.1 988 3659.3 789 2922.2 839 3107.4 1 889 3292.6 939 3477.8 989 3663.0 790 2925.9 840 3111.1 890 3296.3 940 3481.5 990 3666.7 791 2929.6 841 3114.8 891 3300.0 941 3485 2 991 3670.4 792 2933.3 842 3118.5 892 3303.7 : 942 3488.9 992 3674.1 793 2937.0 843 3122.2 893 3307.4 ' 943 3492.6 993 3677.8 794 2940.7 844 3125 9 : 894 3311.1 944 3496.3 994 3681.5 795 2944.4 845 3129.6 895 3314.8 , 945 3r,oo.o 995 3685.2 796 2948.2 846 3133.3 896 3318.5 946 3503.7 996 3688.9 797 2951.9 847 3137.0 i 897 3322.2 947 3507.4 997 3692.6 798 2955.6 848 3140.7 ! 898 3325.9 948 3511.1 998 3696.3 799 2959.3 849 3144.4 i 899 3329.6 949 3514.8 999 3700.0 800 2963.0 850 3148.2 ! 900 3333.3 950 3518.5 1000 3703.7 353 TABLE XVI. CONVERSION OF ENGLISH INCHES INTO CENTIMETRES. Ins. 1 2 3 4 5 6 rt 8 9 Cm. Cm. Cm. Cm. Cm. Cm. Cm. Cm. Cm. Cm. 0.000 2.540 5.080 7.620 10.16 12.70 15.24 17.7.8 20.32 22.86 10 25.40 27.94 30.48 33.02 35.56 38.10 40.64 43.18 45.72 48.26 20 50.80 53.34J 55.88 58.42 60.96 63.50 66.04 68.58 71.12 73.66 30 76.20 78.74! 81.28 83.82 86 36 88.90 91.44 93.98 96.52 99.06 40 101.60 104.14! 106.68 109.22 111.76 114.30 116.84 119.38 121.92 124.46 50 127.00 129.54 132.08 134.62 137.16 139.70 142.24 144.78 147.32 149.86 60 152.40 154.94 157.48 160.02 162.56 165.10 167.64 170.18 172.72 175.26 70 177.80 180.34! 182.88 185.42 187.96 190.50 193.04 195.58 198.12 200.96 80 203.20 205.74 208.28 210.82 213.36 215.90 218.44 220.98 223.52 226.06 90 228.60 231. 14i 233.68 236.22 238.76 241 . 30 243. 84 1246.38 248.92 251.46 100 254.00 256. 54 ; 259.08 261.62 264 10 266.70 269 241271.78 274. 32 27'(>.8'.i CONVERSION OF CENTIMETRES INTO ENGLISH INCHES. Cm. 1 2 3 4 5 6 7 8 9 Ins. Ins. Ins. Ins. Ins. Ins. Ins. Ins. Ins. Ins. 0.000 0.394 0.787; 1.181 1.575 1.969 2.362 2.756 3.150 3.543 10 3.937 4.331 4.742 5.118 5.512 5.906 6.299 6.693 7.087i 7.480 20 7.874 8.268 8.662 9.055 9.449 9.843 10.236 10.630 11.024 ill. 418 30 11.811 12.205 12.599 12.992 13.386 13. 780 14. 1V3 14. 567114. 961115. 855 40 15.748 16.142 16.530 16.929 17.323 17. 71718. Ill 18.504,18.898 19.292 50 60 19.685 23.622 20.079 20.473 20.867 24.016! 24.410 24.804 21.260 25.197 21.654122.04822 441 1 22.835|23.229 25 . 591 25 . 985 26 . 378 -J(j . 772 27 . 1 66 70 27.560 27.953, 28.347 28.741 29.134 29 . 528 29 . 922 30 . 3 1 6 30 . 701) 31 . 103 80 31.497 31.890 82.284 32.678 33.071 33.465 33.859 34.253 34.646 35.040 90 35.434 35.827! 36.221 36.615 37.009 37 . 402 37 . 796 38 . 1 90 38 . 583 38 . 977 100 39.370) 39.764 1 40.158 40.552 40.945 41.33941.73342.126 42.52042.914 CONVERSION OF ENGLISH FEET INTO METRES. Feet. 1 2 3 4 5 6 7 8 9 Met. Met. Met. Met. Met. Met. Met. Met. Met. Met. 0.000 0.3048 0.6096 0.9144 1.2192 1.5239 1.82872.1335 2.4383 2.7431 10 3.0479 3.3527 3.6575 3.9623 4.2671 4.5719 4.87675.1815 5.4863 5.7911 20 6.0359 6.4006 6.7055 7.0102 7.3150 7.61987.92468.2294 8.5342 8.8390 30 9.1438 9.4486 9.7534 10.058 10.363 10.66810.97211.277 11.582 11.887 40 12.192 12.496 12.801 13.106 13.411 13.71614.02014.325 14.630 14.935 50 15.239 15.544 15.849 16.154 16.459 16.763 17.06817.373 17.678 17.983 60 18.287 18.592 18.897 19.202 19.507 19.811J20.11620.421 20.726 21.031 70 21.335 21.640 21.945 22.250 22.555 22.85923.16423.46'.) 23 774 24.079 80 24.383 24.688 24.993 25.298 25.602 25.907:26.21226.517 26.82227.126 90 27.431 27.736 28.041 28.346 28.651 28. 955 '29. 260 29. 565 29. 870130. 174 100 30.479 30.784 31.089 31.394 31.698 3200332.30832.613 32. 91 8 133.222 CONVERSION OF METRES INTO ENGLISH FEET. Met. JL_LJ_ * 4 5 6 7 8 9 Feet. Feet. Feet. Feet. Feet. Feet. Feet. Feet. Feet. Feet. 0.000 3.2809 6.5618 9.8427 13.123 16.404 19.685 22.906 26.247 29.528 10 32.809 36.090 39.371 42.651 45 . 932 49.21352.494 55.775 59.056 62:337 20 65.618 68.899 72.179 75.461 78.741 82.02285.303 88.584 91.865 95.146 30 98.427! 101.71 104.99 108.27 111.55 114 831118.11 121.39 124.67 127.96 40 131.24i 134.52 137.80 141.08 144.36 147.64 150 92 154.20 157.48 160.76 50 164.041 167.33 170 61 173.89 177.17 180.45183.73 187.01 190.29 193.57 60 196.85 200.13 203.42 206.70 209.98 213.26216.54 219.82 223.10 226.38 70 229.66 232.94 236.22 239.51 242.79 246.07249.35 252 63 255.91 259.19 80 262.47 265.75 269.03 272.31 27'5.60 278.88:282.16 285.44 288 72 292.00 - 90 295.28 298.56 391.84 305.12 308.40 311.69314.97 518.25 321.53 324.81 100 328.09 331.37 334.65! 337.93 341.21 344.49;347.78 351.06 354.34 357.62 354 TABLE XVH. CONVERSION OF ENGLISH STATUTE-MILES INTO KILOMETRES. Miles. | 1 2 3 4 5 6 T 1 S 9 Kilo. Kilo. Kilo. Kilo. Kilo. Kilo. Kilo. Kilo. Kilo. Kilo. 0.0000 1.6093 3.2186 4.82796.4372 8.0465 9.6558 11.2652 12.8745 14.4818 10 16.093 17.702 19.312 20.921 22 530 24.139 25.749J 27.358 28.967 30.577 20 32.186 33.795 35.405 37.014 38.623 40.232 41.8421 43 451 45.060 46.670 30 48.279 49.88851.498 53.107 54.716; 56.325 57.935 59.544 61.153 62.763 40 64.372 65.981 67.591 69.200 70.809 72.418 74.028 75.637 77.246 78.856 50 80.46582.07483.684 85.293 86.902 88.511 90.121 91 730 93.339 94.949 60 96.558 98.167 99 777 101.39 102.99 104.60 106 21 107.82 109.43 111.04 70 112.65 114.26115.87 117.48 119.08 120.69 122.801 123.91 125.52 127.13 80 128 74 130.35 131.96 133 57 135.17 136.78 138.39 140.00 141.61 143.22 90 144.85146.44 148.05 149.66 151.26 152.87 154.48 156.09 157.70! 159.31 100 160.93162.53164 14 165 75 167.35 168.96 170.57 172.18 173.79! 175.40 t \r\ vr\7 L^ OO>T/^ XT /~^TJ* T/"TT r\ vr T7